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\begin{document}
\title{On the algebraic connectivity of token graphs
\thanks{This research of C. Dalf\'o and M. A. Fiol has been partially supported by
AGAUR from the Catalan Government under project 2017SGR1087 and by MICINN from the Spanish Government under project PGC2018-095471-B-I00. The research of C. Dalf\'o has also been supported by MICINN from the Spanish Government under project MTM2017-83271-R.}
}
\author{C. Dalf\'o$^a$, M. A. Fiol$^b$,\\
\\
{\small $^a$Dept. de Matem\`atica, Universitat de Lleida, Igualada (Barcelona), Catalonia}\\
{\small {\tt [email protected]}}\\
{\small $^{b}$Dept. de Matem\`atiques, Universitat Polit\`ecnica de Catalunya, Barcelona, Catalonia} \\
{\small Barcelona Graduate School of Mathematics} \\
{\small Institut de Matem\`atiques de la UPC-BarcelonaTech (IMTech)}\\
{\small {\tt [email protected]} }\\
}
\date{}
\maketitle
\begin{abstract}
We study the algebraic connectivity (or second Laplacian eigenvalue) of token graphs, also called symmetric powers of graphs. The $k$-token graph $F_k(G)$ of a graph $G$ is the
graph whose vertices are the $k$-subsets of vertices from $G$, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in $G$.
Recently, it was conjectured that the algebraic connectivity of $F_k(G)$ equals the algebraic connectivity of $G$. In this paper, we prove the conjecture for new infinite families of graphs, such as trees and graphs with maximum degree large enough.
\end{abstract}
\noindent{\em Keywords:} Token graph, Laplacian spectrum, Algebraic connectivity, Binomial matrix.
\noindent{\em MSC2010:} 05C15, 05C10, 05C50.
\section{Introduction}
\label{sec:-1}
Let $G$ be a simple graph with vertex set $V(G)=\{1,2,\ldots,n\}$ and edge set $E(G)$. Let $\Delta(G)$ denote the maximum degree of $G$. For a given integer $k$ such that
$1\le k \le n$, the {\em $k$-token graph} $F_k(G)$ of $G$ is the graph whose vertex set $V (F_k(G))$ consists of the ${n \choose k}$
$k$-subsets of vertices of $G$, and two vertices $A$ and $B$
of $F_k(G)$ are adjacent whenever their symmetric difference $A \bigtriangleup B$ is a pair $\{a,b\}$ such that $a\in A$, $b\in B$, and $(a,b)\in E(G)$.
The naming `token graph'
comes from an observation in
Fabila-Monroy, Flores-Pe\~{n}aloza, Huemer, Hurtado, Urrutia, and Wood \cite{ffhhuw12}, that vertices of $F_k(G)$ correspond to configurations
of $k$ indistinguishable tokens placed at distinct vertices of $G$, where
two configurations are adjacent whenever one configuration can be reached
from the other by moving one token along an edge from its current position
to an unoccupied vertex. Thus,
the maximum degree of $F_k(G)$ satisfies
\begin{equation}
\label{DeltaFk}
\Delta(F_k(G))\le k\Delta(G).
\end{equation}
In Figure \ref{fig1}, we show the 2-token graph of the cycle $C_9$ on 9 vertices.
Note that if $k=1$, then $F_1(G)\cong G$; and if $G$ is the complete graph $K_n$, then $F_k(K_n)\cong J(n,k)$, where $J(n,k)$ denotes the Johnson graph~\cite{ffhhuw12}.
\begin{figure}
\caption{The $2$-token graph $F_2(C_9)$ of the cycle graph, with vertex set $V(C_9)=\{0,1,\ldots,8\}$.
The vertices on the circumference of radius $r_{\ell}$, with $\ell=1,2,3,4$ and $r_1>r_2>r_3>r_4$ are $\{i,j\}$ with $\dist(i,j)=\ell$ in $C_9$.}
\label{fig1}
\end{figure}
Token graphs have some applications in physics. For instance, a
relationship between token graphs and the exchange of Hamiltonian operators in
quantum mechanics is given in Audenaert, Godsil, Royle, and Rudolph \cite{agrr07}.
Recently, it was conjectured by Dalf\'o, Duque, Fabila-Monroy, Fiol, Huemer, Trujillo-Negrete, and Zaragoza Mart\'{\i}nez \cite{ddffhtz21} that the algebraic connectivity of $F_k(G)$ equals the algebraic connectivity of $G$. In this paper, we prove the conjecture for new infinite families of graphs, such as trees, multipartite complete graphs, and graphs with large enough maximum degree.
\section{Known results}
Let us first introduce some notation and known results that are used throughout the paper.
The transpose of a matrix $\mbox{\boldmath $M$}$ is denoted by $\mbox{\boldmath $M$}^\top$, the
identity matrix by $\mbox{\boldmath $I$}$, the all-$1$ vector $(1,..., 1)^{\top}$ by $\mbox{\boldmath $1$}$, the all-$1$ (universal) matrix by $\mbox{\boldmath $J$}$, and the all-$0$ vector and all-$0$ matrix by $\mbox{\boldmath $0$}$
and $\mbox{\boldmath $O$}$, respectively.
Let $[n]:=\{1,\ldots,n\}$ and ${[n]\choose k}$ denote the set of $k$-subsets of $[n]$, which is the set of vertices of the $k$-token graph.
For our purpose, it is convenient to denote by $W_n$ the set of all column vectors $\mbox{\boldmath $v$}$ such that $\mbox{\boldmath $v$}^{\top }\mbox{\boldmath $1$} = 0$.
Recall that any square matrix $\mbox{\boldmath $M$}$ with all zero row sums has an eigenvalue $0$ with corresponding eigenvector $\mbox{\boldmath $1$}$.
\subsection{The algebraic connectivity of token graphs}
When $\mbox{\boldmath $M$}=\mbox{\boldmath $L$}(G)$, the Laplacian matrix of a graph $G$, the matrix is positive semidefinite, with eigenvalues $(0=)\lambda_1\le \lambda_2\le \cdots \le \lambda_n$. Its second smallest eigenvalue $\lambda_2$ is known as the {\em algebraic connectivity} of $G$ (see Fiedler \cite{fi73}), and we denote it by $\alpha(G)$.
The spectral radius $\lambda_{\max}(G)=\lambda_n$ satisfies several lower and upper bounds (see
Patra and Sahoo \cite{ps17} for a survey). Here, we will use the following ones in terms of the maximum degree of $G$:
\begin{equation}
\label{bound-sp-rad}
1+\Delta(G)\le \lambda_{\max}(G)\le 2\Delta(G).
\end{equation}
The upper bound is due to Fiedler \cite{fi73}, whereas the lower bound was proved by Grone and Merris in \cite{gm94}.
In this paper, we want to study the algebraic connectivity of token graphs. As far as we know, this study was initiated by Dalf\'o, Duque, Fabila-Monroy, Fiol, Huemer, Trujillo-Negrete, and Zaragoza Mart\'{\i}nez in \cite{ddffhtz21}, where they proved the following result.
\begin{lemma}[\cite{ddffhtz21}]
\label{coro:LkL1}
Let $G$ be a graph with Laplacian matrix $L_1$. Let $F_k=F_k(G)$ be its token graph with Laplacian $L_k$.
Let $\mbox{\boldmath $B$}$ be the so-called $(n;k)$-\emph{binomial matrix}, which is an ${n \choose k}\times n$ matrix whose rows are the characteristic vectors of the $k$-subsets of $[n]$ in a given order.
Then, the following holds:
\begin{itemize}
\item[$(i)$]
If $\mbox{\boldmath $v$}$ is a $\lambda$-eigenvector of $\mbox{\boldmath $L$}_1$, then $\mbox{\boldmath $B$}\mbox{\boldmath $v$}$ is a $\lambda$-eigenvector of $\mbox{\boldmath $L$}_k$.
Thus, the Laplacian spectrum (eigenvalues and their multiplicities) of $\mbox{\boldmath $L$}_1$ is contained in the Laplacian spectrum of $\mbox{\boldmath $L$}_k$.
\item[$(ii)$]
If $\mbox{\boldmath $u$}$ is a $\lambda$-eigenvector of $\mbox{\boldmath $L$}_k$ such that $\mbox{\boldmath $B$}^{\top}\mbox{\boldmath $u$}\neq \mbox{\boldmath $0$}$, then $\mbox{\boldmath $B$}^{\top}\mbox{\boldmath $u$}$
is a $\lambda$-eigenvector of $\mbox{\boldmath $L$}_1$.
\end{itemize}
\end{lemma}
Given two integers $n,k$ such that $k\in [n]$, the {\em Johnson graph} $J(n,k)$ can be defined as the $k$-token graph of the complete graph $K_n$ , $F_k(K_n)\cong J(n,k)$. It is known that these graphs are antipodal (but not bipartite) distance-regular graphs, with degree $d=k(n-k)$, diameter $D=\min\{k,n-k\}$, and Laplacian spectrum
(eigenvalues and multiplicities)
\begin{equation}
\label{spJ(n,k)}
\lambda_j=d-\mu_j=j(n+1-j)\qquad \mbox{and} \qquad m_j={n\choose j}-{n\choose j-1},\qquad j=0,1,\ldots,D.
\end{equation}
(See again \cite{ddffhtz21}).
For example, $F_2(K_{4})\cong J(4,2)$ is a $2$-regular graph with $n=6$ vertices, diameter $D=2$, and Laplacian spectrum
$
S( F_2(K_{4}))=\{0^1, 4^{3}, 6^{2}\}$.
Let us consider a graph $G$ and its complement $\overline{G}$, with respective Laplacian matrices $L_G$ and $L_{\overline{G}}$.
Since $L_G+L_{\overline{G}}=n\mbox{\boldmath $I$}-\mbox{\boldmath $J$}$, the Laplacian spectrum of $\overline{G}$ is the complement of the Laplacian spectrum of $G$ with respect to the Laplacian spectrum of the complete graph $K_n$. We represent this as
$$
\spec G \oplus \spec \overline{G} = \spec K_n,
$$
where each eigenvalue of $G$ and each eigenvalue of $\overline{G}$ are used once.
In \cite{ddffhtz21}, is was shown that a similar relationship holds between the Laplacian spectra of the $k$-token of $G$ and the $k$-token of $\overline{G}$, but now with respect to the Laplacian spectrum of the Johnson graph.
\begin{theorem}[\cite{ddffhtz21}]
\label{theo:pairing}
Let $G=(V,E)$ be a graph on $n=|V|$ vertices, and let $\overline{G}$ be its complement. For a given $k$, with $1\leq k\le n-1$, let us consider the token graphs $F_k(G)$ and $F_k(\overline{G})$. Then, the Laplacian spectrum of $F_k(\overline{G})$ is the complement of the Laplacian spectrum of $F_k(G)$ with respect to the Laplacian spectrum of the Johnson graph $J(n,k)=F_k(K_n)$.
That is, every eigenvalue $\lambda_J$ of $J(n,k)$ is the sum of one eigenvalue $\lambda_{F_k(G)}$ of $F_k(G)$ and one eigenvalue $\lambda_{F_k(\overline{G})}$ of $F_k(\overline{G})$, where each $\lambda_{F_k({G})}$ and each $\lambda_{F_k(\overline{G})}$ is used once:
\begin{equation}
\label{spFk(G)-sp(Fk(noG))}
\lambda_{F_k({G})}+\lambda_{F_k(\overline{G})}=\lambda_J.
\end{equation}
\end{theorem}
As a consequence of Lemma \ref{coro:LkL1}, the spectrum of $J(n,k)$ in \eqref{spJ(n,k)}, and Theorem \ref{theo:pairing}, we can state the following lemma where, for simplicity, we assume that both $G$ and $\overline{G}$ are connected.
\begin{lemma}
\label{partition}
Let $\Lambda$ be the set of pairs $(\lambda,\overline{\lambda})$ of eigenvalues of $F_k(G)$ and $F_k(\overline{G})$, with $k\le n/2$, sharing both the same eigenvector $\mbox{\boldmath $v$}$ with $J(n,k)$. Then $\Lambda$ can be partitioned into the sets $\Lambda_0,\Lambda_1,\ldots,\Lambda_k$ such that
$\Lambda_0=\{(0,0)\}$, and
$\Lambda_j=\{(\lambda,\overline{\lambda}):\lambda+\overline{\lambda}=j(n+1-j)\}$ for $j=1,\ldots,k$.
Moreover, the eigenvectors and eigenvalues of each set satisfy: $\mbox{\boldmath $v$}=\mbox{\boldmath $1$}$ in $\Lambda_0$; the eigenvalues in $\Lambda_1$ correspond to the eigenvalues of $G=F_1(G)$ and $\overline{G}=F_1(\overline{G})$; the eigenvalues in $\Lambda_j$ come from $\spec F_j(G)\setminus \spec F_{j-1}(G)$ and $\spec F_j(\overline{G})\setminus \spec F_{j-1}(\overline{G})$ with eigenvectors $\mbox{\boldmath $v$}$ such that $\mbox{\boldmath $B$}^{\top}\mbox{\boldmath $v$}=\mbox{\boldmath $0$}$, for $j=2,\ldots, k$.
\end{lemma}
Let us show an example of the results in Theorem \ref{theo:pairing} and Lemma \ref{partition}.
\begin{example}
\label{exemple}
Consider the graph $G$ and its complement graph $\overline{G}$ of Figure \ref{fig2}.
\begin{figure}
\caption{The graph $G$ and its complement graph $\overline{G}$ of Example \ref{exemple}.}
\label{fig2}
\end{figure}
The spectra of $G$, $\overline{G}$, and their $k$-tokens for $k=2,3$ are the following:
\begin{align*}
\spec G &=\{0,2,4^{[3]},6\}\subset \spec F_2(G) =\{0,2,4^{[5]},6^{[4]},8^{[3]},10\}\\
&\subset \spec F_3(G) =\{0,2,4^{[6]},6^{[4]},8^{[5]},10^{[3]}\}.\\
\spec \overline{G} &=\{0^{[2]},2^{[3]},4\}\subset \spec F_2(\overline{G})=\{0^{[3]},2^{[6]},4^{[4]},6^{[2]}\} \subset
\spec F_3(\overline{G}) =\{0^{[3]},2^{[8]},4^{[6]},6^{[2]}\}.
\end{align*}
Then, as shown in Table \ref{table1}, there is a pairing between the eigenvalues of $F_3(G)$ and the eigenvalues of $F_3(\overline{G})$ satisfying Theorem \ref{theo:pairing}.
Namely,
$$
\spec F_3(G)\oplus \spec F_3(\overline{G}) = \spec J(6,3)=\{0, 6^{[5]}, 10^{[9]}, 12^{[5]}\}.
$$
Thus, the pairs of $\Lambda_0$, $\Lambda_1$, $\Lambda_2$, and $\Lambda_3$ add up to 0, 6, 10, and 12, respectively.
\begin{table}
\begin{center}
\scriptsize
\setlength\tabcolsep{3pt}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$F_3(\overline{G})\setminus F_3(G)$
& 0 & 2 & 4 & 4 & 4 & $\mathit{4}$ & $\mathit{4}$ & $\mathbf{4}$ & $\mathit{6}$ & $\mathit{6}$ & $\mathit{6}$ & $\mathit{6}$ & $\mathit{8}$ & $\mathit{8}$ & $\mathit{8}$ & $\mathbf{8}$ & $\mathbf{8}$ & $\mathit{10}$ & $\mathbf{10}$ & $\mathbf{10}$\\
\hline\hline
0
& 0 & & & & & & & & & & & & & & & & & & & \\
\hline
0
& & & & & & & & & 6 & & & & & & & & & & & \\
\hline
$\mathit{0}$
& & & & & & & & & & & & & & & & & & $\mathit{10}$ & & \\
\hline
2
& & & 6 & & & & & & & & & & & & & & & & & \\
\hline
2
& & & & 6 & & & & & & & & & & & & & & & & \\
\hline
2
& & & & & 6 & & & & & & & & & & & & & & & \\
\hline
$\mathit{2}$
& & & & & & & & & & & & & $\mathit{10}$ & & & & & & & \\
\hline
$\mathit{2}$
& & & & & & & & & & & & & & $\mathit{10}$ & & & & & & \\
\hline
$\mathit{2}$
& & & & & & & & & & & & & & & $\mathit{10}$ & & & & & \\
\hline
$\mathbf{2}$
& & & & & & & & & & & & & & & & & & & $\mathbf{12}$ & \\
\hline
$\mathbf{2}$
& & & & & & & & & & & & & & & & & & & & $\mathbf{12}$ \\
\hline
4
& & 6 & & & & & & & & & & & & & & & & & & \\
\hline
$\mathit{4}$
& & & & & & & & & & $\mathit{10}$ & & & & & & & & & & \\
\hline
$\mathit{4}$
& & & & & & & & & & & $\mathit{10}$ & & & & & & & & & \\
\hline
$\mathit{4}$
& & & & & & & & & & & & $\mathit{10}$ & & & & & & & & \\
\hline
$\mathbf{4}$
& & & & & & & & & & & & & & & & $\mathbf{12}$ & & & & \\
\hline
$\mathbf{4}$
& & & & & & & & & & & & & & & & & $\mathbf{12}$ & & & \\
\hline
$\mathit{6}$
& & & & & & $\mathit{10}$ & & & & & & & & & & & & & & \\
\hline
$\mathit{6}$
& & & & & & & $\mathit{10}$ & & & & & & & & & & & & & \\
\hline
$\mathbf{8}$
& & & & & & & & $\mathbf{12}$ & & & & & & & & & & & & \\
\hline
\end{tabular}
\end{center}
\caption{The spectra of $F_3(G)$ and $F_3(\overline{G})$ giving the spectrum of $J(6,3)$ by addition. The eigenvalues of $\Lambda_2$ are in italics. The eigenvalues of $\Lambda_3$ are in boldface. The corresponding additions, giving the eigenvalues of $J(6,3)$, are written accordingly.}
\label{table1}
\end{table}
\end{example}
Concerning the algebraic connectivity of token graphs,
the authors in \cite{ddffhtz21} proposed the following conjecture.
\begin{conjecture}[\cite{ddffhtz21}]
\label{conjecture}
Let $G$ be a graph on $n$ vertices. Then, for every $k=1,\ldots,n-1$, the algebraic connectivity of its token graph $F_k(G)$ equals the algebraic connectivity of $G$.
\end{conjecture}
As a consequence of Lemma \ref{coro:LkL1}$(i)$, and that $F_k(G)=F_{n-k}(G)$, the conjecture only needs to be proved for the case $k=\lfloor n/2 \rfloor$.
Moreover, it was noted that the conjecture
also holds when the graph $G$ is disconnected and for those graphs whose token graphs are regular, which are $K_n$, $S_n$ (with even $n$ and $k=n/2$), and their complements.
Also, computer exploration showed that $\alpha(F_2(G))=\alpha(G)$ for all graphs with at most $8$ vertices.
Moreover, it was shown that the conjecture also holds for the following infinite families of graphs.
\begin{theorem}[\cite{ddffhtz21}]
\label{theo:alg-connec-antic}
For each of the following classes of graphs, the algebraic connectivity of a token graph $F_k(G)$ equals the algebraic connectivity of $G$.
\begin{itemize}
\item[$(i)$]
Let $G=K_n$ be the complete graph on $n$ vertices. Then,
$\alpha(F_k(G))=\alpha(G)=n$ for every $n$ and $k=1,\ldots,n-1$.
\item[$(ii)$]
Let $G=S_n$ be the star graph on $n$ vertices. Then,
$\alpha(F_k(G))=\alpha(G)=1$ for every $n$ and $k=1,\ldots,n-1$.
\item[$(iii)$]
Let $G=P_n$ be the path graph on $n$ vertices. Then, $\alpha(F_k(G))=\alpha(G)\linebreak =2(1-\cos(\pi/n))$ for every $n$ and $k=1,\ldots, n-1$.
\item[$(iv)$]
Let $G= K_{n_1,n_2}$ be the complete bipartite graph on $n=n_1+n_2$ vertices, with $n_1\le n_2$. Then, $\alpha(F_k(G))=\alpha(G)=n_1$ for every $n_1,n_2$ and $k=1,\ldots,n-1$.
\end{itemize}
\end{theorem}
\section{New results}
\label{sec:-7}
In this section, we prove more results supporting Conjecture \ref{conjecture}. In our proofs, we use the following concepts and lemma.
Given a graph $G=(V,E)$ of order $n$,
we say that a vector $\mbox{\boldmath $v$}\in \mathbb{R}^n$ is an \textit{embedding} of $G$ if $\mbox{\boldmath $v$}\in W_n$.
Note that if $\mbox{\boldmath $v$}$ is a $\lambda$-eigenvector of $G$, with $\lambda>0$, then it is an embedding of $G$.
For a graph $G$ with Laplacian matrix $\mbox{\boldmath $L$}(G)$, and an embedding $\mbox{\boldmath $v$}$ of $G$, let
\begin{equation}
\label{rayleigh-quotient}
\lambda_G(\mbox{\boldmath $v$}):=\frac{\mbox{\boldmath $v$}^{\top}\mbox{\boldmath $L$}(G)\mbox{\boldmath $v$}}{\mbox{\boldmath $v$}^{\top}\mbox{\boldmath $v$}}=\frac{\sum\limits_{(i,j)\in E}[\mbox{\boldmath $v$}(i)-\mbox{\boldmath $v$}(j)]^2}{\sum\limits_{i\in V}\mbox{\boldmath $v$}^2(i)}
\end{equation}
where
$\mbox{\boldmath $v$}(i)$ denotes the entry of $\mbox{\boldmath $v$}$ corresponding to the vertex $i\in V(G)$.
The value of $\lambda_G(\mbox{\boldmath $v$})$ is known as
the {\em Rayleigh quotient}.
If $\mbox{\boldmath $v$}$ is an eigenvector of $G$, then its corresponding eigenvalue is $\lambda(\mbox{\boldmath $v$})$.
Moreover, for an embedding $\mbox{\boldmath $v$}$ of $G$, we have
\begin{equation}
\label{bound-lambda(v)}
\alpha(G)\le \lambda_G(\mbox{\boldmath $v$}),
\end{equation}
and we have equality when $\mbox{\boldmath $v$}$ is an $\alpha(G)$-eigenvector of $G$.
\\
\begin{lemma}
\label{lem:-vertex}
Let $G^+=(V^+,E^+)$ be a graph on the vertex set $V=\{1,2,\ldots,n+1\}$, having a vertex of degree $1$, say the vertex $n+1$ that is adjacent to $n$.
Let $G=(V,E)$ be the graph obtained from $G^+$ by deleting the vertex $n+1$.
Then,
$$
\alpha(G)\ge \alpha(G^+),
$$
with equality if and only if the $\alpha(G)$-eigenvector $\mbox{\boldmath $v$}$ of $G$ has entry $\mbox{\boldmath $v$}(n)=0$.
\end{lemma}
\begin{proof}
Let $\mbox{\boldmath $v$}\in W_n$ be an eigenvector of $G$ with eigenvalue $\alpha(G)$ and norm $\|\mbox{\boldmath $v$}\|=1$, so that
\begin{equation}
\label{lambda(v)=alpha}
\lambda(\mbox{\boldmath $v$})=\sum\limits_{(i,j)\in E}[\mbox{\boldmath $v$}(i)-\mbox{\boldmath $v$}(j)]^2=\alpha(G).
\end{equation}
Let $\mbox{\boldmath $w$}\in \mathbb R^{n+1}$ be the vector with components $\mbox{\boldmath $w$}(i)=\mbox{\boldmath $v$}(i)-\frac{\mbox{\boldmath $v$}(n)}{n+1}$ for $i=1,\ldots,n$ and $\mbox{\boldmath $w$}(n+1)=\mbox{\boldmath $w$}(n)=\frac{n\mbox{\boldmath $v$}(n)}{n+1}$. Note that $\mbox{\boldmath $w$}$ is an embedding of $G^+$ since
$$
\sum_{i=1}^{n+1}\mbox{\boldmath $w$}(i)=\sum_{i=1}^n \left(\mbox{\boldmath $v$}(i)-\frac{\mbox{\boldmath $v$}(n)}{n+1}\right)+\mbox{\boldmath $w$}(n+1)=1-\frac{n\mbox{\boldmath $v$}(n)}{n+1}+\mbox{\boldmath $w$}(n)=1.
$$
Then, from \eqref{bound-lambda(v)},
\begin{align*}
\alpha(G^+) & \le \lambda(\mbox{\boldmath $w$})=\frac{\sum\limits_{(i,j)\in E^+}[\mbox{\boldmath $w$}(i)-\mbox{\boldmath $w$}(j)]^2}{\sum\limits_{i\in V^+}\mbox{\boldmath $w$}^2(i)}
=\frac{\sum\limits_{(i,j)\in E}[\mbox{\boldmath $v$}(i)-\mbox{\boldmath $v$}(j)]^2}{\sum\limits_{i\in V}[\mbox{\boldmath $v$}(i)-\frac{\mbox{\boldmath $v$}_n}{n+1}]^2+\mbox{\boldmath $w$}_{n+1}^2}\le \alpha(G)
\end{align*}
where the last inequality comes from \eqref{lambda(v)=alpha} since, as $\mbox{\boldmath $v$}$ is an embedding of $G$,
$$
\sum\limits_{i\in V}\left[\mbox{\boldmath $v$}(i)-\frac{\mbox{\boldmath $v$}(n)}{n+1}\right]^2=\sum\limits_{i\in V}\left[\mbox{\boldmath $v$}(i)^2-2\mbox{\boldmath $v$}(i)\frac{\mbox{\boldmath $v$}(n)}{n+1}+\frac{\mbox{\boldmath $v$}(n)^2}{(n+1)^2}\right]=1+\frac{\mbox{\boldmath $v$}(n)^2}{(n+1)^2}\ge 1.
$$
Finally, the equality $\alpha(G^+)=\alpha(G)$ holds if and only if $\mbox{\boldmath $v$}(n)=0$.
\end{proof}
Let $G$ be a graph with $k$-token graph $F_k(G)$.
For a vertex $a\in V(G)$, let $S_a:=\{A\in V(F_k(G)):a\in A\}$ and $S'_a:=\{B\in V( F_k(G)):a\not\in B\}$.
Let $H_a$ and $H'_a$ be the subgraphs of $F_k(G)$ induced by $S_a$ and $S'_a$, respectively.
Note that $H_a\cong F_{k-1}(G\setminus \{a\})$ and $H'_a\cong F_k(G\setminus \{a\})$.
\begin{lemma}
\label{lem:eigenvectors}
Given a vertex $a\in G$ and an eigenvector $\mbox{\boldmath $v$}$ of $F_k(G)$ such that $\mbox{\boldmath $B$}^{\top}\mbox{\boldmath $v$}=\mbox{\boldmath $0$}$, let
\[
\mbox{\boldmath $w$}_a:=\restr{\mbox{\boldmath $v$}}{S_a} \text{ and \quad } \mbox{\boldmath $w$}'_a:=\restr{\mbox{\boldmath $v$}}{S'_a}.
\]
Then, $\mbox{\boldmath $w$}_a$ and $\mbox{\boldmath $w$}'_a$ are embeddings of $H_a$ and $H'_a$, respectively.
\end{lemma}
\begin{proof}
Assume that the matrix $\mbox{\boldmath $B$}^{\top}$ has the first row indexed by $a\in V(G)$. Then, we have
$$
\mbox{\boldmath $0$}=
\mbox{\boldmath $B$}^{\top}\mbox{\boldmath $v$}=
\left(
\begin{array}{cc}
\mbox{\boldmath $1$}^{\top} & \mbox{\boldmath $0$}^{\top}\\
\mbox{\boldmath $B$}_1 & \mbox{\boldmath $B$}_2
\end{array}
\right)
\left(
\begin{array}{c}
\mbox{\boldmath $w$}_a \\
\mbox{\boldmath $w$}'_a
\end{array}
\right)=
\left(
\begin{array}{c}
\mbox{\boldmath $1$}^{\top}\mbox{\boldmath $w$}_a \\
\mbox{\boldmath $B$}_1\mbox{\boldmath $w$}_a+\mbox{\boldmath $B$}_2\mbox{\boldmath $w$}'_a
\end{array}
\right),
$$
where $\mbox{\boldmath $1$}^{\top}$ is a row ${n-1\choose k-1}$-vector, $\mbox{\boldmath $0$}$ is a row ${n-1\choose k}$-vector, $\mbox{\boldmath $B$}_1=\mbox{\boldmath $B$}(n-1,k-1)^{\top}$, and $\mbox{\boldmath $B$}_2=\mbox{\boldmath $B$}(n-1,k)^{\top}$.
Then, $\mbox{\boldmath $1$}^{\top}\mbox{\boldmath $w$}_a=0$, so that $\mbox{\boldmath $w$}_a$ is an embedding of $H_a$. Furthermore, since $\mbox{\boldmath $v$}$ is an embedding of $G$, we have $\mbox{\boldmath $1$}^{\top}\mbox{\boldmath $v$}=\mbox{\boldmath $1$}^{\top}\mbox{\boldmath $w$}_a+\mbox{\boldmath $1$}^{\top}\mbox{\boldmath $w$}'_a=0$ (with the appropriate dimensions of the all-1 vectors). Hence, it must be $\mbox{\boldmath $1$}^{\top}\mbox{\boldmath $w$}'_a =0$, and $\mbox{\boldmath $w$}'_a$ is an embedding of $H'_a$.
\end{proof}
Now, we introduce some new results related to Conjecture \ref{conjecture}.
\begin{theorem}
\label{theo:alg-connec}
For each of the following classes of graphs, the algebraic connectivity of a token graph $F_k(G)$ satisfies the following.
\begin{itemize}
\item[$(i)$]
Let $T_n$ be a tree on $n$ vertices. Then,
$\alpha(F_k(T_n))=\alpha(T_n)$ for every $n$ and $k=1,\ldots,n-1$.
\item[$(ii)$]
Let $G$ be a graph such that $\alpha(F_k(G))=\alpha(G)$. Let $T_G$ be a graph where each vertex of $G$ is the root vertex of some (possibly empty) tree. Then
$\alpha(F_k(T_G))=\alpha(T_G)$.
\end{itemize}
\end{theorem}
\begin{proof}
To prove $(i)$, let $V(T_n)=[n]$. From previous comments, we can assume that $T_n$ is connected. Then, the result is readily checked for $n\le 4$ and $k=1,2$. Now, we proceed by induction.
Suppose $n>4$ and $k>1$. To our aim, by Lemma \ref{coro:LkL1}$(ii)$, it suffices to show that if $\mbox{\boldmath $v$}$ with a given norm, say $\mbox{\boldmath $v$}^{\top}\mbox{\boldmath $v$}=1$, is an eigenvector of $F_k:=F_k(T_n)$, with $\mbox{\boldmath $B$}^{\top}\mbox{\boldmath $v$}=\mbox{\boldmath $0$}$, then $\lambda(\mbox{\boldmath $v$})\geq \alpha(T_n)$.
Let $i\in [n]$. As defined before, let $S_i:=\{A\in V(F_k):i\in A \}$ and $S'_i:=\{B\in V(F_k):i\not\in B\}$.
Let $H_i$ and $H'_i$ be the subgraphs of $F_k$ induced by $S_i$ and $S'_i$, respectively. We have $H_i\cong F_{k-1}(T_{n-1})$ and $H'_i\cong F_k(T_{n-1})$, where $T_{n-1}=T\setminus i$. Moreover, note that if vertex $i$ is of degree $1$ in $T_n$, then $T_{n-1}$ is also connected. Let $\mbox{\boldmath $w$}_i:=\restr{\mbox{\boldmath $v$}}{S_i}$ and $\mbox{\boldmath $w$}'_i:=\restr{\mbox{\boldmath $v$}}{S'_i}$, by Lemma~\ref{lem:eigenvectors},
we know that $\mbox{\boldmath $w$}_i$ and $\mbox{\boldmath $w$}'_i$ are embeddings of $H_i$ and $H'_i$, respectively. By the induction hypothesis, we have
\[
\lambda(\mbox{\boldmath $w$}_i)=\frac{\sum\limits_{(A,B)\in E(H_i)} [\mbox{\boldmath $w$}_i(A)-\mbox{\boldmath $w$}_i(B)]^2}{\sum\limits_{A\in V(H_i)}\mbox{\boldmath $w$}_i(A)^2}\geq \alpha(T_{n-1}),
\]
and
\[
\lambda(\mbox{\boldmath $w$}'_i)=\frac{\sum\limits_{(A,B)\in E(H'_i)}[\mbox{\boldmath $w$}'_i(A)-\mbox{\boldmath $w$}'_i(B)]^2}{\sum\limits_{A\in V(H'_i)}\mbox{\boldmath $w$}'_i(A)^2}\geq \alpha(T_{n-1}).
\]
Since $V(H_i)\cup V(H'_i)=V(F_k)$ and $\mbox{\boldmath $v$}^{\top}\mbox{\boldmath $v$}=1$, we have
\begin{align}
\lambda(\mbox{\boldmath $v$})&=\sum\limits_{(A,B)\in E(F_k)}[\mbox{\boldmath $v$}(A)-\mbox{\boldmath $v$}(B)]^2 \nonumber\\
& \geq \sum\limits_{(A,B)\in E(H_i)}[\mbox{\boldmath $w$}_i(A)-\mbox{\boldmath $w$}_i(B)]^2 + \sum\limits_{(A,B)\in E(H'_i)}[\mbox{\boldmath $w$}'_i(A)-\mbox{\boldmath $w$}'_i(B)]^2 \nonumber\\
& \geq \alpha(T_{n-1})\Big[\sum\limits_{A\in V(H_i)}\mbox{\boldmath $w$}_i(A)^2 + \sum\limits_{B\in V(H'_i)}\mbox{\boldmath $w$}'_i(B)^2\Big] \nonumber \\
& = \alpha(T_{n-1})\Big[ \sum\limits_{A\in V(H_i)}\mbox{\boldmath $v$}(A)^2 + \sum\limits_{B\in V(H'_i)}\mbox{\boldmath $v$}(B)^2\Big] \nonumber \\
& = \alpha(T_{n-1}) > \alpha(T_n), \label{eq:paths-1}
\end{align}
where (\ref{eq:paths-1}) follows from Lemma \ref{lem:-vertex}. (Notice that, since $i$ has degree $1$, collapsing the edge of which $i$ is an end-vertex is equivalent to removing $i$, so obtaining $T_{n-1}$.)
Furthermore, since $\lambda(\mbox{\boldmath $v$})>\alpha(T_n)$, we get that $\alpha(T_n)$ is an eigenvalue of
both $T_n$ and $F_k(T_n)$ with the same multiplicity.
Regarding $(ii)$, it could be seen as a generalization of $(i)$. Thus, it is proved in the same way by induction on the number of vertices not in $G$ (that is, the non-root vertices of the trees), and starting from $G$.
(The other way around, proved $(ii)$, the result in $(i)$ is a corollary when we start with $G=K_1$ or $G=K_2$.)
\end{proof}
The last step in (\ref{eq:paths-1}) also can be seen as a consequence of the following theorem
by Patra and Lal \cite[Th. 3.1]{pl08}.
\begin{theorem}[\cite{pl08}]
\label{th:pl08}
Let $e=(u, v)$ be an edge of a tree $T$. Let $\widetilde{T}$ be the tree obtained from $T$ by `collapsing' the edge $e$ (that is, deleting $e$ and identifying $u$ and $v$). Then $\alpha(\widetilde{T})\ge \alpha(T)$.
\end{theorem}
Note
that the result of Theorem \ref{theo:alg-connec}$(i)$ implies the ones of Theorem \ref{theo:alg-connec-antic}$(ii)$ and $(iii)$.
\begin{theorem}
\label{theo:Delta}
Let $G$ be a graph on $n$ vertices satisfying $\alpha(F_{k-1}(G))=\alpha(G)$ and maximum degree
\begin{equation}
\label{boundDelta}
\Delta(G)\ge \phi(k)=\frac{k(n+k-3)}{2k-1}
\end{equation}
for some integer $k=1,\ldots, \lfloor n/2\rfloor$. Then, the algebraic connectivity of its $k$-token graph equals the algebraic connectivity of $G$,
$$
\alpha(F_k(G))=\alpha(G).
$$
\end{theorem}
\begin{proof}
When we `go' from the spectra of $\{F_{k-1}(G)$, $F_{k-1}(\overline{G})\}$ to the spectra of $\{F_{k}(G)$, $F_{k}(\overline{G})\}$,
all the eigenvalues of $\Lambda_0,\ldots,\Lambda_{k-1}$ `reappear' (with eigenvectors $\mbox{\boldmath $v$}$ such that $\mbox{\boldmath $B$}^{\top}\mbox{\boldmath $v$}\neq \mbox{\boldmath $0$}$),
together with `new' eigenvalues belonging to $\Lambda_k$ (with eigenvectors $\mbox{\boldmath $v$}$ such that $\mbox{\boldmath $B$}^{\top}\mbox{\boldmath $v$}= \mbox{\boldmath $0$}$). Moreover, the hypothesis $\alpha(F_{k-1}(G))=\alpha(G)$ implies that, in $\spec F_{k}(G)$, all
the eigenvalues of $F_k(G)$ that are in $\Lambda_1,\ldots,\Lambda_{k-1}$ must be greater than or equal to $\alpha(G)$.
Reasoning by contradiction, if $\alpha(F_k(G))<\alpha(G)$, then the eigenvalue $\alpha(F_k(G))$ must belong to
$\Lambda_k$. Then, the eigenvalue $\lambda_{F_k(G)}=\alpha(F_k(G))$ must be paired with one eigenvalue $\lambda_{F_k(\overline{G})}$ of
$F_k(\overline{G})$ belonging also to $\Lambda_k$ (both eigenvalues sharing the same eigenvector $\mbox{\boldmath $v$}$ with $J(n,k)$), so that
$$
\alpha(G)+\lambda_{F_k(\overline{G})}>\alpha(F_k(G))+\lambda_{F_k(\overline{G})}=k(n-k+1)
$$
Thus,
using that $\alpha(G)=n-\lambda_{\max}(\overline{G})$,
$$
\lambda_{\max}(F_k(\overline{G}))\ge \lambda_{F_k(\overline{G})}> k(n-k+1)-\alpha(G)=k(n-k+1)-n+\lambda_{\max}(\overline{G}).
$$
However, from the upper and lower bounds in \eqref{bound-sp-rad} for the spectral radius of a graph, together with \eqref{DeltaFk}, we get
$$
2k\Delta(\overline{G})\ge \lambda_{\max}(F_k(\overline{G}))>k(n-k+1)-n+\lambda_{\max}(\overline{G})\ge (k-1)(n-k)+\Delta(\overline{G})+1,
$$
or, in terms of $\Delta(G)$,
$$
n-1-\Delta(G)=\Delta(\overline{G})>\frac{(k-1)(n-k)+1}{2k-1}.
$$
Hence, $\Delta(G)<n-1-\frac{(k-1)(n-k)+1}{2k-1}=\frac{k(n+k-3)}{2k-1}$, contradicting the hypothesis.
\end{proof}
For the two extreme cases $k=2$ and $k=n/2$, we get the following consequences.
\begin{corollary}
Let $G$ be a graph on $n$ vertices and maximum degree $\Delta(G)$.
\begin{itemize}
\item[$(i)$]
If $\Delta(G)\ge \frac{2}{3}(n-1)$, then $\alpha(F_2(G))=\alpha(G)$.
\item[$(ii)$]
If $\Delta(G)\ge \frac{3}{4}n$, then $G$ satisfies the Conjecture \ref{conjecture}. That is,
$\alpha(F_k(G))=\alpha(G)$ for every $k=1,\ldots,n-1$.
\end{itemize}
\end{corollary}
\begin{proof}
$(i)$ With $k=2$, the condition \eqref{boundDelta} becomes $\Delta(G)\ge \frac{3}{2}(n-1)$. Then, since $\alpha(F_1(G))=\alpha(G)$, Theorem \ref{theo:Delta} gives the result.
$(ii)$ Assuming that $n$ is even (the odd case is similar), it is enough to prove the result for $k=n/2$. In this case, the condition \eqref{boundDelta} becomes
$\Delta(G)\ge \phi(n/2)=\frac{n(3n-6)}{4(n-1)}$. It is readily checked that $\frac{3}{4}n>\phi(n/2)>\phi(k)$ for every $k=2,\ldots,\frac{n}{2}-1$. So, we can use induction from the
case $(i)$ to prove the hypotheses in Theorem \ref{theo:Delta} hold for every $k$.
\end{proof}
Some examples of known graphs satisfying Conjecture \ref{conjecture} are:
\begin{itemize}
\item
With maximum degree $n-1$, the complete and the star graphs (already mentioned), and the wheel graphs.
\item
With degree $n-2$, the cocktail party graph (obtained from the complete graph with even number of vertices minus a matching).
\item
With degree $n-3$, the complement $\overline{C_n}$ of the cycle with $n\ge 12$ vertices.
\item
The complete $r$-partite graph $G=K_{n_1,n_2,\ldots,n_r}\ne K_r$ for $r>2$, with number of vertices $n=n_1+n_2+\cdots +n_r$, for $n_1\le n_2\le \cdots \le n_r$ and $n\ge 4n_r$.
\end{itemize}
\section*{Acknowledgments} The authors are grateful to Clemens Huemer for his valuable comments.
\end{document} | arXiv |
\begin{document}
\title{Supporting Information for ``A synthetic data integration framework to leverage external summary-level information from heterogeneous populations"}
\linespread{2}
\section{Baseline Characteristics of High-Grade Prostate Cancer Datasets}
\begin{table}[H] \centering \begin{adjustbox}{width=1\textwidth} \begin{tabular}{llllll} \hline
& & \textbf{Internal data} & \textbf{Validation data} & \textbf{PCPThg} & \textbf{ERSPC} \\ \hline \multicolumn{2}{l}{Sample size} & 678 & 1174 & 5519 & 3616 \\ \multicolumn{2}{l}{Number of case (prevalence)} & 179 (26.4\%) & 214 (18.2\%) & 257 (4.7\%) & 313 (35.4) \\ \hline Median (range) & PSA (ng/mL) & 5.1 (0.3-460.4) & 4.6 (0.1-290.0) & 1.5 (0.3–287.0) & 4.3 (0.1–316.0) \\ \hline \multirow{3}{*}{Mean (SD)} & Age (years) & 62.6 (8.4) & 64.0 (8.9) & 69.7 (Unknown) & 65.5 (Unknown) \\ & PCA3 & 38.2 (40.3) & 45.2 (65.7) & Unknown & Unknown \\ & T2:ERG & 77.7 (325.9) & 72.6 (365.0) & Unknown & Unknown \\ \hline \multirow{3}{*}{Count (\%)} & Abnormal DRE & 96 (14.2) & 283 (24.1) & Unknown & 1280 (35.4) \\ & Prior biopsy & 167 (24.6) & 235 (20.0) & 753 (13.6) & Unknown \\ & African American & 70 (10.3) & 68 (5.8) & 175 (3.2) & Unknown \\ \hline \end{tabular} \end{adjustbox} \end{table}
\section{Assumptions of the proposed approach}
To implement the multiple imputation in step 2, some covariate information need to be shared across populations, since the missing covariates are completely unobserved in one population, also known as block-wise missing structure. Specifically, assumption 2 in Figure 3 in the main text states that the conditional distribution of the missing covariates to be the same across populations given the observed covariates (i.e., $\rm \mathbf{X}_{miss} \perp\!\!\!\perp [I_0,I_1,...,I_K] | \mathbf{X}_{obs}$; and $\rm \mathbf{B} \perp\!\!\!\perp [I_0,I_1,...,I_K]|\mathbf{X}$). Therefore, the imputation models are $\rm f(X_{miss}|\mathbf{X}_{obs})$ and $\rm f(B|\mathbf{X})$ for missing $\rm \mathbf{X}$ and missing $\rm \mathbf{B}$, respectively (e.g., $\rm \mathbf{X}_{miss}=[X_2, X_3]$ and $\rm X_{obs}=X_1$ in Figure 1 in the main text).
\section{Deriving the Initial Estimates for the External Populations}
In this section, we will show how to obtain the initial parameter estimates of external population k. Let $\rm (\hat{\gamma}_0, \boldsymbol{\rm \hat{\gamma}}_X^{T}, \boldsymbol{\rm \hat{\gamma}}_B^{T})^T$ be the internal data only estimates of $\rm Y|\mathbf{X, B}, I_0$ using internal data only. For external population k, we know the parameter estimates $\rm \hat{\boldsymbol{\beta}}_k=(\hat{\beta}_0, \boldsymbol{\rm \hat{\beta}}_{X_k}^T)^T$ from the fitted model $\rm Y|\mathbf{X}_k; \boldsymbol{\beta}_k$. We assume that all predictors, $\rm \mathbf{X}$ and $\rm \mathbf{B}$, are centered, and the true target model parameter for the external population k is $\rm (\gamma_0^{k}, \boldsymbol{\rm \gamma}_X^{kT}, \boldsymbol{\rm \gamma}_B^{T})^T$, assuming the coefficient of the unobserved variable $\rm \mathbf{B}$ is the same as the internal population.
The goal of estimating $\rm \gamma_0^{k}$ and $\rm \boldsymbol{\rm \gamma}_X^{k}$ from model $\rm Y|\mathbf{X, B}, I_k; \boldsymbol{\gamma}_{I_k}$ is equivalent to correcting the bias of $\rm \hat{\boldsymbol{\beta}}_k$ in the reduced model $\rm Y|\mathbf{X}_k; \boldsymbol{\beta}_k$ considering covariates $\rm \mathbf{X}_{(-k)}$ and $\rm \mathbf{B}$ as omitted. To simplify notation, we assume $\rm \mathbf{B}$ is the only omitted covariate in the derivation below. Neuhaus and Jewell (1993) provided a Taylor-series-expansion approximation to show that the ratio of coefficients remains constant in both the reduced and the full model when the omitted $\rm \mathbf{B}$ is independent of the observed $\rm \mathbf{X}$, i.e. $\rm \frac{\gamma_{X_1}}{\gamma_{X_2}} \approx \frac{\beta_{X_1}}{\beta_{X_2}}$, indicating that the relative effect size among regression coefficients remains consistent across models. In their Table 3 and equation 9 [1] provided the algebraic relationship between $\rm \boldsymbol{\rm \gamma}_X^{k}$ and $\rm \boldsymbol{\rm \beta}_X$ for exponential family when the omitted $\mathbf{B}$ and the observed $\mathbf{X}$ are correlated. In the subsequent paragraphs, we will explain in detail how to estimate $\rm \gamma_0^{k}$ and $\rm \boldsymbol{\rm \gamma}_X^{k}$ in linear regression (continuous Y) and logistic regression (binary Y), respectively.
\textbf{1. Linear Regression:} Suppose E($\rm \mathbf{B} | \rm \mathbf{X}; \boldsymbol\theta$)= $\rm \boldsymbol{\rm \theta X}$. We start by replacing $\rm \mathbf{B}$ with the conditional expected value E($\rm \mathbf{B} | \rm \mathbf{X}; \boldsymbol\theta$) in the mean profile of the target model: \begin{align*}
\rm E(Y|\mathbf{X, B}; \boldsymbol\gamma) &= \rm \gamma_0^{k} + \boldsymbol{\rm \gamma}_X^{k} \mathbf{X} + \boldsymbol{\rm \gamma}_B\mathbf{B}
= \rm \gamma_0^{k} + \boldsymbol{\rm \gamma}_X^{k} \mathbf{X} + \boldsymbol{\rm \gamma}_B\boldsymbol{\theta} \mathbf{X}
= \rm E(Y|\mathbf{X}; \boldsymbol{\gamma}, \boldsymbol{\theta}) \end{align*}
Since $\rm \hat{E}(Y|\mathbf{X}; \boldsymbol{\beta}) = \hat{\beta}_0 + \boldsymbol{\rm \hat{\beta}}_X \mathbf{X}$ is available through the externally fitted model, we can obtain the estimation of $\rm \gamma_0^{k}$ and $\rm \boldsymbol{\rm \gamma}_X^{k}$ by matching the intercept and $\rm \mathbf{X}$ coefficient between $\rm \hat{E}(Y|\mathbf{X};\boldsymbol{\gamma}, \boldsymbol{\theta})$ and $\rm \hat{E}(Y|\mathbf{X}; \boldsymbol{\beta})$, respectively: $\rm \hat{\gamma}_0^{k} = \hat{\beta}_0$ and $\rm \boldsymbol{\rm \hat{\gamma}}_X^{k} = \boldsymbol{\rm \hat{\beta}}_{X_k} - \boldsymbol{\rm \theta}^T\boldsymbol{\rm \hat{\gamma}}_B $. In a special case where the internal and the external population only differ in intercept, we can directly obtain the initial estimates $\rm \hat{\boldsymbol{\rm \gamma}}_{I_k}=(\rm \hat{\beta}_0, \hat{\boldsymbol{\rm \gamma}}_X^{T}, \hat{\boldsymbol{\rm \gamma}}_B^{T})^T$.
\textbf{2. Logistic Regression:}
In logistic regression where g() is the logit link function, we connect the intercepts $\rm \beta_0$ and $\rm \gamma_0^{k}$ through the equation $\rm logit^{-1}(\beta_0) = E_{B|X}(\mu_0^{k})$, where $\rm \mu_{0}^{k}=g^{-1}(Y|\mathbf{X}, \mathbf{B}; \boldsymbol{\rm\gamma}_X^{k}=0)=logit^{-1}(\gamma_0^{k}+\mathbf{B}^T\boldsymbol{\rm \gamma}_B)$. For the right hand side, we expand $\rm \mathbf{B}$, a vector of length Q, at $\rm E(\mathbf{B|X})$ using the third-order Taylor series expansion as follows:
\begin{align*}
\rm E_{B|X}(\mu_0^{k}) &= \rm E_{B|X}[logit^{-1}(\gamma_0^{k}+\mathbf{B}^T\boldsymbol{\rm \gamma}_B)]
\\
&\approx \rm logit^{-1}(w) \Big\{1+\frac{1}{2}\frac{1-e^{w}}{(1+e^{w})^2}\sum_{i=1}^Q\sum_{j=1}^Q\gamma_{B_i}\gamma_{B_j} E_{B|X}\Big[\Big(B_i-E(B_i|\mathbf{X})\Big)\Big(B_j-E(B_j|\mathbf{X})\Big)\Big]\Big\}
\\
&= \rm logit^{-1}(w) \Big[1+\frac{1}{2}\frac{1-e^w}{(1+e^w)^2}Var\Big(\sum_{i=1}^Q\gamma_{B_i}B_i|\mathbf{X}\Big)\Big] \numberthis \label{taylor_expansion_logit-1} \end{align*}
where $\rm w = \gamma_0^{k}+E(\mathbf{B}^T|\mathbf{X})\boldsymbol{\rm \gamma}_B$. Given $\rm \hat{\beta}_0, \boldsymbol{\rm \hat{\gamma}}_B, \hat{E}(\mathbf{B|X})$ and $\rm \hat{V}ar(\mathbf{B|X})$, we can easily obtain $\rm \hat{\gamma}_0^{k}$ by solving the equation $\rm E_{B|X}(\mu_0^{k}) - logit^{-1}(\hat{\beta}_0)=0$.
After obtaining $\rm \boldsymbol\gamma_0^{k}$, we then estimate $\rm \boldsymbol\gamma_X^{k}=(\gamma_{X_1}^{k},...,\gamma_{X_{P_k}}^{k})^T$ according to the following equation provided in Neuhaus and Jewell (1993):
\setlength{\abovedisplayskip}{0pt} \setlength{\abovedisplayshortskip}{0pt} \begin{equation*}
\rm \boldsymbol{\beta}_{X_p} = \rm \Big\{\boldsymbol{\gamma}_{X_p}^{k}+\Big[E(\mathbf{B}^T|\mathbf{X}+1_p)-E(\mathbf{B}^T|\mathbf{X})\Big]\boldsymbol{\gamma}_B \Big\}\Big\{1-\frac{Var_{B|X}(\mu_0^{k})}{1-E_{B|X}(\mu_0^{k})[1-E_{B|X}(\mu_0^{k})]}\Big\} \end{equation*}
where $\rm 1_p$ is a zero vector with the $\rm p^{th}$ term equals to 1 and $\rm p \in \{1,...,P_k\}$. Similar to equation (\ref{taylor_expansion_logit-1}), we can also obtain the Taylor-series-expansion estimation for $\rm E_{B|X}[(\mu_0^{k})^2]=E_{B|X}[logit^{-2}(\gamma_0^{k}+\boldsymbol{\rm B}^T\boldsymbol{\rm \gamma}_B)] \approx \rm \frac{e^{2w}}{(1+e^w)2} \Big[1+\frac{1}{2}\frac{2-e^w}{(1+e^2)^2}\sum_{i=1}^Q\sum_{j=1}^Q \gamma_{B_i} \gamma_{B_j} Cov(B_i,B_j|\mathbf{X})\Big]$, together with $\rm E_{B|X}(\mu_0^{k})$, we then obtain an approximation of $\rm V_{B|X}(\mu_0^{k})=E_{B|X}[(\mu_0^{k})^2]-E_{B|X}(\mu_0^{k})$. Given $\rm \boldsymbol{\rm \hat{\beta}}_X$, $\rm \boldsymbol{\rm \hat{\gamma}}_B$, $\rm \hat{E}(\mathbf{B|X})$, and $\rm \hat{\gamma}_0^{k}$, we can obtain $\rm \boldsymbol{\rm \hat{\gamma}}_X^{k}=\boldsymbol{\rm \hat{\beta}}_X(1-\frac{\hat{V}_{B|X}(\mu_0^{k})}{\hat{E}_{B|X}(\mu_0^{k})[1-\hat{E}_{B|X}(\mu_0^{k})]})^{-1}-\Big[\hat{E}(\mathbf{B}^T|\mathbf{X}+1_p)-\hat{E}(\mathbf{B}^T|\mathbf{X})\Big]\hat{\boldsymbol{\gamma}}_B$.
Note that we estimate $\rm E(\mathbf{B|X};\boldsymbol{\theta})=\rm g'^{-1}(\boldsymbol\theta\mathbf{X})$ and $\rm Var(\mathbf{B|X};\boldsymbol{\theta})=g'^{-1}(\boldsymbol\theta\mathbf{X})\Big[1-g'^{-1}(\boldsymbol\theta\mathbf{X})\Big]$ using the internal data by regressing each B on $\rm \mathbf{X}$ with appropriate link function $\rm g'()$ based on the type of B, e.g., when B is continuous, linear regression and identity link is used; when B is binary, logistic regression and logit link is used. Given $\rm \hat{\boldsymbol{\theta}}$, $\rm \hat{E}(\mathbf{B|X})=\hat{\boldsymbol{\theta}}^T E(\mathbf{X})$ is used.
\section{Additional Simulation Results} In this section, we show the results of additional simulations to assess the performance of the proposed strategy for point estimates and variance estimation.
\subsection{Continuous outcome Y (a supplement to Simulation I in the main manuscript)} \textbf{Goal:} To examine the proposed method when the outcome is continuous and the target model is linear regression. \\ \noindent \textbf{Simulation setup:} This simulation is the same as Simulation I in the main manuscript, except the generative outcome model now follows Gaussian distribution: \begin{align*}
\begin{cases}
\text{Internal: } &\rm Y|\mathbf{X, B} \sim N(-1 - X_1 - X_2 - B_1 - B_2, 1);\\
\text{External 1: } &\rm Y|\mathbf{X, B} \sim N(1 - X_1 - X_2 - B_1 - B_2, 1);\\
\text{External 2: } &\rm Y|\mathbf{X, B} \sim N(3 - X_1 - X_2 - B_1 - B_2, 1).
\end{cases}
\end{align*} The target outcome model (model 2 in the main manuscript) is now a linear regression: \begin{equation*}
\rm E(Y|\mathbf{X, B, S}) = \rm \gamma_{0} + \sum_{k=1}^2 \gamma_0^{S_k}S_k + \sum_{p=1}^2 \gamma_{X_p} X_p + \sum_{q=1}^2 \gamma_{B_q} B_q, \end{equation*}
\noindent \textbf{Results:} Figure \ref{simS1} shows similar pattern as those in Simulation I in the main manuscript, where the proposed method (red dotted curve) has the smallest bias among all for all covariates (Figure \ref{simS1_A}), largest precision gain compared with others (Figure \ref{simS1_B}), and the closest variance estimation to the Monte Carlo empirical variance (Figure \ref{simS1_C}).
\begin{figure}
\caption{Point estimates}
\label{simS1_A}
\caption{Variance estimator vs. the empirical variance}
\label{simS1_B}
\caption{Different variance estimators of the proposed method}
\label{simS1_C}
\caption{Results of Simulation 1.1 over increasing synthetic data size (a) point estimates (b)variance estimation vs. Monte Carlo empirical variance (c) different variance estimators of the proposed method.}
\label{simS1}
\end{figure}
\subsection{Smaller covariate effect (a modification to Simulation I in the main manuscript)} \textbf{Goal:} To assess our approach when the magnitude and the difference of covariate effects are small across different populations in the target outcome model. \\ \noindent \textbf{Simulation setup:} This simulation is the same as Simulation I in the main manuscript, except the coefficient effect is now -0.5 instead of -1, and the intercept difference is smaller among populations: \begin{align*}
\begin{cases}
\text{Internal: } &\rm logit[Pr(Y=1|\mathbf{X, B})] = -1 - 0.5\big(X_1 + X_2 + B_1 + B_2\big), \text{ prevalence=0.28};\\
\text{External 1: } &\rm logit[Pr(Y=1|\mathbf{X, B})] = -0.5 - 0.5\big(X_1 + X_2 + B_1 + B_2\big), \text{ prevalence=0.36};\\
\text{External 2: } &\rm logit[Pr(Y=1|\mathbf{X, B})] = 0 - 0.5\big(X_1 + X_2 + B_1 + B_2\big), \text{ prevalence=0.45}.
\end{cases}
\end{align*} \\ \noindent \textbf{Results:} Figure \ref{simS2_A} shows that compared with larger covariate effects in Simulation I in the main manuscript, when the X covariate effect is small, FCS and IMB have smaller bias in estimating X coefficients but still lack the ability to identify population-specific effects (i.e. intercepts of external populations). Similarly, Figure \ref{simS2_B} shows smaller bias of variance estimation. Note that the Rubin's rule variance estimator in Figure \ref{simS2_B} is too large (the pink curve) so that it falls outside of the range of the figure.
\begin{figure}
\caption{Point estimates}
\label{simS2_A}
\caption{Different variance estimators of the proposed method}
\label{simS2_B}
\caption{Results of Simulation 1.2 over increasing synthetic data size (a) point estimates (b) different variance estimators of the proposed method.}
\label{simS2}
\end{figure}
\subsection{Different X covariate effects in the outcome model (a more flexible outcome model compared with Simulation I in the main manuscript)} \textbf{Goal:} In the main manuscript, we only present the simulation results allowing the target model's intercept to differ across populations. In this simulation, we additionally show the performance of the proposed method when all possible X covariates coefficients are allowed to differ across populations (similar to model 1 or ``different intercept and covariates" model in the real data example in the main manuscript).
\noindent \textbf{Simulation setup:} This simulation is the same as Simulation I in the main manuscript except now that the generative outcome models are as follows: \begin{align*}
\begin{cases}
\text{Internal: } &\rm logit[Pr(Y=1|\mathbf{X, B})] = -1 - X_1 - X_2 - B_1 - B_2, \text{ prevelance= 0.3};\\
\text{External 1: } &\rm logit[Pr(Y=1|\mathbf{X, B})] = 1 + X_1 - X_2 - B_1 - B_2, \text{ prevelance= 0.58};\\
\text{External 2: } &\rm logit[Pr(Y=1|\mathbf{X, B})] = 3 + 3X_1 + 3X_2 - B_1 - B_2,
\text{ prevelance= 0.70}.
\end{cases}
\end{align*}
\noindent \textbf{Results:} Similar to the results of Simulation I in the main manuscript, the results in Figure \ref{simS3} shows outstanding performance of the proposed method in both point estimates and variance estimation compared with others. For example, the proposed method has small bias less than 0.02 when estimating $\rm X_2 $ in population S=1 while the bias in FCS and IMB can go up to 0.78 (i.e. almost 40 times of the proposed method).
\begin{figure}
\caption{Point estimates}
\label{simS3_A}
\caption{Different variance estimators of the proposed method}
\label{simS3_B}
\caption{Results of Simulation 1.3 over increasing synthetic data size (a) point estimates (b) different variance estimators of the proposed method.}
\label{simS3}
\end{figure}
\subsection{Violation of transportability assumption}
\textbf{Goal:} To examine the proposed method when Assumption 2 ($\rm X_{miss}|X_{obs}$ and $\rm B|X$ are transportable between the internal and the external populations) is violated. We present two examples where the violation only causes ignorable bias in case 1 while it has larger impact in case 2.
\subsubsection{\textbf{Case 1: different $\rm B|X$ distribution in external population 2}} \textbf{Simulation setup:} This simulation is the same as Simulation I in the main manuscript except that now the external model 2 has different marginal $\rm B_1$ distribution and different conditional distribution $\rm B_2|X_1, X_2, B_1$: \begin{itemize}
\item $\rm B_1$ has mean 1.5 and standard deviation 1.5 in external population 2 while in other populations $\rm B_1$ has mean 0 and standard deviation 1;
\item $\rm B_2|\mathbf{X},B_1 \sim Ber\{[1+exp^{-1}(0.2X_1+0.3X_2+0.4B_1)]\}$ in external population 2 while in other populations $\rm B_2|\mathbf{X},B_1 \sim Ber\{[1+exp^{-1}(0.1X_1+0.2X_2+0.3B_1)]\}$. \end{itemize}
Note that both $\rm B_1$ and $\rm B_2$ are only observed in the internal study and multiple imputations are needed for them, where $\rm B_2|\mathbf{X}$ and $\rm B_2|\mathbf{X},B_1$ should be the same across populations according to Assumption 2.
\noindent \textbf{Results:} Figure \ref{simS4.1_A} indicates that the violation of transportability assumption in the proposed method has limited impact of point estimation with ignorable bias while Figure \ref{simS4.1_B} shows similar pattern of variance estimations as before.
\begin{figure}
\caption{Point estimates}
\label{simS4.1_A}
\caption{Different variance estimators of the proposed method}
\label{simS4.1_B}
\caption{Results of Simulation 1.4.1 over increasing synthetic data size (a) point estimates (b) different variance estimators of the proposed method.}
\label{simS4.1}
\end{figure}
\subsubsection{\textbf{Case 2: different marginal $\rm X_1$ distribution in external populations}} \textbf{Simulation setup:} This simulation is the same as Simulation I in the main manuscript except now that in the external studies, $\rm X_1 \sim N(1, 1.5)$ while in the internal study $\rm X_1 \sim N(0, 1)$. This will lead to different distribution conditional on $\rm X_1$ and thus violates Assumption 2.
\noindent \textbf{Results:} Figure \ref{simS4.2_A} shows that such violation leads to some bias of estimated coefficient $\rm X_1$, i.e., 0.2 absolute bias. Besides that, the proposed method has nearly unbiased point estimates for other parameter (i.e. up to 0.014 absolute bias) while the bias in FCS and IMB can be up to 15 times the bias of the proposed method. Similarly, Figure \ref{simS4.2_B} shown unbiased variance estimation of the proposed bootstrap estimator.
\begin{figure}
\caption{Point estimates}
\label{simS4.2_A}
\caption{Different variance estimators of the proposed method}
\label{simS4.2_B}
\caption{Results of Simulation 1.4.2 over increasing synthetic data size (a) point estimates (b) different variance estimators of the proposed method.}
\label{simS4.2}
\end{figure}
\end{document} | arXiv |
{{#invoke:Hatnote|hatnote}} {{ safesubst:#invoke:Unsubst||$N=Refimprove |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} In mathematics, the root mean square (abbreviated RMS or rms), also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity. It is especially useful when variates are positive and negative, e.g., sinusoids. In the field of electrical engineering, the RMS value of a periodic current is equal to the direct current (DC) that delivers the same average power to a resistor as the periodic current.[1]
It can be calculated for a series of discrete values or for a continuously varying function. Its name comes from its definition as the square root of the mean of the squares of the values. It is a special case of the generalized mean with the exponent p = 2.
2 RMS of common waveforms
3.1 Average electrical power
3.2 Root-mean-square speed
3.3 Root-mean-square error
4 RMS in frequency domain
5 Relationship to the arithmetic mean and the standard deviation
The RMS value of a set of values (or a continuous-time waveform) is the square root of the arithmetic mean of the squares of the original values (or the square of the function that defines the continuous waveform).
In the case of a set of n values { x 1 , x 2 , … , x n } {\displaystyle \{x_{1},x_{2},\dots ,x_{n}\}} , the RMS
x r m s = 1 n ( x 1 2 + x 2 2 + ⋯ + x n 2 ) . {\displaystyle x_{\mathrm {rms} }={\sqrt {{\frac {1}{n}}\left(x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}\right)}}.}
The corresponding formula for a continuous function (or waveform) f(t) defined over the interval T 1 ≤ t ≤ T 2 {\displaystyle T_{1}\leq t\leq T_{2}} is
f r m s = 1 T 2 − T 1 ∫ T 1 T 2 [ f ( t ) ] 2 d t , {\displaystyle f_{\mathrm {rms} }={\sqrt {{1 \over {T_{2}-T_{1}}}{\int _{T_{1}}^{T_{2}}{[f(t)]}^{2}\,dt}}},}
and the RMS for a function over all time is
f r m s = lim T → ∞ 1 T ∫ 0 T [ f ( t ) ] 2 d t . {\displaystyle f_{\mathrm {rms} }=\lim _{T\rightarrow \infty }{\sqrt {{1 \over {T}}{\int _{0}^{T}{[f(t)]}^{2}\,dt}}}.}
The RMS over all time of a periodic function is equal to the RMS of one period of the function. The RMS value of a continuous function or signal can be approximated by taking the RMS of a series of equally spaced samples. Additionally, the RMS value of various waveforms can also be determined without calculus, as shown by Cartwright.[2]
In the case of the RMS statistic of a random process, the expected value is used instead of the mean.
RMS of common waveforms
Sine, square, triangle, and sawtooth waveforms
A rectangular pulse wave of duty cycle D, the ratio between the pulse duration ( τ {\displaystyle \tau } ) and the period (T); illustrated here with a = 1
Graph of a sine wave's voltage vs. time (in degrees), showing RMS, peak, and peak-to-peak voltages
DC, constant y = a {\displaystyle y=a\,} a {\displaystyle a\,}
Sine wave y = a sin ( 2 π f t ) {\displaystyle y=a\sin(2\pi ft)\,} a 2 {\displaystyle {\frac {a}{\sqrt {2}}}}
Square wave y = { a { f t } < 0.5 − a { f t } > 0.5 {\displaystyle y={\begin{cases}a&\{ft\}<0.5\\-a&\{ft\}>0.5\end{cases}}} a {\displaystyle a\,}
DC-shifted square wave y = { a + D C { f t } < 0.5 − a + D C { f t } > 0.5 {\displaystyle y={\begin{cases}a+DC&\{ft\}<0.5\\-a+DC&\{ft\}>0.5\end{cases}}} D C 2 + a 2 {\displaystyle {\sqrt {DC^{2}+a^{2}}}\,}
Modified square wave y = { 0 { f t } < 0.25 a 0.25 < { f t } < 0.5 0 0.5 < { f t } < 0.75 − a { f t } > 0.75 {\displaystyle y={\begin{cases}0&\{ft\}<0.25\\a&0.25<\{ft\}<0.5\\0&0.5<\{ft\}<0.75\\-a&\{ft\}>0.75\end{cases}}} a 2 {\displaystyle {\frac {a}{\sqrt {2}}}}
Triangle wave y = | 2 a { f t } − a | {\displaystyle y=|2a\{ft\}-a\,|} a 3 {\displaystyle a \over {\sqrt {3}}}
Sawtooth wave y = 2 a { f t } − a {\displaystyle y=2a\{ft\}-a\,} a 3 {\displaystyle a \over {\sqrt {3}}}
Pulse train y = { a { f t } < D 0 { f t } > D {\displaystyle y={\begin{cases}a&\{ft\}<D\\0&\{ft\}>D\end{cases}}} a D {\displaystyle a{\sqrt {D}}}
Phase-to-phase voltage y = a sin ( t ) − a sin ( t − 2 π 3 ) {\displaystyle y=a\sin(t)-a\sin(t-{\frac {2\pi }{3}})\,} a 3 2 {\displaystyle a{\sqrt {\frac {3}{2}}}}
t is time
f is frequency
a is amplitude (peak value)
D is the duty cycle or the percent(%) spent high of the period (1/f)
{r} is the fractional part of r
Waveforms made by summing known simple waveforms have an RMS that is the root of the sum of squares of the component RMS values, if the component waveforms are orthogonal (that is, if the average of the product of one simple waveform with another is zero for all pairs other than a waveform times itself).
R M S T o t a l = R M S 1 2 + R M S 2 2 + ⋯ + R M S n 2 {\displaystyle RMS_{Total}={\sqrt {{RMS_{1}}^{2}+{RMS_{2}}^{2}+\cdots +{RMS_{n}}^{2}}}}
A special case of this, particularly helpful in electrical engineering, is
R M S T o t a l = R M S D C 2 + R M S A C 2 {\displaystyle RMS_{Total}={\sqrt {{RMS_{DC}}^{2}+{RMS_{AC}}^{2}}}}
where R M S D C {\displaystyle RMS_{DC}} refers to the DC component of the signal and R M S A C {\displaystyle RMS_{AC}} is the AC component of the signal.
The RMS value of a function is often used in physics and electrical engineering.
Average electrical power
{{#invoke:main|main}}
Electrical engineers often need to know the power, P, dissipated by an electrical resistance, R. It is easy to do the calculation when there is a constant current, I, through the resistance. For a load of R ohms, power is defined simply as:
P = I 2 R . {\displaystyle P=I^{2}R.}
However, if the current is a time-varying function, I(t), this formula must be extended to reflect the fact that the current (and thus the instantaneous power) is varying over time. If the function is periodic (such as household AC power), it is still meaningful to discuss the average power dissipated over time, which is calculated by taking the average power dissipation:
P a v g {\displaystyle P_{\mathrm {avg} }\,\!} = ⟨ I ( t ) 2 R ⟩ {\displaystyle =\langle I(t)^{2}R\rangle \,\!} (where ⟨ … ⟩ {\displaystyle \langle \ldots \rangle } denotes the mean of a function)
= R ⟨ I ( t ) 2 ⟩ {\displaystyle =R\langle I(t)^{2}\rangle \,\!} (as R does not vary over time, it can be factored out)
= ( I R M S ) 2 R {\displaystyle =(I_{\mathrm {RMS} })^{2}R\,\!} (by definition of RMS)
So, the RMS value, IRMS, of the function I(t) is the constant current that yields the same power dissipation as the time-averaged power dissipation of the current I(t).
Average power can also be found using the same method that in the case of a time-varying voltage, V(t), with RMS value VRMS,
P a v g = ( V R M S ) 2 R . {\displaystyle P_{\mathrm {avg} }={(V_{\mathrm {RMS} })^{2} \over R}.}
This equation can be used for any periodic waveform, such as a sinusoidal or sawtooth waveform, allowing us to calculate the mean power delivered into a specified load.
By taking the square root of both these equations and multiplying them together, the power is found to be:
P a v g = V R M S I R M S . {\displaystyle P_{\mathrm {avg} }=V_{\mathrm {RMS} }I_{\mathrm {RMS} }.}
Both derivations depend on voltage and current being proportional (i.e., the load, R, is purely resistive). Reactive loads (i.e., loads capable of not just dissipating energy but also storing it) are discussed under the topic of AC power.
In the common case of alternating current when I(t) is a sinusoidal current, as is approximately true for mains power, the RMS value is easy to calculate from the continuous case equation above. If Ip is defined to be the peak current, then:
I R M S = 1 T 2 − T 1 ∫ T 1 T 2 ( I p sin ( ω t ) ) 2 d t . {\displaystyle I_{\mathrm {RMS} }={\sqrt {{1 \over {T_{2}-T_{1}}}{\int _{T_{1}}^{T_{2}}{(I_{\mathrm {p} }\sin(\omega t)}\,})^{2}dt}}.\,\!}
where t is time and ω is the angular frequency (ω = 2π/T, where T is the period of the wave).
Since Ip is a positive constant:
I R M S = I p 1 T 2 − T 1 ∫ T 1 T 2 sin 2 ( ω t ) d t . {\displaystyle I_{\mathrm {RMS} }=I_{\mathrm {p} }{\sqrt {{1 \over {T_{2}-T_{1}}}{\int _{T_{1}}^{T_{2}}{\sin ^{2}(\omega t)}\,dt}}}.}
Using a trigonometric identity to eliminate squaring of trig function:
I R M S = I p 1 T 2 − T 1 ∫ T 1 T 2 1 − cos ( 2 ω t ) 2 d t {\displaystyle I_{\mathrm {RMS} }=I_{\mathrm {p} }{\sqrt {{1 \over {T_{2}-T_{1}}}{\int _{T_{1}}^{T_{2}}{1-\cos(2\omega t) \over 2}\,dt}}}}
I R M S = I p 1 T 2 − T 1 [ t 2 − sin ( 2 ω t ) 4 ω ] T 1 T 2 {\displaystyle I_{\mathrm {RMS} }=I_{\mathrm {p} }{\sqrt {{1 \over {T_{2}-T_{1}}}\left[{{t \over 2}-{\sin(2\omega t) \over 4\omega }}\right]_{T_{1}}^{T_{2}}}}}
but since the interval is a whole number of complete cycles (per definition of RMS), the sin terms will cancel out, leaving:
I R M S = I p 1 T 2 − T 1 [ t 2 ] T 1 T 2 = I p 1 T 2 − T 1 T 2 − T 1 2 = I p 2 . {\displaystyle I_{\mathrm {RMS} }=I_{\mathrm {p} }{\sqrt {{1 \over {T_{2}-T_{1}}}\left[{t \over 2}\right]_{T_{1}}^{T_{2}}}}=I_{\mathrm {p} }{\sqrt {{1 \over {T_{2}-T_{1}}}{{T_{2}-T_{1}} \over 2}}}={I_{\mathrm {p} } \over {\sqrt {2}}}.}
A similar analysis leads to the analogous equation for sinusoidal voltage:
V R M S = V p 2 . {\displaystyle V_{\mathrm {RMS} }={V_{\mathrm {p} } \over {\sqrt {2}}}.}
Where IP represents the peak current and VP represents the peak voltage.
Because of their usefulness in carrying out power calculations, listed voltages for power outlets, e.g. 120 V (USA) or 230 V (Europe), are almost always quoted in RMS values, and not peak values. Peak values can be calculated from RMS values from the above formula, which implies Vp = VRMS × √2, assuming the source is a pure sine wave. Thus the peak value of the mains voltage in the USA is about 120 × √2, or about 170 volts. The peak-to-peak voltage, being twice this, is about 340 volts. A similar calculation indicates that the peak-to-peak mains voltage in Europe is about 650 volts.
RMS quantities such as electric current are usually calculated over one cycle. However for some purposes the RMS current over a longer period is required when calculating transmission power losses. The same principle applies, and (for example) a current of 10 amps used for 6 hours each day represents an RMS current of 5 amps in the long term.
The term "RMS power" is sometimes used in the audio industry as a synonym for "mean power" or "average power" (it is proportional to the square of the RMS voltage or RMS current in a resistive load). For a discussion of audio power measurements and their shortcomings, see Audio power.
Root-mean-square speed
In the physics of gas molecules, the root-mean-square speed is defined as the square root of the average squared-speed. The RMS speed of an ideal gas is calculated using the following equation:
v R M S = 3 R T M {\displaystyle {v_{\mathrm {RMS} }}={\sqrt {3RT \over {M}}}}
where R represents the ideal gas constant, 8.314 J/(mol·K), T is the temperature of the gas in kelvins, and M is the molar mass of the gas in kilograms. The generally accepted terminology for speed as compared to velocity is that the former is the scalar magnitude of the latter. Therefore, although the average speed is between zero and the RMS speed, the average velocity for a stationary gas is zero.
Root-mean-square error
{{#invoke:main|main}} When two data sets—one set from theoretical prediction and the other from actual measurement of some physical variable, for instance—are compared, the RMS of the pairwise differences of the two data sets can serve as a measure how far on average the error is from 0.
The mean of the pairwise differences does not measure the variability of the difference, and the variability as indicated by the standard deviation is around the mean instead of 0. Therefore, the RMS of the differences is a meaningful measure of the error.
RMS in frequency domain
The RMS can be computed in the frequency domain, using Parseval's theorem. For a sampled signal,
∑ n x 2 ( t ) = ∑ n | X ( f ) | 2 n {\displaystyle \sum \limits _{n}{{{x}^{2}}(t)}={\frac {\sum \limits _{n}{{\left|X(f)\right|}^{2}}}{n}}} ,
where X ( f ) = F F T { x ( t ) } {\displaystyle X(f)=\mathrm {FFT} \{x(t)\}} and n is number of x(t) samples.
In this case, the RMS computed in the time domain is the same as in the frequency domain:
R M S = 1 n ∑ n x 2 ( t ) = 1 n 2 ∑ n | X ( f ) | 2 = ∑ n | X ( f ) n | 2 . {\displaystyle \mathrm {RMS} ={\sqrt {{\frac {1}{n}}\sum \limits _{n}{{{x}^{2}}(t)}}}={\sqrt {{\frac {1}{n^{2}}}\sum \limits _{n}{{\left|X(f)\right|}^{2}}}}={\sqrt {\sum \limits _{n}{\left|{\frac {X(f)}{n}}\right|^{2}}}}.}
Relationship to the arithmetic mean and the standard deviation
If x ¯ {\displaystyle {\bar {x}}} is the arithmetic mean and σ x {\displaystyle \sigma _{x}} is the standard deviation of a population or a waveform then:[3]
x r m s 2 = x ¯ 2 + σ x 2 = x 2 ¯ . {\displaystyle x_{\mathrm {rms} }^{2}={\bar {x}}^{2}+\sigma _{x}^{2}={\overline {x^{2}}}.}
From this it is clear that the RMS value is always greater than or equal to the average, in that the RMS includes the "error" / square deviation as well.
Physical scientists often use the term "root mean square" as a synonym for standard deviation when referring to the square root of the mean squared deviation of a signal from a given baseline or fit.[4] [5] This is useful for electrical engineers in calculating the "AC only" RMS of a signal. Standard deviation being the root mean square of a signal's variation about the mean, rather than about 0, the DC component is removed (i.e. RMS(signal) = Stdev(signal) if the mean signal is 0).
Central moment
Geometric mean
L2 norm
Least squares
Mean squared displacement
Root mean square deviation
Table of mathematical symbols
True RMS converter
A case for why RMS is a misnomer when applied to audio power
RMS, Peak and Average for some waveforms
A Java applet on learning RMS
Retrieved from "https://en.formulasearchengine.com/index.php?title=Root_mean_square&oldid=223539"
Statistical deviation and dispersion | CommonCrawl |
Connor Harris
I am a policy analyst at the Manhattan Institute for Policy Research. My job entails statistical analysis with R, as well as webscraping and data processing (with Python and Unix command-line tools). I have a BA in mathematics and physics from Harvard University, class of 2016.
I am no longer answering questions on Stack Exchange, in protest of the treatment of Monica Cellio.
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Finite volume method
The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations.[1] In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many computational fluid dynamics packages. "Finite volume" refers to the small volume surrounding each node point on a mesh.[2]
Differential equations
Scope
Fields
• Natural sciences
• Engineering
• Astronomy
• Physics
• Chemistry
• Biology
• Geology
Applied mathematics
• Continuum mechanics
• Chaos theory
• Dynamical systems
Social sciences
• Economics
• Population dynamics
List of named differential equations
Classification
Types
• Ordinary
• Partial
• Differential-algebraic
• Integro-differential
• Fractional
• Linear
• Non-linear
By variable type
• Dependent and independent variables
• Autonomous
• Coupled / Decoupled
• Exact
• Homogeneous / Nonhomogeneous
Features
• Order
• Operator
• Notation
Relation to processes
• Difference (discrete analogue)
• Stochastic
• Stochastic partial
• Delay
Solution
Existence and uniqueness
• Picard–Lindelöf theorem
• Peano existence theorem
• Carathéodory's existence theorem
• Cauchy–Kowalevski theorem
General topics
• Initial conditions
• Boundary values
• Dirichlet
• Neumann
• Robin
• Cauchy problem
• Wronskian
• Phase portrait
• Lyapunov / Asymptotic / Exponential stability
• Rate of convergence
• Series / Integral solutions
• Numerical integration
• Dirac delta function
Solution methods
• Inspection
• Method of characteristics
• Euler
• Exponential response formula
• Finite difference (Crank–Nicolson)
• Finite element
• Infinite element
• Finite volume
• Galerkin
• Petrov–Galerkin
• Green's function
• Integrating factor
• Integral transforms
• Perturbation theory
• Runge–Kutta
• Separation of variables
• Undetermined coefficients
• Variation of parameters
People
List
• Isaac Newton
• Gottfried Leibniz
• Jacob Bernoulli
• Leonhard Euler
• Józef Maria Hoene-Wroński
• Joseph Fourier
• Augustin-Louis Cauchy
• George Green
• Carl David Tolmé Runge
• Martin Kutta
• Rudolf Lipschitz
• Ernst Lindelöf
• Émile Picard
• Phyllis Nicolson
• John Crank
Finite volume methods can be compared and contrasted with the finite difference methods, which approximate derivatives using nodal values, or finite element methods, which create local approximations of a solution using local data, and construct a global approximation by stitching them together. In contrast a finite volume method evaluates exact expressions for the average value of the solution over some volume, and uses this data to construct approximations of the solution within cells.[3][4]
Example
Consider a simple 1D advection problem:
${\frac {\partial \rho }{\partial t}}+{\frac {\partial f}{\partial x}}=0,\quad t\geq 0.$
(1)
Here, $\rho =\rho \left(x,t\right)$ represents the state variable and $f=f\left(\rho \left(x,t\right)\right)$ represents the flux or flow of $\rho $. Conventionally, positive $f$ represents flow to the right while negative $f$ represents flow to the left. If we assume that equation (1) represents a flowing medium of constant area, we can sub-divide the spatial domain, $x$, into finite volumes or cells with cell centers indexed as $i$. For a particular cell, $i$, we can define the volume average value of ${\rho }_{i}\left(t\right)=\rho \left(x,t\right)$ at time ${t=t_{1}}$ and ${x\in \left[x_{i-{\frac {1}{2}}},x_{i+{\frac {1}{2}}}\right]}$, as
${\bar {\rho }}_{i}\left(t_{1}\right)={\frac {1}{x_{i+{\frac {1}{2}}}-x_{i-{\frac {1}{2}}}}}\int _{x_{i-{\frac {1}{2}}}}^{x_{i+{\frac {1}{2}}}}\rho \left(x,t_{1}\right)\,dx,$
(2)
and at time $t=t_{2}$ as,
${\bar {\rho }}_{i}\left(t_{2}\right)={\frac {1}{x_{i+{\frac {1}{2}}}-x_{i-{\frac {1}{2}}}}}\int _{x_{i-{\frac {1}{2}}}}^{x_{i+{\frac {1}{2}}}}\rho \left(x,t_{2}\right)\,dx,$
(3)
where $x_{i-{\frac {1}{2}}}$ and $x_{i+{\frac {1}{2}}}$ represent locations of the upstream and downstream faces or edges respectively of the $i^{\text{th}}$ cell.
Integrating equation (1) in time, we have:
$\rho \left(x,t_{2}\right)=\rho \left(x,t_{1}\right)-\int _{t_{1}}^{t_{2}}f_{x}\left(x,t\right)\,dt,$
(4)
where $f_{x}={\frac {\partial f}{\partial x}}$.
To obtain the volume average of $\rho \left(x,t\right)$ at time $t=t_{2}$, we integrate $\rho \left(x,t_{2}\right)$ over the cell volume, $\left[x_{i-{\frac {1}{2}}},x_{i+{\frac {1}{2}}}\right]$ and divide the result by $\Delta x_{i}=x_{i+{\frac {1}{2}}}-x_{i-{\frac {1}{2}}}$, i.e.
${\bar {\rho }}_{i}\left(t_{2}\right)={\frac {1}{\Delta x_{i}}}\int _{x_{i-{\frac {1}{2}}}}^{x_{i+{\frac {1}{2}}}}\left\{\rho \left(x,t_{1}\right)-\int _{t_{1}}^{t_{2}}f_{x}\left(x,t\right)dt\right\}dx.$
(5)
We assume that $f\ $ is well behaved and that we can reverse the order of integration. Also, recall that flow is normal to the unit area of the cell. Now, since in one dimension $f_{x}\triangleq \nabla \cdot f$, we can apply the divergence theorem, i.e. $\oint _{v}\nabla \cdot fdv=\oint _{S}f\,dS$, and substitute for the volume integral of the divergence with the values of $f(x)$ evaluated at the cell surface (edges $x_{i-{\frac {1}{2}}}$ and $x_{i+{\frac {1}{2}}}$) of the finite volume as follows:
${\bar {\rho }}_{i}\left(t_{2}\right)={\bar {\rho }}_{i}\left(t_{1}\right)-{\frac {1}{\Delta x_{i}}}\left(\int _{t_{1}}^{t_{2}}f_{i+{\frac {1}{2}}}dt-\int _{t_{1}}^{t_{2}}f_{i-{\frac {1}{2}}}dt\right).$
(6)
where $f_{i\pm {\frac {1}{2}}}=f\left(x_{i\pm {\frac {1}{2}}},t\right)$.
We can therefore derive a semi-discrete numerical scheme for the above problem with cell centers indexed as $i$, and with cell edge fluxes indexed as $i\pm {\frac {1}{2}}$, by differentiating (6) with respect to time to obtain:
${\frac {d{\bar {\rho }}_{i}}{dt}}+{\frac {1}{\Delta x_{i}}}\left[f_{i+{\frac {1}{2}}}-f_{i-{\frac {1}{2}}}\right]=0,$
(7)
where values for the edge fluxes, $f_{i\pm {\frac {1}{2}}}$, can be reconstructed by interpolation or extrapolation of the cell averages. Equation (7) is exact for the volume averages; i.e., no approximations have been made during its derivation.
This method can also be applied to a 2D situation by considering the north and south faces along with the east and west faces around a node.
General conservation law
We can also consider the general conservation law problem, represented by the following PDE,
${\frac {\partial \mathbf {u} }{\partial t}}+\nabla \cdot {\mathbf {f} }\left({\mathbf {u} }\right)={\mathbf {0} }.$
(8)
Here, $\mathbf {u} $ represents a vector of states and $\mathbf {f} $ represents the corresponding flux tensor. Again we can sub-divide the spatial domain into finite volumes or cells. For a particular cell, $i$, we take the volume integral over the total volume of the cell, $v_{i}$, which gives,
$\int _{v_{i}}{\frac {\partial \mathbf {u} }{\partial t}}\,dv+\int _{v_{i}}\nabla \cdot {\mathbf {f} }\left({\mathbf {u} }\right)\,dv={\mathbf {0} }.$
(9)
On integrating the first term to get the volume average and applying the divergence theorem to the second, this yields
$v_{i}{{d{\mathbf {\bar {u}} }_{i}} \over dt}+\oint _{S_{i}}{\mathbf {f} }\left({\mathbf {u} }\right)\cdot {\mathbf {n} }\ dS={\mathbf {0} },$
(10)
where $S_{i}$ represents the total surface area of the cell and ${\mathbf {n} }$ is a unit vector normal to the surface and pointing outward. So, finally, we are able to present the general result equivalent to (8), i.e.
${{d{\mathbf {\bar {u}} }_{i}} \over {dt}}+{{1} \over {v_{i}}}\oint _{S_{i}}{\mathbf {f} }\left({\mathbf {u} }\right)\cdot {\mathbf {n} }\ dS={\mathbf {0} }.$
(11)
Again, values for the edge fluxes can be reconstructed by interpolation or extrapolation of the cell averages. The actual numerical scheme will depend upon problem geometry and mesh construction. MUSCL reconstruction is often used in high resolution schemes where shocks or discontinuities are present in the solution.
Finite volume schemes are conservative as cell averages change through the edge fluxes. In other words, one cell's loss is always another cell's gain!
See also
• Finite element method
• Flux limiter
• Godunov's scheme
• Godunov's theorem
• High-resolution scheme
• KIVA (Software)
• MIT General Circulation Model
• MUSCL scheme
• Sergei K. Godunov
• Total variation diminishing
• Finite volume method for unsteady flow
References
1. LeVeque, Randall (2002). Finite Volume Methods for Hyperbolic Problems. ISBN 9780511791253.
2. Wanta, D.; Smolik, W. T.; Kryszyn, J.; Wróblewski, P.; Midura, M. (October 2021). "A Finite Volume Method using a Quadtree Non-Uniform Structured Mesh for Modeling in Electrical Capacitance Tomography". Proceedings of the National Academy of Sciences, India Section A: Physical Sciences. 92 (3): 443–452. doi:10.1007/s40010-021-00748-7.
3. Fallah, N. A.; Bailey, C.; Cross, M.; Taylor, G. A. (2000-06-01). "Comparison of finite element and finite volume methods application in geometrically nonlinear stress analysis". Applied Mathematical Modelling. 24 (7): 439–455. doi:10.1016/S0307-904X(99)00047-5. ISSN 0307-904X.
4. Ranganayakulu, C. (Chennu) (2 February 2018). "Chapter 3, Section 3.1". Compact heat exchangers : analysis, design and optimization using FEM and CFD approach. Seetharamu, K. N. Hoboken, NJ. ISBN 978-1-119-42435-2. OCLC 1006524487.{{cite book}}: CS1 maint: location missing publisher (link)
Further reading
• Eymard, R. Gallouët, T. R., Herbin, R. (2000) The finite volume method Handbook of Numerical Analysis, Vol. VII, 2000, p. 713–1020. Editors: P.G. Ciarlet and J.L. Lions.
• Hirsch, C. (1990), Numerical Computation of Internal and External Flows, Volume 2: Computational Methods for Inviscid and Viscous Flows, Wiley.
• Laney, Culbert B. (1998), Computational Gas Dynamics, Cambridge University Press.
• LeVeque, Randall (1990), Numerical Methods for Conservation Laws, ETH Lectures in Mathematics Series, Birkhauser-Verlag.
• LeVeque, Randall (2002), Finite Volume Methods for Hyperbolic Problems, Cambridge University Press.
• Patankar, Suhas V. (1980), Numerical Heat Transfer and Fluid Flow, Hemisphere.
• Tannehill, John C., et al., (1997), Computational Fluid mechanics and Heat Transfer, 2nd Ed., Taylor and Francis.
• Toro, E. F. (1999), Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag.
• Wesseling, Pieter (2001), Principles of Computational Fluid Dynamics, Springer-Verlag.
External links
• Finite volume methods by R. Eymard, T Gallouët and R. Herbin, update of the article published in Handbook of Numerical Analysis, 2000
• Rübenkönig, Oliver. "The Finite Volume Method (FVM) – An introduction". Archived from the original on 2009-10-02. {{cite journal}}: Cite journal requires |journal= (help), available under the GFDL.
• FiPy: A Finite Volume PDE Solver Using Python from NIST.
• CLAWPACK: a software package designed to compute numerical solutions to hyperbolic partial differential equations using a wave propagation approach
Numerical methods for partial differential equations
Finite difference
Parabolic
• Forward-time central-space (FTCS)
• Crank–Nicolson
Hyperbolic
• Lax–Friedrichs
• Lax–Wendroff
• MacCormack
• Upwind
• Method of characteristics
Others
• Alternating direction-implicit (ADI)
• Finite-difference time-domain (FDTD)
Finite volume
• Godunov
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• Monotonic upstream-centered (MUSCL)
• Advection upstream-splitting (AUSM)
• Riemann solver
• Essentially non-oscillatory (ENO)
• Weighted essentially non-oscillatory (WENO)
Finite element
• hp-FEM
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• Spectral element (SEM)
• Mortar
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• Loubignac iteration
• Smoothed (S-FEM)
Meshless/Meshfree
• Smoothed-particle hydrodynamics (SPH)
• Peridynamics (PD)
• Moving particle semi-implicit method (MPS)
• Material point method (MPM)
• Particle-in-cell (PIC)
Domain decomposition
• Schur complement
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• additive
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• Tearing and interconnect (FETI)
• FETI-DP
Others
• Spectral
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• Method of lines
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• Level-set
• Boundary element
• Method of moments
• Immersed boundary
• Analytic element
• Isogeometric analysis
• Infinite difference method
• Infinite element method
• Galerkin method
• Petrov–Galerkin method
• Validated numerics
• Computer-assisted proof
• Integrable algorithm
• Method of fundamental solutions
Differential equations
Classification
Operations
• Differential operator
• Notation for differentiation
• Ordinary
• Partial
• Differential-algebraic
• Integro-differential
• Fractional
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• Non-linear
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Attributes of variables
• Dependent and independent variables
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• Nonhomogeneous
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• On jet bundles
Relation to processes
• Difference (discrete analogue)
• Stochastic
• Stochastic partial
• Delay
Solutions
Existence/uniqueness
• Picard–Lindelöf theorem
• Peano existence theorem
• Carathéodory's existence theorem
• Cauchy–Kowalevski theorem
Solution topics
• Wronskian
• Phase portrait
• Phase space
• Lyapunov stability
• Asymptotic stability
• Exponential stability
• Rate of convergence
• Series solutions
• Integral solutions
• Numerical integration
• Dirac delta function
Solution methods
• Inspection
• Substitution
• Separation of variables
• Method of undetermined coefficients
• Variation of parameters
• Integrating factor
• Integral transforms
• Euler method
• Finite difference method
• Crank–Nicolson method
• Runge–Kutta methods
• Finite element method
• Finite volume method
• Galerkin method
• Perturbation theory
Applications
• List of named differential equations
Mathematicians
• Isaac Newton
• Gottfried Wilhelm Leibniz
• Leonhard Euler
• Jacob Bernoulli
• Émile Picard
• Józef Maria Hoene-Wroński
• Ernst Lindelöf
• Rudolf Lipschitz
• Joseph-Louis Lagrange
• Augustin-Louis Cauchy
• John Crank
• Phyllis Nicolson
• Carl David Tolmé Runge
• Martin Kutta
• Sofya Kovalevskaya
| Wikipedia |
Difference between revisions of "Past Probability Seminars Spring 2020"
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= Fall 2018 =
= Spring 2020 =
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted.
<b>We usually end for questions at 3:15 PM.</b>
If you would like to sign up for the email list to receive seminar announcements then please send an email to
[mailto:[email protected] [email protected]]
== January 23, 2020, [https://www.math.wisc.edu/~seppalai/ Timo Seppalainen] (UW Madison) ==
'''Non-existence of bi-infinite geodesics in the exponential corner growth model
Whether bi-infinite geodesics exist has been a significant open problem in first- and last-passage percolation since the mid-80s. A non-existence proof in the case of directed planar last-passage percolation with exponential weights was posted by Basu, Hoffman and Sly in November 2018. Their proof utilizes estimates from integrable probability. This talk describes an independent proof completed 10 months later that relies on couplings, coarse graining, and control of geodesics through planarity and increment-stationary last-passage percolation. Joint work with Marton Balazs and Ofer Busani (Bristol).
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==
== January 30, 2020, [https://www.math.wisc.edu/people/vv-prof-directory Scott Smith] (UW Madison) ==
'''Quasi-linear parabolic equations with singular forcing'''
The classical solution theory for stochastic ODE's is centered around Ito's stochastic integral. By intertwining ideas from analysis and probability, this approach extends to many PDE's, a canonical example being multiplicative stochastic heat equations driven by space-time white noise. In both the ODE and PDE settings, the solution theory is beyond the scope of classical deterministic theory because of the ambiguity in multiplying a function with a white noise. The theory of rough paths and regularity structures provides a more quantitative understanding of this difficulty, leading to a more refined solution theory which efficiently divides the analytic and probabilistic aspects of the problem, and remarkably, even has an algebraic component.
Title: '''The distribution of sandpile groups of random regular graphs'''
In this talk, we will discuss a new application of these ideas to stochastic heat equations where the strength of the diffusion is not constant but random, as it depends locally on the solution. These are known as quasi-linear equations. Our main result yields the deterministic side of a solution theory for these PDE's, modulo a suitable renormalization. Along the way, we identify a formally infinite series expansion of the solution which guides our analysis, reveals a nice algebraic structure, and encodes the counter-terms in the PDE. This is joint work with Felix Otto, Jonas Sauer, and Hendrik Weber.
== February 6, 2020, [https://sites.google.com/site/cyleeken/ Cheuk-Yin Lee] (Michigan State) ==
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.
'''Sample path properties of stochastic partial differential equations: modulus of continuity and multiple points'''
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.
In this talk, we will discuss sample path properties of stochastic partial differential equations (SPDEs). We will present a sharp regularity result for the stochastic wave equation driven by an additive Gaussian noise that is white in time and colored in space. We prove the exact modulus of continuity via the property of local nondeterminism. We will also discuss the existence problem for multiple points (or self-intersections) of the sample paths of SPDEs. Our result shows that multiple points do not exist in the critical dimension for a large class of Gaussian random fields including the solution of a linear system of stochastic heat or wave equations.
<!-- ==September 13, TBA == -->
== February 13, 2020, [http://www.jelena-diakonikolas.com/ Jelena Diakonikolas] (UW Madison) ==
'''Langevin Monte Carlo Without Smoothness'''
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==
Langevin Monte Carlo (LMC) is an iterative algorithm used to generate samples from a distribution that is known only up to a normalizing constant. The nonasymptotic dependence of its mixing time on the dimension and target accuracy is understood mainly in the setting of smooth (gradient-Lipschitz) log-densities, a serious limitation for applications in machine learning. We remove this limitation by providing polynomial-time convergence guarantees for a variant of LMC in the setting of non-smooth log-concave distributions. At a high level, our results follow by leveraging the implicit smoothing of the log-density that comes from a small Gaussian perturbation that we add to the iterates of the algorithm and while controlling the bias and variance that are induced by this perturbation.
Based on joint work with Niladri Chatterji, Michael I. Jordan, and Peter L. Bartlett.
Title: '''Stochastic quantization of Yang-Mills'''
== February 20, 2020, [https://math.berkeley.edu/~pmwood/ Philip Matchett Wood] (UC Berkeley) ==
'''A replacement principle for perturbations of non-normal matrices'''
There are certain non-normal matrices whose eigenvalues can change dramatically when a small perturbation is added. However, when that perturbation is an iid random matrix, it appears that the eigenvalues become stable after perturbation and only change slightly when further small perturbations are added. Much of the work is this situation has focused on iid random gaussian perturbations. In this talk, we will discuss work on a universality result that allows for consideration of non-gaussian perturbations, and that shows that all perturbations satisfying certain conditions will produce the same limiting eigenvalue measure. Interestingly, this even allows for deterministic perturbations to be considered. Joint work with Sean O'Rourke.
"Stochastic quantization" refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].
== February 27, 2020, No seminar ==
''' '''
== March 5, 2020, [https://www.ias.edu/scholars/jiaoyang-huang Jiaoyang Huang] (IAS) ==
''' Large Deviation Principles via Spherical Integrals'''
==September 27, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] [https://www.math.wisc.edu/ UW-Madison] ==
In this talk, I'll explain a framework to study the large deviation principle for matrix models and their quantized versions, by tilting the measures using the asymptotics of spherical integrals obtained by Guionnet and Zeitouni. As examples, we obtain
Title:'''Random walk in random environment and the Kardar-Parisi-Zhang class'''
1) the large deviation principle for the empirical distribution of the diagonal entries of $UB_NU^*$, for a sequence of $N\times N$ diagonal matrices $B_N$ and unitary/orthogonal Haar distributed matrices $U$;
Abstract:This talk concerns a relationship between two much-studied classes of models of motion in a random medium, namely random walk in random environment (RWRE) and the Kardar-Parisi-Zhang (KPZ) universality class. Barraquand and Corwin (Columbia) discovered that in 1+1 dimensional RWRE in a dynamical beta environment the correction to the quenched large deviation principle obeys KPZ behavior. In this talk we condition the beta walk to escape at an atypical velocity and show that the resulting Doob-transformed RWRE obeys the KPZ wandering exponent 2/3. Based on joint work with Márton Balázs (Bristol) and Firas Rassoul-Agha (Utah).
2) the large deviation upper bound for the empirical eigenvalue distribution of $A_N+UB_NU^*$, for two sequences of $N\times N$ diagonal matrices $A_N, B_N$, and their complementary lower bounds at "good" probability distributions;
==October 4, [https://people.math.osu.edu/paquette.30/ Elliot Paquette], [https://math.osu.edu/ OSU] ==
3) the large deviation principle for the Kostka number $K_{\lambda_N \eta_N}$, for two sequences of partitions $\lambda_N, \eta_N$ with at most $N$ rows;
Title: '''Distributional approximation of the characteristic polynomial of a Gaussian beta-ensemble'''
4) the large deviation upper bound for the Littlewood-Richardson coefficients $c_{\lambda_N \eta_N}^{\kappa_N}$, for three sequences of partitions $\lambda_N, \eta_N, \kappa_N$ with at most $N$ rows, and their complementary lower bounds at "good" probability distributions.
This is a joint work with Belinschi and Guionnet.
The characteristic polynomial of the Gaussian beta--ensemble can be represented, via its tridiagonal model, as an entry in a product of independent random two--by--two matrices. For a point z in the complex plane, at which the transfer matrix is to be evaluated, this product of transfer matrices splits into three independent factors, each of which can be understood as a different dynamical system in the complex plane. Conjecturally, we show that the characteristic polynomial is always represented as product of at most three terms, an exponential of a Gaussian field, the stochastic Airy function, and a diffusion similar to the stochastic sine equation.
We explain the origins of this decomposition, and we show partial progress in establishing part of it.
Joint work with Diane Holcomb and Gaultier Lambert.
== March 12, 2020, No seminar ==
==October 11, [https://www.math.utah.edu/~janjigia/ Chris Janjigian], [https://www.math.utah.edu/ University of Utah] ==
== March 19, 2020, Spring break ==
== March 26, 2020, CANCELLED, [https://math.cornell.edu/philippe-sosoe Philippe Sosoe] (Cornell) ==
Title: '''Busemann functions and Gibbs measures in directed polymer models on Z^2'''
== April 2, 2020, CANCELLED, [http://pages.cs.wisc.edu/~tl/ Tianyu Liu] (UW Madison)==
Abstract: We consider the model of a nearest-neighbor random walk on the planar square lattice in a general iid space-time potential, which is also known as a directed polymer in a random environment. We prove results on existence, uniqueness (and non-uniqueness), and the law of large numbers for semi-infinite path measures. Our main tools are the Busemann functions, which are families of stochastic processes obtained through limits of ratios of partition functions.
== April 9, 2020, CANCELLED, [http://stanford.edu/~ajdunl2/ Alexander Dunlap] (Stanford) ==
Based on joint work with Firas Rassoul-Agha
== April 16, 2020, CANCELLED, [https://statistics.wharton.upenn.edu/profile/dingjian/ Jian Ding] (University of Pennsylvania) ==
==October 18-20, [http://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium], No Seminar ==
== April 22-24, 2020, CANCELLED, [http://frg.int-prob.org/ FRG Integrable Probability] meeting ==
==October 25, [http://stat.columbia.edu/department-directory/name/promit-ghosal/ Promit Ghosal], Columbia ==
3-day event in Van Vleck 911
== April 23, 2020, CANCELLED, [http://www.hairer.org/ Martin Hairer] (Imperial College) ==
Title: '''Tails of the KPZ equation'''
[https://www.math.wisc.edu/wiki/index.php/Colloquia Wolfgang Wasow Lecture] at 4pm in Van Vleck 911
Abstract: The KPZ equation is a fundamental stochastic PDE related to modeling random growth processes, Burgers turbulence, interacting particle system, random polymers etc. It is related to another important SPDE, namely, the stochastic heat equation (SHE). In this talk, we focus on the tail probabilities of the solution of the KPZ equation. For instance, we investigate the probability of the solution being smaller or larger than the expected value. Our analysis is based on an exact identity between the KPZ equation and the Airy point process (which arises at the edge of the spectrum of the random Hermitian matrices) and the Brownian Gibbs property of the KPZ line ensemble.
This talk will be based on a joint work with my advisor Prof. Ivan Corwin.
==November 1, TBA ==
== April 30, 2020, CANCELLED, [http://willperkins.org/ Will Perkins] (University of Illinois at Chicago) ==
==November 8, [https://cims.nyu.edu/~thomasl/ Thomas Leblé], NYU ==
==November 15, TBA ==
==November 22, [https://en.wikipedia.org/wiki/Thanksgiving Thanksgiving] Break, No Seminar ==
==December 6, TBA ==
[[Past Seminars]]
January 23, 2020, Timo Seppalainen (UW Madison)
Non-existence of bi-infinite geodesics in the exponential corner growth model
January 30, 2020, Scott Smith (UW Madison)
Quasi-linear parabolic equations with singular forcing
February 6, 2020, Cheuk-Yin Lee (Michigan State)
Sample path properties of stochastic partial differential equations: modulus of continuity and multiple points
February 13, 2020, Jelena Diakonikolas (UW Madison)
Langevin Monte Carlo Without Smoothness
Langevin Monte Carlo (LMC) is an iterative algorithm used to generate samples from a distribution that is known only up to a normalizing constant. The nonasymptotic dependence of its mixing time on the dimension and target accuracy is understood mainly in the setting of smooth (gradient-Lipschitz) log-densities, a serious limitation for applications in machine learning. We remove this limitation by providing polynomial-time convergence guarantees for a variant of LMC in the setting of non-smooth log-concave distributions. At a high level, our results follow by leveraging the implicit smoothing of the log-density that comes from a small Gaussian perturbation that we add to the iterates of the algorithm and while controlling the bias and variance that are induced by this perturbation. Based on joint work with Niladri Chatterji, Michael I. Jordan, and Peter L. Bartlett.
February 20, 2020, Philip Matchett Wood (UC Berkeley)
A replacement principle for perturbations of non-normal matrices
February 27, 2020, No seminar
March 5, 2020, Jiaoyang Huang (IAS)
Large Deviation Principles via Spherical Integrals
March 12, 2020, No seminar
March 19, 2020, Spring break
March 26, 2020, CANCELLED, Philippe Sosoe (Cornell)
April 2, 2020, CANCELLED, Tianyu Liu (UW Madison)
April 9, 2020, CANCELLED, Alexander Dunlap (Stanford)
April 16, 2020, CANCELLED, Jian Ding (University of Pennsylvania)
April 22-24, 2020, CANCELLED, FRG Integrable Probability meeting
April 23, 2020, CANCELLED, Martin Hairer (Imperial College)
Wolfgang Wasow Lecture at 4pm in Van Vleck 911
April 30, 2020, CANCELLED, Will Perkins (University of Illinois at Chicago) | CommonCrawl |
Resonant decomposition and the $I$-method for the two-dimensional Zakharov system
DCDS Home
Stability of travelling waves of a reaction-diffusion system for the acidic nitrate-ferroin reaction
September 2013, 33(9): 4071-4093. doi: 10.3934/dcds.2013.33.4071
Heteroclinic limit cycles in competitive Kolmogorov systems
Zhanyuan Hou 1, and Stephen Baigent 2,
School of Computing, London Metropolitan University, 166-220 Holloway Road, London N7 8DB, United Kingdom
Department of Mathematics, UCL, Gower Street, London WC1E 6BT, United Kingdom
Received May 2012 Revised January 2013 Published March 2013
A notion of global attraction and repulsion of heteroclinic limit cycles is introduced for strongly competitive Kolmogorov systems. Conditions are obtained for the existence of cycles linking the full set of axial equilibria and their global asymptotic behaviour on the carrying simplex. The global dynamics of systems with a heteroclinic limit cycle is studied. Results are also obtained for Kolmogorov systems where some components vanish as $t\rightarrow \pm \infty$.
Keywords: Lotka-Volterra systems, Kolmogorov systems, global attractors, heteroclinic limit cycles., global repellers.
Mathematics Subject Classification: Primary: 34C37, 37C29; Secondary: 34D05, 34D45, 37C7.
Citation: Zhanyuan Hou, Stephen Baigent. Heteroclinic limit cycles in competitive Kolmogorov systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4071-4093. doi: 10.3934/dcds.2013.33.4071
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Zhanyuan Hou Stephen Baigent | CommonCrawl |
\begin{document}
\title[Optimal decay for compressible MHD equations]{Optimal decay for the compressible MHD equations in the critical regularity framework}
\author[Q. Bie]{Qunyi Bie} \address[Q. Bie]{College of Science $\&$ Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, PR China} \email{\mailto{[email protected]}}
\author[Q. Wang]{Qiru Wang} \address[Q. Wang]{School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, PR China} \email{\mailto{[email protected]}}
\author[Z.-A. Yao]{Zheng-an Yao} \address[Z.-A. Yao]{School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, PR China} \email{\mailto{[email protected]}}
\date\today \keywords{} \subjclass[2010]{}
\begin{abstract} In this paper, we study the large time behavior of solutions to the compressible magnetohydrodynamic equations in the $L^p$-type critical Besov spaces. Precisely, we show that if the initial data in the low frequencies additionally belong to some Besov space $\dot{B}_{2,\infty}^{-\sigma_1}$ with $\sigma_1\in (1-N/2, 2N/p-N/2]$, then the $\dot{B}_{p,1}^0$ norm of the critical global solutions presents the optimal decay $t^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{p})-\frac{\sigma_1}{2}}$ for $t\rightarrow+\infty$. The pure energy argument without the spectral analysis is performed, which allows us to remove the usual smallness assumption of low frequencies.
\end{abstract}
\maketitle
\section{Introduction} Magnetohydrodynamics (MHD) is concerned with the motion of conducting fluids in an electromagnetic field and has a very wide range of applications. In view of the dynamic motion of field and the magnetic field interacting strongly on each other, both the hydrodynamic and electrodynamic effects must be considered. The compressible viscous MHD equations in the isentropic case take the form (see, e.g.,\cite{cabannes1970theoretical,kulikovskiy1965,laudau1984}) \begin{equation}\label{1.1}
\left\{\begin{array}{ll}\displaystyle\partial_t \rho+{\rm div}(\rho{\bf u}) =0,\\[1ex]
\displaystyle\partial_t(\rho{\bf u})+{\rm div}(\rho{\bf u}\otimes{\bf u})+\nabla P(\rho)\\[1ex]
\quad={\bf B}\cdot\nabla {\bf B}-\frac{1}{2}\nabla(|{\bf B}|^2)+{\rm div}(2\mu D({\bf u})+\lambda{\rm div}{\bf u}\, {\rm Id}),\\[1ex]
\displaystyle\partial_t{{\bf B}}+({\rm div}{\bf u}){\bf B}+{\bf u}\cdot\nabla{\bf B}
-{\bf B}\cdot\nabla{\bf u}=\theta\Delta{\bf B},\,\,\,\,{\rm div}{\bf B}=0,
\end{array}
\right.
\end{equation} for $(t,x)\in \mathbb{R}_+\times\mathbb{R}^N\,(N\geq 2)$. Here $\rho=\rho(t,x)\in \mathbb{R}_+$ is the density function of the fluid, ${\bf u}={\bf u}(t,x)\in \mathbb{R}^N$ is the velocity, and ${\bf B}={\bf B}(t,x)\in\mathbb{R}^{N}$ represents the magnetic field. The scalar function $P(\rho)\in\mathbb{R}$ is the pressure, which is an increasing and convex function in $\rho$. The notation $D({\bf u})\equ \frac{1}{2}(\nabla{\bf u}+\nabla{\bf u}^T)$ stands for the deformation tensor. The density-dependent functions $\lambda$ and $\mu$ (the bulk and shear viscosities) are supposed to be smooth enough and to satisfy $\mu>0$ and $\lambda+2\mu>0$. The constant $\theta>0$ stands for the magnetic diffusivity acting as a magnetic diffusion coefficient of the magnetic field. The symbol $\otimes$ denotes the Kronecker tensor product such that ${\bf u}\otimes{\bf u}=({\bf u}_i{\bf u}_j)_{1\leq i,j \leq N}$. System \eqref{1.1} is supplemented with the initial data \begin{equation}\label{1.2}
(\rho, {\bf u}, {\bf B})|_{t=0}=(\rho_0(x), {\bf u}_0(x), {\bf B}_0(x)), \,\,x\in \mathbb{R}^N, \end{equation} and we focus on solutions that are close to some constant state $(\rho^\ast, {\bf 0}, {\bf B}^\ast)$ with $\rho^\ast>0$ and the nonzero vector ${\bf B}^\ast\in \mathbb{R}^N$, at infinity.
There have been a lot of works on MHD by many physicists and mathematicians due to its physical importance and mathematical challenges, see for example \cite{chen2002global, chen2003existence, ducomet2006the, fan2007vanishing, freistuhler1995existence, hoff2005uniqueness, kawashima1982smooth,wang2003large} and the references therein. By exploiting an energy method in Fourier spaces, Umeda, Kawashima and Shizuta \cite{umeda1984on} first investigated a rather general class of symmetric hyperbolic-parabolic systems, and found that the dissipative mechanism inducing the optimal decay rates are just the same as that of heat kernel. As a direct application, they obtained such decay rate of solutions to system \eqref{1.1}-\eqref{1.2} (near the equilibrium state $(\rho^\ast, {\bf 0}, {\bf B}^\ast)$). Subsequently, Kawashima \cite{kawashima1984systems} in his doctoral dissertation proved the global existence of smooth solutions to \eqref{1.1}-\eqref{1.2} in the condition that the initial data are small in $H^3(\mathbb{R}^3)$. In addition, the author also derived the following fundamental $L^q$-$L^2$ decay estimate in $H^3(\mathbb{R}^3)\cap L^q(\mathbb{R}^3)\, (1\leq q<2)$: \begin{equation}\label{1.10}
\|(\rho-\rho^\ast, {\bf u}, {\bf B}-{\bf B}^\ast)\|_{L^2{\mathbb{R}^3}}\leq C(1+t)^{-\frac{3}{2}(\frac{1}{q}-\frac{1}{2})}. \end{equation} Later on, still for data with high Sobolev regularity, there are a number of works on the long-time behavior of solution to the compressible MHD equations, see for example \cite{chentan2010global,gao2015long, li2011optimal, tan2013optimal, zhang2010some} and the references therein.
As regards global-in-time results, \emph{scaling invariance} plays a fundamental role. Here we observe that system \eqref{1.1} is invariant by the transformation \begin{equation}\nonumber \tilde{\rho}(t,x)=\rho(l^2t, lx),~~\tilde{\bf u}(t, x)=l{\bf u}(l^2t, lx),~~\tilde{\bf B}(t, x)=l{\bf B}(l^2t, lx), \end{equation}
up to a change of the pressure law $\tilde{P}=l^2P$. A critical space is a space in which the norm is invariant under the scaling $ (\tilde{e},\tilde{\bf f}, \tilde{\bf g})(x)=(e(lx), l{\bf f}(lx), l{\bf g}(lx)). $
When ${\bf B}\equiv {\bf 0}$, system \eqref{1.1} becomes the compressible Navier-Stokes equations. In the critical framework, there have been a lot of results for the compressible (or incompressible) Navier-Stokes equations, see for example \cite{cannone1997a, charve2010global, chen2010global, danchin2000global, danchin2015fourier, danchin2016incompressible, danchin2016optimal, fujita1964on, haspot2011existence, kozono1994semilinear, okita2014optimal, xin2018optimal, xu2019a}. In particular, regarding the large time asymptotic behavior of strong solutions for the compressible Navier-Stokes equations, Okati \cite{okita2014optimal} performed low and high frequency decompositions and proved the time decay rate for strong solutions in the $L^2$ critical framework and in dimension $N\geq 3$. In the survey paper \cite{danchin2015fourier},
Danchin proposed another description of the time decay which allows to proceed with dimension $N\geq 2$ in the $L^2$ critical framework. Recently, Danchin and Xu \cite{danchin2016optimal} extended the method of \cite{danchin2015fourier} to get optimal time decay rate in the
general $L^p$ type critical spaces and in any dimension $N\geq 2$. Later on, Xu \cite{xu2019a} developed a general low-frequency condition for optimal decay estimates, where the regularity $\sigma_1$ of $\dot{B}_{2,\infty}^{-\sigma_1}$
belongs to a whole range $(1-\frac{N}{2}, \frac{2N}{p}-\frac{N}{2}]$, and the proof mainly depends on the refined time-weighted energy approach in the Fourier semi-group framework. Very recently, originated from the idea as in \cite{guo2012decay, strain2006almost}, Xin and Xu \cite{xin2018optimal} developed a new energy argument to remove the usual smallness assumption of low frequencies studied in \cite{danchin2016optimal}.
As for system \eqref{1.1}-\eqref{1.2} with ${\bf B}^\ast={\bf 0}$, Hao \cite{hao2011well} obtained the global well-posedness of strong solutions in $L^2$-type critical Besov spaces. Consequently, the authors in \cite{bian2013well, bian2016local, jia2016well} studied the local existence and uniqueness of solutions in the critical $L^p$ framework. Very recently, Shi and Xu \cite{shi2019global} considered the perturbation around the constant equilibrium $(\rho^\ast, {\bf 0}, {\bf B}^\ast)$ with ${\bf B}^\ast\neq {\bf 0}$ and obtained the local and global well-posedness results in the critical $L^p$ framework, and here we list the global well-posedness of strong solutions to system \eqref{1.1} as follows. \begin{theo}\label{th1.1} {\rm(}\cite{shi2019global}{\rm)} Let $N\geq 2$ and $p$ fulfill \begin{equation}\label{1.3}
2\leq p\leq \min (4, 2N/(N-2))\,\,\, { and}, \,additionally, \,p\neq 4\,\,{if}\,\, N=2. \end{equation} Suppose that {\rm div}${\bf B}_0=0$, $P^\prime(\rho^\ast)>0$ and that \eqref{1.2} is satisfied. There exists a small positive constant $c=c(p, \mu, \lambda, \theta, P, \rho^\ast, {\bf B}^\ast)$ and a universal integer $j_0\in \mathbb{Z}$ such that if $a_0\equ \rho_0-\rho^\ast\in \dot{B}_{p,1}^{\frac{N}{p}}$, ${\bf H}_0\equ {\bf B}_0-{\bf B}^\ast\in \dot{B}_{p,1}^{\frac{N}{p}-1}$ and if in addition $(a_0^\ell, {\bf u}_0^\ell, {\bf H}_0^\ell)\in \dot{B}_{2,1}^{\frac{N}{2}-1}$ (with the notation $z^\ell\equ \dot{S}_{k_0+1}z$ and $z^h=z-z^\ell$) with $$
\mathcal{X}_{p,0}\equ \|(a_0, {\bf u}_0, {\bf H}_0)\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell
+\|(\nabla a_0, {\bf u}_0, {\bf H}_0)\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h\leq c, $$ then the Cauchy problem \eqref{1.1}-\eqref{1.2} admits a unique global-in-time solution $(\rho, {\bf u}, {\bf B})$ with $\rho=\rho^\ast+a$, ${\bf B}={\bf B}^\ast+{\bf H}$ and $(a, {\bf u}, {\bf H})$ in the space $X_p$ defined by \begin{equation}\nonumber \left.\begin{array}{ll} (a,{\bf u},{\bf H})^\ell\in \widetilde{\mathcal{C}_b}(\mathbb{R}_+; \dot{B}_{2,1}^{\frac{N}{2}-1})\cap L^1(\mathbb{R}_+; \dot{B}_{2,1}^{\frac{N}{2}+1}), \,\,\,a^h\in \widetilde{\mathcal{C}_b}(\mathbb{R}_+; \dot{B}_{p,1}^{\frac{N}{p}})\cap L^1(\mathbb{R}_+; \dot{B}_{p,1}^{\frac{N}{p}}),\\[1ex] ({\bf u}, {\bf H})^h\in \widetilde{\mathcal{C}_b}(\mathbb{R}_+; \dot{B}_{p,1}^{\frac{N}{p}-1})\cap L^1(\mathbb{R}_+; \dot{B}_{p,1}^{\frac{N}{p}+1}), \end{array} \right. \end{equation} where $s\in \mathbb{R}, 1\leq q\leq \infty$.
Furthermore, we get for some constant $C=C(p,\mu,\lambda, \theta, P, \rho^\ast, {\bf B}^\ast)$, $$ \mathcal{X}_p(t)\leq C\mathcal{X}_{p,0}, $$ for any $t>0$, where \begin{equation}\label{1.4} \begin{split}
\mathcal{X}_p(t)&\equ \|(a, {\bf u}, {\bf H})\|_{\widetilde{L}^\infty(\dot{B}_{2,1}^{\frac{N}{2}-1})}^\ell+
\|(a, {\bf u}, {\bf H})\|_{{L}^1(\dot{B}_{2,1}^{\frac{N}{2}+1})}^\ell
+\|a\|_{\widetilde{L}^\infty(\dot{B}_{p,1}^{\frac{N}{p}})}^h
+\|a\|_{{L}^1(\dot{B}_{p,1}^{\frac{N}{p}})}^h\\[1ex]
&\quad+\|({\bf u},{\bf H})\|_{\widetilde{L}^\infty(\dot{B}_{p,1}^{\frac{N}{p}-1})}^h
+\|({\bf u},{\bf H})\|_{{L}^1(\dot{B}_{p,1}^{\frac{N}{p}+1})}^h. \end{split} \end{equation} \end{theo}
The natural next problem is to explore the large time asymptotic behavior of global solutions constructed above. Shi and Xu \cite{shi2018large} applied Fourier analysis techniques to give precise description for the large time asymptotic behavior of solutions, not only in Lebesgue spaces but also in a full family of Besov spaces with negative regularity indexes. In this paper, motivated by the works \cite{guo2012decay, shi2018large, strain2006almost, xin2018optimal}, we intend to establish the optimal decay for the compressible MHD equations in the $L^p$ type critical framework without the smallness assumption of low frequencies.
\section{Main results}\label{s:2} \setcounter{equation}{0}\setcounter{section}{2}\indent
Let us first rewrite system \eqref{1.1} as the nonlinear perturbation form of constant equilibrium state $(\rho^\ast, {\bf 0}, {\bf B}^\ast)$, looking at the nonlinearities as source terms. To simplify the statement of main results, we assume that $\rho^\ast=1$, ${\bf B}^\ast=I$ ($I$ is an arbitrary nonzero constant vector satisfying $|I|=1$), $P^\prime(\rho^\ast)=1$, $\theta=1$ and $\nu^\ast\equ 2\mu^\ast+\lambda^\ast=1$ (with $\mu^\ast\equ \mu(\rho^\ast)$ and $\lambda^\ast\equ \lambda(\rho^\ast)$). Consequently, in term of the new variables $(a, {\bf u}, {\bf H})$, system \eqref{1.1} becomes \begin{equation}\label{1.5} \left\{\begin{array}{ll} \partial_ta+{\rm div}{\bf u}=f,\\[1ex] \partial_t{\bf u}-\mathcal{A}{\bf u}+\nabla a+\nabla(I\cdot{\bf H})-I\cdot\nabla{\bf H}={\bf g},\\[1ex] \partial_t{\bf H}-\Delta {\bf H}+({\rm div}{\bf u})I-I\cdot\nabla{\bf u}={\bf m},\\[1ex] {\rm div}{\bf H}=0, \end{array} \right. \end{equation} where \begin{equation}\nonumber \begin{split} \left. \begin{array}{ll} &f\equ -{\rm div}(a{\bf u}),\\[2ex] &{\bf g}\equ -{\bf u}\cdot\nabla{\bf u}-\pi_1(a)\mathcal{A}{\bf u}-\pi_2(a)\nabla a+\frac{1}{1+a}{\rm div}(2\widetilde\mu(a) D({\bf u})+\widetilde\lambda(a){\rm div}{\bf u}\,{\rm Id})\\[1.5ex]\displaystyle &\quad\quad+\pi_1(a)(\nabla(I\cdot{\bf H})-I\cdot\nabla{\bf H})-\frac{1}{1+a}(\frac{1}{2}\nabla
|{\bf H}|^2-{\bf H}\cdot\nabla{\bf H}),\\[2ex] &{\bf m}\equ -{\bf H}({\rm div}{\bf u})+{\bf H}\cdot\nabla{\bf u}-{\bf u}\cdot\nabla{\bf H}, \end{array} \right. \end{split} \end{equation} with \begin{equation}\nonumber \begin{array}{ll}\displaystyle \mathcal{A}\equ \mu^\ast\Delta+(\lambda^\ast+\mu^\ast)\nabla{\rm div},\,\,{\rm here}\,\,2\mu^\ast +\lambda^\ast=1\,\,{\rm and}\,\,\mu^\ast>0,\\[1ex]\displaystyle \pi_1(a)\equ \frac{a}{1+a},\,\,\,\pi_2(a)\equ \frac{P^\prime(1+a)}{1+a}-1,\\[2ex]\displaystyle \widetilde\mu(a)\equ \mu(1+a)-\mu(1),\,\,\,\widetilde\lambda(a)\equ \lambda(1+a)-\lambda(1). \end{array} \end{equation} Note that $\pi_1, \pi_2, \widetilde\mu$ and $\widetilde\lambda$ are smooth functions satisfying $$ \pi_1(0)=\pi_2(0)=\widetilde\mu(0)=\widetilde\lambda(0)=0. $$
Denote $\Lambda^sf\equ \mathcal{F}^{-1}(|\xi|^s\mathcal{F}f)$ for $s\in\mathbb{R}$. Now, we state the main results as follows. \begin{theo}\label{th1.2} Let $N\geq 2$ and $p$ satisfy assumption \eqref{1.3}. Let $(\rho,{\bf u}, {\bf B})$ be the global solution addressed by Theorem \ref{th1.1}. If in addition $(a_0, {\bf u}_0, {\bf H}_0)^\ell\in \dot{B}_{2,\infty}^{-\sigma_1}\,(1-\frac{N}{2}<\sigma_1\leq \sigma_0
\equ \frac{2N}{p}-\frac{N}{2})$ such that $\|(a,{\bf u}_0,{\bf H}_0)\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell$ is bounded, then we have \begin{equation}\label{1.6}
\|(a,{\bf u},{\bf H})\|_{\dot{B}_{p,1}^\sigma}\lesssim (1+t)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{p})-\frac{\sigma+\sigma_1}{2}}, \end{equation} where $-\sigma_1 -\frac{N}{2}+\frac{N}{p}<\sigma\leq \frac{N}{p}-1$ for all $t\geq 0$. \end{theo}
By applying improved Gagliardo-Nirenberg inequalities, the optimal decay estimates of $\dot{B}_{2,\infty}^{-\sigma_1}$-$L^r$ type could be deduced as follows. \begin{col}\label{col1} Let those assumptions of Theorem \ref{th1.2} be fulfilled. Then the corresponding solution $(a,{\bf u},{\bf H})$ admits \begin{equation}\label{1.7}
\|\Lambda^l(a,{\bf u},{\bf H})\|_{L^r}\lesssim (1+t)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{r})-\frac{l+\sigma_1}{2}}, \end{equation} where $-\sigma_1 -\frac{N}{2}+\frac{N}{p}<l+\frac{N}{p}-\frac{N}{r}\leq \frac{N}{p}-1$ for $p\leq r\leq \infty$ and $t\geq 0$. \end{col}
In the following, we give some comments.
\begin{re} {\rm The low-frequency assumption of initial data in \cite{shi2018large} is at the endpoint $\sigma_0$ and the corresponding norm needs to be small enough, i.e., there exists a positive constant $c=c(p,\mu, \lambda, P, {\bf B}^\ast)$ such that $
\|(a,{\bf u}_0,{\bf H}_0)\|_{\dot{B}_{2,\infty}^{-\sigma_0}}^\ell\leq c\,\,\,{\rm with}\,\,\,\sigma_0\equ \frac{2N}{p}-\frac{N}{2}. $ Here, the new lower bound $1-\frac{N}{2}<\sigma_1\leq \sigma_0 $ enables us to enjoy larger freedom on the choice of $\sigma_1$, which allows to obtain more optimal decay estimates in the $L^p$ framework. In addition, the smallness of low frequencies is no longer needed in Theorem \ref{th1.2} and Corollary \ref{col1}.} \end{re} \begin{re}
{\rm In \cite{shi2018large}, there is a little loss on decay rates due to the use of different Sobolev embeddings at low (or high) frequencies. For example, when $\sigma_1=\sigma_0$, the result in \cite{shi2018large} presents that the solution itself decays to equilibrium in $L^p$ norm with the rate of $O(t^{-\frac{N}{p}+\frac{N}{4}})$, which is no faster than that of $O(t^{-\frac{N}{2p}})$ derived from Corollary \ref{col1} above. }
\end{re}
\begin{re}
{\rm Condition \eqref{1.3} may allow us to consider the case $p>N$, so that the regularity index $\frac{N}{p}-1$ of $({\bf u},{\bf H})$ becomes negative in physical dimensions $N=2, 3$. Our result thus applies to large highly oscillating initial velocities
and magnetic fields (see \cite{charve2010global, chen2010global} for more details). }
\end{re}
Let us give some illustration on the proof of main results. Based on the works of \cite{danchin2000global, guo2012decay, haspot2011existence, strain2006almost}, Xin and Xu \cite{xin2018optimal} developed a pure energy argument to establish the optimal decay for the barotropic compressible Navier-Stokes equations in the $L^p$ critical framework. Although the current proofs are in spirit of the works mentioned above,
we have some new observations. More precisely, as pointed out in \cite{xin2018optimal}, the nonlinear estimates in the low frequencies (that is $\|(f,{\bf g}, {\bf m})\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell$) play an important role in the process of proving Theorem \ref{th1.2}. They employed different Sobolev embeddings and interpolations to deal with the nonlinear terms in the non oscillation case $(2\leq p \leq N)$ and in the oscillation case $(p>N)$, respectively. Here, we develop a new non-classical product estimate in the low frequencies (see \eqref{4.2} below), which enables us to unify the estimates in the non oscillation case and the oscillation one. On the other hand, compared with \cite{xin2018optimal}, due to the appearance of the magnetic field, we need to take care of the nonlinear estimates for those terms including the magnetic field. To the end, we make full use of the structure of the MHD equations itself. For example, regarding the estimate of trinomial term $\frac{1}{1+a}(\frac{1}{2}\nabla |{\bf H}|^2-{\bf H}\cdot \nabla {\bf H})$, we are going to take full advantage of its symmetrical structure (see \eqref{4.200}-\eqref{4.44}, \eqref{5.37} and \eqref{5.38} below).
The rest of this paper is structured as follows. In Section \ref{s:3}, we recall some basic properties of the homogeneous Besov spaces. In Section \ref{s:4}, making use of the pure energy arguments, we investigate the low-frequency and high-frequency estimates of solutions. Section \ref{s:5} is devoted to the estimation of $L^2$-type Besov norms at low frequencies, which plays the key role in deriving the Lyapunov-type inequality for energy norms. Section \ref{s:6}, i.e., the last section presents the proofs of Theorem \ref{th1.2} and Corollary \ref{col1}.
Throughout the paper, $C$ stands for a harmless ``constant", and we sometimes write $A\lesssim B$ as an equivalent to $A\leq CB$. The notation $A\approx B$ means that $A\lesssim B$ and $B\lesssim A$. For any Banach space $X$ and $u, v\in X$, we agree that $\|(u,v)\|_X\equ \|u\|_X+\|v\|_X$. For $p\in [1,+\infty]$ and $T>0$, the notation $L^p(0,T;X)$ or $L^p_T(X)$ designates the set of measurable functions $f:[0,T]\rightarrow X$ with $t\mapsto
\|f(t)\|_X$ in $L^p(0,T)$, endowed with the norm $$
\|f\|_{L^p_T(X)}:=\bigl{\|}\|f\|_X\bigr{\|}_{L^p(0,T)}. $$ We agree that $\mathcal{C}([0,T];X)$ denotes the set of continuous functions from $[0,T]$ to $X$. \section{Preliminaries}\label{s:3}
\setcounter{equation}{0}\setcounter{section}{3}\indent We first recall the definition of homogeneous Besov spaces. They could be defined by using a dyadic partition of unity in Fourier variables called homogeneous Littlewood-Paley decomposition. To this end, choose a radial function $\varphi\in \mathcal{S}(\mathbb{R}^N)$ supported in $\mathcal{C}=\{\xi\in\mathbb{R}^N, \frac{3}{4}\leq |\xi|\leq \frac{8}{3}\}$ such that $ \sum_{j\in\mathbb{Z}}\varphi(2^{-j}\xi)=1\quad\!\!{\rm if}\quad\!\!\xi\neq 0. $ The homogeneous frequency localization operator $\dot{\Delta}_j$ and $\dot{S}_j$ are defined by $$ \dot{\Delta}_j u=\varphi (2^{-j}D)u, \quad\,\dot{S}_j u=\sum_{k\leq j-1}\dot{\Delta}_k u\quad\,{\rm for}\quad\,j\in\mathbb{Z}. $$ With our choice of $\varphi$, it is easy to see that \begin{equation}\label{1.8}
\dot{\Delta}_j\dot{\Delta}_kf=0\,\,\,{\rm if}\,\,\,|j-k|\geq 2,\,\,\,\,\,{\rm and}\,\,\,\,\dot{\Delta}_j(\dot{S}_{k-1}\dot{\Delta}_kf=0)
\,\,\,{\rm if}\,\,\,|j-k|\geq 5. \end{equation}
Let us denote the space $\mathcal{Y}^\prime(\mathbb{R}^N)$ by the quotient space of $\mathcal{S}^\prime(\mathbb{R}^N)/\mathcal{P}$ with the polynomials space $\mathcal{P}$. The formal equality $ u=\sum_{k\in\mathbb{Z}}\dot{\Delta}_k u $ holds true for $u\in \mathcal{Y}^\prime(\mathbb{R}^N)$ and is called the homogeneous Littlewood-Paley decomposition.
We then define, for $s\in\mathbb{R}$, $1\leq p, r\leq +\infty$, the homogeneous Besov space $$
\dot{B}_{p,r}^s={\Big\{}f\in\mathcal{Y}^\prime(\mathbb{R}^N): \|f\|_{\dot{B}_{p,r}^s}<+\infty{\Big\}}, $$ where $$
\|f\|_{\dot{B}_{p,r}^s}:=\|2^{ks}\|\dot{\Delta}_k f\|_{L^p}\|_{\ell^r}. $$
When employing parabolic estimates in Besov spaces, it is somehow natural to take the time-Lebesgue norm before performing the summation for computing the Besov norm. So we next introduce the following Besov-Chemin-Lerner space $\widetilde{L}_T^\rho(\dot{B}_{p,r}^s)$ (see\,\cite{chemin1995flot}): $$ \widetilde{L}_T^\rho(\dot{B}_{p,r}^s)={\Big\{}f\in (0,+\infty)\times\mathcal{Y}^\prime(\mathbb{R}^N):
\|f\|_{\widetilde{L}_T^\rho(\dot{B}_{p,r}^s)}<+\infty{\Big\}}, $$ where $$
\|f\|_{\widetilde{L}_T^\rho(\dot{B}_{p,r}^s)}:=\bigl{\|}2^{ks}\|\dot{\Delta}_k f(t)\|_{L^\rho(0,T;L^p)}\bigr{\|}_{\ell^r}. $$ The index $T$ will be omitted if $T=+\infty$ and we shall denote by $\widetilde{\mathcal{C}}_b([0,T]; \dot{B}^s_{p,r})$ the subset of functions of $\widetilde{L}^\infty_T(\dot{B}^s_{p,r})$ which are also continuous from $[0,T]$ to $\dot{B}^s_{p,r}$.
A direct application of Minkowski's inequality implies that $$ L_T^\rho(\dot{B}_{p,r}^s)\hookrightarrow \widetilde{L}_T^\rho(\dot{B}_{p,r}^s)\,\,\,{\rm if}\,\,\,r\geq \rho, \quad\,{\rm and}\quad\, \widetilde{L}_T^\rho(\dot{B}_{p,r}^s)\hookrightarrow {L}_T^\rho(\dot{B}_{p,r}^s)\,\,\,{\rm if}\,\,\,\rho\geq r. $$ We will repeatedly use the following Bernstein's inequality throughout the paper: \begin{lem}\label{le2.1} {\rm(}see {\rm \cite{chemin1998perfect}}{\rm )} Let $\mathcal{C}$ be an annulus and $\mathcal{B}$ a ball, $1\leq p\leq q\leq +\infty$. Assume that $f\in L^p(\mathbb{R}^N)$, then for any nonnegative integer $k$, there exists constant $C$ independent of $f$, $k$ such that $$
{\rm supp} \hat{f}\subset\lambda \mathcal{B}\Rightarrow\|D^k f\|_{L^q(\mathbb{R}^N)}:=\sup_{|\alpha|=k}\|\partial^\alpha f\|_{L^q(\mathbb{R}^N)}\leq C^{k+1}\lambda^{k+N(\frac{1}{p}-\frac{1}{q})}\|f\|_{L^p(\mathbb{R}^N)}, $$ \begin{equation}\nonumber
{\rm supp} \hat{f}\subset\lambda\mathcal{C}\Rightarrow C^{-k-1}\lambda^k\|f\|_{L^p(\mathbb{R}^N)}\leq \|D^k f\|_{L^p(\mathbb{R}^N)}\leq C^{k+1}\lambda^k\|f\|_{L^p(\mathbb{R}^N)}. \end{equation} \end{lem}
More generally, if $v$ satisfies ${\rm Supp}\mathcal{F}v\subset\{\xi\in\mathbb{R}^N: R_1\lambda\leq |\xi|\leq R_2\lambda\}$ for some $0< R_1<R_2$ and $\lambda>0$, then for any smooth homogeneous of degree $m$ function $A$ on $\mathbb{R}^N\backslash\{0\}$ and $1\leq q\leq \infty$, it holds that (see e.g. Lemma 2.2 in \cite{bahouri2011fourier}): \begin{equation}\label{2.100}
\|A(D)v\|_{L^q}\lesssim \lambda^m\|v\|_{L^q}. \end{equation}
The following nonlinear generalization of \eqref{2.100} will be applied (see Lemma 8 in \cite{danchin2010well-posedness}):
\begin{prop}\label{pr2.3}
If ${\rm Supp}\mathcal{F}f\subset\{\xi\in\mathbb{R}^N: R_1\lambda\leq |\xi|\leq R_2\lambda\}$ then there exists $c$ depending only on $N, R_1$ and $R_2$ so that for all $1<p<\infty$, $$
c\lambda^2\left(\frac{p-1}{p^2}\right)\int_{\mathbb{R}^N}|f|^pdx\leq (p-1)\int_{\mathbb{R}^N}|\nabla f|^2|f|^{p-2}dx
=-\int_{\mathbb{R}^N}\Delta f|f|^{p-2}fdx. $$ \end{prop}
Let us now state some classical properties for the Besov spaces. \begin{prop}\label{pr2.1} The following properties hold true:
{\rm 1)} Derivation: There exists a universal constant $C$ such that $$
C^{-1}\|f\|_{\dot{B}_{p,r}^s}\leq \|\nabla f\|_{\dot{B}_{p,r}^{s-1}}\leq C\|f\|_{\dot{B}_{p,r}^s}. $$
{\rm 2)} Sobolev embedding: If $1\leq p_1\leq p_2\leq\infty$ and $1\leq r_1\leq r_2\leq\infty$, then $\dot{B}_{p_1, r_1}^s\hookrightarrow \dot{B}_{p_2, r_2}^{s-\frac{N}{p_1}+\frac{N}{p_2}}$.
{\rm 3)} Real interpolation: $\|f\|_{\dot{B}_{p,r}^{\theta s_1+(1-\theta)s_2}}\leq \|f\|_{\dot{B}_{p,r}^{s_1}}^{\theta}\|f\|_{\dot{B}_{p,r}^{s_2}}^{1-\theta}$.
{\rm 4)} Algebraic properties: for $s>0$, $\dot{B}_{p,1}^s\cap L^\infty$ is an algebra.
{\rm 5)} Scaling properties:
\quad\quad {\rm(a)} for all $\lambda>0$ and $f\in\dot{B}_{p,1}^s$, we have $$
\|f(\lambda\cdot)\|_{\dot{B}_{p,1}^s}\approx \lambda^{s-\frac{N}{p}}\|f\|_{\dot{B}_{p,1}^s}, $$
\quad\quad {\rm(b)} for $f=f(t, x)$ in $L^r(0, T; \dot{B}_{p,1}^s)$, we have $$
\|f(\lambda^a\cdot, \lambda^b\cdot)\|_{L_T^r(\dot{B}_{p,1}^s)}\approx
\lambda^{b(s-\frac{N}{p})-\frac{a}{r}}\|f\|_{L_{\lambda^aT}^r(\dot{B}_{p,1}^s)}. $$ \end{prop} Next we recall a few nonlinear estimates in Besov spaces which may be obtained by means of paradifferential calculus. Firstly introduced by Bony in \cite{bony1981calcul}, the paraproduct between $f$ and $g$ is defined by $$ T_f g=\sum_{q\in\mathbb{Z}}\dot{S}_{q-1}f\dot{\Delta}_q g, $$ and the remainder is given by $$ R(f,g)=\sum_{q\in\mathbb{Z}}\dot{\Delta}_q f\widetilde{\dot{\Delta}}_q g\,\,\,\,{\rm with}\,\,\,\,\widetilde{\dot{\Delta}}_q g:=(\dot{\Delta}_{q-1}+\dot{\Delta}_{q}+\dot{\Delta}_{q+1})g. $$ We have the following so-called Bony's decomposition: \begin{equation}\label{2.3} fg=T_g f+T_f g+R(f,g). \end{equation}
The paraproduct $T$ and the remainder $R$ operators satisfy the following continuous properties (see e.g. \cite{bahouri2011fourier}).
\begin{prop}\label{pr2.2} Suppose that $s\in\mathbb{R}, \sigma>0,$ and $1\leq p, p_1, p_2, r, r_1, r_2\leq \infty$. Then we have
{\rm 1)} The paraproduct $T$ is a bilinear, continuous operator from $L^\infty\times\dot{B}_{p,r}^s$ to $\dot{B}_{p,r}^s$, and from $\dot{B}_{\infty, r_1}^{-\sigma}\times\dot{B}_{p,r_2}^s$ to $\dot{B}_{p,r}^{s-\sigma}$ with $\frac{1}{r}=\min\{1, \frac{1}{r_1}+\frac{1}{r_2}\}$.
{\rm 2)} The remainder $R$ is bilinear continuous from $\dot{B}_{p_1,r_1}^{s_1}\times\dot{B}_{p_2,r_2}^{s_2}$ to $\dot{B}_{p,r}^{s_1+s_2}$ with $s_1+s_2>0$, $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}\leq 1$, and $\frac{1}{r}=\frac{1}{r_1}+\frac{1}{r_2}\leq 1$.
\end{prop}
\medbreak From \eqref{2.3} and Proposition \ref{pr2.2}, we may deduce the following two corollaries concerning the product estimates. \begin{col}\label{co2.1} {\rm(}\cite{bahouri2011fourier}, \cite{danchin2002zero}{\rm )} {\rm\,(i)} Let $s>0$ and $1\leq p,r\leq \infty$. Then $\dot{B}_{p,r}^s\cap L^\infty$ is an algebra and $$
\|uv\|_{\dot{B}_{p,r}^s}\lesssim \|u\|_{L^\infty}\|v\|_{\dot{B}_{p,r}^s}+\|v\|_{L^\infty}\|u\|_{\dot{B}_{p,r}^s}. $$
{\rm (ii)}\,If $u\in\dot{B}_{p_1,1}^{s_1}$ and $v\in\dot{B}_{p_2,1}^{s_2}$ with $1\leq p_1\leq p_2\leq \infty,~s_1\leq \frac{N}{p_1},~s_2\leq \frac{N}{p_2}$ and $s_1+s_2>0$, then $uv\in\dot{B}_{p_2,1}^{s_1+s_2-\frac{N}{p_1}}$ and there exists a constant $C$, depending only on $N, s_1, s_2, p_1$ and $p_2$, such that \begin{equation}\label{2.1}
\|uv\|_{\dot{B}_{p_2,1}^{s_1+s_2-\frac{N}{p_1}}}\leq C\|u\|_{\dot{B}_{p_1,1}^{s_1}}
\|v\|_{\dot{B}_{p_2,1}^{s_2}}. \end{equation} \end{col}
\begin{col}\label{co2.2} Let $\sigma_1$ and $p$ satisfy the conditions as in Theorem \ref{th1.2}, that is, $1-\frac{N}{2}<\sigma_1\leq\frac{2N}{p}-\frac{N}{2}\,(N\geq 2)$ and $p$ fulfills \eqref{1.3}, then we have \begin{equation}\nonumber
\|fg\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}\lesssim\|f\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\|g\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}, \end{equation} and \begin{equation}\nonumber
\|fg\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}\lesssim\|f\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\|g\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}. \end{equation} \end{col} Here, the estimates in Corollary \ref{co2.1} are classical, and the non-classical estimates in Corollary \ref{co2.2} are used to establish the evolution of Besov norms at low frequencies in our paper.
We also need the following composition lemma (see \cite{bahouri2011fourier,danchin2000global, runst1996sobolev}). \begin{prop}\label{pra.4}
Let $F:\mathbb{R}\rightarrow \mathbb{R}$ be smooth with $F(0)=0$. For all $1\leq p,r\leq \infty$ and $s>0$, it holds that $F(u)\in \dot{B}_{p,r}^s\cap L^\infty$ for $u\in\dot{B}_{p,r}^s\cap L^\infty$, and $$
\|F(u)\|_{\dot{B}_{p,r}^s}\leq C\|u\|_{\dot{B}_{p,r}^s} $$
with $C$ depending only on $\|u\|_{L^\infty}$, $F^\prime$ (and higher derivatives), $s, p$ and $N$.
In the case $s>-\min(\frac{N}{p}, \frac{N}{p^\prime})$, then $u\in\dot{B}_{p,r}^s\cap\dot{B}_{p,1}^{\frac{N}{p}}$ implies that $F(u)\in\dot{B}_{p,r}^s\cap\dot{B}_{p,1}^{\frac{N}{p}}$, and $$
\|F(u)\|_{\dot{B}_{p,r}^s}\leq C(1+\|u\|_{\dot{B}_{p,1}^{\frac{N}{p}}})\|u\|_{\dot{B}_{p,r}^s}, $$ where $\frac{1}{p}+\frac{1}{p^\prime}=1$. \end{prop} The following commutator estimates (see \cite{danchin2016optimal}) have been employed in the high-frequency estimate for proving Theorem \ref{th1.2}. \begin{prop}\label{pra.5} Let $1\leq p, p_1\leq \infty$ and \begin{equation}\label{A4} -\min\Big{\{}\frac{N}{p_1}, \frac{N}{p^\prime}\Big{\}}<\sigma\leq 1+ \min\Big{\{}\frac{N}{p}, \frac{N}{p_1}\Big{\}}. \end{equation} There exists a constant $C>0$ depending only on $\sigma$ such that for all $j\in\mathbb{Z}$ and $i\in \{1,\cdots, N\}$, we have \begin{equation}\label{A5}
\|[v\cdot\nabla,\partial_i{\dot{\Delta}_j}]a\|_{L^p}\leq Cc_j 2^{-j(\sigma-1)}
\|\nabla v\|_{\dot{B}_{p_1,1}^{\frac{N}{p_1}}}\|\nabla a\| _{\dot{B}_{p,1}^{\sigma-1}}, \end{equation} where the commutator $[\cdot,\cdot]$ is defined by $[f,g]=fg-gf$, and
$(c_j)_{j\in\mathbb{Z}}$ denotes a sequence such that $\|(c_j)\|_{\ell^1}\leq 1$ and $\frac{1}{p^\prime}+\frac{1}{p}=1$. \end{prop} Finally, we list the optimal regularity estimates for the heat equation (see e.g. \cite{bahouri2011fourier}). \begin{prop}\label{pra.6} Let $\sigma\in\mathbb{R},\,\, (p,r)\in [1,\infty]^2$ and $1\leq \rho_2\leq \rho_1\leq \infty$. Let $u$ satisfy \begin{equation}\label{A6} \left\{\begin{array}{ll}
\displaystyle
\partial_t u-\mu \Delta u=f,\\ \displaystyle u|_{t=0}=u_0. \end{array} \right. \end{equation} Then for all $T>0$, the following a prior estimate is satisfied: \begin{equation}\label{A7}
\mu^{\frac{1}{\rho_1}}\|u\|_{\widetilde{L}_T^{\rho_1}(\dot{B}_{p,r}^{\sigma+\frac{2}{\rho_1}})}
\lesssim \|u_0\|_{\dot{B}_{p,r}^\sigma}+\mu^{\frac{1}{\rho_2}-1}
\|f\|_{\widetilde{L}_T^{\rho_2}(\dot{B}_{p,r}^{\sigma-2+\frac{2}{\rho_2}})}. \end{equation} \end{prop}
\section{Low-frequency and high-frequency estimates}\label{s:4} In this section, we derive the low-frequency and high-frequency estimates to system \eqref{1.5}. Based on this, a Lyapunov-type inequality for energy norms could be deduced in next section. \subsection{Low-frequency estimates} \begin{lemma}\label{le1} Let $k_0$ be some integer. Then it holds that for all $t\geq 0$, \begin{equation}\label{3.100}
\frac{d}{dt}\|(a,{\bf u},{\bf H})\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell
+\|(a,{\bf u},{\bf H})\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell\lesssim \|(f,{\bf g},{\bf m})\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell \end{equation} where \begin{equation}\nonumber
\|z\|_{\dot{B}_{2,1}^s}^\ell\equ \sum_{k\leq k_0}2^{ks}\|\dot{\Delta}_kz\|_{L^2}\,\,\,{\rm for}\,\,\,s\in\mathbb{R}. \end{equation} \end{lemma}
\begin{proof} The proof of Lemma \ref{le1} is similar to that in \cite{shi2019global}. Set \begin{equation}\label{3.104} \omega=\Lambda^{-1}{\rm div}{\bf u},\,\, {\bf \Omega}=\Lambda^{-1}{\rm curl}{\bf u}, \,\,\,{\rm and}\,\,\, {\bf E}=\Lambda^{-1}{\rm curl}{\bf H}, \end{equation} where ${\rm curl{\bf v}}\equ (\partial_jv_i-\partial_iv_j)_{ij}$
is $N\times N$ matrix and Let $\Lambda^sz\equ \mathcal{F}^{-1}(|\xi|^s\mathcal{F}z)\,(s\in\mathbb{R})$. So system \eqref{1.5} becomes \begin{equation}\label{3.1} \left\{ \begin{array}{ll} \partial_t a+\Lambda\omega=F,\\[1ex] \partial_t\omega-\Delta\omega-\Lambda a-I\cdot{\rm div}{\bf E}=G,\\[1ex] \partial_t{\bf \Omega}-\mu^\ast\Delta {\bf \Omega}-I\cdot\nabla {\bf E}={\bf L},\\[1ex] \partial_t{\bf E}-\Delta{\bf E}+{\rm curl}(\omega I)-I\cdot\nabla{\bf \Omega}={\bf M},\\[1ex] {\bf u}=-\Lambda^{-1}\nabla\omega+\Lambda^{-1}{\rm div}{\bf\Omega},\,\,{\bf H}=\Lambda^{-1}{\rm div}{\bf E},\,\, {\rm div}{\bf H}=0, \end{array} \right. \end{equation} where \begin{equation}\label{3.101} F=f, \,\,\,G=\Lambda^{-1}{\rm div}{\bf g},\,\,\,{\bf L}=\Lambda^{-1}{\rm curl}{\bf g},\,\,\,{\bf M}=\Lambda^{-1}{\rm curl}{\bf m}. \end{equation} Applying the operator $\dot{\Delta}_k$ to \eqref{3.1} and denoting $n_k\equ \dot{\Delta}_kn$, one has for all $k\in \mathbb{Z}$, \begin{equation}\label{3.2} \left\{ \begin{array}{ll} \partial_t a_k+\Lambda\omega_k=F_k,\\[1ex] \partial_t\omega_k-\Delta\omega_k-\Lambda a_k-I\cdot{\rm div}{\bf E}_k=G_k,\\[1ex] \partial_t{\bf \Omega}_k-\mu^\ast\Delta {\bf \Omega}_k-I\cdot\nabla {\bf E}_k={\bf L}_k,\\[1ex] \partial_t{\bf E}_k-\Delta{\bf E}_k+{\rm curl}(\omega_k I)-I\cdot\nabla{\bf \Omega_k}={\bf M}_k. \end{array} \right. \end{equation}
Taking the $L^2$ scalar product of \eqref{3.2}$_1$ with $a_k$, \eqref{3.2}$_2$ with $\omega_k$,
\eqref{3.2}$_3$ with ${\bf\Omega}_k$, and \eqref{3.2}$_4$ with ${\bf E}_k$, we derive that \begin{equation}\label{3.3}
\frac{1}{2}\frac{d}{dt}\|a_k\|_{L^2}^2+(\Lambda\omega_k,a_k)=(F_k, a_k), \end{equation} \begin{equation}\label{3.4}
\frac{1}{2}\frac{d}{dt}\|\omega_k\|_{L^2}^2+\|\Lambda \omega_k\|_{L^2}^2-(\Lambda a_k,\omega_k) -(I\cdot{\rm div}{\bf E}_k, \omega_k)=(G_k, \omega_k), \end{equation} \begin{equation}\label{3.5}
\frac{1}{2}\frac{d}{dt}\|{\bf \Omega}_k\|_{L^2}^2+\mu^\ast\|\Lambda {\bf \Omega}_k\|_{L^2}^2 -(I\cdot\nabla{\bf E}_k, {\bf\Omega}_k)=({\bf L}_k, {\bf \Omega}_k), \end{equation} \begin{equation}\label{3.6}
\frac{1}{2}\frac{d}{dt}\|{\bf E}_k\|_{L^2}^2+\|\Lambda E_k\|_{L^2}^2+({\rm curl}(\omega_kI), {\bf E}_k) -(I\cdot\nabla{\bf \Omega}_k, {\bf E}_k)=({\bf M}_k, {\bf E}_k). \end{equation} Noticing that $$ (\Lambda \omega_k, a_k)=(\Lambda a_k, \omega_k),\,\,({\rm curl}(\omega_kI), {\bf E}_k) =2(I\cdot{\rm div}{\bf E}_k, \omega_k)\,\, {\rm and}\,\,(I\cdot\nabla{\bf E}_k, {\bf\Omega}_k) =-(I\cdot\nabla{\bf \Omega}_k, {\bf E}_k). $$ Combing \eqref{3.3} to \eqref{3.6}, we have \begin{equation}\label{3.7} \begin{split}
&\quad\frac{1}{2}\frac{d}{dt}(\|a_k\|_{L^2}^2+\|\omega_k\|_{L^2}^2+\frac{1}{2}\|{\bf \Omega}_k\|_
{L^2}^2+\frac{1}{2}\|{\bf E}_k\|_{L^2}^2)+\|\Lambda\omega_k\|_{L^2}^2+\frac{1}{2}\mu^\ast\|\Lambda{\bf \Omega}_k\|_
{L^2}^2+\frac{1}{2}\|\Lambda{\bf E}_k\|_ {L^2}^2\\[1ex] &=(F_k, a_k)+(G_k, \omega_k)+\frac{1}{2}({\bf L}_k, {\bf \Omega}_k)+\frac{1}{2}({\bf M}_k, {\bf E}_k). \end{split} \end{equation}
Taking the $L^2$ scalar product of \eqref{3.2}$_1$ with $\Lambda\omega_k$, \eqref{3.2}$_2$ with $\Lambda a_k$, and \eqref{3.2}$_1$ with $\Lambda^2 a_k$, we obtain, respectively, that \begin{equation}\nonumber \left. \begin{array}{ll}
(\partial_t a_k,\Lambda\omega_k)+\|\Lambda\omega_k\|_{L^2}^2=(F_k,\Lambda\omega_k),\\[1ex]
(\partial_t\omega_k, \Lambda a_k)+(\Lambda_k^2\omega_k, \Lambda a_k)-\|\Lambda a_k\|_{L^2}^2-(I\cdot{\rm div}{\bf E}_k,\Lambda a_k)=(G_k,\Lambda a_k),\\[1ex]\displaystyle
\frac{1}{2}\frac{d}{dt}\|\Lambda a_k\|_{L^2}^2+(\Lambda \omega_k, \Lambda^2a_k)=(\Lambda F_k, \Lambda a_k) \end{array} \right. \end{equation} which yields \begin{equation}\label{3.8} \begin{split}
&\quad\frac{1}{2}\frac{d}{dt}\left(\|\Lambda a_k\|_{L^2}^2-2(a_k,\Lambda\omega_k)\right)+\|\Lambda a_k\|_
{L^2}^2-\|\Lambda\omega_k\|_{L^2}^2+(I\cdot{\rm div}{\bf E}_k, \Lambda a_k)\\[1ex] &=(\Lambda F_k, \Lambda a_k)-(F_k, \Lambda \omega_k)-(G_k, \Lambda a_k). \end{split} \end{equation} Set $$
\mathcal{J}_k^2(t)\equ \|a_k\|_{L^2}^2+\|\omega_k\|_{L^2}^2+\frac{1}{2}\|{\bf \Omega}_k\|_{L^2}^2
+\frac{1}{2}\|{\bf E}_k\|_{L^2}^2+\gamma\left(\|\Lambda a_k\|_{L^2}^2-2(a_k, \Lambda \omega_k)\right) $$ for some $\gamma>0$, we get from \eqref{3.7} and \eqref{3.8} that \begin{equation}\label{3.105} \begin{split}
&\quad\frac{1}{2}\frac{d}{dt}\mathcal{J}_k^2(t)+(1-\gamma)\|\Lambda\omega_k\|_{L^2}^2
+\frac{1}{2}\mu^\ast\|\Lambda{\bf \Omega}_k\|_{L^2}^2+\frac{1}{2}\|\Lambda{\bf E}_k\|_{L^2}^2
+\gamma\left(\|\Lambda a_k\|_{L^2}^2+(I\cdot{\rm div}{\bf E}_k, \Lambda a_k)\right)\\[1ex] &=(F_k, a_k)+(G_k, \omega_k)+\frac{1}{2}({\bf L}_k, {\bf \Omega}_k)+\frac{1}{2}({\bf M}_k, {\bf E}_k) +\gamma\Big[(\Lambda F_k, \Lambda a_k)-(F_k, \Lambda \omega_k)-(G_k, \Lambda a_k)\Big]. \end{split} \end{equation} It follows from Young's inequality that for $k\leq k_0$ \begin{equation}\label{3.103}
\mathcal{J}_k^2(t)\thickapprox\|(a_k, \Lambda a_k, \omega_k, {\bf \Omega}_k, {\bf E}_k)\|_{L^2}^2
\thickapprox\|(a_k, \omega_k, {\bf \Omega}_k, {\bf E}_k)\|_{L^2}^2. \end{equation} Consequently, in the low-frequency case, we get from \eqref{3.105} that \begin{equation}\label{3.9}
\frac{1}{2}\frac{d}{dt}\mathcal{J}_k^2+2^{2k}\mathcal{J}_k^2\lesssim\|(F_k, G_k, {\bf L}_k, {\bf M}_k)\|_{L^2}\mathcal{J}_k, \end{equation} which implies that \begin{equation}\label{3.10}
\frac{d}{dt}\mathcal{J}_k+2^{2k}\mathcal{J}_k\lesssim\|(F_k, G_k, {\bf L}_k, {\bf M}_k)\|_{L^2} \end{equation} for $k\leq k_0$. Therefore, multiplying both sides by $2^{k(N/2-1)}$, summing up on $k\leq k_0$ and using \eqref{3.101} yield \eqref{3.100}. \end{proof} \subsection{High-frequency estimates} In the high-frequency regime, the term ${\rm div}(au)$ would cause a loss of one derivative as there is no smoothing effect for $a$. To get around this difficulty, as in \cite{haspot2011existence}, we introduce the effective velocity \begin{equation}\label{3.102} {\bf w}\equ \nabla (-\Delta)^{-1}(a-{\rm div}{\bf u}). \end{equation} \begin{lemma}\label{le2} Let $k_0$ be chosen suitably large. Then it holds that for all $t\geq 0$, \begin{equation}\label{3.11} \begin{split}
&\quad\frac{d}{dt}\|(\nabla a,{\bf u},{\bf H})\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h+\Big(\|\nabla a\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h
+\|({\bf u},{\bf H})\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\Big)\\[1ex]
&\lesssim \|f\|_{\dot{B}_{p,1}^{\frac{N}{p}-2}}^h+\|({\bf g},{\bf m})\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h
+\|\nabla{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}, \end{split} \end{equation} where \begin{equation}\nonumber
\|z\|_{\dot{B}_{2,1}^s}^h\equ \sum_{k\geq k_0+1}2^{ks}\|\dot{\Delta}_kz\|_{L^2}\,\,\,{\rm for}\,\,\,s\in\mathbb{R}. \end{equation} \end{lemma} \begin{proof} Let $\mathcal{P}\equ {\rm Id}+\nabla(-\Delta)^{-1}{\rm div}$ be the Leray projector onto divergence-free vector fields, and ${\bf w}$ be defined in \eqref{3.102}. Then from system \eqref{1.5}, we get that $\mathcal{P}{\bf u}, {\bf H}$ and ${\bf w}$ satisfy a heat equation, and $a$ satisfies a damped transport equation as follows. \begin{equation}\label{3.12} \left\{ \begin{array}{ll} \partial_t\mathcal{P}{\bf u}-\mu^\ast\Delta\mathcal{P}{\bf u}=\mathcal{P}{\bf g}+I\cdot\nabla{\bf H},\\[1ex] \partial_t{\bf H}-\Delta{\bf H}={\bf m}-({\rm div}{\bf w})I+I\cdot\nabla{\bf w}-aI-I\cdot\nabla^2(-\Delta)^{-1}a+I\cdot\nabla\mathcal{P}{\bf u},\\[1ex] \partial_t{\bf w}-\Delta{\bf w}=\nabla(-\Delta)^{-1}(f-{\rm div}{\bf g})+{\bf w}-(-\Delta)^{-1}\nabla a-\nabla(I\cdot{\bf H}),\\[1ex] \partial_t a+a=-{\rm div}(a{\bf u})-{\rm div}{\bf w}. \end{array} \right. \end{equation} Applying $\dot{\Delta}_k$ to \eqref{3.12}$_1$ yields for all $k\in\mathbb{Z}$, $$ \partial_t\mathcal{P}{\bf u}_k-\mu^\ast\Delta\mathcal{P}{\bf u}_k=\mathcal{P}{\bf g}_k+I\cdot\nabla{\bf H}_k. $$
Then, multiplying each component of the above equation by $|(\mathcal{P}u_k)^i|^{p-2}(\mathcal{P}u_k)^i$ and integrating over $\mathbb{R}^N$ gives for $i=1,2,\cdots,N$, \begin{equation}\nonumber \begin{split}
&\quad\frac{1}{p}\frac{d}{dt}\|\mathcal{P}u_k^i\|_{L^p}^p-\mu^\ast
\int_{\mathbb{R}^N}\Delta(\mathcal{P}u_k)^i|(\mathcal{P}u_k)^i|^{p-2}(\mathcal{P}u_k)^idx\\[1ex]
&=\int_{\mathbb{R}^N}|(\mathcal{P}u_k)^i|^{p-2}(\mathcal{P}u_k)^i(\mathcal{P}g_k^i+I_j\partial_j H_k^i)dx. \end{split} \end{equation} Applying Proposition \ref{pr2.3} and summing on $i=1,2,\cdots,N$, we get for some constant $c_p$ depending only on $p$ that \begin{equation}\nonumber
\frac{1}{p}\frac{d}{dt}\|\mathcal{P}{\bf u}_k\|_{L^p}^p+c_p\mu^\ast2^{2k}\|\mathcal{P}{\bf u}_k\|_{L^p}^p\leq
(\|\mathcal{P}{\bf g}_k\|_{L^p}+C2^k\|{\bf H}_k\|_{L^p})\|\mathcal{P}{\bf u}_k\|_{L^p}^{p-1} \end{equation} which leads to \begin{equation}\label{3.13}
\frac{d}{dt}\|\mathcal{P}{\bf u}_k\|_{L^p}+c_p\mu^\ast2^{2k}\|\mathcal{P}{\bf u}_k\|_{L^p}\leq
\|\mathcal{P}{\bf g}_k\|_{L^p}+C2^k\|{\bf H}_k\|_{L^p}. \end{equation} On the other hand, from \eqref{3.12}$_2$ and \eqref{3.12}$_3$, we argue exactly as for proving \eqref{3.13} and obtain that \begin{equation}\label{3.14}
\frac{d}{dt}\|{\bf H}_k\|_{L^p}+c_p2^{2k}\|{\bf H}_k\|_{L^p}\leq
\|{\bf m}_k\|_{L^p}+C2^k\|({\bf w}_k,\mathcal{P}{\bf u}_k)\|_{L^p}+C2^{-k}\|\nabla a_k\|_{L^p} \end{equation} and \begin{equation}\label{3.15}
\frac{d}{dt}\|{\bf w}_k\|_{L^p}+c_p2^{2k}\|{\bf w}_k\|_{L^p}\leq C2^{-k}\|{f}_k\|_{L^p}+\|({\bf g}_k,{\bf w}_k)\|_{L^p}+C2^k\|{\bf H}_k\|_{L^p}+C2^{-2k}\|\nabla a_k\|_{L^p}. \end{equation} Since the function $a$ fulfills the damped transport equation \eqref{3.12}$_4$, then performing the operator $\partial_i\dot{\Delta}_k$ to \eqref{3.12}$_4$ and denoting $R_k^i\equ [{\bf u}\cdot\nabla, \partial_i\dot{\Delta}_k]a$, one has \begin{equation}\label{3.16} \partial_t\partial_ia_k+{\bf u}\cdot\nabla\partial_ia_k+\partial_ia_k=-\partial_i\dot{\Delta}_k(a{\rm div}{\bf u}) -\partial_i{\rm div}{\bf w}_k+R_k^i,\,\,\,i=1,2,\cdots,N.
\end{equation} Multiplying both sides of \eqref{3.16} by $|\partial_ia_k|^{p-2}\partial_ia_k$, integrating on $\mathbb{R}^N$, and performing an integration by parts in the second term, we arrive at \begin{equation}\nonumber \begin{split}
\frac{1}{p}\frac{d}{dt}\|\partial_ia_k\|_{L^p}^p+\|\partial_ia_k\|_{L^p}^p=\frac{1}{p}&\int_{\mathbb{R}^N}
{\rm div}{\bf u}|\partial_ia_k|^pdx\\[1ex] &\quad+\int_{\mathbb{R}^N}(R_k^i-\partial_i\dot{\Delta}_k(a{\rm div}{\bf u})
-\partial_i{\rm div}{\bf w}_k)|\partial_ia_k|^{p-2}\partial_ia_kdx. \end{split} \end{equation} Summing up on $i=1,2,\cdots,N$ and applying H\"{o}lder and Bernstein inequalities imply \begin{equation}\label{3.17} \begin{split}
\frac{1}{p}\frac{d}{dt}\|\nabla a_k\|_{L^p}^p+\|\nabla a_k\|_{L^p}^p\leq &\Big(\frac{1}{p}\|{\rm div}{\bf u}\|_{L^\infty}
\|\nabla a_k\|_{L^p}+\|\nabla\dot{\Delta}_k(a{\rm div}{\bf u})\|_{L^p}\\[1ex]
&\quad\quad\quad\quad\quad\quad+C2^{2k}\|{\bf w}_k\|_{L^p}+\|R_k\|_{L^p}
\Big)\|\nabla a_k\|_{L^p}^{p-1}, \end{split} \end{equation} which leads to \begin{equation}\label{3.18} \begin{split}
&\quad\frac{1}{p}\frac{d}{dt}\|\nabla a_k\|_{L^p}+\|\nabla a_k\|_{L^p}\\[1ex]
&\leq \frac{1}{p}\|{\rm div}{\bf u}\|_{L^\infty}
\|\nabla a_k\|_{L^p}+\|\nabla\dot{\Delta}_k(a{\rm div}{\bf u})\|_{L^p}
+C2^{2k}\|{\bf w}_k\|_{L^p}+\|R_k\|_{L^p}. \end{split} \end{equation} Adding \eqref{3.18} (multiplying by $\beta c_p$ for some $\beta>0$), \eqref{3.13}, \eqref{3.14} and \eqref{3.15} together gives \begin{equation}\nonumber \begin{split}
&\quad\frac{d}{dt}\left(\|(\mathcal{P}{\bf u}_k, {\bf w}_k, {\bf H}_k)\|_{L^p}+\beta c_p\|\nabla a_k\|_{L^p}\right)
+c_p2^{2k}(\mu^\ast\|\mathcal{P}{\bf u}_k\|_{L^p}+\|({\bf w}_k, {\bf H}_k)\|_{L^p})+\beta c_p\|\nabla a_k\|_{L^p}\\[1ex]
&\leq\|\mathcal{P}{\bf g}_k\|_{L^p}+C2^k\|{\bf H}_k\|_{L^p}+\|{\bf m}_k\|_{L^p}+C2^k\|({\bf w}_k,\mathcal{P}{\bf u}_k)\|_{L^p}+C2^{-k}\|\nabla a_k\|_{L^p}\\[1ex]
&\quad+\beta c_p\left(\frac{1}{p}\|{\rm div}{\bf u}\|_{L^\infty}
\|\nabla a_k\|_{L^p}+\|\nabla\dot{\Delta}_k(a{\rm div}{\bf u})\|_{L^p}
+C2^{2k}\|{\bf w}_k\|_{L^p}+\|R_k\|_{L^p}\right)\\[1ex]
&\quad+C2^{-k}\|{f}_k\|_{L^p}+\|({\bf g}_k,{\bf w}_k)\|_{L^p}+C2^{-2k}\|\nabla a_k\|_{L^p}. \end{split} \end{equation} Choosing $k_0$ suitably large and $\beta$ sufficiently small, we deduce that there exists a constant $c_0>0$ such that for all $k\geq k_0+1$, \begin{equation}\nonumber \begin{split}
&\quad\frac{d}{dt}\|(\mathcal{P}{\bf u}_k, {\bf w}_k, {\bf H}_k,\nabla a_k)\|_{L^p}+c_0\left(2^{2k}\|(\mathcal{P}{\bf u}_k,{\bf w}_k,{\bf H}_k)\|_{L^p}
+\|\nabla a_k\|_{L^p}\right)\\[1ex]
&\lesssim 2^{-k}\|f_k\|_{L^p}+\|({\bf m}_k, {\bf g}_k)\|_{L^p}+\|{\rm div}{\bf u}\|_{L^\infty}
\|\nabla a_k\|_{L^p}+\|\nabla\dot{\Delta}_k(a{\rm div}{\bf u})\|_{L^p}
+\|R_k\|_{L^p}. \end{split} \end{equation} Since $$ {\bf u}={\bf w}-\nabla (-\Delta)^{-1}a+\mathcal{P}{\bf u}, $$ it follows that \begin{equation}\nonumber \begin{split}
&\quad\frac{d}{dt}\|(\nabla a_k, {\bf u}_k, {\bf H}_k)\|_{L^p}+c_0\|(\nabla a_k, 2^{2k}{\bf u}_k, 2^{2k}{\bf H}_k)\|_{L^p}\\[1ex]
&\lesssim \|(2^{-k}f_k, {\bf m}_k, {\bf g}_k)\|_{L^p}+\|{\rm div}{\bf u}\|_{L^\infty}
\|\nabla a_k\|_{L^p}+\|\nabla\dot{\Delta}_k(a{\rm div}{\bf u})\|_{L^p}
+\|R_k\|_{L^p}. \end{split} \end{equation} Thus, multiplying by $2^{k(\frac{N}{p}-1)}$, summing up over $k\geq k_0+1$ and applying Corollary \ref{co2.1} and Proposition \ref{pra.5}, we conclude \eqref{3.11}. \end{proof} \section{Estimation of $L^2$-type Besov norms at low frequencies}\label{s:5} \begin{prop}\label{pr4.1} Let $1-\frac{N}{2}<\sigma_1\leq \frac{2N}{p}-\frac{N}{2} (N\geq 2)$ and $p$ satisfy \eqref{1.3}. Then the following two estimates hold true: \begin{equation}\label{4.1}
\|fg\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim\|f\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\|g\|_{\dot{B}_{2,\infty}^{-\sigma_1}}, \end{equation} and \begin{equation}\label{4.2}
\|fg\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell\lesssim\|f\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\left(\|g\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}
+\|g\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}\right). \end{equation} \end{prop} \begin{proof} Denote $p^\ast\equ\frac{2p}{p-2}$, i.e., $\frac{1}{p}+\frac{1}{p^\ast}=\frac{1}{2}$. By \eqref{2.3}, we decompose $fg$ into $T_fg+R(f,g)+T_gf$.
Firstly, we prove \eqref{4.1}. Thanks to \eqref{1.8}, we have \begin{equation}\label{4.3} \begin{split}
\|\dot{\Delta}_j(T_fg)\|_{L^2}&=\|\sum_{|k-j|\leq 4}\dot{\Delta}_j(\dot{S}_{k-1}f\dot{\Delta}_kg)\|_{L^2}
=\|\sum_{|k-j|\leq 4}\sum_{k^\prime\leq k-2}\dot{\Delta}_j(\dot{\Delta}_{k^\prime}f\dot{\Delta}_kg)\|_{L^2}\\[1ex]
&\lesssim\sum_{|k-j|\leq 4}\sum_{k^\prime\leq k-2}\|\dot{\Delta}_{k^\prime}f\|_{L^\infty}\|\dot{\Delta}_kg\|_{L^2}\\[1ex]
&\lesssim\sum_{|k-j|\leq 4}\sum_{k^\prime\leq k-2}2^{k^\prime\frac{N}{p}}\|\dot{\Delta}_{k^\prime}f\|_{L^p}2^{k\sigma_1}2^{-k\sigma_1}\|\dot{\Delta}_kg\|_{L^2}\\[1ex]
&\lesssim 2^{j\sigma_1}\|f\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\|g\|_{\dot{B}_{2,\infty}^{-\sigma_1}}. \end{split} \end{equation} For the remainder term, one gets \begin{equation}\label{4.4} \begin{split}
\|\dot{\Delta}_jR(f,g)\|_{L^2}&=\|\sum_{k\geq j-3}\sum_{|k-k^\prime|\leq 1}\dot{\Delta}_j(\dot{\Delta}_{k}f\dot{\Delta}_{k^\prime}g)\|_{L^2}\leq \sum_{k\geq j-3}\sum_{|k-k^\prime|\leq 1}
\|\dot{\Delta}_j(\dot{\Delta}_{k}f\dot{\Delta}_{k^\prime}g)\|_{L^2}\\[1ex]
&\lesssim 2^{j\frac{N}{p}}\sum_{k\geq j-3}\sum_{|k-k^\prime|\leq 1}\|\dot{\Delta}_{k}f\dot{\Delta}_{k^\prime}g\|_{L^{\frac{2p}{p+2}}}\\[1ex]
&\lesssim 2^{j\frac{N}{p}}\sum_{k\geq j-3}\sum_{|k-k^\prime|\leq 1}2^{-k\frac{N}{p}}2^{k\frac{N}{p}}\|\dot{\Delta}_{k}f\|_{L^{p}}2^{k^\prime\sigma_1}
2^{-k^\prime\sigma_1}\|\dot{\Delta}_{k^\prime}g\|_{L^{2}}\\[1ex]
&\lesssim2^{j\frac{N}{p}}\sum_{k\geq j-3}2^{k(\sigma_1-\frac{N}{p})}c(k)\|f\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\|g\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim2^{j\sigma_1}\|f\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\|g\|_{\dot{B}_{2,\infty}^{-\sigma_1}}, \end{split} \end{equation}
here $\|c(k)\|_{l^1}=1$ and we used that $\sigma_1-\frac{N}{p}\leq 0$ as $\sigma_1\leq \frac{2N}{p}-\frac{N}{2}\leq \frac{N}{p}$ in the last inequality.
For the term $T_gf$, it follows that \begin{equation}\label{4.5} \begin{split}
\|\dot{\Delta}_j(T_gf)\|_{L^2}&=\|\sum_{|k-j|\leq 4}\dot{\Delta}_j(\dot{S}_{k-1}g\dot{\Delta}_kf)\|_{L^2}
=\|\sum_{|k-j|\leq 4}\sum_{k^\prime\leq k-2}\dot{\Delta}_j(\dot{\Delta}_{k^\prime}g\dot{\Delta}_kf)\|_{L^2}\\[1ex]
&\lesssim\sum_{|k-j|\leq 4}\sum_{k^\prime\leq k-2}\|\dot{\Delta}_{k^\prime}g\|_{L^{p^\ast}}\|\dot{\Delta}_kf\|_{L^p}\\[1ex]
&\lesssim\sum_{|k-j|\leq 4}\sum_{k^\prime\leq k-2}2^{k^\prime(\frac{N}{p}+\sigma_1)}2^{-k^\prime\sigma_1}\|\dot{\Delta}_{k^\prime}g\|_{L^2}2^{-k\frac{N}{p}}2^{k\frac{N}{p}}\|\dot{\Delta}_kf\|_{L^p}\\[1ex]
&\lesssim 2^{j\sigma_1}\|f\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\|g\|_{\dot{B}_{2,\infty}^{-\sigma_1}}, \end{split} \end{equation} here $\sigma_1+\frac{N}{p}>0$ since $\sigma_1>1-\frac{N}{2}\geq -\frac{N}{p}$ if $p\leq \frac{2N}{N-2}$. Combining \eqref{4.3}, \eqref{4.4} and \eqref{4.5}, we finish the proof of \eqref{4.1}.
Now, we are in a position to prove \eqref{4.2}. For the paraproduct term $T_fg$, we have \begin{equation}\label{4.6} \begin{split}
\|\dot{\Delta}_j(T_fg)\|_{L^2}
&\leq\sum_{|k-j|\leq 4}\sum_{k^\prime\leq k-2}\|\dot{\Delta}_j(\dot{\Delta}_{k^\prime}f\dot{\Delta}_kg)\|_{L^2}\lesssim\sum_{|k-j|\leq 4}\sum_{k^\prime\leq k-2}\|\dot{\Delta}_{k^\prime}f\|_{L^{p^\ast}}\|\dot{\Delta}_kg\|_{L^p}\\[1ex]
&\lesssim\sum_{|k-j|\leq 4}\sum_{k^\prime\leq k-2}2^{k^\prime(\frac{2N}{p}-\frac{N}{2})}
\|\dot{\Delta}_{k^\prime}f\|_{L^p}\|\dot{\Delta}_kg\|_{L^p}\\[1ex]
&\lesssim\sum_{|k-j|\leq 4}\sum_{k^\prime\leq k-2}2^{k^\prime(\frac{2N}{p}-\frac{N}{2}+1-\frac{N}{p})}2^{k^\prime(\frac{N}{p}-1)}
\|\dot{\Delta}_{k^\prime}f\|_{L^p}\\[1ex]
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\times 2^{k(\sigma_1-\frac{N}{p}+\frac{N}{2}-1)}2^{-k(\sigma_1-\frac{N}{p}+\frac{N}{2}-1)}\|\dot{\Delta}_kg\|_{L^p}\\[1ex]
&\lesssim 2^{j\sigma_1}\|f\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}\|g\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}, \end{split} \end{equation} where we have used that $1+\frac{N}{p}-\frac{N}{2}\geq 0$ and $p^\ast\geq p$ as $p$ fulfills $2\leq p\leq \min(4,\frac{2N}{N-2})$.
For the remainder term, one gets \begin{equation}\label{4.7} \begin{split}
\|\dot{\Delta}_jR(f,g)\|_{L^2}&\leq \sum_{k\geq j-3}\sum_{|k-k^\prime|\leq 1}
\|\dot{\Delta}_j(\dot{\Delta}_{k}f\dot{\Delta}_{k^\prime}g)\|_{L^2}\\[1ex]
&\lesssim 2^{j(\frac{2N}{p}-\frac{N}{2})}\sum_{k\geq j-3}\sum_{|k-k^\prime|\leq 1}2^{k(1-\frac{N}{p})}2^{k(\frac{N}{p}-1)}\|\dot{\Delta}_{k}f\|_{L^{p}}\\[1ex] &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\times2^{k^\prime(\sigma_1-\frac{N}{p}+\frac{N}{2}-1)}
2^{-k^\prime(\sigma_1-\frac{N}{p}+\frac{N}{2}-1)}\|\dot{\Delta}_{k^\prime}g\|_{L^{p}}\\[1ex]
&\lesssim2^{j(\frac{2N}{p}-\frac{N}{2})}\sum_{k\geq j-3}2^{k(\sigma_1-\frac{2N}{p}+\frac{N}{2})}c(k)\|f\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\|g\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}\\[1ex] &
\lesssim2^{j\sigma_1}\|f\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}\|g\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}, \end{split} \end{equation}
here $\|c(k)\|_{l^1}=1$ and we have used the condition $\sigma_1\leq \frac{2N}{p}-\frac{N}{2}$ in the last inequality.
For the term $T_gf$, we could obtain \begin{equation}\label{4.8} \begin{split}
\|\dot{\Delta}_j(T_gf)\|_{L^2}&\leq \sum_{|k-j|\leq 4}\sum_{k^\prime\leq k-2}\|\dot{\Delta}_j(\dot{\Delta}_{k^\prime}g\dot{\Delta}_kf)\|_{L^2}\leq\sum_{|k-j|\leq 4}\sum_{k^\prime\leq k-2}\|\dot{\Delta}_{k^\prime}g\|_{L^{p^\ast}}\|\dot{\Delta}_kf\|_{L^p}\\[1ex]
&\lesssim\sum_{|k-j|\leq 4}\sum_{k^\prime\leq k-2}2^{k^\prime(\frac{2N}{p}-\frac{N}{2})}\|\dot{\Delta}_{k^\prime}g\|_{L^p}\|\dot{\Delta}_kf\|_{L^p}\\[1ex]
&\lesssim\sum_{|k-j|\leq 4}\sum_{k^\prime\leq k-2}2^{k^\prime(\frac{2N}{p}-\frac{N}{2}+\sigma_1-\frac{2N}{p}+N-1)}2^{k^\prime(-\sigma_1+\frac{2N}{p}-N+1)}
\|\dot{\Delta}_{k^\prime}g\|_{L^p}\\[1ex] &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \times2^{k(1-\frac{N}{p})}2^{k(\frac{N}{p}-1)}
\|\dot{\Delta}_kf\|_{L^p}\\[1ex] &\lesssim 2^{j(\sigma_1+\frac{N}{2}-\frac{N}{p})}
\|f\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}\|g\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}, \end{split} \end{equation} where we used that $\sigma_1>1-\frac{N}{2}$ in the last inequality.
From \eqref{4.6} and \eqref{4.7}, we deduce that \begin{equation}\label{4.9}
\|T_fg+R(f,g)\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim\|f\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}\|g\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}} \end{equation} and from \eqref{4.8}, we get for $p\geq 2$ that \begin{equation}\label{4.10}
\|T_gf\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell\leq
\|T_gf\|_{\dot{B}_{2,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}}}^\ell
\lesssim\|f\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}\|g\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}. \end{equation} Combining \eqref{4.9} and \eqref{4.10}, we get \eqref{4.2}. \end{proof}
Next, we begin to estimate the $L^2$-type Besov norms at low frequencies, which is the main ingredient in the proof of Theorem \ref{th1.2}. \begin{lemma}\label{le3} Let $1-\frac{N}{2}<\sigma_1\leq \frac{2N}{p}-\frac{N}{2}$ and $p$ satisfy \eqref{1.3}, it holds that \begin{equation}\label{4.11} \begin{split}
&\Big(\|(a, {\bf u},{\bf H})(t)\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell\Big)^2\lesssim
\Big(\|(a_0, {\bf u}_0,{\bf H}_0)\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell\Big)^2\\[1ex]
&\quad\quad\quad\quad+\int_0^tA_1(\tau)\Big(\|(a, {\bf u},{\bf H})(\tau)\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell\Big)^2d\tau
+\int_0^tA_2(\tau)\|(a, {\bf u},{\bf H})(\tau)\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell d\tau, \end{split} \end{equation} where \begin{equation}\nonumber \begin{split}
A_1(t)&\equ\|(a,{\bf u},{\bf H})\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell+\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h
+\|({\bf u}, {\bf H})\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\\[1ex]
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^2+\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\|({\bf u}, {\bf H})\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}+\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\|{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}}} \end{split} \end{equation} and \begin{equation}\nonumber \begin{split}
A_2(t)&\equ\Big(\|(a,{\bf u},{\bf H})\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\Big)^2
+\|({\bf u},{\bf H})\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h
+\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^2\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\\[1ex]
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\|({\bf u}, {\bf H})\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h
+\Big(\|{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\Big)^2\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}. \end{split} \end{equation} \end{lemma} \begin{proof}
From \eqref{3.103} and \eqref{3.9}, we have for $k\leq k_0$,
\begin{equation}\label{4.12}
\begin{split}
&\quad\frac{1}{2}\frac{d}{dt}\|(a_k, \omega_k, {\bf \Omega}_k, {\bf E}_k)\|_{L^2}^2+\|(\Lambda a_k,\Lambda \omega_k, \Lambda {\bf \Omega}_k, \Lambda {\bf E}_k)\|_{L^2}^2\\[1ex]
&\lesssim \|(F_k, G_k, {\bf L}_k, {\bf M}_k)\|_{L^2}\|(a_k, \omega_k, {\bf \Omega}_k, {\bf E}_k)\|_{L^2}.
\end{split}
\end{equation} Multiplying $2^{2k(-\sigma_1)}$ on both sides of \eqref{4.12}, taking supremum in terms of $k\leq k_0$, integrating over $[0, t]$ and noticing that \eqref{3.104} and \eqref{3.101}, we arrive at \begin{equation}\label{4.13} \begin{split}
&\quad\Big(\|(a, {\bf u},{\bf H})(t)\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell\Big)^2\\[1ex] &\lesssim
\Big(\|(a_0, {\bf u}_0,{\bf H}_0)\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell\Big)^2+\int_0^t\|(f, {\bf g},{\bf m})(\tau)\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell\|(a, {\bf u},{\bf H})(\tau)\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell d\tau. \end{split} \end{equation}
Next, we focus on the estimates of $\|(f, {\bf g},{\bf m})\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell$. Firstly, we deal with the term $f=-{\rm div}(a{\bf u})=-a{\rm div}{\bf u}-{\bf u}\cdot\nabla a$.
\underline{Estimate of $a{\rm div}{\bf u}$}. We decompose $$a{\rm div}{\bf u}=a^\ell{\rm div}{\bf u}+a^h{\rm div}{\bf u}^\ell+a^h{\rm div}{\bf u}^h.$$ Making use of \eqref{4.1}, we deduce \begin{equation}\label{4.14}
\|a^\ell{\rm div}{\bf u}\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim \|{\rm div}{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\|a\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell\lesssim \Big(\|{\bf u}\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell+\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\Big)
\|a\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell \end{equation} and \begin{equation}\label{4.15}
\|a^h{\rm div}{\bf u}^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim \|a^h\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\|{\rm div}{\bf u}^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim \|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h
\|{\bf u}\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell. \end{equation} By virtue of \eqref{4.2}, one gets \begin{equation}\label{4.16} \begin{split}
\|a^h{\rm div}{\bf u}^h\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell&\lesssim \|a^h\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\Big(\|{\rm div}{\bf u}^h\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}+\|{\rm div}{\bf u}^h\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}\Big)\\[1ex]
&\lesssim\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h, \end{split} \end{equation} where we used that $ -\sigma_1+\frac{2N}{p}-N+2\leq -\sigma_1+\frac{N}{p}-\frac{N}{2}+2<\frac{N}{p}+1 $ since $\sigma_1>1-\frac{N}{2}$ and $p\geq 2$.
\underline{Estimate of ${\bf u}\cdot\nabla a$}. Decomposing ${\bf u}\cdot\nabla a={\bf u}^\ell\cdot\nabla a^\ell+{\bf u}^h\cdot\nabla a^\ell+{\bf u}^\ell\cdot\nabla a^h+{\bf u}^h\cdot\nabla a^h$, we deduce from \eqref{4.1} that \begin{equation}\label{4.17}
\|{\bf u}^\ell\nabla a^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim\|\nabla a^\ell\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\|{\bf u}^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim\|a\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell
\|{\bf u}\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell, \end{equation} and \begin{equation}\label{4.18}
\|{\bf u}^h\nabla a^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim\|{\bf u}^h\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\|\nabla a^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h
\|a\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell. \end{equation} From \eqref{4.2}, one arrives at \begin{equation}\label{4.19} \begin{split}
\|{\bf u}^\ell\nabla a^h\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell&\lesssim\|\nabla a^h\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\Big(\|{\bf u}^\ell\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}
+\|{\bf u}^\ell\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}\Big)\\[1ex]
&\lesssim\|a^h\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\|{\bf u}^\ell\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}
\lesssim\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\|{\bf u}\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell, \end{split}
\end{equation} where we used that $-\sigma_1+\frac{2N}{p}-N+1\leq -\sigma_1+\frac{N}{p}-\frac{N}{2}+1$ in the second inequality and $\|{\bf u}\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell
\hookrightarrow\|{\bf u}\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}^\ell$ when $2\leq p\leq \frac{2N}{N-2}$ in the last inequality. For the term ${\bf u}^h\nabla a^h$, also by \eqref{4.2}, it follows that \begin{equation}\label{4.20} \begin{split}
\|{\bf u}^h\nabla a^h\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell&\lesssim\|\nabla a^h\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\Big(\|{\bf u}^h\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}
+\|{\bf u}^h\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}\Big)\\[1ex]
&\lesssim\|a^h\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\|{\bf u}^h\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}
\lesssim\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h, \end{split} \end{equation} where we have applied that $ -\sigma_1+\frac{2N}{p}-N+1\leq -\sigma_1+\frac{N}{p}-\frac{N}{2}+1\leq \frac{N}{p}+1, $ since $\sigma_1> 1-\frac{N}{2}$ and $p\geq 2$.
In what follows, we estimate $\|{\bf g}\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell$. Recall that \begin{equation}\nonumber \begin{split} &{\bf g}\equ -{\bf u}\cdot\nabla{\bf u}-\pi_1(a)\mathcal{A}{\bf u}-\pi_2(a)\nabla a+\frac{1}{1+a}{\rm div}\Big(2\widetilde\mu(a) D({\bf u})+\widetilde\lambda(a){\rm div}{\bf u}\,{\rm Id}\Big)\\[1ex] &\quad\quad+\pi_1(a)(\nabla(I\cdot{\bf H})-I\cdot\nabla{\bf H})-\frac{1}{1+a}\Big(\frac{1}{2}\nabla
|{\bf H}|^2-{\bf H}\cdot\nabla{\bf H}\Big). \end{split} \end{equation}
\underline{Estimate of ${\bf u}\cdot\nabla{\bf u}$}. Decompose ${\bf u}\cdot\nabla{\bf u}={\bf u}^\ell\cdot\nabla{\bf u}^\ell+{\bf u}^\ell\cdot\nabla{\bf u}^h +{\bf u}^h\cdot\nabla{\bf u}^\ell+{\bf u}^h\cdot\nabla{\bf u}^h$. It holds from \eqref{4.1} that \begin{equation}\label{4.23}
\|{\bf u}^\ell\cdot\nabla {\bf u}^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim\|\nabla {\bf u}^\ell\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\|{\bf u}^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim\|{\bf u}\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell
\|{\bf u}\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell, \end{equation} \begin{equation}\label{4.24}
\|{\bf u}^h\cdot\nabla {\bf u}^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim\|{\bf u}^h\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\|\nabla{\bf u}^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h
\|{\bf u}\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell. \end{equation} In a similar way as deriving \eqref{4.19} and \eqref{4.20}, one has by \eqref{4.2} that \begin{equation}\label{4.124} \begin{split}
\|{\bf u}^\ell\cdot\nabla {\bf u}^h\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell&\lesssim\|\nabla {\bf u}^h\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\Big(\|{\bf u}^\ell\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}
+\|{\bf u}^\ell\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}\Big)\\[1ex]
&\lesssim\|{\bf u}^h\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\|{\bf u}^\ell\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}
\lesssim\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\|{\bf u}\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell, \end{split} \end{equation} and \begin{equation}\label{4.25} \begin{split}
\|{\bf u}^h\cdot\nabla {\bf u}^h\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell&\lesssim\|\nabla {\bf u}^h\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\Big(\|{\bf u}^h\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}
+\|{\bf u}^h\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}\Big)\\[1ex]
&\lesssim\|{\bf u}^h\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\|{\bf u}^h\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}
\lesssim\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h. \end{split} \end{equation}
\underline{Estimate of $\pi_1(a)\mathcal{A}{\bf u}$}. Keeping in mind that $\pi_1(0)=0$, one may write $$ \pi_1(a)=\pi_1^\prime(0)a+\bar{\pi}_1(a)a $$ for some smooth function $\bar{\pi}_1$ vanishing at $0$. Thus, through \eqref{4.1} again, we have \begin{equation}\label{4.26}
\|a^\ell\mathcal{A}{\bf u}^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim\|\mathcal{A}{\bf u}^\ell\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\|a^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim\|{\bf u}\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell
\|a\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell, \end{equation} and \begin{equation}\label{4.27}
\|a^h\mathcal{A}{\bf u}^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim\|a^h\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\|\mathcal{A}{\bf u}^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h
\|{\bf u}\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell. \end{equation} Arguing similarly as \eqref{4.19} and \eqref{4.20}, one has \begin{equation}\label{4.28} \begin{split}
\|a^\ell\mathcal{A} {\bf u}^h\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell&\lesssim\|\mathcal{A} {\bf u}^h\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\Big(\|a^\ell\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}
+\|a^\ell\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}\Big)\\[1ex]
&\lesssim\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\|a^\ell\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}
\lesssim\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\|a\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell, \end{split} \end{equation} and \begin{equation}\label{4.29} \begin{split}
\|a^h\mathcal{A}{\bf u}^h\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell&\lesssim\|\mathcal{A}{\bf u}^h\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\Big(\|a^h\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}
+\|a^h\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}\Big)\\[1ex]
&\lesssim\|{\bf u}^h\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}\|a^h\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}
\lesssim\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h. \end{split} \end{equation} On the other hand, from \eqref{4.1}, \eqref{4.2}, Proposition \ref{pra.4} and Corollaries \ref{co2.1} and \ref{co2.2}, we have \begin{equation}\label{4.30}
\|\bar{\pi}_1(a)a\mathcal{A}{\bf u}^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}
\lesssim\|\bar{\pi}_1(a)a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\|\mathcal{A}{\bf u}^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}
\lesssim\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^2\|{\bf u}\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell, \end{equation} and \begin{equation}\label{4.31} \begin{split}
\|\bar{\pi}_1(a)a\mathcal{A}{\bf u}^h\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell
&\lesssim\|\mathcal{A}{\bf u}^h\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\Big(\|\bar{\pi}_1(a)a\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}
+\|\bar{\pi}_1(a)a\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}\Big)\\[1ex]
&\lesssim\|{\bf u}^h\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}\|\bar{\pi}_1(a)\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\Big(\|a\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}
+\|a\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}\Big)\\[1ex]
&\lesssim\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h
\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\Big(\|a\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}^h
+\|a\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}^\ell\Big)\\[1ex]
&\lesssim\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h
\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\Big(\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h
+\|a\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell\Big). \end{split} \end{equation}
\underline{Estimate of $\pi_2(a)\nabla a$}. In view of $\pi_2(0)=0$, we may write $\pi_2(a)=\pi_2^\prime(0)a+\bar{\pi}_2(a)a$, here $\bar{\pi}_2$ is a smooth function fulfilling $\bar{\pi}_2(0)=0$. For the term $a\nabla a$, we have \begin{equation}\label{4.32}
\|a^\ell\nabla a^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim\|\nabla a^\ell\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\|a^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim\|a\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell
\|a\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell, \end{equation} and \begin{equation}\label{4.33}
\|a^h\nabla a^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim\|a^h\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\|\nabla a^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h
\|a\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell. \end{equation} Arguing similarly as \eqref{4.28} and \eqref{4.29}, one has \begin{equation}\label{4.34} \begin{split}
\|a^\ell\nabla a^h\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell&\lesssim\|\nabla a^h\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\Big(\|a^\ell\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}
+\|a^\ell\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}\Big)\\[1ex]
&\lesssim\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\|a^\ell\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}
\lesssim\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\|a\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell, \end{split} \end{equation} and \begin{equation}\label{4.35} \begin{split}
\|a^h\nabla a^h\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell&\lesssim\|\nabla a^h\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\Big(\|a^h\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}
+\|a^h\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}\Big)\\[1ex]
&\lesssim\|a^h\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\|a^h\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}
\lesssim\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h. \end{split} \end{equation} As for the term $\bar{\pi}_2(a)a\nabla a$, we use the decomposition $\bar{\pi}_2(a)a\nabla a=\bar{\pi}_2(a)a\nabla a^\ell+\bar{\pi}_2(a)a\nabla a^h$ and get from \eqref{4.1}-\eqref{4.2}, Corollary \ref{co2.2} and Proposition \ref{pra.4} again that \begin{equation}\label{4.36} \begin{split}
\|\bar{\pi}_2(a)a\nabla a^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim\|\bar{\pi}_2(a)a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\|\nabla a\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell\lesssim\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^2
\|a\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell, \end{split} \end{equation} and \begin{equation}\label{4.37} \begin{split}
\|\bar{\pi}_2(a)a\nabla a^h\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell&\lesssim\|\nabla a^h\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\Big(\|\bar{\pi}_2(a)a\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}
+\|\bar{\pi}_2(a)a\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}\Big)\\[1ex]
&\lesssim\|a^h\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\|\bar{\pi}_2(a)\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\Big(\|a\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}
+\|a\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}\Big)\\[1ex]
&\lesssim\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^2\Big(\|a\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell
+\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\Big). \end{split} \end{equation}
\underline{Estimate of $\frac{1}{1+a}(2\widetilde{\mu}(a){\rm div}D({\bf u})+\widetilde{\lambda}(a)\nabla{\rm div}{\bf u})$}. The estimate of this term could be similarly handled as the term $\pi_1(a)\mathcal{A}{\bf u}$ and we omit it here.
\underline{Estimate of $\frac{1}{1+a}(2\widetilde{\mu}^\prime(a)D({\bf u})\cdot\nabla a+\widetilde{\lambda}^\prime(a){\rm div}{\bf u}\nabla a)$}. We only deal with the term $\frac{2\widetilde{\mu}^\prime(a)}{1+a}D({\bf u})\cdot\nabla a$ and the remainder term could be similarly handled. Denote by $J(a)$ the smooth function fulfilling \begin{equation}\label{4.1000} J^\prime(a)=\frac{2\mu^\prime(a)}{1+a} \,\,\,{\rm and}\,\,\, J(0)=0, \,\,\,{\rm so\,\, that}\,\,\, \nabla J(a)=\frac{2\mu^\prime(a)}{1+a}\nabla a. \end{equation} Decomposing $J(a)=J^\prime(0)a+\bar{J}(a)a$ implies $\nabla J(a)=J^\prime(0)\nabla a+\nabla (\bar{J}(a)a)$. Then we have from \eqref{4.1} and \eqref{4.2} that \begin{equation}\label{4.38}
\|\nabla a^\ell D({\bf u})^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim\|\nabla a^\ell\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\|D({\bf u})^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim\|a\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell
\|{\bf u}\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell, \end{equation} \begin{equation}\label{4.39}
\|\nabla a^\ell D({\bf u})^h\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim\|D({\bf u})^h\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\|\nabla a^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h
\|a\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell, \end{equation} and \begin{equation}\label{4.40} \begin{split}
\|\nabla a^h D({\bf u})\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell&\lesssim\|\nabla a^h\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\Big(\|D({\bf u})\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}
+\|D({\bf u})\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}\Big)\\[1ex]
&\lesssim\|a^h\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\Big(\|{\bf u}\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+2}}^h
+\|{\bf u}\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+2}}^\ell\Big)\\[1ex]
&\lesssim\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\Big(\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h+\|{\bf u}\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell \Big). \end{split} \end{equation}
\underline{Estimate of $\pi_1(a)(\nabla (I\cdot {\bf H})-I\cdot \nabla {\bf H})$}. Similar as above, we decompose $\pi_1(a)=\pi_1^\prime(0)a+\bar{\pi}_1(a)a$. Firstly, the estimate of $a(\nabla (I\cdot {\bf H})-I\cdot \nabla {\bf H})$ is similar to that of $a{\rm div}{\bf u}$ and we omit it here. The remaining term may be estimated as follows. \begin{equation}\label{4.41}
\|\bar{\pi}_1(a) a(\nabla(I\cdot{\bf H})-I\cdot\nabla{\bf H})^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim\|\bar{\pi}_1(a) a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\|\nabla {\bf H}^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^2
\|{\bf H}\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell, \end{equation} and \begin{equation}\label{4.42} \begin{split}
&\quad\|\bar{\pi}_1(a) a(\nabla(I\cdot{\bf H})-I\cdot\nabla{\bf H})^h\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell\\[1ex]
&\lesssim\|\nabla {\bf H}^h\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\Big(\|\bar{\pi}_1(a)a\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}
+\|\bar{\pi}_1(a)a\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}\Big)\\[1ex]
&\lesssim\|{\bf H}^h\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}\|\bar{\pi}_1(a)\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\Big(\|a\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}
+\|a\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}\Big)\\[1ex]
&\lesssim\|{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\Big(\|a\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell
+\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\Big). \end{split} \end{equation}
\underline{Estimate of $\frac{1}{1+a}(\frac{1}{2}\nabla |{\bf H}|^2-{\bf H}\cdot \nabla {\bf H})$}. Since $\frac{1}{1+a}=1-\pi_1(a)$, it follows that \begin{equation}\label{4.200} \begin{split}
\frac{1}{1+a}\Big(\frac{1}{2}\nabla |{\bf H}|^2-{\bf H}\cdot \nabla {\bf H}\Big)&=\Big(\frac{1}{2}\nabla |{\bf H}|^2-{\bf H}\cdot \nabla {\bf H}\Big)
-\pi_1(a)\Big(\frac{1}{2}\nabla |{\bf H}|^2-{\bf H}\cdot \nabla {\bf H}\Big)\\[1ex] &={\bf H}\cdot((\nabla {\bf H})^T-\nabla {\bf H}) -\pi_1(a){\bf H}\cdot((\nabla {\bf H})^T-\nabla {\bf H}), \end{split} \end{equation} where the superscript $T$ represents the transpose of a matrix.
For the term with ${\bf H}\cdot((\nabla {\bf H})^T-\nabla {\bf H})$, we can handle it similar to the term ${\bf u}\cdot\nabla{\bf u}$, while regarding the term with $\pi_1(a){\bf H}\cdot((\nabla {\bf H})^T-\nabla {\bf H})$, we have from \eqref{4.1} and \eqref{4.2} again that \begin{equation}\label{4.43}
\|\pi_1(a){\bf H}\cdot(\nabla{\bf H})^\ell\|_{\dot{B}_{2,\infty}^{-\sigma_1}}\lesssim\|\pi_1(a){\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\|\nabla{\bf H}\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell\lesssim \|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\|{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\|{\bf H}\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell, \end{equation} and \begin{equation}\label{4.44} \begin{split}
\|\pi_1(a){\bf H}\cdot(\nabla{\bf H})^h\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell
&\lesssim\|\nabla {\bf H}^h\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\Big(\|{\pi}_1(a){\bf H}\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}
+\|{\pi}_1(a){\bf H}\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}\Big)\\[1ex]
&\lesssim\|{\bf H}^h\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\|{\pi}_1(a)\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\Big(\|{\bf H}\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{N}{p}-\frac{N}{2}+1}}^h
+\|{\bf H}\|_{\dot{B}_{p,\infty}^{-\sigma_1+\frac{2N}{p}-N+1}}^\ell\Big)\\[1ex]
&\lesssim\|{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\Big(
\|{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h+\|{\bf H}\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell\Big). \end{split} \end{equation}
\underline{Estimate of ${\bf m}$}. Since $${\bf m}\equ -{\bf H}({\rm div}{\bf u})+{\bf H}\cdot\nabla{\bf u}-{\bf u}\cdot\nabla{\bf H},$$
then its estimation is similar to that of ${\bf u}\cdot\nabla {\bf u}$ and we omit it here. Finally, inserting all estimates above into \eqref{4.13}, we complete the proof of \eqref{4.11}. \end{proof}
By the definition of $\mathcal{X}_p(t)$ in Theorem \ref{th1.1}, one has \begin{equation}\nonumber \begin{split}
\|(a^\ell, {\bf u}^\ell, {\bf H}^\ell)\|_{L_t^2(\dot{B}_{p,1}^{\frac{N}{p}})}&\lesssim\|(a^\ell, {\bf u}^\ell, {\bf H}^\ell)\|_{L_t^\infty(\dot{B}_{p,1}^{\frac{N}{p}-1})}^{\frac{1}{2}}\|(a^\ell, {\bf u}^\ell, {\bf H}^\ell)\|_{L_t^1(\dot{B}_{p,1}^{\frac{N}{p}+1})}^{\frac{1}{2}}\\[1ex]
&\lesssim\Big(\|(a, {\bf u}, {\bf H})\|_{L_t^\infty(\dot{B}_{2,1}^{\frac{N}{2}-1})}^\ell\Big)^{\frac{1}{2}}\Big(\|(a, {\bf u}, {\bf H})\|_{L_t^1(\dot{B}_{2,1}^{\frac{N}{2}+1})}^\ell\Big)^{\frac{1}{2}}, \end{split} \end{equation} \begin{equation}\nonumber
\|a^h\|_{L_t^2(\dot{B}_{p,1}^{\frac{N}{p}})}\lesssim\Big(\|a\|_{L_t^\infty(\dot{B}_{p,1}^{\frac{N}{p}})}^h\Big)^{\frac{1}{2}}
\Big(\|a\|_{L_t^1(\dot{B}_{p,1}^{\frac{N}{p}})}^h\Big)^{\frac{1}{2}}, \end{equation} and \begin{equation}\nonumber
\|({\bf u}^h,{\bf H}^h)\|_{L_t^2(\dot{B}_{p,1}^{\frac{N}{p}})}\lesssim\Big(\|({\bf u},{\bf H})\|_{L_t^\infty(\dot{B}_{p,1}^{\frac{N}{p}-1})}^h\Big)^{\frac{1}{2}}
\Big(\|({\bf u},{\bf H})\|_{L_t^1(\dot{B}_{p,1}^{\frac{N}{p}+1})}^h\Big)^{\frac{1}{2}}. \end{equation} On the other hand, it follows that \begin{equation}\nonumber
\|a\|_{L_t^\infty(\dot{B}_{p,1}^{\frac{N}{p}})}\lesssim\|a\|_{L_t^\infty(\dot{B}_{p,1}^{\frac{N}{p}})}^\ell
+\|a\|_{L_t^\infty(\dot{B}_{p,1}^{\frac{N}{p}})}^h\lesssim\|a\|_{L_t^\infty(\dot{B}_{2,1}^{\frac{N}{2}-1})}^\ell
+\|a\|_{L_t^\infty(\dot{B}_{p,1}^{\frac{N}{p}})}^h. \end{equation} Then, we have \begin{equation}\label{4.45} \int_0^t(A_1(\tau)+A_2(\tau))d\tau\leq \mathcal{X}_p+\mathcal{X}_p^2+\mathcal{X}_p^3\leq C\mathcal{X}_{p,0}, \end{equation} which yields from Gronwall's inequality (see for example, Page 360 of \cite{mitrinovic1994inequalities}) that \begin{equation}\label{4.46}
\|(a,{\bf u},{\bf H})\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell\leq C_0
\end{equation} for all $t\geq 0$, where $C_0>0$ depends on the norm $\|(a_0, {\bf u}_0, {\bf H}_0)\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell$. \section{Proofs of main results}\label{s:6} This section is devoted to proving Theorem \ref{th1.2} and Corollary \ref{col1}. \subsection{Proof of Theorem \ref{th1.2}} From Lemmas \ref{le1} and \ref{le2}, one deduces that \begin{equation}\label{5.1} \begin{split}
&\quad\frac{d}{dt}\Big(\|(a, {\bf u}, {\bf H})\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell+\|(\nabla a, {\bf u}, {\bf H})\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h\Big)\\[1ex]
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
+\Big(\|(a, {\bf u}, {\bf H})\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell
+\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h+\|({\bf u}, {\bf H})\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\Big)\\[1ex]
&\lesssim\|(f, {\bf g}, {\bf m})\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell+\|f\|_{\dot{B}_{p,1}^{\frac{N}{p}-2}}^h
+\|({\bf g}, {\bf m})\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h+\|\nabla{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}. \end{split} \end{equation} In what follows, we deal with the terms in the right hand of \eqref{5.1} one by one. Firstly, for the last term, we have \begin{equation}\label{5.2} \begin{split}
\|\nabla{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
&\lesssim\Big(\|a\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell+\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\Big)
\Big(\|{\bf u}\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell+\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\Big)\\[1ex]
&\lesssim\mathcal{X}_p(t)\Big(\|{\bf u}\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell+\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\Big). \end{split} \end{equation} Next, notice that $$
\|f\|_{\dot{B}_{p,1}^{\frac{N}{p}-2}}^h\lesssim \|a{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h. $$ Decomposing $a{\bf u}=a^\ell{\bf u}^\ell+a^\ell{\bf u}^h+a^h{\bf u}$, we have $$
\|a^\ell{\bf u}^h\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h\lesssim\|a^\ell\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\|{\bf u}^h\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\lesssim\|a\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell
\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\lesssim\mathcal{X}_p(t)\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h, $$ and $$
\|a^h{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h\lesssim\|a^h\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}\lesssim\mathcal{X}_p(t)\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h. $$ It follows from Corollary \ref{co2.1} and Bernstein inequality that $$
\|a^\ell{\bf u}^\ell\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h\lesssim\|a^\ell{\bf u}^\ell\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}
\lesssim\|a^\ell\|_{L^\infty}\|{\bf u}^\ell\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}
+\|{\bf u}^\ell\|_{L^\infty}\|a^\ell\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}
\lesssim\mathcal{X}_p(t)\|(a,{\bf u})\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell. $$ Therefore, we conclude that \begin{equation}\label{5.3}
\|f\|_{\dot{B}_{p,1}^{\frac{N}{p}-2}}^h\lesssim\mathcal{X}_p(t)\Big(\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h
+\|(a,{\bf u})\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell+\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\Big). \end{equation}
Now we are in a position to bound $\|({\bf g}, {\bf m})\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h$ and the tools are mainly involved with Corollaries \ref{co2.1} and \ref{co2.2}, Proposition \ref{pra.4} and Bernstein inequality. \begin{equation}\label{5.4}
\|{\bf u}\cdot\nabla {\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h\lesssim\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\|\nabla{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\lesssim\mathcal{X}_p(t)\Big(\|{\bf u}\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell+\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\Big). \end{equation} \begin{equation}\label{5.5} \begin{split}
\|\pi_1(a)\mathcal{A}{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h&\lesssim\|\pi_1(a)\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\|\mathcal{A}{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}\\[1ex] &
\lesssim\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\Big(\|{\bf u}\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell
+\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\Big)
\lesssim\mathcal{X}_p(t)\Big(\|{\bf u}\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell+\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\Big). \end{split} \end{equation} \begin{equation}\label{5.6} \begin{split}
&\|\pi_2(a)\nabla a\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h\lesssim\|\pi_2(a)\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\|\nabla a\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}\lesssim\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^2
\lesssim\|a^\ell\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^2
+\|a^h\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^2\\[1ex]
&\quad\quad\quad\quad\lesssim\|a^\ell\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}\|a^\ell\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}
+\|a^h\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\|a^h\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\lesssim\mathcal{X}_p(t)\Big(\|a\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell+\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\Big). \end{split} \end{equation}
The term with $\frac{1}{1+a}\Big(2\widetilde{\mu}(a){\rm div}D({\bf u})+\widetilde{\lambda}(a)\nabla{\rm div}{\bf u}\Big)$ could be similarly handled as the term $\pi_1(a)\mathcal{A}{\bf u}$ and we omit it here.
Regarding the term with $\frac{1}{1+a}(2\widetilde{\mu}^\prime(a)D({\bf u})\cdot\nabla a+\widetilde{\lambda}^\prime(a){\rm div}{\bf u}\nabla a)$, as before we only perform the term $\frac{2\widetilde{\mu}^\prime(a)}{1+a}D({\bf u})\cdot\nabla a$ and the other could be handled similarly. Denote by $J(a)$ the smooth function fulfilling $J^\prime(a)=\frac{2\mu^\prime(a)}{1+a}$ and $J(0)=0$, so that $\nabla J(a)=\frac{2\mu^\prime(a)}{1+a}\nabla a$. Then we have \begin{equation}\label{5.7} \begin{split}
&\|\nabla J(a)D({\bf u})\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h\lesssim\|D({\bf u})\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\|\nabla J(a)\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}\\[1ex]
&\quad\quad\quad\quad\quad\quad\lesssim\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\Big(\|{\bf u}\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell
+\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\Big)
\lesssim\mathcal{X}_p(t)\Big(\|{\bf u}\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell+\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\Big). \end{split} \end{equation}
For the term $\pi_1(a)(\nabla (I\cdot {\bf H})-I\cdot \nabla {\bf H})$, it follows that \begin{equation}\label{5.8} \begin{split}
&\|\pi_1(a)(\nabla (I\cdot {\bf H})-I\cdot \nabla {\bf H})\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h\lesssim\|\pi_1(a)\nabla {\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h
\lesssim\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\|\nabla{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\\[1ex]
&\quad\quad\quad\quad\quad\lesssim\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\Big(\|{\bf H}\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell
+\|{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\Big)
\lesssim\mathcal{X}_p(t)\Big(\|{\bf H}\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell+\|{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\Big). \end{split}
\end{equation} As for the last term $\frac{1}{1+a}\Big(\frac{1}{2}\nabla |{\bf H}|^2-{\bf H}\cdot \nabla {\bf H}\Big)$ in ${\bf g}$, we also apply the decomposition \eqref{4.200} to yield that \begin{equation}\label{5.9}
\|{\bf H}\cdot((\nabla {\bf H})^T-\nabla {\bf H})\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h
\lesssim\|{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\|\nabla{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\lesssim\mathcal{X}_p(t)
\Big(\|{\bf H}\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell+\|{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\Big) \end{equation} and \begin{equation}\label{5.10} \begin{split}
\|\pi_1(a){\bf H}\cdot((\nabla {\bf H})^T-\nabla {\bf H})\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h
&\lesssim\|\pi_1(a)\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\|{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\|\nabla{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\\[1ex] &\lesssim\mathcal{X}_p(t)
\Big(\|{\bf H}\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell+\|{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\Big). \end{split} \end{equation} Finally, for terms in ${\bf m}$, it holds that \begin{equation}\label{5.11} \begin{split}
&\|-{\bf H}({\rm div}{\bf u})+{\bf H}\cdot\nabla{\bf u}-{\bf u}\cdot\nabla{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h
\lesssim\|({\bf u},{\bf H})\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\|(\nabla{\bf u},\nabla{\bf H})\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\\[1ex] &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\lesssim\mathcal{X}_p(t)
\Big(\|({\bf u},{\bf H})\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell+\|({\bf u},{\bf H})\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\Big). \end{split} \end{equation} Combining \eqref{5.4}-\eqref{5.11}, we end up with \begin{equation}\label{5.12}
\|({\bf g}, {\bf m})\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h\lesssim\mathcal{X}_p(t)
\Big(\|(a,{\bf u},{\bf H})\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell
+\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h+\|({\bf u},{\bf H})\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\Big). \end{equation}
In what follows, we bound the low frequency term $\|(f,{\bf g}, {\bf m})\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell$ in the right hand of \eqref{5.1}, which has a little bit more difficult. Let us first introduce the following two inequalities: \begin{equation}\label{5.14}
\|T_fg\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}\lesssim\|f\|_{\dot{B}_{p,1}^{s}}\|{ g}\|_{\dot{B}_{p,1}^{-s+\frac{2N}{p}-1}} \end{equation} if $s\leq \frac{2N}{p}-\frac{N}{2}$ and $2\leq p\leq 4$, and \begin{equation}\label{5.15}
\|R(f,g)\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}\lesssim\|f\|_{\dot{B}_{p,1}^{s}}\|{ g}\|_{\dot{B}_{p,1}^{-s+\frac{2N}{p}-1}} \end{equation} if $N\geq 2$ and $2\leq p\leq 4$. \begin{proof}[Proof of \eqref{5.14}]
Set $\frac{1}{p^\ast}+\frac{1}{p}=1$ and $\|c(j)\|_{l^1}=1$. From the definition of $T_fg$, we obtain \begin{equation}\label{5.16} \begin{split}
\|\dot{\Delta}_j(T_fg)\|_{L^2}&\leq \sum_{|k-j|\leq 4}\sum_{k^\prime\leq k-2}\|\dot{\Delta}_j(\dot{\Delta}_{k^\prime}f\dot{\Delta}_kg)\|_{L^2}\leq\sum_{|k-j|\leq 4}\sum_{k^\prime\leq k-2}\|\dot{\Delta}_{k^\prime}f\|_{L^{p^\ast}}\|\dot{\Delta}_kg\|_{L^p}\\[1ex]
&\lesssim\sum_{|k-j|\leq 4}\sum_{k^\prime\leq k-2}2^{k^\prime(\frac{2N}{p}-\frac{N}{2})}\|\dot{\Delta}_{k^\prime}f\|_{L^p}\|\dot{\Delta}_kg\|_{L^p}\\[1ex]
&\lesssim\sum_{|k-j|\leq 4}\sum_{k^\prime\leq k-2}2^{k^\prime(\frac{2N}{p}-\frac{N}{2}-s)}2^{k^\prime s}
\|\dot{\Delta}_{k^\prime}f\|_{L^p}2^{k(s-\frac{2N}{p}+1)}2^{k(-s+\frac{2N}{p}-1)}
\|\dot{\Delta}_kg\|_{L^p}\\[1ex] &\lesssim c(j)2^{j(1-\frac{N}{2})}
\|f\|_{\dot{B}_{p,1}^{s}}\|g\|_{\dot{B}_{p,1}^{-s+\frac{2N}{p}-1}}, \end{split} \end{equation} which yields \eqref{5.14}. Where we used that $p^\ast\geq p$ if $2\leq p\leq 4$ in the third inequality, and the condition $\frac{2N}{p}-\frac{N}{2}-s\geq 0$ in the last inequality. \end{proof}
\begin{proof}[Proof of \eqref{5.15}] From the definition of $R(f,g)$, it follows that \begin{equation}\label{5.17} \begin{split}
&\quad\|\dot{\Delta}_jR(f,g)\|_{L^2}\leq \sum_{k\geq j-3}\sum_{|k-k^\prime|\leq 1}
\|\dot{\Delta}_j(\dot{\Delta}_{k}f\dot{\Delta}_{k^\prime}g)\|_{L^2}\\[1ex]
&\lesssim 2^{j(\frac{2N}{p}-\frac{N}{2})}\sum_{k\geq j-3}\sum_{|k-k^\prime|\leq 1}2^{k(-s)}2^{ks}\|\dot{\Delta}_{k}f\|_{L^{p}}2^{k^\prime(s-\frac{2N}{p}+1)}
2^{k^\prime(-s+\frac{2N}{p}-1)}\|\dot{\Delta}_{k^\prime}g\|_{L^{p}}\\[1ex]
&\lesssim2^{j(\frac{2N}{p}-\frac{N}{2})}\sum_{k\geq j-3}2^{k(-\frac{2N}{p}+1)}c^2(k)\|f\|_{\dot{B}_{p,1}^{s}}
\|g\|_{\dot{B}_{p,1}^{-s+\frac{2N}{p}-1}}\\[1ex]
&\lesssim c(j)2^{j(1-\frac{N}{2})}\|f\|_{\dot{B}_{p,1}^{s}}
\|g\|_{\dot{B}_{p,1}^{-s+\frac{2N}{p}-1}}, \end{split} \end{equation} which yields \eqref{5.15}. Where we use that $1\leq \frac{p}{2}\leq 2$ in the second inequality and $1-\frac{2N}{p}\leq 0$ in the last inequality. \end{proof}
We claim that \begin{equation}\label{5.13}
\|(f,{\bf g}, {\bf m})\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell\lesssim\mathcal{X}_p(t)\Big(\|(a,{\bf u},{\bf H})\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell
+\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h+\|({\bf u},{\bf H})\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\Big). \end{equation} In what follows, we will prove \eqref{5.13} and inequalities \eqref{5.14} and \eqref{5.15} are often used for the purpose.
\underline{Estimate of $a{\rm div}{\bf u}$}. Decomposing $$a{\rm div}{\bf u}=T_a{\rm div}{\bf u}+R(a,{\rm div}{\bf u})+T_{{\rm div}{\bf u}}a^\ell+T_{{\rm div}{\bf u}}a^h,$$ one has \begin{equation}\label{5.18}
\|T_a{\rm div}{\bf u}+R(a,{\rm div}{\bf u})\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell\lesssim\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\|{\rm div}{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\lesssim\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}. \end{equation} \begin{equation}\label{5.19}
\|T_{{\rm div}{\bf u}}a^\ell\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell\lesssim\|{\rm div}{\bf u}\|_{L^\infty}
\|a^\ell\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}\lesssim\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}
\|a\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell. \end{equation} To handle the last term in the decomposition of $a{\rm div}{\bf u}$, we observe that owing to the spectral cut-off, there exists a universal integer $N_0$ such that $$
(T_{{\rm div}{\bf u}}a^h)^\ell=\dot{S}_{k_0+1}\Big(\sum_{|k-k_0|\leq N_0}\dot{S}_{k-1}({\rm div}{\bf u})\dot{\Delta}_ka^h\Big). $$ Thus, one has \begin{equation}\label{5.20} \begin{split}
&\quad\|T_{{\rm div}{\bf u}}a^h\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell\approx 2^{k_0(\frac{N}{2}-1)}\sum_{|k-k_0|\leq N_0}
\|\dot{S}_{k-1}({\rm div}{\bf u})\dot{\Delta}_ka^h\|_{L^2}\\[1ex]
&\lesssim 2^{k_0(\frac{N}{2}-1)}\sum_{|k-k_0|\leq N_0}\|\dot{S}_{k-1}({\rm div}{\bf u})\|_{L^{p^\ast}}\|\dot{\Delta}_ka^h\|_{L^p}\\[1ex]
&\lesssim 2^{k_0(\frac{N}{2}-1)}\sum_{|k-k_0|\leq N_0}\sum_{k^\prime\leq k-2}\|\dot{\Delta}_{k^\prime}({\rm div}{\bf u})\|_{L^{p^\ast}}\|\dot{\Delta}_ka^h\|_{L^p}\\[1ex]
&\lesssim 2^{k_0(\frac{N}{2}-1)}\sum_{|k-k_0|\leq N_0}\sum_{k^\prime\leq k-2}2^{k^\prime(\frac{2N}{p}-\frac{N}{2})}
\|\dot{\Delta}_{k^\prime}({\rm div}{\bf u})\|_{L^{p}}\|\dot{\Delta}_ka^h\|_{L^p}\\[1ex]
&\lesssim 2^{k_0(\frac{N}{2}-1)}\sum_{|k-k_0|\leq N_0}\sum_{k^\prime\leq k-2}2^{k^\prime(\frac{2N}{p}-\frac{N}{2}+2-\frac{N}{p})} 2^{k^\prime(\frac{N}{p}-2)}
\|\dot{\Delta}_{k^\prime}({\rm div}{\bf u})\|_{L^{p}}2^{k(-\frac{N}{p})}2^{k\frac{N}{p}}\|\dot{\Delta}_ka^h\|_{L^p}\\[1ex]
&\lesssim 2^{k_0}\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h, \end{split} \end{equation} where we have used that $2+\frac{N}{p}-\frac{N}{2}>0$ in the last inequality since $p\leq \frac{2N}{N-2}$.
\underline{Estimate of ${\bf u}\cdot\nabla a$}. We also decompose $${\bf u}\cdot\nabla a=T_{{\bf u}}\nabla a^\ell+T_{{\bf u}}\nabla a^h+R({\bf u},\nabla a)+T_{\nabla a}{\bf u},$$ and obtain from \eqref{5.14} and \eqref{5.15} that \begin{equation}\label{5.21} \begin{split}
\|T_{\nabla a}{\bf u}+R({\bf u},\nabla a)\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell\lesssim\|\nabla a\|_{\dot{B}_{p,1}^{\frac{N}{p}-2}}
\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}\lesssim\Big(\|a\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell
+\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\Big)\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}, \end{split} \end{equation} \begin{equation}\label{5.22} \begin{split}
\|T_{{\bf u}}\nabla a^\ell\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}\lesssim\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\|\nabla a^\ell\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\lesssim\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\|a\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell. \end{split} \end{equation} Similar to \eqref{5.20}, it holds that \begin{equation}\label{5.23} \begin{split}
&\quad\|T_{{\bf u}}\nabla a^h\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell\approx 2^{k_0(\frac{N}{2}-1)}\sum_{|k-k_0|\leq N_0}
\|\dot{S}_{k-1}{\bf u}\dot{\Delta}_k\nabla a^h\|_{L^2}\\[1ex]
&\lesssim 2^{k_0(\frac{N}{2}-1)}\sum_{|k-k_0|\leq N_0}\|\dot{S}_{k-1}{\bf u}\|_{L^{p^\ast}}\|\dot{\Delta}_k\nabla a^h\|_{L^p}\\[1ex]
&\lesssim 2^{k_0(\frac{N}{2}-1)}\sum_{|k-k_0|\leq N_0}\sum_{k^\prime\leq k-2}2^{k^\prime(\frac{2N}{p}-\frac{N}{2}+1-\frac{N}{p})} 2^{k^\prime(\frac{N}{p}-1)}
\|\dot{\Delta}_{k^\prime}{\bf u}\|_{L^{p}}2^{k(1-\frac{N}{p})}2^{k(\frac{N}{p}-1)}\|\dot{\Delta}_k\nabla a^h\|_{L^p}\\[1ex]
&\lesssim 2^{k_0}\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h, \end{split} \end{equation} where we have used that $1+\frac{N}{p}-\frac{N}{2}\geq 0$ in the last inequality since $p\leq \frac{2N}{N-2}$.
\underline{Estimate of ${\bf u}\cdot\nabla {\bf u}$}. Similarly, we decompose ${\bf u}\cdot\nabla {\bf u}=T_{{\bf u}}\nabla {\bf u}+R({\bf u},\nabla{\bf u})+T_{\nabla{\bf u}}{\bf u}$ and get that \begin{equation}\label{5.24} \begin{split}
\|T_{{\bf u}}\nabla{\bf u}+R({\bf u},\nabla {\bf u})\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell\lesssim\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\|\nabla{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\lesssim\Big(\|{\bf u}\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell
+\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h\Big)\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}, \end{split} \end{equation} \begin{equation}\label{5.25} \begin{split}
\|T_{\nabla{\bf u}}{\bf u}\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}\lesssim\|\nabla{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}-2}}
\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}\lesssim\Big(\|{\bf u}\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell
+\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h\Big)\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}. \end{split} \end{equation}
\underline{Estimate of $\pi_1(a)\mathcal{A}{\bf u}$}. Decomposing $$ \pi_1(a)\mathcal{A}{\bf u}=T_{\mathcal{A}{\bf u}}\pi_1(a) +R(\pi_1(a), \mathcal{A}{\bf u})+T_{\pi_1(a)}\mathcal{A}{\bf u}^\ell+T_{\pi_1(a)}\mathcal{A}{\bf u}^h, $$ we have \begin{equation}\label{5.26} \begin{split}
\|T_{\mathcal{A}{\bf u}}\pi_1(a)+R(\mathcal{A}{\bf u},\pi_1(a)\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell\lesssim\|\mathcal{A}{\bf u} \|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\|\pi_1(a)\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\lesssim\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}, \end{split} \end{equation} \begin{equation}\label{5.27} \begin{split}
\|T_{\pi_1(a)}\mathcal{A}{\bf u}^\ell\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}\lesssim
\|\pi_1(a)\|_{L^\infty}\|\mathcal{A}{\bf u} \|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell\lesssim\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\|{\bf u}\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell, \end{split} \end{equation} Similar to \eqref{5.20} again, it follows that \begin{equation}\label{5.28} \begin{split}
&\quad\|T_{\pi_1(a)}\mathcal{A}{\bf u}^h\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell\approx 2^{k_0(\frac{N}{2}-1)}\sum_{|k-k_0|\leq N_0}
\|\dot{S}_{k-1}\pi_1(a)\dot{\Delta}_k\mathcal{A}{\bf u}^h\|_{L^2}\\[1ex]
&\lesssim 2^{k_0(\frac{N}{2}-1)}\sum_{|k-k_0|\leq N_0}\sum_{k^\prime\leq k-2}\|\dot{\Delta}_{k^\prime}\pi_1(a)\|_{L^{p^\ast}}\|\dot{\Delta}_k\mathcal{A}{\bf u}^h\|_{L^p}\\[1ex]
&\lesssim 2^{k_0(\frac{N}{2}-1)}\sum_{|k-k_0|\leq N_0}\sum_{k^\prime\leq k-2}2^{k^\prime(\frac{2N}{p}-\frac{N}{2}+1-\frac{N}{p})} 2^{k^\prime(\frac{N}{p}-1)}
\|\dot{\Delta}_{k^\prime}\pi_1(a)\|_{L^{p}}2^{k(1-\frac{N}{p})}2^{k(\frac{N}{p}-1)}\|\dot{\Delta}_k\mathcal{A}{\bf u}^h\|_{L^p}\\[1ex]
&\lesssim 2^{k_0}\|\pi_1(a)\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}\|\mathcal{A}{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h
\lesssim \Big(1+\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\Big)\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\\[1ex]
&\lesssim\Big(1+\|a\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell+\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\Big)
\Big(\|a\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell+\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\Big)
\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h. \end{split} \end{equation}
\underline{Estimate of $\pi_2(a)\nabla a$}. Decomposing $$ \pi_2(a)\nabla a=T_{\nabla a}\pi_2(a) +R(\pi_2(a), \nabla a)+T_{\pi_2(a)}\nabla a^\ell+T_{\pi_2(a)}\nabla a^h, $$ we obtain \begin{equation}\label{5.29} \begin{split}
&\quad\|T_{\nabla a}\pi_2(a)+R(\nabla a,\pi_2(a)\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}\lesssim\|\nabla a \|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\|\pi_2(a)\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\\[1ex]
&\lesssim\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^2\lesssim
\Big(\|a\|_{\dot{B}_{2,1}^{\frac{N}{2}}}^\ell\Big)^2+\Big(\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\Big)^2
\lesssim \|a\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell \|a\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell+\Big(\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\Big)^2, \end{split} \end{equation} \begin{equation}\label{5.30} \begin{split}
\|T_{\pi_2(a)}\nabla a^\ell\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}\lesssim
\|\pi_2(a)\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}\|\nabla a^\ell\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\lesssim\Big(1+\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\Big)\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\|a\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell. \end{split} \end{equation} Similar to \eqref{5.28}, it follows that \begin{equation}\label{5.31} \begin{split}
&\quad\|T_{\pi_2(a)}\nabla a^h\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell\approx 2^{k_0(\frac{N}{2}-1)}\sum_{|k-k_0|\leq N_0}
\|\dot{S}_{k-1}\pi_2(a)\dot{\Delta}_k\nabla a^h\|_{L^2}\\[1ex]
&\lesssim 2^{k_0(\frac{N}{2}-1)}\sum_{|k-k_0|\leq N_0}\sum_{k^\prime\leq k-2}2^{k^\prime(\frac{2N}{p}-\frac{N}{2}+1-\frac{N}{p})} 2^{k^\prime(\frac{N}{p}-1)}
\|\dot{\Delta}_{k^\prime}\pi_2(a)\|_{L^{p}}2^{k(1-\frac{N}{p})}2^{k(\frac{N}{p}-1)}\|\dot{\Delta}_k\nabla a^h\|_{L^p}\\[1ex]
&\lesssim 2^{k_0}\|\pi_2(a)\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}\|\nabla a\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h
\lesssim \Big(1+\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\Big)\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\\[1ex]
&\lesssim\Big(1+\|a\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell+\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\Big)
\Big(\|a\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell+\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\Big)
\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h. \end{split} \end{equation}
\underline{Estimate of $\frac{1}{1+a}(2\widetilde{\mu}(a){\rm div}D({\bf u})+\widetilde{\lambda}(a)\nabla{\rm div}{\bf u})$}. The estimate of this term could be performed similar to that of $\pi_1(a)\mathcal{A}{\bf u}$ and the details are omitted here.
\underline{Estimate of $\frac{1}{1+a}(2\widetilde{\mu}^\prime(a)D({\bf u})\cdot\nabla a+\widetilde{\lambda}^\prime(a){\rm div}{\bf u}\nabla a)$}. We only deal with the term $\frac{2\widetilde{\mu}^\prime(a)}{1+a}D({\bf u})\cdot\nabla a$ and the remainder term could be similarly handled. Recalling \eqref{4.1000}, we derive \begin{equation}\label{5.32} \begin{split}
&\quad\|T_{\nabla J(a)}D{\bf u}+R(\nabla (J(a)),D{\bf u})\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}\lesssim\|\nabla J(a) \|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\|D{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\lesssim
\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}, \end{split} \end{equation} \begin{equation}\label{5.33} \begin{split}
&\quad\|T_{D{\bf u}}\nabla J(a)^\ell\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}\lesssim\|D{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}-2}}\|\nabla J(a)\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^\ell \lesssim
\|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\|a\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell, \end{split} \end{equation} and \begin{equation}\label{5.34} \begin{split}
&\quad\|T_{D{\bf u}}\nabla J(a)^h\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell\approx 2^{k_0(\frac{N}{2}-1)}\sum_{|k-k_0|\leq N_0}
\|\dot{S}_{k-1}D{\bf u}\dot{\Delta}_k\nabla J(a)^h\|_{L^2}\\[1ex]
&\lesssim 2^{k_0(\frac{N}{2}-1)}\sum_{|k-k_0|\leq N_0}\sum_{k^\prime\leq k-2}\|\dot{\Delta}_{k^\prime}D{\bf u}\|_{L^{p^\ast}}\|\dot{\Delta}_k\nabla J(a)^h\|_{L^p}\\[1ex]
&\lesssim 2^{k_0(\frac{N}{2}-1)}\sum_{|k-k_0|\leq N_0}\sum_{k^\prime\leq k-2}2^{k^\prime(\frac{2N}{p}-\frac{N}{2}+2-\frac{N}{p})} 2^{k^\prime(\frac{N}{p}-2)}
\|\dot{\Delta}_{k^\prime}D{\bf u}\|_{L^{p}}2^{k(1-\frac{N}{p})}2^{k(\frac{N}{p}-1)}\|\dot{\Delta}_k\nabla J(a)^h\|_{L^p}\\[1ex]
&\lesssim 2^{k_0}\|D{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}-2}}\|\nabla J(a)\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h
\lesssim \|{\bf u}\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h. \end{split} \end{equation}
\underline{Estimate of $\pi_1(a)(\nabla (I\cdot {\bf H})-I\cdot \nabla {\bf H})$}. Recall that we decompose $\pi_1(a)=\pi_1^\prime(0)a+\bar{\pi}_1(a)a$. The estimate of $a(\nabla (I\cdot {\bf H})-I\cdot \nabla {\bf H})$ is similar to that of $a{\rm div}{\bf u}$ and we omit it here. The remaining term can be estimated as follows. \begin{equation}\label{5.35} \begin{split}
\|T_{\bar{\pi}_1(a)a}\nabla {\bf H}+R(\bar{\pi}_1(a)a,\nabla {\bf H})\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}
&\lesssim\|\bar{\pi}_1(a)a\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\|\nabla{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\\[1ex] &\lesssim
\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\|{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}, \end{split} \end{equation} \begin{equation}\label{5.36} \begin{split}
&\quad\|T_{\nabla{\bf H}}\bar{\pi}_1(a)a\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}} \lesssim
\|\nabla{\bf H}\|_{L^\infty}\|\bar{\pi}_1(a)a\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}\\[1ex] &\lesssim
\|\nabla{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\Big(\|T_{\bar{\pi}_1(a)}a+R(\bar{\pi}_1(a), a)\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}+\|T_{a}\bar{\pi}_1(a)\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}\Big)\\[1ex]
&\lesssim\|{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}\Big(\|\bar{\pi}_1(a)\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
+\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}\|\bar{\pi}_1(a)\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\Big)\\[1ex]
&\lesssim\|{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}\Big(1+\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\Big). \end{split} \end{equation}
\underline{Estimate of $\frac{1}{1+a}(\frac{1}{2}\nabla |{\bf H}|^2-{\bf H}\cdot \nabla {\bf H})$}. By \eqref{4.200}, the term with ${\bf H}\cdot((\nabla {\bf H})^T-\nabla {\bf H})$ may be handled similar to ${\bf u}\cdot\nabla{\bf u}$, and the term with $\pi_1(a){\bf H}\cdot((\nabla {\bf H})^T-\nabla {\bf H})$ would be estimated as follows. \begin{equation}\label{5.37} \begin{split}
\|T_{{\pi}_1(a){\bf H}}\nabla {\bf H}+R({\pi}_1(a){\bf H},\nabla {\bf H})\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}
&\lesssim\|{\pi}_1(a){\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\|\nabla{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\\[1ex] &\lesssim
\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\|{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}
\|{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}, \end{split} \end{equation} \begin{equation}\label{5.38} \begin{split}
\|T_{\nabla{\bf H}}{\pi}_1(a){\bf H}\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}} &\lesssim
\|\nabla{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}\|{\pi}_1(a){\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}}}\\[1ex]
&\lesssim\|{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^2\|{\pi}_1(a)\|_{\dot{B}_{p,1}^{\frac{N}{p}}}
\lesssim \|{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}\|{\bf H}\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}. \end{split} \end{equation}
\underline{Estimate of ${\bf m}$}. The estimation of ${\bf m}$ is similar to that of ${\bf u}\cdot\nabla {\bf u}$ and the details are omitted. So far, the inequality \eqref{5.13} is proved.
Inserting \eqref{5.2},\eqref{5.3},\eqref{5.12} and \eqref{5.13} into \eqref{5.1} and applying the fact that $\mathcal{X}_p(t)\lesssim\mathcal{X}_{p,0}\ll 1$ for all $t\geq 0$, we end up with \begin{equation}\label{5.39} \begin{split}
&\quad\frac{d}{dt}\Big(\|(a, {\bf u}, {\bf H})\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell+\|(\nabla a, {\bf u}, {\bf H})\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h\Big)\\[1ex]
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
+\Big(\|(a, {\bf u}, {\bf H})\|_{\dot{B}_{2,1}^{\frac{N}{2}+1}}^\ell
+\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h+\|({\bf u}, {\bf H})\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\Big)\leq 0. \end{split} \end{equation}
In what follows, we will employ the following interpolation inequalities: \begin{prop}\label{pra.7}(\cite{xin2018optimal}) Suppose that $m\neq \rho$. Then it holds that $$
\|f\|_{\dot{B}_{p,1}^j}^\ell\lesssim (\|f\|_{\dot{B}_{r,\infty}^m}^\ell)^{1-\theta}(\|f\|_{\dot{B}_{r,\infty}^{\rho}}^\ell)^{\theta},\,\,\,\,\,\,
\|f\|_{\dot{B}_{p,1}^j}^h\lesssim (\|f\|_{\dot{B}_{r,\infty}^m}^h)^{1-\theta}(\|f\|_{\dot{B}_{r,\infty}^{\rho}}^h)^{\theta} $$ where $j+N(\frac{1}{r}-\frac{1}{p})=m(1-\theta)+\rho\theta$ for $0<\theta<1$ and $1\leq r\leq p\leq \infty$. \end{prop}
Due to $-\sigma_1<\frac{N}{2}-1\leq \frac{N}{p}<\frac{N}{2}+1$, it follows from Proposition \ref{pra.7} that \begin{equation}\label{5.40}
\|(a,{\bf u},{\bf H})\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell
\leq C\Big(\|(a,{\bf u},{\bf H})\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell\Big)^{\theta_0}
\Big(\|(a,{\bf u},{\bf H})\|_{\dot{B}_{2,\infty}^{\frac{N}{2}+1}}^\ell\Big)^{1-\theta_0}, \end{equation} where $\theta_0=\frac{2}{N/2+1+\sigma_1}\in (0,1)$. In view of \eqref{4.46}, we have $$
\|(a,{\bf u},{\bf H})\|_{\dot{B}_{2,\infty}^{\frac{N}{2}+1}}^\ell\geq c_0\Big(\|(a,{\bf u},{\bf H})\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell\Big)^{\frac{1}{1-\theta_0}} $$ where $c_0=C^{-\frac{1}{1-\theta_0}}C_0^{-\frac{\theta_0}{1-\theta_0}}$. Moreover, it follows from
$\|(\nabla a, {\bf u}, {\bf H})\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h\leq \mathcal{X}_p(t)\lesssim \mathcal{X}_{p,0}\ll 1$ for all $t\geq 0$ that $$
\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\geq \Big(\|a\|_{\dot{B}_{p,1}^{\frac{N}{p}}}^h\Big)^{\frac{1}{1-\theta_0}},\,\,\,\,\,
\|({\bf u}, {\bf H})\|_{\dot{B}_{p,1}^{\frac{N}{p}+1}}^h\geq \Big(\|({\bf u}, {\bf H})\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h\Big)^{\frac{1}{1-\theta_0}}. $$ Thus, there exists a constant $\tilde{c}_0>0$ such that the following Lyapunov-type inequality holds: \begin{equation}\label{5.41} \begin{split}
&\quad\frac{d}{dt}\Big(\|(a, {\bf u}, {\bf H})\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell+\|(\nabla a, {\bf u}, {\bf H})\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h\Big)\\[1ex]
&\quad\quad\quad\quad\quad\quad\quad
+\tilde{c}_0\Big(\|(a, {\bf u}, {\bf H})\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell
+\|(\nabla a, {\bf u}, {\bf H})\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h\Big)^{1+\frac{2}{N/2-1+\sigma_1}}\leq 0. \end{split} \end{equation} Solving \eqref{5.41} yields \begin{equation}\label{5.42} \begin{split}
&\quad\|(a, {\bf u}, {\bf H})(t)\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell+\|(\nabla a, {\bf u}, {\bf H})(t)\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h\\[1ex] &\leq\Big(\mathcal{X}_{p,0}^{-\frac{2}{N/2-1+\sigma_1}} +\frac{2\tilde{c}_0t}{N/2-1+\sigma_1}\Big)^{-\frac{N/2-1+\sigma_1}{2}} \lesssim (1+t)^{-\frac{N/2-1+\sigma_1}{2}} \end{split} \end{equation} for all $t\geq 0$. Through the embedding properties in Proposition \ref{pr2.1}, we arrive at \begin{equation}\label{5.43} \begin{split}
\|(a, {\bf u}, {\bf H})(t)\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}\lesssim\|(a, {\bf u}, {\bf H})(t)\|_{\dot{B}_{2,1}^{\frac{N}{2}-1}}^\ell+\|(\nabla a, {\bf u}, {\bf H})(t)\|_{\dot{B}_{p,1}^{\frac{N}{p}-1}}^h \lesssim (1+t)^{-\frac{N/2-1+\sigma_1}{2}}. \end{split} \end{equation} In addition, if $\sigma\in (-\sigma_1-N(\frac{1}{2}-\frac{1}{p}), \frac{N}{p}-1)$, then employing Proposition \ref{pra.7} once again implies that \begin{equation}\label{5.44} \begin{split}
\|(a, {\bf u}, {\bf H})(t)\|_{\dot{B}_{p,1}^{\sigma}}^\ell\lesssim\|(a, {\bf u}, {\bf H})(t)\|_{\dot{B}_{2,1}^{\sigma+N(\frac{1}{2}-\frac{1}{p})}}^\ell
\lesssim\Big(\|(a,{\bf u},{\bf H})\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell\Big)^{\theta_1}
\Big(\|(a,{\bf u},{\bf H})\|_{\dot{B}_{2,\infty}^{\frac{N}{2}-1}}^\ell\Big)^{1-\theta_1}, \end{split} \end{equation} where $$ \theta_1=\frac{\frac{N}{p}-1-\sigma}{\frac{N}{2}-1+\sigma_1}\in(0,1). $$ Note that $$
\|(a,{\bf u},{\bf H})\|_{\dot{B}_{2,\infty}^{-\sigma_1}}^\ell\leq C_0 $$ for all $t\geq 0$. From \eqref{5.42} and \eqref{5.44}, we deduce that \begin{equation}\label{5.45} \begin{split}
\|(a, {\bf u}, {\bf H})(t)\|_{\dot{B}_{p,1}^{\sigma}}^\ell\lesssim \Big[(1+t)^{-\frac{N/2-1+\sigma_1}{2}}\Big]^{1-\theta_1} =(1+t)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{p})-\frac{\sigma+\sigma_1}{2}} \end{split} \end{equation} for all $t\geq 0$, which leads to \begin{equation}\label{5.46} \begin{split}
\|(a, {\bf u}, {\bf H})(t)\|_{\dot{B}_{p,1}^{\sigma}}\lesssim
\|(a, {\bf u}, {\bf H})(t)\|_{\dot{B}_{p,1}^{\sigma}}^\ell+\|(a, {\bf u}, {\bf H})(t)\|_{\dot{B}_{p,1}^{\sigma}}^h \lesssim (1+t)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{p})-\frac{\sigma+\sigma_1}{2}} \end{split} \end{equation} for $\sigma\in (-\sigma_1-N(\frac{1}{2}-\frac{1}{p}), \frac{N}{p}-1)$. So far, the proof of Theorem \ref{th1.2} is completed. \subsection{Proof of Corollary \ref{col1}} In fact, Corollary \ref{col1} can be regarded as the direct consequence of the following interpolation inequality: \begin{prop}\label{pra.8}(\cite{bahouri2011fourier}) The following interpolation inequality holds true: $$
\|\Lambda^lf\|_{L^r}\lesssim \|\Lambda^mf\|_{L^q}^{1-\theta}\|\Lambda^kf\|_{L^q}^\theta, $$ whenever $0\leq \theta\leq 1, 1\leq q\leq r\leq \infty$ and $$ l+N\Big(\frac{1}{q}-\frac{1}{r}\Big)=m(1-\theta)+k\theta. $$ \end{prop}
With the aid of Proposition \ref{pra.8}, we define $\theta_2$ by the relation $$ m(1-\theta_2)+k\theta_2=l+N\Big(\frac{1}{p}-\frac{1}{r}\Big), $$ where $m=\frac{N}{p}-1$ and $k=-\sigma_1-N(\frac{1}{2}-\frac{1}{p})+\varepsilon$ with $\varepsilon>0$ small enough. It is easy to see that $\theta_2\in(0,1)$ if $\varepsilon>0$ is small enough. As a consequence, we conclude by $\dot{B}_{p,1}^0\hookrightarrow L^p$ that \begin{equation}\label{5.47} \begin{split}
&\quad\|\Lambda^l(a,{\bf u},{\bf H})\|_{L^r}\lesssim
\|\Lambda^m(a,{\bf u},{\bf H})\|_{L^p}^{1-\theta_2}\|\Lambda^k(a,{\bf u},{\bf H})\|_{L^p}^{\theta_2}\\[1ex] &\lesssim \left[(1+t)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{p})-\frac{m+\sigma_1}{2}}\right]^{1-\theta_2} \left[(1+t)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{p})-\frac{k+\sigma_1}{2}}\right]^{\theta_2} =(1+t)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{r})-\frac{l+\sigma_1}{2}} \end{split} \end{equation} for $p\leq r\leq \infty$ and $l\in\mathbb{R}$ satisfying $-\sigma_1 -\frac{N}{2}+\frac{N}{p}<l+\frac{N}{p}-\frac{N}{r}\leq \frac{N}{p}-1$. Thus, we finish the proof of Corollary \ref{col1}.
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December 2011, 31(4): 1469-1477. doi: 10.3934/dcds.2011.31.1469
Hyers--Ulam--Rassias stability of derivations in proper Jordan $CQ^{*}$-algebras
Golamreza Zamani Eskandani 1, and Hamid Vaezi 1,
Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran, Iran
Received October 2009 Revised February 2010 Published September 2011
In this paper, we investigate derivation in proper Jordan $CQ^{*}$-algebras associated with the following Pexiderized Jensen type functional equation \[kf(\frac{x+y}{k}) = f_{0}(x)+ f_{1} (y).\] This is applied to investigate derivations and their Hyers--Ulam--Rassias stability in proper Jordan $CQ^{*}$-algebras.
Keywords: proper Jordan $CQ^{*}$-algebra, Hyers-Ulam-Rassias stability, Jordan derivations..
Mathematics Subject Classification: 17B40, 39B52, 47N50, 47L60, 46B0.
Citation: Golamreza Zamani Eskandani, Hamid Vaezi. Hyers--Ulam--Rassias stability of derivations in proper Jordan $CQ^{*}$-algebras. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1469-1477. doi: 10.3934/dcds.2011.31.1469
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How do I determine the molecular vibrations of linear molecules?
I know how to do it for just about every other point group, but the $D_{\infty \mathrm h}$ and $C_{\infty \mathrm v}$ character tables aren't as straightforward. In particular, I'm interested in the vibrational modes of carbon dioxide, $\ce{CO2}$.
$$\begin{array}{c|cccc|cc} \hline C_{\infty\mathrm{v}} & E & 2C_\infty^\phi & \cdots & \infty\sigma_\mathrm{v} & & \\ \hline \mathrm{A_1} \equiv \Sigma^+ & 1 & 1 & \cdots & 1 & z & x^2 + y^2, z^2 \\ \mathrm{A_2} \equiv \Sigma^- & 1 & 1 & \cdots & -1 & R_z & \\ \mathrm{E_1} \equiv \Pi & 2 & 2 \cos\phi & \cdots & 0 & (x,y), (R_x,R_y) & (xz,yz) \\ \mathrm{E_2} \equiv \Delta & 2 & 2 \cos 2\phi & \cdots & 0 & & (x^2-y^2,xy) \\ \mathrm{E_3} \equiv \Phi & 2 & 2 \cos 3\phi & \cdots & 0 & & \\ \vdots & \vdots & \vdots & \ddots & \vdots & & \\ \hline \end{array}$$
$\,$
$$\small \begin{array}{c|cccccccc|cc} \hline D_{\infty\mathrm{h}} & E & 2C_\infty^\phi & \cdots & \infty\sigma_\mathrm{v} & i & 2S_\infty^\phi & \cdots & \infty C_2 & \\ \hline \mathrm{A_{1g}} \equiv \Sigma^+_{\mathrm{g}} & 1 & 1 & \cdots & 1 & 1 & 1 & \cdots & 1 & & x^2 + y^2, z^2 \\ \mathrm{A_{2g}} \equiv \Sigma^-_{\mathrm{g}} & 1 & 1 & \cdots & -1 & 1 & 1 & \cdots & -1 & R_z & \\ \mathrm{E_{1g}} \equiv \Pi_{\mathrm{g}} & 2 & 2\cos\phi & \cdots & 0 & 2 & -2\cos\phi & \cdots & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{E_{2g}} \equiv \Delta_{\mathrm{g}} & 2 & 2\cos 2\phi & \cdots & 0 & 2 & 2\cos 2\phi & \cdots & 0 & & (x^2-y^2,xy) \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & & \\ \mathrm{A_{1u}} \equiv \Sigma^+_{\mathrm{u}} & 1 & 1 & \cdots & 1 & -1 & -1 & \cdots & -1 & z & \\ \mathrm{A_{2u}} \equiv \Sigma^-_{\mathrm{u}} & 1 & 1 & \cdots & -1 & -1 & -1 & \cdots & 1 & & \\ \mathrm{E_{1u}} \equiv \Pi_{\mathrm{u}} & 2 & 2\cos\phi & \cdots & 0 & -2 & 2\cos\phi & \cdots & 0 & (x,y) & \\ \mathrm{E_{2u}} \equiv \Delta_{\mathrm{u}} & 2 & 2\cos 2\phi & \cdots & 0 & -2 & -2\cos 2\phi & \cdots & 0 & & \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & & \\ \hline \end{array}$$
spectroscopy group-theory
stephen zilchstephen zilch
$\begingroup$ Perhaps not the answer you are looking for, but the most straightforward way is just to learn what they are instead of figuring them out. It is quite logical, there are two stretching modes (symmetric $\Sigma_g^+$ + antisymmetric $\Sigma_u^+$) and two degenerate bending modes $\Pi_u$. You can visualise them here: chemtube3d.com/vibrationsCO2.htm $\endgroup$ – orthocresol♦ Mar 20 '17 at 18:23
$\begingroup$ @orthocresol True, but that only gives an answer for the case where it is $D_{\infty h}$ with 3 atoms. This is the case that we see most often, but it is useful to have a general way to derive it rather than memorizing cases. $\endgroup$ – Tyberius Mar 20 '17 at 22:15
As Tyberius noted, the projection formula does not work for infinite order groups (this is because the Hermitian form on characters is defined to be G-invariant by averaging over all elements in a group. That is, it works because it is possible to hit all the elements of the group in some order).
How can one work in the infinite dimensional groups then? One property that still holds for these particular infinite groups (though not in general 1) is Maschke's Theorem, that every representation is the direct sum of irreducible representations. So we can still determine what irreps make up a rep by adding them up.
First we need to find the representation we wish to reduce. I find that the easiest way to do this is to start with just the atoms (instead of the vectors) and then include the vector contribution afterwards. For the sake of a new example, I will do the linear $\ce{I4^{2-}}$ anion instead of $\ce{CO2}$.
$$\small \begin{array}{c|cccccccc} \hline D_{\infty\mathrm{h}} & E & 2C_\infty^\phi & \cdots & \infty\sigma_\mathrm{v} & i & 2S_\infty^\phi & \cdots & \infty C_2 & \\ \hline \Gamma_\text{atoms} & 4 & 4 & \cdots & 4 & 0 & 0 & \cdots & 0 &\\ \end{array}$$
Now, we multiply by the vector's contributions to the character. $$\begin{array}{c|c}E&3\\\hline C_2&-1\\\hline \sigma&1\\\hline i&-3\\\hline C_n&1+2\cos(\frac{2\pi}{n})=1+2\cos\theta\\\hline S_n&-1+2\cos(\frac{2\pi}{n})=-1+2\cos\theta\end{array}$$
These come from taking the trace of the matrix representations of each symmetry operation. For example, a rotation about the z-axis by an angle $\theta$ is given by: $$\left[\begin{array}{ccc}\cos\theta&\sin\theta&0\\-\sin\theta&\cos\theta&0\\0&0&1\end{array}\right]$$
$$\small \begin{array}{c|cccccccc} \hline D_{\infty\mathrm{h}} & E & 2C_\infty^\phi & \cdots & \infty\sigma_\mathrm{v} & i & 2S_\infty^\phi & \cdots & \infty C_2 & \\ \hline \Gamma_\text{xyz} & 12 & 4+8\cos\theta & \cdots & 4 & 0 & 0 & \cdots & 0 &\\ \end{array}$$
Now, we are only interested in the vibrational modes so we must subtract off the translational and rotational degrees of freedom. We can check the character table for these (they are the ones which correspond to $z$, $(x,y)$ and $(R_x, R_y)$, $R_z$ doesn't count because we define the molecular axis to be the $z$-axis. $$\small \begin{array}{c|cccccccc} \hline D_{\infty\mathrm{h}} & E & 2C_\infty^\phi & \cdots & \infty\sigma_\mathrm{v} & i & 2S_\infty^\phi & \cdots & \infty C_2 & \\ \hline \Gamma_\text{xyz} & 12 & 4+8\cos\theta & \cdots & 4 & 0 & 0 & \cdots & 0 &\\ -\Gamma_\text{rot} & 2 & 2\cos\theta&\cdots&0 & 2 & -2\cos\theta&\cdots&0 \\ -\Gamma_\text{trans} & 3 & 1+2\cos\theta&\cdots& 1 & -3 & -1+2\cos\theta&\cdots&-1 \\ \hline\Gamma_\text{vib} & 7 & 3+4\cos\theta&\cdots& 3 & 1 & 1&\cdots&1 \\ \end{array}$$
So now we have the actual representation of the vibrations. How do we go about decomposing it? We want to eventually subtract off all of the irreps until we have the zero vector so we consider what effect each irrep has and try to move our remaining rep closer to zero. The first thing to note is that only $\Pi_g$ and $\Pi_u$ irreps have $2\cos\theta$. $\Pi_u$ adds $2\cos\theta$ under both proper and improper rotations, while $\Pi_g$ adds to proper and subtracts from improper. We note that in our rep, we only have $\cos\theta$ terms under proper rotations and so we need to subtract off enough $\Pi_u$'s until there are the same number of $\cos\theta$ (with different signs) under proper and improper rotations, at which point we should subtract off $\Pi_g$s until there are no $\cos\theta$ terms.
$$\small \begin{array}{c|cccccccc} \hline D_{\infty\mathrm{h}} & E & 2C_\infty^\phi & \cdots & \infty\sigma_\mathrm{v} & i & 2S_\infty^\phi & \cdots & \infty C_2 & \\ \hline \Gamma_\text{vib} & 7 & 3+4\cos\theta&\cdots& 3 & 1 & 1&\cdots&1 \\ -\Pi_u & 2 & 2\cos\theta&\cdots&0 & -2 & 2\cos\theta&\cdots&1 \\ -\Pi_g & 2 & 2\cos\theta&\cdots&0 & 2 & -2\cos\theta&\cdots&1 \\ \hline\Gamma & 3 & 3 &\cdots& 3 & 1 & 1&\cdots&1 \\ \end{array}$$
Similar to the $\Pi_g/\Pi_u$, the $\Sigma_g^+$ shift adds 1 to all positions, while $\Sigma_u^+$ adds 1 to all positions before $i$ and subtracts from all after. We subtract enough $\Sigma_g^+$ until all the positions before and after $i$ have equal magnitude. $$\small \begin{array}{c|cccccccc} \hline D_{\infty\mathrm{h}} & E & 2C_\infty^\phi & \cdots & \infty\sigma_\mathrm{v} & i & 2S_\infty^\phi & \cdots & \infty C_2 & \\ \hline \Gamma & 3 & 3 &\cdots& 3 & 1 & 1&\cdots&1 \\ -2\Sigma_g^+ & 2 & 2 &\cdots&2 & 2 & 2&\cdots&2 \\ -\Sigma_u^+ & 1 & 1 &\cdots&1 & -1 & -1&\cdots&-1 \\ \hline \Gamma& 0 & 0 &\cdots&0 & 0 & 0&\cdots&0 \end{array}$$
Thus, we have found that the vibrational modes of $\ce{I_4^{2-}}$ to be $2\Sigma_g^+ +\Sigma_u^+ + \Pi_u + \Pi_g$.
We can check this with the formulas given in Group Theory And Chemistry by David M. Bishop as noted by Feodoran. One can also follow Tyberius's suggestion and work in $D_{2h}$ which will give the decomposition $2A_g+B_{1u}+B_{2u}+B_{3u}+B_{2g}+B_{3g}$.
It is important to note that correlation from the $D_{2h}$ point group to $D_{\infty h}$ works fine for vibrations because you cannot have terms higher than $\Pi$ and so there is no ambiguity. For things like electronic states, though, in which direct products must be decomposed, you will run into some problems. If the $D_{2h}$ result includes $A_g+B_{1g}$ does that correlate with a $\Sigma_g^+ +\Sigma_g^-$ or a $\Delta_g$ state in $D_{\infty h}$? Working in the parent infinite group may be more difficult without the reduction formula, but these problems are avoided.
The reason going to lower symmetry works is because all characters under the infinitely many proper and improper rotations are the same for each irrep (namely $2\cos n\theta$) and so we can represent the behavior of all of them by the behavior of one of them. Hence, we pick $\theta=\pi$ and ignore all the others and this produces the $D_{2h}$ point group (the two improper rotation axes are equivalent to $i$ and the two proper axes are equivalent to one $C_2$ about $z$). This is origin of the ambiguity. In $D_{\infty h}$, $\Sigma_g^+ + \Sigma_g^-$ are equal to $\Delta_g$ at $\theta=\pi$, but not in general.
Having now identified the origin of the ambiguity, one can tune how fast these ambiguities occur though. The problem occurs when the $\theta$ chosen to represent all $\theta$ has the property $1=\cos n\theta$ where $n$ represents the period of the ambiguity. So for $D_{2h}$, $\theta=\pi$ and $1=\cos2\theta$ and the period of the ambiguity is 2. That is, every other term will be ambiguously identical ($\Sigma, \Delta, \Gamma, \ldots$ are the same and $\Pi,\Phi,\ldots$ are the same). By dropping to $D_{4h}$ instead, $\theta=\pi/2$ and $1=\cos4\theta$. The ambiguity period is longer and we can distinguish $\Delta$ from $\Sigma$ by correlation from the $D_{4h}$ point group. We can thus select the point group to descend to based on how much resolving power we expect to need in the parent point group (though we should stick to even numbers to maintain inversion symmetry for $D_{\infty h}$). Of course we could also stay in the parent point group and never have a problem, but also have no reduction formula.
answered Jul 8 '17 at 7:01
levinedslevineds
Instead of trying to use symmetry tables it is possible to calculate the vibrational normal modes directly using the equations of motion, rather as would be done for a double pendulum for example.
It is normally assumed that the vibrational motion of the atoms is harmonic, i.e. Hook's law applies, and we additionally assume that the only force constants are those between adjacent atoms; this is equivalent to assuming that the valence bond model describes the bonding. A molecular orbital model would consider forces constants between one atom and every other atom.
The starting point is to calculate the potential energy and, from this, the forces on the atoms. As carbon dioxide is symmetrical, there is only one force constant, k, but as the masses are different it is found that the mass-weighted force constants are needed in the calculation and these are different for each type of atom.
Stretching vibrations
The normal modes for a simple molecule can easily be sketched and as the molecule is linear there are $3N − 5$ or four modes in total, two are stretches and the other two are the degenerate bending motion in the plane of the page. One where the carbon atom moves down and the two oxygen atoms move up and then vice versa. The other bending normal mode is the similar motion but perpendicular to the plane of the page. Only the stretching modes are considered here.
Placing a vector $s_1,\,s_2,\,s_3$ on each atom and pointing in the same direction and along the long axis of the molecule, the potential energy V is the sum of terms for the stretching of each bond; therefore,
$$V(s_1,s_2,s_3)=\frac{k}{2}(s_1-s_2)^2+\frac{k}{2}(s_2-s_3)^2 $$
These equations are just for the stretching motion, a second calculation is needed with vectors perpendicular to the molecular axis for the bending modes.
The forces are the negative derivatives with respect to the displacements
$$-\frac{dV}{ds_1}=-k(s_1-s_2); \quad -\frac{dV}{ds_2}=k(s_1-s_2)-k(s_2-s_3);\quad -\frac{dV}{ds_3}=k(s_2-s_3)$$
With f as the force, placing these equations into matrix form gives
$$\begin{bmatrix} f_1\\ f_2\\f_3 \end{bmatrix} = \begin{bmatrix} -k & k & 0 \\ k & -2k & k\\ 0 & k & -k \end{bmatrix} \begin{bmatrix} s_1\\ s_2\\s_3 \end{bmatrix}$$
but, because the masses are different, we must change to mass weighted force constants, using the formula $K_{i,j}=k_{i,j}/\sqrt{m_im_j}$ where i and j are the atom indices; for example $K_{1,2} = k\sqrt{ m_Om_C}$, is the mass-weighted force constant between atoms 1 and 2, if $m_O$ is the oxygen mass and $m_C$ that of the carbon. The matrix of force constants becomes
$$\boldsymbol{K} = \begin{bmatrix} \displaystyle\frac{-k}{m_O} & \displaystyle\frac{k}{\sqrt{m_Om_C}} & 0 \\ \displaystyle\frac{k}{\sqrt{m_Om_C}} & \displaystyle\frac{-2k}{m_C} & \displaystyle\frac{k}{\sqrt{m_Om_C}}\\ 0 & \displaystyle\frac{k}{\sqrt{m_Om_C}} &\displaystyle \frac{-k}{m_O} \end{bmatrix}$$
which is solved as a secular determinant with eigenvalues $\lambda$,
$$ \lambda_1=-k/m_O; \quad \lambda_2= -k\frac{2m_O+m_C}{m_Om_C}; \quad \lambda_3 = 0 $$
The frequency of each vibration is $\omega^2 = −\lambda$ so the square of the normal mode frequencies are
$$\omega_1^2=k/m_O; \quad \omega_2^2 = k\frac{2m_O+m_C}{m_Om_C} ; \quad \omega_3=0$$
Using the measured stretching vibrational frequencies for $\ce{CO2}$, for example, which are 1337 & 2349 cm$^{-1}$, the force constants are 1680 & 1418 N/m respectively. Note that the frequency $\omega = 2\pi\nu c$ with $\nu$ in cm$^{-1}$.
The normalised eigenvector matrix x is, after some rearranging and with the total mass as $M=2m_O+m_C$
$$\boldsymbol {x}= \frac{1}{\sqrt{2M}} \begin{bmatrix} -\sqrt{M} & \sqrt{m_C} & \sqrt{2m_O} \\ 0 & -2\sqrt{m_O} & \sqrt{2m_C}\\ \sqrt{M}& \sqrt{m_C} & \sqrt{2m_O} \end{bmatrix}$$
Notice that the eigenvectors do not depend on the force constants; this matrix is used to produce the geometry that is related to the symmetry of the vibrations, and cannot depend on the value of the force constants. The normal modes depend only on the geometry because the 'springs' connecting the atoms can only vibrate in certain patterns governed by the geometry or symmetry of the molecule; the frequency and size of extension depend on the force constants.
The normal mode coordinates are calculated using $\boldsymbol{Q = x^Tq}$ where $\boldsymbol{x}^T$ is the transpose of the eigenvector matrix and q the vector of mass weighted coordinates with $q=s\sqrt{m}$.
$$\begin{bmatrix} Q_1\\Q_2\\Q_3 \end{bmatrix} = \begin{bmatrix} -\sqrt{M} & 0 & \sqrt{M} \\ \sqrt{m_C} & -2\sqrt{m_O} & \sqrt{m_C}\\ \sqrt{2m_O}& \sqrt{2m_C} & \sqrt{2m_O} \end{bmatrix} \begin{bmatrix} s_1\sqrt{m_O}\\s_2\sqrt{m_C}\\s_3\sqrt{m_O} \end{bmatrix}$$
and the individual modes are
$$Q_1=\sqrt{m_O/2}(-s_1+s_3); \quad Q_2 = \sqrt{\frac{m_om_C}{2M}}(s_1-2s_2+s_3);\\ \quad Q_3=(m_Os_1+m_Cs_2+m_Os_3)/\sqrt{M}$$
The symmetry of the normal mode can be seen from the displacements s. The last, $Q_3$, is just a translation of the molecule as all the s vectors point in the same direction and has frequency $\omega=0$. $Q_1$ would correspond to a symmetric stretch and therefore $Q_2$ to the assymetric stretch, both O atoms move in one direction and the carbom moves the opposite way to keep the centre of gravity fixed in space.
We can choose any Q to be zero to find the s displacements
(i) Suppose we choose $Q_1 = Q_2 = 0$, then all the atom displacements are the same, $s_1 = s_2 = s_3$ and are $Q_3/\sqrt{M}$. As each atom is moving in the same direction this cannot be a vibrational normal mode as the bonds are neither stretched nor compressed, but represents a translation, and clearly, this corresponds to the zero frequency eigenvalue $\omega_3 = 0$. Note that if $Q_3 = 0$ this must mean that the centre of mass does not change, no translation occurs, and so we expect to produce normal modes with this condition.
(ii) If $Q_1=Q_3=0$ then $\displaystyle s_1=s_3= \frac{m_C}{2m_OM}Q_2$ and $\displaystyle s_2= - \frac{2m_O}{2m_OM}Q_2$ In this case this is the asymmetric stretch with the oxygen atoms moving in the same direction and opposed to that of the carbon, and this is $\omega_2$. The relative motion is $s_1 = s_3 = 0.115Q_2$ to $s_2 = −0.308Q_2$ so the carbon atom moves further than the oxygen atoms do, which is not surprising, because the centre of mass has to be held constant and the C atom has to compensate for the motion of two O atoms.
(iii) Finally, if $Q_2 = Q_3 = 0$ then the equations are solved when $s_2 = 0$ and $s_1 = −s_3$, which is the symmetric stretch $\omega_1$ with displacements $\pm Q_1/\sqrt{2m_O}$ or $0.176Q_1$.
There is a very general method with which to do these types of calculations called the GFG matrix method or some similar name equivalent to this. As this is a matrix method it is easy to calculate equations using symbolic algebra programmes such as SymPy or to do the calculation numerically using for example Python. (Both SymPy & Python are free to use).
The secular determinant to be solved is
$$ | \boldsymbol{GFG} -\omega^2\boldsymbol{I} |=0$$
where I is a unit diagonal matrix, $\omega$ the normal mode frequencies and the matrices are G is a matrix zero everywhere except on the diagonal where it has values $1/\sqrt{m}$, i.e.
$$\boldsymbol{G} = diag \left [ 1/\sqrt{m_i}\right ] $$
and the F matrix is that of the force constants; $$\boldsymbol{F} = \begin{bmatrix} k_{11} & k_{12} & k_{13} & \cdots\\ k_{21} & k_{22} &\cdots & \cdots \\ \vdots & \vdots & & k_{nn} \end{bmatrix}$$
and the product is just the mass weighted matrix of force constants
$$\boldsymbol{GFG} = \begin{bmatrix} k_{11}/m_1 & k_{12}/\sqrt{m_1m_2} & k_{13}/\sqrt{m_1m_3} & \cdots\\ k_{21}/\sqrt{m_1m_2} & k_{22}/m_2 &\cdots & \cdots \\ \vdots & \vdots & & k_{nn}/\sqrt{m_n} \end{bmatrix}$$
The eigenvectors x can be used to produce the normal mode displacements as $\boldsymbol{Q=x^Tq}$ where $\boldsymbol{q=G^{-1}s}$ so that each element is $q_i= s_i\sqrt{m_i}$ and the coordinate displacements via, $\boldsymbol{ s =GxQ}$.
Bending vibrations
In the case of bending vibrations in a molecule with masses $m_{1,2,3}$, such as HCN, the potential is written as
$$V= k_br_1r_2(\delta\theta)^2/2$$
where the bond lengths are $r_{1,2}$ and the bending force constant $k_b=k_\theta r_1r_2$ which keeps its units in N/m. For small angle bends $\displaystyle \delta\theta = \frac{(s_1 - s_2)}{r_1} - \frac{(s_3 - s_2)}{r_2}$ (where vector s is perpendicular to the internuclear axis just as s was along the axis) and the mass weighted K matrix of force constants becomes
$$K = \begin{bmatrix} \displaystyle \frac{k_3}{m_1}\left(\frac{r_2}{r_1}\right) & \displaystyle\frac{-k_3}{\sqrt{m_1m_2}}\left(1+\frac{r_2}{r_1}\right) & \displaystyle \frac{k_3}{\sqrt{m_1m_3}} \\ \displaystyle\frac{-k_3}{\sqrt{m_1m_2}}\left(1+\frac{r_2}{r_1}\right) & \displaystyle\frac{k_3}{m_2}(2+\frac{r_2}{r_1}+\frac{r_1}{r_2}) & \displaystyle \frac{-k_3}{\sqrt{m_2m_3}}\left(1+\frac{r_1}{r_2}\right)\\ \displaystyle \frac{k_3}{\sqrt{m_1m_3}} & \displaystyle\frac{-k_3}{\sqrt{m_2m_3}}\left(1+\frac{r_1}{r_2}\right) & \displaystyle\frac{k_3}{m_3}\left(\frac{r_1}{r_2} \right) \end{bmatrix}$$
which means that the ratio of bond lengths must be known if they are different from 1.
The algebraic eigenvalues are immensely complex so it is necessary to calculate them numerically except for the case of $D_{\infty h}$ point group. In this case the force constants are zero, or $2k_3(2/m_2 +1/m_1)$ where $m_2$ is the central atom.
In the case of $\mathrm{CO}_2$, $m_1 = 16$, $m_2 = 12$ and the bond length $r=0.116$ nm. The only non zero (and doubly degenerate) frequency is $\omega^2 =2k_3(2/m_2 +1/m_1)$ . In $\mathrm{CO}_2$ the bending vibration has a frequency of $667$ cm$^{-1}$ and then $k_3 = 57$ N/m.
(source. The arguments presented here follow closely that in Beddard 'Applying Maths in the Chemical & Biomolecular Sciences, an example based approach' publ OUP, and where there is a more detailed description. Herzberg gives several examples (Vol II, chapter 2) and Wilson, Decius & Cross 'Molecular Vibrations describe the method in great detail.)
porphyrinporphyrin
I actually didn't know how to do this either, but I've found how to work with these infinite symmetry point groups. You have noticed that the reduction formula would not work properly for these groups. So, what we can do to solve for $D_\mathrm {\infty h}$ is solve for the irreducible representation in a lower order symmetry group, say $D_\mathrm{2h}$, and then correlate the results to the higher order symmetry group using the tables below.
$$\begin{array}{c|cccccc} C_\mathrm{\infty v} & \mathrm{A_1 = \Sigma^+} & \mathrm{A_2 = \Sigma^-} & \mathrm{E_1 = \Pi} & \mathrm{E_2 = \Delta} & \cdots \\ \hline C_\mathrm{2v} & \mathrm{A_1} & \mathrm{A_2} & \mathrm{B_1 + B_2} & \mathrm{A_1 + A_2} & \cdots \end{array}$$
$$\begin{array}{c|cccccc} D_\mathrm{\infty h} & \Sigma_\mathrm{g}^+ & \Sigma_\mathrm{g}^- & \Pi_\mathrm{g} & \Delta_\mathrm{g} & \cdots & \Sigma_\mathrm{u}^+ & \Sigma_\mathrm{u}^- & \Pi_\mathrm{u} & \Delta_\mathrm{u} & \cdots \\ \hline D_\mathrm{2h} & \mathrm{A_g} & \mathrm{B_{1g}} & \mathrm{B_{2g} + B_{3g}} & \mathrm{A_g + B_{1g}} & \cdots & \mathrm{B_{1u}} & \mathrm{A_u} & \mathrm{B_{2u} + B_{3u}} & \mathrm{A_u + B_{1u}} & \cdots \end{array}$$
I got these tables, and a lot of information I didn't know about point groups, from this page from the University of Massachusetts Boston.
I also found a nice walk-through on Wikipedia, which explains how to do this for $\ce{CO2}$, which is identical in terms of symmetry grouping.
TyberiusTyberius
Again only part of an answer. As a reference I used the book Group Theory And Chemistry by David M. Bishop. There it is stated (Chapter 9-8) that linear molecules are special cases, and they give generalized results for $N$ atomic molecules:
asymmetric molecules ($C_{\infty v}$) have:
$N-1$ longitudinal vibrations belonging to $\Sigma^+$
$N-2$ pairs of transverse vibrations belongig to $\Pi$
symmetric molecules ($D_{\infty h}$) have for even $N$:
$\frac{N}{2}$ longitudinal vibrations belonging to $\Sigma^+_g$
$\frac{N}{2}-1$ longitudinal vibrations belonging to $\Sigma^+_u$
$\frac{N}{2}-1$ pairs of transverse vibrations belonging to $\Pi_g$
$\frac{N}{2}-1$ pairs of transverse vibrations belonging to $\Pi_u$
symmetric molecules ($D_{\infty h}$) have for odd $N$:
$\frac{N-1}{2}$ longitudinal vibrations belonging to $\Sigma^+_g$
$\frac{N-1}{2}$ longitudinal vibrations belonging to $\Sigma^+_u$
$\frac{N-3}{2}$ pairs of transverse vibrations belonging to $\Pi_g$
$\frac{N-1}{2}$ pairs of transverse vibrations belonging to $\Pi_u$
I tried the $\ce{CO2}$ example, but didn't get very far: The representation we need to reduce is (Chapter 9-6 in the above book explains how to do this):
\begin{array}{c|cccccccc|cc} D_{\infty\mathrm{h}} & E & 2C_\infty^\phi & \cdots & \infty\sigma_\mathrm{v} & i & 2S_\infty^\phi & \cdots & \infty C_2 & \\ \hline \Gamma^{red} & 9 & 3+6\cos\phi & \cdots & 3 & -3 & -1+2\cos\phi & \cdots & -1 \\ \end{array}
As already stated in Tyberius' answer, the problem is to apply the reduction formula. Here we need to sum over infinitely many $C_\infty^\phi$ axes. We can do this by integration, for example in the reduction formula for the IRREP $A_{1g}$ we get the following term:
\begin{equation} 2\times 1\times\int\limits_0^{2\pi}(3+6\cos\phi)\mathrm{d}\phi = 6\pi \end{equation}
In a similar way for the improper rotational axes $S_\infty^\phi$ we get $-4\pi$. What I don't know yet is how to treat the infinit many elements of the $\sigma_v$ and $C_2$ classes.
edited Jul 9 '17 at 7:04
levineds
FeodoranFeodoran
$\begingroup$ The reduction formula cannot be applied to infinite dimensional groups. $\endgroup$ – levineds Jul 8 '17 at 7:02
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Definable real number
Informally, a definable real number is a real number that can be uniquely specified by its description. The description may be expressed as a construction or as a formula of a formal language. For example, the positive square root of 2, ${\sqrt {2}}$, can be defined as the unique positive solution to the equation $x^{2}=2$, and it can be constructed with a compass and straightedge.
Different choices of a formal language or its interpretation give rise to different notions of definability. Specific varieties of definable numbers include the constructible numbers of geometry, the algebraic numbers, and the computable numbers. Because formal languages can have only countably many formulas, every notion of definable numbers has at most countably many definable real numbers. However, by Cantor's diagonal argument, there are uncountably many real numbers, so almost every real number is undefinable.
Constructible numbers
Main article: Constructible number
One way of specifying a real number uses geometric techniques. A real number $r$ is a constructible number if there is a method to construct a line segment of length $r$ using a compass and straightedge, beginning with a fixed line segment of length 1.
Each positive integer, and each positive rational number, is constructible. The positive square root of 2 is constructible. However, the cube root of 2 is not constructible; this is related to the impossibility of doubling the cube.
Real algebraic numbers
A real number $r$ is called a real algebraic number if there is a polynomial $p(x)$, with only integer coefficients, so that $r$ is a root of $p$, that is, $p(r)=0$. Each real algebraic number can be defined individually using the order relation on the reals. For example, if a polynomial $q(x)$ has 5 real roots, the third one can be defined as the unique $r$ such that $q(r)=0$ and such that there are two distinct numbers less than $r$ at which $q$ is zero.
All rational numbers are algebraic, and all constructible numbers are algebraic. There are numbers such as the cube root of 2 which are algebraic but not constructible.
The real algebraic numbers form a subfield of the real numbers. This means that 0 and 1 are algebraic numbers and, moreover, if $a$ and $b$ are algebraic numbers, then so are $a+b$, $a-b$, $ab$ and, if $b$ is nonzero, $a/b$.
The real algebraic numbers also have the property, which goes beyond being a subfield of the reals, that for each positive integer $n$ and each real algebraic number $a$, all of the $n$th roots of $a$ that are real numbers are also algebraic.
There are only countably many algebraic numbers, but there are uncountably many real numbers, so in the sense of cardinality most real numbers are not algebraic. This nonconstructive proof that not all real numbers are algebraic was first published by Georg Cantor in his 1874 paper "On a Property of the Collection of All Real Algebraic Numbers".
Non-algebraic numbers are called transcendental numbers. The best known transcendental numbers are π and e.
Computable real numbers
A real number is a computable number if there is an algorithm that, given a natural number $n$, produces a decimal expansion for the number accurate to $n$ decimal places. This notion was introduced by Alan Turing in 1936.[1]
The computable numbers include the algebraic numbers along with many transcendental numbers including $\pi $ and $e$. Like the algebraic numbers, the computable numbers also form a subfield of the real numbers, and the positive computable numbers are closed under taking $n$th roots for each positive $n$.
Not all real numbers are computable. Specific examples of noncomputable real numbers include the limits of Specker sequences, and algorithmically random real numbers such as Chaitin's Ω numbers.
Definability in arithmetic
Another notion of definability comes from the formal theories of arithmetic, such as Peano arithmetic. The language of arithmetic has symbols for 0, 1, the successor operation, addition, and multiplication, intended to be interpreted in the usual way over the natural numbers. Because no variables of this language range over the real numbers, a different sort of definability is needed to refer to real numbers. A real number $a$ is definable in the language of arithmetic (or arithmetical) if its Dedekind cut can be defined as a predicate in that language; that is, if there is a first-order formula $\varphi $ in the language of arithmetic, with three free variables, such that
$\forall m\,\forall n\,\forall p\left(\varphi (n,m,p)\iff {\frac {(-1)^{p}\cdot n}{m+1}}<a\right).$
Here m, n, and p range over nonnegative integers.
The second-order language of arithmetic is the same as the first-order language, except that variables and quantifiers are allowed to range over sets of naturals. A real that is second-order definable in the language of arithmetic is called analytical.
Every computable real number is arithmetical, and the arithmetical numbers form a subfield of the reals, as do the analytical numbers. Every arithmetical number is analytical, but not every analytical number is arithmetical. Because there are only countably many analytical numbers, most real numbers are not analytical, and thus also not arithmetical.
Every computable number is arithmetical, but not every arithmetical number is computable. For example, the limit of a Specker sequence is an arithmetical number that is not computable.
The definitions of arithmetical and analytical reals can be stratified into the arithmetical hierarchy and analytical hierarchy. In general, a real is computable if and only if its Dedekind cut is at level $\Delta _{1}^{0}$ of the arithmetical hierarchy, one of the lowest levels. Similarly, the reals with arithmetical Dedekind cuts form the lowest level of the analytical hierarchy.
Definability in models of ZFC
A real number $a$ is first-order definable in the language of set theory, without parameters, if there is a formula $\varphi $ in the language of set theory, with one free variable, such that $a$ is the unique real number such that $\varphi (a)$ holds.[2] This notion cannot be expressed as a formula in the language of set theory.
All analytical numbers, and in particular all computable numbers, are definable in the language of set theory. Thus the real numbers definable in the language of set theory include all familiar real numbers such as 0, 1, $\pi $, $e$, et cetera, along with all algebraic numbers. Assuming that they form a set in the model, the real numbers definable in the language of set theory over a particular model of ZFC form a field.
Each set model $M$ of ZFC set theory that contains uncountably many real numbers must contain real numbers that are not definable within $M$ (without parameters). This follows from the fact that there are only countably many formulas, and so only countably many elements of $M$ can be definable over $M$. Thus, if $M$ has uncountably many real numbers, one can prove from "outside" $M$ that not every real number of $M$ is definable over $M$.
This argument becomes more problematic if it is applied to class models of ZFC, such as the von Neumann universe. The assertion "the real number $x$ is definable over the class model $N$" cannot be expressed as a formula of ZFC.[3][4] Similarly, the question of whether the von Neumann universe contains real numbers that it cannot define cannot be expressed as a sentence in the language of ZFC. Moreover, there are countable models of ZFC in which all real numbers, all sets of real numbers, functions on the reals, etc. are definable.[3][4]
See also
• Berry's paradox
• Constructible universe
• Entscheidungsproblem
• Ordinal definable set
• Tarski's undefinability theorem
References
1. Turing, A. M. (1937), "On Computable Numbers, with an Application to the Entscheidungsproblem", Proceedings of the London Mathematical Society, 2, 42 (1): 230–65, doi:10.1112/plms/s2-42.1.230, S2CID 73712
2. Kunen, Kenneth (1980), Set Theory: An Introduction to Independence Proofs, Amsterdam: North-Holland, p. 153, ISBN 978-0-444-85401-8
3. Hamkins, Joel David; Linetsky, David; Reitz, Jonas (2013), "Pointwise Definable Models of Set Theory", Journal of Symbolic Logic, 78 (1): 139–156, arXiv:1105.4597, doi:10.2178/jsl.7801090, S2CID 43689192
4. Tsirelson, Boris (2020), "Can each number be specified by a finite text?", WikiJournal of Science, vol. 3, no. 1, p. 8, arXiv:1909.11149, doi:10.15347/WJS/2020.008, S2CID 202749952
Number systems
Sets of definable numbers
• Natural numbers ($\mathbb {N} $)
• Integers ($\mathbb {Z} $)
• Rational numbers ($\mathbb {Q} $)
• Constructible numbers
• Algebraic numbers ($\mathbb {A} $)
• Closed-form numbers
• Periods
• Computable numbers
• Arithmetical numbers
• Set-theoretically definable numbers
• Gaussian integers
Composition algebras
• Division algebras: Real numbers ($\mathbb {R} $)
• Complex numbers ($\mathbb {C} $)
• Quaternions ($\mathbb {H} $)
• Octonions ($\mathbb {O} $)
Split
types
• Over $\mathbb {R} $:
• Split-complex numbers
• Split-quaternions
• Split-octonions
Over $\mathbb {C} $:
• Bicomplex numbers
• Biquaternions
• Bioctonions
Other hypercomplex
• Dual numbers
• Dual quaternions
• Dual-complex numbers
• Hyperbolic quaternions
• Sedenions ($\mathbb {S} $)
• Split-biquaternions
• Multicomplex numbers
• Geometric algebra/Clifford algebra
• Algebra of physical space
• Spacetime algebra
Other types
• Cardinal numbers
• Extended natural numbers
• Irrational numbers
• Fuzzy numbers
• Hyperreal numbers
• Levi-Civita field
• Surreal numbers
• Transcendental numbers
• Ordinal numbers
• p-adic numbers (p-adic solenoids)
• Supernatural numbers
• Profinite integers
• Superreal numbers
• Normal numbers
• Classification
• List
| Wikipedia |
Qian Yang, Hailong Liu, Pengfei Lin, Yiwen Li. Kuroshio intrusion in the Luzon Strait in an eddy-resolving ocean model and air-sea coupled model[J]. Acta Oceanologica Sinica, 2020, 39(11): 52-68. doi: 10.1007/s13131-020-1670-5
Citation: Qian Yang, Hailong Liu, Pengfei Lin, Yiwen Li. Kuroshio intrusion in the Luzon Strait in an eddy-resolving ocean model and air-sea coupled model[J]. Acta Oceanologica Sinica, 2020, 39(11): 52-68. doi: 10.1007/s13131-020-1670-5
Kuroshio intrusion in the Luzon Strait in an eddy-resolving ocean model and air-sea coupled model
Qian Yang1, 2 ,
Hailong Liu1, 2 , , ,
Pengfei Lin1, 2 ,
Yiwen Li1, 2
State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China
College of Earth Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
Funds: The National Key R&D Program for Developing Basic Sciences under contract Nos 2018YFA0605703, 2016YFC1401401 and 2016YFC1401601; the Strategic Priority Research Program of Chinese Academy of Sciences under contract No. XDB42010404; the National Natural Science Foundation of China under contract Nos 41976026, 41776030, 41931183, 41931182 and 41576026.
Corresponding author: E-mail: [email protected]
The Kuroshio intrusion in a quasi-global eddy-resolving model (LICOMH) and a fully air-sea coupled model (LICOMHC) was evaluated against observations. We found that the Kuroshio intrusion was exaggerated in the former, while biases were significantly attenuated in the latter. Luzon Strait transport (LST) in winter was reduced from –8.8×106 m3/s in LICOMH to –6.0×106 m3/s in LICOMHC. Further analysis showed that different LST values could be explained by different large-scale and local surface wind stresses and the eddies east to the Luzon Strait as well. The relatively stronger cyclonic eddies in LICOMH northeast of the Luzon Island led to weak Kuroshio transport and strong intrusion through the Luzon Strait. The summed transport of all three factors was approximately 2.0×106 m3/s, which was comparable with the difference in LST between the two experiments. The EKE budget showed that strong EKE transport and the baroclinic transformation term led to strong cyclonic eddies east of the Kuroshio in LICOMH, while surface winds contributed little to the differences in the eddies.
Kuroshio intrusion,
South China Sea,
eddy-resolving model,
air-sea coupled model
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Figures(15) / Tables(4)
Qian Yang1, 2,
Hailong Liu1, 2, , ,
Pengfei Lin1, 2,
1. State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China
2. College of Earth Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
Kuroshio intrusion /
South China Sea /
eddy-resolving model /
Abstract: The Kuroshio intrusion in a quasi-global eddy-resolving model (LICOMH) and a fully air-sea coupled model (LICOMHC) was evaluated against observations. We found that the Kuroshio intrusion was exaggerated in the former, while biases were significantly attenuated in the latter. Luzon Strait transport (LST) in winter was reduced from –8.8×106 m3/s in LICOMH to –6.0×106 m3/s in LICOMHC. Further analysis showed that different LST values could be explained by different large-scale and local surface wind stresses and the eddies east to the Luzon Strait as well. The relatively stronger cyclonic eddies in LICOMH northeast of the Luzon Island led to weak Kuroshio transport and strong intrusion through the Luzon Strait. The summed transport of all three factors was approximately 2.0×106 m3/s, which was comparable with the difference in LST between the two experiments. The EKE budget showed that strong EKE transport and the baroclinic transformation term led to strong cyclonic eddies east of the Kuroshio in LICOMH, while surface winds contributed little to the differences in the eddies.
The Luzon Strait (LS), located between the Luzon Island and the Taiwan Island, is a primary gap of the South China Sea (SCS) that forms a connection with the western North Pacific. The Kuroshio, the northward western boundary current of the subtropical gyre of the North Pacific, commonly intrudes westward into the SCS through the LS via various pathways. Sometimes, the intrusion may induce a loop current in the LS, and Kuroshio water flows out of the SCS through the northern part of the LS. The Kuroshio intrusion not only affects stratification (Metzger and Hurlburt, 1996; Qu et al., 2000; Xu and Su, 2000), circulation (Metzger and Hurlburt, 1996; Qu et al., 2000; Xu and Su, 2000; Tian et al., 2006), and mesoscale eddies (Sun et al., 2016) in the northern SCS but also affects the mass, heat and salt budgets of the whole SCS basin (Metzger and Hurlburt, 1996; Qu et al., 2000; Xu and Su, 2000).
Recently, some major features of the Kuroshio intrusion have been gradually identified due to an increasing number of in situ measurements, high resolution satellite data and numerical products, including intrusion types (Hu et al., 2000; Caruso et al., 2006; Nan et al., 2011a; Nan et al., 2015; Huang et al., 2016), water exchange in the LS (Lan et al., 2004; Tian et al., 2006; Shu et al., 2014), and interactions between the Kuroshio and mesoscale processes (Yuan et al., 2006; Sheu et al., 2010; Zhao and Luo, 2010; Nan et al., 2011b; Lu and Liu, 2013; Lien et al., 2014; Chang et al., 2015; Nan et al., 2015; Kuo et al., 2017). The Kuroshio intrusion into the SCS also has multiscale variability, ranging from seasonal (Qu et al., 2000, 2004; Xu and Su, 2000; Lan et al., 2004; Yang et al., 2013; Huang et al., 2017) to interannual (Kim et al., 2004; Qu et al., 2004; Wang et al., 2006a; Wu, 2013) to decadal (Nan et al., 2013) timescales. All relevant works prior to 2014 were well documented in the review paper by Nan et al. (2015).
The Kuroshio intrusion is usually estimated by Luzon Strait transport (LST) in the upper 400 m or 1 000 m (Qu et al., 2004; Nan et al., 2013) water layer from 18.5°N to 22.0°N along 120.75°E due to the relatively large number of observations. However, LST cannot describe which path the Kuroshio intrusion takes in the northern SCS. Hu et al. (2000) and Caruso et al. (2006) concluded that there were four and five types of paths, respectively. Nan et al. (2011a) and Nan et al. (2015) recently proposed an area-average geostrophic vorticity method (based on satellite data in the southwestern region of Taiwan) to classify the Kuroshio intrusion. Three different types were identified: the leaping path, the looping path and the leaking path. Huang et al. (2016, 2017) further refined the method into two subindices, called the double index (DI). These methods have been effective for identifying Kuroshio intrusion paths in both observational and modeling studies (Nan et al., 2013; Huang et al., 2017).
In addition to observational studies, many theoretical and numerical models have also been used to study the Kuroshio intrusion (Metzger and Hurlburt, 1996, 2001; Sheremet, 2001; Xue et al., 2004; Yuan and Wang, 2011; Wu and Hsin, 2012; Jan et al., 2017; Kuo et al., 2017). However, great diversity exists in these modeling results due to the complexity of the processes involved as well as biases in the physics used in the models and/or in the forcing datasets. The annual mean LST in the models summarized by Nan et al. (2015) ranged from –0.6×106 m3/s (Metzger, 2003) to –10.2×106 m3/s (Song, 2006). Metzger and Hurlbert (2001) and Huang et al. (2017) also pointed out that the modeled LST and intrusion path were very sensitive to the resolved topography in the LS. Finer grids and more resolved islands led to smaller LST in the models and less looping or leaking intrusion paths.
With the tremendous increases in computing resources in recent years, more studies have begun to use global- or basin-scale eddy-resolving models either with (Nan et al., 2011b) or without data assimilations (Huang et al., 2017) instead of regional or coarse resolution models due to the large impacts of mesoscale eddy activities on the Kuroshio intrusion (Yuan et al., 2006; Sheu et al., 2010; Zhao and Luo, 2010; Nan et al., 2011b; Lien et al., 2014; Chang et al., 2015; Nan et al., 2015; Kuo et al., 2017). It is likely that circulation becomes more complicated when the models can resolve mesoscale eddies. Therefore, it is necessary to systematically evaluate the Kuroshio intrusion in eddy-resolving models.
Moreover, eddy-resolving models generally tend to simulate stronger eddy activities globally (Masumoto et al., 2004; Yu et al., 2012) and around the LS (Lin et al., 2015; Sun et al., 2016), which is always attributed to insufficient subgrid scale diffusion. The recent works of Renault et al. (2016), Ma et al. (2016) and Feng et al. (2017) all suggested that eddy kinetic energy is significantly reduced in air-sea coupled models due to current feedback or eddy potential energy (EPE) reduction (or both). However, we note that the processes considered in these studies greatly differ. It is also necessary to investigate the simulation of the Kuroshio intrusion in a coupled model.
In the present study, the Kuroshio intrusion in a quasi-global eddy-resolving model and an air-sea coupled model was systemically compared and evaluated against observations. The LST biases in the coupled model were significantly reduced. Differences in the Kuroshio intrusion between the two models could be explained primarily by differences in large-scale surface winds over the Pacific, local surface wind around the LS, and eddy fields. The rest of the paper is organized as follows. Section 2 introduces the datasets. Section 3 presents the Kuroshio intrusion in the two models. Section 4 and Section 5 focus on possible mechanisms to explain the differences between the two models, including remote and local surface wind stress, the pressure gradient between the Pacific and SCS, and mesoscale eddies. In Section 6, we further investigate the mechanisms leading to different eddy fields in terms of budgets. Finally, the discussion and conclusions are provided in Section 7.
2. The numerical models, experiments and observational datasets
2.1. Models and experiments
The eddy-resolving oceanic general circulation model (OGCM) that was used in the present study was the State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics/Institute of Atmospheric Physics (LASG/IAP) Climate System Ocean Model version 2.0 (LICOM2.0) (Liu et al., 2012). To avoid the singularity of the North Pole in the longitude-latitude grid, the model domain extended from 79°S to 66°N. The horizontal grid spacing was uniformly 0.1° with 55 vertical levels. The upper 300 m consisted of 36 uneven levels that increased with depth.
A 60-year hindcast was conducted after a 12-year spinup using climatological forcing from the Ocean Model Intercomparison Project (OMIP; Röske, 2001). For this 60-year period, the model was forced by using the daily Coordinated Ocean-Ice Reference Experiments (COREs) algorithm and data from 1948 to 2007 (Large and Yeager, 2004). Owing to the lack of a sea ice module in LICOM2.0, the sea ice concentration was derived from the observational dataset from the Hadley Center (HadISST, https://climatedataguide.ucar.edu/climate-data/sea-ice-concentration-data-hadisst).
The air-sea coupled model used here mainly consisted of the abovementioned eddy-resolving version of LICOM2.0 and the atmospheric model CAM4 (Community Atmosphere Model version 4), with a 25 km horizontal resolution and 36 vertical levels. The land and sea ice models used in the coupled model were CLM4 (Community Land Model version 4) and CICE4 (Community Ice Code version 4), respectively. The four components were coupled with NCAR Flux Coupler version 7 (Craig et al., 2012). The coupling frequency between the atmosphere and the ocean was 6 h. Because the ocean model excludes the Arctic Ocean, the salinity and potential temperature at the northern boundary are nudged toward the observation of the World Ocean Atlas 2005 (WOA, https://www.nodc.noaa.gov/). The low resolution of this coupled model was described and evaluated in Lin et al. (2016). Here, we called the ocean-only and coupled experiments LICOMH and LICOMHC, respectively. The fully coupled experiment was conducted for 7 a from a quasi-steady state of LICOMH and a CAM spinup. The results from the last 5 years of both experiments were chosen for our analysis.
2.2. Observational datasets
Merged satellite altimeter products from the archiving, validation and interpretation of satellite oceanographic (AVISO, http://www.aviso.oceanobs.com) dataset were used in this study for the period of 2003–2007 to match the period of the ocean model simulation. The spatial resolution of the data was 0.25°×0.25°. We averaged the AVISO daily product every 5 d to obtain the pentad averaged data. The absolute dynamic topography (ADT) was calculated by adding the sea level anomaly (SLA) to the mean dynamic topography. The surface currents were obtained with the geostrophic relation. Water depths less than 200 m were excluded from the diagnosis due to errors caused by tidal signals (Yuan et al., 2006; Nan et al., 2011a).
3. The simulations of the Kuroshio intrusion in LICOMH and LICOMHC
Figures 1a–c compare the mean SSH field and related surface geostrophic currents near the LS among the AVISO observation, LICOMH and LICOMHC. The red lines in Fig. 1 are the Kuroshio axis, defined as the contours of zero geostrophic vorticity. In the observation, the Kuroshio axis extended westward to approximately 120.5°E. The annual mean Kuroshio intrusion in the LICOMH reached approximately 119°E, approximately 1.5° west of the observation. The westward velocity vectors around the LS in the LICOMH were also overestimated, consistent with the exaggerated intrusion. However, the Kuroshio intrusion was greatly reduced in the coupled experiment, LICOMHC, in which the Kuroshio axis extended only to approximately 120.5°E.
Figure 1. Annual mean sea level height (shading) and related surface geostrophic currents (vector) near the Luzon Strait for AVISO (a), LICOMH (b) and LICOMHC (c), and the same as a–c but for winter (d–f). The red curve denotes the Kuroshio axis determined by the zero contour of geostrophic vorticity (GV).
The Kuroshio intrusion has evident seasonal variability, being strong in winter and weak in summer (Nan et al., 2015) (Table 1). We further compared the winter intrusion, defined as the average from December to February, in Figs 1d–f. Clearly, the surface current and LST (Table 1) are stronger in winter than the annual mean, while the Kuroshio intrusion was more exaggerated in LICOMH than in LICOMC. The Kuroshio axis in LICOMH extended to approximately 117.5°E, compared to 119°E in LICOMHC. The surface currents were also significantly stronger in the two modeled experiments in winter than the annual mean.
LST/106 m3·s–1 LST from Island rule/106 m3·s–1
Annual Winter Annual Winter
LICOMH –5.4 –8.8 –7.1 —
LICOMHC –4.4 –6.0 –6.0 —
Long-term in situ Obs. –12.41) to –0.52) –13.73) to –2.754) –4.05) to –12.46) —
Numerical models –10.27) to –0.68) –12.29) to –3.110) — —
Note: 1) Wang et al. (2006b); 2) Wyrtki (1961); 3) Chu and Li (2000); 4) Wyrtki (1961); 5) Qu et al. (2004); 6) Wang et al. (2006b); 7) Song (2006); 8) Metzger (2003); 9) Song (2006); 10) Rong et al. (2007). — means not available.
Table 1. The annual and winter mean Luzon Strait transport (LST), defined as the upper 400 m-integrated zonal velocity along 120.75°E from 18.5°N to 22.0°N and the Island rule that was proposed by Godfrey (1989, details in Part 4); and the LST ranges for both the observations and models from Nan et al. (2015)
The upper 400 m-integrated currents (both vectors and magnitudes) from these two experiments are also presented (Fig. 2). The annual and winter mean velocity vectors (Figs 2a–b and 2d–e, respectively) resembled the surface geostrophic current shown in Figs 1b–c and 1e–f, confirming the more exaggerated Kuroshio intrusion in LICOMH than in LICOMHC for the upper 400 m. Circulations east of the LS also differed between these two experiments. Generally, the eastward Subtropical Counter Current (STCC) could not be clearly found in LICOMH in the annual and winter mean fields (Figs 2a and d), while this current was clearly located at approximately 20.5°N in LICOMHC, which was not the usual location that was noted in the observation. Furthermore, the STCC was only found in LICOMH in the winter mean field east to 130°E, at a location that was approximately 1° north of that in LICOMHC (not shown). The NEC seems to be a bit more northward in LICOMH, so that the Kuroshio intrusion should be weaker in LICOMHC.
Figure 2. Annual mean upper 400 m-integrated currents for LICOMH (a) and LICOMHC (b); their differences (LICOMH minus LICOMHC) (c), and the same as a–c but for winter (d–f). The color represents the upper 400 m-integrated zonal velocity.
The annual and winter circulation differences between LICOMH and LICOMHC (LICOMH minus LICOMHC) are also shown in Figs 2c and f. The annual and winter spatial mean patterns were similar because winter circulation is dominant in this region. The clockwise circulation difference in southwestern Taiwan and weaker upstream Kuroshio both confirmed the stronger intrusion in LICOMH than in LICOMHC. Moreover, the counterclockwise circulation difference at about 18°–21°N east of the LS was related to the missing STCC and weak North Equatorial Current (NEC) at approximately 18°N in LICOMH.
To quantify the intrusion in these two models, we further showed the seasonal cycle of LST in the upper 400 m for these two experiments (Fig. 3). Positive (negative) values represented eastward (westward) transport. The results did not change when the depth was further integrated to the upper 1 000 m (not shown). Table 1 compares the modeled LST with the observed ranges that were tabulated in Nan et al. (2015). The 5-year mean modeled LST values were –5.4×106 m3/s and –4.4×106 m3/s in LICOMH and LICOMHC, respectively, both of which were within the observed range, –0.5×106 m3/s to –12.4×106 m3/s (Table 1). The most notable total difference occurred in winter, which was approximately 2.8×106 m3/s. The LST in LICOMHC (–6.0×106 m3/s) was smaller than that in LICOMH (–8.8×106 m3/s) for the upper 400 m (Table 1).
Figure 3. Seasonal cycle of Luzon Strait transport (LST) for LICOMH (red curve) and LICOMHC (blue curve). Here, LST is defined as the upper 400 m-integrated zonal transport along 120.75°E from 18.5°N to 22.0°N. Positive values represent eastward transport.
We further investigated the various paths of the Kuroshio intrusion using the DI (the detailed method is presented in Appendix A). The frequencies associated with the three types of Kuroshio paths for LICOMH and LICOMHC and their corresponding LST values are listed in Table 2. We find out that the leaking path dominated in winter. The probability of occurrence of leaking path and looping path in LICOMH is 63.3% and 28.9%, but 54.4% and 33.3% for LICOMHC. This finding indicates that the Kuroshio in winter likely penetrated the SCS (Nan et al., 2015). We further calculated the annual (winter) mean transport when the Kuroshio takes its looping path and leaking path in LICOMH and LICOMHC, respectively. According to Nan et al. (2015), this phenomenon explains why the most notable LST difference between LICOMH and LICOMHC occurs in winter. It is also evident that the looping frequency is increased in LICOMHC, which may be caused by the changes of the local surface wind or circulations within the SCS.
Model Looping frequency/LST Leaping frequency/LST Leaking frequency/LST
Annual mean LICOMH 57(15.6%)/–7.2 48(13.2%)/–4.1 260(71.2%)/–5.3
LICOMHC 73(20%)/–6.0 51(14.0%)/–4.6 241(66.0%)/–3.9
Winter mean LICOMH 26(28.9%)/–8.3 7(7.8%)/–7.7 57(63.3%)/–9.1
LICOMHC 30(33.3%)/–6.3 11(12.2%)/–6.8 49(54.5%)/–5.6
Note: The value in parentheses is the percentage of the frequency for each of the three paths.
Table 2. Occurrence (in front of the slash), which means the total number of occurrence days during the 5 years, and mean LST values (after the slash, 106 m/s) for the three types of Kuroshio intrusion paths based on the double index (DI; Huang et al., 2016) method
Previous studies have suggested that when the Kuroshio intrusion is relatively weaker, the eddy activity southwest of Taiwan becomes weaker (Nan et al., 2013; Sun et al., 2016) and vice versa. This scenario is also consistent with the weaker intrusion in LICOMHC than in LICOMH, as shown in the surface eddy kinetic energy (EKE) fields (Fig. 4), although both simulations showed much stronger EKE throughout the study region regardless of season compared with the observation (Figs 4a and d). Here, the EKE was calculated as $ ({u}^{'2}+{v}^{'2}) /{2}$ , where $ {u}' $ and $ {v}' $ are the geostrophic velocity anomalies deduced from the SLA.
Figure 4. Annual mean eddy kinetic energy (EKE) for AVISO (a), LICOMH (b), LICOMHC (c), and the same as a–c but for winter (d–f).
In summary, we found that the Kuroshio intrusion was exaggerated in the ocean-only model (LICOMH), while the biases were reduced in the fully coupled model (LICOMHC). Since the ocean component in the coupled model was identical to the stand-alone ocean-only model, the changes in the Kuroshio intrusion may have been related to differences in large-scale surface winds (both local and remote) or the pressure gradient between the Pacific Ocean and the SCS between these two experiments. Because the changes of sea surface temperature and currents will affect the atmospheric state in the fully coupled model, including the atmosphere temperature and surface wind. That is, the air-sea interactions tend to dump the large changes of the ocean. In addition, mesoscale eddies in the vicinity of the LS, which are related to intrinsic ocean processes, may also have affected the intrusion, mainly through modifying upstream Kuroshio transport (Lien et al., 2014; Nan et al., 2015). The different EKE fields shown in Fig. 4 suggest the different impacts of mesoscale eddies in the two experiments, especially during winter. We will further investigate these potential contributions next.
4. Impact of large-scale and local winds on the Kuroshio intrusion
According to Qu et al. (2000), the impact of large-scale wind stress on the Kuroshio intrusion can be explained by the "island rule" that was proposed by Godfrey (1989). Thus, LST can be expressed as follows:
where, $ {\tau }^{\left(\mathrm{l}\right)} $ is the wind stress along ABCD (Fig. 5a). The Coriolis parameters of A and D are fA and fD, respectively, A is located at the southern tip of Mindanao at approximately 4.75°N, and D is located at the northernmost Luzon at approximately 18.75°N. B and C represent two points at the same latitudes as A and D on the American coast; ρ0 is the reference sea-water density (1 030 kg/m3).
Figure 5. Annual mean surface wind stress (vector) and wind stress curl (shading) for LICOMH (a) and LICOMHC (b). The ABCD box is the region of integration for the island rule. A is located at the southern tip of Mindanao near 4.75°N, and D is at the northernmost Luzon near 18.75°N; B and C represent two points at the same latitudes as A and D on the American coast.
Figure 5 shows the annual mean surface wind stress (vectors) and wind stress curl (WSC) (shaded) for both LICOMH and LICOMHC. The patterns of wind stress were similar to each other in both experiments but the magnitudes were slightly larger in LICOMH. The averaged LST calculated using Eq. (1) was –7.1×106 m3/s in LICOMH, slightly stronger than –6.0×106 m3/s in LICOMHC (Table 1), and this difference can reasonably explain a large part of the modeled LST difference between these two experiments. Although the island rule cannot be used for studying seasonal changes (Yang et al., 2013), it is also important in explaining the winter differences between the two experiments. Therefore, large-scale wind was used in this study to explain the difference in annual mean LST between LICOMH and LICOMHC.
Furthermore, we calculated the volume transport of the simulated upstream Kuroshio along the western coastat around 19.0°N. Other latitudes have also been tested, and the results are basically the same. We found that the transport for the coupled mode (19.01×106 m3/s at 19.0°N) is larger than that for the ocean model (18.78×106 m3/s at 19.0°N) and this will lead to stronger Kuroshio intrusion in LICOMH. The results are basically similar to Fig. 2. Moreover, the WSC can almost describe the NEC bifurcation latitude, so that is also qualitatively same as the results from the island rule.
Different local winds could also have contributed to the LST difference between LICOMH and LICOMHC (Qu et al., 2000; Nan et al., 2013). Figures 6a and b show the annual mean and winter surface wind stress differences (LICOMH-LICOMHC), respectively. Most of the surface wind differences still existed during winter. Strong southward (or southeastward) differences in the LS may drive westward Ekman transport. Therefore, westward Ekman transport in LICOMH may tend to strengthen the LST. We computed the Ekman transport differences along 120.75°E for these two experiments. The annual mean (winter) transport differences reached 0.2×106 m3/s (0.44×106 m3/s), which accounted for approximately 15%–20% of the total transport difference.
Figure 6. Differences in surface wind stress (vector) and its magnitude (shading) between LICOMH and LICOMHC (LICOMH minus LICOMHC) for the annual mean (a) and winter mean (b); and the difference in wind stress curl for the annual mean (c) and winter mean (d). The underlined area represents values above the 95% significance level.
In addition to Ekman transport and the island rule, local WSC can also affect the pressure head between the Pacific Ocean and SCS or change the local gradients of the pressure on both sides of the Kuroshio, which can also affect the intrusion through changing the velocity of the upstream Kuroshio. Figures 6c and d show the annual and winter mean surface WSC difference between these two experiments (LICOMH-LICOMHC), with the hatched area representing values above the 95% significance level. The positive differences throughout the LS suggest that sea surface height anomalies tend to be lower in LICOMH due to Ekman pumping, which compensates for the gradient of sea surface height between the open ocean and the SCS, thus potentially reducing LST. So that, the local wind stress was not responsible for the large intrusion bias in LICOMH.
These results suggest that the excessive intrusion in LICOMH can be largely explained by the remote effect of surface winds through the island rule, with 1.1×106 m3/s for the annual mean. Ekman transport associated with surface winds also contributed approximately 0.2×106 m3/s (or 0.44×106 m3/s during winter) to the biases, approximately 20% of the LST difference.
5. Effect of eddies on the Kuroshio intrusion
Although the NEC bifurcation is an important candidate affecting the Kuroshio intrusion, the mesoscale eddies can also contribute to part of the changes of KI. Previous studies suggested that cyclonic eddies (CEs) and anticyclonic eddies (AEs) east of Luzon Island may weaken and strengthen the velocity of Kuroshio east of the LS, respectively (Chang et al., 2015). Weaker (stronger) Kuroshio transport will consequently result in a looping (leaping) path (Sheu et al., 2010) and lead to a stronger (weaker) intrusion. Figure 4 shows that the eddy activity east of Luzon was more intense in LICOMH, thus potentially influencing the Kuroshio intrusion.
We use the eddy detection and tracking method of Lin et al. (2007) to categorize eddies in the present study (detailed in Appendix B). Here, we focused on eddies with lifetimes longer than 5 weeks and amplitudes greater than 3 cm. Figure 7 shows the annual and winter mean EKE due to CEs and AEs for LICOMH and LICOMHC in a 1° box. The EKE in the region just west of 121°E and at approximately 124°E was significantly larger in LICOMH than in LICOMHC, particularly during winter. In the upstream region of (18.5°–20.5°N, 123.5°–125.5°E), called northeast of Luzon (NEL), the winter average EKE values due to CEs were 598 cm2/s2 and 220 cm2/s2 in LICOMH and LICOMHC, respectively (Table 3). Intensive CEs in the NEL region reduced upstream Kuroshio transport and increased the intrusion in LICOMH. However, the EKE due to AEs in the NEL region still differed between these two experiments, but the magnitude was less than that of CEs, especially in winter, as shown in Table 3.
EKE/ cm2·s–2 Occurrence number Amplitude/cm LST change/106 m3·s–1
LICOMH AE 352/221 1 095/255 18/14 0.2/0.7
CE 471/598 1 210/305 23/23 –0.1/–1.2
LICOMHC AE 259/206 1 135/190 17/19 –0.4/0.4
Table 3. The area-averaged EKE, total occurrence number, mean amplitude and LST change due to both AEs and CEs throughout the key region (18.5°–20.5°N, 123.5°–125.5°E) northeast of the Luzon Strait during the 5 years (in front of the slash) and winters (after the slash) for the two models
Figure 7. Composite mean EKE due to cyclonic (shading) and anticyclonic eddies (contours) in LICOMH (a) and LICOMHC (b), and the same as a and b but for winter (c and d).
The larger EKE in LICOMH may likely be due to the much stronger intensity and frequency of eddies. Figures 8 and 9 show the annual and winter mean occurrence number and intensity of CEs and AEs for the two experiments. The occurrence number of CEs had some differences in general, and the intensity in LICOMH in the NEL region was much higher than that in LICOMHC. The AE results were also consistent, but the difference was small between the two models. During the 5-year integration, we found 305 (1 210) CEs in the NEL region in LICOMH and 220 (1 430) in LICOMHC in winter (5 a) (Table 3). According to the trajectories of these eddies, most were propagated from the STCC region east of the LS (not shown). The location of the large difference due to CEs was colocated with cyclonic circulation anomalies of the mean flow (Figs 2c and f), indicating the westward propagation of CEs in LICOMH is much stronger.
Figure 8. Occurrence number of cyclonic eddies for LICOMH (a) and LICOMHC (b) during winter (shading) and for the 5 years (contours); and the amplitude of cyclonic eddies (cm) for LICOMH (c) and LICOMHC (d) during winter (shading) and for the 5 years (contours). The interval of the contours in a and b is 40, the interval of the contours is 5 cm, and the 15-cm contours have been highlighted in c and d.
Figure 9. Occurrence number of anticyclonic eddies for LICOMH (a) and LICOMHC (b) during winter (shading) and for the 5 years (contours); and the amplitude of cyclonic eddies (cm) for LICOMH (c) and LICOMHC (d) during winter (shading) and for the 5 years (contours). The interval of the contours in a and b is 40, the interval of the contours is 5 cm and the 15-cm contours have been highlighted in c and d.
The difference was much more evident in intensity: approximately 23 cm (23 cm) and 15 cm (19 cm) in the NEL region in winter (composite mean during 5 a) for LICOMH and LICOMHC, respectively (Figs 8c and d, Table 3). Moreover, the 15 cm contours in the NEL region were more westward in LICOMH (approximately 123.5°E for 5 a and 121.5°E in winter) than in LICOMHC (approximately 125°E for 5 a and 123°E in winter) (Figs 8c and d). This result suggests that stronger CEs in LICOMH contributed to the stronger modeled Kuroshio intrusion.
To further analyze the impacts of eddies on the Kuroshio intrusion, we performed a composite SLA and corresponding surface geostrophic current analysis according to the polarity of eddies occurring in the NEL region (Table 3). Figures 10 and 11 show the composite SLA and associated surface geostrophic currents when AEs and CEs occurred in the NEL region, during winter and the 5 years, respectively, and the seasonal circulation was subtracted. We confirmed that eddies affected the Kuroshio intrusion in the models, and the intrusions were significantly stronger when CEs occurred than when AEs occurred, especially in winter. From Fig. 10, the anomaly circulation of Kuroshio was the same as the weak looping path when AEs occurred in the NEL region, while strong transport occurred all the way to the entrance of the SCS (similar to the leaking path) when CEs occurred in LICOMH. However, in LICOMHC, a weak looping pattern was found when CEs occurred, and the intrusion seemed to be weaker than the mean state in Figs 1f and 2f. The pattern looked similar to the leaping path during AE occurrence, and transport anomalies entered the SCS at the northern part of the LS at approximately 22°N. Strong CE circulation in the SCS during AEs occurred in the NEL region in both LICOMH and LICOMHC, hindering the Kuroshio from intruding through the LS. Thus, CEs east of the LS strengthened the Kuroshio intrusion, while AEs weakened it (Zhao and Luo, 2010; Lien et al., 2014; Chang et al., 2015; Nan et al., 2015). The results for the composite mean (Fig. 11) were basically the same, but the enhancement (reduction) of intrusion brought by CEs (AEs) was obviously weaker. It is reasonable to compare the effects related to mesoscale eddies through LST. In this way, we calculated the change in LST when mesoscale eddies occurred in the NEL region (Table 3). The difference in LST caused by eddies in LICOMH was 0.2×106 m3/s stronger than that in LICOMHC in the composite mean state during 5 a, and the difference was 1.2×106 m3/s stronger during winter.
Figure 10. Composite SLA (shading) and geostrophic current anomalies (vector) (subtracting the climatology) when CEs (a) and AEs (b) occurred in the key region (18.5°–20.5°N, 123.5°–125.5°E) northeast of the Luzon Strait for LICOMH during winter, and the same as a and b but for LICOMHC (c and d).
Figure 11. Composite mean SLA (shading) and geostrophic current anomalies (vector) (subtracting the climatology) when CEs (a) and AEs (b) occurred in the key region (18.5°–20.5°N, 123.5°–125.5°E) northeast of the Luzon Strait throughout the 5-year analysis period for LICOMH; and the same as a and b but for LICOMHC (c and d).
In addition, we investigated the evolution of LST when CEs occurred in the NEL region during winter. The LICOMH and LICOMHC scenarios are shown in Figs 12 and 13, respectively. When comparing with the winter mean state, the spatial scale for the westward transport caused by CEs in LICOMH was apparently stronger than that in LICOMHC. The LST anomalies due to CEs were –1.1×106 m3/s for LICOMH and –0.5×106 m3/s for LICOMHC for the study cases. Thus far, we may certainly conclude that different CEs east of the LS between the two models contributed significantly to the difference in LST between the two models.
Figure 12. Winter mean zonal transport along 120.75°E for LICOMH (a); the zonal transport anomalies along 120.75°E when CEs occurred in the key region (18.5°–20.5°N, 123.5°–125.5°E) from January 31 to February 20, 2004 (b–h). The number at the upper right corner of each panel represents an LST anomaly.
Figure 13. Winter mean zonal transportalong 120.75°E for LICOMHC (a); the zonal transport anomalies along 120.75°E when CEs occurred in the key region (18.5°–20.5°N, 123.5°–125.5°E) from December 7, 2004 to January 6, 2005. The number at the upper right corner of each panel represents an LST anomaly.
The above analysis showed that due to the difference in mesoscale eddies simulated in LICOMH and LICOMHC, the enhancement of the Kuroshio intrusion caused by mesoscale eddies in LICOMH was stronger, which could partly explain the difference in intrusion between the two modes.
6. EKE budget
As mentioned previously, the strengthening effect of CEs on the Kuroshio intrusion was stronger in LICOMH than in LICOMHC. This result implies that less EKE was dissipated, more kinetic energy was transferred from the mean flow or more kinetic energy was transported from other regions. To understand the difference in eddies, the EKE budget for each CE in the NEL region was calculated following Chen et al. (2014), which can be used to explain the transport and transformation of EKE change. The details of the method are shown in Appendix C. Here, we primarily investigated the rate of EKE due to the redistribution caused by horizontal advection (AKE), the energy transformation between eddy potential energy (EAPE) and EKE (DKE), which is also called baroclinic production if the DKE is positive (von Storch et al., 2012), and eddy momentum fluxes (MKE) through barotropic instability.
The annual and winter mean vertically integrated AKE, DKE and MKE in the upper 400 m for LICOMH and LICOMHC are shown in Fig. 14. The area-averaged values of the three terms throughout the NEL region are listed in Table 4.
AKE/mW·m−2 DKE/mW·m−2 MKE/mW·m−2
Annual Winter Annual Winter Annual Winter
LICOMH 2.87 4.41 0.30 3.60 –0.53 0.89
LICOMHC –1.06 –0.44 1.43 0.74 –0.73 –0.55
Table 4. The upper 400 m-integrated change rate in EKE due to horizontal advection (AKE), baroclinic instability (DKE) and eddy momentum flux (MKE) for the two models averaged throughout the key region (18.5°–20.5°N, 123.5°–125.5°E) northeast of the Luzon Strait in the composite mean and winter throughout the 5-year analysis period
Figure 14. Upper 400 m-integrated change rate in EKE for each CE in the NEL region due to horizontal advection (AKE, a), baroclinic instability (DKE, c) and eddy momentum flux (MKE, e) for LICOMH during winter (shading), and the contours denote the composite mean values in the 5 years; the same as a, c and e, but for LICOMHC (b, d and f) .
We found that large values of these terms occurred either along the boundary or along the Kuroshio. In the LS, the balances were between the rate of EKE advection and the transformation of barotropic instability, but the rates of EKE due to baroclinic instability were relatively small (Fig. 14). That is, the EKE generated by horizontal velocity shear was transported by currents outside this region. The comparison between the two models showed that there was more EKE transport outside the LS in LICOMH than in LICOMHC.
However, in the NEL region, which is outside of the western boundary, the patches of negative and positive values were scattered throughout the region. We computed the area averages of the three terms in the NEL region during winter (Table 4). The EKE balances in the two experiments differed. In LICOMH, EKE was mainly transported into the region through advection and was generated through baroclinic instability, while the barotropic term was relatively weak. However, in LICOMHC, EKE was generated through the transformation from EAPE and transported outside and dissipated through barotropic processes. We also found that the absolute magnitudes of the main terms, AKE and DKE, were larger in LICOMH than in LICOMHC, indicating strong eddies in LICOMH. The composite mean results in the 5 a were similar, except that EKE dissipation was caused by the barotropic term in LICOMH.
In addition, it has been demonstrated that winds can also dissipate eddy energy (Ferrari and Wunsch, 2009) and can significantly influence mesoscale ocean eddies, especially in energetic western boundary current regions (Xu et al., 2016). Gaube et al. (2015) proposed a formula of the eddy attenuation time scale (Te):
where Te mainly depends on the vertical scale of the mesoscale eddy D, and the wind speed Ua; ρ0 (1 020 kg/m3), ρa (1.2 kg/m3), and CD (1.3×10–3) are surface seawater density, surface air density and the drag coefficient, respectively. Because the differences in D in the two models were very small, we compared the eddy attenuation time scales in LICOMH and LICOMHC by comparing the wind speed. By ignoring the sea surface velocity, Te is inversely proportional to the reciprocal of the square root of wind stress. That is, the smaller the surface wind stress, the longer the eddy attenuation time scale and the weaker the dumping effect.
Figure 15 shows the reciprocal of the square root of wind stress for LICOMH and LICOMHC and their differences. The attenuation times for eddies in the annual mean in LICOMH and LICOMHC were approximately the same, but with a relatively large magnitude in LICOMH during winter. We calculated the area average of ${1}\big/{\sqrt{\left|\vec{\tau }\right|}}$ differences throughout the NEL region. These differences could be neglected compared with the effects of wind stress in LICOMH and LICOMHC (accounting for approximately 5%). Therefore, local wind was not responsible for the differences in eddies between the two models.
Figure 15. The reciprocal of the mean square root of wind stress for LICOMH (a) and LICOMHC (b) during winter (shading) and the annual mean (contours), and their difference (LICOMH minus LICOMHC) (c).
In general, the strong horizontal transport of EKE and strong baroclinic effects in LICOMH led to strong eddies in LICOMH, while surface winds contributed little to the differences in the eddies. The advection term was opposite in the coupled model, and thus, the effect of the air-sea interaction on the EKE balance needs further analysis.
7. Discussion and conclusions
In the present study, the Kuroshio intrusion in a quasi-global eddy-resolving model and an air-sea coupled model was compared and evaluated against observations. The causes of the differences in the Kuroshio intrusion between the two models were explained by differences in large-scale and local surface winds over the tropical Pacific and around the LS, respectively, as well as eddy simulations between the two models. Finally, the balance of EKE was determined to understand differences in the simulated eddies. Interestingly, the Kuroshio intrusion was exaggerated in the ocean-only model, while the biases were significantly attenuated in the fully coupled model. Although the annual mean LST was not significantly reduced in LICOMHC, LST in winter was reduced from –8.8×106 m3/s in LICOMH to –6.0×106 m3/s in LICOMHC, approximately 2.8×106 m3/s less. The transport due to the looping path in LICOMHC was also reduced by approximately 20%, from –8.3×106 m3/s in LICOMH to –6.3×106 m3/s in LICOMHC. The reduction in the Kuroshio intrusion was also demonstrated by other variables, such as the EKE west to the LS.
The analysis of surface winds showed that the large intrusion could be largely explained by the remote effect of surface winds through the island rule. In addition, local wind stress partially contributed to the difference in LST through Ekman transport and directly changed the upstream speed of the Kuroshio. In addition, weaker CEs in the NEL region in LICOMHC led to a relatively strong Kuroshio and subsequently weakened intrusion, also indicating that a change in the eddy simulation changed the Kuroshio intrusion. Although the remote effect of wind could not be quantified seasonally by using the island rule, the sum of all three factors was approximately 2×106 m3/s, which was comparable with the difference in LST between the two runs in winter.
Further assessment of the EKE budget showed that strong EKE transport, and a strong baroclinic production led to strong CEs in winter east of the Kuroshio in LICOMH. Additionally, the EKE balances east of the Kuroshio were basically different between the two models. The advection and barotropic terms were EKE sinks in the coupled model, while the barotropic effect was slightly weak in LICOMH compared with the other terms. That is, the EKE east of the Kuroshio in LICOMH was transferred from the mean flow and the energy from EAPE. These results were similar to those of Ma et al. (2016), who found that air-sea coupling led to a weak mean flow state via the change in the EKE budget. However, some detailed mechanisms and processes are needed for further investigation.
We also found that there were large differences in the EKE and eddy intensity due to CEs between the two models in the southeastern region of Taiwan (Figs 7 and 8). LST in the presence of CEs east of Taiwan (22.5°–23.5°N, 122.5°–123.5°E) was also computed in the two models (figures not shown). We found that the CEs east of Taiwan in LICOMH led to a more westward intrusion (almost 1°), but LST was slightly weaker than that in LICOMHC. Therefore, the CEs east of Taiwan may have contributed little to part of the Kuroshio intrusion, but these eddies were not a primary factor. This result was different from the study of Chang et al. (2015), who demonstrated that larger numbers of CEs east of Taiwan strengthened the Kuroshio in the LS because of upstream convergence and downstream divergence. However, this finding is also worth further investigation using observational data.
The results also indicate there are biases in the surface wind of CORE dataset in the tropic Pacific Ocean. Recently, Sun et al. (2019) has pointed out that the corrections of surface wind made by CORE lead to a weak NECC in almost all the CORE experiments. The biases in surface wind would result in biases in the volume transports in the tropic Pacific region, and then might affect the transport of Kuroshio and the Kuroshio intrusion in the Luzon Strait. The processes related with the surface wind biases and the exaggerate Kuroshio intrusion need further investigating.
We appreciate the constructive comments of Yu-heng Tseng, and acknowledge the technical support from the National Key Scientific and Technological Infrastructure Project "Earth System Science Numerical Simulator Facility" (EarthLab).
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Research article | Open | Published: 13 October 2017
Effects of age and gender on reference levels of biomarkers comprising the pediatric Renal Activity Index for Lupus Nephritis (p-RAIL)
Michael R. Bennett ORCID: orcid.org/0000-0002-2022-11051,
Qing Ma1,
Jun Ying2,
Prasad Devarajan1 &
Hermine Brunner3
Pediatric Rheumatologyvolume 15, Article number: 74 (2017) | Download Citation
Systemic Lupus Erythematosus (SLE) is a multisystem autoimmune disease that disproportionately effects women and children of minorities. Renal involvement (lupus nephritis, or LN) occurs in up to 80% of children with SLE and is a major determinant of poor prognosis. We have developed a non-invasive pediatric Renal Activity Index for Lupus (p-RAIL) that consists of laboratory measures that reflect histologic LN activity. These markers are neutrophil gelatinase associated lipocalin (NGAL), kidney injury molecule-1 (KIM-1), monocyte chemotactic protein (MCP-1), adiponectin (APN), ceruloplasmin (CP) and hemopexin (HPX). A major gap in the knowledge base and a barrier to clinical utility is how these markers behave in healthy children. We set out to establish a reference range for the p-RAIL markers in a population of healthy children, and to determine if levels of these markers fluctuate with age or gender.
Urine was collected from 368 healthy children presenting to Cincinnati Children's primary care clinic for well child visits and assayed for NGAL, KIM-1, MCP-1, APN, CP and HPX using commercially available kits or assay materials.
Specimens were grouped by age (0–5 years (n = 94); 5–10 (n = 89); 10–15 (n = 93); 15–20 (n = 91)) and gender (M = 184, F = 184). For age and gender comparisons, values were log transformed prior to analysis. The medians (minimums, maximums) of each marker in the combined population were as follows: NGAL 6.65 (0.004, 391.52) ng/ml, KIM-1416.84 (6.22, 2512.43) pg/ml, MCP-1209.36 (9.49, 2237.06) pg/ml, APN 8.05 (0.07, 124.50) ng/ml, CP 465.15 (8.02, 7827.00) ng/ml, HPX 588.70 (6.85, 17,658.40)ng/ml. All p-RAIL biomarkers but adiponectin had weak but significant positive correlations with age, with NGAL being the strongest (r = 0.33, p < 0.001). For gender comparisons, NGAL, CP and HPX were elevated in females vs males (86%, p < 0.0001; 3%, p = 0.007, and 5%, p = 0.0005 elevation of the log transformed mean, respectively).
We have established a reference range for the p-RAIL biomarkers and have highlighted age and gender differences. This information is essential for rational interpretation of studies and clinical trials utilizing the p-RAIL algorithm.
Systemic lupus erythematosus (SLE) is an inflammatory autoimmune disease with multi-organ involvement. Renal involvement in the form of lupus nephritis (LN) is one of the main determining factors in poor prognosis [1]. Childhood-onset SLE (cSLE) [2] typically presents with more severe multi-system disease, including the development of LN in up to 80% of patients, a 10–30% higher proportion than in adult SLE [3,4,5,6].
The gold standard of diagnosis of renal involvement in SLE remains histological findings on kidney biopsy [7]. The 3 main patterns of injury used in histological diagnosis and characterization include, mesangial, endothelial and epithelial. These findings are the basis for categorization in the International Society of Nephrology/Renal Pathology Society (INS/RPS) classification system [8]. Due to invasiveness and cost considerations, it is not often practical to perform serial biopsies to track changes in LN such as worsening disease or response to treatment [9]. As a result, conventional laboratory measures are employed, such as changes in proteinuria, complement levels, and anti-ds DNA levels. These measures are not responsive enough to changes, and cannot differentiate activity from damage. Therefore, they are not well suited to direct treatment [10,11,12].
In response to the shortcomings of conventional measures, we and others have described novel urinary biomarkers (UBMs) that can assist with LN diagnosis, anticipation of flares, [13,14,15,16] and correlate with specific histologic changes associated with LN [17]. We recently developed a Renal Activity Index for Lupus Nephritis (RAIL), which was able to predict LN National Institutes of Health Activity Index (NIH-AI) scores with >92% accuracy and tubulointerstitial activity index (TIAI) scores with >80% accuracy [18, 19]. The biomarkers comprising the RAIL include neutrophil-gelatinase associated lipocalin (NGAL), kidney injury molecule 1 (KIM-1), monocyte chemotactic protein 1 (MCP-1), ceruloplasmin (CP), adiponectin (ADP) and hemopexin (HPX). In order to provide greater clinical utility of this panel of markers, we must first understand how these markers behave in healthy individuals, and whether their concentrations change with age or gender. We have previously published reference range and age/gender data for NGAL and KIM-1 [20]. In this study, we set out to establish normative values for MCP-1, CP, ADP and HPX in addition to NGAL and KIM-1 and to determine the effects of age and gender on these normal concentrations.
This study was approved by the Cincinnati Children's Hospital Medical Center Internal Review Board and was carried out in accordance with the Declaration of Helsinki. Similar to our previous study [20], samples included were from the Cincinnati Genomic Control Cohort (CGCC). Inclusion criteria were as follows: between 3 years and 18 years of age (prior to 18th birthday) at the time of enrollment, willingness of family to participate and give consent to participate in this project, willingness for participants aged 11 years of age and older to provide assent to participate in this project, ability to complete the introductory medical history, willingness to be contacted annually for future medical history updates, willingness to consent to long-term storage and future analysis of DNA. Subjects were excluded if they met any of the following criteria: presence of known genetic diseases or severe chronic medical conditions, such as chromosomal abnormality, first degree relative participating in the project, unwillingness to complete family and personal health history or allow storage or genetic testing of samples, and adopted, without full contact with biological parent(s) to be able to obtain family history information. Specific exclusion criteria for the subset of patients used in our study was a history of kidney injury or disease, including, but not limited to IGA Nephropathy, kidney stones, abnormal bladder, urinary reflux and ureteral re-implantation.
Recruits were obtained through a marketing plan developed to ensure community based participation, designed with the help of the Clinical Trials Office. Census tract monitoring was used to ensure both the diversity of cohort as well as the representativeness (an equal number of males and females, and approximately 85% white non-Hispanic, 12% African-American, and 3% Asian, Hispanic and other minorities, which represents the population distribution of the 7 counties of Northern Kentucky and Ohio that comprise Greater Cincinnati).
Potential subjects recruited from the community were screened by telephone to ensure eligibility and scheduled for an approximately 2–4 h visit. At this visit, a questionnaire was administered by the clinical research coordinator, a brief physical exam from a licensed physician was performed and samples (blood, urine, hair) were collected. Random urine samples were collected in 4 oz sterile specimen containers. The specimen was then given to the lab 15–60 min after collection, where it was centrifuged briefly to settle particulate matter and aliquoted prior to storage at -80o C. Samples were collected from 2007 to 2010 and stored until measurement in 2015. All measurements were performed in one batch in a period of one week.
Biomarker measurements
The urine NGAL ELISA was performed using a commercially available assay (NGAL ELISA Kit 036; Bioporto, Grusbakken, Denmark) that specifically detects human NGAL [21]. The intra-assay coefficient of variation (CV's) was 2.1% and inter-assay variation was 9.1%. The urine KIM-1 ELISA was constructed using commercially available reagents (Duoset DY1750, R&D Systems, Minneapolis, MN) as described previously [22]. Intra and inter-assay CV's for KIM-1 was 2% and 7.8%, respectively. Adiponectin [CV inter/intra: 4.0%/9.9%] was measured using a commercially available ELISA Kit (R&D Systems, Minneapolis, MN); ceruloplasmin [CV inter/intra: 4.1% /7.1%] and hemopexin [CV inter/intra: 4.8%/7.3%] were also measured by commercially available ELISA kits (Assaypro, St.Charles, MO). MCP-1 was measured by ELISA (R&D Systems, Minneapolis, MN). Intra and interassay CVs for MCP-1 were 5.0% and 5.9%, respectively. Urine creatinine measurements were made using an enzymatic assay, and microalbumin (MALB) was measured by immunoturbidimetry, both on a Dimension RXL plus HM Clinical Analyzer (Siemens, Munich, Germany). Coefficients of variability for the creatinine measurements were 2.4% (intra) and 4.2% (total), and 2.9% (intra) and 5.9% (inter) for MALB.
Means and 95% confidence intervals were calculated from the non-transformed biomarker values using Sigmaplot 13.0 (Systat Software, Inc., San Jose, CA). All biomarkers showed right skewed in their empirical distributions and were corrected using log transformation before analysis, same methods as showed in other publications [13, 16]. For each biomarker, its means were compared between male and female using a two sample t-test. The relationship between the biomarker and the age was assessed using a Pearson correlation coefficient. In addition, aone-way ANOVA model was performed to test the variations of the biomarker between categorized age groups. For a biomarker that showed associations to both age and gender, we firstly estimated the parameters of intercept and slopes using a linear regression model; and then developed an adjusted biomarker that is invariant to age and gender using the estimates. Considering some of the biomarkers showed a pattern of invert U shape in the initial analysis, we started a model (Model 1) with a Age2 in the independent variable to fit the shape. The model is proposed in the following where \( \widehat{Y} \) denotes a predicted value of the biomarker Y:
$$ \widehat{Y}=a+{b}_1\times Age+{b}_2\times Male+{b}_3\times Age\times Male+{b}_4\times {Age}^2+{b}_5\times {Age}^2\times Male $$
Notice Model1 can also be illustrated for male and female respectively in the following:
$$ \widehat{Y}\mid Female=a+{b}_1\times Age+{b}_4\times {Age}^2 $$
$$ \widehat{Y}\mid Male=\left(a+{b}_2\right)+\left({b}_1+{b}_3\right)\times Age+\left({b}_4+{b}_5\right)\times {Age}^2 $$
If both b 4 and b 5 are insignificant, we conclude the biomarker is more likely linearly related to Age and the Model1 will be replaced by Model2 in the following:
$$ \widehat{Y}=a+{b}_1\times Age+{b}_2\times Male+{b}_3\times Age\times Male $$
Again similarly, the Model 2 can be illustrated for male and female individually in the following:
$$ \widehat{Y}\mid Female=a+{b}_1\times Age $$
$$ \widehat{Y}\mid Male=\left(a+{b}_2\right)+\left({b}_1+{b}_3\right)\times Age $$
The Model2 can be further reduced to Model3 should both b 1 and b 3 are insignificant in the estimation.
$$ \widehat{Y}=a+{b}_1\times Age $$
The age and gender adjusted biomarker Y* is calculated in the following:
$$ {Y}^{\ast }=Y-\widehat{Y} $$
The validation of the adjusted biomarker was performed after randomly stratifying the entire data into two subsets of training data (75% of the entire data) and testing data (25% of the entire data). Models (1) and (2) were repeated in the training data to estimate the intercepts and slopes. Then the estimates were used in the testing data to develop the adjusted biomarkers. The adjusted biomarkers were tested of the associations to age and gender using the correlation coefficients, ANOVA models, and t-tests respectively.
All statistical tests were performed using SAS 9.4 software (SAS, Cary, NC). Two-sided p-values <0.05 were considered statistically significant.
Urine was collected from 368 children from the Cincinnati Genomic Control Cohort and assayed for NGAL, KIM-1, MCP-1, APN, CP and HPX. Patient demographics can be seen in Table 1. Specimens were grouped by age (0–5 years (n = 94); 5–10 (n = 89); 10–15 (n = 93); 15–20 (n = 91)) and sex (M = 184, F = 184). For age and gender comparisons, values were log transformed prior to analysis. The medians (minimums, maximums) of each marker in the combined population were as follows: NGAL 6.65 (0.004, 391.52) ng/ml, KIM-1416.84 (6.22, 2512.43) pg/ml, MCP-1209.36 (9.49, 2237.06) pg/ml, APN 8.05 (0.07, 124.50) ng/ml, CP 465.15 (8.02, 7827.00) ng/ml, HPX 588.70 (6.85, 17,658.40)ng/ml.
Table 1 Patient demographics
In order to determine if there were gender differences between the RAIL biomarkers, raw values were log transformed (natural log) and subjected to a 2-way Student's t-test. Results can be seen in Table 2. As reported previously [20], NGAL was significantly higher in females than in males (2.52 vs 1.3, p = 0.0001). HPX and CP were also higher in females than males (6.58 vs 6.26, p = 0.0005; 5.98 vs. 6.16, p = 0.007, respectively). These results indicate a need to adjust these markers based on gender.
Table 2 Biomarker vs. sex
To analyze the effects of age on biomarker levels, we subjected the natural log of the means in each age grouping to an ANOVA model and F-test of variance. Results can be seen in Table 3. To summarize, all biomarkers except HPX had a significant association with age, though not always in a predictable direction. Only NGAL, KIM-1 and MCP-1 steadily increased with each age group. While this indicates that age needs to be taken into account when adjusting the RAIL biomarkers clinically, individual ages may need to be taken into account as opposed to simple groupings. To investigate further, Pearson's tests of correlation was performed between the natural log of biomarker values and the real age (continuous variable) of the subjects (Table 4). NGAL had the strongest positive correlation with age (r = 0.33; p < 0.0001). All other markers had weak positive correlations with age (r = 0.12–0.13), except LFABP and Adiponectin, which both had weak negative correlations with age (r = −0.10, p = 0.05; r = −0.12, p = 0.04, respectively).
Table 3 Biomarker vs. age group
Table 4 Biomarker correlation with real age (continuous variable)
Due to statistically significant associations between the RAIL biomarkers and both gender and age, we developed a method for adjusting the levels for males and females as a function of real age (continuous variable). These parameters can be seen in Table 5. For biomarkers of KIM-1, NGAL, MCP-1, HPX, and MALB, their slopes of Age2 (or Age x Age) were not significant in either gender. Hence Model 2 fit better for these markers. For the rest of biomarkers of CP, ADP and Creatinine, their slopes of Age2 were significant in boys and hence Model1 was preferred for these biomarkers. To illustrate how we calculate an adjusted biomarker, we use KIM-1 as an example for a boy, aged 10 years. Table 5 shows Model 2 is preferred in prediction. The predicted KIM-1 level for this boy will be 5.42 + (0.05 × 10) = 5.97. To return this natural log to a raw value: exp. (5.97) = 391.5 pg/ml. This formula would differ with the same KIM-1 level if the patient were female. With a female patient: 5.42 + (0.03 × 10) = 5.72. To return the natural log to a raw value: exp. (5.72) = 304.9 pg/ml.
Table 5 Parameters used for adjusting biomarkers
We set out in this manuscript to describe the normative values of the novel urinary biomarkers we established as a pediatric Renal Activity Index for Lupus Nephritis (pRAIL) [18, 19]. We also wanted to determine whether gender and age affected the normative values of these markers. While we and others have reported on reference ranges for urine NGAL and KIM-1 [20, 22,23,24,25], this is to our knowledge, the first study to establish urinary reference ranges for MCP-1, ADP, CP and HPX. The need for reference values is important for establishing this panel, not only in children, but adults as well [26].
Most studies investigating biomarkers in lupus nephritis and other chronic conditions use disease controls, such as juvenile idiopathic arthritis (JIA) or SLE patients without renal involvement [13, 27, 28]. Normal behavior of proteins in the healthy population is an important metric in establishing the clinical utility of laboratory tests. Just as proteins will present variability between relatively similar individuals, normal groupings by age and gender often exhibit greater variability and must be taken into account when establishing clinical diagnostic algorithms [20]. In particular, urine proteins have been shown to differ to a greater degree between males and females than in other body fluids, such as cerebrospinal fluid [29].
In this study, we took advantage of the availability of a large cohort of healthy pediatric patients, namely the Cincinnati Genomic Control Cohort. The goal of the development of the cohort was to obtain a population representative sample which could be utilized as controls for a diverse set of projects. Our results showed significant associations of our biomarkers with both age and gender. In concordance with our earlier studies and with published literature, NGAL was significantly higher in females than in males in both pediatric and adult populations [20, 24, 25, 30, 31]. HPX was also found to be significantly higher in females. This has not, to the best of our knowledge, been previously reported. HPX is the primary binder of free heme in the blood. Adult females naturally have lower hemoglobin and associated blood levels of heme due to menstruation, which results in lower HPX levels. In the kidneys, HPX is produced primarily in the renal cortex in the setting of nephrotoxic insults, and acts to protect the tubules from free heme radicals [19, 32]. The reason urine HPX would be elevated in healthy pediatric females would be speculation at this time.
Urine levels of CP were also found to be higher in females. It has been long known that serum levels of CP, a carrier of copper in the blood, are higher in healthy adult females than males and has also been found to increase in older women [33]. While we found an association with age and ceruloplasmin in the pediatric population, it was more of an inverted U shaped association, increasing from age 3 - < 5 up to 10 - < 15, but then decreasing in our oldest grouping, 15 - < 18. In urine, CP increased in response to infections, acting as a molecular source of copper which can inhibit bacterial growth [34]. CP is also a ferroxidase, which can transform ferrous iron, which is toxic to renal tubules, to its nontoxic ferric state. While speculative, CP may be naturally upregulated in females as a host defense due to their greater incidence of UTI compared to males [35].
All of the RAIL proteins showed associations with age, except hemopexin. It is important to note, however, that only NGAL, KIM-1 and MCP-1 had direct positive correlations with age. We have previously shown NGAL and KIM-1 to increase as a function of age [20]. While serum MCP-1 increases with aging in adults, and increases with risk of cardiovascular disease [36], increases in urine MCP-1 in developing children have not been reported to our knowledge. MCP-1 is expressed at high levels in the tubular epithelium with oxidative stress [37] and is predictive of LN flares and LN severity [14, 38]. It would be plausible that there are small increases in subclinical oxidative stress due to environmental exposures in a developing child/adolescent that could potentially lead to increases in tubular expression of MCP-1 in a healthy individual.
"Normalizing" the data for hydration status with urine using creatinine represents a difficult position with a maturing pediatric population. In the growing child, urine creatinine increases as a function of age and maturity [39]. Therefore any "correction" for creatinine applied to our biomarker levels would nullify any increase as a function of age. Indeed, in our population, using a Spearman correlation, urine creatinine demonstrated a significant positive correlation with age (r = 0.54, p < 0.001). Also, since the pRAIL proteins do not all have a direct positive correlation with age,, creatinine normalization would present additional problems. Since creatinine is dependent upon age in this population, it would not be useful as a normalization tool. It would be important to note that not only does creatinine have an intrinsic relationship with age, it also displays a significant relationship with gender. It is well known that creatinine is higher in males, especially after puberty, than females [40, 41]. As a result, creatinine correction would amplify the difference between males and females in terms of biomarker levels.
The strengths of our study include a large representative healthy pediatric population and established laboratory expertise with the specific methodology used in the study [18, 19]. This study is not without its weaknesses. Our study is from a single center cohort that is representative of the population of a mid-size US city (Cincinnati, Ohio). Demographics from this cohort would differ from other regions and as a result, our results may not be generalizable to the global population. For instance, 85% of our study population classified as White/Non-Hispanic, leaving numbers too small to study race as a variable. Also, while our assays are well established, none of them are standardized assays available on a clinical platform for measurement in urine. As a result, our results may vary from those utilizing different assays. Since the cohort was not originally designed for renal or lupus research, certain pertinent data such as glomerular filtration rate, are not available for these patients. It may be interesting in future studies to investigate other variables, such as hemoglobin levels, which could potentially account for underlying differences between HPX levels in males vs. females.
Our previous work has demonstrated the utility of the RAIL biomarkers to monitor LN activity in both the pediatric and adult population [18, 19]. The current manuscript has elucidated specific gender and age related associations of the RAIL biomarkers in a population of healthy children. These results have enabled us to develop a method to adjust the levels of the biomarkers for individual patients based on age and sex, to increase the accuracy of our RAIL algorithm. These improvements will increase the clinical utility of the RAIL algorithm and may lead to more effective and personalized treatments for patients with lupus nephritis.
ADP:
Adiponectin
Ceruloplasmin
HPX:
Hemopexin
KIM-1:
Kidney injury molecule – 1
LN:
MALB:
Microalbumin
MCP-1:
Monocyte chemotactic protein
NGAL:
Neutrophil gelatinase-associated lipocalin
pRAIL:
Pediatric renal activity index for lupus nephritis
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The Authors would like to thank Christopher Haffner, MS, MBA for his help with laboratory measurements.
Funding for this project was provided by NIH P50 DK096418 to HB and PD. This research was supported in part by the Cincinnati Children's Research Foundation and its Cincinnati Genomic Control Cohort.
Please contact author for data requests.
Division Nephrology and Hypertension, Cincinnati Children's Hospital Medical Center, Cincinnati, OH, USA
Michael R. Bennett
, Qing Ma
& Prasad Devarajan
Environmental Health, University of Cincinnati College of Medicine, Cincinnati, OH, USA
Jun Ying
Rheumatology, Cincinnati Children's Hospital Medical Center, Cincinnati, OH, USA
Hermine Brunner
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MB – study design, data collection, statistical analysis, result interpretation and manuscript preparation; QM – data collection, manuscript preparation; JY – statistical analysis, manuscript preparation; PD and HB– study design, result interpretation, manuscript preparation. All authors read and approved the final manuscript.
Correspondence to Michael R. Bennett.
This study was approved by the Cincinnati Children's Hospital Medical Center Internal Review Board and was carried out in accordance with the Declaration of Helsinki. Families gave consent for their children to participate, and children 11 years of age and older gave assent to participate.
PD, HB and MB have patent applications regarding the use of the RAIL panel as a diagnostic tool.
Reference range
NGAL
Kim-1
MCP-1 | CommonCrawl |
\begin{document}
\begin{abstract}
We prove the Abelian/non-Abelian Correspondence with bundles for target spaces that are partial flag bundles, combining and generalising results by Ciocan-Fontanine--Kim--Sabbah, Brown, and Oh. From this we deduce how genus-zero Gromov--Witten invariants change when a smooth projective variety $X$ is blown up in a complete intersection defined by convex line bundles. In the case where the blow-up is Fano, our result gives closed-form expressions for certain genus-zero invariants of the blow-up in terms of invariants of $X$. We also give a reformulation of the Abelian/non-Abelian Correspondence in terms of Givental's formalism, which may be of independent interest. \end{abstract} \maketitle \section{Introduction}
Gromov--Witten invariants, roughly speaking, count the number of curves in a projective variety $X$ that are constrained to pass through various cycles. They play an essential role in mirror symmetry, and have been the focus of intense activity in symplectic and algebraic geometry over the last 25 years. Despite this, there are few effective tools for computing the Gromov--Witten invariants of blow-ups. In this paper we improve the situation somewhat: we determine how genus-zero Gromov--Witten invariants change when a smooth projective variety $X$ is blown up in a complete intersection of convex line bundles. In the case where the blow-up $\tilde{X}$ is Fano, a special case of our result gives closed-form expressions for genus-zero one-point descendant invariants of $\tilde{X}$ in terms of invariants of $X$, and hence determines the small $J$-function of $\tilde{X}$.
Suppose that $Z \subset X$ is the zero locus of a regular section of a direct sum of convex (or nef) line bundles $$E = L_0 \oplus \cdots \oplus L_r \to X$$ and that $\tilde{X}$ is the blow-up of $X$ in $Z$. To determine the genus-zero Gromov--Witten invariants of $\tilde{X}$, we proceed in two steps. First, we exhibit $\tilde{X}$ as the zero locus of a section of a convex vector bundle on the bundle of Grassmannians $\Gr(r,E^\vee) \to X$: this is Theorem~\ref{step one} below. We then establish a version of the Abelian/non-Abelian Correspondence~\cite{CFKS2008} that determines the genus-zero Gromov--Witten invariants of such zero loci. This is the Abelian/non-Abelian Correspondence with bundles, for target spaces that are partial flag bundles -- see Theorem~\ref{step two}. It builds on and generalises results by Ciocan-Fontanine--Kim--Sabbah~\cite[\S6]{CFKS2008}, Brown~\cite{Brown2014}, and Oh~\cite{Oh2016}.
\begin{theorem}[see Proposition~\ref{geometricconstruction} below for a more general result] \label{step one}
Let $X$ be a smooth projective variety, let $E = L_0 \oplus \cdots \oplus L_r \to X$ be a direct sum of line bundles, and let $Z \subset X$ be the zero locus of a regular section $s$ of $E$. Let $\pi \colon \Gr(r,E^\vee) \to X$ be the Grassmann bundle of subspaces and let $S \to \Gr(r,E^\vee)$ be the tautological subbundle. Then the composition $$S \hookrightarrow \pi^*E^\vee \xrightarrow{\pi^* s^\vee} \cO $$ defines a regular section of $S^\vee$, and the zero locus of this section is the blow-up $\tilde{X} = \Bl_Z X$. \end{theorem}
\noindent If the line bundles $L_i$ are convex, then the bundle $S^\vee$ is also convex. The fact that $\tilde{X}$ is regularly embedded into $\Gr(r, E^\vee)\cong \PP(E)$ (where $\PP(E)$ is the projective bundle of lines) is well-known and true in more generality, see for example \cite[Appendix B8.2]{FultonIntersection} and \cite[Lemma $2.1$]{AluffiChernClasses}. However, to apply the Abelian/non-Abelian correpondence, the crucial point is that $\tilde{X}$ is cut out by a regular section of an explicit representation-theoretic bundle on $\Gr(r, E^\vee)$. Although this should be well-known to experts, we have been unable to find a reference for this.\\ To apply Theorem~\ref{step one} to Gromov--Witten theory, and to state the Abelian/non-Abelian Correspondence, we will use Givental's formalism~\cite{Givental2004}. This is a language for working with Gromov--Witten invariants and operations on them, in terms of linear symplectic geometry. We give details in \S\ref{givental formalism} below, but the key ingredients are, for each smooth projective variety $Y$, an infinite-dimensional symplectic vector space $\cH_Y$ called the Givental space and a Lagrangian submanifold $\cL_Y \subset \cH_Y$. Genus-zero Gromov--Witten invariants of $Y$ determine and are determined by $\cL_Y$.
We will also consider \emph{twisted} Gromov--Witten invariants~\cite{CoatesGivental2007}. These are invariants of a projective variety $Y$ which depend also on a bundle $F \to Y$ and a characteristic class $\mathbf{c}$. For us, this characteristic class will always be the equivariant Euler class (or total Chern class)\begin{align} \label{intro equivariant Euler}
\mathbf{c}(V) = \sum_{k=0}^d \lambda^{d-k} c_k(V) && \text{where $d$ is the rank of the vector bundle $V$.} \end{align} The parameter $\lambda$ here can be thought of as the generator for the $S^1$-equivariant cohomology of a point. There is a Lagrangian submanifold $\cL_{F_\lambda} \subset \cH_Y$ that encodes genus-zero Euler-twisted invariants of $Y$; the Quantum Riemann--Roch theorem~\cite{CoatesGivental2007} implies that \[
\Delta_{F_\lambda} \cL_Y = \cL_{F_\lambda} \] where $\Delta_{F_\lambda} \colon \cH_Y \to \cH_Y$ is a certain linear symplectomorphism. This gives a family of Lagrangian submanifolds $\lambda \mapsto \cL_{F_\lambda}$ defined over $\QQ(\lambda)$, that is, a meromorphic family of Lagrangian submanifolds parameterised by $\lambda$. When $F$ satisfies a positivity condition called convexity, the family $\lambda \mapsto \cL_\lambda$ extends analytically across $\lambda=0$ and the limit $\cL_{F_0}$ exists. This limiting submanifold $\cL_{F_0} \subset \cH_Y$ determines genus-zero Gromov--Witten invariants of the subvariety of $Y$ cut out by a generic section of $F$~\cite{CoatesGivental2007,Coates2014}. Theorem~\ref{step one} therefore allows us to determine genus-zero Gromov--Witten invariants of the blow-up $\tilde{X}$, by analyzing the limiting submanifold~$\cL_{S^\vee_0}$.
Our second main result, Theorem~\ref{step two}, applies to the Grassmann bundle $\Gr(r,E^\vee) \to X$ considered in Theorem~\ref{step one}, and more generally to any partial flag bundle $\Fl(E) \to X$ induced by $E$. Such a partial flag bundle can be expressed as a GIT quotient $A /\!\!/ G$, where $G$ is a product of general linear groups, and so any representation $\rho$ of $G$ on a vector space $V$ induces a vector bundle $V^G \to \Fl(E)$ with fiber $V$. See \S\ref{flag} for details of the construction. We give an explicit family of elements of $\cH_{\Fl(E)}$, \begin{align} \label{intro twisted I}
(t, \tau) \mapsto I_{\GM}(t, \tau, z) && \text{$t \in \CC^R$ for some $R$, $\tau \in H^\bullet(X)$} \end{align} defined in terms of genus-zero Gromov--Witten invariants of $X$ and explicit hypergeometric functions, and show that this family, after changing the sign of $z$, lies on the Lagrangian submanifold that determines Euler-twisted Gromov--Witten invariants of $\Fl(E)$ with respect to~$V^G$.
\begin{theorem}[see Definition~\ref{IGM definition special case} and Theorem~\ref{IGM on twisted cone}]\label{step two}
For all $t \in \CC^R$ and $\tau \in H^\bullet(X)$,
$$I_{\GM}(t, \tau, {-z}) \in \cL_{V^G_\lambda}$$ \end{theorem}
\noindent Under an ampleness condition -- which holds, for example, whenever the blow-up $\tilde{X}$ in Theorem~\ref{step one} is Fano -- the family \eqref{intro twisted I} takes a particularly simple form $$ I_{\GM}(t, \tau, z) = z \left(1 + o(z^{-1})\right) $$ and standard techniques in Givental formalism allow us to determine genus-zero twisted Gromov--Witten invariants of $\Fl(E)$ explicitly: see Corollaries~\ref{I=J} and~\ref{explicit Gr}. Applying this in the setting of Theorem~\ref{step one}, we recover genus-zero Gromov--Witten invariants of the blow-up $\tilde{X}$ by taking the non-equivariant limit $\lambda \to 0$.
The reader who is focussed on blow-ups can stop reading here, jumping to the end of the Introduction for connections to previous work, \S\ref{flag} for basic setup, Corollary~\ref{explicit Gr} for the key Gromov--Witten theoretic result, and then to \S\ref{examples} for worked examples. In the rest of the Introduction, we explain how Theorem~\ref{step two} should be regarded as an instance of the Abelian/non-Abelian Correspondence~\cite{CFKS2008}.
\pagebreak
The Abelian/non-Abelian Correspondence relates the genus-zero Gromov--Witten theory of quotients $A /\!\!/ G$ and $A /\!\!/ T$, where $A$ is a smooth quasiprojective variety equipped with the action of a reductive Lie group $G$, and $T$ is its maximal torus. We fix a linearisation of this action such that the stable and semistable loci coincide and we suppose that the quotients $A /\!\!/ G$ and $A /\!\!/ T$ are smooth. In our setting the non-Abelian quotient $A /\!\!/ G$ will be a partial flag bundle or Grassmann bundle over $X$, and the Abelian quotient $A /\!\!/ T$ will be a bundle of toric varieties over $X$, that is, a toric bundle in the sense of Brown~\cite{Brown2014}. To reformulate the Abelian/non-Abelian Correspondence of~\cite{CFKS2008} in terms of Givental's formalism, however, we pass to the following more general situation. Let $W$ denote the Weyl group of $T$ in $G$. A theorem of Martin (Theorem~\ref{thm:Martin} below) expresses the cohomology of the non-Abelian quotient $H^\bullet(A /\!\!/ G)$ as a quotient of the Weyl-invariant part of the cohomology of the Abelian quotient $H^\bullet(A /\!\!/ T)^W$ by an appropriate ideal, so there is a quotient map \begin{equation} \label{quotient}
H^\bullet(A /\!\!/ T)^W \to H^\bullet(A /\!\!/ G). \end{equation} The Abelian/non-Abelian Correspondence, in the form that we state it below, asserts that this map also controls the relationship between the quantum cohomology of $A /\!\!/ G$ and $A /\!\!/ T$.
When comparing the quantum cohomology algebras of $A /\!\!/ G$ and $A /\!\!/ T$, or when comparing the Givental spaces of $A /\!\!/ G$ and $A /\!\!/ T$, we need to account for the fact that there are fewer curve classes on $A /\!\!/ G$ than there are on $A /\!\!/ T$. We do this as follows. The Givental space $\cH_Y$ discussed above is defined using cohomology groups $H^\bullet(Y;\Lambda)$ where $\Lambda$ is the Novikov ring for~$Y$: see \S\ref{givental formalism}. The Novikov ring contains formal linear combinations of terms $Q^d$ where $d$ is a curve class on~$Y$. The quotient map \eqref{quotient} induces an isomorphism $H^2(A /\!\!/ T)^W \cong H^2(A /\!\!/ G)$, and by duality this gives a map $\varrho \colon \NE(A /\!\!/ T) \rightarrow \NE (A /\!\!/ G)$ where $\NE$ denotes the Mori cone: see Proposition~\ref{maponmori}. Combining the quotient map \eqref{quotient} with the map on Novikov rings induced by $\varrho$ gives a map \begin{equation} \label{quotientH}
p \colon \cH^W_{A /\!\!/ T} \to \cH_{A /\!\!/ G} \end{equation} between the Weyl-invariant part of the Givental space for the Abelian quotient and the Givental space for the non-Abelian quotient. Here, and also below when we discuss Weyl-invariant functions, we consider the Weyl group $W$ to act on $\cH_{A /\!\!/ T}$ through the combination of its action on cohomology classes and its action on the Novikov ring.
We consider now an appropriate twisted Gromov--Witten theory of $A /\!\!/ T$. For each root $\rho$ of~$G$, write $L_\rho \to A /\!\!/ T$ for the line bundle determined by $\rho$, and let $\Phi = \oplus_\rho L_\rho$ where the sum runs over all roots. Consider the Lagrangian submanifold~$\cL_{\Phi_\lambda}$ that encodes genus-zero twisted Gromov--Witten invariants of $A /\!\!/ T$. The bundle $\Phi$ is very far from convex, so one cannot expect the non-equivariant limit of $\cL_{\Phi_\lambda}$ to exist. Nonetheless, the projection along \eqref{quotientH} of the Weyl-invariant part of this Lagrangian submanifold does have a non-equivariant limit.
\begin{theorem}(see Corollary~\ref{GMlimit no bundle})
The limit as $\lambda \to 0$ of $p \Big( \cL_{\Phi_\lambda} \cap \cH^W_{A /\!\!/ T}\Big)$ exists. \end{theorem}
\noindent We call this non-equivariant limit the \emph{Givental--Martin cone\footnote{We have not emphasised this point, but the Lagrangian submanifolds $\cL_Y$, $\cL_{F_{\lambda}}$, etc.~are in fact cones~\cite{Givental2004}.}} $\cL_{\GM} \subset \cH_{A/\!\!/ G}$.
\begin{conjecture}[The Abelian/non-Abelian Correspondence] \label{AnA}
$\cL_{\GM} = \cL_{A /\!\!/ G}$. \end{conjecture}
\noindent This is a reformulation of \cite[Conjecture~3.7.1]{CFKS2008}. The analogous statement for twisted Gromov--Witten invariants is the Abelian/non-Abelian Correspondence with bundles; this is a reformulation of \cite[Conjecture~6.1.1]{CFKS2008}. Fix a representation $\rho$ of $G$, and consider the vector bundles $V^G \to A /\!\!/ G$ and $V^T \to A /\!\!/ T$ induced by~$\rho$. Consider the Lagrangian submanifold $\cL_{\Phi_{\lambda} \oplus V^T_{\mu}}$ that encodes genus-zero twisted Gromov–Witten invariants of $A /\!\!/ T$, where for the twist by the root bundle $\Phi$ we use the equivariant Euler class \eqref{intro equivariant Euler} with parameter $\lambda$ and for the twist by $V^T$ we use the equivariant Euler class with a different parameter $\mu$. As before, the projection along \eqref{quotientH} of the Weyl-invariant part of this Lagrangian submanifold has a non-equivariant limit with respect to $\lambda$. \begin{theorem}(see Theorem~\ref{GMlimit})
The limit as $\lambda \to 0$ of $p \Big(\cL_{\Phi_\lambda \oplus V^T_\mu} \cap \cH^W_{A /\!\!/ T}\Big)$ exists. \end{theorem}
\noindent Let us call this limit the \emph{twisted Givental--Martin cone} $\cL_{\GM,V^T_\mu} \subset \cH_{A /\!\!/ G}$.
\begin{conjecture}[The Abelian/non-Abelian Correspondence with bundles]\label{AnA bundles}
$\cL_{\GM, V^T_\mu} = \cL_{V^G_\mu}$. \end{conjecture}
As in~\cite{CFKS2008}, the Abelian/non-Abelian Correspondence implies the Abelian/non-Abelian Correspondence with bundles.
\begin{proposition}
Conjectures~\ref{AnA} and~\ref{AnA bundles} are equivalent. \end{proposition}
\begin{proof}
Conjecture~\ref{AnA} is the special case of Conjecture~\ref{AnA bundles} where the vector bundles involved have rank zero. To see that Conjecture~\ref{AnA} implies Conjecture~\ref{AnA bundles}, observe that the projection of the Quantum Riemann--Roch operator $\Delta_{V^T_\mu}$ under the map \eqref{quotientH} is $\Delta_{V^G_\mu}$: see Definition~\ref{delta}. Now apply the Quantum Riemann--Roch theorem~\cite{CoatesGivental2007}. \end{proof}
The following reformulations will also be useful. Given any Weyl-invariant family \begin{align*}
t \mapsto I(t) \in \cH^W_{A /\!\!/ T}
&& \text{of the form} &&
I(t) = \sum_{d \in \NE(A /\!\!/ T)} Q^d I_d(t) \end{align*} we define its \emph{Weyl modification} $t \mapsto \widetilde{I}(t) \in \cH^W_{A /\!\!/ T}$ to be $$ \widetilde{I}(t) = \sum_{d \in \NE(A /\!\!/ T)} Q^d W_d I_d(t) $$ where $W_d$ is an explicit hypergeometric factor that depends on $\lambda$ -- see~\eqref{modg}. We prove in Lemma~\ref{IGMexists} below that, for a Weyl-invariant family $t \mapsto I(t)$ the image under \eqref{quotientH} of the Weyl modification $t \mapsto p(\widetilde{I}(t))$ has a well-defined limit as $\lambda \to 0$. We call this limit the \emph{Givental--Martin modification} of $t \mapsto I(t)$ and denote it by $t \mapsto I_{\GM}(t)$; it is a family of elements of $\cH_{A /\!\!/ G}$. Furthermore, if $t \mapsto I(t)$ satisfies the Divisor Equation in the sense of equation~\eqref{divisor equation}, then: \begin{itemize}[itemsep=0.5ex]
\item if $t \mapsto I(t)$ is a family of elements of $\cL_{A /\!\!/ T}$ then $t \mapsto I_{\GM}(t)$ is a family of elements on the Givental--Martin cone $\cL_{\GM}$; and
\item if $t \mapsto I(t)$ is a family of elements of the twisted cone $\cL_{V^T_\mu}$ then $t \mapsto I_{\GM}(t)$ is a family of elements on the twisted Givental--Martin cone $\cL_{\GM,V^T_\mu}$. \end{itemize} The first statement here is Corollary~\ref{IGMonLGM} with $F'=0$; the second statement is Corollary~\ref{IGMonLGM}. This lets us reformulate the Abelian/non-Abelian Correspondence in more concrete terms.
\begin{conjecture}[a reformulation of Conjecture~\ref{AnA}] \label{AnA family}
Let $t \mapsto I(t)$ be a Weyl-invariant family of elements of $\cL_{A /\!\!/ T}$ that satisfies the Divisor Equation. Then the Givental--Martin modification $t \mapsto I_{\GM}(t)$ is a family of elements of $\cL_{A /\!\!/ G}$. \end{conjecture}
\begin{conjecture}[a reformulation of Conjecture~\ref{AnA bundles}] \label{AnA bundles family}
Let $t \mapsto I(t)$ be a Weyl-invariant family of elements of $\cL_{V^T_\mu}$ that satisfies the Divisor Equation. Then the Givental--Martin modification $t \mapsto I_{\GM}(t)$ is a family of elements of $\cL_{V^G_\mu}$. \end{conjecture}
Let us now specialise to the case of partial flag bundles, as in \S\ref{notation} and the rest of the paper, so that $A /\!\!/ G$ is a partial flag bundle $\Fl(E) \to X$ and $A /\!\!/ T$ is a toric bundle $\Fl(E)_T \to X$. Theorem~\ref{brownoh AnA} below establishes the statement of Conjecture~\ref{AnA family} not for an arbitrary Weyl-invariant family $t \mapsto I(t)$ on $\cL_{A /\!\!/ T}$, but for a specific such family called the \emph{Brown $I$-function}. As we recall in Theorems~\ref{ohI} and~\ref{brown2014gromov}, Brown and Oh have defined families $t \mapsto I_{\Fl(E)_T}(t)$ and $t \mapsto I_{\Fl(E)}(t)$, given in terms of genus-zero Gromov--Witten invariants of $X$ and explicit hypergeometric functions, and have shown~\cite{Brown2014, Oh2016} that $I_{\Fl(E)_T}(t) \in \cL_{\Fl(E)_T}$ and $I_{\Fl(E)}(t) \in \cL_{\Fl(E)}$.
\begin{theorem}[see Proposition~\ref{brownoh} for details] \label{brintroh} \label{brownoh AnA}
The Givental--Martin modification of the Brown $I$-function $t \mapsto I_{\Fl(E)_T}$ is $t \mapsto I_{\Fl(E)}(t)$. \end{theorem}
\noindent The main result of this paper is the analogue of Theorem~\ref{brownoh AnA} for twisted Gromov--Witten invariants. We define a twisted version $t \mapsto I_{V^T_\mu}(t)$ of the Brown $I$-function and prove:
\begin{theorem}[see Definition~\ref{IGM definition special case} and Corollary~\ref{IGM on twisted cone} for details] \ \label{step three}
\begin{enumerate}
\item the twisted Brown $I$-function $t \mapsto I_{V^T_\mu}(t)$ is a Weyl-invariant family of elements of~$\cL_{V^T_\mu}$ that satisfies the Divisor Equation;
\item the Givental--Martin modification $t \mapsto I_{\GM}(t)$ of this family satisfies $I_{\GM}(t) \in \cL_{V^G_\mu}$.
\end{enumerate} \end{theorem}
\noindent This establishes the statement of Conjecture~\ref{AnA bundles family}, not for an arbitrary Weyl-invariant family, but for the specific such family $t \mapsto I_{V^T_\mu}(t)$. Theorem~\ref{step three} follows from the Quantum Riemann--Roch theorem~\cite{CoatesGivental2007} together with the results of Brown~\cite{Brown2014} and Oh~\cite{Oh2016}, using a ``twisting the $I$-function'' argument as in~\cite{CCIT2019}.
As we will now explain, Theorem~\ref{brownoh AnA} is quite close to a proof of Conjecture~\ref{AnA family} in the flag bundle case, and similarly Theorem~\ref{step three} is close to a proof of Conjecture~\ref{AnA bundles family}. We will discuss only the former, as the latter is very similar. Theorem~\ref{brownoh AnA} implies that \begin{equation}
\label{AnA intermediate}
\text{the Givental--Martin modification $t \mapsto I_{\GM}(t)$ lies in $\cL_{\Fl(E)}$} \end{equation} for the family $t \mapsto I(t)$ given by the Brown I-function, because the Givental--Martin modification of the Brown $I$-function is the Oh $I$-function $t \mapsto I_{\Fl(E)}(t)$. If Oh's $I$-function were a \emph{big $I$-function}, in the sense of~\cite{CFK2016}, then Conjecture~\ref{AnA family} would follow. The special geometric properties of the Lagrangian submanifold $\cL_Y$ described in~\cite{Givental2004} and~\cite[Appendix B]{CCIT2009Computing}, taking $Y = \Fl(E)$, would then imply that any family $t \mapsto I(t)$ such that $I(t) \in \cL_{\Fl(E)}$ can be written as \begin{equation}
\label{special form}
I(t) = I_{\Fl(E)}(\tau(t)) + \sum_\alpha C_\alpha(t, z) z \frac{\partial I_{\Fl(E)}}{\partial \tau_\alpha}(\tau(t)) \end{equation} for some coefficients $C_\alpha(t, z)$ that depend polynomially on $z$ and some change of variables $t \mapsto \tau(t)$. Furthermore the same geometric properties imply that any family of the form \eqref{special form} satisfies $I(t) \in \cL_{\Fl(E)}$. But $\cL_{\GM}$ has the same special geometric properties as $\cL_Y$ -- it inherits them from the Weyl-invariant part of $\cL_{\Phi_\lambda}$ by projection along \eqref{quotientH} followed by taking the non-equivariant limit -- and so if $t \mapsto I_{\Fl(E)}$ is a big $I$-function then any family of elements $t \mapsto I^\dagger(t)$ on $\cL_{\GM}$ can be written as \begin{equation*}
I^\dagger(t) = I_{\Fl(E)}(\tau^\dagger(t)) + \sum_\alpha C^\dagger_\alpha(t, z) z \frac{\partial I_{\Fl(E)}}{\partial \tau_\alpha}(\tau^\dagger(t)) \end{equation*} That is, $I^\dagger(t)$ can be written in the form \eqref{special form}. It follows that $I^\dagger(t) \in \cL_{\Fl(E)}$. Applying this with $I^\dagger = I_{\GM}$ from Conjecture~\ref{AnA family} proves that Conjecture; note that we know that the family $t \mapsto I_{\GM}(t)$ here lies in $\cL_{\GM}$ by Corollary~\ref{IGMonLGM}.
If the Brown and Oh $I$-functions were big $I$-functions then Theorem~\ref{brownoh AnA} would continue to hold (with the same proof) and Conjecture~\ref{AnA family} would therefore follow. In reality the Brown and Oh $I$-functions are only small $I$-functions, not big $I$-functions, but Ciocan-Fontanine--Kim have explained in \cite[\S5]{CFK2016} how to pass from small $I$-functions to big $I$-functions, whenever the target space is the GIT quotient of a vector space. To apply their argument, and hence prove Conjecture~\ref{AnA family} for partial flag bundles, one would need to check that the Brown $I$-function arises from torus localization on an appropriate quasimap graph space~\cite[\S7.2]{CFKM2014}. The analogous result for the Oh $I$-function is~\cite[Proposition~5.1]{Oh2016}.
Webb has proved a `big $I$-function' version of the Abelian/non-Abelian Correspondence for target spaces that are GIT quotients of vector spaces~\cite{Webb2018}, and this immediately implies Conjectures~\ref{AnA family} and~\ref{AnA bundles family}.
\begin{proposition}
Conjecture~\ref{AnA family} holds when $A$ is a vector space and $G$ acts on $A$ via a representation $G \mapsto \GL(A)$. \end{proposition}
\begin{proof}
Combining \cite[Corollary~6.3.1]{Webb2018} with \cite[Theorem~3.3]{CFK2016} shows that there are big $I$-functions $t \mapsto I_{A /\!\!/ T}(t)$ and $t \mapsto I_{A /\!\!/ G}(t)$ such that $I_{A /\!\!/ T}(t) \in \cL_{A /\!\!/ T}$ and $I_{A /\!\!/ G}(t) \in \cL_{A /\!\!/ G}$. Furthermore it is clear from \cite[equation 62]{Webb2018} that the Givental--Martin modification of the Weyl-invariant part of $t \mapsto I_{A /\!\!/ T}(t)$ is $t \mapsto I_{A /\!\!/ G}(t)$. Now argue as above. \end{proof}
\subsection*{Connection to Earlier Work} Our formulation of the Abelian/non-Abelian Correspondence very roughly says that, for genus-zero Gromov--Witten theory, passing from an Abelian quotient $A /\!\!/ T$ to the corresponding non-Abelian quotient $A /\!\!/ G$ is almost the same as twisting by the non-convex bundle $\Phi \to A /\!\!/ T$ defined by the roots of $G$. This idea goes back to the earliest work on the subject, by Bertram--Ciocan-Fontanine--Kim, and indeed our Conjecture is very much in the spirit of the discussion in~\cite[\S4]{BCFK2008}. These ideas were given a precise form in~\cite{CFKS2008}, in terms of Frobenius manifolds and Saito's period mapping; the main difference with the approach that we take here is that in~\cite{CFKS2008} the authors realise the cohomology $H^\bullet(A /\!\!/ G)$ as the Weyl-anti-invariant subalgebra of the cohomology of the Abelian quotient $A /\!\!/ T$, whereas we realise it as a quotient of the Weyl-invariant part of $H^\bullet(A/\!\!/ T)$. The latter approach seems to fit better with Givental's formalism.
Ruan was the first to realise that there is a close connection between quantum cohomology (or more generally Gromov--Witten theory) and birational geometry~\cite{Ruan1999}, and the change in Gromov--Witten invariants under blow-up forms an important testing ground for these ideas. Despite the importance of the topic, however, Gromov--Witten invariants of blow-ups have been understood in rather few situations. Early work here focussed on blow-ups in points, and on exploiting structural properties of quantum cohomology such as the WDVV equations and Reconstruction Theorems~\cite{Gathmann1996, GottschePandharipande1998, Gathmann2001}. Subsequent approaches used symplectic methods pioneered by Li--Ruan~\cite{LiRuan2001,HuLiRuan2008,Hu2000,Hu2001}, or the Degeneration Formula following Maulik--Pandharipande~\cite{MaulikPandharipande2006,HeHuKeQi2018,ChenDuWang2020}, or a direct analysis of the moduli spaces involved and virtual birationality arguments~\cite{Manolache2012,Lai2009,AbramovichWise2018}. In each case the aim was to prove `birational invariance': that certain specific Gromov--Witten invariants remain invariant under blow-up. We take a different approach. Rather than deform the target space, or study the geometry of moduli spaces of stable maps explicitly, we give an elementary construction of the blow-up $\tilde{X} \to X$ in terms that are compatible with modern tools for computing Gromov--Witten invariants, and extend these tools so that they cover the cases we need. This idea -- of reworking classical constructions in birational geometry to make them amenable to computations using Givental formalism -- was pioneered in \cite{CCGK16}, and indeed Lemma~E.1 there gives the codimension-two case of our Theorem~\ref{step one}.
Compared to explicit invariance statements \[
\langle \pi^*\phi_{i_1}, \ldots, \pi^*\phi_{i_n} \rangle^{\tilde{X}}_{0,n,\pi^! \beta} =\langle \phi_{i_1}, \ldots, \phi_{i_n} \rangle^{X}_{0,n,\beta} \] as in \cite[Theorem 1.4]{Lai2009}, we pay a price for our increased abstraction: the range of invariants for which we can extract closed-form expressions is different (see Corollary~\ref{I=J}) and in general does not overlap with Lai's. But we also gain a lot by taking a more structural approach: our results determine, via a Birkhoff factorization procedure as in~\cite{CoatesGivental2007, CFK2014}, genus-zero Gromov--Witten invariants of the blow-up $\tilde{X}$ for curves of arbitrary degree (not just proper transforms of curves in the base) and with a wide range of insertions that can include gravitional descendant classes. See Remark~\ref{what can we compute blow up}. Furthermore in general one should not expect Gromov--Witten invariants to remain invariant under blow-ups. The correct statement -- cf.~Ruan's Crepant Resolution Conjecture~\cite{CCIT2009WallCrossings, CoatesRuan, Iritani2008, Iritani2009} and its generalisation by Iritani~\cite{Iritani2020} -- is believed to involve analytic continuation of Givental cones, and we hope that our formulation here will be a step towards this.
After the first version of this paper appeared on the arXiv, Fenglong You pointed us to the work~\cite{LeeLinWang2017} in which Lee, Lin, and Wang sketch a construction of blow-ups that is very similar to Theorem~\ref{step one}, and use this to compute Gromov--Witten invariants of blow-ups in complete intersections. The methods they use are different: they rely on a very interesting extension of the Quantum Lefschetz theorem to certain non-split bundles, which they will prove in forthcoming work~\cite{LeeLinWangForthcoming}. At first sight, their result~\cite[Theorem 5.1]{LeeLinWang2017} is both more general and less explicit than our results. In fact, we believe neither is true. Their theorem as stated applies to blow-ups in complete intersections defined by arbitrary line bundles whereas we require these line bundles to be convex; however, discussions with the authors suggest that both results apply under the same conditions, and the convexity hypothesis was omitted from~\cite[Theorem~5.1]{LeeLinWang2017} in error. Furthermore, Lee, Lin, and Wang extract genus-zero Gromov--Witten invariants by combining their generalised Quantum Lefschetz theorem with an inexplicit Birkhoff factorisation procedure whereas we use the formalism of Givental cones. We believe, though, that one can rephrase their argument entirely in terms of Givental's formalism, and after doing so their results become explicit in exactly the same range as ours. The explicit formulas are different, however, and it would be interesting to see if one can derive non-trivial identities from this. Note that Proposition~\ref{geometricconstruction} below is more general than the construction in~\cite[Section~5]{LeeLinWang2017}: the fact that we consider Grassmann bundles rather than projective bundles allows us to treat blow-ups in certain degeneracy loci. Combining this with the methods in Section~\ref{examples} allows one to compute genus-zero Gromov--Witten invariants of blow-ups in such degeneracy loci.
One of the most striking features of Givental's formalism is that relationships between higher-genus Gromov--Witten invariants of different spaces can often be expressed as the quantisation, in a precise sense, of the corresponding relationship between the Lagrangian cones that encode genus-zero invariants~\cite{Givental2004}. Our version of the Abelian/non-Abelian Correspondence hints, therefore, at a higher-genus generalisation. It would be very interesting to develop and prove a higher-genus analog of Conjecture~\ref{AnA}.
\section{GIT Quotients and Flag Bundles}
\subsection{The topology of quotients by a non-Abelian group and its maximal torus}\label{topology}
Let $G$ be a complex reductive group acting on a smooth quasi-projective variety $A$ with polarisation given by a linearised ample line bundle $L$. Let $T \subset G$ be a maximal torus. One can then form the GIT-quotients $A /\!\!/ G$ and $A /\!\!/ T$. We will assume that the stable and semistable points with respect to these linearisations coincide, and that all the isotropy groups of the stable points are trivial; this ensures that the quotients $A /\!\!/ G$ and $A /\!\!/ T$ are smooth projective varieties. The Abelian/non-Abelian Correspondence \cite{CFKS2008} relates the genus zero Gromov--Witten invariants of these two quotients. Let $A^{s}(G)$, and respectively $A^s(T)$, denote the subsets of $A$ consisting of points that are stable for the action of $G$, and respectively $T$. The two geometric quotients $A /\!\!/ G$ and $A /\!\!/ T$ fit into a diagram \begin{equation}
\label{vbongit}
\begin{tikzcd}
A /\!\!/ T & A^{s}(G)/T \arrow[d, "q"] \arrow[l, "j"', hook'] \\
& A /\!\!/ G
\end{tikzcd} \end{equation} where $j$ is the natural inclusion and $\pi$ the natural projection.
A representation $\rho \colon G \to \GL(V)$ induces a vector bundle $V(\rho)$ on $A /\!\!/ G$ with fiber $V$. Explicitly, $V(\rho)=(A\times V)/\!\!/ G$ where $G$ acts as \[ g\colon (a,v) \mapsto (ag, \rho(g^{-1}) v). \]
Similarly, the restriction $\rho|_T$ of the representation $\rho$ induces a vector bundle $V(\rho|_T)$ over $A /\!\!/ T$. Note that since $T$ is Abelian, $V(\rho|_T)$ splits as a direct sum of line bundles, $V(\rho|_T)=L_1 \oplus \dots \oplus L_k$ These bundles satisfy \begin{equation}\label{(21)}
j^*V({\rho |_T}) \cong q^*V(\rho). \end{equation}
When the representation $\rho\colon G \rightarrow \GL(V)$ is clear from context, we will suppress it from the notation, writing $V^G$ for $V(\rho)$ and $V^T$ for $V(\rho|_T)$.
We will now describe the relationship between the cohomology rings of $A /\!\!/ G$ and $A /\!\!/ T$, following \cite{Martin2000}. Let $W$ be the Weyl group of $G$. $W$ acts on $A /\!\!/ T$ and hence on the cohomology ring $H^\bullet(A /\!\!/ T)$. Restricting the adjoint representation $\rho \colon G \to \GL(\mathfrak{g})$ to $T$, we obtain a splitting $\rho|_T=\oplus_{\alpha} \rho_\alpha$ into $1$-dimensional representations, i.e.~characters, of $T$. The set $\Delta$ of characters appearing in this decomposition is the set of roots of $G$, and forms a root system. Write $L_\alpha$ for the line bundle on $A /\!\!/ T$ corresponding to a root $\alpha$. Fix a set of positive roots $\Phi^+$ and define \[
\omega=\prod_{\alpha \in \Phi^+} c_1(L_\alpha). \] \begin{theorem}[Martin]
\label{thm:Martin}
There is a natural ring homomorphism
\[
H^{\bullet}(A /\!\!/ G) \cong\frac{ H^\bullet(A /\!\!/ T)^W}{ \Ann(\omega)}
\]
under which $x \in H^\bullet(A /\!\!/ G)$ maps to $\tilde{x} \in H^\bullet(A /\!\!/ T)$ if and only if $q^*x=j^*\tilde{x}$. \end{theorem} \noindent Theorem~\ref{thm:Martin} shows that any cohomology class $\tilde{x}\in H^\bullet(A /\!\!/ T)^W$ is a lift of a class ${x} \in H^\bullet(A /\!\!/ G)$, with $\tilde{x}$ unique up to an element of $\mathrm{Ann}(\omega)$.
\begin{assumption} Throughout this paper, we will assume that the $G$-unstable locus $A \setminus A^s(G)$ has codimension at least $2$. \end{assumption}
This implies that elements of $H^2(A /\!\!/ G)$ can be lifted uniquely: \begin{proposition}\label{maponmori} Pullback via $q$ gives an isomorphism $H^2(A /\!\!/ G) \cong H^2(A /\!\!/ T)^W$, and induces a map $\varrho \colon \NE(A /\!\!/ T) \rightarrow \NE (A /\!\!/ G)$ where $\NE$ denotes the Mori cone. \end{proposition} \begin{proof} The assumption that $A \setminus A^s(G)$ has codimension at least $2$ implies that $A^s(T)/T \setminus A^s(G)/T$ has codimension at least $2$, so $j$ induces an isomorphism $\Pic(A^s(G)/T) \cong \Pic(A^s(T)/T)$. This gives an isomorphism $H^2(A^s(G)/T) \cong H^2(A^s(T)/T)$ since the cycle class map is an isomorphism for both spaces. Since $q^*$ always induces an isomorphism between $H^2(A /\!\!/ G)$ and $H^2(A^s(G)/T)^W$ \cite{Borel1953}, the first claim follows. Consequently, the lifting of divisor classes is unique and can be identified with the pullback map $q^* \colon \Pic(A /\!\!/ G) \rightarrow \Pic(A^s(G)/T)$. Since the pullback of a nef divisor class along a proper map is nef, we obtain by duality a map $\varrho: \NE(A/\!\!/ T) \rightarrow \NE(A /\!\!/ G)$. \end{proof} \begin{definition} We say that $\tilde{\beta} \in \NE(A/\!\!/ T)$ lifts $\beta \in \NE(A /\!\!/ G)$ if $\varrho(\tilde{\beta}) = \beta$. Note that any effective $\beta$ has finitely many lifts. \end{definition} \subsection{Partial flag varieties and partial flag bundles}\label{flag} \subsubsection{Notation}\label{notation} We will now specialise to the case of flag bundles and introduce notation used in the rest of the paper. Fix once and for all: \label{setup} \begin{itemize} \item a positive integer $n$ and a sequence of positive integers $r_1 < \dots < r_{\ell} < r_{\ell+1}=n$; \item a vector bundle $E \rightarrow X$ of rank $n$ on a smooth projective variety $X$ which splits as a direct sum of line bundles $E=L_1 \oplus \dots \oplus L_n$. \end{itemize} We write $\Fl$ for the partial flag manifold $\Fl(r_1, \dots, r_{\ell};n)$, and $\Fl(E)$ for the partial flag bundle $\Fl(r_1, \dots, r_{\ell};E)$.
Set $N=\sum_{i=1}^\ell r_i r_{i+1}$ and $R=r_1+\dots+r_\ell$ It will be convenient to use the indexing $\{(1,1), \dots (1, r_1), (2, 1), \dots, (\ell, r_\ell)\}$ for the set of positive integers smaller or equal than $R$.
\subsubsection{Partial flag varieties and partial flag bundles as GIT quotients} The partial flag manifold $\Fl$ arises as a GIT quotient, as follows. Consider $\CC^N$ as the space of homomorphisms \begin{equation}
\label{eq:hom}
\bigoplus_{i=1}^\ell \Hom\left(\CC^{r_{i}}, \CC^{r_{i+1}}\right). \end{equation} The group $G = \prod_{i=1}^{\ell} \mathrm{GL}_{r_i}(\CC)$ acts on $\CC^N$ by \[
(g_1, \dots, g_{\ell}) \cdot (A_1, \dots, A_{\ell}) = (g_2^{-1} A_1 g_1, \dots,g_{\ell}^{-1}A_{\ell -1}g_{\ell-1}, A_{\ell}g_\ell). \] Let $\rho_i \colon G \rightarrow \GL_{r_i}(\CC)$ be the representation which is the identity on the $i$th factor and trivial on all other factors. Choosing the linearisation $\chi=\bigotimes_{i=1}^{\ell} \det(\rho_i)$, we have that $\CC^N /\!\!/_\chi G$ is the partial flag manifold $\Fl$. More generally, the partial flag bundle also arises as a GIT quotient, of the total space of the bundle of homomorphisms \begin{equation}
\label{eq:hom_bundle}
\bigoplus_{i=1}^{\ell-1} \Hom\left(\cO^{\oplus r_{i}}, \cO^{\oplus r_{i+1}} \right)
\oplus \Hom \left(\cO^{\oplus r_{\ell}},E \right) \end{equation} with respect to the same group $G$ and the same linearisation. $\Fl(E)$ carries $\ell$ tautological bundles of ranks $r_1, \dots, r_{\ell}$, which we will denote $S_1, \dots, S_{\ell}$. These bundles restrict to the usual tautological bundles on $\Fl$ on each fibre. The bundle $S_i$ is induced by the representation~$\rho_i$. \begin{definition} Let \[
p_i(t)=t^{r_i}-c_1(S_i)t^{r_i-1}+\dots + (-1)^{r_i} c_{r_i}(S_i) \] be the Chern polynomial of $S_i^\vee$. We denote the roots of $p_i$ by $H_{i,j}$,~$1 \leq j \leq r_i$. The $H_{i,j}$ are in general only defined over an appropriate ring extension of $H^\bullet(\Fl(E), \CC)$, but symmetric polynomials in the $H_{i,j}$ give well-defined elements of $H^\bullet(\Fl(E), \CC)$. \end{definition} The maximal torus $T\subset G$ is isomorphic to $(\CC^\times)^{R}$. The corresponding Abelian quotient \[
\Fl(E)_T \coloneqq \Hom\big(\cdots\big) /\!\!/_\chi (\CC^\times)^{R}, \] where $\Hom\big(\cdots\big)$ is the bundle of homomorphisms \eqref{eq:hom_bundle}, is a fibre bundle over $X$ with general fibre isomorphic to the toric variety $\Fl_T:= \CC^N /\!\!/_\chi (\CC^\times)^R$. The space $\Fl(E)_T$ also carries natural cohomology classes: \begin{definition} Let $\rho_{i,j}\colon (\CC^\times)^{R} \rightarrow \GL_1(\CC)$ be the dual of the one-dimensional representation of $(\CC^\times)^{R}$ given by projection to the $(i,j)$th factor $\CC^{\times} = \GL_1(\CC)$; here we use the indexing of the set $\{1,2,\ldots,R\}$ specified in \S\ref{notation}. We define ${L}_{i,j} \in H^2(\Fl_T,\CC)$ to be the line bundle on $\Fl(E)_T$ induced by $\rho_{i,j}$ and denote its first Chern class by $\tilde{H}_{i,j}$. Similarly, we define $h_{i,j}$ to be the first Chern class of the line bundle on $\Fl_T$ induced by the represenation $\rho_{i,j}$. Equivalently, $h_{i,j}$ is the restriction of $\tilde{H}_{i,j}$ to a general fibre $\Fl_T$ of $\Fl(E)_T$. \end{definition} Recall that, for a representation $\rho$ of $G$, the corresponding vector bundle $V^T$ splits as a direct sum of line bundles $F_1 \oplus \cdots \oplus F_k$. It is a general fact that if $f$ is a symmetric polynomial in the $c_1(F_i)$, then $f$ can be written as a polynomial in the elementary symmetric polynomials $e_r(c_1(F_1), \dots, c_1(F_k))$, that is, in the Chern classes $c_r(V^T)$. By \eqref{(21)} we have that $j^*c_r(V^T)=q^*c_r(V^G)$, and so replacing any occurrence of $c_r(V^T)$ by $c_r(V^G)$ gives an expression $g \in H^\bullet(A /\!\!/ G)$ which satisfies $q^*g=j^*f$. That is, $f$ is a lift of $g$. Applying this to the dual of the standard representation $\rho_i$ of the $i$th factor of $G$ shows that any polynomial $p$ which is symmetric in each of the sets $\tilde{H}_{i,j}$ for fixed $i$ projects to the same expression in $H^\bullet(\Fl(E))$ with any occurrence of $\tilde{H}_{i,j}$ replaced by the corresponding Chern root $H_{i,j}$.
\begin{lemma}\label{torus_invariant_divisors} Let $(\CC^\times)^R$ act on $\CC^N$, arrange the weights for this action in an $R \times N$-matrix~$(m_{i,k})$ and consider $E=L_1 \oplus \dots \oplus L_N \xrightarrow{\pi} X$ a direct sum of line bundles. Form the associated toric fibration $E /\!\!/ (\CC^\times)^R$ with general fibre $\CC^N /\!\!/ (\CC^\times)^R$ and let $h_i$ (respectively $H_i$) be the first Chern class of the line bundle on $\CC^N /\!\!/ (\CC^\times)^R$ (respectively on $E /\!\!/ (\CC^\times)^R$ induced by the dual of the representation which is standard on the $i$th factor of $(\CC^\times)^R$ and trivial on the other factors. Then \begin{itemize} \item the Poincar\'e duals $u_k$ of the torus invariant divisors of the toric variety $\CC^N /\!\!/ (\CC^\times)^R$ are: $$u_k=\sum_{k=1}^Rm_{i,k}h_i$$ \item
the Poincar\'e duals $U_k$ of the torus invariant divisors of the total space of the toric fibration $E /\!\!/ (\CC^\times)^R\xrightarrow{\pi} X $ are: $$U_k=\sum_{k=1}^Rm_{i,k}H_i+\pi^*c_1(L_k)$$ \end{itemize} \end{lemma}
When applying Lemma~\ref{torus_invariant_divisors} to our situation \eqref{eq:hom_bundle} it will be convenient to define $H_{\ell+1, j}:=\pi^*c_1(L_j^\vee)$. Then the set of torus invariant divisors is \begin{align*} H_{i,j} - H_{i+1,j'} && 1 \leq i \leq \ell, \, 1 \leq j \leq r_{i}, \, 1 \leq j' \leq r_{i+1} \end{align*}
We will also need to know about the ample cone of a toric variety $\CC^N /\!\!/ (\CC^\times)^R$. This is most easily described in terms of the secondary fan, that is, by the wall-and-chamber decomposition of $\Pic(\CC^N /\!\!/ (\CC^\times)^R) \otimes \RR \cong \RR^R$ given by the cones spanned by size $R-1$ subsets of columns of the weight matrix. The ample cone of $\CC^N /\!\!/ (\CC^\times)^R$ is then the chamber that contains the stability condition $\chi$. Moreover, for a subset $\alpha \subset \{1, \dots, N\}$ of size $R$ the cone in the secondary fan spanned by the classes $u_k$, $k\in \alpha$, contains the stability condition (and therefore also the ample cone) iff the intersection $u_\alpha=\bigcap_{k \notin \alpha} u_k$ is nonempty. In this case, $U_\alpha=\bigcap_{k \notin \alpha} U_k$ restricts to a torus fixed point on every fibre and, since $E$ splits as a direct sum of line bundles, $U_\alpha$ is the image of a section of the toric fibration $\pi$. We denote this section by $s_\alpha$. By construction, the torus invariant divisors $U_{k}$,~$k \in \alpha$, do not meet $U_\alpha$, so that $s_\alpha^*(U_k)=0$ for all $k \in \alpha$. For the toric variety $\Fl_T$ one can easily write down the set of $R$-dimensional cones containing $\chi=(1, \dots, 1)$. For each index $(i,j)$, choose some $j' \in \{1, \dots, r_{\ell+1}\}$. Then the cone spanned by \begin{align}\label{ampleconegen}
h_{i,j}-h_{i+1,j'} && 1 \leq i < \ell-1, \, 1 \leq j \leq r_i && h_{\ell, j}, \, 1 \leq j \leq r_\ell \end{align} contains $\chi$ and every cone containing $\chi$ is of that form.
\section{Givental's Formalism} \label{givental formalism}
In this section we review Givental's geometric formalism for Gromov--Witten theory, concentrating on the genus-zero case. The main reference for this is \cite{Givental2004}. Let $Y$ be a smooth projective variety and consider $$\cH_Y=H^\bullet(Y, \Lambda)[z, z^{-1}] \! ]=\Big\{ \sum_{k= -\infty}^{m} a_i z^i \colon \text{$a_i \in H^{\bullet}(Y,\Lambda)$, $m \in \ZZ$}\Big\}$$ where $z$ is an indeterminate and $\Lambda$ is the Novikov ring for $Y$. After picking a basis $\{\phi_1, \dots, \phi_N\}$ for $H^\bullet(Y;\CC)$ with $\phi_1 = 1$ and writing $\{\phi^1, \dots, \phi^N\}$ for the Poincare dual basis, we can write elements of $\cH_Y$ as \begin{align} \sum_{i=0}^m \sum_{\alpha=1}^N q_i^{\alpha}\phi_{\alpha}z^i+ \sum_{i=0}^\infty \sum_{\alpha=1}^N p_{i,\alpha}\phi^{\alpha}(-z)^{-1-i} \label{eq11} \end{align} where $q_i^{\alpha}$,~$p_{i,\alpha} \in \Lambda$. The $q_i^{\alpha}$,~$p_{i,\alpha}$ then provide coordinates on $\cH_Y$. The space $\cH_Y$ carries a symplectic form \begin{align*} \Omega\colon \cH_Y \otimes \cH_Y &\rightarrow \Lambda\\ f \otimes g &\rightarrow \text{Res}_{z=0}(f(-z), g(z)) \, dz \end{align*} where $(\cdot , \cdot )$ denotes the Poincar\'e pairing, extended $\CC[z, z^{-1}]\!]$-linearly to $\cH_Y$. By construction, $\Omega$ is in Darboux form with respect to our coordinates: \[
\Omega=\sum_i \sum_\alpha dp_{i,\alpha} \wedge dq_i^{\alpha} \] We fix a Lagrangian polarisation of $\cH$ as $\cH_Y=\cH_+ \oplus \cH_-$, where $$\cH_+=H^\bullet(Y;\Lambda)[z], \quad \cH_-=z^{-1}H^\bullet(Y;\Lambda)[\![z^{-1}]\!]$$ This polarisation $\cH_Y = \cH_+ \oplus \cH_-$ identifies $\cH_Y$ with $T^* \cH_+$. We now relate this to Gromov--Witten theory.
\begin{definition} The \emph{genus-zero descendant potential} is a generating function for genus-zero Gromov--Witten invariants:
$$\mathcal{F}_{Y}^{0} = \sum_{n = 0}^\infty \sum_{d \in \NE(Y)} \frac{Q^d}{n!} t^{\alpha_1}_{i_1} \dots t^{\alpha_{n}}_{i_n}\langle \phi_{\alpha_1}\psi^{i_1}, \dots, \phi_{\alpha_n}\psi^{i_n} \rangle_{0,n,d}$$ Here $t_i^\alpha$ is a formal variable, $\NE(Y)$ denotes the Mori cone of $Y$, and Einstein summation is used for repeated lower and upper indices. \end{definition}
\noindent After setting \begin{equation}
\label{eq:dilaton}
t^{\alpha}_{i} = q^{\alpha}_{i} + \delta^{i}_{1}\delta^{1}_{\alpha}, \end{equation} where $\delta_i^j$ denotes the Kronecker delta, we obtain a (formal germ of a) function $\mathcal{F}^{0}_Y \colon \cH_+ \rightarrow \Lambda$. \begin{definition} The Givental cone $\cL_Y$ of $Y$ is the graph of the differential of $\mathcal{F}_{Y}^{0}\colon \cH_+ \rightarrow \Lambda$:
$$\cL_Y = \left\{(\mathbf{q,p}) \in T^*\cH_Y= \cH_+ \oplus \cH_- \colon p_{i, \alpha} = \frac{\partial \mathcal{F}^0_{Y}}{\partial q^{\alpha}_i}\right\}$$
Note that $\cL_Y$ is Lagrangian by virtue of being the graph of the differential of a function. Moreover, it has the following special geometric properties \cite{Givental2004, CCIT2009Computing, CoatesGivental2007}
\begin{itemize}
\item $\cL$ is preserved by scalar multiplication, i.e. it is (the formal germ of) a cone
\item the tangent space $T_f$ of $\cL_Y$ at $f \in \cL_Y$ is tangent to $\cL$ exactly along $z T_f$. This means:
\begin{enumerate}
\item $zT_f \subset \cL_Y$
\item for $g \in zT_f$, we have $T_g = T_f$
\item $T_f \cap \cL_Y = z T_f$
\end{enumerate}
\end{itemize} \end{definition}\label{J} A general point of $\cL_Y$ can be written, in view of the dilaton shift \eqref{eq:dilaton}, as \begin{align*}
&{-z} + \sum_{i = 0}^\infty t^{\alpha}_i \phi_{\alpha}z^i + \sum_{n = 0}^\infty \sum_{d \in \NE(Y)} \frac{Q^d}{n!} t^{\alpha_1}_{i_1} \dots t^{\alpha_{n}}_{i_n}\langle \phi_{\alpha_1}\psi^{i_1}, \dots, \phi_{\alpha_n}\psi^{i_n}, \phi_{\alpha}\psi^{i} \rangle_{0,n+1,d} \phi^{\alpha}(-z)^{-i-1} \\
= &{-z} + \sum_{i = 0}^\infty t^{\alpha}_i \phi_{\alpha}z^i + \sum_{n = 0}^\infty \sum_{d \in \NE(Y)} \frac{Q^d}{n!} t^{\alpha_1}_{i_1} \dots t^{\alpha_{n}}_{i_n}\langle \phi_{\alpha_1}\psi^{i_1}, \dots, \phi_{\alpha_n}\psi^{i_n}, \frac{\phi_{\alpha}}{-z - \psi} \rangle_{0,n+1,d} \phi^{\alpha} \end{align*} Thus knowing $\cL_Y$ is equivalent to knowing all genus-zero Gromov--Witten invariants of $Y$. Setting $t_k^\alpha=0$ for all $k>0$, we obtain the \emph{$J$-function} of $ Y$: \begin{equation*}
J(\tau,-z) = -z + \tau + \sum_{n = 0}^\infty \sum_{d \in \mathrm{\NE(X)}} \frac{Q^d}{n!} \left\langle \tau, \dots \tau, \frac{\phi_{\alpha}}{-z - \psi} \right\rangle_{0,n+1,d} \phi^{\alpha} \end{equation*} where $\tau = t^1_0 \phi_1 + \dots t^N_0 \phi_N \in H^\bullet(Y)$. The $J$-function is the unique family of elements $\tau \mapsto J(\tau,-z)$ on the Lagrangian cone such that \begin{equation*}
J(\tau, -z) = -z + \tau + O(z^{-1}). \end{equation*}
We will need a generalisation of all of this to twisted Gromov--Witten invariants~\cite{CoatesGivental2007}. Let~$F$ be a vector bundle on $Y$ and consider the universal family over the moduli space of stable maps $$\begin{tikzcd} {C_{0,n,d}} \arrow[d, "\pi"'] \arrow[r, "f"] & Y \\ {Y_{0,n,d}} & \end{tikzcd}$$ Let $\pi_!$ be the pushforward in $K$-theory. We define $$F_{0,n,d} = \pi_{!}f^*F=R^{0}\pi_{*}f^*F-R^{1}\pi_{*}f^*F$$ (the higher derived functors vanish). In general $F_{0,n,d}$ is a class in $K$-theory and not an honest vector bundle. This means that in order to evaluate a characteristic class $\mathbf{c}(\cdot)$ on $F_{0,n,d}$ we need $\mathbf{c}(\cdot)$ to be \emph{multiplicative} and \emph{invertible}. We can then set \[
\mathbf{c}(F_{0,n,d}) = \mathbf{c}(R^{0}\pi_{*}f^* F) \cup \mathbf{c}(R^{1}\pi_{*}f^* F)^{-1} \] where $\mathbf{c}(R^{i}\pi_{*}f^* F)$ is defined using an appropriate locally free resolution. \begin{definition} \label{equivarianteuler} Let $F$ be a vector bundle on $Y$ and let $\mathbf{c}(\cdot)$ be an invertible multiplicative characteristic class. We will refer to the pair $(F, {\bf c})$ as twisting data. Define $(F, {\bf c})$-twisted Gromov--Witten invariants as
\begin{equation*}
\langle \alpha_1 \psi_1^{i_1}, \dots \alpha_n \psi_{n}^{i_n} \rangle_{0,n,d}^{F, {\bf c}} = \int_{[Y_{0,n,d}]^{\mathrm{vir}} \cap \mathbf{c}(F_{0,n,d})} \ev_1^{*}\alpha_1 \cup \dots \cup \ev_n^{*}\alpha_n \cup \psi_1^{i_1} \cup \dots \cup \psi_{n}^{i_n}
\end{equation*} \end{definition} Any multiplicative invertible characteristic class can be written as $\mathbf{c}(\cdot ) = \exp(\sum_{k \geq 0} s_k \ch_k(\cdot))$, where $\ch_k$ is the $k$th component of the Chern character and $s_0$,~$s_1$,~\ldots are appropriate coefficients. So we work with cohomology groups $H^{\bullet}(X, \Lambda_s)$, where $\Lambda_s$ is the completion of $\Lambda[s_0, s_1, \dots]$ with respect to the valuation \begin{equation*}
v(Q^d) = \big\langle c_1(\cO(1)), d \big \rangle, \quad v(s_k) = k+1. \end{equation*} Most of the definitions from before now carry over. We have the twisted Poincar\'e pairing $(\alpha,\beta)^{F, {\bf c}} = \int_Y \mathbf{c}(F) \cup \alpha \cup \beta $ which defines the basis $\phi^1, \dots \phi^N$ dual to our chosen basis $1 =\phi_1, \dots, \phi_N$ for $H^\bullet(Y)$. The Givental space becomes $\cH_Y = H^{\bullet}(Y,\Lambda_s) \, \otimes \, \CC[z,z^{-1}]\!]$ with the twisted symplectic form $$\Omega^{F, {\bf c}}(f(z), g(z)) = \mathrm{Res}_{z=0}\big(f(-z),g(z)\big)^{F, {\bf c}}dz.$$ This form admits Darboux coordinates as before which give a Lagrangian polarisation of $\cH_Y$. Then the twisted Lagrangian cone $\cL_{F, {\bf c}}$ is defined, via the dilaton shift \eqref{eq:dilaton}, as the graph of the differential of the generating function $\mathcal{F}^{0,F, {\bf c}}_Y$ for genus zero \textit{twisted} Gromov--Witten invariants. Finally, just as before, we can define a twisted $J$-function: \begin{definition} \label{twisted J} Given twisting data $(F, {\bf c})$ for $Y$, the twisted $J$-function is: \begin{equation*}
J_{F, {\bf c}}(\tau,{-z}) = {-z} + \tau + \sum_{n = 0}^\infty \sum_{d \in \NE(Y)} \frac{Q^d}{n!} \left\langle \tau, \dots \tau, \frac{\phi_{\alpha}}{-z - \psi} \right\rangle^{F, {\bf c}}_{0,n+1,d} \phi^{\alpha} \end{equation*} \end{definition} \noindent This is once again characterised as the unique family $\tau \mapsto J_{F, {\bf c}}(\tau,-z)$ of elements of the twisted Lagrangian cone of the form \begin{equation*}
J_{F, {\bf c}}(\tau, -z) = -z + \tau + O(z^{-1}) \end{equation*} Note that we can recover the untwisted theory by setting $\mathbf{c}=1$.
In what follows we take $\mathbf{c}$ to be the $\CC^\times$-equivariant Euler class \eqref{intro equivariant Euler}, which is multiplicative and invertible. The $\CC^\times$-action here is the canonical $\CC^\times$-action on any vector bundle given by rescaling the fibres. We write $F_\lambda$ for the twisting data $(F, \mathbf{c})$, where $F$ is equipped with the $\CC^\times$-action given by rescaling the fibres with equivariant parameter $\lambda$. In this setting, Gromov--Witten invariants (and the coefficients $s_k$) take values in the fraction field $\CC(\lambda)$ of the $\CC^\times$-equivariant cohomology of a point. Here $\lambda$ is the hyperplane class on $\mathbb{CP}^{\infty}$, so that $H^{\bullet}_{\CC^\times}(\{\mathrm{pt}\}) = \CC[\lambda ]$, and we work over the field $\CC(\lambda)$.
\begin{remark}
As we have set things up, the twisted cone $\cL_{F_\lambda}$ is a Lagrangian submanifold of the symplectic vector space $\big(\cH_Y, \Omega^{F_\lambda}\big)$, so as $\lambda$ varies both the Lagrangian submanifold and the ambient symplectic space change. To obtain the picture described in the Introduction, where all the Lagrangian submanifolds $\cL_{F_\lambda}$ lie in a single symplectic vector space $\big(\cH_Y, \Omega \big)$, one can identify $\big(\cH_Y, \Omega \big)$ with $\big(\cH_Y, \Omega^{F_\lambda} \big)$ by multiplication by the square root of the equivariant Euler class of $F$. See~\cite[\S8]{CoatesGivental2007} for details. \end{remark}
\subsection{Twisting the $I$-function}\label{I-functions} We will now prove a general result following an argument from \cite{CCIT2009Computing}. We say that a family $\tau \mapsto I(\tau)$ of elements of $\cH_Y$ \emph{satisfies the Divisor Equation} if the parameter domain for $\tau$ is a product $U \times H^2(Y)$ and $I(\tau)$ takes the form $$ I(\tau) = \sum_{\beta \in \NE(Y)}Q^{\beta} I_{\beta}(\tau,z) $$ where \begin{align}\label{divisor equation}
z\nabla_{\rho}I_{\beta}= \big(\rho + \langle \rho,\beta \rangle z\big) I_{\beta}
&&
\text{for all $\rho \in H^2(Y)$.} \end{align} Here $\nabla_\rho$ is the directional derivative along $\rho$. Let $F'$ be a vector bundle on $Y$, and consider any family $\tau \mapsto I(\tau) \in \cL_{F'_\mu}$ that satisfies the Divisor Equation. Given another vector bundle $F$ which splits as a direct sum of line bundles $F=F_1 \oplus \dots \oplus F_k$, we explain how to modify the family $\tau \mapsto I(\tau)$ by introducing explicit hypergeometric factors that depend on $F$. We prove that (1) this modified family can be written in terms of the {\it Quantum Riemann-Roch operator} and the original family; and (2) the modified family lies on the twisted Lagrangian cone $\cL_{F_\lambda \oplus F'_\mu}$.
\begin{definition} Define the element $G(x,z) \in \cH_Y$ by $$G(x,z) := \sum_{l=0}^\infty \sum_{m=0}^\infty s_{l + m - 1} \frac{B_m}{m!}\frac{x^l}{l!}z^{m-1}$$ where $B_m$ are the Bernoulli numbers and the $s_k$ are the coefficients obtained by writing the $\CC^\times$-equivariant Euler class \eqref{intro equivariant Euler} in the form $\exp\big(\sum_{k \geq 0} s_k \ch_k(\cdot)\big)$. \end{definition}
\begin{remark}
The discussion in this section is valid for any invertible multiplicative characteristic class, not just the equivariant Euler class, but we will neither need nor emphasize this. \end{remark}
\begin{definition}\label{delta} Let $F$ be a vector bundle -- not necessarily split -- and let $f_i$ be the Chern roots of $F$. Define the \textit{Quantum Riemann-Roch operator}, $\Delta_{F_\lambda} \colon \cH_{Y} \rightarrow \cH_{Y}$ as multiplication by
\begin{equation*}
\Delta_{F_\lambda} = \prod_{i=1}^{k} \exp(G(f_i, z))
\end{equation*} \end{definition} \begin{theorem}[\cite{CoatesGivental2007}] \label{QRR} \ $\Delta_{F_\lambda}$ gives a linear symplectomorphism of $(\cH_{Y},\Omega_{Y})$ with $(\cH_{Y}, \Omega_{Y}^{F_\lambda})$ such that $$\Delta_{F_\lambda}(\cL_{Y}) = \cL_{F_\lambda}$$ \end{theorem}
Since $\Delta_{F_\lambda} \circ \Delta_{F'_\mu}=\Delta_{F_\lambda \oplus F'_\mu}$, it follows immediately that $$\Delta_{F_\lambda}(\cL_{F'_\mu}) = \cL_{F_\lambda \oplus F'_\mu}.$$
\begin{lemma} \label{(3)}
Let $F$ be a vector bundle and let $f_1, \ldots, f_k$ be the Chern roots of $F$. Let
$$D_{F_\lambda}=\prod_{i=1}^{k} \exp\big({-G}(z\nabla_{f_i},z)\big)$$
and suppose that $\tau \mapsto I(\tau)$ is a family of elements of $\cL_{F'_\mu}$. Then $\tau \mapsto D_{F_\lambda}(I(\tau))$ is also a family of elements of $\cL_{F'_\mu}$. \end{lemma} \begin{proof}
This follows \cite[Theorem~4.6]{CCIT2009Computing}. Let $h = -z + \sum_{i=0}^m t_i z^i + \sum_{j=0}^{\infty} p_{j}(-z)^{-j-1}$ be a point on $\cH_{Y}$. The Lagrangian cone $\cL_{F'_\mu}$ is defined by the equations $E_j=0$,~$j=0,1,2,\dots$ where $$E_j(h) = p_j - \sum_{n \geq 0} \sum_{d \in \NE(Y)} \frac{Q^d}{n!} t^{\alpha_1}_{i_1} \dots t^{\alpha_{n}}_{i_n}\langle \phi_{\alpha_1}\psi^{i_1}, \dots, \phi_{\alpha_n}\psi^{i_n}, \phi_{\alpha}\psi^{j} \rangle_{0,n+1,d} \phi^{\alpha}$$
We need to show that $E_j(D_{F_\lambda}(I)) = 0$. Note that $D_{F_\lambda}(I) = \prod_{i=1}^{k} \exp(-G(z\nabla_{f_i},z))I$ depends on the parameters $s_i$. For notational simplicity assume that $k=1$, so that $$D_{F_\lambda}(I) = \exp\big({-G}(z\nabla_{f},z)\big)I$$ Set $\deg s_i = i+1$. We will prove the result by inducting on degree. Note that if $s_0 = s_1= \dots = 0$ then $D_{F_\lambda}(I) = I$ so that $E_j(D_{F_\lambda}(I)) = 0$. Assume by induction that $E_j(D_{F_\lambda}(I))$ vanishes up to degree $n$ in the variables $s_0, s_1, s_2, \dots$ Then $$\frac{\partial}{\partial s_i} E_j(D_{F_\lambda}(I)) = d_{D_{F_\lambda}(I)}E_j (z^{-1}P_{i}(z \nabla_{f},z)D_{F_\lambda}(I))$$ where $$P_{i}(z \nabla_{f},z) = \sum_{m=0}^{i+1} \frac{1}{m!(i+1-m)!}z^{m}B_m (z \nabla_{f})^{i+1-m}$$
By induction there exists $D_{F_\lambda}(I)' \in \cL_{F'_\mu}$ such that $$\frac{\partial}{\partial s_i} E_j(D_{F_\lambda}(I)) = d_{D_{F_\lambda}(I)'}E_j (z^{-1}P_{i}(z \nabla_{f},z)D_{F_\lambda}(I)')$$
up to degree $n$. But the right hand side of this expression is zero, since the term in brackets lies in the tangent space to the Lagrangian cone. Indeed, applying $\nabla_f$ to $D_{F_\lambda}(I_{Y})'$ -- or to any family lying on the cone -- takes it to the tangent space of the cone at the point. And then applying $z\nabla_f$ preserves that tangent space. \end{proof} \begin{corollary}\label{Ithm}
Let $\tau \mapsto I(\tau)$ be a family of elements of $\cL_{F'_\mu}$. Then $\tau \mapsto \Delta_{F_\lambda}(D_{F_\lambda}(I(\tau)))$ is a family of elements of $\cL_{F_\lambda \oplus F'_\mu}$. \end{corollary} \begin{proof} This follows immediately by combining \ref{QRR} and \ref{(3)} \end{proof} Corollary \ref{Ithm} produces a family of elements on the twisted Lagrangian cone $\cL_{F_\lambda \oplus F'_\mu}$, but in general it is not obvious whether the nonequivariant limit $\lambda \rightarrow 0$ of this family exists. However, in the case when $F$ is split and $\tau \mapsto I(\tau)$ satisfies the Divisor Equation we will show that the family $\Delta_{F_\lambda}(D_{F_\lambda}(I(\tau, -z)))$ is equal to the \emph{twisted $I$-function} $I_{F'_\mu \oplus {F_\lambda}}$ given in Definition~\ref{twistedI}. This has an explicit expression, which makes it easy to check whether the nonequivariant limit exists. We make the following definitions.
\begin{definition}\label{modification} \label{twistedI}
Let $\tau \mapsto I(\tau)$ be a family of elements of $\cL_{F'_\mu}$. Let $F= F_1 \oplus \dots \oplus F_k$ be a direct sum of line bundles, and let $f_i=c_1(F_i)$. For $\beta \in \NE(Y)$, we define the modification factor
$$M_{\beta}(z) = \prod_{i=1}^{k} \frac{\prod_{m=-\infty}^{\langle f_{i}, \beta \rangle} \lambda + f_{i} + mz }{\prod_{m=-\infty}^{0} \lambda + f_{i} + mz }$$
The associated \textit{twisted $I$-function} is
\begin{equation*}
I^{\text{\rm tw}}(\tau) = \sum_{\beta \in \NE(Y)} Q^{\beta} I_{\beta}(\tau,z) \cdot M_{\beta}(z)
\end{equation*} \end{definition}
To relate $M_{\beta}(z)$ to the Quantum Riemann--Roch operator we will need the following Lemma: \begin{lemma}\label{delta-M}
$$M_{\beta}(-z) = \Delta_{F_\lambda}\left(\prod_{i=1}^{k}\exp(- G(f_{i} - \langle f_{i}, \beta \rangle z, z))\right)$$ \end{lemma} \begin{proof}
Define $$\mathbf{s}(x) = \sum_{k \geq 0} s_k \frac{x^k}{k!}$$
By \cite[equation 13]{CCIT2009Computing} we have that
\begin{equation}\label{gamma}
G(x+z,z) = G(x,z) + \mathbf{s}(x)
\end{equation}
We can rewrite
$$M_{\beta}(z) = \prod_{i=1}^{k} \frac{\prod_{m=-\infty}^{\langle f_{i}, \beta \rangle} \lambda + f_{i} + mz }{\prod_{m=-\infty}^{0} \lambda + f_{i} + mz } = \prod_{i=1}^{k} \frac{\prod_{m=-\infty}^{\langle f_{i}, \beta \rangle} \exp[\mathbf{s}(f_{i} + mz)] }{\prod_{m=-\infty}^{0} \exp[\mathbf{s}(f_{i} + mz)] }$$
and so
\begin{align*}
M_{\beta}(-z) =& \prod_{i=1}^{k}\exp\left(\sum_{m=-\infty}^{\langle f_i, \beta \rangle} \mathbf{s}(f_{i} - mz) - \sum_{m=-\infty}^{0} \mathbf{s}(f_{i} - mz))\right) \\
=& \prod_{i=1}^{k} \exp(G(f_i,z) - G(f_i - \langle f_i, \beta \rangle z, z)
\end{align*}
where for the second equality we used \eqref{gamma}. \end{proof} \begin{proposition}\label{twisted=Deltad} Let $\tau \mapsto I(\tau)$ be a family of elements of $\cL_{F'_\mu}$ that satisfies the Divisor Equation, and let $F=F_1 \oplus \dots \oplus F_k$ be a direct sum of line bundles. Then \begin{equation}\label{twisted=Deltadeq} I^{\text{\rm tw}}=\Delta_{F_\lambda}(D_{F_\lambda}(I)). \end{equation} As a consequence, $\tau \mapsto I^{\text{\rm tw}}(\tau)$ is a family of elements on the cone $\cL_{F_\lambda \oplus F'_\mu}$. \end{proposition} \begin{proof} Lemma \ref{delta-M} shows that \begin{equation}\label{imagebeta} I^{\text{\rm tw}}(\tau) = \Delta_{F_\lambda}\left(\sum_{\beta \in \NE(Y)}\prod_{i=1}^{k} \exp(-G(f_i - \langle f_i, \beta \rangle z, z))I_{\beta}(\tau,z)\right) \end{equation} Applying the Divisor Equation, we can rewrite this as \begin{equation} I^{\text{\rm tw}}=\Delta_{F_\lambda}(D_{F_\lambda}(I)) \end{equation} as required. The rest is immediate from \ref{Ithm}. \end{proof}
\begin{proposition}\label{nonequiexists} If the line bundles $F_i$ are nef, then the nonequivariant limit $\lambda \rightarrow 0$ of $I^{\text{\rm tw}}(\tau)$ exists. \end{proposition}
\begin{proof}
This is immediate from Definition~\ref{twistedI}. \end{proof}
\section{The Givental--Martin cone}\label{giventalmartincones} We now restrict to the situation described in the Introduction, where the action of a reductive Lie group $G$ on a smooth quasiprojective variety $A$ leads to smooth GIT quotients $A /\!\!/ G$ and~$A /\!\!/ T$. As discussed, the roots of $G$ define a vector bundle $\Phi = \oplus_\rho L_\rho \to Y$, where $Y = A /\!\!/ T$, and we consider twisting data $(\Phi, \mathbf{c})$ for $Y$ where $\mathbf{c}$ is the $\CC^\times$-equivariant Euler class. We call the modification factor in this setting the \emph{Weyl modification factor}, and denote it as \begin{equation}\label{modg}
W_{\beta}(z) = \prod_{\alpha} \frac{\prod_{m=-\infty}^{\langle c_{1}(L_{\alpha}), \beta \rangle} c_{1}(L_{\alpha}) + \lambda + mz}{\prod_{m=-\infty}^{0} c_{1}(L_{\alpha}) + \lambda + mz} \end{equation} where the product runs over all roots $\alpha$. For any family $\tau \mapsto I(\tau)= \sum_{\beta \in \NE(Y)}Q^{\beta}I_{\beta}(\tau,z)$ of elements of $\cH_Y$, the corresponding twisted $I$-function is \begin{equation} \label{general Weyl twist}
I^{\text{\rm tw}}(\tau) = \sum_{\beta \in \NE(Y)}Q^{\beta}I_{\beta}(\tau,z) \cdot W_{\beta}(z) \end{equation} Since the roots bundle $\Phi$ is not convex, in general the non-equivariant limit $\lambda \to 0$ of $I^{\text{\rm tw}}$ will not exist. Recall from \eqref{quotientH}, however, the map $p \colon \cH^W_{A /\!\!/ T} \to \cH_{A /\!\!/ G}$. \begin{lemma}\label{IGMexists}
Suppose that $I$ is Weyl-invariant. Then $p \circ I^\text{\rm tw}$ has a well-defined limit as $\lambda \rightarrow 0$. \end{lemma} \begin{proof}
The map $p$ is given by the composition of the map on Novikov rings induced by $$\varrho \colon \NE(A /\!\!/ T) \to \NE(A /\!\!/ G)$$ (see Proposition~\ref{maponmori}) with the projection map $H^\bullet(A /\!\!/ T; \CC)^W \to H^\bullet(A /\!\!/ G; \CC)$ (see Theorem~\ref{thm:Martin}). Since $I(\tau)$ is Weyl-invariant, $I^\text{\rm tw}(\tau)$ is also Weyl invariant and so, after applying $\varrho$, the coefficient of each Novikov term $Q^\beta$ in $\tau \mapsto I^\text{\rm tw}(\tau)$ lies in $H^\bullet(A /\!\!/ T; \CC)^W$. The composition $p \circ I^{\text{\rm tw}}$ is therefore well-defined.
The Weyl modification \eqref{modg} contains many factors
$$
\frac{c_1(L_\alpha) + \lambda + m z}{- c_1(L_\alpha) + \lambda - m z}
$$
which arise by combining the terms involving roots $\alpha$ and $-\alpha$. Such factors have a well-defined limit, $-1$, as $\lambda \to 0$. Therefore the limit of $p \circ I^\text{\rm tw}$ as $\lambda \to 0$ is well-defined if and only if the limit of
\begin{equation} \label{p of I intermediate}
p \left( \sum_{\beta \in \NE(Y)}Q^{\beta}I_{\beta}(\tau,z) \cdot (-1)^{\epsilon(\beta)}\prod_{\alpha \in \Phi^+} \frac{ c_1(L_\alpha) \pm \lambda + \langle c_1(L_\alpha), \beta \rangle z}{c_1(L_\alpha) \mp \lambda} \right)
\end{equation}
as $\lambda \to 0$ is well-defined, and the two limits coincide. Here $\Phi^+$ is the set of positive roots of $G$, and $\epsilon(\beta) = \sum_{\alpha \in \Phi^+} \langle c_1(L_\alpha), \beta \rangle$; cf.~\cite[equation 3.2.1]{CFKS2008}. The limit $\lambda \to 0$ of the denominator terms
$$
\prod_{\alpha \in \Phi^+} \big(c_1(L_\alpha) - \lambda\big)
$$
in \eqref{p of I intermediate} is the fundamental Weyl-anti-invariant class $\omega$ from the discussion before Theorem~\ref{thm:Martin}. Furthermore
$$ \sum_{\beta \in \NE(Y)}Q^{\beta}I_{\beta}(\tau,z) \cdot (-1)^{\epsilon(\beta)}\prod_{\alpha \in \Phi^+} \big(c_1(L_\alpha) + \lambda + \langle c_1(L_\alpha), \beta \rangle z\big)$$
has a well-defined limit as $\lambda \to 0$ which, as it is Weyl-anti-invariant, is divisible by $\omega$. The quotient here is unique up to an element of $\Ann(\omega)$, and therefore the projection of the quotient along Martin's map $H^\bullet(A /\!\!/ T; \CC)^W \to H^\bullet(A /\!\!/ G; \CC)$ is unique. It follows that the limit as $\lambda \to 0$ of $p \circ I^\text{\rm tw}$ is well-defined. \end{proof}
\begin{definition} \label{IGM definition}
Let $\tau \mapsto I(\tau)$ be a Weyl-invariant family of elements of $\cH_Y$ and let $I^{\text{\rm tw}}$ denote the twisted $I$-function as above. We call the nonequivariant limit of $\tau \mapsto p \big(I^{\text{\rm tw}}(\tau)\big)$ the \emph{Givental--Martin modification} of the family $\tau \mapsto I(\tau)$, and denote it by $\tau \mapsto I_{\GM}(\tau)$ \end{definition}
Recall that we have fixed a representation $\rho$ of $G$ on a vector space $V$, and that this induces vector bundles $V^T \to A /\!\!/ T$ and $V^G \to A /\!\!/ G$. Since the bundle $\Phi \to A /\!\!/ T$ is not convex, one cannot expect the non-equivariant limit of $\cL_{ \Phi_\lambda \oplus V^T_\mu}$ to exist. Nonetheless, the projection along \eqref{quotientH} of the Weyl-invariant part of $\cL_{ \Phi_\lambda \oplus V^T_\mu}$ does admit a non-equivariant limit. \begin{theorem}\label{GMlimit}
The non-equivariant limit $\lambda \to 0$ of $p \left(\cL_{\Phi_\lambda \oplus V^T_\mu} \cap \cH^{W}_{A /\!\!/ T} \right)$ exists. \end{theorem} \noindent We call this non-equivariant limit the \emph{twisted Givental--Martin cone} $\cL_{\GM, V^T_\mu} \subset \cH^{W}_{A/\!\!/ T}$.
\begin{proof}[Proof of Theorem~\ref{GMlimit}]
Recall the twisted $J$-function $J_{V^T_\mu}(\tau,{-z})$ from Definition~\ref{twisted J}. By~\cite{CoatesGivental2007} a general point
$$ {-z} + t_0 + t_1 z + \cdots + O(z^{-1}) $$
on $\cL_{V^T_\mu}$ can be written as
$$ J_{V^T_\mu} \big(\tau({\bf t}),{-z}\big) + \sum_{\alpha=1}^N C_\alpha({\bf t}, z) z \frac{\partial J_{V^T_\mu}}{\partial \tau^\alpha}\big(\tau({\bf t}), {-z}\big)
$$
for some coefficients $C_\alpha({\bf t}, z)$ that depend polynomially on $z$ and some $H^\bullet(A /\!\!/ T)$-valued
function $\tau({\bf t})$ of ${\bf t} = (t_0,t_1,\ldots)$. The Weyl modification $\tau \mapsto I^\text{\rm tw}(\tau)$ of $\tau \mapsto J_{V^T_\mu}(\tau,-z)$ satisfies $I^\text{\rm tw}(\tau) \equiv J_{V^T_\mu}(\tau,{-z})$ modulo Novikov variables, and $I^\text{\rm tw}(\tau) \in \cL_{\Phi_\lambda \oplus V^T_\mu}$ by Proposition~\ref{twisted=Deltad}, so a general point
\begin{equation} \label{general point on equivariant Weyl}
{-z} + t_0 + t_1 z + \cdots + O(z^{-1})
\end{equation}
on $\cL_{\Phi_\lambda \oplus V^T_\mu}$ can be written as
$$ I^\text{\rm tw} \big(\tau({\bf t})^\dagger,{-z}\big) + \sum_{\alpha=1}^N C_\alpha({\bf t}, z)^\dagger z \frac{\partial I^\text{\rm tw}}{\partial \tau^\alpha}\big(\tau({\bf t})^\dagger, {-z}\big) $$
for some coefficients $C_\alpha({\bf t}, z)^\dagger$ that depend polynomially on $z$ and some $H^\bullet(A /\!\!/ T)$-valued
function $\tau({\bf t})^\dagger$. Since the twisted $J$-function is Weyl-invariant, so is $I^\text{\rm tw}(\tau)$, and thus if \eqref{general point on equivariant Weyl} is Weyl-invariant then we may take $C_\alpha({\bf t}, z)^\dagger$ to be such that $\sum_\alpha C_\alpha({\bf t}, z)^\dagger \phi_\alpha$ is Weyl-invariant. Projecting along \eqref{quotientH} we see that a general point
\begin{equation} \label{general point on equivariant GM}
{-z} + t_0 + t_1 z + \cdots + O(z^{-1})
\end{equation}
on $p \left(\cL_{\Phi_\lambda \oplus V^T_\mu} \cap \cH^{W}_{A /\!\!/ T} \right)$ can be written as
$$ p \circ I^\text{\rm tw} \big(\tau({\bf t})^\ddagger,{-z}\big) + \sum_{\alpha=1}^N C_\alpha({\bf t}, z)^\ddagger z \frac{\partial (p \circ I^\text{\rm tw})}{\partial \tau^\alpha}\big(\tau({\bf t})^\ddagger, {-z}\big) $$
for some coefficients $C_\alpha({\bf t}, z)^\ddagger$ that depend polynomially on $z$ and some $H^\bullet(A /\!\!/ T)$-valued
function $\tau({\bf t})^\ddagger$. Furthermore, since $p \circ I^\text{\rm tw}(\tau)$ has a well-defined non-equivariant limit $I_{\GM}(\tau)$, we see that $C_{\alpha}({\bf t}, z)^\ddagger$ also admits a non-equivariant limit. Hence a general point \eqref{general point on equivariant GM} on $p \left(\cL_{\Phi_\lambda \oplus V^T_\mu} \cap \cH^{W}_{A /\!\!/ T} \right)$ has a well-defined limit as $\lambda \to 0$. \end{proof}
\begin{corollary} \label{GMlimit no bundle}
The non-equivariant limit $\lambda \to 0$ of $p \left(\cL_{\Phi_\lambda} \cap \cH^{W}_{A /\!\!/ T} \right)$ exists. \end{corollary}
\noindent We call this non-equivariant limit the \emph{Givental--Martin cone} $\cL_{\GM} \subset \cH^{W}_{A/\!\!/ T}$.
\begin{proof}
Take the vector bundle $V^T$ in Theorem~\ref{GMlimit} to have rank zero. \end{proof}
\begin{corollary}\label{IGMonLGM}
If $\tau \mapsto I(\tau)$ is a Weyl-invariant family of elements of $\cL_{V^T_\mu}$ that satisfies the Divisor Equation \eqref{divisor equation} then the Givental--Martin modification $\tau \mapsto I_{\GM}(\tau)$ is a family of elements of~$\cL_{\GM, V^T_\mu}$ \end{corollary} \begin{proof}
Proposition~\ref{twisted=Deltad} implies that $\tau \mapsto I^\text{\rm tw}(\tau, -z)$ is a family of elements on $\cL_{ \Phi_{\lambda} \oplus V^T_\mu}$. Projecting along \eqref{quotientH} and taking the limit $\lambda \rightarrow 0$, which exists by Lemma~\ref{IGMexists}, proves the result. \end{proof}
This completes the results required to state the Abelian/non-Abelian Correspondence (Conjectures~\ref{AnA} and~\ref{AnA family}) and the Abelian/non-Abelian Correspondence with bundles (Conjectures~\ref{AnA bundles} and~\ref{AnA bundles family}).
\section{The Abelian/non-Abelian Correspondence for Flag Bundles}
\subsection{The Work of Brown and Oh}\label{brownohwn} In this section we will review results by Brown~\cite{Brown2014} and Oh~\cite{Oh2016}, and situate their work in terms of the Abelian/non-Abelian Correspondence (Conjecture~\ref{AnA family}). In particular, we show that the Givental--Martin modification of the Brown $I$-function is the Oh $I$-function. We freely use the notation introduced in Section~\ref{notation}.
Let $X$ be a smooth projective variety. We will decompose the $J$-function of $X$, defined in~\S\ref{J}, into contributions from different degrees: \begin{equation} \label{JX by degrees}
J_{X}(\tau,z)= \sum_{D \in \NE(X)} J_X^{D}(\tau,z) Q^{D}. \end{equation} Recall that we have a direct sum of line bundles $E = L_1 \oplus \dots \oplus L_n \xrightarrow{\pi} X$, and that $\Fl(E) = \Fl(r_1, \dots, r_\ell,E) = A /\!\!/ G$ is the partial flag bundle associated to $E$. As in \S\ref{flag}, we form the toric fibration $\Fl(E)_T = A /\!\!/ T$ with general fibre $\CC^N /\!\!/ (\CC^\times)^R$. We denote both projection maps $\Fl(E) \to X$ and $\Fl(E)_T \to X$ by $\pi$. For the sake of clarity, we will denote homology and cohomology classes on $\Fl(E)_T$ with a tilde and classes on $\Fl(E)$ without. Recall the cohomology classes $\tilde{H}_{\ell+1, j}=-\pi^*c_1(L_j)$ on $\Fl(E)_T$, and $H_{\ell + 1,j}=-\pi^*c_1(L_j)$ on $\Fl(E)$. For a fixed homology class $\tilde{\beta}$ on $\Fl(E)_T$ define $d_{\ell +1,j} = \langle -\pi^*c_{1}(L_j), \tilde{\beta} \rangle$, and for a fixed homology class $\beta$ on $\Fl(E)$ define $d_{\ell +1,j} = \langle -\pi^*c_{1}(L_j), \beta \rangle$. We use the indexing of the set $\{1, \dots, R\}$ defined in Section~\ref{notation}, and denote the components of a vector $\underline{d} \in \ZZ^R$ by $d_{i,j}$. Similarly, we denote components of a vector $\underline{d} \in \ZZ^\ell$ by $d_i$.
In \cite{Oh2016}, the author proves that a certain generating function, the $I$-function of $\Fl(E)$, lies on the Lagrangian cone for $\Fl(E)$. \begin{theorem}\label{ohI} Let $\tau \in H^{\bullet}(X)$, $t=\sum_{i}t_{i}c_1(S_i^\vee)$, and define the $I$-function of $\Fl(E)$ to be \begin{multline*}
I_{\Fl(E)}(t, \tau,z) = \\
e^{\frac{t}{z}} \sum_{\beta \in {\NE}(\Fl(E))}
Q^{\beta} e^{\langle \beta, t \rangle} \pi^{*}J_{X}^{\pi_*\beta}(\tau,z)
\sum_{\substack{\underline{d} \in \ZZ^R\colon \\
\forall i \sum_j d_{i,j}=\langle \beta, c_1(S_i^\vee) \rangle}}
\prod_{i=1}^{ \ell}
\prod_{j=1}^{r_{i}}\prod_{j'=1}^{r_{i+1}} \frac{\prod_{m=-\infty}^{0}H_{i,j} - H_{i+1,j'} + mz}{\prod_{m=-\infty}^{d_{i,j} - d_{i+1,j'}}H_{i,j} - H_{i+1,j'} + mz}\\
\times \prod_{i=1}^\ell \prod_{j \neq j'}
\frac{\prod_{m=-\infty}^{d_{i,j} - d_{i,j'}}H_{i,j} - H_{i,j'} + mz}{\prod_{m=-\infty}^{0} H_{i,j} -H_{i,j'} + mz} \end{multline*} Then $I_{\Fl(E)}(t,\tau,-z) \in \cL_{\Fl(E)}$ for all $t$ and $\tau$. \end{theorem} In \cite{Brown2014}, the author proves an analogous result for the corresponding Abelian quotient $\Fl(E)_T$.
\begin{theorem}\label{brown2014gromov} Let $\tau \in H^{\bullet}(X)$, $t=\sum_{i,j}t_{i,j}\tilde{H}_{i,j}$, and define the Brown $I$-function of $\Fl(E)_T$ to be \begin{multline*} I_{\Fl(E)_T}(t, \tau,z) = \\ e^{\frac{t}{z}} \sum_{\tilde{\beta} \in H_2\Fl(E)_T } Q^{\tilde{\beta}} e^{\langle \tilde{\beta}, t \rangle} \pi^{*}J_{X}^{\pi_* \tilde{\beta}}(\tau, z) \prod_{i=1}^{\ell} \prod_{j=1}^{r_i}\prod_{j'=1}^{r_{i+1}}\frac{ \prod_{m=-\infty}^{0}\tilde{H}_{i,j} - \tilde{H}_{i+1,j'} + mz} {\prod_{m=-\infty}^{\langle \tilde{\beta}, \tilde{H}_{i,j} - \tilde{H} _{i+1,j'} \rangle }\tilde{H}_{i,j} - \tilde{H}_{i+1,j'} + mz} \end{multline*} Then $I_{\Fl(E)_T}(t,\tau,-z) \in \cL_{\Fl(E)_T}$ for all $t$ and $\tau$.
\end{theorem} \begin{remark}\label{novi} We have chosen to state Theorem~\ref{brown2014gromov} in a different form than in Brown's original paper. The equivalence of the two versions follows from Lemma~\ref{effsummationrange} below. The classes $H_{i,j}$ here were denoted in~\cite{Brown2014} by $P_i$, and the classes $H_{i,j}-H_{i+1,j'}$ here were denoted there by~$U_k$. \end{remark} \begin{lemma}\label{effsummationrange} Writing $I_{\Fl(E)_T}=\sum_{\tilde{\beta}} I_{\Fl(E)_T}^{\tilde{\beta}} Q^{\tilde{\beta}}$, any nonzero $I^{\tilde{\beta}}$ must have $\tilde{\beta} \in \NE(\Fl(E)_T)$. \end{lemma} \begin{proof}
To see this we temporarily adopt the notation of Brown and denote the torus invariant divisors by $U_k$, as in Lemma~\ref{torus_invariant_divisors}. Then $I_{\Fl(E)_T}$ takes the form \[
I_{\Fl(E)_T}=\sum_{\substack{\tilde{\beta} \in H_2\Fl(E)_T \colon \\ \pi_*\tilde{\beta} \in \NE(X)}}(\dots)\prod_{k=1}^N \frac{\prod_{m=-\infty}^{0}U_k+mz}{\prod_{m=-\infty}^{\langle \tilde{\beta}, U_k \rangle}U_k+mz} \] Let $\alpha \subset \{1, \dots N\}$ be a subset of size $R$ which defines a section of the toric fibration as in Section~\ref{flag}. We have that \[
s_\alpha^*I_{\Fl(E)_T}=(\dots)
\prod_{k \in \alpha} \frac{\prod_{m=-\infty}^{0}(0) + mz}{\prod_{m=-\infty}^{\langle \tilde{\beta}, U_k \rangle}(0)+ mz}\prod_{k \notin \alpha} \frac{\prod_{m=-\infty}^{0}s_\alpha^*U_k+ mz}{\prod_{m=-\infty}^{\langle \tilde{\beta}, U_k \rangle}s_\alpha^*U_k+ mz} \] since $s_\alpha^*(U_k)=0$ if $k \in \alpha$. Therefore, if $\langle \tilde{\beta}, U_k \rangle < 0$ for some $k \in \alpha$, the numerator contains a term $(0)$ and vanishes. We conclude that any $\tilde{\beta} \in H_2\Fl(E)_T$ which gives a nonzero contribution to $s_\alpha^*I_{\Fl(E)_T}$ must satisfy the conditions \[
\pi_*\tilde{\beta} \in \NE(X), \langle \tilde{\beta}, U_k \rangle \geq 0 \, \forall k \in \alpha. \] The section $s_\alpha$ gives a splitting $H_2(\Fl(E)_T)=H_2(X)\oplus H_2(\Fl_T)$, via which we may write $\tilde{\beta}=s_{\alpha_*}D+\iota_*d$ where $\iota$ is the inclusion of a fibre. We have \[
\langle \tilde{\beta}, U_k \rangle= \langle D, s_\alpha^*U_k \rangle+ \langle d, \iota^*U_k \rangle=\langle d, \iota^*U_k \rangle
\geq 0 \] for all $k \in \alpha$. However, the cone in the secondary fan spanned by the line bundles $\iota^*U_k$ contains the ample cone of $\Fl_T$ (see Section \ref{flag}), so this implies $d \in \NE(\Fl_T)$. It follows that any $\tilde{\beta}$ which gives a nonzero contribution to $s_\alpha^*I_{\Fl(E)_T}$ is effective. We now use the Atiyah-Bott localization formula \[
I_{\Fl(E)_T}=\sum_{\alpha} s_{\alpha_*}\left(\frac{s_\alpha^*I_{\Fl(E)_T}}{e^\alpha}\right), \quad \text{where} \; e^\alpha=\prod_{k \notin \alpha} s_\alpha^*U_k \] where $\alpha$ ranges over the torus fixed point sections of the fibration, to conclude that the same is true for $I_{\Fl(E)_T}$. \end{proof}
\begin{lemma}\label{Brown I satisfies Divisor Equation} Brown's $I$-function satisfies the Divisor Equation. That is, $$z\nabla_{\rho}I_{\Fl(E)_T}^{\tilde{\beta}} = (\rho + \langle \rho,\tilde{\beta} \rangle z ) I_{\Fl(E)_T}^{\tilde{\beta}}$$ for any $\rho \in H^2(\Fl(E)_T)$. \end{lemma} \begin{proof} Decompose $\rho=\rho_F + \pi^*\rho_B$ into fibre and base part.
Basic differentiation and the divisor equation for $J_X$ show that
$$ z\nabla_{\rho}I_{\Fl(E)_T}^{\tilde{\beta}} = \, \left(\rho_F + \langle \rho_F, \tilde{\beta} \rangle z+(\pi^*\rho_{B} + \langle \pi^*\rho_B, \tilde{\beta}\rangle z)\right) e^{t/z} e^{\langle \tilde{\beta}, t \rangle} \pi^{*}J_{X}^{\pi_{*}\tilde{\beta}}(\tau,z) \cdot \mathbf{H} $$
where $\mathbf{H}$ is a hypergeometric factor with no dependence on $t$ or $\tau$. The right-hand simplifies to
$$(\rho+ \langle \rho, \tilde{\beta} \rangle z)I_{\Fl(E)_T}^{\tilde{\beta}}$$ as required. \end{proof} \begin{lemma}\label{weylinvariant}
If we restrict $t$ to lie in the Weyl-invariant locus $H^2(\Fl(E)_T)^W \subset H^2(\Fl(E)_T)$ then $(t,\tau) \mapsto I_{\Fl(E)_T}(t,\tau,z)$ takes values in $H^\bullet(\Fl(E)_T)^W$. \end{lemma} \begin{proof}
This is immediate from the definition of $I_{\Fl(E)_T}(t,\tau,z)$, in Theorem~\ref{brown2014gromov}. \end{proof}
\begin{proposition}\label{brownoh}
Restrict $t$ to lie in the Weyl-invariant locus $H^2(\Fl(E)_T)^W \subset H^2(\Fl(E)_T)$ and consider the Brown $I$-function $(t, \tau) \mapsto I_{\Fl(E)_T}(t, \tau, z)$. The Givental--Martin modification $I_{\GM}(t, \tau)$ of this family is equal to Oh's $I$-function $I_{\Fl(E)}(t, \tau)$.
\end{proposition}
\begin{proof}
Lemma~\ref{weylinvariant} and Lemma~\ref{IGMexists} imply that the Givental--Martin modification $I_{\GM}(t, \tau)$ exists. We need to compute it. Note that the restrictions to the fibre of the classes $\tilde{H}_{i,j}$ form a basis for~$H^{2}(\Fl_T)$. Since the general fibre $\Fl_T$ of $\Fl(E)_T$ has vanishing first homology, the Leray--Hirsch theorem gives an identification $\QQ[H_2(\Fl(E)_T, \ZZ)]=\QQ[H_2(X,\ZZ)][q_{1,1}, \dots, q_{\ell, r_\ell}]$ via the map
\begin{equation}\label{noviab}
Q^{\tilde{\beta}} \mapsto Q^{\pi_*{\tilde{\beta}}}\prod_{i,j}q_{i,j}^{\langle \tilde{H}_{i,j}, \tilde{\beta} \rangle}
\end{equation}
By Lemma~\ref{effsummationrange}, the summation range in the sum defining $I_{\Fl(E)_T}$ is contained in $\NE(\Fl(E)_T)$. We can therefore write the corresponding twisted $I$-function \eqref{general Weyl twist} as
\begin{align*}
I^{\text{\rm tw}}(t, \tau,z)= e^{\frac{t}{z}} \sum_{\substack{D \in \NE(X) \\ \underline{d} \in \ZZ^R}} Q^{D} \prod_{i,j}q_{i,j}^{d_{i,j}}e^{t \cdot \underline{d}} \pi^{*}J_{X}^{D}(\tau, z)
\prod_{i=1}^{\ell} \prod_{j=1}^{r_i}\prod_{j'=1}^{r_{i+1}}\frac{ \prod_{m=-\infty}^{0}\tilde{H}_{i,j} - \tilde{H}_{i+1,j'} + mz}
{\prod_{m=-\infty}^{d_{i,j} - d_{i+1,j'}}\tilde{H}_{i,j} - \tilde{H}_{i+1,j'} + mz} \\
\times \prod_{i=1}^{\ell} \prod_{j \neq j'} \frac{\prod_{m=-\infty}^{d_{i,j} - d_{i,j'}}\tilde{H}_{i,j} - \tilde{H}_{i,j'} + \lambda + mz}{\prod_{m=-\infty}^{0} \tilde{H}_{i,j} -\tilde{H}_{i,j'} + \lambda + mz}
\end{align*}
where the $t_{i,j} \in \CC$, $t = \sum_{i=1}^{\ell}\sum_{j=1}^{r_i} t_{i,j} \tilde{H}_{i,j}$, and $t \cdot \underline{d}=\sum_{i,j}t_{i,j}d_{i,j}$. For the Weyl modification factor we used the fact that the roots of $G$ are given by $\rho_{i,j} \rho_{i,j'}^{-1}$, where the character $\rho_{i,j}$ was defined in section \ref{flag}. By Lemma~\ref{effsummationrange} the effective summation range for the vector $\underline{d}$ here is contained in the set $S \subset \ZZ^R$ consisting of $\underline{d}$ such that $\langle \tilde{\beta}, \tilde{H}_{i,j} \rangle=d_{i,j}$ for some $\tilde{\beta} \in \NE(\Fl(E)_T)$.
We can identify the group ring $\QQ[H_2(\Fl(E))]$ with $\QQ[H_2(X,\ZZ)][q_{1}, \dots, q_{\ell}]$ via the map
\begin{equation} \label{novinonab}
Q^\beta \mapsto Q^{\pi_*\beta}\prod_{i}q_{i}^{\langle c_1(S_i^\vee), \beta \rangle}
\end{equation}
Via \eqref{noviab} and \eqref{novinonab} the map on Mori cones $\varrho: \NE(\Fl(E)_T) \rightarrow \NE(\Fl(E))$ becomes
$$Q^D\prod_{i,j}q_{i,j}^{d_{i,j}} \mapsto Q^D\prod_{i}q_{i}^{\sum_j d_{i,j}}$$
Restricting $t$ to the Weyl-invariant locus $H^2(\Fl(E)_T)^W$ corresponds to setting $t_{i,j}=t_i$ for all~$i$ and~$j$, which gives $e^{t \cdot \underline{d}}=e^{\sum_i t_i d_i}$ where $d_i=\sum_j d_{i,j}$. The identification $H^2(\Fl(E)_T)^W \cong H^2(\Fl(E))$ sends $\sum_{i,j} t_i \tilde{H}_{i,j}$ to $\sum_i t_i c_1(S_i^\vee)$, so projecting along \eqref{quotientH} and taking the limit as $\lambda = 0$ we obtain
\begin{align*}
e^{\frac{t}{z}} \sum_{\substack{D \in \NE(X)\\\underline{\delta} \in \ZZ^\ell}} Q^D\prod_{i}q_{i}^{\delta_i} e^{t \cdot \underline{\delta}} \pi^{*}J_{X}^D(\tau, z) \sum_{\substack{\underline{d} \in \ZZ^R\colon \\
\forall i \sum_j d_{i,j}=\delta_i}}\prod_{i=1}^{\ell} \prod_{j=1}^{r_i}\prod_{j'=1}^{r_{i+1}}\frac{ \prod_{m=-\infty}^{0}{H}_{i,j} - {H}_{i+1,j'} + mz}{\prod_{m=-\infty}^{d_{i,j} - d_{i+1,j'}}{H}_{i,j} - {H}_{i+1,j'} + mz} \\
\times \prod_{i=1}^{\ell} \prod_{j \neq j'} \frac{\prod_{m=-\infty}^{d_{i,j} - d_{i,j'}}H_{i,j} - H_{i,j'} + mz}{\prod_{m=-\infty}^{0} H_{i,j} -H_{i,j'} + mz}
\end{align*}
where now $t=\sum_i t_i c_1(S_i^\vee)$. The effective summation range here is contained in $\NE(\Fl(E))$ by construction.
Using \eqref{novinonab} again we may rewrite this as
\begin{align*}
e^{\frac{t}{z}} \sum_{\substack{\beta \in \NE(\Fl(E))}} Q^{\beta} e^{\langle \beta, t \rangle} \pi^{*}J_{X}^{\pi_*\beta}(\tau, z) \sum_{\substack{\underline{d} \in \ZZ^R\colon \\ \forall i \sum_j d_{i,j}=\langle \beta, c_1(S_i^\vee) \rangle}}\prod_{i=1}^{\ell} \prod_{j=1}^{r_i}\prod_{j'=1}^{r_{i+1}}\frac{ \prod_{m=-\infty}^{0}{H}_{i,j} - {H}_{i+1,j'} + mz}{\prod_{m=-\infty}^{d_{i,j} - d_{i+1,j'}}{H}_{i,j} - {H}_{i+1,j'} + mz} \\
\times \prod_{i=1}^{\ell} \prod_{j \neq j'} \frac{\prod_{m=-\infty}^{d_{i,j} - d_{i,j'}}H_{i,j} - H_{i,j'} + mz}{\prod_{m=-\infty}^{0} H_{i,j} -H_{i,j'} + mz}
\end{align*}
This is $I_{\Fl(E)}(t, \tau, z)$, as required.
\end{proof}
\begin{remark}\label{effectivesummationrange} In view of \eqref{ampleconegen}, we see that the effective summation range in $I_{\Fl(E)}$ is contained in the subset of vectors satisfying \[
d_{i,j} \geq \min_{j'}d_{\ell+1,j'} \; \forall \, i, j \] This will prove useful in calculations in Section \ref{examples}. \end{remark}
\subsection{The Abelian/non-Abelian Correspondence with bundles}\label{AnAwithbundles}
We are now ready to prove Theorem~\ref{step two}. Recall from the Introduction that we have fixed a representation $\rho\colon G \rightarrow \GL(V)$ where $G=\prod_i \GL_{r_i}(\CC)$, and that this determines vector bundles $V^G \to \Fl(E)$ and $V^T \to \Fl(E)_T$. Since $T$ is Abelian, $V^T$ splits as a direct sum of line bundles $$V^T=F_1 \oplus \dots \oplus F_k$$ The Brown $I$-function gives a family \begin{align*}
&(t,\tau) \mapsto I_{\Fl(E)_T}(t,\tau,{-z}) & \text{$t \in H^2(\Fl(E)_T)^W$, $\tau \in H^\bullet(X)$}
\intertext{of elements of $\cH_{\Fl(E)_T}$, and Theorem~\ref{brown2014gromov} shows that $I_{\Fl(E)_T}(t,\tau,{-z}) \in \cL_{\Fl(E)_T}$. Twisting by $(F, \mathbf{c})$ where $\mathbf{c}$ is the $\CC^\times$-equivariant Euler class with parameter $\mu$ gives a twisted $I$-function, as in Definition~\ref{twistedI}, which we denote by}
&(t, \tau) \mapsto I_{V^T_\mu}(t, \tau,{-z}) & \text{$t \in H^2(\Fl(E)_T)^W$, $\tau \in H^\bullet(X)$}
\intertext{Applying Proposition~\ref{twisted=Deltad} shows that $I_{V^T_\mu}(t, \tau, {-z}) \in \cL_{V^T_\mu}$. Twisting again, by $(\Phi, \mathbf{c'})$ where $\Phi \to \Fl(E)_T$ is the roots bundle from the Introduction and $\mathbf{c'}$ is the $\CC^\times$-equivariant Euler class with parameter $\lambda$ gives a twisted $I$-function, as in Definition~\ref{twistedI}, which we denote by}
& (t, \tau) \mapsto I_{\Phi_\lambda \oplus V^T_\mu}(t, \tau,{-z}) & \text{$t \in H^2(\Fl(E)_T)^W$, $\tau \in H^\bullet(X)$}
\intertext{Applying Proposition~\ref{twisted=Deltad} again shows that $I_{\Phi_\lambda \oplus V^T_\mu}(t, \tau, {-z}) \in \cL_{\Phi_\lambda \oplus V^T_\mu}$. We now project along \eqref{quotientH} and take the non-equivariant limit $\lambda \to 0$, obtaining the Givental--Martin modification of~$I_{V^T_\mu}$. This is a family}
& (t, \tau) \mapsto I_{\GM}(t, \tau, {-z}) & \text{$t \in H^2(\Fl(E)_T)^W$, $\tau \in H^\bullet(X)$} \end{align*} of elements of $\cH_{\Fl(E)}$. Explicitly: \begin{definition}[which is a specialisation of Definition~\ref{IGM definition} to the situation at hand] \label{IGM definition special case}
\begin{align*}
& I_{\GM}(t, \tau, z) = \\
& e^{\frac{t}{z}} \sum_{\substack{\beta \in \NE(\Fl(E))}} Q^{\beta} e^{\langle \beta, t \rangle} \pi^{*}J_{X}^{\pi_*\beta}(\tau, z)
\sum_{\substack{\underline{d} \in \ZZ^R\colon \\ \forall i \sum_j d_{i,j}=\langle \beta, c_1(S_i^\vee) \rangle}}\prod_{i=1}^{\ell} \prod_{j=1}^{r_i}\prod_{j'=1}^{r_{i+1}}\frac{ \prod_{m=-\infty}^{0}{H}_{i,j} - {H}_{i+1,j'} + mz}{\prod_{m=-\infty}^{d_{i,j} - d_{i+1,j'}}{H}_{i,j} - {H}_{i+1,j'} + mz} \\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\times \prod_{i=1}^{\ell} \prod_{j \neq j'} \frac{\prod_{m=-\infty}^{d_{i,j} - d_{i,j'}}H_{i,j} - H_{i,j'} + mz}{\prod_{m=-\infty}^{0} H_{i,j} -H_{i,j'} + mz}
\prod_{s=1}^{k} \frac{\prod_{m=-\infty}^{f_s \cdot \underline{d}} f_s + \mu + mz}{\prod_{m=-\infty}^{0} f_s + \mu + mz}
\end{align*}
Here $J^D_X(\tau, z)$ is as in \eqref{JX by degrees}, $f_s \cdot \underline{d} = \sum_{i,j} f_{s,i,j} d_{i,j}$, and $f_s = \sum_{i,j} f_{s,i,j} H_{i,j}$, where $$c_1(F_s) = \sum_{i=1}^\ell \sum_{j=1}^{r_i}f_{s,i,j} \tilde{H}_{i,j}$$ \end{definition}
\noindent Lemma~\ref{IGMexists} shows that this expression is well-defined despite the presence of $$\omega = \textstyle \prod_i \prod_{j < j'} (H_{i,j} - H_{i,j'})$$ in the denominator. Corollary~\ref{IGMonLGM} shows that $I_{\GM}(t, \tau, {-z}) \in \cL_{\GM, V^T_\mu}$. Note that $I_{\GM}(t, \tau)$ is \emph{not} the $V^G$-twist of Oh's $I$-function $I_{\Fl(E)}$. Indeed $V^G$ need not be a split bundle, so the twist may not even be defined.
\begin{theorem} \label{IGM on twisted cone}
Let $I_{\GM}$ be as in Definition~\ref{IGM definition special case}. Then:
\begin{align*}
I_{\GM}(t,\tau,-z) \in \cL_{V^G_\mu} &&
\text{for all $t \in H^2(\Fl(E)_T)^W$, $\tau \in H^\bullet(X)$.}
\end{align*} \end{theorem}
\begin{proof}
Before projecting and taking the non-equivariant limit, we have
$$
I_{\Phi_\lambda \oplus V^T_\mu} = \Delta_{V^T_\mu} \big( D_{V^T_\mu} \big( I_{\Phi_\lambda}\big)\big)
$$
by Proposition~\ref{twisted=Deltadeq}. Projecting along \eqref{quotientH} gives
$$
p \circ I_{\Phi_\lambda \oplus V^T_\mu} = \Delta_{V^G_\mu} \big( D_{V^G_\mu} \big( p \circ I_{\Phi_\lambda}\big)\big)
$$
and taking the limit $\lambda \to 0$, which is well-defined by Lemma~\ref{IGMexists}, gives
$$
I_{\GM} = \Delta_{V^G_\mu} \big( D_{V^G_\mu} \big( I_{\Fl(E)}\big)\big)
$$
by Proposition~\ref{brownoh}. The result now follows from Proposition~\ref{twisted=Deltad}. \end{proof}
Exactly the same argument proves:
\begin{corollary} \label{extra line bundle}
Let $L \to X$ be a line bundle with first Chern class $\rho$, and define the vector bundle $F \to \Fl(E)$ to be $F = V^G \otimes \pi^* L$. Let $I_{\GM}$ be as in Definition~\ref{IGM definition special case}, except that the factor
\begin{align*}
\prod_{s=1}^{k} \frac{\prod_{m=-\infty}^{f_s \cdot \underline{d}} f_s + \mu + mz}{\prod_{m=-\infty}^{0} f_s + \mu + mz}
&& \text{is replaced by} &&
\prod_{s=1}^{k} \frac{\prod_{m=-\infty}^{f_s \cdot \underline{d} + \langle \rho, \pi_* \beta \rangle} f_s + \pi^* \rho + \mu + mz}{\prod_{m=-\infty}^{0} f_s + \pi^* \rho + \mu + mz}
\end{align*}
Then:
\begin{align*}
I_{\GM}(t,\tau,-z) \in \cL_{F_\mu} &&
\text{for all $t \in H^2(\Fl(E)_T)^W$, $\tau \in H^\bullet(X)$.}
\end{align*} \end{corollary}
The following Corollary gives a closed-form expression for genus-zero Gromov--Witten invariants of the zero locus of a generic section $Z$ of $F$ in terms of invariants of $X$.
\begin{corollary} \label{I=J}
With notation as in Corollary~\ref{extra line bundle}, let $Z$ be the zero locus of a generic section of $F \rightarrow \Fl(E)$. Suppose that ${-K_Z}$ is the restriction of an ample class on $\Fl(E)$ and that $\tau \in H^2(X)$. Then
$$J_{F_\mu}(t+\tau, z)=e^{-C(t)/z}I_{\GM}(t,\tau, z)$$
where
$$ C(t)=\sum_{\beta} n_\beta Q^\beta e^{\langle \beta, t \rangle} $$
for some constants $n_\beta \in \QQ$ and the sum runs over the finite set
$$S=\{ \beta \in \NE(\Fl(E)) : \langle {-K}_{\Fl(E)} - c_1(F), \beta \rangle = 1\}$$
If $Z$ is of Fano index two or more then this set is empty and $C(t) \equiv 0$. Regardless, if the vector bundle $F$ is convex then the non-equivariant limit $\mu \to 0$ of $J_{F_\mu}$ exists and
$$
J_Z\big(i^* t + i^* \tau, z \big) = i^* J_{F_0}(t+\tau, z)
$$
where $i \colon Z \to \Fl(E)$ is the inclusion map. \end{corollary} \begin{proof}[Proof of Corollary~\ref{I=J}]
The statement about Fano index two or more follows immediately from the Adjunction Formula
$$ K_Z=\big(K_{\Fl(E)}+c_1(F)\big)\big|_Z$$
We need to show that
\begin{equation} \label{IGM asymptotics}
I_{\GM}(t, \tau, z) = z + t + \tau + C(t) + O(z^{-1})
\end{equation}
Everything else then follows from the characterisation of the twisted $J$-function just below Definition~\ref{twisted J}, the String Equation
\begin{align*}
J_{F_\mu}(\tau + a, z) = e^{a/z} J_{F_\mu}(\tau,z) &&
a \in H^0(\Fl(E))
\end{align*}
and~\cite{Coates2014}. To establish \eqref{IGM asymptotics}, it will be convenient to set $\deg(z) = \deg(\mu) = 1$, $\deg(\phi)=k$ for $\phi \in H^{2k}(\Fl(E))$, and $\deg(Q^\beta)=\langle -K_X, \beta \rangle$ if $\beta \in H_2(X)$. The degree axiom for Gromov--Witten invariants then shows that $J_X^{\pi_*\beta}$ is homogeneous of degree $\langle K_X, \pi_*\beta \rangle +1$. Write
$$I_{\GM}(t, \tau,z) = e^{\frac{t}{z}} \sum_{\beta \in {\NE}(\Fl(E))}
Q^{\beta} e^{\langle \beta, t \rangle} \pi^{*}J_{X}^{\pi_*\beta}(\tau, z) \times {I}_{\beta}(z) \times M_\beta(z)
$$
where
$$ M_\beta(z) = \prod_{s=1}^{k} \frac{\prod_{m=-\infty}^{f_s \cdot \underline{d} + \langle \rho, \pi_* \beta \rangle} f_s + \pi^* \rho + \mu + mz}{\prod_{m=-\infty}^{0} f_s + \pi^* \rho + \mu + mz}
$$
A straightforward calculation shows that
\begin{align*}
{I}_{\beta}(z)&=z^{\langle K_{\Fl(E)}-\pi^*K_X, \beta \rangle}{i}_{\beta}(z)\\
M_\beta(z)&=z^{\langle c_1(F), \beta \rangle}m_\beta(z)
\end{align*}
where $i_\beta(z), m_\beta(z) \in \cH_{\Fl(E)}$ are homogeneous of degree $0$.
It follows that $\pi^{*}J_{X}^{\pi_*\beta}(\tau, z) \times {I}_{\beta}(z) \times M_\beta(z)$ is homogeneous of degree $\langle K_{\Fl(E)}+c_1(F), \beta \rangle+1$ which is nonpositive for $\beta \neq 0$ by the assumptions on $-K_Z$. Since $\tau \in H^2(X)$, any negative contribution to the homogenous degree must come from a negative power of $z$, so that $\pi^{*}J_{X}^{\pi_*\beta}(\tau, z) \times {I}_{\beta}(z) \times M_\beta(z)$ is $O(z^{-1})$, unless $\beta=0$ or $\beta \in S$. In the latter case, the expression has homogeneous degree $0$ and is therefore of the form $c_0+\tfrac{c_1}{z}+O(z^{-2})$ with $c_i$ independent of $z$ and of degree $i$. Relabeling $n_\beta=c_0$ and expanding $I_{\GM}$ in powers of $z$, we obtain
\begin{multline*}
I_{\GM}(t, \tau,z) =
\big(1+t z^{-1}+O(z^{-2})\big) \big(\pi^*J_X^0 \times I_{0} \times M_0 +\Big(
\sum_{\beta \in S}
n_\beta Q^{\beta} e^{\langle \beta, t \rangle} +O(z^{-1})\Big) +\sum_{0 \neq \beta \notin S} O(z^{-1})\big)\\
=(z+\tau+t+C(t)+O(z^{-1}))
\end{multline*}
where $C(t)$ is as claimed. This proves \eqref{IGM asymptotics}, and the result follows. \end{proof}
We restate Corollary~\ref{I=J} in the case where the flag bundle is a Grassmann bundle, i.e $\ell=1$, relabelling $H_{1,j}=H_j$, $d_{1,j} = d_j$ and $r_1=r$. The rest of the notation here is as in \S\ref{notation}.
\begin{corollary}\label{explicit Gr} Let $V^G\rightarrow \Gr(r, E)$ be a vector bundle induced by a representation of $G$, let $L \to X$ be a line bundle with first Chern class $\rho$, and let $F = V^G \otimes \pi^* L$. Let $Z$ be the zero locus of a generic section of $F$. Suppose that $F$ is convex, that $-K_{\Gr(E,r)} - c_1(F)$ is ample, and that $\tau \in H^2(\Gr(r,E))$. Then the non-equivariant limit $\mu \to 0$ of the twisted $J$-function $J_{F_\mu}$ exists and satisfies $$ J_Z\big(i^* t + i^*\tau, z \big) = i^* J_{F_0}(t+\tau, z) $$ where $i \colon Z \to \Gr(r, E)$ is the inclusion map. Furthermore \begin{multline} \label{explicit J for Gr}
J_{F_0}(t+\tau, z) = e^{\frac{t - C(t)}{z}} \sum_{\substack{\beta \in \NE(\Gr(r, E))}} Q^{\beta} e^{\langle \beta, t \rangle} \pi^{*}J_{X}^{\pi_*\beta}(\tau, z) \\
\sum_{\substack{\underline{d} \in \ZZ^r\colon \\ d_1 + \cdots + d_r = \langle \beta, c_1(S^\vee) \rangle}} (-1)^{\epsilon(\underline{d})}
\prod_{i=1}^{r}\prod_{j=1}^{n}\frac{ \prod_{m=-\infty}^{0}{H}_i + \pi^* c_1(L_j) + mz}{\prod_{m=-\infty}^{d_i + \langle \pi_* \beta, c_1(L_j) \rangle} H_i + \pi^* c_1(L_j) + mz} \\
\\
\times \prod_{i < j} \frac{H_i - H_j + (d_i - d_j)z}{H_i -H_j}
\times \prod_{s=1}^{k} \prod_{m=1}^{f_s \cdot \underline{d} + \langle \rho, \pi_* \beta \rangle} \big( f_s + \pi^* \rho + mz \big) \end{multline} Here the Abelianised bundle $V^T$ splits as a direct sum of line bundles $F_1 \oplus \cdots \oplus F_k$ with first Chern classes that we write as $c_1(F_s) = \sum_{i=1}^{r}f_{s,i} \tilde{H}_i$, $J^D_X(\tau,z )$ is as in \eqref{JX by degrees}, $\epsilon(\underline{d}) = \sum_{i<j} d_i - d_j$, $f_s \cdot \underline{d} = \sum_{i} f_{s,i} d_i$, $f_s = \sum_i f_{s,i} H_i$, and $C(t) \in H^0(\Gr(r, E), \Lambda)$ is the unique expression such that the right-hand side of \eqref{explicit J for Gr} has the form $z + t + \tau + O(z^{-1})$. \end{corollary}
\begin{remark} \label{effectivesummationrange Gr}
For a more explicit formula for $C(t)$, see Corollary~\ref{I=J}; in particular if $Z$ has Fano index two or greater then $C(t) \equiv 0$. By Remark~\ref{effectivesummationrange} the summand in \eqref{explicit J for Gr} is zero unless for each $i$ there exists a $j$ such that $d_i + \langle \pi_* \beta, c_1(L_j) \rangle \geq 0$ \end{remark}
\begin{proof}[Proof of Corollary~\ref{explicit Gr}]
We cancelled terms in the Weyl modification factor, as in the proof of Lemma~\ref{IGMexists}, and took the non-equivariant limit $\mu \to 0$. \end{proof}
\begin{remark}
The relationship between $I$-functions (or generating functions for genus-zero quasimap invariants) and $J$-functions (which are generating functions for genus-zero Gromov--Witten invariants) is particularly simple in the Fano case~\cite{Givental1996toric}~\cite[\S1.4]{CFK2014}, and for the same reason Corollary~\ref{I=J} holds without the restriction $\tau \in H^2(X)$ if $Z \to X$ is relatively Fano\footnote{That is, if the relative anticanonical bundle ${-K}_{Z/X}$ is ample.}. This never happens for blow-ups $\tilde{X} \to X$, however, and it is hard to construct examples where $Z \to X$ is relatively Fano and the rest of the conditions of Corollary~\ref{I=J} hold. We do not know of any such examples. \end{remark}
\begin{remark} \label{what can we compute}
Corollary~\ref{I=J} gives a closed-form expression for the small $J$-function of $Z$ -- or, equivalently, for one-point gravitional descendant invariants of $Z$ -- in the case where $Z$ is Fano. But in general (that is, without the Fano condition on $Z$) one can use Birkhoff factorization, as in~\cite{CoatesGivental2007, CFK2014} and~\cite[\S3.8]{CCIT2019}, to compute any twisted genus-zero gravitional descendant invariant of $\Fl(E)$ in terms of genus-zero descendant invariants of $X$. The twisting here is with respect to the $\CC^\times$-equivariant Euler class and the vector bundle $F$. Thus Corollary~\ref{I=J} determines the Lagrangian submanifold $\cL_{F_\mu}$ that encodes twisted Gromov--Witten invariants. Applying~\cite[Theorem~1.1]{Coates2014}, we see that Corollary~\ref{I=J} together with Birkhoff factorization allows us to compute any genus-zero Gromov--Witten invariant of the zero locus $Z$ of the form
\begin{equation} \label{what can we compute GW}
\langle \theta_1 \psi^{i_1}, \dots, \theta_n \psi^{i_n} \rangle_{0,n,d}
\end{equation}
where all but one of the cohomology classes $\theta_i$ lie in $\image(i^*) \subset H^\bullet(Z)$ and the remaining $\theta_i$ is an arbitrary element of $H^\bullet(Z)$. Here $i \colon Z \to \Fl(E)$ is the inclusion map. \end{remark}
\begin{remark} \label{what can we compute blow up}
Applying Remark~\ref{what can we compute} to the blow-up $\tilde{X} \to X$ considered in the introduction, we see that Corollary~\ref{I=J} together with Birkhoff factorization allows us to compute arbitrary invariants of $\tilde{X}$ of the form \eqref{what can we compute GW} in terms of genus-zero gravitional descendants of $X$. In this case $\image(i^*) \subset H^\bullet(\tilde{X})$ contains all classes from $H^\bullet(X)$ and also the class of the exceptional divisor. \end{remark} \section{The Main Geometric Construction} \label{geometric}
\subsection{Main Geometric Construction} Let $F$ be a locally free sheaf on a variety $X$. We denote by $F(x)$ its fibre over $x$, a vector space over the residue field $\kappa(x)$. A morphism $\varphi$ of locally free sheaves induces a linear map on fibres, denoted by $\varphi(x)$. We make the following definition: \begin{definition} Let $\varphi \colon E^m \rightarrow F^n$ a morphism of locally free sheaves of rank $m$ and $n$ respectively.
The $k$-th degeneracy locus is the subvariety of $X$ defined by $$D_k(\varphi)=\big \{ x \in X\colon \rk \, \varphi(x) \leq k \big\}$$ Note that $D_k(\varphi)=X$ if $k \geq \min\{m,n\}$; if $k=\min\{m,n\}-1$ we simply call $D_k(\varphi)$ the degeneracy locus of $\varphi$. \end{definition} We have the following results: \begin{itemize}
\item Scheme-theoretically, $D_k(\varphi)$ may be defined as the zero locus of the section $\wedge^k\varphi$; this shows that locally the ideal of $D_k(\varphi)$ is defined by the $(k+1) \times (k+1)$-minors of $\varphi$.
\item If $E^\vee \otimes F$ is globally generated, then $D_k(\varphi)$ of a generic $\varphi$ is either empty or has expected codimension $(m-k)(n-k)$, and the singular locus of $D_k(\varphi)$ is contained in $D_{k-1}(\varphi)$. In particular, if $\varphi$ is generic and $\dim X < (m-k+1)(n-k+1)$, then $D_k(\varphi)$ is smooth \cite[Theorem 2.8]{Ottaviani1995}.
\item We may freely assume that $m \geq n$ in what follows, since we can always replace $\varphi$ with its dual map whose degeneracy locus is the same. \end{itemize}
\begin{proposition}\label{geometricconstruction} Let X be a smooth variety, and $\varphi\colon E^m \rightarrow F^n$ a generic morphism of locally free sheaves on $X$. Suppose that $m \geq n$ and write $r=m-n$. Let $Y=D_{n-1}(\varphi)$ be the degeneracy locus of $\varphi$, and assume that $\varphi$ has generically full rank, that $Y$ has the expected codimension $m-n+1$ and that $Y$ is smooth. Let $\pi\colon \Gr(r,E) \rightarrow X$ be the Grassmann bundle of $E$ on $X$. Then the blow-up $Bl_Y(X)$ of $X$ along $Y$ is a subvariety of $\Gr(r,E)$, cut out as the zero locus of the regular section $s \in \Gamma(\Hom(S, \pi^*F))$ defined by the composition $$S \hookrightarrow \pi^*E \xrightarrow{\pi^*\varphi} \pi^*F $$ where the first map is the canonical inclusion. \end{proposition} \begin{proof} We write points in $\Gr(r,E)$ as $(p, V)$, where $p \in X$ and $V$ is a $r$-dimensional subspace of the fibre $E(x)$. At $(p, V)$, the section $s$ is given by the composition $$V \hookrightarrow E(x) \xrightarrow{\varphi(x)} F(x)$$ so $s$ vanishes at $(p, V)$ if and only if $V \subset \ker\varphi(x)$.
The statement is local on $X$, so fix a point $P \in X$ and a Zariski open neighbourhood $U=\Spec(A)$ with trivialisations $E |_U=A^m, F |_U=A^n $. We will show that the equations of $Z(s) \cap U$ and $Bl_{U \cap Y}U$ agree. Under these identifications $\varphi$ is given by a $n \times m$ matrix with entries in $A$. Since $\varphi$ has generically maximal rank and $Y$ is nonsingular, after performing row and column operations and shrinking $U$ if necessary, we may assume that $\varphi$ is given by the matrix $$\begin{pmatrix} x_0&\dots &x_{r}&0&0&\dots&0\\ 0& \dots &0&1&0&\dots&0\\ 0& \dots &0&0&1&\dots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 0&\dots&0&0&\dots&0&1\\ \end{pmatrix}$$ Note that the ideal of the minors of this matrix is just $I=(x_0, \dots x_{r})$ and that $x_0, \dots, x_r$ form part of a regular system of parameters around $P$, so we may assume that $n=1, m=r+1$. Writing $y_i$ for the basis of sections of $S^\vee$ on $\Gr(r,A^{r+1})$, we see that $Z(s)$ is given by the equation \begin{align*} x_0y_0+\dots+x_ry_r&=0 \end{align*} Under the Pl\"ucker isomorphism $$\Gr(r,A^{r+1}) \rightarrow \PP(\wedge^{r}A^{r+1})\cong U \times \PP^r_{y_0,\dots, y_r}$$ $Z(s)$ maps to the variety cut out by the minors of the matrix $$\begin{pmatrix} x_0&\dots& x_{r}\\ {y}_0& \dots &{y}_{r} \end{pmatrix}$$ i.e the blowup of $Y \cap U$ in $U$. \end{proof}
\section{Examples} \label{examples}
We close by presenting three example computations that use Theorems~\ref{step one} and~\ref{step two}, calculating genus-zero Gromov--Witten invariants of blow-ups of projective spaces in various high-codimension complete intersections. Recall, as we will need it below, that if $E \to X$ is a vector bundle of rank $n$ then the anticanonical divisor of $\Gr(r,E)$ is \begin{equation}
\label{-K}
{-K}_{\Gr(r,E)}=\pi^*\left(-K_X+r(\det E)\right) +n(\det S^\vee) \end{equation} where $S \to \Gr(r,E)$ is the tautological subbundle. Recall too that the \emph{regularised quantum period} of a Fano manifold $Z$ is the generating function $$ \widehat{G}_Z(x) = 1 + \sum_{d = 2}^\infty d! c_d x^d $$ for genus-zero Gromov--Witten invariants of $Z$, where \begin{align*}
c_d = \sum_{\beta} \langle \theta \psi_1^{d-2} \rangle_{0,1,\beta}
&& \text{for $\theta \in H^{\text{top}}(Z)$ the class of a volume form} \end{align*} and the sum runs over effective classes $\beta$ such that $\langle \beta, {-K}_Z\rangle = d$.
\begin{example} We will compute the regularised quantum period of $\tilde{X}=\Bl_Y\PP^4$ where $Y$ is a plane conic. Consider the situation as in \S\ref{notation} with: \begin{itemize}
\item $X=\PP^4$
\item $E= \cO \oplus \cO \oplus \cO(-1)$
\item $G = \GL_{2}(\CC)$, $T= (\CC^\times)^2 \subset G$ \end{itemize} Then $A /\!\!/ G$ is $\Gr(2, E)$, and $A /\!\!/ T$ is the $\PP^2 \times \PP^2$-bundle $\PP(E) \times_{\PP^4} \PP(E) \to \PP^4$. By Proposition \ref{geometricconstruction} the zero locus $\tilde{X}$ of a section of $S^\vee \otimes \pi^*(\OO(1))$ on $\Gr(2,E)$ is the blowup of $\PP^4$ along the complete intersection of two hyperplanes and a quadric. We identify the group ring $\QQ[H_2(A /\!\!/ T,\ZZ)]$ with $\QQ[Q,Q_1,Q_2]$, where $Q$ corresponds to the pullback of the hyperplane class of $\PP^4$ and $Q_i$ corresponds to $\tilde{H}_i$. Similarly, we identify $\QQ[H_2(A /\!\!/ G,\ZZ)]$ with $\QQ[Q, q]$, where again $Q$ corresponds to the pullback of the hyperplane class of $\PP^4$ and $q$ corresponds to the first Chern class of $S^\vee$.
We will need Givental's formula~\cite{Givental1996equivariant} for the $J$-function of $\PP^4$: \begin{align*}
J_{\PP^4}(\tau, z)=z e^{\tau/z}\sum_{D = 0}^\infty \frac{Q^D e^{D\tau}}{\prod_{m=1}^D (H+mz)^5} && \tau \in H^2(\PP^4) \end{align*} In the notation of \S\ref{notation}, we have $\ell=1$, $r_\ell=r_1=2$, $r_{\ell+1}=3$. We relabel $\tilde{H}_{\ell,j}=\tilde{H}_j$ and $d_{\ell, j}=d_j$. We have that $\tilde{H}_{\ell+1, 1}=\tilde{H}_{\ell+1, 2}=0$, $\tilde{H}_{\ell+1, 3}=\pi^*H$ and $d_{\ell+1, 1}=d_{\ell+1, 2}=0$, $d_{\ell+1, 3}=D$. Write $F = S^\vee \otimes \pi^* \cO(1)$. Corollary~\ref{explicit Gr} and Remark~\ref{effectivesummationrange Gr} give \begin{multline*} J_{F_0}(t, \tau,z) = z e^{\frac{t + \tau}{z}}\sum_{D=0}^\infty \sum_{d_1=0}^\infty \sum_{d_2=0}^\infty \frac{(-1)^{d_1-d_2} Q^{D} q^{d_1+d_2} e^{D\tau}e^{(d_1 +d_2)t} \prod_{i=1}^{2} \prod_{m=1}^{d_i + D} (H_i + H + mz)}{\prod_{m=1}^{D} (H+ mz)^5 \prod_{m=1}^{d_1}(H_1+mz)^2\prod_{m=1}^{d_2}(H_2+mz)^2} \\ \times \prod_{i=1}^2\frac{\prod_{m=-\infty}^{0} (H_i -H + mz)}{\prod_{m=-\infty}^{d_i-D}(H_i -H + mz)}
\frac{(H_1 - H_2 + z(d_1 - d_2))}{H_1 - H_2} \end{multline*} To obtain the quantum period we need to calculate the anticanonical bundle of $\tilde{X}$. Equation \eqref{-K} and the adjunction formula give $$-K_{\widetilde{X}}=3H+3\det S^\vee-(2H+\det S^\vee)=H+2 \det S^\vee.$$ To extract the quantum period from the non-equivariant limit $J_{F_0}$ of the twisted $J$-function, we take the component along the unit class $1 \in H^\bullet(A /\!\!/ G; \QQ)$, set $z=1$, and set $Q^\beta=x^{\langle \beta, -K_{\tilde{X}} \rangle}$. That is, we set $\lambda = 0$, $t=0$, $\tau=0$, $z=1$, $q=x^2$, $Q=x$, and take the component along the unit class, obtaining \begin{multline*}
G_{\tilde{X}}(x)= \sum_{n=0}^\infty \sum_{l=n+1}^\infty \sum_{m=l}^\infty \textstyle (-1)^{l+m-1}x^{l+2m+2n}
\frac{(l+n)!(l+m)!(l-n-1)!}{(l!)^5(m!)^2(n!)^2(n-l)!}(n-m)\\
+ \sum_{l=0}^\infty \sum_{m=l}^\infty \sum_{n=l}^\infty \textstyle (-1)^{m+n}x^{l+2m+2n}
\frac{(l+n)!(l+m)!}{(l!)^5(m!)^2(n!)^2(n-l)!(m-l)!}
\Big(1+(n-m)(-2H_{n}+H_{l+n}-H_{n-l}) \Big) \end{multline*} Thus the first few terms of the regularized quantum period are: \begin{multline*}
\widehat{G}_{\tilde{X}}(x)=1+12x^3+120x^5+540x^6+20160x^8+33600x^9+113400x^{10} \\ +2772000x^{11}+2425500x^{12}+\cdots \end{multline*} This strongly suggests that $\tilde{X}$ coincides with the quiver flag zero locus with ID 15 in~\cite{Kalashnikov2019}, although this is not obvious from the constructions. \end{example}
\begin{example} We will compute the regularised quantum period of $\tilde{X}=\Bl_Y\PP^6$, where $Y$ is a 3-fold given by the intersection of a hyperplane and two quadric hypersurfaces. Consider the situation as in \S\ref{notation} with: \begin{itemize}
\item $X=\PP^6$
\item $E= \cO \oplus \cO \oplus \cO(1)$
\item $G = \GL_{2}(\CC)$, $T= (\CC^\times)^2 \subset G$ \end{itemize} Then $A /\!\!/ G$ is $\Gr(2, E)$, and $A /\!\!/ T$ is the $\PP^2 \times \PP^2$-bundle $\PP(E) \times_{\PP^6} \PP(E) \to \PP^6$. By Proposition \ref{geometricconstruction} the zero locus $\tilde{X}$ of a section of $S^\vee \otimes \pi^*(\OO(2))$ on $\Gr(2,E)$ is the blowup of $\PP^6$ along the complete intersection of a hyperplane and two quadrics. We identify the group ring $\QQ[H_2(A /\!\!/ T,\ZZ)]$ here with $\QQ[Q,Q_1,Q_2]$, where $Q$ corresponds to the pullback of the hyperplane class of $\PP^6$ and $Q_i$ corresponds to $\tilde{H}_i$. Similarly, we identify $\QQ[H_2(A /\!\!/ G,\ZZ)]$ with $\QQ[Q, q]$, where again $Q$ corresponds to the pullback of the hyperplane class of $\PP^6$ and $q$ corresponds to the first Chern class of $S^\vee$.
The $J$-function of $\PP^6$ is~\cite{Givental1996equivariant}: \begin{align*}
J_{\PP^6}(\tau, z)=z e^{\tau/z}\sum_{D = 0}^\infty \frac{Q^D e^{D\tau}}{\prod_{m=1}^D (H+mz)^7} && \tau \in H^2(\PP^6) \end{align*} In the notation of \S\ref{notation}, we have $\ell=1$, $r_\ell=r_1=2$, $r_{\ell+1}=3$. We relabel $\tilde{H}_{\ell,j}=\tilde{H}_j$ and $d_{\ell, j}=d_j$. We have that $\tilde{H}_{\ell+1, 1}=\tilde{H}_{\ell+1, 2}=0$, $\tilde{H}_{\ell+1, 3}=- \pi^*H$ and $d_{\ell+1, 1}=d_{\ell+1, 2}=0$, $d_{\ell+1, 3}=-D$. Write $F = S^\vee \otimes \pi^* \cO(2)$. Corollary~\ref{explicit Gr} and Remark~\ref{effectivesummationrange Gr} give \begin{multline*} J_{F_0}(t, \tau, z) = z e^{\frac{t+\tau}{z}} \sum_{D=0}^\infty \sum_{d_1 = -D}^\infty \sum_{d_2= -D}^\infty \frac{Q^D q^{d_1+d_2} e^{D\tau}e^{(d_1+d_2)t}}{\prod_{m=1}^D(H+mz)^7} \prod_{i=1}^2 \frac{\prod_{m=-\infty}^{0}(H_i+mz)^2} {\prod_{m=-\infty}^{d_i}(H_i+mz)^2}\\ \times \prod_{i=1}^2 \frac{\prod_{m=1}^{d_i+2D}(H_i+2H+mz)}{\prod_{m=1}^{d_i + D} (H_i + H + mz)} (-1)^{d_1-d_2}\frac{(H_1-H_2+z(d_1-d_2))}{H_1-H_2} \end{multline*} Again we will need the anticanonical bundle of $\tilde{X}$, which by \eqref{-K} and the adjunction formula is $$-K_{\widetilde{X}}=9H + 3 \det(S^*)-(4H+\det(S^*))=5H + 2\det(S^*).$$ To extract the quantum period from $J_{F_0}$, we take the component along the unit class $1 \in H^\bullet(A /\!\!/ G; \QQ)$, set $z=1$, and set $Q^\beta=x^{\langle \beta, -K_{\tilde{X}} \rangle}$. That is, we set $\lambda = 0$, $t=0$, $\tau=0$, $z=1$, $q=x^2$, $Q=x^5$, and take the component along the unit class, obtaining
\begin{multline*}
G_{\tilde{X}}(x)=\sum_{D=0}^\infty \sum_{d_1=0}^\infty \sum_{d_2=0}^\infty (-1)^{d_1 + d_2}x^{5D+2d_1+2d_2}
\frac{(d_1 + 2D)! (d_2 + 2D)!}{(D!)^7(d_1!)^2(d_2!)^2(d_1 +D)!(d_2 + D)!} \\
\times \Big(1+(d_1-d_2)(-2H_{d_1}+H_{d_1+2D}-H_{d_1+D}) \Big)
\end{multline*}
The first few terms of the regularized quantum period are:
$$\widehat{G}_{\tilde{X}}(x) = 1+ 480x^5 + 5040 x^7 + 4082400 x^{10} + 119750400 x^{12} + 681080400 x^{14} + \cdots$$ \end{example}
\begin{example} We will compute the regularised quantum period of $\tilde{X}=\Bl_Y\PP^6$, where $Y$ is a quadric surface given by the intersection of 3 generic hyperplanes and a quadric hypersurface. Consider the situation as in \S\ref{notation} with: \begin{itemize}
\item $X=\PP^6$
\item $E= \cO \oplus \cO \oplus \cO \oplus \cO(2)$
\item $G = \GL_{3}(\CC)$, $T= (\CC^\times)^3 \subset G$ \end{itemize} Then $A /\!\!/ G$ is $\Gr(3, E)$, and $A /\!\!/ T$ is $\PP(E) \times_{\PP^6} \PP(E) \times_{\PP^6} \PP(E) \to \PP^6$. By Proposition \ref{geometricconstruction} the zero locus $\tilde{X}$ of a section of $S^\vee \otimes \pi^*(\OO(1))$ on $\Gr(3,E)$ is the blowup of $\PP^6$ along the complete intersection of three hyperplanes and a quadric. We identify the group ring $\QQ[H_2(A /\!\!/ T,\ZZ)]$ with $\QQ[Q,Q_1,Q_2,Q_3]$, where $Q$ corresponds to the pullback of the hyperplane class of $\PP^6$ and $Q_i$ corresponds to $\tilde{H}_i$. Similarly, we identify $\QQ[H_2(A /\!\!/ G,\ZZ)]$ with $\QQ[Q, q]$, where again $Q$ corresponds to the pullback of the hyperplane class of $\PP^6$ and $q$ corresponds the first Chern class of $S^\vee$.
In the notation of \S\ref{notation}, we have $\ell=1, r_\ell=r_1=3, r_{\ell+1}=4$. We relabel $\tilde{H}_{\ell,j}=\tilde{H}_j$ and $d_{\ell, j}=d_j$. We have that $\tilde{H}_{\ell+1, 1}=\tilde{H}_{\ell+1, 2} = \tilde{H}_{\ell + 1,3}=0$, $\tilde{H}_{\ell+1, 4}=- \pi^*2H$ and $d_{\ell+1, 1}=d_{\ell+1, 2}=d_{\ell + 1,3} = 0$, $d_{\ell+1, 4}=-2D$. Write $F = S^\vee \otimes \pi^* \cO(1)$. Corollary~\ref{explicit Gr} and Remark~\ref{effectivesummationrange Gr} give \begin{multline*} J^{F_0}(t, \tau, z) = z e^{\frac{t+\tau}{z}} \sum_{D = 0}^\infty \sum_{d_1 = -2D}^\infty \sum_{d_2 = -2D}^\infty \sum_{d_3 = -2D}^\infty \frac{Q^{D}q^{d_1+d_2+d_3} e^{D\tau}e^{(d_1+d_2+d_3) t}}{\prod_{m=1}^D(H+mz)^7}\\ \times \prod_{i=1}^3 \frac{\prod_{m=-\infty}^{0}(H_i+mz)^3} {\prod_{m=-\infty}^{d_i}(H_i+mz)^3} \prod_{i=1}^3 \frac{1} {\prod_{m=1}^{d_i+2D} (H_i + 2H+ mz)} \prod_{i=1}^3 \frac{\prod_{m=-\infty}^{d_i+D}(H_i+H+mz)}{\prod_{m=-\infty}^{0}(H_i+H+mz)} \\ \times \frac{(H_1-H_2+z(d_1-d_2))}{H_1-H_2} \frac{(H_1-H_3+z(d_1-d_3))}{H_1-H_3}\frac{(H_2-H_3+z(d_2-d_3))}{H_2-H_3} \end{multline*} Arguing as before, $$-K_{\widetilde{X}}=11H + 4 \det(S^*)-(3H+\det(S^*))=8H + 3\det(S^*).$$ To extract the quantum period from $J_{F_0}$, we set $\lambda = 0$, $t=0$, $\tau=0$, $z=1$, $q=x^3$, $Q=x^8$, and take the component along the unit class. The first few terms of the regularised quantum period are: \begin{multline*}
\widehat{G}_{\tilde{X}}(x) = 1+ 108x^3 + 17820 x^6 + 5040 x^{8} + 5473440 x^{9} + 56364000 x^{11} + 1766526300 x^{12} \\ + 117076459500 x^{14} + 672012949608 x^{15} + \cdots \end{multline*} \end{example}
\begin{remark}
Strictly speaking the use of Theorem~\ref{step two} in the examples just presented was not necessary. Whenever the base space $X$ is a projective space, or more generally a Fano complete intersection in a toric variety or flag bundle, then one can replace our use of Theorem~\ref{step two} (but not Theorem~\ref{step one}) by~\cite[Corollary~6.3.1]{CFKS2008}. However there are many examples that genuinely require both Theorem~\ref{step one} and Theorem~\ref{step two}: for instance when $X$ is a toric complete intersection but the line bundles that define the center of the blow-up do not arise by restriction from line bundles on the ambient space. (For a specific such example one could take $X$ to be the three-dimensional Fano manifold $\mathrm{MM}_{3\text{--}9}$: see~\cite[\S62]{CCGK16}.) For notational simplicity we chose to present examples with $X = \PP^N$, but the approach that we used applies without change to more general situations. \end{remark}
\end{document} | arXiv |
From Dedekind to Gödel
From Dedekind to Gödel pp 119-142 | Cite as
Frege's Principle
Richard G. HeckJr.
Part of the Synthese Library book series (SYLI, volume 251)
In his Grundgesetze der Arithmetik,1 Frege does indeed prove the "simplest laws of Numbers", the axioms of arithmetic being among these laws. However, as is well known, Frege does not do so "by logical means alone", since his proofs appeal to an axiom which is not only not a logical truth but a logical falsehood. The axiom in question is Frege's Axiom V, which governs terms of the form "
$$\mathop \varepsilon \limits^, $$
.Φ(ε)", terms which purport to refer to what Frege calls 'value-ranges'. For present purposes, Axiom V may be written:2
$$\mathop \varepsilon \limits^, .F\varepsilon = \mathop \varepsilon \limits^, .G\varepsilon \equiv \forall \left( {Fx \equiv Gx} \right).$$
The formal theory of Grundgesetze, like any (full)3 second-order theory containing this sentence, is thus inconsistent, since Russell's Paradox is derivable from Axiom V in (full) second-order logic.
In my Grundlagen der Arithmetik, I sought to make it plausible that arithmetic is a branch of logic and need not borrow any ground of proof whatever from either experience or intuition. In the present book this shall now be confirmed, by the derivation of the simplest laws of Numbers by logical means alone (Gg I §0).
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© Springer Science+Business Media Dordrecht 1995
1.Harvard UniversityUSA
Heck R.G. (1995) Frege's Principle. In: Hintikka J. (eds) From Dedekind to Gödel. Synthese Library (Studies in Epistemology, Logic, Methodology, and Philosophy of Science), vol 251. Springer, Dordrecht
DOI https://doi.org/10.1007/978-94-015-8478-4_6
Publisher Name Springer, Dordrecht
Print ISBN 978-90-481-4554-6
Online ISBN 978-94-015-8478-4 | CommonCrawl |
Bertram Kostant
Bertram Kostant (May 24, 1928 – February 2, 2017)[1] was an American mathematician who worked in representation theory, differential geometry, and mathematical physics.
Bertram Kostant
Bertram Kostant at a workshop on “Enveloping Algebras and Geometric Representation Theory” in Oberwolfach, 2009
Born(1928-05-24)May 24, 1928
Brooklyn, New York
DiedFebruary 2, 2017(2017-02-02) (aged 88)
Roslindale, Massachusetts
NationalityAmerican
Alma materPurdue University
University of Chicago (PhD)
Known forKostant's convexity theorem
Kostant partition function
Kostant polynomial
Geometric quantization
Kostant–Parthasarathy–Ranga Rao–Varadarajan determinants
Hochschild-Kostant-Rosenberg theorem
AwardsWigner Medal (2016)
Scientific career
FieldsMathematics
InstitutionsMassachusetts Institute of Technology
University of California, Berkeley
ThesisRepresentations of a Lie algebra and its enveloping algebra on a Hilbert space
Doctoral advisorIrving Segal
Doctoral students
• Marjorie Batchelor
• James Lepowsky
• Arlie Petters
• Stephen Rallis
• James Harris Simons
• Birgit Speh
• Moss Sweedler
• David Vogan
Early life and education
Kostant grew up in New York City, where he graduated from Stuyvesant High School in 1945.[2] He went on to obtain an undergraduate degree in mathematics from Purdue University in 1950. He earned his Ph.D. from the University of Chicago in 1954, under the direction of Irving Segal, where he wrote a dissertation on representations of Lie groups.
Career in mathematics
After time at the Institute for Advanced Study, Princeton University, and the University of California, Berkeley, he joined the faculty at the Massachusetts Institute of Technology, where he remained until his retirement in 1993. Kostant's work has involved representation theory, Lie groups, Lie algebras, homogeneous spaces, differential geometry and mathematical physics, particularly symplectic geometry. He has given several lectures on the Lie group E8.[3] He has been one of the principal developers of the theory of geometric quantization. His introduction of the theory of prequantization has led to the theory of quantum Toda lattices. The Kostant partition function is named after him. With Gerhard Hochschild and Alex F. T. W. Rosenberg, he is one of the namesakes of the Hochschild–Kostant–Rosenberg theorem which describes the Hochschild homology of some algebras.[4]
His students include James Harris Simons, James Lepowsky, Moss Sweedler, David Vogan, and Birgit Speh. At present he has more than 100 mathematical descendants.
Awards and honors
Kostant was a Guggenheim Fellow in 1959-60 (in Paris), and a Sloan Fellow in 1961-63. In 1962 he was elected to the American Academy of Arts and Sciences, and in 1978 to the National Academy of Sciences. In 1982 he was a fellow of the Sackler Institute for Advanced Studies at Tel Aviv University. In 1990 he was awarded the Steele Prize of the American Mathematical Society, in recognition of his 1975 paper, “On the existence and irreducibility of certain series of representations.”
In 2001, Kostant was a Chern Lecturer and Chern Visiting Professor at Berkeley. He received honorary degrees from the University of Córdoba in Argentina in 1989, the University of Salamanca in Spain in 1992, and Purdue University in 1997. The latter, from his alma mater, was an honorary Doctor of Science degree, citing his fundamental contributions to mathematics and the inspiration he and his work provided to generations of researchers.
In May 2008, the Pacific Institute for Mathematical Sciences hosted a conference: “Lie Theory and Geometry: the Mathematical Legacy of Bertram Kostant,” at the University of British Columbia, celebrating the life and work of Kostant in his 80th year. In 2012, he was elected to the inaugural class of fellows of the American Mathematical Society. In the last year of his life, Kostant traveled to Rio de Janeiro for the Colloquium on Group Theoretical Methods in Physics, where he received the prestigious Wigner Medal, “for his fundamental contributions to representation theory that led to new branches of mathematics and physics.”
Selected publications
• Kostant, Bertram (1955). "Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold". Trans. Amer. Math. Soc. 80 (2): 528–542. doi:10.1090/S0002-9947-1955-0084825-8.
• Kostant, Bertram (1959). "A formula for the multiplicity of a weight". Trans. Amer. Math. Soc. 93 (6): 53–73. doi:10.1090/S0002-9947-1959-0109192-6. PMC 528626. PMID 16590246.
• Kostant, Bertram (1961). "Lie algebra cohomology and the generalized Borel-Weil theorem" (PDF). Annals of Mathematics. 74 (2): 329–387. doi:10.2307/1970237. hdl:2027/mdp.39015095258318. JSTOR 1970237.
• Kostant, Bertram (1963). "Lie group representations on polynomial rings". American Journal of Mathematics. 85 (3): 327–404. doi:10.2307/2373130. JSTOR 2373130.
• Kostant, Bertram (1969). "On the existence and irreducibility of certain series of representations". Bulletin of the American Mathematical Society. 75 (4): 627–642. doi:10.1090/S0002-9904-1969-12235-4.
• Kostant, Bertram (1970). "Quantization and unitary representations". In: Lectures in modern analysis and applications III. Lecture Notes in Mathematics 170. Vol. 170. pp. 87–208. doi:10.1007/BFb0079068. ISBN 978-3-540-05284-5.
• with Louis Auslander: Auslander, L.; Kostant, B. (1971). "Polarization and unitary representations of solvable Lie groups". Inventiones Mathematicae. 14 (4): 255–354. Bibcode:1971InMat..14..255A. doi:10.1007/BF01389744. S2CID 122009744.
• with Stephen Rallis: Kostant, B.; Rallis, S. (1971). "Orbits and representations associated with symmetric spaces". American Journal of Mathematics. 93 (3): 753–809. doi:10.2307/2373470. JSTOR 2373470.
• Kostant, Bertram (1973). "On convexity, the Weyl group and the Iwasawa decomposition". Annales Scientifiques de l'École Normale Supérieure. 6 (4): 413–455. doi:10.24033/asens.1254.
• Kostant, Bertram (1977). "Graded manifolds, graded Lie theory, and prequantization". In: Differential Geometrical Methods in Mathematical Physics. Lecture Notes in Math 570. Vol. 570. pp. 177–306. doi:10.1007/Bfb0087788. ISBN 978-3-540-08068-8.
• Kostant, Bertram (1978). "On Whittaker vectors and representation theory". Inventiones Mathematicae. 48 (2): 101–184. Bibcode:1978InMat..48..101K. doi:10.1007/BF01390249. S2CID 122765132.
• with David Kazhdan and Shlomo Sternberg: Kazhdan, D.; Kostant, B.; Sternberg, S. (1978). "Hamiltonian group actions and dynamical systems of Calogero type". Communications on Pure and Applied Mathematics. 31 (4): 481–507. doi:10.1002/cpa.3160310405.
• Kostant, Bertram (1979). "The solution to a generalized Toda lattice and representation theory". Advances in Mathematics. 34 (3): 195–338. doi:10.1016/0001-8708(79)90057-4.
• with Shrawan Kumar: Kostant, Bertram; Kumar, Shrawan (1986). "The nil Hecke ring and cohomology of GP for a Kac-Moody group G". Advances in Mathematics. 62 (3): 187–237. doi:10.1016/0001-8708(86)90101-5. PMC 323118. PMID 16593661.
• with Shlomo Sternberg: Kostant, Bertram; Sternberg, Shlomo (1987), "Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras", Annals of Physics, 176 (1): 49–113, Bibcode:1987AnPhy.176...49K, doi:10.1016/0003-4916(87)90178-3
• "On Laguerre polynomials, Bessel functions, Hankel transform and a series in the unitary dual of the simply-connected covering group of ${\mbox{SL}}(2,\mathbb {R} )$". Represent. Theory. 4: 181–224. 2000. doi:10.1090/S1088-4165-00-00096-0.
• with Gerhard Hochschild and Alex Rosenberg: Hochschild, G.; Kostant, Bertram; Rosenberg, Alex (2009). "Differential forms on regular affine algebras". In: Collected Papers. New York: Springer. pp. 265–290. doi:10.1007/b94535_14. ISBN 978-0-387-09582-0. (Reprinted from Trans. Amer. Math. Soc., vol. 102, no. 3, March 1962, pp. 383–408)
• Kostant, Bertram (2009). "The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group". In: Collected Papers. pp. 130–189. doi:10.1007/b94535_11. ISBN 978-0-387-09582-0. (Reprinted from the Amer. J. Math., vol. 81, no. 4, Oct. 1959)
See also
• Chern's conjecture (affine geometry)
• Supermanifold
• Symplectic spinor bundle
Notes
1. "Bertram Kostant, professor emeritus of mathematics, dies at 88". MIT News. 16 February 2017.
2. "Professor Kostant's Homepage". MIT Math Department. Retrieved 2007-10-31.
3. Bertram Kostant (2008-02-12). "On Some Mathematics in Garrett Lisi's 'E8 Theory of Everything'". UC Riverside mathematics colloquium. Retrieved 2008-06-15.
4. Porter, Tim (April 8, 2014), "Hochschild-Kostant-Rosenberg theorem", nLab.
References
• Kostant's home page at MIT
• Bertram Kostant at the Mathematics Genealogy Project
• Brylinski, Jean-Luc, ed. (1994). Lie Theory and Geometry, In Honor of Bertram Kostant. Boston, Birkhäuser. ISBN 0-8176-3761-3.
Authority control
International
• ISNI
• VIAF
National
• Norway
• Germany
• Israel
• United States
• Czech Republic
• Netherlands
Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
People
• Deutsche Biographie
Other
• IdRef
| Wikipedia |
\begin{definition}[Definition:American Currency/Half Dollar]
The '''half dollar''' is a unit of the American monetary system.
{{begin-eqn}}
{{eqn | o =
| r = 1
| c = '''half dollar'''
}}
{{eqn | r = 50
| c = cents
}}
{{eqn | r = 10
| c = nickels
}}
{{eqn | r = 5
| c = dimes
}}
{{eqn | r = 2
| c = quarters
}}
{{end-eqn}}
Category:Definitions/American Currency
\end{definition} | ProofWiki |
List of regular polytopes and compounds
This article lists the regular polytopes and regular polytope compounds in Euclidean, spherical and hyperbolic spaces.
Example regular polytopes
Regular (2D) polygons
Convex Star
{5}
{5/2}
Regular (3D) polyhedra
Convex Star
{5,3}
{5/2,5}
Regular 4D polytopes
Convex Star
{5,3,3}
{5/2,5,3}
Regular 2D tessellations
Euclidean Hyperbolic
{4,4}
{5,4}
Regular 3D tessellations
Euclidean Hyperbolic
{4,3,4}
{5,3,4}
The Schläfli symbol describes every regular tessellation of an n-sphere, Euclidean and hyperbolic spaces. A Schläfli symbol describing an n-polytope equivalently describes a tessellation of an (n − 1)-sphere. In addition, the symmetry of a regular polytope or tessellation is expressed as a Coxeter group, which Coxeter expressed identically to the Schläfli symbol, except delimiting by square brackets, a notation that is called Coxeter notation. Another related symbol is the Coxeter–Dynkin diagram which represents a symmetry group with no rings, and the represents regular polytope or tessellation with a ring on the first node. For example, the cube has Schläfli symbol {4,3}, and with its octahedral symmetry, [4,3] or , it is represented by Coxeter diagram .
The regular polytopes are grouped by dimension and subgrouped by convex, nonconvex and infinite forms. Nonconvex forms use the same vertices as the convex forms, but have intersecting facets. Infinite forms tessellate a one-lower-dimensional Euclidean space.
Infinite forms can be extended to tessellate a hyperbolic space. Hyperbolic space is like normal space at a small scale, but parallel lines diverge at a distance. This allows vertex figures to have negative angle defects, like making a vertex with seven equilateral triangles and allowing it to lie flat. It cannot be done in a regular plane, but can be at the right scale of a hyperbolic plane.
A more general definition of regular polytopes which do not have simple Schläfli symbols includes regular skew polytopes and regular skew apeirotopes with nonplanar facets or vertex figures.
Overview
This table shows a summary of regular polytope counts by dimension.
Note that the Euclidean and hyperbolic tilings are given one dimension more than what would be expected. This is because of an analogy with finite polytopes: a convex regular n-polytope can be seen as a tessellation of (n−1)-dimensional spherical space. Thus the three regular tilings of the Euclidean plane (by triangles, squares, and hexagons) are listed under dimension three rather than two.
Dim. Finite Euclidean Hyperbolic Compounds
Compact Paracompact
Convex Star Skew Convex Convex Star Convex Convex Star
1 1nonenone1nonenonenonenonenone
2 $\infty $$\infty $$\infty $11nonenone$\infty $$\infty $
3 54?3$\infty $$\infty $$\infty $5none
4 610?14none112620
5 3none?3542nonenone
6 3none?1nonenone5nonenone
7 3none?1nonenonenone3none
8 3none?1nonenonenone6none
9+ 3none?1nonenonenone [lower-alpha 1] none
1. ${\begin{cases}2,&{\text{if the number of dimensions is of the form }}2^{k}\\1,&{\text{if the number of dimensions is of the form }}2^{k}-1\\0,&{\text{otherwise}}\\\end{cases}}$
There are no Euclidean regular star tessellations in any number of dimensions.
One dimension
A Coxeter diagram represent mirror "planes" as nodes, and puts a ring around a node if a point is not on the plane. A dion { }, , is a point p and its mirror image point p', and the line segment between them.
A one-dimensional polytope or 1-polytope is a closed line segment, bounded by its two endpoints. A 1-polytope is regular by definition and is represented by Schläfli symbol { },[1][2] or a Coxeter diagram with a single ringed node, . Norman Johnson calls it a dion[3] and gives it the Schläfli symbol { }.
Although trivial as a polytope, it appears as the edges of polygons and other higher dimensional polytopes.[4] It is used in the definition of uniform prisms like Schläfli symbol { }×{p}, or Coxeter diagram as a Cartesian product of a line segment and a regular polygon.[5]
Two dimensions (polygons)
The two-dimensional polytopes are called polygons. Regular polygons are equilateral and cyclic. A p-gonal regular polygon is represented by Schläfli symbol {p}.
Usually only convex polygons are considered regular, but star polygons, like the pentagram, can also be considered regular. They use the same vertices as the convex forms, but connect in an alternate connectivity which passes around the circle more than once to be completed.
Star polygons should be called nonconvex rather than concave because the intersecting edges do not generate new vertices and all the vertices exist on a circle boundary.
Convex
The Schläfli symbol {p} represents a regular p-gon.
Name Triangle
(2-simplex)
Square
(2-orthoplex)
(2-cube)
Pentagon
(2-pentagonal
polytope
)
Hexagon Heptagon Octagon
Schläfli {3} {4} {5} {6} {7} {8}
Symmetry D3, [3]D4, [4]D5, [5]D6, [6]D7, [7]D8, [8]
Coxeter
Image
Name Nonagon
(Enneagon)
Decagon Hendecagon Dodecagon Tridecagon Tetradecagon
Schläfli {9} {10} {11} {12} {13} {14}
Symmetry D9, [9]D10, [10]D11, [11]D12, [12]D13, [13]D14, [14]
Dynkin
Image
Name Pentadecagon Hexadecagon Heptadecagon Octadecagon Enneadecagon Icosagon ...p-gon
Schläfli {15} {16} {17} {18} {19} {20} {p}
Symmetry D15, [15]D16, [16]D17, [17]D18, [18]D19, [19]D20, [20]Dp, [p]
Dynkin
Image
Spherical
The regular digon {2} can be considered to be a degenerate regular polygon. It can be realized non-degenerately in some non-Euclidean spaces, such as on the surface of a sphere or torus. For example, digon can be realised non-degenerately as a spherical lune. A monogon {1} could also be realised on the sphere as a single point with a great circle through it.[6] However, a monogon is not a valid abstract polytope because its single edge is incident to only one vertex rather than two.
Name Monogon Digon
Schläfli symbol {1} {2}
Symmetry D1, [ ] D2, [2]
Coxeter diagram or
Image
Stars
There exist infinitely many regular star polytopes in two dimensions, whose Schläfli symbols consist of rational numbers {n/m}. They are called star polygons and share the same vertex arrangements of the convex regular polygons.
In general, for any natural number n, there are n-pointed star regular polygonal stars with Schläfli symbols {n/m} for all m such that m < n/2 (strictly speaking {n/m}={n/(n−m)}) and m and n are coprime (as such, all stellations of a polygon with a prime number of sides will be regular stars). Cases where m and n are not coprime are called compound polygons.
Name Pentagram Heptagrams Octagram Enneagrams Decagram ...n-grams
Schläfli {5/2} {7/2} {7/3} {8/3} {9/2} {9/4} {10/3} {p/q}
Symmetry D5, [5]D7, [7]D8, [8]D9, [9],D10, [10]Dp, [p]
Coxeter
Image
Regular star polygons up to 20 sides
{11/2}
{11/3}
{11/4}
{11/5}
{12/5}
{13/2}
{13/3}
{13/4}
{13/5}
{13/6}
{14/3}
{14/5}
{15/2}
{15/4}
{15/7}
{16/3}
{16/5}
{16/7}
{17/2}
{17/3}
{17/4}
{17/5}
{17/6}
{17/7}
{17/8}
{18/5}
{18/7}
{19/2}
{19/3}
{19/4}
{19/5}
{19/6}
{19/7}
{19/8}
{19/9}
{20/3}
{20/7}
{20/9}
Star polygons that can only exist as spherical tilings, similarly to the monogon and digon, may exist (for example: {3/2}, {5/3}, {5/4}, {7/4}, {9/5}), however these do not appear to have been studied in detail.
There also exist failed star polygons, such as the piangle, which do not cover the surface of a circle finitely many times.[7]
Skew polygons
In 3-dimensional space, a regular skew polygon is called an antiprismatic polygon, with the vertex arrangement of an antiprism, and a subset of edges, zig-zagging between top and bottom polygons.
Example regular skew zig-zag polygons
Hexagon Octagon Decagons
D3d, [2+,6] D4d, [2+,8] D5d, [2+,10]
{3}#{ } {4}#{ } {5}#{ } {5/2}#{ } {5/3}#{ }
In 4-dimensions a regular skew polygon can have vertices on a Clifford torus and related by a Clifford displacement. Unlike antiprismatic skew polygons, skew polygons on double rotations can include an odd-number of sides.
They can be seen in the Petrie polygons of the convex regular 4-polytopes, seen as regular plane polygons in the perimeter of Coxeter plane projection:
Pentagon Octagon Dodecagon Triacontagon
5-cell
16-cell
24-cell
600-cell
Three dimensions (polyhedra)
In three dimensions, polytopes are called polyhedra:
A regular polyhedron with Schläfli symbol {p,q}, Coxeter diagrams , has a regular face type {p}, and regular vertex figure {q}.
A vertex figure (of a polyhedron) is a polygon, seen by connecting those vertices which are one edge away from a given vertex. For regular polyhedra, this vertex figure is always a regular (and planar) polygon.
Existence of a regular polyhedron {p,q} is constrained by an inequality, related to the vertex figure's angle defect:
${\begin{aligned}&{\frac {1}{p}}+{\frac {1}{q}}>{\frac {1}{2}}:{\text{Polyhedron (existing in Euclidean 3-space)}}\\[6pt]&{\frac {1}{p}}+{\frac {1}{q}}={\frac {1}{2}}:{\text{Euclidean plane tiling}}\\[6pt]&{\frac {1}{p}}+{\frac {1}{q}}<{\frac {1}{2}}:{\text{Hyperbolic plane tiling}}\end{aligned}}$
By enumerating the permutations, we find five convex forms, four star forms and three plane tilings, all with polygons {p} and {q} limited to: {3}, {4}, {5}, {5/2}, and {6}.
Beyond Euclidean space, there is an infinite set of regular hyperbolic tilings.
Convex
The five convex regular polyhedra are called the Platonic solids. The vertex figure is given with each vertex count. All these polyhedra have an Euler characteristic (χ) of 2.
Name Schläfli
{p,q}
Coxeter
Image
(solid)
Image
(sphere)
Faces
{p}
Edges Vertices
{q}
Symmetry Dual
Tetrahedron
(3-simplex)
{3,3} 4
{3}
6 4
{3}
Td
[3,3]
(*332)
(self)
Hexahedron
Cube
(3-cube)
{4,3} 6
{4}
12 8
{3}
Oh
[4,3]
(*432)
Octahedron
Octahedron
(3-orthoplex)
{3,4} 8
{3}
12 6
{4}
Oh
[4,3]
(*432)
Cube
Dodecahedron {5,3} 12
{5}
30 20
{3}
Ih
[5,3]
(*532)
Icosahedron
Icosahedron {3,5} 20
{3}
30 12
{5}
Ih
[5,3]
(*532)
Dodecahedron
Spherical
In spherical geometry, regular spherical polyhedra (tilings of the sphere) exist that would otherwise be degenerate as polytopes. These are the hosohedra {2,n} and their dual dihedra {n,2}. Coxeter calls these cases "improper" tessellations.[8]
The first few cases (n from 2 to 6) are listed below.
Hosohedra
Name Schläfli
{2,p}
Coxeter
diagram
Image
(sphere)
Faces
{2}π/p
Edges Vertices
{p}
Symmetry Dual
Digonal hosohedron {2,2} 2
{2}π/2
2 2
{2}π/2
D2h
[2,2]
(*222)
Self
Trigonal hosohedron {2,3} 3
{2}π/3
3 2
{3}
D3h
[2,3]
(*322)
Trigonal dihedron
Square hosohedron {2,4} 4
{2}π/4
4 2
{4}
D4h
[2,4]
(*422)
Square dihedron
Pentagonal hosohedron {2,5} 5
{2}π/5
5 2
{5}
D5h
[2,5]
(*522)
Pentagonal dihedron
Hexagonal hosohedron {2,6} 6
{2}π/6
6 2
{6}
D6h
[2,6]
(*622)
Hexagonal dihedron
Dihedra
Name Schläfli
{p,2}
Coxeter
diagram
Image
(sphere)
Faces
{p}
Edges Vertices
{2}
Symmetry Dual
Digonal dihedron {2,2} 2
{2}π/2
2 2
{2}π/2
D2h
[2,2]
(*222)
Self
Trigonal dihedron {3,2} 2
{3}
3 3
{2}π/3
D3h
[3,2]
(*322)
Trigonal hosohedron
Square dihedron {4,2} 2
{4}
4 4
{2}π/4
D4h
[4,2]
(*422)
Square hosohedron
Pentagonal dihedron {5,2} 2
{5}
5 5
{2}π/5
D5h
[5,2]
(*522)
Pentagonal hosohedron
Hexagonal dihedron {6,2} 2
{6}
6 6
{2}π/6
D6h
[6,2]
(*622)
Hexagonal hosohedron
Star-dihedra and hosohedra {p/q,2} and {2,p/q} also exist for any star polygon {p/q}.
Stars
The regular star polyhedra are called the Kepler–Poinsot polyhedra and there are four of them, based on the vertex arrangements of the dodecahedron {5,3} and icosahedron {3,5}:
As spherical tilings, these star forms overlap the sphere multiple times, called its density, being 3 or 7 for these forms. The tiling images show a single spherical polygon face in yellow.
Name Image
(skeletonic)
Image
(solid)
Image
(sphere)
Stellation
diagram
Schläfli
{p,q} and
Coxeter
Faces
{p}
Edges Vertices
{q}
verf.
χ Density Symmetry Dual
Small stellated dodecahedron {5/2,5}
12
{5/2}
3012
{5}
−63Ih
[5,3]
(*532)
Great dodecahedron
Great dodecahedron {5,5/2}
12
{5}
3012
{5/2}
−63Ih
[5,3]
(*532)
Small stellated dodecahedron
Great stellated dodecahedron {5/2,3}
12
{5/2}
3020
{3}
27Ih
[5,3]
(*532)
Great icosahedron
Great icosahedron {3,5/2}
20
{3}
3012
{5/2}
27Ih
[5,3]
(*532)
Great stellated dodecahedron
There are infinitely many failed star polyhedra. These are also spherical tilings with star polygons in their Schläfli symbols, but they do not cover a sphere finitely many times. Some examples are {5/2,4}, {5/2,9}, {7/2,3}, {5/2,5/2}, {7/2,7/3}, {4,5/2}, and {3,7/3}.
Skew polyhedra
Regular skew polyhedra are generalizations to the set of regular polyhedron which include the possibility of nonplanar vertex figures.
For 4-dimensional skew polyhedra, Coxeter offered a modified Schläfli symbol {l,m|n} for these figures, with {l,m} implying the vertex figure, m l-gons around a vertex, and n-gonal holes. Their vertex figures are skew polygons, zig-zagging between two planes.
The regular skew polyhedra, represented by {l,m|n}, follow this equation:
2 sin(π/l) sin(π/m) = cos(π/n)
Four of them can be seen in 4-dimensions as a subset of faces of four regular 4-polytopes, sharing the same vertex arrangement and edge arrangement:
{4, 6 | 3} {6, 4 | 3} {4, 8 | 3} {8, 4 | 3}
Four dimensions
Regular 4-polytopes with Schläfli symbol $\{p,q,r\}$ have cells of type $\{p,q\}$, faces of type $\{p\}$, edge figures $\{r\}$, and vertex figures $\{q,r\}$.
• A vertex figure (of a 4-polytope) is a polyhedron, seen by the arrangement of neighboring vertices around a given vertex. For regular 4-polytopes, this vertex figure is a regular polyhedron.
• An edge figure is a polygon, seen by the arrangement of faces around an edge. For regular 4-polytopes, this edge figure will always be a regular polygon.
The existence of a regular 4-polytope $\{p,q,r\}$ is constrained by the existence of the regular polyhedra $\{p,q\},\{q,r\}$. A suggested name for 4-polytopes is "polychoron".[9]
Each will exist in a space dependent upon this expression:
$\sin \left({\frac {\pi }{p}}\right)\sin \left({\frac {\pi }{r}}\right)-\cos \left({\frac {\pi }{q}}\right)$
$>0$ : Hyperspherical 3-space honeycomb or 4-polytope
$=0$ : Euclidean 3-space honeycomb
$<0$ : Hyperbolic 3-space honeycomb
These constraints allow for 21 forms: 6 are convex, 10 are nonconvex, one is a Euclidean 3-space honeycomb, and 4 are hyperbolic honeycombs.
The Euler characteristic $\chi $ for convex 4-polytopes is zero: $\chi =V+F-E-C=0$
Convex
The 6 convex regular 4-polytopes are shown in the table below. All these 4-polytopes have an Euler characteristic (χ) of 0.
Name
Schläfli
{p,q,r}
Coxeter
Cells
{p,q}
Faces
{p}
Edges
{r}
Vertices
{q,r}
Dual
{r,q,p}
5-cell
(4-simplex)
{3,3,3} 5
{3,3}
10
{3}
10
{3}
5
{3,3}
(self)
8-cell
(4-cube)
(Tesseract)
{4,3,3} 8
{4,3}
24
{4}
32
{3}
16
{3,3}
16-cell
16-cell
(4-orthoplex)
{3,3,4} 16
{3,3}
32
{3}
24
{4}
8
{3,4}
Tesseract
24-cell {3,4,3} 24
{3,4}
96
{3}
96
{3}
24
{4,3}
(self)
120-cell {5,3,3} 120
{5,3}
720
{5}
1200
{3}
600
{3,3}
600-cell
600-cell {3,3,5} 600
{3,3}
1200
{3}
720
{5}
120
{3,5}
120-cell
5-cell8-cell16-cell24-cell120-cell600-cell
{3,3,3}{4,3,3}{3,3,4}{3,4,3}{5,3,3}{3,3,5}
Wireframe (Petrie polygon) skew orthographic projections
Solid orthographic projections
tetrahedral
envelope
(cell/
vertex-centered)
cubic envelope
(cell-centered)
cubic envelope
(cell-centered)
cuboctahedral
envelope
(cell-centered)
truncated rhombic
triacontahedron
envelope
(cell-centered)
Pentakis
icosidodecahedral
envelope
(vertex-centered)
Wireframe Schlegel diagrams (Perspective projection)
(cell-centered)
(cell-centered)
(cell-centered)
(cell-centered)
(cell-centered)
(vertex-centered)
Wireframe stereographic projections (Hyperspherical)
Spherical
Di-4-topes and hoso-4-topes exist as regular tessellations of the 3-sphere.
Regular di-4-topes (2 facets) include: {3,3,2}, {3,4,2}, {4,3,2}, {5,3,2}, {3,5,2}, {p,2,2}, and their hoso-4-tope duals (2 vertices): {2,3,3}, {2,4,3}, {2,3,4}, {2,3,5}, {2,5,3}, {2,2,p}. 4-polytopes of the form {2,p,2} are the same as {2,2,p}. There are also the cases {p,2,q} which have dihedral cells and hosohedral vertex figures.
Regular hoso-4-topes as 3-sphere honeycombs
Schläfli
{2,p,q}
Coxeter
Cells
{2,p}π/q
Faces
{2}π/p,π/q
Edges Vertices Vertex figure
{p,q}
Symmetry Dual
{2,3,3} 4
{2,3}π/3
6
{2}π/3,π/3
4 2 {3,3}
[2,3,3] {3,3,2}
{2,4,3} 6
{2,4}π/3
12
{2}π/4,π/3
8 2 {4,3}
[2,4,3] {3,4,2}
{2,3,4} 8
{2,3}π/4
12
{2}π/3,π/4
6 2 {3,4}
[2,4,3] {4,3,2}
{2,5,3} 12
{2,5}π/3
30
{2}π/5,π/3
20 2 {5,3}
[2,5,3] {3,5,2}
{2,3,5} 20
{2,3}π/5
30
{2}π/3,π/5
12 2 {3,5}
[2,5,3] {5,3,2}
Stars
There are ten regular star 4-polytopes, which are called the Schläfli–Hess 4-polytopes. Their vertices are based on the convex 120-cell {5,3,3} and 600-cell {3,3,5}.
Ludwig Schläfli found four of them and skipped the last six because he would not allow forms that failed the Euler characteristic on cells or vertex figures (for zero-hole tori: F+V−E=2). Edmund Hess (1843–1903) completed the full list of ten in his German book Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder (1883).
There are 4 unique edge arrangements and 7 unique face arrangements from these 10 regular star 4-polytopes, shown as orthogonal projections:
Name
Wireframe Solid Schläfli
{p, q, r}
Coxeter
Cells
{p, q}
Faces
{p}
Edges
{r}
Vertices
{q, r}
Density χ Symmetry group Dual
{r, q,p}
Icosahedral 120-cell
(faceted 600-cell)
{3,5,5/2}
120
{3,5}
1200
{3}
720
{5/2}
120
{5,5/2}
4 480 H4
[5,3,3]
Small stellated 120-cell
Small stellated 120-cell {5/2,5,3}
120
{5/2,5}
720
{5/2}
1200
{3}
120
{5,3}
4 −480 H4
[5,3,3]
Icosahedral 120-cell
Great 120-cell {5,5/2,5}
120
{5,5/2}
720
{5}
720
{5}
120
{5/2,5}
6 0 H4
[5,3,3]
Self-dual
Grand 120-cell {5,3,5/2}
120
{5,3}
720
{5}
720
{5/2}
120
{3,5/2}
20 0 H4
[5,3,3]
Great stellated 120-cell
Great stellated 120-cell {5/2,3,5}
120
{5/2,3}
720
{5/2}
720
{5}
120
{3,5}
20 0 H4
[5,3,3]
Grand 120-cell
Grand stellated 120-cell {5/2,5,5/2}
120
{5/2,5}
720
{5/2}
720
{5/2}
120
{5,5/2}
66 0 H4
[5,3,3]
Self-dual
Great grand 120-cell {5,5/2,3}
120
{5,5/2}
720
{5}
1200
{3}
120
{5/2,3}
76 −480 H4
[5,3,3]
Great icosahedral 120-cell
Great icosahedral 120-cell
(great faceted 600-cell)
{3,5/2,5}
120
{3,5/2}
1200
{3}
720
{5}
120
{5/2,5}
76 480 H4
[5,3,3]
Great grand 120-cell
Grand 600-cell {3,3,5/2}
600
{3,3}
1200
{3}
720
{5/2}
120
{3,5/2}
191 0 H4
[5,3,3]
Great grand stellated 120-cell
Great grand stellated 120-cell {5/2,3,3}
120
{5/2,3}
720
{5/2}
1200
{3}
600
{3,3}
191 0 H4
[5,3,3]
Grand 600-cell
There are 4 failed potential regular star 4-polytopes permutations: {3,5/2,3}, {4,3,5/2}, {5/2,3,4}, {5/2,3,5/2}. Their cells and vertex figures exist, but they do not cover a hypersphere with a finite number of repetitions.
Five and more dimensions
In five dimensions, a regular polytope can be named as $\{p,q,r,s\}$ where $\{p,q,r\}$ is the 4-face type, $\{p,q\}$ is the cell type, $\{p\}$ is the face type, and $\{s\}$ is the face figure, $\{r,s\}$ is the edge figure, and $\{q,r,s\}$ is the vertex figure.
A vertex figure (of a 5-polytope) is a 4-polytope, seen by the arrangement of neighboring vertices to each vertex.
An edge figure (of a 5-polytope) is a polyhedron, seen by the arrangement of faces around each edge.
A face figure (of a 5-polytope) is a polygon, seen by the arrangement of cells around each face.
A regular 5-polytope $\{p,q,r,s\}$ exists only if $\{p,q,r\}$ and $\{q,r,s\}$ are regular 4-polytopes.
The space it fits in is based on the expression:
${\frac {\cos ^{2}\left({\frac {\pi }{q}}\right)}{\sin ^{2}\left({\frac {\pi }{p}}\right)}}+{\frac {\cos ^{2}\left({\frac {\pi }{r}}\right)}{\sin ^{2}\left({\frac {\pi }{s}}\right)}}$
$<1$ : Spherical 4-space tessellation or 5-space polytope
$=1$ : Euclidean 4-space tessellation
$>1$ : hyperbolic 4-space tessellation
Enumeration of these constraints produce 3 convex polytopes, zero nonconvex polytopes, 3 4-space tessellations, and 5 hyperbolic 4-space tessellations. There are no non-convex regular polytopes in five dimensions or higher.
Convex
In dimensions 5 and higher, there are only three kinds of convex regular polytopes.[10]
Name Schläfli
Symbol
{p1,...,pn−1}
Coxeter k-faces Facet
type
Vertex
figure
Dual
n-simplex{3n−1}...${{n+1} \choose {k+1}}${3n−2}{3n−2}Self-dual
n-cube{4,3n−2}...$2^{n-k}{n \choose k}${4,3n−3}{3n−2}n-orthoplex
n-orthoplex{3n−2,4}...$2^{k+1}{n \choose {k+1}}${3n−2}{3n−3,4}n-cube
There are also improper cases where some numbers in the Schläfli symbol are 2. For example, {p,q,r,...2} is an improper regular spherical polytope whenever {p,q,r...} is a regular spherical polytope, and {2,...p,q,r} is an improper regular spherical polytope whenever {...p,q,r} is a regular spherical polytope. Such polytopes may also be used as facets, yielding forms such as {p,q,...2...y,z}.
5 dimensions
Name Schläfli
Symbol
{p,q,r,s}
Coxeter
Facets
{p,q,r}
Cells
{p,q}
Faces
{p}
Edges Vertices Face
figure
{s}
Edge
figure
{r,s}
Vertex
figure
{q,r,s}
5-simplex {3,3,3,3}
6
{3,3,3}
15
{3,3}
20
{3}
156{3}{3,3}{3,3,3}
5-cube {4,3,3,3}
10
{4,3,3}
40
{4,3}
80
{4}
8032{3}{3,3}{3,3,3}
5-orthoplex {3,3,3,4}
32
{3,3,3}
80
{3,3}
80
{3}
4010{4}{3,4}{3,3,4}
5-simplex
5-cube
5-orthoplex
6 dimensions
NameSchläfliVerticesEdgesFacesCells4-faces5-facesχ
6-simplex{3,3,3,3,3}72135352170
6-cube{4,3,3,3,3}6419224016060120
6-orthoplex{3,3,3,3,4}1260160240192640
6-simplex
6-cube
6-orthoplex
7 dimensions
NameSchläfliVerticesEdgesFacesCells4-faces5-faces6-facesχ
7-simplex{3,3,3,3,3,3}8285670562882
7-cube{4,3,3,3,3,3}12844867256028084142
7-orthoplex{3,3,3,3,3,4}14842805606724481282
7-simplex
7-cube
7-orthoplex
8 dimensions
NameSchläfliVerticesEdgesFacesCells4-faces5-faces6-faces7-facesχ
8-simplex{3,3,3,3,3,3,3}93684126126843690
8-cube{4,3,3,3,3,3,3}2561024179217921120448112160
8-orthoplex{3,3,3,3,3,3,4}1611244811201792179210242560
8-simplex
8-cube
8-orthoplex
9 dimensions
NameSchläfliVerticesEdgesFacesCells4-faces5-faces6-faces7-faces8-facesχ
9-simplex{38}104512021025221012045102
9-cube{4,37}51223044608537640322016672144182
9-orthoplex{37,4}18144672201640325376460823045122
9-simplex
9-cube
9-orthoplex
10 dimensions
NameSchläfliVerticesEdgesFacesCells4-faces5-faces6-faces7-faces8-faces9-facesχ
10-simplex{39}115516533046246233016555110
10-cube{4,38}1024512011520153601344080643360960180200
10-orthoplex{38,4}2018096033608064134401536011520512010240
10-simplex
10-cube
10-orthoplex
...
Non-convex
There are no non-convex regular polytopes in five dimensions or higher, excluding hosotopes formed from lower-dimensional non-convex regular polytopes.
Regular projective polytopes
A projective regular (n+1)-polytope exists when an original regular n-spherical tessellation, {p,q,...}, is centrally symmetric. Such a polytope is named hemi-{p,q,...}, and contain half as many elements. Coxeter gives a symbol {p,q,...}/2, while McMullen writes {p,q,...}h/2 with h as the coxeter number.[11]
Even-sided regular polygons have hemi-2n-gon projective polygons, {2p}/2.
There are 4 regular projective polyhedra related to 4 of 5 Platonic solids.
The hemi-cube and hemi-octahedron generalize as hemi-n-cubes and hemi-n-orthoplexes in any dimensions.
Regular projective polyhedra
3-dimensional regular hemi-polytopes
NameCoxeter
McMullen
ImageFacesEdgesVerticesχ
Hemi-cube{4,3}/2
{4,3}3
3641
Hemi-octahedron{3,4}/2
{3,4}3
4631
Hemi-dodecahedron{5,3}/2
{5,3}5
615101
Hemi-icosahedron{3,5}/2
{3,5}5
101561
Regular projective 4-polytopes
In 4-dimensions 5 of 6 convex regular 4-polytopes generate projective 4-polytopes. The 3 special cases are hemi-24-cell, hemi-600-cell, and hemi-120-cell.
4-dimensional regular hemi-polytopes
NameCoxeter
symbol
McMullen
Symbol
CellsFacesEdgesVerticesχ
Hemi-tesseract {4,3,3}/2{4,3,3}4 4121680
Hemi-16-cell {3,3,4}/2{3,3,4}4 8161240
Hemi-24-cell {3,4,3}/2{3,4,3}6 124848120
Hemi-120-cell {5,3,3}/2{5,3,3}15 603606003000
Hemi-600-cell {3,3,5}/2{3,3,5}15 300600360600
Regular projective 5-polytopes
There are only 2 convex regular projective hemi-polytopes in dimensions 5 or higher: they are the hemi versions of the regular hypercube and orthoplex. They are tabulated below in dimension 5, for example:
Name Schläfli4-facesCellsFacesEdgesVerticesχ
hemi-penteract {4,3,3,3}/25204040161
hemi-pentacross {3,3,3,4}/21640402051
Apeirotopes
An apeirotope or infinite polytope is a polytope which has infinitely many facets. An n-apeirotope is an infinite n-polytope: a 2-apeirotope or apeirogon is an infinite polygon, a 3-apeirotope or apeirohedron is an infinite polyhedron, etc.
There are two main geometric classes of apeirotope:[12]
• Regular honeycombs in n dimensions, which completely fill an n-dimensional space.
• Regular skew apeirotopes, comprising an n-dimensional manifold in a higher space.
One dimension (apeirogons)
The straight apeirogon is a regular tessellation of the line, subdividing it into infinitely many equal segments. It has infinitely many vertices and edges. Its Schläfli symbol is {∞}, and Coxeter diagram .
......
It exists as the limit of the p-gon as p tends to infinity, as follows:
Name Monogon Digon Triangle Square Pentagon Hexagon Heptagon p-gon Apeirogon
Schläfli {1} {2} {3} {4} {5} {6} {7} {p} {∞}
Symmetry D1, [ ] D2, [2] D3, [3]D4, [4]D5, [5]D6, [6]D7, [7][p]
Coxeter or
Image
Apeirogons in the hyperbolic plane, most notably the regular apeirogon, {∞}, can have a curvature just like finite polygons of the Euclidean plane, with the vertices circumscribed by horocycles or hypercycles rather than circles.
Regular apeirogons that are scaled to converge at infinity have the symbol {∞} and exist on horocycles, while more generally they can exist on hypercycles.
{∞} {πi/λ}
Apeirogon on horocycle
Apeirogon on hypercycle
Above are two regular hyperbolic apeirogons in the Poincaré disk model, the right one shows perpendicular reflection lines of divergent fundamental domains, separated by length λ.
Skew apeirogons
A skew apeirogon in two dimensions forms a zig-zag line in the plane. If the zig-zag is even and symmetrical, then the apeirogon is regular.
Skew apeirogons can be constructed in any number of dimensions. In three dimensions, a regular skew apeirogon traces out a helical spiral and may be either left- or right-handed.
2-dimensions 3-dimensions
Zig-zag apeirogon
Helix apeirogon
Euclidean tilings
There are three regular tessellations of the plane. All three have an Euler characteristic (χ) of 0.
Name Square tiling
(quadrille)
Triangular tiling
(deltille)
Hexagonal tiling
(hextille)
Symmetry p4m, [4,4], (*442) p6m, [6,3], (*632)
Schläfli {p,q} {4,4} {3,6} {6,3}
Coxeter diagram
Image
There are two improper regular tilings: {∞,2}, an apeirogonal dihedron, made from two apeirogons, each filling half the plane; and secondly, its dual, {2,∞}, an apeirogonal hosohedron, seen as an infinite set of parallel lines.
{∞,2},
{2,∞},
Euclidean star-tilings
There are no regular plane tilings of star polygons. There are many enumerations that fit in the plane (1/p + 1/q = 1/2), like {8/3,8}, {10/3,5}, {5/2,10}, {12/5,12}, etc., but none repeat periodically.
Hyperbolic tilings
Tessellations of hyperbolic 2-space are hyperbolic tilings. There are infinitely many regular tilings in H2. As stated above, every positive integer pair {p,q} such that 1/p + 1/q < 1/2 gives a hyperbolic tiling. In fact, for the general Schwarz triangle (p, q, r) the same holds true for 1/p + 1/q + 1/r < 1.
There are a number of different ways to display the hyperbolic plane, including the Poincaré disc model which maps the plane into a circle, as shown below. It should be recognized that all of the polygon faces in the tilings below are equal-sized and only appear to get smaller near the edges due to the projection applied, very similar to the effect of a camera fisheye lens.
There are infinitely many flat regular 3-apeirotopes (apeirohedra) as regular tilings of the hyperbolic plane, of the form {p,q}, with p+q<pq/2. (previously listed above as tessellations)
• {3,7}, {3,8}, {3,9} ... {3,∞}
• {4,5}, {4,6}, {4,7} ... {4,∞}
• {5,4}, {5,5}, {5,6} ... {5,∞}
• {6,4}, {6,5}, {6,6} ... {6,∞}
• {7,3}, {7,4}, {7,5} ... {7,∞}
• {8,3}, {8,4}, {8,5} ... {8,∞}
• {9,3}, {9,4}, {9,5} ... {9,∞}
• ...
• {∞,3}, {∞,4}, {∞,5} ... {∞,∞}
A sampling:
Regular hyperbolic tiling table
Spherical (improper/Platonic)/Euclidean/hyperbolic (Poincaré disc: compact/paracompact/noncompact) tessellations with their Schläfli symbol
p \ q 2 3 4 5 6 7 8 ... ∞ ... iπ/λ
2
{2,2}
{2,3}
{2,4}
{2,5}
{2,6}
{2,7}
{2,8}
{2,∞}
{2,iπ/λ}
3
{3,2}
(tetrahedron)
{3,3}
(octahedron)
{3,4}
(icosahedron)
{3,5}
(deltille)
{3,6}
{3,7}
{3,8}
{3,∞}
{3,iπ/λ}
4
{4,2}
(cube)
{4,3}
(quadrille)
{4,4}
{4,5}
{4,6}
{4,7}
{4,8}
{4,∞}
{4,iπ/λ}
5
{5,2}
(dodecahedron)
{5,3}
{5,4}
{5,5}
{5,6}
{5,7}
{5,8}
{5,∞}
{5,iπ/λ}
6
{6,2}
(hextille)
{6,3}
{6,4}
{6,5}
{6,6}
{6,7}
{6,8}
{6,∞}
{6,iπ/λ}
7 {7,2}
{7,3}
{7,4}
{7,5}
{7,6}
{7,7}
{7,8}
{7,∞}
{7,iπ/λ}
8 {8,2}
{8,3}
{8,4}
{8,5}
{8,6}
{8,7}
{8,8}
{8,∞}
{8,iπ/λ}
...
∞
{∞,2}
{∞,3}
{∞,4}
{∞,5}
{∞,6}
{∞,7}
{∞,8}
{∞,∞}
{∞,iπ/λ}
...
iπ/λ
{iπ/λ,2}
{iπ/λ,3}
{iπ/λ,4}
{iπ/λ,5}
{iπ/λ,6}
{iπ/λ,7}
{iπ/λ,8}
{iπ/λ,∞}
{iπ/λ, iπ/λ}
The tilings {p, ∞} have ideal vertices, on the edge of the Poincaré disc model. Their duals {∞, p} have ideal apeirogonal faces, meaning that they are inscribed in horocycles. One could go further (as is done in the table above) and find tilings with ultra-ideal vertices, outside the Poincaré disc, which are dual to tiles inscribed in hypercycles; in what is symbolised {p, iπ/λ} above, infinitely many tiles still fit around each ultra-ideal vertex.[13] (Parallel lines in extended hyperbolic space meet at an ideal point; ultraparallel lines meet at an ultra-ideal point.)[14]
Hyperbolic star-tilings
There are 2 infinite forms of hyperbolic tilings whose faces or vertex figures are star polygons: {m/2, m} and their duals {m, m/2} with m = 7, 9, 11, .... The {m/2, m} tilings are stellations of the {m, 3} tilings while the {m, m/2} dual tilings are facetings of the {3, m} tilings and greatenings of the {m, 3} tilings.
The patterns {m/2, m} and {m, m/2} continue for odd m < 7 as polyhedra: when m = 5, we obtain the small stellated dodecahedron and great dodecahedron, and when m = 3, the case degenerates to a tetrahedron. The other two Kepler–Poinsot polyhedra (the great stellated dodecahedron and great icosahedron) do not have regular hyperbolic tiling analogues. If m is even, depending on how we choose to define {m/2}, we can either obtain degenerate double covers of other tilings or compound tilings.
Name Schläfli Coxeter diagram Image Face type
{p}
Vertex figure
{q}
Density Symmetry Dual
Order-7 heptagrammic tiling {7/2,7} {7/2}
{7}
3 *732
[7,3]
Heptagrammic-order heptagonal tiling
Heptagrammic-order heptagonal tiling {7,7/2} {7}
{7/2}
3 *732
[7,3]
Order-7 heptagrammic tiling
Order-9 enneagrammic tiling {9/2,9} {9/2}
{9}
3 *932
[9,3]
Enneagrammic-order enneagonal tiling
Enneagrammic-order enneagonal tiling {9,9/2} {9}
{9/2}
3 *932
[9,3]
Order-9 enneagrammic tiling
Order-11 hendecagrammic tiling {11/2,11} {11/2}
{11}
3 *11.3.2
[11,3]
Hendecagrammic-order hendecagonal tiling
Hendecagrammic-order hendecagonal tiling {11,11/2} {11}
{11/2}
3 *11.3.2
[11,3]
Order-11 hendecagrammic tiling
Order-p p-grammic tiling {p/2,p} {p/2} {p} 3 *p32
[p,3]
p-grammic-order p-gonal tiling
p-grammic-order p-gonal tiling {p,p/2} {p} {p/2} 3 *p32
[p,3]
Order-p p-grammic tiling
Skew apeirohedra in Euclidean 3-space
There are three regular skew apeirohedra in Euclidean 3-space, with regular skew polygon vertex figures.[15][16][17] They share the same vertex arrangement and edge arrangement of 3 convex uniform honeycombs.
• 6 squares around each vertex: {4,6|4}
• 4 hexagons around each vertex: {6,4|4}
• 6 hexagons around each vertex: {6,6|3}
Regular skew polyhedra
{4,6|4}
{6,4|4}
{6,6|3}
There are thirty regular apeirohedra in Euclidean 3-space.[19] These include those listed above, as well as 8 other "pure" apeirohedra, all related to the cubic honeycomb, {4,3,4}, with others having skew polygon faces: {6,6}4, {4,6}4, {6,4}6, {∞,3}a, {∞,3}b, {∞,4}.*3, {∞,4}6,4, {∞,6}4,4, and {∞,6}6,3.
Skew apeirohedra in hyperbolic 3-space
There are 31 regular skew apeirohedra in hyperbolic 3-space:[20]
• 14 are compact: {8,10|3}, {10,8|3}, {10,4|3}, {4,10|3}, {6,4|5}, {4,6|5}, {10,6|3}, {6,10|3}, {8,8|3}, {6,6|4}, {10,10|3},{6,6|5}, {8,6|3}, and {6,8|3}.
• 17 are paracompact: {12,10|3}, {10,12|3}, {12,4|3}, {4,12|3}, {6,4|6}, {4,6|6}, {8,4|4}, {4,8|4}, {12,6|3}, {6,12|3}, {12,12|3}, {6,6|6}, {8,6|4}, {6,8|4}, {12,8|3}, {8,12|3}, and {8,8|4}.
Tessellations of Euclidean 3-space
There is only one non-degenerate regular tessellation of 3-space (honeycombs), {4, 3, 4}:[21]
Name Schläfli
{p,q,r}
Coxeter
Cell
type
{p,q}
Face
type
{p}
Edge
figure
{r}
Vertex
figure
{q,r}
χ Dual
Cubic honeycomb{4,3,4}{4,3}{4}{4}{3,4}0Self-dual
Improper tessellations of Euclidean 3-space
There are six improper regular tessellations, pairs based on the three regular Euclidean tilings. Their cells and vertex figures are all regular hosohedra {2,n}, dihedra, {n,2}, and Euclidean tilings. These improper regular tilings are constructionally related to prismatic uniform honeycombs by truncation operations. They are higher-dimensional analogues of the order-2 apeirogonal tiling and apeirogonal hosohedron.
Schläfli
{p,q,r}
Coxeter
diagram
Cell
type
{p,q}
Face
type
{p}
Edge
figure
{r}
Vertex
figure
{q,r}
{2,4,4}{2,4}{2}{4}{4,4}
{2,3,6}{2,3}{2}{6}{3,6}
{2,6,3}{2,6}{2}{3}{6,3}
{4,4,2}{4,4}{4}{2}{4,2}
{3,6,2}{3,6}{3}{2}{6,2}
{6,3,2}{6,3}{6}{2}{3,2}
Tessellations of hyperbolic 3-space
There are ten flat regular honeycombs of hyperbolic 3-space:[22] (previously listed above as tessellations)
• 4 are compact: {3,5,3}, {4,3,5}, {5,3,4}, and {5,3,5}
• while 6 are paracompact: {3,3,6}, {6,3,3}, {3,4,4}, {4,4,3}, {3,6,3}, {4,3,6}, {6,3,4}, {4,4,4}, {5,3,6}, {6,3,5}, and {6,3,6}.
4 compact regular honeycombs
{5,3,4}
{5,3,5}
{4,3,5}
{3,5,3}
4 of 11 paracompact regular honeycombs
{3,4,4}
{3,6,3}
{4,4,3}
{4,4,4}
Tessellations of hyperbolic 3-space can be called hyperbolic honeycombs. There are 15 hyperbolic honeycombs in H3, 4 compact and 11 paracompact.
4 compact regular honeycombs
Name Schläfli
Symbol
{p,q,r}
Coxeter
Cell
type
{p,q}
Face
type
{p}
Edge
figure
{r}
Vertex
figure
{q,r}
χ Dual
Icosahedral honeycomb{3,5,3}{3,5}{3}{3}{5,3}0Self-dual
Order-5 cubic honeycomb{4,3,5}{4,3}{4}{5}{3,5}0{5,3,4}
Order-4 dodecahedral honeycomb{5,3,4}{5,3}{5}{4}{3,4}0{4,3,5}
Order-5 dodecahedral honeycomb{5,3,5}{5,3}{5}{5}{3,5}0Self-dual
There are also 11 paracompact H3 honeycombs (those with infinite (Euclidean) cells and/or vertex figures): {3,3,6}, {6,3,3}, {3,4,4}, {4,4,3}, {3,6,3}, {4,3,6}, {6,3,4}, {4,4,4}, {5,3,6}, {6,3,5}, and {6,3,6}.
11 paracompact regular honeycombs
Name Schläfli
Symbol
{p,q,r}
Coxeter
Cell
type
{p,q}
Face
type
{p}
Edge
figure
{r}
Vertex
figure
{q,r}
χ Dual
Order-6 tetrahedral honeycomb{3,3,6}{3,3}{3}{6}{3,6}0{6,3,3}
Hexagonal tiling honeycomb{6,3,3}{6,3}{6}{3}{3,3}0{3,3,6}
Order-4 octahedral honeycomb{3,4,4}{3,4}{3}{4}{4,4}0{4,4,3}
Square tiling honeycomb{4,4,3}{4,4}{4}{3}{4,3}0{3,3,4}
Triangular tiling honeycomb{3,6,3}{3,6}{3}{3}{6,3}0Self-dual
Order-6 cubic honeycomb{4,3,6}{4,3}{4}{4}{3,6}0{6,3,4}
Order-4 hexagonal tiling honeycomb{6,3,4}{6,3}{6}{4}{3,4}0{4,3,6}
Order-4 square tiling honeycomb{4,4,4}{4,4}{4}{4}{4,4}0Self-dual
Order-6 dodecahedral honeycomb{5,3,6}{5,3}{5}{5}{3,6}0{6,3,5}
Order-5 hexagonal tiling honeycomb{6,3,5}{6,3}{6}{5}{3,5}0{5,3,6}
Order-6 hexagonal tiling honeycomb{6,3,6}{6,3}{6}{6}{3,6}0Self-dual
Noncompact solutions exist as Lorentzian Coxeter groups, and can be visualized with open domains in hyperbolic space (the fundamental tetrahedron having ultra-ideal vertices). All honeycombs with hyperbolic cells or vertex figures and do not have 2 in their Schläfli symbol are noncompact.
Spherical (improper/Platonic)/Euclidean/hyperbolic(compact/paracompact/noncompact) honeycombs {p,3,r}
{p,3} \ r 2345678 ... ∞
{2,3}
{2,3,2}
{2,3,3} {2,3,4} {2,3,5} {2,3,6} {2,3,7} {2,3,8} {2,3,∞}
{3,3}
{3,3,2}
{3,3,3}
{3,3,4}
{3,3,5}
{3,3,6}
{3,3,7}
{3,3,8}
{3,3,∞}
{4,3}
{4,3,2}
{4,3,3}
{4,3,4}
{4,3,5}
{4,3,6}
{4,3,7}
{4,3,8}
{4,3,∞}
{5,3}
{5,3,2}
{5,3,3}
{5,3,4}
{5,3,5}
{5,3,6}
{5,3,7}
{5,3,8}
{5,3,∞}
{6,3}
{6,3,2}
{6,3,3}
{6,3,4}
{6,3,5}
{6,3,6}
{6,3,7}
{6,3,8}
{6,3,∞}
{7,3}
{7,3,2}
{7,3,3}
{7,3,4}
{7,3,5}
{7,3,6}
{7,3,7}
{7,3,8}
{7,3,∞}
{8,3}
{8,3,2}
{8,3,3}
{8,3,4}
{8,3,5}
{8,3,6}
{8,3,7}
{8,3,8}
{8,3,∞}
... {∞,3}
{∞,3,2}
{∞,3,3}
{∞,3,4}
{∞,3,5}
{∞,3,6}
{∞,3,7}
{∞,3,8}
{∞,3,∞}
{p,4,r}
{p,4} \ r 23456∞
{2,4}
{2,4,2}
{2,4,3}
{2,4,4}
{2,4,5} {2,4,6} {2,4,∞}
{3,4}
{3,4,2}
{3,4,3}
{3,4,4}
{3,4,5}
{3,4,6}
{3,4,∞}
{4,4}
{4,4,2}
{4,4,3}
{4,4,4}
{4,4,5}
{4,4,6}
{4,4,∞}
{5,4}
{5,4,2}
{5,4,3}
{5,4,4}
{5,4,5}
{5,4,6}
{5,4,∞}
{6,4}
{6,4,2}
{6,4,3}
{6,4,4}
{6,4,5}
{6,4,6}
{6,4,∞}
{∞,4}
{∞,4,2}
{∞,4,3}
{∞,4,4}
{∞,4,5}
{∞,4,6}
{∞,4,∞}
{p,5,r}
{p,5} \ r 23456∞
{2,5}
{2,5,2}
{2,5,3} {2,5,4} {2,5,5} {2,5,6} {2,5,∞}
{3,5}
{3,5,2}
{3,5,3}
{3,5,4}
{3,5,5}
{3,5,6}
{3,5,∞}
{4,5}
{4,5,2}
{4,5,3}
{4,5,4}
{4,5,5}
{4,5,6}
{4,5,∞}
{5,5}
{5,5,2}
{5,5,3}
{5,5,4}
{5,5,5}
{5,5,6}
{5,5,∞}
{6,5}
{6,5,2}
{6,5,3}
{6,5,4}
{6,5,5}
{6,5,6}
{6,5,∞}
{∞,5}
{∞,5,2}
{∞,5,3}
{∞,5,4}
{∞,5,5}
{∞,5,6}
{∞,5,∞}
{p,6,r}
{p,6} \ r 23456∞
{2,6}
{2,6,2}
{2,6,3} {2,6,4} {2,6,5} {2,6,6} {2,6,∞}
{3,6}
{3,6,2}
{3,6,3}
{3,6,4}
{3,6,5}
{3,6,6}
{3,6,∞}
{4,6}
{4,6,2}
{4,6,3}
{4,6,4}
{4,6,5}
{4,6,6}
{4,6,∞}
{5,6}
{5,6,2}
{5,6,3}
{5,6,4}
{5,6,5}
{5,6,6}
{5,6,∞}
{6,6}
{6,6,2}
{6,6,3}
{6,6,4}
{6,6,5}
{6,6,6}
{6,6,∞}
{∞,6}
{∞,6,2}
{∞,6,3}
{∞,6,4}
{∞,6,5}
{∞,6,6}
{∞,6,∞}
{p,7,r}
{p,7} \ r 23456∞
{2,7}
{2,7,2}
{2,7,3} {2,7,4} {2,7,5} {2,7,6} {2,7,∞}
{3,7}
{3,7,2}
{3,7,3}
{3,7,4}
{3,7,5}
{3,7,6}
{3,7,∞}
{4,7}
{4,7,2}
{4,7,3}
{4,7,4}
{4,7,5}
{4,7,6}
{4,7,∞}
{5,7}
{5,7,2}
{5,7,3}
{5,7,4}
{5,7,5}
{5,7,6}
{5,7,∞}
{6,7}
{6,7,2}
{6,7,3}
{6,7,4}
{6,7,5}
{6,7,6}
{6,7,∞}
{∞,7}
{∞,7,2}
{∞,7,3}
{∞,7,4}
{∞,7,5}
{∞,7,6}
{∞,7,∞}
{p,8,r}
{p,8} \ r 23456∞
{2,8}
{2,8,2}
{2,8,3} {2,8,4} {2,8,5} {2,8,6} {2,8,∞}
{3,8}
{3,8,2}
{3,8,3}
{3,8,4}
{3,8,5}
{3,8,6}
{3,8,∞}
{4,8}
{4,8,2}
{4,8,3}
{4,8,4}
{4,8,5}
{4,8,6}
{4,8,∞}
{5,8}
{5,8,2}
{5,8,3}
{5,8,4}
{5,8,5}
{5,8,6}
{5,8,∞}
{6,8}
{6,8,2}
{6,8,3}
{6,8,4}
{6,8,5}
{6,8,6}
{6,8,∞}
{∞,8}
{∞,8,2}
{∞,8,3}
{∞,8,4}
{∞,8,5}
{∞,8,6}
{∞,8,∞}
{p,∞,r}
{p,∞} \ r 23456∞
{2,∞}
{2,∞,2}
{2,∞,3} {2,∞,4} {2,∞,5} {2,∞,6} {2,∞,∞}
{3,∞}
{3,∞,2}
{3,∞,3}
{3,∞,4}
{3,∞,5}
{3,∞,6}
{3,∞,∞}
{4,∞}
{4,∞,2}
{4,∞,3}
{4,∞,4}
{4,∞,5}
{4,∞,6}
{4,∞,∞}
{5,∞}
{5,∞,2}
{5,∞,3}
{5,∞,4}
{5,∞,5}
{5,∞,6}
{5,∞,∞}
{6,∞}
{6,∞,2}
{6,∞,3}
{6,∞,4}
{6,∞,5}
{6,∞,6}
{6,∞,∞}
{∞,∞}
{∞,∞,2}
{∞,∞,3}
{∞,∞,4}
{∞,∞,5}
{∞,∞,6}
{∞,∞,∞}
There are no regular hyperbolic star-honeycombs in H3: all forms with a regular star polyhedron as cell, vertex figure or both end up being spherical.
Ideal vertices now appear when the vertex figure is a Euclidean tiling, becoming inscribable in a horosphere rather than a sphere. They are dual to ideal cells (Euclidean tilings rather than finite polyhedra). As the last number in the Schläfli symbol rises further, the vertex figure becomes hyperbolic, and vertices become ultra-ideal (so the edges do not meet within hyperbolic space). In honeycombs {p, q, ∞} the edges intersect the Poincaré ball only in one ideal point; the rest of the edge has become ultra-ideal. Continuing further would lead to edges that are completely ultra-ideal, both for the honeycomb and for the fundamental simplex (though still infinitely many {p, q} would meet at such edges). In general, when the last number of the Schläfli symbol becomes ∞, faces of codimension two intersect the Poincaré hyperball only in one ideal point.[13]
Tessellations of Euclidean 4-space
There are three kinds of infinite regular tessellations (honeycombs) that can tessellate Euclidean four-dimensional space:
3 regular Euclidean honeycombs
Name Schläfli
Symbol
{p,q,r,s}
Facet
type
{p,q,r}
Cell
type
{p,q}
Face
type
{p}
Face
figure
{s}
Edge
figure
{r,s}
Vertex
figure
{q,r,s}
Dual
Tesseractic honeycomb{4,3,3,4}{4,3,3}{4,3}{4}{4}{3,4}{3,3,4}Self-dual
16-cell honeycomb{3,3,4,3}{3,3,4}{3,3}{3}{3}{4,3}{3,4,3}{3,4,3,3}
24-cell honeycomb{3,4,3,3}{3,4,3}{3,4}{3}{3}{3,3}{4,3,3}{3,3,4,3}
Projected portion of {4,3,3,4}
(Tesseractic honeycomb)
Projected portion of {3,3,4,3}
(16-cell honeycomb)
Projected portion of {3,4,3,3}
(24-cell honeycomb)
There are also the two improper cases {4,3,4,2} and {2,4,3,4}.
There are three flat regular honeycombs of Euclidean 4-space:[21]
• {4,3,3,4}, {3,3,4,3}, and {3,4,3,3}.
There are seven flat regular convex honeycombs of hyperbolic 4-space:[22]
• 5 are compact: {3,3,3,5}, {5,3,3,3}, {4,3,3,5}, {5,3,3,4}, {5,3,3,5}
• 2 are paracompact: {3,4,3,4}, and {4,3,4,3}.
There are four flat regular star honeycombs of hyperbolic 4-space:[22]
• {5/2,5,3,3}, {3,3,5,5/2}, {3,5,5/2,5}, and {5,5/2,5,3}.
Tessellations of hyperbolic 4-space
There are seven convex regular honeycombs and four star-honeycombs in H4 space.[23] Five convex ones are compact, and two are paracompact.
Five compact regular honeycombs in H4:
5 compact regular honeycombs
Name Schläfli
Symbol
{p,q,r,s}
Facet
type
{p,q,r}
Cell
type
{p,q}
Face
type
{p}
Face
figure
{s}
Edge
figure
{r,s}
Vertex
figure
{q,r,s}
Dual
Order-5 5-cell honeycomb{3,3,3,5}{3,3,3}{3,3}{3}{5}{3,5}{3,3,5}{5,3,3,3}
120-cell honeycomb{5,3,3,3}{5,3,3}{5,3}{5}{3}{3,3}{3,3,3}{3,3,3,5}
Order-5 tesseractic honeycomb{4,3,3,5}{4,3,3}{4,3}{4}{5}{3,5}{3,3,5}{5,3,3,4}
Order-4 120-cell honeycomb{5,3,3,4}{5,3,3}{5,3}{5}{4}{3,4}{3,3,4}{4,3,3,5}
Order-5 120-cell honeycomb{5,3,3,5}{5,3,3}{5,3}{5}{5}{3,5}{3,3,5}Self-dual
The two paracompact regular H4 honeycombs are: {3,4,3,4}, {4,3,4,3}.
2 paracompact regular honeycombs
Name Schläfli
Symbol
{p,q,r,s}
Facet
type
{p,q,r}
Cell
type
{p,q}
Face
type
{p}
Face
figure
{s}
Edge
figure
{r,s}
Vertex
figure
{q,r,s}
Dual
Order-4 24-cell honeycomb{3,4,3,4}{3,4,3}{3,4}{3}{4}{3,4}{4,3,4}{4,3,4,3}
Cubic honeycomb honeycomb{4,3,4,3}{4,3,4}{4,3}{4}{3}{4,3}{3,4,3}{3,4,3,4}
Noncompact solutions exist as Lorentzian Coxeter groups, and can be visualized with open domains in hyperbolic space (the fundamental 5-cell having some parts inaccessible beyond infinity). All honeycombs which are not shown in the set of tables below and do not have 2 in their Schläfli symbol are noncompact.
Spherical/Euclidean/hyperbolic(compact/paracompact/noncompact) honeycombs {p,q,r,s}
q=3, s=3
p \ r 3 4 5
3
{3,3,3,3}
{3,3,4,3}
{3,3,5,3}
4
{4,3,3,3}
{4,3,4,3}
{4,3,5,3}
5
{5,3,3,3}
{5,3,4,3}
{5,3,5,3}
q=3, s=4
p \ r 3 4
3
{3,3,3,4}
{3,3,4,4}
4
{4,3,3,4}
{4,3,4,4}
5
{5,3,3,4}
{5,3,4,4}
q=3, s=5
p \ r 3 4
3
{3,3,3,5}
{3,3,4,5}
4
{4,3,3,5}
{4,3,4,5}
5
{5,3,3,5}
{5,3,4,5}
q=4, s=3
p \ r 3 4
3
{3,4,3,3}
{3,4,4,3}
4
{4,4,3,3}
{4,4,4,3}
q=4, s=4
p \ r 3 4
3
{3,4,3,4}
{3,4,4,4}
4
{4,4,3,4}
{4,4,4,4}
q=4, s=5
p \ r 3 4
3
{3,4,3,5}
{3,4,4,5}
4
{4,4,3,5}
{4,4,4,5}
q=5, s=3
p \ r 3 4
3
{3,5,3,3}
{3,5,4,3}
4
{4,5,3,3}
{4,5,4,3}
Star tessellations of hyperbolic 4-space
There are four regular star-honeycombs in H4 space, all compact:
4 compact regular star-honeycombs
Name Schläfli
Symbol
{p,q,r,s}
Facet
type
{p,q,r}
Cell
type
{p,q}
Face
type
{p}
Face
figure
{s}
Edge
figure
{r,s}
Vertex
figure
{q,r,s}
Dual Density
Small stellated 120-cell honeycomb{5/2,5,3,3}{5/2,5,3}{5/2,5}{5/2}{3}{3,3}{5,3,3}{3,3,5,5/2}5
Pentagrammic-order 600-cell honeycomb{3,3,5,5/2}{3,3,5}{3,3}{3}{5/2}{5,5/2}{3,5,5/2}{5/2,5,3,3}5
Order-5 icosahedral 120-cell honeycomb{3,5,5/2,5}{3,5,5/2}{3,5}{3}{5}{5/2,5}{5,5/2,5}{5,5/2,5,3}10
Great 120-cell honeycomb{5,5/2,5,3}{5,5/2,5}{5,5/2}{5}{3}{5,3}{5/2,5,3}{3,5,5/2,5}10
Five dimensions (6-apeirotopes)
There is only one flat regular honeycomb of Euclidean 5-space: (previously listed above as tessellations)[21]
• {4,3,3,3,4}
There are five flat regular regular honeycombs of hyperbolic 5-space, all paracompact: (previously listed above as tessellations)[22]
• {3,3,3,4,3}, {3,4,3,3,3}, {3,3,4,3,3}, {3,4,3,3,4}, and {4,3,3,4,3}
Tessellations of Euclidean 5-space
The hypercubic honeycomb is the only family of regular honeycombs that can tessellate each dimension, five or higher, formed by hypercube facets, four around every ridge.
Name Schläfli
{p1, p2, ..., pn−1}
Facet
type
Vertex
figure
Dual
Square tiling{4,4}{4}{4}Self-dual
Cubic honeycomb{4,3,4}{4,3}{3,4}Self-dual
Tesseractic honeycomb{4,32,4}{4,32}{32,4}Self-dual
5-cube honeycomb{4,33,4}{4,33}{33,4}Self-dual
6-cube honeycomb{4,34,4}{4,34}{34,4}Self-dual
7-cube honeycomb{4,35,4}{4,35}{35,4}Self-dual
8-cube honeycomb{4,36,4}{4,36}{36,4}Self-dual
n-hypercubic honeycomb{4,3n−2,4}{4,3n−2}{3n−2,4}Self-dual
In E5, there are also the improper cases {4,3,3,4,2}, {2,4,3,3,4}, {3,3,4,3,2}, {2,3,3,4,3}, {3,4,3,3,2}, and {2,3,4,3,3}. In En, {4,3n−3,4,2} and {2,4,3n−3,4} are always improper Euclidean tessellations.
Tessellations of hyperbolic 5-space
There are 5 regular honeycombs in H5, all paracompact, which include infinite (Euclidean) facets or vertex figures: {3,4,3,3,3}, {3,3,4,3,3}, {3,3,3,4,3}, {3,4,3,3,4}, and {4,3,3,4,3}.
There are no compact regular tessellations of hyperbolic space of dimension 5 or higher and no paracompact regular tessellations in hyperbolic space of dimension 6 or higher.
5 paracompact regular honeycombs
Name Schläfli
Symbol
{p,q,r,s,t}
Facet
type
{p,q,r,s}
4-face
type
{p,q,r}
Cell
type
{p,q}
Face
type
{p}
Cell
figure
{t}
Face
figure
{s,t}
Edge
figure
{r,s,t}
Vertex
figure
{q,r,s,t}
Dual
5-orthoplex honeycomb{3,3,3,4,3}{3,3,3,4}{3,3,3}{3,3}{3}{3}{4,3}{3,4,3}{3,3,4,3}{3,4,3,3,3}
24-cell honeycomb honeycomb{3,4,3,3,3}{3,4,3,3}{3,4,3}{3,4}{3}{3}{3,3}{3,3,3}{4,3,3,3}{3,3,3,4,3}
16-cell honeycomb honeycomb{3,3,4,3,3}{3,3,4,3}{3,3,4}{3,3}{3}{3}{3,3}{4,3,3}{3,4,3,3}self-dual
Order-4 24-cell honeycomb honeycomb{3,4,3,3,4}{3,4,3,3}{3,4,3}{3,4}{3}{4}{3,4}{3,3,4}{4,3,3,4}{4,3,3,4,3}
Tesseractic honeycomb honeycomb{4,3,3,4,3}{4,3,3,4}{4,3,3}{4,3}{4}{3}{4,3}{3,4,3}{3,3,4,3}{3,4,3,3,4}
Since there are no regular star n-polytopes for n ≥ 5, that could be potential cells or vertex figures, there are no more hyperbolic star honeycombs in Hn for n ≥ 5.
Tessellations of hyperbolic 6-space and higher
There are no regular compact or paracompact tessellations of hyperbolic space of dimension 6 or higher. However, any Schläfli symbol of the form {p,q,r,s,...} not covered above (p,q,r,s,... natural numbers above 2, or infinity) will form a noncompact tessellation of hyperbolic n-space.[13]
Compound polytopes
Main article: Polytope compound
Two dimensional compounds
For any natural number n, there are n-pointed star regular polygonal stars with Schläfli symbols {n/m} for all m such that m < n/2 (strictly speaking {n/m}={n/(n−m)}) and m and n are coprime. When m and n are not coprime, the star polygon obtained will be a regular polygon with n/m sides. A new figure is obtained by rotating these regular n/m-gons one vertex to the left on the original polygon until the number of vertices rotated equals n/m minus one, and combining these figures. An extreme case of this is where n/m is 2, producing a figure consisting of n/2 straight line segments; this is called a degenerate star polygon.
In other cases where n and m have a common factor, a star polygon for a lower n is obtained, and rotated versions can be combined. These figures are called star figures, improper star polygons or compound polygons. The same notation {n/m} is often used for them, although authorities such as Grünbaum (1994) regard (with some justification) the form k{n} as being more correct, where usually k = m.
A further complication comes when we compound two or more star polygons, as for example two pentagrams, differing by a rotation of 36°, inscribed in a decagon. This is correctly written in the form k{n/m}, as 2{5/2}, rather than the commonly used {10/4}.
Coxeter's extended notation for compounds is of the form c{m,n,...}[d{p,q,...}]e{s,t,...}, indicating that d distinct {p,q,...}'s together cover the vertices of {m,n,...} c times and the facets of {s,t,...} e times. If no regular {m,n,...} exists, the first part of the notation is removed, leaving [d{p,q,...}]e{s,t,...}; the opposite holds if no regular {s,t,...} exists. The dual of c{m,n,...}[d{p,q,...}]e{s,t,...} is e{t,s,...}[d{q,p,...}]c{n,m,...}. If c or e are 1, they may be omitted. For compound polygons, this notation reduces to {nk}[k{n/m}]{nk}: for example, the hexagram may be written thus as {6}[2{3}]{6}.
Examples for n=2..10, nk≤30
2{2}
3{2}
4{2}
5{2}
6{2}
7{2}
8{2}
9{2}
10{2}
11{2}
12{2}
13{2}
14{2}
15{2}
2{3}
3{3}
4{3}
5{3}
6{3}
7{3}
8{3}
9{3}
10{3}
2{4}
3{4}
4{4}
5{4}
6{4}
7{4}
2{5}
3{5}
4{5}
5{5}
6{5}
2{5/2}
3{5/2}
4{5/2}
5{5/2}
6{5/2}
2{6}
3{6}
4{6}
5{6}
2{7}
3{7}
4{7}
2{7/2}
3{7/2}
4{7/2}
2{7/3}
3{7/3}
4{7/3}
2{8}
3{8}
2{8/3}
3{8/3}
2{9}
3{9}
2{9/2}
3{9/2}
2{9/4}
3{9/4}
2{10}
3{10}
2{10/3}
3{10/3}
2{11}
2{11/2}
2{11/3}
2{11/4}
2{11/5}
2{12}
2{12/5}
2{13}
2{13/2}
2{13/3}
2{13/4}
2{13/5}
2{13/6}
2{14}
2{14/3}
2{14/5}
2{15}
2{15/2}
2{15/4}
2{15/7}
Regular skew polygons also create compounds, seen in the edges of prismatic compound of antiprisms, for instance:
Regular compound skew polygon
Compound
skew squares
Compound
skew hexagons
Compound
skew decagons
Two {2}#{ } Three {2}#{ } Two {3}#{ } Two {5/3}#{ }
Three dimensional compounds
A regular polyhedron compound can be defined as a compound which, like a regular polyhedron, is vertex-transitive, edge-transitive, and face-transitive. With this definition there are 5 regular compounds.
Symmetry [4,3], Oh [5,3]+, I [5,3], Ih
Duality Self-dual Dual pairs
Image
Spherical
Polyhedra 2 {3,3} 5 {3,3} 10 {3,3} 5 {4,3} 5 {3,4}
Coxeter {4,3}[2{3,3}]{3,4} {5,3}[5{3,3}]{3,5} 2{5,3}[10{3,3}]2{3,5} 2{5,3}[5{4,3}] [5{3,4}]2{3,5}
Coxeter's notation for regular compounds is given in the table above, incorporating Schläfli symbols. The material inside the square brackets, [d{p,q}], denotes the components of the compound: d separate {p,q}'s. The material before the square brackets denotes the vertex arrangement of the compound: c{m,n}[d{p,q}] is a compound of d {p,q}'s sharing the vertices of an {m,n} counted c times. The material after the square brackets denotes the facet arrangement of the compound: [d{p,q}]e{s,t} is a compound of d {p,q}'s sharing the faces of {s,t} counted e times. These may be combined: thus c{m,n}[d{p,q}]e{s,t} is a compound of d {p,q}'s sharing the vertices of {m,n} counted c times and the faces of {s,t} counted e times. This notation can be generalised to compounds in any number of dimensions.[24]
Euclidean and hyperbolic plane compounds
There are eighteen two-parameter families of regular compound tessellations of the Euclidean plane. In the hyperbolic plane, five one-parameter families and seventeen isolated cases are known, but the completeness of this listing has not yet been proven.
The Euclidean and hyperbolic compound families 2 {p,p} (4 ≤ p ≤ ∞, p an integer) are analogous to the spherical stella octangula, 2 {3,3}.
A few examples of Euclidean and hyperbolic regular compounds
Self-dual Duals Self-dual
2 {4,4} 2 {6,3} 2 {3,6} 2 {∞,∞}
{{4,4}} or a{4,4} or {4,4}[2{4,4}]{4,4}
+ or
[2{6,3}]{3,6} a{6,3} or {6,3}[2{3,6}]
+ or
{{∞,∞}} or a{∞,∞} or {4,∞}[2{∞,∞}]{∞,4}
+ or
3 {6,3} 3 {3,6} 3 {∞,∞}
2{3,6}[3{6,3}]{6,3} {3,6}[3{3,6}]2{6,3}
+ +
+ +
Four dimensional compounds
Orthogonal projections
75 {4,3,3} 75 {3,3,4}
Coxeter lists 32 regular compounds of regular 4-polytopes in his book Regular Polytopes.[25] McMullen adds six in his paper New Regular Compounds of 4-Polytopes, in which he also proves that the list is now complete.[26] In the following tables, the superscript (var) indicates that the labeled compounds are distinct from the other compounds with the same symbols.
Self-dual regular compounds
Compound Constituent Symmetry Vertex arrangement Cell arrangement
120 {3,3,3}5-cell[5,3,3], order 14400[25]{5,3,3}{3,3,5}
120 {3,3,3}(var)5-cellorder 1200[26]{5,3,3}{3,3,5}
720 {3,3,3}5-cell[5,3,3], order 14400[26]6{5,3,3}6{3,3,5}
5 {3,4,3}24-cell[5,3,3], order 14400[25]{3,3,5}{5,3,3}
Regular compounds as dual pairs
Compound 1 Compound 2 Symmetry Vertex arrangement (1) Cell arrangement (1) Vertex arrangement (2) Cell arrangement (2)
3 {3,3,4}[27]3 {4,3,3}[3,4,3], order 1152[25]{3,4,3}2{3,4,3}2{3,4,3}{3,4,3}
15 {3,3,4}15 {4,3,3}[5,3,3], order 14400[25]{3,3,5}2{5,3,3}2{3,3,5}{5,3,3}
75 {3,3,4}75 {4,3,3}[5,3,3], order 14400[25]5{3,3,5}10{5,3,3}10{3,3,5}5{5,3,3}
75 {3,3,4}75 {4,3,3}[5,3,3], order 14400[25]{5,3,3}2{3,3,5}2{5,3,3}{3,3,5}
75 {3,3,4}75 {4,3,3}order 600[26]{5,3,3}2{3,3,5}2{5,3,3}{3,3,5}
300 {3,3,4}300 {4,3,3}[5,3,3]+, order 7200[25]4{5,3,3}8{3,3,5}8{5,3,3}4{3,3,5}
600 {3,3,4}600 {4,3,3}[5,3,3], order 14400[25]8{5,3,3}16{3,3,5}16{5,3,3}8{3,3,5}
25 {3,4,3}25 {3,4,3}[5,3,3], order 14400[25]{5,3,3}5{5,3,3}5{3,3,5}{3,3,5}
There are two different compounds of 75 tesseracts: one shares the vertices of a 120-cell, while the other shares the vertices of a 600-cell. It immediately follows therefore that the corresponding dual compounds of 75 16-cells are also different.
Self-dual star compounds
Compound Symmetry Vertex arrangement Cell arrangement
5 {5,5/2,5}[5,3,3]+, order 7200[25]{5,3,3}{3,3,5}
10 {5,5/2,5}[5,3,3], order 14400[25]2{5,3,3}2{3,3,5}
5 {5/2,5,5/2}[5,3,3]+, order 7200[25]{5,3,3}{3,3,5}
10 {5/2,5,5/2}[5,3,3], order 14400[25]2{5,3,3}2{3,3,5}
Regular star compounds as dual pairs
Compound 1 Compound 2 Symmetry Vertex arrangement (1) Cell arrangement (1) Vertex arrangement (2) Cell arrangement (2)
5 {3,5,5/2}5 {5/2,5,3}[5,3,3]+, order 7200[25]{5,3,3}{3,3,5}{5,3,3}{3,3,5}
10 {3,5,5/2}10 {5/2,5,3}[5,3,3], order 14400[25]2{5,3,3}2{3,3,5}2{5,3,3}2{3,3,5}
5 {5,5/2,3}5 {3,5/2,5}[5,3,3]+, order 7200[25]{5,3,3}{3,3,5}{5,3,3}{3,3,5}
10 {5,5/2,3}10 {3,5/2,5}[5,3,3], order 14400[25]2{5,3,3}2{3,3,5}2{5,3,3}2{3,3,5}
5 {5/2,3,5}5 {5,3,5/2}[5,3,3]+, order 7200[25]{5,3,3}{3,3,5}{5,3,3}{3,3,5}
10 {5/2,3,5}10 {5,3,5/2}[5,3,3], order 14400[25]2{5,3,3}2{3,3,5}2{5,3,3}2{3,3,5}
There are also fourteen partially regular compounds, that are either vertex-transitive or cell-transitive but not both. The seven vertex-transitive partially regular compounds are the duals of the seven cell-transitive partially regular compounds.
Partially regular compounds as dual pairs
Compound 1
Vertex-transitive
Compound 2
Cell-transitive
Symmetry
2 16-cells[28]2 tesseracts[4,3,3], order 384[25]
25 24-cell(var)25 24-cell(var)order 600[26]
100 24-cell100 24-cell[5,3,3]+, order 7200[25]
200 24-cell200 24-cell[5,3,3], order 14400[25]
5 600-cell5 120-cell[5,3,3]+, order 7200[25]
10 600-cell10 120-cell[5,3,3], order 14400[25]
Partially regular star compounds as dual pairs
Compound 1
Vertex-transitive
Compound 2
Cell-transitive
Symmetry
5 {3,3,5/2}5 {5/2,3,3}[5,3,3]+, order 7200[25]
10 {3,3,5/2}10 {5/2,3,3}[5,3,3], order 14400[25]
Although the 5-cell and 24-cell are both self-dual, their dual compounds (the compound of two 5-cells and compound of two 24-cells) are not considered to be regular, unlike the compound of two tetrahedra and the various dual polygon compounds, because they are neither vertex-regular nor cell-regular: they are not facetings or stellations of any regular 4-polytope. However, they are vertex-, edge-, face-, and cell-transitive.
Euclidean 3-space compounds
The only regular Euclidean compound honeycombs are an infinite family of compounds of cubic honeycombs, all sharing vertices and faces with another cubic honeycomb. This compound can have any number of cubic honeycombs. The Coxeter notation is {4,3,4}[d{4,3,4}]{4,3,4}.
Five dimensions and higher compounds
There are no regular compounds in five or six dimensions. There are three known seven-dimensional compounds (16, 240, or 480 7-simplices), and six known eight-dimensional ones (16, 240, or 480 8-cubes or 8-orthoplexes). There is also one compound of n-simplices in n-dimensional space provided that n is one less than a power of two, and also two compounds (one of n-cubes and a dual one of n-orthoplexes) in n-dimensional space if n is a power of two.
The Coxeter notation for these compounds are (using αn = {3n−1}, βn = {3n−2,4}, γn = {4,3n−2}):
• 7-simplexes: cγ7[16cα7]cβ7, where c = 1, 15, or 30
• 8-orthoplexes: cγ8[16cβ8]
• 8-cubes: [16cγ8]cβ8
The general cases (where n = 2k and d = 22k − k − 1, k = 2, 3, 4, ...):
• Simplexes: γn−1[dαn−1]βn−1
• Orthoplexes: γn[dβn]
• Hypercubes: [dγn]βn
Euclidean honeycomb compounds
A known family of regular Euclidean compound honeycombs in five or more dimensions is an infinite family of compounds of hypercubic honeycombs, all sharing vertices and faces with another hypercubic honeycomb. This compound can have any number of hypercubic honeycombs. The Coxeter notation is δn[dδn]δn where δn = {∞} when n = 2 and {4,3n−3,4} when n ≥ 3.
Abstract polytopes
The abstract polytopes arose out of an attempt to study polytopes apart from the geometrical space they are embedded in. They include the tessellations of spherical, Euclidean and hyperbolic space, tessellations of other manifolds, and many other objects that do not have a well-defined topology, but instead may be characterised by their "local" topology. There are infinitely many in every dimension. See this atlas for a sample. Some notable examples of abstract regular polytopes that do not appear elsewhere in this list are the 11-cell, {3,5,3}, and the 57-cell, {5,3,5}, which have regular projective polyhedra as cells and vertex figures.
The elements of an abstract polyhedron are its body (the maximal element), its faces, edges, vertices and the null polytope or empty set. These abstract elements can be mapped into ordinary space or realised as geometrical figures. Some abstract polyhedra have well-formed or faithful realisations, others do not. A flag is a connected set of elements of each dimension - for a polyhedron that is the body, a face, an edge of the face, a vertex of the edge, and the null polytope. An abstract polytope is said to be regular if its combinatorial symmetries are transitive on its flags - that is to say, that any flag can be mapped onto any other under a symmetry of the polyhedron. Abstract regular polytopes remain an active area of research.
Five such regular abstract polyhedra, which can not be realised faithfully, were identified by H. S. M. Coxeter in his book Regular Polytopes (1977) and again by J. M. Wills in his paper "The combinatorially regular polyhedra of index 2" (1987).[29] They are all topologically equivalent to toroids. Their construction, by arranging n faces around each vertex, can be repeated indefinitely as tilings of the hyperbolic plane. In the diagrams below, the hyperbolic tiling images have colors corresponding to those of the polyhedra images.
Polyhedron
Medial rhombic triacontahedron
Dodecadodecahedron
Medial triambic icosahedron
Ditrigonal dodecadodecahedron
Excavated dodecahedron
Vertex figure {5}, {5/2}
(5.5/2)2
{5}, {5/2}
(5.5/3)3
Faces 30 rhombi
12 pentagons
12 pentagrams
20 hexagons
12 pentagons
12 pentagrams
20 hexagrams
Tiling
{4, 5}
{5, 4}
{6, 5}
{5, 6}
{6, 6}
χ −6 −6 −16 −16 −20
These occur as dual pairs as follows:
• The medial rhombic triacontahedron and dodecadodecahedron are dual to each other.
• The medial triambic icosahedron and ditrigonal dodecadodecahedron are dual to each other.
• The excavated dodecahedron is self-dual.
See also
• Polygon
• Regular polygon
• Star polygon
• Polyhedron
• Regular polyhedron (5 regular Platonic solids and 4 Kepler–Poinsot solids)
• Uniform polyhedron
• Petrial
• 4-polytope
• Regular 4-polytope (16 regular 4-polytopes, 4 convex and 10 star (Schläfli–Hess))
• Uniform 4-polytope
• Tessellation
• Tilings of regular polygons
• Convex uniform honeycomb
• Regular polytope
• Uniform polytope
• Regular map (graph theory)
Notes
1. Coxeter (1973), p. 129.
2. McMullen & Schulte (2002), p. 30.
3. Johnson, N.W. (2018). "Chapter 11: Finite symmetry groups". Geometries and Transformations. 11.1 Polytopes and Honeycombs, p. 224. ISBN 978-1-107-10340-5.
4. Coxeter (1973), p. 120.
5. Coxeter (1973), p. 124.
6. Coxeter, Regular Complex Polytopes, p. 9
7. Duncan, Hugh (28 September 2017). "Between a square rock and a hard pentagon: Fractional polygons". chalkdust.
8. Coxeter (1973), pp. 66–67.
9. Abstracts (PDF). Convex and Abstract Polytopes (May 19–21, 2005) and Polytopes Day in Calgary (May 22, 2005).
10. Coxeter (1973), Table I: Regular polytopes, (iii) The three regular polytopes in n dimensions (n>=5), pp. 294–295.
11. McMullen & Schulte (2002), "6C Projective Regular Polytopes" pp. 162-165.
12. Grünbaum, B. (1977). "Regular Polyhedra—Old and New". Aequationes Mathematicae. 16 (1–2): 1–20. doi:10.1007/BF01836414. S2CID 125049930.
13. Roice Nelson and Henry Segerman, Visualizing Hyperbolic Honeycombs
14. Irving Adler, A New Look at Geometry (2012 Dover edition), p.233
15. Coxeter, H.S.M. (1938). "Regular Skew Polyhedra in Three and Four Dimensions". Proc. London Math. Soc. 2. 43: 33–62. doi:10.1112/plms/s2-43.1.33.
16. Coxeter, H.S.M. (1985). "Regular and semi-regular polytopes II". Mathematische Zeitschrift. 188 (4): 559–591. doi:10.1007/BF01161657. S2CID 120429557.
17. Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). "Chapter 23: Objects with Primary Symmetry, Infinite Platonic Polyhedra". The Symmetries of Things. Taylor & Francis. pp. 333–335. ISBN 978-1-568-81220-5.
18. McMullen & Schulte (2002), p. 224.
19. McMullen & Schulte (2002), Section 7E.
20. Garner, C.W.L. (1967). "Regular Skew Polyhedra in Hyperbolic Three-Space". Can. J. Math. 19: 1179–1186. doi:10.4153/CJM-1967-106-9. S2CID 124086497. Note: His paper says there are 32, but one is self-dual, leaving 31.
21. Coxeter (1973), Table II: Regular honeycombs, p. 296.
22. Coxeter (1999), "Chapter 10".
23. Coxeter (1999), "Chapter 10" Table IV, p. 213.
24. Coxeter (1973), p. 48.
25. Coxeter (1973). Table VII, p. 305
26. McMullen (2018).
27. Klitzing, Richard. "Uniform compound stellated icositetrachoron".
28. Klitzing, Richard. "Uniform compound demidistesseract".
29. David A. Richter. "The Regular Polyhedra (of index two)".
References
• Coxeter, H. S. M. (1999), "Chapter 10: Regular Honeycombs in Hyperbolic Space", The Beauty of Geometry: Twelve Essays, Mineola, NY: Dover Publications, Inc., pp. 199–214, ISBN 0-486-40919-8, LCCN 99035678, MR 1717154. See in particular Summary Tables II,III,IV,V, pp. 212–213.
• Originally published in Coxeter, H. S. M. (1956), "Regular honeycombs in hyperbolic space" (PDF), Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, vol. III, Amsterdam: North-Holland Publishing Co., pp. 155–169, MR 0087114, archived from the original (PDF) on 2015-04-02.
• Coxeter, H. S. M. (1973) [1948]. Regular Polytopes (Third ed.). New York: Dover Publications. ISBN 0-486-61480-8. MR 0370327. OCLC 798003. See in particular Tables I and II: Regular polytopes and honeycombs, pp. 294–296.
• Johnson, Norman W. (2012), "Regular inversive polytopes" (PDF), International Conference on Mathematics of Distances and Applications (July 2–5, 2012, Varna, Bulgaria), pp. 85–95 Paper 27
• McMullen, Peter; Schulte, Egon (2002), Abstract Regular Polytopes, Encyclopedia of Mathematics and its Applications, vol. 92, Cambridge: Cambridge University Press, doi:10.1017/CBO9780511546686, ISBN 0-521-81496-0, MR 1965665, S2CID 115688843
• McMullen, Peter (2018), "New Regular Compounds of 4-Polytopes", New Trends in Intuitive Geometry, Bolyai Society Mathematical Studies, 27: 307–320, doi:10.1007/978-3-662-57413-3_12, ISBN 978-3-662-57412-6.
• Nelson, Roice; Segerman, Henry (2015). "Visualizing Hyperbolic Honeycombs". arXiv:1511.02851 [math.HO]. hyperbolichoneycombs.org/
• Sommerville, D. M. Y. (1958), An Introduction to the Geometry of n Dimensions, New York: Dover Publications, Inc., MR 0100239. Reprint of 1930 ed., published by E. P. Dutton. See in particular Chapter X: The Regular Polytopes.
External links
• The Platonic Solids
• Kepler-Poinsot Polyhedra
• Regular 4d Polytope Foldouts
• Multidimensional Glossary (Look up Hexacosichoron and Hecatonicosachoron)
• Polytope Viewer
• Polytopes and optimal packing of p points in n dimensional spheres
• An atlas of small regular polytopes
• Regular polyhedra through time I. Hubard, Polytopes, Maps and their Symmetries
• Regular Star Polytopes, Nan Ma
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
| Wikipedia |
Fourier transform of periodic distributions
Following M. Ruzhansky and V. Turunen's book Pseudo-Differential Operators and Symmetries, in Chapter 3, Definition 3.1.25 (page 304), the space of periodic distributions is defined as follows (paraphrasing):
Definition 3.1.25 (Periodic Distributions) The space of periodic distributions is defined as dual space $\mathcal{D}'(\mathbb{T}^n) = \mathcal{L}(C^{\infty}(\mathbb{T}^n),\mathbb{C})$ (i.e. of continuous linear operators $u:C^{\infty}(\mathbb{T}^n) \to \mathbb{C}$), where we write $u(\varphi) = \langle u , \varphi\rangle$ for any $\varphi\in C^{\infty}(\mathbb{T}^n)$. Furthermore, for any $\psi\in C^\infty(\mathbb{T}^n)$ or $L^p(\mathbb{T}^n)$, define the associated distribution to $\psi$ in the canonical sense, i.e. $u_{\psi} \in \mathcal{D}'(\mathbb{T}^n)$ where for any $\varphi \in C^\infty(\mathbb{T^n})$ \begin{align}u_{\psi}(\varphi) = \langle u_{\psi}, \varphi\rangle=\int_{\mathbb{T}^n}\psi(x) \varphi(x) dx.\end{align}
Given the above definition, as with standard distributions, we can define a Fourier transform of periodic distributions with respect to Fourier transforms of functions.
Definition 3.1.8 (Periodic Fourier Transform) Let $\mathcal{F}_{\mathbb{T}^n} : C^{\infty}(\mathbb{T^n}) \to \mathcal{S}(\mathbb{Z}^n) $ be defined as $$ \begin{align} (\mathcal{F}_{\mathbb{T}^n}f)(\xi):=\int_{\mathbb{T}^n} f(x) e^{-2\pi i x \cdot \xi} dx \end{align}$$ for $f \in C^{\infty}({\mathbb{T}^n})$, and its inverse be defined as $$\begin{align} (\mathcal{F}_{\mathbb{T}^n}^{-1}h)(x) = \sum_{\xi \in \mathbb{Z}^n} h(\xi) e^{2\pi i x \cdot \xi}\end{align}$$ for $h\in\mathcal{S}({\mathbb{Z}^n})$.
Definition 3.1.27 (Fourier Transform of Periodic Distributions) For any $u\in\mathcal{D}'(\mathbb{T}^n)$, define the Fourier transform $\mathcal{F}_{\mathbb{T}^n}:\mathcal{D}'(\mathbb{T}^n) \to \mathcal{S}'(\mathbb{Z}^n)$ as $$\begin{align}\langle \mathcal{F}_{\mathbb{T}^n} u, \varphi \rangle = \langle u, \mathcal{F}_{\mathbb{T}^n}^{-1} \varphi(-\cdot) \rangle\end{align} \tag{1}$$ for any $\varphi \in C^{\infty}(\mathbb{T}^n)$
My question is, is the definition of the Fourier transform on periodic distributions, as given in Definition 3.1.27, correct? I feel that the definition should not involve the inverse Fourier transform, but instead be defined as $$\begin{align} \langle \mathcal{F}_{\mathbb{T}^n} u, \varphi \rangle = \langle u, \mathcal{F}_{\mathbb{T}^n} \varphi \rangle. \tag{2} \end{align}$$
The reason I feel that this should be the case is because, if we consider the periodic Dirac distribution at zero, defined as $\delta_0 \in \mathcal{D}'(\mathbb{T}^n)$ where $$ \begin{align}\langle \delta_0, \varphi\rangle = \varphi(0)\end{align} $$ for any $\varphi \in C^{\infty}(\mathbb{T}^n)$, then its Fourier transform using $(1)$ gives $$ \begin{align} \underbrace{\langle \mathcal{F}_{\mathbb{T}^n} \delta_0, \varphi \rangle}_{\in \mathbb{C}} &= \langle \delta_0, \mathcal{F}_{\mathbb{T}^n}^{-1} \varphi (-\cdot) \rangle \\ &= \mathcal{F}_{\mathbb{T}^n}^{-1} \varphi (0) \\ &= \sum_{\xi \in \mathbb{Z}^n} \varphi(\xi), \end{align} $$ where the right hand side is infinite since $\varphi$ is periodic. So this doesn't make sense. If we instead use $(2)$, then we obtain the following: $$ \begin{align} \underbrace{\langle \mathcal{F}_{\mathbb{T}^n} \delta_0, \varphi \rangle}_{\in \mathbb{C}} &= \langle \delta_0, \mathcal{F}_{\mathbb{T}^n} \varphi \rangle \\ &= (\mathcal{F}_{\mathbb{T}^n} \varphi) (0)\\ &= \int_{\mathbb{T}^n} \varphi(x) dx \\ &= \langle 1, \varphi \rangle \end{align} $$ i.e. $\mathcal{F}_{\mathbb{T}^n} \delta_0 = 1$ in the sense of periodic distributions, as we would expect.
real-analysis ca.classical-analysis-and-odes fourier-analysis fourier-transform schwartz-distributions
YCor
spacemanspaceman
Actually, the definition you gave in the post differs from the one in the book. The test function $\varphi$ should lie in $\mathcal S(\mathbb Z^n)$, not in $C^\infty(\mathbb T^n)$, since the $\mathcal F_{\mathbb T^n}$ maps the second of these spaces into the first, so for $\varphi\in C^\infty(\mathbb T^n)$ the expression $\mathcal F_{\mathbb T^n}^{-1}\varphi$ is undefined, as you correctly noticed. So, the correct formula would be $$ \langle\mathcal F_{\mathbb T^n}u,\varphi \rangle=\langle u,\iota\,\circ\,\mathcal F_{\mathbb T^n}^{-1}\varphi \rangle, $$ where $\iota(f)(x)=f(-x)$ and $\varphi \in \mathcal S(\mathbb Z^n)$.
Alexander KalmyninAlexander Kalmynin
$\begingroup$ The fact that the dual group for $\mathbb R^n$ is isomorphic to $mathbb R^n$ has caused confusion. $\endgroup$
– Gerald Edgar
$\begingroup$ Ah yes, I see! And indeed, following the representation of continuous linear functionals on $\mathcal{S}(\mathbb{Z}^n$ from Exercise 3.1.7, i.e. as $\varphi \mapsto \langle u, \varphi\rangle = \sum_{\xi \in \mathbb{Z}^n} \varphi(\xi) u(\xi)$, we obtain that $\mathcal{F}_{\mathbb{T}^n}\delta_0 = 1$ in the sense of distributions. $\endgroup$
– spaceman
$\begingroup$ Given this definition, out of curiosity how would one define the inverse Fourier transform of these periodic distributions? $\endgroup$
$\begingroup$ Would it be given by $\mathcal{F}_{\mathbb{T}^n}^{-1} : \mathcal{S}'(\mathbb{Z}^n) \to \mathcal{D}'(\mathbb{T}^n)$: where for $v \in \mathcal{S}'(\mathbb{Z}^n)$ $$ \langle \mathcal{F}_{\mathbb{T}^n}^{-1} v, \varphi\rangle = \langle v, \iota \circ \mathcal{F}_{\mathbb{T}_n} \varphi \rangle $$ for $\varphi \in C^{\infty}(\mathbb{T}^n)$? $\endgroup$
$\begingroup$ The last identity holds due to Fourier inversion theorem, which in this case is essentially the representation of a smooth function as a Fourier series $\endgroup$
– Alexander Kalmynin
This is a comment rather than an answer but it will be too long. There is a confusion in your statement which I find rather irritating and which has not, as far as I can see, been addressed here. It is manifested in the formula where you write $\phi(0)$ despite the fact that $\phi$ is a smooth function on the torus. Here is my take:
The periodic distributions can be visualised in two (mathematically equivalent) ways: as distributions on the line which satisfy the usual periodicity condition or as distributions on the circle (= $1$-dimensional torus). (For simplicity, I will confine myself to the one-dimensional case). Under this identification, the $\delta$-distribution at the point $1$ on the circle (embedded into the plane) corresponds to the periodic $\delta$-distribution on the line.
Every distribution on the torus has a Fourier series, with coefficients in the space of slowly increasing sequences indexed by the whole numbers (usually written $s´$ rather than $\cal S´$ as you write. (This corresponds to the group duality between the line and the whole numbers).
Every periodic distribution on the line is a tempered distribution and so has a Fourier transform (which is also a periodic (tempered) distribution). In terms of harmonic analysis, this corresponds to the self-duality of the line.
In a certain sense these two F.T.´s coincide--thus the periodic delta distribution on the line has F.T. $\sum_n a_n e^{2 \pi i n x}$ whereas its version on the torus has Fourier series $\sum a_n z^n$ (both sums over the whole numbers--we use the complex variable on the torus to avoid confusion but see below), up to the usual constant. For example the case of the delta function on the torus, centred at the complex number $1$, is the case where all of the coefficients are $1$, exactly as for its disguise as the periodic delta function. Note that the fact that the latter is an infinite sum is of no consequence since if we regard it is a functional, then there are no convergence problems--it is applied to functions of rapid decrease.
All of this can be found in Schwartz´ original monograph. The fact that the Fourier series on the torus looks like a Laurent series is of some significance and was examined in detail by Köthe in his seminal work on distributions as boundary values of holomorphic functions.
memorialmemorial
In addition to other good comments and answers:
There is indeed a technical issue, which perhaps has scant intuitive content, but can matter regarding precision. This is touched upon in other comments/answers, but which I think deserves clarification. Namely, while "periodic distributions" on $\mathbb R$ are those which descend to the circle, and, oppositely, distributions on the circle compose with the projection of the line to the circle, giving "periodic distributions", there is a hitch with test functions or Schwartz functions.
The obvious, but also obviously harmless, point is that a smooth function on the circle, pulled back to the line, is not Schwartz. So it is possible to give flawed definitions of "periodic distributions", if one tries hard enough.
But/and the truly relevant point is that (already) the averaging map from test functions on $\mathbb R$ to smooth functions on the circle, by $f\to \sum_n f(x+n)$ is a surjection. This is what actually does identify periodic distributions with distributions on the circle. (Yes, we can notate this with suitable symbols.)
A sort of marginally-relevant artifact is that the Fourier transform of a periodic function (under mild hypotheses) is a periodic distribution supported on integers (or $2\pi$ times, etc.) That is, "Fourier series" of periodic functions, at least as tempered distributions, are also Fourier transforms... So, yes, everything is compatible. Yes, it's easy to say literally incorrect things. Mercifully, the stuff works really well. :)
paul garrettpaul garrett
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Fourier coefficients of a periodic distribution? | CommonCrawl |
Non-commutative Analysis Seminar Ical Atom
Seminar on operator algebras, operator theory and non-commutative analysis in general, e.g. non-commutative dynamics and non-commutative function theory.
The seminar meets on Tuesdays, 11:00-12:00, in seminar room -101
2021–22–A meetings
2021–22–A (Current)
Oct 26 Generalized Powers' averaging for commutative crossed products Tattwamasi Amrutam (BGU)
In 1975, Powers proved that the free group on two generators is a $C^{\star}$-simple group. The key insight in Powers's proof of the $C^\star$-simplicity is that the left regular representation of $\mathbb{F}_2$ satisfies Dixmier type averaging property. Using the pioneering work of Kalantar-Kennedy, it was shown by Haagerup and Kennedy independently that the $C^\star$-simplicity of the group $\Gamma$ is equivalent to the group having Powers' averaging property. In this talk, we introduce a generalized version of Powers' averaging property for commutative crossed products. Using the notion of generalized Furstenberg boundary introduced by Kawabe and Naghavi (independently), we show that the simplicity of the commutative crossed products $C(X)\rtimes_r\Gamma$ (for minimal $\Gamma$-spaces $X$) is equivalent to the crossed product having generalized Powers' averaging. As an application, we will show that every intermediate $C^\star$-subalgebra $\mathcal{A}$ of the form $C(Y)\rtimes_r\Gamma\subseteq\mathcal{A}\subseteq C(X)\rtimes_r\Gamma$ is simple for an inclusion $C(Y)\subset C(X)$ of minimal $\Gamma$-spaces whenever $C(Y)\rtimes_r\Gamma$ is simple. This is a joint work with Dan Ursu.
Mon, Nov 15, 14:30–16:00, In -101 On Operators In The Cowen-Douglas Class And Homogeneity (part 1) Prahllad Deb (BGU)
Nov 23, In -101 On Operators In The Cowen-Douglas Class And Homogeneity (part 2) Prahllad Deb (BGU)
Nov 30, In 72/123 Graded isomorphism problems for graph algebras Adam Dor-On (Munster)
In a seminal 1973 paper, Williams recast conjugacy and eventual conjugacy for subshifts of finite type purely in terms of equivalence relations between adjacency matrices of the directed graphs. Williams expected these two notions to be the same, but after around 20 years the last hope for a positive answer, even under the most restrictive conditions, was extinguished by Kim and Roush.
In this talk, we will discuss operator algebras associated with adjacency matrices / directed graphs, which are naturally $\mathbb{Z}$-graded algebras. These operator algebras were first introduced by Cuntz and Krieger in tandem with early attacks on Williams' problem, and manifest several natural properties of subshifts through their classification up to various kinds of isomorphisms.
The works on Cuntz-Krieger algebras later inspired a systematic study of purely algebraic versions called Leavitt path algebras, promoting new interactions between pure algebra and analysis. A well-known conjecture of Hazrat claims that two Leavitt path algebras are graded isomorphic if and only if their unital graded Grothendieck K0 groups are isomorphic. The topological version of this problem asks for a characterization of graded (stable) isomorphisms between Cuntz-Krieger algebras in terms of equivariant K-theory.
A solution to these problems has been sought after by many, and although substantial progress has been made, a proof is still missing in general. In joint work with Carlsen and Eilers we were able to discover subtle obstructions to certain algebraic methods of proof for the latter conjecture, by building on the counterexamples of Kim and Roush
Mon, Dec 6, 15:00–16:00 Non-commutative measures and Non-commutative Function Theory in the unit row-ball Robert Martin (Manitoba)
Mon, Dec 13, 14:30–15:30 Isometric dilations, von Neumann inequality and refined von Neumann inequality(part 1) Sibaprasad Barik (BGU)
Dec 21 Isometric dilations, von Neumann inequality and refined von Neumann inequality (part 2) Sibaprasad Barik (BGU)
Mon, Dec 27, 14:30–15:30 Bratteli diagrams, dynamics, and classification beyond the minimal case Paul Herstedt (BGU)
Earlier this year, we discovered a new class of zero-dimensional dynamical systems, which we call "fiberwise essentially minimal", that are of importance to operator algebras because of the nice properties, in particular K-theoretic classification, of the crossed product. Today, we discuss the Bratteli diagrams associated to these systems, and extend the K-theoretic classification to include a dynamical condition called "strong orbit equivalence", extending the existing result in the minimal case due to Giordano-Putnam-Skau.
Jan 4 Bratteli diagrams, dynamics, and classification beyond the minimal case (part 2) Paul Herstedt (BGU)
Jan 11 TBA Ilan Hirshberg (BGU)
Seminar run by Dr. Daniel Markiewicz, Prof. Victor Vinnikov, Dr. Eli Shamovich and Prof. Ilan Hirshberg
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\begin{document}
\title{Variance-based sensitivity analysis and orthogonal approximations for stochastic models} \begin{abstract} We develop new unbiased estimators of a number of quantities defined for functions of conditional moments, like conditional expectations and variances, of functions of two independent random variables given the first variable, including certain outputs of stochastic models given the models parameters. These quantities include variance-based sensitivity indices, mean squared error of approximation with functions of the first variable, orthogonal projection coefficients, and newly defined nonlinearity coefficients. We define the above estimators and analyze their performance in Monte Carlo procedures using generalized concept of an estimation scheme and its inefficiency constant. In numerical simulations of chemical reaction networks, using the Gillespie's direct and random time change methods, the new schemes for sensitivity indices of conditional expectations in some cases outperformed the ones proposed previously, and variances of some estimators significantly depended on the simulation method being applied.
\end{abstract} \tableofcontents \chapter*{Introduction} \addcontentsline{toc}{chapter}{Introduction} Stochastic models have proven to be useful for describing a variety of physical systems, like chemical reaction networks involving few particle numbers of certain species \cite{Pahle2009, vanKampen_B07}, including gene regulatory networks \cite{Arkin1999, Rao_Wolf_Arkin_2002} and signaling pathways \cite{Lipniacki2007, Tay_2010}. A popular stochastic model for a well-stirred chemical reaction network is a continuous-time Markov chain model of reaction network dynamics (MR) \cite{mcquarrie}, which can be simulated for example using the
Gillespie's direct (GD) method \cite{Gillespie1976}. A number of other stochastic formalisms have also been used to model chemical reactions, like chemical Langevin equation, $\tau$-leaping, or hybrid stochastic-deterministic methods \cite{Pahle2009}.
Sensitivity analysis is a procedure yielding sensitivity indices, which can be thought of as certain measures of importance of arguments in influencing the values of functions.
As such functions one often takes outputs of deterministic models whose arguments are model parameters, e. g. in ordinary differential equation models of chemical kinetics \cite{Pahle2009, goutsias_2010} one can consider concentrations of different chemical species at a given moment of time in function of kinetic rates and initial concentrations of the species. In stochastic models, the stochastic outputs for given parameters are not constants but random variables with distribution specified by the parameters. For instance for an MR a stochastic output can be the number of particles of selected species at a given moment of time, and the parameters can be the initial particle numbers and kinetic rates. Thus as functions for the sensitivity analysis in stochastic models one usually considers parameters of conditional distribution of stochastic outputs, like conditional expectation \cite{Rathinam_2010}, variance, \cite{Barmassa} or histograms \cite{Degasperi2008}, given the model parameters. Sensitivity analysis has been used in a variety of fields, including chemical kinetics \cite{Rabitz_Kramer_Dacol_1983, Turanyi_1990,Saltelli2005}, biochemical reaction networks \cite{Van_Riel_2006, Zhang2009}, nuclear safety \cite{Iooss2008}, environmental science \cite{Tarantola_Giglioli_Jesinghaus_Saltelli_2002}, and molecular dynamics \cite{Cooke_Schmidler_2008}, applications in chemical kinetics including parameter estimation \cite{Kim_Spencer_Albeck_Burke_Sorger_Gaudet_Kim_2010, Juillet2009} and model simplification \cite{Cristaldi_2011, Okino_Mavrovouniotis_1998, Liu_Swihart_Neelamegham_2005, Degenring2004}.
Variance-based sensitivity analysis (VBSA) is a well-established sensitivity analysis method, dating back to applications in chemical kinetics in the seventies \cite{Saltelli2008, Cukier1973}. \mbox{Variance-ba-} sed sensitivity indices provide quantitative answers to questions like \textit{what average reduction of uncertainty of the model output, measured by its variance, can be achieved if some uncertain model parameters are determined e. g. in an experiment}, or \textit{what average error is caused by fixing a parameter for instance to simplify the model} \cite{Sobol2007, badowski2011}. Recently, VBSA has supported parameter estimation in a linear compartmental biochemical model \cite{Juillet2009} and simplification of a model of synthesis of an antiparasitic drug Ivermectin \cite{Cristaldi_2011}.
In many applications outputs of physical models are being approximated by linear combinations of functions of few model parameters. Such approximations are used for example
for potential energies in molecular dynamics simulations \cite{lelievre2010free}. A number of approximations of this type has become known under the common name of high dimensional model representations \cite{rabitz_1999}. Approximations using linear combinations of products of the first few orthogonal functions of model parameters, like polynomials or trigonometric functions,
have proven to accurately imitate outputs of a number of complex models with many parameters, and the coefficients of the approximating linear combinations have been used to estimate variance-based sensitivity indices as an alternative strategy to their Monte Carlo estimation on which we focus on in this work \cite{Cukier1973, Li2002b, Li2002, blatman_2010, goutsias_2010}. In the case of orthogonal polynomials being used, such approximations have also been called polynomial chaos expansions \cite{Khammash_2012, blatman_2010}.
Correlation coefficient is a popular measure of the strength of linear relationship between random variables
\cite{terrell2010mathematical, bickel2000mathematical}.
With a few exceptions, the above mentioned indices and coefficients have predominantly been used for the analysis of input-output relationships in deterministic models, but
they also have a potential for analogous applications to stochastic models. For instance VBSA can be useful for determining parameters whose measurement would on average most reduce the uncertainty of a given parameter of conditional distribution of the model output, like conditional expectation or variance, while polynomial approximations and correlations can provide useful information about the relationship between variance or mean particle numbers and model parameters, which has been of great interest e. g. in the analysis of gene circuits \cite{Stelling2010, Becksei_2000, vanOud2002},
Outputs of stochastic models used in computer simulations can be represented as functions of two independent random variables - the first being random vector of parameters of the model and the second a sequence of random variables used to generate the random trajectories of its stochastic process. In the first work \cite{Degasperi2008} in which VBSA of parameters of conditional distribution of stochastic outputs was considered, conditional histograms were chosen as outputs for sensitivity analysis and a grid-based method was used providing no error estimates of the results \cite{badowski2011}.
In our Master's Thesis in Computer Science \cite{badowski2011} we proposed unbiased estimators of variance-based sensitivity indices of conditional expectations of functions of two independent random variables given the first random variable, which can be used in a MC procedure yielding error estimates. In a numerical experiment such procedure led to lower mean squared error of approximation of the indices
than a grid-based method analogous to that in \cite{Degasperi2008}.
In this thesis we provide for the first time unbiased estimators of variance-based sensitivity indices of a large class of functions of conditional moments, including all conditional moments and central moments, like conditional variance, of functions of two independent random variables given the first variable. We also introduce new unbiased estimators of sensitivity indices of conditional expectations. Furthermore, we derive first unbiased estimators of means of functions of conditional moments, of products and covariances of these moments with functions of the first variable, and estimators of normalized sensitivity indices and correlation coefficients of functions of conditional moments and the first variable. We also introduce different unbiased estimators of coefficients of orthogonal projection of functions of conditional moments onto linear combinations of orthogonal functions of the first variable. We show that in a Hilbert space,
squared error of approximation with a linear combination of orthonormal elements using unbiased estimates of orthogonal projection coefficients, averaged over distribution of the estimates, is a sum of variances of the estimators plus squared error of the approximation with orthogonal projection onto span of the elements. We use this fact to numerically compare average mean squared errors of approximation of conditional expectations and variances of stochastic models outputs by linear combinations of orthogonal functions of model parameters with coefficients obtained using different estimators. We also provide unbiased estimators of mean squared errors of approximation of functions of conditional moments using functions of the first variable, which can be used for the above approximations with linear combinations of orthogonal functions with fixed coefficients. Approximations of conditional expectations and variances of outputs of stochastic models using orthogonal polynomials of model parameters have already been constructed in \cite{Khammash_2012} using double-loop sampling and convex optimization techniques.
As we discuss in a detail in Conclusions, an interesting idea for the future research would be to compare the error
of different methods of approximation of functions of conditional moments using orthogonal functions of the first variable, like those from \cite{Khammash_2012} and this work. We also define nonlinearity coefficients of random independent arguments of a function, which can be used for obtaining lower bounds on probabilities of certain localizations of functions values changes, corresponding to some perturbations of their independent arguments. We also provide unbiased estimators of these coefficients.
In \cite{badowski2011} we introduced the concept of an estimation scheme which is useful for defining generalized estimators acting not only on random variables, but also on functions, as the ones discussed in this work. We also defined inefficiency constant of such a scheme, equal to the product of variance of the corresponding estimator and the number of function evaluations needed to compute it, so that the ratio of such constants for different schemes is equal to the ratio of variances of the final MC estimators for the same number of function evaluations carried out in MC procedures using the schemes. Thus it can be used for quantifying the inefficiency of using unbiased estimation schemes in MC procedures if function evaluations are the most time-consuming elements of the procedures.
Here we formalize and generalize the above concepts of a scheme and its inefficiency constant to be useful for defining and comparing efficiency of the corresponding estimators of estimands depending on many functions and having vector-valued outputs, like vector of coefficients of orthogonal projection of a function of a random variable onto orthogonal functions of the variable. One of the defined schemes, called $SVar$, allows for simultaneous estimation of most of the above mentioned sensitivity indices and coefficients for conditional expectation and variance, including all sensitivity indices with respect to individual coordinates of the first variable and orthogonal projection coefficients onto these coordinates and constant vectors. We derive a number of inequalities between the inefficiency constants of the introduced schemes and schemes from \cite{badowski2011}. For instance we show that subschemes of $SVar$ for estimation of variance-based sensitivity indices of conditional expectation can have no more than four times higher and three times lower inefficiency constants than the best schemes for these indices from \cite{badowski2011}.
We carried out numerical experiments testing estimators introduced in this work for the case of conditional expectations and variances of particle numbers at a given moment of time in a MR simulated using
GD \cite{Gillespie1976} and random time change (RTC) \cite{Rathinam_2010} methods. In some of our numerical experiments the subschemes of scheme $SVar$ for estimation of sensitivity indices of conditional expectation with respect to certain model parameters
had over two times lower inefficiency constants than the best schemes from \cite{badowski2011}. Furthermore, the order of estimators of orthogonal projection coefficients with respect to the average mean squared errors of approximations of conditional expectations and variances constructed using them varied from model to model. The numerical experiments also demonstrated significant dependence of variances of some of the introduced estimators on whether the GD or the RTC method is used, and on the order of reactions in the GD method. We discuss the relationship of these effects with analogous phenomena observed for different estimators in \cite{badowski2011} and \cite{Rathinam_2010}.
The structure of this work is as follows. In Chapter \ref{overview} we give an overwiev of less common definitions and results from the literature, mostly from our previous master's thesis \cite{badowski2011}. Throughout this chapter we frequently make improvements in the definitions, correct errors in the constructions or theorems, generalize the latter, and provide more precise and comprehensive descriptions than in \cite{badowski2011}. Chapter \ref{chapOwnRes} presents our new results. More common mathematical definitions and complex proofs or calculations are provided in the appendices. Readers not acquainted with probability theory are referred to standard textbooks like \cite{Durrett} and \cite{billingsley1979}. Some basic definitions from this theory are also provided in Appendix \ref{appMath}.
\chapter{Literature overview \label{overview}}
\section{Chemical reaction network}\label{secCRN} In this section we repeat selected definitions from Section 1.1 of \cite{badowski2011}, improving some of them, in particular simplifying the formal definition of a chemical reaction network $RN$ and specifying the domain of reaction rates. We shall model the time evolution of a reaction network by a continuous-time Markov chain defined in the next section. Suppose that we are given $N \in \mathbb{N}_+$ chemical species with symbols $X_1, \ldots, X_N$. The state of $RN$ at a given moment of time is described by a vector of natural numbers $x = (x_1,\ldots,x_N)$ from the state space $E = \mathbb{N}^N$, where the $i$th coordinate of $x$ describes the number of particles of the $i$th species. $L$ chemical reactions $(R_1,\ldots,R_L)$ can occur, the $l$th reaction being described by a stoichiometric formula \begin{equation}\label{reac} \underline{s}_{l,1} X_{1} + ... + \underline{s}_{l,N} X_{N} \rightarrow \overline{s}_{l,1} X_{1} + ... + \overline{s}_{l,N} X_{N}. \end{equation} We call vector $\underline{s}_l = (\underline{s}_{l,i})_{i=1}^{N}$ the stoichiometric vector of reactants and $ \overline{s}_l = (\overline{s}_{l,i})_{i=1}^{N}$ of products of the $l$th reaction. In this whole work for $n \in \mathbb{N}_+$ we denote $I_n = \{1,\ldots, n\}$ and for vectors $v,w \in \mathbb{R}^n$, we write $v \geq w$ if $v_i \geq w_i$, $i \in I_n$ (where the last notation means for each $i \in I_n$). We require that $\underline{s}_l \geq 0 $ and $\overline{s}_l \geq 0$, where by $0$ we mean here $(0,\ldots,0) \in \mathbb{R}^N$. We define transition vector of the $l$th reaction as $s_{l} = \overline{s_l} - \underline{s_l}$. In the model of dynamics of reaction network discussed in the next section occurrence of the $l$th reaction will make the system at state $x$ move to state $x + s_l$. There is given a set $B_{RN} \in \mathbb{R}^m$ for some $m \in \mathbb{N}_+$, called the set of admissible reaction rates of $RN$. We define a measurable space $\mc{S}_{RN}=(B_{RN},\mc{B}(B_{RN}))$ (see Section \ref{appMath}).
For each $l \in I_l$, there is given a real nonnegative function $a_l$ measurable on $\mc{S}_{RN,E} = (B_{RN,E},\mc{B}_{RN,E}) = \mc{S}_{RN}\otimes\mc{S}(E)$, called reaction rate of the $l$th reaction. Intuitively speaking, in the mathematical model we discuss in the next section $a_l(k,x)$ describes how quickly the $l$th reaction is proceeding in the state $x$ and for the rate constants $k$. We require that $a_l(k,x) = 0$ if for some $i\in I_N$, $x_i < \underline{s}_i $, which means that there are too few particles of a certain reactant in the system for the reaction to occur. For example in the stochastic version of mass action kinetics \cite{Kurtz1986}, for each $k = (k_i)_{i=1}^{L} \geq 0$ and $l \in I_L$, \begin{equation}\label{alkx} a_l(k,x) = k_l\prod_{i=1}^N{x_i \choose \underline{s}_{l,i} }, \end{equation} which is rate constant of the $l$th reaction times the number of possible ways in which the reactants can collide for the $l$th reaction to happen. Formally, we define the chemical reaction $R_l$, $l \in I_L$, to be a triple \begin{equation} R_l(k) = (a_l(k,\cdot), \underline{s}_l, \overline{s}_l), \end{equation} and the chemical reaction network $RN$ is defined as a sequence of reactions \begin{equation}\label{NTfun} RN(k) = (R_l(k))_{l=1}^L, \end{equation} both being functions of rate constants $k \in B_{RN}$.
\section{\label{secMRCP}Continuous-time Markov chain model of reaction network dynamics with constant parameters (MRCP)} See Appendix \ref{appHMC} for an introduction to stochastic processes, including continuous-time homogeneous Markov chains (HMC). Below we repeat the definition of a discrete stochastic chemical reaction network with constant parameters from our previous work \cite{badowski2011}, calling it this time continuous-time Markov chain model of reaction network dynamics with constant parameters or shortly MRCP. Let the reaction network $RN$ and othernotations be as in Section \ref{secCRN}, let $p = (k,c) \in B_{RN, E}$, and $T= [0,\infty)$. \begin{defin} MRCP corresponding to $RN$ and $p$ is defined as a nonexplosive HMC on $E$ with times $T=[0,\infty)$, with deterministic initial distribution $\delta_{c}$ and $Q$-matrix with intensities equal to, for each $x,y \in E$, $x \neq y,$ \begin{equation}\label{qxyEq} q_{x,y} = \sum_{l: \ y = x + s_l}\ a_l(k,x), \end{equation} where we used the convention that sum over an empty set is zero. \end{defin} Unfortunately, for some $p$ and $RN$ as above such nonexplosive HMC may not exist \cite{KurtzReview2010}. A useful criterion for its existence shall be provided in Section \ref{secMRNew}. We denote the distribution of a nonexplosive MRCP corresponding to $p = (k, c)$ and chemical reaction network $RN$ for which it exists as $\mu_{MRCP}(RN(k), c)$. Let $RN$ be some reaction network as in Section \ref{secCRN} and $p = (k,c) \in B_{RN, E}$ be some its parameters. Below we describe two constructions of processes which yield MRCPs corresponding to $RN$ and $p$, if any such MRCP exists. The description of these constructions is similar as in \cite{badowski2011} in Section 1.3, but we do it in a more formal way and correct a number of oversights we made in \cite{badowski2011}, like overlooking the case when the set of reactions with positive rates is empty in the second construction. The first construction corresponds to the GD method for simulating MRCP introduced in \cite{Gillespie1976}, while the second to the RTC algorithm from \cite{Rathinam_2010}, and is a special case of the random time change representation of Markov processes due to Kurtz (\cite{Kurtz1986}, Section 6.4). In the below constructions we inductively define the initial jump chain $(Z_n)_{n \geq 0} $ and initial jump times $(J_n)_{n \geq 0}$. Let the initial explosion time $\zeta$ be defined as in (\ref{expTime}) in Appendix \ref{appHMC} using the initial jump times. We assume without explicitly writing this in the constructions that $Z_0 = c$ and $J_0 = 0$, and that each construction ends by changing, for some arbitrary $c_1 \in E$, and on each elementary event $\omega$ for
which $\zeta(\omega) < \infty$, all the initial jump chain and times variables with positive indices to $c_1$ and $\infty$, respectively, so that we receive final jump chain and times $Z_n'$, $J_n'$, that are jump times and chain of some unique
nonexplosive process $Y$. If there exists any MRCP corresponding to $RN$ and $p$, then $\zeta = \infty$ a. s. and $Y$ is such an MRCP.
For convenience in the constructions below the dependence on the elementary event $\omega$ is omitted. \begin{constr}[GD construction]\label{GCon} Let $U_1,U_2,\ldots $ be independent identically distributed (i. i. d.), $U_1 \sim \U(0,1)$, and $E_1, E_2,\ldots$ i. i. d., $E_1 \sim \Exp(1)$ (see Appendix \ref{appMath}).
Suppose that $Z_i$ and $J_i$ have been defined for $i\in \mathbb{N}$. Let \begin{equation} q = \sum_{l=1}^L a_l(k,Z_i). \end{equation} If $q = 0$, then we define \begin{equation}
J_{i+1} = \infty,\ Z_{i+1} = Z_{i}, \end{equation} otherwise we take $J_{i+1} = J_i + \frac{E_{i+1}}{q}$, and for \begin{equation}
l = \min\{m \in I_{L}: \frac{1}{q}\sum_{n=1}^{m} a_{n}(k)(Z_i) \geq \U_{i}\}, \end{equation} we set \begin{equation} Z_{i + 1} = x + s_l. \end{equation} \end{constr} For $k \in B_{RN}$ and $x \in E$, we denote $B(k, x) = \{l \in I_L:\ a_l(k, x) > 0\}$ - the set of indices of reactions with positive rates in state $x$ and for the rate constants $k$. \begin{constr}[RTC construction]\label{RTCCon} Let us consider $L$ independent Poisson processes $(N_l)_{l=1}^L$ with unit rates (see Appendix \ref{appHMC}). The initial jump times and chain in this construction are the jump times and chain of any right-continuous process $Y$ satisfying \begin{equation}\label{intEqu} Y_t = c + \sum_{l=1}^L s_l N_l(\int_0^t \! a_l(k, Y_s) \, \mathrm{d}s), \end{equation} for $t < \zeta$ (\cite{Kurtz1986} Section 6 Theorem 4.1 a)).
Suppose that the $i$th call of function $N_l.next$ returns the $i$th holding time of $N_l$. We set for $l\in I_L$ \begin{equation} \tau_{0,l} = N_l.next. \end{equation} Let us assume that $Z_{i}$, $J_{i}$ and $\{\tau_{i ,l} \}_{l \in I_L}$ have been defined for $i \in \mathbb{N}$. If $B(k,Z_i)$ is empty, then we set \begin{equation} J_{i+1} = \infty,\ Z_{i+1} = Z_i, \end{equation} and finish the inductive step. Otherwise, we set \begin{equation} J_{i+1} = J_i + \min_{l \in B(k,Z_i)} \left\{ \frac{\tau_{i,l}}{a_l(k, Z_i)} \right\}, \end{equation} and for a certain $l$ realizing the above minimum, \begin{equation}
Z_{i+1} = Z_i + s_l,\quad \tau_{i+1,l} = N_l.next. \end{equation} Furthermore, for each $m \in B(k,Z_i),\ m \neq l$, we set \begin{equation}
\tau_{m,i+1} = \tau_{m,i} - a_m(k, Z_i) (J_{i+1} - J_i) \end{equation} and for reaction indices $l \notin B(k, Z_i)$, \begin{equation}
\tau_{l,i+1} = \tau_{l,i}. \end{equation} \end{constr}
As we discussed in \cite{badowski2011}, in all constructions of processes corresponding to some stochastic simulation algorithm one uses a random variable $R$, which in the algorithm is generated e. g. using a random number generator, to build the random trajectories of the process. For instance
for the first construction of MRCP above we have $R = (U_i, E_i)_{i \geq 0}$, while for the second one $R = (N_i)_{i=1}^L$. As in \cite{badowski2011}, we call $R$ artificial noise variable or simply noise variable. For some construction of MRCP as above, let $\mc{S}_R = (B_R,\mc{B}_R)$ be the measurable space of possible values of the noise variable from this construction. We define a function $h$ from $B_{RN,E}\times B_R$ to $E^T$ to be such that for each $p \in B_{RN, E}$ and $r \in B_R$, \begin{equation}\label{MRCPFun} h(p,r) \end{equation} is equal to the trajectory $Y(\omega)$ of the process built in the construction using parameters $p$ and for $\omega\in \Omega$ and noise variable $R$ such that
$R(\omega)=r$. In particular, process $Y$ created in the construction using some $p$ and $R$ is equal to $h(p,R)$ on $\Omega$.
\section{\label{genParSec}Models with random parameters and their outputs} As we discussed in Section 2.1 of \cite{badowski2011}, there are many situations when one may want to treat parameters of a model as random variables $(P_i)_{i=1}^N$ rather than constants. Shortly, when the parameters represent uncertain quantities, there are two types of such variables distinguished in the literature - stochastic and epistemic ones. Stochastic variables are changeable in the modelled system, like particle numbers in equilibrium distribution of a reaction network, and their uncertainty, measured e. g. by their variance, cannot be reduced by gaining further knowledge about the system. Epistemic variables are constants in the modelled system, whose exact values are unknown, which is often the case for reaction rates. Distribution of epistemic variables reflects our best judgement about their possible values, based e. g. on the uncertainty estimates of experimental measurements, and uncertainty of these variables can be reduced by gaining further knowledge about the system, like performing more precise experiments.
The definition of a continuous-time Markov chain model of reaction network dynamics (MR) and its construction we provide below are more precise and general than the ones we proposed in \cite{badowski2011}, e. g. because we specify the domain of distribution of parameters in the definition and do not require the existence of an MRCP for each value of the parameters. Let $RN$, $E$ and $T$ be as in the previous sections and let $\mu_0$ be some probability distribution on $\mc{S}(E^T)$ and $\nu$ on $\mc{S}_{RN,E}$. Let $\wt{\mu}: B_{RN, E} \times \mc{B}(E^T)\rightarrow \mathbb{R}$ be such that for each $p=(k,c) \in B_{RN,E}$ for which an MRCP corresponding to $RN$ and $p$ exists $\wt{\mu}(p,\cdot)$ is equal to $\mu_{MRCP}(RN(k), c)$ i. e. the distribution of such MRCP, and for other values of $p$ it is equal to $\mu_0$. \begin{defin}\label{MRdef} We say that a pair $M = (P,Y)$ is an MR corresponding to a chemical reaction network $RN$ and (distribution of parameters) $\nu$, if for $\nu$ almost every (a. e.) $p$, a
MRCP corresponding to $p$ and $RN$ exists, $P$ is a random vector with $\mu_P=\nu$, $Y$ is a right-continuous nonexplosive process on $E$ with times $T$, and $\wt{\mu}$ is a conditional distribution (see Definition \ref{defMu}) of $Y$ given $P$. $P$ is called the parameters and $Y$ the process of $M$.
\end{defin}
If for $\nu$ a. e. $p =(k,c)$ MRCP corresponding to $RN$ and parameters $p$ exists, then MR $(P,Y)$ corresponding to $\nu$ and $RN$ can be constructed similarly as for the previous less general definition of MR in \cite{badowski2011}. For some $P = (K, C) \sim \nu$ independent of the artificial noise variable $R$ used by one of the constructions of MRCP from the previous section, one sets $k= K(\omega)$ and $c = C(\omega)$ at the beginning of this construction and then proceeds with it. Using function $h$ (\ref{MRCPFun}) corresponding to the construction of MRCP, the process of MR we defined above can be written as \begin{equation}\label{formP} Y = h(P,R). \end{equation} $Y$ conforms to the definition of a process of MR with parameters $P$ due to Theorem \ref{aveSecFin} in Appendix \ref{appMath}.
Analogously we can define constant and random parameter versions $M=(P,Y)$, as well as constructions in form of a function $h(P,R)$ of independent parameters $P$ and some noise term $R$ of other stochastic or deterministic models used in computer simulations, where for deterministic models $R$ can be chosen constant. In particular this applies to models used for simulation of chemical kinetics, like Euler-Maruyama approximation of solutions of the chemical Langevin equation \cite{Wilkinson2006}, Euler scheme for ordinary differential equations of chemical kinetics, or hybrid stochastic-deterministic methods \cite{Pahle2009}.
By an output of such a model $M = (P,Y)$ we mean a random variable $g(M)$ for some function $g$, measurable from the product measurable space of the image of $M$ to $\mathbb{R}^n$, for some $n \in \mathbb{N}_+$.
For an MR the output can be e. g. the number of particles of the $i$th species at the moment $t$, while for a deterministic model of chemical kinetics this can be concentration of some species at a given time. One can also consider vector-valued outputs, like vectors of particle numbers of different species or single-sample histograms of numbers of a given particle which we define below.
As discussed in the introduction, for outputs of stochastic models, like particle numbers at a given time for MR, we shall be interested in their certain parameters of conditional distribution, like conditional expectation given the model parameters. According to the above definition of an output, such parameters of conditional distribution are themselves model outputs, which can be expressed as functions of only the model parameters.
Conditional expectation of an integrable random variable $Z$ given a random variable $X$, denoted as $\mathbb{E}(Z|X)$, is a random variable $f(X)$ for a certain function $f$, where $f(x)$ can be informally thought of as the mean value of $Z$ on the set $X = x$ (see Appendix \ref{appMath} for a precise Definition \ref{condDef}). As in \cite{badowski2011}, for $p>0$ we define
$L^p_n(\mu)$ to be the space of classes of equivalence of the relation of being equal $\mu$ a. e. considered on random vectors $X= (X_i)_{i=1}^n$ such that $X_i \in L^p(\mu)$ for each $i \in {I_n}$
(see Appendix \ref{appMath} for more details on $L^p(\mu)$ spaces including the associated notational conventions, which we by analogy extend to $L^p_n(\mu)$ spaces, in particular if $\mu=\mathbb{P}$ is the implicit probability measure, then $L^p_n(\mathbb{P})$ is denoted simply as $L^p_n$).
For an $\mathbb{R}^n$-valued random vector $Z = (Z_i)_{i=1}^n$, we define $\mathbb{E}(Z) = (\mathbb{E}(Z_i))_{i=1}^n$. If $Z\in L^1_n$ and $X$ is a random variable, then we define the conditional expectation of $Z$ given $X$ as \begin{equation} \label{genCond}
\mathbb{E}(Z|X) = (\mathbb{E}(Z_i|X))_{i=1}^n. \end{equation} From the fact that conditional expectation is contraction in $L^p$ (see Theorem \ref{contrac} in Appendix \ref{appMath})
it follows that $\mathbb{E}(Z|X) \in L^p_n$ if $Z \in L^p_n$. For $k \in \mathbb{N}_{+}$ and numbers $x_{min}$, $x_{max}$ such that $L = x_{max} -x_{min} >0$, let $B_i = \left[x_{min}+ \frac{(i-1)L}{k}, x_{min}+ \frac{iL}{k}\right)$, $i \in I_k$. The corresponding histogram function $\hist$ is defined for $x \in \mathbb{R}$ as \begin{equation} \hist(x) = \left(\mathbb{1}_{B_i}(x)\right)_{i=1}^k. \end{equation} A (single-sample) histogram corresponding to a real-valued random variable $Z$ is defined as $\hist(Z)$. Note that $\hist(Z) \in L^p_{k}$ for each $p > 0$.
Conditional histogram of $Z$ given some random variable $X$ is defined as $\mathbb{E}(\hist(Z)|X)$ and mean histogram as $\mathbb{E}(\hist(Z))$. For a random vector $X = (X_1,\ldots, X_N)$ and any $J \subset I_N$, we denote $X_J = (X_i)_{i \in J}$. For $J=\emptyset$, we define $X_J=\emptyset$. For $Z$ integrable, for each $J \subset K \subset I_N$, we have the following iterated expectation property \cite{Durrett} \begin{equation}\label{doubleCond}
\mathbb{E}(\mathbb{E}(Z|X_K)|X_J) = \mathbb{E}(Z|X_J), \end{equation}
where for $Y \in L^1$, by $\mathbb{E}(Y|\emptyset)$ we mean $\mathbb{E}(Y)$. For a stochastic model $M= (P,Y)$ whose process $Y$ has form $h(P,R)$ for some variable $R$ independent of $P$ as in (\ref{formP}), a stochastic output $g(M)$ is equal to $f(P,R)$ where $f$ is defined by formula \begin{equation}\label{obsForm} f(p,r) = g(p, h(p,r)),\quad p \in B_{RN, E}, r \in B_R. \end{equation} In such case, thanks to Theorem \ref{indepCond} we have \begin{equation}
\mathbb{E}((f(P,R)|P) = \mathbb{E}(f(p,R))_{p=P}. \end{equation}
\section{Variance for random vectors}\label{secOrthog} In this Section we mainly reformulate some theory from Section 2.5 of \cite{badowski2011}. Reader not acquainted with Hilbert space theory is referred to Appendix \ref{appHilb}. An example of a Hilbert space is $L^2$ with scalar product given by \begin{equation}\label{scall2} (X,Y) = \mathbb{E}(XY). \end{equation}
As in \cite{badowski2011}, we denote the norm it induces as $||\cdot ||$ and the metric $d$. Let $n \in \mathbb{N}_{+}$, $<,>$ be a scalar product in $\mathbb{R}^n$, and $(a_{ij})_{i,j \in I_n}$ be the real numbers such that for each $x,y \in \mathbb{R}^n$, \begin{equation}\label{scalAij} <x,y> = \sum_{i,j\in I_n} a_{ij} x_iy_j. \end{equation} For the standard scalar product we have $a_{ij} = \delta_{ij}$ ($\delta_{ij}$ being the Kronecker delta).
The norm induced by $<,>$ is denoted as $|\cdot|$ and the distance as $\wt{d}$. $L^2_n$ with scalar product $(,)_{n}$, defined for $X,Y \in L^2_n$ as \begin{equation}\label{scalarFun} (X,Y)_n = \mathbb{E}(<X,Y>)= \sum_{i,j\in I_n}a_{ij}(X_i,Y_j), \end{equation} is a Hilbert space equal to the direct sum of $L^2$ given by $<,>$ (see definition in Theorem \ref{genDSpace} in Appendix \ref{appHilb}) and denoted as $\bigoplus_{<,>}L^2$.
As in \cite{badowski2011}, the norm induced by the scalar product $(,)_n$ is denoted as $||\cdot ||_{n}$ and the metric as $d_{n}$. For some random variable $X$ and $p > 0$, let $L^p_{X}$ be the subspace of $L^p$ consisting of all its classes of random variables containing an element $f(X)$ for some measurable real-valued
function $f$, and $L^p_{n,X}$ be an analogous subspace of $L^p_n$ but for functions $f$ with values in $\mathbb{R}^n$. For $p \geq 1$, $L^p_{X}$ is a closed subspace of $L^p$, because from the change of variable Theorem \ref{thchvar} the map $[f(X)]_{\mathbb{P}} \rightarrow [f]_{\mu_X}$ (see Appendix \ref{appMath}) is a linear isometry between $L^p_{X}$ and the complete space $L^p(\mu_X)$. In particular $L^2_{X}$ is a Hilbert space and $L^2_{n,X}$ with scalar product $(,)_n$ is equal to the direct sum of $L^2_{X}$ given by $<,>$.
Conditional expectation $\mathbb{E}(\cdot|X)$ is an orthogonal projection from $L^2(\mathbb{P})$ onto $L^2_{X}$ (see Lemma \ref{condort} in Appendix \ref{appHilb}), so that from Theorem \ref{projDS} in Appendix \ref{appHilb}
it follows that the generalized conditional expectation $\mathbb{E}(\cdot|X)$ given by (\ref{genCond}) is orthogonal projection from $L^2_n$ to $L^2_{n,X}$.
In particular, $\mathbb{E}(Z|X)$ is the best approximation of $Z$ in $L^2_{n,X}$ and the squared error of this approximation fulfills \begin{equation}\label{condError}
d^2_n(Z, \mathbb{E}(Z|X)) = ||Z||_n^2 - ||\mathbb{E}(Z|X)||_n^2. \end{equation} Similarly as in \cite{badowski2011}, we define variance of a random vector $Z \in L^2_n$, $n \geq 2$, as follows, using for it informally the same notation as for one-dimensional variance, \begin{equation}\label{varGen} \begin{split}
\Var(Z) &= d^2_n(Z, \mathbb{E}(Z)) = ||Z||^2_n - ||\mathbb{E}(Z)||^2_n \\
&= \mathbb{E}(|Z|^2) - |\mathbb{E}(Z)|^2. \end{split} \end{equation}
When the probability measure considered is $\mu$ rather than $\mathbb{P}$, we write $\Var_{\mu}(Z)$ instead of $\Var(Z)$. Standard deviation of $Z$ is defined as \begin{equation} \sigma(Z) = \sqrt{\Var(Z)}. \end{equation} As in \cite{badowski2011}, conditional variance of $Z$ given $X$ is defined as \begin{equation}\label{condGen} \begin{split}
\Var(Z|X) &= \mathbb{E}(\wt{d}^2_n(Z, \mathbb{E}(Z|X))|X) \\
&= \mathbb{E}(|Z|^2 + |\mathbb{E}(Z|X)|^2 - 2<Z, \mathbb{E}(Z|X)>|X) \\
&=\mathbb{E}(|Z|^2|X) - |\mathbb{E}(Z|X)|^2, \end{split} \end{equation}
where in the second equality we used the fact that $\mathbb{E}(<Z, \mathbb{E}(Z|X)>|X) = |\mathbb{E}(Z|X)|^2$, which follows from Theorem \ref{condexpX} from Appendix \ref{appMath} and from (\ref{scalAij}). We have \begin{equation}\label{d2n}
\mathbb{E}(\Var(Z|X)) = ||Z||^2_n - ||\mathbb{E}(Z|X)||^2_n = d^2_n(Z, \mathbb{E}(Z|X)), \end{equation} where in the first equality we used the iterated expectation property (\ref{doubleCond}) applied to the last term in (\ref{condGen}), and in the second equality from (\ref{condError}). From (\ref{d2n}) and the third term in (\ref{varGen}) we receive a formula already derived in \cite{badowski2011}, \begin{equation} \label{aveVarError}
\Var(Z) = \mathbb{E}(\Var(Z|X)) + \Var(\mathbb{E}(Z|X)). \end{equation} As we shall prove in Section \ref{secCondMoms} for $f$ measurable such that $f(Z) \in L^2_n$,
\begin{equation}\label{varmuzx}
\Var(f(Z)|X) = \Var_{\mu_{Z|X}(X,\cdot)}(f). \end{equation}
\section{\label{secVBSA}ANOVA decomposition and variance-based sensitivity indices} In this section we mainly reformulate some definitions and theorems from sections 3.1-3.3 of \cite{badowski2011}. Let $X=(X_i)_{i=1}^N$ be a random vector with $N \in \mathbb{N}_{+}$ independent coordinates. Let $I = I_N$ and $J \subset I$. We define $\sim J = I \setminus J$, and $X_J$ as in Section \ref{genParSec}. For $J \neq \emptyset$, we denote $\mu_J=\mu_{X_J}$, and define $L^2_{n,X_J}$ as in Section \ref{secOrthog}. For $J=\{i\}$ we write $i$ rather than $\{i\}$ in the above and below introduced notations. $L^2_{n,X_{\emptyset}}$ is defined to consist of classes from $L^2_{n}$ containing constant $\mathbb{R}^n$-valued random vectors.
For each $J \subset I$, we define $L^2_{n,J}$ to be the subspace of $L^2_{n,X_J}$ consisting of its classes containing variables $Z$ such that for each $i \in J$, \begin{equation}\label{zeroExp}
\mathbb{E}(Z|X_{\sim i}) = 0. \end{equation} Note that $L^2_{n,\emptyset} = L^2_{n,X_{\emptyset}}$ and that due to Theorem \ref{indepCond}, for $Z = g(X_J)$ for a measurable function $g$, (\ref{zeroExp}) is equivalent to \begin{equation}\label{zeroInt} \int \! g(X_{J \setminus \{i\}}, x_i) \, d\mu_i = 0, \end{equation} where we used a convenient notation for integrating $X_i$ out over its distribution. From (\ref{zeroExp}) and iterated expectation property it follows that for each nonempty $ J \subset I$ and variable $Z \in L^2_{n,J}$, it holds \begin{equation}\label{zj0} \mathbb{E}(Z) = 0. \end{equation} In \cite{badowski2011} we proved as Theorem 5 the following theorem (see Definition \ref{defHilb} of a direct sum in a Hilbert space).
\begin{theorem} For $n \in \mathbb{N}_+$ and Hilbert space $L^2_{n,X}$ with certain scalar product $(,)_n$ defined as in Section \ref{secOrthog}, it holds \begin{equation} L^2_{n,X} = \bigoplus_{J \subset I} L^2_{n,J}. \end{equation} \end{theorem} In the proof of Theorem 5 in \cite{badowski2011} we also showed that if for some measurable $f$, $Z=f(X)\in L^2_{n,X}$, then there exist measurable functions $f_J$ such that $f_J(X_J) \in L^2_{n,J}$, $J \subset I$, and \begin{equation}\label{anovaDec} f(X) = \sum_{J\subset I}f_J(X_J). \end{equation} Random variables $f_J(X)$, $J \subset I$, are uniquely determined a. s. and we call $(f_J(X_J))_{J \subset I}$ the ANOVA decomposition of $f(X)$, as such decompositions for the case of $n=1$ were used under this name in the literature (see \cite{badowski2011} for references). From (\ref{zeroInt}), (\ref{indepCond}), and Fubini's theorem, for each $K \subset I$ we have \begin{equation}\label{subAnova}
\mathbb{E}(f(X)|X_K) = \sum_{J\subset K}f_J(X_J). \end{equation}
Denoting for $J \subset I$, \begin{equation} V_J = \Var(f_J(X_J)), \end{equation} and using (\ref{zj0}), (\ref{anovaDec}), and orthogonality of the elements of ANOVA decomposition, we receive for $D= \Var(f(X))$, \begin{equation}\label{sumvar} D = \sum_{K \subset I} V_K. \end{equation}
For $|J|>1$, $V_J$ has been called an interaction index between the variables with indices in $J$ in the literature \cite{Saltelli2005}, and as we proved in \cite{badowski2011} $V_J$ it can be interpreted as difference of squared errors of the best approximation of $f(X)$ using linear combinations of functions of proper subvectors of $X_J$,
and of the whole vector $X_J$. We define Sobol's indices $S_J =\frac{V_J}{D}$, $J \subset I$. We have \begin{equation}\label{sumSob} 1 = \sum_{K \subset I} S_K \geq \sum_{i \subset I} S_i, \end{equation} equality in the rhs inequality meaning that \begin{equation} f(X)= \sum_{i=1}^N f_i(X_i). \end{equation}
For some $n \in \mathbb{N}_{+}$, let $Z \in L^2_n$, $D=\Var(Z) > 0$, and let now $X = (X_i)_{i=1}^N$ be a random vector (with not necessarily independent coordinates).
The main sensitivity index of $Z$ given $X_J$ is defined as \begin{equation}\label{VXJ}
V_{X_J} = \Var(\mathbb{E}(Z|X_J)). \end{equation} From (\ref{d2n}) and (\ref{aveVarError}) it follows that $D - V_{X_J}$ is equal to the squared error of the best approximation of $Z$ in $L^2_{n,X_J}$.
Suppose that $Z= f(X)$ for a certain measurable function $f$. The total sensitivity index of $f(X)$ with respect to $X_J$ is defined as \begin{equation}\label{VXJtot} V_{X_J}^{tot} = D - V_{X_{\sim J}}. \end{equation} From (\ref{aveVarError}), \begin{equation}\label{evarz}
V_{X_J}^{tot} = \mathbb{E}(\Var(f(X)|X_{\sim J})), \end{equation} so using further (\ref{d2n}) we receive that $V_{X_J}^{tot}$ is the squared error of the best approximation of $f(X)$ in $L^2_{n,X_{\sim J}}$. Sensitivity indices $V_{X_J}$ and $V_{X_J}^{tot}$ divided by $D$ are called Sobol's main and total sensitivity indices or normalized sensitivity indices, and denoted $S_{X_J}$ and $S_{X_J}^{tot}$. Let us assume that coordinates of $X$ are independent so that we can apply the ANOVA decomposition. Then $V_i = V_{X_i}$, $i \in I$, and using (\ref{subAnova}) and (\ref{sumvar}) we receive that $V_{X_J}$ is a sum of all main and interaction indices $V_{K}$, $K \subset J$, and $V_{X_J}^{tot}$ is a sum of indices $V_K$, $K \cap J \neq \emptyset$, which provides some intuition for the words main and total in the names of the indices and from which it follows that \begin{equation} 0\leq S_{X_J} \leq S_{X_J}^{tot} \leq 1. \end{equation} Furthermore, we then have from (\ref{evarz}), (\ref{condGen}), and Theorem \ref{indepCond} that
\begin{equation} V_{X_J}^{tot} = \mathbb{E}((\Var(f(X_J,z)))_{z=X_{\sim J}}),
\end{equation} so in a sense given by this formula $V_{X_J}^{tot}$ can be thought of as an average variance of $f(X)$ with respect to $X_J$.
Let us consider an output $Z = g(M) \in L^2_n$ of an MR $M=(P,Y)$ with parameters $P = (P_i)_{i=1}^{N_P}$ and corresponding to a reaction network $RN$. Note that conditional distribution of $M$ given $P$ is specified by Definition \ref{MRdef} and thus from formula (\ref{condCond}) and iterated expectation property
the distributions of $\mathbb{E}(Z|P_J)$ for different $J \subset I$ are specified by $RN$, $g$, and $\mu_P$. Thus the main sensitivity index with respect to $P_J$, denoted as $V_{P_J}$,
$D = \Var(Z)$, and $Ave=\mathbb{E}(Z)$ are all determined by this data.
As discussed in Section \ref{genParSec}, for a given construction of an MR using the noise variable $R$ one can provide construction of $g(M)$ of form $f(P,R)$
for which some further sensitivity indices can be considered, like \begin{equation}\label{VRTot} V_{R}^{tot} = D - V_P. \end{equation} Its value, by inspection of the rhs of (\ref{VRTot}), is also determined by $RN$, $g$, and $\mu_P$,
and from (\ref{aveVarError}) it is equal to $AveVar = \mathbb{E}(\Var(f(P,R)|P))$. We denote the main sensitivity index with respect to $P_J$ of conditional expectation
$\wt{g}(P) = \mathbb{E}(Z|P)$,
as $VE_{P_J}$ or $\wt{V}_{P_J}$ and such total sensitivity index as $VE_{P_J}^{tot}$ or $\wt{V}_{P_J}^{tot}$.
From the iterated expectation property it follows that $\mathbb{E}(\wt{g}(P)|P_J) = \mathbb{E}(Z|P_J)$, and therefore
\begin{equation}\label{tVCJ}
\wt{V}_{P_J} = \Var(\mathbb{E} (\wt{g}(P)|P_J)) = V_{P_J} \end{equation} and \begin{equation}\label{tildeVP} \wt{V}_{P_J}^{tot} = \wt{V}_{P} - \wt{V}_{P_{\sim J}} = V_{P} - V_{P_{\sim J}}. \end{equation}
For the special case of $J = \{i\}$, we often write $i$ in place of $P_J$ in the above notations. Analogous observations about sensitivity indices can be made and notations introduced also for other types of stochastic models.
\section{Application of VBSA to selection of parameters for determination} \label{varRedSec} Certain possible applications of VBSA were described in our previous work \cite{badowski2011} and include identifying parameters which can be fixed in order to simplify the model, computing measures of average dispersion of stochastic models, as well as planning experiments, but, as discussed in the introduction, the indices have been used also for other purposes, like to assist the process of parameter estimation. In this section we describe in a detailed and novel way the possibility of application of main sensitivity indices to comparing the average decreases of the model output uncertainty resulting from determination of values of uncertain model parameters, e. g. through a measurement, which can be useful in planning of experiments.
See Section 3.5 in \cite{badowski2011} or \cite{Saltelli2008} for alternative descriptions.
For some $n, N \in \mathbb{N}_+$, let us consider some model $M=(P,Y)$ whose output is $g(M) \in L^2_n$ for some measurable function $g$.
The uncertainty of model output can be quantified using the output variance $D = \Var(g(M))$ for some variance for random vectors as in Section \ref{secOrthog}.
Let us assume that the subvector $P_J$ of parameters $P$ consists of epistemic parameters of the model and we can determine their values exactly, for instance by measuring them, which can be a useful idealisation when the uncertainty of these parameters after
the measurement is negligibly small.
For some conditional distribution $\mu_{M|P_J}$ of $M$ given $P_J$, if the determined value of $P_J$
is $p_J$, we update $M$ to a new model $M'$ with distribution equal to $\mu_{M|P_J}(p_J, \cdot)$. In case of $M$ being an MR we can take $M'=(P',Y')$ to be an MR with the same reaction network but distribution of parameters
$\mu(P|P_J)(p_J,\cdot)$, which for $P$ with independent parameters can be taken to be the distribution of $P' = (p_J, P_{\sim J})$. The variance of output of the new model fulfills \begin{equation}
\Var(g(M')) = \Var_{\mu_{M|P_J}(p_J, \cdot)}(g). \end{equation} We received $p_J$ as an outcome of determination, e. g. through a measurement, of value of the initially uncertain random vector $P_J$, so the expected decrease of variance from the initial one $D$ can be obtained by averaging over such possible outcomes as follows \begin{equation}
\mathbb{E}(D - \Var_{\mu_{M|P_J}(P_J, \cdot)}(g)) = D - \mathbb{E}(\Var(g(M)|P_J)) = V_{P_J}, \end{equation} where in the first equality we used (\ref{varmuzx}) and in the last (\ref{aveVarError}). We received the main sensitivity index of $g(M)$ given $P_J$, thus $S_{P_J}$ tells by what fraction the model output variance is reduced on average if we determine the value of $P_J$. Note that since the main sensitivity indices of the output of a model and of its conditional expectation given the parameters are the same (see (\ref{tVCJ})), then so are their average decreases of variances. Note also that for stochastic outputs which are not functions of the parameters, like particle numbers in an MR, even if all the parameters are epistemic and are determined there will still be remaining average output variance $D-V_P$. From comparing values of $V_{P_J}$ or $S_{P_J}$ for different subvectors $P_J$ consisting of epistemic parameters one can get to know determining which of them leads on average to higher reduction of variance of the model output. This knowledge can assist the decision what parameters should be determined next, e. g. in an experiment, if the goal is to improve the precision of the model predictions. After some parameters are determined, the above procedure can be repeated with the updated model $M'$ as above. For an $\mathbb{R}^n$-valued output $g(M)$ for $n >1$, like a vector of different particle numbers or their conditional expectations at a given time, the scalar product $<,>$ used in the definition of its variance as in Section \ref{secOrthog} can be given for example by numbers $a_{ij} = c_{i}\delta_{ij}$ as in (\ref{scalAij}), where $c_{i}$ is a weight describing how important it is to be able to predict the $i$th coordinate of $g(M)$ more precisely using the model.
\section{\label{secSchemesPrev}Estimands on pairs and their unbiased estimation schemes}
Let us recall certain concepts from Section 4.3 of \cite{badowski2011} like generalized estimands, which we call here estimands on pairs, and their unbiased estimation schemes. We make numerous changes to correct errors in the previous definitions and to increase their compatibility with future generalizations in Section \ref{secUnbiased}. See Appendix \ref{appStatMC} for an introduction to statistics including standard definitions of estimands and estimators.
For a measure $\mu$ we denote its measurable space as \begin{equation}\label{smudef} \mc{S}_{\mu}= (B_\mu, \mc{B}_\mu). \end{equation} Let $N \in \mathbb{N}_+$. For a sequence of measures $\mu = (\mu_i)_{i=1}^N$, we denote $\mc{S}_{\mu} = (\mc{S}_{\mu_i})_{i=1}^N$, $B_{\mu} = (B_{\mu_i})_{i=1}^N$, and $\mc{B}_{\mu} = (\mc{B}_{\mu_i})_{i=1}^N$. Furthermore, for a vector $v \in \mathbb{N}_+^{N}$ and a sequence of sets $B = (B_i)_{i=1}^N$, we define $B^v = \prod_{i=1}^NB_i^{v_i}$, sequence of probability distributions $\mu = (\mu_i)_{i=1}^N$, $\mu^v = \bigotimes_{i=1}^N\mu_i^{v_i}$, and of measurable spaces $S = (S_i)_{i=1}^N$, $\mc{S}^v = \bigotimes_{i=1}^N \mc{S}_i^{v_i}$, For a vector $x=((x_{i,j})_{j=1}^{v_i})_{i=1}^N$ we often use a C-like notation $x_{i,j} = x_i[j-1]$, $j \in I_{v_i}$, $i \in I_N$. For $N \in \mathbb{N}_+$ let $\mathcal{R}_N$ be the class of all pairs $(\mu,f)$ such that $\mu = (\mu_i)_{i=1}^N$ is a sequence of probability measures and $f$ is a measurable real-valued function on $\bigotimes_{i=1}^NS_{\mu_i}$. Subsets $\mc{V}\subset\mathcal{R}_N$ are called admissible pairs with $N$ distributions. Set $\mc{V}_1$ is defined to consist of all $\mu$ such that there exists an $f$ such that $(\mu,f) \in \mc{V}$ and $\mc{V}_2$ is defined to consist of all $f$ for which there exists a $\mu$ such that $(\mu,f) \in \mc{V}$. By an estimand on $\mc{V}$ we mean a real-valued function $G$ on it. Note that in fact $\mathcal{R}_N$ is too large to be a set - it is a class so that $\mc{V}$ as above may also not be a set and thus $G$ may not be a function in the set theoretic sense but rather an operation, but we further on ignore such disctinction. In particular we use
notation $\mc{V} = D_G$ as for domain of a function. For some $N\in \mathbb{N}_+$ and $K \subset I_N$, let us consider the total sensitivity index $V_{X_K}^{tot}$ defined in Section \ref{secVBSA} for $Z=f(X) \in L^2(\mathbb{P})$, for a random vector $X = (X_i)_{i=1}^N$ with independent coordinates such that $X_i \sim \mu_i$, $i \in I_N$. Value of $V_{X_K}^{tot}$ is determined by $\mu= (\mu_i)_{i=1}^N$ and $f$, and thus we can an shall treat $V_{X_K}^{tot}$ as an estimand on $\mc{V} = \{((\mu_i)_{i=1}^N,f)\in \mc{R}_N: f \in L^2(\bigotimes_{i=1}^N\mu_i)\}$. We analogously define estimands corresponding to the main sensitivity index $V_{X_K}$ of $f(X)$ with respect to $X_K$ or variance $D= \Var(f(X))$, both being defined on the same admissible pairs as the total sensitivity index, and estimand $Ave = \mathbb{E}(f(X))$ on such pairs but with a less restrictive condition $f \in L^1(\bigotimes_{i=1}^N\mu_i)$ in their definition. Let $\mc{V}$ be some admissible pairs with $N$ distributions. Let us define a new as compared to \cite{badowski2011} helper concept of a real-valued statistic $\phi$ for $\mc{V}$ with dimensions of arguments $v \in \mathbb{N}_+^N$. Such $\phi$ is defined as a function on $\mc{V}_2$, such that for each $\alpha=(\mu, f) \in \mc{V}$, $\phi(f)$ is a real-valued measurable function on $\mc{S}_{\mu}^v$. We denote \begin{equation} Q_\alpha(\phi)= Q_{\mu^v}(\phi(f)) \end{equation} for $Q=\mathbb{E}$ or $Q=\Var$ whenever these expressions make sense. Let $G$ be an estimand on $\mc{V}$. An unbiased estimator of $G$ is a statistic for $\mc{V}$ with some dimensions of arguments $v$ such that for each $\alpha = (\mu,f) \in \mc{V}$,
\begin{equation}
\mathbb{E}_{\mu^v}(\phi(f)) = G(\alpha), \end{equation} i. e. $\phi(f)$ is an unbiased estimator of $G(\alpha)$ for $\mu^v$. Let $w \in \mathbb{N}_+^N$. We define $I_w = \prod_{i=1}^NI_{w_i}$. For each $x = ((x_{i,j})_{j=1}^{w_i})_{i=1}^N \in B^{w}$ and $v= (v_i)_{i=1}^N \in I_w$, we denote \begin{equation} x_v = (x_{i,v_i})_{i=1}^N. \end{equation} Let $A$ be nonempty subset of $\mathbb{N}_+^N$, called set of evaluation vectors for $N$. We define \begin{equation} (A)_i = \{j_i: j \in A\}, \end{equation} \begin{equation}\label{nAi} n_{A,i} = \max\{k: k \in (A)_i\}, \end{equation} and $n_A = (n_{A,i})_{i=1}^N$. For $v \in A$ we define evaluation operator or simply evaluation $g_{\mc{V},A,v}$ to be a real-valued statistic for $\mc{V}$ with dimensions of arguments $n_A$ such that for each $(\mu,f) \in \mc{V}$ and $x \in B_\mu^{n_A}$, \begin{equation}\label{gjDef}
g_{\mc{V},A,v}(f)(x) = f(x_v). \end{equation} For a nonempty $I \subset \mathbb{N}_+$ and a finite nonempty set $D \subset \mathbb{N}_+^I$,
let for $j \in I_{|D|}$, $\psi_{D}(j)$ denote the lexicographically $j$th element of $D$. Since for $I =\{1\}$ we identify $\mathbb{N}_+^I$ with $\mathbb{N}_+$, in such case $D \subset \mathbb{N}_+$. For each set $C$ and its finite subset indexed by $D$,
$\{y_v\in C: v \in D\}$, we define a vector from $C^{|D|}$ as follows \begin{equation}\label{ordnot}
(y_v)_{|v \in D} = (y_{\psi_D(j)})_{j=1}^{|D|}. \end{equation} We define \begin{equation}
g_{\mc{V},A} = (g_{\mc{V},A,j})_{|j \in A}. \end{equation} For $\mc{V}$ and $A$ being known from the context, we denote $g_{\mc{V}, A, v}$ shortly as $g_{v}$ or using a convenient C-array like notation \begin{equation}\label{clikegprev} g[v_1-1]\ldots[v_l-1]. \end{equation} A scheme for $N$ is a pair $\kappa=(t, A)$ for some
set of evaluation vectors $A$ for $N$ as above and $t$ being a real-valued measurable function on $\mathbb{R}^{|A|}$. A statistic given by $\kappa$ and $\mc{V}$ is defined as \begin{equation}\label{phiAF} \phi_{\kappa,\mc{V}} = t(g_{\mc{V},A}). \end{equation} Let $G$ be an estimand on $\mc{V}$. $\kappa$ is called an unbiased estimation scheme for $G$ if $\phi_{\kappa,\mc{V}}$ is unbiased estimator of $G$.
Let $G = (G_i)_{i=1}^n$ be a sequence of estimands, each on some (possibly different) admissible pairs but all with the same number of distributions $N$. Let us assume that $\kappa_i$ is an unbiased estimation scheme for $G_i$, $i \in I_n$, in which case we call $\kappa=(\kappa_i)_{i=1}^n$ an unbiased (many-dimensional) estimation scheme for $G$.
We denote $\wh{G}_{\kappa,i}=\phi_{\kappa_i,D_{G_i}}$, $i \in I_n$. For $G$ being known from the context and $G_i \neq G_j, i \neq j$, $i, j \in I_n$, we call $\kappa_i$ the subscheme of $\kappa$ for estimation of $\lambda=G_i$ and denote $\wh{G}_{\kappa,i}$ as $\wh{\lambda}_{\kappa}$, $i \in I_n$.
We further need the following theorem, generalizing Theorem 6 in \cite{badowski2011}. \begin{theorem}\label{thCond} Let us consider random variables $X = (X_1,X_2)$ and $Y_2$ such that $Y_2\sim X_2$ and $Y_2$ is independent of $X$. Let $g$ and $h$ be measurable real-valued functions such that $g(X),h(X)$, and $g(X)h(X_1,Y_2)$ are integrable. Then it holds \begin{equation}\label{condthcond}
\mathbb{E}(g(X)h(X_1,Y_2)|X_1) = \mathbb{E}(g(X)|X_1)\mathbb{E}(h(X)|X_1). \end{equation} In particular, applying expected values to both sides of (\ref{condthcond}) and using the iterated expectation property, we have \begin{equation}
\mathbb{E}(g(X)h(X_1,Y_2)) = \mathbb{E}(\mathbb{E}(g(X)|X_1)\mathbb{E}(h(X)|X_1)). \end{equation} Using this for $g(X) = h(X)$ we receive the well-known formula \cite{Saltelli_2002} \begin{equation}\label{ggxy}
\mathbb{E}(g(X)g(X_1,Y_2)) = \mathbb{E}((\mathbb{E}(g(X)|X_1))^2), \end{equation} and the fact that \begin{equation}
\Cov(g(X),g(X_1,Y_2)) = \Var(\mathbb{E}(g(X)|X_1)). \end{equation} \end{theorem} \begin{proof} It holds \begin{equation} \begin{split}
\mathbb{E}(g(X)h(X_1,Y_2)|X_1) &= (\mathbb{E}(g(x_1, X_2)h(x_1,Y_2)))_{x_1 = X_1} \\ & = (\mathbb{E}(g(x_1, X_2)))_{x_1 = X_1}(\mathbb{E}(h(x_1,Y_2)))_{x_1 = X_1}\\
& = \mathbb{E}(g(X)|X_1)\mathbb{E}(h(X)|X_1), \end{split} \end{equation} where in the first and last equality we used Theorem \ref{indepCond} and in the second independence of $X_2$ and $Y_2$ and that from Fubini's theorem functions under the expectations are integrable for $\mu_{X_1}$ a. e. $x_1$. \end{proof}
From the above theorem it easily follows that for $X$ and $Y_2$ as in it and $f(X) \in L^2_n$ with some scalar product as in Section (\ref{secOrthog}), we have \begin{equation}\label{thCondVect} \begin{split}
(f(X) ,f(X_1,Y_2))_n = ||\mathbb{E} (f(X)|X_{1})||_n^2 \end{split} \end{equation} (see (3.41) in \cite{badowski2011} for a proof).
For example for the estimand $V_1^{tot}$ we introduced earlier in this section for $N=2$, the unbiased estimation scheme $a2=(t, A)$ was defined in \cite{badowski2011} by taking $A= \{(1,1), (2,1)\}$ and \begin{equation} t(x_{(1,1)}, x_{(2,1)}) = x_{(1,1)}^2 - x_{(1,1)}x_{(2,1)}. \end{equation}
Using notation (\ref{clikegprev}), the estimator given by $a_2$ can be written as \begin{equation}\label{V1a2totg} \widehat{V}_{1,a2}^{tot} = g[0][0](g[0][0] - g[1][0]). \end{equation} The fact that this is an unbiased estimation scheme for $V_1^{tot}$ is a consequence of Theorem \ref{thCond} and the fact that observable of this estimator corresponding to function $f$ and observable $\wt{X} = (\wt{X}_{1}[j]_{j=0}^{1}, \wt{X}_{2}[0]) \sim \mu^{n_A}$ is \begin{equation}\label{obsVitot} f(\wt{X}_1[0],\wt{X}_2[0])(f(\wt{X}_1[0],\wt{X}_2[0]) - f(\wt{X}_1[1],\wt{X}_2[0])). \end{equation}
Similarly as in \cite{badowski2011} we shall often use formulas for estimators like (\ref{V1a2totg}) to concisely define previously undefined schemes, in particular for the mentioned formula retrieving scheme $a2$. Scheme given by a formula like (\ref{V1a2totg}) for estimator $\wh{\lambda}_{\kappa}$ of a certain estimand $\lambda$ on some admissible pairs $\mc{V}$, is a pair $\kappa = (t, A)$, where $A$ consists of indices $v$ of different $g_{v}$ appearing on the rhs of the formula, and $t$ acts on its arguments in the same way as the function of different $g_{v}$ given by the rhs of the formula does. By estimator defined by such a formula we mean $\phi_{\kappa,\mc{V}}$. We can group such received schemes from many formulas for estimators of different estimands in a sequence to get a many-dimensional estimation scheme for a sequence of estimands, an example of which we shall see in the next section. In the next section and further on we often define estimands $F_i, i \in I_n,$ and unbiased estimation schemes $\gamma_i$ for $F_i$, $i \in I_n$, and say that many dimensional scheme $\kappa$ consisting of $\gamma_i, i \in I_n,$ is unbiased for estimation of a sequence of estimands $G$ consisting of $F_1, \ldots, F_n,$ without specifying the order of $F_i$ or $\gamma_i$, $i \in I_n$, in sequences $\kappa$ and $G$, so that one can assume that for some arbitrary permutation $\pi$ of $I_n$, we have
$G=(F_{\pi(i)})_{i=1}^n$
and $\kappa = (\gamma_{\pi(i)})_{i=1}^n$.
\section{\label{secMany}Schemes for sensitivity indices of conditional expectations} We recall here the unbiased estimation schemes for sensitivity indices of conditional expectation from Section 4.5 of \cite{badowski2011}, which will be needed to derive certain new schemes in Section \ref{secPolynEst}. Suppose that for $N_P \in \mathbb{N}_+$, $P= (P_i)_{i=1}^{N_P}$ is a random vector with independent coordinates and $R$ is a random variable independent of $P$. Let us consider a measurable function $f$ from the product measurable space of the image of $(P,R)$ to $\mathbb{R}$. $f(P,R)$ can be for instance an output of an MR corresponding to some of its constructions as discussed in Section \ref{genParSec}. Let us consider quantities $V_k=\wt{V}_k$, $\wt{V}^{tot}_{k}$, $k \in I_{N_P}$, $D$, $V_P$, and $AveVar = V_R^{tot}$ defined for $f(P,R) \in L^2$, and $Ave=AveE$ for $f(P,R) \in L^1$, in the same way as at the end of Section \ref{secVBSA} treating $Z=f(P,R)$ as an output of an MR.
Let $\mc{V}$ be admissible pairs consisting of $\alpha_{\mu_P,\mu_R,f} = ((\mu_i)_{i=1}^{N_P+1}, f),$ such that $\mu_i \sim P_i, i \in I_{N_P},$ and $\mu_{N_P+1} \sim R$ for different $f,P,$ and $R$ as above. We will from now on interpret each of the above sensitivity indices or averages as estimands on $\mc{V}$, whose values on each $\alpha_{\mu_P,\mu_R,f}$ as above are the same as previously for the corresponding $f$, $P$, and $R$.
For $i, j \in \mathbb{N}$, we denote $s[i][j] = g[v_1]\ldots[v_{N_P+1}]$ where $v_{N_P +1} = j$ and $v_n = i$ for $n \in I_{N_P}$. For $i \in \{0,1\}, j \in \mathbb{N},$ and $k \in I_{N_P}$, we denote $s_{k}[i][j] =g[v_1]\ldots[v_{N_P+1}]$, where $v_{N_P +1} = j$ and for $n \in I_{N_P}$, $n \neq k$, $v_n = i$, while for $n = k$, $v_n = 1-i$.
For some $f$, $P$ and $R$ as above, let $\wt{P}= (\wt{P}_k)_{k=1}^{N_P}$ have independent coordinates, where $\wt{P}_k= (\wt{P}_{k,i})_{i=1}^2 \sim \mu_{P_k}^2, k \in I_{N_P}$. We denote $\wt{P}[i] = (\wt{P}_{k,i})_{k=1}^{N_P}$, $i \in \{0,1\}$. Let further for $k \in I_{N_P}$, $\wt{P}_{(k)}[i]$ be equal to vector $\wt{P}[i]$ with $k$th coordinate replaced by $\wt{P}_{k,1-i}$, and let $\wt{R} \sim \mu_R^2$ be independent of $\wt{P}$. Assuming admissible pairs $\mc{V}$ as for some of the above estimands and the set of evaluation vectors $A$ equal to set of all $v$ from evaluations $g_v$ equal to $s[i][j]$ and $s_k[i][j]$, $i,j \in \{0,1\}, k \in I_{N_P}$, we have, identifying $(\wt{P}_1,\ldots,\wt{P}_{N_P},\wt{R})$ with $(\wt{P},\wt{R})$,
\begin{equation}\label{sij} s[i][j](f)(\wt{P},\wt{R}) = f(\wt{P}[i],\wt{R}[j]), \end{equation} and \begin{equation}\label{skij} s_k[i][j](f)(\wt{P},\wt{R}) = f(\wt{P}_{(k)}[i],\wt{R}[j]). \end{equation} Formulas below, defining unbiased estimators of the above estimands are taken from Section 4.5 in \cite{badowski2011}, and the fact they are unbiased is an easy consequence of formula (\ref{ggxy}) in Theorem \ref{thCond} and formulas (\ref{tVCJ}) and (\ref{tildeVP}). We call the scheme these formulas yield scheme $SE$ (in \cite{badowski2011} we called it scheme $E$ but the new name is needed for consistency with notations introduced in Section \ref{secPolynEst}). \begin{equation}\label{estVkE} \widehat{V}_{k,SE} = \frac{1}{4}\sum_{i=0}^{1}(s[i][0] - s_k[i][0])(s_k[1-i][1] - s[1-i][1]),
\end{equation} \begin{equation}\label{estVkTotE}
\widehat{V}^{tot}_{k,SE} = \frac{1}{4} \sum_{i=0}^{1}(s[i][0] - s_{k}[i][0])(s[i][1] - s_{k}[i][1]),
\end{equation} \begin{equation}\label{VEst} \begin{split} \widehat{D}_{SE} &= \frac{1}{4(N_P + 1)}\sum_{i=0}^{1}\sum_{j=0}^{1}(s[i][j](s[i][j] - s[1-i][1-j]), \\ &+ \sum_{k=1}^{N_P}s_k[i][j](s_k[i][j] - s_k[1-i][1-j])), \end{split} \end{equation} \begin{equation}\label{VP} \begin{split} \widehat{V}_{P,SE} &= \frac{1}{4(N_P + 1)}\sum_{i=0}^{1}\sum_{j=0}^{1}(s[i][j](s[i][1-j] - s[1-i][1-j]) \\ & + \sum_{k=1}^{N_P}s_k[i][j](s_k[i][1-j] - s_k[1-i][1-j])), \end{split} \end{equation} \begin{equation}\label{VRtot} \widehat{AveVar}_{SE} = \widehat{V}_{R,SE}^{tot} = \widehat{D}_{SE} - \widehat{V}_{P,SE}, \end{equation} \begin{equation} \widehat{Ave}_{SE} = \frac{1}{4(N_P + 1)}\sum_{i=0}^{1}\sum_{j=0}^{1}(s[i][j] + \sum_{k=1}^{N_P}s_k[i][j]). \end{equation}
Using the same evaluations we can also construct estimation schemes for many further indices, among others for $\wt{V}_{(P_{i},P_{j})}$ and $\widetilde{V}_{(P_{i},P_{j})}^{tot}$ (see \cite{badowski2011}), $i, j \in I_{N_P}$, $i \neq j$.
It is easy to see using Schwartz inequality that it is sufficient that $f(P,R) \in L^4$ for the above estimators and further ones in this section to have finite second moments and thus variances when applied to the corresponding $f,\wt{P}$, and $\wt{R}$. For $\widehat{Ave}_{SE}$ it is even sufficient that $f(P,R) \in L^2$.
In \cite{badowski2011} we also introduced scheme $EM$ consisting of subschemes for estimation of $V_k$, for $k \in I_{N_P}$, \begin{equation}\label{estVkEM} \wh{V}_{k,EM} = \frac{1}{2}(s[0][0] - s_k[0][0])(s_k[1][1] - s[1][1]). \end{equation}
Similarly as in \cite{badowski2011}, we define scheme $ET$ containing subschemes given by formulas \begin{equation}\label{estVkTotET} \widehat{\widetilde{V}}^{tot}_{k,ET} = \frac{1}{2}(s[0][0] - s_k[0][0])(s[0][1] - s_k[0][1]),\ k \in I_{N_P}. \end{equation}
As discussed in \cite{badowski2011} schemes in this section can be generalized to variables $f(P,R)\in L_n^2$ like conditional histograms by using appropriate scalar product of vectors instead of function multiplication in the formulas for estimators, which is a consequence of expression (\ref{thCondVect}) after the proof of Theorem \ref{thCond}.
\section{\label{secMCIneff}Inefficiency constants of MC procedures} See Appendix \ref{appStatMC} for an introduction to Monte Carlo method and associated notations we use, like $Var_s$ and $\Var_f(n)=\Var_f$ for the variances of singles step and final $n$-step MC estimators, respectively, fulfilling \begin{equation}
\Var_f = \frac{\Var_s}{n}. \end{equation} Let us consider a sequence of MC procedures estimating $\lambda \in \mathbb{R}$, indexed by $n \in \mathbb{N}_+$, such that the $n$-th one is an $n$-step MC procedure and its average duration $\tau_f(n)$, e. g. when run on a computer, fulfills \begin{equation}\label{nfs} \tau_f(n) = n\tau_s,\quad n \in \mathbb{N}_+, \end{equation} where $\tau_s \in \mathbb{R}_+$ is called the average duration of a single MC step of this sequence. Assumption (\ref{nfs}) is a good approximation for many sequences of MC procedures run on a computer, especially ones for which the $n$-th procedure consists of $n$ repeated computationally identical single MC steps, each lasting on average $\tau_s$, $n \in \mathbb{N}_+$. Similarly as in Section 4.2 in \cite{badowski2011} we define the inefficiency constant of a sequence of MC procedures as above by formula \begin{equation}\label{cdef} c = \tau_{s} \Var_{s}, \end{equation} so that from (\ref{varfsn}) and (\ref{nfs}), for each $n \in \mathbb{N}_+$, \begin{equation}\label{cntau} c = \tau_{f}(n) \Var_{f}(n). \end{equation} For two different sequences of MC procedures as above for estimating $\lambda$, their inefficiency constants can be used for comparing their efficiency \cite{asmussen2007stochastic, badowski2011}, which can be justified by different interpretations of these constants. We shall provide below a correction of an interpretation from Section 4.2 in \cite{badowski2011} in which we used an incorrect asymmetric definition of $\delta$-approximate inequality. Two new interpretations shall be provided in Section \ref{secStat}. See Chapter 3, Section 10 in \cite{asmussen2007stochastic} for yet another interpretation. For $x,y\in \mathbb{R}_+$ and $\delta \geq 0$, we say that $x$ and $y$ are $\delta$-approximately equal, which we denote as $x \approx_{\delta} y$, if
$\frac{|x-y|}{\min(|x|,|y|)}\leq \delta$; in particular for $\delta=0$ this is equivalent to $x = y$. If for some sequence of MC procedures as above and another one, also for estimating $\lambda$, for which we have the same assumptions and use the same notations but with a prim, we have $\delta$-approximate equality of their respective average duration times for some $n$ and $n'$, that is \begin{equation} \tau_{f}(n) \approx_{\delta} \tau_{f}'(n'), \end{equation} then from (\ref{cntau}) the ratio of variances of their respective final MC estimators is $\delta$-approximately equal to the ratio of their inefficiency constants, i. e. \begin{equation}\label{varAiRatio} \frac{\Var_{f}(n)}{\Var_{f}'(n')} = \frac{c\tau_{f}'(n')}{c'\tau_{f}(n)} \approx_{\delta} \frac{c}{c'}. \end{equation}
\section{\label{secIneffSchemes}Inefficiency constants of schemes} Let us reformulate the theory of inefficiency constants from sections 4.3 and 4.5 in \cite{badowski2011} in a more precise way. Let us consider a sequence of estimands $G=(G_i)_{i=1}^n$ such that $\mc{V} = \bigcap_{i=1}^n D_{G_i} \neq \emptyset$, called estimands on common admissible pairs $\mc{V}$ with $N$ distributions. We denote $D_G=\mc{V}$.
Suppose that $\kappa=(\kappa_i)_{i=1}^n= (t_i,A_i)_{i=1}^n$ is an unbiased estimation scheme for $G$. $A_\kappa = \bigcup_{i=1}^n A_i$ is called the set of evaluation vectors of $\kappa$.
$\kappa$ can be used to generate estimates of coordinates of $G(\alpha)$ for some $\alpha =(\mu,f) \in \mc{V}$ as follows. For a random vector $X \sim \mu^{n_A}$, one computes the quantities $g_{\mc{V},A_i,v}(f)(X_{n_{A_i}})=f(X_v)$, $i \in I_n$, $v \in A_{i}$, considering that they are equal for the same $v$ and different $i$ so that they are computed only once, and then one evaluates $t_i$ on $g_{\mc{V},A_i}(f)(X_{n_{A_i}})$ to get an estimate of $G_i(\alpha)$, $i \in I_n$.
$|A|$ is the total number of evaluations of $f$ in such a computation.
If for some $\alpha \in \mc{V}$ we have $\Var_{\alpha}(\phi_{\kappa_i,\mc{V}})<\infty$, $i \in I_n,$ then the above computation can be performed to get unbiased estimates of coordinates of $G(\alpha)$ in a single step of a MC procedure. We define an inefficiency constant $d_{G,i,\kappa}$ of $\kappa$ for estimating $G_i$ as a function $\mc{V}\rightarrow \overline{\mathbb{R}}$ such that \begin{equation}\label{dIneffprev}
d_{G,i,\kappa}(\alpha) = \Var_{\alpha}(\phi_{\kappa_i,\mc{V}})|A_{\kappa}|. \end{equation}
Let $\kappa'$ be an unbiased estimation scheme for estimands $G'=(G')_{i=1}^{n'}$ on some common admissible pairs $\mc{V}'$ for which we shall use the same notations as for $\kappa$, $G$, and $\mc{V}$ but with prims. Let for some $\alpha=(\mu,f)\in \mc{V}$ , $\alpha' = (\mu',f') \in \mc{V}'$, $i \in I_n$, and $i' \in I_n'$, it hold $G_{i}(\alpha)=G_{i'}(\alpha')$. Suppose that the ratio of positive average durations $\tau_{s}$ to $\tau'_{s}$ of single steps of sequences of MC procedures (see Section \ref{secMCIneff}) using $\kappa$ and $\kappa'$, computing $G(\alpha)$ and $G'(\alpha')$ as above is for some $\delta>0$, $\delta$-approximately equal to ratio of positive numbers of evaluation vectors $A_\kappa$ and $A_\kappa'$ in these schemes, that is \begin{equation}\label{tauAratio}
\frac{\tau_s}{\tau'_s} \approx_\delta \frac{|A_{\kappa}|}{|A_{\kappa'}|}. \end{equation} This can be the case for small $\delta$ e. g. when the most time-consuming part of both sequences of MC procedures are calls to implementations of $f$ and $f'$, respectively, taking on average approximately the same time to compute. As we demonstrate in Section \ref{secImpl}, such approximate proportionality and even its more general version discussed in Section \ref{secineffgen} holds in our numerical experiments using different estimation schemes for variance-based sensitivity indices and some further estimands, in which $f(P,R)$ and $f'(P,R')$ for some parameters $P$ and noise variables $R$ and $R'$, stand for some outputs of an MR, constructed using the GD and RTC methods or two times one of them (see (\ref{obsForm})). From (\ref{tauAratio}), the ratio of inefficiency constant \begin{equation}\label{cscheme} c = \Var_{\alpha}(\phi_{\kappa_i,\mc{V}})\tau_s \end{equation} of a sequence of MC procedures estimating quantities $G_i(\alpha)$, performing computations with scheme $\kappa$ (see (\ref{cdef})) to an analogous constant $c'$ for $\kappa'$, computing $G_{i'}(\alpha')$, fulfills (assuming both constants are finite), \begin{equation} \frac{c}{c'} = \frac{\Var_{\alpha}(\phi_{\kappa_i,\mc{V}})\tau_s}{\Var_{\alpha'}(\phi_{\kappa'_{i'},\mc{V}})\tau_s'} \approx_{\delta} \frac{d_{G,i,\kappa}(\alpha)}{d_{G,i',\kappa'}(\alpha')}, \end{equation} which we already noticed in \cite{badowski2011} but with equality rather than $\delta$-approximate equality in (\ref{tauAratio}). Similarly as for inefficiency constants of sequences of MC procedures in Section \ref{secMCIneff}, one proves that the ratio of positive
real values of inefficiency constants (\ref{dIneffprev}) of $\kappa$ and $\kappa'$ for estimating $G_i(\alpha)$ and $G'_i(\alpha')$ as above is $\delta$-approximately equal to the ratio of variances of the appropriate final MC estimators for $\delta$-approximately the same number of $i$th and $i'$th functions evaluations made in the respective MC procedures. If $G$ is known from the context, and $G_l \neq G_m$ for $l \neq m$, $l,m \in I_n$, then we denote $d_{G,i,\kappa}$ simply as $d_{G_i,\kappa}$
\section{\label{secSymIneq}Symmetrisation of schemes and inequalities between inefficiency constants} Let us recall some definitions and facts from Section 4.4 of \cite{badowski2011} on symmetrisation of schemes, changing them for compatibility with future generalizations in sections \ref{secAveSchemes} and \ref{secGenIneq}. Let $\Theta$ be the group of all bijections of $\mathbb{N}_+$. For $N \in \mathbb{N}_+$, we define $\Theta^N = \{(\pi_i)_{i=1}^N:\pi_i \in \Theta, i \in I_N\}$ and endow it with a structure of a direct product group by defining for each $\pi, \pi'\in \Theta^N$ their product as $\pi\pi'= (\pi_i(\pi_i'))_{i=1}^N$. For $\pi \in \Theta^N$ we define $\wh{\pi}:\mathbb{N}_+^N\rightarrow\mathbb{N}_+^N: \wh{\pi}(v)=(\pi_i(v_i))_{i=1}^N$. Let $\Pi$ be a subgroup of $\Theta^N$. For $A \subset \mathbb{N}_+^N$, we denote its symmetrisation given by $\Pi$ as \begin{equation} \wh{\Pi}[A] = \bigcup_{\pi\in \Pi} \wh{\pi}[A] = \{\wh{\pi}(v): \pi \in \Pi, v \in A\}. \end{equation}
For a function $t: \mathbb{R}^{|A|} \rightarrow \mathbb{R}$, its symmetrisation given by $\Pi$ and $A$ is defined to be a function
$\ave_{A,\Pi}(t): \mathbb{R}^{|\wh{\Pi}[A]|}\rightarrow\mathbb{R}$ such that for each $z=(y_j)_{|j \in \wh{\Pi}[A]} \in \mathbb{R}^{|\wh{\Pi}[A]|}$ \begin{equation} \label{symOp}
\ave_{A,\Pi}(t)(z) = \frac{1}{|\Pi|} \sum_{\pi \in \Pi} t((y_{\wh{\pi}(v)})_{|v \in A}) \end{equation} (see \ref{ordnot}). Let $\kappa =(t,A)$ be a scheme for $N$. Its symmetrisation given by $\Pi$ is defined as \begin{equation}\label{avePiKO} \ave_{\Pi}(\kappa) = (\ave_{A,\Pi},\wh{\Pi}[A]). \end{equation} Let $\mc{V}$ be admissible pairs with $N$ distributions, $\alpha \in \mc{V}$, $\alpha = (\mu,f) \in \mc{V}$, and $X \sim \mu^{n_{\wh{\Pi}[A]}}$. Then the corresponding observable of and estimator given by a symmetrised scheme fulfills \begin{equation}
\phi_{\ave_{\Pi}(\kappa),\mc{V}}(f)(X) = \frac{1}{|\Pi|} \sum_{\pi \in \Pi} t((f(X_\pi(v)))_{v \in A}), \end{equation} which for $Y \sim \mu^{n_A}$ is equal to a sum of random variables with the same distribution as $\phi_{\kappa,\mc{V}}(f)(Y)$. Therefore if $\kappa$ is unbiased for estimation of some estimand $G$ on $\mc{V}$, then so is $\ave_{\Pi}(\kappa)$ and from Lemma \ref{lemVarAve} in Appendix \ref{appStatMC} we have for each $\alpha \in \mc{V}$, \begin{equation}
\Var_\alpha(\phi_{\ave_{\Pi}(\kappa),\mc{V}}) \leq \Var_\alpha(\phi_{\kappa,\mc{V}}). \end{equation} Let $I$ be a nonempty subset of $I_N$. For $\theta \in \Theta$, we define $\pi_{N,I,\theta} \in \Theta^N$ to be such that $\pi_{N,I,\theta,i} = \theta$, $i \in I$, and $\pi_{N,I,\theta,i}= \id_{\mathbb{N}_+}$, $i \in I_N\setminus I$. For $m \in \mathbb{N}$, we call \begin{equation}\label{defthetam} \Theta_m=\{\theta\in\Theta: \theta(i) = i \text{ for } i > m \} \end{equation} the subgroup of $\Theta$ of permutations of the first $m$ indices. Let \begin{equation}\label{thetanim}
\Theta_{N,I,m} = \{\pi_{N,I,\theta}: \theta \in \Theta_m\}. \end{equation} Symmetrisation of a scheme for $N$ w. r. t.
$\Theta_{N,I,m}$ is called its symmetrisation in the argument given by $I$, in $m$ dimensions. If $n_{A,i} = m$, $i \in J$, then we call it simply symmetrisation in the argument given by $I$ and if $n_{A,i} = 1$, $i \in J$, we call it symmetrisation from one to $m$ dimensions. For $I = \{j\}$ we say of symmetrisation in the $j$th argument, in which case we write $j$ instead of $I$ in the subscript.
After symmetrising the scheme given by (\ref{V1a2totg}) in the first argument in two dimensions as in \cite{badowski2011} we receive a scheme given by
\begin{equation}\label{vTotMin} \wh{V}_{1,s2}^{tot} = \frac{1}{2}(g[0][0] - g[1][0])^2, \end{equation} and we conclude that \begin{equation}\label{ineqs2a2} \Var_{\alpha}(\wh{V}_{1,s2}^{tot}) \leq \Var_{\alpha}(\wh{V}_{1,a2}^{tot}),\ \alpha \in D_{V_{1}}. \end{equation}
As both schemes use the same number of function evaluations, we also have \begin{equation} d_{V_{1}^{tot},s2} \leq d_{V_{1}^{tot},a2}, \end{equation} which should be understood as holding for each $\alpha \in D_{V_1^{tot}}$. From expression (\ref{ggxy}) in Theorem \ref{thCond} we receive that the following formula defines an unbiased estimator of the main sensitivity index $V_1$ for $N=2$, already mentioned in \cite{badowski2011}, \begin{equation} \widehat{V}_{1,a3} = g[0][0](g[0][1] - g[1][1]). \end{equation} Similarly as in \cite{badowski2011}, from symmetrising its scheme in the first argument we receive a scheme given by the formula \begin{equation}\label{V1s4} \widehat{V}_{1,s4} = \frac{1}{2}(g[0][0] - g[1][0])(g[0][1] - g[1][1]), \end{equation} which uses 4 rather than 3 evaluation vectors, so that their respective inefficiency constants fulfill \begin{equation}\label{ineqV1} d_{V_1,s4} \leq \frac{4}{3} d_{V_1,a3}. \end{equation} The following theorem is a slight generalization of Theorem 12 from \cite{badowski2011}, the proof of which is analogous as in \cite{badowski2011}, and which shall follow from a more general Theorem \ref{thIneqds} in Section \ref{secGenIneq}. \begin{theorem}\label{thineqOld} Let $\kappa_1$ be unbiased estimation scheme of some estimand $G$ on adissible pairs $\mc{V}$ with $N$ distributions, and let the scheme $\kappa_2$ be created from $\kappa_1$ by its symmetrisation in the argument given by $I \subset I_N$ from one to two dimensions. Then \begin{equation}\label{d12} d_{G,\kappa_1} \leq d_{G, \kappa_2} \leq 2 d_{G,\kappa_1}. \end{equation} which should be understood as holding for each $\alpha \in \mc{V}$. \end{theorem} For an illustration let us consider an estimation scheme for $V^{tot}_1$, for $N=2$, given by the formula \begin{equation}\label{VTot1s4} \widehat{V}^{tot}_{1, s4} = \frac{1}{4}\sum_{i=0}^1(g[0][i] - g[1][i])^2. \end{equation} As in \cite{badowski2011} let us notice that scheme (\ref{VTot1s4}) is received from (\ref{vTotMin}) by symmetrisation in the second argument from one to two dimensions, so from the above theorem \begin{equation}\label{compDi}
d_{V^{tot}_{1},s2} \leq d_{V^{tot}_{1},s4} \leq 2d_{V^{tot}_{1},s2}. \end{equation} As we noticed in \cite{badowski2011}, scheme given by formula (\ref{estVkTotE}) for $\wh{V}_{k,SE}^{tot}$ is symmetrisation of the one given by formula (\ref{estVkTotET}) for $\wh{\wt{V}}_{k,ET}^{tot}$ from $1$ to $2$ dimensions in the argument corresponding to $P_{\sim k}$, so from the above theorem and the fact that the ratio of number of evaluation vectors used by the individual subschemes of $SE$ and $ET$ for the total sensitivity indices and the whole schemes is the same (and equal 2), we receive \begin{equation}\label{VitotComp} d_{\wt{V}_i^{tot},ET} \leq d_{\wt{V}_i^{tot},SE} \leq 2d_{\wt{V}_i^{tot},ET}. \end{equation} In \cite{badowski2011} we also proved the following Theorem 13 which more precisely than originally can be formulated as follows. \begin{theorem}\label{thdEMEComp} For each $\alpha = \alpha_{\mu_P,\mu_R,f} \in \mc{D}_{V_{k}}$ for some $f$, $P$, $R$ as in Section \ref{secMany} such that
$f(P,R) \in L^4$, the inefficiency constants of schemes $EM$ and $SE$ for estimation of $V_k$ fulfill, for $N_P > 2$, \begin{equation}\label{EMComp} d_{V_k,EM}(\alpha) \leq d_{V_k,SE}(\alpha) \leq 2d_{V_k,EM}(\alpha). \end{equation} \end{theorem}
\section{\label{secVarDiff}Variances of estimators using the GD and RTC methods} Let $h(p,R)$ denote certain construction of an MRCP corresponding to parameters $p\in \mathbb{R}^n$ and a reaction network $RN$ (see (\ref{MRCPFun})). For $f(p,R) = g(h(p,R))\in L^2$ denoting the number of particles of certain species at a given moment of time, in \cite{Rathinam_2010} and \cite{Anderson2013} the mean finite difference \begin{equation}\label{fidi} \frac{1}{s} \mathbb{E}(f(p + se_i, R) - f(p, R)) \end{equation} for some $i \in I_n$ and $s \in \mathbb{R}_+$, approximating the partial derivative $\partial_i\mathbb{E}(f(p,R))$, was estimated in a MC procedure evaluating an independent copy of $f(p + s e_i,R) - f(p, R)$ in each step. Using the RTC construction of the output in such a MC procedure for finite difference has been called common reaction path method, while using the GD construction - common random number method \cite{Rathinam_2010}, both names stressing the fact that the same noise variable $R$ is used to construct the initial and perturbed outputs in each MC step. The estimates of variance of $f(p + se_i, R) - f(p, R)$ determining the variance of the final MC estimators of (\ref{fidi}) for the models considered in \cite{Rathinam_2010} was much lower when performing the simulations with the RTC than the GD method, however in Section E of \cite{Anderson2013} an example was provided with an opposite inequality. Note that the variance mentioned being higher for one of the above methods than the other is equivalent to the following mean squared difference \begin{equation}\label{errSqr} \msd(p_1, p_2) = \mathbb{E}((f(p_1, R) - f(p_2, R))^2) \end{equation} being higher for $p_1 =p$, $p_2 = p+se_i$, or $\mathbb{E}(f(p_1, R)f(p_2, R))$ or $\Cov(f(p_1, R),f(p_2, R))$
being lower for such $p_1$ and $p_2$ for one method than the other. Let us consider an output $f(P,R)=g(h(P,R))$ of an MR $(P,h(P,R))$ constructed using the RTC or the GD method. In the numerical experiments in \cite{badowski2011} we observed that for such outputs being particle numbers of some species at a given time, the estimates of the variances of estimators of main and total sensitivity indices given by schemes $EM$, $ET$, and $SE$, corresponding to admissible pairs $\alpha_{\mu_P,\mu_R,f}$ (see Section \ref{secMany}) were in some cases much higher when using the RTC than the GD method. The variances of these estimators also varied with the order of reactions in the GD method. In an experiment for many births - many deaths model which we describe in Section \ref{MBMDPrev}, grouping reactions with similar effect on the considered output together in the sequence of reactions resulted in lower estimates of variance of the above estimators using GD method than when reactions with different effects appeared one after another in the sequence.
Note that reordering the reactions in the RTC method results in reordering of the Poisson processes used in the construction, which causes no change of variance of the above discussed estimators using this method. For some of the above estimators one can show that, given $f(P,R)\in L^2$, if for some measurable set $A$ such that $\mu_P(A) = 1$, for all pairs of parameter values $(p_1,p_2) \in A^2,$ $\msd(p_1, p_2)$ is not higher for one construction of MR than another, e. g. for the GD method than the RTC method or for GD methods but with different orders of reactions, then variance of the considered estimator corresponding to such $f$, $P$, $R$ should be either not lower or not higher for one construction than the other. For instance, as discussed in \cite{badowski2011}, for estimators $\wh{V}_{k,EM}$ and $\wh{V}^{tot}_{k,ET}$, using notations as in Section \ref{secMany} and $\alpha=\alpha_{\mu_P,\mu_R,f}$, from the equalities
\begin{equation}
4\mathbb{E}_{\alpha}((\wh{V}_{k,EM})^2) = \mathbb{E}(\msd(\wt{P}_{(k)}[0], \wt{P}[0])\msd(\wt{P}[1], \wt{P}_{(k)}[1]))\\
\end{equation}
and
\begin{equation}
4\mathbb{E}_{\alpha}((\wh{V}^{tot}_{k,ET})^2) = \mathbb{E}(\msd^2(\wt{P}_{(k)}[0], \wt{P}[0]))
\end{equation} it follows that the inequalities between the variances of estimators $\wh{V}_{k,EM}$ and $\wh{V}^{tot}_{k,ET}$ when using different constructions should be the same as the inequalities between the quantities $\msd(p_1, p_2)$ for all $p_1$, $p_2$ as above.
\section{\label{secExSoft}Existing software used} In \cite{badowski2011} we run experiments using a program written in the C++ programming language and a personal computer with 1GB RAM, 2-core 2.10 GHz processor, and Linux operating system. We shortly describe this program below, see Section 4.7 in \cite{badowski2011} for details. For the numerical experiments in this work we made some extensions to this program, as described in Section \ref{secImpl}. For specifying the reaction network, distribution of parameters, and output of the MR considered in computations we used SBML (Systems Biology Markup Language) \cite{SBML} configuration files. We used GNU Scientific Library \cite{Galassi_2003} implementation of the Mersenne twister random number generator \cite{MATSUMOTO_NISHIMURA_1998}
and simple implementations of the RTC and GD constructions.
Values of the model output were generated by running a given simulation algorithm starting with the selected parameters and reusing the values of independent copies of the noise term $R$, e. g. for the schemes from Section \ref{secMany} when evaluating $s[i][j]$ and $s_{(k)}[i][j]$ for the same $j$. This reusing was implemented by storing the noise variables in lists, in the GD method using a single list for each noise variable, while in the RTC method a different one for each Poisson process in the construction.
\section{Models used} In this work we use certain mathematical models from the literature, which we briefly describe below. \subsection{\label{SBPrev}Simple birth (SB) model}
SB model is a very simple model from \cite{badowski2011}, for which, as opposed to the further models, many sensitivity indices and coefficients of our interest can be computed analytically.
The reaction network of the model consists of a single reaction involving only one species $X$ \begin{equation} R_1:\ \emptyset \rightarrow X. \end{equation} The only kinetic rate of this reaction fulfills $a_1(K,x) = K_1 + K_2 + K_3$, where $K = (K_1,K_2, K_3)$ is a random vector with independent coordinates with respective distributions $U(0.3,0.9)$, $U(0.85, 1.15)$, and $U(0.07, 0.13)$. Random initial number $C$ of particles of $X$ is independent of $K$ and has uniform discrete distribution $U_d(30, 90)$. As the output for analysis we take the number of particles of species $X$ at time $t=100$.
\subsection{\label{GTSPrev}Genetic toggle-switch (GTS) model} Let us consider a model of a genetic toggle switch which is a simplified stochastic version of the model from \cite{Gardner2000}, first analyzed in \cite{Rathinam_2010} and later also in \cite{badowski2011}.
In the model, two species $U$ and $V$ are produced and degraded in the following four reactions \[R_1:\ \emptyset \rightarrow U, \quad R_2:\ U \rightarrow \emptyset,\] \[R_3:\ \emptyset \rightarrow V,\quad R_4:\ V \rightarrow \emptyset.\] For $x = (x_1,x_2)$ being the vector of numbers of species $U$ and $V$, and the rate constants vector equal to $K = (\alpha_1, \alpha_2, \beta, \gamma)$, the rates of the above reactions are \[a_{1}(K,x) = \frac{\alpha_1}{1 + x_2^{\beta}}, \quad a_{2}(K,x) = x_1, \] \[a_{3}(K,x) = \frac{\alpha_2}{1 + x_1^{\gamma}}, \quad a_{4}(K,x) = x_2. \] The second and fourth rates describe degradation with a speed proportional to the current number of particles of a given species. The first and third rates describe the fact that each species is a repressor of the promoter transcribing the other species, that is it inhibits the production of the opposing repressor by attaching itself to the DNA sequence preceding the region coding the other repressor. The value of the rate constants vector $K$ in \cite{Rathinam_2010} was deterministic and equal to $(50, 16, 2.5, 1)$. However, similarly as in \cite{badowski2011}, we consider $K$ to be a random vector whose each coordinate which in \cite{Rathinam_2010} had a fixed value $v$ is considered to be a random variable with distribution U($0.8v, 1.2v$) and independent of the other coordinates. Similarly as in \cite{Rathinam_2010}, the initial particle numbers of both species are zero and we thus consider the vector of random parameters to be equal to $K$. As in \cite{Rathinam_2010} and \cite{badowski2011}, the model output considered for sensitivity analysis is the number of particles of the species $U$ at time $t=10$. Using this model in \cite{badowski2011} we observed lower variances for the RTC than the GD method for all estimators of main and total sensitivity indices of conditional expectation from Section \ref{secMany}.
\subsection{\label{MBMDPrev}Many births - many deaths (MBMD) model} Let us now consider the MBMD model from \cite{badowski2011}, whose reaction network contains one species $X$ and the following $5$ different birth and death reactions can occur \begin{equation} R_{bi}:\ \emptyset \rightarrow X,\ R_{di}:\ X\rightarrow \emptyset, \quad i \in I_5. \end{equation} The order of these reactions is \begin{equation} R_i = R_{bi},\ R_{5 + i} = R_{di}, \quad i \in I_5. \end{equation} The rate constants vector is $K = (K_{b1},\ldots,K_{b5},K_{d1},\ldots,K_{d5})$, and has independent coordinates with $K_{di} \sim \U(0.010,\ 0.040)$ and $K_{bi} \sim \U(0.10,\ 0.40)$, $i \in I_5$. The rates of birth reactions are $a_{bi}(K, x) = K_{bi}$, and of death reactions $a_{di}(K,x) = K_{di}x$, $i \in I_5$. The initial number $C$ of particles of $X$ has distribution $U_d(5, 15)$ and all the parameters are independent. The considered output is as in \cite{badowski2011} the number of particles of species $X$ at time $t=5$.
In the numerical experiments in \cite{badowski2011} we considered three different constructions. The first two are the RTC and GD methods for the above initial reaction network and distribution of parameters, abbreviated shortly as RTC and GDI methods, while the third construction is the GD construction for such distribution of parameters but for a reaction network with reordered sequence of reactions \begin{equation}\label{newOrder} R_{2i-1} = R_{bi},\ R_{2i} = R_{di},\quad i \in I_5, \end{equation} abbreviated as GDR method. The intuition behind such reordering in \cite{badowski2011} was to increase the frequency of switching between the birth and death reactions in a given step of the GD construction for different values of the model parameters $p_1$, $p_2$ and the same noise variable so as to increase the value of the function $\msd(p_1,p_2)$ defined by (\ref{errSqr}). In \cite{badowski2011} the estimates of variances of estimators from the schemes $EM$ for the main and and $ET$ for the total sensitivity indices were lowest when using the RTC method, followed by the GDI and GDR methods.
\chapter{\label{chapOwnRes}Own research} \section{\label{secMRNew}New theorems for MRCP and MR} Below we provide two new theorems giving criteria for existence of MRCP and MR and finiteness of higher moments of particle numbers in the latter model. For $a, b \in \mathbb{R}^N$, we denote
$ab = \sum_{i=1}^Na_ib_i$. Let us consider a reaction network $RN$ as in Section \ref{secCRN}.
The following theorem, which we prove in Appendix \ref{appProc}, gives a useful sufficient condition for the existence of an MRCP. \begin{theorem}\label{thNonexpl} Let $k \in B_{RN}$. If there exists a vector $m =(m_i)_{i=1}^N \in \mathbb{R}^N$ with positive coordinates, such that \begin{equation}\label{supmx} \sup_{x \in E} (\sum_{l=1}^L(ms_la_l(k,x)) - mx) < \infty, \end{equation} then for each $c \in E$, an MRCP corresponding to $RN$ and $p=(c,k)$ exists. \end{theorem} Vector $m$ can often be chosen such that for $i \in I_N$, $m_i$ is mass of the $i$th species, in which case $ms_l$ is the mass increase in the $l$th reaction and $mx$ is the total mass of all species in the system in state $x$.
For real-valued outputs $g(M)$ in this work
we often encounter the requirement that $\mathbb{E}(|g(M)|^n)<\infty$ for some $n \in \mathbb{N}_+$. For $Y_{t,i}$ denoting the number of particles of the $i$th species at the moment $t$ in a MR, we have a following criterion. \begin{theorem}\label{thMomsExist} Let $\nu$ be a probability distribution on $\mc{S}_{RN,E}$. Suppose that there exists a vector $m =(m_i)_{i=1}^N \in \mathbb{R}_+^N$, such that for $L_m = \{l \in I_L: s_{l}m > 0\}$, for the function
\begin{equation} A(k) = \max\{ \sup_{x \in E, l \in L_m}(a_l(k, x)), 0\}, \end{equation} for each $(K, C) \sim \nu$, and certain $n \in \mathbb{N}_{+}$, it holds $\mathbb{E}(A(K)^n) < \infty$ and $\mathbb{E}(C_i^n) < \infty$, $i \in I_N$. Then for $\nu$ a. e. $p$, MRCP given by $RN$ and $p$ exists. Moreover, for a process $Y$ of an MR corresponding to $RN$ and $\nu$, for each $i \in I_N$ and $t \in T$, it holds $\mathbb{E}(Y_{t,i}^n) < \infty$. \end{theorem} Proof of the above theorem is provided in Appendix \ref{appProc}. All the moments of each parameter in the SB, GTS, and MBMD models from sections \ref{SBPrev}, \ref{GTSPrev}, and \ref{MBMDPrev} exist, so given the form of reaction rates of these models assumptions of Theorem \ref{thMomsExist} are satisfied for all $n \in \mathbb{N}_+$ if we take $m_i$ equal to one for each $i$th species.
Thus all moments of each particle numbers at each time instant in these models exist, which makes it possible to use the MC method for estimating the sensitivity indices and various coefficient defined further on using the schemes from the previous and further sections for output being the particle numbers as above.
\section{\label{secCondMoms}Functions of conditional moments} For a real-valued random variable $Z$ on the probability space with a probability measure $\mu$ and $n \in \mathbb{N}_+$, we define the $n$th moment of $Z$ for $\mu$ to be the element of $\overline{\mathbb{R}}$ defined as \begin{equation}\label{Mn} M_{n}(\mu,Z) = \mathbb{E}_{\mu}(Z^n) \end{equation} and the $n$th central moment of $Z$ for $\mu$ the element of $\overline{\mathbb{R}}$ defined as \begin{equation}\label{CMn} CM_{n}(\mu,Z) = \mathbb{E}_{\mu}((Z - \mathbb{E}_\mu(Z))^n), \end{equation} whenever these expressions make sense (that is in the second case
$\mathbb{E}_\mu (|Z|)< \infty$ and in both cases the functions appearing under the outer expectations must have their positive or negative parts Lebesgue integrable with respect to $\mu$).
We shall consider the $n$th moment, denoted as $M_n$, or such central moment $CM_n$ to be a certain function $Q$ whose domain $D_Q$ are pairs $(\mu,Z)$ for which respective expression (\ref{Mn}) or (\ref{CMn}) makes sense, and for each $\alpha=(\mu,Z) \in D_Q$, $Q(\alpha)$, also denoted $Q_\mu(Z)$, is given by the rhs of (\ref{Mn}) or (\ref{CMn}), respectively. Both the $n$th moment and central moment $Q$ restricted to the class \begin{equation}\label{defMCRn} \mc{T}_n = \{(\mu,Z):\text{$\mu$ is a probability measure and } Z \in L^n(\mu)\} \end{equation}
is a real-valued function, equal to some measurable function $f_Q:\mathbb{R}^n\rightarrow \mathbb{R}$
applied to a vector of the first $n$ moments of $Z$, each restricted to $\mc{T}_n$, that is \begin{equation}\label{qfq}
Q_{|\mc{T}_n}= f_Q((M_{k|\mc{T}_n})_{k=1}^n), \end{equation} or equivalently
\begin{equation}\label{qz} Q_\mu(Z) = f_Q((\mathbb{E}_{\mu}(Z^i))_{i=1}^{n}),\ (\mu,Z) \in \mc{T}_n.
\end{equation} The first moment is expectation for which $Q=\mathbb{E}$, $n=1$, and $f_E = \id_\mathbb{R}$. The second central moment is variance for which $Q=\Var$, and for each $(\mu,Z) \in \mc{T}_1$,
\begin{equation}\label{varDef}
\Var_{\mu}(Z) = \mathbb{E}_{\mu}((Z - \mathbb{E}_{\mu}(Z))^2) = \mathbb{E}_{\mu}(Z^2) - \mathbb{E}_{\mu}^2(Z).
\end{equation} We have (\ref{qfq}) for $n=2$ and \begin{equation}\label{deffvar} f_{Var}(x_1, x_2) = x_2 - x^2_1 \end{equation} (note that we write $E$ and $Var$ instead of $\mathbb{E}$ and $\Var$ in the subscripts). In general, let $Q$ be some function whose domain $D_Q$ contains $\mc{T}_n$ and there exists a measurable function $f_Q:\mathbb{R}^n\rightarrow\mathbb{R}$ such that (\ref{qfq}) holds.
This is the case e. g. for $Q$ equal to the $k$th moment or central moment, $k \leq n$, or arbitrary product or linear combination of such moments restricted to $\mc{T}_n$. Similarly as above for moments, for $\alpha = (\mu, Z) \in D_Q$, $Q(\alpha)$ is also denoted as $Q_{\mu}(Z)$ or simply $Q(Z)$ if $\mu = \mathbb{P}$.
Note that the function $f_Q$ for $Q$ as above is unique for $n=1$ while for $n\geq 2$ it is not
since from $\sqrt{\mathbb{E}(X^2)}\geq\mathbb{E}(|X|) \geq \mathbb{E}(X)$ for $X \in L^2$, (see Theorem \ref{leqpq}) the value of $f_Q$ can be changed on some $x \in \mathbb{R}^n$ such that $x_2< x_1^2$ with (\ref{qfq}) still being true.
We denote $f_Q$ one of the possible choices of the required function for $Q$, taking for $Q=\Var$, $f_{Var}$ as in (\ref{deffvar}).
If $Z \in L^n$ and $X$ is a random variable, then we define the function of the first $n$ conditional moments of $Z$ given $X$ and corresponding to $Q$ as
\begin{equation}\label{qxz}
Q(Z|X) = f_Q((\mathbb{E}(Z^i|X))_{i=1}^{n}).
\end{equation}
In particular the function of conditional moments of $Z$ given $X$ and corresponding to $\Var$ is equal to \begin{equation}\label{defCondVar}
\begin{split}
\Var(Z|X) &= \mathbb{E} (Z^2|X) - \mathbb{E}^2(Z|X)\\
&= \mathbb{E}((Z - \mathbb{E}(Z|X))^2|X), \end{split} \end{equation}
and we call it conditional variance of $Z$ given $X$.
If $\mu_{Z|X}$ is conditional distribution of $Z$ given $X$ and $\phi(Z) \in L^1$ for some measurable function $\phi$, then from (\ref{emu}) and (\ref{condCond}) it follows that \begin{equation}
\mathbb{E}(\phi(Z)|X) = \mathbb{E}_{\mu_{Z|X}(X, \cdot)}(\phi). \end{equation} Thus if $\alpha_i(Z) \in L^1$ for some measurable functions $\alpha_i$, $i \in I_n$, then for any function $\beta:\mathbb{R}^n \rightarrow \mathbb{R}$ we have a. s. \begin{equation}\label{fezx}
\beta((\mathbb{E}(\alpha_i(Z)|X))_{i=1}^{n}) = \beta((\mathbb{E}_{\mu_{Z|X}(X,\cdot)}(\alpha_i))_{i=1}^n). \end{equation} In particular for $\phi(Z)\in L^n$, $\beta = f_Q$, and $\alpha_i = \phi^i$, $i \in I_n$, we receive from (\ref{qxz}) and (\ref{fezx}) that \begin{equation}\label{condFun}
Q(\phi(Z)|X)= Q_{\mu_{Z|X}(X,\cdot)}(\phi). \end{equation} The formula (\ref{varmuzx}) from the end of Section \ref{secOrthog}
is obtained for $\alpha_i = f_i$ for $i \in I_n$, $\alpha_{n+1} = |f|^2$,
and $\beta((x_i)_{i=1}^{n+1}) = x_{n+1}^2 - |(x_i)_{i=1}^n|^2$, using (\ref{fezx}) and the last terms in (\ref{varGen}) and (\ref{condGen}).
When $f(P,R) \in L^n$ for some $P$ and $R$ independent and $f$ measurable (from appropriate product measurable space to $\mathbb{R}$), like for stochastic outputs corresponding to certain constructions of MR in Section \ref{genParSec},
we have an intuitive formula
\begin{equation} \label{CQKP}
\begin{split}
Q(f(P,R)|P) &= f_Q((\mathbb{E}(f^i(P,R)|P))_{i=1}^n)\\
&= f_Q(((\mathbb{E}(f^i(p,R)))_{p=P})_{i=1}^n)\\
&= (Q(f(p,R)))_{p=P},
\end{split}
\end{equation} where in the second equality we used Theorem \ref{indepCond} from Appendix \ref{appMath}, and in the third the fact that from Fubini's theorem \cite{rudin1970}, $f(p,R) \in L^n$ for $\mu_P$ a. e. $p$.
Note that from (\ref{condFun}), for $Z \in L^1$, $\phi = \id_\mathbb{R}$, and when
$\mu_{Z|X}(x,\cdot)$ exists and is uniquely determined for $\mu_X$ a. e. $x$,
which holds for a large class of random variables $Z$ and $X$ (see Appendix \ref{appMath}), or from (\ref{CQKP}) when $Z=f(P,R)$ and $X=P$ for some $f$, $P$, and $R$ as above, it follows that
for different choices of the function $f_Q$ corresponding to $Q$ which we used to define $Q(Z|X)$ in (\ref{qxz}),
the resulting $Q(Z|X)$ are a. s. equal.
Let $Z = g(M) \in L^n$ be an output of an MR $M=(P,Y)$ with parameters $P = (P_i)_{i=1}^{N_P}$ and corresponding to a reaction network $RN$. Since conditional distribution of $M$ given $P$ is specified by Definition \ref{MRdef}, from formula (\ref{condFun}) it follows that distributions of functions of certain $n$ first conditional
moments $Q(Z|P)$, like conditional variance, are determined by $RN$, $g$, and $\mu_P$.
Therefore, if $Q(Z|P) \in L^1$, then the values of its mean \begin{equation}\label{defAveQ}
AveQ = \mathbb{E}(Q(Z|P)) \end{equation}
(which for $Q=E$ is equal to $Ave=\mathbb{E}(Z)$ by iterated expectation property), and if $Q(Z|P) \in L^2$, also the values of the main sensitivity indices \begin{equation}
VQ_{P_J} = \Var(\mathbb{E}(Q(g(Y)|P)|P_J)) \end{equation} of these functions of conditional moments are determined by this data, and so are the total sensitivity indices \begin{equation} VQ_{P_J}^{tot} = VQ_{P} - VQ_{P_{\sim J}}, \end{equation} where $\sim J = I_{N_P} \setminus J$, $J \subset I_{N_P}$. The Sobol's main and total sensitivity indices, created by dividing the above indices by $VQ_P$, are denoted as $SQ_{P_J}$ and $SQ_{P_J}^{tot}$, respectively. Similarly as for the special case of $Q=\mathbb{E}$ in Section \ref{secVBSA}, for $J = \{i\}$, we usually write $i$ in place of $P_J$ in the above notations.
\section{Covariance and some properties of variance of random vectors} Covariance of random vectors $U, Z \in L^2_n$ is defined as \begin{equation}\label{covGen} \Cov(U,Z) = (U - \mathbb{E}(U),Z - \mathbb{E}(Z))_n. \end{equation} Let for some $m \in \mathbb{N}_+$, $X_i \in L^2_n,$ $i \in I_m$. We have an easy to prove formula \begin{equation}\label{sumcov} \Var(\sum_{i=1}^m X_i) = \sum_{i=1}^{m}\Var(X_i) + 2\sum_{1\leq i<j \leq n}\Cov(X_i,X_j), \end{equation} which is well-known for $n=1$. If $X_i\in L^2_n$, $i \in I_m,$ are i. i. d., then from (\ref{scalarFun}) and (\ref{covGen}), $\Cov(X_i,X_j)=0$, $i\neq j,$ so from (\ref{sumcov}) we receive \begin{equation}\label{varAveVec} \Var(\frac{1}{m}\sum_{i=1}^m X_i) = \frac{1}{m}\Var(X_1). \end{equation}
\section{Output approximations, correlations, and nonlinearity coefficients}\label{secAppr} Let us consider a set $\Phi = \{v_i\}_{i=1}^l$ of $l \in \mathbb{N}_{+}$ linearly independent elements of a Hilbert space $H$
with some scalar product $(,)$, inducing norm $||\cdot||$ and distance $d$. The linear subspace \begin{equation} V = \text{span}(\Phi) = \{\sum_{i=1}^l a_iv_i: a_i \in \mathbb{R}, i \in I_l \} \end{equation} is closed in $H$ (\cite{rudin1970} Section 4.15), and thus for each $x \in H$ there exists a unique element of $V$ minimizing the distance from $x$ - the orthogonal projection $P_V(x)$ of $x$ onto $V$ (see Appendix \ref{appHilb}). Denoting $y_i = (x,v_i)$ and $g_{ij} =(v_i,v_j)$, the coefficients $(b_i)_{i=1}^l$ such that \begin{equation} P_V(x) = \sum_{i=1}^l b_i v_i, \end{equation} can be computed from the following set of equations \cite{rudin1970} \begin{equation}\label{gijEqu} \{\sum_{j=1}^l g_{ij}b_j = y_i\}_{i=1}^l. \end{equation} In particular, if elements of $\Phi$ are orthonormal (see Appendix \ref{appHilb}), then from (\ref{gijEqu}) it holds $b_i = y_i$, $i \in I_l$. In such case $(b_i)_{i=1}^l$ are known as Fourier's coefficients \cite{rudin1970} of $x$ relative to the elements of $\Phi$, and distance between $x$ and $P_V(x)$ fulfills \begin{equation}\label{dxp}
d(x, P_V(x))^2 = ||x||^2 - \sum_{i=1}^l {b_i^2}. \end{equation}
Let us consider the special case of $H= L^2_n$ with some scalar product $(,)_n$ corresponding to a scalar product $<,>$ on $\mathbb{R}^n$ as in Section \ref{secOrthog}, and let $e_j$, $j \in I_n$, be the elements of some orthonormal base of $\mathbb{R}^n$ with respect to $<,>$, e. g. for the standard scalar product we can take the standard base of $\mathbb{R}^n$. For some $k \in \mathbb{N}_+$, let $l=n+k$ and $\{v_i \in L^2_n: i \in I_{k+n}\}$ be a nonzero orthogonal set (see Appendix \ref{appHilb}) with $v_{k+i} = e_i$, $i \in I_n$. Then \begin{equation}\label{eequzero} \mathbb{E}(v_i) = 0,\quad i \in I_k, \end{equation} since $(v_i,e_j)_n = <\mathbb{E}(v_i),e_j> = 0$, $j \in I_n$.
We normalize $\Phi$ to get an orthonormal set $\Phi'= \{v_i'\}_{i=1}^{k+n}$, $v_i' = \frac{v_i}{\sigma(v_i)}$, $i \in I_k$, $v'_{k+i} = v_{k+i},i \in I_n$.
Let $(b_i)_{i=1}^{k+n}$ denote the coefficients of $P_V(x)$ as above relative to $\Phi$, and $(c_i)_{i=1}^{k+n}$ relative to $\Phi'$. We have
\begin{equation}\label{hieqp} c_i = (x,v_i')_n,\quad i \in I_l,
\end{equation}
\begin{equation}\label{hieq} b_i = \frac{(x,v_i)_n}{\Var(v_i)}= \frac{c_i}{\sigma(v_i)},\quad i \in I_k, \end{equation} and $b_{k+i} = c_{k+i}$, $i \in I_n$.
Let $U, Z \in L^2_n$ have nonzero variances. We define their correlation as \begin{equation} \corr(U,Z) = \frac{\Cov(U,Z)}{\sigma(U)\sigma(Z)}. \end{equation} Correlation is a popular measure of strength of the linear relationship between $U$ and $Z$ for $n=1$, due to its properties which we discuss and prove below for arbitrary $n$. Using (\ref{eequzero}), we have
$(x,v_i)_n = \Cov(x,v_i), i \in I_k,$
and thus from (\ref{hieqp}), $c_i = \Cov(x,v_i')$, $i \in I_k$. Furthermore, if $\Var(x) > 0$, then \begin{equation}\label{correqu} \corr(x,v_i) =\corr(x, v_i') = \frac{c_i}{\sigma(x)} = \frac{b_i\sigma(v_i)}{\sigma(x)},\quad i \in I_k. \end{equation} From discussion in Section \ref{secOrthog}, $\mathbb{E}(x)$ is orthogonal projection of $x$ onto span of constant random vectors, so that from Lemma \ref{lemP1P2} it easily follows that \begin{equation} \sum_{i=1}^n b_{k+i}e_i = \mathbb{E}(x). \end{equation} Thus, from (\ref{dxp}) and (\ref{correqu}) we receive
\begin{equation}\label{dcorr} d(x, c_iv_i' + \mathbb{E}(x))^2 =\Var(x) - c_i^2= \Var(x)(1 - \corr(x,v_i)^2)\geq 0,\quad i \in I_k. \end{equation} In particular, for $U$ and $Z$ as above, taking $k=1$, $x=Z$, and $v_1 = U - \mathbb{E}(U)$, and using the fact that $\corr(Z,U) = \corr(x,v_1)$, we receive \begin{equation} -1\leq \corr(Z,U)\leq 1, \end{equation} with equality in either of the above inequalities implying the linear relationship \begin{equation} Z= b_1v_1 +\mathbb{E}(Z) = b_1U - b_1\mathbb{E}(U) + \mathbb{E}(Z), \end{equation} with the sign of $b_1$ being due to (\ref{correqu}) the same as of the correlation.
In Section \ref{secApprCoeff} we discuss some general methods for estimating the coefficients in the above projection and correlations for
the case of $x=f(X)$
corresponding to different functions of conditional moments of functions of two independent variables given the first variable, like conditional variances of stochastic model outputs given the model parameters, and $v_i = \phi_i(X)$, $i \in I_k$, as above, being some functions of the first variable. However, in the numerical experiments and the discussion below we consider only the coefficients of orthogonal projection of $f(X)$ onto span of constant vectors and independent coordinates of $X$, which describe the linear part of the relationship of $f(X)$ and the coordinates. Let us assume that $Z= f(X) \in L^2_n$, $X \in L^2_N$, and $\Var(X_i) > 0$, $i \in I_N$. Elements of the set $\Phi = \{(X_i-\mathbb{E}(X_i))e_j\}_{i\in I_N, j\in I_n}$ are orthogonal, and for $X'= \left(\frac{X_i-\mathbb{E}(X_i)}{\sigma(X_i)}\right)_{i=1}^N$, elements of $\Phi' = \{X'_ie_j\}_{i\in I_N, j\in I_n}$ are orthonormal with respect to $(,)_n$. Denoting by $W$ the space of constant $\mathbb{R}^n$-valued random vectors, we define $V = \text{span}(\Phi\cup W) = \text{span}(\Phi' \cup W)$. Coefficients of the respective elements of $\Phi$ in the orthogonal projection of $Z$ onto $V$ (in $L^2_n$) fulfill \begin{equation} b_{i,j} = \frac{(Z,\Phi_{i,j})_n}{\Var(X_i)} = \frac{\Cov(Z, X_ie_j)}{\Var(X_i)}, \end{equation} and for coefficients of elements of $\Phi'$ in this projection we have \begin{equation} c_{i,j} = (Z,\Phi_{i,j}')_n = \frac{\Cov(Z, X_ie_j)}{\sigma(X_j)}. \end{equation} We denote for $i \in I_N$, $c_i = \sum_{j=1}^n c_{i,j}e_j$, for $J \subset I_N$, $c_J = (c_j)_{j\in J}$, $c = c_{I_N}$, \begin{equation} c_J^2 = \sum_{i\in J, j \in I_n}c^2_{i,j}, \end{equation} and analogously for coefficients $b_{i,j}$. For $J \subset I_N$, we define the space of functions of $X$ linear in $X_J$ to be
$V_J = \overline{\text{span}}(L^2_{n,X_{\sim J}} \cup \{\Phi_{i,j}\}_{i\in J,j \in I_n})$,
so that $V_{I_N}=V$. One can easily verify that the orthogonal projection of $Z$ onto functions linear in $X_J$ is equal to \begin{equation}
P_{V_J}(Z) = \mathbb{E}(Z|X_{\sim J}) + c_{J}X'_J, \end{equation} where $c_{J}X'_J = \sum_{i \in J}c_{i}X'_i$. We define the nonlinearity coefficient of $Z$ in $X_J$ as \begin{equation}\label{DNJ} \begin{split}
DN_J &= ||Z - P_{V_J}(Z)||_n^2\\
&= ||Z||_n^2 - ||\mathbb{E}(Z|X_{\sim J})||_n^2 - c_{J}^2 \\ &= V_{X_J}^{tot}- c_{J}^2. \end{split} \end{equation}
It holds $0 \leq DN_J \leq V_{X_J}^{tot}$, equality on the left meaning that $Z$ is linear in $X_J$ and on the right that $c_J^2 = 0$, that is knowledge of the linear part of dependence of $Z$ on $X_J$ is of no help in approximating it. For $V_{X_J}^{tot}>0$ one can also consider the normalized nonlinearity coefficient \begin{equation} dN_J = \frac{DN_J}{V_{X_J}^{tot}},
\end{equation} which fulfills $0 \leq dN_J \leq 1$, and is equal to the ratio of squared errors of the best approximation of $Z$ with functions linear in $X_J$ and another one with functions of $X_{\sim J}$. The nonlinearity coefficient of $Z$ in $X$, \begin{equation}{\label{dnc}} DN = DN_{I_N} = \Var(Z^2) - c^2, \end{equation} is equal to the squared error of the best approximation of $f(X)$ in $V$, and \begin{equation} dN = \frac{DN}{\Var(f(X))} \end{equation} tells what its ratio is to the squared error of the best approximation of $f(X)$ using constant vectors. We call $dN$ the relative error of the best linear approximation of $f(X)$. We have focused on nonlinearity coefficients, because they appear directly in our estimates of probabilities of localizations of functions values changes due to perturbations of their independent arguments, discussed in the next section, but similarly one can define linearity coefficients, like such normalized coefficient \begin{equation} dL_J = 1 - dN_J = \frac{c_J^2}{V_{X_J}^{tot}}.
\end{equation} Let us define, for $J \subset I_N$, $g_J$ to be a measurable function such that \begin{equation} \begin{split} g_{J}(X) &= \sum_{K \subset I_N : K \cap J \neq \emptyset} f_K(X_K) \\
&= f(X) - \mathbb{E}(f(X)|X_{\sim J}) \end{split} \end{equation} (for $J = \{j\}$ we simply write $j$ in the subscript), where we have used ANOVA decomposition (\ref{anovaDec}). It holds \begin{equation}
||g_{J}(X)||^2_n = V_{X_J}^{tot}. \end{equation} If $V_i^{tot}\neq 0$ and $n=1$, then let us define a coefficient which we call linear correlation of $f(X)$ in $X_i$, and which could be used as a measure of strength of linearity of $f(X)$ in $X_i$, \begin{equation} \corrL_i = \corr(g_{i}(X),X_i') = \frac{c_i}{\sqrt{V_i^{tot}}}. \end{equation} We have $\corrL_i^2 = dL_J$ and $-1 \leq\corrL_i \leq 1$, with either of the equalities in the inequalities meaning that $f(X)$ is linear in $X_i$ and it holds \begin{equation}
f(X)=c_iX_i' + \mathbb{E}(f(X)|X_{\sim J}) \end{equation} with the sign of $c_i$ being the same as of $\corrL_i$.
\section{\label{secInterv}Interventions into systems with uncertain parameters} Let for some $N,n \in \mathbb{N}_+$ $X= (X_i)_{i=1}^N$ be an $\mathbb{R}^N$-valued random vector and $f$ be a measurable function from $\mathbb{R}^N$ to $\mathbb{R}^n$. The change of $f(X)$ due to a perturbation $\Delta \in \mathbb{R}^N$ of $X$ is defined as \begin{equation}\label{hX} h(X)=f(X + \Delta) - f(X), \end{equation} for any measurable function $h$ from $\mathbb{R}^N$ to $\mathbb{R}^n$ such that this equality holds. $X$ may be for instance uncertain parameters of some model and $f(X)$ can be some its output, like a vector of concentrations of some species at some moment of time for a deterministic chemical model, or vector of certain conditional moments of different particle numbers at a given time or their conditional histogram given the model parameters for a stochastic model. Perturbation $\Delta$ of the model parameters can imitate adding a given amount of some species to the chemical system, e. g. as a pharmaceutical intervention. When planning which uncertain parameters of a model to perturb to receive a desirable effect on the output it might be useful to know the probability that the change of output will belong to a given area , e. g. be positive or negative. We describe here a method for obtaining lower bounds on certain such probabilities for appropriate $f$ and $X$, using only total sensitivity indices and orthogonal projection coefficients.
Let us assume that the coordinates of $Y\in L^2_N$ are independent and have uniform or uniform discrete distributions on $\mathbb{R}$, and let $B_Y$ be the support of $\mu_Y$ (see Appendix \ref{appMath}). We assume that $f(Y) \in L^2_n$, random vector $X$ takes values in a measurable set $B_X\subset B_Y$ satisfying $\mu_Y(B_X)>0$, and for
each measurable $D\subset \mathbb{R}^n$, \begin{equation}\label{muAssum} \mu_{X}(D) = \frac{\mu_Y(D\cap B_X)}{\mu_Y(B_X)}. \end{equation} In particular if $B_X = B_Y$, then $\mu_{X} = \mu_{Y}$.
For a measurable function $s$ such that $s(X)$ is integrable, one can easily prove that \begin{equation} \frac{\mathbb{E}(\mathbb{1}_{Y\in B_X}s(Y))}{\mu_Y(B_X)} = \mathbb{E}(s(X)). \end{equation} If $s$ is further nonnegative, then we receive \begin{equation}\label{geqmuy} \frac{\mathbb{E}(s(Y))}{\mu_Y(B_X)} \geq \mathbb{E}(s(X)). \end{equation}
For a perturbation $\Delta = (\Delta_i)_{i=1}^N \neq 0$, let $J = \{i \in I_N: \Delta_{i} \neq 0\}$ and $\Delta_J = (\Delta_i)_{i\in J}$. We denote $A = \{X + \Delta \in B_Y\}$, which is the event that the perturbed arguments are in $B_Y$. In particular, if $\Delta + B_X = \{x +\Delta:x\in B_X\} \subset B_Y$, then $\mathbb{P}(A) = 1$. We further use notations introduced in the previous section, like coefficients $b_{i,j}$ and $c_{i,j}$, variables $Y'$, sequence $\Phi'$, product $Y_Jb_J$, nonlinearity coefficient $DN_J$, $g_J(Y)$ etc. defined identically but with $X$ replaced by $Y$ in the definitions. Let \begin{equation}
\delta(x) = h(x) - b\Delta. \end{equation} We have \begin{equation} \mathbb{1}_A \delta(X) = \mathbb{1}_A(g_{J}(X + \Delta) - g_J(X) - b\Delta). \end{equation} For a function \begin{equation}\label{axdef} a(x) = g_J(x) - b_J(x_J - \mathbb{E}(Y_J)), \end{equation} it holds \begin{equation}\label{diffA} \mathbb{1}_A\delta(X) = \mathbb{1}_A(a(X + \Delta) - a(X)). \end{equation} Using Lemma \ref{lemP1P2} it is easy to prove that $b_J(Y_J-\mathbb{E}(Y_J)) = c_JY'_J$ is an orthogonal projection of $g_J(Y)$ onto
span($\{Y'_i e_j\}_{i \in J, j\in I_n}$) and $||c_JY'_J||^2_n = c_J^2$, so that from (\ref{dxp}) we have \begin{equation}\label{aY2}
||a(Y)||_n^2 = V_{Y_J}^{tot} - c^2_{J} = DN_J. \end{equation} We have the following easy generalization of Chebyshev's inequality \cite{billingsley1979}. \begin{lemma}\label{Chebyshev} For $Z \in L^2_n$, $\epsilon \in \mathbb{R}_{+}$, and each event $B$ it holds \begin{equation}
||\mathbb{1}_BZ||_n^2 = \mathbb{E} (\mathbb{1}_B|Z|^2) \geq \mathbb{P}(B,|Z| \geq \epsilon)\epsilon^2. \end{equation} \end{lemma} Using it we obtain \begin{equation}\label{prAdelta}
\mathbb{P}(A,|\delta(X)| \geq \epsilon) \leq \frac{||\mathbb{1}_A\delta(X)||^2_n}{\epsilon^2}. \end{equation} Applying triangle inequality \cite{rudin1970} to (\ref{diffA}) we receive \begin{equation}\label{deltaXm}
||\mathbb{1}_A\delta(X)||_n^2 \leq (||\mathbb{1}_A a(X)||_n + ||\mathbb{1}_A a(X + \Delta)||_n)^2. \end{equation} We estimate \begin{equation}\label{Aax} \begin{split}
||\mathbb{1}_A a(X)||^2_n &\leq \mathbb{E}(|a(X)|^2) \leq \frac{\mathbb{E}(|a(Y)|^2)}{\mu_Y(B_X)} \\
& = \frac{||a(Y)||^2_{n}}{\mu_{Y}(B_X)}, \end{split} \end{equation} where in the second inequality we used (\ref{geqmuy}). Furthermore, \begin{equation}\label{xdbaxd} \begin{split}
||\mathbb{1}_{X + \Delta \in B_Y} a(X + \Delta)||^2_n &\leq \frac{\mathbb{E}(\mathbb{1}_{Y + \Delta \in B_Y}|a(Y + \Delta)|^2)}{\mu_Y(B_X)}\\
& \leq \frac{||a(Y)||^2_{n}}{\mu_{Y}(B_X)}, \end{split} \end{equation} where in the first inequality we used (\ref{geqmuy}) and in the last one the assumption of independence and uniform distributions of coordinates of $Y$. From (\ref{deltaXm}), (\ref{Aax}), and (\ref{xdbaxd}), we receive \begin{equation}\label{deltaXmFin}
||\mathbb{1}_A\delta(X)||_n^2 \leq \frac{4||a(Y)||^2_n}{\mu_Y(B_X)} = \frac{4DN_J}{\mu_Y(B_X)}. \end{equation} For $p,\ r \in \mathbb{R}^n$, we define a ball with center $p$ and radius $r$ as \begin{equation} B(p,r) = \{x\in\mathbb{R}^N:\wt{d}(x,p)<r \}. \end{equation} We have the following lower bound on the probability that the effect of perturbation lies in a ball with center $b_J\Delta_J$ and radius $\epsilon >0$ \begin{equation}\label{longhB} \begin{split}
\mathbb{P}(h(X) \in B(b_J\Delta_J,\epsilon)) &= \mathbb{P}(|\delta(X)| < \epsilon) \leq \mathbb{P}(A,|\delta(X)| < \epsilon) \\
&= \mathbb{P}(A) - \mathbb{P}(A, |\delta(X)| \geq \epsilon) \\ & \geq \mathbb{P}(A) - \frac{4DN_{J}}{\mu_Y(B_X)\epsilon^2}, \end{split} \end{equation} where in the last equality we used (\ref{prAdelta}) and (\ref{deltaXmFin}). In particular if $P(A)=1$ and $f(Y)$ is linear in $Y_J$, so that $DN_J=0$, then we receive $h(X) = b_J\Delta_J$,
which also follows from the fact that in such case $f(Y) = b_JY_J + \mathbb{E}(f(Y)|Y_{\sim J})$.
If $n = 1$ and $b\Delta$ is positive (negative), then the probability that the effect of perturbation $\Delta$ on the output is positive (negative) is bounded from below by \begin{equation}\label{probChange} \mathbb{P}(h(X) \in B(b\Delta,b\Delta)) \geq \mathbb{P}(A) - \frac{4DN_J}{ \mu_Y(B_X)(b_J\Delta_J)^2}. \end{equation} We apply the above theory to the GTS model at the end of Section \ref{secGTS}.
\section{\label{secStat}Statistics, Monte Carlo procedures, and inefficiency constants - some new definitions, generalizations and interpretations}
If $\mathcal{P}$ consists of all probability distributions on $\mathbb{R}$ with finite $n$th moments for some $n \in \mathbb{N}_+$, then for $Q$ whose restriction to $\mc{T}_n$ is a function of the first $n$ so restricted moments as in Section \ref{secCondMoms}, we define estimand $G_Q$ on $\mc{P}$ to be such that for each $\mu \in \mc{P}$, \begin{equation}\label{defgqmu} G_Q(\mu) = Q_\mu(\id_\mathbb{R}), \end{equation} or equivalently $G_Q(\mu)=Q(X)$, $X \sim \mu$. In particular, for $Q= \mathbb{E}$ and $\Var$ we receive estimands $G_E$ and $G_{Var}$ from Appendix \ref{appStatMC}. Degree of an estimand $G$ is defined as the smallest $n \in \mathbb{N}_+$ for which there exists an unbiased estimator of $G$ in $n$ dimensions (see Appendix \ref{appStatMC}), assuming that for some $n$ such estimator exists \cite{lehmann1998theory, Halmos_1946}. In other words, it is the minimum value of $n$ for which there exists a measurable real-valued $\phi$ on $\mc{S}^n$ such that
for each $\mu \in \mc{P}$ and $X_1, \ldots, X_n$ i. i. d., $X_1 \sim \mu$, it holds \begin{equation} G(\mu) = \mathbb{E}(\phi(X_1, \ldots, X_n)). \end{equation}
It was proved in \cite{Halmos_1946} that for admissible distributions $\mathcal{P}$ on $\mathbb{R}$ containing all finite discrete distributions on $\{0,1\}$ (see Appendix \ref{appStatMC}) and possibly some other distributions with finite $n$th moments for some $n \in \mathbb{N}_+$, for $Q$ being the $n$th moment or central moment, $G_Q$ restricted to $\mathcal{P}$ has degree exactly $n$.
Let $n \in \mathbb{N}_+$, and $G_i$ be an estimand for $\mc{P}$, $i \in I_n.$ Then we call $G=(G_i)_{i=1}^n$ an $n$-dimensional or if $n$ is left unspecified simply vector-valued estimand. If $\phi_i$ is an [unbiased] estimator of $G_i$, $i \in I_n,$ then we call $\phi=(\phi_i)_{i=1}^n$ an [unbiased] estimator of $G$, where the words in square brackets in a sentence should be either all read or omitted. Error of approximation of $G$ by its unbiased estimator $\phi$ for some $\mu \in \mc{P}$ can be quantified by $\Var_\mu(\phi)$ for some variance for random vectors as in Section \ref{secOrthog}. The suitable scalar product in the definition of such a variance can depend on the estimation problem at hand. In Section \ref{secApprErr} we shall discuss a problem for which the standard scalar product is a natural choice.
Let us assume that similarly as for $m=1$ in Appendix \ref{appStatMC}, to estimate some $\lambda_1,\ldots,\lambda_m \in \mathbb{R}$ for some $m \in \mathbb{N}_+$ we carry out $n$-step MC procedures using the same variable $X\sim \mu$ and single-step MC estimators $\phi_i$ of $\lambda_i$ for $\mu$, $i \in I_m$. Then we say that these quantities are estimated in the same MC procedure. For $i \in I_m$, for the subprocedure estimating $\lambda_i$ we use notations analogous as in Appendix \ref{appStatMC} but with a subscript $i$, like $W_{i,j}$ for the $j$th observable of the $i$th single step estimator as well as $\overline{W}_i$ for the observable and $\wt{\lambda}_i$ for its observed value,
$\Var_{f,i}$ for variance and $\sigma_{f,i}$ for the standard deviation of the $i$th final estimator $\phi_{f,i}$.
Then $\phi=(\phi_i)_{i=1}^m$ is called a single step MC estimator of $\lambda=(\lambda_i)_{i=1}^m$ for $\mu$, and $\phi_f = (\phi_{f,i})_{i=1}^m$ the final or $n$-step one.
We define the variance $\Var_s$ of a single step MC estimator and such variance $\Var_f$ of the final MC estimator using the same formulas as for $m=1$ in Appendix \ref{appStatMC} but with $\Var$ symbol denoting some variance for random variables as in Section \ref{secOrthog}. Note that from (\ref{varAveVec}) we still have \begin{equation}\label{varfsn} \Var_{f} = \frac{\Var_s}{n}. \end{equation} We can define inefficiency constants for sequences of MC procedures for estimating $\lambda \in \mathbb{R}^m$ identically as in Section \ref{secMCIneff} for $m=1$ and thanks to (\ref{varfsn}) they enjoy the same interpretation as in this section - if we have $\delta$-approximate equality of average duration times of two MC procedures then the ratio of the final MC variances is $\delta$-approximately equal to the ratio of their inefficiency constants. Let us notice two further interpretations of the inefficiency constants using notations as in Section \ref{secMCIneff}. The first is that if we have $\delta$-approximate equality of variances of the final estimators of the MC procedures \begin{equation}\label{vdeltav} \Var_{f}(n) \approx_{\delta} \Var_{f}'(n'), \end{equation} then the ratio of their average durations is $\delta$-approximately the same as of the inefficiency constants \begin{equation}\label{varTiRatio} \frac{\tau_{f}(n)}{\tau_{f}'(n')} \approx_{\delta} \frac{c}{c'}. \end{equation} Secondly, consider the approach to estimating $\lambda$ using a sequence of MC procedures in which for some target accuracy threshold $\epsilon>0$, one carries out the MC procedure with the smallest number $n(\epsilon)$ of MC steps for which variance of the final MC estimator $\Var_f(n(\epsilon))$ is below $\epsilon$. In practice one usually does not know $\Var_f(n(\epsilon))$, but can approximate it using values of estimator (\ref{varEst}). For $x \in \mathbb{R}$, let $\lceil x\rceil$ denote the smallest integer $l$ such that $x \leq l$. It holds $n(\epsilon) = \left\lceil \frac{\Var_s}{\epsilon} \right\rceil$ and $\Var_f = \frac{\Var_s}{n({\epsilon})}$, and similarly for the primed sequence. We have \begin{equation} \frac{\Var_{f}(n(\epsilon))}{\Var_{f}'(n'(\epsilon))} = \frac{\Var_s\left\lceil\frac{\Var_s'}{\epsilon}\right\rceil}{{\Var_s'}\left\lceil \frac{\Var_s}{\epsilon} \right\rceil}, \end{equation} which tends to one as $\epsilon$ goes to zero, and thus from (\ref{cntau}) the ratio $\frac{\tau_f(n(\epsilon))}{\tau_f'(n'(\epsilon))}$ of average durations of these procedures tends to $\frac{c}{c'}$.
\section{Testing methodology}
We shall use what we call $k$-$\sigma$ test for each of the null hypotheses that for some $b,\lambda \in \mathbb{R}$, $\lambda = b$, $\lambda \geq b$, or $\lambda \leq b$, in which for $\wt{\lambda}$ denoting observed value of the final MC estimator and $\wt{\sigma}_f$ estimate of its standard deviation as in Appendix \ref{appStatMC}, one
rejects the hypothesis if $|\wt{\lambda} -b| > k\wt{\sigma}_f$, $b-\wt{\lambda} > k\wt{\sigma}_f$, or $\wt{\lambda} -b > k\wt{\sigma}_f$, respectively. For sufficiently large $n$
the significance level (upper bound on the probability of rejecting wrongly the hypothesis if it is correct) for such $k$-$\sigma$ test can be chosen arbitrarily close to $2(1 - \Phi(k))$ for the equality and $1 - \Phi(k)$ for the inequalities hypotheses for $\Phi(k)$ being the cumulative distribution function of the standard normal distribution (see Appendix \ref{appStatMC}). Such significance levels are called asymptotic. Let the coordinates of $\lambda \in \mathbb{R}^2$ be estimated in the same $n$-step MC procedure and let $\overline{W}_d = \overline{W}_1 - \overline{W}_2$ and $\sigma_d = \sigma(\overline{W}_d)$. From the inequality $\sigma(X + Y) \leq \sigma(X) + \sigma(Y)$ for $X,Y \in L^2$, which follows from triangle inequality \cite{rudin1970}, we have $\sigma_d \leq \sigma_{f,1} + \sigma_{f,2}$. Furthermore, from CLT applied to the sequence $W_{1,j} - W_{2,j}, j \in I_n,$ for $n$
going to infinity $\sqrt{n}\overline{W}_d$ converges in distribution to $\ND(\lambda_1 - \lambda_2, \Var(W_{1,1} - W_{2,1}))$. Thus if for the estimand $\lambda_i$ we obtained a final MC estimate $\wt{\lambda}_i \pm \wt{\sigma}_{f,i}$, $i \in I_2$, one can use $k$-$\sigma$ test rejecting the hypothesis $\lambda_1 = \lambda_2$
if $|\wt{\lambda}_1 - \wt{\lambda}_2| > k(\wt{\sigma}_{f,1} + \wt{\sigma}_{f,2})$ or the hypothesis $\lambda_1 \geq \lambda_2$ if $\wt{\lambda}_2 - \wt{\lambda}_1 > k(\wt{\sigma}_{f,1} + \wt{\sigma}_{f,2})$, with the same asymptotic significance levels as for the equalities and inequalities hypotheses discussed above. For two independently run $n_i$-step MC procedures estimating $\lambda_i$ and with observables of the final MC estimators $\overline{W}_i$ with variances $\Var_{f,i}, i \in I_2$, from the Lindeberg CLT \cite{billingsley1979}, \begin{equation} \frac{(\overline{W}_1 - \lambda_1) + (\overline{W}_2 -\lambda_2)}{\sqrt{\Var_{f,1} + \Var_{f,2}}} \end{equation} converges in distribution to $\ND(0,1)$ for $n_1$ and $n_2$ going to infinity. Thus using analogous notations as above
one can use $\sqrt{\wt{\sigma}_{f,1}^2 + \wt{\sigma}_{f,2}^2}$ instead of $\wt{\sigma}_{f,1} + \wt{\sigma}_{f,2}$ in the above tests with the same asymptotic significance levels for $n_1$ and $n_2$ going to infinity as above for the same $k$. We often make statements about the results of our numerical experiments like that the estimate $\wt{\lambda}_1 \pm \wt{\sigma}_{f,1}$ is (statistically significantly) greater than $\wt{\lambda}_2 \pm \wt{\sigma}_{f,2}$ by which we mean that the null hypothesis $\lambda_1 \leq \lambda_2$ can be rejected in a $k$-$\sigma$ test as above for some $k \geq 3$. \newline
\section{\label{secUnbiased}Generalization of estimands on pairs and their estimation schemes to many functions case} In this section we among others generalize the concepts from Section \ref{secSchemesPrev}, like of admissible pairs, estimands, statistics, estimators, and estimation schemes, so that they can be used for problems of estimation of certain quantities defined for several functions of different sequences of random arguments.
These concepts shall be used in their full generality in Section \ref{secApprCoeff} e. g. when dealing with orthogonal projection coefficients onto orthogonal functions of the first variable of functions of conditional moments given the first variable, like conditional variance, of functions of two independent random variables.
Unfortunately, giving only the number of distributions as before is not sufficient to specify the type of the more general admissible pairs we need so we introduce a helper definition of signature containing such specification. \begin{defin}\label{defSign} We call $Sg=(N,k,J,\mc{H})$ a signature (of some admissible pairs) if $N, k \in \mathbb{N}_{+}$, sequence $J = (J_i)_{i=1}^k$ consists of nonempty subsets of $I_N$ such that \begin{equation}\label{inkji} I_N = \bigcup_{i=1}^kJ_i, \end{equation} and coordinates of $\mc{H} = (\mc{H}_i)_{i=1}^k$ are measurable spaces $\mc{H}_i = (C_i, \mc{C}_i)$, $i \in I_k$. \end{defin}
\begin{defin}\label{defAPairs} We call $\mc{V}$ admissible pairs with signature $Sg$ as in Definition \ref{defSign} or equivalently admissible pairs of $N$ distributions and $k$ functions with values spaces $\mc{H}$ and sets of arguments' indices $J$ as in this definition if it is a nonempty class consisting of pairs $(\mu, f) = ((\mu_i)_{i=1}^N,(f_i)_{i=1}^k)$ such that $\mu_i$ is a probability measure, $i \in I_N$, and $f_i$ is a measurable function from $\bigotimes_{i \in J_i}\mc{S}_{\mu_i}$ to $\mc{H}_i$, $i \in I_k$.
\end{defin}
We identify each one-element sequence $(x)$ with $x$ (see Appendix \ref{appMath}), so that for $N=1$ the first coordinate in each pair from $\mc{V}$ in the above definition is a measure, while for $k=1$, its second coordinate is a function and from (\ref{inkji}) we have $J = I_N$. Thus, for $k=1$ and $\mc{H}= \mc{S}(\mathbb{R})$, the above definition reduces to definition of admissible pairs with $N$ distributions from Section \ref{secSchemesPrev}. Note that the class $\mc{T}_n$ (see (\ref{defMCRn})) is an example of admissible pairs of single distributions and single real-valued functions. Similarly as in Section \ref{secSchemesPrev}, an estimand on admissible pairs $\mc{V}$ is any real-valued function on it. For instance for $Q$ such that restricted to $\mc{T}_n$ it is a real-valued function of the first $n$ so restricted moments
as in Section \ref{secCondMoms}, e. g. for the $n$th moment or central moment, $Q_{|\mc{T}_n}$ is an estimand on $\mc{T}_n$. We define estimand $PR$ on the admissible pairs $\mc{V}$ of single distributions and two real-valued functions consisting of all possible $\alpha = (\mu,(f_1,f_2))$ such that $f_1f_2 \in L^1(\mu)$, in which case $PR(\alpha) = \mathbb{E}_{\mu}(f_1f_2)$.
We now describe and illustrate by example a method for obtaining vectors of estimands, which will be frequently used in Section \ref{secApprCoeff}. Let us consider a signature $Sg$ as in Definition \ref{defSign}, signatures $Sg' =(Sg'_i)_{i=1}^n$, such that $Sg_i' = (N,k_i,(\mc{H}_{i,j})_{j=1}^{k_i}, (J_{i,j})_{j=1}^{k_i}), i \in I_n,$ and $\psi=(\psi_i)_{i=1}^n$, where $\psi_i:I_{k_i} \rightarrow \mathbb{N}_+$ are $1$-$1$ functions, $i \in I_n$. We say that such $Sg$ is received from $Sg'$ using $\psi$ if $J_{\psi_i(j)}=J_{i,j}$, $\mc{H}_{\psi_i(j)}=H_{i,j}$, $j \in I_{k_i}$, $i \in I_{n}$, and $\bigcup_{i=1}^n\psi_i[I_{k_i}]=I_k$. Let $G'=(G'_i)_{i=1}^n$ be such that $G'_i$ is an estimand on admissible pairs $\mc{V}'_i$ with signature $Sg'_i$, $i \in I_n,$ and $Sg$ be received from $Sg'$ using some $\psi$ as above. We say that $G=(G_i)_{i=1}^n$ are trivial extensions of $G'$ using $\psi$ if for each $i \in I_n$, $G_i$ is an estimand on pairs $\mc{V}_i$ with signature $Sg$ and consisting of all possible $(\mu,f) = ((\mu_j)_{j=1}^N,(f_i)_{i=1}^k)$ such that for some $\beta =(\mu, (g_1, \ldots, g_{k_i})) \in \mc{V}_i'$, it holds $f_{\psi(j)} = g_j, j \in I_{k_i}$, in which case $G_i(\alpha) = G_i'(\beta)$. As an example of the above construction we define estimands $PR^n = (PR_i)_{i=1}^n$ (identifying $PR^1$ with $PR$) to be trivial extensions of $(PR)_{i=1}^n$ using $\psi$ such that $\psi_i(1)=i$ and $\psi_i(2)=n+1$, $i \in I_n$. The resulting $PR^n$ are estimands on common admissible pairs (defined as at the beginning of
Section \ref{secIneffSchemes}) consisting of $\alpha= (\mu,(f_i)_{i=1}^{n+1})$ such that $(\mu,(f_i,f_{n+1}))\in D_{PR}$, $i \in I_n$, for which $PR^n_i(\alpha) = \mathbb{E}_\mu(f_if_{n+1})$, $i \in I_n$.
Note that if $f_{n+1}\in L^2(\mu)$ and $\Phi = \{f_i\}_{i=1}^n$ is an orthonormal set in $L^2(\mu)$, then for $\alpha$ as above, $PR^n_i(\alpha)$ is the coefficient of $f_i$ in the orthogonal projection of $f_{n+1}$ onto span$(\Phi)$, $i \in I_n$.
For $N \in \mathbb{N}_+$, let us consider a nonempty finite set $K \subset I_N \times \mathbb{N}_+$,
called arguments' indices for $N$. For a sequence of measurable spaces $\mc{S} = (\mc{S}_i)_{i=1}^N$, we define
$\mc{S}^K = \bigotimes_{(i,j)\in K} \mc{S}_i$,
of sets $B = (B_i)_{i=1}^N$,
$B^K = \prod_{(i,j)\in K} B_i$,
and of probability distributions $\mu = (\mu_i)_{i=1}^N$,
$\mu^K = \bigotimes_{(i,j)\in K}\mu_i$.
Note that $\wt{X} \sim \mu^K$ means that $\wt{X} = (\wt{X}_{i,j})_{(i,j)\in K}$, random variables $\wt{X}_{i,j} \sim \mu_i,$ $(i,j) \in K$ being independent. Let $v \in \mathbb{N}_+^N$. We define $K_v = \{(i,j): i \in I_N, j \in I_{v_i}\}$. We identify sequences $((x_{i,j})_{j=1}^{v_i})_{i=1}^N$ and $(x_{\beta})_{\beta \in K_v}$. In particular for $K=K_v$, $\wt{X}$ as above is identified with $((\wt{X}_{i,j})_{j=1}^{v_i})_{i=1}^N$, while for $B$, $\mc{S}$, and $\mu$ as above,
$B^{K}$ is identified with $B^v$, $\mc{S}^{K}$ with $\mc{S}^v$ , and $\mu^{K_v}$ with $\mu^v$, defined in Section \ref{secSchemesPrev}.
Let $\mc{V}$ be some admissible pairs as in Definition \ref{defAPairs} and $K$ be arguments' indices for $N$. Sets $\mc{V}_1$ and $\mc{V}_2$ are defined analogously as in Section \ref{secSchemesPrev}.
For a measurable space $\mc{H}$, a $\mc{H}$-valued
statistic $\phi$ for $\mc{V}$ with (arguments) indices $K$ is a function on $\mc{V}_2$ such that for each $(\mu, f) \in \mc{V}$, $\phi(f)$ is a measurable function from $\mc{S}_{\mu}^K$ to $\mc{H}$. For $k=1$, $K=K_v$ for some $v$, and $\mc{H}=\mc{S}(\mathbb{R})$ this coincides with the definition of statistic for $\mc{V}$ with dimensions of arguments $v$
from Section \ref{secSchemesPrev}. Analogously as in Section \ref{secSchemesPrev}, for a real-valued statistic $\phi$ for $\mc{V}$ with indices $K$, and some $Q$ as in Section \ref{secCondMoms} like variance $\Var$ or expectation $\mathbb{E}$, we denote for $\alpha=(\mu,f) \in \mc{V}$, \begin{equation}\label{Qmuf} Q_{\alpha}(\phi) = Q_{\mu^{K}}(\phi(f)), \end{equation} whenever the expression on the right makes sense. If $\phi$ is an $\mathbb{R}^n$-valued statistic for $\mc{V}$ with indices $K$ then we shall also use notation (\ref{Qmuf}) for $Q=\mathbb{E}$ when $\phi(f) \in L^1_n(\mu^{K})$ or for $Q=\Var$ for some variance for random vectors as in Section \ref{secOrthog} and $\phi(f) \in L^2_n(\mu^{K})$. Let $G$ be an estimand on $\mc{V}$.
We call any real-valued statistic $\phi$ for $\mc{V}$ with indices $K$ estimator of $G$ if for each $\alpha = (\mu,f)\in \mc{V}$, we consider values of $\phi(f)(X)$ for each $X \sim \mu^K$ to be certain approximations of $G(\mu,f)$, and analogously as in Section \ref{secSchemesPrev} such $\phi$ is further called unbiased if \begin{equation}\label{genUnb} \mathbb{E}_{\alpha}(\phi) = G(\alpha), \quad \alpha \in \mc{V}. \end{equation} We shall now introduce a number of notations needed to define estimation schemes for the above estimands. Let us consider some signature $Sg$ as in Definition \ref{defAPairs}. A sequence of finite sets $A =(A_i)_{i=1}^k$ such that $A_i \subset \mathbb{N}_{+}^{J_i}$, $i \in I_k$, and at least one of these sets is nonempty is called sets of evaluation vectors. For $k =1$this reduces to evaluation vectors for $N$ from Section \ref{secSchemesPrev}.
We define the arguments' indices of $A$ as \begin{equation}\label{pal} \begin{split} p_A &= \{(i,j) \in I_N\times\mathbb{N}_+: \text{ for some } l \in I_k \text{ such that } i \in J_l,\\
&\text{ there exists } v \in A_l \text{ such that } v_i = j \}. \end{split} \end{equation}
Let $\mc{V}$ be admissible pairs with signature $Sg$. For each $i \in I_k$ and $v \in A_i$, we define evaluation operator $g_{\mc{V},A,i,v}$ to be a $\mc{H}_i$-valued statistic for $\mc{V}$ with indices $p_A$ such that for each $(\mu,f) \in \mc{V}$ and $x \in B_\mu^{p_{A}}$, it holds
\begin{equation} g_{\mc{V},A,i,v}(f)(x) = f_i(x_v), \end{equation} where \begin{equation}\label{xveq} x_v = (x_{l,v_l})_{l\in J_i}. \end{equation} For $k=1$ we omit subscript $i$ in the above or below alternative notations for evaluation operators, so that if further $p_A = K_w$ for some $w \in \mathbb{N}_+^N$ and $\mc{H}=\mc{S}(\mathbb{R})$, the new $g_{\mc{V},A,v}$ coincides with the definition of evaluation operator from Section \ref{secSchemesPrev}. Similarly as in Section \ref{secSchemesPrev}, $\mc{V}$ and $A$ in the subscripts are omitted when known from the context.
If for some $l \in \mathbb{N}_+, l \leq N$, it holds $J_i = I_l$, then we use a C-array-like notation \begin{equation}\label{clikeg} g_i[v_1-1]\ldots[v_l-1] = g_{i,v}, \end{equation} while for $J_i = \{i\}$ we use notation \begin{equation}\label{ridef} r_i[v_i-1] = g_{i,v}. \end{equation}
For each $i \in I_k$ for which $A_i$ is nonempty, we define the following $\mc{H}_i^{|A_i|}$-valued statistic for $\mc{V}$ with indices $p_A$,
$g_{\mc{V},A,i} = (g_{\mc{V},A,i,v})_{|v \in A_i}$ (see \ref{ordnot}).
Let
$\delta(A) = |\{i\in I_k: A_i \neq \emptyset\}|$, that is the number of nonempty coordinates of $A$,
and for each $i \in I_{\delta(A)}$, let
$\gamma_A(i)$ be the index of the $i$th nonempty coordinate of $A$.
Let \begin{equation}
\mc{H}_A = (C_A, \mc{C}_A) =\bigotimes_{i=1}^{\delta(A)}\mc{H}_{\gamma_A(i)}^{|A_{\gamma_A(i)}|}. \end{equation}
We define the following $\mc{H}_A$-valued statistic for $\mc{V}$ with indices $p_A$, $g_{\mc{V},A}=(g_{\mc{V},A,\gamma_A(i)})_{i=1}^{\delta(A)}$.
For a signature $Sg$, let $A$ be sets of evaluation vectors for $Sg$ and $t$ be a measurable real-valued function on $\mc{H}_A$. Let $\kappa=(t, A)$, which we call a scheme for $Sg$. This coincides with the previous definition of a scheme for $k=1$ and $\mc{H}= \mc{S}(\mathbb{R})$.
We define arguments' indices of $\kappa$ as $p_\kappa = p_A$.
The statistic $\phi_{\kappa,\mc{V}}$ given by $\kappa$ and $\mc{V}$ is defined using the same formula (\ref{phiAF}) as in Section \ref{secSchemesPrev}. Let $G$ be an estimand on $\mc{V}$. Similarly as in Section \ref{secSchemesPrev} $\kappa$ is called an [unbiased] (estimation) scheme for $G$ if $\phi_{\kappa,\mc{V}}$ is an [unbiased] estimator of $G$.
Let now for some $n \in \mathbb{N}_+$, $\kappa = (\kappa_i)_{i=1}^n=(t_i,A_i)_{i=1}^n$ be a sequence of schemes for $Sg$, called an ($n$-dimensional) scheme for $Sg$. We define the vector of sets of evaluation vectors of $\kappa$ as \begin{equation}\label{Akappa} A_\kappa = (\bigcup_{i=1}^n A_{i,j})_{j=1}^k. \end{equation} Let $N \in \mathbb{N}_+$, $K$ be arguments' indices for $N$, $L \subset K$, $L\neq \emptyset$, and $B=(B_1,\ldots,B_N)$ be a sequence of nonepmty sets. For $x \in B^K$, we define \begin{equation}\label{xL} x_L = (x_\beta)_{\beta \in L}, \end{equation}
while for $x \in B^K$ and $L=\emptyset$, we define $x_L=\emptyset$. We also define arguments' indices $p_\kappa$ of $\kappa$ to be equal to $p_{A_\kappa}$ defined as in (\ref{pal}). A statistic given by $\kappa$ and $\mc{V}$, denoted as $\phi_{\kappa,\mc{V}}$,
is defined as an $\mathbb{R}^n$-valued statistic for $\mc{V}$ with indices $p_{A_\kappa}$ such that for each $(\mu,f) \in \mc{V}$ and $x \in B_\mu^{p_{A_\kappa}}$ \begin{equation} \label{phikappamany} \phi_{\kappa,\mc{V}}(f)(x) = (\phi_{\kappa_i,\mc{V}}(f)(x_{p_{\kappa_i}}))_{i=1}^n, \end{equation} which for $n=1$ coincides with the previous definition. Let $G = (G_i)_{i=1}^n$ be a sequence of estimands, each on some (possibly different) admissible pairs but all with the same signature $Sg$. Let us assume that $\kappa_i$ is an [unbiased] estimation scheme for $G_i$, $i \in I_n$, in which case we call the above $\kappa$ an [unbiased] estimation scheme for $G$.
Similarly as in Section \ref{secSchemesPrev} we denote $\wh{G}_{\kappa,i}=\phi_{\kappa_i,D_{G_i}}$, $i \in I_n$, and use for it analogous shorthand notations in that section in analogous situations.
Let us now move on to examples. For an estimand $Q_{|\mc{T}_n}$ as above, if there exists an estimator $\phi_{Q}$ of $G_Q$ in $m$ dimensions (see Section \ref{secStat}),
then an unbiased estimation scheme $SQR=(t,A)$ for $Q_{|\mc{T}_n}$ is given by $t = \phi_Q$ and $A = I_m$. Using notation (\ref{clikeg}), the estimator of this scheme can be written as \begin{equation}\label{gqsq} \wh{G}_{Q,SQR}= \phi_Q((g[i])_{i=0}^{m-1}). \end{equation} The fact that this estimator is unbiased follows from the fact that for each $(\mu,f) \in \mc{T}_n$,
\begin{equation} \begin{split} \mathbb{E}_{\mu^m}(\wh{G}_{Q,SQR}(f))&= \mathbb{E}_{(\mu f^{-1})^m}(\phi_Q) = G_Q(\mu f^{-1}) \\ &= Q(\mu f^{-1},\id_\mathbb{R}) = Q(\mu,f), \\ \end{split} \end{equation} where in the first equality we used the change of variable Theorem \ref{thchvar}, in the second and third the definitions of $\phi_Q$ and $G_Q$ (see (\ref{defgqmu})), respectively, and in the last (\ref{qz}) and again Theorem \ref{thchvar}. An unbiased estimation scheme $SPR=(t,(A_i)_{i=1}^2)$ for $PR$ is given by $A_{1} = A_{2}=\{1\}$ and $t(x_1,x_2) = x_1x_2$, so that, using notation (\ref{clikeg}), its estimator can be written as \begin{equation}\label{defspr} \wh{PR}_{SPR} = g_1[0]g_{2}[0], \end{equation} and for each $(\mu, f) \in \mc{V}$ and $X \sim \mu$, it holds \begin{equation} \wh{PR}_{SPR}(f)(X) = f_1(X)f_{2}(X). \end{equation}
If $Sg$ is received from $Sg'$ using $\psi$ as above and we are given schemes $\kappa' = (\kappa'_i)_{i=1}^n$ such that $\kappa'_i$ is a scheme for $Sg'_i$, $i \in I_n$, then trivial extensions of $\kappa'$ using $\psi$ are defined as a scheme $\kappa = (\kappa_i)_{i=1}^n$ for $Sg$ such that for each $i \in I_n$, $t_{\kappa_i} = t_{\kappa'_i}$ and for $j \in I_k$, if $j \in \psi_i[I_{k_i}]$, then $A_{\kappa_i,j} = A_{\kappa'_i,\psi_i^{-1}(j)}$, and otherwise $A_{\kappa_i,j} = \emptyset$. It is easy to check that if $\kappa'_i$ is an unbiased scheme for estimation $G'_i$, $i \in I_n$, as above, and $G$ are trivial extensions of $G'$ using $\psi$, then $\kappa$ is an unbiased estimation scheme for $G$. An unbiased estimation scheme $SPR^n$ for $PR^n$ is defined as trivial extensions of $(SPR)_{i=1}^n$ using the same $\psi$ as when extending $(PR)_{i=1}^n$ to $PR^n$. With the help of notation (\ref{clikeg}), estimator of its $i$th subscheme can be written as \begin{equation}\label{E1g} \wh{PR}^n_{i,SPR^n} = g_i[0]g_{n+1}[0]. \end{equation} We shall use formulas for estimators like (\ref{gqsq}) and (\ref{E1g}) to define previously undefined schemes analogously as in Section \ref{secSchemesPrev}.
\section{\label{secineffgen}Generalization of the inefficiency constants of schemes} Let us make some generalizations of the definitions of inefficiency constants of schemes from Section \ref{secIneffSchemes} so that they can be used for the more general schemes from the previous section and for quantifying the inefficiency of estimation of several estimands in the same sequence of Monte Carlo procedures using a given scheme. If $\kappa=(\kappa_i)_{i=1}^n= (t_i,A_i)_{i=1}^n$ is an estimation scheme for estimands $G=(G_i)_{i=1}^n$ on some common admissible pairs $\mc{V}$ as in Definition \ref{defAPairs}, then $\kappa$ can be used to generate estimates of coordinates of $G(\alpha)$ for some $\alpha =(\mu,f) \in \mc{V}$ as follows. For a $X \sim \mu^{p_\kappa}$, one computes the quantities $g_{\mc{V},A_i,j,v}(f)(\wt{X}_{p_{A_i}})=f_j(X_v)$, $i \in I_n$, $j \in I_k$, $v \in A_{i,j}$, bearing in mind that they are equal for the same $j$ and $v$ and different $i$, so that they are computed only once, and one evaluates $t_i$ on $g_{\mc{V},A_i}(f)(\wt{X}_{p_{A_i}})$ to obtain an estimate of $G_i(\alpha)$, $i \in I_n$.
Note that this time, for each $i \in I_k$, $|A_{\kappa,i}|$ (see (\ref{Akappa})) is the number of all evaluations of $f_i$ made in such a computation. Let further $\kappa$ be unbiased for estimation of $G$ and $\Var_{\alpha}(\phi_{\kappa_i,\mc{V}})<\infty$, $i \in I_n$. Then we can use the above estimate of $G(\alpha)$ in a single step of a MC procedure. Let $J \subset I_n$ be nonempty. We define subvector of $G$ consisting of its estimands with indices in $J$,
as $G_J=(G_j)_{|j\in J}$ and an analogous subvector of $\kappa$ as $\kappa_J=(\kappa_j)_{|j \in J}$. Note that from (\ref{phikappamany}) and discussion below (\ref{Qmuf}), quantity
$\Var_{\alpha}(\phi_{\kappa_J,\mc{V}}) \in \overline{\mathbb{R}}$ is well-defined for $|J|=1$ for all $\alpha \in \mc{V}$,
while for $|J|>1$, for which symbol $\Var$ in this quantity is some variance for random vectors as in Section \ref{secOrthog}, it is well-defined only for $\alpha \in \mc{V}$ for which $\Var_{\alpha}(\phi_{\kappa_i,\mc{V}})<\infty$, $i \in J$. We define an inefficiency constant $d_{G,J,i,\kappa}$ of $\kappa$ with respect to the $i$th function for estimating the subvector of $G$ with indices in $J$ to be an $\overline{\mathbb{R}}$-valued function defined for each $\alpha \in \mc{V}$ for which $\Var_{\alpha}(\phi_{\kappa_J,\mc{V}})$ is well-defined, in which case it is given by formula \begin{equation}\label{dIneffgen}
d_{G,J,i,\kappa}(\alpha) = \Var_{\alpha}(\phi_{\kappa_J,\mc{V}})|A_{\kappa,i}|. \end{equation}
This is an extension of the definition from Section \ref{secIneffSchemes}
which coincides with the above one for $k = 1$ and $|J|=1$. When $|J|=1$ and the index $i$ of the function
is known from the context and omitted in the subscript, we shall use the same simplified notations as in Section \ref{secIneffSchemes}. The above defined inefficiency constants have analogous interpretation as the less general ones in Section \ref{secIneffSchemes}. However, using notations as in this section,
one now needs to assume that for estimands $G$ and $G'$ it holds $(G_{J})_j(\alpha)=(G_{J'}')_{j}(\alpha'),$ $j=1,\ldots,|J|$, and that the ratio of positive average durations $\tau_{s}$ to $\tau'_{s}$ of single steps of sequences of MC procedures using $\kappa$ and $\kappa'$, computing $G(\alpha)$ and $G'(\alpha')$ fulfills \begin{equation}\label{tauAratiogen}
\frac{\tau_s}{\tau'_s} \approx_\delta \frac{|A_{\kappa,i}|}{|A_{\kappa',i'}|}, \end{equation} which can be the case for small $\delta$ e. g. when the most time-consuming part of both sequences of MC procedures are computations of only the $i$th and $i'$th functions. Similarly as in Section \ref{secSchemesPrev} in our numerical experiments these functions will be constructions of outputs of MRs. Then we receive that the ratio of inefficiency constant $c = \Var_{\alpha}(\phi_{\kappa_J,\mc{V}})\tau_s$ for estimation of $G_J(\alpha)$ (see Section \ref{secStat}) using $\kappa_J$ to an analogous constant for the primed procedure, fulfills \begin{equation} \frac{c}{c'} \approx_{\delta} \frac{d_{G,J,i,\kappa}(\alpha)}{d_{G,J',i',\kappa'}(\alpha')}. \end{equation}
Similarly as for the inefficiency constants of sequences of MC procedures in Section \ref{secStat}, the ratio of positive real values of inefficiency constants (\ref{dIneffgen}) of $\kappa$ and $\kappa'$ for estimating the subvectors of $G(\alpha)$ and $G'(\alpha')$ with indices $J$ and $J'$ as as above, is $\delta$-approximately equal to the ratio of variances of the appropriate final MC estimators for $\delta$-approximately the same number of $i$th and $i'$th functions evaluations made in the respective MC procedures or to the ratio of the number of these functions evaluations in the MC procedures for $\delta$-approximately equal variances of the final MC estimators, and it is also equal to the limit of ratios of minimum numbers of respective functions evaluations needed for the variances of the final MC estimators to be below $\epsilon$ for $\epsilon$ tending to zero.
\section{The possibility of a better performance of translation-invariant estimators} In this section we provide certain criteria for verifying that some estimators of estimands on pairs which are in a sense invariant under translations can in some situations significantly outperform their certain counterparts without this property. Let $\mc{V}$ be some admissible pairs as in Definition \ref{defAPairs} such that $\mc{H}_i = \mc{S}(\mathbb{R})$ for some $i \in I_k$. For $f=(f_j)_{j=1}^k \in \mc{V}_2$ and $c\in \mathbb{R}$, we denote $\tr_{i}(f,c)=(f_1,\ldots f_i+c,\ldots,f_k)$. \begin{defin}\label{definvestimand} We say that an estimand $G$ on $\mc{V}$ is translation-invariant in the $i$th function (or simply translation-invariant if $k=1$), if for each $(\mu,f) \in \mc{V}$ and $c \in \mathbb{R}$ such that $(\mu,\tr_{i}(f,c)) \in \mc{V}$, it holds $G(\mu,f) = G(\mu,\tr_{i}(f,c))$. \end{defin} \begin{lemma}\label{invLem} For an estimand $G$ on $\mc{V}$, translation-invariant in the $i$th function, suppose that there exists $\alpha = (\mu,f) \in \mc{V}$ and a real sequence
$(c_l)_{l=1}^\infty$, $\lim_{l \to \infty}|c_l| = \infty$ such that for each $l \in \mathbb{N}_+$, $(\mu,\tr_{i}(f,c_l)) \in \mc{V}$. Suppose further that for some unbiased estimator $\phi$ of $G$ with indices $K$ and each $\wt{X} \sim \mu^K$, there exist $n \in \mathbb{N}_+$ and $Z_j \in L^2$, $j = 0,\ldots, n$, where $\mathbb{E}(Z_n^2)>0$,
such that for each $l \in \mathbb{N}_+$, \begin{equation}
R(c_l) = \phi(\tr_{i}(f,c_l))(\wt{X}) =\sum_{j=0}^n c^j_l Z_j \end{equation} a. s. Then \begin{equation}\label{limnVar} \lim_{l \to\infty} \Var_{\mu,\tr_{i}(f,c_l)}(\phi) = \infty. \end{equation} \end{lemma} \begin{proof} For certain random variables $W_1, \ldots, W_{2n-1} \in L^2$, it holds a. s. \begin{equation} R^2(c_l) = c^{2n}_l Z_n^2 + \sum_{j=0}^{2n -1} c^j_lW_j. \end{equation} Thus, from $\mathbb{E}(Z_n^2)>0$, we receive \begin{equation} \lim_{l \to \infty}\mathbb{E}(R(c_l)^2) = \infty \end{equation} and (\ref{limnVar}) follows from the fact that \begin{equation} \Var_{\mu,\tr_{i}(f,c_l)}(\phi) = \mathbb{E}(R^2(c_l)) - G^2(\alpha). \end{equation}
\end{proof} In all situations in which we use the above lemma its assumptions are satisfied for each unbounded real sequence $(c_l)_{l=1}^{\infty}$, so we further only specify the required $\alpha$. \begin{defin}\label{definvestimator} A statistic for $\mc{V}$ with indices $K$ is translation-invariant in the $i$th function or simply translation-invariant if $k=1$, if for each $(\mu,f) \in \mc{V}$ and $c \in \mathbb{R}$ such that $(\mu,\tr_{i}(f,c)) \in \mc{V}$, and each $\wt{X} \sim \mu^K$, it holds \begin{equation}\label{thesame} \phi(f)(\wt{X}) = \phi(\tr_{i}(f,c))(\wt{X}). \end{equation} \end{defin} Note that if an unbiased estimator of an estimand $G$ is translation-invariant in the $i$th function, then $G$ must also be translation-invariant in this function. \begin{theorem}\label{thtransl} Let $G$ be an estimand on $\mc{V}$. Let $\phi$ be an unbiased estimator of $G$, translation-invariant in the $i$th function, and let the unbiased estimator $\phi'$ of $G$ satisfy the assumptions of Lemma \ref{invLem}. Then for each $(c_l)_{l=1}^{\infty}$ and $\alpha = (\mu,f)$ as in this lemma for which further $\Var_{\alpha}(\phi)$ is finite, for each $B > 0$, there exists $n\in \mathbb{N}_+$ such that for $\alpha = (\mu,\tr_{i}(f,c))$, \begin{equation}\label{isworse} \Var_{\alpha_n}(\phi') > \Var_{\alpha_n}(\phi) + B. \end{equation} In particular, both the difference and ratio of variances of $\phi'$ and $\phi$ can be arbitrarily large. \end{theorem} \begin{proof} From (\ref{thesame}), $\Var_{\alpha_n}(\phi)=\Var_{\alpha}(\phi)$, $n \in \mathbb{N}_+$, while from Lemma \ref{invLem}, as $n$ goes to infinity, the lhs of (\ref{isworse}) goes to infinity. \end{proof}
Let us apply the above theory to certain estimators defined in Section \ref{secSymIneq}. Estimator $\widehat{V}_{1,s2}^{tot}$ is translation-invariant. For estimator $\wh{V}_{1,a2}^{tot}$ let us take $\alpha=(\mu,f) \in D_{V_1^{tot}}$ such that for each random variables $X_i \sim \mu_i, i \in I_2,$ we have $X_i \in L^2$, $i \in I_2$, $\Var(X_1)>0$, $\mathbb{E}(X_2^2)>0$, and $f(X_1,X_2) = X_1X_2$. Then for $\wt{X} \sim \mu^{p_{a2}}$, the assumptions of Lemma \ref{invLem} are satisfied for $n = 1$ and $Z_1 = (\wt{X}_1[0]-\wt{X}_1[1])\wt{X}_2$, since $\mathbb{E}(Z_1^2) = 2\Var(X_1)\mathbb{E}(X^2_2) > 0$. Thus $\widehat{V}_{1,a2}^{tot}$ can have much higher variance than $\widehat{V}_{1,s2}^{tot}$ in the sense of Theorem \ref{thtransl}, or equivalently $d_{V_1^{tot},a2}$ can be much higher than $d_{V_1^{tot},s2}$ (in the above sense). Notice that $\wh{V}_{1,s4}$ is translation-invariant and $\wh{V}_{1,a3}$ satisfies the conditions of Lemma \ref{invLem} for some $(\mu,f) \in D_{V_1}$ such that for each $X_i \sim \mu_i$, $i \in I_2$, $f(X_1,X_2) = X_1$, $X_1 \in L^2$, and $\Var(X_1)> 0$, since then for $\wt{X} \sim \mu^{p_{a3}}$ we have in Lemma \ref{invLem}, $n = 1$ and $\mathbb{E}(Z^2_1) = \mathbb{E}((\wt{X}_1[0] - \wt{X}_1[1])^2) = 2\Var(X_1)>0$. Thus $d_{V_1,a3}$ can be much higher than $d_{V_1,s4}$.
\section{\label{secAveSchemes}Averaging of estimators and schemes} Let $\mc{V}$ be some admissible pairs with a signature $Sg$ as in Definition \ref{defAPairs}.
Let $\pi \in \Theta^N$ (see Section \ref{secSymIneq}). We define a function $\wt{\pi}$ on $I_N\times\mathbb{N}_+$ by formula $\wt{\pi}(i,j) = (i,\pi_i(j))$. Let $K$ be some arguments' indices for $N$. The image under $\wt{\pi}$ (see Appendix \ref{appMath}) of $K$ is \begin{equation} \wt{\pi}[K] = \{\wt{\pi}(\beta):\beta \in K\}. \end{equation} Let $B = (B_i)_{i=1}^N$ be a sequence of nonempty sets and the function $\sigma_{B,K,\pi}:B^{K}\rightarrow B^{\wt{\pi}[K]}$ be such that for each $x \in B^K$ and $\beta \in K$, \begin{equation}\label{sigmadef} (\sigma_{B,K,\pi}(x))_{\wt{\pi}(\beta)} = x_{\beta}. \end{equation} Note that for each $\mu \in \mc{V}_1$ and $X \sim \mu^K$, we have \begin{equation}\label{sigmapix} \sigma_{B_\mu,K,\pi}(X) \sim \mu^{\wt{\pi}[K]}. \end{equation} For a statistic $\phi$ for $\mc{V}$ with indices $K$, a permutation of $\phi$ given by $\pi$, denoted as $\A_\pi(\phi)$, is defined as a statistic for $\mc{V}$ with indices $\wt{\pi}[K]$ such that for each $(\mu,f) \in \mc{V}$ and $x \in B_\mu^{\wt{\pi}[K]}$, \begin{equation}\label{apidef} \A_\pi(\phi)(f)(x) = \phi(f)(\sigma_{B_\mu,K,\pi}^{-1}(x)). \end{equation} From (\ref{sigmapix}) and (\ref{apidef}) it follows that for each $(\mu,f) \in \mc{V}$, $X \sim \mu^K$, and $Y \sim \mu^{\wt{\pi}[K]}$,
\begin{equation}\label{sameDistr} \A_\pi(\phi)(f)(Y) \sim \phi(f)(X). \end{equation}
For a function $h:\mathbb{R}^n\rightarrow\mathbb{R}$, like e. g. summation $h(x)= \sum_{i=1}^nx_i$, and real-valued statistics $\phi_i$ for $\mc{V}$ with indices $K_i$, $i \in I_n$, we define $h((\phi_i)_{i=1}^n)$ to be a real-valued statistic for $\mc{V}$ with indices $K = \bigcup_{i=1}^n K_i$ such that for each $(\mu,f) \in \mc{V}$ and $x \in B_\mu^K$, \begin{equation} \label{hstat} h((\phi_i)_{i=1}^n)(f)(x) = h((\phi_i(f)(x_{K_i}))_{i=1}^n) \end{equation} (see (\ref{xL})). Let $K$ be some arguments' indices for $N$ and $\Pi$ be a nonempty finite subset of $\Theta^N$. We define \begin{equation} \wt{\Pi}[K] = \bigcup_{\pi \in \Pi}\wt{\pi}[K]. \end{equation} Let $\phi$ be an $\mathbb{R}^n$-valued statistic for $\mc{V}$ with indices $K$. We define an average of $\phi$ given by $\Pi$ as the following statistic for $\mc{V}$ with indices $\wt{\Pi}[K]$, \begin{equation}\label{defAvestat}
\A_{\Pi}(\phi) = \frac{1}{|\Pi|}\left(\sum_{\pi \in \Pi} \A_\pi(\phi)\right). \end{equation} From (\ref{sameDistr}) it follows that for each $(\mu,f) \in \mc{V}$, $X \sim \mu^{\wt{\Pi}[K]}$, and
$Y \sim \mu^{K}$, $\A_{\Pi}(\phi)(f)(X)$ is an average of $|\Pi|$ random variables with the same distribution as $\phi(f)(Y)$.
In particular, if $\phi$ is an estimator of some estimand $G$ on $\mc{V}$, then so is $\A_{\Pi}(\phi)$. For $p > 0$, we write $\phi \in L^p(\mc{V})$ if $\phi(f) \in L^p(\mu^K)$ for each $(\mu,f) \in \mc{V}$. From Lemma \ref{lemVarAve} in Appendix \ref{appStatMC} it follows that for each $\phi \in L^1(\mc{V})$, $\A_{\Pi}(\phi)$ has uniformly not higher variance than $\phi$, that is for each $\alpha \in \mc{V}$, \begin{equation}\label{uniformalpha} \Var_{\alpha}(\A_{\Pi}(\phi)) \leq \Var_{\alpha}(\phi). \end{equation} Let $\pi \in \Theta^N$. For each nonempty $I \subset I_N$, we identify each sequence $v = (v_i)_{i \in I} \in \mathbb{N}_+^{I}$ with the set $\{(i,v_i):i \in I\} \subset I_N \times \mathbb{N}_+$, so that $\wt{\pi}[v] = (\pi_i(v_i))_{i \in I}$. In particular for $v \in \mathbb{N}_{+}^N$ we receive $\wt{\pi}[v] = \wh{\pi}(v)$ (see Section \ref{secSymIneq}). Let $W \subset \mc{P}(I_N\times\mathbb{N}_+)$ (see Appendix \ref{appMath}). For $\wt{\pi}^{\rightarrow}$ denoting the image function of $\wt{\pi}$ (see Appendix \ref{appMath}), we have \begin{equation} \wt{\pi}^{\rightarrow}[W] = \{\wt{\pi}[v]:v \in W\}. \end{equation} For $\Pi$ as above we define \begin{equation} \wt{\Pi}^{\rightarrow}[W] = \bigcup_{\pi \in \Pi} \wt{\pi}^{\rightarrow}[W]. \end{equation} In particular for $W \subset \mathbb{N}_{+}^N$ we receive $\wt{\Pi}^{\rightarrow}[W] = \wh{\Pi}[W]$ (see Section \ref{secSymIneq}). Let $A=(A_i)_{i=1}^k$ be some sets of evaluation vectors for $Sg$. We define \begin{equation} \wt{\Pi}^{\rightarrow}[A] = (\wt{\Pi}^{\rightarrow}[A_i])_{i=1}^k. \end{equation} Let $\pi \in \Theta^N$. We denote $\wt{\{\pi\}}^{\rightarrow}[A]$ simply as $\wt{\pi}^{\rightarrow}[A]$. For convenience we shall write $\delta$ and $\gamma$ instead of $\delta(A)$ and $\gamma_A$ defined in the previous section. We define function $\rho_{C,A,\pi}:C_A \rightarrow C_{A}$ to be such that for each \begin{equation}\label{z1example}
z = ((y_{i,v})_{|v \in (\wt{\pi}^{\rightarrow}[A])_{\gamma(i)}})_{i=1}^\delta \in C_{\wt{\pi}^{\rightarrow}[A]} =C_{A}, \end{equation} it holds \begin{equation}\label{defRho}
\rho_{C,A,\pi}(z) = ((y_{i,\wt{\pi}[v]})_{|v \in A_{\gamma(i)}})_{i=1}^\delta. \end{equation} Let $t: C_{A} \rightarrow \mathbb{R}$. We define function $\ave_{C,A,\Pi}(t): C_{A} \rightarrow \mathbb{R}$, called permutation of $t$ given by $\pi$ and $A$, to be such that for each $z$ as in (\ref{z1example}), \begin{equation} \ave_{C,A,\pi}(t)(z) = t(\rho_{C,A,\pi}(z)). \end{equation} Let further $\eta_{C,A,\Pi,\pi}:C_{\wt{\Pi}^{\rightarrow}[A]} \rightarrow C_A$ be such that for each \begin{equation}\label{zexample}
z = ((y_{i,v})_{|v \in (\wt{\Pi}^{\rightarrow}[A])_{\gamma(i)}})_{i=1}^\delta \in C_{\wt{\Pi}^{\rightarrow}[A]}, \end{equation} it holds \begin{equation}
\eta_{C,A,\Pi,\pi}(z) = ((y_{i,v})_{|v \in (\wt{\pi}^{\rightarrow}[A])_{\gamma(i)}})_{i=1}^\delta. \end{equation} We define $\ave_{C,A,\Pi}(t):C_{\wt{\Pi}^{\rightarrow}[A]} \rightarrow \mathbb{R}$, called average of $t$ given by $\Pi$ and $A$, to be such that for each $z$ as in (\ref{zexample}), \begin{equation}\label{tcap}
\ave_{C,A,\Pi}(t)(z) = \frac{1}{|\Pi|} \sum_{\pi \in \Pi} \ave_{C,A,\pi}(t)(\eta_{C,A,\Pi,\pi}(z)). \end{equation} For the special case of $k=1$, $\mc{H} = \mc{S}(\mathbb{R}),$ and $\Pi$ being a subgroup of $\Theta^N$, $\ave_{C,A,\Pi}(t)$ is equal to $\ave_{A,\Pi}(t)$ given by formula (\ref{symOp}) from Section \ref{secSymIneq}. Let $\kappa = (t,A)$ be a scheme for $Sg$. Its average given by $\Pi$ is defined as a scheme \begin{equation} \ave_{\Pi}(\kappa) = (\ave_{C,A,\Pi}(t),\wt{\Pi}^{\rightarrow}[A]). \end{equation} This coincides with definition (\ref{avePiKO}) from Section \ref{secSymIneq} for the same special case as discussed below (\ref{tcap}). When $\Pi = \{\pi\}$, $\ave_{\Pi}(\kappa)$ is denoted as $\ave_{\pi}(\kappa)$ and called permutation of $\kappa$ given by $\pi$. We have a following theorem, which we prove in Appendix \ref{appcoeffsLem}.
\begin{theorem}\label{thEquAves} Under the preceding assumptions, \begin{equation}\label{avephikappa} \phi_{\ave_{\Pi}(\kappa),\mc{V}} = \A_{\Pi}(\phi_{\kappa,\mc{V}}). \end{equation} \end{theorem} For an $n$-dimensional scheme $\kappa = (\kappa_i)_{i=1}^n$ for $Sg$, we define its average as $\ave_{\Pi}(\kappa) = (\ave_{\Pi}(\kappa_i))_{i=1}^n$. If $\Pi$ is a subgroup of $\Theta^N$, then an average of a scheme or a statistic given by $\Pi$ is called their symmetrisation.
From Theorem \ref{thEquAves} and a similar fact concerning estimators stated above, it follows that an average of an unbiased estimation scheme for some estimand $G$ remains an unbiased scheme for its estimation and its estimator has uniformly not higher variance.
Let us consider an $n$-dimensional scheme $\kappa = (t_i,A_i)_{i=1}^n$ for $\mc{V}$, and $m \in \mathbb{N}_+$. We define an $m$-step MC scheme $\kappa(m)$ using scheme $\kappa=(\kappa_i)_{i=1}^n$
to be an average of $\kappa$ given by any $\Pi \subset \Theta^N$, $|\Pi| = m$, such that for each $\pi_1, \pi_2 \in \Pi, \pi_1 \neq \pi_2,$ schemes $\A_{\pi_i}(\kappa), i \in I_2$, have disjoint arguments' indices, that is
$\wt{\pi_1}(p_{\kappa}) \cap \wt{\pi_2}({p_\kappa}) = \emptyset$. Note that $|A_{\kappa(m),i}| = m |A_{\kappa,i}|, i \in I_k$, and for each $\alpha = (\mu,f) \in \mc{V}$, $i \in I_n$, $X \sim \mu^{p_{\kappa_i}}$, and $Y \sim \mu^{p_{\kappa(m)_i}}$, $\phi_{\kappa(m)_i,\mc{V}}(f)(Y)$ is an average of $m$ independent random variables with the same distribution as $\phi_{\kappa_i,\mc{V}}(f)(X)$. Let us further assume that $\Var_\alpha(\phi_{\kappa_i,\mc{V}}) < \infty$, $i \in I_n$, so that $\Var_\alpha(\phi_{\kappa(m)_i,\mc{V}}) = \frac{\Var_\alpha(\phi_{\kappa_i,\mc{V}})}{m},$ $i \in I_n$. If $\kappa$ is further an unbiased estimation scheme for estimands $G=(G_i)_{i=1}^n$ with common admissible pairs $\mc{V}$, then $\phi_{\kappa_i,\mc{V}}(f)$ and $\phi_{\kappa(m)_i,\mc{V}}(f)$ can be identified with the single-step and final MC estimators of $G_i(\alpha), i \in I_n,$ respectively, and we have equality of inefficiency constants of the schemes \begin{equation}\label{equdIneffMC} d_{G,i,j,\kappa} = d_{G,i,j,\kappa(m)},\ i \in I_n,\ j \in I_k. \end{equation}
\section{\label{secGenIneq}Some general inequalities between variances of estimators and inefficiency constants of schemes} We will now prove some general inequalities between variances of estimators of estimands on pairs and inefficiency constants of schemes, the latter including as a special case the inequality from Theorem \ref{thineqOld},
but first we need some helper facts and definitions. \begin{lemma}\label{lemcomb} For $N \in \mathbb{N}_+$, let $X= (X_i)_{i=1}^N$ be a random vector with independent coordinates, and let us consider independent random variables $Y_{i,j} \sim X_i, i \in I_N, j \in \mathbb{N}_+$. For $v \in \mathbb{N}_+^N$, let $Y_v=(Y_{i,v_i})_{i=1}^N$. For some measurable function $f$ such that $Z = f(X) \in L^2$ and a finite nonempty set $A \subset \mathbb{N}_+^N$, let \begin{equation}
\overline{Z}= \frac{1}{|A|}\sum_{v \in A}f(Y_v). \end{equation} Then it holds \begin{equation}\label{ineqaz}
\frac{1}{|A|}\Var(Z) \leq \Var(\overline{Z}) \leq \Var(Z). \end{equation} \end{lemma} \begin{proof} We have \begin{equation}
\Var(\overline{Z}) = \left(\frac{1}{|A|^2}\sum_{v,w \in A}\Cov(f(Y_v),f(Y_w)\right). \end{equation} For $v,w \in \mathbb{N}_+^N$, let $c(v,w) = \{i \in I_N:v_i = w_i\}$, and $Y_{v,w} = (Y_{i,v_i})_{i \in c(v,w)}$. Then from Theorem
\ref{thCond}, for each $v,w \in A$, $\Cov(f(Y_v),f(Y_w)) = \Var(\mathbb{E}(f(Y_v)|Y_{v,w}))$. Inequalities (\ref{ineqaz}) follow from the fact
that $\Var(\mathbb{E}(f(Y_v)|Y_{v,w}))$ is nonnegative and from (\ref{aveVarError}) it is not higher than $\Var(Z)$ and equal to it for $v=w$. \end{proof}
Let $\mc{V}$ be some admissible pairs as in Definition \ref{defAPairs}. For each arguments' indices $K$ for $N$, we define $n_K$ to be a vector from $\mathbb{N}^N$ whose $i$th coordinate is \begin{equation} n_{K,i} = \max(\{j: (i,j) \in K\} \cup \{0\}). \end{equation} For sets of evaluation vectors $A$ or a scheme $\kappa$ for $Sg$, we define $n_{A} = n_{p_A}$ and $n_{\kappa} = n_{p_\kappa}$, and for a statistic $\phi$ for $\mc{V}$ with indices $K$, $n_\phi = n_K$.
Let $I$ be a nonempty subset of $I_N$ and $m\in \mathbb{N}_+$ For symmetrisations given by $\Theta_{N,I,m}$ (see (\ref{thetanim})) e. g. of some scheme for $Sg$ or a statistic for $\mc{V}$, we use the same nomenclature as for the less general schemes in Section \ref{secSymIneq}.
For some finite subgroup $\Pi \subset \Theta^N$, we say that a scheme for $Sg$ or a statistic for $\mc{V}$ is $\Pi$-symmetric if it is equal to its symmetrisation given by $\Pi$. Suppose that $\psi$ is a statistic for $\mc{V}$ or a scheme for $Sg$ such that $n_{\psi,i}=n$, $i \in I$, and $\psi$ is $\Theta_{N,I,n}$-symmetric. Then for each $n'\in \mathbb{N}_+$, $n'\geq n$, symmetrisation of $\psi$ given by $\Theta_{N,I,n'}$ is called its symmetrisation from $n$ to $n'$ dimensions (simply symmetrisation if $n=n'$) in the argument given by $I$ (or in the $i$th argument if $I= \{i\}$).
For some arguments' indices $K$ for $N$, sequence of sets $B= (B_i)_{i=1}^n$, and $x \in B^K$, for
$L \subset I_N$ and $K_L = \{(i,j)\in K:i \in L\}$, we denote $x_L = x_{K_L}$ (see (\ref{xL})), while for $L \subset \mathbb{N}_+$ and $K:L = \{(i,j)\in K:j \in L\}$, we denote $x:L = x_{K:L}$. For arguments' indices $K_i$ for $N$, $i \in I_2$, $K = K_1\cup K_2$, and $K_1\cap K_2 = \emptyset$, in the proof of the below theorem we identify $x_K$ with $(x_{K_1},x_{K_2})$.
\begin{theorem}\label{thIneqEsts}
Let $\phi'$ be a symmetrisation of a statistic $\phi \in L^1(\mc{V})$ from $n$ to $n'$ dimensions in the argument given by some $I$ as above.
Then for each $\alpha \in \mc{V}$ such that $\Var_{\alpha}(\phi)< \infty$, \begin{equation}\label{VarN1N2} \frac{1}{{n' \choose n}} \Var_{\alpha}(\phi) \leq \Var_{\alpha}(\phi') \leq \Var_{\alpha}(\phi). \end{equation}
\end{theorem} \begin{proof}
Since $\phi$ is $\Theta_{N,I,n}$-symmetric statistic,
for each $\theta_1, \theta_2 \in \Theta_{n'}$ such that $\theta_1[I_n] = \theta_2[I_n]$, or equivalently $\theta_2^{-1}\theta_1[I_n]= I_n$, it holds $\A_{\pi_{N,I,\theta_2^{-1}\theta_1}}(\phi)=\phi$, and thus $\A_{\pi_{N,I,\theta_2^{-1}}}\A_{\pi_{N,I,\theta_1}}(\phi)=\phi$ and $\A_{\pi_{N,I,\theta_1}}(\phi)=\A_{\pi_{N,I,\theta_2}}(\phi)$. We denote $\sim I= I_N \setminus I$. Let $\alpha = (\mu,f) \in \mc{V}$, $K' = \wt{\Theta}_{N,I,n'}[K]$ be the arguments indices of $\phi'$, $X \sim \mu_{K'}$, $U = X_{\sim I}$, and $V = X_I$, so that $X= (U,V)$. We denote $\wt{\phi}'= \phi'(f)$.
Let further $\mc{W} = \{L \subset I_{n'}, |L|= n\}$. For each $L \in \mc{W}$, let us choose certain $\theta_L \in \Theta_{n'}$ such that $\theta_L[I_n] = L$, and denote $\wt{\phi}_L= \A_{\pi_{N,I,\theta_L}}(\phi)(f)$. From (\ref{defAvestat}) and the above remarks we have \begin{equation}\label{phiprimform} \wt{\phi}'(X) = \frac{1}{{n' \choose n}} \sum_{L \in \mc{W}}\wt{\phi}_L(U, V:L). \end{equation} Thus (\ref{VarN1N2}) follows from Lemma \ref{lemcomb}. \end{proof}
\begin{theorem}\label{thIneqds} Let $G=(G_i)_{i=1}^m$ be estimands with common admissible pairs $\mc{V}$. If scheme $\kappa'$ is created from an unbiased estimation scheme $\kappa = (\kappa_i)_{i=1}^m$ for $G$ by its symmetrisation from $n$ to $n'$ dimensions in the argument corresponding to some $I$ then for each $j \in I_m$, $i \in I_k$ such that $J_i \cap I\neq \emptyset$, and $\alpha \in D_{G_j}$ for which $d_{G,j,i,\kappa}(\alpha)< \infty$, it holds \begin{equation}\label{d1d2n} \frac{\frac{n'}{n}}{{n' \choose n}} d_{G,j,i,\kappa}(\alpha) \leq d_{G,j,i,\kappa'}(\alpha) \leq \frac{n'}{n} d_{G,j,i,\kappa}(\alpha). \end{equation} \end{theorem} \begin{proof} Since $\kappa$ is $\Theta_{N,I,n}$-symmetric, it holds for each $l \in I\cap J_i$, \begin{equation}\label{A1A2n}
\frac{|A_{\kappa,i}|}{n} = |\{v \in A_{\kappa,i}: v_l = 1\}| = \frac{|A_{\kappa',i}|}{n'}. \end{equation} Now (\ref{d1d2n}) follows from (\ref{A1A2n}), the fact that for each $j \in I_m$, $\phi_{\kappa_j',\mc{V}}$ is a symmetrisation of $\phi_{\kappa_j,\mc{V}}$ from $n$ to $n'$ dimensions, Theorem \ref{thIneqEsts}, and formula (\ref{dIneffgen}) defining an inefficiency constant. \end{proof} Taking $k=m=n=1$ and $n'=2$, we receive the thesis of Theorem \ref{thineqOld}.
\section{\label{secPolynEst}Schemes for the sensitivity indices of functions of conditional moments} For some $n \in \mathbb{N}_+$, let us consider a function $f$ and independent random variables $P=(P_i)_{i=1}^{N_P}$ and $R$ as at the beginning of Section \ref{secMany} but with $f(P,R) \in L^n$.
Let $Q$ restricted to $\mc{T}_n$ be a function of the first $n$ so restricted moments as in Section \ref{secCondMoms}, like $Q = \Var$ for $n=2$.
Suppose that there exists an unbiased estimator $\phi_Q$ of $G_Q$ in $m \in \mathbb{N}_+$ dimensions (see Appendix \ref{appStatMC}).
For $\wt{R} \sim \mu_R^m$ and independent of $P$, let \begin{equation}\label{hQ} h_Q(f)(P,\wt{R}) = \phi_Q((f(P,\wt{R}_i))_{i=1}^m) \end{equation} and let us assume that $h_Q(f)(P,\wt{R}) \in L^1$. Then it holds
\begin{equation}\label{phiqcq} \begin{split}
Q(f(P,R)|P) &= (Q(f(p,R)))_{p=P}\\ &=(\mathbb{E}(\phi_Q((f(p,\wt{R}_i))_{i=1}^{m})))_{p=P}\\
&= \mathbb{E}(h_Q(f)(P,\wt{R})|P),
\end{split} \end{equation} where in the first equality we used expression (\ref{CQKP}), in the second the fact that $\phi_Q$ is an unbiased estimator of $G_Q$ and that from Fubini's theorem \cite{rudin1970} $f(p,\wt{R}_i) \in L^n$ for $\mu_P$ a. e. $p$, and in the last Theorem \ref{indepCond} and (\ref{hQ}). In particular, expected values and variance-based sensitivity
indices of $Q(f(P,R)|P)$ and $\mathbb{E}(h_Q(f)(P,\wt{R})|P)$ coincide (whenever both are well-defined). Since the latter is a conditional expectation of the function $h_Q(f)$ of independent random variables $P$, $\wt{R}$ given the first variable, its sensitivity indices can be estimated with the help of estimators from Section \ref{secMany}, e. g. in a way we describe below. Let $r_Q$ be the degree of $G_Q$ and as $\phi_Q$ let us take the unique symmetric estimator of $G_Q$ in $r_Q$ dimensions (see Section \ref{secStat}). For instance for $Q = \Var$, we have $r_{Var}=2$ and \begin{equation} \phi_{Var}(x_1,x_{2}) = \frac{1}{2}(x_1 - x_2)^2, \end{equation} so that \begin{equation}\label{hvar} h_{Var}(f)(P,\wt{R}) = \frac{1}{2}(f(P,\wt{R}_1) -f(P,\wt{R}_2))^2. \end{equation} Let us now reinterpret different quantities from the end of Section \ref{secCondMoms} like $AveQ$, $VQ_k$, or $VQ_k^{tot}, k \in I_{N_P}$, as estimands on admissible pairs $\alpha_{\mu_P,\mu_R,f}$ defined analogously as
in Section \ref{secMany}, but for $f(P,R) \in L^n$ and $h_Q(f)(P,\wt{R})\in L^p$, where $p =1$ for $AveQ$ and $p=2$ for other estimands (this condition will be needed for our estimators to be integrable). The values of such estimands on such $\alpha_{\mu_P,\mu_R,f}$ are defined identically as in Section \ref{secCondMoms} treating $Z=f(P,R)$ as output an MR.
For $l \in \mathbb{N}_+$, we call a pair $\pi = (J_1, J_2)$ equal partition of the set $I_{2l}$, if
for $i \in I_2$, we have $J_i \subset I_{2l}$, $|J_1| = |J_2| = l$, $J_1 \cap J_2 = \emptyset$, and $1 \in J_1$ (note that $J_1\cup J_2 = I_{2l}$).
Let $\Psi_Q$ be the set of all equal partitions of
$I_{2r_Q}$. We have $|\Psi_Q| = \frac{{2r_Q \choose r_Q}}{2}$.
Consider $\wt{P}$ corresponding to $P$ as in Section \ref{secMany},
and let $\wt{R} \sim \mu_R^{2r_Q}$ be independent of $\wt{P}$.
For a partition $\psi = (\psi_1, \psi_2) \in \Psi_Q$,
we denote $\wt{R}_{\psi} = (\wt{R}_{\psi_1}, \wt{R}_{\psi_2})$, where $\wt{R}_{\psi_i} = (\wt{R}_{j})_{j \in \psi_i}, i \in I_2$. We shall now define a scheme $SQ$ whose subschemes yield estimators $\wh{\lambda Q}_{SQ}$ for different estimands $\lambda Q$ for $Q$, like $AveQ$, $VQ_k$, and $VQ^{tot}_k$, corresponding to such estimands $\lambda E$ for $E$. These estimators
evaluated on each appropriate function $f$ and random vector $(\wt{P}, \wt{R})$ as above are equal to the average over $\psi \in \Psi_Q$ of the corresponding estimators $\wh{\lambda E}_{SE}$ from Section \ref{secMany} evaluated on the function $h_Q(f)$ and random vector $(\wt{P}, \wt{R}_{\psi})$, that is
\begin{equation}\label{lQ}
\widehat{\lambda Q}_{SQ}(f)(\wt{P},\wt{R}) = \frac{1}{|\Psi_Q|} \sum_{\psi \in \Psi_Q}\widehat{\lambda E}_{SE}(h_Q(f))(\wt{P}, \wt{R}_{\psi}).
\end{equation} For instance for the main sensitivity index and $Q = \Var$ we have
\begin{equation}
\widehat{VVar}_{k,SVar}(f)(\wt{P},\wt{R}) = \frac{1}{3} \sum_{\psi \in \Psi_{Var}}\widehat{V}_{k,SE}(h_{Var}(f))(\wt{P}, \wt{R}_{\psi}).
\end{equation} Formulas like (\ref{lQ}) for different estimands $\lambda Q$ for some $Q$ can be easily expanded in terms of evaluation operators $s[i][j]$ and $s_k[i][j], i \in I_2, j \in I_{2r_Q}$, from Section \ref{secMany}, in which form they define the sought scheme $SQ$ in the sense discussed at the end of Section \ref{secSchemesPrev}. From Schwartz inequality, it is sufficient that $h_Q(f)(P,\wt{R}) \in L^4$ for the estimators of subschemes of $SQ$ to have finite variance. In particular, from (\ref{hvar}), for scheme $SVar$ it is sufficient that $f(P,R) \in L^8$. Such defined scheme $SQ$ uses together $4r_Q(N_P +1)$ evaluation vectors for $N_P > 2$. For scheme $SVar$ this is $8(N_P + 1)$, that is two times more than for scheme $SE$ for the same $N_P$. By analogy to discussion in Appendix C of \cite{badowski2011} for scheme $SE$, for $N_P=3$ one can construct schemes with lower inefficiency constants
for estimation of sensitivity indices of $Q(f(P,R)|P)$ than for the subschemes of $SQ$.
For some $Q$ and $Q'$ as above, such that $r = r_{Q} \leq r' = r_{Q'}$, an unbiased $n$-dimensional estimation scheme $SQ = (SQ_i)_{i=1}^n$ and an $n'$-dimensional one $SQ'= (SQ'_i)_{i=1}^{n'}$ for sequences of estimands $G=(G_i)_{i=1}^n$ and $G'=(G_i)_{i=1}^{n'}$, respectively, one can add symmetrisation of certain subscheme $SQ_i$ of $SQ$ from
$2r$ to $2r'$ dimensions in the argument given by $R$ as the $n+1$st subscheme to $SQ'$ and $G_i$ as such $n+1$st estimand to $G'$. We then have the following inequality of inefficiency constants of schemes in the sense of Theorem \ref{thIneqds}, \begin{equation}\label{CQd} \frac{\frac{r'}{r}}{{2r' \choose 2r}}d_{G,i,SQ} \leq d_{G',n+1,SQ'} \leq \frac{r'}{r} d_{G,i,SQ} \end{equation} and analogously for the inefficiency constants of the subschemes due to proportionality of the number of evaluation vectors used by the subschemes and the whole schemes. Let us add in this way to scheme $SVar$ all subschemes of $SE$, like ones for estimation of $Ave$, $AveVar$, as well as $V_k$ and $\wt{V}_k^{tot}$ for $k \in I_{N_P}$, symmetrised from two to four dimensions in the argument given by $R$. Then for each of such estimands $\lambda$, we have from (\ref{CQd}) for $r = 1$ and $r' = 2$,
\begin{equation}\label{ineqDVar} \frac{1}{3}d_{\lambda,SE} \leq d_{\lambda,SVar} \leq 2 d_{\lambda,SE}. \end{equation} Considering in addition to relations (\ref{ineqDVar}) also inequalities (\ref{EMComp}) and (\ref{VitotComp}), we receive for $N_P >2$ and $\lambda$ equal to $V_k$ (and arguments for which these relations were proved), \begin{equation}\label{compVkVar} \frac{1}{3}d_{V_k,EM} \leq d_{V_k,SVar} \leq 4 d_{V_k,EM}, \end{equation} while for $\lambda=\wt{V}_k^{tot}$ we obtain \begin{equation}\label{compVkTotVar} \frac{1}{3}d_{\wt{V}_k^{tot},ET} \leq d_{\wt{V}_k^{tot},SVar} \leq 4 d_{\wt{V}_k^{tot},ET}. \end{equation}
We shall compute numerical estimates of Sobol's sensitivity indices $SQ_{k}$ and $SQ_{k}^{tot}$ for $Q$ equal to $\Var$ and $\mathbb{E}$, defined in Section \ref{secVBSA},
using scheme
$SVar$
by inserting the final MC estimates obtained using the above defined subschemes for estimands like $VQ_k$, $VQ_k^{tot}$, and $VQ_P$ instead of exact values into appropriate definitions.
\section{\label{secApprCoeff}Schemes for products, covariances, and orthogonal projection coefficients}
For some $n \in \mathbb{N}_+$, let $\psi=(\psi)_{i=1}^{n+1}$ be functions such that $\psi_i: I_2 \rightarrow\mathbb{N}_+,\psi_i(1)=i$, $\psi_i(2)=n+1$, $i \in I_n$, and $\psi_{n+1}:\{1\}\rightarrow\mathbb{N}_+:\psi_{n+1}(1)=n+1$.
For the estimand $PR$ defined in Section \ref{secUnbiased} and $Ave = \mathbb{E}_{|\mc{T}_1}$ (that is estimand $Ave$ from Section \ref{secSchemesPrev} for $N=1$), estimands $PRA = (PRA)_{i=1}^{n+1}$ are defined as trivial extensions (see Section \ref{secUnbiased}) of $n+1$ estimands $(PR,\ldots,PR, Ave)$ ($PR$ appearing $n$-times at the beginning of this sequence) using $\psi$. Informally, this means that $PRA$ is equal to $PR^n$ from Section \ref{secUnbiased} extended by adding to it average of the $n+1$st function as the last estimand. Let $\wt{Cov}$ be an estimand on admissible pairs $\mc{V}$ of single distributions and two functions consisting of all possible $\alpha= (\mu, (f_1, f_2))$ such that $f_1$, $f_2$, $f_1f_2 \in L^1(\mu)$, in which case for any $X \sim \mu$, $\wt{Cov}(\alpha) = \Cov(f_1(X),f_2(X))$. We define $CovA=(CovA_i)_{i=1}^{n+1}$ as trivial extensions of $n+1$ estimands $(\wt{Cov}, \ldots, \wt{Cov},Ave)$ using $\psi$. We define estimands $b=(b_i)_{i=1}^{n+1}$ to be equal to $CovA$ or equivalently $PRA$ with each coordinate restricted to admissible pairs of single distributions and $n+1$ functions consisting of $\alpha = (\mu,(f_i)_{i=1}^{n+1}),$ such that for $X \sim \mu$, $\{f_i(X)\}_{i=1}^n$ is nonzero orthogonal in $L^2$, $\mathbb{E}(f_i(X))=0$, $i \in I_{n}$, and $f_{n+1}(X) \in L^2$. From discussion in Section \ref{secAppr}, $b_i(\alpha)$ is the coefficient of $f_i'(X) = \frac{f_i(X)}{\Var(f_i(X))}$, $i \in I_n$, and $b_{n+1}(\alpha)$ of $\mathbb{1}$ (see Appendix \ref{appMath}), in the orthogonal projection of $f(X)$ onto span($\{f_i'(X)\}_{i=1}^n\cup\{\mathbb{1}\}$). We define $c$ to be a restriction of $b$ to admissible pairs $\alpha$ as above, except that for each above $X$ the set $\{f_i(X)\}_{i=1}^n$ is orthonormal in $L^2$. Note that each unbiased estimation scheme for $PRA$ or for $CovA$ is also an unbiased estimation scheme for $b$ and $c$.
Let us consider an unbiased estimation scheme $SAve$ for $Ave$ given by the formula for estimator \begin{equation}\label{save} \wh{Ave}_{SAve} = g[0] \end{equation} and the following formula for estimator giving an $N$-step MC scheme $SAve(N)$ using $SAve$ \begin{equation}\label{saven} \wh{Ave}_{SAve(N)} = \frac{1}{N}\sum_{i=0}^{N-1}g[i]. \end{equation} For $N \in \mathbb{N}, N>1,$ let us consider scheme $SCov(N)$ given by the following formula for estimator of $\wt{Cov}$ \begin{equation}\label{scov} \wh{\wt{Cov}}_{SCov(N)} = \frac{1}{N-1}\sum_{i=0}^{N-1}g_{1}[i]g_2[i] - \frac{1}{N(N-1)} (\sum_{i=0}^{N-1} g_{1}[i]\sum_{i=0}^{N-1}g_2[i]). \end{equation} We define an unbiased estimation scheme $P1$ for $PRA$ as trivial extensions of $n+1$ schemes $(SPR,\ldots,SPR,SAve)$ using $\psi$, and an unbiased estimation scheme $C1(N)$ for $CovA$ as trivial extensions of $n+1$ schemes $(SCov(N),\ldots, SCov(N), SAve(N))$ also using $\psi$. Each estimand $CovA_k$ and estimator $\wh{CovA}_{k,C1(N)}, k \in I_{n},$ is translation-invariant in all functions (see Definitions \ref{definvestimand} and \ref{definvestimator}), while for each $k \in I_{n},$ estimator $\wh{c}_{k,P1}$ satisfies the assumptions of Lemma \ref{invLem} (and so does $\wh{b}_{k,P1}$) in the $n+1$st function for $n=1$ in this lemma and each $(\mu,f) \in D_{CovA}$, since for $X \sim \mu$, we have $Z_1 = f_k(X)$ and $\mathbb{E}(Z_1^2) = 1$. Thus $P1$ can have much higher inefficiency constant than $C1(N)$ for estimation of $c_k$ and $b_k,$ $k \in I_{n}$, in the sense of Theorem \ref{thtransl}.
However, as we shall now show, for each $k \in I_{n}$ and $N \in \mathbb{N}_+, N > 2$, there exists $\alpha \in \mc{V}$ such that \begin{equation}\label{dckpless} d_{c_k,P1}(\alpha) < d_{c_k,C1(N)}(\alpha). \end{equation} For a MC scheme $P1(N)$ using scheme $P1$ in $N$ steps, we have from (\ref{equdIneffMC}) that (\ref{dckpless}) is equivalent to $d_{c_k,P1(N)}(\alpha) < d_{c_k,C1(N)}(\alpha)$, and since both schemes use the same number of evaluation vectors for the $n+1$st function and both are unbiased, this is equivalent to \begin{equation}\label{MCleqC1N} \mathbb{E}_{\alpha}(\wh{c}^2_{k,P1(N)}) < \mathbb{E}_{\alpha}(\wh{c}^2_{k,C1(N)}). \end{equation} We will need the following lemma which we prove in Appendix \ref{appcoeffsLem}. \begin{lemma}\label{lemPrBetterCov} For a random variable $X \in L^4$, $\mathbb{E}(X) = 0$, $0<\mathbb{E}(X^2)$, let $Y \sim \mu_X^N$. Let us denote for $l \in I_2$, \begin{equation} \overline{Y^l} = \frac{1}{N}\left(\sum_{i=1}^NY_i^l\right). \end{equation} Then it holds \begin{equation} \mathbb{E}(\overline{Y^2}^2) < \left(\frac{N}{N-1}\right)^2\mathbb{E}((\overline{Y^2} - \overline{Y}^2)^2). \end{equation} \end{lemma} Thus for (\ref{MCleqC1N}) to hold it is sufficient to take $\alpha= (\mu,(f_i)_{i=1}^{n+1}) \in D_c$ such that for $X$ as in the above lemma, for which further $\mathbb{E}(X^2)=1$ (e. g. $\mathbb{P}(X=1)=\mathbb{P}(X=-1)= \frac{1}{2}$), it holds $X \sim \mu$ and $f_{k}(X)=f_{n+1}(X)= X$.
For $m \in \mathbb{N}_+$, let us consider a $Q$ whose restriction to $\mc{T}_m$ is a function of the first $m$ so restricted moments as in Section \ref{secCondMoms} and such that $G_Q$ has degree $r_Q$. We define estimand $PRQ$ on admissible pairs of two distributions and two real-valued functions with sets of arguments' indices $(\{1\},\{1,2\})$, consisting of all possible $\alpha = ((\mu_1,\mu_2), (f_1,f_2))$, such that for $P \sim \mu_1$ and $R \sim \mu_2$ it holds $f_2(P,R) \in L^m$ and for $h_Q$ corresponding to the symmetric unbiased estimator of $G_Q$ in $r_Q$ dimensions as in (\ref{hQ}), for $\wt{R} \sim \mu_R^{r_Q}$ and independent of $P$, it holds \begin{equation}\label{hql2} h_Q(f_2)(P,\wt{R})f_1(P), h_Q(f_2)(P,\wt{R}) \in L^1 \end{equation} (this will be needed for our estimators to be integrable), in which case \begin{equation}
PRQ(\alpha) = \mathbb{E}(f_1(P)Q(f_2(P,R)|P)). \end{equation} We also define estimand $CovQ$ on pairs $\alpha$ as above for which additionally for the above $P$ it holds $f_1(P) \in L^1$, in which case \begin{equation}
CovQ(\alpha)= \Cov(f_1(P),Q(f_2(P,R)|P)). \end{equation} Let the estimand $AveQ$ be defined as in the previous section, but for $N_P=1$, for which it is an estimand on admissible pairs of two distributions and single functions. We define $n+1$ estimands $PRAQ$ as trivial extensions of $n+1$ estimands $(PRQ,\ldots,PRQ,AveQ)$ using the above $\psi$, and $n+1$ estimands $CovAQ$ as trivial extensions of $n+1$ estimands $(CovQ,\ldots,CovQ,AveQ)$ also using $\psi$. We define $n+1$ estimands $bQ$ and $cQ$ whose coordinates are equal to these of coordinates of $PRAQ$ or equivalently of $CovAQ$, restricted to $\alpha=((\mu_1,\mu_2),(f_i)_{i=1}^{n+1})$ such that for $P\sim \mu_1$ and $R\sim\mu_2$, the set $\{f_i(P)\}_{i \in I_{n}}$ is nonzero orthogonal in $L^2$ for $bQ$ or orthonormal for $cQ$, $\mathbb{E}(f_i(P)) = 0$, $i \in I_{n}$, and
$Q(f_{n+1}(P,R)|P) \in L^2$,
so that $bQ_i(\alpha)$ is the coefficient of $f_i'(P) = \frac{f_i(P)}{\Var(f_i(P))}$, $i \in I_n$, and
$bQ_{n+1}(\alpha)$ of $\mathbb{1}$, in the orthogonal projection of $Q(f_{n+1}(P,R)|P)$ onto span($\{f'_i(P)\}_{i=1}^n\cup\{\mathbb{1}\}$).
Let us consider an unbiased estimation scheme $SCovE(N)$ for $CovE$, such scheme $SPRE$ for $PRE$, as well as $SAveE$ and $SAveE(N)$ for $AveE$, which are counterparts of the above schemes $SCov$, $SPR$, $SAve$, and $SAve(N)$ and whose formulas for their respective estimators are analogous as for their counterparts but with $g_{2}[i]$ on the rhs of (\ref{scov}) and (\ref{defspr}) replaced by $g_2[i][i]$ for $SCovE(N)$ and $SPRE$, respectively, and with each $g[i]$ on the rhs of (\ref{save}) and (\ref{saven}) replaced by $g[i][i]$ for $SAveE$ and $SAveE(N)$. The fact that such schemes are unbiased is an easy consequence of Theorem \ref{condexpX} and (\ref{hql2}) (note that $h_E(f_2)(P,R) = f_2(P,R)$).
We also define counterparts of schemes $P1$ and $C1(N)$ - an unbiased estimation scheme $P1E$ for $PRAE$ defined as trivial extensions of $n+1$ schemes $(SPRE,\ldots,SPRE,SAveE)$ and scheme $C1E(N)$ for $CovAE$ as such extensions of $n+1$ schemes $(SCovE(N),\ldots,SCovE(N),SAveE(N))$, both using $\psi$. We define another unbiased estimation scheme $SCov2E$ for $CovE$ given by the formula for estimator \begin{equation}\label{CovC2} \wh{CovE}_{SCov2E} = \frac{1}{2}(g_2[0][0]-g_2[1][0])(g_1[0] - g_1[1]), \end{equation} and a scheme $SAve2E$ for $AveE$ given by formula \begin{equation}\label{Ave2E} \wh{AveE}_{SAve2E} = \frac{1}{2}(g[1][0] + g[0][0]). \end{equation} Scheme $SAve2E$ is a symmetrisation of scheme $SAveE$ in the first argument from one to two dimensions and thus from Theorem \ref{thIneqds}, \begin{equation}\label{dAve2E} d_{AveE,SAveE} \leq d_{AveE,SAve2E} \leq 2d_{AveE, SAveE}. \end{equation}
We define an unbiased estimation scheme $C2E$ for $CovAE$ as trivial extensions of $n+1$ schemes $(SCov2E,\ldots,SCov2E,SAve2E)$ using $\psi$.
Analogously as above for schemes $P1$ and $C1(N)$, by arguments based on Theorem \ref{thtransl} one shows that scheme $P1E$ can have much higher inefficiency constants for estimation of $cE_k$ (and thus also $bE_k$) than schemes $C1E(N)$ and $C2E$ do, and also by an analogous argument as for the former schemes there exist $\alpha \in D_{cE_k}$ such that \begin{equation} d_{cE_k,P1E}(\alpha) < d_{cE_k,C1E(N)}(\alpha). \end{equation}
We will now prove that scheme $C1E(N)$ can have arbitrarily higher inefficiency constant for estimation of $cE_k$ (and thus also $bE_k$ and $CovAE$), $k \in I_n$, than scheme $C2E$, from which it also follows that scheme $SCovE(N)$ can have arbitrarily higher inefficiency constant for estimation of $CovE$ than $SCov2E$. We have the following lemma, the proof of which is given in Appendix \ref{appcoeffsLem}. \begin{lemma}\label{lemC1C2} For some $k \in I_n$, let us consider random variables $P$ and $R$, function $f_P \in L^2(\mu_P)$, and functions $f_{R,l} \in L^2(\mu_R), l\in \mathbb{N}_+$, such that $\lim_{l \to \infty}\Var(f_{R,l}(R)) = \infty$ and for each $l \in \mathbb{N}_+,$ there exists $\alpha_l = ((\mu_P, \mu_R), f_l) \in D_{cE}$, such that $f_{l,k} = f_P$ and $f_{l,n+1}(P,R) = f_P(P) + f_{R,l}(R)$.
Then we have $\lim_{l \to \infty}\mathbb{E}_{\alpha_l}(\wh{cE}_{k,C1E(N)}^2) = \infty$. \end{lemma} For notations as in the above lemma we have from independence of $f_P(P)$ and $f_{R,l}(R)$ and $\mathbb{E}(f_P(P))=0$ that $cE_{k}(\alpha_l)=\mathbb{E}(f_P^2(P))$ and it does not depend on $l$, so that \begin{equation} \lim_{l \to \infty}\Var_{\alpha_l}(\wh{cE}_{k,C1E(N)})=\infty. \end{equation} On the other hand the value of $\Var_{\alpha_l}(\wh{cE}_{k,C2E})$ does not depend on $l$ as the evaluations of $f_{R,l}$ cancel out when evaluating its estimator.
For some $Q$ as above, distributions $\mu_1$, $\mu_2$, and some $m_1, m_2 \in \mathbb{N}_+$, let us define the corresponding independent random vectors with i. i. d. coordinates $\wt{P} \sim \mu_1^{m_1}$ and $\wt{R} \sim \mu_2^{m_2r_Q}$, and denote $\wt{R}_{Q} = ((\wt{R}_{r_Q i+l})_{l=1}^{r_Q})_{i=0}^{m_2-1}$. Analogously as when defining the subschemes of $SQ$ in the previous section, let us define an unbiased estimation scheme $SPRQ$ for $PRQ$ such that for each $\wt{P}$, $\wt{R}$, and $\wt{R}_Q$ corresponding to $m_1=1$, $m_2=1$ and each $((\mu_1,\mu_2),(f_1,f_2)) \in D_{PRQ}$, the estimator given by $SPRQ$ fulfills \begin{equation}\label{gencq} \wh{PRQ}_{SPRQ}(f_1,f_2)(\wt{P},\wt{R}) = \wh{Cov}_{k,SPRE}(f_1,h_Q(f_2))(\wt{P},\wt{R}_Q). \end{equation} We analogously define unbiased scheme $SCovQ(N)$ for $CovQ$ but for $m_1 = m_2=N \in \mathbb{N}_+,$ $N >1$, and using $SCovE$ on the rhs of condition analogous to (\ref{gencq}), scheme $SCov2Q$ for $CovQ$, for $m_1 = 2, m_2=1$, and using $SCov2E$ on the rhs of such condition, and the following schemes for $AveQ$ - scheme $SAveQ$ for $m_1 = m_2 = 1$ and using $SAveE$, $SAveQ(N)$ for $m_1 = m_2 = N$ and using $SAveE(N)$, and $SAve2Q$ for $m_1 =2, m_2 = 1$ and using $SAve2E$ in the condition. The fact that the above defined schemes are unbiased for estimation of $PRQ$ or $CovQ$ is an easy consequence of (\ref{phiqcq}), Theorem \ref{condexpX}, and (\ref{hql2}), while for $AveQ$ it is consequence of (\ref{phiqcq}) and the iterated expectation property. Unbiased schemes $P1Q$ for $PRAQ$, and such schemes $C1Q(N)$ and $C2Q$ for $CovAQ$ are defined as trivial extensions using $\psi$ of $n+1$ schemes $(SPRQ, \ldots, SPRQ, SAveQ)$, $(SCovQ(N), \ldots, SCovQ(N), SAveQ(N))$, and $(SCov2Q, \ldots, SCov2Q, SAve2Q)$, respectively. We have a generalization of inequality of inefficiency constants analogous to (\ref{dAve2E}) and with the same justification \begin{equation}\label{dAve2Q} d_{AveQ,SAveQ} \leq d_{AveQ,SAve2Q} \leq 2d_{AveQ, SAveQ}. \end{equation}
For some $Q$ as above let us now consider a random vector with independent coordinates $P = (P_i)_{i=1}^{N_P}$, $N_P \in \mathbb{N}_+$, $0< \Var(P_i) < \infty$, random variable $R$ independent of $P$, $f$ measurable with $f(P,R) \in L^m$, and $h_Q(f)(P,\wt{R}) \in L^2$ for $\wt{R} \sim \mu_R^{r_Q}$ and independent of $P$(\ref{hql2}).
In our numerical experiments we will be using different schemes defined below for estimation of coefficients of elements of the orthogonal set $\Phi = \{P_i - \mathbb{E}(P_i)\}_{i=1}^N \cup \{\mathbb{1}\}$ and the orthonormal one $\Phi' = \left \{\frac{P_i - \mathbb{E}(P_i)}{\sigma(P_i)}\right\}_{i=1}^{N_P} \cup \{\mathbb{1}\}$ in the orthogonal projection
of $Q(f(P,R)|P)$ onto span($\Phi$) for $Q$ equal to $\mathbb{E}$ and $\Var$. As these schemes are unbiased for estimation of some more general estimands we shall start by introducing them.
Let us define $N_P+1$-dimensional vectors of estimands $\wt{PRAQ}$ and $\wt{CovAQ}$ whose each $i$th coordinate $\wt{\lambda}$ corresponding to such $i$th coordinate $\lambda$ of $CovAQ$ or $PRAQ$, respectively, for $n = N_P$, is such that $\wt{\lambda}$ is defined on all admissible pairs $\wt{\alpha} = ((\mu_{1,1},\ldots,\mu_{1,N_P},\mu_2),(f_i)_{i=1}^{N_P+1})$ of $N_P+1$ distributions and $N_P+1$ functions with sets of arguments' indices $(\{1\},\{2\},\ldots,\{N_P\},I_{N_P+1})$ such that for $\mu_1 = \bigotimes_{i=1}^{N_P} \mu_{1,i}$ and $\pi_i$ denoting projection from $B_{\mu_1}\times\ldots\times B_{\mu_N}$ onto the $i$th coordinate, $i \in I_{N_P}$, it holds $\alpha = ((\mu_1,\mu_2),(f_1(\pi_1),\ldots,f_{N_P}(\pi_{N_P}), f_{N_P+1})) \in D_{\lambda}$, in which case \begin{equation} \label{wtlambda} \wt{\lambda}(\wt{\alpha}) = \lambda(\alpha). \end{equation}
We analogously define estimands $\wt{bQ}$ and $\wt{cQ}$ corresponding to $bQ$ and $cQ$, respectively.
For $\mu_i \sim P_i$, $i \in I_{N_P}$, and $\mu = (\mu_i)_{i=1}^{N_P}$, the coefficients of respective elements
of $\Phi$ as above in the orthogonal
projection of $Q(f(P,R)|P)$ onto span($\Phi$) are equal to the consecutive coordinates of $\wt{bQ}(\alpha)$ for $\alpha = (\mu,(\phi_1, \ldots, \phi_{N_P},f))$, $\phi_i(x) = \frac{x - \mathbb{E}(P_i)}{\Var(P_i)}, i \in I_{N_P}$, and the coefficients of such elements $\Phi'$ in this projection are equal to the coordinates of $\wt{cQ}(\alpha')$ for \begin{equation}\label{alphaprim} \alpha' = (\mu,(\phi_1', \ldots, \phi_{N_P}',f)), \end{equation} where $\phi_i(x)' = \frac{x - \mathbb{E}(P_i)}{\sigma(P_i)}, i \in I_{N_P}$.
We define evaluation vectors $s_{(l)}[i][j]$ and $s_{(l),k}[i][j]$ as $s[i][j]$ and $s_{k}[i][j]$ in Section \ref{secMany} but using $g_l[v_1]\ldots[v_{N_P+1}]$ rather than $g[v_1]\ldots[v_{N_P+1}]$ for the same $v$ on the right hand sides of expressions defining them. We define unbiased estimation schemes $\wt{P1Q}$ for $\wt{PRAQ}$ as well as $\wt{C1Q}(N)$ and $\wt{C2Q}$ for $\wt{CovAQ}$ (and thus all three also unbiased for $\wt{bQ}$ and $\wt{cQ}$) as obvious modifications of the schemes $P1Q,$ $C1Q(N)$, and $C2Q$, respectively, whose formulas for estimators have each occurrence of $g_l[i]$ replaced by $r_{l}[i]$ (see (\ref{ridef})), $l \in I_{N_P}$, and $g_{N_P+1}[i][j]$ by $s_{(N_P+1)}[i][j]$ (see Section \ref{secMany}). For instance for some $i \in I_{N_P}$, the estimator of $CovAQ_i$ given by $\wt{C2Q}$ is \begin{equation} \wh{\wt{CovAQ}}_{i, \wt{C2Q}} = \frac{1}{2}(r_{i}[1]-r_{i}[0])(s_{(N_P + 1)}[1][0]-s_{(N_P + 1)}[0][0]). \end{equation} We shall now introduce a new unbiased estimation scheme $SQCov$ for $(\wt{CovAQ}_i)_{i=1}^{N_P}$, that is the first $N_P$ coordinates of $\wt{CovAQ}$. Let $\phi_{Q,t}$ be the unbiased symmetric estimator of $G_Q$ in $t = 2r_Q$ dimensions, where $r_Q$ denotes the degree of $G_Q$ as in the previous section. For $k \in I_{N_P}$, let \begin{equation}
\wh{\wt{CovAQ}}_{k,0} = \frac{1}{2}(\phi_{Q,t}((s_{(N_P + 1)}[0][j])_{j=1}^{t}) -\phi_{Q,t}((s_{(N_P + 1)}[1][j])_{j=1}^{t}))(r_k[0] - r_k[1])
\end{equation} and introducing a C language-like notation \begin{equation}\label{CNot} (a == b)?c:d = \begin{cases}
c & \text{ if $a=b$,} \\
d & \text{otherwise,} \\
\end{cases} \end{equation} for $l \in I_{N_P}$, let \begin{equation}
\wh{\wt{CovAQ}}_{k,l} = \frac{1}{2}(\phi_{Q,t}((s_{(N_P + 1),l}[0][j])_{j=1}^{t}) -\phi_{Q,t}((s_{(N_P + 1),l}[1][j])_{j=1}^{t}))(r_k[0] - r_k[1])(k==l?-1:1).
\end{equation} The unbiased subscheme of $SQCov$ for estimation of $CovAQ_{k}$, $k \in I_{N_P}$, is given by the formula for estimator \begin{equation} \wh{\wt{CovAQ}}_{k,SQCov} = \frac{1}{N_P + 1}\sum_{l=0}^{N_P}\wh{\wt{CovAQ}}_{k,l}. \end{equation} We define scheme $\wt{SQ}$ as a one consisting of trivial extensions of subschemes from $SQ$ from the previous section, for which coordinates of $\psi$ defining the extensions are equal to $\phi_1:\{1\}\rightarrow \mathbb{N}_+:\phi_1(1) = N_P+1$, and also of subchemes of $SQCov$ for which such coordinates are equal to $\phi_2=\id_{I_{N_P+1}}$. Intuitively, scheme $\wt{SQ}$ is created by adding to $SQCov$ subschemes of $SQ$ applied to the $N_P+1$st function. Such scheme is unbiased for estimation of estimands $\wt{\lambda}$ also created by trivial extensions using the above $\psi$ of the corresponding estimands $\lambda$ of scheme $SQ$ for which coordinates of $\psi$ are $\phi_1$ and estimands $(\wt{CovAQ}_{k})_{k=1}^{N_P}$ for which these coordinates are $\phi_2$. Similarly as in the previous section, let us further add to $\wt{SVar}$ subschemes from $SECov$ for estimation of $\wt{CovAE}_l$, $l \in I_{N_P},$ symmetrised from two to four dimensions in the $N_P +1$st argument.
Let us consider the following set of symmetries in different first $N_P$ arguments in two dimensions $\Pi_1 = \bigcup_{j=1}^{N_P}\Theta_{{N_P+1},j,2}$ and set of symmetries in the $N_P+1$st argument in four dimensions $\Pi_2 = \Theta_{{N_P+1},{N_P+1},4}$ (see definitions below (\ref{defthetam})). Subschemes of $\wt{SVar}$ for estimation of $\wt{CovAE}_l, l \in I_{N_P+1}$, (note that $\wt{CovAE}_{N_P+1} = \wt{Ave}$) are averages of subschemes of $\wt{C2E}$ with respect to $\Pi_1\Pi_2 = \{\pi_1\pi_2:\pi_1\in \Pi_1, \pi_2\in\Pi_2\}$, and they use both individually and together $4(N_P + 1)$ times more evaluation vectors for the last function than the latter, so that we have \begin{equation}\label{ineqCovd} d_{\wt{CovAE}_l,\wt{SVar}} \leq 4(N_P + 1)d_{\wt{CovAE}_l, \wt{C2E}},\quad l \in I_{N_P +1}. \end{equation} Subscheme of $\wt{SVar}$ for estimation of $\wt{Ave}$ uses $8(N_P + 1)$ times more evaluation vectors for the last function than the $N_P+1$st subscheme of $\wt{P1E}$ using one such vector, and it is also an average of the latter with respect to $\Theta_{{N_P+1},I_{N_P},2}\Pi_1\Pi_2$, so that from Lemma \ref{lemcomb} it easily follows that \begin{equation}\label{ineqCovd2} d_{\wt{Ave}, \wt{P1E}} \leq d_{\wt{Ave},\wt{SVar}} \leq 8(N_P + 1)d_{\wt{Ave}, \wt{P1E}}. \end{equation} An estimand corresponding to the
nonlinearity coefficient (\ref{DNJ}) of $Q(f(P,R)|P)$ in $P_k$, $k \in I_{N_P}$, for $Q = \Var, \mathbb{E}$, is defined as \begin{equation} DNQ_k = \wt{VQ}_k^{tot} - \wt{cQ}_k^2 \end{equation} for arguments $\alpha'$ as in (\ref{alphaprim}).
We use for its estimation an unbiased scheme which can be treated as an additional subscheme of a scheme $\wt{SVar}(2)$ using $\wt{SVar}$ in two independent steps, $\wt{SVar}_i = \A_{\pi_i}(\wt{SVar})$ (see Section \ref{secAveSchemes}), $i \in I_2$, and which is given by formula \begin{equation} \wh{DNQ}_{k,\wt{SVar}(2)} = \frac{1}{2}\sum_{i=1}^2\wh{VQ}^{tot}_{k,\wt{SVar}_i} - \Pi_{i=1}^2\wh{cQ}_{k,\wt{SVar}_i}. \end{equation}
For the estimand corresponding to the nonlinearity coefficient (\ref{dnc}) of $Q(f(P,R)|P)$ in all coordinates of $P$, $DNQ = \wt{VQ}_P - \sum_{i=1}^{N_P}\wt{cQ}_i^2$ for $Q = \Var, \mathbb{E}$, we use a scheme given by \begin{equation} \wh{DNQ}_{\wt{SVar}(2)} = \frac{1}{2}\sum_{i=1}^2\wh{VQ}^{tot}_{P,\wt{SVar}_i} - \sum_{k=1}^{N_P}\Pi_{i=1}^2\wh{cQ}_{k,\wt{SVar}_i}. \end{equation} In our numerical experiments the above schemes for nonlinearity coefficients were used to obtain estimates once per each two steps of a MC procedure using scheme $\wt{SVar}$ and thus the final MC estimator was computed by averaging over two times fewer estimates than for subschemes of $\wt{SVar}$.
Correlations between a given function of conditional moments $Q(f(P,R))$ and coordinates of $P$ are $\mathbb{N}_P$ estimands $corrQ=(corrQ_i)_{i=1}^{N_P}$ on common admissible pairs, such that (see (\ref{correqu})) \begin{equation} corrQ_i = \frac{\wt{bQ}_i}{\wt{VQ_P}}, \quad i \in I_{N_P}, \end{equation} is defined on the intersection of domains of the divided estimands. One can compute the estimates of it for $Q=\mathbb{E}$ or $\Var$ e. g. by dividing the final MC estimates of $bQ_i$ and $DQ$ obtained using scheme $\wt{SVar}$ and one can use analogously defined schemes $\wt{SQ'}$ for estimating $corrQ_i$ for other $Q$ and $Q'$. Note that similarly as for schemes for variance-based sensitivity indices of conditional expectation in Section \ref{secMany}, estimation schemes for estimands like $PRA$, $CovA$, $PRE$, $CovE$, and $DNE_k$ can be easily generalized to functions with values in $\mathbb{R}^m$, $m \in \mathbb{N}_+$, by using appropriate scalar product of vectors instead of function multiplication in the formulas for estimators. When this should not cause any misunderstandings, to simplify notations we often drop the tilde sign over the symbols of schemes or estimands introduced in this section, e. g. write $SVar$ instead of $\wt{SVar}$.
\section{\label{secApprErr}Schemes for the mean squared error of approximation}
Let us consider a Hilbert space $H$ with some scalar product $<,>$, inducing norm $|\cdot|$. Let $v \in H$, $\Psi = \{\psi_i\}_{i=1}^l$ be an orthonormal set in $H$, $V = \text{span}(\Psi)$, $P_{V}$ be an orthogonal projection from $H$ onto $V$, and $c=(c_i)_{i=1}^l$ be the Fourier's coefficients of $v$ relative to $\Psi$, that is \begin{equation} P_V(v) = \sum_{i=1}^lc_i\psi_i. \end{equation}
For example we can have $H= L^2$, with scalar product (\ref{scall2}), $v= Q(f(P,R)|P)$ for $f(P,R)$ being some construction of an output of an MR and $Q$ being a function which restricted to $\mc{T}_m$ is a function of the first $m$ so restricted moments as in Section \ref{secCondMoms}, and $\psi_i = \phi_i(P)$ for some functions $\phi_i$, $i \in I_l$, orthonormal in $L^2(\mu_P)$. Let $h=(h_i)_{i=1}^l \in \mathbb{R}^l$. Squared error of the approximation of $v$ using $\Psi\cdot h = \sum_{i=1}^{l}h_i\psi_i$ in $H$, denoted as $\err(h)$, fulfills \begin{equation}\label{errh} \begin{split}
\err(h) &= |v - \Psi\cdot h|^2 \\
&= |P_V(v) - \Psi\cdot h|^2 + |v - P_V(v)|^2, \end{split} \end{equation} where in the second equality we used the fact that $v - P_V(v)$ is orthogonal to $V$ and in the last the fact that $\Psi$ is orthonormal. Let us consider an unbiased estimator $w = (w_i)_{i=1}^l$ of $c$ for some distribution $\nu$ so that $\mathbb{E}_\nu(w)=c$ and let us further assume that $w_i \in L^2(\nu), i \in I_n$. For instance for the above example, some unbiased estimation scheme $\kappa = (\kappa_i)_{i=1}^{l}$, $l >1$, for $cQ$ for $n = l-1$ as in the previous section, $\mc{V}=D_{cQ}$, and for $(\mu,g) \in \mc{V}$ such that $\mu=(\mu_P,\mu_R)$ and $g =(\phi_1,\ldots,\phi_n,f)$, we can take $\nu = \mu^{p_{\kappa}}$ and $w = \phi_{\kappa,\mc{V}}(g)$ (see (\ref{phikappamany})).
From (\ref{errh}), the average squared error of approximation of $v$ using estimates of $c$ given by $w(X)$, $X \sim \nu$, fulfills
\begin{equation} \begin{split}
\mathbb{E}(\err(w(X))) &= \sum_{i=1}^l\Var(w_i(X)) + |v - P_V(v)|^2 \\
&= \Var(w(X)) + |v - P_V(v)|^2, \end{split} \end{equation} where by variance in the last term we mean variance for random vectors defined as in Section \ref{secOrthog} using the standard scalar product in $\mathbb{R}^l$. Since for a fixed $v$ and orthonormal set
$\Psi$, $|v - P_V(v)|$ is constant, we get lower mean squared approximation error when using estimator of orthogonal projection coefficients onto $\Psi$ with lower sum of variances of its coordinates. Thus standard scalar product is here a natural choice for defining variance used to quantify error of approximation of $c$ by $w$ for $\nu$.
We define an estimand $ErrQ$ on all admissible pairs $\alpha = (\mu,s)=((\mu_1,\mu_2), (s_1,s_2))$ of two distributions and two functions with sets of arguments' indices $(\{1,2\},\{1\})$, such that for each $P \sim \mu_1$ and $R \sim \mu_2$, it holds $s_1(P,R) \in L^m$, $s_2(P) \in L^2$, and for $\wt{R} \sim \mu_2^{r_Q}$ and independent of $P$, it holds $h_Q(s_1)(P,\wt{R})\in L^2$, in which case \begin{equation}
ErrQ(\alpha) = \mathbb{E}((Q(s_1(P,R)|P) - s_2(P))^2). \end{equation}
For the above discussed example in which $v= Q(f(P,R)|P)$, for $s_1 = f$, $h \in \mathbb{R}^l$, and $s_2(P) = \sum_{i=1}^lh_i\phi_i(P)$, $ErrQ(\alpha)$ is equal to $\err(h)$. Let us consider an unbiased estimation scheme $SErrE$ for $ErrE$, defined by formula \begin{equation} \wh{ErrE}_{SErrE} = (g_1[0][0] - g_2[0])(g_1[0][1] - g_2[0]). \end{equation} The fact that it is unbiased follows from formula (\ref{ggxy}) in Theorem \ref{thCond} since for $\alpha = (\mu,s)\in D_{ErrE}$, $X \sim \mu^{p_{SErrE}}$, and $p(x_1,x_2) = s_1(x_1,x_2) -s_2(x_1)$, we have \begin{equation} \begin{split} \wh{ErrE}_{SErrE}(f)(X) &= \mathbb{E}(p(X_1,X_{2}[0])p(X_1,X_{2}[1])) \\
&=\mathbb{E}((\mathbb{E}(p(X_1,X_{2}[0])|X_1))^2)\\
&= \mathbb{E}((\mathbb{E}(s_1(X_1,X_{2}[0])|X_1) - s_2(X_1))^2). \end{split} \end{equation} Analogously in Section \ref{secPolynEst}, we define scheme $SErrQ$ giving an unbiased estimator of $ErrQ$ such that for $(\mu,s) \in D_{ErrQ}$, $P\sim\mu_1$, $R\sim\mu_2$, and $\wt{R}\sim \mu_2^{2r_Q}$ and independent of $P$, it holds
\begin{equation}
\widehat{ErrQ}_{SErrQ}(s)(P,\wt{R}) = \frac{1}{|\Psi_Q|} \sum_{\pi \in \Psi_Q}\widehat{ErrE}_{SErr}(h_Q(s_1),s_2)(P, \wt{R}_{\pi}), \end{equation} and we add to scheme $SErrVar$ subschemes for estimation of $ErrE$ which are created by symmetrisation of subschemes from $SErrE$ from $2r_{E} = 2$ to $2r_{Var} = 4$ dimensions in the second argument.
\section{\label{secVarDiffNew}Variances of the new estimators for the RTC and GD methods} In our numerical experiments which we describe in the further sections, the estimates of variances of various unbiased estimators of main and total sensitivity indices of conditional variance from Section \ref{secPolynEst} as well as such estimators of orthogonal projection coefficients of conditional variance and expectation onto the span of model parameters and constants significantly depended on whether the GD or the RTC method was used and on the order of reactions in the GD method. Using notations as in Section \ref{secVarDiff}, for $s$ such that $\alpha = ((\mu_P,\mu_R),(s,f)) \in D_{CovE}$ and
$\wt{P}\sim \mu_P^2$ and independent of $R$, it holds \begin{equation} \begin{split} 4\mathbb{E}_{\alpha}(\wh{CovE}^2_{SCov2E}) &= \mathbb{E}((f(\wt{P}[0],R)- f(\wt{P}[1],R))^2(s(\wt{P}[0]) - s(\wt{P}[1]))^2)\\ &= \mathbb{E}(\msd(\wt{P}[0],\wt{P}[1])(s(\wt{P}[0]) - s(\wt{P}[1]))^2), \\ \end{split} \end{equation} so the inequalities between the variances $\Var_\alpha(\wh{CovE}_{SCov2E})$ should also be the same as for $\msd(p_1, p_2)$ for all appropriate $p_1$, $p_2$ as in Section \ref{secVarDiff} depending on the method used. By an analogous argument the same applies to the variances of estimators $\wh{CovAE}_{k,C2E}$,
$k \in I_n$ for appropriate admissible pairs. For some $l, n, m \in \mathbb{N}_+$, let us now consider two functions $\phi: I_l\rightarrow I_n$ and $\psi: I_l\rightarrow I_m$, a random vector with not necessarily independent coordinates $P' = (P'_i)_{i=0}^{n}, P'_i \sim P, i \in I_n$, and a random vector $\wt{R} \sim \mu_R^m$, independent of $P'$. We have \begin{equation} \mathbb{E}((\sum_{i=1}^{l}f(P_{\phi(i)}',R_{\psi(i)}))^2) = \sum_{i, j \in I_l}\mathbb{E}(f(P_{\phi(i)}',R_{\psi(i)})f(P_{\phi(j)}', R_{\psi(j)})). \end{equation} By a proof similar as of Theorem \ref{thCond}, for $\psi(i)\neq\psi(j)$, it holds \begin{equation}\label{efpij}
\mathbb{E}(f(P_{\phi(i)}',R_{\psi(i)})f(P_{\phi(j)}', R_{\psi(j)})) = \mathbb{E}(\mathbb{E}(f(P_{\phi(i)}',R)|P_{\phi(i)}')\mathbb{E}(f(P_{\phi(j)}',R)|P_{\phi(j)}')), \end{equation} which, given $f(P,R)=g(h(P,R))$, is determined by the distribution of $(P_{\phi(i)}',P_{\phi(j)}')$, $g$, and the reaction network $RN$ used in the definition of MR, and thus its value should not depend on the construction of MR being used. For $\psi(i)=\psi(j)$, \begin{equation} \mathbb{E}(f(P_{\phi(i)}',R_{\psi(i)})f(P_{\phi(j)}', R_{\psi(j)}) = \mathbb{E}(f(P,R)^2) - \frac{1}{2}\msd(P_{\phi(i)}',P_{\phi(j)}'). \end{equation} From the above calculations it easily follows that the inequality between the variances of estimators like $\wh{AveE}_{SAve2E}$, $\wh{CovAE}_{n+1,C2E}$, and $\wh{Ave}_{\wt{SVar}},$ for the appropriate admissible pairs corresponding to $f$, $P$, $R$, for different constructions should be opposite than the inequalities between $\msd(p_1,p_2)$ for all $p_1$ and $p_2$ as in Section \ref{secVarDiff}. In our numerical experiments discussed in the next section the estimates of variances of estimators $\wh{cE}_{k,C2E}$ were often much lower and of estimators $\wh{CovAE}_{n+1,C2E}$ and $\wh{Ave}_{\wt{SVar}}$ higher when using the RTC than the GD method. However, for the MBMD model, as discussed in Section \ref{secMBMD}, some estimates of variance of $\wh{cE}_{k,C2E}$ in our experiments were statistically significantly higher while of $\wh{CovAE}_{n+1,C2E}$ (denoted there as $\wh{Ave}_{C2E}$) smaller for the RTC than the GD method, from which it follows that for this model and its output, similarly as for the ones from Section E of \cite{Anderson2013}, there exist parameters $p_1$, $p_2$, for which $\msd(p_1,p_2)$ is higher for the RTC than the GD method.
\section{Numerical experiments}\label{chapNumExp} \subsection{Implementation extensions and tests of validity of the inefficiency constants of schemes}\label{secImpl} The numerical experiments in this work were run using the same hardware and operating system as described in Section \ref{secExSoft}. The program described in that section was extended by adding implementations of MC procedures using the new estimators described in sections \ref{secPolynEst}, \ref{secApprCoeff}, and \ref{secApprErr}. Figure \ref{pic1} describes basic specification process and the corresponding results of computations with our extended program. We carried out a numerical experiment comparing the average execution times of MC procedures using schemes $\wt{SVar}$, $SE$, $C1E$, and $C2E$, and the same number of simulations of the RTC or GD methods for the outputs of the SB, GTS, and MBMD models defined in sections \ref{SBPrev}, \ref{GTSPrev}, and \ref{MBMDPrev}. For $N_P$ denoting the number of parameters of a given model, $k=50$ for the GTS model and $k=500$ for the other models, for each model we performed a $50$-step MC procedure measuring in each step the execution time of $k$ MC-steps using scheme $\wt{SVar}$, $2k$ MC-steps of scheme $SE$, one MC-step of $\wt{C1E}(8(N_P+1)k)$, and $4(N_P +1)k$ MC-steps of scheme $\wt{C2E}$. The computed mean execution times are presented in Table \ref{tabTimeDiff}. From the table we can see that the mean execution times of our implementations of the procedures using different schemes and simulation methods and a given model for the same number of process simulations are comparable. For this reason and to make our analysis independent of the implementation or computer architecture used, rather than comparing the estimates of inefficiency constants of sequences of MC procedures, in our numerical experiments we shall focus on comparing the estimates of variances of the final MC estimators for the same number of process simulations carried out in the MC procedures, the ratio of such variances being equal to the ratio of appropriate inefficiency constants of the schemes used, as discussed in sections \ref{secIneffSchemes} and \ref{secineffgen}. \begin{figure}
\caption{ Diagram describing the basic specification process and the corresponding results of computations carried out with our program. See Section \ref{secExSoft} for details on specification of MR and sections \ref{secMany}, \ref{secPolynEst}, \ref{secApprCoeff}, and \ref{secApprErr} for definitions of the above schemes and estimands.}
\label{pic1}
\end{figure}
\begin{table}[h]
\begin{tabular}{|l|c|c|c|c|} \hline {\multirow{2}{*}{MR}} & $SE$ & $SVar$ & $C1E$& $C2E$\\ \cline{2-5}
& \multicolumn{4}{c|}{RTC} \\ \cline{2-5} \hline SB &$1.8845 \pm 0.0014$&$1.9893 \pm 0.0020$&$2.1302 \pm 0.0015$&$1.9872 \pm 0.0015$\\ GTS &$2.616 \pm 0.014$&$2.612 \pm 0.010$&$2.6934 \pm 0.0045$&$2.6419 \pm 0.0058$\\ MBMD &$2.1798 \pm 0.0068$&$2.5835 \pm 0.0048$&$2.2885 \pm 0.0039$&$2.2357 \pm 0.0049$\\
\hline MR & \multicolumn{4}{c|}{GD} \\ \hline SB &$1.9782 \pm 0.0029$&$2.1013 \pm 0.0023$&$2.4037 \pm 0.0028$&$2.1594 \pm 0.0028$\\ GTS & $2.734 \pm 0.010$&$2.705 \pm 0.016$&$2.8295 \pm 0.0070$&$2.7709 \pm 0.0079$\\ MBMD & $2.1564 \pm 0.0090$&$2.6173 \pm 0.0088$&$2.2450 \pm 0.0081$&$2.2058 \pm 0.0082$\\
\hline \end{tabular}
\caption{\label{tabTimeDiff} Estimates of mean execution times in seconds computed from 50 runs of the MC procedures involving the same number of simulations of the GD or RTC constructions for each model and using different schemes as explained in Section \ref{secImpl}.} \end{table}
\subsection{SB model}\label{secSB} Let us consider the SB model and its output from Section \ref{SBPrev}. See \cite{badowski2011} and Appendix \ref{appd} for derivation of some analytical expressions for the sensitivity indices and orthogonal projection coefficients in this model. Some values obtained from these expressions are presented in Table \ref{exactVal} and the main Sobol's indices of conditional expectation and variance are also shown on pie charts in Figure \ref{pieSB}.
For computations with this model we used only the RTC method since for a reaction network with one reaction there is no difference in variance of our estimators using the GD and RTC methods. We performed a one-million-step MC procedure using scheme $SVar$. The computed sensitivity indices, orthogonal projection coefficients, and nonlinearity coefficients are presented in Table \ref{tabSBRTC}, while the mean value and average variance of the model output are given in Table \ref{tabDisp}. The results of computations are in good agreement with the analytically computed values in Table \ref{exactVal} and Appendix \ref{appd}.
We performed a ten-million-step MC procedure using scheme $SErrVar$ to estimate the mean squared error of approximation of the conditional expectation and conditional variance of the output using linear combinations of elements of the set of centered model parameters and constant one, that is the set $\Phi = \{P_i - \mathbb{E}(P_i)\}_{i=1}^{N_P}\cup\{\mathbb{1}\}$,
using as coefficients the estimates of
$(bE_1, \ldots, bE_{N_P}, Ave)$ from Tables \ref{tabSBRTC} and \ref{tabDisp} when approximating the conditional expectation, and
estimates of $(bVar_1, \ldots, bVar_{N_P}, AveVar)$ from these tables when approximating the conditional variance. We obtained estimates of mean squared approximation error $0.027 \pm 0.022$ for the conditional expectation and $1 \pm 15$ for the variance, both being in good agreement with the values of these errors we computed analytically, approximately equal to $0.00051$ and $0.0062$, respectively. We also performed a numerical experiment comparing the estimates of variances of the final MC estimators of different indices using scheme $SVar$ in $25000$ steps, scheme $SE$ in $50000$, and schemes $EM$ and $ET$ in $100000$ steps, so that each above MC procedure used the same number of one million process simulations. We ran each above procedure five times collecting in each run the estimate of variance of the final MC estimator (\ref{varEst}), and finally computing the estimate of mean and standard deviation of the estimates of variances as described in Appendix \ref{appStatMC}. The results are presented in Table \ref{tabCompareMB} and in Figure \ref{barSBV2}, from which we can see that the estimates of variances of the final MC estimators given by scheme $EM$ are approximately two times lower than for scheme $SE$ and four times lower than for scheme $SVar$ for the main sensitivity indices of all parameters except $K3$ and analogously for schemes $ET$, $SE$, and $SVar$ for the total sensitivity indices of these parameters. Such proportions correspond to equalities in the rhs inequalities of relations (\ref{EMComp}), (\ref{ineqDVar}), and (\ref{compVkVar}) for the main as well as in the relations (\ref{VitotComp}), (\ref{ineqDVar}), and (\ref{compVkTotVar}) for the total sensitivity indices of conditional expectation. For $\Phi' = \{\frac{P_i - \mathbb{E}(P_i)}{\sigma(P_i)}\}_{i=1}^{N_P}\cup\{\mathbb{1}\}$, that is the set of normalized centred parameters and constant one, we also performed a numerical experiment comparing the variances of MC methods estimating the coefficients of orthogonal projection of the conditional expectation and conditional variance onto span($\Phi'$). We used scheme $SVar$ in $k = 100$ steps for estimating the orthogonal projection coefficients of both conditional expectation and conditional variance. For $N_P = 4$ denoting the number of parameters, we also carried out MC procedures using $4(N_P+1)k$ steps of scheme $C2E$, $8(N_P+1)k$ steps of $P1E$, and a single step of $C1E(8(N_P+1)k)$ for the conditional expectation. For the conditional variance we applied besides scheme $SVar$ also $4(N_P+1)k$ steps of scheme $P1Var$, single step of $C1Var(4(N_P+1)k)$, and $2(N_P+1)k$ steps of $C2Var$. The same number of $4000$ process evaluations was used in each above method. We performed a $200$ step procedure to compute the mean variances of the final MC estimators. For schemes $C1E$ and $C1Var$, the variance in each step was computed using unbiased estimator of variance computed from a sample of ten runs of the method, while for the other methods this was an estimate of variance of the mean computed in the method using estimator (\ref{phiAveVar}). Let $\Sigma Q$ for $Q = Var,E$ be defined as a sum of variances
of the final MC estimators of all the coefficients of orthogonal projection of $Q(f(P,R)|P)$ onto span($\Phi'$) given by certain scheme. From discussion in Section \ref{secApprErr}, using the coefficients computed with a scheme with lower value of $\Sigma Q$ should
lead to lower average error of approximation of $Q(f(P,R)|P)$. Furthermore, from discussion in sections \ref{secineffgen} and \ref{secApprErr} the ratio of values of $\Sigma Q$ when using different schemes and the same number $l$ of process simulations is equal to the ratio of inefficiency constants of these schemes for estimating the vector of projection coefficients, with variances in definitions of the constants being given by standard scalar product. In each of the above $200$ steps we also obtained estimates of $\Sigma Q$ for different schemes by summing the estimates of variances of the estimators of the coefficients and then computed the mean from all steps. The results of the above numerical experiment are presented in Table \ref{tabCovSB} and Figure \ref{figSBBar}. From the table we can see that $\Sigma E$ is lowest for scheme $C1E$, followed by $C2E$, $SVar$, and $P1E$, while $\Sigma Var$ is lowest for $SVar$ followed by $P1Var$, $C2Var$, and $C1Var$. The reader can easily confirm that the results in Table \ref{tabCovSB} are in good agreement with various inequalities between variances of estimators of orthogonal projection coefficients given in Section \ref{secApprCoeff}.
\begin{table}[h]
\begin{tabular}{|l|c|c|c|c|c|} \hline $i $ & $\widetilde{V}_i$ & $\widetilde{V}_i^{tot} $ &$\widetilde{S}_i$ &$\widetilde{S}_i^{tot}$ & $bE_i$\\ \hline $C $ &$ 310 $& $310$ &$0.451$&$0.451$ & $ 1 $\\ $K_1 $ &$ 300 $& $300$ & $0.436 $ & $0.436 $&$ 100$\\ $K_2 $ &$ 75 $& $75$ & $0.109 $ & $0.109 $&$ 100 $\\ $K_3 $ &$ 3 $& $3$ & $0.0044 $ & $0.0044 $&$ 100 $\\ \hline $i$& $V_i$ & $V_i^{tot} $ & $S_i$ & $S_i^{tot}$ &\multicolumn{1}{c}{} \\ \cline{1-5} $P $ & $ 688 $& & $0.80$ & &\multicolumn{1}{c}{} \\ $R $ & &$170$ & &$0.20$ &\multicolumn{1}{c}{}\\ $P,R $ &$ 858 $& $ 858 $ & $1$ & $0$ &\multicolumn{1}{c}{}\\ \hline $i$ & $VVar_i$ &$VVar^{tot}_i$ & $SVar_i$ &$SVar_i^{tot}$ & $bVar_i$\\ \hline $C $& $0$ & $0$ &$0$ & $0$ & $0$ \\ $K_1 $&$300$ & $300$ & $0.794$&$0.794$&$ 100 $\\ $K_2 $&$75$& $75$& $0.198 $& $0.198 $ &$ 100$\\ $K_3 $&$3$& $3$&$ 0.00794$&$0.00794$ & $ 100$\\ \hline $K_3 $&$3$& $3$&$ 0.00794$&$0.00794$ & $ 100$\\ \hline
$P $ & $ 378 $& $1$& $1$ & $1$&\multicolumn{1}{c}{} \\ \cline{1-5}
\end{tabular} \caption{\label{exactVal}Values of sensitivity indices and orthogonal projection coefficients onto model parameters of conditional expectation and variance in the SB model, obtained using analytical expressions
from Appendix \ref{appd} and \cite{badowski2011}.} \end{table}
\begin{figure}
\caption{The proportion of the total arc length occupied by a sector of a pie chart is equal to the main Sobol's sensitivity index of conditional expectation in (a) or conditional variance in (b) of the output of the SB model with respect to its given parameter.}
\label{pieSB}
\end{figure}
\begin{table}[h]
\begin{tabular}{|l|c|c|c|c|c|c|} \hline $i $ & $\widetilde{V}_i$ & $\widetilde{V}_i^{tot} $ &$\widetilde{S}_i$ &$\widetilde{S}_i^{tot}$& $bE_i$ & $ DNE_i$\\ \hline $C $& $309.83 \pm 0.37$ & $309.83 \pm 0.37$ &$ 0.45 $&$ 0.45$& $0.9994 \pm 0.0012$ & $-0.45 \pm 0.72$ \\ $K1 $& $299.80 \pm 0.36$ & $299.80 \pm 0.36$ &$ 0.44 $&$ 0.44$& $99.93 \pm 0.12$ & $-0.05 \pm 0.70$ \\ $K2 $& $74.895 \pm 0.091$ & $74.895 \pm 0.091$ &$ 0.11 $&$ 0.11$& $99.83 \pm 0.13$ & $-0.16 \pm 0.18$ \\ $K3 $& $2.9991 \pm 0.0039$ & $2.9999 \pm 0.0039$ &$ 0.0044 $&$ 0.0044$& $100.15 \pm 0.33$ & $-0.0173 \pm 0.0073$ \\ \hline $i $ & $V_i$ & $V_i^{tot} $ & $S_i$ & $S_i^{tot}$ &$ DNE $& $-0.27 \pm 0.56$\\ \hline $P $& $687.43 \pm 0.54$ & $698.59 \pm 0.55$ &$ 0.8 $&$ 0.81$&\multicolumn{2}{c}{}\\ $R $& $158.79 \pm 0.13$ & $169.95 \pm 0.13$ &$ 0.19 $&$ 0.2$& \multicolumn{2}{c}{} \\ $P, R$& $857.38 \pm 0.56$ & $857.38 \pm 0.56$ &$ 1 $&$ 1$ & \multicolumn{2}{c}{} \\ \hline $i$ & $VVar_i$ & $VVar^{tot}_i$ & $SVar_i$ &$SVar_i^{tot}$& $bVar_i$ & $DNVar_i$\\ \hline $C $& $0$ & $0$ &$ 0 $&$ 0$& $-7.2 \pm 6.5\cdot 10^{-4}$ & $0$ \\ $K1 $& $301.5 \pm 5.6$ & $301.1 \pm 5.6$ &$ 0.79 $&$ 0.79$& $99.62 \pm 0.36$ & $2.9 \pm 7.8$ \\ $K2 $& $76.2 \pm 2.4$ & $75.5 \pm 2.4$ &$ 0.2 $&$ 0.2$& $100.34 \pm 0.44$ & $0.7 \pm 3.1$ \\ $K3 $& $3.27 \pm 0.42$ & $3.08 \pm 0.43$ &$ 0.0086 $&$ 0.0081$& $101.85 \pm 0.90$ & $-0.08 \pm 0.53$ \\ \cline{6-7} $P$& $380.7 \pm 6.1$ & $380.7 \pm 6.1$ &$ 1 $&$ 1$& $DNVar $ & $4.1 \pm 4.3$ \\ \hline
\end{tabular} \caption{\label{tabSBRTC} Estimates of various indices and coefficients
for the SB model computed in a one-million-step MC procedure using the RTC algorithm and scheme $SVar$.} \end{table}
\begin{table}[h]
\begin{tabular}{|l|c|c|c|c|} \hline {\multirow{2}{*}{i}}
&$SE$ & $EM$ & $ET$ & $SVar$ \\ \cline{2-5}
& \multicolumn{4}{c|}{$V_i$ } \\ \cline{2-5} \hline $C$ &$2.6922 \pm 0.0061$ & $1.3437 \pm 0.0019$ & &$5.356 \pm 0.024$\\ $K_1$ &$2.696 \pm 0.012$ & $1.3687 \pm 0.0053$ & &$5.221 \pm 0.023$\\ $K_2$ &$0.1754 \pm 0.0016$ & $0.09212 \pm 0.00037$ & &$0.33392 \pm 0.00043$\\ $K_3$ &$3.839 \pm 0.037\cdot 10^{-4}$ &$2.490 \pm 0.010\cdot 10^{-4}$ & &$6.279 \pm 0.024\cdot 10^{-4}$\\ \hline
$i$ & \multicolumn{4}{c|}{$\widetilde{V}^{tot}_i$ } \\ \hline $C$ &$2.6922 \pm 0.0061$ & &$1.3474 \pm 0.0039$ &$5.356 \pm 0.024$\\ $K_1$ &$2.696 \pm 0.012$ & &$1.3684 \pm 0.0030$ &$5.221 \pm 0.024$\\ $K_2$ &$0.1754 \pm 0.0016$ & &$0.09266 \pm 0.00016$ &$0.33393 \pm 0.00046$\\ $K_3$ &$3.842 \pm 0.033\cdot 10^{-4}$ & & $2.498 \pm 0.012\cdot 10^{-4}$ &$6.276 \pm 0.031\cdot 10^{-4}$\\
\hline \end{tabular} \caption{\label{tabCompareMB} Estimates of variances of the final MC estimators of the sensitivity indices of conditional expectation
using different schemes and the SB model.} \end{table}
\begin{figure}
\caption{Chart (a) shows the ratios of estimates of variances of estimators of the main sensitivity indices given by schemes $EM$ and $SE$ to the estimate of variance of such estimator given by scheme $SVar$ for the SB model output, and chart (b) of estimates of variances of estimators of the total sensitivity indices given by schemes $ET$ and $SE$ to such estimate for scheme $SVar$. See Section \ref{secSB} for details.}
\label{barSBV2}
\end{figure}
\begin{table}[h]
\begin{tabular}{|l|c|c|c|c|} \hline {\multirow{2}{*}{i}}
& \multicolumn{4}{c|}{$cE_i$ } \\ \cline{2-5}
& $P1E$ & $C1E$ & $C2E$ & $SVar$ \\
\hline $ C $ &$13.414 \pm 0.014$ &$0.2039 \pm 0.0073$ & $0.4107 \pm 0.0015$ & $4.473 \pm 0.065$\\ $ K1 $ &$13.417 \pm 0.014$ &$0.1913 \pm 0.0068$ & $0.4118 \pm 0.0015$ & $4.366 \pm 0.060$\\ $ K2 $ &$13.456 \pm 0.014$ &$0.2158 \pm 0.0079$ & $0.3635 \pm 0.0014$ & $1.318 \pm 0.022$\\ $ K3 $ &$13.436 \pm 0.015$ &$0.2198 \pm 0.0075$ & $0.3491 \pm 0.0013$ & $0.3232 \pm 0.0057$\\ $Ave$ &$2.1403 \pm 0.0033\cdot 10^{-1}$ &$0.1930 \pm 0.0064$ & $2.5423 \pm 0.0055\cdot 10^{-1}$ & $3.851 \pm 0.034$\\ $\Sigma E$ &$53.936 \pm 0.033$ &$1.024 \pm 0.016$ & $1.7892 \pm 0.0043$ & $14.33 \pm 0.11$\\ \hline {\multirow{2}{*}{i}}
& \multicolumn{4}{c|}{$cVar_i$ } \\ \cline{2-5} & $P1Var$ & $C1Var$ & $C2Var$ & $SVar$ \\ \hline $ C $ &$232.2 \pm 1.3$ &$4.60 \pm 0.15\cdot 10^{4}$ &$279.1 \pm 2.0$ &$1.308 \pm 0.038$\\ $ K1 $ &$145.06 \pm 0.94$ &$2.753 \pm 0.095$ &$137.8 \pm 1.2$ &$38.1 \pm 1.1$\\ $ K2 $ &$144.96 \pm 0.86$ &$0.669 \pm 0.023$ &$134.1 \pm 1.1$ &$13.88 \pm 0.43$\\ $ K3 $ &$144.48 \pm 0.91$ &$2.820 \pm 0.092\cdot 10^{-2}$ &$133.7 \pm 1.1$ &$2.443 \pm 0.076$\\ $AV$ &$92.01 \pm 0.44$ &$90.1 \pm 2.8$ &$116.94 \pm 0.95$ &$184.6 \pm 3.0$\\ $\Sigma V$ &$758.7 \pm 3.8$ &$4.61 \pm 0.15\cdot 10^{4}$ &$801.7 \pm 4.5$ &$240.3 \pm 3.4$\\ \hline \end{tabular} \caption{\label{tabCovSB} Estimates of variances of the final MC estimators of orthogonal projection coefficients of conditional expectation and conditional variance for the SB model as explained in Section \ref{secSB}. $AV$ is an abbreviation for $AveVar$, and $\Sigma V$ for $\Sigma Var$.} \end{table}
\begin{figure}
\caption{Estimates for the SB model of quantities $\Sigma E$ in chart (a) and $\Sigma Var$ in chart (b) for different estimation schemes
of orthogonal projection coefficients of conditional expectation in (a) and variance in (b) for the computations described in Section \ref{secSB}.}
\label{figSBBar}
\end{figure}
\begin{table}[h]
\begin{tabular}{|l|c|c|}
\hline
MR& Ave & AveVar \\
\hline
SB & $230.004 \pm 0.020$ & $169.95 \pm 0.13$ \\
MBMD& $10.0298 \pm 0.0019$ & $7.1035 \pm 0.0051$ \\
GTS & $30.244 \pm 0.021$ & $370.05 \pm 0.40$\\
\hline
\end{tabular}
\caption{\label{tabDisp} Estimates of the means and average variances
of outputs of the SB and MBMD models computed in a one-million-step and of the GTS model in a $250000$-step MC procedure using scheme $SVar$ and the RTC algorithm.}
\end{table}
\subsection{GTS model}\label{secGTS}
Let us consider the GTS model and its output from Section \ref{GTSPrev}. We performed a $250000$ step MC procedure using scheme $SVar$ and the RTC method. The estimates of different coefficients and sensitivity indices obtained in this procedure are given in Table \ref{tabGTS} and figures \ref{pieGTS} and \ref{barGTSVVtot}. The estimates of mean and average variance of the output from the procedure are given in Table \ref{tabDisp}. Note that the sum of Sobol's interaction indices, equal to the proportion of total arc lenght of each pie chart occupied by the empty sector in Figure \ref{pieGTS} (see also formula \ref{sumSob}), is much higher for the conditional variance than the conditional expectation. From Table \ref{tabGTS} and Figure \ref{barGTSVVtot} we can also see that the total sensitivity indices of conditional variance are significantly higher than the main sensitivity indices, especially for the parameters $\gamma$ and $\alpha_2$, and that the order of the parameters with respect to the total indices of conditional variance is different than with respect to its main indices.
We performed a $2.5$-million-step MC procedure using scheme $SErrVar$ to estimate the mean squared error of approximation of conditional expectation and variance of the output using linear combinations of centered parameters and constant one as in the previous section, taking as coefficients the estimates of $bE_i$ and $bVar_i$ from Table \ref{tabGTSComp} and estimates of mean and mean variance from Table \ref{tabDisp}. We obtained estimates of error for conditional expectation $1.779 \pm 0.100$ and for variance $2.27 \pm 0.14\cdot 10^{3}$, both being significantly higher than zero and not significantly different from the estimates of squares of the best possible linear approximation errors, equal to the values of $DNE$ and $DNVar$ given in Table \ref{tabGTS}. In Table \ref{tabGTSComp} we present estimates of variances of the final MC estimators of the procedures using the RTC and the GD methods and $1000$ steps of scheme $SVar$, $2000$ of $SE$, and $4000$ of $EM$ and $ET$, so that the variances are computed for the same number of process simulations used by the schemes. The mean estimates of variances for each method were computed analogously as in the previous section, except that fifty rather than five runs of each procedure were carried out to compute the means and standard deviations. Note that the estimates of variances of estimators from scheme $SVar$ for estimation of some main sensitivity indices in Table \ref{tabGTSComp} are significantly lower than these of the subschemes of scheme $EM$ introduced in \cite{badowski2011} and analogously for the total sensitivity indices and scheme $ET$. For instance the estimate of variance of the total sensitivity index with respect to the parameter $\beta$ computed using scheme $SVar$ is about $2.45$ times lower than the one from scheme $ET$, both using the GD method, which is not far from the theoretical bound of $3$ corresponding to equality in the lhs of relation (\ref{compVkTotVar}). From Table \ref{tabGTSComp} we can also see that the estimates of variances of estimators from scheme $SVar$ are lower for the RTC than the GD method for all the main and total indices of the conditional expectation. They are even over $4$ times lower for the total and main sensitivity index with respect to the parameter $\beta$.
We carried out a numerical experiment comparing the variances of estimation schemes for orthogonal projection coefficients which was analogous as in the previous section, except that here for schemes $C2E$, $C2Var$, and $SVar$ we tested the GD and RTC methods separately. The results are presented in Table \ref{tabCovGTS} and values of $\Sigma E$ and $\Sigma Var$ also in Figure \ref{figGTSBar}. From Table \ref{tabCovGTS} we can see that the estimates of variances of orthogonal projection coefficients onto normalized centred parameters computed with the use of schemes $C2E$ and $SVar$ for conditional expectation and $C2Var$ for conditional variance are significantly lower for the RTC than the GD method for most coefficients. On the other hand, these variances are higher for the RTC than the GD method for all the coefficients of projections onto constant one (that is the averages $AveE$ and $AveVar$), for these schemes.
From Table \ref{tabCovGTS} and Figure \ref{figGTSBar} we can also see that $\Sigma E$ is similarly as for the SB model lowest for the scheme $C1E$, followed by scheme $C2E$ using RTC and then GD methods, but in contrast to the SB model now next comes scheme $P1E$ and then $SVar$ using RTC and GD methods. As opposed to the SB model, for the GTS model $\Sigma Var$ is lowest for scheme $C2Var$ for the RTC method and for scheme $P1Var$, followed by $C2Var$ for the GD method, $SVar$ for the RTC and GD methods, and finally $C1Var$.
Let us now illustrate the theory from Section \ref{secInterv}, using notations from there. Let the distribution of vector $Y$ be as of the parameter vector of the GTS model defined above and let the new parameter vector $X$ have distributions of all coordinates as in $Y$, except for the $i$th coordinate, for certain
$i \in I_4$, which has distribution U($0.8v_i, 1v_i$) for $v_i$ equal to the fixed value of that parameter in \cite{Rathinam_2010}. We have $\mu_Y(B_X) = \frac{1}{2}$ and $P(A)=1$. Using inequality (\ref{probChange}) for a perturbation $\Delta_i = 0.2v_i$ of only the $i$th parameter and values of $DNE_i$ and $bE_i$ from Table \ref{tabGTS}, we receive an estimate of the lower bound on the probability that the effect of this perturbation on the mean number of particles has the same sign as $bE_i$, equal to $95\%$, $84\%$, $89\%$, and $88\%$, for the consecutive $i \in I_4$. Let now $Y$ and $X$ both have distributions as the parameter vector of the GTS model and consider a perturbation $\Delta_i = 0.1v_i$ only of the $i$th parameter. We now have $\mu_Y(B_X)=1, P(A) = \frac{3}{4}$, and the estimates of bounds on the probabilities as above are equal to $64\%$, $43\%$, $52\%$, and $50\%$, for the consecutive $i \in I_4$.
\begin{figure}
\caption{Pie charts analogous as in Figure \ref{pieSB} but for the GTS model.}
\label{pieGTS}
\end{figure}
\begin{figure}
\caption{Estimates of the main and total sensitivity indices of conditional expectation in chart (a) and variance in chart (b) of the GTS model output as discussed in Section \ref{secGTS}.}
\label{barGTSVVtot}
\end{figure}
\begin{table}[h]
\begin{tabular}{|l|c|c|c|c|c|c|} \hline $i $ & $\widetilde{V}_i$ & $\widetilde{V}_i^{tot} $ &$\widetilde{S}_i$ &$\widetilde{S}_i^{tot}$& $bE_i$ & $DNE_i $\\ \hline $\alpha_1 $& $42.41 \pm 0.17$ & $42.64 \pm 0.17$ &$ 0.43 $&$ 0.44$& $1.1255 \pm 0.0031$ & $0.86 \pm 0.25$ \\ $\alpha_2 $& $13.628 \pm 0.097$ & $14.289 \pm 0.100$ &$ 0.14 $&$ 0.15$& $-1.9959 \pm 0.0067$ & $0.81 \pm 0.13$ \\ $\beta $& $2.780 \pm 0.034$ & $2.859 \pm 0.035$ &$ 0.028 $&$ 0.029$& $-5.738 \pm 0.028$ & $0.116 \pm 0.047$ \\ $\gamma $& $38.14 \pm 0.18$ & $38.92 \pm 0.18$ &$ 0.39 $&$ 0.4$& $52.90 \pm 0.16$ & $1.68 \pm 0.26$ \\ \hline $i $ & $V_i$ & $V_i^{tot} $ & $S_i$ & $S_i^{tot}$ &$ DNE $& $1.45 \pm 0.13$\\ \hline $P $& $97.73 \pm 0.26$ & $248.77 \pm 0.36$ &$ 0.21 $&$ 0.53$& \multicolumn{2}{c}{} \\ $R $& $219.01 \pm 0.42$ & $370.05 \pm 0.40$ &$ 0.47 $&$ 0.79$& \multicolumn{2}{c}{} \\ $P, R$& $467.79 \pm 0.44$ & $467.79 \pm 0.44$ &$ 1 $&$ 1$ & \multicolumn{2}{c}{} \\ \hline $i$ & $VVar_i$ & $VVar^{tot}_i$ & $SVar_i$ &$SVar_i^{tot}$& $bVar_i$ & $DNVar_i$\\ \hline $\alpha_1 $& $3.64 \pm 0.12\cdot 10^{3}$ & $4.78 \pm 0.13\cdot 10^{3}$ &$ 0.45 $&$ 0.59$& $10.385 \pm 0.058$ & $1.25 \pm 0.15\cdot 10^{3}$ \\ $\alpha_2 $& $1.02 \pm 0.10\cdot 10^{3}$ & $2.38 \pm 0.12\cdot 10^{3}$ &$ 0.13 $&$ 0.29$& $17.88 \pm 0.15$ & $1.29 \pm 0.13\cdot 10^{3}$ \\ $\beta $& $1116 \pm 50$ & $1487 \pm 61$ &$ 0.14 $&$ 0.18$& $109.98 \pm 0.71$ & $467 \pm 67$ \\ $\gamma $& $2.1 \pm 1.3\cdot 10^{2}$ & $1.71 \pm 0.15\cdot 10^{3}$ &$ 0.025 $&$ 0.21$& $-33.3 \pm 3.0$ & $1.75 \pm 0.18\cdot 10^{3}$ \\ \cline{6-7} $P$& $8.16 \pm 0.16\cdot 10^{3}$ & $8.16 \pm 0.16\cdot 10^{3}$ &$ 1 $&$ 1$& $DNVar $ & $2.63 \pm 0.10\cdot 10^{3}$ \\ \hline \end{tabular} \caption{\label{tabGTS} Estimates of different sensitivity indices and coefficients for the GTS model computed in a $250000$-step MC procedure using scheme $SVar$.} \end{table}
\begin{table}[h] \resizebox{17cm}{!} {
\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \hline
{\multirow{3}{*}{i}}& \multicolumn{2}{c|}{$SE$} & \multicolumn{2}{c|}{$EM$}
& \multicolumn{2}{c|}{$ET$} & \multicolumn{2}{c|}{$SVar$} \\ \cline{2-9} & GD & RTC & GD & RTC & GD & RTC & GD & RTC \\ \cline{2-9}
& \multicolumn{8}{c|}{$\widetilde{V}_i$} \\ \hline $ \alpha_1$ &$16.05 \pm 0.14$ &$7.697 \pm 0.086$ &$13.820 \pm 0.089$ &$6.307 \pm 0.053$ & & &$11.56 \pm 0.14$ &$7.086 \pm 0.087$ \\ $ \alpha_2$ &$5.350 \pm 0.068$ &$4.067 \pm 0.050$ &$5.098 \pm 0.053$ &$3.636 \pm 0.034$ & & &$2.876 \pm 0.051$ &$2.428 \pm 0.039$ \\ $ \beta$ &$3.118 \pm 0.058$ &$0.640 \pm 0.022$ &$2.960 \pm 0.043$ &$0.586 \pm 0.016$ & & &$1.239 \pm 0.021$ &$0.283 \pm 0.013$ \\ $ \gamma$ &$11.41 \pm 0.11$ &$9.950 \pm 0.098$ &$9.733 \pm 0.074$ &$7.635 \pm 0.051$ & & &$8.69 \pm 0.13$ &$8.28 \pm 0.10$ \\
\hline
i & \multicolumn{8}{c|}{$\widetilde{V}_i^{tot}$} \\
\hline $ \alpha_1 $ &$16.27 \pm 0.15$ &$7.721 \pm 0.085$ & & &$13.755 \pm 0.082$ &$6.349 \pm 0.058$ &$11.60 \pm 0.14$ &$7.110 \pm 0.090$\\ $ \alpha_2 $ &$5.720 \pm 0.075$ &$4.223 \pm 0.052$ & & &$5.463 \pm 0.054$ &$3.869 \pm 0.036$ &$3.076 \pm 0.053$ &$2.536 \pm 0.043$\\ $ \beta $ &$3.311 \pm 0.058$ &$0.658 \pm 0.025$ & & &$3.188 \pm 0.053$ &$0.582 \pm 0.019$ &$1.301 \pm 0.024$ &$0.303 \pm 0.014$\\ $ \gamma $ &$11.92 \pm 0.11$ &$10.315 \pm 0.099$ & & &$10.181 \pm 0.070$ &$7.942 \pm 0.061$ &$9.09 \pm 0.12$ &$8.53 \pm 0.11$\\
\hline
\end{tabular} }
\caption{\label{tabGTSComp} Estimates of variances of the final MC estimators of the sensitivity indices of conditional expectation for the GTS model, computed using the RTC and GD methods and different schemes as described in Section \ref{secGTS}.} \end{table}
\begin{table}[h] \resizebox{17cm}{!} {
\begin{tabular}{|l|c|c|c|c|c|c|} \hline {\multirow{2}{*}{i}}
& \multicolumn{6}{c|}{$cE_i$ } \\ \cline{2-7} &P1ERTC & C1ERTC & C2EGD & C2ERTC & SVarGD & SVarRTC \\ \hline $ \alpha_1 $ &$3.4223 \pm 0.0085\cdot 10^{-1}$ &$0.1284 \pm 0.0082$ & $1.7937 \pm 0.0098\cdot 10^{-1}$ & $1.3828 \pm 0.0099\cdot 10^{-1}$ & $0.914 \pm 0.026$ & $0.762 \pm 0.019$\\ $ \alpha_2 $ &$3.4342 \pm 0.0082\cdot 10^{-1}$ &$0.1168 \pm 0.0085$ & $1.6615 \pm 0.0098\cdot 10^{-1}$ & $1.3328 \pm 0.0080\cdot 10^{-1}$ & $0.404 \pm 0.014$ & $0.402 \pm 0.011$\\ $ \beta $ &$3.4620 \pm 0.0083\cdot 10^{-1}$ &$0.1156 \pm 0.0086$ & $1.6423 \pm 0.0098\cdot 10^{-1}$ & $1.2710 \pm 0.0074\cdot 10^{-1}$ & $0.2342 \pm 0.0081$ & $0.1478 \pm 0.0051$\\ $ \gamma $ &$3.2824 \pm 0.0077\cdot 10^{-1}$ &$0.1047 \pm 0.0072$ & $1.7466 \pm 0.0094\cdot 10^{-1}$ & $1.4427 \pm 0.0072\cdot 10^{-1}$ & $0.795 \pm 0.022$ & $0.777 \pm 0.022$\\ $A $&$1.1706 \pm 0.0015\cdot 10^{-1}$ &$0.1060 \pm 0.0076$ & $1.5464 \pm 0.0037\cdot 10^{-1}$ & $1.7194 \pm 0.0031\cdot 10^{-1}$ & $1.015 \pm 0.015$ & $1.144 \pm 0.015$\\ $\Sigma E$&$1.4772 \pm 0.0025$ &$0.572 \pm 0.019$ & $0.8390 \pm 0.0025$ & $0.7149 \pm 0.0023$ & $3.362 \pm 0.036$ & $3.232 \pm 0.038$\\
\hline {\multirow{2}{*}{i}}
& \multicolumn{6}{c|}{$cVar_i$} \\ \cline{2-7} &P1VarRTC & C1VarRTC & C2VarGD & C2VarRTC & SVarGD & SVarRTC \\ \hline $ \alpha_1 $ &$186.2 \pm 1.4$ &$4.36 \pm 0.33\cdot 10^{3}$ &$210.5 \pm 2.7$ &$173.3 \pm 2.1$ &$335 \pm 11$ &$264.4 \pm 9.0$\\ $ \alpha_2 $ &$180.9 \pm 1.3$ &$438 \pm 31$ &$197.8 \pm 1.7$ &$163.7 \pm 1.7$ &$176.4 \pm 7.1$ &$196.2 \pm 6.0$\\ $ \beta $ &$181.5 \pm 1.2$ &$8.27 \pm 0.52$ &$192.9 \pm 2.2$ &$157.5 \pm 1.8$ &$154.6 \pm 5.7$ &$98.8 \pm 4.6$\\ $ \gamma $ &$177.9 \pm 1.2$ &$1.383 \pm 0.094$ &$199.7 \pm 2.3$ &$174.1 \pm 1.9$ &$262.8 \pm 8.2$ &$285.5 \pm 9.4$\\ $AV $&$117.70 \pm 0.59$ &$120.3 \pm 7.5$ &$141.94 \pm 0.82$ &$159.2 \pm 1.0$ &$340.0 \pm 4.9$ &$395.1 \pm 6.2$\\ $\Sigma V$ &$844.3 \pm 4.6$ &$4.93 \pm 0.33\cdot 10^{3}$ &$942.8 \pm 6.8$ &$827.8 \pm 6.0$ &$1269 \pm 17$ &$1240 \pm 16$\\
\hline \end{tabular} } \caption{\label{tabCovGTS} Estimates of variances of the final estimators of the orthogonal projection coefficients of conditional expectation and conditional variance given the parameters using different schemes for the GTS model as described in Section \ref{secGTS}. The suffix RTC or GD of the scheme means that the RTC or the GD method was applied.} \end{table} \begin{figure}
\caption{ Estimates for the GTS model of quantities $\Sigma E$ in chart (a) and $\Sigma Var$ in chart (b) for different estimation schemes
of orthogonal projection coefficients of conditional expectation in (a) and variance in (b) for the computations described in Section \ref{secGTS}. The suffix RTC or GD of the scheme means that the RTC or the GD method was applied.}
\label{figGTSBar}
\end{figure}
\subsection{MBMD model}\label{secMBMD} Let us finally consider the MBMD model and its output from Section \ref{MBMDPrev}. We performed a one-million-step MC procedure computing various indices and coefficients using the RTC method and scheme $SVar$. The results are presented in Table \ref{tabMBMD} and on Figure \ref{pieMBMD}. We carried out a ten-million-step MC procedure using scheme $SErrVar$ to estimate the mean squared error of approximation of conditional expectation and variance using linear combinations of centred parameters and constants and estimates of $bE_i$ and $bVar_i$ from Table \ref{tabMBMD} and mean and mean variance from Table \ref{tabDisp} as coefficients, analogously as in the previous sections. We obtained estimates of error for conditional expectation $3.091 \pm 0.094\cdot 10^{-2}$ and for variance $0.000 \pm 0.027$, which are not significantly different from estimates of the squared best theoretical errors $DNE$ and $DNVar$ in Table \ref{tabMBMD}. We carried out $500$ independent runs of $250$-step MC procedures using scheme $SVar$ and RTC, GDI, and GDR methods described in Section \ref{MBMDPrev} to get estimates of variances of the final MC estimators of the sensitivity indices of conditional variances from this scheme, analogously as for the indices of conditional expectations in the previous sections. The results are presented in Table \ref{tabCompMB} and Figure \ref{fig3SVarMBMD}. For all the parameters except $C$ the estimates of variances are lowest for the RTC method, followed by the GDI, and then the GDR method, while for $C$ they are lower for the GDR than the GDI method, with the RTC method still yielding the smallest variance. The estimate of variance of the final MC estimator of the main sensitivity index of conditional variance with respect to $K_{d1}$ is even about 48 times higher for the RTC than the GDI method. Qualitatively the same results were obtained for variances of estimators of total sensitivity indices using this scheme (data not shown).
We also performed an experiment comparing the variances of estimators of orthogonal projection coefficients from 200 independent runs of MC procedures using different above constructions of the MBMD model and $100$ runs of scheme $SVar$ and procedures using schemes $C2E$ and $C2Var$ using the same number of process evaluations, similarly as in the previous sections. The results are presented in Table \ref{tabCompMBCov} and Figure \ref{figSigmaMBMD}. We can see that for schemes $C2E$ and $SVar$ for the coefficients of conditional expectation, as well as for scheme $C2Var$ for the coefficients of conditional variance, the GDR method yields highest variance of the estimators of coefficients of orthogonal projection onto normalized centered parameters and the lowest variance for the averages for both conditional expectations and variances. For all of the schemes, using the GDR method leads to highest estimates of $\Sigma E$ and $\Sigma Var$, followed by the RTC method, and finally by the GDI method. Note that for the parameters $K_{b1}$ and $K_{d1}$ the estimates of variances of estimators of the orthogonal projection coefficients of conditional expectation from scheme $C2E$ are statistically significantly higher when using the RTC than the GDI method, while the opposite sharp inequality holds for the estimand $Ave$, which, as discussed in Section \ref{secVarDiff}, shows that for this model the value of $\msd(p_1,p_2)$ defined by (\ref{errSqr}) must be higher for certain parameter values when using the RTC than the GD method, both with the initial order of indices.
\begin{table}[h] \resizebox{16cm}{!} {
\begin{tabular}{|l|c|c|c|c|c|c|} \hline $i $ & $\widetilde{V}_i$ & $\widetilde{V}_i^{tot} $ &$\widetilde{S}_i$ &$\widetilde{S}_i^{tot}$& $bE_i$ & $DNE_i $\\ \hline $C $& $2.8913 \pm 0.0038$ & $2.9183 \pm 0.0038$ &$ 0.73 $&$ 0.74$& $5.3781 \pm 0.0070\cdot 10^{-1}$ & $0.0304 \pm 0.0069$ \\ $K_{b1} $& $1.0351 \pm 0.0026\cdot 10^{-1}$ & $1.0371 \pm 0.0026\cdot 10^{-1}$ &$ 0.026 $&$ 0.026$& $3.724 \pm 0.017$ & $5.891 \pm 380.717\cdot 10^{-6}$ \\ $K_{d1} $& $1.0507 \pm 0.0025\cdot 10^{-1}$ & $1.1129 \pm 0.0026\cdot 10^{-1}$ &$ 0.026 $&$ 0.028$& $-37.26 \pm 0.17$ & $6.10 \pm 0.36\cdot 10^{-3}$ \\ \hline $i $ & $V_i$ & $V_i^{tot} $ & $S_i$ & $S_i^{tot}$ &$ DNE $& $0.0306 \pm 0.0038$\\ \hline $P $& $3.9668 \pm 0.0049$ & $6.0810 \pm 0.0052$ &$ 0.36 $&$ 0.55$& \multicolumn{2}{c}{} \\ $R $& $4.9893 \pm 0.0050$ & $7.1035 \pm 0.0051$ &$ 0.45 $&$ 0.64$& \multicolumn{2}{c}{} \\ $P, R$& $11.0703 \pm 0.0071$ & $11.0703 \pm 0.0071$ &$ 1 $&$ 1$ & \multicolumn{2}{c}{} \\ \hline $i$ & $VVar_i$ & $VVar^{tot}_i$ & $SVar_i$ &$SVar_i^{tot}$& $bVar_i$ & $DNVar_i$\\ \hline $C $& $0.606 \pm 0.012$ & $0.602 \pm 0.012$ &$ 0.53 $&$ 0.53$& $0.2454 \pm 0.0011$ & $-0.012 \pm 0.016$ \\ $K_{b1} $& $0.1066 \pm 0.0051$ & $0.1075 \pm 0.0052$ &$ 0.093 $&$ 0.094$& $3.750 \pm 0.037$ & $0.0109 \pm 0.0070$ \\ $K_{d1} $& $0.0016 \pm 0.0041$ & $-0.0023 \pm 0.0044$ &$ 0.0014 $&$ -0.002$& $-7.88 \pm 0.36$ & $-0.0064 \pm 0.0057$ \\ \cline{6-7} $P$& $1.147 \pm 0.028$ & $1.147 \pm 0.028$ &$ 1 $&$ 1$& $DNVar $ & $0.043 \pm 0.016$ \\ \hline
\end{tabular} } \caption{\label{tabMBMD} Estimates of different sensitivity indices and coefficients in the MBMD model computed in a one-million-step MC procedure using the RTC method and scheme $SVar$.} \end{table}
\begin{figure}
\caption{Pie charts analogous as in Figure \ref{pieSB} but for the MBMD model. The portions of total arc lengths occupied by segments with symbols $\Sigma_b$ and $\Sigma_d$ are equal to the sums of the Sobol's main sensitivity indices with respect to all parameters $K_{b,i}$, $i\in I_5$, for $\Sigma_b$ and $K_{b,i}$, $i\in I_5$, for $\Sigma_d$.}
\label{pieMBMD}
\end{figure}
\begin{table}[h]
\begin{tabular}{|l|c|c|c|} \hline {\multirow{2}{*}{i}} & GDR & GDI & RTC \\ \cline{2-4}
& \multicolumn{3}{c|}{$VVar_i$, $SVar$ } \\ \hline $C $ & $2.160 \pm 0.045$ & $2.523 \pm 0.078$ & $0.580 \pm 0.019$ \\ $K_{b1}$ & $3.573 \pm 0.084$ & $0.3063 \pm 0.0072$ & $0.1057 \pm 0.0037$ \\ $K_{d1} $ & $3.223 \pm 0.081$ & $0.2335 \pm 0.0077$ & $0.0669 \pm 0.0021$ \\ \hline \end{tabular} \caption{Estimates of variances of the final MC estimators of the main sensitivity indices of conditional variance using scheme $SVar$ and the GDR, GDI, and RTC methods as described in Section \ref{secMBMD}.}
\label{tabCompMB} \end{table}
\begin{figure}
\caption{Chart illustrating data from Table \ref{tabCompMB}.}
\label{fig3SVarMBMD}
\end{figure}
\begin{table}[h]
\begin{tabular}{|l|c|c|c|} \hline {\multirow{2}{*}{i}} & GDR & GDI & RTC \\ \cline{2-4}
& \multicolumn{3}{c|}{$\wh{cE}_{i,C2E}$ } \\ \hline $ C $& $ 2.3531 \pm 0.0012$&$1.64792 \pm 0.00088$ & $1.54710 \pm 0.00078$ \\ $ K_{b1} $ &$2.1750 \pm 0.0011$ &$1.27239 \pm 0.00069$& $1.30307 \pm 0.00067$ \\ $ K_{d1} $ &$2.1300 \pm 0.0011$ &$1.26781 \pm 0.00068$& $1.29614 \pm 0.00066$ \\ $Ave $ &$1.26760 \pm 0.00037$&$1.68266 \pm 0.00050$ & $1.67421 \pm 0.00050$\\ $\Sigma E $ &$24.8995 \pm 0.0078$&$16.0320 \pm 0.0050$ & $16.2203 \pm 0.0048$\\
\hline i & \multicolumn{3}{c|}{$\wh{cE}_{i,SVar} $ } \\ \hline $C$& $51.76 \pm 0.17$& $50.15 \pm 0.16$ & $48.73 \pm 0.15$\\ $K_{b1}$& $26.233 \pm 0.093$ &$20.747 \pm 0.074$& $21.669 \pm 0.076$\\ $K_{d1}$& $25.602 \pm 0.091$ & $20.655 \pm 0.070$& $21.434 \pm 0.073$\\ $Ave$& $27.023 \pm 0.052$& $34.461 \pm 0.070$ & $34.654 \pm 0.069$\\ $\Sigma E$& $330.69 \pm 0.64$ & $291.88 \pm 0.56$ &$298.35 \pm 0.57$\\
\hline i & \multicolumn{3}{c|}{$\wh{cVar}_{i,C2Var} $ } \\ \hline $ C $ &$69.004 \pm 0.096$&$57.896 \pm 0.083$ &$59.382 \pm 0.083$ \\ $ K_{b1} $ &$59.941 \pm 0.082$&$36.303 \pm 0.051$ &$40.662 \pm 0.055$ \\ $ K_{d1} $ &$59.550 \pm 0.080$ &$36.014 \pm 0.051$&$40.422 \pm 0.055$ \\ $AV $&$32.483 \pm 0.029$ &$44.040 \pm 0.047$&$42.132 \pm 0.045$ \\ $\Sigma V$&$697.29 \pm 0.62$&$464.07 \pm 0.43$ &$507.02 \pm 0.46$ \\
\hline i & \multicolumn{3}{c|}{$\wh{cVar}_{i,SVar} $ } \\ \hline $C$&$122.89 \pm 0.65$&$159.87 \pm 0.98$ &$125.83 \pm 0.71$\\ $K_{b1}$&$127.88 \pm 0.65$ &$75.79 \pm 0.45$&$102.64 \pm 0.56$\\ $K_{d1}$&$121.75 \pm 0.64$ &$69.65 \pm 0.41$&$95.29 \pm 0.52$\\ $AV$&$102.30 \pm 0.35$ &$250.70 \pm 0.98$&$258.81 \pm 0.97$\\ $\Sigma V$&$1400.4 \pm 4.1$ &$1140.6 \pm 3.8$&$1367.8 \pm 4.2$\\ \hline \end{tabular} \caption{\label{tabCompMBCov} Estimates of variances of the final MC estimators of orthogonal projection coefficients of conditional expectation and variance using different schemes and the GDR, GDI, and RTC methods as described in Section \ref{secMBMD}.} \end{table}
\begin{figure}
\caption{ Estimates for the MBMD model of quantities $\Sigma E$ in chart (a) and $\Sigma Var$ in chart (b) for different schemes and methods for the estimation of the orthogonal projection coefficients of conditional expectation in (a) and variance in (b) as described in Section \ref{secMBMD}.
}
\label{figSigmaMBMD}
\end{figure}
\chapter*{Conclusions} \addcontentsline{toc}{chapter}{Conclusions}
In this work we formalized and generalized the former concept of an estimation scheme from our master's thesis in computer sciene \cite{badowski2011}, making it a convenient tool for defining estimators of vector-valued estimands depending on a number of functions. We also defined inefficiency constant of such a scheme, which can be useful for comparing the efficiency of unbiased estimation schemes when used in MC procedures.
We developed new estimation schemes for various quantities defined for functions of two independent random variables, which can be outputs of stochastic models in function of the model parameters and a noise variable used to construct the random trajectories of the model process. In particular, we provided such first unbiased estimation schemes for the variance-based sensitivity indices of a large class of functions of conditional moments other than conditional expectation, like conditional variance, of functions of two independent random variables given the
first variable, and developed some new schemes for the case of conditional expectation. We also provided first unbiased estimation schemes for covariances and products of functions of conditional moments and functions of the first variable, for coefficients of orthogonal projection of functions of conditional moments onto orthogonal functions of the first variable, and of the mean squared error of approximation of functions of conditional moments using functions of this variable. Furthermore, we derived estimation schemes for normalized variance-based sensitivity indices and correlations between functions of conditional moments and functions of the first variable. We defined a new nonlinearity coefficient which can be used for obtaining lower bounds on the probabilities of certain localizations of functions values changes, caused by perturbations of their independent arguments. We also provided unbiased estimation schemes for nonlinearity coefficients with respect to all independent arguments and computed these coefficients numerically for the GTS model. One of the proposed schemes, called scheme $SVar$, allows to estimate most of the above mentioned indices and coefficients for conditional expectation and variance, such as variance-based sensitivity indices and coefficients of orthogonal projection onto linear combinations of coordinates of the first variable and constants.
It can be also easily extended to allow for the
estimation of coefficients of orthogonal projection onto higher polynomials of the first variable. Thus, it may be an efficient and diverse tool for the analysis of outputs of stochastic models. We derived a number of inequalities between the inefficiency constants of the proposed schemes. We tested the introduced schemes and the relationships between their inefficiency constants using outputs of three continuous-time Markov chain models of the reaction network dynamics. In particular, we proved that the inefficiency constant of scheme $SVar$ for the estimation of the sensitivity indices of conditional expectation is no more than four times higher and three times lower, and in numerical experiments using the GTS model we showed that it can be more than two times lower than for the best schemes introduced in \cite{badowski2011}. In our numerical tests the order of estimators of orthogonal projection coefficients with respect to the mean squared errors of the corresponding approximations of conditional expectation and variance was different for different models. We also demonstrated significant dependence of variances of the proposed estimators on the simulation algorithm used, as well as on the order of reactions in the GD method. We discussed the relationship of this effect with similar ones reported in \cite{badowski2011} and \cite{Rathinam_2010}.
In practice, one can choose the simulation algorithm and the scheme adaptively using preliminary simulations to estimate the inefficiency constants of the corresponding MC sequences.
An interesting topic for the future research is to compare the error when using different methods of approximation of functions of conditional moments of functions of two independent random variables given the first variable, using orthogonal functions of the first variable, like approximation error of conditional expectation or variance of some output of an MR given the model parameters. One can consider the method of direct estimation of the coefficients of orthogonal projection proposed here and different methods based on double-loop sampling, or using the least squares method possibly with some regularization and constraints \cite{Li2002, Khammash_2012}. The estimates of mean squared error of approximation of functions of conditional moments for different methods could be computed using the corresponding schemes from Section \ref{secApprErr}, like $SErrVar$ in the case of conditional variances being approximated. Coefficients of orthogonal projections obtained using the above methods can be used to estimate the variance-based sensitivity indices similarly as in \cite{Li2002b, Li2002, blatman_2010, goutsias_2010}, and an interesting question is if the obtained estimates could be more accurate than the ones received using estimators from this work for the same computation time and for any stochastic model of practical importance.
\begin{appendices}
\chapter{\label{appMath}Mathematical background}
For a finite set $A$, we denote by $|A|$ the number of its elements. For a set $B$,
we denote by $\id_{B}$ the identity function on $B$. We assume that the set of natural numbers $\mathbb{N}$ contains zero, and by $\mathbb{N}_{+}$ we denote the positive natural numbers. For $n \in \mathbb{N}_+$, we define $I_n = \{1, \ldots, n\}$ and for $n=0$, $I_n = \emptyset$. We denote by $\mathbb{R}_+$ positive real numbers, and by extended real line we mean $\overline{\mathbb{R}} = \mathbb{R}\cup\{-\infty\}\cup\{+\infty\}$. For $a,b \in \overline{\mathbb{R}}$, we write $a \leq b$ not only when $a, b \in \mathbb{R}$ and $a \leq b$, but also when
$b= \infty$ or $a = -\infty$. We assume an infimum over an empty set to be plus and supremum minus infinity. For sets $X$ and $Y$, we denote by $Y^X$ the set of all functions from $X$ to $Y$. Let $f \in Y^X$, which we also denote $f:X\rightarrow Y$. Domain of $f$, denoted as $D_f$, is the set $X$, and the image of some $A \subset X$ under $f$, denoted as $f[A]$, is the set $\{f(x): x \in A\}$. $f[X]$ is called the image of $f$. If $B\subset Y$, then preimage of $B$ under $f$, denoted as $f^{-1}[B]$, is the set $\{ x\in X: f(x)\in B\}$. Let for a set $A$, $\mc{P}(A)$ be its power set, that is the set of its all subsets. The image function of $f$ is a function $f^{\rightarrow}:\mc{P}(X)\rightarrow\mc{P}(Y)$ such that for each $C \in \mc{P}(X)$, $f^{\rightarrow}(C) = f[C]$. If $X$ is a subset of $\mathbb{N}$, we often write $f_l$ rather than $f(l)$ for $l \in X$, and use notation $(f_l)_{l \in X}$ for $f$. When $X = I_n$ for some $n \in \mathbb{N}_+$, we often denote $f$ as $(f_1,\ldots,f_n)$. Measurable space is a pair $(B,\mc{B})$ consisting of a set $B$ and a $\sigma$-field $\mc{B}$ of its subsets. By default, the $\sigma$-field we associate with a set $B$ with default topology, like $B \subset \mathbb{R}^n$ for some $n \in \mathbb{N}_+$ with topology generated by the Euclidean distance or some countable space like $\mathbb{N}$ with discrete topology, is its Borel $\sigma$-field $\mc{B}(B)$, that is the smallest $\sigma$-field generated by open sets, and the default measurable space for $B$ is $\mc{S}(B)=(B,\mc{B}(B))$.
For measurable spaces $S_i = (B_i,\mc{B}_i)$, $i \in I_2$, a function from $B_1$ to $B_2$ is said to be measurable from $\mc{S}_1$ to $\mc{S}_2$ if for each $A \in \mc{B}_2$, $f^{-1}[A] \in \mc{B}_1$. If $\mc{S}$ is the default measurable space for $B$, then we often use $B$ in place of $\mc{S}$, e. g. we say that a function is measurable from or to $B$.
Suppose that $J$ is a countable nonempty set. For a family of sets $\{B_i \subset B: i \in J\}$, we define their Cartesian product $\prod_{i \in J}B_i$ to be the set of functions $f$ from $J$ to $B$ such that for each $i \in J$ it holds $f(i) \in B_i$. For $B_i$ all equal to $B$, it holds $\prod_{i \in J}B_i = B^J$. For $N \in \mathbb{N}_+$, we denote $B^{I_N}$ simply as as $B^N$, and informally identify $B^1$ with the set $B$. For some measurable spaces $\mc{S}_i = (B_i,\mc{B}_i), i\in J$, the product measurable space $\mc{S} = \bigotimes_{i\in J}\mc{S}_i $ is defined to be a measurable space $(\Pi_{i\in J} B_i, \bigotimes_{i\in J} \mc{B}_i)$, where the product $\sigma$-field $\bigotimes_{i \in J} \mc{B}_i$ is defined as the one generated by the family $\mc{T} = \{\Pi_{i\in J}A_i: A_i \in \mc{B}_i\text{ for } i \in J \text{, and only for finite number of $i \in J$, $A_i \neq B_i$}\}$. For probability distributions $\mu_i$ on $\mc{S}_i$, $i \in J$, their product $\nu = \bigotimes_{i \in J} \mu_i$ is defined as the unique probability distribution on $\bigotimes_{i \in J}\mc{S}_i$ such that for each $D= \Pi_{i\in J}A_i \in \mc{T}$, we have $\nu(D) = \prod_{i\in J}\mu_i(A_i)$.
For a measurable space $\mc{S}$, let $\mc{F}(\mc{S})$ be the set of measurable functions $f$ from $\mc{S}$ to $\mathbb{R}$. For a measure $\mu$ on $\mc{S}$, let $[f]_{\mu}$ be the class of equivalence of $f \in \mc{F}(\mc{S})$
with respect to relation $g \sim h$ if $f = g$, $\mu$ almost everywhere (a. e.).
For $p > 0 $, by $L^p(\mu)$ we denote the linear space $\{[f]_\mu: f \in\mc{F}(\mc{S}),\ \int |f|^p \mathrm{d}\mu <\infty\}$
(see \cite{rudin1970} Section 3.10 for more details). As common in the literature \cite{rudin1970}, for convenience we informally identify classes from $L^p(\mu)$ with their elements, e. g. by writing $f \in L^p(\mu)$ for $f \in \mc{F}(\mc{S})$, when it holds $[f]_\mu \in L^p(\mu)$. We say that that $f\in \mc{F}(\mc{S})$ is integrable with respect to a measure $\mu$ on $\mc{S}$ if if $f \in L^1(\mu)$ and square-integrable if $f \in L^2(\mu)$.
For measurable spaces $\mc{S}_i = (B_i,\mc{B}_i)$, for $i \in I_2$, let the function $T$ be measurable from $\mc{S}_1$ to $\mc{S}_2$. For a measure $\mu$ on $\mc{S}_1$ we define measure $\mu T^{-1}$ on $\mc{S}_2$ by \begin{equation}\label{immes} \mu T^{-1}(A) = \mu (T^{-1}(A)),\ A \in \mc{B}_2. \end{equation} Below we present a change of variable theorem (\cite{billingsley1979}, Theorem 16.12) \begin{theorem}\label{thchvar} $f$ is integrable with respect to $\mu T^{-1}$ if and only if $fT$ is integrable with respect to $\mu$, in which case \begin{equation} \int_{B_1}\! f(T(x))\, \mu(dx) = \int_{B_2}\! f(y)\, \mu T^{-1}(dy). \end{equation} \end{theorem}
Probability space is denoted by default as
$(\Omega, \mathcal{F}, \mathbb{P})$ \cite{Durrett}, and
$L^p(\mathbb{P})$ is denoted simply as $L^p$. For a measurable space $\mc{S}$, an $\mc{S}$-valued random variable is a measurable function from $(\Omega, \mathcal{F})$ to $\mc{S}$.
\begin{defin}\label{distrDef} Distribution of an $\mc{S}$-valued random variable $X$, denoted as $\mu_X$, is a probability distribution on $\mc{S}$ defined as $\mathbb{P} X^{-1}$. In other words, for each $A \in \mc{B}$, $\mu_X(A) = \mathbb{P}(X \in A)$. \end{defin} For two random variables $X$ and $Y$, by $X \sim Y$ we mean that $\mu_X=\mu_Y$ and for a probability distribution $\Lambda$, $X \sim \Lambda$ denotes $\mu_X=\Lambda$.
For $N \in \mathbb{N}_+$, random variable $X=(X_i)_{i=1}^N$ with values in a product $\mc{S}= \prod_{i=1}^N\mc{S}_i$ of measurable spaces $\mc{S}_i$, $i\in I_N$, is also called an $\mc{S}$-valued random vector.
The expected value of a real-valued random variable $\phi$ on the probability space with probability $\mu$ is defined as \begin{equation}\label{emu} \mathbb{E}_{\mu}(\phi) = \int\! \phi(x)\, \mu(dx), \end{equation} where the integral on the rhs is Lebesgue integral and the subscript $\mu$ in $\mathbb{E}_\mu$ is usually omitted if $\mu$ is the default $\mathbb{P}$. We say that a real-valued random variable $Z$ is integrable if it is integrable with respect to $\mathbb{P}$, and analogously for the square-integrability. U($a,b$) denotes uniform distribution on the interval $[a,b]$ and $\Exp(\lambda)$ is exponential distribution with parameter $\lambda$ \cite{Norris1998}. \begin{defin}\label{defUD} We say that a random variable $X$ has uniform discrete distribution and denote it $X\sim U_d(a,b)$ if $a,b \in \mathbb{Z}$, $a \leq b$, and for each $c \in \mathbb{Z}$, $a \leq c \leq b$, \begin{equation} \mathbb{P}(X = c) = \frac{1}{b - a + 1}. \end{equation} \end{defin} \begin{defin}\label{defSupp} The support \cite{lehmann1998theory} of a probability measure $\mathbb{P}$ on $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n))$ is the set $\{ x\in R^n: \mathbb{P}(A)>0 \text{ for every open rectangle $A$ containing }x\} $. \end{defin}
For $A \in \mathcal{F}$, we denote by $\mathbb{1}_{A}$ the indicator of $A$, that is $\mathbb{1}_{A}(\omega) = 1$ if $\omega \in A$ and $0$ otherwise, and we denote $\mathbb{1} = \mathbb{1}_{\Omega}$. \begin{defin}\label{condDef} Let $X$ be a random variable taking values in a measurable space $(B,\mc{B})$.
Conditional expectation $\mathbb{E}(Y|X)$ of an integrable random variable $Y$ given $X$ is a random variable such that (cf. \cite{Durrett} Section 4.1 and \cite{billingsley1979}, Theorem 20.1 ii)) \begin{enumerate}
\item $\mathbb{E}(Y|X)$ is equal to $f(X)$ for some measurable function $f$ from $(B,\mc{B})$ to $\mathbb{R}$,
\item for each ${A \in \mc{B}},\ \mathbb{E}(Y\mathbb{1}_A(X)) = \mathbb{E}(\mathbb{E}(Y|X)\mathbb{1}_A(X))$. \end{enumerate} \end{defin} Conditional expectation always exists, however, function $f$ yielding it is uniquely defined only up to sets of measure $\mu_X$ zero. Thus equalities in the theorems below hold almost surely (a. s.) \cite{Durrett}, but for convenience we omit writing this, and so we do often in the main text.
\begin{theorem}\label{indepCond} For a measurable function $f$ and independent random variables $X$, $Y$ such that $f(X,Y) $ is integrable, it holds (\cite{Durrett} Section 4.1 Example 1.5) \begin{equation}
\mathbb{E}(f(X,Y)|X) = (\mathbb{E} (f(x, Y)))_{x = X}. \end{equation} \end{theorem} \begin{theorem}\label{condexpX} For random variables $X$, $Z$, and a measurable function $f$, such that $Z$ and $f(X)Z$ are integrable, it holds (\cite{Durrett} Section 4.1 Theorem 1.3) \begin{equation}
\mathbb{E}(f(X)Z|X) = f(X)\mathbb{E}(Z|X). \end{equation} In particular \begin{equation}\label{equexpconds}
\mathbb{E}(f(X)Z) = \mathbb{E}(f(X)\mathbb{E}(Z|X)). \end{equation} \end{theorem} For $s,t \in (0,\infty)$, $\frac{1}{s} + \frac{1}{t} = 1$, $X \in L^s$, and $Y \in L^t$, we have the following
H\"{o}lder's inequality \cite{rudin1970} (called Schwartz inequality for $s = t = 2$) \begin{equation}\label{hold}
\mathbb{E}(|XY|) \leq \sqrt[s]{\mathbb{E}(|X|^s)}\sqrt[t]{\mathbb{E}(|Y|^t)}. \end{equation} \begin{theorem}\label{leqpq} For $p\geq q > 0$ and $Z \in L^p$, it holds \begin{equation}
\sqrt[q]{\mathbb{E}(|Z|^q)} \leq \sqrt[p]{\mathbb{E}(|Z|^p)}. \end{equation} In particular, $Z \in L^q$. \end{theorem} \begin{proof} It is sufficient to take $X = 1$, $Y = Z^q$ and $t = \frac{p}{q}$ in H\"{o}lder's inequality. \end{proof}
\begin{theorem}\label{jensen} If $\phi$ is convex and $\phi(Y)$ and $Y$ are integrable, then for each random variable $X$ we have the following Jensen's inequality for conditional expectations (\cite{Durrett} 4.1.1 (d)). \begin{equation}
\mathbb{E}(\phi(Y)|X) \geq \phi(\mathbb{E}(Y|X)). \end{equation}
\end{theorem} The following well-known theorem states that conditional expectation is contraction in $L^p$ for $p \geq 1$. \begin{theorem}\label{contrac} For $Y^p$ integrable for $p \geq 1$ it holds \begin{equation}
\mathbb{E}(|Y|^p) \geq \mathbb{E}(|\mathbb{E}(Y|X)|^p). \end{equation} \end{theorem} \begin{proof}
It follows from Theorem \ref{leqpq}, Theorem \ref{jensen} for $\phi(x) = |x|^p$, and the iterated expectation property (\ref{doubleCond}). \end{proof}
Conditional probability of an event $B \subset \Omega$ given a random variable $X$ is defined as \begin{equation}\label{PBX}
\mathbb{P}(B|X) = \mathbb{E}(\mathbb{1}_{B}|X). \end{equation}
Below we provide a definition of conditional distribution which is convenient for our needs (cf. \cite{borovkov1999mathematical}, Chapter 20, definitions 1 and 2). \begin{defin}\label{defMu} For two random variables $X_i, i \in I_2$,
with values in measurable spaces $\mc{S}_i =(B_i, \mathcal{B}_i), i \in I_2$,
respectively, we call $\mu_{X_2|X_1}:B_1 \times \mathcal{B}_2 \rightarrow [0,1]$ conditional distribution of $X_2$ given $X_1$
if the following conditions are satisfied: \begin{enumerate}
\item for each $x \in B_1$, $\mu_{X_2|X_1}(x, \cdot)$ is a probability measure on $\mathcal{B}_2$,
\item for each $A \in \mathcal{B}_2$, function $x \rightarrow \mu_{X_2|X_1}(x , A) $ is measurable from $\mc{S}_1$ to $\mathbb{R}$,
\item for each $A \in \mathcal{B}_2$, $\mu_{X_2|X_1}(X_1 , A) $ is a version of $\mathbb{P}(X_2 \in A|X_1)$. \end{enumerate}
\end{defin} It turns out that for random variables $X_i, i \in I_2,$ with values in standard Borel spaces \cite{Ikeda1981} such as complete spaces (including $\mathbb{R}^n$ with Euclidean distance) with
Borel $\sigma$-field, conditional distribution $\mu_{Y|X}$ of $Y$ given $X$ exists and $\mu_{Y|X}(x,\cdot)$ is uniquely determined for $\mu_X$ a. e. $x$, which follows from Theorem 3.3 in Chapter 1 in \cite{Ikeda1981}.
For a random variable $Y$ with values in measurable space $\mc{S}$, a real-valued measurable function $g$ on $\mc{S}$ such that $g(Y) \in L^1$, and any random variable $X$ such that
$\mu_{Y|X}$ exists, it holds (cf. \cite{borovkov1999mathematical}, Section 20, Theorem 1) \begin{equation}\label{condCond}
\mathbb{E}(g(Y)|X) = \int\! g(y)\, \mu_{Y|X}(X,dy). \end{equation}
\chapter{\label{appHMC}Continuous-time Markov chains} Let $T = [0,\infty)$ and $E$ be a countable set with discrete topology, called state space. Let $Y$ be a stochastic process on $E$ with times $T$, that is a sequence of random variables $(Y_t)_{t \in T}$ with values in $E$, where variable $Y_t$ describes the random state of the process at time $t$.
By $\mc{B}(E^T)$ we denote the $\sigma$-field of subsets of $E^T$ generated by the family of sets $\{\{f \in \mathbb{E}^T: f(t) = i\}:t \in T, i \in E\}$. Process $Y$ can be identified with a random variable taking values in the measurable space $\mc{S}(E^T)=(E^T, \mc{B}(E^T))$, whose values $Y(\omega)$, known as trajectories of the process, are functions of time given by $Y(\omega)(t) = Y_t(\omega)$, $t \in T$, and they describe evolution of the process
in time corresponding to the elementary event $\omega \in \Omega$. Distribution $\mu_Y$ of a process $Y$ is defined as for any random variable (see Definition \ref{distrDef} in Appendix \ref{appMath}). Let us assume that $Y$ is a right-continuous process, which means that its trajectories are right-continuous functions of time for each $\omega \in \Omega$, so that we can define its jump times, jump chain, and holding times, the names being adopted from \cite{Norris1998}. See Section 1.2 of our previous work \cite{badowski2011} or \cite{Norris1998} for intuitive informal descriptions of these objects. We define jump times $J_0, J_1, \ldots$ of $Y$ inductively as \begin{equation} \begin{split} J_{0} &= 0, \\ J_{n+1} &= \begin{cases}
\inf\{t > J_n: Y_t \neq Y_{J_n}\} & \text{if} \ J_n < \infty, \\
\infty & \text{otherwise,} \\ \end{cases} \end{split} \end{equation} its jump chain $Z_0, Z_1, \ldots$ as
$Z_{n} = X_{J_{m(n)}}$, where $m(n) = \max\{k: k\leq n,\ J_k < \infty\}$, and its holding times $S_1, S_2 \ldots$ as \begin{equation} \begin{split} S_{n} &= \begin{cases}
J_n - J_{n-1} & \text{if} \ J_n < \infty, \\
\infty & \text{otherwise.} \\ \end{cases} \end{split} \end{equation} The moment of explosion $\zeta$ of $Y$ is defined as the moment when $Y$ makes infinitely many jumps for the first time, that is \begin{equation}\label{expTime} \zeta =\sup_{n} J_{n}. \end{equation} We say that $Y$ is nonexplosive if $\zeta = \infty$. We say that a matrix $Q = (q_{x,y})_{x,y \in E}$ is a $Q$-matrix (on $E$) if for each $x, y \in E, x \neq y$, $0 \leq q_{x,y} < \infty$, and for each $x \in E$, \begin{equation}\label{qxx} -q_{x,x} = \sum_{y \in E}\ q_{x,y} < \infty. \end{equation} Entries of a $Q$-matrix are called intensities, and thanks to (\ref{qxx}) it is sufficient to specify the off-diagonal intensities to specify the whole $Q$-matrix. Continuous-time homogeneous Markov chain (HMC) \cite{pierre1999markov} $Y$ with $Q$-matrix $Q$ on $E$ with times $T$ and initial distribution $\Lambda$ is a right-continuous stochastic process with such $E$ and $T$, such that $Y_0 \sim \Lambda$, and for certain function $p$ fulfilling for each $x ,y \in E$ and $h \geq 0$, \begin{equation} p(x,y,h) = q_{x,y}h + o(h), \end{equation}
for each $h \geq 0$, $k \in \mathbb{N}_{+}$, $x_1, x_2, \ldots, x_{k+1} \in E$, and $0 \leq t_1 \leq \ldots \leq t_{k}$, it holds \begin{equation}
\mathbb{P}(Y_{t_k+h}= x_{k+1}|Y_{t_1}=x_1, \ldots, Y_{t_k}=x_k) = p(x_k,x_{k+1},h), \end{equation} whenever the event we condition on has positive probability. Distribution of each nonexplosive HMC with a $Q$-matrix $Q = (q_{x,y})_{x,y \in E}$ and initial distribution $Y_0 \sim \Lambda$ is uniquely determined by $Q$ and $\Lambda$.
Poisson process $N$ with rate $\lambda > 0$ is defined as a nonexplosive HMC on state space $\mathbb{N}$ whose jump chain fulfills $Z_n = n$ for $n \in \mathbb{N}$ and whose holding times $S_1, S_2, \ldots$ are i. i. d., $S_1 \sim \Exp(\lambda)$.
\chapter{\label{appProc}Proofs of new theorems for MRCP and MR} To prove Theorem \ref{thNonexpl} we need the following easy consequence of Theorem 4.3.6 from \cite{stroock05markov}. \begin{theorem}\label{thStroock} For a state space $E$, let $(F_N)_{N=1}^{\infty}$ be finite sets such that $F_N \subset E$, $F_N \subset F_{N+1}$ and $\bigcup_{N=1}^{\infty}F_N = E$. If there exists a nonnegative function $u$ on $E$, such that
$\inf_{j \notin F_N}u(j) \rightarrow \infty$ as $N \rightarrow \infty$, and for some $\alpha > 0$, for a $Q$-matrix $Q = (q_{ij})_{i,j \in E}$, for each $i \in E$, \begin{equation} \sum_{j \in E, j \neq i}q_{ij}(u(j) - u(i)) \leq \alpha u(i), \end{equation} then for each probability distribution $\Lambda$ on $E$ there exists a nonexplosive HMC with initial distribution $\Lambda$ and $Q$-matrix $Q$.
\end{theorem} Below we provide the proof of Theorem \ref{thNonexpl}. \begin{proof} For $m$ as in Theorem \ref{thNonexpl}, for assumptions of Theorem \ref{thStroock} to be fulfilled it is sufficient to take $F_N = \{x \in E: mx \leq N\}$, $\alpha = 1$, and for $A$ denoting the lhs of (\ref{supmx}), $u(x) = \max\{A,0\} + mx$. \end{proof}
\begin{theorem}\label{aveSecFin} Using notations as in Section \ref{genParSec}, if for $\mu_P$ a. e. $p = (k, c)$ we have \mbox{$h(p, R) \sim \mu_{MRCP}(RN(k), c)$}, then $\wt{\mu}$ is conditional distribution of $h(P,R)$ given $P$.
\end{theorem} \begin{proof} Point 1 in Definition \ref{defMu} obviously holds. Let $\zeta(p,R)$ denote the initial explosion time of a process given by the considered construction of MRCP using noise variable $R$ and parameters $p \in B_{RN,E} $. The set $B \subset B_{E,RN}$ on which MRCP exists consists of $p$ such that
$\mathbb{P}(\zeta(p,R) = \infty)=1$ and hence from measurability of $\zeta$ (which is measurable as a supremum of measurable initial jump times), we have $B \in \mc{B}_{RN,E}$. Point 2 now follows from the fact that $h$ is measurable and for each $A \subset \mc{B}(E^T)$, it holds \begin{equation} \wt{\mu}(p, A) = \mathbb{1}_B(p)\mu_0(A) + \mathbb{1}_{B_{RN, E}\setminus B}(p) \mathbb{P}(h(p,R) \in A). \end{equation}
Proof of point 3 is analogous as such proof of a less general Theorem 18 in \cite{badowski2011}. For each $A \in \mc{B}(E^T),$ \begin{equation} \begin{split}
\mathbb{P}(h(P,R) \in A |P) &= \mathbb{E}(\mathbb{1}_{A}(h(P,R))|P) \\ &= (\mathbb{P}(h(p,R) \in A))_{p = P} \\ &= \wt{\mu}(P,A),\\ \end{split} \end{equation} where in the second equality we used Theorem \ref{indepCond} and in the third the assumption of this theorem. \end{proof} Below we provide the proof of Theorem \ref{thMomsExist}. \begin{proof} Let $P = (K,C) \sim \nu$. From $ \mathbb{E}(A(K)^n) < \infty$ it follows that $A(K)$ is finite a. s. The assumptions of Theorem \ref{thNonexpl} are satisfied for $\mu_K$ a. e. $k$ with the same $m$ as here, as the lhs of (\ref{supmx}) is bounded from above by \begin{equation} L\max(\{0\}\cup\{ms_l:l \in I_L\})A(k). \end{equation}
Thus MRCP corresponding to RN and $p$ exists for $\nu$ a. e. $p$.
For an MR $(P,Y)$ corresponding to RN, with $P$ as above and $Y$
built with the help of the RTC construction it holds a. s. for each $t \in T$ and $i \in I_N$ (see formula \ref{intEqu}) \begin{equation} m_iY_{t,i} \leq mY_{t} \leq Cm + \sum_{l \in L_m}s_lm N_l(tA(K)). \end{equation} From Minkowski's inequality \cite{rudin1970},
for $\mathbb{E}(Y_{t,i}^n) < \infty$ to hold it is therefore sufficient that $\mathbb{E}(C_i^n) < \infty$ for $i \in I_N$ and for any unit rate Poisson process $N_1$, $\mathbb{E}(N_1(tA(K)^n) < \infty.$
For $i \in \mathbb{N}_+$ we define polynomial $x^{\underline{i}} = x(x-1)\ldots(x-i+1)$ and let the sequence $(b_i)_{i=1}^n$ be such that \begin{equation} x^n = \sum_{i=1}^nb_i x^{\underline{i}}. \end{equation} For each $\lambda \geq 0$, it holds \begin{equation} \begin{split} \mathbb{E}(N_1^n(\lambda)) &= \sum_{k=0}^{\infty} k^n\frac{\lambda^k}{k!}e^{-\lambda} \\ &= \sum_{k=0}^{\infty} \sum_{i=1}^n(b_i k^{\underline{i}})\frac{\lambda^k}{k!}e^{-\lambda} \\ &= \sum_{i=1}^n(b_i\lambda^i e^{-\lambda}\sum_{k=0}^{\infty}\frac{k^{\underline{i}}}{k!}\lambda^{k-i}) \\ &= \sum_{i=1}^n b_i\lambda^i,\\ \end{split} \end{equation} where in the fourth equality we used the fact that for $i \in \mathbb{N}_+$, \begin{equation} \begin{split} \sum_{k=0}^{\infty}\frac{k^{\underline{i}}}{k!}\lambda^{k-i}) &= \sum_{k=i}^{\infty}\frac{1}{(k-i)!}\lambda^{k-i} \\ &= \sum_{l=0}^{\infty}\frac{1}{l!}\lambda^{l} \\ &= e^{\lambda}. \\ \end{split} \end{equation} Thus, from $\mathbb{E}(A(K)^n) < \infty$ we have \begin{equation}\label{Nln} \begin{split} \mathbb{E}(N_1(tA(K))^n) &= \mathbb{E}((\mathbb{E}(N^n_1(tA(k))))_{k=K}) \\
& \leq \mathbb{E}((\mathbb{E}(\sum_{i=1}^n |b_i| (tA(k))^i))_{k=K}) \\
&= \sum_{i=1}^n |b_i|\mathbb{E}((tA(K))^i) < \infty, \end{split} \end{equation} where in the first and last equalities we used Fubini's theorem, and in the last inequality Theorem \ref{leqpq}. \end{proof}
\chapter{\label{appHilb}Hilbert spaces}
We introduce below some definitions and facts from Hilbert space theory, which are used in the main text (see \cite{rudin1970} and \cite{KolmogorovFomin60} for proofs and more details) and prove some new facts. Hilbert space is a pair $(H, (,))$ consisting of a linear space $H$ and a scalar product $(,)$ in it, such that
for metric $d$ and norm $||\cdot||$ defined as \begin{equation}
d(x,y) = ||x-y|| = \sqrt{(x -y,x-y)}, \end{equation}
$(H,d)$ is a complete metric space. For simplicity we also say that $H$ is a Hilbert space (with scalar product $(,)$).
We say that a set $\{v_i \in H: i \in I_n\}$ is orthogonal in $H$ if $(v_i,v_j) =0$, $i,j \in I_n$, $i \neq j$, nonzero orthogonal if further $v_i \neq 0$, $i \in I_n$, and
orthonormal if it is orthogonal with $||v_i||=1$, $i \in I_n$. For linear subspaces $W_1,\ldots,W_n$ of a certain linear space, we define \begin{equation} \sum_{i=1}^n W_i = \{\sum_{i=1}^{n}w_i:\ \forall i \in I_n,\ w_i \in W_i\}. \end{equation} \begin{defin}\label{defHilb}
Hilbert space $H$ is direct sum of its linear subspaces $H_1,\ldots, H_n$, which we denote \begin{equation} H = \bigoplus_{i=1}^n H_i = H_1 \oplus \ldots \oplus H_n \end{equation}
if the following conditions are fulfilled \begin{enumerate}
\item subspaces $H_1, \ldots, H_n$ are closed in $H$,
\item \begin{equation} H = \sum_{i=1}^n H_i, \end{equation}
\item these subspaces are mutually orthogonal, which means that for each $i,j \in I_n$, $i \neq j$ for each $v_i \in H_i$ and $v_j \in H_j$ \begin{equation}
(v_i,v_j) = 0. \end{equation} \end{enumerate} \end{defin} From point 3 it follows that for $v \in H$, elements $v_i \in H_i$ for $i \in I_n$ such that \begin{equation} v = \sum_{i=1}^n v_i \end{equation} are uniquely determined.
Let $<,>$ be a scalar product in $\mathbb{R}^n$ and $(a_{ij})_{i,j \in I_n}$ be real numbers for which, for each $x,y \in \mathbb{R}^n$, it holds \begin{equation}\label{scpr} <x,y> = \sum_{i,j \in I_n}a_{ij}x_iy_j. \end{equation}
We say that 2 norms $|\cdot|_1, |\cdot|_2$ on a linear space $V$ are equivalent, if there exist $\alpha$ and $\beta$ real positive such that for each $x \in V$ \begin{equation}
\alpha|x|_1 \leq |x|_2 \leq \beta|x|_1. \end{equation} \begin{theorem}\label{genDSpace} For a Hilbert space $H$ with a scalar product $(,)$, and for a scalar product $<,>$ in $\mathbb{R}^n$ as in (\ref{scpr}), the Cartesian product space
$H^n = \{(v_i)_{i=1}^n: v_i \in H,\ i \in I_n\} $
with function $(,)_n:H^n\times H^n\rightarrow\mathbb{R}$ \begin{equation}\label{scDir}
(v,w)_n = \sum_{i,j \in I_n} a_{ij}(v_i,w_j) \end{equation} is a Hilbert space, which we call the direct sum of $H$ given by $<,>$ and denote by $\bigoplus_{<,>}H$.
Norms $||\cdot ||_n$ induced by scalar products (\ref{scDir}) corresponding to different scalar products $<,>$ in $\mathbb{R}^n$ are equivalent. \end{theorem}
\begin{proof} For $<,>$ equal to the standard scalar product on $\mathbb{R}^n$, $\bigoplus_{<,>}H$ is the $n$-fold direct sum of Hilbert spaces known from the literature \cite{KolmogorovFomin60}, which is a Hilbert space, and
whose norm let us denote $||\cdot||_{st}$. For general $<,>$ function $(,)_n$ defined by \ref{scDir} is bilinear and symmetric so for the thesis to hold it is sufficient to show that it is positive definite and
function $||\cdot||_n$ given by $||x||_n = \sqrt{(x,x)}$ is a norm equivalent to $||\cdot||_{st}$.
Since the matrix $A = (a_{ij})_{i,j \in I_n}$ is real symmetric and positive definite, there exists an orthogonal matrix $B = (b_{ij})_{i,j \in I_n}$ and diagonal matrix $C = (c_{ij})_{i,j \in I_n}$ such that $c_{ii} > 0$ for $i \in I_n$ and $A = B^TCB$ (\cite{strang2006linear}, sections 5.6 and 6.2). For each $v \in H^n$, we have \begin{equation}\label{vn2} \begin{split} (v,v)= \sum_{i,j \in I_n} a_{ij}(v_i,v_j) = \sum_{i,j,k \in I_n} b_{ki}c_{kk}b_{kj}(v_i,v_j) \\
= \sum_{k=1}^n c_{kk} ||\sum_{i=1}^n b_{ki} v_i ||^2
\end{split} \end{equation} From orthogonality of $B$, $\sum _{k}b_{ki}b_{kj} = \delta_{ij}$, so that \begin{equation}\label{vst2} \begin{split}
\sum_{k=1}^n ||\sum_{i=1}^n b_{ki} v_i ||^2 = \sum_{i,j,k \in I_n} b_{ki}b_{kj}(v_i,v_j) \\
= \sum_{i,j \in I_n}(v_i, v_j) = ||v||_{st}^2. \end{split} \end{equation} From (\ref{vn2}) and (\ref{vst2}) it holds \begin{equation}
\min_{i \in I_n}(c_{ii}) ||v||^2_{st} \leq ||v||^2_n \leq \max_{i \in I_n}(c_{ii}) ||v||^2_{st}, \end{equation} which completes the proof. \end{proof} If $M$ is a closed subspace of $H$, then the orthogonal complement of $M$ in $H$, defined as
$ M^{\perp} = \{v \in H: \forall w \in M\quad v \perp w\}$ is closed and it holds \begin{equation}\label{mDirect} H = M \oplus M^{\perp}. \end{equation} Projection $P$ onto $M$ in the above direct sum is called orthogonal. For $v \in H$, $P(v)$ is the unique element of $M$ minimizing the distance from $v$, which we also call error of approximation of $v$, \begin{equation}\label{infProp}
d(v,P(v)) = \inf_{w \in M} ||v-w||. \end{equation} Furthermore, it holds \begin{equation}\label{distP}
||v||^2 = ||v - P(v)||^2 + ||P(v)||^2. \end{equation} \begin{lemma}\label{lemP1P2} If $M_2\subset M_1$ are closed subspaces of $H$ and $P_i$ is orthogonal projection from $H$ onto $M_i$, $i \in I_2$, then \begin{equation} P_2P_1= P_1P_2 = P_2. \end{equation}
\end{lemma}
\begin{proof}
Denoting $M_2^{\perp_{1}}$ the orthogonal complement of $M_2$ in $M_1$, one can easily check that
$H= M_1^{\perp}\oplus M_2^{\perp_{1}} \oplus M_2$, from which the thesis easily follows.
\end{proof} A well-known example of orthogonal projection is conditional expectation, which we prove below for the reader's convenience
(cf. \cite{Durrett}, Section 4.1 Theorem 1.4). \begin{lemma}\label{condort} Conditional expectation given $X$ is an orthogonal projection from Hilbert space $L^2$ onto $L^2_X$ (defined in Section \ref{secOrthog}). \end{lemma} \begin{proof}
From the definition of conditional expectation and Theorem \ref{contrac}, $\mathbb{E}(Z|X) \in L^2_X$. It is sufficient to prove that for each random variable $Z \in L^2$, $Z-\mathbb{E}(Z|X) \in (L^2_X)^{\perp}$. For each $f(X) \in L^2$ for some measurable $f$, we have from Schwartz inequality $Zf(X),\mathbb{E}(Z|X)f(X)\in L^1$. Thus, from Theorem \ref{condexpX}, \begin{equation}
\mathbb{E}((Z -\mathbb{E}(Z|X))f(X)) = 0. \end{equation} \end{proof} Let $M$ be a closed subspace of $H$, then $\bigoplus_{<,>}M$ is a complete space,
so it is a closed subspace of $\bigoplus_{<,>}H$. \begin{theorem}\label{projDS} If $P$ is orthogonal projection of $H$ onto $M$, then the function $P_n:H^n\rightarrow H^n,$ given by $P_n(v) = (P(v_i))_{i=1}^n$ is an orthogonal projection from $\bigoplus_{<,>}H$ onto $\bigoplus_{<,>}M$. \end{theorem} \begin{proof} For each $v \in \bigoplus_{<,>}H$ we have $P_n(v) \in \bigoplus_{<,>}M$. Furthermore, for each $w \in \bigoplus_{<,>}M$, it holds \begin{equation}
(v - P_n(v),w)_n = \sum_{i,j\in I_n} a_{ij}(v_i - P(v_i),w_j) = 0,
\end{equation} since for each $i \in I_n$ it holds $v_i - P(v_i)\in M^{\perp}$. Thus $v - P_n(v) \in (\bigoplus_{<,>}M)^{\perp}$. \end{proof}
\chapter{\label{appStatMC}Statistics and Monte Carlo background} In this section we introduce certain definitions and facts from statistics and Monte Carlo simulations (cf. \cite{lehmann1998theory, asmussen2007stochastic, badowski2011}), which are used throughout the text. Let us consider a nonempty set of probability distributions $\mathcal{P}$ defined on the same measurable space $\mc{S}$, called (set of) admissible distributions (on $\mc{S}$, cf. \cite{kolmogorov1992selected}, Section 38). For a measurable space $\mc{H}$, a measurable function from $\mathcal{S}$ to $\mc{H}$ is called $\mc{H}$-valued (simply real-valued if $\mc{H} = \mc{S}(\mathbb{R})$) statistic for $\mc{P}$. For a given $\mu \in \mathcal{P}$, random variable $X \sim \mu$ is called an observable. A real-valued function $G$ on $\mathcal{P}$ is called an estimand on $\mc{P}$. We say that a probability distribution $\mu$ on $\mathbb{R}$ has finite $n$-th moment, $n \in \mathbb{N}_+$, if \begin{equation}
\int |x|^n\! d\mu\, < \infty. \end{equation} Let us define estimand $G_E$ on all probability distributions $\mu$ on $\mathbb{R}$ with finite first moments, for which $G_E(X)= \mathbb{E}(X), X \sim \mu$, and
estimand $G_{Var}$ on distributions $\mu$ on $\mathbb{R}$ with finite second moments, for which $G_{Var}(\mu)=\Var(X)$, $X \sim \mu$. We say that a real-valued statistic $\phi$ for $\mc{P}$ is an estimator of an estimand $G$ on $\mc{P}$ if for each $\mu \in \mathcal{P}$, for observables $X \sim \mu$, we think of random values of $\phi(X)$ as estimates of $G(\mu)$, that is its certain approximations. For each $\mu \in \mc{P}$, average error of this approximation can be measured using mean squared error \begin{equation}\label{meanSqrErr} \mathbb{E}_{\mu} ((\phi - G(\mu))^2). \end{equation} Let estimator $\phi$ of $G$ be unbiased, that is for each $\mu \in \mathcal{P}$, \begin{equation}\label{muPhi} \mathbb{E}_\mu(\phi) = G(\mu). \end{equation} Then from (\ref{muPhi}) we have that for $\mu \in \mc{P}$, variance $\Var_\mu(\phi)$ of $\phi$ is equal to the mean squared error (\ref{meanSqrErr}). For $n \in \mathbb{N}_+$, we define \begin{equation}\label{mcpn} \mc{P}^n = \{\mu^n:\mu \in \mc{P}\}, \end{equation} where $\mu^n$ is the $n$-fold product of distribution $\mu$. For an estimand $G$ on $\mathcal{P}$ and $n \in \mathbb{N}_+$, we define estimand $G_n$ on $\mc{P}^n$ by formula $G_n(\mu^n) = G(\mu)$, and call it $G$ in $n$ dimensions.
Let $n \in \mathbb{N}_+$ and $\Pi_n$ denote the group of all permutations of $I_n$. For some set $B$ and function $\phi$ from $B^n$ to $\mathbb{R}$, we define symmetrisation of $\phi$ to be a function from $B^n$ to $\mathbb{R}$ such that for each $x \in B^n$, \begin{equation} \Sym(\phi)(x) = \frac{1}{n!}\sum_{\pi \in \Pi_n}\phi((x_{\pi(i)})_{i=1}^n). \end{equation} We say that $\phi$ as above is symmetric if it is equal to its symmetrisation. For some admissible distributions $\mc{P}$, for each $\mu \in \mc{P}$, $X \sim \mu^n$, and $\phi$ being a real-valued statistic for $\mc{P}^n$,
$\Sym(\phi)(X)$ is an average of random variables with the same distribution as $\phi(X)$. In particular, if $\phi$ is an estimator of some estimand $G$ on $\mc{P}^n$, then so is $\Sym(\phi)$, and from the lemma below it immediately follows that it has uniformly not higher variance, that is for each $\mu \in \mc{P}$ it holds \begin{equation} \Var_{\mu^n}(\Sym(\phi)) \leq \Var_{\mu^n}(\phi). \end{equation} We proved the below lemma as Theorem 11 in \cite{badowski2011}, but this time we provide a different simpler proof. \begin{lemma} \label{lemVarAve} For some $n \in \mathbb{N}_+$, let $X_1, \ldots, X_n$ be real-valued square-integrable random variables with the same distribution. Then \begin{equation}\label{varSum} \Var\left(\frac{1}{n}\sum_{i=1}^{n}X_i\right) \leq \Var(X_1), \end{equation} and equality in (\ref{varSum}) holds if and only if for each $i,j \in I_n$, $X_i=X_j$ a. s. \end{lemma} \begin{proof} For $x_1,\ldots,x_n$ real positive, from the well-known inequality between arithmetic and quadratic means we have \begin{equation}\label{avesqr} \left(\frac{1}{n}\sum_{i=1}^{n}x_i\right)^2 \leq \frac{1}{n}\sum_{i=1}^{n}x_i^2, \end{equation} which is equivalent to \begin{equation} \sum_{1 \leq i<j \leq n}(x_i -x_j)^2 \geq 0, \end{equation} so equality in (\ref{avesqr}) holds only if all $x_i$ are equal. Replacing $x_i$ by $X_i$ in (\ref{avesqr}), taking expected value of both sides, and using the fact that each $X_i$ has the same expected value as their average, we receive the thesis. \end{proof} We say that a distribution $\mu$ on a measurable space $(B, \mc{B})$ is finite discrete (on $D$) if for some finite set $D \in \mc{B}$, $\mu(D) = 1$. In \cite{Halmos_1946} it was proved that if $\mc{P}$ contains all finite discrete distributions on $\mathbb{R}$, and if $\phi$ is an unbiased estimator of some estimand $G$ on $\mc{P}^n$, then $\Sym(\phi)$ is the unique symmetric unbiased estimator of $G$. In particular for any other unbiased estimator $\phi'$ of $G$ we have $\Sym(\phi)=\Sym(\phi')$, so $\Sym(\phi)$ has uniformly not higher variance than $\phi'$. For instance, the unique symmetric unbiased estimator of $G_E$ in $n \geq 1$ dimensions is given for each $x \in \mathbb{R}^n$ by formula \begin{equation}\label{phiAve} \phi_{E,n}(x) = \frac{1}{n} \sum_{i=1}^n x_i, \end{equation} and of $G_{Var}$ in $n \geq 2$ dimensions by formula \begin{equation}\label{phiVar} \begin{split} \phi_{Var,n}(x) = \frac{1}{n-1} (\sum_{i=1}^n x_i^2 -n(\phi_{E,n}(x))^2). \end{split} \end{equation} For admissible distributions $\mc{P}$ consisting of all probability distributions on $\mathbb{R}$ having second moments and $n \geq 2$, we define estimand $G_{VarAve,n}$ of variance of the mean on $\mc{P}^n$ by formula $G_{VarAve,n}(\mu^n) = \Var_{\mu^n}(\phi_{E,n})$. Its symmetric unbiased estimator is given by formula \begin{equation}\label{phiAveVar} \phi_{VarAve,n}(x) = \frac{\phi_{Var,n}(x)}{n}. \end{equation} For admissible distributions $\mc{P}=\{\mu\}$, let $\phi$ be an unbiased estimator of an estimand $G_{\lambda,\mu}$ on $\mc{P}$ defined by $G_{\lambda,\mu}(\mu)=\lambda$. We call such $\phi$ unbiased estimator of $\lambda$ (for $\mu$). If further $\phi \in L^2(\mu)$,
we call it a single-step MC estimator of $\lambda$.
For some $n \in \mathbb{N}_{+}$, for a random vector $X \sim \mu^n$, i. e. one with independent coordinates with distribution $\mu$, in each $i$th step of an $n$-step MC procedure one computes a value of a random variable $W_i = \phi(X_i)$, called the $i$th observable of the single-step MC estimator.
For $W = (W_i)_{i=1}^n$, we use the values of \begin{equation}\label{meanMC} \overline{W} = \phi_{E,n}(W) \end{equation} as final MC estimates of $\lambda$. Function given by formula \begin{equation} \phi_{f}(x) = \phi_{E,n}((\phi(x_i))_{i=1}^n), \ x \in B^n, \end{equation} for which we have $\overline{W}=\phi_{f}(X)$, is an unbiased estimator of $G_{\lambda,\mu}$ in $n$ dimensions, and we call it an $n$-step or final MC estimator of $\lambda$ (for $\mu$) and call (\ref{meanMC}) its observable.
Let us denote variance of the single-step estimator as $\Var_s = \Var_\mu(\phi)$ and its standard deviation as $\sigma_s = \sqrt{\Var_s}$, while for the $n$-step estimator as $\Var_{f} = \Var_f(n) = \Var_{\mu^n}(\phi_{n})$ and $\sigma_{f} =\sigma_{f}(n)= \sqrt{\Var_f}$. It holds \begin{equation}\label{varfs} \Var_{f} = \frac{\Var_s}{n}. \end{equation}
We use the values of \begin{equation}\label{varEst} \phi_{VarAve,n}(W) \end{equation} as estimates of $\Var_{f}$ for $n \geq 2$, and the values of \begin{equation}\label{sigmaEst} \widehat{\sigma}_{f,n}(W)= \sqrt{\phi_{VarAve,n}(W)} \end{equation} as such estimates of $\sigma_f$. For some such obtained estimates $\wt{\lambda}=\overline{W}(\omega)$ of $\lambda$, and $\wt{\sigma}_f = \widehat{\sigma}_{f,n}(W)(\omega)$ of $\sigma_f$, we report the results of a MC procedure in form $\wt{\lambda} \pm \wt{\sigma}_f$ (cf. Chapter 3, Section 1 in \cite{asmussen2007stochastic}). From the central limit theorem (CLT) \cite{billingsley1979}, as $n$ goes to infinity in the above described MC procedure, $\sqrt{n}(\overline{W} - \lambda)$ converges in distribution to $\ND(\lambda, \Var_s)$, that is normal distribution with mean $\lambda$ and variance $\Var_s$, and from the law of large numbers $\sqrt{n}\widehat{\sigma}_{f,n}(W)$ converges a. s. and thus in probability to $\sigma_s$. In particular for $k > 0$, and $\Phi$ being the cumulative distribution function of standard normal distribution, i. e. $\Phi(x) = P(Z \leq x)$,
$Z \sim N(0,1)$, the probability $\mathbb{P}(|\overline{W} - \lambda| < k\widehat{\sigma}_{f,n}(W))$ converges to $2(1 - \Phi(k))$, which is approximately $68\%$ for $k=1$ and $99,73\%$ for $k=3$.
\chapter{\label{appcoeffsLem}Proofs of Theorem \ref{thEquAves} and lemmas \ref{lemPrBetterCov} and \ref{lemC1C2}}
Below we provide a proof of Theorem \ref{thEquAves}. \begin{proof} Let $(\mu,f) \in \mc{V}$ and $x \in B_\mu^{\wt{\Pi}[p_A]}.$ We have \begin{equation} \begin{split} \phi_{\ave_{\Pi}(\kappa),\mc{V}}(x) &= \ave_{C,A,\Pi}(t)(g_{\mc{V},\wt{\Pi}^{\rightarrow}[A]}(f)(x))\\
&= \frac{1}{|\Pi|} \sum_{\pi \in \Pi} \ave_{C,A,\pi}(t)(\eta_{C,A,\Pi,\pi}(g_{\mc{V},\wt{\Pi}^{\rightarrow}[A]}(f)(x))).\\ \end{split} \end{equation} We denote $x_\pi = x_{\wt{\pi}[p_A]},$ $\pi \in \Pi$. For $\pi \in \Pi$ we have \begin{equation}\label{equvec} \begin{split} \ave_{C,A,\pi}(t)(\eta_{C,A,\Pi,\pi}(g_{\mc{V},\wt{\Pi}^{\rightarrow}[A]}(f)(x))) &= \ave_{C,A,\pi}(t)(g_{\mc{V},{\wt{\pi}^{\rightarrow}[A]}}(f)(x_{\pi})) \\ &= t(\rho_{C,A,\pi}(g_{\mc{V},\wt{\pi}^{\rightarrow}[A]}(f)(x_\pi))) \\
&=t(((g_{\mc{V}, \wt{\pi}^{\rightarrow}[A],i,\wt{\pi}[v]}(f)(x_\pi))_{|v \in A_{\gamma(i)}})_{i=1}^\delta) \\
&=t(((f_i(x_{\wt{\pi}[v]}))_{|v \in A_{\gamma(i)}})_{i=1}^\delta). \\ \end{split} \end{equation} On the other hand, \begin{equation}
\A_{\Pi}(\phi_{\kappa,\mc{V}})(x) =\frac{1}{|\Pi|} \sum_{\pi \in \Pi} \A_{\pi}(\phi_{\kappa,\mc{V}})(f)(x_{\pi}) \\ \end{equation} and for $\pi \in \Pi$ we have \begin{equation}\label{equapik} \begin{split} \A_{\pi}(\phi_{\kappa,\mc{V}})(f)(x_{\pi}) &= \phi_{\kappa,\mc{V}}(f)(\sigma_{B_\mu,\wt{\pi}[p_A],\pi}^{-1}(x_{\pi}))\\
&= t(((f_i((\sigma_{B_\mu,\wt{\pi}[p_A],\pi}^{-1}(x))_v))_{|v \in A_{\gamma(i)}})_{i=1}^\delta). \end{split} \end{equation} Let $i \in I_k$ be such that $A_i \neq \emptyset$, and let $v \in A_i$. Comparing the last terms in (\ref{equvec}) and (\ref{equapik}) we can see that it is sufficient to prove that \begin{equation} (\sigma_{B_\mu,\wt{\pi}[p_A],\pi}^{-1}(x))_v = x_{\wt{\pi}[v]}. \end{equation}
Indeed, for $l \in J_i$ it holds \begin{equation} \begin{split} ((\sigma_{B_\mu,\wt{\pi}[p_A],\pi}^{-1}(x))_v)_l &= (\sigma_{B_\mu,\wt{\pi}[p_A],\pi}^{-1}(x))_{(l,v_l)}\\ &= x_{(l,\pi_l(v_l))}\\ &= (x_{\wt{\pi}[v]})_l.\\ \end{split} \end{equation} \end{proof}
Below we provide the proof of Lemma \ref{lemPrBetterCov}. \begin{proof} The thesis of is equivalent to \begin{equation}\label{sharp0E1C1} 0 < \frac{2N-1}{N^2} \mathbb{E}(\overline{Y^2}^2) - 2\mathbb{E}(\overline{Y^2}\overline{Y}^2) + \mathbb{E}(\overline{Y}^4). \end{equation} We have \begin{equation}\label{ey22} \mathbb{E}(\overline{Y^2}^2) = \frac{1}{N^2}(N\mathbb{E}(X^4) + N(N-1)\mathbb{E}^2(X^2)). \end{equation} Using the fact that $\mathbb{E}(X) = 0$ we receive \begin{equation}\label{ey2y2} \begin{split} \mathbb{E}(\overline{Y^2}\overline{Y}^2) &= \frac{1}{N^3}\mathbb{E}(NY_1^2(Y_1^2 +(N-1)(Y_2^2))) \\ &= \frac{1}{N^2}(\mathbb{E}(X^4) + (N-1)\mathbb{E}^2(X^2)) \end{split} \end{equation} and \begin{equation}\label{ey4}
\mathbb{E}(\overline{Y}^4) = \frac{1}{N^4}(N\mathbb{E}(X^4) + N3(N-1)\mathbb{E}^2(X^2)), \end{equation} where the coefficient $N3(N-1)$ appears since to get a product of squares $Y_i^2Y_j^2$ for some $i \neq j$ when performing multiplication in $(\sum_{i=1}^N Y_i)^4$ one can choose some $i$th of $N$ summands from the first sum, the same summand from one of three other sums, and some $j$th of $N-1$ remaining summands in the two remaining sums. Substituting (\ref{ey22}), (\ref{ey2y2}), and (\ref{ey4}) into (\ref{sharp0E1C1}), we receive \begin{equation} \begin{split} 0 &< \left(\frac{2N-1}{N^3} -\frac{2}{N^2} + \frac{1}{N^3}\right)\mathbb{E}(X^4) \\ &+ \left(\frac{2N-1}{N^2}\frac{N-1}{N} - \frac{2(N-1)}{N^2} + \frac{3(N-1)}{N^3}\right)\mathbb{E}^2(X^2), \end{split} \end{equation} which is equivalent to \begin{equation} 0<\frac{2(N-1)}{N^3}\mathbb{E}^2(X^2). \end{equation} \end{proof}
Below we provide the proof of Lemma \ref{lemC1C2}. \begin{proof} Let $\wt{P} \sim \mu_P^N$, $\wt{R} \sim \mu_R^N$, and for $i \in I_N$,
\begin{equation}\label{bpi}
B_{P,i} = f_P(\wt{P}_i)- \frac{1}{N}\sum_{j=1}^{N} f_P(\wt{P}_j),
\end{equation}
and
\begin{equation}
B_{R,i,l} = f_{R,l}(\wt{R}_i)- \frac{1}{N}\sum_{j=1}^{N} f_{R,l}(\wt{R}_j).
\end{equation}
We have
\begin{equation}
\wh{cE}_{k,C1E(N)}(f_l)(\wt{P}, \wt{R}) = \frac{1}{N-1}\sum_{i=0}^{N-1} (B_{P,i} + B_{R,i,l})B_{P,i},
\end{equation}
and
\begin{equation}\label{covl2}
\begin{split}
\mathbb{E}(\wh{cE}_{k,C1E(N)}^2(\wt{P}, \wt{R})) &\geq \frac{1}{(N-1)^2}\mathbb{E}(\sum_{i,j \in I_N}B_{P,i}B_{R,i,l}B_{P,j}B_{R,j,l}) \\
& = \frac{1}{N-1}\Var(f_P(P))\Var(f_{R,l}(R)),
\end{split}
\end{equation}
where in the first inequality we used the fact that
$\mathbb{E}(B_P[i]^2B_P[j]B_R[j]) = 0$, $i,j \in I_N,$ and in the last equality the easy to check equalities
$\mathbb{E}(B_P[i]^2) = \frac{N-1}{N}\Var(f_P(P))$, $i \in I_N$, and $\mathbb{E}(B_P[i]B_P[j]) = \frac{1}{N}\Var(f_P(P))$, $i,j \in I_N, i \neq j$.
If $\Var(f_{R,l}(R))\rightarrow \infty$ as $l \rightarrow \infty$, then so does the rhs of (\ref{covl2}). \end{proof}
\chapter{\label{appd}New analytical expressions for the SB model} As we justified in Appendix D in \cite{badowski2011}, in the SB model one can replace the considered one birth process with rate equal to the sum of coordinates of random vector $K = (K_i)_{i=1}^3$ with three birth processes with rates equal to its consecutive coordinates without changing the conditional distribution of the model output given the parameters and thus the quantities computed here. We will perform the computations using construction of a process of MR given by integral equation (\ref{intEqu}) but with random parameters \begin{equation}\label{sumIndep} Y_t = C + \sum_{i=1}^3 N_i(K_it). \end{equation}
The values of the main and total sensitivity indices of conditional expectation of output given the parameters were computed in Appendix D of \cite{badowski2011} and we provide them along with results of below computations in Table \ref{exactVal}. For each $\lambda > 0$, \begin{equation} \mathbb{E}(N(\lambda)) = \lambda \end{equation}
and
\begin{equation}\label{Poiss2}
\mathbb{E}(N(\lambda)^2) = \lambda^2 + \lambda.
\end{equation} Furthermore, from Theorem \ref{indepCond}, for $i \in I_3$, \begin{equation}\label{NCond}
\mathbb{E}(N_i(K_i t)|K_i) = (\mathbb{E}(N_i(k_it)))_{k_i = K_i} = K_it, \end{equation} and thus \begin{equation}\label{condexpSB}
\mathbb{E}(Y_t|P) = C + t\sum_{i=1}^3 K_i. \end{equation} We can see that the conditional expectation is linear in the model parameters, so its nonlinearity coefficients with respect to all subvectors of the parameter vector are zero. From the iterated expectation property we have \begin{equation} \mathbb{E}(Y_t) = \mathbb{E}(C) + t\sum_{i=1}^3\mathbb{E}(K_i) = 60 + 100(0.6 + 1 + 0.1) = 230. \end{equation}
From (\ref{condexpSB}), the coefficient of $C - \mathbb{E}(C)$ in the orthogonal projection of the mean output onto span of the centred parameters and constants fulfills \begin{equation}
bE_{C} = \frac{\Cov(\mathbb{E}(Y_t|P),C)}{\Var(C)} =1 \end{equation} and for the kinetic rates we have
\begin{equation}
\Cov(\mathbb{E}(Y_t|P), K_i) = \Cov(tK_i,K_i) = t\Var(K_i)
\end{equation} and thus \begin{equation} bE_{K_i} = t =100,\ i \in I_3. \end{equation}
Due to (\ref{CQKP}) and (\ref{Poiss2}), the conditional variance of $Y_t$ given $P$ is equal to \begin{equation}
\begin{split}
\Var(Y_t|P) &= (\Var(c + \sum_{i=1}^3N_i(k_it)))_{p = P} \\ = \sum_{i=1}^3K_it.
\end{split} \end{equation} Thus, similarly as for the conditional expectation, the nonlinearity coefficients of the conditional variance with respect to all subvectors of the parameter vector are equal zero.
Furthermore, $AveVar = \mathbb{E}(\Var(Y_t|P)) = 170$, $VVar_C =VVar_C^{tot} = bVar_C = 0$, and from (\ref{condexpSB}), $VVar_{K_i} = VVar_{K_i}^{tot} = V_{K_i},\ i \in I_3$. Using the values of $\wt{V}_{K_i}, i \in I_3$, computed in \cite{badowski2011} (see Table \ref{exactVal}), we receive \begin{equation} VVar_{P} = \sum_{i=1}^{3} V_{K_i} = 378. \end{equation}
We also have $\Cov(K_i,\Var(Y_t|P)) = t\Var(K_i),$ and thus $bVar_{K_i} = 100$, $i \in I_3$. \end{appendices}
\end{document} | arXiv |
\begin{definition}[Definition:Cluster Point of Filter]
Let $S$ be a set.
Let $\powerset S$ denote the power set of $S$.
Let $\FF \subset \powerset X$ be a filter on $S$.
Let $x \in S$ be an element of every set in $\FF$:
:$x \in X: \forall U \in \FF: x \in U$
Then $x$ is a '''cluster point of $\FF$'''.
\end{definition} | ProofWiki |
Attention, Perception, & Psychophysics
Sensory uncertainty leads to systematic misperception of the direction of motion in depth
Jacqueline M. Fulvio
Monica L. Rosen
Bas Rokers
Although we have made major advances in understanding motion perception based on the processing of lateral (2D) motion signals on computer displays, the majority of motion in the real (3D) world occurs outside of the plane of fixation, and motion directly toward or away from observers has particular behavioral relevance. Previous work has reported a systematic lateral bias in the perception of 3D motion, such that an object on a collision course with an observer's head is frequently judged to miss it, with obvious negative consequences. To better understand this bias, we systematically investigated the accuracy of 3D motion perception while manipulating sensory noise by varying the contrast of a moving target and its position in depth relative to fixation. Inconsistent with previous work, we found little bias under low sensory noise conditions. With increased sensory noise, however, we revealed a novel perceptual phenomenon: observers demonstrated a surprising tendency to confuse the direction of motion-in-depth, such that approaching objects were reported to be receding and vice versa. Subsequent analysis revealed that the lateral and motion-in-depth components of observers' reports are similarly affected, but that the effects on the motion-in-depth component (i.e., the motion-in-depth confusions) are much more apparent than those on the lateral component. In addition to revealing this novel visual phenomenon, these results shed new light on errors that can occur in motion perception and provide a basis for continued development of motion perception models. Finally, our findings suggest methods to evaluate the effectiveness of 3D visualization environments, such as 3D movies and virtual reality devices.
Motion: in depth 3D perception: *other Binocular vision: neural mechanisms and models
The online version of this article (doi: 10.3758/s13414-015-0881-x) contains supplementary material, which is available to authorized users.
The accurate perception of object motion is critical to survival. Although we have made major advances in our understanding of motion perception based on the processing of lateral (2D) motion signals on computer displays, the vast majority of motion in the real (3D) world occurs outside of the plane of fixation, and motion directly towards or away from the observer tends to have particular behavioral relevance.
One insight gained from the study of 2D motion perception is that when two objects with physically identical speeds but different contrasts translate on a computer screen, observers tend to report that the lower contrast object moves more slowly (Thompson, 1982; Stone & Thompson, 1992). This phenomenon has been explained in terms of perceptual inference, whereby the reduction in contrast and the associated increase in sensory uncertainty, result in a proportionally larger contribution of prior expectations. Because prior experience tells us that most objects in our environment tend to be stationary or move slowly, poorly visible stimuli therefore appear to move more slowly (Yuille & Grzywacz, 1988; Weiss, Simoncelli, & Adelson, 2002; Stocker & Simoncelli, 2006).
Since misperception of the motion of an approaching object can have serious consequences, we would like to know if similar effects occur for perception of 3D motion. Indeed, previous work has reported systematic biases in the estimation of both real and virtual object motion in depth, such that objects appear to move more sideways (Harris & Dean, 2003; Welchman, Tuck, & Harris, 2004; Harris & Drga, 2005; Gray, Regan, Castaneda, & Sieffert, 2006; Poljac, Neggers, & van den Berg, 2006; Lages, 2006; Rushton & Duke, 2007; Welchman, Lam, & Bülthoff, 2008; Duke & Rushton, 2012). This lateral bias is thought to arise based on the geometry of 3D motion perception and the mechanism for 2D speed perception described above (Welchman et al., 2008).
However, such bias is somewhat puzzling. Everyday behavior does not seem to be routinely affected by laterally biased estimates of object motion. The goal of the current study, then, is to revisit this bias and systematically investigate the accuracy of 3D motion perception.
We first assessed performance under relatively optimal conditions, and subsequently investigated changes in performance under two manipulations of sensory noise. In the first manipulation, we increased sensory noise through reductions in stimulus contrast, similar to manipulations employed in the study of biases in 2D motion perception. In the second manipulation, we varied the target's position in depth relative to fixation. The shift of target position in depth does not affect the sensory uncertainty associated with the two retinal images, but we reasoned that it should impact sensory uncertainty due to reduced sensitivity to stimulus properties, such as binocular disparity away from the plane of fixation (Westheimer & Tanzman, 1956; Blakemore, 1970; Schumer & Julesz, 1984; Landers & Cormack, 1997). We further reasoned that if both sensory noise manipulations have similar consequences from a perceptual inference perspective, they should have similar impacts on behavioral performance.
To anticipate, we find little evidence for the previously reported bias in 3D motion perception. Instead, we find relatively accurate performance under optimal conditions. In addition, we find a surprising novel phenomenon, whereby observers systematically confuse the motion in depth direction of a target's motion (i.e., the observer will report approaching motion when the object is in fact receding and vice versa). Because these confusions become more prevalent with manipulations of both contrast and target position, we conclude that sensory uncertainty in general leads to these systematic confusions of the direction of motion in depth.
Experiment 1: Behavioral performance in a 3D motion extrapolation task ("3D Pong")
The goal of Experiment 1 was to establish performance under relatively optimal, low-sensory noise conditions. We measured performance with high-contrast targets moving near the fixation plane. Because our initial observations were inconsistent with previously reported biases in the perception of 3D motion, we verified the robustness of this tendency across five additional levels of relatively high target contrast.
Five experienced observers, including one author (JMF), and four inexperienced observers participated in the experiment. All had normal or corrected-to-normal vision. The experiments were performed in accordance with the guidelines of The University of Wisconsin - Madison Institutional Review Board, and all observers gave informed written consent.
The experiments were performed using Matlab and the Psychophysics Toolbox (Brainard, 1997; Pelli, 1997) on a Windows 7 computer with an Nvidia Quadro 4000 video card. All stimuli were presented on a 29-cm × 51-cm 3D LCD display (Planar, 120 Hz, 1920 × 1080 pixels) at a viewing distance of 90 cm (29.54° × 16.62° of visual angle) such that 1 pixel subtended 0.015° of visual angle. Stimuli were anti-aliased to achieve subpixel resolution. Observers viewed the display through Nvidia 3D shutter glasses, which were synched with the refresh rate of the display (60 Hz per eye). The experiment was conducted in a dark room, with the display being the only source of illumination. The luminance of the display was linearized using standard gamma-correction procedures with mean luminance = 3.85 cd/m2 when viewed through the shutter glasses. Observers used a keyboard to make responses and completed the trials at their own pace.
Observers essentially played a 3D version of the video game Pong. All stimuli were rendered according to the laws of projective geometry (a visual scene was rendered for the left- and right-eye separately using an asymmetric camera frustum in OpenGL). This meant that the visual stimuli contained correct monocular cues (size, looming, perspective) as well as binocular cues (disparity, interocular velocity). However, given that we presented all stimuli on a computer screen, the display lacked correct accommodative/blur cues when stimulus elements moved outside of the fixation plane. We used a chin-rest to maintain head position. To maintain fixation and aid vergence, stimuli were presented within a circular aperture (7.5° in radius), surrounded by a 1/f noise pattern that was identical in both eyes. The background seen through the aperture was black (0.013 cd/m2; Fig. 1a). In addition, a set of Nonius lines was presented within a small 1/f noise patch at the center of the display to further aid vergence and fixation.
Details of all three experiments. a Screenshot of the visual stimulus. Observers wore shutter glasses, so that the left- and right-eye images could be fused. The display consisted of a central fixation patch and Nonius lines, a surround 1/f noise texture, a white target (not shown), and an adjustable textured "paddle" (depicted). As the paddle's position was adjusted by the observer, it rotated about the target's start position, in the x-z plane. Note that in Experiments 2 & 3, the aperture was mid-gray in color (see those sections below). b Schematic of the experimental task. Observers fixated the center of the display. A white target of variable contrast (depicted black for demonstration purposes) appeared and moved in a random direction in 360° ("Observation"). After moving for 1 s, the target disappeared and a textured paddle appeared. Observers were asked to adjust the location of the paddle around an invisible circle that "orbited" the target's origin of motion so that it would intercept the target if it continued along its trajectory ("Estimation"). Feedback was not provided. c Computation of setting judgment error. The judgment error on each trial was computed as the circular distance between the midpoint of the paddle setting ("reported direction") and the true target trajectory endpoint ("presented direction"). d Classification of judgment errors. When observers' reported directions are plotted against the presented directions, the positive diagonal corresponds to accurate performance. Data points that fall directly on the negative diagonal correspond to reports where the lateral component of the target's motion is judged accurately, but the motion in depth component is confused. Left plot: Data points that fall within the red regions correspond to incorrect reports of the direction (i.e., approaching (downward facing arrows) vs. receding (upward facing arrows)) of the target's motion in depth. Right plot: Data points that fall within the red squares correspond to incorrect reports of the target's lateral motion direction (left- vs. rightward)
On each trial, a 0.43-cm diameter dot ("target"), 0.25° at the 90-cm viewing distance, appeared at fixation (i.e., at the midpoint of the screen plane). The target was rendered with one of six contrast levels (Weber fractions: 434.34, 62.16, 44.87, 38.85, 18.55, and 10.28), which corresponded to luminance values of 5.79 cd/m2, 0.84 cd/m2, 0.61 cd/m2, 0.53 cd/m2, 0.26 cd/m2, and 0.15 cd/m2 when viewed through the shutter glasses. (A fully white object was 5.79 cd/m2 when viewed through the shutter glasses). Note that these Weber fractions are all >1, meaning that all contrast levels in this experiment were significantly larger than the corresponding Michelson contrast (100 %) typically used in 2D motion experiments. We will turn to the effect of Weber fractions <1 in Experiment 2.
The target followed a random trajectory defined by independently chosen random speeds in the x (lateral) direction and the z (motion in depth) direction, with no change in y (vertical direction) before disappearing. Velocities in x and z were independently chosen from uniform distributions with magnitudes ranging between (0.02:6.2 cm/s). The velocities were then given a random sign, so that for approximately 50 % of the trials, the stimulus approached/receded. Given the stimulus presentation time of 1 second, the target's motion produced an average maximum binocular disparity of 0.27°. The independently chosen x- and z-velocity components were critical for preventing observers from adopting strategies that bypass the need for accurate estimation of both x- and z-velocity components from the retinal signals.
A 3D rectangular block (paddle) whose faces also consisted of a 1/f noise pattern was positioned within the display at a simulated radial distance of 12.4 cm from the target's initial start position. During adjustment, the paddle moved along a circular path around the target's initial start position in the x-z plane with the same side of the paddle facing the target's start position at all times. No translation occurred in y. The 1-cm wide paddle was positioned 12.4 cm from the target's initial start position, so that it was at a sufficient distance to keep the target's motion within the fuseable area while providing appreciable depth to the displays. A movie demonstrating the stimulus trial sequence can be viewed in the Supplementary Material
Observers were first provided with written and oral instructions from the experimenter. They then performed 10-15 practice trials in the presence of the experimenter to become familiar with the apparatus. During this time, the observer also was asked to note whether the target was approaching or receding. If observers were unable to make these judgments, reported difficulty themselves, or were judged to have difficulty viewing the stimulus in any other way, they were excluded from further participation (this amounted to <10 % of potential observers for the entire study). All remaining participants completed the experimental trials in two sessions (Experiment 1) and one session (Experiments 2 & 3), and all data collected were included in the subsequent analyses. No feedback was provided for either the practice or experimental trials.
On each trial, the observer fixated the region between the Nonius lines at the center of the screen. The target appeared at fixation and followed a linear trajectory defined by the random independent velocities in x and z chosen for that trial, sometimes appearing to come out of the screen ("approaching"), sometimes appearing to move back into the screen ("receding"). After 1 second, the target disappeared and the paddle immediately appeared. On the first trial, the paddle appeared to the far right of fixation (i.e., at 0°). On subsequent trials, the paddle appeared at the last location set by the observer. The observer was instructed to extrapolate the visible portion of the target's trajectory and adjust the paddle's position so that it would have intercepted the target if the target had continued along its trajectory. We asked observers to extrapolate the trajectory so that responses would be based on perceived motion direction rather than some heuristic, such as the location of target disappearance. Explicit visual and verbal instructions, as well as the task familiarization phase validated that observers understood the task as an extrapolation task. During the setting phase of the trial, the observer's eyes were free to move about the display. When the observer was satisfied with the paddle setting, he resumed fixation and pressed the spacebar to initiate a new trial (Fig. 1b).
The angle corresponding to the midpoint of the paddle setting (reported direction) on each trial provided the observer's estimate of the target motion direction (presented direction) on that trial. To quantify overall performance, we computed the error for each trial as the circular distance between the presented motion trajectory and the reported paddle position (Fig. 1c) using the CircStats toolbox for MATLAB (Berens & Velasco, 2009). To draw comparisons to the existing literature that has consistently reported lateral biases in observer reports, we took the direction (sign) of the observer's report relative to the physical trajectory into account (Fig. 1c). Settings that were laterally biased as in previous studies, i.e., biased towards the fixation plane, were assigned positive values. Conversely, settings that were medially biased, i.e., biased towards the vertical plane through the viewing direction (the midsagittal plane) were assigned negative values. Thus, the circular mean of these signed errors not only provide a measure of each observer's overall accuracy for the particular condition, but also a measure of any systematic biases (lateral or medial) in their estimates—negative values correspond to medial bias and positive values correspond to lateral bias.
To summarize performance as a function of sensory uncertainty, we used two basic summary measures of performance: 1) confusions in the motion in depth direction of the target's motion (i.e., approaching vs. receding); 2) confusions in the lateral direction of the target's motion (Fig. 1d).
We begin by considering performance in the high target contrast condition (Weber fraction = 434.34). Each of the five individual observers' reported motion direction is plotted as a function of the presented directions in Fig. 2a (O2 is JMF). The between-subject mean signed error in the reports (see Methods section) was 3.39° [bootstrapped 95 % confidence interval (CI) 1.79, 4.91°], revealing a small, but significant, lateral bias. The paddle itself covered a width of 4.63° around the circular path, so that the observers' paddle intercepted the target on average.
Experiment 1 results: Reported 3D motion direction as a function of presented direction for all five observers for the high contrast target condition (Weber Fraction = 434.34). O2 = JMF. a The reported direction based on the midpoint of the paddle setting is plotted against the trajectory direction the target followed on each trial across the full 360° stimulus space (n = 200 per plot). The box inset represents a range of approaching stimuli [−64:64° relative to straight, head-on motion] used in several previous studies. b A closer look at performance within the inset from a, with the data plotted in the same format
To compare performance to previous reports (Welchman et al., 2004; Welchman et al., 2008), we also briefly consider performance in response to target motion within the range of −64:64° relative to head-on motion (270°), denoted by the inset in Fig. 2a and the data in Fig. 2b. The between-subject mean signed error was −4.25° [bootstrapped 95 % CI −5.30, −3.19°], indicating a small but medial bias within this subset of presented target motion—that is, the settings were closer to the midline than the true target trajectories, revealing an inconsistency with the pattern reported in previous studies.
Next, we investigated performance across five additional levels of reduced, but still relatively high, contrast levels. The reported directions for all five observers in each of the six total target contrast conditions are plotted as a function of the presented directions in Fig. 3a. Reduced target contrast is associated with a significant increase in mean signed error (repeated-measures ANOVA, F(5,20) = 6.8, p < 0.001) with bias occurring in the medial direction on average for the higher contrast targets and in the lateral direction on average for the lower contrast targets. However, this effect is driven largely by performance in the lowest contrast condition—removing this condition from the analysis reveals a marginal nonsignificant effect of reduced target contrast on mean signed error (F(4,16) = 1.72, p = 0.19). When the lowest target contrast condition is excluded and the between-subject mean signed error is computed for the remaining five target contrast conditions combined, the result is a mean error of 0.10° [bootstrapped 95 % CI −0.62, 0.81°] and does not reveal a significant bias. Thus, the small bias reported for the highest target contrast condition above may simply be due to random variability in performance.
Combined observer performance in 3D motion in depth estimation for six target contrast conditions. a Reported direction plotted as a function of presented direction in the same format as Fig. 2 for the six target contrast conditions expressed in Weber fractions as viewed through the shutter glasses (n = 1,000 per plot, combined data for 5 observers). The overlaid pale red regions correspond to the regions of confusion when reporting motion in depth direction (left column) and the regions of confusion when reporting lateral motion direction (right column). See Methods and Fig. 1 for more detail. b Mean percentage of trials in which observers confused the target's direction of motion in depth (i.e., approaching vs. receding). c Mean percentage of trials in which observers confused the target's lateral motion direction. Error bars correspond to bootstrapped 95 % CIs
Due to the decline in performance for the lowest target contrast condition, reduced target contrast is associated with a significant increase in the tendency to confuse whether the target is approaching or receding (Fig. 3b; repeated-measures ANOVA, F(5,20) = 10.2, p < 0.001). On the other hand, there is no effect of reduction in target contrast in the proportion of trials in which observers confuse the lateral direction of motion in depth (Fig. 3c; repeated-measures ANOVA, F(5,20) = 0.73, p = 0.61).
In contrast to previous work, these results reveal highly accurate performance under high-contrast conditions, with little evidence for systematic biases in the judgment of 3D motion direction. When contrast is reduced, the pattern of responses indicates that observers begin to confuse motion in depth, but not lateral motion, revealing a previously undocumented illusion in the perception of 3D motion. Rather than a lateral bias, observers seem to confuse approaching and receding motion. Given the potentially problematic nature of such confusions in the real world, we further explore this phenomenon in the next two experiments. As an aside, O3 & O4 seem to be confusing the direction of motion in depth, for receding, but not approaching, motion even under optimal conditions. We believe this is due to a separate bias in judging the position of stimuli as a function of contrast. We will return to this point in the discussion.
Experiment 2: Manipulating target contrast
Experiment 1 demonstrated that, over a broad range of target contrasts, 3D motion in depth trajectory extrapolation performance is relatively accurate. However, for the lowest contrast level, larger errors did emerge so that all observers became more likely to confuse the direction of the target's motion in depth (i.e., reporting that the target was approaching, when it was in fact receding) even when observers' ability to identify the lateral direction of the target's motion was not impacted. Having established the type of impact target contrast has on 3D trajectory extrapolation, we designed Experiment 2 to explore the impact of target contrast by further reducing target contrast. This was achieved by changing the stimulus aperture to mid-gray. We then used three target Weber contrast levels <1.
Five observers from the UW-Madison community with intact stereovision as determined by the task acclimation phase (General Methods) participated in the experiment. Two were authors (J.F. and M.R.) and three were new and naïve to the purpose of the experiment. The authors' performance did not differ from the naïve observers and thus their data are combined in the analyses.
The stimuli were identical to those of Experiment 1 with the exception that the aperture was set to mid-gray with luminance = 3.66 cd/m2 when viewed through the shutter glasses. The target was rendered with one of three Weber fraction contrast levels (0.58, 0.33, 0.17), which corresponded to luminance values of 5.79 cd/m2, 4.87 cd/m2, and 4.24 cd/m2 when viewed through the shutter glasses. The three target contrast levels were counterbalanced and presented in pseudorandom order. Each contrast level was tested 200 times for a total of 600 trials per observer.
The reported trajectory directions for the three target contrast conditions for all observers as a group are plotted in Fig. 4a. With reductions in target contrast, there is a significant increase in the proportion of trials in which observers confuse the direction of motion in depth (Fig. 4b; repeated-measures ANOVA; F(2,8) = 64.08, p < 0.01). Although there also appears to be a slight increase in the proportion of trials in which observers confuse the lateral direction of motion in depth with reduction in target contrast, the effect is not significant (Fig. 4c; repeated-measures ANOVA; F(2,8) = 2.62, p = 0.13). We will discuss this apparent pattern in more detail after the next experiment.
Combined observer performance in 3D motion in depth estimation for three contrast conditions. a Reported direction plotted as a function of presented direction in the same format as Fig. 2 for the three target contrast conditions expressed in Weber fractions as viewed through the shutter glasses (n = 1,000 per plot, combined data for 5 observers). Left: 0.58; Middle: 0.33; Right: 0.17. b Mean percentage of trials in which observers confused the target's direction of motion in depth (i.e., approaching vs. receding) as a function of target contrast following the order of a. c Mean percentage of trials in which observers confused the target's lateral motion direction in the same format as b. Error bars correspond to ± −1 SEM
Experiment 3: Manipulating position-in-depth
In the preceding experiments, we demonstrated that reductions in target contrast were associated with (i) an increased tendency to confuse the motion in depth direction of the target's motion, and (ii) no significant impact on the ability to estimate the lateral motion direction of the target's motion. We argued that these impacts resulted from increased target motion uncertainty due to increased sensory noise, rather than the contrast of the target per se. If our reasoning is correct, the specific source of the sensory uncertainty should be irrelevant, and other manipulations that increase sensory noise should be associated with similar impact on observer performance. To test this assertion, we fixed the target contrast to the highest level used in Experiment 2 but varied the reliability of binocular information through shifts in the target's position in depth relative to the plane of fixation.
Ten new, naïve observers from the UW-Madison community with intact stereovision as determined by the task acclimation phase (General Methods) participated in the second experiment along with two of the authors (J.F. and M.R.).
The stimuli were identical to those of Experiment 2, with the exception that the target's start position occurred at one of two new locations (Fig. 5a). For the "in front of fixation" (Near) condition, the target's start position was shifted 6.2 cm "out of" the display. For the "behind fixation" (Far) condition, the depth component of the target's start position was shifted 6.2 cm "into" the display. There was no change in the horizontal component of the target's start position. The fixation point remained at the midpoint of the display. The paddle's position was shifted in depth so that it continued to circle the origin of the target's motion. The target was always presented at the maximum Weber contrast fraction value of 0.58 used in Experiment 2.
Combined observer performance in 3D motion in depth estimation for three positions in depth configurations. a Reported direction plotted as a function of presented direction in the same format as Fig. 3 for the three start positions: near, at fixation, and far (n = 1,400 per plot for Near and Far; n = 1,000 for At Fixation). Note that the At Fixation data are the data from the high contrast condition of Experiment 2 for comparison. b Mean percentage of trials in which observers confused the target's direction of motion in depth (i.e., approaching vs. receding). c Mean percentage of trials in which observers confused the target's lateral motion direction. Error bars correspond to ± −1 SEM
Each of the ten naïve observers were assigned randomly to one of the two start position conditions (5 per group). Each completed 200 trials. The two authors participated in both conditions. The authors' performance did not differ from the naïve observers, and thus their data are included in the analyses, yielding a total of seven observers per group. All other experimental details are identical to Experiment 2.
Figure 5a shows the reported direction as a function of the presented direction for all seven observers in each of the two target start position conditions. For comparison, the nonshifted ("At Fixation" start position) data for the five observers from Experiment 2 also are shown. As expected, we find a significant increase in the proportion of trials in which the motion in depth direction of the target was confused for the two shifted target position conditions (F(2,16) = 8.243, p < 0.01; Fig. 5b). The shift in the target position relative to fixation did not significantly alter the proportion of trials in which observers confuse the lateral motion component (F(2,16) = 0.464, p = 0.64; Fig. 5c).
Quantifying the impact of uncertainty on 3D motion estimation
In three experiments, we have provided new insight into the impact of sensory uncertainty on 3D motion estimation. One of the consequences of increased uncertainty as revealed by performance in our task is a tendency for observers to misreport the direction of the motion-in-depth component under conditions of greater uncertainty. Whereas reported lateral motion did not differ significantly across levels of uncertainty, this may simply have been due to the low proportion of lateral motion confusions in our data (<3 %). We therefore wanted to test if the 3D motion estimation process breaks down more generally with increased sensory uncertainty, by nonetheless revealing an effect of uncertainty on reported lateral motion.
Towards this end, we compared the relative reliability of observers' estimates of the two motion components (x and z) in our task. Observers' responses reflect the combination of the perceived lateral and motion in depth speed components of the target's motion on each trial, so we cannot test the reliability of those estimates separately. However, we can index the uncertainty in-motion, in-depth component on a trial-by-trial basis according to whether or not the observer reported the correct direction of motion in depth. We assume that sensory uncertainty was lower for trials in which the observer reported the correct direction of motion and that sensory uncertainty was higher for trials in which the observer reported the incorrect direction of motion. This binary measure is simply a categorization of a continuum of uncertainty. The lateral judgment error variance conditioned on the accuracy of the corresponding depth judgment provides a measure of uncertainty in lateral motion component estimation.
The extent to which motion-in-depth and lateral measures of sensory uncertainty vary together is indicative of the nature of break down in the 3D motion estimation process. The common source of sensory information (i.e., the retinal motion) used to derive these estimates predicts a correlation between the measures such that uncertainty in the motion in depth direction should be associated with greater variability in lateral judgments. Previous work has reported that observers do not rely on the z-component of motion when reporting trajectory direction (Harris & Dean, 2003; Harris & Drga, 2005). This would result in the two measures not being correlated. Such an outcome would suggest that estimation of motion in depth and lateral motion for the targets in our task share no common mechanisms (which could be the case, for example, if lateral motion is velocity-derived and motion in depth is disparity-derived).
Lateral judgment errors were computed as the horizontal distance between the location at which the true target trajectory would intercept the circular "orbit" on each trial projected directly on the fixation plane, denoted x i , and the midpoint of the paddle setting projected on the fixation plane denoted \( {\widehat{x}}_i \) (Fig. 6a). The lateral judgment error variance, conditioned on the accuracy of the corresponding depth judgments was then computed as:
Uncertainty in z-estimation is associated with increased uncertainty in x-estimation. a Lateral judgment errors. Top left: Schematic showing the projection of the observer's paddle setting (pink arrow) and the intersection point of the true trajectory (red arrow) to the fixation plane. Lateral judgment error is defined as the difference in the positions on the fixation plane. Top right: If uncertainty in the estimates of the two motion components is independent, then the variance in lateral judgment errors should be equivalent on trials in which the observer correctly reports the target's depth direction (red arrow; low uncertainty in motion in depth) with trials in which the observer incorrectly reports the target's depth direction (pink arrow; high uncertainty in motion in depth). Bottom left: If there is an increase in uncertainty in the depth component estimation associated with an increase in uncertainty in lateral component estimation, the variance of lateral judgment errors will be larger on trials in which the observer incorrectly reports the target's direction in depth (pink arrow) than on trials when the direction in depth is correctly reported (red arrow). Bottom right: If the observer relies on an estimate of target motion direction on trials in which uncertainty in motion in depth component estimation is low but abandons such estimation and relies on a predictable default location when uncertainty in motion in depth component estimation is high, lower variance of judgment errors is expected for trials in which the observer misreports the depth direction (pink arrow). b Results of the analysis. The lateral judgment error variance for incorrectly reported depth trials is plotted against the lateral judgment error variance for correctly reported depth trials for each observer in each of the conditions for the experiments described above. The symbol colors correspond to the Weber contrast fraction. Note that the Near and Far conditions of Experiment 3 have Weber fraction = 0.58, the largest used in Experiment 2 (denoted by the 'x' symbols). See the individual experimental sections for the corresponding luminance values. The majority of the data points fall above the identity line, including the group mean (± −1 SEM), which is consistent with the predicted results if the uncertainty in the estimates of the two velocity components is correlated and both factor into the variance in judgment errors
$$ RMS=\sqrt{\frac{1}{n}{\displaystyle \sum_{i=1}^n{\left({\widehat{x}}_i-{x}_i\right)}^2}} $$
The lateral judgment variance for the two categories of trials (i.e., correct/incorrect motion in depth direction reports) for all observers and experimental conditions are plotted in Fig. 6b. The majority of the data points fall above the identity line, including the group mean, indicating that the lateral judgment error variance was larger when the motion-in-depth direction was judged incorrectly compared to when it was judged correctly under all levels of uncertainty. These results suggest that estimation of both an object's lateral motion component and motion in depth component are subject to a common source of sensory uncertainty. We previously did not find significant differences in lateral motion confusions with manipulations of sensory uncertainty. However, relating this finding back to the stereotypical pattern of results that emerged across these experiments and highlighting the motion in depth direction confusions, these results can be interpreted as providing direct evidence that trial-to-trial sensory uncertainty as revealed by motion in depth confusions has a similar, albeit much smaller, effect on lateral motion estimates.
We can rule out that the motion in depth direction confusions are the result of a random approaching versus receding response choice, independent of the lateral setting, as would be revealed by the data points falling along the identity line. Lastly, we can further rule out the use of a simple response heuristic in which observers provide a default response on trials in which they are uncertain about the motion in depth direction, as would be revealed by the data points falling below the identity line.
To conclude, the results indicate that the lateral and motion in depth components of the observer's report are affected by a common source of sensory uncertainty. The effect on the motion in depth component is simply much more apparent than the effect on the lateral component.
In the current study, we examined observer reports of the direction of targets moving in depth under manipulations of sensory noise. We uncovered a novel and surprising tendency to confuse the direction of target motion (i.e., confusing approaching and receding motion). These behavioral effects were consistent and systematic across conditions of increased sensory noise, suggesting that confusion in the direction of motion in depth is due to fundamental limitations of the sensory inference of 3D motion, rather than specific properties of the stimuli. These results shed new light on failures of human motion perception and provide a criterion to assess the quality of 3D visualization environments, such as 3D movie or virtual reality devices.
Although initial investigation indicated that the tendency to confuse the lateral component of the target's motion did not significantly increase with sensory noise, further investigation revealed that these effects are in fact related, such that an increase in uncertainty in motion in depth component estimation is associated with an increase in uncertainty in lateral motion component estimation on a trial by trial basis. Thus, the two motion components share a common source. It simply seems the case that the impact of sensory uncertainty is much more apparent in the observer's estimation of the z (depth) motion component.
These results contribute to the understanding of the types of errors that occur in 3D motion processing. It has been shown previously that 2D motion appears slower under high-noise conditions, but that even under low-noise conditions 3D motion appears to be systematically laterally biased—i.e., observers report motion as moving more sideways (Harris & Dean, 2003; Welchman et al., 2004; Harris & Drga, 2005; Gray et al., 2006; Poljac et al., 2006; Lages, 2006; Rushton & Duke, 2007; Welchman et al., 2008; Duke & Rushton, 2012). We do not find evidence for such bias in our data. The systematic decline of observer performance under the noisy conditions in our study supports the overall notion that uncertainty in 3D target motion gives rise to errors in the reported direction of motion (Welchman et al., 2008; Lages, 2006), but this seems to have altogether different effects than an increase in lateral bias.
Why might we observe results different from those reported previously? One distinction between our task and those used in most previous studies is that the full 360° space was utilized for both the stimuli and responses (but see Lages, 2006). In previous work, observers were often provided with a response space that was markedly inconsistent with the true set of stimuli. Specifically, the expected stimulus space comprised a wider (i.e., more lateral) range than the actual stimulus space. Observers were typically told that the stimulus could approach at any angle across the 180° range, whereas the stimuli in actuality were constrained within smaller ranges around the midsagittal plane. We hypothesize that invoking prior beliefs about the distribution of possible target motion trajectories is a strategy observers may use in overcoming the uncertainty in estimating motion in depth. We hypothesize that such a mismatch between one's prior beliefs and the true distribution of possible target motion trajectories may have been a factor in the lateral estimation biases reported previously. In the context of our study, the range of presented trajectory directions did span the full 360° space; however, we note that the use of uniformly distributed x- and z-velocities establishes anisotropies in the range such that presented trajectories near the 45/135/225/315° directions were more likely than those closer to the cardinal directions. Thus, although we do not see strong evidence in observers' settings, our observers may nevertheless have developed a prior for such a distribution. Future work focusing specifically on the range of presented trajectory directions, including work that utilizes a truly isotropic distribution of presented trajectory directions, might provide greater insight into the role and impact of prior beliefs on 3D motion direction estimation.
The directional confusions we report may have been observed previously in motion-in-depth estimation tasks. Lages (2006) established errors in the reported quadrant of the 360° space as a criterion to screen out observers before experimental participation. That is, directional confusions previously have been treated as indications that observers are unable to perform the task. We interpreted these confusions not as failures of stereovision per se, but rather as an index of uncertainty in estimation of the depth component. All our observers easily met the Lages criterion in that they correctly reported the motion in depth direction of the target on 75 % or more of trials for the majority of the contrast levels used. That observers (inexperienced and authors alike) can perform the task and meet the criterion under the best conditions, but start to confuse the direction of motion in depth under poor conditions, indicates that the confusions are a real phenomenon arising from increased sensory uncertainty rather than a fixed characteristic of an observer's stereomotion acuity.
Secondary to our main effects, observers seemed more likely to report receding motion as approaching under high-contrast conditions and approaching motion as receding under low-contrast conditions. This pattern of results is consistent with a "brighter is closer" heuristic. This heuristic mirrors the relationship between contrast and depth in natural scenes, sometimes known as proximity-luminance covariance (Coules, 1955; Schwartz & Sperling, 1983; Dosher, Sperling, & Wurst, 1986) and recently has been shown to bias observer reports of perceived depth in natural images (Cooper & Norcia, 2014). Thus, in the context of our study, observers may be influenced by a prior expectation of target position in addition to any prior expectations of target motion, especially in cases where sensory uncertainty is already high.
Finally, one could propose that these motion-in-depth confusions are due to observers not seeing the target on certain trials. However, if observers did in fact miss the target on a particular trial, the response on that trial should be random. Random responses would not only yield motion-in-depth direction confusions, but also lateral motion confusions (i.e., misreports of whether the target was traveling leftward or rightward). Importantly, we saw few lateral motion confusions in our data (<3 % of all trials), ruling out that potential explanation for these surprising errors.
This work extends our understanding of motion perception and contributes to a principled approach to understanding errors that arise in 3D motion perception. We have shown that the human ability to estimate motion in depth, in particular, the direction of motion in depth (i.e., approaching vs. receding) is limited by sensory uncertainty, which may arise from a variety of sensory noise sources.
This work was supported by the Netherlands Organization for Scientific Research (NWO) Veni Grant 451-09-030 to B. Rokers. Additional support was provided by NSF award SMA-1004961. The authors thank Andrew Welchman, Michael Landy, and one anonymous reviewer for helpful comments on a previous version.
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© The Psychonomic Society, Inc. 2015
1.Psychology DepartmentUniversity of Wisconsin-MadisonMadisonUSA
2.Psychology DepartmentUniversity of Central FloridaOrlandoUSA
3.Experimental PsychologyUtrecht UniversityUtrechtThe Netherlands
4.Department of PsychologyUniversity of Wisconsin - MadisonMadisonUSA
Fulvio, J.M., Rosen, M.L. & Rokers, B. Atten Percept Psychophys (2015) 77: 1685. https://doi.org/10.3758/s13414-015-0881-x
DOI https://doi.org/10.3758/s13414-015-0881-x
Published in cooperation with
The Psychonomic Society | CommonCrawl |
On the weight distribution of codes over finite rings
AMC Home
May 2011, 5(2): 407-416. doi: 10.3934/amc.2011.5.407
The number of invariant subspaces under a linear operator on finite vector spaces
Harald Fripertinger 1,
Institut für Mathematik, Karl-Franzens Universität Graz, Heinrichstr. 36/4, A-8010 Graz, Austria
Received March 2010 Revised April 2011 Published May 2011
Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb F$q and $T$ a linear operator on $V$. For each $k\in\{1,\ldots,n\}$ we determine the number of $k$-dimensional $T$-invariant subspaces of $V$. Finally, this method is applied for the enumeration of all monomially nonisometric linear $(n,k)$-codes over $\mathbb F$q.
Keywords: cyclic vector space, monomially nonisometric linear codes., enumeration, nite eld, Invariant subspaces, nite vector space.
Mathematics Subject Classification: Primary: 05E18; Secondary: 47A4.
Citation: Harald Fripertinger. The number of invariant subspaces under a linear operator on finite vector spaces. Advances in Mathematics of Communications, 2011, 5 (2) : 407-416. doi: 10.3934/amc.2011.5.407
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Harald Fripertinger | CommonCrawl |
Fermat's Last Theorem with complex powers, wrapped in a story every mathematician can relate to
by Robert J Low and Thierry Platini. Published on 6 March 2017.
Fermat's Last Theorem has been a source of fascination and the motivation for an enormous amount of mathematics over the last few centuries, both in attempts (eventually successful) to prove it and as the inspiration for other related questions.
This is the story of how an algebraic question inspired by Fermat's Last Theorem morphed into an analytic question, which subsequently turned out to be expressed best as a geometric question, which could be answered using basic methods of plane geometry.
Fit the first: curiosity
A common strategy in mathematical research is to consider something that is already known, and to try to generalise or vary it hoping for something interesting to appear in the process. Our initial question was motivated by Fermat's Last Theorem: are there solutions to
$x^{\,n\mathrm{i}}+y^{\,n\mathrm{i}}=z^{\,n\mathrm{i}}$
where $x, y, z$ and $n$ are positive integers and $\mathrm{i}^{2}=-1$?
An obvious simplification is to replace $x^{n}$, $y^{n}$ and $z^{n}$ by $x$, $y$ and $z$: if there are no solutions to
$x^{\,\mathrm{i}} + y^{\,\mathrm{i}} = z^{\,\mathrm{i}}$
for (non-zero) integers $x$, $y$ and $z$, then there are clearly none to the original problem, where $x$, $y$ and $z$ have to be $n$th powers.
So let's assume that $x,y,z$ satisfy this relationship. Now, any complex number $a+\mathrm{i}b$ can be written in the form $r\,(\cos\theta+\mathrm{i}\sin\theta)$, where $r$ is called the modulus and $\theta$ the argument of the complex number (see figure below). In this case, $z^{\,\mathrm{i}} = \mathrm{e}^{i\ln z} = \cos(\ln z)+\mathrm{i}\sin(\ln z)$, so $z^{\,\mathrm{i}}$ has modulus $1$ and argument $\ln z$.
The modulus, $r$, and argument, $\theta$, of a complex number. Using properties of right angled triangles, it can be shown that $r^2=a^2+b^2$ and $\tan\theta=b / a$
So now we know that
$(x^{\,\mathrm{i}}+y^{\,\mathrm{i}})(x^{-\mathrm{i}}+y^{-\mathrm{i}}) = z^{\,\mathrm{i}} z^{-\mathrm{i}} = 1$
and expanding the brackets gives
$2+(x/y)^{\,\mathrm{i}}+(x/y)^{-\mathrm{i}}=1,$
which rearranges to
$(x/y)^{\,\mathrm{i}}+(x/y)^{-\mathrm{i}}=-1.$
Using $(x/y)^{\,\mathrm{i}} = \cos(\ln (x/y))+\mathrm{i}\sin(\ln(x/y))$, we obtain
$2\cos(\ln(x/y))=-1,$
$\cos(\ln(x/y))= -\tfrac{1}{2}.$
Therefore an integer $k$ should exist such that $\ln(x/y)=2\pi k \pm 2\pi/3$ or, in other words,
$x/y= \mathrm{e}^{\,2\pi k \pm 2\pi/3}.$
So, remembering that $x$ and $y$ are integers, all we need to do now is to check whether $\mathrm{e}^{\,2\pi k \pm 2\pi/3}$ can be rational. It certainly seems unlikely to be rational: however, there are unlikely truths in mathematics. This seemed like a good time to climb up onto the shoulders of giants, and we did a little digging into transcendental number theory (using Transcendental Number Theory by A Baker). A number is rational if it is a zero of an expression of the form $mx+n$, where $m$ and $n$ are integers, ie it is a root of a linear polynomial with integer coefficients; it is algebraic if it is a root of any polynomial with integer coefficients; and it is transcendental if it is not algebraic. Our search showed us that $\mathrm{e}^{\,\pi}$ is called Gelfond's constant, which is known (from the Gelfond–Schneider theorem) to be transcendental. Furthermore, any algebraic function of it is transcendental, and so we can now conclude that $\mathrm{e}^{\,2\pi k \pm 2\pi/3}$ is transcendental for all integer $k$.
It then follows that there are no integer solutions to the equation
and so there are certainly none where $x$, $y$ and $z$ are all $n$th powers, ie there are no integer solutions to
$x^{\,n\mathrm{i}} + y^{\,n\mathrm{i}} = z^{\,n\mathrm{i}}$
for any positive integer values of $n$.
Fit the second: despair
We were initially happy to conclude this. We had found something interesting. Unfortunately, while looking up transcendental number theory, we found out that (by much the same argument) it was already well known from this theory that there are no solutions to
$x^{\,m+\mathrm{i} n}+y^{\,m+\mathrm{i} n}=z^{\,m+\mathrm{i} n}$
for positive rational numbers $x, y, z$ and exponents of the form $m+\mathrm{i} n$, where $m,n \in \mathbb{Z}$ with $n \neq 0$. So we had managed to prove a special case of a known result—and not for the first time.
For a while, we found this sufficiently off-putting that we stopped thinking about the problem. But finally we rallied, and did what mathematicians do. We changed the question.
Fit the third: enthusiasm
We asked a new question instead: are there positive real solutions to
$x^{\,\mathrm{i}}+y^{\,\mathrm{i}}=z^{\,\mathrm{i}}$
and, if so, is there anything interesting to say about them?
In fact, we already saw the clue as to how to solve this: for any positive real number $x$, the number $x^{\,\mathrm{i}} = \cos(\ln x)+\mathrm{i}\sin(\ln x)$ is a complex number of unit modulus. So now, instead of considering $x, y, z$, we realise that we are looking for three complex numbers $a\; (=x^{\,\mathrm{i}})$, $b\;(=y^{\,\mathrm{i}})$, $c\;(=z^{\,\mathrm{i}})$, such that
$a+b=c \quad \text{and} \quad|a|=|b|=|c| = 1.$
Algebra becomes geometry
At first sight, this has not helped a great deal. But interpreted geometrically, the problem suddenly becomes easy. Given $a$, since $a+b=c$, we immediately know that $|a+b|=1$. $|a+b|$ is the distance between $a$ and $-b$, which is 1. Similarly, we can use $a-c=-b$ to see that $|a-c|$, the distance from $a$ to $c$, must also be $1$. So we need to find a pair of points on the unit circle, centred at the origin, both a distance $1$ from $a$, as shown in the figure to the right.
We should note that $-b$ and $c$ both being a distance $1$ from $a$ is necessary; we must also check that it is sufficient. But it is now clear from the geometry that the segment connecting $a$ to $c$ is parallel to that connecting $-b$ to $0$, so we do indeed have $a+b=c$.
We can now use this geometry to find $b$ and $c$ given $a$: for if we let
$\alpha,\beta,\gamma$ be the arguments of $a,b,c$ (or equivalently, the natural
logarithms of $x,y,z$), the relationship between $\alpha, \beta$ and $\gamma$ is easily seen. Since each of $a,b,c$ has unit modulus, the triangle with vertices at $0$, $a$ and $c$ is equilateral, and so all internal angles are $\pi /3$. It immediately follows that $c$ has argument $\gamma=\alpha+\pi/3$ and that $b$ has argument $\alpha+2\pi/3$.
From a similar picture, with the roles of $b$ and $c$ reversed, the other solution is given by $\gamma=\alpha-\pi/3$ and $\beta=\alpha-2\pi/3$.
Assembling this, we see that given an arbitrary positive real number $x=e^\alpha$, there is a pair of solutions to
$y_\pm = \mathrm{e}^{\,\alpha\pm2\pi/3}=x\mathrm{e}^{\,\pm2\pi/3},\qquad z_\pm=\mathrm{e}^{\,\alpha\pm\pi/3}=x\mathrm{e}^{\,\pm\pi/3}.
Given this, we can now see that these real solutions have the following properties, which we found interesting:
Any solution $x,y,z$ satisfies the relationship $xy=z^{2}$.
Since $\mathrm{e}^\pi$ is transcendental, it follows that the ratio of any two of
$x,y,z$ is transcendental, and so at most one of $x,y,z$ is algebraic.
Furthermore, since there is a solution for any positive choice of $x$, and since there are only countably many algebraic numbers, for almost all solutions the three values $x,y,z$ are all transcendental.
Finally, this tells us that not only are there no integer solutions, which is equivalent to there being no rational solutions, but that the real solutions are not only irrational, they are very irrational, in the sense that they are (almost all) transcendental.
Fit the fourth: the mathematical endeavour
In closing, let's think about how this small investigation fits into what mathematicians do all day. Some people think of mathematicians as people who try to solve problems using mathematics; others think of them as people who try to prove theorems. Very crudely speaking, we could think of these activities as applied and pure mathematics respectively. But both of these are really different approaches to a bigger objective: finding out something interesting. What we went through above is a small version of this, which shows the typical features. You start off with something that you want to understand better, you find out about it—and sometimes what you find out is that somebody else has already understood it—and you refine your question until you have something that you can understand better. Then you share it with other people who find it interesting. At least, I hope we've done the last part!
Cipra B 1999 Fermat's Theorem—at last! What's Happening in the Mathematical Sciences 3 American Mathematical Society
Ribenboim P 2000 Fermat's last theorem for amateurs, Springer
Zuehlke JA 1999 Fermat's last theorem for Gaussian integer exponents American Mathematical Monthly 106 (49)
Baker A 1975 Transcendental number theory Cambridge University Press
[Pictures: Banner adapted from one of Henry Holiday's original illustrations to The Hunting of the Snark (An Agony in 8 Fits) by Lewis Carroll; other pictures by Chalkdust]
Robert J Low
Rob teaches mathematics at Coventry University.
@RobJLow + More articles by Robert
Thierry Platini
Thierry also teaches mathematics at Coventry University.
+ More articles by Thierry
Why knot?
You can un-knot a knot, by cutting it not?
Mathematics for the three-fingered mathematician
Robert J Low flips one upside down.
← Slide rules: the early calculators
Debugging insect dynamics → | CommonCrawl |
\begin{document}
\normalem \begin{abstract} In this paper we investigate the application of pseudo-transient-continuation ({\sf PTC}) schemes for the numerical solution of semilinear elliptic partial differential equations, with possible singular perturbations. We will outline a residual reduction analysis within the framework of general Hilbert spaces, and, subsequently, employ the {\sf PTC}-methodology in the context of finite element discretizations of semilinear boundary value problems. Our approach combines both a prediction-type {\sf PTC}-method (for infinite dimensional problems) and an adaptive finite element discretization (based on a robust \emph{a posteriori} residual analysis), thereby leading to a \emph{fully adaptive {\sf PTC}-Galerkin scheme}. Numerical experiments underline the robustness and reliability of the proposed approach for different examples. \end{abstract}
\keywords{Adaptive pseudo transient continuation method, dynamical system, steady states, semilinear elliptic problems, singularly perturbed problems, adaptive finite element methods.}
\subjclass[2010]{49M15,58C15,65N30}
\maketitle
\section{Introduction and Problem Formulation}
The focus of this paper is on the numerical approximation of semilinear elliptic partial differential equations (PDE), with possible singular perturbations. More precisely, for a fixed parameter~$\varepsilon>0$ (possibly with~$\varepsilon\ll 1$), and a continuously differentiable function $f:\,\mathbb{R}\to\mathbb{R}$, we consider the problem of finding a solution function~$u:\,\Omega\to\mathbb{R}$ which satisfies \begin{equation}\label{poisson} \begin{aligned} -\varepsilon \Delta u &=f(u) \text{ in } \Omega,\qquad u=0 \text{ on } \partial \Omega. \end{aligned} \end{equation} Here, $\Omega\subset\mathbb{R}^d$, with $d=1$ or $d=2$, is an open and bounded 1d interval or a 2d Lipschitz polygon, respectively. Problems of this type appear in a wide range of applications including, e.g., nonlinear reaction-diffusion in ecology and chemical models~\cite{CaCo03,Ed05,Fr08,Ni11,OkLe01}, economy~\cite{BaBu95}, or classical and quantum physics~\cite{BeLi83,St77}.
From an analysis point of view, semilinear elliptic boundary value problems~\eqref{poisson} have been studied in detail by a number of authors over the last few decades; we refer, e.g., to the monographs~\cite{AmMa06,Ra86,Sm94} and the references therein. In particular, solutions of~\eqref{poisson} are known to be typically not unique (even infinitely many solutions may exist), and, in the singularly perturbed case, to exhibit boundary layers, interior shocks, and (multiple) spikes. The existence of multiple solutions due to the nonlinearity of the problem and/or the appearance of singular effects are challenging issues when solving problems of this type numerically; see, e.g., \cite{RoStTo08,Verhulst}.
\subsubsection*{Linearized Galerkin Methods} There are, in general, two approaches when solving nonlinear differential equations numerically: Either the nonlinear PDE problem to be solved is first discretized; this leads to a nonlinear algebraic system. Or, alternatively, a local linearization procedure, resulting in a sequence of linear PDE problems, is applied; these linear problems are subsequently discretized by a suitable numerical approximation scheme. We emphasize that the latter approach enables the use of the large body of existing numerical analysis and computational techniques for \emph{linear} problems (such as, e.g., the development of classical residual-based error bounds). The concept of approximating infinite dimensional nonlinear problems by appropriate \emph{linear discretization schemes} has been studied by several authors in the recent past. For example, the approach presented in~\cite{CongreveWihler:15} (see also the work~\cite{GarauMorinZuppa:11,ChaillouSuri:07}) combines fixed point linearization methods and Galerkin approximations in the context of strictly monotone problems. Similarly, in~\cite{AmreinMelenkWihler:16,AmreinWihler:14,AmreinWihler:15} (see also~\cite{El-AlaouiErnVohralik:11}), the nonlinear PDE problems at hand are linearized by an (adaptive) Newton technique, and, subsequently, discretized by a linear finite element method. On a related note, the discretization of a sequence of linearized problems resulting from the local approximation of semilinear evolutionary problems has been investigated in~\cite{AmreinWihlerTime:15}. In all of the works~\cite{AmreinMelenkWihler:16,AmreinWihler:14,AmreinWihler:15,AmreinWihlerTime:15,CongreveWihler:15}, the key idea in obtaining fully adaptive discretization schemes is to provide a suitable interplay between the underlying linearization procedure and (adaptive) Galerkin methods; this is based on investing computational time into whichever of these two aspects is currently dominant.
\subsubsection*{{\sf PTC}-Approach} In contrast to the classical Newton linearization method, the approach to be discussed in this work relies on a pseudo transient continuation procedure (see, e.g.,~\cite[\S6.4]{5} for finite dimensional problems). The basis of this idea is to first interpret any solution $u$ of the nonlinear equation $ \mathsf{F}(u)=~0$, where~$\mathsf{F}$ is a given operator, as a steady state of the initial value problem \[ \dot{u}=\mathsf{F}(u),\qquad u(0)=u_{0}, \] and, then, to discretize the dynamical system in time by means of the backward Euler method. Furthermore, the resulting sequence of nonlinear problems, $u_{n+1}=u_{n}+t_{n}\mathsf{F}(u_{n+1})$, $n\ge 0$, where~$t_n>0$ is a given time step, is linearized with the aid of the Newton method. This scheme is termed {\sf PTC}-method. On a local level, i.e., whenever the iteration is close enough to a solution point, the {\sf PTC}-method turns into the standard Newton method. Otherwise, if the iteration is far away from a solution point, then the scheme can be interpreted as a continuation method. In a certain sense, the {\sf PTC}-method can also be understood as an inexact Newton method. Following the methodology developed in the articles~\cite{AmreinWihler:14,AmreinWihler:15,AmreinWihlerTime:15,CongreveWihler:15}, the present paper employs the idea of combining the {\sf PTC}-linearization approach with adaptive~$\mathbb{P}_1$-finite element methods (FEM). Our analysis will proceed along the lines of~\cite[\S6.4]{5}, with the aim to provide an optimal residual reduction procedure in the local linearization process. Moreover, in order to address the issue of devising $\varepsilon$-robust \emph{a posteriori} error estimates for the Galerkin discretizations, we employ the approach presented in~\cite{Verfuerth}.
\subsubsection*{Outline} The outline of this paper is as follows. In Section \ref{sec:prediction} we study the {\sf PTC}-method within the context of general Hilbert spaces, and derive a residual reduction analysis. Subsequently, the purpose of Section~\ref{sc:Well-Posedness-FEM} is the discretization of the resulting sequence of {\em linear} problems by the finite element method, and the development of an $\varepsilon$-robust \emph{a posteriori} error analysis. The final estimate (Theorem~\ref{thm:1}) bounds the residual in terms of the (elementwise) finite element approximation (FEM-error) and the error caused by the linearization of the original problem. Then, in order to define a fully adaptive {\sf PTC}-Galerkin scheme, we propose an interplay between the adaptive {\sf PTC}-method and the adaptive finite element approach: More precisely, as the adaptive procedure is running, we either perform a {\sf PTC}-step in accordance with the suggested prediction strategy (Section~\ref{sec:prediction}) or refine the current finite element mesh based on the {\em a posteriori residual} estimate (Section~\ref{sc:Well-Posedness-FEM}); this is carried out depending on which of the errors (FEM-error or {\sf PTC}-error) is more dominant in the present iteration step. In Section~\ref{sec:numerics} we provide a series of numerical experiments which show that the proposed scheme is reliable and $\varepsilon$-robust for reasonable choices of initial guesses. Finally, we add a few concluding remarks in Section~\ref{sc:concl}.
\subsubsection*{Problem Formulation} In this paper, we suppose that a (not necessarily unique) solution~$u\in X:=H^1_0(\Omega)$ of~\eqref{poisson} exists; here, we denote by $H^1_0(\Omega)$ the standard Sobolev space of functions in~$H^1(\Omega)=W^{1,2}(\Omega)$ with zero trace on~$\partial\Omega$. Furthermore, signifying by~$X'=H^{-1}(\Omega)$ the dual space of~$X$, and upon defining the map $\mathsf{F}_{\varepsilon}: X\rightarrow X'$ through \begin{equation}\label{eq:Fweak} \left \langle \mathsf{F}_{\varepsilon}(u),v \right \rangle := \int_{\Omega}{\left\{f(u)v-\varepsilon \nabla u\cdot \nabla v\right\}}\,\mathsf{d}\bm x \qquad \forall v\in X, \end{equation} where $\left\langle\cdot,\cdot\right\rangle$ signifies the dual pairing in~$X'\times X$, the above problem~\eqref{poisson} can be written as a nonlinear operator equation in~$X'$: \[ \mathsf{F}_\varepsilon(u)=0, \] for an unknown zero~$u\in X$. For the purpose of defining the Newton linearization later on in this manuscript, we note that the Fr\'echet-derivative of $\mathsf{F}_{\varepsilon}$ at $u \in X$ is given by \[ \left \langle \mathsf{F}'_{\varepsilon}(u)w,v\right \rangle =\int_{\Omega}{\{f'(u)wv-\varepsilon \nabla w\nabla v\}\,\mathsf{d}\bm x}, \quad v,w,\in X, \] where we write~$f'(u)=\partial_u f(u)$. In addition, we introduce the inner product \[ (u,v)_{X}:=\int_{\Omega}{\{uv+\varepsilon \nabla u \cdot \nabla v\}\,\mathsf{d}\bm x},\qquad u,v\in V, \] with induced norm on $X$ given by \[ \NN{u}_{\varepsilon,D}:=\Bigl(\varepsilon\norm{\nabla u}_{0,D}^2 +\norm{u}_{0,D}^2 \Bigr)^{\nicefrac{1}{2}},\qquad u\in H^1(D), \]
where $\|\cdot\|_{0,D}$ denotes the $L^2$-norm on~$D$. Frequently, for~$D=\Omega$, the subindex~`$D$' will be omitted. Note that, in the case of $ f(u)=-u+g$, with $g \in L^{2}(\Omega)$, i.e., when \eqref{poisson} is linear and strongly elliptic, the norm $ \NN{\cdot}_{\varepsilon,\Omega}$ is a natural energy norm on $X$. As usual, for any $\varphi \in X'$, the dual norm is given by \[ \norm{\varphi}_{X'}=\sup_{x\in X\setminus\{0\}}{\frac{\left\langle \varphi,x\right \rangle}{\NN{x}_{\varepsilon}}}. \]
In what follows we shall use the abbreviation $ x\preccurlyeq y $ to mean $x\leq cy $, for a constant $c>0$ independent of the mesh size $h$ and of~$ \varepsilon>0$.
\section{Abstract Framework in Hilbert Spaces} \label{sec:prediction} In this section we briefly revisit a possible derivation of the {\sf PTC}-scheme. Moreover, following along the lines of \cite{5} we will discuss how residual reduction, based on a {\sf PTC}-iteration-scheme, can be achieved within the context of general Hilbert spaces. To this end, let $ X $ be a real Hilbert space with inner product $(\cdot,\cdot)_{X}$ and induced norm $(x,x)_{X}^{\nicefrac{1}{2}}=\norm{x}_{X}$. Furthermore, by $\mathcal{L}(X;X')$, we signify the space of all bounded linear operators from $X$ into $X'$, with norm \[
\|\mathsf{L}\|_{\mathcal{L}(X;X')}=\sup_{\genfrac{}{}{0pt}{}{x\in X}{\|x\|_X=1}}\|\mathsf{L}(x)\|_{X'}, \] for any~$\mathsf{L}\in\mathcal{L}(X;X')$.
\subsection{{\sf PTC}-Scheme} We take the view of dynamical systems, i.e., given a possibly nonlinear operator \[ \mathsf{F}:X\rightarrow X', \] we interpret any zero $u_{\infty}\in X$ of~$\mathsf{F}$, i.e., $\mathsf{F}(u_\infty)=0$, as a \emph{steady state} of the dynamical system \begin{align}\label{eq:dynamical-system} u(0)=u_{0},\qquad(\dot{u}(t),v)_{X}=\left\langle \mathsf{F}(u(t)),v\right\rangle\quad\forall v\in X, t>0, \end{align} where we denote by~$\langle\cdot,\cdot\rangle$ the dual pairing in~$X'\times X$ as before, and~$u_0\in X$ is a given initial guess. More precisely, we suppose that there exists a solution~$u:\,[0,\infty)\to X$ of~\eqref{eq:dynamical-system} with~$\lim_{t\to\infty}u(t)=u_\infty$ in a suitable sense. Then, we discretize \eqref{eq:dynamical-system} in time using the backward Euler method, i.e., \begin{equation}\label{eq:Euler} (u_{n+1},v)_{X}=(u_{n},v)_{X}+k_{n}\left\langle \mathsf{F}(u_{n+1}),v\right\rangle \quad \forall v\in X,\qquad n\geq 0, \end{equation} where $k_{n}>0 $ signifies the (possibly adaptively chosen) temporal step size. Introducing, for~$n\ge 0$, an operator~$\mathsf{G}_n:\,X\to X'$ by \[ \left\langle \mathsf{G}_n(u),v \right\rangle:=(u-u_{n},v)_{X}-k_{n}\left\langle \mathsf{F}(u),v\right\rangle\quad\forall v\in X, \] we see that the zeros of $ \mathsf{G}_n $ define the next update $u_{n+1}$ in \eqref{eq:Euler}. Then, applying Newton's method to $\mathsf{G}_n$ yields a linear equation for an unknown increment~$\widetilde\delta_n\in X$ such that \[ \left\langle \mathsf{G}_n'(u_{n})\widetilde\delta_{n},v \right\rangle=-\left\langle \mathsf{G}_n(u_{n}),v \right\rangle\quad\forall v\in X, \] and the update \[ u_{n+1}=u_{n}+\widetilde\delta_{n}, \] where~$\mathsf{G}_n'$ denotes the Fr\'echet derivative of~$\mathsf{G}_n$. Equivalently, upon rescaling~$\delta_n=k_n^{-1}\widetilde\delta_n$, we have \begin{equation} \label{eq:linear-implicit-scheme} (\delta_{n},v)_{X}-k_{n}\left\langle \mathsf{F}'(u_{n})\delta_{n},v \right\rangle=\left\langle \mathsf{F}(u_{n}),v \right\rangle,\qquad u_{n+1}=u_{n}+k_{n}\delta_{n}. \end{equation} Incidentally, with~$\delta_n\rightharpoonup 0$ weakly in~$X$, we obtain Newton's method as applied to~$\mathsf{F}$. In order to simplify notation we introduce, for given~$u\in X$ and~$t>0$, an additional operator \[ \mathsf{A}[t;u]:\,X \to X', \] which is defined by \begin{equation} \label{eq:operator-A} x\in X:\qquad \left\langle \mathsf{A}[t;u]x,v \right\rangle:=(x,v)_{X}-t\left\langle \mathsf{F}'(u)x,v \right \rangle\quad\forall v\in X. \end{equation} We can then rewrite~\eqref{eq:linear-implicit-scheme} as \begin{equation} \label{eq:ptc} \left\langle \mathsf{A}[k_n;u_{n}]\delta_n,v \right\rangle=\left\langle \mathsf{F}(u_{n}),v \right\rangle\quad\forall v\in X,\qquad u_{n+1}=u_{n}+k_n\delta_n. \end{equation} For~$n=0,1,2,\ldots$, with a given initial guess~$u_0\in X$, this iteration defines the~{\sf PTC}-scheme for the approximation of a zero of~$\mathsf{F}$. Evidently, in order to be able to solve for $\delta_n $ in \eqref{eq:ptc}, the operator $\mathsf{A}[k_n;u_{n}] $ needs to be invertible.
\subsection{Residual Analysis} The aim of this section is to derive a residual estimate which paves the way for a residual reduction time stepping strategy. It is based on the following structural assumptions on the derivative of~$\mathsf{F}$: \begin{enumerate}[(a)] \item For given~$u_0\in X$, there exists a constant~$\mu=\mu(u_0)>0$ such that \begin{equation} \label{eq:A1}\tag{A.1}
\sup_{\genfrac{}{}{0pt}{}{x\in X}{\|x\|_X=1}}\left \langle \mathsf{F}'(u_0)x,x \right \rangle \leq -\mu. \end{equation} \item There is a constant~$L\ge 0$ such that there holds the Lipschitz property \begin{equation} \label{eq:A2}\tag{A.2} \norm{\mathsf{F}'(x)-\mathsf{F}'(y)}_{\mathcal{L}(X;X')}\leq L\norm{x-y}_{X}\qquad\forall x,y\in X. \end{equation} \end{enumerate} \begin{proposition}\label{pr:lm} Let~$u_0\in X$ such that $\mathsf{F}'(u_0)\in\mathcal{L}(X;X')$. If~\eqref{eq:A1} holds, and if~$\mathsf{F}(u_0)\in X'$, then the linear problem \begin{equation}\label{eq:sys} \mathsf{A}[t;u_0](u(t)-u_0)=t\mathsf{F}(u_0) \end{equation} has a unique solution~$u(t)\in X$ for any~$t>0$. \end{proposition}
\begin{proof} We apply the Lax-Milgram Lemma. In particular, we show that~$\mathsf{A}[t;u_0]$ is coercive and bounded on~$X$. Indeed, for all~$v\in X$, we have \begin{equation}\label{eq:coercive} \langle\mathsf{A}[t;u_0]v,v\rangle
=(v,v)_{X}-t\left\langle \mathsf{F}'(u_0)v,v \right \rangle\ge (1+\mu t)\|v\|_X^2, \end{equation} which proves coercivity. Moreover, for~$v,w\in X$, we have \begin{align*} \abs{\langle\mathsf{A}[t;u_0]v,w\rangle}
&\le\|v\|_X\|w\|_X+t\|\mathsf{F}'(u_0)v\|_{X'}\|w\|_X. \end{align*} Since~$\mathsf{F}'(u_0)$ is bounded, we deduce the boundedness of~$\mathsf{A}[t;u_0]$. This completes the proof. \end{proof}
In order to devise a residual reduction analysis, we insert two preparatory results.
\begin{lemma}\label{lm:Lemma1} If~\eqref{eq:A1} is satisfied, then we have \begin{equation}\label{eq:2nd-estimate} \norm{\mathsf{A}[t;u_0]^{-1}}_{\mathcal{L}(X';X)}\leq \frac{1}{1+t\mu}. \end{equation} Moreover, if~$\mathsf{F}(u_0)\in X'$, the estimate \begin{equation} \label{eq:first-estimate} \norm{u(t)-u_0}_{X}\leq \frac{t}{1+t\mu}\norm{\mathsf{F}(u_0)}_{X'} \end{equation} holds true, where~ $u(t)$, $t\ge 0$, is the solution from~\eqref{eq:sys}. \end{lemma}
\begin{proof} From~\eqref{eq:coercive}, we readily arrive at \[ \norm{\mathsf{A}[t;u_0]v}_{X'}\geq \norm{v}_{X}(1+t\mu)\quad\forall v \in X, \] from which we deduce~\eqref{eq:2nd-estimate}. Furthermore, the second bound results by definition of~$\mathsf{A}[t;u_0]$ in~\eqref{eq:operator-A} with~$v=u(t)-u_0$, and from~\eqref{eq:A1}. Indeed, \begin{equation} \label{eq:use1} \begin{split} \norm{u(t)-u_0}_{X}^{2} &=\langle\mathsf{A}[t;u_0](u(t)-u_0),u(t)-u_0\rangle+t\langle\mathsf{F}'(u_0)(u(t)-u_0),u(t)-u_0\rangle\\ &=t\left \langle \mathsf{F}(u_0),u(t)-u_0\right \rangle+t\left\langle \mathsf{F}'(u_0)(u(t)-u_0),u(t)-u_0\right\rangle \\ &\leq t\norm{\mathsf{F}(u_0)}_{X'}\norm{u(t)-u_0}_{X}-t\mu\norm{u(t)-u_0}_{X}^{2}, \end{split} \end{equation} which immediately implies~\eqref{eq:first-estimate}. \end{proof}
\begin{lemma}\label{lm:Lemma2} If~$u$ from~\eqref{eq:sys} is differentiable for any~$t\ge 0$, then it holds that \begin{equation}\label{eq:id1} \mathsf{A}[t;u_0]\dot{u}(t)=\mathsf{F}(u_0)+\mathsf{F}'(u_0)(u(t)-u_0), \end{equation} as well as \begin{equation} \label{eq:derivative-control} t\mathsf{A}[t;u_{0}]\dot{u}(t) =(u(t)-u_0,\cdot)_{X} \end{equation} in~$X'$. \end{lemma}
\begin{proof} Recalling~\eqref{eq:operator-A}, we observe that \[ \frac{\mathsf{d}}{\,\mathsf{d} t}\mathsf{A}[t;u_{0}]v=-\mathsf{F}'(u_0)v\qquad\forall v\in X, \] in~$X'$. Then, differentiating~\eqref{eq:sys} with respect to $t$ implies \[ \mathsf{A}[t;u_0]\dot{u}(t)-\mathsf{F}'(u_0)(u(t)-u_0) = \mathsf{F}(u_0), \] which yields~\eqref{eq:id1}. Furthermore, multiplying this equality by~$t$, and applying the definition of~$\mathsf{A}[t;u_0]$ from~\eqref{eq:operator-A}, it follows that \[ t\mathsf{A}[t;u_0]\dot{u}(t) =t\mathsf{F}(u_0)+(u(t)-u_0,\cdot)_X-\mathsf{A}[t;u_0](u(t)-u_0) \] in~$X'$. Using~\eqref{eq:sys} gives~\eqref{eq:derivative-control}. \end{proof}
Following along the lines of~\cite{5} there holds the ensuing residual reduction result.
\begin{theorem}\label{thm:res} Under the assumptions in Lemmas~\ref{lm:Lemma1} and~\ref{lm:Lemma2}, and if~\eqref{eq:A2} holds, then we have \begin{equation}\label{eq:res} \norm{\mathsf{F}(u(t))}_{X'}\leq \gamma(t)\norm{\mathsf{F}(u_{0})}_{X'}, \end{equation} with \[ \gamma(t):=\frac{1}{1+t\mu}\left(1+\frac{Lt^{2}}{2(1+t\mu)}\norm{\mathsf{F}(u_0)}_{X'}\right)>0, \] for any~$t\ge 0$. \end{theorem}
\begin{proof} For~$t\ge 0$, there holds \begin{align*} \mathsf{F}(u(t))&=\mathsf{F}(u_{0})+\int_{0}^{t}{\mathsf{F}'(u(s))\dot{u}(s)\,\mathsf{d} s}\\ &=\mathsf{F}(u_{0})+\mathsf{F}'(u_0)(u(t)-u_0)+\int_{0}^{t}{(\mathsf{F}'(u(s))-\mathsf{F}'(u_0))\dot{u}(s)\,\mathsf{d} s}. \end{align*} Involving~\eqref{eq:id1}, we infer that \begin{align*} \mathsf{F}(u(t)) &=\mathsf{A}[t;u_0]\dot{u}(t)+\int_{0}^{t}{(\mathsf{F}'(u(s))-\mathsf{F}'(u_0))\dot{u}(s)\,\mathsf{d} s}. \end{align*} Therefore, we have \[ \norm{\mathsf{F}(u(t))}_{X'} \leq \norm{\mathsf{A}[t;u_0]\dot{u}(t)}_{X'}+\int_{0}^{t}{\norm{(\mathsf{F}'(u(s))-\mathsf{F}'(u_0))\dot{u}(s)}_{X'}\,\mathsf{d} s}. \] Employing \eqref{eq:A2} and applying~\eqref{eq:derivative-control}, we arrive at \begin{equation} \label{eq:step1} t\norm{\mathsf{F}(u(t))}_{X'}\leq \norm{u(t)-u_0}_{X}+Lt\int_{0}^{t}{\norm{u(s)-u_0}_{X}\norm{\dot{u}(s)}_{X}\,\mathsf{d} s}. \end{equation} Moreover, again from~\eqref{eq:derivative-control}, we notice that \begin{equation}\label{eq:aux3} s\norm{\dot{u}(s)}_{X}\leq \norm{\mathsf{A}[s;u_0]^{-1}}_{\mathcal{L}(X';X)}\norm{u(s)-u_0}_{X},\qquad s\ge 0, \end{equation} and, hence, by virtue of Lemma~\ref{lm:Lemma1}, we obtain \[ s\norm{u(s)-u_0}_{X}\norm{\dot{u}(s)}_{X}
\le \frac{1}{1+s\mu}\|u(s)-u_0\|_X^2 \leq \frac{s^2}{(1+s\mu)^3}\norm{\mathsf{F}(u_0)}_{X'}^{2}. \] Combining this with~\eqref{eq:step1}, and using Lemma~\ref{lm:Lemma1} once more, leads to \begin{align*} t\norm{\mathsf{F}(u(t))}_{X'}&\leq \frac{t}{1+t\mu}\norm{\mathsf{F}(u_0)}_{X'}+Lt\norm{\mathsf{F}(u_0)}_{X'}^2\int_{0}^{t}{\frac{s}{(1+s\mu)^{3}}\,\mathsf{d} s}\\ &=\frac{t}{1+t\mu}\norm{\mathsf{F}(u_0)}_{X'}+\frac{Lt^3}{2(1+\mu t)^2}\norm{\mathsf{F}(u_0)}_{X'}^2. \end{align*} This completes the proof. \end{proof}
\begin{remark} Referring to~\eqref{eq:use1}, we see that \[ \norm{u(t)-u_0}_{X}^{2}\leq t\left\langle \mathsf{F}(u_0),u(t)-u_0\right\rangle - t\mu \norm{u(t)-u_0}_{X}^{2}. \] Hence, whenever there holds $ t\left\langle \mathsf{F}(u_0),u(t)-u_0\right\rangle \leq \norm{u(t)-u_0}_{X}^{2}$, it follows that~$ \mu \leq 0 $ (as long as there is~$t>0$ with~$u(t)\neq u_0$). In particular, assumption \eqref{eq:A1} is not fulfilled in this case. We may therefore assume that \[ \norm{u(t)-u_0}_{X}^{2} < t\left\langle \mathsf{F}(u_0),u(t)-u_0\right\rangle, \] for~$t>0$, and~$u(t)\neq u_0$. \end{remark}
From~\eqref{eq:res} it follows that the residual decreases, i.e., $\|\mathsf{F}(u(t))\|_{X'}<\|F(u_0)\|_{X'}$, if~$\gamma(t)\in(0,1)$. For~$t>0$, this happens if there holds \begin{equation}\label{eq:t} \left(\frac{L}{2}\norm{\mathsf{F}(u_{0})}_{X'}-\mu^2\right)t<\mu\qquad (t>0). \end{equation} Therefore, if \[ \frac{L}{2}\norm{\mathsf{F}(u_{0})}_{X'}\le \mu^2, \] then any value of~$t>0$ will lead to a reduction of the residual. Otherwise, \eqref{eq:t} can be satisfied as long as~$t$ is chosen sufficiently small; in the special case that $L\norm{\mathsf{F}(u_0)}_{X'}>\mu^2 $, it is elementary to verify that $\gamma(t) $ attains its minimum for \[ t^\star=\frac{\mu}{L\norm{\mathsf{F}(u_0)}_{X'}-\mu^2}. \]
\subsection{Pseudo Time Stepping} In terms of the {\sf PTC}-scheme~\eqref{eq:ptc}, for~$n\ge 0$, our previous discussion translates into \[
\|\mathsf{F}(u_{n+1})\|_{X'}\le\gamma_n\|\mathsf{F}(u_n)\|_{X'}, \] with a reduction constant \[ \gamma_n=\frac{1}{1+k_n\mu}\left(1+\frac{Lk_n^{2}}{2(1+k_n\mu)}\norm{\mathsf{F}(u_n)}_{X'}\right)>0; \] cf. Theorem~\ref{thm:res}. If \[ \frac{L}{2}\norm{\mathsf{F}(u_{n})}_{X'}\le\mu^2, \] then any choice of~$k_n>0$ will imply that~$\gamma_n\in(0,1)$. Otherwise, for~$k_n$ sufficiently small so that \[ \left(\frac{L}{2}\norm{\mathsf{F}(u_{0})}_{X'}-\mu^2\right)k_n<\mu, \] it holds that~$\gamma_n<1$. In particular, if~$L\norm{\mathsf{F}(u_n)}_{X'}>\mu^2$, then \begin{equation}\label{eq:k*} k_n^\star=\frac{\mu}{L\norm{\mathsf{F}(u_n)}_{X'}-\mu^2} \end{equation} results in a minimal value of~$\gamma_n$. For this value of~$k_n$, we apply~\eqref{eq:first-estimate} to infer the bound \[ \norm{u_{n+1}-u_n}_{X}\leq \frac{k_n^\star}{1+k_n^\star\mu}\norm{\mathsf{F}(u_n)}_{X'}=\frac{\mu}{L}. \] Letting~$\delta_n=\nicefrac{(u_{n+1}-u_n)}{k_n^\star}$ be the increment in the {\sf PTC}-iteration~\eqref{eq:linear-implicit-scheme}, this leads to~$\norm{\delta_n}_X\le \nicefrac{\mu}{(k{_n^\star}L)}$, and, therefore, \begin{equation} \label{eq:computable} k_n^\star \leq \frac{\mu}{L\norm{\delta_n}_{X}}. \end{equation} This upper bound does not contain any dual norms, and can, thus, be employed as an approximation of~$k_n^\star$ in practice.
\begin{remark} In an effort to replace~\eqref{eq:computable} by a computationally even more feasible bound (not involving the possibly unspecified constants~$\mu$ and~$L$), we proceed again along the lines of~\cite{5}. As in~\eqref{eq:use1}, for~$k_n>0$, we have \[
\|\delta_n\|^2_X\le \langle\mathsf{F}(u_n),\delta_n\rangle-\mu k_n\|\delta_n\|^2_X. \] This motivates to define the computable quantity \[ \bm{\mu}_n:=\frac{\left\langle \mathsf{F}(u_n),\delta_n\right\rangle-\norm{\delta_n}_{X}^2}{k_n\norm{\delta_n}_{X}^2}\geq \mu>0. \] Furthermore, similarly as in the proof of Theorem~\ref{thm:res}, we note that \[
\langle\mathsf{F}(u_{n+1}),\delta_n\rangle=\|\delta_n\|_X^2+\int_{t_{n}}^{t_{n+1}}\langle(\mathsf{F}'(u(s))-\mathsf{F}'(u_n))\dot{u}(s),\delta_n\rangle\,\mathsf{d} s, \] where, for~$i\ge 1$, we let~$t_i=\sum_{j=0}^{i-1} k_j$. Then, by means of~\eqref{eq:A2}, it follows that \begin{align*}
\frac{|\left\langle \mathsf{F}(u_{n+1}),\delta_n \right\rangle -\norm{\delta_n}_{X}^2|}{\norm{\delta_n}_{X}} &\leq \int_{t_{n}}^{t_{n+1}}{\norm{(\mathsf{F}'(u(s))-\mathsf{F}'(u_n))\dot{u}(s)}_{X'}\,\mathsf{d} s}\\ &\leq L \int_{t_{n}}^{t_{n+1}}{\norm{u(s)-u_n}_{X}\norm{\dot{u}(s)}_{X}\,\mathsf{d} s}. \end{align*} Furthermore, using~\eqref{eq:aux3} and employing~Lemma~\ref{lm:Lemma1}, this transforms into \begin{align*}
\frac{|\left\langle \mathsf{F}(u_{n+1}),\delta_n \right\rangle -\norm{\delta_n}_{X}^2|}{\norm{\delta_n}_{X}}
&\leq L\int_{t_{n}}^{t_{n+1}}\frac{(s-t_{n})^{-1}}{1+(s-t_n)\mu}\|u(s)-u_n\|_{X}^2\,\mathsf{d} s. \end{align*} Approximating the integral with the aid of the trapezoidal rule, and recalling that~$u_{n+1}-u_n=k_n\delta_n$, cf.~\eqref{eq:linear-implicit-scheme}, yields \[
\frac{|\left\langle \mathsf{F}(u_{n+1}),\delta_n \right\rangle -\norm{\delta_n}_{X}^2|}{\norm{\delta_n}_{X}}\le \frac{L}{2}k_n^2\|\delta_n\|_X^2+\mathcal{O}(k_n^4). \] We then define \[
\bm{L}_n:=\frac{2|\left\langle \mathsf{F}(u_{n+1}),\delta_n \right\rangle -\norm{\delta_n}_{X}^2|}{k_n^2\norm{\delta_n}_{X}^3}\leq L+\mathcal{O}(k_n^2). \] Replacing~$\mu$ and~$L$ in~\eqref{eq:computable} by~$\bm\mu_n$ and~$\bm L_n$, respectively, we are led to introduce the following pseudo time step \begin{equation} \label{eq:stepsize-control} \bm{k}_n^\star=\frac{k_n}{2}\cdot \abs{\frac{ \left \langle \mathsf{F}(u_n),\delta_n \right \rangle-\norm{\delta_n}_{X}^{2}}{\left \langle \mathsf{F}(u_{n+1}),\delta_n \right \rangle -\norm{\delta_n}_{X}^2}}, \end{equation} which does not require explicit knowledge on~$\mu$ and~$L$. \end{remark}
\section{Application to Semilinear Problems}\label{sc:Well-Posedness-FEM}
In this section, we will apply the abstract setting from the previous section to the semilinear problem~\eqref{poisson}, with~$\mathsf{F}=\mathsf{F}_\varepsilon$ from~\eqref{eq:Fweak}.
\subsection{{\sf PTC}-Linearization} For~$u_n\in X$ and~$k_n>0$, the {\sf PTC}-method~\eqref{eq:linear-implicit-scheme} is to find~$\delta_n \in X$ such that \begin{equation}\label{eq:weak-formulation} a_\varepsilon(u_{n},k_n;\delta_n,v) = \ell_\varepsilon(u_{n};v)\qquad\forall v\in X, \end{equation} and~$u_{n+1}=u_n+k_n\delta_n$, where, for {\em fixed}~$u\in X$, $t>0$, we consider the bilinear form \begin{equation*} \begin{aligned} a_{\varepsilon}(u,t;\delta,v)&:=(\delta, v)_{X} -t\int_{\Omega}{\{f'(u)\delta v-\varepsilon \nabla \delta \cdot \nabla v\}\,\mathsf{d}\bm x},\qquad\delta,v\in X, \end{aligned} \end{equation*} as well as the linear form \begin{align*} \ell_{\varepsilon}(u;v)&:=\int_{\Omega}{\{f(u)v-\varepsilon \nabla u \cdot \nabla v\}\,\mathsf{d}\bm x},\qquad v\in X. \end{align*}
Throughout, for given~$u_n$, $n\ge 0$, we assume that \eqref{eq:linear-implicit-scheme} has a unique solution~$u_{n+1}$. In fact, this property can be made rigorous if certain assumptions on the nonlinearity~$f$ are satisfied. This will be addressed in the ensuing two propositions.
\begin{proposition}\label{pr:f} If~$\sigma_f:=\sup_{x\in\mathbb{R}}f'(x)<\varepsilon C_{\tt Poinc}^{-2}$, where~$C_{\tt Poinc}=C_{\tt Poinc}(\Omega)$ is the constant in the Poincar\'e inequality on $\Omega$, \begin{equation} \label{eq:Poincare} \norm{w}_{0}\leq C_{\tt Poinc}\norm{\nabla w}_{0}, \quad \forall w \in X, \end{equation} then~\eqref{eq:A1} is satisfied with \[ \mu=\frac{\varepsilon C_{\tt Poinc}^{-2}-\sigma_f}{\varepsilon C_{\tt Poinc}^{-2}+1}>0. \] \end{proposition}
\begin{proof} Let us set \[ \zeta:=\frac{1+\sigma_f}{\varepsilon C_{\tt Poinc}^{-2}+1}. \] By our assumptions, there holds that~$\zeta<1$. Then, with~\eqref{eq:Poincare}, for~$u,v\in X$, we have \begin{align*}
\langle\mathsf{F}_\varepsilon'(u)v,v\rangle&=(\zeta-1)\varepsilon\|\nabla v\|_0^2-\zeta\varepsilon\|\nabla v\|_0^2+\int_\Omega f'(u)v^2\,\mathsf{d}\bm x\\
&\le(\zeta-1)\varepsilon\|\nabla v\|_0^2+\int_\Omega \{f'(u)-\zeta C_{\tt Poinc}^{-2}\varepsilon\}v^2\,\mathsf{d}\bm x\\ &\le(\zeta-1)\NN{v}^2_\varepsilon =-\mu \NN{v}^2_\varepsilon. \end{align*} Hence, \eqref{eq:A1} is verified. \end{proof}
\begin{remark} Within a given {\sf PTC}-iteration, for~$n\ge 0$, the proof of the above result reveals that~$\sigma_f$ can be replaced by the possibly sharper value~$\sigma_f:=\sup_{\Omega}f'(u_n)$. \end{remark}
\begin{proposition}\label{pr:f2} If $f'$ is globally Lipschitz continuous with Lipschitz constant $L_{f'}$, that is, \begin{equation}\label{eq:f'Lip}
|f'(u_1)-f'(u_2)|\le L_{f'}|u_1-u_2|\qquad\forall u_1,u_2\in\mathbb{R}, \end{equation} then~\eqref{eq:A2} is fulfilled with $L:=CL_{f'}\varepsilon^{-1}$, where~$C>0$ is a constant only depending on~$\Omega$. \end{proposition}
\begin{proof} For $u_1,u_2,w,v \in X$ there holds \[
|\left\langle (\mathsf{F}'(u_1)-\mathsf{F}'(u_2))w,v \right \rangle| \le\norm{(f'(u_1)-f'(u_2))wv}_{L^1(\Omega)}. \] Employing~\cite[Lemma~A.1]{AmreinWihler:15}, and applying the Lipschitz continuity~\eqref{eq:f'Lip}, we obtain \begin{align*}
|\left\langle (\mathsf{F}'(u_1)-\mathsf{F}'(u_2))w,v \right \rangle|
&\le C\|f'(u_1)-f'(u_2)\|_{0}\|\nabla w\|_0\|\nabla v\|_0\\
&\le CL_{f'}\|u_1-u_2\|_{0}\|\nabla w\|_0\|\nabla v\|_0\\ &\le CL_{f'}\varepsilon^{-1}\NN{u_1-u_2}_\varepsilon\NN{w}_\varepsilon\NN{v}_\varepsilon, \end{align*} for a constant~$C>0$ only depending on~$\Omega$. This verifies~\eqref{eq:A2}. \end{proof}
\begin{remark} If the assumptions in the above Propositions~\ref{pr:f} and~\ref{pr:f2} are satisfied, and if for given $u_n\in X$ there holds $f(u_{n}) \in L^{2}(\Omega)$, then the linear problem \eqref{eq:weak-formulation} has a unique solution $\delta_{n}\in X$; cf.~Proposition~\ref{pr:lm}. \end{remark}
\subsection{{\sf PTC}-Galerkin Discretization} In order to provide a numerical approximation of~\eqref{poisson}, we will discretize the \emph{linear} weak formulation~\eqref{eq:weak-formulation} by means of a finite element method, which, in combination with the {\sf PTC}-iteration, constitutes a {\sf PTC}-Galerkin approximation scheme. Furthermore, we shall derive {\em a posteriori} residual estimates for the finite element discretization which allow for an adaptive refinement of the meshes in each {\sf PTC}-step. This, together with the adaptive prediction strategy from Section~\ref{sec:prediction}, leads to a fully adaptive {\sf PTC}-Galerkin discretization method for~\eqref{poisson}.
\subsubsection{Finite Element Meshes and Spaces} Let $ \mathcal{T}^h=\{T\}_{T\in\mathcal{T}^h}$ be a regular and shape-regular mesh partition of $\Omega $ into disjoint open simplices, i.e., any~$T\in\mathcal{T}^h$ is an affine image of the (open) reference simplex~$\widehat T=\{\widehat x\in\mathbb{R}_+^d:\,\sum_{i=1}^d\widehat x_i<1\}$. By~$h_T=\mathrm{diam}(T)$ we signify the element diameter of~$T\in\mathcal{T}^h$, and by $h=\max_{T\in\mathcal{T}^h}h_T$ the mesh size. Furthermore, by $\mathcal{E}^h$ we denote the set of all interior mesh nodes for~$d=1$ and interior (open) edges for~$d=2$ in~$\mathcal{T}^h$. In addition, for~$T\in\mathcal{T}^h$, we let~$\mathcal{E}^h(T)=\{E\in\mathcal{E}^h:\,E\subset\partial T\}$. For~$E\in\mathcal{E}^h$, we let~$h_E$ be the mean of the lengths of the adjacent elements in 1d, and the length of~$E$ in~2d. Let us also define the following two quantities: \begin{equation}\label{boundary} \begin{aligned} \alpha_T&:=\min(1,\varepsilon^{-\nicefrac12}h_T),\qquad \alpha_E:=\min(1,\varepsilon^{-\nicefrac12}h_E), \end{aligned} \end{equation} for~$T\in\mathcal{T}^h$ and~$E\in\mathcal{E}^h$, respectively.
We consider the finite element space of continuous, piecewise linear functions on $\mathcal{T}^h$ with zero trace on~$\partial\Omega$, given by \begin{equation*}
V_{0}^{h}:=\{\varphi\in H^1_0(\Omega):\,\varphi|_{T} \in \mathbb{P}_{1}(T) \, \forall T \in \mathcal{T}^h\}, \end{equation*} respectively, where~$\mathbb{P}_1(T)$ is the standard space of all linear polynomial functions on~$T$.
\subsubsection{Linear Finite Element Discretization} For given~$k_n>0$ and $u_n^{h}\in V_{0}^{h}$, $n\ge 0$, we consider the finite element approximation of~\eqref{eq:weak-formulation}, which is to find~$\delta_{n}^h\in V_{0}^h$ such that \begin{equation}\label{eq:fem} a_\varepsilon(u_n^h,k_{n};\delta_{n}^h,v)=\ell_\varepsilon(u_n^h;v)\qquad\forall v\in V_{0}^{h}; \end{equation} for $n=0$, the function~$u_0^{h}\in V_{0}^{h}$ is a prescribed initial guess. Introducing the linearization operator \[ \mathsf{T}_{f}(u):=f(u_{n}^{h})+f'(u_{n}^{h})(u-u_{n}^{h}), \] as well as \[ u_{n+1}^{h}:=u_{n}^{h}+k_n\delta_{n}^{h}, \] and rearranging terms, \eqref{eq:fem} can be rewritten as \begin{equation} \label{eq:start} \begin{aligned} \int_{\Omega}{\varepsilon\nabla u_{n+1}^{h} \cdot \nabla v\,\mathsf{d}\bm x} = \int_{\Omega}{(\mathsf{T}_{f}(u_{n+1}^{h})-\delta_{n}^{h}) v}\,\mathsf{d}\bm x, \end{aligned} \end{equation} for any~$v \in V_{0}^h$.
\subsection{{\em A Posteriori} Residual Analysis} The aim of this section is to derive {\em a posteriori} residual bounds for the linearized FEM~\eqref{eq:fem}.
\subsubsection{\emph{A Posteriori} Residual Bound} In order to measure the discrepancy between the finite element discretization~\eqref{eq:fem} and the original problem~\eqref{poisson}, a natural quantity to bound is the residual~$\mathsf{F}_{\varepsilon}(u_{n+1}^{h})$ in~$X'$. Let $\mathcal{I}^h:\,H_{0}^{1}(\Omega)\rightarrow V_{0}^{h} $ be the quasi-interpolation operator of Cl\'ement (see, e.g., \cite[Corollary~4.2]{AmreinWihler:15}). Then, testing~\eqref{eq:start} with~$\mathcal{I}^h v\in V^h_0$, for an arbitrary~$v\in X$, implies that \[ \int_{\Omega}{\varepsilon\nabla u_{n+1}^{h} \cdot \nabla \mathcal{I}^h v\,\mathsf{d}\bm x} = \int_{\Omega}{(\mathsf{T}_{f}(u_{n+1}^{h}) -\delta_{n}^{h})\mathcal{I}^h v\,\mathsf{d}\bm x}. \] Then, there holds the identity \begin{align*} \left \langle \mathsf{F}_{\varepsilon}(u_{n+1}^{h}),v \right \rangle &=\int_\Omega {\varepsilon\nabla u_{n+1}^{h}\cdot\nabla (\mathcal{I}^h v-v)}\,\mathsf{d}\bm x +\int_\Omega (\mathsf{T}_{f}(u_{n+1}^h)-\delta_{n}^{h}) (v-\mathcal{I}^h v)\,\mathsf{d}\bm x\\ &\quad-\int_\Omega\left\{\mathsf{T}_{f}(u_{n+1}^h)-\delta_{n}^{h}-f(u_{n+1}^{h})\right\}v\,\mathsf{d}\bm x, \end{align*} for any~$v\in X$. Integrating by parts in the first term on the right-hand side, recalling the fact that~$(v-\mathcal{I}^h v)=0$ on~$\partial\Omega$, and applying some elementary calculations, yields that \[ \left \langle \mathsf{F}_{\varepsilon}(u_{n+1}^{h}),v \right \rangle =\sum_{E\in\mathcal{E}^h}a_E+\sum_{T\in\mathcal{T}^h}(b_T-c_T), \] where \begin{equation} \begin{aligned} a_{E}&:= \int_{E}\varepsilon\jmp{\nabla u_{n+1}^{h}}(\mathcal{I}^h v-v)\,\mathsf{d} s,\qquad c_{T}:=\int_{T}{\left\{\mathsf{T}_{f}(u_{n+1}^h)-\delta_{n}^{h}-f(u_{n+1}^{h})\right\}v}\,\mathsf{d}\bm x,\nonumber \\ b_{T}&:=\int_{T}{\left\{\varepsilon \Delta u_{n+1}^{h}+\mathsf{T}_{f}(u_{n+1}^h)-\delta_{n}^{h}\right\}(v-\mathcal{I}^h v)}\,\mathsf{d}\bm x, \end{aligned} \end{equation} with~$E\in\mathcal{E}^h$, $T\in\mathcal{T}^h$. Here, for any edge $ E=\partial T^\sharp\cap \partial T^\flat \in \mathcal{E}^h $ shared by two neighboring elements~$T^\sharp, T^\flat\in\mathcal{T}^h$, where $\bm n^\sharp$ and~$\bm n^\flat$ signify the unit outward vectors on~$\partial T^\sharp$ and~$\partial T^\flat$, respectively, we denote by \[ \jmp{\nabla u_{n+1}^h}(\bm x)=\lim_{t\to 0^+}\nabla u_{n+1}^h(\bm x+t\bm n^\sharp)\cdot\bm n^\sharp+\lim_{t\to 0^+}\nabla u_{n+1}^h(\bm x+t\bm n^\flat)\cdot\bm n^\flat,\qquad \bm x\in E, \] the jump across~$E$. Then, for~$T\in\mathcal{T}^h$, defining the linearization residual \begin{equation} \label{linearizationerror} R_{n,T}:=\norm{\mathsf{T}_{f}(u_{n+1}^h)-\delta_{n}^{h}-f(u_{n+1}^{h})}_{0,T}, \end{equation} as well as the FEM approximation residual \begin{equation} \label{Femerror} \eta_{n,T}^2:= \alpha_{T}^2 \norm{\varepsilon \Delta u_{n+1}^{h}+\mathsf{T}_f(u_{n+1}^h)-\delta_{n}^{h}}_{0,T}^2+\frac{1}{2}\sum_{E\in \mathcal{E}^{h}(T)}{\varepsilon^{-\nicefrac12}\alpha_E\norm{\varepsilon \jmp{\nabla u_{n+1}^{h} }}_{0,E}^2}, \end{equation} with~$\alpha_T$ and~$\alpha_E$ from~\eqref{boundary}, we proceeding along the lines of the proof of~\cite[Theorem~4.4]{AmreinWihler:15} in order to obtain the following result.
\begin{theorem} \label{thm:1} For~$n\ge 0$ there holds the upper \emph{a posteriori} residual bound \begin{equation} \label{eq:upperbound} \norm{\mathsf{F}(u_{n+1}^{h})}_{X'}^2 \preccurlyeq R_{n,\Omega}^2+\sum_{T\in \mathcal{T}^h}{\eta_{n,T}^{2}}, \end{equation} with~$R_{n,\Omega}$ and~$\eta_{n,T}$, $T\in\mathcal{T}^h$, from~\eqref{linearizationerror} and~\eqref{Femerror}, respectively. \end{theorem}
\begin{remark} Following our approach in~\cite[Theorem~4.5]{AmreinWihler:15}, under certain conditions on the nonlinearity~$f$, it can be shown that the right-hand side of the above bound~\eqref{eq:upperbound} is equivalent to the error norm~$\NN{u-u_{n+1}^{h}}_{\varepsilon,\Omega}$. \end{remark}
\begin{remark} In addition to the upper estimate in the above Theorem~\ref{thm:1}, we notice that local lower \emph{a posteriori} residual bounds can be established for the proposed {\sf PTC}-Galerkin method as well. Indeed, this can be accomplished similarly to our analysis in~\cite[\S4.4.2]{AmreinWihler:15} (see also~\cite{Verfuerth}), which is based on the application of standard bubble function techniques. \end{remark}
\subsection{A Fully Adaptive {\sf PTC}-Galerkin Algorithm} We will now propose a procedure that will combine the {\sf PTC}-method presented in~Section~\ref{sec:prediction} with an automatic finite element mesh refinement strategy. More precisely, based on the \emph{a posteriori} residual bound from Theorem~\ref{thm:1}, the main idea of our approach is to provide an interplay between {\sf PTC}-iterations and adaptive mesh refinements which is based on monitoring the two residuals in~\eqref{linearizationerror} and~\eqref{Femerror}, and on acting according to whatever quantity is dominant in the current computations. We make the assumption that the {\sf PTC}-Galerkin sequence $\left\{u_{n+1}^{h}\right\}_{n\ge 0} $ given by~\eqref{eq:fem}, with step size~$k_n^\star$ from~\eqref{eq:k*} (or~$\bm{k}^\star_n$ from \eqref{eq:stepsize-control}) is well-defined as long as the iterations are being performed. The individual computational steps are summarized in Algorithm~\ref{al:full}.
\begin{algorithm} \caption{Fully-adaptive {\sf PTC}-Galerkin method} \label{al:full} \begin{algorithmic}[1] \State Given a parameter~$\theta>0$, a (coarse starting) triangulation $ \mathcal{T}^h $ of~$\Omega$, an initial step size $k_0>0$, a maximal number of degrees of freedom $\text{DOF}_{\text{max}}$, and an initial guess $ u_{0}^{h} \in V_{0}^{h} $. Set~$n\gets 0$. \While {$\text{DOF}\le\text{DOF}_{\text{max}}$}
\myState {Compute the FEM solution~$u_{n+1}^h$ from~\eqref{eq:fem} on the mesh~$\mathcal{T}^h$.}
\myState {Evaluate the corresponding residual indicators $ \eta_{n,T} $, $ T \in \mathcal{T}^h $, and $R_{n,\Omega} $ from~\eqref{linearizationerror} and~\eqref{Femerror}, respectively. }
\If{ \[ R_{n,\Omega}^2\le \theta\sum_{T\in \mathcal{T}^h}{\eta_{n,T}^2} \] \quad} \myStateDouble{refine the mesh $ T \in \mathcal{T}^h $ adaptively based on the elementwise residual indicators~$\eta_{n,T}$, $T\in\mathcal{T}^h$ from Theorem~\ref{thm:1}, and go back to step~({\footnotesize 2:}) with the previously computed solution~$u_{n+1}^{h}$ as interpolated on the refined mesh;} \Else \myStateDouble{perform another {\sf PTC}-step based on the new step size $k_n=\bm{k}_n^\star$ as proposed in~\eqref{eq:stepsize-control} and go back to~({\footnotesize 3:}).} \EndIf \myState{set~$n\leftarrow n+1$.} \EndWhile \end{algorithmic} \end{algorithm}
\subsection{Numerical Experiments} \label{sec:numerics} We will now illustrate and test the above fully adaptive Algorithm~\ref{al:full} with two numerical experiments in 2d. The linear systems resulting from the finite element discretization~\eqref{eq:start} are solved by means of a direct solver.
\begin{example}\label{ex:1} Let us consider first the Sine-Gordon type problem \begin{equation*}
-\varepsilon\Delta u = -\sin(u)-u+1,\ \text{in }\Omega=(0,1)^2,\qquad u=0\ \text{on }\partial\Omega. \end{equation*} Here, $f(u)=-\sin(u)-u+1$, and~$f'(u)=-\cos(u)-1$. In particular, by application of Proposition~\ref{pr:f}, we observe that the structural assumptions~\eqref{eq:A1} and~\eqref{eq:A2} are fulfilled. Neclecting the boundary conditions for a moment, one observes that the unique positive zero $ u \approx 0.51$ of $ f(u) $ is a solution of the PDE. We therefore expect boundary layers along $\partial \Omega$; see Figure \ref{Sine-Gordon-Mesh} (right). Moreover, the focus of this experiment is on the robustness of the \emph{a posteriori} residual bound \eqref{eq:upperbound} with respect to the singular perturbation paramater $\varepsilon$ as $\varepsilon \to 0$. Starting from the initial mesh depicted in Figure \ref{Sine-Gordon-Mesh} (left) with $u_{0}^{h}(\nicefrac{1}{2},\nicefrac{1}{2})\approx \nicefrac{1}{2}$, we test the fully adaptive {\sf PTC}-Galerkin Algorithm \ref{al:full} for different choices of $\varepsilon=\{10^{-i}\}_{i=0}^{9}$. In Algorithm \ref{al:full} the parameters are chosen to be $\theta = 0.5$ and $ k_0=1$. As $\varepsilon\to 0 $ the resulting solutions feature ever stronger boundary layers; see Figure \ref{Sine-Gordon-Mesh} (right). The performance data in Figure \ref{Performance-Data} (left) shows that the residuals decay, firstly, robust in $\varepsilon$, and, secondly, of (optimal) order~$\nicefrac{1}{2}$ with respect to the number of degrees of freedom. \end{example}
\begin{figure}
\caption{Example~\ref{ex:1} for $\varepsilon=10^{-7}$: Initial mesh (left), and the adaptively refined mesh resolving the solution (right).}
\label{Sine-Gordon-Mesh}
\end{figure}
\begin{figure}
\caption{Estimated residuals for different choices of $\varepsilon$. On the left for Example~\ref{ex:1} and on the right for Example~\ref{ex:2}.}
\label{Performance-Data}
\end{figure}
\begin{example}
\label{ex:2} Finally, we turn to the well-known nonlinear Ginzburg-Landau equation on the square~$\Omega=(-1,1)^2$ given by \begin{equation*}
\begin{aligned} -\varepsilon\Delta u&= u(1- u^2) \ \text{in } \Omega,\qquad u = 0 \ \text{on } \partial \Omega. \end{aligned} \end{equation*} Clearly $ u\equiv 0 $ is a solution. In addition, any solution~$u$ appears pairwise as $-u$ is obviously a solution also. Again, neglecting the boundary conditions for a moment, we observe that $u\equiv1$ and $u \equiv-1$ are solutions of the PDE. We therefore expect boundary layers along $ \partial \Omega$, and possibly within the domain~$\Omega$; see Figure~\ref{Ginzburg-Landau-Mesh} (right). Here we always start from the initial mesh depicted in Figure~\ref{Ginzburg-Landau-Mesh} (left) with $u_{0}^{h}\equiv 1$ on the interior nodes. Again we test the fully adaptive {\sf PTC}-Galerkin Algorithm~\ref{al:full} for different choices of $\varepsilon=\{10^{-i}\}_{i=0}^{9}$. The parameters are still chosen to be $\theta = 0.5$ and $ k_0=1$. As in Example~\ref{ex:1}, for $\varepsilon\to 0 $ the resulting solution feature ever stronger boundary layers; see Figure \ref{Ginzburg-Landau-Mesh} (right). In addition, from the performance data given in Figure \ref{Performance-Data} (right) we observe that the residuals decay again robust in $\varepsilon$. Finally we notice convergence of (optimal) order~$\nicefrac12$ with respect to the number of degrees of freedom. We remark that, although~\eqref{eq:A1} and~\eqref{eq:A2} are not necessarily satisfied for this problem, our fully adaptive {\sf PTC}-Galerkin approach still delivers good results. \begin{figure}
\caption{Example~\ref{ex:2} for $\varepsilon =10^{-7}$: Initial mesh (left), and the adaptively refined mesh resolvong the solution (right).}
\label{Ginzburg-Landau-Mesh}
\end{figure} \end{example}
\section{Conclusions}\label{sc:concl} The aim of this paper was to introduce a reliable and computationally feasible procedure for the numerical solution of semilinear elliptic boundary value problems, with possible singular perturbations. The key idea is to combine adaptive step size control for the {\sf PTC}-metod with an automatic mesh refinement finite element procedure. Furthermore, the sequence of linear problems resulting from the application of pseudo transient continuation and Galerkin discretization is treated by means of a robust (with respect to the singular perturbations) {\em a posteriori} residual analysis and a corresponding adaptive mesh refinement process. Our numerical experiments clearly illustrate the ability of our approach to reliably find solutions reasonably close to the initial guesses, and to robustly resolve the singular perturbations at an optimal rate.
\end{document} | arXiv |
\begin{document}
\IEEEoverridecommandlockouts
\title{G\"odel Logic: from Natural Deduction to Parallel Computation}
\author{\IEEEauthorblockN{Federico Aschieri } \IEEEauthorblockA{Institute of Discrete Mathematics and Geometry\\TU Wien, Austria\\
} \and \IEEEauthorblockN{Agata Ciabattoni
} \IEEEauthorblockA{Theory and Logic Group\\TU Wien, Austria\\
} \and \IEEEauthorblockN{Francesco A. Genco
} \IEEEauthorblockA{Theory and Logic Group\\TU Wien, Austria\\
} \thanks{Supported by FWF: grants M 1930--N35, Y544-N2, and W1255-N23.} }
\maketitle
\begin{abstract} Propositional G\"{o}del logic $\logic{G}$ extends intuitionistic logic with the non-constructive principle of linearity \mbox{$(A \rightarrow B ) \vee (B \rightarrow A)$}. We introduce a Curry--Howard correspondence for $\logic{G}$ and show that a simple natural deduction calculus can be used as a typing system. The resulting functional language extends the simply typed $\lambda$-calculus via a synchronous communication mechanism between parallel processes, which increases its expressive power. The normalization proof employs original termination arguments and proof transformations implementing forms of code mobility. Our results provide a computational interpretation of $\logic{G}$, thus proving A.\ Avron's 1991 thesis. \end{abstract}
\IEEEpeerreviewmaketitle
\section{Introduction}
Logical proofs are static. Computations are dynamic. It is a striking discovery that the two coincide: formulas correspond to types in a programming language, logical proofs to programs of the corresponding types and removing detours from proofs to
evaluation of programs. This correspondence, known as Curry--Howard isomorphism, was first discovered for constructive proofs, and in particular for intuitionistic natural deduction and typed $\lambda$-calculus \cite{Howard} and later extended to classical proofs, despite their use of non-constructive principles, such as the excluded middle \cite{deGrooteex, AschieriZH} or reductio ad absurdum \cite{Griffin, Parigot}.
Nowadays various different logics (linear~\cite{CP2010}, modal~\cite{MCHP} ...) have been related to many different notions of computation; the list is long, and we refer the reader to \cite{Wadler}.
\subsection*{G\"odel logic, Avron's conjecture and previous attempts} Twenty-five years have gone by since Avron conjectured in \cite{Avron91} that G\"odel logic $\logic{G}$ \cite{goedel} -- one of the most useful and interesting logics intermediate between intuitionistic and classical logic -- might provide a basis for parallel $\lambda$-calculi. Despite the interest of the conjecture and despite various attempts, no Curry--Howard correspondence has so far been provided for $\logic{G}$. The main obstacle has been the lack of an adequate natural deduction calculus. Well designed natural deduction inferences can indeed be naturally interpreted as program instructions, in particular as typed $\lambda$-terms. Normalization~\cite{Prawitz}, which corresponds to the execution of the resulting programs, can then be used to obtain proofs only containing formulas that are subformulas of some of the hypotheses or of the conclusion. However the problem of finding a natural deduction for $\logic{G}$ with this property, called analyticity, looked hopeless for decades.
All approaches explored so far to provide a precise formalization of $\logic{G}$ as a logic for parallelism, either sacrificed analyticity~\cite{ Aschieri2016}
or tried to devise forms of natural deduction whose structures mirror {\em hypersequents} -- which are sequents operating in parallel \cite{Avron96}. Hypersequents were indeed successfully used in \cite{Avron91} to define an analytic calculus for $\logic{G}$ and were intuitively connected to parallel computations: the key rule introduced by Avron to capture the linearity axiom -- called {\em communication} -- enables sequents to exchange their information and hence to ``communicate''. The first analytic natural
deduction calculus proposed for $\logic{G}$ \cite{HyperAgata} uses indeed parallel intuitionistic derivations
joined together by the hypersequent separator. Normalization is obtained
there only by translation into Avron's calculus: no reduction rules for deductions and no corresponding $\lambda$-calculus were provided. The former task was carried out in \cite{BP2015}, that contains a propositional hyper natural deduction with a normalization procedure. The definition of a corresponding $\lambda$-calculus and Curry--Howard correspondence are left as an open problem, which might have a complex solution due to the elaborated structure of hyper deductions. Another attempt along the ``hyper line'' has been made in \cite{Hirai}. However, not only the proposed proof system is not shown to be analytic, but the associated $\lambda$-calculus is not a Curry--Howard isomorphism: the computation rules of the $\lambda$-calculus are not related to proof transformations, i.e.\ {\em Subject Reduction} does not hold. \subsection*{$\bblambda_{\mathrm{G}}$: Our Curry--Howard Interpretation of G\"odel Logic} We introduce a natural deduction and a Curry--Howard correspondence for propositional $\logic{G}$. We add to the $\lambda$-calculus an operator that, from the programming viewpoint, represents parallel computations and communications between them; from the logical viewpoint, the linearity axiom; and from the proof theory viewpoint, the hypersequent separator among sequents. We call the resulting calculus $\bblambda_{\mathrm{G}}$: parallel $\lambda$-calculus for $\logic{G}$. $\bblambda_{\mathrm{G}}$ relates to the natural deduction $\mathbf{NG}$ for $\logic{G}$ as typed $\lambda$-calculus relates to the natural deduction ${\bf NJ}$ for intuitionistic logic $\mathbf{IL}$: \tikzstyle{crc}=[circle, minimum size=7mm, inner sep=0pt, draw] \begin{center} \begin{tikzpicture}[node distance=1.5cm,auto,>=latex', scale=0.3]
\node [crc, align=center] (1) at (-10,3.5) {$\mathbf{IL}$};
\node[crc, align=center] (2) at (0,3.5) {${\bf NJ}$};
\node [crc,align=center] (3) at (10,3.5) {$\lambda$};
\node [crc, align=center] (4) at (-10,0) {$\logic{G}$};
\node[crc, align=center] (5) at (0,0) {$\mathbf{NG}$};
\node [crc,align=center] (6) at (10,0) {$\bblambda_{\mathrm{G}}$};
\path[<->] (1) edge [thick, align=center] node {}(2);
\path[<->] (2) edge [double, thick, align=center] node {}(3);
\path[<->] (4) edge [thick, align=center] node {\emph{Soundness and}\\\emph{Completeness}}(5);
\path[<->] (5) edge [double, thick, align=center] node {\emph{Curry--Howard}\\ \emph{correspondence}}(6);
\end{tikzpicture} \end{center} We prove: the perfect match between computation steps and proof reductions in the Subject Reduction Theorem; the Normalization Theorem, by providing a terminating reduction strategy for $\bblambda_{\mathrm{G}}$; the Subformula Property, as corollary. The expressive power of $\bblambda_{\mathrm{G}}$ is illustrated through examples of programs and connections with the $\pi$-calculus~\cite{Milner, sangiorgiwalker2003}.
The natural deduction calculus $\mathbf{NG}$ that we use as type system for $\bblambda_{\mathrm{G}}$ is particularly simple: it extends ${\bf NJ}$ with the $(\mathsf{com})$ rule (its typed version is displayed below), which was first considered in \cite{L1982} to define a natural deduction calculus for $\logic{G}$, but with no normalization procedure. The calculus $\mathbf{NG}$ follows the basic principle of natural deduction that \emph{new axioms require new computational reductions}; this contrasts with the basic principle of sequent calculus employed in the ``hyper approach'', that \emph{new axioms require new deduction structures}. Hence we keep the calculus simple and deal with the complexity of the hypersequent structure at the operational side. Consequently, the programs corresponding to $\mathbf{NG}$ proofs maintain the syntactical simplicity of $\lambda$-calculus. The normalization procedure for $\mathbf{NG}$ extends Prawitz's method with ideas inspired by hypersequent cut-elimination, by normalization in classical logic \cite{AschieriZH} and by the embedding in \cite{CG2016} between hypersequents and systems of rules \cite{Negri:2014}; the latter shows that $(\mathsf{com})$ reformulates Avron's communication rule.
The inference rules of $\mathbf{NG}$ are decorated with $\bblambda_{\mathrm{G}}$-terms, so that we can directly read proofs as typed programs. The decoration of the ${\bf NJ}$ inferences is standard and the typed version of $(\mathsf{com})$ is \[ \infer[\D]{u \parallel_{a} v: C}{\infer*{u:C}{[a^{\scriptscriptstyle A\rightarrow B}: A\rightarrow B]} && \infer*{v:C}{[a^{\scriptscriptstyle B\rightarrow A}: B\rightarrow A]}}\]
Inspired by~\cite{Aschieri2016}, we use the variable $a$ to represent a {\em private} communication channel between the processes $u$ and $v$. The computational reductions associated to $\parallel_{a}$ -- \emph{cross reductions} -- enjoy a natural interpretation in terms of higher-order process passing, a feature which is not directly rendered through communication by reference~\cite{perez2015} and is also present in higher-order $\pi$-calculus~\cite{sangiorgiwalker2003}. Nonetheless cross reductions handle more subtle migration issues. In particular, a cross reduction can be activated whenever a communication channel $a$ is ready to transfer information between two parallel processes: \[ \mathcal{C}[a\, u]\parallel_{a} \mathcal{D}[a\, v] \] Here $\mathcal{C}$ is a process containing a fragment of code $u$, and $\mathcal{D}$ is a process containing a fragment of code $v$. Moreover, $\mathcal{C}$ has to send $u$ through the channel $a$ to $\mathcal{D}$, which in turn needs to send $v$ through $a$ to $\mathcal{C}$. In general we cannot simply send the programs $u$ and $v$: some resources in the computational environment that are used by $u$ and $v$ may become inaccessible from the new locations~\cite{codemobil}. Cross reductions solve the problem by exchanging the location of $u$ and $v$ and creating a new communication channel for their resources. Technically, the channel takes care of \emph{closures} -- the contexts containing the definitions of the variables used in a function's body \cite{landin}. Several programming languages such as JavaScript, Ruby or Swift provide mechanisms to support and handle closures. In our case, they are the basis of a \emph{process migration mechanism} handling the bindings between code fragments and their computational environments. Cross reductions also improve the efficiency of programs by facilitating partial evaluation of open processes (see Example~\ref{ex:code_mobility}).
\section{Preliminaries on G\"odel logic} Also known as G\"odel--Dummett logic~\cite{dummett}, G\"odel logic $\logic{G}$ naturally turns up in a number of different contexts; among them, due to the natural interpretation of its connectives as functions over the real interval $[0, 1]$, $\logic{G}$ is one of the best known `fuzzy logics', e.g.~\cite{MGO}.
Although propositional $\logic{G}$ is obtained by adding the linearity axiom $(lin) $ $(A \rightarrow B) \vee (B \rightarrow A)$ to any proof calculus for intuitionistic logic, analytic calculi for $\logic{G}$ have only been defined in formalisms {\em extending} the sequent calculus. Among them, arguably, the hypersequent calculus in \cite{Avron91} is the most successful one, see, e.g.,~\cite{MGO}. In general a hypersequent calculus is defined by incorporating a sequent calculus (Gentzen's {\em LJ}, in case of $\logic{G}$) as a sub-calculus and allowing sequents to live in the context of finite multisets of sequents. \begin{definition} A \textbf{hypersequent} is a multiset of sequents, written as $ \Gamma_1 \Rightarrow \Pi_1 \ |\ \dots \ |\ \Gamma_n \Rightarrow \Pi_n$ where, for all $i = 1, \dots n,$ $\Gamma_i \Rightarrow \Pi_i$ is an ordinary sequent. \end{definition}
The symbol ``$|$'' is a meta-level disjunction; this is reflected by the presence in the calculus of the external structural rules of weakening and contraction, operating on whole sequents, rather than on formulas. The hypersequent design opens the possibility of defining new rules that allow the ``exchange of information'' between different sequents. It is this type of rules which increases the expressive power of hypersequent calculi compared to sequent calculi. The additional rule employed in Avron's calculus for $\logic{G}$ \cite{Avron91} is the so called {\em communication rule}, below presented in a slightly reformulated version (as usual $G$ stands for a possibly empty hypersequent):
{\small \[ \infer{G\ |\ \Gamma_1, A \Rightarrow C \ |\ \Gamma_2, B \Rightarrow D }{G \ |\ \Gamma_1, B \Rightarrow C & G\ |\ \Gamma_2, A \Rightarrow D}\] }
\section{Natural Deduction}\label{section-ND} The very first step in the design of a Curry--Howard correspondence is to lay a solid logical foundation. No architectural mistake is allowed at this stage: the natural deduction must be structurally simple and the reduction rules as elementary as possible. We present such a natural deduction system $\mathbf{NG}$ for G\"odel logic. $\mathbf{NG}$ extends Gentzen's propositional natural deduction ${\bf NJ}$ (see~\cite{Prawitz}) with a rule accounting for axiom $(lin)$. We describe the reduction rules for transforming every $\mathbf{NG}$ deduction into an analytic one and present the ideas behind the Normalization Theorem, which is proved in the $\lambda$-calculus framework in Section \ref{section-normalization}.
$\mathbf{NG}$ is the natural deduction version of the sequent calculus with systems of rules in~\cite{Negri:2014}; the latter embeds (into) Avron's hypersequent calculus for $\logic{G}$. Indeed~\cite{CG2016} introduces a mapping from (and into) derivations in Avron's calculus into (and from) derivations in the {\em LJ} sequent calculus for intuitionistic logic with the addition of the system of rules {\small \[ \infer[(com_{end})]{\Gamma \Rightarrow \Pi}{\infer*{\Gamma \Rightarrow \Pi}{\infer[(com_{1})]{A , \Gamma_{1} \Rightarrow C}{B, \Gamma _{1} \Rightarrow C}} & \infer*{\Gamma \Rightarrow \Pi}{\infer[(com_{2})]{B, \Gamma _{2} \Rightarrow D}{A, \Gamma _{2} \Rightarrow D}}} \] } where $(com_{1}) , (com_{2})$ can only be applied (possibly many times) above respectively the left and right premise of $(com_{end})$. The above system, that reformulates Avron's {\em communication} rule, immediately translates into the natural deduction rule below, whose addition to ${\bf NJ}$ leads to a natural deduction calculus for $\logic{G}$ \begin{small} \[ \infer[\D]{C}{ \infer*{C} {\infer[\mathsf{com}_{l}]{ A } { \deduce{ B } {\vdots}} } & \infer*{ C } {\infer[\mathsf{com}_{r}]{ B } { \deduce{ A } {\vdots} } } } \] \end{small} Not all the branches of a derivation containing the above rule are ${\bf NJ}$ derivations. To avoid that, and to keep the proof of the Subformula Property (Theorem~\ref{theorem-subformula}) as simple as possible, we use the equivalent rule below, first considered in \cite{L1982}. \begin{definition}[$\mathbf{NG}$] \label{def:NG} The natural deduction calculus $\mathbf{NG}$ extends ${\bf NJ}$ with the $(\mathsf{com})$ rule: \[\infer[\D]{C}{\infer*{C}{[A \rightarrow B]} & \infer*{C}{[B \rightarrow A]}}\] \end{definition}
Let $\vdash_{\mathbf{NG}}$ and $\vdash_{\logic{G}}$ indicate the derivability relations in $\mathbf{NG}$ and in ${\bf NJ} + (\textit{lin})$, respectively. \begin{theorem}[Soundness and Completeness] \label{th:sound} For any set $\Pi$ of formulas and formula $A$,
$\Pi \vdash_{\mathbf{NG}} A$ if and only if $\Pi \vdash_{\logic{G}} A$. \end{theorem} \begin{proof} ($\Rightarrow)$ Applications of $(\mathsf{com})$ can be simulated by $\vee$ eliminations having as major premiss an instance of $(\textit{lin})$. ($\Leftarrow$) Easily follows by the following derivation:
\[ \infer[\mathsf{com}^{1}]{(A \rightarrow B) \vee (B \rightarrow A)}{\infer
{(A \rightarrow
B) \vee (B \rightarrow A)}{{[A \rightarrow
B]^{1}}} &&& \infer
{(A \rightarrow B)
\vee (B \rightarrow A)}{{[B \rightarrow
A]^{1}}}} \]\end{proof} \noindent {\em Notation}. To shorten derivations henceforth we will use \[ \AxiomC{$A$} \RightLabel{$\mathsf{com}_{l}$} \UnaryInfC{$B$} \DisplayProof \qquad \AxiomC{$B$} \RightLabel{$\mathsf{com}_{r}$} \UnaryInfC{$A$} \DisplayProof \quad \text{ as abbreviations for }\] \[ \AxiomC{$[A\rightarrow B]$} \AxiomC{$A$} \BinaryInfC{$B$} \DisplayProof \qquad \AxiomC{$[B\rightarrow A]$} \AxiomC{$B$} \BinaryInfC{$A$} \DisplayProof \] respectively, and call them \textbf{communication inferences}.
As usual, we will use $\neg A$ and $\top$ as shorthand for $A\rightarrow\bot$ and $\bot \rightarrow \bot$. Moreover, we exploit the equivalence of $A \lor B$ and $((A\rightarrow B)\rightarrow B)\ \land\ ((B\rightarrow A)\rightarrow A)$ in $\logic{G}$ (see \cite{dummett}) and treat $\vee$ as a defined connective.
\subsection{Reduction Rules and Normalization}
A normal deduction in $\mathbf{NG}$ should have two essential features: every intuitionistic Prawitz-style reduction should have been carried out and the Subformula Property should hold. Due to the $(\mathsf{com})$ rule, the former is not always enough to guarantee the latter. Here we present the main ideas behind the normalization procedure for $\mathbf{NG}$ and the needed reduction rules. The computational interpretation of the rules will be carried out through the $\bblambda_{\mathrm{G}}$ calculus in Section \ref{section-system}.
The main steps of the normalization procedure are as follows: \begin{itemize} \item We permute down all applications of $(\mathsf{com})$. \end{itemize} The resulting deduction -- we call it in {\em parallel form} -- consists of purely intuitionistic subderivations joined together by consecutive $(\mathsf{com})$ inferences occurring immediately above the root. This transformation is a key tool in the embedding
between hypersequents and systems of rules \cite{CG2016}. The needed reductions are instances of Prawitz-style permutations for $\vee$ elimination. Their list can be obtained by translating into natural deduction the permutations in Fig.~\ref{fig:red}.
Once obtained a parallel form, we interleave the following two steps. \begin{itemize} \item We apply the standard intuitionistic reductions (\cite{Prawitz}) to the parallel branches of the derivation. \end{itemize} This way we normalize each single intuitionistic derivation, and this can be done in parallel. The resulting derivation, however, need not satisfy yet the Subformula Property. Intuitively, the problem is that communications may discharge hypotheses that have nothing to do with their conclusion. \begin{itemize} \item We apply specific reductions to replace the $(\mathsf{com})$ applications that violate the Subformula Property. \end{itemize} These reductions -- called \emph{cross reductions} -- account for the hypersequent cut-elimination. \label{sec:cross_explained} They allow to get rid of the new detours that appear in configurations like the one below on the left. To remove these detours, a first idea would be to simultaneously move the deduction $\mathcal{D}_1$ to the right and $\mathcal{D}_2$ to the left thus obtaining the derivation below right: \[ \infer[\D]{C}{\infer*{C}{\infer[\mathsf{com}_{l}]{B}{\deduce{A}{\mathcal{D}_{1}}}}&\infer*{C}{\infer[\mathsf{com}_{r}
]{A}{\deduce{B}{\mathcal{D}_{2}}}}} \quad \quad \quad \quad \quad \quad \infer{C}{\infer*{C}{\deduce{B}{\mathcal{D}_{2}}}&\infer*{C}{\deduce{A}{\mathcal{D}_{1}}}}\] In fact, in the context of Krivine's realizability, Danos and Krivine [9] studied the linearity axiom as a theorem of classical logic and discovered that its realizers implement a \emph{restricted} version of this transformation. Their transformation does not lead however to the subformula property for NG. The unrestricted transformation above, on the other hand, cannot work; indeed $\mathcal{D}_1$ might contain the hypothesis $A\rightarrow B$ and hence it cannot be moved on the right. Even worse, $\mathcal{D}_1$ may depend on hypotheses that are locally opened, but discharged below $B$ but above $C$. Again, it is not possible to move $\mathcal{D}_1$ on the right as naively thought, otherwise new global hypotheses would be created.
We overcome these barriers by our cross reductions. Let us highlight $\Gamma$ and $\Delta$, the hypotheses of ${\cal D}_{1}$ and ${\cal D}_{2}$ that are respectively discharged below $B$ and $A$ but above the application of $(\mathsf{com})$. Assume moreover, that $A\rightarrow B$ does not occur in ${\cal D}_{1}$ and $B\rightarrow A$ does not occur in ${\cal D}_{2}$ as hypotheses discharged by $(\mathsf{com})$.
A cross reduction transforms the deduction below left into the deduction below right (if $(\mathsf{com})$ in the original proof discharges in each branch exactly one occurrence of the hypotheses, and $\Gamma$ and $\Delta$ are formulas) \[\infer[\D]{C}{\infer*{C}{\infer[\mathsf{com}_{l}]{B}{\deduce{A}{\deduce{\mathcal{D}_{1}}{\Gamma}}}}&\infer*{C}{\infer[\mathsf{com}_{r} ]{A}{\deduce{B}{\deduce{\mathcal{D}_{2}}{\Delta}}}}} \quad \quad \quad \quad \infer[\mathsf{com}] {C}{ \infer*{ C} {\deduce{A} {\deduce{
{\cal D}_{1}}{
\infer[\mathsf{com}_{l} ]{\Gamma} {\Delta} } } } &
\infer*{C} {\deduce{B} { \deduce{
{\cal D}_{2}}{
\infer[\mathsf{com}_{r}]{\Delta}{\Gamma} } } } } \] and into the following deduction, in the general case \[ \vcenter{ \infer[\mathsf{com} ^{3}] {C} { \infer[\mathsf{com} ^{1}] {C} {\infer*{C} {\infer[\mathsf{com}_{l}^{1}]{ B } {\deduce{A} {\deduce{
{\cal D}_{1}}{
\Gamma}} } } & \infer*{ C} {\deduce{A} {\deduce{
{\cal D}_{1}}{
\infer=[\mathsf{com}_{l} ^{3}]{\Gamma} {\Delta} } } }} && \infer[\mathsf{com} ^{2}] {C} { \infer*{C} {\deduce{B} { \deduce{
{\cal D}_{2}}{
\infer=[\mathsf{com}_{r} ^{3}]{\Delta}{\Gamma} } } } & \infer*{C } {\infer[\mathsf{com}_{r} ^{2}]{A } { \deduce{B} { \deduce{
{\cal D}_{2}}{
\Delta} } } } } }}
\] where the double bar notation stands for an application of $(\mathsf{com})$ between {\em sets of
hypotheses} $\Gamma$ and $\Delta$, which means to prove from $\Gamma$
the conjunction of the formulas of $\Gamma$, then to prove the
conjunction of the formulas of $\Delta$ by means of a communication
inference and finally obtain each formula of $\Delta$ by a series of
$\wedge$ eliminations, and vice versa.
Mindless applications of the cross reductions might lead to dangerous loops, see e.g. Example~ \ref{example-nonterm}. To avoid them we will allow cross reductions to be performed only when the proof is not analytic. Thanks to this and to other restrictions, we will prove termination and thus the Normalization Theorem.
\section{The $\bblambda_{\mathrm{G}}$-Calculus} \label{section-system} We introduce $\bblambda_{\mathrm{G}}$, our parallel $\lambda$-calculus for $\logic{G}$. $\bblambda_{\mathrm{G}}$ extends the standard Curry--Howard correspondence~\cite{Wadler} for intuitionistic natural deduction with a parallel operator that interprets the inference for the linearity axiom. We describe $\bblambda_{\mathrm{G}}$-terms and their computational behavior, proving as main result of the section the Subject Reduction Theorem, stating that the reduction rules preserve the type.
{\footnotesize
\hrule \begin{description} \comment{We also assume that the term formation rules are applied in such a way that in each term $t$, if $t$ contains $\Wit{a}{P}{{\alpha}}$ or $\Hyp{a}{P}{{\alpha}}$ and $t$ contains $\Wit{a}{Q}{{\alpha}}$ or $\Wit{a}{Q}{{\alpha}}$, then $\mathsf{P}=\mathsf{Q}$.} \comment{or $\wit \beta$ (for some individual variable $\beta$) and $A_1,\ldots, A_n$ formulas of $\Language$.} \item[Axioms]\hspace{10 pt}
$\begin{array}{c} x^A: A\qquad \end{array}\ \ \ \ $
\item[Conjunction] \hspace{30 pt} $\vcenter{\infer{ \langle u,t\rangle: A \wedge B}{u:A & t:B}} \qquad \vcenter{\infer{u\,\pi_0: A}{u: A\wedge B}} \qquad \vcenter{\infer{u\,\pi_1: B}{u: A\wedge B}}$ \\\\
\item[Implication] \hspace{40 pt} $ \vcenter{\infer{\lambda x^{A} u: A\rightarrow B}{\infer*{u:B}{[x^{A}: A]}}} \qquad \vcenter{\infer{tu:B}{ t: A\rightarrow B & u:A}} $ \\\\
\comment{
\item[Disjunction Introduction] $\begin{array}{c} u: A\\ \hline \inj_{0}(u): A\vee B \end{array}\ \ \ \ $ $\begin{array}{c} u: B\\ \hline \inj_{1}(u): A\vee B \end{array}$\\\\
\item[Disjunction Elimination] \AxiomC{$u: A\lor B$} \AxiomC{$[x^{A}: A]$} \noLine \UnaryInfC{$\vdots$} \noLine \UnaryInfC{$w_{1}: C$} \AxiomC{$[y^{B}: B]$} \noLine \UnaryInfC{$\vdots$} \noLine \UnaryInfC{$w_{2}: C$} \TrinaryInfC{$u\, [x^{A}.w_{1}, y^{B}.w_{2}]: C$} \DisplayProof \\ \item[Universal Quantification] $\begin{array}{c} u:\forall \alpha\, A\\ \hline u m: A[m/\alpha] \end{array} $ $\begin{array}{c} u: A\\ \hline \lambda \alpha\, u: \forall \alpha A \end{array}$\\
where $m$ is any term of the language $\Language$ and $\alpha$ does not occur free in the type $B$ of any free variable $x^{B}$ of $u$.\\
\item[Existential Quantification] $\begin{array}{c} u: A[m/\alpha]\\ \hline ( m,u): \exists \alpha A \end{array}\ \ \ $ \AxiomC{$\exists \alpha\, A$} \AxiomC{$[x^{A}: A]$} \noLine \UnaryInfC{$\vdots$} \noLine \UnaryInfC{$t: C$} \BinaryInfC{$u\, [(\alpha, x^{A}). t]: C$} \DisplayProof \\ where $\alpha$ is not free in $C$ nor in the type $B$ of any free variable of $t$.\\ }
\item[Linearity Axiom]\hspace{50 pt} $\vcenter{ \infer{u \parallel_{a} v: C}{\infer*{u:C}{[a^{\scriptscriptstyle A\rightarrow B}: A\rightarrow B]}&\infer*{v:C}{[a^{\scriptscriptstyle B\rightarrow A}: B\rightarrow A]}} }$ \\
\item[Ex Falso Quodlibet] \hspace{50 pt}$\vcenter{\infer{\Gamma\vdash \efq{P}{u}: P}{\Gamma \vdash u: \bot}}\qquad \text{with $P$ atomic, $P \neq \bot$.}$ \end{description}} \hrule
The table above
defines a type assignment for
$\bblambda_{\mathrm{G}}$-terms, called \textbf{proof terms} and denoted by $t, u, v \dots$, which is isomorphic to $\mathbf{NG}$. The typing rules for axioms, implication, conjunction and ex-falso-quodlibet are standard and give rise to the simply typed $\lambda$-calculus, while parallelism is introduced by the rule for the linearity axiom.
Proof terms may contain variables $x_{0}^{A}, x_{1}^{A}, x_{2}^{A}, \ldots$ of type $A$ for every formula $A$; these variables are denoted as $x^{A},$ $ y^{A}, $ $ z^{A} , \ldots,$ $ a^{A}, b^{A}, c^{A}$ and whenever the type is not important simply as $x, y, z, \ldots, a, b$. For clarity, the variables introduced by the $(\mathsf{com})$ rule will be often denoted with letters $a, b, c, \ldots$, but they are not in a syntactic category apart. A variable $x^{A}$ that occurs in a term of the form $\lambda x^{A} u$ is called \textbf{$\lambda$-variable} and a variable $a$ that occurs in a term $u\parallel_{a} v$ is called \textbf{communication variable} and represents a \emph{private} communication channel between the parallel processes $u$ and $v$.
The free and bound variables of a proof term are defined as usual and for the new term $\Ecrom{a}{u}{v}$, all the free occurrences of $a$ in $u$ and $v$ are bound in $\Ecrom{a}{u}{v}$. In the following we assume the standard renaming rules and alpha equivalences that are used to avoid capture of variables in the reduction rules.
\textbf{Notation}. The connective $\rightarrow$ associates to the right and
\comment{so that
\begin{gather*}
A_{1}\rightarrow A_{2}\rightarrow\ldots \rightarrow A_{n} \\
= \\
A_{1}\rightarrow (A_{2}\rightarrow (\ldots (A_{n-1}\rightarrow
A_{n})\ldots))
\end{gather*}} by $\langle t_{1}, t_{2}, \ldots, t_{n}\rangle $ we denote the term
$\langle t_{1}, \langle t_{2}, \ldots \langle t_{n-1}, t_{n}\rangle\ldots \rangle\rangle$ and by $\proj_{i}$, for $i=0, \ldots, n$, the sequence of projections $\pi_{1}\ldots \pi_{1} \pi_{0}$ selecting the $(i+1)$th element of the sequence. Therefore, for every formula sequence $A_{1} , \dots ,A_{n}$ the expression $A_{1} \wedge \dots \wedge A_{n}$ denotes $( A_{1} \wedge ( A_{2} \wedge \dots ( A_{n-1} \wedge A_{n})\dots ))$ or $\top$ if $n=0$.
Often, when $\Gamma= x_{1}: A_{1}, \ldots, x_{n}: A_{n}$ and the list $x_{1}, \ldots, x_{n}$ includes all the free variables of a proof term $t: A$, we shall write $\Gamma\vdash t: A$. From the logical point of view, $t$ represents a natural deduction of $A$ from the hypotheses $A_{1}, \ldots, A_{n}$. We shall write $\logic{G}\vdash t: A$ whenever $\vdash t: A$, and the notation means provability of $A$ in propositional G\"odel logic. If the symbol $\parallel$ does not occur in it, then $t$ is a \textbf{simply typed $\lambda$-term} representing an intuitionistic deduction.
We define as usual the notion of context $\mathcal{C}[\ ]$ as the part of a proof term that surrounds a hole, represented by some fixed variable. In the expression $\mathcal{C}[u]$ we denote a particular occurrence of a subterm $u$ in the whole term $\mathcal{C}[u]$. We shall just need those particularly simple contexts which happen to be simply typed $\lambda$-terms.
\begin{definition}[Simple Contexts]\label{defi-simplec} A \textbf{simple context} $\mathcal{C}[\ ]$ is a simply typed $\lambda$-term with some fixed variable $[]$ occurring exactly once. For any proof term $u$ of the same type of $[]$, $\mathcal{C}[u]$ denotes the term obtained replacing $[]$ with $u$ in $\mathcal{C}[\ ]$, \emph{without renaming of any bound variable}. \end{definition}
As an example, the expression $\mathcal{C}[\ ]:= \lambda x\, z\, ([])$ is a simple context and
the term $\lambda x\, z\, (x\, z)$ can be written as $\mathcal{C}[xz]$.
\comment{ \begin{definition}[Parallel Contexts]\label{defi-parallelc} Omitting parentheses, a \textbf{parallel context} $\mathcal{C}[\ ]$ is an expression of the form $$u_{1}\parallel_{a_{1}} u_{2}\parallel_{a_{2}}\ldots u_{i} \parallel_{a_{i}} [] \parallel_{a_{i+1}}u_{i+1}\parallel_{a_{i+2}}\ldots \parallel_{a_{n}} u_{n}$$ where $[]$ is a placeholder and $u_{1}, u_{2}, \ldots, u_{n}$ are proof terms. For any proof term $u$, $\mathcal{C}[u]$ denotes the replacement in $\mathcal{C}[\ ]$ of the placeholder $[]$ with $u$: $$u_{1}\parallel_{a_{1}} u_{2}\parallel_{a_{2}}\ldots u_{i} \parallel_{a_{i}} u \parallel_{a_{i+1}}u_{i+1}\parallel_{a_{i+2}}\ldots \parallel_{a_{n}} u_{n}$$ \end{definition}} We define below the notion of stack, corresponding to Krivine stack \cite{Krivine} and known as \emph{continuation} because it embodies a series of tasks that wait to be carried out. A stack represents, from the logical perspective, a series of elimination rules; from the $\lambda$-calculus perspective, a series of either operations or arguments. \begin{definition}[Stack]\label{definition-stack} A \textbf{stack} is a sequence \mbox{$\sigma = \sigma_{1}\sigma_{2}\ldots \sigma_{n} $} such that for every $ 1\leq i\leq n$, exactly one of the following holds: either $\sigma_{i}=t$, with $t$ proof term or $\sigma_{i}=\pi_{j}$, with $j\in\{0,1\}$. We will denote the \emph{empty sequence} with $\epsilon$ and with $\xi, \xi', \ldots$ the stacks of length $1$. If $t$ is a proof term, as usual $t\, \sigma$ denotes the term $(((t\, \sigma_{1})\,\sigma_{2})\ldots \sigma_{n})$.
\end{definition}
We define now the notion of \emph{strong subformula}, which is essential for defining the reduction rules of the $\bblambda_{\mathrm{G}}$-calculus and for proving Normalization. The technical motivations will become clear in Sections~\ref{section-subformula} and~\ref{section-normalization}, but the intuition is that the new types created by cross reductions must be always strong subformulas of already existing types. To define the concept of strong subformula we also need the following definition. \begin{definition}[Prime Formulas and Factors \cite{Krivine1}]
A formula is said to be \textbf{prime} if it is not a
conjunction. Every formula is
a conjunction of prime
formulas, called \textbf{prime factors}. \end{definition}
\begin{definition}[Strong Subformula]\label{definition-strongsubf} $B$ is said to be a \textbf{strong subformula} of a formula $A$, if $B$ is a proper subformula of some prime proper subformula of $A$. \end{definition}
Note that in the present context, prime formulas are either atomic formulas or arrow formulas, so a strong subformula of $A$ must be actually a proper subformula of an arrow proper subformula of $A$. The following characterization of the strong subformula relation will be often used. \begin{proposition}[Characterization of Strong Subformulas]\label{proposition-strongsubf} Suppose $B$ is any {strong subformula} of $A$. Then: \begin{itemize} \item If $A=A_{1}\land \ldots \land A_{n}$, with $n>0$ and $A_{1}, \ldots, A_{n}$ are prime, then $B$ is a proper subformula of one among $A_{1}, \ldots, A_{n}$. \item If $A=C\rightarrow D$, then $B$ is a proper subformula of a prime factor of $C$ or $D$. \end{itemize} \end{proposition} \begin{proof}\mbox{}
See the Appendix.\end{proof}
\begin{definition}[Multiple Substitution]\label{defi-multsubst} Let $u$ be a proof term, $\sq{x}=x_{0}^{A_{0}}, \ldots, x_{n}^{A_{n}}$ a sequence of variables and $v: A_{0}\land \ldots \land A_{n}$. The substitution $u^{v/ \sq x}:=u[v\,\proj_{0}/x_{0}^{A_{0}} \ldots \,v\,\proj_{n}/x_{n}^{A_{n}}]$ replaces each variable $x_{i}^{A_{i}}$ of any term $u$ with the $i$th projection of $v$. \end{definition}
We now seek a measure for determining how complex the communication channel $a$ of a term $u\parallel_{a} v$ is. Logic will be our guide. First, it makes sense to consider the types $B, C$ such that $a$ occurs with type $B\rightarrow C$ in $u$ and thus with type $C\rightarrow B$ in $v$. Moreover, assume $u\parallel_{a} v$ has type $A$ and its free variables are $x_{1}^{A_{1}}, \ldots, x_{n}^{A_{n}}$. The Subformula Property tells us that, no matter what our notion of computation will turn out to be, when the computation is done, no object of type more complex than the types of the inputs and the output should appear. Hence, if the prime factors of the types $B$ and $C$ are not subformulas of $A_{1}, \ldots, A_{n}, A$, then these prime factors should be taken into account in the complexity measure we are looking for. The actual definition is the following.
\begin{definition}[Communication Complexity]\label{definition-comcomplexity} Let $u\parallel_{a} v: A$ a proof term with free variables $x_{1}^{A_{1}}, \ldots, x_{n}^{A_{n}}$. Assume that $a^{B\rightarrow C}$ occurs in $u$ and thus $a^{C\rightarrow B}$ in $v$. \begin{itemize} \item The pair $B, C$ is called the \textbf{communication kind} of $a$. \item The \textbf{communication complexity} of $a$ is the maximum among $0$ and the numbers of symbols of the prime factors of $B$ or $C$ that are neither proper subformulas of $A$ nor strong subformulas of one among $A_{1}, \ldots, A_{n}$. \end{itemize} \end{definition}
We explain now the basic reduction rules for the proof terms of $\bblambda_{\mathrm{G}}$, which are given in Figure \ref{fig:red}. As usual, we also have the reduction scheme: $\mathcal{E}[t]\mapsto \mathcal{E}[u]$, whenever $t\mapsto u$ and for any context $\mathcal{E}$. With $\mapsto^{*}$ we shall denote the reflexive and transitive closure of the one-step reduction $\mapsto$.
\textbf{Intuitionistic Reductions}. These are the very familiar computational rules for the simply typed $\lambda$-calculus, representing the operations of applying a function and taking a component of a pair \cite{Girard}. From the logical point of view, they are the standard Prawitz reductions \cite{Prawitz} for ${\bf NJ}$.
\textbf{Cross Reductions}. The reduction rules for $(\mathsf{com})$
model a communication mechanism between parallel processes. In order to apply a cross reduction to a term
$$ \mathcal{C}[a\, u]\parallel_{a} \mathcal{D}[a\, v]$$ several conditions have to be met. These conditions are both natural \emph{and} needed for the termination of computations.\\ \emph{First}, we require the communication complexity of $a$ to be greater than $0$; again, this is a warning that the Subformula Property does not hold. Here we use a logical property as a \emph{computational criterion} for answering the question: when should computation stop? An answer is crucial here, because, as shown in Example \ref{example-nonterm}, unrestricted cross reductions do not always terminate. In $\lambda$-calculi the Subformula Property fares pretty well as a stopping criterion. In a sense, it detects all the \emph{essential} operations that really have to be done. For example, in simply typed $\lambda$-calculus, a closed term that has the Subformula Property must be a \emph{value}, that is, of the form $\lambda x\, u$, or $\langle u, v \rangle$. Indeed a closed term which is a not a value, must be of the form $h\, \sigma$, for some stack $\sigma$ (see Definition \ref{definition-stack}), where $h$ is a redex $(\lambda y\, u)t$ or $\langle u, v\rangle\, \pi_{i}$; but $(\lambda y\, u)$ and $\langle u, v\rangle$ would have a more complex type than the type of the whole term, contradicting the Subformula Property. \\ \emph{Second}, we require $\mathcal{C}[a\, u], \mathcal{D}[a\, v]$ to be normal simply typed $\lambda$-terms. Simply typed $\lambda$-terms, because they are easier to execute in parallel; normal, because we want their computations to go on until they are really stuck and communication is unavoidable. \emph{Third}, we require the variable $a$ to be as rightmost as possible and that is
needed for logical soundness: how could otherwise the term $u$ be moved to the right, e.g., if it contains $a$?
Assuming that all the conditions above are satisfied, we can now start to explain the cross reduction \begin{small} \[\mathcal{C}[a\, u]\parallel_{a} \mathcal{D}[a\, v] \mapsto ( \mathcal{D}[u^{b\langle \sq{z}\rangle / \sq{y}}] \parallel_{a} \mathcal{C}[a\, u] ) \parallel_{b} (\mathcal{C}[v^{b\langle \sq{y}\rangle / \sq{z}}]\parallel_{a} \mathcal{D}[a\, v])\] \end{small} Here, the communication channel $a$ has been activated, because the processes $\mathcal{C}$ and $\mathcal{D}$ are synchronized and ready to transfer respectively $u$ and $v$. The parallel operator $\parallel_{a}$ let the two occurrences of $a$ communicate: the term $u$ travels to the right in order to replace $a\, v$ and $v$ travels to the left in order to replace $a\, u$. If $u$ and $v$ were data, like numbers or constants, everything would be simple and they could be sent as they are; but in general, this is not possible. The problem is that the free variables $\sq{y}$ of $u$ which are bound in $\mathcal{C}[a\, u]$ by some $\lambda$ cannot be permitted to become free; otherwise, the connection between the binders $\lambda\, \sq{y}$ and the occurrences of the variables $\sq{y}$ would be lost and they could be no more replaced by actual values when the inputs for the $\lambda\, \sq{y}$ are available. Symmetrically, the variables $\sq{z}$ cannot become free. For example, we could have $u=u'\, \sq{y}$ and $v=v'\,\sq{z}$ and\begin{small}
\[\mathcal{C}[a\, u]=w_{1}\, (\lambda
\sq{y}\, a\, (u'\, \sq{y}))\qquad\ \mathcal{D}[a\, v]=w_{2}\,
(\lambda \sq{z}\, a\, (v'\, \sq{z}))\] \end{small}and the transformation \begin{small}
$w_{1}\, (\lambda \sq{y}\, a\, (u'\, \sq{y}))\parallel_{a} w_{2}\,
(\lambda \sq{z}\, a\, (v'\, \sq{z}))$ $\mapsto$ $w_{1}\, (\lambda
\sq{y}\, v'\, \sq{z} )\parallel_{a} w_{2}\, (\lambda \sq{z}\, u'\,
\sq{y})$ \end{small} would just be wrong: the term $v'\, \sq{z}$ will never get back actual values for the variables $\sq{z}$ when they will become available.
\noindent These issues are typical of \emph{process migration}, and can be solved by the concepts of \emph{code mobility}~\cite{codemobil} and \emph{closure}~\cite{landin}. Informally, code mobility is defined as the capability to dynamically change the bindings between code fragments and the locations where they are executed. Indeed, in order to be executed, a piece of code needs a computational environment and its resources, like data, program counters or global variables. In our case the contexts $\mathcal{C}[\ ]$ and $\mathcal{D}[\ ]$ are the computational environments or \emph{closures} of the processes $u$ and $v$ and the variables $\sq{y}, \sq{z}$ are the resources they need. Now, moving a process outside its environment always requires extreme care: the bindings between a process and the environment resources must be preserved. This is the task of the \emph{migration mechanisms}, which allow a migrating process to resume correctly its execution in the new location. Our migration mechanism creates a new communication channel $b$ between the programs that have been exchanged. Here we see the code fragments $u$ and $v$, with their original bindings to the global variables $\sq{y}$ and $\sq{z}$. \begin{center}
\includegraphics[width=0.25\textwidth]{codemobility_before} \end{center} The change of variables $u^{b\langle \sq{z}\rangle / \sq{y}}$ and $ v^{b\langle \sq{y}\rangle / \sq{z}}$ has the effect of reconnecting $u$ and $v$ to their old inputs: \begin{center}
\includegraphics[width=0.25\textwidth]{codemobility_after} \end{center} In this way, when they will become available, the data $\sq{y}$ will be sent to $u$ and the data $\sq{z}$ will be sent to $v$ through the channel $b$. Note that in the result of the cross reduction the processes $\mathcal{C}[a\, u]$ and $\mathcal{D}[a\, v]$ are \emph{cloned}, because their code fragments can be needed again. Thus $a$ behaves as a \emph{replicated input} and \emph{replicated output channel}. E.g., in \cite{CP2010}, replicated input is coded by the bang operator of linear logic:
$$x\langle y\rangle. Q\, |\, !x(z).P \mapsto Q\, |\, P[y/z]\, |\, !x(z).P$$ With symmetrical message passing and a ``!" also in front of $x\langle y\rangle. Q$, one would obtain a version of our cross reduction. Finally, as detailed in Ex.~\ref{ex:pi_calc}, whenever $u$ and $v$ are closed terms the cross reduction is simpler and only maintains the first two of the four processes produced in the general case.
\begin{example}[\textbf{$\parallel_{a}$ in $\bblambda_{\mathrm{G}}$ and $\mid$ in the $\pi$-calculus}] \label{Pi} A private channel $u\parallel_{a} v$ is rendered in the
$\pi$-calculus~\cite{Milner, sangiorgiwalker2003} by the restriction operator $\nu$, as $\nu a\, (u\ |\ v)$. Recall that the $\pi$-calculus term $u \mid v$ represents two processes that run in parallel. The corresponding $\bblambda_{\mathrm{G}}$ term $\langle e, u \rangle \parallel_{e} \langle e, v \rangle$ is defined using a fresh channel $e$ with communication kind $A, A$. As no cross reduction outside $u$ and $v$ can be applied, the whole term reduces neither to $\langle e, u \rangle$ nor to $\langle e, v\rangle$, so that $u$ and $v$ can run in parallel. \end{example}
\begin{example}\label{example-nonterm} Let $y$ and $z$ be bound variables occurring in the normal terms $\mathcal{C}[a\,y]$ and $\mathcal{D}[a\, z]$. Without the condition on the communication complexity $c$ of $a$, a loop could be generated:\begin{small} \[\mathcal{C}[a\, y]\parallel_{a} \mathcal{D}[a\, z] \mapsto (\mathcal{D}[y^{b\langle {z}\rangle / {y}}] \parallel_{a} \mathcal{C}[a\, y] ) \parallel_{b} (\mathcal{C}[z^{b\langle {y}\rangle / {z}}]\parallel_{a} \mathcal{D}[a\, z])\] \[=(\mathcal{D}[b\,{z}] \parallel_{a} \mathcal{C}[a\, y]) \parallel_{b} (\mathcal{C}[b\, {y}]\parallel_{a} \mathcal{D}[a\, z])\mapsto^{*} \mathcal{D}[b\,{z}] \parallel_{b} \mathcal{C}[b\, {y}]\]\end{small}In Sec.~\ref{section-normalization} we show that if $c > 0$, this reduction sequence would terminate. What is then happening here? Intuitively, $\mathcal{C}[a\,y]$ and $\mathcal{D}[a\, z]$ are normal simply typed $\lambda$-terms, which forces $c=0$. \end{example}
\textbf{Permutation Reductions}. They regulate the interaction between parallel operators and the other computational constructs. The first four reductions are the Prawitz-style permutation rules \cite{Prawitz} between parallel operators and eliminations. We also add two other groups of reductions: three permutations between parallel operators and introductions, two permutations between parallel operators themselves. The first group will be needed to rewrite any proof term into a parallel composition of simply typed $\lambda$-terms (Proposition \ref{proposition-parallelform}). The second group is needed to address the \emph{scope extrusion} issue of private channels \cite{Milner}. We point out that a parallel operator $\parallel_{a}$ is allowed to commute with other parallel operators only when it is strictly necessary, that is, when the communication complexity of $a$ is greater than $0$ and thus signaling a violation of the Subformula Property. \begin{example}[\textbf{Scope extrusion (and $\pi$-calculus)}] \label{SE} As example of scope extrusion, let us consider the term \[(v\parallel_{a} \mathcal{C}[b\, a] )\parallel_{b} w \]
Here the process $\mathcal{C}[b\, a]$ wishes to send the channel $a$ to $w$ along the channel $b$, but this is not possible being the channel $a$ private. This issue is solved in the $\pi$-calculus using the congruence $\nu a (P\, |\, Q)\, |\, R \equiv \nu a (P\, |\, Q\, |\, R) $,
provided that $a$ does not occur in $R$, condition that can always be satisfied by $\alpha$-conversion. G\"odel logic offers and actually forces a different solution, which is not just permuting $w$ inward but also duplicating it: \[( v \parallel_{a} \mathcal{C}[b\, a])\parallel_{b} w \mapsto (v \parallel_{b} w)\parallel_{a} (\mathcal{C}[b\, a]\parallel_{b} w) \] After this reduction $\mathcal{C}[b\, a]$ can send $a$ to $w$. If $a$ does not occur in $v$, we have a further simplification step: $$(v \parallel_{b} w)\parallel_{a} (\mathcal{C}[b\, a]\parallel_{b} w)\mapsto v \parallel_{a} (\mathcal{C}[b\, a]\parallel_{b} w)$$ obtaining associativity of composition as in $\pi$-calculus. However, if $b$ occurs in $v$, this last reduction step is not possible and we keep both copies of $w$. It is indeed natural to allow both $v$ and $\mathcal{C}[b\,a]$ to communicate with $w$. \end{example}
Everything works as expected: the reductions steps in Fig.~\ref{fig:red} preserve the type at the level of proof terms, i.e., they correspond to logically sound proof transformations. Indeed
\begin{theorem}[Subject Reduction]\label{subjectred} If $t : A$ and $t \mapsto u$, then $u : A$ and all the free variables of $u$ appear among those of $t$. \end{theorem} \begin{proof} It is enough to prove the theorem for basic reductions: if $ t : A$ and $t \mapsto u$, then $u : A$. The proof that the intuitionistic reductions and the permutation rules preserve the type is completely standard
(full proof in the Appendix). Cross reductions require straightforward considerations as well. Indeed suppose \begin{small}
\begin{gather*}
\mathcal{C}[a^{\scriptscriptstyle A\rightarrow B}\,
u]\parallel_{a} \mathcal{D}[a^{\scriptscriptstyle B\rightarrow
A}\, v]\ \\ \mapsto
\\
( \mathcal{D}[u^{b^{\scriptscriptstyle D\rightarrow C}\langle
\sq{z}\rangle / \sq{y}}]
\parallel_{a} \mathcal{C}[a^{\scriptscriptstyle A\rightarrow B}\,
u] ) \parallel_{b} (\mathcal{C}[v^{b^{\scriptscriptstyle
C\rightarrow D}\langle \sq{y}\rangle / \sq{z}}]\parallel_{a}
\mathcal{D}[a^{\scriptscriptstyle B\rightarrow A}\, v])
\end{gather*} \end{small}Since $
\langle \sq{y}\rangle: C := C_{0}\land \ldots\land C_{n} \; \mbox{and} \;
\langle \sq{z}\rangle: D := D_{0}\land \ldots\land D_{m} $, $b^{\scriptscriptstyle
D\rightarrow C}\langle \sq{z}\rangle$ and $b^{\scriptscriptstyle
C\rightarrow D}\langle \sq{y}\rangle$ are correct
terms. Therefore $u^{b^{\scriptscriptstyle D\rightarrow C}\langle
\sq{z}\rangle / \sq{y}}$ and $v^{b^{\scriptscriptstyle
C\rightarrow D}\langle \sq{y}\rangle / \sq{z}}$, by Definition ~\ref{defi-multsubst},
are correct as well. The assumptions are that $\sq{y}=y_{0}^{C_{0}}, \ldots, y_{n}^{C_{n}}$ is the sequence of the free variables of $u$ which are bound in $\mathcal{C}[a^{\scriptscriptstyle A\rightarrow B}u]$, $\sq{z}=z_{0}^{D_{0}}, \ldots, z_{m}^{D_{m}}$ is the sequence of the free variables of $v$ which are bound in $\mathcal{D}[a^{\scriptscriptstyle B\rightarrow A}v]$, $a$ does not occur neither in $u$ nor in $v$ and $b$ is fresh.\ Therefore, by construction all the variables $\sq{z}$ are bound in $\mathcal{D}[u^{b^{\scriptscriptstyle D\rightarrow C}\langle \sq{z}\rangle / \sq{y}}]$ and all the variables $\sq{y}$ are bound in $\mathcal{C}[v^{b^{\scriptscriptstyle C\rightarrow D}\langle \sq{y}\rangle / \sq{z}}]$. Hence, no new free variable is created.\end{proof}
\begin{figure*}\label{fig:red}
\end{figure*}
\begin{definition}[Normal Forms and Normalizable Terms]\mbox{}
\begin{itemize} \item A \textbf{redex} is a term $u$ such that $u\mapsto v$ for some $v$ and basic reduction of Figure \ref{fig:red}. A term $t$ is called a \textbf{normal form} or, simply, \textbf{normal}, if there is no $t'$ such that $t\mapsto t'$. We define $\mathsf{NF}$ to be the set of normal $\bblambda_{\mathrm{G}}$-terms. \item A sequence, finite or infinite, of proof terms $u_1,u_2,\ldots,u_n,\ldots$ is said to be a reduction of $t$, if $t=u_1$, and for all $i$, $u_i \mapsto u_{i+1}$.
A proof term $u$ of $\bblambda_{\mathrm{G}}$ is \textbf{normalizable} if there is a finite reduction of $u$ whose last term is a normal form. \end{itemize} \end{definition}
\begin{definition}[Parallel Form]\label{definition-head} A term $t$ is a \textbf{parallel form}
whenever, removing the parentheses, it can be written as $$t = t_{1}\parallel_{a_{1}} t_{2}\parallel_{a_{2}}\ldots \parallel_{a_{n}} t_{n+1}$$ where each $t_{i}$, for $1\leq i\leq n+1$, is a simply typed $\lambda$-term.
\end{definition}
\comment{ We conclude the treatment of reduction rules with another example, that also suggests that some other reduction rules, considered in \cite{AschieriZH}, might be useful. The reductions are:
\begin{description} \item[Optional Reduction Rules for $\D$] \[u\parallel_{a}v \mapsto u \mbox{ if $a$ does not occur in $u$}\] \[u\parallel_{a}v \mapsto v \mbox{ if $a$ does not occur in $u$}\] \end{description} Whereas in $\cite{AschieriZH}$ the first reduction is essential, in this setting both the first and the second are useless for all the main results of the paper. However, they can be used to show that the type $A\lor B$ is redundant in the logic $\logic{G}$. This follows of course from the fact that $A\lor B$ is definable by means of $\rightarrow$ and $\land$, as noticed by Dummett \cite{dummett}. We can define: $$(A\lor B)^{*}:= (A\rightarrow B)\rightarrow B \land (B\rightarrow A)\rightarrow A$$ Then, if $u: A$, we define $$(\iota_{0}(u))^{*}:= \langle \lambda y^{{\scriptscriptstyle A\rightarrow B}}\, y\, u, \lambda z^{{\scriptscriptstyle B\rightarrow A}} u\rangle$$ and if $u: B$, we define $$(\iota_{1}(u))^{*}:= \langle \lambda z^{{\scriptscriptstyle A\rightarrow B}} u, \lambda y^{{\scriptscriptstyle B\rightarrow A}}\, y\, u\rangle$$ where $z$ is a dummy variable not occurring in $u$. Finally, we define $$(t\, [x_{0}^{\sml{A}}.u_{0}, x_{1}^{\sml{B}}.u_{1}])^{*}:= (\lambda x_{1}^{\sml{B}}\, u_{1})(t\, \pi_{0}\, a^{{\scriptscriptstyle A\rightarrow B}})\parallel_{a} (\lambda x_{0}^{\sml{A}}\, u_{1})(t\, \pi_{1}\, a^{\sml{B\rightarrow A}})$$ Then $$((\iota_{0}(u))^{*} [x_{0}^{\sml{A}}.u_{0}, x_{1}^{\sml{B}}.u_{1}])^{*}$$ }
\comment{ As usual in $\lambda$-calculus, a value represents the result of the computation: a function for arrow and universal types, a pair for product types, a Boolean for sum types and a witness for existential types and in our case also the abort operator.
\begin{definition}[Values, Neutrality]\label{definition-value}\mbox{} \begin{itemize} \item A proof term is a \textbf{value} if it is of the form $\lambda x\, u$ or $\lambda \alpha\, u$ or $\pair{u}{t}$ or $\inj_{i}(u)$ or $(m, u)$ or $\efq{}{u}$ or $\abort$. \item A proof term is \textbf{neutral} if it is neither a value nor of the form $u\parallel_{a} v$. \end{itemize} \end{definition} }
\section{ The Subformula Property}\label{section-subformula}
We show that normal $\bblambda_{\mathrm{G}}$-terms satisfy the important Subformula Property (Theorem \ref{theorem-subformula}). This, in turn, implies that our Curry--Howard correspondence for $\bblambda_{\mathrm{G}}$ is meaningful from the logical perspective and produces analytic $\mathbf{NG}$ proofs.
We start by establishing an elementary property of simply typed $\lambda$-terms, which will turn out to be crucial for our normalization proof. It ensures that every bound hypothesis appearing in a normal intuitionistic proof is a strong subformula of one the premises or a proper subformula of the conclusion. This property sheds light on the complexity of cross reductions, because it implies that the new formulas introduced by these operations are always smaller than the local premises.
\begin{proposition}[Bound Hypothesis Property]\label{proposition-boundhyp} Suppose $$x_{1}^{A_{1}}, \ldots, x_{n}^{A_{n}}\vdash t: A$$ $t\in\mathsf{NF}$ is a simply typed $\lambda$-term and $z: B$ a variable occurring bound in $t$. Then one of the following holds: \begin{enumerate} \item $B$ is a proper subformula of a prime factor of $A$. \item $B$ is a strong subformula of one among $A_{1},\ldots, A_{n}$.
\end{enumerate} \end{proposition}
\begin{proof} By induction on $t$.
See the Appendix. \end{proof}
The next proposition says that each occurrence of any hypothesis of a normal intuitionistic proof must be followed by an elimination rule, whenever the hypothesis is neither $\bot$ nor a subformula of the conclusion nor a proper subformula of some other premise.
\begin{proposition}\label{proposition-app} Let $t\in \mathsf{NF}$ be a simply typed $\lambda$-term and $$x_{1}^{A_{1}}, \ldots, x_{n}^{A_{n}}, z^{B}\vdash t: A$$ One of the following holds:
\begin{enumerate}
\item Every occurrence of $z^{B}$ in $t$ is of the form $z^{B}\, \xi$ for some proof term or projection $\xi$.
\item $B=\bot$ or $B$ is a subformula of $A$ or a proper subformula of one among the formulas $A_{1}, \ldots, A_{n}$. \end{enumerate} \end{proposition} \begin{proof}
Easy structural induction on the term.
See the Appendix.\end{proof}
\begin{proposition}[Parallel Normal Form Property] \label{proposition-parallelform} If $t\in \mathsf{NF}$ is a $\bblambda_{\mathrm{G}}$-term, then it is in parallel form. \end{proposition} \begin{proof}
Easy structural induction on $t$ using the permutation
reductions.
See the Appendix.\end{proof}
We finally prove the Subformula Property: a normal proof does not contain concepts that do not already appear in the premises or in the conclusion.
\begin{theorem}[Subformula Property]\label{theorem-subformula} Suppose $$x_{1}^{A_{1}}, \ldots, x_{n}^{A_{n}}\vdash t: A \quad \mbox{and} \quad t\in \mathsf{NF}. \quad \mbox{Then}:$$ \begin{enumerate} \item For each communication variable $a$ occurring bound in $t$ and with communication kind $B, C$, the prime factors of $B$ and $C$ are proper subformulas of $A_{1}, \ldots, A_{n}, A$. \item The type of any subterm of $t$ which is not a bound communication variable is either a subformula or a conjunction of subformulas of the formulas $A_{1}, \ldots, A_{n}, A$. \end{enumerate}
\end{theorem} \begin{proof} We proceed by induction on $t$. By Proposition \ref{proposition-parallelform} $t = t_{1}\parallel_{a_{1}} t_{2}\parallel_{a_{2}}\ldots \parallel_{a_{n}} t_{n+1}$ and each $t_{i}$, for $1\leq i\leq n+1$, is a simply typed $\lambda$-term. We only show the case $t= u_{1}\parallel_{b} u_{2}$. Let $C, D$ be the communication kind of $b$, we first show that the communication complexity of $b$ is $0$.
We reason by contradiction and assume that it is greater than $0$.
$u_{1}$ and $u_{2}$ are either simply typed $\lambda$-terms or of the form $v\parallel_{c} w$. The second case is not possible, otherwise a permutation reduction could be applied to $t\in \mathsf{NF}$. Thus $u_{1}$ and $u_{2}$ are simply typed $\lambda$-terms. Since the communication complexity of $b$ is greater than $0$, the types $C\rightarrow D$ and $D\rightarrow C$ are not subformulas of $A_{1}, \ldots, A_{n}, A$. By Prop.~\ref{proposition-app}, every occurrence of $b^{C\rightarrow D}$ in $u_{1}$ is of the form $b^{C\rightarrow D} v$ and every occurrence of $b^{D\rightarrow C}$ in $u_{2}$ is of the form $b^{D\rightarrow C} w$. Hence, we can write $$u_{1}=\mathcal{C}[b^{C\rightarrow D} v] \qquad u_{2}=\mathcal{D}[b^{D\rightarrow C} w]$$ where $\mathcal{C}, \mathcal{D}$ are simple contexts and $b$ is rightmost. Hence a cross reduction of $t$ can be performed, which contradicts the fact that $t\in\mathsf{NF}$. Since we have established that the communication complexity of $b$ is $0$, the prime factors of $C$ and $D$ must be proper subformulas of $A_{1}, \ldots, A_{n}, A$. Now, by induction hypothesis applied to $u_{1}: A$ and $u_{2}: A$, for each communication variable $a^{F\rightarrow G}$ occurring bound in $t$, the prime factors of $F$ and $G$ are proper subformulas of the formulas $ A_{1}, \ldots, A_{n}, A, C\rightarrow D, D\rightarrow C$ and thus of the formulas $A_{1}, \ldots, A_{n}, A$; moreover, the type of any subterm of $u_{1}$ or $u_{2}$ which is not a communication variable is either a subformula or a conjunction of subformulas of the formulas $ A_{1}, \ldots, A_{n}, C\rightarrow D, D\rightarrow C$ and thus of $A_{1}, \ldots, A_{n}, A$.\end{proof}
\begin{remark} Our statement of the Subformula Property is slightly different from the usual one. However the latter can be easily recovered using the communication rule $(com_{end})$ of Section~\ref{section-ND} and additional reduction rules. As the resulting derivations would be isomorphic but more complicated, we prefer the current statement.
\end{remark}
\section{The Normalization Theorem}\label{section-normalization}
Our goal is to prove the Normalization Theorem for $\bblambda_{\mathrm{G}}$: every proof term of $\bblambda_{\mathrm{G}}$ reduces in a finite number of steps to a normal form. By Subject Reduction, this implies that $\mathbf{NG}$ proofs normalize. We shall define a reduction strategy for terms of $\bblambda_{\mathrm{G}}$: a recipe for selecting, in any given term, the subterm to which apply one of our basic reductions. We remark that the permutations between communications have been adopted to simplify the normalization proof, but at the same time, they undermine strong normalization, because they enable silly loops, like in cut-elimination for sequent calculi. Further restrictions of the permutations might be enough to prove strong normalization, but we leave this as an open problem.
The idea behind our normalization strategy is to employ a suitable complexity measure for terms $u\parallel_{a} v$ and, each time a reduction has to be performed, to choose the term of maximal complexity. Since cross reductions can be applied as long as there is a violation of the Subformula Property, the natural approach is to define the complexity measure as a function of some fixed set of formulas, representing the formulas that can be safely used without violating the Subformula Property.
\begin{definition}[Complexity of Parallel
Terms]\label{defi-acomplexity} Let $\mathcal{A}$ be a finite set of formulas. The $\mathcal{A}$-\textbf{complexity} of the term $u\parallel_{a} v$ is the sequence $(c, d, l, o)$ of natural numbers, where: \begin{enumerate} \item
if the communication kind of $a$ is $B, C$, then $c$ is the maximum among $0$ and the number of symbols of the prime factors of $B$ or $C$ that are not subformulas of some formula in $\mathcal{A}$; \item $d$ is the number of occurrences of $\parallel$ in $u$ and $v$; \item $l$ is the sum of the lengths of the intuitionistic reductions of $u$ and $v$ to reach intuitionistic normal form; \item $o$ is the number of occurrences of $a$ in $u$ and $v$. \end{enumerate} The $\mathcal{A}$-\textbf{communication-complexity} of $u\parallel_{a} v$ is $c$.
\end{definition} For clarity, we define the recursive normalization algorithm that represents the constructive content of the proofs of Prop.~\ref{proposition-normpar} and \ref{proposition-normcomp}, which are used to prove the Normalization Theorem. Essentially, our master reduction strategy consists in iterating the basic reduction relation $\succ$ defined below, whose goal is to permute the smallest redex $u\parallel_{a} v$ of maximal complexity until $u$ and $v$ are simply typed $\lambda$-terms, then normalize them and finally apply the cross reductions.
\begin{definition}[Side Reduction Strategy]\label{defi-redstrategy} Let $t: A$ be a term with free variables $x_{1}^{A_{1}},\ldots, x_{n}^{A_{n}}$ and $\mathcal{A}$ be the set of the proper subformulas of $A$ and the strong subformulas of the formulas $A_{1}, \ldots, A_{n}$. Let $u\parallel_{a} v$ be the \emph{smallest subterm} of $t$, if any, among those of \emph{maximal} $\mathcal{A}$-complexity and let $(c, d, l, o)$ be its $\mathcal{A}$-complexity. We write\[t\succ t'\]whenever $t'$ has been obtained from $t$ by applying to $u\parallel_{a} v$: \begin{enumerate} \item a permutation reduction $$(u_{1}\parallel_{b} u_{2})\parallel_{a} v \mapsto (u_{1}\parallel_{a} v)\parallel_{b} (u_{2}\parallel_{a} v)$$ $$u \parallel_{a} (v_{1}\parallel_{b} v_{2}) \mapsto (u\parallel_{a} v_{1})\parallel_{b} (u\parallel_{a} v_{2})$$
if $d>0$ and $u=u_{1}\parallel_{b}u_{2}$ or $v=v_{1}\parallel_{b}v_{2}$; \item a sequence of intuitionistic reductions normalizing both $u$ and $v$, if $d=0$ and $l>0$; \item a cross reduction if $d=l=0$ and $c>0$, immediately followed by intuitionistic reductions normalizing the newly generated simply typed $\lambda$-terms and, if possible, by applications of the cross reductions $u_{1} \parallel_{b} v_{1} \mapsto u_{1}$ and $u_{1} \parallel_{b} v_{1} \mapsto v_{1}$ to the whole term. \item a cross reduction $u \parallel_{a} v \mapsto u$ and $u \parallel_{a} v \mapsto v$ if $d=l=c=0$. \end{enumerate} \end{definition}
\begin{definition}[Master Reduction Strategy]\label{defi-mastredstrategy} We define a normalization algorithm $\nor{N}(t)$ taking as input a typed term $t$ and producing a term $t'$ such that $t\mapsto^{*} t'$. Assume that the free variables of $t$ are $x_{1}^{A_{1}},\ldots, x_{n}^{A_{n}}$ and let $\mathcal{A}$ be the set of the proper subformulas of $A$ and the strong subformulas of the formulas $A_{1}, \ldots, A_{n}$. The algorithm performs the following operations. \begin{enumerate} \item If $t$ is not in parallel form, then, using permutation reductions, $t$ is reduced to a $t'$ which is in parallel form and $\nor{N}(t')$ is recursively executed.
\item \label{dot:simply} If $t$ is a simply typed $\lambda$-term, it is normalized and returned. If $t=u\parallel_{a} v$ is not a redex, then let $\nor{N}(u)=u'$ and $\nor{N}(v)=v'$. If $u'\parallel_{a} v'$ is normal, it is returned. Otherwise, $\nor{N}(u'\parallel_{a} v')$ is recursively executed.
\item If $t$ is a redex, we select the \emph{smallest} subterm $w$ of $t$ having maximal $\mathcal{A}$-communication-complexity $r$. A sequence of terms is produced \[w\succ w_{1}\succ w_{2}\succ \ldots \succ w_{n}\] such that $w_{n}$ has $\mathcal{A}$-communication-complexity strictly smaller than $r$. We substitute $w_n$ for $w$ in $t$ obtaining $t'$ and recursively execute $\nor{N}(t')$.
\end{enumerate} \noindent We observe that in the step~\ref{dot:simply} of the algorithm $\nor{N}$, by construction $u\parallel_{a} v$ is not a redex. After $u$ and $v$ are normalized respectively to $u'$ and $v'$, it can still be the case that $u'\parallel_{a} v'$ is not normal, because some free variables of $u$ and $v$ may disappear during the normalization, causing a new violation of the Subformula Property that transforms $u'\parallel_{a} v'$ into a redex, even though $u\parallel_{a} v$ was not.
\end{definition}
\comment{ \begin{lemma} Let $t: A$ be a term with free variables among $$x_{1}^{A_{1}},\ldots, x_{n}^{A_{n}}$$ and $\mathcal{A}=\{A_{1}, \ldots, A_{n}, A\}$. Then $\succ$ is normalizing, that is, there are no infinite reduction chains $$t\succ t_{0}\succ t_{1}\succ \ldots \succ t_{n}\succ\ldots$$ \end{lemma} }
The first step of the normalization proof consists in showing that any term can be reduced to a parallel form.
\begin{proposition}\label{proposition-normpar} Let $t: A$ be any term. Then $t\mapsto^{*} t'$, where $t'$ is a parallel form. \end{proposition} \begin{proof} Easy structural induction on $t$.
See the Appendix.\end{proof}
We now prove that any term in parallel form can be normalized with the help of the algorithm $\nor{N}$. \begin{lemma}\label{lem:uebermensch} Let $t: A$ be a term in parallel form which is not a simply typed $\lambda$-term and $\mathcal{A}$ a set of formulas containing all proper subformulas of $A$ and closed under subformulas. Assume that $r > 0$ is the maximum among the $\mathcal{A}$-communication-complexities of the subterms of $t$. Assume that the free variables $x_{1}^{A_{1}},\ldots, x_{n}^{A_{n}}$ of $t$ are such that for every ${i}$, either each strong subformula of $A_{i}$ is in $ \mathcal{A}$, or each proper prime subformula of $A_{i}$ which has more than $r$ symbols is in $ \mathcal{A}$.
Suppose moreover that no subterm $u\parallel_{a} v$ with $\mathcal{A}$-communication-complexity $r$ contains a subterm of the same $\mathcal{A}$-communication-complexity. Then there exists $t'$ such that $t \succ^* t' $ and the maximum among the $\mathcal{A}$-communication-complexities of the subterms of $t'$ is strictly smaller than $r$. \end{lemma} \begin{proof} We prove the lemma by lexicographic induction on the pair \[(\rho , k)\] where $k$ is the number of subterms of $t$ with maximal $\mathcal{A}$-complexity $\rho$ among those with $\mathcal{A}$-communication-complexity $r$.
Let $u\parallel_a v$ be the \emph{smallest} subterm of $t$ having $\mathcal{A}$-complexity $\rho$. Four cases can occur.
(a) $\rho =(r, d, l, o)$, with $d>0$. We first show that the term $u\parallel_a v$ is a redex. Assume that the free variables of $u\parallel_a v$ are among $x_{1}^{A_{1}},\ldots, x_{n}^{A_{n}}, a_{1}^{B_{1}\rightarrow C_{1}},\ldots, a_{m}^{B_{m}\rightarrow C_{m}}$ and that the communication kind of $a$ is $C, D$.
Suppose by contradiction that all the prime factors of $C$ and $D$ are proper subformulas of $A$ or strong subformulas of one among $A_1, \ldots, A_n, B_1\rightarrow C_1, \ldots, B_m\rightarrow C_m$. Given that $r>0$ there is a prime factor $P$ of $C$ or $D$ such that $P$ has $r$ symbols and does not belong to $ \mathcal{A}$. The possible cases are two: (i) $P$ is a proper subformula of a prime proper subformula $A'_i $ of $A_i$ such that $ A'_i \notin \mathcal{A}$; (ii) $P$, by Proposition~\ref{proposition-strongsubf}, is a proper subformula of a prime factor of $B_i$ or $C_i$. Suppose (i). Since $A'_{i}\notin \mathcal{A}$, by hypothesis the number of symbols of $A_i'$ is less than or equal to $r$, so $P$ cannot be a proper subfomula of $A_i'$, which is a contradiction. Suppose (ii). Then there is a prime factor of $B_i$ or $C_i$ having a number of symbols greater than $r$. Since by hypothesis $a_i^{B_i\rightarrow C_i}$ is bound in $t$, we conclude that there is a subterm $w_1\parallel_{a_i} w_2$ of $t$ having $\mathcal{A}$-complexity greater than $\rho$, which is absurd.
Now, since $d>0$, we may assume $u=w_1\parallel_b w_2$ (the case $v=w_1\parallel_b w_2$ is symmetric). The term \[(w_1\parallel_b w_2)\parallel_a v\] is then a redex of $t$ and by replacing it with \begin{equation}\label{eq:1} (w_1\parallel_a v)\parallel_b (w_2\parallel_a v) \end{equation} we obtain from $t$ a term $t'$ such that $t\succ t'$ according to Def.~\ref{defi-redstrategy}. We must verify that we can apply to $t'$ the main induction hypothesis. Indeed, the reduction $t\succ t'$ duplicates all the subterms of $v$, but all of their $\mathcal{A}$-complexities are smaller than $r$, because $u\parallel_a v$ by choice is the smallest subterm of $t$ having maximal $\mathcal{A}$-complexity $\rho$. The two terms $w_1\parallel_a v$ and $w_2\parallel_a v$ have smaller $\mathcal{A}$-complexity than $\rho$, because they have numbers of occurrences of the symbol $\parallel$ strictly smaller than in $u\parallel_{a} v$. Moreover, the terms in $t'$ with~\eqref{eq:1} as a subterm have, by hypothesis, $\mathcal{A}$-communication-complexity smaller than $r$ and hence $\mathcal{A}$-complexity smaller than $\rho$. Assuming that the communication kind of $b$ is $F, G$, the prime factors of $F$ and $G$ that are not in $\mathcal{A}$ must have fewer symbols than the prime factors of $C$ and $D$ that are not in $\mathcal{A}$, again because $u\parallel_a v$ by choice is the smallest subterm of $t$ having maximal $\mathcal{A}$-complexity $\rho$; hence, the $\mathcal{A}$-complexity of~\eqref{eq:1} is smaller than $\rho$. Therefore the number of subterms of $t'$ with $\mathcal{A}$-complexity $\rho$ is strictly smaller than $k$. By induction hypothesis, $t'\succ^{*} t''$, where $t''$ satisfies the thesis.\\
(b) $\rho=(r, d, l, o)$, with $d=0$ and $l>0$. Since $d=0$, $u$ and $v$ are simply typed $\lambda$-terms -- and thus strongly normalizable \cite{Girard} -- so we may assume $u\mapsto^{*} u'\in \mathsf{NF}$ and $v\mapsto^{*} v'\in \mathsf{NF}$ by a sequence of intuitionistic reduction rules. By replacing in $t$ the subterm $u\parallel_a v$ with $u'\parallel_a v'$, we obtain a term $t'$ such that $t\succ t'$ according to Definition \ref{defi-redstrategy}. Moreover, the terms in $t'$ with $u'\parallel_a v'$ as a subterm have, by hypothesis, $\mathcal{A}$-communication-complexity smaller than $r$ and hence $\mathcal{A}$-complexity is smaller than $\rho$. By induction hypothesis, $t'\succ^{*} t''$, where $t''$ satisfies the thesis.\\
(c) $\rho=(r, d, l, o)$, with $d=l=0$. Since $d=0$, $u$ and $v$ are simply typed $\lambda$-terms. Since $l=0$, $u$ and $v$ are in normal form and thus satisfy conditions 1. and 2. of Proposition~\ref{proposition-boundhyp}. We need to check that $u\parallel_a v$ is a redex, in particular that the communication complexity of $a$ is greater than $0$. Assume that the free variables of $u\parallel_a v$ are among $x_{1}^{A_{1}},\ldots, x_{n}^{A_{n}}, a_{1}^{B_{1}\rightarrow C_{1}},\ldots, a_{m}^{B_{m}\rightarrow C_{m}}$ and that the communication kind of $a$ is $C, D$. As we argued above, we obtain that not all the prime factors of $C$ and $D$ are proper subformulas of $A$ or strong subformulas of one among $A_1, \ldots, A_n, B_1\rightarrow C_1, \ldots, B_m\rightarrow C_m$. By Definition \ref{definition-comcomplexity}, $u\parallel_a v$ is a redex.
We now prove that every occurrence of $a$ in $u$ and $v$ is of the form $a\, \xi$ for some term or projection $\xi$. First of all, $a$ occurs with arrow type both in $u$ and $v$. Moreover, $u: A$ and $v: A$, since $t: A$ and $t$ is a parallel form; hence, the types $C\rightarrow D$ and $D\rightarrow C$ cannot be subformulas of $A$, otherwise $r=0$, and cannot be proper subformulas of one among $A_{1}, \ldots, A_{n}, B_{1}\rightarrow C_{1}, \ldots, B_{n}\rightarrow C_{n}$, otherwise the prime factors of $C, D$ would be strong subformulas of one among $A_1, \ldots, A_n, B_1\rightarrow C_1, \ldots, B_m\rightarrow C_m$. Thus by Prop.~\ref{proposition-app} we are done. Two cases can occur.
\begin{itemize} \item $a$ does not occur in $u$ or $v$: to fix ideas, let us say it does not occur in $u$. By performing a cross reduction, we replace in $t$ the term $u\parallel_{a} v$ with $u$ and obtain a term $t'$ such that $t\succ t'$ according to Def.~\ref{defi-redstrategy}. After the replacement, the number of subterms having maximal $\mathcal{A}$-complexity $\rho$ in $t'$ is strictly smaller than the number of such subterms in $t$. By induction hypothesis, $t'\succ^{*} t''$, where $t''$ satisfies the thesis.
\item $a$ occurs in $u$ and in $v$. Let $u=\mathcal{C}[a\, w_1\, \sigma]$ and $v=\mathcal{D}[a\, w_2\,\tau]$ where the displayed occurrences of $a$ are the rightmost in $u$ and $v$ and $\sigma , \tau$ are the stacks of \emph{all} terms or projections $a$ is applied to. By applying a cross reduction to $\mathcal{C}[a\, w_{1}\,\sigma]\parallel_{a} \mathcal{D}[a\, w_{2}\,\tau]$ we obtain the term $(\ast)$ $$ \small ( \mathcal{D}[w_1^{b\langle \sq{z}\rangle / \sq{y}}\,\tau] \parallel_{a} \mathcal{C}[a\, w_1])\, \parallel_{b}\, (\mathcal{C}[w_2^{b\langle \sq{y}\rangle / \sq{z}}\,\sigma]\parallel_{a} \mathcal{D}[a\, w_2])$$ By hypothesis, $\sq{y}$ is the sequence of the free variables of $w_1$ which are bound in $\mathcal{C}[a\, w_1\,\sigma]$ and $\sq{z}$ is the sequence of the free variables of $w_2$ which are bound in $\mathcal{D}[a\, w_2\,\tau]$ and $a$ does not occur neither in $w_1$ nor in $w_2$. Since $u, v$ satisfy conditions 1. and 2. of Proposition~\ref{proposition-boundhyp} the types $Y_{1}, \ldots, Y_{i}$ and $Z_{1}, \ldots, Z_{j}$ of respectively the variables $\sq{y}$ and $\sq{z}$ are proper subformulas of $A$ or strong subformulas of the formulas $A_{1}, \ldots, A_{n}, B_{1}\rightarrow C_{1}, \ldots, B_{m}\rightarrow C_{m}$. Hence, the types among $Y_{1}, \ldots, Y_{i}, Z_{1}, \ldots, Z_{j}$ which are not in $\mathcal{A}$ are strictly smaller than all the prime factors of the formulas $B_1, C_1, \ldots, B_m, C_m$. Since the communication kind of $b$ is $Y_{1}\land\ldots \land Y_{i}, Z_{1}\land \ldots\land Z_{j}$, by Definition \ref{defi-acomplexity} either the $\mathcal{A}$-complexity of the term $(\ast)$ above is strictly smaller than the $\mathcal{A}$-complexity $\rho$ of $u\parallel_{a} v$, or the communication kind of $b$ is $\top$. In the latter case we apply a cross reduction $u_1 \parallel_{b} v_1 \mapsto u_1$ or $u_1 \parallel_{b} v_1 \mapsto v_1$ and obtain a term with $\mathcal{A}$-complexity strictly smaller than $\rho$.
In the former case, let $w_{1}', w_{2}'$ be simply typed $\lambda$-terms such that $$w_1^{b\langle \sq{z}\rangle / \sq{y}}\,\tau\mapsto^{*} w_{1}'\in\mathsf{NF} \; \mbox{and} \; w_2^{b\langle \sq{y}\rangle / \sq{z}}\,\sigma\mapsto^{*} w_{2}'\in \mathsf{NF}$$ By hypothesis, $a$ does not occur in $w_{1}, w_{2}, \sigma, \tau$ and thus neither in $w_{1}'$ nor in $w_{2}'$. Moreover, by the assumptions on $\sigma$ and $\tau$ and since $\mathcal{C}[a\, w_1\,\sigma]$ and $\mathcal{D}[a\, w_2\,\tau]$ are normal simply typed $\lambda$-terms, $\mathcal{C}[w_{2}']$ and $\mathcal{D}[ w_1']$ are normal too and contain respectively one fewer occurrence of $a$ than the former terms. Hence, the $\mathcal{A}$-complexity of the terms $$ \mathcal{D}[w_1'] \parallel_{a} \mathcal{C}[a\, w_1] \quad \mbox{and} \quad \mathcal{C}[w_2']\parallel_{a} \mathcal{D}[a\, w_2]$$ is strictly smaller than the $\mathcal{A}$-complexity $\rho$ of $u\parallel_{a} v$. Let now $t'$ be the term obtained from $t$ by replacing the term $\mathcal{C}[a\, w_{1}\,\sigma]\parallel_{a} \mathcal{D}[a\, w_{2}\,\tau]$ with \begin{equation}\label{eq:very_big}
(\mathcal{D}[w_1'] \parallel_{a} \mathcal{C}[a\, w_1])\, \parallel_{b}\,(\mathcal{C}[w_2']\parallel_{a} \mathcal{D}[a\, w_2]) \end{equation} By construction $t\succ t'$. Moreover, the terms in $t'$ with~\eqref{eq:very_big} as a subterm have, by hypothesis, $\mathcal{A}$-communication-complexity smaller than $r$ and hence $\mathcal{A}$-complexity is smaller than $\rho$. Hence, we can apply the main induction hypothesis to $t'$ and obtain by induction hypothesis, $t'\succ^{*} t''$, where $t''$ satisfies the thesis. \end{itemize}
(d) $\rho =(r, d, l, o)$, with $d=l=o=0$. Since $o=0$, $u\parallel_a v$ is a redex. To fix ideas, let us say $a$ does not occur in $u$. By performing a cross reduction, we replace $u\parallel_a v$ with $u$ so that $ u\parallel_a v \succ u$ according to Def.~\ref{defi-redstrategy}. Hence, by induction hypothesis, $t'\succ^{*} t''$, where $t''$ satisfies the thesis. \end{proof}
\begin{proposition}\label{proposition-normcomp} Let $t: A$ be any term in parallel form. Then $t\mapsto^{*} t'$, where $t'$ is a parallel normal form. \end{proposition} \begin{proof} Assume that the free variables of $t$ are $x_{1}^{A_{1}},\ldots, x_{n}^{A_{n}}$ and let $\mathcal{A}$ be the set of the proper subformulas of $A$ and the strong subformulas of the formulas $A_{1}, \ldots, A_{n}$. We prove the theorem by lexicographic induction on the quadruple
\[(|\mathcal{A}|, r, k, s)\] where $|\mathcal{A}|$ is the cardinality of $\mathcal{A}$, $r$ is the maximal $\mathcal{A}$-communication-complexity of the subterms of $t$, $k$ is the number of subterms of $t$ having maximal $\mathcal{A}$-communication-complexity $r$ and $s$ is the size of $t$. If $t$ is a simply typed $\lambda$-term, it has a normal form \cite{Girard} and we are done; so we assume $t$ is not. There are two main cases. \\
\noindent \textit{First case}: $t$ \emph{is not a redex}. Let $t=u\parallel_{a} v$ and let $B, C$ be the communication kind of $a$. Then, the communication complexity of $a$ is $0$ and by Def.~\ref{definition-comcomplexity} every prime factor of $B$ or $C$ belongs to $\mathcal{A}$. Let $\mathcal{A}'$ be the set of the proper subformulas of $A$ and the strong subformulas of the formulas $A_{1}, \ldots, A_{n}, B\rightarrow C$; let $\mathcal{A}''$ be the set of the proper subformulas of $A$ and the strong subformulas of the formulas $A_{1}, \ldots, A_{n}, C\rightarrow B$. By Prop.~\ref{proposition-strongsubf}, every strong subformula of $B\rightarrow C$ or $C\rightarrow B$ is a proper subformula of a prime factor of $B$ or $C$, and this prime factor is in $\mathcal{A}$. Hence, $\mathcal{A}'\subseteq \mathcal{A}$ and $\mathcal{A}''\subseteq \mathcal{A}$.
If $\mathcal{A}'= \mathcal{A}$, then the maximal $\mathcal{A}'$-communication-complexity of the terms of $u$ is less than or equal to $r$ and the number of terms having maximal $\mathcal{A}'$-communication-complexity is less than or equal to $k$; since the size of $u$ is strictly smaller than that of $t$, by induction hypothesis $u\mapsto^{*} u'$, where $u'$ is a normal parallel form.
If $\mathcal{A}'\subset\mathcal{A}$, again by induction hypothesis $u\mapsto^{*} u'$, where $u'$ is a normal parallel form.
The very same argument on $\mathcal{A}''$ shows that $v\mapsto^{*} v'$, where $v'$ is a normal parallel form.
Let now $t'=u'\parallel_{a} v'$, so that $t\mapsto^{*} t'$. If $t'$ is normal, we are done. If $t'$ is not normal, since $u'$ and $v'$ are normal, the only possible redex remaining in $t'$ is the whole term itself, i.e., $u'\parallel_{a} v'$: that happens only if the free variables of $t'$ are fewer than those of $t$; w.l.o.g., assume they are $x_{1}^{A_{1}}, \ldots, x_{i}^{A_{i}}$, with $i< n$. Let $\mathcal{B}$ be the set of the proper subformulas of $A$ and the strong subformulas of the formulas $A_{1}, \ldots, A_{i}$. Since $t'$ is a redex, the communication complexity of $a$ is greater than $0$; by Definition \ref{definition-comcomplexity}, a prime factor of $B$ or $C$ is not in $\mathcal{B}$, so we have $\mathcal{B}\subset \mathcal{A}$. By induction hypothesis, $t'\mapsto^{*}t''$, where $t''$ is a parallel normal form.
\noindent \textit{Second case}: $t$ \emph{is a redex}. Let $u\parallel_a v$ be the \emph{smallest} subterm of $t$ having $\mathcal{A}$-communication-complexity $r$. The free variables of $u\parallel_a v$ satisfy the hypotheses of Lemma~\ref{lem:uebermensch} either because have type $A_i$ and $\mathcal{A}$ contains all the strong subformulas of $A_i$, or because the prime proper subformulas of their type have at most $r$ symbols, by maximality of $r$. By Lemma~\ref{lem:uebermensch} $ u\parallel_a v \succ^* w $ where the maximal among the $\mathcal{A}$-communication-complexity of the subterms of $w$ is strictly smaller than $r$. Let $t'$ be the term obtained replacing $w$ for $u\parallel_a v$ in $t$. We can now apply the induction hypothesis and obtain $t' \mapsto^*t''$ with $t''$ in parallel normal form.
\end{proof} The normalization for $\bblambda_{\mathrm{G}}$, and thus for $\mathbf{NG}$, easily follows.
\begin{theorem}\label{theorem-normalization} Suppose that $ t: A$ is a proof term of $\logic{G}$. Then $t\mapsto^{*} t': A$, where $t'$ is a normal parallel form. \end{theorem}
\section{Computing with $\bblambda_{\mathrm{G}}$} We illustrate the expressive power of $\bblambda_{\mathrm{G}}$ by a few examples. All the examples employ the normalization algorithm in Definition \ref{defi-mastredstrategy}; to limit its non-determinism, when we have to reduce $u\parallel_{a}v$ because $a$ does not occur neither in $u$ nor in $v$, we always use the reduction $u\parallel_{a}v \mapsto u$.
Henceforth we use the types $\mathbb{N}$ for natural numbers, $\mathsf{Bool}$ for the Boolean values and $\mathsf{String}$ for strings.
We start by showing that $\bblambda_{\mathrm{G}}$ is more expressive than simply typed $\lambda$-calculus. \begin{example}[\textbf{Parallel or}]\label{ex:parallel_or} Berry's sequentiality theorem (see \cite{Girard}) implies that there is no $\lambda$-term $\mathsf{O}: \mathsf{Bool} \rightarrow \mathsf{Bool} \rightarrow\mathsf{Bool} $ such that $\mathsf{O}\mathsf{F}\mathsf{F} \mapsto \mathsf{F}$, $\mathsf{O}u\mathsf{T} \mapsto \mathsf{T}$, $\mathsf{O}\mathsf{T}u \mapsto \mathsf{T}$, where $u$ is an arbitrary normal term, and thus possibly a variable.
$\mathsf{O}$ can instead be defined in Boudol's parallel $\lambda$-calculus~\cite{Boudol89}.
The $\bblambda_{\mathrm{G}}$ term for such parallel or is (as usual the term ``$\mathsf{if} \, u \, \mathsf{then} \, s \, \mathsf{else} \, t$'' reduces to $s$ if $u = \mathsf{T}$, and to $t$ if $u = \mathsf{F}$):
\begin{footnotesize}
\[ \mathsf{O} := \lambda x^{\mathsf{Bool}} \, \lambda
y^{\mathsf{Bool}} \, (\mathsf{if} \, x \, \mathsf{then} \,
(\lambda z \, \lambda k \, z ) \, \mathsf{else} \, (\lambda z \,
\lambda k \, k ) ) \mathsf{T} (ax) \]
\[{ } \hspace{58pt} \parallel_{a} (\mathsf{if} \, y \, \mathsf{then} \, (\lambda
z \, \lambda k \, z ) \, \mathsf{else} \, (\lambda z \, \lambda k
\, k ) ) \mathsf{T} (ay) \] \end{footnotesize}where the communication kind of $a$ is $ \mathsf{Bool} , \mathsf{Bool}$. Now \begin{small}\begin{align*}
\mathsf{O}\, u \,\mathsf{T} \mapsto^{*} & (\mathsf{if} \, u \, \mathsf{then} \, (\lambda z \, \lambda k \,
z ) \, \mathsf{else} \, (\lambda z \, \lambda k \, k )
)\mathsf{T}(au) \\ & \parallel_{a} (\mathsf{if} \, \mathsf{T} \, \mathsf{then} \,
(\lambda z \, \lambda k \, z ) \, \mathsf{else} \, (\lambda z \,
\lambda k \, k ) )\mathsf{T}(a \mathsf{T})
\\ \mapsto^{*} & (\mathsf{if} \, u \, \mathsf{then} \, (\lambda z
\, \lambda k \, z ) \, \mathsf{else} \, (\lambda z \, \lambda k \,
k ) )\mathsf{T}(au ) \parallel_{a} \mathsf{T} \, \mapsto \,
\mathsf{T}
\end{align*} \end{small}And symmetrically $\mathsf{O}\, \mathsf{T}\, u \, \mapsto^{*} \, \mathsf{T} $. On the other hand\begin{small}
\begin{align*} \mathsf{O} \, \mathsf{F} \,\mathsf{F} \mapsto^{*} & \quad
(\lambda z \, \lambda k \, k ) \mathsf{T}(a \mathsf{F})
\parallel_{a} (\lambda z \, \lambda k \, k ) \mathsf{T}( a
\mathsf{F}) \\ \mapsto^{*} & \quad ( a \mathsf{F})
\parallel_{a} ( a \mathsf{F} ) \\ \mapsto^{*} & \quad
(\mathsf{F}
\parallel_{a} ( a \mathsf{F})
)\parallel _{b} (\mathsf{F} \parallel_{a} ( a \mathsf{F}) ) \quad \mapsto^{*} \quad \mathsf{F}
\end{align*}\end{small}
\end{example}
\begin{example}[\textbf{Data passing}]\label{ex:pi_calc} As in the previous example, if the messages sent during a cross reduction are closed terms, for example data, the outcome is a simple unidirectional message passing. Indeed, the newly introduced communication is void and is always removed: $$\mathcal{C}[a\, u]\parallel_{a} \mathcal{D}[a \, v] \\ \mapsto$$ $$(\mathcal{D}[u] \parallel_{a} \mathcal{C}[a\, u]) \parallel_{b} (\mathcal{C}[v]\parallel_{a} \mathcal{D}[a\, v]) \, \mapsto \, \mathcal{D}[u] \parallel_{a} \mathcal{C}[a\, u]$$ If we want a process $s$ to transmit a message $m:B$ to a process $t$ without $ t$ passing anything back, we can use the following term ($a$ has communication kind $(B\rightarrow F)\rightarrow F, F\rightarrow F$): \begin{small}
\[(a \lambda z^{A\rightarrow F }
\, zm)s \parallel_{a} (a \lambda y^{F} \, y)(\lambda x^{B} \, t)
\; \mapsto \]\[
( (\lambda z \, zm)(\lambda x\, t)
\parallel_{a} (a \lambda z \, zm)s )
\parallel_{e}
((\lambda y\, y)s \parallel_{a} (a \lambda
y \, y)(\lambda x\, t))\]
\[ \mapsto^{*} \quad (\lambda z \, zm)(\lambda x\,
t)\parallel_{e} (\lambda y\, y)s \quad \mapsto^{*} \quad t[m/x]\parallel_{e} s\] \end{small} \noindent This reduction resembles indeed the unidirectional communication $\overline{a}\langle m\rangle.P \mid a(x).Q \mapsto P \mid Q[m/x]$ in the $\pi$-calculus \cite{Milner, sangiorgiwalker2003}, assuming $a$ does not occur in $P$ and $Q$. \end{example}
In the following example, similar to that in \cite{CP2010}, we simulates the communication needed to conclude an online sale. \begin{example}[\textbf{Buyer and vendor}]\label{ex:sale} We model the following transaction: a buyer tells a vendor a product name $\mathsf{prod}:\mathsf{String}$, the vendor computes the value $\mathsf{price}:\mathbb{N}$ of $\mathsf{prod}$ and sends it to the buyer, the buyer sends back the credit card number $\mathsf{card}: \mathsf{String}$ which is used to pay.
We introduce the following functions: $\mathsf{cost} : \mathsf{String}\rightarrow \mathbb{N} $ with input a product name $\mathsf{prod}$ and output its cost $\mathsf{price}$; $\mathsf{pay\_for}: \mathbb{N}\rightarrow \mathsf{String} $ with input a $\mathsf{price}$ and output a credit card number $\mathsf{card}$; $\mathsf{use}: \mathsf{String}\rightarrow \mathbb{N}$ that obtains money using as input a credit card number $\mathsf{card}: {\mathsf{String}} $. The buyer and the vendor are the contexts $\mathcal{B}$ and $\mathcal{V}$ of type $\mathsf{Bool}$. Notice that the terms representing buyer and vendor exchange their position at each cross reduction. For $a$ of kind $\mathsf{String} , \mathbb{N}$, the program is: \begin{small}\begin{align*} & \mathcal{B}[a( \mathsf{pay\_for}(a(\mathsf{prod})))] \parallel_{a} \mathcal{V}[\mathsf{use}( a(
\mathsf{cost}(a\, 0)))] \\ &\mapsto^{*} \mathcal{V}[\mathsf{use}( a( \mathsf{cost}(
\mathsf{prod} ))) ] \parallel_{a}
\mathcal{B}[a( \mathsf{pay\_for}(a(\mathsf{prod})))] \\ &\mapsto \mathcal{V}[\mathsf{use}( a( \mathsf{price})) ]\parallel_{a} \mathcal{B}[a( \mathsf{pay\_for}(a(\mathsf{prod})))] \\ &\mapsto^{*} \mathcal{B}[a( \mathsf{pay\_for}(\mathsf{price})) ] \parallel_{a} \mathcal{V}[\mathsf{use}( a( \mathsf{price}))] \\ &\mapsto \; \mathcal{B}[a (\mathsf{card})] \parallel_{a} \mathcal{V}[\mathsf{use}( a( \mathsf{price}))] \, \mapsto^{*} \, \mathcal{V}[\mathsf{use}( \mathsf{card})] \parallel_{a} \mathcal{B}[a( \mathsf{card})]
\end{align*}\end{small}Finally $ \mapsto \mathcal{V}[\mathsf{use}( \mathsf{card})]$: the buyer has performed its duty and the vendor uses the card number to obtain the due payment. \end{example}
We show that although more complicated than sending data, sending open processes can enhance efficiency. \begin{example}[\textbf{Efficiency via cross reductions}]\label{ex:code_mobility} Given three processes $M \parallel_{d} ( P \parallel_{a} Q)$. Assume that $Q$ wants to send a process to $P$, but one of the process' parameters is not available because $M$ first needs many time-consuming steps to produce it and only afterwards can send it to $Q$. Cross reductions make it possible to fully exploit parallelism and improve the program efficiency: $Q$ does not need to wait that much and can send the process directly to $P$, which can begin to partially evaluate it with no further delay. After having computed the data, $M$ sends it to $Q$ which in turn forwards it to $P$.
For a concrete example, assume that \begin{small}
\begin{align*} M& \quad \mapsto^{*}\; d \, (\lambda k^{\mathbb{N} \rightarrow \mathbb{N} \rightarrow \mathbb{N}} \, k \, 7 \, 0 ) \\ P & \quad = \quad d \, 0 \, (\lambda j^{\mathbb{N}} \, \lambda x^{\mathbb{N}} \, (ax)5s) \\ Q & \quad = \quad d \, 0 \, (\lambda y^{\mathbb{N}} \, \lambda l^{\mathbb{N}} \, a(\lambda z^{\mathbb{N}} \, \lambda i ^{\sigma} \, h \langle g(z),y \rangle ))
\end{align*} \end{small}where \mbox{$h : \mathbb{N} \wedge \mathbb{N} \rightarrow \mathbb{N}$}, \mbox{$g : \mathbb{N} \rightarrow \mathbb{N}$}, the communication kind of $d$ is $ (\mathbb{N} \rightarrow \mathbb{N} \rightarrow \mathbb{N}) \rightarrow \mathbb{N} , \mathbb{N}$, and the communication kind of $a$ is $\mathbb{N} , \mathbb{N} \rightarrow \sigma \rightarrow \mathbb{N}$ with $\sigma$ arbitrary type of high complexity. Here $Q$ wants to send $\lambda z^{\mathbb{N}} \, \lambda i ^{\sigma} \, h \langle g(z),y \rangle $ to $P$, but the value $7$ of the parameter $y$ is computed and transmitted to $Q$ by $M$ only later. On the other hand, $P$ waits for the process from $Q$ in order to instantiate $z$ with $5$ and compute $h \langle g(5), 7 \rangle$.
Without a special mechanism for sending open terms, $P$ must wait for $M$ to normalize. Afterwards $M$ passes $(\lambda k \, k \, 7 \, 0)$ through $d$ to $P$ and $Q$ with the following computation: \begin{footnotesize}
\begin{align*}
M \parallel_{d} ( P \parallel_{a} Q) \mapsto^{*} & (\lambda k \, k \, 7 \, 0) (\lambda j \, \lambda x\, (ax)5s)
\parallel_{a} \\ &
(\lambda k \, k \, 7 \, 0) (\lambda y \,
\lambda l \, a(\lambda z \, \lambda i \, h \langle g(z),y \rangle )) \mapsto^{*} \\ (a\,0)5s \parallel _{a} a(\lambda z \, \lambda i \, h \langle g & (z),7 \rangle ) \mapsto^{*} (\lambda z \, \lambda i \, h \langle g(z),7 \rangle )5s \mapsto^{*} h \langle g(5),7 \rangle
\end{align*} \end{footnotesize} Our normalization algorithm allows instead $Q$ to directly send $\lambda z^{\mathbb{N}} \, \lambda i ^{\sigma} \, h \langle g(z),y \rangle $ to $P$ by executing first a cross reduction: \begin{footnotesize}\begin{align*}
& M
\parallel_{d} \Big( d \, 0 (\lambda j \, \lambda
x \, (ax)5s) \parallel_{a} d \, 0 (\lambda y\, \lambda l \, a(\lambda z \, \lambda i \, h \langle g(z),y\rangle)) \Big)
\\
\mapsto &
M
\parallel_{d} \Big(
(d \, 0 (\lambda y\, \lambda l \, by
) ||_{a} P ) \parallel_{b} \big(d \, 0 (\lambda j \, \lambda
x \, ( \lambda z \, \lambda i \, h \langle g(z),bx \rangle )5s) \parallel_{a} Q\big)\Big) \\
\mapsto&^{*}
M
\parallel_{d} \Big( d \, 0 (\lambda y\, \lambda l \, by ) \parallel_{b} d \, 0 (\lambda j \, \lambda
x \, ( \lambda z \, \lambda i \, h \langle g(z),bx \rangle )5s) \Big)
\end{align*}
\end{footnotesize} where the communication $b$ (of kind $\mathbb{N} , \mathbb{N}$) redirects the data $x$ and $y$. Then $P$ instantiates $z$ with $5$ and can compute for example $g(5)=9$ without having to evaluate $h \langle g(5),7 \rangle $ all at once. When $M$ terminates the computation, sends $7$ to the new location of the partially evaluated processes $P$ and $Q$ via $\parallel_{b}$:\begin{small}
\begin{align*} \mapsto^{*} &
M
\parallel_{d} \Big( d \, 0 (\lambda y\, \lambda l \, by ) \parallel_{b} d \, 0 (\lambda j \,\lambda
x \, h \langle g(5),bx \rangle )
\Big) \\ \mapsto^{*}& \Big( M \parallel_{d} d \, 0 (\lambda y\, \lambda l \, by )\Big) \parallel_{b} \Big( M
\parallel_{d} d \, 0 (\lambda j \,\lambda
x \, h \langle 9,bx \rangle ) \Big) \\ \mapsto^{*} & (\lambda k \, k \, 7 \, 0) (\lambda y\, \lambda l \, by ) \parallel_{b} (\lambda k \, k \, 7 \, 0) (\lambda
j \, \lambda
x \, h \langle 9,bx \rangle ) \\ \mapsto^{*}& \quad b7 \parallel_{b} h \langle 9,b\, 0 \rangle \quad \mapsto^{*} \quad h \langle 9,7 \rangle \end{align*} \end{small} \end{example}
\noindent {\bf Final Remark} The Curry--Howard isomorphism for $\bblambda_{\mathrm{G}}$ interprets G\"odel logic in terms of communication between parallel processes. In addition to revealing this connection, our results pave the way towards a more general computational interpretation of the intermediate logics formalized by hypersequent calculi. These logics are characterized by disjunctive axioms of a suitable form~\cite{CGT08} -- containing all the disjunctive tautologies of~\cite{DanosKrivine} -- and likely correspond to other communication mechanisms between parallel processes.
\appendix
\section{Appendix}
\noindent \textbf{Propostition~\ref{proposition-strongsubf}}~(Characterization of Strong Subformulas)\textbf{.}
Suppose $B$ is any {strong subformula} of $A$. Then: \begin{itemize} \item If $A=A_{1}\land \ldots \land A_{n}$, with $n>0$ and $A_{1}, \ldots, A_{n}$ are prime, then $B$ is a proper subformula of one among $A_{1}, \ldots, A_{n}$. \item If $A=C\rightarrow D$, then $B$ is a proper subformula of a prime factor of $C$ or $D$. \end{itemize}
\begin{proof}\mbox{} \begin{itemize} \item Suppose $A=A_{1}\land \ldots \land A_{n}$, with $n>0$ and $A_{1}, \ldots, A_{n}$ are prime. Any prime proper subformula of $A$ is a subformula of one among $A_{1}, \ldots, A_{n}$, so $B$ must be a proper subformula of one among $A_{1}, \ldots, A_{n}$.
\item Suppose $A=C\rightarrow D$. Any prime proper subformula $\mathcal{X}$ of $A$ is first of all a subformula of $C$ or $D$. Assume now $C=C_{1}\land \ldots \land C_{n}$ and $D=D_{1}\land \ldots \land D_{m}$, with $C_{1}, \ldots, C_{n}, D_{1}, \ldots, D_{m}$ prime. Since $\mathcal{X}$ is prime, it must be a subformula of one among $C_{1}, \ldots, C_{n}, D_{1}, \ldots, D_{m}$ and since $B$ is a proper subformula of $\mathcal{X}$, it must be a proper subformula of one among $C_{1}, \ldots, C_{n}, D_{1}, \ldots, D_{m}$, QED. \end{itemize} \end{proof}
\noindent \textbf{Theorem}~\ref{subjectred}~(Subject Reduction)\textbf{.}
If $t : A$ and $t \mapsto u$, then
$u : A$ and all the free variables of $u$ appear among those of
$t$.
\begin{proof}
\begin{enumerate} \item $(\lambda x\, u)t\mapsto u[t/x]$.
A term $(\lambda x\, u)t$ corresponds to a derivation of the form \[\infer{(\lambda x\, u)t : B}{\infer[^{1}]{\lambda x\, u: A \rightarrow B}{\infer*{u:B}{\Gamma_{0} , [x:A]^{1}}} & & \deduce{t: A}{\deduce{{\cal P}}{\Gamma_{1}}}}\] where $\Gamma_{0}$ and $\Gamma_{1}$ are the open hypotheses. Hence, the term $u[t/x]$ corresponds to \[ \infer*{u:B} { \Gamma_{0} & \deduce{t: A}{\deduce{{\cal P}}{\Gamma_{1}}} }\] where all occurrences of $[x:A]^{1}$ are replaced by the derivation ${\cal P}$ of $t: A$. It is easy to prove, by induction on the length of the derivation from $\Gamma_{0} , x:A$ to $B$, that this is a derivation of $B$. The two derivations have the same open hypotheses $\Gamma_{0}$ and $\Gamma_{1}$, and hence the two terms $(\lambda x\, u)t$ and $u[t/x]$ have the same free variables. Indeed, it is easy to show that if a term corresponds to a derivation, each free variable of such term corresponds to an open hypothesis of the derivation.
\item $ \pair{u_0}{u_1}\pi_{i}\mapsto u_i$, for $i=0,1$.
A term $\pair{u_0}{u_1}\pi_{i}$ corresponds to a derivation of the form \[\infer{\pair{u_0}{u_1}\pi_{i} : A_{i} }{\infer{\pair{u_0}{u_1}: A_{0} \wedge A_{1}} { \infer*{u_{0}: A_{0}}{\Gamma_{0}} & & \infer*{u_{1}: A_{1}}{\Gamma_{1}} }} \] where $\Gamma_{0}$ and $\Gamma_{1}$ are the open hypotheses. Hence, the term $u_i$ corresponds to \[ \infer*{u_{i}:A_{i}} { \Gamma_{i}}\] Clearly, all open hypotheses of the derivation of $u_{i}:A_{i}$ also occur in the derivation of $\pair{u_0}{u_1}\pi_{i} : A_{i}$, hence the free variables of $u_{i}$ are a subset of those of $\pair{u_0}{u_1}\pi_{i}$.
\item $(\Ecrom{a}{u}{v}) w \mapsto \Ecrom{a}{uw}{vw}$, if $a$ does not occur free in $w$.
A term $(\Ecrom{a}{u}{v}) w$ corresponds to a derivation of the form \[\infer{(\Ecrom{a}{u}{v}) w : D }{\infer[^{1}]{ \Ecrom{a}{u}{v}:C \rightarrow D}{\infer*{u:C \rightarrow D}{\Gamma_{0} , [a:A \rightarrow B]^{1} } && \infer*{v:C \rightarrow D}{\Gamma_{1} , [a:B \rightarrow A]^{1} }} & & \infer*{w: C}{\Gamma_{2}} } \] where $\Gamma_{0} , \Gamma_{1}$ and $\Gamma_{2}$ are the open hypotheses. Hence, the term $ \Ecrom{a}{uw}{vw}$ corresponds to \[\infer[^{1}]
{
\Ecrom{a}{uw}{vw} : D
}
{
\infer{uw:D}
{
\infer*{u:C \rightarrow
D}
{
\Gamma_{0} , [a:A \rightarrow B]^{1}
}
&&
\infer*{w: C}
{\Gamma_{2}}
}
&&
\infer{vw:D}
{
\infer*{v:C \rightarrow D}{\Gamma_{1} , [a:B \rightarrow A]^{1} }
&&
\infer*{w: C}{\Gamma_{2}}
}
} \] Notice that the formulas corresponding to the terms $a$ are discharged above the appropriate premise of the $(\mathsf{com})$ rule application. All open hypotheses of the second derivation also occur in the first derivation, hence the free variables of $(\Ecrom{a}{u}{v}) w$ are also free variables of $\Ecrom{a}{uw}{vw}$.
\item $w(\Ecrom{a}{u}{v}) \mapsto \Ecrom{a}{wu}{wv}$, if $a$ does not occur free in $w$.
A term $w(\Ecrom{a}{u}{v})$ corresponds to a derivation of the form \[\infer{w(\Ecrom{a}{u}{v}): D }{\infer*{w: C \rightarrow D}{\Gamma_{0}} && \infer[^{1}]{ \Ecrom{a}{u}{v}:C }{\infer*{u:C }{\Gamma_{1} , [a:A \rightarrow B]^{1} } && \infer*{v:C}{\Gamma_{2} , [a:B \rightarrow A]^{1} }}} \]
where $\Gamma_{0} , \Gamma_{1}$ and $\Gamma_{2}$ are the open hypotheses. Hence, the term $ \Ecrom{a}{wu}{wv}$ corresponds to \[\infer[^{1}]
{
\Ecrom{a}{wu}{wv} : D
}
{
\infer{wu:D}
{
\infer*{w: C\rightarrow D}{\Gamma_{0}} &
\infer*{u:C}{\deduce{[a:A \rightarrow B]^{1}}{\Gamma_{1} ,} }
} &
\infer{wv:D}
{
\infer*{w: C \rightarrow
D}{\Gamma_{0}} &
\infer*{v:C}{\deduce{[a:B \rightarrow A]^{1}}{\Gamma_{2} ,} }
}
} \] In which the formulas corresponding to the terms $a$ are discharged above the appropriate premise of the $(\mathsf{com})$ rule application. All open hypotheses of the second derivation also occur in the first derivation, and thus the free variables of $\Ecrom{a}{wu}{wv}$ are also free variables of $w(\Ecrom{a}{u}{v})$.
\item $\efq{P}{\Ecrom{a}{w_{1}}{w_{2}}} \mapsto \Ecrom{a}{\efq{P}{w_{1}}}{\efq{P}{w_{2}}}$, where $P$ is atomic.
A term $\efq{P}{\Ecrom{a}{w_{1}}{w_{2}}} $ corresponds to a derivation of the form \[\vcenter{ \infer{\efq{P}{\Ecrom{a}{w_{1}}{w_{2}}}: P}{ \infer{\Ecrom{a}{w_{1}}{w_{2}} : \bot}{\infer*{w_{1}: \bot}{\Gamma
_{1} , [a:A \rightarrow B]^{1}} && \infer*{w_{2}: \bot}{\Gamma _{2} , [a:B \rightarrow A]^{1}}}} }\]
where $\Gamma_{1}$ and $\Gamma_{2}$ are the open hypotheses. Hence, the term $\Ecrom{a}{\efq{P}{w_{1}}}{\efq{P}{w_{2}}}$ corresponds to \[ \vcenter{\infer[^{1}]{\Ecrom{a}{\efq{P}{w_{1}}}{\efq{P}{w_{2}}} :
P}{\infer{\efq{P}{w_{1}} : P}{\infer*{\bot}{\Gamma _{1} , [a:A \rightarrow B]^{1}}} &&& \infer{\efq{P}{w_{1}}:P}{\infer*{\bot}{\Gamma _{2} , [a:B \rightarrow A]^{1}}}}}\] In which the formulas corresponding to the terms $a$ are discharged above the appropriate premise of the $(\mathsf{com})$ rule application. All open hypotheses of the second derivation also occur in the first derivation, and thus $\Ecrom{a}{\efq{P}{w_{1}}}{\efq{P}{w_{2}}}$ has the same free variables as $\efq{P}{\Ecrom{a}{w_{1}}{w_{2}}} $.
\item $(\Ecrom{a}{u}{v})\pi_{i} \mapsto \Ecrom{a}{u\pi_{i}}{v\pi_{i}}$, for $i=0,1$.
A term $(\Ecrom{a}{u}{v})\pi_{i}$ corresponds to a derivation of the form \[\infer{ (\Ecrom{a}{u}{v})\pi_{i} : A_{i}}{ \infer[^{1}]{\Ecrom{a}{u}{v} : C_{0} \wedge C_{1}}{\infer*{u:C_{0} \wedge C_{1}}{\Gamma _{1} , [a:A \rightarrow B]^{1}} & \infer*{v:C_{0} \wedge C_{1}}{\Gamma _{2} , [a:B \rightarrow A]^{1}} }} \] where $\Gamma_{1}$ and $\Gamma_{2}$ are the open hypotheses. Hence, the term $\Ecrom{a}{u\pi_{i}}{v\pi_{i}}$ corresponds to \[ \infer{\Ecrom{a}{u\pi_{i}}{v\pi_{i}}: C_{i}}{ \infer{u\pi _{i}:C_{i}}{\infer*{u:C_{0} \wedge C_{1}}{\Gamma _{1} , [a:A \rightarrow B]^{1}}} & \infer{v\pi _{i}:C_{i}}{\infer*{v:C_{0} \wedge C_{1}}{\Gamma _{2} , [a:B \rightarrow A]^{1}}}} \] In which the formulas corresponding to the terms $a$ are discharged above the appropriate premise of the $(\mathsf{com})$ rule application. All open hypotheses of the second derivation also occur in the first derivation, and thus the free variables of $ (\Ecrom{a}{u}{v})\pi_{i} $ and of $ \Ecrom{a}{u\pi_{i}}{v\pi_{i}} $ are the same.
\item $\lambda x\,(\Ecrom{a}{u}{v}) \mapsto \Ecrom{a}{\lambda x\,u}{\lambda x\, v} $.
A term $\lambda x\,(\Ecrom{a}{u}{v})$ corresponds to a derivation \[ \vcenter{\infer[^{2}]{\lambda x\,(\Ecrom{a}{u}{v}): C\rightarrow D}{\infer[^{1}]{\Ecrom{a}{u}{v}:D}{\infer*{u:D}{\Gamma _{1} , [x:C]^{2}, [a:A \rightarrow B]^{1} } &&& \infer*{v:D}{\Gamma _{2} , [x:C]^{2}, [a:B \rightarrow A]^{1}}}} }\] where $\Gamma_{1}$ and $\Gamma_{2}$ are the open hypotheses. Hence, the term
$\Ecrom{a}{\lambda x\,u}{\lambda x\, v} $ corresponds to \[\infer[^{3}]{\Ecrom{a}{\lambda x\,u}{\lambda x\, v} : C\rightarrow D}{ \infer[^{1}] {\lambda x \, u: C\rightarrow D} {\infer*{u:D}{\Gamma _{1} , [x:C]^{1}, [a:A \rightarrow B]^{3} }} & \infer[^{2}]{\lambda x \, v: C \rightarrow D}{\infer*{v:D}{\Gamma _{2} , [x:C]^{2}, [a:B \rightarrow A]^{3}}}} \]
In which the formulas corresponding to the terms $a$ are discharged above the appropriate premise of the $(\mathsf{com})$ rule application. All open hypotheses of the second derivation also occur in the first derivation, and thus $\lambda x\,(\Ecrom{a}{u}{v})$ and $ \Ecrom{a}{\lambda x\,u}{\lambda x\, v} $ have the same free variables.
\item $\langle u \parallel_{a} v,\, w\rangle \mapsto \langle u, w\rangle \parallel_{a} \langle v, w\rangle$, if $a$ does not occur free in $w$.
A term $\langle u \parallel_{a} v,\, w\rangle$ corresponds to a derivation \[ \infer{\langle u \parallel_{a} v,\, w\rangle : C \wedge D } {
\infer[^{1}]{ u \parallel_{a} v : C}{ \infer*{u:C}{\Gamma _{0}, [a:A \rightarrow B]^{1}} & \infer*{v:C}{\Gamma _{1}, [a:B \rightarrow A]^{1}}} && \infer*{w:D}{\Gamma _{2}}}\] where $\Gamma_{0}, \Gamma_{1}$ and $\Gamma_{2}$ are the open hypotheses. Hence, the term $\langle u, w\rangle \parallel_{a} \langle v, w\rangle$ corresponds to \[ \infer[^{1}]{ \langle u, w\rangle \parallel_{a} \langle v, w\rangle : C \wedge D} { \infer{\langle u, w\rangle:C \wedge D} { \infer*{u:C}{\Gamma _{0}, [a:A \rightarrow B]^{1}} & \infer*{w:D}{\Gamma _{2}} } & \infer{\langle v, w\rangle: C \wedge D} { \infer*{v:C}{\Gamma _{1}, [a:B \rightarrow A]^{1}} & \infer*{w:D}{\Gamma _{2}}} } \]
In which the formulas corresponding to the terms $a$ are discharged above the appropriate premise of the $(\mathsf{com})$ rule applications. All open hypotheses of the second derivation also occur in the first derivation, and thus the free variables of the terms
$\langle u \parallel_{a} v,\, w\rangle$ and $ \langle u, w\rangle \parallel_{a} \langle v, w\rangle$ are the same.
\item $\langle w, \,u \parallel_{a} v\rangle \mapsto \langle w, u\rangle \parallel_{a} \langle w, v\rangle, \mbox{ if $a$ does not occur free in $w$}$.
The case is symmetric to the previous one.
\item $(u\parallel_{a} v)\parallel_{b} w \mapsto (u\parallel_{b} w)\parallel_{a} (v\parallel_{b} w),$ if the communication complexity of $b$ is greater than $0$.
A term $(u\parallel_{a} v)\parallel_{b} w$ corresponds to a derivation
\[ \vcenter{\infer[^{2}]{ (u \parallel_{a} v) \parallel_{b} w : E }
{\deduce{u \parallel_{a} v : E}{{\cal D }_{1}} &&&
\deduce{w:E}{{\cal P}_{2}}} } \qquad \text{where}\] \[{\cal P}_{1} \equiv \qquad \vcenter{\infer[^{1}]{ u \parallel_{a} v : E}{ \infer*{u:E}{\deduce{ [b:C \rightarrow D]^{2}}{\Gamma _{0},
[a:A \rightarrow B]^{1},}} &&& \infer*{v:E} {\deduce{[b:C \rightarrow D]^{2}}{\Gamma
_{1}, [a:B \rightarrow A]^{1},} }}} \]
\[{\cal P}_{2} \quad \equiv \qquad \quad \vcenter{\infer*{w:E}{\Gamma _{2}, [b:D \rightarrow C]^{2}}}\]
where $\Gamma_{0}, \Gamma_{1}$ and $\Gamma_{2}$ are the open hypotheses. Hence, the term $ (u\parallel_{b} w)\parallel_{a} (v\parallel_{b} w)$ corresponds to
\[ \infer[^{3}]{ (u\parallel_{b} w)\parallel_{a} (v\parallel_{b} w)
: E } {{\cal P}_{1} &&&& {\cal P}_{2}}\] where ${\cal P}_{1} \equiv $ \[ \infer[^{1}]{ u\parallel_{b} w : E}{ \infer*{u:E}{\Gamma
_{0}, [a:A \rightarrow B]^{3}, [b:C \rightarrow D]^{1}} &
\infer*{w:E}{\Gamma _{2}, [b:D \rightarrow C]^{1}}} \] and ${\cal P}_{2} \equiv$
\[ \infer[^{2}]{v\parallel_{b} w:E}{ \infer*{v:E}{\Gamma _{1}, [a:B \rightarrow
A]^{3}, [b:C \rightarrow D]^{2}} & \infer*{w:E}{\Gamma _{2}, [b:D
\rightarrow C]^{2}} }\]
In which the formulas corresponding to the terms $a$ and $b$ are discharged above the appropriate premise of the $(\mathsf{com})$ rule applications. All open hypotheses of the second derivation also occur in the first derivation, and thus the free variables of $(u\parallel_{a} v)\parallel_{b} w$
and $ (u\parallel_{b} w)\parallel_{a} (v\parallel_{b} w)$ are the same.
\item $w \parallel_{b} (u\parallel_{a} v) \mapsto (w\parallel_{b} u)\parallel_{a} (w\parallel_{b} v)$, if the communication complexity of $b$ is greater than $0$.
A term $w \parallel_{b} (u\parallel_{a} v)$ corresponds to a derivation
\[ \vcenter{\infer[^{2}]{ w \parallel_{b} (u\parallel_{a} v) : E }
{\deduce{w:E}{{\cal P}_{1}} &&&&& \deduce{u \parallel_{a} v :
E}{{\cal P}_{2}}}} \qquad \text{where}\] \[ {\cal P}_{1} \quad \equiv \quad\vcenter{ \infer*{w:E}{\Gamma _{2},
[b:D \rightarrow C]^{2}}} \] \[{\cal P}_{2} \equiv \qquad \vcenter{\infer[^{1}]{u \parallel_{a} v : E}{ \infer*{u:E}{\deduce{ [b:C \rightarrow D]^{2}}{\Gamma _{0}, [a:A \rightarrow B]^{1},}} &&& \infer*{v:E}{\deduce{[b:C \rightarrow D]^{2}}{\Gamma _{1}, [a:B \rightarrow A]^{1},}}}}\] Here we represent by $\Gamma_{0}, \Gamma_{1}$ and $\Gamma_{2}$ the open hypotheses of the derivation. Hence, the term $(w\parallel_{b} u)\parallel_{a} (w\parallel_{b} v)$ corresponds to
\[ \vcenter{\infer[^{3}]{ (w\parallel_{b} u)\parallel_{a} (w\parallel_{b} v)
: E } {{\cal P}_{1} && {\cal P}_{2}}} \qquad \text{where}\]
\[{\cal P}_{1} \; \equiv \; \vcenter{\infer[^{1}]{ w\parallel_{b} u : E}{
\infer*{w:E}{\Gamma _{2}, [b:D \rightarrow C]^{1}} & \infer*{u:E}{\Gamma
_{0}, [a:A \rightarrow B]^{3}, [b:C \rightarrow D]^{1}}}} \]
\[{\cal P}_{2} \; \equiv \; \vcenter{ \infer[^{2}]{w\parallel_{b} v:E}{ \infer*{w:E}{\Gamma _{2}, [b:D
\rightarrow C]^{2}} & \infer*{v:E}{\Gamma _{1}, [a:B \rightarrow
A]^{3}, [b:C \rightarrow D]^{2}}} }\]
In which the formulas corresponding to the terms $a$ and $b$ are discharged above the appropriate premise of the $(\mathsf{com})$ rule application. The open hypotheses of the second derivation also occur in the first derivation, and thus the free variables of the terms $w \parallel_{b} (u\parallel_{a} v)$ and $ (w\parallel_{b} u)\parallel_{a} (w\parallel_{b} v)$ are the same.
\item $u\parallel_{a}v \mapsto u$, if $a$ does not occur in $u$.
A term $u\parallel_{a}v$ corresponds to a derivation \[\infer[^{1}]{ u\parallel_{a} v : C}{ \infer*{u:C}{\Gamma _{1}} & \infer*{v:C}{\Gamma _{2}, [a:B \rightarrow A]^{1}}} \] where $\Gamma _{1}$ and $\Gamma_{2}$ are all open hypotheses and no hypothesis $a: A \rightarrow B$ is discharged above $u: C$ by the considered $(\mathsf{com})$ application because $a$ does not occur in $u$. The term $u$ corresponds to the
branch rooted at $u:C$, the open hypotheses of which are clearly a subset of the open hypotheses of the whole derivation. Indeed the $(\mathsf{com})$ application in the derivation corresponding to $u\parallel_{a}v$ is redundant because $C$ can be derived from the hypotheses $\Gamma _{1}$ alone. Thus, the free variables of $u$ are a subset of the free variables of $u\parallel_{a}v$.
\item $u\parallel_{a}v \mapsto v$, if $a$ does not occur in $v$.
This case is symmetric to the previous one.
\item $\mathcal{C}[a\, u]\parallel_{a} \mathcal{D}[a\, v]\ \mapsto\ (\mathcal{D}[u^{b\langle \sq{z}\rangle / \sq{y}}] \parallel_{a} \mathcal{C}[a\, u] )\, \parallel_{b}\, (\mathcal{C}[v^{b\langle \sq{y}\rangle / \sq{z}}]\parallel_{a} \mathcal{D}[a\, v])$ where $\mathcal{C}, \mathcal{D}$ are simple contexts; the displayed occurrences of $a$ are the rightmost both in $\mathcal{C}[a\,u]$ and in $\mathcal{D}[a\,v ]$ and $b$ is fresh;
$\sq{y}$ the sequence of the free variables of $u$ which are bound in $\mathcal{C}[au]$; $\sq{z}$ the sequence of the free variables of $v$ which are bound in $\mathcal{D}[av]$; the communication complexity of $a$ is greater than $0$.
We introduce some notation in order to compactly represent the handling of conjunctions of hypotheses corresponding to $\sq{y}$ and $\sq{z}$.
\noindent \emph{Notation}. For any finite set of formulas $\Theta = \{\theta_{1} , \dots , \theta_{k}\}$ we define
\[ \vcenter{ \infer=[\mathrm{I}]{\bigwedge\Theta}{\Theta} } \qquad \qquad \text{as} \qquad \qquad \vcenter{ \infer{ \bigwedge\Theta } { \infer* { } { \infer { } { \dots & \infer { \theta_{3} \wedge \theta_{2} \wedge \theta_{1} } { \theta_{3} & \infer { \theta_{2} \wedge \theta_{1} } { \theta_{2} & \theta_{1} } } } } } }
\] and $\vcenter{\infer=[\mathrm{E}]{{\cal
P}}{\bigwedge\Theta}}$ as the derivation ${\cal P}$ in which all hypotheses contained in $\Theta$ are derived as follows
\[ \infer{\theta_{1}}{\bigwedge\Theta} \qquad \cdots \qquad \infer{\theta_{k}}{\infer*{\theta_{k-1} \wedge \theta_{k}}{\infer{\theta_{2} \wedge \dots \wedge \theta_{k}}{\bigwedge\Theta}}}
\]
A term $\mathcal{C}[a\, u]\parallel_{a} \mathcal{D}[a\, v]$ corresponds to a derivation of the form \[
\infer[^{1}]{\mathcal{C}[a\, u]\parallel_{a} \mathcal{D}[a\, v]: E}{
\deduce{\mathcal{C}[a\, u] : E}{
\deduce{
{\cal P}_{3}
}
{
\infer{
au:B
}
{
[a: A \rightarrow B]^{1} & \deduce{
u:A
}
{
\deduce{
{\cal P}_{1}
}
{
\sq{y} :\Gamma
}
}
}
}
} &&&& \deduce{
\mathcal{D}[a\, v]:E } {
\deduce{
{\cal P}_{4}
}
{
\infer{
av:A
}
{
[a: B \rightarrow A] ^{1} & \deduce{
v:B
}
{
\deduce{
{\cal P}_{2}
}
{
\sq{z} : \Delta
}
}
}
} } } \]
where $\Gamma$ and $\Delta$ are the open hypotheses of ${\cal P}_{1}$ discharged in ${\cal P}_{3}$, respectively, the open hypotheses of ${\cal P}_{2}$ discharged in ${\cal P}_{4}$; thus $\Gamma$ corresponds to the sequence $\sq{y}$ that contains the free variables of $u$ which are bound in $\mathcal{C}[au]$, while $\Delta$ corresponds to the sequence $\sq{z}$ of all the free variables of $v$ which are bound in $\mathcal{D}[av]$. No hypotheses $a: A \rightarrow B$ and $a: B \rightarrow A$ are discharged neither in ${\cal P}_{1}$ nor in ${\cal P}_{2}$ by the considered $(\mathsf{com})$ application because $a$ does not occur neither in $u$ nor in $v$.
The term $(\mathcal{D}[u^{b\langle \sq{z}\rangle / \sq{y}}] \parallel_{a} \mathcal{C}[a\, u])\, \parallel_{b}\, (\mathcal{C}[v^{b\langle \sq{y}\rangle / \sq{z}}]\parallel_{a} \mathcal{D}[a\, v])$ then corresponds to the derivation
\[ \infer[^{3}] {( \mathcal{D}[u^{b\langle \sq{z}\rangle / \sq{y}}] \parallel_{a} \mathcal{C}[a\, u] )\, \parallel_{b}\, (\mathcal{C}[v^{b\langle \sq{y}\rangle / \sq{z}}]\parallel_{a} \mathcal{D}[a\, v]) : E} {\deduce{\mathcal{D}[u^{b\langle \sq{z}\rangle / \sq{y}}] \parallel_{a} \mathcal{C}[a\, u] : E}{{\cal P}_{1}} &&& \deduce{\mathcal{C}[v^{b\langle \sq{y}\rangle /
\sq{z}}]\parallel_{a} \mathcal{D}[a\, v] : E}{{\cal P}_{2}}}
\] where ${\cal P}_{1} \equiv$
\[\infer[^{1}] {\mathcal{C}[a\, u]\parallel_{a}
\mathcal{D}[u^{b\langle \sq{z}\rangle / \sq{y}}] : E}
{
\deduce{\mathcal{D}[u^{b\langle \sq{z}\rangle / \sq{y}}] :
E}{\deduce{{\cal P}_{4}}{\deduce{u^{b\langle \sq{z}\rangle /
\sq{y}}:A}
{\infer=[\mathrm{E}]{{\cal P}_{1}^{b\langle \sq{z} \rangle , \sq{y}}}
{\infer{b\langle \sq{z} \rangle:\bigwedge\Gamma}{[b:
\bigwedge\Delta \rightarrow \bigwedge\Gamma]^{3} &
\infer=[\mathrm{I}]{\langle \sq{z}
\rangle:\bigwedge\Delta}{\sq{z}: \Delta} }}}}} & \deduce{\mathcal{C}[a\, u] : E} { \deduce{ {\cal P}_{3} } {
\infer{ au:B } { [a: A \rightarrow B]^{1} & \deduce{ u:A } {
\deduce{ {\cal P}_{1} } { \sq{y}:\Gamma } } } } } }
\] and ${\cal P}_{2} \equiv$
\[ \infer[^{2}] {\mathcal{C}[v^{b\langle \sq{y}\rangle /
\sq{z}}]\parallel_{a} \mathcal{D}[a\, v] : E} {
\deduce{\mathcal{C}[v^{b\langle \sq{y}\rangle / \sq{z}}]:E}
{\deduce{{\cal P}_{3}}{\deduce{v^{b\langle \sq{y}\rangle /
\sq{z}}:B} {\infer=[\mathrm{E}]{ {\cal P}_{2}^{b\langle \sq{y} \rangle , \sq{z}} } {\infer{b \langle \sq{y} \rangle : \bigwedge\Delta}{ [b:
\bigwedge\Gamma \rightarrow \bigwedge\Delta]^{3} &
\infer=[\mathrm{I}]{\langle \sq{y} \rangle
:\bigwedge\Gamma}{\sq{y}:\Gamma} } } } } } &
\deduce{ \mathcal{D}[a\, v]:E } { \deduce{ {\cal P}_{4} } {
\infer{av:A } { [a: B \rightarrow A] ^{2} & \deduce{ v:B } {
\deduce{ {\cal P}_{2} } { \sq{z}:\Delta } } } } }}
\]
The resulting derivation is well defined and only needs the open hypotheses of the derivation corresponding to the term $\mathcal{C}[a\, u]\parallel_{a} \mathcal{D}[a\, v]$. First, the derivations corresponding to $\mathcal{C}[a\, u]$ and $\mathcal{D}[a\, v]$ are the same as before the reduction. Consider then the derivations corresponding to $\mathcal{D}[u^{b\langle \sq{z}\rangle / \sq{y}}]$ and $\mathcal{C}[v^{b\langle \sq{y}\rangle / \sq{z}}]$. The hypothesis needed in $ {\cal P}_{1}$ and ${\cal P}_{2}$ can be derived by $(e\wedge)$ from $\bigwedge\Gamma$ and $\bigwedge\Delta$, respectively. The formulas $\bigwedge\Gamma$ and $\bigwedge\Delta$ can in turn be derived using $b:\bigwedge\Delta \rightarrow \bigwedge\Gamma $ and $b: \bigwedge\Gamma \rightarrow \bigwedge\Delta$ along with the formulas $\bigwedge\Delta$ and $\bigwedge\Gamma$, in the respective order. Thus it is possible to discharge the elements of $\Gamma$ and $\Delta$ in the derivations ${\cal P}_{3}$ and ${\cal P}_{4}$, respectively, exactly as in the redex derivation corresponding to $\mathcal{C}[a\, u]\parallel_{a} \mathcal{D}[a\, v]$. \end{enumerate} \end{proof}
\noindent \textbf{Proposition~\ref{proposition-boundhyp}}~(Bound Hypothesis Property)\textbf{.}
Suppose $$x_{1}^{A_{1}}, \ldots, x_{n}^{A_{n}}\vdash t: A$$ $t\in\mathsf{NF}$ is a simply typed $\lambda$-term and $z: B$ a variable occurring bound in $t$. Then one of the following holds: \begin{enumerate} \item $B$ is a proper subformula of a prime factor of $A$. \item $B$ is a strong subformula of one among $A_{1},\ldots, A_{n}$.
\end{enumerate}
\begin{proof} By induction on $t$. \begin{itemize} \item $t=x_{i}^{A_{i}}$, with $1\leq i\leq n$. Since by hypothesis $z$ must occur bound in $t$, this case is impossible.
\item $t=\lambda x^{T} u$, with $A=T\rightarrow U$. If $z=x^{T}$, since $A$ is a prime factor of itself, we are done. If $z\neq x^{T}$, then $z$ occurs bound in $u$ and by induction hypothesis applied to $u: U$, we have two possibilities: i) $B$ is a proper subformula of a prime factor of $U$ and thus a proper subformula of a prime factor -- $A$ itself -- of $A$; ii) $B$ already satisfies 2., and we are done, or $B$ is a strong subformula of $T$, and thus it satisfies 1.
\item $t=\langle u_{1}, u_{2}\rangle$, with $A=T_{1}\land T_{2}$. Then $z$ occurs bound in $u$ or $v$ and, by induction hypothesis applied to $u_{1}: T_{1}$ and $u_{2}: T_{2}$, we have two possibilities: i) $B$ is a proper subformula of a prime factor of $T_{1}$ or $T_{2}$, and thus $B$ is a proper subformula of a prime factor of $A$ as well; ii) $B$ satisfies 2. and we are done.
\item $t=\efq{P}{u}$, with $A=P$. Then $z$ occurs bound in $u$. Since $\bot$ has no proper subformula, by induction hypothesis applied to $u: \bot$, we have that $B$ satisfies 2. and we are done.
\item $t=x_{i}^{A_{i}}\, \xi_{1}\ldots \xi_{m}$, where $m>0$ and each $\xi_{j}$ is either a term or a projection $\pi_{k}$. Since $z$ occurs bound in $t$, it occurs bound in some term $\xi_{j}: T$, where $T$ is a proper subformula of $A_{i}$. By induction hypothesis applied to $\xi_{j}$, we have two possibilities: i) $B$ is a proper subformula of a prime factor of $T$ and, by Definition \ref{definition-strongsubf}, $B$ is a strong subformula of $A_{i}$.
ii) $B$ satisfies 2. and we are done. \end{itemize} \end{proof}
\noindent \textbf{Proposition~\ref{proposition-app}}\textbf{.}
Suppose that $t\in \mathsf{NF}$ is a simply typed $\lambda$-term and $$x_{1}^{A_{1}}, \ldots, x_{n}^{A_{n}}, z^{B}\vdash t: A$$ Then one of the following holds:
\begin{enumerate} \item \emph{Every occurrence of $z^{B}$ in $t$ is of the form
$z^{B}\, \xi$ for some proof term or projection $\xi$.} \item \emph{$B=\bot$ or $B$ is a subformula of $A$ or a proper
subformula of one among the formulas $A_{1}, \ldots, A_{n}$.} \end{enumerate}
\begin{proof} By induction on $t$.
\begin{itemize} \item $t=x_{i}^{A_{i}}$, with $1\leq i\leq n$. Trivial.
\item $t=z^{B}$. This means that $B=A$, and we are done.
\item $t=\lambda x^{T} u$, with $A=T\rightarrow U$. By induction hypothesis applied to $u: U$, we have two possibilities: i) every occurrence of $z^{B}$ in $u$ is of the form $z^{B}\, \xi$, and we are done; ii) $B=\bot$ or $B$ is a subformula of $U$, and hence of $A$, or a proper subformula of one among the formulas $A_{1}, \ldots, A_{n}$, and we are done again.
\item $t=\langle u_{1}, u_{2}\rangle$, with $A=T_{1}\land T_{2}$. By induction hypothesis applied to $u_{1}: T_{1}$ and $u_{2}: T_{2}$, we have two possibilities: i) every occurrence of $z^{B}$ in $u_{1}$ and $u_{2}$ is of the form $z^{B}\, \xi$, and we are done; ii) $B=\bot$ or $B$ is a subformula of $T_{1}$ or $T_{2}$, and hence of $A$, or a proper subformula of one among the formulas $A_{1}, \ldots, A_{n}$, and we are done again.
\item $t=\efq{P}{u}$, with $A=P$. By induction hypothesis applied to $u: \bot$, we have two possibilities: i) every occurrence of $z^{B}$ in $u$ is of the form $z^{B}\, \xi$, and we are done; ii) $B=\bot$ or a proper subformula of one among
$A_{1}, \ldots, A_{n}$, and we are done again.
\item $t=x_{i}^{A_{i}}\, \xi_{1}\ldots \xi_{m}$, where $m>0$ and each $\xi_{j}$ is either a term or a projection $\pi_{k}$. Suppose there is an $i$ such that in the term $\xi_{j}: T_{j}$ not every occurrence of $z^{B}$ in $u$ is of the form $z^{B}\, \xi$. If $B=\bot$, we are done. If not, then by induction hypothesis $B$ is a subformula of $T_{j}$ or a proper subformula of one among $A_{1}, \ldots, A_{n}$. Since $T_{j}$ is a proper subformula of $A_{i}$, in both cases $B$ is a proper subformula of one among $A_{1}, \ldots, A_{n}$.
\item $t=z^{B}\, \xi_{1}\ldots \xi_{m}$, where $m>0$ and each $\xi_{i}$ is either a term or a projection $\pi_{j}$. Suppose there is an $i$ such that in the term $\xi_{i}: T_{i}$ not every occurrence of $z^{B}$ in $u$ is of the form $z^{B}\, \xi$. If $B=\bot$, we are done. If not, then by induction hypothesis $B$ is a subformula of $T_{i}$ or a proper subformula of one among $A_{1}, \ldots, A_{n}$. But the former case is not possible, since $T_{i}$ is a proper subformula of $B$, hence the latter holds. \end{itemize} \end{proof}
\noindent \textbf{Proposition~\ref{proposition-parallelform}}~(Parallel Normal Form Property)\textbf{.}
Suppose $t\in \mathsf{NF}$, then it is parallel form.
\begin{enumerate}
\item \emph{Every occurrence of $z^{B}$ in $t$ is of the form
$z^{B}\, \xi$ for some proof term or projection $\xi$.}
\item \emph{$B=\bot$ or $B$ is a subformula of $A$ or a proper
subformula of one among the formulas $A_{1}, \ldots, A_{n}$.}
\end{enumerate}
\begin{proof} By induction on $t$.
\begin{itemize}
\item $t$ is a variable $x$. Trivial.
\item $t=\lambda x\, u$. Since $t$ is normal, $u$ cannot be of the form $v\parallel_{a} w$, otherwise one could apply the permutation $$t=\lambda x\,(\Ecrom{a}{v}{w}) \mapsto \Ecrom{a}{\lambda x\,v}{\lambda x\, w} $$ and $t$ would not be in normal form. Hence, by induction hypothesis $u$ must be a simply typed $\lambda$-term, QED. \\
\item $t=\langle u_{1}, u_{2}\rangle$. Since $t$ is normal, neither $u_{1}$ nor $u_{2}$ can be of the form $v\parallel_{a} w$, otherwise one could apply one of the permutations $$\langle v \parallel_{a} w,\, u_{2}\rangle \mapsto \langle v, u_{2}\rangle \parallel_{a} \langle w, u_{2}\rangle$$ $$\langle u_{1}, \,v \parallel_{a} w\rangle \mapsto \langle u_{1}, v\rangle \parallel_{a} \langle u_{1}, w\rangle$$ and $t$ would not be in normal form. Hence, by induction hypothesis $u_{1}$ and $u_{2}$ must
be simply typed $\lambda$-terms, QED.
\item $t=u\, v$. Since $t$ is normal, neither $u$ nor $v$ can be of the form $w_{1}\parallel_{a} w_{2}$, otherwise one could apply one of the permutations $$(\Ecrom{a}{w_{1}}{w_{2}})\, v \mapsto \Ecrom{a}{w_{1}\, v}{w_{2}\, v}$$ $$u\, (\Ecrom{a}{w_{1}}{w_{2}}) \mapsto \Ecrom{a}{u\, w_{1}}{u\, w_{2}}$$ and $t$ would not be in normal form. Hence, by induction hypothesis $u_{1}$ and $u_{2}$ must
be simply typed $\lambda$-terms, QED.
\item $t=\efq{P}{u}$. Since $t$ is normal, $u$ cannot be of the form $w_{1}\parallel_{a} w_{2}$, otherwise one could apply the permutation $$\efq{P}{\Ecrom{a}{w_{1}}{w_{2}}} \mapsto \Ecrom{a}{\efq{P}{w_{1}}}{\efq{P}{w_{2}}}$$ and $t$ would not be in normal form. Hence, by induction hypothesis $u_{1}$ and $u_{2}$ must
be simply typed $\lambda$-terms, QED.
\item $t=u\, \pi_{i}$. Since $t$ is normal, $u$ cannot be of the form $v\parallel_{a} w$, otherwise one could apply the permutation $$(\Ecrom{a}{v}{w})\pi_{i} \mapsto \Ecrom{a}{v\pi_{i}}{w\pi_{i}} $$ and $t$ would not be in normal form. Hence, by induction hypothesis $u$ must be a simply typed $\lambda$-term, which is the thesis. \item $t=u\parallel_{a} v$. By induction hypothesis the thesis holds for $u$ and $v$ and hence trivially for $t$. \end{itemize} \end{proof}
\noindent \textbf{Theorem~\ref{theorem-subformula}}~(Subformula Property)\textbf{.}
Suppose $$x_{1}^{A_{1}}, \ldots, x_{n}^{A_{n}}\vdash t: A$$ and $t\in \mathsf{NF}$. Then: \begin{enumerate} \item For each communication variable $a$ occurring bound in $t$ and with communication kind $B, C$, the prime factors of $B$ and $C$ are proper subformulas of the formulas $A_{1}, \ldots, A_{n}, A$. \item The type of any subterm of $t$ which is not a bound communication variable is either a subformula or a conjunction of subformulas of the formulas $A_{1}, \ldots, A_{n}, A$. \end{enumerate}
\begin{proof} By Proposition \ref{proposition-parallelform} $t = t_{1}\parallel_{a_{1}} t_{2}\parallel_{a_{2}}\ldots \parallel_{a_{n}} t_{n+1}$ and each term $t_{i}$, for $1\leq i\leq n+1$, is a simply typed $\lambda$-term. By induction on $t$.
\begin{itemize} \item $t=x_{i}^{A_{i}}$, with $1\leq i\leq n$. Trivial.
\item $t=\lambda x^{T} u$, with $A=T\rightarrow U$. By Proposition \ref{proposition-parallelform}, $t$ is a simply typed $\lambda$-term, so $t$ contains no bound communication variable. Moreover, by induction hypothesis applied to $u: U$, the type of any subterm of $u$ which is not a bound communication variable is either a subformula or a conjunction of subformulas of the formulas $T, A_{1}, \ldots, A_{n}, U$ and therefore of the formulas $A_{1}, \ldots, A_{n}, A$.
\item $t=\langle u_{1}, u_{2}\rangle$, with $A=T_{1}\land T_{2}$. By Proposition \ref{proposition-parallelform}, $t$ is a simply typed $\lambda$-term, so $t$ contains no bound communication variable. Moreover, by induction hypothesis applied to $u_{1}: T_{1}$ and $u_{2}: T_{2}$, the type of any subterm of $u$ which is not a bound communication variable is either a subformula or a conjunction of subformulas of the formulas $ A_{1}, \ldots, A_{n}, T_{1}, T_{2}$ and hence of the formulas $A_{1}, \ldots, A_{n}, A$.
\item $t= u_{1}\parallel_{b} u_{2}$. Let $C, D$ be the communication
kind of $b$, we first show that the communication
complexity of $b$ is
$0$.
We reason by contradiction and assume that it is greater than $0$. Since $t\in\mathsf{NF}$ by hypothesis, it follows
from Prop.~\ref{proposition-parallelform} that $u_{1}$ and $u_{2}$ are either simply typed $\lambda$-terms or of the form $v\parallel_{c} w$. The second case is not possible, otherwise a permutation reduction could be applied to $t\in \mathsf{NF}$. Thus $u_{1}$ and $u_{2}$ are simply typed $\lambda$-terms. Since the communication complexity of $b$ is greater than $0$, the types $C\rightarrow D$ and $D\rightarrow C$ are not subformulas of $A_{1}, \ldots, A_{n}, A$. By Prop.~\ref{proposition-app}, every occurrence of $b^{C\rightarrow D}$ in $u_{1}$ is of the form $b^{C\rightarrow D} v$ and every occurrence of $b^{D\rightarrow C}$ in $u_{2}$ is of the form $b^{D\rightarrow C} w$. Hence, we can write $$u_{1}=\mathcal{C}[b^{C\rightarrow D} v] \qquad u_{2}=\mathcal{D}[b^{D\rightarrow C} w]$$ where $\mathcal{C}, \mathcal{D}$ are simple contexts and $b$ is rightmost. Hence a cross reduction of $t$ can be performed, which contradicts the fact that $t\in\mathsf{NF}$. Since we have established that the communication complexity of $b$ is $0$, the prime factors of $C$ and $D$ must be proper subformulas of $A_{1}, \ldots, A_{n}, A$. Now, by induction hypothesis applied to $u_{1}: A$ and $u_{2}: A$, for each communication variable $a^{F\rightarrow G}$ occurring bound in $t$, the prime factors of $F$ and $G$ are proper subformulas of the formulas $ A_{1}, \ldots, A_{n}, A, C\rightarrow D, D\rightarrow C$ and thus of the formulas $A_{1}, \ldots, A_{n}, A$; moreover, the type of any subterm of $u_{1}$ or $u_{2}$ which is not a communication variable is either a subformula or a conjunction of subformulas of the formulas $ A_{1}, \ldots, A_{n}, C\rightarrow D, D\rightarrow C$ and thus of the formulas $A_{1}, \ldots, A_{n}, A$.
\item $t=x_{i}^{A_{i}}\, \xi_{1}\ldots \xi_{m}$, where $m>0$ and each $\xi_{j}: T_{j}$ is either a term or a projection $\pi_{k}$ and $T_{j}$ is a subformula of $A_{i}$. By Proposition \ref{proposition-parallelform}, $t$ is a simply typed $\lambda$-term, so $t$ contains no bound communication variable. By induction hypothesis applied to each $\xi_{j}: T_{j}$, the type of any subterm of $t$ which is not a bound communication variable is either a subformula or a conjunction of subformulas of the formulas $ A_{1}, \ldots, A_{n}, T_{1}, \ldots, T_{m}$ and thus of the formulas $A_{1}, \ldots, A_{n}, A$.
\item $t=\efq{P}{u}$, with $A=P$. Then $u=x_{i}^{A_{i}}\, \xi_{1}\ldots \xi_{m}$, where $m>0$ and each $\xi_{j}$ is either a term or a projection $\pi_{k}$. Hence, $\bot$ is a subformula of $A_{i}$. Finally, by the proof of the previous case, we obtain the thesis for $t$. \end{itemize} \end{proof}
\noindent \textbf{Proposition~\ref{proposition-normpar}}\textbf{.}
Let $t: A$ be any term. Then $t\mapsto^{*} t'$, where $t'$ is a parallel form.
\begin{proof} By induction on $t$.
\begin{itemize} \item $t$ is a variable $x$. Trivial. \item $t=\lambda x\, u$. By induction hypothesis, $$u\mapsto^{*} u_{1}\parallel_{a_{1}} u_{2}\parallel_{a_{2}}\ldots \parallel_{a_{n}} u_{n+1}$$ and each term $u_{i}$, for $1\leq i\leq n+1$, is a simply typed $\lambda$-term. Applying $n$ times the permutations
we obtain $$t\mapsto^{*} \lambda x\, u_{1}\parallel_{a_{1}}\lambda x\, u_{2}\parallel_{a_{2}}\ldots \parallel_{a_{n}}\lambda x\, u_{n+1}$$ which is the thesis. \item $t=u\, v$. By induction hypothesis, $$u\mapsto^{*} u_{1}\parallel_{a_{1}} u_{2}\parallel_{a_{2}}\ldots \parallel_{a_{n}} u_{n+1}$$ $$v\mapsto^{*} v_{1}\parallel_{b_{1}} v_{2}\parallel_{b_{2}}\ldots \parallel_{b_{m}} v_{m+1}$$ and each term $u_{i}$ and $v_{i}$, for $1\leq i\leq n+1, m+1$, is a simply typed $\lambda$-term. Applying $n+m$ times the permutations
we obtain \[ \begin{aligned} t &\mapsto^{*} (u_{1}\parallel_{a_{1}} u_{2}\parallel_{a_{2}}\ldots \parallel_{a_{n}} u_{n+1})\, v \\ &\mapsto^{*} u_{1}\, v \parallel_{a_{1}} u_{2}\, v \parallel_{a_{2}}\ldots \parallel_{a_{n}} u_{n+1}\, v\\ &\mapsto^{*} u_{1}\, v_{1} \parallel_{b_{1}} u_{1}\, v_{2}\parallel_{b_{2}}\ldots \parallel_{b_{m}} u_{1}\, v_{m+1} \parallel_{a_{1}} \ldots \\ & \qquad \, \ldots \parallel_{a_{n}} u_{n+1}\, v_{1} \parallel_{b_{1}} u_{n+1}\, v_{2} \parallel_{b_{2}}\ldots
\parallel_{b_{m}} u_{n+1}\, v_{m+1} \end{aligned} \]
\item $t=\langle u, v\rangle$. By induction hypothesis, $$u\mapsto^{*} u_{1}\parallel_{a_{1}} u_{2}\parallel_{a_{2}}\ldots \parallel_{a_{n}} u_{n+1}$$ $$v\mapsto^{*} v_{1}\parallel_{b_{1}} v_{2}\parallel_{b_{2}}\ldots \parallel_{b_{m}} v_{m+1}$$ and each term $u_{i}$ and $v_{i}$, for $1\leq i\leq n+1, m+1$, is a simply typed $\lambda$-term. Applying $n+m$ times the permutations
we obtain
\[
\begin{aligned}
t &\mapsto^{*}\langle u_{1}\parallel_{a_{1}} u_{2}\parallel_{a_{2}}\ldots \parallel_{a_{n}} u_{n+1},\, v \rangle\\
&\mapsto^{*} \langle u_{1}, v\rangle \parallel_{a_{1}} \langle u_{2}, v \rangle\parallel_{a_{2}}\ldots \parallel_{a_{n}} \langle u_{n+1}, v\rangle\\
&\mapsto^{*} \langle u_{1}, v_{1}\rangle \parallel_{b_{1}}
\langle u_{1}, v_{2}
\rangle\parallel_{b_{2}}\ldots \parallel_{b_{m}} \langle u_{1},
v_{m+1}\rangle \parallel_{a_{1}} \ldots
\\
& \qquad \, \ldots
\parallel_{a_{n}} \langle u_{n+1},
v_{1}\rangle \parallel_{b_{1}} \langle u_{n+1}, v_{2}
\rangle\parallel_{b_{2}}\ldots
\\
& \qquad \, \ldots \parallel_{b_{m}} \langle u_{n+1},
v_{m+1}\rangle
\end{aligned}
\]
\item $t=u\, \pi_{i}$. By induction hypothesis, $$u\mapsto^{*} u_{1}\parallel_{a_{1}} u_{2}\parallel_{a_{2}}\ldots \parallel_{a_{n}} u_{n+1}$$ and each term $u_{i}$, for $1\leq i\leq n+1$, is a simply typed $\lambda$-term. Applying $n$ times the permutations
we obtain $$t\mapsto^{*} u_{1}\, \pi_{i}\parallel_{a_{1}} u_{2}\, \pi_{i}\parallel_{a_{2}}\ldots \parallel_{a_{n}} u_{n+1} \, \pi_{i}.$$
\item $t=\efq{P}{u}$. By induction hypothesis, $$u\mapsto^{*} u_{1}\parallel_{a_{1}} u_{2}\parallel_{a_{2}}\ldots \parallel_{a_{n}} u_{n+1}$$ and each term $u_{i}$, for $1\leq i\leq n+1$, is a simply typed $\lambda$-term. Applying $n$ times the permutations
we obtain $$t\mapsto^{*} \efq{P}{u_{1}}\parallel_{a_{1}} \efq{P}{u_{2}}\parallel_{a_{2}}\ldots \parallel_{a_{n}} \efq{P}{u_{n+1}}$$ QED \end{itemize} \end{proof}
\end{document} | arXiv |
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Martial Arts Anthropology for Sport Pedagogy and Physical Education
Wojciech J. CYNARSKI,Kazimierz OBODYNSKI,Howard Z. ZENG
Revista Romaneasca pentru Educatie Multidimensionala , 2012,
Abstract: The aim of this paper is to discuss the subject as well as the problem of corporeality and spirituality in the anthropology of martial arts. The authors attempt to show the vision of a new psychophysical education on the way of martial arts and the taking of personal patterns here. Analysies are made in the perspective of the holistic pedagogy and humanistic theory of martial arts.Qualitative methods, such as studying literature, direct interview and long-term participant observation were used. The authors wish to begin with the concept of corporeality as it is found in the available literature on the subject.The first author has been active in an environment of martial arts for over 30 years. Interviews and discussions were conducted with 9 martial arts masters of the highest rank.Results and conclusions. Psychophysical system of self-realization is an educational programme – a way which relates to spiritual development through physical and mental exercise, according to teaching by a particular master-teacher. Within the context of martial arts being used as a psycho-educational form of education, the body fulfills, above all, the role of a tool to be used on the way towards enlightenment and wisdom. It is utilized specifically in spiritual progress. Improving one's physical abilities is therefore an ascetic journey of physical perfectionism and technical accomplishment all towards achieving spiritual mastery. In some cases, spiritual development is described in terms of energy (qi, ki) and connected with the capacity of one's health.Traditional understanding of martial arts is often mixed with combat sports or systems of meditation are numbered among movement forms. The opportunity to avoid similar mistakes is to adopt a theoretical perspective of the anthropology of psychophysical progress. Paradigm of systematic approach and integral outlook on the human allow for understanding of the sense of being involved in ascetic and psychophysical practices.Utilitarian values of some martial arts causes, that the martial arts are very useful for PE curriculum. Falls and other exercises, elements of self defence techniques are beneficial for both for more safe teaching and future safe life.
Spontaneous focal activation of invariant natural killer T (iNKT) cells in mouse liver and kidney
Jia Zeng, Jonathan C Howard
BMC Biology , 2010, DOI: 10.1186/1741-7007-8-142
Abstract: We used an interferon (IFN)-γ-inducible cytoplasmic protein, Irga6, as a histological marker for local IFN-γ production. Irga6 was intensely expressed in small foci of liver parenchymal cells and kidney tubular epithelium. Focal Irga6 expression was unaffected by germ-free status or loss of TLR signalling and was totally dependent on IFN-γ secreted by T cells in the centres of expression foci. These were shown to be iNKT cells by diagnostic T cell receptor usage and their activity was lost in both CD1 d and Jα-deficient mice.This is the first report that supplies direct evidence for explicit activation events of NKT cells in vivo and raises issues about the triggering mechanism and consequences for immune functions in liver and kidney.Invariant natural killer-like T (iNKT) cells are placed ambiguously between adaptive and innate immune systems (reviewed in [1,2]). Derived from the thymus, expressing rearranged T-cell receptor (TCR) alpha and beta chains, they seem to belong to the adaptive immune system, while their receptor homogeneity, their continuous state of activation, their rapid secretion of large amounts of interferon (IFN)-γ and interleukin (IL)-4, their presumed recognition of invariant glycolipid self-ligands associated with the non-classical major histocompatibility (MHC) class I molecule, CD1 d, recall various aspects of innate immune recognition. Many features of iNKT cell behaviour are puzzling: their thymic ontogeny and relation to the classical pathways of T-cell mediated differentiation; the relative importance of endogenous and exogenous ligands in activation; and the polarity of their cytokine profile towards IFN-γ or IL-4 in relation to the activating ligand. However, in this report we address the basis for another characteristic of these enigmatic cells, namely their constitutive state of readiness to respond with massive cytokine production (reviewed in [3]). Using a sensitive endogenous reporter for IFN-γ production, we show that iNKT cells
Dynamic Monitoring of Plant Cover and Soil Erosion Using Remote Sensing, Mathematical Modeling, Computer Simulation and GIS Techniques [PDF]
Z. Y. Zeng, J. Z. Cao, Z. J. Gu, Z. L. Zhang, W. Zheng, Y. Q. Cao, H. Y. Peng
American Journal of Plant Sciences (AJPS) , 2013, DOI: 10.4236/ajps.2013.47180
Dynamic monitoring of plant cover and soil erosion often uses remote sensing data, especially for estimating the plant cover rate (vegetation coverage) by vegetation index. However, the latter is influenced by atmospheric effects and methods for correcting them are still imperfect and disputed. This research supposed and practiced an indirect, fast, and operational method to conduct atmospheric correction of images for getting comparable vegetation index values in different times. It tries to find a variable free from atmospheric effects, e.g., the mean vegetation coverage value of the whole study area, as a basis to reduce atmospheric correction parameters by establishing mathematical models and conducting simulation calculations. Using these parameters, the images can be atmospherically corrected. And then, the vegetation index and corresponding vegetation coverage values for all pixels, the vegetation coverage maps and coverage grade maps for different years were calculated, i.e., the plant cover monitoring was realized. Using the vegetation coverage grade maps and the ground slope grade map from a DEM to generate soil erosion grade maps for different years, the soil erosion monitoring was also realized. The results show that in the study area the vegetation coverage was the lowest in 1976, much better in 1989, but a bit worse again in 2001. Towards the soil erosion, it had been mitigated continuously from 1976 to 1989 and then to 2001. It is interesting that a little decrease of vegetation coverage from 1989 to 2001 did not lead to increase of soil erosion. The reason is that the decrease of vegetation coverage was chiefly caused by urbanization and thus mainly occurred in very gentle terrains, where soil erosion was naturally slight. The results clearly indicate the details of plant cover and soil erosion change in 25 years and also offer a scientific foundation for plant and soil conservation.
Microwave Anisotropies from Texture Seeded Structure Formation
R. Durrer,A. Howard,Z. -H. Zhou
Physics , 1993, DOI: 10.1103/PhysRevD.49.681
Abstract: The cosmic microwave anisotropies in a scenario of large scale structure formation with cold dark matter and texture are discussed and compared with recent observational results of the COBE satellite. A couple of important statistical parameters are determined. The fluctuations are slightly non gaussian. The quadrupole anisotropy is $1.5\pm 1.2\times 10^{-5}$ and the fluctuations on a angular scale of 10 degrees are $ (3.8\pm 2.6)\times 10^{-5}$. The COBE are within about one standard deviation of the typical texture + CDM model discussed in this paper. Furthermore, we calculate fluctuations on intermediate scales (about 2 degrees) with the result $\De T/T(\theta \sim 2^o) = 3.9\pm 0.8)\times 10^{-5}$. Collapsing textures are modeled by spherically symmetric field configurations. This leads to uncertainties of about a factor of~2.
Asymptotics of Reaction-Diffusion Fronts with One Static and One Diffusing Reactant
Martin Z. Bazant,Howard A. Stone
Physics , 1999, DOI: 10.1016/S0167-2789(00)00140-8
Abstract: The long-time behavior of a reaction-diffusion front between one static (e.g. porous solid) reactant A and one initially separated diffusing reactant B is analyzed for the mean-field reaction-rate density R(\rho_A,\rho_B) = k\rho_A^m\rho_B^n. A uniformly valid asymptotic approximation is constructed from matched self-similar solutions in a reaction front (of width w \sim t^\alpha where R \sim t^\beta enters the dominant balance) and a diffusion layer (of width W \sim t^{1/2} where R is negligible). The limiting solution exists if and only if m, n \geq 1, in which case the scaling exponents are uniquely given by \alpha = (m-1)/2(m+1) and \beta = m/(m+1). In the diffusion layer, the common ad hoc approximation of neglecting reactions is given mathematical justification, and the exact transient decay of the reaction rate is derived. The physical effects of higher-order kinetics (m, n > 1), such as the broadening of the reaction front and the slowing of transients, are also discussed.
Ground-state OH molecule in combined electric and magnetic fields: Analytic solution of the effective Hamiltonian
M. Bhattacharya,Z. Howard,M. Kleinert
Physics , 2013, DOI: 10.1103/PhysRevA.88.012503
Abstract: The OH molecule is currently of great interest from the perspective of ultracold chemistry, quantum fluids, precision measurement and quantum computation. Crucial to these applications are the slowing, guiding, confinement and state control of OH, using electric and magnetic fields. In this article, we show that the corresponding eight-dimensional effective ground state Stark-Zeeman Hamiltonian is exactly solvable and explicitly identify the underlying chiral symmetry. Our analytical solution opens the way to insightful characterization of the magnetoelectrostatic manipulation of ground state OH. Based on our results, we also discuss a possible application to the quantum simulation of an imbalanced Ising magnet.
A Dedicated Promoter Drives Constitutive Expression of the Cell-Autonomous Immune Resistance GTPase, Irga6 (IIGP1) in Mouse Liver
Jia Zeng, Iana Angelova Parvanova, Jonathan C. Howard
Abstract: Background In general, immune effector molecules are induced by infection. Methodology and Principal Findings However, strong constitutive expression of the cell-autonomous resistance GTPase, Irga6 (IIGP1), was found in mouse liver, contrasting with previous evidence that expression of this protein is exclusively dependent on induction by IFNγ. Constitutive and IFNγ-inducible expression of Irga6 in the liver were shown to be dependent on transcription initiated from two independent untranslated 5′ exons, which splice alternatively into the long exon encoding the full-length protein sequence. Irga6 is expressed constitutively in freshly isolated hepatocytes and is competent in these cells to accumulate on the parasitophorous vacuole membrane of infecting Toxoplasma gondii tachyzoites. Conclusions and Significance The role of constitutive hepatocyte expression of Irga6 in resistance to parasites invading from the gut via the hepatic portal system is discussed.
Phase Separation Dynamics in Isotropic Ion-Intercalation Particles
Yi Zeng,Martin Z. Bazant
Physics , 2013,
Abstract: Lithium-ion batteries exhibit complex nonlinear dynamics, resulting from diffusion and phase transformations coupled to ion intercalation reactions. Using the recently developed Cahn-Hilliard reaction (CHR) theory, we investigate a simple mathematical model of ion intercalation in a spherical solid nanoparticle, which predicts transitions from solid-solution radial diffusion to two-phase shrinking-core dynamics. This general approach extends previous Li-ion battery models, which either neglect phase separation or postulate a spherical shrinking-core phase boundary, by predicting phase separation only under appropriate circumstances. The effect of the applied current is captured by generalized Butler-Volmer kinetics, formulated in terms of diffusional chemical potentials, and the model consistently links the evolving concentration profile to the battery voltage. We examine sources of charge/discharge asymmetry, such as asymmetric charge transfer and surface "wetting" by ions within the solid, which can lead to three distinct phase regions. In order to solve the fourth-order nonlinear CHR initial-boundary-value problem, a control-volume discretization is developed in spherical coordinates. The basic physics are illustrated by simulating many representative cases, including a simple model of the popular cathode material, lithium iron phosphate (neglecting crystal anisotropy and coherency strain). Analytical approximations are also derived for the voltage plateau as a function of the applied current.
Research on the Software of Track Reconstruction in Vertex Chamber of BESII
Y. Zeng,Z. P. Mao
Abstract: The software of track reconstruction of the vertex chamber of BESII-VCJULI was studied when it was transplanted from the HP-Unix platform to PC-Linux. The problems of distinct dictionary storage and precision treatment in these two different platforms were found and settled in the modified software. Then the obvious differences of the candidate track number in an event and some track parameters caused by them were reduced from 74% to 0.02% and from 5% to 0.5%, respectively. As a result, the quality of the track finding was greatly improved and the CPU time saved.
Delocalization and conductance quantization in one-dimensional systems
Z. Y. Zeng,F. Claro
Physics , 2001, DOI: 10.1103/PhysRevB.65.193405
Abstract: We investigate the delocalization and conductance quantization in finite one-dimensional chains with only off-diagonal disorder coupled to leads. It is shown that the appearence of delocalized states at the middle of the band under correlated disorder is strongly dependent upon the even-odd parity of the number of sites in the system. In samples with inversion symmetry the conductance equals $2e^{2}/h$ for odd samples, and is smaller for even parity. This result suggests that this even-odd behaviour found previously in the presence of electron correlations may be unrelated to charging effects in the sample. | CommonCrawl |
Larry Guth
Lawrence David Guth (born 1977) is a professor of mathematics at the Massachusetts Institute of Technology.[1]
Larry Guth
Born
Lawrence David Guth
1977 (age 45–46)
NationalityAmerican
Alma mater
• Yale University
• MIT
Awards
• Salem Prize (2013)
• Clay Research Award (2015)
• New Horizons in Mathematics Prize (2016)
• Bôcher Memorial Prize (2020)
• Maryam Mirzakhani Prize (2020)
Scientific career
FieldsMathematics
Institutions
• Stanford
• University of Toronto
• New York University
• MIT
ThesisArea-contracting maps between rectangles (2005)
Doctoral advisorTomasz Mrowka
Websitemath.mit.edu/~lguth/
Education and career
Guth graduated from Yale in 2000, with BS in mathematics.[2]
In 2005, he got his PhD in mathematics from the Massachusetts Institute of Technology, where he studied geometry of objects with random shapes under the supervision of Tomasz Mrowka.[3][4]
After MIT, Guth went to Stanford as a postdoc, and later to the University of Toronto as an Assistant Professor.[5]
In 2011, New York University's Courant Institute of Mathematical Sciences hired Guth as a professor, listing his areas of interest as "metric geometry, harmonic analysis, and geometric combinatorics."[5]
In 2012, Guth moved to MIT, where he is Claude Shannon Professor of Mathematics.[6]
Research
In his research, Guth has strengthened Gromov's systolic inequality for essential manifolds[7] and, along with Nets Katz, found a solution to the Erdős distinct distances problem.[8] His wide-ranging interests include the Kakeya conjecture and the systolic inequality.
Recognition
Guth won an Alfred P. Sloan Fellowship in 2010.[9] He was an invited speaker at the International Congress of Mathematicians in India in 2010, where he spoke about systolic geometry.[10][11]
In 2013, the American Mathematical Society awarded Guth its annual Salem Prize, citing his "major contributions to geometry and combinatorics."[12]
In 2014 he received a Simons Investigator Award.[13] In 2015, he received the Clay Research Award.[14]
He was included in the 2019 class of fellows of the American Mathematical Society "for contributions to harmonic analysis, combinatorics and geometry, and for exposition of high level mathematics".[15]
On February 20, 2020, the National Academy of Sciences announced that Guth is the first winner of their new $20,000 Maryam Mirzakhani Prize in Mathematics for mid-career mathematicians. The citation states that his award is "for developing surprising, original, and deep connections between geometry, analysis, topology, and combinatorics, which have led to the solution of, or major advances on, many outstanding problems in these fields."[16][17] In 2021, he was elected member of the US National Academy of Sciences.[18]
Personal
He is the son of Alan Guth, a physicist known for the theory of inflation in cosmology.[4]
Work
• Metaphors in systolic geometry: the video
• Guth, Larry (2011), "Volumes of balls in large Riemannian manifolds", Annals of Mathematics, 2nd ser., 173 (1): 51–76, arXiv:math.DG/0610212, doi:10.4007/annals.2011.173.1.2, MR 2753599, S2CID 1392012.
• Guth, Larry; Katz, Nets Hawk (2010), "Algebraic methods in discrete analogs of the Kakeya problem", Advances in Mathematics, 225 (5): 2828–2839, arXiv:0812.1043, doi:10.1016/j.aim.2010.05.015, MR 2680185, S2CID 15590454.
• Guth, Larry (2010), "Systolic inequalities and minimal hypersurfaces", Geometric and Functional Analysis, 19 (6): 1688–1692, arXiv:0903.5299, doi:10.1007/s00039-010-0052-0, MR 2594618, S2CID 17827200.
• Guth, Larry (2010), "The endpoint case of the Bennett–Carbery–Tao multilinear Kakeya conjecture", Acta Mathematica, 205 (2): 263–286, arXiv:0811.2251, doi:10.1007/s11511-010-0055-6, MR 2746348, S2CID 16258342.
• Guth, Larry (2009), "Minimax problems related to cup powers and Steenrod squares", Geometric and Functional Analysis, 18 (6): 1917–1987, arXiv:math/0702066, doi:10.1007/s00039-009-0710-2, MR 2491695, S2CID 10402235.
• Guth, Larry (2008), "Symplectic embeddings of polydisks", Inventiones Mathematicae, 172 (3): 477–489, arXiv:0709.1957, Bibcode:2008InMat.172..477G, doi:10.1007/s00222-007-0103-9, MR 2393077, S2CID 18065526.
• Guth, Larry (2007), "The width-volume inequality", Geometric and Functional Analysis, 17 (4): 1139–1179, arXiv:math/0609569, doi:10.1007/s00039-007-0628-5, MR 2373013, S2CID 16014518.
• Guth, Larry; Katz, Nets Hawk (2015), "On the Erdős distinct distance problem on the plane", Annals of Mathematics, 181 (1): 155–190, arXiv:1011.4105, doi:10.4007/annals.2015.181.1.2, MR 3272924, S2CID 43051852
• Guth, Larry (2016). Polynomial Methods in Combinatorics. American Mathematical Society. ISBN 978-1-4704-2890-7.[19]
References
1. https://math.mit.edu/directory/profile.php?pid=1461 MIT Math Department profile of Larry Guth
2. "Curriculum Vitae Larry Guth" (PDF). MIT Mathematics Department. 2020. Retrieved February 20, 2020. B.S. Mathematics, Yale University, 2000
3. Lawrence Guth at the Mathematics Genealogy Project.
4. Knight, Helen (May 27, 2014). "Like father, like son". MIT News. Retrieved February 20, 2020. Guth moved to MIT as a graduate student, where he began studying geometry under the supervision of mathematics professor Tomasz Mrwoka.
5. "New Faculty: Larry Guth" (PDF). Courant Institute of Mathematical Sciences at NYU. 2011. Retrieved February 20, 2020. He was a postdoc at Stanford and an Assistant Professor at the University of Toronto. He received a Sloan fellowship in 2010.
6. "Larry Guth". MIT Mathematics Department. 2020. Retrieved February 20, 2020. In 2020, Larry received the Bôcher Memorial Prize of the AMS, for his "deep and influential development of algebraic and topological methods for partitioning the Euclidean space and multi-scale organization of data, and his powerful applications of these tools in harmonic analysis, incidence geometry, analytic number theory, and partial differential equations." Larry wrote about this technique in his book "Polynomial Methods in Combinatorics." He also received the newly named Maryam Mirzakhani Prize in Mathematics (formerly the NAS Award in Mathematics)
7. Guth's approach to Gromov's systolic inequality, Shmuel Weinberger, July 18, 2009.
8. "Distinct Distance Problem in the Plane" Solved, Math In The News, Mathematical Association of America, March 2, 2011.
9. "February 19, 2010 — Five U of T scientists awarded prestigious Sloan Fellowships — Faculty of Arts & Science". Archived from the original on June 16, 2012. Retrieved October 23, 2011.
10. Fall newsletter 2010, Univ. of Toronto mathematics department, retrieved May 26, 2011.
11. ICM listing of invited speakers Archived July 17, 2011, at the Wayback Machine, retrieved May 26, 2011.
12. "Guth Awarded 2013 Salem Prize" (PDF). Notices of the AMS. 2014. Retrieved February 20, 2020. Lawrence Guth of the Massachusetts Institute of Technology has been awarded the 2013 Salem Prize for his "major contributions to geometry and combinatorics. His brilliant insights led to the solution of old problems and the introduction of powerful new techniques," according to the prize citation.
13. "Simons Investigator Awardees". Simons Foundation. Archived from the original on August 6, 2017. Retrieved September 11, 2017.
14. Clay Research Award 2015
15. 2019 Class of the Fellows of the AMS, American Mathematical Society, retrieved November 7, 2018
16. @theNASciences (February 20, 2020). "The inaugural recipient for the Maryam Mirzakhani Prize in Mathematics, Larry Guth of @MIT" (Tweet). Retrieved February 20, 2020 – via Twitter.
17. "2020 Maryam Mirzakhani Prize in Mathematics". NAS. February 20, 2020. Retrieved February 20, 2020. Guth is receiving the $20,000 prize 'for developing surprising, original, and deep connections between geometry, analysis, topology, and combinatorics, which have led to the solution of, or major advances on, many outstanding problems in these fields.' The Mirzakhani prize honors exceptional contributions to the mathematical sciences by a mid-career mathematician
18. "News from the National Academy of Sciences". April 26, 2021. Retrieved July 2, 2021. Newly elected members and their affiliations at the time of election are: ... Guth, Larry; Claude Shannon Professor, department of mathematics, Massachusetts Institute of Technology, Cambridge
19. Tao, Terence (2018). "Review of Polynomial methods in combinatorics by Larry Guth". 55: 103–107. doi:10.1090/bull/1586. {{cite journal}}: Cite journal requires |journal= (help)
Authority control
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| Wikipedia |
\begin{document}
\title{An inequality between finite analogues of rank and crank moments}
\author{Pramod Eyyunni, Bibekananda Maji and Garima Sood}\thanks{2010 \textit{Mathematics Subject Classification.} Primary 11P80, 11P81, 11P82; Secondary 05A17\\ \textit{Keywords and phrases.} partitions, finite analogues, smallest parts function, moments inequality, symmetrized moments} \address{Discipline of Mathematics, Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar 382355, Gujarat, India} \email{[email protected], [email protected]\\[email protected]} \maketitle \begin{center} {\it Dedicated to Professor Bruce C. Berndt on the occasion of his 80th birthday} \end{center} \begin{abstract}
The inequality between rank and crank moments was conjectured and later proved by Garvan himself in 2011. Recently, Dixit and the authors introduced finite analogues of rank and crank moments for vector partitions while deriving a finite analogue of Andrews' famous identity for smallest parts function. In the same paper, they also conjectured an inequality between finite analogues of rank and crank moments, analogous to Garvan's conjecture. In the present paper, we give a proof of this conjecture.
\end{abstract}
\section{Introduction} Let $p(n)$ denote the number of unrestricted partitions of a positive integer $n$. To give a combinatorial explanation of the famous congruences of Ramanujan for the partition function $p(n)$, namely, for $ m\geq 0,$ \begin{align*} p(5m + 4) & \equiv 0 \pmod 5,\\ p(7m + 5) & \equiv 0 \pmod 7, \end{align*} Dyson \cite{dys} defined the rank of a partition as the largest part minus the number of parts. He also conjectured that there must be another statistic, which he named `crank', that would explain Ramanujan's third congruence, namely,
\begin{equation*}
p(11m + 6) \equiv 0 \pmod{11}.
\end{equation*}
After a decade, Atkin and Swinnerton-Dyer \cite{atkin2} confirmed Dyson's observations for the first two congruences for $p(n)$. Also, in 1988, `crank' was discovered by Andrews and Garvan \cite{andrewsgarvan88}. An interesting thing to note is that by using the partition statistic `crank', Andrews and Garvan were able to explain not only the third congruence but also the first two. Atkin and Garvan \cite{atkin1} found that the moments of ranks and cranks were important in the study of further partition congruences. In particular, they defined the ${k}^{th}$ moments of rank and crank, respectively as, \begin{align*} N_{k}(n)=\sum_{m=-\infty}^{\infty} m^{k} N(m,n),\\ M_{k} (n) = \sum_{m=-\infty}^{\infty}m^{k} M(m, n), \end{align*} where $N(m,n)$ and $M(m,n)$ denote the number of partitions of $n$ with rank $m$ and crank $m$ respectively. In 2008, Andrews \cite{andrews08} introduced the smallest parts function $\mathrm{spt}(n)$ as the total number of appearances of the smallest parts in all partitions of $n$ and showed that \begin{equation*} \text{spt}{(n)}=np(n)-\frac{1}{2}N_2(n). \end{equation*}
Using Dyson's identity \cite[Theorem 5]{dyson89}, i.e., $np(n)=\frac{1}{2}M_2(n) $, we can rewrite this as \begin{equation}\label{Andrews_spt_identity} \text{spt}{(n)}=\frac{1}{2}M_2(n)-\frac{1}{2}N_2(n). \end{equation} From this result, it is immediate that $M_2(n)>N_2(n)$. Garvan \cite[Conjecture (1.1)]{garvan10} conjectured that \begin{equation}\label{garvan's conjecture} M_{2k} (n)>N_{2k} (n), \end{equation} for all $k>1$ and $n\geq1$. Studying the asymptotic behavior of the difference $M_{2k}(n)-N_{2k}(n)$, Bringmann and Mahlburg \cite{BM09} proved \eqref{garvan's conjecture} for $k=2,4$, and subsequently, for each fixed $k$, the inequality was proved for sufficiently large $n$ by Bringmann, Mahlburg and Rhoades \cite{BMR11}. Later, Garvan \cite{garvan10} himself proved his conjecture for all $n$ and $k$ with the help of a combinatorial interpretation for the difference between symmetrized crank and rank moments. Andrews \cite{andrews07} defined the $k^{th}$ symmetrized rank moment as \begin{equation*} \eta_{k}(n):=\sum_{m=-n}^{n} \left(\begin{matrix} m+ \lfloor \frac{k-1}{2} \rfloor \\ k\end{matrix}\right)N(m,n). \end{equation*} Andrews \cite[Theorem 2]{andrews07} showed that the odd moments are all identically zero and also obtained the generating function for even moments $\eta_{2k}(n)$, that is, for any $k \geq 1$, we have \begin{align} \sum_{n=1}^\infty \eta_{2k}(n) q^n & = \frac{1}{(q)_\infty} \sum_{n=1}^\infty \frac{(-1)^{n-1}(1+q^n) q^{\frac{n(3n-1)}{2}+kn}}{(1-q^n)^{2k}} \label{Andrews_genfn_symmetrized_rank}\\ & = \frac{1}{(q)_\infty} \sum_{\substack{ n=-\infty\\ n\neq 0}}^\infty \frac{(-1)^{n-1} q^{\frac{n(3n+1)}{2}+kn}}{(1-q^n)^{2k}}.\label{andrews_symm_rank2} \end{align} Analogous to the symmetrized rank moments $\eta_k(n)$, Garvan \cite{garvan11} introduced the $k^{th}$ symmetrized crank moment $\mu_k(n)$ in the study of the higher order $\text{spt}$-function $\text{spt}_k(n)$. To be more specific, \begin{align*}{} \mu_{k}(n):=\sum_{m=-n}^{n} \left(\begin{matrix} m+ \lfloor \frac{k-1}{2} \rfloor \\ k\end{matrix}\right)M(m,n). \end{align*} Analogous to \eqref{Andrews_genfn_symmetrized_rank} and \eqref{andrews_symm_rank2}, the generating function for the symmetrized crank moments was given by Garvan \cite[Theorem (2.2)]{garvan11}, that is,
for any $k \geq 1$, we have \begin{align} \sum_{n=1}^{\infty} \mu_{2k}(n) q^n & = \frac{1}{(q)_\infty } \sum_{n=1}^\infty \frac{(-1)^{n-1}(1+q^n) q^{\frac{n(n-1)}{2}+kn}}{(1-q^n)^{2k}} \label{garvan_symmcrank_1}\\ & = \frac{1}{(q)_\infty} \sum_{\substack{ n=-\infty\\ n\neq 0}}^\infty \frac{(-1)^{n-1} q^{\frac{n(n+1)}{2}+kn}}{(1-q^n)^{2k}}.\label{garvan_symmcrank_2} \end{align} Garvan \cite[Equation (1.4)]{garvan11} also gave the following generating function for the symmetrized crank moments: \begin{align}\label{mugf_alternate} \sum_{n=1}^{\infty} \mu_{2k}(n) q^n = \frac{1}{(q)_\infty }\sum_{ n_k\geq \cdots \geq n_2\geq1}\frac{q^{n_1+n_2+ \cdots +n_k} }{(1-q^{n_1})^2(1-q^{n_2})^2 \cdots (1-q^{n_k})^2}. \end{align} One of the main results in \cite{garvan11}, due to Garvan \cite[Equation (1.3)]{garvan11}, which was instrumental in proving the inequality between rank and crank moments is as follows: \begin{align}\label{difference_gen_garvan} \sum_{n=1}^{\infty}( \mu_{2k}(n)- \eta_{2k}(n)) q^n = \sum_{ n_k\geq \cdots \geq n_2\geq1}\frac{q^{n_1+n_2+ \cdots +n_k} }{(1-q^{n_1})^2(1-q^{n_2})^2 \cdots (1-q^{n_k})^2 (q^{n_1+1} ; q)_\infty}, \end{align} for any $k \geq 1$.
One can easily check that for $k=1$, the above theorem reduces to \eqref{Andrews_spt_identity}. After this observation, Garvan defined higher order $\text{spt}$-function $\text{spt}_k(n)$ as \begin{equation*} \text{spt}_k(n) := \mu_{2k}(n)-\eta_{2k}(n), \end{equation*} for all $k\geq1$ and $n\geq1$. He also gave a combinatorial interpretation of $\text{spt}_{k}(n)$.
In the next subsection, we shall describe recent developments related to Andrews' identity \eqref{Andrews_spt_identity} for the smallest parts function $\text{spt}(n)$. \subsection{Finite analogue of Andrews' spt-identity} Ramanujan's identities are a constant source of inspiration for everyone and motivate us to do beautiful mathematics. Recently, Dixit and Maji \cite{dixitmaji18} found a generalization of a $q$-series identity \cite[p.~354]{ramanujantifr}, \cite[p. 263, Entry 3]{bcbramforthnote} of Ramanujan and derived many partition theoretic implications from this generalization. They also established a new identity \cite[Theorem 2.8]{dixitmaji18} involving Fine's function $F(a,b;t)$ \cite[p.~1]{fine} from which they were able to derive Andrews' identity \eqref{Andrews_spt_identity} for $\text{spt}(n)$. Very recently, together with Dixit, the authors found a finite analogue \cite[Theorem 1.1]{DEMS} of the aforementioned generalization of Dixit and Maji \cite[Theorem 2.1]{dixitmaji18}, whose special case gave a finite analogue of Andrews' $\text{spt}$-identity, namely, \begin{theorem}\cite[Theorem 2.4]{DEMS}\label{finite analogue_Andrews_spt}
For any natural numbers $n, N$, we have
\begin{align*}
\textup{spt}(n, N)= \frac{1}{2} \left( M_{2,N}(n) - N_{2,N}(n) \right),
\end{align*}
where $\textup{spt}(n, N )$ is the number of smallest parts in all partitions of $n$ whose corresponding largest parts are less than or equal to $N$, and $M_{2,N}(n)$ and $N_{2,N}(n)$ are defined below. \end{theorem} In \cite[p.~9, Equations (2.9), (2.10)]{DEMS}, for $k\geq 1$, we defined finite analogues of rank and crank moments for vector partitions as \begin{align} N_{k, N}(n) &:= \sum_{m=-\infty}^{\infty} m^k N_{S_1}(m, n),\label{finrankmom}\\ M_{k,N}(n)&:= \sum_{m=-\infty}^{\infty} m^k M_{S_2}(m, n),\label{fincrankmom} \end{align} where $N_{S_1}(m, n)$ and $M_{S_2}(m, n)$ are defined below in \eqref{ns1} and \eqref{ms1} respectively. From Theorem \ref{finite analogue_Andrews_spt}, it is immediate that $M_{2,N}(n) > N_{2,N}(n)$. Analogous to Garvan's conjecture \eqref{garvan's conjecture}, we gave the following conjecture on the inequality between the finite analogues of $k^{\textup{th}}$ rank and crank moments, that is, \begin{conjecture}\cite[Conjecture 10.1]{DEMS}\label{finiteanalogcrankrankconjecture} For any fixed natural number $N$ and even $k\geq 2$, \begin{equation*} M_{k,N}(n) > N_{k,N}(n) \quad \mathrm{for \,\, all} \,\, n\geq 1. \end{equation*} \end{conjecture} In the present paper, our main goal is to prove the above conjecture. We have already mentioned in this introduction that the theory of symmetrized rank and crank moments was developed by Andrews \cite{andrews07} and Garvan \cite{garvan11} respectively. Here, to prove the above conjecture we define finite analogues of symmetrized rank and crank moments and their generating functions. We follow similar techniques as employed by Garvan \cite{garvan11}.
\section{Main Results: Finite analogues of the $k^{th}$ symmetrized rank and crank moments}\label{S2} Before defining finite analogues of symmetrized rank and crank moments we need to recall certain definitions from \cite[p.~7]{DEMS}. For the sake of completeness we reproduce them below.
Let $V_{1}=\mathcal{D}\times\mathcal{P}$ denote a set of vector partitions. So an element $\vec{\pi}$ of $V_1$ is of the form $(\pi_1, \pi_2)$, where the magnitude of $\vec{\pi}$ is
given by $|\vec{\pi}|:=|\pi_1|+|\pi_2|$. Let $N$ be a positive integer. Then for any positive integer $j$ with $1\leq j \leq N$, set \begin{align*}\label{S1}
S_{1}&:=\bigg\{\vec{\pi}\in V_1: \pi_1\hspace{0.5mm}\text{is either an empty partition or such that its parts lie in $[N-j+1, N]$ }\nonumber \\ & \quad\quad \text{and}\,\, \pi_2\hspace{1mm}\text{is an } \text{unrestricted partition with Durfee square of size $j$}\bigg\}. \end{align*} For a vector partition $\vec{\pi}=(\pi_1, \pi_2)$ in $V_1$, let $w_r(\vec{\pi}) := (-1)^{\#(\pi_1)}$ be its weight and $\textup{rank}(\vec{\pi}):=\textup{rank}(\pi_2)$, its vector rank. Now define \begin{equation}\label{ns1} N_{S_1}(m, n):=\sum_{j=1}^N N_{S_{1}}\left(m, n; \boxed{j}\right), \end{equation} where \begin{equation*}
N_{S_{1}}\left(m, n; \boxed{j}\right):= \sum_{\vec{\pi} \in S_{1}, |\vec{\pi}|=n \atop \mathrm{rank}(\vec{\pi})=m} w_r(\vec{\pi}). \end{equation*} As observed in \cite{DEMS}, as $N \rightarrow \infty$, $N_{S_1}(m, n)$ equals $N(m, n)$, the number of ordinary partitions of $n$ with rank $m$.
We are now ready to define the finite analogue of the $k^{th}$ symmetrized rank function. Let $k, N$ be positive integers. Then for any $n \geq 1$, \begin{equation}\label{FR} \eta_{k,N}(n):=\sum_{m=-n}^{n} \left(\begin{matrix} m+ \lfloor \frac{k-1}{2} \rfloor \\ k\end{matrix}\right)N_{S_1}(m,n). \end{equation} \begin{proposition} Let $N$ be a positive integer and $k$ be an odd positive integer. Then $\eta_{k,N}(n)=0$ for all $n\geq1$. \end{proposition} This is straightforward from the fact that the finite analogues of all the odd rank moments $N_{k, N}(n)$ are zero. We now give an expression for the generating function of $\eta_{k,N}(n)$ for \emph{even} $k$. \begin{theorem}\label{gffinsymmetrizedrankfn} Let $N \in \mathbb{N}$. Then for any positive integer $\nu$, we have \begin{align} \sum_{n=1}^{\infty}\eta_{2\nu,N}(n)q^n&=(q)_N \sum_{\substack {n=1}}^{N}\frac{(-1)^{n-1} q^{\frac{n(3n-1)}{2}+\nu n}(1+q^n) }{(q)_{N+n}(q)_{N-n}(1-q^{n})^{2\nu}} \label{gf_symm_rank2}\\
&=(q)_N\sum_{\substack {n=-N\\n\neq 0}}^{N}\frac{(-1)^{n-1} q^{\frac{n(3n+1)}{2}+\nu n} }{(q)_{N+n}(q)_{N-n}(1-q^n)^{2\nu}}. \label{gfsymrankmoment} \end{align} \end{theorem} Letting $N \rightarrow \infty$, we obtain the generating functions for the symmetrized rank moment, namely, \eqref{Andrews_genfn_symmetrized_rank} and \eqref{andrews_symm_rank2}. Next, we are going to define the finite analogue of the symmetrized crank moments. Again, for convenience, we recollect some definitions from \cite[p.~8-9]{DEMS}.
Let $V_{2}$ denote the set of vector partitions $\mathcal{D}\times\mathcal{P}\times\mathcal{P}$. Denote an element $\vec{\pi}$ of $V_2$ by
$(\pi_1, \pi_2, \pi_3)$ so that the magnitude of $\vec{\pi}$ is $|\vec{\pi}|=|\pi_1|+|\pi_2|+|\pi_3|$.
For any positive integer $N$, we define the following set: \begin{align*}{} S_2:=\{\vec{\pi}\in V_2 : l(\pi_1), l(\pi_2), l(\pi_3) \leq N \}. \end{align*} Define $w_c(\vec{\pi}):=(-1)^{\#(\pi_1)}$ to be the weight of the vector partition $\vec{\pi}=(\pi_1, \pi_2, \pi_3)$ and crank$(\vec{\pi}):= \#(\pi_2)-\#(\pi_3)$ be its vector crank. We define \begin{equation}\label{ms1}
M_{S_2}(m, n) := \sum_{\vec{\pi} \in S_2, |\vec{\pi}|=n \atop \mathrm{crank}(\vec{\pi})=m} w_c(\vec{\pi}). \end{equation} Letting $N \rightarrow \infty$ we see that $S_2$ approaches the whole set $V_2$ and consequently $M_{S_2}(m, n)$ approaches
$\sum_{\vec{\pi} \in V_2, |\vec{\pi}|=n \atop \mathrm{crank}(\vec{\pi})=m} w_c(\vec{\pi})$, which is the total number of weighted vector partitions of $n$ with vector crank $m$, a quantity first studied by Garvan (See \cite[p.~50]{garvan88}). By the work of Andrews and Garvan \cite[Theorem 1]{andrewsgarvan88}, we know that this equals $M(m, n)$, the number of integer partitions of $n$ with crank $m$.
We now define a finite analogue of the $k^{th}$ symmetrized crank moment. Let $k, N$ be positive integers. Then for any $n \geq 1$, \begin{equation}\label{SCM} \mu_{k,N}(n):=\sum_{m=-n}^{n} \left(\begin{matrix} m+ \lfloor \frac{k-1}{2} \rfloor \\ k\end{matrix}\right)M_{S_2}(m,n). \end{equation}
\begin{proposition} For any odd positive integer $k$, we have $\mu_{k,N}(n)=0$ for all $n\geq1$. \end{proposition} This easily follows because all the odd crank moments $M_{k, N}(n)$ are zero. Analogous to Theorem \ref{gffinsymmetrizedrankfn} above, we derive the following result for the generating function of $\mu_{k, N}(n)$ for \emph{even} $k$.
\begin{theorem}\label{theoremmugf}
Let $N \in \mathbb{N}$. Then for any positive integer $\nu$, one has \begin{align} \sum_{n=1}^{\infty}\mu_{2\nu,N}(n)q^n &=(q)_N \sum_ {n=1}^{N}\frac{(-1)^{n-1} q^{\frac{n(n-1)}{2}+\nu n}(1+q^n)}{(q)_{N+n}(q)_{N-n}(1-q^n)^{2\nu}} \label{mugf} \\
&= (q)_N\sum_{\substack {n=-N\\n\neq 0}}^{N}\frac{(-1)^{n-1} q^{\frac{n(n+1)}{2}+\nu n} }{(q)_{N+n}(q)_{N-n}(1-q^n)^{2\nu}} \label{fin_symm_crank_2} \end{align} \end{theorem} One can easily observe that this result is a finite analogue of the equations \eqref{garvan_symmcrank_1} and \eqref{garvan_symmcrank_2} by letting $N\rightarrow \infty$. The next result provides us information about the generating function of the difference between finite analogues of symmetrized crank and rank moments. \begin{theorem}\label{difference_gen_fin_rank_crank} Let $N \in \mathbb{N}$. Then for any positive integer $k$, we have \begin{equation}\label{gen_fin_rank_crank} \sum_{n=1}^{\infty}(\mu_{2k,N}(n)-\eta_{2k,N}(n))q^n=\frac{1}{(q)_N}\sum_{N\geq n_k\geq ...\geq n_1\geq1}\frac{(q)_{n_1}q^{n_1+n_2+...+n_k}}{(1-q^{n_1})^2(1-q^{n_2})^2...(1-q^{n_k})^2}. \end{equation} \end{theorem} This is a finite analogue of Garvan's result \eqref{difference_gen_garvan} for the generating function of the difference between symmetrized crank and rank moments. \begin{remark}\label{remark_fin_higher_spt} If we substitute $k=1$ in the above result, then we can obtain Theorem \textup{\ref{finite analogue_Andrews_spt}}. Thus we have $\mu_{2,N}(n)-\eta_{2,N}(n)=\textup{spt}(n,N)$. This suggests us to define a finite analogue of higher order spt-function as $\textup{spt}_k(n,N):= \mu_{2k,N}(n)-\eta_{2k,N}(n)$. \end{remark} The remainder of this paper is organized as follows. In the next section we collect all necessary results which will be useful throughout the paper. The generating functions of the finite analogues of the symmetrized rank and crank moments are proved in Section \ref{main_theorems}. In Section \ref{baily_pairs_proof_conj}, we derive important results using Bailey's lemma and give a proof of Conjecture \ref{finiteanalogcrankrankconjecture}. We conclude the paper, by discussing further questions in Section 6. \section{Preliminaries} In \cite[Theorem 2.2]{DEMS}, Dixit et al. noted that the generating function of $N_{S_1}(m, n)$ is \begin{align}\label{RS1zqdefinition} R_{S_1}(z;q):=\sum_{n=1}^{\infty}\sum_{m=-\infty}^{\infty}N_{S_1}(m,n)z^mq^n=\sum_{j=1}^N \left[\begin{matrix} N \\ j \end{matrix}\right] \frac{q^{j^2} (q)_j }{(z q)_j (z^{-1} q)_j}. \end{align} We call \eqref{RS1zqdefinition} as the finite analogue of the rank generating function, for, letting $N \rightarrow \infty$ on both sides, gives the well-known result for the rank generating function (for more details, see \cite[p. 8]{DEMS}), \begin{equation*} \sum_{n=1}^{\infty} \sum_{m=-\infty}^{\infty} N\left(m, n\right) z^m q^n = \sum_{j=1}^\infty \frac{q^{j^2} }{(z q)_j (z^{-1} q)_j}. \end{equation*} Again, in \cite[p.~252, Theorem 2.1]{andpar}, \cite[Equation (12.2.2), p.~263]{abramlostI}, Andrews showed that \begin{align}\label{frgfbil} \sum_{n=0}^{N}\left[\begin{matrix} N \\ n \end{matrix}\right]\frac{(q)_nq^{n^2}}{(zq)_n(z^{-1}q)_n}=\frac{1}{(q)_N}+(1-z)\sum_{n=1}^{N}\left[\begin{matrix} N \\ n \end{matrix}\right]\frac{(-1)^n(q)_nq^{n(3n+1)/2}}{(q)_{N+n}}\left(\frac{1}{1-zq^n}-\frac{1}{z-q^n}\right). \end{align} Now we recall the crank generating function, that is, \begin{equation}\label{crank_gen_func} \frac{(q)_{\infty}}{(zq)_\infty(z^{-1}q)_\infty} = \sum_{n=0}^{\infty} \sum_{m=-\infty}^{\infty} M(m, n) z^m q^n, \end{equation} where $M(m,n)$ is the number of partitions of $n$ with crank $m$.
In \cite[Theorem 2.3]{DEMS}, Dixit et al. proved that the generating function of $M_{S_2}(m,n)$ is \begin{equation}\label{fincrankgf}
C_{S_2}(z;q) := \sum_{n=0}^{\infty}\sum_{m=-\infty}^{\infty}M_{S_2}(m,n)z^mq^n= \frac{(q)_N}{(zq)_N (z^{-1}q)_N}, \end{equation} which is the finite analogue of \eqref{crank_gen_func}. Andrews \cite[p.~258, Theorem 4.1]{andpar} showed that \begin{equation}\label{andrewsCS2} \frac{(q)_{N}}{(zq)_{N}(z^{-1}q)_{N}}=\frac{1}{(q)_N}+(1-z)\sum_{n=1}^{N}\left[\begin{matrix} N \\ n \end{matrix}\right]\frac{(-1)^n(q)_nq^{n(n+1)/2}}{(q)_{n+N}}\left(\frac{1}{1-zq^n}-\frac{1}{z-q^n}\right). \end{equation} Now we collect some useful facts about Bailey pairs, see \cite[p. 582]{andrewsaskey99}. A pair of sequences $(\alpha_n(a, q), \beta_n(a, q))$ is called a Bailey pair with parameters $(a, q)$ if, for each non-negative integer $n$, \begin{equation}\label{Baileydefn} \beta_n(a, q) = \sum_{r=0}^{n}\frac{\alpha_r(a, q)}{(q;q)_{n-r} (aq;q)_{n+r}}. \end{equation} \begin{theorem}[Bailey's Lemma]\label{Baileylemma} Suppose $(\alpha_n(a, q), \beta_n(a, q))$ is a Bailey pair with parameters $(a, q)$. Then $(\alpha'_n(a, q), \beta^{'}_n(a, q))$ is another Bailey pair with parameters $(a, q)$, where $$ \alpha'_n(a, q)=\frac{(\rho_1, \rho_2;q)_n}{(aq/\rho_1, aq/\rho_2;q)_n}\left(\frac{aq}{\rho_1\rho_2} \right)^n \alpha_n(a, q) $$ and $$ \beta^{'}_n(a, q)=\sum_{k=0}^{n}\frac{(\rho_1, \rho_2;q)_k (aq/\rho_1\rho_2;q)_{n-k}}{(aq/\rho_1, aq/\rho_2;q)_n (q;q)_{n-k}}\left(\frac{aq}{\rho_1 \rho_2}\right)^k\beta_k(a, q). $$ \end{theorem} We also require the following result: \begin{equation}\label{limitresultp1p2} \lim_{\rho_2 \rightarrow 1} \lim_{\rho_1 \rightarrow 1}\frac{1}{(1-\rho_1)(1-\rho_2)}\left( 1-\frac{(q)_k (q/\rho_1 \rho_2)_k}{(q/\rho_1)_k(q/\rho_2)_k}\right)=\sum_{j=1}^{k}\frac{q^j}{(1-q^j)^2}. \end{equation}
\section{Proofs of Theorem \ref{gffinsymmetrizedrankfn} and Theorem \ref{theoremmugf} }\label{main_theorems} \begin{proof}[Theorem \textup{\ref{gffinsymmetrizedrankfn}}][] By definition \eqref{FR} of $\eta_{k, N}(n)$, we know that \begin{equation*} \eta_{2\nu,N}(n):=\sum_{m=-n}^{n} \left(\begin{matrix} m+ \nu-1\\ 2\nu\end{matrix}\right)N_{S_1}(m,n). \end{equation*} From the definition \eqref{RS1zqdefinition} of $R_{S_1}(z;q)$, it follows at once that \begin{equation*}
\left(\frac{d^{2\nu}}{dz^{2\nu}}z^{\nu-1}R_{S_1}(z;q)\right)\Big|_{z=1} = \sum_{n=1}^{\infty}\sum_{m=-\infty}^{\infty}(m+\nu-1)(m+\nu-2)\cdots(m-\nu+1)(m-\nu)N_{S_1}(m,n)q^n. \end{equation*} In other words, \begin{equation*}
\left(\frac{d^{2\nu}}{dz^{2\nu}}z^{\nu-1}R_{S_1}(z;q)\right)\Big|_{z=1} = (2\nu)!\sum_{n=1}^{\infty}\eta_{2\nu,N}(n)q^n. \end{equation*} Using Leibniz's chain rule, we get \begin{align}\label{LBR} \sum_{n=1}^{\infty}\eta_{2\nu,N}(n)q^n=\frac{1}{(2\nu)!}\sum_{j=0}^{\nu-1}\left(\begin{matrix} 2\nu\\ j\end{matrix}\right)(\nu-1)(\nu-2)...(\nu-j)R_{S_1}^{(2\nu-j)}(1;q). \end{align} It will be sufficient for us to find the derivatives of $R_{S_1}(z;q)$ with respect to $z$. To this end, we wish to write $R_{S_1}(z;q)$ in a suitable form. Using \eqref{frgfbil} in the right-most expression of \eqref{RS1zqdefinition}, we deduce that \begin{align*} R_{S_1}(z;q)&= \frac{1}{(q)_N} - 1 + (1-z)\sum_{n=1}^{N}\left[\begin{matrix} N \\ n \end{matrix}\right]\frac{(-1)^n(q)_nq^{n(3n+1)/2}}{(q)_{N+n}}\left(\frac{1}{1-zq^n}-\frac{1}{z-q^n}\right) \\
&= -1 + \frac{1}{(q)_N}\left(1+\sum_{n=1}^{N}\frac{(-1)^n (q)^2_N q^{\frac{n(3n+1)}{2}}}{(q)_{N+n}(q)_{N-n}}\left( \frac{1-z}{1-zq^n}+\frac{1-z^{-1}}{1-z^{-1}q^n}\right)\right). \end{align*} Splitting the summation in the right hand side above, we get \begin{align*} 1+R_{S_1}(z;q)=\frac{1}{(q)_N}\left(1+\sum_{n=1}^{N}\frac{(-1)^n (q)^2_N q^{\frac{n(3n+1)}{2}}}{(q)_{N+n}(q)_{N-n}} \left(\frac{1-z}{1-zq^n}\right)+\sum_{n=1}^{N}\frac{(-1)^n (q)^2_N q^{\frac{n(3n+1)}{2}} }{(q)_{N+n}(q)_{N-n}} \left(\frac{1-z^{-1}}{1-z^{-1}q^n}\right)\right). \end{align*} Making a change of variable from $n$ to $-n$ in the rightmost summation above, we arrive at \begin{align*} 1+R_{S_1}(z;q)=\frac{1}{(q)_N}\sum_{n=-N}^{N}\frac{(-1)^n (q)^2_N q^{\frac{n(3n+1)}{2}}}{(q)_{N+n}(q)_{N-n}} \left(\frac{1-z}{1-zq^n}\right). \end{align*} We now take the derivatives of $1+R_{S_1}(z;q)$ with respect to $z$. Firstly, we obtain \begin{align*}{} R'_{S_1}(z;q)=\frac{-1}{(q)_N}\sum_{\substack {n=-N\\n\neq 0}}^{N}\frac{(-1)^n (q)^2_N q^{\frac{n(3n+1)}{2}} }{(q)_{N+n}(q)_{N-n}} \left(\frac{1-q^n}{(1-zq^n)^2}\right) \end{align*} and so for $j\geq1$, \begin{align}{\label{RS1}} R^{(j)}_{S_1}(z;q)=\frac{-j!}{(q)_N}\sum_{\substack {n=-N\\n\neq 0}}^{N}\frac{(-1)^n (q)^2_N q^{\frac{n(3n-1)}{2}+jn} }{(q)_{N+n}(q)_{N-n}} \left(\frac{1-q^n}{(1-zq^n)^{j+1}}\right). \end{align} Putting \eqref{RS1} in the right hand side of \eqref{LBR}, we have \begin{align*} \sum_{n=1}^{\infty}\eta_{2\nu,N}(n)q^n &=(q)_N\sum_{j=0}^{\nu-1}\left(\begin{matrix} \nu-1\\ j\end{matrix}\right)\sum_{\substack {n=-N\\n\neq 0}}^{N}\frac{(-1)^{n-1} q^{\frac{n(3n-1)}{2}+(2\nu-j)n} }{(q)_{N+n}(q)_{N-n}} \left(\frac{1-q^n}{(1-q^n)^{2\nu-j+1}}\right)\\ &=(q)_N\sum_{\substack {n=-N\\n\neq 0}}^{N}\frac{(-1)^{n-1} q^{\frac{n(3n-1)}{2}+2\nu n} }{(q)_{N+n}(q)_{N-n}(1-q^n)^{2\nu}} \sum_{j=0}^{\nu-1}\left(\begin{matrix} \nu-1\\ j\end{matrix}\right)\frac{q^{-nj}}{(1-q^n)^{-j}}\\ &=(q)_N\sum_{\substack {n=-N\\n\neq 0}}^{N}\frac{(-1)^{n-1} q^{\frac{n(3n+1)}{2}+\nu n} }{(q)_{N+n}(q)_{N-n}(1-q^n)^{2\nu}}, \end{align*} by an application of binomial theorem to the inner sum in the second step. Therefore, \begin{equation*} \sum_{n=1}^{\infty}\eta_{2\nu,N}(n)q^n=(q)_N\sum_{\substack {n=-N\\n\neq 0}}^{N}\frac{(-1)^{n-1} q^{\frac{n(3n+1)}{2}+\nu n} }{(q)_{N+n}(q)_{N-n}(1-q^n)^{2\nu}}. \end{equation*} We split the sum on the right side into two parts, namely, from $1$ to $N$ and from $-N$ to $-1$. \begin{align*}{} \sum_{n=1}^{\infty}\eta_{2\nu,N}(n)q^n=(q)_N \left(\sum_{\substack {n=1}}^{N}\frac{(-1)^{n-1} q^{\frac{n(3n+1)}{2}+\nu n} }{(q)_{N+n}(q)_{N-n}(1-q^n)^{2\nu}}+\sum_{\substack {n=-1}}^{-N}\frac{(-1)^{n-1} q^{\frac{n(3n+1)}{2}+\nu n} }{(q)_{N+n}(q)_{N-n}(1-q^n)^{2\nu}}\right). \end{align*} Replace $n$ by $-n$ in the rightmost sum to get \begin{align*}
\sum_{n=1}^{\infty}\eta_{2\nu,N}(n)q^n=&(q)_N \sum_{\substack {n=1}}^{N}\frac{(-1)^{n-1} q^{\frac{n(3n-1)}{2}+\nu n}(1+q^n) }{(q)_{N+n}(q)_{N-n}(1-q^{n})^{2\nu}}, \end{align*} which is nothing but \eqref{gf_symm_rank2}. \end{proof}
\begin{proof}[Theorem \textup{\ref{theoremmugf}}][] We know from \eqref{SCM} that \begin{equation*}
\mu_{2\nu,N}(n):=\sum_{m=-n}^{n} \left(\begin{matrix} m+ \nu-1 \\ 2\nu \end{matrix}\right)M_{S_2}(m,n). \end{equation*} It follows, from the definition \eqref{fincrankgf} of $C_{S_2}(z;q)$ and by an application of Leibniz's rule, that \begin{equation}\label{gfmuintermsofCS2} \sum_{n=1}^{\infty}\mu_{2\nu,N}(n)q^n=\frac{1}{(2\nu)!}\sum_{j=0}^{\nu-1}\left(\begin{matrix} 2\nu\\ j\end{matrix}\right)(\nu-1)(\nu-2)...(\nu-j)C_{S_2}^{(2\nu-j)}(1;q). \end{equation} Using \eqref{andrewsCS2} in \eqref{fincrankgf}, we get \begin{align*} C_{S_2}(z;q)&=\frac{1}{(q)_N}+(1-z)\sum_{n=1}^{N}\left[\begin{matrix} N \\ n \end{matrix}\right]\frac{(-1)^n(q)_nq^{n(n+1)/2}}{(q)_{n+N}}\left(\frac{1}{1-zq^n}-\frac{1}{z-q^n}\right)\\ &=\frac{1}{(q)_N}\left(1+\sum_{n=1}^{N}\frac{(-1)^n (q)^2_N q^{\frac{n(n+1)}{2}}}{(q)_{N+n}(q)_{N-n}}\left( \frac{1-z}{1-zq^n}+\frac{1-z^{-1}}{1-z^{-1}q^n}\right)\right). \end{align*} Making a change of variable as in Theorem \ref{gffinsymmetrizedrankfn}, we finally get \begin{equation*}
C_{S_2}(z;q)=\frac{1}{(q)_N}\sum_{n=-N}^{N}\frac{(-1)^n (q)^2_N q^{\frac{n(n+1)}{2}}}{(q)_{N+n}(q)_{N-n}} \left(\frac{1-z}{1-zq^n}\right). \end{equation*} Hence, for $j \geq 1$, we have \begin{align*}\label{crankgfderivatives} C^{(j)}_{S_2}(z;q)=\frac{-j!}{(q)_N}\sum_{\substack {n=-N\\n\neq 0}}^{N}\frac{(-1)^n (q)^2_N q^{\frac{n(n-1)}{2}+jn} }{(q)_{N+n}(q)_{N-n}} \left(\frac{1-q^n}{(1-zq^n)^{j+1}}\right). \end{align*} Substituting these derivative expressions in \eqref{gfmuintermsofCS2} and then by an application of binomial theorem, we obtain \begin{equation*} \sum_{n=1}^{\infty}\mu_{2\nu,N}(n)q^n=(q)_N\sum_{\substack {n=-N\\n\neq 0}}^{N}\frac{(-1)^{n-1} q^{\frac{n(n+1)}{2}+\nu n} }{(q)_{N+n}(q)_{N-n}(1-q^n)^{2\nu}}. \end{equation*} Splitting the sum into the ranges $1$ to $N$ and $-N$ to $-1$ and then making a variable change, we get \begin{align*}{} \sum_{n=1}^{\infty}\mu_{2\nu,N}(n)q^n=(q)_N \sum_{\substack {n=1}}^{N}\frac{(-1)^{n-1} q^{\frac{n(n-1)}{2}+\nu n}(1+q^n) }{(q)_{N+n}(q)_{N-n}(1-q^{n})^{2\nu}}. \end{align*} \end{proof}
\section{Proof of Theorem \ref{difference_gen_fin_rank_crank} and Conjecture \ref{finiteanalogcrankrankconjecture}}\label{baily_pairs_proof_conj}
Using Bailey's lemma, i.e., Theorem \ref{Baileylemma}, we give a result which is essential for the proof of Conjecture \ref{finiteanalogcrankrankconjecture}. \begin{proposition}\label{generalBaileypair} Let $(\alpha_n(a, q), \beta_n(a, q))$ be a Bailey pair with $a=1$ and $\alpha_0=\beta_0=1$. We then have \begin{align*} \sum_{N\geq n_k\geq ...\geq n_1\geq1} \frac{(q)^2_{n_1}q^{n_1+...+n_k} \beta_{n_1}}{(1-q^{n_k})^2 (1-q^{n_{k-1}})^2...(1-q^{n_1})^2}&=\sum_{N\geq n_k\geq ...\geq n_1\geq1}\frac{q^{n_1+...+n_k} }{(1-q^{n_k})^2 (1-q^{n_{k-1}})^2...(1-q^{n_1})^2} \nonumber \\ &+\sum_{r=1}^{N}\frac{(q)^2_N}{(q)_{N-r}(q)_{N+r}}\frac{q^{kr}\alpha_r}{(1-q^r)^{2k}}. \end{align*}
\end{proposition}
\begin{proof} Since $(\alpha_n(a, q), \beta_n(a, q))$ form a Bailey pair with $a=1$, we have the relation $$\beta_n(1, q) = \sum_{r=0}^{n}\frac{\alpha_r(1, q)}{(q)_{n-r} (q)_{n+r}}.$$ By Bailey's Lemma, $(\alpha'_n(a, q), \beta^{'}_n(a, q))$ is also a Bailey pair with parameters $(1, q)$. Hence, by \eqref{Baileydefn}, $$ \beta^{'}_n(1, q) = \sum_{r=0}^{n}\frac{\alpha'_r(1, q)}{(q)_{n-r} (q)_{n+r}}. $$ Substituting the values of $\alpha'_n(a, q)$ and $\beta^{'}_n(a, q)$ from Theorem \ref{Baileylemma}, we get $$ \sum_{k=0}^{n}\frac{(\rho_1)_k (\rho_2)_k (q/\rho_1\rho_2)_{n-k}}{(q/\rho_1)_n (q/\rho_2)_n (q)_{n-k}}\left(\frac{q}{\rho_1 \rho_2}\right)^k\beta_k(1, q) =\sum_{k=0}^{n} \frac{(\rho_1)_k (\rho_2)_k}{(q/\rho_1)_k (q/\rho_2)_k(q)_{n-k} (q)_{n+k}}\left(\frac{q}{\rho_1\rho_2} \right)^k \alpha_k(1, q). $$ Separating the terms corresponding to $k=0$ in both the summations and multiplying throughout by $(q/\rho_1)_n (q/\rho_2)_n$, \begin{align*} \sum_{k=1}^{n}\frac{(\rho_1)_k(\rho_2)_k (q/\rho_1\rho_2)_{n-k} (q/\rho_1 \rho_2)^k}{(q)_{n-k}}\beta_k(1, q)&=\frac{(q/\rho_1)_n (q/\rho_2)_n}{(q)_{n}^2}\left(1-\frac{(q)_n(q/\rho_1\rho_2)_n}{(q/\rho_1)_n (q/\rho_2)_n}\right) + \\ & (q/\rho_1)_n (q/\rho_2)_n \sum_{k=1}^{n}\frac{(\rho_1)_k (\rho_2)_k(q/\rho_1 \rho_2)^k}{(q/\rho_1)_k (q/\rho_2)_k(q)_{n-k} (q)_{n+k}} \alpha_k(1, q). \end{align*} Dividing both sides by $(1-\rho_1)(1-\rho_2)$, then letting $\rho_1\rightarrow1$, $\rho_2\rightarrow1$ and using \eqref{limitresultp1p2}, we get $$ \sum_{k=1}^{n}(q)^2_{k-1}q^k \beta_k=\sum_{k=1}^{n}\frac{q^k}{(1-q^k)^2}+\sum_{k=1}^{n}\frac{(q)^2_{n}q^{k}\alpha_k}{(q)_{n-k}(q)_{n+k}(1-q^k)^2}. $$ This is the $k=1$ case of the theorem. We are going to prove the theorem using induction. To this end, suppose that the theorem holds for $k=\ell-1$. This means that
\begin{align}\label{l-1induction} \sum_{N\geq n_\ell\geq ...\geq n_2\geq1} \frac{(q)^2_{n_2}q^{n_2+...+n_\ell} \beta_{n_2}}{(1-q^{n_2})^2...(1-q^{n_\ell})^2}=\sum_{N\geq n_\ell\geq ...\geq n_2\geq1}\frac{q^{n_2+...+n_\ell} }{(1-q^{n_2})^2...(1-q^{n_\ell})^2} \nonumber \\ +\sum_{r=1}^{N}\frac{(q)^2_N}{(q)_{N-r}(q)_{N+r}}\frac{q^{(\ell-1)r}\alpha_r}{(1-q^{r})^{2(\ell-1)}}. \end{align}
This equation is true for any Bailey pair $(\alpha_n(a, q), \beta_n(a, q))$ with $a=1$ and $\alpha_0=\beta_0=1$. Note that, since $\alpha'_0=\alpha_0=1$ and $\beta^{'}_0=\beta_0=1$, \eqref{l-1induction} also holds for the Bailey pair $(\alpha'_n(a, q), \beta^{'}_n(a, q))$. So, we replace $(\alpha_n, \beta_n)$ by $(\alpha'_n, \beta^{'}_n)$ in \eqref{l-1induction} to get \begin{align*} \sum_{N\geq n_\ell\geq ...\geq n_2\geq1} \frac{(q)^2_{n_2}q^{n_2+...+n_\ell} \beta^{'}_{n_2}}{(1-q^{n_2})^2...(1-q^{n_\ell})^2}=\sum_{N\geq n_\ell\geq ...\geq n_2\geq1}\frac{q^{n_2+...+n_\ell} }{(1-q^{n_2})^2...(1-q^{n_\ell})^2} \nonumber \\ +\sum_{r=1}^{N}\frac{(q)^2_N}{(q)_{N-r}(q)_{N+r}}\frac{q^{(\ell-1)r}\alpha'_r}{(1-q^{r})^{2(\ell-1)}}. \end{align*} We now substitute for $\alpha'_n$ and $\beta^{'}_n$ in terms of $\alpha_n$ and $\beta_n$ using Bailey's Lemma, \begin{align*} &\sum_{\substack{N\geq n_\ell\geq ...\geq n_2\geq1,\\n_2\geq n_1\geq 0}} \frac{(q)^2_{n_2}q^{n_2+...+n_\ell}}{(1-q^{n_2})^2...(1-q^{n_\ell})^2}\frac{(\rho_1)_{n_1}(\rho_2)_{n_1}(q/\rho_1\rho_2)_{n_2-n_1}(q/\rho_1\rho_2)^{n_1}\beta_{n_1}}{(q/\rho_1)_{n_2}(q/\rho_2)_{n_2}(q)_{n_2-n_1}}\\ &=\sum_{N\geq n_\ell\geq ...\geq n_2\geq1}\frac{q^{n_2+...+n_\ell} }{(1-q^{n_2})^2...(1-q^{n_\ell})^2}+\sum_{r=1}^{N}\frac{(q)^2_N}{(q)_{N-r}(q)_{N+r}}\frac{q^{(\ell-1)r}(\rho_1)_r(\rho_2)_r(q/\rho_1\rho_2)^r\alpha_r}{(1-q^{r})^{2(\ell-1)}(q/\rho_1)_r(q/\rho_2)_r}. \end{align*} Again, separating the terms corresponding to $n_1=0$ from the sum on the left side, then dividing both sides by $(1-\rho_1)(1-\rho_2)$, letting $\rho_1\rightarrow1$, $\rho_2\rightarrow1$ and using \eqref{limitresultp1p2}, we obtain \begin{align*} \sum_{N\geq n_\ell\geq ...\geq n_1\geq1} \frac{(q)^2_{n_1}q^{n_1+n_2+...+n_\ell} \beta_{n_1}}{(1-q^{n_1})^2(1-q^{n_2})^2...(1-q^{n_\ell})^2}=&\sum_{N\geq n_\ell\geq ...\geq n_2\geq n_1\geq1}\frac{q^{n_1+n_2+...+n_\ell}}{(1-q^{n_1})^2(1-q^{n_2})^2...(1-q^{n_\ell})^2}. \nonumber \\ &+\sum_{r=1}^{N}\frac{(q)^2_Nq^{\ell r}\alpha_r}{(q)_{N-r}(q)_{N+r}(1-q^{r})^{2\ell}}. \end{align*} This concludes the proof of the theorem by induction. \end{proof}
\begin{corollary}\label{mugfcorollary} \begin{equation*} (q)^2_N\sum_{r=1}^{N}\frac{(-1)^{r-1} q^{\frac{r(r-1)}{2}+kr} (1+q^r)}{(q)_{N-r}(q)_{N+r}(1-q^{r})^{2k}}= \sum_{N\geq n_k\geq ...\geq n_1\geq1}\frac{q^{n_1+n_2+...+n_k} }{(1-q^{n_1})^2(1-q^{n_2})^2...(1-q^{n_k})^2}. \end{equation*} \end{corollary} \begin{proof} Consider the well known Bailey pair below (\cite[pp. ~27-28]{andrewsqseries}), \begin{equation*} \alpha_n= \begin{cases} 1, &\; \text{if} \;\;n=0, \\ (-1)^n q^{\frac{n(n-1)}{2}} (1+q^n),& \;\text{if}\;\;n\geq1, \end{cases} \end{equation*} and \begin{equation*} \beta_{n}= \begin{cases} 1, &\; \text{if} \;\;n=0, \\ 0,& \;\text{if}\;\;n\geq1. \end{cases} \end{equation*} Substituting the above Bailey pair in Theorem \ref{generalBaileypair}, we get \begin{align*} 0=\sum_{N\geq n_k\geq ...\geq n_1\geq1}\frac{q^{n_1+n_2+...+n_k} }{(1-q^{n_1})^2(1-q^{n_2})^2...(1-q^{n_k})^2} +(q)^2_N\sum_{r=1}^{N}\frac{(-1)^r q^{\frac{r(r-1)}{2}+kr} (1+q^r)}{(q)_{N-r}(q)_{N+r}(1-q^{r})^{2k}}. \end{align*} \end{proof} \begin{corollary} \begin{align}\label{mugfalternate} \sum_{n=1}^{\infty}\mu_{2k,N}(n)q^n= \frac{1}{(q)_N}\sum_{N\geq n_k\geq ...\geq n_1\geq1}\frac{q^{n_1+n_2+...+n_k} }{(1-q^{n_1})^2(1-q^{n_2})^2...(1-q^{n_k})^2}. \end{align} \end{corollary} \begin{proof} Using equation \eqref{mugf} from Theorem \ref{theoremmugf} along with Corollary \ref{mugfcorollary}, we get this result. Note that this is a finite analogue of \eqref{mugf_alternate}. \end{proof}
\begin{corollary}\label{crudediffsymmoments} \begin{align*} &\sum_{N\geq n_k\geq ...\geq n_1\geq1} \frac{(q)_{n_1}q^{n_1+n_2+...+n_k}}{(1-q^{n_1})^2(1-q^{n_2})^2...(1-q^{n_k})^2}= \nonumber\\ &\sum_{N\geq n_k\geq ...\geq n_1\geq1}\frac{q^{n_1+n_2+...+n_k} }{(1-q^{n_1})^2(1-q^{n_2})^2...(1-q^{n_k})^2} +(q)^2_N\sum_{r=1}^{N}\frac{(-1)^r q^{\frac{r(3r-1)}{2}+kr} (1+q^r)}{(q)_{N-r}(q)_{N+r}(1-q^{r})^{2k}}. \end{align*} \end{corollary}
\begin{proof} Again we use a well known Bailey pair (\cite[p.~28]{andrewsqseries}), \begin{equation*} \alpha_n= \begin{cases} 1, &\; \text{if} \;\;n=0, \\ (-1)^n q^{\frac{n(3n-1)}{2}} (1+q^n),& \;\text{if}\;\;n\geq1, \end{cases}
\ \ \text{and} \ \ \beta_n=\frac{1}{(q)_n}. \end{equation*} Putting the values of $\alpha_n$ and $\beta_n$ in Theorem \ref{generalBaileypair}, we get the result.
\end{proof} Now we are ready to give a proof of Theorem \ref{difference_gen_fin_rank_crank}. \begin{proof}[Theorem \textup{\ref{difference_gen_fin_rank_crank}}][] Divide both sides of \eqref{crudediffsymmoments} by $(q)_N$ to get
\begin{align*} &\frac{1}{(q)_N}\sum_{N\geq n_k\geq ...\geq n_1\geq1}\frac{(q)_{n_1}q^{n_1+n_2+...+n_k}}{(1-q^{n_1})^2(1-q^{n_2})^2...(1-q^{n_k})^2}=\\ &\frac{1}{(q)_N}\sum_{N\geq n_k\geq ...\geq n_1\geq1}\frac{q^{n_1+n_2+...+n_k} }{(1-q^{n_1})^2(1-q^{n_2})^2...(1-q^{n_k})^2} +(q)_N\sum_{r=1}^{N}\frac{(-1)^r q^{\frac{r(3r-1)}{2}+kr} (1+q^r)}{(q)_{N-r}(q)_{N+r}(1-q^{r})^{2k}}.
\end{align*} Using \eqref{mugfalternate} and equation \eqref{gfsymrankmoment} from Theorem \ref{gffinsymmetrizedrankfn}, we get the desired result. \end{proof}
Before going to the proof of Conjecture \ref{finiteanalogcrankrankconjecture}, we require one more concept, an analogue of Stirling numbers of the second kind, defined by Garvan \cite{garvan11}. He defined a sequence of polynomials \begin{align*} g_k(x)=\prod_{j=0}^{k-1}(x^2-j^2), \;\;\; \text{for} \;\;k\geq1 \end{align*} and a sequence of numbers $S^*(n,k)$ such that, for $n\geq1$, \begin{equation}\label{polynomialSnk} x^{2n}=\sum_{k=1}^{n}S^*(n,k)g_k(x). \end{equation} \textbf{Definition} \cite[p.~249]{garvan11}: Define the sequence $S^*(n,k)$, for $1 \leq k \leq n$, recursively by\\ (i) $S^*(1,1)=1$,\\ (ii) $S^*(n,k)=0$ if $k\leq0$ or $k>n$,\\ (iii) $S^*(n+1,k)=S^*(n,k-1)+k^2S^*(n,k)$, for $1\leq k\leq n+1$.\\
From this definition, Garvan showed that the relation \eqref{polynomialSnk} indeed holds (\cite[Lemma 4.2]{garvan11}).
Next, we link the finite analogues of rank and crank moments with their symmetrized counterparts via the numbers $S^*(n,k)$. \begin{proposition} For any two positive integers $k$ and $N$, \begin{align} \mu_{2k,N}(n)&=\frac{1}{(2k)!}\sum_{m=-n}^{n}g_k(m)M_{S_2}(m,n),\label{fg1}\\ \eta_{2k,N}(n)&=\frac{1}{(2k)!}\sum_{m=-n}^{n}g_k(m)N_{S_1}(m,n),\label{fg2}\\ M_{2k,N}(n)&=\sum_{j=1}^{k}(2j)!S^*(k,j)\;\mu_{2j,N}(n),\label{fg3}\\ N_{2k,N}(n)&=\sum_{j=1}^{k}(2j)!S^*(k,j)\;\eta_{2j,N}(n).\label{fg4} \end{align} \end{proposition} \begin{proof} By the definition of finite analogue of $k^{th}$ symmetrized crank moment, we know that \begin{align*} \mu_{2k,N}(n)&=\sum_{m=-n}^{n} \left(\begin{matrix} m+k-1 \\ 2k\end{matrix}\right)M_{S_2}(m,n)\\
&=\frac{1}{(2k)!}\sum_{m=-n}^{n}\left(m^2-(k-1)^2\right) \left(m^2-(k-2)^2\right) \dots\big(m^2- 1^2\big) m(m-k)M_{S_2}(m,n). \end{align*} By the definition of the polynomials $g_k$, this may be written as \begin{align*} \mu_{2k,N}(n)=\frac{1}{(2k)!}\sum_{m=-n}^{n}g_k(m)M_{S_2}(m,n)-\frac{k}{(2k)!}\sum_{m=-n}^{n}\left(m^2-(k-1)^2\right)\dots\big(m^2-1^2\big)mM_{S_2}(m,n). \end{align*} Since $M_{S_2}(m,n)=M_{S_2}(-m,n)$ \cite[p.~9]{DEMS}, the rightmost sum vanishes and we get \eqref{fg1}. Similarly one can prove \eqref{fg2}. Now for the proof of \eqref{fg3}, we start with the definition of $M_{2k,N}(n)$ \eqref{fincrankmom}, namely, \begin{align*} M_{2k,N}(n)=\sum_{m=-n}^{n}m^{2k}M_{S_2}(m,n).
\end{align*} We use \eqref{polynomialSnk} to substitute for $m^{2k}$ and obtain \begin{align*} M_{2k,N}(n)&=\sum_{m=-n}^{n}\left(\sum_{j=1}^kS^*(k,j)g_j(m)\right)M_{S_2}(m,n)\\ &= \sum_{j=1}^{k}S^*(k,j)\sum_{m=-n}^{n}g_j(m)M_{S_2}(m,n)\\ &=\sum_{j=1}^k(2j)!S^*(k,j)\mu_{2j,N}(n), \end{align*} the last step following from \eqref{fg1}. This completes the proof of \eqref{fg3} and on similar lines we can prove \eqref{fg4}. \end{proof} We are now ready to prove the inequality for the finite analogues of rank and crank moments. \begin{proof}[Conjecture \textup{\ref{finiteanalogcrankrankconjecture}}][] From \eqref{fg3} and \eqref{fg4}, we get \begin{equation}\label{diffmoments} M_{2k,N}(n)-N_{2k,N}(n)=\sum_{j=1}^{k}(2j)!S^*(k,j)\;(\mu_{2j,N}(n)-\eta_{2j,N}(n)). \end{equation} From Theorem \ref{difference_gen_fin_rank_crank}, we know $$\sum_{n=1}^{\infty}(\mu_{2t,N}(n)q^n-\eta_{2t,N}(n))q^n=\sum_{N\geq n_t\geq ...\geq n_1\geq1}\frac{q^{n_1+n_2+...+n_t}}{(1-q^{n_1})^2(1-q^{n_2})^2...(1-q^{n_t})^2(q^{n_1 + 1})_{N-n_1}}.$$ From the generating function, we infer that, $\mu_{2t,N}(n)-\eta_{2t,N}(n)\geq 0$ for $n, t, N \geq 1$. Moreover, the numbers $S^*(k,j)$ are all positive, so from \eqref{diffmoments}, we can write $$M_{2k,N}(n)-N_{2k,N}(n) \geq 2(\mu_{2,N}(n)-\eta_{2,N}(n)) = 2\textup{spt}(n, N)>0,$$ where the last equality follows from Remark \ref{remark_fin_higher_spt}. This finishes the proof of the conjecture. \end{proof}
\section{Concluding Remarks} In Remark \ref{remark_fin_higher_spt}, we defined a finite analogue of higher order spt-function as $\textup{spt}_k(n,N):= \mu_{2k,N}(n)-\eta_{2k,N}(n)$. A combinatorial interpretation of the higher order spt-function $\text{spt}_{k}(n)$ was described by Garvan \cite[p.~252]{garvan11}. Looking at the generating function \eqref{gen_fin_rank_crank} of the difference between finite analogues of the symmetrized moments and comparing it with \eqref{difference_gen_garvan}, one can give a combinatorial interpretation of $\text{spt}_k(n,N)$ on similar lines as that of Garvan's for $\text{spt}_k(n)$, the only restriction being that the largest parts of the corresponding partitions are less than or equal to $N$.
Bringmann, Mahlburg and Rhoades \cite{BMR11} showed that, for any $k \geq 1$, as $n \rightarrow \infty$, \begin{align*} & M_{2k}(n) \sim N_{2k}(n) \sim \alpha_{2k} n^k p(n),\\ & M_{2k}(n)-N_{2k}(n) \sim \beta_{2k} n^{k- \frac{1}{2}} p(n), \end{align*} where $\alpha_{2k}, \beta_{2k}$ are certain explicitly computable constants (see \cite[p. 665, Corollary 1.4]{BMR11}). Since in this paper, we have proved the inequality for the finite analogues of rank and crank moments, it would be fascinating to find the asymptotic behavior of the finite analogues and their difference.
Given any prime $p>3$ and for fixed positive integers $k$ and $j$, Bringmann, Garvan and Mahlburg \cite[Corollary 1.3]{BGM09} established that there are infinitely many arithmetic progressions $A n + B$ such that $\eta_{2k}(An+B) \equiv 0 \pmod {p^j}$. It would be worthwhile to see if such congruences exist for $\eta_{2k,N}(n)$.
A number of explicit congruences for higher order $\text{spt}$-functions were proved by Garvan \cite[Theorem 6.1--6.3]{garvan11}. It would also be interesting to see if there exists a refinement of these congruences for $\text{spt}_{k}(n,N)$.
\textbf{Acknowledgements} We would like to thank Prof.~Atul Dixit for going through the manuscript and giving valuable suggestions. The first author wishes to thank Harish-Chandra Research Institute and IIT Gandhinagar for the conducive environment. The second author is a SERB National Post Doctoral Fellow (NPDF) supported by the fellowship PDF/2017/000370. The third author is supported partially by IIT Gandhinagar and by SERB ECR grant ECR/2015/000070 of Prof. Atul Dixit.
\end{document} | arXiv |
Forum — Daily Challenge
Why isn't the area of a triangle just side × side ÷ 2?
Module 0 Day 1 Challenge Part 1
debbie ADMIN M0★ M1 M5 last edited by debbie
Someone little today gave me several answers today to the question, "What's the area of this triangle?"
At first I was a bit shocked, appalled and dismayed at this answer... However, after thinking about it a while, I guess it's a very natural thing for a little person to say!
A triangle looks like it should be \(\color{red}\text{easy}\) to calculate the area of. It's an easy shape, in the same way that a rectangle or a square is "easy":
But if I could just multiply two of the sides of a triangle to get its area, then what if I draw the triangles in a weird way? What if I draw a very short and skinny triangle, like this?
Does it really seem like the area of a triangle is related to the product of its sides? I can draw a triangle with the same two sides, but skinnier and skinnier until its area is almost \(0.\)
My friend sort of "remembered" the formula for the area of a triangle, except she remembered the formula for a right triangle's area. She understood that a right triangle is half of the bounding rectangle that surrounds it. But she didn't understand exactly why the formula was true, so had trouble relating it to a non-right triangle.
$$ \textcolor{red}{\text{Area of a right triangle}} = \frac{1}{2} \text{ } \text{ base } \times \text{ height} $$
Now, why is the area of the second triangle equal to the first triangle? She cuts the first right triangle and says it's because the two mini-triangles are the same size.
Is that really true?
No, it's not! They look the same, but their lengths are actually different. The longest side of the left triangle is a diagonal, which is longer than the longest side of the right triangle, which is just the long side of the rectangle. Also, the altitudes (heights) of the triangle are different. The altitude on the left, colored yellow, is shorter than the height of the rectangle, so it is shorter than the altitude of the right triangle.
There's a different reason why both triangles inscribed in the box have the same area. It's because you can split the box into two smaller boxes, each of which has half taken up by a mini-triangle.
Thus we say the area of a triangle is equal to \( \color{red}\frac{1}{2} \times \text{ } \text{ base } \times \text{ height}, \) where the height is the height of the triangle's bounding rectangle.
We can interpret any of the three sides of the triangle as the base; just remember to draw a bounding rectangle and find the height of this rectangle. This altitude is always perpendicular ( \(\perp\) ) to the base.
Since each of the three sides can be a different base, there are three formulas for the area of a triangle.
Daily Challenge | Terms | COPPA | CommonCrawl |
Witness (mathematics)
In mathematical logic, a witness is a specific value t to be substituted for variable x of an existential statement of the form ∃x φ(x) such that φ(t) is true.
Examples
For example, a theory T of arithmetic is said to be inconsistent if there exists a proof in T of the formula "0 = 1". The formula I(T), which says that T is inconsistent, is thus an existential formula. A witness for the inconsistency of T is a particular proof of "0 = 1" in T.
Boolos, Burgess, and Jeffrey (2002:81) define the notion of a witness with the example, in which S is an n-place relation on natural numbers, R is an (n+1)-place recursive relation, and ↔ indicates logical equivalence (if and only if):
S(x1, ..., xn) ↔ ∃y R(x1, . . ., xn, y)
"A y such that R holds of the xi may be called a 'witness' to the relation S holding of the xi (provided we understand that when the witness is a number rather than a person, a witness only testifies to what is true)."
In this particular example, the authors defined s to be (positively) recursively semidecidable, or simply semirecursive.
Henkin witnesses
In predicate calculus, a Henkin witness for a sentence $\exists x\,\varphi (x)$ in a theory T is a term c such that T proves φ(c) (Hinman 2005:196). The use of such witnesses is a key technique in the proof of Gödel's completeness theorem presented by Leon Henkin in 1949.
Relation to game semantics
The notion of witness leads to the more general idea of game semantics. In the case of sentence $\exists x\,\varphi (x)$ the winning strategy for the verifier is to pick a witness for $\varphi $. For more complex formulas involving universal quantifiers, the existence of a winning strategy for the verifier depends on the existence of appropriate Skolem functions. For example, if S denotes $\forall x\,\exists y\,\varphi (x,y)$ then an equisatisfiable statement for S is $\exists f\,\forall x\,\varphi (x,f(x))$. The Skolem function f (if it exists) actually codifies a winning strategy for the verifier of S by returning a witness for the existential sub-formula for every choice of x the falsifier might make.
See also
• Certificate (complexity), an analogous concept in computational complexity theory
References
• George S. Boolos, John P. Burgess, and Richard C. Jeffrey, 2002, Computability and Logic: Fourth Edition, Cambridge University Press, ISBN 0-521-00758-5.
• Leon Henkin, 1949, "The completeness of the first-order functional calculus", Journal of Symbolic Logic v. 14 n. 3, pp. 159–166.
• Peter G. Hinman, 2005, Fundamentals of mathematical logic, A.K. Peters, ISBN 1-56881-262-0.
• J. Hintikka and G. Sandu, 2009, "Game-Theoretical Semantics" in Keith Allan (ed.) Concise Encyclopedia of Semantics, Elsevier, ISBN 0-08095-968-7, pp. 341–343
| Wikipedia |
\begin{document}
\title{ Perron-Frobenius theory for kernels and Crump-Mode-Jagers processes with macro-individuals
} \author{Serik Sagitov \\Chalmers University of Technology and University of Gothenburg } \maketitle \begin{abstract} Perron-Frobenius theory developed for irreducible non-negative kernels deals with so-called $R$-positive recurrent kernels. If kernel $M$ is $R$-positive recurrent, then the main result determines the limit of the scaled kernel iterations $R^nM^n$ as $n\to\infty$. In the Nummelin's monograph \cite{N} this important result is proven using a regeneration method whose major focus is on $M$ having an atom. In the special case when $M=P$ is a stochastic kernel with an atom, the regeneration method has an elegant explanation in terms of an associated split chain.
In this paper we give a new probabilistic interpretation of the general regeneration method in terms of multi-type Galton-Watson processes producing clusters of particles. Treating clusters as macro-individuals, we arrive at a single-type Crump-Mode-Jagers process with a naturally embedded renewal structure.
\end{abstract} Keywords: irreducible non-negative kernels, multi-type Galton-Watson process, $R$-positive recurrent kernel
\section{Introduction} A Galton-Watson (GW) process describes random fluctuations of the numbers of independently reproducing particles counted generation-wise, see \cite{AN}. Given a measurable type space $(E,\mathcal E)$, the multi-type GW process is defined as a measure-valued Markov chain $\{\Xi_n\}_{n=0}^\infty $, where $\Xi_n(A)$ gives the number of $n$-th generation particles whose types lie in the set $A\in \mathcal E$, see \cite[Ch 3]{Ha}. Given a current state \[\Xi_n=\sum_{i=1}^{Z_n}\delta_{x_i},\quad \delta_x(A):=1_{\{x\in A\}},\] where $Z_n=\Xi_n(E)$ is the number of particles in the $n$-th generation and $x_1,x_2,\ldots$ are the types of these particles, the next state of the Markov chain is determined in terms of the offspring to $Z_n$ particles \[\Xi_{n+1}=\sum_{i=1}^{Z_n}\Xi^{(x_i)}_{i,n},\quad \Xi^{(x_i)}_{i,n}\stackrel{d}{=}\Xi^{(x_i)}.\] The random measure $\Xi^{(x_i)}_{i,n}$, describing the allocation of a group of siblings over the type space $E$, is assumed to be independent of everything else except for the maternal type $x_i$.
A key characteristic of the multi-type GW process is its {\it reproduction kernel} $M$ defining the expected number of offspring found in a given subset of the type space \begin{equation}\label{rke} M(x,A)=\mathrm{E}[ \Xi^{(x)}(A)],\quad x\in E,\quad A\in \mathcal E, \end{equation} as a function of the maternal type $x$. Denote by $M^n$ the iterations of the reproduction kernel: \begin{equation}\label{Mndef} M^0(x,A)=\delta_x(A),\quad M^{n}(x,A)=\int M^{n-1}(y,A)M(x,\mathrm{d} y),\quad n\ge1, \end{equation} here and elsewhere in this paper, the integrals are taken over the whole type space $E$, unless specified otherwise. Then, for the multi-type GW process with the initial state $\Xi_0$, we get $$\mathrm{E} [\Xi_n(A)]=\int M^n(x,A)\mu_0(\mathrm{d} x),\qquad \mu_0= \mathrm{E} [\Xi_0].$$
The asymptotic properties of the multi-type GW processes are studied on the basis of the Perron-Frobenius theorem dealing with the limiting behaviour of the expectation kernels and producing an asymptotic formula of the form $$M^n(x,A)\sim \rho^n\dfrac{h(x)\pi(A)}{\int h(y)\pi(dy)},\quad n\to\infty,$$ see \cite[Ch 6]{Mode}. In the classical case of finitely many types, $M$ is a matrix and $\rho$ is its largest, the so-called Perron-Frobenius eigenvalue. Depending on whether $\rho<1$, $\rho=1$, or $\rho>1$, we distinguish among subcritical, critical, or supercritical GW processes.
The Perron-Frobenius theory for the irreducible non-negative kernels is build around the so-called regeneration method, see \cite{AN1} and especially \cite{N}. A key step of the regeneration method deals with $M$ having an atom, see Section \ref{SPF} for key definitions.
In the special case, when $M=P$ is a stochastic kernel with an atom, one can write \begin{equation}\label{spl} P(x,A)=p(x,A)+g(x)\gamma(A), \end{equation} where $\gamma(E)=1$, $0\le g(x)\le1$ and $p(x,E)=1-g(x)$ for all $x\in E$. The transition probabilities defined by such a kernel $P(x,dy)$ describe a {\it split chain}, whose transition from a given state $x$ is governed either by $\gamma( \mathrm{d} y)$ or $p(x, \mathrm{d} y)/(1-g(x))$ depending on a random outcome of a $g(x)$-coin tossing \cite[Ch. 4.4]{N}. After each $\gamma$-transition step, the future evolution of the split chain becomes independent from the past and present states, so that the sequence of such regeneration events forms a renewal process with a delay. Then, it remains to apply the basic renewal theory to establish the Perron-Frobenius theorem for stochastic kernels.
In this paper we suggest a probabilistic interpretation of the general regeneration method (when kernel $M$ is not necessarily stochastic) in terms of a certain class of multi-type GW processes which we call GW processes with clusters, see Section \ref{xip}.
In Section \ref{eCrump-Mode-Jagers } we show that a GW process with clusters has an intrinsic structure of the single-type Crump-Mode-Jagers (CMJ) process with discrete time \cite{JS}. In Sections \ref{sec} and \ref{genf} we give a proof of a suitable version of the Perron-Frobenius theorem for the kernels with an atom, see Theorem \ref{PF}, using the regeneration property of the renewal process embedded into the CMJ process. Section \ref{xmpl} contains an illuminating example of a GW process with clusters.
\section{ Irreducible kernels } \label{SPF} In this section we give a summary of basic definitions and results presented in \cite{N}, including Theorems 2.1, 5.1, 5.2, and Propositions 2.4, 2.8, 3.4.
Consider a measurable type space $(E,\mathcal E)$ assuming that $\sigma$-algebra $\mathcal E$ is countably generated. We denote by $\mathcal M_+$ the set of $\sigma$-finite measures $\phi$ on $(E,\mathcal E)$, and write $\phi\in\mathcal M^+$ if $\phi\in\mathcal M_+$ and $\phi(E)\in(0,\infty]$.
\begin{definition} A (non-negative) kernel on $(E,\mathcal E)$ is a map $M:E\times\mathcal E\to [0,\infty)$ such that for any fixed $A\in \mathcal E$, the function $M(\cdot,A)$ is measurable, and on the other hand, $M(x,\cdot)\in\mathcal M_+$ for any fixed $x\in E$. For a pair $(x,A)\in(E,\mathcal E)$, we write $x\to A$ if
\[M^n(x,A)>0 \text{ for some } n\ge1.\]
Kernel $M$ is called irreducible, if there is such a measure $\phi\in\mathcal M^+$, that for any $x\in E$, we have $x\to A$ whenever $\phi(A)>0$. Measure $\phi$ is then called an irreducibility measure for $M$. \end{definition}
If measure $\phi'\in\mathcal M^+$ is absolutely continuous with respect to an irreducibility measure $\phi$, then $\phi'$ is itself an irreducibility measure.
For an irreducible kernel $M$, there always exists a {\it maximal irreducible measure} $\psi$ such that any other irreducibility measure $\phi$ is absolutely continuous with respect to $\psi$.
For an irreducible kernel $M$ with a maximal irreducible measure $\psi$, there is a decomposition of the form \begin{equation}\label{m0}
M^{n_0} (x,A)= m(x,A) + g(x)\gamma(A),\quad \text{for all }x\in E, A\in \mathcal E,
\end{equation} where \begin{quote}
$\gamma$ is an irreducibility measure for $M$, \\
$g$ is a measurable non-negative function such that $\int g(x)\psi(dx)>0$, \\ $m$ is a another kernel on $(E,\mathcal E)$, \\
$n_0$ is a positive integer number. \end{quote} \begin{definition}
If \eqref{m0} holds with $n_0=1$, so that \begin{equation}\label{mrg}
M (x,A)= m(x,A) + g(x)\gamma(A),\quad x\in E,\quad A\in \mathcal E,
\end{equation} then the kernel $M$ is said to have an atom $(g,\gamma)$. \end{definition} Given \eqref{m0}, put \begin{equation}\label{eqR}
F(s)= \sum_{n=1}^\infty F_ns^{n},\quad F_n= \iint g(y)M^{n-1}(x,\mathrm{d} y)\gamma(d x). \end{equation}
\begin{definition}\label{defR}
Define the convergence parameter $R\in[0,\infty)$ of the irreducible kernel $M$ by \[F(s)<\infty \text{ for } s<R, \text{ and }F(s)=\infty \text{ for } s>R.\] If $F(R)<\infty$, then kernel $M$ is called $R$-transient, if $F(R)=\infty$, then kernel $M$ is called $R$-recurrent. \end{definition}
\begin{definition}
A non-negative measurable function $h$ which is not identically infinite is called $R$-subinvariant for $M$ if
\[h(x)\ge R\int h(y)M(x,\mathrm{d} y),\quad \text{for all } x\in E.\]
An $R$-subinvariant function is called $R$-invariant if
\[h(x)= R\int h(y)M(x,\mathrm{d} y),\quad \text{for all } x\in E.\]
A measure $\pi\in\mathcal M^+$ such that $\int g(y)\pi (dy)\in(0,\infty)$ is called $R$-subinvariant for $M$ if
\[\pi(A)\ge R\int M(x,A)\pi(dx),\quad \text{for all } A\in \mathcal E.\]
An $R$-subinvariant meaure is called $R$-invariant if
\[\pi(A)= R\int M(x,A)\pi(dx),\quad \text{for all } A\in \mathcal E.\]
\end{definition}
Suppose $M$ is $R$-recurrent. The function $h$ and the measure $\pi$ defined by \begin{align} h(x)=\sum_{n=1}^\infty R^{nn_0}\int g(y)m^{n-1}(x,\mathrm{d} y),\qquad \pi(A)=\sum_{n=1}^\infty R^{nn_0}\int m^{n-1}(x,A)\gamma(\mathrm{d} x) \label{Lus} \end{align} are $R$-invariant for $M$, scaled in such a way that \begin{align}\label{earl} \int h(x)\gamma(\mathrm{d} x)= \int g(y)\pi(\mathrm{d} y)=1. \end{align} For any $R$-subinvariant function $\tilde h$ satisfying $\int \tilde h(x)\gamma(\mathrm{d} x)=1$, we have \[\tilde h=h\quad \psi\text{-everywhere} \quad \text{and}\quad \tilde h\ge h\quad \text{everywhere}.\] The measure $\pi$ is the unique $R$-subinvariant measure satisfying \eqref{earl}.
\begin{definition}\label{posi} An $R$-recurrent kernel $M$ is called $R$-positive recurrent if the $R$-invariant function and measure $(h,\pi)$ satisfy $ \int h(y)\pi(dy)<\infty$. If $ \int h(y)\pi(dy)=\infty$, then $M$ is called $R$-null recurrent. \end{definition}
\begin{definition}
Kernel $M$ has period $d$ if $d$ is the smallest positive integer such that there is a sequence of non-empty disjoint sets $(D_0,D_1,\ldots D_{d-1})$ having the following property
\[ \text{if } x\in D_i, \text{ then } M(x,E\setminus D_j)=0 \quad \text{for }j=i+1\ ({\rm mod}\ d), \quad\ i=0,\ldots, d-1.\]
We call kernel $M$ aperiodic if its period $d=1$. \end{definition} In the periodic case with $d\ge2$, provided $M$ is irreducible and satisfies \eqref{m0}, there is an index $i$, $0\le i\le d-1$, such that $g=0$ over all $D_j$ except $D_i$. Furthermore, \[\gamma(E\setminus D_j)=0 \text{ for } j=i+n_0\ ({\rm mod}\ d).\]
\section{ GW processes with clusters } \label{xip} As will be explained later in this section, the following definition yields the above mentioned split chain construction in the particular case when $\mathrm{P}(\Xi^{(x)}(E)=1)=1$. \begin{definition}\label{split} Consider a multi-type GW process $\{Z_n\}_{n=0}^\infty $ whose reproduction measure can be decomposed into a sum of a random number of integer-valued random measures
\begin{equation}\label{deco} \Xi^{(x)}=\xi^{(x)}+\sum_{i=1}^{N^{(x)}}\tau_i.
\end{equation} Let each $\tau_i$ be independent of everything else and have a common distribution $ \tau_i\stackrel{d}=\tau$.
(i) Such a multi-type GW process will be called a GW process with clusters.
(ii) Each group of particles behind a measure $\tau_i$ in \eqref{deco} will be called a cluster, so that $N^{(x)}$ gives the number of clusters produced by a single particle of type $x$. Simple clusters correspond to the case $\mathrm{P}(\tau(E)=1)=1$.
(iii) A multi-type GW process with the reproduction measure $\xi^{(x)}$
will be called a stem process. \end{definition} Given \eqref{deco} and \begin{equation}\label{decc} \mathrm{E} \xi^{(x)}(A)=m(x,A),\quad \mathrm{E} N^{(x)}=g(x),\quad \mathrm{E} \tau(A)=\gamma(A),
\end{equation} by the total expectation formula, we see that the kernel \eqref{rke} satisfies \eqref{mrg}. Note that we allow for dependence between $\xi^{(x)}$ and $N^{(x)}$.
Definition \ref{split} puts no restrictions on the reproduction kernel $m$ of the stem process.
The example from Section \ref{xmpl} presents a case with $E=[0, \infty)$, where the kernel $m$ is reducible, in that for any ordered pair of types $(x,y)$, where $x<y$, type $x$ particles (within the stem process) may produce type $y$ particles but not otherwise.
Consider a GW process with simple clusters such that \begin{align*} & \mathrm{P}(\xi^{(x)}(E)=0)=g(x),\quad \mathrm{P}(\xi^{(x)}(E)=1)=1-g(x), \quad g(x)\in[0,1],\\
&N^{(x)}=1_{\{\xi^{(x)}(E)=0\}},\quad x\in E,\\
&\tau=\delta_{Y},\quad \mathrm{P}(Y\in A)=\gamma(A). \end{align*} In this case each particle produces exactly one offspring, and the GW process tracks the type of the regenerating particle. Using \eqref{mrg}, we find that $M=P$ is a stochastic kernel satisfying \eqref{spl} with
$$p(x,A)=(1-g(x))\mathrm{P}(\xi^{(x)}(A)=1|\xi^{(x)}(E)=1).$$ As a result we get a split chain corresponding to a stochastic kernel. Notice that the associated stem process is a pure death multi-type GW process.
An important family of GW processes with simple clusters is formed by linear-fractional multi-type GW processes, see \cite{LS, Sa}. This family is framed by the following additional conditions \begin{align*} & \mathrm{P}(\xi^{(x)}(E)=0)+ \mathrm{P}(\xi^{(x)}(E)=1)=1,\\
&N^{(x)}=N\cdot 1_{\{\xi^{(x)}(E)=1\}},\qquad \text{ where $N$ has a geometric distribution},\\
&\tau=\delta_{Y}. \end{align*} In this case \eqref{mrg} holds with \[m(x,A)=\mathrm{P}(\xi^{(x)}(A)=1),\quad g(x)=\mathrm{E} N\cdot \mathrm{P}(\xi^{(x)}(A)=1),\quad \gamma(A)=\mathrm{P}(Y\in A).\] Here again, the stem process is a pure death multi-type GW process.
\section{Embedded CMJ process } \label{eCrump-Mode-Jagers }
The key assumption of Definition \ref{split} guarantees that the procreation of particles constituting a cluster is independent of the other parts of the GW process with clusters. The main idea of this paper is to treat each cluster as a newborn CMJ individual, which reminds the construction of macro-individuals in the sibling dependence setting of \cite{Ol}.
Consider the stem process starting from a single cluster at time 0 and denote by $L\in[1,\infty]$ its extinction time. Put $X_0=1$ and let $X_n$ stand for the number of new clusters generated at time $n$ by the particles in the stem process born at time $n-1$, $n\ge1$. Observe that
\begin{align*} f_n&:= \mathrm{E}(X_n)=\iint g(y)m^{n-1}(x,d y)\gamma(d x). \end{align*} We treat the random vector $(X_1,\ldots, X_{L})$ as the life record of the initial individual in an embedded CMJ process, see Figure \ref{F1}. A CMJ individual during its life of length $L$ at different ages produces random numbers of offspring, cf \cite{JS}. Such independently reproducing CMJ individuals build a population with overlapping generations (in contrast to GW particles living one unit of time, so that there is no time overlap between generations).
\begin{figure}
\caption{Embedding a CMJ individual into a multi-type GW process stemming from a single cluster of size $Z_0=3$.
{\it Left panel}.
Solid lines represent the lineages of the stem process which dies out by time $L=6$. Dashed lines delineate the daughter clusters directly generated by the stem process. We see that $X_1=3$ with $\tau_1(E)=0$, $\tau_2(E)=1$, $\tau_3(E)=3$. {\it Right panel}. The summary of the individual life: $(X_1,\ldots,X_{L})=(3,2,2,0,2,1)$. }
\label{F1}
\end{figure}
Throughout this paper we assume \begin{equation}\label{rpo} f(s_0)\in( 0 , \infty) \text{ for some }s_0>0, \text{ where }f(s)= \sum_{n=1}^\infty f_ns^{n}, \end{equation} so that on one hand, that $ f_n>0$ for some $n\ge1$, and on the other hand, the radius of convergence \[r= \inf \{ s \ge 0 \colon f(s) = \infty \}\]
is positive. The assumption $r>0$ prohibits very fast growing sequences of the type $f_n=e^{n^2}$. \begin{definition}\label{dR} Given \eqref{rpo}, define a parameter $R\in(0,\infty)$ as $R=r$ if $f(r)<1$, and as the unique positive solution of the equation $f(R)=1$ if $f(r)\ge1$. \end{definition} Since $f(R)\le1$, the sequence $(f_nR^{n})$ can be viewed as a (possibly defective) distribution on the lattice $\{1,2,\ldots\}$. This is the distribution of the inter-arrival time for the renewal process naturally embedded into the CMJ process defined above. The renewal process is interpreted as the consecutive ages at childbearing as one tracks a single ancestral lineage backwards in time.
Given $f(R)=1$, the mean inter-arrival time for the embedded renewal process equals \[\sum_{n = 1}^\infty nf_nR^{n}=Rf'(R),\] and is interpreted as the average age at childbearing or the mean generation length for the CMJ process, see \cite{J}.
Focussing on the current waiting time of such a discrete renewal process, we get an irreducible Markov chain with the state space $\{0, 1,\ldots\}$. The following observation concerning this Markov chain is straightforward. \begin{proposition} The embedded renewal process is transient if $f(r)<1$, and recurrent if $f(r)\ge1$. Let $R$ be defined by Definition \ref{dR}. If $f(r)> 1$, then $R\in(0,r)$, $f'(R)<\infty$, and the embedded renewal process is positive recurrent. If $f(r)=1$, then the embedded renewal process is either positive recurrent or null recurrent depending on whether $f'(r)<\infty$ or $f'(r)=\infty$. \end{proposition} Let $W_n$ be the number of newborn individuals at time $n$ in the embedded CMJ process started from a single newborn individual, or in other words, the total number of clusters emerging at time $n$ in the original GW process starting from a single cluster. Clearly, \begin{align*} F_n&:= \mathrm{E}(W_n)= \iint g(y)M^{n-1}(x,\mathrm{d} y)\gamma(d x). \end{align*} \begin{theorem}\label{iv} Consider a kernel $M$ with atom $(g,\gamma)$. Parameter $R$ from Definition \ref{dR} coincides with the convergence parameter of the kernel $M$. Moreover,
(i) if $f(r)<1$, then $R=r$, $f(R)<1$, and $F(R)<\infty$, so that $M$ is $R$-transient,
(ii) if $f(r)\ge1$, then $f(R)=1$ and $F(R)=\infty$, so that $M$ is $R$-recurrent,
(iii) if $f(R)=1$, then either $f'(R)=\infty$ so that $M$ is $R$-null recurrent, or $f'(R)\in(0,\infty)$, so that $M$ is $R$-positive recurrent. \end{theorem} \begin{proof} Using the law of total expectation it is easy to justify the following recursion $$F_n=f_n+f_{n-1}F_1+\ldots+f_1F_{n-1}.$$ This leads to the equality for generating functions $$F(s)=f(s)+f(s)F(s),$$ which yields \begin{equation}\label{kFf}
F(s)={f(s)\over 1-f(s)}\quad \text{for $s$ such that } f(s)<1. \end{equation} From here and in view of Definition \ref{defR}, it is obvious that the first statement is valid. Parts (i) and (ii) follow immediately. Part (iii) is proven in Section \ref{sec}. \end{proof}
\noindent{\bf Remark. } For a general starting configuration of particles $Z_0$, putting $\mu_0=\mathrm{E} Z_0$, we get \begin{align*} \tilde f_n&:=\mathrm{E} X_n=\iint g(y)m^{n-1}(x,d y)\mu_0(d x),\\ \tilde F_n&:= \mathrm{E} Y_n=\iint g(y)M^{n-1}(x,\mathrm{d} y)\mu_0(d x). \end{align*} The corresponding generating functions $$\tilde f(s)= \sum_{n=1}^\infty \tilde f_ns^{n},\quad \tilde F(s)= \sum_{n=1}^\infty \tilde F_ns^{n},$$ are connected by \begin{equation}\label{tFf} \tilde F(s)={\tilde f(s)\over 1-f(s)}\quad \text{for $s$ such that } f(s)<1. \end{equation} (To obtain this relation, observe that $$\tilde F_n=\tilde f_n+\tilde f_{n-1}F_1+\ldots+\tilde f_1F_{n-1},$$ which gives $\tilde F(s)=\tilde f(s)(1+F(s))$, and it remains to apply \eqref{kFf}.)
As mentioned above, under the special initial condition $Z_0\stackrel{d}{=}\tau$, the embedded CMJ process starts from a single newborn individual. For a general $Z_0$, the embedded CMJ process has an immigration component characterised by the generating function $\tilde f(s)$. By immigration we mean the inflow of new clusters generated by the stem process starting from $Z_0$ particles.
\section{Null and positive recurrence of a kernel with atom}\label{sec}
Consider a non-negative kernel $M$ with atom $(g,\gamma)$, and put \[M_s(x,A)=\sum_{n=1}^\infty s^nM^{n-1}(x,A),\quad m_s(x,A)=\sum_{n=1}^\infty s^nm^{n-1}(x,A),\quad s\ge0,\] so that the earlier introduced generating functions $F$ and $f$ can be presented as
\begin{align*} F(s)=\iint g(y)M_s(x,\mathrm{d} y)\gamma(\mathrm{d} x),\quad f(s)=\iint g(y)m_s(x,\mathrm{d} y)\gamma(\mathrm{d} x). \end{align*} Denote \begin{align} h_s(x)=\int g(y)m_s(x,\mathrm{d} y),\qquad \pi_s(A)=\int m_s(x,A)\gamma(\mathrm{d} x),\label{Kus} \end{align} and observe that \begin{align*} \int h_s(x)\gamma(\mathrm{d} x)= \int g(y)\pi_s(\mathrm{d} y)=f(s),\quad \int h_s(y)\pi_s(\mathrm{d} y)=s^2f'(s). \end{align*} The latter equality requires the following argument \begin{align*}
\int h_s(x)\pi_s(\mathrm{d} x)&=\iiint g(y)m_s(x,\mathrm{d} y)m_s(z,\mathrm{d} x)\gamma(\mathrm{d} z)
\\&=\iint g(y)m_s^2(z,\mathrm{d} y)\gamma(\mathrm{d} z) =\sum_{n=1}^\infty ns^{n+1}f_{n}=s^2f'(s), \end{align*} where we used the relation \begin{align*} s^{-2}m_s^2(y,A)&= \int s^{-1}m_s(x,A)s^{-1}m_s(y,\mathrm{d} x)=\sum_{n=0}^\infty\sum_{k=0}^\infty\int s^n m^{n}(x,A)s^km^{k}(y,\mathrm{d} x)\\
&=\sum_{n=0}^\infty\sum_{k=0}^\infty s^{n+k}m^{n+k}(y,A)=\sum_{j=0}^\infty (j+1)s^{j}m^{j}(y,A). \end{align*}
\begin{lemma}\label{eige} Consider a kernel with atom \eqref{mrg}. If a positive $s$ is such that $f(s)\le1$, then the function $h_s$ and the measure $\pi_s$, defined by \eqref{Kus}, satisfy
\begin{align} \int h_s(y)M(x,\mathrm{d} y)&=s^{-1}h_s(x)-(1-f(s))g(x),\label{st2}\\ \int M(y,A)\pi_s(\mathrm{d} y)&=s^{-1}\pi_s(A)-(1-f(s))\gamma(A),\label{st1} \end{align} so that they are $s$-subinvariant function and measure for the kernel $M$. \end{lemma}
\begin{proof} By \eqref{mrg}, we have
\begin{align*} \int m_s(y,A)M(x,\mathrm{d} y)&=\sum_{n=1}^\infty s^{n}m^{n}(x,A)+g(x)\int m_s(y,A)\gamma(\mathrm{d} y)\\ &=s^{-1}m_s(x,A)-\delta_x(A)+g(x)\pi_s(A), \end{align*} which implies relation \eqref{st2}:
\begin{align*} \int h_s(y)M(x,\mathrm{d} y)&=\iint g(w)m_s(y,\mathrm{d} w)M(x,\mathrm{d} y)=s^{-1}h_s(x)-g(x)+g(x)f(s). \end{align*} Similarly, from
\begin{align*} \int M(y,A)m_s(x,\mathrm{d} y)&=\sum_{n=1}^\infty s^{n}m^{n}(x,A)+\gamma(A)\int g(y)m_s(x,\mathrm{d} y) \\ &=s^{-1}m_s(x,A)-\delta_x(A)+\gamma(A)h_s(x), \end{align*} we arrive at relation \eqref{st1}.
\end{proof}
Lemma \ref{eige} yields the following statement which in turn provides the proof of part (iii) of Theorem \ref{iv} (recall Definition \ref{posi}). \begin{corollary} Consider an $R$-recurrent kernel $M$ with atom $(g,\gamma)$.
If $f(R)=1$, then $h=h_R$ and $\pi=\pi_R$ are $R$-invariant function and measure satisfying relation
\eqref{Lus} with $n_0=1$, relation \eqref{earl}, as well as $$
\int h(y)\pi(\mathrm{d} y)=R^2f'(R). $$
\end{corollary} Observe that \begin{equation}\label{hx}
h(x)=\sum_{n=1}^\infty R^n\int g(y)m^{n-1}(x,d y) \end{equation} is the expected $R$-discounted number of clusters ever produced by the stem process starting from a single particle of type $x$.
From this angle, $h(x)$ can be interpreted as the reproductive value of type $x$. On the other hand, \begin{equation}\label{pia} \pi(A)=\sum_{n=1}^\infty R^n\int m^{n-1}(x,A)\gamma(\mathrm{d} x). \end{equation}
is the expected $R$-discounted number of particles whose type belongs to $A$ and which appear in the stem process starting from a single cluster of particles. As shown next, see Theorem \ref{PF}, the measure $\pi$ can be viewed as an asymptotically stable distribution for the types of particles in the GW process with clusters.
\section{Perron-Frobenius theorem for kernels with atom}\label{genf} \begin{theorem}\label{PF} Consider an aperiodic $R$-positive recurrent kernel $M$ with atom $(g,\gamma)$. Let $h$ and $\pi$ be given by \eqref{hx} and \eqref{pia}. If $(x,A)$ are such that
\begin{equation}\label{extra}
R^{n}m^n(x,A)\to0,\quad n\to\infty, \end{equation}
then \begin{equation}\label{pefr}
R^{n}M^n(x,A)\to \dfrac {h(x)\pi(A)}{R^2f'(R)} ,\quad n\to\infty. \end{equation}
If $h(x)<\infty$, then condition \eqref{extra} holds for any $A$ such that \begin{equation}\label{eps} A\subset\{y: g(y)\ge\epsilon\} \text{ for some }\epsilon>0. \end{equation} \end{theorem} To prove this result we need two lemmas. In the end of this section we give a remark addressing condition \eqref{extra}.
\begin{lemma} Consider a kernel $M$ with atom $(g,\gamma)$. If $s>0$ is such that $f(s)<1$, then
\begin{align} M_s(x,A)&=m_s(x,A)+{h_s(x)\pi_s(A)\over 1-f(s)}\quad \text{for all }x\in E, A\in\mathcal E. \label{Ks} \end{align} \end{lemma} \begin{proof}
By \eqref{mrg}, we have the recursion
\begin{align*} M^n(x,A)&=g(x)\int M^{n-1}(y,A)\gamma(\mathrm{d} y)+\int M^{n-1}(y,A)m(x,\mathrm{d} y) \\
&=g(x)\int M^{n-1}(y,A)\gamma(\mathrm{d} y)+\int g(y)m(x,\mathrm{d} y)\int M^{n-2}(z,A)\gamma(\mathrm{d} z)\\ &\quad +\int M^{n-2}(z,A)m^2(x,dz)\\
&=\sum_{i=1}^{n}\int g(y)m^{i-1}(x,\mathrm{d} y)\int M^{n-i}(y,A)\gamma(\mathrm{d} y)+m^{n}(x,A), \end{align*} which
in terms of generating functions gives
\begin{align*} M_s(x,A)&=m_s(x,A)+h_s(x)\int M_s(y,A)\gamma(\mathrm{d} y), \end{align*} and after integration,
\begin{align*} \int M_s(x,A)\gamma(\mathrm{d} x)&={\pi_s(A)\over 1-f(s)}. \end{align*} Combining the last two relations we get \eqref{Ks}. Observe also that the last formula yields \eqref{kFf}.
\end{proof}
\begin{lemma} \label{Fe} Let $$a(s)=\sum_{n=0}^\infty a_ns^n,\quad b(s)=\sum_{n=0}^\infty b_ns^n,\quad c(s)=\sum_{n=0}^\infty c_ns^n,$$ be three generating functions for non-negative sequences connected by $$c(s)={b(s)\over1-a(s)}.$$ If sequence $\{a_n\}$ is aperiodic with $a(1)=1$, $a'(1)\in(0,\infty)$, then
\[c_n\to
\dfrac {b(1)}{a'(1)},\quad n\to\infty. \]
\end{lemma} \begin{proof}
This is a well-known result from Chapter XIII.4 in \cite{Fe1}. \end{proof}
\
\noindent{\sc Proof of Theorem \ref{PF}}. $R$-positive recurrence implies $f(R)=1$ and $f'(R)\in(0,\infty)$. Due to $f(R)=1$, we can rewrite \eqref{Ks} as \[M_{\hat s}(x,A)-m_{\hat s}(x,A)={b(s)\over1-a(s)},\]
where $\hat s=sR$ and \[a(s)=f(sR),\quad b(s)=h_{\hat s}(x)\pi_{\hat s}(A),\] so that $a'(1)=Rf'(R)$, $b(1)=h(x)\pi(A)$. Applying Lemma \ref{Fe}, we find that as $n\to\infty$,
\[R^{n}(M^{n}(x,A)-m^{n}(x,A))\to \dfrac {h(x)\pi(A)}{R^2f'(R)}. \] which combined with condition \eqref{extra} yields the main assertion. The stated sufficient condition for \eqref{extra} is verified using \[\sum_{n=1}^\infty R^nm^{n-1}(x,A)\le \sum_{n=1}^\infty R^n\int1_{\{y:g(y)>\epsilon\}}m^{n-1}(x,dy)\le\epsilon^{-1}h(x)<\infty. \]
$\Box$
\noindent{\bf Remark}. To illustrate the role of the condition \eqref{extra}, consider the kernel \eqref{mrg} with $$m(x,A)=g_1(x)\gamma_1(A),$$ assuming $$\int g_1(x)\gamma_1(dx)=a_1,\quad \int g(x)\gamma(dx)=a,\quad \int g_1(x)\gamma(dx)= \int g(x)\gamma_1(dx)=0,$$ where $a_1>a>0$. In this particular case, we have $$M^n(x,A)=m^n(x,A)+a^ng(x)\gamma(A),\quad m^n(x,A)=a_1^ng_1(x)\gamma_1(A),$$ and clearly, \[ M^n(x,A)\sim\left\{ \begin{array}{ll}
a_1^ng_1(x)\gamma_1(A), & \text{if }g_1(x)\gamma_1(A)>0, \\
a^ng(x)\gamma(A), & \text{if }g_1(x)\gamma_1(A)=0 \text{ and }g(x)\gamma(A)>0.
\end{array} \right. \] Turning to the generating function defined by \eqref{eqR} we find
\[ F_n= \iint g(y)M^{n-1}(x,\mathrm{d} y)\gamma(d x)=a^{n+1},\quad F(s)={a^2s\over 1-as}. \] This yields $R=a^{-1}$ and we see that condition \eqref{extra} is not valid for $(x,A)$ such that $g_1(x)\gamma_1(A)>0$. On the other hand, if $g(x)<\infty$ and $A$ satisfies \eqref{eps}, then \[0=\int g(x)\gamma_1(dx)\ge \int_A g(x)\gamma_1(dx)\ge \epsilon \gamma_1(A), \] so that $\gamma_1(A)=0$ and therefore $R^nM^n(x,A)\to g(x)\gamma(A)$.
\section{3-parameter GW process with clusters } \label{xmpl}
Here we construct a transparent example of a GW process with clusters having the type space $E = [0, \infty)$. Its positive recurrent reproduction kernel is fully specified by just three parameters $a,c\in(0,\infty)$, and $b\in(-1,\infty)$: \[M(x,\mathrm{d} y)=ae^{x-y}1_{\{y\ge x\}}\mathrm{d} y+ce^{-bx}\delta_0(\mathrm{d} y).\] This kernel satisfies \eqref{mrg} with \begin{equation}\label{emgg}
m(x,\mathrm{d} y)=ae^{x-y}1_{\{y\ge x\}}\mathrm{d} y,\quad g(x)=ce^{-bx},\quad \gamma(A) = \delta_0(A), \end{equation} implying that each cluster consists of a single particle of type 0.
The full specification of our example refers to a continuous time Markov branching process modeling the size of a population of {\it Markov particles} having the unit life-length mean and offspring mean $a$. The main idea is to count the Markov particles generation-wise, and to define the type of a Galton-Watson particle as the birth-time of the corresponding Markov particle. The corresponding stem process $\{\xi_n\}_{n\ge0}$ is defined by \begin{quote} $\xi_n(A)=$ the number of $n$-generation Markov particles born in the time period $A$, \end{quote} so that its conditionally on the parent's birth time $x$,
\[m^n(x,[0,t])= a^n \mathbb P(x+T_1+\ldots+T_n\le t)= a^n \mathbb P(N_{t-x}\ge n), \quad \text{for }t>x,
\]
where $T_i$ are independent exponentials with unit mean and $\{N_t\}_{t\ge0}$ is the standard Poisson process.
\begin{proposition} Consider the above described multi-type GW process with clusters characterised by \eqref{emgg}. Then we have
\begin{align}\label{tre} f(s) ={rcs\over r- s}, \qquad r={1+b\over a}, \qquad R={r\over 1+cr}. \end{align} The process is supercritical if $c>{r-1\over r}$, critical if $c={r-1\over r}$, and subcritical if $c<{r-1\over r}$.
Convergence \eqref{pefr} holds for $A=[0,t]$, $t\in [0,\infty)$, with the right hand side equal to
\[e^{-bx}(R\delta_0(\mathrm{d} y)+aR^2e^{(aR-1)y}\mathrm{d} y).\] If $Ra<1$, then \eqref{pefr} holds even for $A=E$ with the right hand side equal to ${Re^{-bx}\over 1-aR}$.
\end{proposition} \begin{proof} Referring to the underlying Poisson process, we find that for $s\ne1/a$, \begin{align*} m_s(0,[0,t])&= s\sum _{n=0}^\infty s^n a^{n}\sum _{k=n}^\infty \mathbb P(N_{t}=k)= s \sum _{k=0}^\infty \mathbb P(N_{t}=k){ 1-(as)^{k+1} \over 1-as}\\ & = { s \over 1-as}(1- as\mathbb E(as)^{N_t}) = { s \over 1-as}(1- ase^{t(as-1)}). \end{align*} More generally, we have $$ m_s(x,[0,t])= m_s(0,[0,t-x])= { s \over 1-as}(1- ase^{(t-x)(as-1)})1_{\{t\ge x\}} , $$ so that \begin{align*} m_s(x,\mathrm{d} y)=s\delta_x(\mathrm{d} y)+as^2e^{(as-1)(y-x)}1_{\{y\ge x\}}\mathrm{d} y. \end{align*}
By \eqref{Kus} \begin{align*} h_s(x)=\int g(y)m_s(x,\mathrm{d} y)= sce^{-bx}+ cas^2\int_{ 0}^\infty e^{h(as-1)}e^{-b(x+u)}du
=f(s)e^{-bx}, \end{align*} where $f(s)$ satisfies \eqref{tre}. Since $f(r)=\infty$, the stated value $R={r\over 1+cr}$ is found from the equation $f(R)=1$.
Applying once again \eqref{Kus}, we find \begin{align*} \pi_s(\mathrm{d} y)=\int m_s(x,\mathrm{d} y)\gamma(\mathrm{d} x)=m_s(0,\mathrm{d} y)=s\delta_0(\mathrm{d} y)+as^2e^{(as-1)y}\mathrm{d} y. \end{align*} To check this and previously obtained expressions, we verify the general formula for the integral \begin{align*}
\int h_s(x)\pi_s(\mathrm{d} x)&=sf(s)+ f(s)as^2\int e^{-(1+b)x}e^{asx}\mathrm{d} x={rsf(s)\over r-s}={r^2s^2\over (r-s)^2}=s^2f'(s).
\end{align*}
With \begin{align*} h(x)&= e^{-bx},\qquad \pi(\mathrm{d} x)=R\delta_0(\mathrm{d} x)+aR^2e^{(aR-1)x}\mathrm{d} x, \end{align*} Theorem \ref{PF} specialised to the current example says that for $t\in [0,\infty)$, \begin{align*} R^{n}M^n(x,[0,t])&\to e^{-bx}(R+aR^2\int_0^t e^{(aR-1)y}\mathrm{d} y)\\ &=e^{-bx}(R+ {aR^2\over aR-1}(e^{(aR-1)t}-1))=e^{-bx}{aR^2e^{(aR-1)t}-R\over aR-1},\quad n\to\infty.
\end{align*} If $aR<1$ and $A=E$, then condition \eqref{extra} holds since $$\pi(E)=R+{aR^2\over 1-aR}={R\over 1-aR}<\infty,$$ and $R^{n}m^n(x,E)=(Ra)^n\to0$. \end{proof}
\noindent {\bf Remark.} If we further specialize this example by letting the stem process to be the Yule process, then we have $a=2$. If furthermore, $b=2$ and $c<{r-1\over r}={1\over 3}$, then the corresponding GW process with clusters is subcritical, despite the total number of particles in the Yule process is infinite.\\
\noindent {\bf Acknowledgements.} The author is grateful for critical remarks of an anonymous reviewer of an earlier version of the paper.
\end{document} | arXiv |
\begin{document} \title{Uniqueness of smooth extensions of generalized cohomology theories} \begin{abstract} We provide an axiomatic framework for the study of smooth extensions of generalized cohomology theories. Our main results are about the uniqeness of smooth extensions, and the identification of the flat theory with the
associated cohomology theory with $\mathbb{R}/\mathbb{Z}$-coefficients.
\textcolor{black}{In particular, we show that there is a unique smooth extension of K-theory and of MU-cobordism with a unique multiplication, and that the flat theory in these cases is naturally isomorphic to the homotopy theorist's version of the cohomology theory with $\mathbb{R}/\mathbb{Z}$-coefficients. For this we only require a small set of natural compatibility conditions.} \end{abstract}
\tableofcontents
\section{Axioms}
A smooth extension of a generalized cohomology theory $E$ is a refinement $\hat E$ of the restriction of $E$ to the category of smooth manifolds. The functor $\hat E$ is no longer homotopy invariant. A class $\hat x\in \hat E(M)$ which refines the underlying topological class $I(\hat x)\in E^*(M)$ contains the information about a closed differential form
$R(\hat x)\in \Omega_{cl}^*(M,E^*\otimes_\mathbb{Z}\mathbb{R})$ which represents the image of $I(\hat x)$ under the natural map ${\mathbf{ch}}\colon E^*(M)\to H^*(M;E^*\otimes_\mathbb{Z}\mathbb{R})$. The deviation of $\hat E$ from homotopy invariance is described by a homotopy formula (Lemma \ref{udqwdqwdqw1}). Let $\hat x\in \hat E^{*+1}([0,1]\times M)$ and $f_0,f_1\colon M\to [0,1]\times M$ be the inclusions of the endpoints. Then \begin{equation}\label{udqwdqwdqw1} f_1^*\hat x-f_0^*\hat x=a(\int_{[0,1]\times M/M} R(\hat x))\ , \end{equation} where $a$ is the natural transformation \ref{ddd1}.A.4.
A typical and motivating example is the smooth extension $\widehat{H\mathbb{Z}}^*$ of integral cohomology. The group $\widehat{H\mathbb{Z}}^2(M)$ can be identified with the group of isomorphism classes $[L,h^L,\nabla^L]$ of hermitean line bundles on $M$ with unitary connections with the tensor product as the group operation. We have $I([L,h^L,\nabla^L])=c_1(L)\in H\mathbb{Z}^2(M)$, the first Chern class of $L$, and $R([L,h^L,\nabla^L])=-\frac{1}{2\pi i}R^{\nabla^L}\in \Omega^2_{cl}(M)$, the first Chern form. Unlike the first Chern class $c_1(L)\in H\mathbb{Z}^2(M)$, the class $[L,h^L,\nabla^L]\in \widehat{H\mathbb{Z}}^2(M)$ captures secondary information, e.g.\ the holonomy of $\nabla^L$ which might be non-trivial even if $L$ is trivial and $\nabla^L$ is flat. Refined characteristic classes for flat bundles were one of the first motivations for the introduction of $\widehat{H\mathbb{Z}}$ in \cite{MR827262}.
The space of hermitean line bundles with unitary connections is the configuration space of Maxwell field theory, i.e. the gauge theory with structure group $U(1)$. In this field theory the field strength is a closed two-form which satisfies the following quantization condition: The
integral of the field strength over cycles is required to be integral. In the past decade the discussion of models of string theory with branes lead to field theories with $p$-form field strength. Furthermore, the quantization conditions motivated the consideration of underlying cohomology theories different from ordinary cohomology theory like $K$-theory, see e.g. \cite{freed-2000}, \cite{freed-2000-0005}, \cite{moore-2000-0005}.
The use of smoothly extended cohomology groups as configuration spaces in field theories, the topological considerations in \cite{MR2192936}, and further developments on secondary invariants (see e.g. \cite{bunke-2002} and the literature cited therein) lead to the development of this circle of ideas to a mathematical theory. The present paper contributes to this theory by presenting axioms for smooth extensions and showing that they imply uniqeness results in many interesting cases.
We consider a generalized cohomology theory $E$ represented by a spectrum ${\bf E}$. It gives rise to the $\mathbb{Z}$-graded abelian group $E^*:=E^*(*)=\pi_{-*}{\bf E}$, and we define the $\mathbb{Z}$-graded $\mathbb{R}$-vector space ${\tt V}:=E^*\otimes_\mathbb{Z} \mathbb{R}$. For a smooth manifold $M$ we define $\Omega^*(M,{\tt V}):=C^\infty(M,\Lambda^*T^*M\otimes_\mathbb{R} {\tt V})$ with the $\mathbb{Z}$-grading by the total degree. To be more precise, in the case of infinite-dimensional ${\tt V}^n$ we topologize ${\tt V}^n$ as a colimit of its finite-dimensional subspaces with their canonical real vector space topologies. Locally an element of $\Omega^*(M,{\tt V}^n)$ can then be written as a finite sum $\sum_{j} \omega_j\otimes v_j$ for collections of forms $\omega_j\in \Omega^*(M)$ and elements $v_j\in {\tt V}^n$. We let $d\colon \Omega^*(M,{\tt V})\to \Omega^{*+1}(M,{\tt V})$ be the de Rham differential, and we write $\Omega_{cl}^{*}(M,{\tt V}):=\ker(d\colon \Omega^*(M,{\tt V})\to \Omega^{*+1}(M,{\tt V}))$ for the subspace of closed forms. We identify $H^*(M;{\tt V})$ with the singular cohomology of $M$ with coefficients in ${\tt V}$. Integration over simplices induces the natural transformation $${\tt Rham}\colon \Omega_{cl}^{*}(M,{\tt V})\to H^*(M;{\tt V})\ .$$ It induces an isomorphism $H^*_{dR}(M;{\tt V})\stackrel{\sim}{\to} H^*(M;{\tt V})$. Furthermore, there is a canonical natural transformation of cohomology theories $${\mathbf{ch}}\colon E^*(X)\to H^*(X;{\tt V})\ .$$
\begin{ddd}\label{ddd1} A \textbf{smooth extension} of the generalized cohomology theory $E$ is a quadruple $(\hat E,R,I,a)$, where \begin{enumerate} \item[A.1] $\hat E$ is a contravariant functor from the category of smooth manifolds to $\mathbb{Z}$-graded abelian groups. Sometimes we will consider a version defined only on the category of compact manifolds (possibly with boundary). \item[A.2] $R$ is a natural transformation of $\mathbb{Z}$-graded abelian group-valued functors $$R\colon \hat E^*(M)\to \Omega^*_{cl}(M,{\tt V})\ .$$ \item[A.3] $I$ is a natural transformation of $\mathbb{Z}$-graded abelian group-valued functors $$I\colon \hat E^*(M)\to E^*(M)\ .$$ \item[A.4] $a$ is a natural transformation of $\mathbb{Z}$-graded abelian group-valued functors $$a\colon \Omega^{*-1}(M,{\tt V})/{\tt im}(d)\to \hat E^{*}(M)\ .$$ \end{enumerate} These objects have to satisfy the following relations: \begin{enumerate} \item[R.1] $R\circ a=d$
\item[R.2] For all manifolds $M$ the diagram $$\xymatrix{\hat E^*(M)\ar[d]^{I}\ar[r]^{R}&\Omega_{cl}^*(M,{\tt V})\ar[d]^{{\tt Rham}}\\E^*(M)\ar[r]^{{\mathbf{ch}}}&H^*(M;{\tt V})}$$ commutes. \item[R.3] For all manifolds $M$ the sequence \begin{equation}\label{ddd13}E^{*-1}(M)\stackrel{{\mathbf{ch}}}{\to} \Omega^{*-1}(M)/{\tt im}(d)\stackrel{a}{\to} \hat E^*(M)\stackrel{I}{\to} E^*(M)\to 0 \end{equation} is exact. \end{enumerate} \end{ddd}
We now consider two smooth extensions $(\hat E,R,I,a)$ and $(\hat E^\prime,R^\prime,I^\prime,a^\prime)$ of $E$. \begin{ddd} A \textbf{natural transformation of smooth extensions} is a natural transformation of $\mathbb{Z}$-graded abelian group valued functors $\Phi\colon \hat E^*\to \hat E^{\prime *}$ such that the following diagram commutes for every manifold $M$: $$\xymatrix{\Omega^{*-1}(M,{\tt V})\ar[r]^a\ar@{=}[d]&\hat E^*(M)\ar[d]^{\Phi}\ar[r]^{I}\ar@/^1cm/[rr]^R&E^*(M)\ar@{=}[d]&\Omega^*_{cl}(M,E)\ar@{=}[d]\\\Omega^{*-1}(M,{\tt V})\ar[r]^{a^\prime}& \hat E^{\prime*}(M)\ar[r]^{I^\prime}\ar@/_1cm/[rr]^{R^\prime}&E^*(M)&\Omega^*_{cl}(M,E)} \ .$$ \end{ddd}
We consider the inclusion of the base point $*\to S^1$. It induces an embedding $i\colon M\to S^1\times M$ for every manifold $M$. Let $p\colon S^1\times M\to M$ be the projection onto the second factor. Since $p\circ i={\tt id}_{M}$ we get splittings $$\hat E^*(S^1\times M)\cong {\tt im}(p^*)\oplus \ker(i^*) ,\quad E^*(S^1\times M)\cong {\tt im}(p^*)\oplus \ker(i^*)\ .$$ Let $\Sigma M_+$ be the suspension which is a space, not a manifold. There is a natural projection $q\colon S^1\times M\to \Sigma M_+$ of spaces. It induces an isomorphism $$q^*\colon \tilde E^*(\Sigma M_+)\stackrel{\sim}{\to} \ker(i^*)\subseteq E^*(S^1\times M)\ .$$ Furthermore, there is the suspension isomorphism $$\sigma\colon E^{*-1}(M)\stackrel{\sim}{\to} \tilde E^*(\Sigma M_+)\ .$$ Composing these isomorphisms with the projection onto $\ker(i)$ we get the integration map $$\int\colon E^{*+1}(S^1\times M)\stackrel{{\tt pr}}{\to} \ker(i^*)\stackrel{(q^{*})^{-1}}{\to} \tilde E^{*+1}(\Sigma M_+)\stackrel{\sigma^{-1}}{\to} E^{*}(M)$$ for the generalized cohomology theory $E$.
We introduce the notation $SF(M):=F(S^1\times M)$ for a functor $F$ defined on manifolds. The integration $$\int\colon SE^{*+1}\to E^*$$ just defined is complemented by an integration map $$\int\colon S\Omega^{*+1}(\dots,{\tt V})\to \Omega^{*}(\dots,{\tt V})$$ for differential forms which preserves the image and kernel of $d$.
\begin{ddd}\label{ddd4} A smooth extension \textbf{with integration} of $E$ is a quintuple $(\hat E,R,I,a,\int)$, where $(\hat E,R,I,a)$ is a smooth extension of $E$ and $\int$ is a natural transformation $$\int\colon S\hat E^{*+1}\to \hat E^*$$ such that \begin{enumerate} \item $\int \circ (t^*\times {\tt id})^*=-\int$, where $t\colon S^1\to S^1$ is given by $t(z):=\bar z$. \item \label{ddd41} $\int\circ p^*=0$ and \item the diagram
$$\xymatrix{S\Omega^{*}(M,{\tt V})\ar[r]^a\ar[d]^\int&S\hat E^{*+1}(M)\ar[d]^{\int}\ar[r]^{I}\ar@/^1cm/[rr]^R&SE^{*+1}(M)\ar[d]^\int&S\Omega^{*+1}_{cl}(M,E)\ar[d]^\int\\\Omega^{*-1}(M,{\tt V})\ar[r]^{a}& \hat E^{*}(M)\ar[r]^{I}\ar@/_1cm/[rr]^{R}&E^*(M)&\Omega^*_{cl}(M,E)} $$ commutes for all manifolds $M$.\end{enumerate} \end{ddd}
We now consider two smooth extensions with integration $(\hat E,R,I,a,\int)$ and $(\hat E^\prime,R^\prime,I^\prime,a^\prime,\int^\prime)$ of $E$. \begin{ddd} A \textbf{natural transfomation between two extensions with integration} of $E$ is a natural transfomation between smooth extensions $\Phi\colon \hat E^*\to \hat E^{\prime,*}$ such that $$\xymatrix{S\hat E^{*+1}(M)\ar[d]^\int\ar[r]^\Phi&S\hat E^{\prime*+1}(M)\ar[d]^{\int^\prime}\\\hat E^*(M)\ar[r]^{\Phi}&\hat E^{\prime*}(M)}$$ commutes for all manifolds $M$. \end{ddd}
Assume now that $E$ is a multiplicative cohomology theory. Then the functor $E^*$ has values in graded commutative rings. In particular, $E^*$ is a $\mathbb{Z}$-graded ring, and $H^*(M;{\tt V} )$ and $\Omega^*(M,{\tt V} )$ are $\mathbb{Z}$-graded rings as well. In this case we can make the following definition. \begin{ddd} A \textbf{multiplicative smooth extension} is a smooth extension $(\hat E,R,I,a)$ such that $\hat E^*$ takes values in $\mathbb{Z}$-graded commutative rings, the transformations $R$ and $I$ are multiplicative, and the identity $$x\cup a(\alpha)=a(R(x)\wedge \alpha)\:\:\footnote{Observe, that $R(x)\wedge d\alpha=d(R(x)\wedge \alpha)$ so that the right-hand side is well-defined.}\ ,\quad x\in \hat E^*(M)\ , \quad \alpha\in \Omega^{*}(M,{\tt V} )/{\tt im}(d)$$ holds true. \end{ddd}
For every generalized cohomology theory $E^*$ represented by a spectrum ${\bf E}$ a smooth extension $(\hat E,R,I,a)$ exists by the construction of Hopkins-Singer \cite{MR2192936}.
The historically first example of a smooth extension was constructed by Cheeger and Simons in \cite{MR827262} for ordinary integral cohomology $H\mathbb{Z}$ (and more generally for $H R$ for discrete subrings $R$ of $\mathbb{R}$). These extensions of ordinary cohomology are multiplicative. The classes in $\widehat{H R}(M)$ were realized in \cite{MR827262} as differential characters. By now there are various different constructions of the smooth extension of ordinary cohomology, e.g by sheaf theory \cite{MR1197353} (under the name smooth Deligne cohomology), using geometric cycles Gajer \cite{MR1423029}, cubical chains \cite{MR2179587}, or stratifold bordisms \cite{bks}. With differential characters the integration over $S^1$ as in Definition \ref{ddd4}, but also for general proper submersions $p\colon M\to B$ is simple, but the product is complicated. In cochain models both structures are involved, while in the sheaf-theoretic Deligne cohomology the product is easy, but integration is complicated. In the stratifold bordism model both structures are straight forward and explicit, and therefore this model is predestinated for the verification of the projection formula $\int_p (y\cup p^*x)=(\int_p y)\cup x$, where $y\in \widehat{H R}^*(M)$, $x\in \widehat{H R}^*(B)$.
In view of the variety of constructions of a smooth extension of ordinary cohomology it is a natural question whether all give equivalent results. This has been answered by \cite{MR2365651}, though a set of slightly different axioms is used\footnote{In \cite{MR2365651} the additional requirement is that the flat theory (Definition \ref{uiqwdqwdwqd}) is topological (Definition \ref{udiqdwqdwqd54545}).}. Uniqueness also follows from the axioms stated above by Theorem \ref{main1} if one takes the integration into account. This has first been observed by
Moritz Wiethaup (2006/2007). In both cases the smooth extension is unique up to unique isomorphism. Moreover, we have uniqueness of the product by \cite[Thm. 1.3]{MR2365651} or Theorem \ref{main23}.
In \cite{bunke-2007} we give a geometric construction of smooth extensions of bordism theories. We developed the details in the case of complex bordism $MU$. The method also applies to other bordism theories, e.g. oriented bordism, $MSpin$ or $MSpin^c$-bordism or framed bordism $S$. In all these cases we obtain a multiplicative extension and a theory of integration for suitably oriented proper submersions. The particular importance of the case of complex bordism theory comes from the Landweber exact functor theorem \cite{MR0423332}. It allows to construct a multiplicative smooth extension for every complex oriented Landweber exact cohomology theory. Examples are complex $K$-theory and certain elliptic cohomology theories.
Using methods of local index theory in \cite{bunke-20071} we have constructed a Dirac operator model of smooth $K$-theory which is again multiplicative and has a nice integration theory for smoothly $K$-oriented proper submersions.
The presence of different constructions (at least two in the case of bordism theories \cite{MR2192936} and \cite{bunke-2007}, and three in the case of $K$-theory \cite{MR2192936}, \cite{bunke-20071}, \cite{bunke-2007}) raises again the question whether they are equivalent. Moreover, for applications to topology, e.g. constructions of secondary invariants, of particular importance is the identification of the associated flat theory with the corresponding $\mathbb{R}/\mathbb{Z}$-theory.
To answer these questions is the main motivation and result of the present paper. Note that all these examples are rationally even (Definition \ref{uzdiqwdqwdwqd666}). The examples constructed from $MU$ by the Landweber exact functor theorem are only defined on the category of compact manifolds. This is the reason for considering this case in the present paper, too. Observe that the coefficients of a Landweber exact theory are torsion-free. Therefore a rationally even Landweber exact cohomology theory is even. This is exactly the additional assumption made in the statement of our uniqueness Theorems \ref{main1} and \ref{main23} in order to cover smooth extensions which are only defined on compact manifolds.
Let us now formulate the main results of the present paper. \begin{theorem}[Thm. \ref{main1}] Let $E$ be a rationally even generalized cohomology theory represented by a spectrum ${\bf E}$. Let $(\hat E,R,I,a,\int)$ and $(\hat E^\prime,R^\prime,I^\prime,a^\prime,\int^\prime)$ be two smooth extensions with integration. We assume that either the smooth extensions are defined on the category of all smooth manifolds and the coefficients $E^m$ are countably generated for all $m\in \mathbb{Z}$, or that $E^m=0$ for all odd $m\in \mathbb{Z}$ and $E^m$ is finitely generated for all even $m\in \mathbb{Z}$. Then there is a unique natural isomorphism $$\Phi\colon \hat E\to \hat E^\prime$$ of smooth extensions with integration. \end{theorem}
A multiplicative smooth extension of a rationally even cohomology theory has a canonical integration by Corollary \ref{uifqfqfqf}.
\begin{theorem}[Cor. \ref{main2}] Assume that $(\hat E,R,I,a)$ and $(\hat E^\prime,R^\prime,I^\prime,a^\prime)$ are two multiplicative extensions of a rationally even generalized cohomology theory with countably generated coefficients such that either both are defined on the category of all smooth manifolds or $E^*$ is even and degree-wise finitely generated. Then there is a unique natural isomorphism between these smooth extensions preserving the canonical integration. This transformation is multiplicative. \end{theorem} At the moment we have no feeling how important the condition of $E$ beeing rationally even is. This theorem applies e.g. to multiplicative smooth extensions of ordinary cohomology, all the bordism theories listed above and complex or real $K$-theory (for complex $K$-theory also to the version defined on compact manifolds).
The flat theory $$\hat E^*_{flat}(M):=\ker\left(R\colon \hat E^*(M)\to \Omega_{cl}^*(M,{\tt V} )\right)$$ is a homotopy invariant functor on smooth manifolds with values $\mathbb{Z}$-graded abelian groups. \begin{theorem}[Thm. \ref{zuddwqdqwdqw}] Assume that $E$ is rationally even with countably generated coefficients. If $(\hat E,R,I,a,\int)$ is a smooth extension of $E$ with integration which is defined on all smooth manifolds (or alternatively, on all compact manifolds and $E^*$ is even and degree-wise finitely generated), then there exists an isomorphism $$\Phi_{flat}\colon \hat E^*_{flat}\stackrel{\sim}{\to} E\mathbb{R}/\mathbb{Z}^{*-1}\ .$$ \end{theorem} For a more precise statement and the notation see Section \ref{uciacascc}. This theorem implies that the additional axiom in \cite{MR2365651} follows from our axioms together with the presence of integration. Theorem \ref{zuddwqdqwdqw} in particular states that the flat theory $\hat E^*_{flat}$ is a generalized cohomology theory. The essential additional datum turning a homotopy invariant functor into a generalized cohomology theory is the boundary operator of a long exact Mayer-Vietoris sequence. Theorem \ref{zuddwqdqwdqw} is proven by a comparison with the Hopkins-Singer theory \cite{MR2192936}. In Section \ref{e89wfoewfwefq} we give an independent construction of the Mayer-Vietoris sequence. \begin{theorem}[Thm. \ref{udidowqdiwqdwqd}]\label{uiewefwef}If $(\hat E,R,I,a,\int)$ is a smooth extension of $E$ with integration, then $\hat E^*_{flat}$ has a natural long exact Mayer-Vietoris sequence. Its restriction to compact manifolds is equivalent to the restriction to compact manifolds of a generalized cohomology \textcolor{black}{theory} represented by a spectrum. \end{theorem} Note that Theorem \ref{uiewefwef} does not require any additional assumptions, but it also does not state that $\hat E^*_{flat}$ is equivalent to $E\mathbb{R}/\mathbb{Z}^{*-1}$. This equivalence can be derived again under additional assumptions, now independently from \cite{MR2192936}, but at the cost of restricting to compact manifolds. \begin{theorem}[Thm. \ref{udqiwdqwdqwddwqdqwd1}]\label{udqiwdqwdqwddwqdqwd} If $(\hat E,R,I,a,\int)$ is a smooth extension of $E$ with integration and $E^*$ is torsion free, then we have a natural equivalence $\hat E^*_{flat}\cong E\mathbb{R}/\mathbb{Z}^{*-1}$ of functors restricted to the category of compact manifolds. \end{theorem}
\textit{Acknowledgement: The basic questions for the present paper have been the topic of a PhD project of Moritz Wiethaup (G\"ottingen). Some basic ideas and first results are due to him. In this paper we work out in detail and further develop the results of fruitful mathematical discussions in G\"ottingen arround 2006/2007.}
\section{Approximation of spaces by manifolds}\label{uziedwqeqw} \newcommand{{\mathcal{T}op}}{{\mathcal{T}op}}
The main technical problem of the present paper is the construction of a natural transformation between two smooth extensions $\hat E^k,\hat E^{\prime k}$ of a generalized cohomology $E^k$. This requires to define a homomorphism $\hat E^k(M)\to\hat E^{\prime
k}(M)$ for every smooth manifold in a natural way. If the underlying topological functor $E^k$ for fixed $k\in\mathbb{Z}$ would be represented by a manifold ${\bf E}$, naturality of the construction could be obtained by making one universal choice $\hat E^k({\bf E})\to\hat E^{\prime k}({\bf E})$, only. Unfortunately, a generalized cohomology functor $E^k$ is rarely represented by a finite-dimensional manifold ${\bf E}$. The idea for nevertheless cutting down the amount of choices in order to secure naturality is to approximate the classifying space ${\bf E}$ by a sequence of manifolds. Since in some examples our smooth extensions are only defined on compact manifolds we take care of the case where such an approximation can be realized by compact manifolds.
Recall that a map $f\colon X\to Y$ between pointed topological spaces is called $n$-connected if $f_*\colon \pi_k(X)\to \pi_k(Y)$ is an isomorphism for $k< n$ and surjective for $k=n$.
\begin{prop}\label{duwidwdwd}
Let ${\bf E}$ be a connected pointed topological space. If ${\bf E}$ is simply connected and $\pi_k({\bf E})$ is finitely generated for all $k\ge 2$, then there exist a sequence of compact pointed manifolds with boundary
$(\mathcal{E}_i)_{i\in \mathbb{N}}$ together with pointed maps $\kappa_i\colon \mathcal{E}_i\to \mathcal{E}_{i+1}$, $x_i\colon \mathcal{E}_{i}\to {\bf E}$ for all $i\in \mathbb{N}$ such that \begin{enumerate} \item\label{pr00} $\mathcal{E}_i$ is homotopy equivalent to an $i$-dimensional $CW$-complex, \item\label{pr01} the map $x_i$ is $i$-connected, \item\label{pr02} $\kappa_i\colon \mathcal{E}_i\hookrightarrow \mathcal{E}_{i+1}$ is an embedding of a submanifold, \item\label{pr03} the diagram $$\xymatrix{\mathcal{E}_{i}\ar[dr]_{x_i}\ar[rr]^{\kappa_i}&&\mathcal{E}_{i+1}\ar[dl]^{x_{i+1}}\\&{\bf E}&}$$ commutes, \item \label{la}for all finite-dimensional pointed $CW$-complexes $X$ the induced map $${\tt colim} ( [X,\mathcal{E}_{i}])\to [X,{\bf E}]$$ is an isomorphism. \end{enumerate} \end{prop} {\it Proof.$\:\:\:\:$} We first construct a similar sequence $(E_i)_{i\in \mathbb{N} }$ of connected finite \textcolor{black}{CW-}complexes together with maps $y_i\colon E_i\to {\bf E}$ and $\sigma_i\colon E_i\to E_{i+1}$ so that \begin{enumerate} \item $E_i$ is an $i$-dimensional \textcolor{black}{CW-}complex, \item the map $y_i$ is $i$-connected \item\label{pr2} $\sigma_i\colon E_i\hookrightarrow E_{i+1}$ is a cofibration, \item\label{pr3} the diagram $$\xymatrix{E_{i}\ar[dr]_{y_i}\ar[rr]^{\sigma_i}&&E_{i+1}\ar[dl]^{y_{i+1}}\\&{\bf E}&}$$ commutes. \end{enumerate} We let $E_{1}$ be a point. Assume that we have constructed $E_j$ for all $1\le j<i$ such that $\pi_{j}(E_{j})$ is finitely generated. Then we have a surjective map $$y_{i-1,*}\colon \pi_{i-1}(E_{i-1})\to \pi_{i-1}({\bf E})\ .$$
We claim that $$\ker \left(y_{i-1,*}\colon \pi_{i-1}(E_{i-1})\to \pi_{i-1}({\bf E})\right)$$ is finitely generated. We have $\pi_{1}(E_{1})=0$. For $j\ge 2$ we know that $\pi_1(E_j)=0$ by our inductive assumption. The homotopy groups of a finite simply-connected $CW$-complex are finitely generated (see e.g.~\cite{hatcherAT}). Our kernel is finitely generated, since it is a subgroup of a finitely generated abelian group. This finishes the verification of the claim.
Let $(\gamma_\iota\colon S^{i-1}\to {\bf E}_{i-1})_{\iota\in I}$ be a finite family of representatives of generators of the kernel of $y_{i-1,*}\colon \pi_{i-1}(E_{i-1})\to \pi_{i-1}({\bf E})$.
Then we define $$\tilde E_i:=E_{i-1}\cup_{(\gamma_\iota)_{\iota\in I}}\bigsqcup_{\iota\in I}D^i\ .$$ The map $y_{i-1}$ has an extension $\tilde y_i\colon \tilde E_i\to {\bf E}$. Note that $\tilde y_{i,*}\colon \pi_j(\tilde E_i)\to \pi_j({\bf E})$ is now an isomorphism for $j<i$.
Let $(\theta_\lambda\colon S^i\to {\bf E})_{\lambda\in L}$ be a finite set of generators for $\pi_i({\bf E})$. We form $$E_i:=\tilde E_i\vee \bigvee_{\lambda\in L} S^i\ ,$$ and we extend $\tilde y_i$ to $y_i\colon E_i\to {\bf E}$ using the maps $\theta_\lambda$. Then $y_{j,*}\colon \pi_j(E_i)\to \pi_j({\bf E})$ is an isomorphism for $j< i$ and surjective for $j=i$.
We let $\sigma_{i-1}\colon E_{i-1}\to E_i$ be the inclusion.
We now construct the desired sequence of manifolds together with homotopy equivalences $z_i\colon \mathcal{E}_i\to E_i$ such that $$\xymatrix{\mathcal{E}_i\ar[d]^{z_i}\ar[r]^{\kappa_i}&\mathcal{E}_{i+1}\ar[d]^{z_{i+1}}\\E_i\ar[r]^{\sigma_i}&E_{i+1}}$$ commutes.
We again start with a point $\mathcal{E}_{1}:=*$. Assume that we have constructed $\mathcal{E}_{i-1}$. We choose an embedding
$(\mathcal{E}_{i-1},\partial \mathcal{E}_{i-1})\hookrightarrow (\mathbb{R}^n_+,\mathbb{R}^{n-1})$, where $\mathbb{R}_+:=\{(x_1,\dots,x_n)|x_n\ge 0\}$, $\mathbb{R}^{n-1}\subset \mathbb{R}^n$ is identified with the boundary $\{x_n=0\}$, and $n$ is sufficiently large.
We let $p\colon U\to \mathcal{E}_{i-1}$ be the projection from a tubular neighbourhood. We choose smooth maps $\tilde \gamma_\iota\colon S^{i-1}\to \partial U$ so that $z_{i-1}\circ p\circ\tilde \gamma_\iota\sim \gamma_\iota$. If we take $n\ge 2i$, then we can assume after a homotopy that these extend to a smooth embedding \begin{equation}\label{eqq1}\sqcup_{\iota\in I} \tilde \gamma_\iota\colon \bigsqcup_{\iota\in I} S^{i-1}\times D^{n-i}\to \partial U\setminus \mathbb{R}^{n-1}\ .\end{equation} We use these maps in order to define $$\tilde \mathcal{E}_i:=U\cup_{\sqcup_{\iota\in I} \tilde \gamma_\iota } \bigsqcup_{\iota\in I} D^i\times D^{n-i}$$ (smooth out corners). The map $z_{i-1}$ has a natural (up to homotopy) extension to $\tilde z_i\colon \tilde \mathcal{E}_i\to \tilde E_i$ which is again a homotopy equivalence. By a similar procedure we form the boundary connected sum $\mathcal{E}_i:=\tilde \mathcal{E}_i\bigsqcup \sqcup_{\lambda\in L} S^i\times D^{n-i}$ and extend $\tilde z_i$ to $z_i\colon \mathcal{E}_i\to E_i$ which is again a homotopy equivalence.
The map $\kappa_{i-1}\colon \mathcal{E}_{i-1}\to \mathcal{E}_i$ is the inclusion. Furthermore we let $x_i\colon \mathcal{E}_{i}\to {\bf E}$ be the composition $x_i\colon \mathcal{E}_i\stackrel{z_i}{\to}E_i\stackrel{y_i}{\to} {\bf E}$. By construction, our sequence satisfies \ref{pr00}..\ref{pr03}.
\begin{lem} Property \ref{la} holds true. \end{lem} We work in the model category structure on pointed topological spaces ${\mathcal{T}op}_*$ where \begin{enumerate} \item weak equivalences are $\pi_*$-equivalences, \item fibrations are Serre fibrations, \item cofibrations are defined by the left lifting property. \end{enumerate} It is known that retracts of relative $CW$-extensions are cofibrations \cite[Thm. 2.4.19]{MR1650134}. We consider the poset $\mathbb{N}$ as a category. The diagram category ${\mathcal{T}op}_*^\mathbb{N}$ has a model category structure with \begin{enumerate} \item level weak equivalences, \item cofibrations $(X_i)\to (Y_i)$ are characterized by the latching space condition, which in this case reduces to the property that $X_i\sqcup_{X_{i-1}}Y_{i-1}\to Y_i$ is a cofibration for all $i\ge 1$, \item fibrations are level fibrations. \end{enumerate} We refer to \cite[Ch. 5.1]{MR1650134} for details. It follows that a system $(X_i)$ is cofibrant if all maps $X_{i-1}\to X_i$ are cofibrations.
The homotopy colimit ${\tt hocolim\:}\colon {\mathcal{T}op}^\mathbb{N}_* \to {\mathcal{T}op}_*$ is the left derived functor of the colimit ${\tt colim}\colon {\mathcal{T}op}_*^{\mathbb{N}}\to {\mathcal{T}op}_*$.
By construction we have a weak equivalence ${\tt colim} (E_i)\to {\bf E}$. Since the structure maps $E_i\to E_{i+1}$ are all cofibrations (since they are $CW$-extensions) the whole system $(E_i)$ is cofibrant in ${\mathcal{T}op}_*^\mathbb{N}$, and there is a homotopy equivalence ${\tt hocolim\:} (E_i)\to {\tt colim} (E_i)$. The map $(\mathcal{E}_i) \to (E_i)$ is a level equivalence. If $X$ is a finite-dimensional $CW$-complex, then we have naturally induced isomorphisms $${\tt colim} ([X,\mathcal{E}_i])\cong {\tt colim}([X,E_i])\stackrel{!}{\cong} [X,{\tt hocolim\:} (E_i)]\cong [X,{\tt colim} (E_i)]\cong [X,{\bf E}]\ .$$ At the marked isomorphism we use the cellular approximation theorem and that $X$ is finite-dimensional. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
The property that ${\bf E}$ is simply connected has been used in order to conclude from the fact that $E_i$ is a finite simply-connected $CW$-complex that $\pi_i(E_i)$ is finitely generated. Finiteness is needed since we want to approximate by compact manifolds. If we allow non-compact manifolds, then essentially the same proof gives the following:
\begin{prop}\label{duwidwdwd1}
Let ${\bf E}$ be a topological space with countably many connected components such that the groups $\pi_k({\bf E},x)$ are countably generated for all $k\ge 1$ and $x\in {\bf E}$. Then there exist a sequence of pointed manifolds
$(\mathcal{E}_i)_{i\in \mathbb{N}_0}$ together with pointed maps $\kappa_i\colon \mathcal{E}_i\to \mathcal{E}_{i+1}$, $x_i\colon \mathcal{E}_{i}\to {\bf E}$ for all $i\in \mathbb{N}_0$ such that \begin{enumerate} \item\label{pr000} $\mathcal{E}_i$ is homotopy equivalent to an $i$-dimensional \textcolor{black}{CW-}complex, \item\label{pr010} the map $x_i$ is $(i-1)$-connected, \item\label{pr020} $\kappa_i\colon \mathcal{E}_i\hookrightarrow \mathcal{E}_{i+1}$ is an embedding of a submanifold, \item\label{pr030} the diagram $$\xymatrix{\mathcal{E}_{i}\ar[dr]_{x_i}\ar[rr]^{\kappa_i}&&\mathcal{E}_{i+1}\ar[dl]^{x_{i+1}}\\&{\bf E}&}$$ commutes\ , \item \label{la0} for all finite-dimensional pointed $CW$-complexes $X$ the induced map $${\tt colim} ([X,\mathcal{E}_{i}])\to [X,{\bf E}]$$ is an isomorphism. \end{enumerate} \end{prop} {\it Proof.$\:\:\:\:$} If ${\bf E}$ is not connected, then we can approximate the countably many components of ${\bf E}$ separately. In the connected case we construct the $CW$-approximations $(E_i,\sigma_i,y_i)$ as before, but starting the induction with $i=0$ and $E_0:=*$. Then we proceed with the construction of the family $(\mathcal{E}_i,\kappa_i,x_i)$. The only modification is as follows. In order to find the embedding (\ref{eqq1}) we compose the embedding $(\mathcal{E}_{i-1},\partial \mathcal{E}_{i-1})\to (\mathbb{R}^n_+,\mathbb{R}^{n-1})$ with an embedding $(\mathbb{R}^n_+,\mathbb{R}^{n-1})\to (\mathbb{R}^{n+1}_+,\mathbb{R}^{n})$, and we use the extra dimension in order to separate the images of the $\tilde \gamma_{\iota}$. If we want the manifolds $\mathcal{E}_i$ without boundary we can just \textcolor{black}{remove} the boundary \textcolor{black}{without changing the homotopy type}. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
In the following we discuss further properties of the approximations found in Proposition \ref{duwidwdwd} or \ref{duwidwdwd1}. The first is a certain Mittag-Leffler condition. For $j>i$ let $\kappa_i^{j}:=\kappa_{j-1}\circ \dots \circ \kappa_i\colon \mathcal{E}_i\to \mathcal{E}_j$. We set $\kappa_i^i:={\tt id}_{\mathcal{E}_i}$. Let ${\tt V} :=\bigoplus_{n\in \mathbb{Z}} {\tt V}^n$ be some $\mathbb{Z}$-graded abelian group and let $$H^*(X;{\tt V} ):=\prod_{s+t=*}H^{s}(X;{\tt V}^t)$$ denote the ordinary cohomology of the space $X$ with coefficients in ${\tt V} $. \begin{lem}\label{cikdcdc} For all $i\in \mathbb{N}_0$ and all $l\ge 1$ we have the equality of subgroups of $H^k(\mathcal{E}_{i};{\tt V} )$ $$\kappa_i^{i+l,*}H^k(\mathcal{E}_{i+l};{\tt V} ) = \kappa_i^{*}H^k(\mathcal{E}_{i+1};{\tt V} )\ .$$ \end{lem} {\it Proof.$\:\:\:\:$} Note that an $i$-connected map $f\colon X\to Y$ induces an isomorphism in cohomology $f^*\colon H^s(Y;G)\to H^s(X;G)$ for $s\le i-1$, and an injection for $s=i$, where $G$ is an arbitrary abelian group. Since $\mathcal{E}_i$ is $i$-dimensional we have $$H^k(\mathcal{E}_i;{\tt V} )\cong \bigoplus_{s+t=k,s\le i}H^s(\mathcal{E}_i;{\tt V}^t)\ .$$ Note that $\kappa_i^{i+l}$ is $i$-connected for all $l\ge 0$. We therefore have for $s\le i$, $j>i$, $l\ge 1$ $H^s(\mathcal{E}_j;{\tt V}^t)\cong \kappa_j^{j+l\textcolor{black}{,*}} H^s(\mathcal{E}_{j+l};{\tt V}^t)$. This implies for all $l\ge 1$ that $$\kappa_i^{*}H^k(\mathcal{E}_{i+1};{\tt V} )\cong \kappa_i^* \circ \kappa_{i+1}^{i+l,*}H^k(\mathcal{E}_{i+l};{\tt V} )\cong \kappa_i^{i+l,*} H^k(\mathcal{E}_{i+l};{\tt V} )\ .$$ \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\begin{prop}\label{fudif}Let $V$ be as above and
${\bf E}$, $(\mathcal{E}_i)_{i\ge 0}$ as in Proposition \ref{duwidwdwd1}.
We consider a class $u\in H^*({\bf E};{\tt V} )$. There exists a sequence of forms $\omega_i\in \Omega_{cl}^*(\mathcal{E}_i,{\tt V} )$ such that ${\tt Rham}(\omega_i)=x_i^*u$ and $\kappa_i^*\omega_{i+1}=\omega_i$ for all $i\ge 0$. \end{prop} {\it Proof.$\:\:\:\:$} We construct $\omega_i$ inductively. Assume that $\omega_j$ has been constructed for all $j\le i$. There exists $\tilde \omega_{i+1}\in \Omega_{cl}^*(\mathcal{E}_{i+1},{\tt V} )$ such that ${\tt Rham}(\tilde \omega_{i+1})= x_{i+1}^*u$. Since $\kappa_i^*{\tt Rham}(\tilde \omega_{i+1})=\kappa_i^*x_{i+1}^*u=x_i^*u$ there exists furthermore a form $\alpha\in \Omega^{*-1}(\mathcal{E}_{i},{\tt V} )$ such that $\kappa_i^*\tilde\omega_{i+1}+d\alpha=\omega_i$. Since $\kappa_i$ is a closed embedding of a submanifold there exists an extension $\tilde \alpha\in \Omega^{*-1}(\mathcal{E}_{i+1},{\tt V} )$ such that $\kappa_i^*\tilde \alpha= \alpha$. We then define $\omega_{i+1}=\tilde \omega_{i+1}+d\tilde \alpha$.\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
Let $(\hat E,R,I,a)$ be a smooth extension of a cohomology theory $E$.
We assume that this smooth extension is defined on all smooth manifold, or we assume that our approximation $(\mathcal{E}_i,x_i,\kappa_i)$ of ${\bf E}$ is by compact manifolds. We consider the $\mathbb{Z}$-graded vector space ${\tt V} :=E^*\otimes \mathbb{R}$.
By Proposition \ref{fudif} we choose a sequence of closed forms $\omega_i\in \Omega^*_{cl}(\mathcal{E}_i,{\tt V} )$ such that ${\tt Rham}(\omega_i)={\mathbf{ch}}(x_i^*u)$ and $\kappa_i^*\omega_{i+1}=\omega_i$ for all $i\ge 0$. \begin{prop}\label{89fwefwefewfwe} There exists a sequence $\hat u_i\in \hat E^*(\mathcal{E}_i)$ such that $R(\hat u_i)=\omega_i$, $I(\hat u_i)=x_i^*u$ and $\kappa_i^*\hat u_{i+1}=\hat u_i$ for all $i\ge 0$. \end{prop} {\it Proof.$\:\:\:\:$}
First we choose for each $i$ independently a class $ \tilde u_i\in \hat E^*(\mathcal{E}_i)$ such that $R(\tilde u_i)=\omega_i$ and $I(\tilde u_i)=x_i^*u$. Note that $\tilde u_i$ is unique up to addition of a term $a(\alpha_i)$ for $\alpha_i\in H^{*-1}(\mathcal{E}_i,{\tt V} )$.
We now argue by induction over $i$ using the Mittag-Leffler condition \ref{cikdcdc} that we can modify our choice by $\hat u_i:=\tilde u_i+a(\alpha_i)$ so that $\kappa_i^*\hat u_{i+1}=\hat u_i$.
Let us assume by induction that we have made the choice of $\hat u_i$, and that we have already chosen $\hat u_{i+1}^\prime\in \hat E^*(\mathcal{E}_{i+1})$ such that $\kappa_i^*(\hat u_{i+1}^\prime)=\hat u_{i}$.
Then $\kappa_{i+1}^*\tilde u_{i+2}-\hat u_{i+1}^\prime=a(\alpha)$ with $\alpha\in H^{*-1}(\mathcal{E}_{i+1};{\tt V} )$. It follows that $\kappa_{i}^{i+2,*} \tilde u_{i+2}-\hat u_i=a(\kappa_i^*\alpha)$.
By \ref{cikdcdc} we can find a class $\tilde \alpha\in H^{*-1}(\mathcal{E}_{i+2};{\tt V} )$ such that $\kappa^{i+1,*}_{i}(\tilde\alpha)=\kappa^{*}_{i}(\alpha)$. We set $\hat u_{i+1}:= \hat u_{i+1}^\prime+a(\alpha)-a(\kappa_{i+1}^*\tilde\alpha)$ and $\hat u_{i+2}^\prime:=\tilde u_{i+2}-a(\tilde \alpha)$. Then we have $\kappa_{i}^*\hat u_{i+1}=\hat u_{i}$ and $\kappa_{i+1}^*(\hat u_{i+2}^\prime)=\hat u_{i+1}$. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\section{The natural transformation $\Phi$}
We consider a generalized cohomology theory $E$ represented by a spectrum ${\bf E}$. We consider two smooth extensions $(\hat E,R,I,a)$ and $(\hat E^\prime,R^\prime,I^\prime,a^\prime)$ of $E$. Let us fix a degree $k\in \mathbb{Z}$ and a classifying space ${\bf E}_k$ for the homotopy functor $X\mapsto E^k(X)$, e.g. the $k$-th space of the spectrum ${\bf E}$ if the latter is an $\Omega$-spectrum. In the present section we give a construction of a natural transformation $\Phi\colon \hat E^k\to \hat E^{\prime k}$ such that the following diagram commutes: $$\xymatrix{\Omega^{k-1}(M,{\tt V} )\ar[r]^a\ar@{=}[d]&\hat E^k(M)\ar[d]^{\Phi}\ar[r]^{I}\ar@/^1cm/[rr]^R&E^k(M)\ar@{=}[d]&\Omega^k_{cl}(M,{\tt V} )\ar@{=}[d]\\\Omega^{k-1}(M,{\tt V} )\ar[r]^{a^\prime}& \hat E^{\prime k}(M)\ar[r]^{I^\prime}\ar@/_1cm/[rr]^{R^\prime}&E^k(M)&\Omega^k_{cl}(M,{\tt V} )} \ .$$ We make one of the following \textcolor{black}{two} assumptions: \begin{ass}\label{uwidwqudqdqwidwqd} \begin{enumerate} \item\label{udwidw} $E^{k-1}=\pi_1({\bf E}_k)=0$, the abelian groups $E^m$ are finitely generated for all $m\le k$, and the smooth extensions are defined on the category of compact manifolds, or \item\label{udwidw1} the smooth extensions are defined on the category of all manifolds and
the abelian group $E^m$ is countably generated for all $m\le k$. \end{enumerate} \end{ass} Note that $\pi_i({\bf E}_k)\cong E^{k-i}$. We choose an approximation $(\mathcal{E}_i,x_i,\kappa_i)$ of ${\bf E}_k$ by smooth manifolds as in Propositions \ref{duwidwdwd} or \ref{duwidwdwd1}. In case \ref{udwidw} we can assume that the $\mathcal{E}_i$ are compact.
Let $u\in E^k({\bf E}_k)$ be the tautological class represented by the identity ${\tt id}\in [{\bf E}_k,{\bf E}_k]$. We choose a family of closed forms $\omega_i\in \Omega^k_{cl}(\mathcal{E}_i,{\tt V} )$ as in Proposition \ref{fudif} such that ${\tt Rham}(\omega_i)={\mathbf{ch}}(x_i^*u)$ and $\kappa_i^*\omega_{i+1}=\omega_i$ for all $i\ge 0$. Then by Proposition \ref{89fwefwefewfwe} we choose families of classes $\hat u_i\in \hat E^k(\mathcal{E}_i)$, $\hat u_i^\prime\in \hat E^{\prime k}(\mathcal{E}_i)$ such that $R(\hat u_i)=R^\prime(\hat u_i^\prime)=\omega_i$ and $I(\hat u_i)= I^\prime(\hat u_i^\prime)=x_i^*u$ and $\kappa_i^*\hat u_{i+1}=\hat u_i$, $\kappa_i^*\hat u_{i+1}^\prime=\hat u_i^\prime$ for all $i\ge 0$.
Now we can define the transformation $\Phi$, which may depend on the choices made above. Let $M$ be a (compact in case \ref{udwidw}) manifold and $\hat v\in \hat E^k(M)$. Note that $I(\hat v)=f^*u$ for some $f\in [M,{\bf E}_k]$. By Property \ref{la0} there exists an $i\in \mathbb{N}$ and a smooth map $f_i\colon M\to \mathcal{E}_i$ such that $f_i^*x_i^*u=I(\hat v)$. Therefore there exists a unique $\alpha \in \Omega^{k-1}(M,{\tt V} )/{\tt im}({\mathbf{ch}})$ such that $a(\alpha)+f_i^*(\hat u_i)=\hat v$. We set $$\Phi(\hat v):=a^\prime(\alpha)+f_i^*(\hat u_i^\prime)\ .$$
\begin{lem} $\Phi$ is well-defined. \end{lem} {\it Proof.$\:\:\:\:$} The only choice involved is the map $f_i$. We can increase the index $i$ to $j$ without changing $\Phi^k(\hat v)$ by replacing $f_i$ by $f_j:=\kappa^{j}_i\circ f_i$. Given $f_i$ and $f_{i^\prime}^\prime$ then by property \ref{duwidwdwd}.\ref{la0} (or \ref{duwidwdwd1}.\ref{la0}, resp.) there exists $j\ge \max\{i,i^\prime\}$ such that $\kappa^j_i\circ f_i$ and $\kappa_{i^\prime}^j\circ f_{i^\prime}^\prime$ are homotopic.
Thus let us assume that we have $f_j$ and $f_j^\prime$ which are homotopic. Let $H\colon I\times M\to \mathcal{E}_j$ denote the homotopy. We define $\beta:=\int_{I\times M/M} H^*\omega_j\in \Omega_{cl}^{k-1}(M,{\tt V} )/{\tt im}({\mathbf{ch}})$. Then by the homotopy formula (\ref{udqwdqwdqw1}) we have $f^{\prime,*}_j(\hat u_j)=f^*_j(\hat u_j)+a(\beta)$, but also $f^{\prime,*}_j(\hat u_j^\prime)=f^*_j(\hat u_j^\prime)+a^\prime(\beta)$. If $\alpha^\prime$ and $\Phi^{\prime}(\hat v)$ denote the result for $\alpha$ and $\Phi(\hat v)$ for the choice $f_j^\prime$, then we have $\alpha^\prime=\alpha-\beta$. But this implies $$\Phi^{\prime}(\hat v)=a^\prime(\alpha^\prime)+f^{\prime,*}_j(\hat u^\prime_j)=a^\prime(\alpha)-a^\prime(\beta)+f^{\prime,*}_j(\hat u_j^\prime)= a^\prime(\alpha)+f_i^*(\hat u_i^\prime)=\Phi(\hat v)\ .$$\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\begin{lem} We have by construction $R^\prime\circ \Phi=R$, $I^\prime\circ \Phi=I$, and $\Phi\circ a=a^\prime$.
\end{lem}
{\it Proof.$\:\:\:\:$} Straightforward verifications. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\begin{lem} $\Phi$ is natural. \end{lem} {\it Proof.$\:\:\:\:$} For a moment we write $\Phi_M$, $\Phi_N$ in order to indicate a possible dependence on the underlying manifold. Let $g\colon N\to M$ be a smooth map between manifolds. Given $\hat v\in \hat E^k(M)$ we must show that $g^*\Phi_M(\hat v)=\Phi_N(g^*\hat v)$. Note that $\Phi_M(\hat v)=a^\prime(\alpha)+f_i^*(\hat u_i^\prime)$, $g^*\hat v=a(g^*\alpha)+g^*f_i^*(\hat u_i)$, and therefore $\Phi_N(g^*\hat v)=a^\prime(g^*(\alpha))+g^*f_i^*(\hat u_i^\prime)=g^*\Phi_M(\hat v)$. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
Note that $\Phi$ as defined above is a natural transformation of set-valued functors. In general it does not preserve the group structures.
The deviation from additivity is a priory a natural transformation $$\tilde B\colon \hat E^k\times \hat E^k\to \hat E^{\prime k}$$ such that \begin{equation}\label{ezwu} \Phi(\hat v+\hat w)=\Phi(\hat v)+\Phi(\hat w)+\tilde B(\hat v,\hat w)\ . \end{equation} which satistifies the cocycle condition and symmetry $$\tilde B(\hat u,\hat v+\hat w)+\tilde B(\hat v,\hat w)=\tilde B(\hat u,\hat v)+\tilde B(\hat u+\hat v,\hat w)\ , \quad \tilde B(\hat u,\hat v)=\tilde B(\hat v,\hat u)\ .$$ Because of the identities
\begin{eqnarray}
0&=&\Phi(\hat v+a(\alpha))-a^\prime(\alpha)-\Phi(\hat v)=\tilde B(\hat v,a(\alpha))\nonumber\\
0&=&R^\prime(\Phi(\hat v+\hat w)-\Phi(\hat v)-\Phi(\hat w))=R^\prime(\tilde
B(\hat v,\hat w))\label{ide1}\\ 0&=&I^\prime(\Phi(\hat v+\hat w)-\Phi(\hat
v)-\Phi(\hat w))=I^\prime(\tilde B(\hat v,\hat w))\nonumber
\end{eqnarray} it factors over a natural transformation \begin{equation}\label{uewiqeeqw} B\colon E^k(M)\times E^k(M)\to H^{k-1}(M; {\tt V} )/{\tt im}({\mathbf{ch}})\ . \end{equation}
\begin{ddd}\label{uzdiqwdqwdwqd666} We call the cohomology \textcolor{black}{theory} $E$ \textbf{rationally even}, if $E^m\otimes_\mathbb{Z}\mathbb{Q}=0$ for all odd $m\in \mathbb{Z}$. \end{ddd}
\begin{theorem}\label{hjsasas} Let $k\in \mathbb{Z}$ be even. If $E^*$ is rationally even and one of \ref{uwidwqudqdqwidwqd}.\ref{udwidw} or \ref{uwidwqudqdqwidwqd}.\ref{udwidw1} is satisfied, then the transformation $\Phi\colon \hat E^k\to \hat E^{k,\prime}$ is additive. \end{theorem} {\it Proof.$\:\:\:\:$} The family $(\mathcal{E}_i\times \mathcal{E}_i,\kappa_i\times \kappa_i)_{i\ge 0}$ of manifolds gives rise to a system of abelian groups $(H^{k-1}(\mathcal{E}_i\times \mathcal{E}_i;{\tt V} ),(\kappa_i\times \kappa_i)^*)_{i\ge 0}$ indexed by $\mathbb{N}^{op}$. The natural transformation $B$ induces a class $$\hat B\in \lim_i ( H^{k-1}(\mathcal{E}_i\times \mathcal{E}_i;{\tt V} )/{\tt im}({\mathbf{ch}}))\ .$$ In detail, the $i$th component is the class $$\hat B_i:=B({\tt pr}_1^*x_i^*u,{\tt pr}_2^*x_i^*u)\in H^{k-1}(\mathcal{E}_i\times \mathcal{E}_i;{\tt V} )/{\tt im}({\mathbf{ch}})\ .$$
\begin{prop} We have $\hat B=0$. \end{prop} {\it Proof.$\:\:\:\:$} We show this result by showing that $\lim ( H^{k-1}(\mathcal{E}_i\times \mathcal{E}_i;{\tt V} )/{\tt im}({\mathbf{ch}}))=0$.
We first show a refinement of the Mittag-Leffler condition. We start with the following general fact. \begin{lem}\label{ghdgfd} Let $X$ be a $CW$-complex such that $\pi_{2i+1}(X)\otimes_\mathbb{Z} \mathbb{Q}=0$\footnote{If $X$ is not connected, then we require this for all its components.} for $i=0,\dots,n$. Then $H_{2i+1}(X;\mathbb{Q})=0$ for $i=0,\dots,n$. \end{lem} {\it Proof.$\:\:\:\:$} We first show that $H^{2i+1}(X;\mathbb{Q})=0$ for $i=0,\dots,n$. We assume the contrary. Let $k\in \{0,\dots,n\}$ be the smallest number such that $H^{2k+1}(X;\mathbb{Q})\not=0$. We have the Postnikov tower
$$X\langle 2k+2\rangle \to X\langle 2k+1\rangle \to \dots\to X\langle 2\rangle \to X\langle 1\rangle \to X\ .$$ The fibre of $X\langle 2j\rangle \to X\langle 2j-1\rangle $ is equivalent to an Eilenberg-MacLane space $K(\pi_{2j-1}(X),2j-2)$ which is rationally acyclic since $\pi_{2j-1}(X)\otimes_\mathbb{Z} \mathbb{Q}=0$. Therefore $X\langle 2j\rangle \to X\langle 2j-1\rangle $ induces a rational cohomology equivalence. The fibre of $X\langle 2j+1\rangle \to X\langle 2j\rangle $ is an Eilenberg-MacLane space $K(\pi_{2j}(X),2j-1)$. With the Serre spectral sequence $$H^*(X\langle 2j\rangle ;H^*(K(\pi_{2j}(X),2j-1);\mathbb{Q}))\Longrightarrow H^*(X\langle 2j+1\rangle ;\mathbb{Q})$$ and $$H^l(K(\pi_{2j}(X),2j-1),\mathbb{Q})=\left\{\begin{array}{cc} 0&l\not\in \{0,2j-1\} \\ \mathbb{Q}&l=0,2j-1\end{array} \right.$$ we conclude that $$H^{2k+1}(X;\mathbb{Q})\to H^{2k+1}(X\langle 1\rangle ;\mathbb{Q})\to \dots\to H^{2k+1}(X\langle 2k+2\rangle ;\mathbb{Q})=0$$ is injective. This is a contradiction.
It now follows by duality that $H_{2i+1}(X;\mathbb{Q})\cong 0$ for $i\in \{0,\dots ,n\}$. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\begin{lem}\label{zuwqeqe} For every $i\in \mathbb{N}$ and all $r\ge i+1$ we have \begin{equation*} (\kappa^{i+r}_{i}\times \kappa_i^{i+r})^* H^{k-1}(\mathcal{E}_{i+r}\times \mathcal{E}_{i+r};{\tt V} )=0\ . \end{equation*} \end{lem} {\it Proof.$\:\:\:\:$} For $r\ge i+1$ the map $x_{i+r}\colon \mathcal{E}_{i+r}\to {\bf E}_k$ is $2i+1$-connected so that $\pi_{l}(\mathcal{E}_{i+r})\cong \pi_l( {\bf E}_k)\cong E^{k-l}$ for all $l\le 2i$. In particular, $\pi_{2l-1}(\mathcal{E}_{i+r})\otimes_\mathbb{Z}\mathbb{Q}=0$ and hence $\pi_{2l-1}(\mathcal{E}_{i+r}\times \mathcal{E}_{i+r})\otimes_\mathbb{Z}\mathbb{Q}=0$ for all $l\in \mathbb{N}$ such that $2l-1\le 2i$.
It follows from Lemma \ref{ghdgfd} that $H^{2l-1}(\mathcal{E}_{i+r}\times \mathcal{E}_{i+r};\mathbb{Q})=0$ for all $l\in \mathbb{N}$ such that $2l-1\le 2 i$. Since $\mathcal{E}_i\times \mathcal{E}_i$ is homotopy equivalent to a $2i$-dimensional complex and ${\tt V}^j=0$ for odd $j$
we now have \begin{eqnarray*}\lefteqn{ (\kappa^{i+r}_{i}\times \kappa_i^{i+r})^* H^{k-1}(\mathcal{E}_{i+r}\times \mathcal{E}_{i+r};{\tt V} )}&&\\&\cong & (\kappa^{i+r}_{i}\times \kappa_i^{i+r})^* \bigoplus_{2l-1\le 2i} H^{2l-1}(\mathcal{E}_{i+r}\times \mathcal{E}_{i+r};{\tt V}^{k-2l})\\ &=&0\ . \end{eqnarray*} \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
We now consider the system of exact sequences $$0\to {\mathbf{ch}}(E^{k-1}(\mathcal{E}_i\times \mathcal{E}_i))\to H^{k-1}(\mathcal{E}_i\times \mathcal{E}_i;{\tt V} )\to\frac{H^{k-1}(\mathcal{E}_i\times \mathcal{E}_i;{\tt V} )}{ {\mathbf{ch}}(E^{k-1}(\mathcal{E}_i\times \mathcal{E}_i))}\to 0$$ indexed by $\mathbb{N}^{op}$. We apply $\lim$ and get the exact sequence $$\lim_i (H^{k-1}(\mathcal{E}_i\times \mathcal{E}_i;{\tt V} )) \to \lim_i \left(\frac{H^{k-1}(\mathcal{E}_i\times \mathcal{E}_i;{\tt V} )}{ {\mathbf{ch}}(E^{k-1}(\mathcal{E}_i\times \mathcal{E}_i))}\right)\to {\lim_i}^1 ({\mathbf{ch}}(E^{k-1}(\mathcal{E}_i\times \mathcal{E}_i)))\ .$$ Now we have $\lim_i ( H^{k-1}(\mathcal{E}_i\times \mathcal{E}_i;{\tt V} ))=0$ because of Lemma \ref{zuwqeqe}. The same lemma implies that the subsystem ${\mathbf{ch}}(E^{k-1}(\mathcal{E}_i\times \mathcal{E}_i))\subseteq H^{k-1}(\mathcal{E}_i\times \mathcal{E}_i;{\tt V} )$ satisfies the Mittag-Leffler condition so that $\lim_i^1 ({\mathbf{ch}}(E^{k-1}(\mathcal{E}_i\times \mathcal{E}_i)))=0$. This implies $$\lim_i \left(\frac{H^{k-1}(\mathcal{E}_i\times \mathcal{E}_i;{\tt V} )}{ {\mathbf{ch}}(E^{k-1}(\mathcal{E}_i\times \mathcal{E}_i))}\right)=0$$ as required. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
We now show Theorem \ref{hjsasas}. Let ${ \hat v }, { \hat w }\in \hat E^k(M)$. We choose $i$ sufficiently large such that there exist $f_{ \hat v },f_{ \hat w }\colon M\to \mathcal{E}_i$ with $f_{ \hat v }^*(x_i^*u)=I({ \hat v })$ and $f_{ \hat w }^*(x_i^*u)=I({ \hat w })$. Let $j\ge i$ be such that there exists $\mu\colon \mathcal{E}_i\times \mathcal{E}_i\to \mathcal{E}_j$ with $\mu^*(x_j^*u)={\tt pr}_0^*x_i^*u+{\tt pr}_1^*x_i^*u$. We choose $\alpha\in \Omega^{k-1}(\mathcal{E}_i\times \mathcal{E}_i,{\tt V} )$ such that $a(\alpha)+\mu^*(\hat u_j)={\tt pr}_0^*\hat u_i+{\tt pr}_1^*\hat u_i$.
We further can choose $f_{{ \hat v }+{ \hat w }}\colon M\to \mathcal{E}_j$ as the composition $f_{{ \hat v }+{ \hat w }}=\mu\circ (f_{ \hat v },f_{ \hat w })$ so that $f_{{ \hat v }+{ \hat w }}^*(x_j^*u)=I({ \hat v }+{ \hat w })$.
We now choose $\alpha_{ \hat v },\alpha_{ \hat w },\alpha_{{ \hat v }+{ \hat w }}\in \Omega^{k-1}(M,{\tt V} )$ such that $a(\alpha_{ \hat v })+f_{ \hat v }^*(\hat u_i)={ \hat v }$, $a(\alpha_{ \hat w })+f_{ \hat w }^*(\hat u_i)={ \hat w }$, and $a(\alpha_{{ \hat v }+{ \hat w }})+f_{{ \hat v }+{ \hat w }}^*(\hat u_j)={ \hat v }+{ \hat w }$.
Then we have $\Phi({ \hat v })=a^\prime(\alpha_{ \hat v })+f_{ \hat v }^*( \hat u_i^\prime)$, $\Phi({ \hat w })=a^\prime(\alpha_{ \hat w })+f_{ \hat w }^*(\hat u_i^\prime)$, and $\Phi({ \hat v }+{ \hat w })=a^\prime(\alpha_{{ \hat v }+{ \hat w }})+f_{{ \hat v }+{ \hat w }}^*(\hat u_j^\prime)$. We now calculate using $\Phi(\hat u_i)=\hat u_i^\prime$ and $$0=B({\tt pr}_0^*x_i^*u,{\tt pr}_1^*x_i^*u)=\Phi({\tt pr}_0^*\hat u_i+{\tt pr}_1^*\hat u_i)-\Phi({\tt pr}_0^*\hat u_i)-\Phi({\tt pr}_1^*\hat u_i)$$ at the marked equality \begin{eqnarray*} \lefteqn{\Phi({ \hat v }+{ \hat w })-\Phi({ \hat v })-\Phi({ \hat w })}&&\\&=&a^\prime(\alpha_{{ \hat v }+{ \hat w }})+f_{{ \hat v }+{ \hat w }}^*(\hat u_j^\prime)- a^\prime(\alpha_{ \hat v })-f_{ \hat v }^*( \hat u_i^\prime)-a^\prime(\alpha_{ \hat w })-f_{ \hat w }^*(\hat u_i^\prime)\\ &=&a^\prime(\alpha_{{ \hat v }+{ \hat w }})+(f_{ \hat v },f_{ \hat w })^*\mu^*(\hat u_j^\prime)- a^\prime(\alpha_{ \hat v })-f_{ \hat v }^*( \hat u_i^\prime)-a^\prime(\alpha_{ \hat w })-f_{ \hat w }^*(\hat u_i^\prime)\\&=& a^\prime(\alpha_{{ \hat v }+{ \hat w }}-\alpha_{ \hat v }-\alpha_{ \hat w })+(f_{ \hat v },f_{ \hat w })^*\left(\Phi({\tt pr}_0^*\hat u_i+{\tt pr}_1^*\hat u_i)-a^\prime(\alpha)\right)\\&&-
f_{ \hat v }^*( \hat u_i^\prime)-f_{ \hat w }^*(\hat u_i^\prime)\\ &\stackrel{!}{=}& a^\prime(\alpha_{{ \hat v }+{ \hat w }}-\alpha_{ \hat v }-\alpha_{ \hat w })+(f_{ \hat v },f_{ \hat w })^*\left(\Phi({\tt pr}_0^*\hat u_i)+\Phi({\tt pr}_1^*\hat u_i)-a^\prime(\alpha)\right)\\&&-
f_{ \hat v }^*( \hat u_i^\prime)-f_{ \hat w }^*(\hat u_i^\prime)\\ &=&a^\prime(\alpha_{{ \hat v }+{ \hat w }}-\alpha_{ \hat v }-\alpha_{ \hat w })+f_{ \hat v }^*\hat u_i^\prime+f_{ \hat w }^*\hat u_i^\prime -(f_{ \hat v },f_{ \hat w })^* a^\prime(\alpha)\\&&-
f_{ \hat v }^*( \hat u_i^\prime)-f_{ \hat w }^*(\hat u_i^\prime)\\ &=&a^\prime(\alpha_{{ \hat v }+{ \hat w }}-\alpha_{ \hat v }-\alpha_{ \hat w }-(f_{ \hat v },f_{ \hat w })^*\alpha)\ . \end{eqnarray*} Doing the same calculation starting with $0=({ \hat v }+{ \hat w })-{ \hat v }-{ \hat w }$ (leave out the symbols $\Phi$ and ${}^\prime$) we get $0=a(\alpha_{{ \hat v }+{ \hat w }}-\alpha_{ \hat v }-\alpha_{ \hat w }-(f_{ \hat v },f_{ \hat w })^*\alpha)$. Since $\ker(a)=\ker(a^\prime)$ we conclude that $$\Phi({ \hat v }+{ \hat w })-\Phi({ \hat v })-\Phi({ \hat w })=0\ .$$ \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\begin{theorem}\label{main1} Let $E$ be a rationally even generalized cohomology theory which is represented by a spectrum ${\bf E}$. Let $(\hat E,R,I,a,\int)$ and $(\hat E^\prime,R^\prime,I^\prime,a^\prime,\int^\prime)$ be two smooth extensions with integration. We assume that either the smooth extensions are defined on the category of all smooth manifolds and the groups $E^m$ are countably generated for all $m\in \mathbb{Z}$, or that $E^m=0$ for all odd $m\in \mathbb{Z}$ and $E^m$ is finitely generated for even $m\in \mathbb{Z}$. Then there is a unique natural isomorphism $$\Phi\colon \hat E\to \hat E^\prime$$ of smooth extensions with integration. \end{theorem} {\it Proof.$\:\:\:\:$} Let us first show the existence of a natural transformation. We let $\Phi^k\colon \hat E^k\to \hat E^{\prime k}$ denote the component in degree $k$. Let $\Phi^{2k}$ be the transformation obtained in Theorem \ref{hjsasas}. We extend $\Phi$ to odd degrees using integration.
Let $i\colon M\to S^1\times M$ be the embedding induced by a point in $S^1$ and $p\colon S^1\times M\to M$ be the projection. Because of $p\circ i={\tt id}$ we have a splitting $$E^*(S^1\times M)\cong p^* E^*(M)\oplus \ker(i^*)\ .$$ Let $q\colon S^1\times M\to \Sigma M_+$ be the projection onto the suspension. It induces an identification $\ker(i^*)\cong q^*\tilde E^*(\Sigma M_+)\cong E^{*-1}(M)$ using the suspension isomorphism. We let $\sigma\colon E^{*-1}(M)\to E^*(S^1\times M)$ be the inclusion of this summand.
Assume that we already have constructed $\Phi^{k}$. This is in particular the case for even $k$. Let $\hat x\in \hat E^{k-1}(M)$ be given. Then we choose $\tilde x\in \hat E^{k}(S^1\times M)$ such that \begin{enumerate} \item \label{uierwer0} $\int(\tilde x)=\hat x$, \item\label{uierwer1} $R(\tilde x)=dt\wedge {\tt pr}^*R(\hat x)$, \item\label{uierwer2} $I(\tilde x)=\sigma(I(\hat x))$. \end{enumerate} The following procedure shows that this choice can be made. We first choose a lift $\tilde x\in \hat E^{k}(S^1\times M)$ of $\sigma( I(\hat x))\in E^{k}(S^1\times M)$ so that \ref{uierwer2} is satisfied. Then we add $a(\alpha)$ for a suitable $\alpha\in \Omega^{k-1}(S^1\times M,{\tt V} )$ in order to satisfy \ref{uierwer1}. Then $\hat x-\int(\tilde x)\in a(H^{k-1}(M;{\tt V} ))$. We can kill this difference by modifying $\alpha$ suitably. Here we use that $\int(a(\alpha))=a(\int(\alpha))$ and $\int\colon \Omega^*(S^1\times M,{\tt V} )\to \Omega^{*-1}(M,{\tt V} )$ is surjective.
We now define $$\Phi^{k-1}(\hat x):=\int^\prime(\Phi^{k}(\tilde x))\ .$$ Let us check that this is well-defined. Note that another choice $\tilde x^\prime$ satisfies $\tilde x^\prime-\tilde x=a(\alpha)$ with $\int(a(\alpha))=0$. Furthermore, $$\int^\prime(\Phi^{k}(\tilde x^\prime))=\int^\prime(\Phi^{k}(\tilde x+a(\alpha)))=\int^\prime(\Phi^{k}(\tilde x)+a(\alpha))=\int^\prime( \Phi^{k}(\tilde x))\ . $$ Naturality of $\Phi^{k-1}$ follows from naturality of the integration maps. Indeed, let $f\colon N\to M$ be a smooth map and $\hat y:=f^*\hat x$. Then we can choose $\tilde y:=({\tt id}_{S^1}\times f)^*\tilde x$. We get $$\Phi^{k-1}(\hat y)=\int^\prime(\Phi^{k}(f^*\tilde x))=\int^\prime ({\tt id}_{S^1}\times f)^*\Phi^{k}(\tilde x)= f^* \int^\prime (\Phi^{k}(\tilde x))=f^*\Phi^{k-1}(\hat x)\ .$$
Let us now discuss uniqueness of $\Phi^{2k}$. Assume that $\Psi\colon \hat E^{2k}\to \hat E^{\prime 2k}$ is a second natural transformation of group-valued functors which is compatible with the transformations $R,I,a$ in the sense that $$R^\prime \circ \Psi=R\ ,\quad I^\prime\circ \Psi=I\ , \quad a^\prime = \Psi\circ a\ .$$ Then we consider the difference $\Delta:=\Phi^{2k}-\Psi$. Compatibility with $R$ shows that $\Delta$ takes values in $\hat E^{\prime 2k}_{flat}$ (see Definition \ref{uiqwdqwdwqd}). Compatibility with $I$ in addition shows that $\Delta$ takes values in the subfunctor $a^\prime(H^{2k-1}(\dots;{\tt V} )/{\tt im}({\mathbf{ch}}))\subseteq \hat E^{\prime 2k}$. Finally, compatibility with $a$ implies that $\Delta$ descends to a natural transformation $$\Delta\colon E^{2k}\to H^{2k-1}(\dots;{\tt V} )/{\tt im}({\mathbf{ch}})\ .$$ We get an element $$(\Delta(x_i^*u))\in \lim_i( H^{2k-1}\left(\mathcal{E}_i;{\tt V} )/{\tt im}({\mathbf{ch}}))\right)\ .$$ The same argument as for Lemma \ref{zuwqeqe} shows that the target group vanishes so that $\Delta(x_i^*u)=0$ for all $i\in \mathbb{N}$. But this implies that $\Delta=0$. Indeed, if $M$ is some manifold and $x\in E^{2k}(M)$, then there exists an $i\in \mathbb{N}$ and $f\colon M\to \mathcal{E}_i$ such that $x=f^*x_i^*u$. We get $\Delta(x)=f^*\Delta(x_i^*u)=0$.
Recall that we have defined $\Phi^{2k}$ using Theorem \ref{hjsasas}. Then we have extended the construction to odd degrees such that $\Phi^{2k-1}$ is chracterized by \begin{equation}\label{duidqwdqwd} \int^\prime \circ\:\: \Phi^{2k}=\Phi^{2k-1}\circ \int\ . \end{equation} We could use the construction above in order to construct another transformation $\Psi\colon \hat E^{2k}\to \hat E^{\prime 2k}$ starting from $\Phi^{2k+1}$ so that $$\int^\prime\circ \:\: \Phi^{2k+1}=\Psi\circ \int\ .$$ By the uniqueness result we see that $\Psi=\Phi^{2k}$.
Therefore we have constructed a natural transformation of smooth extensions with integration. As such it is unique on the even part. Since the integration $\int\colon \hat E^{2k}(S^1\times M)\to \hat E^{2k-1}(M)$ is surjective, the compatibility (\ref{duidqwdqwd}) implies uniqueness on the odd part, too. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\section{Multiplicative structures} In this section we assume that $E$ is a multiplicative cohomology theory. Let $(\hat E,R,I,a)$ be a smooth extension of $E$. We fix the unique $e\in \ker(i^*)\subseteq E^1(S^1)$ such that $\int_{E}(e)=1\in E^0$, the unit of the ring $E^*$. We let $\omega_e\in \Omega^1_{cl}(S^1;{\tt V}^0 )$ be the unique rotationally invariant closed form such that ${\tt Rham}(\omega_e) ={\mathbf{ch}}(e)$. In coordinates $\omega_e=dt\otimes 1$. Note that $\int_{S^1} \omega_e=1$. Then we choose a lift $\hat e\in \hat E^1(S^1)$ such that $I(\hat e)=e$ and $R(\hat e)=\omega_e$. Note that $\hat e$ is determined uniquely up to elements of the form $a(\alpha)$ with $\alpha\in H^{0}(S^1;{\tt V} )/{\tt im}({\mathbf{ch}})$. We can assume that the representative $\alpha$ is constant. We see that $\hat e$ is determined uniquely up to a choice in ${\tt V}^0/E^0\oplus {\tt V}^{-1}/E^{-1}$.
We now want to modify the class $\hat e$ such that $q^*\hat e=-\hat e$, where $q\colon S^1\to S^1$ is given by $q(z)=\bar z$. Apriori $R(q^*\hat e+\hat e)=0$ and $I(q^*\hat e+\hat e)=0$. Therefore $q^*\hat e+\hat e=a(\rho)$ for $\rho\in H^0(S^1;{\tt V} )$. We write $\rho=\rho_0+\rho_1\in H^0(S^1;{\tt V}^0)\oplus H^1(S^1;{\tt V}^{-1})$. Since $a(q^*\rho-\rho)=0$, $q^*\rho_0=\rho_0$ and $q^*\rho_1=-\rho_1$ we conclude that $2{\tt Rham}(\rho_1)= {\mathbf{ch}}(r)$ for some $r\in E^0(S^1)$.
Then $$q^*(\hat e-a(\frac{1}{2}\rho))+ \hat e-a(\frac{1}{2}\rho)=a(\rho-\frac{1}{2}\rho-\frac{1}{2}q^*\rho)=a(\frac{1}{2}(\rho-q^*\rho))=a(\rho_1)=a(\frac{1}{2}{\mathbf{ch}}(r))\ .$$ If we replace $\hat e$ by $\hat e-a(\frac{1}{2}\rho)$, then the new $\hat e$ satisfies $q^*\hat e+\hat e=a(\rho_1)$. The $2$-torsion class $[{\tt Rham}(\rho_1)] \in {\tt V}^{-1}/E^{-1}$ is independent of the choices. Indeed, if we change $\hat e$ by $\hat e+a(\alpha)$ for $\alpha\in H^0(S^1;{\tt V})$, then we can take the same $\rho_1$.
Recall the inclusion $i\colon M\to S^1\times M$ and the projection $p\colon S^1\times M\to M$. We finally consider the condition $i^*\hat e=0$. Of course, we have $i^*\hat e=a(\theta)$ for some $\theta\in H^0(*;{\tt V})\cong {\tt V}^0$. If we replace $\hat e$ by $\hat e-p^*\theta$, then we get $i^*\hat e=0$ retaining all the other conditions. The ramaining choice for $\hat e$ is ${\tt V}^{-1}/E^{-1}$.
\begin{ddd} The class ${\bf o}_{\hat E}:=[{\tt Rham}(\rho_1)]\in {\tt V}^{-1}/E^{-1}$ is called the \textbf{obstruction class}. \end{ddd}
The obstruction class vanishes exactly if we can choose $\hat e\in \hat E^1(S^1)$ such that $$R(\hat e)=dt\otimes 1\ , \quad I(\hat e)=e\ ,\quad i^*\hat e=0\ ,\quad q^*\hat e=-\hat e\ .$$ In this case $\hat e$ is unique up to an element in ${\tt V}^{-1}/E^{-1}$. The obstruction vanishes e.g. if $E^{-1}$ is torsion, and in this case $\hat e$ is unique. We do not have any example of a smooth extension with non-trivial obstruction class.
\begin{prop}\label{uzqwiwqdqd} If $(\hat E,R,I,a)$ is a multiplicative smooth extension with vanishing obstruction class ${\bf o}_{\hat E}$, then it has an integration. \end{prop}
{\it Proof.$\:\:\:\:$}
Using the class $e$ we can make the decomposition $$E^{*+1}(S^1\times M)\cong {\tt im}(p^*)\oplus ¸\ker(i^*)\cong E^{*+1}(M)\oplus E^*(M)$$ more explicit. Namely, we can write the class $x\in E^{*+1}(S^1\times M)$ uniqely in the form $x=p^*u\oplus e\times y$ with $u\in E^{*+1}(M)$ and $y\in E^*(M)$. Note that $y=\int x$.
We now use the decomposition
$$\hat E^{*+1}(S^1\times M)={\tt im}(p^*)\oplus \ker(i^*)$$ in order to define an integration which factors as $$\int \colon \hat E^{*+1}(S^1\times M)\stackrel{{\tt pr}}{\to} \ker(i^*) \to \hat E^*(M)\ .$$ It obviously satisfies the second condition \ref{ddd4}.\ref{ddd41} $$\int\circ\:\: p^*=0\ .$$ Let $\hat x\in \ker(i^*)$. Then we can write $I(\hat x)=e\times y$ for a unique $y\in E^*(M)$. We choose a smooth lift $\hat y\in \hat E^*(M)$ and a form $\rho\in \Omega^{*-1}(S^1\times M,{\tt V})$ such that $\hat x=\hat e\times \hat y +a(\rho)$. Then we define $$\int \hat x:= \hat y+a(\int \rho)\ .$$ Let us show that $\int$ is well-defined. If we choose another smooth lift $\hat y^\prime$, then $\hat y^\prime=\hat y+a(\theta)$ for some form $\theta\in \Omega^{*-1}(M,{\tt V})$. Since $\hat e\times \hat y^\prime=\hat e\times \hat y+\hat e\times a(\theta)=\hat e\times \hat y+a(\omega_e\times \theta)$ we can choose $\rho^\prime=\rho-\omega_e\times \theta$ and our construction produces $$\hat y^\prime + a(\int \rho^\prime)=\hat y+a(\theta)+a(\int \rho)-a(\int \omega_e\wedge\theta)=\hat y+a(\int \rho)\\ .$$ Next we show that the construction is independent of the choice of $\rho$. If we choose $\rho^\prime$, then $\rho^\prime=\rho+\alpha$ such that $\alpha\in \Omega^{*-1}(S^1\times M,{\tt V})$ is closed and ${\tt Rham}(\alpha)={\mathbf{ch}}(u)$ for some $u\in E^{*}(S^1\times M)$. But then our construction produces $$\hat y+a(\int \rho^\prime)=\hat y+a(\int \rho)+a(\int {\mathbf{ch}}(u))=\hat y+a(\int \rho)+a({\mathbf{ch}}(\int u))=\hat y+a(\int \rho)\ .$$
We have $$I(\int \hat x)=I(\hat y+a(\int \rho))=I(\hat y)=y=\int I(\hat x)\ .$$ Furthermore $$R(\int \hat x)=R(\hat y+a(\int\rho))=\int(\omega_e\times R(\hat y))+d\int\rho=\int(\omega_e\times R(\hat y)+d\rho)=\int R(\hat x)\ .$$
Next we check that the integration is linear. Let $\hat x_i\in \ker(i^*)$, $i=0,1$. We choose $y_i\in E^*(M)$ such that $I(x_i)=e\times y_i$ and smooth lifts $\hat y_i\in \hat E^*(M)$. Then we find $\rho_i\in \Omega^{*-1}(S^1\times M,{\tt V})$ such that $\hat x_i=\hat e\times \hat y_i+a(\rho_i)$ for $i=0,1$. Note that $\hat x_0+\hat x_1=\hat e\times(\hat y_0+\hat y_1)+a(\rho_0+\rho_1)$. It follows $$\int (\hat x_0+\hat x_1)=\hat y_0+\hat y_1+a(\int(\rho_0+\rho_1))=\int (\hat x_0)+\int (\hat x_1)\ .$$
If $\hat x=\hat e\times \hat y+a(\rho)$, then $$(q\times {\tt id}_M)^*\hat x=q^*\hat e\times \hat y+a((q\times {\tt id}_M)^*\rho)=\hat e\times (-\hat y)+a((q\times {\tt id}_M)^*\rho)\ .$$ We get $$\int((q\times {\tt id}_M)^*\hat x)=-\hat y+a(\int (q\times {\tt id}_M)^*\rho) =-\hat y-a(\int \rho )=-\int \hat x \ .$$
Let us finally show that the integration is natural. Let $f\colon N\to M$ be a smooth map. Then we can write $f^*\hat x=\hat e\times f^*\hat y+({\tt id}_{S^1}\times f)^* \rho$. It follows that $$\int (f^*\hat x)=f^*\hat y+a(f^*\int \rho)=f^*\int (\hat x)\ .$$ \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
If we change $\hat e$ to $\hat e^\prime=\hat e+a(u)$, $u\in H^1(S^1;{\tt V}^{-1})/{\tt im}({\mathbf{ch}})\cap H^1(S^1;{\tt V}^{-1})\cong {\tt V}^{-1}/E^{-1}$, then we have $$\hat x=\hat e^\prime\times\hat y+a(\rho^\prime)=\hat e\times \hat y+a(u\times R(\hat y)+\rho^\prime)\ .$$ For the corresponding integration we get, using ${\tt Rham}(R(\hat y))={\mathbf{ch}}(\int(I(\hat x)))$, $$\int\hat x=\int^\prime\hat x+a(\int (u\times R(\hat y)))=\int^\prime\hat x+a(u\times {\mathbf{ch}}(\int I(\hat x) ))\ .$$
If ${\tt V}^{-1}\not=0$, then we indeed may change the integration by modifying the choice of $\hat e$. If $E$ is rationally even, or more generally, if only $E^{-1}$ is a torsion group, then of course ${\tt V}^{-1}=0$. \begin{kor}\label{uifqfqfqf} If $(\hat E,R,I,a)$ is a multiplicative extension of a generalized cohomology theory such that $E^{-1}$ is a torsion group, then there is a canonical choice of an integration. \end{kor}
We can now apply Theorem \ref{main1}. \begin{kor}\label{main2} Let $(\hat E,R,I,a)$ and $(\hat E^\prime,R^\prime,I^\prime,a^\prime)$ be two multiplicative extensions of a rationally even generalized cohomology theory. We assume that either both extensions are defined on the category of all smooth manifolds and the groups $E^m$ are countably generated for all $m\in \mathbb{Z}$, or they are defined on the category of compact manifolds , $E^m=0$ for all odd $m\in \mathbb{Z}$ and $E^m$ is finitely generated for even $m\in \mathbb{Z}$. Then there is a unique natural isomorphism between these smooth extensions preserving the canonical integration. \end{kor}
\begin{lem}\label{dqedwqdwqd} The integration defined above satisfies the projection formula $$\int (\hat x\cup p^*\hat z)=(\int\hat x)\cup \hat z\ ,\quad \hat x\in \hat E^*(S^1\times M)\ ,\quad \hat z\in \hat E^*(M)\ .$$ \end{lem} {\it Proof.$\:\:\:\:$} We write $\hat x=p^*\hat u+\hat e\times \hat y+ a(\rho)$. Then by construction $\int \hat x=\hat y+a(\int\rho)$. Furthermore $$\hat x\cup p^*\hat z=p^*(\hat u\cup \hat z)+\hat e\times (\hat y\cup \hat z)+a(\rho\wedge R(p^*\hat z))\ .$$ It follows that $$\int(\hat x\cup p^*\hat z)=\hat y\cup \hat z+a(\int\rho\wedge p^*R(\hat z))=\hat y\cup \hat z+a((\int\rho) \wedge R(\hat z))=(\int \hat x)\cup \hat z\ .$$ \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\begin{theorem}\label{main23} The unique natural transformation of Corollary \ref{main2} is multiplicative. \end{theorem} {\it Proof.$\:\:\:\:$} The argument is similar to the proof of Theorem \ref{hjsasas}. We transport the $\cup$-product of $\hat E^\prime$ to $\hat E$ using the natural transformation $\Phi\colon \hat E\to \hat E^\prime$. Thus we define $$\cup^\prime\colon \hat E^{ev}(M)\otimes \hat E^{ev}(M)\to \hat E^{ev}(M)$$ by $$\hat x\cup^\prime \hat y:=\Phi^{-1}(\Phi(\hat x)\cup \Phi(\hat y))\ .$$ The difference $$\Delta(\hat x,\hat y):=\hat x\cup \hat y-\hat x\cup^\prime \hat y$$ is a natural transformation of set-valued functors $$\hat E^{ev}\times \hat E^{ev}\to \hat E^{ev}\ .$$ Since both $\cup$-products are compatible with $R$ and $I$ we conclude that $\Delta$ actually has values in the subfunctor $a(H^{odd}(\dots;{\tt V})/{\tt im}({\mathbf{ch}}))$. Furthermore, since both cup-products are compatible with $a$ the transformation factors as $$\Delta\colon E^{ev}\times E^{ev }\to a(H^{odd}(\dots;{\tt V})/{\tt im}({\mathbf{ch}}))\ .$$ We approximate ${\bf E}_k$ by a family of manifolds $\mathcal{E}_{k,i}$ as in Proposition \ref{duwidwdwd}. Then we consider compatible families $\hat u_{k,i}\in \hat E^k(\mathcal{E}_{k,i})$ as in Proposition \ref{89fwefwefewfwe}. For even $k,l$ we get a family of elements $$(\Delta(\hat u_{k,i},\hat u_{l,i}))_{i\ge 0}\in \lim\limits_{i} (H^{odd}(\mathcal{E}_{k,i}\times \mathcal{E}_{l,i})/{\tt im}({\mathbf{ch}}))\ .$$ The analog of Lemma \ref{zuwqeqe} shows that this limit is trivial, so that $\Delta(\hat u_{k,i},\hat u_{l,i})=0$ for all $i$. As in the proof of Theorem \ref{main1} we conclude that this implies $\Delta=0$.
We now discuss multiplicativity in general. Let $k$ be odd and $l$ be even, $\hat x\in \hat E^{k}(M)$ and $\hat y\in \hat E^l(M)$. Then we have, using the projection formula \ref{dqedwqdwqd} and the compatibility of $\Phi$ with integration, \begin{eqnarray*} \Phi(\hat x\cup \hat y)&=&\Phi(\int(\hat e\times (\hat x\cup \hat y)))\\&=&\int (\Phi(\hat e\times (\hat x\cup \hat y)))\\&=&\int(\Phi((\hat e\times \hat x) \cup p^* \hat y))\\&=& \int(\Phi(\hat e\times \hat x)\cup \Phi(p^*\hat y))\\&=&\int(\Phi(\hat e\times \hat x) \cup p^*\Phi( \hat y))\\&=& (\int\Phi(\hat e\times \hat x))\cup \Phi(\hat y)\\ &=&\Phi(\int(\hat e\times \hat x))\cup \Phi(\hat y)\\ &=&\Phi(\hat x)\cup \Phi(\hat y) \ . \end{eqnarray*} Similarly, if $k$ and $l$ are odd, then again using the projection formula and the case just shown \begin{eqnarray*} \Phi(\hat x\cup \hat y)&=&\Phi((\int\hat e\times \hat x)\cup \hat y)\\ &=&\Phi(\int((\hat e\times \hat x)\cup p^*\hat y))\\ &=&\int(\Phi((\hat e\times \hat x)\cup p^*\hat y))\\ &=&\int(\Phi(\hat e\times \hat x)\cup \Phi(p^*\hat y))\\ &=&\int(\Phi(\hat e\times \hat x)\cup p^*\Phi(\hat y))\\ &=&(\int\Phi(\hat e\times \hat x))\cup\Phi(\hat y)\\ &=& \Phi(\int(\hat e\times \hat x))\cup\Phi(\hat y)\\ &=&\Phi( \hat x)\cup\Phi(\hat y)\ . \end{eqnarray*} \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\section{The flat theory}\label{uciacascc}
Let $(\hat E,R,I,a)$ be a smooth extension of a generalized cohomology theory $E$. Let $\hat x\in \hat E^{*+1}([0,1]\times M)$ and $f_0,f_1\colon M\to [0,1]\times M$ be induced by the inclusions of the endpoints. Then we have the following homotopy formula. \begin{lem}\label{udqwdqwdqw} We have $$f_1^*\hat x-f_0^*\hat x=a\left(\int_{[0,1]\times M/M} R(\hat x)\right)\ .$$ \end{lem} {\it Proof.$\:\:\:\:$} Let $p\colon [0,1]\times M\to M$ denote the projection. Then there exists a form $\rho\in \Omega^{*}([0,1]\times M,{\tt V})$ such that $\hat x=p^*f_0^*\hat x+a(\rho)$. We have $R(\hat x)=p^*f_0^*R(\hat x)+d\rho$. Let us write $\rho=\rho_0+dt\wedge \rho_1$ for $t$-dependent forms $\rho_0,\rho_1$ on $M$, where $t$ is the coordinate of the interval $[0,1]$. We get
$$i_{\partial t} R(\hat x)=\partial_t \rho_0-d\rho_1\ .$$ Integrating we get
$$\int_{[0,1]\times M/M}R(\hat x)=(\rho_0)_{|t=1}-(\rho_0)_{|t=0}-d\int_0^1 \rho_1dt\ .$$ We get
$$f_1^*\hat x-f_0^*\hat x=a(f_1^* \rho-f_0^*\rho)=a((\rho_0)_{|t=1}-(\rho_0)_{|t=0})=a\left(\int_{[0,1]\times M/M}R(\hat x)\right)\ .$$ \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\begin{ddd}\label{uiqwdqwdwqd} We define the \textbf{flat theory} as the subfunctor $$\hat E_{flat}^*(M):=\ker(R\colon \hat E^*(M)\to \Omega_{cl}^*(M,E))\ .$$ \end{ddd} Recall that a functor on manifolds is called homotopy invariant if it maps smoothly homotopic smooth maps to the same morphisms. As an immediate consequence of the homotopy formula we get: \begin{kor} The flat theory $\hat E_{flat}$ is a homotopy invariant functor. \end{kor} As a direct consequence of \ref{ddd1}.R.3 and \ref{ddd1}.R.1 the flat theory fits into the following natural long exact sequence: $$\dots\ \stackrel{I}{\to} E^{*-1}(M)\stackrel{{\mathbf{ch}}}{\to} H^{*-1}(M;{\tt V})\stackrel{a}{\to} \hat E^*_{flat}(M)\stackrel{I}{\to} E^*(M)\stackrel{{\mathbf{ch}}}{\to } H^{*}(M;{\tt V})\stackrel{a}{\to}\dots \ .$$ There is a natural topological candidate for the flat subfunctor which fits into a similar sequence, see (\ref{ufifwefwefewf}).
Recall the construction of the Moore spectrum ${\bf M} G$ for an abelian group $G$. We choose a free resolution $$0\to \bigoplus_{v\in V} \mathbb{Z} v\stackrel{\alpha}{\to} \bigoplus_{w\in W} \mathbb{Z} w\to G\to 0$$ for suitable sets $V,W$. Then we define a map of spectra $$\hat \alpha\colon \bigvee_{v\in V}\mathbf{S} \to \bigvee_{w\in W} \mathbf{S}$$ which realizes $\alpha$ in reduced integral homology, where we identify $H\mathbb{Z}_*(\bigvee_{v\in V}\mathbf{S})\cong \bigoplus_{v\in V}\mathbb{Z} v$ and $ H\mathbb{Z} _*(\bigvee_{w\in W}\mathbf{S})\cong \bigoplus_{w\in W}\mathbb{Z} w$. The isomorphism class of the Moore spectrum ${\bf M} G$ is defined to fit into the distinguished triangle of the stable homotopy category $$\bigvee_{v\in V}\mathbf{S} \stackrel{\hat \alpha}{\to} \bigvee_{w\in W} \mathbf{S}\to {\bf M} G\to \Sigma \bigvee_{v\in V}\mathbf{S}\ .$$ Note that we can and will take ${\bf M} \mathbb{Z}:=\mathbf{S}$. We fix an element $1\in G$. We assume that there is one generator $w_0\in W$ which maps to $1\in G$. Then we let ${\bf M} \mathbb{Z}\to {\bf M} G$ be the map given by the composition $\mathbf{S}\to \bigvee_{w\in W} \mathbf{S}\to {\bf M} G$, where the first map is the inclusion of the component with label $w_0$.
For a spectrum ${\bf E}$ we define ${\bf E} G:=E\wedge {\bf M} G$. There is a natural identification ${\bf E}\cong {\bf E} \mathbb{Z}$ and an induced morphism ${\bf E}\to {\bf E} \mathbb{R}$. The spectrum ${\bf E}\mathbb{R}$ also represents a cohomology theory which admits a canonical isomorphism $$E\mathbb{R}^*(X)\cong H^*(X;{\tt V})\ .$$
We extend the natural map ${\bf E}\to {\bf E}\mathbb{R}$ to an exact triangle $$\Sigma^{-1} {\bf E} \mathbb{R}/\mathbb{Z}\to {\bf E}\to {\bf E}\mathbb{R}\to {\bf E}\mathbb{R}/\mathbb{Z}$$ thus defining a spectrum ${\bf E}\mathbb{R}/\mathbb{Z}$. Note that ${\bf E}\mathbb{R}/\mathbb{Z}\cong {\bf E}\wedge {\bf M}\mathbb{R}/\mathbb{Z}$ so that our notation is consistent. The fibre sequence induces a long exact sequence in cohomology \begin{equation}\label{ufifwefwefewf} \dots\ \to E^{*-1}(M)\to E \mathbb{R}^{*-1}(M)\to E\mathbb{R}/\mathbb{Z}^{*-1}(M)\to E^*(M)\to E\mathbb{R}^*(M)\to\dots \ . \end{equation} In other words, it is very natural to conjecture that there is a natural transformation $\Phi_{flat}\colon \hat E_{flat}^*(M)\to E\mathbb{R}/\mathbb{Z}^{*-1}(M)$ so that the following diagram commutes \begin{eqnarray}\label{udidwqdwqd} \xymatrix{E^{*-1}(M)\ar@{=}[d]\ar[r]&E \mathbb{R}^{*-1}(M)\ \ar[r]^\alpha&E\mathbb{R}/\mathbb{Z}^{*-1}(M)\ar[r]& E^*(M)\ar@{=}[d]\ar[r]&E\mathbb{R}^*(M)\\E^{*-1}(M)\ar[r]^{\mathbf{ch}}&H^{*-1}(M;{\tt V})\ar[u]^\cong\ar[r]^{a}&\hat E_{flat}^*(M)\ar[r]^I\ar[u]^{\Phi_{flat}}&E^*(M)\ar[r]^{ {\mathbf{ch}}}&H^*(M;{\tt V})\ar[u]^\cong} \end{eqnarray} Such a transformation automatically would be an isomorphism by the Five Lemma. \begin{ddd}\label{udiqdwqdwqd54545} We say that the flat theory is \textbf{topological} if such a natural transformation $\Phi_{flat}$ exists. \end{ddd}
Recall that, given a cohomology theory represented by a spectrum ${\bf E}$, in \cite[ Definition 4.34]{MR2192936} Hopkins and Singer have constructed a smooth extension $(\hat E_{HS},R_{HS},I_{HS},a_{HS})$. Moreover they have shown that $\hat E_{HS,flat}^*$ is topological. We use the notation $$\Phi^*_{HS,flat}\colon \hat E^{*}_{HS,flat}(M)\stackrel{\sim}{\to}E\mathbb{R}/\mathbb{Z}^{*-1}(M)$$ for the natural isomorphism coming from \cite[ Equation (4.57)]{MR2192936}. \begin{theorem}\label{zuddwqdqwdqw} Assume that the cohomology \textcolor{black}{theory} $E$ is rationally even, and that $E^m$ is countably generated for all $m\in \mathbb{Z}$. If $(\hat E,R,I,a,\int)$ is a smooth extension of $E$ with integration which is defined on all smooth manifolds (or alternatively, on all compact manifolds and $E$ is even with $E^{2m}$ finitely generated for all $m\in \mathbb{Z}$), then the flat functor $\hat E_{flat}$ is topological. \end{theorem} {\it Proof.$\:\:\:\:$} If we could assume that both theories have an integration then we could employ Theorem \ref{main1}. Unfortunately an integration for the Hopkins-Singer example has not been worked out yet in detail with all the properties we need. Therefore we follow a different path. We start with Theorem \ref{hjsasas} which gives for all $n\in \mathbb{Z}$ an isomorphism $$\Phi^{2n}\colon \hat E^{2n}\stackrel{\sim}{\to} \hat E_{HS}^{2n}\ .$$ It restricts to an isomorphism $$\Phi^{2n}\colon \hat E_{flat}^{2n}\stackrel{\sim}{\to} \hat E_{HS,flat}^{2n}$$ of flat theories. We get the required $\Phi_{flat}^{2n}\colon \hat E^{2n}_{flat}(M)\to E\mathbb{R}/\mathbb{Z}^{2n-1}(M)$ as the composition $$\Phi_{flat}^{2n}\colon \hat E_{flat}^{2n}(M)\stackrel{\Phi^{2n}}{\to}\hat E^{2n}_{HS,flat}(M)\stackrel{\Phi^{2n}_{flat,HS}}{\to }E\mathbb{R}/\mathbb{Z}^{2n-1}(M)\ .$$
We want to extend this to odd degrees $2n-1$ so that the diagram $$\xymatrix{\hat E_{flat}^{2n}(S^1\times M)\ar@/^1cm/@{.>}[rr]_{\Phi^{2n}_{flat}}\ar[r]^{\Phi^{2n}}\ar[d]^\int&\hat E_{HS, flat}^{2n} (S^1\times M)\ar[r]^\cong_{\Phi_{HS,flat}^{2n}}&E\mathbb{R}/\mathbb{Z}^{2n-1}(S^1\times M)\ar[d]^\int\\\hat E_{flat}^{2n-1}(M) \ar@{.>}[rr]^{\Phi_{flat}^{2n-1}}&&E\mathbb{R}/\mathbb{Z}^{2n-2}(M)}$$ commutes. \begin{lem}\label{zdqwudqwdqw} For all $m\in \mathbb{N}$ $$\int\colon \hat E_{flat}^{m}(S^1\times M)\to \hat E_{flat}^{m-1}(M)$$ is surjective. \end{lem} {\it Proof.$\:\:\:\:$} Let $\hat x\in \hat E_{flat}^{m-1}(M)$. Then we first consider $y:=q^*\sigma(I(\hat x))\in E^{m}(S^1\times M)$, where $q\colon S^1\times M\to\Sigma M_+$ is the projection and $\sigma\colon E^{m-1}(M)\to \tilde E^{m}(\Sigma M_+)$ is the suspension isomorphism. Since $0={\tt Rham}\circ R(\hat x)={\mathbf{ch}}\circ I(\hat x)$ we see that $I(\hat x)$ is a torsion element. Let $0<N\in \mathbb{N}$ be such that $NI(\hat x)=0$. We choose a lift $\hat y\in \hat E^{m}(S^1\times M)$ so that $I(\hat y)=y$. Then $NI(\hat y)=0$ and thus $N \hat y=a(\rho)$ for some $\rho\in \Omega^{m-1}(S^1\times M,{\tt V})$. We now replace $\hat y$ by $\hat y-a(N^{-1}\rho)$. Then still $I(\hat y)=y$, but in addition $R(\hat y)=0$.
We have $I(\int \hat y)=\int I(\hat y )=\int y=I(\hat x)$. Therefore $\int \hat y-\hat x=a(\theta)$ for some $\theta \in \Omega^{m-2}(M,{\tt V})$. Moreover, $d\theta=R(\int (\hat y))-R(\hat x)=0$. Note that $\int(dt\times \theta)=\theta$ and $d(dt\times \theta)=0$. If we further replace $\hat y$ by $\hat y-a(dt\times \theta)$, then $\int \hat y=\hat x$. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
We now construct $\Phi^{2n-1}_{flat}\colon \hat E^{2n-1}_{flat}(M)\textcolor{black}{\to E\mathbb{R}/\mathbb{Z}^{2n-2}(M)}$. Let $\hat x\in \hat E^{2n-1}_{flat}(M)$. Then we choose by Lemma \ref{zdqwudqwdqw} an $\hat y\in \hat E^{2n}_{flat}(S^1\times M)$ such that $\int(\hat y)=\hat x$. Then we define $$\Phi^{2n-1}_{flat}(\hat x):=\int(\Phi^{2n}_{flat}(\hat y))\ .$$ We must show that this is well-defined. Let $\hat y^\prime\in \hat E^{2n}_{flat}(S^1\times M)$ be such that $\int \hat y^\prime=\hat x$. Then we must show that $\int(\Phi^{2n}_{flat}(\hat y))=\int(\Phi^{2n}_{flat}(\hat y^\prime))$. Let $\hat u=\hat y-\hat y^\prime$. It follows from $\int \hat u=0$, that $I(\hat u)= p^*v$ for some $v\in E^{2n}(M)$. Since $\hat u$ is flat we know that $I(\hat u)$ is torsion. Since $p^*\colon E^{2n}(M)\to E^{2n}(S^1\times M)$ is injective we conclude that $v$ is torsion. Therefore we can choose a lift $\hat v\in \hat E_{flat}^{2n}(M)$. We further find a form $\theta\in \Omega^{2n-1}(S^1\times M,{\tt V})$ such that $p^*\hat v+a(\theta)=\hat u$. If we apply $R$ to this equality, then we get
$d\theta=0$. Furthermore, from $\int \hat u=0$ we get
$a(\int \theta)=0$. Therefore ${\tt Rham}(\int \theta)={\mathbf{ch}}(z)$ for some $z\in E^{2n-2}(M)$. We choose $w\in E^{2n-1}(S^1\times M)$ such that $\int(w)=z$. Then $$\int {\tt Rham}(\theta)={\tt Rham}(\int \theta)={\mathbf{ch}}(\int w)=\int {\mathbf{ch}}(w)\ .$$ We now calculate \begin{eqnarray*} \Phi^{2n}_{flat}(\hat u)&=&\int(\Phi^{2n}_{HS,flat}(\Phi^{2n}(p^*\hat v+a(\theta))))\\ &=&\int(\Phi^{2n}_{HS,flat}(p^*\Phi^{2n}(\hat v)+a_{HS}(\theta)))\\ &=&\int(p^*\Phi^{2n}_{HS,flat}( \Phi^{2n}(\hat v))+\Phi^{2n}_{HS,flat}(a_{HS}(\theta)))\\ &=&\int(a({\tt Rham}(\theta)))\\ &=&a(\int({\mathbf{ch}}(w)))\\ &=&0\ . \end{eqnarray*} This finishes the proof that $\Phi^{2n-1}_{flat}$ is well-defined.
Let us check that $$\xymatrix{H^{2n-2}(M;{\tt V})\ar[r]^a\ar[d]^\rho_\cong&\hat E_{flat}^{2n-1}(M)\ar[r]^I\ar[d]^{\Phi_{flat}^{2n-1}}&E^{2n-1}(M)\ar@{=}[d]\\ E\mathbb{R}^{2n-2}(M)\ar[r]^\alpha &E\mathbb{R}/\mathbb{Z}^{2n-2}(M)\ar[r]^c&E^{2n-1}(M)}$$ commutes.
First we consider $x\in H^{2n-2}(M;{\tt V})$. We must show that $$\Phi_{flat}^{2n-1}\circ a(x)= \alpha\circ\rho(x)\ .$$ Let $x={\tt Rham}(\omega)$ for some $\omega\in \Omega^{2n-2}_{cl}(M,{\tt V})$. Then we take $dt\times \omega\in \Omega^{2n-1}_{cl}(S^1\times M,{\tt V})$ so that $\int (a(dt\times \omega))=a(\omega)$. Therefore \begin{eqnarray*} \Phi_{flat}^{2n-1}(a(x))&=&\int(\Phi_{HS,flat}^{2n}(\Phi^{2n}(a(dt\times \omega))))\\ &=&\int(\Phi_{HS,flat}^{2n}(a_{HS}(dt\times \omega)))\\ &=&\int(\alpha(\rho({\tt Rham}(dt\times \omega))))\\ &=&\alpha(\int(\rho({\tt Rham}(dt\times \omega))))\\ &=& \alpha(\rho(\int({\tt Rham}(dt\times \omega))))\\ &=&\alpha(\rho({\tt Rham}(\omega)))\\ &=&\alpha(\rho(x))\ .
\end{eqnarray*} Next we consider $\hat x\in \hat E_{flat}^{2n-1}(M)$. We can choose a lift $\hat y\in \hat E_{flat}^{2n}(S^1\times M)$ such that $\int (\hat y)=\hat x$ and $i^*\hat y=0$. Then we calculate \begin{eqnarray*} c(\Phi^{2n-1}_{flat}(\hat x))&=& c(\int(\Phi^{2n}_{HS,flat}(\Phi^{2n}(\hat y))))\\ &=&\int(c(\Phi^{2n}_{HS,flat}(\Phi^{2n}(\hat y))))\\ &=&\int(I_{HS}(\Phi^{2n}(\hat y)))\\ &=&\int (I(\hat y))\\&=&I(\hat x)\ . \end{eqnarray*}
In Section \ref{dlede} we will show independently \textcolor{black}{of} the Hopkins-Singer construction that the flat theory of a smooth extension with integration gives rise to a generalized cohomology theory which can be compared to $E\mathbb{R}/\mathbb{Z}$.
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\section{Exotic additive structures}
In Theorem \ref{main1} we have seen that a smooth extension of a rationally even cohomology theory with integration is unique up to unique isomorphism. In this section we show by example that if one disregards the integration there might be many different smooth extensions.
As an example we dicuss $K$-theory. This cohomology theory is even. The
even part of a smooth extension is unique up to isomorphism because of Theorem \ref{hjsasas}. In the present section we show that one can change the additive structure of $\hat K^1$ in a non-trivial way. For simplicity we consider $K$-theory and its smooth extension as two-periodic theories. We start with a first example. Then we show that there are actually infinitely many non-equivalent additive structures on $\hat K^1$.
Let us start with some smooth extension $(\hat K,R,I,a)$ of complex $K$-theory with additive structure denoted by $+$. We define a new additive structure on $\hat K^1(M)$ by $$\hat u*\hat v:=\hat u+\hat v+a(\frac{1}{2}{\mathbf{ch}}(I(\hat u)\cup I(\hat v)))\ ,\quad \hat u,\hat v\in \hat K^1(M)\ .$$ Let us verify associativity and commutativity. \begin{eqnarray*} (\hat u*\hat v)*\hat w&=&(\hat u+\hat v+a(\frac{1}{2}{\mathbf{ch}}(I(\hat u)\cup I(\hat v))))*\hat w\\ &=&\hat u+\hat v+a(\frac{1}{2}{\mathbf{ch}}(I(\hat u)\cup I(\hat v))+\hat w+a(\frac{1}{2}{\mathbf{ch}}(I(\hat u*\hat v)\cup I(\hat w))))\\ &=&\hat u+\hat v+\hat w+a(\frac{1}{2}\left({\mathbf{ch}}(I(\hat u)\cup I(\hat v))+{\mathbf{ch}}((I(\hat u)+I(\hat v))\cup I(\hat w))\right))\\ &=&\hat u+\hat v+\hat w+a(\frac{1}{2}{\mathbf{ch}}(I(\hat u)\cup I(\hat v) +I(\hat u)\cup I(\hat w)+I(\hat v)\cup I(\hat w)))\\ &\dots&\\&=& \hat u*(\hat v*\hat w)\\ \hat u*\hat v&=&\hat u+\hat v+a(\frac{1}{2}{\mathbf{ch}}(I(\hat u)\cup I(\hat v)))\\ &=&\hat v+\hat u+a(\frac{1}{2}{\mathbf{ch}}(I(\hat v)\cup I(\hat u)))+a({\mathbf{ch}}(I(\hat u)\cup I(\hat v)))\\ &=&\hat v*\hat u\ . \end{eqnarray*} Observe that this new additive structure is still compatible with the structure maps $R,I,a$. In fact, the additional term $a(\frac{1}{2}{\mathbf{ch}}(I(\hat u)\cup I(\hat v)))$ is annihilated by $R$ and $I$, and it vanishes if e.g. $\hat u=a(\omega)$. Therefore, $\hat K^1$ with this new additive structure together with the old $\hat K^0$ and the old structure maps gives rise to a smooth extension $(\hat K^\prime,R,I,a)$ of $K$-theory.
\begin{prop}\label{jdhbwqdqwdwqdwqdwd} The smooth extensions $(\hat K ,R,I,a)$ and $(\hat K^\prime,R,I,a)$ are not equivalent. \end{prop} {\it Proof.$\:\:\:\:$} Assume that there was a natural isomorphism $\Phi\colon \hat K^1\to \hat K^{\prime 1}$ which is compatible with $R$, $I$ and $a$ and such that \begin{equation}\label{zduqwdqwd} \Phi(u+v)=\Phi(u)*\Phi(v) \end{equation} Note that $K^1=K^{\prime 1}$ as set-valued functors. We define $\hat \delta\colon \hat K^1\to \hat K^1$ by $$\Phi(\hat u)= \hat u+\hat \delta(\hat u)\ .$$ Since $\Phi$ preserves $R$ and $I$ we see that $\hat \delta$ has values in $a(H^{ev}(M;\mathbb{R})/{\tt im}({\mathbf{ch}}))$. Furthermore, since $\Phi$ is compatible with $a$ we conclude that $\hat \delta$ comes from a natural transformation $$\delta\colon K^1(M)\to H^{ev}(M;\mathbb{R})/{\tt im}({\mathbf{ch}})\ , \quad a(\delta(I(\hat u)))=\hat \delta(\hat u)\ .$$ Equation (\ref{zduqwdqwd}) gives \begin{equation}\label{udiwqdwqdqd} \delta(u+v)-\delta(u)-\delta( v)=a(\frac{1}{2}{\mathbf{ch}}( u\cup v))\ . \end{equation} We now consider the smooth manifold $T^2=S^1\times S^1$. Let $e\in K^1(S^1)\cong \mathbb{Z}$ be the generator. Let $p,q\colon T^2\to S^1$ be the two projections. Then we define $ u:=p^* e$ and $ v:=q^* e$. Note that $H^{ev}(T^2;\mathbb{R})\cong \mathbb{R}^2$ with basis $\{1,{\mathrm{ori}}\}$, where ${\mathrm{ori}}\in H^2(T^2;\mathbb{R})$ is normalized such that $\langle{\mathrm{ori}} ,[T^2]\rangle=1$. Then the image of the Chern character is the lattice ${\mathbf{ch}}(K^0(T^2))= \mathbb{Z}\langle1,{\mathrm{ori}}\rangle\subset \mathbb{R}\langle1,{\mathrm{ori}}\rangle$. We have ${\mathbf{ch}}(u\cup v)={\mathrm{ori}}$. In particular, $\frac{1}{2}{\mathbf{ch}}(u\cup v)\not\in {\tt im}({\mathbf{ch}})$ so that \begin{equation}\label{zuqicqwcqwcwq} a(\frac{1}{2}{\mathbf{ch}}(u\cup v))\not=0\ .
\end{equation} Let $*\in T^2$ be a point. Then $a(\frac{1}{2}{\mathbf{ch}}(u\cup v))_{|*}=0$. We define the smooth map $r:=pq\colon T^2\to S^1$ using the group structure of $S^1$. Then we have the identity of $K$-theory classes $u+v=r^*e$. Since $\delta$ is a natural transformation we furthermore get $\delta(u)=\delta(p^*e)=p^*\delta(e)$. Note that $\delta(e)\in H^0(S^1;\mathbb{R})/{\tt im}({\mathbf{ch}})\cong \mathbb{R}/\mathbb{Z}$. It follows that $\delta(u)=c 1$ for some constant $c\in \mathbb{R}/\mathbb{Z}$. In the same way $\delta(v)=c 1$ and $\delta(u+v)=c 1$. Then we have $\delta(u+v)-\delta(u)-\delta( v)=-c1$.
It follows from (\ref{udiwqdwqdqd}) by considering the restriction to a point that $c=0\in \mathbb{R}/\mathbb{Z}$. But then $\delta=0$, and this contradicts (\ref{zuqicqwcqwcwq}). \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\begin{theorem}\label{udqiwdwqdqwd} There are infinitely many non-isomorphic smooth extensions of complex $K$-theory. \end{theorem} {\it Proof.$\:\:\:\:$} We start with a smooth extension $(\hat K,R,I,a,\int)$ with integration, e.g the multiplicative smooth extension \cite{bunke-20071} with the integration given by Proposition \ref{uzqwiwqdqd}.
Since $K$-theory is even, the associated flat theory is topological by Theorem \ref{zuddwqdqwdqw}, i.e. there is a natural isomorphism $$\Phi_{flat}\colon \hat K_{flat}^*(M)\stackrel{\sim}{\to} K\mathbb{R}/\mathbb{Z}^{*-1}(M)\ .$$ In the following, in order to simplify the notation, we will actually identify the flat $K$-theory with the $\mathbb{R}/\mathbb{Z}$-K-theory and will not write the isomorphism explicitly.
Different smooth extensions will be obtained by modifying the additive structure on $\hat K^1$. Any other group structure $*\colon \hat K^1(M)\times \hat K^1(M)\to \hat K^1(M)$ determines and is determined by a natural transformation $$\hat B\colon \hat K^1(M)\times \hat K^1(M)\to \hat K^1(M)\ ,\quad \hat x*\hat y=\hat x+\hat y+\hat B(\hat x,\hat y)\ .$$ Compatibility with $R,I$ and $a$ implies that $\hat B$ comes from a transformation $$B\colon K^1(M)\times K^1(M)\to H^{ev}(M;\mathbb{R})/{\tt im}({\mathbf{ch}})\ , \quad a(B(I(\hat u),I(\hat v)))=\hat B(\hat u,\hat v)\ . $$ Associativity and commutativity of $*$ are equivalent to the conditions \begin{equation}\label{u7qiwdqwdqwd} B(u,v+w)+ B(v,w)= B(u,v)+ B(u+v,w)\end{equation} and \begin{equation}\label{u7qiwdqwdqwd1}B(u,v)=B(v,u)\ .\end{equation}
\begin{def} \textcolor{green}{ Let ${\tt Z}$ be the group of natural transformations of functors $$B\colon K^1(M)\times K^1(M)\to H^{ev}(M;\mathbb{R})/{\tt im}({\mathbf{ch}})$$
which satisfy the two conditions \eqref{u7qiwdqwdqwd} and \eqref{u7qiwdqwdqwd1} for all manifolds $M$ and $u,v,w\in K^1(M)$.} \end{def} Given $B\in {\tt Z}$, the new additive structure $*$ is given by $$\hat u*\hat v:=\hat u+\hat v+a(B(I(\hat u),I(\hat v)))\ , \quad \hat u,\hat v\in \hat K^1(M)\ .$$
As in the proof of Proposition \ref{jdhbwqdqwdwqdwqdwd} we will write a natural transformation $\Phi\colon \hat K^1\to \hat K^1$ with \begin{equation}\label{udiwqdqwdopwqd21} \Phi(\hat x+\hat y)=\Phi(\hat x)*\Phi(\hat y) \end{equation} in the form $\Phi(\hat x)=\hat x+\hat \delta(\hat x)$ for a natural transformation $$\hat \delta\colon \hat K^1(M)\to \hat K^1(M)\ .$$ Since $\Phi$ must preserve $R,I$ and is compatible with $a$ we again conclude that $\hat \delta$ comes from $$\delta\colon K^1(M)\to H^{ev}(M;\mathbb{R})/{\tt im}({\mathbf{ch}})\ ,\quad a(\delta(I(\hat u)))=\hat \delta(\hat u)\ .$$ The equation (\ref{udiwqdqwdopwqd21}) translates to \begin{equation}\label{zuew}B(u,v)=\delta(u+v)-\delta(u)-\delta(v)\ ,\end{equation} the analog of (\ref{udiwqdwqdqd}). Note that $\Phi$ is automatically an isomorphism by the Five-Lemma.
\begin{def} \textcolor{green}{ Let ${\tt T}\subseteq {\tt Z}$ be the group of transformations of the form
(\ref{zuew}). Then the set of isomorphism classes of additive extensions of
$K^1$ is in bijection with the quotient ${\tt Z}/{\tt T}$.} \end{def}
For $i,j\in \{0,1,2\}$ let ${\tt pr}_{ij}\colon {\bf K}_1\times {\bf K}_1\times {\bf K}_1\to {\bf K}_1\times {\bf K}_1$ be the projection onto the $(i,j)$ component and $s_{01},s_{12}\colon {\bf K}_1\times {\bf K}_1\times {\bf K}_1\to {\bf K}_1\times {\bf K}_1$ the map which adds the first two or the last two factors using the $H$-space structure representing the additive structure of $K^1(X)$. Let $G$ be a group valued functor on spaces which can be applied to ${\bf K}_1$ and the products with itself. The examples which we have in mind are $G(X):=H^{ev}(X;\mathbb{R})/{\tt im}({\mathbf{ch}})$ or $G(X):=H^{ev}(X;\mathbb{R})$. The cocycle condition for $r\in G({\bf K}_1\times {\bf K}_1)$ is defined to be \begin{equation}\label{coc} s_{12}^*r+{\tt pr}_{12}^*r={\tt pr}_{01}^*r+s_{01}^*r\ . \end{equation} Furthermore, let $F\colon {\bf K}_1\times {\bf K}_1\to {\bf K}_1\times {\bf K}_1$ be the flip. The symmetry condition for $r\in G({\bf K}_1\times {\bf K}_1)$ is \begin{equation}\label{sym}F^*r=r\ . \end{equation} \begin{lem} There is a canonical inclusion $$i\colon {\tt Z}\hookrightarrow H^{ev}({\bf K}_1\times {\bf K}_1;\mathbb{R})/{\tt im}({\mathbf{ch}})$$ as the subgroup of all elements which satisfy the cocycle and symmetry conditions (\ref{coc}) and (\ref{sym}). \end{lem} {\it Proof.$\:\:\:\:$} We consider $H^{ev}(M;\mathbb{R})/{\tt im}({\mathbf{ch}})\subseteq K\mathbb{R}/\mathbb{Z}^0(M)$ in the natural way. Let $(\mathcal{K}_i)_{i\in \mathbb{N}}$ be the approximation of ${\bf K}_1$ as in Proposition \ref{duwidwdwd}. The family of products $(\mathcal{K}_i\times \mathcal{K}_i)_{i\in \mathbb{N}}$ is then a similar approximation of ${\bf K}_1\times {\bf K}_1$. We let $U\in K^1({\bf K}_1)$ be the tautological element. Let $B\in {\tt Z}$. The family $(B_i)_{i\in \mathbb{N}}:=(B({\tt pr}_0^*( x_i^* U),{\tt pr}_1^*( x_i^*U)))_{i\in \mathbb{N}}$ is an element $(B_i)\in \lim ( K\mathbb{R}/\mathbb{Z}^0(\mathcal{K}_i\times \mathcal{K}_i))$. In view of the Milnor sequence $$0\to \lim\hspace{-2pt}^1 ( K\mathbb{R}/\mathbb{Z}^{-1}(\mathcal{K}_i\times \mathcal{K}_i) )\to K\mathbb{R}/\mathbb{Z}^0({\bf K}_1\times {\bf K}_1)\to \lim ( K\mathbb{R}/\mathbb{Z}^0(\mathcal{K}_i\times \mathcal{K}_i))\to 0$$ we can choose a preimage $\tilde B\in K\mathbb{R}/\mathbb{Z}^0({\bf K}_1\times {\bf K}_1)$. This preimage is unique up to phantom elements (i.e. elements coming from the $\lim^1$-term, compare Definition \ref{dqwudqwdqw}), and therefore unique by Corollary \ref{uoewqeqwe}. We have exact sequences, natural in the space $X$, $$\dots\to K^0(X)\to K\mathbb{R}^0(X)\stackrel{\alpha}{\to} K\mathbb{R}/\mathbb{Z}^0(X) \stackrel{\beta}{\to} K^1(X)\to \dots\ .$$ By construction we know that $B_i \in K\mathbb{R}/\mathbb{Z}^0(\mathcal{K}_i\times \mathcal{K}_i)$ is in the image of $\alpha$. Therefore $\beta((B_i))=0$, and this implies that $\beta(\tilde B)\in K^1_{phantom}({\bf K}_1\times {\bf K}_1)\subseteq K^1({\bf K}_1\times {\bf K}_1)$ (see Definition \ref{dqwudqwdqw}). From Corollary \ref{hdwqdqwdwq} we get $\beta(\tilde B)=0$ so that $\tilde B=\alpha (\bar B)$ for some $\bar B\in K\mathbb{R}^0({\bf K}_1\times {\bf K}_1)$ which is well-defined up to elements coming from $K^0({\bf K}_1\times {\bf K}_1)$. If we apply the Chern character, then we get a well-defined element $$i(B)={\mathbf{ch}}(\bar B)\in H^{ev}({\bf K}_1\times {\bf K}_1;\mathbb{R})/{\tt im}({\mathbf{ch}})\ .$$ In this way we construct a map $${\tt Z}\to H^{ev}({\bf K}_1\times {\bf K}_1;\mathbb{R})/{\tt im}({\mathbf{ch}})\ .$$ We now discuss injectivity. Assume that $i(B)=0$. Then $\tilde B=0$, and this implies that $(B_i)=0$, i.e. $B_i=0$ for all $i\in \mathbb{N}$. We now show that this in turn implies $B=0$.
Indeed, let $M$ be a manifold and $u,v\in K^1(M)$. Then there exists an $i\in \mathbb{N}$ and maps $f_u,f_v\colon M\to \mathcal{K}_i$ such that $u=f_u^*(x_i^*U)$ and $v=f_v^*(x_i^*U)$. But then by naturality of $B$ $B(u,v)=(f_u,f_v)^*B_i=0$.
We now show that every element ${\tt B}\in H^{ev}({\bf K}_1\times {\bf K}_1;\mathbb{R})/{\tt im}({\mathbf{ch}})$ which satisfies the cocycle and symmetry conditions (\ref{coc}) and (\ref{sym}) is in the image of $i$. Indeed, ${\tt B}$ induces a natural transformation $B$ by $B(u,v):=(u,v)^*{\tt B}$ for all compact manifolds $M$ and $u,v\in K^1(M)$ considered here as homotopy classes of maps $M\to {\bf K}_1$. One checks that $B$ satisfies the cocycle and symmetry conditions, and that $i(B)={\tt B}$. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
Let $s\colon {\bf K}_1\times {\bf K}_1\to {\bf K}_1$ be the H-space operation, i.e. the map which represents the additive structure on $K^1(X)$. \begin{lem} Under $i\colon {\tt Z} \to H^{ev}({\bf K}_1\times {\bf K}_1;\mathbb{R})/{\tt im}({\mathbf{ch}})$ the subgroup ${\tt T}\subseteq {\tt Z}$ corresponds to the subgroup of classes of the form $s^*(x)-{\tt pr}_0^*x-{\tt pr}_1^*x$ for some $x\in H^{ev}({\bf K}_1;\mathbb{R})/{\tt im}({\mathbf{ch}})$. \end{lem} {\it Proof.$\:\:\:\:$} We consider a natural transformation $$\delta\colon K^1(M)\to H^{ev}(M;\mathbb{R})/{\tt im}({\mathbf{ch}})\subseteq K\mathbb{R}/\mathbb{Z}^0(M)$$ of functors on manifolds. The family $(\delta_i)_{i\in \mathbb{N}}$, $\delta_i:=\delta(x_i)\in K\mathbb{R}/\mathbb{Z}^0(\mathcal{K}_i)$ gives rise to an element $(\delta_i)\in \lim ( K^0_{\mathbb{R}/\mathbb{Z}}(\mathcal{K}_i))$. It again has a unique lift $\tilde \delta\in K\mathbb{R}/\mathbb{Z}^0({\bf K}_1)$ which is mapped to a phantom class under $K\mathbb{R}/\mathbb{Z}^0 ({\bf K}_1)\to K^1({\bf K}_1)$. From Corollary \ref{hdwqdqwdwq} we conclude that it belongs to the subgroup ${\tt im}(K\mathbb{R}^0 ({\bf K}_1) \to K\mathbb{R}/\mathbb{Z}^0 ({\bf K}_1))\subseteq K\mathbb{R}/\mathbb{Z}^0 ({\bf K}_1)$. If we apply the Chern character we get a well-defined element $\bar \delta\in H^{ev}({\bf K}_1;\mathbb{R})/{\tt im}({\mathbf{ch}})$. Let $B_\delta$ denote the transformation given by (\ref{zuew}). Then $i(B_\delta)=s^*(\bar \delta)-{\tt pr}_0^*\bar \delta-{\tt pr}_1^*\bar \delta$. Commutativity of $s$ means $F^*s^*=s^*$. Furthermore, associativity of $s$ can be expressed as $s_{01}^*s^*=s_{12}^*s^*$. These two relations together with some obvious identities for the projections imply that the image of $s^*-{\tt pr}_0^*-{\tt pr}_1^*$ satisfies the cocycle and symmetry conditions and therefore belongs to $i({\tt Z})$. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
Since ${\bf K}$ is a ring spectrum with free $H_*({\bf K}_*;\mathbb{Z})$ we get a Hopf ring $H_*({\bf K}_*;\mathbb{Z})$ which has been calculated e.g. in \cite{MR1870619}. In particular, the underlying graded ring $$H_*({\bf K}_1;\mathbb{Z})=\Lambda[b_1,b_3,\dots]$$ is an exterior algebra with generators $b_{2k-1}\in H_{2k-1}({\bf K}_1;\mathbb{Z})$. It is free over $\mathbb{Z}$ so that we have by the K\"unneth formula $H_*({\bf K}_1\times {\bf K}_1;\mathbb{Z})\cong H_*({\bf K}_1;\mathbb{Z})\otimes_\mathbb{Z} H_*({\bf K}_1;\mathbb{Z})$. We let $$c_{2k+1}\in H^{2k+1}({\bf K}_1;\mathbb{Z})\cong {\tt Hom}_{\mathrm{Ab}}(H_*({\bf K}_1;\mathbb{Z}),\mathbb{Z})^{2k+1}$$ be dual to $b_{2k+1}$ and so that it annihilates all decomposeables. Then we know that $H^*({\bf K}_1;\mathbb{Z})\cong \Lambda[[c_1,c_3,\dots]]$. Furthermore, we have the K\"unneth formula $$H^*({\bf K}_1\times {\bf K}_1)\cong \Lambda[[c_1,c_3,\dots]]\hat \otimes \Lambda[[c_1,c_3,\dots]]\ .$$ We actually do not have to complete here since the cohomology is finitely generated in every single degree. The sum $s\colon {\bf K}_1\times {\bf K}_1\to {\bf K}_1$ gives a coproduct $$s^*\colon H^*({\bf K}_1;\mathbb{Z})\to H^*({\bf K}_1;\mathbb{Z})\hat\otimes H^*({\bf K}_1;\mathbb{Z})\ ,$$ and by construction the generators $c_i$ are primitive, i.e. they satisfy $$s^*(c_i)=c_i\otimes 1+1\otimes c_i\ .$$
\begin{lem}\label{udiqwedqw} If $x\in H^{*}({\bf K}_1;\mathbb{Z})$ is primitive, then $x\otimes x\in H^{2*}({\bf K}_1\times {\bf K}_1;\mathbb{Z})$ satisfies the cocycle condition (\ref{coc}). \end{lem} {\it Proof.$\:\:\:\:$}
If we insert $r:=x\otimes x$ into the left hand side of (\ref{coc}) we get using primitivity of $x$ $$x\otimes x\otimes 1+x\otimes 1\otimes x+1\otimes x\otimes x\ .$$ The right-hand side of (\ref{coc}) yields $$x\otimes x\otimes 1+x\otimes 1\otimes x+1\otimes x\otimes x\ .$$ \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
The following proposition immediately implies Theorem \ref{udqiwdwqdqwd}. \begin{prop}\label{dioqwdqwd} The group ${\tt Z}/{\tt T}$ has infinite order.
\end{prop} {\it Proof.$\:\:\:\:$} Recall that $H^*(BU;\mathbb{Z})\cong \mathbb{Z}[[c_2,c_4,\dots]]$, where we index the Chern classes by their degree. We furthermore have an injection $H^*(BU;\mathbb{Z})\subset H^*(BU;\mathbb{Q})\cong \mathbb{Q}[[c_2,c_4,\dots]]$. On the integral and rational cohomology of a space $X$ we consider the compatible decreasing filtrations $$F^kH^*(X;\mathbb{Z}):=\prod_{i\ge k}H^i(X;\mathbb{Z})\ ,\quad F^kH^*(X;\mathbb{Q}):=\prod_{i\ge k}H^i(X;\mathbb{Z})\ .$$
Let ${\mathbf{ch}}_{k}(u)\in H^{k}(X;\mathbb{Q})$ denote the degree $k$-component of the Chern character ${\mathbf{ch}}(u)\in H^{*}(X;\mathbb{Q})$ of $u\in K^*(X)$.
If $y\in K^0(BU)$ satisfies ${\mathbf{ch}}(y)\in F^{2k}H^*(BU;\mathbb{Q})$, then we know that ${\mathbf{ch}}_{2k}(y)\in H^{2k}(BU;\mathbb{Z})$. In fact, more is known.
\begin{lem} For all $x\in H^{2k}(BU;\mathbb{Z})$ there exists $u\in K^0(BU)$ such that ${\mathbf{ch}}(u)\in F^{2k}H^{*}(BU;\mathbb{Z})$ and ${\mathbf{ch}}_{2k}(u)=x$. \end{lem} {\it Proof.$\:\:\:\:$} It has been shown by \cite{MR1443417} that for all $1\le k\le n$ we have ${\mathbf{ch}}(\rho_{k,n})\in F^{2k}H^*(BU(k);\mathbb{Q})$ and ${\mathbf{ch}}_{2k}(\rho_{k,n})=c_{2k}$, where $\rho_{k,n}=\lambda^k(\rho_n-n)$, $\rho_n$ is the class in $K^0(BU(n))$ of the universal bundle over $BU(n)$, and $\lambda^k$ is the $k^{\text{th}}$ $\lambda$-operation. Under the map $BU(n-1)\to BU(n)$ the restriction of $\rho_n-n$ is $\rho_{n-1}-(n-1)$. Therefore the family of classes $(\rho_n-n)_{n\in \mathbb{N}}$ defines an element $\rho\in \lim ( K^0(BU(n))\cong K^0(BU))$. We have ${\mathbf{ch}}(\lambda^k\rho)\in F^{2k}H^*(BU;\mathbb{Q})$ and ${\mathbf{ch}}_{2k}(\lambda^k \rho)=c_{2k}$. The class $x\in H^{2k}(BU;\mathbb{Z})$ can be written as $x=p(c_2,c_4,\dots)$, where $p$ is a homogeneous integral polynomial in the classes $c_{2i}$. Using that ${\mathbf{ch}}$ is a ring homomorphism we see that we can choose $u:=p(\lambda^1\rho,\lambda^2\rho,\dots)$. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
In the following we take the infinite unitary group $U:={\tt colim} (U(n))$ as a model for ${\bf K}_1$. The universal bundle $U\to EU\stackrel{\pi}{\to} BU$ gives rise to a transgression $T_E\colon \tilde E^*(BU)\to \tilde E^{*-1}(U)$ for every generalized cohomology theory $E$. Let us describe the transgression geometrically. We assume that all spaces have base points. Let ${\bf E}$ be an $\Omega$-spectrum representing $E$. Let $f\colon BU\to {\bf E}_k$ represent a class $x\in \tilde E^k(BU)$. Since $\pi^*x=0$ the composition $f\circ \pi$ admits a zero homotopy $H\colon I\times EU\to {\bf E}_k$. We identify $U$ with the fibre of $\pi$ over the base point of $BU$. Then the restriction of $H$ to $I\times U$ closes up and defines a map $\Sigma U\to {\bf E}_k$, or equivalently, a map $U\to \Omega {\bf E}_k\cong {\bf E}_{k-1}$. This latter map represents the class $T_E(x)\in \tilde E^{k-1}(U)$. The transgression is natural with respect to transformations of cohomology theories $\phi\colon E^*\to F^*$, i.e we have $\phi\circ T_E=T_F\circ \phi$.
For all $k\in \mathbb{N}$ we choose classes $u_{2k}\in K^0(BU)$ such that ${\mathbf{ch}}(u_{2k})\in F^{2k}H^*(BU;\mathbb{Q})$ and ${\mathbf{ch}}_{2k}(u_{2k})=c_{2k}$. We further define $v_{2k-1}:=T_K(u_{2k})\in K^{-1}(U)\cong K^1(U)$. Since ${\mathbf{ch}}\colon K\to H\mathbb{Q}$ is a natural transformation of cohomology theories we have ${\mathbf{ch}}(v_{2k-1})=T_{H\mathbb{Q}}{\mathbf{ch}}(u_{2k})$. The generators of $$H^*(U;\mathbb{Z})=\Lambda[[c_1,c_3,\dots]]$$ are given by $c_{2k-1}=T_{H\mathbb{Z}}(c_{2k})$. Note that $T(F^k H^*(BU;\mathbb{Q}))\subseteq F^{k-1}H^*(U;\mathbb{Q})$. We conclude that ${\mathbf{ch}}(v_{2k-1})\in F^{2k-1}H^*(U;\mathbb{Q})$ and ${\mathbf{ch}}_{2k-1}(v_{2k-1})=c_{2k-1}$, \textcolor{black}{compare also the appendix of \cite{BS}}.
It is a well-known fact that the transgression $T(x)\in H^{*-1}(U;\mathbb{Q})$ of $x\in H^*(BU;\mathbb{Q})$ is primitive. In order to see this one could consider a dual transgression in homology. One easily observes geometrically that the transgression of a non-trivial Pontrjagin product of homology classes gets annihilated. This implies that $T(x)$ annihilates all non-trivial Pontrjagin products of homology classes. This property is equivalent to primitivity.
We consider the differential\footnote{The $H$-space ${\bf K}_1$ gives rise to the simplicial space $B^\bullet{\bf K}_1$ and the corresponding cobar complex $(H^*(B^\bullet{\bf K}_1;\mathbb{R}),d)$ in cohomology. It is for this reason that we call $d$ a differential.} $d\colon H^*({\bf K}_1;\mathbb{R})\to H^*({\bf K}_1\times {\bf K}_1;\mathbb{R})$ given by $$d(x):=s^* x-1\times x-x\times 1=s^*x-{\tt pr}_0^*x-{\tt pr}_1^*x \ ,$$ where $s\colon {\bf K}_1\times {\bf K}_1 \to {\bf K}_1$ is the $H$-space structure as above. Note that $H^*({\bf K}_1;\mathbb{R})\cong\Lambda_\mathbb{R}[[c_1,c_3,\dots]]$ is generated by primitive elements. We consider $H^*({\bf K}_1;\mathbb{Z})\cong \Lambda_\mathbb{Z}[[c_1,c_3,\dots]]\subset H^*({\bf K}_1;\mathbb{R})$ as a subspace in the natural way. \begin{lem}\label{duqwidqwd} For $i>0$ the differential $d\colon H^{2i}({\bf K}_1;\mathbb{R})\to H^{2i}({\bf K}_1\times {\bf K}_1;\mathbb{R})$ is injective. If $x\in H^{2i}({\bf K}_1;\mathbb{R})$ and $d(x)\in H^{2i}({\bf K}_1\times {\bf K}_1;\mathbb{Z})$, then $x\in H^{2i}({\bf K}_1;\mathbb{Z})$. \end{lem} {\it Proof.$\:\:\:\:$} Let $I:=\{i_1<i_2<\dots<i_{r}\}$ be a sequence of odd integers. It determines a monomial $m_I:=c_{i_1}\dots c_{i_{r}}\in H^*({\bf K}_1;\mathbb{R})$. Using the primitivity of the $c_i$ we observe that $$s^* (m_I)=\sum_{P\sqcup Q} l(P,Q) m_P\otimes m_Q\ ,$$ where the sum is taken over all partitions $P\sqcup Q=\{i_1,\dots i_r\}$ and $l(P,Q)\in \{1,-1\}$ is the sign of the permutation determined by $P,Q$. It follows that $$d m_I=\sum_{P\sqcup Q=I, P,Q\not=\emptyset} l(P,Q) m_P\otimes m_Q\ .$$ If $x=\sum_{I\not=\emptyset} a_Im_I\in H^{2i}({\bf K}_1;\mathbb{R})$ with $a_I\in \mathbb{R}$, then we have \begin{equation}\label{iudoqwdqwd} d(x)=\sum_{I\not=\emptyset}a_I\sum_{P\sqcup Q=I, P,Q\not=\emptyset} l(P,Q) m_P\otimes m_Q\ . \end{equation} As $x$ is an even cohomology class, each $I$ in this sum with $a_I\not=0$ has at least two elements so that there exists at least one partition $P\sqcup Q=I$ with $P$ and $Q$ nonempty. We now see that we can recover the $a_I$ from the right-hand side of (\ref{iudoqwdqwd}). In particular, if the right-hand side is integral, all the coefficients $a_I$ must be integral. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\begin{lem} The images $w_{4k-2} \in H^{ev}({\bf K}_1\times {\bf K}_1;\mathbb{R})/{\tt im}({\mathbf{ch}})$ of the classes $$\frac{1}{2} {\mathbf{ch}}(v_{2k-1})\times {\mathbf{ch}}(v_{2k-1})\in H^{ev}({\bf K}_1\times {\bf K}_1;\mathbb{R})$$
generate an infinite sum of copies of $\mathbb{Z}/2\mathbb{Z}$ in ${\tt Z}/{\tt T}$. \end{lem} {\it Proof.$\:\:\:\:$} Since ${\mathbf{ch}}(v_{2k-1})=T({\mathbf{ch}}(u_{2k}))$ is primitive, the class $w_{4k-2}$ satisfies the cocycle condition (\ref{coc}) by Lemma \ref{udiqwedqw}. Since ${\mathbf{ch}}(v_{2k-1})$ is odd and ${\mathbf{ch}}(v_{2k-1})\times {\mathbf{ch}}(v_{2k-1})={\mathbf{ch}}(v_{2k-1}\times v_{2k-1})\in {\tt im}({\mathbf{ch}})$ and in addition $2w_{4k-2}=0$ we see that $w_{4k-2}$ satisfies the symmetry condition (\ref{sym}). It therefore remains to show that the $w_{4k-2}$ are independent.
Assume that $\sum_{i=1}^r a_i w_{4i-2}=0$. We must show that all $a_i$ are even. We proceed by induction. Let us assume that we have shown that $a_i=2b_i$ for $i=1,\dots k-1$. Then $0=\sum_{i=k}^r a_i w_{4i-2}$ so that there are $x\in K^0({\bf K}_1\times {\bf K}_1)$ and $y\in H^*({\bf K}_1;\mathbb{R})$ such that $$\sum_{i=k}^r \frac{a_i}{2} {\mathbf{ch}}(v_{2i-1})\times {\mathbf{ch}}(v_{2i-1})=d(y)+{\mathbf{ch}}(x)\ .$$ Note that the left-hand side belongs to $F^{4k-2}H^*({\bf K}_1\times {\bf K}_1;\mathbb{R})$, and that the lowest term is $\frac{a_k}{2}c_{2k-1}\times c_{2k-1}$. We claim that we can adjust $y$ such that $d(y)\in F^{4k-2}H^*({\bf K}_1\times {\bf K}_1;\mathbb{R})$. Let $j\in \mathbb{N}$ be minimal such that $d(y)\in F^jH^*({\bf K}_1\times {\bf K}_1;\mathbb{R})$. By the first assertion of Lemma \ref{duqwidqwd} the lowest term of $d(y)$ is given by $d(y)_j=d(y_j)$, where $y_j$ is the component of $y$ in degree $j$. If $j<4k-2$, then
${\mathbf{ch}}(x)\in F^jH^*({\bf K}_1\times {\bf K}_1;\mathbb{R})$, and $d(y_j)=-{\mathbf{ch}}_{j}(x)$. We conclude that $d(y_j)$ is integral. By the second assertion of Lemma \ref{duqwidqwd} we see that $y_j$ is integral. Hence there exists $u\in K^0({\bf K}_1)$ such that ${\mathbf{ch}}_j(u)=y_j$. If we replace $y$ by $y^\prime:=y-{\mathbf{ch}}(u)$ and $x$ by $x^\prime:=x+d(u)$, then we have $d(y)+{\mathbf{ch}}(x)=d(y^\prime)+{\mathbf{ch}}(x^\prime)$, and we have $d(y^\prime)\in F^{j+1}H^*({\bf K}_1\times {\bf K}_1;\mathbb{R})$. If $j=4k-2$, then $$\frac{a_k}{2}c_{2k-1}\times c_{2k-1}=d(y_{4k-2})+{\mathbf{ch}}_{4k-2}(x)\ .$$ Since now ${\mathbf{ch}}(x)\in F^{4k-2}H^*({\bf K}_1\times {\bf K}_1;\mathbb{Q})$ we see that ${\mathbf{ch}}_{4k-2}(x)$ is integral. In particular, the coefficent of ${\mathbf{ch}}_{4k-2}(x)$ in front of the monomial $c_{2k-1}\times c_{2k-1}$ must be integral. Since this monomial does not occur in $d(y)$ by the calculation (\ref{iudoqwdqwd}) we see that $\frac{a_k}{2}$ must be integral, too. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent This finishes also the proof of Proposition \ref{dioqwdqwd} and therefore of Theorem \ref{udqiwdwqdqwd}. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\section{Mayer-Vietoris sequence}\label{dlede}\label{e89wfoewfwefq}
We consider a smooth extension with a natural integration $(\hat E,R,I,a,\int)$ of a generalized cohomology theory $E$. It gives rise to the flat theory $\hat E_{flat}$ (Definition \ref{uiqwdqwdwqd}) which is a homotopy invariant functor on the category of manifolds. In this section we show that it is a generalized cohomology theory by constructing a Mayer-Vietoris sequence. If $\hat E_{flat}$ is topological, then it is clearly a generalized cohomology theory. The point of the present section is to give a construction of the Mayer-Vietoris sequence independently of Theorem \ref{zuddwqdqwdqw} and hence of the Hopkins-Singer theory.
In the present section we assume that the smooth extension is defined on the category of all smooth manifolds. There is a version of the theory for smooth extensions defined on compact manifolds (with boundary). In this case we must replace the words manifold by compact manifold and finite-dimensional countable $CW$-complex by finite $CW$-complex at the appropriate places. We indicate some further modifications as footnotes.
We consider a manifold $M$ which is decomposed as a union of open submanifolds \begin{equation}\label{udiqwdqwdqwdqdqwd} M=U\cup V\ ,\quad A:=U\cap V\ . \footnote{The modification in the case of a smooth extension defined on compact manifolds is as follows. We assume that $U$ and $V$ are closed (with boundary), and that there are deformation retracts of $U$ and $V$ onto compact $U^\prime\subset {\tt int}(U)$ and $V^\prime\subset {\tt int} V$.}\end{equation}
We choose a smooth function $\chi\colon M\to [-1,1]$ such that
$$\chi_{M\setminus V}=-1\ ,\quad \chi_{|M\setminus U}=1\footnote{If we must work with compact manifolds, then we require these condition with $U^\prime,V^\prime$ in place of $U,V$.}\ .$$ Let $\Sigma^uA:=[-1,1]\times A/\sim$ denote the unreduced suspension, where $(t,a)\sim (t^\prime,a^\prime)$ if and only if $t=t^\prime=1$ or $t=t^\prime=-1$ or $t=t^\prime$ and $a=a^\prime$. The equivalence classes in the first two cases will be denoted by $*_+$ and $*_-$. We define a projection $p\colon M\to \Sigma^u A$ by \begin{equation}\label{zuiudwqdqwdqd64646} p(m)=\left\{ \begin{array}{ccc} (\chi(m),m)&m\in A\\ *_-&m\in U\setminus A\\ *_+&m\in V\setminus A \end{array}\right\}\footnote{If we must work with compact manifolds, then the modified formula would involve the retractions of $U$ \textcolor{black}{onto} $U^\prime$ and $V$ \textcolor{black}{onto} $V^\prime$.} \end{equation}
The restriction $p_{|U}$ is zero-homotopic. Let us give the zero homotopy $p_U\colon I\times U\to \Sigma^uA$ by an explicit formula: $$p_U(t,m):=(t+(1-t)\chi(m),m)\ .$$
We define the zero homotopy $p_V\colon I\times V\to \Sigma^u A$ of $p_{|V}$ by a similar formula. The decomposition (\ref{udiqwdqwdqwdqdqwd}) gives rise to a Mayer-Vietoris sequence $$\dots \to E^k(M)\to E^k(U)\oplus E^k(V)\to E^k(A)\stackrel{\partial}{\to} E^{k+1}(M)\to \dots\ .$$ An explicit description of the boundary operator is as follows. We consider a class $x\in E^k(A)$ which we assume to be represented by a map $g\colon A\to {\bf E}_k$. Then using the projection $\Sigma^u{\bf E}_k\to \Sigma {\bf E}_k$ and the structure map $\Sigma{\bf E}_k\to {\bf E}_{k+1}$ we define $\Sigma g\colon \Sigma^u A\to \Sigma^u {\bf E}_k\to \Sigma {\bf E}_k \to {\bf E}_{k+1}$. The composition $h:=\Sigma g\circ p\colon M\to {\bf E}_{k+1}$ represents $\partial x\in E^{k+1}(M)$. It comes with zero homotopies
$h_U:=\Sigma g\circ p_U$ and $h_V:=\Sigma g\circ p_V$ of $h_{|U}$ and $h_{|V}$. Over $A$ we can glue these zero homotopies to a map $$h_U^{op}\sharp h_V\colon S^1\times A\cong \frac{[-1,0]\times A\sqcup [0,1]\times A}{\sim}\stackrel{ h_U^{op}\sqcup h_V}{\to} {\bf E}_{k+1}\ ,$$ where $h_U^{op}(t,m)=h_U(-t,m)$. This map represents a class $\tilde x\in E^{k+1}(S^1\times A)$ such that $\int\tilde x=x$.
\begin{lem}\label{dwqdqwdw}
Let $x\in E^k(M)$ be such that $x_{|U}=0$ and $x_{|V}=0$. Then there exists a based manifold $N$, a class $y\in E^k(N)$ with trivial restriction to the base point, a map $f\colon M\to N$ such that $x=f^*y$, and zero homotopies $f_U$ and $f_V$ of
$f_{|U}$ and $f_{|V}$.
In addition, if $z\in E^{k-1}(A)$ is such that $x=\partial z$ then we can choose $f\colon M\to N$, $y\in E^k(N)$ and the zero homotopies $f_U$ and $f_V$ such that $\int l^*y=z$, where $l:=f_U^{op}\sharp f_V$.
Even more specifically, if $\partial z=x=0$, then we can choose these objects such that in addition there exists a zero homotopy $f_M$ of $f$. \end{lem} {\it Proof.$\:\:\:\:$} Let $g\colon M\to {\bf E}_k$ represent the class $x$. Then there are zero homotopies
$g_U\colon I\times U\to {\bf E}_k$ and $g_V\colon I\times V\to {\bf E}_k$ of $g_{|U}$ and $g_{|V}$. Since $M$, $I\times U$ and $I\times V$ are finite-dimensional manifolds there exists a countable finite-dimensional subcomplex $X\subseteq {\bf E}_k$ containing the images of the maps $g,g_U$ and $g_V$. Every countable finite-dimensional $CW$-complex is homotopy equivalent to a smooth manifold. We choose such a manifold $N$ with base point together with mutually inverse homotopy equivalences $s\colon N\to X$ and $r\colon X\to N$, and we define $y\in E^k(N)$ as the class represented by the composition $N\stackrel{s}{\to} X\to {\bf E}_k$. We further set $\tilde f:=r\circ g$,
$\tilde f_U:= r\circ g_U$, and $\tilde f_V:= r\circ g_V$. Finally we first replace $\tilde f$ by a homotopic smooth map. In the second step we replace the zero homotopies $\tilde f_U, \tilde f_V$ by smooth zero homotopies $f_U,f_V$ of $f_{|U} , f_{|V}$.
If $x=\partial z$, then we can choose a map $j\colon A\to {\bf E}_{k-1}$ representing $z$ and the map $g$ as the composition $$g\colon M\stackrel{p}{\to} \Sigma^u A\stackrel{\Sigma j}{\to} \Sigma^u {\bf E}_{k-1}\to {\bf E}_k\ .$$ In this case the homotopies $g_U,g_V$ are induced by the homotopies $p_U,p_V$. Then we proceed as above.
Finally, if $x=0$, then the map $g$ admits a zero homotopy $g_M$. We choose the finite subcomplex $X$ sufficiently large to capture the image of $g_M$, too. Then we proceed as above and let $f_M$ be induced by $g_M$. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\begin{lem}\label{duiqdwqdqwd} Let $u\in E^{k-1}(U)$. Then there exists a based manifold $N$, a class $y \in E^{k}(N )$ vanishing on the base point, and a map $l \colon \Sigma^u U\to N $ such that $l^*y=\sigma(u)$. We can assume that $l $ is constant near the singular points of the unreduced suspension and smooth elsewhere, \textcolor{black}{and} that the singular points $*_\pm$ are mapped to the base point $*$ of $N $. \end{lem} {\it Proof.$\:\:\:\:$} Let ${\bf E}_k$ be as in the proof of Lemma \ref{dwqdqwdw}. We choose a map $g\colon \Sigma^u U\to {\bf E}_k$ which represents the class $\sigma(u)$ and maps $*_\pm$ to the base point. The map $g$ factors over a finite-dimensional countable subcomplex $X$ which we approximate by a smooth manifold $s\colon N\to X$, $r\colon X\to N$ such that the composition $N\stackrel{s}{\to} X\to {\bf E}_k$ represents $y\in E^k(N)$
and maps the basepoint $*\in N$ to the base point of ${\bf E}_k$. We set $\tilde l :=r¸\circ g$. Finally we replace this by a homotopic map with the required properties. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\begin{lem}\label{7eifdwqedqfqd}
Let $u\in E^{k-1}(U)$ and $v\in E^{k-1}(V)$ be such that $u_{|A}=v_{|A}$. Then there exists a based smooth manifold $N$, a class $y\in E^k(N)$ vanishing on the base point, and maps $f\colon U\to N$ and $g\colon V\to N$ such that $u=f^*y$, $v=g^*y$, and there is a homotopy $f_{|A}\sim g_{|A}$. \end{lem} {\it Proof.$\:\:\:\:$}
Let ${\bf E}_k$ be as in the proof of Lemma \ref{dwqdqwdw}. Furthermore, let
$a\colon U\to {\bf E}_k$ and $b\colon V\to {\bf E}_k$ represent the classes $u$ and $v$. Then there exists a homotopy $h\colon a_{|A}\sim b_{|A}$. We choose a finite-dimensional countable $X\subseteq{\bf E}_k$ over which the maps $a,b$ and the homotopy $h$ factor. Then we choose a smooth approximation $s\colon N\to X$, $r\colon X\to N$ by homotopy equivalences, let $y\in E^k(N)$ be represented by $N\stackrel{s}{\to} X\to {\bf E}_k$,
$\tilde f:=r\circ a$, $\tilde g:=r\circ b$, and $\tilde h:=r\circ h$. Then we first approximate $f$ and $g$ by smooth maps, and then choose the smooth homotopy between $f_{|A}$ and $g_{|A}$ by adapting $\tilde h$. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
Let us now construct the boundary operator $$\hat \partial\colon \hat E_{flat}^k(A)\to \hat E^{k+1}_{flat}(M)\ .$$
Let $\hat x\in \hat E_{flat}^k(A)$ be given. We set $x:=I(\hat x)$. Then by Lemma \ref{dwqdqwdw} there is a pointed smooth manifold $N$, a class $y\in E^{k+1}(N)$ with $y_{|*}=0$, and a smooth map $f\colon M\to N$ such that \begin{enumerate} \item $f^*y=\partial x$, \item there are zero homotopies $f_{U}\colon I\times U\to N$ and $f_V\colon I\times V\to N$
of $f_{|U}$ and $f_{|V}$, \item with $l:=f_U^{op}\sharp f_V\colon S^1\times A\to N$ we have $\int l^*y=x$. \end{enumerate} We choose a smooth lift $\hat y\in \hat E^{k+1}(N)$ which restricts trivially to the base point. Indeed, if we choose some class $\hat y\in \hat E^{k+1}(N)$ such that $I(\hat y)=y$, then
$\hat y_{|*}=a(c)$ for some $c\in \Omega^k(*,{\tt V})\cong {\tt V}^k $. We denote the constant zero form with value $c$ on $N$ by the same symbol. If we replace $\hat y$ by $\hat y-a(c)$, then the restriction of the new $\hat y$ to the base point vanishes.
We furthermore choose
a form $\rho\in \Omega^{k-1}(A,{\tt V})$ such that $$\int l^*\hat y+a(\rho)=\hat x\ .$$ Finally, we choose forms $\rho_U\in \Omega^{k-1}(U,{\tt V})$ and $\rho_V\in \Omega^{k-1}(V,{\tt V})$ such that
$\rho=\rho_{V|A}-\rho_{U|A}$. Using these choices we define a form $\kappa\in \Omega^{k}(M,{\tt V})$ by the description
$$\kappa_{|U}:=\int_{I\times U/U} f_U^* R(\hat y)+d\rho_U\ ,\quad \kappa_{|V}:=\int_{I\times V/V} f_V^* R(\hat y)+d\rho_V\ .$$ Note that $\kappa$ is well-defined on $A$ since \begin{eqnarray*}\lefteqn{
\left(\int_{I\times V/V} f_V^* R(\hat y)+d\rho_V\right)_{|A}- \left(\int_{I\times U/U} f_U^* R(\hat y)+d\rho_U\right)_{|A}}&&\\&=& \int_{S^1\times A/A} l^* R(\hat y)+d\rho\\&=&R(\hat x)\\&=&0\ . \end{eqnarray*} We define $$\hat \partial \hat x:=f^*\hat y+a(\kappa)\ .$$
\begin{lem} $\hat \partial \hat x$ is well-defined. \end{lem} {\it Proof.$\:\:\:\:$} We show step by step that if we alter the choices going into the construction of $\hat \partial\hat x$ we get the same result. We indicate the changed objects by a prime. \begin{enumerate} \item If we choose $\rho_U^\prime$ and $\rho_V^\prime$ such that
$\rho^\prime_{V|A}-\rho^\prime_{U|A}=\rho$, then there exists a form $\theta\in \Omega^{k-1}(M,{\tt V})$ such that
$\theta_{|U}=\rho_U-\rho_{U}^\prime$ and $\theta_{|V}=\rho_V-\rho_{V}^\prime$. We thus get $\kappa^\prime=\kappa-d\theta$ and hence $$\hat \partial^\prime \hat x=f^*\hat y +a(\kappa^\prime)=f^*\hat y +a(\kappa)-a(d\theta)=f^*\hat y +a(\kappa)=\hat \partial \hat x\ .$$ \item Let us choose another form $\rho^\prime$ such that $\int l^*\hat y+a(\rho^\prime)=\hat x$. Then $\rho^\prime=\rho+\theta$ with $d\theta=0$ and
${\tt Rham}(\theta)={\mathbf{ch}}(u)$ for some $u\in E^{k-1}(A)$. As for $\rho$ we choose a decomposition $$\theta_U\in \Omega^{k-1}(U,{\tt V})\ ,\quad \theta_V\in \Omega^{k-1}(V,{\tt V})\ ,\quad \theta_{V|A}-\theta_{U|A}=\theta\ .$$ Furthermore, we set $\rho_U^\prime:=\rho_U+\theta_U$ and $\rho_V^\prime:=\rho_V+\theta_V$. Then $\kappa^\prime=\kappa+\lambda$, where $\lambda\in \Omega^k(M,{\tt V})$ is the closed form determined by $\lambda_{|U}=d\theta_U$ and
$\lambda_{|V}=d\theta_V$. Its cohomology class is given by $${\tt Rham}(\lambda)=\partial {\tt Rham}(\theta)=\partial {\mathbf{ch}}(u)={\mathbf{ch}}(\partial u)\ .$$ Therefore we have $a(\kappa)=a(\kappa^\prime)$. This implies $\hat \partial^\prime \hat x=\hat \partial \hat x$. \item If we choose another lift $\hat y^\prime$, then $\hat y^\prime=\hat y+a(\theta)$ for $\theta\in \Omega^{k}(N,{\tt V})$. We get $$\int_{I\times U/U} f_U^*R(\hat y^\prime)=\int_{I\times U/U} \left(f_U^*R(\hat y)+ f_U^*d\theta\right)=\int_{I\times U/U} f_U^*R(\hat y)
-d\textcolor{black}{(}\int_{I\times U/U}f_U^*\theta\textcolor{black}{)} -f_{|U}^*\theta\ .$$ A similar formula holds true over $V$. Note that $\int l^*\hat y^\prime-\int l^*\hat y=a(\int l^*\theta)$ and therefore $\rho^\prime-\rho= \int l^*\theta$. We can take $\rho_{U}^\prime:=\rho_U+\int_{I\times U/U} f_U^*\theta$ and a similar formula over $V$. We see that $\kappa^\prime=\kappa- f^*\theta$ and hence $$\hat \partial^\prime\hat x=f^*\hat y^\prime +a(\kappa^\prime)=f^*\hat y+a(f^*\theta) +a(\kappa-f^*\theta)=f^*\hat y+a(\kappa)=\hat \partial \hat x\ .$$ \item\label{ttwer2} Let $N^\prime$ be a smooth manifold with class $y^\prime\in E^{k+1}(N)$ and smooth map $u\colon N\to N^\prime$ such that $u^* y^\prime= y$. If $f^\prime=u\circ f$ and $f_U^\prime=u\circ f_U$ and $f_V^\prime=u\circ f_V$, and we choose a lift $\hat y^\prime\in \hat E^{k+1}(N^\prime)$ and take $\hat y:=u^*\hat y^\prime$, then we get $ \hat \partial^\prime\hat x=\hat \partial \hat x$. \item\label{ttwer1} We now assume that we have a fixed manifold $N$ with class $y\in E^{k+1}(N)$ and homotopic choices $f\sim f^\prime$ and compatible homotopic choices of homotopies $f_U\sim f_U^\prime$ and $f_V\sim f_V^\prime$.
We then consider the decomposition $I\times M=I\times U\cup I\times V$. The homotopies give maps $F\colon I\times M\to N$ and zero homotopies $F_U\colon I\times I\times U\to N$ and $F_V\colon I\times I\times V\to N$.
Applying the construction of the boundary operator in this case gives a class $\hat \partial ({\tt pr}_A^*\hat x)\in \hat E^{k+1}_{flat}(I\times M)$ which restricts to the classes $\hat \partial\hat x$ and $\hat \partial^\prime\hat x$ at the two boundary components. By the homotopy invariance of the functor $E^{k+1}_{flat}$ we conclude that
these two classes coincide. \item Finally assume that we have two choices $(N_j,y_j,f_j,f_{j,U},f_{j,V})$, $j=0,1$. Then by a similar argument as in the proof of Lemma \ref{dwqdqwdw} there exists a third choice $(N,y,f,f_{U},f_{V})$ together with maps $u_j\colon N_j\to N$ and homotopies $u_j^*f\sim f_j$ and compatible homotopic choices of homotopies $u^*_jf_{U}\sim f_{j,U}$ and $u^*_jf_{V}\sim f_{j,V}$. By a combination of \ref{ttwer1} and \ref{ttwer2} we conclude that the result for $\hat \partial \hat x$ constructed with $j=0$ coincides with the result for $j=1$. \end{enumerate}
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\begin{lem} The boundary operator $\hat \partial\colon E^k_{flat}(A)\to E^{k+1}_{flat}(M)$ is additive. \end{lem} {\it Proof.$\:\:\:\:$} Let $\hat x_i\in E^k_{flat}(A)$, $i=0,1$ be given. We do the construction of $\hat \partial \hat x_i$ based on the choice $f_i\colon M\to N_i$, $\hat y_i\in \hat E^{k+1}(N_i)$. Then we can use the choice $N:=N_0\times N_1$ with the class $\hat y:={\tt pr}_0^*\hat y_0+{\tt pr}_1^*\hat y_1$, the map $f:=(f_0,f_1)$ and the zero homotopies $f_{U}=(f_{0,U},f_{1,U})$ and $f_{V}=(f_{0,V},f_{1,V})$ for the construction of $\hat \partial (\hat x_1+\hat x_1)$. If we have choosen $\rho_{i,U},\rho_{i,V}$, then $\rho_U:=\rho_{0,U}+\rho_{1,U}$ and $\rho_V:=\rho_{0,V}+\rho_{1,V}$ is an appropriate choice for the sum. We get $\kappa=\kappa_0+\kappa_1$ and finally $\hat \partial \hat x=\hat \partial \hat x_0+\hat \partial \hat x_1$.
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\begin{lem}\label{zqidqwdwqdqw} The boundary operator is natural. \end{lem} {\it Proof.$\:\:\:\:$} Let $M^\sharp =U^\sharp\cup V^\sharp$ be a decomposition of another manifold with a smooth map $\phi\colon M^\sharp\to M$ such that $\phi(U^\sharp)\subseteq U$ and $\phi(V^\sharp)\subseteq V$. Let $A^\sharp:=U^\sharp\cap V^\sharp$. If $\hat x\in \hat E_{flat}^k(A)$, then we must show that
$$\hat \partial^\sharp \phi_{|A^\sharp}^*\hat x=\phi^*\hat \partial \hat x\ .$$ We use the notation introduced in the construction of $\hat \partial$.
We choose $f^\sharp:=f\circ \phi\colon M^\sharp\to N$. Then $$f^\sharp y=\phi^*f^*y =\phi^*\partial x=\partial \phi^*_{|A^\sharp} x\ .$$
For the zero homotopies we choose $f^\sharp_U:=f_U\circ \phi_{|U^\sharp}$ and
$f^\sharp_V:=f_V\circ \phi_{|V^\sharp}$. Then for the loop we get
$l^\sharp=l\circ ({\tt id}_{S^1}\times \phi_{|A^\sharp})\colon S^1\times A\to N$. It follows that
$$\int l^{\sharp,*} y=\int ({\tt id}_{S^1}\times \phi_{|A^\sharp})^* l^*y=
\phi_{|A^\sharp}^* \int l^*y=\phi_{|A^\sharp}^* (x)\ .$$ Hence we construct $\hat \partial^\sharp$ using $f^\sharp\colon M^\sharp\to N$, the homotopies $f^\sharp_U,f^\sharp_V$ and the class $\hat y$.
Indeed, we can choose $\rho^\sharp:=\phi_{|A^\sharp}^*\rho$ since
$$\int l^{\sharp,*} \hat y+a(\phi_{|A^\sharp}^*\rho)=
\phi_{|A^\sharp}^*\int l^* \hat y +\phi_{|A^\sharp}^* a(\rho)=\phi_{|A^\sharp}^*\hat x\ .$$ Furthermore we can take
$\rho^\sharp_U:=\phi_{|U^\sharp}^*\rho_U$ and $\rho^\sharp_V:=\phi_{|V^\sharp}^*\rho_V$ which leads to $\kappa^\sharp=\phi^*\kappa$. After all this we see that
$$\hat \partial^\sharp \phi_{|A^\sharp}^*\hat x=\phi^*\hat \partial \hat x\ .$$ \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\begin{prop} We have a long exact Mayer-Vietoris sequence $$\dots \hat E_{flat}^k(M)\stackrel{c}{\to} \hat E_{flat}^k(U)\oplus \hat E_{flat}^k(V)\stackrel{b}{\to} \hat E_{flat}^k(A)\stackrel{\hat \partial}{\to} \hat E_{flat}^{k+1}(M)\to \dots\ .$$ \end{prop} {\it Proof.$\:\:\:\:$} The map $c$ is given by
$$c(\hat x):=\hat x_{|U}\oplus \hat x_{|V}\ .$$ The map $b$ is defined by
$$b(\hat x\oplus\hat y):=\hat y_{|A}-\hat x_{|A} \ .$$ We first show that this is a complex. \begin{enumerate} \item It is clear that $b\circ c=0$. \item \label{uiwdqwdqw} Next we show that $c\circ \hat \partial=0$. We have, using the homotopy formula Lemma \ref{udqwdqwdqw} and the vanishing of $\hat y$ on the base point,
$$\hat \partial \hat x_{|U}=(f^*\hat y)_{|U} +a(\kappa)_{|U}=-a(\int_{I\times U/U} f_U^*R(\hat y))+a(\kappa_{|U})=a(d\rho_U)=0\ .$$ \item Next we verify that $\hat \partial \circ b=0$. Let for example $\hat x\in \hat E^{k}_{flat}(U)$ be given. Then we set $U^\sharp:=U$ and $V^\sharp=M$ and consider the second decomposition $M=U^\sharp\cup V^\sharp$ with $A^\sharp=U$. The map $\phi={\tt id}\colon M\to M$ respects these decompositions. By \ref{uiwdqwdqw}. we have \textcolor{black}{on the one hand}
$\hat \partial^\sharp \hat x=(\hat \partial^\sharp \hat x)_{|V^\sharp}=0$. On the other hand, by Lemma \ref{zqidqwdwqdqw} $$\hat \partial\hat x=\hat \partial\phi^*\hat x=\phi^*\hat \partial^\sharp \hat x=0\ .$$
\end{enumerate}
We now verify exactness. \begin{enumerate} \item Let $\hat x\in \hat E^{k}_{flat}(A)$ be such that $\hat \partial \hat x=0$. In this case $f^*\hat y+a(\kappa)=0$. By Lemma \ref{dwqdqwdw} we can in addition assume that there is a zero homotopy $f_M$. We get by the homotopy formula Lemma \ref{udqwdqwdqw} and the vanishing of $\hat y$ on the base point $$0=-a(\int_{I\times M/M} f_M^* R(\hat y))+a(\kappa)\ .$$ Hence there is a class $\hat m\in \hat E^{k-1}(M)$ such that \begin{equation}\label{uoiqdqwdqwwqd} \kappa-\int_{I\times M/M} f_M^* R(\hat y)=R(\hat m)\ . \end{equation} First we will make a modification which allows us to assume that $R(\hat m)=0$. To this end by Lemma \ref{duiqdwqdqwd} we choose a manifold $N_M$, a class $y_M\in E^{k-1}(N_M)$, and a smooth map $l_M\colon \Sigma^u M\to N_M$ such that $l_M^*y_M=-\sigma(I(\hat m))$. We can assume that $l_M$ is constant near the singular points and smooth elsewhere. We choose a smooth lift $\hat y_M\in \hat E^{k-1}(N_M)$. Then we set $N^\prime:=N\times N_M$ and $\hat y^\prime:={\tt pr}_N^*\hat y+{\tt pr}_{N_M}^*\hat y_M$. We further define $f^\prime:=f\times *$, $f_U^\prime:=f_U\times*$, $f_V^\prime:=f_V\times *$, and we let $f^\prime_M$ be the concatenation of the homotopy $f_M\times *$ with the loop $*\times l_M\circ p$, where $p\colon I\times M\to \Sigma^u M$ is the obvious projection. Then we have $$\int_{I\times M/M} f^{\prime,*}_M R(\hat y^\prime)=\int_{I\times M/M} f_M^* R(\hat y)+ \int_{I\times M/M} l_M^* R(\hat y_M)\ .$$ The closed form
$\int l_M^* R(\hat y_M)$ represents the class $-{\mathbf{ch}}(I(\hat m))$. Therefore, if we replace $f_M$ by $f_M^\prime$, then we can improve (\ref{uoiqdqwdqwwqd}) to $$ \kappa-\int_{I\times M/M} f_M^* R(\hat y)=d\sigma $$ for some form $\sigma\in \Omega^{k-2}(M,{\tt V})$. If we change the forms $\rho_U$ and $\rho_V$ in the construction of $\kappa$ to
$ \rho_U-\sigma_{|U}$ and $ \rho_V-\sigma_{|V}$, then with the resulting new choice for $\kappa$ we get $$ \kappa-\int_{I\times M/M} f_M^* R(\hat y)=0\ . $$ We form the loops
$l_V:= f_{M|V}^{op}\sharp f_V \colon S^1\times V\to N$ and $l_U:=f_{M|U}^{op}\sharp f_U\colon S^1\times U\to N$. We define $$\hat u:=\int l_U^*\hat y +a(\rho_U)\ ,\quad \hat v:=\int l_V^* \hat y+a(\rho_V)\ .$$ Then we get \begin{eqnarray*} R(\hat u)&=&\int l_U^* R(\hat y)+ d\rho_U\\&=&
\int_{I\times U/U} f_U^* R(\hat y)-\int_{I\times U/U} f_{M|U}^* R(\hat y)+d\rho_U\\&=&\kappa_{|U}-\int_{I\times U/U} f_{M|U}^* R(\hat y)\\&=&0\\ R(\hat v)&=&0\ . \end{eqnarray*} Hence $$\hat u\in \hat E^{k-1}_{flat}(U)\ ,\quad \hat v\in \hat E^{k-1}_{flat}(V)\ .$$ We now verify that
$$\hat v_{|A}-\hat u_{|A}=\hat x\ .$$ We have
$$\hat v_{|A}-\hat u_{|A}=\left(\int l_V^*\hat y\right)_{|A}- \left(\int l_U^* \hat y\right)_{|A}+a(\rho)=\left(\int l_V^*\hat y\right)_{|A}- \left(\int l_U^* \hat y\right)_{|A}+ \hat x-\int l^*\hat y\ .$$ It therefore suffices to show the following Lemma. \begin{lem} \begin{equation}\label{udiwqdqwdqwd}
\left(\int l_V^*\hat y\right)_{|A}- \left(\int l_U^* \hat y\right)_{|A} =\int l^*\hat y\ . \end{equation} \end{lem} {\it Proof.$\:\:\:\:$}
We choose an embedding of $S^1\tilde \vee S^1:=S^1\cup_{\{0\}=*}[0,1]\cup_{\{1\}=*}S^1$ into $\mathbb{R}^2$ which is smooth on the two copies of $S^1$ and the interval and such that the interval intersects the circles \textcolor{black}{transversally}. A smooth function on $S^1\tilde \vee S^1$ is one which extends to a smooth function on $\mathbb{R}^2$. We thus have the notion of a smooth map from $S^1\tilde \vee S^1$ to a manifold. Moreover, a map $W\to S^1\tilde \vee S^1$ from a manifold is smooth if the composition $W\to S^1\tilde \vee S^1\to \mathbb{R}^2$ is smooth.
We furthermore choose an open neighbourhood $W\subset \mathbb{R}^2$ of $S^1\tilde \vee S^1$ which admits a smooth projection $\pi\colon W\to S^1\tilde \vee S^1$ which is a homotopy equivalence.
We define $S^1\times A\tilde \vee_A S^1\times A:=(S^1\tilde \vee S^1\times A)$ and set $$\widetilde{l_U^{op}\vee l_V}\colon W\times A\stackrel{\pi\times {\tt id}}{\to} S^1\times A\tilde \vee_A S^1\times A\stackrel{l_U^{op}\tilde \vee l_V}{\to} N\ ,$$ where the smooth map $l_U^{op}\tilde \vee l_V$ maps the part $ [0,1]\times A$ to the base point and is given by $l_U^{op}$ and $l_V$ on the left and right copies of $S^1\times A$, respectively. We have a diagram $$\xymatrix{&S^1\times A\sqcup S^1\times A\ar[d]^{j\sqcup k}&\\S^1\times A\ar@/^4cm/[rr]^e\ar[r]^s\ar@/^0.2cm/[rd]^{\tilde a}\ar@/_0.2cm/[dr]_{\tilde b}&W\times A\ar[r]^{\widetilde{l_U^{op}\vee l_V}}\ar@/^0.5cm/[d]^a\ar@/_0.5cm/[d]^b&N\\&S^1\times A\ar@/^0.2cm/[ur]^{l_U^{op}}\ar@/_0.2cm/[ru]_{l_V}&} $$ where $$a,b\colon W\times A\stackrel{\pi}{\to}S^1\times A\tilde \vee_A S^1\times A\to S^1\times A$$ are the projections which contract the left or right summand, respectively, and $$j,k\colon S^1\times A\to S^1\times A\tilde \vee_A S^1\times A\to W\times A$$ are the embeddings of the left and right summand. The usual coproduct map $S^1\to S^1\vee S^1$ gives rise to a smooth map $$s\colon S^1\times A\to S^1\times A\tilde \vee_A S^1\times A\to W\times A\ .$$
We first observe that $j^*\oplus k^*\colon E^*(W\times A)\to E^*(S^1\times A)\oplus E^*(S^1\times A)$ and $j^*\oplus k^*\colon {\tt im}(\pi^*)\to \Omega^*(S^1\times A,{\tt V})\oplus \Omega^*(S^1\times A,{\tt V})$ are injective, where \begin{equation}\label{udqiwdqwdwqdqwdqwdwqd} {\tt im}(\pi^*):=a^*\Omega^*(S^1\times A,{\tt V})+b^*\Omega^*(S^1\times A,{\tt V})\ . \end{equation} Note that \textcolor{black}{the} definition of ${\tt im}(\pi^*)$ is a slight abuse of notation. Next we show that $j^*\oplus k^*\colon \hat E^*(\textcolor{black}{W}
\times A)\to \hat E^*(S^1\times A)\oplus \hat E^*(S^1\times A)$ possesses a certain injectivity, too. Let $\hat r\in \hat E^*(W)$ be such that $R(\hat r)\in {\tt im}(\pi^*)$. If $(j^*\oplus k^*)(\hat r)=0$, then $R(\hat r)=0$ and $I(\hat r)=0$. Therefore we can assume that $\hat r=a(\rho)$ for some $\rho\in \Omega^{*-1}_{cl}(W)$. Since $j^*a(\rho)=0$ and $k^*a(\rho)=0$ there exist classes $s,t\in E^{*-1}(S^1\times A)$ such that ${\mathbf{ch}}(s)={\tt Rham}(j^*\rho)$ and ${\mathbf{ch}}(t)={\tt Rham}(k^*\rho)$.
Let $i\colon A\to S^1\times A$ be induced by the base point of $S^1$. Since $j\circ i$ is homotopic to $k\circ i$ we have $i^*j^*{\tt Rham}(\rho)=i^*k^*{\tt Rham}(\rho)$. We therefore can in addition assume after modifying e.g. $t$ by a torsion class coming from $A$ that $i^*s=i^*t$. But then there exists a class $w\in E^{*-1}(W\times A)$ such that $j^*w=s$ and $k^*w=t$. It follows that $j^*{\mathbf{ch}}(w)=j^*{\tt Rham}(\rho)$ and $k^*{\mathbf{ch}}(w)=k^*{\tt Rham}(\rho)$. This implies that ${\mathbf{ch}}(w)-{\tt Rham}(\rho)=0$ and hence $a(\rho)=0$.
The composition $e$ is homotopic to the loop $l$ by a homotopy $H$. Indeed, the loop $e$ is the concatenation
$$f_{U|A}^{op}\sharp f_{M|A}^{op}\sharp f_{M|A}\sharp f_{|V|A}\ ,$$
where $F^{op}$ is the homotopy $F$ run in the opposite direction. The homotopy $H$ to $f_{U|A}^{op}\sharp f_{|V|A}$ can thus be arranged symmetrically so that $\int_{I\times S^1\times A/S^1\times A}H^*\omega=0$ for every $\omega\in \Omega(N)$.
Furthermore, the compositions $l_U^{op}\circ \tilde a,l_V\circ \tilde b$ are homotopic to $l_U^{op}$ and $l_V$ by homotopies of the form $G_U=g_U\times {\tt id}_A$ and $G_V=g_V\times {\tt id}_A$, where $g_U,g_V\colon I\times S^1\to S^1$. In the following we use the symbol $w$ in order to denote various constant maps to the base point of $N$. We have \begin{equation}\label{uidqwdwqdqwdwqdfffef} (\widetilde{l_U^{op}\vee l_V})^*\hat y-(l_U^{op}\circ a)^*\hat y-(l_V\circ b)^*\hat y=-w^*\hat y\ . \end{equation} Indeed, if we apply $j^*$ to the left-hand side we get $$j^*(\widetilde{l_U^{op}\vee l_V})^*\hat y-j^*(l_U^{op}\circ a)^*\hat y-j^*(l_V\circ b)^*\hat y=- w^*\hat y\ .$$ Here we use $$l_U^{op}\circ a=j^* (\widetilde{l_U^{op}\vee l_V}) \ ,\quad w=l_V\circ b\circ j\ .$$ Similarly we get $$k^*(\widetilde{l_U^{op}\vee l_V})^*\hat y-k^*(l_U^{op}\circ a)^*\hat y-k^*(l_V\circ b)^*\hat y=-w^*\hat y$$ We now use the injectivity of $j^*\oplus k^*$. Note that the curvature of both sides of (\ref{uidqwdwqdqwdwqdfffef}) are in ${\tt im}(\pi^*)$ as defined in (\ref{udqiwdqwdwqdqwdqwdwqd}).
Since the constant map $w\colon S^1\times A\to N$ factors over the projection $S^1\times A\to A$ we have $\int w^* \hat y=0$. We calculate $$e^*\hat y=s^*(\widetilde{l_U^{op}\vee l_V} )^*\hat y=l^*\hat y + a(\int_{I\times S^1\times A/S^1\times A} H^*R(\hat y))=l^*\hat y \ .$$ Furthermore, $$e^*\hat y=s^*(l_U^{op}\circ a)^*\hat y+s^*(l_V\circ b)^*\hat y-w^*\hat y=\tilde b^*l_V^*\hat y -\tilde a^*l_U^*\hat y-w^*\hat y\ ,$$ hence $$e^*\hat y=l_V^*\hat y+l_U^{op,*}\hat y+a(\int_{I\times S^1\times A/S^1\times A} G_V^*R(\hat y)) -a(\int_{I\times S^1\times A/S^1\times A} G_U^*R(\hat y))-w^*\hat y\ .$$ We now apply $\int$ and observe using $\int l_U^{op,*}\hat y =-\int l_U^*\hat y$ and $\int w^*\hat y=0$ that it suffices to show that $$\int \int_{I\times S^1\times A/S^1\times A} G_V^*R(\hat y)=0\ ,\quad \int\int_{I\times S^1\times A/S^1\times A} G_U^*R(\hat y) =0\ .$$ Because of the special form of the homotopies $G_U$ and $G_V$ these integrals indeed vanish. This finishes the verification of (\ref{udiwqdqwdqwd}). \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\item
Let $\hat x\in \hat E^k_{flat}(M)$ be such that $c(\hat x)=0$. This means that $\hat x_{|U}=0$ and $\hat x_{|V}=0$. Let $x:=I(\hat x)$. We choose a based manifold $N$, a class $y\in E^k(N)$ with trivial restriction to the base point, and a smooth map $f\colon M\to N$ such that $f^*y=x$ and there are zero homotopies $f_U\colon I\times U\to N$, $f_V\colon I\times V\to N$. We further choose a smooth lift $\hat y$ with trivial restriction to the base point and a form $\lambda\in \Omega^{k-1}(M,{\tt V})$ such that $f^*\hat y=\hat x+a(\lambda)$. >From the homotopy formula Lemma \ref{udqwdqwdqw} we get \begin{eqnarray*}
0&=&\hat x_{|U}=-a(\int_{ I\times U/U} f_U^*R(\hat y)) +a(\lambda_{|U})\\0&=&\hat x_{|V}=-a(\int_{I\times V/V} f_V^*R(\hat y)) +a(\lambda_{|V})\ . \end{eqnarray*} Hence there exists classes $\hat u\in \hat E^{k-1}(U)$ and $\hat v\in\hat E^{k-1}(V)$ such that \begin{equation}\label{zdqiwdqwd}
\int_{I\times V/V} f_V^*R(\hat y)+\lambda_{|V}=R(\hat v)\ ,\quad \int_{I\times U/U} f_U^*R(\hat y)+\lambda_{|U}=R(\hat u)\ . \end{equation} We now show that by modifying the choices of $f\colon M\to N$, $y$ and the homotopies we can assume that $R(\hat u)$ and $R(\hat v)$ are exact. By Lemma \ref{duiqdwqdqwd} we choose a based manifold $N_U$, a class $y_U$ which vanishes on the base point, and a map $l_U\colon \Sigma^u U\to N_U$ such that $\sigma(I(\hat u))=-l_U^*y_U$. We can assume that $l_U$ is constant near the singular points of the unreduced suspension. We adopt a similar choice for $V$. We further choose smooth lifts $\hat y_U$ and $\hat y_V$ again vanishing on the base points. We consider $N^\prime:=N\times N_U\times N_V$ with the class ${\tt pr}_N^*\hat y+{\tt pr}_{N_U}^*\hat y_U+{\tt pr}_{N_V}^*\hat y_V$ and the map $f^\prime:=f\times *\times *$. We further concatenate the homotopy
$(f_U\times *\times *)$ with the loop $*\times l_U\times *$ in order to get a new zero homotopy $f^\prime_U$ of $f^\prime_{|U}$. We define $f^\prime_V$ in a similar manner. We now observe that $$\int_{I\times V/V} f^{\prime,*}_V R(\hat y^\prime)=\int_{I\times V/V} f_V^*R(\hat y)+\int_{I\times V/V} l_V^*R(\hat y_V)$$ and the closed form $\int_{I\times V/V} l_V^*R(\hat y_V)$ is in the cohomology class of $-R(\hat v)$. A similar calculation holds for $U$.
If we replace the old choices by the new choices we can now improve (\ref{zdqiwdqwd}) to \begin{equation}\label{zdqiwdqwd1}
\int_{I\times V/V} f_V^*R(\hat y)+\lambda_{|V}= -d\rho_V\ ,\quad \int_{I\times U/U} f_U^*R(\hat y)+\lambda_{|U}=-d\rho_U\ . \end{equation}
We define $\rho:=\rho_{V|A}-\rho_{U|A}$ and set $$\hat z:=\int l^*\hat y +a(\rho)\ .$$ We calculate \begin{eqnarray*} R(\hat z)&=&\int l^* R(\hat y)+d\rho\\
&=&\left(\int_{I\times V/V} f_V^*R(\hat y)\right)_{|A}+d\rho_{V|A}-\left(\int_{I\times U/U} f_V^*R(\hat y)\right)_{|A}-d\rho_{U|A}\\&=&
-\lambda_{|V|A}+\lambda_{|U|A}\\&=&0 \ . \end{eqnarray*} Hence $\hat z\in \hat E^{k-1}_{flat}(A)$. Furthermore, if we construct $\hat \partial \hat z$ using the choices fixed above we get
$$\kappa_{|U}=\int_{I\times U/U} f_U^*R(\hat y)+d\rho_U=-\lambda_{|U}\ ,\quad \kappa_{|V}=-\lambda_V\ .$$ This gives $$\hat \partial \hat z=f^*\hat y-a(\lambda)=\hat x\ .$$ \item Finally we show exactness at $\hat E^k_{flat}(U)\oplus \hat E^k_{flat}(V)$. Let $$\hat u\in \hat E^k_{flat}(U)\ ,\quad \hat v\in \hat E^k_{flat}(V)$$
be such that $\hat u_{|A}=\hat v_{|A}$. Let $u:=I(\hat u)$ and $v:=I(\hat v)$. By Lemma \ref{7eifdwqedqfqd} we choose a manifold $N$ with a class $y\in E^k(N)$, smooth maps
$f\colon U\to N$, $g\colon V\to N$ such that $f^*y=u$ and $g^*y=v$, and there is a homotopy
$f_{|A}\sim g_{|A}$ which we denote by $h$.
In a first step we show that we can choose a map $e\colon M\to N$ such that \begin{equation}\label{uasidasd}
e^*_{|U}y=u\ ,\quad e^*_{|V}y=v\ . \end{equation} In fact we can define \begin{equation}\label{uasidasd1}e(m):=\left(\begin{array}{cc}f(m)&m\in M\setminus V\\ h(\frac{\chi(m)+1}{2},m)&m\in A\\ g(m)&m\in M\setminus U \end{array}\right)\ .\end{equation}
The relations (\ref{uasidasd}) hold true since there are homotopies $e_{|U}\sim f$ and $e_{|V}\sim g$.
We choose $\alpha_U\in \Omega^{k-1}(U,{\tt V})$ and $\alpha_V\in \Omega^{k-1}(V,{\tt V})$ such that
$$e_{|U}^*\hat y+a(\alpha_U)=\hat u\ ,\quad e_{|V}^*\hat y+a(\alpha_V)=\hat v\ .$$ We then have
$a(\alpha_{V|A}-\alpha_{U|A})=0$ so that
$\alpha_{V|A}-\alpha_{U|A}=R(\hat w)$ for some $\hat w\in \hat E^{k-1}(A)$. In the following we show that by modifying the homotopy $h$ we can assume that $R(\hat w)$ is exact.
Using Lemma \ref{duiqdwqdqwd} we choose a based manifold $N^\prime$, a class $y^\prime\in E^{k-1}(N^\prime)$, and a map $l\colon \Sigma^u A\to N^\prime$ such that $l^*y^\prime=-\sigma (I(\hat w))$. We can assume that $l$ maps $(t,a)\in \Sigma^u A$ to the base point if $t\in [0,1/4]$ or $t\in [3/4,1]$.
Without loss of generality we can assume that the homotopy $h\colon I\times A\to N$ is constant on the part $[1/4,3/4]\times A$. We now replace $N$ by $\tilde N:=N\times N^\prime$, $y$ by $\tilde y={\tt pr}_N^*y+{\tt pr}_{N^\prime}^*y^\prime$, $f$ by $\tilde f:=f\times *$, $g$ by $\tilde g:=g\times *$ and $h$ by $\tilde h\colon I\times A\to \tilde N$ given by $$\tilde h(t,a):=\left(\begin{array}{cc} (h(t,a),*)&t\in [0,1/4]\\ (h(1/2,a),l(t,a))&t\in [1/4,3/4]\\ (h(t,a),*)&t\in [3/4,1] \end{array}\right) \ .$$ Let $\tilde e\colon M\to \tilde N$ be the resulting map (\ref{uasidasd1}). Note that there is a homotopy $d_U$ from
$\tilde e_{|U}$ to $e_{|U}\times *$ and a similar homotopy $d_V$ from $\tilde e_{|V}$ to $e_{|V}\times *$.
We furthermore set $\hat {\tilde y}:={\tt pr}_N^*\hat y+{\tt pr}_{N^\prime}^*\hat y^\prime$
for some smooth lift $\hat y^\prime\in \hat E^{k-1}(N^\prime)$ which we arrange such that $\hat y^\prime_{|*}=0$. Since $(e_{|U}\times *)^*\hat{\tilde y}=e_{|U}^*\hat y$ we can choose $$\tilde \alpha_U:=\alpha_U+\int_{I\times U/U} d_U^*R(\hat {\tilde y})\ , \quad \tilde \alpha_V:=\alpha_V+\int_{I\times V/V} d_V^*R(\hat {\tilde y}) .$$ We get
$$\tilde \alpha_{V|A}-\tilde \alpha_{U|A}=R(\hat w)+\int L^*R(\hat {\tilde y})\ ,$$ where $L\colon S^1\times A\to \tilde N$ is the loop $L:=d_U^{op}\sharp d_V$. Note that we can choose $d_U$ and $d_V$ such that ${\tt pr}_N\circ L$ is constant. Therefore $$\int L^*R(\hat {\tilde y})=\int L^*{\tt pr}_{N^\prime}^* R( \hat y^\prime)\ .$$ We get for the cohomology classes $${\tt Rham}(\int L^*{\tt pr}_{N^\prime}^* R(\hat y^\prime))=\int {\mathbf{ch}}(\sigma(I(\hat w)))=-{\mathbf{ch}}(\hat w)\ .$$
If we replace the objects without a tilde by the objects with the tilde decoration, then we can assume that $\alpha_{V|A}-\alpha_{U|A}=d\sigma$. We choose $\sigma_U\in \Omega^{k-2}(U,{\tt V})$ and $\sigma_V\in \Omega^{k-2}(V,{\tt V})$ such that
$\sigma_{V|A}-\sigma_{U|A}=\sigma$. Then we replace $\alpha_U$ by $\alpha_U-d\sigma_U$ and $\alpha_V$ by $\alpha_V-d\sigma_V$. After these changes we can assume that
$\alpha_{V|A}=\alpha_{U|A}$, hence $\alpha_U$ and $\alpha_V$ are restrictions of a global $\alpha\in \Omega^{k-1}(M,{\tt V})$. We define $$\hat x:=e^*\hat y+ a(\alpha)\ .$$
Then $$\hat u=\hat x_{|U}\ ,\quad \hat v=\hat x_{|V}\ .$$ It also follows that $\hat x\in E^k_{flat}(M)$. This provides the required preimage of the sum $\hat u\oplus \hat v$. \end{enumerate} \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
We now have a homotopy invariant functor $\hat E^*_{flat}$ defined on smooth manifolds with a natural Mayer-Vietoris sequence. It gives rise to a similar functor on the category of finite-dimensional countable $CW$-complexes by the following proposition.
\begin{prop}\label{uidwqdqwd} A homotopy invariant functor $$H\colon \{\mbox{\tt smooth manifolds}\}\to \{\mathbb{Z}-\mbox{\tt graded abelian groups}\}$$
with a natural Mayer-Vietoris sequence gives extends uniquely to a homotopy invariant functor $$h\colon \{\mbox{\tt finite-dimensional countable $CW$-complexes}\}\to \{\mathbb{Z}-\mbox{\tt graded abelian groups}\}$$ with a natural Mayer-Vietoris sequence. \end{prop} {\it Proof.$\:\:\:\:$} For a proof we refer to \cite{ks}. It uses the fact that diagrams of maps between countable finite-dimensional $CW$-complexes can be approximated up to homotopy by corresponding diagrams of manifolds. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
Also the following lemma is well-known. \begin{lem}\label{uoqwdqwdwqd} A functor $$h\colon \{\mbox{\tt finite $CW$-complexes}\}\to \{\mathbb{Z}-\mbox{\tt graded abelian groups}\}$$ with a natural Mayer-Vietoris sequence gives rise to a reduced cohomology theory $\tilde h$ on the category of pointed finite $CW$-complexes. \end{lem} {\it Proof.$\:\:\:\:$} For a finite pointed $CW$-complex $X$ we define $$\tilde h^*(X):=\ker(h^*(X)\to h^*(*))\ .$$ To each map $f\colon X\to Y$ of pointed $CW$-complexes we get an induced map $f^*\colon \tilde h^*(Y)\to \tilde h^*(X)$. Let $C(X):=[0,1]\times X/\{1\}\times X$ be the cone over $X$ with its natural $CW$-structure. Then we can write the unreduced suspension as \begin{equation}\label{udioqwdwqd} \Sigma^uX=C(X)\cup_X C(X)\ . \end{equation} The suspension isomorphism $$\sigma\colon \tilde h^*(X)\to \tilde h^*(\Sigma^u(X))$$ is given by the boundary operator in the Mayer-Vietoris sequence associated to the decomposed $CW$-complex (\ref{udioqwdwqd}). It is obviously natural. Finally, for each subcomplex $A\subseteq X$ the mapping cone sequence $$A\to X\to X\cup_AC(A)$$ gives rise to an exact sequence $$\tilde h^*( X\cup_AC(A))\to \tilde h^*(X)\to \tilde h^*(A)\ .$$ Indeed, this is a part of the Mayer-Vietoris sequence for the decomposition $X\cup_AC(A)$ since $\tilde h^*(CA)=0$. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
If $\tilde h$ is a reduced cohomology theory on the category of pointed finite $CW$-complexes, then by \cite[Thm. 9.27]{MR1886843} there exists a spectrum ${\mathbf{h}}$ which represents $\tilde h$. The isomorphism class of this spectrum is well-defined. Furthermore by \cite[Thm. 9.27]{MR1886843}, a natural transformation $\tilde h\to \tilde h^\prime$ of reduced cohomology theories on finite $CW$-complexes can be represented by a map of spectra ${\mathbf{h}}\to {\mathbf{h}}^\prime$, which might be not uniquely determined.
If we apply these topological results Proposition \ref{uidwqdqwd}, Lemma \ref{uoqwdqwdwqd} to the flat theory $\hat E_{flat}^*$, then we get a reduced cohomology theory $\tilde U^{*+1}$ on the category of pointed finite $CW$-complexes which we can represent by a spectrum ${\mathbf{U}}$ whose isomorphism class is well-defined. Since every compact manifold has the structure of a finite $CW$-complex we can restrict the theory $\tilde U^*$ again to compact manifolds. We thus have shown:
\begin{theorem} \label{udidowqdiwqdwqd} If $(\hat E,R,I,a,\int)$ is a smooth extension of $E$ with integration, then $\hat E^*_{flat}$ has a natural long exact Mayer-Vietoris sequence. Its restriction to compact manifolds is equivalent to the restriction to compact manifolds of a generalized cohomology theory represented by a spectrum. \end{theorem}
We now compare $\hat E_{flat}^{*-1}$ with $E\mathbb{R}/\mathbb{Z}^*$. The natural transformation $H^{*-1}(M;{\tt V})\to \hat E^{*}_{flat}(M)$ induced by $a$ gives a natural transformation $\tilde E\mathbb{R}^{*}\to \tilde U^*$ which can be represented by a map of spectra ${\bf E}\mathbb{R}\to {\mathbf{U}}$.
We now consider the diagram of distinguished triangles in the stable homotopy category $$\xymatrix{{\bf F} ibre\ar[r]&{\bf E}\mathbb{R}\ar[r]\ar@{=}[d]& {\mathbf{U}}\ar[r]&\Sigma{\bf F} ibre\\ {\bf E}\ar@{-->}[u]\ar[r]\ar@{.>}[rru]&{\bf E}\mathbb{R}\ar[r]&{\bf E}\mathbb{R}/\mathbb{Z}\ar@{-->}[u]\ar[r]&\Sigma {\bf E}\ar@{-->}[u]}\ .$$ The fact that $$E^*(M)\stackrel{{\mathbf{ch}}}{\to} H^*(M;{\tt V})\stackrel{a}{\to} \hat E(M)$$ vanishes implies that the dotted arrow is trivial. This gives the dashed factorization ${\bf E}\mathbb{R}/\mathbb{Z}\to {\mathbf{U}}$ which we extend to a map of triangles. Note that the dashed maps are not necessarily unique.
\begin{theorem}\label{udqiwdqwdqwddwqdqwd1} Assume that $(\hat E,R,I,a \int)$ is a smooth extension of a generalized cohomology with integration. If $E^*$ is torsion-free, then there exists a natural isomorphism (not necessarily unique) of functors on compact manifolds $E\mathbb{R}/\mathbb{Z}^*(M)\to \hat E^{*-1}_{flat}(M)$ so that $$\xymatrix{H^{*-1}(M;{\tt V})\ar[r]^a\ar[d]^\cong&\hat E^{*}_{flat}(M)\ar[r]&E^*(M)\ar[d]\\ E\mathbb{R}^{*-1}(M)\ar[r]&E\mathbb{R}/\mathbb{Z}^{*-1}(M)\ar[u]^\cong\ar[r]&E^*(M)}$$ commutes \end{theorem} {\it Proof.$\:\:\:\:$} We know by Theorem \ref{udidowqdiwqdwqd} that there is a natural isomorphism $ \hat E^{*}_{flat}(M)\cong U^{*-1}(M)$.
It suffices to check that the transformation ${\bf E}\mathbb{R}/\mathbb{Z}\to {\mathbf{U}}$ is an equivalence by working on the level of homotopy groups. In other words, we must show that it induces an isomorphism on coefficients. We know that $${\tt coker}(a\colon E\mathbb{R}^k(M)\to \hat E^{k+1}_{flat}(M))\cong E^{k+1}_{tors}(M),\quad \textcolor{black}{\ker(a\colon E\mathbb{R}^n(M)\to \hat E^{n+1}_{flat}(M))={\tt im}({\mathbf{ch}})}\ .$$ \textcolor{black}{Therefore} we have a morphism of exact sequences $$\xymatrix{E^k\ar[r]^{\mathbf{ch}}\ar@{=}[d]&E\mathbb{R}^k\ar[r]\ar@{=}[d]& U^k\ar[r]&E^{k+1}_{tors}\ar[r]&0\\ E^k\ar[r]^{\mathbf{ch}}&E\mathbb{R}^k\ar[r]&E\mathbb{R}/\mathbb{Z}^k\ar[r]\ar[u]&E^{k+1}_{tors}\ar@{=}[u]\ar[r]&0}\ .$$ If $E^{k+1}_{tors}=0$, then the morphism $E\mathbb{R}/\mathbb{Z}^k\to U^k$ is an isomorphism \textcolor{black}{by the Five-lemma}.
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\section{Absence of Phantoms}
In the proof of Proposition \ref{udqiwdwqdqwd} we have used the fact that certain generalized cohomology groups are \textcolor{black}{free of
phantoms}. The absence of phantoms might be interesting also in other cases where an approximation of an infinite loop space by manifolds is invoked. Therefore we add this section. The results are probably well known, but we couldn't find appropriate references.
In the following we assume that $E$ is a cohomology theory represented by a commutative ring spectrum ${\bf E}$. Let $Z$ be a $CW$-complex. \begin{ddd}\label{dqwudqwdqw} We define the subspace of \textbf{phantom classes} $E^*_{phantom}(Z)\subseteq E^*(Z)$ to be the subspace of all classes $\phi\in E^*(Z)$ such that $f^*\phi=0$ for all maps $f\colon X\to Z$ and finite complexes $X$. \end{ddd} In the following we discuss various conditions implying the absence of non-trivial phantom classes.
\begin{prop}\label{pr0} If $E_*(Z)$ is a free $E^*$-module, then $E^*_{phantom}(Z)\cong 0$. \end{prop} {\it Proof.$\:\:\:\:$} We equip $E^*(Z)$ with the profinite filtration topology induced by the submodules $F^aE^*(Z):=\ker\left(E^*(Z)\to E^*(Z_a)\right)$, where $(Z_a)$ is the system of all finite subcomplexes of $Z$. On the other hand we equip $DE_*(Z):={\tt Hom}_{E^*}(E_*(Z),E^*)$ with the dual finite topology generated by the submodules $\ker(D E_*(Z)\to DL_b)$, where $(L_b)$ runs over the system of all finitely generated submodules of $E_*(Z)$. With this topology the $E^*$-module $DE_*(Z)$ is complete and Hausdorff. By \cite[Thm. 4.14]{MR1361899} the evaluation $E^*(Z)\otimes E_*(Z)\to E^*$ induces a topological isomorphism $E^*(Z)\to D E_*(Z)$. The fact that the profinite filtration topology on $E^*(Z)$ is Hausdorff is equivalent to the absence of phantom classes. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\begin{lem}\label{zueweq} $E^*_{Phantom}({\bf E}_k)\cong 0$ for ${\bf E}={\mathbf{MU}}$ or even $k$ and ${\bf E}={\bf K}$. \end{lem} {\it Proof.$\:\:\:\:$} The cohomology theories $MU^*$ and $K^*$ are represented by ring spectra. We first consider the case $MU$. In \cite[Sec. 4]{MR0356030} it \textcolor{black}{is} shown that $MU_*({\mathbf{MU}}_k)$ is a free $MU^*$-module (it actually has been calculated completely). We can therefore apply Proposition \ref{pr0}.
For $K$-theory we we first note that ${\bf K}_0\cong \mathbb{Z}\times BU$, and that $K_*(BU,\mathbb{Z})$ is a free $\mathbb{Z}$-module on even generators (\cite[Prop. 4.3.3 (d)]{MR1407034}). This implies that $K_*({\bf K}_0)$ is a free $K^*$-module, and we can again apply Proposition \ref{pr0}. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\begin{prop}\label{pr1} If $E_*(Z)$ is a free $E^*$-module, then $E^*_{Phantom}(Z\times Z)\cong 0$. \end{prop} {\it Proof.$\:\:\:\:$} If $E_*(Z)$ is a free $E^*$-module, then so is $E_*(Z\times Z)$. In fact, for every complex $X$ we have the K\"unneth isomorphism $$E_*(Z)\otimes_{E^*}E_*(X)\stackrel{\sim}{\to} E_*(Z\times X) \ .$$ This follows from the usual observation that $E_*(Z)\otimes_{E^*}E_*(\dots) \to E_*(Z\times \dots)$ is a natural transformation of homology theories which coincide on the point. Finally, we use the fact that the tensor product of two free modules is again free.\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\begin{kor}\label{hdwqdqwdwq} If $k$ is even, then $$E^{*}_{Phantom}({\bf E}_k\times {\bf E}_k)\cong 0$$ holds true for ${\bf E}\in \{{\mathbf{MU}},{\bf K}\}$. \end{kor}
If $X\mapsto E^*(X)$ is a cohomology theory represented by a spectrum ${\bf E}$, then let $X\mapsto E_*(X)$ denote the associated homology theory.
We define $$E_\mathbb{R}^*(X):={\tt Hom}_{{\mathrm{Ab}}}(E_*(X),\mathbb{R})\ ,\quad E_{\mathbb{R}/\mathbb{Z}}^*(X):={\tt Hom}_{{\mathrm{Ab}}}(E_*(X),\mathbb{R}/\mathbb{Z})\ .$$ Since $\mathbb{R}$ and $\mathbb{R}/\mathbb{Z}$ are injective abelian groups these constructions define cohomology theories on the category of all topological
spaces. Since they satisfy in addition the wedge axiom they can be represented by spectra which we denote by ${\bf E}_\mathbb{R}$ and ${\bf E}_{\mathbb{R}/\mathbb{Z}}$.
\begin{lem}\label{dqduqwdwqdwq} For every $CW$-complex $X$ we have $E^*_{\mathbb{R},phantom}(X)\cong 0$ and $E^*_{\mathbb{R}/\mathbb{Z},phantom}(X)\cong 0$. \end{lem} {\it Proof.$\:\:\:\:$}
Let us discuss the case of $E_\mathbb{R}$. The case of $E_{\mathbb{R}/\mathbb{Z}}$ is similar. It suffices to show that $$E^*_{\mathbb{R}}(X)\cong \lim_a E^*_{\mathbb{R}}(X_a)\ ,$$ where $(X_a)$ is the system of finite subcomplexes of $X$. We have $X\cong {\tt colim}_a\: X_a$. Since homology is compatible with colimits and the ${\tt Hom}_{\mathrm{Ab}}(\dots,\mathbb{R})$-functor turns colimits in the first argument into limits we get (compare \cite[A.9]{frlo}) \begin{eqnarray*} E^*_{\mathbb{R}}(X)&=&{\tt Hom}_{\mathrm{Ab}}(E_*(X),\mathbb{R})\\ &\cong&{\tt Hom}_{\mathrm{Ab}}(E_*({\tt colim}_a \: X_a),\mathbb{R})\\ &\cong&{\tt Hom}_{{\mathrm{Ab}}}({\tt colim}_a\: E_*(X_a),\mathbb{R})\\ &\cong&\lim_a\: {\tt Hom}_{{\mathrm{Ab}}}(E_*(X_a),\mathbb{R})\ . \end{eqnarray*}
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
The projection $\mathbb{R}\to \mathbb{R}/\mathbb{Z}$ induces a natural transformation of cohomology theories $E_\mathbb{R}^*(X)\to E^*_{\mathbb{R}/\mathbb{Z}}(X)$. It is given by a morphism of representing spectra ${\bf E}_\mathbb{R}\to {\bf E}_{\mathbb{R}/\mathbb{Z}}$. \begin{ddd} The Andersen dual $D(E)$ of the cohomology theory $E$ is defined as the cohomology theory represented by the spectrum $D({\bf E})$ obtained by the extension of the map ${\bf E}_\mathbb{R}\to {\bf E}_{\mathbb{R}/\mathbb{Z}}$ to a distinguished triangle in the stable homotopy catgeory $$D({\bf E})\to {\bf E}_\mathbb{R}\to {\bf E}_{\mathbb{R}/\mathbb{Z}}\to \Sigma D({\bf E})\ .$$ \end{ddd}
In \cite[p.~244]{MR704613} a morphism of distinguished triangles in the stable homotopy category $$\xymatrix{D({\bf E})\ar[r]\ar@{=}[d]&D({\bf E})\mathbb{R}\ar[r]\ar[d]^\cong& D({\bf E})\mathbb{R}/\mathbb{Z}\ar[d]\ar[d]^\cong\ar[r]&\Sigma D({\bf E})\ar[d]_{-1}^\cong \\D({\bf E})\ar[r]&{\bf E}_\mathbb{R}\ar[r]&{\bf E}_{\mathbb{R}/\mathbb{Z}}\ar[r]&\Sigma D({\bf E})} $$ has been constructed so that the vertical maps are equivalences.
We now assume that $E^k$ is finitely generated for every $k\in \mathbb{Z}$. Since $D$ is a duality on cohomology theories with finitely generated coefficients \cite[Thm.~33]{MR704613} we get by inserting $D({\bf E})$ in place of ${\bf E}$ and using $D(D({\bf E}))\cong {\bf E}$ that
$$\xymatrix{{\bf E}\ar[r]\ar@{=}[d]&{\bf E}\mathbb{R}\ar[r]\ar[d]^\cong&
{\bf E}\mathbb{R}/\mathbb{Z}\ar[d]^\cong\ar[r]&\Sigma {\bf E}\ar[d]^\cong_{-1}\\ {\bf E}\ar[r]&D({\bf E})_\mathbb{R}\ar[r]&D({\bf E})_{\mathbb{R}/\mathbb{Z}}\ar[r]&{\bf E}}\ , $$
i.e.~\textcolor{black}{we get} in particular isomorphisms \begin{equation}\label{zdqwduqwdd} {\bf E}\mathbb{R}\cong D({\bf E})_\mathbb{R} ,\quad {\bf E}\mathbb{R}/\mathbb{Z}\cong D({\bf E})_{\mathbb{R}/\mathbb{Z}}\ . \end{equation}
Combining (\ref{zdqwduqwdd}) with Lemma \ref{dqduqwdwqdwq} (applied to $D(E)$ in the place of $E$) we get \begin{kor} If $E$ is a cohomology theory represented by a spectrum such that $E^k$ is finitely generated for all $k\in \mathbb{Z}$, then for every $CW$-complex $X$ we have $$E\mathbb{R}^*_{Phantom}(X)=0\ ,\quad E\mathbb{R}/\mathbb{Z}^*_{Phantom}(X)=0\ .$$ \end{kor}
\begin{kor}\label{uoewqeqwe} We have $K\mathbb{R}/\mathbb{Z}^0_{Phantom}({\bf K}_1\times {\bf K}_1)=0$. \end{kor}
\end{document} | arXiv |
\begin{document}
\title{Incompleteness and jump hierarchies}
\author{Patrick Lutz} \address{Department of Mathematics, University of California, Berkeley} \email{[email protected]}
\author{James Walsh} \address{Group in Logic and the Methodology of Science, University of California, Berkeley} \email{[email protected]}
\subjclass[2010]{Primary 03F35, 03D55; Secondary 03F40}
\commby{Heike Mildenberger}
\thanks{Thanks to Antonio Montalb\'an, Ted Slaman, and an anonymous referee for their comments and suggestions. Also, thanks to Jun Le Goh for alerting us to errors in the statements of theorem \ref{ms2} and corollary \ref{ms3}.}
\begin{abstract} This paper is an investigation of the relationship between G\"odel's second incompleteness theorem and the well-foundedness of jump hierarchies. It follows from a classic theorem of Spector's that the relation $\{(A,B) \in \mathbb{R}^2 : \mathcal{O}^A \leq_H B\}$ is well-founded. We provide an alternative proof of this fact that uses G\"odel's second incompleteness theorem instead of the theory of admissible ordinals. We then use this result to derive a semantic version of the second incompleteness theorem, originally due to Mummert and Simpson. Finally, we turn to the calculation of the ranks of reals in this well-founded relation. We prove that, for any $A\in\mathbb{R}$, if the rank of $A$ is $\alpha$, then $\omega_1^A$ is the $(1 + \alpha)^{\text{th}}$ admissible ordinal. It follows, assuming suitable large cardinal hypotheses, that, on a cone, the rank of $X$ is $\omega_1^X$. \end{abstract}
\maketitle
\section{Introduction}
In this paper we explore a connection between G\"odel's second incompleteness theorem and recursion-theoretic jump hierarchies. Our primary technical contribution is a method for proving the well-foundedness of jump hierarchies; this method crucially involves the second incompleteness theorem. We use this technique to provide a proof of the following theorem:
\begin{restatable}{theorem}{main} \label{thm-main} There is no sequence $(A_n)_{n<\omega}$ of reals such that, for each $n$, the hyperjump of $A_{n+1}$ is hyperarithmetical in $A_n$. \end{restatable}
This theorem is an immediate consequence of a result of Spector's, namely that if $\mathcal{O}^A \leq_H B$ then $\omega_1^A < \omega_1^B$ (so the existence of such a sequence $(A_n)_{n<\omega}$ would imply the existence of a descending sequence $\omega_1^{A_0} > \omega_1^{A_1} > \ldots$ in the ordinals). We provide an alternative proof that makes no mention of admissible ordinals, and which has the additional benefit of showing the theorem is provable in $\mathsf{ACA}_0$.
Here is a brief sketch of how our alternative proof works: Consider the theory $\mathsf{ACA}_0 + \mathsf{DS}$ where $\mathsf{DS}$ is a sentence asserting the existence of a sequence of reals as described in Theorem \ref{thm-main}. We work \emph{inside} the theory and let $A_0, A_1, \ldots$ be such a sequence. $\mathsf{ACA}_0$ proves that if the hyperjump of a real exists then there is a $\beta$-model (a model that is correct for $\Sigma^1_1$ sentences) containing it. In this case $\mathcal{O}^{A_1}$ exists so there is a $\beta$-model containing $A_1$. Moreover, since all $A_n$'s for $n \geq 1$ are hyperarithmetical in $A_1$, the $\beta$-model will contain all of them. All $\beta$-models are models of $\mathsf{ACA}_0$ (in fact, $\mathsf{ATR}_0$) so it appears this model is a model of the theory $\mathsf{ACA}_0 + \mathsf{DS}$, meaning that the theory proves its own consistency. By G\"odel's second incompleteness theorem, this implies that $\mathsf{ACA}_0$ proves $\neg \mathsf{DS}$.
There is one problem, however. Just because the model contains all the elements of the sequence $(A_n)_{n \geq 1}$ does not mean it contains the sequence itself (here we are thinking of the sequence as a single real whose slices are the $A_n$'s). Indeed, the sequence itself could be much more complicated than any single real in the sequence. In our proof, we overcome this flaw by showing that if there is a descending sequence then there is a descending sequence that is relatively simple---in fact there is one that is hyperarithmetic relative to $A_1$. This means the $\beta$-model above really does contain a descending sequence.
In \cite{friedman1976uniformly}, H. Friedman uses similar ideas to prove the following theorem originally due to Steel:
\begin{restatable}[Steel]{theorem}{Steel} \label{Steel} Let $P\subset \mathbb{R}^2$ be arithmetic. Then there is no sequence $(A_n)_{n<\omega}$ such that for every $n$, \begin{itemize} \item[(i)] $A_n\geq_T A'_{n+1}$ and \item[(ii)] $A_{n+1}$ is the unique $B$ such that $P(A_n,B)$. \end{itemize} \end{restatable}
In these proofs we move from the second incompleteness theorem to the well-foundedness (or near well-foundedness) of recursion-theoretic jump hierarchies. In fact, the implication goes in both directions: the well-foundedness of appropriate jump hierarchies entails semantic versions of the second incompleteness theorem. For example, theorem \ref{thm-main} yields a simple and direct proof of the following semantic version of the second incompleteness theorem originally due to Mummert and Simpson (recall that $\mathcal{L}_2$ is the standard two-sorted language of second order arithmetic):
\begin{restatable}[Mummert--Simpson]{theorem}{ms} \label{ms} Let $T$ be a recursively axiomatized $\mathcal{L}_2$ theory. For each $n \geq 1$, if there is a $\beta_n$-model of $T$ then there is a $\beta_n$-model of $T$ which contains no countable coded $\beta_n$-models of $T$. \end{restatable}
In fact, our proof sharpens the Mummert-Simpson result somewhat by dropping the requirement that $T$ be recursively axiomatized.
A different semantic version of the second incompleteness theorem also follows from theorem \ref{Steel}, as observed by Steel in \cite{steel1975descending}. Namely, the following:
\begin{restatable}[Steel]{theorem}{Steelincompleteness} \label{Steel2} Let $T$ be an arithmetically axiomatized $\mathcal{L}_2$ theory extending $\mathsf{ACA}_0$. If $T$ has an $\omega$-model then $T$ has an $\omega$-model which contains no countable coded $\omega$-models of $T$. \end{restatable}
These results all point to a general connection between incompleteness and well-foundedness. Elucidating this connection is the central goal of this paper. Though many of the theorems we prove could also be proved from the application of known methods, we believe that the new techniques are more conducive to achieving our central goal. Additionally, our techniques are able to prove somewhat sharper results than the original methods.
We also investigate directly the well-founded hierarchy at the center of theorem \ref{thm-main}. It follows from that theorem that the relation $A \prec B$ defined by $\mathcal{O}^A \leq_H B$ is a well founded partial order. We call the $\prec$ rank of a real its \emph{Spector rank}. There is a recursion-theoretically natural characterization of the Spector ranks of reals:
\begin{restatable}{theorem}{ranks} \label{thm-ranks} For any real $A$, the Spector rank of $A$ is $\alpha$ just in case $\omega_1^A$ is the $(1 + \alpha)^{\text{th}}$ admissible ordinal. \end{restatable}
It follows, assuming suitable large cardinal hypotheses, that, on a cone, the Spector rank of $X$ is $\omega_1^X$.
Here is our plan for the rest of the paper. In \textsection{\ref{well_foundedness}} we describe related research. In \textsection{\ref{proof}} we prove the main theorem. In \textsection{\ref{incompleteness}} we provide an alternative proof of the Mummert-Simpson theorem. In \textsection{\ref{ranks}} we turn to the calculation of Spector ranks.
\section{Second Incompleteness \& Well-Foundedness} \label{well_foundedness}
The second incompleteness theorem implies the well-foundedness of various structures (in particular, sequences of models). In turn, the well-foundedness of structures sometimes yields a semantic version of the second incompleteness theorem (in the form of a minimum model theorem). It is worth emphasizing that the former argument does not rely on the theory of transfinite ordinals and the latter argument does not rely on self-reference or fixed point constructions. This point allows us to sharpen certain results. Because we avoid the use of ordinals, we can verify that Theorem \ref{thm-main} is provable in $\mathsf{ACA}_0$; because we avoid self-reference, we can drop the restriction in the statement of Theorem \ref{ms} that $T$ be recursively axiomatized.
We will now describe both types of arguments, describe their historical antecedents, and point to related research.
\subsection{Well-foundedness via incompleteness}
\leavevmode
To derive well-foundedness from incompleteness we work in the theory $T$ + ``there is a descending sequence,'' where $T$ is sound and sufficiently strong. We build a model of $T$ containing a tail of the sequence, yielding a consistency proof of $T$ + ``there is a descending sequence'' \emph{within the theory} $T$ + ``there is a descending sequence.'' By the second incompleteness theorem, this means that $T$ proves that there are no descending sequences.
The main difficulties lie in building a model that is correct enough that if a descending sequence is in the model, the model knows it is descending and in finding a $T$ that is strong enough to prove the model exists but weak enough that the model built satisfies it.
As far as we know, the first arguments of this type are due to H. Friedman. We were inspired, in particular, by H. Friedman's \cite{friedman1976uniformly} proof of a theorem originally due to Steel \cite{steel1975descending}.
\Steel*
Steel's proof is purely recursion-theoretic, whereas Friedman's proof appeals to the second incompleteness theorem. In particular, Friedman supposes that there is an arithmetic counter-example $P$ to Steel's Theorem. He then works in the theory $T:= \mathsf{RCA} +\textrm{``$P$ produces a descending sequence}$'' and uses $P$ to build $\omega$-models of arbitrarily large fragments of $T$. This yields a proof of $\mathsf{Con}(T)$ in $T$, whence $T$ is inconsistent by G\"odel's second incompleteness theorem.
Recently, Pakhomov and the second named author developed proof-theoretic applications of this technique in \cite{pakhomov2018reflection}. They show that there is no sequence $(T_n)_{n<\omega}$ of $\Pi^1_1$ sound extensions of $\mathsf{ACA}_0$ such that, for each $n$, $T_n$ proves the $\Pi^1_1$ soundness of $T_{n+1}$. This result is proved by appeal to the second incompleteness theorem, though it could be proved by showing that a descending sequence $(T_n)_{n<\omega}$ of theories would induce a descending sequence in the ordinals (namely, the associated sequence of proof-theoretic ordinals). They also show that, ``on a cone,'' the rank of a theory in this well-founded ordering coincides with its proof-theoretic ordinal. These results are strikingly similar to the main theorems of this paper.
\subsection{Incompleteness via well-foundedness}
\leavevmode
Here is an informal argument for incompleteness via well-foundedness. Suppose that second incompleteness fails, i.e.\ that a consistent $T$ proves its own consistency. If $T$ also proves the completeness theorem, then every model $\mathfrak{M}$ of $T$ has (what it is by the lights of $\mathfrak{M}$) a model within it. This produces a nested sequence of models. If these models can be indexed by ordinals, then this produces a descending sequence of ordinals. So the well-foundedness of the ordinals produces some form of the second incompleteness theorem. If we know that the models form a well-founded structure, we can argue directly, without the detour through the ordinals.
To sharpen this argument one must know that the objects that are ``models of $T$'' in the sense of $\mathfrak{M}$ are genuinely models of $T$. So one must restrict one's attention to structures that are sufficiently correct. In addition, one must clarify the relation by which the models are being compared and prove that it is well-founded.
An early argument of this sort is attributed to Kuratowski (see \cite{kennedy2015incompleteness, kripke2009collapse}). Set theory cannot prove the following strong form of the consistency of set theory: that there is an $\alpha$ such that $V_\alpha$ is a model of set theory. For if it does then there is $\alpha$ such that $V_\alpha$ is a model of set theory. Since $V_\alpha$ is a model of set theory, there is also a $\beta<\alpha$ such that $V_\beta$ is a model of set theory. Iterating this argument produces an infinite descending sequence of ordinals. Contradiction.
Steel has also developed an argument of this sort. Using his Theorem \ref{Steel}, he demonstrates that if an arithmetically axiomatized theory of second order arithmetic extends $\mathsf{ACA}_0$ and has an $\omega$-model then it has an $\omega$-model which contains no countable coded $\omega$-models of the theory.
\subsection{Kripke structures}
\leavevmode
We conclude this discussion of related work with the following observation. Formalized in the language of modal logic, the statement of G\"odel's second incompleteness theorem characterizes well-founded Kripke frames. Indeed, the formalization corresponds to the least element principle: $$\diamondsuit\varphi \rightarrow \diamondsuit(\varphi\wedge\neg\diamondsuit\varphi).$$ Its contrapositive (writing $\psi$ for $\neg\varphi$) is a modal formalization of L\"ob's theorem which corresponds to induction\footnote{Note that since we replaced $\varphi$ with $\lnot \varphi$ before taking the contrapositive, the two modal statements are equivalent only as \emph{schemas}.}: $$\Box (\Box\psi \rightarrow \psi) \rightarrow \Box \psi$$ Beklemishev has suggested that this observation is connected with ordinal analysis. In \cite{beklemishev2004provability}, he uses a modal logic of provability known as $\mathsf{GLP}$ to develop both an ordinal notation system for $\varepsilon_0$ and a novel consistency proof of $\mathsf{PA}$.
\section{The Main Theorem} \label{proof}
In this section we provide our alternative proof of Theorem \ref{thm-main}.
\subsection{Outline of proof}
\leavevmode
In broad strokes, here is our strategy. We will consider a statement $\mathsf{DS}$ which states that there \emph{is} a descending sequence in the hyperjump hierarchy. We then work in the theory $\mathsf{ACA}_0 + \mathsf{DS}$ and derive the statement $\mathsf{Con}(\mathsf{ACA}_0 + \mathsf{DS})$. By G\"odel's second incompleteness theorem, this implies that there is a proof of $\neg\mathsf{DS}$ in $\mathsf{ACA}_0$.
To derive $\mathsf{Con}(\mathsf{ACA}_0 + \mathsf{DS})$ in $\mathsf{ACA}_0 + \mathsf{DS}$, we use the hyperjump of a real to construct a coded $\beta$-model of $\mathsf{ACA}_0$ containing that real. In particular, if we are given a descending sequence then we can use the existence of the hyperjump of the second real in the sequence to find a $\beta$-model containing all the elements of the tail of the sequence. The point is that the tail of a descending sequence is again a descending sequence and $\beta$-models are correct enough to verify this.
The only problem is that while the $\beta$-model we found contains all the elements of the tail it may not contain the tail itself (i.e.\ it may not contain the recursive join of all the elements of the tail). Our strategy to fix this is to show that there is a family of descending sequences which is arithmetically definable relative to some parameter whose hyperjump exists. A $\beta$-model containing this parameter must contain an element of this family (because $\beta$-models contain witnesses to all $\Sigma^1_1$ statements).
For the parameter, we will use a countable coded $\beta$-model which contains each element of a tail of the original descending sequence. The arithmetic formula will then essentially say that the $\beta$-model believes each step along the sequence is descending. The point is that we have replaced a $\Pi^1_1$ formula saying the sequence is descending by an arithmetic formula talking about the truth predicate of some coded model and $\beta$-models are correct enough that this does not cause any errors.
The $\beta$-model will just come from the existence of the hyperjump of some element of the original sequence, and we can guarantee the hyperjump of the model exists by taking one more step down the original descending sequence.
\subsection{Useful facts}
\leavevmode
In this section, we record the facts about $\beta$-models that we will use in the proof of the main theorem. Unless otherwise noted, proofs of all propositions in this section can be found in \cite{simpson_2009}.
\begin{definition} A $\beta$-model is an $\omega$-model $\mathfrak{M}$ of second order arithmetic such that for any $\Sigma^1_1$ sentence $\varphi$ with parameters in $\mathfrak{M}$, $\mathfrak{M} \vDash \varphi$ if and only if $\varphi$ is true. \end{definition}
\begin{proposition}[\cite{simpson_2009}, Lemma VII.2.4, Theorem VII.2.7] \label{beta} Provably in $\mathsf{ACA}_0$, all countable coded $\beta$-models satisfy $\mathsf{ATR}_0$ (and hence also $\mathsf{ACA}_0$). \end{proposition}
\begin{proposition}[\cite{simpson_2009}, Lemma VII.2.9] \label{hyperjump} Provably in $\mathsf{ACA}_0$, for any $X$, $\mathcal{O}^X$ exists if and only if there is a countable coded $\beta$-model containing $X$. \end{proposition}
\begin{proposition} \label{absoluteness} All of the following can be written as Boolean combinations of $\Sigma^1_1$ formulas and hence are absolute between $\beta$-models \begin{enumerate}
\item $A$ is the hyperjump of $B$.
\item $A \leq_H B$
\item $M$ is a countable coded $\beta$-model. \end{enumerate} \end{proposition}
\subsection{Proof of the main theorem}
\leavevmode
\main*
\begin{proof} It suffices to prove the inconsistency of the theory $\mathsf{ACA}_0+\mathsf{DS}$, where \[ \mathsf{DS} := \exists X \forall n (\mathcal{O}^{X_{n + 1}} \text{ exists and } \mathcal{O}^{X_{n+1}}\leq_H X_n). \] To do this, we reason in $\mathsf{ACA}_0+\mathsf{DS}$ and derive $\mathsf{Con}(\mathsf{ACA}_0+\mathsf{DS})$. The inconsistency of $\mathsf{ACA}_0+\mathsf{DS}$ then follows from G\"odel's second incompleteness theorem.
\noindent\textbf{Reasoning in $\mathsf{ACA}_0+\mathsf{DS}$:}
Let $A$ witness $\mathsf{DS}$. That is, for all $n$, $\mathcal{O}^{A_{n + 1}}$ exists and $\mathcal{O}^{A_{n+1}}\leq_H A_n$. Our goal is now to show there is a model of $\mathsf{ACA}_0 + \mathsf{DS}$.
\begin{claim} There is a countable coded $\beta$-model $\mathfrak{M}$ coded by $M$ such that $\mathcal{O}^{M}$ exists and $\mathfrak{M}$ contains $A_n$ for all sufficiently large $n$. \end{claim}
The proof of Proposition \ref{hyperjump} in \cite{simpson_2009} actually shows that for any $X$, if $\mathcal{O}^X$ exists then $X$ is contained in a countable coded $\beta$-model which is coded by a real that is recursive in $\mathcal{O}^X$. So $A_2$ is contained in some countable coded $\beta$-model $\mathfrak{M}$, coded by $M$, such that $M \leq_T \mathcal{O}^{A_2} \leq_H A_1$. Hence $\mathcal{O}^M \leq_T \mathcal{O}^{A_1}$. Since $\mathcal{O}^{A_1}$ exists, so does $\mathcal{O}^{M}$. And since $\mathfrak{M}$ is closed under hyperarithmetic reducibility, $\mathfrak{M}$ contains $A_n$ for all $n \geq 2$.
\begin{claim} There is an arithmetic formula $\varphi$ such that \begin{enumerate}
\item[(i)] $\exists X\, \varphi(M, X)$
\item[(ii)] For any $X$, if $\varphi(M, X)$ holds then $X$ is a witness of $\mathsf{DS}$ \end{enumerate} where $M$ is as in the previous claim. \end{claim}
Basically $\varphi(M, X)$ says that $X$ is a sequence of reals whose elements are in $\mathfrak{M}$ and for each $n$, $\mathfrak{M}$ believes that $\mathcal{O}^{X_{n + 1}}$ exists and is hyperarithmetical in $X_n$. More precisely $\varphi(M, X)$ is the sentence \[ \forall n\, (X_{n + 1}, X_n \in \mathfrak{M} \land \mathfrak{M} \vDash ``\exists Y\, [Y = \mathcal{O}^{X_{n + 1}} \land Y \leq_H X_n]"). \]
To see why $\varphi(M, X)$ has a solution, recall that $\mathfrak{M}$ contains $A_n$ for all $n$ sufficiently large. Let $X$ be the sequence $A$ but with the first few elements removed so that $\mathfrak{M}$ contains all elements in $X$. For each $n$, the fact that $A$ is a witness of $\mathsf{DS}$ guarantees that there is some $Y$ such that $\mathcal{O}^{X_{n + 1}} = Y$ and $Y \leq_H X_n$. Since $\mathfrak{M}$ contains $X_n$ and since $\beta$-models are closed under hyperarithmetic reducibility, $\mathfrak{M}$ contains $Y$. And by proposition \ref{absoluteness}, $\beta$-models are sufficiently correct that $\mathfrak{M} \vDash ``Y = \mathcal{O}^{X_{n + 1}} \land Y \leq_H X_n."$
Suppose $X$ is a sequence such that $\varphi(M, X)$ holds. Then for each $n$ there is a $Y$ such that $\mathfrak{M} \vDash ``Y = \mathcal{O}^{X_{n + 1}} \land Y \leq_H X_n."$ By proposition $\ref{absoluteness}$, both clauses of the conjunction are absolute between $\beta$-models. Hence $\mathcal{O}^{X_{n + 1}}$ exists and is hyperarithmetical in $X_n$. So $X$ is a witness of $\mathsf{DS}$.
\begin{claim} There is a model of $\mathsf{ACA}_0 + \mathsf{DS}$. \end{claim}
By proposition \ref{hyperjump}, there is a $\beta$-model $\mathfrak{N}$ that contains $M$. Since $\mathfrak{N}$ is a $\beta$-model, by proposition \ref{beta}, it is a model of $\mathsf{ACA}_0$.
Since the $\Sigma^1_1$ formula $\exists X\, \varphi(M, X)$ holds and $\mathfrak{N}$ is correct for $\Sigma^1_1$ formulas with parameters from $\mathfrak{N}$, there is some $X$ in $\mathfrak{N}$ such that $\mathfrak{N} \vDash \varphi(M, X)$. And since $\mathfrak{N}$ is a $\beta$-model, it is correct about this fact---that is, $\varphi(M, X)$ really does hold. Since $\varphi(M, X)$ holds, $X$ is a witness to $\mathsf{DS}$. The point now is just that $\mathfrak{N}$ is correct enough to see that $X$ is a witness to $\mathsf{DS}$. In detail: for each $n$, $\mathcal{O}^{X_{n + 1}}$ exists and is hyperarithmetical in $X_n$. Since $X_n$ is in $\mathfrak{N}$, this means $\mathcal{O}^{X_{n + 1}}$ is in $\mathfrak{N}$. And by proposition \ref{absoluteness}, $\mathfrak{N}$ agrees that it is the hyperjump of $X_{n + 1}$ and that it is hyperarithmetical in $X_n$. Therefore $\mathfrak{N}$ agrees that $X$ is a witness to $\mathsf{DS}$. \end{proof}
\begin{remark} The previous proof actually demonstrates that $\mathsf{ACA}_0$ proves Theorem \ref{thm-main}. The original Spector proof relies on the theory of admissible ordinals, so it is unlikely to be formalizable in systems weaker than $\mathsf{ATR}_0$. \end{remark}
\section{Semantic Incompleteness Theorems} \label{incompleteness}
Steel derives the following theorem as a corollary of his Theorem \ref{Steel}.
\Steelincompleteness*
Because $\omega$-models are correct for arithmetic statements, we can restate this as
\begin{corollary} Let $T$ be an arithmetically axiomatized $\mathcal{L}_2$ theory extending $\mathsf{ACA}_0$. If there is an $\omega$-model of $T$ then there is an $\omega$-model of \[ T + \textrm{``there is no $\omega$-model of T''}. \] \end{corollary}
Similarly, we can use Theorem \ref{thm-main} to prove a stronger version of a theorem originally proved by Mummert and Simpson in \cite{mummert2004}. Note that in our version we do not need to assume that $T$ is recursively axiomatized, only that it has a $\beta$-model.
\begin{theorem}\label{minimalmodel} Let $T$ be an $\mathcal{L}_2$ theory. If there is a $\beta$-model of $T$ then there is a $\beta$-model of $T$ that contains no countable coded $\beta$-models of $T$. \end{theorem}
\begin{proof} Suppose not. Then every $\beta$-model of $T$ contains a countable coded $\beta$-model of $T$. Let $\mathfrak{M}$ be a $\beta$-model of $T$. So $\mathfrak{M}$ contains some countable coded $\beta$-model $\mathfrak{N}_0$ coded by a real $N_0$. Similarly $\mathfrak{N}_0$ contains a countable coded $\beta$-model of $T$, $\mathfrak{N}_1$, coded by a real $N_1$. In this manner we can define a sequence of countable $\beta$-models of $T$, $\mathfrak{N}_0, \mathfrak{N}_1, \mathfrak{N}_2, \ldots$ along with their codes $N_0, N_1, N_2, \ldots$
But for each $n$, $N_{n + 1} \in \mathfrak{N}_n$ and since $\mathfrak{N}_n$ is a $\beta$-model it is correct about all $\Pi^1_1$ facts about $N_{n + 1}$. In other words, $\mathcal{O}^{N_{n + 1}}$ is arithmetic in $N_n$. So $N_0, N_1, \ldots$ provides an example of the type of descending sequence in the hyperdegrees shown not to exist in theorem \ref{thm-main}. \end{proof}
In fact, this same proof actually yields a seemingly stronger result. A $\beta_n$-model is defined to be an $\omega$-model of second order arithmetic which is correct for all $\Sigma^1_n$ statements with parameters from the model. The same proof as above proves the theorem mentioned in the introduction (where once again our new proof shows that the assumption that $T$ is recursively axiomatized can be dropped):
\ms*
Since the statement that a real is the code for a $\beta_n$-model is $\Pi^1_n$, $\beta_n$-models are correct about such statements. And if $T$ is a $\Sigma^1_n$ axiomatized theory then $\beta_n$-models are also correct about which formulas are in $T$. Thus for $\Sigma^1_n$ axiomatized theories we can restate the above theorem to get the following theorem of Mummert and Simpson:
\begin{theorem}\label{ms2} Let $T$ be a $\Sigma^1_n$ axiomatized $\mathcal{L}_2$ theory. If there is a $\beta_n$-model of $T$, then there is a $\beta_n$-model of $$\textrm{$T$+``there is no countable coded $\beta_n$-model of $T$.''}$$ \end{theorem}
\begin{proof} Let $\mathfrak{M}$ be a $\beta_n$-model of $T$ which contains no countable coded $\beta_n$-models of $T$. For any $N \in \mathfrak{M}$, the statement that $N$ is not a countable coded $\beta_n$-model of $T$ is a true $\Sigma^1_n$ sentence and thus is satisfied by $\mathfrak{M}$. Therefore $\mathfrak{M}$ satisfies the statement ``there is no countable coded $\beta_n$-model of $T$.'' \end{proof}
From this we immediately infer the following corollary, a strengthened version of Mummert and Simpson's Corollary 2.4 from \cite{mummert2004}:
\begin{corollary}\label{ms3} Let $T$ be a $\Sigma^1_n$ axiomatized $\mathcal{L}_2$ theory. If $T$ has a $\beta_n$-model then $T$ has a $\beta_n$ model that is not a $\beta_{n+1}$ model. \end{corollary}
\begin{proof} Let $T$ be a $\Sigma^1_n$ axiomatized $\mathcal{L}_2$ theory with a $\beta_n$ model. By Theorem \ref{ms2}, there is a $\beta_n$ model $\mathfrak{M}$ of $\textrm{$T$+``there is no countable coded $\beta_n$-model of $T$.''}$ The latter is a false $\Pi^1_{n+1}$ sentence, whence $\mathfrak{M}$ is not a $\beta_{n+1}$ model. \end{proof}
\begin{remark} The definability restriction on $T$ in Theorem \ref{ms2} and Corollary \ref{ms3} can be relaxed slightly to the requirement that $T$ have an axiomatization definable by a formula consisting of a series of first order quantifiers over a Boolean combination of $\Sigma^1_n$-formulas. The proof is essentially the same. \end{remark}
\subsection{Further discussion of definability restrictions} When this paper was first published, Theorem \ref{ms2} and Corollary \ref{ms3} were missing the definability restriction on $T$. We will now address whether such a restriction is necessary. Before doing so, it will be helpful to state a result implicit in the proof of Theorem \ref{minimalmodel}, since we will use it several times below.
\begin{theorem}\label{minimalmodel2} The relation ``contains a countable coded model isomorphic to'' is a well-order on $\beta$-models. \end{theorem}
If the definability restriction is removed completely from Theorem \ref{ms2}, it is not even clear what the statement means. The most reasonable interpretation is to assume that $T$ is definable in second order arithmetic (but with no bound on the complexity of the definition) and to use the formula defining $T$ to write ``there is no countable coded $\beta_n$-model of $T$'' as an $\mathcal{L}_2$ sentence. Unfortunately, this interpretation is false, as the following proposition demonstrates.
\begin{proposition} There is an $\mathcal{L}_2$ theory $T$ and a $\Sigma^1_2$ formula $\psi$ which defines $T$ (relative to some fixed, standard coding of $\mathcal{L}_2$ sentences) such that $T$ has a $\beta$-model but every $\beta$-model of $T$ satisfies \[ \text{``there is a countable coded $\beta$-model of the theory defined by $\psi$.''} \] \end{proposition}
\begin{proof} Let $\varphi_1$ be the sentence ``there is a countable coded $\beta$-model'' and let $\varphi_2$ be the sentence ``there is a countable coded $\beta$-model which contains a countable coded $\beta$-model.'' Note that $\varphi_2$ is a true $\Sigma^1_2$-sentence. Let $T$ be the theory $\{\varphi_1\land\lnot\varphi_2\}$. By Theorem \ref{minimalmodel2}, there is a $\beta$-model of $T$.
Now assume that $\langle\theta_n\rangle_{n \in \mathbb{N}}$ is a fixed, standard enumeration of all $\mathcal{L}_2$-sentences. Let $\psi(n)$ be the formula \[ \varphi_2\land (\text{$\theta_n =$ ``$\varphi_1\land\lnot\varphi_2$''}). \] In other words, if $\varphi_2$ holds then $\psi$ defines $T$ and if $\varphi_2$ does not hold then $\psi$ defines the empty theory. Since $\varphi_2$ is true, $\psi$ is a $\Sigma^1_2$ definition of $T$.
Now let $\mathfrak{M}$ be any $\beta$-model of $T$. Since $\mathfrak{M} \models \varphi_1$, $\mathfrak{M}$ believes there is a countable coded $\beta$-model. And since $\mathfrak{M} \models \lnot\varphi_2$, $\mathfrak{M}$ believes that $\psi$ defines the empty theory and thus that the theory defined by $\psi$ is satisfied in every model. Thus $\mathfrak{M}$ believes there is a countable coded $\beta$-model of the theory defined by $\psi$. \end{proof}
Thus the definability restriction on Theorem \ref{ms2} cannot even be relaxed to include all $\Sigma^1_{n + 1}$ axiomatized theories. The situation for Corollary \ref{ms3} is less clear. In particular, the authors do not currently know if the result holds when the definability restriction is removed, and consider this to be an interesting question. The following two results demonstrate why finding a counterexample may not be easy.
\begin{proposition}\label{separation} Let $T$ be an $\mathcal{L}_2$ theory. If $T$ has a $\beta_n$-model that contains $T$ then $T$ has a $\beta_n$-model that is not a $\beta_{n + 1}$-model. \end{proposition}
\begin{proof} By Theorem \ref{minimalmodel2}, we can find a $\beta_n$-model $\mathfrak{M}$ of $T$ which contains $T$ but which contains no countable coded $\beta_n$-model of $T$ containing $T$. Hence the statement \[ \text{``there is a countable coded $\beta_n$-model of $T$''} \] does not hold in $\mathfrak{M}$. Since it is a true $\Sigma^1_{n + 1}$ statement with parameters from $\mathfrak{M}$ (since $\mathfrak{M}$ contains $T$), $\mathfrak{M}$ is not a $\beta_{n + 1}$-model. \end{proof}
\begin{corollary} Let $T$ be a true $\mathcal{L}_2$ theory. Then for all $n \ge 1$, $T$ has a $\beta_n$-model that is not a $\beta_{n + 1}$-model. \end{corollary}
\begin{proof} If $T$ is a true theory, then for every $n$ there is a $\beta_n$-model containing $T$, so we may apply Proposition \ref{separation}. \end{proof}
\section{Spector Ranks} \label{ranks}
Define a relation $\prec$ on pairs of reals by $A \prec B$ iff $\mathcal{O}^A \leq_H B$. By theorem \ref{thm-main}, this relation is well-founded and therefore reals can be assigned ordinal ranks according to it. Let's refer to the $\prec$-rank of a real as its \emph{Spector rank}. In this section we will calculate the Spector ranks of reals, showing that we get the same ranks as those induced by the $\omega_1$'s of reals.
We will need to use the following theorems:
\begin{theorem}[Spector] \label{spector} For any reals $A$ and $B$: \begin{enumerate}
\item If $\mathcal{O}^B \leq_H A$ then $\omega_1^B < \omega_1^A.$
\item If $B \leq_H A$ and $\omega_1^B < \omega_1^A$ then $\mathcal{O}^B \leq_H A.$ \end{enumerate} \end{theorem}
\begin{theorem}[Sacks] \label{sacks} If $\lambda$ is an admissible ordinal greater than $\omega$ and $X$ is a real such that $X$ computes a presentation of $\lambda$ (i.e.\ $\lambda < \omega_1^X$) then there is a real $Y$ that is hyperarithmetical in $X$ such that $\omega_1^Y = \lambda$. \end{theorem}
\begin{remark}
Theorem \ref{sacks} is typically stated without the requirement that $Y$ is hyperarithmetical in $X$, though this is implicit in all or nearly all extant proofs of the theorem. For instance, in \cite{steel1978} Steel uses the method of forcing with tagged trees to prove Sacks' theorem. In that case, the real $Y$ is obtained as the reduct of a generic filter over $L_\lambda$. Since any presentation of $\lambda$ can hypercompute such a generic (if you can compute a presentation of $\lambda$ then it just takes $\omega\cdot(\lambda + 1)$ jumps to compute the theory of $L_\lambda$), $X$ can hypercompute a $Y$ witnessing Sacks' theorem. \end{remark}
The calculation of Spector ranks now follows relatively easily.
\ranks*
\begin{remark} The only reason we need to say $(1 + \alpha)^\text{th}$ admissible rather than $\alpha^\text{th}$ admissible is that the way admissible is usually defined, $\omega$ is an admissible ordinal but unlike all other countable admissible ordinals, it is not the $\omega_1$ of any real. \end{remark}
\begin{proof} We will argue by induction on $\alpha$ that for any $A$ if $\rank(A) > \alpha$ then $\omega_1^A$ is greater than the $(1 + \alpha)^\text{th}$ admissible ordinal and conversely that if $\omega_1^A$ is greater than the $(1 + \alpha)^\text{th}$ admissible then $\rank(A) > \alpha$.
First suppose $\rank(A) > \alpha$. So there is some $B$ of rank $\alpha$ such that $\mathcal{O}^B \leq_H A$. By Spector's result, theorem \ref{spector}, this implies $\omega_1^B < \omega_1^A$. And by the induction assumption, $\omega_1^B$ is at least the $(1 + \alpha)^\text{th}$ admissible so $\omega_1^A$ is greater than the $(1 + \alpha)^\text{th}$ admissible.
Now suppose that $\omega_1^A$ is greater than the $(1 + \alpha)^\text{th}$ admissible. Let $\lambda$ denote the $(1 + \alpha)^\text{th}$ admissible. By Sacks' theorem, there is some $B$ hyperarithmetical in $A$ such that $\omega_1^B = \lambda$. Since $\omega_1^B < \omega_1^A$, Spector's theorem implies that $\mathcal{O}^B \leq_H A$ and hence $\rank(B) < \rank(A)$. By the induction assumption, $\rank(B)$ is at least $\alpha$, so $\rank(A) > \alpha$. \end{proof}
\begin{theorem}[Silver] If $\alpha$ is admissible relative to $0^\sharp$ then $\alpha$ is a cardinal in $L$. \end{theorem}
Hence if $X$ is a real in the cone above $0^\sharp$ then $\omega_1^X$ is a cardinal in $L$. Suppose that $\omega_1^X$ is the $\alpha^\text{th}$ admissible. Since $\omega_1^X$ is a cardinal in $L$, it follows that actually $\alpha = \omega_1^X = \omega_\alpha^{CK}$. So if $0^\sharp$ exists then on a cone, the Spector rank of a real $X$ is equal to $\omega_1^X$.
\begin{theorem} If $0^\sharp$ exists, then for all $A$ on a cone, the Spector rank of $A$ is $\omega_1^A$. \end{theorem}
Alternatively, one can infer the previous theorem from the following proposition due to Martin.
\begin{proposition}[Martin] Assuming appropriate determinacy hypotheses, if $F$ is a degree invariant function from reals to (presentations of) ordinals such that $F(A)\leq \omega_1^A$, then either $F$ is constant on a cone or $F(A)=\omega_1^A$ on a cone. \end{proposition}
One could also consider the analogous relation given by replacing hyperarithmetic reducibility and the hyperjump with Turing reducibility and the Turing jump. Namely, define $\prec_T$ by $A \prec_T B$ iff $A' \leq_T B$. By results of Harrison (see \cite{harrison1968}), this relation is not well-founded. However, it \emph{is} well-founded if we restrict ourselves to the hyperarithmetic reals, as shown by Putnam and Enderton in \cite{enderton1970}. In that paper, Putnam and Enderton also show that the rank of a hyperarithmetic real $A$ in this relation is ``within $2$'' of the least $\alpha$ such that $A$ cannot compute $0^{(\alpha)}$. More precisely, if the rank of $A$ is $\alpha$ then $A$ cannot compute $0^{(\alpha + 1)}$ and if $A$ cannot compute $0^{(\alpha)}$ then the rank of $A$ is at most $\alpha + 2$.
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2021, 17: 529-555. doi: 10.3934/jmd.2021018
On Furstenberg systems of aperiodic multiplicative functions of Matomäki, Radziwiłł, and Tao
Alexander Gomilko 1, , Mariusz Lemańczyk 1, and Thierry de la Rue 2,
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopin street 12/18, 87-100 Toruń, Poland
Laboratoire de Mathématiques Raphaël Salem, Université de Rouen Normandie, CNRS - Avenue de l'Université - 76801, Saint Étienne du Rouvray, France
Received August 25, 2020 Revised August 24, 2021 Published November 2021
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It is shown that in a class of counterexamples to Elliott's conjecture by Matomäki, Radziwiłł, and Tao [23] the Chowla conjecture holds along a subsequence.
Keywords: Aperiodic multiplicative function, Chowla conjecture, Archimedean characters, Furstenberg system of an arithmetic function, transformations with quasi-discrete spectrum, strongly stationary processes.
Mathematics Subject Classification: Primary: 11N37, 37A44; Secondary: 37A50.
Citation: Alexander Gomilko, Mariusz Lemańczyk, Thierry de la Rue. On Furstenberg systems of aperiodic multiplicative functions of Matomäki, Radziwiłł, and Tao. Journal of Modern Dynamics, 2021, 17: 529-555. doi: 10.3934/jmd.2021018
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Alexander Gomilko Mariusz Lemańczyk Thierry de la Rue
Article outline | CommonCrawl |
\begin{document}
\title[]{Operator-valued Kirchberg Theory}
\author[]{Jian Liang and Sepideh Rezvani}
\address{Department of Mathematics, University of Illinois, Urbana, IL 61801, U.S.A.} \email{liang41\@@illinois.edu,rezvani2\@@illinois.edu} \date{\today}
\keywords{QWEP, Hilbert C$^*$-module.} \subjclass[2010]{46L06, 46L08, 46L10, 46L07.}
\begin{abstract} In this paper, we will follow Kirchberg's categorical perspective to establish new notions of WEP and QWEP relative to a C$^*$-algebra, and develop similar properties as in the classical WEP and QWEP. Also we will show some examples of relative WEP and QWEP to illustrate the relations with the classical cases. Finally we will apply our notions to recent results on C$^*$-norms. \end{abstract}
\maketitle
\section{Introduction}
In this paper, we investigate a new notion of operator-valued WEP and QWEP. Let us recall that the weak expectation property (abbreviated as WEP) was introduced by E. Christopher Lance in his paper \cite{La2} of 1973, as a generalization of nuclearity of C$^*$-algebras. In 1993, Eberhard Kirchberg \cite{Ki1} revealed remarkable connections between tensor products of C$^*$-algebras and Lance's weak expectation property. He defined the notion of QWEP as a quotient of a C$^*$-algebra with the WEP, and formulated the famous QWEP conjecture that all C$^*$-algebras are QWEP. He showed a vast amount of equivalences between various open problems in operator algebras. In particular, he showed that the QWEP conjecture is equivalent to an affirmative answer to the Connes' Embedding Problem.
The motivation of our research is to generalize the notion of WEP and QWEP in the setting of Hilbert C$^*$- modules, in which the inner product of a Hilbert space is replaced by a C$^*$-valued inner product. Hilbert C$^*$-modules were first introduced in the work of Irving Kaplansky in 1953 \cite{Kap53}, which developed the theory for commutative, unital algebras. In the 1970s the theory was extended to noncommutative C$^*$- algebras independently by William Lindall Paschke \cite{Pas73} and Marc Rieffel\cite{Rie74}. The latter used Hilbert C$^*$-modules to construct a theory of induced representations of C$^*$-algebras. Hilbert C$^*$-modules are crucial to Kasparov's formulation of KK-theory \cite{Kas80}, and provide the right framework to extend the notion of Morita equivalence to C*-algebras \cite{Rie82}. They can be viewed as the generalization of vector bundles to noncommutative C*-algebras and as such play an important role in noncommutative geometry, notably in C$^*$-algebraic quantum group theory and groupoid C$^*$-algebras.
Another motivation of our research is from the relation with amenable correspondence. The notion of correspondence of two von Neumann algebras has been introduced by Alain Connes \cite{CJ}, as a very useful tool for the study of type II$_1$ factors. Later Sorin Popa systematically developed this point of view to get some new insight in this area \cite{Pop}. Among many interesting results and remarks, he discussed Connes' classical work on the injective II$_1$ factor in the framework of correspondences, and he defined and studied a natural notion of amenability for a finite von Neumann algebra relative to a von Neumann subalgebra using conditional expectations. As Lance was inspired by Tomiyama's work on conditional expectations, we are interested in weak conditional expectations relative to a C$^*$-algebra.
The main results of this paper are inspired by Kirchberg's seminal work on non-semisplit extensions. In Section 3, we define two notions of WEP relative to a C$^*$-algebra $D$. Let $E_D$ be a Hilbert $D$-module, and $\mathcal{L}(E_D)$ be the C$^*$-algebra of bounded adjointable linear operators on $E_D$. Also let $E_{D^{**}}$ be the weakly closed Hilbert $D^{**}$-module, and $\mathcal{L}^w(E_{D^{**}})$ be the von Neumann algebra of bounded adjointable linear operators on $E_{D^{**}}$. We say that a C$^*$-algebra $A$ has the $D$WEP$_1$ if it is relatively weakly injective in $\mathcal{L}(E_D)$, i.e. for a faithful representation $A \subset \mathcal{L}(E_D)$, there exists a ucp map $\mathcal{L}(E_D) \to A^{**}$, which preserves the identity on $A$. Respectively we define the $D$WEP$_2$ to be the relatively weak injectivity in $\mathcal{L}(E_{D^{**}})$. We show that $D$WEP$_1$ implies $D$WEP$_2$, but the converse is not true. After investigating some basic properties, we establish a tensor product characterization of $D$WEP. Let $max^D_1$ be the tensor norm on $A\otimes C^*\mathbb F_{\infty}$ induced from the inclusion $A\otimes C^*\mathbb F_{\infty} \subseteq \mathcal{L}(E^u_D)\otimes_{max}C^*\mathbb F_{\infty}$ for some universal Hilbert $D$-module $E^u_D$ and $A\subset \mathcal{L}(E^u_D) $. Then a
C$^*$-algebra $A$ has the $D$WEP$_1$, if and only if \[
A\underset{max^D_1}\otimes C^*\mathbb F_{\infty} = A \underset{max}\otimes C^*\mathbb F_{\infty}. \] We have the similar result for $D$WEP$_2$ with respect to some universal weakly closed $D^{**}$-module $E^u_{D^{**}}$.
In Section 4, we define two notions of relative QWEP, derived from relative WEP. After developing basic properties of relative QWEP, we show that the two notions are equivalent, unlike the case in the relative WEP. Similarly, we establish a tensor product characterization of relative QWEP. In Section 5, we investigate some properties of WEP and QWEP relative to some special classes of C$^*$-algebras, and illustrate the relations with classical results in the WEP and QWEP theory.
In Section 6, we discuss some application of our tensor product characterization result for $D$WEP from Section 3 in the setting of C$^*$-norms. As we know, the algebraic tensor product $A\otimes B$ of two C$^*$-algebras may admit distinct norms, for
instance, the minimal and maximal norms. A C$^*$-algeba $A$ such that $\|\cdot\|_{\min}=\|\cdot\|_{\max}$ on $A\otimes B$ for any other C$^*$-algebra $B$ is called nuclear. In particular, Simon Wassermann \cite{Was76} shows that ${\mathbb B}({\mathcal H})$ is not nuclear, and
later Gilles Pisier and Marius Junge \cite{JP95} show that $\|\cdot\|_{\min}\neq\|\cdot\|_{\max}$ on ${\mathbb B}({\mathcal H})\otimes {\mathbb B}({\mathcal H})$. Recently, Pisier and Ozawa \cite{OP14} showed that there is at least a continuum of different C$^*$-norms on ${\mathbb B}(\ell_2)\otimes {\mathbb B}(\ell_2)$. In Section 6, we adopt the adea in their paper to construct a new C$^*$-norm on $A\otimes B$ by using the notion of $D$WEP and the $max^D_1$ norm which we constructed in Section 3, and provide the conditions which make it
neither min nor max norm, and distinct from the continuum norms constructed by Pisier and Ozawa. We also give a concrete example, satisfying the conditions and hence with four distinct tensor norms. These conditions will give us a new way to distinguish norms on C$^*$-algebras.
We would like to thank Marius Junge, for having extensive inspiring discussions on this topic.
\section{Preliminaries}
\subsection{WEP and QWEP}
The notion of WEP is from Lance \cite{La2}, and it is inspired by Tomiyama's extensive work on conditional expectations. Kirchberg in \cite{Ki1} raises the famous QWEP conjecture and establishes its several equivalences. Here we list some useful results for readers' convenience. Most of the results and proofs can be found in Ozawa's survey paper \cite{Oz}.
\begin{definition}
Let $A$ be a unital C$^*$-subalgebra of a unital C$^*$-algebra $B$. We say $A$ is
relatively weakly injective (short as r.w.i.) in $B$, if there
is a ucp map $\varphi: B \to A^{**}$ such that $\varphi|_A = \operatorname{id}_{A}$. \end{definition}
For von Neumann algebras $M\subset N$, the relative weak injectivity is equivalent to the existence of a (non-normal) conditional expectation from $N$ to $M$.
We say a C$^*$-algebra $A$ has the weak expectation property (short as WEP), if it is relatively weakly injective in $\mathbb{B}(\mathcal{H})$ for a faithful representation $A\subset {\mathbb B}({\mathcal H})$.
Since ${\mathbb B}({\mathcal H})$ is injective, the notion of WEP does not depend on the choice of a faithful representation of $A$. We say a C$^*$-algebra is QWEP if it is a quotient of a C$^*$-algebra with the WEP. The QWEP conjecture raised by Kirchberg in \cite{Ki1} states that all C$^*$-algebras are QWEP.
From the definition of \emph{r.w.i.}, it is easy to see the following transitivity property.
\begin{lemma} \label{wri wri}
For C$^*$-algebras $A_0 \subseteq A_1 \subseteq A$, such that $A_0$ is relatively weakly injective
in $A_1$, $A_1$ is relatively weakly injective in $A$, then $A_0$ is relatively weakly injective in $A$. \end{lemma}
The property of \emph{r.w.i.} is also closed under direct product. \begin{lemma}
\label{prod wri} If $(A_i)_{i\in I}$ is a net of $C^*$-algebras such that $A_i$ is relatively weakly injective in $B_i$ for all $i\in I$, then $\Pi_{i\in I}A_i$ is relatively weakly injective in $\Pi_{i\in I}B_i$. \end{lemma}
In \cite{La2}, Lance establishes the following tensor product characterization of the WEP. The proof of the theorem is called \emph{The Trick}, and we will be using this throughout the paper. In the following, let $\mathbb{F}_{\infty}$ denote the free group with countably infinite generators, and $C^*\mathbb{F}_{\infty}$ be the full group C$^*$-algebra of $\mathbb{F}_{\infty}$.
\begin{theorem} \label{tensor wri} A C$^*$-algebra $A$ has the WEP, if and only if \[
A \underset{max}\otimes C^*\mathbb{F}_\infty = A \underset{min}\otimes C^*\mathbb{F}_\infty. \] \end{theorem}
As a consequence of the above theorem, we have the following result.
\begin{corollary} \label{wepwri} A C$^*$-algebra $A$ has the WEP if and only if for any inclusion $A\subseteq B$, $A$ is relatively weakly injective in $B$. \end{corollary}
Similar to the WEP, the QWEP is also preserved by the relatively weak injectivity as following.
\begin{lemma}
If a C$^*$-algebra $A$ is relatively weakly injective in a QWEP C$^*$-algebra, then it is QWEP. \end{lemma}
Although the WEP does not pass to the double dual, the QWEP is more flexible.
\begin{proposition}\label{dd}
A C$^*$-algebra $A$ is QWEP if and only if $A^{**}$ is QWEP. \end{proposition}
As a corollary of the above proposition, ${\mathbb B}({\mathcal H})^{**}$ is QWEP. Moreover we have the following equivalence.
\begin{corollary}\label{bl2qwep}
A C$^*$-algebra $A$ is QWEP if and only if $A$ is relatively weakly injective in ${\mathbb B}({\mathcal H})^{**}$. \end{corollary}
\subsection{Hilbert C$^*$-Modules}
The notion of Hilbert C$^*$-modules first appeared in a paper by Irving Kaplansky in 1953 \cite{Kap53}. The theory was then developed by the work of William Lindall Paschke in \cite{Pas73} . In this section we give a brief introduction to Hilbert C$^*$-modules and present some of their fundamental properties which we are going to use throughout this paper.
\begin{definition}
Let $D$ be a C$^*$-algebra. An inner-product $D$-module is a linear space $E$ which is a right $D$-module with compatible scalar multiplication: $\lambda(xa)=(\lambda x)a= x(\lambda a)$, for $x\in E$, $a\in D$, $\lambda\in \mathbb{C}$, and a map $(x,y)\longmapsto \langle x,y\rangle:E\times E\to D$ with the following properties:
\begin{enumerate}
\item $\langle x, \alpha y+\beta z\rangle=\alpha\langle x,y\rangle+\beta \langle x,z\rangle\quad$ for $x$, $y$, $z\in E$ and $\alpha$, $\beta\in \mathbb{C}$;
\item $\langle x,ya\rangle=\langle x,y \rangle a\quad$ for $x$, $y\in E$ and $a\in D$;
\item $\langle y,x\rangle=\langle x,y\rangle^*\quad$ for $x$, $y\in E$;
\item $\langle x,x \rangle \geq 0$; if $\langle x,x \rangle=0$, then $x=0$. \end{enumerate} \end{definition}
For $x\in E$, we let $\|x\|=\|\langle x,x\rangle\|^{1/2}$. It is easy to check that if $E$ is an inner-product $D$-module, then $\|\cdot\|$ is a norm on $E$.
\begin{definition}
An inner-product $D$-module which is complete with respect to its norm is called a Hilbert $D$-module or a Hilbert C$^*$-module over the C$^*$-algebra $D$. \end{definition}
Note that any C$^*$-algebra $D$ is a Hilbert $D$-module itself with the inner product $\langle x,y \rangle = x^*y$ for $x$ and $y$ in $D$. Another important example of a Hilbert C$^*$-module is the following.
\begin{example} \normalfont
Let $\mathcal{H}$ be a Hilbert space. Then the algebraic tensor product $\mathcal{H}\otimes_{alg}D$ can be equipped with a $D$-valued inner-product: \[ \langle \xi\otimes a, \eta \otimes b\rangle=\langle\xi,\eta \rangle a^*b \qquad (\xi,\eta \in \mathcal{H}, a,b\in D). \] Let $\mathcal{H}_D=\mathcal{H}\otimes D$ be the completion of $\mathcal{H}\otimes_{alg}D$ with respect to the induced norm. Then $\mathcal{H}_D$ is a Hilbert $D$-module. \end{example}
Let $E$ and $F$ be Hilbert $D$-modules. Let $t$ be an adjointable map from $E$ to $F$, i.e. there exists a map $t^*$ from $F$ to $E$ such that \[ \langle tx,y \rangle=\langle x,t^*y\rangle, \qquad \text{for }x\in E\text{ and } y\in F. \] One can easily see that $t$ must be right $D$-linear, that is, $t$ is linear and $t(xa)=t(x)a$ for all $x\in E$ and $a\in D$. It follows that any adjointable map is bounded, but the converse is not true -- a bounded $D$-linear map need not be adjointable. Let $\mathcal{L}(E,F)$ be the set of all adjointable maps from $E$ to $F$, and we abbreviate $\mathcal{L}(E,E)$ to $\mathcal{L}(E)$. Note that $\mathcal{L}(E)$ is a C$^*$-algebra equipped with the operator norm.
Now we review the notion of compact operators on Hilbert $D$-modules, as an analogue to the compact operators on a Hilbert space. Let $E$ and $F$ be Hilbert $D$-modules. For every $x$ in $E$ and $y$ in $F$, define the map $\theta_{x,y}:E\to F$ by \[ \theta_{x,y}(z)=y\langle x,z \rangle \qquad \text{for }z\in E. \] One can check that $\theta_{x,y}\in \mathcal{L}(E,F)$ and $\theta_{x,y}^*=\theta_{y,x}$. We denote by $\mathcal{K}(E,F)$ the closed linear subspace of $\mathcal{L}(E,F)$ spanned by $\{\theta_{x,y}: x\in E, y\in F\}$, and we abbreviate $\mathcal{K}(E,E)$ to $\mathcal{K}(E)$. We call the elements of $\mathcal{K}(E,F)$ \emph{compact} operators.
Let $E$ be a Hilbert $D$-module and $Z$ be a subset of $E$. We say that $Z$ is a \emph{generating set} for $E$ if the closed submodule of $E$ generated by $Z$ is the whole of $E$. If $E$ has a countable generating set, we say that $E$ is countably generated.
In \cite{Kas80}, Kasparov proves the following theorem known as the \emph{absorption theorem}, which shows the universality of ${\mathcal H}_D$ in the category of Hilbert $D$-modules. \begin{theorem}
\label{absorption} Let $D$ be a C$^*$-algebra and $E$ be a countably generated Hilbert $D$-module. Then $E\oplus \mathcal{H}_D \approx \mathcal{H}_D$, i.e. there exists an element $u\in\mathcal{L}(E\oplus {\mathcal H}_D,{\mathcal H}_D)$ such that $u^*u=1_{E\oplus {\mathcal H}_D}$ and $uu^*=1_{{\mathcal H}_D}$. \end{theorem}
\begin{remark} \label{universal} \normalfont Using the absorption theorem, for an arbitrary Hilbert $D$-module $E$, we have $\mathcal{L}(E\oplus \mathcal{H}_D) \simeq \mathcal{L(H}_D)$. Hence we have an embedding of $\mathcal{L}(E)$ in $\mathcal{L(H}_D)$ and a conditional expectation from $\mathcal{L(H}_D)$ to $\mathcal{L}(E)$. \end{remark}
Before we proceed to the main results of Hilbert C$^*$-modules, let us recall the notion of \emph{multiplier algebra} of a C$^*$-algebra.
\begin{definition}
Let $A$ and $B$ be C$^*$-algebras. If $A$ is an ideal in $B$, we call $A$ an essential ideal if there is no nonzero ideal of $B$ that has zero intersection with $A$. Or equivalently if $b\in B$ and $bA=\{0\}$, then $b=0$. \end{definition}
It can be shown that for any C$^*$-algebra $A$, there is a unique (up to isomorphism) maximal C$^*$-algebra which contains $A$ as an essential ideal. This algebra is called the multiplier algebra of $A$ and is denoted by $\mathcal{M}(A)$.
\begin{theorem} \label{mult}
If $E$ is a Hilbert $D$-module, then $\mathcal{L}(E)=\mathcal{M(K}(E))$. \end{theorem}
Note that if $E=D$ for a unital C$^*$-algebra $D$, then $D=\mathcal{K}(D)$ and $\mathcal{L}(D)=\mathcal{M}(D)$.
In the special case where $E=\mathcal{H}_D$, we have \[ \mathcal{K(H}_D)\simeq \mathcal{K(H)}\underset{min}\otimes D = {\mathcal K} \otimes D, \] where ${\mathcal K} = \mathcal{K(H)}$ is the C$^*$-algebra of the compact operators. Therefore, by Theorem ~\ref{mult} we have \[ \mathcal{L(H}_D)\simeq \mathcal{M}({\mathcal K}\otimes D). \]
In \cite{Kas80} Kasparov introduces a GNS type of construction in the context of Hilbert C$^*$-modules, known as the KSGNS construction (for Kasparov, Stinespring, Gelfand, Neimark, Segal) as follows.
\begin{theorem}
\label{GNS}
Let $A$ be a C$^*$-algebra, $E$ be a Hilbert $D$-module and let $\rho:A\to \mathcal{L}(E)$ be a completely positive map. There exists a Hilbert $D$-module $E_{\rho}$, a $^*$-homomorphism $\pi_{\rho}:A\to \mathcal{L}(E_{\rho})$ and an element $v_{\rho}$ of $\mathcal{L}(E,E_{\rho})$, such that \begin{align*} \rho(a) =v^*_{\rho}\pi_{\rho}(a)v_{\rho} \qquad(a\in A), \\ \pi_{\rho}(A)v_{\rho}E\quad \text{is dense in}\quad E_{\rho}. \end{align*} \end{theorem}
As a consequence of the above theorem, Kasparov shows that given a C$^*$-algebra $D$, any separable $C^*$-algebra can be considered as a C$^*$-subalgebra of $\mathcal{L(H}_D)$. This indicates that $\mathcal{L(H}_D)$ plays the similar role in the category of Hilbert C$^*$-modules to that of ${\mathbb B}({\mathcal H})$ in the category of C$^*$-algebras.
\begin{proposition}
Let $A$ be a separable C$^*$-algebra. Then there exists a faithful nondegenerate $^*$-homomorphism $\pi: A\to \mathcal{L(H}_D)$. \end{proposition}
As we see, $\mathcal{L}({\mathcal H}_D)$ plays the role of ${\mathbb B}({\mathcal H})$. Note that ${\mathbb B}({\mathcal H})$ is also a von Neumann algebra, but $\mathcal{L}({\mathcal H}_D)$ is not in general. Paschke in \cite{Pas73} introduces self-dual Hilbert C$^*$-modules to play the similar role in the von Neumann algebra context.
Let $E$ be a Hilbert $D$-module. Each $x\in E$ gives rise to a bounded $D$-module map $\hat {x}:E\to D$ defined by $\hat x(y)=\langle y,x \rangle$ for $y\in E$. We will call $E$ \emph{self-dual} if every bounded $D$-module map of $E$ into $D$ arises by taking $D$-valued inner products with some $x\in E$. For instance, if $D$ is unital, then it is a self-dual Hilbert $D$-module. Any self-dual Hilbert C$^*$-module is complete, but the converse is not true.
For von Neumann algebra $N$, it is natural to consider the the self-dual Hilbert $N$-module $E_N$, because of the following theorem from \cite{JS}.
\begin{theorem} For a Hilbert C$^*$-module $E$ over a von Neumann algebra $N$, the following conditions are equivalent: \begin{enumerate} \item The unit ball of $E$ is strongly closed; \item $E$ is principal, or equivalently, $E$ is an ultraweak direct sum of Hilbert C$^*$-modules $q_{\alpha}N$, for some projections $q_{\alpha}$; \item $E$ is self-dual; \item The unit ball of $E$ is weakly closed. \end{enumerate} \end{theorem}
We denote the algebra of adjointable maps on $E_N$ closed in the weak operator topology by $\mathcal{L}^w(E_N)$.
\begin{remark} \label{e} \normalfont
According to \cite{Pas73} and the absorption theorem, for a von Neuamann algebra $N$, we have that $\mathcal{L}^w(E_N) = e {\mathbb B}({\mathcal H})\bar{\otimes} N e$ for some projection $e$. \end{remark}
\begin{remark} \label{unitization} \normalfont
Let $N$ be a von Neumann subalgeba of $M$, such that $N=zM$ for some central projection $z\in M$. Then one can unitize the inclusion map $\iota: {\mathbb B}(\ell_2)\bar{\otimes}N\hookrightarrow {\mathbb B}(\ell_2)\bar{\otimes}M$. Indeed since ${\mathbb B}(\ell_2)$ is a type I$_{\infty}$ factor, the projection $1\otimes z: {\mathbb B}(\ell_2)\bar\otimes M \to {\mathbb B}(\ell_2)\bar\otimes N$ is properly infinite, and hence it is equivalent to identity on ${\mathbb B}(\ell_2)\bar \otimes M$ \cite{Tak}. Let $1\otimes z = v^*v$, and $\operatorname{id}_{{\mathbb B}(\ell_2)\bar \otimes M}=vv^*$. Note that $(1\otimes z)\circ \iota=\operatorname{id}_{{\mathbb B}(\ell_2)\bar \otimes N}$. Multiplying by $v$ from left and by $v^*$ from right, we get $v\iota v^*=\operatorname{id}_{{\mathbb B}(\ell_2)\bar \otimes N}$. \end{remark}
\subsection{Kirchberg's observations on the multiplier algebra}
In this section, we explore Kirchberg's seminal paper on non-semisplit extensions in detail. In particular we show the factorization property explicitly for readers' convenience.
Let $A$, $B$ and $C$ be C$^*$-algebras. We say a map $h:A\to B$ \emph{factors through $C$ approximately via ucp maps in point-norm topology} if there exist ucp maps $\varphi_n:A\to C$ and $\psi_n: C\to B$ such that the following diagram commutes approximately in point-norm topology. \[ \xymatrix{
A\ar [dr]_{\varphi_n} \ar [rr]^{h} && B\\
&C\ar [ur]_{\psi_n}&\\
} \]
i.e. $\|(\psi_n\circ\varphi_n)(x)-h(x)\|\to 0$ for all $x\in A$.
\begin{theorem} \label{factor1} Let $A$ be a C$^*$-algebra and $\mathcal{M}(A)$ be its multiplier algebra. Then the identity map on $\mathcal{M}(A)$
factors through $\ell_{\infty}(A)$ approximately via ucp maps in point-norm topology. \end{theorem}
\begin{proof}[Sketch of proof] Let us fix an approximate unit $\{h_i\}$ for $A$. Assume $B$ is a separable unital subalgebra of $\mathcal{M}(A)/A$. Let $\pi: {\mathcal M}(A) \to {\mathcal M}(A)/A$ be the quotient map, and $D$ be a separable unital subalgebra of $\mathcal{M}(A)$ such that $\pi(D)=B$ and $D$ contains the approximate unit. Using these objects, Kirchberg constructed unital completely positive contractions $V_n:\ell_{\infty}(A) \to \mathcal{M}(A)$, and linear maps $W_n:\mathcal{M}(A)\odot C_0(0,1]\to\ell_{\infty}(A)\otimes C_0(0,1]$ as follows:
Let $(u_i)$ be a dense sequence in the unitary group of $D$. One can find finite convex combinations $\{g_i\}$ of elements $\{h_i\}$ such that \begin{align} g_ng_{n+1}&=g_n, \\ g_nh_n&=h_n, \\
\|g_nu_j-u_jg_n\|&\leq 2^{-2n}, \quad \text{if} \quad j\leq n. \end{align}
Now we define $d_0^{(n)}:=g_n^{1/2}$ and $d_k^{(n)}:=(g_{k+n}-g_{k+n-1})$ for $k\geq 1$, and put $V_n(b_1,b_2,...)=\sum_{k=0}^{\infty}d_k^{(n)}b_kd_k^{(n)}$, where the sum is taken in $A^{**}$. Define \[ X_n:=(x_1^{(n)},x_2^{(n)},...) \]
where
\[
x_1^{(n)}=g_{n+2}, \quad x_k^{(n)}=g_{k+n+2}-g_{k+n+3}
\] Let $\Delta: {\mathcal M}(A) \to \ell_\infty({\mathcal M}(A))$ be defined by $\Delta(a)=(a,a,...)$ for $a\in \mathcal{M}(A)$. Let $f_0\in C_0(0,1]$ be such that $f_0(x)=x$ for $x\in (0,1]$. Now define $W_n(a\otimes f)=f(\lambda_n)(\Delta(a)\otimes id)$, where $\lambda_n:=X_n\otimes f_0\in A\otimes C_0(0,1]$ for $a\in \mathcal{M}(A)$ and $f\in C_0(0,1]$. It is easy to check that the maps $V_n$ and $W_n$ have the following properties: \begin{enumerate} \item $(V_n\otimes \operatorname{id}) (W_n(b))=(V_n\otimes \operatorname{id})(\Delta \otimes \operatorname{id} )(b)$ for $b\in \mathcal{M}(A)\odot C_0(0,1]$. (Use (2.1) and (2.2).)
\item The map \[ a\in \mathcal{M}(A)\longmapsto W_n(a\otimes f)+(c_0(A)\otimes C_0(0,1]) \] lifts to a completely contractive map \[ U_{n,f}:\mathcal{M}(A)\to \ell_{\infty}(A) \otimes C_0(0,1] . \]
\item $\lim_{n\to \infty} V_n(a)=a$, for $a\in {\pi}^{-1}(B)$. (Use (2.3).) \end{enumerate}
Moreover for $f\in C_0(0,1]$ such that $0\leq f\leq 1$ and $f(1) =1$, let $g=\sqrt f$. One can define $U_{n,f}(a)=g(\lambda _n)(\Delta (a)\otimes \operatorname{id})g(\lambda _n)$. Therefore, we have the following commutative diagram: \[ \xymatrix{
\mathcal{M}(A)\ar [drr]_{U_{n,f}} \ar [r]^{\operatorname{id}\otimes 1\qquad} & \mathcal{M}(A)\odot C_0(0,1] \ar [dr]^{W_n}\ar [rr]^{\operatorname{id}}&& \mathcal{M}(A)\odot C_0(0,1] \ar [r] ^{\qquad{\operatorname{id}\otimes\operatorname{ev_1}}} &\mathcal{M}(A)\\
&&\ell_{\infty}(A) \otimes C_0(0,1] \ar [ur]^{V_n\otimes \operatorname{id}}&&\\
} \] where $\operatorname{ev_1}$ is the evaluation of $f(x)$ in $C_0(0,1]$ at 1. From the properties above, we have \begin{align*} \lim_{n\to \infty}(\operatorname{id}\otimes\operatorname{ev_1})(V_n\otimes \operatorname{id})(W_n(b))&=\lim_{n\to \infty}(\operatorname{id}\otimes\operatorname{ev_1})(V_n\otimes \operatorname{id}(\Delta(a)\otimes \operatorname{id}(f)))\\ &=\lim_{n\to \infty}V_n(a)f(1)=a. \end{align*} Since $\ell_{\infty}(A) \otimes C_0(0,1]$ factors though $\ell_{\infty}(A)$, this proves that $\mathcal{M}(A)$ factors through $\ell_{\infty}(A)$ approximately via ucp maps. \end{proof} Using the above theorem, we can establish the following result on the relation between ${\mathcal M}(A)$ and $A^{**}$. \begin{corollary}\label{ma}
Suppose $A$ is a C$^*$-algebra and $\mathcal{M}(A)$ is its multiplier algebra. Then $\mathcal{M}(A)$ is relatively weakly injective in $A^{**}$. \end{corollary}
\begin{proof}
With the notation of the previous theorem, since $U_{n,f}$ lifts $W_n$, we have \[\lim_{n\to \infty}(\operatorname{id}\otimes\operatorname{ev_1})(V_n\otimes \operatorname{id})(U_{n,f}(a))= a,\] for $a\in \pi^{-1}(B)$.
Since there is a natural inclusion $\mathcal{M}(A)\subset A^{**}$, we can define $\tilde {U}_{n,f}:A^{**}\to \ell_{\infty}(A^{**}) \otimes C_0(0,1]$ as an extension of ${U_{n,f}}$, by
$\tilde{U}_{n,f}(a)=g(\lambda _n)(\Delta (a)\otimes \operatorname{id})g(\lambda _n)$ for all $a\in A^{**}$. Using the fact that $\ell_{\infty}(A) \otimes C_0(0,1]$ is \emph{r.w.i.} in $\ell_{\infty}(A^{**}) \otimes C_0(0,1]$, we get the following diagram \[ \xymatrix{
\mathcal{M}(A)\ar@ {^{(}->} [dd] \ar [drr]_{U_{n,f}} \ar [r] & \mathcal{M}(A)\odot C_0(0,1] \ar [dr]^{W_n}\ar [rr]^{\operatorname{id}}&& \mathcal{M}(A)\odot C_0(0,1] \ar [r] ^{\qquad{\operatorname{id}\otimes\operatorname{ev_1}}} &\mathcal{M}(A)\\
&&\ell_{\infty}(A) \otimes C_0(0,1] \ar[ur]^{V_n\otimes \operatorname{id}}&&\\
A^{**} \ar [rr]_{\tilde{U}_{n,f}}&& \ell_{\infty}(A^{**}) \otimes C_0(0,1] \ar [u] &
} \] which commutes locally: let $\varepsilon$ be arbitrary, $F$ a finite-dimensional subspace of $A^{**}$, and $F_0=F\cap {\mathcal M}(A)$. Then we get a net $\Lambda=(F_0, F, \varepsilon)$. Define $\psi_{n,\lambda}: A^{**}\to \mathcal{M}(A)$ locally, as the composition of the following maps \[
F\overset{\tilde{U}_{n,f}}\hookrightarrow A^{**}\to \ell_\infty(A^{**})\otimes C_0(0,1]\to \ell_\infty(A)\otimes C_0(0,1]\overset{V_n\otimes id}\longrightarrow\mathcal{M}(A)\odot C_0(0,1]\overset{\operatorname{id}\otimes\operatorname{ev_1}}\longrightarrow\mathcal{M}(A). \] Then we have \[ \lim_{n, \lambda}\psi_{n,\lambda}(1)=1. \] Let $\psi:=\lim_{n, \lambda}\psi_{n,\lambda}: A^{**}\to \mathcal{M}(A)^{**}$ in the weak $^*$-topology. Then $\psi$ gives the required conditional expectation, and this proves that $\mathcal{M}(A)$ is \emph{r.w.i.} in $A^{**}$. \end{proof}
\section{Module version of the weak expectation property}
The notion of \emph{r.w.i.} is a paired relation between a C$^*$-subalgebra and its parent C$^*$-algebra. If the parent C$^*$-algebra is ${\mathbb B}({\mathcal H})$, the \emph{r.w.i.} property is equivalent to the WEP. By carefully choosing a parent C$^*$-algebra, we can define the notion of WEP relative to a C$^*$-algebra.
Let $\mathcal{C}$ be a collection of inclusions of unital C$^*$-algebras $\{(A\subseteq X)\}$.
For a C$^*$-algebra $D$, there are two classes of objects that we will discuss throughout this paper. \begin{enumerate}
\item $\mathcal{C}_1 = \{ A\subseteq \mathcal{L}(E_D) \}$, where $E_D$ is a Hilbert $D$-module.
\item $\mathcal{C}_2 = \{ A\subseteq \mathcal{L}^w(E{_{D^{**}})}\}$, where $E_{D^{**}}$ is a self dual Hilbert $D^{**}$-module.
\end{enumerate}
\begin{definition}
A C$^*$-algebra $A$ is said to have the $D$WEP$_i$ for $i=1,2$, if there exists a pair of inclusions $A\subseteq X $ in $\mathcal{C}_i$ such that $A$ is relatively weakly injective in $X$. \end{definition}
Notice that the notion of $D$WEP is a \emph{r.w.i.} property. By Corollary ~\ref{wepwri}, the WEP implies the $D$WEP$_i$, for $i=1,2$. Also, inherited from \emph{r.w.i.} property, we have the following lemmas for $D$WEP.
\begin{lemma}
\label{wri wep} Let $A_0$ and $A_1$ be C$^*$-algebras such that $A_0$ is relatively weakly injective in $A_1$. If $A_1$ has the $D$WEP$_i$ for $i=1,2$, then so does $A_0$. \end{lemma}
\begin{proof}
Since $A_1$ has the $D$WEP$_i$, there exists a pair of inclusions $A_1\subseteq X$ in $\mathcal{C}_i$ such that $A_1$ is \emph{r.w.i.} in $X$, for $i=1,2$. By Lemma ~\ref{wri wri}, $A_0$ is \emph{r.w.i.} in $X$. Therefore the result follows. \end{proof}
\begin{remark} \normalfont
By the absorption theorem and Remark ~\ref{universal} and ~\ref{e}, $\mathcal{L}(E_D)$ is \emph{r.w.i.} in some $\mathcal{L}({\mathcal H}_D)$ and $\mathcal{L}^w(E{_{D^{**}})}$ is \emph{r.w.i.} in some ${\mathbb B}({\mathcal H})\bar\otimes D^{**}$. Sometimes it is more convenient to consider the $D$WEP$_1$ as the relatively weak injectivity in $\mathcal{L}({\mathcal H}_D)$, and the $D$WEP$_2$ as the relatively weak injectivity in ${\mathbb B}({\mathcal H})\bar\otimes D^{**}$, because of the concrete structures. \end{remark}
\begin{example}\label{kd} \normalfont
From the above, all WEP algebras have $D$WEP$_i$ for arbitrary C$^*$-algebra $D$. Also, $D$ has the $D$WEP$_i$ trivially for $1$-dimensional Hilbert space ${\mathcal H}$. Our first nontrivial example of $D$WEP$_i$ is $\mathcal{K} \otimes D$. For the first class $\mathcal C_1$, $\mathcal{K} \otimes D$ is a principle ideal of $\mathcal{L}(\mathcal{H}_D)$, and thus is \emph{r.w.i.} in $\mathcal{L}(\mathcal{H}_D)$. For the second class $\mathcal C_2$, note that $(\mathcal{K} \otimes D)^{**} = {\mathbb B}(\mathcal{H}) \bar{\otimes} D^{**}$, so $\mathcal{K} \otimes D$ is \emph{r.w.i.} in ${\mathbb B}(\mathcal{H}) \bar{\otimes} D^{**}$. By universality of $\mathcal{L}(\mathcal{H}_D)$ and ${\mathbb B}(\mathcal{H}) \bar{\otimes} D^{**}$, $\mathcal{K} \otimes D$ has the $D$WEP$_i$ for both $i=1,2$.
\end{example}
Because of the injectivity of ${\mathbb B}({\mathcal H})$, we see that the notion of WEP does not depend on the representation $A\subseteq {\mathbb B}(\mathcal{H})$. By constructing a universal object in the classes $\mathcal C_i$, we can define the $D$WEP$_i$ independent of inclusions.
\begin{lemma} A C$^*$-algebra $A$ has the $D$WEP$_i$ for some inclusion $A\subseteq X $ in $\mathcal{C}_i$, if and only if there exists a universal object $X^u$ and $A\subseteq X^u$ in $\mathcal C_i$, such that \begin{enumerate}
\item $A$ is relatively weakly injective in $X^u$;
\item If $A$ is relatively weakly injective in some $X$, then there exists a ucp map from $X^u$ to $X$, which is identity on $A$. \end{enumerate}
\end{lemma}
\begin{proof}
The ``if'' part is trivial. For the ``only if'' part, take $\mathcal C_1$ for example. The proof of the other case is similar. For all ucp maps $\rho: A\to \mathcal{L}(E_D)$, by KSGNS construction there exists a Hilbert $D$-module $E_\rho$ and a $^*$-homomorphism $\pi_\rho: A \to \mathcal{L}(E_\rho)$. Let $E^u_D = \bigoplus_{\rho} E_\rho$. Then any $\mathcal{L}(E_D)$ containing $A$ can be embedded into $\mathcal{L}(E^u_D)$, and there exists a truncation $\mathcal{L}(E^u_D) \to \mathcal{L}(E_D)$. Now suppose $A$ is \emph{r.w.i.} in some $\mathcal{L}(E_D)$. Then it is also \emph{r.w.i.}
in $\mathcal{L}(E^u_D)$. Hence we complete the proof. \end{proof}
Following Lance's tensor product characterization Theorem ~\ref{tensor wri}, we have a similar result for the $D$WEP$_i$, for $i=1,2$. We only present the result for the first class. The other case can be proved similarly.
Let $A\subseteq\mathcal{L}(E^u_D)$ be the universal representation. We define a tensor norm $max^D_1$ on $A\otimes C^*\mathbb F_{\infty}$ to be the norm induced from the inclusion $A\otimes C^*\mathbb F_{\infty} \subseteq \mathcal{L}(E^u_D)\otimes_{max}C^*\mathbb F_{\infty}$ isometrically. This induced norm is categorical in the sense that if $\phi$ is a ucp map from $A$ to $B$, then $\phi\otimes \operatorname{id}$ extends a ucp map from $A\otimes_{max_1^D} C^*\mathbb F_{\infty}$ to $B\otimes_{max_1^D} C^*\mathbb F_{\infty}$. Indeed, let $\iota$ be the inclusion map from $B$ to its universal representation $\mathcal{L}^B(E^u_D)$, then $\iota \circ \phi$ is a ucp map from $A$ to $\mathcal{L}^B(E^u_D)$. By KSGNS and the construction of $\mathcal{L}^A(E^u_D)$, there exists a ucp map from $\mathcal{L}^A(E^u_D)$ to $\mathcal{L}^B(E^u_D)$ extending the map $\iota \circ \phi$. Hence we have a composition of ucp maps \[
A\underset{max^D_1}\otimes C^*\mathbb F_{\infty} \subseteq \mathcal{L}^A(E^u_D)\underset{max}\otimes C^*\mathbb F_{\infty} \to \mathcal{L}^B(E^u_D)\underset{max}\otimes C^*\mathbb F_{\infty}, \] whose image is $B\otimes_{max_1^D} C^*\mathbb F_{\infty}$.
\begin{theorem} \label{tensor wep}
A C$^*$-algebra $A$ has the the $D$WEP$_1$, if and only if \[
A\underset{max^D_1}\otimes C^*\mathbb F_{\infty} = A\underset{max} \otimes C^*\mathbb F_{\infty}. \] \end{theorem}
\begin{proof}
First, suppose $A$ has the $D$WEP$_1$, then $A$ is \emph{r.w.i.} in $\mathcal{L}(E^u_D)$. That is, there exists a ucp map $\varphi : \mathcal{L}(E^u_D) \to A^{**}$ such that $\varphi |_A = \operatorname{id}_A$.
Then $\varphi \otimes \operatorname{id}$ gives a ucp map from $\mathcal{L}(E^u_D) \otimes_{max} C^*\mathbb{F}_\infty$ to $A^{**}\otimes_{max} C^*\mathbb{F}_\infty$. Therefore the map $\varphi |_A \otimes \operatorname{id}$, defined on the algebraic tensor product, extends to a ucp map from $A\otimes_{max_1^D} C^{*}\mathbb{F}_{\infty}$ to $A^{**}\otimes_{max} C^{*}\mathbb{F}_{\infty}$, whose image is $A\otimes_{max} C^{*}\mathbb{F}_{\infty}$.
To prove the other direction, suppose $A\otimes_{max_1^D} C^{*}\mathbb{F}_{\infty}=A\otimes_{max} C^{*}\mathbb{F}_{\infty}$, and let $A\subseteq {\mathbb B}(\mathcal{H})$be the universal representation, i.e. $A''=A^{**}$. Let $\pi$ be a representation of $C^*\mathbb F_\infty$ to ${\mathbb B}(\mathcal{H})$ with $\pi(C^*\mathbb{F}_\infty)'' = A'$. Then there exists a ucp map $A\otimes_{max_1^D} C^{*}\mathbb{F}_{\infty}=A\otimes_{max} C^{*}\mathbb{F}_{\infty} \to {\mathbb B}(\mathcal{H})$. Since $A\otimes _{max_1^D} C^{*}\mathbb{F}_{\infty}\subseteq \mathcal{L}(E^u_D)\otimes_{max} C^{*}\mathbb{F}_{\infty}$, by Arveson's extension theorem, we can extend the above ucp map to $\Psi: \mathcal{L}(E^u_D)\otimes_{max} C^{*}\mathbb{F}_{\infty}\to {\mathbb B}(\mathcal{H})$. Now define a map $\psi$ on $\mathcal{L}(E^u_D)$ by $\psi (T):=\Psi (T \otimes 1)$, for $T\in \mathcal{L}(E^u_D)$. Then $\psi$ is a ucp extension of $\operatorname{id}_A$. Since $\Psi$ is a $C^*\mathbb{F}_\infty$-bimodule map, we have $\psi(T) \pi (x) = \Psi(T\otimes x) = \pi(x)\psi(T)$ for $T\in \mathcal{L}(E^u_D)$ and $x\in C^*\mathbb{F}_\infty$, i.e., $\psi(T) \in \pi(C^*\mathbb{F}_\infty)' = A^{**}$. This completes the proof. \end{proof}
It is natural to explore the relationship between $D$WEP$_1$ and $D$WEP$_2$. We have the following.
\begin{theorem}\label{12} If a C$^*$-algebra $A$ has the $D$WEP$_1$, then it also has the $D$WEP$_2$. \end{theorem}
In fact, the converse of the above theorem is not true, and we will give a counterexample in Section 5.
To prove the above theorem, we need following lemmas.
\begin{lemma}
\label{ucp} Suppose that the identity map on a C$^*$-algebra $A$ factors through a C$^*$-algebra $B$ approximately via ucp maps in point-norm topology, i.e. there exist
two nets of ucp maps $\phi_i: A \to B $ and $\psi_i: B \to A$, such that $\|\psi_i \circ \phi_i (x) - x \| \to 0$ for $x\in A$. If $B$ has the $D$WEP$_i$, then so does $A$. \end{lemma}
\begin{proof} Following Kirchberg's method, it suffices to show $A\otimes_{max_i^D} C^{*}\mathbb{F}_{\infty} = A\otimes_{max} C^{*}\mathbb{F}_{\infty}$. Since we have ucp maps
$\phi_i: A \to B $ and $\psi_i: B \to A$, such that $\|\psi_i \circ \phi_i (x) - x \| \to 0$ for $x\in A$, we have ucp maps $\phi_i \otimes \operatorname{id} : A\otimes_{max_i^D} C^{*}\mathbb{F}_{\infty} \to B \otimes_{max_i^D} C^{*}\mathbb{F}_{\infty}$ and ucp maps $\psi_i \otimes \operatorname{id} : B\otimes_{max} C^{*}\mathbb{F}_{\infty} \to A \otimes_{max} C^{*}\mathbb{F}_{\infty}$. Since $B$ has the $D$WEP$_i$, by Theorem ~\ref{tensor wep}, we have $B\otimes_{max_i^D} C^{*}\mathbb{F}_{\infty} = B\otimes_{max} C^{*}\mathbb{F}_{\infty}$. Therefore we have ucp maps $A\otimes_{max_i^D} C^{*}\mathbb{F}_{\infty} \to A\otimes_{max} C^{*}\mathbb{F}_{\infty}$ defined by the composition of the maps according to the following diagram \[ A\underset{max^D_i}\otimes C^{*}\mathbb{F}_{\infty}\overset{\phi_i\otimes \operatorname{id}}\longrightarrow B \underset{max^D_i}\otimes C^{*}\mathbb{F}_{\infty}=B \underset{max}\otimes C^{*}\mathbb{F}_{\infty}\overset{\psi_i\otimes \operatorname{id}}\longrightarrow A\underset{max}\otimes C^{*}\mathbb{F}_{\infty}. \] This net of maps converges to the identity. Hence we get the result. \end{proof}
Another lemma we need is that the $D$WEP$_i$ property is preserved under the direct product.
\begin{lemma}
\label{prod wep} If $(A_i)_{i \in I}$ is a net of $C^*$-algebras with the $D$WEP$_i$, then $\prod _{i\in I} A_i$ has the $D$WEP$_i$. \end{lemma}
\begin{proof} We will prove the result for the $D$WEP$_1$. The proof of the other case is similar.
Since each $A_i$ has the $D$WEP$_1$, there exists an inclusion $A_i \subseteq \mathcal{L}((E_i)_D) $ such that $A_i$ is \emph{r.w.i.} in $\mathcal{L}((E_i)_D)$. By Lemma ~\ref{prod wri}, $\prod _{i\in I} A_i$ is \emph{r.w.i.} in $\prod_{i\in I} \mathcal{L}((E_i)_D)$. Since $\mathcal{L}(\oplus_{i \in I} (E_i)_D)$ contains $\prod_{i\in I} \mathcal{L}((E_i)_D)$ and it has a conditional expectation onto $\prod \mathcal{L}((E_i)_D)$, $\prod _{i\in I} A_i$ is also \emph{r.w.i.} in $\mathcal{L}(\oplus_{i\in I} (E_i)_D)$. Therefore $\prod _{i\in I} A_i$ has the $D$WEP$_i$. \end{proof}
Kirchberg\cite{Ki1} shows that for a C$^*$-algebra $A$, the multiplier algebra ${\mathcal M}(A)$ factors through $\ell_{\infty}(A)$ approximately by ucp maps (Theorem ~\ref{factor1}). Using this fact, we have the following.
\begin{corollary}\label{md}
Suppose that the C$^*$-algebra $A$ has the $D$WEP$_i$, for $i = 1, 2$. Then the multiplier algebra ${\mathcal M}(A)$ also has the $D$WEP$_i$, for $i = 1, 2$. \end{corollary}
\begin{proof}
By Theorem ~\ref{factor1}, ${\mathcal M}(A)$ factors through $\ell_\infty(A)$ approximately via ucp maps in point-norm topology. Since $A$ has the $D$WEP$_i$, $\ell_\infty(A)$ has $D$WEP$_i$ by Lemma ~\ref{prod wep}. Therefore by Lemma ~\ref{ucp}, ${\mathcal M}(A)$ also has the $D$WEP$_i$. \end{proof}
Now we are ready to see the proof of the theorem.
\begin{proof} [Proof of Theorem ~\ref{12}] It suffices to show that $\mathcal{L}(\mathcal{H}_D)$ has the $D$WEP$_2$. Notice that $\mathcal{L}(\mathcal{H}_D) = {\mathcal M}(\mathcal{K}\otimes D)$, and also ${\mathcal M}(\mathcal{K}\otimes D)$ factors through $\ell_\infty(\mathcal{K}\otimes D)$ approximately via ucp maps in point-norm topology. Since $\mathcal{K}\otimes D$ has the $D$WEP$_2$ by Remark ~\ref{kd}, and hence $\ell_\infty(\mathcal{K}\otimes D)$ by Lemma ~\ref{prod wep}. By Corollary ~\ref{md}, ${\mathcal M}(\mathcal{K}\otimes D)$ has the $D$WEP$_2$. \end{proof}
\begin{remark} \label{vn wep} \normalfont Note that $D^{**}$WEP$_1$ implies $D$WEP$_2$. Indeed having $D^{**}$WEP$_1$ is equivalent to being \emph{r.w.i.} in ${\mathcal L}({\mathcal H}_{D^{**}})={\mathcal M}({\mathcal K}\otimes D^{**})$, and having $D$WEP$_2$ is equivalent to being \emph{r.w.i.} in ${\mathbb B}({\mathcal H})\bar\otimes D^{**}$. Note that ${\mathcal K}\otimes D^{**}$ is \emph{r.w.i.} in ${\mathbb B}({\mathcal H})\bar\otimes D^{**}$. By Corollary ~\ref{md}, we have ${\mathcal M}({\mathcal K}\otimes D^{**})$ has the $D$WEP$_2$ as well. \end{remark}
Now we investigate some properties of module WEP. The first result is that the module WEP is stable under tensoring with a nuclear C$^*$-algebra, similar to the classical case.
\begin{proposition} \label{min wep}
For a C$^*$-algebra $D$, the following properties hold: \begin{enumerate}
\item If a $C^*$-algebra $A$ has the $D$WEP$_1$, and $B$ is a nuclear C$^*$-algebra, then $A\otimes_{min}B$ has the $D$WEP$_1$ as well.
\item If von Neumann algebras $M$ and $N$ have the $C$WEP$_2$ and $D$WEP$_2$ respectively, then $M \bar{\otimes} N$ has the $(C\otimes_{min}D)$WEP$_2$.
\end{enumerate} \end{proposition}
\begin{proof}
(1) Since $A$ has the $D$WEP$_1$ and $B$ is a nuclear, we have $A$ is \emph{r.w.i.} in $\mathcal{L}(\mathcal{H}_D)$, and $B$ is \emph{r.w.i.} in ${\mathbb B}({\mathcal H})$. Therefore we have ucp maps $A\otimes_{min}B \to \mathcal{L}({\mathcal H}_D) \otimes_{min} {\mathbb B}({\mathcal H}) =\mathcal{L}(\tilde{{\mathcal H}}_D) \to A^{**}\otimes_{min}B^{**}$. Note that $B$ is nuclear and hence exact, so the inclusion map $A^{**} \otimes B^{**} \hookrightarrow (A\otimes_{min}B)^{**}$ is min-continuous. Therefore $A\otimes_{min} B$ is \emph{r.w.i.} in $\mathcal{L}(\tilde{{\mathcal H}}_D)$.
(2) Since $M$ is \emph{r.w.i.} in $\mathcal{L}^w({\mathcal H}_{C^{**}})$ and $N$ is \emph{r.w.i.} in $\mathcal{L}^w({\mathcal H}_{D^{**}})$, we have ucp maps $M\bar{\otimes}N \to \mathcal{L}^w({\mathcal H}_{C^{**}}) \bar{\otimes} \mathcal{L}^w({\mathcal H}_{D^{**}}) = \mathcal{L}^w({\mathcal H}_{C^{**}\bar{\otimes}D^{**}})\to M\bar{\otimes}N \to (M\bar{\otimes}N)^{**}$. Note that $C\otimes_{min} D$ is weak $^*$-dense in $C^{**}\bar{\otimes}D^{**}$. Therefore we have a normal conditional expectation $(C\otimes_{min}D)^{**} \to C^{**}\bar{\otimes}D^{**}$, and hence $C^{**}\bar{\otimes}D^{**}$ is \emph{r.w.i.} in $(C\otimes_{min}D)^{**}$. Therefore $\mathcal{L}^w({\mathcal H}_{C^{**}\bar{\otimes}D^{**}})$ is \emph{r.w.i.} in $\mathcal{L}^w({\mathcal H}_{(C\otimes_{min}D)^{**}})$, and hence $M\bar{\otimes}N$ is \emph{r.w.i.} in $\mathcal{L}^w({\mathcal H}_{(C\otimes_{min}D)^{**}})$. \end{proof}
As a consequence of Corollary ~\ref{md}, we have the transitivity property of $D$WEP.
\begin{proposition}\label{tran}
If $A$ has the $B$WEP$_i$, and $B$ has the $C$WEP$_i$, then $A$ has the $C$WEP$_i$, for $i=1,2$. \end{proposition}
\begin{proof} Since $B$ has the $C$WEP$_i$, then so does $\mathcal{K}\otimes B$, and hence so does ${\mathcal M}(\mathcal{K}\otimes B)$ by Corollary ~\ref{md}. Since $A$ has the $B$WEP$_i$, $A$ is \emph{r.w.i.} in some $\mathcal{L}(\mathcal{H}_B) = {\mathcal M}(\mathcal{K}\otimes B)$. By the transitivity of \emph{r.w.i.}, we conclude that $A$ has the $C$WEP$_i$, for $i=1,2$. \end{proof}
\begin{corollary}\label{trand}
If $A$ has the $D$WEP$_1$, and $D$ has the WEP, then $A$ has the WEP. \end{corollary}
\begin{proof}
It suffices to show that ${\mathcal L}({\mathcal H}_D)={\mathcal M}({\mathcal K}\otimes D)$ has the WEP. This is obvious since if $D$ has the WEP, then so does ${\mathcal K}\otimes D$ and hence ${\mathcal M}({\mathcal K}\otimes D)$ has the WEP. \end{proof}
\begin{remark}
\normalfont The previous result is not necessarily true for the WEP$_2$ case, since ${\mathbb B}(\ell_2)\bar \otimes D^{**}$ may not have the WEP, for instance for $D={\mathbb B}(\ell_2)$. See Example ~\ref{Bl2} for the proof. \end{remark}
In \cite{Ju96}, Junge shows the following finite dimensional characterization of the WEP.
\begin{theorem}
The C$^*$-algebra $A$ has the WEP if and only if for arbitrary finite dimensional subspaces $F\subset A$ and $G\subset A^*$, and $\varepsilon > 0$, there exist matrix algebra $M_m$ and ucp maps $u: F\to M_m$, $v: M_m \to A/G^{\perp}$, such that \[
\|v\circ u-q_G \circ \iota_F\|<\varepsilon, \] where $\iota_F: F\to A$ is the inclusion map and $q_G: A \to A/G^{\perp}$ is the quotient map. \end{theorem}
We have a similar result for the module WEP as follows.
\begin{theorem} The C$^*$-algebra $A$ has the $D$WEP$_1$ if and only if for arbitrary finite dimensional subspaces $F\subset A$ and $G\subset A^*$, and $\varepsilon > 0$, there exist matrix algebra $M_m(D)$ and ucp maps $u: A\to M_m(D)$, $v:M_m(D) \to A/G^{\perp}$, such that \[
\|v\circ u|_F-q_G \circ \iota_F\|<\varepsilon, \] where $\iota_F: F\to A$ is the inclusion map and $q_G: A \to A/G^{\perp}$ is the quotient map. \end{theorem}
For the $D$WEP$_2$ case, we will replace the matrix algebra $M_m(D)$ by $M_m(D^{**})$.
\begin{proof}
$\Leftarrow$: From the assumption, we get a net of maps $u$ and $v$ over $(F, G, \varepsilon)$. Taking the direct product of all such $u$, and one w$^*$-limit of $v$, we have ucp maps $A \to \Pi_{(F,G,\varepsilon)} M_m(D) \to A^{**}$, whose composition is identity on $A$, and hence $A$ is \emph{r.w.i.} in $\Pi_{(F,G,\varepsilon)} M_m(D)$. By Lemma ~\ref{prod wep}, $\Pi_{(F,G,\varepsilon)} M_m(D)$ has the $D$WEP$_1$ since $M_m(D)$ does. Therefore $A$ has the $D$WEP$_1$.
$\Rightarrow$: $A$ has the $D$WEP$_1$, and hence we have $A \to \mathcal{L}({\mathcal H}_D) \to A^{**}$. Let $\sigma_I$ be the composition of the inclusion maps \[ A\hookrightarrow \mathcal{L}({\mathcal H}_{D})\hookrightarrow\Pi_I (M_{m(i)}(D))\hookrightarrow\Pi_{I}(M_{m(i)}(D^{**})). \] Note that each of the inclusions above is \emph{r.w.i.}. By taking the duals, we have \[
\varphi_I: A^* \overset{r.w.i.}\hookrightarrow \mathcal{L}({\mathcal H}_{D})^* \overset{r.w.i.}\hookrightarrow \Pi_I (M_{m(i)}(D))^* \overset{r.w.i.}\hookrightarrow \Pi_I (M_{m(i)}(D^{**}))^* = \ell_1^I(\mathcal{S}_1^{m(i)}(D^*))^{**}. \] By the local reflexivity principle, for arbitrary $F$, $G$ and $\varepsilon$ as in the theorem, there exists a map $\alpha^{\varepsilon}_I : G \to \ell_1(I, \mathcal{S}_1(D^*))$, such that \[
|\langle \alpha^{\varepsilon}_I(g), \sigma_I(f)\rangle - \langle \varphi_I(g), \sigma_I(f)\rangle| < \varepsilon \|f\|\|g\|, \] for $f\in F$ and $g\in G$. By carefully choosing an Auerbach basis for the finite dimensional spaces, we can have the above relation on a finite subset $I_0 \subset I$, i.e. \[
|\langle \alpha^{\varepsilon}_{I_0}(g), \sigma_{I_0}(f)\rangle - \langle \varphi_{I_0}(g), \sigma_{I_0}(f)\rangle| < \varepsilon \|f\|\|g\|. \] By the \emph{r.w.i.} property of $\sigma_I$ we have $\langle \varphi_I(g), \sigma_I(f)\rangle = \langle g, f\rangle$.
Therefore for $f$ and $g$ with norm $1$, we have $|\langle \alpha^{\varepsilon}_{I_0}(g), \sigma_{I_0}(f)\rangle-\langle g, f\rangle|<\varepsilon$, and hence $|\langle g, {\alpha^{\varepsilon*}_{I_0}} \circ \sigma_{I_0}(f) - f\rangle |< \varepsilon$. Let $u = \sigma_{I_0}$ and $v = {\alpha^{\varepsilon*}_{I_0}}$. Then we have the desired result. \end{proof}
\section{Module version of QWEP}
\begin{definition} A $C^*$-algebra $B$ is said to be $D$QWEP$_i$ if it is the quotient of a $C^*$-algebra $A$ with $D$WEP$_i$ for $i=1,2$. \end{definition}
Similar to the $D$WEP$_i$, we have a tensor characterization for $D$QWEP$_i$ for $i=1,2$ as follows. First we need the following result due to Kirchberg.
\begin{lemma}[\cite{Ki1} Corollary 3.2 (v)] \label{ki md} If $\phi:A\to B^{**}$ is a ucp map such that $\phi$ maps the multiplicative domain $\operatorname{md}(\phi)$ of $\phi$ onto a C$^*$-subalgerba $C$ of $B^{**}$ containing $B$ as a subalgebra, then the C$^*$-algebra $\operatorname{md}(\phi)\cap \phi^{-1}(B)$ is relatively weakly injective in $A$. \end{lemma}
We only prove the tensor characterization for $D$QWEP$_1$. The proof of the other case is similar.
\begin{theorem} \label{tensor qwep}
Let $C^*\F_\infty\subset {\mathcal L}({\mathcal H}^u_D)$ be the universal representation. The following statements are equivalent: \begin{enumerate}
\item[(i)] A $C^*$-algebra $B$ is $D$QWEP$_1$;
\item[(ii)] For any ucp map $u: C^*\mathbb F_{\infty} \to B$, the map $u \otimes \operatorname{id}$ extends to a continuous map from $C^*\mathbb F_{\infty}\otimes_{max_1^D}C^*\mathbb F_{\infty}$ to $B \otimes_{max}C^*\mathbb F_{\infty}$, where $max_1^D$ is the induced norm from the inclusion $C^*\mathbb F_{\infty}\otimes C^*\mathbb F_{\infty}\subseteq\mathcal{L(H}^u_D)\otimes_{max}C^*\mathbb F_{\infty}$.
\end{enumerate} \end{theorem}
\begin{proof}
(i)$\Rightarrow$(ii): Suppose $B$ is $D$QWEP$_1$. Then $B=A/J$ for some $C^*$-algebra $A$ with $D$WEP$_1$. Let $u: C^*\mathbb F_{\infty} \to B$ be a ucp map, and $\pi:A\to B$ be the quotient map. Since $C^*\mathbb F_{\infty}$ has the lifting property, there exists a ucp map $\varphi: C^*\mathbb F_{\infty} \to A$ which lifts $u$, i.e. the following diagram commutes \[ \xymatrix{
C^*\mathbb F_{\infty}\ar [r]^{u} \ar[d]_{\varphi} & B\\
A\ar [ur]_{\pi} &
} \]
By Theorem ~\ref{tensor wep}, we have $A\otimes_{max_1^D} C^*\mathbb F_{\infty} = A\otimes_{max} C^*\mathbb F_{\infty}$. Therefore, we have the following continuous maps
\[
C^*\mathbb F_{\infty} \underset{max^D_1}\otimes C^*\mathbb F_{\infty} \overset{\varphi\otimes \operatorname{id}}\longrightarrow A \underset{max^D_1}\otimes C^*\mathbb F_{\infty} = A \underset{max}\otimes C^*\mathbb F_{\infty}\overset{\pi\otimes \operatorname{id}}\longrightarrow B \underset{max}\otimes C^*\mathbb F_{\infty}. \]
Note that $(\pi\otimes \operatorname{id})\circ(\varphi\otimes \operatorname{id})|_{C^*\mathbb F_{\infty}\otimes 1_{C^*\mathbb F_{\infty}}}=u$ by the lifting property. Therefore, $u\otimes \operatorname{id}$ extends to a continuous map from $C^*\mathbb F_{\infty}\otimes_{max_1^D}C^*\mathbb F_{\infty}$ to $B \otimes_{max}C^*\mathbb F_{\infty}$.
(ii)$\Rightarrow$(i): Let $u: C^*\mathbb F_{\infty} \to B$ be the quotient map. We have the following diagram
\[ \xymatrix{
C^*\mathbb F_{\infty} \underset{max_1^D}\otimes C^*\mathbb F_{\infty} \ar [r]^{\qquad{u\otimes id}} \ar@ {^{(}->}[d]&B\underset{max}\otimes C^*\mathbb F_{\infty} \ar [r] & \mathbb{B}({\mathcal H})\\
\mathcal{L(H}^u_D)\underset{max}\otimes C^*\mathbb F_{\infty} \ar [urr], &
} \] where ${\mathbb B}({\mathcal H})$ is the universal representation of $B$. By Arveson's extension theorem, there exists a ucp map $\Phi:\mathcal{L(H}^u_D)\otimes_{max} C^*\mathbb F_{\infty}\to\mathbb{B}({\mathcal H})$. Using \emph{The Trick} (see proof of Theorem ~\ref{tensor wep}), we get a map $\phi: \mathcal{L(H}^u_D)\to B^{**}$. Let $\operatorname{md}(\phi)$ be the multiplicative domain of $\phi$. Note that $C^*\mathbb F_{\infty}\subset \operatorname{md}(\phi)$. Therefore, $\phi$ maps $\operatorname{md}(\phi)$ onto a $C^*$-subalgebra of $B^{**}$ containing $B$. Let $A=\operatorname{md}(\phi)\cap \phi^{-1}(B)$. Then by Corollary ~\ref{ki md}, $A$ is \emph{r.w.i.} in $\mathcal{L(H}^u_D)$, so $A$ has the $D$WEP$_1$. Hence $B$ as a quotient of $A$ is $D$QWEP$_1$. \end{proof}
\begin{remark} \normalfont In the proof of the above Theorem, we showed that the second statement is equivalent to the statement that for any ucp map $u: C^*\mathbb F_{\infty} \to B$, $w:C^*\mathbb F_{\infty} \to B^{\operatorname{op}}$, the map
$u \otimes w$ extends to a continuous map from $C^*\mathbb F_{\infty}\otimes_{max_1^D}C^*\mathbb F_{\infty}$ to $B \otimes_{max} B^{\operatorname{op}}$. \end{remark}
Now let us investigate some basic properties of the $D$QWEP. We have the following proposition similar to the $D$WEP case. \begin{proposition} \label{min qwep} The following hold:
\begin{enumerate}\item If a $C^*$-algebra $B$ is $D$QWEP$_1$ and $C$ is nuclear, then $C\otimes_{min}B$ is also $D$QWEP$_1$.
\item If von Neumann algebras $M$ and $N$ are $C$QWEP$_2$ and $D$QWEP$_2$, respectively, then $M \bar{\otimes} N$ is $(C\otimes_{min}D)$QWEP$_2$.
\end{enumerate} \end{proposition}
\begin{proof} (1) Suppose $B$ is $D$QWEP$_1$, then $B=A/J$ for some C$^*$-algebra $A$ with the $D$WEP$_1$. Since $C$ is nuclear, it is also exact. Therefore, we have \[ C\underset{min}\otimes B=C\underset{min}\otimes (A/J)\cong \frac{C\otimes_{min} A}{{C\otimes_{min} J}}. \] But $C\otimes_{min} A$ has the $D$WEP$_1$ by Proposition ~\ref{min wep}(1). Therefore, $C\otimes_{min}B$ is $D$QWEP$_1$.
(2) Since $M$ is $C$QWEP$_2$, it is \emph{r.w.i.} in $\mathcal{L}^w({\mathcal H}_{C^{**}})^{**}$. Similarly, $N$ is \emph{r.w.i.} in $\mathcal{L}^w({\mathcal H}_{D^{**}})^{**}$. Therefore, we have ucp maps \[ M\bar{\otimes}N \overset{r.w.i.}\hookrightarrow \mathcal{L}^w({\mathcal H}_{C^{**}})^{**} \bar{\otimes} \mathcal{L}^w({\mathcal H}_{D^{**}})^{**} \overset{r.w.i.}\hookrightarrow \mathcal{L}^w({\mathcal H}_{C^{**}\bar{\otimes}D^{**}})^{**}. \]\ Note that by the same argument as in the proof of Proposition ~\ref{min wep} (2), $\mathcal{L}^w({\mathcal H}_{C^{**}\bar\otimes D^{**}})^{**}$ is \emph{r.w.i.} in $\mathcal{L}^w({\mathcal H}_{(C\otimes_{min} D)^{**}})^{**}$. Hence $M \bar{\otimes} N$ is \emph{r.w.i.} in $\mathcal{L}^w({\mathcal H}_{(C\otimes_{min} D)^{**}})^{**}$. Therefore, $M \bar{\otimes} N$ is $(C\otimes_{min}D)$QWEP$_2$. \end{proof}
By Theorem ~\ref{12}, $D$WEP$_1$ implies $D$WEP$_2$, and hence $D$QWEP$_1$ implies $D$QWEP$_2$. In Section 5 we will show that there exist $C^*$-algebras with $D$WEP$_2$ which do not have $D$WEP$_1$. However in the QWEP context, the two concepts coincide. To see this, we need the following lemmas in which we use Kirchberg's categorical method.
\begin{remark} \label{wepqwep} \normalfont If a C$^*$-algebra $A$ has the $D$WEP$_2$, then it is $D^{**}$QWEP$_1$. Indeed since $A$ has the $D$WEP$_2$, it is \emph{r.w.i.} in ${\mathbb B}(\ell_2)\bar\otimes D^{**}=(\mathcal{K}\otimes D)^{**}$. Now since $D$ is $D^{**}$QWEP$_1$, so is $\mathcal{K}\otimes D$ and therefore, so is $(\mathcal{K}\otimes D)^{**}$. Hence $A$ is $D^{**}$QWEP$_1$. \end{remark}
The next lemma shows that $D$QWEP$_i$, for $i=1,2$, is stable under the direct products.
\begin{lemma} \label{prod qwep} Suppose $(B_i)_{i\in I}$ is a net of $C^*$-algebras in ${\mathbb B}(\mathcal{H})$. If $B_i$ is $D$QWEP$_i$, for all $i\in I$, then so is $\Pi_{i\in I} B_i$. \end{lemma}
\begin{proof} Since $B_i$ is $D$QWEP$_i$, it is a quotient of a $C^*$-algebra $A_i$ with $D$WEP$_i$. By Lemma ~\ref{prod wep}, $\Pi_{i\in I} A_i$ has the $D$WEP$_i$. Therefore, $\Pi_{i\in I} B_i$ is $D$QWEP$_i$. \end{proof}
\begin{lemma} \label{wri qwep} Let $B$ be a $D$QWEP$_i$ C$^*$-algebra, for $i=1,2$, and $B_0$ a C$^*$-subalgebra of $B$ which is relatively weakly injective in $B$. Then $B_0$ is also a $D$QWEP$_i$ C$^*$-algebra. \end{lemma}
\begin{proof} If $B$ is $D$QWEP$_i$, then it is a quotient of a $C^*$-algebra $A$ with $D$WEP$_i$. Let $\pi:A\to B$ be the quotient map, $B=A/J$ and $A_0=\pi^{-1}(B_0)$. Then $A_0$ is \emph{r.w.i.} in $A$. In fact this follows from the fact that \[
A_0^{**}=J^{**}\oplus B_0^{**}\subset J^{**}\oplus B^{**}=A^{**}. \] Now by Lemma ~\ref{wri wep}, $A_0=\pi^{-1}(B_0)$ has the $D$WEP$_i$. Hence $B_0$ is $D$QWEP$_i$. \end{proof}
\begin{lemma} \label{uball} Let $A$ and $B$ be unital C$^*$-algebras. Suppose there exists a map $\psi :A\to B$ which maps the closed unit ball of $A$ onto the closed unit ball of $B$. If $A$ has the $D$WEP$_i$, then $B$ is $D$QWEP$_i$, for $i=1,2$. \end{lemma}
\begin{proof}
Let $A_0 \subset A$ be the multiplicative domain of $\psi$. Since $\psi$ maps the closed unit ball of $A$ onto that of $B$, the restriction of $\psi$ on $A_0$ is a surjective $*$-homomorphism onto $B$. Let $\pi=\psi|_{A_0}$.
By Lemma ~\ref{ki md}, we have $A_0$ is \emph{r.w.i.} in $A$ and hence it has the $D$WEP$_i$ by Lemma ~\ref{wri wep}. Since $B$ is a quotient of $A_0$, $B$ is $D$-QWEP$_i$. \end{proof}
\begin{corollary} \label{quball} Let $B$ and $C$ be $C^*$-algebras. Suppose $B$ is $D$QWEP$_i$, and $\psi: B\to C$ is a ucp map that maps the closed unit ball of $B$ onto that of $C$. Then $C$ is $D$QWEP$_i$. \end{corollary}
\begin{proof}
Since $B$ is $D$QWEP$_i$, there exists a $C^*$-algebra $A$ with the $D$WEP$_i$, and a surjective $^*$-homomorphism $\pi: A \to B$. Notice that $\pi$ maps closed unit ball of $A$ onto that of $B$. Hence the composition $\psi \circ \pi$ maps the closed unit ball of $A$ onto that of $C$. By Lemma ~\ref{uball}, $C$ is $D$QWEP$_i$. \end{proof}
\begin{lemma} \label{dc} Suppose $(B_i)_{i\in I}$ is an increasing net of $C^*$-algebras in ${\mathbb B}(\mathcal{H})$. If all $B_i$ are $D$QWEP$_i$, then $\overline{\cup B _i}$ and $(\cup B_i)''$ are $D$QWEP$_i$. \end{lemma}
\begin{proof} Let $B=\cup B_i$. It suffices to show that $B''$ is $D$QWEP$_i$. Since $B_i$ is $D$QWEP$_i$, there exists a $C^*$-algebra $A_i$ with $D$WEP$_i$, and a surjective $^*$-homomorphism $\pi_i: A_i\to B_i$. Let $J$ be a directed set containing $I$. By Lemma \ref{prod wep}, $\prod_{j\in J} A_j$ has the $D$WEP$_i$. Fix a free ultrafilter $\mathcal{U}$ on the net $J$. Define a ucp map $\varphi: \prod_{j\in J} A_j\to B''$ by $\varphi((x_j)_{j\in J}) = \lim_{j \to \mathcal{U}} \pi(x_j)$ in the ultraweak topology. By Kaplansky's density theorem, if $J$ is large enough, then $\varphi$ maps the closed unit ball of $\prod_{j\in J} A_j$ onto that of $B''$. Now by Lemma \ref{uball}, $B''$ is $D$QWEP$_i$. \end{proof}
The next corollary shows that unlike the $D$WEP case, the $D$QWEP of a C$^*$-algebra and its double dual are equivalent. \begin{corollary} \label{ds} A $C^*$-algebra $B$ is $D$QWEP$_i$ if and only if $B^{**}$ is $D$QWEP$_i$ for $i=1,2$. \end{corollary}
\begin{proof} The ``if'' direction follows directly from Lemma ~\ref{wri qwep} since $B$ is \emph{r.w.i.} in $B^{**}$. For the other direction, we can apply Lemma ~\ref{dc} to $B$ together with its universal representation. \end{proof}
\begin{lemma} \label{factor qwep} Suppose $B$ and $C$ are $C^*$-algebras, and $B$ factors through $C$ approximately via ucp maps in the point-weak$^*$ topology. If $C$ is $D$QWEP$_i$, then so is $B$. \end{lemma}
\begin{proof} Since $B$ factors through $C$, there are families of ucp maps $\alpha_i: B\to C$ and $\beta_i: C\to B$, $i\in I$ such that $\beta_i\circ \alpha_i$ converges to the identity map on $B$ in the point-weak$^*$ topology, i.e. \[ \lim_{x,\mathcal{U}} (\beta_i \circ \alpha_i)(x)(x^*)= x^*(x) \] for $x\in B$, $x^*\in B^*$ and an ultrafilter $\mathcal{U}$. Define $\alpha: B\to \prod_{i\in I} C$ by $\alpha(x)=(\alpha_i(x))_{i\in I}$, for $x\in B$. Let $\beta: \prod_{i\in I}C\to B^{**}$, $\beta=\lim_{i\to \mathcal{U}}\beta_i$. Define $\beta^{\#}:B^*\to \prod _{\mathcal{U}}C^*$,
by $\beta^{\#}(x^*)=(\beta^*(x^*))$. In fact $\beta^{\#}=\beta^{*}|_{B^*}$. Now the following map gives the identity on $B$:
\[ \xymatrix{ B\ar [r]^{\alpha} &\prod C\ar [r]^{(\beta^{\#})^*}& B^{**}. } \] Taking the duals, we have \[ \xymatrix{ B^*\ar [r]^{\beta^{\#}} &(\prod_{\mathcal{U}} C)^*\ar [r]^{\alpha^*}& B^{*}. } \]
This gives a conditional expectation from $C^{**}$ to $B^{**}$ which is identity on $B$. Therefore, $B$ is \emph{r.w.i.} in $C^{**}$. By Corollary ~\ref{ds}, $C^{**}$ is $D$QWEP$_i$. Hence so is $B$. \end{proof}
\begin{corollary} If a $C^*$-algebra $B$ is $D$QWEP$_i$, for $i=1,2$, then so is $\mathcal{M}(B)$. \end{corollary}
\begin{proof} Note that by Theorem ~\ref{factor1}, the identity map on ${\mathcal M}(B)$ factors through $\ell_{\infty}(B)$ approximately via ucp maps in point weak $^*$-topology. Since $B$ is $D$QWEP$_i$, by Lemma ~\ref{prod qwep}, so is $\ell_{\infty}(B)$. Therefore, by Lemma ~\ref{factor qwep}, ${\mathcal M}(B)$ is $D$QWEP$_i$. \end{proof}
We have the following transitivity result for $D$QWEP$_i$. We only show the $D$QWEP$_1$ case. The proof of the other case is similar. First we need the following lemma.
\begin{lemma} \label{module} Let $D$ be a $C^*$-algebra. If $D$ is $C$QWEP$_i$ for $i=1,2$, then so are $\mathcal{L(H}_D)$ and $\mathcal{L}^w({\mathcal H}_{D^{**}})$. \end{lemma}
\begin{proof}
If $D$ is $C$QWEP$_1$, then by Proposition ~\ref{min qwep}(1), so is $\mathcal{K}\otimes D$. By Theorem ~\ref{factor1}, $\mathcal{L(H}_D)=\mathcal{M}(\mathcal{K}\otimes D)$ factors through $\ell_{\infty}(\mathcal{K}\otimes D)$, and therefore, it is $D$QWEP$_1$, by Lemma ~\ref{factor qwep}. Hence it is also $D$QWEP$_2$.
For the other case, it suffices to show that ${\mathbb B}({\mathcal H})\bar\otimes D^{**}$ is $D$QWEP$_1$. Note that ${\mathbb B}({\mathcal H})\bar\otimes D^{**}=({\mathcal K}\otimes D)^{**}$ and ${\mathcal K}\otimes D$ is $D$QWEP$_1$. By Corollary ~\ref{ds}, $({\mathcal K}\otimes D)^{**}$ is $D$QWEP$_1$, and hence it is $D$QWEP$_2$. \end{proof}
The following result shows the transitivity of the $D$QWEP$_i$ for $i=1,2$.
\begin{corollary} Let $B$, $C$ and $D$ be $C^*$-algebras such that $B$ is $D$QWEP$_i$, and $D$ is $C$QWEP$_i$. Then $B$ is $C$QWEP$_i$. \end{corollary}
\begin{proof} We only show this for $i=1$. Let $C^*\F_\infty\subset {\mathcal L}({\mathcal H}^u_D)$ be the universal representation. Since $B$ is $D$QWEP$_1$, by Theorem ~\ref{tensor qwep}, for all ucp maps $u: C^*\mathbb F_{\infty}\to B$, the map $u\otimes \operatorname{id}: C^*\mathbb F_{\infty}\otimes_{max_1^D} C^*\mathbb F_{\infty}\to B\otimes_{max}C^*\mathbb F_{\infty}$ is continuous, where $max_1^D$ is the norm induced from the inclusion $C^*\mathbb F_{\infty}\otimes C^*\mathbb F_{\infty}\subset \mathcal{L(H}^u_D)\otimes_{max} C^*\mathbb F_{\infty}$. Since $D$ is $C$QWEP$_1$, so is $\mathcal{L(H}^u_D)$ by Lemma ~\ref{module}. Now by the tensor characterization of $C$QWEP$_1$, the map $w\otimes id: C^*\mathbb F_{\infty}\otimes_{max_1^C}C^*\mathbb F_{\infty}\to \mathcal{L(H}^u_D)\otimes_{max}C^*\mathbb F_{\infty}$ is continuous for all ucp maps $w:C^*\mathbb F_{\infty}\to\mathcal{L(H}^u_D)$. Now let $w$ be a faithful representation $C^*\mathbb F_{\infty}\to\mathcal{L(H}^u_D)$. Then we have following diagram
\[ \xymatrix{
C^*\mathbb F_{\infty} \underset{max_1^D}\otimes C^*\mathbb F_{\infty} \ar@ {^{(}->}[r] \ar [d]_{u\otimes \operatorname{id}}& \mathcal{L(H}^u_D) \underset{max}\otimes C^*\mathbb F_{\infty}\\
B \underset{max}\otimes C^*\mathbb F_{\infty} & C^*\mathbb F_{\infty} \underset{max_1^C}\otimes C^*\mathbb F_{\infty}\ar[u]_{w\otimes \operatorname{id}} \ar@ {^{-}->}[ul]
} \] Note that the image of $w\otimes \operatorname{id}$ is $C^*\mathbb F_{\infty} \otimes_{max_1^D} C^*\mathbb F_{\infty}$. Therefore, we get a continuous map from $C^*\mathbb F_{\infty} \otimes_{max_1^C} C^*\mathbb F_{\infty}$ to $B \otimes_{max} C^*\mathbb F_{\infty}$. This proves that $B$ is $C$QWEP$_1$. \end{proof}
Now we are ready to establish the equivalence between the $D$QWEP notions by observing the following result.
\begin{theorem}
\label{equi} For a $C^*$-algebra $B$, the following conditions are equivalent: \begin{enumerate}
\item $B$ is $D$QWEP$_1$;
\item $B$ is $D$QWEP$_2$;
\item $B^{**}$ is $D^{**}$QWEP$_1$;
\item $B^{**}$ is $D^{**}$QWEP$_2$. \end{enumerate} \end{theorem}
\begin{proof}
(1)$\Rightarrow$(2): This follows from the fact that $D$WEP$_1$ implies $D$WEP$_2$.
(2)$\Rightarrow$(3): Suppose $B$ is $D$QWEP$_2$. Therefore, $B$ is the quotient of a $C^*$-algebra $A$ which is \emph{r.w.i.} in $\mathcal{L}^w(E_{D^{**}})$. By Remark \ref{wepqwep}, since $\mathcal{L}^w(E_{D^{**}})$ has the $D^{**}$WEP$_1$, it is $D^{**}$QWEP$_1$. Hence $A$ is $D^{**}$QWEP$_1$, and therefore, $B$ is $D^{**}$QWEP$_1$.
(3)$\Rightarrow$(4): Follows from (1)$\Rightarrow$(2).
(4)$\Rightarrow$(1): Suppose $B^{**}$ is $D^{**}$QWEP$_2$, and therefore so is $B$ by Corollary ~\ref{ds}. Then $B$ is the quotient of a $C^*$-algebra $A$ which is \emph{r.w.i.} in $\mathcal{L}^w(E_{D^{****}})$. We have
\[
A \overset{r.w.i.}\subset\mathcal{L}^w(E_{D^{****}})\overset{r.w.i.}\subset\mathbb{B}(\ell_2)\bar{\otimes} D^{****}=(\mathcal{K}\underset{min}\otimes D^{**})^{**}.
\] Therefore, it suffices to show that $\mathcal{K}\otimes_{min} D^{**}$ is $D$QWEP$_1$. Notice that $\mathcal{K}\otimes_{min} D^{**}$ factors through $\prod_n M_n(D^{**})$ approximately via ucp maps in point-norm topology, since $\cup M_n(D^{**})$ is norm-dense in $\mathcal{K}\otimes_{min} D^{**}$. Now since $D$ has the $D$WEP$_1$, $D^{**}$ is $D$QWEP$_1$. Therefore, by Proposition ~\ref{min qwep}, so is $M_n(D^{**})=M_n\otimes_{min}D^{**}$. Hence by Lemma ~\ref{factor qwep}, $\mathcal{K}\otimes_{min} D^{**}$ is $D$QWEP$_1$. This finishes the proof.
\end{proof}
\section{Illustrations}
In Section 3, we showed that $D$WEP$_1$ implies $D$WEP$_2$. Our first example will show the converse is not true, and hence the two notions of $D$WEP are not equivalent.
\begin{example} \normalfont \label{Bl2} Let $D = {\mathbb B}(\ell_2)$. Note that $\mathcal{L(H}_D) = {\mathcal M}({\mathcal K}\otimes {\mathbb B}(\ell_2))$, and ${\mathcal K}\otimes {\mathbb B}(\ell_2)$ has the WEP, and so does ${\mathcal M}({\mathcal K}\otimes {\mathbb B}(\ell_2))$. Therefore the two notions of $D$WEP$_1$ and WEP coincide. On the other hand, the $D$WEP$_2$ of a $C^*$-algebra is the same as being \emph{r.w.i.} in ${\mathbb B}({\mathcal H})\bar{\otimes} {{\mathbb B}(\ell_2)}^{**}$. Notice that ${\mathbb B}(\mathcal{H})\bar{\otimes} {\mathbb B}(\ell_2)^{**}=(\mathcal{K} \otimes {\mathbb B}(\ell_2))^{**}$ is QWEP. Therefore by Proposition ~\ref{bl2qwep}, $D$WEP$_2$ is equivalent to QWEP. Hence if $A$ is a QWEP $C^*$-algebra without the WEP, for instance $C^*_{\lambda}\mathbb F_n$, then $A$ has the $D$WEP$_2$ but not the $D$WEP$_1$, for $D = {\mathbb B}(\ell_2)$. \end{example}
Now we are ready to see some examples of relative WEP and QWEP over special classes of C$^*$-algebras.
\begin{proposition} \label{ncl}
Let $D$ be a nuclear $C^*$-algebra. Then a $C^*$-algebra $A$ has the $D$WEP$_i$ for $i=1,2$ if and only if it has the WEP. \end{proposition}
\begin{proof}
Suppose $A$ has the WEP. Therefore $A$ has the $D$WEP$_1$, and hence the $D$WEP$_2$.
Now assume $A$ has the $D$WEP$_2$, i.e. it is \emph{r.w.i.} in ${\mathbb B}(\ell_2)\bar{\otimes} D^{**}$. Since $D$ is nuclear, $D^{**}$ is injective. Hence we have $D^{**} \subseteq {\mathbb B}(\mathcal{H})\overset{\mathbb E}\to D^{**}$, where $\mathbb E$ is a conditional expectation. Let $CB(A,B)$ be the space of completely bounded maps from $A$ to $B$. Therefore we have \[
CB(S_1, D^{**})\overset{\pi} \hookrightarrow CB(S_1, {\mathbb B}(\mathcal{H}))\overset{\varphi}\to CB(S_1, D^{**}), \] where $S_1$ is the algebra of trace class operators, $\pi$ is a $^*$-homomorphism, and $\varphi$ acts by composing the maps in $CB(S_1, {\mathbb B}(\mathcal{H}))$ and $\mathbb E$. Note that by operator space theory $CB(S_1, D^{**})\simeq{\mathbb B}(\ell_2)\bar{\otimes} D^{**}$ and $CB(S_1, {\mathbb B}(\mathcal{H}))\simeq{\mathbb B}(\ell_2) \bar{\otimes} {\mathbb B}(\mathcal{H})={\mathbb B}(\ell_2 \otimes \mathcal{H})$. Hence we have the maps ${\mathbb B}(\ell_2)\bar{\otimes} D^{**}\overset{\pi}\to {\mathbb B}(\ell_2)\bar{\otimes}{\mathbb B}(\mathcal{H})= {\mathbb B}(\ell_2 \otimes \mathcal{H})\overset{\varphi}\to {\mathbb B}(\ell_2)\bar{\otimes} D^{**}$. Now by Remark ~\ref{unitization} we can unitize these two maps. Therefore $A$ is \emph{r.w.i.} in ${\mathbb B}(\ell_2\otimes\mathcal{H})$, and hence it has the WEP. \end{proof}
After nuclear C$^*$-algebras, it is natural to consider the relative WEP for an exact $C^*$-algebra $D$. For convenience, we consider the following stronger version of weak exactness property. A von Neumann algebra $M\subseteq {\mathbb B}(\mathcal{H})$ is said to be \emph{algebraically weakly exact}, (a.w.e. for short), if there exists a weakly dense exact $C^*$-algebra $D$ in $M$. By \cite{Ki2}, we know that the a.w.e. implies the weak exactness.
Notice that the unitization trick works better in $\mathcal{C}_2$ category, and hence we have the following.
\begin{proposition} \label{exact}
A $C^*$-algebra has the $D$WEP$_2$ for some exact $C^*$-algebra $D$ if and only if it is relatively weakly injective in an a.w.e. von Neumann algebra. \end{proposition}
\begin{proof}
Suppose a $C^*$-algebra $A$ has the $D$WEP$_2$, then $A$ is \emph{r.w.i.} in ${\mathbb B}(\mathcal{H})\bar{\otimes}D^{**}$. Since both ${\mathcal K}$ and $D$ are exact C$^*$-algebras, so is ${\mathcal K}\otimes D$. Note that ${\mathcal K}\otimes D$ is weakly dense in $({\mathcal K}\otimes D)^{**}={\mathbb B}(\mathcal{H})\bar{\otimes}D^{**}$. We have ${\mathbb B}(\mathcal{H})\bar{\otimes}D^{**}$ is a.w.e.
For the other direction, suppose $A$ is \emph{r.w.i.} in an a.w.e von Neumann algebra $M$. Let $D$ be an exact $C^*$-algebra with $D''=M$. Then there exists a central projection $z$ in $D^{**}$ such that $M=zD^{**}$. Hence we have completely positive maps $M \hookrightarrow D^{**} \to M$, which preserves the identity on $M$. Therefore by unitization $M$ is \emph{r.w.i.} in ${\mathbb B}(\mathcal{H})\bar{\otimes}D^{**}$ for some infinite dimensional Hilbert space ${\mathcal H}$. Hence if $A$ is \emph{r.w.i.} in $M$, then it is also \emph{r.w.i.} in ${\mathbb B}(\mathcal{H})\bar{\otimes}D^{**}$, and therefore it has the $D$WEP$_2$. \end{proof}
As we see, the nuclear-WEP is equivalent to the WEP. But the exact-WEP is different.
\begin{example} \normalfont Let $\mathbb F_2$ be the free group of two generators. Then it is exact and hence $C^*_{\lambda}\mathbb F_2$ is exact and $L\mathbb F_2$ is weakly exact. Since $C^*_{\lambda}\mathbb F_2$ is \emph{r.w.i.} in $L\mathbb F_2$, by Proposition ~\ref{exact}, $C^*_{\lambda}\mathbb F_2$ has the $D$WEP$_2$ for $D=C^*_{\lambda}\mathbb F_2$. But $C^*_{\lambda}\mathbb F_2$ does not have the WEP, since the WEP of a reduced group C$^*$-algebra is equivalent to the amenability of the group (see Proposition 3.6.9 in \cite{BrOz}). \end{example}
Now we consider the full group $C^*$-algebra of free group $C^*\mathbb F_\infty$. Since it is universal in the sense that for any unital separable $C^*$-algebra $A$, we have a quotient map $q: C^*\mathbb F_\infty \to A$. By the unitization trick, we have the following.
\begin{proposition}
Let $A$ be a unital separable $C^*$-algebra. Then it has the $D$WEP$_2$ for $D=C^*\mathbb F_\infty$. \end{proposition}
\begin{proof}
Since we have a quotient map $q: C^*\mathbb F_\infty \to A$, there exists a central projection $z$ in ${C^*\mathbb F_\infty}^{**}$ such that $A^{**}=z{C^*\mathbb F_\infty}^{**}$. Hence we have an embedding $A^{**}\hookrightarrow {\mathbb B}(\mathcal{H})\bar{\otimes}{C^*\mathbb F_\infty}^{**}$ with a completely positive map from ${\mathbb B}(\mathcal{H})\bar{\otimes}{C^*\mathbb F_\infty}^{**}$ to $A^{**}$ by multiplying $1\otimes z$. By the unitization trick in Remark ~\ref{unitization}, $A^{**}$ has the $D$WEP$_2$ for $D=C^*\mathbb F_\infty$ and so does $A$, since $A$ is \emph{r.w.i.} in $A^{**}$. \end{proof}
It is natural and even more interesting to ask whether the full group C$^*$-algebra $C^*\mathbb F_\infty$ has $D$WEP, for $D$ is the reduced group C$^*$-algebra $C^*_{\lambda}\mathbb F_2$. In fact, this is related to the QWEP conjecture. If $C^*\F_\infty$ has the $D$WEP$_1$ for some WEP algebra $D$, then it has the WEP by Corollary ~\ref{trand} of transitivity. If $C^*\F_\infty$ does not have the $D$WEP$_1$ for some C$^*$-algebra $D$, then it does not have the WEP either. At the time of writing, we do not have an answer for this question.
Now let us discuss some properties of being module QWEP relative to some special classes of C$^*$-algebras. In the rest of this section, we will examine the relation between one of the equivalent statements of Theorem ~\ref{equi} (for example statement (1), $B$ is $D$QWEP$_i$), and the statement that $B^{**}$ is $D^{**}$WEP$_i$, for either $i=1$ or $2$.
\begin{proposition}
Let $B$ be a C$^*$-algebra. If $B^{**}$ has the $D^{**}$WEP$_i$, then $B$ is $D$QWEP$_i$, for $i=1,2$. \end{proposition}
\begin{proof}
Suppose $B^{**}$ has the $D^{**}$WEP$_i$, and hence $B^{**}$ is $D^{**}$QWEP$_i$ by the trivial quotient. By Theorem \ref{equi}, $B$ is $D$QWEP$_i$. \end{proof}
For some C$^*$-algebra $D$, the four equivalent statements in Theorem \ref{equi} are equivalent to the statement that $B^{**}$ has the $D^{**}$WEP$_i$. But this is not true in general. We will show examples of both circumstances.
\begin{example} \normalfont Let $D={\mathbb B}(\ell_2)$. Then a $C^*$-algebra $B$ is $D$QWEP$_i$ if and only if $B^{**}$ has the $D^{**}$WEP$_i$, since they are both equivalent to $B$ being QWEP. Indeed, if $B$ is $D$QWEP$_1$, then $B=A/J$ and $A$ has the $D$WEP$_1$. Since $\mathcal{L(H}_D)$ has the WEP as shown in Example \ref{Bl2}, so does $A$, and hence $B$ is QWEP. On the other hand, having ${\mathbb B}(\ell_2)^{**}$WEP$_1$ is equivalent to being \emph{r.w.i.} in ${\mathcal M}({\mathcal K}\otimes {\mathbb B}(\ell_2)^{**})$, which is QWEP. Hence $B^{**}$ is QWEP. By Proposition ~\ref{dd}, $B$ is QWEP as well. \end{example}
\begin{example} \normalfont \label{nclq}
Let $D$ be a nuclear C$^*$-algebra. Then the above statements are not equivalent. Indeed, it follows from Proposition ~\ref{ncl} that a C$^*$-algebra is $D$QWEP$_i$ if and only if it is QWEP. On the other hand, assume that $B^{**}$ has the $D^{**}$WEP$_1$. Note that $D^{**}$WEP$_1$ implies $D$WEP$_2$ by Remark ~\ref{vn wep}, which is equivalent to WEP by Proposition ~\ref{ncl}, and $B^{**}$ has the WEP if and only if it is injective. Therefore the fact that a C$^*$-algebra $B$ is $D$QWEP$_i$ does not imply that $B^{**}$ has the $D^{**}$WEP$_1$. \end{example}
\begin{example} \normalfont
For a von Neumann algebra $M$, let us compare the properties $M$QWEP$_1$ of $B$ and the $M^{**}$WEP$_1$ of $B^{**}$. We have the following partial results.
Case (i): $M$ is of type I$_n$. Then $M$ is subhomogeneous, which is equivalent to nuclearity. By Example ~\ref{nclq}, these two statements are not equivalent.
Case (ii): $M$ is of type I$_{\infty}$, then ${\mathbb B}(\ell_2)\bar{\otimes}M$ is \emph{r.w.i.} in $M$. Suppose $B$ is $M$QWEP$_1$, then $B$ is a quotient of a $C^*$-algebra $A$ which is \emph{r.w.i.} in ${\mathbb B}(\ell_2)\bar{\otimes}M$. Hence $B^{**}$ is \emph{r.w.i.} in $A^{**}$ and hence in $({\mathbb B}(\ell_2)\bar{\otimes}M)^{**}$, and hence in $M^{**}$. Since $M^{**}$ is isomorphic to ${\mathcal L}({\mathcal H}_{M^{**}})$ for 1-dimensional Hilbert space $\mathcal{H}$, it follows that $B^{**}$ has the $M^{**}$WEP$_1$.
Case (iii): $M$ is of type II$_\infty$ or III, then ${\mathbb B}(\ell_2)\bar{\otimes}M \simeq M$. By a similar argument to that of Case (ii), we have the same conclusion.
Case (iv): $M$ is of type II$_1$ and a McDuff factor, i.e. $M\bar{\otimes}R \simeq M$. Then we have \[
M\simeq M\bar{\otimes}R \simeq M\bar{\otimes}R\bar{\otimes}R \supseteq M\bar{\otimes}R \bar{\otimes} L_{\infty}[0,1] \supseteq M\bar{\otimes} \prod_{n=1}^{\infty} M_n \supseteq M\bar{\otimes} {\mathbb B}(\ell_2) \] with conditional expectation from the larger algebra to the smaller for each inclusion. Hence $M\bar{\otimes} {\mathbb B}(\ell_2)$ is \emph{r.w.i.} in $M$. By the same argument above, the equivalence is established.
Problem: $M$ is a non-McDuff II$_1$ factor. At the time of writing, we do not have an affirmative answer for this case. \end{example}
\section{Application to C$^*$-norms}
In this section, we will discuss some application of our tensor norm $max^D$ constructed in Section 3, to C$^*$-norms. We will follow the approach in \cite{OP14} to construct norms on $A\otimes B$ for C$^*$-algebras $A$ and $B$.
Let $E$ be a $n$-dimensional subspace in $B$, and $C^*\langle E\rangle$ be the separable unital C$^*$-subalgeba of $B$ generated by $E$, which contains $E$ completely isometrically. For free group of countably infinite generators $\mathbb F_\infty$, we have a quotient map $C^*\F_\infty \to B$. Let $q: C^*\langle E\rangle \ast C^*\mathbb F_\infty \to B$ denote the free product of the inclusion $C^*\langle E\rangle \subset B$ and the quotient map $C^*\mathbb F_\infty \to B$, and let $I=\ker(q)$, so that we have $B \simeq (C^*\langle E\rangle \ast C^*\F_\infty) /I$. Following \cite{OP14}, let \begin{align}\label{enorm}
A\underset{E}\otimes B = \displaystyle\frac{A\otimes_{min} (C^*\langle E\rangle \ast C^*\F_\infty)}{A\otimes_{min} I}. \end{align}
Similarly, we can construct a new norm using the $max^D_1$ norm defined in Section 3. Recall that for a universal inclusion $A\subset {\mathcal L}({\mathcal H}^u_D)$, the $max^D_1$ norm is the induced tensor norm from the inclusion $A\otimes C \subset {\mathcal L}({\mathcal H}^u_D) \otimes_{max} C$. Now we define \begin{align}\label{ednorm}
A\underset{D,E}\otimes B = \displaystyle\frac{A\otimes_{max^D_1} (C^*\langle E\rangle \ast C^*\F_\infty)}{A\otimes_{max^D_1} I}. \end{align}
By their constructions, it is easy to see the following continuous inclusions \[
A\underset{min}\otimes B \supseteq A\underset{E}\otimes B \supseteq A\underset{D,E}\otimes B \supseteq A\underset{max}\otimes B. \] The goal of this section is to determine the conditions which distinguish the above norms, and make the inclusions strict.
We will follow the notations in \cite{OP14}. Let us first recall the operator space duality $F^*\otimes_{min} E \subset CB(F, E)$ isometrically (see Theorem B.13 in \cite{BrOz}). This gives us a correspondence between a tensor $x = \sum_k f^*_k\otimes e_k \in F^*\otimes E$, and a map
$\varphi_x: F \to E$ given by $\varphi_x(f) = \sum_k f^*_k (f)e_k$, with $\|x\|_{\min} = \|\varphi_x\|_{\operatorname{cb}}$. For finite dimensional operator space $E$, we denote by $t_E$ the ``identity'' element in $E^* \otimes E$.
Note that $\|t_E\|_{\min} = 1$ and that any norm of $t_E$ is independent of embeddings $E^*\hookrightarrow {\mathbb B}(\ell_2)$ and $E\hookrightarrow {\mathbb B}(\ell_2)$.
For any $n\in \mathbb{N}$, let $\mathcal{OS}_n$ denote the metric space of all $n$-dimensional operator spaces, equipped with the completely bounded Banach-Mazur distance. Note that by \cite{JP95}, $\mathcal{OS}_n$ is non-separable for $n\geq 3$. If $A$ is a separable C$^*$-algebra, then the set $\mathcal{OS}_n(A)$ of all $n$-dimensional operator subspaces of $A$ is a separable subset of $\mathcal{OS}_n$.
The first lemma will help us distinguish $\|\cdot\|_{E,D}$ and $\|\cdot\|_{\min}$. \begin{lemma}\label{cbfact}
Let $E$ and $F$ be subspaces of C$^*$-algebra $B$, and $E^*$, $F^*$ be subspaces of $C^*$-algebra $A$.
Then $\|t_F\|_{E,D} \geq d_{cb}(F, \mathcal{OS}_n(C^*\F_\infty))$, where $d_{cb}(F, \mathcal{OS}_n(C^*\F_\infty)) = \inf \{d_{cb}(F, G)\text{ }|\text{ } G\in\mathcal{OS}_n(C^*\F_\infty)\}$. \end{lemma}
\begin{proof}
By their construction, we have the following diagram
\[ \xymatrix{ A\underset{max^D_1}\otimes (C^*\langle E\rangle\astC^*\F_\infty) \ar [d]_{q} \ar@ {^{(}->}[r] & {\mathcal L}({\mathcal H}^u_D) \underset{max}\otimes (C^*\langle E\rangle\astC^*\F_\infty) & {\mathcal L}({\mathcal H}^u_D) \underset{max}\otimes C^*\F_\infty\ar [l]_{\qquad{\pi}}\ar@ {^{(}->}[d]^{\iota} \ar@ {^{(}->}[d]^{\iota} \\ A\underset{E,D}\otimes B && {\mathcal L}({\mathcal H}^u_D) \underset{min}\otimes C^*\F_\infty } \]
where $\pi$ is induced from a quotient map $C^*\F_\infty \to C^*\langle E\rangle\ast C^*\F_\infty$, and $\iota$ is a continuous inclusion.
Note that for finite dimensional $F^* \subset A$ and $F\subset B$, we can lift $F$ to a subspace $G \subset C^*\langle E\rangle\astC^*\F_\infty$, and then a $\tilde{G}\subset C^*\F_\infty$. Therefore the identity map $t_F$ on $F^*\otimes F \subset A\otimes_{E,D} B$ admits a lifting $\xi \in F^*\otimes \tilde{G}\subset {\mathcal L}({\mathcal H}^u_D) \otimes_{min} C^*\F_\infty$ which corresponds to a map $\alpha: F\to \tilde{G}$. Hence we have a factorization \[
F \overset{\alpha} \longrightarrow \tilde{G} \overset{q\circ\pi}\longrightarrow B, \] such that the composition is the inclusion $F\subset B$. Therefore the image $\alpha(F)$ in $\tilde{G}$ is isomorphic to $F$. Hence we have \[ \xymatrix{
t_F: F\ar[r] & \alpha(F) \ar@ {^{(}->}[d]\ar[r] & F\\
& C^*\F_\infty& } \]
and therefore, $\|t_F\|_{E,D} \geq d_{cb}(F, \mathcal{OS}_n(C^*\F_\infty))$. \end{proof}
The next lemma will help us distinguish $\|\cdot\|_{E,D}$ and $\|\cdot\|_{\max}$.
\begin{lemma}\label{max}
Let $A\subset {\mathcal L}({\mathcal H}^u_D)$ be the universal representation of $A$. Also let $\pi$ be a surjective ucp from $B_1$ to $B$ with kernal $I$, such that $(A\otimes_{max^D_1} B_1) /(A\otimes_{max^D_1} I) \simeq A\otimes_{max} B$. If there exists a surjective completely positive map $\sigma: B \to A^{\operatorname{op}}$, then $A$ has the $D$WEP$_1$. \end{lemma}
\begin{proof}
From the assumptions, we have the following diagram
\[ \xymatrix{
A \underset{max^D_1} \otimes B_1 \ar@ {^{(}->}[d]\ar [r]& A\underset{max}\otimes B \ar [r] & A\underset{max}\otimes A^{\operatorname{op}} \ar [r] & {\mathbb B}(L_2(A^{**}))\\
{\mathcal L}({\mathcal H}^u_D)\otimes_{max} B_1\ar [urrr] } \] By Arveson's extension theorem, there exists a ucp map $\Phi: {\mathcal L}({\mathcal H}^u_D)\otimes_{max} B_1 \to {\mathbb B}(L_2(A^{**}))$. Applying \emph{the Trick}, we obtain a ucp map $\phi: {\mathcal L}({\mathcal H}^u_D) \to A^{**}$, which is identity on $A$, and hence $A$ has the $D$WEP$_1$. \end{proof}
Now we are ready to give the conditions which distinguish the norms.
\begin{theorem}\label{norms}
For the continuous inclusions \[
A\underset{min}\otimes B \underset{(a)}\supseteq A\underset{E}\otimes B \underset{(b)}\supseteq A\underset{D,E}\otimes B \underset{(c)}\supseteq A\underset{max}\otimes B, \] the strict inclusion holds for \begin{enumerate}
\item[(a),] if there exists $n$-dimensional subspaces $F^*\subset A$ and $F\subset B$, such that $F\not\in \mathcal{OS}_n(C^*\langle E\rangle\astC^*\F_\infty)$;
\item[(b),] if $E\not\in \mathcal{OS}_n(C^*\F_\infty)$;
\item[(c),] if there exists a surjective ucp $B\to A^{\operatorname{op}}$, and $A$ does not have the $D$WEP$_1$. \end{enumerate} Moreover, for $n$-dimensional subspaces $E$, $F$ in $B$, and $E^*$, $F^*$ in $A$, we have $A\otimes_{D,E} B \neq A\otimes_F B$, if $F\not\in \mathcal{OS}_n(C^*\F_\infty)$. Therefore $A\otimes_{D,E} B$ gives us a new norm on $A\otimes B$, distinct from the continuum norms constructed in \cite{OP14}. \end{theorem}
\begin{proof}
(a) is proved in \cite{OP14}. Indeed, if such $F$ and $F^*$ exist, then the identity map $t_F$ on $F^* \otimes F \subset A\otimes_{min} B$ has norm $1$. On the other hand, notice that the norm of
$t_F$ in $A\otimes_{E} B$ is greater than $1$. Indeed if $\|t_F\|_{E} = 1$, then by the construction of $A\otimes_{E} B$,
it lifts to an element $\xi \in F^* \otimes (C^*\langle E\rangle\ast C^*\F_\infty)$ with $\|\xi\|_{\min} = 1$. This corresponds to a completely isometric mapping $F \to C^*\langle E\rangle\astC^*\F_\infty$, showing that $F$ is completely isometric to a subspace of $C^*\langle E\rangle\astC^*\F_\infty$, which contradicts the condition
$F\not\in \mathcal{OS}_n(C^*\langle E\rangle\astC^*\F_\infty)$. Hence $\|t_F\|_E > \|t_F\|_{\min} = 1$.
(b) By Lemma ~\ref{cbfact}, $\|t_E\|_{E,D} \geq d_{\operatorname{cb}}(E, \mathcal{OS}_n(C^*\F_\infty))$. If $E\not\in \mathcal{OS}_n(C^*\F_\infty)$, then we have $d_{\operatorname{cb}}(E, \mathcal{OS}_n(C^*\F_\infty)) > 1$,
and so is $\|t_E\|_{E,D}$. Therefore $\|t_E\|_{E,D} > \|t_E\|_{E} = 1$.
(c) Apply Lemma ~\ref{max} to $B_1 = C^*\langle E\rangle\astC^*\F_\infty$. Then by the construction we have $A\otimes_{D,E} B = (A\otimes_{max^D_1} B_1) /(A\otimes_{max^D_1} I)$. If $A\otimes_{D,E} B = A\otimes_{max} B$, then by Lemma ~\ref{max}, $A$ has the $D$WEP$_1$, which contradicts the condition.
Moreover, similar to the proof of (b), Lemma ~\ref{cbfact} shows that $A\otimes_{D,E} B \neq A\otimes_F B$, if $F\not\in \mathcal{OS}_n(C^*\F_\infty)$. \end{proof}
Now we will construct C$^*$-algebras $A$ and $B$, to give a concrete example with above distinct norms. Our goal is to construct a C$^*$-algebra $A$ such that $A\simeq A^{\operatorname{op}}$ without $D$WEP$_1$, and let $B = A$.
Recall that for operator spaces $E$ and $F$, $C^*\langle E\rangle\ast C^*\langle F\rangle \simeq C^*\langle E\otimes_{h} F\rangle$, where $E\otimes_{h} F$ is the Haagerup tensor product, and also that $C^*\langle E^{\operatorname{op}}\rangle \simeq C^*\langle E\rangle^{\operatorname{op}}$.
\begin{lemma}\label{haar}
Let $C = C^*\langle E\otimes_{h} E^{\operatorname{op}}\rangle$. Then $C\simeq C^{\operatorname{op}}$. \end{lemma}
\begin{proof}
Let $\pi: C^*\F_\infty \to C^*\langle E\rangle$ be the quotient map, then so is $\pi^{\operatorname{op}}: C^*\F_\infty^{\operatorname{op}} \to C^*\langle E\rangle^{\operatorname{op}}$. Then we have a quotient map $C^*\F_\infty \ast C^*\F_\infty^{\operatorname{op}} \to C^*\langle E\rangle \ast C^*\langle E\rangle^{\operatorname{op}}$, which maps the unitaries to unitaries. Notice that for $I$ the index set of $\mathbb F_{\infty}$, we have the following isomorphism given by \begin{align*}
C^*\F_\infty \ast C^*\F_\infty^{\operatorname{op}} && \simeq && C^*\mathbb F_{I\times I} &&\simeq && (C^*\mathbb F_{I\times I})^{\operatorname{op}} \\
g_i \ast 1 &&\longmapsto && 1\times g_i^{-1} &&\longmapsto && (1\times g_i)^{\operatorname{op}} \\
1 \ast h_i && \longmapsto && h_i^{-1}\times 1 &&\longmapsto && (h_i\times 1)^{\operatorname{op}} \\ \end{align*}
Let $\pi(g_i) = x$ and $\pi^{\operatorname{op}}(h_i) = y^{\operatorname{op}}$. Define the map $C^*\langle E\rangle \ast C^*\langle E\rangle^{\operatorname{op}} \to (C^*\langle E\rangle \ast C^*\langle E\rangle^{\operatorname{op}})^{\operatorname{op}}$, by $x\ast 1 \to (1\ast x^{\operatorname{op}})^{\operatorname{op}}$, and $1\ast y^{\operatorname{op}} \to (y\ast 1)^{\operatorname{op}}$. Then it is easy to check that this is an isomorphism following from the isomorphism $C^*\F_\infty \ast C^*\F_\infty^{\operatorname{op}} \to (C^*\mathbb F_{I\times I})^{\operatorname{op}}$. \end{proof}
Now we are ready to construct the example. For $n$-dimensional operator spaces $E$ and $F$ satisfying the conditions (a) and (b) in Theorem ~\ref{norms}, let \[
D = C^*\langle (E\oplus E^* \oplus F \oplus F^*) \otimes_{h}(E\oplus E^* \oplus F \oplus F^*)^{\operatorname{op}}\rangle, \] where the direct sum is in $\ell_{\infty}$. Then by Lemma \ref{haar}, $D\simeq D^{\operatorname{op}}$.
Let $A = D \otimes_{min} C^*_{\lambda}\mathbb{F}_2$. Then we have \[ A^{\operatorname{op}} = (D \underset{min}\otimes C^*_{\lambda}\mathbb{F}_2)^{\operatorname{op}} \simeq D^{\operatorname{op}}\underset{min}\otimes C^*_{\lambda}\mathbb{F}_2^{\operatorname{op}} \simeq D \underset{min}\otimes C^*_{\lambda}\mathbb{F}_2 = A. \] Let $B = A$, and hence we have a surjective ucp $B\to A^{\operatorname{op}}$. Also since $C^*_{\lambda}\mathbb{F}_2$ does not have the WEP, the faithful representation for $C^*_{\lambda}\mathbb{F}_2 \subset {\mathbb B}({\mathcal H})$ induces an inclusion $A = D\otimes_{min} C^*_{\lambda}\mathbb{F}_2\hookrightarrow D\otimes_{min} {\mathbb B}({\mathcal H})$ which is not \emph{r.w.i.}. Notice that this is not equivalent to $D$WEP. However if we construct the $max^D$ norm from the inclusion $A\subseteq D\otimes_{min} {\mathbb B}({\mathcal H})$ instead of $A\subseteq {\mathcal L}({\mathcal H}^u_D)$, we will have the same conclusion that the four norms are distinct.
\begin{corollary}
Let $D$ be as above, and $A= D \otimes_{min} C^*_{\lambda}\mathbb{F}_2$. For a faithful representation $C^*_{\lambda}\mathbb{F}_2 \subset {\mathbb B}({\mathcal H})$, define the $max^D$ norm on $A\otimes C$ to be the tensor norm induced from the inclusion $A\otimes C \subseteq (D\otimes_{min} {\mathbb B}({\mathcal H})) \otimes_{max} C$. Let $B=A$. Define the quotient norms $A\otimes_E B$ as in ~(\ref{enorm}) and $A\otimes_{D,E}B$ as in ~(\ref{ednorm}) with the new $max^D$ norm as follows \[
A\underset{D,E}\otimes B = \displaystyle\frac{A\otimes_{max^D} (C^*\langle E\rangle \ast C^*\F_\infty)}{A\otimes_{max^D} I}. \] Then we have the following strict inclusions \[
A\underset{min}\otimes B \supset A\underset{E}\otimes B \supset A\underset{D,E}\otimes B \supset A\underset{max}\otimes B. \] \end{corollary}
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\end{document} | arXiv |
\begin{document}
\begin{abstract} This is a survey article on the currently very active research area of free (=non-commutative) real algebra and geometry. We first review some of the important results from the commutative theory, and then explain similarities and differences as well as some important results in the free setup. \end{abstract}
\title{Free Semialgebraic Geometry} \section{Classical Semialgebraic Geometry} In this section we briefly review some concepts and results from commutative real algebra and geometry. For details and proofs see for example \cite{pd, m, bcr, netznote, habil}.
Important objects of study in classical (=commutative) real algebra and geometry are semialgebraic sets. A {\it basic closed semialgebraic set} is of the form $$W(p_1,\ldots, p_r):=\left\{ a\in\R^d\mid p_1(a)\geq 0, \ldots, p_r(a)\geq 0\right\}$$ where $p_1,\ldots, p_r\in\R[x_1,\ldots, x_d]$ are polynomials. A general {\it semialgebraic set} is a (finite) Boolean combination of basic closed semialgebraic sets. \begin{figure}
\caption{Two (basic closed) semialgebraic sets}
\end{figure} An important and foundational result in real algebraic geometry is the following: \begin{Thm}[Projection Theorem] Projections of semialgebraic sets are again semialgebraic. \end{Thm} The Projection Theorem is not easy to proof. It also has some strong implications for logic and model theory of real closed fields. Since projections correspond to existential quantifiers, it leads to quantifier elimination in the theory of real closed fields, which lies at the core of almost any Positivstellensatz in real algebra. It also proves decidability of the theory of real closed fields. Although not easy, the Projection Theorem admits constructive proofs. In practice, finding a semialgebraic description of a projection can be very challenging, however.
In classical algebraic geometry, affine varieties are classified via polynomial functions that vanish on them. These functions are described algebraically by Hilbert's Nullstellensatz. For semialgebraic sets, one considers {\it nonnegative} polynomials, and {\it Positivstellens\"atze} provide algebraic characterization of such polynomials. The role of ideals is taken by {\it preorderings}. The preordering generated by polynomials $p_1,\ldots, p_r\in\R[x_1,\ldots, x_d]$ arises from the $p_i$ and sums of squares of polynomials, by addition and multiplication:
$${\mathcal P}(p_1,\ldots, p_r):=\left\{ \sum_{e\in\{0,1\}^r} \sigma_e p_1^{e_1}\cdots p_r^{e_r} \mid \sigma_e\ \mbox{ sums of squares of polynomials}\right\}.$$ Note that polynomials from the preordering are obviously nonnegative on $W(p_1,\ldots, p_r)$.
\begin{Thm}[Nichtnegativstellensatz] For $p,p_1,\ldots, p_r\in\R[x_1,\ldots, x_d]$, the following are equivalent: \begin{itemize} \item[($i$)] $p\geq 0$ on $W(p_1,\ldots, p_r)$. \item[($ii$)] $t_1p=p^{2n}+t_2$ for some $t_1,t_2\in {\mathcal P}(p_1,\ldots, p_r)$, $n\in\N$. \end{itemize} \end{Thm}
The direction ($ii$)$\Rightarrow$($i$) is straightforward to see, so ($ii$) is an algebraic certificate for nonnegativity of $p$. The factor $t_1$ in ($ii$) is often called a {\it denominator}. Over the field $\mathbb R(x_1,\ldots, x_d)$ it can be brought to the other side, providing a representation with rational functions. The case $r=1$ and $p_1=1$ is precisely Hilbert's 17th Problem: every globally nonnegative polynomial is a sum of squares of rational functions. One can get rid of the denominator only under additional assumptions. The first and most important such Positivstellensatz without denominators is the following, where the conditions of boundedness and strict positivity is necessary for the theorem to hold: \begin{Thm}[Schm\"udgen's Positivstellensatz] Let $p_1,\ldots, p_r\in\R[x_1,\ldots, x_d]$ and assume $W(p_1,\ldots,p_r)$ is bounded. Then for any $p\in \R[x_1,\ldots, x_d]$ $$p>0 \mbox{ on } W(p_1,\ldots, p_r) \quad\Rightarrow\quad p\in {\mathcal P}(p_1,\ldots, p_r).$$ \end{Thm} After this very brief introduction, let us now pass to the non-commutative setup.
\section{Free Real Algebra and Geometry}
Semialgebraic sets are defined by polynomial inequalities. So before we can talk about non-commutative semialgebraic sets, we introduce non-commutative polynomials. In the non-commutative setup we will always use an involution (in fact the involution is also there in the classical case, however invisible since it is just the identity on real polynomials). In presence of an involution we can use complex numbers as our ground field and take Hermitian elements as the "real"\ objects. Using complex numbers is often more convenient and allows for cleaner proofs.
So let $\C\langle z_1,\ldots, z_d\rangle$ denote the algebra of polynomials in the {\it non-commuting variables} $z_1,\ldots, z_d$. Their elements are $\C$-linear combinations of {\it words} in the variables. Since the variables do not commute, words like $z_1z_2$ and $z_2z_1$ are different, where they would coincide in the commutative case. We use the involution $*$ on $\C\langle z_1,\ldots, z_d\rangle$ that fixes the variables, i.e.\ $z_i^*=z_i$ holds for all $i$, but reverses the order in each word and acts as complex conjugation on the coefficients. For example, we have $$\left(z_1^3-z_1z_2+i\right)^*=z_1^3-z_2z_1-i.$$ Let $$\C\langle z_1,\ldots, z_d\rangle_h:=\left\{ p\in \C\langle z_1,\ldots, z_d\rangle\mid p^*=p\right\}$$ be the set of {\it Hermitian elements}. They form a real vectorspace, but not an algebra (in case $d\geq 2)$. Note that Hermitian elements do not necessarily have real coefficients, and polynomials with real coefficients are not necessarily Hermitian.
Into a non-commutative polynomial $p\in \C\langle z_1,\ldots, z_d\rangle$ we can plug in a $d$-tuple of elements from any complex algebra, and obtain an element from this algebra as the result. We will restrict ourselves to matrix algebras here, i.e. we take $(A_1,\ldots, A_d)\in {\rm Mat}_s(\C)^d$ for some $s\geq 1$ and obtain $$p(A_1,\ldots, A_d)\in {\rm Mat}_s(\C).$$ The need for an involution and Hermitian elements becomes clear when trying to capture real phenomena. Indeed if $A_1,\ldots,A_d\in {\rm Her}_s(\C)$ are Hermitian matrices and $p\in \C\langle z_1,\ldots, z_d\rangle_h$ is Hermitian as well, then so is the result: $$p(A_1,\ldots, A_d)\in{\rm Her}_s(\C).$$ A Hermitian matrix is {\it positive semidefinite} if all of its Eigenvalues are nonnegative; we denote this by $\geqslant 0$. This is the right notion of positivity in our setup, so if $$p(A_1,\ldots, A_d)\geqslant 0$$ we say that $p$ is nonnegative at the (non-commutative) point $(A_1,\ldots, A_d)\in{\rm Her}_s(\C)^d$.
It is obvious that every matrix $A\in{\rm Her}_s(\C)$ that can be written as an {\it Hermitian square} $$A=B^*B$$ for some $B\in {\rm Mat}_s(\C)$ is positive semidefinite; in fact every positive semidefinite matrix is of that form. So if $$p=\sum_{i=1}^nq_i^*q_i$$ for certain $q_1,\ldots, q_n\in \C\langle z_1,\ldots, z_d\rangle,$ we obtain $$p(A_1,\ldots, A_d)=\sum_i q_i(A_1,\ldots, A_d)^*q_i(A_1,\ldots, A_d)\geqslant 0$$ for any $(A_1,\ldots, A_d)\in{\rm Her}_s(\C)^d$. So the set of {\it sums of Hermitian squares} $$\Sigma\C\langle z_1,\ldots, z_d\rangle^2:=\left\{ \sum_{i=1}^n q_i^*q_i\mid n\in\N, q_i\in \C\langle z_1,\ldots, z_d\rangle\right\} \subseteq \C\langle z_1,\ldots, z_d\rangle_h$$ only contains polynomials that are positive semidefinite on each Hermitian matrix tuple. The first surprising result, a global Positivstellensatz and a non-commutative analogue of Hilbert's 17th Problem, is due to Helton:
\begin{Thm}[\cite{he0}]\label{hth} Let $p\in\C\langle z_1,\ldots, z_d\rangle_h$ and assume $$p(A_1,\ldots, A_d)\geqslant 0$$ for all $(A_1,\ldots, A_d)\in{\rm Her}_s(\C)^d$ and all $s\geq 1$. Then $$p\in\Sigma\C\langle z_1,\ldots, z_d\rangle^2.$$ \end{Thm}
In contrast to the commutative result, no denominator is needed in Helton's theorem. However, the natural notion of positivity is much stronger here than in Hilbert's 17th Problem, where positivity is only assumed on matrices of size $1$, instead of matrices of all sizes. Note however that also in Helton's result one can bound the matrix size, depending only on $d$ and the degree of $p$.
Let us now define free basic closed semialgebraic sets. In analogy to the above described commutative setup, we define for $p_1,\ldots, p_r\in\C\langle z_1,\ldots, z_d\rangle_h$ and $s\geq 1$ $$W_s(p_1,\ldots, p_r):=\left\{ (A_1,\ldots, A_d)\in{\rm Her}_s(\C)^d\mid p_i(A_1,\ldots, A_d)\geqslant 0, i=1,\ldots, r\right\}.$$ A guiding principle in non-commutative geometry is to not consider matrices of one size alone, but all sizes at once. We thus define the {\it free basic closed semialgebraic set} defined by $p_1,\ldots, p_r$ as $${\rm F}W(p_1,\ldots, p_r):=\left(W_s(p_1,\ldots, p_r)\right)_{s=1}^\infty.$$ There is no known Positivstellensatz for positivity on general free basic closed semialgebraic sets, however certain results in special cases. One of them deals with the {\it matrix cube}, see for example \cite{aleknetzthom}:
\begin{Thm}\label{box} Assume $p\in\C\langle z_1,\ldots, z_d\rangle_h$ is nonnegative on $${\rm F}W\left(1-z_1^2,\ldots, 1-z_d^2\right).$$ Then there exists a representation $$p=\sum_{i} q_i^*q_i + \sum_{i,j}q_{ij}^*\left(1-z_j^2\right)q_{ij}$$ for certain $q_i,q_{ij}\in\C\langle z_1,\ldots, z_d\rangle_h.$ \end{Thm} Another such Positivstellensatz is explained in the next section, and we also refer to \cite{aleknetzthom} for more examples and unified proofs.
A notion of free semialgebraic sets beyond free basic closed semialgebraic sets has not been established in the literature so far. Boolean combination of basic closed sets will surely have to be allowed, but maybe that is not yet the best possible notion. This becomes clear when trying to prove a free projection theorem. For any $s\geq 1$ we apply the projection map \begin{align*}\pi_s\colon {\rm Her}_s(\C)^d&\to{\rm Her}_s(\C)^{d-1} \\ (A_1,\ldots, A_d)&\mapsto (A_1,\ldots, A_{d-1})\end{align*} to $W_s(p_1,\ldots, p_r)$ and altogether obtain $$\pi({\rm F}W(p_1,\ldots, p_r)):=\left(\pi_s(W_s(p_1,\ldots, p_r))\right)_{s=1}^\infty.$$How does such a projected set look like, is it a Boolean combination of free basic closed sets? The answer to this question is no, and the whole topic does not look too encouraging. For example (see \cite{dreschnetzthom}), using free basic closed semialgebraic sets, Boolean combinations and projections, one can construct the set $$\left(S_s\right)_{s=1}^\infty, \qquad S_s=\left\{\begin{array}{cc} {\rm Her}_s(\C) & s \mbox{ prime} \\ \emptyset & \mbox{ else.} \end{array} \right.$$ This set cannot be defined without projections, even if the language is enlarged by using trace, determinant and many other functions. Even more discouraging is the following result from \cite{dreschnetzthom}:
\begin{Thm} It is undecidable whether a set constructed from free basic closed semialgebraic sets, Boolean combinations and projections is empty (at each level). \end{Thm}
A very recent positive result is \cite{kltr}. Without going too much into the details, it states that quantifiers in non-commutative formulas can be eliminated, as long as the formula is evaluated at matrix tuples of {\it fixed size}. This is not a trivial result, since the variables in such formulas do not refer to the single matrix entries (where the result would follow from classical (commutative) quantifier elimination), but to matrices as a whole. However, the formula without quantifiers will depend on the matrix size. So the result does not imply a general (size independent) projection theorem.
A much more fruitful concept is {\it free convexity}, as we now explain in our last section.
\section{Free Convexity}
In this section we define the notion of a non-commutative convex set. Since definitions and results become cleaner for convex cones instead of convex sets, we restrict ourselves to cones here. As above we consider free sets $$S=\left(S_s\right)_{s=1}^\infty, \quad S_s\subseteq {\rm Her}_s(\C)^d \ \mbox{ for all } s\geq 1.$$ Matrix convexity of $S$ is defined via two properties. A very reasonable assumption, even fulfilled for all free basic closed semialgebraic sets, is {\it closedness under direct sums}. For $\underline A=(A_1,\ldots, A_d)\in{\rm Her}_s(\C)^d, \underline B=(B_1,\ldots, B_d)\in{\rm Her}_t(\C)^d$ we define $$\underline A\oplus\underline B:=\left(\left(\begin{array}{c|c}A_1 & 0 \\\hline0 & B_1\end{array}\right), \ldots,\left(\begin{array}{c|c}A_d & 0 \\\hline0 & B_d\end{array}\right)\right) \in {\rm Her}_{s+t}(\C)^d.$$ $S$ is closed under direct sums if \begin{equation}\tag{C1}\underline A\in S_s, \underline B\in S_t\ \Rightarrow\ \underline A\oplus \underline B \in S_{s+t}.\end{equation} The second condition resembles scaling with positive reals, but even connects the different levels of $S$. For $V\in {\rm Mat}_{s,t}(\C)$ and $\underline A\in{\rm Her}_s(\C)^d$ we define $$V^*\underline AV:= \left(V^*A_1V,\ldots, V^*A_dV \right)\in {\rm Her}_t(\C)^d.$$ The second condition then reads \begin{equation}\tag{C2}\underline A\in S_s, V\in {\rm Mat}_{s,t}(\C)\ \Rightarrow\ V^*\underline A V\in S_t.\end{equation} If $S$ fulfills (C1) and (C2), it is called a {\it matrix convex cone}. It is easily checked that each $S_s$ is a classical convex cone in the real vector space ${\rm Her}_s(\C)^d$ in this case. However, matrix convexity is a stronger condition in general, connecting the different levels of $S$ via (C2). Also note that a matrix convex cone is almost the same as an {\it abstract operator system} \cite{fnt, pau}, which only requires all $S_s$ to be closed and salient with nonempty interior, additionally.
The most basic examples of matrix convex cones are {\it free spectrahedral cones} (or operator systems with a finite-dimensional realization, equivalently). For $M_1,\ldots, M_d\in {\rm Her}_r(\C)$ define $$S_s(M_1,\ldots, M_d):=\left\{ (A_1,\ldots, A_d)\in{\rm Her}_s(\C)^d\mid M_1\otimes A_1+\cdots +M_d\otimes A_d\geqslant 0\right\}$$ and $${\rm F}S(M_1,\ldots, M_d)=\left( S_s(M_1,\ldots, M_d)\right)_{s=1}^\infty.$$ Here, $\otimes$ denotes the Kronecker-/tensorproduct of matrices. The set $S_1(M_1,\ldots, M_d)$ is known as a classical {\it spectrahedron}. Such sets are precisely the feasible sets of semidefinite programming. The free spectrahedron ${\rm F}S(M_1,\ldots, M_d)$ is a non-commutative extension, precisely in the spirit as above. For free spectrahedra, there exists a nice Positivstellensatz. As in Theorems \ref{hth} and \ref{box} above, we see that the natural notion of positivity in the non-commutative setup is strong enough to provide the best possible algebraic certificate (we do not cite the most general result and suppress some minor technical details for better readability):
\begin{Thm}[\cite{he2}] Let $M_1,\ldots, M_d\in {\rm Her}_r(\C)$ and $p\in \C\langle z_1,\ldots, z_d\rangle_h$. If $$p(\underline A)\geqslant 0$$ for all $\underline A\in S_s(M_1,\ldots, M_d)$ and all $s\geq 1$, in other words if $p$ is nonnegative on the free spectrahedron ${\rm F}S(M_1,\ldots, M_d)$, then there exists a representation $$p=\sum_i q_i^*q_i+ \sum_j f_j^*Mf_j $$ where $ q_i\in \C\langle z_1,\ldots, z_d\rangle, f_j\in \C\langle z_1,\ldots, z_d\rangle^r$ and $$M:=z_1M_1+\cdots +z_dM_d\in {\rm Her}_r\left(\C\langle z_1,\ldots, z_d\rangle\right).$$ \end{Thm}
Sometimes facts about classical spectrahedra can only be learned by extending them to the non-commutative setup. One such instance is the {\it containment problem} for spectrahedra, a problem appearing in different areas of (applied) mathematics \cite{ktt}. Given $M_1,\ldots, M_d\in {\rm Her}_r(\C)$ and $N_1,\ldots, N_d\in {\rm Her}_t(\C)$, how can one check efficiently whether \begin{equation}\tag{A}S_1(M_1,\ldots, M_d)\subseteq S_1(N_1,\ldots, N_d)\end{equation} holds? Since spectrahedra are generalizations of polyhedra (which appear in the case of commuting coefficient matrices), this includes the problem of polyhedral containment. An important algorithm to solve this problem was proposed in \cite{bt}. Instead of checking (A) once checks \begin{equation}\tag{B} \exists V_1,\ldots, V_n\in {\rm Mat}_{r,t}(\C)\colon\quad \sum_jV_j^*M_iV_j=N_i \mbox{ for } i=1,\ldots, d.\end{equation} It is obvious that (B) implies (A). Condition (B) can be transformed into a semidefinite optimization problem, and thus often solved efficiently. It was however known that (A) and (B) are not equivalent, so the answer to (B) could be {\it no}, where the answer to (A) is {\it yes}. A much better understanding of the method was gained through the following result (again we suppress some minor technical details):
\begin{Thm}[\cite{hkinf}]\label{hkkk} Condition (B) is equivalent to \begin{equation}\tag{A'}{\rm FS}(M_1,\ldots, M_d)\subseteq {\rm FS}(N_1,\ldots, N_d).\end{equation} Inclusion is meant level-wise here, i.e. $S_s(M_1,\ldots, M_d)\subseteq S_s(N_1,\ldots, N_d)$ for all $s\geq 1$. \end{Thm}
This result mostly relies on Choi's characterization of completely positive maps between matrix algebras \cite{choi}. The insight of Theorem \ref{hkkk} can now be used to determine instances in which (A) and (B) are equivalent nonetheless. For this let $C\subseteq \R^d$ be a convex cone. There is one smallest and one largest matrix convex set with $C$ at level one. Indeed define $$C^{\min}_s:=\left\{ \sum_i c_i^t\otimes P_i\mid c_i\in C, P_i\in {\rm Her}_s(\C), P_i\geqslant 0\right\}$$ and $$C^{\max}_s:=\left\{ \underline A\in {\rm Her}_s(\C)^d\mid v^*\underline Av\in C\ \mbox{ for all } v\in \C^s\right\}.$$ Then $$C^{\min}:=\left(C^{\min}_s\right)_{s=1}^\infty\quad \mbox{and}\quad C^{\max}:=\left(C^{\max}_s\right)_{s=1}^\infty$$ are easily checked to be the smallest/largest such matrix convex set. Now assume $$C=S_1(M_1,\ldots, M_d)\subseteq \R^d$$ is a (classical) spectrahedral cone with ${\rm F}S(M_1,\ldots, M_d)= C^{\min}$. In that case, condition (A) implies $${\rm F}S(M_1,\ldots, M_d)=C^{\min} \subseteq {\rm F}S(N_1,\ldots, N_d)$$ and thus (B), by Theorem \ref{hkkk}. On the other hand, if $$C^{\min}\subsetneq {\rm F}S(M_1,\ldots, M_d)$$ it can be shown by the non-commutative separation theorem from \cite{ew}, that there exist matrices $N_1,\ldots, N_d$ with $$C=S_1(M_1,\ldots, M_d)=S_1(N_1,\ldots, N_d)$$ and $$ {\rm F}S(M_1,\ldots, M_d) \nsubseteq{\rm F}S(N_1,\ldots, N_d).$$ In such an instance the answer to (B) is no, whereas the answer to (A) is yes. So the method from \cite{bt} works reliably if any only if ${\rm F}S(M_1,\ldots, M_d)$ is the smallest matrix convex cone over the classical spectrahedron $S_1(M_1,\ldots, M_d)$. Unfortunately, this happens very rarely, already for polyhedral cones:
\begin{Thm}[\cite{fnt}]\label{op} Assume $C=S_1(M_1,\ldots, M_d)\subseteq\R^d$ is polyhedral. Then $${\rm F}S(M_1,\ldots, M_d)= C^{\min}$$ if and only if $C$ is a simplex cone, i.e. has only $d$ extremal rays. \end{Thm}
The last theorem also has some surprising application in theoretical quantum physics. The state of a bipartite quantum system is usually described by a positive semidefinite matrix $$\rho\in{\rm Mat}_r(\C)\otimes{\rm Mat}_s(\C)\cong {\rm Mat}_{rs}(\C).$$ So $\rho$ can be written as $$0\leqslant\rho=\sum_{i=1}^n M_i\otimes A_i$$ with $M_i\in {\rm Mat}_r(\C), A_i\in{\rm Mat}_s(\C).$ Although $\rho$ is supposed to be positive semidefinite and in particular Hermitian, this is not necessarily true for the $M_i, A_i$. If there exists a different such representation where all the $M_i,A_i$ are positive semidefinite as well, then $\rho$ is called {\it separable}, otherwise it is {\it entangled}. The smallest possible $n$ in the representation of $\rho$ above is called the {\it tensor rank} of $\rho$. A corollary of Theorem \ref{op} now reads as follows:
\begin{Thm}[\cite{ddn}]Every bipartite quantum state of tensor rank $2$ is separable. \end{Thm}
In fact $$0\leqslant\rho=M_1\otimes A_1+M_2\otimes A_2$$ just means that $(A_1,A_2)\in S_s(M_1,M_2)$. Now since the convex cone $$C=S_1(M_1,M_2)\subseteq \R^2$$ is automatically a simplex cone, we obtain $(A_1,A_2)\in C_s^{\min}$ from Theorem \ref{op}. Writing down a representation in this smallest matrix convex cone and using bilinearity of the tensor product immediately implies the result.
Let us close with a result about non-commutative polytopes and polyhedra. The theorem of Minkowski-Weyl (see for example \cite{schr}) states that every polyhedral cone $C\subseteq \R^d$ is finitely generated, and vice versa. In other words, the notions {\it polyhedral} and {\it polytopal} coincide for convex cones. Now a short contemplation reveals that $C^{\min}$ is a good generalization of the notion {\it polytope/finitely generated} to the non-commutative setup, whereas $C^{\max}$ corresponds to the {\it polyhedral} notion. Interestingly, these two notions differ almost always, already at the first level of non-commutativity:
\begin{Thm}[\cite{fnt,sh,hn}] Let $C\subseteq \R^d$ be a convex cone.
($i$) If $C$ is a simplex cone, then $C^{\min}=C^{\max}$. Otherwise $C^{\min}\neq C^{\max}$.
($ii$) If $C$ is polyhedral but not a simplex cone, then $C_2^{\min}\subsetneq C_2^{\max}.$ \end{Thm}
As a concluding remark, we note that the methods used in the non-commutative setup differ quite strongly from the ones in the commutative theory. Many of the results are proven by functional-analytic methods, such as GNS-constructions, dilations, and the theory of completely positive maps and operator algebras. Sometimes results and examples from group theory and the theory of $C^*$-algebras can be useful. All in all, the whole area is not yet mature, many interesting results and methods are hopefully developed in the coming years.
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Volume 19 Supplement 9
Selected articles from the 13th International Symposium on Bioinformatics Research and Applications (ISBRA 2017): bioinformatics
BMCMDA: a novel model for predicting human microbe-disease associations via binary matrix completion
Jian-Yu Shi1,
Hua Huang2,
Yan-Ning Zhang3,
Jiang-Bo Cao1 &
Siu-Ming Yiu4
BMC Bioinformatics volume 19, Article number: 281 (2018) Cite this article
Human Microbiome Project reveals the significant mutualistic influence between human body and microbes living in it. Such an influence lead to an interesting phenomenon that many noninfectious diseases are closely associated with diverse microbes. However, the identification of microbe-noninfectious disease associations (MDAs) is still a challenging task, because of both the high cost and the limitation of microbe cultivation. Thus, there is a need to develop fast approaches to screen potential MDAs. The growing number of validated MDAs enables us to meet the demand in a new insight. Computational approaches, especially machine learning, are promising to predict MDA candidates rapidly among a large number of microbe-disease pairs with the advantage of no limitation on microbe cultivation. Nevertheless, a few computational efforts at predicting MDAs are made so far.
In this paper, grouping a set of MDAs into a binary MDA matrix, we propose a novel predictive approach (BMCMDA) based on Binary Matrix Completion to predict potential MDAs. The proposed BMCMDA assumes that the incomplete observed MDA matrix is the summation of a latent parameterizing matrix and a noising matrix. It also assumes that the independently occurring subscripts of observed entries in the MDA matrix follows a binomial model. Adopting a standard mean-zero Gaussian distribution for the nosing matrix, we model the relationship between the parameterizing matrix and the MDA matrix under the observed microbe-disease pairs as a probit regression. With the recovered parameterizing matrix, BMCMDA deduces how likely a microbe would be associated with a particular disease. In the experiment under leave-one-out cross-validation, it exhibits the inspiring performance (AUC = 0.906, AUPR =0.526) and demonstrates its superiority by ~ 7% and ~ 5% improvements in terms of AUC and AUPR respectively in the comparison with the pioneering approach KATZHMDA.
Our BMCMDA provides an effective approach for predicting MDAs and can be also extended to other similar predicting tasks of binary relationship (e.g. protein-protein interaction, drug-target interaction).
Human intestine provides a nutrient-rich and temperature-constant habitat for microbes, such that the microbes have a mutualistic association with their host [1]. Diverse communities of microbes, especially bacteria, are found by sequencing techniques (e.g. 16S ribosomal RNA sequencing) in human bodies [2]. It is surprising that the number of genes in human microbiome is up to 5 million [3]. Both these genes and their products are participating in a diverse range of biological activities, such as metabolic capabilities, pathogens, immune system, and gastrointestinal development [4]. It can be said that they somehow serve as a physiological complement in the human body. Meanwhile, both communities and populations of microbes can be significantly influenced by their dynamic habitat in the human body. Diverse environmental variables, such as season [5], host diet [6], smoking [7], hygiene [3] and use of antibiotics [8], may change the habitat of microbes frequently. This kind of mutualistic associations between human host and its microbiota would cause the modifications of transcriptomic, proteomic and metabolic profiles in the human body. However, some of the modifications could be harmful.
Beyond the fact that microbe is the main player in the pathogenic mechanism of infectious diseases, an increasing number of clinical studies have demonstrated that the microbiota in human body is strongly associated with a wide range of human non-infectious diseases, such as cancer [9], obesity [10, 11], diabetes [12, 13], kidney stones [14] and systemic inflammatory response syndrome [15]. Nevertheless, people have only a limited understanding of what microbes cause the diseases and how they do.
Fortunately, the increasing number of experimentally validated associations between human non-infectious diseases and microbes enable us to perform a systematic analysis on microbe-disease associations (MDAs). For example, Ma et al. recently published the first database of MDA, Human Microbe-Disease Association Database (HMDAD), by collecting a large number of MDAs from previously published literature [16]. The MDA entries in HMDAD mainly focuses on experimentally supported associations between diverse microbes and non-infectious diseases, and all of them are experimentally supported with sufficient samples. The systematic analysis on a large scale of MDAs provides a new insight to discover the mechanism of microbe-related non-infectious diseases [17]. As one of the most important steps towards that goal, the identification of MDA is helpful to understand how non-infectious diseases develop and exploit novel methods for disease diagnosis and therapy. However, traditional experiment-based approaches for discovering MDAs are time-consuming and costly. Even worse, many bacteria cannot be cultivated at all by current culturing bio-techniques [18].
As the complement of biological experiment-based approaches, computational approaches are promising to rapidly screen MDA candidates, such that the further biological validation reduces the cost and time significantly. More importantly, they are expected to output the MDA candidates involving uncultivable microbes. A few efforts have been made to develop computational models for the large-scale MDA prediction. Recently, a pioneering work developed an approach, KATZHMDA, for predicting potential MDAs on a large scale [19]. After constructing an MDA network based on HMDAD, KATZHMDA models MDA prediction as link prediction on the network.
In this work, by modeling MDA prediction as a problem of matrix completion (Fig. 1), we propose a new predictive approach based on Binary Matrix Completion (BMCMDA) to predict potential MDAs on a large scale by only using a set of approved microbe-disease associations. The following sections are organized as follows. Section Method first introduces the basic idea to model MDA prediction, then represents the algorithm of binary matrix completion. Section Experiments briefly describes the benchmark dataset of MDA, shows how to tune the parameters in the proposed model, and demonstrates the ability of BMCMDA by the comparison with other state-of-the-art approaches. The final section draws our conclusion. In addition, human non-infectious diseases are termed as 'diseases' and their microbes in the body are termed as 'microbes' in the following texts for concision.
A Toy MDA Example for Matrix Completion. The left matrix is an observed matrix, in which xij are the observed MDA entries and '?'s denote the unobserved microbe-disease pairs. The right matrix is the expected matrix with fully observed entries
Problem formulation
Given p kinds of microbes M = {mi}, q types of diseases D = {dj}, and a set of associations between them, we aim to deduce or predict new potential associations among them. Those microbe-disease associations can be organized into a p × q binary adjacent matrix A = {aij}, where aij = + 1 and aij = − 1 account for whether mi is associated with dj or not respectively, and aij = ? if the association between mi and dj is NOT observed. Our problem is to deduce how likely those unobserved entries are MDAs (Fig. 1).
Matrix completion is one of the popular techniques to deduce the relationship between two types of objects (i.e. users and items) in recommendation system. However, the standard algorithms of matrix completion working on real-valued or categorical observations fail to infer the binary relationship between the objects [20], such as MDA prediction. Therefore, we adopted a different technique in the next section.
Binary matrix completion
We state the problem as a matrix completion with 1-bit observation, in which each observed entry represents a positive (yes) or negative (no) response to MDA. Such a binary matrix completion can be defined as a generalized linear model,
$$ {a}_{ij}=\left\{\begin{array}{cc}+1& {x}_{ij}+{z}_{ij}\ge 0\\ {}-1& {x}_{ij}+{z}_{ij}<0\end{array}\right. $$
where only a subset Ω of entries of A is observed, X = {xij} is a low-rank parameterizing distribution matrix of A, and Z = {zij} is a stochastic matrix containing noise. The recovery of matrix X is usually transformed to another form to solve as follows [21].
Given an incomplete observed MDA matrix A ∈ ℝp × q, a subset of its observed entry subscripts Ω ⊂ [p] × [q] and a differentiable function f : ℝ → [0, 1], we observe
$$ {a}_{ij}=\left\{\begin{array}{cc}+1& \mathrm{with}\ \mathrm{the}\ \mathrm{probability}\kern0.5em f\left({x}_{ij}\right)\\ {}-1& \mathrm{with}\ \mathrm{the}\ \mathrm{probability}\kern0.5em 1-f\left({x}_{ij}\right)\end{array}\right.\kern1em \mathrm{for}\forall \left(i,j\right)\in \Omega $$
where [d] denotes the set of integers {1,..,d}. In other words, the entries of A depend on a p × q underlying low-rank preference matrix X = {xij} ∈ ℝp × q somehow (Fig. 2).
Binary matrix completion. The left matrix is the latent preference matrix. The right matrix is the observed matrix, in which the observed entries are labelled with '+ 1' if an MDA is found, with '-1' if a non-MDA is found, and '?' if the entry is not observed
We assume that the subscript subset Ω follows a binomial model, in which the subscript (i, j) ∈ [p] × [q] of each observed entry in A occurs with probability m/(pq) independently, where m is the cardinality (the number of observed entries) of Ω. The assumption reflects p×q independent experiments, of which each determines microbe-disease associations with m/(pq) success probability.
In addition, if we suppose that the entries of the underlying noising matrix Z are independently and identically drawn from the distribution, whose cumulative distribution function (CDF) is given by FZ(x) = P(z ≤ x) = 1 − f(−x), then the model in Formula (2) reduces to its special case in Formula (1). In such a sense, the selection of CDF f is equivalent to that of Z. Thus, X can be also viewed as a parameter of a distribution.
Since our aim is to determine the likelihood that a microbe would be associated with a particular disease, we naturally model MDA prediction as the problem that recovers the latent low-rank matrix X.
When defining the CDF f(xij) = 1 − Φ(−xij/σ) = Φ(xij/σ), where Φ is the cumulative distribution function of a standard Gaussian (a standard mean-zero Gaussian with variance σ2 for the noising matrix Z), Formula (2) captures a probit regression model. Thus, the recovery of X can be achieved by solving the following optimization problem [21],
$$ {\displaystyle \begin{array}{l}\widehat{\mathbf{X}}=\underset{\mathbf{X}}{\arg \max }{F}_{\Omega, \mathbf{A}}\left(\mathbf{X}\right)\\ {}{F}_{\Omega, \mathbf{A}}\left(\mathbf{X}\right)=\sum \limits_{\left(i,j\right)\in \Omega}\left(B\left({a}_{ij}=+1\right)\log \left(f\left({x}_{ij}\right)\right)+B\left({a}_{ij}=-1\right)\log \left(1-f\left({x}_{ij}\right)\right)\right)\\ {}f\left({x}_{ij}\right)=1-\Phi \left(-{x}_{ij}/\sigma \right)=\Phi \left({x}_{ij}/\sigma \right)\\ {}s.t.\kern1em {\left\Vert \mathbf{X}\right\Vert}_{\ast}\le \sqrt{rpq}\end{array}} $$
where B(ε) is the binary indicator function for an event ε(i.e. B(ε) = 1 if ε occurs and 0 otherwise), Φ(xij/σ) ∈ ℝ → [0, 1] is the cumulative distribution function of a standard Gaussian distribution with variance σ2, and r is the expected rank of X.
Consider that Formula (3) is just a special instance of the general formulation
$$ \underset{\mathbf{x}}{\min}\kern1em f\left(\mathbf{x}\right)\kern1em \mathrm{subject}\ \mathrm{to}\kern1.25em \mathbf{x}\in \boldsymbol{C} $$
where f(x) is a smooth convex function from ℝn → ℝ, and C is a closed convex set in ℝn. In particular, defining V as the bijective linear mapping that vectorizes ℝp × q to ℝpq, we have f(x) = − FΩ, A(V−1x) and C = V({X : ‖X‖∗ ≤ τ}). Therefore, non-monotone Spectral Projected Gradient (SPG) can be applied to solve the above optimization [22]. It is an iterative algorithm, which requires at each iteration the evaluation of f(x), its gradient g(x) = ∇f(x) and an orthogonal projection PC(v) onto C, PC(v) = arg min ‖x − v‖2 subject to x ∈ C. Since the orthogonal projection onto the nuclear-norm ball C amounts to singular-value soft thresholding [23], the projection is equivalent to
$$ {\boldsymbol{P}}_{\boldsymbol{C}}\left(\mathbf{X}\right)={\boldsymbol{S}}_{\lambda}\left(\mathbf{X}\right):= \mathbf{U}\max \left\{\boldsymbol{\Sigma} -\lambda \mathbf{I},0\right\}{\mathbf{V}}^T $$
where \( \mathbf{X}\overset{SVD}{=}{\mathbf{U}\boldsymbol{\Sigma } \mathbf{V}}^T \),Σ = diag (σ1, …, σn), the maximum operation is taken entry-wise and λ ≥ 0 is the smallest value for which \( {\sum}_{i=1}^n\max \left\{{\sigma}_i-\lambda \right\}\le \tau \).
Cross validation
As a standard technique, cross-validation (CV) is popularly adopted to evaluate the performance of machine learning models and estimate their power of generalization on future samples. Usually, there are two kinds of CV, k-fold cross-validation (k-CV) and leave-one-out cross-validation (LOOCV).
In the scheme of k-CV, all the observed samples are randomly split into k subsets of approximately equal size. Among them, one subset is taken as the testing set, in which the samples are masked as unobserved. Meanwhile, the remaining k-1 subsets are merged as the training set, in which the observed samples are used to train a predicting model. Once the training is done, the predicting model is performed on the testing set and outputs the confidence scores of being observed samples for all the masked samples. This procedure repeats k times by taking each subset as the testing set in turn. In each round of k-CV, the performance of the predicting model is measured and recorded. Its final performance is defined as the average of the performance in all the rounds.
LOOCV can be regarded as an extreme case of k-CV, where k is equal to the number of observed samples. In each step of LOOCV, each observed sample is blinded as an unobserved one and the remaining observed samples are used to build the predicting model. The procedure of LOOCV takes each of the observed samples as the testing sample in turn. When the number of samples is enough large, the results of k-CV and LOOCV have no significant difference in statistics.
The performance of MDA prediction is measured by Receiver Operating Characteristic (ROC) curve as well as Precision-Recall (PR) curve. Two measuring metrics adopted are both the Area Under ROC curve (AUC) and the Area Under PR curve (AUPR). One could easily obtain other metrics, such as true positive rate (TPR, Recall, or Sensitivity) and false positive rate (FPR, 1-Specificity), by setting thresholds on ROC or PR curves.
We adopted the same dataset of MDAs as that in [19]. The dataset was originally collected from the Human Microbe-Disease Association Database (HMDAD, http://www.cuilab.cn/hmdad), which was built in 2016 and published in 2017 [16]. HMDAD collected MDA entries from 61 publications in microbiome studies based on 16s RNA sequencing. Each entry is an experimentally supported association between diverse microbes and non-infectious diseases with sufficient samples. HMDAD provides a benchmark source for developing prediction model [19].
Originally, there are 483 MDAs, including 292 microbes and 39 human diseases in the dataset. After removing the duplicate MDAs, which come from different experiments, Chen et al. [19] give 450 distinct MDAs among those microbes and diseases, and organizes them into a 292×39 association matrix. The corresponding MDA network is shown in Fig. 3.
The Network of Microbe-Disease Associations. Blue triangles and red circles denote microbes and diseases respectively. Lines between nodes are the associations between them. The minimum, the median, the mean, and the maximum of microbe degrees are 1, 1, 1.54 and 11, while those of disease degrees are 1, 3, 11.54 and 167 respectively
Parameter tuning
In this section, we investigated the influence of two important parameters in Formula 2, the standard derivation σ and the estimated rank r. First, we tuned it from the list {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0}. Since the maximum rank rmax of the underlying matrix is equal to min(p, q), we then tuned r from the ratio list of \( \left\{\frac{1}{10},\frac{1}{9},\frac{1}{8},\frac{1}{7},\frac{1}{6},\frac{1}{5},\frac{1}{4},\frac{1}{3},\frac{1}{2},1\right\} \) w.r.t rmax and searched the best values on the 10 × 10 grid expanded by both σ and r.
Considering that AUPR is a better metric than AUC when the number of positive samples is significantly less than that of negative samples [24], we recorded the performance of BMCMDA for each pairwise value of (σ, r) under 5-CV in terms of AUPR (Fig. 4). When running BMCMDA, all the parameters (e.g. the number of iterations and the tolerance of stopping iteration) in SPG were set to their default values.
Illustration of Determining the Best Value Pair of (σ, r). The position w.r.t (σ∗, r∗) is highlighted by a white circle
Finally, we picked up the pair of \( \left({\sigma}^{\ast },{r}^{\ast}\right)=\left(0.2,\frac{1}{3}{r}_{max}\ \right) \), which achieves the highest one among 100 values of AUPR, as the best value of (σ, r), and further applied them in all the following experiments.
Comparison with the state-of-the-art approach
With the best pair (σ∗, r∗), we compared BMCMDA with three approaches, including one baseline approach and two state-of-the-art approaches, RKNNMDA [25] and KATZHMDA [19]. The baseline approach directly applies singular value decomposition (SVD) on the MDA adjacency matrix with missing entries and uses the product of two unitary matrices and the rectangle diagonal matrix to recover the missing values. RKNNMDA was originally designed for miRNA-disease associations [25]. It performs MDA prediction by directly applying a ranking-based KNN on the MDA prediction [19]. KATZHMDA also constructs a heterogeneous network, which consists of the known MDA network and two MDA-induced networks [19]. The first MDA-induced network indicates a microbe similarity network, while the second one accounts for a disease similarity network. Both of them are derived from the MDA network by Gaussian interaction profile kernel. By leveraging KATZ index to calculate similarities between microbe nodes and disease nodes in the heterogeneous network, KATZHMDA infers the potential association between a microbe node and a disease node if the value of their KATZ index is large. The comparison was performed with the exactly same dataset under LOOCV as mentioned in [19]. The results in Fig. 5. show that BMCMDA wins the best and outperforms those approaches significantly.
Comparison with state-of-the-art approaches
Furthermore, we selected the second best approach KATZHMDA to make a detailed comparison. Considering the fact that AUPR is a better metric than AUC when the number of positive samples is significantly less than that of negative samples [24], we measured the prediction by not only ROC curves but also PR curves. The results illustrated in Fig. 6 show that BMCMDA, compared with KATZHMDA, achieves a significant improvement of both ~ 7% increment in terms of AUC and ~ 5% increment in terms of AUPR.
Comparison between BMCMDA and KATZHMDA in terms of ROC curve and PR curve
As the complement of biological experiments, computational methods have a potential to be a promising approach, which predicts MDA candidates rapidly among a plenty of microbe-disease pairs with the advantage of no limitation on microbe cultivation.
In this paper, we have modeled MDA prediction in a novel sight, which utilizes an underlying real-valued matrix to reflect the magnitude of MDAs and regards the binary MDA adjacent matrix as its incomplete and noisy observation. Upon this model, we have proposed a new approach based on Binary Matrix Completion (BMCMDA) to predict potential MDAs among a large scale of microbe-disease pairs. The comparison with other state-of-the-art approaches demonstrates the superiority of BMCMDA for predicting microbe-disease associations on a large scale and also validates that the assumption we adopted is reasonable. Obviously, BMCMDA can be directly applied to other similar forms of problems in bioinformatics, including the inference of the binary relationship between mono-partite objects (e.g. protein-protein interaction, drug-drug interaction [26, 27] and drug combination [28]) or that between bi-partite objects (e.g. drug-target interaction [29, 30], gene-disease association, RNA-disease association [31]).
In addition, we consider the possible improvement of BMCMDA. First, we may enhance the MDA prediction by integrating additional and independent microbe/disease similarities or features with BMCMDA. Secondly, as suggested in [31], we may generalize BMCMDA to be appropriate in more predicting scenarios, including the prediction of the associations between newly-found microbes (having no known MDA) and existing diseases, the prediction of the associations between existing microbes and newly-concerned diseases (having no known MDA), and the prediction of the associations between newly-found microbes and newly-concerned diseases.
AUC:
The area under the receiver operating characteristic curve
AUPR:
The area under precision-recall curve
BMCMDA:
Binary Matrix Completion for predicting human Microbe-Disease Associations
CDF:
Cumulative distribution function
HMDAD:
Human Microbe-Disease Association Database
LOOCV:
Leave-one-out cross-validation
MDA:
Microbe-disease association
SPG:
Spectral Projected Gradient
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Shi J-Y, Huang H, Li J-X, Lei P, Zhang Y-N, Yiu S-M. Predicting comprehensive drug-drug interactions for new drugs via triple matrix factorization. In: IWBBIO: 2017; Spain. Lecture notes in computer science: bioinformatics and biomedical engineering. Granada: Springer; 2017. p. 108–17.
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The abridged 2-page abstract of this work was previously published in the Proceedings of the 13th International Symposium on Bioinformatics Research and Applications (ISBRA 2017), Lecture Notes in Computer Science: Bioinformatics Research and Applications [32].
This work was supported by RGC Collaborative Research Fund (CRF) of Hong Kong (C1008-16G), National High Technology Research and Development Program of China (No. 2015AA016008), the Fundamental Research Funds for the Central Universities of China (No. 3102015ZY081), the Program of Peak Experience of NWPU (2016), and China National Training Programs of Innovation and Entrepreneurship for Undergraduates (No. 201710699330). The publication charge was funded by China National Training Programs of Innovation and Entrepreneurship for Undergraduates (No. 201710699330).
The dataset of MDA used in this work can be download from https://github.com/JustinShi2016/ISBRA2017
About this supplement
This article has been published as part of BMC Bioinformatics Volume 19 Supplement 9, 2018: Selected articles from the 13th International Symposium on Bioinformatics Research and Applications (ISBRA 2017): bioinformatics. The full contents of the supplement are available online at https://bmcbioinformatics.biomedcentral.com/articles/supplements/volume-19-supplement-9.
School of Life Sciences, Northwestern Polytechnical University, Xi'an, 70072, China
Jian-Yu Shi
& Jiang-Bo Cao
School of Software and Microelectronics, Northwestern Polytechnical University, Xi'an, 70072, China
Hua Huang
School of Computer Science, Northwestern Polytechnical University, Xi'an, 70072, China
Yan-Ning Zhang
Department of Computer Science, The University of Hong Kong, Hong Kong, 999077, China
Siu-Ming Yiu
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JYS and YNZ conceived, designed and carried out the experiments. JYS and SMY drafted the manuscript. HH collected the heterogeneous data. JYS performed the experiments. JBC answers the final round of textual comments. JYS and SMY analysed the data. JYS and HH developed the codes used in the analysis. All authors read and approved the final manuscript.
Correspondence to Jian-Yu Shi.
Shi, J., Huang, H., Zhang, Y. et al. BMCMDA: a novel model for predicting human microbe-disease associations via binary matrix completion. BMC Bioinformatics 19, 281 (2018) doi:10.1186/s12859-018-2274-3
Matrix completion
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What is the value of the 25th term of the arithmetic sequence $2,
5, 8, \ldots$?
The common difference is $5 - 2 = 3$, so the $25^{\text{th}}$ term is $2 + 3 \cdot 24 = \boxed{74}$. | Math Dataset |
\begin{document}
\title{ Characterization of hypersurfaces in four dimensional product spaces via two different $\Spinc$ structures} \begin{abstract}
The Riemannian product $\mathbb M_1(c_1) \times \mathbb M_2(c_2)$, where $\mathbb M_i(c_i)$ denotes the $2$-dimensional space form of constant sectional curvature $c_i \in \mathbb R$, has two different $\mathrm{Spin^c}$ structures carrying each a parallel spinor. The restriction of these two parallel spinor fields to a $3$-dimensional hypersurface $M$ characterizes the isometric immersion of $M$ into $\mathbb M_1(c_1) \times \mathbb M_2(c_2)$. As an application, we prove that totally umbilical hypersurfaces of $\mathbb M_1(c_1) \times \mathbb M_1(c_1)$ and totally umbilical hypersurfaces of $\mathbb M_1(c_1) \times \mathbb M_2(c_2)$ ($c_1 \neq c_2$) having a local structure product, are of constant mean curvature. \end{abstract} {\bf Keywords:} $\mathrm{Spin^c}$ structures on hypersurfaces, totally umbilcal hypersurfaces, parallel $\mathrm{Spin^c}$ spinors, generalized Killing $\mathrm{Spin^c}$ spinors, K\"ahler manifolds.\\\\ {\bf Mathematics subject classifications (2010):} 53C27, 53C40, 53C80.
\section{Introduction}
Over the past years, the real spinorial ($\mathrm{Spin}$ geometry) and the complex spinorial ($\mathrm{Spin^c}$ geometry) approaches have been used fruitfully to characterize (\cite{Fr, Mo, Rot, blr, brj, blr13, LR, NR12, nr17} and references therein) submanifolds of some special ambient manifolds. These approaches allowed also to study the geometry and topology of submanifolds and solve naturally some extrinsic problems. For instance, elementary proofs of the Alexandrov theorem in the Euclidean space \cite{hmz}, in the hyperbolic space \cite{hmr} and in the Minkowski spacetime \cite{hmr} were obtained (see also \cite{HM13, HM14}). In 2006, O. Hijazi, S. Montiel and F. Urbano \cite{omu} constructed on K\"ahler-Einstein manifolds with positive scalar curvature, a $\mathrm{Spin^c}$ structure carrying K\"ahlerian Killing spinors. The restriction of these spinors to minimal Lagrangian submanifolds provides topological and geometric restrictions on these submanifolds. The authors \cite{nr17, nakadroth, NR12}, and by restricting $\mathrm{Spin^c}$ spinors, gave an elementary $\mathrm{Spin^c}$ proof for a Lawson type correspondence between constant mean curvature surfaces of $3$-dimensional homogeneous manifolds with $4$-dimensional isometry group \cite{Da2}. Furthermore, they gave necessary and sufficient geometric conditions to immerse a $3$-dimensional Sasaki manifold and a complex/Lagrangian surface into the complex projective space of complex dimension $2$.\\
The main idea behind characterizing hypersurfaces of $\mathrm{Spin}$ or $\mathrm{Spin^c}$ manifolds is the restriction to the hypersurface of a special spinor field (parallel, real Killing, imaginary Killing, K\"ahlerian Killing...). For example, the restriction $\phi$ of a parallel spinor field on a Riemannian $\mathrm{Spin}$ or $\mathrm{Spin^c}$ manifold to an oriented hypersurface $M$ is a solution of the generalized Killing equation \begin{eqnarray} \nabla_X\phi= -\frac 12 \gamma(II X) \phi, \end{eqnarray} where $\gamma$ and $\nabla$ are respectively the Clifford multiplication and the $\mathrm{Spin}$ or $\mathrm{Spin^c}$ connection on $M$, the tensor $II$ is the Weingarten tensor of the immersion and $X$ any vector field on $M$. Conversely and in the two-dimensional case, the existence of a generalized Killing $\mathrm{Spin}$ spinor field allows to immerse $M$ in $\mathbb R^3$ \cite{Fr}. This characterization has been extended to surfaces of other $3$-dimensonal (pseudo-) Riemannian manifolds \cite{Mo, Rot, LR11}. Moreover, the existence of a generalized Killing $\mathrm{Spin^c}$ spinor on a surface $M$ allows to immerse $M$ in the $3$-dimensional homogeneous manifolds with 4-dimensional isometry group \cite{NR12}. All these previous results are the geometrical invariant versions of previous works on the spinorial Weierstrass representation by R. Kusner and N. Schmidt, B. Konoplechenko, I. Taimanov and many others (see \cite{kono, sch, ta}). \\
In the three dimensional case, having a generalized Killing $\mathrm{Spin}$ or $\mathrm{Spin^c}$ spinor is not sufficient to characterize the immersion of $M$ in the desired $4$-dimensional manifold. The problem is that unlike in the $2$-dimensional case, the spinor bundle of a $3$-dimensional manifold does not decompose into subbundles of positive and negative half-spinors. In fact, Morel \cite{Mo} proved that the existence of a Codazzi generalized Killing $\mathrm{Spin}$ spinor on a $3$-dimensional manifold $M$ is equivalent to immerse $M$ in $\mathbb R^4$. But it was proved in \cite{blr13, roth14} that restricting a $\mathrm{Spin}$ structure with a spinor field having non-vanishing positive and negative parts is required to get the integrability condition of an immersion in the desired $4$-dimensional target space. Hence, Morel's result has been reformulated for hypersurfaces of $\mathbb R^4$ \cite{LR} because $\mathbb R^4$ has a $\mathrm{Spin}$ structure with positive and negative parallel spinors. The restriction of both spinors to $M$ gives two generalized Killing spinors which, conversely, allow to characterize the immersion of $M$ in $\mathbb R^4$. This result has been extended to other 4-dimensional space forms and product spaces, that is $\mathbb S^4$, $\mathbb H^4$, $\mathbb S^3 \times \mathbb R$ and $\mathbb H^3\times \mathbb R$ \cite{LR}. In the $\mathrm{Spin^c}$ setting, the existence of a Codazzi generalized Killing $\mathrm{Spin^c}$ spinor on a $3$-dimensional manifold $M$ is equivalent to immerse $M$ in the 2-dimensional complex space form $\mathcal M_2 (k)$ of constant holomorphic sectional curvature $4k$ \cite{NR12}. However here, the condition ``Codazzi" cannot be dropped as in the $\mathrm{Spin}$ case, because $\mathcal M_2 (k)$ has only two different $\mathrm{Spin^c}$ structures (the canonical and the anti-canonical $\mathrm{Spin^c}$ structures) carrying each one parallel spinor lying in the positive half-part of the corresponding $\mathrm{Spin^c}$ bundles. \\
The aim of the present article is to use $\mathrm{Spin^c}$ geometry to characterize hypersurfaces of the Riemannian product $\mathbb M_1(c_1) \times \mathbb M_2(c_2)$, where $\mathbb M_i(c_i)$ denotes the $2$-dimensional space form of constant sectional curvature $c_i \in \mathbb R$. The key starting point is that this product has two different $\mathrm{Spin^c}$ structures carrying each a non-vanishing parallel spinor. The first structure $S_1$ is the product of the canonical $\mathrm{Spin^c}$ structure on $\mathbb M_1(c_1)$ with the canonical $\mathrm{Spin^c}$ structure on $\mathbb M_2(c_2)$ and it has a non-vanishing parallel spinor lying in the positive half-part of the $\mathrm{Spin^c}$ bundle. The second structure $S_2$ is the product of the canonical $\mathrm{Spin^c}$ structure on $\mathbb M_1(c_1)$ with the anti-canonical $\mathrm{Spin^c}$ structure on $\mathbb M_2(c_2)$ and it has a non-vanishing parallel spinor lying in the negative half-part of the $\mathrm{Spin^c}$ bundle. Having said that one could expect that restricting both structures $S_1$ and $S_2$, and hence both parallel spinors, to a hypersurface $M$ of $\mathbb M_1(c_1) \times \mathbb M_2(c_2)$ could allow to characterize the immersion. \\
We denote by $\nabla^j$, $\gamma_j$ and $i\Omega^j$ respectively the Clifford multiplication, the $\mathrm{Spin^c}$ connection and the curvature of the auxiliary line bundle on the hypersurface $M$ obtained after restricting the $\mathrm{Spin^c}$ structure $S_j$ on $\mathbb M_{1} (c_1) \times \mathbb M_2(c_2)$ (here $j \in \{1, 2\}$). The main theorem of the paper is: \begin{thm}\label{thmCM} Let $\big(M^3,g=(.,.)\big)$ be a simply connected oriented Riemannian manifold endowed with an almost contact metric structure
$(\mathfrak{X},\xi,\eta)$. Let $E$ be a field of symmetric endomorphisms on $M$, $h$ a function on $M$ and $V$ a vector field on $M$. Then, the following statements are equivalent: \begin{enumerate} \item There exists an isometric immersion of $(M^3,g)$ into $\mathbb M_{1} (c_1) \times \mathbb M_2(c_2)$ with shape operator $E$ and so that, over $M$, the complex structure of $\mathbb M_{1} (c_1) \times \mathbb M_2(c_2)$ is given by $J=\mathfrak{X}+\eta(\cdot)\nu$, where $\nu$ is the unit normal vector of the immersion and the product structure is given by $F = f + (V, \cdot) \nu$ for some endomorphism $f$ on $M$. \item There exists two $\mathrm{Spin^c}$ structures on $M$ carrying each one a non-trivial spinor $\varphi_1$ and $\varphi_2$ satisfying $$\nabla^1_X\varphi_1=-\frac{1}{2}\gamma_1(EX)\varphi_1\ \ \ \text{and}\ \ \ \gamma_1(\xi)\varphi_1= - i\varphi_1.$$ $$\nabla^2_X\varphi_2=\frac{1}{2}\gamma_2(EX)\varphi_2\ \ \ \text{and}\ \ \ \ \gamma_2(V) \varphi_2 = -i \gamma_2(\xi) \varphi_2 + h \varphi_2.$$ The curvature 2-form $i\Omega^j$ of the connection on the auxiliary bundle associated with each $\mathrm{Spin^c}$ structure is given by ($j \in \{1, 2\}$) $$\left\{ \begin{array}{l} \Omega^j(e_1, e_2) = \frac 12 (-1)^{j-1}c_1 (h-1) -\frac 12 c_2 (h+1),\\ \Omega^j(e_1, \xi) = \frac 12 \Big( (-1)^{j-1} c_1 - c_2\Big) (e_1, V),\\ \Omega^j (e_2, \xi) = \frac 12 \Big( (-1)^{j-1} c_1 - c_2\Big) (e_2, V), \end{array} \right. $$ in the basis $\{e_1,e_2=\mathfrak{X} e_1,e_3=\xi\}$.
\end{enumerate} \end{thm}
Again, these two $\mathrm{Spin^c}$ structures (resp. two generalized Killing $\mathrm{Spin^c}$ spinors) on $M$ comes from the restriction of the two $\mathrm{Spin^c}$ structures $S_1$ and $S_2$ (resp. the two parallel spinors) on $\mathbb M_{1} (c_1) \times \mathbb M_2(c_2)$. Needless to say, when $c_1 = c_2=0$, these two $\mathrm{Spin^c}$ structures on $M$ coincide and it is in fact the $\mathrm{Spin}$ structure coming from the restriction of the unique $\mathrm{Spin}$ structure on $\mathbb R^4$ having positive and negative parallel spinors. When $c_1\neq 0$ or $c_2 \neq 0$, the two structures in $M$ are different because they are the restriction of the two different structures $S_1$ and $S_2$ .\\
As an application of Theorem \ref{thmCM}, we prove that totally umbilical hypersurfaces of $\mathbb M_1(c_1) \times \mathbb M_1(c_1)$ and totally umbilical hypersurfaces of $\mathbb M_1(c_1) \times \mathbb M_2(c_2)$ ($c_1 \neq c_2$) having a local structure product are of constant mean curvature (see Proposition \ref{pr1} and Proposition \ref{pr2}).
\section{Preliminaries} In this section we briefly introduce basic facts about $\mathrm{Spin^c}$ geometry of hypersurfaces (see \cite{LM, montiel, fr1, r1, r2}). Then we describe hypersurfaces of the Riemannian product $\mathbb M_1(c_1) \times \mathbb M_2(c_2)$ \cite{Ko, LTV}, where $\mathbb M_i(c_i)$ denotes the $2$-dimensional space form of constant sectional curvature $c_i \in \mathbb R$.
\subsection{Hypersurfaces and induced Spin$^c$ structures}
\underline{{\bf Spin$^c$ structures on manifolds:}} Let $(N^{n+1}, \overline g)$ be a Riemannian $\mathrm{Spin^c}$ manifold of dimension $n+1\geq 2$ without boundary. On such a manifold, we have a Hermitian complex vector bundle $\Sigma N$ endowed with a natural scalar product $(., .)$ and with a connection $\nabla^N $ which parallelizes the metric. This complex vector bundle, called the $\mathrm{Spin^c}$ bundle, is endowed with a Clifford multiplication denoted by $``\cdot"$, $\cdot: TN \rightarrow \mathrm{End}_{\mathbb C} (\Sigma N)$, such that at every point $x \in N$, defines an irreducible representation of the corresponding Clifford algebra. Hence, the complex rank of $\Sigma N$ is $2^{[\frac {n+1}{2}]}$. Given a $\mathrm{Spin^c}$ structure on $(N^{n+1}, \overline g)$, one can prove that the determinant line bundle $\mathrm{det} (\Sigma N)$ has a root of index $2^{[\frac{n+1}{2}]-1}$. We denote by $L^N$ this root line bundle over $N$ and call it the auxiliary line bundle associated with the $\mathrm{Spin^c}$ structure. Locally, a Spin structure always exists. We denote by $\Sigma' N$ the (possibly globally non-existent) spinor bundle. Moreover, the square root of the auxiliary line bundle $L^N$ always exists locally. But, $\Sigma N = \Sigma' N \otimes (L^N)^{\frac 12}$ exists globally. This essentially means that, while the spinor bundle and $(L^N)^{\frac 12}$ may not exist globally, their tensor product (the $\mathrm{Spin^c}$ bundle) is defined globally. Thus, the connection $\nabla^N$ on $\Sigma N$ is the twisted connection of the one on the spinor bundle (coming from the Levi-Civita connection, also denoted by $\nabla^N$) and a fixed connection on $L^N$. \\
We may now define the Dirac operator $D^N$ acting on the space of smooth sections of $\Sigma N$ by the composition of the metric connection and the Clifford multiplication. In local coordinates this reads as \begin{align*} D^N =\sum_{j=1}^{n+1} e_j \cdot \nabla^N_{e_j},
\end{align*} where $\{e_1,\ldots,e_{n+1}\}$ is a local oriented orthonormal tangent frame. It is a first order elliptic operator, formally self-adjoint with respect to the $L^2$-scalar product and satisfies, for any spinor field $\psi$, the Schr\"odinger-Lichnerowicz formula \begin{eqnarray} (D^N)^2\psi= (\nabla^N)^* \nabla^N\psi+\frac{1}{4}S\psi+\frac{i}{2}\Omega^N\cdot\psi,
\label{bochner} \end{eqnarray} where $S$ is the scalar curvature of $N$, $ (\nabla^N)^*$ is the adjoint of $\nabla^N$ with respect to the $L^2$ scalar product, $i\Omega^N$ is the curvature of the fixed connection on the auxiliary line bundle $L^N$ ($\Omega^N$ is a real $2$-form on $N$) and $\Omega^N\cdot$ is the extension of the Clifford multiplication to differential forms. For any $X \in \Gamma(TN)$, the Ricci identity is given by \begin{eqnarray} \sum_{k=1}^{n+1} e_k \cdot \mathcal{R}^N(e_k,X)\psi = \frac 12 \mathrm{Ric}^N(X) \cdot \psi -\frac i2 (X\lrcorner\Omega^N)\cdot\psi, \label{RRicci} \end{eqnarray} where $\mathrm{Ric}^N$ is the Ricci curvature of $(N^{n+1}, \overline g)$ and $\mathcal{R}^N$ is the curvature tensor of the spinorial connection $\nabla^N$. In odd dimension, the volume form $\omega_{\mathbb{C}} := i^{[\frac{n+2}{2}]} e_1 \cdot...\cdot e_{n+1}$ acts on $\Sigma N$ as the identity, i.e., $\omega_\mathbb{C} \cdot\psi = \psi$ for any spinor $\psi \in \Gamma(\Sigma N)$. Besides, in even dimension, we have $\omega_\mathbb{C}^2 =1$. We denote by $\Sigma^\pm N$ the eigenbundles corresponding to the eigenvalues $\pm 1$, hence $\Sigma N= \Sigma^+ N\oplus \Sigma^- N$ and a spinor field $\psi$ can be written $\psi = \psi^+ + \psi^-$. The conjugate $\overline \psi$ of $\psi$ is defined by $\overline \psi = \psi^+ - \psi^-$.\\
Every $\mathrm{Spin}$ manifold has a trivial $\mathrm{Spin^c}$ structure \cite{fr1}. In fact, we choose the trivial line bundle with the trivial connection whose curvature is zero. Also every K\"ahler manifold $(N, J, \overline g)$ of complex dimension $m$ ($n+1=2m$) has a canonical $\mathrm{Spin^c}$ structure coming from the complex structure $J$. Let $\ltimes$ be the K\"{a}hler form defined by the complex structure $J$, i.e. $\ltimes (X, Y)= \overline g(JX, Y)$ for all vector fields $X,Y\in \Gamma(TN).$ The complexified tangent bundle $T^\mathbb{C} N =TN \otimes_\mathbb{R} \mathbb{C}$ decomposes into $$T^\mathbb{C} N = T_{1,0} N\oplus T_{0,1} N,$$ where $T_{1,0} N$ (resp. $T_{0,1} N$) is the $i$-eigenbundle (resp. $-i$-eigenbundle) of the complex linear extension of the complex structure. Indeed,
$$T_{1,0}N = \overline{T_{0,1}N} = \{ X- iJX\ \ | X\in \Gamma(TN)\}.$$ Thus, the spinor bundle of the canonical $\mathrm{Spin^c}$ structure is given by $$\Sigma N = \Lambda^{0,*} N =\bigoplus_{r=0}^m \Lambda^r (T_{0,1}^* N),$$ where $T_{0,1}^*N$ is the dual space of $T_{0,1} N$. The auxiliary bundle of this canonical $\mathrm{Spin^c}$ structure is given by $L^N = (K_{N})^{-1}= \Lambda^m (T_{0,1}^* N)$, where $K_{N}= \Lambda^m (T_{1,0}^* N)$ is the canonical bundle of $N$ \cite{fr1}. This line bundle $ L^N$ has a canonical holomorphic connection
induced from the Levi-Civita connection whose curvature form is given by $i\Omega^N = -i\rho$, where $\rho$ is the Ricci form given by $\rho(X, Y) = \mathrm{Ric}^N(JX, Y)$ for all $X, Y \in \Gamma(TN)$. Hence, this $\mathrm{Spin^c}$ structure carries parallel spinors (the constant complex functions) lying in the set of complex functions $\Lambda^{0, 0}N\subset \Lambda^{0, *} N$ \cite{Moro1}. Of course, we can define another $\mathrm{Spin^c}$ structure for which the spinor bundle is given by $\Lambda^{*, 0} N =\bigoplus_{r=0}^m \Lambda^r (T_{1, 0}^* N)$ and the auxiliary line bundle by $K_{N}$. This $\mathrm{Spin^c}$ structure will be called the anti-canonical $\mathrm{Spin^c}$ structure \cite{fr1} and it carries also parallel spinors (the constant complex functions) lying in the set of complex functions $\Lambda^{0, 0}N\subset \Lambda^{0, *} N$ \cite{Moro1}.\\
For any other $\mathrm{Spin^c}$ structure on the K\"ahler manifold $N$, the spinorial bundle can be written as \cite{fr1, omu}: $$\Sigma N = \Lambda^{0,*}N \otimes\mathfrak L,$$ where $\mathfrak L^2 = K_{N}\otimes L^N$ and $L^N$ is the auxiliary bundle associated with this $\mathrm{Spin^c}$ structure. In this case, the $2$-form $\ltimes$ can be considered as an endomorphism of $\Sigma N$ via
Clifford multiplication and it acts on a spinor field $\psi$ locally by \cite{kirch, fr1}: $$\ltimes\cdot\psi = \frac 12 \sum_{j=1}^{m} e_j\cdot Je_j\cdot\psi.$$ Hence, we have the well-known orthogonal splitting \begin{align}\label{splitt}\Sigma N = \bigoplus_{r=0}^{m}\Sigma_r N, \end{align} where $\Sigma_r N$ denotes the eigensubbundle corresponding to the eigenvalue $i(m-2r)$ of $\ltimes$, with complex rank $\Big(^m_k\Big)$. The bundle $\Sigma_r N$ corresponds to $\Lambda^{0, r}N \otimes\mathfrak L$. Moreover, $$\Sigma^+N = \bigoplus_{r \ \text{even}} \Sigma_r N\ \ \ \text{and} \ \ \ \Sigma^-N = \bigoplus_{r \ \text{odd}} \Sigma_r N.$$ For the canonical (resp. the anti-canonical) $\mathrm{Spin^c}$ structure, the subbundle $\Sigma_0 N$ (resp. $\Sigma_m N$) is trivial, i.e., $\Sigma_0 N = \Lambda^{0, 0} N \subset \Sigma^+N $ (resp. $\Sigma_m N= \Lambda^{0, 0}N$ which is in $\Sigma^+ N$ if $m$ is even and in $\Sigma^-N$ if $m$ is odd). \\
The product $N_1 \times N_2 $ of two K\"ahler $\mathrm{Spin^c}$ manifolds is again a $\mathrm{Spin^c}$ manifold. We denote by $m_1$ (resp. $m_2$) the complex dimension of $N_1$ (resp. $N_2$). The spinor bundle is identified by $$\Sigma (N_1\times N_2) \simeq \Sigma N_1\otimes \Sigma N_2,$$ via the Clifford multiplication denoted also by ``$\cdot$": $$(X_1 + X_2)\cdot(\psi_1 \otimes \psi_2) = X_1\cdot\psi_1 \otimes \psi_2 + \overline \psi_1 \otimes X_2\cdot\psi_2, $$ where $X_1 \in \Gamma(TM_1)$, $X_1 \in \Gamma(TM_2)$, $\psi_1 \in \Gamma (\Sigma M_1)$ and $\psi_2 \in \Gamma (\Sigma M_2)$. We consider the decomposition (\ref{splitt}) of $\Sigma N_1$ and $\Sigma N_2$ with respect to their K\"ahler forms $\ltimes^{N_1}$ and $\ltimes^{N_2}$. Then, the corresponding decomposition of $\Sigma (N_1\times N_2)$ into eigenbundles of $\ltimes^{N_1\times N_2}= \ltimes^{N_1} + \ltimes^{N_2}$ is: $$\Sigma (N_1\times N_2) \simeq \bigoplus_{k=0}^{m_1+m_2}\Sigma_r(N_1\times N_2),$$ with $$\Sigma_r (N_1\times N_2) \simeq \bigoplus_{k=0}^{r} \Sigma_kN_1 \otimes \Sigma_{r-k}N_2,$$ since the K\"ahler form $\ltimes^{N_1\times N_2}$ acts on a section of $\Sigma_kN_1 \otimes \Sigma_{r-k}N_2$ as $$\ltimes^{N_1\times N_2}(\psi_1\otimes \psi_2) = \ltimes^{N_1}\cdot \psi_1 \otimes\psi_2 +\psi_1\otimes \ltimes^{N_2} \cdot\psi_2 = i (m_1+m_2-2r)\psi_1\otimes\psi_2.$$
\underline{{\bf Spin$^c$ hypersurfaces and the Gauss formula:}} Let $N$ be an oriented ($n+1$)-dimensional Riemannian $\mathrm{Spin^c}$ manifold and $M\subset N$ be an oriented hypersurface. The manifold $M$ inherits a $\mathrm{Spin^c}$ structure induced from the one on $N$, and we have \cite{r2} $$ \Sigma M\simeq \left\{ \begin{array}{l}
\Sigma N_{|_M} \ \ \ \ \ \ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if\ $n$ is even,} \\\\
\Sigma^+ N_{|_M} \ \text{or}\ \ \Sigma^- N_{|_M}\ \text{\ if\ $n$ is odd.} \end{array} \right. $$ Moreover the Clifford multiplication by a vector field $X$, tangent to $M$, is denoted by $\gamma$ and given by \begin{eqnarray}
\gamma(X)\phi = (X\cdot\nu\cdot \psi)_{|_M}, \label{Clifford} \end{eqnarray} where $\psi \in \Gamma(\Sigma N)$ (or $\psi \in \Gamma(\Sigma^+ N)$ if $n$ is odd), $\phi$ is the restriction of $\psi$ to $M$, ``$\cdot$'' is the Clifford multiplication on $N$, $\gamma$ that on $M$ and $\nu$ is the unit inner normal vector. If $\psi \in \Gamma(\Sigma^- N)$ when $n$ is odd, then we have \begin{eqnarray}
\gamma(X)\phi = - (X\cdot\nu\cdot \psi)_{|_M}. \label{Clifford-} \end{eqnarray}
The curvature 2-form $i\Omega$ on the auxiliary line bundle $L=L^N_{\vert_M}$ defining the $\mathrm{Spin^c}$ structure on $M$ is given by $i\Omega= {i\Omega^N}_{|_M}$. For every $\psi \in \Gamma(\Sigma N)$ ($\psi \in \Gamma(\Sigma^+ N)$ if $n$ is odd), the real 2-forms $\Omega$ and $\Omega^N$ are related by \cite{r2}: \begin{eqnarray}
(\Omega^N \cdot\psi)_{|_M} = \gamma(\Omega)\phi - \gamma(\nu\lrcorner\Omega^N)\phi. \label{glucose} \end{eqnarray} When $\psi \in \Gamma(\Sigma^-N)$ if $n$ is odd, we have \begin{eqnarray}
(\Omega^N \cdot\psi)_{|_M} = \gamma(\Omega)\phi + \gamma(\nu\lrcorner\Omega^N)\phi. \label{g-} \end{eqnarray} We denote by $\nabla$ the spinorial Levi-Civita connection on $\Sigma M$. For all $X\in \Gamma(TM)$ and $\psi \in \Gamma(\Sigma^+N)$, we have the $\mathrm{Spin^c}$ Gauss formula \cite{r2}: \begin{equation}
(\nabla^{\Sigma N}_X\psi)_{|_M} = \nabla_X \phi + \frac 12 \gamma(II X)\phi, \label{spingauss} \end{equation} where $II$ denotes the Weingarten map of the hypersurface. If $\psi \in \Gamma(\Sigma^-N)$, we have \begin{equation}
(\nabla^{\Sigma N}_X\psi)_{|_M} = \nabla_X \phi - \frac 12 \gamma( II X)\phi, \label{spingauss-} \end{equation} for all $X \in \Gamma(TM)$.
\subsection{Basic facts about $\mathbb M_1(c_1) \times \mathbb M_2(c_2)$ and their real hypersurfaces} \label{cm} Let ($\mathbb M_1(c_1) \times \mathbb M_2(c_2), \overline g)$ be the Riemannian product of $\mathbb M_1(c_1)$ and $ \mathbb M_2(c_2)$, where $M_i(c_i)$ denotes the space form of constant sectional curvature $c_i$ and $\overline g$ denotes the product metric. Consider $\big(M^3, g=(.,.)\big)$ an oriented real hypersurface of $\mathbb M_1(c_1) \times \mathbb M_2(c_2)$ endowed with the metric $g := (\cdot, \cdot)$ induced by $\overline{g}$. The product structure of $\mathbb P:= \mathbb M_1(c_1) \times \mathbb M_2(c_2)$ is given by the map $F : T \mathbb P \rightarrow T \mathbb P$ defined, for $X_1 \in \Gamma(T \mathbb M_1(c_1))$ and $X_2 \in \Gamma(\mathbb TM_2(c_2))$, by \begin{eqnarray} F(X_1 + X_2) = X_1 - X_2. \end{eqnarray} The map $F$ satisfies $F^2 = \mathrm{Id}_{T\mathbb P}, F \neq \mathrm{Id}_{T\mathbb P}$, where $\mathrm{Id}_{T\mathbb P}$ denotes the identity map on $T\mathbb P$. Denoting the Levi-Civita connection on $\mathbb P$ by $\nabla^{\mathbb P}$, we have $\nabla^{\mathbb P} F =0$ and for any $X, Y \in \Gamma(T\mathbb P)$, we have $\overline g(FX, Y) = \overline g (X, FY)$.
The product structure $F$ induces the existence on $M$ of a vector $V \in \Gamma(TM)$, a function $h: M \rightarrow \mathbb R$ and an endomorphism $f: TM \rightarrow TM$ such that, for all $X \in \Gamma(TM)$, \begin{eqnarray} FX = f X + (V, X) \nu \ \ \ \text{and}\ \ \ \ \ \ F\nu = V +h\nu, \end{eqnarray} where $\nu$ is the unit normal vector of the immersion. \begin{lem}\label{lem1}
The function $f$ is symmetric. Moreover, for any $X \in \Gamma(TM)$, we have \begin{eqnarray}f^2X + (V, X)V = X,\label{Structure1} \end{eqnarray} \begin{eqnarray}\label{Structure2} f V = -hV, \end{eqnarray} \begin{eqnarray} \label{Structure3} h^2 + \Vert V \Vert^2 = 1.\end{eqnarray}
\end{lem} \begin{proof} First of all, for any $X, Y \in \Gamma(TM)$, we have \begin{eqnarray*} (fX, Y) &=& \overline g(fX, Y)= \overline g\big(FX - (V, X)\nu, Y\big) = \overline g(FX, Y) \\ &=& \overline g(X, FY) = \overline g(X, fY + (V, Y)\nu) = (X, fY). \end{eqnarray*} Hence $f$ is symmetric. For any $X \in \Gamma(TM)$, $F^2 X = X$. This means that $$(f+ (V, X) \nu)^2 (X) = X,$$ and hence $$\left \{ \begin{array}{l} f^2 X + (V, X) V = X,\\ (V, fX) + h(V, X) = 0, \end{array} \right. $$ which are Equation (\ref{Structure1}) and Equation (\ref{Structure2}). We also have $F^2 \nu = \nu$. Thus, $$V + (V, V) \nu + h V + h^2 \nu = \nu.$$ This gives $\Vert V \Vert^2 + h^2 = 1$ which is Equation (\ref{Structure3}). \end{proof} Moreover, the complex structure $J=J_1+J_2$ on $\mathbb P$ (where $J_i$ denotes the complex structure on $\mathbb M_i(c_i)$) induces on $M$ an almost contact metric structure $\big(\mathfrak{X}, \xi, \eta, g=(.,.)\big)$, where $\mathfrak{X}$ is the $(1,1)$-tensor defined, for all $X,Y\in \Gamma(TM)$ by $$(\mathfrak{X} X,Y)=\overline{g}(JX,Y).$$
The tangent vector field $\xi$ and the $1$-form $\eta$ associated with $\xi$ satisfy
$$\xi=-J\nu\ \ \ \ \text{and}\ \ \ \ \ \ \eta(X)=(\xi,X),$$ for all $X\in\Gamma(TM)$. Then, we can easily see that, for all $X\in\Gamma(TM)$, the following holds: \begin{eqnarray} J X = \mathfrak{X} X + \eta(X) \nu, \end{eqnarray} \beqt \mathfrak{X}^2X=-X+\eta(X)\xi,\quad g(\xi,\xi)=1,\quad\text{and}\quad \mathfrak{X}\xi=0. \label{chi} \eeqt Here, we recall that given an almost contact metric structure $(\mathfrak{X}, \xi, \eta, g)$ one can define a $2$-form $\varTheta$ by $\varTheta(X, Y) = g(X, \mathfrak{X} Y)$ for all $X, Y\in \Gamma(TM)$. Now, $(\mathfrak{X}, \xi, \eta, g)$ is said to satisfy the contact condition if $-2 \varTheta =d\eta$ and if it is the case, $(\mathfrak{X}, \xi, \eta, g)$ is called a contact metric structure on $M$. A contact metric structure $(\mathfrak{X}, \xi, \eta, g)$ is called a Sasakian structure (and $M$ a Sasaki manifold) if $\xi$ is a Killing vector field (or equivalently, $\mathfrak{X} = \nabla \xi$) and $$(\nabla_X\mathfrak{X} )Y = \eta (Y) X - g(X, Y) \xi, \ \ \text{for all} \ \ X, Y \in \Gamma(TM).$$ For $\mathbb P$, one can choose a local orthonormal frame $\{e_1, e_2 = \mathfrak{X} e_1, \xi, \nu\}$ where $\{e_1, e_2 = \mathfrak{X} e_1, \xi\}$ denotes a local orthonormal frame of $M$. \begin{lem}\label{Fo} We have
\begin{enumerate}[label=(\rm{\roman{*}}), ref=\rm{\roman{*}}] \item\label{as1} $\mathfrak{X}$ is antisymmetric on $TM$, i.e. $(\mathfrak{X} e_1, e_2) = -(e_1, \mathfrak{X} e_2)$ and $\mathfrak{X} \xi = 0$ \item\label{as2} $J \circ F = F \circ J$ \item\label{as3} $(V, \mathfrak{X} X) + \eta(X) h = \eta(fX)$ \item \label{as4}$f\mathfrak{X} X + \eta (X)V = \mathfrak{X} fX - (V, X)\xi$ \item \label{as5}$\eta (V) = 0$ \item \label{as6}$f\xi = h\xi - \mathfrak{X} V$ \item \label{as7}$\eta (fV) = 0$ \item \label{as8}$(fe_1, e_2) = 0$ and $(fe_1, e_1) = (fe_2, e_2) = -h$ \item \label{as9}$JV = \mathfrak{X} V$ \item \label{as10}$F\xi = f \xi$
\end{enumerate} \end{lem} \begin{proof} For any $X, Y \in \Gamma(TM)$, wer have $(\mathfrak{X} X, \mathfrak{X} Y) = (X, Y ) - \eta(X)\eta(Y ).$ Thus, $(\mathfrak{X} e_1, e_2) = (\mathfrak{X}^2 e_1, \mathfrak{X} e_2) = -(e_1, \mathfrak{X} e_2)$. It is evident that $\mathfrak{X} \xi = 0$. This proves (\ref{as1}). Now, for any $X_1+X_2 \in \Gamma(T\mathbb P)$, we have \begin{eqnarray*} J \circ F (X_1 + X_2) &=& J(X_1 - X_2)= J_1 X_1 - J_2 X_2 \\ &=&F(J_1 X_1 + J_2 X_2)= F \circ J (X_1 + X_2). \end{eqnarray*} This proves (\ref{as2}). From $J \circ F = F \circ J$, and using that $f$ is symmetric and (\ref{chi}), we have for any $X \in \Gamma (TM)$, $$\left \{ \begin{array}{l} f \mathfrak{X} X + \eta(X) V = \mathfrak{X} fX - (V, X) \xi,\\ (V, \mathfrak{X} X) + h \eta (X) = \eta (fX). \end{array} \right. $$ This proves (\ref{as3}) and (\ref{as4}). We also have $J (F \nu) = \ F(J \nu)$. Thus, $$\mathfrak{X} V + \eta (V) \nu - h \xi = -f\xi-(V, \xi) \nu.$$ This implies $$\left \{ \begin{array}{l} f\xi = h \xi - \mathfrak{X} V, \\ \eta (V) = -(V, \xi) = 0. \end{array} \right. $$ This proves (\ref{as5}) and (\ref{as6}). From $(V, \mathfrak{X} X) + h \eta (X) = \eta (fX)$ and for $X = V$, we get
$$\eta(fV) = h \eta (V) + {\underbrace{(V, \mathfrak{X} V) }_{=0}}= 0,$$
which is (\ref{as7}). We calculate
\begin{eqnarray*}
(fe_1, e_2) &=& -(f\mathfrak{X} e_2, e_2) = (-\mathfrak{X} f e_2 + (V, e_2)\xi, e_2) \\
&=& -(\mathfrak{X} fe_2, e_2) = (fe_2, \mathfrak{X} e_2) = -(fe_2, e_1).
\end{eqnarray*}
Since $f$ symmetric, it implies that $(fe_1, e_2) = 0$. Moreover, we have
\begin{eqnarray*}
(fe_1, e_1) &=& -(f\mathfrak{X} e_2, e_1) = (-\mathfrak{X} f e_2 + (V, e_2)\xi, e_1) \\
&=& -(\mathfrak{X} fe_2, e_1) = (fe_2, \mathfrak{X} e_1) = (fe_2, e_2).
\end{eqnarray*}
We know that $\mathrm{tr} (F) = 0$. Thus,
\begin{eqnarray*}
0 &=& (Fe_1, e_1) + (Fe_2, e_2) + (F\xi, \xi) + (F\nu, \nu)\\
&=& (fe_1, e_1) + (fe_2, e_2) + {\underbrace{(f\xi, \xi)}_{= h}}+{\underbrace{(V + h\nu, \nu)}_{h+0=h}}
\\ &=& (fe_1, e_1) + (fe_2, e_2).
\end{eqnarray*} Thus, $(fe_1, e_1) = (fe_2, e_2) = -h$. This proves (\ref{as8}). Since $(V, \xi) = 0$, it is clear that $F \xi = f\xi$ and from $J = \mathfrak{X} + \eta (\cdot)\nu$, we have $JV =\mathfrak{X} V$. This proves (\ref{as9}) and (\ref{as10}).
\end{proof} For all $X, Y, Z \in \Gamma(TM)$, the Gauss equation for the hypersurface $M$ of $\mathbb P$ can be written as \beq\label{gausssr} \mathcal R(X,Y)Z&=& \frac {c_1}{4}\Big(( X+fX) \wedge (Y +fY)\Big) + \frac {c_2}{4}\Big(( X-fX) \wedge( Y -fY)\Big)\nonumber\\ &&+g(IIY,Z)IIX -g(IIX,Z)IIY, \eeq where $\mathcal R$ denotes the Riemann curvature tensor. The Codazzi equation is \beq\label{codazzisr} d^{\nabla}II(X,Y) &=& \frac {c_1}{4} \Big(g(fY, Z)g(V, X)-g(fX, Z)g(V, Y) \nonumber \\ && + g(Y, Z)g(V, X) - g(X, Z)g(V, Y)\Big) \nonumber \\ && - \frac{c_2}{4} \Big( g(Y, Z)g(V, X)-g(Y, fZ)g(V, X) \nonumber \\ && - g(X, Z)g(Y, V) + g(X, fZ)g(V, Y)\Big) \eeq Now, we ask if the Gauss equation \eqref{gausssr} and the Codazzi equation \eqref{codazzisr} are sufficient to get an isometric immersion of $(M,g)$ into $\mathbb P= \mathbb M_1(c_1) \times \mathbb M_2(c_2)$. \begin{defn}[{\bf Compatibility equations}]\label{compp} Let $(M^3,g)$ be a simply connected oriented Riemannian manifold endowed with an almost contact metric structure $(\mathfrak{X}, \xi, \eta)$ and $E$ be a field of symmetric endomorphisms on $M$. We say that $(M, g, E)$ satisfies the compatibility equations for $\mathbb M_1(c_1) \times \mathbb M_2(c_2)$ if and only if for any $X,Y,Z\in\Gamma(TM)$, we have \begin{eqnarray} \label{gaussSR} \mathcal R(X,Y)Z&=& \frac {c_1}{4}\Big(( X+fX) \wedge (Y +fY)\Big) + \frac {c_2}{4}\Big(( X-fX) \wedge( Y -fY)\Big)\nonumber\\ &&+g(EY,Z)EX -g(EX,Z)EY, \end{eqnarray} \begin{eqnarray} \label{codazziSR} d^{\nabla}E(X,Y) &=& \frac {c_1}{4} \Big(g(fY, Z)g(V, X)-g(fX, Z)g(V, Y) \nonumber \\ && + g(Y, Z)g(V, X) - g(X, Z)g(V, Y)\Big) \nonumber \\ && - \frac{c_2}{4} \Big( g(Y, Z)g(V, X)-g(Y, fZ)g(V, X) \nonumber \\ && - g(X, Z)g(Y, V) + g(X, fZ)g(V, Y)\Big), \end{eqnarray} \begin{eqnarray} \label{Structure4} (\nabla_Xf)Y = g(Y, V) EX + g(EX, Y)V, \end{eqnarray} \begin{eqnarray} \label{Structure5} \nabla_XV = -f (EX) + h EX, \end{eqnarray} \begin{eqnarray}\label{Structure6} \nabla f = -2 EV. \end{eqnarray} \end{defn} In \cite{Ko, LTV}, Kowalczyk and De Lira-Tojeiro-Vit\'orio proved independently that that the Gauss equation \eqref{gaussSR} and the Codazzi equation \eqref{codazziSR} together with (\ref{Structure1}), (\ref{Structure2}), (\ref{Structure3}), (\ref{Structure4}), (\ref{Structure5}), (\ref{Structure6}) and if $\frac{F \pm \mathrm{Id}}{2}$ are of rank $2$, where $F$ is given by $F =
\left( \begin{array}{cc} f & V\\ V & h \end{array} \right) $, are necessary and sufficient for the existence of an isometric immersion from $M$ into $\mathbb M_1(c_1) \times \mathbb M_2(c_2)$ such that the complex structure of $\mathbb M_1(c_1) \times \mathbb M_2(c_2)$ over $M$ is given by $J= \mathfrak{X} + \eta(\cdot) \nu$ , $E$ as second fundamental form and such that the product structure coincides with $F$ over $M$. This immersion is global if $M$ is simply connected. Note that this was previously proven in a more abstract way by Piccione and Tausk \cite{PT}.
\section{Isometric immersions into $\mathbb{M}_1(c_1) \times \mathbb{M}_2(c_2)$ via spinors}
In this section, we consider two different $\mathrm{Spin^c}$ structures on $\mathbb P= \mathbb{M}_1(c_1) \times \mathbb{M}_2(c_2)$ carrying parallel spinor fields. For the first structure, the parallel spinor $\psi$ is lying in $\Sigma^+ \mathbb P$ and for the second $\mathrm{Spin^c}$ structure the parallel spinor field $\Psi$ is lying in $\Sigma^- \mathbb P$. The restriction of these two $\mathrm{Spin^c}$ structures to any hypersurface $M^3$ defines two $\mathrm{Spin^c}$ structures on $M$, each one with a generalized Killing spinor field. These spinor fields will characterize the isometric immersion of $M$ into $\mathbb P= \mathbb{M}_1(c_1) \times \mathbb{M}_2(c_2)$. \\
We denote by $\pi_i(X)$ the projection of a vector $X$ on $T \mathbb M_i(c_i)$. We have
\begin{eqnarray}\label{Sys1} \left \{ \begin{array}{l} \pi_1 (V) = \frac{(1-h)V + \Vert V\Vert^2 \nu}{2} \\\\ \pi_2 (V) = \frac{(h+1)V - \Vert V\Vert^2 \nu}{2} \\\\ \pi_1 (\xi) = - \pi_1(J\nu) = -J (\pi_1(\nu)) \\\\ \pi_2 (\xi) = - \pi_2(J\nu) = -J(\pi_2(\nu))\\\\ \pi_1 (\nu) = \frac{(h+1)\nu + V }{2}\\\\ \pi_2(\nu) = \frac{(1-h)\nu -V }{2} \end{array} \right. \end{eqnarray}
\subsection{A first $\mathrm{Spin^c}$ structure on $\mathbb{M}_1(c_1) \times \mathbb{M}_2(c_2)$ and its restriction to hypersurfaces} Assume that there exists an isometric immersion of $(M^3,g)$ into $\mathbb{M}_1(c_1) \times \mathbb{M}_2(c_2)$ with shape operator $II$. By Section \ref{cm}, we know that $M$ has an almost contact metric structure $(\mathfrak{X},\xi,\eta)$ such that $\mathfrak{X} X = JX -\eta(X) \nu$ for every $X\in \Gamma(TM)$ and the product structure $F$ on $\mathbb{M}_1(c_1) \times \mathbb{M}_2(c_2)$ will be restricted via $f, V$ and $h$. Consider the product of the canonical $\mathrm{Spin^c}$ structure on $\mathbb{M}_1(c_1) $ with the canonical one on $ \mathbb{M}_2(c_2)$ . It has a parallel spinor $\psi = \psi_1^+ \otimes \psi_2^+$ lying in $\Sigma_0 (\mathbb{M}_1(c_1) \times \mathbb{M}_2(c_2) ) = \Sigma_0(\mathbb{M}_1(c_1) ) \otimes \Sigma_0 (\mathbb{M}_2(c_2) ) \subset \Sigma^+ (\mathbb{M}_1(c_1) \times \mathbb{M}_2(c_2))$. First of all, using (\ref{splitt}), we have for any $X \in \Gamma(TM)$, $$J(\pi_2(X))\cdot \pi_2(X)\cdot \psi_2^+ = i \vert \pi_2(X)\vert^2 \psi_2^+ \ \ \text{and}\ \ \
J(\pi_1(X))\cdot \pi_1(X)\cdot \psi_1^+ = i \vert \pi_1(X)\vert^2 \psi_1^+.$$
\begin{lem}\label{lemaCM} We have
$$ - \pi_1(\nu)\cdot \psi_1^+ \otimes \pi_2(\xi) \cdot \psi_2^+ + \pi_1 (\xi)\cdot \psi_1^+ \otimes \pi_2(\nu)\cdot \psi_2^+ = 0$$
\end{lem} \begin{proof}
Using that $ i\pi_2(\nu) \cdot \psi_2^+ = J(\pi_2(\nu)) \cdot \psi_2^+$ and $ i\pi_1(\nu)\cdot \psi_2^+ = J(\pi_1(\nu))\cdot \psi_2^+$, we have
\begin{eqnarray*}
&& - \pi_1(\nu) \cdot \psi_1^+ \otimes \pi_2(\xi)\cdot \psi_2^+ + \pi_1 (\xi)\cdot \psi_1^+ \otimes \pi_2(\nu)\cdot \psi_2^+ \\ &=& i
\pi_1(\nu) \cdot \psi_1^+ \otimes \pi_2(\nu)\cdot \psi_2^+ - i \pi_1(\nu) \cdot \psi_1^+ \otimes \pi_2(\nu)\cdot \psi_2^+\\ &=& 0.
\end{eqnarray*}
\end{proof} \begin{lem}
The restriction $\varphi_1$ of the parallel spinor $\psi$ on $\mathbb{M}_1(c_1) \times \mathbb{M}_2(c_2)$ is a solution of the generalized Killing equation \begin{eqnarray}
\nabla^1_X\varphi_1 + \frac 12 \gamma_1 (II X)\varphi_1=0, \end{eqnarray} where $\nabla^1$ (resp. $\gamma_1$) denotes the $\mathrm{Spin^c}$ Levi-Civita connection (resp. the Clifford multiplication) on the induced $\mathrm{Spin^c}$ bundle. Moreover, $\varphi_1$ satisfies $\gamma_1(\xi)\varphi_1= -i\varphi_1$. The curvature 2-form $i\Omega^1$ of the auxiliary line bundle associated with the induced $\mathrm{Spin^c}$ structure is given in the basis $\{e_1, e_2 = \mathfrak{X} e_1, \xi\}$ by $$\Omega^1(e_1, e_2) = \frac {c_1}{2} (h-1) - \frac{c_2}{2} (h+1),$$ $$\Omega^1 (e_1, \xi) = \frac {c_1-c_2}{2} (e_1, V),$$ $$\Omega^1 (e_2, \xi) = \frac {c_1-c_2}{2} (e_2, V).$$
\end{lem} \begin{proof}
By the Gauss formula (\ref{spingauss}), the restriction $\varphi_1$ of the parallel spinor $\psi$ on $\mathbb P$ satisfies $$\nabla^1_X\varphi_1 =- \frac 12 \gamma_1(II )\varphi_1.$$
Now, for any $X, Y \in \Gamma(TM)$, we have \begin{eqnarray*} \Omega^1 (X, Y) &=& \Omega^{\mathbb{M}_1 (c_1) \times \mathbb{M}_2(c_2)} (X, Y) \\&=& -\mathrm{Ric}^{\mathbb{M}_1 (c_1) } (J\pi_1X, \pi_1Y) - \mathrm{Ric}^{\mathbb{M}_2 (c_2) } (J\pi_2X, \pi_2Y) \\&=& -\frac{c_1}{4} g\Big(\mathfrak{X} X +\eta(X) \nu + \mathfrak{X} fX +\eta(fX)\nu -(V, X)\xi, Y+fY+(V, Y)\nu \Big) \\&& -\frac{c_2}{4} g\Big(\mathfrak{X} X +\eta(X) \nu - \mathfrak{X} fX -\eta(fX)\nu +(V, X)\xi, Y - fY-(V, Y)\nu \Big). \end{eqnarray*} Using Lemma \ref{Fo}, we have $$\Omega^1(e_1, e_2) = \frac{c_1}{2} (h-1) - \frac{c_2}{2} (h+1),$$ $$ \Omega^1(e_1, \xi) = \frac{c_1- c_2}{2} (V, e_1),$$ $$ \Omega^1(e_2, \xi) = \frac{c_1- c_2}{2} (V, e_2).$$ Now, we have \begin{eqnarray*} \gamma_1(\xi) (\varphi_1) &=& {\xi \cdot \nu \cdot (\psi_1^+ \otimes \psi_2^+)}_{\vert_M} \\ &=& {[\pi_1 (\xi)\cdot \pi_1(\nu) \cdot \psi_1^+ \otimes \psi_2^+ - \pi_1(\nu) \cdot \psi_1^+ \otimes \pi_2(\xi)\cdot \psi_2^+]}_{\vert_M} \\ && + {[\pi_1 (\xi) \cdot \psi_1^+ \otimes \pi_2(\nu)\cdot \psi_2^+ + \psi_1^+ \otimes \pi_2(\xi) \cdot \pi_2(\nu)\cdot \psi_2^+ ]}_{\vert_M} \end{eqnarray*}
Thus,
\begin{eqnarray*} \gamma_1(\xi) (\varphi_1) &=& {\xi \cdot \nu \cdot (\psi_1^+ \otimes \psi_2^+)}_{\vert_M} \\ &=& [-i \vert \pi_1(\nu) \vert ^2 - i \vert \pi_2 (\nu) \vert^2 ]\varphi_1 - {[\pi_1(\nu) \cdot \psi_1^+ \otimes \pi_2(\xi)\cdot \psi_2^+ + \pi_1 (\xi)\cdot \psi_1^+ \otimes \pi_2(\nu) \cdot \psi_2^+]}_{\vert_M} \\ &=& - i \varphi_1 + {\underbrace{{[- \pi_1(\nu) \cdot \psi_1^+ \otimes \pi_2(\xi) \cdot \psi_2^+ + \pi_1 (\xi)\cdot \psi_1^+ \otimes \pi_2(\nu)\cdot \psi_2^+]}_{\vert_M} }_{=0 \ \ \text{by Lemma} \ \ \ref{lemaCM}}} \\&=& -i \varphi_1. \end{eqnarray*} \end{proof} \subsection{A second $\mathrm{Spin^c}$ structure on $\mathbb{M}_1 (c_1) \times \mathbb{M}_2(c_2)$ and its restriction to hypersurfaces} One can also endow $\mathbb{M}_1 (c_1) \times \mathbb{M}_2(c_2)$ with another $\mathrm{Spin^c}$ structure. Mainly, the one coming from the product of the anticanonical $\mathrm{Spin^c}$ on $\mathbb{M}_1 (c_1)$ with the canonical $\mathrm{Spin^c}$ structure on $ \mathbb{M}_2(c_2)$ which carries also a parallel spinor $\Psi = \psi_1^- \otimes \psi_2^+ $. The parallel spinor $\Psi$ lies in $ \Sigma_1 (\mathbb M_1(c_1)) \otimes \Sigma_0( \mathbb M_2(c_2)) \subset \Sigma^-(\mathbb{M}_1 (c_1) \times \mathbb{M}_2(c_2))$. Using (\ref{splitt}), we have for any $X \in \Gamma(TM)$
$$J(\pi_2(X))\cdot \pi_2(X)\cdot \psi_2^+ = i \vert \pi_2(X)\vert^2 \psi_2^+ \ \ \text{and}\ \ \ J(\pi_1(X))\cdot \pi_1(X)\cdot \psi_1^- = - \vert \pi_1(X)\vert^2i \psi_1^-.$$
\begin{lem} \label{lemacl} We have $$\pi_1(\nu) \cdot \psi_1^- \otimes (\pi_2(V) + i \pi_2(\xi))\cdot \psi_2^+ - (\pi_1(V) +i \pi_1(\xi)) \cdot \psi_1^- \otimes \pi_2 (\nu)\cdot \psi_2^+ = 0$$ \end{lem} \begin{proof} Using that $i\pi_2(\nu) \cdot \psi_2^+ = J(\pi_2(\nu))\cdot \psi_2^+$ and $i\pi_1(\nu)\cdot \psi_1^- = -J(\pi_1(\nu))\cdot \psi_1^-$, we have \begin{eqnarray*} && \pi_1(\nu) \cdot\psi_1^- \otimes (\pi_2(V) + i \pi_2(\xi)) \cdot\psi_2^+ - (\pi_1(V) +i \pi_1(\xi)) \cdot\psi_1^- \otimes \pi_2 (\nu) \cdot\psi_2^+ \\&=& {\underbrace{ 2 \pi_1(\nu)\cdot\psi_1^- \otimes \pi_2(\nu)\cdot\psi_2^+}_{A}} + {\underbrace{ \pi_1(\nu)\cdot\psi_1^- \otimes \pi_2(V)\cdot\psi_2^+}_{B}} -{\underbrace{ \pi_1(V)\cdot\psi_1^- \otimes \pi_2(\nu)\cdot\psi_2^+}_{C}} \end{eqnarray*} Let's calculate each term of the last identity. First we have \begin{eqnarray*} A &= & 2 \pi_1(\nu)\cdot\psi_1^- \otimes \pi_2(\nu)\cdot\psi_2^+ \\ &=& \frac{1-h^2}{2} (\nu \cdot\psi_1^- \otimes \nu \cdot\psi_2^+) - \frac{h+1}{2} (\nu \cdot\psi_1^- \otimes V\cdot \psi_2^+) \\&& + \frac{1-h}{2} (V \cdot\psi_1^- \otimes \nu\cdot \psi_2^+) - \frac 12 (V\cdot \psi_1^- \otimes V\cdot \psi_2^+). \end{eqnarray*} Next, we have \begin{eqnarray*} B &= & \pi_1(\nu)\cdot\psi_1^- \otimes \pi_2(V)\cdot\psi_2^+ \\ &=& \frac{(h+1)^2}{4} (\nu \cdot\psi_1^- \otimes V\cdot \psi_2^+) - \frac{h+1}{4}\Vert V\Vert^2 (\nu\cdot \psi_1^- \otimes \nu \cdot\psi_2^+) \\&& + \frac{h+1}{4} (V\cdot \psi_1^- \otimes V\cdot \psi_2^+) - \frac 14 \Vert V \Vert^2 (V \cdot\psi_1^- \otimes \nu \cdot\psi_2^+), \end{eqnarray*} and \begin{eqnarray*} C &= & \pi_1(V)\cdot\psi_1^- \otimes \pi_2(\nu)\cdot\psi_2^+ \\ &=& \frac{(1-h)^2}{4} (V\cdot \psi_1^- \otimes \nu\cdot \psi_2^+) - \frac{1-h}{4} (V \cdot\psi_1^- \otimes V\cdot \psi_2^+) \\&& + \frac{1-h}{4} \Vert V \Vert^2 (\nu\cdot \psi_1^- \otimes \nu\cdot \psi_2^+) - \frac 14 \Vert V \Vert^2 (\nu \cdot\psi_1^- \otimes V \cdot\psi_2^+). \end{eqnarray*} It's clear that $A+B+C = 0$. \end{proof} \begin{lem}
The restriction $\varphi_2$ of the parallel spinor $\Psi$ (for the $\mathrm{Spin^c}$ structure described above) on $\mathbb M_1(c_1) \times \mathbb M_2(c_2)$ is a solution of the generalized Killing equation \begin{eqnarray}
\nabla^2_X\varphi_2 =\frac 12 \gamma_2(II X)\varphi_2, \end{eqnarray} where $\nabla^2$ (resp. $\gamma_2$) denotes the $\mathrm{Spin^c}$ connection (resp. the Clifford multiplication) on the induced $\mathrm{Spin^c}$ bundle. Moreover, $\varphi_2$ satisfies $\gamma_2(V) \varphi_2 = - i \gamma_2(\xi)\varphi_2 + h \varphi_2$. The curvature 2-form of the auxiliary line bundle associated with the induced $\mathrm{Spin^c}$ structure is given in the basis $\{e_1, e_2 = \mathfrak{X} e_1, \xi\}$ by $$\Omega^2(e_1, e_2) = -\frac {c_1}{2} (h-1) - \frac{c_2}{2} (h+1),$$ $$\Omega^2 (e_1, \xi) = -\frac {c_1+c_2}{2} (e_1, V),$$ $$\Omega^2 (e_2, \xi) = -\frac {c_1+c_2}{2} (e_2, V).$$ Moreover, we have
\begin{eqnarray} \label{Ide1}
0 = (\gamma_2 (V) \varphi_2, \varphi_2),
\end{eqnarray}
\begin{eqnarray}\label{Ide2}
g(V, e_1) = -i (\gamma_2 (e_2)\varphi_2, \varphi_2),
\end{eqnarray}
\begin{eqnarray}\label{Ide3}
g(V, e_2) = i (\gamma_2 (e_1) \varphi_2, \varphi_2),
\end{eqnarray}
\begin{eqnarray}\label{Ide4}
h = i (\gamma_2 (\xi) \varphi_2, \varphi_2).
\end{eqnarray}
\end{lem} \begin{proof}
By the Gauss formula (\ref{spingauss}), the restriction $\varphi_2$ of the parallel spinor $\Psi$ on $\mathbb M_{1}(c_1) \times \mathbb M_2(c_2)$ satisfies $$\nabla^2_X\varphi_2 = \frac 12 \gamma_2(II X)\varphi_2.$$
Now, for any $X, Y \in \Gamma(TM)$, we have \begin{eqnarray*} \Omega (X, Y) &=& \Omega^{\mathbb{M}_1 (c_1) \times \mathbb{M}_2(c_2)} (X, Y) \\&=& \mathrm{Ric}^{\mathbb{M}_1 (c_1) } (J\pi_1X, \pi_1Y) - \mathrm{Ric}^{\mathbb{M}_2 (c_2) } (J\pi_2X, \pi_2Y) \\&=& \frac{c_1}{4} g(\mathfrak{X} X +\eta(X) \nu + \mathfrak{X} fX +\eta(fX)\nu -(V, X)\xi, Y+fY+(V, Y)\nu ) \\&& -\frac{c_2}{4} g(\mathfrak{X} X +\eta(X) \nu - \mathfrak{X} fX -\eta(fX)\nu +(V, X)\xi, Y - fY-(V, Y)\nu). \end{eqnarray*} In the basis $\{e_1, e_2 = \mathfrak{X} e_1, \xi\}$, we have $$\Omega(e_1, e_2) = -\frac{c_1}{2} (h-1) - \frac{c_2}{2} (h+1),$$ $$ \Omega(e_1, \xi) = -\frac{c_1+ c_2}{2} (V, e_1),$$ $$ \Omega(e_2, \xi) = -\frac{c_1+ c_2}{2} (V, e_2).$$ Now, let's calculate \begin{eqnarray*} -\gamma_2(\xi) (\varphi_2) &=& {[\xi \cdot \nu \cdot (\psi_1^- \otimes \psi_2^+) ]}_{\vert_M}\\ &=& {[\pi_1 (\xi) \cdot \pi_1(\nu) \cdot \psi_1^- \otimes \psi_2^+ + \pi_1(\nu) \cdot \psi_1^- \otimes \pi_2(\xi)\cdot \psi_2^+]}_{\vert_M} \\ && - [{\pi_1 (\xi) \cdot\psi_1^- \otimes \pi_2(\nu) \cdot \psi_2^+ + \psi_1^- \otimes \pi_2(\xi)\cdot \pi_2(\nu) \cdot\psi_2^+]}_{\vert_M}. \end{eqnarray*} Thus, we get \begin{eqnarray*} -\gamma_2(\xi) (\varphi_2) &=& [{\xi \cdot \nu \cdot (\psi_1^- \otimes \psi_2^+)]}_{\vert_M} \\ &=&i (\vert\pi_1(\nu)\vert^2 - \vert \pi_2(\nu)\vert^2) \varphi_2
+ {[\pi_1(\nu)\cdot\psi_1^- \otimes \pi_2(\xi)\cdot \psi_2^+ - \pi_1 (\xi)\cdot \psi_1^- \otimes \pi_2(\nu)\cdot \psi_2^+]}_{\vert_M} \\ &=&
ih \varphi_2 + {[\pi_1(\nu) \cdot \psi_1^- \otimes \pi_2(\xi)\cdot \psi_2^+ - \pi_1 (\xi)\cdot \psi_1^- \otimes \pi_2(\nu) \cdot \psi_2^+]}_{\vert_M}. \end{eqnarray*} In a similar way, we have \begin{eqnarray*} -\gamma_2(V) (\varphi_2) &=&{[ V \cdot \nu \cdot (\psi_1^- \otimes \psi_2^+) ]}_{\vert_M}\\ &=& {[\pi_1 (V)\cdot \pi_1(\nu)\cdot \psi_1^- \otimes \psi_2^+ + \pi_1(\nu)\cdot \psi_1^- \otimes \pi_2(V) \cdot\psi_2^+ ]}_{\vert_M}\\ && - {[\pi_1 (V) \cdot\psi_1^- \otimes \pi_2(\nu)\cdot \psi_2^+ + \psi_1^- \otimes \pi_2(V)\cdot \pi_2(\nu)\cdot \psi_2^+ ]}_{\vert_M}\\ &=& {[\pi_1(\nu)\cdot \psi_1^- \otimes \pi_2(V)\cdot \psi_2^+ - \pi_1 (V)\cdot \psi_1^- \otimes \pi_2(\nu)\cdot \psi_2^+]}_{\vert_M}. \end{eqnarray*} Now, we have \begin{eqnarray*} && [-\gamma_2(V) - i \gamma_2(\xi) ] (\varphi_2) \\ &=& {[\pi_1(\nu)\cdot \psi_1^- \otimes \pi_2(V)\cdot \psi_2^+ - \pi_1 (V) \cdot\psi_1^- \otimes \pi_2(\nu) \cdot\psi_2^+ ]}_{\vert_M}\\&& -h \varphi_2 +{[ i \pi_1(\nu) \cdot\psi_1^- \otimes \pi_2(\xi)\cdot \psi_2^+ - i \pi_1 (\xi) \cdot\psi_1^- \otimes \pi_2(\nu)\cdot \psi_2^+]}_{\vert_M} \\ &=& -h \varphi_2 \\ && {\underbrace{+{[\pi_1(\nu)\cdot \psi_1^- \otimes (\pi_2(V) + i \pi_2(\xi))\cdot \psi_2^+ - (\pi_1(V) +i \pi_1(\xi)) \cdot\psi_1^- \otimes \pi_2 (\nu)\cdot \psi_2^+]}_{\vert_M}}_{= 0 \ \ \text{by Lemma}\ \ \ref{lemacl}}} \\ &=& - h \varphi_2. \end{eqnarray*}
Taking the scalar product of the last identity with $\varphi_2$, then the real part of the scalar product with $\gamma_2 (e_1) \varphi_2$, then with $\gamma_2 (e_2) \varphi_2$, we get (\ref{Ide1}), (\ref{Ide2}), (\ref{Ide3}) and (\ref{Ide4}). \end{proof}
\section{Generalized Killing $\mathrm{Spin^c}$ spinors and isometric immersions}
\begin{lem} \cite{NR12} Let $E$ be a field of symmetric endomorphisms on a $\mathrm{Spin^c}$ manifold $M^3$ of dimension $3$, then \begin{eqnarray}
\gamma(E(e_i))\gamma( E(e_j)) - \gamma(E(e_j)) \gamma( E(e_i ))&=& 2 (a_{j3} a_{i2} - a_{j2}a_{i3}) e_1 \nonumber \\&& +2(a_{i3}a_{j1} - a_{i1}a_{j3}) e_2 \nonumber \\ && + 2(a_{i1}a_{j2} - a_{i2}a_{j1})e_3, \end{eqnarray} where $(a_{ij})_{i,j}$ is the matrix of $E$ written in any local orthonormal frame of $TM$. \label{aij} \end{lem} \begin{prop} Let $(M^3,g)$ be a Riemannian $\mathrm{Spin^c}$ manifold endowed with an almost contact metric structure $(\mathfrak{X},\xi,\eta)$. Assume that there exists a vector $V$ and a function $h$ and a $\mathrm{Spin^c}$ structure with non-trivial spinor $\varphi_1$ satisfying $$\nabla^1_X\varphi_1=-\frac{1}{2}\gamma_1(EX)\varphi_1 \ \ \ \text{and}\ \ \ \ \gamma_1(\xi)\varphi_1=-i\varphi_1,$$ where $E$ is a field of symmetric endomorphisms on $M$. Moreover, we suppose that the curvature 2-form of the connection on the auxiliary line bundle associated with the $\mathrm{Spin^c}$ structure is given by $$\Omega^1(e_1, e_2) = \frac{c_1}{2} (h-1) - \frac{c_2}{2} (h+1),$$ $$ \Omega^1(e_1, \xi) = \frac{c_1- c_2}{2} (V, e_1),$$ $$ \Omega^1(e_2, \xi) = \frac{c_1- c_2}{2} (V, e_2),$$
in the basis $\{e_1,e_2=\mathfrak{X} e_1,e_3=\xi\}$. Hence, the Gauss equation is satisfied for $\mathbb M_{1} (c_1) \times \mathbb M_2(c_2)$ if and only if the Codazzi equation for $\mathbb M_{1} (c_1) \times \mathbb M_2(c_2)$ is satisfied. \label{GGCC} \end{prop} \begin{proof} We compute the spinorial curvature $\mathcal{R}^1$ on $\varphi_1$, we get $$\mathcal{R}^1_{X, Y} \varphi_1= -\frac 12 \gamma_1(d{^\nabla} E(X, Y))\varphi_1 + \frac 14 \big(\gamma_1(EY)\gamma_1(EX) - \gamma_1(EX)\gamma_1( EY)\big)\varphi_1.$$ In the basis $\{e_1, e_2 = \mathfrak{X} e_1, e_3 = \xi\}$, the Ricci identity (\ref{RRicci}) gives that \begin{eqnarray*}
\frac 12 \gamma_1(\mathrm{Ric}(X))\varphi_1 -\frac i2 \gamma_1(X\lrcorner\Omega^1)\varphi_1 &=& \frac 14 \sum_{k=1}^3 \gamma_1(e_k) \big(\gamma_1(EX)\gamma_1( Ee_k) -\gamma_1( Ee_k)\gamma_1(EX)\big)\varphi_1 \\&& -\frac 12 \sum_{k=1}^3 \gamma_1(e_k)\gamma_1( d^{\nabla} E(e_k, X ))\varphi_1. \end{eqnarray*} By Lemma \ref{aij} and for $X= e_1$, the last identity becomes \begin{eqnarray}
&\ &(\mathrm{R}_{1221} +\mathrm{R}_{1331} -a_{11}a_{33} - a_{11}a_{22} + a_{13}^2 +a_{12}^2 + \frac {c_1}{2}(h-1)-\frac{c_2}{2}(h+1) )\gamma_1(e_1 )\varphi_1 \nonumber\\ && +( \mathrm{R}_{1332} -a_{12}a_{33} + a_{32}a_{13})\gamma_1(e_2) \varphi_1 \nonumber \\ && + (\mathrm{R}_{1223} -a_{22}a_{13} + a_{32}a_{12})\gamma_1(e_3) \varphi_1 \nonumber\\
&&-\frac {c_1-c_2}{2} (V, e_1) \varphi_1
\\ &=& - \gamma_1(e_2)\gamma_1( d^{\nabla}E (e_2, e_1))\varphi_1- \gamma_1(e_3)\gamma_1( d^{\nabla}E (e_3, e_1))\varphi_1.\nonumber \label {e1varphi} \end{eqnarray} Since $\vert \varphi\vert$ is constant ($\vert \varphi \vert =1$), the set $\{\varphi_1, \gamma_1(e_1)\varphi_1, \gamma_1(e_2)\varphi_1, \gamma_1(e_3)\varphi_1\}$ is an orthonormal frame of $\Sigma M$ with respect to the real scalar product $\Re e ( ., .)$. Hence, from Equation (\ref{e1varphi}) we deduce \begin{eqnarray*} \mathrm{R}_{1221} +\mathrm{R}_{1331} -(a_{11}a_{33} + a_{11}a_{22} - a_{13}^2 -a_{12}^2 ) &+& \frac {c_2}{2} (h-1) -\frac{c_2}{2} (h+1) \\ &=& g(d^\nabla E(e_1, e_2), e_3) - g(d^\nabla E(e_1, e_3), e_2)
\end{eqnarray*}
\begin{eqnarray*} \mathrm{R}_{1332} -(a_{12}a_{33} - a_{32}a_{13}) &=& g(d^\nabla E(e_1, e_3), e_1)\\ \mathrm{R}_{1223} -(a_{22}a_{13} - a_{32}a_{12})&=& -g(d^\nabla E(e_1, e_2), e_1)\\
- \frac {(c_1-c_2)}{2} (V, e_1) &= & g(d^\nabla E(e_2, e_1), e_2) + g(d^\nabla E(e_3, e_1), e_3) \end{eqnarray*} The same computation holds for the unit vector fields $e_2$ and $e_3$ and we get \begin{eqnarray*} \mathrm{R}_{2331} -(a_{12}a_{33} - a_{13}a_{23})& =& - g(d^\nabla E(e_2, e_3), e_2)\\ \mathrm{R}_{2332} + \mathrm{R}_{2112} -(a_{22}a_{33} + a_{22}a_{11} - a_{13}^2 - a_{12}^2) && + \frac {c_1}{2}(h-1) - \frac{c_2}{2} (h+1) \\ &=& g(d^\nabla E(e_2, e_3), e_1) +g(d^\nabla E(e_1, e_2), e_3)\\ \mathrm{R}_{2113} -(a_{23}a_{11} - a_{12}a_{13}) &=& - g(d^\nabla E(e_1, e_2), e_2)\\
-\frac {(c_1-c_2)}{2} (V, e_2)&=& g(d^\nabla E(e_1, e_2), e_1) + g(d^\nabla E(e_3, e_2), e_3)\\ \mathrm{R}_{3221} -(a_{13}a_{22} - a_{23}a_{21}) -\frac {(c_1-c_2)}{2} (V, e_2) &=& - g(d^\nabla E(e_2, e_3), e_3)\\ \mathrm{R}_{3112} - (a_{32}a_{11} - a_{31}a_{12}) + \frac {(c_1-c_2)}{2} (V, e_1) &=& g(d^\nabla E(e_1, e_3), e_3)\\ \mathrm{R}_{3113}+ \mathrm{R}_{3223}-(a_{22}a_{33} - a_{11}a_{33} + a_{13}^2 + a_{23}^2) &= & g(d^\nabla E(e_2, e_3), e_1) - g(d^\nabla E(e_1, e_3), e_2)\\ g(d^\nabla E(e_2, e_3), e_2) &=& - g(d^\nabla E(e_1, e_3), e_1) \end{eqnarray*} The last twelve equations will be called System 1 and it is clear that the Gauss equation for $\mathbb M_1(c_1) \times \mathbb M_2(c_2)$ is satisfied if and only if the Codazzi equation
for $\mathbb M_1(c_1) \times \mathbb M_2(c_2)$ is satisfied.
\end{proof} \begin{lem} Under the same condition as Proposition \ref{GGCC}, we have $\nabla_X \xi = \mathfrak{X} EX$.
\label{chii} \end{lem} \begin{proof} In fact, we simply compute the derivative of $\gamma_1(\xi)\varphi_1= -i \varphi_1$ in the direction of $X\in \Gamma (TM)$ to get
\begin{eqnarray*}
\gamma_1(\nabla_X \xi)\varphi &=& \frac i2 \gamma_1(EX)\varphi _1+ \frac 12 \gamma_1(\xi)\gamma_1( EX)\varphi_1 \end{eqnarray*} Using that $-i \gamma_1(e_2)\varphi_1 = \gamma_1(e_1)\varphi_1$, the last equation reduces to $$\gamma_1(\nabla_X\xi)\varphi_1 - g(EX, e_1) \gamma_1(e_2)\varphi_1 + g(EX, e_2) \gamma_1(e_1)\varphi_1 =0.$$ Finally $\nabla_X \xi= \mathfrak{X} EX$. \end{proof}
\begin{prop} Let $(M^3,g)$ be a Riemannian $\mathrm{Spin^c}$ manifold endowed with an almost contact metric structure $(\mathfrak{X},\xi,\eta)$. Assume that there exist a nonzero vector field $V$ and a function $h$ such that there exists a $\mathrm{Spin^c}$ structure with non-trivial spinor $\varphi$ satisfying $$\nabla^2_X\varphi=\frac{1}{2}\gamma_2(EX)\varphi \ \ \ \text{and}\ \ \ \ \ \gamma_2(V)\varphi_2 = -i \gamma_2(\xi)\varphi_2 + h \varphi_2,$$ where $E$ is a field of symmetric endomorphisms on $M$. Moreover, we suppose that the curvature 2-form of the connection on the auxiliary line bundle associated with the $\mathrm{Spin^c}$ structure is given by $$\Omega^2(e_1, e_2) = -\frac {c_1}{2} (h-1) - \frac{c_2}{2} (h+1)$$ $$\Omega^2 (e_1, \xi) =- \frac {(c_1+c_2)}{2}(e_1, V)$$ $$\Omega^2 (e_2, \xi) = -\frac {(c_1+c_2)}{2} (e_2, V)$$
in the basis $\{e_1,e_2=\mathfrak{X} e_1,e_3=\xi\}$. The Gauss equation for $\mathbb M_1(c_1) \times \mathbb M_2(c_2)$ is satisfied if and only if the Codazzi equation
for $\mathbb M_1(c_1) \times \mathbb M_2(c_2)$ is satisfied. \label{GGCC2} \end{prop} \begin{proof} First, from $ \gamma_2(V)\varphi_2 = -i \gamma_2(\xi)\varphi_2 + h \varphi_2$, we have that (\ref{Ide1}), (\ref{Ide2}), (\ref{Ide3}) and (\ref{Ide4}) are satisfied. We compute the spinorial curvature $\mathcal{R}^2$ on $\varphi_2$, we get $$\mathcal{R}^2_{X, Y} \varphi_2= \frac 12 \gamma_2(d{^\nabla} E(X, Y))\varphi_2 + \frac 14 \big(\gamma_2(EY)\gamma_2( EX) - \gamma_2(EX) \gamma_2(EY)\Big)\varphi_2.$$ In the basis $\{e_1, e_2 = \mathfrak{X} e_1, e_3 = \xi\}$, the Ricci identity (\ref{RRicci}) gives that \begin{eqnarray*}
\frac 12 \gamma_2(\mathrm{Ric}(X))\varphi_2 -\frac i2 \gamma_2(X\lrcorner\Omega^2)\varphi_2 &=& \frac 14 \sum_{k=1}^3 \gamma_2(e_k)\big( \gamma_2(EX)\gamma_2( Ee_k) - \gamma_2(Ee_k)\gamma_2( EX)\big)\varphi_2\\&+& \frac 12 \sum_{k=1}^3 \gamma_2(e_k) \gamma_2(d^{\nabla} E(e_k, X ))\varphi_2. \end{eqnarray*} By Lemma \ref{aij} and for $X= e_1$, the last identity becomes \begin{eqnarray}
&\ &(\mathrm{R}_{1221} +\mathrm{R}_{1331} -a_{11}a_{33} - a_{11}a_{22} + a_{13}^2 +a_{12}^2) \gamma_2(e_1) \varphi_2 \\ && + \frac i2 (c_1+c_2)(V, e_1) \gamma_2(\xi)\varphi_2 + \frac i2 [c_1(h-1)+c_2(h+1)] \gamma_2(e_2) \varphi_2 \nonumber\\ && +( \mathrm{R}_{1332} -a_{12}a_{33} + a_{32}a_{13})\gamma_2(e_2) \varphi_2 \nonumber \\ && + (\mathrm{R}_{1223} -a_{22}a_{13} + a_{32}a_{12})\gamma_2(e_3)\varphi_2 \nonumber\\
&=& \gamma_2( e_2) \gamma_2(d^{\nabla}E (e_2, e_1))\varphi_2 + \gamma_2(e_3)\gamma_2(d^{\nabla}E (e_3, e_1))\varphi_2. \nonumber \label {e1varphi} \end{eqnarray} Since $\vert \varphi\vert$ is constant ($\vert \varphi \vert =1$), the set $\{\varphi_2, \gamma_2(e_1)\varphi_2, \gamma_2(e_2) \varphi_2, \gamma_2(e_3)\varphi_2\}$ is an orthonormal frame of $\Sigma M$ with respect to the real scalar product $\Re e ( ., .)$. Hence, from Equation (\ref{e1varphi}) we deduce \begin{eqnarray*}
&& \mathrm{R}_{1221} +\mathrm{R}_{1331} -(a_{11}a_{33} + a_{11}a_{22} - a_{13}^2 -a_{12}^2 ) \\ & -&
\frac 12 (c_1 +c_2) (V, e_1)^2 - \frac 12 h\Big( c_1(h-1)+c_2(h+1)\Big) \\ &=& g(d^\nabla E(e_2, e_1), e_3) - g(d^\nabla E(e_3, e_1), e_2)
\end{eqnarray*}
\begin{eqnarray*} \mathrm{R}_{1332} -(a_{12}a_{33} - a_{32}a_{13}) - \frac 12 (c_1 +c_2)(V, e_1)(V, e_2) &=& - g(d^\nabla E(e_1, e_3), e_1)\\ \mathrm{R}_{1223} -(a_{22}a_{13} - a_{32}a_{12}) + \frac 12 \Big(c_1(h-1)+c_2(h+1)\Big) (V, e_2)&=& -g(d^\nabla E(e_2, e_1), e_1)\\
\frac 1 2 (c_1 +c_2)h(V, e_1)- \frac 12 \Big( c_1(h-1)+c_2(h+1)\Big) (V, e_1) &= & -g(d^\nabla E(e_2, e_1), e_2) \\ && - g(d^\nabla E(e_3, e_1), e_3) \end{eqnarray*} The same computation holds for the unit vector fields $e_2$ and $e_3$ and we get \begin{eqnarray*} \mathrm{R}_{2331} -(a_{12}a_{33} - a_{13}a_{23}) -\frac 12 (c_1 +c_2)(V, e_2) (V, e_1) & =& g(d^\nabla E(e_2, e_3), e_2) \end{eqnarray*} \begin{eqnarray*} && \mathrm{R}_{2332} + \mathrm{R}_{2112} -(a_{22}a_{33} + a_{22}a_{11} - a_{13}^2 - a_{12}^2) \\ & -& \frac 12(c_1+c_2) (e_2, V)^2 - \frac 12 h\Big(c_1(h-1)+c_2(h+1)\Big) \\ &=& -g(d^\nabla E(e_2, e_3), e_1) -g(d^\nabla E(e_1, e_2), e_3) \end{eqnarray*} \begin{eqnarray*} \mathrm{R}_{2113} -(a_{23}a_{11} - a_{12}a_{13}) -\frac 12 \Big(c_1(h-1)+c_2(h+1)\Big) (V, e_1) &=& g(d^\nabla E(e_1, e_2), e_2)\\
-\frac 12 \Big( c_1(h-1) +c_2(h+1)\Big) (V, e_2) + \frac 12(c_1 +c_2) h (V, e_2) &=& - g(d^\nabla E(e_1, e_2), e_1) \\ && - g(d^\nabla E(e_3, e_2), e_3)\\ \mathrm{R}_{3221} -(a_{13}a_{22} - a_{23}a_{21}) +\frac 12 (c_1 +c_2)h(V, e_2) &=& g(d^\nabla E(e_2, e_3), e_3)\\ \mathrm{R}_{3112} - (a_{32}a_{11} - a_{31}a_{12}) - \frac 12 (c_1 +c_2)h (V, e_1) &=& -g(d^\nabla E(e_1, e_3), e_3)\end{eqnarray*} \begin{eqnarray*} && \mathrm{R}_{3113}+ \mathrm{R}_{3223}-(a_{22}a_{33} - a_{11}a_{33} + a_{13}^2 + a_{23}^2) \\ &-& \frac 12 (c_1 +c_2)(e_1, V)^2 - \frac 12 (c_1 +c_2)(V, e_2)^2 \\ &= & -g(d^\nabla E(e_2, e_3), e_1) +g(d^\nabla E(e_1, e_3), e_2)\end{eqnarray*} \begin{eqnarray*}
{\underbrace{-\frac i2 (e_1, V) (\gamma_2(e_1) \varphi_2, \varphi_2) - \frac i2 (V, e_2) (\gamma_2(e_2) \varphi_2, \varphi_2)}_{(\gamma_2(V)\varphi_2, \varphi_2)= 0}} &=& - g(d^\nabla E(e_2, e_3), e_2) - g(d^\nabla E(e_1, e_3), e_1) \end{eqnarray*} The last twelve equations will be called System 2 and it is clear that the Gauss equation for $\mathbb M_{1} (c_1) \times \mathbb M_2(c_2)$ is satisfied if and only if the Codazzi equation
for $\mathbb M_{1} (c_1) \times \mathbb M_2(c_2)$ is satisfied. \end{proof}
\subsection{Spinorial characterization of hypersurfaces of $\mathbb M_{1} (c_1) \times \mathbb M_2(c_2)$} The main goal of this section is to prove Theorem \ref{thmCM}. \begin{proof}[Proof of Theorem \ref{thmCM}]
It is clear from the previous section that Assesrtion 1 implies Assertion 2. Now, Assume that Assertion 2 holds.
We have to establish the compatibility equation (\ref{gaussSR}), (\ref{codazziSR}), (\ref{Structure1}), (\ref{Structure2}), (\ref{Structure3}), (\ref{Structure4}), (\ref{Structure5}) and (\ref{Structure6}).
First we define $f: TM \rightarrow TM$ by $(fe_1, e_1) =(fe_2, e_2)= -h, (fe_1, e_2) = 0$ and $(f\xi, e_1) = (V, e_2)$, $(f\xi, e_2) = -(V, e_1)$. Since $\gamma_2(V) \varphi_2 = -i \gamma_2(\xi) \varphi_2 + h \varphi_2$. It is clear that $$\left\{ \begin{array}{l} h^2 + \Vert V \Vert^2 = 1\\ (V, e_1) = -i (\gamma_2(e_2)\varphi_2, \varphi_2)\\ (V, e_2) = i (\gamma_2(e_1)\varphi_2, \varphi_2)\\ (V, \xi) = 0\\ fV = -hV\\ f^2 = \mathrm{Id} - (V, \cdot) V \end{array} \right. $$ So Equations (\ref{Structure1}), (\ref{Structure2}), (\ref{Structure3}) are satisfied. Moreover, by Lemma \ref{chii}, we have \begin{eqnarray*} (\nabla_{e_1} V , \xi) &=& (V, e_1) (\nabla_{e_1} e_1, \xi) + (V, e_2) (\nabla_{e_1}e_2, \xi) \\ &=& -(V, e_1) (e_1 \mathfrak{X} Ee_1) - (V, e_2) (e_2, \mathfrak{X} Ee_1) \\ &=& (V, e_1) E_{12} - (V, e_2) E_{11} + E_{11} (V, e_1) - E_{12} (V, e_2). \end{eqnarray*} By a similar computation, we get that Equation (\ref{Structure5}) is satisfied. Now, Equation (\ref{Structure6}) is also satisfied because \begin{eqnarray*} X(h) &=& (i \gamma_2({\underbrace{\nabla_X\xi}_{=\mathfrak{X} EX}})\varphi_2, \varphi_2) + \frac i2 (\gamma_2(\xi) \gamma_2(EX)\varphi_2, \varphi_2) - \frac i2 (\gamma_2(EX) \gamma_2(\xi )\varphi_2, \varphi_2) \\ &=& i E(X, e_1) (\gamma_2(e_2)\varphi_2, \varphi_2) - i E(X, e_2) (\gamma_2(e_1 )\varphi_2, \varphi_2) \\ && \frac i2 E(X, e_2) (\gamma_2(\xi)\gamma_2(e_1)\varphi_2, \varphi_2) - \frac i2 E(X, e_1) (\gamma_2(e_1 )\gamma_2( \xi) \varphi_2, \varphi_2) \\ && +\frac i2 E(X, e_2) (\gamma_2(\xi)\gamma_2( e_2)\varphi_2, \varphi_2) - \frac i2 E(X, e_2) (\gamma_2(e_2) \gamma_2(\xi) \varphi_2, \varphi_2) \\ && - \frac i2 E(X, \xi) \vert \varphi_2\vert^2 + \frac i2 E(X, \xi) \vert \varphi_2\vert^2 \\ &=& -(V, e_1) E(X, e_1) - E(X, e_2) (V, e_2) \\ && \frac i2 E(X, e_1) (\gamma_2(e_2 ) \varphi_2, \varphi_2) + \frac i2 E(X, e_1) (\gamma_2(e_2)\varphi_2, \varphi_2) \\ && - \frac i2 E(X, e_2) (\gamma_2(e_1) \varphi_2, \varphi_2) - \frac i2 E(X, e_2) (\gamma_2(e_1) \varphi_2, \varphi_2) \\ &=& - 2 E(X, e_2) (V, e_2) - 2 E(X, e_1) (V, e_1) \\ &=& -2 (EV, X). \end{eqnarray*} Now, we have \begin{eqnarray*} (\nabla_{e_1} fe_1, e_1 ) &=& e_1 (fe_1, e_1) + (fe_1, \xi) (\nabla_{e_1}\xi, e_1) \\ &=& e_1 (-h) + (V, e_2) (\mathfrak{X} Ee_1, e_1) \\ &=& 2 (EV, e_1)- (V, e_2) E_{12} \\ &=& 2 E_{11} (e_1, V) - (e_2, V) E_{12}, \end{eqnarray*} and \begin{eqnarray*} (f(\nabla_{e_1} e_1) , e_1) &=& (\nabla_{e_1} e_1, fe_1) \\ &=& (fe_1, e_2) (\nabla_{e_1}{e_1}, e_2 ) + (fe_1, \xi) (\nabla_{e_1}e_1, \xi) \\ &=& -(\nabla_{e_1}{\xi}, e_1) (V, e_2) = -(\mathfrak{X} Ee_1, e_1) (V, e_2) = E_{12} (V, e_2) \end{eqnarray*} Thus, $((\nabla_{e_1}f) e_1, e_1 ) = 2 E_{11} (e_1, V) $. By a similar computation, one can get (\ref{Structure4}). Solving System 1 and System 2 simultaneously gives the Gauss and the Codazzi equations. Finally, we have to check that $\frac{F +\mathrm{Id}}{2}$ and $\frac{F -\mathrm{Id}}{2}$ are of rank $2$. In fact, in the basis $\{e_1, e_2 = \mathfrak{X} e_1, \xi, \nu\}$, the matrix $\frac{F +\mathrm{Id}}{2}$ can be written as
\[ \frac 12 \left( \begin{array}{ccccc} -h+1 & 0 & (V, e_2) & (V, e_1) \\ 0 & -h+1 & (V, e_1)& (V, e_2) \\ (V, e_2) & -(V, e_1) & h+1 & 0 \\
(V, e_1) & (V, e_2) & 0 & h+1 \end{array} \right)\] Using that $h^2 + \Vert V \Vert^2 = 1$, one can check that it is of rank $2$. Same holds for $\frac{F -\mathrm{Id}}{2}$. \end{proof} \begin{rem} Before giving some applications, we want to mention that both equivalent assertions of Theorem \ref{thmCM} are also equivalent to a third one described in terms of the Dirac operators $D^1$ and $D^2$, and the energy-momentum tensors associated to $\varphi_1$ and $\varphi_2$. We recall that the energy-momentum tensors $Q_{\varphi_j}$, $j=1,2$, associated to the spinors field $\varphi_j$ are the $(2,0)$-tensors respectively defined by
$$Q_{\varphi_j}(X,Y)=\frac{1}{2}\Re e(\gamma_j(X)\cdot\nabla^j_{Y}\varphi+\gamma_j(Y)\cdot\nabla^j_{X}\varphi,\frac{\varphi}{|\varphi|^2}).$$ This third assertion can be written as:\\\\ {\it 3. There exists 2 $\mathrm{Spin^c}$ structures on $M$ carrying each one a non-trivial spinor $\varphi_1$ and $\varphi_2$ of constant norms and satisfying $$D^1\varphi_1=\frac{3}{2}H\varphi_1\ \ \ \text{and} \ \ \ \gamma_1(\xi)\varphi_1= - i\varphi_1,$$ $$D^2\varphi_2=-\frac{3}{2}H\varphi_2\ \ \ \text{and} \ \ \ \gamma_2(V) \varphi_2 = -i \gamma_2(\xi) \varphi_2 + h \varphi_2,$$ so that their energy-momentum tensors $Q_{\varphi_1}$ and $Q_{\varphi_2}$ are the same. Moreover, the curvature 2-form of the connection on the auxiliary bundle associated with these two $\mathrm{Spin^c}$ structure are given by ($j \in \{1, 2\}$) $$\left\{ \begin{array}{l} \Omega^j(e_1, e_2) = \frac 12 (-1)^{j-1}c_1 (h-1) -\frac 12 c_2 (h+1),\\ \Omega^j(e_1, \xi) = \frac 12 \Big( (-1)^{j-1} c_1 - c_2\Big) (e_1, V),\\ \Omega^j (e_2, \xi) = \frac 12 \Big( (-1)^{j-1} c_1 - c_2\Big) (e_2, V), \end{array} \right. $$ in the basis $\{e_1,e_2=\mathfrak{X} e_1,e_3=\xi\}$.}\\
Indeed, clearly, the second assertion of Theorem \ref{thmCM} implies assertion $3.$. Reciprocally, as proven in \cite{LR}, $D^1\varphi_1=\frac{3}{2}H\varphi_1$ with $\varphi_1$ of constant norm implies that $\nabla^1_X\varphi_1=-\frac{1}{2}\gamma_1(E_1X)\varphi_1$ with $E_1=Q_{\varphi_1}$. Similarly, we also get $\nabla^2_X\varphi_2=\frac{1}{2}\gamma_1(E_2X)\varphi_2$ with $E_2=Q_{\varphi_2}$. Now, since $Q_{\varphi_1} = Q_{\varphi_2}$, this gives assertion 2 of Theorem \ref{thmCM}. \end{rem}
\section{Totally geodesic and totally umbilical hypersurfaces of $\mathbb M_{1} (c_1) \times \mathbb M_2(c_2)$} In this section, we use our main result, Theorem \ref{thmCM}, to give some geometric results on totally geodesic and umbilical hypersurfaces of $\mathbb M_{1} (c_1) \times \mathbb M_2(c_2)$.
\begin{lem} Let $\big(M^3, g=(.,.)\big)$ be a totally umbilical hypersurface of $\mathbb M_{1} (c_1) \times \mathbb M_2(c_2)$. Then, \begin{eqnarray}\label{meanV} \Vert V \Vert \vert c_1 -c_2\vert = 4 \Vert dH \Vert \end{eqnarray}
\end{lem}
{\bf Proof.} From Theorem \ref{thmCM}, we know that there exists 2 $\mathrm{Spin^c}$ structures on $M$ carrying each one a non-trivial spinor $\varphi_1$ and $\varphi_2$ satisfying $$\nabla^1_X\varphi_1=-\frac{1}{2}\gamma_1(EX)\varphi_1 = -\frac H2 \gamma_1(X)\varphi_1\ \ \ \text{and}\ \ \ \gamma_1(\xi)\varphi_1= - i\varphi_1.$$ $$\nabla^2_X\varphi_2=\frac{1}{2}\gamma_2(EX)\varphi_2= \frac H2 \gamma_2(X)\varphi_2\ \ \ \text{and}\ \ \ \ \gamma_2(V) \varphi_2 = -i \gamma_2(\xi) \varphi_2 + h \varphi_2.$$ The curvature 2-form of the connection on the auxiliary bundle associated with these two $\mathrm{Spin^c}$ structure are given by ($j \in \{1, 2\}$) \begin{eqnarray}\label{Ome1} \Omega^j(e_1, e_2) = \frac 12 (-1)^{j-1}c_1 (h-1) -\frac 12 c_2 (h+1), \end{eqnarray} \begin{eqnarray}\Omega^i (e_1, \xi) = \frac 12 \Big( (-1)^{j-1} c_1 - c_2\Big) (e_1, V),\label{omega2}\end{eqnarray} \begin{eqnarray}\Omega^j (e_2, \xi) = \frac 12 \Big( (-1)^{j-1} c_1 - c_2\Big) (e_2, V), \label{Ome3}\end{eqnarray} in the basis $\{e_1,e_2=\mathfrak{X} e_1,e_3=\xi\}$.\\\\ {\bf For the second Spin$^c$ structure:} The Ricci identity for $M$ can be written as $$\frac 12 \gamma_2(\mathrm{Ric}^M (X)) \varphi_2 - \frac i2 \gamma_2(X \lrcorner \Omega^2) \varphi_2 = \frac 12 \gamma_2 (dH)\gamma_2(X)\varphi_2 +\frac 32 dH(X) \varphi_2 + H^2 \gamma_2(X) \varphi_2 .$$ For $X =\xi$, the \ of the scalar product of the previous identity with $\varphi_2$ gives \begin{eqnarray*} -\frac i2 \Omega^2 (\xi, e_1) (\gamma_2(e_1)\varphi_2, \varphi_2) &-& \frac i2 \Omega^2 (\xi, e_2) (\gamma_2(e_2)\varphi_2, \varphi_2) \\&=& \frac 12 \Re e(\gamma_2 (dH)\gamma_2(\xi)\varphi_2, \varphi_2)+ \frac 32 dH(\xi) \end{eqnarray*} Using that $-\gamma_2(e_1)\gamma_2(e_2)\gamma_2(\xi)\varphi_2 = \varphi_2$, we get
$$-\frac i2 \Omega^2 (\xi, e_1) (\gamma_2(e_1)\varphi_2, \varphi_2) - \frac i2 \Omega^2 (\xi, e_2) (\gamma_2(e_2)\varphi_2, \varphi_2) = dH(\xi)$$ Finally, using (\ref{Ide2}), (\ref{Ide3}), (\ref{omega2}) and (\ref{Ome3}), we obtain $dH(\xi) = 0$. In a similar way, for $X = e_1$ the real part of the scalar product with $\varphi_2$ of the Ricci identity gives \begin{eqnarray*} -\frac i2 \Omega^2 (e_1, e_2) (\gamma_2(e_2)\varphi_2, \varphi_2) &-& \frac i2 \Omega^2 (e_1, \xi) (\gamma_2(\xi)\varphi_2, \varphi_2) \\&=& \frac 12 \Re e(\gamma_2 (dH)\gamma_2(e_1)\varphi_2, \varphi_2) + \frac 32 dH(e_1). \end{eqnarray*} Using that $-\gamma_2(e_1)\gamma_2(e_2)\gamma_2(\xi)\varphi_2 = \varphi_2$ and $dH(\xi) = 0$, we get \begin{eqnarray*} -\frac i2 \Omega^2 (e_1, e_2) (\gamma_2(e_2)\varphi_2, \varphi_2) - \frac i2 \Omega^2 (e_1, \xi) (\gamma_2(\xi)\varphi_2, \varphi_2) = dH(e_1). \end{eqnarray*} Finally, using (\ref{Ide2}), (\ref{Ide3}), (\ref{omega2}) and (\ref{Ome1}), we obtain $dH(e_1) = \frac {c_1-c_2}{4} g(V, e_1) $. In a similar way we can get $dH(e_2) = \frac {c_1-c_2}{4} g(V, e_2) $. Hence, we have $\Vert dH\Vert^2 = \frac{(c_1-c_2)^2}{16}\Vert V \Vert^2.$ For consistency, one can also take the first Spin$^c$ structure and check that a similar identity can be obtained. \begin{prop} \label{pr1} Let $M$ be a totally umbilical hypersurface of in $\mathbb M_{1} (c_1) \times \mathbb M_1(c_1)$. Then $M$ is totally geodesic or an extrinsic hypersphere. Moreover, if $c_1 \neq 0$, the universal cover of $M$ is a Non-Einstein Sasaki manifold or a product of a K\"ahler manifold (of complex dimension 1) with $\mathbb R$. If $c_1 = 0$, then $M$ is a $\mathrm{Spin}$ manifold with a parallel or Killing spin spinor. \end{prop} \begin{proof}Let $M$ be a totally umbilical hypersurface of $\mathbb M_{1} (c_1) \times \mathbb M_1(c_1)$. We have from (\ref{meanV}) that $dH = 0$, so $H$ is constant. Assume that $c_1 \neq 0$. If this constant $H$ is 0, $M$ has a parallel Spin$^c$ spinor and if $H \neq 0$, then $M$ has a Killing Spin$^c$ spinor. Form the classification of parallel and Killing Spin$^c$ spinors \cite{Moro1}, we get the desired result. If $c_1 =0$, then the curvature of the auxiliary line bundle defining the Spin$^c$ structure is zero and hence $M$ is a $\mathrm{Spin}$ manifold with parallel or Killing spin spinor. \end{proof} \begin{prop} \label{pr2} Let M be a totally umbilical hypersurface of in $\mathbb M_{1} (c_1) \times \mathbb M_2(c_2)$ ($c_1 \neq c_2$) having a local product structure. Then $M$ is totally geodesic or an extrinsic hypersphere. If $c_1 \neq c_2\neq 0$. the universal cover of $M$ is a non-Einstein Sasaki manifold or a product of a K\"ahler manifold (of complex dimension $1$) with $\mathbb R$ \end{prop} \begin{proof} Let $M$ be a totally umbilical hypersurface of $\mathbb M_{1} (c_1) \times \mathbb M_2(c_2)$. Since $V= 0$, we have from (\ref{meanV}) that $dH = 0$, so $H$ is constant. If this constant is $0$, $M$ has a parallel Spin$^c$ spinor and if $H \neq 0$, then $M$ has a Killing Spin$^c$ spinor. From the classification of parallel and Killing Spin$^c$ spinors \cite{Moro1}, we get the desired result. \end{proof} Using also Theorem \ref{thmCM}, one can also prove the following: \begin{prop}\label{pr3} Simply connected 3-dimensional homogeneous manifolds $\mathbb{E}(\kappa,\tau)$ ($\tau \neq 0$), with 4-dimensional isometry group cannot be immersed in $\mathbb M_{1} (c_1) \times \mathbb M_1(c_1)$ as totally umbilical hypersurfaces. \end{prop} \begin{proof} For $c_1=0$, this has been proved by Lawn and Roth \cite{LR}, even without assuming the umbilicity. Assume that $c_1 \neq 0$ and $\mathbb{E}(\kappa,\tau)$ can be immersed in a totally umbilical way in $\mathbb M_{1} (c_1) \times \mathbb M_1(c_1)$. By Proposition \ref{pr1}, we have that $H$ is constant and by Theorm \ref{thmCM}, $\mathbb{E}(\kappa,\tau)$ has two $\mathrm{Spin^c}$ structures carrying each one a non-trivial spinor $\varphi_1$ and $\varphi_2$ satisfying $$\nabla^1_X\varphi_1=-\frac{H}{2}\gamma_1(X)\varphi_1\ \ \ \text{and}\ \ \ \gamma_1(\xi)\varphi_1= - i\varphi_1.$$ $$\nabla^2_X\varphi_2=\frac{H}{2}\gamma_2(X)\varphi_2\ \ \ \text{and}\ \ \ \ \gamma_2(V) \varphi_2 = -i \gamma_2(\xi) \varphi_2 + h \varphi_2.$$ The curvature 2-form of the connection on the auxiliary bundle associated with these two $\mathrm{Spin^c}$ structure are given by ($j \in \{1, 2\}$) $$\Omega^j(e_1, e_2) = \frac 12 (-1)^{j-1}c_1 (h-1) -\frac 12 c_2 (h+1),$$ $$\Omega^i (e_1, \xi) = \frac 12 \Big( (-1)^{j-1} c_1 - c_2\Big) (e_1, V),$$ $$\Omega^j (e_2, \xi) = \frac 12 \Big( (-1)^{j-1} c_1 - c_2\Big) (e_2, V),$$ in the basis $\{e_1,e_2=\mathfrak{X} e_1,e_3=\xi\}$. We will call the first one $\mathrm{Spin^c}$ structure $t_1$ and the second one $t_2$. Since $H$ is constant, these two spinors are in fact real Killing spinors. But, it is known \cite{NR12} that the manifold $\mathbb{E}(\kappa,\tau)$ has only two $\mathrm{Spin^c}$ structures carrying Killing spinors. The first one (call it $t_3$) carries a Killing spinor $\varphi$ with Killing constant $\frac {\tau}{2}$ and for which $\gamma_3(\xi)\varphi = -i \varphi, \Omega^3(e_1, e_2) = -(\kappa-4\tau^2)$ and $\xi\lrcorner\Omega^3 = 0$, where we denote by $\gamma_3$ and $i\Omega^3$ the Clifford multiplication and the curvature 2-form of the auxiliary line bundle associated to the structure $t_3$. The second one (let's call it $t_4$) also carries a Killing spinor $\varphi$ with Killing constant $\frac{\tau}{2}$ for which $\gamma_4(\xi)\varphi = i \varphi, \Omega^4(e_1, e_2) = (\kappa-4\tau^2)$ and $\xi\lrcorner\Omega^4 = 0$, where we denote by $\gamma_4$ and $i\Omega^4$ the Clifford multiplication and the curvature 2-form of the auxiliary line bundle associated to the structure $t_4$. By comparison, we must have that $t_1 = t_3$ and $t_2 = t_4$. Thus we get $$\gamma_2(V) \varphi_2 = -i \gamma_2(\xi) \varphi_2 + h \varphi_2 = (h+1)\varphi_2.$$ Hence $V=0$. This means that $\mathbb{E}(\kappa,\tau)$ has a local product, which is a contradiction. \end{proof} {\bf Acknowledgement:} This work was initiated in 2017 during the one month research stay of the first author at the ``Laboratoire d'Analyse et de Math\'ematiques Appliqu\'ees" (UMR 8050) of the University of Paris-Est Marne-la-Vall\'ee. The first author gratefully acknowledges the support and hospitality of the University of Paris-Est Marne-la-Vall\'ee.
\end{document} | arXiv |
Applied Network Science
Evolving network representation learning based on random walks
Farzaneh Heidari1 &
Manos Papagelis1
Applied Network Science volume 5, Article number: 18 (2020) Cite this article
Large-scale network mining and analysis is key to revealing the underlying dynamics of networks, not easily observable before. Lately, there is a fast-growing interest in learning low-dimensional continuous representations of networks that can be utilized to perform highly accurate and scalable graph mining tasks. A family of these methods is based on performing random walks on a network to learn its structural features and providing the sequence of random walks as input to a deep learning architecture to learn a network embedding. While these methods perform well, they can only operate on static networks. However, in real-world, networks are evolving, as nodes and edges are continuously added or deleted. As a result, any previously obtained network representation will now be outdated having an adverse effect on the accuracy of the network mining task at stake. The naive approach to address this problem is to re-apply the embedding method of choice every time there is an update to the network. But this approach has serious drawbacks. First, it is inefficient, because the embedding method itself is computationally expensive. Then, the network mining task outcome obtained by the subsequent network representations are not directly comparable to each other, due to the randomness involved in the new set of random walks involved each time. In this paper, we propose EvoNRL, a random-walk based method for learning representations of evolving networks. The key idea of our approach is to first obtain a set of random walks on the current state of network. Then, while changes occur in the evolving network's topology, to dynamically update the random walks in reserve, so they do not introduce any bias. That way we are in position of utilizing the updated set of random walks to continuously learn accurate mappings from the evolving network to a low-dimension network representation. Moreover, we present an analytical method for determining the right time to obtain a new representation of the evolving network that balances accuracy and time performance. A thorough experimental evaluation is performed that demonstrates the effectiveness of our method against sensible baselines and varying conditions.
Network science, built on the mathematics of graph theory, leverage network structures to model and analyze pairwise relationships between objects (or people) (Newman 2003). With a growing number of networks — social, technological, biological — becoming available and representing an ever increasing amount of information, the ability to easily and effectively perform large-scale network mining and analysis is key to revealing the underlying dynamics of these networks, not easily observable before. Traditional approaches to network mining and analysis inherit a number of limitations. First, networks are typically represented as adjacency matrices, which suffer from high-dimensionality and data sparsity issues. Then, network analysis typically requires domain-knowledge in order to carry out the various steps of network data modeling and processing that is involved, before (multiple iterations of) analysis can take place. An ineffective network representation along with a requirement for domain expertise, render the whole process of network mining cumbersome for non-experts and limits their applicability to smaller networks.
To address the aforementioned limitations, there is a growing interest in learning low-dimensional representations of networks, also known as network embeddings. These representations are learned in an agnostic way (without domain-expertise) and have the potential to improve the performance of many downstream network mining tasks that now only need to operate in lower dimensions. Example tasks include node classification, link prediction and graph reconstruction (Wang et al. 2016), to name a few. Network representation learning methods are typically based on either a graph factorization or a random-walk based approach. The graph factorization ones (e.g., GraRep (Cao et al. 2015), TADW (Yang et al. 2015), HOPE (Ou et al. 2016)) are known to be memory intensive and computationally expensive, so they don't scale well. On the other hand, random-walk based methods (e.g., DeepWalk (Perozzi et al. 2014), node2vec (Grover and Leskovec 2016)) are known to be able to scale to large networks. A comprehensive coverage of the different methods can be found in the following surveys (Cai et al. 2018; Hamilton et al. 2017; Zhang et al. 2018).
A major shortcoming of these network representation learning methods is that they can only be applied on static networks. However, in real-world, networks are continuously evolving, as nodes and edges are added or deleted over time. As a result, any previously obtained network representation will now be outdated having an adverse effect on the accuracy of the data mining task at stake. In fact, the more significant the network topology changes are, the more likely it is for the mining task to perform poorly. One would expect though that network representation learning should account for continuous changes in the network, in an online mode. That way, (i) the low-dimensional network representation could continue being employed for downstream data mining tasks, and (ii) the results of the mining tasks obtained by the subsequent network representations would be comparable to each other. Going one step further, one would expect that while obtaining the network representation at any moment is possible, the evolving network representation learning framework suggest the best time to obtain the representation based on the upcoming changes in the network.
The main objective of this paper is to develop methods for learning representations of evolving networks. The focus of our work is on random-walk based methods that are known to scale well. The naive approach to address this problem is to re-apply the random-walk based network representation learning method of choice every time there is an update to the network. But this approach has serious drawbacks. First, it will be very inefficient, because the embedding method is computationally expensive and it needs to run again and again. Then, the data mining results obtained by the subsequent network representations are not directly comparable to each other, due to the differences involved between the previous and the new set of random walks, as well as, the non-deterministic nature of the deep learning process itself (see "Background and motivation" section for a detailed discussion). Therefore the naive approach would be inadequate for learning representations of evolving networks.
In contrast to the naive approach, we propose a novel random-walk based method for learning representations of evolving networks. The key idea of our approach is to design efficient methods that are incrementally updating the original set of random walks in such a way that it always respects the changes that occurred in the evolving network. As a result, we are able to continuously learn a new mapping function from the evolving network to a low-dimension network representation, by only updating a small number of random walks required to re-obtain the network embedding. The advantages of this approach are multifold. First, since the changes that occur in the network topology are typically local, only a small number of the original set of random walks will be affected, giving rise to substantial time performance gains. In addition, since the network representation will now be continuously informed, the accuracy performance of the network mining task will be improved. Furthermore, since the original set of random walks is maintained as much as possible, subsequent results of the mining tasks will be comparable to each other. In summary, the major contributions of this work include:
a systematic analysis that illustrates the instability of the random-walk based network representation methods and motivates our work.
an algorithmic framework for efficiently maintaining a set of random walks that respect the changes that occur in the evolving network topology. The framework treats random walks as "documents" that are indexed using an open-source distributed indexing and searching library. Then, the index allows for efficient ad hoc querying and update of the collection of random walks in hand.
a novel algorithm, EVONRL, for Evolving Network Representation Learning based on random walks, which offers substantial time performance gains without loss of accuracy. The method is generic, so it can accommodate the needs of different domains and applications.
an analytical method for determining the right time to obtain a new representation of the evolving network. The method is based on adaptive evaluation of the degree of divergence between the most recent random-walk set and the random-walk set utilized in the most recent network embedding. The method is tunable so it can be adjusted to meet the accuracy/sensitivity requirement of different domains, therefore can provide support for a number of real-world applications.
a thorough experimental evaluation on synthetic and real data sets that demonstrates the effectiveness of our method against sensible baselines, for a varying range of conditions.
An earlier version of this work appeared in the proceedings of the International Conference on Complex Networks and their Applications 2018 (Heidari and Papagelis 2018). The conference version addressed only the case of adding new edges. The current version extends the problem to the cases of deleting existing edges, adding new nodes and deleting existing nodes. In addition, it provides an analytical method that aims to provide support to the decision making process of when to obtain a new network embedding. This decision is critical as it can effectively balance accuracy versus time performance of the method extending its applicability in domains of diverse sensitivity. In addition, it provides further experiments for the additional cases that offer substantial, new insights of the problem's complexity and the performance of our EVONRL method.
The remainder of this paper is organized as follows: "Background and motivation" section provides background and motivates our problem. "Problem definition" section formalizes the problem of efficiently indexing and maintaining a set of random walks defined on the evolving network and "Algorithmic framework of dynamic random walks" section presents our algorithmic framework for addressing it. Our evolving network representation method and analytical method for obtaining new representations of the evolving network are presented in "Evolving network representation learning" section. "Experimental evaluation" section presents the experimental evaluation of our methods and "Extensions and variants" section discusses interesting variants and future directions. After reviewing the related work in "Related work" section, we conclude in "Conclusions" section.
Background and motivation
As mentioned earlier, there are many different approaches for static network embedding. A family of these methods is based on performing random walks on a network. Random-walk based methods, inspired by the word2vec's skip-gram model of producing word embeddings (Mikolov et al. 2013b), try to establish an analogy between a network and a document. While a document is an ordered sequence of words, a network can effectively be described by a set of random walks (i.e., ordered sequences of nodes). Typical examples of these algorithms include DeepWalk (Perozzi et al. 2014) and node2vec (Grover and Leskovec 2016). In fact, the latter can be seen as a generalization of the former, as node2vec can be configured to behave as DeepWalk. We collectively refer to these methods as StaticNRL for the rest of the manuscript. A typical StaticNRL method, is operating in two steps:
(i) a set of random walks, say walks, is collected by performing r random walks of length l starting at each node in the network (typical values are r=10,l=80).
(ii) walks are provided as input to an optimization problem that is solved using variants of Stochastic Gradient Descent using a deep neural network architecture (Bengio et al. 2013). The context size employed in the deep learning phase is k (typical value is k=5). The outcome is a set of d-dimensional representations, one for each node.
These representations are learned in an unsupervised way and can be employed for a number of predictive tasks. It is important to note that there are many possible strategies for performing random walks on nodes of a network, resulting in different learned feature representations and different strategies might work better for specific prediction tasks. The methods we will be presenting in this paper are orthogonal to what features the random walks aim to learn, therefore they can accommodate most of the existing random-walk based network representation learning methods.
Evaluation of the stability of StaticNRL methods
In this paragraph, we present a systematic evaluation of the stability of the StaticNRL methods, similar to the one presented in (Antoniak and Mimno 2018). The evaluation aims to motivate our approach to address the problem of interest. Intuitively, a stable embedding method is one in which successive runs of it on the same network would learn the same (or similar) embedding. Our interest for such an evaluation is stemming from the fact that StaticNRL methods are to a great degree dependent on two random processes: (i) the set of random walks collected, and (ii) the initialization of the parameters of the optimization method. Both factors can be a source of instability for the StaticNRL method.
Comparing two embeddings can happen either by measuring their similarity or by measuring their distance. Let us introduce the following measures of instability:
Cosine Similarity: Cosine similarity is a popular similarity measure for real-valued vector space models. It can also been used to compare two network embeddings using the pairwise cosine similarity on the learned d-dimensional representations (Kim et al. 2014; Hamilton et al. 2016). Formally, given the vector representations ni and \(n_{i}^{\prime }\) of the same node ni in two different network embeddings obtained at two different attempts, their cosine similarity is represented as:
$$sim(n_{i}, n_{i}^{\prime}) = cos(\theta)=\frac{\mathbf{n_{i}} \cdot \mathbf{n_{i}^{\prime}}}{ \|\mathbf{n_{i}} \|\|\mathbf{n_{i}^{\prime}} \|} $$
We can extend the similarity to two network embeddings E and E′ by summing and normalizing over all nodes:
$$sim(E, E^{\prime}) = \frac{\sum_{i \in V}sim\left(n_{i}, n_{i}^{\prime}\right)}{|V|} $$
Matrix Distance: Another possible way is to obtain the distance between two network embeddings by subtracting the matrices that represent the embeddings of all nodes, similarly to the approach followed in (Goyal et al. 2018). Formally, given a graph G=(V,E), a network embedding is a mapping \(f: V \rightarrow \mathbb {R}^{d}\), where d≪|V|. Let \(F(V) \in \mathbb {R}^{|V| \times d}\) be the matrix of all node representations. Then, we can define the following distance measure for the two network embeddings E, E′:
$$distance(E, E^{\prime}) = ||F^{\prime}(V) - F(V)||_F $$
Experimental scenario: We design a controlled experiment on two real-world networks, namely Protein-Protein-Interaction (PPI) (Breitkreutz et al. 2007) and a collaboration network, Digital Bibliography Library & Project (dblp) (Yang and Leskovec 2015) that aims to evaluate the effect of the two random processes in the final network embeddings. In these experiments, we have three settings. For each setting, we run StaticNRL on a network (using parameter values: r=10,l=10,k=5) two consecutive times, say t and t+1, and compute the cosine similarity and the matrix distance of the two network embeddings Et,Et+1 obtained. We repeat the experiment 10 times and report averages. The three settings are:
StaticNRL: Each run collects independent random walks and random weights are used in the initialization phase.
StaticNRL-i: Each run collects independent random walks but employs the same set of weights for the initialization phase, over all runs. The purpose is to eliminate one of the random processes.
StaticNRL-rw-i: Each run employs the same set of random walks and the same set of weights for the initialization phase, over all runs. The purpose is to eliminate both random processes.
Results: The results of the experiment are shown in Fig. 1a (cosine similarity) and Fig. 1 (matrix distance). They show that the set of random walks and the randomized initialization of the deep learning process have a significant role in moving the embedding despite the fact that there is no actual change in the topology of the network. As a matter of fact, when the same set of random walks and the same initialization is used then consecutive runs of StaticNRL result in the same embedding (as depicted by the sim(·,·)=1 in Fig. 1a or distance(·,·)=0 in Fig. 1b). However, when the set of random walks is independent or both the random walks and the initialization are independent then substantial differences are illustrated in consecutive runs of the StaticNRL methods.
Instability of the StaticNRL methods. Controlled experiments on running StaticNRL multiple times on the same network depict that the network representations learned are not stable, as a result of random initialization and random walks collected. When any of these random processes are fixed, then the network representations learned become more stable. a cosine similarity and b matrix distance
Implications: Let us start by noting that the implications of the experiment is not that StaticNRL is not useful. In fact, it has been shown to work very well. The problem is that while each independent embedding is inherently correct and has approximately same performance in downstream data mining task, these embeddings are not directly comparable to each other. In reality, the embeddings will be approximately equivalent if we are able to rotationally align them — most of similar work in the literature correct this problem by applying an alignment method (Hamilton et al. 2016). While alignment methods can bring independent embeddings closer and eliminate the effect of different embeddings, this approach won't work well in random walk based models. The main reason for that is that as we have showed in the experiment, consecutive runs suffer from instability that is introduced by the random processes. Therefore, in the case of evolving networks (which is the focus of this work), changes that occur in the network topology will not be easily interpretable in the changes observed in the network embedding (since differences might incorporate changes due to the two random processes). However, changes in the evolving network need to be proportional to the changes in the learned network representation. For instance, minor changes in the network topology should cause small changes in the representation, and significant changes in the network topology should cause large changes in the network representation.
Key idea: This motivated us to consider learning representations of evolving networks by efficiently maintaining a set of random walks that consistently respect the network topology changes. At the same time, we eliminate the effect of the random processes by, first, preserving, as much as possible, the original random walks that haven't been affected by the network changes. Then, by initializing the model with a previous run's initialization (Kim et al. 2014). There are two main advantages in doing so. Changes to the network representations of successive instances of an evolving network will be more interpretable and data mining task results will be more comparable to each other. In addition, it is possible to detect anomalies in the evolving network or extract laws of change in domain-specific networks (e.g., a social network) that explain which nodes are more vulnerable to change, similar to research in linguistics (Hamilton et al. 2016). Furthermore, our framework makes it possible to quantify the importance of any occurring changes in the network topology and therefore obtain a new network representation at an optimal time or when is really needed.
In "Background and motivation" section, we have established the instability of random walk based methods even when they are repeatedly applied to the same static network. That observation alone highlights the main challenge of employing these methods for learning representations of evolving networks. We have also introduced our key idea to address this problem. Stemming from our key idea, in this Section, we present a few definitions that allow to formally define the problem of interest in this paper.
(simple random walk or unbiased random walk on a graph) A simple random walk or unbiased random walk on a graph is a stochastic process that describes a path in a mathematical space (Pearson 1905), where the random walker transits from its current state (node) to one of its potential new states (neighboring nodes) with an equal probability. For instance, assume a graph G=(V,E) and a source node v0∈V. We uniformly at random select a node v1 to visit from the set Γ(v0)of all neighbors of v0. Then, we uniformly at random select a node v2 to visit from the set Γ(v1) of all neighbors of v1, and so on. Apparently, the sequence of vertices v0,v1,...,vk,... forms a simple random walk or an unbiased random walk on G. Formally, at every step k, we have a random variable Xk taking values on V, and the random sequence X0,X1,...,Xk,... is a discrete time stochastic process defined on the state space V. Assuming that at time k we are at node vi, we select to uniformly at random move to one of its adjacent nodes vj∈Γ(vi) based on the following transition probability:
$${} p_{v_{i}, v_{j}}=P(X_{k+i} = v_{j} | X_{k} = v_{i}) = \left\{ \begin{array}{ll} \frac{1}{d_{v_{i}}}, \quad if \ (v_{i}, v_{j}) \in E\\ 0, \quad otherwise \end{array}\right. $$
where \(d_{v_{i}}\) is the degree of node vi.
(biased random walk) A biased random walk is a stochastic process on graph, where the random walker jumps from its current state (node) to one of its potential new states (neighboring nodes) with unequal probability. Formally, assuming that at time k we are at node vi, we select to move to one of its adjacent nodes vj∈Γ(vi) based on the following transition probability:
$${} p_{v_{i}, v_{j}}=P(X_{k+i} = v_{j} | X_{k} = v_{i}) = \left\{ \begin{array}{ll} p, \quad if \ (v_{i}, v_{j}) \in E\\ 0, \quad otherwise \end{array}\right. $$
where p is unequal for each of the neighbours vj∈Γ(vi).
(evolving graph) Assume a connected, unweighted and undirected graph Gt=(Vt,Et) where Vt denotes the node set of Gt and Et denotes the edge set of Gt at time t. Since all nodes are connected to at least another node it holds that ∀u∈Vt it is du≥1. Now assume that at time t+1 a change occurs in the network topology of Gt forming Gt+1=(Vt+1,Et+1). This change can occur due to the following events:
a new edge (u′,v′)∉Et is added in Gt; then Et+1=Et∪(u′,v′).
an existing edge (u,v)∈Et of Gt is deleted; then, Et+1=Et∖(u,v).
a new node u′∉Vt is added in Gt; then Vt+1=Vt∪u′.
an existing node u∈Vt of Gt is deleted; then, Vt+1=Vt∖u.
Note that since we have assumed that the graph is connected, the events of adding a new nodeu′∉Vt in Gt or deleting an existing nodeu∈Vt from Gt can be treated as instances of edge addition and edge deletion, respectively. We discuss construction of these cases in "Algorithmic framework of dynamic random walks" section. Over time, nodes and edges are added to and/or deleted from the graph at time t′=t+i, i∈[1,2,...,+∞) forming an evolving graph\(G_{t}^{\prime }\).
(a valid set of random walks) A set of random walks RWt at time t is valid, if and only if, every random walk in RWt is an unbiased random walk on Gt.
(maintaining a valid set of random walks on an evolving network) Let a connected, unweighted and undirected graph Gt=(Vt,Et) where Vt denotes the node set of Gt and Et denotes the edge set of Gt at time t. Assume a valid set of random walks RWt are obtained on Gt at time t. As new edges are added to and/or deleted to the evolving graph, at any time t′=t+i, i∈[1,2,...,+∞) forming \(G_{t}^{\prime }\), the original set of random walks RWt will soon be rendered invalid, since many of its random walks will begin introducing a bias. We would like to design and develop methods for efficiently updating any biased random walk in \(RW_{t}^{\prime }\) with an unbiased random walk, so that \(RW_{t}^{\prime }\) always represents a valid set of random walks of \(G_{t}^{\prime }\).
The premise is that if we are able to solve Problem1 efficiently, then we will be in a position to obtain an accurate representation of the evolving network at anytime.
Algorithmic framework of dynamic random walks
In this Section, we describe a general algorithmic framework and novel methods for incrementally updating the set of random walks in reserve, obtained on the original network Gt(Vt,Et) at time t, so that they respect the updated network \(G_{t}^{\prime } \left (V_{t}^{\prime }, E_{t}^{\prime }\right)\) at time t′, where t′>t, and do not introduce any bias. Recall that these are random walks that could have been obtained directly by performing random walks on \(G_{t}^{\prime }\). The framework we describe is generic and can be used in any random walk-based embedding method. The first part of the Section presents algorithms for incrementally updating the set of random walks in hand, as edges and/or nodes are added to and/or deleted from the evolving network. The second part, presents an indexing mechanism that supports the efficient storage and retrieval (i.e., query, insert, update, deletion operations) of the set of random walks used for learning subsequent representations of the evolving network. A summary of notations is provided in Table 1.
Table 1 Summary of notations used in the dynamic random walk framework
Incremental update of random walks
Given a network Gt=(Vt,Et) at time t, we employ a standard StaticNRL methodFootnote 1 to simulate random walks. This method is configured to perform r random walks per node, each of length l (default values are r=10 and l=80). Let RWt be the set of random walks obtained, where |RWt|=|Vt|×r. We store the random walks in memory, using a data structure that provides random access to its elements (i.e., a 2-Dnumpy matrixFootnote 2). In practice, each finite-length random walk is stored as a row of a matrix, and each matrix element represents a single node of the network that is traversed by a random walk.
As we monitor changes in the evolving network, there are four distinct events that need to be addressed: i) edge addition, ii) edge deletion, iii) node addition, and iv) node deletion. These events can affect the network topology (and the set of random walks in hand) in different ways, therefore they need to be studied separately. First, we provide details of the edge addition and edge deletion events. This will bring up the challenges that need to be addressed in updating random walks and will introduce our main methods. Then, we visit node addition and node deletion and show that they can be treated as special cases of edge addition and edge deletion, respectively.
Edge addition
Assume that a single new edge eij=(nodei,nodej) arrives in the network at time t+1, so Et+1=Et∪(nodei,nodej). There are two operations that need to take place in order to properly update the set RWt of the random walks in hand:
Operation 1: contain the new edge to existing random walks in RWt.
Operation 2: discard obsolete parts of random walks of RWt and replace them with new random walks to form the new RWt+1.
Details of each operation are provided in the next paragraphs.
Operation 1: contain a new edge in RW We want to update the set RWt to contain the new edge (nodei,nodej). The update should occur in a way that it represents an instance of a possible random walk on Gt+1, and at the same time, it preserves the previous set of random walks RWt, as much as possible (to maintain network embedding stability). Note that due to the way that the original set of random walks was obtained, both nodei and nodej will occur in a number of random walks of RWt. We explain the update process for nodei; the same process is followed for nodej. First, we need to find all the random walks walksi∈RWt that include nodei. Then, we need to update them so as to reflect the existence of the new edge (nodei,nodej). In practice, the new edge offers a new possibility for each random walk in Gt+1 that reaches nodei to traverse nodej in the next step. The number of these random walks that include (nodei,nodej) depends on the node degree of nodei and it is critical for correctly updating random walks in RW. Formally, if the node degree of nodei in Gt is dt then in Gt+1 it will be incremented by one, dt+1=dt+1. Effectively, a random walk that visits nodei in Gt+1 would have a probability \(\frac {1}{d_{t+1}}\) to traverse nodej. This means that if there are freqi occurrences of nodei in RWt, then \(\frac {freq_{i}}{d_{t+1}}\) edges (nodei,nodej) need to be contained, by setting the next node of nodei to be nodej, in the current random walk. If nodei is the last node in a random walk then, there is no need to update the new edge in that random walk.
Naive approach: The naive approach to perform the updates is to visit all freqi occurrences of nodei in walksi∈RW and for each of them to decide whether to perform an update of the random walk (or not), by setting the next node to be nodej. The decision is based on tossing a biased coin, where with probability \(p_{{success}}=\frac {1}{d_{t+1}}\) we update the random walk, and with probability pfailure=1−psuccess we do not. While this method is accurate, it is not efficient as all occurrences of nodei need to be examined, when only a portion of them needs to be updated.
Faster approach: A more efficient way is to find all the freqi occurrences of nodei, and then to uniformly at random sample \(\frac {freq_{i}}{d_{t+1}}\) of them and update them by setting the next node to be nodej. While this method will be faster than the naive approach, it still resides on finding all the freqi occurrences of nodei in the set of random walks RW, which is an expensive operation. We will soon describe how this method can be accelerated by using an efficient indexing library that allows for fast querying and retrieval of all occurrences a node in random walks.
Operation 2: replace obsolete random walks Once a new edge (nodei,nodej) is contained in an existing random walk, it renders the rest of it obsolete, so it is best to be avoided. Our approach is to replace the remainder of the random walk by simulating a new random walk on the updated network Gt+1. The random walk starts at nodej and has a length lsim=l−(Indi+1), where Indi,0≤Indi≤l−1, is the index of nodei in the random walk that is currently updated. Once updates for nodei have been performed, the updates that are due to nodej are computed and performed.
Figure 2a presents an illustrative example of how updates of random walks work, in the case of a single incoming edge on a simple network. First, a set of random walks RWt are obtained (say 5 as illustrated by the upper lists of random walks). Let us assume that a new edge (1,4) arrives. Note that now, the degree of node 1 and node 2 will increase by 1 (dt+1=dt+1). Because of the new edge, some random walks need to be updated to account for the change in the topology. To perform the updates, we first search for all occurrences of i, freqi. Then, we uniformly at random sample \(\frac {freq_{i}}{d_{t+1}} = 2 / 2 = 1\) of them to determine where to contain the new edge. In the example, node 4 is listed after node 1 (i.e., the second node in the random walk #4 is now updated). The rest of the current random walk is obsolete, so it needs to be replaced. To perform the replacement a new random walk is simulated on the updated network Gt+1 that starts at node 4 and has a length of lsim=l−(Ind1+1)=10−(0+1)=9. The same process is repeated for node 4 of the added edge (1,4) (see the updates in random walks #2 and #5, respectively).
Illustrative example of EVONRL updates for edge addition and edge deletion (colored). a Example addition of a new edge (1;4). Random walks in reserve need to be updated to adhere to the change in the network topology. Our method guarantees that the new edge is equally represented in the updated set of random walks. b Example deletion of an existing edge (1;4). Random walks in reserve need to be updated to adhere to the change in the network topology. In this example, random walk #2 and #4 traverse edge (1;4) and need to be updated
The details of the proposed algorithm are described in Algorithm 1. Lines ?? and ?? of the algorithm invoke a Query operator. This operator is responsible for searching and retrieving information about all the occurrences of nodei in the set of the random walks RWt. In addition, lines ?? and ?? of the algorithm invoke a UpdateRandomWalks operator. This operator is responsible for updating any obsolete random walks of RWt with the updated ones to form the new valid set of random walks RWt+1, related to Gt+1. However, these operators are very computationally expensive, especially for larger networks, and therefore will perform very poorly. In paragraph 1, we describe how these two slow operators, UpdateRandomWalks and Query, can be replaced by similar operators offered off-the-shelf by high performance indexing and searching open-source technologies. In addition, so far, we have relied on maintaining the set of random walks RWt in memory. However, this is unrealistic for larger networks — while storing a network in memory as an edge list requires O(E), storing the set of random walks requires O(V·r·l) that is typically much larger for sparse networks. The indexing and searching technologies we will employ are very fast and at the same time are designed to scale to very large number of documents. Therefore, they are in position to scale well to very large number of random walks, as we discuss in "Extensions and variants" section.
To accommodate a set of new edges E+, the same algorithm needs to be applied repeatedly. The main assumption is that edges become available in a temporal order (a stream of edges), which is a common assumption for evolving networks. The premise of our method is that every time, only a small portion of the random walks need to be updated, therefore large performance gains are possible, without any loss in accuracy. In fact, the number of random walks affected depends on the node centrality of the nodes nodei and nodej that form the new edge (nodei,nodej). While our approach suggests that a new representation is required every time a single change occurs in the network that is not the case in real-world use cases. In fact, in paragraph 1, we provide an analytical method for determining the right time to obtain a new representation of the evolving network. As will see the method is based on an adaptive evaluation of the degree of divergence between the most recent random-walk set and the random-walk set utilized in the most recent network embedding. The method is tunable so it can be adjusted to meet the accuracy/sensitivity requirement of different domains, therefore can provide support for a number of real-world applications. We discuss also the implications of this issue to the time performance of the method in "Experimental evaluation" section.
Edge deletion
Assume a single existing edge eij=(nodei,nodej) is deleted from the network. Similar to edge addition, there are two operations that need to take place:
Operation 1: delete the existing edge from current random walks in RWt by removing any consecutive occurrence of edge's endpoints in the set.
Operation 1: delete an existing edge from RW In edge deletion, unlike with the case of edge addition (where we had to sample over all the occurrences of a specific node), all the walks that have traversed the existing edge (nodei,nodej) should be modified because all of them are now invalid. Other than that, the rest of the process is similar to that of edge addition. First, all random walks that have occurrences of (nodei,nodej) and (nodej,nodei) need to be retrieved. Then, the retrieved random walks need to be modified according to the method described in 1. Algorithm 2 describes this procedure in detail. Figure 2b presents an illustrative example of updates that need to take place due to a single edge deletion. First, a set of random walks are obtained. Let us assume that a new edge (1,4) is deleted, therefore random walks that traverse it, need to be updated. First, we retrieve random walks where node 1 and node 4 occur the one right after the other. For example, in random walk #4 of Fig. 2b, node 4 appears right after 1. Since now that edge doesn't exist anymore in the network, we need to update the random walk so as to allow an existing neighbor of node 1 to appear after node 4. This action is performed in operation 2.
Operation 2: replace obsolete random walks This operation is similar to the one in the case of adding a new edge. We just need to replace the remainder of any random walk affected by the Operation 1 by simulating a new random walk on the updated network Gt+1 of the right length. Following up with the running example, to perform the replacement of the obsolete random walk, a new random walk is simulated on network Gt+1 that starts at node 1 and has a length of lsim=l−(Ind1+1)=10−(0+1)=9.
A Note About Disconnected Nodes: During the process of deleting edges, any of the edge nodes might be disconnected from the rest of the network, forming isolated nodes. In that case, all r random walks in RW that start from an isolated node need to be deleted. In the case that only one of the nodes of a deleted edge becomes isolated, then the simulated random walk is obtained by starting a random walk from the node that remains connected in the network.
Node addition
Assume that a new node nodei is added to the network at time t+1, so Vt+1=Vt∪{nodei}. Initially, this node forms an isolated node (i.e., \(d_{i}^{t+1} = 0\)) and therefore there is no need to update the set of random walks RW. Now, assume that at a later time the node connects to the rest of the network through an edge (nodei,nodej). In that case, we treat the new edge as described earlier in paragraph 1. In addition to that we need to simulate a set of r new random walks, each of length l, all of which start from the new node nodei (recall that our original set of random walks consisted of r random walks of length l for each node in the graph). The newly obtained random walks are appended to RWt (i.e., it is |RWt+1|=|RWt|+r) and are utilized in subsequent network embeddings. There is also a special case where two isolated nodes are connected. In that case we need to simulate r random walks of length l starting from each node of nodei and nodej, respectively and append them to RWt.
Node deletion
Assume that an existing node nodei is deleted from the network at time t+1, so Vt+1=Vt∖{nodei}. In this case, first we obtain the set of neighbors Γ(nodei) of nodei. For each nodej∈Γ(nodei) there is an edge (nodei,nodej) in the network that needs to be deleted. We delete each of these edges as described earlier in paragraph 1 and obtain the updated set RW. The deletes occur in an arbitrary order, without any side effect. Eventually, this process forms an isolated node, which is removed from the graph. Deletion of the isolated node itself doesn't further affect the set RW.
Efficient storage and retrieval of random walks
The methods of updating random walks presented in the previous paragraph are accurate. However, they depend on operators Query and UpdateRandomWalks that are computationally expensive and cannot scale to larger networks. The most expensive operation is to search the random walks RWt to find occurrences of nodei and nodej of the new edge (nodei,nodej). In addition, updates of random walks can be expensive as large number of existing random walks might need to be updated.
To address these shortcomings, our framework of efficiently updating random walks relies on popular open-source indexing and searching technologies. These technologies offer operations for efficiently indexing and searching large collections of documents. For example, they support efficient full-text search capabilities where given a query term q, all documents in the collection that contain q are retrieved. In our framework we treat each random walk as a text "document". Therefore, each node visited by a random walk would be represented as a text "term", and all random walks would represent "a collection of documents". Using this analogy, we build an inverted random walk index, IRW. IRW is an index data structure that stores a mapping from nodes (terms) to random walks (documents). The purpose of IRW is to enable fast querying of nodes in random walks, and fast updates of random walks that can inform Algorithm 1. Figure 3 provides an illustrative example of a small inverted random walk index. In addition, we briefly describe how to create the index and use it in our setting.
Example inverted random walk index. Given a graph, five random walks are performed. Each random walk is treated as a document and is indexed using an open-source distributed indexing and searching library. The result is an inverted index that provides information about the frequency of any node in the random walks and information about where in the random walk the node is found
Indexing Random Walks: We obtain the initial set of random walks RWt at time t by performing random walks on the original network, similarly to the process followed in standard StaticNRL methods. Each random walk is transformed to a document by properly concatenating the ids of the nodes in the walk. For example, a short walk (x→y→z) over nodes x, y and z, will be represented as a document with content "x y z". These random walks are indexed to create IRW. It is important to note that once an index is available, there is no need to maintain the random walks in memory any more.
Querying Random Walks: We rely on the index IRW to perform any Query operation. Note, however, that there are additional advantages on using an efficient index. Besides searching and retrieving all random walks that contain a specific nodei, the index IRW can be configured to provide more quantities of interest. Specifically, we configure IRW so that every query retrieves additional information about the frequency of nodei,freqi and the position Indi of nodei in a retrieved random walk (see Fig. 3). The first quantity (freqi) is used to determine the number of updates that are required as discussed earlier. The second (Indi), is used to inform the operator Position in Algorithm 1 (lines ?? and ??). Note that there is a slight variation of how the Query operation is configured in the case of the edge deletion. Recall that in that event we need to retrieve random walks where the two nodes nodei and nodej are found the one right after the other (i.e., they form a step of the random walk). To accommodate this case we just need to configure the Query operation to retrieve all random walks that contain the bigram " nodeinodej". A bigram is a pair of contiguous sequence of words in a document or, following the analogy, a pair of contiguous sequence of nodes in a random walk. The indexing and searching technology we employ can handily support such queries.
Updating Random Walks: We rely on the index IRW for any UpdateRandomWalk operation. An update of a random walk is analogous to an update of a document in the index. In practice, any update of the index IRW is equivalent to deleting an old random walk and then indexing a new random walk. While querying using an inverted index is a fast process, updating an index is a slower process. Therefore, the performance of our methods is dominated by the number of random walks updates required. Still, our methods would perform multitude of times faster than StaticNRL methods. A detailed analysis of this issue is provided in "Experimental evaluation" section. Following the discussion about the edge deletion/addition, special care is required when these events involve isolated nodes. In particular, if a new edge connects a previously isolated node nodei to the network, then r new random walks need to be added in the index, each of which starts from nodei. The process of indexing the new random walks is similar to the process described in paragraph 1. Similarly, if an edge deletion event resulted in a node nodei being isolated, then all the r random walks that start from nodei need to be removed from the index. Removing a random walk from the index is analogous to deleting a document from the index.
Bulk updates: Additional optimizations are available as a result of employing an inverted index for the random walks. For example, we can take advantage of bulk updates, where the index need only be updated when a number of new edges have arrived. This means that changes of single incoming edges won't be reflected in IRW right away. While this optimization has the premise to make our methods faster (since updates occur once in a while), it risks harming its accuracy. In practice, it offers an interesting trade-off between accuracy and time performance that domain-specific applications need to tune. Experiments in "Experimental evaluation" section demonstrate this tradeoff.
Evolving network representation learning
So far we have described our framework for maintaining an always valid set of random walks RWt at time t. Recall that our final objective is to be able to learn a representation of this evolving network. For the embedding process we resort to the same embedding of standard StaticNRL methods. Below we describe how embeddings of the evolving network are obtained, given a set of random walks RWt. Then, a general strategy for obtaining an embedding only when it is mostly needed.
Learning embeddings
Given a general network, Gt=(Vt,Et), our goal is to learn the network representation f(Vt) using the skip-gram model. f(Vt) is a |Vt|×d matrix where d is the network representation dimension and each row is the vector representation of a node. At the first time-stamp, the node vector representations (neural network's weights) are initialized randomly and we use this initialization for other timestamps' training. The training objective function is to maximize the log-probability of the nodes appearing in the context of the node ni. Context of each node ni is provided by the valid set of random walks RWt, similarly to the process described in previous work (Perozzi et al. 2014; Grover and Leskovec 2016). Using the approximate objective, skip-gram with negative sampling (Mikolov et al. 2013a), these embeddings are optimized by stochastic gradient decent so that:
$$ Pr(\mathit{n_{j}}|\mathbf{n_{i}}) \propto \exp{\left(\mathbf{n_{j}^{T}}\mathbf{n_{i}}\right)} $$
where ni is the vector representation of a node ni(f(ni)=ni). Pr(nj|ni) is the probability of the observation of neighbor node nj, within the window-size given that the window contains ni. In our experiments, we use the gensim implementation of the skip-gram modelFootnote 3. We set our context-size to k=5 and the number of dimensions to d=128, unless otherwise stated.
Analytical method for determining the timing of a network embedding
EVONRL has the overhead of first indexing the set of initial random walks RW. At that time, we randomly initialize the skip-gram model and keep these initialization weights for the learning phase of subsequent times. As new edges/nodes are added/deleted, EVONRL performs the necessary updates as described earlier. At each time step a valid set of random walks is available that can be used to obtain a network embedding. As we show in "Experimental evaluation" section an embedding obtained by our incrementally updated set of random walks effectively represents embeddings obtained by applying a StaticNRL method directly on the updated network. However, while re-embedding the network every time a change occurs in it will result in accurate embeddings, this process is very expensive and risks to render the method non-applicable in real-world scenarios. Therefore, and depending on the domain, it is reasonable to assume that only a limited number of re-embeddings be obtained. This introduces a new problem: when is the right time to obtain a network embedding? In fact, this decision process demonstrates an interesting tradeoff between accuracy and time performance of the method proposed. In the rest of the paragraph we introduce two strategies for determining the time to obtain network embeddings.
PERIODIC: This is a sensible baseline where, as the name reveals, obtains embeddings periodically, every q time steps. Depending on the sensitivity of the domain we operate on, the period can be shorter or longer. This method is easy to implement, but it is obtaining network embedding being agnostic of the different changes that occur in the network and whether they are significant (or not).
ADAPTIVE: We introduce an analytical method for determining the right timing of obtaining a network embedding. The key idea of the method is to continuously monitor the changes that occur in the network. Then, if significant changes are detected it obtains a new network embedding. In fact, we monitor two conditions, the first is able to detect occurrence of a critical change (e.g., addition of a very important edge) and is based on the idea of peak detection; the second is able to evaluate cumulative effects due to a number changes. We discuss the structure of these conditions in the following paragraphs.
Peak detection: We start by providing background of a z-score. A z-score (or standard score) is a popular statistical measure that indicates how many standard deviations away an observation is from its mean. When the population mean and the population standard deviation are unknown, the standard score may be calculated using the sample mean and sample standard deviation as estimates of the population values. In that case, the z-score of observed values x can be calculated from the following formula:
$$ z = \frac{x - \hat{x}}{\hat{\sigma}} $$
where \(\hat {x}\) is the mean of the sample and \(\hat {\sigma }\) is the standard deviation of the sample.
In our setting, we want to detect when important changes occur in the network, so as to obtain a timely network representation. As we described earlier a good proxy for what consists an important change in a network is the number of random walks that are affected because of the change (edge addition/deletion, node addition/deletion). We can utilize the z-score of Eq. (4) to detect peaks. A peak or spike is a generic term which describes a sudden increase or outburst in a sequenced data (Barnett and Lewis 1974). In our problem, the number of random walk changes are monitored and peaks represent significant changes in the number of random walks affected. Formally, let lag be the number of changes observed in the sample. The observation window is spanning from t−lag to t and we compute the mean of the sample at t as avg[t]. In a similar way, we calculate the standard deviation of the sample at t to be std[t]. Let N[t] be the observation at time t that represents the number of random walks that have been updated due to a network change. Now, given N[t],avg[t],std[t] and a threshold τ, a peak occurs at time t if the following condition holds:
$$ N[t] > \tau \times std[t] + avg[t] $$
If the condition of Eq. (5) holds, then we know that a significant change has occurred and we decide to obtain a new network representation. The details of the procedure are shown in Algorithm 3. Notations used in this algorithm are summarized in Table 2. Figure 4 provides an illustrative example of the peak detection method. In this example we set lag=10 and τ=3. The figure shows the results of the peak detection method for 100 changes occurring in a network (BlogCatalog network, edge addition; edges are added one by one and are randomly selected from the potential edges of the network). Our peak detection algorithm detects a total of 6 peaks occurring at t={13,19,48,53,57,60}.
Example peak detection method for the case of adding edges in the BlogCatalog network. The upper plot shows the number of random walks that are updated in RW as a function of new edges added. It is evident that some edges have a larger effect in RW as depicted by higher values. The middle plot, shows the mean (middle almost straight line), as well as the boundaries defined by the current threshold of τ×std (the two lines above and below the mean line). The bottom plot provides the signal for decision making; every time that the current change at time t is outside the threshold it signals that a network embedding should be obtained. In the example this is the case for five times t={13,19,48,53,57,60}
Table 2 Summary of notations used in decision-making algorithm
Cut-off score: Sometimes, changes in the network can be smooth, without any acute changes. In that case the peak detection method will fail to obtain any embedding as peaks (almost) never occur. To avoid these cases, besides the peak detection method, we employ an additional metric that monitors the cumulative effect of all the changes since the last embedding was obtained. Formally, let N[t] be the observation at time t that represents the number of random walks that have been updated due to a network change. Then, the total number of random walks that have been changed between the time that the last embedding told was obtained and the current time t is given by:
$$ \#RW_{t_{{old}}}^{t} = \sum_{t=t^{old}}^{t} N[t] $$
Now, given \(\#RW_{t_{{old}}}^{t}\) and a threshold cutoff, we monitor the following condition:
$$ \#RW_{t_{{old}}}^{t} > cutoff $$
If at any time t Eq. (7) holds, then we know that significant cumulative changes have occurred in the network and we decide to obtain a new network representation.
As we show in "Experimental evaluation" section combining both conditions of Eqs. (5) and (7) gives the best results, as it balances locally significant as well as cumulative effect of changes.
Experimental evaluation
In this Section, we experimentally evaluate the performance of our dynamic random walk framework and EVONRLFootnote 4. In particular, we aim to answer the following questions:
Q1 effect of network topology How the topology of the network affects the number of random walks that need to be updated?
Q2 effect of arriving edge importance How edges of different importance affect the overall random walk update time?
Q3 accuracy performance ofEVONRL What is the accuracy performance of EvoNRL compared to the ground truth provided by StaticNRL methods?
Q4 classification performance ofEVONRL What is the accuracy performance of EvoNRL in a downstream data-mining task?
Q5 time performance ofEVONRL What is the time performance of EvoNRL?
Q6 decision-making performance ofEVONRL How well does the strategy of EvoNRL for obtaining network representations work?
Q1 and Q2 aim to shed light on the behavior of our generic computational framework for dynamically updating random walks in various settings. Q3, Q4, Q5 and Q6 aim to demonstrate how EVONRL performs. Before presenting the results, we provide details of the computational environment and the data sets employed.
Environment: All experiments are conducted on a workstation with 8x Intel(R) Core(TM) i7-7700 CPU @ 3.60GHz and 64GB memory. Python 3.6 is used and the static graph calculations use the state-of-the-art algorithms for the relevant metrics provided by the NetworkX network library.
Data: For the needs of our experiments both synthetic data and real data sets have been employed.
Protein-Protein Interactions (PPI): We use a subgraph of PPI for Homo Sapiens and use the labels from the preprocessed data used in (Grover and Leskovec 2016). The network consists of 3890 nodes, 76584 edges and 50 different labels.
BlogCatalog (Reza and Huan): BlogCatalog is a social network of blogers which each edge indicates a social interaction among them. This network consists of 10312 nodes, 333983 edges and 39 different labels.
Facebook Ego Network (Leskovec and Krevl 2014): Facebook ego network is the combined ego network of each node. There is an edge from a node to each of its friends. This network consists of 4039 nodes, 88234 edges.
Arxiv HEP-TH (Leskovec and Krevl 2014): Arxiv HEP-TH (high energy physics theory) network is the citation network from e-print Arxiv. If paper i cites paper j, there is a directed edge from i to j. This network consists of 27770 nodes, 352807 edges.
Synthetic Networks: We create a set of Watts-Strogatz (Newman 2003) random networks of different sizes (n={1000,10000}) and different rewiring probabilities (p={0,0.5,1.0}). The rewiring probability is used to create representative Lattice (p=0), Small-world (p=0.5) and Erdos-Reyni (p=1) networks, respectively.
Q1 effect of network topology
We evaluate the effect of randomly adding a number of new edges in networks of different topologies, but same size. For each case, we report the number of the random walks that need to be updated. Figure 5 shows the results, where it becomes clear that as more new edges are added, more random walks are affected. The effect is more stressed in the case of the Small-world and Erdos-Reyni networks. This is to be expected, since these networks are known to have small diameter, therefore every node is easily accessible from any other node. As a result, every node has a high chance to appear in any random walk. In contrast, Lattices are known to have larger diameter, therefore only a small number of nodes (out of all nodes in the network) can be accessible by any random walk. As a result, nodes are more equally distributed in all random walks.
Effect of network topology (the axis of #RW affected is in logarithmic scale). As more new edges are added, more random walks are affected. The effect is more stressed in the case of the Small-world and Erdos-Reyni networks, than the Lattice network
Q2 effect of arriving edge importance
By answering Q1, it becomes evident that even a single new edge can have a dramatic effect in the number of random walks that need to be updated. Eventually, the number of random walks affected, will have an effect to the time performance of updating these random walks in our framework. In this set of experiments we perform a systematic analysis of the effect of the importance of an arriving edge to the time required for the update to occur. Importance of an incoming edge \(e_{{ij}}^{t+1} = (n_{i}, n_j)\) at time t+1 in a network can be defined in different ways. Here, we define three metrics of edge importance, based on properties of the endpointsni,nj of the arriving edge:
Sum of frequencies of edge endpoints inRWt.
Sum of the node degrees of edge endpoints inGt.
Sum of the node-betweenness of edge endpoints inGt.
Results of the different experiments are presented in Fig. 6. The first observation is that important incoming edges are more expensive to update, sometimes up to three or four times (1.6sec vs 0.4sec). This is expected, as more random walks need to be updated. However, the majority of the edges are of least importance (lower left dense areas in Fig. 6a, b, and c), so fast updates are more common. Finally, the behavior of sum of node frequencies (Fig. 6a) and sum of node degrees (Fig. 6b) of the edge endpoints are correlated. This is because the node degree is known to be directly related to the number of random walks that traverse it. On the other hand, node-betweenness demonstrates more unstable behavior since it is mostly related to shortest paths and not just paths (which are related to random walks).
Dependency of EVONRL running time on importance of added edge as described by various metrics on PPI Network. a frequency of the new edge endpoints, b node degree of the new edge endpoints, and c node betweenness of the new edge endpoints
Q3 accuracy performance of EVONRL
In this set of experiments we evaluate the accuracy performance of EVONRL and show that it is very accurate. At this point, it is important to note that evidence of our EVONRL performing well is provided by demonstrating it obtains similar representations to the ground truth provided by running StaticNRL on different instances of the evolving network. This is because the objective of our method is to resemble as much as possible what the actual changes in the original network are by incrementally maintaining a set of random walks and monitoring the changes. In practice, we aim to show that our proposed algorithm is able to update random walks in reserve such that they are always representing unbiased random walks that could have been obtained by running StaticNRL on the updated network. In these experiments, we show the representation learned by EvoNRL and the ground truth provided by the StaticNRL are similar to each other by using a representational similarity metric.
Similarity of two representations
Our goal here is to compare the representations learned by the neural network and show that EvoNRL results in a similar representations to ground truth provided by StaticNRL methods. Comparing representations in neural networks is difficult as the representations vary even across the neural networks trained on the same input data with the same task (Raghu et al. 2017). In this paper, representations are weights of the representation learned by either our EvoNRL method or the StaticNRL method, and they represent the representation learned by a skip-gram neural network. In order to determine the correspondence between these representations, we use the recent similarity measures of neural networks studied in (Morcos et al. 2018) and (Kornblith et al. 2019). Dynamics of neural networks call for a similarity metric that is invariant to orthogonal transformation and invariant to isotropic scaling. Assuming two representations \(X \in \mathbb {R}^{n \times d}\) and \(Y \in \mathbb {R}^{n \times d}\), we are concerned about a scalar similarity index s(X,Y) which can be used to compare the two neural network representations. There are many methods for comparing two finite set of vectors and measure the similarity between them. The simplest approach is to employ a dot-product based similarity. By summing the square dot-product of each corresponding pair of vectors in X and Y, we can have a similarity index between matrices X and Y. This approach is not practical as representations of the neural networks can be described on two different basis and result in a misleadingly similarity index. Therefore invariance to linear transforms is crucial in neural network representational similarity metrics. Recently, Canonical Correlation Analysis (CCA) (Hotelling 1992) is used as a tool to compare representations across networks. Canonical Correlation Analysis has been widely used to evaluate the similarity between computing models and brain activity. CCA can find similarity between representations where they are superficially dissimilar. Its invariance to linear transforms makes CCA a useful tool to quantify the similarity of EvoNRL and StaticNRL representations (Morcos et al. 2018).
Canonical correlation analysis (CCA): Canonical Correlation Analysis (Hotelling 1992) is a statistical technique to measure the linear relationship between two multidimensional set of vectors. Ordinary Correlation analysis is highly dependent on the basis which the vectors are described on. The important property of CCA is that it is invariant to affine transformations of the variables which makes it a proper tool to measure representation similarity by. If we have two sets of matrices \(X \in \mathbb {R}^{n \times d}\) and \(Y \in \mathbb {R}^{n \times d}\), Canonical Correlation Analysis will find two bases, one for X and one for Y such that after their projections into these bases, their correlation will be maximized. for 1≤i≤d, the ith, canonical correlation coefficient is given by:
$$ \begin{aligned} \rho_{i} = \max_{w^{i}_{X}, w^{i}_{Y}} corr\left(Xw^{i}_{X}, Yw^{i}_{Y}\right)\\ {subject to} \; \forall_{j < i} \: Xw^{i}_{X} \bot Xw^{j}_{X}\\ \forall_{j < i} \: Yw^{i}_{Y} \bot Yw^{j}_{Y} \end{aligned} $$
where the vectors \(w^{i}_{X} \in \mathbb {R}^{d}\) and \(w^{i}_{Y} \in \mathbb {R}^{d}\) transform the original matrices into canonical variables \(Xw^{i}_{X}\) and \(Yw^{i}_{Y}\).
$$ R^{2}_{{CCA}} = \frac{\Sigma_{i=1}^{d} \rho_{i}^{2}}{d} $$
The mean squared CCA correlation (J. Ramsay et al. 1984), \(R^{2}_{{CCA}}\) reports the sum of the squared canonical correlations. This sum is a metric that shows the similarity of the two multidimensional sets of vector.
Experimental scenario: In these experiments, the original network is the initial network at the beginning. We simulate random walks on this network and learn its representation. After that, we sequentially make changes (add edges, remove edges, add nodes and remove nodes) to the initial network and keep the random walks updated using EvoNRL. In certain points (for example after every 1000 edge addition in the PPI network), we learn the network representation in two ways. One is by simulating new random walks on the updated network (original network with new edges/nodes or missing edges/nodes) and second is learning the representation using EvoNRL. Now we have two representations of the same network and the goal is to compare them to see how similar EvoNRL is to StaticNRL. Note that StaticNRL simulates walks on the updated networks while EvoNRL has been updating the original random walk set. Representations obtained by StaticNRL are results of simulating random walks on the network. Because of the randomness involved in the process, it is typical that two differnet StaticNRL representations of the same network are not identical. We can measure, the similarity of the different representations using CCA. In our evaluation, we aim to demonstrate that EvoNRL is as similar to StaticNRL and that this similarity is comparable to the similarity obtained by applying StaticNRL multiple times on the same network. At any stage of the change (edge addition, edge deletion, node addition, node deletion) in the network, EvoNRL is updating the random walk set in a way that it is representing the network. First, we run StaticNRL multiple times (x5) on a network. Each StaticNRL is simulating a random walk set on the evolving network at certain times. Representations are two finite sets of vectors in d-dimensional space and compare how similar these two sets are.
Adding edges: Given a network G=(V,E), we can add a new edge by randomly picking two nodes in the network that are not currently connected and connect them. Adding new edges to the network should have an effect on the network embedding. By adding edges, as the network diverges from its original state, the embedding will diverge from the original network as well. Figure 7 shows the accuracy results of EvoNRL. We observe that the CCA similarity index of EVONRL follows the same trend as the StaticNRL in all the networks: BlogCatalog (Fig. 7a) and the PPI (Fig. 7b), Facebook (Fig. 7c) and Cit-HepTh (Fig. 7d) networks. The similarity of the two methods remains consistent as more edges are added (up to 12% of the number of edges in the original PPI; up to 14% of the number of edges in the original BlogCatalog, Facebook and Cit-HepTh). In Fig. 7, there are two sorts of comparison. First, The similarity of EvoNRL and the Original Network (The network before changes occur to it) is measured. The decreasing trend in orange stars in Fig. 7 shows that the EvoNRL is updating the set of random walks and the representations of the updated networks are diverging from the representation of the original network. On the other hand, we see that EvoNRL is more correlated to the original set of the random walk (orange stars), compared to StaticNRL (Blue Triangles). Blue Triangles are the average of canonical correlation of the original network with 4 different runs of StaticNRL. It shows that the representation of the evolving network is diverging from the original network. So far we have showed that EvoNRL is consistently updating the original set of random walks and makes difference in the network's representation. The question is are these updates accurate? To answer this question we add edges step by step to the original network. Using EvoNRL we keep updating a set of random walk and get the representation of the network in a certain points. On the other hand, we run StaticNRL on the updated network at the same certain points. Because of the randomness of the random walks we repeat StaticNRL 4 times. We compare the StaticNRL representations obtained from the same network with each other to have a baseline of the similarity metric. The red squares showing as 'StaticNRL vs StaticNRL' in Fig. 7 are showing the average similarity of representations of StaticNRL compared to each other 2 by 2. Our goal is to show, EvoNRL keeps updating the random walk set in an accurate way and the representation obtained by EvoNRL is as accurate as StaticNRL. To show this, we measure the canonical correlation of EvoNRL representation and the StaticNRl. We observe that (green circles) EvoNRL representations is very similar to the StaticNRL representations and can be an instance on StaticNRL.
Accuracy performance of EVONRL — adding edges. a BlogCatalog, b PPI, c Facebook, d Cit-HepTh
Removing edges: Given a network G=(V,E), we can remove an edge by randomly choosing an existing edge e∈E and remove it from the network. Removing existent edges should have an effect in the network embedding. Figure 8 show the accuracy results of edge deletion. Similar to edge addition, We observe that the CCA similarity of EVONRL follows the same trend as the StaticNRL in all the networks: BlogCatalog (Fig. 8a) and the PPI (Fig. 8b), Facebook (Fig. 8c) and Cit-HepTh (Fig. 8d) networks.
Accuracy performance of EVONRL — removing edges. a BlogCatalog, b PPI, c Cit-HepTh, d Facebook
Adding nodes: As we described in "Evolving network representation learning" section node addition can be treated as a special case of edge addition. This is because whenever a node is added in a network, a number of edges attached to that node need to be added as well. To emulate this process, given a network G=(V,E), first we create a network G′=(V′,E′), where V′⊆V,E′⊆E as follows. We uniformly at random sample nodes V′⊆V from G and then remove these nodes and all their attached edges E′⊆E from G, forming G′. Following that process, we obtain a new network for BlogCatalog with V′=8312 and a new network for PPI with V′=3390 nodes, respectively. Then, we start adding the nodes v∈V′′=V∖V′ that have been removed from G, one by one. Whenever, a node v∈V′′ is added to G′, any edge between v and nodes existing in the current state of network G′ are added as well. Adding nodes to the network should have an effect in the network embedding. Figure 9 shows the accuracy results of node addition. CCA compares two sets of vectors with the same cardinality. Because the number of the nodes and therefore the number of the vectors in the representation are variant, we can not compare the updated representations with the original network. In these experiments we show that EvoNRL and StaticNRL on the same network are very similar to each other and EvoNRL is an accurate instance of StaticNRL.
Accuracy performance of EVONRL — adding nodes. a BlogCatalog, b PPI, c Cit-HepTh, d Facebook
Removing nodes: As we described in "Evolving network representation learning" section node deletion can be treated as a special case of edge deletion. Given a network G=(V,E), we start removing nodes v∈V from the network, one by one. When a node is removed all the edges connecting this node to the network are removed as well. The process of removing nodes will result in a new network G′(V′,E′), where V′⊆V and E′⊆E. Removing existing nodes from the networ effect in the network embedding. Figure 10 shows the accuracy result of node deletion. In the evolving network, nodes are removed from the network sequentially and EvoNRL always maintains a valid set of random walks. we show that the representations obtained from these random walks are similar to StaticNRL representations. Same as node addition, because the number of the nodes are changing, we can not compare the representations with the original network's representation. The experiments above provides strong evidence that our random walk updates are correct and can incrementally maintain a set of random walks that is their corresponding representations are similar to that of obtained by StaticNRL.
Accuracy performance of EVONRL — removing nodes. a BlogCatalog, b PPI, c Cit-HepTh, d Facebook
Q4 classification performance of EVONRL
In this set of experiments we evaluate the accuracy performance of EVONRL and show that it is very accurate. At this point, it is important to note that evidence of our EVONRL performing well is provided by demonstrating it has similar accuracy to StaticNRL, for the various aspects of the evaluation (and not by demonstrating loss/gains in accuracy). This is because the objective of our method is to resemble as much as possible what the actual changes in the original network are by incrementally maintaining a set of random walks and monitoring the changes. In practice, we aim to show that our proposed algorithm is able to update random walks in reserve such that they are always representing unbiased random walks that could have been obtained by running StaticNRL on the updated network.
Experimental scenario: To evaluate our random walk update algorithm, we resort to accuracy experiments performed on a downstream data mining task: multi-label classification. The network topology of many real-world networks can change over time due to either adding/removing edges or adding/removing nodes in the network. In our experimental scenario, given a network we simulate and monitor network topology changes. Then, we run StaticNRL multiple times, one time after each network change and learn multiple network representations over time. The same process is followed for EVONRL but this time we only need to update the random walks RWt at each time t and use these for learning multiple network representations over time. In multi-label classification each node has one or more labels from a finite set of labels. In our experiments, we see 50% of nodes and their labels in the training phase and the goal is to predict labels of the rest of the nodes. We use node vector representations as input to a one-vs-rest logistic regression classifier with L2 regularization. Finally, we report the Macro−F1 accuracy of the multi-label classification of StaticNRL and EVONRL as a function of the fraction of the network changes. For StaticNRL, since it is sensitive to the fresh set of random walks obtained every time, we run multiple times (10x) and report the averages. We experiment with the BlogCatalog and PPI networks. In the following paragraphs we present and discuss the results for each of the interesting cases (adding/removing edges, adding/removing nodes).
Adding edges: Given a network G=(V,E), we can add a new edge by randomly picking two nodes in the network that are not currently connected and connect them. Adding new edges to the network should have an effect on the network embedding and thus in the overall accuracy of the classification results. Figure 7 shows the results. We observe that the Macro-F1 accuracy of EVONRL follows the same trend as the one of StaticNRL in both the BlogCatalog (Fig. 11a) and the PPI (Fig. 11b) networks. The accuracy of the two methods remains consistent as more edges are added (up to 12% of the number of edges in the original PPI; up to 14% of the number of edges in the original BlogCatalog). This provides strong evidence that our random walk updates are correct and can incrementally maintain a set of random walks that is similar to that obtained by StaticNRL when applied in an updated network.
Accuracy performance of EVONRL — adding new edges. a BlogCatalog, b PPI
Removing edges: Given a network G=(V,E), we can remove an edge by randomly choosing an existing edge e∈E and remove it from the network. Removing existent edges should have an effect in the network embedding and thus in the overall accuracy of the classification results. We evaluate the random walk update algorithm for the case of edge deletion in a way similar to that of adding edges. The only difference is that every time an edge is deleted at t we update random walks to obtain RWt. Then, the updated RWt can be used for obtaining a network representation. Same setting is used in multi-label classification. Figure 12 shows the results. Again we observe that the Macro-F1 accuracy of EVONRL follows the same trend as the one of StaticNRL in both the BlogCatalog (Fig. 12a) and the PPI (Fig. 12b) networks.
Accuracy performance of EVONRL — removing edges. a BlogCatalog, b PPI
Adding nodes: As we described in "Evolving network representation learning" section node addition can be treated as a special case of edge addition. This is because whenever a node is added in a network, a number of edges attached to that node need to be added as well. To emulate this process, given a network G=(V,E), first we create a network G′=(V′,E′), where V′⊆V,E′⊆E as follows. We uniformly at random sample nodes V′⊆V from G and then remove these nodes and all their attached edges E′⊆E from G, forming G′. Following that process, we obtain a new network for BlogCatalog with V′=8312 and a new network for PPI with V′=3390 nodes, respectively. Then, we start adding the nodes v∈V′′=V∖V′ that have been removed from G, one by one. Whenever, a node v∈V′′ is added to G′, any edge between v and nodes existing in the current state of network G′ are added as well. Adding nodes to the network should have an effect in the network embedding and thus in the overall accuracy of the classification results. We evaluate the random walk update algorithm for the case of node addition in a way similar to that of adding edges. The only difference is that every time a node is added at t we update random walks to obtain RWt, by adding a number of edges. Then, the updated RWt can be used for obtaining a network representation. Figure 13 shows the results. Again we observe that the Macro-F1 accuracy of EVONRL follows the same trend as the one of StaticNRL in both the BlogCatalog (Fig. 13a) and the PPI (Fig. 13b) networks.
Accuracy performance of EVONRL — adding new nodes. a BlogCatalog, b PPI
Removing nodes: As we described in "Evolving network representation learning" section node deletion can be treated as a special case of edge deletion. Given a network G=(V,E), we start removing nodes v∈V from the network, one by one. When a node is removed all the edges connecting this node to the network are removed as well. The process of removing nodes will result in a new network G′(V′,E′), where V′⊆V and E′⊆E. Removing existing nodes from the network should have an effect in the network embedding and thus in the overall accuracy of the classification results. We evaluate the random walk update algorithm for the case of node deletion in a way similar to that of deleting edges. The only difference is that every time a node is deleted at t we update random walks to obtain RWt, by removing a number of edges. Then, the updated RWt can be used for obtaining a network representation. Figure 14 shows the results. Again we observe that the Macro-F1 accuracy of EVONRL follows the same trend as the one of StaticNRL in both the BlogCatalog (Fig. 14a) and the PPI (Fig. 14b) networks.
Accuracy performance of EVONRL — removing new nodes. a BlogCatalog, b PPI
Discussion about accuracy value fluctuations: While we have demonstrated that EVONRL is able to resemble the accuracy performance obtained by StaticNRL, one can observe that in some cases the accuracy values of the methods can substantially fluctuate. This behavior can be explained by the sensitivity of the StaticNRL methods to the set of random walks obtained from the network, as discussed in the motivating example of "Evaluation of the stability of StaticNRL methods" section. EVONRL would also inherit this problem, as it depends on an initially obtained set of random walks that is subsequently updated at every network topology change. To demonstrate this sensitivity effect, we run control experiments on the PPI network for the case of adding new nodes in the network G, similar to the experiment in Fig. 13b. However, this time, instead of reporting the average over a number of runs for the StaticNRL method, we report all its instances (ref(Fig. 15)). In particular, as we add more nodes (the number of nodes increases from 3390 to 3990) a new network is obtained. We report the accuracy values obtained by running StaticNRL multiple times (40x) on the same network. We also depict the values of two different runs for EVONRL. Each run obtains an initial set of random walks that is incrementally updated in subsequent network topology changes. It becomes evident that the StaticNRL values can significantly fluctuate due to the sensitivity to the set of random walks obtained. It is important to note that EVONRL manages to fall within the range of these fluctuations.
Accuracy values obtained by running StaticNRL multiple times on the same network. The values are significantly fluctuating due to sensitivity to the set of random walks obtained. Similarly, EVONRL is sensitive to the initial set of random walks obtained. Two instances of EVONRL are shown, each of which operates on a different initial set of random walks
Q5 time performance of EVONRL
In this set of experiments we evaluate the time performance of our method and show that EVONRL is very fast. We run experiments on two Small-world networks (Watts-Strogatz (p=0.5)), with two different number of nodes (|V|=1000 and |V|=10000). We evaluate EVONRL against a standard StaticNRL method from the literature (Grover and Leskovec 2016). Both algorithms start with the same set of random walks RW. As new edges are arriving, StaticNRL needs to learn a new network representation by resimulating a new set of walks every time. On the other hand, EVONRL has the overhead of first indexing the set of initial random walks RW. Then, for every new edge that is arriving it just needs to perform the necessary updates as described earlier. Figure 16 shows the results. It can be seen that the performance of StaticNRL is linear to the number of new edges, since it has to run again and again for every new edge. At the same time, EVONRL is able to accommodate the changes more than 100 times faster than StaticNRL. This behavior is even more stressed in the larger network (where the number of nodes is larger). By increasing the number of nodes, running StaticNRL becomes significantly slower, because by design it needs to simulate larger amount of random walks. On the other hand, EVONRL has a larger initialization overhead, but after that it can easily accommodate new edges. This is because every update is only related to the number of random walks affected and not the size of the network. This is an important observation, as it means that the benefit of EVONRL will be more stressed in larger networks.
EVONRL scalability (running time axis is in logarithmic scale). StaticNRL scales linearly to the number of new edges added in the network, since it has to run again and again for every new edge. At the same time, EVONRL is able to accommodate the changes more than 100 times faster than StaticNRL. This behavior is even more stressed in the larger network (where the number of nodes is larger)
Q6 decision-making performance of EVONRL
In this experiment, we compare the two different strategies for deciding when to obtain a network representation, PERIODIC and ADAPTIVE. The experiment is performed using the BlogCatalog network and the changes in the network are related to edge addition. For presentation purposes, we limit the experiment to 1000 edges. The evaluation of this experiment is based on the number of random walk changes \(RW^{t}_{t_{{old}}}\) between a random walk set obtained at time t (one edge is added at each time) and a previously obtained network representation as defined by each strategy. Results are shown in Fig. 17. The PERIODIC strategy represents a "blind" strategy where new embeddings are obtained periodically (every 50 times steps or every 100 time steps). On the other hand, the ADAPTIVE method is able to make informed decisions as it monitors the importance of every edge added in the network. The ADAPTIVE method is basing its decisions on the a peak detection method (τ=3.5) and a method that monitors cumulative effects due to a number of changes (cutoff=4000). As a result, ADAPTIVE is able to perform much better, as depicted by many very low values in the \(RW^{t}_{t_{{old}}}\).
Comparative analysis of different strategies for determining when to obtain a network representation. The PERIODIC methods will obtain a new representation every 50 or 100 time steps (i.e., network changes). Our proposed method, ADAPTIVE, is combining a peak detection method and a cumulative changes cut-off method to determine the time to obtain a new network representation. As a result it is able to make more informed decisions and perform better. This is depicted by smaller (on average) changes of the \(RW_{t_{old}}^{t}\), which implies that a more accurate network representation is available for down-stream network mining tasks
Extensions and variants
While our algorithms have been described and evaluated on a single machine, they have been designed with scalability in mind. Recall that our indexing and searching of random walks is supported by ElasticsearchFootnote 5, which itself is based on Apache LuceneFootnote 6. Elasticsearch is a distributed index and search engine that can naturally scale to very large number of documents (i.e., a very large number of random walks in our setting). There are a couple of basic concepts that make a distributed index and search engine scalable enough to be suitable for the needs of our problem:
Index sharding: One of the great features of a distributed index is that it's designed from the ground up to be horizontally scalable, meaning that more nodes can be added to the cluster to match the capacity required by the problem. It achieves horizontal scalability by sharding its index and assigning each shard to a node in the cluster. This allows each node to have to deal with only part of the full random walk index. Furthermore, it also has the concept of replicas (copies of shards) to allow fault tolerance and redundancy, as well as an increased throughput.
Distributed search: Searching a distributed index is done in two phases:
Querying: Each query q is sent to all shards of the distributed index and each shard returns a list of the matching random walks. Then, the lists are merged, sorted and returned along with the random walk ids.
Fetching: Each random walk is fetched by the shard that owns it using the random walk id information. Random walks that lie in different shards can be processed in parallel by the method requesting them.
Therefore, while our algorithms are demonstrated in smaller networks for clarity of coverage and better representation of the algorithmic comparison, in practice they can be easily and naturally expanded to very large graphs. Extensions of the algorithms to a distributed environment are out of the scope of this work.
Our work is mostly related to research in the area of static network representations learning and dynamic network representation learning. It is also related to research in random walks.
Static network representations learning: Starting with Deepwalk (Perozzi et al. 2014), these methods use finite length random walks as their sampling strategy and inspired by word2vec (Mikolov et al. 2013b) use skip-gram model to maximize likelihood of observing a node's neighborhood given its low dimensional vector. This neighborhood is based on random walks. LINE (Tang et al. 2015) proposes a breadth-first sampling strategy which captures first-order proximity of nodes. In (Grover and Leskovec 2016), authors presented node2vec that combines LINE and Deepwalk as it provides a flexible control of random walk sampling strategy. HARP (Chen et al. 2017) extends random walks by performing them in a repeated hierarchical manner. Also there have been further extensions to the random walk embeddings by generalizing either the embeddings or random walks (Chamberlain et al. 2017;Perozzi et al. 2016). Role2Vec (Ahmed et al. 2018) maps nodes to their type-functions and generalizes other random walk based embeddings. Our work is focusing on how many of the above methods introduced for static networks (the ones that use random walks) can be extended to the case of evolving networks.
Dynamic network representation learning: Existing work on embedding dynamic networks often apply static embedding to each snapshot of the network and then rotationally align the static embedding across each time-stamp (Hamilton et al. 2016). Graph factorization approaches attempted to learn the embedding of dynamic graphs by explicitly smoothing over consecutive snapshots (Ahmed et al. 2013). DANE (Li et al. 2017) is a dynamic attributed network representation framework which first proposes an offline embedding method, then updates the embedding results based on the changes in the attributed evolving network. Know-Evolve (Trivedi et al. 2017) proposes an evolving network embedding method in a knowledge-graph for entity embeddings based on multivariate event detection. EvoNRL is a more general method which extracts the network representation without using node features or explicit use of events. CTDN (Nguyen et al. 2018) is a random walk-based continuous-time dynamic network embedding. Our work is different from this paper in two aspects. First the random walk in CTDN is a temporal random walk and second CTDN is not an online framework and you need to have all the snapshots of the network before embedding it. HTNE (Zuo et al. 2018) tries to model the temporal network as a self-excited system and using Hawkes process model neighbourhood formation in the network and optimize the embedding based on point-time process. HTNE is an online dynamic network embedding framework which is different from EvoNRL as it uses history in its optimization and it needs to be tuned for history in each step. NetWalk (Yu et al. 2018) is a random walk based clique embedding. The random walk update in that paper is different from EvoNRL. First in NetWalk, the reservoir is in memory which finds the next step based on the reservoir and it doesn't benefit the sampling method used in EvoNRL which is based on node degrees. Also, EvoNRL leverages the speed of the inverted-indexing tools. In (Du et al. 2018), authors propose a dynamic skip-gram framework which is orthogonal to our work. Moreover, (Rudolph and Blei 2018) proposes a dynamic word embedding which uses Gaussian random walks to project the vector representations of words over time. The random walks in that work are based on vector representations and are defined over time-series, which is different to our approach.
Random walks: Our work is also related to general concept of random walks on networks (Lovász 1993) and its applications (Craswell and Szummer 2007;Page et al. 1999). READS (Jiang et al. 2017) is an indexing scheme for Simrank computation in dynamic graphs which keeps an online set of reverse-random walks and re-simulates the walks on all of the instances of the node queries. Our proposed method, keeps a set of finite-length random walks which is different from pagerank random walks and has a different sampling strategy and application compared to READS. Another aspects of random walk used in streaming data are continuous-time random walks. Continuous Time Random Walks (CTRW) (Kenkre et al. 1973) are widely studied in time-series analysis and has applications in Finance (Paul and Baschnagel 2010). CTRW is orthogonal to our work as we are not using time-variant random walks and our random walks do not jump over time.
Our focus in this paper is on learning representations of evolving networks. To extend static random walk based network representation methods to evolving networks, we proposed a general framework for updating random walks as new edges and nodes are arriving in the network. The updated random walks leverage time and space efficiency of inverted indexing methods. By indexing an initial set of random walks in the network and efficiently updating it based on the occurring network topology changes, we manage to always keep a valid set of random walks with minimum possible divergence from the initial random walk set. Our proposed method, EVONRL, utilizes the always valid set of random walks to obtain new network representations that respect the changes that occurred in the network. We demonstrated that our proposed method, EVONRL is both accurate and fast. We also discussed the interesting trade-off between time performance and accuracy when obtaining subsequent network representations. Determining the right time for obtaining a network embedding is a challenging problem. We demonstrated that simple strategies for monitoring the changes that occur in the network can provide support in decision making. Overall, the methods presented are easy to understand and simple to implement. They can also be easily adopted in diverse domains and applications of graph/network mining.
Reproducibility: We make source code and data sets used in the experiments publicly availableFootnote 7 to encourage reproducibility of results.
node2vec — code is available at https://github.com/aditya-grover/node2vec
NumPy — https://www.numpy.org/
https://github.com/RaRe-Technologies/gensim
code is available at https://github.com/farzana0/EvoNRL
Elasticsearch: https://www.elastic.co
Apache Lucene: http://lucene.apache.org/core/
https://github.com/farzana0/EvoNRL
CCA:
Canonical correlation analysis
Cit-HepTh:
High energy physics theory citation network
CTDN:
Continious-time dynamic network embedding
CTRW:
Continuous time random walks
Dynamic attributed network embedding
Digital bibliography &library project
EvoNRL:
HOPE:
High-order proximity preserved embedding
HTNE:
Hawkes process based temporal network embedding
NSERC:
Natural sciences and engineering research council of Canada PPI: Protein-protein interaction
READS:
Randomized efficient accurate dynamic SimRank computation
StaticNRL:
Static network representation learning
TADW:
Text-associated DeepWalk
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This research has been supported by a Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant (#RGPIN-2017-05680).
York University, Toronto, M3J1P3, ON, Canada
Farzaneh Heidari & Manos Papagelis
Farzaneh Heidari
Manos Papagelis
FH has made substantial contributions to the design of the work; the acquisition, analysis, and interpretation of data; the creation of new software used in the research; has drafted and revised the work. MP has made substantial contributions to the conception and design of the work; interpretation of data and results; has drafted and revised the work. The author(s) read and approved the final manuscript.
Correspondence to Farzaneh Heidari.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Heidari, F., Papagelis, M. Evolving network representation learning based on random walks. Appl Netw Sci 5, 18 (2020). https://doi.org/10.1007/s41109-020-00257-3
Network representation learning
Evolving networks
Dynamic random walks
Dynamic graph embedding
Machine Learning with Graphs | CommonCrawl |
2 July 2014, Theoretical Computer Science Seminar, Anke van Zuylen
Speaker: Anke van Zuylen (College of William & Mary, USA)
Title: On some recent MAX SAT approximation algorithms
Date: Wednesday 2 July 2014
Location: CWI room L016, Science Park 123, Amsterdam
Recently a number of randomized 3/4-approximation algorithms for MAX SAT have been proposed that all work in the same way: given a fixed ordering of the variables the algorithm makes a random assignment to each variable in sequence, in which the probability of assigning each variable true or false depends on the current set of satisfied (or unsatisfied) clauses. To our knowledge, the first such algorithm was proposed by Poloczek and Schnitger (2011); Van Zuylen (2011) subsequently gave an algorithm that set the probabilities differently and had a simpler analysis. Buchbinder, Feldman, Naor, and Schwartz (2012), as a special case of their work on maximizing submodular functions, also give a randomized $\frac{3}{4}$-approximation algorithm for MAX SAT with the same structure as these previous algorithms. In this talk, we give an even simpler version of the algorithm and analysis that was proposed by Buchbinder et al and we show that in fact it is equivalent to the algorithm proposed by Van Zuylen. We also show how it extends to a deterministic LP rounding algorithm, and we show that this same algorithm was also given by Van Zuylen. This is based on joint work with Matthias Poloczek and David Williamson. _______________________________________________
For more information, please contact c.schaffner at uva.nl. | CommonCrawl |
Difference between revisions of "Mathematics and Statistics for Data Science"
From Sinfronteras
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(Tag: Blanking)
* '''Descriptive Data Analysis:'''
::* Rather than find hidden information in the data, descriptive data analysis looks to summarize the dataset.
::* They are commonly implemented measures included in the descriptive data analysis:
:::* Central tendency (Mean, Mode, Median)
:::* Variability (Standard deviation, Min/Max)
==Central tendency==
https://statistics.laerd.com/statistical-guides/measures-central-tendency-mean-mode-median.php
A central tendency (or measure of central tendency) is a single value that attempts to describe a set of data by identifying the central position within that set of data.
'''The mean''' (often called the average) is most likely the measure of central tendency that you are most familiar with, but there are others, such as the median and the mode.
'''The mean, median and mode''' are all valid measures of central tendency, but under different conditions, some measures of central tendency become more appropriate to use than others. In the following sections, we will look at the mean, mode and median, and learn how to calculate them and under what conditions they are most appropriate to be used.
===Mean===
Mean (Arithmetic)
The mean (or average) is the most popular and well known measure of central tendency.
The mean is equal to the sum of all the values in the data set divided by the number of values in the data set.
So, if we have <math>n</math> values in a data set and they have values <math>x_1, x_2, ..., x_n,</math>the sample mean, usually denoted by <math>\bar{x}</math> (pronounced x bar), is:
<math>\bar{x} = \frac{(x_1 + x_2 +...+ x_n)}{n} = \frac{\sum x}{n}</math>
The mean is essentially a model of your data set. It is the value that is most common. You will notice, however, that the mean is not often one of the actual values that you have observed in your data set. However, one of its important properties is that it minimises error in the prediction of any one value in your data set. That is, it is the value that produces the lowest amount of error from all other values in the data set.
An important property of the mean is that it includes every value in your data set as part of the calculation. In addition, the mean is the only measure of central tendency where the sum of the deviations of each value from the mean is always zero.
====When not to use the mean====
The mean has one main disadvantage: it is particularly susceptible to the influence of outliers. These are values that are unusual compared to the rest of the data set by being especially small or large in numerical value. For example, consider the wages of staff at a factory below:
!Staff
!1
!'''Salary'''
!<math>15k</math>
The mean salary for these ten staff is $30.7k. However, inspecting the raw data suggests that this mean value might not be the best way to accurately reflect the typical salary of a worker, as most workers have salaries in the $12k to 18k range. The mean is being skewed by the two large salaries. Therefore, in this situation, we would like to have a better measure of central tendency. As we will find out later, taking the median would be a better measure of central tendency in this situation.
Another time when we usually prefer the median over the mean (or mode) is when our data is skewed (i.e., the frequency distribution for our data is skewed). If we consider the normal distribution - as this is the most frequently assessed in statistics - when the data is perfectly normal, the mean, median and mode are identical. Moreover, they all represent the most typical value in the data set. However, as the data becomes skewed the mean loses its ability to provide the best central location for the data because the skewed data is dragging it away from the typical value. However, the median best retains this position and is not as strongly influenced by the skewed values. This is explained in more detail in the skewed distribution section later in this guide.
====Mean in R====
mean(iris$Sepal.Width)
===Median===
The median is the middle score for a set of data that has been arranged in order of magnitude. The median is less affected by outliers and skewed data. In order to calculate the median, suppose we have the data below:
We first need to rearrange that data into order of magnitude (smallest first):
!<span style="color:#FF0000">56</span>
Our median mark is the middle mark - in this case, 56. It is the middle mark because there are 5 scores before it and 5 scores after it. This works fine when you have an odd number of scores, but what happens when you have an even number of scores? What if you had only 10 scores? Well, you simply have to take the middle two scores and average the result. So, if we look at the example below:
We again rearrange that data into order of magnitude (smallest first):
Only now we have to take the 5th and 6th score in our data set and average them to get a median of 55.5.
====Median in R====
median(iris$Sepal.Length)
===Mode===
The mode is the most frequent score in our data set. On a histogram it represents the highest bar in a bar chart or histogram. You can, therefore, sometimes consider the mode as being the most popular option. An example of a mode is presented below:
[[File:Mode-1.png|center|thumb|359x359px]]
Normally, the mode is used for categorical data where we wish to know which is the most common category, as illustrated below:
[[File:Mode-1a.png|center|thumb|380x380px]]
We can see above that the most common form of transport, in this particular data set, is the bus. However, one of the problems with the mode is that it is not unique, so it leaves us with problems when we have two or more values that share the highest frequency, such as below:
We are now stuck as to which mode best describes the central tendency of the data. This is particularly problematic when we have continuous data because we are more likely not to have any one value that is more frequent than the other. For example, consider measuring 30 peoples' weight (to the nearest 0.1 kg). How likely is it that we will find two or more people with '''exactly''' the same weight (e.g., 67.4 kg)? The answer, is probably very unlikely - many people might be close, but with such a small sample (30 people) and a large range of possible weights, you are unlikely to find two people with exactly the same weight; that is, to the nearest 0.1 kg. This is why the mode is very rarely used with continuous data.
Another problem with the mode is that it will not provide us with a very good measure of central tendency when the most common mark is far away from the rest of the data in the data set, as depicted in the diagram below:
[[File:Mode-3.png|center|thumb|379px]]
In the above diagram the mode has a value of 2. We can clearly see, however, that the mode is not representative of the data, which is mostly concentrated around the 20 to 30 value range. To use the mode to describe the central tendency of this data set would be misleading.
====To get the Mode in R====
install.packages("modeest")
library(modeest)
> mfv(iris$Sepal.Width, method = "mfv")
===Skewed Distributions and the Mean and Median===
We often test whether our data is normally distributed because this is a common assumption underlying many statistical tests. An example of a normally distributed set of data is presented below:
[[File:Skewed-1.png|center|thumb|379px]]
When you have a normally distributed sample you can legitimately use both the mean or the median as your measure of central tendency. In fact, in any symmetrical distribution the mean, median and mode are equal. However, in this situation, the mean is widely preferred as the best measure of central tendency because it is the measure that includes all the values in the data set for its calculation, and any change in any of the scores will affect the value of the mean. This is not the case with the median or mode.
However, when our data is skewed, for example, as with the right-skewed data set below:
we find that the mean is being dragged in the direct of the skew. In these situations, the median is generally considered to be the best representative of the central location of the data. The more skewed the distribution, the greater the difference between the median and mean, and the greater emphasis should be placed on using the median as opposed to the mean. A classic example of the above right-skewed distribution is income (salary), where higher-earners provide a false representation of the typical income if expressed as a mean and not a median.
If dealing with a normal distribution, and tests of normality show that the data is non-normal, it is customary to use the median instead of the mean. However, this is more a rule of thumb than a strict guideline. Sometimes, researchers wish to report the mean of a skewed distribution if the median and mean are not appreciably different (a subjective assessment), and if it allows easier comparisons to previous research to be made.
===Summary of when to use the mean, median and mode===
Please use the following summary table to know what the best measure of central tendency is with respect to the different types of variable:
!'''Type of Variable'''
!'''Best measure of central tendency'''
|Nominal
|Mode
|Ordinal
|Median
|Interval/Ratio (not skewed)
|Mean
|Interval/Ratio (skewed)
For answers to frequently asked questions about measures of central tendency, please go to: https://statistics.laerd.com/statistical-guides/measures-central-tendency-mean-mode-median-faqs.php
==Measures of Variation==
===Range===
The Range just simply shows the min and max value of a variable.
In R:
> min(iris$Sepal.Width)
> max(iris$Sepal.Width)
> range(iris$Sepal.Width)
Range can be used on '''''Ordinal, Ratio and Interval''''' scales
===Quartile===
https://statistics.laerd.com/statistical-guides/measures-of-spread-range-quartiles.php
Quartiles tell us about the spread of a data set by breaking the data set into quarters, just like the median breaks it in half.
For example, consider the marks of the 100 students, which have been ordered from the lowest to the highest scores.
*'''The first quartile (Q1):''' Lies between the 25th and 26th student's marks.
**So, if the 25th and 26th student's marks are 45 and 45, respectively:
***(Q1) = (45 + 45) ÷ 2 = 45
*'''The second quartile (Q2):''' Lies between the 50th and 51st student's marks.
**If the 50th and 51th student's marks are 58 and 59, respectively:
***(Q2) = (58 + 59) ÷ 2 = 58.5
*'''The third quartile (Q3):''' Lies between the 75th and 76th student's marks.
In the above example, we have an even number of scores (100 students, rather than an odd number, such as 99 students). This means that when we calculate the quartiles, we take the sum of the two scores around each quartile and then half them (hence Q1= (45 + 45) ÷ 2 = 45) . However, if we had an odd number of scores (say, 99 students), we would only need to take one score for each quartile (that is, the 25th, 50th and 75th scores). You should recognize that the second quartile is also the median.
Quartiles are a useful measure of spread because they are much less affected by outliers or a skewed data set than the equivalent measures of mean and standard deviation. For this reason, quartiles are often reported along with the median as the best choice of measure of spread and central tendency, respectively, when dealing with skewed and/or data with outliers. A common way of expressing quartiles is as an interquartile range. The interquartile range describes the difference between the third quartile (Q3) and the first quartile (Q1), telling us about the range of the middle half of the scores in the distribution. Hence, for our 100 students:
<math>Interquartile\ range = Q3 - Q1 = 71 - 45 = 26</math>
However, it should be noted that in journals and other publications you will usually see the interquartile range reported as 45 to 71, rather than the calculated '''<math>Interquartile\ range.</math>'''
A slight variation on this is the <math>semi{\text{-}}interquartile range,</math>which is half the <math>Interquartile\ range. </math>Hence, for our 100 students:
<math>Semi{\text{-}}Interquartile\ range = \frac{Q3 - Q1}{2} = \frac{71 - 45}{2} = 13</math>
====Quartile in R====
quantile(iris$Sepal.Length)
0% and 100% are equivalent to min max values.
===Box Plots===
boxplot(iris$Sepal.Length,
col = "blue",
main="iris dataset",
ylab = "Sepal Length")
===Variance===
https://statistics.laerd.com/statistical-guides/measures-of-spread-absolute-deviation-variance.php
Another method for calculating the deviation of a group of scores from the mean, such as the 100 students we used earlier, is to use the variance. Unlike the absolute deviation, which uses the absolute value of the deviation in order to "rid itself" of the negative values, the variance achieves positive values by squaring each of the deviations instead. Adding up these squared deviations gives us the sum of squares, which we can then divide by the total number of scores in our group of data (in other words, 100 because there are 100 students) to find the variance (see below). Therefore, for our 100 students, the variance is 211.89, as shown below:
<math>variance = \sigma = \frac{\sum(X - \mu)^2}{N}</math>
<math>\mu: \text{Mean};\ \ \ X: \text{Score};\ \ \ N: \text{Number of scores}</math>
*Variance describes the spread of the data.
*It is a measure of deviation of a variable from the arithmetic mean.
*The technical definition is the average of the squared differences from the mean.
*A value of zero means that there is no variability; All the numbers in the data set are the same.
*A higher number would indicate a large variety of numbers. <br />
====Variance in R====
var(iris$Sepal.Length)
===Standard Deviation===
https://statistics.laerd.com/statistical-guides/measures-of-spread-standard-deviation.php
The standard deviation is a measure of the spread of scores within a set of data. Usually, we are interested in the standard deviation of a population. However, as we are often presented with data from a sample only, we can estimate the population standard deviation from a sample standard deviation. These two standard deviations - sample and population standard deviations - are calculated differently. In statistics, we are usually presented with having to calculate sample standard deviations, and so this is what this article will focus on, although the formula for a population standard deviation will also be shown.
The '''sample standard deviation formula''' is:
<math>s = \sqrt{\frac{\sum(X - \bar{X})^2}{n -1}}</math>
<math>\bar{X}: \text{Sample mean};\ \ \ n: \text{Number of scores in the sample}</math>
The '''population standard deviation''' formula is:
<math>\sigma = \sqrt{\frac{\sum(X - \mu)^2}{n}}</math>
<math>\mu: \text{population mean}</math>
*The Standard Deviation is the square root of the variance.
*This measure is the most widely used to express deviation from the mean in a variable.
*The higher the value the more widely distributed are the variable data values around the mean.
*Assuming the frequency distributions approximately normal, about 68% of all observations are within +/- 1 standard deviation.
*Approximately 95% of all observations fall within two standard deviations of the mean (if data is normally distributed).
====Standard Deviation in R====
sd(iris$Sepal.Length)
=== Z Score ===
* z-score represents how far from the mean a particular value is based on the number of standard deviations.
* z-scores are also known as standardized residuals
* Note: mean and standard deviation are sensitive to outliers
> x <-((iris$Sepal.Width) - mean(iris$Sepal.Width))/sd(iris$Sepal.Width)
> x
> x[77] #choose a single row # or this
> x <-((iris$Sepal.Width[77]) - mean(iris$Sepal.Width))/sd(iris$Sepal.Width)
==Shape of Distribution==
===Skewness===
* '''Skewness''' is a method for quantifying the lack of symmetry in the distribution of a variable.
* '''Skewness''' value of zero indicates that the variable is distributed symmetrically. Positive number indicate asymmetry to the left, negative number indicates asymmetry to the right.
[[File:Skewness.png|400px|thumb|center|]]
====Skewness in R====
> install.packages("moments") and library(moments)
> skewness(iris$Sepal.Width)
===Histograms in R===
> hist(iris$Petal.Width)
===Kurtosis===
* '''Kurtosis''' is a measure that gives indication in terms of the peak of the distribution.
* Variables with a pronounced peak toward the mean have a high ''Kurtosis'' score and variables with a flat peak have a low ''Kurtosis'' score.
====Kurtosis in R====
> kurtosis(iris$Sepal.Length)
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We've already seen how to multiply a one digit number with another one digit number. So what do we do when it's a two digit number multiplied with another $2$2 digit number?
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\begin{document}
\vskip 3mm
\noindent \textbf{Strongly mixed random errors in Mann's iteration algorithm for a contractive real function} \vskip 3mm
\vskip 5mm \noindent Hassina ARROUDJ$^{1}$, Idir ARAB$^{2}$ and Abdelnasser DAHMANI$^{3}$
\noindent $^{1}$Laboratoire de Math\'{e}matiques Appliqu\'{e}es, Facult\'{e} des Sciences Exactes, Universit\'{e} A. Mira Bejaia, Algerie.\\ \noindent $^{2}$CMUC, Department of Mathematics, University of Coimbra, Portugal\\ \noindent $^{3}$Centre Universitaire de Tamanrasset.
\noindent $^{1}$e-mail: has\[email protected]\\
$^{2}$ [email protected]\\ $^{3}$a\[email protected]
\vskip 3mm \noindent \textbf{Keywords:} Fixed point; Fuk-Nagaev's inequalities; Mann's iteration algorithm; $\alpha $-mixing; Rate of convergence; confidence domain. \vskip 3mm
\noindent \textbf{Abstract}
This work deals with the Mann's stochastic iteration algorithm under $\alpha -$ mixing random errors. We establish the Fuk-Nagaev's inequalities that enable us to prove the almost complete convergence with its corresponding rate of convergence. Moreover, these inequalities give us the possibility of constructing a confidence interval for the unique fixed point. Finally, to check the feasibility and validity of our theoretical results, we consider some numerical examples, namely a classical example from astronomy.
\vskip 4mm
\noindent 1. \textbf{Introduction}
In many mathematical problems arising from various domains, the existence of a solution is the same as the existence of a fixed point by some appropriate transformation of the problem. The most known problem in that framework is the root existence which can be tackled easily as the existence of a fixed point and vice versa. Therefore, the fixed point theory is of paramount importance in engineering sciences and many areas of mathematics. Fixed point theory provides conditions under which maps have the existence and uniqueness of solutions. Over the last decades, that theory has been revealed as one of the most significant tool in the study of nonlinear problems. In particular, in many fields, equilibria or stability are fundamental concepts that can be described in terms of fixed points. For example, in economics, a Nash equilibrium of a game is a fixed point of the game's best response correspondence. However, in informatics, programming language compilers use fixed point computations for program analysis, for example in data-flow analysis, which is often required for code optimization. The vector of PageRank values of all web pages is the fixed point of a linear transformation derived from the World Wide Web's link structure. In astronomy, the eccentric anomaly $E$ of a given planet is related to a fixed point equation that cannot not be solved analytically, this will be well described in example (\ref{astronomy_example}) and many examples could be found in engineering sciences such as physics, geology, chemistry, biology, mechanical statistics, etc.\\ Mathematically, a fixed point problem is presented under the following form
\begin{equation}\label{Fixed-Point} \text{Find }x\in X\text{ such that }Fx=x \end{equation} Where $F$ is an operator, defined on a space $X$. The solutions of that equation if they exist are called "fixed points" of the mapping $F$. The classical result in fixed point theory is the Banach fixed-point theorem \cite{banac}; it ensures the existence and uniqueness of a fixed point of certain self-maps of a metric space. Additionally, it provides a constructive numerical method to approximate the fixed point.
After verifying the existence and uniqueness conditions, it is necessary to find (or approximate) the unique fixed point of the problem (\ref{Fixed-Point}). This leads to find the unique root of $F-id_X$ where $id_X$ denotes the identity operator on $X$. Analytically, to find that root, one has to reverse the operator $F-id_X$, and one could immediately think about the difficulty when dealing with inversion and most of the time that task is impossible. Alternatively, numerical methods become the most appropriate and have attracted many researches these last decades. The pioneering work after Picard's iterative method was introduced by Mann \cite{Man} to remedy the problem of convergence while using the Picard's method for approximating the fixed point of nonexpansive mapping. Later, many modified algorithms were introduced, by considering the stochastic part, i.e., considering the errors generated by the numerical evaluation of the algorithm. For an account of relevant literature on that topic, see (\cite{Ber,Ceg,cha2,Hus,Ish,Kan,Kim,Liu3}).
In the framework of this paper, we consider the Mann iterative algorithm as described in (\ref{smi}) by taking into account the committed errors at each evaluation of the approximated fixed point $x_n$. These errors are supposed to be random and modeled by random variables and we suppose them to be strong mixing. Recall that a sequence $\left( \xi _{i}\right) $ is said to be strong mixing or $\alpha -$mixing if the following condition is satisfied: \begin{equation} \alpha \left( n\right) =\sup_{A\in \mathcal{F}_{-\infty }^{k},B\in \mathcal{F }_{k+n}^{+\infty }}\left\vert \mathbb{P}\left( A\cap B\right) -\mathbb{P} \left( A\right) \mathbb{P}\left( B\right) \right\vert \underset{ n\longrightarrow +\infty }{\longrightarrow }0 \label{mf} \end{equation} where $\mathcal{F}_{l}^{m}$ denotes the $\sigma $-algebra engendered by events of the form $\left\{ \left( \xi _{i_{1}},\cdots ,\xi _{i_{k}}\right) \in E\right\} $, where $l\leq i_{1}<i_{2}<\cdots <i_{k}\leq m$ and $E$ is a Borel set.
The notion of $\alpha $-mixing was firstly introduced by Rosenbaltt in 1956 \cite{Ros} and the central limit theorem has been established. The strong mixing random variables have many interest in linear processes and found many application in finance, for more examples and properties concerning the mixing notions, see \cite{Dou,Ibr}.
In this paper, we establish Fuk-Nagaev inequalities. These inequalities allow us to prove the almost complete convergence of Mann's algorithm to the fixed point, with convergence rate and the possibility of giving the a confidence interval. To strengthen the obtained theoretical results, some numerical examples are considered.\newline The rest of the paper is organized as follows: In section 2 the statement of the problem is described and some known results are recalled. In section 3, some new results were established by using stochastic methods. In section 4, the validity of our approach is checked up by considering some numerical examples.\\
\noindent 2. \textbf{Preliminaries and known results} \\
Let $\left( \Omega ,\mathcal{F},\mathbb{P}\right) $ be a probability space and $f:\mathbb{R\rightarrow R}$ \ a non-linear function. We consider the stochastic Mann's iteration algorithm: \begin{equation} x_{n+1}=\left( 1-a_{n}\right) x_{n}+b_{n}f\left( x_{n}\right) +c_{n}\xi _{n}, \label{smi} \end{equation} where the sequences of positive numbers $(a_{n})_{n\geq 1}$ and $(b_{n})_{n\geq 1}$ satisfy the following conditions \begin{eqnarray*} \lim_{n\rightarrow +\infty}b_{n}=\lim_{n\rightarrow +\infty}a_{n}&=&0\text{ and }\sum_{n=1}^{+\infty }b_{n}=\sum_{n=1}^{+\infty}a_{n}=+\infty \\ \sum_{n=1}^{+\infty }c_{n}^{2} &<&+\infty . \end{eqnarray*}
Without loss of generality, we take \begin{equation*} a_{n}=b_{n}=\frac{a}{n}\ \text{ and } \ c_{n}=\frac{a}{n^{2}}, a>0. \end{equation*}
Hence, the stochastic Mann's iteration algorithm (\ref{smi}) takes the following form: \begin{equation} x_{n+1}=\left( 1-\frac{a}{n}\right) x_{n}+\frac{a}{n}\left[ f\left( x_{n}\right) +\frac{1}{n}\xi _{n}\right] . \label{mannstocha} \end{equation} We now introduce some classical hypothesis that will be useful tools for the proof of established results in the sequel:
(H1) : The fixed point $x^{\ast }$ satisfies \begin{equation} \exists \ N>0,\ \left\vert x_{1}-x^{\ast }\right\vert \leq N<+\infty . \label{apriori} \end{equation}
(H2) : The function $f$ is contractive, i.e, it satisfies the following property: \begin{equation} \forall \ x,y\in \mathbb{R},\left\vert f\left( x\right) -f\left( y\right) \right\vert \leq c\left\vert x-y\right\vert ,c\in \left( 0,1\right) . \label{contraction} \end{equation}
(H3) : The random variables $(\xi _{i})$ fulfill the condition of uniform decrease, that is \begin{equation} \exists \ p>2, \begin{array}{c} \end{array} \ \forall \ t>0,\ \mathbb{P}\left\{ \left\vert \xi _{i}\right\vert >t\right\} \leq t^{-p}. \label{queues} \end{equation}
(H4) : The coefficients of the $\alpha $-mixing sequence $\left( \xi _{n}\right) _{n}$ satisfy the following arithmetic decay condition: \begin{equation} \exists \ d\geq 1,\ \exists \ \beta >1:\alpha \left( n\right) \leq d \ n^{-\beta },\forall \ n\in \mathbb{N}^{\ast }. \label{cm} \end{equation}
(H5) : The $\alpha $-mixing coefficients satisfy the following condition \begin{equation} \exists \ \rho >0,\ \rho \frac{\left( \beta +1\right) p}{\beta +p}>2. \label{dec} \end{equation}
\begin{remark} The assumption (H1) is classical. Arbitrary choice of $x_{1}$ and the existence of $x^{\ast }$ allows us to assume such supposition. The contraction's assumption (H2) ensures the existence and uniqueness of the fixed point $x^{*}$ of the function $f$ according to Banach's theorem for fixed point. When the function $f$ is derivable, the condition (H2) is equivalent to the boundness of the derivative $f^{\prime}$, i.e, $ \exists \ c >0, \sup\limits_{x}\left\vert f^{\prime }\left( x\right) \right\vert \leq c<1$. The hypothesis (H3) is satisfied for all bounded random variables and Gaussian ones. Assumption (H4) is used in order to characterize the dependence structure of errors. Moreover, combined to (H3), the assumption (H4) allows the obtention of Fuk-Nagaev's inequalities \cite{Rio}, which ensures the almost complete convergence result. As a particular example, the geometric $\alpha $-mixing sequence $\left( \xi _{i}\right) _{i}$ and its mixing coefficients are defined as follows \begin{equation*} \exists \ d_{0}>0,\exists \ \kappa \in \left( 0,1\right) :\alpha \left( n\right) \leq d_{0}\ \kappa ^{n},\forall \ n\ \in \mathbb{N}^{\ast }. \end{equation*}
The assumption (H5) will be useful for specifying the rate of almost complete convergence of the stochastic Mann's iteration algorithm. That condition is classical, see \cite{Ait,Fer}. \end{remark} First, we state the following theorem which will be used in the sequel during the proof of the main result. \begin{theorem} \label{Fuk-Nagaev}Let $\left( \xi _{i}\right) _{i\in \mathbb{N}^{\ast }}$ be a \ centered sequence of real-valued random variables and $\left( \alpha _{n}\right) _{n\in \mathbb{N}^{\ast }}$ the corresponding sequence of mixing coefficients as defined in (\ref{mf}) such that the hypothesis (H2) and (H4) are satisfied. Let us set \begin{equation*} s_{n}^{2}=\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{n}\left\vert Cov\left(\xi_{i},\xi_{k}\right) \right\vert . \end{equation*} Then, for every real numbers $r\geq 1$ and $\lambda >0,$ we have \begin{equation*} \mathbb{P}\left\{ \sup_{k\in \left[ 1,n\right] }\left\vert \sum_{i=1}^{k}\xi _{i} \right\vert \geq 4\lambda \right\} \leq 4\left( 1+\frac{\lambda ^{2}}{rs_{n}^{2}}\right) ^{ \frac{-r}{2}}+2cnr^{-1}\left( \frac{2r}{\lambda }\right) ^{\frac{\left( \beta +1\right) p}{\left( \beta +p\right) }}. \end{equation*} \end{theorem}
\begin{proof} The proof is well detailed in \cite{Rio}, page 84 to 87. \end{proof}
\begin{lemma} Using the hypothesis (H1), we get the following inequality: \begin{equation} \left\vert x_{n+1}-x^{\ast }\right\vert \leq N\prod\limits_{i=1}^{n}\left( 1-\frac{a\left( 1-c\right) }{i}\right) +\sum\limits_{i=1}^{n}\frac{a}{i^{2}} \prod\limits_{j=i+1}^{n}\left( 1-\frac{a\left( 1-c\right) }{j}\right) \left\vert \xi _{i}\right\vert . \label{formule} \end{equation} \end{lemma}
\begin{proof} The proof is straightforward by induction on $n$.
\end{proof}
\begin{lemma} For all constants $a<1$, we have the following inequalities \begin{equation} \prod\limits_{j=i+1}^{n}\left( 1-\frac{a\left( 1-c\right) }{j}\right) \leq \left( \frac{i+1}{n+1}\right) ^{a\left( 1-c\right) }, \label{produit} \end{equation} and, \begin{equation} \sum_{i=1}^{n}\frac{a}{i^{2}}\prod\limits_{j=i+1}^{n}\left( 1-\frac{a\left( 1-c\right) }{j}\right) \leq \frac{aS}{\left( n+1\right) ^{a\left( 1-c\right) }}. \label{somprod} \end{equation} \end{lemma}
\begin{proof} We have, \begin{equation*} \prod\limits_{j=i+1}^{n}\left( 1-\frac{a\left( 1-c\right) }{j}\right) \leq \exp \left( -a\left( 1-c\right) \sum_{j=i+1}^{n}\frac{1}{j}\right) \leq \left( \frac{i+1}{n+1}\right) ^{a\left( 1-c\right) }. \end{equation*} The second inequality follows immediately from inequality (\ref{produit}) by setting $S$ the sum of the convergent series $\frac{\left( i+1\right) }{i^{2}}^{a\left(1-c\right) }$. \end{proof}
\vskip 3mm
\noindent 3. \textbf{Convergence of Mann iterative algorithm}
\begin{theorem} Under the assumptions (H1)--(H5), we have for any real positive $\rho $ such that $\frac{2(\beta+p)}{p(\beta+1)}<\rho <a\left( 1-c\right) <1$, we have: \begin{equation} x_{n+1}-x^{\ast }=\mathcal{O}\left(\frac{\sqrt{\ln n}}{n^{a\left( c-1\right) -\rho }}\right)\ \ \ \ a.co.\label{Rate} \end{equation} \end{theorem}
\begin{proof} Using the inequality (\ref{formule}), we have \begin{eqnarray} &&\mathbb{P}\left\{ \left\vert x_{n+1}-x^{\ast }\right\vert >\varepsilon \right\} \notag \\ &\leq &\mathbb{P}\left\{ N\prod\limits_{i=1}^{n}\left( 1-\frac{a\left( 1-c\right) }{i}\right) +\sum\limits_{i=1}^{n}\frac{a}{i^{2}} \prod\limits_{j=i+1}^{n}\left( 1-\frac{a\left( 1-c\right) }{j}\right) \left\vert \xi _{i}\right\vert >\varepsilon \right\} \notag \\ &\leq &\mathbb{P}\left\{ N\prod\limits_{i=1}^{n}\left( 1-\frac{a\left( 1-c\right) }{i}\right) +\sum\limits_{i=1}^{n}\frac{a}{i^{2}} \prod\limits_{j=i+1}^{n}\left( 1-\frac{a\left( 1-c\right) }{j}\right) \mathbb{E}\left\vert \xi _{i}\right\vert \geq \frac{\varepsilon }{2}\right\} \notag \\ &&+\mathbb{P}\left\{ \sum\limits_{i=1}^{n}\frac{a}{i^{2}} \prod\limits_{j=i+1}^{n}\left( 1-\frac{a\left( 1-c\right) }{j}\right) \left( \left\vert \xi _{i}\right\vert -\mathbb{E}\left\vert \xi _{i}\right\vert \right) >\frac{\varepsilon }{2}\right\} . \label{sommation} \end{eqnarray}
Firstly, we have \begin{equation} \mathbb{P}\left\{ N\prod\limits_{i=1}^{n}\left( 1-\frac{a\left( 1-c\right) }{i}\right) +\sum\limits_{i=1}^{n}\frac{a}{i^{2}}\prod\limits_{j=i+1}^{n} \left( 1-\frac{a\left( 1-c\right) }{j}\right) \mathbb{E}\left\vert \xi _{i}\right\vert >\frac{\varepsilon }{2}\right\} \leq K_{1}e^{-n^{2a\left( 1-c\right) }\varepsilon ^{2}}. \label{terme1} \end{equation}
We set \begin{equation*} Z_{i}=\frac{an^{a\left( 1-c\right) }}{i^{2}}\prod\limits_{j=i+1}^{n}\left( 1-\frac{a\left( 1-c\right) }{j}\right) \left( \left\vert \xi _{i}\right\vert -\mathbb{E}\left\vert \xi _{i}\right\vert \right) . \end{equation*} Note that the random variables $(Z_{i})$ are centered and according to (\ref {queues}), we show that, there exists a positive constant $M$ such that, \begin{equation} \forall \ t>0 \begin{array}{c} , \end{array} \mathbb{P}\left\{ \left\vert Z_{i}\right\vert >t\right\} \leq Mt^{-p}. \label{queuesZ} \end{equation} Finally, we notice that if the random errors $(\xi _{i})$ are $\alpha $-mixing, then the random variables $(Z_{i})$ remain also with mixing coefficients less than or equal to those of the sequence $\left( \xi _{i}\right) _{i}$. Thus, applying the Fuk-Nagaev's exponential inequality given by Rio (Theorem \ref{Fuk-Nagaev}) to the variables $(Z_{i})$, we obtain for any $\varepsilon >0$ and $r\geq 1:$ \begin{eqnarray} \mathbb{P}\left\{ \left\vert x_{n+1}-x^{\ast }\right\vert >\varepsilon \right\} &\leq &K_{1}e^{-n^{2a\left( 1-c\right) }\varepsilon ^{2}}+4\left( 1+ \frac{\varepsilon ^{2}n^{2a\left( 1-c\right) }}{4rs_{n}^{2}}\right) ^{\frac{ -r}{2}} \notag \\ &&+4Cnr^{-1}\left( \frac{2r}{\varepsilon n^{a\left( 1-c\right) }}\right) ^{ \frac{\left( \beta +1\right) p}{\left( \beta +p\right) }} \label{Nagaev} \end{eqnarray} where, \begin{equation*} C=2Mp\left( 2p-1\right) ^{-1}\left( 2^{\beta }d\right) ^{\frac{p-1}{\beta +p} }\text{ and }s_{n}^{2}=\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{n}\left\vert Cov\left( Z_{i},Z_{k}\right) \right\vert . \end{equation*}
Let us bound the double sum of covariances $s_{n}^{2}$, we have: \begin{equation*} s_{n}^{2}=\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{n}\left\vert Cov \left( Z_{i},Z_{k}\right) \right\vert =\sum\limits_{i=1}^{n} Var \left( Z_{i}\right) +\sum\limits_{i=1}^{n}\sum\limits_{k\neq i}^{n}\left\vert Cov\left( Z_{i},Z_{k}\right) \right\vert . \end{equation*}
We recall that \begin{equation} \sum\limits_{i=1}^{n} Var\left( Z_{i}\right) \leq \sum\limits_{i=1}^{n}\frac{a^{2}\left( i+1\right) ^{2a\left( 1-c\right) }}{ i^{4}} Var\left( \left\vert \xi _{i}\right\vert \right) \leq S_{v} \label{var} \end{equation} since it is a partial sum of a convergent series with positive terms.
On the other hand, for $i\neq k$, we have \begin{equation} \left\vert Cov\left( Z_{i},Z_{k}\right) \right\vert \leq \frac{ a^{2}\left( i+1\right) ^{a\left( 1-c\right) }}{i^{2}}\frac{\left( k+1\right) ^{a\left( 1-c\right) }}{k^{2}}\left \vert \mathbb{E}\left( \left\vert \xi _{i}\right\vert -\mathbb{E}\left\vert \xi _{i}\right\vert \right) \left( \left\vert \xi _{k}\right\vert -\mathbb{E}\left\vert \xi _{k}\right\vert \right) \right \vert . \end{equation} According to the inequality given by Ibragimov \cite{Ibr} (Theorem 17.2.2 page 307), we obtain: \begin{equation} \left\vert \mathbb{E}\left( \left\vert \xi _{i}\right\vert -\mathbb{E} \left\vert \xi _{i}\right\vert \right) \left( \left\vert \xi _{k}\right\vert -\mathbb{E}\left\vert \xi _{k}\right\vert \right) \right\vert \leq \left( 4+6C\right) \left( \alpha \left( \left\vert i-k\right\vert \right) \right) ^{ \frac{p-2}{p}}, \end{equation} consequently, \begin{equation} \left\vert Cov\left( Z_{i},Z_{k}\right) \right\vert \leq a^{2}\left( 4+6C\right) \frac{\left( i+1\right) ^{a\left( 1-c\right) }}{i^{2} }\frac{\left( k+1\right) ^{a\left( 1-c\right) }}{k^{2}}\left( \alpha \left( \left\vert i-k\right\vert \right) \right) ^{\frac{p-2}{p}}. \label{ras1} \end{equation} Since the mixing coefficients of the sequence $\left( \left\vert \xi _{i}\right\vert -\mathbb{E}\left\vert \xi _{i}\right\vert \right) _{i}$ are less than or equal to those of the sequence $\left( \xi _{i}\right) _{i}$, we get
\begin{eqnarray} &&\ \sum\limits_{i=1}^{n}\sum\limits_{k\neq i}^{n}\left\vert Cov\left( Z_{i},Z_{k}\right) )\right\vert \leq \sum\limits_{i=1}^{n}\sum\limits_{\left\vert i-k\right\vert \leq u_{n}}a^{2}\left( 4+6C\right) \frac{\left( i+1\right) ^{a\left( 1-c\right) } }{i^{2}}\frac{\left( k+1\right) ^{a\left( 1-c\right) }}{k^{2}}\left( \alpha \left( \left\vert i-k\right\vert \right) \right) ^{\frac{p-2}{p}} \notag \\ &&+\sum\limits_{i=1}^{n}\sum\limits_{\left\vert i-k\right\vert >u_{n}}a^{2}\left( 4+6C\right) \frac{\left( i+1\right) ^{a\left( 1-c\right) } }{i^{2}}\frac{\left( k+1\right) ^{a\left( 1-c\right) }}{k^{2}}\left( \alpha \left( \left\vert i-k\right\vert \right) \right) ^{\frac{p-2}{p}}\leq S_{c}. \label{cov} \end{eqnarray} Combining (\ref{var}) and (\ref{cov}), we obtain: \begin{equation} s_{n}^{2}\leq S_{v}+S_{c}=S. \label{sn2} \end{equation} So, from (\ref{sn2}), we have the inequality \begin{equation} \mathbb{P}\left\{ \left\vert x_{n+1}-x^{\ast }\right\vert >\varepsilon \right\} \leq T_{1}+T_{2}+T_{3} \end{equation} where \begin{equation*} T_{1}=K_{1}e^{-n^{2a\left( 1-c\right) }\varepsilon ^{2}},\ T_{2}=4\left( 1+ \frac{n^{2a\left( 1-c\right) }\varepsilon ^{2}}{4rS}\right) ^{\frac{-r}{2}} \text{ and }T_{3}=4Cnr^{-1}\left( \frac{r}{n^{a\left( 1-c\right) }\varepsilon }\right) ^{\frac{\left( \beta +1\right) p}{\beta +p}}. \end{equation*}
For a well chosen positive number $r$ and $\varepsilon$, the quantities $T_{1},T_{2}$ and $T_{3}$ become a general terms of convergent series. Consequently, we obtain \begin{equation*} \sum_{n=1}^{+\infty }\mathbb{P}\left\{ \left\vert x_{n+1}-x^{\ast }\right\vert >\varepsilon \right\} <+\infty \end{equation*} that ensures the almost complete convergence of $\left( x_{n}\right) _{n}$ to the unique fixed point $x^{\ast }.$ The choice of the tuning positive number $r$ will be specified while deriving the corresponding rate of convergence.
Recall that $x_{n}-x^{\ast }=\mathcal{O}\left( \epsilon_{n}\right) $ almost completely $ (a.co),$ where $\left( \epsilon_{n}\right) _{n}$ is a sequence of real positive numbers tending to zero, if there exists a positive constant $k$ such that \begin{equation*} \sum_{n=1}^{+\infty }\mathbb{P}\left\{ \left\vert x_{n}-x^{\ast }\right\vert >k\epsilon_{n}\right\} <+\infty . \end{equation*} Basing on the inequalities obtained above, we take: \begin{equation*} \varepsilon=\varepsilon_{n}=k\epsilon _{n}, \text{ where } k=\sqrt{1+\delta },\text{ }\delta >0\text{ and }\epsilon_{n}=\frac{ \sqrt{\ln n}}{n^{a\left( 1-c\right) -\rho }}. \end{equation*} Hence, we obtain \begin{equation} T_{1}=K_{1}e^{-\left( n+1\right) ^{2a\left( 1-c\right) }\varepsilon ^{2}}\leq K_{1}e^{-\left( 1+\delta \right) \ln n}=\frac{K_{1}}{n^{1+\delta }} . \label{T1} \end{equation} For a suitably chosen $r$ such that $r>\frac{2}{\rho }$, we obtain \begin{equation} T_{2}=4\left( 1+\frac{\left( 1+\delta \right) n^{\rho }}{rS}\right) ^{\frac{ -r}{2}}\leq K_{2}n^{-\rho \frac{r}{2}} \label{T2} \end{equation} where \begin{equation*} K_{2}=\left( \frac{rS}{1+\sigma }\right) ^{r/2}. \end{equation*} With regard to $T_{3}$, we have \begin{eqnarray} T_{3} &\leq &4Cnr^{-1}\left( \frac{r}{\sqrt{1+\delta }n^{\rho }\ln n}\right) ^{\frac{\left( \beta +1\right) p}{\beta +p}} \notag \\ &=&4Cr^{\frac{\left( \beta +1\right) p}{\beta +p}-1}\frac{n}{\left( \sqrt{ 1+\delta }n^{\rho }\ln n\right) ^{\frac{\left( \beta +1\right) p}{\beta +p}}}
\notag \end{eqnarray} With $r$ chosen as in (\ref{T2}), we deduce \begin{equation} T_{3}\leq K_{3}\frac{1}{n^{\rho \frac{\left( \beta +1\right) p}{\beta +p} -1}\left( \ln n\right) ^{\frac{\left( \beta +1\right) p}{\beta +p}}} \label{T3} \end{equation} which is a general term of Bertrand series, it is convergent because of the hypothesis (H5). It leads that: \begin{equation} \mathbb{P}\left\{ \left\vert x_{n+1}-x^{\ast }\right\vert >\sqrt{1+\delta } \frac{\sqrt{\ln n}}{n^{a\left( 1-c\right) -\rho }}\right\} \leq \frac{K_{1}}{ n^{1+\delta }}+\frac{K_{2}}{n^{\rho \frac{r}{2}}}+\frac{K_{3}}{n^{\rho \frac{ \left( \beta +1\right) p}{\beta +p}}\left( \ln n\right) ^{\frac{\left( \beta +1\right) p}{\beta +p}}}. \label{dering} \end{equation} The right-hand side of the last inequality is a term of a convergent series. \end{proof} \begin{remark} Remark that in the obtained rate of convergence given by the formula (\ref{Rate}), more the quantity $\frac{2(\beta+p)}{p(\beta+1)}$ is small, more we have the choice of taking $\rho$ small and consequently the rate of convergence becomes more interesting. \end{remark} \begin{corollary} Under the assumptions (H1)--(H5), for a given level $\sigma $, there exists a natural integer $n_{\sigma }$ for which the fixed point $x^{\ast }$ of the function $f$ belongs to closed interval of center $x_{n_{\sigma}}$ and radius $ \varepsilon $ with a probability greater than or equal to $1-\sigma $. \begin{equation} \forall\ \varepsilon >0,\forall \ \sigma >0,\exists \ n_{\sigma }\in \mathbb{N}: \mathbb{P}\left\{ \left\vert x_{n_{\sigma }}-x^{\ast }\right\vert \leq \varepsilon \right\} \geq 1-\sigma . \label{ic} \end{equation} \end{corollary}
\begin{proof} Indeed, using Kronecker's Lemma, we obtain $\lim\limits_{n\rightarrow +\infty }\alpha \left( n\right) =0$ which implies \begin{equation} \lim_{n\rightarrow +\infty }\frac{K_{1}}{n^{1+\delta }}+\frac{K_{2}}{n^{\rho \frac{r}{2}}}+\frac{K_{3}}{n^{\rho \frac{\left( \beta +1\right) p}{\beta +p} }\left( \ln n\right) ^{\frac{\left( \beta +1\right) p}{\beta +p}}}=0. \end{equation} Since there exists a natural integer $n_{\sigma }$ such that \begin{equation} \forall \ n\in \mathbb{N},n\geq n_{\sigma }-1\Longrightarrow \frac{K_{1}}{ n^{1+\delta }}+\frac{K_{2}}{n^{\rho \frac{r}{2}}}+\frac{K_{3}}{n^{\rho \frac{ \left( \beta +1\right) p}{\beta +p}}\left( \ln n\right) ^{\frac{\left( \beta +1\right) p}{\beta +p}}}\leq \sigma , \label{fin} \end{equation} thus, (\ref{ic}) arises from (\ref{dering}) and (\ref{fin}). \end{proof}
\noindent 4. \textbf{Numerical results}
In this section, a simulation study is proposed to check the validity of our obtained theoretical results. We consider two examples. In the first one, a contractive function where its unique fixed point is known exactly and we compare the fixed point to the approximated ones obtained using the Mann's iterative algorithm. In the second example, we consider a classical problem from astronomy, where the mathematical equation cannot be solved to obtain the exact value of the fixed point and we use the Cauchy's criterium to compare two successive iterates to insure the convergence of the sequence obtained using iterative Mann's algorithm.
\begin{equation*} x_{n+1}=\left( 1-\frac{a}{n}\right) x_{n}+\frac{a}{n}\left[ f\left( x_{n}\right) +\frac{1}{n}\xi _{n}\right] \end{equation*}
$0<a(1-c)<1,$ $\xi _{0}=0,$ $n\in
\mathbb{N}
^{\ast }.$
To characterize the strong mixing random errors $(\xi _{i})$, we consider an autoregressive model $(\xi_{i})_{i}$ of order 1 (see \cite{dahm}) described as follows \begin{equation} \begin{array}{c} \xi _{i+1}=\varphi \xi _{i}+g_{i}, \end{array} \label{regress} \end{equation} where $g_{i}$ is a Gaussian white noise process, $\varphi $ is a constant such that $\left\vert \varphi \right\vert <1$. For the simulation of Gaussian random variables $(g_{i})_{i}$, we use the method of Box-Muller : \begin{equation} \begin{array}{cc} g_{k} & =\sqrt{-2ln(u_{1})}\ cos(2\pi u_{2}) \end{array}
\label{gauss} \end{equation} where $u_{1}$ and $u_{2}$ are uniform distributed random numbers. \begin{example} We consider the following function defined by: \begin{equation*} \begin{array}{cc} f & :\left[ 0,5\right] \rightarrow \left[ 0,5\right] \\ x & \mapsto \sqrt{x+1} \end{array} \end{equation*} The function $f$ is a contractive function with $c=$ $ \underset{x\in \left[ 0,+\infty \right) }{max}\left\vert f^{'} (x)\right\vert =\frac{1}{2}.$ Hence $f$ has a unique fixed point $x^{\ast }=\frac{1+\sqrt{5}}{2}=1,618033988749895$, which is known as golden number. For $x_{1}=1.3,a=\frac{1}{4}$ and $\varphi =0.8,$ the following results are obtained: \begin{equation*}
\begin{tabular}{|l|l|l|} \hline $n$ & $x_{n}$ & $\left\vert x_{n}-x^{\ast }\right\vert $ \\ \hline $10^{3}$ & 1.614142671526978 & 0.003891317222917 \\ \hline $10^{4}$ & 1.615420224332314 & 0.002613764417581 \\ \hline $10^{5}$ & 1.616115916146472 & 0.001918072603423 \\ \hline \end{tabular} \end{equation*} \end{example}
\begin{example}\label{astronomy_example} Most of mathematical problems come from other engineering sciences ( physics, chemistry, geology, astronomy, etc.). When studying some physical problems using the appropriate mathematical models, we obtain an equation or a set of equations and usually cannot be solved analytically because in general, these equations are corrupted by noise or the known mathematical tools do not allow us to solve them. To illustrate this fact, we consider the following \ classical example from astronomy.
Consider a planet in an orbit around the sun as described by the following diagram.
Let $n$ be the mean angular motion of the Mercury's orbit around the sun, $t$ the elapsed time since the planet was last closet to the sun (this is called perifocus or perihelion in astronomy) and $e= \sqrt{1-\frac{b^{2}}{a^{2}}}$, the eccentricity of the planet's elliptical orbit.\ Using Kepler's laws of planetary motion, we obtain the location of the planet at time $t.$ \[ \left\{ \begin{array}{c} x=a\left( \cos \left( E\right) -e\right) \\ y=a\sqrt{1-e^{2}}\sin \left( E\right) \end{array} \right. \] The quantity $E$ is called the eccentric anomaly and is given by the following equation \[ E=nt+e\sin \left( E\right) =M+e\sin \left( E\right). \] where $M$ is called the mean anomaly which increases linearly in time at the rate $n$. Note that $E$ is the fixed point of the function $f,\ $where $f\left( x\right) =M+e\sin \left( x\right) $ for a given time $t$ and the frequency of the orbit $\omega$. In this equation, we cannot find an explicit formula of the eccentric anomaly $E.$ It is easy to check that $f$ is a contraction, moreover, we have \[ \left\vert f\left( x\right) -f\left( y\right) \right\vert \leq e\left\vert x-y\right\vert \] which ensures the existence and uniqueness of the fixed point $E.$
For our simulation, by choosing the planet Mercury, we have its eccentricity $e= 0.20563069$ and the mean anomaly M= $3.05076572$ (The Mann's process is implemented for $a=0.9,\varphi =0.7),$ and given an initial guess $x_{1}=3$, we obtain the following iterates : \begin{equation*}
\begin{tabular}{|l|l|l|l|} \hline $n$ & $x_{n}$ & $\left\vert x_{n}-x_{n-1}\right\vert $&$\left\vert x_{n}-x_{fp}\right\vert $ \\ \hline
100& 3.066277803444744& 5.084582991976561e-06&3.292480386907215e-05 \\ \hline 1000 & 3.066247563732222 & 2.842158499660741e-07&2.685091347043311e-06 \\ \hline $10^{4}$ & 3.066245125153754 & 6.291136500635730e-08&2.465128789985727e-07 \\ \hline $10^{5}$ & 3.066244900326525 & 7.696832948766996e-09&2.168565016447133e-08 \\ \hline \end{tabular} \end{equation*}
\end{example}
\begin{remark} Note that the numerical solution of the equation $f(x)=x$ given by Matlab is $x_{fp}=3.066244878640875$. As we can observe, the used Mann algorithm gives nice approximations of the unique fixed point of the function $f$. Thus, the complementary numerical examples considered above make the obtained theoretical results of convergence well palpable. \end{remark}
\end{document} | arXiv |
\begin{document}
\title{ On blowup of nonendpoint borderline Lorentz norms for the Navier-Stokes equations
} \begin{abstract} \end{abstract} \setcounter{equation}{0} Assuming $T$ is a potential blow up time for the Navier-Stokes system in $\mathbb{R}^3$ or $\mathbb{R}^3_+$, we show that the $L^{3,q}$ Lorentz norm, with $q$ finite, of the velocity field goes to infinity as time $t$ approaches $T$.\\
\setcounter{equation}{0} \section{Introduction} In our paper we consider the Cauchy problem for the Navier-Stokes system in the space-time domain $Q_+=\Omega\times ]0,\infty[$ for vector-valued function $v=(v_1,v_2,v_3)=(v_i)$ and scalar function $q$, satisfying the equations \begin{equation}\label{directsystem} \partial_tv+v\cdot\nabla v-\Delta v=-\nabla q,\qquad\mbox{div}\,v=0 \end{equation} in $Q_+$, the boundary conditions \begin{equation}\label{directbc} v=0\end{equation} on $\partial\Omega\times [0,\infty[$, and the initial conditions \begin{equation}\label{directic} v(\cdot,0)=v_0(\cdot)\in C^\infty_{0,0}(\Omega):=\{v\in C_{0}^{\infty}(\Omega): \rm{div}\, v=0\}\end{equation} in $\Omega$. It is assumed that
$\Omega$ either $\mathbb{R}^3$ or $\mathbb{R}^3_{+}$.
For $\Omega=\mathbb{R}^3$, in the classical paper \cite{Le}, Leray showed the existence of a solution $v$ to the problem (\ref{directsystem})-(\ref{directic}) satisfying the global energy inequality
\begin{equation}\label{energyinequality}
\frac 12\int\limits_{\mathbb{R}^3} |v(x,t)|dx+\int\limits_0^t\int\limits_{\mathbb{R}^3}|\nabla v|^2dxdt'\leqslant \frac 12\int\limits_{\mathbb{R}^3}|v_0|^2 dx.
\end{equation}
Later on, in \cite{Hopf}, Hopf made important contributions when $\Omega$ is a bounded domain with sufficiently smooth boundary.
To recall the modern defition of weak Leray-Hopf solutions, we first introduce certain necessary notation.
Let $Q_{T}:=\Omega\times ]0,T[$, $L_2(\Omega)$ and ${W}^{1}_{2}(\Omega)$ be usual Lebesgue and Lebesgue spaces, respectively. We define further
$\overset{\circ}{J}(\Omega)$ as the closure of $C_{0,0}^{\infty}(\Omega)$ in $L_2(\Omega)$ and
$\overset{\circ}{J}{^{1}_{2}}(\Omega)$ is the closure of the same set
with respect to the Dirichlet metric.
\begin{definition}\label{weakLerayHopf}
Let $\Omega$ be a domain in $\mathbb{R}^3$
and $v_0\in\overset{\circ}{J}(\Omega)$. A weak-Leray Hopf solution to (\ref{directsystem})-(\ref{directic}) is a vector field $v:Q_\infty\rightarrow \mathbb{R}^3$ such that
\begin{equation}\label{vLerayenergyspace}
v\in L_{\infty}(0,\infty;\overset{\circ}{J}(\Omega))\cap L_{2}(0,\infty;\overset{\circ}{J}{^{1}_{2}}(\Omega)); \end{equation}
the function $t\rightarrow \int\limits_{\Omega}v(x,t)\cdot w(x)dx$ is continuous at any point $t\in[0,\infty[$ for any $w\in L_2(\Omega)$;
for any divergence free test function $w\in C_{0}^{\infty}(Q_\infty)$
\begin{equation}\label{distributionalsolution}
\int\limits_{Q^+}(-v\cdot\partial_t w- v\otimes v:\nabla w+\nabla v:\nabla w) dx dt=0;
\end{equation}
for any $t\in[0,\infty[$
\begin{equation}\label{energyleray}
\frac 12\int\limits_{\Omega} |v(x,t)|dx+\int\limits_0^t\int\limits_{\Omega}|\nabla v|^2dxdt'\leqslant \frac 12\int\limits_{\Omega}|v_0|^2 dx;
\end{equation}
\begin{equation}\label{strongcontzero}
\lim_{t\rightarrow 0^+}\|v(\cdot,t)-v_0(\cdot)\|_{L_{2}(\Omega)}=0.
\end{equation}
\end{definition}
The uniqueness or non-uniqueness of Leray-Hopf solutions is a long-standing open problem. It is also known for a long time that smoothness of weak Leray-Hopf solutions implies their uniqueness in the class of weak solutions. For initial data satisfying (\ref{directic}), the velocity field $v$ is smooth and bounded
for some short interval at least, see (incomplete) references \cite{Heywood},\cite{kiselevL}, \cite{L1967} and \cite{Le}. The loss of regularity might happen due to a blowup of the velocity. We define a blowup time $T$ as the first moment of time when
\begin{equation}\label{definitionblowup}
\lim\limits_{t\uparrow T}\|v(\cdot,t)\|_{L_{\infty}(\Omega)}=\infty.\end{equation}
Our aim is to prove a certain necessary conditions for $T$ be a blowup time. Let us first mention several results in this direction.
In \cite{Le} Leray proves necessary conditions for $T$ to be a blowup time for $\Omega=\mathbb{R}^3$ namely: \begin{equation}\label{Lerayrates}
\|v(\cdot,t)\|_{s,\Omega}\geq \frac {c_s}{(T-t)^{\frac {s-3}{2s}}} \end{equation} for any $0<t<T$, for all $s>3$, and for a positive constant $c_s$ depending only on $s$. Later in \cite{Giga1986} Giga proves (\ref{Lerayrates}) for a wide class of domains $\Omega$ including a half space and bounded domains with sufficiently smooth boundaries.
For the case $s=3$ no estimate of the form (\ref{Lerayrates}) is available.
However, in \cite{ESS2003}, it has been proven:
\begin{equation}\label{limsup}
\limsup\limits_{t\uparrow T}\|v(\cdot,t)\|_{L_{3}(\Omega)}=\infty. \end{equation}
In papers \cite{MSh2006} and \cite{S2005}, it has been shown that (\ref{limsup}) remains to be true for for $\Omega=\mathbb R^3_+:=\{x=(x_i)\in \mathbb R^3:\,\,x_3>0\}$ and for $\Omega$ being a bounded domain with sufficiently smooth boundary.
Recent progress has been made in establishing the validity of (\ref{limsup}) for other critical spaces. We refer to $X$, consisting of measurable functions acting on domains in $\mathbb{R}^3$, as critical if for $u\in X$ such that $u_{\lambda}(x)=\lambda u(\lambda x)$ we have
$\|u_{\lambda}\|_{X}=\|u\|_{X}.$ In \cite{Phuc} and \cite{WangZhang1}, it was shown that when $X$ is the non endpoint Lorentz space with $q$ finite $L^{3,q}(\mathbb{R}^3)$ condition (\ref{limsup}) remains to be true. Recently, Gallagher et al proved (\ref{limsup}) holds for $X= \dot{B}^{-1+\frac{3}{ p}}_{p,q}(\mathbb{R}^3)$ in the framework of ''strong'' solutions. Later this was shown for weak Leray-Hopf solutions in \cite{WangZhang1}.
Our work is motivated by the following question: what are the critical spaces $X(\Omega)$ and domains $\Omega$ for which \begin{equation}\label{criticalblowuplimit}
\lim\limits_{t\uparrow T}\|v(\cdot,t)\|_{X(\Omega)}=\infty \end{equation} holds true?
In \cite{Ser12} one of the authors proves this holds for $\Omega=\mathbb{R}^3$ and $X(\Omega)=L_{3}(\Omega)$ using the theory of local energy solutions in \cite {LR1}. Later on both authors showed in \cite{BarkerSer} this remains to be true in the case $\Omega=\mathbb{R}^3_{+}$ and $X(\Omega)=L_{3}(\Omega)$. We now claim the following: \begin{theorem}\label{limlorentz} Let $\Omega=\mathbb{R}$ or $\mathbb{R}^{3}_{+}$. Suppose $3\leqslant q<\infty$. Let $T$ be a blow-up time. Then necessarily
$$\lim\limits_{t\uparrow T}\|u(\cdot,t)\|_{L^{3,q}(\Omega)}=\infty.$$ \end{theorem}
Before commenting further let us define the Lorentz spaces. For a measurable function $f:\Omega\rightarrow\mathbb{R}$ define: \begin{equation}\label{defdist}
d_{f,\Omega}(\alpha):=|\{x\in \Omega : |f(x)|>\alpha\}|. \end{equation}
Given a measurable subset $\Omega\subset\mathbb{R}^{n}$, the Lorentz space $L^{p,q}(\Omega)$, with $p\in ]0,\infty[$, $q\in ]0,\infty]$, is the set of all measurable functions $g$ on $\Omega$ such that the quasinorm $\|g\|_{L^{p,q}(\Omega)}$ is finite. Here:
\begin{equation}\label{Lorentznorm}
\|g\|_{L^{p,q}(\Omega)}:= \Big(p\int\limits_{0}^{\infty}\alpha^{q}d_{g,\Omega}(\alpha)^{\frac{q}{p}}\frac{d\alpha}{\alpha}\Big)^{\frac{1}{q}}, \end{equation} \begin{equation}\label{Lorentznorminfty}
\|g\|_{L^{p,\infty}(\Omega)}:= \sup_{\alpha>0}\alpha d_{g,\Omega}(\alpha)^{\frac{1}{p}}. \end{equation}\\ It is well known that for $q\in ]0,\infty[,\,q_{1}\in ]0,\infty]$ and $q_{2}\in ]0,\infty]$ with $q_{1}\leqslant q_{2}$ we have the following embedding $ L^{p,q_1} \hookrightarrow L^{p,q_2}$ and the inclusion is known to be strict. Roughly speaking, the second index of Lorentz spaces gives information regarding nature of logarithmic bumps. For example, for any $1>\beta>0, q>3$ we have \begin{equation}\label{lorentzlogsing}
|x|^{-1}|\log(|x|^{-1})|^{-\beta}\chi_{|x|< 1}(x)\in L^{3,q}(\mathbb{R}^3)\,\,\,\rm{if\, and\, only\, if}\,\,q>\frac{1}{\beta}. \end{equation}
In this way Theorem \ref{limlorentz} gives a strengthening of the previous result obtained by both authors in \cite{BarkerSer}. It should be stressed that, at the time of writing, (\ref{criticalblowuplimit}) is open for the critical norm $L^{3,\infty}(\Omega)$ that contains $|x|^{-1}$. Furthermore, uniqueness of weak Leray-Hopf solutions in the space $L_{\infty}(0,T; L^{3,\infty}(\Omega))$ remains open. We mention that interior regularity results, that have smallness condition on $L_{\infty}(L^{3,\infty})$ norm, have been obtained in \cite{kimkozono}, \cite{kozono} and \cite{Tsai}, for example.
In order to describe the heuristics behind the proof of Theorem \ref{limlorentz}, we first mention previous work. In \cite{Ser12}, the method used for the statement in $L_{3}(\mathbb{R}^3)$ is, roughly speaking, as follows. It is based on assuming, for contradiction, an increasing sequence of times tending to the blow-up time such that the $L_{3}$-norm of the velocity is bounded. Then a suitable rescaling and limiting procedure is performed which gives the special type of the so-called local energy ancient solutions to the Navier-Stokes equations that coincide with Lemarie-Rieusset solutions to the Cauchy problem for same equations on some finite time interval. Those solutions have been introduced by Lemarie-Rieusset in \cite{LR1}, see also for some definitions in \cite{KS}. Moreover, the ancient solution is non-trivial. The contradiction is then obtained by proving a Liouville type theorem for those solutions based on backward uniqueness. The key point to mention from this is that in order to produce a Lemarie-Rieusset local energy solution to the Cauchy problem on a finite time interval, one needs certain behaviour of the limiting solution near it's initial time. The way to gain this in \cite{Ser12} is to split the rescaled solutions to two parts, one is the heat semigroup with weakly converging initial data and the other contains the non linearity but has zero initial data.
However, in \cite{BarkerSer}, a different approach has been used, for both $L_{3}(\mathbb{R}^3)$ and $L_{3}(\mathbb{R}^3_{+})$. The rescaling and splitting of rescaled solutions is the same as in \cite{Ser12}, the main difference with \cite{Ser12} is that the theory of Lemarie-Rieusset local energy solutions is not used. Hence, heuristically speaking, the behaviour of the limit solution near it's initial time is not necessary in \cite{BarkerSer}. Our observation is that this allows us to decompose the rescaled initial data and solutions in a different way to as was done in \cite{BarkerSer} and \cite{Ser12}. Combining this observation and ideas from \cite{BarkerSer} allows us to strengthen the result of \cite{BarkerSer}, and thus providing Theorem \ref{limlorentz}.
\setcounter{equation}{0} \section{Proof of Theorem 1.2} We make use of the following notation: $$ Q^+_{-A,0}:= \mathbb{R}^3_+\times ]-A,0[,\,\,Q_{\infty}:=\mathbb{R}^3\times ]0,\infty[,\,\,\,Q_{\infty}^+:=\mathbb{R}^3_+\times ]0,\infty[.$$
Furthermore $$B(x_0,R):=\{x: |x-x_0|<R\},\,B^+(x_0,R):=\{x\in B(x_0,R): x_3\geqslant x_{03}\}.$$ Let $T$ be a positive parameter, $\Omega$ a domain in $\mathbb{R}^3$. Then $Q_{T}:=\Omega\times ]0,T[$. Let $L_{m,n}(Q_T)$ be the space of measurable $\mathbb{R}^l$- valued functions with the following norm
$$\|f\|_{L_{m,n}(Q_{T})}:=(\int\limits_0^T\|f(\cdot,t)\|_{L_{m}(\Omega)}^{n} dt)^{\frac 1n},$$ for $n\in[1,\infty[$, and with the usual modification if $n=\infty$.We define the following Sobolev spaces with the mixed norm:
$$ W^{1,0}_{m,n}(Q_{T})=\{ v\in L_{m,n}(Q_{T}): \|v\|_{L_{m,n}(Q_{T})}+$$$$+\|\nabla v\|_{L_{m,n}(Q_{T})}<\infty\},$$
$$ W^{2,1}_{m,n}(Q_{T})=\{ v\in L_{m,n}(Q_{T}): \|v\|_{L_{m,n}(Q_{T})}+$$$$+\|\nabla v\|_{L_{m,n}(Q_{T})}+\|\nabla^{2} v\|_{L_{m,n}(Q_{T})}+\|\partial_{t} v\|_{L_{m,n}(Q_{T})}<\infty\}.$$
Now we state and prove simple (but important) fact about Lorentz spaces concerning a decomposition. This will be formulated as a Lemma. Analogous statement is Lemma II.I proven by Calderon in \cite{Calderon90}. \begin{lemma}\label{Decomp} Take $1< t<r<s\leqslant\infty$, and suppose that $g\in L^{r,\infty}(\Omega)$. For any $N>0$, we let
$g_{N_{-}}:= g\chi_{|g|\leqslant N}$ and $g_{N_{+}}:= g-g_{N_{-}}.$ Then \begin{equation}\label{bddpartg}
\|g_{N_{-}}\|_{L_{s}(\Omega)}^{s}\leqslant\frac{s}{s-r}N^{s-r}\|g\|_{L^{r,\infty}(\Omega)}^{r}-N^{s}d_{g}(N) \end{equation}
if $s<\infty$ and $\|g_{N_{-}}\|_{L_{\infty}(\Omega)}\leqslant N$, and \begin{equation}\label{unbddpartg}
\|g_{N_{+}}\|_{L_{t}(\Omega)}^{t}\leqslant \frac{r}{r-t}N^{t-r}\|g\|_{L^{r,\infty}(\Omega)}^{r}. \end{equation} Moreover for $\Omega=\mathbb{R}^{3}$ or $\mathbb{R}^{3}_{+}$, if $g\in L^{r,p}(\Omega)$ with $1\leqslant p\leq\infty$ and $\rm{div}\,\,g=0$ in weak sense (also $g_3(x',0)= 0$ for half space in weak sense), then $g= g_{1}+g_{2}$ where $g_{1}\in [C_{0,0}^{\infty}(\Omega)]^{L_{s}(\Omega)}$ with \begin{equation}\label{g1divfree}
\|g_{1}\|_{L_{s}(\Omega)}\leqslant C(s,r,p,\|g\|_{L^{r,p}(\Omega)}) \end{equation} and $g_{2}\in [C_{0,0}^{\infty}(\Omega)]^{L_{t}(\Omega)}$ with \begin{equation}\label{g2divfree}
\|g_{2}\|_{L_{t}(\Omega)}\leqslant C(r,t,p,\|g\|_{L^{r,p}(\Omega)}). \end{equation}
\end{lemma} \begin{proof} Proof of decomposition (\ref{bddpartg})-(\ref{unbddpartg}) can be found in \cite{Mccormick}.
Given $g$, satisfying assumptions of the lemma, we can find $g_{1_-}$ and $g_{1_+}$. We then can use the Helmoltz-Weyl decomposition $g_{1_-}=g_1+\nabla q_1$ where $g_1$ belongs to the required space with the estimate
$\|g_1\|_{L_s(\Omega)}\leq c\|g_{1_-}\|_{L_s(\Omega)}$, $\|\nabla q_1\|_{_s(\Omega)}\leq c\|g_{1_-}\|_{L_s(\Omega)}$, and $$\int\limits_\Omega\nabla q_1\cdot \nabla \varphi dx = \int\limits_\Omega g_{1_-}\cdot \nabla \varphi dx,\quad \forall\varphi\in C^\infty_0(\mathbb R^3).$$ The same is true for the second counterpart. So, we have $$\int\limits_\Omega\nabla (q_1+q_2)\cdot \nabla \varphi dx=0\quad \forall\varphi\in C^\infty_0(\mathbb R^3).$$ Using properties of harmonic functions and the above global integrability of $\nabla q_1$ and $\nabla q_2$, we conclude that $\nabla (q_1+q_2)=0$. From this, from estimates (\ref{bddpartg})-(\ref{unbddpartg}), and embedding $L^{r,p}(\Omega)$ into $L^{r,\infty}(\Omega)$, we derive the required estimates (\ref{g1divfree}) and (\ref{g2divfree}).
\end{proof} Now we are in a position to prove Theorem\ref{limlorentz}. Consider the more difficult case $\Omega=\mathbb R^3_+$.
Suppose that the statement of the theorem is false, then there exists an increasing sequence $t_{n}\uparrow T$ such that \begin{equation}\label{Lorentzbdd}
M :=\sup_{n}\|u(\cdot, t_{n})\|_{L^{3,q}(\mathbb{R}^3_{+})}<\infty. \end{equation} We know there exists a singular point $x_{0}=(x'_{0},x_{03})$. Without loss of generality, we may assume that $x_0=0$. The case $x_{03}>0$ is easier.
First examine the profile. It can be shown $u(x,t)\rightarrow u(x,T)$ for a.a $x\in\Omega$. From this and Fatou's lemma we infer \begin{equation}\label{profiledistribution} d_{u(\cdot,T),\Omega}(\alpha)\leqslant \liminf_{n\rightarrow \infty} d_{u(\cdot,t_{n}),\Omega}(\alpha)' \end{equation}
\begin{equation}\label{profileLorentz}
\|u(\cdot,T)\|_{L^{3,q}(\mathbb{R}^3_{+})}\leqslant M. \end{equation} Our rescaling will be as follows: $$ u^{(n)} (y,s):= \lambda_n u(x,t),\, p^{(n)} (y,s):= \lambda_{n}^{2} q(x,t),$$ where $$ x= \lambda_{n} y,\,t=T+\lambda_{n}^{2}s,\,\lambda_{n}=\sqrt{\frac{T-t_{n}}{2}}.$$ So smooth solutions $u^{(n)}$ and $p^{(n)}$ satisfy Navier-Stokes equations in $\mathbb{R}^{3}_{+}$ with $$u^{(n)}_{0}(y,-2)=\lambda_{n}u(\lambda_{n}y,t_{n}).$$ By scale invariance of the norm in the space $L^{3,q}$,
we observe the following: \begin{equation}\label{scaleinvariance}
\sup_{n}\|u^{(n)}_{0}(\cdot,-2)\|_{L^{3,q}(\mathbb{R}^{3}_{+})}=M<\infty. \end{equation} The key observation for Lorentz spaces as as follows. From Lemma \ref{Decomp} (along with fact embedding into weak $L_{3}$) we may decompose: $$u_{0}^{(n)}(\cdot,-2):= u_{0}^{1,(n)}(\cdot,-2)+u_{0}^{2,(n)}(\cdot,-2),$$ where \begin{equation}\label{decobspace} u_{0}^{1,n}(\cdot,-2)\in [C^{\infty}_{0,0}(\mathbb{R}^{3}_{+})]^{L_{\frac{10}{3}}(\mathbb{R}^{3}_{+})},\,u_{0}^{2,n}(\cdot,-2)\in [C^{\infty}_{0,0}(\mathbb{R}^{3}_{+})]^{L_{2}(\mathbb{R}^{3}_{+})} \end{equation} \begin{equation}\label{decompest}
\|u_{0}^{1,(n)}(\cdot,-2)\|_{L_{\frac{10}{3}}(\mathbb{R}^{3}_{+})}+\|u_{0}^{2,(n)}(\cdot,-2)\|_{L_{2}(\mathbb{R}^{3}_{+})}\leqslant c(M,q). \end{equation} We then decompose $u^{(n)}=v^{1,(n)}+v^{2,(n)}$
where $v^{1,(n)}$ and $p^{1,(n)}$ solve the linear problem $$\partial_tv^{1,n}-\Delta v^{1,(n)}=-\nabla p^{1,(n)},\qquad \mbox{div}\,v^{1,n}=0$$ in $Q^+_{-2,0}$, $$v^{1,(n)}(x',0,t)=0$$ for $(x',t)\in\mathbb R^2\times [-2,0]$, $$v^{1,(n)}(\cdot,-2)=u_{0}^{1,(n)}(\cdot,-2)\in L_\frac {10}3(\mathbb R^3_+).$$ Using Solonnikov estimates for the Green function in a half-space one sees that the following estimates are valid for $v^{1,n}$ ($\frac{10}{3}\leqslant l\leqslant \infty, k=1,2,\ldots$): \begin{equation}\label{semigroupest}
\|\nabla^{k}v^{1,(n)}(\cdot,t)\|_{L_{l}(\mathbb{R}^3_{+})}\leqslant \frac{c(M)}{(t+2)^{\frac{k}{2}+\frac{3}{2}(\frac{3}{10}-\frac{1}{l})}}. \end{equation}
Thus $$\|v^{1,(n)}(\cdot,t)\|_{L_{5}(\mathbb{R}^3_{+})}\leqslant \frac{c(M)}{(t+2)^{\frac{3}{20}}},\quad
\|v^{1,(n)}(\cdot,t)\|_{L_{4}(\mathbb{R}^3_{+})}\leqslant \frac{c(M)}{(t+2)^{\frac{3}{40}}}.$$ It is then seen that: \begin{equation}\label{L4,5}
\|v^{1,(n)}\|_{L_{5}(Q_{-2,0}^+)}+\|v^{1,(n)}\|_{L_{4}(Q_{-2,0}^+)}+\sup\limits_{t\in ]-2,0[}\|v^{1,(n)}(\cdot,t)\|_{L_{\frac {10}{ 3}}(R^3_+)}\leqslant c(M). \end{equation}
The second counterpart of $u^{(n)}$ satisfies the non-linear system $$\partial_tv^{2,(n)}+\mbox{div}\,u^{(n)}\otimes u^{(n)}-\Delta v^{2,(n)}=-\nabla p^{2,(n)},\qquad \mbox{div}\,v^{2,(n)}=0$$ in $Q^+_{-2,0}$, the boundary conditions $$v^{2,(n)}(x',0,t)=0$$ for $(x',t)\in\mathbb R^2\times [-2,0]$, and the initial conditions $$v^{2,(n)}(\cdot,-2)= u^{2,n}_{0}(\cdot,-2)$$ in $\mathbb{R}^3_+$.
The standard energy approach to the second system gives
$$ \partial_t\|v^{2,(n)}(\cdot,t)\|^2_{L_{2}(\mathbb R^3_+)}+2\|\nabla v^{2,(n)}(\cdot,t)\|^2_{L_{2}(\mathbb R^3_+)}=$$ $$= 2\int\limits_{\mathbb R^3_+}u^{(n)}\otimes u^{(n)}:\nabla v^{2,(n)}dx ds=I_1+I_2+I_3+I_4,$$ where $$I_1=2\int\limits_{\mathbb R^3_+}v^{1,(n)}\otimes v^{1,(n)}:\nabla v^{2,(n)}dx,\quad I_2=2\int\limits_{\mathbb R^3_+}v^{1,(n)}\otimes v^{2,(n)}:\nabla v^{2,(n)}dx $$ and $I_3=I_4=0$.
Next, let us consequently evaluate terms on the right hand side of the energy identity. For the first term, we have
$$|I_1|\leq c\|v^{1,(n)}(\cdot,t)\|^2_{L_{4}(\mathbb{R}^3_+)}\|\nabla v^{2,(n)}(\cdot,t)\|_{L_{2}(\mathbb R^3_+)}.$$
The second term can be treated as follows:
$$|I_2|\leq c\|v^{1,(n)}(\cdot,t)\otimes v^{2,(n)}(\cdot,t)\|_{L_{2}(\mathbb R^3_+)}\|\nabla v^{2,(n)}(\cdot,t)\|_{L_{2}(\mathbb R^3_+)}\leq$$
$$\leq c\|v^{2,(n)}(\cdot,t)\|_{L_{5}(\mathbb R^3_+)}\|v^{2,(n)}(\cdot,t)\|_{L_{\frac {10}{3}}(\mathbb R^3_+)}\|\nabla v^{2,(n)}(\cdot,t)\|_{L_{2}(\mathbb R^3_+)}. $$ Applying the known multiplicative inequality to the second factor in the right hand side of the latter bound, we find
$$|I_2|\leq c\|v^{1,(n)}(\cdot,t)\|_{L_{5}(\mathbb R^3_+)}\|v^{2,(n)}(\cdot,t)\|^{\frac 25}_{L_{2}(\mathbb R^3_+)}\|\nabla v^{2,(n)}(\cdot,t)\|^\frac 85_{L_{2}(\mathbb R^3_+)}.$$ Letting
$$y(t):=\|v^{2,(n)}(\cdot,t)\|^2_{L_{2}(\mathbb R^3_+)}$$ and using the Young inequality, we find
$$y'(t)+\|\nabla v^{2,(n)}(\cdot,t)\|^2_{L_{2}(\mathbb R^3_+)}\leq c\|v^{1,(n)}(\cdot,t)\|^5_{L_{5}(\mathbb R^3_+)}y(t)+c\|v^{1,(n)}(\cdot,t)\|^4_{L_{4}(\mathbb R^3_+)}.$$ Next, elementary arguments lead to the inequality
$$\Big(y(t)\exp{\Big(-\int\limits^t_{-2}}\|v^{1,(n)}(\cdot,s)\|^5_{L_{5}(\mathbb R^3_+)}ds\Big)\Big)'\leq$$$$\leq c\exp{\Big(-\int\limits^t_{-2}}\|v^{1,(n)}(\cdot,s)\|^5_{L_{5}(\mathbb R^3_+)}ds\Big)\|v^{2,(n)}(\cdot,t)\|^4_{L_{4}(\mathbb R^3_+)}.$$ So,
$$y(t)\leq c\int\limits^t_{-2}\exp{\Big(\int\limits^t_\tau}\|v^{1,(n)}(\cdot,s)\|^5_{L_{5}(\mathbb R^3_+)}ds\Big)\|v^{1,(n)}(\cdot,\tau)\|^4_{L_{4}(\mathbb R^3_+)}d\tau+$$$$+
\|u_{0}^{2,(n)}(\cdot,-2)\|_{L_{2}(\mathbb{R}^{3}_{+})}^{2}$$ Using, (\ref{decompest}) and (\ref{L4,5}) it is easily seen that \begin{equation}\label{bddenergy}
\sup_{-2<t<0}y(t)\leqslant C(M),\,\,\quad\|\nabla v^{2,(n)}\|_{L_{2}(Q_{-2,0}^+)}\leqslant C(M). \end{equation}
From these estimates and from the multiplicative inequality, one can deduce that \begin{equation}\label{globalu2}
\|v^{2,(n)}\|_{L_{s}(Q^+_{-2,0})}\leq C(s,M)\end{equation}with any $s\in [2,\frac {10}3]$. Moreover, \begin{equation}\label{globalu2}
\|u^{(n)}\|_{L_{\frac{10}{3}}(Q^+_{-2,0})}\leq C(M). \end{equation} We now sketch the plan as to how to obtain the contradiction. Full details are found in \cite{BarkerSer}. Let $$f^{(n)}= u^{(n)}.\nabla u^{(n)}+ v^{1,(n)}+v^{2,(n)}.$$ Using (\ref{semigroupest}) and (\ref{bddenergy}), one can decompose $$f^{(n)}=f^{(n)}_1+f^{(n)}_2+f^{(n)}_3$$ with estimates
\begin{equation}\|f^{(n)}_1\|_{L_{\frac 9 8,\frac 3 2}(Q_{-\frac 7 4,0}^+)}+\|f^{(n)}_2\|_{L_{\frac {10} {3}}(Q_{-\frac 7 4,0}^+)}+\|f^{(n)}_3\|_{L_{2}(Q_{-\frac 7 4,0}^+)}\leqslant C(M). \end{equation} Multiplying $u^{(n)}$ and $p^{(n)}$ by a suitable cut off function in time and using of Solonnikov coercive estimates for the linear Stokes system (see \cite{Sol1973}-\cite{Sol2003UMN}), we can decompose $$u^{(n)}= \sum_{i=1}^3 u^{(n)}_{i},\,p^{n}= \sum_{i=1}^3 p^{(n)}_{i}$$ on $Q_{-\frac 3 2,0}^+$ with the following estimates \begin{equation}\label{coerciveest}
\|u^{(n)}_1\|_{W^{2,1}_{\frac 9 8,\frac 3 2}(Q_{-\frac 3 2,0}^+)}+\|u^{(n)}_2\|_{W^{2,1}_{\frac {10} {3}}(Q_{-\frac 3 2,0}^+)}+\|u^{(n)}_3\|_{W^{2,1}_{2}(Q_{-\frac 3 2,0}^+)}+$$$$+
\|\nabla p^{(n)}_1\|_{L_{\frac 9 8,\frac 3 2}(Q_{-\frac 3 2,0}^+)}+\|\nabla p^{(n)}_2\|_{L_{\frac {10} {3}}(Q_{-\frac 3 2,0}^+)}+\|\nabla p^{(n)}_3\|_{L_{2}(Q_{-\frac 3 2,0}^+)} \leqslant C(M). \end{equation} One then shows $(u^{n},p^{n})$ tends to a suitable weak solution $(u,p)$ of the Navier-Stokes system on $Q_{-\frac 3 2,0}^+$ with estimate (\ref{globalu2}) and with analogous decomposition and estimates to (\ref{coerciveest}). One uses suitable epsilon regularity criterion developed in \cite{CKN}, \cite{S3} and \cite{Ser09} to show that $u$ is in fact non trivial in $Q^+(a)$ for all $a$ sufficiently small as a by product of the assumption that the origin is a singular point at time $T$. Moreover, the previously mentioned global estimates for suitable decompositions of $(u,p)$ imply \begin{equation}\label{ubddexterior}
|u(x,t)|+|\nabla u(x,t)|\leq c_1(\delta),
\end{equation} for all $(x,t)\in (\mathbb{R}^3_{+\delta}\setminus B^{+}(R_{1}) \times ]-\frac{5}{4},0[$. Here, $\mathbb{R}^3_{+\delta}:=\mathbb{R}^3_{+}\cap \{x_3>\delta\}$.\\ Let us briefly mention minor difference concerning zero endpoint of the limit solution in the context of Lorentz spaces. It can be shown (as in \cite{BarkerSer}) that $$
\int\limits_{B^{+}(a)}|u^{(n)}(x,0)| dx\rightarrow \int\limits_{B^{+}(a)}|u(x,0)| dx. $$ Using generalized Holder inequality for Lorentz spaces (along with scale invariance), it is not so difficult to show
$$\frac{1}{a^{2}}\int\limits_{B^{+}(a)}|u^{(n)}(x,0)| dx\leqslant c\|u^{(n)}(\cdot,0)\|_{L^{3,q}(B^{+}(a))}=c\|u(\cdot,T)\|_{L^{3,q}(B^{+}(\lambda_{n}a))}.$$ Now $u(\cdot,T)\in L_{3,q}(\mathbb{R}^{3}_{+})$ and obviously $$d_{u(\cdot,T),B^{+}(\lambda_{n}a))}(\alpha)\rightarrow 0.$$
So for $0<q<\infty$, $\|u(\cdot,T)\|_{L^{3,q}(B^{+}(\lambda_{n}a))}\rightarrow 0$.\\ With this in mind one, then considers the vorticity equation $$\partial_{t}\omega-\Delta\omega= \rm{\,div}\,(\omega\otimes u-u\otimes\omega).$$ Then contradiction is then reached by showing that the limit function is trivial as a consequence of showing that $\omega=0$.
This is shown by properties of the limit functions $(u,p)$ together with backward uniqueness and unique continuation through spatial boundaries for the heat operator with lower order terms as in \cite{ESS2003}. $\Box$
\end{document} | arXiv |
In triangle $ABC,$ $BC = 32,$ $\tan B = \frac{3}{2},$ and $\tan C = \frac{1}{2}.$ Find the area of the triangle.
Let $\overline{AD}$ be the altitude from $A,$ and let $x = AD.$
[asy]
unitsize (0.15 cm);
pair A, B, C, D;
B = (0,0);
C = (32,0);
A = (8,12);
D = (8,0);
draw(A--B--C--cycle);
draw(A--D);
label("$A$", A, N);
label("$B$", B, SW);
label("$C$", C, SE);
label("$D$", D, S);
label("$x$", (A + D)/2, E);
[/asy]
Then $BD = \frac{x}{3/2} = \frac{2x}{3},$ and $CD = \frac{x}{1/2} = 2x,$ so
\[BC = BD + DC = \frac{2x}{3} + 2x = \frac{8x}{3}.\]Since $BC = 32,$ $x = 12.$
Therefore, $[ABC] = \frac{1}{2} \cdot AD \cdot BC = \frac{1}{2} \cdot 12 \cdot 32 = \boxed{192}.$ | Math Dataset |
\begin{document}
\begin{talk}[Elaine Cohen, Richard Riesenfeld]{Tom Lyche and Georg Muntingh} {Simplex Spline Bases on the Powell-Sabin 12-Split: Part I} {Lyche, Tom}
\begin{my_abstract} We review the construction and a few properties of the S-basis, a simplex spline basis for the $C^1$ quadratic splines on the Powell-Sabin 12-split. \end{my_abstract}
\noindent Piecewise polynomials or splines defined over triangulations form an indispensable tool in the sciences, with applications ranging from scattered data fitting to finding numerical solutions to partial differential equations. In applications like geometric modelling and solving PDEs by isogeometric methods one often desires a low degree spline with $C^1$, $C^2$ or $C^3$ smoothness. For a general triangulation, it is known that the minimal degree of a triangular $C^r$ element is $4r+1$, e.g., degrees 5, 9, 13 for the classes $C^1$, $C^2$ or $C^3$. To obtain smooth splines of lower degree one can split each triangle in the triangulation into several subtriangles. One such split that we consider here is the Powell-Sabin 12-split of a triangle. \begin{center} \includegraphics[scale=0.6]{m1-m10}\\The 12-split with numbering of vertices. \end{center}
Once a space is chosen one determines its dimension. The spaces $\mathcal{S}^1_2(\PSB)$ and $\mathcal{S}^3_5(\PSB)$ of $C^1$ quadratics and $C^3$ quintics on the 12-split $\PSB$ of a single triangle have dimension 12 and 39, respectively. Over a general triangulation $\mathcal{T}$ of a polygonal domain we can 12-split each triangle in $\mathcal{T}$ to obtain a triangulation $\mathcal{T}_{12}$. The dimensions of the corresponding $C^1$ quadratic and $C^2$ quintic spaces (the latter with $C^3$ supersmoothness at the vertices and the interior edges of each macro triangle) are $3|\mathcal{V}| + |\mathcal{E}|$ and $10|\mathcal{V}| + 3|\mathcal{E}|$, respectively, where $|\mathcal{V}|$ and $|\mathcal{E}|$ are the number of vertices and edges in $\mathcal{T}$. Moreover, in addition to giving $C^1$ and $C^2$ spaces on any triangulation these spaces are suitable for multiresolution analysis, see for example \cite{TL_LycheMuntingh14}.
To compute with these spaces one needs a suitable basis. In the univariate case the B-spline basis is an obvious choice. In this talk we consider a bivariate generalization known as simplex splines. We review the construction and a few properties shown in \cite{TL_CohenLycheRiesenfeld13} of the S-basis consisting of $C^1$ quadratic simplex splines in $\mathcal{S}^1_2(\PSB)$. We also introduce some concepts needed in Part II of this talk given by Georg Muntingh.
{\bf A short background on simplex splines}. Let $\boldsymbol{K} = \{\boldsymbol{v}_1\cdots \boldsymbol{v}_{d+s+1}\} \subset \mathbb{R}^s$ be a finite multiset. Consider a simplex $\sigma = [\overline{\boldsymbol{v}}_1, \ldots, \overline{\boldsymbol{v}}_{d+s+1}] \subset \mathbb{R}^{d+s}$ together with a projection $\pi: \sigma \longrightarrow \mathbb{R}^s$ satisfying $\pi(\overline{\boldsymbol{v}}_i) = \boldsymbol{v}_i$. We define the \emph{simplex spline} $ B[\boldsymbol{K}](\boldsymbol{x}) = \text{vol}_d\big(\sigma\cap \pi^{-1}(\boldsymbol{x})\big)/\text{vol}_{d+s} (\sigma)$. For instance, three knots in $\mathbb{R}^1$ define a linear B-spline, four knots in $\mathbb{R}^1$ define a quadratic B-spline, and four knots in $\mathbb{R}^2$ define a linear bivariate simplex spline: \begin{center} \includegraphics[scale=0.7, clip = true, trim = 0 10 25 16]{SimplexSplineDef1}\qquad \includegraphics[scale=0.7, clip = true, trim = 0 10 25 22]{SimplexSplineDef2}\qquad \includegraphics[scale=0.7, clip = true, trim = 0 10 25 21]{SimplexSplineDef3} \end{center}
Simplex splines have all the usual properties of univariate B-splines. This includes continuity which can be controlled locally, a recurrence relation, and differentiation and knot insertion formulas. The support of a simplex spline is the convex hull of its knots, and in $\mathbb{R}^2$ the collection of knotlines is obtained by connecting each knot to all other knots (the complete graph). A simplex spline with $d+3$ knots in $\mathbb{R}^2$ has $d-m+1$ continuous derivatives across a knot line containing $m$ knots counting multiplicites.
{\bf Simplex splines on the 12-split}. Since the knotlines form a complete graph the simplex splines are natural candidates for a $C^r$ basis on this split. A simplex spline on the 12-split will have a knotset of the form $\boldsymbol{K} = \{\boldsymbol{v}_1^{m_1} \cdots \boldsymbol{v}_{10}^{m_{10}}\}$, where $\boldsymbol{v}_1,\ldots,\boldsymbol{v}_{10}$ are the vertices numbered as above, and $m_i\ge 0$ is the multiplicity of $\boldsymbol{v}_i$, i.e., the number of repetitions of $\boldsymbol{v}_i$ in the multiset. A convenient scaling is the \emph{(area normalized) simplex spline} $Q[\boldsymbol{K}]: \mathbb{R}^2 \longrightarrow \mathbb{R}$, recursively defined by \[ Q[\boldsymbol{K}](\boldsymbol{x}) := \left\{ \begin{array}{cl} 0 & \text{if~}\text{area}([\boldsymbol{K}]) = 0,\\
\boldsymbol{1}_{[\boldsymbol{K})}(\boldsymbol{x})\frac{\text{area}(\PSsmall)}{\text{area}([\boldsymbol{K}])} & \text{if~}\text{area}([\boldsymbol{K}]) \neq 0\text{~and~} |\boldsymbol{K}| = 3,\\
\sum_{j = 1}^{10} \beta_j Q[\boldsymbol{K}\backslash \boldsymbol{v}_j](\boldsymbol{x}) & \text{if~}\text{area}([\boldsymbol{K}]) \neq 0\text{~and~} |\boldsymbol{K}| > 3,\\ \end{array} \right. \] with $\boldsymbol{x} = \beta_1 \boldsymbol{v}_1 + \cdots + \beta_{10} \boldsymbol{v}_{10}, \beta_1 + \cdots + \beta_{10} = 1$, and $\beta_i = 0$ whenever $m_i = 0$.
By Theorem 4 in \cite{TL_Micchelli79} this definition is independent of the choice of the $\beta_j$. Whenever $m_7 = m_8 = m_9 = m_{10} = 0$, we use the graphical notation \[ \SimSgeneric := Q[\boldsymbol{v}_1^i \boldsymbol{v}_2^j \boldsymbol{v}_3^k \boldsymbol{v}_4^l \boldsymbol{v}_5^m \boldsymbol{v}_6^n]. \]
{\bf B-splines on the boundary}. It is useful for the simplex splines to restrict to consecutive univariate B-splines on the boundary. For example, on $[\boldsymbol{v}_1,\boldsymbol{v}_2]$ the quadratic simplex splines $\frac14$\SimS{300101}, $\frac12$\SimS{210101},$\frac12$\SimS{120110}, $\frac14$\SimS{030110} restrict to: \vskip 0.1cm
\begin{tabular}{m{11em}c} \includegraphics[scale=0.3]{Bboundary}&$\blue{B[\boldsymbol{v}_1^3,\boldsymbol{v}_4]}$, $\orange{B[\boldsymbol{v}_1^2,\boldsymbol{v}_4,\boldsymbol{v}_2]}$, $\green{B[\boldsymbol{v}_1,\boldsymbol{v}_4,\boldsymbol{v}_2^2]}$, $\red{B[\boldsymbol{v}_4,\boldsymbol{v}_2^3]}$ \end{tabular} \vskip 0.3cm
{\bf Symmetries}. Identifying a triangle with an equilateral triangle, its symmetries \begin{center} \includegraphics[scale=0.55]{PS12-S3-1} \includegraphics[scale=0.55]{PS12-S3-2} \includegraphics[scale=0.55]{PS12-S3-3} \includegraphics[scale=0.55]{PS12-S3-4} \includegraphics[scale=0.55]{PS12-S3-5} \includegraphics[scale=0.55]{PS12-S3-6} \end{center} form a group $S_3$ that acts on the simplex splines by permuting knots. We write \[ [\mathcal{B}]_{S_3} := \{Q[\sigma(\boldsymbol{K})]\,:\, Q[\boldsymbol{K}]\in \mathcal{B},\ \sigma\in S_3\} \] for the set of simplex splines related to $\mathcal{B}$ by a symmetry in $S_3$. Let \begin{equation*} \begin{aligned} & c_4 := \frac{c_1 + c_2}{2},\quad c_5 := \frac{c_2 + c_3}{2},\quad c_6 := \frac{c_1 + c_3}{2},\\ & c_7 := \frac{c_4 + c_6}{2},\quad c_8 := \frac{c_4 + c_5}{2},\quad c_9 := \frac{c_5 + c_6}{2},\quad c_{10} := \frac{c_1 + c_2 + c_3}{3}. \end{aligned} \end{equation*} Via the identification $c_i \leftrightarrow \boldsymbol{v}_i$ with the vertices of $\PSB$, the group $S_3$ acts on polynomials in $c_1,\ldots,c_{10}$ and simplex splines, or combinations of these, e.g., \begin{align*} \left[c_4c_{10}\SimS{110111}\right]_{S_3} & = \left\{c_4c_{10}\SimS{110111}, c_5c_{10}\SimS{011111}, c_6c_{10}\SimS{101111}\right\}. \end{align*} {\bf The quadratic S-basis}. It is given by \[\left[\frac14\SimS{300101}, \frac12\SimS{210101}, \frac34\SimS{110111}\right]_{S_3} = \left\{\frac14\SimS{300101}, \frac14\SimS{030110}, \frac14\SimS{003011},\ldots, \frac34\SimS{011111}, \frac34\SimS{110111} \right\} \] and is the unique simplex spline basis for $\mathcal{S}_2^1(\PSB)$ with local linear independence. Moreover, it is symmetric, reduces to B-splines on the boundary, can be computed by a pyramidal scheme, and has B\'ezier-like smoothness conditions across adjacent macro triangles. Furthermore, it has a barycentric Marsden identity \[
\left( \sum \left[c_1\SimS{211000}\right]_{S_3}\right)^2 =
\sum \left[\frac14 c_1^2\SimS{300101}\right]_{S_3} \cup \left[\frac34 c_4c_{10} \SimS{110111}\right]_{S_3} \cup \left[\frac12 c_1 c_4 \SimS{210101} \right]_{S_3}, \] which yields polynomial reproduction, explicit dual functionals and a simple quasi-interpolant. These show that the S-basis is stable independently of the geometry, which implies an $h^2$ bound on the distance between a spline and its control surface.
\end{talk}
\end{document} | arXiv |
\begin{document}
\title{Regularity up to the boundary for singularly perturbed fully nonlinear elliptic equations}
\begin{abstract} In this article we are interested in studying regularity up to the boundary for one-phase singularly perturbed fully nonlinear elliptic problems, associated to high energy activation potentials, namely $$
F(X, \nabla u^{\varepsilon}, D^2 u^{\varepsilon}) = \zeta_{\epsilon}(u^{\epsilon})
\quad \mbox{in} \quad \Omega \subset \R^n $$ where $\zeta_{\varepsilon}$ behaves asymptotically as the Dirac measure $\delta_{0}$ as $\varepsilon$ goes to zero. We shall establish global gradient bounds independent of the parameter $\varepsilon$.
\noindent \textbf{Keywords:} Fully nonlinear elliptic operators, one-phase problems, regularity up to the boundary, singularly perturbed equations, global gradient bounds.
\noindent \textbf{AMS Subject Classifications 2010: 35B25, 35B65, 35D40, 35J15, 35J60, 35J75, 35R35.}
\end{abstract} \tableofcontents
\section{Introduction}\label{int} \hspace{0.6cm}Throughout the last three decades or so, variational problems involving singular PDEs has received a warm attention as they often come from the theory of critical points of non-differentiable functionals. The pioneering work of Alt-Caffarelli \cite{AC} marks the beginning of such a theory by carrying out the variational analysis of the minimization problem \begin{equation}\tag{{\bf Minimum}}
\displaystyle \min_{} \int_{\Omega} \left( |\nabla v|^2 + \chi_{\{v>0\}} \right) \ dX, \end{equation} among competing functions with the same non-negative Dirichlet boundary condition.
Since the very beginning it has been well established that such discontinuous minimization problems could be treated by penalization methods. Indeed, Lewy-Stampacchia, Kinderlehrer-Nirenberg, Caffarelli among others were the precursors of such an approach to the study of problem $\Delta u^{\epsilon} = \zeta_{\varepsilon}(u^{\varepsilon})$ over of 70s and 80s. Linear problems in non-divergence form was firstly considered by Berestycki \textit{et al} in \cite{BCN}. Teixeira in \cite{Tei} started the journey of investigation into fully nonlinear elliptic equations via singular perturbation methods: \begin{equation}\label{eqRT}
F(X, D^2 u^{\varepsilon}) = \zeta_{\varepsilon}(u^{\varepsilon}) \quad \mbox{in} \quad \Omega, \end{equation} where $\zeta_{\varepsilon} \sim \varepsilon^{-1}\chi_{(0,\varepsilon)}$. The problem appears in nonlinear formulations of high energy activation models, see \cite{RT} and \cite{Tei}. It can also be employed in the analysis of overdetermined problems as follows. Given $\Omega \subset \mathbb{R}^n$ a domain and a non-negative function $\varphi\colon \Omega \rightarrow \mathbb{R}$, it plays a crucial role in Geometry and Mathematical Physics the question of finding a compact hyper-surface $ \partial \Omega' \subset \Omega$ such that the following elliptic boundary value problem \begin{equation} \label{Free} \left\{ \begin{array}{rclcl} F(X, \nabla u, D^2 u) &=& 0 & \mbox{in} & \Omega \backslash \Omega^{\prime}\\ u & = & \varphi & \mbox{on} & \partial \Omega\\ u & = & 0 & \mbox{in} & \Omega^{\prime}\\ u_{\nu} & = & \psi &\mbox{in} & \partial \Omega^{\prime}, \end{array} \right. \end{equation} can be solved. Problems as \eqref{eqRT} became known over the years in the Literature as \textit{cavity type problems}.
\begin{figure}
\caption{Configuration for Free Boundary Problem}
\end{figure}
Hereafter in this paper, $F \colon \Omega \times \mathbb{R}^n \times \textrm{Sym}(n) \rightarrow \mathbb{R}$ is a fully nonlinear uniformly elliptic operator, i.e, there exist constants $\Lambda \geq \lambda > 0$ such that \begin{equation}\label{UE} \tag{{\bf Unif. Ellip.}}
\lambda \|N\| \le F(X,\overrightarrow{ p}, M+N) - F(X,\overrightarrow{ p}, M) \le \Lambda \|N\|, \end{equation} for all $M,N \in \textrm{Sym}(n), N \ge 0, \overrightarrow{ p }\in \mathbb{R}^n \,\ and \,\ X \in \Omega$. As usual $\textrm{Sym}(n)$ denotes the set of all $n \times n$ symmetric matrices. Moreover, we must to observe the mapping $M \mapsto F(X, \overrightarrow{p} ,M)$ is monotone increasing in the natural order on $Sym(n)$ and Lipschitz continuous. Under such a structural condition, the theory of viscosity solutions provides a suitable notion for weak solutions.
\begin{definition}[{\bf Viscosity solution}] For an operator $F\colon \Omega \times \mathbb{R}^n \times \text{Sym}(n) \to \mathbb{R}$, we say a function $u \in C^0(\Omega)$ is a viscosity supersolution (resp. subsolution) to $$
F(X, \nabla u, D^2 u) = f(X) \quad \mbox{in} \quad \Omega, $$ if whenever we touch the graph of $u$ by below (resp. by above) at a point $Y \in \Omega$ by a smooth function $\phi$, there holds $$
F(Y, \nabla \phi(Y), D^2 \phi (Y)) \le f(Y) \quad (\mbox{resp.} \ge f(Y)). $$ Finally, we say $u$ is a viscosity solution if it is simultaneously a viscosity supersolution and subsolution. \end{definition}
\begin{remark} All functions considered in the paper will be assumed continuous in $\overline{\Omega}$, namely $C$-viscosity solutions, see Caffarellli-Cabr\'{e} \cite{CC} and Teixeira \cite{Tei}. However, we also can to consider $L^p$-viscosity notion for such a solutions, see for example Winter \cite{Wint}. \end{remark}
In \cite{RT}, several analytical and geometrical properties of such a fully nonlinear singular problem \eqref{eqRT} were established. Notwithstanding, regularity up to the boundary for approximating solutions has not been proven in the literature yet. This is the key goal of the present article. More precisely, we shall prove a uniform gradient estimate up to the boundary for viscosity solutions of the singular perturbation problem \begin{equation}\label{Equation Pe} \tag{\bf $E_{\varepsilon}$} \left\{ \begin{array}{rclcl} F(X, \nabla u^{\varepsilon}, D^2 u^{\epsilon}) &=& \zeta_{\epsilon}(u^{\epsilon}) & \mbox{in} & \Omega\\ u^{\epsilon} &=&\varphi & \mbox{on} & \partial \Omega, \end{array} \right. \end{equation}
where we have: the singular reaction term $\zeta_{\varepsilon}(s) = \frac{1}{\varepsilon} \zeta \left(\frac{s}{\varepsilon}\right)$ for some non-negative $\zeta \in C^{\infty}_{0}([0,1])$, a parameter $\varepsilon >0$, a non-negative $\varphi \in C^{1, \gamma}(\overline{\Omega})$, with $0<\gamma < 1$, and, a bounded $C^{1,1}$ domain $\Omega$ (or $\partial \Omega$ for short). Throughout this paper we will assume the following bounds: $\|\varphi\|_{C^{1,\gamma}(\overline{\Omega})} \le \mathcal{A}$ and $\|\zeta\|_{L^{\infty}([0, 1])} \le \mathcal{B}$.
\begin{theorem}[{\bf Global uniform Lipschitz estimate}] \label{principal2} Let $u^{\epsilon}$ be a viscosity solution to the singular perturbation problem \eqref{Equation Pe}. Then under the assumptions ${\bf (F1)-(F3)}$ there exists a constant $C(n, \lambda, \Lambda, b, \mathcal{A}, \mathcal{B},\Omega)>0$ independent of $\epsilon$, such that $$
\|\nabla u^{\epsilon}\|_{L^{\infty}(\overline{\Omega})} \le C. $$ \end{theorem}
Our new estimate allows us to obtain existence for corresponding free boundary problem \eqref{Free}, keeping the prescribed boundary value data, see Theorem \ref{limFB}. Finally, we should emphasize our estimate generalizes the local gradient bound proven in \cite{Tei}, see also \cite{RT} for a rather complete local analysis of such a free boundary problem.
Although we have chosen to carry out the global analysis for the homogeneous case, the results presented in this paper can be adapted, under some natural adjustments, for the non-homogeneous case, $$ \left\{ \begin{array}{rclcl}
F(X, \nabla u^{\varepsilon}, D^2 u^{\epsilon}) &=& \zeta_{\epsilon}(u^{\epsilon}) + f_\varepsilon (X) & \mbox{in} & \Omega\\ u^{\epsilon}(X) &=&\varphi(X) & \mbox{on} & \partial \Omega, \end{array} \right. $$ with $0\leq c_0 \le f_\varepsilon \le c_1$.
Our approach follows the pioneering work of Gurevich \cite{Gu}, where it is introduced a new strategy to investigate uniform estimate up to boundary of two-phase singular perturbation problems involving linear elliptic operators of type $\mathcal{L}u = \partial_i(a_{ij}\partial_j u).$ This method has been successfully applied by Karakhanyan in \cite{K} for the one-phase problem in the case involving nonlinear singular/degenerate elliptic operators of the $p$-Laplacian type $\Delta_{p}u^{\epsilon} = \zeta_{\epsilon}(u^{\epsilon}).$
\subsection{Organization of the article}
\hspace{0.6cm}The paper is organized of following way: In Section \ref{NoSta} we shall introduce the notation which will be used throughout of the paper, as well as we set up the structural assumptions for fully nonlinear elliptic operators. In Section \ref{ExisUniq} we discuss about the existence of appropriated notion of weak solutions to problem \eqref{Equation Pe}, namely \textit{Perron's type solutions}, see Theorem \ref{ExistMinSol}. The Section \ref{reglocal} is devoted to prove the main Theorem \ref{principal2}, for this reason it contains several keys Lemmas which are standard in the global regularity theory for elliptic operators in accordance with Gurevich \cite{Gu} and Karakhanyan \cite{K}, as well as Teixeira \cite{Tei} and Ricarte-Teixeira \cite{RT} for the corresponding local fully nonlinear singular perturbation theory. The free boundary problem, namely Theorem \ref{limFB} is obtained as consequence of global Lipschitz regularity. Finally, the last Section \ref{Append} is an Appendix where we prove two technical Lemmas (respectively Lemmas \ref{lemma2.0} and \ref{lemma2.1}) that play an important role in order to prove the main Theorem \ref{principal2} in Section \ref{reglocal}.
\section{Notation and statements}\label{NoSta}
\hspace{0.6cm}We shall introduce some notations and structural assumptions which we will use throughout this paper. \begin{itemize} \item[\checkmark] $n$ indicates the dimension of the Euclidean space. \item[\checkmark] $\mathcal{H}_{+}$ is the half-space $\{X_n >0\}$. \item[\checkmark] $\mathcal{H} \mathrel{\mathop:}= \{X=(X_1, \ldots, X_n) \in \mathbb{R}^n : X_n=0\}$ indicates the hyperplane. \item[\checkmark] $\hat{X}$ is the vertical projection of $X$ on $\mathcal{H}$.
\item[\checkmark] $\mathcal{C}_X \mathrel{\mathop:}= \left\{Y \in H_{+} : |Y-\hat{Y}| \ge \frac{1}{2}|Y-X|\right\}$ is the cone with vertex at point $X \in \mathcal{H}$. \item[\checkmark] $B_r(X)$ is the ball with center at $X$ and radius $r$, and, $B_r$ the ball $B_r(0)$. \item[\checkmark] $B^{+}_{r}(X) \mathrel{\mathop:}= B_{r}(X) \cap \mathcal{H}_{+}$. \item[\checkmark] $B^{\prime}_{r}(X) $ is the ball with center at $X$ and radius $r$ in $\mathcal{H}$. \end{itemize}
\begin{remark} Throughout this article \textit{Universal constants} are the ones depending only on the dimension, ellipticity and structural properties of $F$, i. e., $n, \lambda, \Lambda$ and $b$. \end{remark}
Also, following classical notation, for constants $\Lambda \geq \lambda >0$ we denote by $$
\mathcal{P}^{+}_{\lambda,\Lambda}(M) \mathrel{\mathop:}= \lambda \cdot \sum_{e_i <0} e_i + \Lambda \cdot \sum_{e_i >0} e_i \quad \mbox{and} \quad \mathcal{P}^{-}_{\lambda,\Lambda}(M) \mathrel{\mathop:}= \lambda \cdot \sum_{e_i >0} e_i + \Lambda \cdot \sum_{e_i <0} e_i $$ the \textit{Pucci's extremal operators}, where $e_i = e_i(M)$ are the eigenvalues of $M \in Sym(n)$.
We shall introduce structural conditions that will be frequently used throughout of this paper: \begin{enumerate} \item[{\bf (F1)}](\textit{Ellipticity and Lipschitz regularity condition }) For all $M,N \in \textrm{Sym}(n)$, $\overrightarrow{p},\overrightarrow{q} \in \mathbb{R}^n$, $X \in \Omega$\label{F1} $$
\mathcal{P}^{-}_{\lambda, \Lambda}(M-N)-b|\overrightarrow{p}-\overrightarrow{q}| \le F(X,\overrightarrow{p},M) - F(X,\overrightarrow{q},N) \le \mathcal{P}^{+}_{\lambda, \Lambda}(M-N) + b|\overrightarrow{p}-\overrightarrow{q}| $$ \item[{\bf (F2)}] (\textit{Normalization condition}) We shall suppose that, $$
F(X, 0, 0) = 0 $$ \item[{\bf(F3)}] (\textit{Small oscillation condition}) We must to assume $$
\displaystyle \sup_{X_0 \in \Omega} \Theta_F(X, X_0)\ll 1 $$ where $$
\displaystyle \Theta_F(X, X_0) \mathrel{\mathop:}= \sup_{M \in Sym(n)\backslash \{0\}} \frac{|F(X, 0, M)- F(X_0, 0, M)|}{\|M\|} $$
\end{enumerate}
\begin{remark} Assumption ${\bf (F1)}$ is equivalent to notion of uniform ellipticity \ref{UE} when $\overrightarrow{p} = \overrightarrow{q}$. The assumption ${\bf (F2)}$ is not restrictive, since we can always redefine the operator in order to check it. The smallest regime on oscillation of $F$, namely condition ${\bf (F3)}$, depends only on universal parameters, see \cite{Wint}. \end{remark}
\begin{example}[{\bf Isaacs type operators}]An example which we must have in mind are the Isaacs' operators from stochastic game theory \begin{equation}\label{IO}
\displaystyle F(X, \overrightarrow{p}, M) \mathrel{\mathop:}= \sup_{\alpha \in \mathfrak{A}} \inf_{\beta \in \mathfrak{B}}\left(\textrm{Tr}\left[A^{\alpha, \beta}(X)\cdot M\right] + \left\langle B^{\alpha, \beta}(X), \overrightarrow{p}\right\rangle \right) \quad \left(\mbox{resp.} \,\,\, \inf\limits_{\mathfrak{A}} \,\,\sup_{\mathfrak{B}} (\cdots)\right), \end{equation} where $A^{\alpha, \beta}$ is a family of measurable $n \times n$ real symmetric matrices with small oscillation satisfying $$
\lambda\|\xi\|^2 \leq \xi^TA^{\alpha, \beta}(X)\xi \leq \Lambda \|\xi\|^2, \,\, \forall \,\, \xi \in \R^n \quad \mbox{and} \quad \|B^{\alpha, \beta}\|_{L^{\infty}(\Omega)}\leq b. $$ \end{example}
\section{Existence of solutions}\label{ExisUniq}
\hspace{0.6cm}In this Section we shall comment on the existence of appropriated viscosity solutions to the singularly perturbed problem \eqref{Equation Pe}. Such a solutions are labeled by \textit{Perron's type solutions}.
\begin{theorem}[{\bf Perron's type method}, \cite{RT}]\label{PerMeth} Let $f \in C^{0, 1}([0, \infty)) $ be a bounded function. Suppose that there exist a viscosity sub-solution $\underline{u} \in C(\overline{\Omega}) \cap C^{0, 1}(\Omega)$ (respectively super-solution $\overline{u} \in C(\overline{\Omega}) \cap C^{0, 1}(\Omega)$) to $F(X, \nabla w, D^2w) = f(w)$ satisfying $\underline{u} = \overline{u} = g \in C(\partial \Omega)$. Define the set of functions $$
\mathcal{S} \mathrel{\mathop:}= \left\{ v \in C(\overline{\Omega}) \;\middle|\; \begin{array}{c} v \text{ is a viscosity super-solution to } \\ F(X, \nabla w, D^2w) = f(w) \text{ such that } \underline{u} \le v \le \overline{u} \end{array} \right\}. $$ Then, \begin{equation}\label{2.1}
u(X) \mathrel{\mathop:}= \inf_{v \in \mathcal{S}} v(X), \,\, \mbox{for} \,\, x \in \Omega \end{equation} is a continuous viscosity solution to $F(X, \nabla w, D^2w) = f(w)$ in $\Omega$ with $u=g$ continuously on $\partial \Omega$. \end{theorem}
Existence of Perron's type solution to \eqref{Equation Pe} will follow by choosing $\underline{u} \mathrel{\mathop:}= \underline{u}^{\varepsilon}$ and $\overline{u} \mathrel{\mathop:}= \overline{u}^{\varepsilon}$ as solutions to the boundary value problems: $$ \begin{array}{ccc} \left\{ \begin{array}{rcccl} F(X, \nabla \underline{u}^{\varepsilon}, D^2 \underline{u}^{\varepsilon}) & = & \sup\limits_{[0, \infty)} \zeta_{\varepsilon}(u^{\varepsilon}(X)) & \mbox{in} & \Omega\\ \underline{u}^{\varepsilon}(X) & = & \varphi(X) & \mbox{on} & \partial \Omega \end{array} \right.
& \mbox{and} &
\left\{ \begin{array}{rcccl} F(X, \nabla \overline{u}^{\varepsilon}, D^2 \overline{u}^{\varepsilon}) & = & 0 & \mbox{in} & \Omega\\ \overline{u}^{\varepsilon}(X) & = & \varphi(X) & \mbox{on} & \partial \Omega, \end{array} \right. \end{array} $$
We must note that for each $\varepsilon>0$ fixed, existence of such a $\underline{u}^{\varepsilon}$ and $\overline{u}^{\varepsilon}$ follows as consequence of standard methods of sub and super solutions. Moreover, we have that $\underline{u} \in C(\overline{\Omega})\cap C^{0, 1}(\Omega)$ and $\overline{u} \in C(\overline{\Omega})\cap C^{0, 1}(\Omega)$ are viscosity subsolution and supersolution to \eqref{Equation Pe} respectively. Finally, as consequence of the Theorem \ref{PerMeth} we have the following existence Theorem:
\begin{theorem}[{\bf Existence of Perron's type solutions, \cite{RT}}]\label{ExistMinSol} Given $\Omega \subset \R^n$ be a bounded Lipschitz domain and $g \in C(\partial \Omega)$ be a nonnegative boundary datum. There exists for each $\varepsilon>0$ fixed, a nonnegative Perron's type viscosity solution $u^{\varepsilon} \in C(\overline{\Omega})$ to \eqref{Equation Pe}. \end{theorem}
\section{Optimal Lipschitz regularity}\label{reglocal}
\hspace{0.6cm}In this section, we shall present the proof of Theorem \ref{principal2}. Thus let us assume the assumptions of problem \eqref{Equation Pe}.
We make a pause as to discuss some remarks which will be important throughout this work. Firstly it is important to highlight that is always possible to perform a change of variables to flatten the boundary. Indeed, if $\partial \Omega$ is a $C^{1, 1}$ set, the part of $\Omega$ near $\partial \Omega$ can be covered with a finite collection of regions that can be mapped onto half-balls by diffeomorphisms (with portions of $\partial \Omega$ being mapped onto the ``flat" parts of the boundaries of the half-balls). Hence, we can use a smooth mapping, reducing this way the general case to that one on $B^{+}_{1}$, and, the boundary data would be given on $B^{\prime}_1$.
Previously we start the proof of the global Lipschitz estimative, we need to assure the non-negativity and boundedness of solutions to \eqref{Equation Pe}. This statement is a consequence of the Alexandroff-Bekelman-Pucci Maximum Principle, see \cite{CC} for more details.
\begin{lemma}[{\bf Nonnegativity and boundedness, \cite{RT} and \cite{Tei}}] \label{ABP} Let $u^{\varepsilon}$ be a viscosity solution to \eqref{Equation Pe}. Then there exists a universal constant $C>0$ such that $$
0 \le u^{\varepsilon}(X) \le C\|\varphi\|_{L^{\infty}(\overline{\Omega})} \quad \mbox{in} \quad \Omega. $$ \end{lemma}
We will now establish a universal bound for the Lipschitz norm of $u^{\varepsilon}$ up to the boundary. The proof will be divided in two cases.
\begin{center}
{\large \bf Case 1: Lipschitz regularity up to the boundary in the region $\{0 \le u^{\varepsilon} \le \varepsilon\}$}. \end{center}
\begin{theorem} \label{prop1} Let $u^{\epsilon}$ be a viscosity solution to \eqref{Equation Pe}. For $X \in \{0 \le u^{\varepsilon} \le \varepsilon\} \cap B^{+}_{\frac{1}{2}}$ there exists a universal constant $C_1>0$ independent of $\varepsilon$ such that $$
|\nabla u^{\epsilon}(X)| \le C_1. $$ \end{theorem} \begin{proof}
We denote by $$
\delta(X) \mathrel{\mathop:}= \textrm{dist}(X, \mathcal{H}) $$ the vertical distance. If $\delta(X) \ge \epsilon$, then $B_{\varepsilon}(X) \subset B^{+}_{1}$ for $\varepsilon \ll 1$. Therefore, from local gradient bounds \cite{RT, Tei} , there exists a universal constant $C_0>0$ independent of $\varepsilon$, such that $$
|\nabla u^{\epsilon}(X)| \le C_0. $$ On the other hand, if $\delta(X) < \epsilon$, then it is sufficient to prove that there exists a universal constant $C_0>0$ independent of $\varepsilon$, such that \begin{equation} \label{new}
u^{\epsilon}(\hat{X}) \le C_0 \epsilon. \end{equation} Indeed, suppose that \eqref{new} holds. Consider $h \colon \overline{B}^{+}_{1} \rightarrow \mathbb{R}$ to be the viscosity solution to the Dirichlet problem $$ \left\{ \begin{array}{rclcl} F(Y, \nabla h, D^2 h) &=& 0 & \mbox{in} & B^{+}_{1}\\ h & = & u^{\epsilon} & \mbox{on} & \partial B^{+}_{1}. \end{array} \right. $$ From $C^{1,\alpha}$ regularity estimates up to the boundary (see for instance \cite[Theorem 3.1]{Wint}), we know that $h \in C^{1,\alpha}\left(\overline{B}^{+}_{\frac{3}{4}}\right)$ with the following estimate $$
| \nabla h(Y)| \le c \left(\|h\|_{L^{\infty}(B^{+}_{1})} + \|\varphi\|_{C^{1,\gamma}(B^{\prime}_{1})}\right) \le C \quad \textrm{in} \quad B^{+}_{\frac{3}{4}} $$ and by Comparison Principle we have $$
u^{\epsilon} \le h \quad \mbox{in} \quad B^{+}_{1}. $$ Hence, it follows from assumption \eqref{new} that $$
u^{\epsilon}(Y) \le h(Y) \le h(\hat{X}) + C |Y - \hat{X}| \le C \epsilon \quad \textrm{if} \quad Y \in B^{+}_{2 \epsilon}(\hat{X}) $$ Then, again applying $C^{1,\alpha}$ regularity estimates from \cite{Wint}, we obtain $$
|\nabla u^\varepsilon(X)| \le C_0(n, \lambda, \Lambda, b, \mathcal{B}). $$
In order to prove \eqref{new} suppose, by purpose of contradiction, there exists $\epsilon >0$ such that $$
u^{\epsilon}(\hat{X}) \ge k \epsilon \quad \text{ for } \quad k \gg 1. $$ We shall denote $$
r_0 \mathrel{\mathop:}= \textrm{dist}(\hat{X}, \{0 \le u^{\varepsilon} \le \varepsilon\}). $$ Consider $X_0 \in \{0 \le u^{\varepsilon} \le \varepsilon\} \cap \partial B_{r_0}^{+}(\hat{X})$ a point to which the distance is achieved, i.e., $$
r_0 = |X_0 - \hat{X}|. $$ Thereafter, let $
\mathcal{C}_{\hat{X}} $ be the cone with vertex at $\hat{X} \in \mathcal{H}$. Suppose initially that $X_0 \in \mathcal{C}_{\hat{X}}$ then $B_{\frac{r_0}{2}}(X_0) \subset B^{+}_{1}$ .
Now,
let us define, $v^{\epsilon} : B_1 \rightarrow \mathbb{R}$ by $$
v^{\epsilon}(Y) \mathrel{\mathop:}= \frac{u^{\epsilon}(X_0 + (r_0 /2)Y)}{\epsilon}. $$ Therefore, $v^{\epsilon}$ satisfies in the viscosity sense $$
F_{\varepsilon}(Y, \nabla v^{\varepsilon}, D^2 v^{\epsilon}) = \frac{1}{\epsilon^{2}} \left(\frac{r_0}{2}\right)^{2} \zeta(v^{\epsilon}) \mathrel{\mathop:}= \mathfrak{g}(Y), $$ where \begin{equation} \label{escale} F_{\varepsilon}(Y, \overrightarrow{p}, M) \mathrel{\mathop:}= \frac{1}{\varepsilon} \left(\frac{r_0}{2}\right)^2 F \left(X_0 + \frac{r_0}{2} Y, \frac{2\varepsilon}{r_0} \cdot p, \varepsilon \left(\frac{2}{r_0}\right)^2 M\right). \end{equation}
Now note that $\mathfrak{g} \in L^{\infty}(B_1)$, since $r_0 < \epsilon$ and $F_{\varepsilon}$ satisfies ${\bf (F1)-(F3)}$ with constant $\tilde{b} = \frac{r_0}{2}\cdot b$.
Moreover, since $v^{\epsilon}(0) \le 1$ it follows from Harnack inequality that $$
v^{\epsilon}(Y) \le c \quad \mbox{for} \quad Y \in B_{\frac{1}{2}}, $$ i.e., $$
u^{\epsilon}(X) \le c \epsilon, \quad X \in B_{\frac{r_0}{4}}(X_0). $$ Consider now $Z \in B^{\prime }_{r_0}(\hat{X})$. It follows that $$
\varphi(Z) \ge \varphi(\hat{X}) - \mathcal{A} \cdot |Z-\hat{X}| \ge k \epsilon -r_0 \cdot \mathcal{A} \ge (k-\mathcal{A}) \epsilon, $$ since $r_0 < \epsilon$. Define the scaled function $w^{\epsilon} : B^{+}_{1} \rightarrow \mathbb{R}$, $$
w^{\epsilon}(Y) \mathrel{\mathop:}= \frac{u^{\epsilon}(\hat{X} + r_0 Y)}{\epsilon}. $$ It readily follows that
$$ \left\{ \begin{array}{rcl} F_{\varepsilon}(Y, \nabla w^{\varepsilon}, D^2 w^{\epsilon}) = 0 & \mbox{in} & B^{+}_{1}\\ w^{\epsilon}(Y) \ge k-\mathcal{A} & \mbox{on} & B^{\prime}_{1}, \end{array} \right. $$ where $F_{\varepsilon}$ is as in \eqref{escale}. Therefore according to Lemma \ref{lemma2.0}, $$
w^{\epsilon}(Y) \ge c(k-\mathcal{A}) \quad \mbox{in} \quad B^{+}_{\frac{3}{4}}. $$ In other words, we have reached that $$
u^{\epsilon}(X) \ge c \epsilon (k-\mathcal{A}) \quad \textrm{in} \quad B^{+}_{\frac{3r_0}{4}}(\hat{X}). $$ Hence $$
c \epsilon (k-\mathcal{A}) \le u^{\epsilon}(Z_0) \le c \epsilon, \quad \forall \,\,\,\, Z_0 \in \partial B_{\frac{3r_0}{4}}^{+}(\hat{X}) \cap \partial B_{\frac{r_0}{4}}(X_0), $$ which leads to a contradiction for $k \gg 1$.
On the one hand if $X_0 \not\in \mathcal{C}_{\hat{X}}$, choose $X_1 \in \{0 \le u^{\varepsilon} \le \varepsilon\}$ such that $$
r_1 \mathrel{\mathop:}= \textrm{dist}(\hat{X}_{0}, \{0 \le u^{\varepsilon} \le \varepsilon\}) = |\hat{X}_{0} - X_1|. $$ From triangular inequality and the fact that $r_1 \le \frac{r_0}{2}$ we have $$
|X_1 - \hat{X}| \le |X_1 - \hat{X}_{0}| + |\hat{X}_{0} - \hat{X}| \le r_1 +r_0 \le \frac{r_0}{2} + r_0. $$ If $X_1 \in \mathcal{C}_{\hat{X}_{0}}$ the result follows from previous analysis. Otherwise, let $X_{2}$ be such that $$
r_2 \mathrel{\mathop:}= \textrm{dist}(\hat{X}_{1}, \{0 \le u^{\varepsilon} \le \varepsilon\}) = |\hat{X}_{1} - X_2|. $$ As before we have
$$
|X_2 - \hat{X}| \le |\hat{X}_{1} - X_2| + |\hat{X}_{1} - \hat{X}| \le \frac{r_0}{4} + \frac{r_0}{2} + r_0, $$ since $r_2 \le \frac{r_1}{2} \le \frac{r_0}{4}$. Observe that this process must finish up within a finite number of steps. Indeed, suppose that we have a sequence of points $
X_j \in \partial \{0\le u^{\varepsilon} \le \varepsilon\}, \quad X_{j+1} \not \in \mathcal{C}_{\hat{X}_{j}} \,\,\, (j=1,2,\ldots) $ satisfying, $$
r_{j+1} \mathrel{\mathop:}= \textrm{dist}(\hat{X}_{j}, \{0 \le u^{\varepsilon} \le \varepsilon\}) = |X_{j+1} - \hat{X}_{j}| $$ and \begin{equation} \label{ind}
r_{j+1} \le \frac{r_{j}}{2} \le \frac{r_0}{2^{j+1}}. \end{equation} Thus, it follows from \eqref{ind} that $$
|X_j - \hat{X}| \le r_0 + r_0 \sum_{i=1}^{j} \frac{1}{2^i} \le 2r_0. $$ Therefore, up to a subsequence, $X_j \to X_{\infty} \in B^{\prime}_{2r_0}(\hat{X})$ with $\varphi(X_{\infty}) = \varepsilon$. However, $$
\varphi(X_{\infty}) \ge \varphi(\hat{X}) - \mathcal{A} \cdot | \hat{X}- X_{\infty} | \ge \varepsilon (k-2 \mathcal{A}) \gg \varepsilon
$$ for $k \gg 1$ which drives us to a contradiction, and, hence the assertion \eqref{new} is proved. \end{proof}
\begin{figure}
\caption{Geometric argument for the inductive process.}
\end{figure}
\begin{center} {\large \bf Case 2: \,\, Lipschitz regularity in the region $B^{+}_{1/8} \setminus \{0\le u^{\varepsilon} \le \varepsilon\}$}. \end{center}
\begin{theorem}
Let $u^{\varepsilon}$ be a viscosity solution to \eqref{Equation Pe}. If $X \in B^{+}_{\frac{1}{8}} \cap \{u^{\varepsilon} > \varepsilon\}$, then there exists a constant $C_0 = C_0(n, \lambda, \Lambda, b, \mathcal{A})>0$ such that
$$
|\nabla u^{\varepsilon}(X)| \le C_0.
$$ \end{theorem}
The proof of the theorem consists in analysing three possible cases (Lemmas \ref{prop2}, \ref{prop2.1}, \ref{prop2.2} below). Henceforth we shall use the following notation $$
\delta_{\varepsilon}(X) \mathrel{\mathop:}= \textrm{dist}(X, \{0 \le u^{\varepsilon} \le \varepsilon\}) \quad \textrm{and} \quad\delta(X) \mathrel{\mathop:}= \textrm{dist}(X, \mathcal{H}). $$
The next result is decisive in our approach.
\begin{lemma}\label{l5} Let $u^{\epsilon}$ be a viscosity solution to \eqref{Equation Pe} with $\varphi \in C^{1,\gamma}(B^{\prime}_{1})$. Then, for all $X \in B^{\prime}_{\frac{1}{4}} \cap \{u^{\varepsilon}> \varepsilon\}$, there exists a constant $c_0=c_0(n, \lambda, \Lambda, b) >0$ such that $$
\varphi(X)\le \epsilon + c_0 \cdot \delta_{\varepsilon}(X). $$ \end{lemma} \begin{proof}
Let us suppose for sake of contradiction that there exists an $\epsilon>0$ and
$X_0 \in B^{\prime}_{\frac{1}{4}} \setminus \{0\le u^{\varepsilon} \le \varepsilon\}$ such that $$
\varphi(X_0) \ge \epsilon + k \cdot \delta_{\varepsilon}(X_0) $$ holds for $k \gg 1$, large enough. Let $Z=Z_{\epsilon} \in \partial \{0 \le u^{\varepsilon} \le \varepsilon\}$ be a point to which the distance is achieved, i.e. $$
\delta_{\varepsilon} \mathrel{\mathop:}= \delta_{\epsilon}(X_0) = |X_0 - Z|. $$ We have two cases to analyse: If $Z \in \mathcal{C}_{X_0}$, then the normalized function $v^{\epsilon} \colon B^{+}_{1} \rightarrow \mathbb{R}$ given by $$
v^{\epsilon}(Y) \mathrel{\mathop:}= \frac{u^{\epsilon}(X_0 + \delta_{\epsilon} Y)-\varepsilon}{\delta_{\epsilon}} $$ satisfies $$
F_{\varepsilon}(Y, \nabla v^{\varepsilon}, D^{2} v^{\epsilon}) = 0 \quad \mbox{in} \quad B^{+}_{1} $$
in the viscosity sense, where $$ F_{\varepsilon}(Y, \overrightarrow{p}, M) \mathrel{\mathop:}= \delta_{\varepsilon} F\left(X_0+\delta_{\varepsilon}Y, \overrightarrow{p}, \frac{1}{\delta_{\varepsilon}} M\right). $$ As in Theorem \ref{prop1}, $F_{\varepsilon}$ satisfies ${\bf (F1)-(F3)}$ with constant $\tilde{b} = \delta_{\varepsilon} b$. Moreover, $$
v^{\epsilon}(Y) \ge 0 \quad \mbox{in} \quad B^{+}_{1}. $$ Now, for any $X \in B^{\prime}_{\delta_{\epsilon}}(X_0)$ we should have for $k \gg 1$, \begin{eqnarray*}
\varphi(X) &\ge& \varphi(X_0) - \mathcal{A} \delta_{\varepsilon} \ge \varepsilon + k \delta_{\varepsilon} - \mathcal{A} \delta_{\varepsilon} \\
&\ge& \varepsilon + \frac{k}{2} \delta_{\varepsilon}, \end{eqnarray*} i.e, $$
\frac{\varphi(X_0 + \delta_{\varepsilon} Y) - \varepsilon}{\delta_{\varepsilon}} \ge \frac{k}{2} \quad \textrm{in} \,\,\, B^{\prime}_1. $$ In other words, $$
v^{\epsilon}(Y) \ge c k \quad \forall \,\, Y \in B'_{1}. $$ Hence, from Lemma \ref{lemma2.0} we have that $$ v^{\epsilon} \ge ck \quad \mbox{in} \quad B^{+}_{\frac{3}{4}}. $$ In a more precise manner, \begin{equation} \label{m1}
u^{\epsilon}(X) \ge \epsilon + C k \delta_{\epsilon}, \quad X \in B^{+}_{\frac{3\delta_{\epsilon}}{4}}(X_0). \end{equation}
From now on, let us consider $\tilde{B} \mathrel{\mathop:}= B_{\frac{\delta_{\epsilon}}{4}}(\mathbf{P})$, where $\mathbf{P} = \mathbf{P}_{\epsilon}\mathrel{\mathop:}= Z + \frac{X_0 - Z}{4}$. If we define $\omega^{\varepsilon} \mathrel{\mathop:}= u^{\epsilon}-\epsilon$, then since $Z \in \partial \tilde{B}$, it follows that \begin{eqnarray}
F_{\epsilon}(X, \nabla \omega^{\varepsilon} , D^2 \omega^{\epsilon})=0 \quad \text{ in } \quad \tilde{B}, \label{LL1}\\
\omega^{\epsilon}(Z)=u^{\varepsilon}(Z)-\varepsilon =0, \label{LL2}\\
\frac{\partial \omega^{\varepsilon}}{\partial \nu}(Z) \le |\nabla \omega^{\varepsilon}(Z)| \le C. \label{LL3} \end{eqnarray}
Therefore, from \eqref{LL1}-\eqref{LL3} we can apply Lemma \ref{lemma2.1}, which gives
$$
\omega^{\varepsilon}(\mathbf{P}) \le C_0 \cdot \delta_{\varepsilon},
$$ i.e., \begin{equation}\label{m2}
u^{\epsilon}(\mathbf{P}) \le \epsilon + C \delta_{\epsilon}. \end{equation} At a point $\mathbf{P}$ on $\partial B_{\frac{3 \delta_{\epsilon}}{4}}^{+}(X_0)$ we have (according to \eqref{m1} and \eqref{m2}) $$
\epsilon + k c \delta_{\epsilon} \le u^{\epsilon}(\mathbf{P}) \le \varepsilon + C_0 \delta_{\epsilon} $$ which gives a contradiction if $k$ has been chosen large enough.
The second case, namely $Z \not\in \mathcal{C}_{X_{0}}$, it is treated similarly as in Theorem \ref{prop1} and for this reason we omit the details here. \end{proof}
\begin{lemma}\label{prop2} Let $u^{\varepsilon}$ be a viscosity solution to \eqref{Equation Pe} and $X \in B^{+}_{\frac{1}{8}} \cap \{u^{\varepsilon} > \varepsilon\}$ such that $\delta_{\varepsilon}(X) \le \delta(X)$. Then there exists a universal constant $C_0>0$, such that $$
|\nabla u^{\epsilon}(X)| \le C_0. $$ \end{lemma} \begin{proof} We may assume with no loss of generality that $\delta_{\epsilon}(X) \le \frac{1}{8}$. Otherwise, if we suppose that $\delta_{\epsilon}(X) > \frac{1}{8}$, then the result would follow from \cite{RT, Tei}. From now on, we select $X_{\epsilon} \in \partial \{0 \le u^{\varepsilon} \le \varepsilon\}$ a point which achieves the distance, i.e., $$
\delta_{\epsilon} \mathrel{\mathop:}= \delta_{\varepsilon}(X) = |X-X_{\epsilon}|. $$ Since $$
|X_{\epsilon}| \le |X| + \delta_{\epsilon} \le \frac{1}{4}, $$ we must have that $X_{\varepsilon} \in B^{+}_{\frac{1}{4}} \cap \{0 \le u^{\varepsilon} \le \varepsilon\}$. This way, by applying Theorem \ref{prop1}, there exists a constant $C_1=C(n,\lambda,\Lambda, b, \mathcal{A}, \mathcal{B})>0$ such that $$
|\nabla u^{\epsilon}(X_{\epsilon})| \le C_1. $$ By defining the re-normalized function $v^{\epsilon} : B_1 \rightarrow \mathbb{R}$ as $$
v^{\epsilon}(Y) \mathrel{\mathop:}= \frac{u^{\epsilon}(X + \delta_{\epsilon} Y) - \epsilon}{\delta_{\epsilon}}. $$ Then, as before $v^{\epsilon}$ satisfies \begin{eqnarray}
F_{\epsilon}(Y, \nabla v^{\varepsilon} , D^2 v^{\epsilon})=0 \quad \text{ in } \quad B_1, \label{L1}\\
v^{\epsilon}(Y_{\epsilon})=0, \label{L2}\\
|\nabla v^{\epsilon}(Y_{\epsilon})| \le C_1, \label{L3}\\
v^{\epsilon}(Y) \ge 0 \,\,\, \text{for}\,\,\, Y \in B_1 \label{L4}, \end{eqnarray} where $$
F_{\epsilon}(Y,\overrightarrow{ p} , M) \mathrel{\mathop:}= \delta_{\epsilon}F\left(X+ \delta_{\varepsilon} Y, \overrightarrow{p} , \frac{1}{\delta_{\epsilon}}M\right) \quad \mbox{and} \quad Y_{\epsilon} := \frac{X_{\epsilon} -X}{\delta_{\epsilon}} \in \partial B_1. $$ From \eqref{L1}-\eqref{L4} we are able to apply Lemma \ref{lemma2.1} and conclude that there exists a universal constant $c>0$ such that $$
v^{\varepsilon}(0) \le c. $$ Moreover, from Harnack inequality $$
v^{\varepsilon} \le C_0 \quad \mbox{in} \quad B_{1/2}. $$ Therefore, by $C^{1,\alpha}$ regularity estimates (see for example \cite{CC}) we must have that $$
|\nabla u^{\varepsilon}(X)| = |\nabla v^{\varepsilon}(0)| \le \frac{1}{\delta_{\varepsilon}} \|u^{\varepsilon} - \varepsilon\| \le C_0, $$ and the Lemma is proved. \end{proof}
\begin{lemma}\label{prop2.1} For $X \in B^{+}_{\frac{1}{8}} \cap \{u^{\varepsilon} > \varepsilon\}$ such that $\delta(X) < \delta_{\varepsilon}(X) \le4 \delta(X)$, we have $$
|\nabla u^{\epsilon}(X)| \le C_0 $$ for some constant $C_0 = C_0(n,\lambda,\Lambda, b, \mathcal{A}, \mathcal{B}) >0$. \end{lemma} \begin{proof} Similar to Lemma \ref{prop2}, we may assume that $\delta_{\epsilon} \le \frac{1}{8}$, otherwise, as in Lemma \ref{prop2} the gradient boundedness follows from local estimates \cite{RT, Tei}. Define the scaled function $v^{\epsilon} \colon B_1 \rightarrow \mathbb{R}$ by $$
v^{\epsilon}(Y) \mathrel{\mathop:}= \frac{u^{\epsilon}(X+\delta Y)-\epsilon}{\delta}, $$ where $\delta = \delta(X)$. Clearly $$
F_{\delta}(Y, \nabla v^{\varepsilon}, D^2 v^{\epsilon})=0 \quad \mbox{in} \quad B_1 $$ in the viscosity sense, and, from Harnack inequality $$
v^{\varepsilon} \le C v^{\varepsilon}(0) \sim \frac{1}{\delta} \quad \textrm{in} \,\,\, B_{\frac{1}{2}}. $$ By applying once more $C^{1,\alpha}$ regularity estimates, we obtain \begin{equation}\label{inf}
|\nabla u^{\varepsilon}(X)| = |\nabla v^{\varepsilon}(0)| \le \frac{C}{\delta}. \end{equation}
Therefore, the idea is to find an estimate for $u^{\varepsilon}-\varepsilon$ in terms of the vertical distance $\delta(X).$ To this end, consider $h$ the viscosity solution to the Dirichlet problem \begin{equation} \label{(14)} \left\{ \begin{array}{rclcl}
F(X,\nabla h,D^2 h)& = & 0 & \mbox{in} & B^{+}_{1}\\
h & = & u^{\epsilon} & \mbox{on} & \partial B^{+}_{1} . \end{array} \right. \end{equation} Since $0 \le u^{\epsilon} \le C(n, \lambda, \Lambda, b, \mathcal{B})$, it follows from $C^{1,\alpha}$ estimate up to boundary \cite{Wint} that $h \in C^{1,\alpha}\left(\overline{B}^{+}_{\frac{3}{4}}\right)$. Moreover $$
|\nabla h(X)| \le \overline{C} \left(\|h\|_{L^{\infty}(B^{+}_1)} + \|\varphi\|_{C^{1,\gamma}( B^{\prime}_{1})}\right) \le \overline{C}(C+\mathcal{A})\mathrel{\mathop:}= \mathcal{C}^{\ast}. $$ From Comparison Principle, we have that $$
u^{\epsilon} \le h \quad \mbox{in} \quad B^{+}_{1}. $$ Hence, \begin{equation}\label{(15)}
u^{\epsilon}(X) \le h(X) \le h(\hat{X}) + \mathcal{C}^{\ast}|X-\hat{X}| \le \varphi(\hat{X}) + \mathcal{C}^{\ast}\delta. \end{equation}
Now, we have that $|\hat{X}| \le |X| + \delta \le \frac{1}{4}$ , and, consequently we are able to apply Lemma \ref{l5} which gives \begin{equation}\label{(16)}
\varphi(\hat{X}) \le \epsilon + c_0 \cdot \textrm{dist}(\hat{X}, \{0 \le u^{\varepsilon} \le \varepsilon\}) \le \epsilon + c_0(\delta_{\epsilon} + \delta) \le \epsilon + 5 c_0 \delta. \end{equation} Thus, it follows from \eqref{(15)} and \eqref{(16)} that $$
u^{\epsilon}(X) - \varepsilon \le C_0 \delta, $$ where $C_0 \mathrel{\mathop:}= C(5c_0+\mathcal{C}^{\ast})$. Finally, if we apply $C^{1,\alpha}$ estimate, Harnack inequality and estimate \eqref{inf}, respectively, we end up with $$
|\nabla u^{\varepsilon}(X)| = |\nabla v^{\varepsilon}(0)| \le \frac{1}{\delta} \|u^{\varepsilon} - \varepsilon\|_{L^{\infty}\left(B_{\frac{1}{2}}\right)} \le C_0 $$ which concludes the proof. \end{proof}
\begin{lemma}\label{prop2.2} If $X \in B^{+}_{\frac{1}{8}} \cap \{u^{\varepsilon} > \varepsilon\}$ and $4\delta(X) < \delta_{\varepsilon}(X) $, then there exists a constant $C_0 = C_0(n, \lambda,\Lambda, b, \mathcal{A}, \mathcal{B}) >0$ such that $$
|\nabla u^{\epsilon}(X)| \le C_0. $$ \end{lemma} \begin{proof} Initially we will consider the case when $\delta_{\epsilon} \le \frac{1}{8}$. The following inclusion holds true: $B^{+}_{\frac{\delta_{\epsilon}}{2}}(\hat{X}) \subset B^{+}_{\frac{1}{4}} \setminus \{0 \le u^{\varepsilon} \le \varepsilon\}$. In fact, if $Y \in B^{+}_{\frac{\delta_{\varepsilon}}{2}}(\hat{X})$ then $$
|Y| \le |Y-X| + |X| \le 2 \frac{\delta_{\epsilon}}{2} + |X| \le \frac{1}{4}. $$ Now, using the same argument as in Lemma \ref{prop2.1} (see \eqref{(14)}) we are able to estimate $u^{\epsilon}$ in $B^{+}_{\frac{\delta_{\epsilon}}{2}}(\hat{X})$ as follows $$
u^{\epsilon}(Y) \le u^{\epsilon}(\hat{Y}) + \mathcal{C}^{\ast} \frac{\delta_{\epsilon}}{2} \le \epsilon + c_0 \cdot \textrm{dist}(\hat{Y}, \{0 \le u^{\varepsilon} \le \varepsilon\}) + \mathcal{C}^{\ast} \frac{\delta_{\epsilon}}{2}. $$ Since the distance function is Lipschitz continuous with Lipschitz constant $1$, we have $$
\textrm{dist}(\hat{Y}, \{0 \le u^{\varepsilon} \le \varepsilon\}) \le \delta_{\epsilon} + |\hat{Y}-X| \le 2\delta_{\epsilon}. $$ Therefore, $$
u^{\epsilon}(Y) \le \epsilon + \left(2c_0 + \frac{\mathcal{C}^{\ast}}{2}\right) \delta_{\epsilon} = \epsilon + c \delta_{\epsilon}. $$ By considering the function $v^{\epsilon}(Y) = u^{\epsilon}(Y) - \epsilon$ in $B^{+}_{\frac{\delta_{\epsilon}}{2}}(\hat{X})$, we have that $$
F(Y, \nabla v^{\varepsilon}, D^2 v^{\epsilon}) =0 \quad \textrm{in} \quad B^{+}_{\frac{\delta_{\epsilon}}{2}}(\hat{X}) $$ in the viscosity sense. From $C^{1,\alpha}$ estimate up to boundary and Lemma \ref{ABP}, we have $$
|\nabla u^{\epsilon}(X)| = |\nabla v^{\epsilon}(X)| \le C (c+\mathcal{A}). $$
On the other hand, for the case $\delta_{\epsilon} \ge \frac{1}{8}$ we have the following inclusion $B^{+}_{\frac{1}{16}}(\hat{X}) \subset B_1 \setminus \{0 \leq u^{\varepsilon} \le \varepsilon\}$. In this situation, since $\textrm{supp}(\zeta_{\epsilon})=[0,\epsilon]$, $$ \left\{ \begin{array}{rcl}
F(X, \nabla u^{\varepsilon}, D^2 u^{\epsilon}) = 0 & \mbox{in} & B^{+}_{\frac{1}{16}}(\hat{X})\\ 0 \le u^{\epsilon} = \varphi \le C & \mbox{on} & B^{\prime}_{\frac{1}{16}}(\hat{X}) \end{array} \right. $$ and, consequently, the estimate will follow from $C^{1,\alpha}$ estimates up to the boundary. \end{proof}
An immediate consequence of Theorem \ref{principal2} and Lemma \ref{ABP}is the existence of solutions via compactness in the Lip-Topology for any family $(u^{\varepsilon})_{\varepsilon >0}$ of viscosity solutions to singular perturbation problem \eqref{Equation Pe}. We consequently obtain
\begin{theorem}[{\bf Limiting free boundary problem}]\label{limFB} Let $(u^{\varepsilon})_{\varepsilon >0}$ be a family of solutions to \eqref{Equation Pe}.
For every $\varepsilon_{k} \to 0^{+}$ there exist a subsequence $\varepsilon_{k_j} \to 0^{+}$ and $u_0 \in C^{0, 1}(\overline{\Omega})$ such that \begin{enumerate} \item[{\bf (1)}] $u^{\varepsilon_{k_j}} \to u_0$ uniformly in $\overline{\Omega}$. \item[{\bf (2)}] $F(X,\nabla u_0, D^2 u_0)=0$ in $\overline{\Omega} \cap \{u_0>0\}$ in the viscosity sense. \end{enumerate} \end{theorem}
\section{Appendix}\label{Append}
\hspace{0.6cm}In this final section we are going to give the proof of some technical results, which were temporarily omitted.
\begin{lemma}[{\bf Boundary's estimates propagation Lemma}]\label{lemma2.0} Suppose that $u \ge 0$ is a viscosity solution to $$ \left\{ \begin{array}{rcl}
F(X, \nabla u, D^2 u) = 0 & \mbox{in} & B^{+}_{1}\\
u \ge \sigma >0 & \mbox{on} & B^{\prime}_{1}. \end{array} \right. $$
Then there exists a universal constant $C= C(n,\lambda,\Lambda, b) >0$ such that $$
u(X) \ge C \sigma, \quad X \in B^{+}_{\frac{3}{4}}. $$ \end{lemma}
\begin{proof} First of all consider the following Dirichlet problem \begin{equation} \label{D1} \left\{ \begin{array}{rcccl}
F(X, \nabla w, D^2 w) & = & 0 & \mbox{in} & B^{+}_{1}\\
w & = & \sigma & \mbox{on} & B^{\prime}_1\\ w & = & 0 & \mbox{on} & \partial B_{1} \cap \{X_n >0\}. \end{array} \right. \end{equation} From $C^{1,\alpha}$ regularity estimate, \cite[Theorem 3.1]{Wint} we have $w \in C^{1,\alpha}\left(\overline{B}^{+}_{\frac{3}{4}}\right)$, and, by the Comparison Principle \begin{equation} \label{D3}
0 \le w \le \sigma \quad \textrm{in} \,\,\, B^{+}_{1}. \end{equation} From now on, it is appropriate we define the following reflection $\mathfrak{U} \colon B_1 \rightarrow \mathbb{R}$, \begin{equation} \label{D2} \mathfrak{U}(X) \mathrel{\mathop:}= \left\{ \begin{array}{ccl}
w(X) & \mbox{if} & X \in B^{+}_{1} \cup B^{\prime}_1\\
2 \sigma - w(X_1, \ldots,X_{n-1} -X_n) & \mbox{if} & X \in B_{1} \cap \{X_n <0\}. \end{array} \right. \end{equation} We observe that $\mathfrak{U}$ is a viscosity solution to $$
\mathcal{G}(X, \nabla \mathfrak{U}, D^2 \mathfrak{U})=0 \quad \textrm{in} \quad B_1, $$ where $$ \mathcal{G}(X, \overrightarrow{p}, M) \mathrel{\mathop:}= \left\{ \begin{array}{rcl}
F(X, \overrightarrow{p}, M) & \mbox{if} & X_n \geq 0\\
-F(\widetilde{X}, \overrightarrow{\widetilde{p}}, \widetilde{M}) & \mbox{if} & X_n<0, \end{array} \right. $$ with $$ \begin{array}{rcl}
\widetilde{X} & \mathrel{\mathop:}= & (X_1, \ldots , X_{n-1}, -X_n),\\
\widetilde{p} & \mathrel{\mathop:}= & (-p_1, \ldots, -p_{n-1}, p_n),\\
\widetilde{M} & \mathrel{\mathop:}= & \left\{ \begin{array}{rcl}
-M_{ij} & \mbox{if} & 1 \leq i, j \leq n-1 \,\, \mbox{or} \,\, i=j=n\\
M_{ij} & \mbox{otherwise.} & \end{array} \right. \end{array} $$ Thus, from \eqref{D3}, $$ \sigma \le \mathfrak{U} \le 2 \sigma \quad \mbox{in} \quad B_{1}^{-} $$ Hence, $$ 0 \le \mathfrak{U} \le 2 \sigma \quad \mbox{in} \quad B_1. $$ Moreover, from Harnack inequality we have that $$
\sup_{B_{3/4}} \mathfrak{U} \le c_0 \inf_{B_{3/4}} \mathfrak{U}. $$ Particularly, $$ w(X) \ge c^{-1}_{0} \sigma \quad in \quad B^{+}_{\frac{3}{4}}. $$ Therefore, the proof follows through the previous inequality combined with the Comparison Principle. \end{proof}
\begin{lemma}[{\bf Hopf's type boundary principle}]\label{lemma2.1} Let $u$ be a viscosity solution to $$ \left\{ \begin{array}{rcl}
F(X, \nabla u, D^2 u) = 0 & \mbox{in} & B_{r}(Z)\\
u \ge 0 & \mbox{in} & B_{r}(Z). \end{array} \right. $$ with $r \le 1$. Assume that for some $X_0 \in \partial B_r(Z)$, $$
u(X_0)=0 \quad \textrm{and} \quad \frac{\partial u}{\partial \nu}(X_0) \le \theta, $$ where $\nu$ is the inward normal direction at $X_0$. Then there exists a universal constant $C >0$ such that $$
u(Z) \le C\theta r. $$ \end{lemma}
\begin{proof} By using a scaling argument, we may assume $r=1$. Indeed, it is sufficient to consider the scaled function $v : B_1 \rightarrow \mathbb{R}$ $$
v_r(Y) = \frac{u(Z+r Y)}{r}. $$ As before, $v_r$ is a viscosity solution of $$
F_r(Y,\nabla v_r, D^2 v_r) =0 \quad \textrm{in} \quad B_1, $$ with $$
F_r(Y, \overrightarrow{p}, M) \mathrel{\mathop:}= rF\left(Z+rY, \overrightarrow{p}, \frac{1}{r} M\right) $$ Let $\mathfrak{A} \mathrel{\mathop:}= B_{1} \setminus B_{\frac{1}{2}}$ be an annular region and define $\omega \colon \overline{\mathfrak{A}} \rightarrow \mathbb{R}$ by $$
\omega(Y) \mathrel{\mathop:}= \mu \left(e^{-\delta |Y|^2} - e^{-\delta}\right) $$ where the positive constants $\mu$ and $\delta$ will be chosen \textit{a posteriori}. One can computer the gradient and Hessian of $\omega$ in $\mathfrak{A}$ as follows \begin{eqnarray*}
\partial_{i} \omega(Y) &=& -2 \mu \delta Y_i e^{-\delta |Y|^2},\\
\partial_{ij} \omega(Y) &=& 4 \mu \delta^2 Y_i Y_j e^{-\delta |Y|^2} -2 \mu \delta e^{-\delta |Y|^2} \delta_{ij},\\
|\nabla \omega(Y)| &= & 2\mu \delta e^{-\delta |Y|^2}|Y|. \end{eqnarray*}
In particular, for every $M \in \mathcal{A}_{\lambda,\Lambda}\mathrel{\mathop:}= \left\{A \in \textrm{Sym}(n) \suchthat \lambda \|\xi\|^2 \le \sum\limits_{i,j=1}^{n}A_{ij} \xi_i \xi_j \le \Lambda \|\xi\|^{2}, \, \forall \, \xi \in \mathbb{R}^n\right\}$ we have \begin{eqnarray*}
\textrm{Tr} \left(M \cdot D^2 \omega\right) - b |\nabla \omega| &=& \sum_{i,j=1}^{n} m_{ij} \partial_{ij} \omega - b \cdot \sqrt{\sum_{i=1}^{n} (\partial_{i} \omega)^2} \\
&=& 4 \mu \delta^2 e^{-\delta |Y|^2} \textrm{Tr}(M \cdot Y \otimes Y) - 2 \delta \mu \textrm{Tr}(M) e^{-\delta |Y|^2} - 2 \mu \delta b |Y| e^{-\delta |Y|^2}\\
&\ge& 4 \mu \delta^2 \lambda |Y|^2 e^{-\mu |Y|^2} - 2 \delta \mu n \Lambda e^{-\delta |Y|^2} - 2 \mu \delta b |Y| e^{-\delta |Y|^2}\\
&=& 2 \mu \delta (2 \delta \lambda |Y|^2 - b|Y| - n \Lambda) e^{-\delta |Y|^2} \\
&\ge& 2 \mu \delta \left(\frac{\delta \lambda}{2} - b - n \Lambda\right) e^{-\delta |Y|^2} \quad \textrm{in} \quad \mathfrak{A}, \end{eqnarray*} where $\xi \otimes \xi = (\xi_i \xi_j)_{i,j}$. Choose and fix $\delta \ge \frac{2}{\lambda} (b + n \Lambda)$.
Then, it follows readily that $$
\mathcal{P}_{\lambda, \Lambda}^{-} (D^2 \omega) - b |\nabla \omega| \ge 0 \quad \textrm{in} \quad \mathfrak{A}. $$ Therefore, since $r \le 1$, if $\delta\in \left[\frac{2}{\lambda}(\tilde{b}+ n \Lambda), + \infty\right)$, with $\tilde{b} = r b$, we have $$
F_r(Y, \nabla \omega(Y), D^ 2\omega(Y)) \ge 0 \quad \textrm{in} \quad \mathfrak{A}. $$ Now by Harnack inequality $$
v_r(0) \le \sup_{B_{1/2}} v_r \le c_0 \inf_{B_{1/2}} v_r, $$ Hence $$
v_r(Y) \ge c_o^{-1}v_r(0) \quad \mbox{in} \quad B_{\frac{1}{2}}. $$ By choosing $\mu = \frac{v_r(0)}{c_0\left(e^{-\frac{\delta}{4}} - e^{-\delta}\right)}$ we have $$
\omega \le v_r \quad \mbox{on} \quad \partial \mathfrak{A} $$ and Comparison Principle gives that $$
\omega \le v_r \quad \mbox{in} \quad \mathfrak{A} $$ Thus, if we label $Y_0 \mathrel{\mathop:}= \frac{X_0-Z}{r}$ then $$
\mu\delta e^{- \delta} \le \frac{\partial \omega}{\partial \nu}(Y_0) \le \frac{\partial v_r}{\partial \nu}(Y_0) \leq \theta. $$
Therefore, $$
v_r(0) \le \theta \delta^{-1} c_0 \left(e^{\frac{3\delta}{4}}-1\right), $$ and by returning to the original sentence we can conclude that $$
u(Z) \le c \theta r. $$ \end{proof}
\end{document} | arXiv |
\begin{definition}[Definition:Plane of Reference]
The '''plane of reference''' is a conceptual plane in a spatial context which is arbitrarily distinguished from the others.
Category:Definitions/Solid Geometry
\end{definition} | ProofWiki |
\begin{document}
\title{A note on stochastic Schr\"odinger equations with fractional multiplicative noise}
\begin{abstract}
This work is devoted to non-linear stochastic Schr\"odinger equations with multiplicative fractional noise, where the stochastic integral is defined following the Riemann-Stieljes approach of Z\"ahle. Under the assumptions that the initial condition is in the Sobolev space $H^q(\Rm^n)$ for a dimension $n$ less than three and $q$ an integer greater or equal to zero, that the noise is a $Q-$fractional Brownian motion with Hurst index $H\in(\frac{1}{2},1)$ and spatial regularity $H^{q+4}(\Rm^n)$, as well as appropriate hypotheses on the non-linearity, we obtain the local existence of a unique pathwise solution in ${\mathcal C}^0(0,T,H^q(\Rm^n)) \cap {\mathcal C}^{0,\gamma}(0,T,H^{q-2}(\Rm^n))$, for any $\gamma \in [0,H)$. Contrary to the parabolic case, standard fixed point techniques based on the mild formulation of the SPDE cannot be directly used because of the weak smoothing in time properties of the Schr\"odinger semigroup. We follow here a different route and our proof relies on a change of phase that removes the noise and leads to a Schr\"odinger equation with a magnetic potential that is not differentiable in time. \end{abstract}
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\section{Introduction} \label{sec:intro}
This work is concerned with the existence theory for the stochastic Schr\"odinger equation with fractional multiplicative random noise of the form \begin{equation} \label{SSE} \left\{ \begin{array}{l} d \Psi=i \Delta \Psi dt-i \Psi d B_t^H -ig(\Psi)dt, \qquad t>0, \qquad x \in \Rm^n, \quad n\leq 3\\ \Psi(t=0,\cdot)=\Psi_0, \end{array} \right. \end{equation} where $B_t^H\equiv B^H(t,x)$ is an infinite dimensional fractional Brownian motion in time and smooth in the space variables. The sense of the stochastic integral will be precised later on and the term $g(\Psi)$ is non-linear. We limit ourselves to $n \leq 3$ for physical considerations, but the theory should hold for arbitrary $n$ with adjustments of some hypotheses. Our interest for such a problem is motivated by the study of the propagation of paraxial waves in random media that are both strongly oscillating and slowly decorrelating in the variable associated to the distance of propagation. Such media are encountered for instance in turbulent atmosphere or in the earth's crust \cite{dolan,sidi}. More precisely, it is well-known that the wave equation reduces in the paraxial approximation \cite{TAP} to the Schr\"odinger equation on the enveloppe function $\Psi$, which reads in three dimensions $$ i \partial_z \Psi=-\Delta_\bot \Psi + V(z,x) \Psi, \qquad z \in \Rm^+, \quad x\in \Rm^2, $$ where $z$ is the direction of propagation of the collimated beam, $x=(x_1,x_2)$ is the transverse plane, $\Delta_\bot=\partial_{x_1^2}^2+\partial_{x_2^2}^2$, and $V$ is a random potential accounting for the fluctuations of the refraction index. If $V$ is stationary and its correlation function $R(z,x):=\mathbb E \{V(z+u,x+y) V(u,y)\}$ has the property that $$ R(z,x) \underset{z \to \infty}{\sim} z^{-\alpha} R_0(x), $$ where $R_0$ is a smooth function and $0<\alpha<1$, then the process $V$ presents long-range correlations in the $z$ variable since $R$ is not integrable. Rescaling $V$ as $V \to \varepsilon^{-\frac{\alpha}{2}} V(z/\varepsilon,x)$ and invoking the non-central limit theorem, see e.g. \cite{taqqu}, one may expect formally when $\varepsilon \to 0$ that $$ \frac{1}{\varepsilon^{\frac{\alpha}{2}}} V(\frac{z}{\varepsilon},x) \Psi^\varepsilon(z,x) \tolaw \Psi(z,x) dB^H(z,x), $$ where $B^H$ is a Gaussian process with correlation function \begin{equation} \label{corrB}
\mathbb E\{B^H(z,x+y)B^H(z',x) \}=\frac{1}{2H(2H-1)} (z^{2H}+(z')^{2H}-|z-z'|^{2H})R_0(x), \end{equation} with $H=1-\frac{\alpha}{2} \in (\frac{1}{2},1)$. Proving this fact is an open problem, while the short-range case (when $R$ is integrable) was addressed in \cite{BCF-SIAP-96,Garnier-ITO}, and the limiting wave function $\Psi$ is shown to be a solution to the It\^o-Schr\"odinger equation. Our starting point here is \fref{SSE}, where we added a non-linear term $g(\Psi)$ to account for possible non-linearities arising for instance in non-linear optics.
Let us be more precise now about the nature of the stochastic integral in \fref{SSE}. Since \fref{SSE} is obtained after formal asymptotic limit of a $L^2$ norm preserving Schr\"odinger equation, one may legitimately expect the limiting equation to also preserve the $L^2$ norm. The appropriate stochastic integral should therefore be of Stratonovich type, which in the context of fractional Brownian motions are encountered in the literature as pathwise integrals of various types, e.g. symmetric, forward, or backward, \cite{biagini-book,zahle}. Since in our case of interest the Hurst index $H$ is greater than $\frac{1}{2}$, all integrals are equivalent and can be seen as Riemann-Stieljes integrals of appropriate functions, see \cite{biagini-book,zahle} and section \ref{prelim} for more details. Such an integral is well-defined for instance if both integrands are of H\"older regularity with respective indices $\beta$ and $\gamma$ such that $\beta+\gamma>1$ \cite{zahle}.
In the context of SPDEs, the infinite dimensional character of the Gaussian process is usually addressed within two frameworks, whether for standard or fractional Brownian motions: the $Q-$(fractional) Brownian type, or the cylindrical type, see \cite{daprato}. The first class is more restrictive and requires the correlation operator $Q$ in the space variables to be a positive trace class operator (or even more for fractional Brownian motions, see \cite{mas-nualart}); in the second class, it is only supposed that $Q$ is a positive self-adjoint operator on some Hilbert space with appropriate Hilbert-Schmidt embeddings. As was done in \cite{mas-nualart} for parabolic equations with multiplicative fractional noise, we will assume our noise is of $Q-$fractional type, which yields direct pathwise (almost sure) estimates on $B^H$ in some functional spaces. The cylindrical case is more difficult and our approach does not seem to generalize to it. The $Q-$fractional case actually excludes stationary in $x$ correlation functions of the form \fref{corrB} since they lead to a cylindrical type noise, which is a drawback of our assumptions. This latter case, even in the more favorable situation of parabolic equations, seems to still be open. Standard Brownian motions are more amenable to cylindrical noises since the It\^o isometry holds. In the case of fractional type integrals, the ``It\^o isometry'' involves the Malliavin derivative of the process, which is difficult to handle in the context of SPDEs with multiplicative noise. Hence, an existence theory for the Schr\"odinger equation in some average sense seems more involved to achieve, and we thus focus on a pathwise theory which requires the $Q-$Brownian assumption in our setting.
Stochastic ODEs with fractional Brownian motion were investigated in great generality in \cite{nualart-rasc}. Stochastic PDEs with fractional multiplicative noise are somewhat difficult to study and to the best of our knowledge, the most advanced results in the field are that of Maslowski and al \cite{mas-nualart}, Duncan et al \cite{duncan-mas-mult} or Grecksch et al \cite{grecksch}. The reference \cite{duncan-mas-mult} involves finite dimensional fractional noises, which is a limitation. Several other works deal with additive noise, which is a much more tractable situation as stochastic integrals are seen as Wiener integrals \cite{duncan-mas,tindel,gautier} and cylindrical type noises are allowed. References \cite{mas-nualart,duncan-mas-mult,grecksch} consider variations of parabolic equations of the form \begin{equation} \label{equ} d u= A u dt+u d B_t^H, \end{equation} where $A$ is the generator of an analytic semigroup $S$, and the equation can be complemented with non-linear terms and a time-dependency in $A$ in \cite{duncan-mas-mult}. The noise $B^H$ in these references is a $Q-$fractional Brownian with possibly additional assumptions. The difficulty is naturally to make sense of the term $u d B_t^H$ and to show that $u$ is H\"older in time. In that respect, the analyticity hypothesis is crucial: indeed, the standard technique to analyze \fref{equ} is to use mild solutions of the form $$ u(t)=S(t)u_0+\int_0^t S(t-s)u(s)d B^H_s, $$ and for the integral to exist, one needs the term $S(t-s)u(s)$ to be roughly of H\"older regularity in time with index greater than $1-H$. This means that both $u$ and the semigroup $S$ need such a regularity. While the term $u(s)$ can be treated in the fixed point procedure, the semigroup $S(t-s)$ has to be sufficiently smooth in time, which holds for analytic semigroups, but not in the case of $C_0$ unitary groups generated by $i \Delta$ in the Schr\"odinger equation. In the latter situation, one can ``trade'' some regularity in time for $S$ with some spatial regularity on $u$, but this procedure does not seem to be exploitable in a fixed point procedure. Another possibility could be to take advantage of the regularizing properties of the Schr\"odinger semigroup that provide a gain of almost half a derivative in space, and therefore to almost a quarter of a derivative in time \cite{cazenave}. It looked to us rather delicate to follow such an approach since the smoothing effects hold for particular topologies involving spatial weights which looked fairly intricate to handle in our problem, even by using the classical exchange regularity/decay for the Schr\"odinger equation. The strictly linear case (i.e. when $g=0$ in \fref{SSE}) can likely be treated by somewhat brute force with iterated Wiener integrals and the Hu-Meyer formula, but this approach does not carry on to the non-linear setting.
We propose in this work a different route than the mild formulation and a quite simple remedy based on two direct observations: (i) the usual change of variables formula holds for the pathwise stochastic integral and (ii) using it along with a change of phase removes the noise and leads to a Schr\"odinger equation with magnetic vector potential $A(t,x)=-\nabla B^H(t,x)$. Forgetting for the moment the non-linear term $g(\Psi)$, introducing $\varphi(t,x):=e^{i B^H(t,x)} \Psi(t,x)$ the filtered wavefunction, and supposing without lack of generality that $B^H(0,x)=0$, this yields the system \begin{equation} \label{SEM} \left\{ \begin{array}{l} i \partial_t \varphi = -\Delta_{B_t^H} \varphi, \qquad \Delta_{B_t^H} =e^{i B_t^H}\circ \Delta \circ e^{-i B_t^H}\\ \varphi(t=0,\cdot)=\Psi_0, \end{array} \right. \end{equation} which is a standard Schr\"odinger equation with a time-dependent Hamiltonian. There is a vast literature on the subject, see \cite{RS-80-2, debouard_mag,kato0, katolinear,nakamura, michel,yajima,yajima2,zagrebnov} for a non-exhaustive list. One of the most classical assumptions on $A$ for the existence of an evolution operator generated by the Hamiltonian $\Delta_{B_t^H}$ is that $A$ is a ${\mathcal C}^1$ function in time with values in $H^1(\Rm^n)$. This is of course not verified for the fractional Brownian motion. The price to pay for that is to require additional spatial regularity, and one possibility (likely not optimal) is to suppose that $B^H$ has values in $H^4(\Rm^n)$. Assuming such a strong regularity is naturally a drawback in this approach.
Regarding the treatment of the non-linearity, we suppose that it is invariant by a change of phase, that is $e^{iB^H_t} g(\Psi)=g(\varphi)$, which is verified by power non-linearities of the form $g(\Psi)=|\Psi^\sigma| \Psi$ or by $g(\Psi)=V[\Psi] \Psi$ where $V$ is the Poisson potential. Contrary to the case of non-linear It\^o-Schr\"odinger equations where various tools such as Strichartz estimates or Morawetz estimates have been successfully used to investigate focusing/defocusing phenomena in random and deterministic settings \cite{cazenave,debu2,debu3,debu4}, there are very few available techniques to study \fref{SEM} with a potential vector $A$ not smooth in time and augmented with the term $g$. There are Strichartz estimates in the context of magnetic Schr\"odinger equations, but some require $A$ to be ${\mathcal C}^1$ in time \cite{yajima}, and some others avoid such an hypothesis but assume instead that $A$ is small in some sense \cite{stefanov_mag}, which has no reason to hold here. As a result, we are lead to make rather crude assumptions on $g$ in order to obtain a local existence result. Moreover, the analysis of non-linear Schr\"odinger equations generally relies in a crucial manner on energy methods. In our problem of interest, we are only able to obtain energy conservation for smooth solutions, which turns out to be of no use when trying to obtain a global-in-time result and limits us to local results, unless the non-linearity is globally Lipschitz in the appropriate topology. This is due to the fractional noise that does not allow us to obtain $H^1$ estimates for $\Psi$ via the energy relation, as we explain further in remark \ref{ener}.
The main result of the paper is therefore a local existence result of pathwise solutions to \fref{SSE} with a smooth $Q-$fractional noise $B^H_t$ and appropriate assumptions on the non-linearity $g$. The article is structured as follows: in section \ref{prelim}, we recall basic results on fractional stochastic integration, and present our main result in section \ref{main}; section \ref{secmag} is devoted to the magnetic Schr\"odinger equation \fref{SEM}, while section \ref{back} concerns the proof of our main theorem.
\section{Preliminaries} \label{prelim} \noindent \textbf{Notation.} We denote by $H^k(\Rm^n)$ and $W^{k,q}(\Rm^n)$, $1 \leq n \leq 3$, the standard Sobolev spaces with the convention that $H^0(\Rm^n):=L^2(\Rm^n)$. For a Banach space $V$, $T>0$, and $0<\alpha<1$, $W_{\alpha,1}(0,T,V)$
denote the space or mesurable functions $f: [0,T] \to V$ equipped with the norm $$
\|f \|_{\alpha,1,V}=\int_0^T\left(\frac{\|f(s)\|_V}{s^\alpha}+\int_0^s\frac{\|f(s) -f(\tau)\|_V}{(s-\tau)^{\alpha+1}} d\tau \right) ds. $$
The space ${\mathcal C}^{0,\alpha}(0,T,V)$ denotes the classical H\"older space of functions with values in $V$. When $V=\Cm$ or $\Rm$, we will simply use the notations $W_{\alpha,1}(0,T)$, ${\mathcal C}^{0,\alpha}(0,T)$ and $\|\cdot \|_{\alpha,1}$. Notice that for any $\varepsilon>0$, ${\mathcal C}^{0,\alpha+\varepsilon}(0,T,V) \subset W_{\alpha,1}(0,T,V)$. For two Banach spaces $U$ and $V$, ${\mathcal L}(U,V)$ denotes the space of bounded operators from $U$ to $V$, with the convention ${\mathcal L}(U)={\mathcal L}(U,U)$. The $L^2$ inner product is denoted by $(f, g)=\int_{\Rm^n} \overline{f}g dx$ where $\overline{f}$ is the complex conjugate of $f$.\\
\noindent \textbf{Fractional Brownian motion.} For some positive time $T$, we denote by $\beta^H=\{ \beta^H(t), \; t \in [0,T]\}$ a standard fractional Brownian motion (fBm) over a probability space $(\Omega, {\mathcal F},\mathbb P)$ with Hurst index $H \in (\frac{1}{2},1)$. We will denote by $L^2(\Omega)$ the space of square integrable random variables for the measure $\mathbb P$ and will often omit the dependence of $\beta^H$ on $\omega \in \Omega$ for simplicity. The process $\beta^H$ is a centered Gaussian process with covariance $$
\mathbb E\{ \beta^H_t\beta^H_s \}=\frac{1}{2} (t^{2H}+s^{2H}-|t-s|^{2H}). $$
Since $\mathbb E\{ (\beta^H_t-\beta^H_s)^2\}=|t-s|^{2H}$, $\beta^H$ admits a H\"older continuous version with index strictly less than $H$. In order to definite the infinite dimensional noise $B^H(t,x)$, consider a sequence of independent fBm $(\beta_n^H)_{n \in \Nm}$. Let $Q$ be a positive trace class operator on $L^2(\Rm^n)$ and denote by $(\mu_n, e_n)_{n \in \Nm}$ its spectral elements. For $V=H^{q+4}(\Rm^n)$, $q$ non-negative integer, and $\lambda_n=\sqrt{\mu_n}$, we assume that \begin{equation} \label{assumQ}
\sum_{p \in \Nm} \lambda_p \|e_p\|_V < \infty. \end{equation} The process $B^H(t,x)$ is then formally defined by $$ B^H(t,x):=\sqrt{Q} \sum_{p \in \Nm} e_p(x) \beta^H_p(t)=\sum_{p \in \Nm} \lambda_p e_p(x) \beta^H_p(t). $$ The sum is normally convergent in ${\mathcal C}^{0,\gamma}(0,T,V)$, $\mathbb P$ almost surely for $0\leq \gamma<H$. Indeed, in the same fashion as \cite{mas-nualart}, let
$$
K(\omega)=\sum_{p \in \Nm} \lambda_p \|e_p\|_V \|\beta^H_p(\cdot, \omega)\|_{C^{0,\gamma}(0,T)} $$ so that by monotone convergence $$
\mathbb E K=\sum_{p \in \Nm} \lambda_p \|e_p\|_V \mathbb E \|\beta^H_p\|_{C^{0,\gamma}(0,T)}. $$
According to \cite{nualart-rasc} Lemma 7.4, for every $T>0$ and $\varepsilon>0$, there exists a positive random variable $\eta_{\varepsilon,T,p}$ where $\mathbb E\{|\eta_{\varepsilon,T,p}|^q \}$ is finite for $1 \leq q < \infty$ and independent of $p$ since the $\beta_p^H$ are identically distributed, such that $|\beta^H_p(t)-\beta^H_p(s)| \leq \eta_{\varepsilon,T,p} |t-s|^{H-\varepsilon}$ almost surely. Hence, thanks to \fref{assumQ} and picking $\gamma=H-\varepsilon$, we have $\mathbb E K<\infty$, \begin{equation} \label{defK} K(\omega) < \infty, \quad \mathbb P \quad \textrm{almost surely}, \end{equation} and $B^H$ defines almost surely an element of ${\mathcal C}^{0,\gamma}(0,T,V)$. As a contrast, a cylindrical fractional Brownian motion is defined for a positive self-adjoint $Q$, which does not provide us with almost sure bounds on $B^H$ in ${\mathcal C}^{0,\gamma}(0,T,V)$. Suppose indeed that $\sqrt{Q}$ is a convolution operator of the form $\sqrt{Q} u= g*u$ for some smooth real-valued kernel $q$ and that $(e_p)_{p \in \Nm}$ is a real-valued basis of $L^2(\Rm^n)$. Then, the resulting correlation function is stationary (this follows from the convolution and is motivated by \fref{corrB}) and $$
\mathbb E \{(B^H(t,x)-B^H(s,x))^2\}= |t-s|^{2H}\sum_{p \in \Nm} (g*e_p(x))^2=|t-s|^{2H} \|g\|^2_{L^2} $$ so that $B^H$ belongs to ${\mathcal C}^{0,\gamma}(0,T,L^\infty(\Rm^n, L^2(\Omega)))$ for $0\leq \gamma \leq H$. As explained in the introduction, we are not able to handle such a noise since integration in the probability space is required beforehand in order to get some estimates. This is not an issue in the context of standard Brownian motions or additive fractional noise, but leads to technical difficulties here.\\
\noindent \textbf{Fractional stochastic integration.} We follow the approach of \cite{mas-nualart,nualart-rasc} based on the work of Z\"ahle \cite{zahle} and introduce the so-called Weyl derivatives defined by, for any $\alpha \in (0,1)$ and $t \in (0,T)$: \begin{eqnarray*} D_{0+}^\alpha f(t)&=&\frac{1}{\Gamma(1-\alpha)} \left(\frac{f(t)}{t^\alpha}+\alpha \int_0^t \frac{f(t) -f(s)}{(t-s)^{\alpha+1}} ds \right)\\ D_{T-}^\alpha f(t)&=&\frac{(-1)^\alpha}{\Gamma(1-\alpha)} \left(\frac{f(t)}{(T-t)^\alpha}+\alpha \int_t^T \frac{f(t) -f(s)}{(s-t)^{\alpha+1}} ds \right), \end{eqnarray*} whenever these quantities are finite. Above, $\Gamma$ stands for the Euler function. Following \cite{zahle}, the generalized Stieljes integral of a function $f \in {\mathcal C}^{0,\lambda}(0,T)$ against a function $g \in {\mathcal C}^{0,\gamma}(0,T)$ with $\lambda+\gamma>1$, $\lambda>\alpha$ and $\gamma>1-\alpha$ is defined by \begin{equation} \label{stiel} \int_0^T f dg:=(-1)^\alpha \int_0^T D_{0+}^\alpha f(s) D_{T-}^{1-\alpha} g_{T-}(s) ds, \end{equation} with $g_{T-}(s)=g(s)-g(T-)$. The definition does not depend on $\alpha$ and $$ \int_0^t f dg:=\int_0^T f {\mathbbmss{1}}_{(0,t)} dg. $$ The integral can be extended to different classes of functions since, see \cite{nualart-rasc}, \begin{equation} \label{estimint}
\left|\int_0^T f dg \right| \leq \|f \|_{\alpha,1} \Lambda_\alpha(g), \end{equation} where $$
\Lambda_\alpha(g):=\frac{1}{\Gamma(1-\alpha) \Gamma(\alpha)} \sup_{0<s<t<T} \left(\frac{|g(t)-g(s)|}{(t-s)^{1-\alpha}}+\alpha \int_s^t \frac{|g(\tau) -g(s)|}{(\tau-s)^{2-\alpha}} d\tau\right), $$ so that the integral is well-defined if $f \in W_{\alpha,1}(0,T)$ and $\Lambda_\alpha(g)<\infty$. Besides, the fractional integral satisfies the following change of variables formula, see \cite{zahle}: let $F \in {\mathcal C}^1(\Rm \times [0,T])$, $g \in {\mathcal C}^{0,\lambda}(0,T)$ and $\partial_1 F(g(\cdot),\cdot) \in {\mathcal C}^{0,\gamma}(0,T)$ with $\lambda+\gamma>1$, then \begin{equation} \label{chain} F(g(t),t)-F(g(s),s)=\int_s^t \partial_2 F(g(\tau),\tau) d\tau+ \int_s^t \partial_1 F(g(\tau),\tau) d g(\tau), \end{equation} where $\partial_j F$, $j=1,2$ denotes the partial derivative of $F$ with respect to the $j$ coordinate.
For some Banach space $U$ and an operator-valued random function $F \in W_{\alpha,1}(0,T,{\mathcal L}(V,U))$ almost surely for some $\alpha \in (1-H,\frac{1}{2})$, the stochastic integral of $F$ with respect to $B^H$ is then formally defined by \begin{equation} \label{defintsto} \int_0^t F_s d B_s^H:=\sum_{p \in \Nm} \lambda_p \int_0^t F_s(e_p) d\beta_p^H(s). \end{equation} The integral defines almost surely an element of $U$ for all $t\in[0,T]$ since by Jensen's inequality for the second line \begin{eqnarray*}
\sum_{p \in \Nm} \lambda_p \left\| \int_0^t F_s(e_p) d\beta_p^H(s) \right\|_U &\leq& \sum_{p \in \Nm} \lambda_p \Lambda_\alpha(\beta^H_p) \left\| \|F_s(e_p) \|_{\alpha,1} \right\|_U \\
&\leq& C \|F_s\|_{\alpha,1,{\mathcal L}(U,V)} \sum_{p \in \Nm} \lambda_p \|e_p\|_V \Lambda_\alpha(\beta_p^H) \end{eqnarray*} and as shown in \cite{mas-nualart}, \begin{equation} \label{finitelamb}
\sum_{p \in \Nm} \lambda_p \|e_p\|_V \Lambda_\alpha(\beta_p^H(\cdot,\omega)) <\infty \qquad \mathbb P \quad \textrm{almost surely}. \end{equation} Hence \fref{defintsto} is well-defined and the convergence of the sum has to be understood as the $\mathbb P$ almost sure convergence in $U$.
We will use the following two results: the first Lemma is a generalization of the change of variables formula \fref{chain} to the infinite dimensional setting, and the second a version a the Fubini theorem adapted to the stochastic integral. Their proofs are given in the appendix. Below, $V=H^{q+4}(\Rm^n)$. \begin{lemma} \label{chain2} Let $F: V \times [0,T] \to \Cm$ be a continuously differentiable function. Let $\partial_1 F$ be the differential of $F$ with respect to the first argument and $\partial_2 F$ be its partial derivative with respect to the second. For every $v \in V$ and $B \in {\mathcal C}^{0,\gamma}(0,T,V)$ for any $0\leq
\gamma<H$, let $\phi(t):=\partial_1 F(B_t,t)(v)$. Assume that $\phi \in {\mathcal C}^{0,\lambda}(0,T)$ with $\lambda+\gamma>1$, and that there exists a constant $C_M>0$ such that, for all $B$ with $\|B\|_{{\mathcal C}^{0,\gamma}(0,T,V)} \leq M$: \begin{equation} \label{hypphi}
\|\phi\|_{{\mathcal C}^{0,\lambda}(0,T)} \leq C_M \|v\|_V. \end{equation}
Then, we have the change of variables formula, $\forall (s,t) \in [0,T]^2$, $\mathbb P$ almost surely: $$ F(B_t^H,t)-F(B_s^H,s)=\int_s^t \partial_2 F(B_{\tau}^H,\tau) d\tau+ \sum_{p \in \Nm} \lambda_p \int_s^t \partial_1 F(B_{\tau}^H,\tau)(e_p) d \beta^H_p(\tau). $$ \end{lemma}
\begin{lemma} \label{fubini}Let $F \in W_{\alpha,1}(0,T,{\mathcal L}(V,L^1(\Rm^n)))$ with $1-H<\alpha<\frac{1}{2}$. Then we have:
$$ \sum_{p \in \Nm} \lambda_p \int_s^t \left(\int_{\Rm^n} F_{\tau,x}(e_p) dx \right) d \beta^H_p(\tau) =\int_{\Rm^n} \left(\int_s^t F_{\tau,x} dB_\tau^H \right)dx. $$ \end{lemma}
\section{Main result} \label{main}
We present in this section the main result of the paper. We precise first in which sense \fref{SSE} is understood. We say that $\Psi \in {\mathcal C}^0(0,T,H^q(\Rm^n)) \cap {\mathcal C}^{0,\gamma}(0,T,H^{q-2}(\Rm^n))$, for all $0\leq \gamma<H$, $q$ non-negative integer, is a solution to \fref{SSE} if it verifies for all test function $w\in {\mathcal C}^1(0,T,H^{q+2}(\Rm^n))$, for all $t \in [0,T]$ and $\mathbb P$ almost surely \begin{align} \label{defsol} \nonumber &\left( \Psi(t), w(t) \right)-\left( \Psi_0, w(0) \right) =\int_0^t \left( \Psi(s), \partial_s w(s) \right) ds \\[3mm] & -i\int_0^t \left( \Psi(s),\Delta w(s)\right) ds +i \int_0^t\left(\Psi(s), w(s) d B_s^H\right)+i\int_0^t \left(g(\Psi(s)),w(s) \right) ds, \end{align} where the term involving the stochastic integral is understood as $$ \int_0^t\left(\Psi(s), w(s) d B_s^H \right):=\sum_{p \in \Nm} \lambda_p \int_0^t\left(\Psi(s)e_p, w(s)\right) d \beta^H_p(s). $$ The latter is well-defined since the mapping $F_s: e_p \mapsto (\Psi e_p, w)$ belongs to $\in {\mathcal C}^{0,\gamma}(0,T,{\mathcal L}(V,\Rm))$ thanks to standard Sobolev embeddings for $n \leq 3$. We assume the following hypotheses on the non-linear term $g$:\\
\textbf{H}: We have $g( e^{i \theta(t,x)} \Psi)= e^{i \theta(t,x)} g(\Psi)$ for all real function $\theta$, and for any $\Psi_1, \Psi_2$ in $H^q(\Rm^n)$ with $\|\Psi_i\|_{H^s}\leq M$, $i=1,2$, there exist $p \in \{0,\cdots,q\}$ and positive constants $C_M$ and $C'_M$ such that \begin{eqnarray*}
\| g(\Psi_1) \|_{H^q} &\leq& C_M \|\Psi\|_{H^q}\\
\| g(\Psi_1)-g(\Psi_2) \|_{H^p} &\leq& C'_M \|\Psi_1-\Psi_2\|_{H^p}. \end{eqnarray*}
The main result of this paper is the following: \begin{theorem} \label{th1} Assume that \textbf{H} is satisfied. Suppose moreover that \fref{assumQ} is verified for $V=H^{q+4}(\Rm^n)$, $q$ non-negative integer. Then, for every $\Psi_0 \in H^q(\Rm^n)$, there exists a maximal existence time $T_M>0$ and a unique function $\Psi \in {\mathcal C}^0(0,T_M,H^{q}(\Rm^n)) \cap{\mathcal C}^{0,\gamma}(0,T_M,H^{q-2}(\Rm^n))$, $0\leq \gamma <H$, verifying \fref{defsol} for all $t\in[0,T_M]$ $\mathbb P$ almost surely. Moreover, $\Psi$ admits the following representation formula: \begin{equation} \label{repre_th} \Psi(t)= e^{-i B^H_t} U(t,0) \Psi_0+ e^{-i B^H_t} \int_0^t U(t,s) e^{i B^H_s} g(\Psi(s))ds, \end{equation} where $U=\{U(t,s)\}$ is the evolution operator generated by the operator $$ i\Delta_{B_t^H} =i e^{i B_t^H}\circ \Delta \circ e^{-i B_t^H}. $$ If in addition $\Im g(\Psi) \overline{\Psi}=0$, then for all $t\in[0,T_M]$ the charge conservation holds: $$
\| \Psi(t)\|_{L^2}=\| \Psi(0)\|_{L^2}. $$ If $g$ is globally Lipschitz in $H^q(\Rm^n)$, then the solution exists for all time $T<\infty$. \end{theorem}
When $d=3$, a classical example of a non-linearity satisfying \textbf{H} for $q=p=1$ is $g(\Psi)=V[\Psi] \Psi$, where $V[\Psi]$ is the Poisson potential defined by $$
V[\Psi](x)=\int_{\Rm^3} \frac{|\Psi(y)|^2}{|x-y|}dy. $$ Indeed, $g$ is locally Lipschitz in $H^1(\Rm^3)$: let $\Psi_1, \Psi_2 \in H^1(\Rm^n)$; thanks to the Hardy-Littlewood-Sobolev inequality \cite{RS-80-2}, Chapter IX.4, as well as standard Sobolev embeddings, we have $$
\|\nabla V[\Psi_1] -\nabla V_2[\Psi_2] \|_{L^3} \leq C \| |\Psi_1|^2-|\Psi_2|^2 \|_{L^{\frac{3}{2}}} \leq C \| \Psi_1 -\Psi_2 \|_{L^2}\| \Psi_1 +\Psi_2 \|_{H^1} $$ and direct computations yield $$
\| V [\Psi_1] \|_{L^\infty} \leq C \|\Psi_1\|^2_{L^2}+C\||\Psi_1|^2\|_{L^2}. $$ Hence, \begin{align*}
&\| g(\Psi_1)-g(\Psi_2)\|_{H^1} \\
&\qquad \leq C\|V[\Psi_1] \|_{L^\infty} \|\Psi_1 -\Psi_2\|_{L^2}+ C\|\Psi_1-\Psi_2 \|_{H^1}\|\Psi_1+\Psi_2 \|_{H^1}\|\Psi_2\|_{L^2}\\
&\qquad \qquad +C\|V[\Psi_1] \|_{L^\infty} \|\nabla \Psi_1 - \nabla \Psi_2\|_{L^2}+C\| \Psi_1 -\Psi_2 \|_{L^2}\| \Psi_1 +\Psi_2 \|_{H^1} \|\Psi_2 \|_{L^6}\\
&\qquad \leq C( \| \Psi_1\|^2_{H^1}+ \| \Psi_2\|^2_{H^1}) \|\Psi_1-\Psi_2 \|_{H^1}. \end{align*}
Another example is given by power non-linearities of the form $g(\Psi)=\mu |\Psi|^{2\sigma} \Psi$ for some $\mu \in \Rm$ and $\sigma>0$. A $L^\infty$ bound is needed on $\Psi$ for \textbf{H} to be verified. When $n>1$, we set then $q=2$ and obtain, for all $\sigma\geq \frac{1}{2}$: $$
\| g(\Psi) \|_{H^2} \leq C \| \Psi \|^{2\sigma+1}_{H^2}, \qquad \| g(\Psi_1)-g(\Psi_2) \|_{L^2} \leq C \| \Psi_2 +\Psi_1\|^{2\sigma}_{H^2}\| \Psi_1-\Psi_2 \|_{L^2}, $$ while it can be easily shown that $\textbf{H}$ is verified for $n=1$ and $q=1$ for all $\sigma \geq 0$.
\begin{remark} \label{ener} In order to both lower the spatial regularity assumptions on $B^H$, $\Psi_0$, $g$ and to obtain global-in-time results, it is natural to consider the energy conservation identity (derived formally by multiplying \fref{mag} by $\overline{\partial_t \varphi}$ and integrating, and can be justified for classical solutions when $q \geq 2$ using the regularity of $\varphi$ of Theorem \ref{th_mag} and Lemma \ref{chain2}) that reads for $g=0$ for simplicity: $$
\frac{1}{2} \| \nabla \Psi(t) \|^2_{L^2}=\frac{1}{2} \| \nabla \Psi_0 \|^2_{L^2}-\Im \int_0^t \int_{\Rm^n} \overline{\Psi(s)} \nabla \Psi(s) \cdot \nabla d B^H_s dx. $$
Unfortunately, it is not clear to us how this identity can be used in order to obtain estimates on $\| \nabla \Psi\|_{W_{\alpha,1}(0,T,L^2)}$ for $1-H<\alpha<\frac{1}{2}$ that would depend only on $\| \nabla \Psi_0 \|_{L^2}$ and $\| B^H\|_{{\mathcal C}^{0,\gamma}(0,T,W^{1.\infty})}$, $\frac{1}{2}<\gamma<H$. Indeed, following the lines of the stochastic ODE case of \cite{nualart-rasc} in order to treat the stochastic integral and use the Gronwall Lemma, what can be deduced from the above relation is an estimate of the form $$
\left\| \| \nabla \Psi(t,\cdot) \|^2_{L^2} \right\|_{W_{\alpha,1}(0,T)} \leq C+C \int_0^T f(s) \| \nabla \Psi(s,\cdot) \|^2_{_{W_{\alpha,1}(0,T,L^2)}}ds $$
for some positive integrable function $f$ and where the constant $C$ depends on $\| \nabla \Psi_0 \|_{L^2}$ and $\| B^H\|_{{\mathcal C}^{0,\gamma}(0,T,W^{1.\infty})}$.
This does not yield the desired bound since we cannot control the term $\| \nabla \Psi(s,\cdot) \|^2_{_{W_{\alpha,1}(0,T,L^2)}}$ by $\left\| \| \nabla \Psi(s,\cdot) \|^2_{L^2} \right\|_{W_{\alpha,1}(0,T)}$. Hence, as opposed to the standard Brownian case, energy methods do not provide us here with an $H^1$ global-in-time estimate. \end{remark} \begin{remark} \label{rem2} When $q \geq 2$, then $\Psi$ is a classical solution to \fref{SSE} in the sense that it satisfies for all $t \in [0,T_m]$, $\mathbb P$ a.s., $x$ a.e.: $$ \Psi(t)=\Psi(0)+i \int_0^t \Delta \Psi(s) ds-i \int_0^t \Psi(s) dB_s^H-i\int_0^t g(\Psi(s)) ds. $$ A proof of this result is given in the appendix. \end{remark}
The rest of the paper is devoted to the proof of Theorem \ref{th1}. The starting point is to define $\varphi(t,x)=e^{i B^H(t,x)} \Psi(t,x)$, to use the invariance of $g$ with respect to a change of phase and to formally apply Lemma \ref{chain2} to arrive at \begin{equation} \label{MSE2} i\partial_t \varphi = -\Delta_{B_t^H} \varphi+g(\varphi). \end{equation} Remark that $\Delta_{B_t^H}$ can formally be recast as $$
\Delta_{B_t^H} =\Delta -2i \nabla B_t^H \cdot \nabla -|\nabla B_t^H|^2 -i \Delta B_t^H . $$ In section \ref{secmag}, we construct the evolution operator $U=\{U(t,s)\}$ generated by $i\Delta_{B_t^H}$ and obtain the existence of a unique solution to the latter magnetic Schr\"odinger equation. In section \ref{back}, we use the regularity properties of the function $\varphi$ together with Lemma \ref{chain2} to prove that $\Psi=e^{-i B^H_t}\varphi$ is the unique solution to \fref{SSE}. The existence follows from showing that $e^{-i B^H_t}\varphi$ is a solution to \fref{defsol}. The uniqueness stems from a reverse argument: owing a solution $\Psi$ to $\fref{defsol}$ with the corresponding regularity, we show that $\Psi e^{i B^H_t}$ is a solution to \fref{MSE2}. This requires some regularization since the function $ e^{-i B^H_t} z$ for $z$ smooth cannot be used as a test function in \fref{defsol}, as well as the interpretation of a classical integral involving a full derivative as a fractional integral.
\section{Existence theory of the magnetic Schr\"odinger equation} \label{secmag} The first part of this section consists in constructing the evolution operator $U$. We follow the classical methods of Kato \cite{katolinear} and \cite{pazy}. The second part is devoted to the existence theory for the linear magnetic Schr\"odinger equation, which is then used for the non-linear case.
\subsection{Construction of the evolution operator}
We follow here the construction of \cite{pazy}, Chapter 5. Let $X$ and $Y$ be Banach spaces with norms $\|\cdot \|$ and $\| \cdot \|_Y$, where $Y$ is densely and continuously embedded in $X$. For $t \in [0,T]$, let $A(t)$ be the infinitesimal generator of a $C_0$ semigroup on $X$. Consider the following hypotheses: \begin{itemize} \item[(H1)] $\{ A(t)\}_{t\in [0,T]}$ is such that there are constants $\omega_0$ and $M\geq 1$, where $]\omega_0,\infty[ \subset \rho(A(t))$ for $t \in [0,T]$, $\rho(A(t))$ denoting the resolvent set of $A(t)$, and $$
\left\|\prod_{j=1}^k e^{- s_j A(t_j )}\right \| \leq M e^{\omega_0 \sum_{j=1}^k s_j }, \qquad s_j \geq 0, \qquad 0 \leq t_1 \leq t_2 \leq \dots \leq T. $$ \item[(H2)] There is a family $\{ Q(t)\}_{t\in [0,T]}$ of isomorphisms of $Y$ onto $X$ such that for every $y\in Y$, $Q(t)v$ is continously differentiable in $X$ on $[0,T]$ and $$ Q(t) A(t) Q(t)^{-1}=A(t)+C(t) $$ where $C(t)$, $0 \leq t \leq T$, is a strongly continuous family of bounded operators on $X$. \item[(H3)] For $t\in [0,T]$, $Y \subset D(A(t))$, $A(t)$ is a bounded operator from $Y$ into $X$ and $t \to A(t)$ is continuous in the ${\mathcal L}(Y,X)$ norm. \end{itemize} We then have the following result, see \cite{katolinear}, or \cite{pazy}, Chapter 5, Theorems 2.2 and 4.6: \begin{theorem}\label{kato} Assume that (H-1)-(H-2)-(H-3) are verified. Then, there exists a unique evolution operator $U=\{U(t,s)\}$, defined on the triangle $\Delta_T:T\geq t\geq s\geq 0$ such that \begin{itemize} \item[(a)] $U$ is strongly continuous on $\Delta_T$ to ${\mathcal L}(X)$, with $U(s,s)=I$, \item[(b)] $U(t,r)U(r,s)=U(t,s)$, \item[(c)] $U(t,s) Y \subset Y$, and $U$ is strongly continuous on $\Delta_T$ to ${\mathcal L}(Y)$, \item[(d)] $dU(t,s)/dt=-A(t)U(t,s)$, $dU(t,s)/ds=U(t,s)A(s)$, which exist in the strong sense in ${\mathcal L}(Y,X)$, and are strongly continuous $\Delta_T$ to ${\mathcal L}(Y,X)$. \end{itemize} \end{theorem} In the next result, we show that for suitable functions $B$, the operator $i \Delta_B=i e^{iB} \circ \Delta \circ e^{-iB}$ generates an evolution operator $U$. \begin{proposition} \label{geneevol} Let $X=L^2(\Rm^n)$ and $Y=H^{2k}(\Rm^n)$, $k \geq 1$, and let $B \in {\mathcal C}^0(0,T,H^{2k+2}(\Rm^n))$. Then, the operator $i \Delta_B$ generates an evolution operator $U$ satisfying Theorem \ref{kato} and $U$ is an isometry on $L^2(\Rm^n)$. \end{proposition} \begin{proof} We verify hypotheses (H-1)-(H-2)-(H-3) for $A(t)=i \Delta_B$. Let $\Delta_{B_t} := \Delta +L(t)$ with \begin{equation} \label{defL}
L(t)=-2i \nabla B_t \cdot \nabla -|\nabla B_t|^2 -i \Delta B_t .\end{equation}
First, for $t$ fixed in $[0,T]$, the Kato-Rellich theorem \cite{RS-80-2} yields that $\Delta_{B_t}$ is self-adjoint on $D(\Delta)=H^2(\Rm^n)$. Indeed, using the regularity $B \in {\mathcal C}^0([0,T],H^4(\Rm^n))$, it is straightforward to verify that $L(t)$ is symmetric and $\Delta$-bounded with relative bound strictly less than one. We also obtain that $D(\Delta_{B_t})=H^2(\Rm^n)$, $\forall t \in [0,T]$. Stone's theorem \cite{RS-80-I} then implies that for $t$ fixed, $i \Delta_{B_t}$ is the generator of a $C_0$ unitary group on $X$. Moreover, $-\Delta_{B_t}$ is positive, so that the spectrum of $i \Delta_{B_t}$ lies in $i[0,\infty)$. We therefore conclude that the family $\{i \Delta_{B_t} \}_{t\in [0,T]}$ satisfies hypothesis (H-1).
Regarding (H-2), let $Q=\Delta_{(k)}+I$, where $I$ is the identity operator and $$\Delta_{(k)}=(-1)^k \sum_{j=1}^n \partial^{2k}_{x_j^{2k}}, \qquad k \geq 1.$$The operator $Q$ is a positive definite self-adjoint operator on $H^{2k}(\Rm^n)$, and an isomorphism from $H^{2k}(\Rm^n)$ to $L^2(\Rm^2)$. It is also obviously continuously differentiable since it does not depend on $t$. Moreover, $$ Q \Delta_{B_t} Q^{-1}=\Delta_{B_t}+[Q,\Delta_{B_t}]Q^{-1}, $$ where $[A,B]$ denotes the commutator between two operators $A$ and $B$. We have the following Lemma: \begin{lemma} For $k \geq 1$, let $B \in {\mathcal C}^0([0,T],H^{2k+2}(\Rm^n))$. Then $[Q,\Delta_{B_t}]Q^{-1} \in {\mathcal L}(L^2(\Rm^n))$. \end{lemma}
\begin{proof} We have $[Q,\Delta_{B_t}]Q^{-1}=[\Delta_{(k)},L(t)]Q^{-1}$, and using the product rule \begin{align*} &[\Delta_{(k)},L(t)]=\\ &(-1)^k \sum_{j=1}^n \sum_{p=0}^{2k-1} \left( \begin{array}{c} 2k \\p \end{array} \right)
\left(-2i \{\nabla \partial^{2k-p}_{x^{2k-p}_j} B \}\cdot \nabla \partial^p_{x^p_j} - \{\partial^{2k-p}_{x^{2k-p}_j}|\nabla B|^2 \}\partial^p_{x^p_j}-i\{ \Delta \partial^{2k-p}_{x^{2k-p}_j} B \}\partial^p_{x^p_j} \right) \end{align*} where $\tiny{\left( \begin{array}{c} 2k \\p \end{array} \right)}$ is the binomial coefficient and there are as usual no terms corresponding to $p=2k$ because of the commutator. Using Standard Sobolev embeddings for $H^{2k+2}(\Rm^n)$ when $n \leq 3$, we have for $j=1,\dots,n$ that $\partial^{2k+1}_{x^{2k+1}_j} B \in L^\infty_t L^p_x$ for $p=6$, $p<\infty$ and $p=\infty$ when $n=3,2,1$, respectively, and $\partial^q_{x^q_j} B \in L^\infty_t L^\infty_x$ for $q \leq 2k$. Together with the fact that $Q^{-1}$ is an isomorphism from $L^2(\Rm^n)$ to $H^{2k}(\Rm^n) \subset W^{2k-2,\infty}(\Rm^n)$, this is enough to insure that $[Q,\Delta_B]Q^{-1} \in {\mathcal L}(L^2(\Rm^n))$. \end{proof}
Hypothesis (H-2) is then verified with $C(t)=[Q,\Delta_{B_t}]Q^{-1}$, the strong continuity of $C$ following from the continuity of $B$.
Finally, (H-3) follows easily from $H^{2k}(\Rm^n) \subset D(\Delta_{B_t})=H^2(\Rm^n)$, $k\geq 1$, and that $B \in {\mathcal C}^0(0,T,H^{2k+2}(\Rm^n))$. We can thus apply Theorem \ref{kato} and obtain the existence of an evolution group $U$ generated by $i \Delta_{B_t}$. The fact that $U$ is an isometry on $L^2(\Rm^n)$ is a consequence of $\Re i (\Delta_{B_t} \varphi,\varphi)=0$ for every $\varphi \in H^2(\Rm^n)$. \end{proof}
\begin{remark} When $B \in {\mathcal C}^1(0,T,H^2(\Rm^n))$, a classical choice \cite{pazy} for $Q$ is $Q(t)= \lambda I-A(t)$ for $\lambda$ in the resolvent set of $A$. This allows to lower the spatial regularity of $B$ but is not verified when $B=B_t^H$. Notice that in the case when $B=B_t^H$, Proposition \ref{geneevol} can likely be improved in terms of the required spatial regularity of $B$ since we have not used the H\"older regularity in time of $B_t^H$ at all. \end{remark} \subsection{Application to the magnetic Schr\"odinger equation} We apply now the result of the preceeding section to the differential equation \begin{equation} \label{eqgene} \partial_t u=i \Delta_Bu+f, \qquad 0<t\leq T, \qquad u(0)=v, \end{equation} where $\Delta_B=e^{i B} \circ \Delta \circ \,e^{-iB}$. As for \fref{SSE}, we say that $u \in {\mathcal C}^0(0,T,H^{q}(\Rm^n)) \cap{\mathcal C}^1(0,T,H^{q-2}(\Rm^n))$, $q$ non-negative integer, is a solution to \fref{eqgene} if it verifies for all $w\in {\mathcal C}^1(0,T,H^{q+2}(\Rm^n))$, for all $t \in [0,T]$: \begin{align} \label{defsol2} \nonumber &\left( u(t), w(t) \right)-\left( v, w(0) \right) =\int_0^t \left( u(s), \partial_s w(s) \right) ds \\ & \qquad \qquad -i\int_0^t \left( u(s),\Delta_B w(s)\right) ds +i\int_0^t \left(f,w(s) \right) ds. \end{align}
We have the following result: \begin{proposition} \label{gene_exist} Let $B \in {\mathcal C}^0(0,T,H^{q+4}(\Rm^n))$, $q$ non-negative integer, and denote by $U$ the evolution operator of Proposition \ref{geneevol}. Then, for every $v \in H^{q}(\Rm^n)$ and $f \in {\mathcal C}^0(0,T,H^{q}(\Rm^n))$, the function \begin{equation} \label{rep} u(t)=U(t,0) v+\int_0^t U(t,s) f(s)ds \end{equation} belongs to ${\mathcal C}^0(0,T,H^{q}(\Rm^n)) \cap{\mathcal C}^1(0,T,H^{q-2}(\Rm^n))$ and is the unique solution to \fref{eqgene}. Moreover, $u$ satisfies the estimate, for all $t \in [0,T]$: \begin{equation} \label{estimu}
\|u(t)\|_{H^q} \leq C \| v\|_{H^q}+C\int_0^t \| f(s) \|_{H^q}ds,
\end{equation} where the constant $C$ depends on $\| B\|_{{\mathcal C}^0(0,T,H^{q+4}(\Rm^n))}$ when $q\neq 0$. \end{proposition} \begin{proof} Consider first the case $q=2k$ with $k\geq 1$. The result then follows from \cite{katolinear}, Theorem II and the equation \fref{eqgene} in order to obtain the regularity on $\partial_t u$. The cases $q=2k-1$, $k \geq 1$, and $q=0$ are treated by approximation: choose for instance sequences $B_\varepsilon \in {\mathcal C}^0(0,T,H^{q+9}(\Rm^n))$, $v_\varepsilon\in H^{q+5}(\Rm^n)$ and $f_\varepsilon \in {\mathcal C}^0(0,T,H^{q+5}(\Rm^n))$ such that as $\varepsilon \to 0$: \begin{eqnarray} \label{convB1} B_\varepsilon &\to&B \qquad \textrm{ in} \quad {\mathcal C}^0(0,T,H^{q+4}(\Rm^n))\\ \label{convv} v_\varepsilon &\to&v \qquad \textrm{ in} \quad H^{q}(\Rm^n)\\ \label{convf} f_\varepsilon &\to&f \qquad \textrm{ in} \quad {\mathcal C}^0(0,T,H^{q}(\Rm^n)).
\end{eqnarray} Applying the result when $q=2k$ with $k\geq 1$, the corresponding smooth solution $u_\varepsilon$ to \fref{eqgene} when $q=2k-1$ belongs to ${\mathcal C}^0(0,T,H^{2k+4}(\Rm^n))$ with $\partial_t u_\varepsilon \in {\mathcal C}^1(0,T,H^{2k+2}(\Rm^n))$, with the convention that $k=\frac{1}{2}$ when $q=0$. In order to pass to the limit, it is proven in \cite{katolinear}, Theorem V, that if $i \Delta_{B_\varepsilon}$ converges to $i \Delta_{B}$ in ${\mathcal L}(H^2(\Rm^n),L^2(\Rm^n))$ a.e. $t$, and $\|\Delta_{B_\varepsilon}\|_{{\mathcal L}(H^2(\Rm^n),L^2(\Rm^n))} $ is uniformly bounded in $t$ independently of $\varepsilon$, then \begin{equation} \label{convU} U_\varepsilon(t,s) \to U(t,s) \quad \textrm{in} \quad {\mathcal L}(L^2(\Rm^n)) \quad \textrm{uniformly in } (t,s), \end{equation} where $U$ is the evolution operator associated to $B$. These latter conditions are direcly satisfied because of \fref{convB1}. We then write: \begin{eqnarray*} u_\varepsilon(t)&=& U_\varepsilon(t,0)v_\varepsilon+\int_0^t U_\varepsilon(t,s)f_\varepsilon(s) ds, \qquad \forall t \in [0,T]\\ &=&U(t,0)v+\int_0^t U(t,s)f(s) ds+R^1_\varepsilon+R^2_\varepsilon=u+R^1_\varepsilon+R^2_\varepsilon\\ R^1_\varepsilon&=&U_\varepsilon(t,0)(v_\varepsilon-v)+\int_0^t U_\varepsilon(t,s) (f_\varepsilon(s)-f(s))\\ R^2_\varepsilon&=&(U_\varepsilon(t,0)-U(t,0))v+\int_0^t (U_\varepsilon(t,s)-U(t,s)) f(s)) ds. \end{eqnarray*} Using \fref{convU} and the strong convergence of $v_\varepsilon$ and $f_\varepsilon$, we then obtain that $u_\varepsilon \to u$ in ${\mathcal C}^0(0,T,L^2(\Rm^n))$. Assume first that $q \neq 0$. In order to get the announced better regularity on $u$, we use the fact that $u_\varepsilon \in {\mathcal C}^0(0,T,{\mathcal C}^{2k+2}(\Rm^n))$ and $\partial_t u_\varepsilon \in {\mathcal C}^1(0,T,{\mathcal C}^{2k}(\Rm^n))$ thanks to standard Sobolev embeddings for $n \leq 3$. We can then differentiate equation \fref{geneevol}, and find using the representation formula \begin{equation} \label{eqDbeta}
D^\beta u_\varepsilon(t)= U_\varepsilon(t,0) D^\beta v_\varepsilon+\int_0^t U_\varepsilon(t,s)(D^\beta f_\varepsilon(s)+[D^\beta,L_\varepsilon(s)]u_\varepsilon(s)) ds,
\end{equation} where $1\leq |\beta| \leq q$ and
$$D^\beta:=\frac{\partial^{\beta_1}}{\partial x_1^{\beta_1}} \times \cdots \times \frac{\partial^{\beta_n}}{\partial x_n^{\beta_n}}, \qquad \beta=(\beta_1,\cdots,\beta_n), \quad |\beta|=\beta_1 + \cdots + \beta_n,$$
and $L_\varepsilon(s)$ is defined in \fref{defL} with $B$ replaced by $B_\varepsilon$. Only the term involving the commutator requires some attention. Using \fref{convB1}, we can show that for all $s \in [0,T]$, $$
\| [D^\beta,L_\varepsilon(s)]u_\varepsilon(s)\|_{L^2} \leq C \|u_\varepsilon(s)\|_{H^{|\beta|}}, $$ where the constant $C$ is independent of $\varepsilon$. Together with \fref{convB1}-\fref{convv}-\fref{convf}-\fref{convU}-\fref{eqDbeta} and the Gronwall lemma, this yields a uniform bound for $u_\varepsilon$ in ${\mathcal C}^0(0,T,H^q(\Rm^n))$. Using this latter bound along with \fref{convB1}-\fref{convv}-\fref{convf}-\fref{convU}-\fref{eqDbeta} and equation \fref{eqgene} for the smooth solution $u_\varepsilon$ in order to estimate $\partial_t u_\varepsilon$, it is then not difficult to show that $(u_\varepsilon)_\varepsilon$ is a Cauchy sequence in ${\mathcal C}^0(0,T,H^{q}(\Rm^n)) \cap{\mathcal C}^1(0,T,H^{q-2}(\Rm^n))$, whose limit $u$ satisfies estimate \fref{estimu} and \fref{defsol2}. When $q=0$, it suffices to use equation \fref{eqgene} for the smooth solution $u_\varepsilon$ in order to show that $(\partial_t u_\varepsilon)_\varepsilon$ is Cauchy in ${\mathcal C}^0(0,T,H^{-2}(\Rm^n))$. This proves the existence, the representation formula \fref{rep} and estimate \fref{estimu}.
Uniqueness is straightforward in the case $q \geq 1$ since solutions to \fref{eqgene} are regular enough to be used as test functions and to obtain after an integration by part that $\Im (\nabla (e^{-i B_t} u),\nabla (e^{-i B_t} u ))=0$. When $q=0$, we use the adjoint formulation of \fref{eqgene}. The difference between two solutions to \fref{eqgene} satisfies in the case of a test function $w \in {\mathcal C}^0(0,T,H^2(\Rm^n)) \cap {\mathcal C}^1(0,T,L^2(\Rm^n))$, \begin{align*} &\left( u(t), w(t)\right)=\int_0^t \left ( u(s), \partial_s w(s)+(i\Delta_{B_s})^* w(s) \right) ds, \end{align*} where $(i\Delta_{B_s})^*=-i\Delta_{B_s}$ is the adjoint of $i \Delta_{B_s}$. Let $t \in [0,T]$, pick some $w_0 \in H^2(\Rm^n)$ and let $w(s)=z(t-s)$ where $z(s)$ is the solution to $\partial_s z(s)=(i \Delta_{B_{t-s}})^* z(s)$, $z(0)=w_0$, $0<s<t$. Adapting Proposition \ref{geneevol} to the operator $(i\Delta_{B_s})^*$ , Theorem \ref{kato} yields that $z \in {\mathcal C}^0(0,T,H^2(\Rm^n)) \cap {\mathcal C}^1(0,T,L^2(\Rm^n))$. Hence, $\partial_s w(s)+(i\Delta_{B_s})^* w(s)=0$ $x$ a.e., $w(t)=w_0$ and it comes, for all $t \in [0,T]$: $$ \left( u(t), w_0 \right)=0, \qquad \forall w_0 \in H^2(\Rm^n), $$ so that $u=0$. This ends the proof. \end{proof}
We use the result of the last Proposition to prove that the non-linear magnetic Schr\"odinger equation \begin{equation} \label{mag} \partial_t \varphi=i \Delta_{B^H_t} \varphi +g(\varphi), \qquad 0<t\leq T, \qquad u(0)=\Psi_0, \end{equation} admits a unique solution $\mathbb P$ almost surely in the same sense as \fref{defsol2}: \begin{theorem} \label{th_mag} Assume that \textbf{H} is satisfied. Suppose moreover that \fref{assumQ} is verified for $V=H^{q+4}(\Rm^n)$, $q$ non-negative integer. Then, for every $\Psi_0 \in H^q(\Rm^n)$, there exists a maximal existence time $T_M>0$ and a unique function $\varphi \in {\mathcal C}^0(0,T_M,H^{q}(\Rm^n)) \cap{\mathcal C}^1(0,T_M,H^{q-2}(\Rm^n))$ verifying \fref{mag} for $t\in[0,T_M]$ which admits the following representation formula:
$$ \varphi(t)= U(t,0) \Psi_0+ \int_0^t U(t,s) g(\varphi(s))ds, $$
where $U=\{U(t,s)\}$ is the evolution operator generated by the operator $$ i\Delta_{B_t^H} =i e^{i B_t^H}\circ \Delta \circ e^{-i B_t^H}. $$ If moreover $\Im g(\varphi) \overline{\varphi}=0$, then for all $t\in[0,T_M]$ \begin{equation} \label{charge}
\| \varphi(t)\|_{L^2}=\| \varphi(0)\|_{L^2}. \end{equation} If $g$ is globally Lipschitz on $H^q(\Rm^n)$, then the solution exists for all time $T<\infty$. \end{theorem} \begin{proof} The proof is very classical and relies on Proposition \ref{gene_exist} and a standard fixed point procedure. First of all, \fref{assumQ} insures that $\mathbb P$ almost surely, $B^H \in {\mathcal C}^0(0,T,H^{q+4}(\Rm^n))$, which allows us to define an evolution operator $U$ according to Proposition \ref{geneevol}. The rest of the proof follows the usual arguments of for instance \cite{pazy}, Theorem 1.4, Chapter 6, that we sketch here for completeness. Given $\Psi_0$ in $H^{q}(\Rm^n)$ and $\varphi \in {\mathcal C}^0(0,t_1,H^q(\Rm^n))$ for some $t_1>0$ to be fixed later on, denote by $u:=F(\varphi)$ the solution to $\fref{eqgene}$ in ${\mathcal C}^0(0,t_1,H^{q}(\Rm^n)) \cap{\mathcal C}^1(0,t_1,H^{q-2}(\Rm^n))$ where $f=g(\varphi)$ belongs to ${\mathcal C}^0(0,t_1,H^{q}(\Rm^n))$ thanks to hypothesis \textbf{H}. Using the latter, estimate \fref{estimu}, following the aforementioned theorem of \cite{pazy}, one can establish the existence of $M>0$ and a time $t_1(M)$ such that $F$ maps the ball of radius $M$ of ${\mathcal C}^0(0,t_1,H^{q}(\Rm^n))$ centered at 0 into itself. In this ball, the function $g$ being uniformly Lipschitz on $H^p(\Rm^n)$, $0\leq p\leq q$ according to hypothesis \textbf{H}, existence and uniqueness of a fixed point of $F$ in ${\mathcal C}^0(0,t_1,H^{q}(\Rm^n))$, denoted by $\varphi^\star$, follows from the contraction principle. Moreover, this solution verifies the representation formula \fref{rep} with $f=g(\varphi^\star) \in {\mathcal C}^0(0,t_1,H^{q}(\Rm^n))$ and according to Proposition \ref{gene_exist}, belongs in addition to ${\mathcal C}^1(0,t_1,H^{q+2}(\Rm^n))$ and satisfies \fref{mag}. The existence of a maximal time of existence $T_M$ is established following the same lines as \cite{pazy}. When $g$ is globally Lipschitz in $H^q(\Rm^n)$, then $T_M<\infty$ by the Gronwall Lemma.
Regarding the conservation of charge \fref{charge}, the case $q \geq 1$ is direct since the solution $\varphi^\star$ is regular enough to be used as a test function in \fref{defsol2} (after interpretation of $(\cdot, \cdot )$ as the $H^{-1}-H^1$ duality pairing when $q=1$) and it then suffices to take the imaginary part of the equation. When $q=0$, we use a regularization procedure very similar to that of the proof of Proposition \ref{gene_exist}, the details are left to the reader.
\end{proof}
\section{Back to the stochastic Schr\"odinger equation} \label{back} We apply the result of the last section to prove Theorem \ref{th1}. Owing the solution $\varphi$ of Theorem \ref{th_mag}, it suffices to show (i) that $e^{-i B_t^H} \varphi$ is a solution to \fref{defsol}, which will follow from the regularity of $\varphi$ and Lemma \ref{chain2}, this yields existence; and (ii) that all solutions to \fref{defsol} with the corresponding regularity read $e^{-i B_t^H} u$ where $u$ is a solution to \fref{mag}, which yields uniqueness since \fref{mag} has a unique solution. As explained earlier, the last step requires a regularization procedure since test functions of the form $w=e^{-i B_t^H} z$, with $z$ smooth, are not differentiable in time and cannot be used directly in \fref{defsol}. In the whole proof, $T$ denotes some time $T\geq T_M$.\\
\noindent \textbf{Proof of Theorem \ref{th1}.} \textit{Existence.} Let $\varphi$ be the unique solution to \fref{mag} according to Theorem \ref{th_mag} and define, for any test function $w \in {\mathcal C}^1(0,T_M,H^{q+2}(\Rm^n))$, $$ F(B^H_t,t)=( e^{-i B_t^H} \varphi(t), w(t)). $$ We verify that $F$ satisfies the hypotheses of Lemma \ref{chain2}. First of all, $F$ is clearly continuously differentiable w.r.t. the first variable and for all $v \in H^{q+4}(\Rm^n)$, let $\phi(t):=\partial_1 F(B^H_t,t)(v)=i ( e^{-i B_t^H} \varphi(t) v, w(t))$. Second of all, we need to show that $\phi \in {\mathcal C}^{0,\lambda}(0,T)$ for $\lambda$ verifying $\lambda+\gamma>1$, together with the bound \fref{hypphi}. To this goal, we have for $(t,s) \in [0,T_M]^2$: \begin{eqnarray*} \phi(t)-\phi(s)&=&i\left( (e^{-i B_t^H}-e^{-i B_s^H}) \varphi(t) v, w(t)\right)+i\left(e^{-i B_s^H} (\varphi(t)-\varphi(s)) v, w(t)\right)\\ &&+i\left(e^{-i B_s^H} \varphi(s) v, (w(t)-w(s))\right)\\ &:=&T_1+T_2+T_3. \end{eqnarray*} We treat each term separately. We have, using standard Sobolev embeddings for $n\leq 3$: \begin{eqnarray} \nonumber
|T_1| &\leq& C \|\varphi(t)\|_{L^2} \|w(t)\|_{L^\infty} \|v\|_{L^\infty} \|B_t^H-B_s^H\|_{L^2}\\ \nonumber
&\leq & C (t-s)^{\gamma} \|v\|_{H^{q+4}} \|B_t^H\|_{{\mathcal C}^{0,\gamma}(0,T_M,L^2)}\\
&\leq & C (t-s)^{\gamma} \|v\|_{H^{q+4}}, \label{estimT1} \end{eqnarray} for all $0\leq \gamma<H$. Regarding the term $T_2$, notice that the product $e^{i B_t^H} \overline{v} w$ belongs to $H^{q+2}(\Rm^n)$ when $n \leq 3$, so that since $\partial_t \varphi \in {\mathcal C}^{0}(0,T_M,H^{q-2}(\Rm^n))$, we can write $$ \left(e^{-i B_s^H} (\varphi(t)-\varphi(s)) v, w(t)\right)=\int_s^t \langle \partial_\tau \varphi(\tau) , e^{ i B_s^H} \overline{v} w(t) \rangle_{H^{q-2},H^{q+2}} d\tau, $$ where when $q\geq 2$, the pairing $\langle \cdot, \cdot \rangle_{H^{q-2},H^{q+2}}$ is replaced by the $L^2$ inner product. Hence, $$
|T_2| \leq |t-s| \|w(t)\|_{H^{q+2}} \|v\|_{H^{q+4}} \|B_t^H\|_{{\mathcal C}^{0}(0,T_M,H^{q+4})} \|\partial_t \varphi\|_{{\mathcal C}^{0}(0,T_M,H^{q-2})} \leq C |t-s| \|v\|_{H^{q+4}}. $$ Estimation of $T_3$ is straightforward and leads to a similar estimate as above. This, together with \fref{estimT1} yields that $$
\| \phi\|_{{\mathcal C}^{0,\gamma}(0,T_M)} \leq C \|v\|_{H^{q+4}}. $$ Since $\frac{1}{2}<H$, we can pick $H=\frac{1}{2}+\varepsilon$ and $\gamma=\frac{1}{2}+\frac{\varepsilon}{2}$ such that $2\gamma>1$ and the assumption on $\phi$ of Lemma \ref{chain2} is verified. It remains to show that $\partial_2 F$ exists and is continuous, and this is a consequence of the fact that $\partial_t \varphi \in {\mathcal C}^{0}(0,T_M,H^{q-2}(\Rm^n))$.
Applying Lemma \ref{chain2} then yields \begin{align} \label{eqpphi} &( e^{-i B_t^H} \varphi(t), w(t))-( \Psi_0, w(0))=\int_0^t\langle \partial_\tau \varphi(\tau), e^{i B_\tau^H} w(\tau)\rangle_{H^{q-2},H^{q+2}} d\tau\\ \nonumber &+\int_0^t( e^{-i B_\tau^H} \varphi(\tau), \partial_\tau w(\tau)) d\tau+i \sum_{p \in \Nm} \lambda_p \int_0^t ( e^{-i B_\tau^H} \varphi(\tau)e_p, w(\tau)) d \beta^H_p(\tau). \end{align} In order to conclude, picking $w(t,x)=w(x) \in H^{q+2}(\Rm^n)$ in \fref{defsol2} with $f=g(\varphi)$, it comes that \fref{mag} is verified in $H^{q-2}(\Rm^n)$ for all $t\in [0,T_M]$ and almost surely. This yields $\partial_\tau \varphi= i \Delta_{B_t^H} \varphi-i g(\varphi)$ in $H^{q-2}(\Rm^n)$, and replacing $\partial_\tau \varphi$ by its latter expression in \fref{eqpphi}, setting $\Psi=e^{-i B_t^H} \varphi \in {\mathcal C}^0(0,T_M,H^q(\Rm^n)) \cap {\mathcal C}^{0,\gamma}(0,T_M,H^{q-2}(\Rm^n))$ for all $0\leq \gamma<H$, finally yields \fref{defsol}.
\textit{Uniqueness, Step 1: regularization.} Starting from a solution $\Psi$ to \fref{defsol} with the above regularity, we would like to choose the test function $w=e^{-i B_\tau^H} z$ for some regular function $z$ in order to recover the weak formulation of \fref{mag}, which admits a unique solution. This is not allowed of course since $B^H$ is not differentiable. The solution is to use the H\"older regularity of $\Psi$ in order to reinterpret the term $$ \int_0^t \left( \Psi(s), \partial_s w(s) \right) ds $$ as a fractional integral. To this end, let $$ B^{H,\varepsilon}(t,x):=\sum_{p \in \Nm} \lambda_p e_p(x) \beta_p^{H,\varepsilon}(t) $$
where $\beta^{H,\varepsilon}_p$ is a ${\mathcal C}^1$ regularization of $\beta^{H}_p$ such that $\beta^{H,\varepsilon}_p \to \beta^{H}_p$ in ${\mathcal C}^{0,\gamma}(0,T)$ almost surely for all $p$ and $\|\beta^{H,\varepsilon}_p\|_{{\mathcal C}^{0,\gamma}(0,T)} \leq \|\beta^{H}_p\|_{{\mathcal C}^{0,\gamma}(0,T)}$, $0\leq \gamma <H$. We have \begin{equation} \label{conVV} B^{H,\varepsilon}\to B^{H}\quad \textrm{in}\quad {\mathcal C}^{0,\gamma}(0,T,H^{q+4}(\Rm^n)),\qquad \mathbb P \quad \textrm{almost surely}. \end{equation} Indeed: $$
\| B^{H,\varepsilon}-B^{H}\|_{{\mathcal C}^{0,\gamma}(0,T,H^{q+4})} \leq \sum_{p \in \Nm} \lambda_p \| e_p \|_{H^{q+4}} \|\beta^{H,\varepsilon}_p -\beta^{H}_p\|_{{\mathcal C}^{0,\gamma}(0,T)} $$ and $$
\|\beta^{H,\varepsilon}_p -\beta^{H}_p\|_{{\mathcal C}^{0,\gamma}(0,T)} \leq 2 \|\beta^{H}_p\|_{{\mathcal C}^{0,\gamma}(0,T)}, $$ which, together with the convergence of $\beta_p^{H,\varepsilon}$ to $\beta_p^{H}$ in ${\mathcal C}^{0,\gamma}$, \fref{defK} and the Weierstrass rule gives the desired result. Set $w_\varepsilon=e^{-i B_t^{H,\varepsilon}} z$ where $z \in {\mathcal C}^1(0,T,H^{q+2}(\Rm^n))$. Then \begin{align*} &\int_0^t \left( \Psi(s), \partial_s w_\varepsilon(s) \right) ds\\ &\qquad =\int_0^t \left( \Psi(s)e^{i B_s^{H,\varepsilon}}, \partial_s z(s) \right)ds-i\sum_{p\in \Nm} \lambda_p \int_0^t \left( \Psi(s)e^{i B_s^{H,\varepsilon}},z(s) e_p\right) (\beta_p^{H,\varepsilon}(s))' ds, \end{align*} where all permutations of sum and integrals were permitted since the series defining $B^{H,\varepsilon}$ is normally convergent in ${\mathcal C}^1(0,T,H^{q+4}(\Rm^n))$. We now use the fact that for a continuously differentiable function $f$, $D_{t-}^\alpha f \to f'$ as $\alpha \to 1$, see \cite{zahle} section 1, and that $D_{t-}^{\alpha+\beta} f=D_{t-}^{\alpha} D_{t-}^{\beta} f$, where the operator $D_{t-}^{\alpha}$ is defined in section \ref{prelim}. We then introduce, for $\frac{1}{2}<1-\mu<\gamma$, \begin{eqnarray*} I^\varepsilon_p&:=&\int_0^t \left( \Psi(s)e^{i B_s^{H,\varepsilon}},e_p\right) (\beta_p^{H,\varepsilon}(s)-\beta_p^{H,\varepsilon}(t))' ds\\ &=&\lim_{\alpha \to 1} I^{\varepsilon,\alpha}_p := \lim_{\alpha \to 1} \int_0^t \left( \Psi(s)e^{i B_s^{H,\varepsilon}},e_p z(s)\right) D_{t-}^{\alpha-\mu} D_{t-}^{\mu} (\beta_p^{H,\varepsilon})_{t-}(s)ds, \end{eqnarray*} where $(\beta_p^{H,\varepsilon})_{t-}(s)=\beta_p^{H,\varepsilon}(s)-\beta_p^{H,\varepsilon}(t^-)$. Owing the fact that $( \Psi(s)e^{i B_s^{H,\varepsilon}},z(s) e_p)\in {\mathcal C}^{0,\gamma}(0,T_M)$ for any $0\leq \gamma<H$ and using the fractional integration by part formula of \cite{zahle} section 1, we find $$ I^{\varepsilon,\alpha}_p=(-1)^{\alpha-\mu} \int_0^t D_{0+}^{\alpha-\mu} \left( \Psi(s)e^{i B_s^{H,\varepsilon}},z(s) e_p\right) D_{t-}^{\mu} (\beta_p^{H,\varepsilon})_{t-}(s)ds. $$ Moreover, we can send $\alpha$ to one above thanks to dominated convergence since the term $( \Psi(s)e^{i B_s^{H,\varepsilon}},z(s) e_p)$ belongs to ${\mathcal C}^{0,\gamma}(0,T_M)$ and $1-\mu<\gamma$ in order to obtain $$ I^\varepsilon_p=(-1)^{1-\mu} \int_0^t D_{0+}^{1-\mu} \left( \Psi(s)e^{i B_s^{H,\varepsilon}},z(s) e_p\right) D_{t-}^{\mu} (\beta_p^{H,\varepsilon})_{t-}(s)ds. $$ We derive below some estimates needed to pass to the limit $\varepsilon \to 0$.\\ \noindent \textit{Uniqueness, Step 2: uniform estimates.} Recall that $w_\varepsilon=e^{-i B_t^{H,\varepsilon}} z$ and define first, with $w=e^{-i B_t^{H}} z$: $$\phi^\varepsilon(s)=( \Psi(s),(w_\varepsilon(s)-w(s)) e_p):=( \Psi(s),r_\varepsilon(s)e_p).$$ In order to estimate $D_{0+}^{1-\mu} \phi^\varepsilon$, we write \begin{eqnarray*} \phi^\varepsilon(t)-\phi^\varepsilon(s)&=&( \Psi(t)- \Psi(s),r_\varepsilon(t)e_p)+( \Psi(s),(r_\varepsilon(t)-r_\varepsilon(s))e_p):=T_1+T_2. \end{eqnarray*} For the term $T_1,$ we use the ${\mathcal C}^{0,\gamma}$ regularity of $\Psi$ in $H^{q-2}$, while we use that of $r^\varepsilon$ for $T_2$. It comes with the help of standard Sobolev embeddings: \begin{eqnarray*}
|T_1| &\leq& |t-s|^\gamma \|\Psi\|_{{\mathcal C}^{0,\gamma}(0,T_M,H^{q-2})}\|r_\varepsilon(t)\|_{H^{q+2}} \|e_p\|_{H^{q+2}}\\
|T_2| &\leq& |t-s|^\gamma \|\Psi(t)\|_{L^2} \|e_p\|_{L^\infty} \| r_\varepsilon \|_{{\mathcal C}^{0,\gamma}(0,T,L^{2})}. \end{eqnarray*} Since $1-\mu<\gamma$, this gives \begin{equation} \label{estphiW}
\| \phi^\varepsilon\|_{W_{1-\mu,1}(0,T_M)} \leq C \|e_p\|_{H^{q+4}} \| B^{H,\varepsilon}-B^H \|_{{\mathcal C}^{0,\gamma}(0,T,H^{q+4}(\Rm^n))}. \end{equation} On the other hand, using the notation of section \ref{prelim}, we find \begin{equation} \label{estimbet}
|D_{t-}^{\mu} (\beta_p^{H,\varepsilon})_{t-}(s)| \leq \Lambda_{1-\mu}(\beta^{H,\varepsilon}_p) \leq C \|\beta_p^{H,\varepsilon}\|_{{\mathcal C}^{0,\gamma}(0,T)} \leq C \|\beta_p^{H}\|_{{\mathcal C}^{\gamma}(0,T)} < \infty, \end{equation} and \begin{equation} \label{estimbet2}
|D_{t-}^{\mu} (\beta_p^{H,\varepsilon}-\beta_p^{H})_{t-}(s)| \leq \Lambda_{1-\mu}(\beta^{H,\varepsilon}_p-\beta_p^{H}) \leq C \|\beta_p^{H,\varepsilon}-\beta_p^{H}\|_{{\mathcal C}^{0,\gamma}(0,T)}. \end{equation}
\noindent \textit{Uniqueness, Step 3: passing to the limit.} We have all needed now to pass to the limit in the weak formulation \fref{defsol}. Plugging $w^\varepsilon(t)=e^{-i B_t^{H,\varepsilon}} z \in {\mathcal C}^1(0,T,H^{q+2}(\Rm^n))$ yields \begin{align} \label{defsol3} \nonumber &\left( \Psi(t), w^\varepsilon(t) \right)-\left( \Psi_0, z(0) \right) =\int_0^t \left( \Psi(s), \partial_s w^\varepsilon(s) \right) ds \\[3mm] & -i\int_0^t \left( \Psi(s),\Delta w^\varepsilon(s)\right) ds +i \int_0^t\left(\Psi(s),w^\varepsilon(s) d B_s^H\right)+i\int_0^t \left(g(\Psi(s)),w^\varepsilon(s) \right) ds. \end{align} We have $$ \int_0^t \left( \Psi(s), \partial_s w^\varepsilon(s) \right) ds=\int_0^t \left( \Psi(s)e^{i B_t^{H,\varepsilon}}, \partial_s z(s) \right) ds-i\sum_{p \in \Nm^*} I_p^\varepsilon. $$ Using \fref{estimint}-\fref{conVV}-\fref{estphiW}-\fref{estimbet}-\fref{estimbet2} as well as \fref{defK}, we can pass to the limit in the latter equation and obtain that, $\forall t \in [0,T_M]$: $$ \lim_{\varepsilon \to 0} \int_0^t \left( \Psi(s), \partial_s w^\varepsilon(s) \right) ds=\int_0^t \left( \Psi(s)e^{i B_s^{H}}, \partial_s z(s) \right) ds-i \int_0^t\left(\Psi(s)e^{i B_s^{H}}, z(s) d B_s^H\right). $$ Similar arguments can be employed to pass to the limit in the remaining terms of \fref{defsol3}. The stochastic integrals simplify and we are left with \begin{align} \label{defsol4} \nonumber &\left( \Psi(t)e^{i B_t^{H}}, z(t) \right)-\left( \Psi_0, z(0) \right) =\int_0^t \left( \Psi(s)e^{i B_s^{H}}, \partial_s z(s) \right) ds \\[3mm] & -i\int_0^t \left( \Psi(s),\Delta (e^{-i B_s^{H}}z(s))\right) ds +i\int_0^t \left(g(\Psi(s)),e^{-i B_s^{H}}z(s) \right) ds. \end{align} Hence, $\Psi e^{i B_t^{H}}$ verifies the magnetic Schr\"odinger equation $\fref{defsol2}$ with $f=g(\Psi(s))e^{i B_s^{H}}=g(\Psi(s)e^{i B_s^{H}})$. Since the latter admits a unique solution $\varphi \in {\mathcal C}^0(0,T_M,H^q(\Rm^n)) \cap {\mathcal C}^1(0,T_M,H^{q-2}(\Rm^n))$ according to Theorem \ref{th_mag}, we can conclude that \fref{SSE} admits a unique solution. The representation formula \fref{repre_th} follows then without difficulty with the identification $\Psi e^{i B_t^{H}}=\varphi$ and Theorem \ref{th_mag}. This ends the proof of Theorem \ref{th1}. \section{Appendix} \subsection{Proof of Lemma \ref{chain2}} First of all, we know by section \ref{prelim} that $B_t^{H}$ belongs to $E:={\mathcal C}^{0,\gamma}(0,T,V)$, $\mathbb P$ almost surely for $0\leq \gamma<H$. We proceed by approximation in order to apply the change of variables formula \fref{chain} valid in finite dimensions. Let $$B_t^{H,N}(x):=\sum_{p=0}^N \lambda_p e_p(x) \beta^H_p(t),$$ so that, \begin{equation} \label{convB}B_t^{H,N} \to B_t^{H} \quad \textrm{in } E, \qquad \mathbb P \textrm{ almost surely,} \end{equation}
thanks to \fref{assumQ} and \fref{defK}. We have moreover the bound $\|B_t^{H,N}\|_E \leq \|B_t^{H}\|_E:=M$. Since $F$ is ${\mathcal C}^1$, and $\phi \in {\mathcal C}^{0,\lambda}(0,T)$ with $\lambda+\gamma>1$, we can use \fref{chain} and find, for $0\leq s \leq t \leq T$ fixed: \begin{eqnarray*} F(B_t^{H,N},t)-F(B_s^{H,N},s)=\int_s^t \partial_2 F(B_{\tau}^{H,N},\tau) d\tau+ \sum_{p=0}^N \lambda_n \int_s^t \partial_1 F(B_{\tau}^{H,N},\tau)(e_p)\, d \beta_p^H(\tau). \end{eqnarray*} By continuity of $F$, it is direct to pass to the limit in the left hand side. The same holds for the first term of the right hand side thanks to dominated convergence and the fact that $\partial_2 F$ is continuous and $B_\tau^{H,N}$ is bounded in $E$ independently of $N$. Regarding the last term, let $\phi^N_p(\tau)=\partial_1 F(B_{\tau}^{H,N},\tau)(e_p)$ and $$ f_p^N:=\int_s^t \phi_p^N(\tau) \, d \beta_p^H(\tau)=(-1)^\alpha \int_s^t D_{s+}^\alpha \phi_p^N(\tau) D_{t-}^{1-\alpha} (\beta_p^H)_{t-}(s) d\tau, $$
by \fref{stiel} for some $\alpha$ verifying $\alpha<\lambda$ and $1-\alpha<\gamma$. Since $\|B_t^{H,N}\|_E \leq M$, we have by \fref{hypphi} $$
|\tau-s|^{-\alpha-1}|(\phi_p^N(\tau)-\phi_p^N(s))| \leq |\tau-s|^{\lambda-\alpha-1} \|\phi_p^N \|_{{\mathcal C}^{0,\lambda}(0,T)} \leq C_M |\tau-s|^{\lambda-\alpha-1} \|e_p \|_V, $$
where $\lambda-\alpha>0$. The latter estimate, dominated convergence, \fref{convB} together with the continuity of $\partial_1 F$ yield first that $D_{s+}^\alpha \phi_p^N(\tau) \to D_{s+}^\alpha \phi_p(\tau)$ a.e. where $\phi_p(\tau)=\partial_1 F(B_{\tau}^{H},\tau)(e_p)$. Then, since $|D_{t-}^{1-\alpha} (\beta_p^H)_{t-}(s)| \leq \Lambda_\alpha(\beta^H_p)<\infty$, $\mathbb P$ almost surely, we have \begin{equation} \label{estimDphi}
|D_{s+}^\alpha \phi_p^N(\tau) D_{t-}^{1-\alpha} (\beta_p^H)_{t-}(s) | \leq C |\tau-s|^{\lambda-\alpha-1}\|e_p \|_V \end{equation} so that dominated convergence implies that $f_p^N \to f_p=\int_s^t \phi_p(\tau) \, d \beta_p^H(\tau)$, for all $p \in \Nm$. Finally, since $$
\lambda_p |f_p^N| \leq C \lambda_p \|e_p \|_V, $$ thanks to \fref{estimDphi}, and moreover \fref{assumQ} holds, we can apply the Weierstrass rule and conclude that $\mathbb P$ almost surely: $$ \lim_{N\to \infty} \sum_{p=0}^N \lambda_n f_p^N=\sum_{p=0}^\infty \lambda_n \int_s^t \partial_1 F(B_{\tau}^{H},\tau)(e_p)\, d \beta_p^H(\tau). $$ This ends the proof. \subsection{Proof of Lemma \ref{fubini}} The hypothesis on $F$ show that the integral on the left is well-defined and that $$ \sum_{p \in \Nm} \lambda_p \int_0^t \left(\int_{\Rm^n} F_{\tau,x}(e_p) dx \right) d \beta^H_p(\tau) = \lim_{N\to \infty } \sum_{p=0}^N \lambda_p \int_0^t \left(\int_{\Rm^n} F_{\tau,x}(e_p) dx \right) d \beta^H_p(\tau):=\lim_{N\to \infty } I_N. $$
Morever, for $1-H<\alpha<\frac{1}{2}$, we have that $|D_{t-}^{1-\alpha} (\beta_p^H)_{t-}(s)| \leq \Lambda_\alpha(\beta^H_p)<\infty$, $\mathbb P$ almost surely and $D_{0+}^\alpha F_{t,x}(e_n) \in L^1((0,T)\times \Rm^n)$ since $F_{t,x}(e_p)\in W_{\alpha,1}(0,T,L^1(\Rm^n))$. Hence, using the definition of the stochastic integral and Fubini Theorem, it comes
\begin{eqnarray*} I_N&=&(-1)^{|\alpha|}\sum_{p=0}^N \lambda_p \int_0^t \left[D_{0+}^\alpha \int_{\Rm^n} F_{\tau,x}(e_p)(\tau) dx \right]\left[D_{t-}^{1-\alpha} (\beta_p^H)_{t-}(\tau)\right] d\tau \\ &=&
(-1)^{|\alpha|}\sum_{p=0}^N \lambda_p \int_{\Rm^n} \left(\int_0^t \left[D_{0+}^\alpha F_{\tau,x}(e_p)(\tau) \right] \left[D_{t-}^{1-\alpha} (\beta_p^H)_{t-}(\tau)\right] d\tau \right)dx\\ &=& \int_{\Rm^n} \left(\sum_{p=0}^N \lambda_p \int_0^t F_{t,x}(e_p) d \beta^H_p(\tau) \right) dx:=\int_{\Rm^n} f_N(x) dx. \end{eqnarray*} Moreover, $\mathbb P$ almost surely: \begin{eqnarray*}
\|f_N\|_{L^1} &\leq& \sum_{p=0}^N \lambda_p \left\| \|F_{t,x}(e_p)\|_{W_{\alpha,1}(0,T)} \right\|_{L^1} \Lambda_\alpha(\beta_p^H)\\
& \leq& C \|F\|_{W_{\alpha,1}(0,T,{\mathcal L}(V,L^1))} \sum_{p=0}^N \lambda_p \|e_p\|_V \Lambda_\alpha(\beta_p^H), \end{eqnarray*}
so that thanks to \fref{finitelamb}, the series defining $f_N$ converges strongly in $L^1(\Rm^n)$ and almost surely. This yields $$ \lim_{N\to \infty } I_N=\int_{\Rm^n} \left(\sum_{p=0}^\infty \lambda_p \int_s^t F_{t,x}(e_n) d \beta^H_p(\tau) \right) dx $$ and ends the proof. \subsection{Proof of Remark \ref{rem2}} For $q \geq 2$, using the regularity $\Psi \in {\mathcal C}^0(0,T_M,H^q(\Rm^n)) \cap {\mathcal C}^{0,\gamma}(0,T_M,H^{q-2}(\Rm^n))$ and picking $w \in L^2(\Rm^n)$, we can recast \fref{defsol} as \begin{align*} &\left( \Psi(t), w \right)-\left( \Psi_0, w \right)=
-i\int_0^t \left( \Delta \Psi(s), w\right) ds +i \int_0^t\left(\Psi(s), w d B_s^H\right)+i\int_0^t \left(g(\Psi(s)),w \right) ds. \end{align*} Since the mapping $F: e_p \to \overline{\Psi(s)} w e_p$ belongs to ${\mathcal C}^{0,\gamma}(0,T_M,{\mathcal L}(V,L^1(\Rm^n)))$, we can use Lemma \ref{fubini} together with Fubini theorem to arrive at \begin{align*} &\left( \Psi(t), w \right)-\left( \Psi_0, w \right)= \\[3mm] &-i \left( \int_0^t \Delta \Psi(s) ds, w\right) +i \left( \int_0^t\Psi(s)d B_s^H, w \right)+i\left( \int_0^t g(\Psi(s))ds,w \right), \end{align*} which yields the desired result.
\footnotesize{
}
\end{document}
\end{document} | arXiv |
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Finding distances and midpoints
If we want to find the distance between two points on a number line we use the distance formula:
$$AB=\left | b-a \right |\; or\; \left | a-b \right |$$
Point A is on the coordinate 4 and point B is on the coordinate -1.
$$AB=\left | 4-(-1) \right |=\left | 4+1 \right |=\left | 5 \right |=5$$
If we want to find the distance between two points in a coordinate plane we use a different formula that is based on the Pythagorean Theorem where (x1,y1) and (x2,y2) are the coordinates and d marks the distance:
$$d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$$
The point that is exactly in the middle between two points is called the midpoint and is found by using one of the two following equations.
Method 1: For a number line with the coordinates a and b as endpoints:
$$midpoint=\frac{a+b}{2}$$
Method 2: If we are working in a coordinate plane where the endpoints has the coordinates (x1,y1) and (x2,y2) then the midpoint coordinates is found by using the following formula:
$$midpoint=\left ( \frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2} \right )$$
Find the midpoint of the line segment.
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Stress, health, noise exposures, and injuries among electronic waste recycling workers in Ghana
Katrina N. Burns1,
Stephanie K. Sayler1 &
Richard L. Neitzel ORCID: orcid.org/0000-0001-5500-25891
Journal of Occupational Medicine and Toxicology volume 14, Article number: 1 (2019) Cite this article
Electronic waste (e-waste) recycling workers in low and middle-income countries have the potential for occupational injuries due to the nature of their work at informal e-waste sites. However, limited research exists on stress, noise, occupational injuries, and health risks associated with this work environment. This study evaluated injury experience, noise exposures, and stress risk factors among e-waste workers at the large recycling site in the Agbogbloshie market, Accra, Ghana.
Participants completed a survey addressing their work, health status, stress, exposures to several occupational hazards (including noise), use of personal protective equipment at work, and injury experience. A subset of participants also completed personal noise dosimetry measurements. Poisson regression was used to evaluate the association between the number of injuries experienced by participants and various factors evaluated in the survey.
Forty-six male e-waste workers completed the survey, and 26 completed a noise dosimetry measurement. Participants experienced an average of 9.9 ± 9.6 injuries per person in the previous 6 months (range: 1–40). The majority of injuries were lacerations (65.2%), and the most common injury location was the hand (45.7%). Use of personal protective equipment was rare. The mean time-weighted average noise level was 78.8 ± 5.9 dBA. Higher perceived stress, greater age, poorer health status, not using gloves, and involvement in dismantling activities were associated with an increased number of injuries. After controlling for each of these risk factors, perceived stress level and perceived noise exposure were associated with a significantly greater number of injuries.
Our study identified a large number of injuries among informal e-waste recyclers, and we found that higher levels of perceived stress and perceived noise were associated with an increased number of occupational injuries, even after controlling for other injury risk factors.
Electronic waste (e-waste) consists of discarded cell phones, computers, appliances, and other electrical or electronic products, and electronic waste recycling involves the salvaging of these items for repair or for extraction of valuable metals and components. As of 2016, a total of 44.7 million tons of e-waste had been created globally, and that number is expected to grow to 52.2 million by 2021 [10]. Due to the high consumer demand for the latest generation of electronics, and subsequent discarding of older electronics, the amount of e-waste being created is increasing substantially over time [24, 30, 45]. The high e-waste recycling and disposal costs in high-income countries has led to the exportation of e-waste to low- and middle-income countries (LMICs) where labor costs are lower, but where resources to recycle and dispose of the products in a safe and sustainable manner are limited or absent [44]. The 1989 Basel Convention was developed to prevent the inter-country movement of hazardous waste, but some high-income countries, such as the United States, have not ratified the convention and continue to export used, obsolete, and often unrepairable electronic equipment as "donations" to countries throughout Africa and Asia, where they are recycled by informal workers [50]. This informal recycling scheme creates important economic opportunities for impoverished workers and communities in LMICs and recovers valuable raw materials from e-waste, preventing these materials from being discarded, but the processes used may result in unnecessarily high occupational risks [3, 9]. Unfortunately, informal e-waste recycling sites are not well-regulated by local governments and many may have no occupational health and safety oversight.
Informal e-waste recycling workers collect, sort, and repair or dismantle e-waste using crude de-manufacturing methods and very basic, non-specialized hand tools. Workers often remove plastic or rubber coatings from wires and other valuable metal components for resale (the final goal of e-waste recycling activities) by burning the e-waste materials. All of these tasks are often performed without the benefit of safe working procedures, personal protective equipment (PPE), sanitation facilities, or safety training, all of which are common features in formal work settings in high-income nations [18, 48]. This can place e-waste workers at an increased risk of injury, exposure to noise and subsequent noise-induced hearing loss, and multiple other adverse health effects associated with their work [5, 15, 39, 54]. In addition to the adverse physical health impacts, workers may also suffer from personal and occupational stress due to factors often affecting vulnerable (and frequently migrant) e-waste workers [15]. Regulatory attention and efforts to improve occupational health among e-waste recycling workers could result in safer working and living conditions, a living wage [8, 11, 35, 43], and better access to PPE, but additional research is needed to better characterize occupational health needs among these workers.
The Agbogbloshie market in Accra, Ghana, offers an excellent example of informal e-waste recycling. After over 10 years of accepting and recycling electronic waste from Europe, North America, and Asia [6], Agbogbloshie has become one of the most polluted informal e-waste recycling sites in the world [23]. The informal e-waste site at Agbogbloshie consists of primarily workers who have traveled to the site from northern Ghana and, to a lesser extent, other West African countries in search of work. At the time of our study this site was effectively unregulated; there was no formal oversight for the e-waste recycling workers at the site, and all workers were essentially independent contractors. The payment structure used at the site was based on the amount by weight of valuable metals (primarily copper) that each worker was able to recover from the e-waste. To the best of our knowledge, the Greater Accra Scrap Dealers Association, which provided overall leadership to work operations at the site, but which did not directly employ any e-waste recycling workers, did not advocate for the workers in any formalized capacity or control employment conditions at the site.
In order to address the multiple occupational exposures among this vulnerable population of recycling workers, we examined the relationship between e-waste work activities, stress, noise exposures, and injury experience. While a number of studies have focused on the relationship between noise and injuries [16, 17, 19, 33, 36, 37], there is a paucity of studies on injuries among e-waste recycling workers [18], and those that have been conducted have not examined work activities in detail [27]. Our study had two hypotheses: first, that higher perceived stress levels would be associated with higher injury risk, and second, that higher noise exposures (evaluated both objectively and subjectively) would be associated with higher injury risk. Our study evaluated exposures and injury outcomes without defining or exploring an a priori-defined injury mechanism.
Our previously described data collection methods [15] will be briefly summarized here. All research procedures were approved by the University of Michigan Institutional Review Board (HUM00084062) and the University of Ghana Institutional Review Board at the Noguchi Memorial Institute for Medical Research (NMIMR-IRB CPN 070/13-14). A research team composed of students, staff, and faculty members from the University of Michigan and the University of Ghana-Legon collected the data in May 2014.
E-waste recyclers 18 years of age and older who worked at Agbogbloshie during the study were recruited to participate in the study. We conducted recruitment with the approval and assistance of the Chairman of the Greater Accra Scrap Dealers' Association. Recruiting was conducted with the assistance of hired translators, and scheduled for approximately 1 h in the mornings and 2–3 h in the afternoons of days in which we collected data. Our goal was to recruit a convenience sample of 60 e-waste recycling workers, recruited through word of mouth and by our translators actively approaching individuals onsite to assess their interest. Workers participated in this study over a single day each, but were not required to be actively working on their day of participation. Each participant was approached in person and read a recruitment script and the informed consent in their native language. Interested individuals signed or provided an ink thumbprint on the informed consent form for official enrollment in the study. Participants received 9 GHS (about 3 USD) and a 3 GHS (about 1 USD) snack as a thank-you for their participation.
All participants completed a comprehensive interview administered in the language of their preference by our hired interpreters. The interview consisted of questions on demographics (e.g., age, religion, marital status, education, time living in Agbogbloshie, and income); health-related behaviors and outcomes (e.g., smoking status, self-reported health status); and personal stress factors, as measured by a subscale of Cohen's Perceived Stress Scale (PSS, [21]) with a total of 16 points possible and higher scores indicating greater stress. The PSS survey has been translated and validated in a number of countries, but not, to our knowledge, in Ghana. Occupational information was also collected through the survey, including work activities, work duration, use of personal protective equipment (PPE), information on job demands and working conditions, and the number of injuries received during recent e-waste recycling activities. Work activities were defined after making initial observations of ongoing work at the site and with the assistance of the Chairman of the Greater Accra Scrap Dealers' Association. The final categories developed were: burning of e-waste, collecting e-waste from consumers or businesses, collecting e-waste after burning, dismantling e-waste, lead acid battery recycling, lead smelting, removing wire coverings, repairing e-waste, and sorting e-waste. Occupational stress scores (OSS) were calculated on 28-point scale based on occupational stress-related survey questions; as with the perceived stress scale, higher values indicated higher levels of occupational stress. The frequency of exposure to noise, intended to represent participants' "typical" exposure, was subjectively assessed on a five-point scale, with categories of "Never," "Almost never," "Sometimes," "Fairly often," "Very often," or "Don't Know."
Noise measurements
While the perceived frequency of noise exposure question on the survey provided information about typical exposures, we also objectively measured personal noise exposures using ER-200D personal noise dosimeters (Etymotic Research Inc. Elk Grove Village, IL, USA). The dosimetry data were intended to compliment the subjective data by providing quantified estimates of noise exposure levels overall, as well as during particular work and non-work activities. These data were also collected to allow for assessment of the potential relationship between objective noise levels and injury risk. The devices were configured to measure the equivalent continuous average noise level (LEQ) according to the exposure limit for community noise recommended by the World Health Organization: A-weighting, slow time response, 3 dB time-intensity exchange rate, 70 dB threshold, and 75 dB criterion level [13]. The measurement range of the noise dosimeter was 70–130 dBA. Average and maximum noise exposures were data logged every 3 min 45 s (the default and only data-logging interval length available for these dosimeters) for up to 24 h. Exposures during work activities were compared to the 85 dBA exposure limit used for occupational noise exposures in most countries around the world [49].
Data cleaning and statistical analysis
Study data were cleaned using methods we have described previously [15]. All statistical analyses were performed using SPSS Version 24.0 (IBM SPSS Statistics for Windows, IBM Corp., Armonk, NY, USA). For workers who completed personal noise dosimetry, we computed 8-h time-weighted average (TWA, in dBA) exposures using Eq. 1, where LAEQ is a 3.75-min average equivalent continuous noise level, N is the total number of 3.75-min intervals i in the measured shift, and 128 is the number of 3.75-min intervals in a 480-min shift.
$$ TWA=10\times {\mathit{\log}}_{10}\left[1/128{\int}_{i=1}^N1\times {10}^{L_{AEQ}/10}\right] $$
Descriptive univariate and bivariate analyses were conducted; for all inferential analyses, results were considered statistically significant where p < 0.05. We used multivariable Poisson regression [34] to model the relationship between the number of self-reported e-waste-related injuries and other occupational and non-occupational factors. Eq. 2 depicts the Poisson regression model, where the expected rate (E) of the number of injuries (Y) given the continuous and categorical predictor variables, x, equals the value of the effect on the predictors, α, added to the coefficient β′, or the multiplicative effect on the mean of (Y) as the result of x.
$$ \mathit{\log}\left(E\left(Y|x\right)\right)=\alpha +{\beta}^{\prime }x $$
We ran unadjusted (i.e., single predictor variable) and adjusted Poisson regression models on a number of potential injury risk factors. Variables tested in unadjusted models were chosen based on the participant responses to questions about work activities where they received the most injuries and the body sites reported where workers sustained injuries. Noise-related variables were selected due to the knowledge of the worksite conditions, worker responses to questions about noise exposure and a priori knowledge of noise related occupational injuries [4, 16, 17, 33, 38]. Each variable was tested individually in an unadjusted Poisson model to determine its effect on the outcome and on improvements in model fit (as assessed via the Akaike Information Criterion, AIC). A forward stepwise selection routine was used to select the final adjusted model. Two variables (age and self-reported health status) were forced into our adjusted model based on a priori assumptions derived from previous research [12, 22, 47], though it is important to note that these prior studies were conducted in formal (and not informal) work settings. However, recent studies on informal waste sites in Nigeria demonstrate a significant association of age to adverse outcomes of exposure and injury [41, 42].
We also performed a sensitivity analysis related to noise exposures. We repeated the final multivariable Poisson regression developed from all workers on the subset of workers who completed personal noise dosimetry, and replaced the self-reported noise exposure frequency variable with a variable representing measured noise exposure level in dBA.
Demographics and health status
Demographic characteristics are shown in Table 1 for all participants (N = 46 e-waste workers) and the subset of workers who completed personal noise dosimetry (N = 26). We did not track the total number of potential participants approached to participate, and so cannot report a participation rate. Among workers who declined to participate, the most common reason given was an insufficient financial incentive. No significant differences were noted between the total and subset samples (data not shown). Participants had an average age of about 25 ± 6 years old, and had been a resident at the site for an average of 5 ± 3 years. Slightly over half of had no formal education, and a similar fraction described their general health as fair or poor. The majority of participants were migrants from Northern Ghana who had traveled to Accra to pursue employment opportunities (data not shown). Dagbani was by far the most commonly-used interview language (65% of participants), and a slight majority of interviews were conducted by interviewer 1.
Table 1 Demographics information (n = 46 e-waste recycling workers)
Work-related activities
Participants had worked at the site an average of 5.7 ± 3.3 years, and worked an average of approximately 10 h/day (Table 2). Workers reported participating in a wide variety of activities, and generally did not specialize in any activity. The most commonly-reported activities were dismantling e-waste (reported by 83% of participants) collecting e-waste (80%) and sorting e-waste (78%). The least-commonly reported activities included lead smelting (reported by 39% of participants) and removing wire coverings from e-waste (28%).
Table 2 Work Characteristics, Activities, and Exposures (N = 46 male e-waste workers)a
The average score on the PSS and OSS was over 50% of the total possible points (PSS mean = 9.9 of 16 possible points, OSS mean = 19.7 of 32 possible points). Nineteen percent of participants reported impairments that limited their work abilities, and a minority of workers (43.5%) reported receiving any safety training prior to performing work that resulted in injury. Thirty-eight percent of the participants made less than 10 Ghanaian Cedis (about 2.20 USD) per day; for comparison, the average daily wage in Ghana during the time period of this study was 6 Ghanaian Cedis (about 1.30 USD, [1]). While seniority (i.e., more time on the job) is often associated with greater pay, the correlation between time at the site and pay was poor and insignificant.
Self-reported and measured noise exposures
The vast majority of workers (87%) reported frequent exposure to high noise on the job, and also reported being frequently bothered by this noise (78.3%, Table 2). Among the 26 workers with personal dosimetry, the average TWA for noise was 78.8 ± 5.9 dBA. Approximately 15% of TWA exposures exceeded the recommended 85 dBA exposure limit; all of these overexposures were associated with dismantling activities. Additionally, 73% of the workers reporting frequent exposure to high noise did dismantling. The Spearman correlation coefficient between measured noise level (in dBA) and self-reported noise exposure frequency among the participants who completed personal noise dosimetry was not significant (p = 0.21).
Injury experience and potential injury risk factors
A total of 426 injuries in the prior 6 months were reported across all 46 participants. The average number of injuries was 9.9 ± 9.6 per participant, though the range of injuries was quite large (1–40 injuries per participant, Table 3). The total number of days of work missed due to occupational injuries was 193; the average number of missed workdays for injuries during this period was 4 ± 2.6 days, although few participants were hospitalized for their injuries (6.5%). Most of the reported injuries were lacerations (65.2%), and the most common injury locations was the hand (45.7%). The PPE categories most commonly used were pants and foot protection (defined here as shoes, as opposed to the open sandals commonly used by workers on the site). Few workers reported wearing gloves; only 26% reported glove use during dismantling or sorting of e-waste. Fisher's exact tests of the associations between glove use and age, income, or seniority. No workers reported wearing hearing protection. The work activity associated with the most injuries was dismantling, an activity during which PPE use was rare (Fig. 1). PPE use was uncommon during all tasks, and at least 50% of injuries occurred when participants were not using any PPE.
Table 3 E-waste recycling injuries, activities, and use of personal protective equipment (N = 46 e-waste workers)
Number of injuries by work activity and use of personal protection equipment (N = 46 workers)
When all workers were considered, the Spearman correlation coefficient between PSS and injuries (Fig. 2) was significant, where higher stress levels were associated with more injuries (r = 0.42, p = 0.001). Age, self-reported health status, perceived noise exposure, and measured noise exposure were not significantly correlated with number of injuries. Age and stress were also not significantly correlated (data not shown).
Perceived stress score vs. number of e-waste related injuries (R2 = 0.12, p = 0.01)
Poisson modeling
Our unadjusted Poisson regression results identified a number of variables that were significantly associated with number of injuries. These include: Age, education, perceived noise, impairment that limits work, measured noise, glove use, occupational stress score, perceived health status, perceived stress scale, and training prior to injury. Also, being interviewed in Dagbani and being interviewed by interviewer 2 were both significantly associated with an increased number of injuries. When all 46 participants were included in our adjusted multivariate Poisson regression model with number of injuries as the outcome (adjusted model, Table 4), higher PSS, not using gloves, and higher perceived noise exposure frequency were significantly associated with a higher number of injuries. Not performing dismantling work, greater age, and better self-reported health status were significantly associated with a reduced number of injuries. Interview language and interviewer were not selected in the forward stepwise selection process for the final adjusted model based on lack of improvement in the AIC when these variables were included in the model.
Table 4 Unadjusted (i.e., single predictor variable) and adjusted (i.e., multivariable) Poisson regression models with number of injuries as the outcome (n = 46 e-waste recycling workers)
We completed a sensitivity analysis comparing the number of injuries associated with perceived noise exposure vs the measured noise levels from personal noise dosimetry (i.e., a multivariate Poisson model with the same variables in Table 4 but restricted to the subset of 26 participants with personal noise dosimetry). The subset of 26 workers did not differ significantly from those of the total sample for any of the variables assessed (data not shown). Our sensitivity analysis yielded model results which were generally very similar (data not shown); however, in this sensitivity analysis, higher measured noise levels were associated with a significant decrease in the number reported injuries (β = − 0.05 injuries per dBA, SE 0.01, p = 0.001).
This study is one of the first to evaluate injury risk factors among e-waste recycling workers. The participants in our study had experienced a substantial number of injuries in the 6 months prior to their participation in the study, many of them involving lacerations to the hand, a finding that has also been reported by other authors [9, 54]. Use of certain PPE, such as gloves, was found to be associated with a significantly reduced number of injuries. A number of factors, including participation in e-waste dismantling activities, higher perceived stress and more frequent perceived noise exposure, were associated with an increased number of injuries. These relationships remained strong even when controlling for other factors such as age, use of gloves, perceived health status, work activity, and measured noise levels. These findings suggest the need for increased attention to injury risks faced by e-waste recycling workers, and present possible opportunities for intervention, including increased public awareness, worker training programs, government intervention to address health and safety issues [46, 54], promotion of regulation, and government financing to enforce higher safety measures [52].
Although other studies have examined health-related issues among informal e-waste workers at the Agbogbloshie market site in Accra [23], our study filled an important gap by evaluating several upstream factors related to total worker health and safety outcomes. Our results supported our first hypothesis: greater perceived stress was associated with a higher number of injuries. Cohen's PSS measures perceived lack of control over personal stress associated with day-to-day issues [21], and our results indicated that workers had high levels of perceived stress which could influence their injury rate related to e-waste recycling work. This finding, coupled with the significant independent relationship of self-reported health status and number of injuries, suggests that intervention efforts focused on mental and physical health among e-waste recycling workers may be warranted. Additional studies of the relationship between occupational exposures and working conditions among informal e-waste recycling workers and perceived and objectively assessed health status are needed to confirm these findings [2, 23] and better characterize the environmental health issues associated with e-waste recycling work [2, 7, 25].
Our second hypothesis (that increased noise exposure would be associated with a significantly increased number of injuries) was also supported by our results. Greater perceived noise exposure was associated with a significant increase in number of injuries after controlling for other co-exposures. However, in our sensitivity analysis, higher objectively measured noise levels (quantified via personal noise dosimetry) showed the opposite effect, where increase noised levels were associated with a significant reduction in the number of injuries. It is possible that this is a spurious finding, given the very small sample size of 26 participants for the sensitivity analysis. One important issue that may help explain these divergent results is that our evaluation of noise exposures was not temporally aligned with our injury evaluation, which may have introduced additional bias. Our noise measurements covered a single 24-h period (the maximum duration we deemed possible while still being able to reliably recover our dosimeters), while our perceived noise items addressed frequency of high noise exposure but did not specify a time period, and our injury questions related to injuries experienced in the past 6 months. Given this situation, the perceived noise exposure frequency variable may be considered a more valid measure of long-term noise exposure, since participant reporting may have involved consideration of exposures over a period of weeks or months. Our results, combined with those of others [16, 19, 33, 36, 37], support a relationship between occupational noise exposure and injuries, and suggest a potential route for interventions intended to reduce noise exposure and, perhaps, risk of injury.
The total number of injuries sustained by the e-waste workers in this study exceeded 400 over a 6-month period, including one worker who reported 40 injuries. Work activities, and particularly dismantling, were significantly associated with injuries even after controlling for other co-variates. Most of the injuries reported were lacerations from sharp objects and occurred during dismantling activities. While approximately one-quarter of the participating workers reported using gloves during e-waste recycling work, use of PPE was quite low overall. This likely reflects at least two factors: first, a lack of sufficient individual-level financial resources to individually purchase PPE (necessary, given the absence of a formal employer); and second, a likely lack of access to commercially-available supplies of PPE. However, we did not survey workers with regards to where they obtained gloves and other PPE, so there is substantial uncertainty regarding both PPE origin and pricing. Our results did, however, indicate that glove use was not significantly associated with age, income, or seniority. It is possible that the practice of handwashing before prayer (Wudo) in the Muslim tradition influences the decision to seek out protection for the hands. If worker's hands are not defiled prior to prayer, they do not have to practice Wudo, which can be challenging given the lack of sanitation on the site. Therefore, it is possible that glove use may have more to do with religious practice than with occupational safety for at least some workers, in an effort to avoid having to purchase bottled or sachet water for Wudo.
In addition to injury hazards, participants reported long hours of work. Other studies have documented dangerous physical working conditions [3, 26, 39] and identified a relationship between mental health factors and occupational injury in other work settings [14, 28, 32]. Unfortunately, e-waste recycling workers at Agbogbloshie have previously reported insufficient income to afford the Ghanaian health insurance scheme, and as a result many of their injuries were left untreated [43, 54]. This is an important environmental justice issue; the informal e-waste recycling industry in Ghana has been reported to generate millions of dollars in revenue for the country annually [20], but the benefits to e-waste recycling workers have been limited. The economic opportunities e-waste recycling work presents for unskilled and often undocumented labor has likely hindered formalization of the work processes and, as a result, possibly slowed adoption of improved health and safety practices [29,30,31, 35, 40, 51, 53].
This study has a number of limitations. First is the small sample size and cross-sectional nature of the study. While most participants had worked at the site for several years, and anecdotal reports indicated that conditions and work practices do not appear to have changed substantially over that period, it is nevertheless possible that injury risks could have changed in meaningful ways over our 6-month injury reporting window. This could introduce positive or negative bias into our estimates of injury frequency. Also, because exposures and injury experiences were evaluated simultaneously, we were unable to assess causality; we cannot determine, for example, whether workers who reported poor health status have more injuries as a result of that health status, or vice versa. Second, our assessment of injury risk may have been influenced by the healthy worker effect, and our injury estimates may have been biased negatively by the exclusion of less healthy workers who are no longer doing e-waste recycling work due to illness or injury. Third, our assessment of work activities conducted by the workers was completed using activities reported to us by participants, that we then collapsed into smaller post hoc categories. We used this approach to try to reduce the large number of specific work activities reported by individual workers into a manageable number of broader job categories. However, in doing so, we may have introduced misclassification of work activities, which could bias our understanding of injury risks associated with our defined work activities. Fourth, our participants came from Northern Ghana and other West African countries and spoke a number of languages; it is possible that errors or miscommunication by our translators, differences in the context and interpretation of some questions, or social response biases introduced by perceived or actual social class differences between our participants and translators, may have introduced additional misclassification. This possibility is reflected in the significant univariate Poisson regression results for interview language and interviewer. The cultural appropriateness of the PSS and OSS instruments we used has not been validated in this population. Fifth, the temporal misalignment between our objective and subjective measures of noise exposure may at least partially explain the poor correlation between objective or subjective noise exposures, as well as the inconsistencies in our injury risk models that included a noise exposure variable. Finally, although we emphasized to participants that their responses would be confidential and would never be shared with others, some workers may have purposely misrepresented the number of injuries experienced or health status to avoid perceived adverse impacts on their employment.
Our study is one of the first to evaluate injury experience among informal e-waste recycling workers and highlights the hazardous working conditions present at sites like Agbogbloshie. Our results suggest that these workers have an elevated risk of injury and experience reduced perceived health status and elevated levels of perceived and occupational stress. Additional studies with larger sample sizes and longitudinal study designs are needed to better characterize the injury risk among informal e-waste recycling workers. If future studies also demonstrate elevated risk of injury among these vulnerable workers, international programs and research efforts are needed to enhance the safety and economic sustainability of informal e-waste work.
AIC:
Akaike Information Criterion
A-weighted decibels
Low- or Middle-Income Country
OSS:
Occupational Stress Scale
PPE:
PSS:
Perceived Stress Scale
TWA:
Time-Weighted Average
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The authors are indebted to the participants, without whom this research would not have been possible. The authors also wish to acknowledge Professors Julius Fobil and Emmanuel Assampong, Judith Stephens, Krystin Carlson, Rachel Long, Mozhgon Rajaee, Kwaku Badu-Dwuma, and Rollin Kofi for their assistance with data collection.
This research was supported by a pilot project research training grant from the Center for Occupational Health and Safety Engineering (COHSE) at the University of Michigan, an Education and Research Center supported by training grant No. 2T42OH008455 from the Centers for Disease Control and Prevention/National Institute for Occupational Safety and Health. The contents of this paper are solely the responsibility of the authors and do not represent official views of the National Institute for Occupational Safety and Health. The research was also supported by the University of Michigan Office of Research, the University of Michigan School of Public Health, and the University of Michigan Rackham Graduate School.
The datasets collected and analysed as part of the current study are available from the corresponding author on reasonable request.
Department of Environmental Health Sciences, University of Michigan, 1415 Washington Heights 6611 SPH I, Ann Arbor, MI, 48109, USA
Katrina N. Burns
, Stephanie K. Sayler
& Richard L. Neitzel
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RLN conceived and guided execution of the study; SKS performed onsite data collection and project management; KNB performed data analysis; KNB, SKS, and RLN wrote the paper. All authors read and approved the final manuscript.
Correspondence to Richard L. Neitzel.
All research procedures were performed in accordance with the Declaration of Helsinki and approved by the University of Michigan Institutional Review Board (HUM00084062) and the University of Ghana Institutional Review Board at the Noguchi Memorial Institute for Medical Research (NMIMR-IRB CPN 070/13-14). All participants provided written informed consent prior to participation.
Burns, K.N., Sayler, S.K. & Neitzel, R.L. Stress, health, noise exposures, and injuries among electronic waste recycling workers in Ghana. J Occup Med Toxicol 14, 1 (2019) doi:10.1186/s12995-018-0222-9
Electronic waste recycling | CommonCrawl |
BMC Emergency Medicine
Validation of age-specific survival prediction in pediatric patients with blunt trauma using trauma and injury severity score methodology: a ten-year Nationwide observational study
Chiaki Toida ORCID: orcid.org/0000-0003-4551-48411,2,
Takashi Muguruma2,
Masayasu Gakumazawa2,
Mafumi Shinohara2,
Takeru Abe2,
Ichiro Takeuchi2 &
Naoto Morimura1
BMC Emergency Medicine volume 20, Article number: 91 (2020) Cite this article
In-hospital mortality in trauma patients has decreased recently owing to improved trauma injury prevention systems. However, no study has evaluated the validity of the Trauma and Injury Severity Score (TRISS) in pediatric patients by a detailed classification of patients' age and injury severity in Japan. This retrospective nationwide study evaluated the validity of TRISS in predicting survival in Japanese pediatric patients with blunt trauma by age and injury severity.
Data were obtained from the Japan Trauma Data Bank during 2009–2018. The outcomes were as follows: (1) patients' characteristics and mortality by age groups (neonates/infants aged 0 years, preschool children aged 1–5 years, schoolchildren aged 6–11 years, and adolescents aged 12–18 years), (2) validity of survival probability (Ps) assessed using the TRISS methodology by the four age groups and six Ps-interval groups (0.00–0.25, 0.26–0.50, 0.51–0.75, 0.76–0.90, 0.91–0.95, and 0.96–1.00), and (3) the observed/expected survivor ratio by age- and Ps-interval groups. The validity of TRISS was evaluated by the predictive ability of the TRISS method using the receiver operating characteristic (ROC) curves that present the sensitivity, specificity, positive predictive value, negative predictive value, accuracy, area under the receiver operator characteristic curve (AUC) of TRISS.
In all the age categories considered, the AUC for TRISS demonstrated high performance (0.935, 0.981, 0.979, and 0.977). The AUC for TRISS was 0.865, 0.585, 0.614, 0.585, 0.591, and 0.600 in Ps-interval groups (0.96–1.00), (0.91–0.95), (0.76. − 0.90), (0.51–0.75), (0.26–0.50), and (0.00–0.25), respectively. In all the age categories considered, the observed survivors among patients with Ps interval (0.00–0.25) were 1.5 times or more than the expected survivors calculated using the TRISS method.
The TRISS methodology appears to predict survival accurately in Japanese pediatric patients with blunt trauma; however, there were several problems in adopting the TRISS methodology for younger blunt trauma patients with higher injury severity. In the next step, it may be necessary to develop a simple, high-quality prediction model that is more suitable for pediatric trauma patients than the current TRISS model.
Trauma scoring methods for survival prediction in trauma patients are essential to assess the quality of trauma care because they permit valid comparison of trauma patients who have different anatomical and physiological severities [1]. The Trauma and Injury Severity Score (TRISS) method has been commonly used to calculate the statistical survival probability in trauma patients since its introduction in 1987 by Boyd et al. [2]. After the validation of the revised-version of TRISS by the American College of Surgeons Committee on Trauma coordinated Major Trauma Outcome Study (MTOS) [3, 4] in the Japanese cohort, the TRISS method is reported as a standard technique for estimating survival probability and has commonly been used for evaluating the quality of trauma care [5,6,7,8].
The accuracy of the TRISS method, nevertheless, has various challenges in terms of the investigated area, time, and age. First, previous studies suggested that the TRISS has a low accuracy for survival prediction in patients with higher severity of the injury or younger pediatric patients [9, 10]. Second, previous studies suggested that there is a trend to improve the observed-to-expected mortality ratio in major trauma patients, and therefore, new coefficients should be calculated according to these improvements in trauma care for the TRISS to maintain the accuracy for survival prediction [9, 11]. Finally, there are also studies indicating that the modified TRISS methodology with local database-derived coefficients might enhance the accuracy of survival prediction in all regions except the USA, wherein the original TRISS methodology was developed because there are marked differences by region such as in Asian countries [12, 13].
Although the birth rate and mortality of the Japanese population have changed yearly [8, 14], to the best of our knowledge, no study has evaluated the validity of the TRISS method in a pediatric cohort by detailed classification of patients' age and severity in Japan. Therefore, this study aimed to evaluate the validity of the TRISS method in predicting the survival of Japanese pediatric patients with blunt trauma by detailed classification of age (neonates/infants, preschool children, schoolchildren, and adolescents) and severity of the injury. This study analyzed data obtained from the Japan Trauma Data Bank (JTDB) for the 10-year study period during 2009–2018.
Study design, setting and population
This retrospective, nationwide, observational study analyzed data obtained from the JTDB, which registers data of patients with trauma and/or burn and records prehospitalization and hospital-related information. The JTDB records data of demographics, comorbidities, injury types, mechanism of injury, means of transportation, vital signs, Abbreviated Injury Scale (AIS) score, Injury Severity Score (ISS), prehospital/in-hospital procedures, trauma diagnosis as indicated using the AIS, and clinical outcomes. In most cases, physicians who are trained in AIS coding by using the 1990 revision of AIS [15] undertake the online registration of individual patient data. The JTDB data collection started in 55 hospitals in 2003. The number of participating hospitals in the JTDB registry increased yearly, up to a total of 280 hospitals, including 92% of Japanese government-approved tertiary emergency medical centers in March 2019. The Japan Association for the Surgery of Trauma permits open access and update of existing medical information and the Japan Association for Acute Medicine evaluates the submitted data.
Figure 1 shows a flow diagram of the patient disposition. In this study, we used a JTDB dataset that included information for the period January 1, 2009, to December 31, 2018, which initially yielded the data of 313,643 patients. The inclusion criteria for this study were as follows: the presence of trauma and age 18 years or less. Patients aged 19 years or more, with burns or penetrating trauma, with cardiac arrest on hospital arrival, or with missing data of outcome and TRISS prediction were excluded from this study. Among 26,329 patients with blunt trauma and younger than 18 years, 2480 (9.4%) patients had missing data of survival and 5446 (20.7%) patients had the missing data for TRISS predictor, and hence, the survival probability (Ps) was not calculated using the TRISS method. Furthermore, 683 (2.6%), 1948 (7.4%), 1608 (6.1%), and 3824 (14.5%) patients had missing data of ISS, Glasgow Coma Scale (GCS) score, systolic blood pressure (sBP), and respiratory rate (RR), respectively. Table S1 shows the number of patients who had missing data by age category and each variable.
Flow diagram of the study patient disposition
We collected information on the following variables from the JTDB: age (years), sex, AIS, AIS of the injured region, Revised Trauma Score [3], ISS [10], Ps, and in-hospital mortality. The TRISS ranges from 0 (certain death) to 1 (certain survival), and the survival probability (Ps) is calculated as follows:
$$ \mathrm{TRISS}=\mathrm{Ps}=1/\left(1+{\mathrm{e}}^{\hbox{-} \mathrm{b}}\right) $$
where b = b0 + b1(RTS) + b2(ISS) + b3(age).
RTS is calculated using the GCS score, the sBP, and the RR.
$$ \mathrm{RTS}=0.9368\ast \mathrm{GCS}+0.7326\ast \mathrm{sBP}+0.2908\ast \mathrm{RR} $$
The outcomes were as follows: (1) patients' characteristics and mortality by age groups (neonates/infants aged 0 years, preschool children aged 1–5 years, schoolchildren aged 6–11 years, and adolescents aged 12–18 years), (2) validity of Ps assessed using the TRISS methodology by the four age groups and six Ps-interval groups (0.00–0.25, 0.26–0.50, 0.51–0.75, 0.76–0.90, 0.91–0.95, and 0.96–1.00), and (3) the observed/expected survivor ratio by age- and Ps-interval groups. In the primary analysis, which was conducted to identify the characteristics of pediatric trauma patients during the study period, the Mann–Whitney U test and the Kruskal–Wallis test were used for analyzing continuous variables, whereas, a chi-square test was used for analyzing categorical variables. In the secondary analysis, the validity of TRISS was evaluated by the predictive ability of the TRISS method using the receiver operating characteristic (ROC) curves that present the sensitivity, specificity, positive predictive value, negative predictive value, accuracy, area under the receiver operator characteristic curve (AUC), and its 95% confidence interval (CI) of TRISS and show the ability of TRISS to distinguish between positive and negative outcomes. The AUC varies as < 0.7 (low performance), 0.7–0.9 (moderate performance), and > 0.9 (high performance) [16]. In the third analysis, the expected survival calculated using TRISS Ps was compared with the actual Ps. The expected number of survivors in each Ps-interval group was calculated by integrating mean Ps and the number of patients for six Ps-interval group. The results of these comparisons are expressed as the medians and interquartile ranges (IQRs; 25th–75th percentile) for continuous variables and as the mean and percentages for categorical variables. All statistical analyses were performed using STATA/SE software, version 16.0 (StataCorp; College Station, Texas, USA). A two-tailed P-value of less than 0.05 indicated statistical significance.
During the 10-year study period, the data of 17,745 pediatric patients with blunt trauma were included (Fig. 1). The median age and Ps of the total cohort were 13 years (IQR, 8–17) and 0.99 (IQR, 0.98–0.99), respectively. The overall in-hospital mortality rate was 2.1%.
Table 1 shows the demographic and characteristics and variables by age. There were significant differences in all variables by the age category considered, except for neck injury with AIS ≥ 3. Neonates/infants had the highest percentage of head injury with AIS ≥ 3 (88%), highest mean ISS, lowest RTS, and lowest median Ps compared to those of the other age categories.
Table 1 Comparison of demographic characteristics and variables by age groups
Table 2 shows the accuracy and AUC of TRISS for each age category. In all age categories, the AUC of TRISS demonstrated high performance (0.935, 0.981, 0.979, and 0.977). Table 3 shows the accuracy and AUC of TRISS by each Ps-interval group. The AUC for TRISS was 0.865, 0.585, 0.614, 0.585, 0.591, and 0.600 in Ps-interval groups (0.96–1.00), (0.91–0.95), (0.76. − 0.90), (0.51–0.75), (0.26–0.50), and (0.00–0.25), respectively. The AUC of TRISS demonstrated moderate performance in the Ps-interval (0.96–1.00) group (AUC, 0.865); however, the AUC of TRISS demonstrated low performance in other Ps-interval groups.
Table 2 Validation analysis and AUC of the TRISS model by age groups
Table 3 Validation analysis and AUC of the TRISS model by survival probability interval
Table 4 shows the observed-to-expected survivor ratio in the Ps interval by age category. In all the age categories considered, the observed survivors among patients with Ps interval (0.00–0.25) were 1.5 times or more than the expected survivors calculated using the TRISS method.
Table 4 Observed-to-expected survivor ratio in each Ps interval by age categories
We evaluated the validity of the TRISS method in Japanese pediatric patients with blunt trauma by age-group and severity of injury from the JTDB registry during 2009–2018. This study showed that the performance of the TRISS methodology was lower in the case of survival prediction for pediatric patients with younger age and/or Ps ≤ 0.95, and TRISS underestimated expected survivors in pediatric patients with Ps ≤ 0.25.
Because the accuracy of the TRISS model may reflect the influence of demographic differences in trauma such as the trauma care system or the population structure between the sample area and the USA, wherein the TRISS method was developed, local database-derived coefficients may further enhance the predictive performance of the TRISS [12, 13, 17, 18]. Previous studies based on a Japanese cohort, including children registered in the JTDB during 2005–2008 and 2009–2013 proved that the AUC of TRISS was 0.962 and 0.948 [9, 17]. These Japanese studies focused on pediatric patients with blunt trauma and demonstrated that the TRISS method had a high performance only for Japanese pediatric patients, as in a previous study [9, 18]. Although it is difficult to compare the results between previous studies and this study because of the different periods when the studies were conducted, our results suggest that the TRISS model may be appropriate for Japanese pediatric patients with blunt trauma. However, there is no unified consensus on whether TRISS is a suitable prediction model for pediatric patients. One study recommended the use of the TRISS methodology for both adult and pediatric patients because both TRISS models with and without pediatric coefficients equally predict survival with high performance in pediatric patients with blunt trauma [13, 19]. In the other study, the TRISS model had significantly lower performance than the revised TRISS model based on age-adjusted weights (AUC, 0.785 vs. 0.985, p < 0.05)) [10, 20]. Our results suggest several problems in adopting the TRISS model for pediatric blunt trauma patients of all ages or all severity, although our results showed that TRISS had high performance in the overall pediatric cohort.
In this subclass analysis by age category, the accuracy of the TRISS model for neonates/infants was lower than that of the other age categories. First, the neonate/infant group sustained the largest proportion of severe head injury with ISS ≥ 3 in this study. A previous study showed that the accuracy of the TRISS model for pediatric trauma patients with head injury or younger than 5 years was significantly inferior to that for the other pediatric-specific model [10]. Second, the abovementioned finding might be attributed to the higher proportion of patients with a head injury in neonates /infants than the other age-groups in this study cohort (81% vs 31–39%, P < 0.001). Finally, another reason was considered that the evaluation of physiological status parameters such as GCS, sBP, and RR is challenging owing to their age-related variation and limited verbal communications/motor responses [21, 22]. There is a possibility of bias while evaluating the physiological status in younger pediatric patients and this may be reflected in the result of this study with a large rate of missing data of physiological status parameters in neonate/infant patients than the other age-groups (Table S1). Our results may suggest that a dataset with high-quality and without missing data may contribute to improving the accuracy of TRISS in predicting the survival of pediatric patients; however, improving the trauma database would arguably be difficult to achieve. Moreover, a previous study showed that RR data, which are missing in most cases in the Japan JTDB dataset, might be less needed for the calculation of TRISS Ps accuracy [9, 17, 23] and suggested that it may be effective to reduce the number of parameters or changes in the parameter in the prediction model for improving the accuracy of the model [24]. In the next step of research, therefore, not only modifying the coefficient of the TRISS model but also developing a new different prediction model that requires only easily collected and fewer missing data, may be necessary to improve the accuracy of survival prediction for pediatric trauma patients.
In the subclass analysis by TRISS Ps-interval groups, the accuracy of the performance of the TRISS model for patients with Ps ≤ 0.95 was low and the observed-to-expected mortality ratio in pediatric patients with Ps ≤ 0.25 was 2.15. Previous studies also suggested similar results, which are as follows: TRISS had lower performance in Japanese blunt trauma patients with Ps < 0.9 than those with Ps ≥ 0.9 [9] and TRISS underestimates survival for pediatric trauma patients with TRISS Ps ≤ 91% [10]. Previous studies suggested that the decreasing trend of in-hospital mortality among trauma patients decreased in recent years would lead the TRISS model to be out of calibration [8, 11]. Previous studies conducted using the JTDB data suggested that improvements in trauma care and trauma care systems account for decreasing mortality, especially in major trauma after the Japan Advanced Trauma Evaluation and Care was introduced in 2002 [7, 8, 25]. Therefore, our results may suggest that new coefficients related to injury severity should be calculated periodically to keep up with changes in trauma care in their own country.
Our study had several limitations. First, there was a selection bias because not all Japanese hospitals that treat have registered in the JTDB. Table S1 shows the rate of missing data by age category in the JTDB dataset. The number of neonates/infants with blunt trauma was lowest (N = 771, 2.9% of all), but the proportion of patients with missing data on survival and TRISS prediction was the largest (53.7% of neonate/infants with blunt trauma). These might have an adverse effect on the prediction accuracy of TRISS in neonates/infants. Therefore, a dataset with high-quality and without missing data should be constructed to improve the accuracy of TRISS in predicting the survival of pediatric patients. In addition, the number of participating hospitals differed across the study period. Furthermore, pediatric blunt trauma patients younger than 18 years whose data were registered in the JTDB (N = 7926, 30.1%) had missing data on important variables, although selection bias occurred in the data set with more than 10% missed rate [23]. Although this study population represents the Japanese trauma experience, our results may be nearly close to those obtained in many other Asian countries such as South Korea, Hong Kong, and Thailand, where trauma patient demographics are similar [12, 18, 26]. Our study attempted to utilize cross-validation procedures to assess the validity of the results obtained. In the next step, assessing the quality of the trauma care exactly by using the survival prediction model with higher accuracy than the current TRISS method could be achieved by using the data of each hospitals and type of trauma [1]. Therefore, developing a new regression model that is more suitable to the country's situation, would result in better outcomes of trauma patients in that country, contributing to a decrease in the number of preventable trauma deaths.
This study showed that overall the TRISS methodology appears to accurately predict survival in Japanese pediatric patients with blunt trauma. However, there were several problems in adopting the TRISS model for blunt trauma patients who are younger and/or with higher injury severity. In the future, it may be necessary to consider developing a simple, high-quality prediction model that is more suitable for pediatric trauma patients than the current TRISS model.
The datasets supporting the conclusions of this article are available from the corresponding author on reasonable request.
TRISS:
Trauma and Injury Severity Score
AUC:
Survival probability
MOTS:
Major Trauma Outcome Study
JTDB:
Japan Trauma Data Bank
AIS:
Abbreviated Injury Scale
ISS:
Injury Severity Score
GCS:
Glasgow Coma scale
sBP:
Respiratory rate
Receiver operating characteristic
IQRs:
Interquartile ranges
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The authors acknowledge Editage (https://www.editage.jp) for assistance in English language editing.
Department of Disaster Medical Management, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8655, Japan
Chiaki Toida & Naoto Morimura
Department of Emergency Medicine, Yokohama City University Graduate School of Medicine, 4-57 Urafunecho, Minami-ku, Yokohama, 232-0024, Japan
Chiaki Toida, Takashi Muguruma, Masayasu Gakumazawa, Mafumi Shinohara, Takeru Abe & Ichiro Takeuchi
Chiaki Toida
Takashi Muguruma
Masayasu Gakumazawa
Mafumi Shinohara
Takeru Abe
Ichiro Takeuchi
Naoto Morimura
Conceptualization C.T. and T.M., methodology C.T., software C.T., and T.A., validation C.T. T.M. T.A. M.G. and M.S., formal analysis C.T., investigation C.T. T.M. T.A. M.G., and T.A., resources C.T. and T.A., data curation C.T. and T.A., writing—original draft preparation C.T., writing—review and editing C.T. T.M. T.A. M.G. T.A. I.T. and N.M., visualization C.T., supervision N.M., project administration and funding acquisition C.T. All authors have read and agreed to the final version of the manuscript.
Correspondence to Chiaki Toida.
This study was approved by the Institutional Ethics Committees of Yokohama City University Medical Centre (approval no. B170900003). The approving authority for data access was the Japanese Association for the Surgery of Trauma (Trauma Registry Committee). The need for consent to individual participate was waived by the Institutional Ethics Committees that approved our study due to the observational nature of the study design.
Additional file 1: Supplement 1.
Number of patients with missing data by age group and for each variable.
Toida, C., Muguruma, T., Gakumazawa, M. et al. Validation of age-specific survival prediction in pediatric patients with blunt trauma using trauma and injury severity score methodology: a ten-year Nationwide observational study. BMC Emerg Med 20, 91 (2020). https://doi.org/10.1186/s12873-020-00385-0
Trauma scoring system
Trauma emergency medicine
Submission enquiries: [email protected] | CommonCrawl |
Cantor-Bernstein-Schröder Theorem
This article was Featured Proof between 14 September 2008 and 22 September 2008.
2 Proof 1
4.1 $z$ is an increasing mapping
7.1 $E$ is increasing
8 Law of the Excluded Middle
9 Also known as
11 Source of Name
If a subset of one set is equivalent to the other, and a subset of the other is equivalent to the first, then the two sets are themselves equivalent:
$\forall S, T: T \sim S_1 \subseteq S \land S \sim T_1 \subseteq T \implies S \sim T$
Alternatively, from Equivalence of Definitions of Dominate (Set Theory), this can be expressed as:
$\forall S, T: T \preccurlyeq S \land S \preccurlyeq T \implies S \sim T$
where $T \preccurlyeq S$ denotes the fact that $S$ dominates $T$.
That is:
If $\exists f: S \to T$ and $\exists g: T \to S$ where $f$ and $g$ are both injections, then there exists a bijection from $S$ to $T$.
Proof 1
From the facts that $T \sim S_1$ and $S \sim T_1$, we can set up the two bijections:
$f: S \to T_1$
$g: T \to S_1$
Let:
$S_2 = g \left({f \left({S}\right)}\right) = g \left({T_1}\right) \subseteq S_1$
$T_2 = f \left({g \left({T}\right)}\right) = f \left({S_1}\right) \subseteq T_1$
So $S_2 \subseteq S_1$ and $S_2 \sim S$, while $T_2 \subseteq T_1$, and $T_2 \sim T$.
For each natural number $k$, let $S_{k+2} \subseteq S$ be the image of $S_k$ under the mapping $g \circ f$.
This article, or a section of it, needs explaining, namely:
Why do these superset relations hold?
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this point in more detail, feel free to use the talk page.
If you are able to explain it, then when you have done so you can remove this instance of {{Explain}} from the code.
Then $S \supseteq S_1 \supseteq S_2 \supseteq \ldots \supseteq S_k \supseteq S_{k+1} \ldots$.
Let $\displaystyle D = \bigcap_{k=1}^\infty S_k$.
Why can we represent it so?
Now we can represent $S$ as:
\(\displaystyle S\) \(=\) \(\displaystyle \left({S \setminus S_1}\right) \cup \left({S_1 \setminus S_2}\right) \cup \ldots \cup \left({S_k \setminus S_{k+1} }\right) \cup \ldots \cup D\) $(1)$
where $S \setminus S_1$ denotes set difference.
This article, or a section of it, needs explaining.
Similarly, we can represent $S_1$ as:
\(\displaystyle S_1\) \(=\) \(\displaystyle \left({S_1 \setminus S_2}\right) \cup \left({S_2 \setminus S_3}\right) \cup \ldots \cup \left({S_k \setminus S_{k+1} }\right) \cup \ldots \cup D\) $(2)$
Now let:
\(\displaystyle M\) \(=\) \(\displaystyle \left({S_1 \setminus S_2}\right) \cup \left({S_3 \setminus S_4}\right) \cup \left({S_5 \setminus S_6}\right) \cup \ldots\)
\(\displaystyle N\) \(=\) \(\displaystyle \left({S \setminus S_1}\right) \cup \left({S_2 \setminus S_3}\right) \cup \left({S_4 \setminus S_5}\right) \cup \ldots\)
\(\displaystyle N_1\) \(=\) \(\displaystyle \left({S_2 \setminus S_3}\right) \cup \left({S_4 \setminus S_5}\right) \cup \left({S_6 \setminus S_7}\right) \cup \ldots\)
... and rewrite $(1)$ and $(2)$ as:
\(\displaystyle S\) \(=\) \(\displaystyle D \cup M \cup N\) $(3)$
\(\displaystyle S_1\) \(=\) \(\displaystyle D \cup M \cup N_1\) $(4)$
\(\displaystyle g \circ f \left({S \setminus S_1}\right) = \left({S_2 \setminus S_3}\right)\) \(\implies\) \(\displaystyle \left({S \setminus S_1}\right) \sim \left({S_2 \setminus S_3}\right)\)
\(\displaystyle g \circ f \left({S_2 \setminus S_3}\right) = \left({S_4 \setminus S_5}\right)\) \(\implies\) \(\displaystyle \left({S_2 \setminus S_3}\right) \sim \left({S_4 \setminus S_5}\right)\)
This article contains statements that are justified by handwavery.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding precise reasons why such statements hold.
If you are able to do this, then when you have done so you can remove this instance of {{Handwaving}} from the code.
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So $N \sim N_1$.
It follows from $(3)$ and $(4)$ that a bijection can be set up between $S$ and $S_1$.
But $S_1 \sim T$.
Therefore $S \sim T$.
Suppose $S \preccurlyeq T$ and $T \preccurlyeq S$.
By definition, we have that there exist injections $f: S \to T$ and $g: T \to S$.
We are going to try to build a sequence $t_1, s_1, t_2, s_2, t_3 \ldots$ as follows.
Consider any $t_1 \in T$.
By Law of Excluded Middle there are two choices:
\(\displaystyle \exists s_1 \in S: f \left({s_1}\right)\) \(=\) \(\displaystyle t_1\)
\(\displaystyle \neg \exists s_1 \in S: f \left({s_1}\right)\) \(=\) \(\displaystyle t_1\)
Suppose $\exists s_1 \in S: f \left({s_1}\right) = t_1$.
Because $f$ is injective, such an $s_1$ is unique.
So we can choose $s_1 = f^{-1} \left({t_1}\right)$.
Again, by Law of Excluded Middle there are two further choices:
\(\displaystyle \exists t_2 \in T: g \left({t_2}\right)\) \(=\) \(\displaystyle s_1\)
\(\displaystyle \neg \exists t_2 \in T: g \left({t_2}\right)\) \(=\) \(\displaystyle s_1\)
Suppose $\exists t_2 \in T: g \left({t_2}\right) = s_1$.
Because $f$ is injective, such an $t_2$ is unique.
Similarly, we choose $s_2 = f^{-1} \left({t_2}\right)$, if it exists.
This process goes on until one of the following happens:
We reach some $s_n \in S$ such that $\neg \exists t \in T: g \left({t}\right) = s_n$. This may be possible because $g$ may not be a surjection.
We reach some $t_n \in T$ such that $\neg \exists s \in S: f \left({s}\right) = t_n$.This may be possible because $f$ may not be a surjection.
The process goes on for ever.
For each $t \in T$, then, there is a well-defined process which turns out in one of the above three ways.
We partition $T$ up into three subsets that are mutually disjoint:
Let $T_A = \{$ all $t \in T$ such that the process ends with some $s_n\}$
Let $T_B = \{$ all $t \in T$ such that the process ends with some $t_n\}$
Let $T_C = \{$ all $t \in T$ such that the process goes on for ever$\}$.
We can do exactly the same thing with the elements of $S$:
Let $S_A = \{$ all $s \in S$ such that the process ends with some $s_n\}$
Let $S_B = \{$ all $s \in S$ such that the process ends with some $t_n\}$
Let $S_C = \{$ all $s \in S$ such that the process goes on for ever$\}$.
What we need to do is show that $S \sim T$.
We do this by showing that $S_A \sim T_A$, $S_B \sim T_B$ and $S_C \sim T_C$.
The restriction of $f$ to $S_A$ is a bijection from $S_A$ to $T_A$.
To do this we need to show that:
$s \in S_A \implies f \left ({s}\right) \in T_A$;
$\forall t \in T_A: \exists s \in S_A: f \left ({s}\right) = t$.
Let $s \in S_A$. Then the process applied to $s$ ends in $S$.
Now consider the process applied to $f \left ({s}\right)$. Its first step leads us back to $s$. Then it continues the process, applied to $s$, and ends up in $S$. Thus $f \left ({s}\right) \in T_A$.
Thus $s \in S_A \implies f \left ({s}\right) \in T_A$.
Now suppose $t \in T_A$. Then the process applied to $t$ ends in $S$.
In particular, it must have a first stage, otherwise it would end in $T$ with $t$ itself.
Hence $t = f \left ({s}\right)$ for some $s$.
But the process applied to this $s$ is the same as the continuation of the process applied to $t$, and therefore it ends in $S$.
Thus $s \in S_A$ as required.
Hence the restriction of $f$ to $S_A$ is a bijection from $S_A$ to $T_A$.
We can use the same argument to show that $g: T_B \to S_B$ is also a bijection. Hence $g^{-1}: S_B \to T_B$ is a bijection.
Finally, suppose $t \in T_C$.
Because $f$ is an injection, $t = f \left({s}\right)$ for some $s$, and the process applied to $t$ must start.
And this $s$ must belong to $S_C$, because the process starting from $s$ is the same as the process starting from $t$ after the first step. This never ends, as $t \in T_C$.
Now we can define a bijection $h: S \to T$ as follows:
$\displaystyle h \left({x}\right) = \begin{cases} f \left({x}\right): x \in S_A \\ f \left({x}\right): x \in S_C \\ g^{-1} \left({x}\right): x \in S_B \end{cases}$
The fact that $h$ is a bijection follows from the facts that:
$(1): \quad S_A$, $S_B$ and $S_C$ are mutually disjoint
$(2): \quad T_A$, $T_B$ and $T_C$ are mutually disjoint
$(3): \quad f$, $f$ and $g^{-1}$ are the bijections which respectively do the mappings between them.
Let $S, T$ be sets, and let $\mathcal P \left({S}\right), \mathcal P \left({T}\right)$ be their power sets.
Let $f: S \to T$ and $g: T \to S$ be injections that we know to exist between $S$ and $T$.
Consider the relative complements of elements of $\mathcal P \left({S}\right)$ and $\mathcal P \left({T}\right)$ as mappings:
$\complement_S: \mathcal P \left({S}\right) \to \mathcal P \left({S}\right): \forall X \in \mathcal P \left({S}\right): \complement_S \left({X}\right) = S \setminus X$
$\complement_T: \mathcal P \left({T}\right) \to \mathcal P \left({T}\right): \forall Y \in \mathcal P \left({T}\right): \complement_T \left({Y}\right) = T \setminus Y$
which follow directly from the definition of relative complement.
Let $\alpha$ and $\beta$ denote the mappings induced on $\mathcal P \left({S}\right)$ and $\mathcal P \left({T}\right)$ by $f$ and $g$, respectively.
Consider the mapping $z: \mathcal P \left({S}\right) \to \mathcal P \left({S}\right)$ defined by the composition:
$z = \complement_S \circ \beta \circ \complement_T \circ \alpha$
$z$ is an increasing mapping
Let $A, B \in \mathcal P \left({S}\right)$ with $A \subseteq B$.
\(\displaystyle \alpha \left({A}\right)\) \(\subseteq\) \(\displaystyle \alpha \left({B}\right)\) Image of Subset under Relation is Subset of Image: Corollary 2
\(\displaystyle \implies \ \ \) \(\displaystyle \left({\complement_T \circ \alpha}\right) \left({A}\right)\) \(\supseteq\) \(\displaystyle \left({\complement_T \circ \alpha}\right) \left({B}\right)\) Set Complement inverts Subsets
\(\displaystyle \implies \ \ \) \(\displaystyle \left({\beta \circ \complement_T \circ \alpha}\right) \left({A}\right)\) \(\supseteq\) \(\displaystyle \left({\beta \circ \complement_T \circ \alpha}\right) \left({B}\right)\) Image of Subset under Relation is Subset of Image: Corollary 2
\(\displaystyle \implies \ \ \) \(\displaystyle \left({\complement_S \circ \beta \circ \complement_T \circ \alpha}\right) \left({A}\right)\) \(\subseteq\) \(\displaystyle \left({\complement_S \circ \beta \circ \complement_T \circ \alpha}\right) \left({B}\right)\) Set Complement inverts Subsets
\(\displaystyle \implies \ \ \) \(\displaystyle z \left({A}\right)\) \(\subseteq\) \(\displaystyle z \left({B}\right)\) Definition of $z$
$\Box$
By the Knaster-Tarski Lemma: Power Set, there is a $\mathbb G \in \mathcal P \left({S}\right)$ such that:
$z \left({\mathbb G}\right) = \mathbb G$
From Relative Complement of Relative Complement we have that $\complement_S \circ \complement_S$ is the identity mapping on $\mathcal P \left({S}\right)$.
Thus we obtain:
\(\displaystyle \complement_S \left({\mathbb G}\right)\) \(=\) \(\displaystyle \left({\complement_S \circ z}\right) \left({\mathbb G}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \left({\complement_S \circ \complement_S \circ \beta \circ \complement_T \circ \alpha}\right) \left({\mathbb G}\right)\) Definition of $z$
\(\displaystyle \) \(=\) \(\displaystyle g \left[{\left({\complement_T \circ \alpha}\right] \left({\mathbb G}\right)}\right)\) as $\complement_S \circ \complement_S = I_{\mathcal P \left({S}\right)}$
At this stage, a diagram can be helpful:
Let $h: S \to T$ be the mapping defined as:
$\forall x \in S: h \left({x}\right) = \begin{cases} f \left({x}\right) & : x \in \mathbb G \\ g^{-1} \left({x}\right) & : x \in \complement_S \left({\mathbb G}\right) \end{cases}$
From the above, we have that:
$\complement_S \left({\mathbb G}\right) \subseteq g \left[{T}\right]$
Therefore, as $g$ is an injection, it follows that the preimage $g^{-1} \left({x}\right)$ is a singleton.
So $h$ is a bijection by dint of the injective nature of both $f$ and $g^{-1}$.
From the facts that $T \sim S_1$ and $S \sim T_1$, we can set up the two injections:
$f \left({S}\right) = T_1 \subseteq T$
$g \left({T}\right) = S_1 \subseteq S$
Using $f$ and $g$, $S$ and $T$ are divided into disjoint subsets such that there exists a bijection between the subsets of $S$ and those of $T$, as follows.
Let $a \in S$.
Then we define the elements of $S$:
$\ldots, a_{-2n}, \ldots, a_{-2}, a_0, a_2, \ldots, a_{2n}, \ldots$
and the elements of $T$:
$\ldots, a_{-2n+1}, \ldots, a_{-1}, a_1, \ldots, a_{2n-1}, \ldots$
recursively as follows:
\(\displaystyle a_0\) \(=\) \(\displaystyle a\)
\(\displaystyle a_1\) \(=\) \(\displaystyle f \left({a_0}\right)\)
\(\displaystyle a_2\) \(=\) \(\displaystyle g \left({a_1}\right)\)
\(\displaystyle \) \(\vdots\) \(\displaystyle \)
\(\displaystyle a_{2n-1}\) \(=\) \(\displaystyle f \left({a_{2 n - 2} }\right)\)
\(\displaystyle a_{2n}\) \(=\) \(\displaystyle g \left({a_{2 n - 1} }\right)\)
This construction is valid for all $n \ge 1$, but note that some of the $a_n$'s may coincide with others.
We set up a similar construction for negative integers:
\(\displaystyle a_{-1}\) \(=\) \(\displaystyle g^{-1} \left({a_0}\right)\) if such an element exists in $T$
\(\displaystyle a_{-2}\) \(=\) \(\displaystyle f^{-1} \left({a_{-1} }\right)\) if such an element exists in $S$
\(\displaystyle a_{-2 n + 1}\) \(=\) \(\displaystyle g^{-1} \left({a_{-2 n + 2} }\right)\) if such an element exists in $T$
\(\displaystyle a_{2 n}\) \(=\) \(\displaystyle f^{-1} \left({a_{-2 n + 1} }\right)\) if such an element exists in $S$
As $f$ and $g$ are injections, it follows that if $f^{-1} \left({x}\right)$ and $g^{-1} \left({y}\right)$ exist for any $x \in T$, $y \in S$, then those elements are unique.
the elements of $S$ with an even index be denoted $\left[{a}\right]_S$
the elements of $T$ with an odd index be denoted $\left[{a}\right]_T$
\(\displaystyle \left[{a}\right]_S\) \(=\) \(\displaystyle \left\{ {\ldots, a_{-2 n}, \ldots, a_{-2}, a_0, a_2, \ldots, a_{2 n}, \ldots}\right\} \subseteq S\)
\(\displaystyle \left[{a}\right]_T\) \(=\) \(\displaystyle \left\{ {\ldots, a_{-2 n + 1}, \ldots, a_{-1}, a_1, \ldots, a_{2 n - 1}, \ldots}\right\} \subseteq T\)
We are given that $f$ and $g$ are injections.
So by the definition of $\left[{a}\right]_S$, for any two $a, b \in S$, $\left[{a}\right]_S$ and $\left[{b}\right]_S$ are either disjoint or equal.
The same applies to $\left[{a}\right]_T$ and $\left[{b}\right]_T$ for any $a, b \in T$.
It follows that:
$\mathcal A_S = \left\{{\left[{a}\right]_S: a \in S}\right\}$ is a partition of $S$
$\mathcal A_T = \left\{{\left[{a}\right]_T: a \in S}\right\}$ is a partition of $T$.
every element of $S$ belongs to exactly one element of $\mathcal A_S$
every element of $T$ belongs to exactly one element of $\mathcal A_T$.
So let $a \in S$ and $b \in T$ such that $f \left({a}\right) = b$.
It follows that a bijection can be constructed from $\mathcal A_S$ to $\mathcal A_T$.
Now there are two different kinds of $\left[{a}\right]_S$ sets:
$(1):$ It is possible that no repetition occurs in the sequence $\left \langle {a_{2n}}\right \rangle$.
As a consequence, no repetition occurs in the sequence $\left \langle {a_{2n-1}}\right \rangle$ either.
By the method of construction it can be seen that $\left[{a}\right]_S$ and $\left[{a}\right]_T$ are both countably infinite.
So a bijection can be constructed between $\left[{a}\right]_S$ and $\left[{a}\right]_T$.
$(2):$ There may be a repetition in $\left[{a}\right]_S$.
Suppose such a repetition is $a_{2m} = a_{2n}$ for some $m \ne n$.
Then $a_{2m+1} = a_{2n+1}$ and $a_{2m+2} = a_{2n+2}$ and so on.
In general $a_{2m+k} = a_{2n+k}$ for all $k \in \N$.
But because $f$ and $g$ are injections it follows that $a_{2m-1} = a_{2n-1}$ and $a_{2m-2} = a_{2n-2}$ and so on, where they exist.
So in general $a_{2m-k} = a_{2n-k}$ for all $k \in \N$, where they exist.
Given $m$, we can choose $n$ so that the elements $a_{2m}, a_{2m+2}, \ldots, a_{2n-2}$ are distinct.
Then $a_{2m+1}, a_{2m+3}, \ldots, a_{2n-1}$ are likewise distinct elements of $T$.
Thus we can set up a bijection:
$a_{2m} \mapsto a_{2m+1}, a_{2m+2} \mapsto a_{2m+3}, \ldots, a_{2n-2} \mapsto a_{2n-1}$
between the elements of $\left[{a}\right]_S$ and $\left[{a}\right]_T$.
It follows that a bijection can be constructed between any two elements of the partitions of $S$ and $T$.
These maps then automatically yield a bijection from $S$ to $T$.
Hence the result.
By Injection to Image is Bijection:
$g: T \to g \left({T}\right)$ is a bijection.
Thus $T$ is equivalent to $g \left({T}\right)$.
By Composite of Injections is Injection $g \circ f: S \to g \left({T}\right) \subset S$ is also an injection (to a subset of the domain).
Then by Cantor-Bernstein-Schröder Theorem: Lemma:
There is a bijection $h: S \to g \left({T}\right)$.
Thus $S$ is equivalent to $g \left({T}\right)$.
We already know that $T$ is equivalent to $g \left({T}\right)$.
Thus by Set Equivalence is Equivalence Relation, $S$ is equivalent to $T$.
By the definition of set equivalence:
There is a bijection $\phi: S \to T$.
Let $\mathcal P \left({A}\right)$ be the power set of $A$.
Define a mapping $E: \mathcal P \left({A}\right) \to \mathcal P \left({A}\right)$ thus:
$E \left({S}\right) = A \setminus g \left({B \setminus f \left({S}\right)}\right)$
$E$ is increasing
Let $S, T \in \mathcal P \left({A}\right)$ such that $S \subseteq T$.
\(\displaystyle f \left({S}\right)\) \(\subseteq\) \(\displaystyle f \left({T}\right)\) Image of Subset is Subset of Image
\(\displaystyle \implies \ \ \) \(\displaystyle B \setminus f \left({T}\right)\) \(\subseteq\) \(\displaystyle B \setminus f \left({S}\right)\) Set Difference with Subset is Superset of Set Difference
\(\displaystyle \implies \ \ \) \(\displaystyle g \left({B \setminus f \left({T}\right)}\right)\) \(\subseteq\) \(\displaystyle g \left({B \setminus f \left({S}\right)}\right)\) Image of Subset is Subset of Image
\(\displaystyle \implies \ \ \) \(\displaystyle A \setminus g \left({B \setminus f \left({S}\right)}\right)\) \(\subseteq\) \(\displaystyle A \setminus g \left({B \setminus f \left({T}\right)}\right)\) Set Difference with Subset is Superset of Set Difference
That is, $E \left({S}\right) \subseteq E \left({T}\right)$.
By the Knaster-Tarski Lemma, $E$ has a fixed point $X$.
By the definition of fixed point:
$E \left({X}\right) = X$
Thus by the definition of $E$:
$A \setminus g \left({B \setminus f \left({X}\right)}\right) = X$
$(1): \quad A \setminus \left({A \setminus g \left({B \setminus f \left({X}\right)}\right)}\right) = A \setminus X$
Since $g$ is a mapping into $A$:
$g \left({B \setminus f \left({X}\right)}\right) \subseteq A$
Thus by Relative Complement of Relative Complement:
$A \setminus \left({A \setminus g \left({B \setminus f \left({X}\right)}\right)}\right) = g \left({B \setminus f \left({X}\right)}\right)$
Thus by $(1)$:
$g \left({B \setminus f \left({X}\right)}\right) = A \setminus X$
Let $f' = f \restriction_{X \times f \left({X}\right)}$ be the restriction of $f$ to $X \times f \left({X}\right)$.
Similarly, let $g' = g \restriction_{\left({B \setminus f \left({X}\right)}\right) \times \left({A \setminus X}\right)} = g \restriction_{\left({B \setminus f \left({X}\right)}\right) \times g \left({B \setminus f \left({X}\right)}\right)}$.
By Injection to Image is Bijection, $f'$ and $g'$ are both bijections.
Define a relation $h: A \to B$ by $h = f' \cup {g'}^{-1}$.
We will show that $h$ is a bijection from $A$ onto $B$.
The domain of $f'$ is $X$, which is disjoint from the codomain, $A \setminus X$, of $g'$.
The domain of $g'$ is $B \setminus f \left({X}\right)$, which is disjoint from the codomain, $f \left({X}\right)$, of $f'$.
Let $h = f' \cup {g'}^{-1}$.
By the corollary to Union of Bijections with Disjoint Domains and Codomains is Bijection:
$h$ is a bijection from $X \cup \left({A \setminus X}\right)$ onto $f \left({X}\right) \cup \left({B \setminus f \left({X}\right)}\right)$.
By Union with Relative Complement, $h$ is a bijection from $A$ onto $B$.
Since $f' \subseteq f$ and $g' \subseteq g$, each element of $h$ is an element of $f$ or of $g^{-1}$.
That is, if $y = h \left({x}\right)$ then either $y = f \left({x}\right)$ or $x = g \left({y}\right)$.
Law of the Excluded Middle
This theorem depends on the Law of the Excluded Middle.
This is one of the axioms of logic that was determined by Aristotle, and forms part of the backbone of classical (Aristotelian) logic.
However, the intuitionist school rejects the Law of the Excluded Middle as a valid logical axiom. This in turn invalidates this theorem from an intuitionistic perspective.
The Cantor-Bernstein Theorem
The Cantor-Schroeder-Bernstein Theorem or Cantor-Schröder-Bernstein Theorem
The Schroeder-Bernstein Theorem or Schröder-Bernstein Theorem
This theorem states in set theoretical concepts the "intuitively obvious" fact that if $a \le b$ and $b \le a$ then $a = b$.
Care needs to be taken to make well sure of this, because when considering infinite sets, intuition is frequently misleading.
In order to prove equivalence, a bijection needs to be demonstrated. It can be significantly simpler to demonstrate an injection than a surjection, so proving that there is an injection from $S$ to $T$ and also one from $T$ to $S$ may be a lot less work than proving that there is both an injection and a surjection from $S$ to $T$.
Source of Name
This entry was named for Georg Cantor, Felix Bernstein and Ernst Schröder.
Cantor first attempted to prove this theorem in his 1897 paper. Ernst Schröder had also stated this theorem some time earlier, but his proof, as well as Cantor's, was flawed. It was Felix Bernstein who finally supplied a correct proof in his 1898 PhD thesis.
1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis: $\S 2.6$: Theorem $7$
1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.03$
1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 8$: Theorem $8.2$: Corollary
2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $2.7$
2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Cardinality
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\begin{document}
\title{Some conditions implying stability of graphs}
\author{Ademir Hujdurović} \address{University of Primorska, UP IAM, Muzejski trg 2, 6000 Koper, Slovenia and University of Primorska, UP FAMNIT, Glagolja\v ska 8, 6000 Koper, Slovenia} \email{[email protected]}
\author{Đorđe Mitrović} \address{University of Primorska, UP FAMNIT, Glagolja\v ska 8, 6000 Koper, Slovenia} \email{[email protected]}
\date{\today}
\begin{abstract}
A graph $X$ is said to be {\em unstable} if the direct product $X\times K_2$ (also called the {\em canonical double cover} of $X$) has automorphisms that do not come from automorphisms of its factors $X$ and $K_2$. It is {\em non-trivially unstable} if it is unstable, connected, non-bipartite, and distinct vertices have distinct sets of neighbours.
In this paper, we prove two sufficient conditions for stability of graphs in which every edge lies on a triangle, revising an incorrect claim of Surowski and filling in some gaps in the proof of another one. We also consider triangle-free graphs, and prove that there are no non-trivially unstable triangle-free graphs of diameter 2. An interesting construction of non-trivially unstable graphs is given and several open problems are posed.
\end{abstract}
\maketitle
\section{Introduction}
All graphs considered in this paper are finite, simple and undirected. For a graph $X$, we denote by $V(X)$, $E(X)$ and $\Aut(X)$ the vertex set, the edge set and the automorphism group of $X$, respectively. {\it The canonical bipartite double cover} (also called {\it the bipartite double cover} or {\it the Kronecker cover}) of a graph $X$, denoted by $BX$, is the direct product $X \times K_2$ (where $K_2$ denotes the complete graph on two vertices). This means that $V(BX)=V(X)\times \{0,1\}$ and $E(BX)=\{\{(x,0),(y,1)\}\mid \{x,y\}\in E(X)\}$.
Canonical double covers have proven to play an important role in algebraic graph theory and have been studied by multiple groups of authors from a variety of perspectives \cite{ FengKutnar2fold,Hujdurovic,Krnc,Matsushita, Kneser2017,RegularEmbeddingsNedela,Pacco,WallerDouble,ZelinkaDouble}. It is well-known that $BX$ is connected if and only if $X$ is connected and non-bipartite, see \cite{HIK}. It is easy to see that $\Aut(BX)$ contains a subgroup isomorphic to $\Aut(X)\times S_2$. However, determining the full automorphism group of $BX$ is not as trivial. Hammack and Imrich \cite{HI} investigated vertex-transitivity of the direct product of graphs, and proved that for a non-bipartite graph $X$ and a bipartite graph $Y$, their direct product $X\times Y$ is vertex-transitive if and only if both $BX$ and $Y$ are vertex-transitive. Hence, the problem of vertex-transitivity of the direct product of graphs reduces to the problem of vertex-transitivity of canonical double covers.
If $\Aut(BX)$ is isomorphic to $\Aut(X)\times S_2$ then the graph $X$ is called {\it stable}, otherwise it is called {\it unstable}. This concept was first defined by Maru\v si\v c et al. \cite{maruvsivc1989characterization}. A graph is said to be {\em twin-free} (also called {\em worthy}, {\em vertex-determining} or {\em irreducible}) if distinct vertices have different neighbourhoods. It is not difficult to prove that disconnected graphs, bipartite graphs with non-trivial automorphism groups, and graphs containing twin vertices are unstable. These graphs are considered {\em trivially unstable}. An unstable graph is said to be {\em non-trivially unstable} if it is non-bipartite, connected, and twin-free.
In \cite{WilsonUnExpected}, Wilson gave several sufficient conditions for a graph to be unstable, which he then applied to families of circulants, Rose Window graphs, toroidal graphs and generalized Petersen graphs. The characterization of unstable generalized Petersen graphs was obtained by Qin, Xia and Zhou in \cite{Qin21}. Stability of circulant graphs was studied by Qin, Xia and Zhou in \cite{QinXiaZhou}, where it was proven that there are no non-trivially unstable arc-transitive circulants. In \cite{FernandezHujdurovicCirc}, it was proven by Fernandez and Hujdurovi\' c that there are no non-trivially unstable circulants of odd order. In \cite{morris2021automorphisms}, Morris extended this result by proving that there are no non-trivially unstable Cayley graphs on Abelian groups of odd order. In \cite{HMMAtMost7}, the complete classification of unstable circulants of valency at most $7$ was obtained, while in \cite{HMMAuto} several constructions of unstable circulants were presented, and unstable circlants of order $2p$ (with $p$ a prime) were classified.
Although it is in some sense expected that most graphs are stable, there are not many results giving sufficient conditions for a general graph to be stable. The main motivation for this paper comes from the work of Surowski \cite{SurowskiStabArcTrans} on stability of arc-transitive graphs, where he describes two sufficient stability conditions, namely {\cite[Proposition~2.1]{SurowskiStabArcTrans}} for vertex-transitive graphs of diameter at least 4 and {\cite[Proposition~2.2]{SurowskiStabArcTrans}} for strongly regular graphs. The first of these results has been shown not to hold by Lauri, Mizzi and Scapellato in \cite{LauriMizziScapellato}, where the authors constructed an infinite family of counterexamples. In the same article, the authors pointed out that the proof of the second result seems incomplete, hence asking for a further investigation of stability of strongly regular graphs.
In \cref{SectionTriangles}, we will discuss the original statement of Surowski's first stability criterion and an infinite family of counterexamples by Lauri, Mizzi and Scapellato (see \cref{LMScounterexample}). Then we will explain the mistake in the original proof and fix it by introducing additional assumptions, consequently obtaining a valid stability criterion in \cref{SurowskiDiameterUpdated}, which no longer requires the graph to be vertex-transitive. Then we turn our attention to the second result of Surowski concerning strongly regular graphs, which turned out to be correct, although its original proof was slightly incomplete. We prove a generalized version of this result in \cref{StableTriangleEdgeNonEdge} that applies to a wider class of graphs.
In \cref{SectionTriangleFree}, to contrast the previous results which required every edge of a graph to lie on a triangle, we consider triangle-free graphs, and prove that there are no non-trivially unstable triangle-free graphs of diameter $2$ (see \cref{TriangleFreeStable}).
In \cref{sec:construction}, we present an interesting construction of non-trivially unstable graphs, which shows that every connected, non-bipartite, twin-free graph of order $n$ is an induced subgraph of a non-trivially unstable graph of order $n+4$. With the help of a computer, we check that most of the non-trivially unstable graphs up to 10 vertices can be obtained using this construction.
\section{Preliminaries} \label{sec:preliminaries} Let $X$ be a graph and $BX$ its bipartite double cover. For an automorphism $\varphi$ of $X$, it is easy to see that the function $\overline{\varphi}$, defined by $\overline{\varphi}(x,i)=(\varphi(x),i)$ is an automorphism of $BX$, called \textit{the lift of $\varphi$}. It is straightforward to check that the function $\tau$ defined by $\tau(x,i)=(x,i+1)$ (the second coordinate is calculated modulo 2) is also an automorphism of $BX$.
The subgroup generated by $\tau$ and the lifts $\overline{\varphi}$ with $\varphi\in\Aut(X)$ is isomorphic to $\Aut(X)\times S_2$. Automorphisms of $BX$ that belong to this subgroup are called the {\em expected automorphisms} of $BX$ (see \cite{WilsonUnExpected}). If an automorphism of $BX$ does not belong to this subgroup, it is called {\em unexpected}. Using this terminology, a graph $X$ is stable if and only if every automorphism of $BX$ is expected.
The following result is known and will be used later on for establishing stability of graphs. For the sake of completeness, we provide a short proof.
\begin{lem}\label{lem:stability (x,0)(x,1)} Let $X$ be a connected, non-bipartite graph, and let $\alpha$ be an automorphism of $BX$. Then $\alpha$ is an expected automorphism of $BX$ if and only if for every $x\in V(X)$, it holds that $\alpha(\{(x,0),(x,1)\})=\{(y,0),(y,1)\}$ for some $y\in V(X)$. \end{lem} \begin{proof} Since $X$ is a connected and non-bipartite graph, it follows that $BX$ is connected and bipartite with bipartite sets $V(X)\times \{0\}$ and $V(X)\times \{1\}$. If $\alpha$ interchanges the bipartite sets, then one can consider $\alpha\tau$ instead. Hence, without loss of generality, we can assume that $\alpha$ preserves the sets $V(X)\times \{0\}$ and $V(X)\times \{1\}$. Then $\alpha$ satisfies the condition from Lemma \ref{lem:stability (x,0)(x,1)} if and only if $\alpha(x,i)=(\varphi(x),i)$, for some permutation $\varphi$ of $V(BX)$. It is easy to see that the map $(x,i)\mapsto (\varphi(x),i)$ is an automorphism of $BX$ if and only if $\varphi$ is an automorphism of $X$. \end{proof}
It is not difficult to show that even cycles are unstable, while odd cycles are stable. Moreover, we have the following easy example.
\begin{eg}[Qin-Xia-Zhou,{\cite[Example~2.2]{QinXiaZhou}}]\label{CompStab} The complete graph $K_n$ is unstable if and only if $n = 2$. \end{eg}
Recall that a {\em strongly regular graph} with parameters $(n,k,\lambda,\mu)$ is a $k$-regular connected graph of diameter $2$ on $n$ vertices such that every edge lies on $\lambda$ triangles, and any two non-adjacent vertices have exactly $\mu$ neighbours in common. Note that $\mu>0$ and that a strongly regular graph is twin-free if and only if $k>\mu$.
For a graph $X$ and a vertex $x$ of $X$, we will denote by $X_i(x)$ the set of vertices of $X$ at distance $i$ from $x$. We will also sometimes write $N_X(x)$ for the set of neighbours of $x$, instead of $X_1(x)$.
\section{Stability of graphs with many triangles}\label{SectionTriangles}
In their article \cite{LauriMizziScapellato} on TF-automorphisms of graphs, Lauri, Mizzi and Scapellato describe a method for constructing unstable graphs of an arbitrarily large diameter with the property that every edge lies on a triangle. This contradicts {\cite[Proposition~2.1]{SurowskiStabArcTrans}}, a result of Surowski claiming that a connected, vertex-transitive graph of diameter at least $4$ with every edge on a triangle is stable. The particular family of counterexamples described by Lauri, Mizzi and Scapellato (based on their result {\cite[Theorem~5.1]{LauriMizziScapellato}}) can be reformulated in terms of the lexicographic product of graphs as follows (see \cite[Definition 4.2.1.]{DobsonBook} for the definition of the lexicographic product of graphs).
\begin{prop}\label{LMScounterexample} Let $H$ be a non-trivial vertex-transitive graph that is also twin-free and bipartite. Let $m\geq 8$ be a positive integer. Then $C_m\wr H$ is a connected, non-trivially unstable, vertex-transitive graph of diameter at least $4$ whose every edge lies on a triangle. \end{prop}
A particular example pointed out by Lauri, Mizzi and Scapellato at the end of their article is the graph $C_8\wr C_6$. In the language of \cref{LMScounterexample}, we set $m=8$ and $H=C_6$.
In his proof, Surowski uses the assumption that every edge lies on a triangle to describe a part of the distance partition of $BX$ with respect to a fixed vertex $(x,0)$. In particular, Surowski shows that the vertex $(x,1)$ is at distance $3$ from $(x,0)$. The author then attempts to show that $(x,1)$ is the unique vertex at distance $3$ from $(x,0)$ in $BX$ with no neighbours lying at distance $4$ from $(x,0)$. This would imply that any automorphism of $BX$ fixing $(x,0)$ must also fix $(x,1)$, which is equivalent to $X$ being stable as it is assumed to be vertex-transitive (see for example \cite[Lemma~2.4]{FernandezHujdurovicCirc}).
However, the last claim about $(x,1)$ does not always hold. For example, if $X=C_8\wr C_6$ and $x$ is its arbitrary vertex, then any of the $3$ vertices lying in the same copy of $C_6$ as $x$, that are distinct from $x$ and are not adjacent to it, induce vertices in $BX$ at distance $3$ from $(x,0)$, all of whose neighbours are at distance at most $2$ from $(x,0)$. In particular, it is not possible to distinguish $(x,1)$ from other elements of $(BX)_3(x,0)$ in the manner proposed by Surowski.
We remedy this situation by adding additional assumptions on $X$. On the other hand, we no longer require $X$ to be vertex-transitive.
\begin{prop}\label{SurowskiDiameterUpdated} Let $X$ be a non-trivial connected graph. Assume that $X$ satisfies the following conditions.
\begin{enumerate}
\item \label{SurowskiDiameterUpdated-triangle} Every edge of $X$ lies on a triangle.
\item \label{SurowskiDiameterUpdated-distance} For every $x\in V(X)$, it holds that
\begin{enumerate}
\item \label{SurowskiDiameterUpdated-distance-2} every vertex at distance $2$ from $x$ has a neighbour at distance $3$ from $x$, and
\item \label{SurowskiDiameterUpdated-distance-3} every vertex at distance $3$ from $x$ has a neighbour at distance $4$ from $x$.
\end{enumerate} \end{enumerate}
Then $X$ is stable. \end{prop}
\begin{proof} It is easy to show that if $X_2(x)$ is empty for some $x\in V(X)$, and $X$ satisfies the assumptions, then $X$ is complete and therefore, stable by \cref{CompStab} (assumption \pref{SurowskiDiameterUpdated-triangle} rules out the case $X=K_2$). We can therefore assume that $X_2(x)$ is non-empty for all $x\in V(X)$. Conditions \pref{SurowskiDiameterUpdated-distance}\pref{SurowskiDiameterUpdated-distance-2} and \pref{SurowskiDiameterUpdated-distance}\pref{SurowskiDiameterUpdated-distance-3} then imply that also $X_3(x)$ and $X_4(x)$ are non-empty for all $x\in V(X)$.
\begin{comment} Using the assumption that every edge lies on a triangle, we can derive the following for any $x\in V(X)$. \begin{enumerate}
\item $(BX)_1(x,0)=X_1(x)\times \{1\}$,
\item $(BX)_2(x,0)=X_1(x)\times\{0\}\cup X_2(x)\times \{0\}$,
\item $(x,1)\in (BX)_3(x,0)$,
\item $(BX)_3(x,0)\subseteq X_2(x)\times\{1\}\cup X_3(x)\times\{1\}\cup \{(x,1)\}$. \end{enumerate} \end{comment}
Observe that \[(BX)_2(x,0)=X_1(x)\times \{0\}\cup X_2(x)\times \{0\},\] and \[(x,1)\in (BX)_3(x,0)\subseteq \{(x,1)\}\cup X_2(x)\times \{1\}\cup X_3(x)\times \{1\}.\]
From here, it is clear that all neighbours of $(x,1)$ in $BX$ lie in $(BX)_2(x,0)$. Consequently, $(x,1)$ has no neighbours in $(BX)_4(x,0)$. We will show that $(x,1)$ is the unique element of $(BX)_3(x,0)$ with this property. To show this, we observe the following. \begin{itemize}
\item If $y\in X_2(x)$, then by assumption \pref{SurowskiDiameterUpdated-distance}\pref{SurowskiDiameterUpdated-distance-2}, it has a neighbour $z\in V(X)$ such that $z\in X_3(x)$. Then $(z,0)\in (BX)_4(x,0)$ is a neighbour of $(y,1)$.
\item If $y\in X_3(x)$, then by assumption \pref{SurowskiDiameterUpdated-distance}\pref{SurowskiDiameterUpdated-distance-3}, it has a neighbour ~$z\in V(X)$ such that $z\in X_4(x)$. By the same arguments as before, it follows that $(z,0)\in (BX)_4(x,0)$ is a neighbour of $(y,1)$. \end{itemize}
Let $\alpha\in\Aut(BX)$ be arbitrary. As $X$ is connected and non-bipartite (as it contains triangles), after composing $\alpha$ with $\tau$ if necessary, we can assume that \[\text{$\alpha(V(X)\times \{i\})=V(X)\times \{i\}$ for $i\in \{0,1\}$.}\] Let $x\in V(X)$ be arbitrary and choose $y\in V(X)$ such that $\alpha(x,0)=(y,0)$. As $\alpha$ is an automorphism, it preserves distances, so it follows that \[\alpha(x,1)\in (BX)_3(\alpha(x,0)) = (BX)_3(y,0).\]
As $(x,1)$ has no neighbours in $(BX)_4(x,0)$, it follows that $\alpha(x,1)$ has no neighbours in $(BX)_4(y,0)$. As we have already observed, the only element in $(BX)_3(y,0)$ with this property is $(y,1)$. It follows that $\alpha(x,1)=(y,1)$. The stability of $X$ now follows by Lemma \ref{lem:stability (x,0)(x,1)}. \end{proof}
Our updated criterion has a nice application to distance-regular graphs. Recall that a regular graph $X$ of diameter $d$ is said to be {\em distance-regular} if there exists an array of integers $\{b_0,\ldots,b_{d-1},c_1,\ldots,c_d\}$, called the {\em intersection array}, with the property that for all $1\leq j\leq d$, it holds that for any pair of vertices $x,y\in V(X)$ at distance $j$ in $X$, the number of neighbours of $y$ at distance $j+1$ from $x$ is $b_j$, while the number of neighbours of $y$ at distance $j-1$ from $x$ is $c_j$.
\begin{cor}\label{StableDistanceReg} Let $X$ be a connected distance-regular graph of diameter $d$ with an intersection-array $\{b_0,\ldots,b_{d-1},c_1,\ldots,c_d\}$. If $d\geq 4$, $b_0>b_1+1$ and $b_2,b_3\geq 1$, then $X$ is stable. \end{cor} \begin{proof} The assumption that $b_0>b_1+1$ guarantees that every edge of $X$ lies on a triangle, while the assumption that $b_2,b_3\geq 1$ guarantees that the conditions \pref{SurowskiDiameterUpdated-distance}\pref{SurowskiDiameterUpdated-distance-2} and \pref{SurowskiDiameterUpdated-distance}\pref{SurowskiDiameterUpdated-distance-3} from \cref{SurowskiDiameterUpdated} are satisfied. \end{proof}
We now turn our attention to the following result of Surowski {\cite[Proposition~2.2]{SurowskiStabArcTrans}}. \begin{prop}[Surowski {\cite[Proposition~2.2]{SurowskiStabArcTrans}}]\label{StableSRGLambdaMu} Let $X$ be a strongly regular graph with parameters $(n,k,\lambda,\mu)$. If $k>\mu \neq \lambda \geq 1$, then $X$ is stable. \end{prop}
Lauri, Mizzi and Scapellato observed in \cite{LauriMizziScapellato} that the proof given by Surowski is not complete, and they suggested \cref{StableSRGLambdaMu} should be reviewed. Surowski's original proof implies that $\Aut(BX)_{(x,0)} = \Aut(BX)_{(x,1)}$ for all $x\in V(X)$. This is a necessary, but in general not sufficient condition for a graph to be stable. For example, the Swift graph shown in \cref{SGGraph} satisfies this condition, but is non-trivially unstable (see \cite{WilsonUnExpected} for more details on the Swift graph). We will show that \cref{StableSRGLambdaMu} is correct, by proving a more general result.
\begin{figure}\label{SGGraph}
\end{figure}
\begin{prop}\label{StableTriangleEdgeNonEdge} Let $X$ be a non-trivial connected, twin-free graph such that every edge of $X$ lies on a triangle. If for all pairs $x,y\in V(X)$ of adjacent vertices and all pairs $z,w\in V(X)$ of vertices at distance $2$ from each other, it holds that \begin{equation*}
|N_X(x)\cap N_X(y)| \neq |N_X(z)\cap N_X(w)|, \end{equation*} then $X$ is stable. \end{prop}
\begin{proof} As every edge of $X$ lies on a triangle, for every $x\in V(X)$ it holds that \[(BX)_2(x,0) = X_1(x)\times \{0\}\cup X_2(x)\times \{0\}.\]
Let $\alpha\in \Aut(BX)$ be arbitrary. As $X$ is connected and non-bipartite (as it contains triangles), after possibly composing $\alpha$ with $\tau$, we may assume that $\alpha(V(X)\times \{i\})=V(X)\times\{i\}$ for $i\in \{0,1\}$. Let $x\in V(X)$ and choose $z\in V(X)$ such that $\alpha(x,0)=(z,0)$.
Let $y\in X_1(x)$ and choose $w\in V(X)$ such that $\alpha(y,0)=(w,0)$. As $\alpha$ is an automorphism of $BX$, it preserves distance, so since $(y,0)\in (BX)_2(x,0)$, we have that \[(w,0) \in (BX)_2(\alpha(x,0)) =(BX)_2(z,0) = X_1(z)\times \{0\}\cup X_2(z)\times \{0\}.\]
The number of neighbours of $(w,0)$ lying in $(BX)_1(z,0)$ is by definition $|N_X(z)\cap N_X(w)|$, but as $\alpha$ is an automorphism of $BX$, this number also equals to $|N_X(x)\cap N_X(y)|$. As $\{x,y\}$ is an edge of $X$, if $w\in X_2(z)$, we would arrive at a contradiction with our assumption. Therefore, $w\in X_1(z)$ and $\alpha(y,0)\in X_1(z)\times \{0\}$. We conclude that \[\alpha(X_1(x)\times \{0\})\subseteq X_1(z)\times \{0\}.\]
As $\alpha$ is an automorphism of $BX$, it follows that $(x,0),(z,0),(x,1)$ and $(z,1)$ are all of the same valency, so we have in fact proven that \[N_{BX}(\alpha(x,1))=\alpha(N_{BX}(x,1))=\alpha(X_1(x)\times \{0\})= X_1(z)\times \{0\} = N_{BX}(z,1).\]
Hence, $\alpha(x,1)$ and $(z,1)$ are twins in $BX$, and as $X$ is twin-free, we conclude that $\alpha(x,1)=(z,1)$. By Lemma~\ref{lem:stability (x,0)(x,1)}, it follows that $X$ is stable. \end{proof}
We can now obtain the original result of Surowski as a corollary of the one we just established.
\begin{proof}[Proof of \cref{StableSRGLambdaMu}] First we note that $X$ must be connected, since vertices in distinct connected components would have no neighbours in common, which contradicts the fact that $\mu > 0$. Next, $X$ is twin-free, as twins are non-adjacent vertices that would have $k$ neighbours in common, which contradicts the assumption that $k>\mu$. Every edge of $X$ lies on a triangle since $\lambda\geq 1$.
Finally, let $x,y,z,w\in V(X)$ be such that $\{x,y\}\in E(X)$, and $z$ and $w$ lie at distance $2$. Then $z$ and $w$ are not adjacent and we have that
\[|N_X(x)\cap N_X(y)| = \lambda \neq \mu = |N_X(z)\cap N_X(w)|.\]
It follows that $X$ is stable by \cref{StableTriangleEdgeNonEdge}. \end{proof}
The results we established can be used to analyze the stability of various families of graphs. Below, we will show how they can be used to give a complete classification of unstable Johnson graphs.
\begin{defn}[{\hspace{1sp}\cite[p.~9]{GodsilRoyleAlgGraphTh}}]\label{JohnsonDefn} Let $n\geq k\geq 1$ be positive integers. The \textit{Johnson graph} is the graph $J(n,k)$ with $k$-element subsets of $\{1,\ldots,n\}$ as vertices, which are adjacent if and only if the size of their intersection as sets is $k-1$. \end{defn}
We recall that the Johnson graph $J(n,k)$ is connected with diameter $\min(k,n-k)$. Moreover, the map assigning to each subset of $\{1,\ldots,n\}$ its complement induces a graph isomorphism $J(n,k)\cong J(n,n-k)$.
\begin{thm}\label{JohnsonStable} Let $n\geq k\geq 1$ be positive integers. The Johnson graph $J(n,k)$ is unstable if and only if it is one of the following: \begin{enumerate}
\item \label{JohnsonStable-(2,1)} the complete graph $J(2,1)\cong K_2$,
\item \label{JohnsonStable-(4,2)} the octahedral graph $J(4,2)$,
\item \label{JohnsonStable-(6,2)(6,4)} $J(6,2)\cong J(6,4)$ or
\item \label{JohnsonStable-(6,3)} $J(6,3)$. \end{enumerate} Moreover, $J(2,1)$ is bipartite, while $J(4,2)$ is not twin-free, so both are trivially unstable.~Graphs ~$J(6,2)$ and~$J(6,3)$ are non-trivially unstable with indices of instability $28$ and $2$, respectively. \end{thm} \begin{proof} It is easy to see that the only non-trivial bipartite Johnson graph is $J(2,1)$, while the only Johnson graph admitting twins is $J(4,2)$. Hence, we assume that $(n,k)\not \in \{(2,1),(4,2)\}$.
It can be checked that any two adjacent vertices in $J(n,k)$ have $n-2$ neighbours in common, while any two vertices at distance $2$ have $4$ neighbours in common.
Applying \cref{StableTriangleEdgeNonEdge}, we conclude that $J(n,k)$ is stable whenever $n\neq 6$. Letting $n=6$, we see that $J(6,1)\cong J(6,5)\cong K_6$ is stable by \cref{CompStab}, while it can be verified with the help of a computer (for example using Magma \cite{MAGMA}) that $J(6,2)\cong J(6,4)$ and $J(6,3)$ are unstable with indicated indices of instability. \end{proof}
\begin{rem} Stability of Johnson graphs has been studied by Mirafzal in \cite{Mirafzal}, where it has been incorrectly claimed that all Johnson graphs are stable (see {\cite[Theorem~3.20]{Mirafzal}}). \end{rem}
\section{A stability condition for triangle-free graphs}\label{SectionTriangleFree}
In the previous section, we considered stability of graphs containing many triangles. On the other end of the spectrum, we consider a stability criterion for graphs that do not contain any triangles whatsoever.
\begin{thm}\label{TriangleFreeStable}
Let $X$ be a connected, non-bipartite and twin-free graph. If $X$ is triangle-free of diameter $2$, then $X$ is stable. \end{thm} \begin{proof} Let $x\in V(X)$ be arbitrary and define \[S(x)\coloneqq \{z\in X_2(x)\mid X_1(z)\cap X_2(x)\neq \emptyset\}.\]
We first establish the following \[\text{$(x,1)\in (BX)_5(x,0)\subseteq \{(x,1)\}\cup(X_2(x)\setminus S(x))\times \{1\}.$}\]
Clearly, $(x,1)\not\in (BX)_1(x,0)$ and as $X$ is triangle-free, it also holds that $(x,1)\not\in (BX)_3(x,0)$. Note that as $X$ is non-bipartite, $S(x)$ is not empty (otherwise, $X_1(x)\cup (X_2(x)\cup \{x\})$ would be a bipartition of $X$). Hence, we obtain a path $x\sim y\sim z \sim w$ with $y\in X_1(x)$ and $z,w\in X_2(x)$. We know that $w$ must have a neighbour $v\in X_1(x)$, which is distinct from $y$ as $X$ is triangle-free. Hence, we obtained a $5$-cycle formed by the vertices $x,y,z,w$ and $v$, which implies that $(x,1)\in (BX)_5(x,0)$.
Let $(w,1)\in (BX)_5(x,0)$. We may assume that $w\neq x$. Note that $w\not\in X_1(x)$ as then $(w,1)\in (BX)_1(x,0)$. It follows that $w\in X_2(x)$. If $w$ had a neighbour $z\in X_2(x)$, then using an arbitrary $y\in X_1(x)\cap X_1(z)$, we would be able to construct a path $(x,0)\sim (y,1)\sim (z,0)\sim (w,1)$, showing that $(w,1)$ is at distance at most $3$ from $(x,0)$. It follows that $w\in X_2(x)\setminus S(x)$. This finishes the proof of the claim.
Let $y\in X_2(x)\setminus S(x)$. Then all neighbours of $y$ are contained in $X_1(x)$, by definition of $S(x)$. Since $X$ is twin-free, it follows that $y$ has smaller valency than $x$. Consequently $(y,1)$ has smaller valency than $(x,1)$. We conclude that \[\text{$(x,1)$ is the unique element of $(BX)_5(x,0)$ of the same valency as $(x,0)$}.\]
Let $\alpha\in \Aut(BX)$ be arbitrary. After possibly composing $\alpha$ with $\tau$, we may assume that it preserves the colour classes of $BX$. Let $\alpha(x,0)=(y,0)$ for $x,y\in V(X)$.
As $\alpha$ preserves distances in $BX$, it follows that \[\alpha(x,1)\in (BX)_5(\alpha(x,0))= (BX)_5(y,0).\]
Note that the valency of $\alpha(x,1)$ is equal to the valency of $(y,0)$. We conclude that $\alpha(x,1)=(y,1)$ and by Lemma~\ref{lem:stability (x,0)(x,1)} it follows that $X$ is stable. \end{proof}
This result can be used to slightly extend Surowski's result for strongly regular graphs (see \cref{StableSRGLambdaMu}).
\begin{cor}\label{SRGLambda=0} Let $X$ be an $(n,k,\lambda,\mu)$-strongly regular graph. If $k > \mu $ and $\lambda = 0$, then $X$ is stable. \end{cor} \begin{proof}
We make the following observations. \begin{itemize}
\item As $X$ is a strongly regular graph, it is connected with diameter $2$ by definition.
\item $k>\mu$ implies that $X$ is twin-free.
\item $\lambda = 0$ implies that $X$ is triangle-free.
\item Every edge lies on a $5$-cycle, proving that $X$ is non-bipartite. \end{itemize}
We can now apply \cref{TriangleFreeStable} to conclude that $X$ is stable. \end{proof}
It is worth noting that besides the trivial example given by $K_{n,n}$ with $n\geq 2$, which is $(2n,n,0,n)$-srg, there are only seven other currently known strongly regular graphs that are triangle-free, as explained by Godsil in {\cite{AlgCombProblemsGODSIL}}.
As a corollary of the results we derived so for, we obtain the following stability criterion for strongly regular graphs.
\begin{prop}\label{SRGStable} Let $X$ be an $(n,k,\lambda,\mu)$-strongly regular graph. If $X$ is non-trivially unstable, then $\lambda = \mu > 0$. \end{prop}
\begin{proof} As $X$ is non-trivially unstable, it is connected and twin-free, so $k>\mu>0$. As $X$ is unstable, \cref{SRGLambda=0} shows that $\lambda >0$. Finally, it follows by \cref{StableSRGLambdaMu} that $\mu = \lambda$, as desired. \end{proof}
\begin{rem} In \cite{SurowskiStabArcTrans}, Surowski constructs an infinite family of non-trivially unstable strongly regular graphs with $\lambda =\mu$. \end{rem}
\begin{problem} Characterize non-trivially unstable strongly regular graphs. \end{problem}
\section{The number of unstable graphs} \label{sec:construction}
An interesting question to consider is how dense is the set of all non-trivially unstable graphs in the set of all connected, non-bipartite and twin-free graphs. We will now present an interesting construction of non-trivially unstable graphs which shows that every connected, non-bipartite, twin-free graph of order $n$ is an induced subgraph of a non-trivially unstable graph of order $n+4$.
\begin{construction}\label{constructionX(A,B)} Let $X$ be a graph and $A$ and $B$ be subsets of $V(X)$. Let $X(A,B)$ denote the graph with \begin{itemize}
\item $V(X(A,B))\coloneqq V(X)\cup \{a_1,a_2,b_1,b_2\}$, where $a_1,a_2,b_1$ and $b_2$ are four distinct vertices with the property that $a_1,a_2,b_1,b_2\not\in V(X)$,
\item $E(X(A,B))\coloneqq E(X)\cup\{\{a_1,b_1\},\{a_2,b_2\}\}\cup \{\{a_1,a\},\{a_2,a\}\mid a\in A\}\cup \{\{b_1,b\},\{b_2,b\}\mid b\in B\}$. \end{itemize} In particular, $X(A,B)$ is obtained from $X$ by adding two new edges $\{a_1,b_1\}$ and $\{a_2,b_2\}$ to $X$, and then joining $a_1$ and $a_2$ with every vertex in $A$ and $b_1$ and $b_2$ with every vertex in $B$ (see Figure~\ref{fig:construction}). \end{construction}
\begin{figure}
\caption{The construction of $X(A,B)$ from a graph $X$.}
\label{fig:construction}
\end{figure}
\begin{prop}\label{WX} Let $X$ be a graph and let $A$ and $B$ be subsets of $V(X)$. Then $X(A,B)$ is unstable. Moreover, if $X$ is connected, non-bipartite and twin-free, and at least one of $A$ and $B$ is non-empty, then $X(A,B)$ is non-trivially unstable. \end{prop} \begin{proof} It is easy to check that the permutation \[\gamma^*\coloneqq ((a_1,0),(a_2,0))((b_1,1),(b_2,1))\] of the vertex set of $B(X(A,B))$ swapping $(a_1,0)$ with $(a_2,0)$ and $(b_1,1)$ with $(b_2,1)$, while fixing all other vertices, is an unexpected automorphism of $B(X(A,B))$ of order $2$, showing that $X(A,B)$ is unstable.
Suppose now that $X$ is connected, non-bipartite and twin-free, and that $A$ is a non-empty set of vertices of $X$. It is clear that $X(A,B)$ is connected. Since $X$ is non-bipartite, and $X$ is an induced subgraph of $X(A,B)$ it follows that $X(A,B)$ is also non-bipartite. Observe that it follows from the definition that vertices $a_1,a_2,b_1,b_2$ have different sets of neighbours. Since no two vertices of $X$ have the same sets of neighbours, it follows that $X(A,B)$ is twin-free, showing that $X(A,B)$ is non-trivially unstable. \end{proof}
In Table~\ref{tab:number of graphs}, for a positive integer $n$ between $3$ and $10$, the number given in the second row is the number of connected, non-bipartite, twin-free graphs of order $n$, the number in the third row is the number of non-trivially unstable graphs of order $n$, and the number in the fourth row is the number of non-trivially unstable graphs that can be realized using Construction~\ref{constructionX(A,B)}. The entries of Table~\ref{tab:number of graphs} have been obtained using Magma \cite{MAGMA}.
\begin{table}[h]
\captionsetup{width=\linewidth} \caption{\label{tab:number of graphs} The number of non-trivially unstable graphs up to 10 vertices.} \begin{adjustbox}{max width=\textwidth}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline
$n$ & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
Connected, non-bipartite, twin-free & 1 & 2 & 10 & 56 & 498 & 7.397 & 197.612 & 9.807.191 \\
\hline
Non-trivially unstable & 0 & 0 & 1 & 6 & 43 & 395 & 5.113 & 105.919\\
\hline
$X(A,B)$ & 0 & 0 & 1 & 5 & 37 & 330 & 4.374 & 93.610 \\
\hline \end{tabular} \end{adjustbox} \end{table}
The data in Table~\ref{tab:number of graphs} suggests that the density of non-trivially unstable graphs among all connected, non-bipartite, twin-free graphs goes to $0$ as $n$ tends to infinity. On the other hand, it seems that almost all non-trivially unstable graphs can be constructed using Construction~\ref{constructionX(A,B)}.
We conclude the paper with the following open problem. \begin{problem} Give an approximation formula for the number of non-trivially unstable graphs of order $n$. \end{problem}
\end{document} | arXiv |
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Biology Direct
December 2020 , 15:2 | Cite as
Model-based exploration of the impact of glucose metabolism on the estrous cycle dynamics in dairy cows
Mohamed Omari
Alexander Lange
Julia Plöntzke
Susanna Röblitz
Part of the following topical collections:
Nutrition plays a crucial role in regulating reproductive hormones and follicular development in cattle. This is visible particularly during the time of negative energy balance at the onset of milk production after calving. Here, elongated periods of anovulation have been observed, resulting from alterations in luteinizing hormone concentrations, likely caused by lower glucose and insulin concentrations in the blood. The mechanisms that result in a reduced fertility are not completely understood, although a close relationship to the glucose-insulin metabolism is widely supported.
Following this idea, we developed a mathematical model of the hormonal network combining reproductive hormones and hormones that are coupled to the glucose compartments within the body of the cow. The model is built on ordinary differential equations and relies on previously introduced models on the bovine estrous cycle and the glucose-insulin dynamics. Necessary modifications and coupling mechanisms are thoroughly discussed. Depending on the composition and the amount of feed, in particular the glucose content in the dry matter, the model quantifies reproductive hormones and follicular development over time. Simulation results for different nutritional regimes in lactating and non-lactating dairy cows are examined and compared with experimental studies. The simulations describe realistically the effects of nutritional glucose supply on the ovulatory cycle of dairy cattle.
The mathematical model enables the user to explore the relationship between nutrition and reproduction by running simulations and performing parameter studies. Regarding its applicability, this work is an early attempt towards developing in silico feeding strategies and may eventually help to refine and reduce animal experiments.
This article was reviewed by John McNamara and Tin Pang (nominated by Martin Lercher).
Systems biology Mathematical modelling Ordinary differential equations Metabolism Nutrition Bovine Fertility Reproduction Lactation Hormones Follicles
A few weeks after calving, modern high-yielding dairy cattle in intensive production systems give around 40 liters of milk per day. This is a high amount that comes at a cost. High-producing cows are highly susceptible to diseases, show metabolic disorders and fertility problems [1]. Early culling and smaller lifetime milk production are the consequences [2]. Countermeasures have already been taken, and the trend of breeding cows with ever increasing peak milk yield – prevalent for decades – may have come to an end. Optimizing lifetime milk production has proven to be more beneficial for both economic as well as environmental reasons [3].
The most critical time period for a cow's health and her future performance is the periparturient period and the period of early lactation [4, 5]. During that time, the cow mobilizes body reserves because of her inability to meet energy demands solely from the feed energy consumed. This state is referred to as negative energy balance (NEB) [6].
For ruminants, the energy content in feed cannot be increased without limits due to the fermentative character of the digestive system [7]. High energetic feed with little fiber leads to an imbalance of microbes, rumen acidosis, and may even cause severe illness and death. Nevertheless, targeted feeding strategies are able to extenuate the NEB and to ensure animal health and welfare [5, 8, 9].
A number of experimental and clinical studies were performed to examine the relationship between the metabolic status and the fertility of cows, both, in qualitative and in quantitative manners, e.g., [10, 11]. Reduced nutrition intake was observed to delay the onset of puberty in beef heifers [12, 13, 14], to change the growth pattern of the dominant follicles (maximal diameter, persistence, number of follicular waves) [15], and to increase the period to conception postpartum [16, 17, 18]. Studies in the postpartum period of dairy cows showed that the NEB is strongly correlated with low concentrations of glucose, insulin and IGF-1 in the blood [19, 20, 21]. Changes in the secretion of gonadotropins, caused by low glucose levels, lead to low FSH and LH concentrations [10, 22], whereby missing LH peaks cause anovulation [4]. Non-regular estrous cycles are often associated with low average concentrations of insulin in the blood [23]. On the other hand, it was reported that good feed management, e.g., nutritional manipulation that causes increased insulin, reduces the incidence of non-regular estrous cycles [24].
This paper focuses on glucose, as part of the feed and as one of the main energy sources of the body. The aim is to develop a mathematical model that represents metabolic processes as well as reproductive regulation, thus allowing to analyze the impact of glucose originating from the feed on the reproductive hormones and the follicular development.
Previous modeling efforts mainly focused on either the bovine estrous cycle [25, 26, 27, 28, 29] or the nutritional strategies [30, 31, 32], yet there are a few approaches that combine the two topics. The most recent model, named "Jenny", was developed by McNamara and Shields [33]. It connects the reproductive cycle (given by differential equations from [25, 26]) with nutrition (implemented by a rather sophisticated model called Molly [31]) via the ATP to ADP reduction reaction. Martin et al. [34] introduced an empirical model that includes nutritional effects on the reproduction. Pring et al. [27] modeled different nutritional scenarios by varying parameters in an estrous cycle model. A more conceptual model was suggested by Scaramuzzi et al. [35], where the coupling between nutrition and reproduction is realized by IGF-1, the glucose-insulin system, and leptin.
None of these models, except [33], captures the dynamics between nutrition, hormonal regulation, and milk yield, mechanisms that are of particular interest in cows. The model of McNamara and Shields [33] contains these elements, as it is based not only on Molly but on the BovCycle model [25, 26]. The effort by McNamara and Shields and the effort here are closely related and complementary. However, McNamara and Shields [33] do not include the full reproductive process. The model introduced here aims at understanding the involved interactions and time evolution on a more detailed level. It includes compartments for the nutrient intake, the glucose-insulin system [36], the milk production, and the reproductive hormones [26]. Based on that model, it is analyzed how changes in dietary intake, which usually happen on the time-scale of days, affect the behavior of the estrous cycle on the scale of weeks and months.
The paper is organized as follows. The glucose-insulin model and its coupling to the estrous cycle model are presented in the "Methods" section. The "Results and discussion" section deals with the simulation for non-lactating and lactating cows and compares the outcome with data from literature. Finally, the results are summarized again and limitations of the model are presented in the Conclusion. The model was implemented in MATLAB (release 2014b). The code is available in Additional file 1.
The model that is developed in this section and, later, used for simulations in the "Results and discussion" section is built on two major pillars. The one is the glucose-insulin dynamics in dairy cows, which was modeled in [36] utilizing the Systems Biology Markup Language [37] and CellDesigner [38]. The other is the bovine estrous cycle, modeled by a system of differential equations (BovCycle) that quantifies reproductive hormones and other relevant compartments, representing follicles and corpora lutea [25, 26].
The model here consists entirely of ordinary differential equations (ODEs), which are solved for problem-specific initial conditions and parameter values. One half of the model (Fig. 1 and r.h.s. of Fig. 2) implements the mechanisms explained in [36], which allows for simulating the time-evolution of glucose and insulin for different dietary inputs in lactating as well as non-lactating cows. The other half (l.h.s. of Fig. 2) implements the biological feedback mechanisms between hypothalamus, pituitary gland and ovaries, which produces periodic estrous cycles of constant duration, similar to [26]. However, modifications needed to be implemented as the mechanisms suggested in [26] are not tailored to cows during pregnancy, calving and lactation. In these stages the interaction between hormones is somewhat different. To simulate the onset of lactation, oxytocin is included in the model; this hormone peaks during delivery [39], and it is required for milk ejection [40, 41, 42].
Open image in new window
Schematic representation of the metabolic model. The pink boxes indicate the state variables of the model, gray ellipses indicate sources and sinks. The five compartments of the underlying ODE model are denoted by upper case letters; they have units of concentration or mass (see also Table 2). Rates are denoted by lower case letters; they have units of gram per day (see also Table 3)
Schematic representation of the coupled metabolic-reproductive model. The coupled model links the metabolic model (right hand side) to the bovine estrous cycle model [26] (left hand side). Red arrows depict the sites where both models are coupled. Insulin acts on the site of anterior pituitary influencing LH and FSH release to the blood circulation. Insulin stimulates IGF-1 levels in the blood. Progesterone inhibits IGF-1 secretion which in turn decreases the responsiveness of follicular cells to LH
Metabolic model
The metabolic model to be developed in this section is based on an improved version of the glucose-insulin model in [36]. It involves six components (Glublood,Gluliver,Glustore,Fat,Ins,Gluca; see Table 2) and, as formulated here in terms of ODEs, their explicit interaction over time. Initial conditions are chosen based on the following calculation. For a cow of weight 600 kg and body condition score 3.5, the total body fat can be estimated by 25% of the total body weight [7, 43]. That is, 150 kg is taken as initial value for Fat. Typical physiological ranges for Glublood, Ins and Gluca are listed in Table 1. As long as the initial values are within these ranges, they do not affect the performance of the model.
Physiological ranges of blood plasma glucose, insulin and glucagon levels
Glublood
0.39–0.59 g/L (2.22–3.30 mmol/L)
2–50 mU/L
Gluca
50–120 ng/L
Species in the metabolic model
Glucose concentration in the blood
g/L
Gluliver
Glucose generated in the liver
Glustore
Glucose stored as glycogen
Insulin concentration in the blood
mU/L
Glucagon concentration in the blood
ng/L
The initial values are used to solve the differential equations
Rates in the metabolic model
glufeed−bl
Glucose in the DMI available for direct absorption
glufeed−gng
Glucose generated from glucogenic substances in the DMI
glubl−lv
Glucose absorbed from the blood into liver cells
glust−lv
Glucose generated from glycogen (glycogenolysis)
glulv−st
Glucose stored as glycogen (glycogenesis)
glulv−fat
Glucose converted to triglycerides (lipogenesis)
glufat−lv
Glucose synthesized from glycerol
gluprod
Glucose released from the liver to the blood
glubl−usage
Glucose usage for maintenance and milk production
glulv−usage
Glucose usage for liver metabolism
inssec
Insulin secretion
mU/(L ·d)
insdeg
Insulin degradation
glucasec
Glucagon secretion
ng/(L ·d)
glucadeg
Glucagon degradation
The model only involves the most basic mechanisms that regulate the flow of glucose through the body. It starts with the feed, continues with the digestive system and the blood, and ends up with glucose usage. Glucose and glucogenic substances are ingested with the dry matter intake (DMI). In the liver, the glucogenic substances are converted to glucose via gluconeogenesis. Glucose is used for maintanance and milk production, it is stored as glycogen or, after conversion, as fat. The compartments of the model and their interactions are illustrated in Fig. 1. Flows and regulatory mechanisms are summarized in Table 3 and explained in detail in the following subsections.
Feed intake
The first step involves the quantification of the amount of substances in the DMI that are either available for gluconeogenesis in the liver or directly absorbable as glucose into the blood. There exist empirical formulas that estimate the DMI needed to meet the energy requirements; these formulas are based on the cow's body weight (BW) and the net energy (NE) of the diet; see, e.g., [7]. Throughout the paper, a standard cow with body weight 600 kg is considered, and the value for DMI of 11700 gram per day (g/d) is adopted from [36]. This value also results from a formula in [7], assuming a diet's net energy of 1.32 Mcal/kg.1
Ruminants digestion involves fermentation, which makes consumption of a high-fiber diet possible and necessary [44, 45]. In the default setting, the fraction of glucose and glucogenic substances in the DMI, glupool, is assumed to be 8% of the total DMI,
$$\begin{array}{@{}rcl@{}} \mathit{glu_{pool}} = c_{0}\cdot \mathit{DMI}, \end{array} $$
where c0 is a mass-fraction parameter (with default value c0=0.08) that allows for varying the total amount of glucose and glucogenic substances that can be extracted from DMI. This fraction combines glucose precursor substances such as short chain fatty acids, which are converted to glucose in the liver by gluconeogenesis, as well as glucose that can directly be absorbed from the digestive tract into the blood [45, 46, 47]. In the cow, only very little glucose is available for direct absorption from the digestive tract [48]. From the total amount of glucose and glucogenic substances in the DMI (glupool), the portion of glucose was estimated to be less than 10% [49, 50, 51], whereas up to 90% of glupool are glucogenic substances.
The flow of absorbable glucose that goes directly to the systemic circulation is incorporated into the model via the rate
$$\begin{array}{*{20}l} \mathit{glu_{feed-bl}} &= c_{1}\cdot\mathit{glu_{pool}}\,. \end{array} $$
The flow of glucose precursor substances that are converted to glucose by gluconeogenesis in the liver is incorporated into the model via the rate
$$\begin{array}{*{20}l} \mathit{glu_{feed-gng}} &= (1-c_{1})\cdot\mathit{glu_{pool}}\,. \end{array} $$
The default parameter value is c1=0.08 (cf. Table 4). It is assumed here that there is no loss from the glucose pool (the flows sum up to 1·glupool), i.e., the processes take place with 100% efficiency. If some loss was included here, the simulation results presented further below would be the same but correspond to higher values of c0 (the amount of glucose and glucose precurser substances in the feed).
Values of rate and effect parameters
Relative glucose content in the DMI
Fraction of directly absorbable glucose
Rate constant for insulin secretion
1/d
Rate constant for insulin degradation
Rate constant for glucagon secretion
Rate constant for glucagon degradation
(g ·L)/(mU ·d)
Rate constant for glucose absorption from blood into liver cells
L/(mU ·d)
Rate constant for glycogenesis
Rate constant for lipogenesis
(g ·L)/(ng ·d)
Rate constant for glycogenolysis
Rate constant for gluconeogenesis
L/(ng ·d)
Rate constant for glucose release from the liver to the blood
Glucose usage for maintenance
g/kg
Glucose usage for milk production
[IGF]/d
Basal IGF-1 synthesis rate in the blood
P4- and insulin-regulated IGF-1 synthesis rate
IGF-1 clearance rate
Maximum effect of LH on follicular function
[LH]
Maximum threshold of LH to stimulate follicular function
Maximum effect of insulin on FSH synthesis in the pituitary
Maximum effect of insulin on LH synthesis in the pituitary
[Oxy]/d
Maximum rate of additional oxytocin synthesis during lactation
1/d2
Clearance of additional oxytocin during lactation
Extracellular volume of blood
Insulin and glucagon
The blood glucose concentration is maintained at normal levels primarily through the action of two hormones, namely insulin and glucagon. Any elevation in the blood glucose concentration leads to the production of insulin in the pancreatic beta cells. Insulin promotes glucose uptake in target cells, e.g., those in the liver, muscles and fat tissue, and it promotes the conversion of glucose to glycogen (glycogenesis) in the liver [52]. When the glucose blood concentration is low, the pancreatic alpha-cells produce glucagon. Glucagon increases the plasma glucose concentration by stimulating the generation of glucose from non-carbohydrate substrates (gluconeogenesis) and the breakdown of glycogen to glucose (glycogenolysis) in the liver [52]. In the model here, the dynamics of the blood insulin and glucagon concentrations are determined by their secretion rates (inssec,glucasec) and their degradation rates (insdeg,glucadeg),
$$\begin{array}{*{20}l} \frac{d}{dt}\mathit{Ins} = ins_{sec} - ins_{deg},\quad \frac{d}{dt}\mathit{Gluca} = gluca_{sec} - gluca_{deg}, \end{array} $$
with linear degradation rates
$$\begin{array}{*{20}l} \mathit{ins_{deg}} = c_{3}\cdot Ins,\quad \mathit{gluca_{deg}} = c_{5}\cdot Gluca. \end{array} $$
It is assumed that the insulin secretion rate increases when the glucose concentration in the blood is above a certain threshold value (T1 = 0.5 g/L = 2,77 mmol/L), whereas the glucagon secretion rate decreases whenever the glucose concentration in the blood is above that threshold value (T2 = 0.5 g/L = 2,77 mmol/L),
$$ \begin{aligned} \mathit{ins_{sec}} &= c_{2}\cdot \mathit{H^{+} \left({Glu_{blood}}, T_{1}, 10\right)},\quad\\ \mathit{gluca_{sec}} &= c_{4}\cdot \mathit{H^{-} \left({Glu_{blood}}, T_{2}, 2\right)}. \end{aligned} $$
The symbols H+ and H− denote a positive and a negative Hill function,
$$ \begin{aligned} H^{+}(S,T,n):&=\frac{S^{n}}{S^{n}+T^{n}},\quad\\ H^{-}(S,T,n):&=\frac{T^{n}}{S^{n}+T^{n}}=1-H^{+}(S,T,n), \end{aligned} $$
which are used to model threshold-dependent stimulatory or inhibitory effects. Here, S≥0 denotes the substance, T≥0 the threshold, and n≥1 the Hill coefficient. A Hill function is a sigmoidal function between zero and one that switches at the threshold S=T from one level to the other with a slope specified by n and T. Threshold kinetics were selected to account for rapid adaptivity, which is an important mechanism to keep the plasma glucose concentration within the physiological range. There are no reference values for the individual rate constants c2,3,4,5, but their values were chosen such that a constant glucose blood concentration of Glublood=T1=T2=0.5 g/L (resulting in a Hill function value of 0.5) would give rise to constant insulin and glucagon concentrations that are within the physiological range, namely 0.5·c2/c3=20 mU/L and 0.5·c4/c5=100 ng/L, respectively, compare Table 1.
Glucose production and storage in the liver
When the glucose blood level rises above a certain threshold (T3=0.45 g/L = 2,77 mmol/L), insulin promotes the absorption of glucose from the blood into liver cells (rate glubl−lv),
$$\begin{array}{@{}rcl@{}} \mathit{glu_{bl-lv} = c_{6}\cdot H^{+} \left({Glu_{blood}}, T_{3}, 10\right)\cdot Ins}\,. \end{array} $$
Insulin also stimulates the conversion of glucose available in the liver (Gluliver) to glycogen (glycogenesis rate glulv−st). It is assumed here that this rate decreases when the cow produces more than a certain amount of milk (threshold T4=10 L) per day in order to make more glucose available for milk production. In addition, the rate glulv−st is switched off when the glycogen store, Glustore, reaches the maximal carrying capacity K=1000g2. The equation that describes this process is given by
$$ {\begin{aligned} \mathit{glu_{lv-st} = c_{7}\cdot H^{-} \left({Milk}, T_{4}, 2\right)\cdot \left(1-\frac{Glu_{store}}{K}\right)\cdot Glu_{liver}\cdot Ins}\,. \end{aligned}} $$
In addition, insulin promotes the absorption of glucose into fat cells and its conversion into triglycerides via lipogenesis. It is assumed here that this process is enhanced once the glycogen storage Glustore is full (threshold T6=1000g). Again, similar to the glycogenesis rate glulv−st, the rate is assumed to decrease when the cow produces more than a certain amount of milk (threshold T5=10 L) per day,
$$ {\begin{aligned} \mathit{glu_{lv-fat}= c_{8}\cdot H^{-} \left({Milk}, T_{5}, 1\right)\cdot H^{+} \left({Glu_{store}}, T_{6}, 10\right)\cdot Glu_{liver}\cdot Ins}\,. \end{aligned}} $$
When nutritional supply with glucose is insufficient, the glucagon concentration increases and stimulates the breakdown of glycogen to glucose in the liver (glycogenolysis) to maintain blood glucose homeostasis [55]. This process is assumed to slow down when the glycogen store is below a certain threshold (T7=10g),
$$\begin{array}{@{}rcl@{}} \mathit{glu_{st-lv} = c_{9}\cdot H^{+} \left(Glu_{store}, T_{7}, 10\right)\cdot {Gluca}}\,. \end{array} $$
In this case, i.e., when the glycogen store falls below a threshold (T8=10g), glucagon additionally stimulates the breakdown of lipids into glycerol and free fatty acids (lipolysis) in adipose tissue and the conversion of glycerol into glucose via gluconeogenesis in the liver. This rate is assumed to slowly decrease whenever the total body fat becomes smaller than a certain threshold (T9=150 kg),
$$ {\begin{aligned} \mathit{glu_{fat-lv} = c_{10}\cdot H^{-} \left({Glu_{store}}, T_{8}, 10\right)\cdot H^{+} \left(Fat, T_{9}, 1\right)}\cdot Gluca\,. \end{aligned}} $$
Finally, glucagon stimulates the release of glucose synthesized in the liver (Gluliver) into the blood,
$$\begin{array}{@{}rcl@{}} \mathit{glu_{prod} = c_{11}\cdot \mathit{Glu_{liver}}\cdot Gluca}. \end{array} $$
In the equations above, threshold kinetics were selected for Glustore to differentiate between full end empty store, without modifying the rates in dependence on the actual amount of glycogen in the store.
There are no reference values for the rate constants c6 to c11. They were fitted manually such that the simulation results qualitatively agree with the results reported in literature.
Glucose utilization
All organs and tissues of dairy cows use glucose, except adipose tissue which cannot directly convert glucose to fatty acids [45]. Glucose provides energy for maintenance and production. In the milk producing dairy cow, glucose utilization is dominated by the requirements of the mammary gland for milk synthesis [56]. These requirements increase rapidly right after parturition[57]. Glucose utilization is modeled here in terms of two different sink terms, one from Gluliver,
$$\begin{array}{@{}rcl@{}} \mathit{glu_{lv-usage}} &= c_{14} \cdot \mathit{Glu_{liver}}, \end{array} $$
and one from Glublood,
$$\begin{array}{@{}rcl@{}} \mathit{glu_{bl-usage}} = c_{12}\cdot H^{+} \left({Glu_{blood}}, T_{10}, 10\right) + c_{13}\cdot \mathit{Milk}. \end{array} $$
The sink term from Glublood accounts for maintenance (1st term) and milk production (2nd term). Maintenance refers to glucose utilization by non-mammary tissues including brain and skeletal muscle, but excluding liver. For example, glucose that is stored in skeletal muscle as glycogen cannot be released back into the bloodstream due to the absence of glucose-6-phosphatase. It is assumed here that the glucose consumption for maintenance decreases when the glucose blood level drops below a certain threshold (T10=0.5 g/L = 2,77 mmol/L). The second term accounts for glucose utilized for milk production, including substance and energy. The variable Milk quantifies the daily milk yield in kg/day, whereas the parameter c13=72 g/kg [58] quantifies the amount of glucose (in gram) per kg of milk. Hence, the mammary glucose requirement in a cow with a daily milk yield of 40 kg would be about 3 kg per day. There is no reference value for the non-mammary glucose requirement, but according to the literature [56] this value should be significantly lower (here, c12=1 kg/day was chosen).
The system of differential equations
The final set of ordinary differential equations modeling the dynamics of the glucose exchange reads
$$\begin{array}{*{20}l} V\cdot \frac{d}{dt}\mathit{Glu_{blood}} =\ & \mathit{glu_{feed-bl}} \,+\, \mathit{glu_{prod}} \,-\, \mathit{glu_{bl-lv}} \,-\, \mathit{glu_{bl-usage}}, \end{array} $$
$$\begin{array}{*{20}l} \frac{d}{dt}\mathit{Glu_{liver}} =\ & \mathit{glu_{feed-gng}} - \mathit{glu_{prod}} + \mathit{glu_{bl-lv}} - \mathit{glu_{lv-st}}\notag\\ & \,+\, \mathit{glu_{st-lv}} \,-\, \mathit{glu_{lv-fat}} \,+\, \mathit{glu_{fat-lv}}\,-\, \mathit{glu_{lv-usage}}, \end{array} $$
$$\begin{array}{*{20}l} \frac{d}{dt}\mathit{Glu_{store}} =\ & \mathit{glu_{lv-st}} - \mathit{glu_{st-lv}}, \end{array} $$
$$\begin{array}{*{20}l} \frac{d}{dt}\mathit{Fat} =\ & \mathit{glu_{lv-fat}} - \mathit{glu_{fat-lv}}, \\ \frac{d}{dt}\mathit{Ins} =\ & ins_{sec} - ins_{deg},\notag\\ \frac{d}{dt}\mathit{Gluca} =\ & gluca_{sec} - gluca_{deg},\notag \end{array} $$
where V=22.8 L is the extracellular volume of blood [36]. The ordinary differential equations were solved using the software MATLAB. The parameters and the initial values are listed in Tables 2, 4, and 5, respectively.
Values of threshold parameters
Threshold of glucose in the blood to stimulate insulin secretion
Threshold of glucose in the blood to inhibit glucagon secretion
Threshold of glucose in the blood to stimulate the absorption of glucose into liver cells
Threshold of milk to inhibit glycogenesis
Threshold of milk to inhibit lipogenesis
Threshold of glygogen store to stimulate lipogenesis
Threshold of glycogen store to stimulate glycogenolysis
Threshold of glycogen store to stimulate gluconeogenesis
Threshold of fat to stimulate gluconeogenesis
Threshold of glucose in the blood to stimulate non-mammary utilization
Threshold of P4 to inhibit IGF-1 synthesis
Threshold of insulin to stimulate IGF-1 synthesis
Threshold of LH to stimulate decrease of the follicular function
[IGF]
Threshold of IGF-1 to stimulate the responsiveness of follicles to LH
Threshold of insulin to stimulate FSH synthesis
Threshold of insulin to stimulate LH synthesis
A metabolic-reproductive model
Several studies have shown that the metabolic status has a large influence on growing cattle and on reproductive performance in dairy cows. During negative energy balance, which can be caused, e.g., by dietary restrictions or high milk yield, a remarkable change occurs in the levels of the metabolic components IGF-1, insulin, and glucose in the systemic circulation, which in turn influences the levels of reproductive hormones and follicular development [19, 20, 21]. The aim is to reproduce these observations by coupling the metabolic model and the reproductive model BovCycle introduced in [25, 26]. The initial values for the species in the BovCycle model are listed in Table 6. The flowchart for the coupled model is presented in Fig. 2. Detailed explanations of the coupling mechanisms are given in the three sections below.
Initial values for species in the BovCycle model
GnRH in the hypothalamus
[GnRH]
GnRH in the pituitary
FSH in the pituitary
[FSH]
FSH in the blood
LH in the pituitary
LH in the blood
[Follicle]
PGF 2α
[PGF 2α]
Corpus luteum
[CL]
Inhibin
[Inhibin]
[Enzyme]
Oxytocin (non-lactating case)
[Oxy]
Oxytocin (lactating case)
Insulin-like growth factor 1 (IGF-1)
Intra ovarian factor (IOF)
[IOF]
IGF-1 and insulin
Kawashima et al. [59] reported that IGF-1 is positively correlated with the level of feed intake. The authors argue that the plasma IGF-1 concentration increases transiently during the follicular phase and decreases during the luteal phase of the estrous cycle, i.e., IGF-1 levels decrease when progesterone (P4) increases. On the other hand, IGF-1 is lowest during early lactation when there is no P4 in circulation, and highest in late lactation [60]. In particular, a decrease in blood insulin and glucose concentrations in postpartum cattle is associated with the decrease in IGF-I [21]. In addition, acute dietary restrictions reduce both insulin and IGF-1 concentrations in the blood [4, 61]. Even if these are only empirical observations and evidence for mechanistic relationships is missing, these observations are incorporated into the equation for IGF-1 as follows,
$$ {\begin{aligned} \frac{d}{dt}{IGF} = c_{17} + c_{18}\cdot H^{-} \left(P4, T_{11}, 4\right)\cdot H^{+} \left(Ins, T_{12}, 10\right) - c_{19}\cdot {IGF}, \end{aligned}} $$
where c17 accounts for the basal IGF-1 synthesis rate. The rate constants c17,18,19 were determined such that the simulated IGF-1 concentrations match with the experimental data from 13 Holstein cows [59], see Fig. 3b. Moreover, in order to fit the simulated progesterone concentrations to the data (Fig. 3a), the basal P4 production rate had to be increased from cP4=0 in the original model [26] to cP4=0.1. This is consistent with reports about baseline progesterone levels [62].
Changes in P4 and IGF-1 levels during the estrous cycle. Growth of P4 (a) is correlated to the decay of IGF-1 (b). Data of IGF-1 and P4 from 13 Holstein dairy cows (red dots) were collected and kindly provided by Kawashima et al. [59]
A change in plasma IGF-1 has an impact on follicular cell development and responsiveness to hormonal signals. In particular, experimental studies demonstrated that reduced IGF-1 reduces ovarian responsiveness to LH stimulation [21, 63]. To include this mechanism in the model, the term in [26] that models the follicular cell responsiveness to LH,
$$ H^{+}({LH}_{Bld}) = c_{20} \cdot H^{+} \left(LH, T_{13}, 2\right), $$
was improved as follows. The LH blood concentration that is required for an ovarian response (threshold T13) is made dependent on IGF-1,
$$\begin{array}{@{}rcl@{}} T_{13}:= hm_{IGF} = c_{21}\cdot H^{-} \left(IGF, T_{14}, 2\right). \end{array} $$
Such a dependency was chosen because it allows for LH concentrations to increase in response to IGF-1 being below a certain threshold, T14. This mechanism is essential to ensure appropriate ovarian responses to IGF-1.
Insulin serves as a metabolic signal influencing the release of LH and FSH from the anterior pituitary into the blood [21, 64]. This mechanisms is included in the model by a stimulatory effect of insulin on the synthesis rates of LH and FSH. The equations for LH and FSH in [26] are changed to
$$\begin{array}{@{}rcl@{}} \frac{d}{dt}LH_{Pit} &= LH_{syn}\cdot hp^{LH}_{Ins} - LH_{rel}, \end{array} $$
$$\begin{array}{@{}rcl@{}} \frac{d}{dt}FSH_{Pit} &= FSH_{syn}\cdot hp^{FSH}_{Ins} - FSH_{rel}, \end{array} $$
where LHsyn,FSHsyn,LHrel, and FSHrel are the synthesis and release rates of LH and FSH, respectively, as described in [26]. The Hill functions \(hp^{LH}_{Ins}\) and \(hp^{FSH}_{Ins}\) describe the influence of insulin on LH and FSH pituitary levels, respectively,
$$\begin{array}{@{}rcl@{}} hp^{LH}_{Ins} &= c_{23}\cdot H^{+} \left(Ins, T_{16}, 10\right), \end{array} $$
$$\begin{array}{@{}rcl@{}} hp^{FSH}_{Ins} &= c_{22}\cdot H^{+} \left(Ins, T_{15}, 10\right). \end{array} $$
Hence, if insulin levels drop below a certain threshold (T15=T16=21 mU/L), the synthesis of LH and FSH halts.
Pregnancy and calving are characterized by a complex interplay of hormones. One of these hormones is oxytocin. The release of this hormone and milk yield are positively correlated [41]. Overall as well as peak concentrations of oxytocin decrease over one ongoing lactation [65]; earlier studies reported similar dynamics [66, 67, 68, 69]. According to measurements in those studies, peak concentrations of oxytocin during early lactation are more than twice the magnitude of those during late lactation.
The BovCycle model [26] does not capture changes in oxytocin concentrations during pregnancy and calving. To this end, the model was extended by introducing an additional term Oxylac into the equation of oxytocin,
$$\begin{array}{@{}rcl@{}} \frac{d}{dt}Oxy = Oxy_{lac} + Oxy_{syn} - Oxy_{cle}, \end{array} $$
$$ Oxy_{lac} = c_{24}\cdot \exp(-c_{25}\cdot t^{2}). $$
This is the simplest form of a non-negative decreasing function, namely a Gaussian function, see Fig. 4. The parameter value c25=0.0007 determines the width of the curve and was adopted to the approximate length of the early lactation period, whereas the parameter value c24=1.5 was fitted so that Oxy(t) during early lactation is about twice as high as Oxy(t) during late lactation.
Modelled additional oxytocin during lactation. Plot of the additional time-dependent oxytocin source term during lactation as defined by Eq. (26)
Reparametrization of the BovCycle model
The changes in the equations of the original BovCycle model [26] required changes of some of the original parameter values in order to be able to recover regular estrous cycles. In addition, the original BovCycle model [26] was challenged with the scenario of adding exogenous oxytocin early in the cycle. In a study by Donaldson et. al [70], it was shown that daily oxytocin injections to eight non-lactating cows starting on day two of the cycle reduced the estrous cycle length to nine days. The slow increase in plasma P4 concentration during the first five days of the cycle was not altered significantly, but plasma P4 concentrations decreased again to low values after day five. These results confirmed earlier studies [71, 72]. However, the original BovCycle model [26] did not reproduce these results. Hence, changes were made on parameters that describe the interaction of oxytocin and enzymes with prostaglandin F2α and the interovarian factor such that the recalibrated model correctly reflects the effects of oxytocin administration on the length of the estrus cycle and plasma P4 concentrations. Parameters that required changes are listed in Table 7.
Values of parameters that have been changed compared to [26]
Value in [26]
cLH
LH clearance rate constant
[P4]/d
P4 baseline concentration in the blood
\(\text {ex}^{CL}_{CL}\)
Exponent of CL to stimulate self-growth
\(\text {ex}^{Enz}_{PGF}\)
Exponent of enzyme to stimulate prostaglandin F2 α synthesis
\(\text {ex}^{Oxy}_{PGF}\)
Exponent of oxytocin to stimulate prostaglandin F2 α synthesis
\(\text {ex}^{P4}_{Enz}\)
Exponent of P4 to stimulate enzyme synthesis
\(\text {ex}^{PGF}_{IOF}\)
Exponent of prostaglandin F2 α to stimulate interovarian factor synthesis
\(\text {ex}^{CL}_{IOF}\)
Exponent of CL to stimulate interovarian factor synthesis
\(T^{Follicle}_{FSH}\)
Threshold of FSH to stimulate follicular function
\(\mathrm {T}^{FSH}_{Follicle}\)
Threshold of follicular function to reduce FSH influence on follicular growth
\(\mathrm {T}^{CL}_{CL}\)
Threshold of CL to stimulate self-growth
\(\mathrm {c}^{CL}_{CL}\)
[CL]/d
Maximum rate of CL self-growth
\(\mathrm {c}^{CL}_{LH}\)
Maximum rate of LH stimulated growth of CL
Sensitivity analysis aims at determining the model input parameters which mostly contribute to a quantity of interest depending on the model output. Let us denote the model input parameter vector as \(\mathbf {p}=(p_{1},\ldots,p_{d})\in {\mathbb R}^{d}\). The model here is an ordinary differential equation model of the form
$$ \mathbf{x}'(t)=f(\mathbf{x},\mathbf{p}),\quad \mathbf{x}(0)=\mathbf{x}_{0}\in{\mathbb R}^{n}, $$
and a quantity of interest, y, can be any observable depending on the model output x,
$$ \mathbf{y}=\mathbf{y}(\mathbf{x}(t,\mathbf{p})). $$
This quantity can be for instance the value of a specific output variable xj at a specific time point t, or the variance of xj over a specific time interval. These are examples for scalar outputs. For the sake of simplicity, the study here is restricted to a scalar output y. The sensitivity of y with respect to input parameter pi is given by
$$ S_{y}^{i}=\frac{\partial{y}}{\partial p_{i}}. $$
To account for differences in physical units among variables and parameter, often relative sensitivities are used,
$$ {\hat S}_{y}^{i}=\frac{\partial{y}}{\partial p_{i}}\cdot\frac{|{p_{i}}|}{|{y}|}. $$
If the exact derivative is difficult to compute, the sensitivity can be approximated by a finite difference scheme,
$$ S_{y}^{i}\approx\frac{y(\mathbf{x}(t,\mathbf{p}+\Delta e_{i}))-y(\mathbf{x}(t,\mathbf{p}))}{\Delta}, $$
where Δ is the size of the perturbation and ei is a vector of the canonical base. Often, Δ is a relative perturbation, i.e., Δ=ε·pi for some small number ε (e.g. ε=0.1) corresponds to a perturbation by 10%. In this case, the relative sensitivity is approximated by
$$ {\hat S}_{y}^{i}\approx\frac{y(\mathbf{x}(t,\mathbf{p}+\Delta e_{i}))-y(\mathbf{x}(t,\mathbf{p}))}{\epsilon\cdot |y(\mathbf{x}(t,\mathbf{p}))|}. $$
This is a local sensitivity in the sense that it describes the influence of a specific local perturbation of parameter pi on the model output. Sampling Δ or sampling pairs of input and output variables would allow for a global sensitivity analysis, but this is computationally much more demanding and the results are often difficult to interpret. For details on global sensitivity analysis, the reader is referred to [73].
For the metabolic-reproductive model presented here, sensitivity analysis is performed to determine the model parameters that are most important for the onset of luteal activity after calving. Hence, the observable y is chosen as the earliest time point at which the (relative) P4-level is larger than a threshold TP4=1,
$$ y(\mathbf{x}(t,\mathbf{p})):=\min_{t\geq 0}(P4(t)\geq T_{P4}). $$
The results of this analysis are presented in the following section.
The aim of this study was to analyze the impact of supplied glucose, represented by the parameter c0, on the estrous cycle dynamics in both lactating and non-lactating cows. For this purpose, the model was simulated for different feeding scenarios, including short and long time dietary restrictions. For a cow of 600 kg BW, DMI at maintenance is set to its default value of 11.7 kg/d [36]. This is the reference value corresponding to 100% DMI throughout the following, and variations to this value are stated accordingly.
Non-lactating cows
To model these cows, the value of Milk in Eq. (14) is set to zero. The numerical experiments for acute and chronic dietary restrictions are designed according to three experimental feeding studies from Mackey et al. [74], Murphy et al. [15] and Richards et al. [75]. Since these are studies in beef heifers and anestrus beef cows, respectively, the results are expected to agree only qualitatively, not necessarily quantitatively.
Varying the glucose content in the DMI
The effect of varying glucose content in the DMI on the glucose-insulin dynamics is analyzed by changing the value of the parameter c0 (glucose content in the DMI) between 4%, 8% and 16%. Simulation results are presented in Figs. 5 and 6.
Simulated glucose and insulin dynamics in non-lactating cows for different values of glucose content in the DMI. The glucose content in the DMI is varied with c0={0.04,0.08,0.16}, corresponding to 4, 8, and 16%, whereby 8 % represents the amount required for maintenance. With higher/lower glucose content in the DMI, blood levels of glucose (a), insulin (d), stored glucose (b) and fat (e), and glucose production (f) increase/decrease over time. Glucagon (c) behaves inversely to the glucose blood level (a)
Simulated metabolic rates in non-lactating cows at maintenance. Glucose content in the DMI was fixed at 8%. The figure illustrates glucose input, storage, and usage in terms of the amount of glucose absorbed via the digestive tract (a), glucose generated from glucogenic substances in the feed (b), glucose released from the liver into the blood (c), glucose absorbed into liver cells (f), and glucose used for body maintenance (d) and for metabolic processes in the liver (e). At maintenance intake, the cow is able to cover the daily glucose requirement, which results in stable levels of glucose in the different compartments
At maintenance intake, i.e. c0=0.08, the model calculates the non-mammary usage to be slightly less then 400 gram per day (Fig. 6d). This number is in qualitative agreement with Danfær et al [76], who estimated the amount of glucose required for maintenance in a non-lactating cow with a slightly lower body weight of 500 kg to be 290-380 gram per day. The amount of glucose absorbed from the digestive tract directly into the blood is calculated to be 75 g/d (Fig. 6a). The calculated amount of glucose released from the liver into the blood is about 800 g/d (Fig. 6c). This means that the total amount of glucose available in the blood is around 875 g/d, whereas the glucose uptake into liver cells (Fig. 6f) and the non-mammary usage (Fig. 6d) sum up to the same amount. This balance between input and consumption of glucose leads to stable glucose and insulin levels in the blood (Fig. 5a, d). In addition, this leads to stable glycogen and fat levels in the respective storage components (Fig. 5b, e).
With increasing glucose content in the DMI (c0=0.16), more glucogenic substances are available and lead to an increased gluconeogenesis [45]. This increases glucose and insulin concentrations in the blood, but they are still within their physiological range (Fig. 5a, d). Excess glucose in the system is stored as glycogen or fat reserves (Fig. 5b, e). When the glucose content in the DMI is decreased to 4%, blood glucose and insulin levels decrease towards their lower physiological bounds within two days (Fig. 5a, d), compare Table 1. As a result, the stored glycogen and the fat reserves (Fig. 5b, e) are reduced as well.
Acute nutritional restriction
To simulate the effect of acute nutritional restriction on the estrous cycle, a numerical experiment was designed according to the study of Mackey et al. [74], who reported about the effect of nutritional deprivation for a period of 13–15 days. Heifers with 406 ±5 kg body weight were allocated to a diet with a DMI of 1.2% of body weight for maintenance and then reduced to a diet with a DMI of 0.4% of body weight. In the model here, this reduction to 1/3 of the default diet corresponds to a reduction in the DMI from 11.7 kg/d to 3.84 kg/d.
This acute nutritional restriction is applied immediately after ovulation. The simulation results show increased levels of P4 (Fig. 7d), indicating a failure of luteolysis. Anovulation can be attributed to the absence of LH pulses (Fig. 7a) and lower FSH levels (Fig. 7b), as a result of decreased insulin levels (Fig. 7f). In addition, IGF-1 is decreased during the dietary restriction (Fig. 7e), which negatively influences the responsiveness of follicular cells to LH [20].
Effect of acute dietary restriction on the bovine estrous cycle in non-lactating dairy cows. On day 43, DMI is reduced from 100% (11.7 kg/d) to 33% (3.84 kg/d) for 15 days (the time period bounded by the two red lines). During the restriction period, one can observe a decrease of glucose in the store (g), insulin in the blood (f) and IGF-1 (e), an absence of LH pulses (a), and a decrease of amplitude in the FSH waves (b), leading to anovulation and failure in luteolysis with increasing P4 (d). The cycle re-starts soon after the end of the restriction period
Chronic nutritional restriction
To simulate the effect of chronic nutritional restriction on the estrous cycle, numerical experiments were designed according to the studies of Murphy et al. [15] and Richards et al. [75]. Murphy et al. [15] examined the effect of chronic dietary restriction on the estrous cycle over 10 weeks. In this study, heifers with 375 ±5 kg body weight were allocated to a maintenance diet with an amount of DMI corresponding to 1.2% of the body weight and a reduced diet with 0.7% of the body weight. In the model here, this reduction to 58% of the maintenance diet corresponds to a reduction in the DMI from 11.7 kg/d to 6.79 kg/d. In the experiment by Richards et al. [75], multiparous non-lactating Hereford cows underwent a chronic nutritional restriction for 30 weeks. They were fed to lose 1% of their bodyweight weekly. After the restriction period, the diet was increased to 160% of the maintenance diet.
The simulation was adapted to these two scenarios as follows. The nutritional restriction starts after ovulation. From then on, the model was simulated with 58% of the maintenance DMI within a time interval of 30 weeks. Simulation results (Fig. 8) show that the cow exhibits normal estrous cycles over a period of 15 weeks. During the chronic restriction period, the glycogen store (Fig. 8g) and the insulin in blood (Fig. 8f) decrease. LH (Fig. 8a), FSH (Fig. 8b) and IGF-1 (Fig. 8e) pulses decrease in frequency and amplitude, resulting in cessation of cyclicity after 15 weeks of feed restriction. The fat compartment loses around 10%. After 15 weeks, P4 decreases to a low level for the remaining 15 weeks, indicating the onset of anestrus. FSH and E2 exhibit changes in their wave patterns, that is, the number of waves per cycle increases. A similar tendency was observed in [15].
Effect of chronic dietary restriction on the bovine estrous cycles in non-lactating cows. DMI is reduced to 58% for 30 weeks (period between the red lines) and increased to 160% afterwards. During the restriction period, the glycogen store (g) and insulin in blood (f) decrease. LH (a), FSH (b) and IGF-1 (e) pulses decrease in frequency and amplitude, resulting in cessation of cyclicity after 15 weeks of feed restriction
Murphy et al. [15] examined ultrasound data and serum P4 between week 6 and 9. They found no alteration in CL growth, whereas P4 in restricted cows was numerically higher than in cows on maintenance diet. No anestrus was observed in the first 10 weeks of the restriction period, which is in agreement with the simulation results.
During the first weeks of restriction in the experiment by Richards et al. [75], P4 concentration increased as well. After losing 24.0 ±0.9% of their initial body weight, cows had decreased luteal activity measured via P4, and cessation of the estrous cycle was observed in 54% of the cows after 26 weeks. The authors reported that estrous cycles were re-initiated by week 40 in 64% of the restricted cows, feeding 160% of maintenance diet. The model predicts re-initiation of cyclicity by week 32, feeding 160% of DMI at maintenance.
Lactating cows
To investigate the effect of lactational metabolism and NEB on fertility hormones, different scenarios were simulated with the metabolic-reproductive model. As model input, interpolated time series data of DMI and milk yield from a study by Friggens et al. [77] were used, see Fig. 9. Each kilogram of milk produced requires around 72 gram glucose (parameter c13 in Eq. (14) [58]. Hence, the production of 41 kg milk per day requires about 3 kg of glucose per day. This was confirmed by Reynolds et al. [78], who predicted the glucose usage for milk to be between 2500 g/d and 3000 g/d. Milk production and the provided DMI in this study were 41 kg/d and 21 kg/d, respectively, averaged over 5 Holstein cows with an average body weight of 647 kg.
Model input data of DMI and milk. In this data, the highest milk yield (about 41 kg/d) can be observed 8 weeks postpartum. It coincides with the peak in the DMI (22 kg/d)
Energy balance is usually calculated as energy input minus output, requiring measurements of feed intake and energy output sources (milk, maintenance, activity, growth, and pregnancy)[79]. Alternatively, the energy balance can be calculated based on changes in the body reserves, using body weight and body condition score [79,80]. Since the model presented here does not explicitly calculate the body weight, the change in body fat is considered as an indicator of the energy balance,
$$\Delta_{Fat}= \mathit{glu_{lv-fat}} - \mathit{glu_{fat-lv}}. $$
This approach was also used in [81].
To explore the metabolic processes during lactation, simulations were performed for different values of glucose content in the DMI (parameter c0). The results are compared qualitatively with the studies by Elliot [82] and Reynolds et al. [78]. The changes in the glucose-insulin dynamics, body fat reserves, and metabolic rates are illustrated in Figs. 10, 11, and 12, respectively.
Simulated glucose and insulin levels in lactating dairy cows for different values of glucose content in the DMI. Time series data of milk yield and DMI from Holstein cows [77] are used as input for the model (c). Glucose and insulin dynamics were simulated with different glucose content in the DMI (c0={0.2,0.225,0.25,0.30}). When c0=0.2 (corresponding to 20% glucose content), glucose levels during peak milk drop towards the physiological limit (0.39 g/L) (a). In general, low amounts of glucose lead to a rapid depletion of the store (b), accompanied by a decrease in body fat (e), indicating a negative energy balance due to high milk production
Simulated change in body fat as an indicator of energy balance in lactating dairy cows for different values of glucose content in the DMI. When c0=0.2, energy balance is negative throughout the lactation period (a). When c0=0.225 or higher, the period of negative energy balance becomes shorter (b,c). When c0=0.3, energy balance is positive throughout the lactation period (d)
Simulated metabolic rates in a lactating cow for different values of glucose content in the DMI. Glucose content in the DMI was fixed at 20% (red line) or 30% (black line). During lactation, mammary glucose usage (f) gets prioritized compared to the non-mammary usage (d)
The simulation results clearly show a non-linear relationship between glucose content in the DMI and the values of glucose in blood and storage as well as insulin in blood at peak milk. Decreasing the glucose content in the DMI, starting from c0=0.3, first leads to a slow decrease in glucose and insulin levels, followed by a rapid decrease if c0 approaches the value 0.2.
For a high amount of glucose in the DMI (30%, c0=0.3), glucose and insulin levels in the blood are maintained within their physiological range (Fig. 10a, d). After the peak milk phase, the cow is even able to store glucose and fat (Fig. 10b, e). Consequently, the overall energy balance is positive throughout the lactation period (Fig. 11d). The model calculates the amount of glucose available in the circulation by direct absorption from the digestive tract (rate glufeed−bl) to be between 500 and 600 g/d (Fig. 12a). This is in agreement with Elliot [82], who estimated that for a cow with 600 kg BW and a milk yield of 40 kg/d, the amount of glucose absorbed from the digestive tract is around 600 g glucose per day.
For medium amounts of glucose in the DMI (22.5% or 25%), glucose and insulin levels are still kept within their physiological range (Fig. 10a, d), but the period of negative energy balance is prolonged (Fig. 11b, c).
If the amount of glucose in the DMI is decreased even further (20%, c0=0.2), one can observe an extended phase of negative energy balance with glucose and insulin dropping towards their lower physiological limits around peak milk (Fig. 10a, d). High demand and low input trigger the mobilization of body reserves, represented in the model by glycogen and fat in the store (Fig. 10b, e).
When c0 is varied between 0.2 and 0.3, the calculated amount of glucose released from the liver (gluprod) within the first 83 days post partum is 2500–4400 g/d (Fig. 12c). These numbers are in qualitative agreement with Reynolds et al. [78], who estimated the daily glucose production in the liver within the first 83 days post partum to be between 2700 and 3600 g/d. On can also observe that the mammary glucose usage gets prioritized compared to the non-mammary usage (Fig. 12f, d), and that this effect becomes more pronounced for low glucose diets.
The effect of changing glucose in the DMI on the estrous cycle
The glucose content in the DMI (parameter c0) has an effect on the estrous cycle. In the previous subsection, it was shown that decreasing c0 from 0.3 to 0.2 prolongs the phase of negative energy balance. A decrease in blood glucose and insulin concentrations is associated with a decrease in IGF-I [83–85]. As a consequence, elongated postpartum anestrus periods occur [86–89]. Similarly, Walsh et al. [5] resumed that NEB leads to low insulin concentrations in blood, which in turn prevents an increase in IGF-1 secretion, resulting in delayed resumption of ovarian cyclicity [90].
The simulation results (Fig. 13) agree with those observations. Increasing the relative amount of glucose in the DMI from c0=0.2 to 0.3 increases the IGF-1 concentration. This stimulates the responsiveness of follicles to LH, thereby shortening the postpartum anestrus interval from about 150 to 40 days (Fig. 14). Accordingly, the oxytocin level becomes cyclic again at the end of the anestrus interval, after having significantly decreased over the postpartum period (Fig. 15).
Simulated levels of P4, IGF-1, LH and estradiol during lactation for different values of glucose content in the DMI. Hormonal cycles were simulated over the lactation period for different fractions of glucose in DMI (parameter c0). A lower glucose content results in negative energy balance (Fig. 11), thereby prolonging the anestrus period. A higher glucose content results in an improved energy balance, which leads to increased insulin and IGF-1 levels and an earlier re-start of the estrous cycle
Effect of changing the glucose content in the DMI on the time of first ovulation after calving. Hormonal cycles were simulated over the lactation period for different fractions of glucose in DMI (parameter c0). Simulated data (red dots), which represents the estimated incidence of first ovulation, is determined by the time of first LH peak followed by an increase in progesterone production above baseline. The blue line represents the fitted curve f(x)=a· exp(−b·x)+c to the data with a=45581, b=0.30317, c=35.644. A lower glucose content results in a late ovulation, whereas a higher glucose content results in an early ovulation
Simulated levels of oxytocin during lactation for different values of glucose content in the DMI. Levels of oxytocin, which are very high in early lactation, decrease with ongoing lactation and become cyclic again at the end of the anestrus period
The length of the postpartum anestrus in the simulations agrees with the literature. In studies based on postpartum progesterone profiles, it was demonstrated that 90 to 95% of post partum dairy cows have resumed ovarian cycles by day 50 after calving [91–93]. Hence, a postpartum dairy cow is considered 'normal' if it has resumed ovarian cyclicity by day 50 post partum and continues cycling at regular intervals of approximately 21 days [94].
The simulations also show that estradiol levels at the beginning of the lactation period are within their normal range. This was confirmed by several studies. The authors in [75] found that restricted nutrition leads to anovulation but does not alter estradiol blood concentrations. Although ovulation and luteal development do not occur in anestrus cows, follicular growth is not totally impaired by restricted nutrient intake. In a review, Diskin et al. [10] suggested that NEB in early lactation does not affect the follicle population but does affect the ovulatory fate of the first dominant follicle. The authors summarized that low IGF-I and insulin cumulatively reduce follicular responsiveness to LH and ultimately suppress follicular oestradiol production.
There is evidence that a good management of the diet can reduce the incidence of abnormal estrous cycles [23,24,27]. Improving postpartum nutrition increases the blood concentration of insulin and IGF-I, which ultimately enhance LH pulsatility [19,85]. Higher IGF-1 levels during the first two weeks postpartum lead to an earlier re-start of the estrous cycle [5]. It was demonstrated in a study that providing a diet high in starch promotes an increased insulin release with a subsequent rise from 55% to 90% in the number of cows that ovulated within 50 days postpartum [24], a time interval that is considered to be an indicator for good reproductive performance [91]. In sum, resumption of cyclicity during lactation is crucial for good fertility in dairy farming. It can be influenced by feed intake, but also depends on many other factors such as uterine health, metabolic status, milk yield and overall condition.
The effect of changing model parameters on the estrous cycle
A local sensitivity analysis as described in Eq. 27, was performed to assess the influence of all model parameters on the time of first ovulation after calving, characterized by the onset of luteal activity (increased P4 levels). Throughout the calculations, glucose content in the DMI was fixed at c0=0.25, which resulted in an onset of luteal activity at day 50 post partum. The parameters' impact on the timepoint of ovulation is illustrated in Fig. 16. Figure 16a shows the change in the timepoint of first ovulation after perturbation of single parameters by +10%, whereas Fig. 16b shows the change in the timepoint of first ovulation after perturbation by -10%. Note that in the two subplots (A) and (B) only the numerator of \({\hat S}_{y}^{i}\) is plotted, since the denominator is independent of the parameter index i. The two most influential parameters are T1 (parameter number 91) and \(T_{P4}^{Foll}\) (parameter 33, described in [26]). The first one describes the threshold of glucose in the blood to stimulate insulin secretion, while the second one is the threshold of P4 to stimulate decrease of follicular function. A change of the parameters 91 and 33 by +10% and -10%, respectively, results in a later occurrence of ovulation (Fig. 16c). Indeed, an increase in the value of T1 by 10% limits the secretion rate of insulin. As insulin influences the release of LH, LH pulses are suppressed, which delays the ovulation to day ≈90. On the other hand, a decrease in the value of \(T_{P4}^{Foll}\) by -10% stimulates the decay of follicular function, which causes a prolongation of the anovulatory period to day ≈120.
Sensitivity analysis results for the time of first ovulation post partum. A sensitivity analysis was performed to assess the influence of all model parameters on the time of first ovulation after calving as described by Eq. (27). a shows the change in the timepoint of first ovulation after perturbation of single parameters by +10%, whereas b shows the change in the timepoint of first ovulation after perturbation by -10%. Note that in the two subplots a and b only the numerator of \({\hat S}_{y}^{i}\) is plotted since the denominator is independent of the parameter index i. The two most influential parameters are T1 (parameter 91) and \(T_{P4}^{Foll}\) (parameter 33). A change of the parameter T1 by +10% results in a later occurrence of ovulation (Fig. 16c). On the other hand, a decrease in the value of \(T_{P4}^{Foll}\) by -10% stimulats the decay of follicular function, which causes a prolongation of the anovulatory period to day ≈120
In the previous sections, the relationship between fertility and metabolism was explored based on two validated models [26,36]. These models were slightly modified and coupled to simulate the interplay of follicular development and its hormonal regulation with the glucose-insulin system. Information about the mechanistic interactions between fertility and metabolism, if taken straight from the literature, is sometimes contradictory and/or redundant. Therefore, only a small number of mechanisms was included, sufficient to realize the coupling of the two models.
With the coupled model, acute and chronic dietary restriction scenarios were simulated, intending to reproduce clinical study findings for non-lactating cows [74,75,95]. Furthermore, numerical experiments were run by varying the amount of DMI and the glucose content in the DMI for both lactating and non-lactating cows, and the effect of dietary restrictions on the estrous cycle was analyzed in lactating cows. The simulation results agree with the findings from the clinical studies, at least on a qualitative level.
The graphs presented here show the same qualitative behavior as the graphs supplied by McNamara and Shields [33]. This is not surprising since the model in [33] uses parts of the BovCycle model [25,26] and combines it with the more complicated Molly model. Despite using independent data sets (compared to the study by McNamara and Shields), the parameter values and trends came out very similarly.
The model here has also some limitations. Increasing (decreasing) the glucose content in the DMI, given by the parameter c0, results in the same simulation output as increasing (decreasing) total DMI, because only the product c0·DMI is contained in the model equations but not the individual factors. In reality, this is certainly not true. A way out would be to relate DMI directly or indirectly (e.g., via metabolic activity as in [36]) to one of the other variables. However, this would have complicated the model structure which, from the authors' point of view, is not necessary for the modeling purpose in this paper.
Furthermore, the model presented here only describes processes in a single representative cow. In its current form, the model is not able to display inter- or intra-individual variability. However, since the implemented mechanisms are universal, variability could easily be included by adapting parameter values to individual measurements, once such measurements are available.
One could also criticize the model for its restriction to glucose as the only feed component. Hence, the protein content should be included in addition to glucose and fat to complete energetic composition of DMI. This would provide one with a more realistic nutrient supply, change of body composition and body weight as well as milk production and composition.
In addition, experimental research is gaining more and more insights into the effect of nutrition on follicular development. With an improved follicle model, similar to the one introduced in [96], further simulations can be conducted to explore the effect of nutrition on multiple follicles in more detail.
So far, it is fair to say that the model presented here is only a starting point. It will certainly be modified and improved in future. However, by conducting numerical simulations relying on it, it was confirmed that an appropriate nutritional intake is fundamental in mitigating the effects and the extension of NEB in order to reduce the incidence of metabolic disorders in high producing cows and to avoid subsequent fertility problems [1,5,8,97]. To understand the interaction between nutrition, metabolism and reproduction, a unified approach was followed, similar to [33,98], where these fields of interest are integrated in one mathematical framework. The model here, formulated in terms of differential equations, enables the user to explore the relationship between nutrition and reproduction by performing related parameter studies. The local sensitivity analysis with respect to the onset of luteal activity after calving is just one example for such an analysis, which can easily be extended to other quantities of interest.
Reviewer's comments
We thank the editor and the reviewers for their time and effort to handle our manuscript. We revised our manuscript according to their recommendations.
Reviewer 1, John McNamara, Department of Animal Sciences, Washington State University, US (Authors' response included in italics)
Reviewer summary
The work is very nice in an area sorely in need of a quantitative approach. It is a sound approach to a very complex problem. It is original in that they are the first to put this level of detail into the system, and it is valid and significant to the field of dairy cattle nutrition and reproduction.
Recommendations to authors
Abstract. The line 'simulations confirm that an appropriate nutritional...' really should state 'simulations describe realistically the effects of nutritional glucose supply on the ovulatory cycle of lactating dairy cattle' (no model can confirm what happens in reality, it is the other way around).
That is right, we edited the text accordingly.
Page 4, line 29/30 or 50/51: "none of these models....in dairy cows. Excepting the model of McNamara and Shields, which contains many of the same elements herein, as it was based not only on Molly but on our BovCycle model (ref); however the McNamara and Shields model did not include the full reproductive process model." (I think the authors as well as the authors of the referred to article (McNamara and Shields) should have a more in depth reference, as the efforts are closely related and complementary).
We agree and added a more detailed discussion about the model by McNamara and Shields and its relation to our modeling effort in the introductory section.
Lines 50-51. It is not clear what the 'exact value' refers to here. I think they need to be specific in that 'as long as the initial values are within this range, they do not affect the performance of the model' (which is kind of neat, as that is how reality works too).
We followed this suggestion.
Page 11, line 179. All organs use glucose, except adipose tissue which cannot directly convert glucose to fatty acids.
Thank you for the correction, we adopted it.
Line 286. I don't think the sole purpose was to examine the effect of glucose in the feed, but to describe the role of supplied glucose in affecting the ovulatory cycle, so it should be stated somehow like that.
In fact, it was already stated like this in the manuscript, but the sentence structure was misleading. Hence, we reformulated the sentence.
It is interesting, but not surprising, that the graphs look a lot like the simpler graphs supplied by McNamara and Shields and some mention should be made of this, as again, that model used the BovCycle model but integrated it with the more complicated Molly model. It is a good scientific confirmation that using independent data sets (as the McNamara/Shields study did) from the study here, that the parameter values came out very similarly again, in a simpler way in the McNamara/Shields model and expanded upon very nicely here.
Thank you very much for pointing this out. We included this observation in the conclusion section.
Reviewer 2, Tin Pang, Heinrich-Heine University, Düsseldorf, Germany
The manuscript, submitted by Omari et al., proposes a new phenomenological model of a dairy cow that couples the existing glucose-insulin model and the estrous cycle model, which qualitatively describes the level of hormones and glucose in the cow under different conditions. This new model divides a cow into different "compartments", e.g., liver, blood, etc., with network-like interactions between the level of various hormones in blood and the level of glucose in different compartments. The authors also showed that the model predictions roughly match the experimental results, supporting the conceptual validity of the model. While previous modelling studies have quantified the interaction of hormones and glucose level in different compartments, the proposed model provides a better description at the network level, providing a stepping stone to better understanding of the trade-off between reproduction and milk production of dairy cows, and may serve the general interest of the research community.
No major issues.
In [7], the following formula was proposed for growing, non-lactating Holstein heifers.
$$\begin{array}{@{}rcl@{}} \mathit{DMI} = \big(-0.1128 + 0.2435\cdot\mathit{NE}_{M} - 0.0466\cdot\mathit{NE}_{M}^{2}\big)\cdot\frac{ \mathit{BW}^{0.75}}{\mathit{NE}_{M}}, \end{array} $$
where DMI is in (kg/d), BW is the body weight (kg) and NEM is net energy of diet for maintenance. NE recommendations are stated in the range between 1.24 and 1.55 Mcal/kg.
Berg et al. [53] estimated that 2% of the weight of the muscle tissue is formed by glycogen, and 10% of the liver weight. Is was also estimated that for a cow with 600 kg body-weight the mass of muscle, liver and kidney is around 280 kg, whereof 9 kg is liver weight [54]. According to these numbers, the liver stores about 900 g glycogen.
Supplementary information accompanies this paper at https://doi.org/10.1186/s13062-019-0256-7.
Many thanks to our colleagues, Mascha Berg, Rainald Ehrig, and Jane Knöchel, who have supported this work in various ways.
SR and JP conceived the study. SR obtained the funding. MO, AL and JP analyzed and interpreted the data. JP designed the conceptual model. MO, AL, and SR constructed the mathematical model. MO implemented the model and performed the simulations. All authors wrote the paper, as well as read and approved the final manuscript.
The work of MO, AL, and JP was funded by the German Federal Ministry for Education and Research (BMBF, https://www.bmbf.de), project BovSys (FKZ 031A311), with SR being the grant holder. The work of SR was supported by the Trond Mohn Foundation (BSF, https://www.mohnfoundation.no/), Grant no. BFS2017TMT01. The funder had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
13062_2019_256_MOESM1_ESM.zip (79 kb)
Additional file 1 The Matlab code contains the three m-files BovSys_para.m, BovSys_equa.m and BovSys_run.m. To start simulations, one has to run BovSys_run.m, which guides the user through different simulation scenarios (lactating/non-lactating with different diets). In addition, there are four data files containing the data for IGF (Data_IGF_Hol.mat), P4 (Data_P4_Hol.mat), milk (ML.mat), and DMI (DM.mat), that are needed as input files to run the model.
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© The Author(s) 2020
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1.Computational Systems Biology Group, Zuse Institute BerlinBerlinGermany
2.Department of Applied Biosciences and Process Engineering, Anhalt University of Applied SciencesKöthenGermany
3.Computational Biology Unit, University of Bergen, Department of InformaticsBergenNorway
Omari, M., Lange, A., Plöntzke, J. et al. Biol Direct (2020) 15: 2. https://doi.org/10.1186/s13062-019-0256-7
Received 21 November 2018
Revised 28 November 2019
Accepted 24 December 2019
Publisher Name BioMed Central | CommonCrawl |
\begin{document}
\title{VDM recursive functions in Isabelle/HOL}
\author{Leo Freitas\inst{1} \and Peter Gorm Larsen\inst{2}}
\authorrunning{ }
\institute{ School of Computing, Newcastle University, \\ \email{[email protected]} \and DIGIT, Aarhus University, Department of Engineering, \\
\email{[email protected]} }
\maketitle \setcounter{footnote}{0} \begin{abstract} For recursive functions general principles of induction needs to be applied. Instead of verifying them directly using the Vienna Development Method Specification Language (VDM-SL), we suggest a translation to Isabelle/HOL. In this paper, the challenges of such a translation for recursive functions are presented. This is an extension of an existing translation and a VDM mathematical toolbox in Isabelle/HOL enabling support for recursive functions. \end{abstract}
\keywords{VDM, Isabelle/HOL, Translation, Recursion, VSCode}
\parindent 0pt\parskip 0.5ex
\input{VDMToolkit.tex} \input{RecursiveVDM.tex}
\end{document} | arXiv |
\begin{document}
\title{\LARGE \bf ISS characterization of retarded switching systems with relaxed Lyapunov--Krasovskii functionals
\footnote{This work is supported by a public grant overseen by the Research and Valorization Service (SRV) of ENSEA.} \thispagestyle{empty} \pagestyle{empty}
\begin{abstract} This paper gives further insights about the Lyapunov--Krasovskii characterization of input-to-state stability (ISS) for switching retarded systems on the basis of the results in {\it [I. Haidar and P. Pepe. Lyapunov–krasovskii characterization of the input-to-state stability for switching retarded systems. SIAM Journal on Control and Optimization, 59(4):2997–3016, 2021]}. We give new characterizations of the ISS property through the existence of a relaxed common Lyapunov-Krasovskii functional. More precisely, we show that the existence of a continuous Lyapunov-Krasovskii functional whose upper right-hand Dini derivative satisfies a dissipation inequality almost everywhere is necessary and sufficient for the ISS of switching retarded systems with measurable inputs and measurable switching signals. Different characterization results, using different derivative notions, are also given. \end{abstract}
\textbf{Keywords:} Input-to-state stability; Converse theorems; Lyapunov--Krasovskii functionals; retarded functional differential equations; switching systems.
\section{Introduction} The problem of stability of switching systems has attracted much attention in the literature of control theory (see, e.g., \cite{AL2001, Boscain2002, Liberzon, Liberzon-Morse, Mazenc2018, SSGE, WANG201678, YanOzbay2008} and the references therein). The existence of a common Lyapunov function, i.e., a function which decreases uniformly along the trajectories of individual subsystems, is a sufficient condition for various stability notions like uniform asymptotic, exponential, and input-to-state stability. The existence of a common Lyapunov function is also necessary for switching systems which are uniformly stable. Converse Lyapunov theorems characterizing the stability of a switching system by the existence of a common Lyapunov function have been then developed for various switching dynamics (see, e.g.,~\cite{Dayawansa,MANCILLAAGUILAR200067,Wirth2005}) for finite-dimensional systems,~\cite{haidar2021lyapunov, Hante-Sigalotti} for infinite-dimensional systems, and~\cite{HaidarPepe2020, Haidar-Automatica, HaidarChapter2019, HaidarPepe21} for retarded systems).
In this paper we give a collection of converse Lyapunov theorems for switching retarded systems with measurable switching signals. The novelty of the obtained results lies in the relaxation of the conditions required by a Lyapunov-Krasovskii functional. We show that the ISS property of a switching retarded system can be characterized by the existence of a continuous (instead of Lipschitz on bounded sets) Lyapunov-Krasovskii functional whose upper right-hand Dini derivative satisfies a dissipation inequality almost everywhere. An important technical tool on which our arguments are based is the recent equivalence property given in~\cite[Theroem 1]{HaidarPepe21} proving that a switching retarded system is ISS (with measurable inputs and measurable switching signals) if and only if it is ISS for all piecewise-constant inputs and piecewise-constant switching signals. Recall that, when dealing with a retarded system, the map describing the evolution of the state is simply continuous with respect to time~(see, e.g., \cite[Lemma 2.1]{Hale}). Thus a continuous, or even Lipschitz on bounded sets, Lyapunov--Krasovskii functional $V$ evaluated on the solution of a retarded system will be in general continuous and not absolutely continuous with respect to time. By consequence, the nonpositivity of the upper right-hand Dini derivative of $V$ holding almost everywhere, is not sufficient to conclude about the monotonicity of $V$ along the solutions. Thanks to the equivalence property mentioned above, this problem is overcome by restricting the class of inputs and switching signals to the class of piecewise-constant ones. Indeed, in this case, the nonpositivity of the Dini derivative of $V$ along the solutions holds everywhere instead of almost everywhere permitting to conclude about its monotonicity (see~\cite{Hagood, MironchenkoIto2016}).
Another contribution of this paper is through the ISS characterization of switching retarded systems using different derivative notions of Driver's and Dini's types. Driver's type derivative (see, e.g., ~\cite{Driver-62, Pepe-Automatica-2007}), by contrast to Dini's one, is an appropriate definition of the derivative of a Lyapunov–Krasovskii functional that does not involve the solution. In~\cite{Pepe-Automatica-2007} it is shown that Driver and Dini derivatives coincide for locally Lipschitz Lyapunov-Krasovskii functionals. Here we extend this result to switching retarded systems. Furthermore, we show that the existence of a Lipschitz on bounded sets Lyapunov-Krasovskii functional whose Driver derivative satisfies a dissipation inequality (which is equivalent, by~\cite[Theorem 2]{HaidarPepe21}, to ISS) is equivalent to the existence of a continuous Lyapunov functional having its Dini derivative satisfying a dissipation inequality almost everywhere. Other Dini's type derivative definitions, which are used in the literature of retarded systems (see, e.g., ~\cite{8619545}), are also used in the collection of our converse Lyapunov theorems.
The paper is organized as follows. Section~\ref{sec: not} presents the notation, definitions and assumptions in use. The statements and proofs of our main results are presented in Section~\ref{main section}. The obtained results are discussed in Section~\ref{sec: dis}
\section{Switching retarded systems}\label{sec: not} In this section we list the notation, definitions, and the main assumptions in use.
\subsection{Notation}
Throughout the paper, we adopt the following notation: $\mathbb{R}$ denotes the set of real numbers, $\mathbb{R}_{+}$ the set of non-negative real numbers, and $\overline{\mathbb{R}}$ the extended real line. By $(\mathbb{R}^n, \|\cdot\|)$ we denote the $n$-dimensional Euclidean space, where $n$ is a positive integer and $\|\cdot\|$ is the Euclidean norm. Given $r>0$, $B(0,r)$ denotes the closed ball of $(\mathbb{R}^n, \|\cdot\|)$ of center $0$ and radius $r$.
By ${1}_{I}$ we denote the indicator function of a nonempty subset $I$ of $\mathbb{R}$.
Given $\Delta>0$, $\mathcal{C}:=(\mathcal{C}([-\Delta,0],\mathbb{R}^n),\|\cdot\|_{\infty})$ denotes the Banach space of continuous
functions mapping $[-\Delta,0]$ into $\mathbb{R}^n$, where $\|\cdot\|_{\infty}$ is the norm of uniform convergence. For a function $x:[-\Delta, b)\to\mathbb{R}^n$, with $0< b\leq +\infty$, for $t\in [0,b)$, $x_t:[-\Delta,0]\rightarrow\mathbb{R}^n$
denotes the history function defined by $x_t(\theta)=x(t+\theta)$, $-\Delta\leq\theta\leq0$. For a positive real $H$ and given $\phi\in \mathcal{C}$, $\mathcal{C}_{H}(\phi)$ denotes the subset $\{\psi\in \mathcal{C}: \|\phi-\psi\|_{\infty}\leq H\}$. We simply denote $\mathcal{C}_{H}(0)$ by $\mathcal{C}_{H}$.
A measurable function $u:\mathbb{R}_{+}\to\mathbb{R}^{m}$, $m$ positive integer, is said to be essentially bounded if $ess\sup_{t\geq 0}|u(t)|<+\infty$. We use the symbol $\|\cdot\|_{\infty}$ to indicate the essential supremum norm of an essentially bounded function. For given times $0\leq t_1<t_2$, $u_{[t_1,t_2)}:\mathbb{R}_+\to \mathbb{R}^m$ indicates the function given by $u_{[t_1,t_2)}=u(t){1}_{[t_1,t_2)}(t)$ for $t\geq 0$. A function $u:\mathbb{R}_{+}\to\mathbb{R}^{m}$ is said to be locally essentially bounded if, for any $t>0$, $u_{[0,t)}$ is essentially bounded.
A function $\alpha:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is said to be of class $\mathcal{K}$ if it is continuous, strictly increasing and $\alpha(0)=0$; it is said to be of class $\mathcal{K}_{\infty}$ if it is of class $\mathcal{K}$ and unbounded. A continuous function $\beta:\mathbb{R}_{+}\times\mathbb{R}_{+}\to\mathbb{R}_{+}$ is said to be of class $\mathcal{KL}$ if $\beta(\cdot,t)$ is of class $\mathcal{K}$ for each $t\geq 0$ and, for each $s\geq 0$, $\beta(s,\cdot)$ is nonincreasing and converges to zero as $t$ tends to $+\infty$.
With the symbol $\|\cdot\|_{a}$ we indicate any semi-norm in $\mathcal{C}$ such that, for some positive constants $\underline{\gamma_a}$ and $\overline{\gamma_a}$, the following inequalities hold: \begin{equation*}\label{semi equivalent norm}
\underline{\gamma_a}|\phi(0)|\leq \|\phi\|_{a}\leq \overline{\gamma_a}\|\phi\|_{\infty}, \quad \forall\, \phi\in \mathcal{C}. \end{equation*}
\subsection{Definitions and assumptions} Let us consider the switching control system described by the following retarded functional differential equation \begin{equation*} \Sigma: \begin{array}{llll} \dot x(t)&=&f_{\sigma(t)}(x_t, u(t)), \quad & a.e. ~t\geq 0,\\ x(\theta)&=&x_0(\theta), \quad &\theta\in [-\Delta,0], \end{array} \end{equation*} where: $x(t)\in \mathbb{R}^n$; $n$ is a positive integer; $\Delta$ is a positive real (the maximum involved time delay); $x_0\in \mathcal{C}$ is the initial state; the function $\sigma:\mathbb{R}_{+}\to \mathrm{S}$ is the switching signal; $\mathrm{S}$ is a nonempty set; $u:\mathbb{R}_{+}\to\mathbb{R}^m$, $m$ positive integer, is a Lebesgue measurable locally essentially bounded input signal.
We introduce the following two assumptions:
\begin{assumption}\label{Ass-LBS} For each $s\in \mathrm{S}$, $f_{s}(0,0)=0$. Moreover, $f_{s}(\cdot,\cdot)$ is uniformly (with respect to $s\in \mathrm{S}$) Lipschitz on bounded subsets of $\mathcal{C}\times \mathbb{R}^m$, i.e., for any $H>0$ there exists $L_{H}>0$ such that for every $\varphi,\psi\in \mathcal{C}_{H}$ and $u,v\in B(0,H)$, the following inequality holds for all $s\in \mathrm{S}$ \begin{equation*}
|f_s(\varphi,u)-f_s(\psi,v)|\leq L_{H}\left(\|\varphi-\psi\|_{\infty}+|u-v|\right).
\end{equation*} \end{assumption}
We denote by $\mathcal{U}$ the set of Lebesgue measurable locally essentially bounded inputs from $\mathbb{R}_+$ to $\mathbb{R}^m$ and by $\mathcal{U}^{\mathrm{PC}}$ the subset of right-continuous piecewise-constant ones. We denote also by $\mathcal{S}$ the set of measurable signals $\sigma:\mathbb{R}_+\to \mathrm{S}$ and by $\mathcal{S}^{\mathrm{PC}}$ the subset of right-continuous piecewise-constant ones.
\begin{assumption}\label{Ass-Measurable} For each $\phi\in \mathcal{C}$, $\sigma\in \mathcal{S}$ and $u\in {\cal U}$, the function $t\mapsto f_{\sigma(t)}(\phi,u(t))$, $t\in \mathbb R_+$, is Lebesgue measurable. \end{assumption}
Under Assumption~\ref{Ass-LBS} and Assumption~\ref{Ass-Measurable}, the existence and uniqueness of a solution for system~$\Sigma$ as well as its continuous dependence on the initial state is guaranteed by the theory of systems described by retarded functional differential equations (see, e.g., \cite{Hale, Kolmanovskii}). This can be reformulated by the following lemma.
\begin{lemma}\label{existence-uniqueness} For any $\phi\in \mathcal{C}$, $u\in\mathcal{U}$ and $\sigma\in\mathcal{S}$, there exists, uniquely, a locally absolutely continuous solution $x(t,\phi,u,\sigma)$ of $\Sigma$ in a maximal time interval $[0,b)$, with $0<b\leq +\infty$. If $b<+\infty$, then the solution is unbounded in $[0,b)$. Moreover, for any $\varepsilon>0$, for any $c\in (0,b)$, there exists $\delta>0$ such that, for any $\psi\in \mathcal{C}_{\delta}(\phi)$, the solution $x(t,\psi,u,\sigma)$ exists in $[0,c]$ and, furthermore, the following inequality holds \begin{equation*}
|x(t,\phi,u,\sigma)-x(t,\psi,u,\sigma)|\leq \varepsilon, \quad \forall\,t\in [0,c]. \end{equation*}
\end{lemma}
Let us recall the following definition about Driver's form derivative of a continuous functional $V:\mathcal{C}\to \mathbb{R}_+$. This definition is a variation of the one given in~\cite{Driver-62, Pepe-Automatica-2007,PEPE20061006} for retarded functional differential equations without switching. \begin{definition} For a continuous functional $V:\mathcal{C}\to \mathbb{R}_+$, its Driver's form derivative, $D_{(1)}^{+}V:\mathcal{C}\times \mathbb{R}^m\to \overline{\mathbb{R}}$, is defined, for the switching system $\Sigma$, for $\phi\in \mathcal{C}$ and $u\in \mathbb{R}^m$, as follows, \begin{equation*} D_{(1)}^{+}V(\phi,u)=\sup_{s\in \mathrm{S}}\limsup_{h\to0^{+}}\dfrac{V\left(\phi^{\Sigma,s}_{h,u}\right)-V\left(\phi\right)}{h}, \end{equation*} where $\phi^{\Sigma,s}_{h,u}\in \mathcal{C}$ is defined, for $h\in [0,\Delta)$ and $\theta\in [-\Delta,0]$, as follows \begin{equation*}\label{def-driver} \phi^{\Sigma,s}_{h,u}(\theta)=\left\{ \begin{array}{lll} \phi(\theta+h), \quad &\theta\in [-\Delta,-h)\\ \phi(0)+(\theta+h)f_{s}(\phi,u), \quad &\theta\in [-h,0]. \end{array}\right. \end{equation*} \end{definition}
Let us also recall the following definition about Dini derivative of a continuous functional $V:\mathcal{C}\to \mathbb{R}_+$. This definition is the one given in~\cite{Hale} for retarded functional differential equations without switching.
\begin{definition} Given initial state $\phi\in \mathcal{C}$, $u\in \mathcal{U}$ and $\sigma\in\mathcal{S}$, for a continuous functional $V:\mathcal{C}\to \mathbb{R}_+$ its Dini derivative $D_{(2)}^{+}V:[0,b)\to \overline{\mathbb{R}}$ is defined, for the switching system $\Sigma$, as follows, \begin{equation*} D_{(2)}^{+}V(t)=\limsup_{h\to 0^{+}}\dfrac{V(x_{t+h})-V(x_t)}{h}, \end{equation*} where $x(\cdot)$ is the solution of $\Sigma$ starting from $\phi$ and associated with $u$ and $\sigma$ over a maximal time interval $[0,b)$. \end{definition}
\begin{definition} For a continuous functional $V:\mathcal{C}\to \mathbb{R}_+$, its $\mathcal{S}$-Dini derivative, $D_{(3)}^{+}V:\mathcal{C}\times \mathcal{U}\times\mathcal{S}\to \overline{\mathbb{R}}$, is defined, for the switching system $\Sigma$, for $\phi\in \mathcal{C}$, $u\in \mathcal{U}$ and $\sigma\in\mathcal{S}$, as follows, \begin{equation*} D_{(3)}^{+}V(\phi,u, \sigma)=\limsup_{h\to 0^{+}}\dfrac{V(x_{h}(\phi,u,\sigma))-V(\phi)}{h}, \end{equation*} where $x_h(\phi,u,\sigma)$ is the solution of $\Sigma$ starting from $\phi$ and associated with $u$ and $\sigma$. \end{definition}
\begin{definition} For a continuous functional $V:\mathcal{C}\to \mathbb{R}_+$, its mode-Dini derivative, $D_{(4)}^{+}V:\mathcal{C}\times {\rm I}\!{\rm R}^m\times \mathrm{S}\to \overline{\mathbb{R}}$, is defined, for the switching system $\Sigma$, for $\phi\in \mathcal{C}$, $v\in {\rm I}\!{\rm R}^m$ and $s\in\mathrm{S}$, as follows, \begin{equation*} D_{(4)}^{+}V(\phi,v,s)=\limsup_{h\to 0^{+}}\dfrac{V(x_{h}(\phi,v,s))-V(\phi)}{h}, \end{equation*} where $x_h(\phi,v,\sigma)$ is the solution of $\Sigma$ starting from $\phi$ and associated with $u(t)\equiv v$ and $\sigma(t)\equiv s$, $t\geq 0$. \end{definition}
\begin{definition} For a continuous functional $V:\mathcal{C}\to \mathbb{R}_+$, its sup-mode-Dini derivative, $D_{(5)}^{+}V:\mathcal{C}\times {\rm I}\!{\rm R}^m\to \overline{\mathbb{R}}$, is defined, for the switching system $\Sigma$, for $\phi\in \mathcal{C}$ and $v\in {\rm I}\!{\rm R}^m$, as follows, \begin{equation*} D_{(5)}^{+}V(\phi,v)=\sup_{s\in\mathrm{S}}D_{(4)}^{+}V(\phi,v,s). \end{equation*} \end{definition}
We give in the following the definition of ISS of system $\Sigma$.
\begin{definition}\label{ISS-def} We say that system $\Sigma$ is $\mathrm{M}$-$\mathrm{ISS}$ ($\mathrm{PC}$-$\mathrm{ISS}$, respectively), if there exist a function $\beta\in \mathcal{KL}$ and a class $\mathcal{K}$ function $\gamma$ such that, for any $x_0\in \mathcal{C}$, $u\in \mathcal{U}$ ($\mathcal{U}^{\mathrm{PC}}$, respectively) and $\sigma\in \mathcal{S}$ ($\mathcal{S}^{\mathrm{PC}}$, respectively), the corresponding solution exists in $\mathbb{R}_+$ and, furthermore, satisfies the inequality \begin{equation*}
|x(t, x_0,u,\sigma)|\leq \beta(\|x_0\|_{\infty},t)+\gamma(\|u_{[0,t)}\|_{\infty}), \quad\forall\,t\geq 0. \end{equation*} \end{definition}
\section{Main results}\label{main section} The following theorem gives different characterizations of the input-to-state stability property of system $\Sigma$.
\begin{theorem}\label{ISS-converse theorem-Dini} The following statements are equivalent: \begin{itemize}
\item [1)] System $\Sigma$ is $\mathrm{PC}$-$\mathrm{ISS}$;
\item [2)] System $\Sigma$ is $\mathrm{M}$-$\mathrm{ISS}$;
\item [3)] there exist a Lipschitz on bounded sets functional $V:\mathcal{C}\to \mathbb{R}_+$, functions $\alpha_1, \alpha_2, \alpha_3\in \mathcal{K}_{\infty}$, and $\alpha_{4}\in \mathcal{K}$ such that the following inequalities hold for every $\phi\in\mathcal{C}$ and $u\in \mathbb{R}^m$: \begin{enumerate} \item[(i)]
$\alpha_1(|\phi(0)|)\leq V(\phi)\leq \alpha_2 (\|\phi\|_{a})$,
\item[(ii)]
$D_{(1)}^{+}V(\phi,u)\leq -\alpha_3 (\|\phi\|_{a})+\alpha_{4}(|u|)$;
\end{enumerate}
\item [4)] there exist a continuous functional $V:\mathcal{C}\to \mathbb{R}_+$, functions $\alpha_1, \alpha_2, \alpha_3\in \mathcal{K}_{\infty}$, and $\alpha_{4}\in \mathcal{K}$ such that, for any $\phi\in\mathcal{C}$, any $u\in \mathcal{U}$ and any $\sigma\in \mathcal{S}$, the following inequalities hold: \begin{enumerate} \item[(i)]
$\alpha_1(|\phi(0)|)\leq V(\phi)\leq \alpha_2 (\|\phi\|_{a}),$
\item[(ii)]
$D_{(3)}^{+}V(x_t,\overline u,\overline\sigma)\le-\alpha_3(\|x_t\|_a)+\alpha_4(|u(t)|)$, \\$ a.e.\, t\in [0,b)$,
\\
where $x(\cdot)$ is the solution of $\Sigma$ starting from $\phi$ and associated with $u$ and $\sigma$ over the maximal interval of definition $[0,b)$, $\overline u(\tau)=u(t+\tau)$ and $\overline\sigma(\tau)=\sigma(t+\tau)$, for all $\tau\in [0,b-t)$. \\ Furthermore if $u\in \mathcal{U}^{\mathrm{PC}}$ and $\sigma\in \mathcal{S}^{\mathrm{PC}}$ then
\item[(iii)]
$D_{(3)}^{+}V(x_t,\overline u,\overline\sigma)\le-\alpha_3(\|x_t\|_a)+\alpha_4(|u(t)|)$, \\ $\forall\, t\in [0,b)$,
\\
where $x(\cdot)$ is the solution of $\Sigma$ starting from $\phi$ and associated with $u$ and $\sigma$ over the maximal interval of definition $[0,b)$, $\overline u(\tau)=u(t+\tau)$ and $\overline\sigma(\tau)=\sigma(t+\tau)$, for all $\tau\in [0,b-t)$; \end{enumerate}
\iffalse \item [5)] there exist a continuous functional $V:\mathcal{C}\to \mathbb{R}_+$, functions $\alpha_1, \alpha_2, \alpha_3\in \mathcal{K}_{\infty}$, and $\alpha_{4}\in \mathcal{K}$ such that, for any $\phi\in\mathcal{C}$, any $u\in \mathcal{U}$ and any $\sigma\in \mathcal{S}$, the following inequalities hold: \begin{itemize} \item[(i)]
$\alpha_1(|\phi(0)|)\leq V(\phi)\leq \alpha_2 (\|\phi\|_{a}),$
\item[(ii)]
$D_{(3)}^{+}V(x_t,\bar u,\bar\sigma)\le\\
-\alpha_3(\|x_t\|_a)+\alpha_4\left(\displaystyle\lim_{h\to 0^+}\frac{1}{h}\int_{0}^{h}|u(\tau)|d\tau\right)$; where $x(\cdot)$ is the solution of $\Sigma$ starting from $\phi$ and associated with $u\in$ and $\sigma$ over the maximal interval of definition $[0,b)$, $\overline u(\tau)=u(t+\tau)$ and $\overline\sigma(\tau)=\sigma(t+\tau)$, for all $\tau\in [0,b-t)$; \end{itemize}
\textcolor{red}{ \item[(iii)]
$D_{(3)}^{+}V(x_t,\overline u,\overline\sigma)\le-\alpha_3(\|x_t\|_a)+\alpha_4(|u(t)|)$, \\ for all $t\in [0,b)$,
where $x(\cdot)$ is the solution of $\Sigma$ associated with $u$ and $\sigma$ over the maximal interval of definition $[0,b)$, $\overline u(\tau)=u(t+\tau)$ and $\overline\sigma(\tau)=\sigma(t+\tau)$, for all $\tau\in [0,b-t)$; \end{itemize}} \fi
\item [5)] there exist a continuous functional $V:\mathcal{C}\to \mathbb{R}_+$, functions $\alpha_1, \alpha_2, \alpha_3\in \mathcal{K}_{\infty}$, and $\alpha_{4}\in \mathcal{K}$ such that, for any $\phi\in\mathcal{C}$, any $u\in \mathcal{U}$ and any $\sigma\in \mathcal{S}$, the following inequalities hold: \begin{enumerate} \item[(i)]
$\alpha_1(|\phi(0)|)\leq V(\phi)\leq \alpha_2 (\|\phi\|_{a}),$
\item[(ii)]
$D_{(2)}^{+}V(t)\le-\alpha_3(\|x_t\|_a)+\alpha_4(|u(t)|), \, a.e.\,t\in [0,b)$,
\\ where $x(\cdot)$ is the solution of $\Sigma$ starting from $\phi$ and associated with $u$ and $\sigma$ over the maximal interval of definition $[0,b)$.\\ Furthermore if $u\in \mathcal{U}^{\mathrm{PC}}$ and $\sigma\in \mathcal{S}^{\mathrm{PC}}$ then
\item[(iii)]
$D_{(2)}^{+}V(t)\le-\alpha_3(\|x_t\|_a)+\alpha_4(|u(t)|),\, \forall\,t\in [0,b)$,
\end{enumerate} where $x(\cdot)$ is the solution of $\Sigma$ starting from $\phi$ and associated with $u$ and $\sigma$ over the maximal interval of definition $[0,b)$;
\item [6)] there exist a continuous functional $V:\mathcal{C}\to \mathbb{R}_+$, functions $\alpha_1, \alpha_2, \alpha_3\in \mathcal{K}_{\infty}$, and $\alpha_{4}\in \mathcal{K}$ such that, for any $\phi\in\mathcal{C}$, any $u\in \mathcal{U}^{\mathrm{PC}}$ and any $\sigma\in \mathcal{S}^{\mathrm{PC}}$, the following inequalities hold: \begin{enumerate} \item[(i)]
$\alpha_1(|\phi(0)|)\leq V(\phi)\leq \alpha_2 (\|\phi\|_{a}),$
\item[(ii)]
$D_{(3)}^{+}V(x_t,\bar u,\bar \sigma)\le -\alpha_3(\|\phi\|_a)+\alpha_4(|u(t)|),\\ \forall\,t\in [0,b)$,
\end{enumerate} where $x(\cdot)$ is the solution of $\Sigma$ starting from $\phi$ and associated with $u$ and $\sigma$ over the maximal interval of definition $[0,b)$, $\overline u(\tau)=u(t+\tau)$ and $\overline\sigma(\tau)=\sigma(t+\tau)$, for all $\tau\in [0,b-t)$;
\item [7)] there exist a continuous functional $V:\mathcal{C}\to \mathbb{R}_+$, functions $\alpha_1, \alpha_2, \alpha_3\in \mathcal{K}_{\infty}$, and $\alpha_{4}\in \mathcal{K}$ such that, for any $\phi\in\mathcal{C}$, any $u\in{\rm I}\!{\rm R}^m$ and any $s\in\mathrm{S}$, the following inequalities hold: \begin{enumerate} \item[(i)]
$\alpha_1(|\phi(0)|)\leq V(\phi)\leq \alpha_2 (\|\phi\|_{a}),$
\item[(ii)]
$D_{(4)}^{+}V(\phi,u,s)\le -\alpha_3(\|\phi\|_a)+\alpha_4(|u|)$;
\end{enumerate}
\item [8)] there exist a continuous functional $V:\mathcal{C}\to \mathbb{R}_+$, functions $\alpha_1, \alpha_2, \alpha_3\in \mathcal{K}_{\infty}$, and $\alpha_{4}\in \mathcal{K}$ such that, for any $\phi\in\mathcal{C}$ and any $u\in{\rm I}\!{\rm R}^m$ the following inequalities hold: \begin{enumerate} \item[(i)]
$\alpha_1(|\phi(0)|)\leq V(\phi)\leq \alpha_2 (\|\phi\|_{a}),$
\item[(ii)]
$D_{(5)}^{+}V(\phi,u)\le -\alpha_3(\|\phi\|_a)+\alpha_4(|u|)$.
\end{enumerate} \end{itemize} \end{theorem}
Before giving the proof of Theorem~\ref{ISS-converse theorem-Dini} let us underline what we have mentioned in the introduction concerning the absolute continuity problem of Lyapunov--Krasovskii functionals. In fact, since we deal with a retarded system, the map describing the evolution of the state is simply continuous with respect to time. Thus a continuous (even Lipschitz on bounded sets) Lyapunov--Krasovskii functional evaluated on the solution of such a system will be in general continuous and not absolutely continuous with respect to time (we highlight that this problem is overcome in~\cite{Pepe-Tac-2007} by restricting the class of initial states to continuously differentiable ones; this does not yield any loss of generality because, as it is shown in the same paper, the ISS property holds with continuous initial states if and only if it holds with continuously differentiable ones). By consequence, we cannot directly use the standard comparison lemma~\cite[Lemma 4.4]{LinSontagWang1996} in the proof of the sufficiency parts (i.e., the ones implying the $\mathrm{ISS}$) of Theorem~\ref{ISS-converse theorem-Dini}. Instead, exploiting the equivalence between $\mathrm{M}$-$\mathrm{ISS}$ and $\mathrm{PC}$-$\mathrm{ISS}$ given by Theorem~\ref{ISS-converse theorem-Dini}, one can use the following comparison lemma from~\cite{MironchenkoIto2016}.
\begin{lemma}{\cite[Lemma 1]{MironchenkoIto2016}}\label{relax_Sontag} For each continuous and positive definite function $\alpha$, there exists a class $\mathcal{KL}$ function $\beta_{\alpha}$ with the following property: if, for $T>0$ (or $T=+\infty$), $y:[0,T)\to \mathbb{R}_+$ is a continuous non-negative function which satisfies the inequality \begin{equation}\label{beta0} D^{+}y(t)\leq -\alpha(y(t)), \quad\forall\,t\in[0,T), \end{equation} where $D^{+}y$ denotes the upper-right Dini derivative of $y$, with $y(0)=y_0\in \mathbb{R}_+$, then it holds that \begin{equation}\label{beta1} y(t)\leq \beta_{\alpha}(y_0,t), \quad\,\forall\, t\in[0,T). \end{equation} \end{lemma}
{\it Proof of Theorem~\ref{ISS-converse theorem-Dini}.} The proof of $1) \implies 2)$ is given in \cite[Theorem 3.1]{HaidarPepe21}. The proof of $2)\implies 3)$ is given in \cite[Theorem 3.2]{HaidarPepe21}. Concerning the proof of $3)\implies 4)$, let $V$ be the Lipschitz on bounded sets functional given by point 3). Let $x(\cdot)$ be the solution of $\Sigma$ associated with some $\phi\in\mathcal{C}$, $u\in\mathcal{U}$ and $\sigma\in \mathcal{S}$ over a maximal time interval of definition $[0,b)$. Following the same steps of the proof of~\cite[Theorem 2]{Pepe-Automatica-2007} (see also~\cite{Driver-62}) given for retarded non-switching systems, one can verify that the following equality holds for almost every $t\in [0,b)$ \begin{eqnarray}\label{DD}
&&\displaystyle\limsup_{h\to 0^{+}}\frac{V(x_{h}(x_t,\bar u,\bar\sigma))-V(x_t)}{h}\nonumber\\ && =\displaystyle\limsup_{h\to0^{+}}\frac{V\left((x_t)^{\Sigma,\sigma(t)}_{h,u(t)}\right)-V(x_t)}{h}. \end{eqnarray} Indeed, Observe that \begin{eqnarray*} &&\displaystyle\frac{V\left((x_t)^{\Sigma,\sigma(t)}_{h,u(t)}\right)-V\left(x_t\right)}{h}\\ =&&\displaystyle \frac{V\left((x_t)^{\Sigma,\sigma(t)}_{h,u(t)}\right)-V\left(x_{h}(x_t,\bar u,\bar \sigma)\right)}{h}\\ &&+\displaystyle\frac{V\left(x_{h}(x_t,\bar u,\bar \sigma)\right)-V\left(x_t\right)}{h}, \end{eqnarray*} it is sufficient to prove that for almost every $t\in [0,b)$ we have \begin{eqnarray}\label{DD1} \displaystyle\limsup_{h\to 0^{+}}\dfrac{V\left((x_t)^{\Sigma,\sigma(t)}_{h,u(t)}\right)-V\left(x_{h}(x_t,\bar u,\bar \sigma)\right)}{h}=0. \end{eqnarray} For this, using the fact that $V$ is Lipschitz on bounded sets, there exists $L=L(x_t)$ such that \begin{eqnarray*}
&&\left|V\left((x_t)^{\Sigma,\sigma(t)}_{h,u(t)}\right)-V\left(x_{h}(x_t,\bar u,\bar \sigma)\right)\right|\\
&&\hspace{-0.35cm}\leq L\left\|(x_t)^{\Sigma,\sigma(t)}_{h,u(t)}-x_{h}(x_t,\bar u,\bar \sigma)\right\|_{\infty}\\
&&\hspace{-0.35cm}=L\sup_{\theta\in [0,h]}\left\|\theta f_{\sigma(t)}(x_t,u(t))
-\int_{t}^{t+\theta}f_{\sigma(\tau)}(x_{\tau},u(\tau))d\tau\right\|\\
&&\hspace{-0.35cm}= L\sup_{\theta\in [0,h]}\left\|
\int_{t}^{t+\theta}\left(f_{\sigma(t)}(x_t,u(t))-f_{\sigma(\tau)}(x_{\tau},u(\tau))\right)d\tau\right\|\\ &&\hspace{-0.35cm}\leq L\displaystyle\sup_{\theta\in [0,h]}
\int_{t}^{t+\theta}\left\|f_{\sigma(t)}(x_t,u(t))-f_{\sigma(\tau)}(x_{\tau},u(\tau))\right\|d\tau\\ &&\hspace{-0.35cm}= L
\int_{t}^{t+h}\left\|f_{\sigma(t)}(x_t,u(t))-f_{\sigma(\tau)}(x_{\tau},u(\tau))\right\|d\tau.\\ \end{eqnarray*} Under Assumption~\ref{Ass-Measurable}, using the Lebesgue's Differentiation Theorem it follows that for almost every $t\in [0,b)$ we have \begin{eqnarray*}
\displaystyle\lim_{h\to 0^+}\frac{1}{h}\int_{t}^{t+h}\left|f_{\sigma(t)}(x_t,u(t)-f_{\sigma(\tau)}(x_\tau,u(\tau))d\tau)\right|=0. \end{eqnarray*} Therefore, equality~\eqref{DD} holds for almost every $t\in [0,b)$. Now observe that \begin{eqnarray}\label{DD2}
&\displaystyle\limsup_{h\to0^{+}}\dfrac{V\left((x_t)^{\Sigma,\sigma(t)}_{h,u(t)}\right)-V\left(x_t\right)}{h}\leq D_{(1)}^+V(x_t,u(t))\nonumber\\
& \leq -\alpha_3(\|x_t\|_a)+\alpha_4(|u(t)|). \end{eqnarray}
From~\eqref{DD} together with~\eqref{DD2} it follows that \begin{eqnarray*}
D^+_{(3)}V(x_t,\bar u,\bar\sigma)\leq -\alpha_3(\|x_t\|_a)+\alpha_4(|u(t)|),\, a.e. t\in [0,b). \end{eqnarray*} Hence the proof of $3)\implies 4)$.
Notice that, given any initial state $\phi\in\mathcal{C}$, $u\in\mathcal{U}$ and $\sigma\in\mathcal{S}$, the following equality holds for all $t\in [0,b)$ \begin{equation}
D_{(2)}^{+}V(t)=D_{(3)}^{+} V(x_t,\overline u,\overline\sigma), \end{equation}
the proof of $4)\implies 5)\implies 6)$ is obvious. The proof of $6) \implies 7)$ follows from the fact that, for each $s\in\mathrm{S}$ and $v\in {\rm I}\!{\rm R}^m$, we have $D_{(4)}^{+}V(\phi,v,s)=D_{(3)}^{+}V(\phi,\bar u,\bar\sigma)$ with $u(\cdot)\equiv v$ and $\sigma(\cdot)\equiv s$. The proof of $7) \implies 8)$ is obvious. Concerning the proof of $8)\implies 1)$, let $\phi\in \mathcal{C}$, $u\in\mathcal{U}^{\mathrm{PC}}$, $\sigma\in \mathcal{S}^{\mathrm{PC}}$, and let $x(\cdot)$ be the corresponding solution over a maximal interval of time $[0, b)$, $0<b\leq +\infty$. Let $w: [0,b)\to \mathbb{R}_+$ be the function which is defined by
$$w(t)=V(x_{t}(\phi,u,\sigma)), \quad \forall\,t\in [0,b).$$
Knowing that $u$ and $\sigma$ are piecewise-constants, then for a sufficiently small $h>0$ we have $\sigma_{|_{[t,t+h)}}\equiv \sigma(t)$ and $u_{|_{[t,t+h)}}\equiv u(t)$. By inequality (ii) of point 8), the following holds for every $t\in [0,b)$ \begin{equation*}
D^{+}w(t)\leq -\alpha_{3}(\|x_t(\phi,u(t),\sigma(t))\|_{a})+\alpha_4(|u(t)|). \end{equation*}
Let the input $u(t)$ be such that $\sup_{t\geq 0}|u(t)|=v$, for a suitable $v\geq 0$. By analogous reasoning as in \cite{28018, PEPE20061006}, one can prove the existence of $c\in (0,b]$ such that \begin{eqnarray} &D^{+}w(t)\leq -\alpha(w(t)),\quad &\forall\,t\in [0,c),\label{suf1}\\
&|x(t,\phi,u(t),\sigma(t))|\leq \gamma(v), \quad &\forall\,t\in [c,b),\label{suf2} \end{eqnarray} where $\alpha=\frac{1}{2}\alpha_{3}\circ\alpha_{2}^{-1}$ and $\gamma=\alpha_2\circ\alpha_3^{-1}\circ2\alpha_4$.
Since $t\mapsto w(t)$ is continuous, from Lemma~\ref{relax_Sontag} it holds the existence of a class $\mathcal{KL}$ function $\beta_{\alpha}$ such that \begin{equation*}
|w(t)|\leq \beta_{\alpha}(w(0),t), \quad \forall\, t\in [0,c), \end{equation*} from which it follows that \begin{equation}\label{ISS1}
|x(t,\phi,u,\sigma)|\leq \beta(\|\phi\|_{\infty},t), \quad \forall\, t\in [0,c), \end{equation} with $\beta(r,t)=\alpha_{1}^{-1}\circ\beta_{\alpha}(\alpha_2(\overline{\gamma_a}r),t)$. By consequence, inequalities~\eqref{suf2} and~\eqref{ISS1} lead to the following inequality \begin{equation}\label{ISS}
|x(t,\phi,u,\sigma)|\leq \beta(\|\phi\|_{\infty},t)+\gamma(v), \quad \forall\, t\in [0,b). \end{equation} It follows, from Lemma~\ref{existence-uniqueness}, that $b=+\infty$. By causality arguments, and given the arbitrarity of $\varphi\in \mathcal{C}$, $u\in \mathcal{U}^{\mathrm{PC}}$ and $\sigma\in \mathcal{S}^{\mathrm{PC}}$, the $\mathrm{PC}$-$\mathrm{ISS}$ of system $\Sigma$ is proved.
\ensuremath{\Box}\\
We highlight that, for the cases of nonlinear finite-dimensional and retarded non-switching systems, the result stated in Theorem~\ref{ISS-converse theorem-Dini} concerning the equivalence between items 1) and 2) can be also deduced by~\cite[Theorem 3.3]{Karafyllis2008InputtoOutputSF}, which concerns piecewise-continuous and right-continuous inputs, and by density arguments (see the reasoning used in~\cite[Proposition 3]{Pepe-Tac-2007} for equivalence of ISS with respect to dense sets of initial states).
\iffalse \section{Illustrative example}\label{example-sec} \begin{example} Consider the following example \begin{equation}\label{example2} \begin{array}{llll} \dot x(t)=&Ax(t)+B(x(t-\tau(t)))u(t), & a.e. ~t\geq 0,\\
x(\theta)=&x_0(\theta), & \theta\in [-\Delta,0],
\end{array} \end{equation} where, for $t\geq 0$, $x(t)=(x_1(t),x_2(t))^T\in\mathbb{R}^2$; $\tau:\mathbb{R}_+\to[0,\Delta]$, is a measurable uncertain time delay function; $u:\mathbb{R}_+\to \mathbb{R}$ is a measurable locally essentially bounded input; $A$ is a Hurwitz matrix and $B$ is a uniformly bounded nonlinear matrix function.
Let $\mathrm{S}= [0,\Delta]$. System~\eqref{example2} can be equivalently written as system~\eqref{pure delay-equivalent}, where, for each $s\in \mathrm{S}$, the function $f_{s}: \mathcal{C}\times\mathbb{R}\to \mathbb{R}^2$ is given, for $\phi\in\mathcal{C}$ and $v\in \mathbb{R}$, by
\begin{equation}\label{example2_1} f_s(\phi,v)=A\phi(0)+B(\phi(-s))v. \end{equation} Consider the following Lyapunov--Krasovskii functional \begin{equation*} V(\phi)=\phi(0)^TP\phi(0)+\int_{-\Delta}^{0}{\phi^T(\tau)\left(\frac{-\tau}{\Delta}Q_1+\frac{\tau+\Delta}{\Delta}Q_2\right)\phi(\tau)d\tau}, \end{equation*} where $P, Q_1$ and $Q_2$ are symmetric positive definite matrix with $Q_2-Q_1>0$, which has been introduced in~\cite{PEPE20061006} to study the ISS property of systems like~\eqref{example2} with constant delay. Computing the upper right-hand Dini derivative of $V$ along system~\eqref{example2} (see~\cite{PEPE20061006} for more details) for $Q_1=I, Q_2=2I$, and $P$ such that $A^TP+PA=-4I$ ($I$ is the identity matrix) gives \begin{align*} D^{+}V(\phi,v)
\leq -\min\left\{1,\frac{1}{\Delta}\right\}\|\phi\|^2_{2}+\rho^2|v|^2, \end{align*}
where $\rho$ is sufficiently large such that $\frac{b\|P\|}{\rho}<\min\{1,\frac{1}{\Delta}\}$ and $b$ is the uniform bound of $B$. Thus, by Theorem~\ref{measurable delay systems}, it is proved that system~\eqref{example2} is $\mathrm{M}$-$\mathrm{ISS}$. \end{example} \fi
\section{Conclusions}\label{sec: dis} In this paper we give a collection of converse Lyapunov theorems for ISS of nonlinear switching retarded systems. In particular, we show that the existence of continuous (instead of locally Lipschitz) Lyapunov-Krasovskii functional whose upper right-hand Dini derivative satisfies a dissipation inequality almost everywhere is necessary and sufficient for the ISS of switching retarded systems. This equivalence property is obtained for a very general class of Lebesgue measurable switching signals. Different derivative notions, which are usually used in the literature of retarded systems, are also used to establish our converse theorems. Future developments may concern the problem of the input-to-state stabilization and of the input delay tolerance (see, e.g., \cite{wang2020input} and \cite{zhao2022memoryless}) for switching retarded systems.
\end{document} | arXiv |
\begin{document}
\title{\TheTitle\thanks{Submitted to the editors \today.\funding{\TheFunding}
\begin{abstract}
Bilevel optimization is a comprehensive framework that bridges single- and multi-objective optimization. It encompasses many general formulations, including, but not limited to, standard nonlinear programs. This work demonstrates how elementary proximal gradient iterations can be used to solve a wide class of convex bilevel optimization problems without involving subroutines. Compared to and improving upon existing methods, ours (1) can handle a wider class of problems, including nonsmooth terms in the upper and lower level problems, (2) does not require strong convexity or global Lipschitz gradient continuity assumptions, and (3) provides a systematic adaptive stepsize selection strategy, allowing for the use of large stepsizes while being insensitive to the choice of parameters.
\end{abstract}
\begin{keywords}\TheKeywords \end{keywords}
\begin{AMS}\TheSubjclass \end{AMS}
\section{Introduction}\label{sec:introduction}
Bilevel programs consist of optimization problems with a hierarchical structure where the solution of one optimization problem is sought over the set of solutions of another one. Such problems originally emerged in the framework of game theory and have been studied extensively since the 1950s, see \cite{dempe2002foundations,dempe2020bilevel} for an extensive overview. Recently, they have also found applications in various areas of machine learning such as hyperparameter optimization, meta learning, data poisoning attacks, and reinforcement learning \cite{franceschi2018bilevel,rajeswaran2019meta,hong2023two,grazzi2023bilevel,borsos2020coresets}. Variational inequality variants have also been of much interest in recent years \cite{facchinei2014vi,bigi2022combining,lampariello2022solution,kaushik2021method,pedregosa2016hyperparameter}. The standard approach for addressing bilevel programs consists in solving a series of approximate problems with better regularity properties; refer to \cite{facchinei2014vi,bahraoui1994convergence,attouch1996viscosity} and the references therein. However, it is widely known that these techniques can suffer from many practical issues related to convergence speed and stability.
In this work, we study \emph{simple bilevel programs} which refers to problems where the lower level problem does not have a parametric dependence on the variables of the upper level problem. We split both the upper and the lower cost functions as the sum of differentiable and nonsmooth terms and propose an explicit algorithm without the need to solve any inner minimizations. In particular, we consider structured simple bilevel programs of the form \begin{subequations}\label{eq:P}
\begin{align}
\label{eq:P1}
\minimize_{x\in\R^n}{} &~ \cost_1(x)\coloneqq \f_1(x)+\g_1(x)
\\
\label{eq:P2}
\stt{} &~ x\in\X_2\coloneqq\argmin_{w\in\R^n}\set{\cost_2(w)\coloneqq \f_2(w)+\g_2(w)}
\end{align} \end{subequations}
where functions \(\f_1,\f_2:\R^n\to \R\) are convex and have \emph{locally} Lipschitz continuous gradients, and \(\g_1, \g_2:\R^n\to \Rinf\) are proper closed convex (potentially nonsmooth) functions. Some notable example applications include regularized problems in machine learning and signal processing, where the regularization can be captured by the upper level functions, e.g., \(\g_1 = \|\cdot\|_p\), \(p\geq 1\), corresponding to \(\ell_p\) regularization, while the loss function can be captured by the lower level function \(\f_2\), and there may be additional constraints such as nonnegativity constraints captured by \(\g_2\). Another notable example is the class of convex nonlinear programs (NLPs) \begin{align*}
\minimize_{x\in\R^n}{} &~ \f_1(x)+\g_1(x)
\\
\stt{} &~ x \in D, \quad Ax = b, \quad h(x) \leq 0, \end{align*}
where \(\f_1, h:\R^n\to \R\) are convex and have locally Lipschitz continuous gradients, \(\g_1:\R^n\to \Rinf\) is a proper closed convex (possibly nonsmooth) function, \(A\in \R^{m\times n}, b\in \R^m\), and \(D\) is a nonempty closed convex set. As noted in \cite{solodov2007explicit}, this problem can be formulated in the form of \eqref{eq:P} by setting \(\g_2 = \indicator_D\) and \(\f_2(x) = \|Ax-b\|^2 + \|\max\set{0, h(x)}\|^2\), where \(\indicator_D\) denotes the indicator function of the set \(D\).
For solving \eqref{eq:P}, we rely on the so-called \emph{diagonal} approach \cite{bahraoui1994convergence,cabot2005proximal,solodov2007explicit}, which involves examining the scaled sum of the upper and lower cost functions. To this end, we define the family of functions \begin{equation}\label{eq:phik}
\fk\coloneqq\sigk\f_1+\f_2,
\quad\gk\coloneqq\sigk\g_1+\g_2, \quad\text{and}\quad
\phik\coloneqq \fk + \gk = \sigk \cost_1 + \cost_2, \end{equation} parametrized with a scalar \(\sigk>0\). The method by Cabot \cite{cabot2005proximal} for solving simple bilevel programs, here dubbed \emph{Cabot's proximal point algorithm} (\cabot), involves iterative proximal maps (see \cref{sec:preliminaries}) \begin{equation*}
x_{k+1} =
\prox_{\alphk*\phik*}(x_k), \end{equation*} where the parameter \(\sigk\) is updated after each iteration. Convergence of \cabot{} was established under the \emph{slow control} condition \( \sigk \searrow 0, \) \( \sum_{k\in \N} \sigk = \infty \).
However, due to its implicit nature, in many applications \cabot{} leads to inner minimizations or matrix inversions.
A notable advancement in this regard was achieved by Solodov in \cite{solodov2007explicit}, who studies \eqref{eq:P} when \(\g_1 \equiv 0\) and \(\g_2\) is an indicator of a closed convex set \(D\). The method, here dubbed \emph{Solodov's explicit descent method} (\solodov), uses explicit oracles (gradients for \(\f_1 ,\f_2\) and projections onto \(D\)) and updates \(\sigk\) after a single step of projected gradient method with Armijo linesearch, without the need to solve any inner minimizations. More specifically, given \(\nu, \eta \in (0,1)\) and some \(\widehat\alpha_0>0\), in each iteration an inverse penalty \(0<\sigk*\leq\sigk\) is chosen and the variable updated as \begin{subequations}\label{eq:solodov}
\begin{align}
x_{k+1}
={} &
\proj_D\left(x_k - \alphk*\nabla\fk*(x_k)\right),
\shortintertext{where \(\alphk*=\widehat\alpha_0\eta^{m_k}\) and \(m_k\in\N\) is the smallest such that}
\label{eq:solodov:LS}
\fk*(x_{k+1})
\leq{} &
\fk*(x_k)
+
\nu
\@ifstar\@innprod\@@innprod{\nabla\fk*(x_k)}{x_{k+1}-x_k}.
\end{align} \end{subequations} The convergence of the method was established in \cite{solodov2007explicit} under the slow control condition. As is typical in linesearch methods, fixing the parameter \(\widehat\alpha_0\) (the stepsize tried first in each iteration) offline renders the behavior of the algorithm very susceptible to this choice. This is also evident in the simulation preview of \cref{fig:backtrack}: high values of \(\widehat\alpha_0\) allow for large stepsizes to be tested and potentially accepted, thereby favoring convergence speed in terms of number of iterations, but may also lead to more backtrackings and function evaluations in the linesearch \eqref{eq:solodov:LS}; conversely, small values of \(\widehat\alpha_0\) reduce the complexity of each iteration by reducing the number of backtrackings at the expense of smaller stepsizes and consequently slower convergence.
\subsection{Proposed methodology}
Inspired by the analysis of \cite{solodov2007explicit,cabot2005proximal}, we propose the proximal-gradient method \refadabim{} that allows for nonsmooth proximable terms \(\g_1, \g_2\) (see the discussion in \cref{sec:problem}), and unlike \cite{solodov2007explicit} does not require any cost evaluation. Most importantly, while it retains the linesearch nature of \solodov{}, compared to the fixed value \(\widehat\alpha_0\) of \eqref{eq:solodov} it provides a dynamic update of the initial stepsize \(\balphk\), cf. \eqref{eq:adabim:alphk*_0}, which is refined over the iterations based on local geometry estimates and on the algorithmic history, yielding much larger stepsizes with considerably fewer backtrackings. This online self-correcting feature renders the proposed algorithm insensitive to parameters chosen at initalization, and is the reason why we dub it \emph{essentially adaptive}, borrowing the ``adaptive'' terminology of single-level optimization schemes such as \cite{malitsky2020adaptive,latafat2023adaptive}. Differently from these methods, however, our proposed \refadabim* is \emph{``essentially'' adaptive}, for further adjustments through a backtracking procedure may be needed.
The numerical evidence that will be presented in \cref{sec:num} demonstrates the efficacy of the proposed adaptive strategy. Before that, we here extracted a snapshot of some simulation instances to substantiate our claims about the benefits of an adaptive strategy. \Cref{fig:backtrack} illustrates the cumulative number of backtrackings per iteration required by \adabim{} compared to \solodov{} with different choices of \(\widehat\alpha_0\), and \cref{fig:gam} reports a comparison of stepsize magnitudes with other proximal-gradient methods.
Similar to \cite{solodov2007explicit}, the linesearch has the twofold benefit of waiving \emph{global} Lipschitz-differentiability assumptions on functions \(\f_1\) and \(\f_2\), as well as the need of a priori knowledge on (local) Lipschitz moduli, both conditions being predominant in related literature. Nevertheless, our proposed linesearch integrated with the adaptive stepsize initialization is not designed to simply cope with the lack of global Lipschitz differentiability, for its employment is beneficial even when such conditions are met. To show this, as a byproduct of our analysis the \emph{static} variant \refstabim{} is proposed that uses nonadaptive (and nevertheless increasing) stepsizes when the differentiable terms are globally Lipschitz differentiable with known moduli.
\refStabim* is on par and competetive with existing works that require global Lipschitz differentiability, yet without strong convexity or smoothness requirements on either level. On top of this, not only is the adaptive variant \refadabim* applicable to the much larger class of problems captured by \cref{ass:basic}, but it also enables much faster convergence, as empirically demonstrated through numerical simulations.
\begin{figure}
\caption{
Preview of some numerical results presented in \cref{sec:num} indicating the advantages of the proposed adaptive algorithm over existing related methods.
\protect\refAdabim* enables much larger stepsizes through a linesearch which, thanks to the tailored adaptive warm starting, most often passes at the first trial.
}
\label{fig:backtrack}
\label{fig:gam}
\end{figure}
\subsection{Algorithmic overview}
\begin{algorithm}[tb]
\caption{
\textbf{\sffamily Ada}ptive \textbf{\sffamily Bi}level \textbf{\sffamily M}ethod (\adabim) using local Lipschitz estimates \eqref{eq:CL}
}
\label{alg:adabim}
\input{TeX/Alg/adabim.tex} \end{algorithm} \footnotetext{ The argument of the square root in the numerator of the second term in \eqref{eq:adabim:alphk*_0} is larger than \(1-\nu >0\) owing to the conditions \(\nicefrac{\sigk*}{\sigk} \geq \nicefrac34\) and \eqref{eq:LS} enforced during the preceding iteration (see \eqref{eq:sqrt>0}). }
The key idea of our adaptive scheme is based on the recent work \cite{latafat2023adaptive} that studies the proximal gradient method, which considers local Lipschitz and (inverse) cocoercivity estimates of the differentiable functions \(\fk=\sigk\f_1+\f_2\) at the previous iterates \(x_k, x_{k-1}\) as \begin{subequations}\label{eq:CL}
\begin{align}
\lk
{}\coloneqq{} &
\frac{
\@ifstar\@innprod\@@innprod{\nabla\fk(x_{k-1})-\nabla\fk(x_k)}{x_{k-1}-x_k}
}{
\|x_{k-1}-x_k\|^2
}
\shortintertext{and}
\ck
{}\coloneqq{} &
\frac{
\|\nabla\fk(x_{k-1})-\nabla\fk(x_k)\|^2
}{
\@ifstar\@innprod\@@innprod{\nabla\fk(x_{k-1})-\nabla\fk(x_k)}{x_{k-1}-x_k}
},
\shortintertext{
and similarly for \(\f_i\) with
}
\label{eq:Lk2}
\lk^{(i)}
{}\coloneqq{} &
\frac{
\@ifstar\@innprod\@@innprod{\nabla\f_i(x_{k-1})-\nabla\f_i(x_k)}{x_{k-1}-x_k}
}{
\|x_{k-1}-x_k\|^2
},
\quad
i=1,2, \end{align} so that \(\lk=\sigk\lk^{(1)}+\lk^{(2)}\). \end{subequations} The proposed \refadabim* can be viewed as an extension of \cite[{\sffamily adaPGM} (Alg. 1)]{latafat2023adaptive} to the simple bilevel setting: if \(\sigk =\sigma\) for all \(k\geq0\) and the linesearch is eliminated, then the algorithm reduces to {\sffamily adaPGM} applied to the problem of minimizing \(\fk+\gk\). One of the main advantages of this approach is that it leads to much larger stepsizes compared to classical methods, which in turn translates to faster convergence in practice. In the stepsize initialization at \cref{state:adabim:alphk*_0}, note that when \(\alphk\ck\leq1\) one has that the second term in \eqref{eq:adabim:alphk*_0} reduces to \(\nicefrac10=\infty\) and the update simplifies as \(
\balphk* =
\min\set{
\sqrt{\tfrac{\sigk}{\sigma_{k-1}}\bigl(1+\rhok\bigr)}
\tfrac{\sigk}{\sigk*}
\alphk
,\,
\alpha_{\rm max}
} \). This is of significant importance since it allows the initialization \(\balphk*\) to strictly increase compared to the stepsize \(\alphk\); for instance, under the standard choice \(\sigk = \nicefrac1{k+1}\), the first term is always larger than \(\alphk\). This standard choice of inverse penalties is the one adopted in the simulations: this is compatible with our requirement on the sequence \(\seq{\sigk}\), a slight refinement of the slow control condition in which the ratio of consecutive parameters must converge to one: \begin{equation}\label{eq:sigk}
\sigk\searrow 0, \quad
\sum_{k\in\N}\sigk=\infty, \quad
\tfrac{\sigk*}{\sigk}\to 1. \end{equation}
As already mentioned, the stepsizes \(\alphk\) can be set offline without the need of a linesearch whenever the smooth functions \(\f_1\) and \(\f_2\) are \emph{globally} Lipschitz differentiable with known moduli. The details are provided in \refstabim, the \emph{static} counterpart of \refadabim*, which is however consistently orders of magnitude slower. This fact is used as a testimony of the importance of the linesearch in combination with the adaptive stepsize initialization strategy.
\begin{algorithm}[t]
\caption{
\textbf{\sffamily Sta}tic \textbf{\sffamily Bi}level \textbf{\sffamily M}ethod (\stabim) when \(\protect\f_i\) are globally \(L_{\protect\f_i}\)-Lipschitz smooth, \(i=1,2\)
}
\label{alg:stabim}
\input{TeX/Alg/stabim.tex} \end{algorithm}
\subsection{Related literature}
In addition to \cite{cabot2005proximal,solodov2007explicit}, there have been several other notable methods for solving simple bilevel problems. The \emph{minimal norm gradient} method (\mng) \cite{beck2014first} is applicable to \eqref{eq:P} when \(\g_1\equiv 0\), \(\g_2\) is the indicator of a closed convex set, and the upper level problem is strongly convex. \Mng{} relies on a cutting plane approach which can lead to inner minimizations. The \emph{bilevel gradient sequential averaging method} (\bigsam) is an explicit method proposed in \cite{sabach2017first} that is based on a viscosity approximation approach \cite{xu2004viscosity,moudafi2000viscosity} and considers problems with Lipschitz differentiable and strongly convex upper level cost functions. In addition to enforcing \eqref{eq:sigk}, \bigsam{} requires \(\sigk \in (0,1]\) (see \cref{sec:bigsam} for further details). Another related algorithm that will be compared against in the simulations is the \emph{iterative regularization via dual diagonal descent} (\itthreeD) \cite{garrigos2018iterative}, which is designed for the iterative regularization of linear inverse problems. A remarkable property of \itthreeD{} is that it does not impose the slow control condition (see \cite[Rem. 10]{garrigos2018iterative}). Finally, the \emph{diagonal gradient scheme} (\dgs) was proposed in \cite{peypouquet2012coupling} for solving smooth simple bilevel programs. It however involves an implicit stepsize rule that is available in closed form only in certain scenarios, such as when a quadratic growth condition holds (see \cite[Assumptions H1-H3 and \S3.2]{peypouquet2012coupling}). A summary of the settings of the aforementioned methods is provided in \cref{table:comparisons}.
The above literature review is focused on \emph{explicit} methods and is not exhaustive. Other recent contributions include \cite{doron2022methodology} that does not require strong convexity or differentiability of the upper level problem, but involves inner subroutines. A cutting-plane strategy that employs conditional gradient--type updates is proposed in \cite{jiang2023conditional}. In \cite{guan2023first} the authors propose a minimal like--norm gradient method under H\"{o}lderian-type assumptions.
Another line of works has focused on \emph{nonsimple} bilevel programs, that is, those in which the lower level problem is allowed to have a parametric dependence on the upper level variable. Most of the existing results operate under the assumption that each parametric problem has a unique solution. Initially developed in \cite{ghadimi2018approximation}, the approach consists of a gradient-type scheme for the upper level problem where the gradient oracle is approximated through an inner procedure; see also \cite{khanduri2021near,franceschi2018bilevel,grazzi2023bilevel,gould2016differentiating,pedregosa2016hyperparameter,luo2020stochastic,hong2023two,arbel2021amortized,chen2022single} for related work and extensions. We also remark that \cite{arbel2022nonconvex} studied nonsimple bilevel programs without the uniqueness assumption through introducing the notion of selection maps, while \cite{liu2021value,sow2022constrained} propose various reformulations as constrained optimization problems.
\begin{table}[htb]
\centering
\begin{minipage}{10.75cm}
\renewcommand{\fnsymbol{footnote}}{\fnsymbol{footnote}}
\newcommand{\ecs}{\multicolumn{2}{c|}{}}
\let\mc\multicolumn
\newcommand{\mr}[1]{\multirow{2}{*}{#1}}
\rowcolors{4}{gray!15}{}
\setlength\extrarowheight{1pt}
\setlength\tabcolsep{3pt}
\begin{tabular}{@{}|c *{6}{c|}@{}}
\cline{3-7}
\ecs & \mc{2}{c|}{upper level $\cost_1$} & \mc{2}{c|}{lower level $\cost_2$} & \mr{explicit}
\\
\ecs & \mc{1}{c}{$\f_1$} & $\g_1$ & \mc{1}{c}{$\f_2$} & $\g_2$ &
\\\hline
\cite{cabot2005proximal} & \cabot
& {\bf\color{Red}\xmark} & & {\bf\color{Red}\xmark} & & {\bf\color{Red}\xmark}
\\
\cite{peypouquet2012coupling} & \dgs
& $C^{1,1}$ & {\bf\color{Red}\xmark} & $C^{1,1}$ & {\bf\color{Red}\xmark} & {\bf\color{Green}\cmark}
\\
\cite{beck2014first} & \mng
& $C^1$, str. cvx & {\bf\color{Red}\xmark} & $C^{1,1}$ & $\indicator_D$ & {\bf\color{Red}\xmark}
\\
\cite{sabach2017first} & \bigsam
& $C^{1,1}$, str. cvx & {\bf\color{Red}\xmark} & $C^{1,1}$, str. cvx & & {\bf\color{Green}\cmark}
\\
\cite{garrigos2018iterative} & \itthreeD\footnotemark
& $C^{1,1}$ & {\bf\color{Red}\xmark} & $C^{1,1}$ & & {\bf\color{Green}\cmark}
\\
\cite{solodov2007explicit} & \solodov
& $C^{1,+}$ & {\bf\color{Red}\xmark} & $C^{1,+}$ & $\indicator_D$ & {\bf\color{Green}\cmark}
\\\rowcolor{Green!15}
Alg. \ref{alg:adabim} & \adabim
& $C^{1,+}$ & & $C^{1,+}$ & & {\bf\color{Green}\cmark}
\\\rowcolor{Green!05}
Alg. \ref{alg:stabim} & \stabim
& $C^{1,1}$ & & $C^{1,1}$ & & {\bf\color{Green}\cmark}
\\\hline
\end{tabular}
\footnotetext{
$^{\text{\fnsymbol{footnote}}}$
It addresses linear inverse problems and the requirements in the table are on the dual formulation that the algorithm addresses (see \S\ref{sec:itthreeD} for a discussion).
}
\end{minipage}
\caption[]{
Comparison between the proposed \refadabim, \refstabim, and other methods for simple convex bilevel programming.
``Explicit'' methods are those not requiring inner minimization subroutines.
Legend:
\(C^{1,1}\): \emph{globally} Lipschitz differentiable;
\(C^{1,+}\): \emph{locally} Lipschitz differentiable;
str. cvx.: strongly convex.
}
\label{table:comparisons} \end{table}
Regarding adaptive methods, we also point out that they have been of growing interest in several areas of optimization. For example, \cite{malitsky2020adaptive} proposed an adaptive gradient descent scheme for smooth unconstrained minimization, later extended in \cite{latafat2023adaptive} to account for proximal terms. Furthermore, \cite{latafat2023adaptive,vladarean2021first,chang2022golden} develope adaptive schemes in the primal-dual setting, while \cite{malitsky2020golden,alacaoglu2022beyond} focus on hemivariational inequalities. The aforementioned works permit increasing stepsizes, striking a noticeable advantage in practice compared to other adaptive schemes such as the ones in \cite{yang2018modified,thong2020self,bot2023relaxed,yang2021self,bohm2022solving} that enforce decreasing stepsizes. We refer the reader to \cite{latafat2023adaptive} for additional references and further discussions on adaptive approaches.
\subsection{Contributions}
The main contribution of the paper is \refadabim{}, a simple scheme based on proximal gradient updates involving a family of parametrized forward and backward terms that vary across iterations. We establish convergence under a carefully designed stepsize rule. The proposed scheme is (i) explicit, in the sense that it does not involve any inner minimization subroutines, (ii) adapts to the local geometry of the differentiable terms allowing for large stepsizes, (iii) is not sensitive to the choice of parameters at initialization, (iv) is parameter agnostic, in the sense that no knowledge of problem parameters such as (local) strong convexity or smoothness moduli is required, (v) does not involve function evaluations, and (vi) can cope with nonsmoothness on both levels without necessitating global Lipschitz differentiability or (not even local) strong convexity.
\subsection{Organization}
After listing some preliminary material and known facts, the next section presents the problem setup and the statement of \cref{thm:BiM:convergence}, our main convergence result; the concluding \cref{sec:notation} therein offers a schematic overview of the notation adopted throughout the paper to ease the reading. The convergence analysis is presented in detail in \cref{sec:cnv}: after some preliminary lemmas related to the adaptive strategy, we derive a quasi-descent inequality in \cref{sec:prelem}; this is used in \cref{sec:recipe} to establish an implicit convergence recipe for proximal gradient iterations that enables a unified treatment of \refadabim* and \refstabim* in the proof of \cref{thm:BiM:convergence} reported in \cref{proofthm:BiM:convergence}. Numerical simulations are carried out in \cref{sec:num}, and \cref{sec:conclusions} concludes the paper.
\section{Problem setting and main results}\label{sec:problem}
\subsection{Preliminaries}\label{sec:preliminaries}
The sets of natural, real, and extended-real numbers are \(\N\), \(\R\coloneqq(-\infty,\infty)\) and \(\Rinf\coloneqq\R\cup\set\infty\), respectively, while the positive and strictly positive reals are \(\R_+\coloneqq[0,\infty)\) and \(\R_{++}\coloneqq(0,\infty)\). We adopt the conventions that \(0\in\N\) and \(1/0=\infty\). Given \(x\in\Rinf\), its positive part is indicated as \([x]_+\coloneqq\max\set{0,x}\). We denote by \(\@ifstar\@innprod\@@innprod{\cdot}{\cdot}\) and \(\|{}\cdot{}\|\) the standard Euclidean inner product and the induced norm, and with \(\id\) the identity function defined on a suitable space.
The \DEF{domain} and \DEF{epigraph} of an extended real--valued function \(\func{h}{\R^n}{\Rinf}\) are, respectively, the sets \(
\dom h \coloneqq
\set{x\in\R^n}[h(x)<\infty] \) and \(
\epi h \coloneqq
\set{(x, c )\in\R^n\times\R}[h(x)\leq c] \). Function \(h\) is said to be \DEF{proper} if \(\dom h\neq\emptyset\), and \DEF{lower semicontinuous (lsc)} if \(\epi h\) is a closed subset of \(\R^{n+1}\). We say that \(h\) is \emph{level bounded} if its \( c \)-sublevel set \(
\lev_{\leq c }h {}\coloneqq{}
\set{x\in\R^n}[
h(x)\leq c
] \) is bounded for all \( c \in\R\).
The \DEF{indicator function} of a set $E\subseteq\R^n$ is denoted by \(\indicator_E\), namely \(\indicator_E(x)=0\) if \(x\in E\) and \(\infty\) otherwise. The projection onto and the distance from $E$ are respectively denoted by \[
\proj_E(x)\coloneqq\argmin_{z\in E}\|z-x\| \quad\text{and}\quad
\dist(x,E)\coloneqq\inf_{z\in E}\|z-x\|. \] The (convex) \DEF{subdifferential} of a proper lsc convex function \(\func{h}{\R^n}{\Rinf}\) at a point \(\bar x\) is the set \(
\partial h(\bar x) \coloneqq
\set{v\in\R^n}[h(x)\geq h(\bar x)+\@ifstar\@innprod\@@innprod{v}{x-\bar x}\ \forall x\in\R^n] \). The \DEF{proximal mapping} of \(h\) is \(\func{\prox_h}{\R^n}{\R^n}\) defined by \[
\prox_h(x) =
\argmin_{w\in\R^n}\set{
h(w)+\tfrac12\|w-x\|^2
}, \] and is characterized by the implicit subdifferential inclusion \cite[Eq. (24.2)]{bauschke2017convex} \begin{equation}\label{eq:proxsubgrad}
x-\prox_h(x) \in
\partial h\bigl(\prox_h(x)\bigr). \end{equation}
\subsection{Assumptions and main result}
We will pattern the frameworks of \cite{solodov2007explicit,cabot2005proximal}, while considering general convex functions \(\g_1\) and \(\g_2\) (as opposed to indicator functions), and differentiable functions \(\f_1, \f_2\) with locally Lipschitz-continuous gradients. Throughout the paper, \refadabim{} is studied under the following assumptions.
\begin{subequations}
\begin{assumption}[basic requirements]\label{ass:basic}
The following hold in problem \eqref{eq:P}:
\begin{enumeratass}
\item \label{ass:f}
\(\f_1,\f_2:\R^n\to \R\) are convex and have locally Lipschitz-continuous gradients;
\item \label{ass:g}
\(\g_1,\g_2:\R^n \to \Rinf\) are proper lsc convex functions;
\item
the upper level problem restricted to \(\dom\cost_2\) is lower bounded:
\begin{equation}\label{eq:costinf_1}
\costinf_1
{}\coloneqq{}
\inf_{x\in\dom\cost_2}\set{\f_1(x)+\g_1(x)}
{}>{}
-\infty;
\end{equation}
\item \label{ass:X1}
the set of solutions \(\X_1\coloneqq\argmin\set{\cost_1(x)}[x\in\argmin\cost_2]\) is nonempty and compact; in particular,
\begin{equation}\label{eq:costinf_2}
\costinf_2
{}\coloneqq{}
\inf_{\fillwidthof[c]{x\in\dom \g_2}{x\in\R^n}}\set{\f_2(x)+\g_2(x)}
{}>{}
-\infty\mathrlap{.}
\end{equation}
\end{enumeratass}
\end{assumption} \end{subequations}
The iterations of our proposed method amount to selecting a stepsize \(\alphk*>0\) together with an (inverse) penalty parameter \(\sigk*\leq\sigk\), followed by one proximal gradient step on the inversely penalized cost function \(\fk+\gk\). For the sake of ``explicitness'', we assume throughout that \(\gk\) is \emph{prox-friendly}, \ie, that its proximal mapping can be evaluated efficiently. We remark that the proximal point method of \cite{cabot2005proximal} involves the same construct, and studies solving proximal minimizations approximately. In contrast, we assume \emph{exact} proximability and focus on splitting the problem into differentiable and nonsmooth terms. Although prox-friendliness is in general not preserved by the sum, in many instances of practical interest the proximal map of \(\gk\) can be evaluated based on that of \(\g_1\) and/or \(\g_2\). Many such instances involve the case in which either one is the \(\ell^1\) norm, or the indicator of a simple set such as a box or an \(\ell^2\)-ball; refer to \cite[\S24]{bauschke2017convex} and \cite[\S6]{beck2017first} for examples and further details of proximable functions. Also the case in which either one is zero has its own appeal: when \(\g_1=0\) we recover (and generalize) other methods, while when \(\g_2=0\) we can allow for an arbitrary prox-friendly term in the upper level, differently from any other explicit method of \cref{table:comparisons}.
The convergence of the proposed schemes is stated in the next theorem, which constitutes the main result of the paper and whose proof is deferred to \cref{proofthm:BiM:convergence}. Similarly to \cite{solodov2007explicit,cabot2005proximal}, it is shown that the distance from the set of solutions \(\seq{\dist(x_k, \X_1)}\) converges to zero. Because of nonemptiness and boundedness of the optimal set \(\X_1\) prescribed by \cref{ass:X1}, this condition is equivalent to existence and optimality of the cluster points.
\begin{theorem}\label{thm:BiM:convergence}
Suppose that \cref{ass:basic} holds and that \(\seq{\sigk}\) complies with \eqref{eq:sigk}.
Then, the sequence \(\seq{x_k}\) generated by \refadabim{} is bounded and all its cluster points are solutions to the bilevel problem \eqref{eq:P}.
The same is true also for \refstabim{} provided that \(\nabla\f_i\) are \(L_{\f_i}\)-Lipschitz continuous, \(i=1,2\). \end{theorem}
\subsection{Notational conventions}\label{sec:notation}
As done above and throughout, we use subscripts for iteration counters, typically \(k\), and bracketed superscripts to indicate the level (either 1 or 2). Other symbols will be introduced for the sake of the convergence proofs, which follow the same conventions and are synopsised in the list detailed in \cref{table:notation} inclusive of references to the respective definitions (those which are local to the scope of individual proofs are omitted from the list). In particular, the uppercase \(\Fk\), \(\Gk\) and \(\Phik\) will be useful for the convergence analysis, and correspond to the respective lowercase symbols scaled by \(\frac{1}{\sigk}\). Keeping in mind that \(\sigk\) is an \emph{inverse} penalty parameter, in the sense that it is driven to 0, we refer to \(\Phik=\cost_1+\frac{1}{\sigk}\cost_2\) as the penalized cost, and to \(\phik=\sigk\cost_1+\cost_2\) as the \emph{inversely} penalized cost of the single-level subproblems.
\begin{table}[hb]
\centering
\footnotesize
\let\mr\multirow
\renewcommand{1.35}{1.35}
\ifsiam
\setlength{\tabcolsep}{4pt}
\else
\setlength{\tabcolsep}{3pt}
\fi
\begin{tabular}[t]{@{}|c@{~~}l|c| c |c@{~~}l|c|@{}}
\multicolumn{3}{@{}c}{\bf upper level \(\cost_1=\f_1+\g_1\)}
& \multicolumn{1}{c}{} &
\multicolumn{3}{c@{}}{\bf lower level \(\cost_2=\f_2+\g_2\)}
\\\cline{1-3}\cline{5-7}
\(\f_1\) & smooth part & \mr{2}{*}{\eqref{eq:P1}}
&&
\(\f_2\) & smooth part & \mr{2}{*}{\eqref{eq:P2}}
\\
\(\g_1\) & proximable part &
&&
\(\g_2\) & proximable part &
\\\cline{1-3}\cline{5-7}
\(\costinf_1\) & \(\inf_{\dom\cost_2}\cost_1\) & \eqref{eq:costinf_1}
&&
\(\costinf_2\) & \(\inf\cost_2\) & \eqref{eq:costinf_2}
\\\cline{1-3}\cline{5-7}
\(\bcost_1\) & \(\cost_1-\costinf_1\) (\(\geq0\) on \(\dom\cost_2\)) & \cref{rem:cost}
&&
\(\bcost_2\) & \(\cost_2-\costinf_2\) (\(\geq0\)) & \cref{rem:cost}
\\\cline{1-3}\cline{5-7}
\(\X_1\) & \(\argmin_{\X_2}\cost_1\) (optimal set) & \cref{ass:X1}
&&
\(\X_2\) & \(\argmin\cost_2\) (feasible set) & \eqref{eq:P2}
\\\cline{1-3}\cline{5-7}
\(\costinf*\) & \(\min_{\X_2}\cost_1\) (optimal cost) & \eqref{eq:phi*}
\\\cline{1-3}
\multicolumn{7}{c}{}\\[-2.5ex]
\multicolumn{3}{@{}c}{\bf single-level inverse-penalty reformulation}
& \multicolumn{1}{c}{} &
\multicolumn{3}{@{}c}{\bf single-level penalty reformulation}
\\\cline{1-3}\cline{5-7}
\(\fk\) & \(\sigk\f_1+\f_2\) smooth part & \mr{3}{*}{\eqref{eq:phik}}
&&
\(\Fk\) & \(\f_1+\frac{1}{\sigk}\f_2\) smooth part & \mr{3}{*}{\eqref{eq:uppercase}}
\\
\(\gk\) & \(\sigk\g_1+\g_2\) proximable part &
&&
\(\Gk\) & \(\g_1+\frac{1}{\sigk}\g_2\) proximable part &
\\
\(\phik\) & \(\fk+\gk=\sigk\cost_1+\cost_2\) &
&&
\(\Phik\) & \(\Fk+\Gk=\cost_1+\frac{1}{\sigk}\cost_2\) &
\\\cline{1-3}\cline{5-7}
\(\bphik\) & \(\sigk\bcost_1+\bcost_2\) (\(\geq0\) on \(\dom\cost_2\)) & \cref{rem:cost}
&&
\(\Delk\) & \(\nicefrac{1}{\sigk}-\nicefrac{1}{\sigma_{k-1}}\) & \cref{thm:ineq}
\\\cline{1-3}\cline{5-7}
\multicolumn{7}{c}{}\\[-2.5ex]
\multicolumn{3}{@{}c}{\bf algorithmic parameters}
& \multicolumn{1}{c}{} &
\multicolumn{3}{@{}c}{\bf adaptive estimates}
\\\cline{1-3}\cline{5-7}
\(\alphk*\) & stepsize & \mr{3}{*}{\cref{alg:adabim}}
&&
\(\lk\) & Lipschitz estimate of \(\nabla\fk\) at \(x_k\) & \mr{3}{*}{\eqref{eq:CL}}
\\\cline{1-2}\cline{5-6}
\(\sigk*\) & (inverse) penalty &
&&
\(\lk^{(i)}\) & Lipschitz estimate of \(\nabla\f_i\) at \(x_k\) &
\\\cline{1-2}\cline{5-6}
\(\rhok*\) & \(\nicefrac{\sigk*\alphk*}{\sigk\alphk}\) &
&&
\(\ck\) & cocoercivity estimate of \(\nabla\fk\) at \(x_k\) &
\\\cline{1-3}\cline{5-7}
\end{tabular}
\caption{
Schematics of the notation adopted in the paper with references to their definitions.
}
\label{table:notation} \end{table}
\section{Convergence analysis}\label{sec:cnv}
In this section we examine the convergence properties of \refadabim* (resp. \refstabim*) for solving problem \eqref{eq:P} under local (resp. global) Lipschitz continuity of the gradients of \(\f_1\) and \(\f_2\). Crucially, regardless of the stepsize selection strategy, our analysis relies on a quasi-descent inequality for the proximal gradient updates \eqref{eq:PG} presented in \cref{thm:PG:descent}. Consequently, a convergence recipe will be provided in \cref{sec:recipe} leading to a unified convergence proof in \cref{proofthm:BiM:convergence}.
We begin by elaborating on some of the notational conventions listed in \cref{table:notation}.
\begin{remark}[Bar notation for minima and shifted costs]\label{rem:cost}
Let us define
\[
\bcost_i\coloneqq\cost_i-\costinf_i, \; i=1,2,
\quad\text{and}\quad
\bphik \coloneqq \sigk \bcost_1 + \bcost_2,
\]
where \(\costinf_i=\inf_{\dom\cost_2}\cost_i\) as in \cref{ass:basic} (restricting to \(\dom\cost_2\) is superfluous for \(i=2\)), and let
\begin{equation}\label{eq:phi*}
\costinf*
\coloneqq
\inf_{\X_2}\cost_1
\end{equation}
be the optimal cost of problem \eqref{eq:P}.
Then,
\begin{enumerate}
\item\label{thm:bar>=0}
\(\bcost_1(x)\geq0\) for any \(x\in\dom\cost_2\) and \(\bcost_2(x)\geq0\) for any \(x\in\R^n\);
\item \label{thm:Phi*}
\( \tfrac1{\sigk}\left(\phik(x_\star)-\costinf_2\right)=\costinf*\) for any \(x_\star\in\X_1\) and \(k\in\N\).
\qedhere
\end{enumerate} \end{remark}
A key step in developing our adaptive scheme is the introduction of the quantities in \eqref{eq:CL} which provide an exact description of the local Lipschitz modulus of not only \(\nabla \fk\) but also that of the forward operator \(\id - \alphk\nabla\fk\), as presented next.
\begin{fact}[{\cite[Lem. 2.1]{latafat2023adaptive}}]
Suppose that \cref{ass:f} holds, and for \(x_{k-1},x_k\in\R^n\) and \(\alphk>0\) let \(\lk\) and \(\ck\) be as in \eqref{eq:CL} and \(\Hk\coloneqq\id-\alphk\nabla\fk\).
Then, the following hold:
\begin{enumerate}
\item \label{thm:CL}
\(
\|\nabla\fk(x_{k-1})-\nabla\fk(x_k)\|^2
{}={}
\ck\lk\|x_{k-1}-x_k\|^2
\).
\item \label{thm:H}
\(
\|\Hk(x_{k-1})-\Hk(x_k)\|^2
{}={}
\bigl(1-\alphk\lk(2-\alphk\ck)\bigr)
\|x_{k-1}-x_k\|^2
\).
\item\label{thm:Lk<=Ck}
\(
\lk
{}\leq{}
\frac{
\|\nabla\fk(x_{k-1})-\nabla\fk(x_k)\|
}{
\|x_{k-1}-x_k\|
}
{}\leq{}
\ck
{}\leq{}
\sigk L_{\f_1,\mathcal V} + L_{\f_2,\mathcal V}
\),
where, for \(i=1,2\), \(L_{\f_i,\mathcal V}\) is a Lipschitz modulus for \(\nabla\f_i\) on a convex set \(\mathcal V\) containing \(x_{k-1}\) and \(x_k\).
\end{enumerate} \end{fact}
\subsection{A quasi-descent inequality}\label{sec:prelem}
Before delving into the convergence analysis, we will present a series of preliminary results pertaining to our adaptive scheme that can be regarded as an extension of the adaptive mechanism of {\sffamily adaPGM} proposed in \cite[Alg. 1]{latafat2023adaptive}. Not only does this adaptive scheme markedly enhance the computational efficiency of our approach, but it also allows us to consider nonsmooth terms \(\g_1, \g_2\), and provides an appropriate initialization.
We proceed to investigate the progress of a single proximal gradient step as described in \eqref{eq:PG} using an arbitrary stepsize \(\alphk*>0\). Departing from \cite{solodov2007explicit,cabot2005proximal}, the key of our convergence analysis, captured in the following lemma, is the adoption of penalized (as opposed to inversely penalized) costs. As already mentioned in the preview of \cref{table:notation}, we adopt an uppercase notation for the penalized cost \begin{equation}\label{eq:uppercase}
\Phik=\Fk+\Gk \quad\text{with}\quad
\Fk=\tfrac1{\sigk}\fk=\f_1+\tfrac1{\sigk}\f_2 \quad\text{and}\quad
\Gk=\tfrac1{\sigk}\gk=\g_1+\tfrac1{\sigk}\g_2, \end{equation} noticing that \(\Phik=\cost_1+\frac{1}{\sigk}\cost_2=\frac{1}{\sigk}\phik\). Doing so allows us to express the difference \begin{equation}
\Phik*-\Phik {}={}
\left(\tfrac1{\sigk*}-\tfrac1{\sigk}\right)\cost_2 \end{equation} as a multiple of the lower-level cost \(\cost_2\), rather than of the upper-level cost \(\cost_1\).
\begin{lemma}\label{thm:ineq}
Suppose that \cref{ass:basic} holds.
Consider iterations \eqref{eq:PG} with \(0<\sigk*\leq\sigk\), and let a solution \(x_\star\in\X_1\) be fixed.
Let \(\bcost_2\coloneqq\cost_2-\costinf_2\geq0\) be as in \cref{rem:cost}, and define
\begin{equation}
p_k
\coloneqq
\phik(x_k)-\phik(x_\star)
=
\sigk\left(\cost_1(x_k)-\costinf*\right)
+
\bcost_2(x_k)
\end{equation}
(which is not necessarily positive), \(\Delk\coloneqq\frac{1}{\sigk}-\frac{1}{\sigma_{k-1}}\), and \(\rhok*\coloneqq\frac{\sigk*\alphk*}{\sigk\alphk}\).
Then, for every \(\betk*\geq0\) and \(\epsk*>0\) the following hold:
\begin{multline*}
\tfrac{1}{2}
\|x_{k+1}-x_\star\|^2
+
\tfrac{\sigk*}{\sigk}\alphk*(1+\betk*)p_k
+
\alphk*\sigk*\Delk*\bcost_2(x_{k+1})
\\
+
\left(
\tfrac{1-\rhok*\epsk*}{2}
-
\alphk*\sigk*\Delk*
\lk*^{(2)}
\right)
\|x_k-x_{k+1}\|^2
\leq
\tfrac{1}{2}
\|x_k-x_\star\|^2
+
\tfrac{\sigk*}{\sigma_{k-1}}\betk*\alphk*p_{k-1}
\\
-
\rhok*
\left(
\betk*(1-\alphk\lk)
-
\tfrac{1-\alphk\lk(2-\alphk\ck)}{2\epsk*}
\right)
\|x_{k-1}-x_k\|^2
+
\alphk*\sigk*\betk*\Delk\bcost_2(x_{k-1}).
\end{multline*}
\begin{proof}
We will prove the claim using the uppercase notation of \eqref{eq:uppercase} and, consistently, set
\(P_k
\coloneqq{\tfrac{1}{\sigk}}p_k
=\Phik(x_k)-\Phik(x_\star)
\).
Being a simple matter of multiplicative constants, note that proximal gradient iterations \eqref{eq:PG} can equivalently be expressed in the penalized cost reformulation \(\Phik*=\Fk*+\Gk*\) up to suitable scaling of the stepsize, namely,
\[
x_{k+1}
=
\prox_{\sigk*\alphk*\Gk*}(x_k-\sigk*\alphk*\nabla\Fk*(x_k)).
\]
The subgradient characterization of the proximal mapping \eqref{eq:proxsubgrad} yields
\begin{equation}\label{eq:subgrad}
\tfrac{\Hk(x_{k-1})-x_k}{\sigk\alphk}
=
\tfrac{x_{k-1}-x_k}{\sigk\alphk}-\nabla\Fk(x_{k-1})
\in
\partial\Gk(x_k),
\end{equation}
where \(\Hk=\id-\alphk\nabla\fk\).
Hence, since \(\partial\Phik=\nabla\Fk+\partial\Gk\),
\begin{subequations}
\begin{align}
\nonumber
0
\leq{} &
\Phik(x_{k-1})
-
\Phik(x_k)
-
\tfrac{1}{\sigk\alphk}
\@ifstar\@innprod\@@innprod{x_{k-1}-x_k}{x_{k-1}-x_k}
+
\@ifstar\@innprod\@@innprod{\nabla\Fk(x_{k-1})-\nabla\Fk(x_k)}{x_{k-1}-x_k}
\\
\nonumber
={} &
\Phik(x_{k-1})
-
\Phik(x_k)
-
\tfrac{1}{\sigk\alphk}
\|x_{k-1}-x_k\|^2
+
\tfrac1{\sigk}\@ifstar\@innprod\@@innprod{\nabla\fk(x_{k-1})-\nabla\fk(x_k)}{x_{k-1}-x_k}
\\
={} &
\Phik(x_{k-1})
-
\Phik(x_k)
-
\tfrac{1-\alphk\lk}{\sigk\alphk}
\|x_{k-1}-x_k\|^2
\label{eq:phikdescent}
\\
={} &
P_{k-1}
-
P_k
+
(\tfrac{1}{\sigk}-\tfrac{1}{\sigma_{k-1}})\bcost_2(x_{k-1})
-
\tfrac{1-\alphk\lk}{\sigk\alphk}\|x_{k-1}-x_k\|^2.
\label{eq:ineqsubgrad}
\end{align}
\end{subequations}
Again from \eqref{eq:subgrad}, this time with \(k\gets k+1\), we have
\begin{subequations}\label{subeq:3ineq}
{\ifarxiv\mathtight[0.6]\fi
\begin{align*}
0
\leq{} &
\Gk*(x_\star)
-
\Gk*(x_{k+1})
+
\@ifstar\@innprod\@@innprod{\nabla\Fk*(x_k)}{x_\star-x_{k+1}}
-
\tfrac{1}{\sigk*\alphk*}
\@ifstar\@innprod\@@innprod{x_k-x_{k+1}}{x_\star-x_{k+1}}
\\
={} &
\Gk*(x_\star)
-
\Gk*(x_{k+1})
+
\@ifstar\@innprod\@@innprod{\nabla\Fk*(x_k)}{x_\star-x_{k+1}}
+
\tfrac{1}{2\sigk*\alphk*}
\|x_k-x_\star\|^2
-
\tfrac{1}{2\sigk*\alphk*}
\|x_\star-x_{k+1}\|^2
\\
&
-
\tfrac{1}{2\sigk*\alphk*}
\|x_k-x_{k+1}\|^2
\\
={} &
\Gk*(x_\star)
-
\Gk*(x_{k+1})
+
\@ifstar\@innprod\@@innprod{\nabla\Fk*(x_k)}{x_\star-x_k}
+
\@ifstar\@innprod\@@innprod{\nabla\Fk*(x_k)}{x_k-x_{k+1}}
+
\tfrac{1}{2\sigk*\alphk*}
\|x_k-x_\star\|^2
\\
&
-
\tfrac{1}{2\sigk*\alphk*}
\|x_\star-x_{k+1}\|^2
-
\tfrac{1}{2\sigk*\alphk*}
\|x_k-x_{k+1}\|^2
\\
\leq{} &
\Gk*(x_\star)
-
\Gk*(x_{k+1})
+
\fillwidthof[c]{
\@ifstar\@innprod\@@innprod{\nabla\Fk*(x_k)}{x_\star-x_k}
}{
\Fk*(x_\star)-\Fk*(x_k)
}
+
\underbracket*{
\@ifstar\@innprod\@@innprod{\nabla\Fk*(x_k)}{x_k-x_{k+1}}
}_{\text{(A)}}
+
\tfrac{1}{2\sigk*\alphk*}
\|x_k-x_\star\|^2
\\
&
-
\tfrac{1}{2\sigk*\alphk*}
\|x_\star-x_{k+1}\|^2
-
\tfrac{1}{2\sigk*\alphk*}
\|x_k-x_{k+1}\|^2,
\numberthis\label{eq:ineq1}
\end{align*}}
where the last inequality uses convexity of \(\Fk*\).
As to term (A), we have
\begin{align*}
\text{(A)}
={} &
\tfrac{1}{\sigk\alphk}
\@ifstar\@innprod\@@innprod{\Hk(x_{k-1})-x_k}{x_{k+1}-x_k}
+
\tfrac{1}{\sigk\alphk}
\@ifstar\@innprod\@@innprod{\Hk(x_{k-1})-x_k+\sigk\alphk\nabla\Fk*(x_k)}{x_k-x_{k+1}}
\\
\overrel[\leq]{\eqref{eq:subgrad}}{} &
\fillwidthof[c]{
\tfrac{1}{\sigk\alphk}
\@ifstar\@innprod\@@innprod{\Hk(x_{k-1})-x_k}{x_{k+1}-x_k}
}{
\Gk(x_{k+1})-\Gk(x_k)
}
+
\@ifstar\@innprod\@@innprod{\nabla\Fk*(x_k)-\nabla\Fk(x_k)}{x_k-x_{k+1}}
\\
&
+
\underbracket[0.5pt]{
\tfrac{1}{\sigk\alphk}
\@ifstar\@innprod\@@innprod{\Hk(x_{k-1})-\Hk(x_k)}{x_k-x_{k+1}}
}_{\text{(B)}}.
\numberthis\label{eq:ineqA}
\end{align*}
Next, we bound the term (B) by \(\epsk*\)-Young's inequality as
\begin{align*}
\text{(B)}
\leq{} &
\tfrac{\epsk*}{2\sigk\alphk}\|x_k-x_{k+1}\|^2
+
\tfrac{1}{2\epsk*\sigk\alphk}\|\Hk(x_{k-1})-\Hk(x_k)\|^2
\\
\overrel{\ref{thm:H}}{} &
\tfrac{\epsk*}{2\sigk\alphk}\|x_k-x_{k+1}\|^2
+
\tfrac{1-\alphk\lk(2-\alphk\ck)}{2\epsk*\sigk\alphk}\|x_{k-1}-x_k\|^2.
\numberthis\label{eq:ineqB}
\end{align*}
\end{subequations}
Combining the three inequalities \eqref{subeq:3ineq} yields
{\ifarxiv\mathtight[0.9]\fi
\begin{align*}
0
\leq{} &
\Phik*(x_\star)
-
\Gk*(x_{k+1})
-
\Fk*(x_k)
+
\Gk(x_{k+1})
-
\Gk(x_k)
+
(\tfrac1{\sigk*}-\tfrac{1}{\sigk})
\@ifstar\@innprod\@@innprod{\nabla \f_2(x_k)}{x_k-x_{k+1}}
\\
&
+
\Biggl\{
\tfrac{1-\alphk\lk(2-\alphk\ck)}{2\epsk*\sigk\alphk}\|x_{k-1}-x_k\|^2
-
\left(
\tfrac{1}{2\sigk*\alphk*}
-
\tfrac{\epsk*}{2\sigk\alphk}
\right)
\|x_k-x_{k+1}\|^2
-
\tfrac{1}{2\sigk*\alphk*}
\|x_\star-x_{k+1}\|^2
\\
&
\hphantom{+\Biggl\{{}}
+
\tfrac{1}{2\sigk*\alphk*}
\|x_k-x_\star\|^2
\Biggr\}
\\
\overrel{\ref{thm:Phi*}}{} &
\costinf*
+
\tfrac1{\sigk*}\costinf_2
-
\Fk(x_k)
-
\Gk(x_k)
+
\bigl(\tfrac1{\sigk*}-\tfrac{1}{\sigk}\bigr)
\vphantom{\f^2}
\left(
\@ifstar\@innprod\@@innprod{\nabla \f_2(x_k)}{x_k-x_{k+1}}
-
\f_2(x_k)
-
\g_2(x_{k+1})
\right)
\\
&
+
\{\cdots\}
\\
={} &
-P_k
+
\overbracket[0.5pt]{
\left(
\tfrac1{\sigk*}-\tfrac{1}{\sigk}
\vphantom{\f^2}
\right)
}^{\geq0}
\,
\overbracket[0.5pt]{
\left(
\@ifstar\@innprod\@@innprod{\nabla \f_2(x_k)}{x_k-x_{k+1}}
-
\f_2(x_k)
-
\g_2(x_{k+1})
+
\costinf_2
\right)
}^{\text{(D)}}
+
\Bigl\{\cdots\Bigr\}.
\end{align*}}
By using convexity of \(\f_2\) we can bound the term (D) as
{\ifarxiv\mathtight[0.75]\fi
\begin{align*}
\text{(D)}
={} &
\@ifstar\@innprod\@@innprod{\nabla \f_2(x_{k+1})}{x_k-x_{k+1}}
-
\f_2(x_k)
-
\g_2(x_{k+1})
+
\@ifstar\@innprod\@@innprod{\nabla \f_2(x_{k+1})-\nabla \f_2(x_k)}{x_{k+1}-x_k}
+
\costinf_2
\\
\leq{} &
\fillwidthof[c]{
\@ifstar\@innprod\@@innprod{\nabla \f_2(x_{k+1})}{x_k-x_{k+1}}
-
\f_2(x_k)
}{
-\f_2(x_{k+1})
}
-
\g_2(x_{k+1})
+
\fillwidthof[c]{
\@ifstar\@innprod\@@innprod{\nabla \f_2(x_{k+1})-\nabla \f_2(x_k)}{x_{k+1}-x_k}
}{
\lk*^{(2)}\|x_{k+1}-x_k\|^2
}
+
\costinf_2,
\end{align*}
}
which plugged in the previous inequality results in
{\ifarxiv\mathtight[0.75]\fi
\begin{align*}
0
\leq{} &
-P_k
-
(\tfrac1{\sigk*}-\tfrac{1}{\sigk})\bcost_2(x_{k+1})
+
(\tfrac1{\sigk*}-\tfrac{1}{\sigk})
\lk*^{(2)}\|x_{k+1}-x_k\|^2
+
\tfrac{1-\alphk\lk(2-\alphk\ck)}{2\epsk*\sigk\alphk}\|x_{k-1}-x_k\|^2
\\
&
-
\left(
\tfrac{1}{2\sigk*\alphk*}
-
\tfrac{\epsk*}{2\sigk\alphk}
\right)
\|x_k-x_{k+1}\|^2
-
\tfrac{1}{2\sigk*\alphk*}
\|x_\star-x_{k+1}\|^2
+
\tfrac{1}{2\sigk*\alphk*}
\|x_k-x_\star\|^2.
\numberthis\label{eq:3ineqs}
\end{align*}}
Summing \eqref{eq:3ineqs}\(+\betk*\)\eqref{eq:ineqsubgrad}, multiplying by \(\sigk*\alphk*\), and rearranging yields the claimed inequality.
\end{proof} \end{lemma}
By selecting \(\betk=\rhok\) and \(\epsk\coloneqq\nicefrac{1}{2\rhok}\) the inequality simplifies as follows.
\begin{corollary}[quasi-descent inequality]\label{thm:LW}
Under \cref{ass:basic}, iterations \eqref{eq:PG} satisfy
\begin{align*}
\mathcal L_{k+1}(x_\star)
-
\mathcal L_k(x_\star)
\leq{} &
-
\left(
\tfrac{1}{4}
+
\rhok*^2\alphk\lk(1-\alphk\ck)
-
\sigk\alphk\lk^{(2)}\Delk
\right)
\|x_{k-1}-x_k\|^2
\\
&
-
\sigk\alphk
\bigl(
1+\rhok-\rhok*^2
\bigr)
\bigl(\cost_1(x_{k-1})-\costinf*\bigr)
\\
&
-
\sigk*\alphk*
\left(
\tfrac{\Delk}{\rhok*}
+
\tfrac{1}{\sigk}\left(
1+\rhok*
-
\rho_{k+2}^2\tfrac{\sigk}{\sigk*}
\right)
\right)
\bcost_2(x_k),
\numberthis\label{eq:puya:muk:descent}
\end{align*}
for every \(k\in\N\) and \(x_\star\in\X_1\), where
\(
\mathcal L_k(x_\star)
\coloneqq
\tfrac12\|x_k-x_\star\|^2
+
W_k
\)
with
\begin{align*}
W_k
\coloneqq{} &
\tfrac{1-4\alphk\sigk\lk^{(2)}\Delk}{4}\|x_k-x_{k-1}\|^2
+
\tfrac{\sigk*}{\sigk}\alphk*\rhok*
\bcost_2(x_{k-1})
\\
&
+
\sigk\alphk\Delk\bcost_2(x_k)
+
\sigk\alphk\left(1+\rhok\right)
\bigl(
\cost_1(x_{k-1})
-
\costinf*
\bigr).
\end{align*} \end{corollary}
By looking at the update rule \eqref{eq:adabim:alphk*_0} for \(\balphk*\), it is apparent that the choice of stepsizes in \cref{alg:adabim} is designed so as to ensure that all the multiplying coefficients on the right-hand side of \eqref{eq:puya:muk:descent} are positive.
Even so, it should be noted that the inequality does not, in general, imply a monotonic decrease of \(\mathcal L_k(x_\star)\) along the iterates, the reason being that the term \(\cost_1(x_{k-1})-\costinf*\) therein is not necessarily positive (by the same argument, \(\mathcal L_k(x_\star)\) is not guaranteed to be positive).
For this reason we talk in terms of \emph{quasi}-descent when referring to inequality \eqref{eq:puya:muk:descent}, a complication that, similarly to the analysis in \cite{solodov2007explicit}, is the culprit of a nonstraightforward derivation of convergence results.
\subsection{Convergence recipe for proximal gradient iterations}\label{sec:recipe}
The convergence of the two proposed algorithms hinges on the behavior of proximal gradient iterations when some implicit conditions are met. This is materialized in the next theorem through a convergence recipe and plays a key role in the subsequent developments in \cref{proofthm:BiM:convergence}.
\begin{theorem}[convergence recipe for proximal gradient iterations]\label{thm:PG:convergence}
Suppose that \cref{ass:basic} holds, and consider proximal gradient iterations
\begin{equation}\label{eq:PG}
x_{k+1}
=
\prox_{\alphk*\gk*}(x_k-\alphk*\nabla\fk*(x_k))
\end{equation}
with \(\seq{\sigk}\) and \(\seq{\alphk}\) selected such that
\begin{enumeratprop}
\item\label{cond:sigk}
the inverse penalties \(\seq{\sigk}\) comply with \eqref{eq:sigk};
\item\label{cond:alphk}
there exists \(\nu\in(0,1)\) such that \(\alphk*\leq\balphk*\) and \(\alphk*\lk*\leq\nu\) hold for every \(k\), where \(\balphk*\) is as in \eqref{eq:adabim:alphk*_0} for some \(\alpha_{\rm max}>0\);
\item\label{cond:alphamin}
if \(\seq{x_k}\) is bounded, then \(\inf_{k\in\N}\alphk>0\).
\end{enumeratprop}
Then, \(\seq{x_k}\) is bounded and all its cluster points are solutions to the bilevel problem \eqref{eq:P}. \end{theorem}
The proof will make use of some properties of proximal gradient iterations in which the stepsizes comply with \cref{cond:sigk,cond:alphk,cond:alphamin}.
\begin{lemma}\label{thm:PG:descent}
Suppose that \cref{ass:basic} holds, and consider the proximal gradient iterations \eqref{eq:PG} with \(\seq{\sigk}\) and \(\seq{\alphk}\) complying with \cref{cond:sigk,cond:alphk}.
Then, the following hold:
\begin{enumerate}
\item \label{thm:adabim:rhomax}
\(
\rhok
\coloneqq
\frac{\sigk\alphk}{\sigma_{k-1}\alpha_{k-1}}
\leq
\rho_{\rm max}
\coloneqq
\max\set{
\frac{\alpha_0}{\alpha_{-1}},\,
\frac{1+\sqrt5}{2}
}
\)
for every \(k\in\N\).
\item \label{thm:adabim:qSD}
For \(\mathcal L_k\) as in \cref{thm:LW} it holds that
\(
\mathcal L_{k+1}(x_\star)
\leq
\mathcal L_k(x_\star)
-
\sigk\alphk(1+ \rhok -\rhok*^2)
\bigl(
\cost_1(x_{k-1})-\costinf*
\bigr)
\)
for all \(k\in\N\) and \(x_\star\in\X_1\).
\item \label{thm:adabim:barPhi}
\(
\bphik*(x_{k+1})
\leq
\bphik(x_k)
-
\tfrac{1-\nu}{\alphk*}\|x_{k+1}-x_k\|^2
\)
holds for every \(k\in\N\).
In particular, \(\|x_k-x_{k-1}\|\to0\) as \(k\to \infty\), \(\seq{\bphik(x_k)}\) is convergent, and \(\seq{\cost_2(x_k)}\) is bounded.
\item \label{thm:adabim:feas}
If \(\seq{x_k}\) is bounded and \cref{cond:alphamin} holds, then \(\bcost_2(x_k)\to0\) as \(k\to\infty\) and all the limit points of \(\seq{x_k}\) belong to \(\X_2\).
\end{enumerate}
\begin{proof}
\begin{proofitemize}
\item \ref{thm:adabim:rhomax}~
Follows from a trivial induction.
\item \ref{thm:adabim:qSD}~
Since \(\alphk\leq\balphk\), from the definition of \(\balphk\) in \eqref{eq:adabim:alphk*_0} it follows that the multiplying coefficients in \eqref{eq:puya:muk:descent} are positive:
\begin{align*}
\mathcal L_{k+1}(x_\star) - \mathcal L_k(x_\star)
\leq{} &
-
\sigk*\alphk*
\left(
\overbracket*{
\tfrac{\Delk}{\rhok*}
}^{\geq0}
+
\tfrac{1}{\sigk}
\overbracket*{
\left(
1+\rhok*
-
\rho_{k+2}^2\tfrac{\sigk}{\sigk*}
\right)
}^{\geq0}
\right)
\bcost_2(x_k)
\\
&
-
\sigk\alphk
\overbracket[0.5pt]{
\bigl(
1+\rhok-\rhok*^2
\bigr)
}^{\geq0}
\bigl(\cost_1(x_{k-1})-\costinf*\bigr)
\\
&
-
\underbracket*{
\left(
\tfrac{1}{4}
+
\rhok*^2\alphk\lk(1-\alphk\ck)
-
\alphk\sigk\lk^{(2)}\Delk
\right)
}_{\geq0}
\|x_{k-1}-x_k\|^2.
\vphantom{
\underbracket[0.5pt]{
\tfrac{1}{4}
}_0
}
\end{align*}
The claim then readily follows from the fact that \(\bcost_2\geq0\), cf. \cref{thm:bar>=0}.
\item \ref{thm:adabim:barPhi}~
The inequality follows from \eqref{eq:phikdescent} and the fact that \(\alphk\lk\leq\nu\), after observing that
\begin{align*}
\bphik*(x_{k+1})
-
\bphik(x_k)
\leq{} &
\bphik*(x_{k+1})
-
\bphik*(x_k)
\\
={} &
\phik*(x_{k+1})
-
\phik*(x_k)
\\
={} &
\sigk*\left(
\Phik*(x_{k+1})
-
\Phik*(x_k)
\right).
\end{align*}
Here, the first inequality uses the fact that \(\sigk\geq\sigk*\), and therefore \(\bphik=\sigk\bcost_1+\bcost_2\) is smaller than \(\bphik*=\sigk*\bcost_1+\bcost_2\) on \(\dom\cost_2\), cf. \cref{thm:bar>=0}, and \(x_k\in\dom\gk=\dom\cost_1\cap\dom\cost_2\subseteq\dom\cost_2\).
Similarly, the other claims in turn follow from the facts that \(\nu<1\), \(\bphik\geq\bcost_2\geq0\) on \(\dom\cost_2\supseteq\seq{x_k}\), and \(\alphk*\leq\alpha_{\rm max}\).
\item \ref{thm:adabim:feas}~
Consider a convergent subsequence \(x_{k_\ell}\to x_\infty\), so that \(x_{k_\ell+1}\to x_\infty\) by \cref{thm:adabim:barPhi}.
Up to further extracting if necessary we have that
\(\alpha_{k_\ell+1}\to\alpha_\infty\geq\alpha_{\rm min}>0\) and \(\sigma_{k_\ell}\to0\).
Observe that
\begin{align*}
x_{k+1}
={} &
\argmin h({}\cdot{};x_k,\alpha_{k+1},\sigk*),
\shortintertext{where}
h(w;x,\alpha,\sigma)
\coloneqq{} &
(\sigma \g_1+\g_2)(w)
+
\tfrac{1}{2\alpha}
\left\|w-x+\alpha\nabla\bigl(\sigma \f_1+\f_2\bigr)(x)\right\|^2
\end{align*}
is level bounded in \(w\) locally uniformly in \((x,\alpha,\sigma)\), as a function from \(\R^n\times\bigl(\R^n\times[\alpha_{\rm min},\infty)\times[0,\infty)\bigr)\) to \(\Rinf\).
Since \(h\) is continuous in \((x,\alpha,\sigma)\), it follows from \cite[Thm. 1.17]{rockafellar2011variational} that
\[
x_\infty
\in
\argmin h({}\cdot{};x_\infty,\alpha_\infty,0)
=
\prox_{\alpha_\infty \g_2}\bigl(x_\infty-\alpha_\infty\nabla \f_2(x_\infty)\bigr),
\]
this condition being equivalent to \(x_\infty\in\argmin(\f_2+\g_2)\defeq\X_2\).
It remains to show that \(\bcost_2(x_k)\to0\).
Since \(\seq{\bcost_1(x_k)}\) is bounded and \(\seq{\sigk} \to 0\), the sequence \(\seq{\sigk \bcost_1(x_k)}\) converges to zero. Combined with \cref{thm:adabim:barPhi}, it follows that \(\bcost_2(x_k) = \bphik (x_k) - \sigk \bcost_1(x_k)\)
is convergent. It suffices to show that \(\seq{\bcost_2(x_k)}\) converges to zero along a subsequence.
The subdifferential characterization of \(x_{k+1}\in\prox_{\alphk*\gk*}(x_k-\alphk*\nabla\fk*(x_k))\)
\[
\tfrac{x_{k-1}-x_k}{\alphk}
-
(\nabla\fk(x_{k-1})-\nabla\fk(x_k))
\in
\partial\phik(x_k)
\]
implies that
\[
\phik(x_\infty)
\geq
\phik(x_k)
+
\@ifstar\@innprod\@@innprod{
\tfrac{x_{k-1}-x_k}{\alphk}
-
(\nabla\fk(x_{k-1})-\nabla\fk(x_k))
}{
x_\infty-x_k
}.
\]
With \(k = k_\ell\) and letting \(\ell\to \infty\) we obtain using assertion \ref{thm:adabim:barPhi} and boundedness of \(\seq{x_k}\) that
\[
\cost_2(x_\infty)
\geq
\limsup_{\ell\to\infty}
\cost_2(x_{k_\ell}),
\]
which combined with lower semicontinuity of \(\cost_2\) implies
\(\lim_{\ell\to \infty} \cost_2(x_{k_\ell}) = \cost_2(x_\infty) = \costinf_2\), completing the proof.
\qedhere
\end{proofitemize}
\end{proof} \end{lemma}
\begin{proof}[Proof of \cref{thm:PG:convergence}]
We pattern the proof structure of \cite[Thm. 3.2]{solodov2007explicit}, thereby considering two mutually exclusive cases.
\begin{proofitemize}
\item
{\it Case 1: \(\cost_1(x_k)\geq\costinf*\) holds for \(k\) large enough.}
In this case, we will actually show that the sequence \(\seq{x_k}\) converges to a solution of \eqref{eq:P}.
Recall the definition of \(W_k\) and \(\mathcal L_k\) in \cref{thm:LW}.
Observing that \(W_k\geq0\) and \(\frac12\|x_k-x_\star\|^2\leq\mathcal L_k(x_\star)\), \cref{thm:adabim:qSD} implies that \(\seq{\mathcal L_k(x_\star)}\) converges and that consequently \(\seq{x_k}\) is bounded.
We first argue that there exists an optimal limit point; to this end, since all limit points are feasible by \cref{thm:adabim:feas} and because of lower semicontinuity it suffices to show that \(\liminf_{k\to\infty}\cost_1(x_k)=\costinf*\).
Boundedness of \(\seq{x_k}\) implies through \cref{cond:alphamin} the existence of \(\alpha_{\rm min}>0\) such that \(\alphk\geq\alpha_{\rm min}\) holds for all \(k\).
A telescoping argument on \cref{thm:adabim:qSD} yields
\begin{equation}
\alpha_{\rm min}\sum_{k\in \N}\sigk(1+\rhok-\rhok*^2)\bigl(\cost_1(x_{k-1})-\costinf*\bigr)
\leq
\sum_{k\in \N}\sigk\alphk(1+\rhok-\rhok*^2)\bigl(\cost_1(x_{k-1})-\costinf*\bigr)
<
\infty.
\end{equation}
Since \(\sum_{k\in\N}\sigk=\infty\), necessarily
\begin{align*}
\liminf_{k\to\infty}(1+\rhok-\rhok*^2)(\cost_1(x_{k-1})-\costinf*)
=
0.
\numberthis\label{eq:liminf:cost1}
\end{align*}
Observe first that
\begin{align*}
\rhok*
=
\tfrac{\sigk*\alphk*}{\sigk\alphk}
\leq
\tfrac{\sigk*\balphk*}{\sigk\alphk}
\leq
\tfrac{\sigk*}{\sigk}
\sqrt{\tfrac{\sigk}{\sigma_{k-1}}\bigl(1+\rhok\bigr)}
\tfrac{\sigk}{\sigk*}
\leq
\sqrt{1+\rhok}
,
\numberthis\label{eq:rhok:plus}
\end{align*}
where the first inequality holds by \cref{cond:alphk}, the second is due to the first term in \eqref{eq:adabim:alphk*_0}, and the last inequality follows from \cref{cond:sigk}.
Therefore, either \(\limsup_{k\to\infty}(1+\rhok-\rhok*^2)>0\), or \(1+\rhok-\rhok*^2 \to 0\).
In the latter case, one necessarily has that \(\liminf_{k\to \infty} \rhok >1\), implying that \(\alphk\sigk\) eventually increases exponentially, contradicting \(\sigk\alphk \leq \sigk\alpha_{\rm max} \to 0\).
Hence, \(\limsup_{k\to\infty}(1+\rhok-\rhok*^2)>0\), which along with \eqref{eq:liminf:cost1} implies that \(\liminf_{k\to\infty}\cost_1(x_{k-1})=\costinf*\).
Therefore, an optimal limit point exists, be it \(x_\infty\).
Observe that
\begin{align*}
W_k
\defeq{} &
\tfrac{1-4\alphk\sigk\lk^{(2)}\Delk}{4}
\overbracket[0.5pt]{
\|x_k-x_{k-1}\|^2
}^{\mathclap{\to 0\text{ by \cref{thm:adabim:barPhi}}}}
+
\tfrac{\sigk*}{\sigk}\alphk*\rhok*
\overbracket[0.5pt]{
\bcost_2(x_{k-1})
}^{\mathclap{\to 0\text{ by \cref{thm:adabim:barPhi}}}}
\\
&
+
\underbracket[0.5pt]{
\sigk\alphk\Delk\bcost_2(x_k)
}_{\to 0}
+
\underbracket[0.5pt]{
\vphantom{\bigl(\cost_1}
\sigk
}_{\mathclap{\to0}}
\,
\underbracket[0.5pt]{
\alphk
\left(1+\rhok\right)
\bigl(
\cost_1(x_{k-1})
-
\costinf*
\bigr)
}_{\text{bounded}}
\to
0
\quad\text{as \(k\to\infty\)},
\end{align*}
where we also used the fact that \(\seq{\alphk}\leq\alpha_{\rm max}\) and \(\seq{\rhok}\leq\rho_{\rm max}\) as in \cref{thm:adabim:rhomax}.
Having shown that \(\seq{W_k}\) converges and since \(\mathcal L_k(x_\infty)\) also does, it follows that \(\frac12\|x_k-x_\infty\|^2 = \mathcal L_k(x_\infty) - W_k\) converges as well.
Since along a subsequence \(\frac12\|x_k-x_\infty\|^2\) converges to zero, necessarily \(\lim_{k\to\infty}\frac12\|x_k-x_\infty\|^2=0\), proving that the entire sequence \(\seq{x_k}\) converges to \(x_\infty\).
\item
{\it Case 2: \(\cost_1(x_k)<\costinf*\) holds infinitely often.}
In this case, for every \(k\) large enough the index
\begin{equation}\label{eq:ik}
i_k
\coloneqq
\max\set{i\leq k}[\cost_1(x_i)<\costinf*]
\end{equation}
is well defined.
We proceed by intermediate claims.
\begin{claims}
\item\label{claim:convergence:ik}
{\it the sequences \(\seq{x_{i_k}}\) and \(\seq{x_{i_k+1}}\) are bounded}.
The set
\(
\X_1
=
\set{x\in\X_2}[\cost_1(x)\leq \costinf*]
=
\set{x}[\cost_1(x)\leq\costinf*,\ \bcost_2(x)\leq0]
\)
is nonempty and bounded, and coincides with a level set of the convex function \(h\coloneqq\max\set{\cost_1-\costinf*,\bcost_2}\).
As such, \(h\) is level bounded; see, e.g., \cite[Prop. 11.13]{bauschke2017convex} or \cite[Lem. 1]{themelis2019acceleration}.
Note that \cref{thm:adabim:barPhi} implies that \(\seq{\bcost_2(x_k)}\) is (upper) bounded, which combined with the fact that \(\cost_1(x_{i_k})<\costinf*\) implies that \(\seq{x_{i_k}}\) lies in a sublevel set of \(h\), and is therefore bounded.
In turn, \cref{thm:adabim:barPhi} implies that so is \(\seq{x_{i_k+1}}\).
\item
{\it the whole sequence \(\seq{x_k}\) is bounded}.
To this end, it remains to show that \(\seq{x_{k'}}[k'\in K']\) is bounded, where
\begin{equation}\label{eq:K'}
K'\coloneqq\set{k'\in\N}[k'\geq i_{k'}+2].
\end{equation}
With \(\rho_{\rm max}\) as in \cref{thm:adabim:rhomax}, for every \(k\in\N\)
\begin{align*}
\mathcal L_{i_k+1}(x_\star)
\leq{} &
\overbracket[0.5pt]{
\tfrac12
\|x_{i_k+1}-x_\star\|^2
\vphantom{\left(\tfrac{\sigma_{i_k+1}}{\sigma_{i_k}}\right)}
}^{\text{bounded by \cref{claim:convergence:ik}}}
+~~
\overbracket*{
\tfrac{1}{4}\|x_{i_k+1}-x_{i_k}\|^2
\vphantom{\left(\tfrac{\sigma_{i_k+1}}{\sigma_{i_k}}\right)}
}^{\mathclap{\to0 \text{ by \cref{thm:adabim:barPhi}}}}
+
\overbracket[0.5pt]{
\sigma_{i_k+1}\alpha_{i_k+1}(1+\rho_{i_k+1})
(\cost_1(x_{i_k})-\costinf*)
\vphantom{\left(\tfrac{\sigma_{i_k+1}}{\sigma_{i_k}}\right)}
}^{<0}
\\
&
+
\underbracket*{
\alpha_{\rm max}\rho_{\rm max}^2
\bcost_2(x_{i_k})
+
\alpha_{\rm max}\left(1-\tfrac{\sigma_{i_k+1}}{\sigma_{i_k}}\right)
\bcost_2(x_{i_k+1})
}_{\text{bounded by \cref{thm:adabim:barPhi}}}.
\end{align*}
In particular, we have
\begin{equation}\label{eq:supL}
\sup_{k\in\N}\mathcal L_{i_k+1}(x_\star)<\infty.
\end{equation}
Let now \(k'\in K'\).
Since \(\cost_1(x_j)-\costinf*\geq0\) for \(j=i_{k'}+1,\dots,k'\) observe that
\begin{equation}\label{eq:Lk'geq}
\tfrac12\|x_{k'}-x_\star\|^2
\leq
\mathcal L_{k'}(x_\star)
\quad
\forall k'\in K',
\end{equation}
and \cref{thm:adabim:qSD} yields that
\begin{align*}
\mathcal L_{k'}(x_\star)
\leq{} &
\mathcal L_{k'-1}(x_\star)
\leq\dots\leq
\mathcal L_{i_{k'}+2}(x_\star)
\numberthis\label{eq:Lk'descent}
\\
\leq{} &
\mathcal L_{i_{k'}+1}(x_\star)
-
\sigma_{i_{k'}+1}\alpha_{i_{k'}+1}(1+\rho_{i_{k'}+1}-\rho_{i_{k'}+2}^2)
\bigl(
\cost_1(x_{i_{k'}})-\costinf*
\bigr)
\\
\leq{} &
\mathcal L_{i_{k'}+1}(x_\star)
+
\underbracket[0.5pt]{
\alpha_{\rm max}(1+\rho_{i_{k'}+1}-\rho_{i_{k'}+2}^2)
\sigma_{i_{k'}+1}
\bigl|
\cost_1(x_{i_{k'}})-\costinf*
\bigr|
}_{\text{bounded by \cref{thm:adabim:barPhi}}}
< \infty
\qquad
\forall k'\in K',
\end{align*}
where \eqref{eq:supL} was used in the last inequality.
Here, boundedness of the under-bracketed term follows from boundedness of \(\seq{x_{i_{k'}}}[k'\in K']\) and lower semicontinuity of \(\cost_1\).
Then, \eqref{eq:Lk'geq} implies that the sequence \(\seq{x_{k'}}[k'\in K']\) is bounded.
Combined with \cref{claim:convergence:ik} and the fact that the index set \(K'\) is the complement of the indices therein, the claim follows.
\item
{\it the limit points of \(\seq{x_{i_k}}\) and \(\seq{x_{i_k+1}}\) are all optimal}.
Having established boundedness of the entire sequence, we may invoke \cref{thm:adabim:feas} to infer that all the limit points of \(\seq{x_{i_k}}\) are feasible.
Moreover, it follows from \eqref{eq:ik} that \(\limsup_{k\to\infty}\cost_1(x_{i_k})\leq\costinf*\), and a lower semicontinuity argument then yields that all the limit points attain the optimal cost and are therefore optimal.
\item
{\it all the limit points of \(\seq{x_k}\) are optimal}.
Having assessed the existence of optimal limit points, the convergence of \(\seq{\bphik(x_k)}\) established in \cref{thm:adabim:barPhi} implies that
\(
\bphik(x_k)
=
\sigk\bcost_1(x_k)+\bcost_2(x_k)
\to
0
\),
and since both summands are positive this means that
\begin{equation}\label{eq:adabim:barPhi0}
\lim_{k\to\infty}\sigk\bcost_1(x_k)
=
\lim_{k\to\infty}\bcost_2(x_k)
=
0.
\end{equation}
By virtue of \cref{thm:adabim:barPhi} and the previous claims, it remains to show that
\(
\dist(x_{k'},\X_1)
\to
0
\)
as \(K'\ni k'\to\infty\).
Recall that \(W_k\) defined in \cref{thm:LW} is independent of \(x_\star\in\X_1\), and observe that \(W_k\) vanishes as \(k\to\infty\) owing to \cref{thm:adabim:barPhi}, \eqref{eq:adabim:barPhi0}, and the fact that \(\sigk\alphk\to0\).
For each \(k\) let \(\bar x_k\coloneqq\proj_{\X_1}\!x_k\), which is well defined since \(\X_1\neq\emptyset\) is closed and convex.
Recalling that \(\mathcal L_k(x_\star)=W_k + \tfrac12\|x_k-x_\star\|^2\), for \(k'\in K'\) we have
\begin{align*}
\tfrac12\dist(x_{k'},\X_1)^2
\leq
\tfrac12\|x_{k'}-\bar x_{i_{k'}+2}\|^2
={} &
\mathcal L_{k'}(\bar x_{i_{k'}+2})-W_{k'}
\\
\dueto{\scriptsize by \eqref{eq:Lk'descent}}
\
\leq{} &
\mathcal L_{i_{k'}+2}(\bar x_{i_{k'}+2})
-
W_{k'}
\\
={} &
\tfrac12\dist(x_{i_{k'}+2},\X_1)^2
+
W_{i_{k'}+2}
-
W_{k'}
\to
0
\quad
\text{ as }K'\ni k'\to\infty,
\end{align*}
where the limit follows from \cref{claim:convergence:ik}, the vanishing of \(\seq{W_k}\), and the fact that \(i_k\to\infty\) as \(k\to\infty\).
\qedhere
\end{claims}
\end{proofitemize} \end{proof}
\proofsection{thm:BiM:convergence}
Once \cref{cond:alphk,,cond:alphamin} are verified, the claims for both \refstabim* and \refadabim* will follow from \cref{thm:PG:convergence}. The respective proofs are presented with the help of two lemmas. The first lemma shows that \refstabim* complies with \cref{cond:alphk}. Notice that \cref{cond:alphamin} is trivially satisfied for \refstabim* due to the underlying global Lipschitz continuity, and since \(\seq{\sigk}\) is decreasing.
The justification for the adaptive method \refadabim* under mere local smoothness requires more elaborate arguments. This involves demonstrating the well definedness of the stepsizes and confirming that \cref{cond:alphamin} is satisfied, and is provided in \cref{thm:adabim:fit}.
\begin{lemma}[stepsize compliance in \refstabim*]\label{thm:stabim:fit}
In addition to \cref{ass:basic}, suppose that \(\nabla\f_i\) is \(L_{\f_i}\)-Lipschitz continuous, \(i=1,2\), and consider the iterates generated by \refstabim{} with \(\sigma_1=\sigma\) for some initial inverse penalty \(\sigma>0\).
Then, for every \(k\) it holds that
\[
\alphk*\ell_{k+1}\leq\nu
\quad\text{and}\quad
\alpha_1\leq\alphk*\leq\balphk*,
\]
where \(\balphk*\) is as in \eqref{eq:adabim:alphk*_0} with \(\alpha_{\rm max}\geq\frac{\nu}{L_{\f_2}}\) and initialization \(\alpha_{-1}=\alpha_0=\alpha_{\rm max}\) and \(\sigma_{-1}=\sigma_0=\sigma\).
\begin{proof}
Since \(\ell_{k+1}\leq\sigk* L_{\f_1}+L_{\f_2}\), the first inequality is obvious.
Similarly, the second inequality follows from the fact that \(\seq{\sigk*}\) is decreasing, and thus \(\seq{\alphk*}\) is increasing.
Notice further that \(\alphk*\leq\frac{\nu}{L_{\f_2}}\leq\alpha_{\rm max}\); moreover, since \(\ck\leq\sigk L_{\f_1}+L_{\f_2}\), one has that \(\alphk\ck\leq\nu<1\), altogether proving that \(\alphk*\) is smaller than both the second and third terms in \eqref{eq:adabim:alphk*_0} (the second one being \(\infty\)).
To conclude, it remains to show that it is also smaller than the first term in the minimum.
To this end, observe that
\[
\rhok*
\defeq
\frac{\sigk*\alphk*}{\sigk\alphk}
=
\frac{1+\frac{1}{\sigk}\frac{L_{\f_2}}{L_{\f_1}}}{1+\frac{1}{\sigk*}\frac{L_{\f_2}}{L_{\f_1}}}
\leq
1,
\]
where the inequality follows from the fact that \(\seq{\sigk*}\) is decreasing.
Similarly,
\[
\rhok*
=
\tfrac{\sigk*\alphk*}{\sigk\alphk}
\geq
\tfrac{\sigk*}{\sigk}
\geq
\tfrac34,
\]
where the first inequality follows from the fact that \(\seq{\alphk*}\) is increasing and the second one from the constraints on \(\sigk*\) prescribed at \cref{alg:stabim:sigk*}.
Overall, it follows that \(\rhok*\in[\nicefrac34,1]\) holds for every \(k\).
Therefore,
\[
\sqrt{
\tfrac{\sigk}{\sigma_{k-1}}\bigl(1+\rhok\bigr)
}
\geq
\sqrt{
\tfrac{3}{4}\left(1+\tfrac{3}{4}\right)
}
>
1
\geq
\rhok*
\defeq
\tfrac{\sigk*\alphk*}{\sigk\alphk},
\]
where the first inequality again follows from the bounds on \(\sigk*\) at \cref{alg:stabim:sigk*}.
Rearranging yields the sought inequality
\(
\alphk*
\leq
\sqrt{\tfrac{\sigk}{\sigma_{k-1}}\bigl(1+\rhok\bigr)}
\tfrac{\sigk}{\sigk*}
\alphk
\).
\end{proof} \end{lemma}
Next we examine the compliance of \refadabim* with the recipe provided in \cref{thm:PG:convergence} showing that \cref{cond:alphamin} holds. In particular, given the stepsize selection of \cref{state:adabim:alphk*_0}, it is shown that the stepsizes are well defined and bounded away from zero whenever the generated sequence is bounded.
\begin{lemma}[stepsize compliance in \refadabim*]\label{thm:adabim:fit}
Suppose that \cref{ass:basic} holds.
For every \(k\geq1\), the following hold for the iterates generated by \refadabim{}:
\begin{enumerate}
\item \label{thm:adabim:WD}
The stepsize \(\alphk*\) at \cref{state:adabim:x+} is well defined and strictly positive.
\item \label{thm:adabim:gammamin}
If \(\seq{x_k}\) is bounded and \(\lim_{k\to\infty}\nicefrac{\sigk*}{\sigk}=1\), then for all \(k\) large enough it holds that
\(
\alphk
\geq
\alpha_{\rm min}
\coloneqq
\min\set{
\frac{1}{\ell_{f_0,\mathcal V}}\big(\tfrac{1 - \nu}{\nu}\big)^{\nicefrac12},\,
\frac{\eta\nu}{\ell_{f_0,\mathcal V}},\,
\alpha_{\rm max}
}
\),
where \(\ell_{f_0,\mathcal V}\) is a Lipschitz modulus for \(\nabla f_0=\sigma_0\nabla\f^1+\nabla\f^2\) on a bounded and convex set \(\mathcal V\) that contains all the sequences \(\seq{x_k}\) and \(\seq{x_k-\balphk*\nabla\fk*(x_k)}\).
\end{enumerate}
\begin{proof}
In what follows, we denote \(\alphk*_i\coloneqq\balphk*\eta^i\) so that in particular \(\alphk*_0=\balphk*\).
Accordingly, we let \(x_{k+1,i}\) be the value of \(x_{k+1}\) with \(\alphk*\gets\balphk*\eta^i\) and similarly for the Lipschitz estimates \(\lk*_i\) and \(\ck*_i\).
\begin{proofitemize}
\item \ref{thm:adabim:WD}~
It suffices to show that \(\alphk*_0\) is (well defined and) strictly positive.
In fact, all the attempts \(\seq{x_{k+1,i}}[i]\) remain confined in a bounded set \(\mathcal V_{k+1}\), over which \(\nabla\fk*\) has finite Lipschitz modulus, be it \(L_{\fk*,\mathcal V_{k+1}}\);
as such, for all \(i\) one has that \(\alphk*_i\lk*_i\leq\alphk*_0\eta^iL_{\fk*,\mathcal V_{k+1}}\to0\) as \(i\to\infty\), implying that condition \eqref{eq:LS} is satisfied for all \(i\) large enough.
We thus proceed by induction to show that \(\alphk*_0>0\).
Equivalently, in view of the update \eqref{eq:adabim:alphk*_0} it suffices to show that \(1-4\left(1-\tfrac{\sigk}{\sigma_{k-1}}\right)\alphk\lk^{(2)}>0\) holds for all \(k\).
For \(k=0\) this is true because \(1-\frac{\sigma_0}{\sigma_{-1}}=0\) by initialization.
Suppose that the claim holds for \(k\); then,
\(
\alphk\lk^{(2)}
\leq
\alphk\bigl(\sigk\lk^{(1)}+\lk^{(2)}\bigr)
=
\alphk\lk
\leq
\nu
\),
where the last inequality holds by inductive hypothesis.
Therefore,
\begin{equation}\label{eq:sqrt>0}
1
-
4\left(1-\tfrac{\sigk}{\sigma_{k-1}}\right)
\alphk\lk^{(2)}
\geq
1
-
4\nu\left(1-\tfrac{\sigk}{\sigma_{k-1}}\right)
\geq
1 - \nu
>
0,
\end{equation}
where the last inequality owes to the bound \(\sigk\geq\frac{3}{4}\sigma_{k-1}\) prescribed at \cref{state:adabim:alphk*_0}.
\item \ref{thm:adabim:gammamin}~
We start by observing that
\begin{equation}\label{eq:alphamaxLB}
\alpha_{\rm max}
>
\tfrac{1}{\ell_0}
\geq
\tfrac{1}{\ell_{f_0,\mathcal V}}
\geq
\tfrac{\alpha_{\rm min}}{\eta\nu}.
\end{equation}
Moreover, since \(\alphk*_i\in(0,\balphk*]\) holds for all \(k\) and \(i\) and \(\mathcal V\) is convex, it follows from the assumptions that not only does \(x_{k+1}\) belong to \(\mathcal V\) for every \(k\), but so do all the attempts \(x_{k+1,i}\) as well.
Combined with the fact that \(\seq{\sigk}\) is decreasing, it then follows that
\begin{equation}\label{eq:clbound}
\lk\leq\ck\leq\ell_{f_0,\mathcal V}
\quad\text{and}\quad
\lk*_i\leq\ck*_i\leq\ell_{f_0,\mathcal V}
\quad\text{hold for every \(i\) and \(k\).}
\end{equation}
\begin{subequations}\label{subeq:rhok*}
In particular, notice that
\begin{equation}\label{eq:rhok*:A}
\alphk\ell_{f_0,\mathcal V}
{}\leq{}
1
\quad\Rightarrow\quad
\alphk*_0
=
\min\set{
\tfrac{\sigk}{\sigk*}
\sqrt{\tfrac{\sigk}{\sigma_{k-1}}(1+\rhok)}\,
\alphk
,\,
\alpha_{\rm max}
},
\end{equation}
since in such a scenario it also holds that \(\alphk\ck\leq1\) making the second term in \eqref{eq:adabim:alphk*_0} infinite (recall the convention \(\nicefrac10 = \infty\)).
Similarly, the second group of inequalities in \eqref{eq:clbound} implies that whenever \(\alphk*_0\leq\tfrac{\nu}{\ell_{f_0,\mathcal V}}\) the initial stepsize \(\alphk*_0\) already complies with \eqref{eq:LS} and will thus not be reduced by the backtracks; otherwise, it is possibly multiplied by \(\eta\) (finitely many times, as shown in assertion \ref{thm:adabim:WD}) only up to when the bound in \eqref{eq:LS} is satisfied: namely,
\begin{equation}\label{eq:alphk0=alphk}
\alphk*_0\leq\tfrac{\nu}{\ell_{f_0,\mathcal V}}
~\Rightarrow~
\alphk*=\alphk*_0
\quad\text{and}\quad
\alphk*_0>\tfrac{\nu}{\ell_{f_0,\mathcal V}}
~\Rightarrow~
\alphk*
{}\geq{}
\tfrac{\eta\nu}{\ell_{f_0,\mathcal V}}.
\end{equation}
\end{subequations}
We now proceed by intermediate steps.
\begin{claims}
\item \label{claim:limsup>0}
{\it \(\hat\alpha\coloneqq\limsup_{k\to\infty}\alphk>0\).}
Contrary to the claim, suppose that \(\alphk\to0\).
Then, eventually \(\alphk<\tfrac{\eta\nu}{\ell_{f_0,\mathcal V}}\) always holds and \eqref{subeq:rhok*} yields that
\[
\alphk*
=
\alphk*_0
=
\min\set{
\tfrac{\sigk}{\sigk*}
\sqrt{\tfrac{\sigk}{\sigma_{k-1}}(1+\rhok)}\,
\alphk
,\,
\alpha_{\rm max}
}
=
\tfrac{\sigk}{\sigk*}
\sqrt{\tfrac{\sigk}{\sigma_{k-1}}(1+\rhok)}\,
\alphk,
\]
where the last identity holds for all \(k\) large enough (since \(\alphk\to0\)).
Hence,
\[
\rhok*
=
\tfrac{\sigk*}{\sigk}
\tfrac{\alphk*}{\alphk}
=
\sqrt{\tfrac{\sigk}{\sigma_{k-1}}(1+\rhok)}
\]
for large enough \(k\).
Since \(\nicefrac{\sigk*}{\sigk}\to1\), one has that \(\rhok\to\frac{1+\sqrt5}{2}>1\), yielding the contradiction \(\alphk\nearrow\infty\).
\item \label{claim:inf>0}
{\it\(\alphk\geq\hat\alpha_{\rm min}\coloneqq\min\set{\alpha_{\rm min},\eta\hat\alpha}\) holds for all \(k\) large enough.}
It follows from \cref{claim:limsup>0} that there exists \(\hat k\in\N\) such that \(\alpha_{\hat k}\geq\eta\hat\alpha\).
We proceed by induction to show that the sought \cref{claim:inf>0} holds for all \(k\geq\hat k\).
The base case is trivial.
For the induction step, suppose that \(\alphk\geq\hat\alpha_{\rm min}\) holds for some \(k\geq\hat k\).
By virtue of \eqref{eq:alphk0=alphk}, we have \(\alpha_{k+1}\geq \min\set{\alpha_{k+1,0}, \tfrac{\eta\nu}{\ell_{f_0,\mathcal V}}}\).
Therefore, since
\(
\frac{\eta\nu}{\ell_{f_{0,\mathcal V}}}
\geq
\alpha_{\rm min}
\geq
\hat\alpha_{\rm min}
\),
it suffices to show that \(\alphk*_0\geq\hat\alpha_{\rm min}\).
If \(\alphk*_0\geq\alphk\) there is nothing to show, since \(\alphk\geq\hat \alpha_{\rm min}\) by inductive hypothesis.
Suppose that \(\alphk*_0<\alphk\).
Note that, since \(\nicefrac{\sigk}{\sigma_{k-1}}\to1\) and up to possibly enlarging \(\hat k\), for all \(k\geq\hat k\) it holds that
\(
\tfrac{\sigk}{\sigma_{k-1}}
\left(
1+\tfrac{\sigk}{\sigma_{k-1}}\tfrac{\hat\alpha_{\rm min}}{\alpha_{\rm max}}
\right)
\geq
1
\).
In particular, since \(\alpha_k \geq \hat\alpha_{\rm min}\)
\begin{equation}\label{eq:hatk}
\sqrt{\tfrac{\sigk}{\sigma_{k-1}}(1+\rhok)\!}\,
=
\sqrt{
\tfrac{\sigk}{\sigma_{k-1}}
\left(
1+\tfrac{\sigk}{\sigma_{k-1}}\tfrac{\alphk}{\alpha_{k-1}}
\right)
\!}\,
\geq
\sqrt{
\tfrac{\sigk}{\sigma_{k-1}}
\left(
1+\tfrac{\sigk}{\sigma_{k-1}}\tfrac{\hat\alpha_{\rm min}}{\alpha_{\rm max}}
\right)
\!}\,
\geq
1.
\end{equation}
Because of \eqref{eq:hatk} and since we consider the case \(\alpha_{k+1,0}<\alpha_k\), it follows from \eqref{eq:adabim:alphk*_0} that the minimum therein is attained at the second element, implying in particular that \(\ck\alphk>1\).
Therefore, we have
\[\mathloose
\bigl(\alphk*_0\ell_{f_0,\mathcal V}\bigr)^2
\overrel[\geq]{\eqref{eq:clbound}}
\bigl(\alphk*_0\ck\bigr)^2
\overrel{\eqref{eq:adabim:alphk*_0}}
{\underbracket*{
\left(
\tfrac{\sigk\vphantom{\lk}}{\sigk*}
\right)
}_{\scriptscriptstyle\geq1}}^2
\bigl(\alphk\ck\bigr)^2
\tfrac{
1
-
4(
1-\nicefrac{\sigk}{\sigma_{k-1}}
)
\alphk\lk^{(2)}
}{
4
\underbracket*{
\scriptstyle
\alphk\lk
}_{\leq\nu}
(\alphk\ck-1)
}
\overrel[\geq]{\eqref{eq:sqrt>0}}
\underbracket*{
\left(
\tfrac{
(\alphk\ck)^2
\vphantom{\lk}
}{
\alphk\ck-1
}
\right)
}_{\scriptscriptstyle\geq4}
\tfrac{1-\nu}{4\nu}
\geq
\tfrac{1-\nu}{\nu},
\]
proving that
\(
\alphk*_0
\geq
\left(\tfrac{1-\nu}{\nu}\right)^{\nicefrac12}\tfrac{1}{\ell_{f_0,\mathcal V}}
\geq
\alpha_{\rm min}
\geq
\hat\alpha_{\rm min}
\).
\item
{\it \(\alphk\geq\alpha_{\rm min}\) holds for all \(k\) large enough.}
Having assessed the validity of \cref{claim:inf>0}, it suffices to show that \(\hat\alpha_{\rm min}=\alpha_{\rm min}\), that is, that \(\hat\alpha\geq\frac1\eta\alpha_{\rm min}\).
To arrive to a contradiction, suppose that \(\hat\alpha<\frac1\eta\alpha_{\rm min}\), and recall that \(\hat\alpha=\limsup_{k\to\infty}\alphk\).
Therefore,
\(
\alphk
\leq
\frac1\eta\alpha_{\rm min}
\leq
\frac{\nu}{\ell_{f_0,\mathcal V}}
<
\alpha_{\rm max}
\)
holds for all \(k\) large enough, where the last inequality owes to \eqref{eq:alphamaxLB}.
In particular, \(\alphk\neq\alpha_{\rm max}\) and \eqref{eq:rhok*:A} then implies that \(\alphk*=\tfrac{\sigk}{\sigk*}\sqrt{\tfrac{\sigk}{\sigma_{k-1}}(1+\rhok)}\,\alphk\).
We may thus tighten the estimate in \eqref{eq:hatk} by using \(\frac1\eta\hat\alpha_{\rm min}\) as upper bound for \(\alpha_{k-1}\) in place of \(\alpha_{\rm max}\), namely
\[
\alphk*
=
\tfrac{\sigk}{\sigk*}
\sqrt{\tfrac{\sigk}{\sigma_{k-1}}(1+\rhok)}\,\alphk
=
\tfrac{\sigk}{\sigk*}
\sqrt{
\tfrac{\sigk}{\sigma_{k-1}}
\left(
1+\tfrac{\sigk\alphk}{\sigma_{k-1}\alpha_{k-1}}
\right)
}\,
\alphk
\geq
\sqrt{
\tfrac{\sigk}{\sigma_{k-1}}
\left(
1+\eta\tfrac{\sigk}{\sigma_{k-1}}
\right)
}\,
\alphk
\quad
\forall k\geq\hat k,
\]
where the last inequality uses the proven fact that \(\alphk\geq\hat\alpha_{\rm min}\) holds for all \(k\geq\hat k\).
Since \(\nicefrac{\sigk}{\sigma_{k-1}}\to1\), one then obtains the contradiction \(\alphk\nearrow\infty\).
\qedhere
\end{claims}
\end{proofitemize}
\end{proof} \end{lemma}
\section{Numerical simulations}\label{sec:num}
In this section the performance of the proposed algorithms is evaluated through a series of simulations on standard problems on both synthetic data and standard datasets from LIBSVM dataset \cite{chang2011libsvm}. All the algorithms are implemented in the Julia programming language and are available online.\footnote{
\url{https://github.com/pylat/adaptive-bilevel-optimization}. } All the ``explicit'' algorithms listed in \cref{table:comparisons} are used whenever applicable. An overview of each one of them is provided in the following subsection.
In accounting for the difference in iteration complexity among the methods, the simulations report the progress against the number of calls to \(\nabla\f_2\), since in all problems \(\nabla\f_1\) and proximal operations have negligible cost. As explained in \cref{sec:solodov}, this criterion favors the method \solodov{}, as it disconsiders the cost of the backtrackings which involve function evaluations. On the contrary, all the backtracking steps included in \refadabim*, which involve gradient evaluations, are fully accounted for in the comparisons.
\subsection{Compared algorithms}
Other than \refadabim* and \refstabim*, the algorithms involved in the simulations are \solodov, \bigsam, and \itthreeD. For \itthreeD{}, \(\sigk=\nicefrac{1}{(k+1)^2}\) was used, whereas a slowly controlled \(\sigk=\nicefrac{1}{k+1}\) was adopted for the rest. Although only \bigsam{} requires \(\sigk\in(0,1]\), this limitation was applied across all methods to maintain more uniform comparisons.
It is also worth noting that \refadabim* is not sensitive to the choice of initial stepsizes \(\alpha_{-1}\) and \(\alpha_0\), as the stepsize is automatically adjusted during the initial iteration. In all the simulations, the parameter \(\nu = 0.99\) was used, and the parameter \(\alpha_{\rm max}\) appearing in \eqref{eq:adabim:alphk*_0} of \refadabim* was set as a large constant; as remarked before, it is only of theoretical significance. As we will see, the same parameter in \solodov{} is instead crucial for dictating the algorithmic performance. These facts are better detailed in the following brief description of the algorithms compared against in the simulations.
\subsubsection{\Solodov-\texorpdfstring{\(\alpha\)}{a}}\label{sec:solodov}
This is Solodov's explicit descent method \cite[Alg. 2.1]{solodov2007explicit} already outlined in \eqref{eq:solodov}.
In the simulations, the suffix ``-\(\alpha\)'' is used to distinguish different choices for the value of \(\widehat\alpha_0\) therein; namely, whenever \(\f^2\) is \(L_{\f_2}\)-Lipschitz differentiable, we set \(\widehat\alpha_0\coloneqq\frac{\alpha}{L_{\f_2}}\).
In addition to \cref{ass:basic}, the algorithm requires:
\begin{itemize}[itemsep=1pt]
\item
\(\g_1=0\);
\item
\(\g_2=\indicator_D\) for a nonempty, closed, and convex set \(D\subseteq\R^n\).
\end{itemize}
Notice that the the backtracks involved for \solodov{} do not require any additional gradient evaluations, but instead require function value evaluations which are not reflected in the comparisons in terms of total number of gradients.
For this reason, in \cref{fig:backtrack} we provided a sample plot for selected three applications demonstrating the higher number of backtracks that it incurs compared to \refadabim*.
As evident in the figures, in practice \solodov{} is sensitive to parameter tuning; while selecting a larger \(\widehat\alpha_0\) can lead to larger stepsizes and consequently faster convergence speed in terms of number of iterations (gradient evaluations), it also results in a higher number of backtracks (each requiring one cost evaluation), and vice versa.
In comparison, our proposed method \refadabim* achieves a better performance while providing a suitable initialization based on the proposed adaptive scheme, insensitive to parameter initialization.
In all the simulations, both for \solodov{} and \refadabim* we used the backtrack parameter \(\eta = \nicefrac12\), and the linesearch related parameter \(\nu=0.98\).
\subsubsection{\Bigsam}\label{sec:bigsam}
Proposed in \cite{sabach2017first}, \bigsam{} addresses strongly convex bilevel methods under global Lipschitz differentiability assumptions.
Specifically, in addition to \cref{ass:basic} the algorithm requires that
\begin{itemize}[itemsep=1pt]
\item
\(\f_1\) is \(L_{\f_1}\)-Lipschitz differentiable and \(\mu_{\f_1}\)-strongly convex;
\item
\(\f_2\) is \(L_{\f_2}\)-Lipschitz differentiable;
\item
\(\g_1=0\).
\end{itemize}
The knowledge of the Lipschitz moduli is also required for selecting the stepsizes (that of the strong convexity only for an optimal stepsize tuning): with \(\alpha^{(1)}\leq\frac{2}{L_{\f_1}+\mu_{\f_1}}\), \(\alpha^{(2)}\leq\frac{1}{L_{\f_2}}\), and \(\seq{\sigk}\) as in \cref{cond:sigk} with \(\sigma_1\leq1\), \bigsam{} iterates
\[
\begin{cases}
\@ifstar\@@xk\@xk*_1 & {}= \@ifstar\@@xk\@xk_1-\alpha^{(1)}\nabla\f_1\left(\@ifstar\@@xk\@xk_1\right)\\
\@ifstar\@@xk\@xk*_2 & {}= \prox_{\alpha^{(2)}\g_2}\left(\@ifstar\@@xk\@xk_1-\alpha^{(2)}\nabla\f_2\left(\@ifstar\@@xk\@xk_2\right)\right)\\
x_{k+1}& {}= \sigk*\@ifstar\@@xk\@xk*_1+(1-\sigk*)\@ifstar\@@xk\@xk*_2.
\end{cases}
\]
A notable advantage of this method is that the operations on the upper and lower level are decoupled, and can therefore potentially be performed in parallel.
\subsubsection{\ItthreeD}\label{sec:itthreeD}
This method, presented in \cite{garrigos2018iterative}, is an algorithm specialized to linear inverse problems and operating on the dual formulation, which complicates the comparison with the other methods.
A strongly convex upper layer cost is required, but the iterations only involve gradient (and not proximal) evaluations on its (Lipschitz-differentiable) conjugate.
Similarly, the lower level cost is an infimal convolution between a prox-friendly and a strongly convex function, making its dual the sum of a prox-friendly and a Lipschitz-differentiable terms.
The simulations report the progress against number of calls to gradient operators, and this is the reason why the problem requirements synopsized in \cref{table:comparisons} are relative to the dual formulation that agrees with the structure of other methods.
The requirements on the primal formulation are roughly as follows:
\begin{itemize}
\item
\(\cost_1\) is \(\mu_{\cost_1}\)-strongly convex;
\item
\(\cost_2\) is a coercive ``data-fit'' function.
\end{itemize}
In referring the reader to \cite{garrigos2018iterative} for a rigorous account on the problem formulation and its requirements, we point out that among our simulations \itthreeD{} is only applicable to the linear inverse problem of \cref{sec:lasso} with \(\ell^2\)-norm upper layer cost.
In that setting, initializing with \(x_0\in\range\trans A\) the method performs the following iterations
\[
x_{k+1}
=
x_k
-
\gamma\nabla\fk(x_k)
=
x_k
-
\trans A(Ax_k - b) - \gamma\sigk x_k,
\]
each increasing the total gradient count by one.
Remarkably, in this setting it does not constrain \(\seq{\sigk}\) to a nonsummable decay; see \cite[Rem. 10]{garrigos2018iterative}.
For this reason, in the simulations inverse penalties \(\sigk=\nicefrac{1}{(k+1)^2}\) were used for \itthreeD, while \(\sigk=\nicefrac{1}{k+1}\) for all other methods.
\subsection{Logistic regression}\label{sec:logreg}
\begin{figure}\label{fig:logreg:sqrL2}
\end{figure}
We consider logistic regression problem \begin{subequations}\label{eq:logreg}
\begin{align}
\minimize_{x\in\R^n}{} &~ \cost_1(x)
\\
\stt{} &~ x \in \argmin_{w\in\R^n}
\set{
\tfrac1m \textstyle\sum_{i=1}^m \left(y_i \log(s_i(w)) + (1-y_i) \log(1-s_i(w))\right)
},
\end{align} \end{subequations}
where \(m,n\) are the number of samples and features, the pair \(a_i\in\R^{n+1}\) denotes the \(i\)-th sample (up to absorbing the bias terms), \(y_i\in\set{-1,1}\) is the associated label, and \(s_i(x) = (1+ \exp(-\trans{a_i}x))^{-1}\) is the logistic sigmoid function. In the simulations we used \(\cost_1 = \|{}\cdot{}\|^2\) (\cref{fig:logreg:sqrL2}) and \(\cost_1 = \|{}\cdot{}\|_1\) (\cref{fig:ell1}, first column); for the latter, only one dataset is reported, as the plots for other ones are very similar.
Note also that in the simulations for \refadabim* and \refstabim* we set \(\g_1 = \cost_1\) and \(\f_1 \equiv0\). For other methods the (smooth) upper level cost is captured using \(\f_1\) with \(L_{\f_1} = 1\) and its strong convexity modulus equal to \(1\) (in the case of \bigsam{}). For methods that require Lipschitz modulus of \(\nabla \f_2\), \(L_{\f_2} = \tfrac1{4m}\|A\|^2\) was used where \(A\) is the data matrix that is the concatenation of \((a_j)_{j=1,\dots,m}\).
\subsection{Linear inverse problems with simulated data}\label{sec:lasso}
In a series of experiments we consider the special cases of the following problem \begin{subequations}\label{eq:lininverse}
\begin{align}
\minimize_{x\in\R^n}{} &~ \cost_1(x)
\\
\stt{} &~ x \in \argmin_{w\in\R^n}\tfrac12\|Aw -b\|^2,
\end{align} \end{subequations} where \(A\in \R^{m\times n}\) and \(b\in \R^n\) are generated based on the procedure described in \cite[\S6]{nesterov2013gradient}, and \(n_\star\) denotes the number of nonzero elements of the solution. For the upper level cost \(\cost_1\), we consider two sets of experiments: \begin{enumerate} \item
\(\cost_1 = \tfrac12\|{}\cdot{}\|^2\), corresponding to the Moore--Penrose solution, see \cref{fig:lasso:sqrl2}; \item
least \(\ell_1\) norm solutions corresponding to \(\cost_1 = \|{}\cdot{}\|_1\), see the last two columns in \cref{fig:ell1} (the behavior of the algorithm is consistent with these plots for other values of \(m,n\)). \end{enumerate}
Note also that in the simulations for \refadabim* and \refstabim* we set \(\g_1 = \cost_1\) and \(\f_1 \equiv0\). For other methods the (smooth) upper level cost is captured using \(\f_1\) with \(L_{\f_1} = 1\) and its strong convexity modulus equal to \(1\) (in the case of \bigsam{}). For methods that require Lipschitz modulus of \(\nabla \f_2\), \(L_{\f_2} = \|A\|^2\).
\begin{figure}\label{fig:lasso:sqrl2}
\end{figure}
\subsection{Solution of integral equations}
We consider the solution of integral equations using the setting described in \cite[Sec. 5.2]{beck2014first}. The corresponding bilevel problem is the following: \begin{subequations}\label{eq:IntEq}
\begin{align}
\minimize_{x\in\R^n}{} &~ \tfrac12\|x\|^2_Q
\\
\stt{} &~ x \in \argmin_{w\geq0}\tfrac12\|Aw -b\|^2.
\end{align} \end{subequations}
The data matrix \(A\) in \eqref{eq:lininverse} is generated using \emph{philips, foxgood, baart} functions. Let \(L\) denote the discrete gradient operator, and let \(Q_1 = \trans LL\) and \(Q = Q_1 + \I\). In the simulations for \bigsam{} and \solodov, the upper level cost is captured using \(\f_1\), while for \refadabim* and \refstabim* we used \(\f_1 = \tfrac12\langle x,Q_1x \rangle\) and \(\g_1 = \tfrac12\|{}\cdot{}\|^2\) (as a rule of thumb, considering formulating the problem using the proximable term is preferable, a tweak that only our method can take advantage of, it being the only one allowing proximable terms in the upper level); using the calculus rule of \cite[Prop. 24.8.(i)]{bauschke2017convex}, the proximal mapping of \(\gk=\tfrac{\sigk}{2}\|{}\cdot{}\|^2+\g_2\) is given by \[
\prox_{\alphk \gk}(u) =
\prox_{\frac{\alphk}{1+\alphk\sigk c}\g_2}\left(\tfrac{1}{1 + \alphk\sigk c}u\right). \]
Notice that multiplications by \(Q_1\) involved in calls to \(\nabla\f_1\) can efficiently be handled through abstract linear operators and are ignored in the gradient calls count for all methods. In order to compare the methods in a fair manner, in addition to the deviation of the upper level cost from \(\costinf*\), we also plot a measure of optimality for the lower level. For example, in the case of \refadabim*, given that \(\g_1 = \tfrac12\|{}\cdot{}\|^2\), it is of immediate verification that \[
\tfrac{1}{\alphk*}(x_k-x_{k+1})-\sigk x_{k+1}+\nabla\f_2(x_{k+1})-\nabla\fk*(x_k)
\in \partial \cost_2(x_{k+1}). \] Similar computation applies to the other methods that are included in the comparisons.
\begin{figure}
\caption{Solution of integral equations}
\label{fig:integralEq}
\end{figure}
\begin{figure}\label{fig:ell1}
\end{figure}
\section{Conclusions}\label{sec:conclusions}
In this paper we considered structured bilevel problems where both the upper and lower level minimizations are split as the sum of a nonsmooth and a locally Lipschitz differentiable function. We showed that proximal gradient updates involving a family of parametrized forward and backward terms that vary across iterations converge, without the need for any inner minimizations. This was achieved by selecting the stepsizes according to a carefully designed adaptive scheme that generalizes the {\sffamily adaPGM} algorithm in \cite[Alg. 1]{latafat2023adaptive} to the bilevel setting, resulting in the proposed \refadabim. While our scheme involves a linesearch, it waives the need for function value evaluations and, remarkably, prescribes a suitable initialization for the linesearch based on estimating the local geometry of the differentiable terms, leading to much larger stepsizes compared to existing methods in practice. Finally, the favorable convergence properties of the method was confirmed through a series of numerical simulations. Future research directions involve designing adaptive strategies for the inverse penalty parameters \(\sigk\), the derivation of stopping criteria, and extensions to non-simple and possibly nonconvex bilevel settings.
\ifsiam
\else
\phantomsection
\addcontentsline{toc}{section}{References}
\fi
\end{document} | arXiv |
KxngVee
A circular-flow diagram is a visual model of the economy.
A macroeconomist, rather than a microeconomist, would study the effects on a market from two firms merging.
A production possibilities frontier has a bowed shape if the opportunity cost is constant at all levels of output.
An outcome is said to be efficient if an economy is conserving the largest possible quantity of its scarce resources while still meeting the basic needs of society.
Two variables that have a positive correlation move in the same direction.
While the scientific method is applicable to studying natural sciences, it is not applicable to studying a nation's economy.
With the resources it has, an economy can produce at any point on or outside the production possibilities frontier, but it cannot produce at points inside the frontier.
If a line passes through the points (20,5) and (10,10), then the slope of the line is 1/2.
If an economy can produce more of one good without giving up any of another good, then the economy's current production point is inefficient.
In the circular-flow diagram, factors of production are the goods and services produced by firms.
The classic tradeoff between "guns and butter" states that when a society spends more on national defense, it has less to spend on consumer goods to raise the standard of living.
With careful planning, we can usually get something that we like without having to give up something else that we like.
Economists regard events from the past as
interesting and valuable, since those events are capable of helping us to understand the past, the present, and the future.
The production possibilities frontier provides an illustration of the principle that
people face trade offs
The scientific method is applicable to studying
both natural sciences and social sciences.
When an economy is operating inside its production possibilities frontier, we know that
there are unused resources or inefficiencies in the economy.
When constructing a production possibilities frontier, which of the following assumptions is not made?
The quantities of the factors of production that are available are increasing over the relevant time period.
Which of the following is not an assumption of the productions possibilities frontier?
There is a fixed quantity of money.
Between the two ordered pairs (3, 6) and (7, 18), the slope is
Econ Hw 2 (Peralta)
ISDS ch 6, 7, 8
Econ 2000 Peralta Practice Exam 1
Ohio law provides for a free education for all children between the ages of $6$ and $21.$ In $1971$, widespread student unrest took place in the public schools of Columbus, Ohio. Students who either participated in, or were present at, demonstrations held on school grounds were suspended. Many suspensions were for a period of ten days. Students were not given a hearing before suspension, although at a later date some students and their parents were given informal conferences with the school principal. A number of students, through their parents, sued the board of education, claiming that their right to due process had been violated when they were suspended without a hearing. In Goss v. Lopez, the U.S. Supreme Court decided that public school students who are suspended for ten days or less are entitled to certain rights before their suspension. These rights include $(1)$ oral or written notice of the charges, $(2)$ an explanation of the evidence against them (if students deny the charges), and $(3)$ an opportunity for students to present their side of the story. The Court stated that in an emergency, students could be sent home immediately, and a hearing could be held at a later date. The Court did not give students a right to a lawyer, a right to call or cross-examine witnesses, or a right to a hearing before an impartial person. In Goss, the Court considered the due process interests of harm, cost, and risk. The Court ruled that reputations were harmed and educational opportunities were lost during the suspension; that an informal hearing would not be overly costly for the schools; and that while most disciplinary decisions were probably correct, an informal hearing would help reduce the risk of error. What happened in the Goss case? What rights did the Supreme Court say the students should be given prior to a brief suspension?
Decide whether the following statement is true or false. Explain your reasoning. The probability of getting heads and tails when you toss a coin is $0$, but the probability of getting heads or tails is $1$.
For each data set, find the median, midrange, and geometric mean. Are they reasonable measures of central tendency? Explain. a. Exam scores (9 students) 42, 55, 65, 67, 68, 75, 76, 78, 94 b. GPAs (8 students) 2.25, 2.55, 2.95, 3.02, 3.04, 3.37, 3.51, 3.66 c. Class absences (12 students) 0, 0, 0, 0, 0, 1, 2, 3, 3, 5, 5, 15
Given the following observations in a simple random sample from a population that is approximately normally distributed, construct and interpret the $95 \%$ and $99 \%$ confidence intervals for the mean: $$ \begin{array}{llllllllll}66 & 34 & 59 & 56 & 51 & 45 & 38 & 58 & 52 & 52 \\ 50 & 34 & 42 & 61 & 53 & 48 & 57 & 47 & 50 & 54\end{array} $$
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\begin{document}
\begin{abstract} In this paper we prove a KAM theorem for small-amplitude solutions of
the non linear beam equation on the d-dimensional torus $$u_{tt}+\Delta^2 u+m u + \partial_u G(x,u)=0\ ,\quad t\in { \mathbb{R}} , \; x\in \ { \mathbb{T}}^d, \qquad \qquad (*) $$
where $G(x,u)=u^4+ O(u^5)$. Namely, we show that, for generic $m$, many of the small amplitude invariant finite dimensional tori of the linear equation $(*)_{G=0}$, written as the system $$ u_t=-v,\quad v_t=\Delta^2 u+mu, $$ persist as invariant tori of the nonlinear equation $(*)$, re-written similarly. The persisted tori are filled in with time-quasiperiodic solutions of $(*)$. If $d\ge2$, then not all the persisted tori are linearly stable, and we construct explicit examples of partially hyperbolic invariant tori. The unstable invariant tori, situated in the vicinity of the origin, create around them some local instabilities, in agreement with the popular belief in the nonlinear physics that small-amplitude solutions of space-multidimensional Hamiltonian PDEs behave in a chaotic way.
\end{abstract}
\subjclass{37K55, 70H08, 70H09, 70K25, 70K43, 70K45, 74H40, 74K10} \keywords{Beam equation, KAM theory, Hamiltonian systems, stable solutions, unstable solutions. }
\maketitle \tableofcontents
\section{Introduction} \subsection{The beam equation and KAM for PDE's} \label{s_1.1} The paper deals with small-amplitude solutions of
the multi-dimensional nonlinear
beam equation on the torus: \begin{equation} \label{beam}u_{tt}+\Delta^2 u+m u = - g(x,u)\,,\quad u=u(t,x), \
t\in \mathbb{R}, \ x\in \mathbb{T} ^d= \mathbb{R}^d/(2\pi \mathbb{Z} )^d, \end{equation}
where $g$ is a real analytic function of $x\in \mathbb{T} ^d$ and of $u$ in the vicinity of the origin in
$ \mathbb{R}$. We shall consider functions $g$ of the form
\begin{equation} \label{g}
g=\partial_u G,\quad G(x,u)=u^4+O(u^5).
\end{equation}
The polynomial $u^4$ is the {\it main part} of $G$ and $O(u^5)$ is its {\it higher order part}. $m$ is the mass parameter and we assume that $m\in[1,2]$.
This equation is interesting by itself. Besides, it is a good model for the Klein--Gordon equation
\begin{equation}\label{KG}
u_{tt} - \Delta u+mu=-\partial_u G(x,u),\qquad x\in \mathbb{T} ^d,
\end{equation}
which is among the most important equations of mathematical physics. We feel confident that the ideas and methods
of our work apply -- with additional technical efforts -- to eq.~\eqref{KG} (but the situation with the nonlinear wave
equation \eqref{KG}${}_{m=0}$, as well as with the zero-mass beam equation, may be quite different).
Our goal is to develop a general KAM-theory for small-amplitude solutions of \eqref{beam}. To do this we compare them
with time-quasi-periodic solution of the linearised at zero equation
\begin{equation}\label{linear}
u_{tt} +\Delta^2 u+ mu=0\,.
\end{equation}
Decomposing real functions $u(x)$ on $ \mathbb{T} ^d$ to Fourier series
$$
u(x)= \sum_{a\in \mathbb{Z} ^d} u_a e^{{\bf i}\langle a, x\rangle }\ +\text{c.c.}
$$
(here c.c. stands for ``complex conjugated"), we write time-quasiperiodic solutions for \eqref{linear}, corresponding
to a finite set of excited wave-vectors $ \mathcal{A} \subset \mathbb{Z} ^d $, as
\begin{equation}\label{sol}
u(t,x) = \sum_{a\in \mathcal{A} } (\xi_a e^{{\bf i}\lambda_a t}+ \eta_ae^{-{\bf i}\lambda_a t}) e^{{\bf i}\langle a, x\rangle }
+ \text{c.c.},
\end{equation}
where $\lambda_a = \sqrt{|a|^4+m}\,$. We examine these solutions and their perturbations in eq.~\eqref{beam}
under the assumption that the action-vector
$
I= \{\tfrac12( |\xi_a|^2 +|\eta_a|^2),\ a\in \mathcal{A} \}\
$
is small.
In our work this goal is achieved provided that
\noindent
- the finite set $ \mathcal{A} $ is typical in a probabilistic sense;
\noindent
- the mass parameter $m$ does not belong to a certain set of zero measure.
The linear stability of the obtained solutions for \eqref{beam} is under control. If $d\ge2$, and $| \mathcal{A} |\ge2$,
then some of them are linearly unstable.
The specific choice of a Hamiltonian PDE with the mass parameter which we work with -- the beam equation \eqref{beam} --
is sufficiently arbitrary. This is simply the easiest non-linear space-multidimensional equation from mathematical
physics for which we can perform our programme of the KAM-study of small-amplitude solutions in space-multidimensional
Hamiltonian PDEs, and obtain for them the results, outlines above.
Before to give exact statement of the result, we discuss the state of affairs in the KAM for PDE theory. The theory
started in late 1980's and originally applied to 1d Hamiltonian PDEs, see in \cite{K87, K93, Cr}. The first works
on this theory treated
a) perturbations of linear Hamiltonian PDE, depending on a vector-parameter of the dimension, equal to
the number of frequencies of the unperturbed quasiperiodic solution of the linear system (for solutions \eqref{sol} this is
$| \mathcal{A} |$). Next the theory was applied to
b) perturbations of integrable Hamiltonian PDE, e.g. of the KdV or Sine-Gordon equations, see \cite{K00}. In paper \cite{BoK}
c) small-amplitude solutions of the 1d Klein-Gordon equation \eqref{KG} with $G(x,u)=-u^4+O(u^4)$
were treated as perturbed solutions of the Sine-Gordon equation, and a singular version of the KAM-theory b) was developed to study them.
(Notice that for suitable $a$ and $b$ we have $mu-u^3+O(u^4) = a\sin bu+O(u^4)$. So the 1d equation \eqref{KG} is the Sine-Gordon equation, perturbed by a small term $O(u^4)$.)
It was proved in \cite{BoK} that for a.a. values of $m$ and for
any finite set $ \mathcal{A} $ most of the small-amplitude solutions \eqref{sol} for the
linear Klein-Gordon equation (with $\lambda_a=\sqrt{|a|^2+m}$) persist as linearly stable time-quasipe\-rio\-dic
solutions for \eqref{KG}. In \cite{KP} it was realised that it can be fruitful in 1d
equations like \eqref{KG}, just as it is in finite-dimensional Hamiltonian systems (see for example \cite{E88}), to study small solutions not as perturbations of solutions for an integrable PDE, but rather as perturbations of solutions
for a Birkhoff--integrable system, after the equation is normalised by a Birkhoff transformation. The paper \cite{KP} deals not with
1d Klein-Gordon
equation \eqref{KG}, but with 1d NLS equation, which is similar to \eqref{KG}
for the problem under discussion; in \cite{P} the method of \cite{KP} was applied to the 1d equation \eqref{KG}.
The approach of \cite{KP} turned out to be very efficient and later was used for many other 1d Hamiltonian PDEs.
In \cite{GY06b} it was applied to the $d$-dimensional beam equation \eqref{beam} with an $x$-independent
nonlinearity $g$ and allowed to treat perturbations of some special solutions \eqref{sol}.
Space-multidimensional KAM for PDE theory started 10 years later with the paper \cite{B1} and, next, publications
\cite{B2}
and \cite{EK10, EK09}. The just mentioned works deal with perturbations of
parameter-depending linear equations (cf. a)\,). The approach of
\cite{EK10, EK09} is different from that of \cite{B1, B2} and allows to analyse the linear stability of the obtained KAM-solutions.
Also see \cite{BB12, BB13}. Since integrable space-multidimensional PDE (practically) do not exist, then no
multi-dimensional analogy of the 1d theory b) is available.
Efforts to create space-multidimensional analogies of the KAM-theory c) were made in \cite{WM} and \cite{PP1, PP2}, using the
KAM-techniques of \cite{B1, B2} and \cite{EK10}, respectively. Both works deal with the NLS equation. Their main
disadvantage compare to the 1d theory c) is severe restrictions on the finite set $ \mathcal{A} $ (i.e. on the class of unperturbed solutions
which the methods allow to perturb).
The result of \cite{WM} gives examples
of some sets $ \mathcal{A} $ for which the KAM-persistence of the corresponding small-amplitude solutions \eqref{sol} holds,
while the result of \cite{PP1, PP2} applies to solutions \eqref{sol}, where the set $ \mathcal{A} $ is nondegenerate in certain very
non-explicit way. The corresponding
notion of non-degeneracy is so complicated that it is not easy to give examples of
non-degenerate sets $ \mathcal{A} $.
Some KAM-theorems for small-amplitude solutions of multidimensional beam equations \eqref{beam}
with typical $m$ were obtained in
\cite{GY06a, GY06b}. Both works treat equations with a constant-coefficient nonlinearity
$g(x,u)=g(u)$, which is significantly easier than the general case (cf. the linear theory, where constant-coefficient
equations may be integrated by the Fourier method). Similar to \cite{WM, PP1, PP2}, the theorems of \cite{GY06a, GY06b}
only allow to perturb solutions \eqref{sol} with very special sets $ \mathcal{A} $ (see also Appendix B). Solutions of \eqref{beam}, constructed in these works,
all are linearly stable.
\subsection{Beam equation in real and complex variables}\label{s_complex}
Introducing $v=u_t\equiv\dot u$ we rewrite
\eqref{beam} as \begin{equation}\label{beam'}
\left\{\begin{array}{ll}
\dot u &= -
v,\\
\dot v &=\Lambda^2 u +g(x,u)\,, \end{array}\right. \end{equation} where $\Lambda=(\Delta^2+m)^{1/2}$. Defining
$
\psi(t,x) =\frac 1{\sqrt 2}(\Lambda^{1/2}u +{\bf i}\Lambda^{-1/2}v) $
we get for the complex function
$\psi(t,x)$ the equation $$ \frac 1{\bf i}\dot \psi =\Lambda \psi+ \frac{1}{\sqrt 2}\Lambda^{-1/2}g\left(x,\Lambda^{-1/2}\left(\frac{\psi+ \bar\psi}{\sqrt 2}\right)\right)\,. $$ Thus, if we endow the space $L^2( \mathbb{T} ^d, \mathbb{C} )$ with the standard real symplectic structure, given by the two-form $\ -{\bf i}d\psi\wedge d\bar \psi, $
then equation
\eqref{beam} becomes a Hamiltonian system $$\dot \psi={\bf i} \,{\partial h} /{\partial \bar\psi} $$ with the Hamiltonian function $$ h(\psi,\bar\psi)=\int_{ \mathbb{T} ^d}(\Lambda \psi)\bar\psi \text{d} x +\int_{ \mathbb{T} ^d}G\left(x,\Lambda^{-1/2}\left(\frac{\psi+\bar\psi}{\sqrt 2}\right)\right) \text{d} x. $$ The linear operator $\Lambda$ is diagonal in the complex Fourier basis $$ \{e_a(x)= {(2\pi)^{-d/2}}e^{{\bf i}\langle a, x\rangle }, \ a\in \mathbb{Z} ^d\}. $$ Namely, $$
\Lambda e_a=\lambda_a e_a,\;\;\lambda_a= \sqrt{|a|^4+m}, \qquad \forall\,a\in \mathbb{Z} ^d\,. $$
Let us decompose $\psi$ and $\bar\psi$ in the basis $\{e_a\}$: $$ \psi=\sum_{a\in \mathbb{Z} ^d}\xi_a e_a,\quad \bar\psi=\sum_{a\in \mathbb{Z} ^d}\eta_a e_{-a}\,. $$ Let \begin{equation}\label{change} \left\{\begin{array}{l} p_a=\frac1{\sqrt2}(\xi_a+\eta_a) \\ q_a=\frac{{\bf i}}{\sqrt2}(\xi_a-\eta_a) \end{array}\right.\end{equation} and denote by $\zeta_a$ the pair $(p_a,q_a)$. \footnote{\ $\zeta_a$ will be considered as a line-vector or a colon-vector according to the context.}
We fix any $m_*>d/2$ and define the Hilbert space \begin{equation}\label{YC} Y = \{\zeta=(p,q)\in \ell^2( \mathbb{Z} ^d, \mathbb{C} )\times\ell^2( \mathbb{Z} ^d, \mathbb{C} ) \mid
\aa{\zeta}^2=\sum_a \langle a\rangle^{2m_*} |\zeta_a|^2 <\infty \}\,, \end{equation}
-- $\langle a\rangle =\max(1, |a|)$ -- corresponding to the decay of Fourier coefficients of complex functions $(\psi(x), \bar\psi(x))$ from the Sobolev space
$ H^{m_*}( \mathbb{T} ^d, \mathbb{C} ^2)$. A vector $\zeta\in Y$ is called {\it real} if all its components are real.
Let us endow $Y$ with the symplectic structure \begin{equation}\label{J} \big(dp \wedge dq\big) (\zeta,\zeta')= \sum_a \langle J\zeta_a,\zeta'_a\rangle,\quad J=\left(\begin{array}{cc} 0&1\\-1&0\end{array}\right)\,, \end{equation}
and consider there the Hamiltonian system \begin{equation} \label{beam2} \dot\zeta_a=J\frac{\partial h}{\partial \zeta_a},\quad a\in \mathbb{Z} ^d\,,
\end{equation} where the Hamiltonian function $h$ equals the quadratic part \begin{equation}\label{H2} h_2=\frac12 \sum_{a\in \mathbb{Z} ^d}\lambda_a (p^2_a+q^2_a)\end{equation} plus the higher order term \begin{equation}\label{H1} h_{\ge4}= \int_{ \mathbb{T} ^d}G\left(x,\sum_{a\in \mathbb{Z} ^d}\frac{(p_a-{\bf i}q_a) e_a+(p_{-a}+{\bf i}q_{-a}) e_a}{2\sqrt{\lambda_a}}\right) \text{d} x.\end{equation} The beam equation \eqref{beam'}, considered in the Sobolev space $\{(u,v) \mid(\psi, \bar\psi) \in H^{m_*}\}$, is
equivalent to the Hamiltonian system \eqref{beam2}.
We will write the Hamiltonian $h$ as \begin{equation}\label{PPP} h = h_2+h_{\ge4} = h_2+h_4+h_{\ge5}\,, \end{equation} where
\begin{equation}\label{quatr}
h_4=\int_{ \mathbb{T} ^d}u^4 \text{d} x= \int_{ \mathbb{T} ^d}\left(\sum_{a\in \mathbb{Z} ^d}\frac{(p_a-{\bf i}q_a) e_a+(p_{-a}+{\bf i}q_{-a}) e_a}{2\sqrt{ \lambda_a}}\right)^4 \text{d} x,
\end{equation}
$h_{\ge5} = O(u^5)$ comprise the remaining higher order terms and $ h_{\ge4} = h_4+h_{\ge5}$.
Note that $h_4$
satisfies the {\it zero momentum condition}, i.e.
$$
h_4=\sum_{a,b,c,d\in \mathbb{Z} ^d}C(a,b,c,d) (\xi_a+\eta_{-a}) ( \xi_b+\eta_{-b}) (\xi_c+\eta_{-c}) (\xi_d+\eta_{-d})\,,
$$
where $C(a,b,c,d)\ne0$ only if $a+b+c+d=0$. This condition turns out to be useful to restrict the set of small divisors that have to be controlled. If the function $G$ does not depend on $x$, then $h$ satisfies a similar property at any order.
\subsection{Invariant tori and admissible sets}
The quadratic Hamiltonian $h_2$ (which is $h$ when $G=0$ in \eqref{beam}) is integrable and its phase-space is foliated into (Lagrangian or isotropic) invariant tori. Indeed, take a finite subset $ \mathcal{A} \subset \mathbb{Z} ^d$ and let
$$ \mathcal{L} = \mathbb{Z} ^d\setminus \mathcal{A} \,.$$
For any subset $X$ of $ \mathbb{Z} ^d$, consider the projection $$\pi_X:( \mathbb{C} ^2)^{ \mathbb{Z} ^d}\to ( \mathbb{C} ^2)^{X}=\{\zeta\in ( \mathbb{C} ^2)^{ \mathbb{Z} ^d}: \zeta_a=0\ \forall a\notin X\}.$$ We can thus write $( \mathbb{C} ^2)^{ \mathbb{Z} ^d}=( \mathbb{C} ^2)^{X}\oplus ( \mathbb{C} ^2)^{ \mathbb{Z} ^d\setminus X}$, $\zeta=(\zeta_X,\zeta_{ \mathbb{Z} ^d\setminus X})$, and when $X$ is finite this gives an injection $$ ( \mathbb{C} ^2)^{\#X}\hookrightarrow ( \mathbb{C} ^2)^{ \mathbb{Z} ^d}$$ whose image is $ ( \mathbb{C} ^2)^{X}$. \footnote{\ we shall frequently, without saying, identify $ ( \mathbb{C} ^2)^{X}$ and $( \mathbb{C} ^2)^{\#X}$}
For any real vector with positive components $I_ \mathcal{A} =(I_a)_{a\in \mathcal{A} }$, the $| \mathcal{A} |$-dimensional torus \begin{equation}\label{ttorus} T_{I_ \mathcal{A} }=
\left\{\begin{array}{lll}
p_a^2+q_a^2 =2I_a& p_a, q_a\in \mathbb{R},\; &a\in \mathcal{A} \\ p_a=q_a=0& & a\in \mathcal{L} \,, \end{array}\right. \end{equation} is invariant under the flow of $h_2$. $T_{I_ \mathcal{A} }$ is the image of the torus \begin{equation}\label{ttorusbis} \mathbb{T} ^ \mathcal{A} =\{r_ \mathcal{A} =0\}\times \{\theta_a\in \mathbb{T} : a\in \mathcal{A} \}\times\{\zeta_ \mathcal{L} =0\}\end{equation}
under the embedding \begin{equation}\label{embedding} U_{I_ \mathcal{A} }:\theta_ \mathcal{A} \mapsto
\left\{\begin{array}{ll} p_a-{\bf i}q_a=\sqrt{2I_a} \, e^{{\bf i} \theta_a} &a\in \mathcal{A} \\ p_a=q_a=0 & a\in \mathcal{L} \,, \end{array}\right. \end{equation} and the pull-back, by $U_{I_ \mathcal{A} }$, of the induced flow is simply the translation \begin{equation}\label{inducedflow} \theta_ \mathcal{A} \mapsto \theta_ \mathcal{A} +t \omega _ \mathcal{A} ,\end{equation} where we have denoted the translation vector (the tangential frequencies) by $ \omega _ \mathcal{A} $, i.e. $\lambda_a= \omega _a$ for $a\in \mathcal{A} $. The parametrised curve $$
t \mapsto U_{I_ \mathcal{A} }(\theta +t\omega)$$ is thus a quasi-periodic solution of the beam equation \eqref{beam2} when $G=0$.
When $G\not=0$ the higher order terms in $h$ give rise to a perturbation of $h_2$ -- a perturbation that gets smaller, the smaller is $I$. Our goal is to prove the persistency of the invariant torus $T^{ \mathcal{A} }_I$, or, more precisely, of the invariant embedding $ U_I^ \mathcal{A} $, for most values of $I$ when the higher order terms are taken into account. The problem doing this for this model is two-fold. First the integrable Hamiltonian $h_2$ is completely degenerate in the sense of KAM-theory: the frequencies $ \omega _ \mathcal{A} $ do not depend on $I$. One can try to improve this by adding to $h_2$ an integrable part of the Birkhoff normal form. This will, in ``generic'' situations, correct this default. However, and that's the second problem, our model is far from ``generic'' since the eigenvalues $\{ \lambda_a : a\in \mathbb{Z} ^d\}$ are very resonant. This has the effect that the Birkhoff normal form is not integrable, and therefore is difficult to use.
An important part of our analysis will be to show that this program can be carried out if we exclude a zero-measure set of masses $m$ and restrict the choice of $ \mathcal{A} $ to {\it admissible} or {\it strongly admissible} sets.
Let $|\cdot|$ denote the euclidean norm in $ \mathbb{R}^d$. For vectors $a,b \in \mathbb{Z} ^d$ we define \begin{equation}\label{ddd}
a \, \angle\, b \quad \text{iff} \quad \#\{x\in \mathbb{Z} ^d \mid |x|=|a| \;\text{and}\;
|x-b| = |a-b|\} \le 2\,.\end{equation} Relation $a \, \angle\, b$ means that
the integer sphere of radius $|b-a|$ with the centre at $b$ intersects the integer sphere
$\{x\in \mathbb{Z} ^d\mid |x|=|a|\}$ in at most two points.
\begin{definition}\label{adm} A finite set $ \mathcal{A} \in \mathbb{Z} ^d$ is called admissible iff $$
a,b\in \mathcal{A} , \ a\ne b
\Rightarrow |a|\neq |b|\,.$$
An admissible set $ \mathcal{A} $ is called strongly admissible iff $$
a,b\in \mathcal{A} , \ a\ne b \Rightarrow a \, \angle\, a+ b\,. $$
\end{definition}
Certainly if $| \mathcal{A} |\le1$, then $ \mathcal{A} $ is admissible, but for $| \mathcal{A} | >1$ this is not true. For $d\le2$ every admissible set is strongly admissible, but in higher dimension this is no longer true: see for example the set \eqref{AAA} in Appendix B.
However, strongly admissible, and hence admissible sets are typical: see Appendix E for a precise formulation and proof of this statement.
We shall define a subset of $ \mathcal{L} $, important for our construction: \begin{equation}\label{L+}
{ \mathcal{L} _f}=\{a\in \mathcal{L} \mid \exists\ b\in \mathcal{A} \text{ such that } |a|=|b| \}. \end{equation} Clearly $ \mathcal{L} _f$ is a finite subset of $ \mathcal{L} $. For example, if $d=1$ and $ \mathcal{A} $ is admissible, then $ \mathcal{A} \cap- \mathcal{A} \subset\{0\}$, so if $d=1$, then ${ \mathcal{L} _f} = -( \mathcal{A} \setminus\{0\})$.
\subsection{The Birkhoff normal form}
In a neighbourhood of an invariant torus $T_{I_ \mathcal{A} }$
we introduce (partial) action-angle variables $(r_ \mathcal{A} ,\theta_ \mathcal{A} ,\xi_ \mathcal{L} ,\eta_ \mathcal{L} )$ by the relation \begin{equation}\label{actionangle}
\frac1{\sqrt2}(p_a-{\bf i}q_a)=\sqrt{I_a+r_a} \, e^{{\bf i} \theta_a},\quad a\in \mathcal{A} . \end{equation}
These variables define a diffeomorphism from a neighbourhood of $ \mathbb{T} ^ \mathcal{A} $ in (the Hilbert manifold) \begin{equation}\label{YCmanifold} \mathbb{C} ^{ \mathcal{A} }\times( \mathbb{C} /2\pi \mathbb{Z} )^{ \mathcal{A} }\times \pi_{ \mathcal{L} }Y\end{equation} to a neighbourhood of $T_{I_ \mathcal{A} }$ in $Y$. It is {\it real} in the sense that it gives real values to real arguments.
The symplectic structure on $Y$ is pull-backed to \begin{equation}\label{Jbis} dr_ \mathcal{A} \wedge d\theta_ \mathcal{A} +d\xi_ \mathcal{L} \wedge d\eta_ \mathcal{L} ,\end{equation} which endows the space \eqref{YCmanifold} with a symplectic structure.
In these variables $h$ will depend on $I$, but its integrable part $h_2$ becomes, up to an additive constant, $$ \sum_{a\in \mathcal{A} } \omega _a r_a+ \frac12\sum_{a\in \mathcal{L} }\lambda_a (p_a^2+q_a^2)$$ which does not depend in $I$. \footnote{\ both $h_2$ and the higher order terms of $h$ depend on the mass $m$.}
The Birkhoff normal form will provide us with an integrable part that does depend on $I$. We shall prove
\begin{theorem}\label{NFTl}
There exists a zero-measure Borel set $ \mathcal{C} \subset[1,2]$ such that for any $m\notin \mathcal{C} $, any
admissible set $ \mathcal{A} $, any $c_*\in(0,1/2]$ and any analytic nonlinearity of the form \eqref{g},
there exist $\nu_0>0$ and $\beta_0>0$ such that for any $0<\nu\le\nu_0$, $0<\beta_{\#}\le\beta_0$ there exists an open set
$Q \subset [\nu c_*,\nu]^ \mathcal{A} $,
$$\operatorname{meas}([\nu c_*,\nu]^ \mathcal{A} \setminus Q)\le C\nu^{\# \mathcal{A} +\beta_{\#}},$$
and for every $I=I_ \mathcal{A} \in Q$ there exists a real symplectic holomorphic
diffeomorphism $\Phi_{I} $, defined in a neighbourhood (that depends on $c_*$ and $\nu$) of $ \mathbb{T} ^ \mathcal{A} $
such that
\begin{equation}\label{transff} \begin{split} & h\circ\Phi_I(r_ \mathcal{A} , \theta_ \mathcal{A} , p_ \mathcal{L} , q_ \mathcal{L} )= \langle\Omega(I), r_ \mathcal{A} \rangle +\frac12\sum_{a\in \mathcal{L} \setminus \mathcal{L} _f } \Lambda_a(I) (p_a^2+q_a^2)+\\ &+ \frac12\sum_{b\in \mathcal{L} _f\setminus \mathcal{F} } \Lambda_b(I) (p_b^2+q_b^2)
+ \langle K(I)\zeta_ \mathcal{F} ,\zeta_ \mathcal{F} \rangle + f_I(r_ \mathcal{A} ,\theta_ \mathcal{A} , p_ \mathcal{L} ,q_ \mathcal{L} )\,, \end{split} \end{equation} where $ \mathcal{F} = \mathcal{F} _I$ is a (possibly empty) subset of $ \mathcal{L} _f$, has the following properties:
i) $\Omega(I)= \omega _ \mathcal{A} + M I$ and the matrix $M$ is invertible;
ii) each $ \Lambda_a (I)$, $a\in \mathcal{L} \setminus \mathcal{L} _f$, is real and close to $ \lambda_a $, $$\ab{ \Lambda_a (I)- \lambda_a }\le C\ab{I} \langle a\rangle^{-2};$$
iii) each $ \Lambda_b (I)$, $b\in \mathcal{L} _f\setminus \mathcal{F} $, is real and non-zero,
$$C^{-1} \ab{I}^{1+ c\beta_{\#}}\le | \Lambda_b (I) |\le C\ab{I}^{1- c\beta_{\#}};$$
iv) the operator $K(I)$ is real symmetric and satisfies $\| K(I)\|\le C \ab{I}^{1-c\beta_{\#}}$. The Hamiltonian operator $JK(I)$ is hyperbolic (unless $ \mathcal{F} _I$ is empty), and the moduli of the real parts of its eigenvalues are bigger than $C^{-1} \ab{I}^{1+\beta_{\#}}$.
v) The function $f_I$ is much smaller than the quadratic part.
Moreover, all objects depend $C^\infty$ on $I$.
\end{theorem}
This result is proven in Part II. For a more precise formulation, giving in particular the domain of definition of $\Phi_I$, the smallness in $f_I$ and estimates of the derivatives with respect to $I$, see Theorem~\ref{NFT}. The matrix $M$ is explicitly defined in \eqref{Om}, and
the functions $ \Lambda_a $ are explicitly defined in \eqref{Lam}. An interesting information is that the mapping $\Phi_I$ and the domain $Q$
only depend on $h_2+h_4$, and that the set $ \mathcal{F} _I$ is empty on some connected components of $Q$.
\subsection{The KAM theorem}
The Hamiltonian $h_I\circ\Phi_I$ \eqref{transff} is much better than $h_I$ since its integrable part depends on $I$ in a non-degenerate way because $M$ is invertible. Does the invariant torus \eqref{ttorusbis} persist under the perturbation $f_I$? \dots and, if so, is the persisted torus reducible?
In finite dimension the answer is yes under very general conditions -- for the first proof in the purely elliptic case see \cite{E88}, and for a more general case see \cite{GY99}. These statements say that, under general conditions, the invariant torus persists and remains reducible under sufficiently small perturbations for a subset of parameters of large Lebesgue measure.
In infinite dimension the situation is more delicate, and results can only be proven under quite severe restrictions on the normal frequencies $\Lambda_a$; see the discussion above in Section~\ref{s_1.1}.
A result for the beam equation (which is a simpler model than the Schr\"odinger and wave equations) was first obtained in \cite{GY06a} and \cite{GY06b}. Here we prove a KAM-theorem which improves on these results
in at least two respects: \begin{itemize} \item We have imposed no ``conservation of momentum'' on the perturbation, which allows us to treat equations \eqref{beam} with
$x$-dependent nonlinearities $g$. This has the effect that our normal form is not diagonal in the purely elliptic directions. In this respect it resembles the normal form obtained in \cite{EK10} for the non-linear Schr\"odinger equation, and where the block diagonal form is the same. \item We have a finite-dimensional, possibly hyperbolic, component, whose treatment requires higher smoothness in the parameters. \end{itemize}
The proof has the structure of a classical KAM-theorem carried out in a complex infinite-dimensional situation. The main part is, as usual, the solution of the homological equation with reasonable estimates. The fact that the block structure is not diagonal complicates a lot: see for example, \cite{EK10} where this difficulty was also encountered. The iteration combines a finite linear iteration with a ``super-quadratic'' infinite iteration. This has become quite common in KAM and was also used in \cite{EK10}.
A technical difference, with respect to \cite{EK10}, is that here we use a different matrix norm which has much better multiplicative properties. This simplifies a lot the functional analysis which is described in Part I.
A special difficulty in our setting is that we are facing a {\it singular perturbation problem}. The perturbation $f_I$ becomes small only by taking $I$ small, but when $I$ gets smaller the integrable part becomes more degenerate. This is seen for example in the lower bounds for $ \Lambda_b (I)$ and for the real parts of the eigenvalues of $JK(I)$. So there is a competition between the smallness condition on the perturbation and the degeneracies of the integrable part which requires quite careful estimates.
A KAM-theorem which is adapted to our beam equation is proven in Part III and formulated in Theorem~\ref{main} and its Corollary~\ref{cMain-bis}.
\subsection{Small amplitude solutions for the beam equation}
Applying to the normal form of Part II, the KAM theorem of Part III, we in Part IV obtain the
main results of this work. To state them we recall that a Borel subset ${\mathfrak J}\subset \mathbb{R}^{ \mathcal{A} }_+$ is said to have
a {\it positive density} at the origin if
\begin{equation}\label{posdens}
\liminf_{\nu\to0}\frac{\operatorname{meas}( \mathfrak J \cap \{x\in \mathbb{R}^{ \mathcal{A} }_+ \ab{x}<\nu\})}{\operatorname{meas}\{x\in \mathbb{R}^{ \mathcal{A} }_+\ab{x}<\nu\}} >0\,.
\end{equation}
The set $ \mathfrak J $ has the {\it density one} at the origin if the $ \liminf$ above equals one (so the ratio of the measures
of the two sets converges to one as $\nu\to0$).
\begin{theorem}\label{t72} There exists a zero-measure
Borel set $ \mathcal{C} \subset[1,2]$ such that for any strongly admissible\ set $ \mathcal{A} \subset \mathbb{Z} ^d $, any $m\notin \mathcal{C} $ and any analytic
nonlinearity \eqref{g}, there exist constants $\aleph_1\in (0,1/16],
\aleph_2>0$, only depending on $ \mathcal{A} $ and $m$, and
a set $ \mathfrak J = \mathfrak J _ \mathcal{A} \subset ]0,1]^{ \mathcal{A} }$, having density one at the origin, with the following property:
\noindent
There exist a constant $C>0$, a real continuous mapping
$\ U'=U'_ \mathcal{A} : \mathbb{T} ^{ \mathcal{A} }\times \mathfrak J \to Y,
$
analytic in the first argument, satisfying
\begin{equation}\label{dist1}
\big|\big| U'(\theta,I)-U_{I}(\theta)
\big|\big|
\le C |I|^{1 -\aleph_1}\
\end{equation}
(see \eqref{embedding})
for all $(\theta,I)\in \mathbb{T} ^ \mathcal{A} \times \mathfrak J $, and a continuous mapping
$ \Omega'= \Omega'_ \mathcal{A} : \mathfrak J \to \mathbb{R}^{ \mathcal{A} }$,
\begin{equation}\label{dist11}
| \Omega'(I) - \omega _ \mathcal{A} - MI| \leq C |I|^{1+ \aleph_2}\,, \end{equation} where the matrix $M$ is the same as in \eqref{transff}, such that:
i) for any $I\in \mathfrak J $ and $\theta\in \mathbb{T} ^{ \mathcal{A} }$ the parametrised curve \begin{equation}\label{solution}
t \mapsto U'(\theta +t \Omega'(I),I) \end{equation} is a solution of the beam equation \eqref{beam2}-\eqref{H1}, and,
accordingly, the analytic torus $U'( \mathbb{T} ^{ \mathcal{A} },I)$ is invariant for this equation;
ii) the set $ \mathfrak J $ may be written as a countable disjoint union of compact sets $ \mathfrak J _j$,
such that the restrictions of the mappings $U'$ and $ \Omega'$ to the sets $ \mathbb{T} ^{ \mathcal{A} }\times \mathfrak J _j$ are $C^1$ Whitney -smooth;
iii) the solution \eqref{solution} is linearly stable if and only if in \eqref{transff} the operator $K(I)$ is
trivial (i.e. the set $ \mathcal{F} = \mathcal{F} _I$ is non-empty).
The set of $I\in \mathfrak J $ such that $K(I)$ is trivial is always of positive measure, and it equals $ \mathfrak J $ if
$d=1$ or $| \mathcal{A} |=1$, but for $d\ge2$ and for some choices of the set $ \mathcal{A} $ its complement
has positive measure.
\end{theorem}
If the set $ \mathcal{A} $ is admissible but not strongly admissible,
then a weaker version of the theorem above is true.
\begin{theorem}\label{t73}
There exists a zero-measure
Borel set $ \mathcal{C} \subset[1,2]$ such that for any admissible set $ \mathcal{A} \subset \mathbb{Z} ^d $, any $m\notin \mathcal{C} $ and any analytic
nonlinearity \eqref{g}, there exist constants $\aleph_1\in (0,1/16], \aleph_2>0$, only
depending on $ \mathcal{A} $ and $m$, and
a set $ \mathfrak J = \mathfrak J _ \mathcal{A} \subset ]0,1]^{ \mathcal{A} }$, having positive density at the origin, such that all assertions of Theorem~\ref{t72} are true.
\end{theorem}
\begin{remark} \label{r_1}
1) The torus $U_{I}( \mathbb{T} ^ \mathcal{A} ,I)$ (see \eqref{ttorus}),
invariant for the linear beam equation \eqref{beam2}${}_{G=0}$,
is of size $\sim\sqrt I$. The constructed invariant torus $U'_ \mathcal{A} ( \mathbb{T} ^{ \mathcal{A} },I)$
of the nonlinear beam equation is its small perturbation
since by \eqref{dist1} the Hausdorff distance between $U'_{ \mathcal{A} }( \mathbb{T} ^{ \mathcal{A} },I)$ and
$U_{I}( \mathbb{T} ^ \mathcal{A} )$ is smaller than $C |I|^{1 -2\aleph_1}\le C |I|^{7/8}$.
2) Denote by $ \mathcal{T} _ \mathcal{A} $ the image of the mapping $U'_ \mathcal{A} $. This set is invariant for the beam equation
and is filled in with its time-quasiperiodic solutions. By the item~ii) of Theorem~\ref{t72} its Hausdorff
dimension equals $2| \mathcal{A} |$. Now consider $ \mathcal{T} = \cup \mathcal{T} _ \mathcal{A} $, where the onion is taken over all
strongly admissible sets $ \mathcal{A} \subset \mathbb{Z} ^d$. This invariant set has infinite Hausdorff dimension. Some
time-quasiperiodic solutions of \eqref{beam}, lying on $ \mathcal{T} $, are linearly stable, while, if $d\ge2$, then
some others are unstable.
3) Our result applies to eq. \eqref{beam} with any $d$. Notice that for $d$ sufficiently large the global in time
well-posedness of this equation is unknown.
4) The construction of solutions \eqref{solution} crucially depends on certain equivalence relation in
$ \mathbb{Z} ^d$, defined in terms of the set $ \mathcal{A} $ (see \eqref{class}).
This equivalence is trivial if $d=1$ or $| \mathcal{A} |=1$ and is non-trivial otherwise.
5) We discuss in
Appendix~B examples of sets $ \mathcal{A} $ for which the operator $K(I)$ is non-trivial for certain values of $I$.
6) The solutions \eqref{solution} of eq. \eqref{beam2}, written in terms of the $u(x)$-variable as solutions $u(t,x)$ of eq.~\eqref{beam},
are $H^{m_*+1}$-smooth as functions of $x$ and analytic as functions of $t$. Here $m_*$ is
a parameter of the construction for which we can take any real number $>d/2$ (see \eqref{YC}). The set $ \mathfrak J $ depends on
$m_*$, so the assertion of the theorem does not imply immediately that the solutions $u(t,x)$
are $C^\infty$--smooth in $x$.
Still, since
$$
-(\Delta^2+m) u = u_{tt}+\partial_u G(x,u),
$$
where $G$ is an analytic function, then the theorems
imply by induction that the solutions $u(t,x)$ define analytic curves
$ \mathbb{R}\to H^m( \mathbb{T} ^d)$, for any $m$. In particular, they are smooth functions.
\end{remark}
\noindent {\bf Structure of text}
The paper consists of Introduction and four parts. Part~I comprises general techniques needed to read the paper. The main Parts~II-III
are independent of each other, and the final Part~IV, containing the proofs of Theorems~\ref{t72},~\ref{t73},
uses only the main theorems of Parts~II-III, and the intermediate results
are not needed to understand it.
\noindent {\bf Some notation and agreements.}
We denote a cardinality of a set $X$ as $|X|$ or as $\,\# X$. For $a\in \mathbb{Z} ^N$ we denote $\langle a\rangle =\max(1, |a|)$.
In any finite-dimensional space $X$ we denote by
$|\cdot|$ the Euclidean norm. For subsets $X$ and $Y$ of a Euclidean space we denote $$
\underline{\text{dist}}\,(X,Y) = \inf_{x\in X, y\in Y} |x-y|\,,\qquad \text{diam}\,(X) = \sup_{x,y\in X}|x-y|\,.$$ The distance on a torus induced by the Euclidean distance (on the tangent space) will be denoted
$|\cdot - \cdot|$.
For any matrix $A$, finite or infinite, we denote by ${}^t\!A$ the transposed matrix.
$I$ stands for the identity matrix of any dimension.
The space of bounded linear operators between Banach spaces $X$ and
$Y$ is denoted $ \mathcal{B} (X,Y)$. Its operator norm will be usually denoted $\|\cdot\|$ without specification the
spaces. If $A$ is a finite matrix, then $\|A\|$ stands for its operator-norm.
We call analytic mappings between domains in complex Banach spaces {\it holomorphic} to reserve the name {\it analytic} for mappings between domains in real Banach spaces. This definition extends from Banach spaces to Banach manifolds.
\noindent{\it Pairings in $l^2$-spaces}. The scalar product on any complex Hilbert space is, by convention, complex anti-linear in the first variable and complex linear in the second variable. For any $l^2$-space $X$ of finite or infinite dimension,
the natural complex-bilinear pairing is denoted \begin{equation}\label{pairing}\langle \zeta,\zeta'\rangle=\langle \bar \zeta,\zeta'\rangle_{l^2},\qquad \zeta,\zeta'\in X.\end{equation} This is a symmetric complex-bilinear mapping.
\noindent{\it Constants}. The €numbers $d$ (the space-dimension) and $\# \mathcal{A} $, as well as $s_*,m_*$ and $\# \mathcal{P} , \# \mathcal{F} $ (that will occur in Part II) will be fixed in this paper. Constants depending only on the numbers and on the choice of finite-dimensional norms are regarded as absolute constants. An absolute constant only depending on $x$ is thus a constant that, besides these factors, only depends on $x$. Arbitrary constants will often be denoted by $Ct., ct.$ and, when they occur as an exponent, by $exp$. Their values may change from line to line. For example we allow ourselves to write $2Ct. \le Ct.$.
\noindent {\bf Acknowledgments.} We acknowledge the support from Agence Nationale de la Recherche through the grant
ANR-10-BLAN~0102.
The third author wishes to thank P.~Milman and V.~\v{S}ver\'ak for
helpful discussions.
\begin{samepage} \centerline{PART I. SOME FUNCTIONAL ANALYSIS}
\section{Matrix algebras and function spaces.} \end{samepage}
\subsection{The phase space}\label{sThePhaseSpace} Let $ \mathcal{A} $ and $ \mathcal{F} $ be two finite sets in $ \mathbb{Z} ^{d}$ and let $ \mathcal{L} _\infty$ be an infinite subset of $ \mathbb{Z} ^{d}$. Let $ \mathcal{L} $ be the disjoint union $ \mathcal{F} \sqcup \mathcal{L} _{\infty}$. \footnote{\ this is a more general setting than in the introduction, where $ \mathcal{L} $ and $ \mathcal{A} $ were two disjoint subsets of $ \mathbb{Z} ^d$} Let $ \mathcal{Z} $ be the disjoint union $ \mathcal{A} \sqcup \mathcal{F} \sqcup \mathcal{L} _{\infty}$ and consider $( \mathbb{C} ^2)^{ \mathcal{Z} }$.
For any subset $X$ of $ \mathcal{Z} $, consider the projection $$\pi_X:( \mathbb{C} ^2)^{ \mathcal{Z} }\to ( \mathbb{C} ^2)^{X}=\{\zeta\in ( \mathbb{C} ^2)^{ \mathcal{Z} }: \zeta_a=0\ \forall a\notin X\}.$$ We can thus write $( \mathbb{C} ^2)^{ \mathcal{Z} }=( \mathbb{C} ^2)^{X}\times ( \mathbb{C} ^2)^{ \mathcal{Z} \setminus X}$, $\zeta=(\zeta_X,\zeta_{ \mathcal{L} \setminus X})$, and when $X$ is finite this gives an injection $$ ( \mathbb{C} ^2)^{\#X}\hookrightarrow ( \mathbb{C} ^2)^{ \mathcal{Z} }$$ whose image is $ ( \mathbb{C} ^2)^{X}$.
In $ \mathbb{R}^2$ we consider the partial ordering $(\gamma _1',\gamma _2')\le (\gamma _1,\gamma _2)$ if, and only if $\gamma _1'\le\gamma _1$ and $\gamma _1'\le \gamma _2'$.
Let $\gamma =(\gamma _1,\gamma _2)\in \mathbb{R}^2$ and let $Y_\gamma $ be the Hilbert space of sequences $\zeta\in ( \mathbb{C} ^2)^{ \mathcal{Z} } $ such that \begin{equation}\label{Ygamma}
||\zeta||_{\gamma }^2=
{ \sum_{a\in \mathcal{Z} } |\zeta_a|^2e^{2\gamma _1|a|}\langle a\rangle^{2 \gamma _2} }<\infty\,, \end{equation} provided with the scalar product
\footnote{\ complex linear in the second variable and complex anti-linear in the first} $$ \langle \zeta, \zeta' \rangle_{\gamma } =
\sum_{a\in \mathcal{Z} } \langle \zeta_a,\zeta_a'\rangle_{ \mathbb{C} ^2} e^{2\gamma _1|a|}\langle a\rangle^{2 \gamma _2}.$$ \noindent If $\gamma _1\ge0$ and $\gamma _2>d/2$, then this space is an algebra with respect to the convolution. If $\gamma _1=0$, this is a classical property of Sobolev spaces. For the case $\gamma _1>0$ see \cite{EK08}, Lemma~1.1. (The space $Y_{(0,m_*)}$ coincides with the space $Y$, defined in \eqref{YC}, while $Y_{(0,0)}$ is the $l^2$-space of complex sequences $( \mathbb{C} ^2)^ \mathcal{Z} $.)
\begin{example}\label{analyt} Let $ \mathcal{A} = \mathcal{F} =\emptyset$, $ \mathcal{L} _\infty= \mathbb{Z} ^d$ and $\varrho>0$. Then any vector $\hat f=(\hat f_a, a\in \mathbb{Z} ^d)\in Y_\varrho$ defines a holomorphic vector-function $f(y) = \sum \hat f_ae^{{\bf i}\langle a, y\rangle}$ on the $\varrho$-vicinity $ \mathbb{T} ^n_\varrho$ of the torus $ \mathbb{T} ^n$, $ \mathbb{T} ^n_\varrho=\{y \in \mathbb{C} ^n/2\pi \mathbb{Z} ^n\mid |\Im y|<\sigma\}$, where its norm is bounded by $C_d\|\hat f\|_\varrho$. Conversely, if $f: \mathbb{T} ^n_\varrho \to \mathbb{C} ^2$ is a bounded holomorphic function, then its Fourier coefficients satisfy $|\hat f_a|\le\,$Const$\, e^{-|a|\varrho}$, so $\hat f\in Y_{\varrho'}$ for any $\varrho'<\varrho$. \end{example}
Write $\zeta_a=(p_a,q_a)$ and let $$ \Omega= \sum_{a\in \mathcal{Z} } dp_a\wedge dq_a.$$ $ \Omega$ is an anti-symmetric bi-linear form which is continuous on $$Y_\gamma \times Y_{-\gamma }\cup Y_{-\gamma }\times Y_{\gamma }\to \mathbb{C} $$ with norm $\aa{ \Omega}= 1$. The subspaces $( \mathbb{C} ^2)^{\{a\}}$ are symplectic subspaces of two (complex) dimensions carrying the canonical symplectic structure.
$ \Omega$ defines (by contraction on the second factor ) a bounded bijective operator $$Y_\gamma \ni \zeta\mapsto \Omega(\cdot,\zeta) \in Y^*_{-\gamma }$$ where $Y^*_{-\gamma }$ denotes the Banach space dual of $Y_{-\gamma }$. (Notice that $\zeta'\mapsto \Omega(\zeta', \zeta)$ is a well-defined bounded linear form on $Y_{-\gamma }\,.$) We shall denote its inverse by $${J_ \Omega}: Y^*_{-\gamma }\to Y_\gamma .$$
We shall also let $J_ \Omega$ act on operators $$J_ \Omega: \mathcal{B} (X, Y^*_{-\gamma })\to \mathcal{B} (X,Y_\gamma )$$ through $(J_ \Omega H)(x)=J_ \Omega (H(x))$ for any bounded operator $H:X\to Y^*_{-\gamma }$.
\begin{remark}
The complex-bilinear pairing \eqref{pairing} on the $l^2$-space $Y_{(0,0)}$
extends to a continuous mapping $Y_\gamma \times Y_{-\gamma } \to \mathbb{C} \,$
which allows to identify $Y_{-\gamma } $ with the dual space $Y_{\gamma }^* $.
Then \begin{equation}\label{oega} \Omega(\zeta, \zeta') = \langle J \zeta, \zeta'\rangle\,, \end{equation} where $J$ here stands for the linear operator $\zeta \mapsto J\zeta$ defined by $$ (J\zeta)_a = J\zeta_a\;\ \; \; \forall a\in \mathcal{Z} ,$$ where the $2\times2$-matrix $J$ (in the right hand side) is defined in \eqref{J}. \footnote{\ sorry for the abuse of notation} Then we have \begin{equation}\label{equal} J_ \Omega \zeta = J\zeta \qquad \forall\, \zeta\in Y^*_{-\gamma }\,, \end{equation} where $\zeta$ in the r.h.s. is regarded as a vector in $Y_\gamma $, and we shall frequently denote the operator $J_ \Omega$ by $J$. (It will be clear from the context which of the two operators $J$ denotes.) \end{remark}
A bijective bounded operator $A:Y_\gamma \to Y_\gamma $, $\gamma \ge(0,0)$, is {\it symplectic} if, and only if, $$ \Omega(A\zeta, A\zeta') = \Omega(\zeta, \zeta')\qquad \forall \ \zeta,\zeta'\in Y_\gamma \,. $$ Writing $ \Omega$ in the form \eqref{oega} we see that $A$ is symplectic if and only if ${}^t AJA=J$. Here ${}^t A$ stands for the operator, symmetric to $A$ with respect to the pairing $\langle \cdot,\cdot\rangle$ (its matrix is transposed to that of $A$).
Let $$ \mathbb{A} ^{ \mathcal{A} } = \mathbb{C} ^{ \mathcal{A} }\times( \mathbb{C} /2\pi \mathbb{Z} )^{ \mathcal{A} }$$ and consider the Hilbert manifold $ \mathbb{A} ^{ \mathcal{A} } \times \pi_{ \mathcal{L} } Y_\gamma $ whose elements are denoted $x=(r,\theta= [z],w)$.
We provide this manifold with the metric $$\aa{x-x'}_\gamma =
\inf_{p\in \mathbb{Z} ^{d}} ||(r, z+2\pi p, w)-(r', z',w')||_\gamma . \quad\footnote{\ using this notation for the metric on the manifold will not confuse it with
the norm on the tangent space, which is also denoted $||\cdot||_\gamma $, we hope} $$
We provide $ \mathbb{A} ^{ \mathcal{A} } \times \pi_{ \mathcal{L} } Y_\gamma $ with the symplectic structure $ \Omega$. To any $C^{1}$-function $f(r,\theta,w)$ on (some open set in) $ \mathbb{A} ^{ \mathcal{A} }\times \pi_{ \mathcal{L} } Y_{\gamma }$ it associates a vector field $X_f= J(df)$ -- the Hamiltonian vector field of $f$ -- which in the coordinates $(r,\theta,w)$ takes the form $$ \left(\begin{array}{c} \dot r_a \\ \dot \theta_a\end{array}\right)=J \left(\begin{array}{c} \frac{\partial}{\partial r_a} f(r,\theta,w)\\ \frac{\partial}{\partial \theta_a} f(r,\theta,w)\end{array}\right) \qquad \left(\begin{array}{c} \dot p_a \\ \dot q_a\end{array}\right)=J \left(\begin{array}{c} \frac{\partial}{\partial p_a} f(r,\theta,w)\\
\frac{\partial}{\partial q_a} f(r,\theta,w)\end{array}\right).
\quad\footnote{\ there is no agreement as to the sign of the Hamiltonian vectorfield -- we've used the choice of Arnold \cite{Arn}} $$
\subsection{A matrix algebra}\label{ssMatrixAlgebra} \ The mapping \begin{equation}\label{pdist}
(a,b)\mapsto [a-b]=\min (|a-b|,|a+b|)\end{equation} is a pseudo-metric on $ \mathbb{Z} ^{d}$, i.e. it verifies all the relations of a metric with the only exception that
$[a-b]$ is $=0$ for some $a\not=b$. This is most easily seen by observing that $[a-b]=\textrm{d}_{ {\operatorname{Hausdorff} } }(\{\pm a\},\{\pm b\})$. We have $ [a-0]=|a|$.
Define, for any $\gamma =(\gamma _1,\gamma _2)\ge(0,0)$ and $ \varkappa \ge0$, \begin{equation}\label{weight} e_{\gamma , \varkappa }(a,b)=Ce^{\gamma _1[a-b]}\max([a-b],1)^{\gamma _2}\min(\langle a\rangle,\langle b\rangle)^ \varkappa .\end{equation}
\begin{lemma}\label{lWeights} \ \begin{itemize} \item[(i)] If $\gamma _1, \gamma _2- \varkappa \ge0$, then $$e_{\gamma , \varkappa }(a,b)\le e_{\gamma ,0}(a,c) e_{\gamma , \varkappa }(c,b),\quad \forall a,b,c,$$ if $C$ is sufficiently large (bounded with $\gamma _2, \varkappa $).
\item[(ii )] If $-\gamma \le\tilde \gamma \le \gamma $, then $$e_{\tilde\gamma , \varkappa }(a,0)\le e_{\gamma , \varkappa }(a,b) e_{\tilde\gamma , \varkappa }(b,0),\quad \forall a,b$$ if $C$ is sufficiently large (bounded with $\gamma _2, \varkappa $). \end{itemize} \end{lemma}
\begin{proof} (i). Since $[a-b]\le [a-c]+[c-b]$ it is sufficient to prove this for $\gamma _1=0$. If $\gamma _2=0$ then the statement holds for any $C\ge1$, so it is sufficient to consider $\gamma _2>0$. This reduces easily to $\gamma _2=1$ and, hence, $ \varkappa \le 1$. Then we want to prove \begin{multline*} \max([a-b],1)\min(\langle a\rangle,\langle b\rangle)^ \varkappa \le C \max([a-c],1)\max([c-b],1) \min(\langle c \rangle,\langle b\rangle)^ \varkappa . \end{multline*} Now $\max([a-b],1)\le \max([a-c],1)+\max([c-b],1)$, $$ \max([c-b],1) \min(\langle c \rangle,\langle b\rangle)^ \varkappa \gtrsim \langle b\rangle^ \varkappa , $$ and $$ \max([a-c],1) \min(\langle c \rangle,\langle b\rangle)^ \varkappa \gtrsim \min(\langle a \rangle,\langle b\rangle)^ \varkappa .$$ This gives the estimate.
(ii) Again it suffices to prove this for $\gamma _1=0$ and $\gamma _2=1$. Then we want to prove $$ \max(\ab{a},1)^{\tilde \gamma _2} \le C \max([a-b],1)\min(\langle a \rangle,\langle b\rangle)^ \varkappa \max(\ab{b},1)^{\tilde \gamma _2}.$$ The inequality is fulfilled with $C\ge1$ if $a$ or $b$ equal $0$. Hence we need to prove $$ \ab{a}^{\tilde \gamma _2} \le C \max([a-b],1)\min(\langle a \rangle,\langle b\rangle)^ \varkappa \ab{b}^{\tilde \gamma _2}.$$ Suppose $\tilde\gamma _2\ge0$. If $\ab{a}\le 2 \ab{b}$ then this holds for any $C\ge2$. If $\ab{a}\ge2\ab{b}$ then $[a-b]\ge \tfrac12\ab{a}$ and the statement holds again for any $C\ge2$.
If instead $\tilde\gamma _2<0$, then we get the same result with $a$ and $b$ interchanged.
\end{proof}
\subsubsection {The space $ \mathcal{M} _{\gamma , \varkappa }$}\label{sM2}
We shall consider matrices $A: \mathcal{Z} \times \mathcal{Z} \to gl(2, \mathbb{C} )$, formed by $2\times2$-blocs, (each $A_a^{b}$ is a complex $2\times2$-matrix). Define \begin{equation}\label{matrixnorm}
|A|_{\gamma , \varkappa }=\max\left\{\begin{array}{l} \sup_a\sum_{b} \aa{A_a^b} e_{\gamma , \varkappa }(a,b)\\ \sup_b\sum_{a} \aa{A_a^b} e_{\gamma , \varkappa }(a,b), \end{array}\right.\end{equation} where the norm on $A_a^{b}$ is the matrix operator norm.
Let $ \mathcal{M} _{\gamma , \varkappa }$ denote the space of all matrices $A$ such that $\ab{A}_{\gamma , \varkappa }<\infty$. Clearly $\ab{\cdot}_{\gamma , \varkappa }$ is a norm on $ \mathcal{M} _{\gamma , \varkappa }$ -- this is indeed true for all $(\gamma _1,\gamma _2, \varkappa )\in \mathbb{R}^3$. It follows by well-known results that $ \mathcal{M} _{\gamma , \varkappa }$, provided with this norm, is a Banach space.
Transposition -- $({}^tA)_a^b={}^t\!A_b^a$ -- and $ \mathbb{C} $-conjugation -- $(\overline{A})_a^b={\overline{A_a^b}})$ -- do not change
this norm.The identity matrix is in $ \mathcal{M} _{\gamma , \varkappa }$ if, and only if, $ \varkappa =0$, and then $|I|_{\gamma ,0}=C$.
\begin{remark*} The ``$l^1$-norm'' used here is a bit
more complicated than the ``sup-norm'' used in \cite{EK10}, but it has, as we shall see, much better multiplicative properties. \end{remark*}
\subsubsection{Matrix multiplication}
We define (formally) the {\it matrix product} $$(AB)_a^b=\sum_{c} A_a^cB_c^b.$$ Notice that complex conjugation, transposition and taking the adjoint behave in the usual way under this formal matrix product.
\begin{proposition}\label{pMatrixProduct} Let $\gamma _2\ge \varkappa $. If $A\in \mathcal{M} _{\gamma ,0}$ and $B\in \mathcal{M} _{\gamma , \varkappa }$, then $AB$ and $BA\in \mathcal{M} _{\gamma , \varkappa }$ and $$ \ab{{AB} }_{\gamma , \varkappa }\ \textrm{and}\ \ab{{BA} }_{\gamma , \varkappa }\le \ab{A}_{\gamma ,0} \ab{B}_{\gamma , \varkappa }.$$
\end{proposition}
\begin{proof} (i) We have, by Lemma~\ref{lWeights}(i), $$\sum_{b} \aa{( AB)_a^b} e_{\gamma , \varkappa }(a,b)\le \sum_{b,c} \aa{ A_a^c} \aa{ B_c^b} e_{\gamma , \varkappa }(a,b)\le $$ $$\le\sum_{b,c}\aa{ A_a^c} \aa{ B_c^b} e_{\gamma ,0}(a,c)e_{\gamma , \varkappa }(c,b) $$ which is $\le \aa{A}_{\gamma ,0} \aa{B}_{\gamma , \varkappa }$. This implies in particular the existence of $(AB)_a^b$.
The sum over $a$ is shown to be $\le \ab{A}_{\gamma ,0} \ab{B}_{\gamma , \varkappa }$ in a similar way. The estimate of $BA$ is the same. \end{proof}
Hence $ \mathcal{M} _{\gamma ,0}$ is a Banach algebra, and $ \mathcal{M} _{\gamma , \varkappa }$ is an ideal in $ \mathcal{M} _{\gamma ,0}$ when $ \varkappa \le\gamma _2$.
\subsubsection{The space $ \mathcal{M} _{\gamma , \varkappa }^b$} We define (formally) on $Y_\gamma $
$$(A\zeta )_a=\sum_{b} A_a^b \zeta _b.$$
\begin{proposition}\label{pMatrixBddOp} Let $-\gamma \le\tilde \gamma \le\gamma $. If $A\in \mathcal{M} _{\gamma , \varkappa }$ and $\zeta \in Y_{\tilde \gamma }$, then $A\zeta \in Y _{\tilde \gamma }$ and $$ \aa{A\zeta }_{\tilde \gamma }\le \ab{A}_{\gamma , \varkappa }\aa{\zeta }_{\tilde\gamma }.$$ \end{proposition}
\begin{proof} Let $\zeta '=A\zeta $. We have $$ \sum_{a} \ab{\zeta '_a}^2e_{\tilde \gamma ,0}(a,0)^2\le\sum_{a} \big(\sum_{b} \aa{A_a^b}\ab{\zeta _{b}} e_{\tilde \gamma ,0 }(a,0)\big)^2.$$ Write $$\aa{A_a^b}\ab{\zeta _{b}}e_{\tilde \gamma ,0 }(a,0) = I\times (I\ab{ \zeta _{b}}e_{\tilde \gamma ,0}(b,0))\times J,$$ where $$I=I_{a,b}=\sqrt{\aa{A_a^b} e_{\gamma , \varkappa }(a,b)}$$ and $$J=J_{a,b}= \frac{e_{\tilde \gamma ,0}(a,0)}{e_{\gamma , \varkappa }(a,b)e_{\tilde\gamma ,0}(b,0)}.$$
Since, by Lemma~\ref{lWeights}(ii), $J=\le 1$ we get, by H\"older, \begin{multline*} \sum_{a} \ab{\zeta' _a}^2e_{\tilde \gamma ,0}(a,0)^2\le \sum_{a} (\sum_{b} I_{a,b}^2)( \sum_{b} I_{a,b}^2\ab{ \zeta _{b}}^2e_{\tilde\gamma ,0}(b,0)^2)\\ \le \ab{A}_{\gamma , \varkappa } \sum_{a,b} I_{a,b}^2\ab{\zeta _{b}}^2e_{\tilde\gamma ,0}(b,0)^2 \le \ab{A}_{\gamma , \varkappa } \sum_{b} \ab{ \zeta _{b}}^2e_{\tilde \gamma ,0}(b,0)^2\sum_{a}I_{a,b}^2\le \end{multline*} $$\le \ab{A}_{\gamma , \varkappa }^2 \aa{\zeta }_{\tilde \gamma }^2. $$ This shows that $y_a$ exists for all $a$, and it also proves the estimate. \end{proof}
We have thus, for any $-\gamma \le \tilde\gamma \le\gamma $, a continuous embedding of $ \mathcal{M} _{\gamma , \varkappa }$, $$ \mathcal{M} _{\gamma , \varkappa }\hookrightarrow \mathcal{M} _{\gamma ,0}\to
\mathcal{B} (Y_{\tilde \gamma },Y_{\tilde \gamma }),$$ into the space of bounded linear operators on $Y_{\tilde \gamma }$. Matrix multiplication in $ \mathcal{M} _{\gamma , \varkappa }$ corresponds to composition of operators.
For our applications (see Lemma~\ref{lemP}) we shall consider a somewhat larger sub algebra of $ \mathcal{B} (Y_\gamma , Y_\gamma )$
with somewhat weaker decay properties. Let \begin{equation}\label{b-space} \mathcal{M} _{\gamma , \varkappa }^b= \mathcal{B} (Y_{\gamma },Y_{\gamma })\cap \mathcal{M} _{(\gamma _1,\gamma _2-m_*), \varkappa }\end{equation} which we provide with the norm \begin{equation}\label{b-matrixnorm} \aa{A}_{\gamma , \varkappa }=\aa{A}_{ \mathcal{B} (Y_{\gamma };Y_{\gamma })}+ \ab{A}_{(\gamma _1,\gamma _2-m_*), \varkappa }.\end{equation} When $\gamma =(\gamma _1,\gamma _2)\ge \gamma _*=(0,m_*+ \varkappa )$,
Proposition \ref{pMatrixProduct} shows that this norm makes $ \mathcal{M} _{\gamma ,0}^b$ into a Banach sub-algebra of $ \mathcal{B} (Y_{\gamma };Y_{\gamma })$ and $ \mathcal{M} _{\gamma , \varkappa }^b$ becomes an ideal in $ \mathcal{M} _{\gamma ,0}^b$.
\subsection{Functions}
For $\sigma,\mu\in (0,1]$ let \begin{equation}\label{domain} \begin{split} \mathcal{O} _\gamma ( \sigma ,\mu)=&\\ \{x=(r_ \mathcal{A} ,\theta_ \mathcal{A} ,w)\in \mathbb{A} ^{ \mathcal{A} } \times\pi_{ \mathcal{L} } Y_{\gamma }: &
|r_ \mathcal{A} |<\mu,\ \mid |\Im \theta_ \mathcal{A} |<\sigma,\ \mid \|w\|_\gamma <\mu \}. \end{split}\end{equation} It is often useful to scale the action variables $r$ by $\mu^2$ and not by $\mu$, but in our case $\mu$ will be $\approx 1$, and then there is no difference (on the contrary, in Section~\ref{s_4.2} we scale $r_ \mathcal{A} $ as $\mu^2$ to simplify the calculations we perform there).
The advantage with our scaling is that the Cauchy estimates becomes simpler.
Let \begin{equation}\label{gamma}\gamma =(\gamma _1,\gamma _2)\ge \gamma _*=(0,m_*+ \varkappa ),\end{equation} We shall consider perturbations $$f: \mathcal{O} _{\gamma _*}( \sigma , \mu)\to \mathbb{C} $$ that are {\it real holomorphic and continuous up to the boundary} (rhcb). This means that it gives real values to real arguments and extends continuously to the closure of $ \mathcal{O} _{\gamma _*}( \sigma , \mu)$. $f$ is clearly also rhcb on $ \mathcal{O} _{\gamma }( \sigma , \mu)$ for any $\gamma \ge\gamma _*$, and $$d f: \mathcal{O} _{\gamma }( \sigma , \mu)\to Y_{\gamma }^* $$ and $$ J_ \Omega d^2f: \mathcal{O} _\gamma ( \sigma , \mu)\to \mathcal{B} (Y_\gamma ,Y_{-\gamma }) $$ are rhcb.
\begin{remark} Identifying $Y_\gamma ^*$ with $Y_{-\gamma }$ via the paring $\langle\cdot,\cdot\rangle$ we will interpret the differential $df(\zeta)$ as a gradient $\nabla f(\zeta)\in Y_{-\gamma }$, $$ df(\zeta) (\zeta') =\langle \nabla f(\zeta), \zeta'\rangle\qquad \forall\, \zeta'\in Y_\gamma \,. $$ As classically, $\nabla f(\zeta)$ is the vector $\nabla f(\zeta)=( \nabla_a f(\zeta), a\in \mathcal{Z} )$, where for $ \zeta=\big( \zeta_a = (p_a,q_a), a\in \mathcal{Z} \big)$, $\nabla_a f$ is the 2-vector $(\partial f/\partial p_a, \partial f/\partial q_a)$.
Similar we will interpret $d^2f$ as the Hessian $\nabla^2 f$, which is an operator
the matrix $ ((\nabla^2 f)_a^b, a,b\in \mathcal{Z} ), $ formed by the $2\times2$-blocks $(\nabla^2 f)_a^b = \nabla_a\nabla_b f$. The Hessian defines bounded linear operators $\nabla^2 f(\zeta) : Y_\gamma \to Y_{-\gamma }$, and $$ d^2 f(\zeta) (\zeta^1, \zeta^2) =\langle \nabla^2 f(\zeta) \zeta^1, \zeta^2\rangle \quad \forall\ \zeta^1, \zeta^2 \in Y_\gamma \,. $$ \end{remark}
We shall require that the mappings $df$ and $d^2f$ posses some extra smoothness:
\begin{itemize}
\item[R1] {\it -- first differential}. There exists a $\gamma \ge\gamma _*$ such that $$ Jd f=J\nabla f : \mathcal{O} _{\gamma '}( \sigma , \mu)\to Y_{\gamma '} $$ is rhcb for any $\gamma _*\le \gamma '\le\gamma $.
\end{itemize} This is a natural smoothness condition on the space of holomorphic functions on $ \mathcal{O} _{\gamma _*}( \sigma , \mu)$, and it implies, in particular, that $Jd^2 f(x)=J\nabla^2 f(x)
\in \mathcal{B} (Y_{\gamma '},Y_{\gamma '})$ for any $x\in \mathcal{O} _{\gamma '}( \sigma , \mu)$. So $$ (\nabla^2 f(x))_a^b
\le {\operatorname{Ct.} } e^{-\gamma '_1\ab{ \ab{a}-\ab{b}}}\min(\frac{\langle a\rangle}{\langle b\rangle}, \frac{\langle b\rangle}{\langle a\rangle})^{\gamma '_2}\qquad \forall\, a,b\in \mathcal{Z} \,. $$
But many Hamiltonian PDE's verify other, and stronger, decay conditions in terms of $[a-b]=\min(\ab{a-b},\ab{a+b}).$
Indeed we shall assume
\begin{itemize}
\item[R2] {\it -- second differential}. $$Jd^2f= J\nabla^2 f : \mathcal{O} _{\gamma '}( \sigma , \mu)\to \mathcal{M} _{\gamma ', \varkappa }^b$$ is rhcb for any $\gamma _*\le \gamma '\le\gamma $.
\end{itemize}
Such decay conditions do not seem to be naturally related to any smoothness condition of $f$, but they
are instrumental in the KAM-theory for multidimensional PDE's: see for example \cite{EK10} where
such conditions were used to build a KAM-theory for some multidimensional non-linear Schr\"odinger equations.
\subsubsection{The function space $ \mathcal{T} _{\gamma , \varkappa } $ }\label{s_2.3.1} Consider the space of functions $f: \mathcal{O} _{\gamma _*}( \sigma , \mu)\to \mathbb{C} $ which are real holomorphic and continuous up to the boundary (rhcb) of $ \mathcal{O} _{\gamma _*}( \sigma , \mu)$. We define $ \mathcal{T} _{\gamma , \varkappa }( \sigma ,\mu) $ to be the space of all such functions which verify $R1$ and $R2$.
We provide $ \mathcal{T} _{\gamma , \varkappa } ( \sigma , \mu)$ with the norm \begin{equation}\label{norm}
|f|_{\begin{subarray}{c} \sigma ,\mu\\ \gamma , \varkappa \end{subarray}}= \max\left\{\begin{array}{l}
\sup_{x\in \mathcal{O} _{\gamma _*}( \sigma , \mu)}|f(x)|\\
\sup_{\gamma _*\le \gamma '\le\gamma }\sup_{x\in \mathcal{O} _{\gamma '}( \sigma , \mu)} ||Jd f(x)||_{\gamma '}=\|\nabla f(x)\|_{\gamma '} \\
\sup_{\gamma _*\le \gamma '\le\gamma }\sup_{x\in \mathcal{O} _{\gamma '}( \sigma , \mu)}||Jd^2f(x)||_{\gamma ', \varkappa } =||\nabla^2f(x)||_{\gamma ', \varkappa } \end{array}\right. \end{equation} making it into a Banach space. (It is even a Banach algebra with the constant function $f=1$ as unit, but we shall be concerned with Poisson products rather than with products.)
This space is relevant for our application because
\begin{lemma}\label{lemP} Let $ \mathcal{Z} = \mathcal{L} _\infty= \mathbb{Z} ^d$ and $ \varkappa =2$. Then the Hamiltonian function $h_{\ge4}$,
defined in \eqref{H1}, belongs to $ \mathcal{T} _{\gamma _g,2}(1, \mu_g) $ for suitable $\mu_g\in(0,1]$ and $\gamma _g>\gamma _*$.
\end{lemma}
The lemma in proven in Appendix A. Notice that we would not have been to prove this if we had used the matrix norm \eqref{matrixnorm} instead of \eqref{b-matrixnorm}.
The higher differentials $d^{k+2}f$ can be estimated by Cauchy estimates on some smaller domain in terms of this norm.
\begin{remark*} The higher order differential $d^{k+2}f(x)$, $x\in \mathcal{O} _{\gamma }( \sigma , \mu)$, is canonically identified with three bounded symmetric multi-linear maps $$\begin{array}{l}
(Y_{\gamma })^{k+2}
\longrightarrow \mathbb{C} \,,\\
(Y_{\gamma })^{k+1} \longrightarrow Y_{\gamma }^*\,,\\ (Y_{\gamma })^{k} \longrightarrow \mathcal{B} (Y_\gamma ,Y_\gamma ^*). \end{array}$$ Due to the smoothing condition R1 the second one takes its values in the subspace $Y_{-\gamma }^*$. Due to the smoothing condition R2 $Jd^{k+2}f(x)$ is a bounded symmetric multi-linear map \begin{equation}\label{dk+2} (Y_{\gamma })^{k} \longrightarrow \mathcal{M} _{\gamma , \varkappa }^b. \end{equation}
Alternatively, identifying $d^2f$ with the hessian $\nabla^2 f$, we may identify $d^{k+2}f$ with a continuous symmetric multilinear mapping of the form \eqref{dk+2}.
\end{remark*}
\subsubsection{The function space $ \mathcal{T} _{\gamma , \varkappa , \mathcal{D}} $ }
Let $ \mathcal{D}$ be an open set in $ \mathbb{R}^{ \mathcal{P} }$. We shall consider functions $$f: \mathcal{O} _{\gamma ^*}( \sigma , \mu)\times \mathcal{D}\to \mathbb{C} $$ which are of class $ \mathcal{C} ^{{s_*}}$ for some integer $s_*\ge0$. We say that $f\in \mathcal{T} _{\gamma , \varkappa , \mathcal{D}}( \sigma ,\mu) $ if, and only if, $$\frac{\partial^j f}{\partial \rho ^j}(\cdot, \rho )\in \mathcal{T} _{\gamma , \varkappa }( \sigma ,\mu) $$ for any $ \rho \in \mathcal{D}$ and any $\ab{j}\le {s_*}$. We provide this space by the norm \begin{equation}\label{normwithparameter}
|f|_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}= \max_{\ab{j}\le {s_*}}\sup_{ \rho \in \mathcal{D}}
|\frac{\partial^j f}{\partial \rho ^j}(\cdot, \rho )|_{\begin{subarray}{c} \sigma ,\mu\\ \gamma , \varkappa \end{subarray}}.\end{equation} This norm makes $ \mathcal{T} _{\gamma , \varkappa , \mathcal{D}}( \sigma ,\mu) $ a Banach space.
\subsubsection{ Jets of functions.} \label{ss5.1}
For any function $f\in \mathcal{T} _{\gamma , \varkappa , \mathcal{D}}( \sigma ,\mu)$ we shall consider the following
Taylor polynomial of $f$ at $r=0$ and $w=0$ \begin{equation} \label{jet} f^T(x)=f(0,\theta,0)+d_r f(0,\theta,0)[r] +d_w f(0,\theta,0)[w]+\frac 1 2 d^2_wf(0,\theta,0)[w,w] \end{equation} Functions of the form $f^T$ will be called {\it jet-functions.}
\begin{proposition}\label{lemma:jet} Let $f\in \mathcal{T} _{\gamma , \varkappa , \mathcal{D}}( \sigma ,\mu)$. Then $f^T\in \mathcal{T} _{\gamma , \varkappa , \mathcal{D}}( \sigma ,\mu)$ and $$
|f^T|_{\begin{subarray}{c} \sigma ,\mu \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}
\leq C |f|_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}.$$ ($C$ is an absolute constant.) \end{proposition}
\begin{proof} The first part follows by general arguments. Look for example on $$g(x)= d^2_wf\circ p(x)[ w,w],\quad x=(r,\theta,w),$$ where $p(x)$ is the projection onto $(0,\theta,0)$. This function $g$ is rhcb on $ \mathcal{O} _{\gamma _*}( \sigma , \mu)$, being a composition of such functions. A bound for its sup-norm is obtained by a Cauchy estimate of $f$: $$\aa{d^2_wf(p(x))}_{ \mathcal{B} (Y_{\gamma _*},Y_{-\gamma _*}^*)}\aa{w}^2_{\gamma '} \le {\operatorname{Ct.} } \frac1{\mu^2} \sup_{ \mathcal{O} _{\gamma _*}( \sigma , \mu)}\ab{f(y)}\aa{w}_{\gamma _*}^2\le {\operatorname{Ct.} } \sup_{y\in \mathcal{O} _{\gamma _*}( \sigma , \mu)}\ab{f(y)}.$$
Since $Jd g(x)[\cdot]$ equals $$\big(J dd^2_w f\circ p(x)[w,w]\big)[dp[\cdot]]+ 2\big(Jd^2_w f\circ p(x)[ w]\big)[\cdot], $$ and $$Jd^2_w f: \mathcal{O} _{\gamma '}( \sigma , \mu)\to \mathcal{B} (Y_{\gamma '},Y_{\gamma '})$$ and $$Jd d^2_w f=J d^2_w df: \mathcal{O} _{\gamma '}( \sigma , \mu)\to \mathcal{B} (Y_{\gamma '}, \mathcal{B} ( Y_{\gamma '},Y_{\gamma '}))$$ are rhcb, it follows that $d g$ verifies R1 and is rhcb. The norm $\aa{Jd g(x)}_{\gamma '}$ is less than $$\aa{J d^2_w d f(p(x))}_{ \mathcal{B} (Y_{\gamma '}, \mathcal{B} ( Y_{\gamma '},Y_{\gamma '}))}\aa{w}^2_{\gamma '} +2\aa{Jd^2_wf(p(x))}_{ \mathcal{B} (Y_{\gamma '},Y_{\gamma '})}\aa{w}_{\gamma '},$$ which is $\le {\operatorname{Ct.} } \sup_{y\in \mathcal{O} _{\gamma '}( \sigma , \mu)}\aa{Jd f(y)}_{\gamma '}$ -- this follows by Cauchy estimates of derivatives of $Jd f$.
Since $Jd^2 g(x)[\cdot,\cdot]$ equals $$\big(J d^2 d_w^2f\circ p(x)[w,w]\big)[dp[\cdot],dp[\cdot]]+ 2J\big( d d^2_w f\circ p(x)[w]\big)[\cdot,dp[\cdot]+ 2Jd_w^2 f\circ p(x)[\cdot,\cdot],$$ and $$Jd^2_w f: \mathcal{O} _{\gamma '}( \sigma , \mu)\to \mathcal{M} _{\gamma ', \varkappa }^b,$$ $$J d d^2_w f=J d^2_w df: \mathcal{O} _{\gamma '}( \sigma , \mu)\to \mathcal{B} (Y_{\gamma '}, \mathcal{M} _{\gamma ', \varkappa }^b)$$ and $$J d^2 d_w^2f=J d_w^2 d^2f : \mathcal{O} _{\gamma '}( \sigma , \mu)\to \mathcal{B} (Y_{\gamma '}, \mathcal{B} (Y_{\gamma '}, \mathcal{M} _{\gamma ', \varkappa }^b))$$ are rhcb, it follows that $Jd g^2$ verifies R2 and is rhcb. The norm $\aa{Jd^2 g}_{\gamma ', \varkappa }$ is less than $$\aa{J d_w^2 d^2f(p(x))}_{ \mathcal{B} (Y_{\gamma '}, \mathcal{B} (Y_{\gamma '}, \mathcal{M} _{\gamma ', \varkappa }^b))}\aa{w}^2_{\gamma '} +2\aa{J d_w d^2f(p(x))}_{ \mathcal{B} (Y_{\gamma '}, \mathcal{M} _{\gamma ', \varkappa }^b)}\aa{w}_{\gamma '}+$$ $$ +2\aa{Jd^2f(x)}_{\gamma ', \varkappa },$$ which is $\le {\operatorname{Ct.} } \sup_{y\in \mathcal{O} _{\gamma '}( \sigma , \mu)}\aa{Jd^2 f(y)}_{\gamma ', \varkappa }$ -- this follows by a Cauchy estimate of $Jd^2 f$.
The derivatives with respect to $ \rho $ are treated alike. \end{proof}
\subsection{Flows} \subsubsection{ Poisson brackets.} \label{ss5.2} The Poisson bracket $\{f,g\}$ of two $ \mathcal{C} ^1$-functions $f$ and $g$ is (formally) defined by $$ \{f,g\}= \Omega(Jdf,Jdg) =\langle J\nabla f, \nabla g\rangle =-df[Jdg]=dg[Jdf]$$ If one of the two functions verify condition R1, this product is well-defined. Moreover, if both $f$ and $g$ are jet-functions, then $\{f,g\}$ is also a jet-function.
\begin{proposition}\label{lemma:poisson} Let $f,g\in \mathcal{T} _{\gamma , \varkappa , \mathcal{D}}( \sigma ,\mu)$, and let $ \sigma '< \sigma $ and $\mu'<\mu\le1$. Then \begin{itemize} \item[(i)] $\{g,f\}\in \mathcal{T} _{\gamma , \varkappa , \mathcal{D}}( \sigma ',\mu')$ and $$ \ab{\{g,f\}}_{\begin{subarray}{c} \sigma ',\mu' \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}\leq C_{ \sigma - \sigma '}^{\mu-\mu'}\ab{g}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}} \ab{f}_{\begin{subarray}{c} \sigma ,\mu \ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}$$ for $$C_{ \sigma - \sigma '}^{\mu-\mu'}=C \big(\frac1{(\sigma-\sigma')} + \frac1{ (\mu-\mu') }\big).$$
\item[(ii)] the n-fold Poisson bracket $ P_g^n f \in \mathcal{T} _{\gamma , \varkappa , \mathcal{D}}( \sigma ,\mu)$ and $$\ab{ P_g^n f }_{\begin{subarray}{c} \sigma ',\mu' \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}\leq \big(C_{ \sigma - \sigma '}^{\mu-\mu'}\ab{g}_{\begin{subarray}{c} \sigma ,\mu \ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}\big)^n \ab{f}_{\begin{subarray}{c} \sigma ,\mu \ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}$$ where $P_g f=\{g,f\}$. \end{itemize} ($C$ is an absolute constant.)
\end{proposition}
\begin{proof} (i) We must first consider the function $h= \Omega(Jdg,Jdf)$ on $ \mathcal{O} _{\gamma _*}( \sigma , \mu)$ Since $J dg,\ J df: \mathcal{O} _{\gamma _*}( \sigma , \mu)\to Y_{\gamma _*}$ are rhcb, it follows that $h: \mathcal{O} _{\gamma _*}( \sigma , \mu)\to \mathbb{C} $ is rhcb, and $$\ab{h(x)}\le \aa{J dg (x)}_{\gamma _*}\aa{J df(x)}_{\gamma _*}.$$
The vector $J d h(x)$ is a sum of $$J \Omega(Jd^2g(x),Jdf(x))=Jd^2g(x)[Jdf(x)]$$ and another term with $g$ and $f$ interchanged. Since $Jd^2g: \mathcal{O} _{\gamma '}( \sigma , \mu)\to \mathcal{B} (Y_{\gamma '},Y_{\gamma '})$ and $Jdg,\ Jdf: \mathcal{O} _{\gamma '}( \sigma , \mu)\to Y_{\gamma '}$ are rhcb, it follows that $Jd h$ verifies R1 and is rhcb. Moreover $$\aa{Jd^2g(x)[Jdf(x),\cdot]}_{\gamma '}\le \aa{J d^2g (x)}_{ \mathcal{B} (Y_{\gamma '},Y_{\gamma '}) }\aa{J df(x)}_{\gamma '}$$ and, by definition of $ \mathcal{M} _{\gamma , \varkappa }^b$ , $$\aa{J d^2g (x)}_{ \mathcal{B} (Y_{\gamma '},Y_{\gamma '}) }\le \aa{J d^2g (x)}_{\gamma ',0}.$$
The operator $Jd^2h(x)=d(Jdh) (x)$ is a sum of $$Jd^3g(x)[Jdf(x)]$$ and $$Jd^2g(x)[Jd^2f(x)]$$ and two other terms with $g$ and $f$ interchanged.
Since $Jd^3g: \mathcal{O} _{\gamma '}( \sigma , \mu)\to \mathcal{B} (Y_{\gamma '}, \mathcal{M} _{\gamma ', \varkappa }^b)$ and $Jdf: \mathcal{O} _{\gamma '}( \sigma , \mu)\to Y_{\gamma '}$ are holomorphic functions, it follows that the first function $ \mathcal{O} _{\gamma '}( \sigma , \mu)\to \mathcal{B} (Y_{\gamma '}, \mathcal{M} _{\gamma ', \varkappa }^b)$ also is holomorphic. It can be estimated on a smaller domain using a Cauchy estimate for $Jd^3g(x)$.
The second term is treated differently. Since $$Jd^2f,\ Jd^2g: \mathcal{O} _{\gamma '}( \sigma , \mu)\to \mathcal{M} _{\gamma , \varkappa }^b$$ are rhcb, and since, by Proposition \ref{pMatrixProduct}, taking products is a bounded bi-linear maps with norm $\le 1$, it follows that the second function $ \mathcal{O} _{\gamma '}( \sigma , \mu)\to \mathcal{M} _{\gamma ', \varkappa }^b$ is rhcb and $$\aa{Jd^2g(x)[Jd^2f(x)]}_{\gamma ', \varkappa }\le \aa{Jd^2g(x)}_{\gamma ', \varkappa }\aa{Jd^2f(x)}_{\gamma ', \varkappa }.$$
The derivatives with respect to $ \rho $ are treated alike.
(ii) That $ g_n=P_g^n f \in \mathcal{T} _{\gamma , \varkappa , \mathcal{D}}( \sigma ',\mu')$ follows from (ii), but the estimate does not follow from the estimate in (ii). The estimate follows instead from Cauchy estimates of $n$-fold product $P_g^n f $ and from the following statement:
for any $n\ge1$ and any $k\ge0$, $\ab{d^k g_n(x)}$, $x\in \mathcal{O} _{\gamma '}( \sigma , \mu)$, is bounded by a sum of terms of the form $$\ab{d^{m_1} g(x)}\dots\ab{d^{m_n} g(x)}\ab{d^{m_{n+1}}f(x)} \quad\footnote{\ in the norms of the appropriate Banach spaces} $$ with $\sum m_j=n+1+k$ and each $m_j\ge1$. The number of terms in the sum is $\le 2^{nk}$. [This is proven above for $n=1$ and $k\le2$. It follows for $k\ge 3$ by the product formula for derivatives. It follows then for all $n\ge2$ and any $k\ge0$ by an easy induction.]
Let now $m'_j=2$ if $m_j\ge3$ and $=m_j$ if $m_j\le2$. Then the term above can be estimated by Cauchy estimates: $$ \le (C_{ \sigma - \sigma '}^{\mu-\mu'})^{\sum (m_j-m'_j)} \ab{d^{m'_1} g(x)}\dots\ab{d^{m'_n} g(x)}\ab{d^{m'_{n_1}}f (x)}\le $$ $$\le (C_{ \sigma - \sigma '}^{\mu-\mu'})^{\sum (m_j-m'_j)} (\ab{g}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}})^n \ab{f}_{\begin{subarray}{c} \sigma ,\mu \ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}$$ The result now follows by observing that $\sum (m_j-m'_j)\le \max(n+k-2,0)$ and taking $k=2$. [Indeed, if $\sum (m_j-m'_j)$ were $\ge n+k-1$, then $\sum m'_j\le \sum m_j-(n+k-1)=2$. Since $m'_j\ge1$ this forces $n$ to be $=1$ and all $m'_j$ to be $=1$. Hence $m_j=m'_j$ and $\sum( m_j- m'_j)=0$.] \end{proof}
\begin{remark} The proof shows that the assumptions can be relaxed when $g$ is a jet function: it suffices then to assume that
$g\in \mathcal{T} _{\gamma ,0, \mathcal{D}}( \sigma ,\mu)$ and $g-\hat g(\cdot,0,\cdot)\in \mathcal{T} _{\gamma , \varkappa , \mathcal{D}}( \sigma ,\mu)$. \footnote{\ $\hat g(\cdot,0,\cdot)$ this is the $0$:th Fourier coefficient of the function $\theta\mapsto g(\cdot,\theta,\cdot)$}
Then $\{g,f\}$ will still be in $ \mathcal{T} _{\gamma , \varkappa , \mathcal{D}}( \sigma ,\mu)$ but with the bound $$ \ab{\{g,f\}}_{\begin{subarray}{c} \sigma ',\mu' \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}\leq C_{ \sigma - \sigma '}^{\mu-\mu'} \big(\ab{g}_{\begin{subarray}{c} \sigma ,\mu \ \ \\ \gamma , 0, \mathcal{D} \end{subarray}} +\ab{g-\hat g(\cdot,0,\cdot)}_{\begin{subarray}{c} \sigma ,\mu \ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}\big) \ab{f}_{\begin{subarray}{c} \sigma ,\mu \ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}.$$
To see this it is enough to consider a jet-function $g$ which does not depend on $\theta$. The only difference with respect to case (i) is for the second differential. The second term is fine since, by Proposition \ref{pMatrixProduct}, $ \mathcal{M} _{\gamma ', \varkappa }^b$ is a two-sided ideal in $ \mathcal{M} _{\gamma ',0}^b$ and $$\aa{Jd^2g(x)[Jd^2f(x)]}_{\gamma ', \varkappa }\le \aa{Jd^2g(x)}_{\gamma ',0}\aa{Jd^2f(x)}_{\gamma ', \varkappa }.$$ For the first term we must consider $Jd^3g(x)[Jdf(x)]$ which, a priori, takes its values in $ \mathcal{M} _{\gamma ',0}^b$ and not in $ \mathcal{M} _{\gamma ', \varkappa }^b$. But since $g$ is a jet-function independent of $\theta$ this term is $=0$.
\end{remark}
\subsubsection {Hamiltonian flows} \label{ss5.3}
The Hamiltonian vector field of a $ \mathcal{C} ^1$-function $g$ on (some open set in) $Y_\gamma $ is $Jdg$.
Without further assumptions it is an element in $Y_{-\gamma }$, but if $g\in \mathcal{T} _{\gamma , \varkappa }$, then it is an element
in $Y_\gamma $ and has a well-defined local flow $\{\Phi_g^t\}$. Clearly
$(d/dt) f(\Phi^t_g) = \{f,g\}\circ \Phi^t_g$ for a $C^1$-smooth function $f$.
\begin{proposition}\label{Summarize} Let $g\in \mathcal{T} _{\gamma , \varkappa , \mathcal{D}}( \sigma ,\mu)$, and let $ \sigma '< \sigma $ and $\mu'<\mu\le1$. If $$ \ab{g}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}\leq\\
\frac1{C}\min( \sigma - \sigma ',\mu-\mu'),$$
then \begin{itemize} \item[(i)] the Hamiltonian flow map $\Phi^t=\Phi^t_g$ is,
for all $\ab{t}\le1$ and all $\gamma _*\le \gamma '\le\gamma $, a $ \mathcal{C} ^{{s_*}}$-map $$ \mathcal{O} _{\gamma '}( \sigma ',\mu')\times \mathcal{D} \to \mathcal{O} _{\gamma '}( \sigma ,\mu)$$ which is real holomorphic and symplectic for any fixed $\rho\in \mathcal{D}$.
Moreover, $$\aa{ \partial_ \rho ^j (\Phi^t(x, \rho )-x)}_{\gamma '}\le C\ab{g}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}},$$ and $$ \aa{ \partial_ \rho ^j(d\Phi^t(x)-I)}_{\gamma ', \varkappa }\le C\ab{g}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}},$$ for any $x\in \mathcal{O} _{\gamma '}( \sigma ',\mu')$, $\gamma _*\le \gamma '\le\gamma $, and $0\le \ab{j}\le {s_*}$.
\item[(ii)] $f\circ \Phi_g^t\in \mathcal{T} _{\gamma , \varkappa }( \sigma ',\mu', \mathcal{D})$ for $\ab{t}\le1$ and $$ \ab{ f\circ \Phi_g^t }_{\begin{subarray}{c} \sigma ',\mu' \ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}\leq C \ab{f}_{\begin{subarray}{c} \sigma ,\mu \ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}$$ for any $f\in \mathcal{T} _{\gamma , \varkappa }( \sigma ,\mu, \mathcal{D})$. \end{itemize}
($C$ is an absolute constant.) \end{proposition}
\begin{proof} It follows by general arguments that the local flow $\Phi=\Phi_g: U\to \mathcal{O} _{\gamma }( \sigma ,\mu)$ is real holomorphic in $(t,\zeta)$ in some $U\subset \mathbb{C} \times \mathcal{O} _{\gamma }( \sigma ,\mu)$, and that it depends smoothly on any smooth parameter in the vector field. Clearly, for $\ab{t}\le 1$ and $x\in \mathcal{O} _{\gamma }( \sigma ',\mu')$ $$\aa{\Phi^t(x, \rho )-x}_{\gamma }\le \sup_{x\in \mathcal{O} _{\gamma }( \sigma ,\mu)}\aa{Jdg(x)}_\gamma \le \ab{g}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , 0, \mathcal{D} \end{subarray}}$$ as long as $\Phi^t(x)$ stays in the domain $ \mathcal{O} _{\gamma }( \sigma ,\mu)$. It follows by classical arguments that this is the case if $$\ab{g}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , 0, \mathcal{D} \end{subarray}} \le {\operatorname{ct.} } \min( \sigma - \sigma ',\mu-\mu').$$
{\it The differential}. We have $$\frac{d}{dt} d\Phi^t(x)=-Jd^2g(\Phi^t(x))d\Phi^t(x)=B(t)d\Phi^t(x),$$ where $B(t)\in \mathcal{M} _{\gamma , \varkappa }^b$. By re-writing this equation in the integral form
$d\Phi^t(x)=\operatorname{Id}+\int_0^t B(s) d\Phi^s(x) \text{d} s$ and iterating this relation, we get that $ d\Phi^t(x)-\operatorname{Id}= B^{\infty}(t)$ with
$$B^{\infty}(t)
=\sum_{k\geq 1}\int_{0}^{t}\int_{0}^{t_{1}}\cdots \int_{0}^{t_{k-1}} \prod_{j=1}^{k}B(t_{j})\text{d}t_{k} \cdots \text{d}t_{2}\,\text{d}t_{1}.$$
We get, by Proposition \ref{pMatrixProduct}, that $d\Phi^t(x)-\operatorname{Id}\in \mathcal{M} _{\gamma , \varkappa }^b$ and, for $\ab{t}\le1$ , $$ \aa{ d\Phi^t(x)-\operatorname{Id}}_{\gamma , \varkappa }\le \sum_{k\geq 1} \aa{Jd^2g(\Phi^t(x))}^k_{\gamma , \varkappa }\frac{t^k}{k!}\le \aa{Jd^2g(\Phi^t(x))}_{\gamma , \varkappa }.$$
In particular, $A=d\Phi^t(x)$ is a bounded bijective operator on $Y_\gamma $. Since $Jd^2g$ is a Hamiltonian vector field we clearly have that $$ \Omega(A\zeta,A\zeta')= \Omega(\zeta,\zeta'),\quad\forall \zeta,\zeta'\in Y_\gamma ,$$ so $A$ is symplectic.
{\it Parameter dependence}. For $\ab{j}=1$, we have $$\frac{d}{dt} Z(t)=\frac{d}{dt} \frac{\partial^j\Phi^t(x, \rho )}{\partial \rho ^j}= B(t, \rho )Z(t) -\frac{\partial^j Jdg(\Phi^t(x, \rho ), \rho )}{\partial \rho ^j} =B(t)Z(t)+A(t).$$
Since $$\aa{A(t)}_\gamma + \aa{B(t)}_{\gamma , \varkappa }\le {\operatorname{Ct.} } \ab{g}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}},$$ it follows by classical arguments, using Gronwall, that $$\aa{Z(t)}_{\gamma ,0}\le {\operatorname{Ct.} } \ab{g}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}\ab{t}.$$ The higher order derivatives (with respect to $\rho$) of $\Phi^t(x, \rho )$, and the derivatives of $d\Phi^t(x, \rho )$ are treated in the same way.
The same argument applies to any $\gamma _*\le\gamma '\le\gamma $.
Since $$
f\circ \Phi_g^t = \sum_{n\ge0}\frac1{n!}t^nP^n_{-g}f ,$$ (ii) is a consequence of Proposition \ref{lemma:poisson}(ii). \end{proof}
\begin{remark}\label{r_sigma} If the set $ \mathcal{Z} $ is such that $ \mathcal{A} = \mathcal{F} =\emptyset$ and $ \mathcal{L} _\infty = \mathbb{Z} ^d$ (so $ \mathcal{Z} = \mathbb{Z} ^d$), then the domains $ \mathcal{O} _\gamma (\sigma,\mu)$ and the functional spaces on these domains which we introduced do not depend on $\sigma$. In this case in our notation we will chose the dumb parameter $\sigma$ to be 1. The assertions of the Propositions~\ref{lemma:poisson} and \ref{Summarize} remain true if we there take $ \sigma = \sigma '=1$ and drop the assumptions, related to $ \sigma $ and $ \sigma '$ (in particular, replace there $\min( \sigma - \sigma ', \mu-\mu')$ by $\mu-\mu'$, and replace $1/( \sigma - \sigma ')$ by 0). \end{remark}
\centerline{PART II. A BIRKHOFF NORMAL FORM} \section{Small divisors} \label{s_2} \subsection{Non resonance of basic frequencies}
In this subsection we assume that the set $ \mathcal{A} \subset \mathbb{Z} ^d$ is admissible, i.e. it only contains integer vectors with different norms (see Definition \ref{adm}).\\ We consider the vector of basic frequencies \begin{equation}\label{-om} \omega \equiv \omega (m)=( \omega _a(m))_{a\in \mathcal{A} }\,, \quad m\in [1,2]\,, \end{equation} where $ \omega _a(m)=\lambda=
\sqrt{|a|^4+m}$. The goal of this section is to prove the following result: \begin{proposition} \label{NRom}
Assume that $ \mathcal{A} $ is an admissible subset of $ \mathbb{Z} ^d$ of cardinality $n$ included in $\{a\in \mathbb{Z} ^d\mid |a|\leq N\}$. Then for any $k\in \mathbb{Z} ^ \mathcal{A} \setminus\{0\}$, any $ \kappa >0$ and any $c\in \mathbb{R}$ we have \begin{equation*}
\operatorname{meas}\ \left\{m\in[1,2]\ \mid \
\left|\sum_{a\in \mathcal{A} } k_a\omega_a(m)+c\right|\leq { \kappa }\right\}\leq C_n \frac{N^{4n^2} \kappa ^{1/n}}{|k|^{1/n}}\,, \end{equation*}
where $|k|:=\sum_{a\in \mathcal{A} }|k_a|$ and $C_n>0$ is a constant, depending only on $n$. \end{proposition}
The proof follows closely that of Theorem 6.5 in \cite{Bam03} (also see \cite{BG06}); a weaker form of the result was obtained earlier in \cite{bou95}. Non of the constants $C_j$ etc. in this section depend on the set $ \mathcal{A} $.
\begin{lemma}\label{l_det} Assume that $ \mathcal{A} \subset\{a\in \mathbb{Z} ^d\mid |a|\leq N\}$. For any $p\leq n= | \mathcal{A} |$, consider $p$ points $a_1,\cdots,a_p$ in $ \mathcal{A} $. Then the modulus of the following determinant \begin{equation*}
D:=\left| \begin{matrix} \frac{d \omega _{a_1}}{dm} & \der \null {a_2} & .& .&.&\der \null {a_p} \\ \frac{d^2 \omega _{a_1}}{dm^2} & \frac{d^2 \omega _{a_2}}{dm^2} & .& .&.&\frac{d^2 \omega _{a_p}}{dm^2} \\ .& .& .& .& .&. \\ .& .& .& .& .&. \\ \der{p}{a_1}& \der{p}{a_ 2}& .& .&.&\der {p}{a_p} \end{matrix}
\right| \end{equation*} is bounded from below: $$
|D|\geq C N^{-3p^2+p}\,, $$ where $C=C(p)>0$ is a constant depending only on $p$. \end{lemma}
\proof First note that, by explicit computation, \begin{equation}\label{-9}
\frac{d^j\omega_i}{dm^j}= (-1)^{j} \Upsilon_j\(|i|^4+m\)^{\frac12 -j}\,, \qquad \Upsilon_j=\prod_{l=0}^{j-1} \frac{2l-1}2\,. \end{equation} Inserting this expression in $D$, we deduce by factoring from each $l-th$ column the term
$(|a_\ell|^4+m)^{-1/2}= \omega _\ell^{-1}$, and from each $j-th$ row the term $\Upsilon_j$
that the determinant, up to a sign, equals \begin{eqnarray*} \left[\prod_{l=1}^{p}\omega_{a_\ell}^{-1}\right]
\left[\prod_{j=1}^{p} \Upsilon_j
\right] \times
\left| \begin{matrix} 1& 1& 1 &. & . & . & 1 \cr x_{a_1}& x_{a_2}& x_{a_3}&.&.&.&x_{a_p} \cr x_{a_1}^2& x_{ a_2}^2& x_{a_3}^2&.&.&.&x_{a_p}^2 \cr .& .& .& .& .&.&. \cr .& .& .& .& .&.&. \cr .& .& .& .& .&.&. \cr x_{a_1}^{p}& x_{a_2}^{p}& x_{a_3}^{p}&.&.&.&x_{a_p}^{p} \end{matrix}
\right|, \end{eqnarray*}
where we denoted $x_{a}:=(|a|^4+m)^{-1}= \omega_{a}^{-2}$. Since $| \omega _{a_k}|\le2|a_k|^2\le 2 N^2$ for every $k$, the first factor is bigger than $(2N^2)^{-p}$. The second is a constant,
while the third is the Vandermond determinant, equal to \begin{equation*} \prod_{1\leq l<k\leq p}(x_{a_\ell}-x_{a_k})=\prod_{1\leq l<k\leq p}
\frac{|a_k|^4-|a_\ell|^4}{\omega_{a_\ell}^2\omega_{a_k}^2} =: V\,. \end{equation*} Since $ \mathcal{A} $ is admissible, then $$
|V| \ge \prod_{1\leq l<k\leq p}
\frac{|a_k|^2+|a_\ell|^2}{\omega_{a_\ell}^2\omega_{a_k}^2} \ge \(\frac14\)^{p(p-1)} N^{-3p(p-1)}\,, $$
where we used that each factor is bigger than $\frac1{16} N^{-6}$ using again that $| \omega _{a_k}|\le2|a_k|^2\le 2 N^2$ for every $k$. This yields the assertion. \endproof
\begin{lemma}\label{m1.1}
Let $u^{(1)},...,u^{(p)}$ be $p$ independent vectors in $ \mathbb{R}^p$ of norm at most one, and
let $w\in \mathbb{R}^p$ be any non-zero vector. Then there exists $i\in[1,...,p]$ such that $$
| \langle u^{(i)}, w\rangle |\geq
C_p |w| |\det(u^{(1)},\ldots,u^{(p)})|\,. $$ \end{lemma} \begin{proof}
Without lost of generality we may assume that $|w|=1$.
Let $| \langle u^{(i)}, w\rangle | \le a$ for all $i$. Consider the $p$-dimensional parallelogram $\Pi$, generated by the vector $u^{(1)},...,u^{(p)}$ in $ \mathbb{R}^p$ (i.e., the set of all linear combinations $\sum x_j u^{(j)}$, where $0\le x_j\le 1$ for all $j$). It lies in the strip of width $2pa$, perpendicular to the vector $w$, and its projection to to the $p-1$-dimensional space, perpendicular to $w$, lies in the ball around zero of radius $p$. Therefore the volume of $\Pi$ is bounded by
$C_p p^{p-1} (2pa)=C_p' a$. Since this volume equals $|\det(u^{(1)},\ldots,u^{(p)})|$, then
$a\ge C_p |\det(u^{(1)},\ldots,u^{(p)})|$. This implies the assertion. \end{proof}
Consider vectors $\frac{d^i\omega}{dm^i}(m)$, $1\le i\le n$, denote
$K_i= | \frac{d^i\omega}{dm^i}(m) |$ and set $$ u^{(i)}= K_i^{-1}\frac{d^i\omega}{dm^i}(m) , \qquad 1\le i\le n\,. $$ From \eqref{-9} we see that\footnote{In this section $C_n$ denotes any positive constant depending only on $n$.} $\ K_i\le C_n$ for all\ $1\le i\le n\, $ (as before, the constant does not depend on the set $ \mathcal{A} $). Combining Lemmas~\ref{l_det} and \ref{m1.1}, we find that for any vector $w$ and any $m\in[1,2]$ there exists $r=r(m)\le n$ such that \begin{equation}\label{m1.2} \begin{split}
\Big| \langle \frac{d^r\omega}{dm^r}(m), w \rangle \Big| =K_r \big| \langle u^{(r)}, w\rangle \big| \ge
K_r C_n |w| (K_1\dots K_n)^{-1}| D|\\
\ge C_n |w| N^{-3n^2+n}\,. \end{split} \end{equation}
Now we need the following result (see Lemma B.1 in \cite{E98}):
\begin{lemma} \label{v.112} Let $g(x)$ be a $C^{n+1}$-smooth function on the segment [1,2]
such that $|g'|_{C^n} =\beta$ and $\max_{1\le k\le n}\min_x|\partial^k g(x)|=\sigma$. Then $$
\operatorname{meas}\{x\mid |g(x)|\le\rho\} \le C_n \(\frac{\beta}{\sigma}+1\) \(\frac{\rho}{\sigma}\)^{1/n}\,. $$ \end{lemma}
Consider the function $g(m)=|k|^{-1}\sum_{a\in \mathcal{A} }k_a \omega _a(m) +|k|^{-1}c$. Then $|g'|_{C^n}\le C'_n$, and
$\max_{1\le k\le n}\min_m|\partial^k g(m)|\ge C_n N^{-3n^2+n}$ in view of \eqref{m1.2}. Therefore, by Lemma~\ref{v.112}, \begin{equation*} \begin{split}
\operatorname{meas}\{m\mid |g(m)|\le \frac{\kappa}{|k|} \} \le C_n N^{3n^2-n} \(\frac{\kappa}{|k|} N^{3n^2-n}\)^{1/n} = C_n N^{3n^2+2n-1}
\(\frac{\kappa}{|k|} \)^{1/n}\,. \end{split} \end{equation*} This implies the assertion of the proposition.
\subsection{Small divisors estimates}
We recall the notation \eqref{L+}, \eqref{-om}, and note the elementary estimates \begin{equation}\label{estimla} \langle a\rangle^2 < \lambda_a (m)< \langle a\rangle^2 + \frac{m}{2 \langle a\rangle ^2}\qquad \forall\, a\in \mathbb{Z} ^d\,,\ m\in[1,2]\,,
\end{equation} where $\langle a\rangle =\max(1,|a|^2)$.
In this section we study four type of linear combinations of the frequencies $ \lambda_a (m)$: \begin{align*} D_0=& \langle \omega , k\rangle, \quad k\in \mathbb{Z} ^ \mathcal{A} \setminus\{0\}\\ D_1=&\langle \omega , k\rangle + \lambda_a , \quad k\in \mathbb{Z} ^ \mathcal{A} ,\; a\in \mathcal{L} \\ D_2^\pm=&\langle \omega , k\rangle+ \lambda_a \pm \lambda_b , \quad k\in \mathbb{Z} ^ \mathcal{A} ,\; a, b\in \mathcal{L} \,. \end{align*} In subsequent sections they will become divisors for our constructions, so we call these linear combinations ``divisors".
\begin{definition}\label{Res-kab} Consider independent formal variables $x_0, x_1, x_2,\dots$. Now take any divisor of the form $D_0$, $D_1$ or $D_2^\pm$, write there each $\omega_a, a\in \mathcal{A} $, as $\lambda_a$, and then replace every $\lambda_a, a\in \mathbb{Z} ^d$, by
$x_{|a|^2}$. Then the divisor is called resonant if the obtained algebraical sum of the variables $x_j, j\ge0$, is zero. Resonant divisors are also called trivial resonances. \end{definition}
Note that a $D_0$-divisor cannot be resonant since $k\ne0$ and the set $ \mathcal{A} $ is admissible; a $D_1$-divisor $(k;a)$ is resonant only if $a\in { \mathcal{L} _f}$, $|k|=1$
and $\langle \omega , k\rangle=-\omega_b$,
where $|a|=|b|$. Finally, a $D_2^+$-divisor or a $D_2^-$ divisor with $k\ne0$ may be resonant only when $(a,b)\in { \mathcal{L} _f}\times { \mathcal{L} _f}$, while the divisors $D_2^-$
of the form $ \lambda_a - \lambda_b $, $|a|=|b|$, all are resonant. So there are finitely many trivial resonances of the form $D_0, D_1, D_2^+$ and of the form $D_2^-$ with $k\ne0$, but infinitely many of them of the form $D_2^-$ with $k=0$.
Our first aim is to remove from the segment $[1,2]=\{m\}$ a small subset to guarantee that for the remaining $m$'s
moduli of all non-resonant divisors admit positive lower bounds.
Below in this section \begin{equation}\label{agreement} \begin{split} &\text{constants $C, C_1$ etc. depend on the admissible set $ \mathcal{A} $,}\\
&\text{while the exponents $c_1, c_2$ etc depend only on $| \mathcal{A} |$. Borel }\\ &\text{sets $ \mathcal{C} _ \kappa $ etc. depend on the indicated arguments and $ \mathcal{A} $.} \end{split} \end{equation}
We begin with the easier divisors
$D_0$, $D_1$ and $D_2^+$.
\begin{proposition}\label{D1D2} Let $1\ge \kappa >0$. There exists a Borel set $ \mathcal{C} _ \kappa \subset[1,2]$ and positive constants $C$ (cf. \eqref{agreement}), satisfying $\ \operatorname{meas}\ \mathcal{C} _ \kappa \leq C \kappa ^{1/(n+2)} , $
such that for all $m\notin \mathcal{C} _ \kappa $, all $k$ and all $a,b \in \mathcal{L} $ we have \begin{equation}\label{D0}
|\langle \omega , k\rangle|\geq \kappa {\langle k\rangle}^{-n^2}, \qquad
\text{ except if } k=0,
\end{equation}
\begin{equation}\label{D1}
|\langle \omega , k\rangle+ \lambda_a |\geq \kappa {\langle k\rangle}^{-3(n+1)^3}, \quad \text{ except if the divisor is a trivial resonance}, \end{equation} \begin{equation} \label{D2}
|\langle \omega , k\rangle + \lambda_a + \lambda_b |\geq \kappa {\langle k\rangle}^{-3(n+2)^3}, \text{ except if
the divisor is a trivial resonance}.
\end{equation} Here
${\langle k\rangle}=\max(|k|,1)$.
Besides, for each $k\ne0$ there exists a set ${\mathfrak A}^k_ \kappa $ whose measure is $\ \le C \kappa ^{1/n}$ such that for $m\notin {\mathfrak A}^k_ \kappa $ we have
\begin{equation}\label{D22}|\langle \omega , k\rangle+j |\geq \kappa {\langle k\rangle}^{-(n+1)n } \text{for all $j\in \mathbb{Z} $ }. \end{equation} \end{proposition} \proof We begin with the divisors \eqref{D0}. By Proposition \ref{NRom} for any non-zero $k$ we have $$
\operatorname{meas} \{m\in[1,2]\mid |\langle \omega , k\rangle|\leq \kappa |k|^{-n^2} \} < C { \kappa ^{1/n}}{|k|^{-n-1/n}} \,. $$ Therefore the relation \eqref{D0} holds for all non-zero $k$ if $m\notin\mathfrak A_0$,
where $\operatorname{meas}\mathfrak A_0\le C \kappa ^{1/n} \sum_{k\ne0} |k|^{-n-1/n} =C \kappa ^{1/n}$.
Let us consider the divisors \eqref{D1}. For $k=0$ the required estimate holds
trivially. If $k\ne0$, then the relation, opposite to \eqref{D1} implies that $| \lambda_a |\le C|k|$. So we may assume that $|a|\le C|k|^{1/2}$. If $|a|\notin\{|s|\mid s\in \mathcal{A} \}$, then Proposition~\ref{NRom} with
$n:=n+1$, $ \mathcal{A} := \mathcal{A} \cup \{a\}$ and $N=C|k|^{1/2}$ implies that \begin{equation*} \begin{split}
\operatorname{meas}& \{m\in[1,2]\mid |\langle \omega , k\rangle+ \lambda_a |\leq \kappa |k|^{-3(n+1)^3}\}\\ \le& C
\kappa ^{1/(n+1)} |k|^{2(n+1)^2- 3(n+1)^2 -\frac1{n+1} }
\leq C\kappa^{1/(n+1)} |k|^{-(n+1)^2}\,. \end{split} \end{equation*} This relation with $n+1$ replaced by $n$
also holds if $|a|=|s|$ for some $s\in \mathcal{A} $, but $\langle \omega , k\rangle+ \lambda_a $ is not a trivial resonant.
Since for fixed $k$ the set$\{ \lambda_a \mid |a|^2\leq C|k| \}$ has cardinality less than $2C|k|$, then the relation $
|\langle \omega , k\rangle+ \lambda_a |\le \kappa |k|^{-3(n+1)^3} $ holds for a fixed $k$ and all $a$ if we remove from [1,2] a set of measure
$\le C \kappa ^{1/(n+1)}|k|^{-(n+1)^2+1}\leq C \kappa ^{1/(n+1)}|k|^{-n-1}$. So we achieve that
the relation \eqref{D1} holds for all $k$ if we remove from $[1,2]$ a set $\mathfrak A_1$ whose
measure is bounded by $C \kappa ^{1/(n+1)} \sum_{k\ne0} |k|^{-n-1} =C \kappa ^{1/(n+1)}$.
For a similar reason there exist a Borel set $\mathfrak A_2$ whose measure is bounded by $C \kappa ^{1/(n+2)}$ and such that \eqref{D2} holds for $m\notin \mathfrak A_2$. Taking $ \mathcal{C} _ \kappa = \mathfrak A_0\cup \mathfrak A_1\cup \mathfrak A_2$ we get \eqref{D0}-\eqref{D2}. Proof of \eqref{D22} is similar. \endproof
Now we control divisors $D_2^-=\langle \omega , k\rangle+ \lambda_a - \lambda_b $.
\begin{proposition}\label{prop-D3} There exist positive constants $C,c, c_-$ and for $0< \kappa $ there is a Borel set $ \mathcal{C} '_ \kappa \subset[1,2]$ (cf. \eqref{agreement}), satisfying \begin{equation}\label{meas-estim2} \operatorname{meas}\ \mathcal{C} '_ \kappa \leq C \kappa ^{c}, \end{equation}
such that for all $m\in [1,2]\setminus \mathcal{C} '_ \kappa $,
all $k\ne0$ and all $a,b \in \mathcal{L} $ we have \begin{equation}\label{D3} R(k;a,b):=
|\langle \omega , k\rangle + \lambda_a - \lambda_b |\geq \kappa |k|^{-c_- }, \end{equation}
except if the divisor is a trivial resonance \end{proposition} \proof We may assume that
$|b|\geq |a|$. We get from \eqref{estimla} that $$
| \lambda_a - \lambda_b -(|a|^2-|b|^2)|\leq {m}{|a|^{-2}}\leq 2 |a|^{-2}. $$ Take any $ \kappa _0\in (0,1]$ and construct the set $\mathfrak A^k_{ \kappa _0}$ as in Proposition~\ref{D1D2}. Then $\operatorname{meas} {\mathfrak A}^k_{ \kappa _0}\le C \kappa _0^{1/n}$ and for any
$m\notin {\mathfrak A}^k_{ \kappa _0}$ we have $$ R:=
R(k;a,b)\ge \big| \langle \omega , k\rangle +|a|^2-|b|^2\big|-2|a|^{-2}\ge
\kappa _0|k|^{-(n+1)n} - 2 |a|^{-2}\,. $$
So $R \ge\tfrac12 \kappa _0|k|^{-(n+1)n}$ and \eqref{D3} holds if $$
|b|^2\ge |a|^2 \ge 4 \kappa _0^{-1}|k|^{(n+1)n}=:Y_1. $$
If $|a|^2\le Y_1$, then $$
R \ge \lambda_b - \lambda_a -C|k| \ge |b|^2-Y_1-C|k|-1. $$
Therefore \eqref{D3} also holds if $|b|^2\ge Y_1 +C|k|+2$, and
it remains to consider the case when $|a|^2\le Y_1 $ and
$|b|^2\le Y_1 +C|k|+2$. That is (for any fixed non-zero $k$), consider the pairs $( \lambda_a , \lambda_b )$, satisfying \begin{equation}\label{above}
|a|^2\le Y_1,\qquad |b|^2\le Y_1+2+C|k| =:Y_2 \,. \end{equation} There are at most $CY_1Y_2$ pairs like that. Since the divisor $\langle \omega , k\rangle + \lambda_a - \lambda_b $
is not resonant, then in view of Proposition~\ref{NRom} with $N =Y_2^{1/2}$
and $| \mathcal{A} |\le n+2$, for any $\tilde \kappa >0$ there exists a set ${\mathfrak B}^k_{\tilde \kappa }\subset [1,2]$, whose measure is bounded by $$
C \tilde \kappa ^{1/(n+2)} \kappa _0^{-c_1} |k|^{c_2},\qquad c_j=c_j(n)>0, $$ such that $R \ge \tilde \kappa $ if $m\notin {\mathfrak B}^k_{\tilde \kappa }\, $ for all pairs $(a,b)$ as in \eqref{above} (and $k$ fixed).
Let us choose $\tilde \kappa = \kappa _0^{2c_1(n+2)}$. Then $
\operatorname{meas} {\mathfrak B}^k_{\tilde \kappa }\le C \kappa _0^{c_1}|k|^{c_2} $ and $R\ge \kappa _0^{2c_1(n+2)}$ for $a,b$ as in \eqref{above}. Denote $ \mathfrak C^k_{ \kappa _0}= \mathfrak A^k_{ \kappa _0}\cup {\mathfrak B}^k_{\tilde \kappa }\, $. Then
$\operatorname{meas} \mathfrak C^k_{ \kappa _0}\le C \( \kappa _0^{1/n} + \kappa _0^{c_1}|k|^{c_2}\)$, and for $m$ outside this set and all $a,b$ (with $k$ fixed)
we have $
R\ge \min\(\tfrac12 \kappa _0|k|^{-(n+1)n}, \kappa _0^{2c_1(n+2)}\)\,. $
We see that if $ \kappa _0= \kappa _0(k)=2 \kappa ^{c_3} |k|^{-c_4}$ with suitable $c_3,c_4>0$, then $$ \operatorname{meas}\( \mathcal{C} '_ \kappa = \cup_{k\ne0} \mathfrak C^k_{ \kappa _0}\) \le C \kappa ^{c_3} \,, $$ and, if $m$ is outside $ \mathcal{C} '_ \kappa $, then
$R(k;a,b)\ge \kappa |k|^{-c_-}$ with a suitable $c_->0$. \endproof
It remains to consider the divisors $D_2^-$ with $k=0$, i.e. $D_2^-= \lambda_a - \lambda_b $. Such a divisor is resonant if $|a|=|b|$. \begin{lemma}\label{lem:D3-k=0}
Let $m\in [1,2]$ and the divisor $D_2^-= \lambda_a - \lambda_b $ is non-resonant, i.e. $|a|\neq|b|$. Then $
\left| { \lambda_a - \lambda_b }\right|\ge \frac1 4.$ \end{lemma} \proof We have \begin{align*}
\left| \lambda_a - \lambda_b \right|= \frac{\left||a|^4-|b|^4\right|}{\sqrt{|a|^4+m}+\sqrt{|b|^4+m}}\geq
\frac{|a|^2+|b|^2}{\sqrt{|a|^4+m}+\sqrt{|b|^4+m}}\ge \frac 1 4. \end{align*} \endproof
By construction the sets $ \mathcal{C} _ \kappa $ and $ \mathcal{C} '_ \kappa $ decrease with $ \kappa $. Let us denote \begin{equation}\label{setC} \mathcal{C} = \bigcap_{ \kappa >0} ( \mathcal{C} _ \kappa \cup \mathcal{C} '_ \kappa )\,. \end{equation} From Propositions \ref{D1D2}, \ref{prop-D3} and Lemma~\ref{lem:D3-k=0} we get:
\begin{proposition}\label{prop-m} The set $ \mathcal{C} $ is a Borel subset of $[1,2]$ of zero measure. For any $m\notin \mathcal{C} $ there exists $ \kappa _0= \kappa _0(m)>0$ such that the relations \eqref{D0}, \eqref{D1}, \eqref{D2} and \eqref{D3} hold with $ \kappa = \kappa _0$. \end{proposition}
In particular, if $m\notin \mathcal{C} $ then any of the divisors $$ \langle \omega , s\rangle,\;\; \langle \omega , s\rangle\pm \lambda_a ,\;\;\langle \omega , s\rangle\pm \lambda_a \pm \lambda_b ,\quad s\in \mathbb{Z} ^d,\; a,b\in \mathcal{L} , $$ vanishes only if this is a trivial resonance. If it is not, then its modulus
admits a qualified estimate from below.
The zero-measure Borel set $ \mathcal{C} $ serves a fixed admissible set $ \mathcal{A} $, $ \mathcal{C} = \mathcal{C} _ \mathcal{A} $. But since the set of all
admissible sets is countable, then replacing $ \mathcal{C} $ by $\cup_ \mathcal{A} \mathcal{C} _ \mathcal{A} $ we obtain a zero-measure Borel set which
suits all admissible sets $ \mathcal{C} $.
For further purposes we modify $ \mathcal{C} $ as follows: \begin{equation}\label{modif} \mathcal{C} =: \mathcal{C} \cup \{\tfrac43, \tfrac 53\}\,. \end{equation}
\section{The Birkhoff normal form. I}\label{BNF}
In Sections \ref{BNF} and \ref{s_4} we construct a symplectic change of variable that puts the Hamiltonian
\eqref{H1} to a normal form.
In Sections \ref{BNF} and \ref{s_4}
constants in the estimates may depend on
\begin{equation}\label{depen}
\text{
$d$, $G$, $ \mathcal{A} $ and constants with lower index $*$ (including $c_*$)
}
\end{equation}
without saying. Their dependence on other parameters will be indicated. This does not contradicts Agreements
(see the end of Introduction) since in these sections the set $ \mathcal{F} $ is defined in terms of $ \mathcal{A} $ and $ \mathcal{P} $ does not
occur.
\subsection{Statement of the result}\label{s3.1}
The goal of this section is to get a normal form for the Hamiltonian $h=h_2+h_4+h_{\ge5}$ of the beam equation, written in the form \eqref{beam2}, in toroidal domains in the space which are complex neighbourhoods of the $n$-dimensional real tori $T_{I_ \mathcal{A} }$ (see \eqref{ttorus}). We scale the parameters $I_ \mathcal{A} $ as $\nu \rho $ where $\nu>0$ is small and $\rho=(\rho_a,a\in \mathcal{A} )$ belongs to the domain \begin{equation}\label{DDD} \mathcal{D}=[c_*,1]^ \mathcal{A} . \end{equation} In this section $c_*\in(0,\tfrac12]$ is regarded as a fixed parameter.
Consider the complex vicinity of the torus $T_{\nu \rho\, \mathcal{A} }$ (see \eqref{ttorus}) \begin{equation}\label{a-a} {\mathbf T}_\rho (\nu,\sigma,{\mu},\gamma)=\{(p_ \mathcal{A} ,q_ \mathcal{A} ,p_ \mathcal{L} ,\zeta_ \mathcal{L} ): \left\{\begin{array}{lll}
| \tfrac12 (p_a^2+q_a^2) -\nu \rho _a|<\nu c_*^2 {\mu}^2 & a\in \mathcal{A} &\\
|\Im\theta_a|<\sigma & a\in \mathcal{A} &\\
\|(p_ \mathcal{L} , q_ \mathcal{L} )\|_\gamma <\nu^{1/2}c_*{\mu}& &, \end{array}\right.\end{equation} where $\theta_a$ is related to $p_a,q_a$ through $\frac{p_a-{\bf i}q_a}{\sqrt{p_a^2+q_a^2}}=e^{{\bf i}\theta_a}$ --- this is well-defined when $\mu\le1$ because then $p_a^2+q_a^2\not=0$ for all $a\in \mathcal{A} $ whenever the point belongs to this vicinity.
In this section we use the complex coordinates $(\xi_a, \eta_a), a\in \mathbb{Z} ^d$, defined in \eqref{change}, denoting $(\xi_a, \eta_a) =\zeta_a$. So we will write points of $ {\mathbf T}_\rho (\nu,\sigma,{\mu},\gamma)$ as $\zeta = (\zeta_ \mathcal{A} , \zeta_ \mathcal{L} )$. We recall (see \eqref{L+}) that we have split the set $ \mathcal{L} = \mathbb{Z} ^d\setminus \mathcal{A} $ into the union $ \mathcal{L} = \mathcal{L} _f\cup \mathcal{L} _\infty$. We will write $\zeta_ \mathcal{L} =(\zeta_f, \zeta_\infty)$ and will use the notation of Section~\ref{sThePhaseSpace} with $ \mathcal{Z} = \mathbb{Z} ^d,\, \mathbb{Z} = \mathcal{A} \cup \mathcal{L} _f\cup \mathcal{L} _\infty $ (i.e. with $ \mathcal{F} = \mathcal{L} _f$).
\begin{proposition}\label{thm-HNF} There exists a zero-measure Borel set $ \mathcal{C} \subset[1,2]$ such that for any admissible set $ \mathcal{A} $, any $c_*\in(0,1/2]$ and $m\notin \mathcal{C} $ we can find
real numbers $\gamma _g>\gamma _*=(0,m_*+2)$ and $\nu_0>0$, where
$\nu_0$ depends on $m$,
with the following property.
For any $0<\nu\le\nu_0$ and $\rho\in [c_*,1]^ \mathcal{A} $ there exists al holomorphic diffeomorphism (onto its image) \begin{equation}\label{mu*} \Phi_\rho: \mathcal{O} _{\gamma _*} \big({\frac 12}, {\mu_*^2} \big)\to {\mathbf T}_\rho (\nu, 1,1,\gamma _*)\,,\qquad {\mu_*}={\tfrac{c_*}{2\sqrt2}}\,, \end{equation} which defines analytic transformations $$ \Phi_\rho: \mathcal{O} _{\gamma } \big({\tfrac 12}, {\mu_*^2} \big)\to {\mathbf T}_\rho (\nu, 1,1,\gamma )\,,\quad \gamma _*\le\gamma \le \gamma _g\,, $$ such that $$ \Phi_ \rho ^*\big(-{\bf i}dp \wedge dq\big)= \nu dr_ \mathcal{A} \wedge d\theta_ \mathcal{A} \ -{\bf i}\ \nu d\xi_ \mathcal{L} \wedge d\eta_ \mathcal{L} , $$ and such that \begin{equation}\label{HNF} \begin{split} \frac1{\nu} h\circ\Phi_\rho(r,\theta,\xi_ \mathcal{L} ,\eta_ \mathcal{L} ) =\langle \Omega(\rho), r\rangle &+ \sum_{a\in \mathcal{L} _\infty }\Lambda_a (\rho)\xi_a\eta_a+\\ &+\frac{\nu}2\, \langle K(\rho) \zeta_f, \zeta_f\rangle + f( r,\theta,\zeta_ \mathcal{L} ;\rho), \end{split} \end{equation} where $h$ is the Hamiltonian \eqref{H2}$+$\eqref{H1}, satisfies:
(i) $\Phi_\rho$ depends smoothly (even analytically) on $\rho$, and \begin{equation}\label{boundPhi} \begin{split} \mid\mid \Phi_ \rho (r,\theta,\xi_ \mathcal{L} ,\eta_ \mathcal{L} )- (\sqrt{\nu \rho }\cos(\theta),&\sqrt{\nu \rho }\sin(\theta),0,0
) \mid\mid_\gamma \le \\ &\le C(\sqrt\nu\ab{r}+\sqrt\nu\aa{(\xi_ \mathcal{L} ,\eta_ \mathcal{L} )}_\gamma +\nu^{\frac32} ) \end{split} \end{equation} for all $(r,\theta,\xi_ \mathcal{L} ,\eta_ \mathcal{L} )\in \mathcal{O} _{\gamma } (\frac 12, \mu_*^2)\cap\{\theta\ \textrm{real}\}$ and all $\gamma _*\le \gamma \le\gamma _g$.
(ii) the vector $ \Omega$ and the scalars $ \Lambda_a , a\in \mathcal{L} _\infty $ are affine functions of $\rho$, explicitly defined by
\eqref{Om} and \eqref{Lam};
(iii) $K$ is a symmetric real matrix. It is a quadratic polynomial of $\sqrt\rho=(\sqrt\rho_1,\dots,\sqrt\rho_n)$, explicitly defined by relation \eqref{K};
(iv) the remaining term $f$ belongs to $ \mathcal{T} _{\gamma _g, \varkappa =2, \mathcal{D}}({\tfrac12}, \mu_*^2)$ and satisfies \begin{equation}\label{est}
|f|_{\begin{subarray}{c}1/2,\mu_*^2 \ \\ \gamma _g, 2, \mathcal{D} \end{subarray}}
\le C\nu \,, \qquad
|f^T|_{\begin{subarray}{c}1/2,\mu_*^2 \ \\ \gamma _g, 2, \mathcal{D} \end{subarray}}
\le C \nu^{3/2} \,. \end{equation}
Finally, $\Phi_ \rho $ is not a real diffeomorphism, but verifies the ``conjugate-reality'' condition: $$\Phi_\rho(r,\theta,\xi_ \mathcal{L} ,\eta_ \mathcal{L} )\quad\textrm{ is real if, and only if},\quad \eta_ \mathcal{L} = \overline {\xi}_ \mathcal{L} .$$
The constant $C$ depends on $m$ (we recall \eqref{depen}) but not on $\nu$. \end{proposition}
\begin{remark}\label{r_p4.1} 1)
$\Phi_\rho$ is
close to the { scaling by the factor $\nu^{1/2}$ } on the $ \mathcal{L} _\infty $-modes
but not on the $( \mathcal{A} \cup { \mathcal{L} _f})$-modes, where it is close to a certain affine
transformation, depending on $\theta$. Moreover
$$\Phi_\rho\big ( \mathcal{O} _{\gamma } ({\tfrac 12}, {\mu_*^2}) \big)\subset {\mathbf T}_\rho (\nu, 1,1,\gamma ),\qquad \gamma _*\le \gamma \le\gamma _g.$$
2) All the objects, involved in this proposition, except the remaining term $f$ in \eqref{HNF}, depend only on the main part $u^4$ of $G$, and not on the higher order correction. \end{remark}
The rest of this section is devoted to the proof of Proposition \ref{thm-HNF}. From now on we arbitrarily enumerate the set $ \mathcal{A} $ of excited modes, i.e. we write $ \mathcal{A} $ as \begin{equation}\label{labA} \mathcal{A} = \{a_1,\dots, a_n\}\,, \end{equation} so that the cardinality of $ \mathcal{A} $ is $n$, and accordingly identify $ \mathbb{R}^{ \mathcal{A} }$ with $ \mathbb{R}^n$ and identify various $ \mathcal{A} $-valued maps with maps, valued in the set $\{1,\dots, n\}$.
\subsection{Resonances and the Birkhoff procedure}\label{s_4.2}
Instead of the domains $ \mathcal{O} _\gamma (\sigma,\mu)$, in this section we will use domains \begin{equation}\label{ddomain}
\mathcal{O} _\gamma (\sigma,\mu^2,\mu) = \{(r,\theta,w): |r| <\mu^2, |\Im\theta|<\sigma, \|w\|_\gamma <\mu\}\,, \end{equation} more convenient for the normal form calculation. The space of functions on $ \mathcal{O} _\gamma (\sigma,\mu^2,\mu)$, defined similar to the space $ \mathcal{T} _{\gamma , \varkappa }(\sigma,\mu)$, will be denoted $ \mathcal{T} _{\gamma , \varkappa }(\sigma,\mu^2,\mu)$. The norm
$ |f|_{\begin{subarray}{c} \sigma ,\mu,\mu^2\\ \gamma , \varkappa \end{subarray}} $ in this space is defined by the relation \eqref{norm}, where the first line is given the weight $\mu^0=1$, the second line -- the weight $\mu^1$, and the third line -- $\mu^2$. Note that \begin{equation}\label{O_relation} \mathcal{O} _\gamma (\sigma,\mu^2,\mu)\subset \mathcal{O} _\gamma (\sigma,\mu)\subset \mathcal{O} _\gamma (\sigma,\mu, \sqrt\mu),
\end{equation} and that $|\cdot |_{\begin{subarray}{c} \sigma ,\mu,\mu^2\\ \gamma , \varkappa \end{subarray}}$
and $|\cdot |_{\begin{subarray}{c} \sigma ,\mu\\ \gamma , \varkappa \end{subarray}}$ are equivalent if $\mu\sim1$.
In the situation of Remark~\ref{r_sigma}, when $ \mathcal{Z} = \mathbb{Z} ^d$ and $ \mathcal{A} = \mathcal{F} =\emptyset$, we have
$ \mathcal{T} _{\gamma , \varkappa }(1,\mu^2,\mu) = \mathcal{T} _{\gamma , \varkappa }(1,\mu)$, and \begin{equation}\label{twonorms1}
| f |_{\begin{subarray}{c}1,\mu,\mu^2\\ \gamma , \varkappa \end{subarray}} \le
|f |_{\begin{subarray}{c}1,\mu\\ \gamma , \varkappa \end{subarray}} \le \mu^{-2}
| f |_{\begin{subarray}{c}1,\mu,\mu^2\\ \gamma , \varkappa \end{subarray}} \end{equation} for any $0<\mu\le1$.
\begin{example}[homogeneous functionals] \label{ex_homog} Let $ \mathcal{Z} = \mathbb{Z} ^d$ and $ \mathcal{A} = \mathcal{F} =\emptyset$ and let $f(w)\in \mathcal{T} _{\gamma , \varkappa }(1,1,1) = \mathcal{T} _{\gamma , \varkappa }(1,1)$ be an $r$-homogeneous function, $r\le 2$ integer. Then $df$ and $d^2f$ are, accordingly, $(r-1)\,$-- and $(r-2)$--homogeneous. So for any $0<\mu\le1$ we have \begin{equation}\label{homog}
|f|_{\begin{subarray}{c}1,\mu,\mu^2\\ \gamma , \varkappa \end{subarray}} = \mu^r
|f|_{\begin{subarray}{c}1,1,1\\ \gamma , \varkappa \end{subarray}} \,. \end{equation} If for $j=1,2\ $ $f_j(w)\in \mathcal{T} _{\gamma , \varkappa }(1,1,1)$ is an $r_j$--homogeneous functional, $r_j\ge2$, then the functional $ \{f_1,f_2\} $ is $r_1+r_2-2$--homogeneous. So the relation above and Proposition~\ref{lemma:poisson} imply that \begin{equation}\label{poiss_homog}
|\{f_1,f_2\} |_{\begin{subarray}{c}1,1,1\\ \gamma , \varkappa \end{subarray}} \le C
|f_1 |_{\begin{subarray}{c}1,1,1\\ \gamma , \varkappa \end{subarray}} \cdot
|f_2|_{\begin{subarray}{c}1,1,1\\ \gamma , \varkappa \end{subarray}} \,. \end{equation} \end{example}
Let us consider the quartic part $h_2 + h_4$ of the Hamiltonian $h$, $$ h_2=
\sum_{a\in \mathbb{Z} ^d}\lambda_a \xi_a\eta_a,\quad h_4= (2\pi)^{-d}\sum_{(i,j,k,\ell)\in \mathcal{J} }\frac{(\xi_i+\eta_{-i})(\xi_j+\eta_{-j})(\xi_k+\eta_{-k})(\xi_\ell+\eta_{-\ell})}{4\sqrt{ \lambda_i \lambda_j \lambda_k \lambda_\ell }}\, $$
(the variables $\xi, \eta$ are defined in \eqref{change}), where
$ \mathcal{J} $ denotes the zero momentum set:$$ \mathcal{J} :=\{(i,j,k,\ell)\subset \mathbb{Z} ^d\mid i+j+k+\ell=0\}.$$
We decompose $h_4=h_{4,0}+h_{4,1}+h_{4,2}$ according to \begin{align*} h_{4,0}=& \frac 1 4 (2\pi)^{-d}\sum_{(i,j,k,\ell)\in \mathcal{J} }\frac{ \xi_i\xi_j\xi_k\xi_\ell +\eta_i\eta_j\eta_k\eta_\ell}{\sqrt{ \lambda_i \lambda_j \lambda_k \lambda_\ell }},\\ h_{4,1}=& (2\pi)^{-d}\sum_{(i,j,k,-\ell)\in \mathcal{J} }\frac{ \xi_i\xi_j\xi_k\eta_\ell +\eta_i\eta_j\eta_k\xi_\ell}{\sqrt{ \lambda_i \lambda_j \lambda_k \lambda_\ell }},\\ h_{4,2}=& \frac 3 2 (2\pi)^{-d}\sum_{(i,j,-k,-\ell)\in \mathcal{J} }\frac{ \xi_i\xi_j\eta_k\eta_\ell }{\sqrt{ \lambda_i \lambda_j \lambda_k \lambda_\ell }}\,, \end{align*} and define $$ \mathcal{J} _2= \{ (i,j,k,\ell)\subset \mathbb{Z} ^d\mid (i,j,-k,-\ell)\in \mathcal{J} , \; \sharp \{i,j,k,\ell\}\cap \mathcal{A} \geq 2\}\,. $$ By Proposition~\ref{prop-m} we have \begin{lemma}\label{res-mon}If $m\notin \mathcal{C} $, then there exists $ \kappa (m)>0$ such that for all $(i,j,k,\ell)\in \mathcal{J} _2$ \begin{align*}
| \lambda_i + \lambda_j + \lambda_k - \lambda_\ell |&\geq \kappa (m)\, ;\\
| \lambda_i + \lambda_j - \lambda_k - \lambda_\ell |&\geq \kappa (m), \quad \text{except if } \{|i|,|j| \}=\{|k|,|\ell| \}\, . \end{align*} \end{lemma}
For $\gamma =(\gamma _1, \gamma _2)$, where $0\le\gamma _1\le1$, $\gamma _2\ge m_*$, and for $ \mathcal{Z} = \mathbb{Z} ^d$ as above consider the space $Y_\gamma $ as in Section~\ref{sThePhaseSpace}, written in terms of the complex coordinates $\zeta_a=(\xi_a, \eta_a), a\in \mathbb{Z} ^d$. In these variables the symplectic from $\Omega$ reads $ \Omega = -i\sum d\xi_a\wedge d\eta_a$. For $0<\mu\le1$ consider the ball
$ \mathcal{O} _\gamma (1,\mu^2,\mu)= \mathcal{O} _\gamma (1,\mu) =\{|\zeta|_\gamma <\mu\}$.
For any vector $\zeta=(\zeta_a=(\xi_a,\eta_a), a\in \mathbb{Z} ^d)$, we will write $\zeta_a^+ = \xi_a$ and $\zeta_a^- = \eta_a$. For an integer $r\ge2$ we abbreviate $a=(a_1,\dots, a_r)\in ( \mathbb{Z} ^d)^r$, $ \varsigma =( \varsigma _1,\dots, \varsigma _r)\in\{+,-\}^r$, and consider a homogeneous polynomial $$ P^r(\zeta) = M\sum_{a\in ( \mathbb{Z} ^d)^r} \sum_{ \varsigma \in \{+,-\}^r} A_a^ \varsigma \, \zeta_{a_1}^{ \varsigma _1}\dots \zeta_{a_r}^{ \varsigma _r}\,. $$ Here $M$ is a positive constant, the moduli of all coefficients $A_a^ \varsigma $ are bounded by 1, and $$ A_a^ \varsigma = 0 \quad\text{unless} \quad a_1 \varsigma _1^0+\dots a_r \varsigma _r^0=0 $$ for some fixed boolean vector $ \varsigma ^0\in\{+,-\}^r$. Denote by $D^-$ the block-diagonal operator \begin{equation}\label{D-}
D^- = \operatorname{diag} \{ |\lambda_a|^{-1/2} I, a\in \mathbb{Z} ^d\}\,,\qquad I\in M(2\times2)\,, \end{equation} and set $Q^r(\zeta) = P^r(D^-\zeta)$.
\begin{lemma}\label{XPanalytic} For any $\gamma $ as above, $Q^r\in \mathcal{T} _{\gamma ,2}(1,1,1)$ and \begin{equation}\label{z.1}
|Q^r|_{\begin{subarray}{c}1,1,1\\ \gamma , 2 \end{subarray}} \le CM\,,\qquad C=C(r)\,. \end{equation} \end{lemma} The lemma is proved in Appendix A.
Note that by this lemma, \eqref{twonorms1} and \eqref{homog}, $
|Q^r |_{\begin{subarray}{c}1,\mu\\ \gamma , 2 \end{subarray}} \le CM \mu^{r-2}. $
So by Lemma~\ref{Summarize} if the function $Q^r$ is real, then
the Hamiltonian flow-maps $\Phi^t = \Phi^t_{Q^r}$, $|t|\le1$,
define real-holomorphic symplectic mappings
\begin{equation}\label{homog_flow}
\begin{split}
\Phi^t: \mathcal{O} _\gamma (1,\mu^2,\mu) \to \mathcal{O} _\gamma (1, 4\mu^2,2\mu)\quad
\text{if $r\ge4$ and $\mu\le \mu_1,\ \mu_1=\mu_1(M)>0$,}\\
\text{ or if $r=3$ and $M>0$ is sufficiently small
}
\end{split}
\end{equation}
(we recall that now
$ \mathcal{O} _\gamma (1,\mu^2,\mu) = \mathcal{O} _\gamma (1,\mu)$).
\begin{proposition}\label{Thm-BNF} For $m\notin \mathcal{C} $ and $\mu_g>0$, $\gamma _g> \gamma _*$ as in Lemma~\ref{lemP}
there exists $\mu\in(0,\mu_g]$ and a real holomorphic symplectomorphism
$$
\tau: \mathcal{O} _{\gamma _*} (1,\mu) =\{|\zeta|_{\gamma _*}<\mu\}
\to \mathcal{O} _{\gamma _*} (1, 2\mu)
$$
which is a diffeomorphism on its image and which for $\gamma _*\le\gamma \le\gamma _g$
defines analytic mappings
$\tau: \mathcal{O} _{\gamma } (1,\mu) \to \mathcal{O} _{\gamma } (1, 2\mu)$, such that
\begin{equation}\label{esti1}
\|\tau^{\pm1}(\zeta)-\zeta\|_{\gamma } \le C \|\zeta\|_{\gamma }^3
\qquad \forall\,\zeta\in \mathcal{O} _{\gamma } (1,\mu) \,.
\end{equation}
It transforms the Hamiltonian $h=h_2+h_4+ h_{\ge5}$
as follows: \begin{equation}\label{trans} h \circ \tau= h_2 + z_4+ q_4^3+ r_6^0 +h_{\ge5}\circ \tau\,, \end{equation} where \begin{align*}
z_4=&\frac 3 2(2\pi)^{-d} \sum_{\substack{(i,j,k,\ell)\in \mathcal{J} _2 \\ \{|i|,|j| \}=\{|k|,|\ell| \}}} \frac{ \xi_i\xi_j\eta_k\eta_\ell }{ \lambda_i \lambda_j }, \end{align*} and $q_4^3=q_{4,1}+q_{4,2}$ with\footnote{The upper index 3 signifies that $q_4^3$ is at least cubic in the transversal directions $\{\zeta_a, a\in \mathcal{L} \}$.} \begin{align*} q_{4,1}=&(2\pi)^{-d} \sum_{(i,j,-k,\ell)\not\in \mathcal{J} _2}\frac{ \xi_i\xi_j\xi_k\eta_\ell +\eta_i\eta_j\eta_k\xi_\ell}{\sqrt{ \lambda_i \lambda_j \lambda_k \lambda_\ell }},\\ q_{4,2}=& \frac 3 2(2\pi)^{-d}\sum_{(i,j,k,\ell)\not\in \mathcal{J} _2}\frac{ \xi_i\xi_j\eta_k\eta_\ell }{\sqrt{ \lambda_i \lambda_j \lambda_k \lambda_\ell }}\,. \end{align*} The functions $z_4, q_4^3, r_6^0, h_{\ge5}\circ\tau$ are real holomorphic on $ \mathcal{O} _{\gamma } (1,\mu)$ for each $\gamma _*\le\gamma \le\gamma _g$. Besides
$r_6^0$ and $h_{\ge5}\circ\tau$ are, respectively, functions of order 6 and 5 at the origin.
For any $0<\mu'\le \mu$ the functions
$z_4, q_4^3, r_6^0$ and $h_{\ge5}\circ\tau$ belong to ${\Tc}_{\gamma _g, 2}(1, (\mu')^2, \mu')$, and \begin{equation}\label{Z4}
|z_4|_{\begin{subarray}{c}1,\mu',(\mu')^2\\ \gamma _g, 2 \end{subarray}} +
|q_4^3|_{\begin{subarray}{c}1,\mu',(\mu')^2\\ \gamma _g, 2 \end{subarray}} \le C(\mu')^4\,,
\end{equation} \begin{equation}\label{R6}
|r^0_6|_{\begin{subarray}{c}1,\mu',(\mu')^2\\ \gamma _g, 2 \end{subarray}} \le C( \mu')^6\,, \end{equation} \begin{equation}\label{R66}
|h_{\ge5}\circ\tau|_{\begin{subarray}{c}1,\mu',(\mu')^2\\ \gamma _g, 2 \end{subarray}} \le C (\mu')^5\,.
\end{equation} The constants $C$ and $\mu$ depend on $m$ (we recall \eqref{depen}). \end{proposition}
\proof We use the classical Birkhoff normal form procedure. We construct the transformation $\tau$ as the time one flow $\Phi^1_{\chi_4}$ of a Hamiltonian $\chi_4$, given by \begin{align}\label{chi4}\begin{split} \chi_4=& -\frac{{\bf i} } 4 (2\pi)^{-d}\sum_{(i,j,k,\ell)\in \mathcal{J} }\frac{ \xi_i\xi_j\xi_k\xi_\ell -\eta_i\eta_j\eta_k\eta_\ell}{( \lambda_i + \lambda_j + \lambda_k + \lambda_\ell )\sqrt{ \lambda_i \lambda_j \lambda_k \lambda_\ell }}\\ &-{\bf i}(2\pi)^{-d}\sum_{(i,j,-k,\ell)\in \mathcal{J} _2}\frac{ \xi_i\xi_j\xi_k\eta_\ell -\eta_i\eta_j\eta_k\xi_\ell}{( \lambda_i + \lambda_j + \lambda_k - \lambda_\ell )\sqrt{ \lambda_i \lambda_j \lambda_k \lambda_\ell }}\\
&- \frac {3{\bf i}} 2 (2\pi)^{-d} \sum_{\substack{(i,j,k,\ell)\in \mathcal{J} _2 \\ \{|i|,|j| \}\neq\{|k|,|\ell| \}}} \frac{ \xi_i\xi_j\eta_k\eta_\ell }{( \lambda_i + \lambda_j - \lambda_k - \lambda_\ell )\sqrt{ \lambda_i \lambda_j \lambda_k \lambda_\ell }} \end{split}\end{align}
The Hamiltonian $\chi_4$ is 4-homogeneous and real (its takes real values if $\xi_a = \bar \eta_a$ for each $a$). If $m\notin \mathcal{C} $, then by Lemma~\ref{XPanalytic} $\chi_4 \in \mathcal{T} _{\gamma ,2}(1,1,1)$, and by Lemma~\ref{Summarize} and \eqref{homog_flow} the time-one flow-map of this Hamiltonian,
$\tau=\Phi^1_{\chi_4}$ is a real holomorphic and symplectic change of coordinates, defined in the $\mu$--neighbourhood of the origin in $Y_{\gamma }$ for any $\gamma _*\le \gamma \le \gamma _g$ and
a suitable positive $\mu=\mu(m)$. The relation \eqref{homog} implies that on $ \mathcal{O} _{\gamma } (1,2\mu)$ the norm of the Hamiltonian vector field is bounded by $C \mu^3$. This implies \eqref{esti1}.
{ Since the Poisson bracket, corresponding to the symplectic form $-{\bf i}d\xi\wedge d\eta$ is $ \{F,G\} ={\bf i}\langle \nabla_\eta F, \nabla_\xi G\rangle -{\bf i} \langle \nabla_\xi F, \nabla_\eta G\rangle, $ and since $\nabla_{\eta_s}h_2=\lambda_s \xi_s$, $\nabla_{\xi_s}h_2=\lambda_s \eta_s$, then} we
calculate \begin{equation}\label{h2chi4} \begin{split} \{\chi_4, h_2\}= & \frac{1 } 4 (2\pi)^{-d}\sum_{(i,j,k,\ell)\in \mathcal{J} }\frac{ \xi_i\xi_j\xi_k\xi_\ell + \eta_i\eta_j\eta_k\eta_\ell}{\sqrt{ \lambda_i \lambda_j \lambda_k \lambda_\ell }}\\ +&(2\pi)^{-d}\sum_{(i,j,-k,\ell)\in \mathcal{J} _2}\frac{ \xi_i\xi_j\xi_k\eta_\ell +\eta_i\eta_j\eta_k\xi_\ell}{\sqrt{ \lambda_i \lambda_j \lambda_k \lambda_\ell }}\\ +& \frac {3} 2 (2\pi)^{-d}
\sum_{\substack{(i,j,k,\ell)\in \mathcal{J} _2 \\ \{|i|,|j| \}\neq\{|k|,|\ell| \}}} \frac{ \xi_i\xi_j\eta_k\eta_\ell }{\sqrt{ \lambda_i \lambda_j \lambda_k \lambda_\ell }}\,. \end{split} \end{equation} Therefore the transformed quartic part of the Hamiltonian $h$, $(h_2+ h_4)\circ \tau$, equals \begin{align*}
h_2+ \big(h_4 + \{\chi_4, h_2\} \big) + \big( \{\chi_4, h_4 \} +&\int_0^1 (1-t) \{\chi_4,\{\chi_4, h_2+ h_4\}\}\circ \Phi_{\chi_4}^t \text{d} t \big)\\ =& h_2 +( z_4+ q_4^3)+ r_6^0 \end{align*} with $z_4$ and $q_4^3$ as in the statement of the proposition and $$ r_6^0= \{\chi_4, h_4 \} +\int_0^1 (1-t) \{\chi_4,\{\chi_4, h_2+ h_4\}\}\circ \Phi_{\chi_4}^t \text{d} t\,.
$$
The reality of the functions $z_4$ and $q_4^3$ follow from the explicit formulas for them, while the inclusion of these functions to ${\Tc}_{\gamma _g, 2}(1, 1, 1)$
and the estimate \eqref{Z4} for any $0<\mu'\le\mu$ hold by Lemma~\ref{XPanalytic} and \eqref{homog}.
To verify \eqref{R6} we first note that $\{\chi_4, h_4\}$ is a 6-homogeneous function, belonging to ${\Tc}_{\gamma _g, 2}(1, 1, 1)=:\Tc$ by \eqref{poiss_homog}. It satisfies the estimate in \eqref{R6} by \eqref{homog}. Next,
$\{\chi_4, h_2\}$ is a 4-homogeneous function, given by \eqref{h2chi4}. By Lemma~\ref{XPanalytic} it belongs to $\Tc$. The function $\{\chi_4, h_4\}$ is 6-homogeneous and belongs to $\Tc$ by \eqref{poiss_homog}. So $ \{\chi_4,\{\chi_4, h_2+ h_4\}\}$ is a sum of a 6\,- and 8-homogeneous functions, belonging to $\Tc$ by \eqref{poiss_homog}. Now the estimate
\eqref{R6} for the second component of $r^0_6$ follows from \eqref{homog_flow}, Lemma~\ref{Summarize}
and \eqref{homog}.
Finally, the estimate \eqref{R66} follows by applying the argument above to homogeneous components of $h_{\ge5}$ and noting that the obtained sum converges, if $\mu$ is sufficiently small. We skip the details. \endproof
Clearly $ {\mathbf T}_\rho (\nu, 1,1,\gamma )\subset \mathcal{O} _\gamma (1,\mu)$ if $\nu\le C^{-1}\mu^2$ (see \eqref{a-a}).
Due to \eqref{esti1}, if $\zeta\in {\mathbf T}_\rho (\nu,1/2,1/2,{\gamma })$ and $\gamma _*\le\gamma \le \gamma _g$,
then $\|\tau^{\pm1}(\zeta)-\zeta\|_\gamma \le C'(m)\nu^{\frac 3 2}$. Therefore \begin{equation}\label{prop1} \tau^{\pm1} ( {\mathbf T}_\rho (\nu,1/2,1/2,\gamma ))\subset {\mathbf T}_\rho (\nu,1 ,1,\gamma ) \subset \mathcal{O} _\gamma (1,\mu)\,, \end{equation} provided that $\nu\le C^{-1}\mu^2$, $\gamma _*\le\gamma \le \gamma _g$ and $\rho\in [c_*, 1]^ \mathcal{A} $.
\subsection{Normal form, corresponding to admissible sets $ \mathcal{A} $} \label{s_3.3} Everywhere below in Sections \ref{BNF}--\ref{s_4}
the set $ \mathcal{A} $ is assumed to be admissible in the sense of Definition~\ref{adm}.
In the domains $ {\mathbf T}_\rho = {\mathbf T}_\rho (\nu,\sigma,{\mu},\gamma)$ we pass from the complex variables $(\zeta_a, a\in \mathcal{A} )$, to the corresponding complex action-angles $(I_a, \theta_a)$, using the relations \begin{equation}\label{ac-an} \xi_a=\sqrt I_a e^{{\bf i}\theta_a},\qquad \eta_a=\sqrt I_a e^{-{\bf i}\theta_a}\,,\quad a\in \mathcal{A} \,. \end{equation} By $ {\mathbf T}_\rho ^{I,\theta}= {\mathbf T}_\rho ^{I,\theta}(\nu,\sigma, \mu, \gamma) $ we will denote a domain $ {\mathbf T}_\rho (\nu,\sigma, \mu, \gamma)$, written in the variables $(I,\theta, \xi_ \mathcal{L} \mathcal{L} , \eta_ \mathcal{L} )$, and will denote by $\iota$ the corresponding change of variables, \begin{equation}\label{iota} \iota: {\mathbf T}_\rho ^{I,\theta} \to {\mathbf T}_\rho ,\qquad (I,\theta, \xi_ \mathcal{L} , \eta_ \mathcal{L} ) \mapsto \zeta. \end{equation} Thus, $ \iota^{-1} T_{\nu\rho\, \mathcal{A} } =\{(I,\theta, 0,0) : I=\nu\rho, \theta\in \mathbb{T} ^n\}\,. $
The Hamiltonian $z_4$ contains the integrable part,
formed by monomials of the form $ \xi_i\xi_j\eta_i\eta_j=I_iI_j$ that only
depend on the actions $I_n=\xi_n\eta_n$, $n\in \mathbb{Z} ^d$. Denote it $z_4^+$ and denote the rest $z_4^-$. It is not hard to see that \begin{equation}\label{Z4+} z_4^+ \circ\iota =\frac 3 2(2\pi)^{-d} \sum_{\ell\in \mathcal{A} ,\ k\in \mathbb{Z} ^d} (4-3\delta_{\ell,k})\frac{I_\ell I_k}{\lambda_\ell\lambda_k}. \end{equation} To calculate $z_4^-$, we decompose it according to the number of indices in $ \mathcal{A} $: a monomial $\xi_i\xi_j\eta_k\eta_\ell $ is in $z_4^{-r}$ ($r=0,1,2,3,4$) if $(i,j,-k,-\ell)\in \mathcal{J} $ and $\sharp \{i,j,k,\ell\}\cap \mathcal{A} =r$. We note that, by construction, $z_4^{-0}=z_4^{-1}=\emptyset$.
Since $ \mathcal{A} $ is admissible, then in view of Lemma~\ref{res-mon} for $m\notin \mathcal{C} $ the set $z_4^{-4}$ is empty. The set $z_4^{-3}$ is empty as well:
\begin{lemma} If $m\notin \mathcal{C} $, then $z_4^{-3}=\emptyset.$ \end{lemma} \proof Consider any term $\xi_i\xi_j\eta_k\eta_\ell \in z_4^{-3}$, i.e. $\{i,j,k,\ell\}\cap \mathcal{A} =3$. Without lost of generality we can assume that $i,j,k\in \mathcal{A} $ and $\ell\in \mathcal{L} $. Furthermore we know that $i+j-k-\ell=0$ and
$\{|i|,|j|\}=\{|k|,|\ell|\}$. In particular we must have $|i|=|k|$ or $|j|=|k|$ and thus, since $ \mathcal{A} $ is admissible, $i=k$ or $j=k$. Let for example, $i=k$. Then $|j|=|\ell|$. Since $i+j=k+\ell$ we conclude that $\ell=j$ which contradicts our hypotheses. \endproof
Recall that the finite set $ { \mathcal{L} _f}\subset \mathcal{L} $ was defined in \eqref{L+}. The mapping \begin{equation}\label{lmap}
\ell: \mathcal{L} _f \to \mathcal{A} ,\quad a\mapsto \ell(a)\in \mathcal{A} \text{ if } \ |a|=|\ell(a)|, \end{equation} is well defined since the set $ \mathcal{A} $ is admissible. Now we define two subsets of $ { \mathcal{L} _f}\times { \mathcal{L} _f}$: \begin{align} \label{L++} ( { \mathcal{L} _f}\times { \mathcal{L} _f})_+=&\{(a,b)\in { \mathcal{L} _f}\times { \mathcal{L} _f}\mid \ell(a)+\ell(b)=a+b\}\\ \label{L+-} ( { \mathcal{L} _f}\times { \mathcal{L} _f})_-=&\{(a,b)\in { \mathcal{L} _f}\times { \mathcal{L} _f}\mid a\neq b \text{ and }\ell(a)-\ell(b)=a-b\}. \end{align} \begin{example}\label{Ex39}
If $d=1$, then $\ell(a)=-a$ and the sets $( { \mathcal{L} _f}\times { \mathcal{L} _f})_\pm$ are empty. If $d$ is any, but $ \mathcal{A} $ is a one-point set $ \mathcal{A} =\{b\}$, then $ \mathcal{L} _f$ is the punched discrete sphere $\{a\in \mathbb{Z} ^d\mid |a|=|b|, a\ne b\}$,
$\ell(a)= b$ for each $a$, and the sets $( { \mathcal{L} _f}\times { \mathcal{L} _f})_\pm$ again are empty. If $d\ge2$ and $| \mathcal{A} |\ge2$, then in general the sets $( { \mathcal{L} _f}\times { \mathcal{L} _f})_\pm$ are non-trivial. See in Appendix~B. \end{example}
Obviously
\begin{equation}\label{obv}
( { \mathcal{L} _f}\times { \mathcal{L} _f})_+\cap ( { \mathcal{L} _f}\times { \mathcal{L} _f})_-=\emptyset\,.
\end{equation} For further reference we note that \begin{lemma}\label{L++-}
If $(a,b)\in ( { \mathcal{L} _f}\times { \mathcal{L} _f})_+\cup ( { \mathcal{L} _f}\times { \mathcal{L} _f})_-$ then $|a|\neq|b|$. \end{lemma}
\proof If $(a,b)\in ( { \mathcal{L} _f}\times { \mathcal{L} _f})_+$ and $|a|=|b|$ then $\ell(a)=\ell(b)$ and we have
$$|a+b|=|2\ell(a)|=2|a|=|a|+|b|$$ which is impossible since $b$ is not proportional to $a$.
If $(a,b)\in ( { \mathcal{L} _f}\times { \mathcal{L} _f})_-$ and $|a|=|b|$ then $\ell(a)=\ell(b)$ and we get $a-b=0$ which is impossible in $ ( { \mathcal{L} _f}\times { \mathcal{L} _f})_-$. \endproof
Our notation now agrees with that of Section \ref{sThePhaseSpace}, where $ \mathcal{Z} = \mathbb{Z} ^d$ is the disjoint union $ \mathbb{Z} ^d = \mathcal{A} \cup \mathcal{L} _f\cup \mathcal{L} _\infty$. Accordingly, the space $Y_\gamma =Y_{\gamma \mathbb{Z} ^d}$ decomposes as \begin{equation}\label{YY} Y_\gamma = Y_{ \mathcal{A} }\oplus Y_{ \mathcal{L} _f}\oplus Y_{\gamma \mathcal{L} _\infty},\qquad Y_\gamma =\{ \zeta = (\zeta_ \mathcal{A} , \zeta_f, \zeta_\infty)\}\,, \end{equation} where $Y_{\gamma \mathcal{A} } = \, \text{span}\, \{\zeta_s, s\in \mathcal{A} \}$, etc. Below in this Section and in Section~\ref{s_4}, the domains $ \mathcal{O} _\gamma ( \sigma , \mu^2, \mu)$ and $ \mathcal{O} _\gamma ( \sigma , \mu)$, as well as the corresponding function spaces, refer the $ \mathcal{Z} $ as above.
\begin{lemma}\label{lem:adm} For $m\notin \mathcal{C} $ the part $z_4^{-2}$ of the Hamiltonian $z_4$ equals \begin{equation}\label{Z421} \begin{split} 3{(2\pi)^{-d}} \Big(& \sum_{(a,b)\in ( { \mathcal{L} _f}\times { \mathcal{L} _f})_+} \frac{ \xi_{\ell(a)}\xi_{\ell(b)}\eta_a\eta_b+ \eta_{\ell(a)}\eta_{\ell(b)}\xi_a\xi_b}{ \lambda_a \lambda_b }\\ + 2&\sum_{(a,b)\in ( { \mathcal{L} _f}\times { \mathcal{L} _f})_-} \frac{ \xi_{a}\xi_{\ell(b)}\eta_{\ell(a)}\eta_b }{ \lambda_a \lambda_b }\Big)\,. \end{split} \end{equation} \end{lemma}
\proof Let $\xi_i\xi_j\eta_k\eta_\ell $ be a monomial in $z_4^{-2}$. We know that $(i,j,-k,-\ell)\in \mathcal{J} $ and $\{|i|,|j|\}=\{|k|,|\ell|\}$.
If $i,j\in \mathcal{A} $ or $k,\ell\in \mathcal{A} $ then we obtain the finitely many monomials as in the first sum in \eqref{Z421}.
Now we assume that
$i,\ell\in \mathcal{A} $ and $ j,k\in \mathcal{L} .$ Then we have that,
{ either } $|i|=|k|$ and
$|j|=|\ell|$ which leads to finitely many monomials as in the second sum in \eqref{Z421}.
{ Or} $i=\ell$ and $|j|=|k|$. In this last case, the zero momentum condition implies that $j=k$ which is not possible in $z_4^{-}$. \endproof
\subsection{Eliminating the non integrable terms} For $\ell\in \mathcal{A} $ we introduce the variables $(I_a, \theta_a, \zeta_ \mathcal{L} )$ as in \eqref{ac-an}, \eqref{iota}. Now the symplectic structure $-{\bf i}d\xi\wedge d\eta$ reads \begin{equation}\label{nsympl} -\sum_{a\in \mathcal{A} } dI_a\wedge d\theta_a -{\bf i} d\xi_ \mathcal{L} \wedge d\eta_ \mathcal{L} \,. \end{equation} In view of \eqref{Z4+}, \eqref{trans} and Lemma~\ref{lem:adm}, for $m\notin \mathcal{C} $ the Hamiltonian $h$, transformed by $\tau\circ\iota$, may be written as \begin{align*} h \circ\tau\circ\iota =&\langle \omega , I\rangle +\sum_{s\in \mathcal{L} } \lambda_s \xi_s\eta_s+ \frac 3 2(2\pi)^{-d} \sum_{\ell\in \mathcal{A} ,\ k\in \mathbb{Z} ^d} (4-3\delta_{\ell,k})\frac{I_\ell \xi_k\eta_k}{\lambda_\ell\lambda_k}\\ +&3(2\pi)^{-d} \Big(\sum_{(a,b)\in ( { \mathcal{L} _f}\times { \mathcal{L} _f})_+} \frac{ \xi_{\ell(a)}\xi_{\ell(b)}\eta_a\eta_b+ \eta_{\ell(a)}\eta_{\ell(b)}\xi_a\xi_b}{ \lambda_a \lambda_b }\\ +&2 \sum_{(a,b)\in ( { \mathcal{L} _f}\times { \mathcal{L} _f})_-} \frac{ \xi_{a}\xi_{\ell(b)}\eta_{\ell(a)}\eta_b }{ \lambda_a \lambda_b }\Big) + q_4^3 \circ\iota+r^0_5\,,
\, \end{align*} where $r^0_5=h_{\ge5}\circ \tau\circ\iota+r^0_6\circ\iota$ (recall that $\omega=(\lambda_a, a\in \mathcal{A} )$). The first line contains the integrable terms. The second and third lines contain the lower-order non integrable terms, depending on the angles $\theta$; there are finitely many of them. The last line contains the remaining high order terms, where $q^3_4$ is of total order (at least) 4 and of order 3 in the {normal directions $\zeta$, while $r^0_5$ is of total order at least 5.
The latter is the sum of $r^0_6\circ\iota$ which comes from the Birkhoff normal form procedure
(and is of order 6) and $h_{\ge5}\circ \tau\circ\iota$ which comes from the term of order 5 in the nonlinearity \eqref{g}. Here $I$ is regarded as a variable of order 2, while $\theta$ has zero order. The terms $q_4^3 \circ\iota$ and $r^0_5$
should be regarded as a perturbation.
To deal with the non integrable terms in the second and third lines, following the works on the finite-dimensional reducibility (see \cite{E01}), we introduce a change of variables $$\Psi: (\tilde I, \tilde\theta, \tilde\xi,\tilde\eta) \mapsto( I, \theta, \xi, \eta)\,,$$
symplectic with respect to \eqref{nsympl}, but such that its differential at the origin is not close to the identity. It is defined by the following relations: \begin{equation*} \begin{split}
& I_\ell=\tilde I_\ell-\sum_{\substack{|a|=|\ell| ,\ a\neq \ell}}{\tilde\xi}_a \tilde\eta_a,\quad \theta_\ell =\tilde \theta_\ell \quad \ell\in \mathcal{A} \,;\\ & \xi_a={\tilde\xi}_a e^{{\bf i} \tilde \theta_{\ell(a)}},\quad \eta_a=\tilde\eta_a e^{-{\bf i} \tilde \theta_{\ell(a)}} \quad a\in { \mathcal{L} _f}\,; \qquad \xi_a={\tilde\xi}_a , \quad \eta_a=\tilde\eta_a \quad a\in \mathcal{L} _\infty. \end{split} \end{equation*}
For any $(\tilde I, \tilde\theta, \tilde\zeta)\in {\mathbf T}_\rho ^{I,\theta}(\nu,\sigma,\mu,\gamma )$ denote by $y=\{y_l, l\in \mathcal{A} \}$ the vector, whose $l$-th component equals
$y_l=\sum_{|a|=|l|\,, a\ne l}\tilde\xi_a\tilde\eta_a$. Then \begin{equation}\label{Inu}
|I- \nu\rho|\le |\tilde I- \nu\rho |+|y|\le c_*^2 \nu\mu^2 +\sum_{a\in \mathcal{L} _f}|\tilde\xi_a\tilde\eta_a|\le 2c_*^2\nu\mu^2\,. \end{equation} This implies that \begin{equation}\label{prop11} \Psi^{\pm1}( {\mathbf T}_\rho ^{I,\theta}\big(\nu, \tfrac12, \frac1{2\sqrt2} ,\gamma \big) ) \subset {\mathbf T}_\rho ^{I,\theta}\big(\nu,\tfrac12, \tfrac12,\gamma \big) =: {\mathbf T}_\rho ^{I,\theta}\,. \end{equation} The transformation $\Psi$ is identity on each torus $\{ (I, \theta, \zeta_ \mathcal{L} ): I=\,$const$, \theta\in \mathbb{T} ^n,\zeta_ \mathcal{L} =0\}$. Writing it as $(I, \theta, \zeta_ \mathcal{L} )\mapsto (\tilde I,\tilde\theta, \tilde\zeta_ \mathcal{L} )$ we see that
\begin{equation}\label{TheRem}
|\tilde I_a-I_a|\leq \| \zeta_ \mathcal{L} \|_\gamma ^2\,,\ a\in \mathcal{A} ,\ \tilde\theta= \theta \text{ and } \| \tilde\zeta_ \mathcal{L} \|_\gamma =\|\zeta_ \mathcal{L} \|_\gamma \,, \end{equation} and that $(\xi, \eta) = \iota (\tilde I,\tilde\theta, \tilde\zeta_ \mathcal{L} )$ satisfies \begin{equation}\label{newxi}
\xi_l=\sqrt{I_l}\,e^{{\bf i}\theta_l} = \sqrt{\tilde I_l}\,e^{{\bf i} \tilde \theta_l} +O (\nu^{-1/2} )\,O(|\zeta_ \mathcal{L} |^2)\,, \quad
l\in \mathcal{A} \,. \end{equation}
Accordingly, dropping the tildes, we write the restriction to $ {\mathbf T}_\rho ^{I,\theta}$ of the
transformed Hamiltonian $h^1= h\circ\tau\circ\iota\circ \Psi$ as \begin{align*} h^1=& \langle \omega , I\rangle +\sum_{a\in \mathcal{L} _\infty} \lambda_a {\xi}_a\eta_a
+6(2\pi)^{-d} \sum_{\ell\in \mathcal{A} ,\ k\in \mathcal{L} } \frac{1}{\lambda_\ell\lambda_k}(I_\ell-\sum_{\substack{|a|=|\ell| \\ a\in { \mathcal{L} _f}}}\xi_a\eta_a) \xi_k\eta_k\\
&+\frac 3 2(2\pi)^{-d} \sum_{\ell,k\in \mathcal{A} } \frac{4-3\delta_{\ell,k}}{\lambda_\ell\lambda_k}(I_\ell-\sum_{\substack{|a|=|\ell| \\ a\in { \mathcal{L} _f}}}\xi_a\eta_a) (I_k-\sum_{\substack{|a|=|k| \\ a\in { \mathcal{L} _f}}}\xi_a\eta_a)\\ &+3(2\pi)^{-d}\sum_{(a,b)\in ( { \mathcal{L} _f}\times { \mathcal{L} _f})_+} \frac{ \sqrt{I_{\ell(a)}I_{\ell(b)}}}{ \lambda_a \lambda_b }(\eta_a\eta_b+
\xi_a\xi_b)\\ &+6(2\pi)^{-d}\sum_{(a,b)\in ( { \mathcal{L} _f}\times { \mathcal{L} _f})_-} \frac{ \sqrt{I_{\ell(a)}I_{\ell(b)}}}{ \lambda_a \lambda_b }\xi_a\eta_b+ q_4^{3'} +r^{0'}_5 +\nu^{-1/2}r^{4'}_5 \,. \end{align*} Here $q^{3'}_4$ and $r^{0'}_5$ are the function $q^{3}_4$ and $r^{0}_5$, transformed by $\Psi$, so the former satisfy the same estimates as the latter, while $r^{4'}_5$ is a function of forth order in the normal variables. The latter comes from re-writing terms like $\xi_{\ell(a)} \xi_{\ell(b)}\eta_a\eta_b$, using \eqref{newxi} and expressing $\eta_a, \eta_b$ via the tilde-variables. Or, after a simplification: \begin{align} \begin{split} \label{H-fin} h^1= &\langle \omega , I\rangle +\sum_{a\in \mathcal{L} _\infty} \lambda_a \xi_a\eta_a +\frac 3 2(2\pi)^{-d} \sum_{\ell,k\in \mathcal{A} } \frac{4-3\delta_{\ell,k}}{\lambda_\ell\lambda_k}I_\ell I_k\\
&+3(2\pi)^{-d} \Big( 2\sum_{\ell\in \mathcal{A} ,\ a\in \mathcal{L} _\infty} \frac{1}{\lambda_\ell\lambda_a}I_\ell \xi_a\eta_a- \sum_{\ell\in \mathcal{A} ,\ a\in { \mathcal{L} _f}} \frac{(2-3\delta_{\ell,|a|})}{\lambda_\ell\lambda_a}I_\ell \xi_a\eta_a\Big)ñ \\ &+3(2\pi)^{-d} \sum_{(a,b)\in ( { \mathcal{L} _f}\times { \mathcal{L} _f})_+} \frac{ \sqrt{I_{\ell(a)}I_{\ell(b)}}}{ \lambda_a \lambda_b }(\eta_a\eta_b+ \xi_a\xi_b)\\ &+6(2\pi)^{-d}\sum_{(a,b)\in ( { \mathcal{L} _f}\times { \mathcal{L} _f})_-} \frac{ \sqrt{I_{\ell(a)}I_{\ell(b)}}}{ \lambda_a \lambda_b }\xi_a\eta_b + q_4^{3'} +r^{0'}_5 +\nu^{-1/2}r^{4'}_5 \,. \end{split}\end{align}
We see that the transformation $\Psi$ removed from $h\circ\tau\circ\iota$ the non-integrable lower-order terms on the price of introducing ``half-integrable" terms which do not depend on the angles $\theta$, but depend on the actions $I$ and quadratically depend on the finitely many variables $\xi_a,\eta_a$ with $a\in { \mathcal{L} _f}$.
The Hamiltonian $h\circ\tau\circ \Psi$ should be regarded as a function of the variables $(I,\theta,\zeta_ \mathcal{L} )$. Abusing notation, below we often drop the lower-index $ \mathcal{L} $ and write $\zeta_ \mathcal{L} = (\xi_ \mathcal{L} ,\eta_ \mathcal{L} )$ as $\zeta= ( \xi,\eta)$.
\subsection{Rescaling the variables and defining the transformation
$\Phi$}
Our aim is to study the transformed Hamiltonian $h^1$ on the domains
$ {\mathbf T}_\rho ^{I,\theta} = {\mathbf T}_\rho ^{I,\theta}(\nu, \frac12, \frac1{2\sqrt2} ,\gamma ) $, $0\le\gamma \le\gamma _g$
(see \eqref{prop11}).
To do this we re-parametrise points of $ {\mathbf T}_\rho ^{I,\theta}$ by mean of the scaling
\begin{equation}\label{scaling}
\chi_\rho : (\tilde r,\tilde\theta,\tilde\xi, \tilde\eta) \mapsto (I,\theta,\xi,\eta)\,,
\end{equation}
where
$\ I=\nu\rho+\nu \tilde r,\quad \theta=\tilde\theta,\quad \xi=\sqrt\nu\, \tilde \xi,\quad \eta=\sqrt\nu\, \tilde\eta\,. $ Clearly, $$ \chi_\rho: \mathcal{O} _{\gamma }(\tfrac 12, \mu_*^2, \mu_* ) \to {\mathbf T}_\rho ^{I,\theta}\, $$ for $0\le\gamma \le\gamma _g$, where $\mu_*$ is defined in \eqref{mu*},
and in the new variables the symplectic structure reads $$ -\nu\sum_{\ell\in \mathcal{A} }\tilde dr_\ell\wedge d\tilde \theta_\ell \ -{\bf i}\ \nu\sum_{a\in \mathcal{L} }d\tilde\xi_a\wedge d\tilde\eta_a. $$ Denoting $$ \Phi =\Phi_\rho=\tau\circ\iota \circ\Psi\circ\chi_\rho, $$ we see that this transformation is analytic in $\rho\in \mathcal{D}$. In view of \eqref{TheRem}, $\zeta=(\xi,\eta) = \Phi(\tilde r, \tilde\theta, \tilde\zeta)$ satisfies $$
\| \zeta- \zeta' \|_\gamma \le C(\sqrt\nu\, (|\tilde r| +\|\zeta\|_\gamma ))\,,\qquad \zeta'=\big( \sqrt{\nu\rho} \,e^{{\bf i}\tilde\theta}, \sqrt{\nu\rho} \,e^{{\bf i}\tilde\theta}, 0\big)\,. $$ This relation and \eqref{esti1} imply \eqref{boundPhi}, so the assertion (i) of the proposition holds.
Dropping the tildes and forgetting the irrelevant constant $\nu\langle \omega , \rho\rangle$, we have \begin{align} \begin{split} \label{H-rescall} h \circ \Phi(r,\theta,\zeta) &=\nu\Big[ \langle \omega , r\rangle + \sum_{a\in \mathcal{L} _\infty} \lambda_a \xi_a\eta_a +(2\pi)^{-d}\nu\, \Big(\, \frac 3 2
\sum_{\ell,k\in \mathcal{A} } \frac{4-3\delta_{\ell,k}}{\lambda_\ell\lambda_k}\rho_\ell r_k\\
+6 &\sum_{\ell\in \mathcal{A} ,\ a\in \mathcal{L} _\infty} \frac{1}{\lambda_\ell\lambda_a}\rho_\ell \xi_a\eta_a-3 \sum_{\ell\in \mathcal{A} ,\ a\in { \mathcal{L} _f}} \frac{(2-3\delta_{\ell,|a|})}{\lambda_\ell\lambda_a}\rho_\ell \xi_a\eta_a\\ +3 &\sum_{(a,b)\in ( { \mathcal{L} _f}\times { \mathcal{L} _f})_+} \frac{ {\sqrt{\rho_{\ell(a)}}\sqrt{\rho_{\ell(b)}}}}{ \lambda_a \lambda_b }(\eta_a\eta_b+ \xi_a\xi_b)\\ +6 & \sum_{(a,b)\in ( { \mathcal{L} _f}\times { \mathcal{L} _f})_-} \frac{ {\sqrt{\rho_{\ell(a)}}\sqrt{\rho_{\ell(b)}}}}{ \lambda_a \lambda_b }\xi_a\eta_b\Big) \Big] \\ &+\Big(\big(q_4^{3'} +r^{0'}_5+\nu^{-1/2}r^{4'}_5\big)(I,\theta,\sqrt\nu\zeta)\Big)\mid_{I=\nu\rho+\nu r}\,, \end{split}\end{align} where $\zeta=\zeta_ \mathcal{L} =(\zeta_a)_{a\in \mathcal{L} },\ \zeta_a=(\xi_a,\eta_a)$, and $\zeta_f=(\zeta_a)_{a\in { \mathcal{L} _f}}$. So, \begin{equation}\label{f} {\nu}^{-1} h \circ \Phi = \tilde h_2+ f\,, \end{equation}
where $f$ is the perturbation, given by the last line in \eqref{H-rescall},
\begin{equation}\label{ff}
f=\nu^{-1}\Big(\big(q_4^{3'} +r^{0'}_5 +\nu^{-1/2}r^{4'}_5\big)(I,\theta,\nu^{1/2}\zeta)\Big)\mid_{I=\nu\rho+\nu r}\,,
\end{equation} and
$\tilde h_2= \tilde h_2( I, \xi,\eta;\rho,\nu)$ is the quadratic part of the Hamiltonian, which is
independent from the angles $\theta$: $$ \tilde h_2=\langle \Omega, r\rangle + \sum_{a\in \mathcal{L} _\infty}\Lambda_a \xi_a\eta_a+\nu \langle K(\rho)\zeta_f,\zeta_f\rangle\,. $$ Here $\Omega=(\Omega_k)_{k\in \mathcal{A} }$ with \begin{align}\label{Om}\Omega_k=\Omega_k(\rho,\nu)&= \omega _k+ \nu\sum_{\ell\in \mathcal{A} } M^\ell_k \rho_l,\quad
M^\ell_k=\frac{3(4-3\delta_{\ell,k})}{(2\pi)^d \lambda_k \lambda_\ell} \,, \\ \label{Lam}\Lambda_a=\Lambda_a(\rho,\nu)&= \lambda_a +6\nu(2\pi)^{-d}\sum_{\ell\in \mathcal{A} }\frac{\rho_\ell}{\lambda_\ell \lambda_a }\,, \end{align} and $K(\rho) $ is a symmetric complex matrix, acting in the space \begin{equation}\label{Yf}
Y_{ \mathcal{L} _f}=\{\zeta_f\}\simeq \mathbb{C} ^{2| \mathcal{L} _f|}\,, \end{equation} such that the corresponding quadratic form is \begin{align} \begin{split} \label{K} \langle K(\rho)\zeta_f,\zeta_f\rangle=\,&3(2\pi)^{-d} \Big(\sum_{\ell\in \mathcal{A} ,\ a\in { \mathcal{L} _f}} \frac{(
3\delta_{\ell,|a|}-2 )}{\lambda_\ell\lambda_a} \rho_\ell \xi_a\eta_a\\ + \sum_{(a,b)\in ( { \mathcal{L} _f}\times { \mathcal{L} _f})_+}& \frac{ {\sqrt{\rho_{\ell(a)}}\sqrt{\rho_{\ell(b)}}}}{ \lambda_a \lambda_b }(\eta_a\eta_b + \xi_a\xi_b)+ \\ 2 \sum_{(a,b)\in ( { \mathcal{L} _f}\times { \mathcal{L} _f})_-} &\frac{ {\sqrt{\rho_{\ell(a)}}\sqrt{\rho_{\ell(b)}}}}{ \lambda_a \lambda_b }\xi_a\eta_b \Big). \end{split}\end{align} Note that the matrix $M$ in \eqref{Om} is invertible since $$ \det M={3^n}{(2\pi)^{-dn}}\big(\Pi_{k\in \mathcal{A} } \lambda_k \big)^{-2}\det \left(4-3\delta_{\ell,k}\right)_{\ell,k\in \mathcal{A} }\ne0\,. $$ The explicit formulas \eqref{Om}-\eqref{K} imply the assertions (ii) and (iii).
The transformations $\Psi\circ\chi_\rho$ and $\tau\circ\iota$ both are real if we use in the spaces $Y_\gamma $ and $Y_{\gamma \, \mathcal{L} }$ the real coordinates $(p_a, q_a)$, see \eqref{change}. This implies the stated ``conjugate-reality" of $\Phi_\rho$.
It remains to verify (iv). By Proposition \ref{Thm-BNF} the function $f$ belongs to the class
$ {\Tc}_{\gamma _g, 2} ({\tfrac 12},\mu_*^2, \mu_*) $.
Since the reminding term $f$ has the form \eqref{ff} then in view of \eqref{Z4}-\eqref{R66} for $(r,\theta,\zeta)\in \mathcal{O} _{\gamma }({\tfrac 12}, \mu_*^2, \mu_* )$ it satisfies the estimates \begin{equation*} \begin{split}
|f|\le C \nu \,,\quad
\| \nabla_\zeta f\|_{\gamma } \le C \nu \,,\quad
\|\nabla^2_\zeta f\|_{\gamma , 2}^b \le C \nu \,. \end{split} \end{equation*} Now consider the $f^T$-component of $f$. Only the second term in \eqref{ff} contributes to it and we have that $$
|f^T| + \| \nabla_\zeta f^T\|_\gamma +\|\nabla_\zeta^2 f^T\|^b_{\gamma ,2}\le C\nu^{3/2}\,. $$
This implies the assertion (iv) of the proposition in view of \eqref{O_relation} and \eqref{twonorms1}. \endproof
We will provide the domains $ \mathcal{O} _{\gamma } \big({\frac 12}, \mu_*^2 \big)\subset \mathcal{O} _{\gamma } \big({\frac 12}, \mu_*^2, \mu_* \big) = \{(r,\theta,\xi,\eta)\}$ with the symplectic structure $- \sum_{\ell\in \mathcal{A} }dr_\ell\wedge d \theta_\ell \ -{\bf i} \sum_{a\in \mathcal{L} }d\xi_a\wedge d\eta_a$. Then the transformed Hamiltonian system, constructed in Proposition~\ref{thm-HNF} has the Hamiltonian, given by the r.h.s. of \eqref{HNF}.
\section{The Birkhoff normal form. II} \label{s_4}
In this section we shall refine the normal form \eqref{HNF} further. We shall construct a $\rho$-dependent transformation which diagonalises the Hamiltonian operator (modulo the term $f$) and shall examine its
smoothness in $\rho$. So here we are concerned with
analysis of the finite-dimensional linear Hamiltonian operator ${\mathbf i}JK( \rho )$ defined by the Hamiltonian
\eqref{K}. To do this we will have to restrict $ \rho $ to some (large) subset $Q\subset \mathcal{D}= [c_*,1]^ \mathcal{A} $.
In this section and below $c_*$ is regarded as a parameter of the construction, belonging to an
interval $(0,\tfrac12 c_0]$, where $c_0>0$ depends on $m$ and on the constants in \eqref{depen}. This $c_0$ is introduced in Lemma~\ref{laK} and is fixed after it.
The parameter $c_*$ will be fixed till Section~\ref{s_10.2} (the last in our work),
where we will vary it.
In this section we shall also shift from the conjugate-reality to the ordinary reality,
thus restoring the original real character of the system.
\begin{theorem}\label{NFT} There exists a zero-measure Borel set $ \mathcal{C} \subset[1,2]$ such that for any admissible set $ \mathcal{A} $ and any $m\notin \mathcal{C} $
there exist
real numbers $\gamma _g>\gamma _*=(0,m_*+2)$ and
$\beta_0, \nu_0,c_0 >0$,
where $c_0$, $\beta_0$, $\nu_0$ depend on $ m$, such that, for any $0<c_*\le c_0$, $0<\nu\le\nu_0$ and $0<\beta_{\#} \le\beta_0$
there exists an open set $Q=Q(c_*,\beta_{\#},\nu) \subset [c_*,1]^ \mathcal{A} $, increasing as $\nu\to0$ and satisfying \begin{equation}\label{mesmes} \operatorname{meas} ([c_*,1]^ \mathcal{A} \setminus Q ) \le C\nu^{\beta_{\#}}\,, \end{equation} with the following property.
For any $ \rho \in Q$ there exists a real holomorphic diffeomorphism (onto its image) \begin{equation}\label{mu*bis} \Phi_\rho: \mathcal{O} _{\gamma _*} \big({\tfrac 12}, {\mu_*^2} \big)\to {\mathbf T}_\rho (\nu, 1,1,\gamma _*)\,,\qquad {\mu_*}={\tfrac{c_*}{2\sqrt2}}, \end{equation} which defines analytic diffeomorphisms $ \Phi_\rho: \mathcal{O} _{\gamma } \big({\frac 12}, {\mu_*^2} \big)\to {\mathbf T}_\rho (\nu, 1,1,\gamma )$, $ \gamma _*\le\gamma \le\gamma _g\,, $ such that \begin{equation}\label{new_symplec} \Phi_\rho^*\big(d\xi\wedge d\eta\big)= \nu dr_ \mathcal{A} \wedge d\theta_ \mathcal{A} \ +\ \nu d p_ \mathcal{L} \wedge d q_ \mathcal{L} , \end{equation} and \begin{equation}\label{HNFbis} \begin{split} \frac1{\nu} \,h\circ&\Phi_\rho(r,\theta,p_ \mathcal{L} ,q_ \mathcal{L} ) =\langle \Omega(\rho), r\rangle + \frac12 \sum_{a\in \mathcal{L} _\infty }\Lambda_a (\rho) ( p_{a}^2 + q_{a}^2)+\\ &+ \frac12\sum_{b\in \mathcal{L} _f\setminus \mathcal{F} } \Lambda_b(\rho) ( p_{b}^2 + q_{b}^2)
+ \nu \langle K(\rho) _ \mathcal{F} , \zeta_ \mathcal{F} \rangle
+ f(r,\theta, \zeta_ \mathcal{L} ; \rho), \end{split} \end{equation} where $ \mathcal{F} = \mathcal{F} _ \rho \subset \mathcal{L} _f$ (only depending on the connected component of $Q$ containing $\rho$), and $h$ is the Hamiltonian \eqref{H2}$+$\eqref{H1}. $\Phi_\rho$ satisfies:
(i) $\Phi_\rho$ depends smoothly on $\rho$ and \begin{equation}\label{boundPhibis} \begin{split} \mid\mid \Phi_ \rho (r,\theta,\xi_ \mathcal{L} ,\eta_ \mathcal{L} )-(\sqrt{\nu \rho }\cos(\theta),&\sqrt{\nu \rho }\sin(\theta),\sqrt{\nu \rho }\xi_ \mathcal{L} ,\sqrt{\nu \rho }\eta_ \mathcal{L} ) \mid\mid_\gamma \le \\ &\le C(\sqrt\nu\ab{r}+\sqrt\nu\aa{(\xi_ \mathcal{L} ,\eta_ \mathcal{L} )}_\gamma +\nu^{\frac32} )\nu^{-\hat c\beta_{\#}} \end{split} \end{equation} for all $(r,\theta,\xi_ \mathcal{L} ,\eta_ \mathcal{L} )\in \mathcal{O} _{\gamma } (\frac 12, \mu_*^2)\cap\{\theta\ \textrm{real}\}$ and all $\gamma _*\le \gamma \le\gamma _g$.
(ii) the vector $ \Omega$ and the scalars $ \Lambda_a , a\in \mathcal{L} _\infty $, are affine functions of $\rho$, explicitly defined \eqref{Om}, \eqref{Lam};
(iii) the functions $\Lambda_b(\rho)$, $b\in \mathcal{L} _f\setminus \mathcal{F} $, are smooth in $Q $, \begin{equation}\label{Lambdab}
\|\Lambda_b\|_{C^j(Q )}\le C_j\nu^{- \beta_{\#}\beta(j)}\nu ,\qquad \forall j\ge0, \end{equation} where $0<\beta(1)\le\beta(2)\le\dots$, and satisfy \eqref{K4}. In some open subset of $[c_*,1]^ \mathcal{A} $ they also satisfy \eqref{aaa}.
(iv) $ K$ is a symmetric real matrix that depends smoothly on $\rho\in Q $, and
\begin{equation}\label{normK}
\sup_{\rho\in Q }
\|\partial^j_\rho K(\rho)\| \le C_j\nu^{-\beta_{\#}\beta(j) },\qquad \forall j\ge0\,. \end{equation}
The set $ \mathcal{F} = \mathcal{F} _ \rho $ is void for some $ \rho $
(in which case the operator $K(\rho)$ is trivial).
(v) the eigenvalues $\{ \pm{\bf i}\Lambda_{a}, a\in \mathcal{F} \}$ of $J K$ are smooth in $Q $,
satisfy \eqref{Lambdab} and \begin{equation}\label{hyperb}
\inf_{\rho\in Q } | \Im \Lambda_a(\rho) | \ge C^{-1} \nu^{ \bar c \beta_{\#}},\qquad \forall a\in \mathcal{F} \,. \end{equation}
(vi) There exists a complex symplectic operator $U(\rho)$ such that $$ U(\rho)^{-1} J K(\rho) U(\rho) ={\bf i}\operatorname{diag} \{\pm \Lambda_{a}(\rho), a\in \mathcal{F} \}\,. $$ The operator $U(\rho)$ smoothly depends on $\rho$ and satisfies \begin{equation}\label{Ubound} \sup_{\rho\in Q }
\big( \|\partial^j_\rho U (\rho)\| + \|\partial^j_\rho U (\rho)^{-1} \|) \le C_j\nu^{-\beta_{\#}\beta(j) },
\qquad \forall j\ge0\,.
\end{equation}
vii) $ f$ belongs to $ \mathcal{T} _{\gamma , \varkappa =2,Q}({\tfrac12}, \mu_*^2)$ and satisfies \begin{equation}\label{estbis}
| f|_{\begin{subarray}{c}1/2,\mu_*^2 \ \\ \gamma _g, 2, Q \end{subarray}}
\le C\nu^{- \hat c \beta_{\#}}\nu \,, \qquad
| f^T|_{\begin{subarray}{c}1/2,\mu_*^2 \ \\ \gamma _g, 2, Q \end{subarray}}
\le C\nu^{- \hat c \beta_{\#}}\nu^{3/2} \,. \end{equation}
The set $Q$ and the matrix $K(\rho)$ do not depend on the function $G$ (having the form \eqref{g}). The constants $C, C_j$ are as in \eqref{depen}, while
the exponents $\bar c, \hat c$ and $\beta(j)$ depend on $m$ (we recall \eqref{depen}). \end{theorem}
\begin{remark*} 1) By \eqref{new_symplec} the transformation $\Phi_\rho$ transforms the beam equation, written in the form \eqref{beam2}, to a system, which has the Hamiltonian \eqref{HNFbis} with respect to the symplectic structure $dr_ \mathcal{A} \wedge d\theta_ \mathcal{A} +\nu d p_ \mathcal{L} \wedge d q_ \mathcal{L} $.
2) We also have $\ \Phi_\rho\big ( \mathcal{O} _{\gamma } ({\frac 12}, {\mu_*^2}) \big)\subset {\mathbf T}_\rho (\nu, 1,1,\gamma )$ for $ \gamma _*\le \gamma \le\gamma _g. $ \end{remark*}
The remaining part of this section is devoted to the proof of this result.
\subsection{Matrix $K(\rho)$} \label{s_3.6} Recalling \eqref{labA} and \eqref{DDD}, we write the
symmetric matrix $K(\rho)$, defined by relation \eqref{K}, as a block-matrix, polynomial in $ \sqrt\rho = (\sqrt\rho_1, \dots, \sqrt\rho_n) \,. $ We write it as $ K(\rho) = K^d(\rho) + K ^{n/d} (\rho)$. Here $ { K}^d $ is the block-diagonal matrix \begin{equation}\label{.1} \begin{split} { K}^d (\rho) &=\text{diag}\,\Big( \left(\begin{array}{ll} 0& \mu(a,\rho)\\ \mu(a,\rho)& 0 \end{array}\right), \ a\in \mathcal{L} _f\Big), \\ \mu(a,\rho) &= C_* \big(\frac32\,\rho_{\ell(a)} \lambda_a ^{-2}- \lambda_a ^{-1} \sum_{l\in \mathcal{A} } \rho_l\lambda_l^{-1}\big)\,,\quad C_*=3(2\pi)^{-d}. \end{split} \end{equation} Note that\footnote{Here and in similar situations below we do not mention the obvious dependence on the parameter $m\in[1,2]$.} \begin{equation}\label{munew}
\mu(a,\rho)\quad \text{is a function of $|a|$ and $\rho$\,. } \end{equation}
The non-diagonal matrix $ K ^{n/d} $ has zero diagonal blocks, while for $a\ne b$ its block $ K ^{n/d} (\rho)_a^b $ equals $$
C_*\frac{\sqrt{ {\rho_{l(a)} \rho_{l(b)}}}}{ \lambda_a \lambda_b } \, \left(
\left(\begin{array}{ll} 1& 0\\ 0& 1 \end{array}\right) \chi^+(a,b)+
\left(\begin{array}{ll} 0& 1\\ 1& 0 \end{array}\right) \chi^-(a,b) \right)\,, $$ where \begin{equation*} \chi^+(a,b)=
\left\{\begin{array}{ll} 1, \;\; (a,b)\in ( \mathcal{L} _f\times \mathcal{L} _f)_+, \\ 0, \;\; \text {otherwise}, \end{array}\right. \end{equation*} and $\chi^-$ is defined similar in terms of the set $ ( \mathcal{L} _f\times \mathcal{L} _f)_-$. In view of \eqref{obv}, $$ \chi^+(a,b)\cdot \chi^-(a,b)\equiv0. $$
Accordingly, the Hamiltonian matrix
$ \mathcal{H} (\rho) =={\bf i}JK(\rho)$ equals $ \big( \mathcal{H} ^d(\rho) + \mathcal{H} ^{n/d}(\rho)\big)$, where \begin{align}\begin{split}\label{diag} \mathcal{H} ^d(\rho) &={\bf i}\,\text{diag}\, \left(
\left(\begin{array}{ll} \mu(a,\rho)& 0\\ 0& -\mu(a,\rho) \end{array}\right) ,\; a\in \mathcal{L} _f \right),\\
\mathcal{H} ^{n/d}(\rho) _a^b &={\bf i}C_* \frac{{\sqrt{\rho_{\ell(a)}} \sqrt{\rho_{\ell(b)}}}}{ \lambda_a \lambda_b }\Big[
J
\chi^+{(a,b)} +\left(\begin{array}{ll} 1& 0\\ 0& -1 \end{array} \right) \chi^-{(a,b)}
\Big] \,. \end{split}\end{align}
Note that all elements of the matrix $ \mathcal{H} (\rho)$ are pure imaginary, and \begin{equation}\label{Lf+=0} \text{ if
$( { \mathcal{L} _f}\times { \mathcal{L} _f})_+=\emptyset$, then $-{\bf i} \mathcal{H} (\rho)$ is real symmetric},
\end{equation}
in which case all eigenvalues of $ \mathcal{H} (\rho)$ are pure imaginary. In Appendix~B we show that if $d\ge2$, then, in general, the set $( { \mathcal{L} _f}\times { \mathcal{L} _f})_+ $ is not empty and the matrix $ \mathcal{H} (\rho)$ may have hyperbolic eigenvalues.
\begin{example}\label{Ex41} In view of Example \ref{Ex39}, if $d=1$ then the operator $ \mathcal{H} ^{n/d}$ vanishes. We see immediately that in this case $ \mathcal{H} ^{d}$ is a diagonal operator with simple spectrum. \end{example}
Let us introduce in $ \mathcal{L} _f$ the relation $\sim$, where \begin{equation}\label{class} a\sim b\;\; \text{if and only if }\;\; a=b\;\;\text{or}\;\; (a,b)\in ( \mathcal{L} _f\times \mathcal{L} _f)_+\cup ( \mathcal{L} _f\times \mathcal{L} _f)_-\,. \end{equation} It is easy to see that this is an equivalence relation. By Lemma~\ref{L++-} \begin{equation}\label{.2}
a\sim b,\ a\ne b \;\Rightarrow \; |a| \ne |b|\,\,. \end{equation} The equivalence $\sim\,$, as well as the sets $ ( \mathcal{L} _f\times \mathcal{L} _f)_\pm$, depends only on the lattice $ \mathbb{Z} ^d$ and the set $ \mathcal{A} $, not on the eigenvalues $ \lambda_a $ and the vector $\rho$. It
is trivial if $d=1$ or $| \mathcal{A} |=1$ (see Example~\ref{Ex39})) and, in general, is non-trivial otherwise. If $d\ge2$ and $| \mathcal{A} |\ge2$
it is rather complicated.
The equivalence relation divides $ \mathcal{L} _f$ into equivalence classes, $ \mathcal{L} _f = \mathcal{L} _f^1\cup\dots \cup \mathcal{L} _f^M\,. $
The set $ \mathcal{L} _f$ is a union of the punched spheres $\Sigma_a=\{b\in \mathbb{Z} ^d\mid |b|=|a|, b\ne a\}$, $a\in \mathcal{A} $, and by \eqref{.2} each equivalence class $ \mathcal{L} _f^j$ intersects every punched sphere $\Sigma_a$
at at most one point.
Let us order the sets $ \mathcal{L} _f^j$ in such a way that for a suitable $0\le M_0\le M$ we have
-- $ \mathcal{L} _f^j = \{b_j\}$ (for a suitable point $b_j\in \mathbb{Z} ^d)$ if $j\le M_0$;
-- $| \mathcal{L} _j^f|=n_j\ge2$ if $j>M_0$.\\
Accordingly the complex space $Y_{ \mathcal{L} _f}$ (see \eqref{YY}) decomposes as \begin{equation}\label{deco} Y_{ \mathcal{L} _f}= Y^{f1}\oplus\dots \oplus Y^{fM},\quad Y^{fj}= \,\text{span}\, \{\zeta_s, s\in \mathcal{L} _f^j\}\,. \end{equation} Since each $\zeta_s, s\in \mathcal{L} _f$, is a 2-vector, then $$
\dim Y^{f j}=2| \mathcal{L} _j^j|:= 2n_j\,,\qquad \dim Y_{ \mathcal{L} _f}=2| \mathcal{L} _f| = 2\sum_{j=1}^M n_j := 2\mathbf N\,. $$ So dim$\,Y^{f j}=2$ for $j\le M_0$ and dim$\,Y^{f j}\ge4$ for $j> M_0$. In view of \eqref{.2}, \begin{equation}\label{.0}
| \mathcal{L} ^j_f | = n_j \le | \mathcal{A} |\qquad \forall\, j\,. \end{equation}
We readily see from the formula for the matrix $ \mathcal{H} (\rho)={\bf i}J K(\rho)$ that the spaces $ Y^{fj}$ are invariant for the operator $ \mathcal{H} (\rho)$. So \begin{equation}\label{decomp} \mathcal{H} (\rho)= \mathcal{H} ^1(\rho)\oplus\dots \oplus \mathcal{H} ^M(\rho) \,,\qquad \mathcal{H} ^j = \mathcal{H} ^{j\,d} + \mathcal{H} ^{j\,n/d}, \end{equation} where $ \mathcal{H} ^j$ operates in the space $Y^{fj}$, so this is a block of the matrix $ \mathcal{H} (\rho)$. The operators $ \mathcal{H} ^{j\,d}$ and $ \mathcal{H} ^{j\,n/d}$ are given by the formulas \eqref{diag} with $a,b\in \mathcal{L} _f^j$. The Hamiltonian operator $ \mathcal{H} ^j(\rho)$ polynomially depends on $\sqrt\rho$, so its eigenvalues form an algebraic function of $\sqrt\rho$. Since the spectrum of $ \mathcal{H} ^j(\rho)$ is an even set, then we can write branches of this algebraic function as $\{ \pm{\bf i}\Lambda_1^j(\rho), \dots, \pm{\bf i}\Lambda_{n_j}^j(\rho)\}$ (the factor ${\bf i}$ is convenient for further purposes). The eigenvalues of $ \mathcal{H} (\rho)$ are given by another algebraic function and we write its branches as
$\{\pm{\bf i} \Lambda_m(\rho), 1\le m\le \mathbf N=| \mathcal{L} _f|\}$. Accordingly, \begin{equation}\label{spectrum} \{\pm \Lambda_1(\rho),\dots,\pm \Lambda_{\mathbf N}(\rho)
\} = \cup_{j\le M} \{\pm \Lambda_k^j(\rho), k\le n_j\}\,,
\end{equation}
and $ \Lambda_j= \Lambda^j_1$ for $j\le M_0$.
The functions $ \Lambda_k$ and $ \Lambda^j_k$ are defined up to multiplication by $\pm1$.\footnote{More precisely, if $ \Lambda_k$
is not real, then well defined is the quadruple $\{\pm \Lambda_k, \pm\bar \Lambda_k\}$; see below Section~\ref{s_bd}.
}
But if $j\le M_0$, then
$ \mathcal{L} ^j_f = \{b_j\}$ and $ \mathcal{H} ^j = \mathcal{H} ^{jd}$, so the spectrum of this operator is $\{\pm{\bf i}\mu(b_j,\rho)\}$,
where $\mu(b_j,\rho)$ is a well defined analytic function of $\rho$, given by the explicit formula \eqref{.1}.
In this case we specify the choice of
$ \Lambda^j_1$:
\begin{equation}\label{normaliz}
\text{ if $\ \mathcal{L} ^j_f=\{b_j\}$, we choose $ \Lambda^j_1(\rho) = \mu(b_j,\rho)$. }
\end{equation}
So for $j\le M_0$,
$ \Lambda_j(\rho) = \mu(b_j,\rho)$ is a polynomial of $\sqrt\rho$, which depends only on $|b_j|$ and $\rho$.
Since the norm of the operator $K(\rho)$ satisfies \eqref{esti1}, then
\begin{equation} \label{N1}
| \Lambda^j_r(\rho)|\le C_2 \quad \forall\,\rho, \ \forall\,r\,,\; \forall\,j\,. \end{equation}
\begin{example}\label{n=1}
In view of \eqref{.0},
if
$ \mathcal{A} =\{a_*\}$, then all sets $| \mathcal{L} ^j_f|$ are one-point. So $M_0=M=\mathbf N$ and
$$
\{\pm \Lambda_1(\rho),\dots,\pm \Lambda_{\mathbf N}(\rho)\} =
\{ \pm\mu(a,\rho)\mid a\in \mathbb{Z} ^d, |a| = |a_*| , a\ne a_*\}.
$$
In this case the spectrum of the Hamiltonian operator $ \mathcal{H} (\rho)$ is pure imaginary and multiple. It analytically
depends on $\rho$.
\end{example}
Let $1\le j_*\le n$ and
$ \mathcal{D}^{j_*}_0$ be the set \begin{equation}\label{DD}
\mathcal{D}^{j_*}_0 = \{\rho=(\rho_1,\dots, \rho_n)\mid c_* \le \rho_l\le c_0 \;\;\text{if}\;\;
l\ne j_* \;\;\text{and}\;\; 1-c_0 \le \rho_{j_*}\le 1\}\,,
\end{equation} where $0<c_*\le\tfrac12 c_0<1/4$. Its measure satisfies $$ \operatorname{meas} \mathcal{D}_0^{j_*} \ge \tfrac12 c_0^n\,. $$ This is a subset of $ \mathcal{D}=[c_*,1]^n$ which lies in the (Const$\,c_0$)-vicinity of the point
$\rho_*=(0,\dots,1,\dots,0)$ in $[0,1]^n$, where 1 stands on the $j_*$-th place. Since
$ K ^{n/d} (\rho_*)=0$, then $K(\rho_*) = K^d(\rho_*)$.
Consider any equivalence class $ \mathcal{L} _f^j$ and enumerate its elements as $b^j_1,\dots,b^j_{n_j}$ $(n_j\le n)$.
For $\rho=\rho_*$ the matrix $ \mathcal{H} ^j(\rho_*)$ is diagonal with the eigenvalues
$\pm{\bf i}\mu(b^j_r,\rho_*), 1\le r\le n_j$. This suggests that for $c_0$ sufficiently small we may uniquely
numerate the
eigenvalues $\{\pm{\bf i}\Lambda^j_r(\rho)\} \ (\rho\in \mathcal{D}^{j_*}_0)$ of the matrix $ \mathcal{H} ^j(\rho)$ in such a way that
$\Lambda^j_r(\rho)$ is close to $\mu(b^j_r,\rho_*)$. Below we justify this possibility.
Take any $b\in \mathcal{L} _f$ and denote $\ell(b)=a_b\in \mathcal{A} $. If $a_b = a_{j_*}$, then \begin{equation}\label{mu1} \mu(b, \rho_*) = C_*(\frac32 \lambda_{a_{j_*}}^{-2}-\lambda_{a_{j_*}}^{-2})= \tfrac12C_* \lambda_{a_{j_*}}^{-2}\,. \end{equation} If $a_b \ne a_{j_*}$, then \begin{equation}\label{mu2} \mu(b, \rho_*) = -C_* \lambda_{a({b})}^{-1} \lambda_{a_{j_*}}^{-1}. \end{equation} If $m\in[1,2]$ is different from $4/3$ and $ 5/3$, then it is easy to see that $2 \lambda_a \ne \pm \lambda_{a'}$
for any $a, a'\in \mathcal{A} $. By \eqref{modif} this
implies that for $m\in[1,2]\setminus \mathcal{C} $ and for $ b, b'\in \mathcal{L} _f$ such that
$|b|\ne |b'|$ we have $$
|\mu(b, \rho_*) | \ge 2c^{\#}(m)>0\,,\quad
|\mu(b, \rho_*) \pm \mu(b', \rho_*) | \ge 2c^{\#}(m)\,,\quad
$$ and \begin{equation}\label{N110}
|\mu(b,\rho)|\ge c^{\#}(m)>0\,,\qquad |\mu(b,\rho) \pm \mu(b',\rho) |\ge c^{\#}(m)\;\;\;\text{for}\;\; \rho\in \mathcal{D}_0^{j_*}\,, \end{equation} if $c_0$ is small. In particular, for each $j$ the spectrum $\pm{\bf i}\mu(b^j_r,\rho_*), 1\le r\le n_j$ of the matrix $ \mathcal{H} ^j(\rho_*)$ is simple.
\begin{lemma}\label{laK} If $c_0 \in(0,1/2)$ is sufficiently small,\footnote{Its smallness only depends on $ \mathcal{A} , m$ and $g(\cdot)$.}
then there exists $c^o=c^o(m)>0$ such that for each $r$ and $j$,
$\Lambda^j_r(\rho)$ is a real analytic function of $\rho\in \mathcal{D}^{j_*}_0$, satisfying \begin{equation}\label{N111} \begin{split}
|\Lambda^j_r(\rho) - \mu(b^j_r,\rho)|
\le C\sqrt{c_0}\qquad \forall\, \rho\in D^{j_*}_0\,,
\end{split} \end{equation} and \begin{equation}\label{N11}
|\Lambda^j_r(\rho)|\ge c^o(m)>0
\;\;\text{and}\;\; |\Lambda^j_r(\rho)\pm\Lambda^j_l(\rho)
|\ge c^o(m)\;\; \forall \, r\ne l, \forall\, j, \forall\, \rho\in D^{j_*}_0\,, \end{equation} \begin{equation}\label{aaa}
| \Lambda^{j_1}_{r_1}(\rho) + \Lambda^{j_2}_{r_2}(\rho) | \ge c^{0}(m)\quad \forall \, j_1, j_2, r_1, r_2 \quad \text{and}\;\; \rho\in D^{j_*}_0\,. \end{equation} In particular, \begin{equation} \label{N3} \Lambda^j_r \not\equiv 0\quad \forall r;\quad \Lambda^j_r \not\equiv \pm\Lambda^j_l \quad \forall\, r\ne l\,. \end{equation} \end{lemma}
The estimate \eqref{N111} assumes that for $\rho\in \mathcal{D}_0^{j_*}$ we fix the sign of the function $\Lambda_r^j$ by the following agreement: \begin{equation}\label{agreem} \Lambda_j^r(\rho) \in \mathbb{R}\;\;\text{and \ \ sign}\, \Lambda_r^j(\rho) = \ \text{sign}\,\mu(b_r^j,\rho)\;\; \forall \rho\in \mathcal{D}_0^{j_*}\,,\; \forall 1\le j_*\le n\,,\; \forall\, r, j\,, \end{equation} see \eqref{mu1}, \eqref{mu2}.
Below we fix any $c_0 =c_0( \mathcal{A} , m, g(\cdot))\in(0,1/2)$ such that the lemma's assertion holds, but the parameter $c_*\in(0, \tfrac12 c_0]$ will vary at the last stage of our proof, in Section~\ref{s_10.2}.
\begin{proof} Since the
spectrum of $ \mathcal{H} ^j(\rho_*)$ is simple and the matrix $ \mathcal{H} ^j(\rho)$ and the numbers $\mu(b^j_r,\rho)$ are polynomials of $\sqrt\rho$, then the basic perturbation theory implies that the functions $\Lambda^j_r(\rho)$ are real analytic in $\sqrt\rho$ in the vicinity of $\rho_*$ and we have $$
|\mu(b^j_r,\rho_*) - \mu(b^j_r,\rho)|\le C\sqrt{c_0}\,,\quad
|\Lambda^j_r(\rho_*) - \Lambda^j_r(\rho)| \le C\sqrt{c_0}\,. $$ So \eqref{N111} holds. It is also clear that the functions $\Lambda^j_r(\rho)$ are analytic in $\rho\in \mathcal{D}_0^{j_*}$. Relations \eqref{N111} and \eqref{N110} (and the fact that $\mu(b,\rho)$ depends only on
$|b|$ and $\rho$) imply \eqref{N11} and \eqref{aaa}
if $c_0>0$ is sufficiently small. \end{proof}
\begin{remark}\label{r_m}
The differences $|2 \lambda_a - \lambda_b |$ can be estimated from below uniformly in $a,b$ in terms of the distance from $m\in[1,2]$ to the points $4/3$ and $5/3$. So the constants $c^{\#}$ and $c^o$ depend only on this distance, and they can be chosen independent from $m$ if the latter belongs to the smaller segment $[1, 5/4]$. \end{remark}
Contrary to \eqref{aaa}, in general a difference of two eigenvalues $\Lambda^{j_1}_{r_1}- \Lambda^{j_2}_{r_2}$ may vanish identically. Indeed, if $j,k\le M_0$, then $ \mathcal{L} ^k_f$ and $ \mathcal{L} ^j_f$ are one-point sets, $ \mathcal{L} ^k_f=\{b_k\}$ and $ \mathcal{L} ^j_f=\{b_j\}$, and $ \Lambda^j_1=\mu(b_j,\cdot)$, $ \Lambda^k_1=\mu(b_k,\cdot)$.
So if $|b_j|=|b_k|$, then $\Lambda^j_1 \equiv \Lambda^k_1$ due to \eqref{munew}. In particular, in
view of Example~\ref{n=1}, if $n=1$ then each $ \mathcal{L} ^j_f$ is a one-point set, corresponding to some point $b_j$ of the same length. In this case all functions $\Lambda_k(\rho)$ coincide identically. But if $j\le M_0 <k$, or if $\max{j,k}>M_0$ and the set $ \mathcal{A} $ is strongly admissible\ (recall that everywhere in this section it is assumed to be admissible), then $\Lambda^{j_1}_{r_1}- \Lambda^{j_2}_{r_2}\not\equiv0$. This is the assertion of the non-degeneracy lemma below, proved in Section~\ref{s_2d}.
\begin{lemma}\label{l_nond} Consider any two spaces $Y^{f\,r_1}$ and $Y^{f\,r_2}$ such that $r_1\le r_2$ and $r_2>M_0$. Then \begin{equation}\label{single}
\Lambda_j^{r_1} \not\equiv \pm\Lambda_k^{r_2} \qquad \forall\, (r_1,j)\ne (r_2,k)\,, \end{equation} provided that either $r_1\le M_0$, or the set $ \mathcal{A} $ is strongly admissible. \end{lemma}
We recall that for $d\le2$ all admissible sets are strongly admissible. For $d\ge3$ non-\!\! strongly admissible\ sets exist. In Appendix~B we give an example \eqref{AAA} of such a set for $d=3$ and show that for it the relation \eqref{single} does not hold.
\subsection{Removing singular values of the parameter $\rho$}\label{rho_sing} We recall that the Hamiltonian operator $ \mathcal{H} (\rho)$ equals ${\bf i}JK(\rho)$; so $\{\Lambda^j_l(\rho)\}$ are the eigenvalues of the real matrix $JK(\rho)$. Accordingly, the numbers $\{\Lambda^j_l(\rho), 1\le l\le n_j\}$, are eigenvalues of the real matrix $\tfrac1{{\bf i}} \mathcal{H} ^j(\rho)=:L^j(\rho)$. Due to Lemma~\ref{laK} we know that for each $j$ the eigenvalues $\{\pm \Lambda_k^j(\rho)$, $k\le n_j\}$,
do not vanish identically in $\rho$ and do not identically coincide. Now our goal is to quantify these statements by removing certain singular values of the parameter $\rho$. To do this let us first denote $P^j(\rho)=(\prod_l\Lambda^j_l(\rho))^2 = \pm \det L^j(\rho)$ and consider the determinant $$
P(\rho)=\prod _jP^j(\rho) =\pm \det JK(\rho)\,.
$$
Recall that for an $R\times R$-matrix with eigenvalues $ \kappa _1,\dots, \kappa _R$ (counted with their multiplicities)
the discriminant of the determinant of this matrix
equals the product $\prod_{i\ne j}( \kappa _1- \kappa _j)$. This is a polynomial of the matrix' elements.
Next we define a ``poly-discriminant" $D(\rho)$, which is another polynomial
of the matrix elements of $JK(\rho)$. Its definition is motivated by Lemma~\ref{l_nond}, and it is
different for the admissible and strongly admissible\ sets $ \mathcal{A} $. Namely, if $ \mathcal{A} $ is strongly admissible, then
-- for $r=1,\dots, M_0$ define $D^r(\rho)$ as the discriminant of the determinant of the matrix
$ L^r(\rho)\oplus L^{M_0+1}(\rho)\oplus\dots\oplus L^M(\rho)$;
-- set $D(\rho) =
D^1(\rho)\cdot\dots\cdot D^{M_0}(\rho)$.
\noindent
This is a polynomial in the matrix coefficients of $JK(\rho)$, so a polynomial of $\sqrt\rho$. It vanishes if and only if
$ \Lambda^r_m(\rho)$ equals $\pm \Lambda^l_k(\rho)$ for some $r, l, m$ and $k$, where either $r,l \ge M_0+1$ and
$m\ne k$ if $r=l$, or $r\le M_0$ and $m=1$.
If $ \mathcal{A} $ is admissible, then we:
-- for $l\le M_0, r\ge M_0+1$ define $D^{l,r}(\rho)$ as the discriminant of the determinant of the matrix
$ L^l(\rho)\oplus L^r(\rho)$;
-- set $D(\rho) =
\prod_{l\le M_0, r\ge M_0+1} D^{l,r}(\rho) $.
\noindent
This is a polynomial in the matrix coefficients of $JK(\rho)$, so a polynomial in $\sqrt\rho$. It
vanishes if and only if
$ \Lambda^r_1(\rho)$ equals $\pm \Lambda^l_k(\rho)$ for some $r\le M_0$, some $l\ge M_0+1$ and some $k$,
or if $ \Lambda^l_k(\rho)$ equals $\pm \Lambda^l_m(\rho)$ for some $l\ge M_0+1$ and some $k\ne m$.
Finally, in the both cases we set
$$
M(\rho) = \prod_{b\in \mathcal{L} _f } \mu(b,\rho)
\prod_{\substack{b,b'\in \mathcal{L} _f \\ |b| \ne |b'| }} \big( \mu(b,\rho) - \mu(b',\rho)\big).
$$
This also is a polynomial in $\sqrt\rho$ which does not vanish identically due to \eqref{N110}.
The set
$$
X=\{\rho\mid P(\rho)\,D(\rho)\, M(\rho)
=0\}
$$
is an algebraic variety, if written in the variable $\sqrt\rho$ (analytically diffeomorphic to the variable
$\rho\in[c_*,1]^ \mathcal{A} $), and is non-trivial by Lemma~\ref{laK}.
The open set $ \mathcal{D}\setminus X$ is dense in $ \mathcal{D}$
and is formed by finitely many connected components. Denote them $Q_1,\dots, Q_L$.
For any component $Q_{l}$ its boundary is a stratified analytic manifold with finitely many smooth analytic components
of dimension $<n$, see \cite{ KrP}. The eigenvalues $\Lambda_j(\rho)$ and the corresponding eigenvectors are locally
analytic functions on the domains $Q_l$, but since some of these domains
may be not simply connected, then the functions may have
non-trivial monodromy, which would be inconvenient for us. But since each $Q_l$ is a domain with a regular boundary, then
by removing from it finitely many smooth closed hyper-surfaces we cut $Q_l$
to a finite system of simply connected domains $Q^1_l, \dots, Q_l^{\hat n_l}$
such that their union has the same measure as
$Q_l$ and each domain $Q_l^\mu$ lies on one side of its boundary.\footnote{For
example, if $n=2$ and $ \tilde{Q}_{l} $
is the annulus $A=\{1<\rho_1^2+\rho_2^2<2\}$, then we remove from $A$ not the interval $\{\rho_2=0, 1<\rho_1<2\}=:J$ (this would lead to a simply connected domain which lies on both parts of the boundary $J$), but two intervals, $J$ and $-J$. } We may realise these cuts (i.e. the
hyper-surfaces) as the zero-sets of certain
polynomial functions of $\rho$. Denote by $R_1(\rho)$ the product of the polynomials, corresponding to the cuts made, and remove from $ \tilde{Q}_{l} \setminus X$ the zero-set of $R_1$. This zero-set contains all the cuts we made (it may be bigger than the union of the cuts),
and still has zero measure. Again, $(\tilde Q_l\setminus X)\setminus \{\text{zero-set of} \ R_1\}$
is a finite union of domains, where each one lies in some domain $Q^r_l$.
Intersections of these new domains with the sets $ \mathcal{D}^{j_*}_0$ (see \eqref{DD}) will be important for us by virtue of Lemma~\ref{laK}, and any fixed set $ \mathcal{D}^{j_*}_0$, say $ \mathcal{D}^{1}_0$, will be sufficient for out analysis. To agree the domains with $ \mathcal{D}^1_0$ we note that the boundary of $ \mathcal{D}^1_0$ in $ \mathcal{D}$ is the zero-set of the polynomial $$ R_2(\rho) = (\rho_1-(1-c_0))(\rho_2-c_0)\dots(\rho_n-c_0)\,, $$ and modify the set $X$ above to the set $\tilde X$, $$ \tilde X = \{\rho\in \mathcal{D}\mid \mathcal{R} (\rho)=0\}\,,\qquad \mathcal{R} (\rho) = P(\rho) D(\rho) M(\rho) R_1(\rho) R_2(\rho)\,. $$ As before, $ \mathcal{D}\setminus \tilde X$ is a finite union of open domains with regular boundary. We still denote them $Q_l$:
\begin{equation}\label{components}
\mathcal{D}\setminus \tilde X = Q_1\cup\dots\cup Q_{\mathbb J}\,,\qquad \,\mathbb J <\infty\,.
\end{equation}
A domain $Q_j$ in \eqref{components} may be non simply connected, but since each $Q_j$
belongs to some domain $Q^r_l$, then the eigenvalues $ \Lambda_a (\rho)$ and the corresponding
eigenvectors define in these domains single-valued analytic functions. Since every domain
$Q_l$ lies either in $ \mathcal{D}^1_0$ or in its complement, we may enumerate the domains $Q_l$ in
such a way that
\begin{equation}\label{newJ} \mathcal{D}_0^1 \setminus \tilde X = Q_1\cup\dots\cup Q_{ \,\mathbb J _1}\,,\quad 1\le \,\mathbb J _1\le \,\mathbb J \,. \end{equation} The domains $Q_l$ with $l\le \,\mathbb J _1$ will play a special role in our argument.
Let us take $c_1 = \tfrac12 c_*$ and consider the complex vicinity $ \mathcal{D}_{c_1}$ of $ \mathcal{D}$,
\begin{equation}\label{Dset}
\mathcal{D}_{c_1}=\{ \rho\in \mathbb{C} ^ \mathcal{A} \mid |\Im\rho_j|<c_1, \, c_*-c_1<\Re\rho_j<1+c_1\ \forall j\in \mathcal{A} \}
\,. \end{equation}
We naturally extend $\tilde X$ to a complex-analytic subset $\tilde X^c$ of $ \mathcal{D}_{c_1}$
(so $\tilde X=\tilde X^c\cap \mathcal{D}$), consider the set $ \mathcal{D}_{c_1}\setminus \tilde X^c$, and for
any $\delta>0$ consider its open sub-domain $ \mathcal{D}_{c_1}(\delta)$, $$
\mathcal{D}_{c_1}(\delta) = \{ \rho\in \mathcal{D}_{c_1}\mid | \mathcal{R} (\rho)| > \delta
\}\subset \mathcal{D}_{c_1}\setminus \tilde X^c\,. $$ Since the factors, forming $ \mathcal{R} $, are polynomials with bounded coefficients, then they are bounded in $ \mathcal{D}_{c_1}$:
\begin{equation}\label{det}
\|P\|_{C^1( \mathcal{D}_{c_1})} \le C_1\,, \dots, \|R_2\|_{C^1( \mathcal{D}_{c_1})} \le C_1\,.
\end{equation} So in the domain $ \mathcal{D}_{c_1}(\delta)$ the norms of the factors $P,\dots, R_2$, making $ \mathcal{R} $, are bounded from below by $C_2\delta$, and similar estimates hold for the factors, making $P$, $D$ and $M$. Therefore, by the Kramer rule \begin{equation}\label{Kram}
\|(JK)^{-1}(\rho)\| \le C_1 \delta^{-1}\qquad \forall \rho\in \mathcal{D}_{c_1}(\delta)\,. \end{equation} Similar for $\rho \in \mathcal{D}_{c_1}(\delta)$ we have \begin{equation}\label{K4}
| \Lambda^j_{k}(\rho) |\ge C^{-1} \delta \qquad \forall j,k\,, \end{equation} \begin{equation}\label{K44}
|\mu(b,\rho) |
\ge C^{-1} \delta\,, \quad
|\mu(b,\rho) - \mu(b', \rho)|
\ge C^{-1} \delta \quad \text{if $\ b,b'\in \mathcal{L} _f\ $ and $|b|\ne |b'|$\,,
} \end{equation} and \begin{equation}\label{K04}
| \Lambda^j_{k_1}(\rho) \pm \Lambda^r_{k_2} (\rho)|\ge C^{-1} \delta \quad \text{where}\;\; (j,k_1)\ne (r,k_2)\,. \end{equation} In \eqref{K04} if the set $ \mathcal{A} $ is strongly admissible, then
the index $j$ is any and $r\ge M_0+1$, while if $ \mathcal{A} $ is admissible, then either $j\le M_0$ (and so
$k_1=1$) and $r\ge M_0+1$, or $j=r\ge M_0+1$. The functions $ \Lambda^j_{k}(\rho)$ are algebraic functions on the complex domain $ \mathcal{D}_{c_1}(\delta)$, but their restrictions to the real parts of these domains split to branches which are well defined analytic functions.
We have \begin{equation}\label{K1} \operatorname{meas} ( \mathcal{D}\setminus \mathcal{D}_{c_1}(\delta))\le C \delta^{\beta_{4}}, \end{equation} for some positive $C$ and $\beta_{4}$ -- this follows easily from Lemma~\ref{lTransv1} and Fubini since $ \mathcal{R} $ is a polynomial in $\sqrt \rho $ (also see Lemma~D.1 in \cite{EGK1}). Denote $c_2=c_1/2$, define set $ \mathcal{D}_{c_2}$ as in \eqref{Dset} but replacing there $c_1$ with $c_2$, and denote $ \mathcal{D}_{c_2}(\delta) = \mathcal{D}_{c_1}(\delta) \cap \mathcal{D}_{c_2}$. Obviously, \begin{equation}\label{two_sets} \text{the set $ \mathcal{D}_{c_2}(2\delta)$ lies in $ \mathcal{D}_{c_1}(\delta)$ with its $C^{-1}\delta$-vicinity\,. } \end{equation}
Consider the eigenvalues $\pm{\bf i} \Lambda_k(\rho)$. They analytically depend on $\rho\in \mathcal{D}_{c_1}(\delta)$, where $| \Lambda_k|\le C_2$ for each $k\le \mathbf N$ by \eqref{N1}. In view of \eqref{two_sets}, \begin{equation}\label{K2}
|\frac{\partial^l}{\partial\rho^l} \Lambda_k (\rho)| \le C_l\delta^{-l}\qquad \forall\,\rho\in \mathcal{D}_{c_2}(2\delta)\,,\ l\ge0\,,\ k\le \mathbf N\,, \end{equation} by the Cauchy estimate.
\subsection{Block-diagonalising and the end of the proof of Theorem \ref{NFT}} \label{s_bd}
We shall block-diagonalise the operator ${\mathbf i}JK(\rho)$ for $\rho \in \mathcal{D}_{c_1}(\delta)$. By \eqref{decomp} this operator is a direct sum of operators, each of which has a simple spectrum with eigenvalues that are separated by $\ge C^{-1}\delta$. Let us denote one of these blocks by ${\mathbf i}JK_1(\rho)$. Let its dimension be $2N$ and let $I(\xi,\eta)=(\bar\eta,\bar\xi)$. Notice that since ${\mathbf i}JK_1(\rho)$ is ``conjugate-real'' we have $${\mathbf i}JK_1(\rho)I( z)=I({{\mathbf i}JK_1(\rho)z}).$$
Fix now a $ \rho _0 \in \mathcal{D}_{c_1}(\delta)$. Then, by \eqref{two_sets} with $\delta$ replaced by $\delta/2$, for $\ab{ \rho - \rho _0}\le C^{-1}\delta^{4N}$ the operator ${\mathbf i}JK_1(\rho)$ has a single spectrum. Consider a (complex) matrix $$ U( \rho )=\big(z_1(\rho), \dots, z_{2N}(\rho)\big), $$ whose column vectors $ \aa{z_j( \rho )}=1 $ are eigenvectors of ${\mathbf i}JK_1(\rho)$. It diagonalises ${\mathbf i}JK_1$: \begin{equation}\label{diago} U(\rho)^{-1}\big( {\mathbf i}JK_1(\rho) \big)U(\rho)={\bf i}\,\text{diag}\,\{\pm \Lambda_1(\rho),\dots,\pm \Lambda_{{N}}(\rho)\}. \end{equation} The operator $U$ is smooth in $ \rho $ with estimates \begin{equation}\label{K8} \sup_{\rho}
\big( \|\partial^j_\rho U(\rho)\| + \|\partial^j_\rho U(\rho)^{-1} \|) \le C_j\delta^{-\beta(j) }
\qquad \forall\, j\ge0\,,
\end{equation}
and
\begin{equation}\label{K8bis} \inf_{\rho} \ab{\det(U(\rho))}\ge \frac1{C_0}\delta^{\beta(0) }\,,
\end{equation} for some $0<\beta(0)\le\beta(1)\le\dots$. See Lemma A.6 in \cite{E98} and Lemma~C.1 in \cite{EGK1}.
Since the spectrum is simple, then the pairing $\langle {\mathbf i}Jz_{k}( \rho ), z_{l}( \rho )\rangle = {}^t\!z_{l}( \rho )\big({\mathbf i}J\big)z_{k}( \rho )$ is zero unless the eigenvalues of $z_{k}( \rho )$ and $z_{l}( \rho )$ are equal but of opposite sign. We therefore enumerate the eigenvectors so that $z_{2j-1}( \rho )$ and $z_{2j}( \rho )$ correspond to eigenvalues of opposite sign. If now $\pi_j(\rho)=\langle {\mathbf i}Jz_{2j-1}(\rho), z_{2j}(\rho)\rangle $, then, for each $j$, $$
\frac1{C_0}\delta^{\beta(0) }\le \ab{\det(U)}=\sqrt{\ab{\det({}^t\!U{\mathbf i}JU)}}=\prod_l \ab{\pi_l}\le \ab{\pi_j}\le 1\,,
$$
since the matrix elements of ${}^t\!U{\mathbf i}JU$ are $\langle {\mathbf i}Jz_{k}( \rho ), z_{l}( \rho )\rangle $.
Replacing each eigenvector $z_{2j}$ by $\frac{1}{\pi_{2j}}z_{2j}$, we can assume without restriction that $U$ verifies \begin{equation}\label{eigenvectorsbis}
\frac1{C_0}\delta^{\beta(0) }\le \aa{z_j( \rho )}\le C_0\delta^{-\beta(0) }\end{equation}
and \eqref{diago}-\eqref{K8bis} (for some choice of constants) and, moreover, \begin{equation}\label{matrixU} {}^t\!U\big({\mathbf i}J\big)U=J.\end{equation}
Suppose now that some $\Lambda_j$, $\Lambda_1$ say, is real. Then $z_2$ and $I(z_1)$ are parallel, so $z_2={\bf i} \alpha I(z_{1})$ for some complex number $\alpha\in \mathbb{C} ^*$ satisfying the bound \eqref{eigenvectorsbis} (for some choice of constants). Since $\langle {\mathbf i}Jz_{1}, z_{2}\rangle =1$, we have that $ \alpha = \langle Jz_{1}, I(z_{1})\rangle^{-1} $ is real, and, by eventually interchanging $z_1$ and $z_2$, we can assume that $\alpha=\beta^2>0$. Replacing now $z_1,z_2$ by $\beta z_1, \frac1{\beta}z_{2}$ we can assume without restriction that $U$ verifies \eqref{diago}-\eqref{matrixU} (for some choice of constants), and $z_2={\bf i} I({z_{1}})$.
Suppose then that some $\Lambda_j$, $\Lambda_1$ again say, is purely imaginary. Then $z_1$ and $I(z_1)$ are parallel, so $z_1=\alpha I( z_1)$ for some unit $\alpha$. Similarly, $z_2=\beta I( z_2)$ for some unit $\beta$. Since $\langle {\mathbf i}Jz_{1}, z_{2}\rangle =1$, we have that $1= \alpha\beta \langle {\mathbf i}JI(z_{1}), I(z_{2})\rangle=\alpha\beta$. Let now $ \alpha =\gamma ^2$, and by replacing $z_1,z_2$ by $\bar \gamma z_1, \frac1{\bar\gamma }z_{2}$ we can assume without restriction that $U$ verifies \eqref{diago}-\eqref{matrixU} (for some choice of constants), and $z_1= I({z_{1}})$ and $z_2= I({z_{2}})$.
Suppose finally that some $\Lambda_j$, $\Lambda_1$ say, is neither real nor purely imaginary. Then $-{\mathbf i} \overline {\Lambda_1}$ also is an eigenvalue,\footnote{An example, considered in Appendix B, shows that quadruples of eigenvalues $\{\pm {\bf i}\Lambda, \pm {\bf i} \bar\Lambda\}$ indeed may occur in the spectra of operators ${\bf i}JK$.}
and, hence, equals to $\pm {\mathbf i}\Lambda_2$ say. Let us assume it is ${\mathbf i}\Lambda_2$, the other case being similar. Then $z_3= \alpha I({z_{1}})$ for some unit $ \alpha $, and $z_2=\beta I({z_{4}})$ for some $\beta\in \mathbb{C} ^*$, both satisfying the bound \eqref{eigenvectorsbis} (for some choice of constants). Since $\langle {\mathbf i}Jz_{1}, z_{2}\rangle = \langle {\mathbf i}Jz_{3}, z_{4}\rangle =1$, $ \alpha \beta$ must be $=1$.
Let now $ \alpha =\gamma ^2$, and by replacing $z_1,z_3$ by $\bar \gamma z_1, \bar\gamma z_{3}$ and $z_2,z_4$ by $\frac1{\bar \gamma } z_2, \frac1{\bar\gamma }z_{4}$ we can assume without restriction that $U$ verifies \eqref{diago}-\eqref{matrixU} (for some choice of constants), and $z_3= I({z_{1}})$ and $z_4= I({z_{2}})$.
Now we define a new matrix $$\tilde U( \rho )=\big(p_1(\rho)\ q_1( \rho )\dots p_{N}(\rho)\ p_{N}(\rho)\big)$$ in the following way. If $\Lambda_1$ is real, then we take $$p_1=-\frac{{\bf i}}{\sqrt2}(z_{1}+{\bf i}z_2),\quad q_1=-\frac{1}{\sqrt2}(z_{1}-{\bf i}z_2),$$ so that $I(p_1)=p_1$, $I(q_1)=q_1$ and $\langle {\mathbf i}Jp_{1}, q_{1}\rangle= 1$. We do similarly for all $\Lambda_j$ real.
If $\Lambda_1$ is purely imaginary, then we take $p_1=z_1$ and $q_1=z_2$, and similarly for all $\Lambda_j$ purely imaginary. If $\Lambda_1$ is neither real nor purely imaginary, and $z_1=I({z_{3}})$ and $z_2=I({z_{4}})$, then $$p_1=-\frac{{\bf i}}{\sqrt2}(z_{1}+{\bf i}z_3),\quad p_2=-\frac{1}{\sqrt2}(z_{1}-{\bf i}z_3)$$ and $$q_1=-\frac{{\bf i}}{\sqrt2}(z_{2}+{\bf i}z_4),\quad q_2=-\frac{1}{\sqrt2}(z_{2}-{\bf i}z_4),$$ similarly for all $\Lambda_j$ neither real nor purely imaginary.
Then the matrix $\tilde U( \rho )$ verifies
\eqref{K8}-\eqref{matrixU} (for some choice of constants) and the mapping
$$w\mapsto \tilde U( \rho )w$$
takes any real vector $w$ into the subspace $\{I(w)=w\}$. By doing this for each ``component'' ${\mathbf i}JK_1( \rho )$ of the operator \eqref{decomp} and taking the direct sum we find a matrix $\hat U( \rho )$ which
transforms the Hamiltonian of ${\mathbf i}JK( \rho )$ to the form \begin{equation}\label{rham} \frac12
\sum_{j=1}^{M_0} \mu(b_j,\rho) \Big( p_{b_j}^2 + q_{b_j}^2\Big)+
\frac12
\sum_{j=M_0+1}^{M_{00}} \Lambda_j(\rho) \Big( p_{b_j}^2+ q_{b_j}^2\Big) + \frac12
\langle \widehat K(\rho) \zeta_h, \zeta_h\rangle\, ,
\end{equation}
where $\zeta_h$ denotes the the remaining $\{(p_{b_j},q_{b_j}): M_{00}+1\le j\le \mathbf{N}\}$. The Hamiltonian operator $J \widehat K(\rho)$ is formed by the hyperbolic eigenvalues of the operator ${\bf i} J\widetilde K(\rho)$.
Since $\Lambda_a (\rho)\xi_a\eta_a$ is transformed to $\frac12 \Lambda_a(\rho) \Big( p_{a}^2+ q_{a}^2\Big)$ by a matrix $\hat U_a$, independent of $ \rho $, that verifies ${}^t\tilde U_a(iJ_a)\tilde U_a=J_a=J$ (see \eqref{change}) , the full Hamiltonian \eqref{HNF} gets transformed to \begin{equation}\label{hhak} \begin{split} \langle \Omega(\rho), r\rangle +\frac12 \sum_{a\in \mathcal{L} _\infty}\Lambda_a (\rho)\Big( p_{a}^2 + q_{a}^2\Big) +\frac12
\sum_{j=1}^{M_0} \mu(b_j,\rho) \Big( p_{b_j}^2 + q_{b_j}^2\Big)+ \\ + \frac12
\sum_{j=M_0+1}^{M_{00}} \Lambda_j(\rho) \Big( p_{b_j}^2+ q_{b_j}^2\Big) + \frac12
\langle \widehat K(\rho) \zeta_h, \zeta_h\rangle \end{split} \end{equation} plus the error term $\tilde f(r,\theta, p_ \mathcal{L} ,q_ \mathcal{L} ; \rho)=f(r,\theta, \xi_ \mathcal{L} ,\eta_ \mathcal{L} ; \rho)$.
Note that in difference with the normal form \eqref{HNF}, the variable $\zeta_h$ belongs to a subspace of the linear space, formed by the vectors $\{(p_a, q_a), a\in \mathcal{L} _f\}$, with the usual reality condition.
We choose any subset $ \mathcal{F} \subset \mathcal{L} _f$
of cardinality $| \mathcal{F} |={\bf N}- M_{00}$, and identify the space, where acts the operator $\hat K(\rho)$, with the space $ \mathcal{L} _ \mathcal{F} =\big\{ \zeta_ \mathcal{F} =\{(p_a,q_a), a\in \mathcal{F} \} \big\} $. We denote the operator $\hat K(\rho)$, re-interpreted as an operator in $ \mathcal{L} _ \mathcal{F} $, as $K(\rho)$. Finally, we identify the set of nodes $\{1,\dots, M_{00}\}$ with $ \mathcal{L} _ \mathcal{F} \setminus \mathcal{F} $, and write the collection of frequencies $ \{\mu(b_j,\rho), 1\le j\le M_{0}\} \cup \{\Lambda_j(\rho), M_0+1\le\rho \le M_{00}\} $ as $\{ \Lambda_b (\rho), b\in \mathcal{L} _ \mathcal{F} \setminus \mathcal{F} \}$. After that the Hamiltonian \eqref{hhak} takes the form \eqref{HNFbis}, required by Theorem~\ref{NFT}. We denote by $\bf\hat U_\rho$ the constructed linear symplectic change of variables which transforms the Hamiltonian \eqref{HNF} to \eqref{HNFbis}
For convenience we denote \begin{equation}\label{barc}\bar c= 1/{\beta_{4}}\quad \text{and} \quad \hat c=\beta(0) \bar c . \end{equation} With an eye on the relation \eqref{K1}, for
$\beta_{\#}>0$ and any $\nu>0$ we denote $\delta(\nu) = C^{\bar c} \nu^{\bar c\beta_{\#}}$. Then \begin{equation}\label{delta} C\delta^{\beta_{4}} = \nu^{\beta_{\#}}\,. \end{equation} For any $\nu>0$ we set $$ Q(c_*,\beta_{\#},\nu) = \mathcal{D}\cap \mathcal{D}_{c_1} (\delta(\nu))\,. $$ This is a monotone in $\nu$
system of subdomains of $ \mathcal{D}$, and $
Q(c_*,\beta_{\#},\nu)
\nearrow( \mathcal{D}\setminus \tilde X)$ as $ \nu\to0$. In view of \eqref{K1} the measures of these domains satisfy \eqref{mesmes}.
For $\rho\in Q(c_*,\beta_{\#},\nu) $ the operator $\tilde \Phi_\rho = \Phi_\rho \circ{\bf\hat U}_\rho$ transforms the Hamiltonian $\nu^{-1} h$ to \eqref{HNFbis}. Re-denoting this transformation back to $\Phi_\rho$, we see that the constructed objects satisfy the assertions (i)-(v) and (vii) of the theorem. To prove (vi) we recall (see \eqref{diago}) that the operator $U(\rho)$ (complex-)diagonalises one block of those, forming the operator ${\bf i}JK(\rho)$. Denote by ${\bf U}(\rho)$ the direct sum of the operators $U(\rho)$, corresponding to all blocks of ${\bf i}JK(\rho)$. It diagonalises the whole operator ${\bf i}JK(\rho)$. Accordingly, the operator $ {\bf U}(\rho) \circ {\bf\hat U}^{-1}(\rho) $ diagonalises $JK(\rho)$. Denoting it $U(\rho)$ we see that this operator satisfies the assertion (vi)
\subsection{Proof of the non-degeneracy Lemma~\ref{l_nond}}\label{s_2d}
Consider the decomposition \eqref{decomp} of the Hamiltonian operator $ \mathcal{H} (\rho)$. To simplify notation, in this section we suspend the agreement that $|L^r_f|=1$ for $r\le M_0$, and changing the order of the direct summands achieve that the indices $r_1$ and $r_2$, involved in \eqref{single}, are $r_1=1$ and $r_2=2$. For $r=1,2$ we will write elements of the set $ \mathcal{L} ^r_f$ as $a^r_j, 1\le j\le n_r$, and vectors of the space $Y^{fr}$ as \begin{equation}\label{vectors} \zeta= \big(\zeta_{a^r_j}=(\xi_{a^r_j} , \eta_{a^r_j} ), 1\le j\le n_r\big) = \big( ( \xi_{a^r_1} , \eta_{a^r_1}),\dots, ( \xi_{a^r_{n_r}} , \eta_{a^r_{n_r}})\big)\,. \end{equation}
Using \eqref{labA} and abusing notation, we will regard the mapping $\ell: \mathcal{L} _f\to \mathcal{A} $ also as a mapping $\ell: \mathcal{L} _f\to \{1,\dots,n\}$. Consider the points $ \ell(a^1_1),\dots, \ell(a^1_{n_1}) $ (they are different by \eqref{.2}). Changing if needed the labelling \eqref{labA} we achieve that \begin{equation}\label{achieve} \{ \ell(a^1_1),\dots, \ell(a^1_{n_1}) \ni 1\,. \end{equation}
We write the operator $ \mathcal{H} ^r$ as $ \mathcal{H} ^r={\bf i}M^r$, where $$ M^r (\rho)= JK^r(\rho) = JK^{r\,d}(\rho) +JK^{r\,n/d}(\rho) =: M^{r\,d}(\rho) +M^{r\,n/d}(\rho)\,, $$ and the real block-matrices $M^{r\, d}={\bf i}^{-1} \mathcal{H} ^{r,\,d}$, $\ M^{r\, n/d}={\bf i}^{-1} \mathcal{H} ^{r,\,n/d}$ are given by \eqref{diag}. Then $\{\pm\Lambda^r_j(\rho)\}$ are the eigenvalues of $M^r(\rho)$, and $$ M^{r\,d}(\rho) = \text{diag}\ \left(
\left(\begin{array}{ll}
\mu(a^r_j,\rho) & 0\\ 0& - \mu(a^r_j,\rho) \end{array}\right) ,\; 1\le j\le n_r \right), $$ where $
\mu(a^r_j, \rho) $ is given by \eqref{.1}.
Renumerating the eigenvalues we achieve that in \eqref{single} (with $r_1=1, r_2=2$) we have
$\Lambda^1_j=\Lambda^1_1$ and $\Lambda^2_k=\Lambda^2_1$.
As in the proof of Lemma~\ref{laK}, consider the vector $\rho_*= (1,0,\dots,0)$. Let us abbreviate $$ \mu(a, \rho_*)= \mu(a)\qquad \forall\, a\,, $$
where $\mu(a)$ depends only on $|a|$ by \eqref{munew}.
In view of \eqref{diag} $M^r(\rho_*)=M^{r\,d}(\rho_*)$ and thus
$\Lambda^1_1(\rho_*)=\mu(a^1_1)$ and $\Lambda^2_1(\rho_*)= \mu(a^2_1)$, if we numerate the elements of
$ \mathcal{L} ^1_f$ and $ \mathcal{L} ^2_f$ accordingly. As in the proof of Lemma~\ref{laK}, $\mu(|a^r_1|)$ equals
$\tfrac12 C_*\lambda^{-2}_{a^r_1}$ or $- C_* \lambda^{-1}_{\ell(a^r_1)} \lambda^{-1}_{a^r_1}$. Therefore
the relation
$\mu(a^1_1) =\pm \mu(a^2_1) $
is possible only if the sign is ``+" and $|a^1_1| = |a^2_1|$. So it remains to verify that under the lemma's assumption
\begin{equation}\label{sin}
\Lambda^1_1(\rho)\not\equiv \Lambda^2_1(\rho)\quad\text{if}\quad |a^1_1| = |a^2_1|\,.
\end{equation}
Since $ |a^1_1| = |a^2_1|$, then $$ \ell(a^1_1) = \ell(a^2_1)=:{a_{j_{\#}}}\in \mathcal{A} \;\; \text{ and } \Lambda^1_1(\rho_*)=\Lambda^2_1(\rho_*)=: \Lambda\,. $$
To prove that $\Lambda^1_1(\rho)\not\equiv \Lambda^2_1(\rho)$
we compare variations of the two functions around $\rho=\rho_*$. To do this it
is convenient to pass from $\rho$ to the new parameter $y=(y_j)_1^n$, defined by
$$
y_j=\sqrt{\rho_{j}}, \quad j=1,\cdots ,n.
$$
Abusing notation we will sometime write $y_{a_j}$ instead of $y_j$.
Take any vector $x=(x_1, \dots,x_n)\in \mathbb{R}^n $, where $x_1=0$ and $x_j > 0$ if $j\ge2$,
and consider the following variation $y(\varepsilon)$ of $y_*=(1,0,\cdots,0)$: \begin{equation}\label{yx}
y_j(\varepsilon) = \begin{cases} 1& \text{ if} \ \ j=1,\\
\varepsilon x_j & \text{ if} \ \ j\ge2.
\end{cases} \end{equation} By \eqref{N11}, for small $\varepsilon$ the real matrix $M^r(\varepsilon):= M^r(\rho(\varepsilon))$ $(r=1,2)$ has a simple eigenvalue $\Lambda^r_1(\varepsilon)$, close to $\Lambda$.
We will show that for a suitable choice of vector $x$ the functions $ \Lambda^1_1(\varepsilon)$ and $ \Lambda^2_1(\varepsilon)$ are different. More specifically, that their jets at zero of sufficiently high order are different.
Let $r$ be 1 or 2. We denote $\Lambda(\varepsilon) = \Lambda^r_1(\rho(\varepsilon))$,
$M(\varepsilon)= M^r(\rho(\varepsilon))$ and denote by $M^d(\varepsilon)$ and $M^{n/d}(\varepsilon)$ the diagonal and non-diagonal parts of $M(\varepsilon)$. The matrix
$M^{n/d}(\varepsilon)$ is formed by $2\times2$-blocks \begin{equation}\label{M} \Big( M^{n/d}(\varepsilon)\Big)^{a^r_j}_{a^r_k} =
C_*
\frac{{ {y_{\ell(a^r_k)} y_{\ell(a^r_j)}}}}{ \lambda_{ a^r_k} \lambda_{a^r_j }} \,\left(
\left(\begin{array}{ll} 0& 1\\ -1& 0 \end{array}\right) \chi^+( a^r_k,a^r_j )+
\left(\begin{array}{ll} 1& 0\\ 0& -1 \end{array}\right) \chi^-( a^r_k,a^r_j ) \right)\,, \end{equation} (note that if $j=k$, then the block vanishes).
For $\varepsilon=0$, $M(0) = M^{rd}(0)$ is a matrix with the single eigenvalue $\Lambda(0) = \mu(a^r_1, \rho_*)$, corresponding to the eigen-vector $\zeta(0) = (1,0,\dots,0)$. For small $\varepsilon$ they analytically extend to a real eigenvector $\zeta(\varepsilon)$
of $M(\varepsilon)$ with the eigenvalue $\Lambda(\varepsilon)$, i.e. $$
M(\varepsilon)\zeta(\varepsilon) = \Lambda(\varepsilon) \zeta(\varepsilon)\,,\qquad |\zeta(\varepsilon)|\equiv 1\,. $$
We abbreviate $\zeta= \zeta(0), M=M(0) $ and define similar $ \dot \zeta, \ddot\zeta, \Lambda,\dot\Lambda\dots$ etc, where the upper dot stands for $d/d\varepsilon$. We have \begin{equation}\label{y1} M=M^d =\text{diag}\big(\mu(a^r_1), -\mu(a^r_1), \dots, -\mu(a^r_{n_r}) \big)\,, \end{equation} \begin{equation}\label{y01}
\dot M^d=0\,. \end{equation}
Since $ (M(\varepsilon) - \Lambda(\varepsilon))\zeta(\varepsilon)\equiv 0$, then \begin{equation}\label{y3} (M(\varepsilon) - \Lambda(\varepsilon)) \dot\zeta(\varepsilon) = -\dot M(\varepsilon) \zeta(\varepsilon) +\dot\Lambda(\varepsilon) \zeta(\varepsilon). \end{equation} Jointly with \eqref{y1} and \eqref{y01} this relation with $\varepsilon=0$ implies that \begin{equation}\label{y4} (M^d-\Lambda)\dot \zeta =-\dot M^{n/d} \zeta +\dot\Lambda \zeta. \end{equation} In view of \eqref{y1} we have $\langle(M^d- \Lambda)\dot\zeta, \zeta\rangle =0$. We derive from here and from \eqref{y4} that \begin{equation}\label{y5} \dot \Lambda=\langle \dot M^{n/d}\zeta,\zeta\rangle=0\,. \end{equation} Let us denote by $\pi$ the linear projection $ \ \pi: \mathbb{R}^{2n_r} \to \mathbb{R}^{2n_r} $ which makes zero the first component of a vector to which it applies. Then $M^d-\Lambda$ is an isomorphism of the space $\pi \mathbb{R}^{2n_r} $, and the vectors $\dot\zeta$ and $-\dot M \zeta+ \dot\Lambda\zeta = \dot M^{n/d}\zeta$ belong to $\pi \mathbb{R}^{2n_r} $. So we get from \eqref{y4} that \begin{equation}\label{y6} \dot\zeta= -(M^d- \Lambda)^{-1} \dot M^{n/d}\zeta\,, \end{equation} where the equality holds in the space $\pi \mathbb{R}^{2n_r} $.
Differentiating \eqref{y3} we find that \begin{equation}\label{y7} (M(\varepsilon)- \Lambda(\varepsilon))\ddot\zeta(\varepsilon)= -\ddot M(\varepsilon) \zeta(\varepsilon) -2\dot M(\varepsilon)\dot\zeta(\varepsilon) +\ddot \Lambda(\varepsilon)\zeta(\varepsilon) +2\dot \Lambda(\varepsilon)\dot\zeta(\varepsilon)\,.
\end{equation} Similar to the derivation of \eqref{y5} (and using that $\langle\zeta, \dot\zeta\rangle=0$ since $|\zeta(\varepsilon)|\equiv1$), we get from \eqref{y7} and \eqref{y5} that \begin{equation}\label{y9} \begin{split} \ddot \Lambda =\langle\ddot M\zeta,\zeta\rangle+2\langle\dot M\dot\zeta,\zeta\rangle =\langle\ddot M\zeta,\zeta\rangle +2\langle(M- \Lambda)^{-1}\dot M^{n/d}\zeta, {}^t(\dot M)\zeta\rangle\,. \end{split} \end{equation} Since for each $\varepsilon$ and every $j$ $$ \frac{d^2}{d\varepsilon^2}\rho_j(\varepsilon)=\frac{d^2}{d\varepsilon^2} y^2_{j} (\varepsilon) =2 x^2_{j}\,, \qquad \frac{d^2}{d\varepsilon^2} y_{1}(\varepsilon) y_j(\varepsilon)=0\,, $$ and since $ \langle \ddot M \zeta, \zeta\rangle =\langle \ddot M^d \zeta, \zeta\rangle $, then \begin{equation}\label{ddot} \langle \ddot M \zeta, \zeta\rangle = \frac{d^2}{d\varepsilon^2} \mu(a^r_1, \rho(\varepsilon))\!\mid_{\varepsilon=0} \,= C_* \lambda^{-1}_{a_{j_{\#}}} \big( 3 \lambda^{-1}_{a_{j_{\#}}} x_{j_{\#}}^2
-2 \sum^{n}_{j=2} x_{j}^2\lambda^{-1}_{a_j}\big)=:k_1\,. \end{equation} Note that $k_1$ does not depend on $r$.
Now consider the second term in the r.h.s.
\eqref{y9}. For any $a,b \in \mathcal{L} ^r_f$ we see that $\
\frac{d}{d\varepsilon} (y_{\ell(a)}(\varepsilon)y_{\ell(b)}(\varepsilon))\mid_{\varepsilon=0} \
$
is non-zero if exactly one of the numbers $\ell(a), \ell(b )$ is $a_1$, and this derivative
equals $x_{\ell (c)}$, where $c\in\{a,b\}$, $\ell (c)\ne a_1$. Therefore, by \eqref{M}, \begin{equation}\label{y100} \begin{split} &(\dot M^{n/d}\zeta )_{{a^r_j}} = \frac{C_*}{\lambda_{a_{j_{\#}}}} (\xi^o_{{a^r_j}}, -\eta^o_{{a^r_j}}),\qquad {a^r_j}\in \mathcal{L} ^r_f\,, \\ & \xi^o_{{a^r_j}}= \frac{ \varphi (a_1^r, a_j^r) }{\lambda_{{a^r_j}}}\chi^-(a^r_1, {a^r_j}), \quad \eta^o_{{a^r_j}}= \frac{ \varphi (a_1^r, a_j^r) }{\lambda_{{a^r_j}}}\chi^+(a^r_1, {a^r_j})\,, \end{split} \end{equation} where $ \varphi (a_1^r, a_1^r) =0$ and for $j\ne1$ $$
\varphi (a_1^r, a_j^r) = \begin{cases} x_{\ell(a^r_j)} & \text{ if} \ \ {j_{\#}}=1\,, \\
x_{{{j_{\#}}}} & \text{ if} \ \ \ell(a^r_j)=a_1 \,,\\
0 & \text{ if} \ \ {j_{\#}}\ne1,\ \ell(a^r_j)\ne a_1\,.
\end{cases} \qquad $$ Since $\chi^\pm(a^r_1, a^r_1)=0$, then $\xi^o_{a^r_1} = \eta^o_{a^r_1}=0$.
In view of \eqref{obv}, at most one of the numbers $\xi^o_{{a^r_j}}, \eta^o_{{a^r_j}}$ is non-zero. By \eqref{y100}, \begin{equation}\label{y11} ((M-\Lambda)^{-1}\dot M^{n/d}\zeta)_{{a^r_j}} = \frac{C_*}{\lambda_{a_{j_{\#}}}} (\xi^{oo}_{{a^r_j}}, \,\eta^{oo}_{{a^r_j}}), \end{equation} where $\xi^{oo}_{{a^r_j}}= \eta^{oo}_{{a^r_j}}=0$ if $j=1$, and otherwise $$
\xi^{oo}_{{a^r_j}}= \frac{ \varphi (a_1^r, a_j^r) \chi^-(a^r_1, {a^r_j}) }{ \lambda_{{a^r_j}} ( \mu({a^r_j})-\mu(a^r_1) )} , \;\;\;\; \eta^{oo}_{{a^r_j}}= \frac{ \varphi (a_1^r, a_j^r) \chi^+(a^r_1, {a^r_j}) }{ \lambda_{{a^r_j}} ( \mu({a^r_j})+\mu(a^r_1))} \,. $$ Here $\ \mu({a^r_j}) =\frac12 C_*\lambda^{-2}_{a_1}$ if $\ell(a^r_j)=a_1$ and $\ \mu(a^r_j) = -C_*\lambda^{-1}_{a^r_l} \lambda^{-1}_{a_1}$
if $\ell(a^r_j)\ne a_1$.
Similar, $$ ({}^t\dot M\zeta )_{{a^r_j}} = \frac{C_*}{\lambda_{a_{j_{\#}}}} (\xi^o_{{a^r_j}}, \eta^o_{{a^r_j}}), $$ so the second term in the r.h.s. of \eqref{y9} equals \begin{equation}\label{k2}
\frac{C_*^2}{\lambda^2_{a_{j_{\#}}}}\sum_{j=2}^{n_r} \frac{ \varphi (a_1^r, a_j^r)^2 }{ \lambda^2_{a^r_j}}
\Big(\frac{\chi^-(a_1^r, a^r_j)}{\mu(a^r_j)-\mu(a^r_1) } +
\frac{\chi^+(a_1^r, a^r_j)}{ \mu(a^r_j)+\mu(a^r_1) }
\Big)=: k_2(r)\ . \end{equation}
Finally, we have seen that $$ \Lambda^r_1(\rho(\varepsilon)) = \Lambda^1_1(\rho_*)+\tfrac12 \varepsilon^2 k_1+ \tfrac12 \varepsilon^2k_2(r) +O(\varepsilon^3),\quad r=1,2, $$ where $k_1$ does not depend on $r$. Since $a_1^r\sim a^r_j$ for each $r$ and each $j$ (see \eqref{class}),
then for $j>1$
at least one of the coefficients $\chi^\pm(a_1^r,a^r_j)$ is non-zero. As $\chi^+\cdot\chi^-\equiv0$, then \begin{equation}\label{non-zer}
\frac{\chi^-(a_1^r, a^r_j)}{\mu(a^r_j)-\mu(a^r_1) } +
\frac{\chi^+(a_1^r, a^r_j)}{ \mu(a^r_j)+\mu(a^r_1) }
\ne0 \qquad \forall\,r,\;\; \forall\,j >1 \,. \end{equation} We see that the sum, defining $k_2(r)$, is a non-trivial quadratic polynomial of the quantities $ \varphi (a_1^r, a_j^r)$ if $n_r\ge2$, and vanishes if $n_r=1$.
The following lemma is crucial for the proof. \begin{lemma}\label{l_triv}
If the set $ \mathcal{A} $ is strongly admissible \ and $|a|=|b|$, $a\ne b$, and $\chi^+(a,a')\ne0$, $\chi^+(b,b')\ne0$, or $\chi^-(a,a')\ne0$, $\chi^-(b,b')\ne0$, then
$|a'|\ne |b'|$. \end{lemma}
\begin{proof}
Let first consider the case when $\chi^+\ne0$. \\
We know that $\ell(a)=\ell(b)=:{a_{j_{\#}}}$. Assume that $|a'|=|b'|$. Then $\ell(a')=\ell(b')=: {{a_{j_\flat}}} \in \mathcal{A} $. Denote ${a_{j_{\#}}}+{{a_{j_\flat}}}=c$. Then $c\ne0$ since the set $ \mathcal{A} $ is admissible. As $(a,a'), (b,b') \in ( \mathcal{L} _f\times \mathcal{L} _f)_+$, then we have $\
|{a_{j_{\#}}}-c| = |a-c| = |b-c|\,. $
As $|{a_{j_{\#}}}| = |a| = |b|$, then the three points ${a_{j_{\#}}}, a$ and $b$ lie in the intersection of two circles, one centred in the origin and another centred in $c ={a_{j_{\#}}}+{{a_{j_\flat}}}$. Since $ \mathcal{A} $ is strongly admissible, then ${a_{j_{\#}}} \, \angle\, c$ (see \eqref{ddd}). So among the
three point two are equal, which is a contradiction. Hence, $|a'|\ne|b'|$ as stated.
The case $\chi^-\ne0$ is similar. \end{proof}
We claim that this lemma implies that \begin{equation}\label{hi} \Lambda_1^1(\rho(\varepsilon))\not\equiv \Lambda^2_1(\rho(\varepsilon)) \quad\text{for a suitable choice of the vector $x$ in } \eqref{yx} \,, \end{equation} so \eqref{sin} is valid and Lemma~\ref{l_nond} holds. To prove \eqref{hi}
we consider two cases.
\noindent {\it Case 1: ${j_{\#}}=1$.} Then $ \varphi (a_1^r, a_j^r) = x_{\ell(a_j^r)}$. Denoting $\
\frac{C_*^2}{\lambda^2_{a_1}} \, \frac{x^2_{\ell(a^r_j)}}{ \lambda^2_{a^r_j}} =: z_{\ell(a^r_j)} $ we see that
$k_2(1)$ and $k_2(2)$ are linear functions of the variables
$z_{a_1},\dots, z_{\ell_n}$.
i) Assume that $\chi^-(a_1^r, a_j^r)=1$ for some $r\in \{1,2\}$ and some $j>1$. Denote $\ell(a_j^r)=a_{{j_*}}$. Then ${j_*}\ne j_{\#}$ and $$ k_2(r) = \frac{z_{a_{{j_*}}}}{\mu(a_j^r) -\mu(a_1^r)} +\dots\,, $$ where $\dots$ is independent from $ z_{{j_*}} $. Now let $r'= \{1,2\}\setminus \{r\}$, and find $j'$ such that $\ell (a^{r'}_{j'}) =a_{{j_*}}$. If such $j'$ does not exist, then $k_2(r')$ does not depend on $z_{{j_*}}$. Accordingly, for a suitable $x$ we have $k_2(r)\ne k_2(r')$, and \eqref{hi} holds. If $n_2=1$, then $r=1$ and $r'=2$. So $j'$ does not exists and \eqref{hi} is established.
If $j'$ exists, then $n_1, n_2\ge2$, so the set $ \mathcal{A} $ is strongly admissible. By Lemma~\ref{l_triv}
$\chi^- (a^{r'}_{1} , a^{r'}_{j'} ) =0 $ since $\chi^-(a_1^r, a_j^r)=1$ and \begin{equation}\label{kkk}
|a_1^r| = |a_1^{r'}|, \qquad |a_j^r| = |a_{j'}^{r'}|\,. \end{equation}
So
$$ k_2(r') = z_{{j_*}}\,\frac{ \chi^+ (a^{r'}_{1} , a^{r'}_{j'} ) }{\mu(a_j^{r'}) +\mu(a_1^{r'})} +\dots\,. $$
Since $\chi^+$ equals 1 or 0, then using again \eqref{kkk} and the fact that $\mu(a)$ only depends on $|a|$, we see that $k_2(r)\ne k_2(r')$ for a suitable $x$, so \eqref{hi} again holds.
ii) If $\chi^-(a_1^r, a_j^r)=0$ for all $j$ and $r$, then $\chi^+(a_1^r, a_j^r)=1$ for some $r$ and $j$. Define $z_{{j_*}}$ as above. Then the coefficient in $k_2(r)$ in
front of $z_{{j_*}}$ is non-zero, while for $k_2(r')$ it vanishes. This is obvious if $n_{r'}=1$. Otherwise $ \mathcal{A} $ is strongly admissible\
and it holds by Lemma~\ref{l_triv} (and since $\chi^-\equiv0$). So \eqref{hi} again holds.
\noindent {\it Case 2: ${j_{\#}}\neq 1$. Then by \eqref{achieve} there exists $a^1_j\in \mathcal{L} ^r_f$ such that $\ell(a^r_j)=a_1$. So $\chi^+(a ^1_1, a ^1_j)\ne 0$ or $\chi^-(a ^1_1, a ^1_j)\ne 0$. } Then $ \varphi (a_1 ^1, a_j ^1) = x_{a_{j_{\#}} }$,
the sum in \eqref{k2} is non-trivial and
for the same reason as in Case~1 \eqref{hi} holds.
This completes the proof of Lemma~\ref{l_nond}.
\centerline{PART III. A KAM THEOREM} \section{KAM normal form Hamiltonians} \subsection{Block decomposition, normal form matrices.} In this subsection we recall two notions introduced in \cite{EK10} for the nonlinear Schr\"odinger equation. They are essential to overcome the problems of small divisors in a multidimensional context. Since the structure of the spectrum for the beam equation,
$\{\sqrt{|a|^4+m},\ a\in \mathbb{Z} ^{d}\}$, is similar to that for the NLS equation,
$\{|a|^2+\hat{V}_a,\ a\in \mathbb{Z} ^{d}\}$, then to study the beam equation we will use
tools, similar to those used to study the NLS equation.
\subsubsection{Partitions} \label{blockdecomp}
For any $\Delta\in \mathbb{N} \cup \{\infty\}$ we define an equivalence relation on $ \mathbb{Z} ^{d}$, generated by the pre-equivalence relation
$$ a\sim b \Longleftrightarrow \left\{\begin{array}{l} |a|=|b| \\ {[a-b]}
\leq \Delta. \end{array}\right.$$ (see \eqref{pdist}). Let $[a]_\Delta$ denote the equivalence class of $a$ -- the {\it block} of $a$. For further references we note that \begin{equation}\label{a-b}
|a|=|b| \text{ and } [a]_{\Delta}\neq [b]_{\Delta} \Rightarrow [a-b]\geq \Delta \end{equation} The crucial fact is that the blocks have a finite maximal ``diameter''
$$d_\Delta=\max_{[a]=[b]} [a-b]$$ which do not depend on $a$ but only on $\Delta$. This is the content of
\begin{proposition}\label{blocks} \begin{equation}\label{block} d_\Delta\leq C \Delta^{\frac{(d+1)!}2}. \end{equation} The constant $C$ only depends on $d$. \end{proposition}
\begin{proof} In \cite{EK10} it was considered the equivalence relation on $ \mathbb{Z} ^{d}$, generated by the pre-equivalence $$
a\approx b\quad\text{if}\quad |a|=|b|\quad \text{and}\quad |a-b|\le\Delta. $$ Denote by $[a]^o_\delta$ and $d^o_\Delta$ the corresponding equivalence class and its diameter (with respect to the usual distance). Since $a\sim b$ if and only if $a\approx b$ or $a\approx -b$, then \begin{equation}\label{union} [a]_\Delta = [a]^o_\Delta \cup -[a]^o_\Delta, \end{equation} provided that the union in the r.h.s. is disjoint. It is proved in \cite{EK10} that
$d_\Delta^o\le D_\Delta=:C \Delta^{\frac{(d+1)!}2}$. Accordingly, if $|a|\ge D_\Delta$, then the union above is disjoint, \eqref{union} holds and diameter of $[a]_\Delta$ satisfies
\eqref{block}. If $|a|< D_\Delta$, then $[a]_\Delta$ is contained in a sphere of radius $< D_\Delta$. So the block's
diameter is at most $2D_\Delta$. This proves \eqref{block} if we replace there $C_{d}$ by $2C_{d}$. \end{proof}
If $\Delta=\infty$ then the block of $a$ is the sphere $\{b: |b|=|a|\}$.
Each block decomposition is a sub-decomposition of the trivial decomposition formed by the spheres $\{|a|=\text{const}\}$.
\subsubsection{Normal form matrices} \label{normalformmatrices} On $ \mathcal{L} _{\infty}\subset \mathbb{Z} ^{d} $ we define the partition $$[a]_\Delta= \left\{\begin{array}{ll}
[a]_\Delta\cap \mathcal{L} _{\infty} & \ \textrm{if}\ a\in \mathcal{L} _{\infty}\ \textrm{and}\ \ab{a}>c\\
\{ b\in \mathcal{L} _{\infty}: \ab{b}\le c\} & \ \textrm{if}\ a\in \mathcal{L} _{\infty} \ \textrm{and}\ \ab{a}\le c.\end{array}\right.$$ On $ \mathcal{L} = \mathcal{F} \sqcup \mathcal{L} _{\infty}$ we define the partition, denoted $ \mathcal{E} _\Delta$, \begin{equation}\label{Fcluster} [a]=[a]_\Delta= \left\{\begin{array}{ll}
[a]_\Delta\cap \mathcal{L} _{\infty} & a\in \mathcal{L} _{\infty}\\ \mathcal{F} & a\in \mathcal{F} .\end{array}\right. \end{equation}
\begin{remark}\label{remark-blocks} Now the diameter of each block $[a]$ is bounded as in \eqref{a-b} if we just let $C \gtrsim \max( \# \mathcal{F} ,c^{d})$. \end{remark}
If $A:\ \mathcal{L} \times \mathcal{L} \to gl(2, \mathbb{C} )$ we define its {\it block components} $$ A_{[a]}^{[b]}:[a]\times[b]\to gl(2, \mathbb{C} )$$ to be the restriction of $A$ to $[a]\times[b]$. $A$ is {\it block diagonal} over $ \mathcal{E} _\Delta$ if, and only if, $A_{[b]}^{[a]}=0$ if $[a]\neq [b]$. Then we simply write $A_{[a]}$ for $A_{[a]}^{[a]}$.
On the space of $2\times 2$ complex matrices we introduce a projection $$ \Pi: gl(2, \mathbb{C} )\to \mathbb{C} I+ \mathbb{C} J, $$
orthogonal with respect to the Hilbert-Schmidt scalar product. Note that $ \mathbb{C} I+ \mathbb{C} J$ is the space of matrices, commuting with the symplectic matrix $J$. \begin{definition}\label{d_31} We say that a matrix $A:\ \mathcal{L} \times \mathcal{L} \to gl(2, \mathbb{C} )$ is on normal form with respect to $\Delta$, $\Delta\in \mathbb{N} \cup \{\infty\}$, and write $A\in \mathcal{NF} _\Delta$, if
\begin{itemize}
\item[(i)] $A$ is real valued,
\item[(ii)] $A$ is symmetric, i.e. $A_b^a\equiv {}^t\hspace{-0,1cm}A_a^b$,
\item[(iii)] $A$ is block diagonal over $ \mathcal{E} _\Delta$,
\item[(iv)] $A$ satisfies $\Pi A^a_b\equiv A^a_b$ for all $a,b\in \mathcal{L} _{\infty}$.
\end{itemize}
\end{definition}
Any real quadratic form ${\mathbf q}(w)= \frac 1 2\langle w,Aw \rangle$, $w=(p,q)$, can be written as $$ \frac 1 2\langle p,A_{11}p \rangle+\langle p,A_{12}q \rangle+\frac 1 2\langle q,A_{22}q \rangle +\frac 1 2\langle w_{ \mathcal{F} }, H( \rho ) w_{ \mathcal{F} }\rangle $$ where $A_{11},\ A_{22}$ and $H$ are real symmetric matrices and $A_{12}$ is a real matrix. We now pass from the real variables $w_a=(p_a,q_a)$ to the complex variables $z_a=(\xi_a,\eta_a)$ by the transformation $w=U z$ defined through \begin{equation}\label{transf} \xi_a=\frac 1 {\sqrt 2} (p_a+{\mathbf i}q_a),\quad \eta_a =\frac 1 {\sqrt 2} (p_a-{\mathbf i}q_a),\end{equation} for $a\in \mathcal{L} _\infty$, and acting like the identity on $ ( \mathbb{C} ^2)^ \mathcal{F} $. Then we have $$ {\mathbf q}(Uz)=\frac 1 2\langle \xi,P\xi\rangle+ \frac 1 2\langle \eta,{\overline P}\eta\rangle+\langle \xi,Q\eta\rangle +\frac 1 2\langle z_{ \mathcal{F} }, H( \rho ) z_{ \mathcal{F} }\rangle ,$$ where $$P=\frac12\Big( (A_{11}-A_{22})-{\mathbf i}(A_{12}+{}^t A_{12})\big)$$ and $$Q=\frac12\Big( (A_{11}+A_{22})+{\mathbf i}(A_{12}-{}^t A_{12})\big).$$ Hence $P$ is a complex symmetric matrix and $Q$ is a Hermitian matrix. If $A$ is on normal form, then $P=0$.
Notice that this change of variables is not symplectic but changes the symplectic form slightly: $$ U^*\Omega={\mathbf i}\sum_{a\in \mathcal{L} }d\xi_a\wedge d\eta_a+\sum_{a\in \mathcal{F} }d\xi_a\wedge d\eta_a.$$
\subsection{The unperturbed Hamiltonian}\label{ssUnperturbed}
Let $h_{\textrm up}(r,w, \rho )$ be a function of the form \begin{equation}\label{unperturbed} \langle r,\Omega_{\textrm up}( \rho )\rangle +\frac12\langle w,A_{\textrm up}( \rho )w \rangle= \langle r,\Omega_{\textrm up}( \rho )\rangle +\frac12\langle w_{ \mathcal{F} },H_{\textrm up}( \rho )w_{ \mathcal{F} } \rangle +\frac12\sum_{a\in \mathcal{L} _{\infty}} \Lambda_a (p_a^2+q_a^2),\end{equation} where $w_a=(p_a,q_a)$ and \begin{equation}\label{properties}\left\{\begin{array}{ll} \Omega_{\textrm up}: \mathcal{D}\to \mathbb{R}^{ \mathcal{A} }&\\ \Lambda_a: \mathcal{D}\to \mathbb{R},&\quad a\in \mathcal{L} _\infty\\ H_{\textrm up}: \mathcal{D}\to gl( \mathbb{R}^{ \mathcal{F} }\times \mathbb{R}^{ \mathcal{F} }),&\quad {}^t\! H_{\textrm up}=H_{\textrm up} \end{array}\right.\end{equation} are $ \mathcal{C} ^{{s_*}}$-functions, ${s_*}\ge 1$. $ \mathcal{D}$ is an open ball or a cube of diameter at most $1$ in the space $ \mathbb{R}^{ \mathcal{P} }$, parametrised by some finite subset $ \mathcal{P} $ of $ \mathbb{Z} ^{d}$.
We can write $$\langle w,A_{\textrm up}( \rho )w\rangle =\langle w_{ \mathcal{F} },H_{\textrm up} ( \rho )w_{ \mathcal{F} } \rangle +\frac12\big( \langle p_{\infty},Q_{\textrm up} ( \rho )p_{\infty} \rangle + \langle q_{\infty},Q_{\textrm up} ( \rho )q_{\infty} \rangle \big)$$ and $$Q_{\textrm up} ( \rho )=\operatorname{diag}\{\Lambda_a( \rho ): a\in \mathcal{L} _\infty\}.$$
\begin{definition}\label{definitionup} A function $h_{\textrm {up}}$ of the form \eqref{unperturbed}+\eqref{properties} will be called un {\it unperturbed Hamiltonian} if it verifies Assumptions A1-3 (given below) described by the positive constants
$$c',c, \delta _0,\beta=(\beta_1,\beta_2,\beta_3),\tau.$$
\end{definition} To formulate these assumptions we shall use the partition $[a]=[a]_{\infty}$ of $ \mathcal{F} \sqcup \mathcal{L} _{\infty}$ defined in \eqref{Fcluster}. Notice that this partition depend on a (possibly quite large) constant $c$.
\subsubsection{A1 -- spectral asymptotics.} There exist a constant $0< c'\le c$ and exponents $\beta_1\ge0,\beta_2>0$ such that for all $ \rho \in \mathcal{D}$:
\begin{equation}\label{la-lb-ter}
|\Lambda_a( \rho )-\ab{a}^{2} |\leq c \frac1{\langle a\rangle^{\beta_1}}\quad a\in \mathcal{L} _{\infty}; \end{equation}
\begin{multline}\label{la-lb}
\qquad |(\Lambda_a( \rho )-\Lambda_b( \rho ))-(\ab{a}^{2}-\ab{b}^{2}) |
\le c\max( \frac1{\langle a\rangle^{\beta_2}},\frac1{\langle b\rangle^{\beta_2}}),\quad a,b\in \mathcal{L} _{\infty}\,;\quad\end{multline}
\begin{equation}\label{laequiv} \left\{\begin{array}{l}
| \Lambda_a( \rho )|\geq c' \ \quad a\in \mathcal{L} _\infty\\
||(JH_{\text{up}}(\rho))^{-1}||\leq \frac1{c'}; \end{array}\right.\end{equation}
\begin{equation}\label{laequiv-bis}
| \Lambda_a( \rho )+\Lambda_b( \rho )|\geq c' \ \quad a,b\in \mathcal{L} _\infty\end{equation}
\begin{equation}\label{la-lb-bis} \left\{\begin{array}{ll}
|(\Lambda_a( \rho )-\Lambda_b( \rho ))) |\ge c' & a,b\in \mathcal{L} _\infty,\ [a]\not=[b]\\
||(\Lambda_a( \rho )I-{\mathbf i}JH_{\text{up}}(\rho))^{-1}||\leq \frac1{c'} & a\in \mathcal{L} _{\infty}, \end{array}\right. \end{equation} Notice that if $\beta_1\ge\beta_2$, then \eqref{la-lb-ter} implies \eqref{la-lb} (if $c$ is large enough).
\subsubsection{A2 -- transversality.} Denote by $(Q_{\textrm up})_{[a]}$ the restriction of the matrix $Q_{\textrm up}$ to $[a]\times [a]$ and let $(Q_{\textrm up})_{[\emptyset]}=0$. Let also $JH_{\text{up}}( \rho )_{[\emptyset]}=0$.
There exists a $1\ge\delta_0>0$ such that
for all $ \mathcal{C} ^{{s_*}}$-functions \begin{equation}\label{o}
\Omega: \mathcal{D}\to \mathbb{R}^n,\quad |\Omega-\Omega_{\textrm up}|_{ \mathcal{C} ^{{s_*}}( \mathcal{D})}<\delta_0,\end{equation} and for all $k\in \mathbb{Z} ^n\setminus 0$
there exists a unit vector ${\mathfrak z}$ such that $$\ab{\partial_{\mathfrak z}\langle k, \Omega( \rho )\rangle} \ge \delta_0, \quad \forall \rho \in \mathcal{D}$$ \footnote{\ $\partial_{\mathfrak z} $ denotes here the directional derivative in the direction ${\mathfrak z}\in \mathbb{R}^p$} and the following dichotomies hold for each $k\in \mathbb{Z} ^n\setminus 0$:
\begin{itemize}
\item[$(i)$] for any $a,b\in \mathcal{L} _\infty\cup \{\emptyset\} $ let $$L( \rho ):X\mapsto \langle k,\Omega( \rho ) \rangle X+(Q_{\textrm up})_{[a]}( \rho )X\pm X(Q_{\textrm up})_{[b]}:$$ then either $L( \rho ) $ is {\it $ \delta _0$-invertible} for all $ \rho \in \mathcal{D}$ , i.e. \begin{equation}\label{invert} \aa{L( \rho )^{-1}}\le\frac1{\delta_0}\qquad\forall \rho \in \mathcal{D},\end{equation} or there exists a unit vector ${\mathfrak z}$ such that $$\ab{\langle v,\partial_{\mathfrak z} L( \rho ) v\rangle} \ge \delta_0, \quad \forall \rho \in \mathcal{D} $$ and for any unit-vector $v$ in the domain of $L( \rho )$ \footnote{\ $L$ is a linear operator acting on $([a]\times[b])$-matrices};
\item[$(ii)$] let $$L( \rho ,\lambda):X\mapsto \langle k,\Omega( \rho ) \rangle X+\lambda X+ {\mathbf i}XJH_{\textrm up}( \rho )$$ and $$P_{\textrm up}( \rho ,\lambda)= \det L( \rho ,\lambda):$$ then either $L( \rho ,\Lambda_a( \rho ))$ is $ \delta _0$-invertible for all $ \rho \in \mathcal{D}$ and $a\in[a]_\infty$, or there exists a unit vector ${\mathfrak z}$ such that, with $m=2\# \mathcal{F} $, $$ \ab{\partial_{\mathfrak z}P_{\textrm up}( \rho ,\Lambda_a( \rho ))+\partial_\lambda P_{\textrm up}( \rho ,\Lambda_a( \rho )) \langle v,\partial_{\mathfrak z} Q_{\textrm up}( \rho ) v\rangle} \ge
\delta_0\aa{L(\cdot,\Lambda_a(\cdot))}_{ \mathcal{C} ^{1}( \mathcal{D})}\aa{L(\cdot,\Lambda_a(\cdot))}_{ \mathcal{C} ^{0}( \mathcal{D})}^{m-2} $$ for all $ \rho \in \mathcal{D}$, $a\in[a]_\infty$ and for any unit-vector $v\in ( \mathbb{C} ^2)^{[a]}$ \footnote{\ $L$ is a linear operator acting on $(1\times m)$-matrices};
\item[$(iii)$] for any $a,b\in \mathcal{F} \cup \{\emptyset\} $ let $$ L( \rho ):X\mapsto \langle k,\Omega( \rho ) \rangle X-{\mathbf i}JH_{\textrm up}( \rho )_{[a]}X+ {\mathbf i}XJH_{\textrm up}( \rho )_{[b]}:$$ then either $L( \rho )$ is $ \delta _0$-invertible for all $ \rho \in \mathcal{D}$, or
there exists a unit vector ${\mathfrak z}$ and an integer $1\le j\le {s_*}$ such that
\begin{equation}\label{altern1} \ab{ \partial_{\mathfrak z}^j \det L( \rho ) }\ge \delta_0 \aa{L}_{ \mathcal{C} ^{j}( \mathcal{D})}\aa{L}_{ \mathcal{C} ^{0}( \mathcal{D})}^{m^2-2}, \quad \forall \rho \in \mathcal{D},\end{equation} where $m^2=(2\# \mathcal{F} )^2$ if both $[a]$ and $[b]$ are $\not=\emptyset$ and $m^2=2\# \mathcal{F} $ if one of $[a]$ and $[b]$ $=\emptyset$ \footnote{\ in the first case $L$ is a linear operator acting on $(m\times m)$-matrices, and in the second case $L$ is a linear operator acting on $(1\times m)$-matrices or $(m\times 1)$-matrices. } \end{itemize}
\begin{remark}\label{remro} The dichotomy in A2 is imposed not only on $\Omega_{\textrm up}$ but also on $ \mathcal{C} ^{s_*}$-perturbations of $\Omega_{\textrm up}$, because, in general, the dichotomy for $\Omega_{\textrm up}$ does not imply that for perturbations.
If, however, any $ \mathcal{C} ^{{s_*}}$ perturbation of $\Omega_{\textrm up}$ can be written as $\Omega_{\textrm up}\circ f$ for some diffeomorphism $ f=id+ \mathcal{O} (\delta_0)$ -- this is for example the case when $\Omega( \rho )= \rho $ -- then the dichotomy on $\Omega$ implies a dichotomy on $ \mathcal{C} ^{{s_*}}$-perturbations. \end{remark}
\subsubsection{A3 -- a Melnikov condition.}
There exist constants $\beta_3,\tau>0$ such that \begin{equation}\label{melnikov}
|\langle k,\Omega(0)\rangle-(\Lambda_a(0)-\Lambda_b(0))) |\ge\frac{\beta_3}{|k|^\tau}\end{equation} for all $k\in \mathbb{Z} ^ \mathcal{P} \setminus 0$ and all $a,b\in \mathcal{L} _{\infty}\setminus [0]$.
\subsection{KAM normal form Hamiltonians}
Consider now an unperturbed Hamiltonian $h_{\textrm {up}}$ defined on the set $ \mathcal{D}$ (see Definition \ref{definitionup}). The essential properties of this function are described by the positive constants
$$c',c, \delta _0,\beta=(\beta_1,\beta_2,\beta_3),\tau$$ (occurring in assumptions A1-3), and by the constant \begin{equation}\label{chi} \chi=
|\nabla_ \rho \Omega_{\textrm up} |_{ \mathcal{C} ^{ {{s_*}}-1 } ( \mathcal{D})}+\sup_{a\in \mathcal{L} _\infty} |\nabla_ \rho \Lambda_a|_{ \mathcal{C} ^{ {{s_*}}-1 } ( \mathcal{D})}
+ ||\nabla_ \rho H_{\textrm up} ||_{ \mathcal{C} ^{ {{s_*}}-1 } ( \mathcal{D}) }.\end{equation} Notice that, by Assumption A2, $\chi\ge \delta_0$, and in order to simplify the estimates a little we shall assume that
\begin{equation}\label{Conv}0<c'\le \delta _0\le\chi\le c.\end{equation}
We shall consider a somewhat larger class of functions.
\begin{definition} A function of the form \begin{equation}\label{normform} h(r,w, \rho )=\langle \Omega( \rho ), r\rangle +\frac 1 2\langle w, A( \rho )w\rangle\end{equation} is said to be on {\it KAM normal form} with respect to the unperturbed Hamiltonian $h_{\textrm {up}}$, satisfying \eqref{Conv}, if
\noindent ({\bf Hypothesis $\Omega$}) $\Omega$ is of class $ \mathcal{C} ^{{s_*}}$ on $ \mathcal{D}$ and \begin{equation}\label{hyp-omega}
|\Omega-\Omega_{\textrm up}|_{ \mathcal{C} ^{{s_*}}( \mathcal{D})}\le \delta. \end{equation}
\noindent ({\bf Hypothesis B}) $A-A_{\textrm up}: \mathcal{D}\to \mathcal{M} _{(0, m_*+ \varkappa ),\varkappa}^b$ is of class $ \mathcal{C} ^{{s_*}}$, $A( \rho )$ is on normal form $\in \mathcal{NF} _{\Delta}$ for all $ \rho \in \mathcal{D}$
and
\begin{equation}\label{hypoB}
|| \partial_ \rho ^j (A( \rho )-A_{\textrm up}( \rho ))_{[a]} || \le \delta\frac{1}{\langle a \rangle^\varkappa} \end{equation} for $ |j| \le {{s_*}}$, $a\in \mathcal{L} $ and $ \rho \in \mathcal{D}$
\footnote{\ here it is important that $||\cdot ||$ is the matrix operator norm}. Here we require that \begin{equation}\label{varkappa} 0<\varkappa.\end{equation}
We denote this property by $$h\in \mathcal{NF} _{\varkappa}(h_{\textrm up},\Delta,\delta).$$ Since the unperturbed Hamiltonian $h_{\textrm up}$ will be fixed in Part III we shall often suppress it, writing simply $h\in \mathcal{NF} _{\varkappa }(\Delta,\delta)$.
\end{definition}
\subsection{The KAM theorem} \ In this section we state an abstract KAM result for perturbations of a certain KAM normal form Hamiltonians.
Let $$h_{\textrm {up}}= h_{\textrm {up}, \chi,c', \delta _0,c}$$ be a fixed unperturbed Hamiltonian satisfying \eqref{Conv}. ($h_{\textrm {up}}$ also depends on $\beta, \tau$ but we shall not track this dependence.)
Let $h$ be a KAM normal form Hamiltonian, $$h\in \mathcal{NF} _{\varkappa}(h_{\textrm {up}, \chi,c', \delta _0,c},\Delta, \delta ),$$ and recall \eqref{varkappa}. We shall also assume $\Delta\ge1$.
The perturbation will belong to $ \mathcal{T} _{\gamma ,\varkappa, \mathcal{D} }( \sigma ,\mu)$ with $$0< \sigma ,\mu,\gamma _1\le 1$$ and (recall \eqref{gamma}) $$\gamma =(\gamma _1,m_*+ \varkappa )> \gamma _*=(0,m_*+ \varkappa ).$$
These bounds will be, often implicitly, assumed in the rest of Part III.
\begin{theorem}\label{main} There exist positive constants $C$, $\alpha$ and $\exp$ such that, for any $h\in \mathcal{NF} _{\varkappa,h_{\textrm up}}(\Delta, \delta )$ and for any $f\in \mathcal{T} _{\gamma ,\varkappa, \mathcal{D} }( \sigma ,\mu)$, $$ \varepsilon=\ab{f^T}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\ \textrm{and}\
\xi=\ab{f}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}},$$ if $$\delta \le \frac1{2C} c'$$ and \begin{equation}\label{epsi} \varepsilon(\log \frac1\varepsilon)^{\exp}\le \frac1{C}\big( \frac { \sigma \mu}{\max(\gamma _1^{-1} ,d_{\Delta})}\frac{c'}{\chi+\xi}\big)^{\exp}c', \end{equation} then there exist a closed subset $ \mathcal{D}'= \mathcal{D}'(h, f)\subset \mathcal{D}$, \begin{equation}\label{measure} \operatorname{meas} ( \mathcal{D}\setminus \mathcal{D}')\leq C\big(\log\frac1{\varepsilon} \frac{ \max(\gamma _1^{-1} ,d_{\Delta})}{ \sigma \mu} \big)^{\exp} \frac{\chi}{\delta_0}((\chi+\xi) \frac{\varepsilon}{\chi})^{\alpha},\end{equation} and a $ \mathcal{C} ^{{s_*}}$ mapping $$\Phi: \mathcal{O} _{ \gamma _*}( \sigma /2,\mu/2)\times \mathcal{D}\to \mathcal{O} _{ \gamma _*}( \sigma ,\mu),$$ real holomorphic and symplectic for each parameter $ \rho \in \mathcal{D}$, such that $$(h+ f)\circ \Phi= h'+f'$$ with \begin{itemize} \item[(i)] $$h'\in \mathcal{NF} _{ \varkappa }(\infty, \delta '),\quad \delta '\le \frac{c'}2,$$
and $$\ab{ h'- h}_{\begin{subarray}{c} \sigma /2,\mu/2\ \ \\ \gamma _*, \varkappa, \mathcal{D} \end{subarray}}\le C;$$
\item[(ii)] for any $x\in \mathcal{O} _{ \gamma _*}( \sigma /2,\mu/2)$, $ \rho \in \mathcal{D}$ and $\ab{j}\le{s_*}$
$$|| \partial_ \rho ^j (\Phi(x, \rho )-x)||_{ \gamma _*}+ \aa{ \partial_ \rho ^j (d\Phi(x, \rho )-I)}_{ \gamma _*, \varkappa } \le C$$ and $\Phi(\cdot, \rho )$ equals the identity for $ \rho $ near the boundary of $ \mathcal{D}$;
\item[(iii)] for $ \rho \in \mathcal{D}'$ and $\zeta=r=0$ $$d_r f'=d_\theta f'= d_{\zeta} f'=d^2_{\zeta} f'=0.$$ \end{itemize}
Moreover, \begin{itemize} \item[(iv)] if $\tilde \rho =(0, \rho _2,\dots, \rho _p)$ and $f^T(\cdot,\tilde \rho )=0$ for all $\tilde \rho $, then $h'=h$ and $\Phi(x,\cdot)=x$
for all $\tilde \rho $. \end{itemize}
The exponent $\alpha$ is a positive constant only depending on $d,s_*, \varkappa $ and $\beta_2 $. The exponent $\exp$ only depends on $d$, $\# \mathcal{A} $ and $\tau,\beta_2, \varkappa $. ${C}$ is an absolute constant that depends on $c,\tau,\beta_2,\beta_3$ and $ \varkappa $. ${C}$ also depend on $\sup_ \mathcal{D}\ab{\Omega_{\textrm up}}$ and $\sup_ \mathcal{D}\ab{H_{\textrm up}}$, but stays bounded when these do. \end{theorem}
The condition on $\Phi$ and $h'-h$ may look bad but it is not.
\begin{corollary}\label{cMain} Under the assumption of Theorem~\ref{main}, let $\varepsilon_*$ be the largest positive number such that \eqref{epsi} holds. Then, for any $ \rho \in \mathcal{D}$ and $\ab{j}\le{s_*}-1$, \begin{itemize} \item[$(i)'$] $$\ab{ \partial_ \rho ^j (h'(\cdot, \rho )- h(\cdot, \rho ))}_{\begin{subarray}{c} \sigma /2,\mu/2\ \ \\ \gamma _*, \varkappa,\ \ \ \ \end{subarray}}\le \frac{C}{\varepsilon_*}\ab{f^T}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}};$$
\item[$(ii)'$]
$$|| \partial_ \rho ^j (\Phi(x, \rho )-x)||_{\gamma _*}+ \aa{ \partial_ \rho ^j (d\Phi(x,r)-I)}_{\gamma ^*, \varkappa } \le \frac{C}{\varepsilon_*}\ab{f^T}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}},$$ for any $x\in \mathcal{O} _{\gamma _*}( \sigma /2,\mu/2)$. \end{itemize} \end{corollary}
\begin{proof} Let us denote $ \rho $ here by $ \rho _1$. If $|f^T|_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}\le \varepsilon_*$, then we can apply the theorem to $\varepsilon f$ for any $|\varepsilon|\le 1$. Let now $ \rho =(\varepsilon, \rho _1)$ and consider $h_{\mathrm up}$, $h$ and $f$ as functions depending on this new parameter $ \rho $ -- they will still verify the assumptions of the theorem, which will provide us with a mapping $\Phi$ with a $ \mathcal{C} ^{{s_*}}$ dependence in $ \rho =(\varepsilon, \rho _1)$ and equal to the identity when $\varepsilon=0$. The bound on the derivative together with assertion $(iv)$ now implies that
$$|| \Phi(x,\varepsilon,\tilde \rho )-x ||_{\gamma _*}\le C\varepsilon\le
\frac C{\varepsilon_*}|f^T|_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}} $$ for any $x\in \mathcal{O} _{\gamma _*}( \sigma /2,\mu/2)$. The same estimate holds for all derivatives with respect to $\tilde \rho $ up to order ${{s_*}}-1$. Take now $\varepsilon=1$ and we get $(ii)'$.
The argument for $h'-h$ is the same. \end{proof}
A special case that will interest us in particular is the following.
\begin{corollary}\label{cMain-bis} Let $h_{\textrm up}=h_{\textrm {up}, \chi,c', \delta _0,c}$ be an unperturbed Hamiltonian, satisfying $$ \;\;a) \qquad\qquad\qquad\qquad \delta _0^{1+\aleph}\le c'\le \delta_0\le\chi \le C' \delta_0^{1-\aleph} \le c, \qquad\qquad $$ and be $f\in \mathcal{T} _{\gamma ,\varkappa, \mathcal{D} }( \sigma ,\mu)$ with $$ \;\; b) \qquad\qquad\qquad\qquad \xi=\ab{f}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\le C'\delta_0^{1-\aleph}. \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad $$
for some $1>\aleph>0$ and $C'>0$.
Then there exist constants $\varepsilon_0>0$, $ \alpha $ and $ \kappa $ -- independent of $c',\delta_0,\chi$ and $\aleph$ -- such that
if $ \varepsilon=\ab{f^T}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}$ satisfies \begin{equation}\label{epsi-bis} \varepsilon(\log \frac1\varepsilon)^{ \kappa }\le \varepsilon_0\delta_0^{1+ \aleph \kappa },\end{equation}
then there exist a closed subset $ \mathcal{D}'= \mathcal{D}'(h, f)\subset \mathcal{D}$, \begin{equation}\label{measure-bis} \operatorname{meas} ( \mathcal{D}\setminus \mathcal{D}')\leq \frac1{\varepsilon_0}\delta_0^{-\aleph \kappa }\varepsilon^{\alpha},\end{equation} and a $ \mathcal{C} ^{{s_*}}$ mapping $\Phi$ $$\Phi: \mathcal{O} _{ \gamma _*}( \sigma /2,\mu/2)\times \mathcal{D}\to \mathcal{O} _{ \gamma _*}( \sigma ,\mu),$$ real holomorphic and symplectic for each parameter $ \rho \in \mathcal{D}$, such that $$(h_{\text{up}}+ f)\circ \Phi(r,w, \rho )= \langle \Omega'( \rho ), r\rangle +\frac 1 2\langle w, A'( \rho )w\rangle+f'(r,w, \rho )$$ with \begin{itemize} \item[$(i)$] the frequency vector $ \Omega'$ satisfies
$$|\Omega'-\Omega_{\textrm up}|_{ \mathcal{C} ^{{s_*-1}}( \mathcal{D})}\le c'$$
and, for each $ |j| \le {{s_*}}$ and $ \rho \in \mathcal{D}$, the matrix $$A'( \rho )=A'_\infty( \rho )\oplus H'( \rho )\in \mathcal{NF} _{\infty}$$ and satisfies
$$ || \partial_ \rho ^j (H'( \rho )-H_{\textrm up}( \rho ) || \le c';$$
\item[$(ii)'$] for any $x\in \mathcal{O} _{\gamma _*}( \sigma /2,\mu/2)$, $ \rho \in \mathcal{D}$ and $\ab{j}\le{s_*}-1$,
$$|| \partial_ \rho ^j (\Phi(x, \rho )-x)||_{\gamma _*}+ \aa{ \partial_ \rho ^j (d\Phi(x,r)-I)}_{\gamma ^*, \varkappa } \le \frac1{\varepsilon_0}\frac{\varepsilon}{\delta_0^{1+ \aleph \kappa }}(\log \frac1{\delta_0})^{ \kappa } $$ and $\Phi(\cdot, \rho )$ equals the identity for $ \rho $ near the boundary of $ \mathcal{D}$; \item[(iii)] for $ \rho \in \mathcal{D}'$ and $\zeta=r=0$ $$d_r f'=d_\theta f'= d_{\zeta} f'=d^2_{\zeta} f'=0.$$ \end{itemize}
The exponent $\alpha$ is a positive constant only depending on $d,s_*, \varkappa $ and $\beta_2 $. The exponent $ \kappa $ also depends on $\# \mathcal{A} $ and $\tau$. The constant ${\varepsilon_0}$ depends on everything except, as already said, $c',\delta_0,\chi$ and $\aleph$.
\end{corollary}
\begin{proof} We apply the theorem with $h=h_{\textrm up}$, i.e. $\delta=0$ and $\Delta=1$. The condition \eqref{epsi} is implied by $$\varepsilon(\log \frac1\varepsilon)^{\exp}\le \frac1{C''}\big(\frac{c'}{\chi+\xi}\big)^{\exp}c'$$ for some $C''$ depending on $C,\gamma _1, \sigma ,\mu$. With the choice of $c',\xi,\chi$ this is now implied by \eqref{epsi-bis} if $ \kappa \ge1+2\exp$.
The estimate of the measure becomes, from \eqref{measure}, $$\frac1{\varepsilon_0}\big(\log\frac1{\varepsilon})^{\exp}\delta_0^{-\aleph(1+\alpha)}\varepsilon^{\alpha} \le \frac1{\varepsilon_0}\delta_0^{-\aleph(1+\alpha)}\varepsilon^{\frac{\alpha}2},$$ which is what is claimed if we replace $\frac{\alpha}2$ by $ \alpha $, and take $ \kappa \ge(1+ \alpha )$.
(i) is just a consequence of $h'\in \mathcal{NF} (\infty,c')$. The bound in $(ii)$ follows from the bound $(ii)'$ in Corollary~\ref{cMain} plus an easy estimate of $\varepsilon_*$. \end{proof}
\section{Small divisors}\label{s4}
Control of the small divisors is essential for solving the homological equation (next section). In this section we shall control these divisors for $k\not=0$ using Assumptions A2 and A3.
For a mapping $L: \mathcal{D}\to gl(\dim, \mathbb{R})$ define, for any $ \kappa >0$,
$$\Sigma(L,\kappa)=\{ \rho \in \mathcal{D}: ||L^{-1}(\rho)| |>\frac1 \kappa \}.$$ Let $$h(r,w, \rho )=\langle r,\Omega( \rho )\rangle+ \frac12\langle w,A( \rho ) w \rangle$$ be a normal form Hamiltonian in $ \mathcal{NF} _{\varkappa}(\Delta,\delta)$. Recall the convention \eqref{Conv} and assume $ \varkappa >0$ and \begin{equation}\label{ass} \delta \le \frac{1}{C}c' ,\end{equation} where $C$ is to be determined.
\begin{lemma}\label{lSmallDiv1} Let $$L_{k}=\langle k,\Omega( \rho )\rangle.$$ There exists a constant $C$ such that if \eqref{ass} holds, then $$\operatorname{meas}\big(\bigcup_{0<\ab{k}\le N} \Sigma(L_k, \kappa )\big) \le C N^{\exp} \frac{ \kappa }{ \delta _0}$$ and $$\operatorname{dist}( \mathcal{D}\setminus \Sigma(L_k, \kappa ),\Sigma(L_k,\frac\ka2))>\frac{1}{C}\frac{ \kappa }{N\chi}$$
\footnote{\ this is assumed to be fulfilled if $\Sigma_{L_k}(\frac\ka2)=\emptyset$} for any $ \kappa >0$ .
(The exponent $\exp$ only depends on $\# \mathcal{A} $. $C$ is an absolute constant.) \end{lemma}
\begin{proof} We only need to consider $ \kappa \le\delta_0$ since otherwise the result is trivial. Since $\delta\le\delta_0$, using Assumption A2$(i)$, with $a=b=\emptyset$, we have, for each $k\not=0$, either that
$$ |\langle\Omega( \rho ), k\rangle|\ge \delta _0\ge \kappa \quad \forall \rho \in \mathcal{D}$$ or that $$ \partial_{\mathfrak z} \langle\Omega( \rho ), k\rangle \ \geq \delta_0\quad \forall \rho \in \mathcal{D} $$ (for some suitable choice of a unit vector $\mathfrak z$). The first case implies $\Sigma(L_{k}, \kappa )=\emptyset$. The second case implies that
$\Sigma(L_k,\kappa)$
has Lebesgue measure $ \lesssim \frac{ \kappa }{ \delta _0}$.
Summing up over all $0<\ab{k}\le N$ gives the first statement. The second statement
follows from the mean value theorem and the bound $$\ab{\nabla_ \rho L_k( \rho )}\le N(\chi+\delta).$$ \end{proof}
\begin{lemma}\label{lSmallDiv2} Let $$L_{k,[a]}=\big( \langle k,\Omega\rangle I - {\mathbf i} J A \big)_{[a]}.$$ There exists a constant $C$ such that if \eqref{ass} holds, then, $$\operatorname{meas}\big(\bigcup_{\begin{subarray}{c} 0<\ab{k}\le N\\ [a] \end{subarray}} \Sigma(L_{k,[a]}( \kappa )\big) \le C N^{\exp} (\frac{ \kappa }{\delta_0})^{\frac1{{s_*}}}$$ and $$\operatorname{dist}( \mathcal{D}\setminus \Sigma(L_{k,[a]}, \kappa ),\Sigma(L_{k,[a]},\frac\ka2))>\frac{1}{C}\frac \kappa {N\chi},$$ for any $ \kappa >0$.
(The exponent $\exp$ only depends on $d$ and $\# \mathcal{A} $. $C$ is an absolute constant that depends on $c$. $C$ also depend on $\sup_ \mathcal{D}\ab{\Omega_{\textrm up}}$ and $\sup_ \mathcal{D}\ab{H_{\textrm up}}$, but stays bounded when these do.)
\end{lemma}
\begin{proof} Consider first $a\in \mathcal{L} _\infty$. Then $L_{k,[a]}$ is conjugate to a sum of two Hermitian operators of the form $$L=\langle k,\Omega\rangle I + Q_{[a]},$$ where $Q_{[a]}$ is the restriction of $Q$ to $[a]\times [a]$ (see the discussion in section \ref{normalformmatrices}) .
If we let $$L_{\textrm up}= \langle k,\Omega\rangle I + (Q_{\textrm up}) _{[a]},$$ where $ Q_{\textrm u p}$ comes from the unperturbed Hamiltonian, then it follows, from \eqref{hypoB} and \eqref{ass}, that $$\aa{L-L_{\textrm up}}_{ \mathcal{C} ^{{1}}( \mathcal{D})} \leq \delta \leq {\operatorname{ct.} } \delta_0.$$ If now $L_{\textrm up}$ is $\delta_0$-invertible, then this implies that $L$ is $\frac{\delta_0}2$-invertible.
Otherwise, by assumption A2$(i)$, there exists a unit vector ${\mathfrak z}$ such that $$\ab{\langle v,\partial_{\mathfrak z} L_{\textrm up}( \rho ) v\rangle}\ge \delta_0$$ for any unit vector $v$. Since $Q_{[a]}$ is Hermitian we have, for any eigenvalue $\Lambda(\rho)$, $ \mathcal{C} ^1$ in the direction ${\mathfrak z}$, and any associated unit eigenvector $v(\rho)$, $$\partial_{\mathfrak z}\big( \langle k,\Omega( \rho )\rangle+\Lambda( \rho )\big)= \langle v( \rho ),\partial_{\mathfrak z} L( \rho ) v( \rho )\rangle
=\langle v( \rho ),\partial_{\mathfrak z} L_{\textrm up}( \rho ) v( \rho )\rangle + \mathcal{O} ( \delta). $$
Hence $$\ab{\partial_{\mathfrak z}\big( \langle k,\Omega( \rho )\rangle+\Lambda( \rho )\big)}\ge \delta_0- {\operatorname{Ct.} } \delta \ge \frac{\delta_0}2,$$ which implies that $\ab{ \langle k,\Omega( \rho )\rangle+\Lambda( \rho )}$ is larger than $ \kappa $ outside a set of Lebesgue measure $ \lesssim \frac \kappa {\delta_0}$. Since $L(\rho)$ is Hermitian this implies that $$\operatorname{meas} \Sigma(L, \kappa )) \lesssim \ab{a}^{d}\frac \kappa {\delta_0}$$ -- the dimension of $L$ is $ \lesssim \ab{a}^{d}$. (This argument is valid if $\Lambda( \rho )$ is $ \mathcal{C} ^1$ in the direction $\mathfrak z$ which can always be assumed when $Q$ is analytic in $ \rho $. The non-analytic case follows by analytical approximation.)
We still have to sum up over, a priori, infinitely many $[a]$'s. However, since
$|\langle k, \Omega( \rho )\rangle | \lesssim \ab{k} \lesssim N$, it follows, by \eqref{la-lb-ter}, that
$$|\langle k, \Omega( \rho )\rangle + \Lambda( \rho )|\geq \ab{\Lambda_a( \rho )}-\delta- {\operatorname{Ct.} } \ab{k} \ge \ab{a}^{2}- c \langle a \rangle ^{-\beta2} -\delta- {\operatorname{Ct.} } \ab{k}$$
for some appropriate $a\in[a]$. Hence $\ab{ \langle k,\Omega( \rho )\rangle+\Lambda( \rho )}$ is larger than $ \kappa $
for $ |a | \gtrsim N^{\frac1{2}}$. Summing up over all $0<\ab{k}\le N$ and
all $ |a | \lesssim N^{\frac1{2}}$ gives a set whose complement $\Sigma$ verifies the estimate.
Consider now $a\in \mathcal{F} $ and let $L( \rho )=\big(\langle k,\Omega\rangle I- {\mathbf i}JH\big)$. It follows, by \eqref{hypoB} and \eqref{ass}, that $$\aa{L-L_{\textrm up}}_{ \mathcal{C} ^{{s_*}}}\le \delta \leq \frac12 \delta_0,$$ where $L_{\textrm up}( \rho )=\big(\langle k,\Omega\rangle I- {\mathbf i}JH_{\textrm up}\big)$ -- now we are not dealing with an Hermitian operator.
If now $L_{\textrm up}$ is $\delta_0$-invertible, then $L$ will be $\frac{\delta_0}2$-invertible. Otherwise, by assumption A2(iii), there exists a unit vector ${\mathfrak z}$ and an integer $1\le j\le {s_*}$ such that $$\ab{ \partial_{\mathfrak z}^j \det L_{\textrm up}( \rho ) }\ge \delta_0 \aa{L_{\textrm up}}_{ \mathcal{C} ^{j}( \mathcal{D})}\aa{L_{\textrm up}}_{ \mathcal{C} ^{0}( \mathcal{D})}^{m-2}, \quad \forall \rho \in \mathcal{D}.$$ Since, by convexity estimates (see \cite{Ho}), $$\ab{ \partial_{\mathfrak z}^j \det L_{\textrm up}( \rho ) }\le {\operatorname{Ct.} } \aa{L_{\textrm up}}_{ \mathcal{C} ^{j}( \mathcal{D})}\aa{L_{\textrm up}}_{ \mathcal{C} ^{0}( \mathcal{D})}^{m-1}$$ and $$\ab{ \partial_{\mathfrak z}^j(\det L( \rho )-\det L_{\textrm up}( \rho ))} \le {\operatorname{Ct.} } \delta\big(\aa{L_{\textrm up}}_{ \mathcal{C} ^{j}}+\delta \big) ( \aa{ L}_{ \mathcal{C} ^{0}( \mathcal{D})} +\delta)^{m-2},$$ this implies that $$\ab{ \partial_{\mathfrak z}^j \det L( \rho ) }\ge (\delta_0- {\operatorname{Ct.} } \delta)\aa{ L_{\textrm up}}_{ \mathcal{C} ^{1}( \mathcal{D})}\aa{ L_{\textrm up}}_{ \mathcal{C} ^{0}( \mathcal{D})}^{m-1}, \quad \forall \rho \in \mathcal{D},$$ which is $\ge\frac{\delta_0}2$ if $\delta$ is sufficiently small.
Then, by Lemma \ref{lTransv1}, $$ {\ab{\det L( \rho )}}\ge \kappa \, { \aa{ L }_{ \mathcal{C} ^{j}}^{m-1} } \,, $$ outside a set of Lebesgue measure $$\le {\operatorname{Ct.} } (\frac \kappa { \delta _0})^{\frac1j}.$$ Hence, by Cramer's rule, $$ \operatorname{meas} \Sigma(L, \kappa )\le {\operatorname{Ct.} } (\frac \kappa { \delta _0})^{\frac1j} \le {\operatorname{Ct.} } (\frac \kappa { \delta _0})^{\frac1j}.$$ Summing up over all $\ab{k}\le N$ gives the first estimate.
The second estimate follows from the mean value theorem and the bound $$\ab{\nabla_ \rho L_{k,[a]}( \rho )}\le N(\chi+\delta).$$ \end{proof}
\begin{lemma}\label{lSmallDiv3} Let $$L_{k,[a],[b]}=(\langle k,\Omega\rangle I-{\mathbf i}\operatorname{ad}_{JA})_{[a]}^{[b]}.$$ There exists a constant $C$ such that if \eqref{ass} holds, then, $$\bigcup_{\begin{subarray}{c}0<\ab{k}\le N\\ [a],[b]\end{subarray}} \Sigma(L_{k,[a],[b]}, \kappa ) \le C (N \Delta)^{\exp}(\frac{ \kappa }{\delta_0})^{\alpha}(\frac{\chi}{\delta_0})^{1-\alpha}$$ and $$\operatorname{dist}( \mathcal{D}\setminus \Sigma(L_{k,[a],[b]}, \kappa ),\Sigma(L_{k,[a],[b]},\frac\ka2))> \frac{1}{C}\frac \kappa {\Delta^{\exp}} N\chi,$$ for any $ \kappa >0$. Here $$\alpha=\min\big(\frac{\beta_2 \varkappa }{\beta_2 \varkappa +2d(\beta_2+ \varkappa )},\frac1{s_*}\big).$$
(The exponent $\exp$ only depends on $d$, $\# \mathcal{A} $ and $\tau,\beta_2, \varkappa $. $C$ is an absolute constant that depends on $c,\tau,\beta_2,\beta_3$ and $ \varkappa $. $C$ also depend on $\sup_ \mathcal{D}\ab{\Omega_{\textrm up}}$ and $\sup_ \mathcal{D}\ab{H_{\textrm up}}$, but stays bounded when these do.)
\end{lemma}
\begin{proof} Consider first $a,b \in \mathcal{F} $. This case is treated as the operator $L( \rho )=\big(\langle k,\Omega\rangle I- {\mathbf i}JH\big)$ in the previous lemma.
Consider then $a\in \mathcal{L} _\infty$ and $b \in \mathcal{F} $. Then $L_{k,[a]}$ is conjugate to a sum of two operators of the form $$X\mapsto \langle k,\Omega( \rho ) \rangle X + Q_{[a]}( \rho )X+X{\mathbf i}JH( \rho )$$ (see the discussion in section \ref{normalformmatrices}). This operator in not Hermitian, but only ``partially'' Hermitian: it decomposes as an orthogonal sum of operators of the form $L( \rho ,\Lambda(\rho))$, where $$L( \rho ,\lambda):X\mapsto \langle k,\Omega( \rho ) \rangle X+\lambda X+ {\mathbf i}XJH( \rho ),$$ and $\Lambda(\rho) $ is an eigenvalue of $Q_{[a]}( \rho )$.
If we let $$L_{\textrm up}( \rho ,\lambda):X\mapsto \langle k,\Omega( \rho ) \rangle X + \lambda X+X{\mathbf i}JH_{\textrm up}( \rho ),$$ then it follows, from \eqref{hypoB} and \eqref{ass}, that $$\aa{L(\cdot,\lambda)-L_{\textrm up}(\cdot,\lambda)}_{ \mathcal{C} ^{{1}}( \mathcal{D})} \leq \delta \leq {\operatorname{ct.} } \delta_0.$$ If $L_{\textrm up}(\rho,\Lambda_a(\rho))$ is $\delta_0$-invertible for all $a\in[a]$, then this implies that, for any eigenvalue $\Lambda(\rho)$ of $Q_{[a]}( \rho )$, $L(\rho,\Lambda(\rho))$ is $\frac{\delta_0}2$-invertible.
Otherwise, by Assumption A2$(ii)$, there exists a unit vector ${\mathfrak z}$ such that $$ \ab{\partial_{\mathfrak z}P_{\textrm up}( \rho ,\Lambda_a( \rho ))+\partial_\lambda P_{\textrm up}( \rho ,\Lambda_a( \rho )) \langle v,\partial_{\mathfrak z} Q_{\textrm up}( \rho ) v\rangle} \ge \delta_0\aa{L_{\textrm up}}_{ \mathcal{C} ^{1}( \mathcal{D})}\aa{L_{\textrm up}}_{ \mathcal{C} ^{0}( \mathcal{D})}^{m-2}$$ for all $ \rho \in \mathcal{D}$, all $a\in[a]$ and for any unit-vector $v\in ( \mathbb{C} ^2)^{[a]}$. If now $$P( \rho ,\lambda)= \det L( \rho ,\lambda),$$ then, for any eigenvalue $\Lambda(\rho)$, $ \mathcal{C} ^1$ in the direction ${\mathfrak z}$, and any associated unit eigenvector $v(\rho)$, $$\frac{d}{d_{\mathfrak z}}P( \rho ,\Lambda( \rho ))= \partial_{\mathfrak z}P( \rho ,\Lambda( \rho ))+\partial_\lambda P( \rho ,\Lambda( \rho ))\langle v( \rho ),\partial_{\mathfrak z}Q( \rho )v( \rho )\rangle=$$ $$=\partial_{\mathfrak z}P_{\textrm up}( \rho ,\Lambda_a( \rho ))+\partial_\lambda P_{\textrm up}( \rho ,\Lambda_a( \rho )) \langle v( \rho ),\partial_{\mathfrak z}Q_{\textrm up}( \rho )v( \rho )\rangle + \mathcal{O} (\delta \aa{L_{\textrm up}}_{ \mathcal{C} ^{1}( \mathcal{D})}\aa{L_{\textrm up}}_{ \mathcal{C} ^{0}( \mathcal{D})}^{m-1}).$$ Hence $$ \ab{\frac{d}{d_{\mathfrak z}}P( \rho ,\Lambda( \rho ))}\ge\frac{\delta_0}2 \aa{L_{\textrm up}}_{ \mathcal{C} ^{1}( \mathcal{D})}\aa{L_{\textrm up}}_{ \mathcal{C} ^{0}( \mathcal{D})}^{m-2}.$$
Then
$$\ab{ \frac {P( \rho ,\Lambda( \rho ))} {|| L||_{ \mathcal{C} ^{0}( \mathcal{D})}^{m-1}}}\ge \kappa $$ outside a set of Lebesgue measure $ \lesssim \frac \kappa { \delta _0}$. Hence, by Cramer's rule, $$ \operatorname{meas} \Sigma(L, \kappa )\le {\operatorname{Ct.} } \frac \kappa { \delta _0}.$$
Since $|\langle k, \Omega( \rho )\rangle | \lesssim \ab{k} \lesssim N$, it follows, by \eqref{la-lb-ter}, that for any eigenvalue $\alpha( \rho )$ of $JH( \rho )$,
$$|\langle k, \Omega( \rho )\rangle + \Lambda( \rho )+\alpha( \rho )|\geq \ab{\Lambda_a( \rho )} -\delta- {\operatorname{Ct.} } \ab{k} \ge \ab{a}^{2}-c \langle a \rangle ^{-\beta_1} -\delta - {\operatorname{Ct.} } \ab{k}$$
for some appropriate $a\in[a]$. Hence,
$\Sigma(L, \kappa )=\emptyset$ for $ |a | \gtrsim N^{\frac1{2}}$.
Summing up over all $0<\ab{k}\le N$ and
all $ |a | \lesssim N^{\frac1{2}}$ gives the first estimate.
Consider finally $a,b \in \mathcal{L} _\infty$. Then $L_{k,[a],[b]}$ is conjugate to a sum of four operators of the forms $$X\mapsto \langle k,\Omega\rangle X + Q_{[a]}X+X{}^tQ_{[b]}$$ and $$X\mapsto \langle k,\Omega\rangle X + Q_{[a]}X-XQ_{[b]}.$$ These operators are Hermitian with respect to the Hilbert-Schmidt norm on the space of matrices $X$. Changing from the operator norm to the Hilbert-Schmidt norm (and conversely) changes any estimate by a factor that depends on the dimension of the space of matrices $X$, which, we recall, is bounded by some power of $\Delta$.
With this modification, the first operator is treated exactly as the operator $X\mapsto \langle k,\Omega\rangle X + Q_{[a]}X$ in the previous lemma, so let us concentrate on the second one, which we shall call $L=L_{k,[a],[b]}$. It follows as in the previous lemma that the Lebesgue measure of $\Sigma(L, \kappa )$ is $ \lesssim (\ab{a}\ab{b})^{d} \frac{ \kappa }{\delta_0}$ -- recall that the operator is of dimension $ \lesssim (\ab{a}\ab{b})^{2d} $.
The problem now is the measure estimate of $\bigcup\Sigma(L_{k,[a],[b]}, \kappa )$ since, a priori, there may be infinitely many $\Sigma(L_{k,[a],[b]}, \kappa )$ that are non-void.
We can assume without restriction that $\ab{a}\le \ab{b}$. Since $|\langle k, \Omega( \rho )\rangle |\le {\operatorname{Ct.} } \ab{k}\le {\operatorname{Ct.} } N$, it is enough to consider $\ab{b}-\ab{a} \le {\operatorname{Ct.} } N$.
Suppose first that $[a]$ and $[b]$ are $\not=[0]$. Let $\alpha(\rho)$ and $\beta(\rho)$ be eigenvalues of $Q_{[a]}(\rho)$ and $Q_{[b]}(\rho)$ respectively, and chose $a,b$ such that $$\ab{\alpha(\rho)-\Lambda_a(\rho)}\le \delta \frac1{\langle a \rangle^\varkappa},\quad \ab{\beta(\rho)-\Lambda_b(\rho)}\le \delta \frac1{\langle b \rangle^\varkappa}.$$
Using Assumption A3 now gives
$$|\langle k,\Omega( \rho )\rangle \ +\alpha( \rho )-\beta( \rho )|\ge
|\langle k,\Omega_{\textrm up}( \rho )\rangle \ +\Lambda_a( \rho )-\Lambda_b( \rho )|-\ab{k} \delta -2 \delta \frac1{\langle a\rangle^ \varkappa } $$
$$
\ge |\langle k,\Omega_{\textrm up}(0)\rangle \ +\Lambda_a(0)-\Lambda_b(0)|- \chi(\ab{k}+2)- \delta (\ab{k}+2)\ge
\frac{\beta_4}{\ab{k}^\tau}-6\ab{k}\chi,$$
and this is $\ge \kappa $ unless
$$\ab{k}\ge K\approx(\frac{\beta_3}{\chi})^{\frac1{\tau+1}}.$$ Recall that $\chi\ge \delta _0$, by convention, and that $ \kappa \le \delta _0$, because otherwise the lemma is trivial.
From now on we only consider $K\le\ab{k}\le N$. By Assumption A2, there exists a unit vector ${\mathfrak z}$ such that $$ \ab{\partial_{\mathfrak z}\langle k,\Omega( \rho )\rangle }\ge \delta _0.$$ Since $\ab{k}\le N$ and $\ab{a}^2-\ab{b}^2$ are integers, it follows that (for any $ \kappa '$)
$$|\langle k,\Omega( \rho )\rangle \ +\ab{a}^2-\ab{b}^2|\ge 2 \kappa '$$
for all $a,b$ and all $ \rho $ outside a set of Lebesgue measure $ \lesssim N\frac{ \kappa '}{ \delta _0}$. Summing up over all
$K\le\ab{k}\le N$ gives a set $\Sigma_1$ of Lebesgue measure
$$ \lesssim N^{\exp}\frac{ \kappa '}{ \delta _0}.$$
By \eqref{la-lb} it follows that, for
$ \rho $ outside of $\Sigma_1$,
$$|\langle k,\Omega( \rho )\rangle \ +\Lambda_a( \rho )-\Lambda_b ( \rho )|\ge \kappa ',$$
if just
$$\ab{a}^{\beta_2}\ge2\frac{c}{ \kappa '}.$$ Then
$$|\langle k,\Omega( \rho )\rangle \ +\alpha( \rho )-\beta( \rho )|\ge
\kappa '-2 \delta \frac1{\langle a\rangle^ \varkappa }$$
which is $\ge \kappa $ if $ \kappa '\ge 2 \kappa $ and
$$\ab{a}^ \varkappa \ge 2(\frac{ \delta }{ \kappa '}).$$
Let
$$M=2\max( (\frac{c}{ \kappa '})^{\frac1{\beta_2}} , (\frac{ \delta _0}{ \kappa '})^{\frac1{ \varkappa }} ).$$
Then it only remains to consider $[a]$ and $[b]$ with $\ab{a}\le M$ and $\ab{b}\le M+ {\operatorname{Ct.} } N$.
We have seen above that the the Lebesgue measure of each $\Sigma(L_{k,[a],[b]}, \kappa )$ is $ \lesssim (\ab{a}\ab{b})^{d} \frac{ \kappa }{\delta_0}$. Summing up over all these $a$ and $b$ gives a set $\Sigma_2$ of Lebesgue measure $$ \lesssim N^{\exp}M^{2d}\frac{ \kappa }{ \delta _0}.$$
Suppose now that $[a]$ or $[b]$ is $=[0]$. Then $\ab{a}$ and $\ab{b}$ are $ \lesssim c+N \lesssim N$. Summing up over all these $a$ and $b$ gives a set $\Sigma_3$ of Lebesgue measure $$ \lesssim N^{\exp}\frac{ \kappa }{ \delta _0}.$$
The union of $\Sigma_1$, $\Sigma_2$ and $\Sigma_3$ has Lebesgue measure $$ \lesssim N^{\exp} \big( \frac{ \kappa '}{ \delta _0} + M^{4d}\frac{ \kappa }{ \delta _0}\big) \lesssim N^{\exp} \big( \frac{ \kappa '}{ \delta _0} + (\frac1{ \kappa '})^\theta\frac{ \kappa }{ \delta _0}\big)\qquad \theta=4d(\frac1{\beta_2}+\frac1{ \varkappa }).$$ Take now $ \kappa '= \kappa ^{\frac1{1+\theta}}$ and observe that $N\chi^{\frac1\tau} \gtrsim 1$ (because $N\ge K$). Then the bound becomes $$ \lesssim N^{\exp} (\frac{ \kappa }{ \delta _0})^{\frac1{1+\theta}} (\frac{\chi}{ \delta _0})^{\frac\theta{1+\theta}} $$
( with a new and larger exponent $\exp$).
\end{proof}
\section{Homological equation}\label{s5}
Let $h$ be a normal form Hamiltonian \eqref{normform}, $$ h(r,w, \rho )=\langle \Omega( \rho ), r\rangle +\frac 1 2\langle w, A( \rho ) w\rangle\in \mathcal{NF} _{\varkappa}(\Delta,\delta)$$ -- recall the convention \eqref{Conv} -- and assume $ \varkappa >0$ and
\begin{equation}\label{ass1}
\delta \le \frac{1}{C}c' ,\end{equation}
where $C$ is to be determined. Let $$\gamma =(\gamma ,m_*)\ge \gamma _*=(0,m_*).$$
\begin{remark}\label{rAbuse} Notice the abuse of notations here. It will be clear from the context when $\gamma $ is a two-vector, like in $\aa{\cdot}_{\gamma , \varkappa }$, and when it is a scalar, like in $e^{\gamma d}$. \end{remark}
Let $f\in \mathcal{T} _{\gamma ,\varkappa, \mathcal{D}}( \sigma ,\mu)$. In this section we shall construct a jet-function $S$ that solves the {\it non-linear \footnote{\ ``non-linear'' because the solution depends non-linearly on $f$}
homological equation} \begin{equation}\label{eqNlHomEq} \{ h,S \}+ \{ f-f^T,S \}^T+f^T=0\end{equation} as good as possible -- the reason for this will be explained in the beginning of the next section. In order to do this we shall start by analysing the {\it homological equation} \begin{equation} \label{eqHomEq} \{ h,S \}+f^T=0. \end{equation} We shall solve this equation modulo some ``cokernel'' and modulo an ``error''.
\subsection{Three components of the homological equation}\label{ssFourComponents}
Let us write $$f^T(\theta,r,w)=f_r(r,\theta)+\langle f_w(\theta),w\rangle+\frac 1 2 \langle f_{ww}(\theta)w,w \rangle$$ and recall that, by Proposition \ref{lemma:jet}, $f^T\in \mathcal{T} _{\gamma ,\varkappa, \mathcal{D}}( \sigma ,\mu)$. Let $$ S(\theta,r,w)=S_r(r,\theta)+\langle S_w(\theta),w\rangle+ \frac 1 2 \langle S_{ww}(\theta)w,w \rangle,$$ where $f_r$ and $S_r$ are affine functions in $r$ -- here we have not indicated the dependence on $ \rho $.
Then the Poisson bracket $\{h, S\}$ equals \begin{multline*} -\big( \partial_{\Omega} S_r(r, \theta) + \langle \partial_{\Omega} S_w(\theta), w\rangle + \frac12 \langle \partial_{\Omega} S_{ww}(\theta),w\rangle + \\ +\langle AJ S_w(\theta),w\rangle + \frac12\langle AJ S_{ww}(\theta)w,w\rangle - \frac12\langle S_{ww}(\theta)JAw,w\rangle \end{multline*} where $\partial_{\Omega}$ denotes the derivative of the angles $\theta$ in direction $\Omega$. Accordingly the homological equation \eqref{eqHomEq}
decomposes into three linear equations: $$\left\{\begin{array}{l} \partial_{\Omega} S_r(r,\theta) =f_r(r,\theta),\\
\partial_{\Omega} S_w(\theta) -AJ S_w(\theta)= f_w(\theta),\\
\partial_{\Omega} S_{ww}(\theta) - AJS_{ww}(\theta) +S_{ww}(\theta)JA=f_{ww}(\theta). \end{array}\right.$$
\subsection{The first equation}\label{homogene}
\begin{lemma}\label{prop:homo12} There exists constant $C$ such that if \eqref{ass1} holds, then,
for any $N\ge1$ and $ \kappa >0$,
there exists a closed set $ \mathcal{D}_1= \mathcal{D}_1(h, \kappa ,N)\subset \mathcal{D}$, satisfying $$\operatorname{meas} ( \mathcal{D}\setminus \mathcal{D}_1)\leq C N^{\exp} \frac{ \kappa }{ \delta _0}$$ and there exist $ \mathcal{C} ^{{s_*}}$ functions $S_r$ and $R_r$ on $ \mathbb{C} ^{ \mathcal{A} }\times \mathbb{T} ^ \mathcal{A} \times \mathcal{D}\to \mathbb{C} $, real holomorphic in $r,\theta$, such that for all $ \rho \in \mathcal{D}_1$
\begin{equation}\label{homo1}
\partial_{\Omega( \rho )}S_r (r,\theta, \rho ) =f_r(r,\theta, \rho )-\hat f_r(r,0, \rho )
-R_r(\theta, \rho ) \quad \footnote{\ $\hat f_r(r,0, \rho )$ is the $0$:th Fourier coefficient, or the mean value, of the function $\theta\mapsto f_r(r,\theta, \rho )$}
\end{equation} and for all $(r,\theta, \rho )\in \mathbb{C} ^{ \mathcal{A} }\times \mathbb{T} ^ \mathcal{A} _{ \sigma '}\times \mathcal{D}$, $\ab{r}<\mu$, $ \sigma '< \sigma $, and $|j|\le{{s_*}}$
\begin{align} \label{homo1S}
|\partial_ \rho ^jS_r(r,\theta, \rho )|\leq &
C \frac{1}{ \kappa ( \sigma - \sigma ')^{n}}\big(N\frac{\chi}{ \kappa }\big)^{|j|}
|f^T|_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}} ,\\ \label{homo1R}
|\partial_ \rho ^j R_r(r,\theta, \rho )|\leq & C\frac{ e^{- ( \sigma - \sigma ')N}} { ( \sigma - \sigma ')^{n}}|f^T|_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}\,. \end{align} Moreover, $S_r(\cdot, \rho )=0$ for $ \rho $ near the boundary of $ \mathcal{D}$.
(The exponent $\exp$ only depends on $n=\# \mathcal{A} $, and $C$ is an absolute constant.) \end{lemma}
\begin{proof} Written in Fourier components the equation \eqref{homo1} then becomes, for $k\in \mathbb{Z} ^{ \mathcal{A} }$, $$L_k( \rho )\hat S(k)=:
\langle k, \Omega( \rho ) \rangle \hat S(k)=-{\mathbf i}(\hat F(k)-\hat R(k))$$ where we have written $S,F$ and $ R$ for $S_r, (f_r-\hat f_r)$ and $R_r$ respectively. Therefore \eqref{homo1} has the (formal) solution $$S(r,\theta, \rho )=\sum\hat S (r,k, \rho ) e^{{\mathbf i}\langle k,\theta\rangle}\quad\textrm{and}\quad R(r,\theta, \rho )=\sum\hat F (r,k, \rho ) e^{{\mathbf i}\langle k,\theta\rangle}$$ with $$\hat S(r,k, \rho )= \left\{\begin{array}{ll}
-L_k( \rho )^{-1}{\mathbf i}\hat F(r,k, \rho ) & \textrm{ if } 0< |k|\le N\\ 0 & \textrm{ if not} \end{array}\right.$$ and $$\hat R(r,k, \rho )= \left\{\begin{array}{ll}
\hat F(r,k, \rho ) & \textrm{ if } |k|> N\\ 0& \textrm{ if not}. \end{array}\right.$$
By Lemma~\ref{lSmallDiv1}
$$ ||(L_k( \rho ))^{-1}||\le \frac1 \kappa \,
$$ for all $ \rho $ outside some set $\Sigma(L_k,\kappa)$ such that $$\operatorname{dist}( \mathcal{D}\setminus \Sigma(L_k, \kappa ),\Sigma(L_k,\frac\ka2))\ge {\operatorname{ct.} } \frac \kappa {N\chi}$$ and
$$ \mathcal{D}_1= \mathcal{D}\setminus \bigcup_{0<|k|\le N}\Sigma(L_k, \kappa )$$ fulfils the estimate of the lemma.
For $ \rho \notin \Sigma(L_k,\frac\kappa2)$ we get
$$ |\hat S (r,k, \rho )| \le {\operatorname{Ct.} } \frac{1}{ \kappa }|\hat F(r,k, \rho )|\,.$$
Differentiating the formula for $\hat S(r,k, \rho )$ once we obtain
$$\partial^j_ \rho \hat S(r,k, \rho )=
\Big(
\ -\frac{{\mathbf i}}{ \langle\Omega, k\rangle}\partial^j_ \rho \hat F(r,k, \rho )+
\ \frac{{\mathbf i}}{ \langle\Omega, k\rangle^2} \langle \partial^j_ \rho \Omega, k\rangle\hat F(r,k, \rho )\Big)
$$
which gives, for $ \rho \notin \Sigma(L_k,\frac\kappa2)$,
$$
|\partial^j_ \rho \hat S(r,k, \rho )|\le {\operatorname{Ct.} } \frac1\kappa( N\frac{\chi}{\kappa})\max_{0\le l\le j}|\partial^l_ \rho \hat F(r,k, \rho )|.
$$
(Here we used that $|\partial_ \rho \Omega(\rho)|\le \chi+ \delta $. ) The higher order derivatives are estimated in the same way and this gives
$$
|\partial_ \rho ^j\hat S(r,k, \rho )|\le {\operatorname{Ct.} } \frac1\kappa(N\frac{\chi}{\kappa})^{ |j |}\max_{0\le l\le j} |\partial^l_ \rho \hat F(r,k, \rho )|
$$
for any $ |j |\le{{s_*}}$, where $ {\operatorname{Ct.} } $ is an absolute constant.
By Lemma \ref{lExtension}, there exists a $ \mathcal{C} ^\infty$-function $g_k: \mathcal{D}\to \mathbb{R}$, being $=1$ outside $\Sigma(L_k, \kappa )$ and $=0$ on $\Sigma(L_k,\frac\ka2)$ and such that for all $j\ge 0$
$$| g_k |_{ \mathcal{C} ^j( \mathcal{D})}\le ( {\operatorname{Ct.} } \frac{N\chi}{ \kappa })^j.$$
Multiplying $\hat S(r,k, \rho )$ with $g_k( \rho )$ gives a $ \mathcal{C} ^{{s_*}}$-extension of $\hat S(r,k, \rho )$ from
$ \mathcal{D}\setminus \Sigma(L_k, \kappa )$ to $ \mathcal{D}$ satisfying the same bound \eqref{homo1S}.
It follows now, by a classical argument, that the formal solution converges and that
$| \partial_ \rho ^j S(r,\theta, \rho )|$ and $|\partial_ \rho ^j R(r,\theta, \rho )|$
fulfils the estimates of the lemma. When summing up the series for $|\partial_ \rho ^j R(r,\theta, \rho )|$ we get a term $e^ {-\frac1C( \sigma - \sigma ')N}$ (because of truncation of Fourier modes), but the factor $\frac1C$ disappears by replacing $N$ by $CN$.
By construction $S$ and $R$ solve equation \eqref{homo1} for any $ \rho \in \mathcal{D}_1$.
If we multiply $\hat S(r,k, \rho )$ by a second $ \mathcal{C} ^\infty$ cut-off function $h_k: \mathcal{D}\to \mathbb{R}$ -- which is $=1$ at a distance $\ge \frac{ \kappa }{N\chi}$ from the boundary of $ \mathcal{D}$ and $=0$ near this boundary -- then the new function will satisfy the bound \eqref{homo1S}, it will solve the equation \eqref{homo1} on a new domain, smaller but still satisfying the measure bound of the Lemma, and it will vanish near the boundary of $ \mathcal{D}$. \end{proof}
\subsection{The second equation}\label{s5.3} Concerning the second component of the homological equation we have
\begin{lemma}\label{prop:homo3} There exists an absolute constant $C$ such that if \eqref{ass1} holds, then, for any $N\ge1$ and $$0< \kappa \le c',$$ there exists a closed set $ \mathcal{D}_2= \mathcal{D}_2(h, \kappa ,N)\subset \mathcal{D}$, satisfying
$$
\operatorname{meas} ( \mathcal{D}\setminus \mathcal{D}_2)\leq C N^{\exp}
(\frac{ \kappa }{\delta_0})^{\frac1{{s_*}}}, $$ and there exist $ \mathcal{C} ^{{s_*}}$-functions $S_w$ and $R_w$ $: \mathbb{T} ^ \mathcal{A} \times \mathcal{D}\to Y_\gamma $, real holomorphic in $\theta$, such that for $ \rho \in \mathcal{D}_2$ \begin{equation}\label{homo2}\partial_{\Omega( \rho )} S_w(\theta, \rho ) -A( \rho ) JS_w(\theta, \rho )= f_w(\theta, \rho )-R_w(\theta, \rho )
\end{equation} and for all $(\theta, \rho )\in \mathbb{T} ^ \mathcal{A} _{ \sigma '}\times \mathcal{D}$, $ \sigma '< \sigma $, and $|j|\le {{s_*}}$
\begin{align} \label{homo2S}
|| \partial_ \rho ^j S_w(\theta, \rho )||_{\gamma }\leq &
C\frac{1}{ \kappa ( \sigma - \sigma ')^{n}}\big( N\frac{\chi }{ \kappa }\big)^{|j|}
|f^T|_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}} \\ \label{homo2R}
|| \partial_ \rho ^j R_w(\theta, \rho )||_{\gamma }\leq & C\frac{ e^{-( \sigma - \sigma ')N} } {( \sigma - \sigma ')^{n}}
|f^T|_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}. \end{align} Moreover, $S_w(\cdot, \rho )=0$ for $ \rho $ near the boundary of $ \mathcal{D}$.
(The exponent $\exp$ only depends on $d$ and $\# \mathcal{A} $. $C$ is an absolute constant that depends on $c$. $C$ also depend on $\sup_ \mathcal{D}\ab{\Omega_{\textrm up}}$ and $\sup_ \mathcal{D}\ab{H_{\textrm up}}$, but stays bounded when these do.) \end{lemma}
\proof Let us re-write \eqref{homo2} in the complex variables $,z=(\xi\eta)$ described in section \ref{ssUnperturbed}. The quadratic form $(1/2)\langle w, A(\rho) w\rangle\ $ gets transformed, by $w=Uz$, to $$ \langle\xi, Q(\rho)\eta\rangle + \frac12 \langle z_{ \mathcal{F} }, H'(\rho) z_{ \mathcal{F} }\rangle,$$ where $Q'$ is a Hermitian matrix and $H'$ is a real symmetric matrix. Then we make in \eqref{homo2} the substitution $ S={}^t\!US_w$, $R={}^t\! UR_w$ and $F={}^t\!Uf_w$, where $S={}^t(S_\xi, S_\eta,S_{ \mathcal{F} })$, etc. In this notation eq.~\eqref{homo2} decouples into the equations \begin{align*}
\partial_{\Omega} S_\xi + {\mathbf i} QS_\xi= F_\xi-R_\xi,\\
\partial_{\Omega} S_\eta- {\mathbf i}{}^t\!QS_\eta= F_\eta-R_\eta\\ \partial_{\Omega} S_{ \mathcal{F} }- HJS_{ \mathcal{F} }= F_{ \mathcal{F} }-R_{ \mathcal{F} }. \end{align*}
Let us consider the first equation. Written in the Fourier components it becomes \begin{equation}\label{homo2.10} ( \langle k, \Omega( \rho ) \rangle I + Q) \hat S_\xi (k)=-{\mathbf i}(\hat F_\xi(k)-\hat R_\xi(k)).\end{equation} This equation decomposes into its ``components'' over the blocks $[a]=[a]_\Delta$ and takes the form \begin{equation}\label{homo2.1}L_{k,[a]}( \rho )\hat S_{[a]}(k)=: ( \langle k, \Omega( \rho ) \rangle + Q_{[a]}) \hat S_{[a]}(k)=-{\mathbf i}(\hat F_{[a]}(k)-\hat R_{[a]}(k))\end{equation} -- the matrix $Q_{[a]}$ being the restriction of $Q_\xi$ to $[a]\times [a]$, the vector $F_{[a]}$ being the restriction of $F_\xi$ to $[a]$ etc.
Equation \eqref{homo2.1} has the (formal) solution $$\hat S_{[a]}(k, \rho )= \left\{\begin{array}{ll}
-(L_{k,[a]}( \rho ))^{-1}{\mathbf i}\hat F_{[a]}(k, \rho ) & \textrm{ if } |k|\le N\\ 0 & \textrm{ if not} \end{array}\right.$$ and $$\hat R_{a}(k, \rho )= \left\{\begin{array}{ll}
\hat F_{a}(k, \rho ) & \textrm{ if } |k|> N\\ 0& \textrm{ if not}. \end{array}\right.$$
For $k\not=0$, by Lemma~\ref{lSmallDiv2},
$$ ||(L_{k,[a]}( \rho ))^{-1}||\le \frac1 \kappa \,
$$ for all $ \rho $ outside some set $\Sigma(L_{k,[a]},\kappa)$ such that $$\operatorname{dist}( \mathcal{D}\setminus \Sigma(L_{k,[a]}, \kappa ),\Sigma(L_{k,[a]},\frac\ka2))\ge {\operatorname{ct.} } \frac \kappa {N\chi}$$ and
$$ \mathcal{D}_2= \mathcal{D}\setminus \bigcup_{\begin{subarray}{c} 0< |k|\le N\\ [a]\end{subarray}}\Sigma_{k,[a]}( \kappa ),$$ fulfils the required estimate.
For $k=0$, it follows by \eqref{ass1} and \eqref{laequiv} that
$$ ||(L_{k,[a]}( \rho ))^{-1}||\le \frac1{c'}\le \frac2{ \kappa }\, . $$
We then get, as in the proof of Lemma~\ref{prop:homo12}, that $\hat S_{[a]}(k,\cdot)$ and $\hat R_{[a]}(k,\cdot)$ have $ \mathcal{C} ^{{s_*}}$-extension to $ \mathcal{D}$ satisfying
$$|| \partial_ \rho ^j \hat S_{[a]} (k, \rho )|| \leq {\operatorname{Ct.} } \frac{1}{ \kappa }\big(N\frac{\chi}{ \kappa }\big)^{|j|}
\max_{0\le l\le j} || \partial_ \rho ^l \hat F_{[a]}(k, \rho )|| $$ and
$$|| \partial_ \rho ^j R_{[a]}(k, \rho )||\leq {\operatorname{Ct.} } || \partial_ \rho ^j \hat F_{[a]}(k, \rho )||,$$ and satisfying \eqref{homo2.1} for $ \rho \in \mathcal{D}_2$.
These estimates imply that
$$|| \partial_ \rho ^j \hat S_\xi(k, \rho )||_\gamma \leq {\operatorname{Ct.} } \frac{1}{ \kappa }\big(N\frac{\chi}{ \kappa }\big)^{|j|}
\max_{0\le l\le j} || \partial_ \rho ^l \hat F_\xi (k, \rho )||_\gamma $$ and
$$|| \partial_ \rho ^j R_\xi (k, \rho )||_\gamma \leq {\operatorname{Ct.} } || \partial_ \rho ^j F_\xi(k, \rho )||_\gamma .$$ Summing up the Fourier series, as in Lemma~\ref{prop:homo12}, we get
$$|| \partial_ \rho ^j S_\xi (\theta, \rho )||_\gamma \leq {\operatorname{Ct.} } \frac{1}{ \kappa ( \sigma - \sigma ')^{n}}\big(N\frac{\chi}{ \kappa }\big)^{|j|}
\max_{0\le l\le j} \sup_{|\Im\theta|< \sigma }|| \partial_ \rho ^l F_\xi(\cdot, \rho )||_\gamma $$ and
$$|| \partial_ \rho ^j R_\xi(\theta, \rho )||_\gamma \leq {\operatorname{Ct.} } \frac{ e^{-\frac1{ {\operatorname{Ct.} } }( \sigma - \sigma ')N} } {( \sigma - \sigma ')^{n}}
\sup_{|\Im\theta|< \sigma }|| \partial_ \rho ^j F_\xi(\cdot, \rho )||_\gamma $$
for $(\theta, \rho )\in \mathbb{T} ^{ \mathcal{A} }_{ \sigma '}\times \mathcal{D}$, $0< \sigma '< \sigma $, and $|j|\le{{s_*}}$. This implies the estimates \eqref{homo2S} and \eqref{homo2R} -- the factor $\frac1{ {\operatorname{Ct.} } }$ disappears by replacing $N$ by $ {\operatorname{Ct.} } N$.
The other two equations are treated in exactly the same way. \endproof
\subsection{The third equation}\label{s5.4}
Concerning the third component of the homological equation, \eqref{eqHomEq}, we have the following result.
\begin{lemma}\label{prop:homo4} There exists an absolute constant $C$ such that if \eqref{ass1} holds, then, for any $N\ge1$, $\Delta'\ge \Delta\ge 1$, and $$ \kappa \le\frac1C c',$$ there exist a closed subset $ \mathcal{D}_3= \mathcal{D}_3(h, \kappa ,N)\subset \mathcal{D}$, satisfying $$\operatorname{meas}( \mathcal{D}\setminus { \mathcal{D}_3})\le C (\Delta N)^{\exp_1} (\frac{ \kappa }{\delta_0})^{\alpha}(\frac{\chi}{\delta_0})^{1-\alpha}$$ and there exist real $ \mathcal{C} ^{{s_*}}$-functions $B_{ww}: \mathcal{D}\to \mathcal{M} _{\gamma ,\varkappa} \cap \mathcal{NF} _{\Delta'} $ and $S_{ww}$, $R_{ww}=R_{ww}^F+R_{ww}^s: \mathbb{T} ^{ \mathcal{A} }\times \mathcal{D}\to \mathcal{M} _{\gamma ,\varkappa}$, real holomorphic in $\theta$, such that for all $ \rho \in \mathcal{D}_3$ \begin{multline}\label{homo3} \partial_{\Omega( \rho )} S_{ww}(\theta, \rho ) -A( \rho )JS_{ww}(\theta, \rho )+ S_{ww}(\theta, \rho )JA( \rho )=\\ f_{ww} (\theta, \rho )-B_{ww}( \rho )-R_{ww}(\theta, \rho )
\end{multline}
and for all $(\theta, \rho )\in \mathbb{T} ^ \mathcal{A} _{ \sigma '}\times \mathcal{D}$, $ \sigma '< \sigma $, and $|j|\le {{s_*}}$
\begin{equation}\label{homo3S} \aa{\partial_ \rho ^j S_{ww}(\theta, \rho )}_{\gamma ,\varkappa}\leq C\Delta'\frac{\Delta^{\exp_2}e^{2\gamma d_\Delta}}{ \kappa ( \sigma - \sigma ')^{n}}
\big(N\frac{\chi+ \delta }{ \kappa }\big)^{|j|} \ab{f^T}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}},\end{equation}
\begin{equation} \label{homo3B} \aa{\partial_ \rho ^j B_{ww}( \rho )}_{\gamma ', \varkappa }\leq C \Delta' \Delta^{\exp_2} \ab{f^T}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}},\end{equation} and \begin{equation}\label{homo3R} \left\{\begin{array}{l} \aa{ \partial_ \rho ^j R_{ww}^F(\theta, \rho )}_{\gamma , \varkappa }\leq C\Delta' \Delta^{\exp_2}\left(\frac{e^{-( \sigma - \sigma ')N}}{ ( \sigma - \sigma ')^{n}}\right) \ab{f^T}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}\\ \aa{ \partial_ \rho ^j R_{ww}^s(\theta, \rho )}_{\gamma ', \varkappa }\leq C\Delta' \Delta^{\exp_2}e^{-(\gamma -\gamma ')\Delta'} \ab{f^T}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}} \end{array}\right.,\end{equation} for any $\gamma _*\le \gamma '\le\gamma $.
Moreover, $S_{ww}(\cdot, \rho )=0$ for $ \rho $ near the boundary of $ \mathcal{D}$.
The exponent $\alpha$ is a positive constant only depending on $d,s_*, \varkappa $ and $\beta_2 $. \footnote{\ $\alpha$ is the exponent of Lemma~\ref{lSmallDiv3}} .
(The exponent $\exp$ only depends on $d$, $n=\# \mathcal{A} $ and $\tau,\beta_2, \varkappa $. The exponent $\exp_2$ only depends on $d,m_*,s_*$. $C$ is an absolute constant that depends on $c,\tau,\beta_2,\beta_3$ and $ \varkappa $. $C$ also depend on $\sup_ \mathcal{D}\ab{\Omega_{\textrm up}}$ and $\sup_ \mathcal{D}\ab{H_{\textrm up}}$, but stays bounded when these do.)
\end{lemma}
\proof It is also enough to find complex solutions $S_{ww}$, $R_{ww}$ and $B_{ww}$ verifying the estimates, because then their real parts will do the job.
As in the previous section, and using the same notation, we re-write \eqref{homo3} in complex variables. So we introduce $S={}^t\! US_{\zeta,\zeta} U$, $R={}^t\! UR_{\zeta,\zeta} U$, $B={}^t\! UB_{\zeta,\zeta} U$ and $F={}^t\! UJf_{\zeta,\zeta} U$. In appropriate notation \eqref{homo3} decouples into the equations \begin{align*} &\partial_{\Omega} S_{\xi\xi} +{\mathbf i} Q S_{\xi\xi}+ {\mathbf i} S_{\xi\xi}\ {}_tQ= F_{\xi\xi}-B_{\xi\xi}- R_{\xi\xi},\\ &\partial_{\Omega} S_{\xi\eta} + {\mathbf i}Q S_{\xi\eta} - {\mathbf i}S_{\xi\eta} Q= F_{\xi\eta}-B_{\xi\eta}- R_{\xi\eta},\\ &\partial_{\Omega} S_{\xi z_{ \mathcal{F} }} +{\mathbf i} Q S_{\xi z_{ \mathcal{F} }}+ S_{\xi z_{ \mathcal{F} }}\ JH = F_{\xi z_{ \mathcal{F} }}-B_{\xi\xi}- R_{\xi z_{ \mathcal{F} }},\\ &\partial_{\Omega} S_{z_{ \mathcal{F} }z_{ \mathcal{F} }} +HJ S_{z_{ \mathcal{F} }z_{ \mathcal{F} }}- S_{z_{ \mathcal{F} }z_{ \mathcal{F} }}JH= F_{z_{ \mathcal{F} }z_{ \mathcal{F} }}- B_{z_{ \mathcal{F} }z_{ \mathcal{F} }}- R_{z_{ \mathcal{F} }z_{ \mathcal{F} }}, \end{align*} and equations for $ S_{\eta\eta},S_{\eta\xi}, S_{z_{ \mathcal{F} }\xi }, S_{\eta z_{ \mathcal{F} }},S_{z_{ \mathcal{F} } \eta}$. Since those latter equations are of the same type as the first four, we shall concentrate on these first.
{\it First equation. }
Written in the Fourier components it becomes \begin{equation}\label{homo3.10} ( \langle k, \Omega( \rho ) \rangle I + Q) \hat S_{\xi\xi} (k)+\hat S_{\xi\xi} (k){}^tQ =-{\mathbf i}(\hat F_{\xi\xi}(k)- \delta _{k,0} B-\hat R_{\xi\xi}(k)).\end{equation} This equation decomposes into its ``components'' over the blocks $[a]\times[b]$, $[a]=[a]_\Delta$, and takes the form \begin{multline}\label{homo3.1} L(k,[a],[b], \rho )\hat S_{[a]}^{[b]}(k)=:\langle k, \Omega( \rho )\rangle \ \hat S_{[a]}^{[b]}(k) + Q_{[a]}( \rho ) \hat S_{[a]}^{[b]}(k)+ \\ \hat S_{[a]}^{[b]}(k)\ {}^tQ_{[b]}( \rho )= -{\mathbf i}(\hat F_{[a]}^{[b]}(k, \rho ) -\hat R_{[a]}^{[b]}(k)- \delta _{k,0} B_{[a]}^{[b]}) \end{multline} -- the matrix $Q_{[a]}$ being the restriction of $Q_{\xi\xi}$ to $[a]\times [a]$, the vector $F_{[a]}^{[b]}$ being the restriction of $F_{\xi\xi}$ to $[a]\times[b]$ etc.
Equation \eqref{homo3.1} has the (formal) solution: $$\hat S_{[a]}^{[b]}(k, \rho )= \left\{\begin{array}{ll} -L(k,[a],[b], \rho )^{-1}{\mathbf i}\hat F_{[a]}^{[b]}(k, \rho ) &
\textrm{ if } \operatorname{dist}([a],[b])\le\Delta'\ \textrm{and}\ \ |k|\le N\\ 0 & \textrm{ if not }, \end{array}\right. $$ $B_{[a]}^{[b]}=0$ and $$\hat R_{a}^{b}(k, \rho )= \left\{\begin{array}{ll}
\hat F_{a}^{b}(k, \rho )& \textrm{ if } \operatorname{dist}([a],[b])\ge\Delta'\ \textrm{or}\ \ |k|>N\\ 0 & \text{ if not}. \end{array}\right. $$ We denote $\hat R_{a}^{b}(k, \rho )$ by $\widehat {(R^s)}_{a}^{b}(k, \rho )$ if $\operatorname{dist}([a],[b])\ge\Delta'$ -- truncation off ``diagonal'' in space modes --
and by $\widehat {(R^F)}_{a}^{b}(k, \rho )$ if $ |k|>N$ -- truncation in Fourier modes.
For $k\not=0$, by Lemma~\ref{lSmallDiv3},
$$ ||(L_{k,[a],[b]}( \rho ))^{-1}||\le \frac1 \kappa \,
$$ for all $ \rho $ outside some set $\Sigma_{k,[a],[b]}(\kappa)$ such that $$\operatorname{dist}( \mathcal{D}\setminus \Sigma_{k,[a],[b]}( \kappa ),\Sigma_{k,[a],[b]}(\frac\ka2))\ge {\operatorname{ct.} } \frac \kappa {N\chi},$$ and
$$ \mathcal{D}_3= \mathcal{D}\setminus \bigcup_{\begin{subarray}{c} 0<|k|\le N\\ [a],[b]\end{subarray}}\Sigma_{k,[a],[b]}( \kappa )$$ fulfils the required estimate. For $k=0$, it follows by \eqref{ass1} and \eqref{laequiv-bis} that
$$ ||(L_{k,[a],[b]}( \rho ))^{-1}||\le \frac1{c'}\le \frac1{ \kappa }\, . $$
We then get, as in the proof of Lemma~\ref{prop:homo12}, that $\hat S_{[a]}^{[b]}(k,\cdot)$ and $\hat R_{[a]}^{[b]}(k,\cdot)$ have $ \mathcal{C} ^{{s_*}}$-extension to $ \mathcal{D}$ satisfying
$$|| \partial_ \rho ^j \hat S_{[a]}^{[b]} (k, \rho )|| \leq {\operatorname{Ct.} } \frac{1}{ \kappa }\big(N\frac{\chi}{ \kappa }\big)^{|j|}
\max_{0\le l\le j} || \partial_ \rho ^l \hat F_{[a]}^{[b]}(k, \rho )|| $$ and
$$|| \partial_ \rho ^j R_{a}^{b}(k, \rho )||\leq {\operatorname{Ct.} } || \partial_ \rho ^j \hat F_{a}^b(k, \rho )||,$$ and satisfying \eqref{homo3.1} for $ \rho \in \mathcal{D}_3$.
These estimates imply that, for any $\gamma _*\le \gamma '\le\gamma $,
$$|| \partial_ \rho ^j \hat S_{\xi\xi}(k, \rho )||_{ \mathcal{B} (Y_{\gamma '},Y_{\gamma '})}\leq {\operatorname{Ct.} } \Delta'\frac{\Delta^{\exp}e^{2\gamma d_\Delta}}{ \kappa }\big(N\frac{\chi}{ \kappa }\big)^{|j|}
\max_{0\le l\le j} || \partial_ \rho ^l \hat F_{\xi\xi} (k, \rho )||_{ \mathcal{B} (Y_{\gamma '},Y_{\gamma '})} $$ and
$$|| \partial_ \rho ^j \hat R_{\xi\xi}(k, \rho )||_{ \mathcal{B} (Y_{\gamma '},Y_{\gamma '})}\leq {\operatorname{Ct.} } \Delta' \Delta^{\exp} || \partial_ \rho ^j \hat F_{\xi\xi} (k, \rho )||_{ \mathcal{B} (Y_{\gamma '},Y_{\gamma '})}.$$ The factor $\Delta^{\exp}e^{2\gamma d_\Delta}$ occurs because the diameter of the blocks $\le d_{\Delta}$ interferes with the exponential decay and influences the equivalence between the $l^1$-norm and the operator-norm. The factor $\Delta' \Delta^{\exp} $ occurs because the truncation $ \lesssim \Delta'+ d_{\Delta}$ of diagonal influences the equivalence between the sup-norm and the operator-norm.
The estimates of the ``block components'' also gives estimates for the matrix norms and, for any $\gamma _*\le \gamma '\le\gamma $,
$$|| \partial_ \rho ^j \hat S_{\xi\xi}(k, \rho )||_{\gamma , \varkappa }\leq {\operatorname{Ct.} } \Delta'\frac{\Delta^{\exp}e^{2\gamma d_\Delta}}{ \kappa }\big(N\frac{\chi}{ \kappa }\big)^{|j|}
\max_{0\le l\le j} || \partial_ \rho ^l \hat F_{\xi\xi} (k, \rho )||_{\gamma , \varkappa } $$ and
$$|| \partial_ \rho ^j R_{\xi\xi} (k, \rho )||_{\gamma , \varkappa }\leq {\operatorname{Ct.} } || \partial_ \rho ^j F_{\xi\xi}(k, \rho )||_{\gamma , \varkappa }.$$
Summing up the Fourier series, as in Lemma~\ref{prop:homo3}, we get that $S_{\xi\xi}(\theta, \rho )$ satisfies the estimate \eqref{homo3S}. $R_{\xi\xi}(\theta, \rho )$ decompose naturally into a sum of a factor $R^F_{\xi\xi}(\theta, \rho )$, which is truncated in Fourier modes and therefore satisfies the first estimate of \eqref{homo3R}, and a factor $R^s_{\xi\xi}(\theta, \rho )$, which is truncated in off ``diagonal'' in space modes and therefore satisfies the second estimate of \eqref{homo3R}.
{\it The third equation. } We write the equation in Fourier components and decompose it into its
``components'' on each product block $[a]\times[b]$, $[b]= \mathcal{F} $: \begin{multline*} L(k,[a],[b],\rho) \hat S_{[a]}^{[b]}(k) := \langle k, \Omega( \rho )\rangle \ \hat S_{[a]}^{[b]}(k) +Q_{[a]}( \rho )\hat S_{[a]}^{[b]}(k) -\\
{\mathbf i}\hat S_{[a]}^{[b]}(k) JH( \rho )= -{\mathbf i}( \hat F_{[a]}^{[b]}(k, \rho )-\delta_{k,0}B_{[a]}^{[b]}-\hat R_{[a]}^{[b]}(k)) \end{multline*} -- here we have suppressed the upper index ${\xi z_{ \mathcal{F} }}$.
The formal solution is the same as in the previous case and it converges to functions verifying \eqref{homo3S} and \eqref{homo3R}, by Lemma~\ref{lSmallDiv3}, and by \eqref{la-lb-bis}.
{\it The fourth equation. } We write the equation in Fourier components: \begin{multline*} L(k,[a],[b],\rho) \hat S_{[a]}^{[b]}(k) := \langle k, \Omega( \rho )\rangle \ \hat S_{[a]}^{[b]}(k) - {\mathbf i}HJ( \rho )\hat S_{[a]}^{[b]}(k) +\\
{\mathbf i}\hat S_{[a]}^{[b]}(k) JH( \rho )= -{\mathbf i}( \hat F_{[a]}^{[b]}(k, \rho )-\delta_{k,0}B_{[a]}^{[b]}-\hat R_{[a]}^{[b]}(k)), \end{multline*} where $[a]=[b]= \mathcal{F} $ -- here we have suppressed the upper index ${z_{ \mathcal{F} } z_{ \mathcal{F} }}$.
The equation is solved (formally) by $$\hat S_{[a]}^{[b]}(k, \rho )= \left\{\begin{array}{ll}
-L(k,[a],[b], \rho )^{-1} {\mathbf i}\hat F_{[a]}^{[b]}(k, \rho ) & \textrm{ if } 0<|k|\le N\\ 0 & \textrm{ if not} , \end{array}\right. $$ $$\hat R_{[a]}^{[b]}(k, \rho )= \left\{\begin{array}{ll}
\hat F_{[a]}^{[b]}(k, \rho ) & \textrm{ if } |k|> N\\ 0& \textrm{ if not}; \end{array}\right. $$ and $$B^{[b]}_{[a]}( \rho )=\hat F_{[a]}^{[b]}(0, \rho ). $$
The formal solution now converges to a solution verifying \eqref{homo3S}, \eqref{homo3B} and \eqref{homo3R} by Lemma~\ref{lSmallDiv3}. The factor $R^s$ is here $=0$.
{\it The second equation. } We write the equation in Fourier components and decompose it into its
``components'' on each product block $[a]\times[b]$: \begin{multline*} L(k,[a],[b], \rho )\hat S_{[a]}^{[b]}(k)=:\langle k, \Omega( \rho )\rangle \ \hat S_{[a]}^{[b]}(k) + Q_{[a]}( \rho ) \hat S_{[a]}^{[b]}(k)- \\ \hat S_{[a]}^{[b]}(k)Q_{[b]}( \rho )= -{\mathbf i}(\hat F_{[a]}^{[b]}(k, \rho ) -\hat R_{[a]}^{[b]}(k)- \delta _{k,0} B_{[a]}^{[b]}) \end{multline*} -- here we have suppressed the upper index $\xi\eta$. This equation is now solved (formally) by $$ S_{[a]}^{[b]}(\theta, \rho )=\sum\hat S_{[a]}^{[b]} (k, \rho ) e^{{\mathbf i}k\cdot \theta} \quad\textrm{and}\quad R_{[a]}^{[b]}(\theta, \rho ) =\sum\hat R_{[a]}^{[b]} (k, \rho ) e^{{\mathbf i}k\cdot \theta},$$ with $$\hat S_{[a]}^{[b]}(k, \rho )= \left\{\begin{array}{ll} L(k,[a],[b], \rho )^{-1}{\mathbf i}\hat F_{[a]}^{[b]}(k, \rho ) &
\textrm{ if } \operatorname{dist}([a],[b])\le\Delta'\ \textrm{and}\ \ 0< |k|\le N\\ 0 & \textrm{ if not }, \end{array}\right. $$ $$B_{a}^{b}( \rho )= \left\{\begin{array}{ll} \hat F_{a}^{b}(0, \rho ) & \textrm{ if } \operatorname{dist}([a],[b])\le\Delta'\ \textrm{and}\ \ k=0\\ 0& \text{ if not} \end{array}\right. $$ and $$\hat R_{a}^{b}(k, \rho )= \left\{\begin{array}{ll}
\hat F_{a}^{b}(k, \rho )& \textrm{ if } \operatorname{dist}([a],[b])\ge\Delta'\ \textrm{or}\ \ |k|>N\\ 0 & \text{ if not}. \end{array}\right. $$ We denote again $\hat R_{a}^{b}(k, \rho )$ by $\widehat {(R^s)}_{a}^{b}(k, \rho )$ if $\operatorname{dist}([a],[b])\ge\Delta'$
and by $\widehat {(R^F)}_{a}^{b}(k, \rho )$ if $ |k|>N$.
We have to distinguish two cases, depending on when $k= 0$ or not.
{\it The case $k\not=0$. }
We have, by Lemma~\ref{lSmallDiv3},
$$ ||(L_{k,[a],[b]}( \rho ))^{-1}||\le \frac1 \kappa \,
$$ for all $ \rho $ outside some set $\Sigma_{k,[a],[b]}(\kappa)$ such that $$\operatorname{dist}( \mathcal{D}\setminus \Sigma_{k,[a],[b]}( \kappa ),\Sigma_{k,[a],[b]}(\frac\ka2))\ge {\operatorname{ct.} } \frac \kappa {N\chi}, $$ and
$$ \mathcal{D}_3= \mathcal{D}\setminus \bigcup_{\begin{subarray}{c} 0<|k|\le N\\ [a],[b]\end{subarray}}\Sigma_{k,[a],[b]}( \kappa )$$ fulfils the required estimate.
{\it The case $k=0$.} In this case we consider the block decomposition $ \mathcal{E} _{\Delta'}$ and we distinguish
whether $|a|=|b|$ or not.
If $|a|>|b|$, we use \eqref{ass1} and \eqref{la-lb-bis} to get
$$|\alpha( \rho )-\beta( \rho )|\geq c'-\frac{ \delta }{\langle a\rangle^\varkappa} -\frac{ \delta }{\langle b\rangle^\varkappa}\geq \frac{c'}{2}\ge \kappa .$$
This estimate allows us to solve the equation by choosing $$B_{[a]}^{[b]}=\hat R_{[a]}^{[b]}(0) =0$$ and $$\hat S_{[a]}^{[b]}(0, \rho )= L(0,[a],[b], \rho )^{-1}\hat F_{[a]}^{[b]}(0, \rho )$$ with $$
||\partial_ \rho ^j\hat S_{[a]}^{[b]}(0, \rho )||\le {\operatorname{Ct.} } \frac{1}{ \kappa } (N\frac{\chi}{\kappa})^{ |j |} \max_{0\le l\le j}\aa{ \partial_ \rho ^l\hat F_{[a]}^{[b]}(0, \rho )},$$ which implies \eqref{homo3S}.
If $|a|=|b|$, we cannot control $|\alpha( \rho )-\beta( \rho )|$ from below, so then we define $$\hat S_{[a]}^{[b]}(0)=0$$ and \begin{align*} B_{a}^{b}( \rho )=\hat F_{a}^{b}(0, \rho )) ,\quad \hat R_{a}^{b}(0) =0\quad &\text{for } [a]_{\Delta'}= [b]_{\Delta'}\\
\hat R_{a}^{b}(0, \rho ) =\hat F_{a}^{b}(0, \rho )\quad B_{a}^{b}=0,\quad& \text{for } [a]_{\Delta'}\not= [b]_{\Delta'}. \end{align*} Clearly $R$ and $B$ verify the estimates \eqref{homo3R} and \eqref{homo3B}.
Hence, the formal solution converges to functions verifying \eqref{homo3S}, \eqref{homo3B} and \eqref{homo3R} by Lemma~\ref{lSmallDiv3}. Moreover, for $ \rho \in \mathcal{D}'$, these functions are a solution of the fourth equation.
\endproof
\subsection{The homological equation}
For simplicity we shall restrict ourselves here to $ \sigma ,\mu,\gamma \le1$.
\begin{lemma}\label{thm-homo} There exists a constant $C$ such that if \eqref{ass1} holds, then, for any $N\ge1$, $\Delta'\ge \Delta\ge 1$ and $$ \kappa \le\frac1C c',$$ there exists a closed subset $ \mathcal{D}'= \mathcal{D}(h, \kappa , N)\subset \mathcal{D}$, satisfying $$\operatorname{meas}( \mathcal{D}\setminus { \mathcal{D}'})\le C (\Delta N)^{\exp_1} (\frac{ \kappa }{\delta_0})^{\alpha}(\frac{\chi}{\delta_0})^{1-\alpha}$$ and there exist real jet-functions $S, R=R^F+R^s\in \mathcal{T} _{\gamma ,\varkappa, \mathcal{D}}( \sigma ,\mu)$ and $h_+$ verifying, for $ \rho \in \mathcal{D}'$, \begin{equation}\label{eqHomEqbis} \{ h,S \}+ f^T= h_++R,\end{equation} and such that $$h+h_+\in \mathcal{NF} _{\varkappa}(\Delta', \delta _+)$$ and, for all $0< \sigma '< \sigma $,
\begin{equation}\label{estim-B} \ab{ h_+}_{\begin{subarray}{c} \sigma ',\mu\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\le X \ab{f^T}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}} \end{equation}
\begin{equation}\label{estim-S} \ab{S}_{\begin{subarray}{c} \sigma ',\mu\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\\ \leq \frac{1} \kappa X (N\frac{\chi}{ \kappa })^{{{s_*}}} \ab{f^T}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}} \end{equation}
and \begin{equation}\label{estim-R} \left\{\begin{array}{l} \ab{R^F}_{\begin{subarray}{c} \sigma ',\mu\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\leq Xe^{-( \sigma - \sigma ')N}\ab{f^T}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\\ \ab{R^s}_{\begin{subarray}{c} \sigma ',\mu\ \ \\ \gamma ', \varkappa, \mathcal{D} \end{subarray}}\leq Xe^{-(\gamma -\gamma ')\Delta'} \ab{f^T}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}} \end{array}\right.,\end{equation} for $\gamma _*\le\gamma '\le \gamma $, where $$ X=C\Delta'\big(\frac{\Delta}{ \sigma - \sigma ' }\big)^{\exp_2} e^{2\gamma d_\Delta}.$$
Moreover, $S_r(\cdot, \rho )=0$ for $ \rho $ near the boundary of $ \mathcal{D}$.
The exponent $\alpha$ is a positive constant only depending on $d,s_*, \varkappa $ and $\beta_2 $.
(The exponent $\exp_1$ only depends on $d$, $n=\# \mathcal{A} $ and $\tau,\beta_2, \varkappa $. The exponent $\exp_2$ only depends on $d,m_*,s_*$. $C$ is an absolute constant that depends on $c,\tau,\beta_2,\beta_3$ and $ \varkappa $. $C$ also depend on $\sup_ \mathcal{D}\ab{\Omega_{\textrm up}}$ and $\sup_ \mathcal{D}\ab{H_{\textrm up}}$, but stays bounded when these do.)
\end{lemma}
\begin{remark} The estimates \eqref{estim-B} provides an estimate of $ \delta _+$. Indeed, let $\frac 1 2 \langle w, Bw\rangle$ denote the quadratic part of $h_+$. Then, for any $a,b\in [a]_{\Delta'}$, $$
\ab{\partial_ \rho ^j B_a^b}\le \frac1 C||\partial_ \rho ^j B||_{(\gamma ,m_*), \varkappa }e_{(\gamma , \varkappa ), \varkappa }(a,b)^{-1}\le {\operatorname{Ct.} } (\Delta')^{ \varkappa } \ab{f^T}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}} \frac1{\langle a\rangle^{ \varkappa }} $$ -- recall the definition of the matrix norm \eqref{b-matrixnorm} and of the exponential weight \eqref{weight}. By
\eqref{estim-B} this is
$$\le {\operatorname{Ct.} } (\Delta')^{ \varkappa } \ab{f^T}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}} \frac1{\langle a\rangle^{ \varkappa }}.$$ Since $\# [a]_{\Delta'} \lesssim (\Delta')^{\exp}$ we get $$
|| \partial_ \rho ^j B( \rho )_{[a]_{\Delta'}} || \le {\operatorname{Ct.} } (\Delta')^{\exp} \ab{f^T}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}} \frac1{\langle a\rangle^{ \varkappa }}. $$ This gives the estimate $$\delta_+-\delta\le {\operatorname{Ct.} } (\Delta')^{\exp} \ab{f^T}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}.$$ \end{remark}
\begin{proof} The set $ \mathcal{D}'$ will now be given by the intersection of the sets in the three previous lemmas of this section. We set $$ h_+(r,w)=\hat f_r(r,0) + \frac 1 2 \langle w, Bw\rangle$$ $$ S(r,\theta,w)=S_r(\theta,r)+\langle S_w(\theta)w\rangle+ \frac 1 2 \langle S_{ww}(\theta)w,w \rangle$$ and $$ R(r,\theta,w)= R_r(r,\theta)+\langle R_w(\theta),w\rangle+ \frac 1 2 \langle R_{ww}(\theta)w,w \rangle,$$ with $R_{ww}=R_{ww}^F+R_{ww}^s$. These functions also depend on $ \rho \in \mathcal{D}$ and they verify equation \eqref{eqHomEqbis} for $ \rho \in \mathcal{D}'$.
If $x=(r,\theta,w)\in \mathcal{O} _{\gamma _*}( \sigma ,\mu)$, then
$$| h_+(x)|\le \ab{f^T}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}
+ \frac 1 2 || Bw ||_{\gamma _*}||w ||_{\gamma _*}.$$ Since $$\aa{B}_{\gamma , \varkappa } \ge \aa{B}_{\gamma _*, \varkappa }\ge \aa{B}_{ \mathcal{B} (Y_{\gamma _*},Y_{\gamma _*})}$$ it follows that
$$| h_+(x)|\le {\operatorname{Ct.} } \ab{f^T}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}.$$
We also have for any $x=(r,\theta,w)\in \mathcal{O} _{\gamma '}( \sigma ,\mu)$, $\gamma _*\le \gamma '\le\gamma $,
$$||Jd h_+(x)||_{\gamma '}\le
{\operatorname{Ct.} } \ab{f^T}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}+
||Bw||_{\gamma '}.$$ Since $$\aa{B}_{\gamma , \varkappa } \ge \aa{B}_{\gamma ', \varkappa }\ge \aa{B}_{ \mathcal{B} (Y_{\gamma '},Y_{\gamma '})}$$ it follows that
$$||Jd h_+(x)||_{\gamma '}\le {\operatorname{Ct.} } \ab{f^T}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}.$$ Finally $Jd^2 h_+(x)$ equals $ JB$ which satisfies the required bound.
The estimates of the derivatives with respect to $ \rho $ are the same and obtained in the same way.
The functions $S(\theta,r,\zeta)$, $R^F(\theta,r,\zeta)$ and $R^s (\theta,r,\zeta)$ are estimated in the same way. \end{proof}
\subsection{The non-linear homological equation}
The equation \eqref{eqNlHomEq} can now be solved easily. We restrict ourselves again to $ \sigma ,\mu,\gamma \le1$.
\begin{proposition}\label{thm-Eq} There exists a constant $C$ such that for any $$h\in \mathcal{NF} _{\varkappa}(\Delta, \delta ),\quad\delta \le \frac1C c',$$ and for any $$N\ge 1,\quad \Delta'\ge \Delta\ge 1,\quad \kappa \le\frac1C c'$$ there exists a closed subset $ \mathcal{D}'= \mathcal{D}(h, \kappa ,N)\subset \mathcal{D}$, satisfying $$\operatorname{meas}( \mathcal{D}\setminus { \mathcal{D}'})\le C (\Delta N)^{\exp_1} (\frac{ \kappa }{\delta_0})^{\alpha}(\frac{\chi}{\delta_0})^{1-\alpha},$$ and, for any $f\in \mathcal{T} _{\gamma ,\varkappa}( \sigma ,\mu, \mathcal{D})$ $$\varepsilon=\ab{f^T}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}\quad \textrm{and}\quad \xi=\ab{f}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}},$$ there exist real jet-functions $S,R=R^F+R^s\in \mathcal{T} _{\gamma ,\varkappa, \mathcal{D}}( \sigma ,\mu)$ and $h_+$ verifying, for $ \rho \in \mathcal{D}'$, \begin{equation}\label{eqNlHomEqbis} \{ h,S \}+\{ f-f^T,S \}^T+ f^T= h_++R\end{equation} and such that $$h+ h_+\in \mathcal{NF} _{\varkappa}(\Delta',\delta_+)$$ and, for all $ \sigma '< \sigma $ and $\mu'<\mu$,
\begin{equation}\label{estim-B2} \ab{ h_+}_{\begin{subarray}{c} \sigma ',\mu'\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\le CXY\varepsilon\end{equation}
\begin{equation}\label{estim-S2} \ab{S}_{\begin{subarray}{c} \sigma ',\mu'\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\\ \leq C \frac1 \kappa XY\varepsilon\end{equation}
and \begin{equation}\label{estim-R2} \left\{\begin{array}{l} \ab{R^F}_{\begin{subarray}{c} \sigma ',\mu'\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\leq Ce^{-( \sigma - \sigma ')N}XY\varepsilon\\ \ab{R^s}_{\begin{subarray}{c} \sigma ',\mu'\ \ \\ \gamma ', \varkappa, \mathcal{D} \end{subarray}}\leq Ce^{-(\gamma -\gamma ')\Delta'}XY\varepsilon, \end{array}\right.\end{equation} for $\gamma _*\le\gamma '\le \gamma $, where $$X=(\frac{N\Delta' e^{\gamma d_\Delta} }{( \sigma - \sigma ')(\mu-\mu')})^{\exp_2}$$ and $$ Y=(\frac{\chi+\xi}{ \kappa })^{4{s_*}+3}.$$
Moreover, $S_r(\cdot, \rho )=0$ for $ \rho $ near the boundary of $ \mathcal{D}$.
Moreover, if $\tilde \rho =(0, \rho _2,\dots, \rho _p)$ and $f^T(\cdot,\tilde \rho )=0$ for all $\tilde \rho $, then $S=R=0$ and $h_+=h$ for all $\tilde \rho $.
The exponent $\alpha$ is a positive constant only depending on $d,s_*, \varkappa $ and $\beta_2 $.
(The exponent $\exp_1$ only depends on $d$, $n=\# \mathcal{A} $ and $\tau,\beta_2, \varkappa $. The exponent $\exp_2$ only depends on $d,m_*,s_*$. $C$ is an absolute constant that depends on $c,\tau,\beta_2,\beta_3$ and $ \varkappa $. $C$ also depend on $\sup_ \mathcal{D}\ab{\Omega_{\textrm up}}$ and $\sup_ \mathcal{D}\ab{H_{\textrm up}}$, but stays bounded when these do.)
\end{proposition}
\begin{remark} Notice that the ``loss'' of $S$ with respect to $ \kappa $ is of ``order'' $4{{s_*}}+4$. However, if $\chi$, $ \delta $ and $\xi= \ab{f}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma ', \varkappa, \mathcal{D} \end{subarray}}$ are of size $ \lesssim \kappa $, then the loss is only of ``order'' 1. \end{remark}
\begin{proof} Let $S=S_0+S_1+S_2$ be a jet-function such that $S_1$ starts with terms of degree $1$ in $r,w$ and $S_2$ starts with terms of degree $2$ in $r,w$ -- jet functions are polynomials in $r,w$ and we give (as is usual) $w$ degree $1$ and $r$ degree $2$.
Let now $ \sigma '= \sigma _5< \sigma _4< \sigma _3< \sigma _2< \sigma _1< \sigma _0= \sigma $ be a (finite) arithmetic progression, i.e. $ \sigma _j- \sigma _{j+1}$ do not depend on $j$, and let
and $\mu'=\mu_5<\mu_4<\mu_3<\mu_2<\mu_1<\mu_0=\mu$ be another arithmetic progressions.
Then $\{ h',S\}+\{ f-f^T,S \}^T+ f^T= h_++R$ decomposes into three homological equations $$\{ h',S_0 \}+ f^T= ( h_+)_{0}+R_0,$$ $$\{ h',S_1 \}+f_1^T= ( h_+)_{1}+R_1,\quad f_1=\{ f-f^T,S_0 \},$$ $$\{ h',S_2 \}+ f_2^T= ( h_+)_{2}+R_2,\quad f_2=\{ f-f^T,S_1 \}.$$
By Lemma~\ref{thm-homo} we have for the first equation $$\ab{( h_+)_0}_{\begin{subarray}{c} \sigma _1,\mu\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\le X\varepsilon, \quad \ab{S_0}_{\begin{subarray}{c} \sigma _1,\mu\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\\ \leq \frac{1} \kappa X Y\varepsilon $$ where $$ X=C\Delta'\big(\frac{5\Delta}{ \sigma - \sigma ' }\big)^{\exp} e^{2\gamma _1 d_\Delta}.$$ and where $Y,Z$ are defined by the right hand sides in the estimates \eqref{estim-S} and \eqref{estim-R}.
By Proposition \ref{lemma:poisson} we have $$ \xi_1=\ab{f_1}_{\begin{subarray}{c} \sigma _2,\mu_2\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\le \frac{1} \kappa X YW\xi \varepsilon$$ where $$W=C \big(\frac5{(\sigma-\sigma')} + \frac5{ (\mu-\mu') }\big).$$ By Proposition \ref{lemma:jet} $\varepsilon_1=\ab{f_1^T}_{\begin{subarray}{c} \sigma _2,\mu_2\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}$ satisfies the same bound as $\xi_1$
By Lemma~\ref{thm-homo} we have for the second equation $$\ab{( h_+)_1}_{\begin{subarray}{c} \sigma _3,\mu_2\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\le X\varepsilon_1,\quad \ab{S_1}_{\begin{subarray}{c} \sigma _3,\mu_2\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\\ \leq \frac{1} \kappa X Y\varepsilon_1. $$
By Propositions \ref{lemma:jet} and \ref{lemma:poisson} we have $$ \xi_2=\ab{f_2}_{\begin{subarray}{c} \sigma _4,\mu_4\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\le \frac1 \kappa X YW\xi_1 \varepsilon_1,$$ and $\varepsilon_2=\ab{f_2^T}_{\begin{subarray}{c} \sigma _4,\mu_4\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}$ satisfies the same bound.
By Lemma~\ref{thm-homo} we have for the third equation $$\ab{( h_+)_2}_{\begin{subarray}{c} \sigma _5,\mu_4\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\le X\varepsilon_2,\quad \ab{S_2}_{\begin{subarray}{c} \sigma _5,\mu_4\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\\ \leq \frac{1} \kappa X Y\varepsilon_2. $$
Putting this together we find that $$\varepsilon+\varepsilon_1+\varepsilon_2 \le (1+\frac1 \kappa X YW\xi)^3\varepsilon= T\varepsilon$$ and $$\ab{h_+}_{\begin{subarray}{c} \sigma ',\mu'\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\le XT\varepsilon,\quad \ab{S}_{\begin{subarray}{c} \sigma ',\mu'\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\\ \leq \frac1 \kappa X YT\varepsilon. $$
Renaming $X$ and $Y$ gives now the estimates for $h_+$ and $S$. $R=R_0+R_1+R_2$ and its estimates follows immediately from the homological equation.
The final statement does not follow from Lemma~\ref{thm-homo}. However, if one follows the whole construction through the proofs of Lemmas \ref{prop:homo12} to \ref{thm-homo} one sees that it holds. For example in Lemma~\ref{prop:homo12} it is seen immediately that this holds for $\tilde \rho \notin \Sigma(L_k,\frac\kappa2)$. The only arbitrariness in the construction is the extension, but we have chosen it so that $S_r$ and $R_r$ are $=0$ on $\Sigma(L_k,\frac\kappa2)$. The construction Lemmas \ref{prop:homo3} and \eqref{prop:homo4} displays the same feature. \end{proof}
\section{Proof of the KAM Theorem}
Theorem~\ref{main} is proved by an infinite sequence of change of variables typical for KAM-theory. The change of variables will be done by the classical Lie transform method which is based on a well-known relation between composition of a function with a Hamiltonian flow $\Phi^t_S$ and Poisson brackets: $$\frac{d}{d t} f\circ \Phi^t_S=\{f,S\}\circ \Phi^t_S $$ from which we derive $$f \circ \Phi^1_S=f+\{f,S\}+\int_0^1 (1-t)\{\{f,S\},S\}\circ \Phi^t_S\ \text{d} t.$$ Given now three functions $ h,k$ and $f$. Then \begin{multline*}( h+k+f )\circ \Phi^1_S=\\
h+k+f +\{ h+k+f ,S\}+\int_0^1 (1-t)\{\{ h+k+f ,S\},S\}\circ \Phi^t_S\ \text{d} t. \end{multline*} If now $S$ is a solution of the equation \begin{equation} \label{eq-homobis} \{ h,S \}+\{ f-f^T,S \}^T+ f^T= h_++R^F+R^s, \end{equation} then $$ ( h+k+f )\circ \Phi^1_S= h+k+h_++f_+ +R^s$$ with \begin{multline}\label{f+} f_+=R^F+ (f-f^T)+\{k+f^T ,S\}+\{f-f^T,S\}-\{f-f^T,S\}^T+\\ +\int_0^1 (1-t)\{\{ h+k+f ,S\},S\}\circ \Phi^t_S\ \text{d} t \end{multline} and \begin{equation} \label{f+T} f_+^T= R^F+ \{k+f^T ,S\}^T+(\int_0^1 (1-t)\{\{h+k+f ,S\},S\}\circ \Phi^t_S\ \text{d} t)^T. \end{equation}
If we assume that $S$ and $R^F$ are ``small as'' $f^T$, then $f_+^T$ is is ``small as'' $k f^T$ -- this is the basis of a linear iteration scheme with (formally) linear convergence. \footnote{\ it was first used by Poincar\'e, credited by him to the astronomer Delauney, and it has been used many times since then in different contexts. } But if also $k$ is of the size $f^T$, then $f^+$ is ``small as'' the square of $f^T$ -- this is the basis of a quadratic iteration scheme with (formally) quadratic convergence. We shall combine both of them.
First we shall give a rigorous version of the change of variables described above. We restrict ourselves to the case when $ \sigma ,\mu,\gamma \le1$.
\subsection{The basic step} Let $h\in \mathcal{NF} _{\varkappa}(\Delta, \delta )$ and assume $ \varkappa >0$ and
\begin{equation}\label{ass2}
\delta \le \frac{1}{C}c' .\end{equation}
Let $$\gamma =(\gamma ,m_*)\ge \gamma _*=(0,m_*)$$ and recall Remark~\ref{rAbuse} and the convention \eqref{Conv}. Let $N\ge 1$, $\Delta'\ge \Delta\ge 1$ and $$ \kappa \le\frac1C c'.$$ The constant $C$ is to be determined.
Proposition \ref{thm-Eq} then gives, for any $f\in \mathcal{T} _{\gamma ,\varkappa, \mathcal{D}}( \sigma ,\mu)$, $$\varepsilon=\ab{f^T}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}\quad \textrm{and}\quad \xi=\ab{f}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}},$$ a set $ \mathcal{D}'= \mathcal{D}'(h, \kappa ,N)\subset \mathcal{D}$ and functions $h_+,S, R=R^F+R^s,$ satisfying \eqref{estim-B2}+\eqref{estim-S2}+\eqref{estim-R2} and solving the equation \eqref{eq-homobis}, $$\{h,S \}+\{ f-f^T,S \}^T+ f^T= h_++R,$$ for any $ \rho \in \mathcal{D}'$. Let now $0< \sigma '= \sigma _4< \sigma _3< \sigma _2< \sigma _1< \sigma _0= \sigma $ and $0<\mu'=\mu_4<\mu_3<\mu_2<\mu_1<\mu_0=\mu$ be (finite) arithmetic progressions.
\noindent{\it The flow $\Phi^t_S$.}\ We have, by \eqref{estim-S2}, $$\ab{S}_{\begin{subarray}{c} \sigma _1,\mu_1\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\\ \leq {\operatorname{Ct.} } \frac1 \kappa XY\varepsilon$$ where $X,Y$ and $ {\operatorname{Ct.} } $ are given in Proposition \ref{thm-Eq}, i.e. $$X=(\frac{\Delta' e^{\gamma d_\Delta}N}{( \sigma _0- \sigma _1)(\mu_0-\mu_1)})^{\exp_2}= (\frac{4^2\Delta' e^{\gamma d_\Delta}N}{( \sigma - \sigma ')(\mu-\mu')})^{\exp_2}\,,\qquad Y=(\frac{\chi+\xi}{ \kappa })^{4{{s_*}}+3}$$ -- we can assume without restriction that $\exp_2\ge 1$.
If \begin{equation}\label{hyp-f1} \varepsilon\leq \frac1C \frac{ \kappa }{X^2Y}, \end{equation} and $C$ is sufficiently large, then we can apply Proposition \ref{Summarize}(i). By this proposition it follows that for any $0\le t\le1$ the Hamiltonian flow map $\Phi^t_S$ is a $ \mathcal{C} ^{{s_*}}$-map $$ \mathcal{O} _{\gamma '}( \sigma _{i+1},\mu_{i+1})\times \mathcal{D}\to \mathcal{O} _{\gamma '}( \sigma _i,\mu_i),\quad \forall \gamma _*\le\gamma '\le\gamma ,\quad i=1,2,3, $$
real holomorphic and symplectic for any fixed $\rho\in \mathcal{D}$. Moreover,
$$|| \partial_ \rho ^j (\Phi^t_S(x,\cdot)-x)||_{\gamma '}\le {\operatorname{Ct.} } \frac1 \kappa XY \varepsilon$$ and $$\aa{ \partial_ \rho ^j (d\Phi^t_S(x,\cdot)-I)}_{\gamma ', \varkappa }\le {\operatorname{Ct.} } \frac1 \kappa XY \varepsilon$$ for any $x\in \mathcal{O} _{\gamma '}( \sigma _2,\mu_2)$, $\gamma _*\le \gamma '\le\gamma $, and $0\le \ab{j}\le {s_*}$.
\noindent{\it A transformation.} \ Let now $k\in \mathcal{T} _{\gamma ,\varkappa, \mathcal{D}}( \sigma ,\mu)$ and set $$\eta=\ab{k}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}.$$
Then we have $$(h+k+f )\circ \Phi^1_S= h+k+h_++f_+ +R$$ where $f_+$ is defined by \eqref{f+}, i.e. \begin{multline*} f_+= (f-f^T)+\{k+f^T ,S\}+\{f-f^T,S\}-\{f-f^T,S\}^T+\\ +\int_0^1 (1-t)\{\{ h+k+f ,S\},S\}\circ \Phi^t_S\ \text{d} t. \end{multline*} The integral term is the sum $$ \int_0^1 (1-t)\{h_++R-f^T,S\}\circ \Phi^t_S\ \text{d} t +\int_0^1 (1-t)\{\{k+f ,S\}-\{f-f^T,S\}^T,S\}\circ \Phi^t_S\ \text{d} t.$$
\noindent{\it The estimates of $\{k+f^T,S \}$ and $\{f-f^T,S \}$.} \ By Proposition \ref{lemma:poisson}(i) $$ \ab{\{k+f^T,S\}}_{\begin{subarray}{c} \sigma _2,\mu_2 \ \\ \gamma , \alpha, \mathcal{D} \end{subarray}} \leq {\operatorname{Ct.} } X \ab{S}_{\begin{subarray}{c} \sigma _1,\mu_1\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}
\ab{k+f^T}_{\begin{subarray}{c} \sigma _1,\mu_1 \ \\ \gamma , \alpha, \mathcal{D} \end{subarray}}.$$
Hence
\begin{equation} \ab{\{k+f^T,S\}}_{\begin{subarray}{c} \sigma _2,\mu_2 \ \\ \gamma , \alpha, \mathcal{D} \end{subarray}}\leq {\operatorname{Ct.} } \frac1 \kappa X^2Y(\eta+\varepsilon) \varepsilon.\end{equation}
Similarly, \begin{equation}
\ab{\{f-f^T,S\}}_{\begin{subarray}{c} \sigma _2,\mu_2 \ \\ \gamma , \alpha, \mathcal{D} \end{subarray}}\leq {\operatorname{Ct.} } \frac1 \kappa X^2Y\xi\varepsilon.\end{equation}
\noindent{\it The estimate of $\{h_+-f^T,S \}\circ\Phi^t_S$.} \ The estimate of $h_+$ is given by \eqref{estim-B2}: $$\ab{ h_+}_{\begin{subarray}{c} \sigma _1,\mu_1\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\le {\operatorname{Ct.} } XY\varepsilon.$$ This gives, again by Proposition \ref{lemma:poisson}(i), $$ \ab{\{h_+-f^T,S\}}_{\begin{subarray}{c} \sigma _2,\mu_2 \ \\ \gamma , \alpha, \mathcal{D} \end{subarray}}\leq {\operatorname{Ct.} } \frac1 \kappa X^3Y^2\varepsilon^2.$$
Let now $F=\{h_+-f^T,S \}$. If $\varepsilon$ verifies \eqref{hyp-f1} for a sufficiently large constant $C$, then we can apply Proposition \ref{Summarize}(ii). By this proposition, for $\ab{t}\le1$, the function $F\circ \Phi_S^t\in \mathcal{T} _{\gamma , \varkappa , \mathcal{D}}( \sigma _3,\mu_3)$ and \begin{equation} \ab{\{h_+-f^T,S \}\circ \Phi_S^t}_{\begin{subarray}{c} \sigma _3,\mu_3 \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}\leq {\operatorname{Ct.} } \frac1 \kappa X^3Y^2\varepsilon^2.\end{equation}
\noindent{\it The estimate of $\{R,S \}\circ\Phi^t_S$.} \ The estimate of $R$ is given by \eqref{estim-R2}. It implies that
$$\ab{R}_{\begin{subarray}{c} \sigma _1,\mu_1\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\leq {\operatorname{Ct.} } XY\varepsilon.$$ Then, as in the previous case, \begin{equation} \ab{\{R,S \}\circ \Phi_S^t}_{\begin{subarray}{c} \sigma _3,\mu_3 \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}\leq {\operatorname{Ct.} } \frac1 \kappa X^3Y^2\varepsilon^2.\end{equation}
\noindent{\it The estimate of $\{\{k+f,S \}-\{f-f^T,S\}^T,S\}\circ\Phi^t_S$.} \ This function is estimated as above. If $F=\{\{k+f,S \}-\{f-f^T,S\}^T,S\}$, then, by Proposition \ref{lemma:jet} and Proposition \ref{lemma:poisson}(i), $$ \ab{F}_{\begin{subarray}{c} \sigma _3,\mu_3 \ \\ \gamma , \alpha, \mathcal{D} \end{subarray}}\leq {\operatorname{Ct.} } (\frac1 \kappa X^2Y)^2(\eta+\xi)\varepsilon^2$$ and by Proposition \ref{Summarize}(ii) \begin{equation} \ab{\{\{k+f,S \}-\{f-f^T\}^T,S\}\circ \Phi_S^t}_{\begin{subarray}{c} \sigma _4,\mu_4 \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}\leq {\operatorname{Ct.} } (\frac1 \kappa X^2Y)^2(\eta+\xi)\varepsilon^2 .\end{equation}
\noindent{\it The estimates of $R^F$ and $R^s$.}\ These estimates are given by \eqref{estim-R2}: $$\ab{R^F}_{\begin{subarray}{c} \sigma _1,\mu_1\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\leq {\operatorname{Ct.} } XY e^{-( \sigma - \sigma ')N}\varepsilon$$ and $$\ab{R^s}_{\begin{subarray}{c} \sigma _1,\mu_1\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\leq {\operatorname{Ct.} } XY e^{-(\gamma -\gamma ')\Delta'}\varepsilon.$$
Renaming now $X$ and $Y$ and denoting $R^s$ by $R_+$ gives the following lemma.
\begin{lemma}\label{basic} There exists an absolute constant $C_1$ such that, for any $$h\in \mathcal{NF} _{\varkappa}(\Delta, \delta ),\quad \varkappa >0,\quad \delta \le \frac1{C_1} c',$$ and for any $$N\ge 1,\quad \Delta'\ge \Delta\ge 1,\quad \kappa \le\frac1{C_1} c',$$ there exists a closed subset $ \mathcal{D}'= \mathcal{D}( h, \kappa ,N)\subset \mathcal{D}$, satisfying $$\operatorname{meas}( \mathcal{D}\setminus { \mathcal{D}'})\le {C_1} (\Delta N)^{\exp_1} (\frac{ \kappa }{\delta_0})^{\alpha}(\frac{\chi}{\delta_0})^{1-\alpha}$$ and, for any $f\in \mathcal{T} _{\gamma ,\varkappa, \mathcal{D}}( \sigma ,\mu)$, $$\varepsilon=\ab{f^T}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}\quad \textrm{and}\quad \xi=[f]_{ \sigma ,\mu, \mathcal{D}}^{\gamma ,\varkappa},$$ satisfying $$\varepsilon \leq \frac1{C_1} \frac{ \kappa }{XY}, \qquad \left\{\begin{array}{ll} X=(\frac{N\Delta' e^{\gamma d_\Delta}}{( \sigma - \sigma ')(\mu-\mu')})^{\exp_1},& \sigma '< \sigma ,\ \mu'<\mu\\ Y= (\frac{\chi+\xi} \kappa )^{\exp_1},&\ \end{array}\right. $$ and for any $k\in \mathcal{T} _{\gamma ,\varkappa, \mathcal{D}}( \sigma ,\mu)$, $$\eta=\ab{k}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}},$$ there exists a $ \mathcal{C} ^{{s_*}}$ mapping $$\Phi: \mathcal{O} _{\gamma '}( \sigma ',\mu')\times \mathcal{D}\to \mathcal{O} _{\gamma '}( \sigma -\frac{ \sigma - \sigma '}2,\mu-\frac{\mu-\mu'}2),\quad \forall \gamma _*\le\gamma '\le\gamma ,$$ real holomorphic and symplectic for each fixed parameter $ \rho \in \mathcal{D}$, and functions $f_+,R_+\in \mathcal{T} _{\gamma ,\varkappa, \mathcal{D}}( \sigma ',\mu')$ and $$h+h_+\in \mathcal{NF} _{\varkappa}(\Delta',\delta_+),$$ such that $$(h+k+f )\circ \Phi= h+k+ h_++f_++R_+,\quad \forall \rho \in \mathcal{D}',$$ and $$\ab{ h_+}_{\begin{subarray}{c} \sigma ',\mu'\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\le
{C_1}XY\varepsilon,$$ $$\ab{ f_+-f}_{\begin{subarray}{c} \sigma ',\mu'\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\le
{C_1}XY(1+\eta+\xi)\varepsilon,$$ $$\ab{ f_+^T}_{\begin{subarray}{c} \sigma ',\mu'\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\le {C_1}\frac1 \kappa XY (\eta+ \kappa e^{-( \sigma - \sigma ')N}+\varepsilon )\varepsilon $$ and $$\ab{ R_+}_{\begin{subarray}{c} \sigma ',\mu'\ \ \\ \gamma ', \varkappa, \mathcal{D} \end{subarray}}\le {C_1} XYe^{-(\gamma -\gamma ')\Delta'}\varepsilon$$ for any $\gamma _*\le\gamma '\le\gamma $.
Moreover,
$$|| \partial_ \rho ^j (\Phi(x, \rho )-x)||_{\gamma '}+ \aa{ \partial_ \rho ^j (d\Phi(x, \rho )-I)}_{\gamma ', \varkappa } \le {C_1}\frac1 \kappa XY \varepsilon$$ for any $x\in \mathcal{O} _{\gamma '}( \sigma ',\mu')$, $\gamma _*\le\gamma '\le\gamma $, $\ab{j}\le{s_*}$ and $ \rho \in \mathcal{D}$, and $\Phi(\cdot, \rho )$ equals the identity for $ \rho $ near the boundary of $ \mathcal{D}$.
Finally, if $\tilde \rho =(0, \rho _2,\dots, \rho _p)$ and $f^T(\cdot,\tilde \rho )=0$ for all $\tilde \rho $, then $f_+-f=R_+=h_+=0$ and $\Phi(x,\cdot)=x$
for all $\tilde \rho $.
\end{lemma}
\begin{remark}\label{constants} The exponent $\alpha$ is a positive constant only depending on $d,s_*, \varkappa $ and $\beta_2 $.
The exponent $\exp_1$ only depends on $d$, $n=\# \mathcal{A} ,s_*$ and $\tau,\beta_2, \varkappa $. ${C_1}$ is an absolute constant that depends on $c,\tau,\beta_2,\beta_3$ and $ \varkappa $. ${C_1}$ also depend on $\sup_ \mathcal{D}\ab{\Omega_{\textrm up}}$ and $\sup_ \mathcal{D}\ab{H_{\textrm up}}$, but stays bounded when these do.
\end{remark}
\subsection{A finite induction} We shall first make a finite iteration without changing the normal form in order to decrease strongly the size of the perturbation. We shall restrict ourselves to the case when $N=\Delta'$.
\begin{lemma}\label{Birkhoff} There exists a constant ${C_2}$ such that, for any $$h\in \mathcal{NF} _{\varkappa}(\Delta, \delta ),\quad \varkappa >0,\quad\delta \le \frac1{C_2} c',$$ and for any $$\Delta'\ge \Delta\ge 1,\quad \kappa \le\frac1{C_2} c',$$ there exists a closed subset $ \mathcal{D}'= \mathcal{D}( h, \kappa ,\Delta')\subset \mathcal{D}$, satisfying $$\operatorname{meas}( \mathcal{D}\setminus { \mathcal{D}'})\le {C_2} (\Delta')^{\exp_2} (\frac{ \kappa }{\delta_0})^{\alpha}(\frac{\chi}{\delta_0})^{1-\alpha}$$ and, for any $f\in \mathcal{T} _{\gamma ,\varkappa, \mathcal{D}}( \sigma ,\mu)$, $$\varepsilon=\ab{f^T}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}\quad \textrm{and}\quad \xi=[f]_{ \sigma ,\mu, \mathcal{D}}^{\gamma ,\varkappa},$$ satisfying $$\varepsilon \leq \frac1{C_2} \frac{ \kappa }{XY},\quad \left\{\begin{array}{ll} X=(\frac{\Delta' e^{\gamma d_\Delta}}{( \sigma - \sigma ')(\mu-\mu')}\log\frac1{\varepsilon})^{\exp_2},& \sigma '< \sigma ,\ \mu'<\mu\\ Y= (\frac{\chi+\xi} \kappa )^{\exp_2},&\ \end{array}\right. $$ there exists a $ \mathcal{C} ^{{s_*}}$ mapping $$\Phi: \mathcal{O} _{\gamma '}( \sigma ',\mu')\times \mathcal{D}\to \mathcal{O} _{\gamma '}( \sigma -\frac{ \sigma - \sigma '}{2},\mu-\frac{\mu-\mu'}{2}), \quad \forall \gamma _*\le\gamma '\le\gamma ,$$ real holomorphic and symplectic for each fixed parameter $ \rho \in \mathcal{D}$, and functions $f'\in \mathcal{T} _{\gamma ,\varkappa, \mathcal{D}}( \sigma ',\mu')$ and $$h'\in \mathcal{NF} _{\varkappa}(\Delta', \delta '),$$ such that $$(h+f )\circ \Phi= h'+f',\quad \forall \rho \in \mathcal{D}',$$ and $$\ab{ h'- h}_{\begin{subarray}{c} \sigma ',\mu'\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\le {C_2} XY \varepsilon,$$ $$\xi'=\ab{ f'}_{\begin{subarray}{c} \sigma ',\mu'\ \ \\ \gamma ', \varkappa, \mathcal{D} \end{subarray}} \le \xi+ C_2XY(1+\xi)\varepsilon$$ and $$ \varepsilon'=\ab{ (f')^T}_{\begin{subarray}{c} \sigma ',\mu'\ \ \\ \gamma ', \varkappa, \mathcal{D} \end{subarray}} \le {C_2} XY(e^{-( \sigma - \sigma ')\Delta'}+ e^{-(\gamma -\gamma ')\Delta'})\varepsilon,$$ for any $\gamma _*\le\gamma '\le \gamma $.
Moreover,
$$|| \partial_ \rho ^j (\Phi(x, \rho )-x)||_{\gamma '}+ \aa{ \partial_ \rho ^j (d\Phi(x, \rho )-I)}_{\gamma ', \varkappa } \le {C_2}\frac1 \kappa XY \varepsilon$$ for any $x\in \mathcal{O} _{\gamma '}( \sigma ',\mu')$, $\gamma _*\le\gamma '\le\gamma $, $\ab{j}\le{s_*}$, and $ \rho \in \mathcal{D}$, and $\Phi(\cdot, \rho )$ equals the identity for $ \rho $ near the boundary of $ \mathcal{D}$.
Finally, if $\tilde \rho =(0, \rho _2,\dots, \rho _p)$ and $f^T(\cdot,\tilde \rho )=0$ for all $\tilde \rho $, then $f'-f=h'=0$ and $\Phi(x,\cdot)=x$
for all $\tilde \rho $.
( The exponents $\alpha$, $\exp_2$ and the constant ${C_2}$ have the same properties as those in Remark
\ref{constants}.)
\end{lemma}
\begin{proof} Let $N=\Delta'$ and $ \kappa \le\frac{c'}{C_1}$. Let $ \sigma _1= \sigma -\frac{ \sigma - \sigma '}2$, $\mu_1=\mu-\frac{\mu-\mu'}2$ and $ \sigma _{K+1}= \sigma '$, $\mu_{K+1}=\mu'$, and let $\{ \sigma _j\}_1^{K+1}$ and $\{\mu_j\}_1^{K+1}$ be arithmetical progressions. Let $$( \sigma - \sigma ')\Delta'\le K\le ( \sigma - \sigma ')\Delta'(\log\frac \kappa {\varepsilon})^{-1}.$$ This implies that
$$ \kappa e^{-( \sigma _{j}- \sigma _{j+1})N}\le \varepsilon.$$
We let $f_1=f$ and $k_1=0$, and we let $\varepsilon_1= [f_1^T]_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}=\varepsilon$, $\xi_1= [f_1]_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}=\xi$, $\delta_1=\delta$ and $\eta_1=[k_1]_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}=0$.
Define now $$\varepsilon_{j+1}=C_1\frac1 \kappa X_jY_j(\eta_j+\varepsilon_1+\varepsilon_j)\varepsilon_j,$$ $$\xi_{j+1}=\xi_j+ C_1 X_jY_j(1+\eta_j+\xi_j) \varepsilon_j,\quad \eta_{j+1}=\eta_j+C_1X_jY_j\varepsilon_j,$$ with $$X_j=(\frac{N\Delta' e^{\gamma d_\Delta}} {( \sigma _j- \sigma _{j+1})(\mu_j-\mu_{j+1})})^{\exp_1},\quad Y_j=(\frac{\chi+\xi_j}{ \kappa })^{\exp_1},$$ where $C_1, \exp_1$ are given in Lemma~\ref{basic}. Notice that $X_j=X_1$.
\begin{sublem*} If $$\varepsilon_1\le\frac 1{C_2}
\frac \kappa { X_1^2Y_1^2},\quad C_2=3eC_1 2^{\exp_1},$$
then, for all $j\ge1$,
$$\varepsilon_j\le \frac1{C_1} \frac \kappa { X_j^2Y_j^2}\quad\textrm{and}\quad\varepsilon_{j}\le (\frac{C_2}2 \frac{ X^2_1Y^2_1} \kappa \varepsilon_1)^{j-1}\varepsilon_1\le e^{-(j-1)}\varepsilon_1 ,$$ $$ \xi_{j} - \xi_1 \le 2C_1 X_1 Y_1(1+\xi_1)\varepsilon_1 \quad\textrm{and}\quad \eta_{j} \le 2C_1 X_1 Y_1\varepsilon_1.$$ \end{sublem*}
This sublemma shows that we can apply Lemma~\ref{basic} K times to get a sequence of mappings $$\Phi_j: \mathcal{O} _{\gamma '}( \sigma _{j+1},\mu_{j+1})\times \mathcal{D}'\to \mathcal{O} _{\gamma '}( \sigma _j-\frac{ \sigma _j- \sigma _{j+1}}2,\mu_j-\frac{\mu_j-\mu_{j+1}}2),\quad \gamma _*\le\gamma '\le\gamma _{j}$$ and functions $f_{j+1}$ and $R_{j+1}$ such that, for $ \rho \in \mathcal{D}'$, $$(h+k_j+f_j )\circ \Phi_j= h+k_{j+1}+ f_{j+1}$$ with $k_{j+1}=k_j+ h_{j+1}+R_{j+1}$.
Let $f'=f_{K+1}+R_1+\dots+R_{K+1}$ and $h'=h_1+\dots+h_{K+1}$. Then $$\ab{h'-h}_{\begin{subarray}{c} \sigma ',\mu'\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\le C_1\sum X_jY_j\varepsilon_j\le\eta_{K+1}\le 2C_1X_1Y_1\varepsilon_1,$$ $$\ab{f'-f}_{\begin{subarray}{c} \sigma ',\mu'\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\le C_1\sum X_jY_j(1+\xi_j+\eta_j)\varepsilon_j \le4C_1X_1Y_1(1+\xi_1)\varepsilon_1$$ and $$\ab{(f')^T}_{\begin{subarray}{c} \sigma ',\mu'\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}}\le \varepsilon_{K+1}+ C_1\sum X_jY_je^{(\gamma -\gamma ')\Delta'}\varepsilon_j\le $$ $$ e^{-K}\varepsilon_1+2C_1X_1Y_1e^{(\gamma -\gamma ')\Delta'}\varepsilon_1\le e^{( \sigma - \sigma ')\Delta'}\varepsilon_1+ 2C_1X_1Y_1e^{(\gamma -\gamma ')\Delta'}\varepsilon_1.$$
We then take $\Phi=\Phi_1\circ\dots\circ \Phi_K$. For the estimates of $\Phi$, write $\Psi_j=\Phi_j\circ\dots\circ \Phi_K$ and $\Psi_{K+1}=id$. For $(x, \rho )\in \mathcal{O} _{\gamma '}( \sigma ',\mu')\times \mathcal{D}$ we then have
$$||\Phi(x, \rho )-x||_{\gamma '}\le
\sum_{j=1}^K ||\Psi_j(x, \rho )-\Psi_{j+1}(x, \rho )||_{\gamma '}.$$ Then $$
||\Psi_j(x, \rho )-\Psi_{j+1}(x, \rho )||_{\gamma '}=||\Phi_j(\Psi_{j+1}(x, \rho ), \rho )-\Psi_{j+1}(x, \rho )||_{\gamma '}$$ is $$ \le C_1 \frac1 \kappa X_jY_j \varepsilon_j.$$ It follows that
$$||\Phi(x, \rho )-x||_{\gamma '}\le 2C_1 \frac1 \kappa X_1Y_1 \varepsilon_1.$$
The estimate of $||d\Phi(x, \rho )-I||_{\gamma '}$ is obtained in the same way.
The derivatives with respect to $ \rho $ depends on higher order differentials which can be estimated by Cauchy estimates.
The result now follows if we take $C_2$ sufficiently large and increases the exponent $\exp_1$. \end{proof}
\noindent{\it Proof of sublemma.}\ The estimates are true for $j=1$ so we proceed by induction on $j$. Let us assume the estimates hold up to $j$. Then, for $k\le j$, $$Y_{k}\le (\frac{\chi+\xi_1+2C_1X_1Y_1(1+\xi_1)\varepsilon_1}{ \kappa })^{\exp_1}= 2^{\exp_1} Y_1$$ and $$\varepsilon_{j+1}\le 2^{\exp_1} \frac{ X_1Y_1} \kappa [2C_1 X_1 Y_1\varepsilon_1+\varepsilon_1+\varepsilon_1]\varepsilon_j\le C'\frac{ X^2_1Y^2_1} \kappa \varepsilon_1\varepsilon_j,$$ $C'=3C_1 2^{\exp_1} $. Then $$\xi_{j+1}-\xi_1\le2^{\exp_1} X_1Y_1(1+\xi_1+4C_1 X_1 Y_1(1+\xi_1)\varepsilon_1 )(\varepsilon_1+\dots+\varepsilon_{j+1})\le $$ $$ 2^{\exp_1} X_1Y_1(1+\xi_1)(1+4C_1 X_1 Y_1\varepsilon_1 )2\varepsilon_1 \le 2^{\exp_1} 4X_1Y_1(1+\xi_1)\varepsilon_1, $$ if $4C_1 X_1 Y_1\varepsilon_1\le1$ and $C'\frac{ X^2_1Y^2_1} \kappa \varepsilon_1\le\frac1e\le\frac12$ -- and similarly for $\eta_{j+1}$. \subsection{The infinite induction}
We are now in position to prove our main result, Theorem~\ref{main}.
Let $h$ be a normal form Hamiltonian in $ \mathcal{NF} _{\varkappa}(\Delta,\delta)$ and let $f\in \mathcal{T} _{\gamma ,\varkappa, \mathcal{D}}( \sigma ,\mu)$ be a perturbation such that $$ 0<\varepsilon= \ab{f^T}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}},\quad \xi=\ab{f}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa , \mathcal{D} \end{subarray}}. $$ We construct the transformation $\Phi$ as the composition of infinitely many transformations $\Phi$ as in Lemma~\ref{Birkhoff}. We first specify the choice of all the parameters for $j\geq 1$.
Let $C_2,\exp_2$ and $\alpha$ be the constants given in Lemma~\ref{Birkhoff}.
\subsubsection{Choice of parameters} We have assumed $\gamma , \sigma ,\mu\le1$ and we take $\Delta\ge1$. By decreasing $\gamma $ or increasing $\Delta$ we can also assume $\gamma =(d_{\Delta})^{-1}$.
We choose for $j\ge1$ $$ \mu_j=\big(\frac 12 +\frac 1 {2^j}\big)\mu\quad\textrm{and}\quad \sigma _{j}=\big(\frac 1 2 +\frac 1 {2^j}\big) \sigma .$$ We define inductively the sequences $\varepsilon_j$, $\Delta_j$, $ \delta _j$ and $\xi_j$ by \begin{equation}\left\{\begin{array}{ll} \varepsilon_{j+1}= \varepsilon^{K_j}\varepsilon & \varepsilon_1=\varepsilon\\ \Delta_{j+1} =4K_j \max(\frac {1} { \sigma _j- \sigma _{j+1}},d_{\Delta_j})\log \frac1{\varepsilon}&\Delta_1=\Delta\\ \gamma _{j+1}=(d_{\Delta_{j+1}})^{-1}& \gamma _1=\gamma \\ \delta _{j+1}= \delta _j+ C_2 X_jY_j\varepsilon_j& \delta _1= \delta \ge0 \\ \xi_{j+1}= \xi_j+C_2X_jY_j(1+\xi_j)\varepsilon_j&\xi_1=\xi, \end{array}\right.\end{equation} where $$\left\{\begin{array}{ll} X_j=(\frac{\Delta_{j+1} e^{\gamma _j d_{\Delta_j}}}{( \sigma _j- \sigma _{j+1})(\mu_j-\mu_{j+1})}\log \frac1{\varepsilon_j})^{\exp_2} &=(\frac{K_j\Delta_{j+1} e4^{j+1}}{ \sigma \mu}\log \frac1{\varepsilon})^{\exp_2}\\ Y_j=( \frac{\chi+\xi_j}{ \kappa _j})^{\exp_2}& \end{array}\right.$$ --for $d_\Delta$ see \eqref{block}.The $ \kappa _j$ is defined implicitly by $$2^j\varepsilon_j=\frac 1{C_2}
\frac{ \kappa _j}{ X_jY_j},$$
These sequences depend on the choice of $K_j$. We shall let $K_j$ increase like $$K_{j}=K^{j}$$ for some $K$ sufficiently large.
\begin{lemma}\label{numerical2} There exist constants $C'$ and $\exp'$ such that, if $$K\ge C' $$ and $$ \varepsilon(\log\frac1\varepsilon)^{\exp'}\le\frac1{C'}\big( \frac{ \sigma \mu}{(\chi+\xi)K\Delta}\big)^{\exp'},$$ then \begin{itemize} \item[(i)] $$\delta_j-\delta,\quad \xi_j-\xi,\quad \kappa _j\ \le\ 2C_2X_1Y_1\varepsilon;$$
\item[(ii)]
$$\varepsilon_{j+1}\ge C_2 X_jY_j(e^{-\frac12( \sigma _j- \sigma _{j+1})\Delta_{j+1}}+ e^{-\frac12(\gamma _j-\gamma _{j+1})\Delta_{j+1}} )\varepsilon_j;$$
\item[(iii)]
$$
\sum_{j\ge1} \Delta_{j+1}^{\exp_2} \kappa _j^\alpha \le 2\Delta_{2}^{\exp_2} \kappa _1^\alpha\le
C'\big(\frac{Kd_{\Delta}\log\frac1\varepsilon}{ \sigma \mu})^{\exp_2}((\chi+\xi)\varepsilon)^\alpha .
$$
\end{itemize}
( The exponents $\alpha$, $\exp'$ and the constant ${C'}$ has the same properties as those in Remark
\ref{constants}.)
\end{lemma}
\begin{proof} $\Delta_{j+1} $ is equal to $$4K_j\max(\frac {1} { \sigma _j- \sigma _{j+1}},d_{\Delta_j})\log \frac1{\varepsilon} \le ( {\operatorname{Ct.} } \frac {1} { \sigma } \log\frac1{\varepsilon})(2K)^{j}\Delta_j^{a},$$ where $a$ is some exponent depending on $d$. By a finite induction one sees that this is $$\le ( {\operatorname{Ct.} } \frac {1} { \sigma } \log\frac1{\varepsilon})(2K)\Delta)^{a^j},$$ if, as we shall assume, $a\ge2$. Now $X_j$ equals $$ (\frac{K_j\Delta_{j+1} e4^{j+1}}{ \sigma \mu}\log \frac1{\varepsilon})^{\exp_2}\le \big(( {\operatorname{Ct.} } \frac {1} { \sigma \mu} \log\frac1{\varepsilon})(4K)^{j^2}\Delta_j^{a}\big)^{2\exp_2}. $$ which, by assumption on $\varepsilon$, is $$\le \big(( {\operatorname{Ct.} } \frac {1} { \sigma \mu} \log\frac1{\varepsilon})K\Delta\big)^{4\exp_2 a^j}\le (\frac1{\varepsilon})^{4\exp_2 a^j},$$ if, as we shall assume, $a\ge3$.
(i) holds trivially for $j=1$, (i) , so assume it holds up to $j-1\ge1$. Then $ \delta _j\le \delta +2C_2X_1Y_1\varepsilon$ and $\xi_j\le\xi+2C_2X_1Y_1\varepsilon$, and hence $$ Y_j\le ( \frac{\chi+\xi+ 2C_2X_1Y_1 \varepsilon}{ \kappa _j})^{\exp_2}\le 2^{\exp_2} Y_1(\frac{ \kappa _1}{ \kappa _j})^{\exp_2}.$$ By definition of $ \kappa _j$, $$ \kappa _j^{1+\exp_2}=2^jC_2X_jY_j\varepsilon_j \kappa _j^{\exp_2} \le 2^{\exp_2} C_2Y_1 \kappa _1^{\exp_2}2^jX_j\varepsilon_j \le 2^jX_j\varepsilon^{K_{j-1}}$$ by assumption on $\varepsilon$. Hence $$2^jC_2X_jY_j\varepsilon_j= \kappa _j\le 2^jX_j\varepsilon^{2b K_{j-1}}\le \varepsilon^{2b K_{j-1}-4\exp_2 a^j-j\log2} ,\quad b=\frac1{2(1+\exp_2)}.$$ If $K$ is large enough -- notice that $j\ge2$ -- this is $\le \varepsilon^{b K_{j-1}}$.
Hence $$ \kappa _j\le \varepsilon^{b K_{j-1}}\le \varepsilon^{b K}\le\varepsilon\le 2C_2X_1Y_1\varepsilon,$$ if $K$ is large enough. Moreover $$\delta_j-\delta=\sum_{k=2}^j C_2 X_kY_k\varepsilon_k \le \varepsilon^{b K_{1}}\le 2C_2X_1Y_1\varepsilon_1$$ if $K$ is large enough. From these estimates one also obtains the required bound for $\xi_j-\xi$ if $K$ is large enough. This concludes the proof of (i).
To see (ii), notice that $$e^{-( \sigma _j- \sigma _{j+1})\Delta_{j+1}}\le e^{-4K_j\log\frac1\varepsilon}\le\varepsilon^{K_j}\varepsilon.$$ Notice also that $\Delta_{j+1}$ is much larger then $\Delta_{j}$ so that $\gamma _{j+1}$ is much smaller than $\gamma _j$ and, hence, $$e^{-(\gamma _j-\gamma _{j+1})\Delta_{j+1}}\le e^{-4K_j\frac{\gamma _j-\gamma _{j+1}}{\gamma _j} \log\frac1\varepsilon}\le\varepsilon^{K_j}\varepsilon.$$ This implies that $$C_2 X_jY_j(e^{-\frac12( \sigma _j- \sigma _{j+1})\Delta_{j+1}}+ e^{-\frac12(\gamma _j-\gamma _{j+1})\Delta_{j+1}} )\varepsilon_j\le \varepsilon^{K_j} \varepsilon=\varepsilon_{j+1}.$$
To see (iii) we have for $j\ge 2$
$$ \Delta_{j+1}^{\exp_2} \kappa _j^{\alpha}\le X_j^{\exp_2} \kappa _j^{\alpha}
\le (\frac1{\varepsilon})^{4\exp_2^2 a^j} \kappa _j^{\alpha}\le
e^{-4\exp_2^2 a^j\log\frac1{\varepsilon}}e^{\alpha bK_{j-1}\log\frac1{\varepsilon}} $$
which is
$$\le \varepsilon^{\frac12b K_{j-1}\alpha }\le 2^{-j}\varepsilon,$$
if $K$ is large enough (depending on $\alpha$). This implies the first inequality in (iii). The second one is a simple computation. \end{proof} \subsubsection{The iteration}
\begin{proposition} There exist positive constants $C_3$, $\alpha$ and $\exp_3$ such that, for any $h\in \mathcal{NF} _{\varkappa}(\Delta, \delta )$ and for any $f\in \mathcal{T} _{\gamma ,\varkappa, \mathcal{D}}( \sigma ,\mu)$, $$ \varepsilon=\ab{f^T}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}},\quad
\xi=\ab{f}_{\begin{subarray}{c} \sigma ,\mu\ \ \\ \gamma , \varkappa, \mathcal{D} \end{subarray}},$$ if $$\delta \le \frac1{C_3} c'$$ and $$ \varepsilon(\log \frac1\varepsilon)^{\exp_3}\le\frac1{C_3}\big( \frac{ \sigma \mu} {(\chi+\xi)\max(\frac1\gamma ,d_{\Delta})}c'\big)^{\exp_3}c',$$ then there exist a closed subset $ \mathcal{D}'= \mathcal{D}'(h, f)\subset \mathcal{D}$, $$\operatorname{meas} ( \mathcal{D}\setminus \mathcal{D}')\leq C_3\big(\frac{\max(\frac1\gamma ,d_{\Delta})\log\frac1\varepsilon}{ \sigma \mu})^{\exp_3}\frac{\chi}{\delta_0}((\chi+\xi)\frac\varepsilon\chi)^\alpha$$ and a $ \mathcal{C} ^{{s_*}}$ mapping $$\Phi : \mathcal{O} _{\gamma _*}( \sigma /2,\mu/2)\times \mathcal{D} \to \mathcal{O} _{\gamma _*}( \sigma ,\mu),$$ real holomorphic and symplectic for given parameter $\rho\in \mathcal{D}$, and $$h'\in \mathcal{NF} _{\varkappa}(\infty, \delta '),\quad \delta '\le \frac{c'}2,$$ such that $$(h+f)\circ \Phi=h'+f'$$ verifies $$\ab{ f'-f}_{\begin{subarray}{c} \sigma /2,\mu/2\ \ \\ \gamma _*, \varkappa, \mathcal{D} \end{subarray}} \le C_3$$ and, for $ \rho \in \mathcal{D} '$, $(f')^T=0$.
Moreover, $$\ab{ h'- h}_{\begin{subarray}{c} \sigma /2,\mu/2\ \ \\ \gamma _*, \varkappa, \mathcal{D} \end{subarray}}\le C_3$$ and
$$|| \partial_ \rho ^j (\Phi(x,\cdot)-x)||_{\gamma _*}+ \aa{ \partial_ \rho ^j (d\Phi(x,\cdot)-I)}_{\gamma _*, \varkappa } \le C_3$$ for any $x\in \mathcal{O} _{(0,m_*)}( \sigma ',\mu')$, $\ab{j}\le{s_*}$, and $ \rho \in \mathcal{D}$, and $\Phi(\cdot, \rho )$ equals the identity for $ \rho $ near the boundary of $ \mathcal{D}$.
Finally, if $\tilde \rho =(0, \rho _2,\dots, \rho _p)$ and $f^T(\cdot,\tilde \rho )=0$ for all $\tilde \rho $, then $h'=h$ and $\Phi(x,\cdot)=x$ for all $\tilde \rho $.
( The exponents $\alpha$, $\exp_3$ and the constant $C_3$ have the same properties as those in Remark
\ref{constants}.)
\end{proposition}
\begin{proof} Assume first that $\gamma =d_\Delta^{-1}$.
Choose the number $\mu_j, \sigma _j,\varepsilon_j,\Delta_j,\gamma _j, \delta _j,\xi_j,X_j,Y_j, \kappa _j$ as above in Lemma~\ref{numerical2} with $K= C' $. Let $h_1=h$, $f_1=f$.
Since $$ \kappa _j,\ \delta_j-\delta\ \le 2C_2X_1Y_1\varepsilon\le\frac1{2C_2}c'$$ by Lemma~\ref{numerical2} and by assumption on $\varepsilon$ we can apply Lemma~\ref{Birkhoff} iteratively. It gives, for all $j\ge1$, a set $ \mathcal{D}_{j}\subset \mathcal{D}$, $$\operatorname{meas} ( \mathcal{D}\setminus \mathcal{D}_{j})\leq {C_2} \Delta_{j+1}^{\exp_2} (\frac{ \kappa _j}{\delta_0})^{\alpha}(\frac{\chi}{\delta_0})^{1-\alpha},$$ and a $ \mathcal{C} ^{{s_*}}$ mapping $$\Phi_{j+1} : \mathcal{O} ^{\gamma '}( \sigma _{j+1},\mu_{j+1})\times \mathcal{D}_{j+1}\to \mathcal{O} ^{\gamma '}( \sigma _j-\frac{ \sigma _j- \sigma _{j+1}}{2},\mu_j-\frac{\mu_j-\mu_{j+1}}{2}),\quad \forall \gamma _*\le\gamma '\le\gamma _{j+1},$$ real holomorphic and symplectic for each fixed parameter $ \rho $, and functions $f_{j+1}\in \mathcal{T} _{\gamma ,\varkappa, \mathcal{D}}( \sigma _{j+1},\mu_{j+1})$ and $$h_{j+1}\in \mathcal{NF} _{\varkappa}(\Delta_{j+1}, \delta _{j+1})$$ such that $$(h_j+f_j)\circ \Phi_{j+1}=h_{j+1}+f_{j+1},\quad\forall \rho \in \mathcal{D}_{j+1},$$ with $$\ab{ f_{j+1}^T}_{\begin{subarray}{c} \sigma _{j+1},\mu_{j+1}\ \ \\ \gamma _{j+1}, \varkappa, \mathcal{D} \end{subarray}} \le \varepsilon_{j+1}$$ and $$\ab{ f_{j+1}}_{\begin{subarray}{c} \sigma _{j+1},\mu_{j+1}\ \ \\ \gamma _{j+1}, \varkappa, \mathcal{D} \end{subarray}} \le \xi_{j+1}. $$
Moreover, $$\ab{ h_{j+1}- h_j}_{\begin{subarray}{c} \sigma _{j+1},\mu_{j+1}\ \ \\ \gamma _{j+1}, \varkappa, \mathcal{D} \end{subarray}} \le C_2 X_jY_{j} \varepsilon_j$$ and
$$|| \partial_ \rho ^l (\Phi_{j+1}(x,\cdot)-x)||_{\gamma '}+ \aa{ \partial_ \rho ^l (d\Phi_{j+1}(x,\cdot)-I)}_{\gamma ', \varkappa } \le C_2\frac1{ \kappa _j} X_jY_j \varepsilon_j$$ for any $x\in \mathcal{O} _{\gamma '}( \sigma _{j+1},\mu_{j+1})$, $\gamma _*\le \gamma '\le\gamma _{j+1}$ and $\ab{l}\le{s_*}$.
We let $h'=\lim h_j$, $f'=\lim f_j$ and $\Phi=\Phi_2\circ\dots\circ \Phi_3\circ\dots $. Then $(h+f)\circ \Phi=h'+f'$ and $h'$ and $f'$ verify the statement. The convergence of $\Phi$ and its estimates follows as in the proof of Lemma~\ref{Birkhoff}.
Let $ \mathcal{D}'=\bigcup \mathcal{D}_j$. Then, by Lemma~\ref{numerical2}, $$\operatorname{meas} ( \mathcal{D}\setminus \mathcal{D}')\leq {C_2} \frac{\chi^{1-\alpha}}{\delta_0}\sum_j \Delta_{j+1}^{\exp_2} \kappa _j^{\alpha}\le C_3\frac{\chi^{1-\alpha}}{\delta_0} \big(\frac{d_{\Delta}\log\frac1\varepsilon}{ \sigma \mu})^{\exp_2}((\chi+\xi)\varepsilon)^\alpha.$$
The last statement is obvious.
If $\gamma <(d_{\Delta})^{-1}$, then we increase $\Delta$ and we obtain the same result. If $\gamma >(d_{\Delta})^{-1}$, then we can just decrease $\gamma $ and we obtain the same result.\end{proof}
Theorem~\ref{main} now follows from this proposition.
\begin{samepage} \centerline{PART IV. SMALL AMPLITUDE SOLUTIONS} \section{Proofs of Theorems \ref{t72}, \ref{t73}}\label{s11} \end{samepage} We shall now treat the beam equation by combining the Birkhoff normal form theorem \ref{NFT} and the KAM theorem \ref{main} or, more precisely, its Corollary~\ref{cMain-bis}. In order to apply
Corollary~\ref{cMain-bis} we need to verify, first that the quadratic part of the Hamiltonian \eqref{HNFbis}
is a KAM normal form Hamiltonian and, second that the perturbation $f$ is sufficiently small.
We recall the agreement about constants made in the introduction.
\subsection{A KAM normal form Hamiltonian}
\
Let $h$ be the Hamiltonian \eqref{H2}$+$\eqref{H1}.
\begin{theorem}\label{p_KAM} There exists a zero-measure Borel set $ \mathcal{C} \subset[1,2]$ such that for any strongly admissible set $ \mathcal{A} $ and any $m\notin \mathcal{C} $
there exist
real numbers $\gamma _g>\gamma _*=(0,m_*+2)$ and
$\beta_0, \nu_0,c_0 >0$, where $c_0$, $\beta_0$, $\nu_0$ depend on $ m$, such that, for any $0<c_*\le c_0$, $0<\beta_{\#}\le\beta_0$ and $0<\nu\le\nu_0$
there exists an open set $Q=Q(c_*,\beta_{\#},\nu) \subset [c_*,1]^ \mathcal{A} $, increasing as $\nu\to0$ and satisfying \begin{equation}\label{} \operatorname{meas} ([c_*,1]^ \mathcal{A} \setminus Q ) \le C\nu^{\beta_{\#}}\,, \end{equation} with the following property:
For any $ \rho \in Q$ there exists a real holomorphic diffeomorphism (onto its image) \begin{equation}\label{} \Psi_\rho: \mathcal{O} _{\gamma _*} \big({\tfrac 12}, {\mu_*^2} \big)\to {\mathbf T}_\rho (\nu, 1,1,\gamma _*)\,,\qquad {\mu_*}={\tfrac{c_*}{2\sqrt2}}, \end{equation} such that $$ \Psi_\rho^*\big(dp\wedge dq\big)= \nu dr_ \mathcal{A} \wedge d\theta_ \mathcal{A} \ +\ \nu d u_ \mathcal{L} \wedge d v_ \mathcal{L} , $$ and such that $$\frac1{\nu} (h\circ\Psi_\rho)=h_{\textrm{up}}+f,$$ \begin{equation}\label{unperturbedbis} h_{\textrm{up}}(r,\theta,p_ \mathcal{L} ,q_ \mathcal{L} ) =\langle \Omega(\rho), r\rangle + \frac12 \sum_{a\in \mathcal{L} _\infty}\Lambda_a (\rho) ( p_{a}^2 + q_{a}^2) + \nu\langle K(\rho) \zeta_ \mathcal{F} , \zeta_ \mathcal{F} \rangle \end{equation} where $ \mathcal{F} = \mathcal{F} _ \rho \subset \mathcal{L} _f$, with the following properties:
(i) $\Psi_\rho$ depends smoothly on $\rho$ and $$ \Psi_\rho\big ( \mathcal{O} _{\gamma } ({\tfrac 12}, {\mu_*^2}) \big)\subset {\mathbf T}_\rho (\nu, 1,1,\gamma ),\qquad \gamma _*\le \gamma \le\gamma _g; $$
(ii) $h_{\textrm{up}}$ satisfies, on any ball (or cube) $ \mathcal{D}\subset Q$, the Hypotheses~A1-A3 of Section~\ref{ssUnperturbed} for some constants $c',c,\delta_0,\beta,\tau$ satisfying \begin{equation}\label{choice1} c' \ge \nu^{1+ \beta_{\#}} \,, \quad c=2 \max\{\langle a\rangle^3, a\in \mathcal{A} \},\quad \beta_1=\beta_2=2\,,\quad \end{equation} \begin{equation}\label{choice2} \delta_0\ge \nu^{1+\beta_{\#} } \,, \quad s_*=4\, (\# \mathcal{F} )^2 \end{equation}
\begin{equation}\label{choice3} \beta_3=\beta_3(m)>0 \,, \quad \tau=\tau(m)>0\,; \end{equation}
(iii) $$ \chi=
|\nabla_ \rho \Omega |_{ \mathcal{C} ^{ {{s_*}}-1 } ( \mathcal{D})}+\sup_{a\in \mathcal{L} _\infty} |\nabla_ \rho \Lambda_a|_{ \mathcal{C} ^{ {{s_*}}-1 } ( \mathcal{D})}
+ ||\nu \nabla_ \rho K ||_{ \mathcal{C} ^{ {{s_*}}-1 } ( \mathcal{D}) }\le
{\color{red} C}\nu^{1-\beta_{\#}};$$
(iv) $ f$ belongs to $ \mathcal{T} _{\gamma , \varkappa =2,Q}({\tfrac12}, \mu_*^2)$ and satisfies $$
\xi=| f|_{\begin{subarray}{c}1/2,\mu_*^2 \ \\ \gamma _g, 2, \mathcal{D} \end{subarray}}
\le C\nu^{1- \beta_{\#}} \,, \qquad
\varepsilon= | f^T|_{\begin{subarray}{c}1/2,\mu_*^2 \ \\ \gamma _g, 2, \mathcal{D} \end{subarray}}
\le C\nu^{3/2-\beta_{\#}} \,.$$
If $ \mathcal{A} $ is admissible but not strongly admissible, then the same thing is true with the difference that $(ii)$ only holds for
balls (or cubes) $ \mathcal{D}\subset Q\cap \mathcal{D}_0$, where $ \mathcal{D}_0\subset [0, 1]^ \mathcal{A} $ is an open set,
independent of $c_*,\beta_{\#}$ and $ \nu$, such that
\begin{equation}\label{hren2} \operatorname{meas} ( \mathcal{D}_0)\ge \tfrac12\, c_0^{\# \mathcal{A} }. \end{equation}
The constant $ C$ depends on $m, c_*,\beta_{\#}$, but not on $\nu$.
\end{theorem}
\begin{proof} We apply Theorem~\ref{NFT} and denote the constructed there
symplectic transformation by $\Psi$. We let $ \mathcal{L} _\infty= \mathcal{L} \setminus \mathcal{F} = ( \mathcal{L} \setminus \mathcal{L} _f)\cup ( \mathcal{L} _f\setminus \mathcal{F} )$ (this is a slight abuse of notation since in Part~II we denoted by $ \mathcal{L} _\infty$ the set $ \mathcal{L} \setminus \mathcal{L} _f$). For $\beta_0, \nu_0$ and $\varepsilon_0$ we take the same constants as in Theorem~\ref{NFT}. If $ \mathcal{A} $ is only admissible, we take for $ \mathcal{D}_0$ the set $ \mathcal{D}_0= \mathcal{D}_0^1$, see \eqref{DD}.
The assertion (i) of the theorem holds by Theorem~\ref{NFT}.
To prove (ii) and (iii) we will first verify (ii) for a smaller $c'$, \begin{equation}\label{choice0} c' \ge \nu^{1+2\beta_{\#}(\beta(0)+\bar c)}\,, \end{equation} and in (iii) will replace the exponent for $\nu$ by a bigger number.
By \eqref{Om}, \eqref{Lam}, \eqref{Lambdab} and \eqref{normK} we have that $$ \chi=
|\nabla_ \rho \Omega |_{ \mathcal{C} ^{ {{s_*}}-1 } (Q)}+\sup_{a\in \mathcal{L} _\infty} |\nabla_ \rho \Lambda_a|_{ \mathcal{C} ^{ {{s_*}}-1 } (Q)}
+ ||\nu \nabla_ \rho K ||_{ \mathcal{C} ^{ {{s_*}}-1 } (Q) }\le
{\operatorname{ct.} } \nu^{1-\beta_{\#} \beta(s_*-1)},
$$
which implies (iii) with a modified exponent.
Now let us consider (ii).
We will check the validity of the three hypotheses A1--A3 (with $c'$ as in \eqref{choice0}).
First we note that using \eqref{Lam}, \eqref{estimla}, \eqref{N1}, \eqref{K4} and \eqref{delta} we get \begin{equation}\label{basic1}
\tfrac12 +\tfrac12 |a|^2\le \Lambda_a \le 2|a|^2 +1\,,\quad
| \Lambda_a - \lambda_a |_{C^j( \mathcal{D}_0)} \le C_3 \nu |a|^{-2}\quad \forall\, j\ge1 \,,\;\forall\, a\in \mathcal{L} \setminus \mathcal{L} _f\,, \end{equation} \begin{equation}\label{basic2}
C_1\nu^{1+\bar c \beta_{\#}} \le | \Lambda_a | \le C_2\nu\qquad \forall\, a\in \mathcal{L} _f\setminus \mathcal{F} \,. \end{equation} It is convenient to re-denote \begin{equation}\label{rel2} \lambda_a =:0\quad\text{if}\quad a\in \mathcal{L} _f\setminus \mathcal{F} \,; \end{equation} then the second relation in \eqref{basic1} holds for all $a$. We recall that the numbers $\{\pm \lambda_a , a\in \mathcal{F} \}$ are the eigenvalues of the operator $JK$. They satisfy the estimates \eqref{hyperb}.
The vector--function $\Omega(\rho)\in \mathbb{R}^n$ is defined in \eqref{Om}, so \begin{equation}\label{ddet} \Omega(\rho) = \omega +\nu M\rho,\qquad \det M\ne0\,, \end{equation} and $K$ is a symmetric real linear operator in the space $Y_ \mathcal{F} $. Its norm satisfies \begin{equation}\label{basic3}
\| {\nu K}(\rho)\|_{C^j}
\le C_j \nu^{1-\beta_{\#} \beta(j)}\,,\qquad j\ge0\,. \end{equation} See Theorem~\ref{NFT}, items (ii)-(iv).
\noindent {\it Hypothesis~A1}. Relations \eqref{la-lb-ter} and \eqref{la-lb} and the first relation in \eqref{laequiv}
immediately follow from \eqref{basic1} and \eqref{basic2}.
To prove the second relation in \eqref{laequiv} note that by Theorem~\ref{NFT}
the operator $U$ conjugates $JK$ with the diagonal operator with the eigenvalues $\pm{\bf i}\Lambda_j^h(\rho)$.
So by \eqref{basic2} and \eqref{Ubound} the norm of $(JH)^{-1}$ is bounded by $C\nu^{-1 -\beta_{\#}(\bar c +2\beta(0)}$,
and the required estimate follows from \eqref{choice0}. The second relation in \eqref{la-lb-bis} follows by the
same argument from \eqref{hyperb}, which implies that the norms of the eigenvalues of
$ \Lambda_a I-{\bf i}JH$ are $\ge C^{-1}\nu^{\bar c\beta_{\#}}$. The first relation in \eqref{la-lb-bis} is a consequence of
\eqref{basic1}, \eqref{basic2} and \eqref{Fcluster}.
Now consider \eqref{laequiv-bis}.\footnote{This is the only condition of Theorem~\ref{main}
which we cannot verify for any $\rho\in Q$ without assuming that the set $ \mathcal{A} $ is strongly admissible.
}
If $a\in \mathcal{L} _\infty$ and $b\in \mathcal{L} \setminus \mathcal{L} _f$, then again the relation follows from \eqref{basic1} and \eqref{basic2}.
Next, let $a,b\in \mathcal{L} _f\setminus \mathcal{F} $. Let us write $ \Lambda_a $ and $ \Lambda_b $ as
$\Lambda^j_r$ and $\Lambda^k_m$, $j\le k$. If $j=k$, then the condition follows from \eqref{K04}, \eqref{delta}
(from \eqref{K4} if $m=r$).
If $j\le M_0<k$, then again it follows from \eqref{K04}. If $j,k\le M_0$,
then $\Lambda^j_r=\Lambda^j_1=\mu(b_j,\rho)$ and $\Lambda^k_m=\mu(b_m,\rho)$, so the relation follows
from \eqref{K44}. Finally, let $j,k> M_0$. Then if the set $ \mathcal{A} $ is strongly admissible, the required
relation follows from \eqref{K04}, while if $\rho \in \mathcal{D}_0= \mathcal{D}_0^1$, then it follows from \eqref{aaa}.
\noindent {\it Hypothesis~A2}. By \eqref{ddet}, $\partial_\mathfrak z\Omega(\rho) = \nu M\mathfrak z$. Choosing \begin{equation}\label{zet}
\mathfrak z= \frac{{}^t\!M k}{ |{}^t\!M k|}
\end{equation} and using that $|\Omega' -\Omega|_{C^{s_*}}\le \delta_0$ we achieve that $\partial_\mathfrak z\langle k, \Omega'(\rho)\rangle \ge C\nu$, so \eqref{o} holds.
To verify (i) we restrict ourselves to the more complicated case when $a,b\ne\emptyset$. Then $L(\rho)$ is a diagonal operator with the eigenvalues $$ \lambda_{a b}^k :=\langle k,\Omega'(\rho)\rangle +\Lambda_a(\rho) \pm \Lambda_b(\rho)\,\quad a\in[a],\; b\in[b]\,. $$ Clearly $$
|\lambda_{a b}^k -( \langle k, \omega\rangle +\lambda_a \pm \lambda_b)| \le C\nu |k|\,. $$ (we recall \eqref{rel2}). Therefore by Propositions \ref{D1D2} and \ref{prop-D3} the first alternative in (i) holds, unless \begin{equation}\label{unless}
|k|\ge C \nu^{-\bar\beta} \end{equation} for some (fixed) $\bar\beta>0$. But if we choose $\mathfrak z$ as in \eqref{zet}, then $\partial_\mathfrak z L(\rho)$ becomes a
diagonal matrix with the diagonal elements bigger than $|{}^tMk| - C\nu |k| - C_1\nu$. So if $k$ satisfies \eqref{unless}, then the second alternative in (i) holds.
To verify (ii) we write $L(\rho, \Lambda_a)$ as the multiplication from the right by the matrix $$ L = (\langle k,\Omega'\rangle +\Lambda_a(\rho) )I +{\bf i}\nu J \widehat K\,. $$
The transformation $U$ conjugates $L$ with the diagonal
operator with the eigenvalues
$\lambda^k_{a j}=: \langle k,\Omega'\rangle +\Lambda_a(\rho) \pm \nu{\bf i}\Lambda^h_j$.
In view of \eqref{hyperb},
$ |\lambda^k_{a j}|\ge |\Im \lambda^k_{a j}|\ge C^{-1} \nu^{ 1+\bar c\beta_{\#}}$. This
implies (ii) by \eqref{Ubound} and \eqref{choice0}.
It remains to verify (iii). As before, we restrict ourselves to the more complicated case $a,b\in \mathcal{F} $. Let us denote
$$
\lambda(\rho) := \langle k,\Omega'(\rho)\rangle = \langle k,\omega\rangle + \nu\langle k, M \rho\rangle +
\langle k,(\Omega' -\Omega)(\rho)\rangle\,,
$$
and write the operator $L(\rho)$ as
$$
L(\rho) = \lambda(\rho) I +L^0(\rho)\,,\quad L^0(\rho) X = [X, iJ{(\nu K)}(\rho)]\,.
$$
In view of \eqref{basic3},
\begin{equation}\label{9.0}
\|L^0\|_{C^j} \le C_j \nu^{1- \beta(j)\beta_{\#}}\qquad\text{for}\; j\ge0\,.
\end{equation}
Now it is easy to see that if
$|\langle k,\omega| \rangle \ge C(\nu^{1-\beta(0)\beta_{\#}} +\nu|k|)$ with a sufficiently big $C$,
then the first alternative in (iii) holds.
So it remains
to consider the case when
\begin{equation}\label{9.1}
|\langle k,\omega \rangle | \le C(\nu^{1-\beta(0)\beta_{\#}} +\nu|k|)\,.
\end{equation}
By Proposition \ref{D1D2} the l.h.s. is bigger than $\kappa|k|^{-n^2}$. Assuming that $\beta_0\ll1$, we derive from this and
\eqref{9.1} that
\begin{equation}\label{9.2}
|k| \geq C \nu^{-1/(1+n^2)}\,.
\end{equation}
In view of \eqref{9.0}-\eqref{9.2}, again if $\beta_0\ll1$, we have:
\begin{equation}\label{9.5}
|\lambda(\rho)
| \le C\nu (\nu^{-\beta(0)\beta_{\#}} +|k|)\le C_1 \nu |k|\,,
\end{equation}
\begin{equation}\label{9.6}
|(\partial_\rho)^j \lambda(\rho)| \le C_j |k| \delta_0,\qquad 2 \le j\le s_*\,,
\end{equation}
\begin{equation}\label{9.3}
\|L\|_{C^j} \le C\nu (\nu^{-\beta(j)\beta_{\#}} + |k|) +C_j|k|\delta_0\,,\qquad j\ge0\,.
\end{equation}
Denote det$\,L(\rho) = D(\rho)$. Then
$$
D(\rho) = \prod_{a,b\in \mathcal{F} } \prod_{\sigma_1, \sigma_2=\pm} \Lambda(\rho;a,b,\sigma_1, \sigma_2)\,,
$$
where $ \Lambda(\rho;a,b,\sigma_1, \sigma_2) =
\lambda(\rho) +\sigma_1 \nu \Lambda_a(\rho) -\sigma_2 \nu \Lambda_b(\rho)\,.
$
Choosing $\mathfrak z$ as in
\eqref{zet} we get
$$
|\Lambda|\le C\nu|k|\,,\;\;\; |\partial_\mathfrak z\Lambda| \ge C^{-1} |k|\nu - |k|\delta_0\ge \tfrac12 C^{-1}|k|\nu\,,\;\;\;
|\partial_\mathfrak z^j \Lambda| \le C_j|k|\delta_0\;\;\text{if}\;\;j\ge2
$$
(that is, these relations hold for all values of the arguments $\rho, a, b, \sigma_1,\sigma_2$).
Recall that $2\,| \mathcal{F} |=m$; then $s_*=m^2$. Chose in \eqref{altern1} $j=s_*=m^2$.
Then, in view of the relations above, we get:
$$
| \partial_\mathfrak z^{s_*} D(\rho)| \ge m^2! \, \big(C^{-1} |k|\nu\big)^{m^2} - C_1 (|k|\nu)^{m^2-1} (|k|\delta_0)
\ge \tfrac12 m^2! \, \big(C^{-1} |k|\nu\big)^{m^2}
\,.
$$ In the same time, by \eqref{9.3} the r.h.s. of \eqref{altern1} is bounded from above by $$
C_m\delta_0 (\nu^{(m^2-1)(1-\beta(m^2) \beta_{\#})} + \nu^{m^2-1} |k|^{m^2-1})\,. $$
In view of \eqref{choice0}, \eqref{choice2} this implies the relation
\eqref{altern1} if we choose
$\beta_{\#}<(\beta(m^2)(1+n^2))^{-1}$ (as always, we decrease $\nu_0$, if needed).
\noindent {\it Hypothesis~A3}. The required inequality follows from Proposition \ref{prop-D3} since the divisor, corresponding to \eqref{melnikov} where $a,b\not\in \mathcal{L} _f$, cannot be resonant.
Finally, let us denote $$ \beta_{\#}^0= \beta_{\#} \max(1, \hat c, 2(\beta(0)+\bar c), \beta(s_*-1))\,. $$ Our argument shows that the assertions (ii), (iii) of the theorem hold with $\beta_{\#}$ replaced by $\beta_{\#}^0$. The assertion (iv) with $\beta_{\#}=:\beta_{\#}^0$ follows from \eqref{estbis}. Now it remains to re-denote $\beta_{\#}^0$ by $\beta_{\#}$. \end{proof}
\subsection{The main result}\label{s_10.2}
We have $c_0,\beta_0,\nu_0$ so small so that Theorem~\ref{p_KAM} applies. Now we shall make them even smaller.
\begin{theorem}\label{thm10.2} There exists a zero-measure Borel set $ \mathcal{C} \subset[1,2]$ such that for any strongly admissible set $ \mathcal{A} $ and any $m\notin \mathcal{C} $ there exist
real numbers $c_0,\beta_0 >0$, depending only on $ \mathcal{A} $, $ m$ and $G$, such that, for any $0<c_*\le c_0$ and $0<\beta_{\#}\le \beta_0$ the following hold.
There exists a $\nu_0$ such that if $\nu\le\nu_0$, then there exist a closed set $Q'=Q'(c_*,\beta_{\#},\nu)\subset Q=Q(c_*,\beta_{\#},\nu)$, and a $ \mathcal{C} ^{{s_*}}$-mapping $\Phi$ $$\Phi: \mathcal{O} _{ \gamma _*}(1/4,\mu_*^2/2)\times Q\to \mathcal{O} _{ \gamma _*}(1/2,\mu_*^2),\qquad {\mu_*}={\tfrac{c_*}{2\sqrt2}},\qquad \gamma _*=(0,m_*+2),$$ real holomorphic and symplectic for each parameter $ \rho \in Q$, such that $$(h_{\textrm{up}}+ f)\circ \Phi(r,w, \rho )= \langle \Omega'( \rho ), r\rangle +\frac 1 2\langle w, A'( \rho )w\rangle+f'(r,w, \rho )$$ with the following properties:
\noindent (i)
the frequency vector $ \Omega'$ satisfies
$$|\Omega'-\Omega|_{ \mathcal{C} ^{{s_*-1}}(Q)}\le \nu^{1+ \aleph}\,, $$ and the matrix $$A'( \rho )=A'_\infty( \rho )\oplus H'( \rho )\in \mathcal{NF} _{\infty}$$
satisfies
$$ || \partial_ \rho ^j (H'( \rho )-\nu K( \rho ) || \le \nu^{1+\aleph} , $$
for $ |j| \le {{s_*}}$ and $ \rho \in Q$;
\noindent (ii) for any $x\in \mathcal{O} _{\gamma _*}(1/4,\mu_*^2/2)$, $ \rho \in Q$ and $\ab{j}\le{s_*}-1$,
$$|| \partial_ \rho ^j (\Phi(x, \rho )-x)||_{\gamma _*}+ \aa{ \partial_ \rho ^j (d\Phi(x,r)-I)}_{\gamma ^*, \varkappa } \le
\nu^{\frac12-\aleph(\kappa+2)} ;$$
\noindent (iii) for $ \rho \in Q'$ and $\zeta=r=0$ $$d_r f'=d_\theta f'= d_{\zeta} f'=d^2_{\zeta} f'=0;$$
\noindent (iv) if $ \mathcal{A} $ is strongly admissible, then
$$ \lim_{\nu\to0} \operatorname{meas} Q'(c_*,\beta_{\#},\nu)= (1-c_*)^{\# \mathcal{A} }. $$ If $ \mathcal{A} $ is admissible but not strongly admissible, then
$$ \liminf_{\nu\to0} \operatorname{meas} Q'(c_*,\beta_{\#},\nu)\ge \tfrac12 c_0^{\# \mathcal{A} }.$$
The exponent $\aleph$ is defined by $\aleph( \kappa +2)=\min(\frac18,\alpha)$ where $\alpha$ and $\kappa$ are given in Corollary~\ref{cMain-bis}. \end{theorem}
\begin{proof} By Proposition \ref{p_KAM} we know that the Hamiltonian $h_{\textrm{up}}$ of \eqref{unperturbedbis} satisfies the Hypotheses~A1-A3 of Section \ref{ssUnperturbed} with the choice of parameters \eqref{choice1}-\eqref{choice3} -- $c',\delta_0 $ are here still to be determined -- on any ball $ \mathcal{D}\subset Q(c_*,\beta_{\#},\nu) \subset [c_*,1]^ \mathcal{A} $ with \begin{equation}\label{} \operatorname{meas} ([c_*,1]^ \mathcal{A} \setminus Q (c_*,\beta_{\#},\nu) )\le C\nu^{\beta_{\#}}\,,\end{equation} In order to apply Corollary~\ref{cMain-bis} to the Hamiltonian $h_{up}+f$ it remains to verify the assumptions a), b) of that corollary, and
\eqref{epsi-bis}.
Choose $\aleph$ so that $\aleph( \kappa +2)=\min(\frac18,\alpha)$. (Here $ \kappa $ and $\alpha$ are given in Corollary~\ref{cMain-bis}.) If we take $\beta_0\le \aleph^2$, then $$\chi,\ \xi \le {\operatorname{Ct.} } \nu^{1-\aleph^2}\quad\textrm{and}\quad \varepsilon\le {\operatorname{Ct.} } (\nu^{1-\aleph^2})^{\frac32}$$ for any $\beta_{\#}\le\beta_0$. By \eqref{choice1} and \eqref{choice2} we have $$c'=\delta_0\ge \nu^{1+\aleph}.$$ Then a) and b) are fulfilled.
The smallness condition \eqref{epsi-bis} in Corollary~\ref{cMain-bis}, is now easily seen hold, by the first assumption on $\aleph$, if we take $\nu$ sufficiently small. (Notice that this bound on $\nu$ depends on $c_*$ through $\mu_*$.) We can therefore apply this corollary: there exists a subset $ \mathcal{D}'(\nu)\subset \mathcal{D}$, with the measure bound \eqref{measure-bis} becomes $$ \operatorname{meas}( \mathcal{D}\setminus \mathcal{D}'(\nu)) \le\frac1{\varepsilon_0}\delta_0^{-\aleph \kappa }\varepsilon^{\alpha}\le \nu^{\aleph},$$ (by the second assumption on $\aleph$); the bound in (ii) follows since $c'\ge \nu^{1+\aleph}$; the bound in (iii) holds if $\nu_0$ is small enough. The diffeomorphism $\Phi$ is trivially extended from $ \mathcal{D}$ to $Q$ since it equals the identity near the boundary of $ \mathcal{D}$.
In order to prove (iv), assume first that $ \mathcal{A} $ is strongly admissible. Then for any $c_*$ the sets $Q_\nu=Q(c_*,\beta_{\#},\nu)$, form
an increasing system of open sets in $[c_*,1]^ \mathcal{A} $ such that their union is of full measure. So for any $\epsilon>0$
we can find $\nu_\epsilon>0$ such that $\operatorname{meas} Q_\nu \ge (1-\epsilon)(1-c_*)^n$ ($n=\# \mathcal{A} $) if $\nu\le\nu_\epsilon$.
Since
$Q_{\nu_\epsilon}$ is open there is a finite disjoint union
$\cup_{j=1}^N \mathcal{D}_j\subset Q_{\nu_\epsilon}$ of open balls (or cubes) whose measure differ from that of $Q_{\nu_\epsilon}$
by at most $\epsilon(1-c_*)^n$. [Use for example the Vitali covering theorem.]
For any $j\ge1$ we construct a closed
set $ \mathcal{D}'_j(\nu)$ as above. Then
$$
\mathcal{D}'_j(\nu)\subset \mathcal{D}_j \subset Q_{\nu_\epsilon} \subset Q_\nu
$$
for any $0<\nu\le\nu_\epsilon$, and $\operatorname{meas} ( \mathcal{D}_j\setminus \mathcal{D}_j'(\nu))\le \nu^{\aleph}$. If now
$
Q'_\nu = \cup _{j=1}^N \mathcal{D}'_j(\nu)
$ and $\nu'_\epsilon\in[0,\nu_\epsilon]$ is sufficiently small, then $\operatorname{meas} (Q_\nu\setminus Q'_\nu)\le 2\epsilon(1-c_*)^n$ for all $0<\nu\le\nu'_\epsilon$. This implies the first assertion in (iv). To prove the second we simply replace in the argument above the cube $[0,1]^ \mathcal{A} $ by the set $ \mathcal{D}_0$ as in \eqref{hren2}. \end{proof}
\subsubsection{Proof of Theorem~\ref{t72} and \ref{t73}}
\begin{proof} Given $\beta_{\#}$. For any $c_*$ and $\nu$, let $Q'(c_*,\nu)\subset Q(c_*,\beta_{\#},\nu)$ be the set defined in Theorem~\ref{thm10.2}. Then, for any $c_*>0$, $$\bigcup_{\nu\in \mathbb{Q} ^*}Q'(c_*,\nu)$$ is of Lebesgue measure: $=(1-c_*)^{\# \mathcal{A} }$ when $ \mathcal{A} $ is strongly admissible; $\ge c_0^{\# \mathcal{A} }$ when $ \mathcal{A} $ is admissible. It follows that the set $$ \tilde \mathfrak J =\{I=\nu \rho : \rho \in \bigcup_{\begin{subarray}{c}c_*,\nu \in \mathbb{Q} ^*\\\nu^{\beta_{\#}}\le c_*\end{subarray}}Q'(c_*,\nu)\}$$ at $I=0$ has: density $=1$ when $ \mathcal{A} $ is strongly admissible; positive density when $ \mathcal{A} $ is admissible.
Chose an enumeration $\{(c_j,\nu_j)\}_j$ of $ \mathbb{Q} ^*\times \mathbb{Q} ^*$ and let $ \tilde \mathfrak J _j=\nu_jQ'(c_j,\nu_j)$ so that $ \tilde \mathfrak J =\bigcup_j \tilde \mathfrak J _j$.
Now we fix $j$ and let $\nu=\nu_j$. We define for any $I\in \tilde \mathfrak J _j$,
$$ U'_j(\theta_ \mathcal{A} ,I=\nu \rho )=\Psi_ \rho \circ\Phi(r_ \mathcal{A} =0,\theta_ \mathcal{A} ,\zeta_ \mathcal{L} =0, \rho ).$$
We have, by Theorem~\ref{thm10.2},
$$ ||\Phi(x, \rho )-x||_{\gamma _*} \le \nu^{\frac12-\aleph( \kappa +2)} ;$$ for any $x\in \mathcal{O} _{\gamma _*}(1/4,\mu_*^2/2)$, $ \rho \in Q(c_j,\beta_{\#},\nu_j)$, and, by Theorem~\ref{NFT}, \begin{equation*} \begin{split} \mid\mid \Psi_ \rho (r,\theta,\xi_ \mathcal{L} ,\eta_ \mathcal{L} )-(\sqrt{\nu \rho }\cos(\theta),&\sqrt{\nu \rho }\sin(\theta),\sqrt{\nu \rho }\xi_ \mathcal{L} ,\sqrt{\nu \rho }\eta_ \mathcal{L} ) \mid\mid_{\gamma _*}\le \\ &\le C(\sqrt\nu\ab{r}+\sqrt\nu\aa{(\xi_ \mathcal{L} ,\eta_ \mathcal{L} )}_{\gamma _*}+\nu^{\frac32} )\nu^{-\tilde c\beta_{\#}} \end{split} \end{equation*} for all $(r,\theta,\xi_ \mathcal{L} ,\eta_ \mathcal{L} )\in \mathcal{O} _{\gamma } (\frac 12, \mu_*^2)\cap\{\theta\ \textrm{real}\}$. Therefore \begin{equation*} \begin{split} \mid\mid U'_j(\theta_ \mathcal{A} ,\nu \rho )-&(\sqrt{\nu \rho }\cos(\theta),\sqrt{\nu \rho }\sin(\theta),0,0 )\mid\mid _{\gamma _*}\le\\ &C(\sqrt\nu\nu^{\frac12-\aleph( \kappa +2)}+\nu^{\frac32} )\nu^{-\tilde c\beta_{\#}}\le C\nu^{1-\aleph( \kappa +2)-\tilde c\beta_{\#}} \le CI^{1-\aleph( \kappa +2)-\tilde c\beta_{\#}-\beta_{\#}} \end{split} \end{equation*} which is $\le C I^{1-\aleph( \kappa +3)}$ if $\beta_{\#}$ is small enough.
Thus $U_j'$ verifies \eqref{dist1}.
Also, by Theorem~\ref{thm10.2}, the frequency vector $ \Omega'_j$ satisfies
$$|\Omega_j'( \rho )-\Omega( \rho )|\le \nu^{1+\aleph}\le C I^{1+\aleph-\beta_{\#}}\le C I^{1+\frac\aleph2}$$ for $ \rho \in Q(c_*,\beta_{\#},\nu)$, and, by Theorem~\ref{NFT}, $$ \Omega( \rho )= \omega _ \mathcal{A} +\nu M \rho .$$ Therefore the vector $ \Omega'_{ \mathcal{A} ,j}(\nu \rho )=\Omega_j'( \rho )$ will satisfy \eqref{dist11}.
Part (i), for $ \rho \in\tilde \mathfrak J _j$ is clear by construction.
If $ \rho $ is such that $ \mathcal{F} = \mathcal{F} _ \rho $ is non-void, then the eigenvalues $\{ \pm{\bf i}\Lambda_{a}( \rho ), a\in \mathcal{F} \}$ of $J K( \rho ) $ verifies (see \eqref{hyperb}) $$
| \Im \Lambda_a(\rho) | \ge C^{-1} \nu^{ \tilde c \beta_{\#}},\qquad \forall a\in \mathcal{F} .$$ Since, by Theorem~\ref{thm10.2},
$$ ||\frac1\nu J H'( \rho )-JK( \rho ) || \le \nu^{\aleph},$$ it follows (see for example Lemma A2 in \cite{E98} and Lemma~C.2 in \cite{EGK1}) that the eigenvalues of the matrix $\frac1\nu JH'( \rho )$, hence those of $JH'( \rho )$, have real parts bounded away from $0$ when $\tilde c \beta_{\#}<\aleph$ and $\nu$ is small enough.This proves (iii).
If the $\tilde \mathfrak J _j$'s were mutually disjoint, the mappings $U'_j$ would extend to a mapping $U'$ on $\tilde \mathfrak J $. But they are not. However there are closed subsets $ \mathfrak J _j$ of $\tilde \mathfrak J _j$, mutually disjoint, such that the density of the set $ \mathfrak J =\bigcup_j \mathfrak J _j$ at $I=0$ is the same as that of the set $\tilde \mathfrak J $. Now we just restrict each $U'_j$ to $ \mathfrak J _j$, and these restrictions extend to a mapping $U'$ on $ \mathfrak J $.
[To see the existence of the sets $ \mathfrak J _j$ we construct, by induction, subsets $ \mathfrak J _j'$ of $\tilde \mathfrak J _j$, mutually disjoint, such that $\bigcup_j \mathfrak J '_j=\tilde \mathfrak J $. The set $ \mathfrak J _j'$ are not closed, but each has a closed subset $ \mathfrak J _j$ such that $\operatorname{meas}( \mathfrak J '_j\setminus \mathfrak J _j)<2^{-j}\operatorname{meas}( \mathfrak J '_j)$. Since each $\tilde \mathfrak J _j$ is separated from $I=0$, it follows that the density of $ \mathfrak J =\bigcup_j \mathfrak J _j$ at $0$ is the same as that of $\tilde \mathfrak J $.] \end{proof}
\appendix
\section{Proofs of Lemmas \ref{lemP} and \ref{XPanalytic}}
For any $\gamma =(\gamma _1, \gamma _2)$ let us denote by $Z_\gamma$ the space of complex sequences $v=(v_s, s\in \mathbb{Z} ^d)$
with the finite norm $\|v\|_\gamma$, defined by the same relation as the norm in the space $Y_\gamma$. By $M_{\gamma ,0}$ we denote the space of complex
$ \mathbb{Z} ^d\times \mathbb{Z} ^d$--matrices, given a norm, defined by the same formula as the norm in $ \mathcal{M} _{\gamma ,0}$,
but with $[a-b]$ replaced by $|a-b|$.
For any vector $v\in Z_\varrho$, $\varrho\ge0$, we will denote by $ \mathcal{F} (v)$ its Fourier-transform: $$ \mathcal{F} (v)= u(x)\; \Leftrightarrow \;
u(x)=\sum v_a e^{{\bf i} \langle a, x\rangle }\,.
$$
By Example~\ref{analyt} if $u(x)$ is a bounded real holomorphic function with the radius of analyticity $\varrho'>0$,
then $ \mathcal{F} ^{-1} u\in Z_\varrho$ for $\varrho<\varrho'$. Finally, for a Banach space $X$ and $r>0$ we denote by $B_r(X)$ the open ball
$ \{x\in X\mid |x|_X< r\}$.
Let $F$ be the Fourier-image of the nonlinearity $g$, regarded as the mapping $u(x)\mapsto g(x,u(x))$, i.e. $\ F(v) = \mathcal{F} ^{-1} g(x, \mathcal{F} (v)(x)). $
\begin{lemma}\label{l1} For sufficiently small $\mu_g>0$, $\gamma _{g1}>0$ and for $\gamma _g=(\gamma _{g1}, \gamma _{g2})$, where $\gamma _{g2}\ge m_*+\varkappa$ we have:
i) $F$ defines a real holomorphic mapping $B_{\mu_g}(Z_{\gamma _g}) \to Z_{\gamma _g}$,
ii) $d F$ defines a real holomorphic mapping $B_{\mu_g}(Z_{\gamma _g}) \to M^b_{\gamma ',0}\,$, where $\gamma '= (\gamma _{g1}, \gamma _{g2}-m_*)$. \end{lemma}
\begin{proof} i) For sufficiently small $\varrho', \mu>0$ the
nonlinearity $g$ defines a real holomorphic function $g: \mathbb{T} ^d_{\varrho'}\times B_\mu ( \mathbb{C} )
\to \mathbb{C} $ and the norm
of this function is bounded by some constant $M$. We may write it as $\ g(x,u) = \sum_{r=3}^\infty g_r(x) u^r\,, $ where $g_r(x) = \frac1{r!} \frac{\partial^r}{\partial u^r}g(x,u)\!\mid_{u=0}$. So $g_r(x)$ is holomorphic in $x\in \mathbb{T} ^d_{\varrho'}$
and by the Cauchy estimate $|g_r|\le M\mu^{-r}$ for all $x\in \mathbb{T} ^d_{\varrho'}$. Accordingly, $$
\| \mathcal{F} ^{-1} g_r\|_{\gamma _g}\le C_{\varrho} M\mu^{-r}\quad \text{if}\quad 0\le\gamma _{g1}\le \varrho\,, $$ for any $\varrho<\varrho'$; cf. Example~\ref{analyt}. We may write $F(v)$ as \begin{equation}\label{b1} F(v) = \sum_{r=3}^\infty ( \mathcal{F} ^{-1} g_r)\star \underbrace {v\star\dots\star v}_{r}=: \sum_{r=3}^\infty F_r(v) \,. \end{equation} Since the space $Z_{\gamma _g}$ is an algebra with respect to the convolution (see Lemma~1.1 in \cite{EK08}), the $r$-th term of the sum is a mapping from $Z_{\gamma _g}$ to itself, whose norm is bounded as follows: \begin{equation}\label{b2}
\| ( \mathcal{F} ^{-1} g_r)\star \underbrace {v\star\dots\star v}_{r}\|_{\gamma _g}\le C_1
C^{r+1} \mu^{-r} \|v\|^r_{\gamma _g}\,. \end{equation} This implies the assertion with a suitable $\mu_g>0$.
ii) The assertion i) and the Cauchy estimate imply that the operator-norm of $dF(v)$ is bounded if $|v|_\gamma <\mu_g$. To estimate $|d F(v)|_{\gamma ', 0}$, for
$r\ge3$ consider the term $F_r(v)$ in \eqref{b1}. This is the Fourier transform of the mapping $u(x)\mapsto g_r(x) u(x)^r$, and its differential $dF_r(v)$ is a linear operator in $Z_{\gamma _g}$ which is the Fourier-image of the operator of multiplication by the function $rg_r(x) u^{r-1}(x)$. So the matrix $\big(\, dF_r(v)^b_a, \,a,b\in \mathbb{Z} ^d\big)$ of the former operator is nothing but the matrix of the latter operator, written in the trigonometric
basis $\{e^{{\bf i}(a,x)}\}$. Therefore $$ (dF_r(v))_a^b = (2\pi)^{-d} \int e^{-{\bf i}\langle b, x\rangle } rg_r(x) u^{r-1} e^{ {\bf i} \langle a, x\rangle }\, dx\,. $$ That is, $(dF_r(v))_a^b = G_r(b-a)$, where $G_r(a)$ is the Fourier transform of the function $r g_r(x) u^{r-1}$. So \begin{equation*} \begin{split}
|d F_r(v)|_{\gamma ', 0} =& \sup_a C\sum_b |(|d F_r(v)^b_a|e^{\gamma _{g1}|a-b|} \langle a-b \rangle^{\gamma _{g2}-m_*}\\
=& \sup_a C\sum_b |(|G_r(a-b)|e^{\gamma _{g1}|a-b|} \langle a-b \rangle^{\gamma _{g2}} \langle a-b \rangle^{-m_*}
\le C' |G_r(\cdot)|_{\gamma _g}\\
\le&C \Big(\sum_c |G_r(c)|^2 e^{2\gamma _{g1} |c|} \langle c\rangle^{2(\gamma _{g2}-m_*)}\Big)^{1/2}
\big( \sum_c \langle c\rangle^{-2m_*}\big)^{1/2}= C' |G_r|_{\gamma _g} \end{split} \end{equation*}
(we recall that $m_*>d/2$). Applying \eqref{b2} with $r$ convolutions instead of $r+1$, we see that $\
|G_r(\cdot)|_{\gamma _g} \le C_2 C^r \mu^{-r} \|v\|_{\gamma _g}^{r-1}\,. $ So $$
|(dF_r(v))|_{\gamma ',0}
\le C_3C^r \mu^{-r} \|v\|_{\gamma _g}^{r-1}\,. $$ Since $d F(v) = \sum_{r\ge3} d F_r(v)$, then the assertion ii) follows, if we replace $\mu_g$ by a smaller positive number. \end{proof}
\noindent \begin{proof}[ Proof of Lemma~\ref{lemP}.] Let us consider the functional $h_{\ge4}(\zeta)$ as in \eqref{H1}, and write it as $\ h_{\ge4}(\zeta) = {\mathbf G}\circ \Upsilon\circ D^{-}\zeta\,. $ Here $D^-$ is defined in \eqref{D-}, $\Upsilon$ is the operator $$ \Upsilon: Y_\gamma \to Z_\gamma ,\qquad \zeta\to v,\;\; v_a= {(\xi_a+\eta_{-a})}/{\sqrt2} \;\;\forall\, a, $$ and ${\mathbf G}(v) = \int g(x,( \mathcal{F} ^{-1}v)(x))\,dx$. Lemma~\ref{l1} with $F$ replaced by ${\mathbf G}$ immediately implies that $p$ is a real holomorphic function on $B_{\mu_g}(Y_\gamma )$
with a suitable $\mu_g>0$.
Next, since $$ \nabla h_{\ge4}(\zeta) =D^{-}\circ{}^t\Upsilon \circ \nabla {\mathbf G}(\Upsilon\circ D^{-}\zeta)\,, $$ where $\nabla {\mathbf G}=F$ is the map in Lemma~\ref{l1}, then $\nabla h_{\ge4}$ defines a real holomorphic mapping $B_{\mu_g}(Y_\gamma )\to Y_{\gamma }$, bounded uniformly in $\gamma _*\le\gamma \le\gamma _g$.
By the Cauchy estimate, for any $0<\mu_g'<\mu$ the Hessian of $h_{\ge4}$ defines an analytic mapping \begin{equation}\label{Phess} \nabla^2 h_{\ge4}: B_{\mu_g'}(Y_\gamma )\to \mathcal{B} ( Y_{\gamma }, Y_\gamma )\,, \end{equation} and $\nabla^2 h_{\ge4}(\zeta)$ is the linear operator $$ \nabla^2 h_{\ge4}(\zeta) =D^{-}({}^t\Upsilon \ \nabla^2 {\mathbf G}(\Upsilon\circ D^{-}\zeta)\ \Upsilon)D^{-}\,. $$ Note that for any infinite matrix $A$ the matrix ${}^t\Upsilon A \Upsilon$ is formed by $2\times2$--blocks and satisfies $$
|({}^t\Upsilon A \Upsilon)^b_a | \le \frac12 \sum_{a'=\pm a, \,b'=\pm b}
|A^{b'}_{a'}|\,. $$
Noting also that for $a' = \pm a$, $b' = \pm b$ we have $[a-b] \le|a'-b'|$, and that $ \min(r_1, r_2)^2 r_1^{-1} r_2^{-1}\le 1 $
if $r_1, r_2\ge1$, we find that the first term which enters the definition of $\nabla^2 h_{\ge4}|_{\gamma ', 2}$ estimates as follows: \begin{equation*}\begin{split}
&\sup_{a\in \mathbb{Z} ^d} \sum_{b\in \mathbb{Z} ^d} |\nabla^2 p|^b_a e^{\gamma _1 [a-b]} \max(1, [a-b])^{\gamma _2 - m_*} \min(\langle a\rangle, \langle b\rangle)^2\\ \le& \sup_{a\in \mathbb{Z} ^d} \frac12 \sum_{b\in \mathbb{Z} ^d} \,\sum_{ a'=\pm a , b'=\pm b}
|\nabla^2 {\mathbf G}|^{b'}_{a'} e^{\gamma _1 |a'-b'|} \max(1, [a'-b'])^{\gamma _2 - m_*}
\frac{ \min( \langle a'\rangle, \langle b'\rangle)^2}{ \langle a'\rangle \, \langle b'\rangle}\\ \le& \sup_{a'\in \mathbb{Z} ^d} 2 \sum_{b'\in \mathbb{Z} ^d}
|\nabla^2 {\mathbf G}|^{b'}_{a'} e^{\gamma _1 |a'-b'|} \max(1, [a'-b'])^{\gamma _2 - m_*} \,
\le \,2 |\nabla^2 {\mathbf G}|_{\gamma ',0} \end{split} \end{equation*} The second term which enters the definition of the norm estimates similar, so \begin{equation}\label{b3}
|\nabla^2 h_{\ge4}(\zeta)|_{\gamma '},2 \le 2 |\nabla^2 {\mathbf G}(v)|_{\gamma '} =
2 | d F(v)|_{\gamma '}\,,
\end{equation} $v=\Upsilon\zeta$. In view of \eqref{Phess} and
item ii) of Lemma~\ref{l1}, the mapping $$ \nabla^2 p: B_{\mu_g'}(Y_\gamma )
\to \mathcal{M} ^{ b}_{\gamma ,2} \,, $$ is real holomorphic and is bounded in norm by a $\gamma $-independent constant. Jointly with \eqref{b3} and Lemma~\ref{l1} this implies the assertion of Lemma~\ref{lemP}, if we replace $\mu_g$ by any smaller positive number.
\end{proof}
\noindent \begin{proof}[ Proof of Lemma~\ref{XPanalytic}.]
The proof is similar to that of Lemma~\ref{lemP} but simpler, and we restrict ourselves to estimating the Hessian of $Q^r$. Let us start with the Hessian of $P^r$. For any $\zeta\in \mathcal{O} (1,1,1)$ we have: \begin{equation}\label{z.2} d^2P^r(\zeta)(\zeta', \zeta') = 2M \sum_a \sum_ \varsigma A^ \varsigma _a ( \zeta_{a_1}^{ \varsigma _1} \dots \zeta_{a_{r-2}}^{ \varsigma _{r-2}}) {\zeta'}_{a_{r-1}}^{ \varsigma _{r-1}}
{\zeta'}_{a_{r}}^{ \varsigma _{r}}+\dots =: R(\zeta)(\zeta', \zeta')+\dots\,. \end{equation} Here the dots $\dots$ stand for similar sums, where the pair $\zeta', \zeta'$ replaces $\zeta,\zeta$ on other $\binom{r}2$ positions. For any $b_1,b_2\in \mathbb{Z} ^d$ the element $(\nabla_1^2P^r(\zeta))_{b_1}^{b_2}$ of the Hessian $(\nabla^2P^r(\zeta))_{b_1}^{b_2}$, coming from the component $R$ of $d^2P^r$, corresponds to the quadratic form $R(\zeta)\Big( 1_{b_1} (\xi,\eta), 1_{b_2} (\xi,\eta)\Big)$, where $1_b$ stands for the $\delta$-function on the lattice $ \mathbb{Z} ^d$, equal one at $b$ at equal zero outside $b$.
Denote by $\tilde\zeta$ the vector $\
\tilde\zeta_a = |\zeta_a| + |\zeta_{-a}|,\; a\in \mathbb{Z} ^d\,. $ Then
$|\zeta_{( \varsigma _j^0\ a_j)} | \le |\tilde\zeta_{a_j}|$, and we see from \eqref{z.2} that $| \nabla_1^2P^r(\zeta)_{b_1}^{b_2}|$ is bounded by \begin{equation*} \begin{split}
2^{r-1} M \sum_{\substack{
a_1+\dots+ a_{r-2} = - \varsigma ^0_{r-1} b_{r-1} - \varsigma ^0_r b_r}} \tilde\zeta_{a_1}\dots \tilde\zeta_{a_{r-2}} = 2^{r-1} M (\tilde\zeta\star \dots\star \tilde\zeta)(- \varsigma ^0_{r-1} b_{1} - \varsigma ^0_rb_2)\,. \end{split} \end{equation*} Since the space $Y_\gamma $ is an algebra with respect to the convolution, then \begin{equation}\label{z.3}
| \tilde\zeta\star \dots\star \tilde\zeta |_\gamma \le C^{r-3}|\tilde\zeta|_\gamma ^{r-2}\,. \end{equation}
As in the proof of Lemma~\ref{lemP}, $\
| \nabla^2Q^r(\zeta)_{b_1}^{b_2}| \le \langle b_1\rangle^{-1} \langle b_2\rangle^{-1}
| \nabla^2P^r(D^-\zeta)_{b_1}^{b_2}|\,. $ Denoting by $\nabla_1^2Q^r$ the component of $\nabla^2Q^r$, corresponding to $\nabla_1^2P^r$,
denoting $b_1' = - \varsigma ^0_{r-1}b_1, \ b'_2 = \varsigma ^0_rb_2$, and using that $[b_1-b_2] \le |b'_1-b'_2|$, we find : \begin{equation*} \begin{split}
&\sup_{b_1}\ \sum_{b_2} | (\nabla_1^2Q^r)_{b_1}^{b_2}| e^{\gamma _1[b_1-b_2]} \max(1, [b_1-b_2])^{\gamma _2-m_*}\min (\langle b_1\rangle , \langle b_2\rangle )^2 \\ &\le C^r M \sup_{b_1}\ \sum_{b_2} (\tilde\zeta\star \dots\star \tilde\zeta)(b'_1-b'_2) e^{\gamma _1[b_1-b_2]} \max(1, [b_1-b_2])^{\gamma _2-m_*}\frac{\min (\langle b_1\rangle , \langle b_2\rangle )^2 } {\langle b_1\rangle \langle b_2\rangle}\\ &\le C^r M \sup_{b'_1}\ \sum_{b'_2} (\tilde\zeta\star \dots\star \tilde\zeta)(b'_1-b'_2) e^{\gamma _1[b_1-b_2]}
\langle b_1-b_2\rangle^{\gamma _2-m_*}
\le {C'}^r M |\tilde\zeta|_\gamma ^{r-2} \le C^r M \end{split} \end{equation*}
(since $|\zeta|_\gamma \le1$). This implies the estimate for $\nabla_1^2 Q^r$, required by the lemma. Other components of $\nabla^2 Q^r$, corresponding to the dots in \eqref{z.2}, may be estimated in the same way. \end{proof}
\section{Examples} In this appendix we discuss some examples of Hamiltonian operators $ \mathcal{H} (\rho)={\bf i}JK(\rho)$ defined in \eqref{diag}, corresponding to various dimensions $d$ and sets $ \mathcal{A} $. In particular we are interested in examples which give rise to partially hyperbolic KAM solutions.
\noindent{\bf Examples with $({ \mathcal{L} _f}\times { \mathcal{L} _f})_+=\emptyset$.}\\ As we noticed in \eqref{Lf+=0}, if $({ \mathcal{L} _f}\times { \mathcal{L} _f})_+=\emptyset$ then $ \mathcal{H} $ is Hermitian, so the constructed
KAM-solutions are linearly stable. This is always the case when $d=1$.\\ When $d=2$ and $ \mathcal{A} =\{(k,0),(0,\ell)\}$ with the additional assumption that neither $k^2$ nor $\ell^2$ can be written as the sum of squares of two natural numbers, we also have $({ \mathcal{L} _f}\times { \mathcal{L} _f})_+=\emptyset$.\\ Similar examples can be constructed in higher dimension, for instance for $d=3$ we can take $ \mathcal{A} =\{(1,0,0),(0,2,0)\}$ or $ \mathcal{A} =\{(1,0,0),(0,2,0),(0,0,3)\}$.\\ We note that in \cite{GY06b} the authors perturb solutions \eqref{sol}, corresponding to
set $ \mathcal{A} $ for which $({ \mathcal{L} _f}\times { \mathcal{L} _f})_+=\emptyset$ and $({ \mathcal{L} _f}\times { \mathcal{L} _f})_-=\emptyset$. This significantly simplifies
the analysis since in that case there is no matrix $K$ in the normal form
\eqref{HNF} and the unperturbed quadratic Hamiltonian is diagonal.
\noindent{\bf Examples with $({ \mathcal{L} _f}\times { \mathcal{L} _f})_+\neq\emptyset$.} In this case hyperbolic directions may appear as we show below.\\ The choice $ \mathcal{A} =\{(j,k),(0,-k)\}$ leads to $((j,-k),(0,k))\in ({ \mathcal{L} _f}\times { \mathcal{L} _f})_+$.\\ Note that this example can be plunged in higher dimensions, e.g. the 3d-set $ \mathcal{A} =\{(j,k,0),(0,-k,0)\}$ leads to a non trivial $({ \mathcal{L} _f}\times { \mathcal{L} _f})_+$.
\noindent{\bf Examples with hyperbolic directions}\\ Here we give examples of normal forms with hyperbolic eigenvalues, first in
dimension two, then -- in higher dimensions. That is, for the beam equation \eqref{beam} we will find
admissible sets $ \mathcal{A} $ such that the corresponding matrices ${\bf i}JK(\rho)$ in the normal form \eqref{HNF} have
unstable directions. Then by Theorem~\ref{t73}
the time-quasiperiodic solutions of \eqref{beam}, constructed in the theorem, are linearly unstable.
We begin with dimension $d=2$. Let $$ \mathcal{A} =\{(0,1),(1,-1)\}\,. $$ We easily compute using \eqref{L++}, \eqref{L+-} that $$ \mathcal{L} _f=\big\{ (0,-1),(1,0),(-1,0),(1,1), (-1,1),(-1,-1)\big)\}\,, $$ and $$ ( { \mathcal{L} _f}\times { \mathcal{L} _f})_+=\{\big( (0,-1),(1,1)\big);\big( (1,1),(0,-1)\big)\}, \qquad ( { \mathcal{L} _f}\times { \mathcal{L} _f})_-=\emptyset. $$ So in this case the decomposition \eqref{decomp} of the Hamiltonian operator $ \mathcal{H} (\rho)={\bf i}JK(\rho)$ reads $$ \mathcal{H} (\rho)= \mathcal{H} _1(\rho)\oplus \mathcal{H} _2(\rho)\oplus \mathcal{H} _3(\rho)\oplus \mathcal{H} _4(\rho)\oplus \mathcal{H} _5(\rho)\,, $$ where $ \mathcal{H} _1(\rho)\oplus \mathcal{H} _2(\rho)\oplus \mathcal{H} _3(\rho)\oplus \mathcal{H} _4(\rho)$ is a diagonal operator with purely imaginary eigenvalues and $ \mathcal{H} _5(\rho)$ is an operator in $ \mathbb{C} ^4$ which may have hyperbolic eigenvalues. That is, now $M=5$ and $M_0=4$. \\
Let us denote $\zeta_1=(\xi_1,\eta_1)$ (reps. $\zeta_2=(\xi_2,\eta_2)$) the $(\xi,\eta)$-variables corresponding to the mode $(0,-1)$ (reps. $(1,1)$). We also denote $\rho_1=\rho_{(1,0)}$, $\rho_2=\rho_{(1,-1)}$, $\lambda_1=\sqrt{1+m}$ and $\lambda_2=\sqrt{4+m}$. By construction $ \mathcal{H} _5(\rho)$ is the restriction of the Hamiltonian $ \langle K(m,\rho)\zeta_f, \zeta_f\rangle$ to the modes $(\xi_1,\eta_1)$ and $(\xi_2,\eta_2)$. We calculate using \eqref{K} that \begin{equation}\label{hr}\langle \mathcal{H} _5(\rho)(\zeta_1,\zeta_2),(\zeta_1,\zeta_2)\rangle=\beta(\rho) \xi_1\eta_1+\gamma (\rho) \xi_2\eta_2+\alpha(\rho) (\eta_1\eta_2+\xi_1\xi_2)\,, \end{equation} where $$ \alpha(\rho)= \frac6{4\pi^{2}} \frac{\sqrt{\rho_1\rho_2}}{\lambda_1\lambda_2}\,,\quad \beta(\rho)= \frac3{4\pi^{2}}\frac1{\lambda_1}\Big( \frac{\rho_1}{\lambda_1}-\frac{2\rho_2}{\lambda_2} \Big) \,,\quad \gamma (\rho)= \frac3{4\pi^{2}} \frac1{\lambda_2}\Big( \frac{\rho_2}{\lambda_2} -\frac{2\rho_1}{\lambda_1}\Big) \,. $$ Thus the linear Hamiltonian system, governing the two modes, reads\footnote{Recall that the symplectic two-form is: $-{\bf i}\sum d\xi\wedge d\eta$.} \begin{equation*} \left\{\begin{array}{ll}
\dot \xi_1 &=-{\bf i}(\beta \xi_1+\alpha \eta_2)\\
\dot \eta_1 &={\bf i}(\beta \eta_1+\alpha \xi_2)\\
\dot \xi_2 &=-{\bf i}(\gamma \xi_2+\alpha \eta_1)\\
\dot \eta_2 &={\bf i}(\gamma \eta_2+\alpha \xi_1). \end{array}\right. \end{equation*} So the Hamiltonian operator $ \mathcal{H} _5$ has the matrix ${\bf i}L$, where \begin{equation*}
L= \left(\begin{array}{cccc}
-\beta &0&0&-\alpha\\
0&\beta &\alpha&0\\ 0&-\alpha& -\gamma &0\\
\alpha &0&0&\gamma \\ \end{array}\right). \end{equation*} We calculate the characteristic polynomial of $L$ and obtain after a factorisation that $$\det (L-\lambda I)=\big(\lambda^2+(\gamma -\beta)\lambda -\beta\gamma +\alpha^2\big)\, \big(\lambda^2-(\gamma -\beta)\lambda -\beta\gamma +\alpha^2\big)\,. $$ Both quadratic polynomials which are the factors in the r.h.s. have the same discriminant $\ \Delta= (\beta+\gamma )^2-4\alpha^2.$ If $\rho_1\sim1$ and $0<\rho_2\ll1$, then $\Delta>0$. So all eigenvalues of $L$ are real, while the eigenvalues of $ \mathcal{H} _5$ and $ \mathcal{H} $ are pure imaginary (in agreement with Lemma~\ref{laK}). But if $\rho_1=\rho_2=\rho$, then $$ \gamma -\beta = \frac{3\rho}{4\pi^2} \Big( \frac1{\lambda_2^2} - \frac1{\lambda_1^2}\big)\,, \quad \beta+\gamma = \frac{3\rho}{4\pi^2} \Big(\frac{1}{\lambda_1^2}+ \frac{1}{\lambda_2^2}-\frac{4}{\lambda_1\lambda_2} \Big),\quad \alpha= \frac{6\rho}{4\pi^2} \frac{1}{\lambda_1\lambda_2}\,, $$ and \begin{align*}\Delta=\frac{9 \rho}{(2\pi)^4}\Big(\frac1{\lambda_1^2}+\frac1{\lambda_2^2}\Big)\Big(\frac1{\lambda_1^2}+\frac1{\lambda_2^2} -\frac{8}{\lambda_1\lambda_2}\Big) \leq \frac{9 \rho}{(2\pi)^4}\Big(\frac1{\lambda_1^2}+\frac1{\lambda_2^2}\Big)\Big(\frac1{\lambda_1^2}-\frac7{\lambda_2^2} \Big)\,. \end{align*} Thus, $\Delta <0$ for all $m\in[1,2]$. Since the eigenvalues of the matrix $L=(1/{\bf i}) \mathcal{H} _5$ are $\pm(\gamma -\beta)\pm \sqrt\Delta$, then all four of them have nontrivial imaginary parts for all values of the parameter $m\in[1,2]$, and accordingly
the operator $ \mathcal{H} $ has 4 hyperbolic directions. By analyticity, for all $m\in[1,2]$ with a possible exception of finitely many points, the real parts of the eigenvalues also are non-zero. In this case the operator $ \mathcal{H} $ has a quadruple of hyperbolic eigenvalues.
This example can be generalised to any dimension $d\geq 3$. Let us do it for $d=3$. Let \begin{equation}\label{AAA} \mathcal{A} =\{(0,1,0),(1,-1,0)\}. \end{equation} We verify that $ \mathcal{L} _f$ contains 16 points, that $( \mathcal{L} _f\times \mathcal{L} _f)_-=\emptyset$ and \begin{align*}( \mathcal{L} _f\times \mathcal{L} _f)_+=\{&((0,-1,0),(1,1,0)); ((1,1,0),(0,-1,0));\\ &((1,0,-1),(0,0,1)); ((0,0,1),(1,0,-1));\\ &((1,0,1),(0,0,-1)); ((0,0,-1),(1,0,1))\}\,. \end{align*} I.e. $( \mathcal{L} _f\times \mathcal{L} _f)_+$ contains three pairs of symmetric couples $(a,b),(b,a)$ which give rise to three non trivial $2\times2$-blocks in the matrix $ \mathcal{H} $. Now $M=13$, $M_0=10$ and the decomposition \eqref{decomp} reads $$ \mathcal{H} (\rho)= \mathcal{H} _1(\rho)\oplus\cdots \oplus \mathcal{H} _{13}(\rho)\,. $$ Here
$ \mathcal{H} _1(\rho)\oplus\cdots \oplus \mathcal{H} _{10}(\rho)$ is the diagonal part of $ \mathcal{H} $ with purely imaginary eigenvalues, while the operators $ \mathcal{H} _{11}(\rho)$, $ \mathcal{H} _{12}(\rho)$, $ \mathcal{H} _{13}(\rho)$ correspond to non-diagonal $4\times4$--matrices.
Denoting
$\rho_1=\rho_{(0,1,0)}$ and $\rho_2=\rho_{(1,-1,0)}$ we find that the restriction of the Hamiltonian $ \langle K(m,\rho)\zeta_f, \zeta_f\rangle$ to the modes $(\xi_1,\eta_1):=(\xi_{(0,-1,0)},\eta_{(0,-1,0)})$ and $(\xi_2,\eta_2):=(\xi_{(1,1,0)},\eta_{(1,1,0)})$ is governed by the Hamiltonian \eqref{hr}, as in the 2d case. Similarly the restrictions of the Hamiltonian $ \langle K(m,\rho)\zeta_f, \zeta_f\rangle$ to the pair of modes $(\xi_{(1,0,-1)},\eta_{(1,0,-1)})$ and $(\xi_{(0,0,1)},\eta_{(0,0,1)})$ and to the pair of modes $(\xi_{(1,0,1)},\eta_{(1,0,1)})$ and $(\xi_{(0,0,-1)},\eta_{(0,0,-1)})$ are given by the same Hamiltonian \eqref{hr}. So $ \mathcal{H} _{11}(\rho)\equiv \mathcal{H} _{12}(\rho)\equiv \mathcal{H} _{13}(\rho)$ and for $\rho_1=\rho_2$ we have 3 hyperbolic directions, one in each block $Y^{f11}$, $Y^{f12}$ and $Y^{f13}$ (see \eqref{deco}) with the same eigenvalues.
We notice that the eigenvalues are identically the same for all three blocks, thus the relation \eqref{single} is violated. This does not contradict Lemma~\ref{l_nond} since the set \eqref{AAA} is not strongly admissible. Indeed, denoting $a=(0,1,0)$, $b=(1,-1,0)$ we see that $c:=a+b=(1,0,0)$. So three points
$(0,-1,0), (0,0,\pm1)\in \{x\mid\, |x|=|a|\}$ all lie at the distance $\sqrt2$ from $c$. Hence, it is not true that $a \, \angle\!\angle\, b$.
\section{Admissible and strongly admissible random $R$-sets}
Given $d$ and $n$, let $B(R)$ be the (round) ball of radius $R$ in $ \mathbb{R}^d$, and $ \mathbf{B} (R)=B(R)\cap \mathbb{Z} ^d$. The family $ \Omega= \Omega(R)$ of $n$-sets $\{a_1,\dots, a_n\}$ in $ \mathbf{B} (R)$, $ \Omega = \mathbf{B} \times \dots\times \mathbf{B} $ ($n$ times) has cardinality of order
$CR^{nd}$.
The family on $n$-sets $\{a_1,\dots, a_n\}$ in $ \Omega$ such that $\ab{a_j}=\ab{a_k}$ for some $j\not=k$ has cardinality $\le C'R^{nd-1}$ (the constant $C'$ as well as all other constants in this section depend, without saying, on $n,d$). Its complement in $ \Omega$ is the set $ \Omega_{\text{adm}} = \Omega_{\text{adm}}(R)$ of admissible $n$-sets in $ \mathbf{B} (R)$. Hence $$ \frac{\# \Omega_{\text{adm}}(R)}{\# \Omega(R)} = 1-O(R^{-1})\,,
\qquad R\to\infty\,. $$ We provide the set $ \Omega$ with the uniform probability measure $ \mathbb{P}\,$ and will call elements of $ \Omega$ {\it $n$-points random $R$-sets}. The calculation above shows that \begin{equation}\label{om+} \mathbb{P}\,( \Omega_{\text{adm}})\to 1 \quad \text{as} \quad R\to\infty\,. \end{equation} That is, admissible $n$-points random $R$-sets with large $R$ are typical.
To consider strongly admissible sets, let $S(R)$ be the sphere of radius $R$ in $ \mathbb{R}^d$, i.e. the boundary of $B(R)$, and let $ \mathbf{S} (R) = S(R)\cap \mathbb{Z} ^d$ (this set is non-empty only if $R^2$ is an integer).
We have that, for any $\varepsilon>0$ there exists $C_\varepsilon>0$ such that
\begin{equation}\label{VC}
\Gamma_{R,d}:=
| \mathbf{S} (R) | \le C_\varepsilon R^{d-2+\varepsilon} \qquad \forall\, R>0.
\end{equation} Indeed, for $d=2$ this is a well-known result from number theory (see \cite{Har}, Theorem~338). For $d\ge 3$ it follows by induction and an easy integration argument. For example for $d=3$, then $$ \mathbf{S} (R)=\{a\in \mathbb{Z} ^3: \ab{a}^2=R^2\}=\bigcup_{a_3^2\le R^2}\{a=(a_1,a_2,a_3)\in \mathbb{Z} ^3: a_1^2+a_2^2=R^2-a_3^2\}$$ so $$
\Gamma_{R,3}=\sum _{n^2\le R^2} \Gamma_{\sqrt{R^2-n^2},2}\le C_\varepsilon \sum _{n^2\le R^2} (R^2-n^2)^{\varepsilon/2} \le
C_\varepsilon R^\varepsilon \sum _{n^2\le R^2} \big(1-(\frac nR)^2\big)^{ \varepsilon/2 }\,,$$
which is
$\
\le C_\varepsilon R^\varepsilon(2R+1)\le C'_\varepsilon R^{1+\varepsilon}.
$
For vectors $a,b \in \mathbb{Z} ^d$ we will write $\ a \, \angle\!\angle\, b \quad \text{iff} \quad a \, \angle\, a+b\,. $ Consider again the ensemble $ \Omega= \Omega(R) $ of $n$-points random $R$-sets, $ \Omega =\{\omega=(a_1,\dots,a_n) \}$, and for $j=1,\dots,n$ define the random variable $\xi_j$ as $\xi_j( \omega ) = a_j$. Consider the event $$ \Omega_{{}_{ \, \angle\!\angle\,}} = \{ \xi^i \, \angle\!\angle\,\xi^j \quad\text{for all}\quad i\ne j\}\,. $$ Then $ \Omega_{\text{s\_adm}} = \Omega_{\text{adm}}\cap \Omega_{{}_{ \, \angle\!\angle\,}} $ is the collection of strongly admissible sets. Clearly \begin{equation}\label{w22} \mathbb{P}\,(\Omega\setminus \Omega_{{}_{ \, \angle\!\angle\,}} ) \le n(n-1) (1- \mathbb{P}\,\{ \xi^1 \, \angle\!\angle\, \xi^2\})\,. \end{equation} So if we prove that \begin{equation}\label{w3}
1- \mathbb{P}\,\{ \xi^1 \, \angle\!\angle\, \xi^2\} \le CR^{- \kappa }\,, \end{equation} then, in view of \eqref{om+}, we would show that \begin{equation}\label{om++} \mathbb{P}\,( \Omega_{\text{s\_adm}})\to 1 \quad \text{as} \quad R\to\infty\,. \end{equation}
Below we restrict ourselves to the case $d=3$ since for higher dimension the argument is similar, but
more cumbersome. We have that
\begin{equation}\label{w4}
1- \mathbb{P}\,\{ \xi^1 \, \angle\!\angle\, \xi^2\} = | \mathbf{B} (R)|^{-2} C^{**}\,,\quad C^{**}=
\# \{ (a,b) \in \mathbf{B} (R) \times \mathbf{B} (R)\mid \;\text{not}\; a \, \angle\!\angle\, b\}\,,
\end{equation}
and, denoting $a+b=c$, that
\begin{equation}\label{w5}
C^{**} \le
\# \{ (a,c) \in \mathbf{B} (2R) \times \mathbf{B} (2R)\mid \;\text{not}\; a \, \angle\, c\}\,.
\end{equation}
Now we will estimate the r.h.s. of \eqref{w5}, re-denoting $2R$ back to $R$. That is, will
estimate the cardinality
of the set
$$
X = \{ (a,b) \in \mathbf{B} (R) \times \mathbf{B} (R)\mid \;\text{not}\; a \, \angle\, b \}\,.
$$
It is clear that $(a,b) \in X$, $a\ne0$, iff there exist points $a', a^{\prime\prime} \in \mathbf{S} (|a|)$ such that $b$ lies in the line
$\Pi_{ a, a', a^{\prime\prime} }$, which is perpendicular to the triangle $( a, a', a^{\prime\prime} )$ and passes through its centre, so it also passes
through the origin. Let $v = v_{ a, a', a^{\prime\prime} }$ be a primitive integer vector in the direction of $\Pi_{ a, a', a^{\prime\prime} }$.
For any $a\in \mathbb{Z} ^d, a\ne0$, denote
$$
\Delta(a) = \big\{\, \{a', a^{\prime\prime} \} \subset \mathbf{S} (|a|)\setminus \{a\}\mid a'\ne a^{\prime\prime} \big\}\,.
$$
Then
$$
|\Delta(a)| < \Gamma_{|a|,3}^2 \le C^2_\theta R^{2\theta},\qquad \theta = \theta_3\,,
$$
see \eqref{VC}. For a fixed $a\in \mathbf{B} (R)\setminus \{0\}$ consider the mapping
$$
\Delta(a) \ni \{a', a^{\prime\prime} \} \mapsto v=v_{ a, a', a^{\prime\prime} }\,.
$$
It is clear that each direction $v=v_{ a, a', a^{\prime\prime} }$ gives rise to at most $2R |v|^{-1}$ points
$b$ such that $(a,b)\in X$. So, denoting
$$
X_a = \{ b\in \mathbf{B} (R) \mid (a,b) \in X\}\,,
$$
we have
$$
|X_a| \le 2R \sum |v_{ a, a', a^{\prime\prime} }|^{-1}\,,\quad \text{if}\; a\ne0\,,
$$
where the summation goes through all different vectors $v$, corresponding to various $\{a', a^{\prime\prime} \}\in\Delta(a)$.
As $|v|^{-1}$ is the bigger the smaller $|v|$ is, we
see that
the r.h.s. is $\,\le 2R\sum_{v\in \mathbf{B} (R')\setminus\{0\}}|v|^{-1} $,
where $R'$ is any number
such that $| \mathbf{B} (R')| \ge |\Delta(a)|$. Since $|\Delta(a)| \le \Gamma_{|a|,3}^2$, then choosing
$R'=R'_a=C\Gamma_{|a|,3}^{2/3}$ we get for any $a\in \mathbf{B} (R)\setminus \{0\}$ that
\begin{equation*}
\begin{split}
|X_a| \le 2CR \sum_{ \mathbf{B} (R'_a)\setminus\{0\}}|v|^{-1} \le C_1 R \int_{ B(R'_a)}|x|^{-1}\,dx
\le C_2 R(R'_a)^2 = C_3 R \,\Gamma_{|a|,3}^{4/3}\,.
\end{split}
\end{equation*}
Since $0 \, \angle\, b$ for any $b$, then $X_0 = \{0\}$ and
$$
|X| = \sum_{a\in \mathbf{B} (R)} |X_a| \le 1+ CR \sum_{a\in \mathbf{B} (R)\setminus\{0\}} \Gamma_{|a|,3}^{4/3}\,.
$$
Evoking the estimate \eqref{VC} we finally get that
\begin{equation*}
\begin{split}
|X| \le C_1R \sum_{a\in \mathbf{B} (R)\setminus\{0\}} |a|^{{\frac43\theta_3}}
\le C_2R \int_{B(R)} |x|^{{\frac43\theta_3}}\,dx\le C_3 R^{1+3+{\frac43\theta_3}} = C_3 R^{5+1/3+\varepsilon'}\,,
\end{split}
\end{equation*}
with any positive $\varepsilon'$.
Jointly with \eqref{w4}, \eqref{w5} and the definition of the set $X$ this
implies the required relation \eqref{w3} with $ \kappa =2/3 - \varepsilon'$, and \eqref{om++} follows.
That is, $n$-points random $R$-sets with large $R$ are typical, for any $d$ and any $n$.
\section{Two lemmas}
\subsubsection{Transversality}\label{ssTransversality} \ \begin{lemma}\label{lTransv1} Let $I$ be an open interval and let $f:I \to \mathbb{R}$ be a $ \mathcal{C} ^{j}$-function whose $j$:th derivative satisfies $$\ab{f^{(j)}(x)}\ge \delta,\quad \forall x\in I.$$ Then, $$ \operatorname{meas} \{x\in I: \ab{f(x)}<\varepsilon\}\le C (\frac\varepsilon{ \delta _0})^{\frac1j}.$$
$C$ is a constant that only depends on $\ab{f}_{ \mathbb{C} ^j(I)}$. \end{lemma}
\begin{proof} It is enough to prove this for $\varepsilon<1$. Let $I_1=I$, $\delta_1=\delta$ and $g_k=f^{(j-k)}$, $k=1,\dots j$. Let $\delta_1,\delta_2,\dots\delta_{j+1}$ be a deceasing sequence of positive numbers.
Since $g_1'=f^{(j)}$ we have $\ab{g'_1(x)}\ge\delta_1$ for all $x\in I_1$ and, hence, the set $$E_1= \{x\in I_1: \ab{g_1(x)}<\delta_2\}$$ has Lebesgue measure $ \lesssim \frac{\delta_2}{\delta_1}$. On $I_2=I_1\setminus E_1$ we have $\ab{g'_2(x)}\ge\delta_2$ for all $x\in I_2$ and, hence, the set $$E_2= \{x\in I_2: \ab{g_2(x)}<\delta_3\}$$ has Lebesgue measure $ \lesssim \frac{\delta_3}{\delta_2}$. Continue this $j$ steps. On $I_j=I_{j-1}\setminus E_{j-1}$ we have $\ab{g'_{j}(x)}\ge\delta_{j}$ for all $x\in I_{j}$ and, hence, the set $$E_j= \{x\in I_j: \ab{g_j(x)}<\delta_{j+1}\}$$ has Lebesgue measure $ \lesssim \frac{\delta_{j+1}}{\delta_{j}}$.
Now the set $\{x\in I: \ab{f(x)}<\delta_{j+1}\}$ is contained in the union of the sets $E_k$ which has measure $$ \lesssim \frac{\delta_{2}}{\delta_{1}}+\dots+\frac{\delta_{j+1}}{\delta_{j}}.$$ Take now $\delta_k=\eta^{k-1}\delta$. Then this measure is $ \lesssim \eta$ and $\delta_{j+1}=\eta^j\delta$. Chose finally $\eta$ so that $\eta^j\delta=\varepsilon$. \end{proof}
\subsubsection{Extension}\label{ssExtension}
\begin{lemma}\label{lExtension} Let $X\subset Y$ be subsets of $ \mathcal{D}_0$ such that $$\operatorname{dist}( \mathcal{D}_0\setminus Y,X)\ge \varepsilon,$$ then there exists a $ \mathcal{C} ^\infty$-function $g: \mathcal{D}_0\to \mathbb{R}$, being $=1$ on $X$ and $=0$ outside $Y$ and such that for all $j\ge 0$
$$| g |_{ \mathcal{C} ^j( \mathcal{D}_0)}\le C(\frac C{\varepsilon})^j.$$ $C$ is an absolute constant.
\end{lemma}
\begin{proof} This is a classical result obtained by convoluting the characteristic function of $X$ with a $ \mathcal{C} ^\infty$-approximation of the Dirac-delta supported in a ball of radius $\le \frac{\varepsilon}2$. \end{proof}
\end{document} | arXiv |
By Xavier Geerinck in Discrete Math — Jun 5, 2015
Fields - Galois Fields
1 Galois Fields
1.1 Construction of Galois Fields
A Galois field has a finite amount of numbers and is written as GF(q) or Fq. Where q = pn.
When knowing all this we can now construct a Galois Field with n dimensions and p elements.
When we want to construct the galois field of F49 then we know that p = 7 and n = 2. Which means that we have 2 dimensions and 7 elements (elements ranging from 0 to 6). So to construct this we will need a grid the exists out of pn elements. Like this:
$$ \begin{matrix} (0,0) & (1,0) & (2,0) & (3,0) & (4,0) & (5,0) & (6,0) \newline (0,1) & (1,1) & (2,1) & (3,1) & (4,1) & (5,1) & (6,1) \newline (0,2) & (1,2) & (2,2) & (3,2) & (4,2) & (5,2) & (6,2) \newline (0,3) & (1,3) & (2,3) & (3,3) & (4,3) & (5,3) & (6,3) \newline (0,4) & (1,4) & (2,4) & (3,4) & (4,4) & (5,4) & (6,4) \newline (0,5) & (1,5) & (2,5) & (3,5) & (4,5) & (5,5) & (6,5) \newline (0,6) & (1,6) & (2,6) & (3,6) & (4,6) & (5,6) & (6,6) \newline \end{matrix} $$
Of course, when n > 2 then we are not able to write this as a grid, but we do now the elements that are in our field.
Now because our field construction is defined as (Fq, ⊕, ⊗) we need to have an additive interaction and a multiplicative interaction.
Our additive interaction is being done by executing the additive operations mod p for every dimension.
The multiplicative interaction needs to have a polynomial at every grid point with grade < n
If we apply this on our field that we calculated then we get:
$$ \begin{matrix} 0 & x & 2x & 3x & 4x & 5x & 6x \newline 1 & x+1 & 2x+1 & 3x+1 & 4x+1 & 5x+1 & 6x+1 \newline 2 & x+2 & 2x+2 & 3x+2 & 4x+2 & 5x+2 & 6x+2 \newline 3 & x+3 & 2x+3 & 3x+3 & 4x+3 & 5x+3 & 6x+3 \newline 4 & x+4 & 2x+4 & 3x+4 & 4x+4 & 5x+4 & 6x+4 \newline 5 & x+5 & 2x+5 & 3x+5 & 4x+5 & 5x+5 & 6x+5 \newline 6 & x+6 & 2x+6 & 3x+6 & 4x+6 & 5x+6 & 6x+6 \end{matrix} $$
1.2 Calculating with polynomials
1.2.1 Multiplying Polynomials
The multiplication is written as: polynomial 1 ⊗ polynomial 2. To perform this multiplication, we do the same as we would do with for example: (2x + 2) * (4x + 3). But this time, we do modulo our prime on the end!
Example: 123x4+76x2+7x+4 ⊗ 196x4+12x3+225x2+4x+76 %251 for prime = 251
Here we can see that the result: $$12x^2+221x^7+152x^6+15x^5+208x^4+170x^3+178x^2+46x+53$$
1.2.2 Dividing Polynomials
Dividing polynomials is also easy, but we have to pay attention. If the coefficient of the divisor it's head grade = 1 then we can do a normal division like we saw when we were 12 years old (we just use some bigger numbers now).
1.2.2.1 Head grade divisor is 1
Example: 12x8+221x7+152x6+25x5+208x4+117x3+150x2+30x+53 / x5+x4+12x3+9x2+7
This results into: $$12x^3+209x^2+50x+120 \otimes x^5+x^4+12x^3+9x^2+7 \oplus 117x^4+151x^3+117x^2+182x+217 \text{ mod 251}$$
which is just the result multiplied by the divisor and the remainder added to it.
1.2.2.2 Head grade divisor > 1
If the head grade > 1, then we have to multiply both the divisor and the polynomal by (head grade) ^ -1 mod p
So this means if we got this division: $$12x^8+221x^7+152x^6+25x^5+208x^4+117x^3+150x^2+30x+53 / 3x^5+x^4+12x^3+9x^2+7$$
$$4x^8+241x^7+218x^6+92x^5+153x^4+39x^3+50x^2+10x+185 / x^5+84x^4+4x^3+3x^2+86$$
On the end we just have to remultiply the remainder by the head coeficient of the divisor, and then we got our result
Steps executed:
1.2.3 The Euclidean Algorithm For Polynomials
The euclidean algorithm was already explained in the first post for discrete math http://desple.com/post/104343618742/discrete-math-fields-prime-fields-part-1 and we are now able to do this with polynomials too.
1.3 Group Tables
We got 2 different group tables, the additive group table and the multiplicative. We will first start by constructing the additive group table since this is the easiest one. To construct this we follow these steps:
Find the elements belonging to the GF being given (Write only the coefficients, easier to calculate with)
When we found the elements, create a table with the X and Y values being the elements from the GF
Now we calculate the sum of the X and Y values and fill in our table
> Note: When filling in the table, don't forget that we are working mod p
Example: We will create the additive table for GF(27)
Basics of the GF:GF(27), p = 3, n = 3, q = 27
1) Elements
$$ \begin{matrix} 0 & 1 & 2 & x & x+1 & x+2 & 2x & 2x+1 & 2x+2 \newline x^2 & x^2+1 & x^2+2 & x^2+x & x^2+x+1 & x^2+x+2 & x^2+2x & x^2+2x+1 & x^2+2x+2 \newline 2x^2 & 2x^2+1 & 2x^2+2 & 2x^2+x & 2x^2+x+1 & 2x^2+x+2 & 2x^2+2x & 2x^2+2x+1 & 2x^2+2x+2 \end{matrix} $$
Just the coefficients:
$$ \begin{matrix} 0 & 1 & 2 & 10 & 11 & 12 & 20 & 21 & 22 \newline 100 & 101 & 102 & 110 & 111 & 112 & 120 & 121 & 122 \newline 200 & 201 & 202 & 210 & 211 & 212 & 220 & 221 & 222 \end{matrix} $$
2) & 3) Create table with X and Y as the elements, and calculate MOD P
If we now want to calculate the multiplicative field, then we just need to add a 0 column and a 0 row in the beginning and then we have our multiplicative result:
1.4 irreducible Polynomials
1.4.1 Number of irreducible Polynomials
We know that there are pn amount of monic polynomials. By using the Mobius Inversion, we can calculate how many there are irreducible.
$$ \frac{1}{n} \sum_{d|n} \mu() * p^{\frac{n}{d}} $$
Example: How many polynomials are irreducible for (F66, ⊕, ⊗)
$$ \frac{6^6 - 6^3 - 6^2 + 6}{6} = 7735 $$
Which means that there are 7735 of the 46656 polynomials that are irreducible for the prime 6 with grade 6
1.4.2 Checking if a Polynomial is irreducible
To check if a Polynomial is irreducible, we have to check if it has not common factors mod p with the polynomials of xpi - x, for every i <=n / 2
Some examples for checking:
Fp2: check xp - x
Fp4: check xp - x and xp2
Fp6: check xp - x , xp2 and xp3
> Note: We wrote - everywhere, this is correct since we still have to do mod p afterwards. So this could become + if we would do this for for p = 2 (since -1 mod 2 = +1)!
Example: Find an irreducible polynomial with n = 7 for F128
128 ==> p = 2, n = 7
Since n = 7, we know that we have to check for x2 + 1, x4 + 1 and x8 + 1
Now we pick a polynomial that could be irreducibel and is monic. For example: x7 + x2 + 1
To check if this is irreducibel, we need to be sure this polynomial can not be divided by the 3 checks that we have placed. We check this by using the Euclidean algorithm and the dividing of Polynomials.
> A lot of irreducible polynomials have been found, we can find those in the FIPS 186 standard (FIPS186-1, FIPS186-2, FIPS186-3, FIPS186-4)
1.5 Primitive Elements
If we found an Irreducible Polynomial, then we have to find a primitive element ω of the multiplicative group. We can do this by using the method that I explained in my first blog post about Discrete math http://desple.com/post/104343618742/discrete-math-fields-prime-fields-part-1
I also refer to this post for finding the discrete logarithm for an index of a primitive root using the baby-step giant-step method.
1.6 Inverse Elements
We can calculate the inverse of an element using the Euclidean Algorithm.
Example: (x4+x3+x2)-1
Following the solution above, we can see that the inverse is equal to: x6 + x3 + x2
1.7 Calculating with indices
1.7.1 Method
To perform caslculations in Finite fields we have to choose between 2 methods
We identify the coordinates in a n-dimensional grid for the polynomial with grade < n
Advantage: Additive Calculation are just additions mod p
Disadvantage: Multiplicative Calculations require inverse elements and are therefor harder
We identify the elements by comparing it's index to a primitive element ω (ω is randomly chosen)
Advantage: We simplify multiplicative calculations to 1 additive calculations modulo(q - 1)
Disadvantage: Additive Calculations are harder, but we can still perform them if we have a list of all it's elements en their indices and we have an additive group table where the elements are identified by it's indices)
1.7.2 Constructing the needed additive group table
Calculate 1 row by using the list of all the elements and their indices. (Easiest on row index = 0)
Calculate the following rows by doing the previous cel +1 mod (q-1) and shift 1 position
Example: Calculate the group table for F4, μ=x2+x+1 and ω=x
1) Calculate the elements in this group (see 1.1) F4 = 22, so 4 elements. These elements are:
$$ \begin{matrix} 0 & 1 \newline 1 & x + 1 \end{matrix} $$
2) Assign indices to every element and put them as the x and y value for the table (or more dimensions if n > 2) (Note: if element is 0, then indice is ∞! (This is because an element with itsself will return 0 after modulo))
I wrote the x and y indices in bold.
||⊕|0|1|2|∞| | :-: | :-: | :---: | :---: | :---: | :---: | |1|0| |x|1| |x+1|2| |0|∞|
3) Once we found the indices, we can just do the addition for the elements
For the first row this becomes:
indice 0 and 0 = ∞, same indices is ∞)
indice 1 and indice 0 = 1 + x = x + 1 = indice 2
indice 2 and indice 0 = x + 1 + 1 = x + 2 = x (since modulo 2) = indice 1
indice ∞ and indice 0 = 0 + 1 = 1 = indice 0
||⊕|0|1|2|∞| | :-: | :-: | :---: | :---: | :---: | :---: | |1|0|∞|2|1|0 |x|1|2|∞|0|1 |x+1|2|1|0|∞|2 |0|∞|0|1|2|∞
1.7.3 Constructing the needed multiplicative table
We do the same steps as the method for constructing the additive group, only this time we will do a multiplicative operation instead of an additive
Example: Multiplicative table for F>sub>4, μ=x2+x+1 and ω=x
2) Once we found the indices (See step 2 for the additive group table in 1.7.2), we can just do the multiplication for the elements
||⊕|0|1|2|∞| | :-: | :-: | :---: | :---: | :---: | :---: | |1|0|0|1|2|∞ |x|1|1|0|0|∞ |x+1|2|2|0|0|∞ |0|∞|∞|∞|∞|∞
> We can clearly see that we got more 0's in this table. This is because when we multiply the indices together, and we get a value that is not in our table. Then we write ∞
Concluding
There is only a field with order q if q is the power of a prime number!
But how can we find finite cyclical groups where we can write the number of elements as pn - 1?
This solution will be provided in part 3
Fields - Elliptical Curves Over Finite Fields
Fields - Prime Fields | CommonCrawl |
Whittaker function
In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by Whittaker (1903) to make the formulas involving the solutions more symmetric. More generally, Jacquet (1966, 1967) introduced Whittaker functions of reductive groups over local fields, where the functions studied by Whittaker are essentially the case where the local field is the real numbers and the group is SL2(R).
Whittaker's equation is
${\frac {d^{2}w}{dz^{2}}}+\left(-{\frac {1}{4}}+{\frac {\kappa }{z}}+{\frac {1/4-\mu ^{2}}{z^{2}}}\right)w=0.$
It has a regular singular point at 0 and an irregular singular point at ∞. Two solutions are given by the Whittaker functions Mκ,μ(z), Wκ,μ(z), defined in terms of Kummer's confluent hypergeometric functions M and U by
$M_{\kappa ,\mu }\left(z\right)=\exp \left(-z/2\right)z^{\mu +{\tfrac {1}{2}}}M\left(\mu -\kappa +{\tfrac {1}{2}},1+2\mu ,z\right)$
$W_{\kappa ,\mu }\left(z\right)=\exp \left(-z/2\right)z^{\mu +{\tfrac {1}{2}}}U\left(\mu -\kappa +{\tfrac {1}{2}},1+2\mu ,z\right).$
The Whittaker function $W_{\kappa ,\mu }(z)$ is the same as those with opposite values of μ, in other words considered as a function of μ at fixed κ and z it is even functions. When κ and z are real, the functions give real values for real and imaginary values of μ. These functions of μ play a role in so-called Kummer spaces.[1]
Whittaker functions appear as coefficients of certain representations of the group SL2(R), called Whittaker models.
References
1. Louis de Branges (1968). Hilbert spaces of entire functions. Prentice-Hall. ASIN B0006BUXNM. Sections 55-57.
• Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 13". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. pp. 504, 537. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253. See also chapter 14.
• Bateman, Harry (1953), Higher transcendental functions (PDF), vol. 1, McGraw-Hill.
• Brychkov, Yu.A.; Prudnikov, A.P. (2001) [1994], "Whittaker function", Encyclopedia of Mathematics, EMS Press.
• Daalhuis, Adri B. Olde (2010), "Whittaker function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
• Jacquet, Hervé (1966), "Une interprétation géométrique et une généralisation P-adique des fonctions de Whittaker en théorie des groupes semi-simples", Comptes Rendus de l'Académie des Sciences, Série A et B, 262: A943–A945, ISSN 0151-0509, MR 0200390
• Jacquet, Hervé (1967), "Fonctions de Whittaker associées aux groupes de Chevalley", Bulletin de la Société Mathématique de France, 95: 243–309, doi:10.24033/bsmf.1654, ISSN 0037-9484, MR 0271275
• Rozov, N.Kh. (2001) [1994], "Whittaker equation", Encyclopedia of Mathematics, EMS Press.
• Slater, Lucy Joan (1960), Confluent hypergeometric functions, Cambridge University Press, MR 0107026.
• Whittaker, Edmund T. (1903), "An expression of certain known functions as generalized hypergeometric functions", Bulletin of the A.M.S., Providence, R.I.: American Mathematical Society, 10 (3): 125–134, doi:10.1090/S0002-9904-1903-01077-5
Further reading
• Hatamzadeh-Varmazyar, Saeed; Masouri, Zahra (2012-11-01). "A fast numerical method for analysis of one- and two-dimensional electromagnetic scattering using a set of cardinal functions". Engineering Analysis with Boundary Elements. 36 (11): 1631–1639. doi:10.1016/j.enganabound.2012.04.014. ISSN 0955-7997.
• Gerasimov, A. A.; Lebedev, Dmitrii R.; Oblezin, Sergei V. (2012). "New integral representations of Whittaker functions for classical Lie groups". Russian Mathematical Surveys. 67 (1): 1–92. arXiv:0705.2886. Bibcode:2012RuMaS..67....1G. doi:10.1070/RM2012v067n01ABEH004776. ISSN 0036-0279.
• Baudoin, Fabrice; O’Connell, Neil (2011). "Exponential functionals of brownian motion and class-one Whittaker functions". Annales de l'Institut Henri Poincaré, Probabilités et Statistiques. 47 (4): 1096–1120. Bibcode:2011AIHPB..47.1096B. doi:10.1214/10-AIHP401. S2CID 113388.
• McKee, Mark (April 2009). "An Infinite Order Whittaker Function". Canadian Journal of Mathematics. 61 (2): 373–381. doi:10.4153/CJM-2009-019-x. ISSN 0008-414X. S2CID 55587239.
• Mathai, A. M.; Pederzoli, Giorgio (1997-03-01). "Some properties of matrix-variate Laplace transforms and matrix-variate Whittaker functions". Linear Algebra and Its Applications. 253 (1): 209–226. doi:10.1016/0024-3795(95)00705-9. ISSN 0024-3795.
• Whittaker, J. M. (May 1927). "On the Cardinal Function of Interpolation Theory". Proceedings of the Edinburgh Mathematical Society. 1 (1): 41–46. doi:10.1017/S0013091500007318. ISSN 1464-3839.
• Cherednik, Ivan (2009). "Whittaker Limits of Difference Spherical Functions". International Mathematics Research Notices. 2009 (20): 3793–3842. arXiv:0807.2155. doi:10.1093/imrn/rnp065. ISSN 1687-0247. S2CID 6253357.
• Slater, L. J. (October 1954). "Expansions of generalized Whittaker functions". Mathematical Proceedings of the Cambridge Philosophical Society. 50 (4): 628–631. Bibcode:1954PCPS...50..628S. doi:10.1017/S0305004100029765. ISSN 1469-8064. S2CID 122348447.
• Etingof, Pavel (1999-01-12). "Whittaker functions on quantum groups and q-deformed Toda operators". arXiv:math/9901053.
• McNamara, Peter J. (2011-01-15). "Metaplectic Whittaker functions and crystal bases". Duke Mathematical Journal. 156 (1): 1–31. arXiv:0907.2675. doi:10.1215/00127094-2010-064. ISSN 0012-7094. S2CID 979197.
• Mathai, A. M.; Pederzoli, Giorgio (1998-01-15). "A whittaker function of matrix argument". Linear Algebra and Its Applications. 269 (1): 91–103. doi:10.1016/S0024-3795(97)00059-1. ISSN 0024-3795.
• Frenkel, E.; Gaitsgory, D.; Kazhdan, D.; Vilonen, K. (1998). "Geometric realization of Whittaker functions and the Langlands conjecture". Journal of the American Mathematical Society. 11 (2): 451–484. doi:10.1090/S0894-0347-98-00260-4. ISSN 0894-0347. S2CID 13221400.
| Wikipedia |
\begin{document}
\title {Rank 4 vector bundles on the quintic threefold} \author{Carlo Madonna \footnote{Dipartimento di Matematica, Universit\`a degli Studi di Roma "La Sapienza", P.le A.Moro 1, 00185 Roma, Italia. email: [email protected]}} \date{} \maketitle \begin{abstract} By the results of the author and Chiantini in \cite{cm1}, on a general quintic threefold $X \subset \Pj^4$ the minimum integer $p$ for which there exists a positive dimensional family of irreducible rank $p$ vector bundles on $X$ without intermediate cohomology is at least three. In this paper we show that $p \leq 4$, by constructing series of positive dimensional families of rank $4$ vector bundles on $X$ without intermediate cohomology. The general member of such family is an indecomposable bundle from the extension class $Ext^1(E,F)$, for a suitable choice of the rank $2$ ACM bundles $E$ and $F$ on $X$. The existence of such bundles of rank $p = 3$ remains under question. \end{abstract}
\section{Introduction} \label{S:intro}
Let $X \subset \Pj^4$ be a smooth quintic hypersurface and let $E$ be a rank $2$ vector bundle without intermediate cohomology, i.e. such that \begin{equation} \label{eq:int} h^i(X,E(n))=0 \end{equation} for all $n \in \ZZ$ and $i=1,2$. In \cite{mad3} we found all the possible Chern classes of an indecomposable rank $2$ vector bundle satisfying condition (\ref{eq:int}). Moreover in \cite{cm1} we showed, when $X$ is general, if such bundles exist then they are all infinitesimally rigid, i.e. $Ext^1(E,E)=0$.
On the other hand it was showed in \cite{BGS} the existence of infinitely many isomorphism classes of irreducible vector bundles without intermediate cohomology on any smooth hypersurface $X_r$ of degree $r \geq 3$ in $\Pj^4$. It can be checked that when the hypersurface is general then the rank of these bundles is $2^3$. Hence we introduced in \cite{cm1} the number $$ BGS(X_r) $$ defined as the minimum positive integer $p$ for which there exists a positive dimensional family of irreducible rank $p$ vector bundles without intermediate cohomology on $X_r$.
Then combining the above quoted results we get, on a general quintic $X$, that \begin{equation} 3 \leq BGS(X) \leq 8. \end{equation}
In this paper we show the following:
\begin{theorem} \label{th:main} If $X$ is general then $BGS(X) \leq 4$. \end{theorem}
We should then answer the following:
\begin{question} Let $X$ be a general quintic hypersurface in $\Pj^4$. Could it be $BGS(X)=3$? \end{question}
To show our main result we give examples of rank $4$ vector bundles without intermediate cohomology, which are not infinitesimally rigid.
The examples are constructed by means of extension classes \begin{equation} 0 \to E_1 \to \E \to E_2 \to 0 \end{equation} i.e. elements in $Ext^1(E_2,E_1)$, where $E_1$ and $E_2$ are rank $2$ bundles on $X$. When the bundles $E_1$ and $E_2$ are not split then $\E$ has not trivial summand. Moreover for a suitable choice of bundles $E_1$ and $E_2$, there exists a non trivial extension class such that the rank $4$ bundle $\E$ which corresponds to this class does not split as a direct sum of two rank $2$ bundles, for reason of Chern classes. Of course if $E_1$ and $E_2$ have no intermediate cohomology it is so also for $\E$. We then conclude by direct calculations to make the right choice of bundles $E_1$ and $E_2$.
\section{Generalities}
We work over the complex numbers $\CC$ and we denote by $X \subset \Pj^4$ a smooth hypersurface of degree $5$ in $\Pj^4$. Since $Pic(X) \cong \ZZ[H]$ is generated by the class of a hyperplane section, given the vector bundle $E$ we identify $c_1(E)$ with the integer number $c_1$ which corresponds to $c_1(E)$ under the above isomorphism. We identify $c_2$ with $\deg c_2(E)=c_2(E)\cdot H$. If $E$ is a rank $k$ vector bundle on $X$ we denote by $E(n)=E \otimes \Oc_X(n)$.
\begin{definition} A rank $k$ vector bundle $E$ is called arithmetically Cohen-Macaulay (ACM for short) if $E$ has no intermediate cohomology, i.e. \begin{equation} h^i(E(n))=0 \end{equation} for all $i=1,2$, and $n \in \ZZ$. \end{definition}
Theorem \ref{th:main} will follow by:
\begin{proposition} \label{prop:main} Let $X$ be a smooth quintic hypersurface in $\Pj^4$. Then, there exist indecomposable rank $2$ vector bundles $E_1$ and $E_2$ on $X$ without intermediate cohomology such that there exists an open subset of a positive dimensional projective space parameterizing extension classes $Ext^1(E_2,E_1)$ which correspond to infinitely many isomorphism classes of irreducible rank $4$ vector bundles $\E$ on $X$ without intermediate cohomology. \end{proposition}
A proof of previous proposition will be given in the next section.
We will frequently use the following version of Riemann-Roch theorem for vector bundles:
\begin{RR} \label{thm:rr} If $\E$ is a rank $2$ vector bundle on a smooth hypersurface $X \subset \Pj^4$ of degree $5$ with Chern classes $c_i(\E)=c_i \in \ZZ$ for $i=1,2$, then \begin{equation} \label{eq:RRk} \begin{aligned} \chi(\E) =\frac56 c_1^3-\frac12 c_1c_2+\frac{25}{6}c_1 \end{aligned} \end{equation} \end{RR}
\section{The examples}
In this section we will give a proof of Proposition \ref{prop:main} which is a direct consequence of Proposition \ref{prop:list} and Theorem \ref{thm:main} below. As in \cite{cm1} given a rank $2$ vector bundle $E$ we introduce the non negative integer \begin{equation} \label{eq:norm} b(E)=\max \{ n \mid h^0(E(-n)) \ne 0 \}. \end{equation} We say that the vector bundle $E$ is {\it normalized} if $b(E)=0$. Notice that changing $E$ by $E(-b)$ we may always assume that $E$ is normalized. The rank two bundle $E$ is {\it semistable} if $2b-c_1 \leq 0$. If $2b-c_1<0$ then $E$ is {\it stable}.
All the possible Chern classes of irreducible rank 2 ACM bundles are listed in the following (see \cite{mad3} and \cite{cm1}):
\begin{proposition} \label{prop:list} Let $E$ be a normalized and indecomposable rank $2$ ACM bundle on a smooth quintic $X$. Then $$ (c_1,c_2) \in A \cup B $$ where $$ A=\{ (-2,1), (-1,2), (0,3), (0,4), (0,5), (1,4), (1,6), (1,8), (4,30) \} $$ and $B=\{(2,\alpha), (3,20) \}$ with $\alpha=11,12,13,14$. When $X$ is general, all the case in $A$ arise on $X$ and moreover for all the pairs $(c_1,c_2)\in A \cup B$ the corresponding rank $2$ ACM bundles are infinitesimally rigid i.e. $Ext^1(E,E)=0$. \end{proposition}
Below we shall construct examples of rank $4$ bundles $\G$ as extensions of type \begin{equation} \label{eq:ext} 0 \to F(m) \to \G \to E \to 0, \end{equation} where $m \le 0$, and $F$ and $E$ are indecomposable and normalized rank 2 ACM bundles on $X$ with Chern classes as in Proposition \ref{prop:list}.
Such nontrivial extensions $\G$ will exist whenever the extension space $Ext^1(E,F(m))$ has positive dimension, i.e. $h^1(F(m) \otimes E^{\vee}) > 0$. By the long exact sequence of cohomology of (\ref{eq:ext}), any such extension $\G$ has vanishing intermediate cohomology since $F$ and $E$ are ACM.
\begin{lemma} Let $E$ and $F$ be two normalized and indecomposable rank 2 ACM bundles on the smooth quintic $X$, and suppose that $h^0(F^{\vee}(c_1(E)-m))=0$ (hence $c_1(E)-c_1(F)-m<0$ since $F$ is normalized). Then for any zero-locus $C \subset X$ of a global section of $E$ $$h^0(\Ii_C(c_1(E)) \otimes F^{\vee}(-m))=0.$$
Moreover, if $h^0(F^{\vee}(-m))=0$ (hence $-m-c_1(F)<0$ since $F$ is normalized) then $$h^3(F(m) \otimes E^{\vee})=h^0(E \otimes F^{\vee}(-m))=0.$$ \end{lemma}
\begin{proof} From the tensored by $F^{\vee}(-m+c_1(E))$ ideal sheaf sequence of $C \subset X$: $$ 0 \to \Ii_C(c_1(E)) \otimes F^{\vee}(-m) \to F^{\vee}(c_1(E)-m) \to \Oc_C(c_1(E)) \otimes F^{\vee}(-m) \to 0 $$ we get $h^0(\Ii_C(c_1(E)) \otimes F^{\vee}(-m)) \leq h^0(F^{\vee}(c_1(E)-m))=0$. The rank 2 bundle $E$ fits in the exact sequence \begin{equation} \label{eq:kosz} 0 \to \Oc_X \to E \to \Ii_C(c_1(E)) \to 0; \end{equation} and after tensoring (\ref{eq:kosz}) by $F^{\vee}(-m)$ we get $$ 0 \to F^{\vee}(-m) \to E \otimes F^{\vee}(-m) \to \Ii_C(c_1(E)) \otimes F^{\vee}(-m) \to 0. $$
Therefore, since $h^0(F^{\vee}(-m))=h^0(\Ii_C(c_1(E)) \otimes F^{\vee}(-m))=0$ then $$ h^0(E \otimes F^{\vee}(-m))=0, $$ and by duality $h^3(F(m) \otimes E^{\vee})=0$. \end{proof}
\begin{remark} \label{rem:l1} \normalfont{ Let $E$ and $F$ be in (\ref{eq:ext}), and suppose that $\chi(F(m) \otimes E^{\vee})<0$. Then by the above lemma, the space of extensions (\ref{eq:ext}) will be no-empty since $h^1(F(m) \otimes E^{\vee})=h^0(F(m) \otimes E^{\vee})+ h^2(F(m) \otimes E^{\vee})-\chi(F(m) \otimes E^{\vee})>0$.
More generally the argument used here works whenever $$ h^3(F(m) \otimes E^{\vee})<-\chi(F(m) \otimes E^{\vee}). $$ In the following table we summarize the cases, which we are interested in, depending on the Chern classes of the bundles $E$ and $F$. To get the value of $\chi(F(m) \otimes E^{\vee})$ we used Schubert package (see \cite{ks}), and then by the Lemma we derived the lower bound for $d$.
\par
\begin{tabular}{llllll} Case & $(c_1(F),c_2(F))$ & $(c_1(E), c_2(E))$ & $\chi(F(m) \otimes E^{\vee})$ & $m$ & $d$ \cr (1) & (4,30) & (1,8) & $-14$ & $0$ & $>14$ \cr (2) & (4,30) & (0,3) & $-6$ & $-1$ & $>6$ \cr (3) & (4,30) & (0,4) & $-8$ & $-1$ & $>8$ \cr (4) & (4,30) & (0,5) & $-10$ & $-1$ & $>10$ \cr (5) & (1,8) & (0,3) & $-1$ & $0$ & $>1$ \cr (6) & (1,8) & (0,4) & $-2$ & $0$ & $>2$ \cr (7) & (1,8) & (0,5) & $-3$ & $0$ & $>3$ \end{tabular} \par
} \end{remark}
We are now ready to show the following:
\begin{theorem} \label{thm:main} Let $X$ be a smooth quintic in $\Pj^4$, and let $E,F,m,d$ be as in the above table. Then in each of the cases $(1)-(7)$ there exists a d--dimensional parameter space of extensions (\ref{eq:ext}), with a general element $\G$ an indecomposable rank $4$ vector bundle on $X$ without intermediate cohomology. \end{theorem}
\begin{proof} For $F,E$ as in the above table, the dimension $d =\dim Ext^1(E,F(m))$ is always $d>1$. Therefore for such $F,E$ there exist nontrivial extensions given by (\ref{eq:ext}), and let $\G$ be one of them.
Since $E$ and $F$ are ACM then by the cohomology sequence of (\ref{eq:ext}) $\G$ is without intermediate cohomology, and by Remark \ref{rem:l1} we need only to show that $\G$ is indecomposable. \par Suppose the contrary, i.e. that $\G$ splits. Then either \par
(i) $\G=\Oc_X(a) \oplus \G_1$ for $a \in \ZZ$ and $\G_1$ a rank 3 bundle without intermediate cohomology, or
(ii) $\G = \G_1 \oplus \G_2$ for two rank 2 ACM bundles $\G_1$ and $\G_2$. \par
We show that under the conditions of the theorem both cases (i) and (ii) are impossible.
\noindent Let us start with case (i). In this case the exact sequence (\ref{eq:ext}) reads as
\begin{equation} \label{eq:ext1} \begin{CD} 0 @>{}>> F(m) @>f>> \Oc_X(a) \oplus \G_1 @>g>> E @>{}>> 0. \end{CD} \end{equation}
We use the following (see below for a proof)
\begin{Lemma} \label{cl:cl1} Under the above conditions either $h^0E(-a)=0$ or $h^0 F(-m-c_1(F)+a)=0$. \end{Lemma}
Suppose $h^0(E(-a))=0$. Then by the exact sequence (\ref{eq:ext1}) tensorized by $\Oc_X(-a)$ we have $h^0(F(m-a))>0$. Let $s$ be a non trivial global section of $F(m-a)$, then we have a map $$ s:\Oc_X(a) \to F(m). $$ Let $j:\Oc_X(a) \oplus \G_1 \to \Oc_X(a)$ be the projection. Then we have the composition map $$ \varphi:=j \circ f \circ s: \Oc_X(a) \to F(m) \to \Oc_X(a). $$ Then $\varphi \in H^0 \Oc_X \cong \CC$ and hence it is either the identity map or the zero map. If this map is the identity then $j \circ f$ is surjective and hence $ker(j\circ f)\cong \Oc_X(b)$ for some $b \in \ZZ$. Then we have exact sequence $$ 0 \to \Oc_X(b) \to F(m) \to \Oc_X(a) \to 0 $$ and $F(m)$, and hence also $F$, splits since $\dim Ext^1(\Oc_X(a),\Oc_X(b))=0$, which is absurd.
Now suppose $\varphi$ is zero. Then $j \circ f$ is zero. Thus the image of $F(m)$ in exact sequence (\ref{eq:ext1}) is contained in $\G_1$. Then the kernel of $g$ is contained in $\G_1$, being equal to the image of $f$. Let $i:\Oc_X(a) \to \Oc_X(a) \oplus \G_1$ be the inclusion. By the assumption the map $g \circ i:\Oc_X(a) \to E$ is the zero map, which means that $\ker g$ is not contained in $\G_1$, which is absurd. \par
Suppose now that $h^0F(-m-c_1(F)+a)=0$ and consider the dual exact sequence of exact sequence (\ref{eq:ext1}) \begin{equation} 0 \to E^{\vee} \to \Oc_X(-a) \oplus \G_1^{\vee} \to F^{\vee}(-m) \to 0. \end{equation} Set $c=c_1(F)$ and $c'=c_1(E)$. Since $E^{\vee}\cong E(-c')$ and $F^{\vee}(-m)\cong F(-c-m)$ the above exact sequence reads as $$ 0 \to E(-c') \to \Oc_X(-a) \oplus \G_1^{\vee} \to F(-c-m) \to 0. $$ This exact sequence tensorized by $\Oc_X(a)$ reads as $$ 0 \to E(-c'+a) \to \Oc_X \oplus \G_1^{\vee}(a) \to F(-c-m+a) \to 0. $$ Then $h^0E(-c'+a)>0$ and a non trivial global section $s$ of $E(-c'+a)$ gives a non zero map $$ s:\Oc_X(-a) \to E(-c'). $$ Arguing as above this implies that $E$ splits which is absurd. \par
Then to finish the proof that case (i) can not arise we have to show the lemma. \par
\noindent{\bf Proof of the Lemma.} If $h^0 E(-a)>0$, since $E$ is normalized then $-a \geq 0$ i.e. $a \leq 0$. Suppose that $h^0F(-m-c+a)>0$. Since $F$ is normalized then $-m-c+a \geq 0$. Then from conditions $-m-c+a \geq 0$ and $-a \geq 0$ we derive condition $c+m \leq 0$ which is absurd since by hypotheses we have condition $c+m>0$ (see the table). \qed \par
To show the theorem it remains now to consider the case (ii) i.e. when $\G$ has an indecomposable summand which is ACM of rank equal to $2$, i.e. when \begin{equation} \label{eq:spl} \G=\G_1 \oplus \G_2 \end{equation} with both $\G_i$ ACM of rank equal to 2. Of course, we may assume that $\G_i$ are both indecomposable otherwise we reduce to the case (i) above. Then we have non trivial extension class \begin{equation} \label{eq:ext2} \begin{CD} 0 @>{}>> F(m) @>f>> \G_2 \oplus \G_1 @>g>> E @>{}>> 0. \end{CD} \end{equation}
The extension class (\ref{eq:ext2}) is non trivial by assumption. Moreover one has \begin{equation} \label{eq:cl} c_1(\G_i) \notin \{c_1(\F(m),c_1(E)\} \end{equation} for $i=1,2$, by the corollary to Lemma 1.2.8 in \cite{OSS}. Indeed, suppose that
$c_1(\G_i) \in \{c_1(F(m)),c_1(E)\}$ for at least on $i=1,2$. Here we note that at least one of the bundles $F(m)$ and $E$ is stable. Hence $\G_i$'s are semistable and one of these is always stable. Then from the above exact sequence we have map between semistable bundles of the same rank with the same first Chern class where at least one is stable. Therefore this map is an isomorphism and hence the extension class is trivial, which is absurd.
Then to show that the splitting of (\ref{eq:spl}) can not arise we will use Proposition \ref{prop:list} and a direct computation on the Chern classes. It will show that the only possibility is that the extension class (\ref{eq:ext1}) is trivial, which is absurd, since by assumption $\G$ is represented by a non trivial class in $Ext^1(E,F(m))$.
To start with, we notice that the bundle $\G$ of (\ref{eq:ext1}) is normalized since so are $F$ and $E$, and $m \leq 0$. In particular also $\G_1$ and $\G_2$ are normalized. \par Then we consider all the possible splitting type of $\G$ under condition (\ref{eq:cl}) in all the cases (1)-(7) of the table. Some of these decomposition are easy to show to be impossible, so we give here only the cases, which require some more computations. \par
Case (1). In this case (see the table) $$ (c_1(\G),c_2(\G))=(5,58). $$ By Proposition \ref{prop:list} and by condition (\ref{eq:cl}) if $\G \cong \G_1\oplus \G_2$ splits then $c_2(\G_1)=20$ and $c_2(\G_2)=\alpha$, with $\alpha=11,12,13,14$, are the only possible cases. A direct calculation on the Chern classes shows in these cases $(c_1(\G),c_2(\G))\ne (5,58)$.\par Case (2). In this case $$ (c_1(\G),c_2(\G))=(2,18). $$ By Proposition \ref{prop:list} and by condition (\ref{eq:cl}) if $\G \cong \G_1 \oplus \G_2$ then we have only two possibilities: either $(c_1(\G_1),c_2(\G_1))=(2,14)$ and $(c_1(\G_2),c_2(\G_2))=(0,4)$ or $(c_1(\G_1),c_2(\G_1))=(2,13)$ and $(c_1(\G_2),c_2(\G_2))=(0,5)$. The first case is impossible since by Riemann-Roch theorem we have $h^0F(-1)+h^0(E)=1<h^0(\G_1)+h^0(\G_2)=2$. The second case is also impossible since one computes $h^0(\G_1)+h^0(\G_2)=3$. \par Case (3). In this case we have $$ (c_1(\G),c_2(\G))=(2,19). $$ If $\G \cong \G_1 \oplus \G_2$ by Proposition \ref{prop:list} and by condition (\ref{eq:cl}) we could have possible cases $(c_1(\G_1),c_2(\G_1))=(1,6)$ and $(c_1(\G_2),c_2(\G_2))=(1,8)$ or $(c_1(\G_1),c_2(\G_1))=(2,14)$ and $(c_1(\G_2),c_2(\G_2))=(0,5)$. In the first case by Riemann-Roch we compute $0=h^0F(-1)+h^0E < h^0\G_1+h^0\G_2=3$. In the second case one concludes in similar way since $h^0\G_1+h^0\G_2=1$. One concludes in similar way for the other cases. \par Case (4). In this case we have $$ (c_1(\G_1),c_2(\G_2))=(2,20). $$ If $\G \cong \G_1 \oplus \G_2$ by Proposition \ref{prop:list} and by condition (\ref{eq:cl}) we could have possible cases $(c_1(\G_1),c_2(\G_1))=(1,\alpha)$ and $(c_1(\G_2),c_2(\G_2))=(1,\alpha')$ with $\alpha,\alpha'=4,6,8$ and $\alpha+\alpha'=15$ which is impossible. \par Cases (5)--(7). In this case we have $$ (c_1(\G_1),c_2(\G_2))=(1,\alpha+8) $$ where $\alpha=3,4,5$. If $\G \cong \G_1 \oplus \G_2$ by Proposition \ref{prop:list} and by condition (\ref{eq:cl}) soon the conclusion follows.
\end{proof}
\end{document} | arXiv |
Defect Formation Mechanisms in Selective Laser Melting: A Review
Bi Zhang ORCID: orcid.org/0000-0001-5738-32661,2,
Yongtao Li1 &
Qian Bai1
Chinese Journal of Mechanical Engineering volume 30, pages515–527(2017)Cite this article
This article has been updated
Defect formation is a common problem in selective laser melting (SLM). This paper provides a review of defect formation mechanisms in SLM. It summarizes the recent research outcomes on defect findings and classification, analyzes formation mechanisms of the common defects, such as porosities, incomplete fusion holes, and cracks. The paper discusses the effect of the process parameters on defect formation and the impact of defect formation on the mechanical properties of a fabricated part. Based on the discussion, the paper proposes strategies for defect suppression and control in SLM.
Additive manufacturing (AM) is an approach in which a part is manufactured layer by layer from the data of a 3D model. AM is a "bottom-up" approach as opposed to the traditional subtractive manufacturing that is often referred to as the "top-down" approach [1, 2]. The AM approach does not require the traditional tools, fixtures and complicated procedures. Therefore, it can offer an advantage of economically fabricating a customized part with complex geometries in a rapid design-to-manufacture cycle. With the development of high energy beams, it becomes possible to manufacture metal parts of high performance. Due to its unique advantages, the AM approach has been widely applied in many industries, such as aerospace, medical devices, military and automobile [3–5].
Selective Laser Melting (SLM) is one of the additive manufacturing processes. It is relatively mature and has been a research focus in manufacturing metallic parts [6]. A schematic layout of a typical SLM setup is presented in Fig. 1 [7]. During the SLM process, data is provided from a CAD model which is then sliced into thin layers. Each sliced layer is further developed with the appropriate scan paths. Through the scanner mirrors, a laser beam selectively scans and melts the powders that are previously paved on the substrate according to the developed scan paths. After a layer is finished, the building platform is lowered by an amount equal to the layer thickness, and a new layer of powders is paved. The process repeats until the completion of the whole part. To date, the SLM process is able to fabricate metallic parts from different material powders, such as titanium alloys [7, 8], nickel-based superalloys [9, 10], aluminum alloys [11, 12] and stainless steels [13, 14].
Schematic layout of a typical SLM setup [7]
Although the SLM process offers a great advantage in manufacturing complex parts at a high material utilization rate [15], it is affected by many factors, such as laser energy input and scan speed, scan strategy, powder material, powder size and morphology. The SLM process consists of complicated physics, such as absorption and transmission of laser energy [16], rapid melting and solidification of material, microstructure evolution [17, 18], flow in a molten pool [19], and materials evaporation [20]. The process is thus affected by the aforementioned factors to form defects of porosities, incomplete fusion holes, cracks, and impurities, etc. These defects are detrimental to a fabricated part in terms of its mechanical and physical properties, which in turn limits the application of SLM [21–24].
Since defect formation is a critical issue in an SLM process, research has been directed towards understanding and suppression of defect formation [7, 24–36]. This paper reviews the recent research outcomes on the types and formation mechanisms of the common SLM defects, such as porosities, incomplete fusion holes, and cracks. The paper also reveals how the SLM defects may affect the mechanical properties of a fabricated part. Other defects, such as metallic inclusions, segregations, residual stresses, metallurgical imperfections may also have a significant impact on the mechanical properties of a fabricated part, their respective formation mechanisms will be reviewed in a separate paper and published elsewhere. Finally, the paper provides a reference for defect suppression and control in the SLM processes.
Defect Types
Many parameters are involved in an SLM process, such as laser power, scan speed, hatch spacing, layer thickness, powder materials and chamber environment. Defects are inevitably introduced if any of these parameters are improperly chosen. The common defects are classified in three types: porosities, incomplete fusion holes, and cracks.
Porosities
A porosity is usually small in size, typically less than 100 μm with an approximately spherical shape, as shown in Fig. 2 by arrows. The formation mechanisms of porosities are described as follows [7, 23, 24, 28].
Optical image of spherical porosities (marked by arrows) in an SLM part [7]
Firstly, if the packing density of metal powders is low, e.g., 50 percent, the gas present between the powder particles may dissolve in the molten pool. Because of the high cooling rate during the solidification process, the dissolved gas cannot come out of the surface of the molten pool before solidification takes place. Porosities are thus formed and remain in the fabricated part. Porosities may also be formed when metal powders of a hollow structure are utilized in an SLM process. On the other hand, the molten pool temperature is generally high due to the intense laser power. At this temperature, gas solubility in the liquid metal is high, making its enrichment easier. Furthermore, in the process of preparing powder materials, gas is inevitably introduced into the powder materials, especially the gas atomized powder materials in the scope of protection by an inert gas, such as argon or helium.
Qiu et al [28] observe that the porosities contain ridges in the internal surfaces and are thus probably associated with the incomplete re-melting of some local surfaces from the previous layers. The ridges form small volumes to which the molten metal is difficult to flow and penetrate. On the other hand, Gong et al [29] attribute these porosities to gas bubbles generated when a high laser energy is applied to the molten pool. Gas bubbles can be induced due to vaporization of low melting point constituents within an alloy. They can be far beneath the surface at the bottom of the molten pool. The high solidification rate of the molten pool does not give gas bubbles sufficient time to rise and escape from the surface. Thus, gas bubbles are trapped in the molten pool, resulting in defect inclusions of regular spherical porosities in the forming part.
It is therefore understood that such spherical porosities are generally resulted from the entrapped gases in the molten pool due to the excessive energy input or unstable process conditions. The spherical porosities are randomly distributed in an SLM fabricated part, and difficult to eliminate completely.
Incomplete Fusion Holes
Incomplete fusion holes, also known as lack-of-fusion (LOF) defects, are mainly due to the lack of energy input during an SLM process. The formation of the LOF defects is because the metal powders are not fully melted to deposit a new layer on the previous layer with a sufficient overlap [24, 29]. An LOF defect may contain numerous un-melted metal powders, as shown in Fig. 3(b) [37]. There are two types of LOF defects: (1) poor bonding defects due to insufficient molten metal during a solidification process, as in Fig. 3(a), and (2) defects with un-melted metal powders in Fig. 3(b).
Optical images of LOF defects in SLM fabricated parts: (a) poor bonding defects; (b) LOF defects with un-melted metal powders [37]
In the SLM process, a laser selectively melts the metal powders point by point, line by line, and layer by layer to complete the whole part. If the laser energy input is low, the width of the molten pool is small, which results in an insufficient overlap between the scan tracks. The insufficient overlap is a cause of formation of the un-melted powders between the scan tracks. In the deposition process of a new layer, it becomes difficult to fully re-melt these powders. As a consequence, incomplete fusion holes are formed and remain in the SLM fabricated part. Furthermore, if the laser energy input is too low to cause an enough penetration depth of the molten pool, LOF defects may be generated due to a poor interlayer bonding [24, 29, 37]. Therefore, LOF defects are usually distributed between the scan tracks and the deposited layers.
Moreover, in a location where defects have been generated, the surface of the location becomes rough. The rough surface directly contributes to the poor flow of the molten metal to form interlayer defects. The interlayer defects may gradually extend and propagate upwards to form large multi-layer defects in a continuous deposition process [38].
For the easily oxidized alloy materials, such as aluminum AlSi10Mg, a layer of oxide film is usually produced at the surface of a part with residual oxygen in the SLM process. Then wettability decreases and molten metal flow is blocked, leading to a poor bonding between the layers to form the incomplete fusion defects [25, 39]. Fig. 4 shows an image of an incomplete fusion defect. From the EDX (Energy Dispersive X-ray Spectrometer) data presented in Table 1, location 2 of an incomplete fusion defect is rich in oxygen, suggesting that this irregular defect should be associated with the presence of an oxide layer which could prevent the progress of bonding [39].
SEM image of an incomplete fusion defect at different locations [39]
Table 1 Element content analysis of the defect (wt. %)
In an SLM process, metal powders experience rapid melting and rapid solidification under a high local laser energy input. The cooling rate of the molten pool reaches 108 K/s [22], which creates a great temperature gradient and correspondingly a large residual thermal stress in the fabricated part. The high temperature gradient combined with the great residual stress often causes crack initiation and propagation in a fabricated part [22, 40, 41]. Fig. 5(a) shows the crack morphology in an SLM fabricated titanium part. Cracks are more prone to initiating from the as-built surface that is adhered with the partially melted metal powders. Fig. 5(b) shows the microstructure on both sides of a crack. It can be observed that elongated needle-type crystal grains are continued on the both sides of the crack, indicating a typical transgranular mode of cracking [40].
SEM images of crack morphology and microstructure from the cross section of an SLM fabricated Ti6Al4V part: (a) crack morphology; (b) microstructure on both sides of the crack [40]
For stainless steels and nickel-based superalloys, because of their low thermal conductivity and high thermal expansion coefficient, they are more vulnerable to generating cracks with high susceptibility to cracking in an SLM process [9, 27, 42, 43]. To solve this problem, pre-heating the substrate and improving the ambient temperature are recommended to reduce the cracks in the SLM fabricated parts [26, 27].
Effect of Process Factors on Defect Formation
Many process factors are involved in an SLM process. Some of the factors are process parameters that can be predetermined, while the others cannot be predetermined since they are generated from the SLM process. As described in Fig. 6, the major process factors can be classified into four types: laser-related, scan-related, powder-related, and temperature-related. Based on the principle that laser selectively melts the powders, the major factors which are related to defect formation in an SLM process are laser energy input, powder material, and scan strategy. Therefore, the following Sections 3.1-3.3 are dedicated to discussing defect formation in terms of the three factors.
Process factors involved in an SLM process [23]
Effect of Laser Energy Input
Laser energy input directly determines the melt condition of metal powders, the flow of molten metal, which has a significant impact on the type and size of the defects in an SLM process. The energy input in the material can be related to the main process parameters, such as laser power, scan speed, hatch spacing, and layer thickness.
At a relatively low scan speed and a high laser power, the energy input is high, more powders are melted at an elevated temperature, porosity defects are created. These defects can be attributed to the entrapped gas originated from the raw material powders in the SLM process as mentioned above. In addition, low melting point constituents, e.g., Al, Mg elements in the alloy, may evaporate into gas to form gas bubbles. During the rapid solidification process in SLM, the gas bubbles do not have sufficient time to escape from the molten pool to the pool surface. They remain within the molten pool to form porosity defects of a spherical shape [29, 44]. On the other hand, the molten pool becomes large if energy input is high, which causes powder denudation around the molten pool. The denudation process results in insufficient molten metal to fill the gap between the adjacent tracks. Large porosities are thus formed [7].
Furthermore, a relatively low scan speed and a high energy input may result in a high residual thermal stress in a rapid melting and solidification process. The higher the energy input, the more severe the contraction of the molten metal in the solidification process. A high residual stress is induced during the solidification process [22, 40, 45]. As shown in Fig. 7(a), with a high energy input, micro-cracks are observed in an SLM CP-Ti part. Conversely, almost no defects are found when an appropriate energy input is utilized, as shown in Fig. 7(b).
Optical images showing the microstructures on the cross-sections of SLM-processed Ti parts fabricated at different energy inputs: (a) cross-section with micro-cracks due to a higher energy input (P = 90 W, v = 100 mm/s); (b) nearly defect-free cross-section due to an appropriate energy input (P = 90 W, v = 200 mm/s) [22]
At a relatively high scan speed and a low laser power, the energy input is too low to fully melt the powders, generating a discontinuous molten pool. This makes it difficult to fully melt the powders between the adjacent tracks to form an effective overlap, resulting in the formation of incomplete fusion defects. In addition, if a large powder thickness causes an insufficient penetration of the laser energy input, an effective overlap may not be developed between layers, causing the formation of interlayer incomplete fusion defects [24, 29, 45, 46].
Fig. 8 shows two different defect types in the SLM fabricated titanium alloys under two different energy input conditions (laser power and other parameters remain constant, but scan speed varies) [47]. Fig. 8(a) describes the regular spherical defects under the higher energy input conditions. Conversely, Fig. 8(b) shows irregular incomplete fusion holes under the lower energy input conditions.
Optical images of defect morphologies at different energy inputs in the SLM-processed Ti6Al4V parts: (a) spherical porosities (P = 120 W, v = 40 mm/s); (b) incomplete fusion holes (P = 120 W, v = 1500 mm/s) [47]
Generally, energy density E is widely used to characterize energy input, which is a measure of the average applied energy per unit volume of the deposited material in an SLM process. Eq. (1) provides a representation of energy density E (J/mm3):
$$E = \frac{P}{v \times h \times t},$$
where P is laser power (W); v is scan speed (mm/s); h is hatch spacing (mm); and t is layer thickness (mm). The parameters in the equation reflect the impact of overlap between tracks, layer thickness and energy input and can easily be determined. Therefore, as a representation of energy input, this equation is used widely in the SLM processes [7, 29, 48].
Fig. 9 shows a scatter plot for both the void and defect fractions (%) as well as crack density that is represented by crack length per unit cross-sectional area (mm/mm2) against energy density E (J/mm3) in the SLM fabrication of high temperature Ni-superalloy and porosity in SLM Titanium parts [27]. As seen from the figure, with an increase in the energy density, more material is melted, void fraction is quickly reduced, especially when the energy density exceeds 70 J/mm3. A similar result can be acquired from the calculation results by different research groups on energy density calculation for the SLM fabricated titanium parts, but the appropriate energy density is different due to different materials [7, 22, 28, 47–52]. Conversely, the crack density shows a slight increase with the increase in energy density due to the large thermal stress caused by the excessive energy input.
Scatter plot of crack density and void fraction in SLM Ni-superalloy parts, and defect fraction in titanium parts against energy density; lines represent data trend
Therefore, as an integrated parameter, energy density represents the combined effect of the major process parameters on defect formation in an SLM process. Energy density is handy to use in selecting the appropriate laser power, scan speed, hatch spacing, layer thickness to minimize the defects and improve the manufacturing efficiency in the SLM process.
Effect of Powder Materials
The morphology and size of metal powders have a significant influence on the powder bed smoothness and powder flowability, thus are strictly required in an SLM process. Metal powders are produced in different methods, such as water atomization, gas atomization, plasma rotating electrode and electrolytic method, which has a diverse effect on defect formation [53–55]. In addition, the gas contained in the powders increases the probability of defect formation.
Wang et al [33] examined the effect of different powder sizes of the 316L stainless steel on the part quality in the SLM process. They reported that the metal powders of a smaller size tended to reduce porosities in the fabricated parts compared to those of a larger size. The relative density reached 99.75% with the average powder size of 26.36 μm, which was compared to 97.50% with the average powder size of 50.81 μm. Li et al [41] explored the densification behavior of gas and water atomized 316L stainless steel powders. As shown in Fig. 10, the gas atomized powders possessed spherical shapes compared to irregular shapes of the water atomized powders. The results demonstrated that the parts fabricated with the gas atomized powders acquired a higher relative density, less porosity compared to those with the water atomized powders, which can be attributed to the differences in morphology, packing density, flowability and oxygen contents between the two powders.
SEM images showing characteristic morphologies of stainless steel powders produced by: (a) gas atomization; (b) water atomization [41]
Effect of Scan Strategy
Scan strategy directly affects the heat transfer, powders melting and solidification, and ultimately defect location and distribution. Generally, three different scan strategies have been utilized in the SLM processes, namely "unidirectional", "zigzag", and "cross-hatching", as shown in Fig. 11 [7]. For the unidirectional and zigzag scan strategies, at the beginning and end of a scan track, laser power is usually unstable and scan speed is gradually reduced, which tends to result in a relatively higher laser energy input and defect formation [20, 56]. In addition, the impurities in the powders may also be pushed to the ends of a track in the densification process to form higher defect density. Actually, incomplete fusion defects are more frequently generated between the scan tracks and layers [57, 58]. Cross-hatching scan strategy can make the entire laser energy input more balanced in the whole layers, which effectively prevents defect accumulation and propagation.
Three different scan strategies: "unidirectional" (left), "zigzag" (center), and "cross-hatching" (right) [7]
The "island scan strategy" has been developed for parts fabrication, as illustrated in Fig. 12 [39]. Firstly, the filled layer is divided into several islands with each island being built randomly and continuously. Then the successive layers are displaced in a certain distance, so as to avoid the accumulation of defects in the same location. Furthermore, the residual thermal stress in the SLM fabricated parts can be more balanced to reduce cracks development. However, due to the problem of potentially unstable laser energy input and the change in scan orders, defects are generally formed at the border of small islands, which needs further improvements for the "island scan strategy".
Schematic illustration of the island scan strategy, (a) each layer is divided into islands and raster scanned; (b) the successive layers are displaced in 1 mm [39]
Yang et al [59, 60] applied the interlayer staggering and the orthogonal scan strategy to reducing the defects formed in the overlapping zone between tracks. As shown in Fig. 13, after one layer is completed, the laser scans the overlapping zone between the adjacent tracks to sufficiently melt powders in the next layer deposition. The orthogonal scan strategy is adopted in the next layer, so that energy input can be more balanced for reducing defects, as previously also mentioned.
Interlayer staggering and orthogonal scan strategy [60]
Influence of Defects on Mechanical Properties
Defects in an SLM process cause stress concentration in the fabricated part, which may lead to the part failure. When stress exceeds the material limit, a crack may form and gradually propagate in the part. The following Sections 4.1-4.2 are dedicated to discussing the influence of defects on the mechanical properties in the SLM parts.
Tensile Properties
As mentioned earlier, the metallic powders suffer a rapid melting and solidification in the SLM process, due to a large cooling rate, which produces a part with a finer grain microstructure and better tensile properties than those made of the traditional wrought counterparts [17, 30, 61, 62]. The tensile strength (TS), ultimate tensile strength (UTS), and elongation of the SLM titanium alloy parts are shown in Table 2. Both TS and UTS of the SLM titanium alloys are higher than their wrought counterpart, generally above 1,000 MPa. Therefore, the SLM titanium alloys can meet the tensile strength requirements for engineering applications. However, the elongations of the SLM titanium alloys are rather low (less than 10%), which may be attributed to the defects in the SLM parts.
Table 2 Tensile Properties of SLM-Fabricated Ti6Al4V Alloys and Wrought Counterpart
Furthermore, the SLM process has a directional effect on the properties of the forming parts due to its basic deposition principle. The directional effect is a direct cause of the severe anisotropy in the mechanical properties of the fabricated part. For a part fabricated based on the orthogonal scan strategy shown in Fig. 13, defects may be formed and distributed in the horizontal direction, resulting in the obvious reduction of the load-bearing cross-section area in the fabricated part [66]. If the loading direction coincides with the building direction, the part is more susceptible to failure, leading to a low strength of the part [24, 37, 67]. In addition, because of the epitaxial growth in the SLM process, the elongated columnar grains in the fabricated part also aggravate anisotropy of the part [28, 34].
Fatigue Properties
For an SLM fabricated part, defects are more detrimental to its fatigue strength due to the points of stress concentration. A defect often serves as a source of crack initiation and propagation, which may greatly reduce the fatigue strength of the part. The stochastic distribution of the defects also aggravates the scattering of fatigue life, which may severely restrict the application of the SLM fabrication.
Fig. 14 shows the results on fatigue life obtained from the literature on the SLM Ti6Al4V parts and their wrought counterparts [68–73]. In Fig. 14, the filled dots represent the parts that were not post-processed ("as-built"), while the unfilled dots were subjected to machining. Due to the presence of both internal and surface defects, the fatigue strength of the "as-built" SLM samples was approximately 200 MPa, far below that of their wrought counterparts. After machining, the SLM samples showed a slightly improved fatigue life, but still lower than that of the wrought counterparts.
Fatigue life of the SLM Ti6Al4V parts and their wrought counterparts
The morphology, number, size and location of defects all have a significant influence on the fatigue life of the SLM fabricated parts. Generally, the spherical defects have less influence on the fatigue life of a part due to their regular shapes and small size. On the other hand, the defects of an irregular shape (e.g., an incomplete fusion hole) promote stress concentration of a part so as to seriously reduce fatigue strength of the part because of the irregular shapes and larger sizes of the defects [37, 47].
Gong et al [47] tested the SLM fabricated parts containing different numbers and types of defects, and found that the spherical defects had less influence on the fatigue life when the level of such defects was less than 1%, as shown in Fig. 15. However, the fatigue life was considerably decreased when these defects were amounted at the level of 5% porosity. Conversely, the irregular defects were found significantly detrimental to the fatigue life even when present in an amount as low as 1% porosity. When a part containing such defects in a higher amount of 5% in porosity, the part tended to have a low fatigue life with a narrow trend of dispersion, which suggests that the defects be so seriously detrimental to the fatigue life of the part even the statistical nature of fatigue life is defeated.
Fatigue performance of the parts containing defects from the SLM process [47]
Kasperovich et al [65] tested the parts that were differently post-processed, namely, "as-built", "surface-machined", "heat-treated", "hot isostatically pressed" (HIP, which is used in the traditional powder metallurgy and foundry technology, allows not only to adjust the microstructure, but also to fuse un-melted particles and generate "kissing bonds"). As shown in Fig. 16, after machining, the surface microcrack sources of parts are removed, enhancing the fatigue cycle of a part. The heat treatment process merely improves the microstructure of the part, but does not reduce or eliminate the defects. In this regard, the process can hardly improve the fatigue life of the parts. However, the HIP process may collapse the defects up to a certain size at an elevated temperature and pressure, reducing the number and size of the defects, and therefore improving the fatigue life of a part.
Fatigue life of the parts subjected to different post-processes [63]
Leuders et al [52, 63, 73] studied the mechanical properties and the growth mechanisms of fatigue cracks in the SLM titanium parts. Their results indicated that defects had a major impact on the fatigue life of the parts, especially at the stage of fatigue crack initiation. Due to the presence of defects, stress concentration could occur, causing crack initiation and consequently a decrease in fatigue strength. Leuders et al also analyzed the effect of defect location on the fatigue strength in their research. When a defect was located near the surface of a part, its fatigue life was shorter in comparison with that located far from the surface, indicating that defect location is critical to the fatigue strength of the part. Surface treatment, such as machining and shot peening, can be adopted to suppress or eliminate the near-surface defects so as to enhance part fatigue strength.
However, since it is difficult to accurately control the type, number, and location of a defect in a fabricated part, the fatigue strength of a part can be in jeopardy. Therefore, the fatigue strength of an SLM fabricated part is still questionable and needs to be improved.
Strategies for Defect Suppression
Defect suppression is a challenging issue in the SLM process. Currently, there are two major strategies to suppress defect formation in the SLM processes, namely online detection and numerical simulation, in addition to machining to reduce or eliminate defects.
Clijsters et al [74] designed a high-speed and real-time molten pool monitoring system, consisting of four modules, namely optical set-up, data processing, reference data and quality estimation. For each layer of deposition, the information of molten pool in the form of a light signal was collected by sensors, then transferred to a data processing module to establish the molten pool image, then used to analyze the location and size of defects compared with reference data to get the characteristics of molten pool to deduce defect formation information. Finally, the analysis results were used for the feedback control to optimize the process, and to reduce defect formation in the SLM fabricated parts.
Panwisawas et al [75] established a mathematical model of thermal fluid dynamics to better understand the morphological evolution of porosity during an SLM process. According to the deposition mechanism of the heating-melting-solidification cycles of metal powders, a thermal fluid dynamics model based on the Navier-Stokes equation, surface tension, capillary force, and Marangoni effect was introduced to explore the evolution of porosity as the scan speed increased. The results showed that for a fixed laser input power, increasing scan speed reduced energy input density, resulting in serious unfused defects in the interlayers.
It is an effective research strategy to combine the three methods, i.e., the traditional optimization test, the numerical simulation calculation and the online detection, for a systematic study on the defect formation and control in the SLM processes. As shown in Fig. 17, online detection is conducted for obtaining information on defect morphology, location and dimensions through detection sensors, data processing, image analysis and feedback control. The detected defects can be eliminated by the successive subtractive process. On the other hand, the strategy should also investigate defect formation and evolution mechanisms, including material melt-flow behavior, solidification and shrinkage, the interaction effect of surface tension, capillary force and gravity, by using the numerical simulation method. Finally, combining the information on defect detection and defect formation mechanisms to further accomplish the process optimization to achieve the objective of defect suppression and control in an SLM process.
Schematic illustration of defect detection and control in an SLM process
Defect formation is a critical problem in the SLM processes. It has a significant influence on the real-world application of the SLM fabricated parts. This paper reviews defect formation mechanisms in SLM processes, discusses the effect of process parameters on defect formation, and proposes a strategy for defect suppression and control. Based on the review, the paper summarizes conclusions:
The common defects are three types, namely spherical porosities, irregularly incomplete fusion holes, and cracks. Spherical porosities are randomly distributed, while incomplete fusion holes are generally distributed between the tracks and layers.
Many process parameters, such as laser power, scan speed, hatch spacing, layer thickness, and scan strategy, have significant influences on the formation of defects. Energy density is an integrated parameter for controlling defect formation; scan strategy has a significant influence on the location distribution of defects, most of the defects distribute at both ends of scan tracks and in between two adjacent tracks.
Defect formation has a significant influence on the mechanical properties of the SLM fabricated parts, especially fatigue strength. Defects play a prominent role in fatigue crack initiation, directly reduce the fatigue life of a part, which restricts the application of the SLM technique.
The quality control in an SLM process relies on defect detection and elimination. For high quality SLM fabrications, defect monitoring, simulation and modeling, as well as real-time defect elimination become necessary. Defect-free SLM fabrications are anticipated in the near future.
An erratum to this article has been published.
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Key Laboratory for Precision and Non-traditional Machining Technology of Ministry of Education, Dalian University of Technology, Dalian, 116024, China
Bi Zhang
, Yongtao Li
& Qian Bai
Department of Mechanical Engineering, University of Connecticut, Storrs, CT, 06269, USA
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Correspondence to Bi Zhang.
Supported by National Natural Science Foundation of China (Grant No. 51605077), Science Challenge Project (Grant No. CKY2016212A506-0101) and Science Fund for Creative Research Groups of NSFC (Grant No. 51621064).
An erratum to this article is available at https://doi.org/10.1007/s10033-017-0184-3.
Zhang, B., Li, Y. & Bai, Q. Defect Formation Mechanisms in Selective Laser Melting: A Review. Chin. J. Mech. Eng. 30, 515–527 (2017) doi:10.1007/s10033-017-0121-5
Revised: 07 February 2017
Issue Date: May 2017 | CommonCrawl |
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Weakening convergence conditions of a potential reduction method for tensor complementarity problems
Xiaofei Liu and Yong Wang ,
School of Mathematics, Tianjin University, Tianjin 300350, China
* Corresponding author: Yong Wang
Received August 2020 Revised January 2021 Early access April 2021
Fund Project: The second author's work was supported by the National Natural Science Foundation of China (grant number 11871051)
Figure(1) / Table(3)
Recently, under the condition that the included tensor in the tensor complementarity problem is a diagonalizable and positive definite tensor, the convergence of a potential reduction method for tensor complementarity problems is verified in [a potential reduction method for tensor complementarity problems. Journal of Industrial and Management Optimization, 2019, 15(2): 429–443]. In this paper, we improve the convergence of this method in the sense that the condition we used is strictly weaker than the one used in the above reference. Preliminary numerical results indicate the effectiveness of the potential reduction method under the new condition.
Keywords: Tensor complementarity problem, potential reduction method, strong P tensor, positive definite tensor, convergence condition.
Mathematics Subject Classification: Primary: 15A69, 68W40, 90C33.
Citation: Xiaofei Liu, Yong Wang. Weakening convergence conditions of a potential reduction method for tensor complementarity problems. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021080
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Figure 1. The relationships among three classes of tensors
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Table 1. Numerical Results for Example 4.1
$ (z^{0})^{\top} $ $ \mathrm{Iter} $ $ \mathrm{Time(s)} $ $ (z^{*})^{\top} $
(1.2, 0.4, 1.6280, 0.1640) 40 0.3584 (0.4642, 0.0000, 0.0000, 0.1000)
(2.4, 0.8, 13.7240, 0.6120) 46 0.3714 (0.4642, 0.0000, 0.0000, 0.1000)
(4.8, 1.6,110.4920, 4.1960) 52 0.4612 (0.4642, 0.0000, 0.0000, 0.1000)
(7.2, 2.4,373.1480, 13.9240) 56 0.4293 (0.4642, 0.0000, 0.0000, 0.1000)
$ {q}^{\top} $ $ (z^{0})^{\top} $ $ \mathrm{Iter} $ $ \mathrm{Time(s)} $ $ (z^{*})^{\top} $
$ (-1, 1) $ (2, 1, 28, 2) 47 0.3207 (1, 0, 0, 1)
$ (-1, 1) $ (4.2, 2.1, 1183.3893, 41.8410) 67 0.4496 (1, 0, 0, 1)
$ (-6, -2) $ (2.8, 1.4,149.9690, 3.3782) 52 0.3200 (1.6438, 1.1487, 0, 0)
$ (-6, -2) $ (8, 4, 29690, 1022) 129 1.3179 (1.6438, 1.1487, 0, 0)
$ (36, -19) $ (4, 2,964, 13) 66 0.4528 (1.8384, 1.8020, 0, 0)
$ (36, -19) $ (24, 12, 7216164, 248813) 760 7.4286 (1.8384, 1.8020, 0, 0)
Kaili Zhang, Haibin Chen, Pengfei Zhao. A potential reduction method for tensor complementarity problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 429-443. doi: 10.3934/jimo.2018049
Mengmeng Zheng, Ying Zhang, Zheng-Hai Huang. Global error bounds for the tensor complementarity problem with a P-tensor. Journal of Industrial & Management Optimization, 2019, 15 (2) : 933-946. doi: 10.3934/jimo.2018078
Wanbin Tong, Hongjin He, Chen Ling, Liqun Qi. A nonmonotone spectral projected gradient method for tensor eigenvalue complementarity problems. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 425-437. doi: 10.3934/naco.2020042
ShiChun Lv, Shou-Qiang Du. A new smoothing spectral conjugate gradient method for solving tensor complementarity problems. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021150
Ya Li, ShouQiang Du, YuanYuan Chen. Modified spectral PRP conjugate gradient method for solving tensor eigenvalue complementarity problems. Journal of Industrial & Management Optimization, 2022, 18 (1) : 157-172. doi: 10.3934/jimo.2020147
Nicolas Van Goethem. The Frank tensor as a boundary condition in intrinsic linearized elasticity. Journal of Geometric Mechanics, 2016, 8 (4) : 391-411. doi: 10.3934/jgm.2016013
Fengming Ma, Yiju Wang, Hongge Zhao. A potential reduction method for the generalized linear complementarity problem over a polyhedral cone. Journal of Industrial & Management Optimization, 2010, 6 (1) : 259-267. doi: 10.3934/jimo.2010.6.259
Jan Boman, Vladimir Sharafutdinov. Stability estimates in tensor tomography. Inverse Problems & Imaging, 2018, 12 (5) : 1245-1262. doi: 10.3934/ipi.2018052
Shenglong Hu. A note on the solvability of a tensor equation. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021146
Michael Anderson, Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas and Michael Taylor. Metric tensor estimates, geometric convergence, and inverse boundary problems. Electronic Research Announcements, 2003, 9: 69-79.
Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617
Yanfei Wang, Dmitry Lukyanenko, Anatoly Yagola. Magnetic parameters inversion method with full tensor gradient data. Inverse Problems & Imaging, 2019, 13 (4) : 745-754. doi: 10.3934/ipi.2019034
H. M. Hastings, S. Silberger, M. T. Weiss, Y. Wu. A twisted tensor product on symbolic dynamical systems and the Ashley's problem. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 549-558. doi: 10.3934/dcds.2003.9.549
Meng Ding, Ting-Zhu Huang, Xi-Le Zhao, Michael K. Ng, Tian-Hui Ma. Tensor train rank minimization with nonlocal self-similarity for tensor completion. Inverse Problems & Imaging, 2021, 15 (3) : 475-498. doi: 10.3934/ipi.2021001
Yiju Wang, Guanglu Zhou, Louis Caccetta. Nonsingular $H$-tensor and its criteria. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1173-1186. doi: 10.3934/jimo.2016.12.1173
Aleksander Denisiuk. On range condition of the tensor x-ray transform in $ \mathbb R^n $. Inverse Problems & Imaging, 2020, 14 (3) : 423-435. doi: 10.3934/ipi.2020020
Mirela Kohr, Sergey E. Mikhailov, Wolfgang L. Wendland. Dirichlet and transmission problems for anisotropic stokes and Navier-Stokes systems with L∞ tensor coefficient under relaxed ellipticity condition. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4421-4460. doi: 10.3934/dcds.2021042
Tomáš Oberhuber, Tomáš Dytrych, Kristina D. Launey, Daniel Langr, Jerry P. Draayer. Transformation of a Nucleon-Nucleon potential operator into its SU(3) tensor form using GPUs. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1111-1122. doi: 10.3934/dcdss.2020383
Tobias Breiten, Sergey Dolgov, Martin Stoll. Solving differential Riccati equations: A nonlinear space-time method using tensor trains. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 407-429. doi: 10.3934/naco.2020034
Henry O. Jacobs, Hiroaki Yoshimura. Tensor products of Dirac structures and interconnection in Lagrangian mechanics. Journal of Geometric Mechanics, 2014, 6 (1) : 67-98. doi: 10.3934/jgm.2014.6.67
Xiaofei Liu Yong Wang | CommonCrawl |
The impact of FMISO hypoxic volume on local control after single dose irradiation in FADU HNSCC in nude mice
Schütze, C.; Bergmann, R.; Mosch, B.; Yaromina, A.; Hessel, F.; Krause, M.; Thames, H. D.; Zips, D.; Mäding, P.; Baumann, M.; Beuthien-Baumann, B.
kein Abstract verfügbar
ICTR 2009 - Fourth International Conference on Translational Research in Radiation Oncology, 11.-13.03.2009, Genf, Schweiz
Permalink: https://www.hzdr.de/publications/Publ-13615
Publ.-Id: 13615
Characterization of ZnO nanostructures: A challenge to positron annihilation spectroscopy and other methods
Brauer, G.; Anwand, W.; Grambole, D.; Egger, W.; Sperr, P.; Beinik, I.; Wang, L.; Teichert, C.; Kuriplach, J.; Lang, J.; Zviagin, S.; Cizmar, E.; Ling, C. C.; Hsu, Y. F.; Xi, Y. Y.; Chen, X.; Djurisic, A. B.; Skorupa, W.
ZnO nanostructures are of special interest for device applications.
However, their structural characterization remains an ongoing challenge.
This paper reviews recent efforts and latest achievements in this direction. Results comprise PAS in the form of Slow Positron Implantation Spectroscopy (SPIS) and Pulsed Low Energy Positron Lifetime Spectroscopy (PLEPS), Nuclear Reaction Analysis (NRA), Atomic Force Microscopy (AFM), conductive AFM (C-AFM), Nuclear Magnetic Resonance (NMR), Electron Spin Resonance (ESR), Photoluminescence (PL) spectroscopy, and latest theoretical investigations of structure-related and positron properties of selected defects. The fundamental importance of a relationship between fabrication conditions, native defect formation, and resulting optical and electronic properties is demonstrated by getting either inferior (nanorods) or significantly improved (tetrapods) optical properties compared to single crystal samples, depending on the nanostructure fabrication method.
Physica Status Solidi (C) 6(2009)11, 2556-2560
Radiolabeled Cdk4/6 inhibitors for molecular imaging of tumors
Graf, F.; Köhler, L.; Mosch, B.; Pietzsch, J.
Overexpression of cell-cycle regulating cyclin-dependent kinases 4 and 6 (Cdk4/6) and deregulation of Cdk4/6-pRb-E2F pathway are common aspects in human tumors. The aim of our study was the evaluation of pyrido[2,3 d]pyrimidin-7-one derivatives (CKIA and CKIE) concerning their efficacy and suitability as small molecule Cdk4/6 inhibitors and, after iodine-124 ([124I]CKIA) or fluorine-18 ([18F]CKIE) radiolabeling, as radiotracers for Cdk4/6 imaging in tumors by positron emission tomography (PET).
CKIA and CKIE were analyzed concerning their biological properties (effects on cell growth, cell cycle distribution, Cdk4/6 mediated pRb-Ser780 phosphorylation, mRNA expression of pRb affected genes E2F-1 and PCNA) and radiopharmacological properties (cellular radiotracer uptake and PET studies) using human tumor cell lines HT-29, a colorectal adenocarcinoma cell line, FaDu, a head and neck squamous cell carcinoma cell line, and THP-1, an acute monocytic leukemia cell line, as well as phorbol ester TPA-activated THP-1 cells, as model of tumor-associated macrophages.
CKIA and CKIE were identified as potent inhibitors of Cdk4/6-pRb-E2F pathway due to decreased Cdk4/6 specific phosphorylation at pRb Ser780 and downregulation of E2F-1 and PCNA mRNA expression in HT-29, FaDu and THP-1 tumor cells. This resulted in arrest of these tumor cell lines in G1 phase of the cell cycle and growth inhibition. Otherwise, in non-proliferating TPA-activated THP-1 macrophages no change of cell-cycle distribution after treatment with CKIA and CKIE was observed. Furthermore, TPA-activated THP-1 macrophages showed lower Cdk4 mRNA and protein levels, than other tumor cell lines. In vitro radiotracer uptake studies using [124I]CKIA and [18F]CKIE demonstrated tumor cell uptake, which could be blocked with both nonradioactive CKIA and CKIE. However, THP-1 macrophages showed similar radiotracer uptake like other tumor cells. Preliminary small animal PET studies in mouse tumor xenograft models further analyzed the hypothesis that radiolabeled Cdk4/6 inhibitors are suitable tracers for molecular imaging of tumors
Abstract in refereed journal
Cancer Microenvironment 2(2009), S185
5th International Conference On Tumor Microenvironment: Progression, Therapy & Prevention, 20.-24.10.2009, Versailles, France
Irradiation-induced changes in metabolism and metastatic properties of melanoma cells
Mosch, B.; Müller, K.; Steinbach, J.; Pietzsch, J.
As it is known that irradiation can influence cellular metabolism it is conceivable that it can induce metabolic changes which lead to a predisposition of certain cells to show enhanced survival, migratory activity and metastasis. The aim of this study was to investigate short term and long term irradiation effects on proliferation and metabolism of melanoma cells in vitro and their ability to form metastases in vivo.
B16-F10 melanoma cells were irradiated with different doses of X-ray irradiation in the range of 1 to 20 Gy. One, two, and three days (short term effects) and, furthermore, 7, 14 and 21 days (long term effects) after treatment cells were analyzed concerning cell growth, proliferation, viability, glucose and amino acid transport. Additionally, we performed in vivo studies in a syngeneic mouse model to analyze the capability of irradiated melanoma cells to form lung metastases.
The analysis of short term effects showed decreased cell growth, viability and arrest in the G2/M phase of the cell cycle while glucose transport is increased. Long term effects involve recovered proliferation, accompanied by increased glucose transport and decreased viability and amino acid transport. In vivo studies showed loss of metastasis immediately after irradiation and reduced metastasis if cells were allowed to recover proliferation before injection.
We conclude that melanoma cells as short term response to irradiation show cell cycle arrest and impairment in growth and viability. Three days after irradiation compensatory mechanisms start, leading to recovered growth within three weeks. Studies concerning metabolic properties indicate that a subpopulation of surviving melanoma cells compensate for the initial irradiation-induced damage possibly by metabolic modulations such as increase in glycolysis. As metastasis in vivo is impaired beyond recovered cell proliferation, the role of adjusted cell metabolism and additional extrinsic factors is strongly suggested.
Cancer Microenvironment 2(2009), S150-S151
Changes in metabolism and metastatic properties of melanoma cells after X-ray irradiation
Background: Malignant melanoma has the ability to form metastases at very early stages and in addition to surgical resection treatment involves immunotherapy, chemotherapy and also radiotherapy. As it is known that irradiation can influence cellular metabolism it is conceivable that it can induce metabolic changes which lead to a predisposition of certain cells to show enhanced survival, migratory activity and metastasis. The aim of this study was to investigate short term and long term irradiation effects on metabolism and proliferation of irradiated melanoma cells in vitro and their ability to form metastases in vivo.
Material and methods: B16-F10 melanoma cells were irradiated with different doses of X-ray irradiation in the range of 1 to 20 Gy. One, two, and three days (short term effects) and, furthermore, 7, 14 and 21 days (long term effects) after treatment cells were analyzed concerning cell growth, viability, proliferation, cell cycle distribution, glucose and amino acid transport. Additionally, we performed in vivo studies in a syngeneic mouse model to analyze the capability of irradiated melanoma cells to form lung metastases.
Results: The analysis of short term effects showed decreased cell growth, viability and arrest in the G2/M phase of the cell cycle. Long term effects involve increase in proliferation, cell growth and glucose uptake but still decreased viability and amino acid transport. Our in vivo studies showed no formation of lung metastases when cells were irradiated before injection. If irradiated cells were allowed to recover for 2 weeks before injection, mice again developed lung metastases although to a lesser extent than control mice.
Conclusions: We conclude that melanoma cells as short term response to irradiation show cell cycle arrest and decrease in cell viability, growth and metabolic properties. One to three weeks after irradiation, the re-start of proliferation and recurrence of metabolic properties such as glucose uptake indicate that a subpopulation of surviving melanoma cells compensate for the initial irradiation-dependent damage possibly by metabolic modulations such as increase in glycolysis. Furthermore, in vivo studies reveal that irradiated melanoma cells are able to resume their metastatic potential within two weeks. As lung metastasis is lower when using recovered cells versus untreated cells, the role of additional mechanisms is strongly suggested.
European Journal of Cancer 7(2009), 587
ECCO 15 - 34th ESMO Multidisciplinary Congress, 20.-24.09.2009, Berlin, D
Influence of irradiation on the metabolism of melanoma cells and metastasis in mice.
Irradiation is a powerful tool for the therapy of solid tumors. But often single cells elude this treatment and constitute a basis for recurrence of the primary tumor and formation of metastases. Until today it is unclear which properties enable some cells to this. One possible explanation could be predicted on irradiation-dependent metabolic changes which lead to a predisposition of certain cells to show enhanced survival and migratory activity. The aim of this study was to investigate metabolic properties and proliferation of irradiated melanoma cells in vitro and their ability to form metastases in vivo.
We applied different single-dose X-ray irradiation (200kV X-rays, 0.5mm Cu, ~ 1.2 Gy min-1; 1, 2, 5, 7, 10, and 20 Gy) to murine B16-F10 melanoma cells. At particular times we analyzed cell viability, growth properties and cell cycle distribution. Furthermore, we analyzed the cellular uptake of the radiotracers 2-[18F]fluoro-2-deoxy-D-glucose ([18F]FDG) and 3-O-methyl-[18F]fluoro-L-DOPA ([18F]OMFD), providing information about the glucose and amino acid metabolism before and after irradiation. Additionally, we performed in vivo studies in a syngeneic mouse model to analyze the capability of irradiated melanoma cells to form lung metastases after injection into the tail vein of NMRI mice.
In a dose-dependent manner we detected a decrease in cell viability and cell growth properties starting 3 days after irradiation. Decreased cell growth persists up to 1 week for 5 Gy irradiated cells and up to 2 weeks for 10 Gy irradiated cells. After this periods growth of irradiated cells is comparable to control cells. Cell cycle analyses showed an increase in G2/M phase cells up to 3 days after X-ray followed by an increase in S phase cells 6 days after X-ray. At this point of time uptake of radiotracers was altered inasmuch as [18F]FDG uptake decreased, whereas [18F]OMFD uptake increased. Our in vivo studies showed a loss of lung metastases when cells were irradiated (10 Gy) before injection. If irradiated cells were allowed to recover for 2 weeks before injection, mice again developed lung metastases although to a lesser extent than control mice.
We conclude that irradiation of melanoma cells leads to a dose-dependent decrease in cell viability, growth properties and glucose uptake. Cell cycle analyses suggest an arrest in the G2/M phase. One week after irradiation compensating mechanisms of these effects seems to start as indicated by the uptake of [18F]OMFD, the increase in S phase cells and recovered growth of low-dose (5 Gy) irradiated cells. Two weeks after irradiation cell growth is completely recovered in vitro. Accordingly, in vivo studies reveal that irradiated melanoma cells are able to resume their metastatic potential within two weeks, even though to a lesser extent than before irradiation. The questions why and how some cells modulate their metabolism and thus re-start proliferation and why metastasis is influenced in vivo although growth properties are recovered in vitro, need to be further investigated.
2nd Workshop on Radiation and Multidrug Resistance via the Tumor Microenvironment, 09.-10.02.2009, Dresden, D
Detection and quantification of hypoxia in xenotransplanted human squamous cell carcinoma
Bergmann, R.; van den Hoff, J.; Pietzsch, J.; Strobel, K.; Mosch, B.; Schütze, C.; Brüchner, K.; Hofheinz, F.; Beuthien-Baumann, B.
World Molecular Imaging Conference, 10.-13.09.2008, Nizza, Frankreich
Pre-treatment FMISO hypoxic volume is a significant prognostic factor for local control after irradiation of FaDu HNSCC xenografts
Schütze, C.; Bergmann, R.; Mosch, B.; Yaromira, A.; Hessel, F.; Krause, M.; Thames, H. D.; Zips, D.; Mäding, P.; Baumann, M.; Beuthien-Baumann, B.
Objective: To investigate whether pre-treatment FMISO hypoxic tumour volume (HV) adds significant information about radiotherapy outcome in FaDu human squamous cell carcinoma (hSCC) in nude mice.
Materials and Methods: The hSCC cell line FaDu was transplanted subcutaneously into the hind leg of NMRI nude mice. Seventy animals entered the study at tumour volumes ranging from 165-343 mm³. [18F]fluoromisonidazole ([18F]FMISO)-PET scanning was performed under anesthesia (9% desflurane in 40% oxygen/air) on a dedicated animal PET scanner (MicroPET® P4, CTI Molecular Imaging Inc, measured attenuation correction, 11 MBq 18FMISO i.v., list mode acquisition for 30 min after 210 min p.i). The regions of interest (ROI) include the FMISO positive hypoxic volume, the mean, the maximum concentration (ROVER software, ABX GmbH, Radeberg, Germany). After an initial FMISO-PET (day 0) the tumours were stratified according to the median hypoxic volume (HV) for single dose irradiation with either 25 Gy (tumour control probability, TCP20) or 35 Gy (TCP80) under normal blood flow conditions using 200 kV X-rays (0.5 mm Cu, ~ 1.2 Gy min-1). The endpoint was time to local failure. Five animals are currently still in follow up.
Results: Tumour local control rate after irradiation with 25 Gy was lower than after irradiation with 35 Gy (22% vs. 69%, log rank p<0.0001). HV ranged from 38-353 mm³. Median HV was 112 mm³ (95%CI: 92; 128 mm³). In tumours with HV less than the median, local control was 33% after 25 Gy vs. 82% after 35 Gy (p=0.001) and in tumours with HV above the median 15% after 25 Gy vs. 53% after 35 Gy (p=0.0005). Multivariate Cox analysis revealed a significant effect of hypoxic volume treated either as a continuous (p=0.009) or a dichotomic variable (stratification by median HV) (p=0.039) when corrected for dose and tumour volume effects. Dose had a significant impact on hazard of recurrence (p<0.0005), whereas total tumour volume showed no effect (p=0.5).
Conclusions: Hypoxic volume is a significant predictor of tumour control after irradiation with high single doses in a single tumour line. This supports the hypothesis that pre-treatment FMISO-PET may provide useful information for heterogeneous radiation dose prescription in sub volumes of individual tumours. Confirmatory investigations using other tumour models and fractionated radiotherapy are warranted.
This work was performed within the 6th framework EU-project BioCare, proposal# 505785.
Radiotherapy and Oncology 88(2008), S102
27th conference of the European Society for Therapeutic Radiology and Oncology (ESTRO), 14.-18.09.2008, Göteborg, Schweden
Neuronal Aneuploidy in Health and Disease: A Cytomic Approach to Understand the Molecular Individuality of Neurons
Arendt, T.; Mosch, B.; Morawski, M.
Structural variation in the human genome is likely to be an important mechanism for neuronal diversity and brain disease. A combination of multiple different forms of aneuploid cells due to loss or gain of whole chromosomes giving rise to cellular diversity at the genomic level have been described in neurons of the normal and diseased adult human brain. Here, we describe recent advances in molecular neuropathology based on the combination of slide-based cytometry with molecular biological techniques that will contribute to the understanding of genetic neuronal heterogeneity in the CNS and its potential impact on Alzheimer´s disease and age-related disorders
Keywords: alu-repeats; Alzheimer´s disease; cell cycle; cell death; chromosomal mosaicism; in situ hybridisation; laser capture microdissection; neurodegeneration; slide-based cytometry
International Journal of Molecular Sciences 10(2009), 1609-1627
Grain growth in Ni-Mn-Ga alloys
Thoss, F.; Poetschke, M.; Gaitzsch, U.; Freudenberger, J.; Anwand, W.; Roth, S.; Rellinghaus, B.; Schultz, L.
The influence of annealing temperature and time on grain growth in polycrystalline Ni-Mn-Ga samples near the stoichiometric composition Ni2MnGa was investigated. Grain growth was only observed for compositions with a Ni content below 50 at.%. The existence of constitutional vacancies as a possible origin for the different grain growth behaviour was excluded by positron annihilation spectroscopy (PAS). In order to activate grain boundary motion and hence grain growth in Ni50Mn29Ga21 the samples were annealed and deformed in situ in compression up to various strain levels. A sharp threshold to initiate grain growth is observed between 8% and 10% of compression strain.
Keywords: Magnetically ordered materials; casting; grain boundaries
Journal of Alloys and Compounds 488(2009), 420-424
Isotopic Comparative Method (ICM) for the determination of variations of the useful ion yields in boron doped silicon as a function of oxygen concentration in the 0 - 10 at% range
Dupuy, J. C.; Dubois, C.; Prudon, G.; Gautier, B.; Kögler, R.; Akhmadaliev, S.; Perrat-Mabilon, A.; Peaucelle, C.
Specific samples containing O-18 and O-16 are used to measure the variations of the relative ion yields of boron, oxygen and silicon as a function of oxygen concentration. O-18 and O-16 are used to implement an Isotopic Comparative Method (ICM) which allows to correct the matrix effects involved by the presence of a high concentration of oxygen in the sample: the near-flat profile of O-18, measured in the 'dilute', linear regime (weak concentration) is used to calculate the real concentration of O-16. The ion yields of B+, O+, Si+, O- and Si- are measured as a function of the oxygen concentration. For B+ ion yield, the variation is important whereas they are weak for Si-+/- and O-+/- ion yields for the range [0-12 at.%]. This ICM applied to oxygen in silicon can be considered as an interesting complementary method of previous 'O-16 implantation method' and of 'O-18 single marker method'.
Keywords: Secondary ion mass spectrometry; SIMS; Isotopic comparative method; ICM; Oxygen; Silicon; Boron
SIMS XVII Toronto, Canada, 13.-17.09.2009, Toronto, Canada
Surface and Interface Analysis 43(2010)1-2, 137-140
DOI: 10.1002/sia.3657
Carrier profiling of individual Si nanowires by scanning spreading resistance microscopy
Ou, X.; Das Kanungo, P.; Kögler, R.; Werner, P.; Skorupa, W.; Gösele, U.; Wang, X.
Individual silicon nanowires (NWs) doped either by ion implantation or by in-situ dopant incorporation during NW growth were investigated by scanning spreading resistance microscopy (SSRM). The carrier profiles across the axial cross sections of the NWs are derived from the measured spreading resistance values and calibrated by the known carrier concentrations of the connected Si substrate or epi-layer. In case of the phosphorus ionimplanted and subsequently annealed NWs the SSRM profiles revealed a radial core-shell distribution of the activated dopants. The carrier concentration close to the surface of a phosphorus-doped NW is found to be by a factor of 6-7 higher than the value in the core and on average only 25% of the implanted phosphorus is electrically active. In contrast, for the insitu boron-doped NW, the activation rate of the boron atoms is significantly higher than for phosphorus atoms. Some specific features of SSRM experiments of Si NWs are discussed including the possibility of three-dimensional measurements.
Keywords: Nanowires; Silicon; Doping; Scanning Spreading Resistance Microscopy; SSRM
Nano Letters 10(2010), 171-175
DOI: 10.1021/nl903228s
Physics and techniques at high pulsed magnetic fields
Herrmannsdörfer, T.
es hat kein Abstract vorgelegen
2nd Workshop of the Resonant Scattering and Diffraction Beamline (P09) at PETRA III, 04.11.2009, Hamburg, Deutschland
Superconductivity in p-doped elemental semiconductors
Herrmannsdoerfer, T.; Heera, V.; Ignatchik, O.; Uhlarz, M.; Muecklich, A.; Posselt, M.; Reuther, H.; Schmidt, B.; Heinig, K. H.; Skorupa, W.; Voelskow, M.; Wuendisch, C.; Skrotzki, R.; Helm, M.; Wosnitza, J.
We report the first observation of superconductivity in heavily p-type doped Germanium at ambient-pressure conditions. Using advanced doping and annealing techniques, we have fabricated a highly Ga-doped Ge (Ge:Ga) layer in near-intrinsic cubic Ge. Depending on the detailed annealing conditions, we demonstrate that superconductivity can be generated and tailored in the p-doped semiconducting Ge host at temperatures as high as 0.5 K. Critical-field measurements reveal the quasi-two-dimensional character of superconductivity in the ~ 60 nm thick Ge:Ga layer. We find critical magnetic in-plane fields up to about 1T, even slightly larger than the Pauli-Clogston limit. There might be interest in the technological potential of on-chip thin-film superconductivity in a semiconducting environment demonstrated here as our preparation method is compatible with state-of-the-art semiconductor processing used nowadays for the mass production of logic circuits. After its finding in Si [1] and diamond [2], our work adds another unexpected observation of superconductivity in doped elemental semiconductors and in one of the few remaining 'islands of the periodic table of elements' on which superconductivity has not been found so far.
The 9th European Conference on Applied Superconductivity, EUCAS 2009, 13.-17.09.2009, Dresden, Deutschland
Workshop: Noval approaches to pairing and condensation, 03.10.2009, Dresden, Deutschland
Workshop "Physics and Metrology at Very Low Temperatures", 10.12.2009, Berlin, Deutschland
Semiconductor quantum structures for quadratic detection at mid-infrared and THz frequencies
Schneider, H.
There is no abstract.
Keywords: intersubband transition; two-photon detection; terahertz
Novel Quantum Structure Detectors for Opto-electronic Conversion 2010, 04.-06.01.2010, Sanya, China
Ultrafast infrared and Terahertz spectroscopy of semiconductor quantum structures
This seminar gives an overview on our recent experimental studies involving ps and fs lasers, including the free-electron laser at FZD. In particular, I briefly discuss nonlinear sideband spectroscopy and time-resolved photoluminescence. Then I will focus on quadratic photocurrent autocorrelation involving intersubband transitions in semiconductor quantum wells at mid-infrared wavelengths, and present our concept for scalable photoconductive Terahertz emitters.
Physics Colloquium, Shanghai Jiao Tong University, 30.12.2009, Shanghai, China
Fulde-Ferrell-Larkin-Ovchinnikov State in the Organic Superconductor κ-(BEDT-TTF)2Cu(NCS)2
Lortz, R.; Wang, Y.; Demuer, A.; Sheikin, I.; Bergk, B.; Wosnitza, J.; Nakazawa, Y.
extraordinary high and the Pauli limit becomes of importance instead. For this case the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state was theoretically predicted. It is now 55 years that this state is being looked for experimentally. The most promising candidates are layered superconductors such as the organic compound κ-(BEDT-TTF)2Cu(NCS)2. When the field is applied parallel to the layers, the orbital limit exceeds the Pauli limit at 22 T. I will present specific heat and magnetic torque experiments up to 30 T and down to 50 mK. The upper critical field transition changes from second to first-order nature above 21 T, the transition line shows a characteristic upturn and a novel transition line separates an unconventional high-field SC phase from the conventional low-field phase. Additional transitions are observed in slightly tilted angles, suggesting a rich physics with a competition between orbital and Pauli limits. These results provide strong evidence for the FFLO state in κ-(BEDT-TTF)2Cu(NCS)2.
9th Conference on Materials and Mechanisms of Superconductivity (M2S), 07.-12.09.2009, Tokyo, Japan
Fermi-Surface Topology of the Iron-Pnictide Compound LaFe2P2
Ignatchik, O.; Bartkowiak, M.; Wosnitza, J.; Blackburn, S.; Pesant, S.; Côté, M.; Seyfarth, G.; Bianchi, A. D.; Capan, C.; Fisk, Z.; Goodrich, R. G.
A thorough knowledge on the electronic band structure of metals is a prerequisite for gaining insight in their electronic properties. This becomes especially true for a better understanding of the occurrence of superconductivity. Here, we present results of a comprehensive de Haas-van Alphen (dHvA) study of LaFe2P2, a non-superconducting parent compound of the recently discovered family of iron-pnictide superconductors. By use of the cantilever torque magnetometry we were able to observe dHvA oscillations in high-quality LaFe2P2 single crystals starting at about 10 T at dilution-refrigerator temperatures. From the angular dependence of the oscillations we find a strongly anisotropic, but three-dimensional Fermi-surface topology. The effective masses of two different bands observed for magnetic fields aligned along the c axis are 1.4 and 1.7 me, where me is the free-electron mass. The dHvA frequencies and the effective masses in-crease considerably when rotating away from the c axis. We compare our results to state-of-the-art band-structure calculations.
9th Conference on Materials and Mechanisms of Superconductivity, 07.-12.09.2009, Tokyo, Japan
Thin-film Superconductivity in Ga-doped Germanium
Herrmannsdörfer, T.; Skrotzki, R.; Heera, V.; Ignatchik, O.; Uhlarz, M.; Mücklich, A.; Posselt, M.; Reuther, H.; Schmidt, B.; Heinig, K.-H.; Skorupa, W.; Voelskow, M.; Wündisch, C.; Helm, M.; Wosnitza, J.
We report the first observation of superconductivity in heavily p-type doped germanium at ambient pressure conditions. Using Ga as dopant, we have produced Ge:Ga samples by ion-beam implantation and subsequent short-term (msec) flash-lamp annealing. The combination of these techniques allows for Ga-doping levels up to 6%, not accessible to any other preparation method so far. The superconducting critical parameters strongly depend on the annealing conditions. Transport measurements reveal Tc up to 0.5 K and anisotropic Bc(T) with a linear temperature dependence reflecting the two-dimensional character of the superconducting state in the ~ 60 nm thin Ge:Ga layer. We find critical magnetic in-plane fields even larger than the Pauli-Clogston limit. After its finding in Si [1] and diamond [2], this work reports another unexpected observation of superconductivity in doped elemental semiconductors. Our fabrication techniques are compatible to industrial semiconductor processing and allow for on-chip superconductivity in integrated circuits.
9th International Conference on Materials and Mechanisms of Superconductivity (M2S), 07.-12.09.2009, Tokyo, Japan
Phase-sensitive terahertz spectroscopy with backward-wave oscillators in reflection mode
Pronin, A. V.; Goncharov, Y. G.; Fischer, T.; Wosnitza, J.
In this article we describe a method which allows accurate measurements of the complex reflection coefficient rˆ=|rˆ| ·exp(iφR) of a solid at frequencies of 1–50 cm−1 (30 GHz–1.5 THz). Backward-wave oscillators are used as sources for monochromatic coherent radiation tunable in frequency. The amplitude of the complex reflection (the reflectivity) is measured in a standard way, while the phase shift, introduced by the reflection from the sample surface, is measured using a Michelson interferometer. This method is particular useful for nontransparent samples, where phase-sensitive transmission measurements are not possible. The method requires no Kramers–Kronig transformation in order to extract the sample's electrodynamic properties (such as the complex dielectric function or complex conductivity). Another area of application of this method is the study of magnetic materials with complex dynamic permeabilities different from unity at the measurement frequencies (for example, colossal-magnetoresistance materials and metamaterials). Measuring both the phase-sensitive transmission and the phase-sensitive reflection allows for a straightforward model-independent determination of the dielectric permittivity and magnetic permeability of such materials.
Review of Scientific Instruments 80(2009), 123904
Short-range correlations in quantum frustrated spin system
Sytcheva, A.; Chiatti, O.; Wosnitza, J.; Zherlitsyn, S.; Zvyagin, A. A.; Coldea, R.; Tylczynski, Z.
We report on results of sound-velocity and sound-attenuation measurements in the low-dimensional spin-1/2 antiferromagnet Cs2CuCl4 (TN = 0.6 K), in external magnetic fields up to 15 T, applied along the b axis, and at temperatures down to 300 mK. The experimental data are analyzed with a theory based on exchange-striction coupling resulting in a qualitative agreement between theoretical results and experimental data.
Spatiotemporal observation of transport in fractured rocks
Kulenkampff, J.; Enzmann, F.; Gründig, M.; Mittmann, H.; Wolf, M.
We apply positron emission tomography (PET) with a high-resolution "small-animal" PET-scanner (ClearPET by Raytest, Straubenhardt) for process observation in rocks. Without affecting its physico-chemical properties, the fluid is labelled with the PET-tracer, a positron-emitting isotope. The annihilation radiation from individual decaying tracer atoms is detected with high sensitivity, and tomographic reconstruction of the recorded events yields a quantitative 3D-image of the tracer concentration. Sequential tomograms during tracer injection are used for the spatiotemporal observation of transport.
The raw data have to be corrected, prevalently with respect to background radiation (randoms) and Compton scattering, which is more significant than in common biomedical applications. Although these effects can be considered exactly in principle, we had to develop and apply simplified correction methods for performance reasons. Deficiencies of these correction algorithms generate some artefacts, that cause a lower limit of the tracer concentration in the order of 1 kBq/µl or about 107 atoms/µl, outranging other methods (e.g. nmr or resistivity tomography) by many orders.
A number of injection experiments in different rocks have been conducted with PET-process-tomography. New 3D-visualizations of the process-tomograms in fractured rocks showed strongly localized and complex flow paths and some unexpected deviations from the fractures that are deducible from µCT-images.
At least, the results demonstrate the large discrepancy between the µCT-derived volume and specific surface area and the hydraulic effective parameters, which also can be analyzed quantitatively with this method. Possibly, these discrepancies and the complexity of the process show the limits of parameter determination methods with model simulations based on structural pore-space models - as long as the simulations are not verified by experimental data.
European Geosciences Union General Assembly 2010, 02.-07.05.2010, Wien, Österreich
Nuclear Engineering Handbook - Chapter 1.4: VVER-type reactors of Russian design
Gado, J.; Rohde, U.; (Editors)
The Russian designed VVER-type reactors represent a special kind of pressurised light water reactors. The most important features of the VVER-440 and VVER-1000 reactor series are presented here by the representatives of the Russian design company OKB "GIDROPRESS", Podolsk, Russian Federation.
Design and basic safety properties are described in details, mentioning the differences between various reactors. VVER-440 and VVER-1000 reactors are presented separately. The successful operational experience of the two series is summarised together with data on decommissioned reactors.
Keywords: VVER-type reactors; pressurized water reactors; basic design; safety features; operation experience; decommissioning
D. G. Cacuci, J. Gado, U. Rohde: Handbook of Nuclear Engineering; Vol.IV - Reactors of Generation III and IV, Chapter 20: VVER-Type Reacctors of Russian Design, Heidelberg: Springer, 2010, 978-0-387-98130-7
Prediction of positron emitter distributions produced during 7Li irradiation
Priegnitz, M.; Fiedler, F.; Kunath, D.; Laube, K.; Parodi, K.; Enghardt, W.
in: GSI Scientific Report 2009, Darmstadt: GSI Helmholtzzentrum für Schwerionenforschung GmbH, 2010, 497
Electromagnetic inspection of a two-phase GaInSn/Argon flow
Terzija, N.; Yin, W.; Gerbeth, G.; Stefani, F.; Timmel, K.; Wondrak, T.; Peyton, A.
Adequate control of steel flow through the submerged entry nozzle during continuous casting is essential for maintaining steel cleanliness and ensuring good surface quality in downstream processing. Monitoring the flow in the nozzle presents a challenge for the instrumentation system because of the high temperature environment and the limited access to the nozzle in between the tundish and the mould. We study the distribution of two-phase liquid metal/gas flows by using a laboratory model of an industrial steel caster and an inductive sensor array. The experiments were performed with GaInSn as an analogue for liquid steel, which has similar conductive properties as molten steel and allows the measurements at the room temperature. A scaled (approx. 10:1) experimental rig consisting of a tundish, stopper rod, nozzle and mould was used. Argon gas was injected through the centre of the stopper rod and the behavior of two phase GaInSn/Argon flows was studied. The electromagnetic system used in our experiments to monitor the behavior of two phase GaInSn/Argon flows consisted of an array of 8 equally spaced inductive coils arranged around the object, a data acquisition system and a host computer. The present system operates at 10 kHz and has a capture rate of 10 frames per second. The results showed clearly that the injection of the Argon gas was distinguishable from the continuous flow by observing the appearance of oscillation patterns in the raw signals. These oscillations become more dominant with the increase of the Argon flow. In some cases two main oscillation patterns were present in the raw signals and our results suggested that those patterns are highly correlated with the level height in the mould and with the pressure in the nozzle.
6th International Conference on Electromagnetic Processing of Materials (EPM), 19.-23.10.2009, Dresden, Germany, 19.-23.10.2009, Dresden, Germany
6th International Conference on Electromagnetic Processing of Materials EPM 2009, Dresden: FZD, 978-3-936104-65-3, 391-394
6th International Conference on Electromagnetic Processing of Materials (EPM), 19.-23.10.2009, Dresden, Germany
The Impact Ionization MOSFET (IMOS) as Low-Voltage Optical Detector
Schlosser, M.; Iskra, P.; Abelein, U.; Lange, H.; Lochner, H.; Sulima, T.; Wiest, F.; Zilbauer, T.; Schmidt, B.; Eisele, I.; Hansch, W.
The avalanche photodiode (APD) is promoted as an alternative to photomultiplier tubes for optical sensing. When operated in Geiger mode, optically generated electron-hole-pairs trigger an avalanche multiplication, releasing typically about 1E6 charge carriers. However, APDs need operating voltages of about 70 V in order to allow cascaded impact ionization, making them unsuitable for several applications, especially in battery-powered devices. We propose a new device concept based on the vertical Impact Ionization MOSFET (IMOS), which could significantly reduce the operating voltage to about 5 V by using gate control and an additional current-enhancing effect.
Keywords: Impact ionization; MOSFET; Optical Detector
11th European Symposium on Semiconductor Detectors, NEW DEVELOPMENTS IN RADIATION DETECTORS, 07.-11.06.2009, Wildbad Kreuth, Germany
Nuclear Instruments and Methods in Physics Research A 624(2010), 524-527
DOI: 10.1016/j.nima.2010.05.060
Atomistic simulations of elastic and plastic properties in amorphous silicon
Talati, M.; Albaret, T.; Tanguy, A.
We present here potential-dependent mechanical properties of amorphous silicon studied through molecular dynamics (MD) at low temperature. On average, the localization of elementary plastic events and the co-ordination defect sites appears to be correlated. For Tersoff potential and SW potential the plastic events centered on defect sites prefer 5-fold defect sites, while for modified Stillinger-Weber potential such plastic events choose 3-fold defect sites. We also analyze the non-affine displacement field in amorphous silicon obtained for different shear regime. The non-affine displacement field localizes when plastic events occur and shows elementary shear band formation at higher shear strains.
EPL - Europhysics Letters 86(2009), 66005
Temperature Effect on Vibrational Properties of La0.7Sr0.3MnO3
Talati, M.; Jha, P. J.
Temperature dependence of phonons spectra and allied properties of rhombohedral La0.7Sr0.3MnO3 are investigated by using the lattice dynamical method. A tendency of phonon mode to instability causing JT lattice distortion is reflected in a softening of the stretching mode in the phonon dispersion curve of La0.7Sr0.3MnO3 at both 1.6 and 300 K. While the A(1g) mode softens because of gradual decrease in MnO6 rotations, the stretching mode hardens upon reduction in temperature. The distinct features of phonon modes at different temperatures are also reflected in the calculated phonon density of states. Other thermal properties such as specific heat, the Debye temperature, and Gruuneisen parameter are also presented. The decrease in the Debye temperature at 300 K indicates the effect of temperature in lattice softening. Anomalously high value of the Gruneisen parameter points out the presence of anharmonic lattice modes.
Keywords: Phonon dispersion; phonon density of states; colossal magnetoresistance; manganites
International Journal of Modern Physics B 23(2009)23, 4767-4777
Fulltext from www.worldscinet.com
Talk on Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray source
Bussmann, M.
In this talk I present the ideas and concepts behind phase contrast imaging with conventional x-ray tubes as presented by F. Pfeifer et al. in their 2006 Nature Physics article (Nature Physics Vol. 2, p. 258-261, April 2006).
Keywords: phase contrast imaging; coherence; grating; x-ray; oncoray journal club
OncoRay Journal Club, 07.01.2010, OncoRay, Dresden, Deutscland
Role of soft-iron impellers on the mode selection in the VKS dynamo experiment
Giesecke, A.; Stefani, F.; Gerbeth, G.
A crucial point for the understanding of the von-Karman-Sodium (VKS) dynamo experiment is the influence of soft-iron impellers. We present numerical simulations of a VKS-like dynamo with a localized permeability distribution that resembles the shape of the flow driving impellers. It is shown that the presence of soft-iron material essentially determines the dynamo process in the VKS experiment. An axisymmetric magnetic field mode can be explained by the combined action of the soft-iron disk and a rather small $\alpha$-effect parametrizing the induction effects of unresolved small scale flow fluctuations.
Keywords: dynamo; VKS; simulation; permeability; magnetohydrodynamics
Physical Review Letters 104(2010)4, 044503
HTGR Fuel Element Depletion Benchmark: Stage Three Results
Fridman, E.; Shwageraus, E.
Recently, a new numerical benchmark exercise for High Temperature Gas Cooled Reactor (HTGR) fuel depletion was defined. The purpose of this benchmark is to provide a comparison basis for different codes and methods applied for burnup analysis of HTGRs. The benchmark specifications include three different models: (1) an infinite lattice of tristructural isotropic (TRISO) fuel particles, (2) an infinite lattice of fuel pebbles, and (3) prismatic fuel including fuel and coolant channels. In this paper, we present the results of the third stage of the benchmark obtained with MCNP based depletion code BGCore and deterministic lattice code HELIOS 1.9. The depletion calculations were performed for three-dimensional model of prismatic fuel with explicitly described TRISO particles as well as for two-dimensional model in which double heterogeneity of the TRISO particles was eliminated using reactivity equivalent physical transformation (RPT). Generally good agreement in the results of calculations obtained by different methods and codes was observed.
PHYSOR 2010 – Advances in Reactor Physics to Power the Nuclear Renaissance, 09.-14.05.2010, Pittsburgh, Pennsylvania, USA
HTGR Fuel Element Depletion Benchmark: Stage One Results
Recently, a new numerical benchmark exercise1 for High Temperature Gas Cooled Reactor (HTGR) fuel depletion was defined. The purpose of this benchmark is to provide a comparison basis for different codes and methods applied for burnup analysis of HTGRs.
The benchmark specifications include three different models: (1) an infinite lattice of tristructural isotropic (TRISO) fuel particles, (2) an infinite lattice of fuel pebbles, and (3) prismatic fuel including fuel and coolant channels.
In this summary, we present the results of the first stage of the benchmark.
2009 ANS Winter Meeting, 15.-19.11.2009, Washington, DC Omni Shoreham Hotel, USA
Ultrafast carrier capture in InGaAs quantum posts
Stehr, D.; Morris, C. M.; Talbayev, D.; Wagner, M.; Kim, H. C.; Taylor, A. J.; Schneider, H.; Petroff, P. M.; Sherwin, M. S.
To explore the capture dynamics of photoexcited carriers in semiconductor quantum posts, optical pump terahertz (THz) probe and time-resolved photoluminescence spectroscopy were performed. The results of the THz experiment show that after ultrafast excitation, electrons relax within a few picoseconds into the quantum posts, which act as efficient traps. The saturation of the quantum post states, probed by photoluminescence, was reached at approximately ten times the quantum post density in the samples. The results imply that quantum posts are highly attractive nanostructures for future device applications.
Keywords: quantum post; terahertz probe; ultrafast; relaxation
Applied Physics Letters 95(2009), 251105
X-ray investigation of the interface structure of free standing InAs nanowires grown on GaAs [(1)over-bar(1)over-bar(1)over-bar](B)
Bauer, J.; Pietsch, U.; Davydok, A.; Biermanns, A.; Grenzer, J.; Gottschalch, V.; Wagner, G.
The heteroepitaxial growth process of InAs nanowires (NW) on GaAs [(1) over bar(1) over bar(1) over bar](B) substrate was investigated by X-ray grazing-incidence diffraction using synchrotron radiation. For crystal growth we applied the vapor-liquid-solid (VLS) growth mechanism via gold seeds. The general sample structure was extracted from various electron microscopic and X-ray diffraction experiments. We found a closed GaxIn1-xAs graduated alloy layer at the substrate to NW interface which was formed in the initial stage of VLS growth with a Au-Ga-In liquid alloy. With ongoing growth time a transition from this VLS layer growth to the conventional VLS NW growth was observed. The structural properties of both VLS grown crystal types were examined. Furthermore, we discuss the VLS layer growth process.
Keywords: MOVPE GROWTH; WHISKERS; ARRAYS; MECHANISM; STRAIN
Applied Physics A 96(2009)4, 851-859
Low temperature silicon dioxide by thermal atomic layer deposition: investigation of material properties
Hiller, D.; Zierold, R.; Bachmann, J.; Alexe, M.; Yang, Y.; Gerlach, J. W.; Stesmans, A.; Jivanescu, M.; Müller, U.; Vogt, J.; Hilmer, H.; Löper, P.; Künle, M.; Munnik, F.; Nielsch, K.; Zacharias, M.
SiO2 is the most widely used dielectric material but its growth or deposition involves high thermal budgets or suffers from shadowing effects. The low-temperature method presented here (150 °C) for the preparation of SiO2 by thermal atomic layer deposition (ALD) provides perfect uniformity and surface coverage even into nanoscale pores, which may well suit recent demands in nanoelectronics and nanotechnology. The ALD reaction based on 3-aminopropyltriethoxysilane (APTES), water and ozone provides outstanding SiO2 quality and is free of catalysts or corrosive by-products. A variety of optical, structural and electrical properties are investigated by means of infrared spectroscopy, UV-VIS spectroscopy, secondary ion mass spectrometry, capacitance-voltage- and current-voltage-measurements, electron spin resonance, Rutherford backscattering, elastic recoil detection analysis, atomic force microscopy, and variable angle spectroscopic ellipsometry. Many features, such as the optical constants (n, k), optical transmission and surface roughness (1.5 Å) are found to be similar to thermal oxide quality. Rapid thermal annealing (RTA) at 1000°C is demonstrated to significantly improve certain properties, in particular by reducing the etch rate in hydrofluoric acid, oxide charges and interface defects. Besides a small amount of OH-groups and a few atomic per mille of nitrogen in the oxide remaining from the growth and curable by RTA no impurities could be traced. Altogether, the data point to a first reliable low temperature ALD-growth process for silicon dioxide.
Journal of Applied Physics 107(2010), 064314
Molecular and Crystal Structures of Plutonyl(VI) Nitrate Complexes with N-Alkylated 2-Pyrrolidone Derivatives: Cocrystallization Potentiality of UVI and PuVI for Uniform MOX Fuel Precursor
Kim, S.-Y.; Takao, K.; Haga, Y.; Yamamoto, E.; Kawata, Y.; Morita, Y.; Nishimura, K.; Ikeda, Y.
Plutonyl(VI) nitrate complexes with N-cyclohexyl-2-pyrrolidone (NCP) and N-neopentyl-2-pyrrolidone (NNpP) were prepared, and their molecular and crystal structures were determined by single crystal X-ray analysis. The obtained compounds have the similar composition, PuO2(NO3)2(NRP)2 (NRP = NCP, NNpP), which are analogous to the corresponding UVI complexes. Both PuO2(NO3)2(NRP)2 complexes show typical structural properties of actinyl(VI) nitrates, i.e., hexagonal-bipyramidal geometry consisting of two NRP molecules and two NO3– ions located in trans positions in the equatorial plane of PuO22+ moiety, Pu=Oax = 1.73 Å, Pu–ONRP = 2.38 Å, Pu–ONO3 = 2.50 Å, and a bond angle between the U–ONRP bond and the carbonyl group of NRP ≈. 135°. The lattice constants and molecular arrangement of PuO2(NO3)2(NCP)2 were completely different from those of UO2(NO3)2(NCP)2. In contrast, these properties of PuO2(NO3)2(NNpP)2 are the same as those of UO2(NO3)2(NNpP)2. These findings provide one of criteria in selection of suitable NRP as a precipitation agent for the spent nuclear fuel reprocessing based on the precipitation method from a viewpoint of crystal engineering.
Keywords: Plutonium nitrate; precipitate; 2-pyrrolidone derivative; reprocessing; single crystal X-ray analysis
Crystal Growth & Design 10(2010)5, 2033-2036
Neptunium Carbonato Complexes in Aqueous Solution: An Electrochemical, Spectroscopic, and Quantum Chemical Study
Ikeda-Ohno, A.; Tsushima, S.; Takao, K.; Rossberg, A.; Funke, H.; Scheinost, A.; Bernhard, G.; Yaita, T.; Hennig, C.
The electrochemical behavior and complex structure of Np carbonato complexes, which are of major concern for the geological disposal of radioactive wastes, have been investigated in aqueous Na2CO3 and Na2CO3/NaOH solutions at different oxidation states by using cyclic voltammetry, X-ray absorption spectroscopy, and density functional theory calculations. The end-member complexes of penta- and hexavalent Np in 15 M Na2CO3 with pH = 11.7 have been determined as a transdioxo neptunyl tricarbonato complex. [NpO2(CO3)(3)](n-) (n=5 for Np-V, and 4 for Np-VI). Hence, the electrochemical reaction of the Np-V/VI redox couple merely results in the shortening/lengthening of bond distances mainly because of the change of the cationic charge of Np, without any structural rearrangement. This explains the observed reversible-like feature on their cyclic voltammograms. In contrast, the electrochemical oxidation of Np-V in a highly basic carbonate solution of 2.0 M Na2CO3/1.0 M NaOH (pH > 13) yielded a stable heptavalent Np complex of [(NpO4)-O-VII(OH)(2)](3-), indicating that the oxidation reaction from Np-V to Np-VII in the carbonate solution involves a drastic structural rearrangement from the transdioxo configuration to a square-planar-tetraoxo configuration, as well as exchanging the coordinating anions from carbonate ions (CO32-) to hydroxide ions (OH-).
Keywords: Neptunium; carbonate; redox; EXAFS; structure
Inorganic Chemistry 48(2009)24, 11779-11787
DOI: 10.1021/ic901838r
Tailoring Magnetic Properties Using Ion Beam Irradiation
Lenz, K.
Magnetism is a collective phenomenon. Hence, local variations on the nanoscale of material properties, which act on the magnetic properties, affect the overall magnetism in an intriguing way. In particular important are the length scales on which a material property changes. These might be related to the exchange length, the domain wall width, a typical roughness correlation length, or a length scale introduced by patterning of the material.
Ion beam erosion can be applied to create well ordered substrate ripples with nanometer periodicity. These artificially created templates serve as a source of a predefined surface morphology and hence allow for the investigation of roughness phenomena. In contrast to that post ion beam irradiation can be used to tailor the magnetic properties of conventional thin films and multilayers. The resulting magnetic properties are neither present in non-implanted nor in homogeneously implanted films. In both cases the magnetic properties depend sensitively on the artificially introduced length scale. Ferromagnetic resonance data of irradiated Py/Ta multilayers as well as Co and Fe thin films on ripple substrates are discussed.
Keywords: FMR; irradiation; ripples; coupling
Division seminar of the condensed matter theory group, 15.10.2009, Irvine, CA, USA
Spin dynamics in ferromagnets: Gilbert damping vs. two-magnon scattering
There exist several quite different damping mechanisms, which might contribute to the magnetic relaxation processes following the dynamic excitation of the spin system by ferromagnetic resonance (FMR). Most of the thin film magnetism community however seems to consider only Gilbert type damping contributions.
Using FMR and microwave frequencies between 1 and 225 GHz I will show how the different relaxation channels, i.e., dissipative, isotropic Gilbert damping G as well as anisotropic two-magnon scattering , are identified and disentangled by frequency and angle dependent FMR. In the case of Fe3Si films the scattering rates due to two-magnon scattering at crystallographic defects for spin waves propagating in [100] and [110] directions, and the Gilbert damping term are determined. Changing the film thickness from 8 to 40 nm and slightly modifying the Fe concentration influences these relaxation channels.
Finally, for the case of Ne+ irradiated Py/Ta multilayers, I will present how ion beam irradiation can be used to tailor the static and dynamic properties determined by FMR and MOKE.
Keywords: FMR; Gilbert Damping; Two-Magnon Scattering
Talk at the electromagnetics group seminar, 13.10.2009, Boulder, Colorado, USA
Talk at the Physics Seminar Series, 12.10.2009, Colorado Springs, USA
Determination of the Saturation Magnetization from Perpendicular Magnetic Anisotropy Measurements of Ion Irradiated Multilayers
Lenz, K.; Markó, D.; Strache, T.; Kaltofen, R.; Fassbender, J.
Ion beam irradiation and ion implantation of ferromagnetic films is a smart technique to tailor their magnetic properties and structural composition of multilayers or nanostructured samples [1,2]. Metals like Ta are commonly used as seed and cap layers in spintronic devices like Giant Magneto-Resistance sensors as Ta is chemically stable. However, it is known that 12% of Ta intermixing in Py leads to magnetically dead layers of 0.6-1.2 nm in thickness [3]. These dead layers make it impossible to determine the correct magnetic volume, which is needed to obtain the saturation magnetization from the magnetic moment measured e.g. by SQUID. This is especially true for multilayer samples which typically have a large number of interfaces.
Here we present a method to determine the saturation magnetization of Py/Ta multilayers from VNA-FMR (Vector Network Analyzer Ferromagnetic Resonance) and MOKE (Magneto-optical Kerr Effect) measurements even in the case of interfacial mixing due to ion irradiation, where SQUID magnetometry fails due to the unknown magnetic volume. Three sets of Py/Ta thin film multilayer systems were sputter-deposited on a Si/SiO2 substrate: (1xPy) is a single 20 nm thick Py layer, (5×Py) a multilayer of the structure 31 nm Ta/[4 nm Py/1 nm Ta]5/2 nm Ta and (10xPy) a multilayer of 30.5 nm Ta/[2 nm Py/0.5 nm Ta]10/2.5 nm Ta. The overall Py amount was always 20 nm and the total Ta thickness including seed and cap layer corresponds to 38 nm. Finally, the films have been irradiated with Ne ions at 40 keV with ion fluences in the range of 5×1013 to 5×1016 Ne/cm2. FMR shows that the FMR frequency vs. field dependence is significantly influenced by the amount of irradiation and number of interfaces (see Fig 1). At fluences above 2.5×1015 Ne/cm2 a significant decrease of the resonance frequency can be observed for the 1xPy samples. For the three unirradiated samples the FMR frequency decreases with an increasing number of Py/Ta repetitions, i.e. increasing number of interfaces. This decrease will be even more pronounced if larger numbers of interfaces are used. This can be explained by the higher number of neighboring Ta atoms in those cases. The deleterious effect of Ta on the ferromagnetic properties is becoming much stronger reducing the effective ferromagnetic film thickness by creating magnetically dead layers close to the interface [1,3-6]. This reduction of the saturation magnetization is directly linked to the resonance frequency. This allow to determine not only the uniaxial in-plane anisotropy field K2|| but also the saturation magnetization μ0Ms from the FMR frequency vs. field dependence. However, this can only be done, if there is only shape anisotropy but no uniaxial out-of-plane anisotropy as it is the case for our Py/Ta multilayers. From polar MOKE loops μ0Ms can be obtained by determining the perpendicular anisotropy field of the samples as well. This feature proves to be very useful here since SQUID magnetometry suffers from a major drawback: It requires the exact effective Py volume to calculate μ0Ms from the magnetic moment, which becomes more and more difficult to determine due to the increasing interfacial mixing induced by the ion irradiation and thinner Py layers. In our case, polar MOKE and FMR represent suitable alternatives since they allow μ0Ms to be determined without the knowledge of any film thickness. The good agreement with the SQUID data of the non-irradiated samples supports this (see Fig 2). ...
Keywords: FMR; MOKE, Irradiation
11th Joint MMM-Intermag Conference, 18.-22.01.2010, Washington DC, USA
Tailoring the Néel- and Interlayer Exchange Coupling of Fe/Cr/Fe Trilayers using Rippled Substrates
Lenz, K.; Körner, M.; Liedke, M. O.; Strache, T.; Dzenisevich, S.; Keller, A.; Facsko, S.; Fassbender, J.
rtificial antiferromagnets made from magnetically coupled trilayer structures are the basis for all types of spintronic devices like MRAM, GMR sensors etc. For years major effort lay on adjusting the coupling strength by changing the spacer thickness or material. Today, nanostructures offer a different approach as they add additional coupling mechanisms like proximity effects or N\'eel orange-peel coupling to the common interlayer exchange coupling (IEC). By means of ion beam erosion techniques it is possible to create well ordered substrate ripples with nanometer periodicity. They are transferred into the films grown on these rippled substrates. Hence, such ripples are a convenient way to induce N\'eel orange-peel coupling [1] and thus allow for tailoring the magnetic properties [2] as well as the coupling strength by varying the ripple periodicity without adjusting the spacer thickness.
We have investigated the influence of rippled vs. flat Si substrates on the interlayer exchange coupling contributions in polycrystalline Fe (4nm)/Cr ($x$ nm)/Fe (4nm) thin film trilayers ($x$=0--5 nm). The substrate surface was periodically modulated (periods of 23 nm and 37 nm) by Ar$^+$ ion beam erosion. The influence of the resulting surface and interface structure on the magnetic properties has been investigated by longitudinal magneto-optical Kerr effect (MOKE) applying a Stoner-Wohlfarth model on the magnetization reversal loops. Using 23 nm period ripples, we find an orange peel type coupling, predicted by N\'eel's theory superimposed on the IEC. In addition due to the morphology of the magnetic layers, a strong uniaxial magnetic anisotropy is induced.
[1] M. Körner, K. Lenz, M.O. Liedke, T. Strache, S. Dzenisevich, A. Keller, S. Facsko, and J. Fassbender, submitted to Phys. Rev. B.
[2] J. Fassbender, T. Strache, M.O. Liedke, D. Mark\'o, S. Wintz, K. Lenz, A. Keller, S. Facsko, I. M\"onch, and J. McCord, submitted to New. J. Phys.
This work is supported by DFG grant FA 314/6-1.
Keywords: MOKE; orange-peel coupling; Neel coupling; interlayer exchange coupling; ripples
Advances in Magnetic Nanostructures, 04.-09.10.2009, Vail, Colorado, USA
Determination of the Saturation Magnetization from Perpendicular Magnetic Anisotropy Measurements
Lenz, K.; Marko, D.; Strache, T.; Fassbender, J.
Over the last years the modification of magnetic parameters of thin films and multilayers by ion irradiation and implantation has become fashionable. However, especially in multilayer structures ion irradiation can lead to interfacial mixing and thus to a significant reduction of the magnetically active volume, i.e. 'dead' layers. Thus, it can be quite difficult---if not at all impossible---to determine the correct effective magnetic volume, e.g. for measurements of the saturation magnetization by SQUID. Here we show how ferromagnetic resonance (FMR) and magneto-optical Kerr effect (MOKE) measurements can be utilized to determine the saturation magnetization of irradiated Py/Ta multilayers instead and how the magnetic properties change upon irradiation.
We prepared thin films and multilayers of Py/Ta with an overall Py thickness of 20 nm and varying number of Py/Ta stacks and irradiated them with Ne ions at various fluences. FMR, MOKE, and SQUID magnetometry were used for investigation. With both, increasing ion fluences and increasing number of Py/Ta interfaces, a decrease of saturation magnetization and an increase of precessional damping can be observed. The uniaxial anisotropy of the samples is only of small magnitude and remains almost unaffected. There is, depending on the number of interfaces, a critical ion fluence at which ferromagnetic order in the multilayers vanishes.
Keywords: FMR; ion irradiation
Spin wave excitations: coupling and damping effects in ultrathin films
Spinwave excitations in thin film ferromagnets control several phenomena starting with spin transfer torque/spin pumping effects, magnetic relaxation and damping, to interlayer exchange coupling (IEC) and its temperature dependence. Most of them are accessible by ferromagnetic resonance (FMR).
There exist several quite different damping mechanisms, which might contribute to the magnetic relaxation processes following the excitation of the spin system. Using a broad range of microwave frequencies, I will show how these different relaxation channels, i.e., dissipative isotropic Gilbert damping as well as anisotropic two-magnon scattering, can be identified and disentangled by frequency and angle dependent FMR on Fe3Si Heusler alloys.
Changing the film thickness from 8 to 40 nm and slightly modifying the Fe concentration influences these relaxation channels [1]. Ion beam irradiation can be used to tailor the damping properties of Py/Ta multilayers after sample preparation [2].
FMR can also be used to study the interlayer exchange coupling. For single crystalline Ni/Cu/Co prototype trilayers the IEC's temperature dependence was investigated [3]. It follows an effective power law AT n, n ≈ 1.5. The results clearly indicate that the dominant contribution to the temperature dependence is due to the excitation of thermal spin waves. This is corroborated by recently developed theory [4].
[1] Kh. Zakeri et al., Phys Rev. B 76, 104416 (2007).
[2] D. Markó, T. Strache, K. Lenz, J. Fassbender, and R. Kaltofen, submitted
[3] S.S. Kalarickal, X.Y. Xu, K. Lenz, W. Kuch, and K. Baberschke, Phys. Rev. B 75, 224429 (2007).
[4] S. Schwieger, J. Kienert, K. Lenz, J. Lindner, K. Baberschke, and W. Nolting, Phys. Rev. Lett. 98, 057205 (2007).
Keywords: Spin waves; Ferromagnetic Resonance; Interlayer exchange coupling
448. WE-Heraeus-Seminar: Excitement in magnetism: Spin-dependent scattering and coupling of excitations in ferromagnets, 22.-25.11.2009, Ringberg, Deutsch
Parallel proximal probe arrays with vertical interconnections
Sarov, Y.; Frank, A.; Ivanov, T.; Zöllner, J.-P.; Ivanova, K.; Volland, B.; Rangelow, I. W.; Brogan, A.; Wilson, R.; Zawierucha, P.; Zielony, M.; Gotszalk, T.; Nikolov, N.; Zier, M.; Schmidt, B.; Kostic, I.
This article presents the fabrication and the characteristics of 8x64, parallel, self-actuated, and independently addressable scanning proximal probes with through-silicon via interconnection passing completely through a silicon wafer. The low-resistance highly doped polysilicon through-wafer electrical interconnects have been integrated with scanning proximal probes (SPPs) to enable back side contacts to the application-specific integrated circuit used as an atomic force microscope control circuitry. Every SPP sensor contains a deflection sensor, thermally driven bimetal (bimorph) actuator, and sharp silicon tip. Dry etching-based silicon on insulator three-dimensional-micromachining technique is employed by the creation of the through-silicon vias and the SPP arrays keeping fully complementary metal-oxide semiconductor compatible process regime. The application of the vertical interconnection technology in large-scale two-dimensional cantilever arrays with off-plane bent cantilevers over the chip's surface, in a combination with the flip-chip packaging technology allow simultaneous approach and parallel scanning of large areas in noncontact mode.
Keywords: AFM probe array; electrical through wafer interconnect; piezoresistive deflection sensor; thermally driven deflection actuator
Journal of Vacuum Science & Technology B 27(2009)6, 3132-3138
Electrical Characterisation of USJs in doped Si
Ogiewa, M.; Zier, M.; Schmidt, B.
An adaption of the DHE technique, using a stepwise oxidation, as an alternative to e.g. SRP methods as a tool to determine active carrier concentration and mobility depth profiles is presented. As known from the literature, a reduction of dopand activation as well as a decrease of mobility with increasing implantation dose above the solubility limit is observed. The aim is to combine the effects known from the literature in a model system. For industrial purpose, this enables one to find a "sweet spot" of high mobility and active dopand concentration while minimizing the negative effects of a high implantation dose.
Keywords: ultra-shallow pn-junction; charge carrier concentration and mobility; dopand activation
DPG Frühjahrstagung der Sektion Kondensierte Materie (SKM), 22.-27.03.2009, Dresden, Germany
EFTEM, EELS, and Cathodoluminescence in Si-implanted SiO2 Layers
Fitting, H.-J.; Fitting Kourkoutis, L.; Salh, R.; Schmidt, B.
Scanning transmission electron microscopy (STEM) in combination with electron energy loss spectroscopy (EELS) and cathodoluminescence (CL) have been used to investigate Si+-implanted amorphous silicon dioxide layers and the formation of Si nanoclusters.
Keywords: Ion implantation; Si nanoclusters
Microscopy and Microanalysis 15(2009)Suppl. 2, 1104-1105
Ion-erosion-induced pattern as template for layers with magnetic anisotropy and coupling
Fassbender, J.; Liedke, M. O.; Körner, M.; Markó, D.; Lenz, K.; Facsko, S.
Ion-erosion-induced ripples are perfect template systems to systematically investigate the influence of a periodic surface modulation on magnetic properties like magnetic anisotropy in the case of single magnetic films or interlayer exchange coupling in the case of multilayer systems. One of the key advantages of these ripples is that their periodicity can easily be varied in the range between 20 and 60 nm. This matches exactly the range where magnetic properties can be affected by a surface modulation. Two different examples will be discussed: i.) ripple-induced magnetic anisotropies in soft magnetic Permalloy films [1,2] and ii.) the appearance of roughness induced magnetic coupling, e.g. Neel coupling, in multilayer systems [3]. In both cases a significant influence of the surface and interface modulation on the magnetic properties is observed, which drastically depends on the ripple periodicity itself.
[1] M.O. Liedke, B. Liedke, A. Keller, B. Hillebrands, A. Mücklich, S. Facsko, J. Fassbender, Phys. Rev. B 75, 220407 (2007).
[2] J. Fassbender, T. Strache, M.O. Liedke, D. Markó, S. Wintz, K. Lenz, S. Facsko, I. Mönch, J. McCord, New J. Phys. in press.
[3] M. Körner, K. Lenz, M.O. Liedke, T. Strache, A. Mücklich, S. Facsko, J. Fassbender, Phys. Rev. B, submitted.
Keywords: magnetism; ion erosion; templates; nanopatterning; magnetic anisotropy; magnetic damping; magnetic coupling
2009 MRS Fall Meeting, 30.11.-04.12.2009, Boston, USA
Designing soft magnetic materials by ion irradiation
McCord, J.; Fassbender, J.
The control of the relevant magnetic material parameters like magnetic anisotropy, saturation magnetization as well as the dynamic magnetic properties in ferromagnetic thin films is of significant importance for applications in spin electronics. Commonly, the magnetic anisotropy in ferromagnetic single or multi-layers is initialized either by applying a magnetic field during film deposition or by annealing a magnetic field, which results in an anisotropy aligned along the applied field direction. Another important magnetic parameter, the saturation magnetization, is mainly determined by the film's composition.
We discuss novel ways of patterning magnetic films in terms of laterally varying magnetic properties. The difference of these hybrid property films with respect to conventional ferromagnetic systems is that the magnetic behaviour is strongly influenced by the direct exchange interaction across the regions of different magnetic behaviour. This makes them comparable to magnetic multilayer structures.
Different samples of anisotropy, exchange bias, and saturation magnetization [4] modulated thin films are prepared by local ion irradiation or implantation [1-5]. The magnetization reversal processes in the two-phase materials exhibit unique features, some of them so far only known from multilayer samples. The main emphasis of the presented work is on the role of the magnetic microstructure in stripe-like magnetic hybrid structures on the overall magnetization properties. Unique effects are derived from magnetic property measurements and magnetic domain imaging.
The presented paths of film preparations provide additional degrees of freedom for the tailoring of magnetic properties and functionality of soft-magnetic thin films. The presented methods allow for a local setting of magnetic properties without irreversible structural and magnetic alterations.
[1] J. Fassbender, J. McCord, Magnetic patterning by means of ion irradiation and implantation, J. Magn. Magn. Mat. 320, 579 (2008)
[2] J. McCord, I. Mönch, J. Fassbender, A. Gerber, E. Quandt, Local setting of magnetic anisotropy in amorphous films by Co ion implantation, J. Phys. D: Appl. Phys. 42, 55006 (2009)
[3] J. McCord, L. Schultz, J. Fassbender, Hybrid soft-magnetic lateral exchange spring films prepared by ion irradiation, Adv. Mat. 20, 2009 (2008)
[4] N. Martin, J. McCord, A. gerber, T. Strache, T. Gemming, I. Mönch, N. Farag, R. Schäfer, J. Fassbender, E. Quandt, L. Schultz, Local stress engineering of magnetic anisotropy in soft magnetic thin films, Appl. Phys. Lett. 94, 62506 (2009)
Keywords: magnetism; ion irradiation; magnetic properties; material modification; patterning; microscopy
XVIII International Materials Research Congress, 16.-20.08.2009, Cancun, Mexiko
Ion Beam Mixing as Basic Technology for a Light-emitting silicon nanocrystal field-effect transistor
Schmidt, B.; Heinig, K.-H.; Beyer, V.; Stegemann, K.-H.
A light emitting field-effect transistor (LEFET) which is based on silicon nanocrystals in the gate oxide is demonstrated. The Si nanocrystals in the gate oxide were optimized for a multi-dot floating-gate nonvolatile memory operation. For this aim, ion irradiation through the MOSFET stack of 50 nm poly-Si/15 nm SiO2/Si substrate was performed with 50 keV Si+ ions. The ion beam mixing of the upper poly-Si/SiO2 interface and the lower SiO2/(001)Si interface leads to Si excess in the gate oxide. Subsequent rapid thermal annealing reforms sharp interfaces and separates the excess Si from SiO2. Adjacent to the recovered interfaces, 3-4 nm thick SiO2 zones denuded completely of excess Si have been found, whereas the more distant tails of excess Si form well-aligned narrow layers of nanocrystals with 2-3 nm diameter. LEFETs with an active gate area of 20x20 µm2 were fabricated as nMOSFET devices in a standard 0.6 µm CMOS process line. An AC voltage was applied to the gate in order to inject charges of both polarities into the lower and upper Si nanocrystal layer from the channel and the poly-Si gate of the transistor, respectively. AC voltage and frequency dependent electroluminescence spectra were recorded in the wavelength region of 400-1000 nm as a function of the annealing conditions. The performance of the LEFETs and further possibilities of optimization of efficient light emission will be discussed.
Keywords: Ion beam mixing; Si/SiO2 interface; Si nanocrystals; electroluminesence; MOS-FFET device
Ionenstrahltreffen 2009, 06.-08.04.2009, Jena, Germany
Ion implantation in AFM cantilever array fabrication
Schmidt, B.; Zier, M.; Potfajova, J.
This paper describes the fabrication of p-type silicon piezoresistive sensing and actuating resistive heater elements monolithically integrated in AFM cantilever arrays using ion implantation for boron doping of all elements including corresponding interconnecting lines between them. Because it has been found that for p-type piezoresistivity the predicted values of the piezoresistive coefficients are approximately two times higher in ultra-shallow boron doped layers with a pn-junction depth < 10 nm than in the silicon p-type bulk material special efforts were done for the realization of ultra-shallow boron profiles using low-energy ion implantation and point defect engineering.
Keywords: Low energy ion implantation; piezoresistor; AFM cantilever array
54th Internationales Wissenschaftliches Kolloquium, Workshop "PRONANO",, 10.09.2009, Ilmenau, Germany
Bildrekonstruktion für die ultraschnelle Limited-Angle-Röntgen-Computertomographie von Zweiphasenströmungen
Bieberle, M.
Die ultraschnelle Röntgentomographie ist eine Messmethode, die speziell für die Untersuchung von transienten Zwei- und Mehrphasenströmungen entwickelt wurde. Speziell angepasste Bildrekonstruktionsalgorithmen ermöglichen die direkte Rekonstruktion von Phasengrenzflächen.
Keywords: Röntgen; Computertomographie; Zweiphasenströmungen; Messtechnik
KOMPOST Doktorandenseminar, 10.12.2009, Dresden, FZD, Deutschland
Ultrafast electron-beam x-ray computed tomography for multi-phase flow measurement
Ultrafast x-ray computed tomography is an imaging technique that has been optimized for transient flow measurements. Special image reconstruction algorithms enable the direct reconstruction of the physical phases of the investigated two-phase-flows.
Keywords: x-ray tomography; flow measurement; image reconstruction
FZD Doktorandenseminar, 16.-18.09.2009, Krögis, Deutschland
Compositional, structural and morphological modifications of N-rich Cu3N films induced by irradiation with Cu8+ at 42 MeV
Gordillo, N.; Rivera, A.; Grötzschel, R.; Munnik, F.; Güttler, D.; Crespillo, M. L.; Agulló-López, F.; González-Arrabal, R.
N-rich Cu3N films were irradiated with Cu8+ at 42 MeV in the fluence range from 4×1011 to 1×1014 cm-2. The radiation-induced changes in the chemical composition, structural phases, surface morphology and optical properties have been characterised as a function of fluence, substrate temperature and angle of incidence of the incoming ion by means of ion beam analysis (IBA), X-ray diffraction (XRD), atomic force microscopy (AFM), profilometry and Fourier transform IR spectrophotometry (IRFT). IBA techniques reveal a very efficient sputtering of N whose yield (5×103 at/ion) is almost independent of substrate temperature (RT-300ºC) but slightly depends on the incidence angle of the incoming ion. The area density of Cu remains essentially constant within the investigated fluence range. All data suggest an electronic mechanism to be responsible for the N depletion. The release of nitrogen and the formation of Cu2O and metallic Cu are discussed on the basis of existing models.
Keywords: Copper nitride; ion beam modification of materials; ion beam mixing; swift heavy ion irradiation; electronic sputtering
Journal of Physics D: Applied Physics 43(2010), 345301
Combined effects of humic matter and surfactants on PAH solubility: Is there a mixed micellization?
Lippold, H.
It has been recognized that solid-liquid distribution and transport of hydrophobic contaminants such as PAH (polycyclic aromatic hydrocarbons) are governed by their interaction with mineral-bound and dissolved humic matter, acting as a sink or a mobilizing agent, respectively. As surface-active compounds, humic substances are often compared to surfactants. Emerging environmental technologies involve a deliberate application of surfactants to enhance the sorption capacity of soils and aquifer materials, or to increase the efficiency of soil washing procedures and pump-and-treat operations for groundwater decontamination. Whereas contaminant binding to humics as well as to surfactants has been extensively studied, there is a notable lack of literature on their combined action in mixed systems. This topic is, however, important because environmental influences of surfactants are inevitably associated with the effects of the ubiquitous natural organics. Since both are amphiphilic, it seems conceivable that mixed micelles can be formed, involving synergistic or antagonistic effects in the solubilization of organic compounds.
In this study, we have examined the joint influence of humic acid and surfactants (cationic, anionic) on the water solubility of pyrene as a representative of PAH, at surfactant concentrations below and above the critical micelle concentration (CMC). In order to detect and characterize interaction processes, we have investigated the octanol-water partitioning of humic acid in the presence of various surfactants, using radiolabelled humic material. In particular, the hypothesis of a micellar nature of dissolved humic substances has been addressed.
2. Materials and Methods: omitted here
3. Results and Discussion
The water solubility of pyrene is increased in the presence of humic acid, which acts as a carrier due to hydrophobic interaction of both components. When adding the cationic surfactant dodecyltrimethylammonium bromide (DTAB), this solubility enhancement was found to be cancelled; the humic colloids were precipitated as a consequence of charge compensation by the organo-cations.
Interestingly, an antagonistic effect was also observed on addition of an anionic surfactant, sodium dodecylsulfate (SDS). While no precipitation was induced in this case, the solubility of pyrene was reduced by half and remained constant on further addition. Only at surfactant concentrations above the CMC, the solubility increased sharply owing to micellar incorporation. The presence of HA did not cause any change in the CMC of SDS, as is normally observed on addition of a second amphiphilic compound. Furthermore, the effects of HA and micellar SDS on pyrene solubility turned out to be strictly additive. Consequently, they are based on distinct processes, occurring independently of each other, i.e., there is no mixed micellization with humic molecules acting as a co-surfactant.
The octanol-water partition ratios of HA changed significantly in the presence surfactants. The partitioning equilibrium was shifted towards the organic phase on addition of cationic surfactants, and towards the aqueous phase on addition of anionic surfactants. Based on these findings, different modes of interaction could be identified, providing an explanation for the decline in pyrene solubilization in systems of HA and SDS. Obviously, a competitive situation arises in the hydrophobic binding of the PAH and the surfactant tail groups. The fact that the pyrene molecules cannot be displaced completely supports the proposition that different binding sites exist in humic colloids: weak near-surface sites and strong inner sites.
The size distribution of the colloids was found to be unaffected by the association with anionic as well as with cationic surfactants. A general micellar character is thus unlikely since a co-aggregation should then entail substantial disruptions and rearrangement processes.
4. Conclusions: omitted here
15th meeting of the International Humic Substances Society (IHSS 15), 27.06.-02.07.2010, Tenerife, Espana
Role of impurities and dislocations for the unintentional n-type conductivity in InN
Darakchieva, V.; Barradas, N. P.; Xie, M.-Y.; Lorenz, K.; Alves, E.; Schubert, M.; Persson, P. O. A.; Giuliani, F.; Munnik, F.; Hsiao, C. L.; Tu, L. W.; Schaff, W. J.
We present a study on the role of dislocations and impurities for the unintentional n-type conductivity in high-quality InN grown by molecular beam epitaxy. The dislocation densities and H profiles in films with free electron concentrations in the low 1017 cm-3 and mid 1018 cm-3 range are measured, and analyzed in a comparative manner. It is shown that dislocations alone could not account for the free electron behavior in the InN films. On the other hand, large concentrations of H sufficient to explain, but exceeding substantially, the observed free electron densities are found. Furthermore, enhanced concentrations of H are revealed at the film surfaces, resembling the free electron behavior with surface electron accumulation. The low-conductive film was found to contain C and it is suggested that C passivates the H donors or acts as an acceptor, producing compensated material in this case. Therefore, it is concluded that the unintentional impurities play an important role for the unintentional n-type conductivity in InN. We suggest a scenario of H incorporation in InN that may reconcile the previously reported observations for the different role of impurities and dislocations for the unintentional n-type conductivity in InN.
Physica B 404(2009), 4476-4481
DOI: 10.1016/j.physb.2009.09.042
Numerical simulation of the insulation material transport in a PWR core under loss of coolant conditions
Höhne, T.; Grahn, A.; Kliem, S.; Rohde, U.; Weiss, F.-P.
In 1992, strainers on the suction side of the ECCS pumps in Barsebäck NPP Unit 2 became partially clogged with mineral wool after a safety valve opened because steam impinged on the thermally-insulated equipment and released mineral wool. This event pointed out that strainer clogging in the course of a loss-of-coolant accident is an issue and induced many investigations to understand and prevent strainer clogging effects.
Modifications of the insulation material, the strainer area and mesh size were carried out in most of the German NPPs. Moreover, back flushing procedures to remove the mineral wool from the strainers and differential pressure measurement were implemented to assure the performance of emergency core cooling during the containment sump recirculation mode.
Nevertheless, it cannot be completely ruled out, that a limited amount of the smaller fractions of insulation material could be transported into the RPV. During a postulated cold leg LOCA with hot leg ECC injection, the fibres enter the upper plenum and can accumulate at the fuel element spacer grids, preferably at the uppermost grid level. This effect might affect the ECC flow into the core and could result in degradation of core cooling.
It was the aim of the numerical simulations presented to study where and how many mineral wool fibers are deposited at the upper spacer grid. The 3D, time dependent, multi-phase flow problem was modelled by applying the CFD code ANSYS CFX.
The spacer grids were modeled as a strainer, which completely retains all the insulation material that reaches the uppermost spacer level. There, the accumulation of the insulation material gives rise to the formation of a compressible fibrous layer, the permeability of which to the coolant flow is calculated in terms of the local amount of deposited material and the local value of the superficial liquid velocity.
Before the switch over of the ECC injection from the flooding mode to the sump mode, the coolant circulates in an inner convection loop in the core extending from the lower plenum to the upper plenum. The CFD simulations have shown that after starting the sump mode, the ECC water injected through the hot legs flows down into the core via so-called "brake through channels" located in the outer core region where the downward leg of the convection role had established. The hotter, lighter coolant rises in the center of the core. As a consequence, the insulation material is preferably deposited at the uppermost spacer grids positioned in the break through zones. This means that the fibres are not uniformly deposited over the core cross section.
When the inner recirculation stops later in the transient, insulation material can also be collected in other regions of the core cross section at the level of the upper spacer grids. Nevertheless, with a total of 2.7 kg fiber material deposited at the uppermost spacer level, the pressure drop over the fiber cake is not higher than 8 kPa and all the ECC water could still enter the core. The CFD calculation does not yet include steam production in the core and also does not include re-suspension of the insulation material during reverse flow. This will certainly further improve the coolability of the core.
Keywords: CFD; Fibre; Core; PWR
18th International Conference on Nuclear Engineering, 17.-21.05.2010, Xi'an, China
Temperature dependence of the crossover between the near-infrared Er and defect-related photoluminescence bands of Ge-rich Er-doped SiO2 layers
Kanjilal, A.; Rebohle, L.; Prucnal, S.; Voelskow, M.; Skorupa, W.; Helm, M.
Temperature-dependent photoluminescence of Ge-rich SiO2 in the presence or absence of Er shows a crossover between defect-related (15–150 K) and Er-related (150–295 K) emission within 1525 and 1440 nm. The origin of the near-infrared defect-related bands is discussed in the light of recombination of localized excitons in luminescence centers at the Ge cluster/SiO2 interface. Time-resolved photoluminescence further enables us to illustrate the observed 1.53 um Er emission above 150 K in terms of a phonon-assisted nonradiative energy-transfer process from the luminescence centers to the Er3+ ions.
Keywords: PL; Er; Ge clusters
Physical Review B 80(2009), 241313 (R)-1-4
Establishing the mechanism of thermally induced degradation of ZnO:Al electrical properties using synchrotron radiation
Vinnichenko, M.; Gago, R.; Cornelius, S.; Shevchenko, N.; Rogozin, A.; Kolitsch, A.; Munnik, F.; Möller, W.
X-ray absorption near edge structure and x-ray diffraction studies with synchrotron radiation have been used to relate the electrical properties of ZnO:Al films to their bonding structure and phase composition. It is found that Al-sites in an insulating metastable homologous (ZnO)3Al2O3 phase are favored above a certain substrate temperature (Ts) leading to deterioration of both the crystallinity and the electrical properties of the films. The higher film resistivity is associated with lower carrier mobility due to increased free electron scattering. Lower metal to oxygen flux ratios during deposition expand the range of Ts at which low-resistivity films are obtained.
Keywords: transparent conductive oxides; Al-doped ZnO; reactive pulsed magnetron sputtering; electrical properties; XANES
Applied Physics Letters 96(2010)14, 141907
DREAMS - a universal AMS facility based on the 6 MV - TandetronTM at FZD in Dresden
Akhmadaliev, S.; Kolitsch, A.; Merchel, S.; Möller, W.
A new accelerator mass spectrometry (AMS) system has been installed at the Forschungszentrum Dresden-Rossendorf (FZD). The system is based on a 6 MV-TandetronTM accelerator produced by High Voltage Engineering Europe (HVEE). The AMS facility is specified for measurements of 10Be, 14C, 26Al, 36Cl, 41Ca and 129I with isotopic ratios of 10-10 - 10-16 and precision better than 0.3% for 14C/12C.
The system uses a bouncer sequential injector with two Cs-sputter ion sources and a 54°-electrostatic analyser (ESA). On the high-energy site it has a 90°-analysing magnet, Faraday-Cups for stable nuclides, a 35°-ESA, a post-stripper foil, and a 30°-vertical magnet for suppression of interfering species, and gas ionisation chamber for detection of radionuclides [1].
The Cockroft-Walton type high voltage generator provides a terminal voltage of up to 6 MV. The system is additionally equipped with a multipurpose ion injector containing a third Cs-sputter ion source and a duoplasmatron for high-energy ion implantation and ion-beam materials analysis.
[1] M. Arnold et al., accepted for Nucl. Instr. and Meth. B (Proceedings
of IBA-2009).
Keywords: Accelerator mass spectrometry; AMS
DPG Frühjahrstagung der Sektion AMOP (SAMOP), 08.-12.03.2010, Hannover, Deutschland
Fulltext from www.dpg-verhandlungen.de
Comparison among MCNP-based depletion codes applied to burnup calculations of pebble-bed HTR lattices
Bomboni, E.; Cerullo, N.; Fridman, E.; Lomonaco, G.; Shwageraus, E.
The double-heterogeneity characterising pebble-bed high temperature reactors (HTRs) makes Monte Carlo based calculation tools the most suitable for detailed core analyses. These codes can be successfully used to predict the isotopic evolution during irradiation of the fuel of this kind of cores. At the moment, there are many computational systems based on MCNP that are available for performing depletion calculation. All these systems use MCNP to supply problem dependent fluxes and/or microscopic cross sections to the depletion module. This latter then calculates the isotopic evolution of the fuel resolving Bateman's equations.
In this paper, a comparative analysis of three different MCNP-based depletion codes is performed: Montburns2.0, MCNPX2.6.0 and BGCore. Monteburns code can be considered as the reference code for HTR calculations, since it has been already verified during HTR-N and HTR-N1 EU project. All calculations have been performed on a reference model representing an infinite lattice of thorium-plutonium fuelled pebbles. The evolution of k-inf as a function of burnup has been compared, as well as the inventory of the important actinides.
The k-inf comparison among the codes shows a good agreement during the entire burnup history with the maximum difference lower than 1%. The actinide inventory prediction agrees well. However significant discrepancy in Am and Cm concentrations calculated by MCNPX as compared to those of Monteburns and BGCore has been observed. This is mainly due to different Am-241 (n,γ) branching ratio utilized by the codes.
The important advantage of BGCore is its significantly lower execution time required to perform considered depletion calculations. While providing reasonably accurate results BGCore runs depletion problem about two times faster than Monteburns and two to five times faster than MCNPX.
Keywords: HTR; Pebble-bed; MCNP; Monte-Carlo depletion codes; MCNPX; Monteburns; BGCore
Nuclear Engineering and Design 240(2010), 918-924
Silicon Cluster Aggregation in Silica Layers
Fitting, H.-J.; Fitting Kourkoutis, L.; Salh, R.; Kolesnikova, E. V.; Zamoryanskaya, M. V.; von Czarnowski, A.; Schmidt, B.
Scanning transmission electron microscopy (STEM) in combination with electron energy loss spectroscopy (EELS) and cathodoluminescence (CL) have been used to investigate Si+-implanted amorphous silicon dioxide layers and the formation of Si nanoclusters. The microstructure of the Si doped silica films was studied by energy filtered transmission electron microscopy (EFTEM) in a 200 kV FEI Tecnai F20 TEM. The samples were amorphous, thermally grown 500 nm SiO2 layers on Si substrate doped by Si+ ions with an energy of 150 keV up to an atomic dopant fraction of about 4 at%. A thermal post-annealing leads to formation of silicon clusters with sizes 1-5 nm and concentrations of about 1018 cm-3. Respective cathodoluminescence spectra in the near IR region indicate such structural changes by appearance of an additional band at 1.35 eV as well as additional emission bands in the visible green-yellow region.
Keywords: Si ion implantation; Nanoclusters; Understoichiometric silica; Cathodoluminescence
Solid State Phenomena 156-158(2010), 529-533
DOI: 10.4028/www.scientific.net/SSP.156-158.529
Parallel Hardware-accelerated Particle in Cell (PiC) Physics using MPI, pThreads and CUDA: Implementing a Prototype
Juckeland, G.; Bussmann, M.
Presentation of the parallel communication scheme implemented by PIConGPU
Keywords: particle-in-cell; algorithm; parallel; pic; simulation; gpu; cluster; network; mpi; threads; pthreads; message-passing-interface; laser; plasma
Herausforderungen des HPC in Deutschland, 29.09.2009, Leogang, Österreich
New approaches investigating production rates of in-situ produced terrestrial cosmogenic nuclides
Merchel, S.; Braucher, R.; Benedetti, L.; Bourlès, D.
In-situ produced cosmogenic nuclides have proved to be valuable tools for environmental and Earth sciences. However, accurate application of this method is only possible, if terrestrial production rates in a certain environment over a certain time period and their depth-dependence within the exposed material are exactly known. Unfortunately, the existing data and models differ up to several tens of percent.
Thus, one of the European project CRONUS-EU goals is the high quality calibration of the 36Cl production rate by spallation at independently dated surfaces. As part of fulfilling this task we have investigated calcite-rich samples from four medieval landslide areas in the Alps: Mont Granier, Le Claps, Dobratsch, and Veliki Vrh (330-1620 m, 1248-1442 AD).
For investigating the depth-dependence of the different nuclear reactions, especially, the muon- and thermal neutron-induced contributions, we have analysed mixtures of carbonates and siliceous conglomerate samples - for 10Be, 26Al, and 36Cl - exposed at different shielding depths and taken from a core drilled in 2005 at La Ciotat, France (from surface to 11 m shielding).
AMS of 36Cl was performed at LLNL and ETH, 10Be and 26Al at ASTER.
Acknowledgments: Thanks to V. Alfimov, M. Arnold, G. Aumaître, J. Borgomano, R. Finkel, I. Mrak, and J.M. Reitner.
Keywords: accelerator mass spectrometry; cosmogenic nuclides; TCN
DPG Frühjahrstagung des Arbeitskreises Atome, Moleküle, Quantenoptik und Plasmen (AMOP), 08.-12.03.2010, Hannover, Deutschland
Die Jagd nach dem Feldrekord - Forschung in hohen Magnetfeldern
Wosnitza, J.
es hat kein Abstract vorgelegen!
Studium Generale "Naturwissenschaft Aktuell", 03.12.2009, Dresden, Deutschland
Was sind Magnetfelder? Wo findet man sie? Wie erzeugt man sie und zu was sind sie nutze? Antworten auf diese Fragen sollen in dem Vortrag durch Vorstellung der weltweiten Bestrebungen, immer höhere Magnetfelder zu erreichen, gegeben werden. Ähnlich wie z. B. Druck und Temperatur haben magnetische Felder einen tief greifenden Einfluss auf den Zustand und Zustandsänderungen der Materie. Untersuchungen von Materialien in hohen Magnetfeldern sind daher mittlerweile Standard und eine Vielzahl von Anwendungen in unserem täglichen Leben sind ohne Magnetfeldeffekte undenkbar. In der Forschung wird der stetig wachsende Bedarf an möglichst großen Magnetfeldstärken durch Hochfeldlaboratorien abgedeckt. In dem neu aufgebauten Hochfeld-Magnetlabor Dresden sollen demnächst gepulste Magnetfelder bis zu 100 Tesla erzeugt werden. Erste Hochfeldmagnete sind in Betrieb und seit 2007 hat neben der Eigenforschung der Nutzerbetrieb begonnen. Der momentane Status des Labors, die Schwierigkeiten, die zur Erzeugung so hoher Magnetfelder überwunden werden müssen, und exemplarische wissenschaftliche Ergebnisse aus Hochfeldstudien sollen vorgestellt werden.
Technische Universität München, 13.11.2009, München - Garching, Deutschland
Research at the new Dresden High Magnetic Field Laboratory
High magnetic fields are one of the most powerful tools available to scientists for the study, modification, and control of the state of matter. The application of magnetic fields, therefore, has become a commonly used instrument for condensed-matter physics. For the observation of many phenomena very high magnetic fields are essential. Consequently, the demand for the highest possible magnetic-field strengths is increasing. At the Dresden High Magnetic Field Laboratory (Hochfeld-Magnetlabor Dresden, HLD), that in 2007 has opened its doors for external users, pulsed magnetic fields up to 70 T are available and a European record field of 87.2 T have been reached. The laboratory has set the ambitious goal of reaching 100 T on a 10 ms timescale. As a unique feature, a free-electron-laser facility next door allows high-brilliance radiation to be fed into the pulsed field cells of the HLD, thus making possible high-field magneto-optical experiments in the range 3-250 µm. Cryotechniques and different sample probes for a broad range of experimental techniques custom designed for the pulsed magnets are readily available for users. In-house research of the HLD focuses on electronic properties of strongly correlated materials at high magnetic fields. Besides introducing some highlights of the HLD experimental infrastructure, some recent scientific research results will be presented. This includes e.g. the detection of Shubnikov-de Haas oscillations in electron-doped high-temperature superconductors that allowed to unravel a drastic change of the Fermi-surface topology upon doping [1]. Furthermore, pulsed-field experiments at the HLD allowed to observe the field-induced conductance switching in single-walled carbon nanotubes
6th International Symposium on High Magnetic Field Spin science in 100T, 07.-09.12.2009, Sendai, Japan
Optical spectroscopy of superconductors in terahertz frequency range
Pronin, A.
Universität Göttingen, 16.11.2009, Göttingen, Deutschland
Exploring the spin-1/2 frustrated square lattice model with high-field magnetization studies
Tsirlin, A. A.; Schmidt, B.; Skourski, Y.; Nath, R.; Geibel, C.; Rosner, H.
We report on high-field magnetization measurements for a number of layered vanadium phosphates that were recently recognized as spin- 1/2 frustrated square lattice compounds with ferromagnetic nearest-neighbor couplings (J1) and antiferromagnetic next-nearest-neighbor couplings (J2). The saturation fields of the materials lie in the range from 4 to 24 T and show excellent agreement with the previous estimates of the exchange couplings deduced from low-field thermodynamic measurements. The consistency of the high-field data with the regular frustrated square lattice model provides experimental evidence for a weak impact of spatial anisotropy on the nearest-neighbor couplings in layered vanadium phosphates. The variation in the J2 /J1 ratio within the compound family facilitates the experimental access to the evolution of the magnetization curve upon the change in the frustration magnitude. Our results support the recent theoretical prediction by Thalmeier et al. [Phys. Rev. B 77, 104441 (2008)] and give evidence for the enhanced bending of the magnetization curves due to the increasing frustration of the underlying spin system
Interplay of frustration and magnetic field in the two-dimensional quantum antiferromagnet Cu(tn)Cl-2
Orendacova, A.; Cizmar, E.; Sedlakova, L.; Hanko, J.; Kajnakova, M.; Orendac, M.; Feher, A.; Xia, J.; Yin, L.; Pajerowski, D.; Meisel, M.; Zelenak, V.; Zvyagin, S.; Wosnitza, J.
Specific heat and ac magnetic susceptibility measurements, spanning low temperatures (T >= 40 mK) and high-magnetic fields (B <= 14 T), have been performed on a two-dimensional (2D) antiferromagnet Cu(tn)Cl-2 (tn=1,3-diaminopropane=C3H10N2). The compound represents a S = 1/2 spatially anisotropic triangular antiferromagnet realized by a square lattice with nearest-neighbor (J/kB = 3 K), frustrating next-nearest-neighbor (0 < J'/J < 0.6), and interlayer (|J''/J| approximate to 10-3) interactions. The absence of long-range magnetic order down to T = 60 mK in B = 0 and the T-2 behavior of the specific heat for T <= 0.4 K and B >= 0 are considered evidence of a high degree of 2D magnetic order. In fields lower than the saturation field, Bsat = 6.6 T, a specific heat anomaly, appearing near 0.8 K, is ascribed to bound vortex-antivortex pairs stabilized by the applied magnetic field. The resulting magnetic phase diagram is remarkably consistent with the one predicted for a square lattice without a frustrating interaction, expect that Bsat is shifted to values lower than expected. Potential explanations for this observation, as well as the possibility of a Berezinski-Kosterlitz-Thouless (BKT) phase transition in a spatially anisotropic triangular magnet with the collinear Neel ground state, are discussed.
Effects of two gaps and paramagnetic pair breaking on the upper critical field of SmFeAsO0.85 and SmFeAsO0.8F0.2 single crystals
Lee, H.-S.; Bartkowiak, M.; Park, J.-H.; Lee, J.-Y.; Kim, J.-Y.; Sung, N.-H.; Cho, B. K.; Jung, C.-H.; Kim, J. S.; Lee, H.-J.
We investigated the temperature dependence of the upper critical field [Hc2(T)] of fluorine-free SmFeAsO0.85 and fluorine-doped SmFeAsO0.8F0.2 single crystals by measuring the resistive transition in low static magnetic fields and in pulsed fields up to 60 T. Both crystals show that Hc2(T)'s along the c axis [Hc2 c(T)] and in an ab-planar direction [Hc2 ab(T)] exhibit a linear and a sublinear increase, respectively, with decreasing temperature below the superconducting transition. Hc2(T)'s in both directions deviate from the conventional one-gap Werthamer-Helfand-Hohenberg theoretical prediction at low temperatures. A two-gap nature and the paramagnetic pair-breaking effect are shown to be responsible for the temperature-dependent behavior of Hc2 c and Hc2 ab, respectively.
Modifications in structural and optical properties of Mn-ion implanted CdS thin films
Chandramohan, S.; Kanjilal, A.; Strache, T.; Tripathi, J. K.; Sarangi, S. N.; Sathyamoorthy, R.; Som, T.
In this paper, we report on modifications in structural and optical properties of CdS thin films due to 190 keV Mn-ion implantation at 573 K. Mn-ion implantation induces disorder in the lattice, but does not lead to the formation of any secondary phase, either in the form of metallic clusters or impurity complexes. The optical band gap was found to decrease with increasing ion fluence. This is explained on the basis of band tailing due to the creation of localized energy states generated by structural disorder. Enhancement in the Raman scattering intensity has been attributed to the enhancement in the surface roughness due to increasing ion fluence. Mn-doped samples exhibit a new band in their photoluminescence spectra at 2.22 eV, which originates from the d–d (4T1 → 6A1) transition of tetrahedrally coordinated Mn2+ ions.
Keywords: CdS thin films; Mn-ion implantation; Structural properties; Optical properties
Applied Surface Science 256(2009)2, 465-468
DOI: 10.1016/j.apsusc.2009.07.015
Ion irradiation of permalloy: From thin magnetic films to lateral exchange spring nanostructures
Strache, T.; Reichel, L.; Wintz, S.; Fritzsche, M.; Mönch, I.; Raabe, J.; Martin, N.; McCord, J.; Körner, M.; Markó, D.; Romstedt, F.; Fassbender, J.
Due to its low coercivity and negligible magnetostriction, permalloy (Ni80Fe20) is one of the most used materials in thin film and micro/nano magnetism. By means of ion irradiation the magnetic properties can be modified [1], and in combination with a lithographically defined mask or the use of a focused ion beam magnetic patterning can be achieved [2]. The changes of the magnetic properties due to ion-solid- interaction must be related to different origins, e.g. direct implantation, surface sputtering and interfacial mixing. Their respective influence depends strongly on the chosen multilayer system as well as on the implantation conditions.
Here we present a systematic study of irradiation of permalloy with common ion species. Special emphasis is put on the separation of the effect of direct implantation from mixing and sputtering. By transferring this knowledge to laterally resolved irradiation, direct exchange coupled magnetic stripes of submicron width are created. The magnetization reversal process of this lateral exchange spring structures depends on the interaction between adjacent soft and hard magnetic stripes. By scaling the stripe sizes down, fundamental questions regarding the maximum domain wall density and the domain–wall interaction may be addressed.
[1] J. Fassbender et al., Physical Review B 73, 184410 (2006).
[2] J. McCord et al., Advanced Materials 20, 2090 (2008).
Workshop Ionenstrahlphysik, 06.-08. April 2009, Friedrich-Schiller-Universität Jena, 06.04.2009, Jena, Deutschland
Tuning Coercivity in CoCrPt-SiO2 Hard Disk Material
Strache, T.; Tibus, S.; Springer, F.; Rohrmann, H.; Albrecht, M.; Lenz, K.; Fassbender, J.
In order to increase the storage density of modern computer disk drives and to push the superparamagnetic limit to the smallest achievable bit sizes further, smaller grains with even larger magnetic anisotropies are required, which are accompanied by large coercive fields obstructing the writing process. One route to overcome this problem is to independently reduce the coercive field without altering anisotropy and remanence by tailoring the intergranular exchange in granular CoCrPt-SiO2 films. Here we demonstrate that by means of ion implantation of Co and Ne a continuous reduction of the coercive field can be achieved without significant modification of the remaining magnetic parameters. In addition to the magnetization reversal behavior of the entire film investigated by magneto-optic Kerr effect and SQUID magnetometry, also the magnetic domain configuration in the demagnetized state is imaged by magnetic force microscopy.
Keywords: magnetism; ion irradiation; magnetic storage; hard disk; coercivity; anisotropy; magnetic domains
DPG Spring Meeting of the Condensed Matter Section, March 22-27, 2009, Dresden, Germany, 25.03.2009, Dresden, Deutschland
Hochintensitätslaser und ihre Anwendungen
Bussmann, M.; Kroll, F.
Der Vortrag gibt einen Überblick über die nichtlinearen optischen Effekte die im Aufbau und der Diagnostik hochintensiver Kurzpulslaser eine Rolle spielen sowie über die Anwendungen moderner Hochleistungslaser in der Strahlenphysik.
The talk gives an overview of those nonlinear optical effects which are important for building high-intensity short-pulse lasers and their diagnostics as well as the application of high-power lasers in beam physics.
Keywords: high-intensity lasers; nonlinear optics; diagnostics; ultra-short laser pulses; particle acceleration; beam physics; radiation sources
Lectures on laser and plasma physics, 16.12.2009, TFH Wildau, Berlin, Germany
Di-Electrons from Resonances in Nucleon-Nucleon Collisions
Kaptari, L. P.; Kämpfer, B.
The contribution of the low-lying nucleon resonances P-33(1232), P-11(1440), D-13(1520), and S-11(1535) to the invariant-mass spectra of di-electrons stemming from the exclusive processes pp -> pp e(+)e(-) and pn -> pn e(+)e(-) is investigated within a fully covariant and gauge-invariant diagrammatical approach. We employ, within the one-boson exchange approximation, effective nucleon-meson interactions including the exchange mesons pi, eta, sigma, omega, and rho as well as excitations and radiative decays of the above low-lying nucleon resonances. The total contribution of these resonances is dominant; however, bremsstrahlung processes in pp and, in particular, pn collisions at beam energies of 1-2 GeV are still significant in certain phase-space regions.
Surface nanostructures by single highly charged ions
Facsko, S.; Heller, R.; El-Said, A.; Meissl, W.; Aumayr, F.
It has recently been demonstrated that the impact of individual, slow but highly charged ions on various surfaces can induce surface modifications with nanometer dimensions. Generally, the size of these surface modifications (blisters, hillocks, craters or pits) increases dramatically with the potential energy of the highly charged ion, while the kinetic energy of the projectile ions seems to be of little importance. This paper presents the currently available experimental evidence and theoretical models and discusses the circumstances and conditions under which nanosized features on different surfaces due to the impact of slow highly charged ions can be produced.
Keywords: nanostructures; highly charge ions; AFM
Journal of Physics: Condensed Matter 21(2009)22, 224012
Interaction of uranium(VI) towards glutathione - an example to study different functional groups in one molecule
Frost, L.; Geipel, G.; Viehweger, K.; Bernhard, G.
Glutathione, the most abundant thiol compound of the cell, has a great binding potential towards heavy metal ions. Hence it might influence the distribution of actinides on a cellular level. The unknown strength of the interaction of uranium(VI) with glutathione at physiologically relevant pH is subject of this paper and was studied with UV-vis spectroscopy and time-resolved laser-induced fluorescence spectroscopy (TRLFS). The complex stability constant of UO2H2GS+ at 0 ionic strength, log β121 , was calculated to be 39.09 ± 0.15 and 39.04 ± 0.02 in case of UV-vis spectroscopy and TRLFS respectively. Therefore the average formation constant for UO22+ + H2GS- = UO2H2GS+ at 0 ionic strength can be assigned to be log K11 = 19.83±0.15.
Furthermore it was demonstrated, that derivatization of the ligand associated with an enhancement of the ligand's spectroscopic properties can be used for the determination of complex stability constants and to assess the coordination chemistry more detailed. Using UV-vis spectroscopy, the stability constant of the complex between UO22+ and glutathione pyruvate S-conjugate, a well absorbing ligand in contrast to glutathione, was calculated to be > 39.24 ± 0.08. Furthermore the interaction of UO22+ with glutathione derivatized with the fluorescent label monobromobimane was examined with femtosecond laser fluorescence spectroscopy. Thereby the stability constant of the 1:1 complex was determined to be > 39.35 ± 0.02. Although the thiol group of glutathione was blocked a strong coordination was found. Thus a significant involvement of the thiol group in the coordination of U(VI) can be excluded.
Keywords: Uranium; Glutathione; Complexation; Derivatization; UV-vis spectroscopy; TRLFS
APSORC'09, 29.11.-04.12.2009, Napa, USA
Proceedings in Radiochemistry 1(2011), 357-362
DOI: 10.1524/rcpr.2011.0063
Phase separation in carbon:transition metal nanocomposite thin films
Berndt, M.
Wissenschaftlich-Technische Berichte / Helmholtz-Zentrum Dresden-Rossendorf; FZD-527 2009
Visualization of freckle formation induced by forced melt convection in solidifying GaIn alloys
Boden, S.; Eckert, S.; Gerbeth, G.
A bottom-up solidification of a Ga-25%In alloy under the influence of buoyancy-driven and electromagnetically driven convection was investigated by X-ray radioscopy. The main effect of the flow on the solidification is determined by the flow-induced redistribution of solute concentration which results in a change of the growth direction of the dendrites and the preference of secondary branches for an accelerated or decelerated growth. The experiments demonstrate how the interdendritic flow contributes to the formation of spacious segregation freckles.
Keywords: directional solidification; dendritic growth; convection; segregation; X-ray radioscopy
Materials Letters 64(2010), 1340-1343
Advanced spectroscopic synchrotron techniques to unravel the intrinsic properties of dilute magnetic oxides: the case of Co:ZnO
Ney, A.; Opel, M.; Kaspar, T. C.; Ney, V.; Ye, S.; Ollefs, K.; Kammermeier, T.; Bauer, S.; Nielsen, K.-W.; Goennenwein, S. T. B.; Engelhard, M. H.; Zhou, S.; Potzger, K.; Simon, J.; Mader, W.; Heald, S. M.; Cezar, J. C.; Wilhelm, F.; Rogalev, A.; Gross, R.; Chambers, S. A.
The use of synchrotron-based spectroscopy has revolutionized the way we look at matter. X-ray absorption spectroscopy (XAS) using linear and circular polarized light offers a powerful toolbox of element-specific structural, electronic, and magnetic probes that is especially well suited for complex materials containing several elements. We use the specific example of Zn1−xCoxO (Co:ZnO) to demonstrate the usefulness of combining these XAS techniques to unravel its intrinsic properties. We demonstrate that as long as phase separation or excessive defect formation is absent, Co:ZnO is paramagnetic. We can establish quantitative thresholds based on four reliable quality indicators using XAS; samples which show ferromagnet-like behaviour fail to meet these quality indicators, and complementary experimental techniques indeed prove phase separation. Careful analysis of XAS spectra is shown to provide quantitative information on the presence and type of dilute secondary phases in a highly sensitive, non-destructive manner.
New Journal of Physics 12(2010), 013020
Capped Colloidal Particular: A Model System for Spin Arangements
Erbe, A.
no abstract submitted
Keywords: colloids; soft matter; model systems
SPP 1296 Workshop, 13.-14.10.2009, Bayreuth, Deutschland
Colloids: Mesoscopic model systems of matter on a nano scale
Leiderer, P.; Erbe, A.
Colloidal suspensions consist of small particles in (mostly) aqueous medium, which allow to model phenomena in condensed matter on a mesoscopic scale. Due to the dominant length and time scales such systems are readily accessible by means of video microscopy. In this talk examples for both the structure of colloidal particle ensembles and transport phenomena in colloidal systems will be discussed. In the case of structure formation, e.g., configurations of particles with a mesoscopic "spin" will be presented, realized by colloidal spheres with permanent magnetic caps, which allow one to model the spin configurations of magnetic clusters. The transport investigations focus on phenomena in narrow channels and on "active swimmers", i.e. capped magnetic particles which are propelled through the surrounding liquid by a catalytic reaction and can be steered by an external magnetic field.
Juelich Soft Matter Days 2009, 10.-13.11.2009, Bonn, Deutschland
Confined longitudinal acoustic phonon modes in free-standing Si membranes coherently excited by femtosecond laser pulses
Hudert, F.; Bruchhausen, A.; Issenmann, D.; Schecker, O.; Waitz, R.; Erbe, A.; Scheer, E.; Dekorsy, T.; Mlayah, A.; Huntzinger, J.-R.
In this Rapid Communication we report the first time-resolved measurements of confined acoustic phonon modes in free-standing Si membranes excited by fs laser pulses. Pump-probe experiments using asynchronous optical sampling reveal the impulsive excitation of discrete acoustic modes up to the 19th harmonic order for membranes of two different thicknesses. The modulation of the membrane thickness is measured with fm resolution. The experimental results are compared with a theoretical model including the electronic deformation potential and thermal stress for the generation mechanism. The detection is modeled by the photoelastic effect and the thickness modulation of the membrane, which is shown to dominate the detection process. The lifetime of the acoustic modes is found to be at least a factor of 4 larger than that expected for bulk Si.
Keywords: Elemental semiconductors and insulators; Time resolved luminescence; Mechanical modes of vibration; Free films
Physical Review B 79(2009), 201307 (R)
Molecular Electronics: A Review of Experimental Results
Erbe, A.; Verleger, S.
Molecular electronics aims for scaling down electronics to its ultimate limits by choosing single molecules as the building blocks of active devices. The advantages of this approach are the high reproducibility of molecular synthesis on the nanometer scale, the ability of molecules to form large structures by self-assembly, and the huge versatility of molecular complexes. On the other hand, conventional contacting techniques cannot form contacts on the single molecule scale and imaging techniques nowadays cannot provide a detailed image of such junctions. Therefore, the fabrication has to rely to some degree on self-organization of the constituents. The proof that a molecule has been contacted successfully can only be given by indirect methods, for example by measuring the current transport through the junctions. Here we give an overview of various techniques that were used successfully to contact molecules and to characterize them electrically. The techniques range from methods to contact single molecules to such which can be used to characterize ensembles of molecules. Especially, the comparison between such different techniques shows that a single measurement is always prone to artefacts originating from the unknown microscopic details of the junctions. It is therefore necessary to perform a statistically relevant number of measurements in order to resolve molecular properties. Various properties of the molecules can be studied. Special examples axe the influence of conformational changes of the molecules, differences between various coupling endgroups of the molecules and effects of light-irradiation onto the molecular junctions.
Keywords: Molecular electronic devices; Nanoelectronic devices; Electronic transport in nanoscale materials and structures
Applying contactless inductive flow tomography to a continuous casting model
Wondrak, T.; Galindo, V.; Gerbeth, G.; Gundrum, T.; Stefani, F.; Timmel, K.
The contactless inductive flow tomography (CIFT) is able to reconstruct the velocity field in electrically conducting melts by measuring the induced magnetic field outside the melt. In this paper, we apply this method to the flow field in a small model of a continuous casting mould. It is shown that the flow structure in general, and the jet position and the intensity in particular, can be reliably determined, using a moderate number of sensors.
Keywords: continuous casting; industrial tomography; liquid metal flow measurement
12th MHD Days, 08.-9.12.2009, Potsdam, Germany
Alfven wave experiments with liquid rubidium
Gundrum, T.; Hüller, J.; Stefani, F.; Gerbeth, G.; Herrmannsdörfer, T.; Putzke, C.; Arnold, F.
We present first experiments at the Dresden High Magnetic Field Laboratory, with liquid rubidium inserted into a high pulsed magntic field and describe the resulting effects when the Alfven velocity crosses the sound speed.
Keywords: Alfven; Rubidium; Alkali metal; High Magnetic Field; MHD
12. MHD-Tag, 08.-9.12.2009, Potsdam, Germany
Cyclin-dependent kinases Cdk4 and Cdk6: targets for cancer treatment and visualization.
Graf, F.; Köhler, L.; Mosch, B.; Steinbach, J.; Pietzsch, J.
Cancerogenesis is closely associated with deregulated cell proliferation and, consequently, aberrant cell cycle control. The first phase of the cell cycle (G1) comprises important steps for initiation of DNA replication and subsequent cell division. The activation and coordination of G1 phase is accomplished by enzymes of serine/threonine kinase family. As members of this protein family and important regulators of early cell cycle machinery, cyclin-dependent kinases 4 and 6 (Cdk4/6) associate with regulatory protein cyclin D, and phosphorylate retinoblastoma protein pRb. Hyperphosphorylated pRb dissociates from E2F transcription factors and triggers transcription of genes required for further G1 phase progression. Hence, Cdk4/6 were identified as essential and critical activators of G1 phase in human embryogenesis, homeostasis, and cancer development; G1 phase progression in cell cycle by phosphorylation of retinoblastoma protein pRb and thua, activation of gene transcription.
In 80% of human tumors the Cdk4/6-cyclin D/ pRb/ E2F pathway is altered provoked by multiple mechanisms. In tumor formation, hyperactivation of Cdk4/6 is often a result of overexpression, silencing, and epigenetic alteration of their regulators or substrates. On the other hand, disruption of Cdk4/6-associated cell cycle control in cancer is also caused directly by mutations and amplification of Cdk4/6 themself. Cdk4/6 protein amplification was found in a wide spectrum of solid tumors and blood cell cancer, e.g., gliomas, lymphomas, melanomas, carcinomas, and leukemias. In consequence, Cdk4/6 were considered to be attractive targets for pharmacological anti-cancer drug development. In the recent years, Cdk4/6 inhibitors of high potency and selectivity against other kinases, especially other cyclin-dependent kinases, were developed and evaluated. One of these compounds, a pyrido[2,3-d]pyrimidine-7-one derivative currently is undergoing clinical trials for cancer therapy.
Though, potent and selective pyrido[2,3-d]pyrimidine-7-one Cdk4/6 inhibitors are not only promising new compounds for cancer therapy, but also for visualization and functional characterization of human tumors. Radiolabeled Cdk4/6 inhibitors could be of particular interest for the assessment of Cdk4/6 protein status and Cdk4/6 activity of human tumors by non-invasive imaging technique positron-emission-tomography (PET). PET affords the opportunity of three-dimensional imaging of physiological processes in vivo. Additionally, PET would provide pharmacological data of radiolabeled Cdk4/6 inhibitors, which may help to estimate the applicability of the compounds for tumor therapy. In this regard, positron-emitting Cdk4/6 inhibitors were designed, synthesized and characterized in our institute for the first time. The radiolabeled compounds and their nonradioactive analogs are based on the lead structure of pyrido[2,3-d]pyrimidine-7-one CKIA.
First, iodine-containing pyrido[2,3-d]pyrimidine-7-one derivatives CKIA and CKIB were evaluated concerning their efficacy and suitability as Cdk4/6 inhibitors in human tumor cell lines. The compounds showed both significant and specific inhibition of tumor cell proliferation and G1 phase arrest by targeting the Cdk4/6-cyclin D/ pRb/ E2F signaling pathway [2]. The iodine substituent of CKIA and CKIB represents an attractive site for an isotopic substitution with the positron emitter iodine-124 (124I). 124I-labeled Cdk4/6 inhibitors [124I]CKIA and [124I]CKIB were evaluated concerning their radiopharmacological properties in cellular radiotracer uptake studies, biodistribution studies, and small animal PET studies in NMRI nu/nu mice bearing the human squamous cell carcinoma tumor FaDu [3]. With 4.18 d half-life, 124I affords extended radiopharmacological evaluation and imaging studies using PET. Nevertheless, high positron energy and minor positron emission (26%) are disadvantages, especially for the resolution of PET images. ....
OncoPost & OncoPeople The official Newspaper of the ECCO 15 – 34th ESMO Congress 3(2009), 10
Applying the contactless inductive flow tomography to a model of continuous steel casting
Stefani, F.; Wondrak, T.; Gundrum, T.; Gerbeth, G.
We utilize the contactless inductive flow tomography (CIFT) for visualizing the flow of GaInSn in a model of continuous steel casting. Since for thin slab casting the velocity can be assumed to be mainly two-dimensional it is sufficient to apply only one external magnetic field and to measure the induced fields at the narrow faces of the mould. In a first step we show that a numerically determined flow field can be reconstructed by CIFT with an empirical correlation coefficient of about 75 per cent. Then we apply the method to various single-phase and two-phase flows in the real model and show that typical flow features can be reliably detected.
GAMM 2010, 22.-26.03.2010, Karlsruhe, Germany
Functionalized Mineral Surfaces: Sorption Mechanisms of Growth Proteins on the Surface of Bone Replacement Materials (BioMin)
Fischer, H.; Koczur, K.; Lindner, M.; Jennissen, H. P.; Meissner, M.; Zurlinden, K.; Mueller-Mai, C.; Seifert, G.; Oliveira, A.; Morawetz, K.; Gemming, S.
In the field of biomaterials, more specifically of materials which are used for medical implants, recent research is focused on the interface between implant material and biological environment. Among crystalline and glassy bone substitutes calcium- and phosphorus-based oxides are of special interest, because their chemical composition can be adjusted to natural bone. Materials containing an appropriate ratio of oxides of calcium, phosphorus and other physiologically compatible constitutents such as silicate and alkalis are bioactive and degradable in vivo. Therefore, the strategy in using such bone substitute implants is that these materials are slowly degraded inside the body and successively substituted by natural bone tissue.
Especially granular media with particles from appropriate calcium alkali orthophosphates, such as [Ca2KNa(PO4)2], exhibit a strongly enhanced biodegradability, but the sponge-like structure of the bone can not be remodelled with granulates. Yet, generative manufacturing techniques (rapid prototyping) nowadays allow to build up even large, three-dimensional structures that are adapted to macro-/microscopic structural bone characteristics.
However, the natural bioactivity of calcium phosphates and related inorganic compounds is limited. bone remodelling is impeded, in particular when larger bone defects are to be restored by this class of material. Yet, the growth of bone tissue can significantly be stimulated by so-called Bone Morphogenetic Proteins (BMPs), proteins that are synthesized during build-up of bone tissue by the human body. Since a couple of years it has become possible to produce BMPs synthetically and couple them to surfaces of bone replacement materials. After the successful bioactivation of metallic Titanium-based prosthetic surfaces with BMP, the present study is devoted to elucidating the mechanism of protein coupling and especially the desorption kinetics of BMP on mineral surfaces.
Those experimental studies are accompanied by a scale-bridging simulation of the BMP sorption process from physiologic solution as a function of the local pH-value and the structure formation at the solid-liquid interface. With this knowledge the project BioMin yields a significant contribution to develop and manufacture tailored bone substitute implants, so that degradation of the substitute material and build-up of new bone tissue can go hand-in-hand in vivo.
Keywords: bone replacement; apatite; phosphate; BMP; molecular modeling
GeoDresden 2009, 30.09.-02.10.2009, Dresden, Deutschland
Simulation of bone replacement materials
Gemming, S.
Keywords: bone replacement; biomaterials; bioglasses; apatite; calcium phosphate; molecular modeling
Seminar on Topical Problems in Theoretical Physics, 16.12.2009, Chemnitz, Deutschland
Density-functional theory in materials science
Simulations of materials behaviour are an important component of materials development when measurements are indirect and gain from theoretical interpretation, when the 'ideal' experiment is impeded by technological limitations, or when novel concepts and possible routes to their practical implementation are explored. Empirical physical models can be specifically tailored and have been employed successfully for the first two tasks, but the transfer to new tasks beyond the original application requires careful parameter reassessment. Quantum mechanical models, on the other hand, afford an a priori parameter-free access to materials properties on the nanoscale. In particular the density-functional theory provides computationally efficient access to the electronic structure of materials in the electronic ground state. Derived quantities such as forces, stresses, and responses to external electric or magnetic fields allow for the calculation of atom arrangements, lattice constants, elastic tensors, polarisabilities, dielectric and piezoelectric constants, or conductivity of nanosized systems or of systems with nanoscale building blocks. After a short introduction to the fundamentals of density-functional theory the applicability and the limitations of the approach for condensed matter systems will be discussed.
Keywords: density-functional theory; materials science; electronic structure calculations
FZD Theory Seminar Series, 17.12.2009, Dresden-Rossendorf, Deutschland
Quasiantiferromagnetic 120° Néel state in two-dimensional clusters of dipole-quadrupole-interacting particles on a hexagonal lattice
Mikuszeit, N.; Baraban, L.; Vedmedenko, E.; Erbe, A.; Leiderer, P.; Wiesendanger, R.
The magnetostatic interactions of colloidal particles, "capped" with radially magnetized Co/Pt multilayers, are modeled. Motivated by experiment the particles are arranged in microscopic two-dimensional clusters on a hexagonal lattice and are free to rotate. The thermodynamically stable states of clusters containing up to 108 particles are calculated theoretically by means of Monte Carlo simulations in the framework of multipole expansion. It is shown analytically that radially magnetized hemispheres have higher-order multipole moments beyond the dipole. Depending on geometrical details also even order moments appear. The even order moments break the inversion symmetry of the magnetic potential of a single particle. For a specific mixing ratio of dipole and quadrupole moments, the experimentally observed antiferromagnetic 120° Néel state in the clusters is found.
Keywords: Magnetostatics; General theory and models of magnetic ordering; Magnetic liquids
Overview of Superconducting Photo Injectors
Arnold, A.; Teichert, J.
The success of most of the proposed electron accelerator projects for future FELs, ERLs or 4th generation light sources is contingent upon the development of an appropriate source to generate the electrons with an unprecedented combination of high-brightness, low emittance and high average current.
An elegant way is to combine the high brightness of RF guns with the superconducting technology. This concept was first proposed at the University of Wuppertal*. In 2002, the successful operation of a SRF photo-injector could be demonstrated at FZD for the 1st time**. However, this type of injectors is still in the R&D phase.
Challenges are the design of the cavity with its specific geometry, the choice of the photocathode type, its life time, a possible cavity contamination, the problems on coupling of high-average power into the cavity and the risk of beam excitation of higher order cavity modes.
During the last years several R&D projects have been launched. Most of them pursue different approaches to deal with these issues. This contribution gives an overview on the progress of the SRF photo-injector development based on the most prominent projects in the world.
Keywords: RF gun combine with superconducting technology; SRF photo injector; photocathode; overview
Overview of Superconducting Photo Injectors, 20.-25.09.2009, Berlin, Deutschland
14th International Workshop on RF Superconductivity (SRF09), 20.-25.09.2009, Berlin, Germany
Overview of Superconducting Photo Injectors, 20-26
Link zum Talk from srf2009.bessy.de
Link zum Paper MOOBAU03 from srf2009.bessy.de
Low Emittance Polarized Electron Source Based on FZD Superconducting RF Gun
Xiang, R.; Arnold, A.; Michel, P.; Murcek, P.; Teichert, J.
The state-of-art DC guns with GaAs type photocathodes have been successfully operated as polarized electron sources for accelerators, but the beam emittance is regretfully very high because of the bunch
compressing after the gun. Though not all of the high energy physics experiments using polarized beams are sensitive to the source emittance, but a new source with lower emittance can simplify the injector
system and lower radiation load during the beam transport. Normal conducting rf gun can produce beams with low emittance, but the limit on vacuum is still an open question for the currently designed RF guns. In
this paper a new type of polarized source is reported: FZD polarized SRF gun, i.e. FZD superconducting rf gun with GaAs-type photocathode. The SRF gun is able to produce 1mA CW beam with 9.5MeV energy
and 1 mm mrad emittance. It has higher accelerating field and thus lower emittance than DC guns, at the same time much better vacuum condition than RF guns. Based on the successful running experience
in last two years, SRF gun applied with GaAs type cathode is considered as a promising alternative for current polarized guns. Some interesting questions will be discussed here, including the back bombardment
on cathode, cathode dielectric loss in strong RF field and the cathode time response.
Keywords: state of art DC guns; GaAs Type photocathodes; SRF gun
The XIIIth International Workshop on Polarized Sources, Targets & Polarimetry, 07.-11.09.2009, Ferrara, Italien
CFD-simulations and experiments on steam condensation in polydisperse bubbly flows
Schmidtke, M.; Krepper, E.; Lucas, D.; Beyer, M.
Bubble condensation in sub-cooled water is a complex process, to which various phenomena contribute. Since the condensation rate depends on the interfacial area density, bubble size distribution changes caused by breakup and coalescence play a crucial role.
To investigate the involved phenomena and their complex interplay, new experiments on steam bubble condensation in vertical co-current steam/water flows have been have been carried out in the TOPFLOW test facility in Dresden, which consists of an 8m long vertical DN200 pipe (inner diameter: 195mm). Steam is injected into the pipe and the development of the bubbly flow is measured at different height levels with a wire mesh sensor. By varying the steam nozzle diameter (1mm or 4 mm) the initial bubble size can be influenced. Larger bubbles come along with a lower interfacial area density and therefore they condensate slower (see figure). In addition to previous experiments (Lucas & Prasser, 2007) also the steam velocity is measured by correlating the signals of two wire mesh sensors installed in a small distance to each other. In the new experiment also the pressure drop along the pipe is measured as well as the temperature at selected points (Lucas et al., 2009). The additional sensors allow for choosing a defined gas inflow pressure as well as a distinct sub-cooling temperature at the injection location. Here steam pressures between 1-2 MPa and sub-cooling temperatures from 2 to 4 Kelvin were applied. Due to the pressure drop along the pipe, the saturation temperature falls towards the upper pipe end. This affects the sub-cooling temperature and can even cause re-evaporation in the upper part of the test section.
In second part of the present contribution, the new TOPFLOW condensation experiments are compared with simulations using an extended MUSIG approach. This approach has been developed in cooperation with ANSYS-CFX for the computation of condensation in polydispersed bubbly flows with CFD. This extended MUSIG approach includes the transition of bubbles to smaller size groups due to condensation as well as the shift of the bubble size distribution due to pressure changes. In the second part of the present contribution, simulations with the extended MUSIG approach are compared with the new TOPFLOW condensation experiments. The new CFD approach is able to catch all relevant phenomena qualitatively, such as bubble condensation and evaporation (if the saturation temperature falls below the water temperature) and radial bubble size segregation. Crucial for the condensation simulations is the modeling of coalescence and breakup, which still needs to be improved. The presented condensation experiments and the extended MUSIG approach are a useful basis for validating models for polydispersed bubbly flows.
Keywords: water/steam flow; condensation; polydisperse; MUSIG; TOPFLOW
Jahrestagung Kerntechnik 2010, 04.-06.05.2010, Berlin, Deutschland
Jahrestagung Kertnechnik 2010, 04.-06.05.2010, Berlin, Deutschland
The Properties of Normal Conducting Cathodes in FZD Superconducting Cavity
Xiang, R.; Arnold, A.; Buettig, H.; Janssen, D.; Justus, M.; Lehnert, U.; Michel, P.; Murcek, P.; Schamlott, A.; Schneider, C.; Schurig, R.; Teichert, J.; Staufenbiel, F.
The superconducting rf photoinjector (SRF photoinjector) is one of the latest application of SC technology in the accelerator field. Since superconducting cathodes with high QE are not available up to now, normal conducting cathode material is the main choice for the SRF photoinjectors. However, the compatibility between the cathode and the cavity is one of the challenges for this concept. The SRF gun with Cs2Te cathode has been successfully operated under the collaboration of BESSY, DESY, FZD, and MBI. In this paper, some experience gained in the gun commissioning will be concluded. The results of the properties of Cs2Te photo cathodes in the cavity will be presented, such as the Q.E., life time, regeneracy, dark current and thermal emittance. At the same time, the cavity quality is showed to be steady before and after the cathode working.
Keywords: Superconducting RF Photoinjector; SC technology; Accelerator; SRF Gun; Cs2Te cathode
14th International Conference on RF Superconductivity (SRF 2009), 20.-25.09.2009, Berlin, Deutschland
Hydrogen depth profiling with nanometre resolution
Munnik, F.; Heller, R.; Neelmeijer, C.
The amount of hydrogen in semiconductors can highly influence electrical, physical and chemical properties on a microscopic as well as on a macroscopic scale. Many applications in the micro electronics industry, in solar cell research and in surface science require the precise knowledge of the actual hydrogen concentration and its concentration distribution near the surface. Nuclear Reactions Analysis (NRA) has been successfully established as a standard technique in chemical analysis within recent decades. In the particular case of hydrogen depth profiling the so-called 15N-method became one of the most successful, non-destructive and standard-free analysing technique. This method is based on the nuclear reaction 15N(1H,αγ)12C, which is characterized by a pronounced resonance at 6.385 MeV. By variation of the initial 15N-ion energy the depth in the sample where the 15N resonance energy is reached can be easily adjusted according to the particular ion stopping in the material. Thus, a depth sensitive measurement of the absolute H-concentration becomes feasible. Using grazing incident angles the depth resolution near the surface can reach 1 nm. Detection limits under optimum conditions are as low as 0.05 at%. Fundamentals of the 15N-method and experimental set-up at the 6 MV accelerator at the FZD as well as particular examples of hydrogen depth profiling in ongoing state-of-the-art experiments are presented.
Keywords: NRA; high-resolution; hydrogen
2nd International Conference on Functional Nanocoatings, 28.-31.03.2010, Dresden, Germany
Enhancement in the photocatalytic nature of nitrogen-doped PVD-grown titanium dioxide thin films
Tavares, C. J.; Marques, S. M.; Viseu, T.; Teixeira, V.; Carneiro, J. O.; Alves, E.; Barradas, N. P.; Munnik, F.; Girardeau, T.; Rivière, J.-P.
Nitrogen-doped titanium dioxide semiconductor photocatalytic thin films have been deposited by unbalanced reactive magnetron physical vapor deposition on glass substrates for self-cleaning applications. In order to increase the photocatalytic efficiency of the titania coatings, it is important to enhance the catalysts absorption of light from the solar spectra. Bearing this fact in mind, a reduction in the titania semiconductor band-gap has been attempted by using nitrogen doping from a coreactive gas mixture of N2:O2 during the titanium sputtering process. Rutherford backscattering spectroscopy was used in order to assess the composition of the titania thin films, whereas heavy-ion elastic recoil detection analysis granted the evaluation of the doping level of nitrogen. X-ray photoelectron spectroscopy provided valuable information about the cation-anion binding within the semiconductor lattice. The as-deposited thin films were mostly amorphous, however, after a thermal annealing in vacuum at 500 °C the crystalline polymorph anatase and rutile phases have been
developed, yielding an enhancement in the crystallinity. Spectroscopic ellipsometry experiments enabled the determination the refractive index of the thin films as a function of the wavelength, while from the optical transmittance it was possible to estimate the semiconductor indirect band-gap of these coatings, which has been proven to decrease as the N-doping increases. The photocatalytic performance of the titania films has been characterized by the degradation rate of an organic reactive dye under UV/visible irradiation. It has been found that for a certain critical limit of 1.19 at. % of nitrogen doping in the titania anatase crystalline lattice enhances the photocatalytic behavior of the thin films and it is in accordance with the observed semiconductor band-gap narrowing to 3.18 eV. By doping the titania lattice with nitrogen, the photocatalytic activity is enhanced under both UV and visible light.
Running Experience of the Superconducting RF Photoinjector at FZD
Xiang, R.; Buettig, H.; Janssen, D.; Justus, M.; Lehnert, U.; Michel, P.; Murcek, P.; Schamlott, A.; Schneider, C.; Arnold, A.; Teichert, J.; Schurig, R.; Kamps, T.; Rudolph, J.; Schenk, M.; Staufenbiel, F.; Will, I.; Klemz, G.
More and more electron accelerator projects for FELs, ERLs or 4th generation light sources require "super" electron beams with high brightness, low emittance, and high average current. Under this background, much attention is paid on the research and development of new electron sources. A Superconducting RF photoinjector within a collaboration of HZB, DESY, FZD, and MBI is designed to improve the beam quality for ELBE IR-FEL users, and at the same time to test this kind of promising injector concept. The main design parameters of this gun are the final electron energy of 9.5 MeV, 1 mA average current, and transverse normalized emittances (rms) of 1 mm mrad at 77 pC and 2.5 mm mrad at 1 nC bunch charge. In this paper the results of the RF and beam parameter measurements with Cs2Te photo cathodes will be presented, and the experience for the gun running gained at the first beam experiment will be concluded, including the life time and the compatibility of the normal conducting photocathode in SC cavity, the cavity properties after the cathode's inserting.
Keywords: electron accelerator for FEL; 4th generation; high brightness; low emittance; high average; new electron source; superconducting RF photoinjector
The 31st International Free Electron Laser Conference FEL 2009,, 23.-28.08.2009, Liverpool, UK
Interstitial-Mediated Diffusion in Germanium under Proton Irradiation
Bracht, H.; Schneider, S.; Klug, J. N.; Liao, C. Y.; Lundsgaard Hansen, J.; Haller, E. E.; Nylandsted Larsen, A.; Bougeard, D.; Posselt, M.; Wündisch, C.
We report experiments on the impact of 2.5 MeV proton irradiation on self-diffusion and dopant diffusion in germanium (Ge). Self-diffusion under irradiation reveals an unusual depth independent broadening of the Ge isotope multilayer structure. This behavior and the observed enhanced diffusion of B and retarded diffusion of P demonstrates that an interstitial-mediated diffusion process dominates in Ge under irradiation. This fundamental finding opens up unique ways to suppress vacancy-mediated diffusion in Ge and to solve the donor deactivation problem that hinders the fabrication of Ge-based nanoelectronic devices.
Keywords: Germanium; dopant diffusion; proton irradiation
Physical Review Letters 103(2009), 255501 | CommonCrawl |
znso4 oxidation number
To make an electrically neutral compound, the copper must be present as a 2+ ion. The oxidation number of a central atom in a coordination entity is the charge it would bear if all the ligands were removed along with the electron pairs that were shared with the central atom (McNaught and Wilkinson, 1997). Solving for x, it is evident that the oxidation number for sulfur is +4. [1ΔH f (ZnSO4 (aq)) + 1ΔH f (Mg (s))] - [1ΔH f (Zn (s)) + 1ΔH f (MgSO4 (aq))] [1(-1063.17) + 1(0)] - [1(0) + 1(-1376.12)] = 312.95 kJ 312.95 kJ (endothermic) Find the Oxidation Numbers KClO. Oxidation number is used to define the state of oxidation of an element. Inter state form of sales tax income tax? 6. Answer to: Zn + H2SO4 arrow ZnSO4 + H2. THis means, Zn is undergoing oxidation or being oxidized or acting as an reducing agent. The oxidation number of O in Li20 and KN03 is -2. The Cu has a decrease oxidation number of +2 going to 0 therefore is being reduced the Zn has an increase in oxidation number going from 0 to +2. In this oxidation state decreases. 1 ; View Full Answer Same with ZnSO4.. but why is it +2 and not +10?? Zinc Sulfate (1:1) Zinc Sulphate ZnSO4 Zinc(II) Sulfate Zinc(2+) Sulfate Zinc Sulfate Anhydrous Zinc Sulfate, Anhydrous Zinc Sulfate (anhydrous) Molar Mass of O4SZn Oxidation State of O4SZn Calculate Reaction Stoichiometry Calculate Limiting Reagent Chemical reaction. Explaining what oxidation states (oxidation numbers) are. Oxidation number for Zn is increased to +2 while H reduce to 0. The oxidation state, sometimes referred to as oxidation number, describes the degree of oxidation (loss of electrons) of an atom in a chemical compound.Conceptually, the oxidation state, which may be positive, negative or zero, is the hypothetical charge that an atom would have if all bonds to atoms of different elements were 100% ionic, with no covalent component. In addition to -2 oxidation state sulfur exhibits +2,+4, and +6 oxidation states respectively. Who is the actress in the saint agur advert? The oxidation state, sometimes referred to as oxidation number, describes the degree of oxidation (loss of electrons) of an atom in a chemical compound.Conceptually, the oxidation state, which may be positive, negative or zero, is the hypothetical charge that an atom would have if all bonds to atoms of different elements were 100% ionic, with no covalent component. Zinc sulfate is an inorganic compound.It is used as a dietary supplement to treat zinc deficiency and to prevent the condition in those at high risk. Oxidation Number Calculator is a free online tool that displays the oxidation number of the given chemical compound. Inter state form of sales tax income tax? In a compound or simple ion: group 1 metals are always +1, group 2 metals are always +2. Preparation of ZnSO4 solution. So, reduction decreases the oxidation number (valency). Any atom by itself is neutral and its oxidation number is zero. How tall are the members of lady antebellum? Oxidation states simplify the whole process of working out what is being oxidised and what is being reduced in redox reactions. Assign an oxidation number of -2 to oxygen (with exceptions). This fits with the charge of the peroxide anion ($2 \times -1 = -2$), and as $\ce{BaO2}$ is a neutral compound, the sum of all oxidation numbers is 0. BYJU'S online oxidation number calculator tool makes the calculation faster and it displays the oxidation number in a fraction of seconds. Facebook. 0 1. sapphire5434. Points, as so4 having a charge of -2 therfore oxidation state of zn is+2… The oxidation number is 2. explaination: SO4 has an oxidation number of -2, and the molecule (MnSO4) has an overall oxidation number of 0. a) O in the compound O2 b) Ag in AgNO3 c) Mn in MnO2 d) Zn in ZnSO4 e) Cl in ClO3- f) C in CO3^2- g) Mn in KMnO4 h) S in MgSO3 i) S in Na2S2O3 Thanks! Sum of the oxidation number of all the atoms present in a neutral molecule is zero. Twitter. Oxidation number of Zn in ZnSO4 is +2, In Zn oxidation state of Zn is 0, When reacted with H2SO4, Zn displaces H to form ZnSO4, whose oxidation state is +2, Since there is an increase in oxidation number therefore Zn is oxidised. Zn has an oxidation number (valency) of 0, and is oxidised to Zn2+, an oxidation number of 2+, ... CuSO4 is reduced to Cu and Zn is oxidised to ZnSO4. Yes. Which species is oxidized? In almost all cases, oxygen atoms have oxidation numbers of -2. Does pumpkin pie need to be refrigerated? What are the disadvantages of primary group? In the compound, ZnS04(aq), the oxidation number of the zinc ion (Zn (aq)) is +2. The oxygen state refers to the number of electrons gained or forfeited by the element in order to achieve a noble gas configuration. Cu goes from ON. Notice that the oxidation state isn't simply counting the charge on the ion (that was true for the first two cases but not for this one). +2 to 0. This is a sneaky one! Oxidation number is 0 when element is in atom or molecule state. Enn. How long will the footprints on the moon last? Where can i find the fuse relay layout for a 1990 vw vanagon or any vw vanagon for the matter? (+3)+(-2) = Z Z=(+1). Amante y aprendiz de las letras. The oxidation stats of Zn(s) is 0 Cu2+ in CuSO4 is +2 Cu(s) is 0 Zn2+ in ZnSO4 is +2. Siento enorme interés por la química supramolecular, la nanotecnología, y los compuestos organometálicos. When did organ music become associated with baseball? ZnSO4: the Zn2+ has a +2 oxidation number and SO42- (which we can keep as a compound) has a -2 oxidation number. 5. In the compound, the oxidation number of the calcium ion (Ca (aq)) is +2. The oxidation number of IA elements (Li, Na, K, Rb, Cs and Fe) is +1 and the oxidation number IIA elements (Be, Mg, Ca, Sr, Ba and Ra) is +2. The oxidation numbers of all atoms in a neutral compound must add up to zero. Side effects of excess supplementation may include abdominal pain, vomiting, headache, and tiredness.. To find this oxidation number, it is important to know that the sum of the oxidation numbers of atoms in compounds that are neutral must equal zero. The oxidation number of SO2 is 0. To find the oxidation number of sulfur, it is simply a matter of using the formula SO2 and writing the oxidation numbers as S = (x) and O2 = 2(-2) = -4. Start studying Oxidation Reduction Quiz. ... An increase in the oxidation number of an atom suggests oxidation brought by the loss of electron/s during the chemical reaction. 2) increase in oxidation number. Sum of oxidation number of all the atoms of a complex ion is equal to the net charge on the ion. The material on this site can not be reproduced, distributed, transmitted, cached or otherwise used, except with prior written permission of Multiply. An oxidation number is defined as the charge an atom would carry if the molecule or polyatomic ion were completely ionic.When calculating the oxidation number of an element in a compound, treat all the elements present as if they are present as ions, EVEN if they are clearly part of a covalent molecule. The oxidation number of IA elements (Li, Na, K, Rb, Cs and Fe) is +1 and the oxidation number IIA elements (Be, Mg, Ca, Sr, Ba and Ra) is +2. There is 4 main aspects to show an element get oxidised: 1) release of electron. What are the reacting proportions? Different ways of displaying oxidation numbers of ethanol and acetic acid. How long will the footprints on the moon last? What is the oxidation state of ZnSO4? Find the oxidation numbers of all atoms in the reaction. Print. oxidation number of … The most common form includes water of crystallization as the heptahydrate, with the formula Zn SO 4 •7H 2 O. The oxidation number of Ba is +II, and the oxidation number of each of the oxygens in the peroxide anion is -I. The oxidation number of H in H20 and CH4 is The oxidation number of F in MgF2 is -1. Oxidation number is 0 when element is in atom or molecule state. To maintain electrical neutrality as required for all compounds, the two nitrogen atoms must have a total oxidation charge of +10, so that each of the two nitrogen atoms has an oxidation number of +5. Why don't libraries smell like bookstores? Explaining what oxidation states (oxidation numbers) are. The oxidation number of Zn in ZnS is +2 and the oxidation number of S in ZnS is -2. The Oxidation State of Oxygen. What is the oxidation state of ZnSO4. Does pumpkin pie need to be refrigerated? The oxidation number of S in CS2 is -2. Since is in column of the periodic table, it will share electrons and use an oxidation state of . 1. OXIDATION STATE APPROACH: IDEAL IONIC BONDS. In the compound sulfur dioxide (SO2), the oxidation number of oxygen is -2. You wrote that its total charge was #-2#.. ... An increase in the oxidation number of an atom suggests oxidation brought by the loss of electron/s during the chemical reaction. How long was Margaret Thatcher Prime Minister? Answer to Mg(s) + ZnSO4(aq)---> MgSO4(aq)+Zn(s) 1. Out of these +4 and +6 are common oxidation states. It is the reducing agent. +2 The proper name for CuSO_4 is copper(II) sulphate - the Roman numerals denote the oxidation state for any atom which has variable oxidation states, such as transition metals. The sum of the oxidation numbers for an ion is equal to the net charge on the ion. Since is in column of the periodic table, it will share electrons and use an oxidation state of . Since is in column of the periodic table, it will share electrons and use an oxidation … Zn has an oxidation number (valency) of 0, and is oxidised to Zn2+, an oxidation number of 2+, So oxidation increases the oxidation number (valency). Sum of oxidation number of all the atoms of a complex ion is equal to the net charge on the ion. H+ has an oxidation number (valency) of +1, and is reduced to an oxidation number (valency) of 0. Who is the actress in the saint agur advert? There are a few exceptions to this rule: When oxygen is in its elemental state (O 2), its oxidation number is 0, as is the case for all elemental atoms. Remember the mnemonic "oil rig": oxidation is loss, reduction is gain. The oxidation number of O is usually -2, unless it's part of a hydroxide (and then it's -1) or bonded to fluorine (and then it's +2). Where can i find the fuse relay layout for a 1990 vw vanagon or any vw vanagon for the matter? There are several ways to work this out. You will know that it is +2 because you know that metals form positive ions, and the oxidation state will simply be the charge on the ion. The oxidation number of each atom can be calculated by subtracting the sum of lone pairs and electrons it gains from bonds from the number of valence electrons. Since sulfate (#"SO"_4^(2-)#) is already … i just don't get oxidation number for example, in CuSO4 O: (-2)x4= -8 S= -2 so shouldn't Cu be +10?? complete transfer of the electron pair to only one atom in the bond.. You wrote #"SO"_4^(2-)#, which is the sulfate polyatomic ion. When did Elizabeth Berkley get a gap between her front teeth? An oxidation number is used to indicate the oxidation state of an atom in a compound. Oxidation Number. I hope this will be helpful. RULES A pure element in atom form has an oxidation number … Balancing chemical equations. How long was Margaret Thatcher Prime Minister? Sum of the oxidation number of all the atoms present in a neutral molecule is zero. Recuperado de: occc.edu; WhatsApp. oxidation numbers. The oxidation numbers of all atoms in … 4) releasing of hydrogen. The oxidation number is the charge of each atom in the compound. The SO4 is just a spectator ion and doesn't participate in the reaction. The positive oxidation state is counting the total number of electrons which have had to be removed - starting from the element. Peroxides include hydrogen peroxide, H2O2. There are a few exceptions to this rule: When oxygen is in its elemental state (O 2), its oxidation number is 0, as is the case for all elemental atoms. dont worry its correct. There are a few exceptions to this rule: When oxygen is in its elemental state (O 2), its oxidation number is 0, as is the case for all elemental atoms. in Zn(s) the oxidation number is 0. oxidation number of a compound in its element state is 0. it changes only when compounds are formed. How tall are the members of lady antebellum? Figure 1. Copyright © 2020 Multiply Media, LLC. +2 The proper name for CuSO_4 is copper(II) sulphate - the Roman numerals denote the oxidation state for any atom which has variable oxidation states, such as transition metals. What are the oxidation numbers of of everything in the system2. ? There is 4 main aspects to show an element get oxidised: 1) release of electron. ; When oxygen is part of a peroxide, its oxidation number is -1. . Assigning Oxidation Numbers. When did organ music become associated with baseball? 5. zn is equal to +2 and so4 is equal to -2. those are the oxidation numbers. reaction: Fe + ZnSO4 = Fe(2)SO4(3) + Zn oxidation #s: 0 , 2+ , 3+ , 0 . Copyright © 2020 Multiply Media, LLC. A: The table for the calculation of number average molecular weight is given below.
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Schouten tensor
In Riemannian geometry the Schouten tensor is a second-order tensor introduced by Jan Arnoldus Schouten defined for n ≥ 3 by:
$P={\frac {1}{n-2}}\left(\mathrm {Ric} -{\frac {R}{2(n-1)}}g\right)\,\Leftrightarrow \mathrm {Ric} =(n-2)P+Jg\,,$
where Ric is the Ricci tensor (defined by contracting the first and third indices of the Riemann tensor), R is the scalar curvature, g is the Riemannian metric, $J={\frac {1}{2(n-1)}}R$ is the trace of P and n is the dimension of the manifold.
The Weyl tensor equals the Riemann curvature tensor minus the Kulkarni–Nomizu product of the Schouten tensor with the metric. In an index notation
$R_{ijkl}=W_{ijkl}+g_{ik}P_{jl}-g_{jk}P_{il}-g_{il}P_{jk}+g_{jl}P_{ik}\,.$
The Schouten tensor often appears in conformal geometry because of its relatively simple conformal transformation law
$g_{ij}\mapsto \Omega ^{2}g_{ij}\Rightarrow P_{ij}\mapsto P_{ij}-\nabla _{i}\Upsilon _{j}+\Upsilon _{i}\Upsilon _{j}-{\frac {1}{2}}\Upsilon _{k}\Upsilon ^{k}g_{ij}\,,$
where $\Upsilon _{i}:=\Omega ^{-1}\partial _{i}\Omega \,.$
Further reading
• Arthur L. Besse, Einstein Manifolds. Springer-Verlag, 2007. See Ch.1 §J "Conformal Changes of Riemannian Metrics."
• Spyros Alexakis, The Decomposition of Global Conformal Invariants. Princeton University Press, 2012. Ch.2, noting in a footnote that the Schouten tensor is a "trace-adjusted Ricci tensor" and may be considered as "essentially the Ricci tensor."
• Wolfgang Kuhnel and Hans-Bert Rademacher, "Conformal diffeomorphisms preserving the Ricci tensor", Proc. Amer. Math. Soc. 123 (1995), no. 9, 2841–2848. Online eprint (pdf).
• T. Bailey, M.G. Eastwood and A.R. Gover, "Thomas's Structure Bundle for Conformal, Projective and Related Structures", Rocky Mountain Journal of Mathematics, vol. 24, Number 4, 1191-1217.
See also
• Weyl–Schouten theorem
• Cotton tensor
| Wikipedia |
\begin{document}
\title{Fundamental Quantum Limits for Practical Devices}
\author{Ryo Namiki}\affiliation{Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan}
\date{\today} \begin{abstract} We present experimentally testable quantum limitations on the phase-insensitive linear amplification and phase conjugation with respect to the transformation of a Gaussian distributed set of coherent states following the footing to assess the success of continuous-variable quantum teleportation and quantum memory devices. The results enable us to compare the real device with the quantum limited device via feasible input of coherent states. \end{abstract}
\maketitle
An important role of theoretical physics is to derive the fundamental limitation on the performance of physical devices for manipulating the states of physical system. The controllability of the physical states over the existence of the quantum noise and its connections to quantum measurement are central objective in wide area of quantum physics
\cite{Cle10}. An elementary operation for signal processing is amplification and its quantum limitation is generally determined based on the canonical commutation relation \cite{amp}. A pertinent approach is optimal cloning of quantum states so as to address the limitation on amplifying quantum information \cite{rmp-clone,Cerf00,Cer05}. Those limitations are thought to be in the reach of experiments
\cite{amp2,Cle10,Sab07}.
Any physical process is described by a completely positive trace-preserving map referred to as quantum channel \cite{NC00,CV-RMP}. We often use the average fidelity as a figure of merit to estimate the performance of the process in quantum information science. The problems to find the quantum limit phase-insensitive linear amplifier and to optimize the cloning map for coherent states are equivalent when the figure of merit is the trace norm \cite{Gut06} or the joint fidelity \cite{Cer05,namiki07}. In the case of the most familiar amplification limit, the figure of merit is the ratio of the signal-to-noise ratios of the input and output fields \cite{amp}.
Besides the amplification, an interesting quantum-state manipulation
is the phase conjugation \cite{Cer01a,Bus03}. It corresponds to the universal not gate for qubit states \cite{Buz99} and to a transposition map for finite and infinite dimensional states \cite{Bus03}.
In contrast to those active signal processing, elementary devices for quantum communication and computation are designed to transfer quantum states in a rather passive manner. Actually, ideal quantum memory or quantum teleportation process is an identical map, which retrieves the input states without disturbance, and main step of quantum computation is to perform the unitary operations, which implies reversible transformation of quantum states. Toward the realization of useful quantum devices, a cornerstone is to prove the quantum coherence of the process by beating the classical limit achieved by the classical measure-and-prepare (MP) schemes \cite{Bra00,Ham05,namiki07,namiki08,Has10,Takano08}. The MP scheme is an entanglement breaking channel \cite{16} and surpassing the classical limit fidelity is a proof of entanglement. It is known that the optimal fidelity of the phase conjugation
can be achieved by a classical device \cite{Buz99,Bus03,Cer01a} and that the Gaussian phase-conjugation (time-reversal) map belongs to the entanglement breaking channel \cite{Hol08}.
To experimentally test the performance of the quantum device, an accessible input state is the coherent state.
It is theoretically simple to determine the classical (or other physical) limitation assuming a uniform set of input states because the figure of merit has a covariant property \cite{Cer05} and the group theoretical treatment is useful \cite{Chi05}. However,
neither testing the input-output relation for any coherent state nor assuming the displacement covariant property for the real device is feasible. In practical, available power of the input field is limited and the linearity of the real device is maintained only on a limited range of the input variable. In the case of optical or atomic continuous-variable quantum information processing \cite{CV-RMP}, the amplitude of the input coherent states has to be much smaller than the total photon number of the so-called local oscillator fields.
To avoid the problem, a Gaussian distribution has been employed to observe the performance on a flat distribution of the coherent-state amplitude, thereby one can determine the device performance by using coherent states with a feasible amount of phase-space displacement \cite{Bra00,Ham05,namiki07}. The classical limit fidelity was initially determined for essentially identical process such as quantum teleportation and quantum memory \cite{Bra00,Ham05}, and generalized for a class of the non-unitary processes by considering a transformation task to take the effect of loss and amplification into account \cite{namiki07}. The classical limit was also determined for a class of multi-mode gates \cite{Takano08}. It is worth noting that the classical capacity for bosonic quantum channels has been derived under the energy constraint \cite{Gio04}.
Remarkably, the quantum limitations on the amplification \cite{amp,Gut06,Cer05,namiki07} and phase-conjugation \cite{Cer01a,Bus03} are presented based on the uniform distribution and covariant property.
In order to give a solid foundation as an experimental science, it is crucial to address the quantum limitations under experimentally testable frameworks. Noting that the well-known quantum teleportation \cite{Furusawa98} and quantum memory \cite{Jul04} protocols serve as amplifiers via the gain control mechanism, it is natural to work with the fidelity-based figure of merit.
In this Letter, we consider the quantum limits of the phase-insensitive linear amplification and phase conjugation in terms of the average fidelity with respect to the Gaussian distributed set of coherent states. We derive a tight quantum limit fidelity for the phase-insensitive amplification task and show that this fundamental limit is achieved by the known Gaussian amplifier. We also derive a tight quantum limit fidelity for the phase conjugation task and show that this limit is achieved by a classical MP device.
In what follows the state vector with the Greek letter ``$\alpha $'' denotes the coherent state and the state vector with the Roman letter ``$n$'' denotes the number state, e.g., we write the coherent state in the number basis as $\ket{ \alpha}= e^{-|\alpha |^2 /2} \sum_{n=0}^\infty \alpha ^n \ket{ n} /\sqrt{n!}$. When we work on the state with two modes, we call the first system $A$ and the second system $B$.
Let us define the average fidelity of the physical process $\mathcal E$ for transformation task on the coherent states $\{|\sqrt N \alpha\rangle \} \to \{\ket{\sqrt \eta \alpha }\} $ with $N, \eta > 0$ by \begin{eqnarray}
F_{N, \eta, \lambda} (\mathcal E) &:=& \int p_\lambda( \alpha )\bra{\sqrt \eta \alpha }\mathcal E \Big(|\sqrt N \alpha \rangle \langle \sqrt N \alpha| \Big) \ket{\sqrt\eta \alpha} d^2 \alpha \nonumber \\ \label{eq1} \end{eqnarray} where the prior distribution of a symmetric Gaussian function with the inverse width of $\lambda >0 $ is given by \begin{eqnarray}
p_\lambda( \alpha ) := \frac{\lambda }{\pi} \exp (- \lambda |\alpha |^2 ).\label{eq2}\end{eqnarray} This distribution describes the uniform distribution in the limit $\lambda \to 0$.
The fidelity represents the average probability that the input state $ | \sqrt N \alpha \rangle $ is exactly transformed into the corresponding target state $ | \sqrt \eta \alpha \rangle $ by the process $\mathcal E$. When $\eta / N \ge 1 $, the transformation task implies the amplification of the coherent-state amplitude with the gain factor $\eta / N $. When $\eta = N =1$, the task is referred to as the unit-gain task and the fidelity estimates how well the input coherent state is retrieved at the output port. When $\eta / N < 1 $ the transformation suggests amplitude dumping. This is the case for practical transmission and storage processes, and the loss of fidelity can be seen as a deviation from the ideal lossy channel. When $N$ and $\eta $ are positive integers the task may be called $N$-to-$\eta$ cloning where the fidelity implies how well the transformation from $N$ copies $\ket{\alpha} ^{\otimes N}$ to $\eta$ copies $\ket{\alpha} ^{\otimes \eta}$ can be achieved. The \textit{quantum limit fidelity} is defined as an upper limit of the average fidelity $F ( \mathcal E)$ achieved by the completely positive trace-preserving map $\mathcal E$. We call the limit is \textit{tight} if the fidelity limit is achieved by a completely positive trace-preserving map. Note that, from Eqs. (\ref{eq1}) and (\ref{eq2}), by changing the integral parameter we can verify the identity: \begin{eqnarray}F_{N, \eta, \lambda} &=& F_{\frac{N}{\eta },1, \frac{\lambda}{\eta }}, = F_{1, \frac{\eta }{N}, \frac{\lambda}{N} }.\label{imm}\end{eqnarray}
\textit{Quantum optimal phase-insensitive linear amplifier.---}
Let us consider the amplification task $\{ |{ \alpha }\rangle \} \to \{ \ket{{\sqrt \eta }\alpha} \} $ with the gain $\eta > 1 $.
In the following we show that the fidelity $F_{1,\eta, \lambda}$ is bounded above by $\frac{1+ \lambda }{ \eta}$ for sufficiently small $\lambda$ and that this bound is achieved by the know Gaussian amplifier. Note that the tight quantum limit fidelity of attenuation task with $\eta \in [0,1]$ is unity \cite{namiki07}.
\textit{Proof.---} Let us consider the following integration \cite{namiki-up}
with the parameters $s \ge 0 $, $0 \le \kappa \le 1 $ and $0\le \xi < 1 $, \begin{eqnarray}
J_{ \mathcal E}(s, \kappa ,\xi ) &:=& \int d^2 \alpha p_s(\alpha ) \bra{ \alpha}_A \bra{\kappa \alpha^*}_B \mathcal\nonumber\\ & & \mathcal E_A \otimes I_B \left( \ket{\psi_\xi}\bra{\psi_\xi} \right ) \ket{\kappa \alpha^*}_B
\ket{ \alpha}_A \label{start} \end{eqnarray} where $\ket{\psi_\xi}= \sqrt{1-\xi ^2} \sum_{n=0}^\infty \xi^n\ket{n}\ket{n}$ is the two-mode squeezed state and $I$ represents the identity process. The integration can be connected to the average fidelity by \begin{eqnarray}
J_{ \mathcal E}(s,\kappa ,\xi ) &=& \frac{s(1-\xi^2)}{\lambda}
F_{N,1, \lambda} (\mathcal E ) \label{jj} \end{eqnarray} where the parameters are supposed to satisfy the following relations \begin{eqnarray} \lambda &=& s+ (1-\xi^2)\kappa ^2, \label{lam}\\ \sqrt N &=& \kappa \xi. \label{n} \end{eqnarray} From the condition $s\ge 0 $ with Eqs. (\ref{lam}) and (\ref{n}), we have \begin{eqnarray} \frac{\lambda}{1-\xi ^2} \le N+ \lambda . \label{scondition} \end{eqnarray}
We proceed to consider the upper bound of $J_{\mathcal E}$ instead of the upper bound of the fidelity $F(\mathcal E)$. For any physical process with the complete positivity and trace-preserving condition, $\rho_{\mathcal E} := \mathcal E \otimes I ( \ketbra{\psi_\xi}{\psi_\xi})$ is a density operator. Then, the maximum of $J_{\mathcal E}$ with respect to the optimization of the process $ {\mathcal E}$ is no larger than the maximum achieved by the optimization of the density operator $\rho_{\mathcal E}$ over the set of the whole physical states. Thus we have
\begin{eqnarray} \sup_{\mathcal E } J_{ \mathcal E}(s,\kappa ,\xi) &\le& \max_{\rho_{\mathcal E}} \tr \left[ \rho M \right] = \| M \| \label{cp1} \end{eqnarray}
where we define \begin{eqnarray} M:= \int p_s(\alpha) \ket{ \alpha }\bra{ \alpha } \otimes \ket{\kappa \alpha ^* }\bra{\kappa \alpha ^* } d^2\alpha \nonumber \end{eqnarray}
and $\| \cdot \|:= \max_{\braket{u}{u}=1} \bra{ u} \cdot \ket{u} $ stands for the maximum eigenvalue.
Since $M $ is a two-mode Gaussian state, its maximum eigenvalue is given from the symplectic eigenvalues of its covariance matrix \cite{Adess07}. Let us define the covariance matrix of a density operator on the two-mode field $ \rho $ \begin{eqnarray} \gamma_\rho := \langle \hat R \hat R^t + (\hat R \hat R^t)^t \rangle _{ \rho} - 2 \langle \hat R \rangle \langle \hat R^t \rangle _{ \rho} \nonumber \end{eqnarray} where $\hat R := (\hat x_A,\hat p_A,\hat x_B, \hat p_B)^t $ is the set of the quadrature operators of the mode $A$ and mode $B$ whose elements satisfy the canonical commutation relations $[\hat x_A, \hat p_A]=i $ and $[\hat x_B, \hat p_B]= i $. Then, the covariance matrix of the operator $M $ is calculated to be \begin{eqnarray} \gamma_{M } = \openone_4 + \frac{2}{s }\left(
\begin{array}{cc}
\openone_2 & \kappa Z \\
\kappa Z & \kappa ^2 \openone_2 \\
\end{array} \right), \nonumber
\end{eqnarray} where $\openone_4 :=\textrm{diag} (1,1,1,1) $, $\openone_2 := \textrm{diag}(1,1)$ and $ Z := \textrm{diag} (1,-1)$.
In order to diagonalize this matrix we define a matrix $U(r)$ corresponding to the two-mode squeezing operator $\hat U_r := e^{-i( \hat x_A \hat p_B +\hat x_B \hat p_A ) r } = e^{( \hat a ^\dagger \hat b^\dagger -\hat a \hat b ) r } $ through the transformation \begin{eqnarray} \hat U^\dagger \hat R \hat U &= &\left(
\begin{array}{cc}
\cosh r \openone_2 & \sinh r Z \\
\sinh r Z & \cosh r \openone_2 \\
\end{array} \right) \hat R
=: U(r) \hat R. \nonumber \end{eqnarray}
When the squeezing parameter satisfies $\tanh{2 r} = 2\kappa /(1+ s + \kappa ^2)$ the covariance matrix is diagonalized as
$U(-r) \gamma_{M } U^t (-r) = \textrm{diag} (\nu_+,\nu_+ ,\nu_-, \nu_-)$
where the symplectic eigenvalues are determined to be \begin{eqnarray} \nu_\pm = \left[ \sqrt{(1+s+ \kappa ^2 )^2 -4\kappa ^2}\pm (1-\kappa ^2) \right] /s. \nonumber \end{eqnarray}
Therefore, the diagonal form of $M $ is the product of the thermal states $T(\bar n _+ )\otimes T(\bar n _- ) $ with the mean photon numbers $\bar n_\pm = (\nu_\pm - 1) /2 $ where the thermal state with the mean photon number $\bar n $ is defined by \begin{eqnarray}
T( \bar n ) := {\frac{1 }{ 1+ \bar n }} \sum_{n =0}^{\infty} \left( \frac{\bar n }{ 1+ \bar n } \right)^{n } |n \rangle \langle n |. \nonumber \label{thermal} \end{eqnarray}
This implies the following form of the maximum eigenvalue with the help of Eqs. (\ref{lam}) and (\ref{n}):
\begin{eqnarray}
\|M \| &=& 4/{[(\nu_+ +1)(\nu_- +1 )] } \nonumber \\ &=& \frac{2s}{ N + \lambda +1 +\sqrt{(N + \lambda +1)^2 -4 N / \xi^2 } } . \nonumber \end{eqnarray}
Using this relation, Eqs. (\ref{jj}), (\ref{scondition}), and (\ref{cp1}) we have \begin{eqnarray}
\sup_{\mathcal E} F_{N,1,\lambda } (\mathcal E) &\le& \frac{ \lambda }{s(1-\xi ^2 )}\| M \| \nonumber \\ &\le & \frac{ 2(N+ \lambda )}{ N+\lambda +1 +\sqrt{(N+\lambda-1) ^2}}\nonumber \\
&=& \left\{ \begin{array}{cc} N+ \lambda & \textrm{if }(N+\lambda ) \le 1 , \\ 1 & \textrm{if } (N+\lambda ) > 1 . \\ \end{array}\right. \end{eqnarray} By taking the replacement $(N, \lambda) \to (1/\eta, \lambda /\eta )$ and using the identity of Eq. (\ref{imm}), we obtain the upper bound of the fidelity for the amplification task, \begin{eqnarray} \sup_{\mathcal E} F_{1,\eta,\lambda} (\mathcal E)
&\le& \left\{ \begin{array}{cc} \frac{1+ \lambda}{\eta } & \textrm{if } \eta \ge 1 +\lambda , \\ 1 & \textrm{if } \eta < 1 +\lambda . \\ \end{array}\right. \label{amplim} \end{eqnarray}
Next we consider the attainability of this bound. The Gaussian amplifier with the gain $g= \cosh ^2 r \ge 1$ is defined by ${\mathcal A}_g (\rho ):= \tr_B [ U_r\rho \otimes \ketbra{0}{0}_B U_r^\dagger ]$. It transforms the coherent state as
${\mathcal A} _g (\ketbra{\alpha }{\alpha })
= \frac{1 }{\pi (g-1 ) }\int e^{-\frac{| \beta |^2 }{g-1 }} \ketbra{\sqrt g \alpha + \beta}{\sqrt g \alpha + \beta } d^2\beta$.
This implies \begin{eqnarray}
F_{1,\eta, \lambda}({ \mathcal A}_g) &=& \frac{\lambda}{\lambda g +| \sqrt g- \sqrt{ \eta }|^2} \nonumber \\ &=& \frac{\lambda}{(\lambda +1) (\sqrt{ g} - \frac{\sqrt \eta}{ \lambda +1 } )^2+ \frac{\lambda \eta}{ \lambda +1 } }
\le \frac{1+ \lambda}{ \eta }, \nonumber \end{eqnarray} where the equality is achieved when
$g= \eta / (1+ \lambda )^2 \ge 1 $. Therefore, the upper one of Ineqs. (\ref{amplim}) is saturated by the Gaussian phase-insensitive amplifier if the distribution is sufficiently flat so as to satisfy $\eta \ge (1+\lambda )^2 $.
$\blacksquare$
In the limit of $\lambda \to 0$, our fidelity reproduces the quantum limit for the case of the uniform distribution $F_o= 1/\eta $
\cite{namiki07,Cer05}. As we can see, the fidelity value $(1+ \lambda )/\eta $ always exceeds the uniform limit $F_o$, and thus a naive comparison of the experimental fidelity with $F_o$ gives an illegal result or an overestimation on how well the experimental device is approximating the quantum limited device. In contrast, our result includes the effect of the finite distribution $\lambda$, and enables a legitimate estimation toward the fundamental quantum limitation.
\textit{Optimal phase conjugator.---}
Let us consider the phase-conjugation task $\{| \sqrt N \alpha \rangle \} \to \{\ket{ \alpha ^* }\}$ with $N>0$ and define the fidelity $F_{N,1,\lambda}^* (\mathcal E ):= \int d^2 \alpha p_\lambda (\alpha )\bra{\alpha^* }\mathcal E(| \sqrt N \alpha\rangle\langle {\sqrt N \alpha}|)\ket{\alpha^* } $. We can show that the optimal fidelity is given by \begin{eqnarray} \sup_{\mathcal E} F_{N,1,\lambda}^* (\mathcal E ) = \frac{N+\lambda }{N+ \lambda + 1 } , \label{pclimit} \end{eqnarray}
and is achieved by the classical MP scheme
\begin{eqnarray} \mathcal E_{MP}^* (\rho ):= \frac{1}{\pi} \int \bra{\alpha }\rho \ket{ \alpha} \ket{\frac{\sqrt{N} \alpha ^*}{N +\lambda} }\bra{\frac{\sqrt{N}\alpha ^* }{N +\lambda} } d^2 \alpha . \label{pceb} \end{eqnarray}
\textit{Proof.---} We start by defining $J_{ \mathcal E}^*(s,\xi,\kappa ) := \int d^2 \alpha p_s(\alpha ) \bra{ \alpha^*}_A \bra{\kappa \alpha^*}_B \mathcal E_A \otimes I_B \left( \ket{\psi_\xi}\bra{\psi_\xi} \right ) \ket{\kappa \alpha^*}_B \ket{ \alpha^*}_A $ similarly to Eq. (\ref{start}). Here, different from the previous case we assume a weaker constraint of $\kappa \ge 0 $. This suggests the phase-conjugation task with either attenuation or amplification.
Similar to Eq. (\ref{jj}) we can confirm the following relation with the help of Eqs. (\ref{lam}) and (\ref{n}): \begin{eqnarray}
J_{ \mathcal E}^*(s,\kappa ,\xi ) &=& \frac{s(1-\xi^2)}{\lambda}
F_{N,1, \lambda}^* (\mathcal E ). \label{jj-2} \end{eqnarray}
An upper bound of $ J_{ \mathcal E}^*(s,\xi,\kappa ) $ is given by the optimization of the density operator $\rho = \mathcal E \otimes I (\ketbra{\psi_ \xi}{\psi_ \xi}) $ over the physically possible states, namely, we have \begin{eqnarray} \sup_{\mathcal E} J_{ \mathcal E}^* (s,\kappa ,\xi ) &=& \max_{\rho } \tr [ \rho M^* ]
\le \| M ^* \| \label{ff} \end{eqnarray} where we define \begin{eqnarray}
M ^* = \int p_s(\alpha) \ket{ \alpha }\bra{ \alpha } \otimes \ket{\kappa \alpha }\bra{\kappa \alpha } d^2\alpha . \end{eqnarray}
This operator is also a two-mode Gaussian state, and its covariance matrix is calculated to be \begin{eqnarray} \gamma_{M^* } = \openone_4 + \frac{2}{s }\left(
\begin{array}{cc}
\openone_2 & \kappa \openone_2 \\
\kappa \openone_2 & \kappa ^2 \openone_2 \\
\end{array} \right). \nonumber
\end{eqnarray} This covariance matrix can be diagonalized by a beamsplitter transformation, and the symplectic eigenvalues are determined to be
$(\nu_+, \nu_-) = (1, 1+2(1+\kappa ^2 )/s )$. Hence, we have \begin{eqnarray}
\| M^* \| =4/ [(\nu_++1) (\nu_-+1) ]= \frac{s }{s+1+ \kappa ^2 }. \label{ongm} \end{eqnarray} Equations (\ref{ff}) and (\ref{ongm}) lead to \begin{eqnarray}
\sup_{\mathcal E } J_{ \mathcal E}^*(s,\kappa ,\xi )
&\le & \frac{s }{s+1+ \kappa ^2 }. \nonumber \end{eqnarray} Using this relation and Eqs. (\ref{lam}), (\ref{n}), (\ref{scondition}),
and, (\ref{jj-2}) we obtain the upper bound of the fidelity for the phase-conjugation task \begin{eqnarray}
\sup_{\mathcal E } F_{N,1, \lambda }^* (\mathcal E ) &\le& \frac{\lambda}{ (1-\xi^2)} \frac{1}{N+ \lambda +1 } \le \frac{N+\lambda }{N+ \lambda + 1 } . \nonumber \end{eqnarray} On the other hand, this bound is achieved by the MP scheme of Eq. (\ref{pceb}), i.e., $F_{N,1, \lambda } (\mathcal E_{MP}^* )= \frac{N+\lambda }{N+ \lambda + 1 }$ holds.
We thus obtain
Eq. (\ref{pclimit}).
$\blacksquare$
The value of the optimal fidelity for the covariant approach \cite{Cer01a,Bus03} is reproduced when we set $N=1$ and take the limit $\lambda \to 0$. Our result shows that the optimality of the classical device for the phase-conjugation task occurs beyond the case of the uniform distribution. The optimality of the classical device suggests the coincidence of the quantum limit and classical limit. Such a coincidence also occurs when the target states are orthogonal to each other \cite{namiki08}. Note that, when the optimization of the state $\rho_{\mathcal E}$ in Eq. (\ref{cp1}) is limited over the positive-partial-transpose states \cite{namiki-up}, the value of the optimal fidelity corresponds to the value of the optimal fidelity for the phase-conjugation task. Hence, for many of the tasks whose target states are given by the transpose of the input states, it is likely that the gap between the quantum limit and classical limit disappears.
In conclusion, we have presented quantum limitations on the phase-insensitive linear amplification and phase conjugation in terms of the average fidelity by assuming transformation tasks on a Gaussian distributed set of coherent states. Thereby, experimental test can be done by using coherent states with a finite amount of phase-space displacement on the same footing as the success criterion for continuous-variable quantum teleportation and quantum memory. It was also shown that both of the fidelity limits can be achieved by the known Gaussian machines and that the known results for the case of the uniform distribution are safely reproduced. The present results give a solid foundation to experimentally observe how well the real device approximates the quantum limited device in a legitimate manner.
R.N. acknowledges support from JSPS.
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Animal Suffering and the Darwinian Problem of Evil
John R. Schneider
John Schneider explores the problem that animal suffering, caused by the inherent nature of Darwinian evolution, poses to belief in theism. Examining the aesthetic aspects of this moral problem, Schneider focuses on the three prevailing approaches to it: that the Fall caused animal suffering in nature (Lapsarian Theodicy), that Darwinian evolution was the only way for God to create an acceptably good and valuable world (Only-Way Theodicy), and that evolution is the source of major, God-justifying beauty (Aesthetic Theodicy). He also uses canonical texts and doctrines from Judaism and Christianity - notably the book of Job, and the doctrines of the incarnation, atonement, and resurrection - to build on insights taken from the non-lapsarian alternative approaches. Schneider thus constructs an original, God-justifying account of God and the evolutionary suffering of animals. His book enables readers to see that the Darwinian configuration of animal suffering unveiled by scientists is not as implausible on Christian theism as commonly supposed.
Reciprocal Space Mapping of Epitaxial Materials Using Position-Sensitive X-ray Detection
S. R. Lee, B. L. Doyle, T. J. Drummond, J. W. Medernach, P. Schneider
Journal: Advances in X-ray Analysis / Volume 38 / 1994
Reciprocal space mapping can be efficiently carried out using a position-sensitive x-ray detector (PSD) coupled to a traditional double-axis diffractometer. The PSD offers parallel measurement of the total scattering angle of all diffracted x-rays during a single rocking-curve scan. As a result, a two-dimensional reciprocal space map can be made in a very short time similar to that of a one-dimensional rocking-curve scan. Fast, efficient reciprocal space mapping offers numerous routine advantages to the x-ray diffraction analyst. Some of these advantages arc the explicit differentiation of lattice strain from crystal orientation effects in strain-relaxed heteroepitaxial layers; the nondestructive characterization of the size, shape and orientation of nanocrystalline domains in ordered-alloy epilayers; and the ability to measure the average size and shape of voids in porous epilayers. Here, the PSD-based diffractometer is described, and specific examples clearly illustrating the advantages of complete reciprocal space analysis are presented.
Probing the high-redshift universe with SPICA: Toward the epoch of reionisation and beyond
Exploring Astronomical Evolution with SPICA
E. Egami, S. Gallerani, R. Schneider, A. Pallottini, L. Vallini, E. Sobacchi, A. Ferrara, S. Bianchi, M. Bocchio, S. Marassi, L. Armus, L. Spinoglio, A. W. Blain, M. Bradford, D. L. Clements, H. Dannerbauer, J. A. Fernández-Ontiveros, E. González-Alfonso, M. J. Griffin, C. Gruppioni, H. Kaneda, K. Kohno, S. C. Madden, H. Matsuhara, F. Najarro, T. Nakagawa, S. Oliver, K. Omukai, T. Onaka, C. Pearson, I. Perez-Fournon, P. G. Pérez-González, D. Schaerer, D. Scott, S. Serjeant, J. D. Smith, F. F. S. van der Tak, T. Wada, H. Yajima
Published online by Cambridge University Press: 26 December 2018, e048
With the recent discovery of a dozen dusty star-forming galaxies and around 30 quasars at z > 5 that are hyper-luminous in the infrared (μ LIR > 1013 L⊙, where μ is a lensing magnification factor), the possibility has opened up for SPICA, the proposed ESA M5 mid-/far-infrared mission, to extend its spectroscopic studies toward the epoch of reionisation and beyond. In this paper, we examine the feasibility and scientific potential of such observations with SPICA's far-infrared spectrometer SAFARI, which will probe a spectral range (35–230 μm) that will be unexplored by ALMA and JWST. Our simulations show that SAFARI is capable of delivering good-quality spectra for hyper-luminous infrared galaxies at z = 5 − 10, allowing us to sample spectral features in the rest-frame mid-infrared and to investigate a host of key scientific issues, such as the relative importance of star formation versus AGN, the hardness of the radiation field, the level of chemical enrichment, and the properties of the molecular gas. From a broader perspective, SAFARI offers the potential to open up a new frontier in the study of the early Universe, providing access to uniquely powerful spectral features for probing first-generation objects, such as the key cooling lines of low-metallicity or metal-free forming galaxies (fine-structure and H2 lines) and emission features of solid compounds freshly synthesised by Population III supernovae. Ultimately, SAFARI's ability to explore the high-redshift Universe will be determined by the availability of sufficiently bright targets (whether intrinsically luminous or gravitationally lensed). With its launch expected around 2030, SPICA is ideally positioned to take full advantage of upcoming wide-field surveys such as LSST, SKA, Euclid, and WFIRST, which are likely to provide extraordinary targets for SAFARI.
2138 Susceptibility to social influence is associated with alcohol self-administration and subjective alcohol effects
Alyssa Schneider, Bethany Stangl, Elgin R. Yalin, Jodi M. Gilman, Vijay Ramchandani
Journal: Journal of Clinical and Translational Science / Volume 2 / Issue S1 / June 2018
Published online by Cambridge University Press: 21 November 2018, pp. 47-48
OBJECTIVES/SPECIFIC AIMS: Peer groups are one of the strongest determinants of alcohol use and misuse. Furthermore, social influence plays a significant role in alcohol use across the lifespan. One of the factors that most consistently predicts successful treatment outcomes for alcohol use disorders is one's ability to change their social network. However, the concept of social influence as defined by suggestibility or susceptibility to social influence has not yet been studied as it relates to drinking behavior and acute subjective response to alcohol. Our objective was to examine the relationship between suggestibility and alcohol consumption and responses, using an intravenous alcohol self-administration (IV-ASA) paradigm in social drinkers. METHODS/STUDY POPULATION: Healthy, social drinkers (n=20) completed a human laboratory session in which they underwent the IV-ASA paradigm. This consisted of an initial 25-minute priming phase, where participants were prompted to push a button to receive individually standardized IV alcohol infusions, followed by a 125-minute phase during which they could push the button for additional infusions. IV-ASA measures included the peak and average breath alcohol concentration (BrAC) and number of button presses. Subjective responses were assessed using the Drug Effects Questionnaire (DEQ) and Alcohol Urge Questionnaire (AUQ) collected serially during the session. Participants completed the Multidimensional Iowa Suggestibility Scale (MISS) to assess suggestibility. The Alcohol Effects Questionnaire (AEFQ) was used to assess alcohol expectancies and the Timeline Followback questionnaire measured recent drinking history. RESULTS/ANTICIPATED RESULTS: After controlling for drinking history, greater suggestibility significantly predicted greater average BrAC, greater peak BrAC, and a greater number of button presses (p=0.03, p=0.02, p=0.04, respectively) during the early open bar phase. Suggestibility significantly predicted subjective alcohol effects following the priming phase which included "Feel," "Want," "High," and "Intoxicated" and was trending for "Like" (p=0.02, p=0.03, p=0.01, p=0.03, p=0.054, respectively) as well as AUQ (p=0.03). After controlling for drinking history, suggestibility significantly predicted "Feel," "Like," "High," and "Intoxicated" peak scores during the open bar phase (p=0.03, p=0.009, p=0.03, p=0.03, respectively). There was no association between suggestibility and "Want More" alcohol. Suggestibility was positively associated with three positive expectancies (global positive; p=0.04, social expressiveness; p=0.005, relaxation; p=0.03), and one negative expectancy (cognitive and physical impairment; p=0.02). DISCUSSION/SIGNIFICANCE OF IMPACT: These results indicate that social drinkers that were more suggestible had higher alcohol consumption, greater acute subjective response to alcohol, and more positive alcohol expectancies. As such, susceptibility to social influence may be an important determinant of alcohol consumption, and may provide insight into harmful drinking behavior such as binge drinking. Future analyses should examine the impact of suggestibility on alcohol-related phenotypes across the spectrum of drinking from social to binge and heavy drinking patterns.
P.002 Exosomal miR-204-5 and miR-632 in CSF are candidate biomarkers for frontotemporal dementia: a GENFI study
R Schneider, P McKeever, T Kim, C Graff, J van Swieten, A Karydas, A Boxer, H Rosen, B Miller, R Laforce, D Galimberti, M Masellis, B Borroni, Z Zhang, L Zinman, JD Rohrer, MC Tartaglia, J Robertson
Background: To determine whether exosomal microRNAs (miRNAs) in CSF of patients with FTD can serve as diagnostic biomarkers, we assessed miRNA expression in the Genetic FTD Initiative (GENFI) cohort and in sporadic FTD. Methods: GENFI participants were either carriers of a pathogenic mutation or at risk of carrying a mutation because a first-degree relative was a symptomatic mutation carrier. Exosomes were isolated from CSF of 23 -pre-symptomatic and 15 symptomatic mutation carriers, and 11 healthy non-mutation carriers. Expression of miRNAs was measured using qPCR arrays. MiRNAs differentially expressed in symptomatic compared to pre-symptomatic mutation carriers were evaluated in 17 patients with sporadic FTD, 13 patients with sporadic Alzheimer's disease (AD), and 10 healthy controls (HCs). Results: In the GENFI cohort, miR-204-5p and miR-632 were significantly decreased in symptomatic compared to pre-symptomatic mutation carriers. Decrease of miR-204-5p and miR-632 revealed receiver operator characteristics with an area of 0.89 [90% CI: 0.79-0.98] and 0.81 [90% CI: 0.68-0.93], and when combined an area of 0.93 [90% CI: 0.87-0.99]. In sporadic FTD, only miR-632 was significantly decreased compared to sporadic AD and HCs. Decrease of miR-632 revealed an area of 0.89 [90% CI: 0.80-0.98]. Conclusions: Exosomal miR-204-5p and miR-632 have potential as diagnostic biomarkers for genetic FTD and miR-632 also for sporadic FTD.
Galaxy Evolution Studies with the SPace IR Telescope for Cosmology and Astrophysics (SPICA): The Power of IR Spectroscopy
L. Spinoglio, A. Alonso-Herrero, L. Armus, M. Baes, J. Bernard-Salas, S. Bianchi, M. Bocchio, A. Bolatto, C. Bradford, J. Braine, F. J. Carrera, L. Ciesla, D. L. Clements, H. Dannerbauer, Y. Doi, A. Efstathiou, E. Egami, J. A. Fernández-Ontiveros, A. Ferrara, J. Fischer, A. Franceschini, S. Gallerani, M. Giard, E. González-Alfonso, C. Gruppioni, P. Guillard, E. Hatziminaoglou, M. Imanishi, D. Ishihara, N. Isobe, H. Kaneda, M. Kawada, K. Kohno, J. Kwon, S. Madden, M. A. Malkan, S. Marassi, H. Matsuhara, M. Matsuura, G. Miniutti, K. Nagamine, T. Nagao, F. Najarro, T. Nakagawa, T. Onaka, S. Oyabu, A. Pallottini, L. Piro, F. Pozzi, G. Rodighiero, P. Roelfsema, I. Sakon, P. Santini, D. Schaerer, R. Schneider, D. Scott, S. Serjeant, H. Shibai, J.-D. T. Smith, E. Sobacchi, E. Sturm, T. Suzuki, L. Vallini, F. van der Tak, C. Vignali, T. Yamada, T. Wada, L. Wang
IR spectroscopy in the range 12–230 μm with the SPace IR telescope for Cosmology and Astrophysics (SPICA) will reveal the physical processes governing the formation and evolution of galaxies and black holes through cosmic time, bridging the gap between the James Webb Space Telescope and the upcoming Extremely Large Telescopes at shorter wavelengths and the Atacama Large Millimeter Array at longer wavelengths. The SPICA, with its 2.5-m telescope actively cooled to below 8 K, will obtain the first spectroscopic determination, in the mid-IR rest-frame, of both the star-formation rate and black hole accretion rate histories of galaxies, reaching lookback times of 12 Gyr, for large statistically significant samples. Densities, temperatures, radiation fields, and gas-phase metallicities will be measured in dust-obscured galaxies and active galactic nuclei, sampling a large range in mass and luminosity, from faint local dwarf galaxies to luminous quasars in the distant Universe. Active galactic nuclei and starburst feedback and feeding mechanisms in distant galaxies will be uncovered through detailed measurements of molecular and atomic line profiles. The SPICA's large-area deep spectrophotometric surveys will provide mid-IR spectra and continuum fluxes for unbiased samples of tens of thousands of galaxies, out to redshifts of z ~ 6.
Behavioral considerations for effective time-varying electricity prices
IAN SCHNEIDER, CASS R. SUNSTEIN
Journal: Behavioural Public Policy / Volume 1 / Issue 2 / November 2017
Wholesale prices for electricity vary significantly due to high fluctuations and low elasticity of short-run demand. End-use customers have typically paid flat retail rates for their electricity consumption, and time-varying prices (TVPs) have been proposed to help reduce peak consumption and lower the overall cost of servicing demand. Unfortunately, the general practice is an opt-in system: a default rule in favor of TVPs would be far better. A behaviorally informed analysis also shows that when transaction costs and decision biases are taken into account, the most cost-reflective policies are not necessarily the most efficient. On reasonable assumptions, real-time prices can result in less peak conservation of manually controlled devices than time-of-use or critical-peak prices. For that reason, the trade-offs between engaging automated and manually controlled loads must be carefully considered in time-varying rate design. The rate type and accompanying program details should be designed with the behavioral biases of consumers in mind, while minimizing price distortions for automated devices.
Glacier change and climate forcing in recent decades at Gran Campo Nevado, southernmost Patagonia
M. Möller, C. Schneider, R. Kilian
Journal: Annals of Glaciology / Volume 46 / 2007
Digital terrain models of the southern Chilean ice cap Gran Campo Nevado reflecting the terrain situations of the years 1984 and 2000 were compared in order to obtain the volumetric glacier changes that had occurred during this period. The result shows a slightly negative mean glacier change of 3.80 m. The outlet glacier tongues show a massive thinning, whereas the centre of the ice cap is characterized by a moderate thickening. Thus a distinct altitudinal variability of the glacier change is noticed. Hypothetically this could be explained by the combined effects of increased precipitation and increased mean annual air temperature. Both to verify and to quantify this pattern of climatic change, the mean glacier change as well as its hypsometric variation are compared with the results of a degree-day model. The observed volumetric glacier change is traced back to possible climate forcing and can be linked to an underlying climate change that must be comparable with the effects of a precipitation offset of at least 7–8% and a temperature offset of around 0.3 K compared to the steady-state conditions in the period 1984–2000.
2276: The impact of social influence and impulsivity on IV alcohol self-administration in non-dependent drinkers
Alyssa Schneider, Bethany L. Stangl, Elgin R. Yalin, Jodi M. Gilman, Vijay Ramchandani
Journal: Journal of Clinical and Translational Science / Volume 1 / Issue S1 / September 2017
Published online by Cambridge University Press: 10 May 2018, pp. 33-34
OBJECTIVES/SPECIFIC AIMS: Impulsivity is a significant predictor of alcohol use and drinking behavior, and has been shown to be a critical trait in those with alcohol use disorder. Suggestibility, or susceptibility to social influence, has been shown to correlate with impulsivity, with highly suggestible individuals being more likely to make impulsive decisions influenced by peer groups. However, the relationship between social influence and drinking behavior is unclear. Our objective was to describe the relationship between social influence and impulsivity traits using the social delayed discounting task and potential differences in intravenous alcohol self-administration (IV-ASA) behavior. METHODS/STUDY POPULATION: Healthy, non-dependent drinkers (n=20) completed a CAIS session, which consisted of an initial 25-minute priming phase, where subjects were prompted to push a button to receive individually standardized IV alcohol infusions, followed by a 125-minute phase during which they could push the button for additional infusions. IV-ASA measures included the peak (PEAK) and average (AVG) BrAC and Number of Button Presses (NBP). Participants completed a social delayed discounting task (SDDT), where participants were presented with the choice of a small, sooner (SS) reward or a large, later (LL) reward. Before starting the task, participants chose peers who selected either the impulsive (SI) or non-impulsive choice (S). Intermittently, the peers' choice was not shown (X) or different choices (D) were selected. Participants also completed the MISS, the Barratt Impulsiveness Scale (BIS-11), UPPS-P Impulsive Behavior Scale, and the NEO personality inventory. RESULTS/ANTICIPATED RESULTS: Participants with higher suggestibility scores had greater NBP, AVG, and PEAK BrAC in the early phase of the IV-ASA session. Higher scores on the MISS were also correlated with higher impulsivity scores including the NEO Neuroticism (N-factor) measure, BIS-11, and UPPS-P. Results also showed that the MISS score was inversely correlated with the percent of impulsive choices in the SDDT, but that this was independent of peers' impulsive or nonimpulsive choices. DISCUSSION/SIGNIFICANCE OF IMPACT: These results indicate that non-dependent drinkers that were more susceptible to social influence had heavier drinking patterns, higher IV-ASA, and higher scores on impulsivity measures. In addition, individuals that were more susceptible to social influence made more impulsive choices in general, but those choices were not affected by peer decisions during the task. As such, susceptibility to social influence may be an important determinant of impulsive choices, particularly in relation to alcohol consumption.
Of Maps and Models: A New Method for Determining the Biological Significance of Sclerobiont Positions on Brachiopod Hosts
Kristina M. Barclay, Chris L. Schneider, Lindsey R. Leighton
Journal: The Paleontological Society Special Publications / Volume 13 / 2014
Published online by Cambridge University Press: 26 July 2017, p. 49
Superior petrosal sinus causing superior canal dehiscence syndrome
S M D Schneiders, J W Rainsbury, E F Hensen, R M Irving
Journal: The Journal of Laryngology & Otology / Volume 131 / Issue 7 / July 2017
To determine signs and symptoms for superior canal dehiscence syndrome caused by the superior petrosal sinus.
A review of the English-language literature on PubMed and Embase databases was conducted, in addition to a multi-centre case series report.
The most common symptoms of 17 patients with superior petrosal sinus related superior canal dehiscence syndrome were: hearing loss (53 per cent), aural fullness (47 per cent), pulsatile tinnitus (41 per cent) and pressure-induced vertigo (41 per cent). The diagnosis was made by demonstration of the characteristic bony groove of the superior petrosal sinus and the 'cookie bite' out of the superior semicircular canal on computed tomography imaging.
Pulsatile tinnitus, hearing loss, aural fullness and pressure-induced vertigo are the most common symptoms in superior petrosal sinus related superior canal dehiscence syndrome. Compared to superior canal dehiscence syndrome caused by the more common apical location of the dehiscence, pulsatile tinnitus and exercise-induced vertigo are more frequent, while sound-induced vertigo and autophony are less frequent. There is, however, considerable overlap between the two subtypes. The distinction cannot as yet be made on clinical signs and symptoms alone, and requires careful analysis of computed tomography imaging.
Network dynamics of HIV risk and prevention in a population-based cohort of young Black men who have sex with men – CORRIGENDUM
J. Schneider, B. Cornwell, A. Jonas, N. Lancki, R. Behler, B. Skaathun, L. E. Young, E. Morgan, S. Michaels, R. Duvoisin, A. S. Khanna, S. Friedman, P. Schumm, E. Laumann, for the uConnect Study Team
Journal: Network Science / Volume 5 / Issue 2 / June 2017
Published online by Cambridge University Press: 20 April 2017, p. 247
The order of the authors in the published article is incorrect. The authors should appear as follows:
J. Schneider, B. Cornwell, A. Jonas, R. Behler, N. Lancki, B. Skaathun, L. E. Young, E. Morgan, S. Michaels, R. Duvoisin, A. S. Khanna, S. Friedman, P. Schumm, E. Laumann, for the uConnect Study Team
The authors regret the error.
Ion angular distribution simulation of the Highly Efficient Multistage Plasma Thruster
Solved and Unsolved problems in Plasma Physics
J. Duras, D. Kahnfeld, G. Bandelow, S. Kemnitz, K. Lüskow, P. Matthias, N. Koch, R. Schneider
Journal: Journal of Plasma Physics / Volume 83 / Issue 1 / February 2017
Published online by Cambridge University Press: 22 February 2017, 595830107
Ion angular current and energy distributions are important parameters for ion thrusters, which are typically measured at a few tens of centimetres to a few metres distance from the thruster exit. However, fully kinetic particle-in-cell (PIC) simulations are not able to simulate such domain sizes due to high computational costs. Therefore, a parallelisation strategy of the code is presented to reduce computational time. The calculated ion beam angular distributions in the plume region are quite sensitive to boundary conditions of the potential, possible additional source contributions (e.g. from secondary electron emission at vessel walls) and charge exchange collisions. Within this work a model for secondary electrons emitted from the vessel wall is included. In order to account for limits of the model due to its limited domain size, a correction of the simulated angular ion energy distribution by the potential boundary is presented to represent the conditions at the location of the experimental measurement in $1~\text{m}$ distance. In addition, a post-processing procedure is suggested to include charge exchange collisions in the plume region not covered by the original PIC simulation domain for the simulation of ion angular distributions measured at $1~\text{m}$ distance.
Network dynamics of HIV risk and prevention in a population-based cohort of young Black men who have sex with men
Journal: Network Science / Volume 5 / Issue 3 / September 2017
Critical to the development of improved HIV elimination efforts is a greater understanding of how social networks and their dynamics are related to HIV risk and prevention. In this paper, we examine network stability of confidant and sexual networks among young black men who have sex with men (YBMSM). We use data from uConnect (2013–2016), a population-based, longitudinal cohort study. We use an innovative approach to measure both sexual and confidant network stability at three time points, and examine the relationship between each type of stability and HIV risk and prevention behaviors. This approach is consistent with a co-evolutionary perspective in which behavior is not only affected by static properties of an individual's network, but may also be associated with changes in the topology of his or her egocentric network. Our results indicate that although confidant and sexual network stability are moderately correlated, their dynamics are distinct with different predictors and differing associations with behavior. Both types of stability are associated with lower rates of risk behaviors, and both are reduced among those who have spent time in jail. Public health awareness and engagement with both types of networks may provide new opportunities for HIV prevention interventions.
Feasibility of common bibliometrics in evaluating translational science
M. Schneider, C. M. Kane, J. Rainwater, L. Guerrero, G. Tong, S. R. Desai, W. Trochim
Journal: Journal of Clinical and Translational Science / Volume 1 / Issue 1 / February 2017
A pilot study by 6 Clinical and Translational Science Awards (CTSAs) explored how bibliometrics can be used to assess research influence.
Evaluators from 6 institutions shared data on publications (4202 total) they supported, and conducted a combined analysis with state-of-the-art tools. This paper presents selected results based on the tools from 2 widely used vendors for bibliometrics: Thomson Reuters and Elsevier.
Both vendors located a high percentage of publications within their proprietary databases (>90%) and provided similar but not equivalent bibliometrics for estimating productivity (number of publications) and influence (citation rates, percentage of papers in the top 10% of citations, observed citations relative to expected citations). A recently available bibliometric from the National Institutes of Health Office of Portfolio Analysis, examined after the initial analysis, showed tremendous potential for use in the CTSA context.
Despite challenges in making cross-CTSA comparisons, bibliometrics can enhance our understanding of the value of CTSA-supported clinical and translational research.
Ice Volcanoes of the Lake Erie Shore Near Dunkirk, New York, U.S.A.
R. K. Fahnestock, D. J. Crowley, M. Wilson, H. Schneider
Journal: Journal of Glaciology / Volume 12 / Issue 64 / 1973
Conical mounds of ice have been observed to form in a few hours during violent winter storms along the edge of shore-fast ice near Dunkirk, New York. They occur in lines which parallel depth contours, and are evenly spaced in the manner of beach cusps. The height and spacing of mounds and number of rows vary from year to year depending on such factors as storm duration and intensity, and the position of the edge of the shore-fast ice at the beginning of the storm.
The evenly sloping conical mounds have central channels which increase in width lakeward. The ice between the channels forms headlands above the lake surface. Spray-formed levees develop along the headlands and slope gently away from the lake margin. Lake marginal walls of ice are usually vertical.
Spray, slush and ice blocks are ejected over the cone as each successive wave is focused by the converging channel walls. Ice blocks, interlayered with frozen slush and dirt, form bedding paralleling the sloping surface of cones, headlands and levees. These features are here termed "ice volcanoes" because their origin is in so many ways analogous to that of true volcanoes.
Conceptual design of initial opacity experiments on the national ignition facility
R. F. Heeter, J. E. Bailey, R. S. Craxton, B. G. DeVolder, E. S. Dodd, E. M. Garcia, E. J. Huffman, C. A. Iglesias, J. A. King, J. L. Kline, D. A. Liedahl, P. W. McKenty, Y. P. Opachich, G. A. Rochau, P. W. Ross, M. B. Schneider, M. E. Sherrill, B. G. Wilson, R. Zhang, T. S. Perry
Published online by Cambridge University Press: 09 January 2017, 595830103
Accurate models of X-ray absorption and re-emission in partly stripped ions are necessary to calculate the structure of stars, the performance of hohlraums for inertial confinement fusion and many other systems in high-energy-density plasma physics. Despite theoretical progress, a persistent discrepancy exists with recent experiments at the Sandia Z facility studying iron in conditions characteristic of the solar radiative–convective transition region. The increased iron opacity measured at Z could help resolve a longstanding issue with the standard solar model, but requires a radical departure for opacity theory. To replicate the Z measurements, an opacity experiment has been designed for the National Facility (NIF). The design uses established techniques scaled to NIF. A laser-heated hohlraum will produce X-ray-heated uniform iron plasmas in local thermodynamic equilibrium (LTE) at temperatures ${\geqslant}150$ eV and electron densities ${\geqslant}7\times 10^{21}~\text{cm}^{-3}$ . The iron will be probed using continuum X-rays emitted in a ${\sim}200$ ps, ${\sim}200~\unicode[STIX]{x03BC}\text{m}$ diameter source from a 2 mm diameter polystyrene (CH) capsule implosion. In this design, $2/3$ of the NIF beams deliver 500 kJ to the ${\sim}6$ mm diameter hohlraum, and the remaining $1/3$ directly drive the CH capsule with 200 kJ. Calculations indicate this capsule backlighter should outshine the iron sample, delivering a point-projection transmission opacity measurement to a time-integrated X-ray spectrometer viewing down the hohlraum axis. Preliminary experiments to develop the backlighter and hohlraum are underway, informing simulated measurements to guide the final design.
Developing mental health research in sub-Saharan Africa: capacity building in the AFFIRM project
M. Schneider, K. Sorsdahl, R. Mayston, J. Ahrens, D. Chibanda, A. Fekadu, C. Hanlon, S. Holzer, S. Musisi, A. Ofori-Atta, G. Thornicroft, M. Prince, A. Alem, E. Susser, C. Lund
Journal: Global Mental Health / Volume 3 / 2016
There remains a large disparity in the quantity, quality and impact of mental health research carried out in sub-Saharan Africa, relative to both the burden and the amount of research carried out in other regions. We lack evidence on the capacity-building activities that are effective in achieving desired aims and appropriate methodologies for evaluating success.
AFFIRM was an NIMH-funded hub project including a capacity-building program with three components open to participants across six countries: (a) fellowships for an M.Phil. program; (b) funding for Ph.D. students conducting research nested within AFFIRM trials; (c) short courses in specialist research skills. We present findings on progression and outputs from the M.Phil. and Ph.D. programs, self-perceived impact of short courses, qualitative data on student experience, and reflections on experiences and lessons learnt from AFFIRM consortium members.
AFFIRM delivered funded research training opportunities to 25 mental health professionals, 90 researchers and five Ph.D. students across 6 countries over a period of 5 years. A number of challenges were identified and suggestions for improving the capacity-building activities explored.
Having protected time for research is a barrier to carrying out research activities for busy clinicians. Funders could support sustainability of capacity-building initiatives through funds for travel and study leave. Adoption of a train-the-trainers model for specialist skills training and strategies for improving the rigor of evaluation of capacity-building activities should be considered.
Emergency department visits for attempted suicide and self harm in the USA: 2006–2013
J. K. Canner, K. Giuliano, S. Selvarajah, E. R. Hammond, E. B. Schneider
Journal: Epidemiology and Psychiatric Sciences / Volume 27 / Issue 1 / February 2018
Published online by Cambridge University Press: 17 November 2016, pp. 94-102
Aims.
To characterise and identify nationwide trends in suicide-related emergency department (ED) visits in the USA from 2006 to 2013.
We used data from the Nationwide Emergency Department Sample (NEDS) from 2006 to 2013. E-codes were used to identify ED visits related to suicide attempts and self-inflicted injury. Visits were characterised by factors such as age, sex, US census region, calendar month, as well as injury severity and mechanism. Injury severity and mechanism were compared between age groups and sex by chi-square tests and Wilcoxon rank-sum tests. Population-based rates were computed using US Census data.
Between 2006 and 2013, a total of 3 567 084 suicide attempt-related ED visits were reported. The total number of visits was stable between 2006 and 2013, with a population-based rate ranging from 163.1 to 173.8 per 100 000 annually. The frequency of these visits peaks during ages 15–19 and plateaus during ages 35–45, with a mean age at presentation of 33.2 years. More visits were by females (57.4%) than by males (42.6%); however, the age patterns for males and females were similar. Visits peaked in late spring (8.9% of all visits occurred in May), with a smaller peak in the fall. The most common mechanism of injury was poisoning (66.5%), followed by cutting and piercing (22.1%). Males were 1.6 times more likely than females to use violent methods to attempt suicide (OR = 1.64; 95% CI = 1.60–1.68; p < 0.001). The vast majority of patients (82.7%) had a concurrent mental disorder. Mood disorders were the most common (42.1%), followed by substance-related disorders (12.1%), alcohol-related disorders (8.9%) and anxiety disorders (6.4%).
Conclusions.
The annual incidence of ED visits for attempted suicide and self-inflicted injury in the NEDS is comparable with figures previously reported from other national databases. We highlighted the value of the NEDS in allowing us to look in depth at age, sex, seasonal and mechanism patterns. Furthermore, using this large national database, we confirmed results from previous smaller studies, including a higher incidence of suicide attempts among women and individuals aged 15–19 years, a large seasonal peak in suicide attempts in the spring, a predominance of poisoning as the mechanism of injury for suicide attempts and a greater use of violent mechanisms in men, suggesting possible avenues for further research into strategies for prevention.
Monitoring and evaluating capacity building activities in low and middle income countries: challenges and opportunities
M. Schneider, T. van de Water, R. Araya, B. B. Bonini, D. J. Pilowsky, C. Pratt, L. Price, G. Rojas, S. Seedat, M. Sharma, E. Susser
Published online by Cambridge University Press: 21 October 2016, e29
Lower and middle income countries (LMICs) are home to >80% of the global population, but mental health researchers and LMIC investigator led publications are concentrated in 10% of LMICs. Increasing research and research outputs, such as in the form of peer reviewed publications, require increased capacity building (CB) opportunities in LMICs. The National Institute of Mental Health (NIMH) initiative, Collaborative Hubs for International Research on Mental Health reaches across five regional 'hubs' established in LMICs, to provide training and support for emerging researchers through hub-specific CB activities. This paper describes the range of CB activities, the process of monitoring, and the early outcomes of CB activities conducted by the five research hubs.
The indicators used to describe the nature, the monitoring, and the early outcomes of CB activities were developed collectively by the members of an inter-hub CB workgroup representing all five hubs. These indicators included but were not limited to courses, publications, and grants.
Results for all indicators demonstrate a wide range of feasible CB activities. The five hubs were successful in providing at least one and the majority several courses; 13 CB recipient-led articles were accepted for publication; and nine grant applications were successful.
The hubs were successful in providing CB recipients with a wide range of CB activities. The challenge remains to ensure ongoing CB of mental health researchers in LMICs, and in particular, to sustain the CB efforts of the five hubs after the termination of NIMH funding. | CommonCrawl |
\begin{document}
\title{Primary Decompositions of Regular Sequences}
\author{Thomas Polstra}
\address{Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487 USA} \email{[email protected]}
\thanks{Polstra was supported in part by NSF Grant DMS \#2101890} \maketitle
\begin{abstract} Let $R$ be a Noetherian ring and $x_1,\ldots,x_t$ a permutable regular sequence of elements in $R$. Then there exists a finite set of primes $\Lambda$ and natural number $C$ so that for all $n_1,\ldots,n_t$ there exists a primary decomposition $(x_1^{n_1},\ldots,x_t^{n_t})=Q_1\cap \cdots \cap Q_\ell$ so that $\sqrt{Q_i}\in \Lambda$ and $\sqrt{Q_i}^{C(n_1+\cdots + n_t)}\subseteq Q_i$ for all $1\leq i\leq \ell$. \end{abstract}
\section{Introduction}
Primary decompositions of ideals in commutative algebra correspond to decompositions of closed subschemes into irreducible subspaces in algebraic geometry. Let $R$ be a Noetherian ring and $I\subseteq R$ an ideal. By the main result of \cite{Swanson} there exists an integer $C$ so that for every $n\in \mathbb{N}$ there exists a primary decomposition \[ I^n=Q_1\cap \cdots \cap Q_\ell \] so that $\sqrt{Q_i}^{Cn}\subseteq Q_i$ for all $1\leq i\leq \ell$. Swanson's theorem has applications to multiplicity theory, \cite{Cutkosky1,Cutkosky2,Cutkosky3,Das,Cid-RuizMontano}, the uniform symbolic power topology problem, \cite{SwansonLinear,EinLazarsfeldSmith,HochsterHuneke,HKV,HK,GHM}, and localizations problems in tight closure theory, \cite{HunekeSaturation,Vraciu,SmithSwanson,Dinh}.
Swanson's proof proceeds by first reducing to the scenario that $I$ is principally generated by a nonzerodivisor. Indeed, the extended Rees algebra $S:=R[It,t^{-1}]$ enjoys the property that $t^{-1}$ is a nonzerodivisor and $I^n=(t^{-1}S)^n\cap R$. If $(t^{-1}S)^n=Q_1\cap \cdots \cap Q_\ell$ is a suitable primary decomposition of $(t^{-1}S)^n$ then $I^n=(Q_1\cap R)\cap \cdots \cap (Q_\ell\cap R)$ is a primary decomposition of $I^n$ with the desired properties. Our main result extends Swanson's Theorem to ideals generated by a permutable regular sequence.
\begin{theorem}
\label{main theorem}
Let $R$ be a Noetherian ring and $x_1,\ldots,x_t$ a permutable regular sequence. There exists a finite set of primes $\Lambda$ and a constant $C$ so that for every $n_1,\ldots, n_t\in\mathbb{N}$ there exists a primary decomposition
\[
(x_1^{n_1},\ldots,x_t^{n_t})=Q_1\cap \cdots \cap Q_\ell
\]
so that $\sqrt{Q_i}\in\Lambda$ and $\sqrt{Q_i}^{C(n_1+\cdots +n_t)}\subseteq Q_i$ for all $1\leq i\leq \ell$. \end{theorem}
The methodology of \cite{Swanson} is akin to the techniques of Huneke's Uniform Artin-Rees Theorem, \cite{HunekeUniform}. Other's have re-proven Swanson's theorem without relying on such technicalities, see \cite{Sharp1,Sharp2,Yao1,Yao2} for more general decomposition statements involving products of powers of ideals $I_1^{n_1}\cdots I_t^{n_t}$ and their integral closures. Similar to their methods, the present article fundamentally depends only upon the standard Artin-Rees Lemma \cite[Proposition~10.9]{AtiyahMacdonald}, and the theory of injective hulls, \cite[Section~3.2]{BrunsHerzog}.
\section{Primary Decompositions of Regular Sequences}
Let $I\subseteq R$ be an ideal and $x\in R$ an element. By the Artin-Rees Lemma there exists a constant $C$ so that $(x)\cap P^{h+C}=((x)\cap P^C)P^h\subseteq xP^h$ for all $h$, see \cite[Proposition~10.9]{AtiyahMacdonald}.
\begin{lemma} \label{lemma: Sharp's lemma} Let $R$ be a Noetherian ring and $x\in R$ a non-unit. Let $P\in \Spec(R)$ and $E=E_R(R/P)$. Let $C$ be chosen such that $(x)\cap P^{h+C}\subseteq xP^h$ for all $h\in \mathbb{N}$. If $\varphi: R\to E$ is an $R$-linear map with the property that $P^h\varphi=0$ then there exists an $R$-linear map $\psi: R\to E$ such that \begin{itemize}
\item $\varphi=x\psi$;
\item $P^{h+C}\psi=0$. \end{itemize} \end{lemma}
\begin{proof} We are assuming that $C$ is chosen such that $(x)\cap P^{h+C}\subseteq xP^h$ for all integers $h$. In particular, there are natural surjections \[ \frac{R}{(x)\cap P^{h+C}}\to \frac{R}{xP^h}. \] Therefore there are inclusions \[ \Hom_R\left(\frac{R}{xP^h}, E\right)\to \Hom_R\left(\frac{R}{(x)\cap P^{h+C}}, E\right). \] Equivalently, \[ (0:_E xP^{h})\subseteq (0:_E ((x)\cap P^{h+C})). \] Even further, there are natural inclusions \[ \frac{R}{(x)\cap P^{h+C}}\subseteq \frac{R}{(x)}\oplus \frac{R}{P^{h+C}}. \] Therefore there are natural surjections \[ (0:_E (x)) + (0:_E P^{h+C})\twoheadrightarrow (0:_E ((x)\cap P^{h+C})). \] In conclusion, if \[ \lambda \in \Hom_R(R/xP^h,E)\cong (0:_ExP^h)\subseteq (0:_E ((x)\cap P^{h+C})) \] then there exists \[ \lambda'\in\Hom_R(R/(x),E) \cong (0:_E (x)) \] and \[ \psi\in \Hom_R(R/P^{h+C},E)\cong (0:_E P^{h+C}) \] such that $\lambda=\lambda'+\psi$.
The module $E$ is injective and therefore there exists $\lambda$ such that $\varphi=x\lambda$, i.e. the following diagram commutes: \[ \begin{xymatrix} { \displaystyle R\ar[r]^{\cdot x}\ar[d]^{\varphi} & R\ar[ld]^{\lambda} \\ E } \end{xymatrix} \] Since $P^h\varphi=0$ we have that $xP^h\lambda =0$. We can therefore write $\lambda= \lambda'+\psi$ so that $x\lambda'=0$ and $P^{h+C}\psi=0$. Therefore $\varphi=x\lambda=x\psi$ and $\psi:R\to E$ enjoys the desired properties. \end{proof}
Adopt the following notation: Let $\underline{x}=x_1,\ldots, x_t$ be a sequence of elements of a Noetherian ring $R$ and $\underline{n}=(n_1,\ldots,n_t)\in \mathbb{N}^{\oplus t}$.
\begin{itemize}
\item $\underline{x}^{\underline{n}}=x_1^{n_1},\ldots, x_t^{n_t}$;
\item $e_i\in \mathbb{N}^{\oplus t}$ is the element with a $1$ in the $i$th coordinate and $0$'s elsewhere;
\item $\underline{1}=(1,\ldots, 1)\in \mathbb{N}^{\oplus t}$;
\item If $\underline{n}'\in \mathbb{N}^{\oplus t}$ then $\underline{n}\cdot \underline{n}'$ denotes the dot product of $\underline{n}$ and $\underline{n}'$. In particular, the element $\underline{n}-(\underline{n}\cdot e_i-1)e_i$ is the element of $\mathbb{N}^{\oplus t}$ obtained by replacing the $i$th coordinate of $\underline{n}$ with the number $1$. \end{itemize} Observe that if $x_1,\ldots,x_t$ is a permutable regular sequence and $\underline{n}\in\mathbb{N}^{\oplus t}$ then $(\underline{x}^{\underline{n}+e_i}):x_i=(\underline{x}^{\underline{n}})$.
\begin{theorem} \label{Thm: embedding regular sequences into injective modules} Let $R$ be a Noetherian ring and $\underline{x}=x_1,\ldots,x_t$ a permutable regular sequence. Fix a finite list of prime ideals $\Lambda$, allowing for the possibility of repeated primes in $\Lambda$, and an embedding \[ \frac{R}{(\underline{x})}\xhookrightarrow{\varphi_{\underline{1}}}\bigoplus_{P\in \Lambda}E_R(R/P). \]
Let $C$ be chosen large enough so that $P^{C|\underline{1}|}\varphi_{\underline{1}}=0$ and $(x_i)\cap P^{h+C}\subseteq x_iP^h$ for all $P\in\Lambda$, $h\in\mathbb{N}$, and $1\leq i\leq t$. Then for all $\underline{n}\in\mathbb{N}^{\oplus t}$ there exists an embedding \[ \frac{R}{(\underline{x}^{\underline{n}})}\xhookrightarrow{\varphi_{\underline{n}}}E_{\underline{n}} \]
such that $E_{\underline{n}}\cong \left(\bigoplus_{P\in \Lambda}E_R(R/P)\right)^{\ell_{\underline{n}}}$ for some integer $\ell_{\underline{n}}$ and $P^{C|\underline{n}|}\varphi_{\underline{n}}=0$ for all $P\in\Lambda$. \end{theorem}
\begin{proof}
By induction, we may suppose that we have constructed the injective module $E_{\underline{n}}$ for all $\underline{n}\leq \underline{n}'$ and maps $\varphi_{\underline{n}}:R/(\underline{x}^{\underline{n}})\hookrightarrow E_{\underline{n}}$ such that $P^{C|\underline{n}|}\varphi=0$ for the purposes of constructing $E_{\underline{n}'+e_i}$ and map $\varphi_{\underline{n}'+e_i}: R/(\underline{x}^{\underline{n}'+e_i})\hookrightarrow E_{\underline{n}'+e_i}$ such that $P^{C|\underline{n}'+e_i|}\varphi_{\underline{n}'+e_i}=0$. Even further, we suppose that $E_{\underline{n}}$ consists of direct sums of $\bigoplus_{P\in \Lambda}E_R(R/P)$ for all $\underline{n}\leq \underline{n}'$.
Because $\underline{x}$ is a permutable regular sequence there exists short exact sequences \[ 0\to \frac{R}{(\underline{x}^{\underline{n}'})}\xrightarrow{\cdot x_i} \frac{R}{(\underline{x}^{\underline{n} +e_i})} \xrightarrow{\pi}\frac{R}{(\underline{x}^{\underline{n}'-(\underline{n}'\cdot e_i-1)e_i})}\to 0. \]
Lemma~\ref{lemma: Sharp's lemma} applied to each of the irreducible direct summands of $E_{\underline{n}'}$ produces a map $\psi_{\underline{n}'}:R/(\underline{x}^{\underline{n}'+e_i})\to E_{\underline{n}'}$ such that $P^{C|\underline{n}'+e_i|}\psi_{\underline{n}'}=0$ and the following diagram commutes: \[ \begin{xymatrix} { 0 \ar[r]& \displaystyle \frac{R}{(\underline{x}^{\underline{n}'})}\ar[rr]^{\cdot x_i}\ar@{^{(}->}[d]^{\varphi_{\underline{n}'}} && \displaystyle \frac{R}{(\underline{x}^{\underline{n}'+e_i})}\ar[lld]^{\psi_{\underline{n}'}}\ar[d]^{(\psi_{\underline{n}'},\varphi_{\underline{n}'-(\underline{n}'\cdot e_i-1)e_i}\circ\pi)} \ar[rr]^{\pi}&& \displaystyle \frac{R}{(\underline{x}^{\underline{n}'-(\underline{n}'\cdot e_i-1)e_i})}\ar[d]^{\varphi_{\underline{n}'-(\underline{n}'\cdot e_i-1)e_i}} \ar[r]& 0 \\ 0 \ar[r]& E_{\underline{n}'}\ar[rr] && E_{\underline{n}'}\oplus E_{\underline{n}'-(\underline{n}'\cdot e_i-1)e_i}\ar[rr] && E_{\underline{n}'-(\underline{n}'\cdot e_i-1)e_i} \ar[r]& 0 } \end{xymatrix} \]
It is straight-forward to verify that $\varphi_{\underline{n}'+e_i}:=(\psi_{\underline{n}'},\varphi_{\underline{n}'-(\underline{n}'\cdot e_i-1)e_i}\circ\pi)$ is an injective map and $P^{C|\underline{n}+e_i|}\varphi_{\underline{n}'+e_i}=0$. \end{proof}
\begin{corollary}[Swanson's Theorem for regular sequences] \label{Corollary: Swanson's Theorem for regular sequences} Let $R$ be a Noetherian ring and $\underline{x}=x_1,\ldots,x_t$ a permutable regular sequence. There exists a finite set of primes $\Lambda$ and a constant $C$ such that for all $\underline{n}\in\mathbb{N}^{\oplus t}$ there exists a primary decomposition \[ (\underline{x}^{\underline{n}})=Q_1\cap \cdots \cap Q_\ell \]
such that $\sqrt{Q_i}\in \Lambda$ and $\sqrt{Q_i}^{C|\underline{n}|}\subseteq Q_i$ for all $1\leq i \leq \ell$. \end{corollary}
\begin{proof} Fix $\underline{n}\in\mathbb{N}^{\oplus t}.$ By Theorem~\ref{Thm: embedding regular sequences into injective modules} there exists a constant $C$, not depending on $\underline{n}\in\mathbb{N}^{\oplus t}$, and a finite set of primes $\Lambda_{\underline{n}}$, allowing for the possibility of repeated primes in $\Lambda_{\underline{n}}$, and an embedding \[ \varphi_{\underline{n}}: \frac{R}{(\underline{x}^{\underline{n}})}\hookrightarrow \bigoplus_{P\in\Lambda_{\underline{n}}}E(R/P) \]
such that $P^{C|\underline{n}|}\varphi_{\underline{n}}=0$ for all $P\in\Lambda_{\underline{n}}$. Let $\pi:R\to R/(\underline{x}^{\underline{n}})$ and $\pi_P:\bigoplus_{P\in\Lambda_{\underline{n}}}E(R/P)\to E(R/P)$ be the natural surjections. Then \[ (\underline{x}^{\underline{n}})=\bigcap_{P\in \Lambda}\Ker(\pi_P\circ \varphi_{\underline{n}}\circ \pi) \] is a primary decomposition of $(\underline{x}^{\underline{n}})$ as there are embeddings \[ \frac{R}{\Ker(\pi_P\circ \varphi_{\underline{n}}\circ \pi)}\hookrightarrow E(R/P). \]
Furthermore, $P^{C|\underline{n}|}\subseteq\Ker(\pi_P\circ \varphi_{\underline{n}}\circ \pi)$ since $P^{C|\underline{n}|}\varphi=0$. \end{proof}
\end{document} | arXiv |
\begin{document}
\begin{frontmatter} \title{
Non-uniform bounds and Edgeworth expansions in self-normalized limit theorems
}
\runtitle{
Edgeworth expansions in self-normalized limit theorems }
\begin{aug} \author[A]{\fnms{Pascal}~\snm{Beckedorf}\ead[label=e1]{[email protected]}} \and \author[A]{\fnms{Angelika}~\snm{Rohde}\ead[label=e2]{[email protected]}}
\address[A]{Albert-Ludwigs-Universit\"at Freiburg \\ \printead[presep={ }]{e1,e2}} \end{aug}
\begin{abstract}
We study Edgeworth expansions in limit theorems for self-normalized sums. Non-uniform bounds for expansions in the central limit theorem are established while only imposing minimal moment conditions. Within this result, we address the case of non-integer moments leading to a reduced remainder. Furthermore, we provide non-uniform bounds for expansions in local limit theorems. The enhanced tail-accuracy of our non-uniform bounds allows for deriving an Edgeworth-type expansion in the entropic central limit theorem as well as a central limit theorem in total variation distance for self-normalized sums. \end{abstract}
\begin{keyword}[class=MSC] \kwd[Primary ]{60F05}
\kwd[; secondary ]{62E20} \end{keyword}
\begin{keyword}
\kwd{Edgeworth expansion}
\kwd{non-uniform bounds}
\kwd{central limit theorem}
\kwd{local limit theorem}
\kwd{entropy}
\kwd{total variation distance}
\kwd{self-normalized sums}
\kwd{rate of convergence} \end{keyword}
\end{frontmatter}
\tableofcontents
\section{Introduction and main results}\label{ch.intro}
Let $X_1,\dots,X_n$ be independent, identically distributed random variables with mean $\mathbb{E} X_1=0$ and variance $\ensuremath{\mathop{\mathrm{Var}}} X_1=1$. Here and throughout this article, \[ S_n=\sum_{j=1}^{n} X_j, \qquad V_n=\sqrt{\sum_{j=1}^{n} X_j^2}, \qquad T_n=\begin{cases} S_n/V_n& \text{ if }V_n>0\\ 0 & \text{ otherwise.} \end{cases} \] The self-normalized sum $T_n$ was shown to be asymptotically normal if and only if $X_1$ belongs to the domain of attraction of the normal law in \cite{GGM97}. This result has been generalized in \cite{CG04}.
The distance of the distribution of $T_n$ to a normal distribution has been investigated in various forms such as Berry--Esseen bounds \cite{BG96,Fri89,Hal88,vZw84} and non-uniform Berry--Esseen bounds \cite{CG03,JSW03,RW05, WJ99}, to mention just a few. For an extensive survey on further self-normalized limit theorems we refer to \cite{SW13Survey}.
Concerning asymptotic expansions of the distribution of self-normalized sums, Chung \cite{Chu46} was the first to prove an Edgeworth expansion for the Student $t$-statistic. Bhattacharya and Ghosh \cite{BG78} provided an approach of Edgeworth expansion including self-normalized sums, relaxing Chung's moment conditions. The moment and smoothness assumptions of \cite{BG78} were further weakened in \cite{BR91,BG88}. \cite{Hal87} attained an expansion for the $t$-statistic under minimal moment conditions. One-term Edgeworth expansions were treated in \cite{BGvZ97,BP99,BP03,HW04,JW10,PvZ98}. Recently, \cite{BG22} used the first two expansion terms to derive a sharper bound in a Berry--Esseen type result. Although not explicitly stated for self-normalized sums, Edgeworth expansions for densities of more general statistics are given in \cite[Section 2.8]{Hal92edgeworth}.
For normalized sums $Z_n=S_n/\sqrt{n}$, Edgeworth expansions with non-integer moment conditions and non-uniform bounds have been investigated for a long time in the CLT and LLT (see e.g. \cite{BR76normal,Pet75sums}). In case of non-integer moments, a CLT for $Z_n$ has been derived in \cite{BCG11} using fractional calculus. These stronger results lay the foundation for limit theorems in more powerful metrics such as expansions in the entropic CLT \cite{BCG13} or related results for other information-theoretic distances \cite{BCG14-2,BCG19,BCK15,BM19}.
For \textit{self-normalized sums} however, non-uniform bounds for Edgeworth expansions have not been studied so far. The main aim of this article is to derive such non-uniform bounds in the CLT and in the LLT for an Edgeworth expansion. These provide substantially stronger accuracy in the tails, enabling us to prove an entropic CLT and a CLT in total variation for self-normalized sums. As compared to normalized sums, the crucial obstacle here is the missing product form of the statistic's characteristic function. Besides, we also treat the case of non-integer moments in the CLT. Subsequently, we give an overview on our results.
Let $\Phi$ and $\phi$ denote the standard normal distribution function and density function, respectively. Write $\mu_k=\mathbb{E} X_1^k$ and $F_n(x)=\mathbb{P}(T_n\le x)$ for $k,n\in\mathbb{N}$ and $x\in\mathbb{R}$. Let $\lfloor x \rfloor$ denote the integer part of $x$. With $\Phi^Q_{m,n}$ we denote the Edgeworth expansion \begin{equation}\label{eq.Phi^Q-def} \Phi^Q_{m,n}(x)=\Phi(x)+\sum_{r=1}^{m-2}Q_{r}(x)n^{-r/2}, \quad x\in\mathbb{R}, \end{equation} with $m-2$ expansion terms of the distribution of $T_n$. Throughout this article, $X_1$ is assumed to be symmetric such that all uneven moments vanish and thus $Q_{r}=0$ for uneven $r\in\mathbb{N}$. In contrast to the approximation functions of $Z_n$, the polynomials $Q_{r}$ generally do not have a closed form. However, for even $r$, $Q_{r}$ provide a useful form which consist of $\phi$ multiplied with an uneven polynomial of degree $2r-1$ in $x$. The coefficients of $Q_{r}$ are functions of the moments $\mu_3,\dots,\mu_{r+2}$. The first approximating functions for the distribution function have the form \begin{equation}\label{eq.Q-2+4} \begin{split} Q_{2}(x)&=-\phi(x) H_{3}(x) \big(- \tfrac{1}{12}\big) \mu_4,\\ Q_{4}(x)&=-\phi(x) \Big( H_{7}(x) \tfrac{1}{288} \mu_4^2 + H_{5}(x) \tfrac{1}{45} \mu_6 + H_{3}(x) \tfrac{1}{12} \big(2\mu_6 +\mu_4 - 3 \mu_4^2\big) \Big), \end{split} \end{equation} where $H_k$ is the $k$-th Hermite polynomial.
Unless stated otherwise, all orders of convergence or divergence in $n$ are understood for $n\to\infty$. Our first result is
\begin{theorem}\label{t.clt}
Assume that $X_1$ is symmetric, the distribution of $X_1$ is non-singular and $\mathbb{E}|X_1|^{s}<\infty$ for some ${s}\ge2$. Then for $m=\lfloor {s} \rfloor$,
\begin{equation}\label{eq.t-clt}
\sup_{x\in\mathbb{R}} \, (1+|x|)^{m}|F_n(x)-\Phi^{Q}_{m,n}(x)|= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-({s}-2)/2} \, (\log n)^{({s}+2m)/2} \big).
\end{equation} \end{theorem}
Note that only the moments used in the expansion have to be finite, thus we only pose minimal moment conditions. The distribution of $X_1$ being non-singular (with respect to the Lebesgue measure) is only slightly stronger than Cram\'er's condition \begin{equation}\label{eq.cramer}
\limsup_{|t|\to\infty}|\mathbb{E}\exp(itX_1)|<1 \end{equation}
which is usually imposed when working with $Z_n$ (see e.g. \cite{BR76normal,Pet75sums}). Due to the denominator $V_n$, the characteristic function of $T_n$ lacks the typical product form, required for an Edgeworth expansion. To overcome this problem, we first condition on $\mathcal{F}_n=\sigma(|X_1|,\dots,|X_n|)$ as in \cite{Hal87}. On the one hand, the independence of the factors then provides a desired structure as a product. On the other hand, the random variables $X_1,\dots,X_n$ conditional on $\mathcal{F}_n$ are discrete (non-lattice) random variables that are not identically distributed any longer. Therefore, the necessary Cram\'er condition \labelcref{eq.cramer} is not fulfilled and the standard theorems for Edgeworth expansions (such as from \cite{BR76normal,Pet75sums}) do not apply. These properties necessitate adjustments in the derivation of an Edgeworth expansion, but lead to rather involved integral remainder terms which have to be bounded thoroughly. At this point, the non-singularity is getting essential.
If $T_n$ has a density, then by taking derivatives in \labelcref{eq.Phi^Q-def}, the Edgeworth expansion of the density of $T_n$ has the form \begin{equation}\label{eq.phi^q-def} \phi^q_{m,n}(x)=\phi(x)+\sum_{r=1}^{m-2}q_{r}(x)n^{-r/2} \end{equation} for all $x\in\mathbb{R}$. By differentiating \labelcref{eq.Q-2+4}, the first approximating functions are \begin{equation}\label{eq.q-2} \begin{split} q_{2}(x)&=\phi(x) H_{4}(x) \big(- \tfrac{1}{12}\big) \mu_4,\\ q_{4}(x)&=\phi(x) \Big( H_{8}(x) \tfrac{1}{288} \mu_4^2 + H_{6}(x) \tfrac{1}{45} \mu_6 + H_{4}(x) \tfrac{1}{12} \big(2\mu_6 +\mu_4 - 3 \mu_4^2\big) \Big). \end{split} \end{equation} In the case that $r$ is even, $q_{r}$ provides a useful form which constitutes of $\phi$ multiplied with an even polynomial of degree $2r$ in $x$. The coefficients of $q_{r}$ are functions of the moments $\mu_3,\dots,\mu_{r+2}$. As above, $q_{r}=0$ for uneven $r\in\mathbb{N}$.
Our second result is an application of our \cref{t.llt} which has weaker but more technical conditions. Further LLTs with more formal but less stringent conditions are provided in \cref{ch.density}. For the classical normalized statistic $Z_n$, such a result has been derived in \cite{BCG11}.
\begin{theorem}\label{t.llt-red}
Assume that $X_1$ is symmetric, has a bounded density and $\mathbb{E}|X_1|^{2m}<\infty$ for some $m\in\mathbb{N}$, $m\ge3$. Then there exists $N\in\mathbb{N}$ such that for all $n\ge N$, $T_{n}$ have densities $f_{n}$ that satisfy
\begin{equation*}
\sup_{x\in\mathbb{R}} \, (1+|x|)^{m}|f_n(x)-\phi^{q}_{m,n}(x)|= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m-3)/2}\big).
\end{equation*}
Moreover, if $\mathbb{E}|X_1|^{2m+2}<\infty$ for some $m\in\mathbb{N}, m\ge2$, the order of convergence reduces to $\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m-1)/2} \, \log n \big)$. \end{theorem}
The route of the proof of \cref{t.llt-red} is related to the proof of \cref{t.clt}. However, conditional on $\mathcal{F}_n$, the self-normalized sum $T_n$ does not have a density as it is discrete. In particular, the Fourier inversion formula is not available any longer. The idea is to tackle this severe obstacle by perturbing $T_n$ with an independent normal random variable with suitably chosen variance. Besides enjoying the properties that tails of density and characteristic function decay in the same magnitude, a Gaussian perturbation does not hinder convergence to the normal distribution. This perturbation trick allows to work with densities and deriving an Edgeworth expansion but comes at the cost of a deconvolution after returning to the unconditional setting. For performing a tight approximation in the final deconvolution step, verifiable conditions have to be imposed that guarantee sufficient smoothness of the density of $T_n$.
Based on \cref{t.llt-red}, we prove an entropic CLT and its rates of convergence. For a random variable $X$ with density $p$, mean $\mu$ and variance $\sigma^2$ we define its entropy (with $0\log 0:=0$) \begin{align*} h(X)=-\int_{-\infty}^{\infty}p(x)\log p(x)\,\mathrm{d}x, \end{align*} and its relative entropy \begin{align}\label{eq.def-rel-entropy}
D(X)=D(X\|Z)=h(Z)-h(X)=\int_{-\infty}^{\infty}p(x)\log \frac{p(x)}{\phi_{\mu,\sigma^2}(x)}\,\mathrm{d}x, \end{align} where $Z$ is $\mathcal{N}(\mu,\sigma^2)$-distributed with density $\phi_{\mu,\sigma^2}$. As the normal distribution maximizes the entropy for given mean and variance, we know $D(X)\ge0$.
Barron \cite{Bar86} was the first to prove an entropic CLT for the classical statistic $Z_n$, i.e. $D(Z_n)\to 0$. Many more results on entropic CLTs can be found in \cite{Johnson04}. Moreover, in \cite{BCG13}, an Edgeworth expansion was used to determine the rates of convergence in the entropic CLT for the classical statistic. We prove such type of result for the self-normalized sum.
\begin{theorem}\label{t.entropy}
Assume that $X_1$ is symmetric, has a bounded density and $\mathbb{E}|X_1|^{2m}<\infty$ for some $m\in\mathbb{N}$, $m\ge3$. Then
\begin{equation*}
D(T_n)=\frac{c_2}{n^2}+\dots+\frac{c_{\lfloor (m-2)/2 \rfloor}}{n^{\lfloor (m-2)/2 \rfloor}}+ \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big((n\log n)^{-(m-2)/2}\,\log n\big),
\end{equation*}
where
\begin{align*}
c_l=\sum_{k=2}^{2l} \frac{(-1)^k}{k(k-1)} \sum \int_{\mathbb{R}} \frac{q_{r_1}(x)\dots q_{r_k}(x)}{\phi(x)^{k-1}} \,\mathrm{d}x, \quad l\in\mathbb{N},
\end{align*}
with the inner sum running over all positive integers $r_1,\dots,r_k$ such that $r_1+\dots+r_k=2l$. \end{theorem}
\cref{t.entropy} also holds under the weaker but more technical conditions of \cref{t.llt}, see \cref{t.entropy-general}.
\begin{remark}Comparing the relative entropy of $T_n$ and $Z_n$ under the conditions of \cref{t.entropy} with $m=6$, we obtain \begin{align*} D(T_n)
&= n^{-2} \tfrac{1}{12} \mu_4^2 + \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big((n\log n)^{-2}\,\log n\big) \end{align*} and (from \cite{BCG13}) \begin{align*} D(Z_n) = n^{-2} \tfrac{1}{48}(\mu_4-3)^2 + \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big((n\log n)^{-2}\big). \end{align*} Note that under these conditions, $\mu_4^2/12 > (\mu_4-3)^2 /48$. The relative entropies are always of the same magnitude and particularly close if $\mu_4$ is close to 1. The limit case $\mu_4=1$ is excluded because $X_1$ has a density. \end{remark}
Our final goal is prove a bound on the total variation. On the linear space of finite signed measures on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$, the total variation norm is defined as $$\Arrowvert\mu\Arrowvert_{TV}=\sup_{B\in\mathcal{B}(\mathbb{R})}\Big(\arrowvert \mu(B)\arrowvert +\arrowvert \mu(B^c)\arrowvert\Big)$$ To circumvent unnecessary notational complexity, we identify finite (signed) measures with their corresponding cumulative distribution functions. If $F_P$ and $F_Q$ have densities, bounding their distance in total variation is equivalent to bounding the $L^1$ norm of the difference of the corresponding densities \begin{equation}\label{eq.TV-L1}
\|F_n-\Phi^{Q}_{m,n}\|_{\textup{TV}}=\|f_n-\phi^{q}_{m,n}\|_{L^1}. \end{equation}
Assume that the conditions of \cref{t.llt-red} are satisfied, in particular $\mathbb{E}|X_1|^{2m}<\infty$ for some $m\in\mathbb{N}$, $m\ge3$. Then the second statement of \cref{t.llt-red} applied with $m-1$ implies \begin{align*}
\|f_n-\phi^{q}_{m-1,n}\|_{L^p}^p
=\int_\mathbb{R}\big|f_n(x)-\phi^{q}_{m-1,n}(x)\big|^p\,\mathrm{d}x
\le\int_{\mathbb{R}}\big|(1+|x|)^{-(m-1)} r_n\big|^p\,\mathrm{d}x \le c \, r_n^p \end{align*} for $p\in[1,\infty)$ and an absolute constant $c>0$. Here, $r_n=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m-2)/2} \, \log n \big)$ and thus \begin{equation}\label{eq.fn-phi-Lp}
\|f_n-\phi^{q}_{m-1,n}\|_{L^p}
= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m-2)/2} \log n\big). \end{equation}
The following theorem shows that we can achieve a stronger order of convergence by methods that are particularly suitable for optimizing the $\textup{TV}$ distance.
\begin{theorem}\label{t.TV}
Assume that $X_1$ is symmetric, has a bounded density and $\mathbb{E}|X_1|^{2m}<\infty$ for some $m\in\mathbb{N}$, $m\ge3$. Then
\begin{equation*}
\|F_n-\Phi^{Q}_{m,n}\|_{\textup{TV}} = \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m-2)/2}\big).
\end{equation*} \end{theorem}
\cref{t.TV} also holds under the weaker but more technical conditions of \cref{t.llt}, see \cref{t.dichte-L1}.
The remainder of this article is organized as follows. In \cref{ch.pre}, we introduce further notation and the formal Edgeworth expansion for self-normalized sums. The CLT-type result (\cref{t.clt}) is proven in \cref{ch.distr}. \cref{ch.density} provides more general LLTs that lead to \cref{t.llt-red}. In \cref{ch.entropy,ch.TV}, the prior results are used to attain the entropic CLT (\cref{t.entropy}) and the CLT in total variation distance (\cref{t.TV}), respectively.
\section{Notation and preliminaries}\label{ch.pre} The notation $\varphi$ is used for the characteristic functions of random variables or the Fourier transforms of functions. Their dependence is indicated by a subscript, e.g. $\varphi_{T_n}$ or $\varphi_{\Phi^{\tilde{P}}_{m,n}}$, while $\varphi_{X_j}$ is abbreviated as $\varphi_j$. For $k\ge0$, $c(k)$ represents different positive constants in different (or even in the same) formulas, only depending on $k$ and the distribution of $X_1$. $\lceil x \rceil$ denotes the smallest integer larger than or equal to $x$.
If a sum $\sum$, product $\prod$ or a maximum $\max$ does not have an index, it is always meant to run over $j=1,\dots,n$. The sum $\sum_{*(k_\cdot,r,u)}$ is carried out over all non-negative integer solutions $(k_1,k_2,\dots,k_r)$ of the equalities \mbox{$k_1+2k_2+\dots+rk_r=r$} and $k_1+k_2+\dots + k_r=u$. In many occurrences, only the variable $r$ is fixed which will be denoted by $\sum_{*(k_\cdot,r,\cdot)}$. Then the $k_i$ only have to satisfy the equation $k_1+2k_2+\dots+rk_r=r$ and $u=u(k_\cdot)$ is defined by $u(k_\cdot)=k_1+k_2+\dots +k_r$. In some situations only the variable $u$ is fixed. In this case we need information on the maximal index of $k_\cdot$. This is denoted by $\sum_{*(k_\cdot,\cdot, u, t)}$ meaning the summation over all non-negative integer solutions $(k_1,k_2,\dots,k_t)$ of the equation $k_1+k_2+\dots +k_t=u$ and $r=r(k_\cdot)$ is defined by $r(k_\cdot)=k_1+2k_2+\dots+tk_t$. Clearly $u\le r$ holds.
We set \[
M_n=\max |X_j| \qquad \text{and} \qquad B_n=V_n/M_n. \] As $V_n=0$ and $M_n=0$ are equivalent, we define $B_n=1$ in this case, whence $1 \le B_n \le \sqrt{n}$ is always satisfied. For $l\ge0$ the $l$-th cumulant of $X_1$ will be denoted by $\kappa_{l}$ and is defined by \begin{equation}\label{eq.cumulant-abl}
\kappa_{l} = i^{-l}\ddT{l} \log\varphi_1(t)\Big|_{t=0} \, . \end{equation}
The conditional expected value with respect to $\mathcal{F}_n$ is shortened as $\tilde{\mathbb{E}}[\ \cdot\ ]=\mathbb{E}[\ \cdot\ |\mathcal{F}_n]$. Correspondingly, the $\sim$-notation is extended to $\mathbb{P}, \mu, \kappa, F, f, p, P, U, \varphi$ for describing the variants conditional on $\mathcal{F}_n$. We define \[ \tilde{L}_{k,n}
= V_n^{-k}\sum|X_j|^{k} \]
for $k\ge2$ integer. Clearly $V_n=0$ and $\sum|X_j|^k=0$ are equivalent. In this case, we define $\tilde{L}_{k,n}=0$. Note that for all $k\ge2$, \begin{equation}\label{eq.Lle1} \tilde{L}_{k,n}
= V_n^{-k}\sum|X_j|^{k}
\le V_n^{-2}\,\frac{M_n^{k-2}}{V_n^{k-2}}\,\sum|X_j|^{2}
\le V_n^{-2}\,\sum|X_j|^{2} =1. \end{equation}
When we state that the sequence $(A_n)$ increases (or decreases, respectively) in polynomial order, we mean that there exist $a,b>0$ (or $a,b<0$, respectively) such that \begin{equation*}
\limsup_{n\to\infty}\Big|\frac{n^a}{A_n}\Big|<\infty \quad\text{and}\quad
\limsup_{n\to\infty}\Big|\frac{A_n}{n^b}\Big|<\infty. \end{equation*}
When writing (probability) densities, we exclusively mean Lebesgue densities. Densities with respect to the counting measure will be denoted as probability mass functions. When mentioning a density (or a function in general), we always refer to the continuous version of it in case it exists.
\subsection{Edgeworth expansions for self-normalized sums}\label{s.pre.self-normalized}
In this subsection, we introduce the Edgeworth expansion for $T_n$ conditional on $\mathcal{F}_n$ and establish a connection to \labelcref{eq.Phi^Q-def,eq.phi^q-def}. In particular, the explicit form of the approximating polynomials is derived for later purposes. To this end, we assume that $X_1$ is symmetric such that $\tilde{\mathbb{P}}(X_j= \pm |X_j|)=1/2$, $j=1,\dots,n$. For $k\in\mathbb{N}$, we denote by $\tilde{\mu}_{k,j}=\tilde{\mathbb{E}}[X_j^k]$ the $k$-th conditional moment of $X_j$. Accordingly, we extend the notation to $\tilde{\kappa}_{k,j}$. Note that \begin{equation}\label{eq.cumulants-bed-ug}
\tilde{\kappa}_{2k-1,j}=\tilde{\mu}_{2k-1,j}=0 \qquad \text{and} \qquad \tilde{\mu}_{2k,j}=|X_j|^{2k}. \end{equation} Although not necessary, we state the absolute value for even powers of $X_j$ to point out that we operate in the conditional setting where the absolute values are deterministic. By explicitly computing the values of the Bell polynomials, the $k$-th cumulant can be expressed in terms of moments up to degree $k$
which yields \begin{equation}\label{eq.tk-2r} \begin{split} \tilde{\kappa}_{2r,j}
=|X_j|^{2r} \sum_{*(k_\cdot,r,\cdot)}(-1)^{u(k_\cdot)-1} (u(k_\cdot)-1)! (2r)! \prod_{l=1}^{r} \frac{1}{k_l!(2l)!^{k_l}} \end{split} \end{equation} for all $r\in\mathbb{N}$ because all summands with one or more $k_{2l-1}>0$ vanish due to \labelcref{eq.cumulants-bed-ug}. In particular, \begin{equation*}
\tilde{\kappa}_{2,j}=|X_j|^{2}, \qquad \tilde{\kappa}_{4,j}
=-2 |X_j|^{4}, \qquad \tilde{\kappa}_{6,j}
=16 |X_j|^{6}. \end{equation*}
Below, we outline the characteristics of the Edgeworth expansion of $T_n$, conditioned on $\mathcal{F}_n$. Clearly, the classical Edgeworth expansion (see e.g. \cite[Section VI.1]{Pet75sums}) needs to be adjusted in terms of approximating polynomials and functions. In particular, for $l,r,m\in\mathbb{N}$ and $t\in\mathbb{R}$, we define \begin{align} \tilde{\lambda}_{l,n}\label{eq.tlambda} &=\Big(\sum\tilde{\mu}_{2,j}\Big)^{-l/2}\sum\tilde{\kappa}_{l,j} =V_n^{-l}\sum\tilde{\kappa}_{l,j} \quad (\tilde{\lambda}_{l,n}=0 \text{ if } V_n=0),\\ \tilde{U}_{r,n}(it)\label{eq.tU} &=\sum_{*(k_\cdot,r,\cdot)} (it)^{r+2u(k_\cdot)} \prod_{l=1}^{r} \frac{1}{k_l!} \Big(\frac{\tilde{\lambda}_{l+2,n}}{(l+2)!}\Big)^{k_l},\\ \tilde{\varphi}_{\Phi^{\tilde{P}}_{m,n}}(t)\label{eq.tphi} &=e^{-t^2/2}\Big(1+\sum_{r=1}^{m-2} \tilde{U}_{r,n}(it)\Big). \end{align}
Next, the Edgeworth expansion for the conditional distribution function of $T_n$ is \begin{equation}\label{eq.Phi^tP} \Phi^{\tilde{P}}_{m,n}(x) =\Phi(x)+\sum_{r=1}^{m-2}\tilde{P}_{r,n}(x), \end{equation} where \begin{equation}\label{eq.tP} \tilde{P}_{r,n}(x) =-\phi(x)\sum_{*(k_\cdot,r,\cdot)} H_{r+2u(k_\cdot)-1}(x) \prod_{l=1}^{r} \frac{1}{k_l!} \Big(\frac{\tilde{\lambda}_{l+2,n}}{(l+2)!}\Big)^{k_l} \end{equation} for all $x\in\mathbb{R}$.
In \labelcref{eq.tU,eq.tP}, all summands with one or more $k_{2l-1}>0$ vanish due to \labelcref{eq.cumulants-bed-ug}. For uneven indices the polynomials only consist of such summands and therefore $\tilde{U}_{2r-1,n}=\tilde{P}_{2r-1,n}=0$ for all $r\in\mathbb{N}$. The even polynomials $\tilde{P}_{2r,n}$ reduce to \begin{align}\label{eq.tP-2+4}
\tilde{P}_{2r,n}(x)&=-\phi(x)\sum_{*(k_\cdot,r,\cdot)} H_{2r+2u(k_\cdot)-1}(x) \prod_{l=1}^{r} \frac{1}{k_l!} \Big(\frac{\tilde{\lambda}_{2l+2,n}}{(2l+2)!}\Big)^{k_l},\nonumber\\ \tilde{P}_{2,n}(x)&=-\phi(x) H_{3}(x) \frac{\tilde{\lambda}_{4,n}}{4!},\\ \tilde{P}_{4,n}(x)&=-\phi(x) \Big( H_{7}(x) \frac{1}{2} \Big(\frac{\tilde{\lambda}_{4,n}}{4!}\Big)^2 + H_{5}(x) \frac{\tilde{\lambda}_{6,n}}{6!}\Big).\nonumber
\end{align}
Taking expectations of $\Phi^{\tilde{P}}_{m,n}$ and $\tilde{P}_{r,n}$ respectively returns us to the unconditional setting. In \cref{r.V_n-lambda}, the expectation of the $\tilde{\lambda}_{l,n}$ appearing in $\tilde{P}_{2,n}$ and $\tilde{P}_{4,n}$ is evaluated -- the only source of randomness in \labelcref{eq.tP-2+4}. For now, let all moments be finite for clear illustration. The evaluation results in \begin{equation}\label{eq.E-tlambda_2k} \begin{split} \mathbb{E}\bigg[\frac{\tilde{\lambda}_{4,n}}{4!}\bigg] &= n^{-1} \big(- \tfrac{1}{12}\big) \mu_4 + n^{-2} \tfrac{1}{12} \big(2\mu_6 +\mu_4 - 3 \mu_4^2\big) + \mathcal{O}\big(n^{-3}\big),\\ \mathbb{E}\bigg[\frac{\tilde{\lambda}_{6,n}}{6!}\bigg] &= n^{-2} \tfrac{1}{45} \mu_6 + n^{-3} \big(- \tfrac{1}{45}\big) (3\mu_8+3\mu_6-6\mu_4\mu_6) + \mathcal{O}\big(n^{-4}\big),\\ \mathbb{E}\bigg[\frac{1}{2} \Big(\frac{\tilde{\lambda}_{4,n}}{4!}\Big)^2\bigg] &=n^{-2} \tfrac{1}{288} \mu_4^2 + n^{-3} \big(- \tfrac{1}{288}\big) (8\mu_6\mu_4-7\mu_4^2- \mu_8) + \mathcal{O}\big(n^{-4}\big). \end{split} \end{equation}
As the expected value of $\tilde{\lambda}_{l,n}$ (and its powers) admit an expansion in powers of $n^{-1}$, it becomes evident that $\mathbb{E}\big[\Phi^{\tilde{P}}_{m,n}\big]$ also admits an expansion in powers of $n^{-1}$. However, we will keep using the common notation of an expansion in powers of $n^{-1/2}$. For $r\in\mathbb{N}$, the approximating polynomial $Q_{r}$ is thus defined as the coefficient of $n^{-r/2}$ in the expansion of \begin{equation}\label{eq.Q-def} \mathbb{E}\big[\Phi^{\tilde{P}}_{m,n}(x)\big] = \Phi(x)+\sum_{r=1}^{m-2}Q_{r}(x)n^{-r/2} + \mathcal{O}\big(n^{-(m-1)/2}\big) \end{equation} for all $x\in\mathbb{R}$. As a result, we may define the approximating functions for the distribution function of $T_n$ by \labelcref{eq.Phi^Q-def}.
Combining \labelcref{eq.tP-2+4,eq.Q-def,eq.E-tlambda_2k}, it is apparent that the first polynomials have the form \labelcref{eq.Q-2+4}. As $\tilde{P}_{2r-1,n}=0$ and the expected value of the relevant $\tilde{\lambda}$ and its powers do not provide terms of orders $n^{-(2r-1)/2}$ for any $r\in\mathbb{N}$, also $Q_{2r-1}=0$ for all $r\in\mathbb{N}$. The approximating polynomials $Q_r$ can be directly calculated by Chung's method \cite{Chu46,FD10}, by the $\delta$-method or by the smooth function model which are both explained in detail in \cite[Chapter 2]{Hal92edgeworth}.
The Edgeworth expansion can also be applied to the density of a normalized sum (see e.g. \cite[Chapter 4]{BR76normal}, \cite{BCG11}, \cite[Section 2.8]{Hal92edgeworth}, \cite[Section VII.3]{Pet75sums}). Concerning this expansion for the density, we state the suitable density-related functions for our setting. By differentiating \labelcref{eq.tP}, for $r\in\mathbb{N}$ we get the approximating polynomials for the conditional density function \begin{equation}\label{eq.tp} \tilde{p}_{r,n}(x)=\phi(x)\sum_{*(k_\cdot,r,\cdot)} H_{r+2u(k_\cdot)}(x) \prod_{l=1}^{r} \frac{1}{k_l!} \Big(\frac{\tilde{\lambda}_{l+2,n}}{(l+2)!}\Big)^{k_l}, \end{equation} (where clearly $\tilde{p}_{2r-1,n}=0$) and thus differentiation \labelcref{eq.Phi^tP} yields the Edgeworth expansion for the density of $T_n$ \begin{equation}\label{eq.phi^tp} \phi^{\tilde{p}}_{m,n}(x)=\phi(x)+\sum_{r=1}^{m-2}\tilde{p}_{r,n}(x) \end{equation} for all $x\in\mathbb{R}$.
As above, for $r\in\mathbb{N}$ we define the approximating polynomials $q_r$ as the coefficient of $n^{-r/2}$ in the expansion of \begin{equation*} \mathbb{E}\big[\phi^{\tilde{p}}_{m,n}(x)\big] = \phi(x)+\sum_{r=1}^{m-2}q_{r}(x)n^{-r/2} + \mathcal{O}\big(n^{-(m-1)/2}\big) \end{equation*} for all $x\in\mathbb{R}$ and the approximating functions for the density of $T_n$ by \labelcref{eq.phi^q-def}. In consequence, the first approximating polynomials $q_r$ have the form \labelcref{eq.q-2}. Note \begin{equation*} \phi^{q}_{m,n}(x)=\ddX{} \Phi^{Q}_{m,n}(x) \quad \text{and} \quad q_{r}(x)=\ddX{} Q_{r}(x). \end{equation*}
As we will not assume all moments to be finite, the remainder terms have to be specified. In view of the non-uniform bounds, their dependence on the argument $x$ is crucial.
\begin{proposition}\label{p.E-tP-Q}
Assume that $X_1$ is symmetric and $\mathbb{E} |X_1|^{{s}}<\infty$ for some ${s}\ge2$. Then for $m=\lfloor {s} \rfloor$,
\begin{equation}\label{eq.E-tP-Q}
\sup_{x\in\mathbb{R}} \,\exp(x^2/4)\, \big|\mathbb{E}\big[\Phi^{\tilde{P}}_{m,n}(x)\big] - \Phi^Q_{m,n}(x)\big| = \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big)
\end{equation}
and
\begin{equation}\label{eq.E-tp-q}
\sup_{x\in\mathbb{R}} \,\exp(x^2/4)\, \big|\mathbb{E}\big[\phi^{\tilde{p}}_{m,n}(x)\big] - \phi^q_{m,n}(x)\big| = \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big).
\end{equation}
\end{proposition}
Although this result could have been expected, its proof (see \cref{app.proofs.pre}) is quite elaborate. Since the expectation of $\tilde{\lambda}_{l,n}$ cannot be explicitly evaluated, we study the expectation of its Taylor expansion around $n^{-l/2}\sum\tilde{\kappa}_{l,j}$. Here, powers of $X_1$ that are higher than $s$ arise. As we need a bound in order of $n$ while controlling the powers of $X_1$, tight bounds are developed on the critical summands. We show that by definition (see \labelcref{eq.Phi^tP} and \labelcref{eq.tP}), $\mathbb{E}\big[\Phi^{\tilde{P}}_{m,n}(x)\big]$ possesses a similar expansion and similar bounds that result in \cref{p.E-tP-Q} when combined with \labelcref{eq.Q-def}. Finally, we need some bounds on the approximation functions.
\begin{remark}
All summands in \labelcref{eq.tU} with $k_1>0$ are 0 as $\tilde{\lambda}_{3,n}=0$. Therefore, in all non-zero summands $k_1=0$ and thus $u(k_\cdot)\le r/2$. Additionally, note that the only $n$-dependence in the following expressions is in $\tilde{\lambda}_{l,n}$, which satisfies $|\tilde{\lambda}_{l,n}|\le c(l)$ for all $l=0,\dots,m$ due to \labelcref{eq.Lle1}. Then from \labelcref{eq.tphi},
\begin{equation}\label{eq.ddtl-tphi}
\Big|\ddT{l}\tilde{\varphi}_{\Phi^{\tilde{p}}_{m,n}}(t)\Big|
=\Big|\ddT{l} \Big(e^{-t^2/2}\Big(1+\sum_{r=1}^{m-2} \tilde{U}_{r,n}(it)\Big)\Big)\Big|
\le c(m) e^{-t^2/2}\big(1+|t|^{2m-4+l}\big)
\end{equation}
for $l=0,\dots,m$ and $t\in\mathbb{R}$. Equivalently, from \labelcref{eq.tP},
\begin{equation}\label{eq.tP-bound}
\big|\tilde{P}_{r,n}(x)\big|
\le c(m) \phi(x) (1+|x|^{2r-1})
\le c(m)
\end{equation}
and from \labelcref{eq.tp},
\begin{equation}\label{eq.tp-bound}
\big|\tilde{p}_{r,n}(x)\big|
\le c(m) \phi(x) (1+|x|^{2r})
\le c(m)
\end{equation}
for $r=0,\dots,m$ and $x\in\mathbb{R}$. Here, the constants $c(m)$ can be chosen uniformly for all $x\in\mathbb{R}$. \end{remark}
\section{Non-uniform bounds for Edgeworth expansions in the central limit theorem for self-normalized sums}\label{ch.distr}
Recall $\Phi^Q_{m,n}(x)=\Phi(x)+\sum_{r=1}^{m-2}Q_{r}(x)n^{-r/2}$ from \labelcref{eq.Phi^Q-def}. For convenience, we restate \cref{t.clt} which is the goal of this section:
\begin{theorem2}
Assume that $X_1$ is symmetric, the distribution of $X_1$ is non-singular and $\mathbb{E}|X_1|^{s}<\infty$ for some ${s}\ge2$. Then for $m=\lfloor {s} \rfloor$,
\begin{equation*}
\sup_{x\in\mathbb{R}} \, (1+|x|)^{m}|F_n(x)-\Phi^{Q}_{m,n}(x)|= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-({s}-2)/2} \, (\log n)^{({s}+2m)/2} \big).
\end{equation*} \end{theorem2}
For $s\in[2,4)$, $\Phi^{Q}_{m,n}$ reduces to $\Phi$. Here, the non-uniform Berry--Esseen bounds in \cite{JSW03,RW05} allow for better bounds in view of the argument $x$ than \cref{t.clt}.
Without the factor $(1+|x|)^{m}$, this result has been derived in \cite{Hal87}, but only for integer ${s}$. For $Z_n$, non-uniform bound in the spirit of ours can be found in \cite[Theorem 2, Chapter VI]{Pet75sums}, for instance.
\paragraph*{Structure of the proof} First, we condition on $\mathcal{F}_n$ and derive a result in the conditional setting (\cref{p.cond}) by showing a bound on the characteristic functions and their derivatives (\cref{p.lemma4}) and using an extension of the Fourier inversion formula. Afterwards we return to the unconditional setting by taking the expectation. On the left-hand side, we employ \cref{p.E-tP-Q} to move from $\mathbb{E}\Phi^{\tilde{P}}_{m,n}$ to $\Phi^Q_{m,n}$ while on the right-hand side, we evaluate the expectation of multiple rather involved random remainder terms (\cref{p.E.I,l.E.tL,l.exp.B_n,l.E.mom}), proving \cref{t.clt}. However, when returning to the unconditional setting by taking expectations, we face similar problems as \cite{Chu46,Hal87} including the expectation of the absolute value of the conditional characteristic function. Same as our precursors, we solve this problem by demanding for non-singularity.
\begin{remark}[Non-singularity]\label{r.non-singularity}
The condition of the distribution of $X_1$ being non-singular with respect to the Lebesgue measure is sometimes also denoted as the distribution of $X_1$ having a non-zero absolutely continuous component.
For $Z_n$ the analogue condition is Cram\'er's condition \labelcref{eq.cramer} which we compare to non-singularity now.
By the Lebesgue decomposition, for every distribution function there exists a unique decomposition
\begin{equation}\label{eq.lebesgue-dec}
F=p_{ac}F_{ac}+p_{d}F_{d}+p_{sc}F_{sc},
\end{equation}
with $p_{ac}+p_{d}+p_{sc}=1$, $p_\cdot\ge0$ where $F_\cdot$ are the distribution functions of an absolutely continuous distribution ($F_{ac}$), a discrete distribution ($F_{d}$) and a singular continuous distribution ($F_{sc}$) respectively. This corresponding decomposition holds for the characteristic function.
By the Riemann--Lebesgue lemma \cite[Theorem 4.1 (iii)]{BR76normal}, $\limsup_{|t|\to\infty}|\varphi_{1,ac}(t)]|=0$ and by \cite{Sch41}, $\limsup_{|t|\to\infty}|\varphi_{1,d}(t)]|=1$ and $\limsup_{|t|\to\infty}|\varphi_{1,sc}(t)]|\in[0,1]$ (the $\limsup$ can in fact take all values in the interval).
That is why non-singularity implies \labelcref{eq.cramer}. If $\limsup_{|t|\to\infty}|\varphi_{1,sc}(t)]|=1$ (for examples see \cite[p. 164]{Durrett05} or \cite{Sch41}) or $p_{sc}=0$, non-singularity and \labelcref{eq.cramer} are equivalent and both imply that the distribution of $X_1$ is not purely discrete.
The reason for our need for the stronger condition of non-singularity is that we need a variant of Cram\'er's condition including the expectation of the absolute value of the conditional characteristic function in \labelcref{eq.cases} in the proof of \cref{p.E.I}. This requires the stronger assumption of non-singularity due to the absolute value within the expectation.
For discrete lattice distributions (which are neither non-singular nor satisfy \labelcref{eq.cramer}), Edgeworth expansions for CLT and LLT of $Z_n$ can be found in \cite[Chapter 5]{BR76normal}. For $T_n$, the recent reference \cite{GvZ21} presents limit theorems and interesting occurring phenomena.
\end{remark}
\subsection{The conditional setting}\label{s.distr.cond}
Recall the definitions $M_n=\max |X_j|$, $B_n=V_n/M_n$, $\tilde{L}_{k,n}=V_n^{-k}\sum|X_j|^k$ and $\tilde{\varphi}_{\Phi^{\tilde{P}}_{m,n}}(t) =e^{-t^2/2}\big(1+\sum_{r=1}^{m-2} \tilde{U}_{r,n}(it)\big)$, where \[ \tilde{U}_{r,n}(it) = \sum_{*(k_\cdot,r,\cdot)} (it)^{r+2u(k_\cdot)} \prod_{l=1}^{r} \frac{1}{k_l!} \Big(\frac{\tilde{\lambda}_{l+2,n}}{(l+2)!}\Big)^{k_l} \] from \labelcref{eq.tU,eq.tphi} and the sum $\sum_{*(k_\cdot,r,\cdot)}$ is introduced in \cref{ch.pre}.
The following proposition examines differences of derivatives of the Fourier--Stieltjes transforms of $\tilde{F}_n$ and $\Phi^{\tilde{P}}_{m,n}$ around 0. Although versions of this proposition are a key part of every CLT or LLT including an Edgeworth expansion, the crucial point here is that the random variables $X_1,\dots,X_n$ conditional on $\mathcal{F}_n$ are discrete and furthermore not identically distributed any longer.
\begin{proposition}\label{p.lemma4}
Assume that $X_1$ is symmetric, $V_n^2>0$ and $m\ge2$ is an integer. Then
\[
\Big|\ddT{k} \Big(\tilde{\varphi}_{T_n}(t)-\tilde{\varphi}_{\Phi^{\tilde{P}}_{m,n}}(t)\Big)\Big|
\le c(m)\tilde{L}_{m+1,n}e^{-t^2/6}\big(|t|^{m+1-k}+|t|^{3m-1+k}\big)
\]
holds for $k=0,\dots, m$ in the interval $|t|< B_n$. \end{proposition}
Although there are many significant differences, the structure of our proof has its origins in the proof of \cite[Lemma 4, Chapter VI]{Pet75sums}. Prior to proving the proposition, we cite Lemma 2 from \cite[Chapter VI]{Pet75sums} in the conditional case. It gives an upper bound on $\tilde{L}_{a,n}$.
\begin{lemma}\label{l.lemma2}
Let $X_1,\dots,X_n$ be independent random variables and $\mathbb{E} X_j=0, \mathbb{E} X_j^2<\infty$ for all $j=1,\dots,n$. If $3\le a\le b$, then
\[
\tilde{L}_{a,n}^{1/(a-2)}\le \tilde{L}_{b,n}^{1/(b-2)}.
\] \end{lemma}
\begin{proof}[Proof of \cref{p.lemma4}]
$X_j$ has the conditional characteristic function
\begin{equation*}
\tilde{\varphi}_j(t)=\tilde{\mathbb{E}}[\exp(itX_j)]=\frac{1}{2} \Big(\exp(it|X_j|)+\exp(-it|X_j|)\Big)
=\cos(t|X_j|)
\end{equation*}
for $j=1,\dots,n$. Clearly $\cos(t|X_j|)=\cos(tX_j)$, but we keep the first notation to highlight that we are operating in the conditional setting where $|X_j|$ is deterministic. By conditional independence we can write for the conditional Fourier transform of $T_n$
\begin{equation}\label{eq.phi_n}
\tilde{\varphi}_{T_n}(t)
=\prod\tilde{\mathbb{E}}\big[\exp(itV_n^{-1}X_j)\big]
=\prod\tilde{\varphi}_j(tV_n^{-1})
=\prod \cos\big(tV_n^{-1}|X_j|\big).
\end{equation}
For applying the logarithm later, we need to bound $\tilde{\varphi}_j(tV_n^{-1})=\cos(tV_n^{-1}|X_j|)$ away from 0 for all $j=1,\dots,n$. So we restrict ourselves on the interval $|t|< B_n$ throughout the rest of the proof. Thus,
\begin{equation}\label{eq.phi_j2}
\tilde{\varphi}_j(tV_n^{-1})=\cos\big(tV_n^{-1}|X_j|\big)\ge \cos(1) > \tfrac12
\end{equation}
for all $j=1,\dots,n$.
Since $X_j$ are symmetric, $\sum\tilde{\kappa}_{1,j} = 0$ and $\sum\tilde{\kappa}_{2,j} = V_n^2$. Recall $\tilde{\lambda}_{l,n}=V_n^{-l}\sum\tilde{\kappa}_{l,j}$ from \labelcref{eq.tlambda}. By taking the logarithm, we thus get (see \labelcref{eq.cumulant-abl})
\begin{align*}
\log \tilde{\varphi}_{T_n}(t)
&=\sum\log\tilde{\varphi}_j(tV_n^{-1})
=\sum\sum_{l=1}^{m}\tilde{\kappa}_{l,j}\frac{(itV_n^{-1})^l}{l!}+\mathcal{O}(t^{m+1})\\
&=\sum_{l=1}^{m-2}\frac{\tilde{\lambda}_{l+2,n}}{(l+2)!}(it)^{l+2}-\frac{t^2}{2}+\mathcal{O}(t^{m+1}).
\end{align*}
The series expansion is valid as all conditional cumulants are finite. To eliminate the normal part (which is $-t^2/2$), we define
\begin{equation}\label{eq.v_n}
v_n(t,z):
=\frac{t^2}{2}+\frac{1}{z^2}\log \tilde{\varphi}_{T_n}(tz)
=\sum_{l=1}^{m-2}\frac{\tilde{\lambda}_{l+2,n}}{(l+2)!}(it)^{l+2}z^l+\mathcal{O}(t^{m+1},z^{m-1})
\end{equation}
for $0<z\le1$.
Now we expand $e^{v_n(t,z)}$ as a series in $z$ (with fixed $t$)
\begin{equation*}
e^{v_n(t,z)}=1+\sum_{r=1}^{m-2}\tilde{U}_{r,n}(it)z^r+\mathcal{O}(z^{m-1})
\end{equation*}
such that
\begin{equation}\label{eq.phi_n^z}
\tilde{\varphi}_{T_n}(tz)^{1/z^2}
=\exp\Big(-\frac{t^2}{2}+v_n(t,z)\Big)
=e^{-t^2/2}\Big(1+\sum_{r=1}^{m-2}\tilde{U}_{r,n}(it)z^r+R_n(t,z)\Big).
\end{equation}
Recalling $\tilde{\varphi}_{\Phi^{\tilde{P}}_{m,n}}(t) =e^{-t^2/2}\big(1+\sum_{r=1}^{m-2} \tilde{U}_{r,n}(it)\big)$, we now have to show that $R_n(t,z)$ and its derivatives are small for $z\to1$.
First, we examine (see e.g. \cite[Theorem 3.3.18]{Durrett05})
\begin{equation}\label{eq.dl-phi}
\begin{split}
\lefteqn{\Big|\ddT{l} \tilde{\varphi}_j(tzV_n^{-1})\Big|}\quad\\
&=\Big|\tilde{\mathbb{E}}\Big[\big(izV_n^{-1}X_j\big)^l\exp(itzV_n^{-1}X_j)\Big]\Big|\\
&\le \tilde{\mathbb{E}}\Big[z^lV_n^{-l}|X_j|^l\Big]
= z^lV_n^{-l}|X_j|^l
\end{split}
\end{equation}
for $j=1,\dots,n$ and $l\in\mathbb{N}$.
According to the chain rule (see e.g. \cite[Lemma 1, Chapter VI]{Pet75sums}), we derive
\begin{equation}\label{eq.dr-logphi}
\begin{split}
&\Big|\ddT{r} \log\tilde{\varphi}_j(tzV_n^{-1})\Big|\\
&\quad=\Big|r!\sum_{*(k_\cdot,r,\cdot)} (-1)^{u(k_\cdot)-1}(u(k_\cdot)-1)!\tilde{\varphi}_j(tzV_n^{-1})^{-u(k_\cdot)} \prod_{l=1}^{r} \frac{1}{k_l!} \Big(\frac{1}{l!}\ddT{l} \tilde{\varphi}(tzV_n^{-1})\Big)^{k_l} \Big|\\
&\quad\le c(r)\sum_{*(k_\cdot,r,\cdot)} \prod_{l=1}^{r}\big(z^lV_n^{-l}|X_j|^l\big)^{k_l}
= c(r)z^rV_n^{-r}|X_j|^r
\end{split}
\end{equation}
for $j=1,\dots,n$ and $r=1,\dots,m+1$, where we used \labelcref{eq.phi_j2,eq.dl-phi}.
By Taylor expansion of $v_n(t,z)$ (at the point $0$, with an intermediate point $u\in\mathbb{R}$ between $0$ and $t$, depending on $v_n(\cdot,z)$ and $t$) using \labelcref{eq.cumulant-abl}, we get
\begin{align*}
v_n(t,z)&=\frac{t^2}{2}+\frac{1}{z^2}\sum\log\tilde{\varphi}_j(tzV_n^{-1})\\
&=v_n(0,z)\\
&\quad+t\Big(0+\frac{1}{z^2}\sum zV_n^{-1}\tilde{\kappa}_{1,j}\,i\Big)\\
&\quad+\frac{t^2}{2}\Big(1+\frac{1}{z^2}\sum (zV_n^{-1})^2\tilde{\kappa}_{2,j}\,i^2\Big)\\
&\quad+\frac{t^3}{6z^2}\sum\ddT{3} \log\tilde{\varphi}_j(tzV_n^{-1})\Big|_{t=u}\\
&=\frac{t^3}{6z^2}\sum\ddT{3} \log\tilde{\varphi}_j(tzV_n^{-1})\Big|_{t=u}\, .
\end{align*}
Now \labelcref{eq.dr-logphi} implies
\begin{equation}\label{eq.v_n-1}
|v_n(t,z)|
\le\frac{|t|^3}{6z^2}\sum c(3)z^3V_n^{-3}|X_j|^3
\le c(3)|t|^3V_n^{-3}\sum|X_j|^3
=c(3)|t|^3\tilde{L}_{3,n}.
\end{equation}
By a Taylor expansion of $\ddT{} v_n(t,z)$ (at the point $0$, with an intermediate point $u\in\mathbb{R}$ between $0$ and $t$ and $u$ depending on $\ddT{} v_n(\cdot,z)$ and $t$), we get in the same manner
\begin{align*}
\ddT{} v_n(t,z)
&=\frac{t^2}{2z^2}\sum\ddT{3} \log\tilde{\varphi}_j(tzV_n^{-1})\Big|_{t=u}
\end{align*}
which leads to
\begin{equation}\label{eq.d1-v_n}
\Big|\ddT{} v_n(t,z)\Big|
\le\frac{t^2}{2z^2}\sum c(3)z^3V_n^{-3}|X_j|^3
\le c(3)|t|^2V_n^{-3}\sum|X_j|^3
=c(3)|t|^2\tilde{L}_{3,n}
\end{equation}
by \labelcref{eq.dr-logphi}. Similarly,
\begin{equation}\label{eq.d2-v_n}
\Big|\ddT{2} v_n(t,z)\Big|\le c(3)|t|V_n^{-3}\sum|X_j|^3=c(3)|t|\tilde{L}_{3,n}.
\end{equation}
For $l=3,\dots,m$, the definition of $v_n(t,z)$ and \labelcref{eq.dr-logphi} directly give
\begin{equation}\label{eq.dl-v_n}
\Big|\ddT{l} v_n(t,z)\Big|
\le\frac{1}{z^2}\sum c(l)z^lV_n^{-l}|X_j|^l
= c(l)\tilde{L}_{l,n}\,z^{l-2}
\le c(l)\tilde{L}_{l,n}.
\end{equation}
The necessity of bounding the next term will become evident in \labelcref{eq.R_n,eq.R_n^1,eq.R_n^2,eq.da-R_n^2}. For ${1\le r\le m \le h}$, the chain rule for powers of functions (see for example \cite[Lemma 3, Chapter VI]{Pet75sums}) and our derived estimates \labelcref{eq.v_n-1,eq.d1-v_n,eq.d2-v_n,eq.dl-v_n} gives
\begin{align*}
\lefteqn{\Big|\frac{1}{h!}\ddT{r} v_n^h(t,z)\Big|
=\Big|\frac{r!\cdot h!}{h!} \sum_{u=1}^{r\land h} \sum_{*(k_\cdot,r,u)} \frac{v_n^{h-u}(t,z)}{(h-u)!} \prod_{l=1}^{r} \frac{1}{k_l!} \Big(\frac{1}{l!}\ddT{l} v_n(t,z)\Big)^{k_l}\Big|}\qquad\\
&\le c(m)\sum_{u=1}^{r} \sum_{*(k_\cdot,r,u)} \frac{|v_n(t,z)|^{h-m}}{(h-m)!}|v_n(t,z)|^{m-u} \prod_{l=1}^{r} \Big|\ddT{l} v_n(t,z)\Big|^{k_l}\\
&\le c(m) \frac{|v_n(t,z)|^{h-m}}{(h-m)!} \sum_{u=1}^{r} \sum_{*(k_\cdot,r,u)}
\big(|t|^3\tilde{L}_{3,n}\big)^{m-u}
\big(|t|^2\tilde{L}_{3,n}\big)^{k_1}
\big(|t|\tilde{L}_{3,n}\big)^{k_2}
\prod_{l=3}^{r} \tilde{L}_{l,n}^{k_l}\\
&= c(m) \frac{|v_n(t,z)|^{h-m}}{(h-m)!} \sum_{u=1}^{r} \sum_{*(k_\cdot,r,u)}
|t|^{3m-3u+2k_1+k_2}
\tilde{L}_{3,n}^{m-u+k_1+k_2}
\prod_{l=3}^{r} \tilde{L}_{l,n}^{k_l}.
\end{align*}
Now, \cref{l.lemma2} (with $b=m+1$) implies that this is bounded by
\begin{align*}
c(m) &\frac{|v_n(t,z)|^{h-m}}{(h-m)!} \sum_{u=1}^r \sum_{*(k_\cdot,r,u)}
|t|^{3m-3u+2k_1+k_2}
\tilde{L}_{m+1,n}^{(m-u+k_1+k_2)/(m-1)}
\prod_{l=3}^{r} \tilde{L}_{m+1,n}^{(l-2)k_l/(m-1)}\\
&= c(m) \frac{|v_n(t,z)|^{h-m}}{(h-m)!} \sum_{u=1}^r \sum_{*(k_\cdot,r,u)}
|t|^{3m-3u+2k_1+k_2}
\tilde{L}_{m+1,n}^{(m-3u+2k_1+k_2+r)/(m-1)}\\
&= c(m) \frac{|v_n(t,z)|^{h-m}}{(h-m)!} \tilde{L}_{m+1,n} |t|^{3m-r-1}\sum_{u=1}^r \sum_{*(k_\cdot,r,u)}
\big(|t|\tilde{L}_{m+1,n}^{1/(m-1)}\big)^{1-3u+2k_1+k_2+r},
\end{align*}
where the last exponent is greater than 0 due to
\[2k_1+k_2+r=3k_1+3k_2+3k_3+4k_4+\dots+rk_r\ge3(k_1+k_2+\dots+k_r)=3u.\]
For $k\ge3$, we can bound
\begin{equation}\label{eq.tL-k-1}
|t|^{k-2}\tilde{L}_{k,n}
\le B_n^{k-2}\tilde{L}_{k,n}
=\frac{V_n^{k-2}\sum|X_j|^k}{V_n^kM_n^{k-2}}
\le\frac{\sum|X_j|^2}{V_n^2}
=1
\end{equation}
and thus
\begin{equation}\label{eq.tL-k-2}
|t|\tilde{L}_{k,n}^{1/(k-2)}\le1
\end{equation}
which in case of $k=m+1$ leads to
\begin{equation}\label{eq.dr-v_n^h}
\Big|\frac{1}{h!}\ddT{r} v_n^h(t,z)\Big|\le c(m) \frac{|v_n(t,z)|^{h-m}}{(h-m)!} \tilde{L}_{m+1,n} |t|^{3m-r-1}
\end{equation}
for $1\le r\le m \le h$.
We look back at \labelcref{eq.phi_n^z} and define
\begin{equation}\label{eq.R_n}
R_n(t,z)=R_{n,1}(t,z)+R_{n,2}(t,z),
\end{equation}
where
\begin{align}\label{eq.R_n^1}
R_{n,1}(t,z)
&= \sum_{u=0}^{m-1}\frac{v_n^u(t,z)}{u!}-\Big(1+\sum_{r=1}^{m-2}\tilde{U}_{r,n}(it)z^r\Big),\\
R_{n,2}(t,z)
&= \sum_{u=m}^{\infty}\frac{v_n^u(t,z)}{u!}.\label{eq.R_n^2}
\end{align}
By \labelcref{eq.dr-v_n^h}, this implies (with $h=u$ and $r=a$)
\begin{equation}\begin{split}\label{eq.da-R_n^2}
\Big|\ddT{a} R_{n,2}(t,z)\Big|
&\le\sum_{u=m}^{\infty}\Big|\frac{1}{u!}\ddT{a} v_n^u(t,z)\Big|\\
&\le\sum_{u=m}^{\infty}c(m) \frac{|v_n(t,z)|^{u-m}}{(u-m)!} \tilde{L}_{m+1,n} |t|^{3m-a-1}\\
&=c(m) e^{|v_n(t,z)|} \tilde{L}_{m+1,n} |t|^{3m-a-1}
\end{split}\end{equation}
for all $a=1,\dots, m$. By \labelcref{eq.v_n-1}, \labelcref{eq.tL-k-1} and \cref{l.lemma2},
\begin{equation}\begin{split}\label{eq.d0-R_n^2}
\Big|R_{n,2}(t,z)\Big|
&\le\sum_{u=m}^{\infty}\Big|\frac{1}{u!} v_n^u(t,z)\Big|
\\
&\leq e^{|v_n(t,z)|}|v_n(t,z)|^{m}
\\
&\le c(m) e^{|v_n(t,z)|} \tilde{L}_{3,n}^{m-1}|t|^{3m-1}\\
&
\le c(m) e^{|v_n(t,z)|} \tilde{L}_{m+1,n}|t|^{3m-1},
\end{split}\end{equation}
so \labelcref{eq.da-R_n^2} also holds for $a=0$.
Let $x>-1$. As
\[
\ddX{} \big(\log(1+x)-(x-x^2)\big) = \frac{1}{1+x}-1+2x
\]
is less than or equal to 0 for $x\in[-1/2,0]$ and greater than or equal to 0 for $x\ge 0$, the function $x\mapsto \log(1+x)-(x-x^2)$ is decreasing for $x\in[-1/2,0]$ and increasing for $x\ge 0$. Additionally, it is continuous and $\log(1+0)-(0-0^2)=0$. Hence, $\log(1+x)-(x-x^2)\ge0$ for $x\ge-1/2$ and therefore $\log(1+x)\ge x-x^2$ for $x\ge-1/2$.
Additionally $\log(1+x)\le x$. Thus, for every $x\ge-1/2$ there exists a $\vartheta$ (depending on $x$) with $0 \le \vartheta \le 1$ such that $\log(1+x) = x-\vartheta x^2$.
By \labelcref{eq.phi_j2}, $\tilde{\varphi}_j(tzV_n^{-1})>1/2$ for all $j=1,\dots,n$ and thus by inserting $x=\tilde{\varphi}_j(tzV_n^{-1})-1$ into the equation above,
\begin{align*}
|v_n(t,z)|
&= \Big|\frac{t^2}{2}+\frac{1}{z^2}\sum\log\tilde{\varphi}_j(tzV_n^{-1})\Big|\\
&= \Big|\frac{t^2}{2}+\frac{1}{z^2} \sum\Big(\big(\cos(tzV_n^{-1}|X_j|)-1\big)+\vartheta_j \big(\tilde{\varphi}_j(tzV_n^{-1})-1\big)^2\Big)\Big|
\end{align*}
for some $\vartheta_j$ (depending on $\tilde{\varphi}_j(tzV_n^{-1})$), where $|\vartheta_j|\le1$ for $j=1,\dots,n$. Additionally, there exists $\vartheta'$ (depending on $x$) with the properties $\cos(x)=1-x^2/2+\vartheta'x^4/(4!)$ and ${0\le\vartheta'\le1}$ and $|\cos(x)-1|\le x^2/2$ for $x\in\mathbb{R}$. By \labelcref{eq.tL-k-1}, we deduce
\begin{align}
|v_n(t,z)|
&= \Big|\frac{t^2}{2}+\frac{1}{z^2} \sum\Big(\big(\cos(tzV_n^{-1}|X_j|)-1\big)+\vartheta_j \big(\tilde{\varphi}_j(tzV_n^{-1})-1\big)^2\Big)\Big|\nonumber\\
&= \Big|\frac{1}{z^2} \sum\Big(\big(\vartheta_j'\tfrac{(tzV_n^{-1}|X_j|)^4}{24}\big)+\vartheta_j \big(\tilde{\varphi}_j(tzV_n^{-1})-1\big)^2\Big)\Big|\nonumber\\
&\le \frac{t^4z^2}{24\,V_n^4} \sum|X_j|^4+\frac{1}{z^2}\sum \big|\tilde{\varphi}_j(tzV_n^{-1})-1\big|^2\label{eq.v_n-3}\\
&\le \frac{t^4z^2}{24} \tilde{L}_{4,n}+\frac{1}{2z^2}\sum \big|\tilde{\varphi}_j(tzV_n^{-1})-1\big|\nonumber\\
&\le \frac{t^2z^2}{24} (t^2\tilde{L}_{4,n})+\frac{1}{2z^2}\sum \frac{(tzV_n^{-1}|X_j|)^2}{2}
\le \frac{t^2}{3}\nonumber
\end{align}
for some $\vartheta_j'$ (depending on $tzV_n^{-1}|X_j|$), where $0\le\vartheta_j'\le1$ for $j=1,\dots,n$.
Putting together \labelcref{eq.da-R_n^2,eq.d0-R_n^2,eq.v_n-3} implies
\begin{equation}\begin{split}\label{eq.da-R_n^2-2}
\Big|\ddT{a} R_{n,2}(t,z)\Big|
\le c(m) e^{|v_n(t,z)|} \tilde{L}_{m+1,n} |t|^{3m-a-1}
\le c(m) \tilde{L}_{m+1,n} |t|^{3m-a-1}e^{t^2/3}
\end{split}\end{equation}
for all $a=0,\dots,m$.
Now to $R_{n,1}$. By \labelcref{eq.v_n}, we have
\begin{equation}\label{eq.v_n-4}
v_n(t,z)=\sum_{k=1}^{m-2}\frac{\tilde{\lambda}_{k+2,n}}{(k+2)!}(it)^{k+2}z^k+r_n(t,z)z^{m-1},
\end{equation}
where $r_n$ is the remainder of a Taylor expansion (at the point $0$, with intermediate point $u\in\mathbb{R}$ between $0$ and $t$ and $u$ depending on $v_n(\cdot,z)$ and $t$)
\begin{equation*}
r_n(t,z)=\frac{t^{m+1}}{(m+1)!z^{m-1}}\;\ddT{m+1} v_n(t,z)\Big|_{t=u}.
\end{equation*}
For $l=0,\dots,m$ and $j=1,\dots,n$, by Taylor expansion of $\ddT{l}\log\tilde{\varphi}_j(tzV_n^{-1})$ (at $0$, with intermediate point $u_j\in\mathbb{R}$ between $0$ and $t$, depending on $\ddT{l}\log\tilde{\varphi}_j(\cdot \,zV_n^{-1})$ and $t$) and \labelcref{eq.cumulant-abl},
\begin{align*}
\lefteqn{\ddT{l}\log\tilde{\varphi}_j(tzV_n^{-1})}\quad\\*
&=\sum_{k=0}^{m-l} \frac{t^{k}}{k!}\;\ddT{l+k}\log\tilde{\varphi}_j(tzV_n^{-1})\Big|_{t=0}
+\frac{t^{m-l+1}}{(m-l+1)!}\;\ddT{m+1}\log\tilde{\varphi}_j(tzV_n^{-1})\Big|_{t=u_j}\\
&=\sum_{k=l}^{m} \frac{\tilde{\kappa}_{k,j}}{V_n^k}z^{k}i^{k} \frac{t^{k-l}}{(k-l)!}
+\frac{t^{m-l+1}}{(m-l+1)!}\;\ddT{m+1}\log\tilde{\varphi}_j(tzV_n^{-1})\Big|_{t=u_j}
\end{align*}
for some $0\le u_j\le t$.
For $l=0,\dots,m$, differentiating \labelcref{eq.v_n-4} and using \labelcref{eq.dr-logphi} implies
\begin{align*}
\lefteqn{\Big|\ddT{l} r_n(t,z)\Big|
}\:\:\\
&= z^{-(m-1)}\Big|\ddT{l}\Big(\frac{t^2}{2}+\frac{1}{z^2}\sum\log\tilde{\varphi}_j(tzV_n^{-1})\Big) -\ddT{l}\sum_{k=1}^{m-2}\frac{\tilde{\lambda}_{k+2,n}}{(k+2)!}(it)^{k+2}z^k\Big|\\
&= z^{-(m-1)}\Big|\Big(\ddT{l}\frac{t^2}{2}\Big)+\Big(\frac{1}{z^2}\sum\ddT{l}\log\tilde{\varphi}_j(tzV_n^{-1})\Big)
-\sum_{k=3\vee l}^{m}\tilde{\lambda}_{k,n}i^{k}\frac{t^{k-l}}{(k-l)!}z^{k-2} \Big|\\
&= z^{-(m-1)}\bigg|\Big(\ddT{l}\frac{t^2}{2}\Big)\\
&\qquad+\Big(\frac{1}{z^2}\sum\Big(\sum_{k=l}^{m} \frac{\tilde{\kappa}_{k,j}}{V_n^k}z^{k}i^{k} \frac{t^{k-l}}{(k-l)!}
+\Big(\frac{t^{m-l+1}}{(m-l+1)!}\;\ddT{m+1}\log\tilde{\varphi}_j(tzV_n^{-1})\Big|_{t=u_j}\Big)\Big)\Big)\\
&\qquad-\sum_{k=l}^{m}\tilde{\lambda}_{k,n}i^{k}\frac{t^{k-l}}{(k-l)!}z^{k-2} +\mathbbm{1}_{\{l\le2\}}\tilde{\lambda}_{2,n}i^{2}\frac{t^{2-l}}{(2-l)!}\,\bigg|\\
&= z^{-(m-1)}\Big|\frac{1}{z^2}\sum\frac{t^{m-l+1}}{(m-l+1)!}\;\ddT{m+1}\log\tilde{\varphi}_j(tzV_n^{-1})\Big|_{t=u_j}\Big|\\
&\le z^{-(m+1)}\frac{|t|^{m-l+1}}{(m-l+1)!}\sum c(m)z^{m+1}V_n^{-(m+1)}|X_j|^{m+1}\\
&= c(m)|t|^{m-l+1}\tilde{L}_{m+1,n}.
\end{align*}
For $r,k=0,\dots,m$ by chain rule, this leads to
\begin{align*}
\Big|\ddT{r} r_n^k(t,z)\Big|
&=\Big|r!\cdot k!\sum_{u=1}^{r\land k} \sum_{*(k_\cdot,r,u)} \frac{r_n^{k-u}(t,z)}{(k-u)!} \prod_{l=1}^{r} \frac{1}{k_l!} \Big(\frac{1}{l!}\ddT{l} r_n(t,z)\Big)^{k_l}\Big|\\
&\le c(m)\sum_{u=1}^{r\land k} \sum_{*(k_\cdot,r,u)} \Big(|t|^{m+1}\tilde{L}_{m+1,n}\Big)^{k-u} \prod_{l=1}^{r} \Big(|t|^{m-l+1}\tilde{L}_{m+1,n}\Big)^{k_l}\\
&= c(m)\sum_{u=1}^{r\land k} \sum_{*(k_\cdot,r,u)} |t|^{(m+1)(k-u)}\tilde{L}_{m+1,n}^{k-u} |t|^{(m+1)u-r}\tilde{L}_{m+1,n}^{u}\\
&= c(m)|t|^{(m+1)k-r}\tilde{L}_{m+1,n}^{k}
\end{align*}
and for $a,k=0,\dots,m$ and $v=0,\dots,3m$ by the Leibniz rule,
\begin{equation}\label{eq.da-mix}
\begin{split}
\Big|\ddT{a} \big(r_n^{k}(t,z)\ t^{v-(m+1)k}\big)\Big|
&\le c(m)\sum_{b=0}^a\Big(|t|^{(m+1)k-b}\tilde{L}_{m+1,n}^{k}\Big)\Big(|t|^{v-(m+1)k-a+b}\Big)\\
&\le c(m)|t|^{v-a}\tilde{L}_{m+1,n}^{k}.\\
\end{split}
\end{equation}
In \labelcref{eq.R_n^1} the summand with $u=0$ of the first sum and the 1 cancel each other out, so we analyse the remaining first sum using \labelcref{eq.v_n-4}
\begin{align*}
\sum_{u=1}^{m-1}\frac{v_n^u(t,z)}{u!}
&=\sum_{u=1}^{m-1}\frac{1}{u!}\Big(
\sum_{l=1}^{m-2}\frac{\tilde{\lambda}_{l+2,n}}{(l+2)!}(it)^{l+2}z^l+r_n(t,z)z^{m-1}
\Big)^u\\
&=\sum_{u=1}^{m-1}\sum_{*(k_\cdot,\cdot,u,m-1)}
\prod_{l=1}^{m-2}\frac{1}{k_l!}\Big(\frac{\tilde{\lambda}_{l+2,n}}{(l+2)!}(it)^{l+2}z^l\Big)^{k_l}\;\frac{1}{k_{m-1}!}\Big(r_n(t,z)z^{m-1}\Big)^{k_{m-1}}\\
&=\sum_{u=1}^{m-1}\sum_{*(k_\cdot,\cdot,u,m-1)}
\prod_{l=1}^{m-2}\frac{1}{k_l!}\Big(\frac{\tilde{\lambda}_{l+2,n}}{(l+2)!}\Big)^{k_l}\;\frac{1}{k_{m-1}!}\Big(r_n(t,z)\Big)^{k_{m-1}} \\
&\qquad\qquad\qquad\qquad \cdot (it)^{r(k_\cdot)+2u-(m+1)k_{m-1}}\,z^{r(k_\cdot)}.
\end{align*}
Let $r(k_\cdot)\le m-2$, so particularly $u\le r\le m-2$. Since $k_r$ is the highest $k_l$ that can be greater 0, also $k_{r+1}=k_{r+2}=\dots =k_{m-1}=0$. After shortening the term, we see that instead of summing over $u$, we can sum over $r$ as well without changing any summand. This yields
\begin{align*}
&\sum_{u=1}^{m-1}\sum_{*(k_\cdot,\cdot,u,m-1)} \mathbbm{1}_{\{r(k_\cdot)\le m-2\}}
\prod_{l=1}^{m-2}\frac{1}{k_l!}\Big(\frac{\tilde{\lambda}_{l+2,n}}{(l+2)!}\Big)^{k_l}\;\frac{1}{k_{m-1}!}\Big(r_n(t,z)\Big)^{k_{m-1}} \\*
&\qquad\qquad\qquad\qquad\qquad\qquad \cdot (it)^{r(k_\cdot)+2u-(m+1)k_{m-1}}\,z^{r(k_\cdot)}\\
&=\sum_{u=1}^{m-2}\sum_{*(k_\cdot,\cdot,u,m-2)} \mathbbm{1}_{\{r(k_\cdot)\le m-2\}}
\prod_{l=1}^{r(k_\cdot)}\frac{1}{k_l!}\Big(\frac{\tilde{\lambda}_{l+2,n}}{(l+2)!}\Big)^{k_l}(it)^{r(k_\cdot)+2u}\,z^{r(k_\cdot)}\\
&=\sum_{u=1}^{m-2}\sum_{r=1}^{m-2}\sum_{*(k_\cdot,r,u)}
\prod_{l=1}^{r}\frac{1}{k_l!}\Big(\frac{\tilde{\lambda}_{l+2,n}}{(l+2)!}\Big)^{k_l}(it)^{r+2u}\,z^{r}\\
&=\sum_{r=1}^{m-2}\sum_{*(k_\cdot,r,\cdot)}
\prod_{l=1}^{r}\frac{1}{k_l!}\Big(\frac{\tilde{\lambda}_{l+2,n}}{(l+2)!}\Big)^{k_l}(it)^{r+2u(k_\cdot)}\,z^r\\
&=\sum_{r=1}^{m-2}\tilde{U}_{r,n}\,z^r.
\end{align*}
As a result, in \labelcref{eq.R_n^1} everything is eliminated except for $c(m)$ many summands from the first sum, having the form
\begin{equation}\label{tlambda-bound}
c(m)\prod_{l=1}^{m-2}\tilde{\lambda}_{l+2,n}^{k_l}\;r_n(t,z)^{k_{m-1}}(it)^{r(k_\cdot)+2u-(m+1)k_{m-1}}\,z^{r(k_\cdot)}
\end{equation}
for some $k_1,\dots, k_{m-1}\ge0, \sum_{l=1}^{m-1}lk_l=r(k_\cdot)\ge m-1, u=\sum_{l=1}^{m-1}k_l, 1\le u\le m-1$. The sum over these $k_i$ will be denoted by $\sum_{*(k_\cdot,\cdot,u,m-1)*}$.
The cumulants are upper bounded by constants times the absolute moments, so by \cref{l.lemma2},
\[\tilde{\lambda}_{l+2,n}
=\frac{\sum\tilde{\kappa}_{l+2,j}}{V_n^{l+2}}
\le c(l)\frac{\sum|X_j|^{l+2}}{V_n^{l+2}}
= c(l)\tilde{L}_{l+2,n}
\le c(l) \tilde{L}_{m+1,n}^{l/(m-1)}
\]
for $l=1,\dots,m-1$. For the second part of \labelcref{tlambda-bound}, we use \labelcref{eq.da-mix} (with $k=k_{m-1}$ and ${v=r+2u}$) and get by \labelcref{eq.tL-k-2}
\begin{equation}\label{eq.da-R_n^1}
\begin{split}
\Big|\ddT{a} R_{n,1}(t,z)\Big|
&\le c(m)\sum_{*(k_\cdot,\cdot,u,m-1)*}\prod_{l=1}^{m-2}\tilde{L}_{m+1,n}^{k_l\cdot l/(m-1)}|t|^{r(k_\cdot)+2u-a}\tilde{L}_{m+1,n}^{k_{m-1}}\,z^{r(k_\cdot)}\\
&\le c(m)\sum_{*(k_\cdot,\cdot,u,m-1)*}\tilde{L}_{m+1,n}^{r(k_\cdot)/(m-1)}|t|^{r(k_\cdot)+2u-a}\\
&\le c(m)\sum_{*(k_\cdot,\cdot,u,m-1)*}\tilde{L}_{m+1,n}\Big(\tilde{L}_{m+1,n}^{1/(m-1)}|t|\Big)^{r(k_\cdot)-(m-1)}|t|^{m-1+2u-a}\\
&\le c(m)\sum_{*(k_\cdot,\cdot,u,m-1)*}\tilde{L}_{m+1,n}|t|^{m-1+2u-a}\\
&\le c(m)\tilde{L}_{m+1,n}\Big(|t|^{m+1-a}+|t|^{3m-3-a}\Big)
\end{split}
\end{equation}
for $a=0,\dots,m$.
Combining \labelcref{eq.R_n,eq.da-R_n^2-2,eq.da-R_n^1} gives
\begin{equation*}
\Big|\ddT{a} R_n(t,z)\Big|
\le c(m) \tilde{L}_{m+1,n} \Big(|t|^{m+1-a}+|t|^{3m-1-a}\Big)e^{t^2/3}
\end{equation*}
for $a=0,\dots,m$ and by the Leibniz rule
\begin{align*}
\lefteqn{\Big|\ddT{k} \Big(e^{-t^2/2}R_n(t,z)\Big)\Big|}\quad\\
&\le c(m)\sum_{l=0}^k (1+|t|^l)e^{-t^2/2}\tilde{L}_{m+1,n} \Big(|t|^{m+1-(k-l)}+|t|^{3m-1-(k-l)}\Big)e^{t^2/3}\\
&\le c(m) \tilde{L}_{m+1,n} e^{-t^2/6} \Big(|t|^{m+1-k}+|t|^{3m-1+k}\Big)
\end{align*}
for $k=0,\dots,m$. By \labelcref{eq.phi_n^z}, we get
\begin{equation*}
\tilde{\varphi}_{T_n}(tz)^{1/z^2}-e^{-t^2/2}\Big(1+\sum_{r=1}^{m-2}\tilde{U}_{r,n}(it)z^r\Big)
=e^{-t^2/2}R_n(t,z)
\end{equation*}
and by differentiating $k$ times and setting $z=1$, this implies
\begin{align*}
&\Big|\ddT{k} \Big(\tilde{\varphi}_{T_n}(t)-e^{-t^2/2}\Big(1+\sum_{r=1}^{m-2}\tilde{U}_{r,n}(it)\Big)\Big)\Big|
\le c(m) \tilde{L}_{m+1,n} e^{-t^2/6} \Big(|t|^{m+1-k}+|t|^{3m-1+k}\Big)
\end{align*}
for $k=0,\dots,m$. \end{proof}
In the next step, we prove the non-uniform bound for our Edgeworth expansion in the conditional setting. Recall the definition $\Phi^{\tilde{P}}_{m,n}(x)=\Phi(x)+\sum_{r=1}^{m-2}\tilde{P}_{r,n}(x)$, where \[ \tilde{P}_{r,n}(x)=-\phi(x)\sum_{*(k_\cdot,r,\cdot)} H_{r+2u(k_\cdot)-1}(x) \prod_{l=1}^{r} \frac{1}{k_l!} \Big(\frac{\tilde{\lambda}_{l+2,n}}{(l+2)!}\Big)^{k_l} \] from \labelcref{eq.tP,eq.Phi^tP}.
\begin{proposition}\label{p.cond}
Assume that $X_1$ is symmetric, $V_n^2>0$ and $m\ge2$ is an integer. Then
\begin{equation}\label{eq.pcond}
\begin{split}
&\sup_{x\in\mathbb{R}} \, (1+|x|)^{m}|\tilde{F}_n(x)-\Phi^{\tilde{P}}_{m,n}(x)|
\le c(m) \Big(\tilde{L}_{m+1,n}
+ I_2
+ \sum_{k=0}^{m} I_{k,2}
+ e^{-B_n^2/4}\Big),
\end{split}
\end{equation}
where $I_2=\int_{B_n\le|t|<\tilde{L}_{m+1,n}^{-1}}|t|^{-1}\big|\tilde{\varphi}_{T_n}(t)\big|\,\mathrm{d}t$ and for $k=0,\dots,m$,
\[I_{k,2}=\int_{B_n\le|t|<\tilde{L}_{m+1,n}^{-1}}|t|^{k-m-1}\Big|\ddT{k} \tilde{\varphi}_{T_n}(t)\Big|\,\mathrm{d}t\].
\end{proposition}
\begin{proof}
First, we use Lemma 8 with $F=\tilde{F}_n$ and $G=\Phi^{\tilde{P}}_{m,n}$, $T=\tilde{L}_{m+1,n}^{-1}$ and Lemma 7 with $G=\tilde{F}_n-\Phi^{\tilde{P}}_{m,n}$ from \cite[Chapter VI]{Pet75sums}, each applied with conditional Fourier transforms and finite $m$-th conditional moment. Next, $\ddX{} \Phi^{\tilde{P}}_{m,n}=\phi^{\tilde{p}}_{m,n}$, thus by \labelcref{eq.phi^tp,eq.tp-bound},
\[
\big(1+|x|\big)^m \Big| \ddX{} \Phi^{\tilde{P}}_{m,n}(x)\Big|
= \big(1+|x|\big)^m \big| \phi^{\tilde{P}}_{m,n}(x)\big|
= \big(1+|x|\big)^m \Big| \phi(x)+\sum_{r=1}^{m-2}\tilde{p}_{r,n}(x) \Big|
\le c(m)
\]
uniformly for all $x$ and $n$ and all conditions of the lemmas are satisfied.
Let
\[\delta_m(t)=\int_{-\infty}^{\infty}e^{itx}\,\mathrm{d}\big(x^m(\tilde{F}_n(x)-\Phi^{\tilde{P}}_{m,n}(x))\big)\,.\]
Now the lemmas yield
\begin{align}\label{eq.pcond-1}
&\big|\tilde{F}_n(x)-\Phi^{\tilde{P}}_{m,n}(x)\big|\nonumber\\*
&\le c(m)(1+|x|)^{-m}\Big(\int_{-\tilde{L}_{m+1,n}^{-1}}^{\tilde{L}_{m+1,n}^{-1}}\Big|\frac{\tilde{\varphi}_{T_n}(t)-\tilde{\varphi}_{\Phi^{\tilde{P}}_{m,n}}(t)}{t}\Big|\,\mathrm{d}t + \int_{-\tilde{L}_{m+1,n}^{-1}}^{\tilde{L}_{m+1,n}^{-1}}\Big|\frac{\delta_m(t)}{t}\Big|\,\mathrm{d}t + \frac{c(m)}{\tilde{L}_{m+1,n}^{-1}}\Big)\nonumber\\
&\le c(m)(1+|x|)^{-m}
\Big(\int_{-\tilde{L}_{m+1,n}^{-1}}^{\tilde{L}_{m+1,n}^{-1}}|t|^{-1}\big|\tilde{\varphi}_{T_n}(t)-\tilde{\varphi}_{\Phi^{\tilde{P}}_{m,n}}(t)\big|\,\mathrm{d}t\nonumber\\*
&\qquad\qquad\qquad\qquad+ \sum_{k=0}^{m}\int_{-\tilde{L}_{m+1,n}^{-1}}^{\tilde{L}_{m+1,n}^{-1}}|t|^{k-m-1}\Big|\ddT{k} \tilde{\varphi}_{T_n}(t)-\tilde{\varphi}_{\Phi^{\tilde{P}}_{m,n}}(t)\Big|\,\mathrm{d}t + c(m) \, \tilde{L}_{m+1,n} \Big)\nonumber\\
&= c(m)(1+|x|)^{-m}
\Big(\underbrace{\int_{-\tilde{L}_{m+1,n}^{-1}}^{\tilde{L}_{m+1,n}^{-1}}|t|^{-1}\big|\tilde{\varphi}_{T_n}(t)-\tilde{\varphi}_{\Phi^{\tilde{P}}_{m,n}}(t)\big|\,\mathrm{d}t}_{=I}\nonumber\\
&\qquad\qquad\qquad\qquad + \sum_{k=0}^{m}\underbrace{\int_{|t|<B_n}|t|^{k-m-1}\Big|\ddT{k} \tilde{\varphi}_{T_n}(t)-\tilde{\varphi}_{\Phi^{\tilde{P}}_{m,n}}(t)\Big|\,\mathrm{d}t}_{=I_{k,1}}\nonumber\\
&\qquad\qquad\qquad\qquad + \sum_{k=0}^{m}\underbrace{\int_{B_n\le|t|<\tilde{L}_{m+1,n}^{-1}}|t|^{k-m-1}\Big|\ddT{k} \tilde{\varphi}_{T_n}(t)\Big|\,\mathrm{d}t}_{=I_{k,2}}\\
&\qquad\qquad\qquad\qquad + \sum_{k=0}^{m}\underbrace{\int_{B_n\le|t|<\tilde{L}_{m+1,n}^{-1}}|t|^{k-m-1}\Big|\ddT{k} \tilde{\varphi}_{\Phi^{\tilde{P}}_{m,n}}(t)\Big|\,\mathrm{d}t}_{=I_{k,3}}\nonumber\\*
&\qquad\qquad\qquad\qquad + c(m) \, \tilde{L}_{m+1,n}\Big)\nonumber
\end{align}
for $x\in\mathbb{R}$ with $\tilde{\varphi}_{\Phi^{\tilde{P}}_{m,n}}$ from \labelcref{eq.tphi}.
Since in our conditional setting all moments exist, we can deduce from \cref{p.lemma4}
\begin{align*}
I_{k,1}
&\le\int_{|t|<B_n}|t|^{k-m-1}c(m)\tilde{L}_{m+1,n}e^{-t^2/6}\big(|t|^{m+1-k}+|t|^{3m-1+k}\big)\,\mathrm{d}t\\
&\le c(m)\tilde{L}_{m+1,n}\int_{|t|<B_n}e^{-t^2/6}\big(1+|t|^{2m-2+2k}\big)\,\mathrm{d}t\le c(m)\tilde{L}_{m+1,n}
\end{align*}
for $k=0,\dots,m$.
Now we examine $I_{k,3}$.
By \labelcref{eq.ddtl-tphi} and since $|t|\ge B_n\ge 1$,
\begin{equation}\label{eq.Ik3}
\begin{split}
I_{k,3}
&=\int_{B_n\le|t|<\tilde{L}_{m+1,n}^{-1}}|t|^{k-m-1}\Big|\ddT{k} \tilde{\varphi}_{\Phi^{\tilde{P}}_{m,n}}(t)\Big|\,\mathrm{d}t\\
&\le c(m) \int_{|t|\ge B_n} |t|^{m-5+2k} e^{-t^2/2} \,\mathrm{d}t
\le c(m) e^{-B_n^2/4}
\end{split}
\end{equation}
for $k=0,\dots,m$, where we use \cref{l.int-t-exp} (with $\beta=1$, $\nu_n=B_n$) in the last step.
In the same manner we proceed for $I$ and get
\begin{align*}
I
&=\int_{-\tilde{L}_{m+1,n}^{-1}}^{\tilde{L}_{m+1,n}^{-1}}|t|^{-1}\big|\tilde{\varphi}_{T_n}(t)-\tilde{\varphi}_{\Phi^{\tilde{P}}_{m,n}}(t)\big|\,\mathrm{d}t\\
&\le\int_{|t|<B_n}c(m)\tilde{L}_{m+1,n}e^{-t^2/6}\big(|t|^{m}+|t|^{3m-2}\big)\,\mathrm{d}t\\
&\quad + \underbrace{\int_{B_n\le|t|<\tilde{L}_{m+1,n}^{-1}}|t|^{-1}\Big|\tilde{\varphi}_{T_n}(t)\Big|\,\mathrm{d}t}_{=I_2} \ +\ \int_{|t|\ge B_n}\Big| e^{-t^2/2}\Big(1+\sum_{r=1}^{m-2} \tilde{U}_{r,n}(it)\Big)\Big|\,\mathrm{d}t\\
&\le c(m)\tilde{L}_{m+1,n} + I_2 + c(m) e^{-B_n^2/4}.
\end{align*}
Inserting everything in \labelcref{eq.pcond-1}, we arrive at
\begin{equation*}
\big|\tilde{F}_n(x)-\Phi^{\tilde{P}}_{m,n}(x)\big|
\le c(m)(1+|x|)^{-m}
\Big(\tilde{L}_{m+1,n}
+ I_2
+ \sum_{k=0}^{m} I_{k,2}
+ e^{-B_n^2/4}\Big)
\end{equation*}
for $x\in\mathbb{R}$, which can be rewritten as \labelcref{eq.pcond}. \end{proof}
\subsection{Proof of Theorem \ref{t.clt}}\label{s.distr.main}
Before proving the main theorem, we formulate some auxiliary propositions in which we bound expectation values of summands from \labelcref{eq.pcond} separately.
The proof of the following proposition involves thorough investigation and precise bounding of integral terms. The crucial point here is that $B_n$ is random and its expectation is smaller in order than $n^{-1/2}$ for our self-normalized sum $T_n$. In contrast, for the classical CLT-statistic $Z_n$, the integration interval starts at a deterministic point of order $n^{-1/2}$. Our larger integration interval complicates the bounding of the integral enormously and finally leads to the logarithmic factor in the remainder of \cref{t.clt}. The randomness in our integral, combined with the expectation results in a proof that differs fundamentally from usual proofs for Edgeworth expansions.
\begin{proposition}\label{p.E.I}
Assume that the distribution of $X_1$ is non-singular and ${\mathbb{E}|X_1|^{s}<\infty}$ for some ${s}\ge2$. Let $\nu_n,\tau_n\to \infty$ in polynomial order, bounded by $\nu_n,\tau_n\le n^{c}$ for some ${c= c(s)}$. Then for $m=\lfloor {s} \rfloor$ and $\zeta \in \mathbb{R}$,
\begin{align*}
&\mathbb{E}\bigg[\mathbbm{1}_{\{\zeta < M_n< \tau_n\}} \int_{B_n\le|u|< \nu_n} \Big|\ddU{r} \prod \cos\big(uV_n^{-1}|X_j|\big)\Big|\,\mathrm{d}u \bigg]\\
&\quad= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-({s}-2)/2} \, (\log n)^{({s}+\max\{2r,1\})/2} \big)
\end{align*}
holds for all $r=0,\dots,m$. \end{proposition}
The following lemma treats the expectation of mixed moments (with an additional term) and is required in the proof of \cref{p.E.I}. Its proof can be found in \cref{app.proofs.distr}.
\begin{lemma}\label{l.E.mom}
Assume that $\mathbb{E} |X_1|^{{s}}<\infty$ for some ${s}\ge2$. Then for all $l=1,\dots, \lfloor s \rfloor$, ${1\le a_i \le {s}}$ for $i=1,\dots,l$, and $1\le \eta \le c(s)$, \begin{align*}
\mathbb{E}\Big[\mathbbm{1}_{\big\{M_n\ge \sqrt{\frac{n}{\log(n)\, \eta}}\big\}} |X_1|^{a_1} \cdots |X_l|^{a_l}\Big] = \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\Big(n^{-({s}-\max\{2,a_1,\dots,a_l\})/2} \, (\log n)^{{s}/2} \, \eta^{{s}/2} \Big). \end{align*} \end{lemma}
\begin{remark}[Moment assumptions]\label{r.moment}
Assuming finite moments ($\mathbb{E} |X_1|^{{s}}<\infty$ for some ${s}\ge2$) provides useful truncation inequalities. We truncate the maximum $M_n$ at $a_n\to\infty$ and get by Markov's inequality
\begin{equation}\label{eq.Mn-an}
\mathbb{P}\big(M_n\ge a_n\big)
\le n \, \mathbb{P}\big(|X_1|\ge a_n\big)
\le n \, a_n^{-{s}} \int_{|x|\ge a_n} |x|^{{s}} \,\mathrm{d}\mathbb{P}_{X_1}(x)
= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix} \big( n \, a_n^{-{s}} \big).
\end{equation}
Note
that
$\mathbb{E}\big[ \mathbbm{1}_{\{|X_1|\ge n^{1/4}\}} |X_1|^s \big] \longrightarrow 0
$
for $n\to\infty$. Let $(\delta_n)$ be a sequence that satisfies $n^{-1/4} \le \delta_n <1$ and $\delta_n\to0$ sufficiently slowly that
\begin{align*}
\delta_n^{-s} \mathbb{E}\Big[ \mathbbm{1}_{\{|X_1|\ge n^{1/4}\}} |X_1|^s \Big] \longrightarrow 0.
\end{align*}
Then,
\begin{align}\label{eq.delta_n-def}
\mathbb{E}\Big[ \mathbbm{1}_{\{|X_1|\ge \delta_n \sqrt{n} \, \}} \delta_n^{-s} |X_1|^s \Big]
\le \delta_n^{-s} \mathbb{E}\Big[ \mathbbm{1}_{\{|X_1|\ge n^{-1/4} \sqrt{n} \, \}} |X_1|^s \Big]
\longrightarrow 0
\end{align}
and thus
\begin{equation}\label{eq.Mn-delta}
\begin{split}
\mathbb{P}\big(M_n\ge \delta_n \sqrt{n} \, \big)
&\le n \, (\delta_n \sqrt{n} \, )^{-{s}} \int_{|x|\ge \delta_n \sqrt{n}} |x|^{{s}} \,\mathrm{d}\mathbb{P}_{X_1}(x)\\
&= n^{-(s-2)/2} \mathbb{E}\Big[ \mathbbm{1}_{\{|X_1|\ge \delta_n \sqrt{n} \, \}} \delta_n^{-s} |X_1|^s \Big]
= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix} \big( n^{-(s-2)/2} \big).
\end{split}
\end{equation}
Additionally by \cite[Theorem 28, Chapter IX]{Pet75sums},
\begin{equation*}
\mathbb{P}\big(|V_n^2-n\mu_2|\ge \varepsilon n\big)
=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big)
\end{equation*}
for all $\varepsilon>0$ and particularly (by choosing $\varepsilon=\mu_2/2=1/2$)
\begin{align}\label{eq.Vn-bound-1/2}
\mathbb{P}\big(V_n^2 \le n/2\big)
&\le\mathbb{P}\big(|V_n^2-n|\ge n/2\big)
=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big).
\end{align} \end{remark}
\begin{proof}[Proof of \cref{p.E.I}]
For the sake of legibility, we suppress the notation of the indicator $\mathbbm{1}_{\{\zeta < M_n< \tau_n\}}$ which is denoted by $\mathbb{E}_*[\,\cdot\,]:=\mathbb{E}[\mathbbm{1}_{\{\zeta < M_n< \tau_n\}}\,\cdot\,]$ throughout this proof. Every time these bounds are used, we will mention it explicitly. In the following set of inequalities, we use the indicator. By $V_n\ge M_n >\zeta$ and Fubini's theorem, we get
\begin{equation}\label{eq.fubini}
\begin{split}
\lefteqn{\mathbb{E}_*\bigg[\int_{B_n\le|u|< \nu_n} \Big|\ddU{r} \prod \cos\big(uV_n^{-1}|X_j|\big)\Big|\,\mathrm{d}u\bigg]}\quad\\
&= \mathbb{E}_*\bigg[\int_{M_n^{-1}\le|u|< \nu_nV_n^{-1}} V_n^{-(r-1)}\Big|\ddU{r} \prod \cos(u|X_j|)\Big|\,\mathrm{d}u\bigg]\\
&\le \mathbb{E}_*\bigg[\int_{0\le|u|< \nu_n\zeta^{-1}} \mathbbm{1}_{\{M_n^{-1}\le|u|\}} V_n^{-(r-1)}\Big|\ddU{r} \prod \cos(u|X_j|)\Big|\,\mathrm{d}u\bigg]\\
&= \int_{0\le|u|< \nu_n\zeta^{-1}}\mathbb{E}_*\Big[\mathbbm{1}_{\{M_n^{-1}\le|u|\}} V_n^{-(r-1)}\Big|\ddU{r} \prod \cos(u|X_j|)\Big|\Big]\,\mathrm{d}u.
\end{split}
\end{equation}
Subsequently, we deduce rather tight $u$-dependent bounds on the $\mathbb{E}_*$ expression within the integral. To this end, we derive precise bounds on the derivatives of the product inside.
The higher derivative of a product can be evaluated by the generalized Leibniz formula which states in our case
\[
\ddU{r} \prod \cos(u|X_j|)
=\sum _{i_{1}+i_{2}+\cdots +i_{n}=r} {r \choose i_{1},\ldots ,i_{n}}\prod\Big(\ddU{i_j}\cos(u|X_j|)\Big)\,.
\]
Here the sum extends over all $n$-tuples $(i_1,...,i_n)$ of non-negative integers with $\sum_{j=1}^ni_j=r$ and
\[
{r \choose i_{1},\ldots ,i_{n}}={\frac {r!}{i_{1}!\,i_{2}!\cdots i_{n}!}}.
\]
Since $X_1,\dots,X_n$ are i.i.d.,
\begin{align*}
\mathbb{E}\Big[ {r \choose i_{\pi(1)},\ldots ,i_{\pi(n)}}\prod\Big(\ddU{i_{\pi(j)}}\cos(u|X_{j}|)\Big) \Big]
= \mathbb{E}\Big[ {r \choose i_{1},\ldots ,i_{n}}\prod\Big(\ddU{i_j}\cos(u|X_j|)\Big) \Big]
\end{align*}
for all (non-random) permutations $\pi$ from $\{1,\dots,n\}$ to itself.
For $l=1,\dots,r$, let $k_l$ be the number of appearances of $l$ in the tuple $(i_1,...,i_n)$ (so $\sum_{l=1}^rlk_l=\sum_{j=1}^ni_j=r$). Then for each tuple $(k_1,\dots,k_r)$, there exist ${n \choose k_{1},\ldots ,k_{r},(n-\sum_{l=1}^rk_l)}$ tuples $(i_1,...,i_n)$ such that $k_l$ is the number of appearances of $l$ in the tuple $(i_1,...,i_n)$. So instead of summing over all $n$-tuples $(i_1,...,i_n)$ with $\sum_{j=1}^ni_j=r$, we can sum over all $r$-tuples $(k_1,...,k_r)$ of non-negative integers with $\sum_{l=1}^rlk_l=r$ and multiply each summand by ${n \choose k_{1},\ldots ,k_{r},(n-\sum_{l=1}^rk_l)}$. We perform the just described summation by using the sum $\sum_{*(k_\cdot,r,\cdot)}$ and $u(k_\cdot)$ which are explained in \cref{ch.pre}. Additionally, we convert
\[
{r \choose i_{1},\ldots ,i_{n}}={\frac {r!}{i_{1}!\,i_{2}!\cdots i_{n}!}}=\frac{r!}{1!^{k_1}\cdots r!^{k_r}}.
\]
Now for any $(k_1,\dots,k_r)$, we identify that element of the corresponding set
\[
\{(i_{\pi(1)},...,i_{\pi(n)}): \pi \text{ permutations from }\{1,\dots,n\}\text{ to itself} \}
\]
for which
\begin{align*}
i_1&=\dots=i_{k_1}=1,\\
i_{k_1+1}&=\dots=i_{k_1+k_2}=2,\\
&\vdots\\
i_{\sum_{l=1}^{r-1}k_l+1}&=\dots=i_{\sum_{l=1}^rk_l}=r,\\
i_{\sum_{l=1}^rk_l+1}&=\dots=i_n=0.
\end{align*}
When summing over $(k_1,\dots,k_r)$, this will be the representative, evaluated within the expectation.
As $V_n^2$ and $M_n$ are invariant under permutation, we convert the above expression \labelcref{eq.fubini} with $A_n:=\mathbbm{1}_{\{M_n^{-1}\le|u|\}} V_n^{-(r-1)}$ into
\begin{align}\label{eq.reordering}
\lefteqn{\mathbb{E}_*\bigg[\mathbbm{1}_{\{M_n^{-1}\le|u|\}} V_n^{-(r-1)}\Big|\ddU{r} \prod \cos(u|X_j|)\Big|\bigg]}\quad\nonumber\\
&=\mathbb{E}_*\bigg[A_n\cdot\Big|\sum_{i_{1}+i_{2}+\cdots +i_{n}=r}{r \choose i_{1},\ldots ,i_{n}}\prod\Big(\ddU{i_j}\cos(u|X_n|)\Big)\Big|\bigg]\\
&=\mathbb{E}_*\bigg[A_n\cdot\Big|\sum_{*(k_\cdot,r,\cdot)} {n \choose k_{1},\ldots ,k_{r},(n-u(k_\cdot))} \frac{r!}{1!^{k_1}\cdots r!^{k_r}}\nonumber\\*
&\qquad\qquad\quad\cdot\prod_{l=1}^r\Big(\prod_{a=1}^{k_l}\Big(\ddU{l}\cos\big(u|X_{\sum_{b=1}^{l-1}k_b+a}|\big)\Big)\Big)\prod_{l=u(k_\cdot)+1}^n\cos(u|X_l|) \Big|\bigg].\nonumber
\end{align}
In the next step we apply the triangle inequality and evaluate the derivates of these cosine-structures which result in multiplying by the respective $|X_j|$ and switching to the sine for uneven derivatives. Overall we get
\begin{equation}\label{eq.triangle}
\begin{split}
&\mathbb{E}_*\bigg[A_n\cdot\Big|\sum_{*(k_\cdot,r,\cdot)} {n \choose k_{1},\ldots ,k_{r},(n-u(k_\cdot))} \frac{r!}{1!^{k_1}\cdots r!^{k_r}}\\
&\qquad\qquad\cdot\prod_{l=1}^r\Big(\prod_{a=1}^{k_l}\ddU{l}\cos\big(u|X_{\sum_{b=1}^{l-1}k_b+a}|\big)\Big)\prod_{l=u(k_\cdot)+1}^n\cos(u|X_l|) \Big|\bigg]\\
&\quad\le \mathbb{E}_*\bigg[A_n\,c(r)\sum_{*(k_\cdot,r,\cdot)} n^{u(k_\cdot)} \prod_{l=1}^r\Big(\prod_{a=1}^{k_l}\Big|\ddU{l}\cos\big(u|X_{\sum_{b=1}^{l-1}k_b+a}|\big)\Big|\Big)\prod_{l=u(k_\cdot)+1}^n\big|\cos(u|X_l|)\big|\bigg]
\end{split}
\end{equation}
\begin{equation*}
\begin{split}
&\quad= c(r)\sum_{*(k_\cdot,r,\cdot)} n^{u(k_\cdot)} \,
\mathbb{E}_*\bigg[A_n
\prod_{l=1}^{\lceil r/2\rceil} \Big(\prod_{a=1}^{k_{2l-1}}\big|X_{\sum_{b=1}^{(2l-1)-1}k_b+a}\big|^{2l-1}\Big|\sin\big(u\big|X_{\sum_{b=1}^{(2l-1)-1}k_b+a}\big|\big)\Big|\Big)\\
&\quad\qquad\cdot\prod_{l=1}^{\lfloor r/2\rfloor} \Big(\prod_{a=1}^{k_{2l}}\big|X_{\sum_{b=1}^{2l-1}k_b+a}\big|^{2l}\Big|\cos\big(u\big|X_{\sum_{b=1}^{2l-1}k_b+a}\big|\big)\Big|\Big)
\prod_{l=u(k_\cdot)+1}^n\big|\cos(u|X_l|)\big|\bigg].
\end{split}
\end{equation*}
Since $|\sin|$ and $|\cos|$ are symmetric around 0, we can only work with positive $u$ and multiply by 2 (which is included in $c(r)$). Thus in \labelcref{eq.fubini,eq.reordering,eq.triangle}, we showed
\begin{equation}\label{eq.E.expanded}
\begin{split}
&\mathbb{E}_*\bigg[\int_{B_n\le|u|< \nu_n} \Big|\ddU{r} \prod \cos\big(uV_n^{-1}|X_j|\big)\Big|\,\mathrm{d}u\bigg]
\le c(r) \sum_{*(k_\cdot,r,\cdot)} n^{u(k_\cdot)} \int_{0\le u< \nu_n\zeta^{-1}}\\
&\qquad \mathbb{E}_*\bigg[\mathbbm{1}_{\{M_n^{-1}\le u\}} V_n^{-(r-1)}\prod_{l=1}^{\lceil r/2\rceil} \Big(\prod_{a=1}^{k_{2l-1}}\big|X_{\sum_{b=1}^{(2l-1)-1}k_b+a}\big|^{2l-1}\Big|\sin\big(u\big|X_{\sum_{b=1}^{(2l-1)-1}k_b+a}\big|\big)\Big|\Big)\\
&\qquad\quad \cdot\prod_{l=1}^{\lfloor r/2\rfloor} \Big(\prod_{a=1}^{k_{2l}}\big|X_{\sum_{b=1}^{2l-1}k_b+a}\big|^{2l}\Big|\cos\big(u\big|X_{\sum_{b=1}^{2l-1}k_b+a}\big|\big)\Big|\Big)
\prod_{l=u(k_\cdot)+1}^n\big|\cos(u|X_l|)\big| \bigg]\,\mathrm{d}u\\
&=:W_1+W_2,
\end{split}
\end{equation}
where $W_1$ represents the integral over $u\in \big[0,\sqrt{n^{-1}\log(n)\, \eta}\big)$ and $W_2$ represents the integral over $u\in \big[\sqrt{n^{-1}\log(n)\, \eta}, \nu_n\zeta^{-1}\big)$ with
\[
\eta := 6\big(2r+{s}+4c\big).
\]
Note $1\le \eta\le c(s)$. Our choice of the split point $\sqrt{n^{-1}\log(n)\, \eta}$ is motivated by the bounds in \labelcref{eq.n^log(n),eq.W_2-finish}.
First, we focus on a bound on $W_1$. Here, our basic idea is to simplify the complex expectation in $W_1$ to allow using \cref{l.E.mom}. The bounds we use have to be particularly tight for summands where $k_1$ and $k_2$ are large (i.e. $k_3=\dots=k_r=0$) as these summands result in the largest powers of $n$. We bound $|\cos(x)|\le 1$ and $\sin(x)\le x$ for $x\ge 0$. Thus,
\begin{align*}
W_1
&\le c(r) \sum_{*(k_\cdot,r,\cdot)} n^{u(k_\cdot)} \int_{0\le u< \sqrt{\frac{\log(n)\, \eta}{n}}} \\
&\qquad\mathbb{E}_*\bigg[\mathbbm{1}_{\{M_n^{-1}\le u\}} V_n^{-(r-1)}\prod_{l=1}^{\lceil r/2\rceil} \Big(\prod_{a=1}^{k_{2l-1}}\big|X_{\sum_{b=1}^{(2l-1)-1}k_b+a}\big|^{2l-1}\Big|\big(u\big|X_{\sum_{b=1}^{(2l-1)-1}k_b+a}\big|\big)\Big|\Big)\\
&\qquad\quad\cdot\prod_{l=1}^{\lfloor r/2\rfloor} \Big(\prod_{a=1}^{k_{2l}}\big|X_{\sum_{b=1}^{2l-1}k_b+a}\big|^{2l} \Big)
\bigg]\,\mathrm{d}u.
\end{align*}
We replace the integral by the supremum of the integrand multiplied with the interval length, which bounds the expression above by
\begin{align*}
&c(r) \sum_{*(k_\cdot,r,\cdot)} n^{u(k_\cdot)} n^{-\frac12 \sum_{l=1}^{\lceil r/2\rceil}k_{2l-1}} \big(\log(n)\, \eta\big)^{\frac12 \sum_{l=1}^{\lceil r/2\rceil} k_{2l-1}} \sqrt{\tfrac{\log(n)\, \eta}{n}} \, \mathbb{E}_*\bigg[\mathbbm{1}_{\big\{M_n^{-1}\le \sqrt{\frac{\log(n)\, \eta}{n}}\big\}} \\*
&\qquad \cdot V_n^{-(r-1)}\prod_{l=1}^{\lceil r/2\rceil} \Big(\prod_{a=1}^{k_{2l-1}}\big|X_{\sum_{b=1}^{(2l-1)-1}k_b+a}\big|^{2l}\Big)
\prod_{l=1}^{\lfloor r/2\rfloor} \Big(\prod_{a=1}^{k_{2l}}\big|X_{\sum_{b=1}^{2l-1}k_b+a}\big|^{2l} \Big)
\bigg]\\
&= c(r) \sum_{*(k_\cdot,r,\cdot)} \hspace{-0.8ex} n^{\frac12 (-1+\sum_{l=1}^{\lceil r/2\rceil}k_{2l-1}+2\sum_{l=1}^{\lfloor r/2\rfloor}k_{2l})} \big(\log(n)\, \eta\big)^{\frac12 (1+\sum_{l=1}^{\lceil r/2\rceil} k_{2l-1})} \hspace{0.2ex} \mathbb{E}_*\bigg[\mathbbm{1}_{\big\{M_n\ge \sqrt{\frac{n}{\log(n)\, \eta}}\big\}}\\*
&\qquad \cdot V_n^{-(r-1)}\prod_{l=1}^{\lceil r/2\rceil} \Big(\prod_{a=1}^{k_{2l-1}}\big|X_{\sum_{b=1}^{(2l-1)-1}k_b+a}\big|^{2l}\Big)
\prod_{l=1}^{\lfloor r/2\rfloor} \Big(\prod_{a=1}^{k_{2l}}\big|X_{\sum_{b=1}^{2l-1}k_b+a}\big|^{2l} \Big)
\bigg]\\*
&=:W_3.
\end{align*}
Hence, $W_1\le W_3$. For $r=0$, $W_3$ reduces significantly. Using \labelcref{eq.Mn-an} and \cref{l.E.mom} ($l=1$, $a_1=2$) yields
\begin{align}\label{eq.W_3-0}
W_3
&= c(0) \sqrt{\tfrac{\log(n)\, \eta}{n}} \; \mathbb{E}_*\Big[\mathbbm{1}_{\big\{M_n\ge \sqrt{\frac{n}{\log(n)\, \eta}}\big\}} V_n \Big]\\
& \le c(0) \sqrt{\tfrac{\log(n)\, \eta}{n}} \sqrt{\tfrac n2} \; \mathbb{E}_*\Big[\mathbbm{1}_{\big\{M_n\ge \sqrt{\frac{n}{\log(n)\, \eta}}\big\}}\mathbbm{1}_{\{V_n^2 \le \frac n2\}} \Big]\nonumber\\*
&\quad+c(0) \sqrt{\tfrac{\log(n)\, \eta}{n}} \sqrt{\tfrac 2n} \; \mathbb{E}_*\Big[\mathbbm{1}_{\big\{M_n\ge \sqrt{\frac{n}{\log(n)\, \eta}}\big\}} \mathbbm{1}_{\{V_n^2 > \frac n2\}} V_n^2\Big]\nonumber\\
& \le c(0) \sqrt{\log(n)\, \eta} \; \mathbb{E}_*\Big[\mathbbm{1}_{\big\{M_n\ge \sqrt{\frac{n}{\log(n)\, \eta}}\big\}} \Big]
+c(0) \sqrt{\log(n)\, \eta} \; \mathbb{E}_*\Big[\mathbbm{1}_{\big\{M_n\ge \sqrt{\frac{n}{\log(n)\, \eta}}\big\}} X_1^2\Big]\nonumber\\
& = \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix} \big( n^{-({s}-2)/2} \, \log(n)^{({s}+1)/2} \big).\nonumber
\end{align}
For $r\ge1$, we can simplify $W_3$ by using $|X_j| \le V_n$,
\begin{align*}
&c(r) \sum_{*(k_\cdot,r,\cdot)} \hspace{-0.9ex} n^{\frac12 (-1+\sum_{l=1}^{\lceil r/2\rceil}k_{2l-1}+2\sum_{l=1}^{\lfloor r/2\rfloor}k_{2l})} \big(\log(n)\, \eta\big)^{\frac12 (1+\sum_{l=1}^{\lceil r/2\rceil} k_{2l-1})} \, \mathbb{E}_*\bigg[\mathbbm{1}_{\big\{M_n\ge \sqrt{\frac{n}{\log(n)\, \eta}}\big\}} \\*
&\qquad \cdot V_n^{-(r-1)} \prod_{l=1}^{\lceil r/2\rceil} \Big(\prod_{a=1}^{k_{2l-1}}\big|X_{\sum_{b=1}^{(2l-1)-1}k_b+a}\big|^{2l}\Big)
\prod_{l=1}^{\lfloor r/2\rfloor} \Big(\prod_{a=1}^{k_{2l}}\big|X_{\sum_{b=1}^{2l-1}k_b+a}\big|^{2l} \Big)
\bigg]\\
&\le c(r) \sum_{*(k_\cdot,r,\cdot)} \hspace{-0.9ex} n^{\frac12 (-1+\sum_{l=1}^{\lceil r/2\rceil}k_{2l-1}+2\sum_{l=1}^{\lfloor r/2\rfloor}k_{2l})} \big(\log(n)\, \eta\big)^{\frac12 (1+\sum_{l=1}^{\lceil r/2\rceil} k_{2l-1})} \, \mathbb{E}_*\bigg[\mathbbm{1}_{\big\{M_n\ge \sqrt{\frac{n}{\log(n)\, \eta}}\big\}} \\*
&\qquad \cdot V_n^{-(r-1)} \prod_{l=1}^{\lceil r/2\rceil} \Big(\prod_{a=1}^{k_{2l-1}}\big|X_{\sum_{b=1}^{(2l-1)-1}k_b+a}\big|^{2} V_n^{2l-2}\Big)
\prod_{l=1}^{\lfloor r/2\rfloor} \Big(\prod_{a=1}^{k_{2l}}\big|X_{\sum_{b=1}^{2l-1}k_b+a}\big|^{2} V_n^{2l-2}\Big)
\bigg].
\end{align*}
The power of $V_n$ in the respective summands is
\begin{align*}
-(r-1)+\sum_{l=1}^{\lceil r/2\rceil} k_{2l-1} (2l-2)+\sum_{l=1}^{\lfloor r/2\rfloor} k_{2l} (2l-2)
&={1-\sum_{l=1}^{\lceil r/2\rceil}k_{2l-1}-2\sum_{l=1}^{\lfloor r/2\rfloor}k_{2l}}
\end{align*}
such that the above expression equals
\begin{align*}
& c(r) \sum_{*(k_\cdot,r,\cdot)} n^{\frac12 (-1+\sum_{l=1}^{\lceil r/2\rceil}k_{2l-1}+2\sum_{l=1}^{\lfloor r/2\rfloor}k_{2l})} \big(\log(n)\, \eta\big)^{\frac12 (1+\sum_{l=1}^{\lceil r/2\rceil} k_{2l-1})} \, \mathbb{E}_*\bigg[\mathbbm{1}_{\big\{M_n\ge \sqrt{\frac{n}{\log(n)\, \eta}}\big\}} \\
&\quad \cdot V_n^{1-\sum_{l=1}^{\lceil r/2\rceil}k_{2l-1}-2\sum_{l=1}^{\lfloor r/2\rfloor}k_{2l}}
\prod_{l=1}^{\lceil r/2\rceil} \Big(\prod_{a=1}^{k_{2l-1}}\big|X_{\sum_{b=1}^{(2l-1)-1}k_b+a}\big|^{2} \Big)
\prod_{l=1}^{\lfloor r/2\rfloor} \Big(\prod_{a=1}^{k_{2l}}\big|X_{\sum_{b=1}^{2l-1}k_b+a}\big|^{2} \Big)
\bigg].
\end{align*}
As $r\ge1$, we know that $k_l>0$ for some $l=1,\dots,r$ such that $V_n$ has a non-positive power. Thus, using $V_n\ge M_n$, we can bound the above expression by
\begin{align*}
&c(r) \sum_{*(k_\cdot,r,\cdot)} n^{\frac12 (-1+\sum_{l=1}^{\lceil r/2\rceil}k_{2l-1}+2\sum_{l=1}^{\lfloor r/2\rfloor}k_{2l})} \big(\log(n)\, \eta\big)^{\frac12 (1+\sum_{l=1}^{\lceil r/2\rceil} k_{2l-1})} \\*
&\qquad \cdot \Big(\sqrt{\tfrac{n}{\log(n)\, \eta}}\,\Big)^{1-\sum_{l=1}^{\lceil r/2\rceil}k_{2l-1}-2\sum_{l=1}^{\lfloor r/2\rfloor}k_{2l}} \, \mathbb{E}_*\bigg[\mathbbm{1}_{\big\{M_n\ge \sqrt{\frac{n}{\log(n)\, \eta}}\big\}}\\*
&\qquad \cdot \prod_{l=1}^{\lceil r/2\rceil} \Big(\prod_{a=1}^{k_{2l-1}}\big|X_{\sum_{b=1}^{(2l-1)-1}k_b+a}\big|^{2} \Big)
\prod_{l=1}^{\lfloor r/2\rfloor} \Big(\prod_{a=1}^{k_{2l}}\big|X_{\sum_{b=1}^{2l-1}k_b+a}\big|^{2} \Big)
\bigg]\\
&= c(r) \sum_{*(k_\cdot,r,\cdot)} \big(\log(n) \, \eta\big)^{u(k_\cdot)} \\*
&\qquad \cdot \mathbb{E}_*\bigg[\mathbbm{1}_{\big\{M_n\ge \sqrt{\frac{n}{\log(n)\, \eta}}\big\}}
\prod_{l=1}^{\lceil r/2\rceil} \Big(\prod_{a=1}^{k_{2l-1}}\big|X_{\sum_{b=1}^{(2l-1)-1}k_b+a}\big|^{2} \Big)
\prod_{l=1}^{\lfloor r/2\rfloor} \Big(\prod_{a=1}^{k_{2l}}\big|X_{\sum_{b=1}^{2l-1}k_b+a}\big|^{2} \Big)
\bigg]\\
&=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-({s}-2)/2} \, (\log n)^{({s}+2r)/2} \big)
\end{align*}
where we used \cref{l.E.mom} ($l=u(k_\cdot),a_1=\dots=a_{u(k_\cdot)}=2$) in the last step. Thus, for $r\ge1$,
\begin{equation}\label{eq.W_3}
W_1\le W_3=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-({s}-2)/2} \, (\log n)^{({s}+2r)/2} \big).
\end{equation}
After having dealt with $W_1$, we direct our attention to bounding the other part of \labelcref{eq.E.expanded}, $W_2$. Due to the area of integration in $W_2$, $u$ now is larger and thus, $\cos(u|X_\cdot|)$ is smaller than in the integral in $W_1$. Therefore, our basic idea is to show that the expectation of the cosines are sufficiently small and to use their $(n-u(k_\cdot))$ times product $\big|\cos(u|X_{u(k_\cdot)+1}|)\big|\cdots \big|\cos(u|X_n|)\big|$ to bound $W_2$. In the process, we use $\sin(x)\le1$ and $\cos(x)\le1$ for all $x\in\mathbb{R}$ and $|X_j|/V_n\le1$ for all $j=1,\dots,n$. This leads to
\begin{align}\label{eq.W_2}
W_2&\le c(r) \sum_{*(k_\cdot,r,\cdot)} n^{u(k_\cdot)} \int_{\sqrt{\frac{\log(n)\, \eta}{n}}\le u< \nu_n\zeta^{-1}}
\mathbb{E}_*\bigg[ V_n \prod_{l=u(k_\cdot)+1}^n\big|\cos(u|X_l|)\big| \bigg]\,\mathrm{d}u\\
&\le c(r) n^{r} \int_{\sqrt{\frac{\log(n)\, \eta}{n}}\le u< \nu_n\zeta^{-1}}
\mathbb{E}_*\bigg[ V_n \prod_{l=r+1}^n\big|\cos(u|X_l|)\big| \bigg]\,\mathrm{d}u.\nonumber
\end{align}
Using the indicator $\mathbbm{1}_{\{\zeta < M_n< \tau_n\}}$ implies $V_n^2\le n M_n^2\le n \tau_n^2$ such that we can eliminate the $V_n$ and since the $X_j$ are i.i.d., by defining $\mathcal{U}_n:=\big[\sqrt{n^{-1}\log(n)\, \eta}, \nu_n\zeta^{-1}\big]$, we get
\begin{align}\label{eq.W_2-2}
\lefteqn{c(r) n^r \int_{\sqrt{\frac{\log(n)\, \eta}{n}}\le u< \nu_n\zeta^{-1}} \mathbb{E}_*\bigg[ V_n \prod_{l=r+1}^n\big|\cos(u|X_l|)\big| \bigg]\,\mathrm{d}u}\quad\nonumber\\*
&\le c(r) n^{(2r+1)/2}\tau_n\,|\mathcal{U}_n|\,\sup_{u\in\mathcal{U}_n}\mathbb{E}_*\bigg[ \prod_{l=r+1}^n\big|\cos(u|X_l|)\big| \bigg]\\
&\le c(r) n^{(2r+1)/2}\tau_n\,\nu_n\zeta^{-1}\,\sup_{u\in\mathcal{U}_n}\prod_{l=r+1}^n\mathbb{E}\Big[\big|\cos(u|X_l|)\big|\Big]\nonumber\\
&\le c(r) n^{(2r+1)/2}\tau_n\,\nu_n\,\sup_{u\in\mathcal{U}_n}\mathbb{E}\Big[\big|\cos(u|X_1|)\big|\Big]^{n-r}.\nonumber
\end{align}
To achieve the aspired rate of convergence for $W_1$, we will show that
\begin{align}\label{eq.cases}
\sup_{u\in\mathcal{U}_n}\mathbb{E}\Big[\big|\cos(u|X_1|)\big|\Big]\le1-\frac{\log(n)\, \eta / 6}{n}
\end{align}
for all $n\in\mathbb{N}$ sufficiently large.
We conduct the proof by contradiction. Suppose \labelcref{eq.cases} does not hold, meaning for every $N\in\mathbb{N}$, there exists some $n\ge N$ for which there exists at least one $u_n\in\mathcal{U}_n$ such that
\begin{align}\label{eq.cases-supp}
\mathbb{E}\Big[\big|\cos\big(u_n|X_1|\big)\big|\Big]>1-\frac{\log(n)\, \eta / 6}{n}.
\end{align}
Let $(n_k)$ be a subsequence of these $n$ such that
\begin{align*}
\mathbb{E}\Big[\big|\cos\big(u_{n_k}|X_1|\big)\big|\Big]>1-\frac{\log(n_k)\, \eta / 6}{n_k}.
\end{align*}
Hence, this sequence $(u_{n_k})$ either has a convergent subsequence $(u_{n_{k_l}})$ with $u_{n_{k_l}}\to u^*$ for some $u^*\in[0,\infty)$ or $u_{n_{k_l}}\to \infty$. Note $u_{n_{k_l}}>0$. For the ease of notation, we still denote the subsubsequence by $(u_n)$. In consequence, we assume that for all members of this sequence, \labelcref{eq.cases-supp} holds. We subsequently distinguish the cases $u^*=0,u^*\in(0,\infty)$ and $u_n\to\infty$.\\
\underline{Case 1 ($u^*=0$):}
Due to $\cos(x)\le 1-x^2/3$ for $x\in(0,\tfrac{\pi}{2})$, we have
\begin{align*}
\mathbb{E}\Big[\big|\cos\big(u_n|X_1|\big)\big|\Big]
\le1-\mathbb{E}\bigg[\mathbbm{1}_{\{0<u_n|X_1|< \pi/2\}} \frac{u_n^2|X_1|^2}{3}\bigg]
=1-\frac{u_n^2\,\mu_{2,n}'}{3},
\end{align*}
where
\[
\mu_{2,n}'
:=\mathbb{E}\Big[\mathbbm{1}_{\{0<u_n|X_1|< \pi/2\}}|X_1|^2\Big]
= \mathbb{E}\Big[\mathbbm{1}_{\{0<|X_1|< \frac{\pi}{2u_n}\}}|X_1|^2\Big]
\longrightarrow 1
\]
by dominated convergence. Therefore, $\mu_{2,n}'\ge 1/2$ for $n$ sufficiently large. Since $u_n \in\mathcal{U}_n$, $u_n\ge \sqrt{n^{-1}\log(n)\, \eta}$ such that
\begin{equation*}
\mathbb{E}\Big[\cos\big(u_n|X_1|\big)\Big]
\le 1-\frac{u_n^2\,\mu_{2,n}'}{3}
\le 1-\frac{\log(n)\, \eta / 6}{n}
\end{equation*}
for $n$ sufficiently large.\\
\underline{Case 2 ($u^*\in(0,\infty)$\,):}
Due to non-singularity, the distribution of $|X_1|$ cannot have mass in $\{j\,\pi/u^*\mid j\in\mathbb{Z}\}$ only. By \cref{l.ana1}, there exists $\beta$ with $0<\beta<1$ such that
\begin{align*}
\mathbb{E}\Big[\big|\cos(u^*|X_1|)\big|\Big]
=\beta.
\end{align*}
So due to continuity of the cosine and dominated convergence, for all $\beta'$ with $\beta<\beta'<1$, there exists $N\in\mathbb{N}$ such that
\[
\mathbb{E}\Big[\big|\cos(u_n|X_1|)\big|\Big]
\le\beta'
\le 1-\frac{\log(n)\, \eta / 6}{n}
\]
holds for all $n\ge N$.\\
\underline{Case 3 ($u_n\to \infty$):}
By non-singularity, we know that $p_{ac}>0$ (see \cref{r.non-singularity}), so we first focus on the absolute continuous part $F_{ac}$ of the distribution of $X_1$.
By the Riemann--Lebesgue lemma,
\[
\Big|\int_\mathbb{R}\exp(iu_nx)\,\mathrm{d} F_{ac}(x)\Big|\too0
\]
for $u_n\to\infty$. Thus for the real part, we know
\[
\int_\mathbb{R}\cos(u_nx)\,\mathrm{d} F_{ac}(x)\too0
\]
for $u_n\to\infty$. Since the cosine is symmetric, this yields
\[
\tfrac12 \int_\mathbb{R}\big(\cos(2u_n|x|)+1 \big)\,\mathrm{d} F_{ac}(x)\longrightarrow \tfrac12
\]
for $u_n\to\infty$. By Jensen's inequality and the trigonometric equation $\cos(2\alpha)= 2 \cos(\alpha)^2-1$ \cite[6.5.10]{Zwillinger},
\[
\Big(\int_\mathbb{R}|\cos(u_n|x|)|\,\mathrm{d} F_{ac}(x)\Big)^2
\le \int_\mathbb{R}\cos(u_n|x|)^2\,\mathrm{d} F_{ac}(x)
=\tfrac12 \int_\mathbb{R}\cos(2u_n|x|)+1 \,\mathrm{d} F_{ac}(x).
\]
The right-hand side converges to $1/2$ for $u_n\to\infty$. Due to the Lebesgue decomposition of the distribution function \labelcref{eq.lebesgue-dec},
\begin{align*}
\lefteqn{\mathbb{E}\big[|\cos(u_n|X_{1}|)|\big]}\enspace\\
&=p_{ac}\int_\mathbb{R}|\cos(u_n|x|)|\,\mathrm{d} F_{ac}(x)+p_{d}\int_\mathbb{R}|\cos(u_n|x|)|\,\mathrm{d} F_{d}(x)+p_{sc}\int_\mathbb{R}|\cos(u_n|x|)|\,\mathrm{d} F_{sc}(x)\\
&\le p_{ac}\sqrt{\tfrac12 \int_\mathbb{R}\cos(2u_n|x|)+1 \,\mathrm{d} F_{ac}(x)}+p_{d}+p_{sc}\longrightarrow \frac{1}{\sqrt{2}}p_{ac}+p_{d}+p_{sc}<1
\end{align*}
for $u_n\to\infty$. Hence, for any $\beta'$ with $p_{ac}/\sqrt{2}+p_{d}+p_{sc}<\beta'<1$, there consequently exists $N\in\mathbb{N}$ such that
\[
\mathbb{E}\big[|\cos(u_n|X_1|)|\big]
\le\beta'
\le 1-\frac{\log(n)\, \eta / 6}{n}
\]
holds for all $n\ge N$.
As demonstrated, \labelcref{eq.cases-supp} leads to a contradiction in all three cases. Thus, we have shown \labelcref{eq.cases} by contradiction.
Since $\log(1-x)\le -x$ for all $x<1$, for $n\ge2m$ and $n>\log(n)\, \eta / 6$, we can write
\begin{equation}\label{eq.n^log(n)}
\begin{split}
\Big(1-\tfrac{\log(n)\, \eta / 6}{n}\Big)^{n-r}
&\le \Big(1-\tfrac{\log(n)\, \eta / 6}{n}\Big)^{n/2}
=\exp\Big(\tfrac n2\, \log\big(1-\tfrac{\log(n)\, \eta / 6}{n}\big)\Big)\\
&\le \exp\Big(\tfrac n2 \big(-\tfrac{\log(n)\, \eta / 6}{n}\big)\Big)
= n^{-\eta / 12}.
\end{split}
\end{equation}
Recall $\nu_n,\tau_n\le n^{c}$ and $\eta = 6(2r+{s}+4c)$. Thus, by \labelcref{eq.W_2,eq.W_2-2,eq.cases,eq.n^log(n)}
\begin{align}\label{eq.W_2-finish}
W_2
&\le c(r) n^{(2r+1)/2}\tau_n\,\nu_n\,\Big(1-\frac{\log(n)\, \eta / 6}{n}\Big)^{n-r}\\
&\le c(r) n^{(2r+1)/2}\tau_n\,\nu_n\,n^{-(2r+{s}+4c) / 2}
=\mathcal{O}\big(n^{-({s}-1)/2}\big).\nonumber
\end{align}
We finish the proof by merging \labelcref{eq.E.expanded,eq.W_3-0,eq.W_3,eq.W_2-finish} to
\begin{align*}
\mathbb{E}_*\bigg[\int_{B_n\le|u|< \nu_n} \Big|\ddU{r} \prod \cos\big(uV_n^{-1}|X_j|\big)\Big|\,\mathrm{d}u\bigg]
&\le W_1+W_2
\le W_3+W_2\\
&= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-({s}-2)/2} \, (\log n)^{({s}+\max\{2r,1\})/2} \big).
\end{align*} \end{proof}
In the next lemma, we bound the expectation of $\tilde{L}_{k,n}$.
\begin{lemma}\label{l.E.tL}
Assume that $\mathbb{E} |X_1|^{{s}}<\infty$ for some ${s}\ge2$. Then for all $k> {s}$,
\begin{equation*} \mathbb{E}[\tilde{L}_{k,n}]=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big). \end{equation*} \end{lemma}
The expectation of the remaining summand, an exponential of $B_n$, is bounded in the lemma below.
\begin{lemma}\label{l.exp.B_n}
Assume that $\mathbb{E} |X_1|^{{s}}<\infty$ for some ${s}\ge2$. Then for all $\eta>0$, \begin{align*} \mathbb{E}\left[ \exp\big(-\eta\,B_n^2\big)\right] = \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-({s}-2)/2} \, (\log n)^{{s}/2} (\eta\wedge 1)^{-s/2} \big). \end{align*} \end{lemma}
The rather straightforward proofs of \cref{l.E.tL,l.exp.B_n} are deferred to \cref{app.proofs.distr}. The following remark is the last required component for the proof of the main theorem.
\begin{remark}\label{r.T_n}
By the Cauchy--Schwarz inequality,
\begin{equation*}
T_n^2=\frac{S_n^2}{V_n^2}=\frac{(\sum X_j)^2}{\sum X_j^2}\le\frac{n\sum X_j^2}{\sum X_j^2}=n,
\end{equation*}
that is, $|T_n|\le\sqrt{n}$. \end{remark}
\begin{proof}[Proof of \cref{t.clt}]
As $|T_n|\le\sqrt{n}$, $F_n(x)=0$ for $x< -\sqrt{n}$ and $F_n(x)=1$ for $x> \sqrt{n}$. Due to the form \labelcref{eq.Phi^Q-def} and the factor $\phi$ in all expansion terms ${Q}_{r}$,
\[
\sup_{x>\sqrt{n}}(1+|x|)^{m}\Big(\big|\Phi^{Q}_{m,n}(-x)\big|+\big|1-\Phi^{Q}_{m,n}(x)\big|\Big)=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big).
\]
Therefore, the left-hand side of \labelcref{eq.t-clt} is of order $\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big)$ for the supremum restricted to $|x|> \sqrt{n}$. That is why we only have to consider the supremum over $|x|\le \sqrt{n}$ throughout this proof.
In order to apply \cref{p.cond}, the left-hand side of \labelcref{eq.t-clt} has to be converted into a form related to the left-hand side of \labelcref{eq.pcond}. By $\mathbb{E}[\tilde{F}_n(x)]=F_n(x)$, Jensen's inequality and \cref{p.E-tP-Q}, we have
\begin{align*}
\lefteqn{\sup_{|x|\le\sqrt{n}} (1+|x|)^{m}|F_n(x)-\Phi^{Q}_{m,n}(x)|}\enspace\\
&\le \sup_{|x|\le\sqrt{n}} (1+|x|)^{m}\left|\mathbb{E}\left[\tilde{F}_n(x)-\Phi^{\tilde{P}}_{m,n}(x)\right]\right| + \sup_{|x|\le\sqrt{n}} (1+|x|)^{m}\left|\mathbb{E}\left[\Phi^{\tilde{P}}_{m,n}(x)\right]-\Phi^{Q}_{m,n}(x)\right|\\
&\le \mathbb{E}\left[\sup_{|x|\le\sqrt{n}} (1+|x|)^{m}\left|\tilde{F}_n(x)-\Phi^{\tilde{P}}_{m,n}(x)\right|\right] +\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big).
\end{align*}
Due to $|x|\le \sqrt{n}$, the maximum order the factor $(1+|x|)^{m}$ can take is $n^{m/2}$. Next, $\Phi^{\tilde{P}}_{m,n}$ from \labelcref{eq.Phi^tP} has the property
\[
\sup_n \|\Phi^{\tilde{P}}_{m,n}\|_{\sup} \le c(m)
\]
due to \labelcref{eq.tP-bound}.
Thus, the absolute difference $\|\tilde{F}_n-\Phi^{\tilde{P}}_{m,n}\|_{\sup}$ is bounded by a constant $c(m)$.
By \labelcref{eq.Mn-an},
\begin{align*}
\mathbb{P}(M_n\ge n)
=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-1)}\big).
\end{align*}
Let $\zeta>0$ such that $\mathbb{P}(|X_1|\le \zeta)<1$. Then,
\[
\mathbb{P} (M_n \le \zeta) = \mathbb{P}(|X_1|\le \zeta)^n = \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m+{s}-2)/2}\big)
\]
in particular. Therefore, decomposing into $\{M_n \le \zeta \},\{\zeta < M_n< n\}$ and $\{M_n \ge n\}$,
\begin{align*}
\lefteqn{\mathbb{E}\left[\sup_{|x|\le\sqrt{n}}(1+|x|)^{m}\left|\tilde{F}_n(x)-\Phi^{\tilde{P}}_{m,n}(x)\right|\right]}\quad\\
&= \mathbb{E}\left[ \mathbbm{1}_{\{\zeta < M_n< n\}} \sup_{|x|\le\sqrt{n}}(1+|x|)^{m}\left|\tilde{F}_n(x)-\Phi^{\tilde{P}}_{m,n}(x)\right| \right]
+\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big).
\end{align*}
Recall $B_n=V_n/M_n$. Applying \cref{p.cond} ($M_n>\zeta$ guarantees $V_n^2>0$) yields
\begin{align*}
\lefteqn{\mathbb{E}\left[\mathbbm{1}_{\{\zeta < M_n< n\}} \sup_{|x|\le\sqrt{n}}(1+|x|)^{m}\left|\tilde{F}_n(x)-\Phi^{\tilde{P}}_{m,n}(x)\right| \right]}\quad\\
&\le c(m) \mathbb{E}\left[ \mathbbm{1}_{\{\zeta < M_n< n\}} \Big(\tilde{L}_{m+1,n}
+ I_2
+ \sum_{k=0}^{m} I_{k,2}
+ e^{-B_n^2/4}\Big)
\right],
\end{align*}
where $I_2=\int_{B_n\le|t|<\tilde{L}_{m+1,n}^{-1}}|t|^{-1}\big|\tilde{\varphi}_{T_n}(t)\big|\,\mathrm{d}t$ and
\[I_{k,2}=\int_{B_n\le|t|<\tilde{L}_{m+1,n}^{-1}}|t|^{k-m-1}\Big|\ddT{k} \tilde{\varphi}_{T_n}(t)\Big|\,\mathrm{d}t.\]
Regarding the first summand,
\begin{equation*}
\mathbb{E}[\tilde{L}_{m+1,n}] = \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big)
\end{equation*}
by \cref{l.E.tL} with $m+1>{s}$.
Concerning $I_2$ and $I_{k,2}$, we first bound the upper endpoint of the integration interval $\tilde{L}_{m+1,n}^{-1}$ by
\begin{align*}
\tilde{L}_{m+1,n}^{-1}
=V_n^{m+1}\Big(\sum|X_j|^{m+1}\Big)^{-1}
\le \big(n \,M_n^2\big)^{(m+1)/2} \big(M_n^{m+1}\big)^{-1}
= n^{(m+1)/2}
=:\nu_n.
\end{align*}
Note that $\nu_n$ is deterministic.
Next, we can leave out negative powers of $|t|$ since $|t|\ge B_n\ge1$. Then we get from \cref{p.E.I} (with $\tau_n=n$)
\begin{align*}
\mathbb{E}[\mathbbm{1}_{\{\zeta < M_n< n\}} I_2]
&\le\mathbb{E}\bigg[\mathbbm{1}_{\{\zeta < M_n< n\}} \int_{B_n\le|t|< \nu_n} \Big| \prod \cos\big(tV_n^{-1}|X_j|\big)\Big|\,\mathrm{d}t\bigg]\\
&= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-({s}-2)/2} \, (\log n)^{({s}+1)/2} \big)
\end{align*}
and
\begin{align*}
\mathbb{E}[\mathbbm{1}_{\{\zeta < M_n< n\}} I_{k,2}]
&\le\mathbb{E}\bigg[\mathbbm{1}_{\{\zeta < M_n< n\}} \int_{B_n\le|t|< \nu_n} \Big|\ddT{k} \prod \cos\big(tV_n^{-1}|X_j|\big)\Big|\,\mathrm{d}t\bigg]\\
&= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-({s}-2)/2} \, (\log n)^{({s}+\max\{2k,1\})/2} \big)
\end{align*}
for all $k=0,\dots,m$.
\cref{l.exp.B_n} yields
\begin{align*}
\mathbb{E}\left[ e^{-B_n^2/4}\right]
= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-({s}-2)/2} \, (\log n)^{{s}/2} \big).
\end{align*}
Therefore \labelcref{eq.t-clt} holds and \cref{t.clt} is proved. \end{proof}
\section{Non-uniform bounds for Edgeworth expansions in local limit theorems for self-normalized sums}\label{ch.density}
This section is devoted to proving \cref{t.llt-red} and more general LLT-type results for self-normalized sums. The crucial obstacle as compared to the CLT-setting is that, conditional on $\mathcal{F}_n$, the self-normalized sum $T_n$ does not have a density with respect to the Lebesgue measure. Instead it is continuous with respect to the counting measure. Therefore, the Fourier inversion formula does not apply any longer.
We introduce the following condition, which will be imposed in \cref{t.llt}. It is particularly satisfied if $X_1$ has a bounded density, see \cref{p.prop}. For instance, it has been successfully applied in \cite{JSZ04} to derive a saddle-point approximation for $T_n$.
\begin{condition}\label{c.density-cf}
The joint characteristic function $\varphi_{(X_1,X_1^2)}$ of $(X_1,X_1^2)$ satisfies
\begin{align*}
\int_{\mathbb{R}^2} |\varphi_{(X_1,X_1^2)}(t)|^c \,\mathrm{d}t < \infty
\end{align*}
for some $c\ge1$. \end{condition}
\cref{c.density-cf} is a smoothness condition for the corresponding density. By \cite[Theorem 19.1]{BR76normal}, it is equivalent to $(\sum_{j=1}^n X_j,\sum_{j=1}^n X_j^2)$
having bounded densities for $n$ sufficiently large. In this case, $p_{d}=0$ (see \labelcref{eq.lebesgue-dec}) holds necessarily because any discrete part cannot not vanish when summing up i.i.d. random variables (see \cite[Proposition 6.1]{JSZ04}). \cref{c.density-cf} implies Cram\'er condition (see \cite[p. 78]{Hal92edgeworth}), but it does not imply non-singularity and neither does non-singularity imply \cref{c.density-cf} (see \cite[p. 192]{BR76normal}).
Recall $\phi^{q}_{m,n}(x)=\phi(x)+\sum_{r=1}^{m-2}q_{r}(x)n^{-r/2}$ from \labelcref{eq.phi^q-def}.
\begin{theorem}\label{t.llt}
Assume that $X_1$ is symmetric, the distribution of $X_1$ is non-singular, \cref{c.density-cf} is satisfied and $\mathbb{E}|X_1|^{2m}<\infty$ for some $m\in\mathbb{N}$, $m\ge3$. Then there exists $N\in\mathbb{N}$ such that for all $n\ge N$, the statistics $T_{n}$ have densities $f_{n}$ that satisfy
\begin{equation}\label{eq.t-llt}
\sup_{x\in\mathbb{R}} \, (1+|x|)^{m}|f_n(x)-\phi^{q}_{m,n}(x)|= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m-3)/2}\big).
\end{equation}
Moreover, if $\mathbb{E} |X_1|^{2m+2}<\infty$ for some $m\in\mathbb{N}, m\ge2$, the order of convergence reduces to $\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m-1)/2} \, \log n \big)$. \end{theorem}
Note that \labelcref{eq.t-llt} implies that the densities $f_n$ are uniformly bounded for $n\ge N$. For the normalized sum $Z_n$, a similar non-uniform bound can be found in \cite{BCG11}. Together with the following \cref{p.prop}, \cref{t.llt} implies \cref{t.llt-red}. Its rather technical proof can be found in \cref{app.proofs.density}.
\begin{proposition}\label{p.prop}
Assume that $X_1$ has a bounded density. Then \labelcref{c.density-cf} holds. \end{proposition}
\paragraph*{Structure of the proof of \cref{t.llt}} As already mentioned, the starting point is again conditioning on $\mathcal{F}_n$, leaving us with discrete $T_n$. Perturbing $T_n$ (see \cref{s.density.cond}) allows to derive an LLT in the conditional setting (\cref{p.dichte-p-c}) by bounding the characteristic functions (\cref{l.lemma4'}) and using the Fourier inversion formula. Afterwards, we return to the unconditional setting by taking expectations (\cref{p.dichte-p}). The necessary deconvolution leads to \cref{p.dichte-general}, which applies under \cref{c.density-cf} and is used in the proof of \cref{t.llt}.
\begin{remark}
The one-dimensional analogue of \cref{c.density-cf}, namely $|\varphi_1|^c$ being integrable for some $c\ge 1$, is equivalent to $Z_n$ having a bounded density $f_{Z_N}$ for some $N\in\mathbb{N}$ (or equivalently for all sufficiently large $n$) by \cite[Theorem 19.1]{BR76normal}. This is necessary for any LLT for $Z_n$ to hold and therefore posed as a condition in all LLTs such as \cite[Theorem 19.2]{BR76normal}, \cite{BCG11} and \cite[Theorem 17, Chapter VII]{Pet75sums}.
\end{remark}
\subsection{The conditional setting}\label{s.density.cond}
Given $|X_1|,\dots,|X_n|$, there are (at most) $2^n$ possible values for $T_n$ as each $|X_j|$ can be included in $S_n$ with positive or negative sign. This is denoted by $\pm_i$, $i=1,\dots,2^n$. Due to the symmetry of $X_1$, the self-normalized sum $T_n$ (conditioned on $|X_1|,\dots,|X_n|$) takes each of these values with probability $2^{-n}$. Thus in the conditional setting, $T_n$ does not have a density but the probability mass function \begin{equation*} \tilde{f}_n(x)
:=f_{T_n\mid|X_1|,\dots,|X_n|}\big(x\mid |X_1| ,\dots,|X_n|\big)
=2^{-n}\sum_{i=1}^{2^n}\mathbbm{1}_{\{V_n^{-1}{\sum \pm_i |X_j|}=x\}}\,. \end{equation*} for all $x\in\mathbb{R}$.
As $\tilde{f}_n$ is not a Lebesgue density, the Fourier inversion formula does not apply. Therefore, we perturb $T_n$ with an independent absolutely continuous random variable $\varepsilon_n$ and examine \[ T_n':=T_n+\varepsilon_n. \] For the distribution of $\varepsilon_n$, the normal distribution seems to be a suitable choice. In contrast to other distributions, the tails of its density and characteristic function decay in the same magnitude. As we operate with both of these functions, their simultaneous exponential decay will minimize the negative effects of perturbation. Furthermore, it does not hinder convergence to a normal distribution.
Hence, we take $\varepsilon_n\sim \mathcal{N}(0,\beta_n)$ with density $\phi_{\beta_n}$, independent of $X_1,\dots,X_n$. Here, ${\beta_n\in(0,1)}$ for all $n\in\mathbb{N}$ and $\beta_n\to 0$ in polynomial order for $n\to\infty$. The precise sequence $(\beta_n)$ will be fixed later.
Next, we derive the distribution of $T_n'$ in the conditional setting. To this end, let $B\in\mathcal{B}(\mathbb{R})$ and $a_1,\dots,a_n\ge0$, then \begin{align*}
&\mathbb{P}\big(T_n+\varepsilon_n\in B \;|\; |X_1|=a_1,\dots,|X_n|=a_n\big)\\*
&\quad=\sum_{i=1}^{2^n}\mathbb{E}\Big[\mathbbm{1}\Big\{T_n=\tfrac{\sum\pm_ia_j}{(\sum a_j^2)^{1/2}}\Big\}\mathbbm{1}\Big\{\varepsilon_n\in B-\tfrac{\sum\pm_ia_j}{(\sum a_j^2)^{1/2}}\Big\} \;\Big|\; |X_1|=a_1,\dots,|X_n|=a_n\Big]. \end{align*}
As $T_n$ and $\varepsilon_n$ are independent given $|X_1|,\dots,|X_n|$, the expression above is equal to \begin{align*}
\sum_{i=1}^{2^n} &\mathbb{P}\Big(T_n=\tfrac{\sum\pm_ia_j}{(\sum a_j^2)^{1/2}} \;\Big|\; |X_1|=a_1,\dots,|X_n|=a_n\Big) \mathbb{P}\Big(\varepsilon_n\in B-\tfrac{\sum\pm_ia_j}{(\sum a_j^2)^{1/2}}\Big)\\
&\quad=\int_{B}2^{-n}\sum_{i=1}^{2^n} \phi_{\beta_n}\Big(y-\tfrac{\sum\pm_ia_j}{(\sum a_j^2)^{1/2}}\Big)\,\mathrm{d}y. \end{align*} Thus, the conditional distribution of $T_n'=T_n+\varepsilon_n$ is a convolution of the two underlying distributions, which will be denoted by $\ast$. Its conditional density has the form
\begin{align*} &\tilde{f}_n'(x)
:=f_{T_n'\mid|X_1|,\dots,|X_n|}(x\mid |X_1| ,\dots,|X_n|) =\tilde{f}_n\ast \phi_{\beta_n}(x)
=2^{-n}\sum_{i=1}^{2^n} \phi_{\beta_n}\Big(x-\tfrac{\sum \pm_i |X_j|}{V_n}\Big). \end{align*}
For the following steps until \labelcref{eq.E-fn'}, assume that the self-normalized sums $T_{n}$ have densities $f_{n}$ for $n\ge N$ for some $N\in\mathbb{N}$, and that we are in the regime $n\ge N$. This assumption is implied by the conditions of \cref{t.llt} (see \cref{s.density.uncond}).
In the unconditional setting, $T_n$ and $\varepsilon_n$ are independent, too. Thus, the distribution of $T_n'=T_n+\varepsilon_n$ has the density
\begin{equation}\label{eq.f_n'-def} f_n'(x)=f_{T_n'}(x)=f_n\ast \phi_{\beta_n}(x)=\int_{-\infty}^{\infty}f_n(x-y) \phi_{\beta_n}(y)\,\mathrm{d}y. \end{equation} By Fubini's theorem, $\tilde{f}_n'$ and $f_n'$ are connected by \begin{align*} \int_B \mathbb{E}\big[\tilde{f}_n'(x)\big]\,\mathrm{d}x &= \mathbb{E}\Big[\int_B \tilde{f}_n'(x)\,\mathrm{d}x\Big]
= \mathbb{P}\big(T_n' \in B \big)= \int_B f_n'(x)\,\mathrm{d}x \end{align*} for all $B\in\mathcal{B}(\mathbb{R})$ and thus $\mathbb{E}\big[\tilde{f}_n'(x)\big]= f_n'(x)$ $\lambda$-a.e. Since $\phi_{\beta_n}$ is continuous, $\tilde{f}_n'$ is also a continuous function. If $x_k\to x$ for $k\to\infty$, then \[ \lim_{k\to\infty} \mathbb{E}\big[\tilde{f}_n'(x_k)\big] =\mathbb{E}\big[\lim_{k\to\infty} \tilde{f}_n'(x_k)\big] =\mathbb{E}\big[\tilde{f}_n'(x)\big] \] by dominated convergence which is applicable since $0\le\tilde{f}_n'(x)\le(2\pi\beta_n)^{-1/2}$ for all $x\in\mathbb{R}$ and $n\in\mathbb{N}$. Thus, $\mathbb{E}\big[\tilde{f}_n'\big]$ is also continuous. As $f_n\in L^1$ and $\phi_{\beta_n}\in L^\infty$, the convolution $f_n'=f_n\ast \phi_{\beta_n}$ also is a bounded and continuous function (see e.g. \cite[Theorem 15.8]{Schilling17}). We can therefore conclude that \begin{equation}\label{eq.E-fn'} \mathbb{E}\big[\tilde{f}_n'(x)\big]= f_n'(x) \qquad\text{ for all }x\in\mathbb{R}. \end{equation}
For $T_n'$, we proceed as described above. First, we need a version of \cref{p.lemma4} that is suitable for $T_n'$. We do not need perturbed approximation functions because we will choose $\beta_n$ sufficiently small such that the approximation functions $\tilde{p}$ and $\tilde{U}$ are still appropriate. Recall the definitions $M_n=\max|X_j|$, $B_n=V_n/M_n$, $\tilde{L}_{k,n}=V_n^{-k}\sum|X_j|^k$ and \[ \tilde{\varphi}_{\Phi^{\tilde{P}}_{m,n}}(t) =e^{-t^2/2}\Big(1+\sum_{r=1}^{m-2} \tilde{U}_{r,n}(it)\Big), \] where \[ \tilde{U}_{r,n}(it) = \sum_{*(k_\cdot,r,\cdot)} (it)^{r+2u(k_\cdot)} \prod_{l=1}^{r} \frac{1}{k_l!} \Big(\frac{\tilde{\lambda}_{l+2,n}}{(l+2)!}\Big)^{k_l} \] from \labelcref{eq.tU,eq.tphi}, and where the sum $\sum_{*(k_\cdot,r,\cdot)}$ is introduced in \cref{ch.pre}. The following lemma is a consequence of \cref{p.lemma4}. Its proof is deferred to \cref{app.proofs.density}.
\begin{lemma}\label{l.lemma4'}
Assume that $X_1$ is symmetric, $V_n^2>0$ and $m\ge2$ is an integer. Then
\begin{equation}\label{eq.lemma4'}
\Big|\ddT{k} \big(\tilde{\varphi}_{T_n'}(t)-\tilde{\varphi}_{\Phi^{\tilde{P}}_{m,n}}(t)\big)\Big|
\le c(m)\big(\tilde{L}_{m+1,n}+\beta_n\big)e^{-t^2/6}\big(1+|t|^{3m+1+k}\big)
\end{equation}
holds for $k=0,\dots, m$ in the interval $|t|<B_n$. \end{lemma}
Next, we prove the non-uniform bound for our Edgeworth expansion of the density of the perturbed distribution in the conditional setting.
Recall the definition $\phi^{\tilde{p}}_{m,n}(x)=\phi(x)+\sum_{r=1}^{m-2}\tilde{p}_{r,n}(x)$, where \[ \tilde{p}_{r,n}(x)=\phi(x)\sum_{*(k_\cdot,r,\cdot)} H_{r+2u(k_\cdot)}(x) \prod_{l=1}^{r} \frac{1}{k_l!} \Big(\frac{\tilde{\lambda}_{l+2,n}}{(l+2)!}\Big)^{k_l} \] from \labelcref{eq.tp,eq.phi^tp}.
\begin{proposition}\label{p.dichte-p-c}
Assume that $X_1$ is symmetric, $V_n^2>0$ and $m\ge2$ is an integer. Then
\begin{equation}\label{eq.p-dichte-p-c}
\begin{split}
&\sup_{x\in\mathbb{R}} \, (1+|x|)^{m}|\tilde{f}_n'(x)-\phi^{\tilde{p}}_{m,n}(x)|\\
&\enspace\le c(m)\Big(\tilde{L}_{m+1,n}
+ \beta_n
+ J_{0,2}
+ J_{m,2}
+ \beta_n^{-(m+2)/2} n^{m-\log(n)/4}
+ e^{-B_n^2/4}\Big),
\end{split}
\end{equation}
where $J_{k,2}=\int_{B_n\le|t|< \nu_n} \big|\ddT{k}\tilde{\varphi}_{T_n'}(t)\big|\,\mathrm{d}t$ for $k=0,\dots,m$ and $\nu_n=\beta_n^{-1/2}\cdot \log(n)$. \end{proposition}
\begin{proof}
By conditional independence and \labelcref{eq.phi_n}, the characteristic function of $T_n'$ has the form
\begin{equation}\label{eq.phi_n'}
\begin{split}
\tilde{\varphi}_{T_n'}(t)
&= \Big( \prod\tilde{\varphi}_{X_j}(tV_n^{-1}) \Big) \cdot \exp(-\tfrac12\beta_nt^2)\\
&= \Big( \prod \cos\big(tV_n^{-1}|X_j|\big) \Big) \cdot \exp(-\tfrac12\beta_nt^2).
\end{split}
\end{equation}
For $k=0,\dots,m$, all derivatives $\ddT{k}\tilde{\varphi}_{T_n'}$ exist and are continuous. Furthermore, $\ddT{k}\tilde{\varphi}_{T_n'}$ is bounded by $c(m)n^k|t|^k\exp(-\beta_nt^2/2)\in L^1$ and thus in $L^1$. The Fourier transform of our approximating function $\tilde{\varphi}_{\Phi^{\tilde{P}}_{m,n}}(t)$ also possesses these properties by \labelcref{eq.ddtl-tphi}. This means that the conditions of the Fourier inversion theorem (see e.g. \cite[Theorem 4.1 (iv,v)]{BR76normal}) are satisfied. This yields
\begin{equation*}
\sup_{x\in\mathbb{R}} |x|^{k}|\tilde{f}_n'(x)-\phi^{\tilde{p}}_{m,n}(x)|
\le \int_{-\infty}^{\infty} \Big|\ddT{k}\big(\tilde{\varphi}_{T_n'}(t)-\tilde{\varphi}_{\Phi^{\tilde{P}}_{m,n}}(t)\big)\Big|\,\mathrm{d}t
\end{equation*}
for $k=0,\dots,m$. As before, we split up the integral at $B_n$ and additionally at $\nu_n$. This leads to
\begin{equation}\label{eq.dichte-int-split}
\begin{split}
\lefteqn{\sup_{x\in\mathbb{R}} |x|^{k}|\tilde{f}_n'(x)-\phi^{\tilde{p}}_{m,n}(x)|}\quad\\
&\le \underbrace{\int_{|t|<B_n} \Big|\ddT{k}\big(\tilde{\varphi}_{T_n'}(t)-\tilde{\varphi}_{\Phi^{\tilde{P}}_{m,n}}(t)\big)\Big|\,\mathrm{d}t}_{=J_{k,1}}
\, + \underbrace{\int_{B_n\le|t|< \nu_n} \Big|\ddT{k}\tilde{\varphi}_{T_n'}(t)\Big|\,\mathrm{d}t}_{=J_{k,2}}\\
&\quad + \underbrace{\int_{|t|\ge \nu_n} \Big|\ddT{k}\tilde{\varphi}_{T_n'}(t)\Big|\,\mathrm{d}t}_{=J_{k,3}}
\, + \underbrace{\int_{|t|\ge B_n} \Big|\ddT{k}\tilde{\varphi}_{\Phi^{\tilde{P}}_{m,n}}(t)\Big|\,\mathrm{d}t}_{=J_{k,4}}.
\end{split}
\end{equation}
From \cref{l.lemma4'} we deduce
\[
J_{k,1}
\le \int_{|t|<B_n} c(m)(\tilde{L}_{m+1,n}+\beta_n\big)e^{-t^2/6}\big(1+|t|^{3m+1+k}\big) \,\mathrm{d}t
\le c(m)(\tilde{L}_{m+1,n}+\beta_n\big),
\]
for all $k=0,\dots,m$.
Next, by the Leibniz rule,
\begin{equation}\label{eq.J_k3}
\begin{split}
J_{k,3}
&=\int_{|t|\ge \nu_n} \Big|\ddT{k} \Big( \prod \cos\big(tV_n^{-1}|X_j|\big) \Big) \cdot \exp(-\tfrac12\beta_nt^2)\Big|\,\mathrm{d}t\\
&\le\int_{|t|\ge \nu_n} \sum_{l=0}^{k}\binom{k}{l}\Big|\ddT{l} \prod \cos\big(tV_n^{-1}|X_j|\big)\Big| \Big|\ddT{k-l}\exp(-\tfrac12\beta_nt^2)\Big|\,\mathrm{d}t.
\end{split}
\end{equation}
As all derivatives of $\cos(x)$ are bounded in absolute value by 1,
\begin{align*}
\Big|\ddT{l} \prod \cos(tV_n^{-1}|X_j|)\Big|
&\le\sum _{i_{1}+i_{2}+\cdots +i_{n}=l} {l \choose i_{1},\ldots ,i_{n}} \prod\Big|\ddT{i_j}\cos(tV_n^{-1}|X_j|)\Big|\\
&\le\sum _{i_{1}+i_{2}+\cdots +i_{n}=l} {l \choose i_{1},\ldots ,i_{n}} \prod\Big|(V_n^{-1}|X_j|)^{i_j}\Big|\\
&\le\sum _{i_{1}+i_{2}+\cdots +i_{n}=l} {l \choose i_{1},\ldots ,i_{n}}=n^l.
\end{align*}
Here, the sum extends over all $n$-tuples $(i_1,...,i_n)$ of non-negative integers with $\sum_{j=1}^ni_j=l$ and
\[
{l \choose i_{1},\ldots ,i_{n}}={\frac {l!}{i_{1}!\,i_{2}!\cdots i_{n}!}}.
\]
Since $\beta_n<1$ and $|t|\ge \nu_n \ge1$, the $l$-th derivative of the exponential term in \labelcref{eq.J_k3} is bounded by $c(k)|t|^k\exp(-\beta_nt^2/2)$ for $l=0,\dots,k$.
Using \cref{l.int-t-exp}, $J_{k,3}$ can be bounded by
\begin{align*}
\lefteqn{\int_{|t|\ge \nu_n} \sum_{l=0}^{k}\binom{k}{l} n^l \, c(k) |t|^{k}\exp(-\beta_nt^2/2)\,\mathrm{d}t}\quad\\
&\le c(k) n^k \beta_n^{-(k+2)/2} \exp(-\beta_n \nu_n^2/4)\\
&\le c(m) n^m \beta_n^{-(m+2)/2} \exp\big(-(\log n)^{2}/4\big)\\
&= c(m) \beta_n^{-(m+2)/2} n^{m- \log(n)/4}
\end{align*}
for all $k=0,\dots,m$.
As in \labelcref{eq.Ik3}, by \labelcref{eq.ddtl-tphi} and since $|t|\ge B_n\ge 1$,
\[
J_{k,4}
=\int_{|t|\ge B_n} \Big|\ddT{k}\tilde{\varphi}_{\Phi^{\tilde{P}}_{m,n}}(t)\Big|\,\mathrm{d}t
\le c(m)e^{-B_n^2/4}
\]
for all $k=0,\dots,m$, where we use \cref{l.int-t-exp} (with $\beta=1$, $\nu_n=B_n$) in the last step.
Setting $k=0$ and $k=m$ in \labelcref{eq.dichte-int-split} now yields \labelcref{eq.p-dichte-p-c}. \end{proof}
\subsection{The unconditional setting}\label{s.density.uncond}
In the next step towards \cref{t.llt}, we apply the expectation to the result of \cref{p.dichte-p-c} and prove the non-uniform bound for our Edgeworth expansion of the density of the perturbed distribution in the unconditional setting. For this purpose, we introduce the following condition \cref{c.density-ex}.
Together with a moment condition on $X_1$, \cref{c.density-cf} implies \cref{c.density-ex} by \cref{t.llt}.
\begin{condition}\label{c.density-ex}
There exists $N\in\mathbb{N}$ such that the statistics $T_{n}$ have densities $f_{n}$
for
all $n\ge N$. \end{condition}
The well-known argument, which allows to conclude that the existence of a density $f_N$ for some $N\in\mathbb{N}$ implies the existence of densities $f_n$ for any $n\ge N$ which holds for the CLT-statistic $Z_n$ does not apply here.
For both statistics $T_n$ and $Z_n$, conditions of this type do not imply that $X_1$ is absolutely continuous or even has a non-zero absolutely continuous component. For a counterexample with $X_1,X_2$ having singular continuous distribution and $X_1+X_2$ being absolutely continuous with bounded density, see \cite[p. 17]{Luk72}.
Note that in contrast to the distribution functions in the proof of \cref{t.clt}, the densities $f_n$ do not exist a priori. Therefore, the requirement \cref{c.density-ex} is crucial for applying \labelcref{eq.E-fn'}.
\begin{proposition}\label{p.dichte-p}
Assume that $X_1$ is symmetric, the distribution of $X_1$ is non-singular and $\mathbb{E}|X_1|^{s}<\infty$ for some ${s}\ge2$. Additionally assume that \cref{c.density-ex} is satisfied and ${\beta_n=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big)}$ in polynomial order. Then for $m=\lfloor {s} \rfloor$,
\begin{equation}\label{eq.p-dichte-p}
\sup_{x\in\mathbb{R}} \, (1+|x|)^{m}|f_n'(x)-\phi^{q}_{m,n}(x)|= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-({s}-2)/2} \, (\log n)^{({s}+2m)/2} \big).
\end{equation} \end{proposition}
\begin{proof}
Let $n\ge N$ with $N$ from \cref{c.density-ex}. Recall $|T_n|\le\sqrt{n}$ from \cref{r.T_n} and thus, $f_n(x)=0$ for all $|x|>\sqrt{n}$.
Let $x>2\sqrt{n}$, then
\begin{align*}
(1+|x|)^{m}f_n'(x)
&=(1+x)^{m}\int_{-\sqrt{n}}^{\sqrt{n}} f_n(y) \phi_{\beta_n}(x-y)\,\mathrm{d}y\\
&\le(1+x)^{m} \phi_{\beta_n}(x-\sqrt{n} \, ) \int_{-\sqrt{n}}^{\sqrt{n}} f_n(y) \,\mathrm{d}y\\
&=(1+x)^{m} (2\pi\beta_n)^{-1/2} \exp\big(-\tfrac12\beta_n^{-1}(x-\sqrt{n} \, )^2\big).
\end{align*}
Due to
\begin{align*}
\lefteqn{\ddX{} (1+x)^{m} \exp\big(-\tfrac12\beta_n^{-1}(x-\sqrt{n} \, )^2\big)}\quad\\
&= \Big(m-(1+x) \beta_n^{-1}(x-\sqrt{n} \, )\Big) (1+x)^{m-1} \exp\big(-\tfrac12\beta_n^{-1}(x-\sqrt{n} \, )^2\big)\\
&\le \Big(m-(1+2\sqrt{n}\,) \beta_n^{-1} \sqrt{n}\Big) (1+x)^{m-1} \exp\big(-\tfrac12\beta_n^{-1}(x-\sqrt{n} \, )^2\big)< 0,
\end{align*}
$(1+x)^{m} \exp\big(-\beta_n^{-1}(x-\sqrt{n} \, )^2/2\big)$ is monotonically decreasing for $x>2\sqrt{n}$ (and $n>m$). Therefore,
\begin{equation}\label{eq.fn'-bound}
\begin{split}
\sup_{x > 2\sqrt{n}} (1+|x|)^{m}f_n'(x)
&\le \sup_{x > 2\sqrt{n}} (1+x)^{m} (2\pi\beta_n)^{-1/2} \exp\big(-\tfrac12\beta_n^{-1}(x-\sqrt{n} \, )^2\big)\\
&= (1+2\sqrt{n} \, )^{m} (2\pi\beta_n)^{-1/2} \exp\big(-\tfrac12\beta_n^{-1}n\big)= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big)
\end{split}
\end{equation}
in particular.
This bound also holds for negative $x$ as $f_n$ and $\phi_{\beta_n}$ are symmetric. Additionally, due to the form \labelcref{eq.phi^q-def} and the factor $\phi$ in all expansion terms $q_r$,
\begin{equation}\label{eq.phi^q-bound}
\sup_{|x|>2\sqrt{n}} (1+|x|)^{m} \big|\phi^{q}_{m,n}(x)\big|=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big).
\end{equation}
Therefore, the left-hand side of \labelcref{eq.p-dichte-p} is of order $\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big)$ for the supremum restricted to $|x|> 2\sqrt{n}$. That is why we only have to consider the supremum over $|x|\le 2\sqrt{n}$ throughout the remainder of this proof.
In order to apply \cref{p.dichte-p-c}, the left-hand side of \labelcref{eq.p-dichte-p} has to be converted into a form related to the left-hand side of \labelcref{eq.p-dichte-p-c}. By \labelcref{eq.E-fn'}, Jensen's inequality and \cref{p.E-tP-Q}, we have
\begin{align*}
\lefteqn{\sup_{|x|\le2\sqrt{n}} (1+|x|)^{m}|f_n'(x)-\phi^{q}_{m,n}(x)|}\;\\
&\le \sup_{|x|\le2\sqrt{n}} (1+|x|)^{m}\left|\mathbb{E}\left[\tilde{f}_n'(x)-\phi^{\tilde{p}}_{m,n}(x)\right]\right| + \sup_{|x|\le2\sqrt{n}} (1+|x|)^{m}\left|\mathbb{E}\left[\phi^{\tilde{p}}_{m,n}(x)\right]-\phi^{q}_{m,n}(x)\right|\\
&\le \mathbb{E}\left[\sup_{|x|\le2\sqrt{n}} (1+|x|)^{m}\left|\tilde{f}_n'(x)-\phi^{\tilde{p}}_{m,n}(x)\right|\right] +\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big).
\end{align*}
Due to the form of $\tilde{f}_n'$ and uniform boundedness of $\phi^{\tilde{p}}_{m,n}$ (see \labelcref{eq.phi^tp,eq.tp-bound}),
\begin{align*}
\lefteqn{\sup_{|x|\le2\sqrt{n}}(1+|x|)^{m}\left|\tilde{f}_n'(x)-\phi^{\tilde{p}}_{m,n}(x)\right|}\quad\\*
&=\sup_{|x|\le2\sqrt{n}}(1+|x|)^{m}\left|2^{-n}\sum_{i=1}^{2^n} \phi_{\beta_n}\Big(x-\tfrac{\sum \pm_i |X_j|}{V_n}\Big)-\phi^{\tilde{p}}_{m,n}(x)\right|\\
&\le \sup_{|x|\le2\sqrt{n}} (2\sqrt{n} \, )^{m}\left((2\pi\beta_n)^{-1/2}+c(m)\right)= \mathcal{O}\big(n^{m/2}\beta_n^{-1/2}\big).
\end{align*}
By \labelcref{eq.Mn-an} for $\tau_n:=n^{(m+s)/(2s)}\beta_n^{-1/(2s)}$,
\begin{align*}
\mathbb{P}(M_n\ge \tau_n)
=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m+s-2)/2}\beta_n^{1/2}\big).
\end{align*}
Let $\zeta>0$ such that $\mathbb{P}(|X_1|\le \zeta)<1$. Then,
\[
n^{m/2}\beta_n^{-1/2} \cdot \mathbb{P} (M_n \le \zeta)
= n^{m/2}\beta_n^{-1/2} \cdot \mathbb{P}(|X_1|\le \zeta)^n
= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big)
\]
in particular. Therefore, decomposing into $\{M_n \le \zeta \},\{\zeta < M_n< \tau_n\}$ and $\{M_n \ge \tau_n\}$,
\begin{align*}
&\mathbb{E}\left[\sup_{|x|\le2\sqrt{n}} (1+|x|)^{m}\left|\tilde{f}_n'(x)-\phi^{\tilde{p}}_{m,n}(x)\right|\right]\\*
&\quad= \mathbb{E}\left[\mathbbm{1}_{\{\zeta < M_n< \tau_n\}} \sup_{|x|\le2\sqrt{n}} (1+|x|)^{m}\left|\tilde{f}_n'(x)-\phi^{\tilde{p}}_{m,n}(x)\right| \right]
+\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big).
\end{align*}
Recall $B_n=V_n/M_n$. Applying \cref{p.dichte-p-c} ($M_n>\zeta$ guarantees $V_n^2>0$) yields
\begin{align*}
&\mathbb{E}\left[\mathbbm{1}_{\{\zeta < M_n< \tau_n\}} \sup_{|x|\le2\sqrt{n}} (1+|x|)^{m}\left|\tilde{f}_n'(x)-\phi^{\tilde{p}}_{m,n}(x)\right| \right]
\le c(m) \mathbb{E}\Big[\mathbbm{1}_{\{\zeta < M_n< \tau_n\}} \\*
&\qquad\cdot \Big(\tilde{L}_{m+1,n}
+ \beta_n
+ J_{0,2}
+ J_{m,2}
+ \beta_n^{-(m+2)/2} n^{m-\log(n)/4}
+ e^{-B_n^2/4}\Big)
\Big],
\end{align*}
where $J_{k,2}=\int_{B_n\le|t|< \nu_n} \big|\ddT{k}\tilde{\varphi}_{T_n'}(t)\big|\,\mathrm{d}t$ for $k=0,\dots,m$ and $\nu_n=\beta_n^{-1/2}\cdot \log(n)$.
For the first summand,
\begin{equation*}
\mathbb{E}[\tilde{L}_{m+1,n}] = \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big).
\end{equation*}
by \cref{l.E.tL} with $m+1>{s}$.
Clearly, for the second summand $\beta_n=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big)$ and for the fifth summand $\beta_n^{-(m+2)/2} n^{m-\log(n)/4}=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big)$ as $\beta_n$ decreases in polynomial order and $n^{-\log(n)}$ is smaller than $n^{-k}$ for every $k>0$ and $n$ sufficiently large.
Now we examine the expectation of $J_{k,2}$. Notice $\beta_n|t|< \beta_n\nu_n=\sqrt{\beta_n}\cdot \log(n)\le 1$ in the domain of the integral for $n$ sufficiently large.
For $B_n\le t< \nu_n$ and $a\ge0$, this yields
\begin{align*}
\Big|\ddT{a}\exp(-\tfrac12 \beta_n t^2)\Big|
\le c(a) \exp(-\tfrac12 \beta_n t^2)
\le c(a).
\end{align*}
Thus by \labelcref{eq.phi_n'} and the Leibniz formula,
\begin{align*}
J_{k,2}
&=\int_{B_n\le|t|< \nu_n} \Big|\sum_{l=0}^k\binom{k}{l}\Big(\ddT{l} \prod \cos\big(tV_n^{-1}|X_j|\big)\Big) \Big(\ddT{k-l}\exp(-\tfrac12\beta_nt^2)\Big)\Big|\,\mathrm{d}t\\
&\le c(k)\sum_{l=0}^{k}\int_{B_n\le|t|< \nu_n} \Big|\ddT{l} \prod \cos\big(tV_n^{-1}|X_j|\big)\Big| \Big|\ddT{k-l}\exp(-\tfrac12\beta_nt^2)\Big|\,\mathrm{d}t\\
&\le c(k)\sum_{l=0}^{k}\int_{B_n\le|t|< \nu_n} \Big|\ddT{l} \prod \cos\big(tV_n^{-1}|X_j|\big)\Big| \,\mathrm{d}t.
\end{align*}
As $\nu_n=\beta_n^{-1/2}\cdot \log(n)$ and $\tau_n=n^{(m+s)/(2s)}\beta_n^{-1/(2s)}$ with $\beta_n=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big)$ grow in polynomial order, by \cref{p.E.I} we get
\begin{align*}
\mathbb{E}\big[\mathbbm{1}_{\{\zeta < M_n< \tau_n\}} J_{k,2}\big]
&\le c(k)\sum_{l=0}^{k}\mathbb{E}\bigg[\mathbbm{1}_{\{\zeta < M_n< \tau_n\}} \int_{B_n\le|t|< \nu_n} \Big|\ddT{l} \prod \cos\big(tV_n^{-1}|X_j|\big)\Big|\,\mathrm{d}t \bigg]\\
&= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-({s}-2)/2} \, (\log n)^{({s}+\max\{2k,1\})/2} \big)
\end{align*}
for $k=0,\dots,m$.
Regarding the sixth summand, \cref{l.exp.B_n} yields
\begin{align*}
\mathbb{E}\left[ e^{-B_n^2/4}\right]
= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-({s}-2)/2} \, (\log n)^{{s}/2} \big).
\end{align*}
Therefore \labelcref{eq.p-dichte-p} holds and \cref{p.dichte-p} is proved. \end{proof}
\cref{p.dichte-p} provides a non-uniform bound for $f_n'$. We will now deconvolute the densities to achieve a similar bound for $f_n$. The H\"older type condition $(iii)$ of the following result seems hard to verify, but it will turn out to be satisfied by the conditions of \cref{t.llt}.
\begin{proposition}\label{p.dichte-general}
Assume that the following conditions are satisfied:
\begin{enumerate}[$(i)$]
\item $X_1$ is symmetric, the distribution of $X_1$ is non-singular, and $\mathbb{E}|X_1|^{s}<\infty$ for some ${s}\ge2$.
\item \cref{c.density-ex} is satisfied and there exists $N_1\in\mathbb{N}$ such that there exist versions of $f_{n}$
that are bounded uniformly over all $n\ge N_1$.
\item There exist $a\in(0,1],b,\alpha>0$ (that are allowed to depend on $s$), $N_2\in\mathbb{N}$ and a sequence $(r_n)$ with $r_n\to 0$ such that for some $m\in\mathbb{N}$ with $2\le m \le {s}$,
\begin{equation}\label{eq.density-Hoelder-1}
(1+|x|)^{m}|f_n(x)-f_n(x-h)|
\le \alpha \, (1+|x|)^{m} |h|^{a} n^{b} + r_n
\end{equation}
holds for all $n\ge N_2$, $|h|<1$ and for all $|x|\le2\sqrt{n}$.
\end{enumerate}
Then,
\begin{equation}\label{eq.p-dichte-general}
\begin{split}
&\sup_{x\in\mathbb{R}} \, (1+|x|)^{m}|f_n(x)-\phi^{q}_{m,n}(x)|\\*
&\quad= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-({s}-2)/2} \, (\log n)^{({s}+2\lfloor {s} \rfloor)/2} \big)
+ \mathcal{O}\big( n^{-\lceil\frac{m-1}2\rceil} \big)
+ r_n.
\end{split}
\end{equation} \end{proposition}
\begin{proof}
Let $n\ge N\vee N_1\vee N_2$ with $N$ from \cref{c.density-ex}. In this proof we decompose the left-hand side of \labelcref{eq.p-dichte-general} such that
\begin{equation}\label{eq.f-split}
\begin{split}
\lefteqn{\sup_{x\in\mathbb{R}} \, (1+|x|)^{m}|f_n(x)-\phi^{q}_{m,n}(x)|}\quad\\*
&\le\sup_{x\in\mathbb{R}} \, (1+|x|)^{m}|f_n(x)-f_n'(x)|
+\sup_{x\in\mathbb{R}} \, (1+|x|)^{\lfloor {s} \rfloor}|f_n'(x)-\phi^{q}_{\lfloor s \rfloor,n}(x)|\\
&\quad +\sup_{x\in\mathbb{R}} \, (1+|x|)^{m}|\phi^{q}_{\lfloor {s} \rfloor,n}(x)-\phi^{q}_{m,n}(x)|.
\end{split}
\end{equation}
Now we fix $\beta_n= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-(2b+m+{s}-2)/a} \big)$ (polynomial in $n$), which satisfies the assumptions of \cref{p.dichte-p}. Thus, the second summand of the right-hand side in \labelcref{eq.f-split} is of order $\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-({s}-2)/2} \, (\log n)^{({s}+2\lfloor {s} \rfloor)/2} \big)$.
By \labelcref{eq.phi^q-def}, the third summand equals
\begin{equation*}
\sup_{x\in\mathbb{R}} \, (1+|x|)^{m}\Big|\sum_{r=m-1}^{\lfloor {s} \rfloor-2}q_{r}(x)n^{-r/2}\Big|
\le c({s}) n^{-\lceil\frac{m-1}2\rceil}
\end{equation*}
for all $x\in\mathbb{R}$. Similar to \labelcref{eq.phi^q-bound}, the bound is due to the factor $\phi$ in all expansion terms $q_{r}$ and the fact $q_r=0$ for uneven $r$.
As a last step, we need to bound $\sup_{x\in\mathbb{R}} (1+|x|)^{m}|f_n(x)-f_n'(x)|$. By \cref{r.T_n}, $|T_n|\le\sqrt{n}$ and thus $f_n(x)=0$ for all $|x|>\sqrt{n}$. Then, \labelcref{eq.fn'-bound} implies
\begin{align*}
\sup_{|x|>2\sqrt{n}}(1+|x|)^{m}|f_n(x)-f_n'(x)|
=\sup_{|x|>2\sqrt{n}}(1+|x|)^{m}f_n'(x)
\le \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big).
\end{align*}
Hence, from now on we only have to consider the supremum over $|x|\le 2\sqrt{n}$. For ${\varepsilon_n\sim \mathcal{N}(0,\beta_n)}$ with density $\phi_{\beta_n}$, we get
\begin{align}\label{eq.fn-fn'}
\lefteqn{\sup_{|x|\le 2\sqrt{n}}(1+|x|)^{m}|f_n(x)-f_n'(x)|}\quad\nonumber\\
&=\sup_{|x|\le 2\sqrt{n}}(1+|x|)^{m}\Big|\int_{-\infty}^{\infty} \big(f_n(x)-f_n(x-y)\big) \phi_{\beta_n}(y)\,\mathrm{d}y\Big|\\
&=\sup_{|x|\le 2\sqrt{n}}(1+|x|)^{m}\Big|\mathbb{E}\big[f_n(x)-f_n(x-\varepsilon_n)\big]\Big|\nonumber\\
&\le\mathbb{E}\Big[\sup_{|x|\le 2\sqrt{n}}(1+|x|)^{m}\big|f_n(x)-f_n(x-\varepsilon_n)\big|\Big].\nonumber
\end{align}
By \labelcref{eq.density-Hoelder-1}, we get for $|\varepsilon_n|<1$
\begin{align*}
\sup_{|x|\le 2\sqrt{n}}(1+|x|)^{m}|f_n(x)-f_n(x-\varepsilon_n)|
&\le \sup_{|x|\le 2\sqrt{n}} \alpha \, (1+|x|)^{m} |\varepsilon_n|^{a} n^{b} + r_n\\
&= c(s) \sup_{|x|\le 2\sqrt{n}} (1+|x|)^{m} |\varepsilon_n|^{a} n^{b} + r_n.
\end{align*}
By uniform boundedness of $f_n$ for $|\varepsilon_n|\ge1$,
\begin{align*}
\sup_{|x|\le 2\sqrt{n}}(1+|x|)^{m} \big|f_n(x)-f_n(x-\varepsilon_n)\big|
&\le \sup_{|x|\le 2\sqrt{n}}(1+|x|)^{m} \cdot 2\sup_{n\ge N_1}\|f_n\|_{\sup}\\
&\le c(s) \sup_{|x|\le 2\sqrt{n}}(1+|x|)^{m} |\varepsilon_n|^{a} n^{b} + r_n.
\end{align*}
Therefore, noting that $\varepsilon_n\sim\mathcal{N}(0,\beta_n)$, we deduce from \labelcref{eq.fn-fn'}
\begin{align*}
\sup_{|x|\le 2\sqrt{n}}(1+|x|)^{m}|f_n(x)-f_n'(x)|
&\le\mathbb{E}\Big[\sup_{|x|\le 2\sqrt{n}}(1+|x|)^{m}\big|f_n(x)-f_n(x-\varepsilon_n)\big|\Big]\\
&\le \mathbb{E}\Big[c(s) \sup_{|x|\le 2\sqrt{n}}(1+|x|)^{m} |\varepsilon_n|^{a} n^{b} + r_n\Big]\\
&\le c(s) n^{(2b+m)/2} \big(\mathbb{E}\varepsilon_n^2\big)^{a/2} + r_n\\
&= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-({s}-2)/2} \big) + r_n.
\end{align*}
\end{proof}
\subsection{The smooth function model}\label{s.density.smooth}
In this subsection, we briefly explain the implications of the smooth function model for self-normalized sums, which we use in the proof of \cref{t.llt}. For a more detailed introduction, see \cite[Sections 2.4 and 2.8]{Hal92edgeworth}. Let $Y_j:=(X_j,X_j^2)$ with mean $\mu_Y=(0,1)$ and covariance matrix \begin{align*} \Sigma:= \ensuremath{\mathop{\mathrm{Cov}}}(Y)= \begin{pmatrix} 1 & \mu_3\\ \mu_3 & \mu_4-1 \end{pmatrix}. \end{align*} Assume that \cref{c.density-cf} holds and that $\mu_4<\infty$. This implies that $(X_1,X_1^2)$ is non-degenerate and thus $\Sigma$ is positive definite, see \cref{r.non-deg}. Let $Y^{(i)}$ denote the $i$-th coordinate of $Y$ and $\overline{Y}:=\tfrac1n \sum Y_j$.
If $\mathbb{E} |X_1|^{2m}<\infty$ holds for some $m\in\mathbb{N}$, $m\ge3$, by \cite[Theorem 19.2]{BR76normal}, for $n$ sufficiently large, the density of $n^{1/2}(\overline{Y}-\mu_Y)$, $g_n$ exists, is bounded and admits the expansion \begin{equation}\label{eq.BR-19.2}
\sup_{y\in\mathbb{R}^2} (1+\|y\|)^{m}\Big|g_n(y)-\sum_{r=0}^{m-2}n^{-r/2}\pi_r(y)\phi_{\Sigma}(y)\Big| =\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m-2)/2}\big), \end{equation} where $\pi_r$ is a bivariate polynomial of degree $3r$ (which includes moments up to order $2r+4$) and $\phi_{\Sigma}$ denotes the 2-dimensional normal density with parameters 0 and $\Sigma$.
Next, we derive a density $f_n$ of $T_n$ in terms of $g_n$. \begin{align*} \mathbb{P}(T_n\le a)
&=\mathbb{P}\Big( n^{1/2} \, \overline{Y^{(1)}} \, \Big(\overline{Y^{(2)}}\Big)^{-1/2} \le a\Big)\\
&=\mathbb{P}\bigg( \Big(n^{1/2} \, \overline{Y^{(1)}}\Big) \, \Big(n^{-1/2}n^{1/2}\Big(\overline{Y^{(2)}}-1\Big)+1\Big)^{-1/2} \le a\bigg)\\ &=\int_{\mathbb{R}^2} \mathbbm{1}\Big\{ u_1 \, \big(n^{-1/2}u_2+1\big)^{-1/2} \le a , \, u_2>-n^{1/2}\Big\} g_n(u) \,\mathrm{d}u\\
&=\int_{-n^{1/2}}^\infty \int_{-\infty}^{a \, (n^{-1/2}u_2+1)^{1/2}} g_n(u_1,u_2) \,\mathrm{d}u_1\,\mathrm{d}u_2.
\intertext{By substituting $u_1=x\big(n^{-1/2}u_2+1\big)^{1/2}$, applying Fubini's theorem, and substituting ${u_2=z-n^{1/2}}$,}
\mathbb{P}(T_n\le a)
&=\int_{-n^{1/2}}^\infty \int_{-\infty}^{a} g_n\big(x \, (n^{-1/2}u_2+1)^{1/2},u_2\big) \big(n^{-1/2}u_2+1\big)^{1/2} \,\mathrm{d}x \,\mathrm{d}u_2\\ &=\int_{-\infty}^{a} \int_{-n^{1/2}}^\infty g_n\big(x \, (n^{-1/2}u_2+1)^{1/2},u_2\big) \big(n^{-1/2}u_2+1\big)^{1/2} \,\mathrm{d}u_2 \,\mathrm{d}x\\
&=\int_{-\infty}^{a} \int_{0}^\infty g_n\big(x \, (n^{-1/2}z)^{1/2},z-n^{1/2}\big) \big(n^{-1/2}z\big)^{1/2} \,\mathrm{d}z \,\mathrm{d}x\\ &=\int_{-\infty}^{a} \int_{0}^\infty g_n\big(\gamma_{x,n}(z)\big) \big(n^{-1/2}z\big)^{1/2} \,\mathrm{d}z \,\mathrm{d}x \end{align*} where $\gamma_{x,n}(z)=\big(x(n^{-1/2}z)^{1/2},z-n^{1/2}\big)$ and thus \begin{align}\label{eq.fn-int-gn} f_n(x) = \int_{0}^\infty g_n\big(\gamma_{x,n}(z)\big) \big(n^{-1/2}z\big)^{1/2} \,\mathrm{d}z. \end{align}
\begin{remark}
In this context, Equation (2.67) on page 79 in \cite{Hal92edgeworth} states that
\begin{equation}\label{eq.fn-int-gn-Hall}
f_n(x) = \int_{S(n,x)} g_n(y) \,\mathrm{d}y
\end{equation}
with
$S(n,x):
=\big\{\gamma_{x,n}(z)\in\mathbb{R}^2\colon z\in(0,\infty)\big\}$.
However, the integral in \labelcref{eq.fn-int-gn-Hall} can neither be understood as an integral with respect to the two-dimensional Lebesgue measure (as it vanishes in this case) nor as a classical curve integral for the parametrization of $S(n,x)$ by $z\mapsto\gamma_{x,n}(z)=\big(x(n^{-1/2}z)^{1/2},z-n^{1/2}\big)$, namely
\begin{align*}
\int_{S(n,x)} g_n\,
=\int_0^\infty g_n(\gamma_{x,n}(z)) \|\dot\gamma_{x,n}(z)\| \,\mathrm{d}z.
\end{align*}
Indeed, adopting the latter strategy for the uniform distribution on the lower triangle of the unit square
$
\bigcup_{x\in[0,1]} S(x)$
with
\begin{align*}
S(x)=\big\{\gamma_{x,n}(t)\in[0,1]\times[0,1] \colon t\in[0,1-x]\big\} \quad \text{and} \quad \gamma_{x,n}(t)=(x+t,t),
\end{align*}
we do not obtain a density:
\begin{align*}
\int_0^1 \Big( \int_{ S(x)} 2 \Big) \,\mathrm{d}x
=\int_0^1 \int_0^{1-x} 2\, \|\dot\gamma_{x,n}(t)\| \,\mathrm{d}t \,\mathrm{d}x
=\sqrt{2}
\ne 1.
\end{align*}
\end{remark}
\subsection{Proof of Theorem \ref{t.llt}}\label{s.density.main}
In the following remark, we bound polynomials that appear in the proof of \cref{t.llt}.
\begin{remark}\label{r.polynome}
In the following derivations, we will often consider polynomials of the form $\pi_r\big(\gamma_{x,n}(z)\big)$, where $\pi_r$ is a bivariate polynomial of degree $3r$, which includes moments of $X_1$ up to order $(2r+4)$ and $\gamma_{x,n}(z)=\big(x(n^{-1/2}z)^{1/2},z-n^{1/2}\big)$. Integration of these polynomials leads to cumbersome calculations which we shall avoid by the subsequent upper bounds.
Let $\pi_{r,1}$ denote the function which arises from $\pi_r$ by replacing all coefficients and both arguments by their absolute values. Then
\begin{equation*}
\big|\pi_r\big(\gamma_{x,n}(z)\big)\big|=\big|\pi_r\big(x(n^{-1/2}z)^{1/2},z-n^{1/2}\big)\big|\le\pi_{r,1}\big(x(n^{-1/2}z)^{1/2},z-n^{1/2}\big).
\end{equation*}
Now, $\pi_{r,1}$ is symmetric in both arguments and thus non-decreasing for growing absolute values of the variables. Hence, replacing the factors $n^{-1/4}$ by 1 and $|x|$ by $(|x|\vee 1)$ yields an upper bound. This yields
\begin{align*}
\pi_{r,1}\big(x(n^{-1/2}z)^{1/2},z-n^{1/2}\big)
&\le \pi_{r,1}\big((|x|\vee 1)z^{1/2},z-n^{1/2}\big)\\
&\le (|x|\vee 1)^{3r}\pi_{r,1}\big(z+1,z-n^{1/2}\big)
\end{align*}
because the degree of the polynomial is at most $3r$. Next,
\begin{equation*}
\pi_{r,1}\big(z+1,z-n^{1/2}\big)
\le c(r) n^{3r/2}\hat{\pi}_{r}(|z|)
\end{equation*}
for some suitable univariate polynomial $\hat{\pi}_{r}$ of degree $3r$.
If $|x|<2\sqrt{n}$, we have
\begin{equation*}
\big|\pi_r\big(\gamma_{x,n}(z)\big)\big|
\le c(r) n^{3r/2}(|x|\vee 1)^{3r}\hat{\pi}_{r}(|z|)
\le c(r) n^{3r} \hat{\pi}_{r}(|z|).
\end{equation*} \end{remark}
Now we are in the position to prove the main theorem.
\begin{proof}[Proof of \cref{t.llt}]
Depending on which moment assumption in \cref{t.llt} is satisfied, let ${w}=m$ or ${w}=m+1$. In any case, ${w}\in\mathbb{N}$, ${w}\ge 3$ and $\mathbb{E} |X_1|^{2{w}}<\infty$.
\cref{c.density-cf} and $\mathbb{E} |X_1|^{2{w}}<\infty$ yield that there exists $N_1\in \mathbb{N}$ such that \labelcref{eq.fn-int-gn,eq.BR-19.2} hold for all $n\ge N_1$. Throughout the remainder of the proof assume $n\ge N_1$. Now \labelcref{eq.fn-int-gn} implies \cref{c.density-ex}. In order to apply \cref{p.dichte-general}, it remains to show that $f_n$ are uniformly bounded over $n\ge N_1$ and that the H\"older-type condition $(iii)$ holds. We start with the latter.
\labelcref{eq.fn-int-gn} and \labelcref{eq.BR-19.2} together with the required notation as introduced in \cref{s.density.smooth} yield
\begin{equation}\label{eq.f_n-von-g_n}
\begin{split}
f_n(x)
&= \int_{0}^\infty g_n\big(\gamma_{x,n}(z)\big) \big(n^{-1/2}z\big)^{1/2} \,\mathrm{d}z\\
&=\int_0^\infty \Big(\sum_{r=0}^{{w}-2}n^{-r/2}\pi_r(\gamma_{x,n}(z))\phi_{\Sigma}(\gamma_{x,n}(z)) \\
&\qquad\qquad+ \big(1+\|\gamma_{x,n}(z)\|\big)^{-{w}} R_n(\gamma_{x,n}(z))\Big) \big(n^{-1/2}z\big)^{1/2} \,\mathrm{d}z,
\end{split}
\end{equation}
where $\pi_r$ is a bivariate polynomial of degree $3r$, which includes moments up to order $2r+4$ and $\phi_{\Sigma}$ denotes the 2-dimensional normal density with parameters 0 and
\begin{align*}
\Sigma=
\begin{pmatrix}
1\enskip & 0\\
0\enskip & \mu_4-1
\end{pmatrix}.
\end{align*}
\cref{c.density-cf} implies that $X_1^2$ is non-degenerate and thus $\Sigma$ is positive definite. Additionally, $\gamma_{x,n}(z)=\big(x(n^{-1/2}z)^{1/2},z-n^{1/2}\big)$ and
\begin{equation}\label{eq.def-Rn'}
R_n':=\sup_{y\in\mathbb{R}^2} |R_n(y)|=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({w}-2)/2}\big).
\end{equation}
Then by the triangle-inequality,
\begin{align}\label{eq.f_n-f_n-Z}
\lefteqn{|f_n(x)-f_n(x-h)|}\quad\\*
&=\bigg|\int_0^\infty \Big(\sum_{r=0}^{{w}-2}n^{-r/2}\pi_r(\gamma_{x,n}(z))\phi_{\Sigma}(\gamma_{x,n}(z)) \big(n^{-1/2}z\big)^{1/2} \nonumber\\
&\qquad+ \big(1+\|\gamma_{x,n}(z)\|\big)^{-{w}} R_n(\gamma_{x,n}(z)) \big(n^{-1/2}z\big)^{1/2} \Big) \nonumber\\
&\quad- \Big(\sum_{r=0}^{{w}-2}n^{-r/2}\pi_r(\gamma_{x-h,n}(z))\phi_{\Sigma}(\gamma_{x-h,n}(z)) \big(n^{-1/2}z\big)^{1/2} \nonumber\\
&\qquad+ \big(1+\|\gamma_{x-h,n}(z)\|\big)^{-{w}} R_n(\gamma_{x-h,n}(z)) \big(n^{-1/2}z\big)^{1/2}\Big)\,\mathrm{d}z \bigg|\nonumber\\
&\le \sum_{r=0}^{{w}-2}n^{-r/2}\int_0^\infty \big|\pi_r(\gamma_{x,n}(z))-\pi_r(\gamma_{x-h,n}(z))\big| \, \phi_{\Sigma}(\gamma_{x,n}(z)) \, \big(n^{-1/2}z\big)^{1/2}\,\mathrm{d}z\nonumber\\
&\quad+ \sum_{r=0}^{{w}-2}n^{-r/2}\int_0^\infty \big|\pi_r(\gamma_{x-h,n}(z))\big| \, \big| \phi_{\Sigma}(\gamma_{x,n}(z))-\phi_{\Sigma}(\gamma_{x-h,n}(z)) \big| \, \big(n^{-1/2}z\big)^{1/2}\,\mathrm{d}z\nonumber\\
&\quad+R_n'\int_0^\infty \big(1+\|\gamma_{x,n}(z)\|\big)^{-{w}} \big(n^{-1/2}z\big)^{1/2}+ \big(1+\|\gamma_{x-h,n}(z)\|\big)^{-{w}} \big(n^{-1/2}z\big)^{1/2}\,\mathrm{d}z\nonumber\\
&=: K_1+K_2+K_3.\nonumber
\end{align}
Recall that we are in the case of $|x|<2\sqrt{n}$, $|h|<1$. All following formulas will be deduced uniformly in this regime.
\cref{r.polynome} provides the bound
\[
\big|\pi_r\big(\gamma_{x,n}(z)\big)\big|
\le c(r) n^{3r} \hat{\pi}_{r}(|z|)
\]
for $z\in\mathbb{R}$. Here $\hat{\pi}_{r}$ being a polynomial of degree $3r$ whose coefficients depend on the coefficients of $\pi_r$ and $r$. Due to
\begin{equation*}
\begin{split}
\big|\pi_r\big(\gamma_{x-h,n}(z)\big)\big|
\le c(r) n^{3r/2}(|x|+1)^{3r}\hat{\pi}_{r}(|z|)
\le c(r) n^{3r} \hat{\pi}_{r}(|z|),
\end{split}
\end{equation*}
the bound also holds if we replace $x$ by $x-h$.
Note that
\begin{align*}
\phi_{\Sigma}(\gamma_{x,n}(z))=\phi\big(x(n^{-1/2}z)^{1/2}\big)\,\phi_{\mu_4-1}\big(z-n^{1/2}\big).
\end{align*}
We also need to bound integrals of the form
\begin{align*}
\int_0^\infty \hat{\pi}_{r}(z) \, z \, \phi_{\mu_4-1}\big(z-n^{1/2}\big) \,\mathrm{d}z
&\le \int_{-\infty}^\infty \hat{\pi}_{r}(z) \, |z| \, \phi_{\mu_4-1}\big(z-n^{1/2}\big) \,\mathrm{d}z\\
&\le \mathbb{E}\Big[\hat{\pi}_{r}(W_n) |W_n| \Big],
\end{align*}
where $W_n\sim\mathcal{N}(n^{1/2},\mu_4-1)$ is a normally distributed random variable with mean $n^{1/2}$ and variance $\mu_4-1$. As the expected value is bounded up to a constant by the sum over the first $(3r+1)$ absolute moments of $W_n$,
\begin{equation}\label{int-pi-bound}
\int_0^\infty \hat{\pi}_{r}(z) \, z \, \phi_{\mu_4-1}\big(z-n^{1/2}\big) \,\mathrm{d}z
\le c(r)n^{(3r+1)/2}.
\end{equation}
Now we expand
\begin{align*}
K_1
&=\sum_{r=0}^{{w}-2}n^{-r/2}\int_0^\infty
\phi\big(x(n^{-1/2}z)^{1/2}\big)\,\phi_{\mu_4-1}\big(z-n^{1/2}\big)
\big(n^{-1/2}z\big)^{1/2}\\
&\qquad \cdot \Big| \pi_r\big(x(n^{-1/2}z)^{1/2},z-n^{1/2}\big) - \pi_r\big((x-h)(n^{-1/2}z)^{1/2},z-n^{1/2}\big) \Big|\,\mathrm{d}z.
\end{align*}
The polynomial $\pi_r\big(x(n^{-1/2}z)^{1/2},z-n^{1/2}\big)$ is a part of $\pi_r\big((x-h)(n^{-1/2}z)^{1/2},\allowbreak z-n^{1/2}\big)$ in the sense that every summand of the first polynomial is also a summand of the second polynomial. The remaining summands of $\pi_r\big((x-h)(n^{-1/2}z)^{1/2},z-n^{1/2}\big)$ all feature the factor $h(n^{-1/2}z)^{1/2}$. Therefore, $h(n^{-1/2}z)^{1/2}$ is a common factor of every summand of $\pi_r\big(x(n^{-1/2}z)^{1/2},\allowbreak z-n^{1/2}\big) - \pi_r\big((x-h)(n^{-1/2}z)^{1/2},z-n^{1/2}\big)$ and thus
\begin{align*}
&\big|\pi_r\big(x(n^{-1/2}z)^{1/2},z-n^{1/2}\big) - \pi_r\big((x-h)(n^{-1/2}z)^{1/2},z-n^{1/2}\big)\big|\\
&\quad\le |h|(n^{-1/2}z)^{1/2} \cdot \pi_{r,2}\big((|x|\vee |x-h|\vee 1)(n^{-1/2}|z|)^{1/2},|z-n^{1/2}|\big).
\end{align*}
for a suitable polynomial $\pi_{r,2}$ with non-negative coefficients and degree at most $3r$.
Similar in spirit to \cref{r.polynome}, we bound
\[
\pi_{r,2}\big((|x|\vee |x-h|\vee 1)(n^{-1/2}|z|)^{1/2},|z-n^{1/2}|\big)
\le c(r) n^{3r} \hat{\pi}_{r}(z)
\]
with $\hat{\pi}_{r}$ being a polynomial of degree $3r$ whose coefficients depend on the coefficients of $\pi_r$ and $r$. Using \labelcref{int-pi-bound}, the expression above is bounded by
\begin{align*}
&\sum_{r=0}^{{w}-2}n^{-r/2}\int_0^\infty \phi_{\mu_4-1}\big(z-n^{1/2}\big) \big(n^{-1/2}z\big)^{1/2}
\cdot |h|(n^{-1/2}z)^{1/2} c(r) n^{3r} \widehat{\pi}_{r}(z) \,\mathrm{d}z\\*
&\quad\le |h|c(w) \sum_{r=0}^{{w}-2}n^{(5r-1)/2}\int_0^\infty \phi_{\mu_4-1}\big(z-n^{1/2}\big) \, z \,
\widehat{\pi}_{r}(z) \,\mathrm{d}z\\
&\quad\le|h| c({w}) \sum_{r=0}^{{w}-2}n^{(5r-1)/2}n^{(3r+1)/2}\\
&\quad\le|h| c({w}) n^{4{w}-8}.
\end{align*}
We conclude that
\begin{equation}\label{eq.k1}
K_1 \le |h| c({w}) n^{4{w}-8}.
\end{equation}
For $K_2$, we use that $\phi$ is Lipschitz continuous with Lipschitz constant $L>0$ and get
\begin{align}\label{eq.k2}
K_2
&=\sum_{r=0}^{{w}-2}n^{-r/2}\int_0^\infty
\big|\pi_r(\gamma_{x-h,n}(z))\big|
\big(n^{-1/2}z\big)^{1/2}
\phi_{\mu_4-1}\big(z-n^{1/2}\big)\nonumber\\
&\qquad\qquad\qquad\quad \cdot \Big| \phi\big(x(n^{-1/2}z)^{1/2}\big) - \phi\big((x-h)(n^{-1/2}z)^{1/2}\big) \Big|\,\mathrm{d}z\nonumber\\
&\le\sum_{r=0}^{{w}-2}n^{-r/2}\int_0^\infty
c(r) n^{3r} \hat{\pi}_{r}(z)
\big(n^{-1/2}z\big)^{1/2}
\phi_{\mu_4-1}\big(z-n^{1/2}\big)\\*
&\qquad\qquad\qquad\quad \cdot L |h|(n^{-1/2}z)^{1/2} \,\mathrm{d}z\nonumber\\
&\le|h|c({w})\sum_{r=0}^{{w}-2}n^{(5r-1)/2}\int_0^\infty
\hat{\pi}_{r}(z)
\, z \,
\phi_{\mu_4-1}\big(z-n^{1/2}\big) \,\mathrm{d}z\nonumber\\
&\le|h| c({w}) n^{4{w}-8}.\nonumber
\end{align}
We are left with
\begin{equation*}
K_3
= R_n'\int_0^\infty \big(1+\|\gamma_{x,n}(z)\|\big)^{-{w}} \big(n^{-1/2}z\big)^{1/2}+ \big(1+\|\gamma_{x-h,n}(z)\|\big)^{-{w}} \big(n^{-1/2}z\big)^{1/2}\,\mathrm{d}z
\end{equation*}
where it is obviously impossible to attain a bound in $h$. Therefore, we also have to take the factor $(1+|x|)^m$ from \labelcref{eq.density-Hoelder-1} into account and find a bound in $n$. We only examine the first summand as the second one can be handled in a similar fashion. Thus,
\begin{align*}
&(1+|x|)^m\int_0^\infty \big(1+\|\gamma_{x,n}(z)\|\big)^{-{w}} \big(n^{-1/2}z\big)^{1/2} \,\mathrm{d}z\\*
&\quad\leq c(m) \int_0^\infty \sqrt{(1+x^2)^{m} \big(1+n^{-1/2}zx^2+(z-n^{1/2})^2\big)^{-{w}} \big(n^{-1/2}z\big) }\,\mathrm{d}z\\
&\quad= c(m) \int_0^\infty \sqrt{ \bigg( \frac{1+x^2}{1+n^{-1/2}zx^2+(z-n^{1/2})^2} \bigg)^{m}}\\*
&\quad\qquad\qquad\quad \cdot \sqrt{\big(1+n^{-1/2}zx^2+(z-n^{1/2})^2\big)^{-({w}-m)} \big(n^{-1/2}z\big) }\,\mathrm{d}z.
\end{align*}
Recalling $w\ge m$ and $|x|<2\sqrt{n}$, we split this integral into three regions that we treat separately. First,
\begin{align*}
&\int_0^{\sqrt{n}/2} \sqrt{ \bigg( \frac{1+x^2}{1+n^{-1/2}zx^2+(z-n^{1/2})^2} \bigg)^{m}}\\*
&\qquad\quad\cdot \sqrt{\big(1+n^{-1/2}zx^2+(z-n^{1/2})^2\big)^{-({w}-m)} \big(n^{-1/2}z\big) }\,\mathrm{d}z\\*
&\quad\le \int_0^{\sqrt{n}/2} \sqrt{ \bigg( \frac{1+x^2}{(\frac{1}{2}\sqrt{n}-n^{1/2})^2} \bigg)^{m} \big((\tfrac{1}{2}\sqrt{n}-n^{1/2})^2\big)^{-({w}-m)} \cdot \tfrac12 }\,\mathrm{d}z\\
&\quad\le \int_0^{\sqrt{n}/2} \sqrt{ \bigg( \frac{1+4n}{n/4} \bigg)^{m} \big(n/4\big)^{-({w}-m)} }\,\mathrm{d}z\\
&\quad\le c(w)n^{(m+1-{w})/2}
\end{align*}
and second by \cref{l.int-a/2},
\begin{align*}
&\int_{\frac{1}{2}\sqrt{n}}^{2\sqrt{n}} \sqrt{ \bigg( \frac{1+x^2}{1+n^{-1/2}zx^2+(z-n^{1/2})^2} \bigg)^{m}}\\*
&\qquad\quad\cdot \sqrt{\big(1+n^{-1/2}zx^2+(z-n^{1/2})^2\big)^{-({w}-m)} \big(n^{-1/2}z\big) }\,\mathrm{d}z\\*
&\quad\le \int_{\frac{1}{2}\sqrt{n}}^{2\sqrt{n}} \sqrt{ \bigg( \frac{1+x^2}{1+x^2/2} \bigg)^{m} \big(1+(z-n^{1/2})^2\big)^{-({w}-m)} \cdot 2 }\,\mathrm{d}z\\
&\quad\le c(m) \int_{\frac{1}{2}\sqrt{n}}^{2\sqrt{n}} \big(1+(z-n^{1/2})^2\big)^{-({w}-m)/2} \,\mathrm{d}z\\
&\quad\le \begin{cases}
c(w) \sqrt{n} , &\text{ if } {w}=m,\\
c(w) \log n , &\text{ if } {w}=m+1.
\end{cases}
\end{align*}
Finally,
\begin{align*}
&\int_{2\sqrt{n}}^\infty \sqrt{ \bigg( \frac{1+x^2}{1+n^{-1/2}zx^2+(z-n^{1/2})^2} \bigg)^{m}}\\* &\qquad\quad\cdot\sqrt{\big(1+n^{-1/2}zx^2+(z-n^{1/2})^2\big)^{-({w}-m)} \big(n^{-1/2}z\big) }\,\mathrm{d}z\\*
&\quad\le \int_{2\sqrt{n}}^\infty \sqrt{ (1+4n)^{m} \big((z-n^{1/2})^2\big)^{-{w}} \big(n^{-1/2}z\big) }\,\mathrm{d}z\\
&\quad\le c(m) n^{m/2} n^{-1/4} \int_{2\sqrt{n}}^\infty \sqrt{ \big((z-z/2)^2\big)^{-{w}} \cdot z}\,\mathrm{d}z\\
&\quad\le c(m) n^{m/2} n^{-1/4} \int_{2\sqrt{n}}^\infty z^{-(2{w}-1)/2} \,\mathrm{d}z\\
&\quad\le c(w) n^{(m+1-{w})/2}.
\end{align*}
This also holds if $x$ is replaced by $x-h$ and thus combining the last four equations and \labelcref{eq.def-Rn'}
\begin{align}\label{eq.k3}
(1+|x|)^m K_3
&= (1+|x|)^m R_n' \int_0^\infty \big(1+\|\gamma_{x,n}(z)\|\big)^{-{w}} \big(n^{-1/2}z\big)^{1/2}\nonumber\\*
&\qquad\qquad\qquad\qquad+ \big(1+\|\gamma_{x-h,n}(z)\|\big)^{-{w}} \big(n^{-1/2}z\big)^{1/2}\,\mathrm{d}z\nonumber\\
&\le R_n' \begin{cases}
c(w) \sqrt{n} , &\text{ if } {w}=m,\\
c(w) \log n , &\text{ if } {w}=m+1,
\end{cases}\\&
\le \begin{cases}
\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m-3)/2}\big), &\text{ if } {w}=m,\nonumber\\
\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m-1)/2} \, \log n \big), &\text{ if } {w}=m+1,
\end{cases}\nonumber\\*
&=:r_n\nonumber
\end{align}
uniformly for all $|x|<2\sqrt{n}$, $|h|<1$. In summary, we get by \labelcref{eq.f_n-f_n-Z,eq.k1,eq.k2,eq.k3},
\begin{equation*}
(1+|x|)^{m}|f_n(x)-f_n(x-h)|
\le c(w) (1+|x|)^{m} |h| n^{4{w}-8} + r_n.
\end{equation*}
Hence, setting $a=1$ and $b=4{w}-8$, the H\"older-type condition $(iii)$ in \cref{p.dichte-general} is satisfied.
It remains to derive a uniform bound on $f_n$. For this aim we subsequently use the identity for the normal density
\begin{align*}
\phi_{\sigma^2}(x)=\phi_{\sigma^2/2}(x)\,\phi_{\sigma^2/2}(x) \sqrt{2/(\pi\sigma^2)}
\end{align*}
for $x\in\mathbb{R}$ and $\sigma^2>0$.
By \labelcref{eq.k3,eq.f_n-von-g_n}, it suffices to bound
\begin{align*}
\lefteqn{\int_0^\infty \sum_{r=0}^{{w}-2}n^{-r/2}\pi_r(\gamma_{x,n}(z))\phi_{\Sigma}(\gamma_{x,n}(z)) \big(n^{-1/2}z\big)^{1/2} \,\mathrm{d}z}\quad\\*
&= \sqrt{2/(\pi(\mu_4-1))} \int_0^\infty \sum_{r=0}^{{w}-2}n^{-r/2}\pi_r(\gamma_{x,n}(z)) \, \phi\big(x(n^{-1/2}z)^{1/2}\big) \, \phi_{(\mu_4-1)/2}\big(z-n^{1/2}\big) \\*
&\qquad \qquad \qquad \qquad \qquad \qquad \cdot \phi_{(\mu_4-1)/2}\big(z-n^{1/2}\big) \, \big(n^{-1/2}z\big)^{1/2} \,\mathrm{d}z\\
&= \sqrt{2/(\pi(\mu_4-1))} \int_0^\infty \sum_{r=0}^{{w}-2}n^{-r/2}\pi_r(\gamma_{x,n}(z)) \phi_{\Sigma_2}(\gamma_{x,n}(z)) \\*
&\qquad \qquad \qquad \qquad \qquad \qquad \cdot \phi_{(\mu_4-1)/2}\big(z-n^{1/2}\big) \, \big(n^{-1/2}z\big)^{1/2} \,\mathrm{d}z\\
&\le c(w) n^{-1/4}\int_0^\infty \phi_{(\mu_4-1)/2}\big(z-n^{1/2}\big) \, z^{1/2} \,\mathrm{d}z
\end{align*}
where
\begin{align*}
\Sigma_2=
\begin{pmatrix}
1\enskip & 0\\
0\enskip & (\mu_4-1)/2
\end{pmatrix}.
\end{align*}
The integral is divided into two parts that are evaluated separately. First,
\begin{align*}
c(w) n^{-1/4}\int_0^{2n^{1/2}} \phi_{(\mu_4-1)/2}\big(z-n^{1/2}\big) \, z^{1/2} \,\mathrm{d}z
&\le c(w) \int_0^{2n^{1/2}} \phi_{(\mu_4-1)/2}\big(z-n^{1/2}\big) \,\mathrm{d}z\\
&\le c(w)
\end{align*}
and second,
\begin{align*}
\lefteqn{c(w) n^{-1/4}\int_{2n^{1/2}}^\infty \phi_{(\mu_4-1)/2}\big(z-n^{1/2}\big) \, z^{1/2} \,\mathrm{d}z}\quad\\
&= c(w) n^{-1/4}\int_{n^{1/2}}^\infty \phi_{(\mu_4-1)/2}(z) \, \big(z+n^{1/2}\big)^{1/2} \,\mathrm{d}z\\
&\le c(w) n^{-1/4}\int_{n^{1/2}}^\infty \phi_{(\mu_4-1)/2}(z) \, z^{2} \,\mathrm{d}z \le c(w).
\end{align*}
Thus, $f_n$ is bounded uniformly over all $n\ge N_1$ and \cref{p.dichte-general} yields
\begin{equation*}
\sup_{x\in\mathbb{R}} \, (1+|x|)^{m}|f_n(x)-\phi^{q}_{m,n}(x)|
= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-({s}-2)/2} \, (\log n)^{({s}+2\lfloor {s} \rfloor)/2} \big)
+ \mathcal{O}\big( n^{- \lceil \frac{m-1}2 \rceil} \big)
+ r_n
\end{equation*}
for both $m =w-1$ and $m=w$, and for all ${s}$ with $m \le {s} \le 2{w}$. We choose ${s}=m+2$ to finally get
\begin{align*}
\sup_{x\in\mathbb{R}} \, (1+|x|)^{m}|f_n(x)-\phi^{q}_{m,n}(x)|
&= \begin{cases}
\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m-3)/2}\big), &\text{ if } {w}=m,\\
\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m-1)/2} \, \log n \big), &\text{ if } {w}=m+1.
\end{cases}
\end{align*} \end{proof}
\section{Rate of convergence and Edgeworth-type expansion in the entropic central limit theorem for self-normalized sums}\label{ch.entropy}
The main result of this section is the following theorem.
\begin{theorem}\label{t.entropy-general}
Assume that $X_1$ is symmetric, the distribution of $X_1$ is non-singular, \cref{c.density-cf} is satisfied and $\mathbb{E}|X_1|^{2m}<\infty$ for some $m\in\mathbb{N}$, $m\ge3$. Then
\begin{equation}\label{eq.t-entropy-general}
D(T_n)=\frac{c_2}{n^2}+\dots+\frac{c_{\lfloor (m-2)/2 \rfloor}}{n^{\lfloor (m-2)/2 \rfloor}}+ \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big((n\log n)^{-(m-2)/2}\,\log n\big),
\end{equation}
where
\begin{align*}
c_l=\sum_{k=2}^{2l} \frac{(-1)^k}{k(k-1)} \sum \int_{\mathbb{R}} \frac{q_{r_1}(x)\dots q_{r_k}(x)}{\phi(x)^{k-1}} \,\mathrm{d}x, \quad l\in\mathbb{N},
\end{align*}
with the inner sum running over all positive integers $r_1,\dots,r_k$ such that $r_1+\dots+r_k=2l$. \end{theorem}
Together with \cref{p.prop}, \cref{t.entropy-general} implies \cref{t.entropy}. Before starting with the proof of \cref{t.entropy-general}, let us record the following observation.
\begin{remark}\label{r.E-T_n-normalized}
The classical statistic $Z_n$ is normalized in the sense that $\mathbb{E} Z_n=0$ and $\ensuremath{\mathop{\mathrm{Var}}} Z_n=1$. The $t$-statistic is not normalized (see \cite[p. 72f.]{Hal92edgeworth}). In general, the self-normalized sum is also not normalized (see \cref{r.E-T_n^2}). If $X_1$ is symmetric however, it is normalized because
\begin{equation*}
\mathbb{E} T_n
=\mathbb{E} \Big[\tilde{\mathbb{E}} \big[ S_n/V_n \big]\Big]
=\mathbb{E} \Big[V_n^{-1}\sum \tilde{\mathbb{E}} \big[ X_j \big] \Big]
=0
\end{equation*}
and
\begin{equation*}
\mathbb{E} T_n^2
=\mathbb{E} \Big[\tilde{\mathbb{E}} \big[ S_n^2/V_n^2 \big]\Big]
=\mathbb{E} \Big[V_n^{-2}\Big(\sum \tilde{\mathbb{E}} \big[ X_j^2 \big] + 2\sum_{j=1}^{n-1}\sum_{k=j+1}^{n} \tilde{\mathbb{E}} \big[ X_jX_k \big]\Big) \Big]
=1.
\end{equation*} \end{remark}
In the following proof, we adapt some ideas from \cite{BCG13}.
\begin{proof}[Proof of \cref{t.entropy-general}]
By \cref{t.llt}, the densities $f_n$ exist and are uniformly bounded by some constant $M$ for $n\ge N$. Throughout the remainder of the proof assume $n\ge N$.
Since $X_1$ is symmetric, $T_n$ is normalized (see \cref{r.E-T_n-normalized}). Therefore, its relative entropy, defined in \labelcref{eq.def-rel-entropy}, includes the standard normal density $\phi_{0,1}=\phi$.
Set $\Delta_{m,n}=(n\log n)^{-(m-2)/2}\,\log n$.
Let $A_n\ge1$, then we split up the integral from the definition
\begin{align}\label{eq.D-ints}
D(T_n)
= \int_{|x|\le A_n} f_n(x)\log \frac{f_n(x)}{\phi(x)}\,\mathrm{d}x
+ \int_{|x|>A_n} f_n(x)\log \frac{f_n(x)}{\phi(x)}\,\mathrm{d}x.
\end{align}
For the second integral, we get the upper bound
\begin{align*}
\int_{|x|>A_n} f_n(x)\log \frac{f_n(x)}{\phi(x)}\,\mathrm{d}x
&\le\int_{|x|>A_n} f_n(x)\log \frac{M}{\phi(x)}\,\mathrm{d}x\\
&=\log \big(M\sqrt{2\pi}\big)\int_{|x|>A_n} f_n(x)\,\mathrm{d}x+\tfrac12 \int_{|x|>A_n} x^2 f_n(x)\,\mathrm{d}x\\
&\leq\Big(\log \big(M\sqrt{2\pi}\big)A_n^{-2}+\tfrac12\Big) \int_{|x|>A_n} x^2 f_n(x)\,\mathrm{d}x.
\end{align*}
By means of the inequality $u\log u\ge u-1$ for $u>0$, we get the lower bound
\begin{align*}
\int_{|x|>A_n} f_n(x)\log \frac{f_n(x)}{\phi(x)}\,\mathrm{d}x
&=\int_{|x|>A_n} \phi(x) \frac{f_n(x)}{\phi(x)} \log \frac{f_n(x)}{\phi(x)}\,\mathrm{d}x\\
&\ge\int_{|x|>A_n} \phi(x) \Big(\frac{f_n(x)}{\phi(x)} -1\Big)\,\mathrm{d}x\\
&=\int_{|x|>A_n} \big( f_n(x)-\phi(x) \big) \,\mathrm{d}x\\
&\ge - \mathbb{P}\{|Z|>A_n\}.
\end{align*}
Therefore,
\begin{align}\label{eq.entropy>A_n}
\bigg|\int_{|x|>A_n} f_n(x)\log \frac{f_n(x)}{\phi(x)}\,\mathrm{d}x\bigg|
\le C \int_{|x|>A_n} x^2 f_n(x)\,\mathrm{d}x
+\mathbb{P}\{|Z|>A_n\},
\end{align}
where $C=\log \big(M\sqrt{2\pi}\big)+1/2$. Before further evaluation of this term, we specify $A_n$ with the intent to balance both integrals in \labelcref{eq.D-ints}.
Set
\begin{align*}
A_n=\sqrt{(m-2)\log n + (m-3)\log\log n + \rho_n},
\end{align*}
where $\rho_n\to \infty$ sufficiently slow and $\rho_n\le\log n$. $(\rho_n)$ is needed to achieve the order $\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}$ and not only $\mathcal{O}$ in the remainder in \labelcref{eq.t-entropy-general}. It will be defined at a later stage, depending on the remainder in \cref{t.llt-red}. For any $k\ge0$, the bound
\begin{equation}\label{eq.A_n-exp}
\begin{split}
A_n^k \, e^{-A_n^{2}/2}
&= \big((m-2)\log n + (m-3)\log\log n + \rho_n\big)^{k/2}\\*
&\quad \cdot n^{-(m-2)/2} (\log n)^{-(m-3)/2} \, e^{-\rho_n/2}\\
&=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m-2)/2} (\log n)^{-(m-3-k)/2}\big)
\end{split}
\end{equation}
is frequently needed.
Recall that $|T_n|\le\sqrt{n}$ by \cref{r.T_n}. We return to \labelcref{eq.entropy>A_n} and derive by partial integration for Stieltjes integrals (see e.g. \cite[Theorem 21.67 (iv)]{HS-Analysis}) using the symmetry of $T_n$,
\begin{align}\label{eq.entropy>A_n-expanded}
\int_{|x|>A_n} x^2 f_n(x)\,\mathrm{d}x
&= 2\int_{A_n}^{\sqrt{n}} x^2 \,\mathrm{d} F_n(x)\nonumber\\
&= 2\Big(\sqrt{n}^2 (F_n(\sqrt{n} \, ))-A_n^2 (F_n(A_n))-\int_{A_n}^{\sqrt{n}} F_n(x) 2x \,\mathrm{d}x \Big)\\
&= 2A_n^2 \big(1-F_n(A_n)\big)+4\int_{A_n}^{\infty} x \big( 1-F_n(x)\big)\,\mathrm{d}x\nonumber
\end{align}
By \cref{t.clt} (with $s=m+1$),
\begin{equation*}
F_n(x) = \Phi^{Q}_{m+1,n}(x) + (1+|x|)^{-(m+1)} R_n(x)
\end{equation*}
for all $x\in\mathbb{R}$ and $n$ sufficiently large. Here, $\sup_{x} |R_n(x)|=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-(m-1)/2} \, (\log n)^{(3m+3)/2} \big)$.
Therefore, we can replace $A_n^2\big(1-F_n(A_n)\big)$ with
\begin{equation*}
A_n^2\big(1-\Phi^{Q}_{m+1,n}(A_n)\big)
\end{equation*}
with an error of magnitude
\begin{align*}
A_n^2(1+A_n)^{-(m+1)}R_n(A_n)
=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-(m-1)/2} \, (\log n)^{(2m+4)/2} \big).
\end{align*}
Recall the definition
\begin{align*}
\Phi^{Q}_{m+1,n}(x)=\Phi(x)+\sum_{r=1}^{m-1}Q_{r}(x)n^{-r/2}
\end{align*}
from \labelcref{eq.Phi^Q-def}. For uneven $r$, $Q_r$ vanishes and for even $r$, $Q_{r}$ has the useful form of $\phi$ multiplied with a polynomial of degree $2r-1$. The coefficients of $Q_{r}$ are functions of the moments $\mu_3,\dots,\mu_{r+2}$ and can thus be bounded by $c(r)$. Now by \labelcref{eq.A_n-exp},
\begin{align*}
\Big|A_n^2\sum_{r=1}^{m-1}Q_{r}(A_n)n^{-r/2}\Big|
\le A_n^2 \, c(m) \, A_n^{2m-3} \phi(A_n) n^{-1/2}
=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m-1)/2} (\log n)^{(m+2)/2}\big).
\end{align*}
For the last part, we use $1-\Phi(x)\le \phi(x)/x$ (for $x>0$) and \labelcref{eq.A_n-exp} to achieve
\begin{align*}
A_n^2\big(1-\Phi(A_n)\big)
\le A_n \phi(A_n)
=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}(\Delta_{m,n})
\end{align*}
and therefore
\begin{align*}
A_n^2\big(1-F_n(A_n)\big)
=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}(\Delta_{m,n}).
\end{align*}
Similarly, in the second part of \labelcref{eq.entropy>A_n-expanded}, we can replace $\int_{A_n}^{\infty} x \big( 1-F_n(x)\big)\,\mathrm{d}x$ with
\begin{align*}
\int_{A_n}^{\infty} x \big( 1-\Phi^{Q}_{m+1,n}(x)\big)\,\mathrm{d}x
\end{align*}
with an error of at most
\begin{align*}
\int_{A_n}^{\infty} x (1+|x|)^{-(m+1)} R_n(x)\,\mathrm{d}x
=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-(m-1)/2} \, (\log n)^{(2m+4)/2} \big).
\end{align*}
As above, using $\alpha=\frac{1}{2(2m-2)}$, \cref{l.int-t-exp} (with $\beta=1$, $\nu_n=n^\alpha$) and \labelcref{eq.A_n-exp},
\begin{align}\label{eq.int-Q}
\lefteqn{\Big|\int_{A_n}^{\infty} x\sum_{r=1}^{m-1}Q_{r}(x)n^{-r/2} \,\mathrm{d}x\Big|}\quad\nonumber\\*
&\leq c(m) n^{-1/2} \Big(\int_{A_n}^{n^\alpha} x^{2m-2} \phi(x) \,\mathrm{d}x + \int_{n^\alpha}^{\infty} x^{2m-2} \phi(x) \,\mathrm{d}x \Big)\nonumber\\
&\le c(m) \int_{A_n}^{n^\alpha} \phi(x) \,\mathrm{d}x + c(m) \exp\big(-\tfrac14 n^{2\alpha}\big)\\
&\le c(m) \phi(A_n) + c(m) \exp\big(-\tfrac14 n^{2\alpha}\big)\nonumber\\
&=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m-2)/2} (\log n)^{-(m-3)/2}\big)\nonumber
\end{align}
and by \labelcref{eq.A_n-exp},
\begin{align*}
\int_{A_n}^{\infty} x \big( 1-\Phi(x)\big)\,\mathrm{d}x
&\le \int_{A_n}^{\infty} \phi(x) \,\mathrm{d}x\le \phi(A_n)
=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m-2)/2} (\log n)^{-(m-3)/2}\big)
\end{align*}
and therefore
\begin{align*}
\int_{A_n}^{\infty} x \big( 1-F_n(x)\big)\,\mathrm{d}x
=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m-2)/2} (\log n)^{-(m-3)/2}\big).
\end{align*}
Hence by \labelcref{eq.entropy>A_n-expanded},
\begin{align}\label{eq.int-x^2f_n}
\int_{|x|>A_n} x^2 f_n(x)\,\mathrm{d}x
=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}(\Delta_{m,n})
\end{align}
and by inserting
\begin{align}\label{eq.P-Z>A_n}
\mathbb{P}\{|Z|>A_n\}
=2\big(1-\Phi(A_n)\big)
\le 2 \phi(A_n)
=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m-2)/2} (\log n)^{-(m-3)/2}\big)
\end{align}
into \labelcref{eq.entropy>A_n}, we obtain
\begin{align*}
\bigg|\int_{|x|>A_n} f_n(x)\log \frac{f_n(x)}{\phi(x)}\,\mathrm{d}x\bigg|
=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}(\Delta_{m,n}).
\end{align*}
In order to evaluate $D(T_n)$, we thus only have to examine (with $L(u):=u\log(u)$)
\begin{equation}\label{eq.entropy<A_n}
\begin{split}
\int_{|x|\le A_n} f_n(x)\log \frac{f_n(x)}{\phi(x)}\,\mathrm{d}x
&= \int_{|x|\le A_n} L\Big(\frac{f_n(x)}{\phi(x)}\Big)\phi(x) \,\mathrm{d}x\\
&= \int_{|x|\le A_n} L\big(1+u_m(x)+v_n(x)\big)\phi(x) \,\mathrm{d}x,
\end{split}
\end{equation}
where
\begin{align*}
u_m(x)=\frac{\phi^{q}_{m-1,n}(x)-\phi(x)}{\phi(x)}
\qquad \text{and}\qquad
v_n(x)=\frac{f_n(x)-\phi^{q}_{m-1,n}(x)}{\phi(x)}.
\end{align*}
By \cref{t.llt-red},
\begin{equation}\label{eq.entropy-llt}
\begin{split}
\big| f_n(x) - \phi^{q}_{m-1,n}(x)\big|
&\le (1+|x|)^{-(m-1)} R_n(x) \\
&\le (1+|x|)^{-(m-1)} n^{-(m-2)/2} \log(n) r_n
\end{split}
\end{equation}
for all $x\in\mathbb{R}$ and $n$ sufficiently large. Here, $r_n=n^{(m-2)/2} (\log n)^{-1} \, \sup_{x} |R_n(x)| =\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}(1)$.
There exists $C_m>0$, such that $y\mapsto (1+y)^{m-1}\phi(y)$ is decreasing for $y\ge C_m$ and thus its inverse is increasing. Additionally, for $|x|\le C_m$, we bound
\[
(1+|x|)^{-(m-1)}\phi(x)^{-1}\le c(m).
\]
Therefore, we estimate
\[
(1+|x|)^{-(m-1)}\phi(x)^{-1}\le (1+A_n)^{-(m-1)}\phi(A_n)^{-1}
\]
for all $|x|\le A_n$ and for $n$ sufficiently large. Hence,
\begin{align}\label{eq.vn-bound}
|v_n(x)|
&\le (1+|x|)^{-(m-1)} \phi(x)^{-1} n^{-(m-2)/2} \log(n) r_n\nonumber\\
&\le c(m) A_n^{-(m-1)} e^{A_n^2/2} \, n^{-(m-2)/2} \log(n) r_n\nonumber\\
&= c(m) \big((m-2)\log n + (m-3)\log\log n + \rho_n\big)^{-(m-1)/2} \big(\log n\big)^{(m-3)/2} \\
&\quad \cdot e^{\rho_n/2} \log(n) r_n\nonumber\\
&\le c(m) e^{\rho_n/2} r_n.\nonumber
\end{align}
Next, we choose the sequence $(\rho_n)$ in a way that the last expression goes to 0. Thus, ${|v_n(x)|<1/4}$ holds for $|x|\le A_n$ and $n$ sufficiently large.
Recall $\phi^{q}_{m-1,n}(x)=\phi(x)+\sum_{r=1}^{m-3}q_{r}(x)n^{-r/2}$ from \labelcref{eq.phi^q-def}. For even $r$, $q_{r}$ factorizes as $\phi$ multiplied with an even polynomial of degree $2r$. The coefficients of $q_{r}$ are functions of the moments $\mu_3,\dots,\mu_{r+2}$ and can thus be bounded by $c(r)$. As above, $q_{r}=0$ for uneven $r\in\mathbb{N}$. So
\begin{equation}\label{eq.u_m}
\begin{split}
|u_m(x)|
=\frac{\big|\phi^{q}_{m-1,n}(x)-\phi(x)\big|}{\phi(x)}
\le c(m)\big(1+|x|^{2m-6}\big)n^{-1/2}
\le c(m)A_n^{2m-6} n^{-1/2},
\end{split}
\end{equation}
where the last inequality only holds for $|x|\le A_n$. In particular $|u_m(x)|<1/4$ holds for $|x|\le A_n$ and $n$ sufficiently large.
Next, by Lemma \ref{Taylor_L},
\begin{equation*}
L(1+u+v)=L(1+u)+v+\vartheta_1 u v+\vartheta_2 v^2
\end{equation*}
for $|u|\le 1/4, |v|\le1/4$ and $|\vartheta_j|\le2$ depending on $u$ and $v$.
Inserting $u=u_m(x)$ and $v=v_n(x)$, shows how to eliminate $v_n(x)$ from \labelcref{eq.entropy<A_n}. This produces an error not exceeding
\[
|D_1|+2D_2+2D_3,
\]
where
\begin{align*}
D_1
&=\int_{|x|\le A_n}\big(f_n(x)-\phi^{q}_{m-1,n}(x)\big)\,\mathrm{d}x,\\
D_2
&=\int_{|x|\le A_n}\big|u_m(x)\big|\big|f_n(x)-\phi^{q}_{m-1,n}(x)\big|\,\mathrm{d}x \\
\intertext{and}
D_3
&=\int_{|x|\le A_n}\frac{\big(f_n(x)-\phi^{q}_{m-1,n}(x)\big)^2}{\phi(x)}\,\mathrm{d}x.
\end{align*}
For even $r\in\mathbb{N}$, the function $q_{r}$ factorizes as $\phi$ multiplied with a sum of Hermite polynomials $H_k$ for $k>0$ (see \labelcref{eq.tp,eq.E-tp-q,eq.q-2}). Now
\begin{align*}
\int_\mathbb{R} \phi(x)H_k(x)\,\mathrm{d}x=\int_\mathbb{R} \phi(x)H_k(x)H_0(x)\,\mathrm{d}x=0
\end{align*}
for all $k>0$ yields
\begin{align}\label{eq.int-q=0}
\int_\mathbb{R} q_{r}(x)\,\mathrm{d}x=0
\end{align}
for all even $r=1,\dots,m-3$. As $q_{r}=0$ for uneven $r\in\mathbb{N}$,
\begin{align*}
\int_\mathbb{R}\ \big(f_n(x)-\phi^{q}_{m-1,n}(x)\big)\,\mathrm{d}x=0.
\end{align*}
Thus,
\begin{equation*}
|D_1|
=\Big|\int_{|x|>A_n}\big(f_n(x)-\phi^{q}_{m-1,n}(x)\big)\,\mathrm{d}x\Big|
\le \int_{|x|>A_n}f_n(x)\,\mathrm{d}x + \int_{|x|>A_n}\big|\phi^{q}_{m-1,n}(x)\big|\,\mathrm{d}x,
\end{equation*}
where the first integral is of order $\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}(\Delta_{m,n})$ by \labelcref{eq.int-x^2f_n}. Using \labelcref{eq.u_m} and \labelcref{eq.P-Z>A_n}, the second integral is bounded by
\begin{equation}\label{eq.int1}
\begin{split}
&\int_{|x|>A_n}\big|\phi^{q}_{m-1,n}(x)-\phi(x)\big|\,\mathrm{d}x +\int_{|x|>A_n}\phi(x)\,\mathrm{d}x\\
&\quad \le c(m)\int_{|x|>A_n}|x|^{2m-6} n^{-1/2} \phi(x)\,\mathrm{d}x +\mathbb{P}\{|Z|>A_n\}\\
&\quad =\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m-2)/2} (\log n)^{-(m-3)/2}\big).
\end{split}
\end{equation}
For $m>3$, we have used \labelcref{eq.int-Q} with $\alpha=\frac{1}{2(2m-6)}$ and for $m=3$ the order is true because in that case the order of the first term is smaller than the second one.
Therefore, ${|D_1|=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}(\Delta_{m,n})}$. By \labelcref{eq.u_m} and \labelcref{eq.entropy-llt},
\begin{align*}
D_2
&\le c(m) \int_{|x|\le A_n} A_n^{2m-6} n^{-1/2} (1+|x|)^{-(m-1)} n^{-(m-2)/2} \log(n) r_n \,\mathrm{d}x\\
&\le c(m) A_n^{2m-5} n^{-(m-1)/2} \log(n) r_n\\
&=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}(n^{-(m-1)/2} (\log n)^{(2m-3)/2}).
\end{align*}
Similarly as in \labelcref{eq.vn-bound}, we obtain for all $|x|\le A_n$
\begin{align*}
D_3
&\le \int_{|x|\le A_n} (1+|x|)^{-2(m-1)} \phi(x)^{-1} n^{-(m-2)} (\log n)^2 r_n^2 \,\mathrm{d}x\\
&\le c(m) A_n^{-(2m-3)} \phi(A_n)^{-1} n^{-(m-2)} (\log n)^2 r_n^2 \\
&= c(m) \big((m-2)\log n + (m-3)\log\log n + \rho_n\big)^{-(2m-3)/2} \big(\log n\big)^{(m-3)/2} \, e^{\rho_n/2} \\
&\quad \cdot n^{-(m-2)/2} (\log n)^2 r_n^2 \\
&\le c(m) (\log n)^{-(m-4)/2} n^{-(m-2)/2} r_n\\
&=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}(\Delta_{m,n}).
\end{align*}
Thus, eliminating $v_n(x)$ from \labelcref{eq.entropy<A_n} leads to
\begin{equation*}
D(T_n)= \int_{|x|\le A_n} L\big(1+u_m(x)\big)\phi(x) \,\mathrm{d}x + \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}(\Delta_{m,n}).
\end{equation*}
For $m=3$, $u_m(x)=0$ and \labelcref{eq.t-entropy-general} holds. Hence, assume $m\ge4$ from now on. By Taylor expansion around $u=0$,
\begin{align*}
L(1+u)
&= u + \sum_{k=2}^{m-2} \frac{(-1)^k}{k(k-1)}u^k+\vartheta u^{m-1}
\end{align*}
for some $\vartheta \ge 0$, depending on $u$ and $m$ that can be bounded by $c(m)$ for all $|u|\le1/4$.
Inserting $u_m$ yields
\begin{align*}
\lefteqn{\int_{|x|\le A_n} L\big(1+u_m(x)\big)\phi(x) \,\mathrm{d}x}\quad\\
&\le\int_{|x|\le A_n} \phi^{q}_{m-1,n}(x)-\phi(x) \,\mathrm{d}x
+ \sum_{k=2}^{m-2} \frac{(-1)^k}{k(k-1)} \int_{|x|\le A_n} u_m(x)^k\phi(x) \,\mathrm{d}x \\
&\quad+ c(m) \int_{|x|\le A_n} \big|u_m(x)\big|^{m-1} \phi(x) \,\mathrm{d}x\\
&=:E_1+E_2+E_3
\end{align*}
for $n$ large enough. By \labelcref{eq.int-q=0} and \labelcref{eq.int1},
\begin{align*}
\lefteqn{|E_1|=\bigg| \int_{|x|\le A_n} \phi^{q}_{m-1,n}(x)-\phi(x) \,\mathrm{d}x \bigg|
= \bigg| \int_{|x|> A_n} \phi^{q}_{m-1,n}(x)-\phi(x) \,\mathrm{d}x \bigg|}\quad\\*
&\le \int_{|x|> A_n} \big|\phi^{q}_{m-1,n}(x)-\phi(x)\big| \,\mathrm{d}x
=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m-2)/2} (\log n)^{-(m-3)/2}\big)
\end{align*}
and by \labelcref{eq.u_m}
\begin{align*}
E_3
&\le \int_\mathbb{R} \big|u_m(x)\big|^{m-1} \phi(x) \,\mathrm{d}x\\
&\le c(m) n^{-(m-1)/2}\int_\mathbb{R} \big(1+|x|^{2m-6}\big)^{m-1} \phi(x) \,\mathrm{d}x=\mathcal{O}\big(n^{-(m-1)/2}\big).
\end{align*}
By \labelcref{eq.u_m} and \labelcref{eq.int-Q} with $\alpha=\frac{1}{2(2m-6)}$, the integral in $E_2$ can be extended to the whole real line at the expense of an error not exceeding
\begin{align*}
\lefteqn{\sum_{k=2}^{m-2} \frac{(-1)^k}{k(k-1)} \int_{|x|> A_n} u_m(x)^k\phi(x) \,\mathrm{d}x}\quad\\
&\le c(m) \sum_{k=2}^{m-2} \frac{(-1)^k}{k(k-1)} n^{-k/2} \int_{|x|> A_n} \big(1+|x|^{2m-6}\big)^k \phi(x) \,\mathrm{d}x\\
&=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m-1)/2} (\log n)^{-(m-3)/2}\big).
\end{align*}
In summary,
\begin{align*}
D(T_n)
&= \int_{|x|\le A_n} L\big(1+u_m(x)\big)\phi(x) \,\mathrm{d}x + \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}(\Delta_{m,n})\\*
&= \sum_{k=2}^{m-2} \frac{(-1)^k}{k(k-1)} \int_{\mathbb{R}} \frac{\big(\phi^{q}_{m-1,n}(x)-\phi(x)\big)^k}{\phi(x)^{k-1}} \,\mathrm{d}x + \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}(\Delta_{m,n}).
\end{align*}
Recalling $\phi^{q}_{m-1,n}(x)-\phi(x)=\sum_{r=1}^{m-3}q_{r}(x)n^{-r/2}$, we get
\begin{align*}
\big(\phi^{q}_{m-1,n}(x)-\phi(x)\big)^k
=\sum_{l=1}^{k(m-3)} n^{-l/2} \sum q_{r_1}(x)\dots q_{r_k}(x)
\end{align*}
where the inner sum runs over all positive integers $r_1,\dots,r_k\le m-3$ with $r_1+\dots+r_k=l$. So
\begin{align*}
D(T_n)= \sum_{k=2}^{m-2} \frac{(-1)^k}{k(k-1)} \sum_{l=1}^{k(m-3)} n^{-l/2} \sum \int_{\mathbb{R}} \frac{q_{r_1}(x)\dots q_{r_k}(x)}{\phi(x)^{k-1}} \,\mathrm{d}x + \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}(\Delta_{m,n}).
\end{align*}
Here, all summands with uneven $l=r_1+\dots+r_k$ vanish as $q_{r}=0$ for uneven $r$.
Additionally, all summands with $l>m-2$ will be absorbed by the remainder $\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}(\Delta_{m,n})$ which yields
\begin{equation}\label{eq.D(Tn)-finish}
\begin{split}
D(T_n)
&= \sum_{k=2}^{m-2} \frac{(-1)^k}{k(k-1)} \sum_{l=1,~l \text{ even}}^{m-2} n^{-l/2} \sum \int_{\mathbb{R}} \frac{q_{r_1}(x)\dots q_{r_k}(x)}{\phi(x)^{k-1}} \,\mathrm{d}x + \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}(\Delta_{m,n})\\*
&= \sum_{k=2}^{m-2} \frac{(-1)^k}{k(k-1)} \sum_{l=1}^{\lfloor (m-2)/2 \rfloor} n^{-l} \sum \int_{\mathbb{R}} \frac{q_{r_1}(x)\dots q_{r_k}(x)}{\phi(x)^{k-1}} \,\mathrm{d}x + \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}(\Delta_{m,n})
\end{split}
\end{equation}
where the inner sum runs over all positive integers $r_1,\dots,r_k\le m-3$ such that $r_1+\dots+r_k=l$ in the first line and $r_1+\dots+r_k=2l$ in the second line. Define
\begin{align*}
c_l
&=\sum_{k=2}^{m-2} \frac{(-1)^k}{k(k-1)} \sum \int_{\mathbb{R}} \frac{q_{r_1}(x)\dots q_{r_k}(x)}{\phi(x)^{k-1}} \,\mathrm{d}x\\*
&=\sum_{k=2}^{2l} \frac{(-1)^k}{k(k-1)} \sum \int_{\mathbb{R}} \frac{q_{r_1}(x)\dots q_{r_k}(x)}{\phi(x)^{k-1}} \,\mathrm{d}x.
\end{align*}
Here, the inner sum in the first line runs over all positive integers $r_1,\dots,r_k\le m-3$ such that $r_1+\dots+r_k=2l$ and the the inner sum in the second line runs over all positive integers $r_1,\dots,r_k$ such that $r_1+\dots+r_k=2l$. The fact $2\le k\le 2l\le m-2$ implies the second identity.
Note that
\begin{equation*}
c_1
= \frac{(-1)^2}{2} \sum_{r_1+r_2=2} \int_{\mathbb{R}} \frac{q_{r_1}(x) q_{r_2}(x)}{\phi(x)} \,\mathrm{d}x
= \frac{1}{2} \int_{\mathbb{R}} \frac{q_1(x) q_1(x)}{\phi(x)} \,\mathrm{d}x
=0
\end{equation*}
and thus, \cref{t.entropy-general} follows from \labelcref{eq.D(Tn)-finish}. \end{proof}
\section{Rate of convergence for Edgeworth expansions in the central limit theorem in total variation distance for self-normalized sums}\label{ch.TV}
The main goal of this section is to prove \cref{t.TV} by examining the total variation distance between $F_n$ and $\Phi^{Q}_{m,n}$. By \labelcref{eq.TV-L1}, this is equivalent to bounding the $L^1$ norm of the difference of the corresponding densities.
\begin{theorem}\label{t.dichte-L1}
Assume that $X_1$ is symmetric, the distribution of $X_1$ is non-singular, \cref{c.density-cf} is satisfied and $\mathbb{E}|X_1|^{2m}<\infty$ for some $m\in\mathbb{N}$, $m\ge3$. Then there exists $N\in\mathbb{N}$ such that for all $n\ge N$, the statistics $T_{n}$ have densities $f_{n}$ that satisfy
\begin{equation}\label{eq.t-dichte-L1}
\|f_n-\phi^{q}_{m,n}\|_{L^1}= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m-2)/2}\big).
\end{equation} \end{theorem}
Together with \cref{p.prop} and \labelcref{eq.TV-L1}, \cref{t.dichte-L1} implies \cref{t.TV}. On our route to proving \cref{t.dichte-L1}, we first show the following proposition. For any $h\in\mathbb{R}$, let $f(\lbullet+h)$ denote the function $x\mapsto f(x+h)$.
\begin{proposition}\label{p.dichte-Lp}
Assume that $X_1$ is symmetric, the distribution of $X_1$ is non-singular and $\mathbb{E}|X_1|^{s}<\infty$ for some ${s}\ge2$. Additionally, assume that \cref{c.density-ex} is satisfied and there exists $p\in[1,\infty)$ and $\alpha\in\mathbb{R}$ with $0<\alpha\le c(s)$ such that $\|f_n\|_{L^p}\le \exp(n^\alpha)$ for all $n$ sufficiently large. Let $m=\lfloor {s} \rfloor$ and $a_n=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}(1)$ with polynomial order. Then
\begin{equation}\label{eq.p-dichte-Lp}
\|f_n-\phi^{q}_{m,n}\|_{L^p} \le \sup_{|h|\le a_n} \big\|f_n(\lbullet)-f_n(\lbullet+h)\big\|_{L^p} + \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-({s}-2)/2} \, (\log n)^{({s}+2m)/2} \big).
\end{equation} \end{proposition} \begin{proof}
Set $\beta_n= a_n^2\cdot n^{-\max\{2\alpha,(s-2)/2\}}$ and recall $f_n'$ from \labelcref{eq.f_n'-def}. Now, \cref{p.dichte-p} yields
\begin{equation*}
\sup_{x\in\mathbb{R}} \, (1+|x|)^{m}|f_n'(x)-\phi^{q}_{m,n}(x)|= r_n
\end{equation*}
for $r_n=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-({s}-2)/2} \, (\log n)^{({s}+2m)/2} \big)$.
By Minkowski's inequality and \cref{t.llt},
\begin{align*}
\|f_n-\phi^{q}_{m,n}\|_{L^p}
&\le \|f_n-f_n'\|_{L^p} + \|f_n'-\phi^{q}_{m,n}\|_{L^p}\\
&\le \|f_n-f_n'\|_{L^p} + \|(1+|\lbullet|)^{-m}\|_{L^p} \cdot r_n\\
&= \|f_n-f_n'\|_{L^p} + \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-({s}-2)/2} \, (\log n)^{({s}+2m)/2} \big).
\end{align*}
Next, for $b_n:=\sqrt{\beta_n}\cdot n^\alpha \le a_n$ and by \cite[4.13(1)]{Alt},
\begin{align*}
\|f_n-f_n'\|_{L^p}
&\le \Big\|\int_{-\infty}^{\infty} \big(f_n(\lbullet)-f_n(\lbullet-y)\big) \phi_{\beta_n}(y) \mathbbm{1}_{\{|y|\le b_n\}} \,\mathrm{d}y\Big\|_{L^p}\\
&\quad+\Big\|\int_{-\infty}^{\infty} \big(f_n(\lbullet)-f_n(\lbullet-y)\big) \phi_{\beta_n}(y) \mathbbm{1}_{\{|y|> b_n\}} \,\mathrm{d}y\Big\|_{L^p}\\
&\le \big\|\phi_{\beta_n}(\lbullet)\mathbbm{1}_{\{|\lbullet|\le b_n\}}\big\|_{L^1} \cdot \sup_{|h|\le b_n} \big\|f_n(\lbullet)-f_n(\lbullet+h)\big\|_{L^p}\\
&\quad + \big\|\phi_{\beta_n}(\lbullet)\mathbbm{1}_{\{|\lbullet|> b_n\}}\big\|_{L^1} \cdot \sup_{|h|> b_n} \big\|f_n(\lbullet)-f_n(\lbullet+h)\big\|_{L^p}.
\end{align*}
As $1-\Phi(x)\le \phi(x)/x$, for $n$ sufficiently large, the expression above is bounded by
\begin{align*}
\lefteqn{ \sup_{|h|\le b_n} \big\|f_n(\lbullet)-f_n(\lbullet+h)\big\|_{L^p}
+ 2\big(1-\Phi\big(b_n/\sqrt{\beta_n}\,\big)\big)\cdot 2 \|f_n\|_{L^p}}\quad\\
&= \sup_{|h|\le b_n} \big\|f_n(\lbullet)-f_n(\lbullet+h)\big\|_{L^p}
+ 2\big(1-\Phi(n^\alpha)\big)\cdot 2 \|f_n\|_{L^p}\\
&\le \sup_{|h|\le a_n} \big\|f_n(\lbullet)-f_n(\lbullet+h)\big\|_{L^p} + 4 \,n^{-\alpha} \exp(-n^{2\alpha}/2+n^\alpha) \\
&= \sup_{|h|\le a_n} \big\|f_n(\lbullet)-f_n(\lbullet+h)\big\|_{L^p} + \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-({s}-2)/2} \, (\log n)^{({s}+2m)/2} \big)
\end{align*}
in particular. \end{proof}
The term \begin{align*}
\sup_{|h|\le a_n} \|f_n(\lbullet)-f_n(\lbullet+h)\|_{L^p} \end{align*} from \labelcref{eq.p-dichte-Lp} is known as the integral (or $L^p$) modulus of continuity. There exists plentiful literature on bounds of this term if the Fourier transform of $f_n$ satisfies various kinds of tail bounds (see e.g. \cite{Cli91,GT12,Tit48}).
Next, we derive a bound on the $L^1$ modulus of continuity of $f_n$.
\begin{proposition}\label{p.L1-modulus}
Assume that $X_1$ is symmetric, the distribution of $X_1$ is non-singular, \cref{c.density-cf} is satisfied and $\mathbb{E}|X_1|^{2m}<\infty$ for some $m\in\mathbb{N}$, $m\ge3$. Let $a_n=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(9m-17)/2}\big)$. Then there exists $N\in\mathbb{N}$ such that for all $n\ge N$, the statistics $T_{n}$ have densities $f_{n}$ that satisfy
\begin{align}\label{eq.p-L1-modulus}
\sup_{|h|\le a_n} \|f_n(\lbullet)-f_n(\lbullet+h)\|_{L^1} = \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m-2)/2}\big).
\end{align} \end{proposition}
\begin{proof}
Analogously to the proof of \cref{t.llt}, \cref{c.density-cf} and $\mathbb{E} |X_1|^{2m}<\infty$ yield that there exists $N\in \mathbb{N}$ such that \labelcref{eq.fn-int-gn,eq.BR-19.2} hold for all $n\ge N$. Throughout the remainder of the proof assume $n\ge N$. Now \labelcref{eq.fn-int-gn} implies that the self-normalized sums $T_{n}$ have densities $f_n$. We expand
\[
|f_n(x)-f_n(x-h)|
\le K_1+K_2+K_3
\]
as in \labelcref{eq.f_n-f_n-Z}, where the terms $K_1$, $K_2$ and $K_3$ are introduced. This time, the $2m$-th moment is assumed to be finite. Therefore with $w=m$, \labelcref{eq.k1,eq.k2} yield
\[
K_1+K_2\le |h| c(m) n^{4m-8},
\]
and thus
\begin{align*}
|f_n(x)-f_n(x-h)|
&\le |h| c(m) n^{4m-8}
+ K_3\\
&= |h| c(m) n^{4m-8}\\
&\quad+ R_n' \int_0^\infty \big(1+\|\gamma_{x,n}(z)\|\big)^{-m} \big(n^{-1/2}z\big)^{1/2}\\
&\qquad\qquad\quad + \big(1+\|\gamma_{x-h,n}(z)\|\big)^{-m} \big(n^{-1/2}z\big)^{1/2}\,\mathrm{d}z
\end{align*}
for all $|x|<2\sqrt{n}$ and $|h|<1$. Here, $R_n'=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m-2)/2}\big)$ and $\gamma_{x,n}(z)=\big(x(n^{-1/2}z)^{1/2}, \allowbreak z-n^{1/2}\big)$. By \cref{r.T_n},
\begin{equation}\label{eq.L1-modulus-zerlegung}
\begin{split}
\lefteqn{\sup_{|h|\le a_n} \|f_n(\lbullet)-f_n(\lbullet+h)\|_{L^1}}\quad\\
&\le\sup_{|h|\le a_n} \int_{-2\sqrt{n}}^{2\sqrt{n}} |h| c(m) n^{4m-8}\,\mathrm{d}x\\
&\quad + \sup_{|h|\le a_n} \int_{-2\sqrt{n}}^{2\sqrt{n}} R_n' \int_0^\infty \big(1+\|\gamma_{x,n}(z)\|\big)^{-m} \big(n^{-1/2}z\big)^{1/2}\\
&\qquad\qquad\qquad\qquad\qquad+ \big(1+\|\gamma_{x-h,n}(z)\|\big)^{-m} \big(n^{-1/2}z\big)^{1/2}\,\mathrm{d}z\,\mathrm{d}x.
\end{split}
\end{equation}
Regarding the first summand,
\begin{equation}\label{eq.L1-K3}
\sup_{|h|\le a_n} \int_{-2\sqrt{n}}^{2\sqrt{n}} |h| c(m) n^{4m-8}\,\mathrm{d}x
\le c(m) a_n \, n^{(8m-15)/2}\\
= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m-2)/2}\big).
\end{equation}
For the second summand, it is obviously impossible to attain a bound in $h$. Therefore, we have to find a bound in $n$.
For $m\ge3$, by Fubini's theorem
\begin{align}\label{eq.int-Z4}
\lefteqn{\int_{-2\sqrt{n}}^{2\sqrt{n}} \int_0^\infty \big(1+\|\gamma_{x,n}(z)\|\big)^{-m} \big(n^{-1/2}z\big)^{1/2} \,\mathrm{d}z \,\mathrm{d}x}\quad\nonumber\\
&\le \int_{-2\sqrt{n}}^{2\sqrt{n}} \int_0^\infty \big(1+\|\gamma_{x,n}(z)\|^2\big)^{-3/2} \big(n^{-1/2}z\big)^{1/2} \,\mathrm{d}z \,\mathrm{d}x\\
&= \int_0^\infty \big(n^{-1/2}z\big)^{1/2} \int_{-2\sqrt{n}}^{2\sqrt{n}} \big(1+n^{-1/2}zx^2+(z-n^{1/2})^2\big)^{-3/2} \,\mathrm{d}x \,\mathrm{d}z.\nonumber
\end{align}
Applying \cref{l.int-ax^2+b} gives
\begin{equation}\label{eq.inner-int}
\begin{split}
\lefteqn{\int_{-2\sqrt{n}}^{2\sqrt{n}} \big(1+n^{-1/2}zx^2+(z-n^{1/2})^2\big)^{-3/2} \,\mathrm{d}x}\quad\\
&= 2\big(1+4n^{1/2}z+(z-n^{1/2})^2\big)^{-1/2} (1+(z-n^{1/2})^2)^{-1} 2 n^{1/2}
\end{split}
\end{equation}
such that we have to examine
\begin{align*}
\int_0^\infty \big(n^{-1/2}z\big)^{1/2} \big(1+4n^{1/2}z+(z-n^{1/2})^2\big)^{-1/2} (1+(z-n^{1/2})^2)^{-1} n^{1/2} \,\mathrm{d}z.
\end{align*}
We split this integral into three regions that we treat separately. First,
\begin{align*}
&\int_0^{\sqrt{n}/2} \big(n^{-1/2}z\big)^{1/2} \big(1+4n^{1/2}z+(z-n^{1/2})^2\big)^{-1/2} (1+(z-n^{1/2})^2)^{-1} n^{1/2} \,\mathrm{d}z\\
&\quad\le c(0) \int_0^{\sqrt{n}/2} 1 \cdot n^{-1/2} \, n^{-1} \, n^{1/2} \,\mathrm{d}z
\le c(0) n^{-1/2}.
\end{align*}
Second, by \cref{l.int-a/2}
\begin{align*}
&\int_{\frac{1}{2}\sqrt{n}}^{2\sqrt{n}} \big(n^{-1/2}z\big)^{1/2} \big(1+4n^{1/2}z+(z-n^{1/2})^2\big)^{-1/2} (1+(z-n^{1/2})^2)^{-1} n^{1/2} \,\mathrm{d}z\\
&\quad\le c(0) \int_{\frac{1}{2}\sqrt{n}}^{2\sqrt{n}} \sqrt{2} \cdot n^{-1/2} \, (1+(z-n^{1/2})^2)^{-1} \, n^{1/2} \,\mathrm{d}z\\
&\quad= c(0) \int_{\frac{1}{2}\sqrt{n}}^{2\sqrt{n}} (1+(z-n^{1/2})^2)^{-1} \,\mathrm{d}z\le c(0).
\end{align*}
Finally,
\begin{align*}
&\int_{2\sqrt{n}}^\infty \big(n^{-1/2}z\big)^{1/2} \big(1+4n^{1/2}z+(z-n^{1/2})^2\big)^{-1/2} (1+(z-n^{1/2})^2)^{-1} n^{1/2} \,\mathrm{d}z\\
&\quad\le c(0) \int_{2\sqrt{n}}^\infty n^{-1/4} z^{1/2} \big(z-n^{1/2}\big)^{-1} \, \big(z-n^{1/2}\big)^{-2} \, n^{1/2} \,\mathrm{d}z\\
&\quad\le c(0) \, n^{1/4} \int_{2\sqrt{n}}^\infty z^{1/2} \big(z-z/2\big)^{-3} \,\mathrm{d}z\\
&\quad= c(0) \, n^{1/4} \int_{2\sqrt{n}}^\infty z^{-5/2} \,\mathrm{d}z= c(0) \, n^{-1/2}.
\end{align*}
Combining \labelcref{eq.int-Z4}, \labelcref{eq.inner-int} and the last three formulas yields
\begin{equation*}
\int_{-2\sqrt{n}}^{2\sqrt{n}} \int_0^\infty \big(1+\|\gamma_{x,n}(z)\|\big)^{-m} \big(n^{-1/2}z\big)^{1/2} \,\mathrm{d}z \,\mathrm{d}x
\le c(0).
\end{equation*}
This remains valid if $x$ is replaced by $x-h$ for any $|h|\le a_n$ and thus
\begin{align*}
& \sup_{|h|\le a_n} \int_{-2\sqrt{n}}^{2\sqrt{n}} R_n' \int_0^\infty \big(1+\|\gamma_{x,n}(z)\|\big)^{-m} \big(n^{-1/2}z\big)^{1/2}\\*
&\qquad\qquad\qquad\qquad+ \big(1+\|\gamma_{x-h,n}(z)\|\big)^{-m} \big(n^{-1/2}z\big)^{1/2}\,\mathrm{d}z \,\mathrm{d}x\\
&\quad\le R_n' \, c(0)
=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(m-2)/2}\big),
\end{align*}
which yields \labelcref{eq.p-L1-modulus} when combined with \labelcref{eq.L1-modulus-zerlegung,eq.L1-K3}. \end{proof}
\begin{proof}[Proof of \cref{t.dichte-L1}]
\cref{t.llt} implies \cref{c.density-ex}.
Due to the form \labelcref{eq.phi^q-def} and the factor $\phi$ in all expansion terms $q_r$, $\|\phi^{q}_{m,n}\|_{L^p}\le c(m)$ for all $p\in[1,\infty)$. Thus by \labelcref{eq.fn-phi-Lp}, $\|f_n\|_{L^p}\le c(m)$ for all $p\in[1,\infty)$ and all $n$ sufficiently large.
Set ${s}=m+1/2$ and $a_n=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(9m-16)/2}\big)$ with polynomial order.
Now \cref{p.dichte-Lp,p.L1-modulus} imply \labelcref{eq.t-dichte-L1}. \end{proof}
\begin{remark}
The bounds in the $L^1$ limit theorem are by $\log n$ tighter than in the LLT. This is because the integral modulus of continuity used in the $L^1$ setting is easier to estimate than the pointwise H\"older-type bound which was the equivalent condition in the LLT setting. As this bound was used in the proof of the entropic CLT, the discrepancy of $\log n$ transfers there. The reason behind the factor $\log n$ is the appearance of different exponents in \cref{l.int-a/2}. \end{remark}
\begin{appendix}
\section{Proofs, remarks and auxiliary lemmas}\label{app.proofs}
\subsection{Proof of Proposition \ref{p.E-tP-Q}}\label{app.proofs.pre}
\begin{remark}\label{r.V_n-lambda}
We calculate the expected values of the $\tilde{\lambda}_{l,n}$ appearing in $\tilde{P}_{2,n}$ and $\tilde{P}_{4,n}$ (see \labelcref{eq.tP-2+4}). For clear illustration, let all moments be finite in the following procedure. Additionally, we will explicitly write $\mu_2$ instead of its fixed value 1 as it helps to keep track of the order of moments. First, we need the expected values of the conditional cumulants divided by powers of $V_n$ for which we expand
\begin{equation}\label{eq.V_n^-2}
\begin{split}
V_n^{-2}
&=n^{-1}\Big(\mu_2+n^{-1} \Big(\sum (X_j^2-\mu_2)\Big)\Big)^{-1}\\
&= n^{-1} \mu_2^{-1}
- n^{-2} \mu_2^{-2} \Big(\sum (X_j^2-\mu_2)\Big) \\
&\quad+ n^{-3} \mu_2^{-3} \Big(\sum (X_j^2-\mu_2)\Big)^2
+ \mathcal{O}_p\big(n^{-3}\big)
\end{split}
\end{equation}
therefore
\begin{align*}
V_n^{-4}
&= n^{-2} \mu_2^{-2}
- n^{-3} 2 \mu_2^{-3} \Big(\sum (X_j^2-\mu_2)\Big)
+ n^{-4} 3 \mu_2^{-4} \Big(\sum (X_j^2-\mu_2)\Big)^2
+ \mathcal{O}_p\big(n^{-4}\big),\\
V_n^{-6}
&= n^{-3} \mu_2^{-3}
- n^{-4} 3 \mu_2^{-4} \Big(\sum (X_j^2-\mu_2)\Big)
+ n^{-5} 6 \mu_2^{-5} \Big(\sum (X_j^2-\mu_2)\Big)^2
+ \mathcal{O}_p\big(n^{-5}\big),\\
V_n^{-8}
&= n^{-4} \mu_2^{-4}
- n^{-5} 4 \mu_2^{-5} \Big(\sum (X_j^2-\mu_2)\Big)
+ n^{-6} 10 \mu_2^{-6} \Big(\sum (X_j^2-\mu_2)\Big)^2
+ \mathcal{O}_p\big(n^{-6}\big).
\end{align*}
Due to $\mathbb{E} X_j^2=\mu_2$, the index of every factor $(X_j^2-\mu_2)$ within
\[
\Big(\sum (X_j^2-\mu_2)\Big)^{k}\sum|X_j|^{2 l}
=\sum_{j_1,\dots,j_{k+1}=1}^n \big(X_{j_1}^2-\mu_2\big) \cdots \big( X_{j_k}^2-\mu_2 \big) |X_{j_{k+1}}|^{2 l}
\]
has to be equal to the index of another factor or the summand vanishes within the expectation. Thus, for $k\in\mathbb{N}$, the expectation of summands with $\big(\sum (X_j^2-\mu_2)\big)^{2k-1}$ and ${\big(\sum (X_j^2-\mu_2)\big)^{2k}}$ produce the same order of $n$. This yields
\begin{align*}
\mathbb{E}\Big[V_n^{-4}\sum|X_j|^4\Big]
&=n^{-1} \mu_2^{-2} \mu_4
- n^{-2} \mu_2^{-4} \big(2\mu_2\mu_6 +\mu_2^2\mu_4 - 3 \mu_4^2\big)
+ \mathcal{O}\big(n^{-3}\big),\\
\mathbb{E}\Big[V_n^{-6}\sum|X_j|^6\Big]
&=n^{-2} \mu_2^{-3} \mu_6
- n^{-3} \mu_2^{-5} (3\mu_8\mu_2+3\mu_6\mu_2^2-6\mu_4\mu_6)
+ \mathcal{O}\big(n^{-4}\big),\\
\mathbb{E}\Big[V_n^{-8}\Big(\sum|X_j|^4\Big)^2\Big]
&=n^{-2} \mu_2^{-4} \mu_4^2
- n^{-3} \mu_2^{-5} (8\mu_6\mu_4-7\mu_4^2\mu_2- \mu_8\mu_2)
+ \mathcal{O}\big(n^{-4}\big)
\end{align*}
such that by \labelcref{eq.tk-2r,eq.tlambda}
\begin{align*}
\mathbb{E}\bigg[\frac{\tilde{\lambda}_{4,n}}{4!}\bigg]
&= n^{-1} \big(- \tfrac{1}{12}\big) \mu_2^{-2} \mu_4
+ n^{-2} \tfrac{1}{12} \mu_2^{-4} \big(2\mu_2\mu_6 +\mu_2^2\mu_4 - 3 \mu_4^2\big)
+ \mathcal{O}\big(n^{-3}\big),\\
\mathbb{E}\bigg[\frac{\tilde{\lambda}_{6,n}}{6!}\bigg]
&= n^{-2} \tfrac{1}{45} \mu_2^{-3} \mu_6
+ n^{-3} \big(- \tfrac{1}{45}\big) \mu_2^{-5} (3\mu_8\mu_2+3\mu_6\mu_2^2-6\mu_4\mu_6)
+ \mathcal{O}\big(n^{-4}\big),\\
\mathbb{E}\bigg[\frac{1}{2} \Big(\frac{\tilde{\lambda}_{4,n}}{4!}\Big)^2\bigg]
&=n^{-2} \tfrac{1}{288} \mu_2^{-4} \mu_4^2
+ n^{-3} \big(- \tfrac{1}{288}\big) \mu_2^{-5} (8\mu_6\mu_4-7\mu_4^2\mu_2- \mu_8\mu_2)
+ \mathcal{O}\big(n^{-4}\big).
\end{align*} \end{remark}
\begin{proof}[Proof of \cref{p.E-tP-Q}]
The connection $\mathbb{E}\big[\Phi^{\tilde{P}}_{m,n}\big]$ and $\Phi^Q_{m,n}$ (the procedure for $\mathbb{E}\big[\phi^{\tilde{p}}_{m,n}\big]$ and $\phi^q_{m,n}$ is the same) is given in \labelcref{eq.Phi^tP}, \labelcref{eq.tP}, \labelcref{eq.Q-def} and \labelcref{eq.Phi^Q-def} and the corresponding computations are conducted in \cref{r.V_n-lambda}. However, we need to address the issue of moments higher than ${s}$ (as they could be infinite) and the factor $\exp(x^2/4)$. For ${s}<4$, there are no expansion terms in \labelcref{eq.E-tP-Q,eq.E-tp-q} and there is nothing to prove. So assume ${s}\ge4$.
As in \cref{r.V_n-lambda}, we first evaluate the expectation of $V_n^{-2\alpha}\sum|X_j|^{2\alpha}$ for $\alpha\in\mathbb{N}$, ${2\le \alpha\le m/2}$. Define ${h}=2\lceil \frac{m-2\alpha+1}{2}\rceil$. Similar to \labelcref{eq.V_n^-2}, we perform a Taylor expansion of $V_n^{-2\alpha}$ up to order ${h}-1$ as follows
\begin{equation}\label{eq.Exp-tL}
\begin{split}
V_n^{-2\alpha}\sum|X_j|^{2\alpha}
&=\sum_{r=0}^{{h}-1} (-1)^r n^{-(\alpha+r)} t_{r,2\alpha} \Big(\sum (X_j^2-1)\Big)^{r}\Big(\sum|X_j|^{2\alpha}\Big)\\
&\quad+(-1)^{{h}} n^{-(\alpha+{h})} t_{{h},2\alpha} \xi^{-(\alpha+{h})} \Big(\sum (X_j^2-1)\Big)^{{h}}\Big(\sum|X_j|^{2\alpha}\Big),
\end{split}
\end{equation}
with (random) intermediate value $\xi$ between $1$ and $n^{-1} V_n^2$. Here, $t_{r,2\alpha}$ are the constant factors which appear in this Taylor expansion and are bounded in absolute value by $c(m)$.
First, we will show that the sum of all terms where $X_\cdot$ appears with a power higher than ${s}$ is of order $\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix} \big(n^{-({s}-2)/2}\big)$. The same is shown for the remainder of the Taylor expansion including the $\xi$.
Note that by \labelcref{eq.Lle1,eq.Mn-delta,eq.Vn-bound-1/2}, we can restrict the expectation of the terms in \labelcref{eq.Exp-tL} to the event
\begin{equation}\label{eq.event}
A_{n,1}\cap A_{n,2} \quad \text{where} \quad A_{n,1}:=\{M_n< \delta_n \sqrt{n} \, \} \quad\text{and} \quad A_{n,2}:=\{V_n^2 > \tfrac{n}{2}\}
\end{equation}
with an error of order $\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix} \big(n^{-({s}-2)/2}\big)$. The sequence $(\delta_n)$ is introduced in \cref{r.moment}.
Therefore,
\begin{equation}\label{eq.E-expanded}
\begin{split}
\lefteqn{\mathbb{E}\Big[ V_n^{-2\alpha}\sum|X_j|^{2\alpha} \Big]}\quad\\
&=\mathbb{E}\Big[ \mathbbm{1}_{A_{n,1}\cap A_{n,2}} V_n^{-2\alpha}\Big(\sum|X_j|^{2\alpha}\Big) \Big] + \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix} \big(n^{-({s}-2)/2}\big)\\
&=\mathbb{E}\bigg[ \mathbbm{1}_{A_{n,1}\cap A_{n,2}} \sum_{r=0}^{{h}-1} (-1)^r n^{-(\alpha+r)} t_{r,2\alpha} \Big(\sum (X_j^2-1)\Big)^{r}\Big(\sum|X_j|^{2\alpha}\Big)\bigg]\\
&\quad+\mathbb{E}\bigg[ \mathbbm{1}_{A_{n,1}\cap A_{n,2}} (-1)^{{h}} n^{-(\alpha+{h})} t_{{h},2\alpha} \xi^{-(\alpha+{h})} \Big(\sum (X_j^2-1)\Big)^{{h}}\Big(\sum|X_j|^{2\alpha}\Big) \bigg] \\
&\quad+ \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix} \big(n^{-({s}-2)/2}\big).
\end{split}
\end{equation}
Note that by \labelcref{eq.event}
\begin{align*}
\lefteqn{\bigg|\mathbb{E}\bigg[\mathbbm{1}_{A_{n,1}\cap A_{n,2}^c} \sum_{r=0}^{{h}-1} (-1)^r n^{-(\alpha+r)} t_{r,2\alpha} \Big(\sum (X_j^2-1)\Big)^{r}\Big(\sum|X_j|^{2\alpha}\Big)\bigg]\bigg|}\quad\\*
&\le c(h) \sum_{r=0}^{{h}-1} \mathbb{E}\Big[\mathbbm{1}_{A_{n,1}\cap A_{n,2}^c} n^{-(\alpha+r)} \big(V_n^2+n\big)^{r} V_n^2 (\delta_n \sqrt{n} \, )^{2\alpha-2}\Big]\\
&\le c(h) \sum_{r=0}^{{h}-1} \mathbb{E}\Big[\mathbbm{1}_{A_{n,2}^c} n^{-(r+1)} \big(\tfrac{n}{2}+n\big)^{r} \tfrac{n}{2}\Big] \\
&\le c(h) \mathbb{P}\big(A_{n,2}^c\big)\\
&= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix} \big(n^{-({s}-2)/2}\big)
\end{align*}
and thus,
\begin{equation}\label{eq.A_{n,2}-raus}
\begin{split}
&\mathbb{E}\bigg[ \mathbbm{1}_{A_{n,1}\cap A_{n,2}} \sum_{r=0}^{{h}-1} (-1)^r n^{-(\alpha+r)} t_{r,2\alpha} \Big(\sum (X_j^2-1)\Big)^{r}\Big(\sum|X_j|^{2\alpha}\Big)\bigg]\\
&= \sum_{r=0}^{{h}-1} (-1)^r n^{-(\alpha+r)} t_{r,2\alpha} \, \mathbb{E}\Big[ \mathbbm{1}_{A_{n,1}} \Big(\sum (X_j^2-1)\Big)^{r}\Big(\sum|X_j|^{2\alpha}\Big)\Big]
+ \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix} \big(n^{-({s}-2)/2}\big).
\end{split}
\end{equation}
Initially, we examine
\[
n^{-(\alpha+r)} \, \mathbb{E}\Big[ \mathbbm{1}_{A_{n,1}} \Big(\sum (X_j^2-1)\Big)^{r}\Big(\sum|X_j|^{2\alpha}\Big)\Big]
\]
for $r=0,\dots,m$.
We expand
\[
\Big(\sum (X_j^2-1)\Big)^{r}
=\sum_{i_1+\dots+i_{n}=r} \binom{r}{i_1,\dots,i_n} \prod (X_j^2-1)^{i_{j}}.
\]
Here the sum extends over all $n$-tuples $(i_1,...,i_n)$ of non-negative integers with $\sum_{j=1}^ni_j=r$ and
\[
{r \choose i_{1},\ldots ,i_{n}}={\frac {r!}{i_{1}!\,i_{2}!\cdots i_{n}!}}.
\]
Since $X_1,\dots,X_n$ are i.i.d.,
\begin{align*}
\mathbb{E}\bigg[ {r \choose i_{\pi(1)},\ldots ,i_{\pi(n)}}\prod (X_j^2-1)^{i_{\pi(j)}} \bigg]
= \mathbb{E}\bigg[ {r \choose i_{1},\ldots ,i_{n}}\prod (X_j^2-1)^{i_{j}} \bigg]
\end{align*}
for all (non-random) permutations $\pi$ from $\{1,\dots,n\}$ to itself.
For $l=1,\dots,r$ let $k_l$ be the number of appearances of $l$ in the tuple $(i_1,...,i_n)$ (so $\sum_{l=1}^{r}lk_l =\sum_{j=1}^ni_j=r$). Then for each tuple $(k_1,\dots,k_r)$, there exist ${n \choose k_{1},\ldots ,k_{r},(n-\sum_{l=1}^{r} k_l)}$ tuples $(i_1,...,i_n)$ such that $k_l$ is the number of appearances of $l$ in the tuple $(i_1,...,i_n)$. So instead of summing over all $n$-tuples $(i_1,...,i_n)$ with $\sum_{j=1}^ni_j=r$, we can sum over all $r$-tuples $(k_1,...,k_r)$ of non-negative integers with $\sum_{l=1}^{r}lk_l =r$ and multiply each summand by ${n \choose k_{1},\ldots ,k_{r},(n-\sum_{l=1}^{r} k_l)}$. We perform the just described summation by using the sum $\sum_{*(k_\cdot,r,\cdot)}$ and $u(k_\cdot)$ which are explained in \cref{ch.pre}. Additionally, we convert
\[
{r \choose i_{1},\ldots ,i_{n}}={\frac {r!}{i_{1}!\,i_{2}!\cdots i_{n}!}}=\frac{r!}{1!^{k_1}\cdots r!^{k_r}}.
\]
Now for any $(k_1,\dots,k_r)$, we identify that element of the corresponding set
\[
\{(i_{\pi(1)},...,i_{\pi(n)}): \pi \text{ permutations from }\{1,\dots,n\}\text{ to itself} \}
\]
for which
\begin{align*}
i_1&=\dots=i_{k_1}=1,\\
i_{k_1+1}&=\dots=i_{k_1+k_2}=2,\\
&\vdots\\
i_{\sum_{l=1}^{r-1}k_l+1}&=\dots=i_{\sum_{l=1}^rk_l}=r,\\
i_{\sum_{l=1}^rk_l+1}&=\dots=i_n=0.
\end{align*}
When summing over $(k_1,\dots,k_r)$, this will be the representative, evaluated within the expectation. This reordering yields
\begin{align*}
&\mathbb{E}\left[\Big(\sum (X_j^2-1)\Big)^{r} \Big(\sum|X_j|^{2\alpha}\Big) \right]\\
&\quad=\mathbb{E}\left[\sum_{i_1+\dots+i_{n}=r} \binom{r}{i_1,\dots,i_n} \Big(\prod (X_j^2-1)^{i_{j}}\Big) \Big(\sum|X_j|^{2\alpha}\Big) \right]\\
&\quad=\mathbb{E}\Bigg[\sum_{*(k_\cdot,r,\cdot)} {n \choose k_{1},\ldots ,k_{r},(n-u(k_\cdot))} \frac{r!}{1!^{k_1}\cdots r!^{k_r}}\\
&\quad\qquad\cdot\bigg( \prod_{l=1}^{r} \prod_{a=1}^{k_l}\Big(X_{\sum_{b=1}^{l-1}k_b+a}^2-1 \Big)^{l} \bigg) \Big(\sum|X_j|^{2\alpha}\Big) \Bigg].
\end{align*}
As $M_n$ is invariant under permutation, the last set of formulas remains valid if $\mathbbm{1}_{A_{n,1}}$ is present in the expectation. So
\begin{equation}\label{eq.expanded}
\begin{split}
&n^{-(\alpha+r)} \, \mathbb{E}\Big[ \mathbbm{1}_{A_{n,1}} \Big(\sum (X_j^2-1)\Big)^{r}\Big(\sum|X_j|^{2\alpha}\Big)\Big]\\
&\quad= n^{-(\alpha+r)} \sum_{*(k_\cdot,r,\cdot)} {n \choose k_{1},\ldots ,k_{r},(n-u(k_\cdot))} \frac{r!}{1!^{k_1}\cdots r!^{k_r}}\\
&\quad\qquad\cdot \sum \mathbb{E}\bigg[ \mathbbm{1}_{A_{n,1}} \bigg( \prod_{l=1}^{r} \prod_{a=1}^{k_l}\Big(X_{\sum_{b=1}^{l-1}k_b+a}^2-1 \Big)^{l} \bigg) |X_j|^{2\alpha} \bigg]
\end{split}
\end{equation}
for all $r=0,\dots,m$.
Below, we distinguish between the values of $j$, which only coincide with the index of another factor for $j\le u(k_\cdot)$, and get
\begin{equation}\label{eq.j-split}
\begin{split}
&\sum \mathbb{E}\bigg[ \mathbbm{1}_{A_{n,1}} \bigg( \prod_{l=1}^{r} \prod_{a=1}^{k_l}\Big(X_{\sum_{b=1}^{l-1}k_b+a}^2-1 \Big)^{l} \bigg) |X_j|^{2\alpha} \bigg]\\
&\quad= \sum_{j=u(k_\cdot)+1}^{n} \mathbb{E}\bigg[ \mathbbm{1}_{A_{n,1}} \bigg( \prod_{l=1}^{r} \prod_{a=1}^{k_l}\Big(X_{\sum_{b=1}^{l-1}k_b+a}^2-1 \Big)^{l} \bigg) |X_j|^{2\alpha} \bigg]\\
&\quad\quad+ \sum_{j=1}^{u(k_\cdot)} \mathbb{E}\bigg[ \mathbbm{1}_{A_{n,1}} \bigg( \prod_{l=1}^{r} \prod_{a=1}^{k_l}\Big(X_{\sum_{b=1}^{l-1}k_b+a}^2-1 \Big)^{l} \bigg) |X_j|^{2\alpha} \bigg].
\end{split}
\end{equation}
Next, we use that $X_1,\dots,X_n$ are i.i.d. and
\begin{equation}\label{eq.A_{n,1}-exp}
A_{n,1}=\{|X_1|<\delta_n \sqrt{n} \, \} \cap \dots \cap \{|X_n|<\delta_n \sqrt{n} \, \}.
\end{equation}
Additionally, we introduce the notation
\[
\mathbb{E}_{\delta_n,n}[\,\cdot\,]:=\mathbb{E}[\mathbbm{1}_{\{|X_1|<\delta_n \sqrt{n} \, \}}\,\cdot\,].
\]
For the upper part of the sum this yields
\begin{align*}
&\sum_{j=u(k_\cdot)+1}^{n} \mathbb{E}\bigg[ \mathbbm{1}_{A_{n,1}} \bigg( \prod_{l=1}^{r} \prod_{a=1}^{k_l}\Big(X_{\sum_{b=1}^{l-1}k_b+a}^2-1 \Big)^{l} \bigg) |X_j|^{2\alpha} \bigg]\\*
&= \big(n-u(k_\cdot)\big) \bigg( \prod_{l=1}^{r}\Big(\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg) \mathbb{E}_{\delta_n,n}\Big[ |X_1|^{2\alpha} \Big] \mathbb{P}\big(\{|X_1|<\delta_n \sqrt{n} \, \}\big)^{n-u(k_\cdot)-1}.
\end{align*}
Assume that $k_l>0$ for at least one $l \ge \lfloor s/2\rfloor+1$ (i.e. $2l>{s}$). Furthermore, in the following formula we use the fact $2\le\alpha\le m/2$. As
\[
{n \choose k_{1},\ldots ,k_{r},(n-u(k_\cdot))}\le c(r) n^{u(k_\cdot)}
\qquad
\text{and}
\qquad
\big(n-u(k_\cdot)\big)\le n,
\]
these summands of $\sum_{*(k_\cdot,r,\cdot)}$ in \labelcref{eq.expanded} can be bounded by
\begin{align*}
&c(r) n^{-\alpha-r+u(k_\cdot)+1}
\bigg( \prod_{l=1}^{r}\Big(\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg)\\
& \qquad \cdot \mathbb{E}_{\delta_n,n}\Big[ |X_1|^{2\alpha} \Big] \mathbb{P}\big(\{|X_1|<\delta_n \sqrt{n} \, \}\big)^{n-u(k_\cdot)-1} \\
&=c(r) n^{1-\alpha}
\bigg( \prod_{l=1}^{r}\Big(n^{1-l}\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg)\\
& \qquad \cdot \mathbb{E}_{\delta_n,n}\Big[ |X_1|^{2\alpha} \Big] \mathbb{P}\big(\{|X_1|<\delta_n \sqrt{n} \, \}\big)^{n-u(k_\cdot)-1} \\
&=c(r) n^{1-\alpha}
\bigg( \prod_{l=1}^{\lfloor s/2\rfloor}\Big(n^{1-l}\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg) \bigg( \prod_{l=\lfloor s/2\rfloor+1}^{r}\Big(n^{1-l}\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg) \\
& \qquad \cdot \mathbb{E}_{\delta_n,n}\Big[ |X_1|^{2\alpha} \Big] \mathbb{P}\big(\{|X_1|<\delta_n \sqrt{n} \, \}\big)^{n-u(k_\cdot)-1} \\
&\le c(r) n^{1-\alpha}
\bigg( \prod_{l=1}^{\lfloor s/2\rfloor}\Big(n^{1-l}\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg) \\
& \qquad \cdot \bigg( \prod_{l=\lfloor s/2\rfloor+1}^{r}\Big(n^{1-\lfloor s/2\rfloor}\delta_n^{2(l-\lfloor s/2\rfloor)} \mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{\lfloor s/2\rfloor} \Big]\Big)^{k_l} \bigg)\\
& \qquad \cdot \mathbb{E}_{\delta_n,n}\Big[ |X_1|^{2\alpha} \Big] \mathbb{P}\big(\{|X_1|<\delta_n \sqrt{n} \, \}\big)^{n-u(k_\cdot)-1} \\
&\le c(r) n^{1-\alpha+1-\lfloor s/2\rfloor} \delta_n^2 \\
&= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix} \big(n^{-({s}-2)/2}\big).
\end{align*}
Thus, the upper part of the sum reduces to
\begin{equation}\label{eq.j-ge}
\begin{split}
& n^{-(\alpha+r)} \sum_{*(k_\cdot,r,\cdot)} {n \choose k_{1},\ldots ,k_{r},(n-u(k_\cdot))} \frac{r!}{1!^{k_1}\cdots r!^{k_r}} \big(n-u(k_\cdot)\big) \mathbbm{1}_{\{k_{\lfloor s/2\rfloor+1}=\dots=k_r=0\}}\\
&\qquad \cdot \bigg( \prod_{l=1}^{r}\Big(\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg) \mathbb{E}_{\delta_n,n}\Big[ |X_1|^{2\alpha} \Big] \mathbb{P}\big(\{|X_1|<\delta_n \sqrt{n} \, \}\big)^{n-u(k_\cdot)-1}\\
&\quad+\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix} \big(n^{-({s}-2)/2}\big).
\end{split}
\end{equation}
Now to the lower part of the sum in \labelcref{eq.j-split}. We treat the summands separately. For any $j$, define $l^*=l^*(j)\in\{1,\dots,r\}$ as the power of the term $(X_{j}^2-1 )$ within the products in \labelcref{eq.j-split}. in turn, for any $l^*$, let $j^*$ be the smallest index with the power $l^*$, i.e. $j^*=j^*(l^*):=\argmin\{j:l^*(j)=l^*\}$. With this assignment $j^*=\sum_{b=1}^{{l^*}-1}k_b+1$.
We use that $X_1,\dots,X_n$ are i.i.d. and \labelcref{eq.A_{n,1}-exp}, which yields
\begin{align*}
\lefteqn{\mathbb{E}\bigg[ \mathbbm{1}_{A_{n,1}} \bigg( \prod_{l=1}^{r} \prod_{a=1}^{k_l}\Big(X_{\sum_{b=1}^{l-1}k_b+a}^2-1 \Big)^{l} \bigg) |X_j|^{2\alpha} \bigg]}\quad\\*
&=\mathbb{E}\bigg[ \mathbbm{1}_{A_{n,1}} \bigg( \prod_{l=1}^{r} \prod_{a=1}^{k_l}\Big(X_{\sum_{b=1}^{l-1}k_b+a}^2-1 \Big)^{l} \bigg) |X_{\sum_{b=1}^{{l^*}-1}k_b+1}|^{2\alpha} \bigg]\\
&= \bigg( \prod_{l=1 , ~ l \ne l^*}^{r}\Big(\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg) \Big(\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l^*} \Big]\Big)^{k_{l^*}-1}\\
& \qquad \cdot \mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l^*}|X_1|^{2\alpha} \Big] \mathbb{P}\big(\{|X_1|<\delta_n \sqrt{n} \, \}\big)^{n-u(k_\cdot)}.
\end{align*}
Let $2\alpha+2l^*>{s}$. These summands of $\sum_{*(k_\cdot,r,\cdot)}$ in \labelcref{eq.expanded} can be bounded by
\begin{align*}
&c(r) n^{-\alpha-r+u(k_\cdot)}
\bigg( \prod_{l=1 , ~ l \ne l^*}^{r}\Big(\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg) \Big(\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l^*} \Big]\Big)^{k_{l^*}-1}\\*
& \qquad \cdot \mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l^*}|X_1|^{2\alpha} \Big] \mathbb{P}\big(\{|X_1|<\delta_n \sqrt{n} \, \}\big)^{n-u(k_\cdot)} \\
&=c(r) n^{-\alpha}
\bigg( \prod_{l=1 , ~ l \ne l^*}^{r}\Big(n^{1-l}\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg) \Big(n^{1-l^*}\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l^*} \Big]\Big)^{k_{l^*}-1}\\
& \qquad \cdot n^{1-l^*} \mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l^*}|X_1|^{2\alpha} \Big] \mathbb{P}\big(\{|X_1|<\delta_n \sqrt{n} \, \}\big)^{n-u(k_\cdot)} \\
&\le c(r) n^{-\alpha}
\bigg( \prod_{l=1 , ~ l \ne l^*}^{r}\Big(\delta_n^{2(l-1)}\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big) \Big]\Big)^{k_l} \bigg) \Big(\delta_n^{2(l^*-1)}\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big) \Big]\Big)^{k_{l^*}-1}\\
& \qquad \cdot n^{1-l^*} \mathbb{E}_{\delta_n,n}\Big[ \big(|X_1|^{2l^*+2\alpha}+|X_1|^{2\alpha} \big) \Big] \mathbb{P}\big(\{|X_1|<\delta_n \sqrt{n} \, \}\big)^{n-u(k_\cdot)} \\
&\le c(r) n^{-\alpha}
\bigg( \prod_{l=1 , ~ l \ne l^*}^{r}\Big(\delta_n^{2(l-1)}\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big) \Big]\Big)^{k_l} \bigg) \Big(\delta_n^{2(l^*-1)}\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big) \Big]\Big)^{k_{l^*}-1}\\
& \qquad \cdot n^{1-l^*} (\delta_n \sqrt{n} \, )^{2l^*+2\alpha-{s}} \mathbb{E}_{\delta_n,n}\Big[ \big(|X_1|^{s}+|X_1|^{{s}-2l^*} \big) \Big] \mathbb{P}\big(\{|X_1|<\delta_n \sqrt{n} \, \}\big)^{n-u(k_\cdot)} \\
&\le c(r) n^{-\alpha+1-l^*+l^*+\alpha-{s}/2} \delta_n^{2l^*+2\alpha-{s}} \\
&= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix} \big(n^{-({s}-2)/2} \big)
\end{align*}
since $\delta_n\to0$.
If $2\alpha+2l^*\le{s}$, a similar procedure and bound as for the upper part of the sum in \labelcref{eq.j-split} apply. To this end, assume that $k_l>0$ for at least one $l \ge \lfloor s/2\rfloor+1$. Furthermore, in the following formula we use the fact $2\le\alpha\le m/2$. With $l^*$ as above, these summands of \labelcref{eq.expanded} can be bounded by
\begin{align*}
&c(r) n^{-\alpha-r+u(k_\cdot)}
\bigg( \prod_{l=1 , ~ l \ne l^*}^{r}\Big(\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg) \Big(\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l^*} \Big]\Big)^{k_{l^*}-1}\\*
& \quad \cdot \mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l^*}|X_1|^{2\alpha} \Big] \mathbb{P}\big(\{|X_1|<\delta_n \sqrt{n} \, \}\big)^{n-u(k_\cdot)} \\*
&=c(r) n^{-\alpha}
\bigg( \prod_{l=1 , ~ l \ne l^*}^{r}\Big(n^{1-l}\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg) \Big(n^{1-l^*}\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l^*} \Big]\Big)^{k_{l^*}-1}\\
& \quad \cdot n^{1-l^*} \mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l^*}|X_1|^{2\alpha} \Big] \mathbb{P}\big(\{|X_1|<\delta_n \sqrt{n} \, \}\big)^{n-u(k_\cdot)} \\
&=c(r) n^{-\alpha}
\bigg( \prod_{l=1 , ~ l \ne l^*}^{\lfloor s/2\rfloor}\Big(n^{1-l}\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg) \Big(n^{1-l^*}\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l^*} \Big]\Big)^{k_{l^*}-1}\\
& \quad \cdot n^{1-l^*} \mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l^*}|X_1|^{2\alpha} \Big] \bigg( \prod_{l=\lfloor s/2\rfloor+1}^{r}\Big(n^{1-l}\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg)\\
& \quad \cdot \mathbb{P}\big(\{|X_1|<\delta_n \sqrt{n} \, \}\big)^{n-u(k_\cdot)} \\
&\le c(r) n^{-\alpha}
\bigg( \prod_{l=1 , ~ l \ne l^*}^{\lfloor s/2\rfloor}\Big(n^{1-l}\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg) \Big(n^{1-l^*}\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l^*} \Big]\Big)^{k_{l^*}-1}\\
& \quad \cdot n^{1-l^*} \mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l^*}|X_1|^{2\alpha} \Big] \\
&\quad \cdot \bigg( \prod_{l=\lfloor s/2\rfloor+1}^{r}\Big(n^{1-\lfloor s/2\rfloor}\delta_n^{2(l-\lfloor s/2\rfloor)} \mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{\lfloor s/2\rfloor} \Big]\Big)^{k_l} \bigg) \mathbb{P}\big(\{|X_1|<\delta_n \sqrt{n} \, \}\big)^{n-u(k_\cdot)} \\
&\le c(r) n^{-\alpha+1-l^*+1-\lfloor s/2\rfloor} \delta_n^2 \\
&= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix} \big(n^{-({s}-2)/2}\big).
\end{align*}
Summarizing, for the summands of $\sum_{*(k_\cdot,r,\cdot)}$ in \labelcref{eq.expanded} with $j\le u(k_\cdot)$ we get
\begin{align}\label{eq.j-le}
& n^{-(\alpha+r)} {n \choose k_{1},\ldots ,k_{r},(n-u(k_\cdot))} \frac{r!}{1!^{k_1}\cdots r!^{k_r}}\nonumber\\
&\quad \qquad \cdot \mathbb{E}\bigg[ \mathbbm{1}_{A_{n,1}} \bigg( \prod_{l=1}^{r} \prod_{a=1}^{k_l}\Big(X_{\sum_{b=1}^{l-1}k_b+a}^2-1 \Big)^{l} \bigg) |X_j|^{2\alpha} \bigg]\nonumber\\
&\quad = n^{-(\alpha+r)} {n \choose k_{1},\ldots ,k_{r},(n-u(k_\cdot))} \frac{r!}{1!^{k_1}\cdots r!^{k_r}} \mathbbm{1}_{\{k_{\lfloor s/2\rfloor+1}=\dots=k_r=0,\, 2\alpha+2l^*\le{s}\}}\\
&\quad \qquad \cdot \bigg( \prod_{l=1 , ~ l \ne l^*}^{r}\Big(\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg) \Big(\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l^*} \Big]\Big)^{k_{l^*}-1}\nonumber\\
&\quad \qquad \cdot \mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l^*}|X_1|^{2\alpha} \Big] \mathbb{P}\big(\{|X_1|<\delta_n \sqrt{n} \, \}\big)^{n-u(k_\cdot)}\nonumber\\
&\quad\quad+\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix} \big(n^{-({s}-2)/2}\big).\nonumber
\end{align}
Putting together \labelcref{eq.j-ge,eq.j-le,eq.j-split,eq.expanded}, we get
\begin{align}\label{eq.together-indicator}
&n^{-(\alpha+r)} \, \mathbb{E}\Big[ \mathbbm{1}_{A_{n,1}} \Big(\sum (X_j^2-1)\Big)^{r}\Big(\sum|X_j|^{2\alpha}\Big)\Big]\\
&=n^{-(\alpha+r)} \sum_{*(k_\cdot,r,\cdot)} {n \choose k_{1},\ldots ,k_{r},(n-u(k_\cdot))} \frac{r!}{1!^{k_1}\cdots r!^{k_r}} \big(n-u(k_\cdot)\big) \mathbbm{1}_{\{k_{\lfloor s/2\rfloor+1}=\dots=k_r=0\}}\nonumber\\*
&\qquad \cdot \bigg( \prod_{l=1}^{r}\Big(\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg) \mathbb{E}_{\delta_n,n}\Big[ |X_1|^{2\alpha} \Big] \mathbb{P}\big(\{|X_1|<\delta_n \sqrt{n} \, \}\big)^{n-u(k_\cdot)-1}\nonumber\\
& \quad + n^{-(\alpha+r)} \sum_{*(k_\cdot,r,\cdot)} {n \choose k_{1},\ldots ,k_{r},(n-u(k_\cdot))} \frac{r!}{1!^{k_1}\cdots r!^{k_r}} \nonumber\\
& \qquad \cdot \Bigg( \sum_{j=1}^{u(k_\cdot)} \mathbbm{1}_{\{k_{\lfloor s/2\rfloor+1}=\dots=k_r=0,\, 2\alpha+2l^*\le{s}\}} \bigg( \prod_{l=1 , ~ l \ne l^*}^{r}\Big(\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg) \nonumber\\
& \qquad \cdot \Big(\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l^*} \Big]\Big)^{k_{l^*}-1} \mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l^*}|X_1|^{2\alpha} \Big] \mathbb{P}\big(\{|X_1|<\delta_n \sqrt{n} \, \}\big)^{n-u(k_\cdot)} \Bigg)\nonumber\\*
& \quad +\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix} \big(n^{-({s}-2)/2}\big).\nonumber
\end{align}
Note that for $a=0,\dots, \lfloor s/2\rfloor$ and $b=0,\dots, \lfloor s/2\rfloor$ with $2a+2b\le{s}$, the expectations including the complementary indicator are bounded by
\begin{align}\label{eq.X_1^a*X_1^b-ge}
\lefteqn{\bigg|\mathbb{E}\Big[ \mathbbm{1}_{\{|X_1|\ge\delta_n \sqrt{n} \, \}}\big(X_1^2-1 \big)^{a} |X_1|^{2b} \Big] \bigg|}\quad\nonumber\\*
&\le \mathbb{E}\Big[ \mathbbm{1}_{\{|X_1|\ge\delta_n \sqrt{n} \, \}}\big(|X_1|^{2a}+1 \big) |X_1|^{2b} \Big] \nonumber\\
&\le \mathbb{E}\Big[ \mathbbm{1}_{\{|X_1|\ge\delta_n \sqrt{n} \, \}} (\delta_n \sqrt{n} \, )^{-(s-2a-2b)} \big(|X_1|^s+|X_1|^{s-2a} \big) \Big] \\
&\le c(0) n^{-(s/2-a-b)} \, \mathbb{E}\Big[ \mathbbm{1}_{\{|X_1|\ge\delta_n \sqrt{n} \, \}} \delta_n ^{-s} |X_1|^s \Big] \nonumber\\
&=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(s/2-a-b)}\big)\nonumber
\end{align}
by \labelcref{eq.delta_n-def}. In particular for $a=b=0$, $\mathbb{P}\big(\{|X_1|\ge\delta_n \sqrt{n} \, \}\big)=\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-s/2}\big)$.
For any $l_0=0,\dots,\lfloor s/2\rfloor$, the presence of a factor $\mathbb{E}_{\delta_n,n}\big[ \big(X_1^2-1 \big)^{l_0} \big]$ in a summand of \labelcref{eq.together-indicator} implies $k_{l_0}\ge1$. Transitioning from $\mathbb{E}_{\delta_n,n}\big[ \big(X_1^2-1 \big)^{l_0} \big]$ to $\mathbb{E}\big[\big(X_1^2-1 \big)^{l_0} \big]$ in one of its appearances leads us to study the additional summand in \labelcref{eq.together-indicator}
\begin{align*}
& n^{-(\alpha+r)} {n \choose k_{1},\ldots ,k_{r},(n-u(k_\cdot))} \frac{r!}{1!^{k_1}\cdots r!^{k_r}} \big(n-u(k_\cdot)\big) \mathbbm{1}_{\{k_{\lfloor s/2\rfloor+1}=\dots=k_r=0\}}\\
&\qquad \cdot \bigg( \prod_{l=1,~l\ne l_0}^{r}\Big(\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg) \mathbb{E}_{\delta_n,n}\Big[ |X_1|^{2\alpha} \Big] \mathbb{P}\big(\{|X_1|<\delta_n \sqrt{n} \, \}\big)^{n-u(k_\cdot)-1}\\
&\qquad \cdot \Big(\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l_0} \Big]\Big)^{k_{l_0}-1}
\bigg(- \mathbb{E}\Big[ \mathbbm{1}_{\{|X_1|\ge\delta_n \sqrt{n} \, \}} \big(X_1^2-1 \big)^{l_0} \Big]\bigg)
\end{align*}
in the upper part and in the lower part correspondingly.
Using \labelcref{eq.X_1^a*X_1^b-ge}, this summand is bounded in absolute value by
\begin{align*}
&c(m) n^{-(\alpha+r)} n^{u(k_\cdot)} n
\mathbb{E}\Big[ \mathbbm{1}_{\{|X_1|\ge\delta_n \sqrt{n} \, \}} \big(X_1^2-1 \big)^{l_0} \Big]\\
&\quad\le c(m) n^{-\alpha-r+u(k_\cdot)+1}
\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-(s/2-l_0)}\big)\\
&\quad= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-\alpha-r+u(k_\cdot)+1-s/2+l_0}\big).
\end{align*}
Using the facts $\alpha\ge 2$ and $k_{l_0}\ge1$, we examine the power of $n$
\begin{align*}
-\alpha-r+u(k_\cdot)+1-s/2+l_0
&\le -2-r+u(k_\cdot)+1-s/2+(l_0-1)k_{l_0}+1\\
&= -s/2-r+(k_1+\dots+k_{l_0}+\dots+k_r)+(l_0-1)k_{l_0}\\
&= -s/2-r+(k_1+\dots+l_0k_{l_0}+\dots+k_r)\\
&\le -s/2.
\end{align*}
Thus, leaving out the indicator in the factor $\mathbb{E}_{\delta_n,n}\big[ \big(X_1^2-1 \big)^{l_0} \big]$ produces an error of order $\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-s/2}\big)$ in the upper part of \labelcref{eq.together-indicator}. In the lower part, the factor $n-u(k_\cdot)$ is missing and the order therefore even smaller.
Repeating this argument consecutively (note that this requires $c(r)$ steps at most) until no $\mathbb{E}_{\delta_n,n}\big[ \big(X_1^2-1 \big)^{l} \big]$ is left anymore, we arrive at
\begin{align*}
&n^{-(\alpha+r)} \sum_{*(k_\cdot,r,\cdot)} {n \choose k_{1},\ldots ,k_{r},(n-u(k_\cdot))} \frac{r!}{1!^{k_1}\cdots r!^{k_r}} \big(n-u(k_\cdot)\big) \mathbbm{1}_{\{k_{\lfloor s/2\rfloor+1}=\dots=k_r=0\}}\\*
&\quad\qquad \cdot \bigg( \prod_{l=1}^{r}\Big(\mathbb{E}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg) \mathbb{E}_{\delta_n,n}\Big[ |X_1|^{2\alpha} \Big] \mathbb{P}\big(\{|X_1|<\delta_n \sqrt{n} \, \}\big)^{n-u(k_\cdot)-1}\\
&\quad \quad + n^{-(\alpha+r)} \sum_{*(k_\cdot,r,\cdot)} {n \choose k_{1},\ldots ,k_{r},(n-u(k_\cdot))} \frac{r!}{1!^{k_1}\cdots r!^{k_r}} \\
&\quad \qquad \cdot \Bigg( \sum_{j=1}^{u(k_\cdot)} \mathbbm{1}_{\{k_{\lfloor s/2\rfloor+1}=\dots=k_r=0,\, 2\alpha+2l^*\le{s}\}} \bigg( \prod_{l=1 , ~ l \ne l^*}^{r}\Big(\mathbb{E}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg) \\
&\quad \qquad \cdot \Big(\mathbb{E}\Big[ \big(X_1^2-1 \big)^{l^*} \Big]\Big)^{k_{l^*}-1} \mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l^*}|X_1|^{2\alpha} \Big] \mathbb{P}\big(\{|X_1|<\delta_n \sqrt{n} \, \}\big)^{n-u(k_\cdot)} \Bigg)\\*
&\quad \quad +\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix} \big(n^{-({s}-2)/2}\big).
\end{align*}
Similar in spirit, we consecutively transition from $\mathbb{P}\big(\{|X_1|\ge\delta_n \sqrt{n} \, \}\big)$ to 1 at the cost of at most $\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix} \big(n^{-{s}/2}\big)$, from $\mathbb{E}_{\delta_n,n}\Big[ |X_1|^{2\alpha} \Big]$ to $\mathbb{E}\Big[ |X_1|^{2\alpha} \Big]$ at the cost of at most $\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix} \big(n^{-({s}-2)/2}\big)$ because
\begin{align*}
-\alpha-r+u(k_\cdot)+1-s/2+\alpha \le -(s-2)/2,
\end{align*}
and from $\mathbb{E}_{\delta_n,n}\Big[ \big(X_1^2-1 \big)^{l^*}|X_1|^{2\alpha} \Big]$ to $\mathbb{E}\Big[ \big(X_1^2-1 \big)^{l^*}|X_1|^{2\alpha} \Big]$ at the cost of at most $\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix} \big(n^{-({s}-2)/2}\big)$ because
\begin{align*}
-\alpha-r+u(k_\cdot)-s/2+{l^*}+\alpha
&\le -s/2-r+(k_1+\dots+k_{l^*}+\dots+k_r)+({l^*}-1)k_{l^*}+1\\
&= -s/2-r+(k_1+\dots+{l^*}k_{l^*}+\dots+k_r)+1\\
&\le -(s-2)/2.
\end{align*}
As a consequence, we get from \labelcref{eq.together-indicator}
\begin{align}\label{eq.together}
\lefteqn{n^{-(\alpha+r)} \mathbb{E}\Big[ \mathbbm{1}_{A_{n,1}} \Big(\sum (X_j^2-1)\Big)^{r}\Big(\sum|X_j|^{2\alpha}\Big)\Big]}\quad\nonumber\\*
&=n^{-(\alpha+r)} \sum_{*(k_\cdot,r,\cdot)} {n \choose k_{1},\ldots ,k_{r},(n-u(k_\cdot))} \frac{r!}{1!^{k_1}\cdots r!^{k_r}} \nonumber\\
&\qquad \cdot \big(n-u(k_\cdot)\big) \mathbbm{1}_{\{k_{\lfloor s/2\rfloor+1}=\dots=k_r=0\}} \bigg( \prod_{l=1}^{r}\Big(\mathbb{E}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg) \mathbb{E}\Big[ |X_1|^{2\alpha} \Big]\nonumber\\
&\quad+ n^{-(\alpha+r)} \sum_{*(k_\cdot,r,\cdot)} {n \choose k_{1},\ldots ,k_{r},(n-u(k_\cdot))} \frac{r!}{1!^{k_1}\cdots r!^{k_r}} \\
& \qquad \cdot \sum_{j=1}^{u(k_\cdot)} \mathbbm{1}_{\{k_{\lfloor s/2\rfloor+1}=\dots=k_r=0,\, 2\alpha+2l^*\le{s}\}} \bigg( \prod_{l=1 , ~ l \ne l^*}^{r}\Big(\mathbb{E}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg) \nonumber\\
& \qquad \cdot \Big(\mathbb{E}\Big[ \big(X_1^2-1 \big)^{l^*} \Big]\Big)^{k_{l^*}-1} \mathbb{E}\Big[ \big(X_1^2-1 \big)^{l^*}|X_1|^{2\alpha} \Big]\nonumber\\*
& \quad +\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix} \big(n^{-({s}-2)/2}\big).\nonumber
\end{align}
Now we return to the remainder of the Taylor expansion in \labelcref{eq.E-expanded} and shall prove that this term is of order $\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big)$. Recall that $\xi$ is a (random) intermediate value between $1$ and $n^{-1} V_n^2$ and ${h}=2\lceil \frac{m-2\alpha+1}{2}\rceil$. Note that $\xi\ge 1/2$ on $A_{n,2}$ and since $2\le\alpha\le m/2$, $\mathbbm{1}_{A_{n,2}}\xi^{-(\alpha+{h})}\le c(m)$. Together with $|t_{{h},2\alpha}|\le c(m)$, this yields
\begin{align*}
&\bigg|\mathbb{E}\Big[ \mathbbm{1}_{A_{n,1}\cap A_{n,2}} (-1)^{{h}} n^{-(\alpha+{h})} t_{{h},2\alpha} \xi^{-(\alpha+{h})} \Big(\sum (X_j^2-1)\Big)^{{h}}\Big(\sum|X_j|^{2\alpha}\Big) \Big]\bigg| \\
&\quad\le c(m) n^{-(\alpha+{h})} \, \mathbb{E}\Big[ \mathbbm{1}_{A_{n,1}} \Big(\sum (X_j^2-1)\Big)^{{h}}\Big(\sum|X_j|^{2\alpha}\Big) \Big].
\end{align*}
If we trace through the arguments above, we see that they are valid for all $r=0,\dots,m$. Thus, all steps are also valid for the remainder of the Taylor expansion, where $r={h}$. By \labelcref{eq.together}, the expression above thus is equal to
\begin{align}\label{eq.remainder-together}
&c(m)\Bigg(n^{-(\alpha+r)} \sum_{*(k_\cdot,r,\cdot)} {n \choose k_{1},\ldots ,k_{r},(n-u(k_\cdot))} \frac{r!}{1!^{k_1}\cdots r!^{k_r}} \nonumber\\*
&\qquad \cdot \big(n-u(k_\cdot)\big) \mathbbm{1}_{\{k_{\lfloor s/2\rfloor+1}=\dots=k_r=0\}} \bigg( \prod_{l=1}^{r}\Big(\mathbb{E}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg) \mathbb{E}\Big[ |X_1|^{2\alpha} \Big]\nonumber\\
&\quad + n^{-(\alpha+r)} \sum_{*(k_\cdot,r,\cdot)} {n \choose k_{1},\ldots ,k_{r},(n-u(k_\cdot))} \frac{r!}{1!^{k_1}\cdots r!^{k_r}} \\
&\qquad \cdot \sum_{j=1}^{u(k_\cdot)} \mathbbm{1}_{\{k_{\lfloor s/2\rfloor+1}=\dots=k_r=0,\, 2\alpha+2l^*\le{s}\}} \bigg( \prod_{l=1 , ~ l \ne l^*}^{r}\Big(\mathbb{E}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg) \nonumber\\
&\qquad \cdot \Big(\mathbb{E}\Big[ \big(X_1^2-1 \big)^{l^*} \Big]\Big)^{k_{l^*}-1} \mathbb{E}\Big[ \big(X_1^2-1 \big)^{l^*}|X_1|^{2\alpha} \Big]\Bigg)\nonumber\\*
&\quad +\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix} \big(n^{-({s}-2)/2}\big)\nonumber
\end{align}
with $r={h}$.
Note that $\mathbb{E} X_1^2=1$. Thus in the first part, all summands with $k_1>0$ vanish and in the second part, all summands with $k_1>1$ vanish. This results in
\[
u(k_\cdot)=k_1+k_2+\dots+k_r=\tfrac12(2k_2+\dots+2k_r)\le r/2
\]
for the first part and
\[
u(k_\cdot)=k_1+k_2+\dots+k_r\le1+\tfrac12(2k_2+\dots+2k_r)\le 1+r/2
\]
for the second part.
Now we want to determine the order in $n$ of the terms in \labelcref{eq.remainder-together}. Note that all appearing moments are finite. Using $r={h}\ge m-2\alpha+1$, we can focus on the power of $n$ which is in the first part
\begin{align*}
-\alpha-r+u(k_\cdot)+1
&\le -\alpha-r+r/2+1
=-\alpha-r/2+1\\
&\le-\alpha-(m-2\alpha+1)/2+1
=-(m-1)/2
\end{align*}
and in the second part
\begin{align*}
-\alpha-r+u(k_\cdot)
\le -\alpha-r+1+r/2
=-\alpha-r/2+1
\le-(m-1)/2.
\end{align*}
Thus, all terms (at most $c(m)$ many) from \labelcref{eq.remainder-together} are of order $\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix} \big(n^{-({s}-2)/2}\big)$.
Ultimately, merging \labelcref{eq.E-expanded,eq.A_{n,2}-raus,eq.together,eq.remainder-together}, we arrive at
\begin{align}\label{eq.all}
\lefteqn{\mathbb{E}\Big[ V_n^{-2\alpha}\sum|X_j|^{2\alpha} \Big]}\quad\nonumber\\
&=\sum_{r=0}^{{h}-1} (-1)^r n^{-(\alpha+r)} t_{r,2\alpha} \sum_{*(k_\cdot,r,\cdot)} {n \choose k_{1},\ldots ,k_{r},(n-u(k_\cdot))} \frac{r!}{1!^{k_1}\cdots r!^{k_r}} \nonumber\\
&\qquad \cdot \big(n-u(k_\cdot)\big) \mathbbm{1}_{\{k_{\lfloor s/2\rfloor+1}=\dots=k_r=0\}} \bigg( \prod_{l=1}^{r}\Big(\mathbb{E}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg) \mathbb{E}\Big[ |X_1|^{2\alpha} \Big]\nonumber\\
&\quad+ \sum_{r=0}^{{h}-1} (-1)^r n^{-(\alpha+r)} t_{r,2\alpha} \sum_{*(k_\cdot,r,\cdot)} {n \choose k_{1},\ldots ,k_{r},(n-u(k_\cdot))} \frac{r!}{1!^{k_1}\cdots r!^{k_r}} \\
&\qquad \cdot \sum_{j=1}^{u(k_\cdot)} \mathbbm{1}_{\{k_{\lfloor s/2\rfloor+1}=\dots=k_r=0,\, 2\alpha+2l^*\le{s}\}} \bigg( \prod_{l=1 , ~ l \ne l^*}^{r}\Big(\mathbb{E}\Big[ \big(X_1^2-1 \big)^{l} \Big]\Big)^{k_l} \bigg) \nonumber\\
&\qquad \cdot \Big(\mathbb{E}\Big[ \big(X_1^2-1 \big)^{l^*} \Big]\Big)^{k_{l^*}-1} \mathbb{E}\Big[ \big(X_1^2-1 \big)^{l^*}|X_1|^{2\alpha} \Big]\nonumber\\
&\quad +\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix} \big(n^{-({s}-2)/2}\big).\nonumber
\end{align}
All expectations in \labelcref{eq.all} can be calculated explicitly as no power of $X_1$ is greater than $s$. Using $u(k_\cdot)\le r, \alpha\in\mathbb{N}$ and $\alpha\ge 2$, this results in an expansion in terms of orders $n^{-1},n^{-2},\dots,n^{-\lfloor m-2/2\rfloor}$ with moments up to order $m$ and a remainder of order $\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix} \big(n^{-({s}-2)/2}\big)$. By \labelcref{eq.tk-2r,eq.tlambda} for $k\ge4$, the expectation of $\tilde{\lambda}_{k,n}$ and its powers admit the same expansion up to the respective coefficient from \labelcref{eq.tk-2r}. In \cref{r.V_n-lambda}, this expansion is performed exemplarily for $k=4$ and $k=6$. By \labelcref{eq.Phi^tP} and \labelcref{eq.tP}, $\mathbb{E}\big[\Phi^{\tilde{P}}_{m,n}(x)\big]$ admits a similar representation. By the definition of $Q_r$ in \labelcref{eq.Q-def}, the expansion terms of order $n^{-r/2}$ are assigned to their respective $Q_r$ (or to the remainder $\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix} \big(n^{-({s}-2)/2}\big)$). Note that $Q_{r}=0$ for uneven $r$. That is, all the non-remainder terms in $$\mathbb{E}\big[\Phi^{\tilde{P}}_{m,n}(x)\big]$$ are exactly the summands appearing in $\Phi^Q_{m,n}$, defined in \labelcref{eq.Phi^Q-def}, and therefore cancel each other out in \labelcref{eq.E-tP-Q,eq.E-tp-q}.
What still needs to be examined is the dependence of the remainder on the argument $x$. In $\tilde{P}_{k,n}$ (as well as in $\tilde{p}_{k,n}$), the $\tilde{\lambda}_{k,n}$ are multiplied by $\phi$ and a Hermite polynomial of order $l\le2m$. For $l\le2m$, we can bound
\[
\sup_{x} \exp(x^2/4)\phi(x)H_l(x)\le c(l).
\]
As this non-random factor remains unaffected by the expectation, it also appears in the remainder such that \labelcref{eq.E-tP-Q,eq.E-tp-q} hold.
\end{proof}
\subsection{Further proofs for Section \ref{ch.distr}}\label{app.proofs.distr}
\begin{proof}[Proof of \cref{l.E.mom}]
We prove this lemma by an approach similar to the one used in the classical proof of Markov's inequality. Without loss of generality let $a_1=\max\{a_1,\dots,a_l\}$. By using $M_{n,l}=\max_{j= l+1,l+2,\dots,n} |X_j|$ and $j_0\in\{1,\dots,n\}$ being the smallest index with $|X_{j_0}|=M_n$, we see \begin{align*}
\lefteqn{ \mathbb{E}\Big[\mathbbm{1}_{\big\{M_n\ge \sqrt{\frac{n}{\log(n)\, \eta}}\big\}} |X_1|^{a_1} \cdots |X_l|^{a_l}\Big]}\quad\\
&= \sum_{j=1}^l \mathbb{E}\Big[ \mathbbm{1}_{\{j_0=j\}}\mathbbm{1}_{\big\{M_n\ge \sqrt{\frac{n}{\log(n)\, \eta}}\big\}} |X_1|^{a_1} \cdots |X_l|^{a_l}\Big]\\
&\quad+ \mathbb{E}\Big[ \sum_{j=l+1}^n \mathbbm{1}_{\{j_0=j\}}\mathbbm{1}_{\big\{M_n\ge \sqrt{\frac{n}{\log(n)\, \eta}}\big\}} |X_1|^{a_1} \cdots |X_l|^{a_l}\Big]\\
&\le l \, \mathbb{E}\Big[\mathbbm{1}_{\{j_0=1\}}\mathbbm{1}_{\big\{M_n\ge \sqrt{\frac{n}{\log(n)\, \eta}}\big\}} |X_1|^{a_1} \cdots |X_l|^{a_l}\Big]\\
&\quad+ \mathbb{E}\Big[\mathbbm{1}_{\{j_0>l\}}\mathbbm{1}_{\big\{M_n\ge \sqrt{\frac{n}{\log(n)\, \eta}}\big\}} |X_1|^{a_1} \cdots |X_l|^{a_l}\Big]\\
&\le \, l \, \big(\tfrac{n}{\log(n)\, \eta}\big)^{-({s}-a_1)/2} \, \mathbb{E}\Big[\mathbbm{1}_{\{j_0=1\}}\mathbbm{1}_{\big\{M_n\ge \sqrt{\frac{n}{\log(n)\, \eta}}\big\}} |X_1|^{{s}} |X_2|^{a_2} \cdots |X_l|^{a_l}\Big]\\
&\quad+ \mathbb{E}\Big[\mathbbm{1}_{\{j_0>l\}}\mathbbm{1}_{\big\{M_{n,l}\ge \sqrt{\frac{n}{\log(n)\, \eta}}\big\}} |X_1|^{a_1} \cdots |X_l|^{a_l}\Big]\\
&\le \, l \, \big(\tfrac{n}{\log(n)\, \eta}\big)^{-({s}-a_1)/2} \, \mathbb{E}\Big[ |X_1|^{{s}} |X_2|^{a_2} \cdots |X_l|^{a_l}\Big]\\
&\quad+ \mathbb{E}\Big[\mathbbm{1}_{\big\{M_{n,l}\ge \sqrt{\frac{n}{\log(n)\, \eta}}\big\}} |X_1|^{a_1} \cdots |X_l|^{a_l}\Big]\\
&= l \, \eta^{({s}-a_1)/2}\, (\log n)^{({s}-a_1)/2} n^{-({s}-a_1)/2} \mathbb{E} |X_1|^{{s}} \mathbb{E} |X_1|^{a_2} \cdots \mathbb{E} |X_1|^{a_l}\\
&\quad+ \mathbb{P}\big(M_{n,l}\ge \sqrt{\tfrac{n}{\log(n)\, \eta}} \, \big) \mathbb{E} |X_1|^{a_1} \cdots \mathbb{E} |X_1|^{a_l}\\
& = \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\Big( n^{-({s}-\max\{2,a_1\})/2} \, (\log n)^{{s}/2} \Big), \end{align*} where we used \labelcref{eq.Mn-an} in the last step. \end{proof}
\begin{proof}[Proof of \cref{l.E.tL}] Using $(\delta_n)$ from \cref{r.moment}, we get by \labelcref{eq.Lle1}, \labelcref{eq.Vn-bound-1/2} and \labelcref{eq.Mn-delta} \begin{align*} \mathbb{E}\left[ \tilde{L}_{k,n}\right]
&\le\mathbb{E}\left[ \mathbbm{1}_{\{V_n^2 \le \frac n2\}}\right]\\ &\quad+\mathbb{E}\left[ \mathbbm{1}_{\{M_n\ge \delta_n \sqrt{n} \, \}}\right]\\
&\quad+\mathbb{E}\left[ \mathbbm{1}_{\{V_n^2 > n/2, M_n< \delta_n \sqrt{n} \, \}} V_n^{-k}M_n^{k-{s}}\sum|X_j|^{{s}} \right]\\
&\le\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big)\\ &\quad+ \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big)\\
&\quad+ 2^{k/2} n^{-({s}-2)/2} \delta_n^{k-{s}} \mathbb{E}\big[ |X_1|^{{s}}\big]\\ &= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big) \end{align*} since $\delta_n\to0$. \end{proof}
\begin{proof}[Proof of \cref{l.exp.B_n}] We split up the expectation and get by \labelcref{eq.Vn-bound-1/2,eq.Mn-an} \begin{align*} \mathbb{E}\left[ \exp\big(-\eta\,B_n^2\big)\right]
&\le\mathbb{E}\left[ \mathbbm{1}_{\{V_n^2 \le \frac n2\}}\right]\\ &\quad+\mathbb{E}\left[ \mathbbm{1}_{\big\{M_n\ge \sqrt{\frac{n\,\eta}{\log(n)\, {s}}}\big\}}\right]\\ &\quad+\mathbb{E}\left[ \mathbbm{1}_{\big\{V_n^2 > n/2, M_n< \sqrt{\frac{n\,\eta}{\log(n)\, {s}}}\big\}} \exp\big(-\eta\,V_n^2\,M_n^{-2}\big) \right]\\ &\le\mathbb{P}\left(V_n^2 \le \tfrac n2\right)\\ &\quad+\mathbb{P}\left(M_n\ge \sqrt{\tfrac{n\,\eta}{\log(n)\, {s}}}\,\right)\\ &\quad+\mathbb{E}\left[ \mathbbm{1}_{\big\{V_n^2 > n/2, M_n< \sqrt{\frac{n\,\eta}{\log(n)\, {s}}}\big\}} \exp\big(-\eta\,\tfrac n2\,\tfrac{\log(n)\,s}{n\,\eta}\big)\right]\\ &\le\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big(n^{-({s}-2)/2}\big)\\ &\quad+\begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-({s}-2)/2} \, (\log n)^{{s}/2} (\eta\wedge 1)^{-s/2} \big)\\ &\quad+n^{-{s}/2}\\ &= \begin{smallmatrix}\!{\mathcal {O}}\!\end{smallmatrix}\big( n^{-({s}-2)/2} \, (\log n)^{{s}/2} (\eta\wedge 1)^{-s/2} \big). \end{align*} \end{proof}
\subsection{Further proofs for Section \ref{ch.density}}\label{app.proofs.density}
\begin{proof}[Proof of \cref{p.prop}] Take $f$ to be the density of $X_1$, $Y_j:=(X_j,X_j^2)$, $B:=B_1\times B_2:=[a_1,b_1]\times[a_2,b_2]$ with $a_1<b_1$, $a_2<b_2$ in $\mathbb{R}$. Then by substitution and Fubini's theorem, \begin{align*} \lefteqn{\mathbf{P}\big(Y_1+Y_2+Y_3 \in B\big)}\quad\\* &=\mathbf{P}\Big(X_1+X_2+X_3 \in B_1, X_1^2+X_2^2+X_3^2 \in B_2\Big)\\ &=2\int_\mathbb{R}\int_\mathbb{R}\int_\mathbb{R} f(x)f(y)f(z)\mathbbm{1}\big\{x+y+z\in B_1, x^2+y^2+z^2\in B_2, x\le y\big\}\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\\ &=2\int_\mathbb{R}\int_\mathbb{R}\int_\mathbb{R} f(s-y-z)f(y)f(z)\\* &\qquad\qquad\quad\cdot\mathbbm{1}\big\{s\in B_1, (s-y-z)^2+y^2+z^2 \in B_2, (s-z)/2 \le y\big\}\,\mathrm{d}s\,\mathrm{d}y\,\mathrm{d}z\\ &=2\int_{B_1}\int_\mathbb{R}\int_\mathbb{R} f(s-y-z)f(y)f(z)\\* &\qquad\qquad\quad\cdot\mathbbm{1}\big\{ (s-y-z)^2+y^2+z^2 \in B_2, (s-z)/2 \le y\big\}\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}s. \end{align*} Define \[ v_{s,z}(y):=(s-y-z)^2+y^2+z^2=\tfrac12 \Big( \big(2y-(s-z)\big)^2+s^2+3z^2-2sz\Big). \] Note that \[ v'_{s,z}(y) = 2 (2y-s+z). \] As we only consider the regime $(s-z)/2 \le y$, the inverse is \[ v_{s,z}^{-1}(v)=\tfrac12 \Big(s-z+\sqrt{2v-s^2-3z^2+2sz}\Big)=\tfrac12 \Big(s-z+\tfrac{1}{\sqrt{3}} \sqrt{6v-2s^2-(3z-s)^2}\Big). \] Thus the expression above is equal to \begin{align*} &2\int_{B_1}\int_\mathbb{R}\int_{(s-z)/2}^\infty \frac{2(2v_{s,z}^{-1}(v_{s,z}(y))-s+z)}{2(2v_{s,z}^{-1}(v_{s,z}(y))-s+z)}f(s-v_{s,z}^{-1}(v_{s,z}(y))-z)f(v_{s,z}^{-1}(v_{s,z}(y)))f(z)\\* &\qquad\qquad\qquad\quad\cdot\mathbbm{1}\big\{ v_{s,z}(y) \in B_2 \big\}\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}s\\* &=2\int_{B_1}\int_\mathbb{R}\int_{v_{s,z}((s-z)/2)}^\infty \frac{1}{2(2v_{s,z}^{-1}(v)-s+z)} f\big(s-v_{s,z}^{-1}(v)-z\big) f(v_{s,z}^{-1}(v)) f(z) \\* &\qquad\qquad\qquad\qquad\qquad\cdot\mathbbm{1}\big\{ v \in B_2\big\}\,\mathrm{d}v\,\mathrm{d}z\,\mathrm{d}s\\ &=\int_{B_1}\int_{B_2}\int_\mathbb{R} \frac{1}{2v_{s,z}^{-1}(v)-s+z} f\big(s-v_{s,z}^{-1}(v)-z\big) f(v_{s,z}^{-1}(v)) f(z)\\* &\qquad\qquad\qquad\cdot\mathbbm{1}\big\{ v_{s,z}\big((s-z)/2\big) \le v \big\}\,\mathrm{d}z\,\mathrm{d}v\,\mathrm{d}s. \end{align*} Hence the random variable $(Y_1+Y_2+Y_3)$ has the density \[ \int_\mathbb{R} \frac{1}{2v_{s,z}^{-1}(v)-s+z} f\big(s-v_{s,z}^{-1}(v)-z\big) f(v_{s,z}^{-1}(v)) f(z) \mathbbm{1}\big\{ v_{s,z}\big((s-z)/2\big) \le v \big\}\,\mathrm{d}z \] for $s,v\in\mathbb{R}$. Note that \[ v_{s,z}((s-z)/2) \le v \quad \Leftrightarrow \quad 6v-2s^2-(3z-s)^2 \ge 0 \quad \Rightarrow \quad 6v-2s^2 \ge 0. \] We examine the integral \begin{align*} \lefteqn{\int_\mathbb{R} \frac{1}{2v_{s,z}^{-1}(v)-s+z} \mathbbm{1}\big\{ v_{s,z}((s-z)/2) \le v \big\}\,\mathrm{d}z}\quad\\* &=\int_\mathbb{R} \frac{\sqrt{3}}{\sqrt{6v-2s^2-(3z-s)^2}} \mathbbm{1}\big\{ \tfrac12 \big( s^2+3z^2-2sz\big) \le v \big\}\,\mathrm{d}z\\ &=\int_\mathbb{R} \frac{\sqrt{3}}{\sqrt{6v-2s^2-(3z-s)^2}} \mathbbm{1}\big\{ \tfrac13 \big(s-\sqrt{6v-2s^2} \big) \le z \le \tfrac13 \big(s+\sqrt{6v-2s^2}\big) \big\}\,\mathrm{d}z\\ &=\int_{\frac13 (s-\sqrt{6v-2s^2})}^{\frac13 (s+\sqrt{6v-2s^2})} \frac{\sqrt{3}}{\sqrt{6v-2s^2-(3z-s)^2}}\,\mathrm{d}z\\ &=\frac{1}{\sqrt{3}}\int_{-\sqrt{6v-2s^2}}^{\sqrt{6v-2s^2}} \frac{1}{\sqrt{6v-2s^2-w^2}}\,\mathrm{d}w=\frac{\pi}{\sqrt{3}} \end{align*} by \cref{l.int-a-w^2}. Therefore for all bounded $f$, $(\sum X_j,\sum X_j^2)$ has a bounded density for $n=3$ (or equivalently, for all $n\ge3$). Thus by \cite[Theorem 19.1]{BR76normal}, \cref{c.density-cf} is fulfilled.
\end{proof}
\begin{proof}[Proof of \cref{l.lemma4'}]
By the triangle inequality and \cref{p.lemma4}, we get for ${k=0,\dots,m}$
\begin{align*}
\lefteqn{\Big|\ddT{k} \Big(\tilde{\varphi}_{T_n'}(t)-\tilde{\varphi}_{\Phi^{\tilde{P}}_{m,n}}(t)\Big)\Big|}\quad\\
&\le \Big|\ddT{k} \Big(\tilde{\varphi}_{T_n'}(t)-\tilde{\varphi}_{T_n}(t)\Big)\Big|+ c(m)\tilde{L}_{m+1,n}e^{-t^2/6}\big(|t|^{m+1-k}+|t|^{3m-1+k}\big),
\end{align*}
where we expand the first summand by the Leibniz rule as
\begin{align*}
\Big|\ddT{k} \Big(\tilde{\varphi}_{T_n'}(t)-\tilde{\varphi}_{T_n}(t)\Big)\Big|
&\le \sum_{l=0}^{k-1} \binom{k}{l}\Big|\ddT{l}\tilde{\varphi}_{T_n}(t)\Big|\Big|\ddT{k-l}\exp(-\tfrac12 \beta_n t^2)\Big|\\
&\quad+\Big|\ddT{k}\tilde{\varphi}_{T_n}(t)\Big|\Big(1-\exp(-\tfrac12 \beta_n t^2)\Big).
\end{align*}
By \cref{p.lemma4}, \labelcref{eq.ddtl-tphi} and \labelcref{eq.Lle1},
\begin{align*}
\Big|\ddT{l}\tilde{\varphi}_{T_n}(t)\Big|
&\le \Big|\ddT{l}\tilde{\varphi}_{\Phi^{\tilde{P}}_{m,n}}(t)\Big|+c(m)\tilde{L}_{m+1,n}e^{-t^2/6}\big(|t|^{m+1-l}+|t|^{3m-1+l}\big)\\
&\le c(m)e^{-t^2/2}\big(1+|t|^{2m-4+l}\big)+c(m)e^{-t^2/6}\big(|t|^{m+1-l}+|t|^{3m-1+l}\big)\\
&\le c(m)e^{-t^2/6}\big(1+|t|^{3m-1+l}\big)
\end{align*}
for $l=0,\dots,m$. Additionally for $k>l$,
\[\Big|\ddT{k-l}\exp(-\tfrac12 \beta_n t^2)\Big|
\le c(k-l) \beta_n (1+|t|^{k-l}) \exp(-\tfrac12 \beta_n t^2)
\]
and
\[\Big(1-\exp(-\tfrac12 \beta_n t^2)\Big)\le\tfrac12 \beta_n t^2.\]
Putting everything together yields
\begin{align*}
\lefteqn{\Big|\ddT{k} \Big(\tilde{\varphi}_{T_n'}(t)-\tilde{\varphi}_{T_n}(t)\Big)\Big|}\quad\\*
&\le \sum_{l=0}^{k-1} \binom{k}{l}\Big|\ddT{l}\tilde{\varphi}_{T_n}(t)\Big|\Big|\ddT{k-l}\exp(-\tfrac12 \beta_n t^2)\Big|\\
&\quad+\Big|\ddT{k}\tilde{\varphi}_{T_n}(t)\Big|\Big(1-\exp(-\tfrac12 \beta_n t^2)\Big)\\
&\le \sum_{l=0}^{k-1} \binom{k}{l}\Big(c(m)e^{-t^2/6}\big(1+|t|^{3m-1+l}\big)\Big)\Big(c(k-l) \beta_n (1+|t|^{k-l}) \exp(-\tfrac12 \beta_n t^2)\Big)\\
&\quad+\Big(c(m)e^{-t^2/6}\big(1+|t|^{3m-1+k}\big)\Big)\Big(\tfrac12 \beta_n t^2\Big)\\
&\le c(m) \beta_n e^{-t^2/6} \big(1+|t|^{3m+1+k}\big),
\end{align*}
which proves \labelcref{eq.lemma4'}. \end{proof}
\subsection{Further remarks}\label{app.proofs.further}
\begin{remark}\label{r.non-deg}
Let $\Sigma$ be the covariance matrix of $(X_1,X_1^2)$ and assume $\mu_4<\infty$. Then the following equivalences hold:\\
$(X_1,X_1^2)$ is non-degenerate\\
$\Leftrightarrow$ $\ensuremath{\mathop{\mathrm{Var}}}\big(a_1 X_1+a_2 X_1^2)>0$ for all $a=(a_1,a_2)\in\mathbb{R}^2$ with $a\ne0$\\
$\Leftrightarrow$ $a^T \Sigma a >0$ for all $a\in\mathbb{R}^2$ with $a\ne0$\\
$\Leftrightarrow$ $\Sigma$ is positive definite. \end{remark}
\begin{remark}\label{r.E-T_n^2} In this remark, we do neither assume that $X_1$ is symmetric, nor that $\mu_2=1$. Under these conditions, we investigate if $T_n$ is normalized, expanding \cref{r.E-T_n-normalized}. For clear illustration however, let all moments be finite in the following procedure. By \labelcref{eq.V_n^-2} we expand
\begin{equation}\label{eq.Tn-exp} \begin{split} T_n^2
&=n^{-1} \mu_2^{-1} S_n^2 - n^{-2} \mu_2^{-2} S_n^2 \Big(\sum (X_j^2-\mu_2)\Big)\\ &\quad+ n^{-3} \mu_2^{-3} S_n^2 \Big(\sum (X_j^2-\mu_2)\Big)^2 + \mathcal{O}_p\big(n^{-2}\big). \end{split} \end{equation} Due to $\mu_1=0$ and $\mathbb{E} X_j^2=\mu_2$, the index of every factor within \[ \Big(\sum X_j\Big)^{2}\Big(\sum (X_j^2-\mu_2)\Big)^{k} =\sum_{j_1,\dots,j_{k+2}=1}^n X_{j_1} X_{j_2} \big(X_{j_3}^2-\mu_2\big) \cdots \big( X_{j_{k+2}}^2-\mu_2 \big) \] has to be equal to the index of another factor or the summand vanishes within the expectation. We take the expectation of each summand of \labelcref{eq.Tn-exp} separately \begin{align*} \mathbb{E} S_n^2 &= n \mu_2,\\ \mathbb{E} \Big[ S_n^2 \Big(\sum (X_j^2-\mu_2)\Big) \Big] &= n (\mu_4-\mu_2^2),\\ \mathbb{E} \Big[ S_n^2 \Big(\sum (X_j^2-\mu_2)\Big)^2 \Big]
&= n^2 \big(\mu_2\mu_4-\mu_2^3+2\mu_3^2\big) + \mathcal{O}(n) \end{align*} which leads to \begin{align*} \mathbb{E} T_n^2
&= 1 + n^{-1} 2 \mu_2^{-3} \mu_3^2 + \mathcal{O}\big(n^{-2}\big). \end{align*} \end{remark}
\subsection{Auxiliary results}\label{app.lemmas}
\begin{lemma}\label{l.int-t-exp}
Let $\nu_n\ge1$, $k=0,1,\dots$ and $0<\beta<4$. Then
\begin{equation*}
\int_{\nu_n}^\infty t^k \exp\big(- \beta t^2 /2 \big)\,\mathrm{d}t
\le c(k)\beta^{-(k+2)/2} \exp\big(- \beta \nu_n^2 /4 \big).
\end{equation*} \end{lemma} \begin{proof}
We examine
\begin{align*}
\int_{\nu_n}^\infty t^k \exp(- \beta t^2 /2)\,\mathrm{d}t
&\le \int_{\nu_n}^\infty t \, (t^2)^{\lceil\frac{k-1}{2}\rceil} \exp(- \beta t^2 /2)\,\mathrm{d}t\\
&= \frac{\lceil\frac{k-1}{2}\rceil!}{(\beta/4 )^{\lceil\frac{k-1}{2}\rceil}} \int_{\nu_n}^\infty t \, \frac{(\beta t^2 /4)^{\lceil\frac{k-1}{2}\rceil}}{\lceil\frac{k-1}{2}\rceil!} \exp(- \beta t^2 /2)\,\mathrm{d}t\\
&\le \frac{k!}{(\beta/4 )^{k/2}} \int_{\nu_n}^\infty t \Big(\sum_{l=0}^\infty \frac{(\beta t^2 /4)^{l}}{l!} \Big) \exp(- \beta t^2 /2)\,\mathrm{d}t\\
&= \frac{k!\,2^k}{\beta^{k/2}} \int_{\nu_n}^\infty t \, \exp(\beta t^2 /4) \exp(- \beta t^2 /2)\,\mathrm{d}t\\
&= \frac{k!\,2^{k+1}}{\beta^{(k+2)/2}} \exp( \beta \nu_n^2 /4).
\end{align*} \end{proof}
\begin{lemma}\label{l.ana1}
Let $f:\mathbb{R}\to\mathbb{R}$ be a measurable function with $0\le f(x)\le1$ for all $x\in\mathbb{R}$ and let $\mu$ be a probability measure on $\mathbb{R}$. Assume that $\mu(\{x:f(x)<1\})>0$. Then
\[
\int_\mathbb{R} f(x)\,\mathrm{d}\mu < 1.
\] \end{lemma} \begin{proof}
For $n\in\mathbb{N}$ we define $A_n:=\{x:f(x)<1-1/n\}$, which are increasing for ${n\to \infty}$. By continuity from below,
\[
\lim_{n\to\infty}\mu(A_n)=\mu\big(\{x:f(x)<1\}\big)>0.
\]
Thus, there exists an $N\in\mathbb{N}$ such that $\mu(A_N)>0$ and therefore,
\begin{align*}
\int_\mathbb{R} f(x)\,\mathrm{d}\mu
=\int_{A_N} f(x)\,\mathrm{d}\mu + \int_{A_N^c} f(x)\,\mathrm{d}\mu
<(1-\tfrac{1}{N}) \mu(A_N) + \mu(A_N^c)
=1-\tfrac{1}{N} \mu(A_N)<1.
\end{align*} \end{proof}
\begin{lemma}\label{l.int-a/2} Let $n\ge2$. Then \begin{align*} \int_{\frac{1}{2}\sqrt{n}}^{2\sqrt{n}} \big(1+(z-n^{1/2})^2\big)^{-a/2} \,\mathrm{d}z \le \begin{cases} c(0) \sqrt{n} , &\text{ if } a=0,\\ c(0) \log n , &\text{ if } a=1,\\ c(0) , &\text{ if } a=2. \end{cases} \end{align*} \end{lemma} \begin{proof} For $a=0$, the statement is trivial. Otherwise, \begin{align*} \int_{\frac{1}{2}\sqrt{n}}^{2\sqrt{n}} \big(1+(z-n^{1/2})^2\big)^{-a/2} \,\mathrm{d}z =\int_{-\frac{1}{2}\sqrt{n}}^{\sqrt{n}} \big(1+z^2\big)^{-a/2} \,\mathrm{d}z \le2\int_{0}^{\sqrt{n}} \big(1+z^2\big)^{-a/2} \,\mathrm{d}z. \end{align*} Now for $a=1$ and the inverse hyperbolic sine $\ensuremath{\mathop{\mathrm{arsinh}}}$, \begin{align*} \int_{0}^{\sqrt{n}} \big(1+z^2\big)^{-1/2} \,\mathrm{d}z = \ensuremath{\mathop{\mathrm{arsinh}}}(\sqrt{n}) = \log(\sqrt{n} + \sqrt{1 + n}) \le c(0) \log(n), \end{align*} and for $a=2$ \begin{align*} \int_{0}^{\sqrt{n}} \big(1+z^2\big)^{-1} \,\mathrm{d}z =\arctan(\sqrt{n}) \le \tfrac \pi 2. \end{align*} \end{proof}
\begin{lemma}\label{Taylor_L} For $L(x)=x\log x$ with $x>0$,
\begin{equation*}
L(1+u+v)=L(1+u)+v+\vartheta_1 u v+\vartheta_2 v^2
\end{equation*}
for $|u|\le 1/4, |v|\le1/4$ and $|\vartheta_j|\le2$ depending on $u$ and $v$. \end{lemma}
\begin{proof}
By Taylor expansion around $v=0$,
\begin{align*}
(1+u)\log(1+u+v)
&=(1+u)\Big(\log(1+u+0) + v (1+u+0)^{-1} \\*
&\qquad\qquad+ \tfrac{v^2}{2} (-1) (1+u+\xi_1v)^{-2} \Big)\\
&=(1+u)\log(1+u) + v + v^2 (1+u) (-\tfrac12) (1+u+\xi_1v)^{-2}
\end{align*}
with $\xi_1\in[0,1]$ depending on $u$ and $v$ and by Taylor expansion around $u+v=0$,
\begin{align*}
v\log(1+u+v)
&=v\Big(\log(1+0) + (u+v) (1+\xi_2(u+v))^{-1} \Big)\\*
&= (uv+v^2) (1+\xi_2(u+v))^{-1}
\end{align*}
with $\xi_2\in[0,1]$ depending on $u$ and $v$.
Here,
\begin{align*}
(1+\xi_2(u+v))^{-1}
&\le (1-1/2)^{-1}
=2,\ \ \
(1+\xi_2(u+v))^{-1}
\ge (1+1/2)^{-1}
= 2/3
\end{align*}
and
\begin{align*}
(1+u) \tfrac{1}{2} (1+u+\xi_1v)^{-2}
&\le (1+u) \tfrac{1}{2} (1+u-1/4)^{-2}
\le (3/4) \tfrac{1}{2} (1/2)^{-2}
= 3/2,\\*
(1+u) \tfrac{1}{2} (1+u+\xi_1v)^{-2}
&\ge (1+u) \tfrac{1}{2} (1+u+1/4)^{-2}
\ge (5/4) \tfrac{1}{2} (3/2)^{-2}
= 5/18.
\end{align*}
Consequently,
$
(1+\xi_2(u+v))^{-1}-(1+u) \tfrac{1}{2} (1+u+\xi_1v)^{-2}
\in [-5/6,31/18]$.
\end{proof}
\begin{lemma}\label{l.int-ax^2+b}
Let $a,b,l>0$. Then
\begin{align*}
\int_{-l}^l (ax^2+b)^{-3/2}\,\mathrm{d}x = 2\big(al^2+b\big)^{-1/2}\,b^{-1}\,l.
\end{align*} \end{lemma} \begin{proof}
\begin{align*}
\int_{-l}^l (ax^2+b)^{-3/2}\,\mathrm{d}x
&=2b^{-3/2} \int_{0}^l (\tfrac ab x^2+1)^{-3/2}\,\mathrm{d}x\\
&=2b^{-3/2}\sqrt{b/a} \int_{0}^{l\sqrt{a/b}} (x^2+1)^{-3/2}\,\mathrm{d}x\\
&=2b^{-3/2}\sqrt{b/a} \, \frac{l\sqrt{a/b}}{\sqrt{l^2a/b+1}} = 2\big(al^2+b\big)^{-1/2}\,b^{-1}\,l.
\end{align*} \end{proof}
\begin{lemma}\label{l.int-a-w^2}
Let $a > 0$. Then
\begin{align*}
\int_{-\sqrt{a}}^{\sqrt{a}} (a-w^2)^{-1/2} \,\mathrm{d}w
= \pi.
\end{align*} \end{lemma} \begin{proof}
\begin{align*}
\int_{-\sqrt{a}}^{\sqrt{a}} (a-w^2)^{-1/2}\,\mathrm{d}w
&=2a^{-1/2} \int_{0}^{\sqrt{a}} (1-w^2/a)^{-1/2}\,\mathrm{d}w\\
&=2a^{-1/2} a^{1/2} \int_{0}^{1} (1-w^2)^{-1/2}\,\mathrm{d}w\\
&= 2 \arcsin(1)=\pi.
\end{align*} \end{proof}
\end{appendix}
\begin{acks}[Acknowledgments] This work was supported by the DFG research units 1735 and 5381. \end{acks}
\end{document} | arXiv |
\begin{document}
\begin{abstract}
The celebrated \emph{Mirror Theorem} states that the genus zero part of the $A$ model (quantum cohomology, rational curves counting) of the Fermat quintic threefold is equivalent to the $B$ model (complex deformation, variation of Hodge structure) of its mirror dual orbifold. In this article, we establish a mirror-dual statement.
Namely, the \emph{$B$ model} of the Fermat quintic threefold is shown to be equivalent to the \emph{$A$ model} of its mirror, and hence establishes the mirror symmetry as a true duality. \end{abstract}
\maketitle
\small \setcounter{tocdepth}{1} \tableofcontents \normalsize
\setcounter{section}{-1}
\section{Introduction} \label{s:0}
\subsection{Mirror Theorem for the Fermat quintic threefold} \label{s:0.1}
Let $M$ be the Fermat quintic threefold defined by \[
M := \{ x_0^5 + x_1^5 + x_3^5 + x_4^5 + x_5^5 = 0 \} \subset \mathbb{P}^4. \] The Greene--Plesser \cite{bGrP} \emph{mirror construction} gives the mirror \emph{orbifold} as the quotient stack \[
\mathcal{W} := [M / \bar{G}], \] where $\bar{G} \cong (\mathbb{Z}/5\mathbb{Z})^3$ is a (finite abelian) subgroup of the big torus of $\mathbb{P}^4$ acting via generators $e_1, e_2, e_3$: \[
\begin{split}
e_1[x_0, x_1, x_2, x_3, x_4] &=
[\zeta x_0, x_1, x_2, x_3, \zeta^{-1} x_4] \\
e_2[x_0, x_1, x_2, x_3, x_4] &=
[ x_0, \zeta x_1, x_2, x_3, \zeta^{-1} x_4] \\
e_3[x_0, x_1, x_2, x_3, x_4] &=
[ x_0, x_1, \zeta x_2, x_3, \zeta^{-1} x_4] .
\end{split} \] Assuming the validity of mirror symmetry for the mirror pair $(M, \mathcal{W})$, Candelas--de la Ossa--Green--Parkes made the celebrated calculation which in particular predicted the number of rational curves in the Fermat quintic of any degree. This calculation was verified in full generality only after many years of works, involving many distinguished mathematicians and culminating in the proof by A.~Givental \cite{aG1} (and Liu--Lian--Yau \cite{LLY}). The mathematical proof of the CDGP Conjecture was termed the \emph{Mirror Theorem} for the Fermat quintic threefold.
In a way, what the Mirror Theorem says is that the invariants from the complex deformations of $\mathcal{W}$ matches those from the K\"ahler deformations of $M$, up to a change of variables termed the \emph{mirror map}. In terms of E.~Witten's terminology \cite{eW}, the above mirror theorem states that the (genus 0) $A$ model of $M$ is equivalent to $B$ model of $\mathcal{W}$. This can be formulated in mathematical terms as saying that the genus zero Gromov--Witten theory (GWT), or quantum cohomology, on $M$ is equal to the variation of Hodge structures (VHS) associated to the complex deformations of $\mathcal{W}$.
The complex deformation of Calabi--Yau's is unobstructed by Bogomolov--Tian--Todorov. The dimension of the Kodaira--Spencer space can be identified as the Hodge number $h^{2,1}$ due to the Calabi--Yau property $K \cong \mathscr{O}$. In this case $h^{2,1}(\mathcal{W})=1$. CDGP chose the following one-dimensional deformation family $\{ \mathcal{W}_{\psi} \} = \{Q_\psi(x) = 0\}$, where \begin{equation} \label{e:0.1}
Q_\psi(x) = x_0^5 + x_1^5 + x_3^5 + x_4^5 + x_5^5
- \psi x_0 x_1 x_2 x_3 x_4 x_5 \end{equation} of hypersurfaces in $[\mathbb{P}^4/ \bar{G}]$, such that $\psi = \infty$ is the maximally degenerate moduli point. We note that it is often convenient to use $t = -5 \log \psi$ as the variable. By local Torelli for Calabi--Yau, the deformation is embedded into VHS, which then gives all information about the complex deformation.
The K\"ahler deformation is given by genus zero GWT along the ``small'' variable $t$, which is the dual coordinate for the hyperplane class $H$. $H^{1,1}(M)_{\mathbb{C}}$ is often called the complexified K\"ahler moduli.
We can rephrase the above in much more precise terms. Both genus zero GWT and VHS can be described by differential systems associated to flat connections. For GWT, it is the \emph{Dubrovin connection}; for VHS the \emph{Gauss--Manin} connection. The definitions can be found in Sections~\ref{s:1} and \ref{s:4} respectively. Therefore, we can phrase the Mirror Theorem for the Fermat quintic in the following form.
\begin{theorem}[$=$ Theorem~\ref{t:MTfull}] \label{t:0.1} The fundamental solutions of the Gauss--Manin connection for $\mathcal{W}_t$ are equivalent, up to a mirror map, to the fundamental solutions of the Dubrovin connection for $M$, when restricted to $H^2(M)$. \end{theorem}
\subsection{Mirror Theorem for the mirror quintic} \label{s:0.2}
Theorem~\ref{t:0.1} can be stated suggestively as \[
\text{$A$ model of $M$ $\equiv$ $B$ model of $\mathcal{W}$.} \] In order for the mirror symmetry to be a true duality, one will also have to show that \[
\text{$B$ model of $M$ $\equiv$ $A$ model of $\mathcal{W}$.} \] This is the task we set for ourselves in this paper.
The first thing we note is that $\mathcal{W}$ is an orbifold. Thus we must replace the singular cohomology by the Chen--Ruan cohomology, and the usual Gromov--Witten theory by the orbifold GWT. These are defined in Section~\ref{s:1}.
Upon a closer look, however, there is a serious technical issue. In the B model of $M$, the Kodaira--Spencer space is of dimension $101$ and the VHS of $H^3(M)$ is a system of rank $204$, thus a calculation of the full Gauss--Manin connection for $M$ is unfeasible. As a first step however, we choose a one-dimensional deformation family $\{ M_t \}$ defined by the vanishing of \eqref{e:0.1}, \emph{reinterpreted as a family in $\mathbb{P}^4$}. Similarly, in the $A$ model of $\mathcal{W}$, we have $h^{1,1}_{CR}(\mathcal{W}) =101$, where the subscript denotes Chen--Ruan cohomology. We choose the one-dimensional subspace of the complexified K\"ahler moduli spanned by the hyperplane class and call the coordinate $t$ as before. These one dimensional families are arguably \emph{the most natural and the most important dimension}.
With these choices, the Gauss--Manin system for $M$ still has rank $204$, but over a one dimensional base. The fundamental solution is a matrix of size $204$ by $204$ in one variable. The Dubrovin connection on $H^{even}_{CR}(\mathcal{W})$ likewise has the fundamental solution matrix of size $204$ by $204$. Here $204 = \dim H^{even}_{CR}(\mathcal{W})$.
The main result of this paper is the following theorem.
\begin{theorem}[$=$ Theorem~\ref{t:MMTfull}] \label{t:0.2} The fundamental solutions of the Gauss--Manin connection for $\{ M_t \}$ are equivalent, up to a mirror map, to the fundamental solutions of the Dubrovin connection for $\mathcal{W}$ restricted to $t \in H^2(\mathcal{W})$. \end{theorem}
\subsection{Outline of the paper} \label{s:0.3} We have in mind the readership with diverse background. For convenience, we have included short introductions in Section~1 and Section~4 to orbifold Gromov--Witten theory and the theory of variation of Hodge structures, recalling only facts pertinent to our presentation. Sections~2 and 3 present the $A$ model calculation for $\mathcal{W}$. We first calculate the genus zero Gromov--Witten theory for $[\mathbb{P}^4/\bar{G}]$ in Section~\ref{s:2}; we then calculate the genus zero Gromov--Witten theory for $\mathcal{W}$ in Section~\ref{s:3}. In Section~5 we present a reformulation of the results from \cite{DGJ}, and summarize our $B$ model calculation for $M_t$. In the last section, we prove our main result, showing the validity of the Mirror-dual statement of the Mirror Theorem. For the benefit of our dual readership, we include a derivation of Theorem~\ref{t:0.1} from the usual statement of the Mirror Theorem.
\section{Quantum orbifold cohomology} \label{s:1} In this section we give a brief review of Chen--Ruan cohomology and quantum orbifold cohomology, with the parallel goal of setting notation. A more detailed general review can be found in \cite{CCLT}.
\begin{conventions} \label{conv:1} We work in the algebraic category. The term \emph{orbifold} means ``smooth separated Deligne--Mumford stack of finite type over $\mathbb{C}$.''
The various dimensions are complex dimensions. On the other hand, the degrees of cohomology are all in real/topological degrees.
Unless otherwise stated all cohomology groups have coefficients in $\mathbb{C}$. \end{conventions}
\subsection{Chen--Ruan cohomology groups} \label{s:1.1}
Let $\mathcal{X}$ be a stack. Its inertia stack $I\mathcal{X}$ is the fiber product \[ \xymatrix{ I\mathcal{X} \ar[r] \ar[d] & \mathcal{X} \ar[d]^\Delta \\ \mathcal{X}\ar[r]^\Delta & \mathcal{X} \times \mathcal{X} \\ } \] where $\Delta$ is the diagonal map. The fiber product is taken in the $2$-category of stacks. One can think of a point of $I\mathcal{X}$ as a pair $(x,g)$ where $x$ is a point of $\mathcal{X}$ and $g \in \operatorname{Aut}_{\mathcal{X}}(x)$. There is an involution $I: I\mathcal{X} \to I\mathcal{X}$ which sends the point $(x,g)$ to $(x,g^{-1})$. It is often convenient to call the components of $I\mathcal{X}$ for which $g \neq e$ the \emph{twisted sectors}.
If $\mathcal{X} = [V/G]$ is a global quotient of a nonsingular variety $V$ by a finite group $G$, $I\mathcal{X}$ takes a particularly simple form. Let $S_G$ denote the set of conjugacy classes $(g)$ in $G$, then \[
I [V/G] = \coprod_{(g) \in S_G} [ V^g/C(g) ]. \]
The \emph{Chen--Ruan orbifold cohomology groups} $H^*_{CR}(X)$ (\cite{CR1}) of a Deligne--Mumford stack $\mathcal{X}$ are the cohomology groups of its inertia stack \[
H_{CR}^*(\mathcal{X}) := H^*(I\mathcal{X}). \]
Let $(x, g)$ be a geometric point in a component $\mathcal{X}_i$ of $I\mathcal{X}$. By definition $g \in \operatorname{Aut}_{\mathcal{X}}(x)$. Let $r$ be the order of $g$. Then the $g$-action on $T_x \mathcal{X}$ decomposes as eigenspaces \[ T_x \mathcal{X} = \bigoplus_{0 \leq j < r} E_j \] where $E_j$ is the subspace of $T_x \mathcal{X}$ on which $g$ acts by multiplication by $\exp(2\pi\sqrt{-1}j/r)$. Define the age of $\mathcal{X}_i$ to be \[
\operatorname{age}(\mathcal{X}_i) := \sum_{j=0}^{r-1} \frac{j}{r} \dim( E_j). \] This is independent of the choice of geometric point $(x,g) \in \mathcal{X}_i$.
Let $\alpha$ be an element in $H^p(\mathcal{X}_i) \subset H^*(I\mathcal{X})$. Define the age-shifted degree of $\alpha$ to be \[
\deg_{CR}(\alpha) := p + 2 \operatorname{age}(\mathcal{X}_i). \] This defines a grading on $H_{CR}(\mathcal{X})$.
When $\mathcal{X}$ is compact the {\em orbifold Poincar\'e pairing} is defined by \[
(\alpha_1, \alpha_2)^{\mathcal{X}}_{CR} := \int_{I\mathcal{X}} \alpha_1 \cup I^*(\alpha_2), \] where $\alpha_1$ and $\alpha_2$ are elements of $H^*_{CR}(\mathcal{X})$. It is easy to see that when $\alpha_1$ and $\alpha_2$ are homogeneous elements, $(\alpha_1, \alpha_2)_{CR} \neq 0$ only if $\deg_{CR} (\alpha_1) + \deg_{CR} (\alpha_2) = 2 \dim (\mathcal{X})$.
\subsection{Orbifold Gromov-Witten theory} \label{s:1.2}
\subsubsection{Orbifold Gromov--Witten invariants}
We follow the standard references \cite{CR2} and \cite{AGV} of orbifold Gromov--Witten theory.
Given an orbifold $\mathcal{X}$, there exists a moduli space $\overline{\mathscr{M}}_{g,n}(\mathcal{X}, d)$ of stable maps from $n$-marked genus $g$ pre-stable orbifold curves to $\mathcal{X}$ of degree $d \in H_2(\mathcal{X}; \mathbb{Q})$. Each source curve $(\mathcal{C}, p_1, \ldots, p_n)$ has non-trivial orbifold structure only at the nodes and marked points: At each (orbifold) marked point it is a cyclic quotient stack and at each node a \emph{balanced} cyclic quotient. That is, \'etale locally isomorphic to \[
\left[ \operatorname{Spec} \left( \frac{\mathbb{C}[x, y]}{(xy)} \right) / \mu_r \right],\] where $\zeta \in \mu_r$ acts as $(x,y) \mapsto (\zeta x, \zeta^{-1} y)$. The maps are required to be representable at each node.
Each marked point $p_i$ is \'etale locally isomorphic to $[\mathbb{C}/ \mu_{r_i}]$. There is an induced homomorphism \[
\mu_{r_i} \to \operatorname{Aut}_{\mathcal{X}}(f(p_i)). \] Maps in $\overline{\mathscr{M}}_{g,n}(\mathcal{X}, d)$ are required be representable, which amounts to saying that these homomorphisms be injective. For each marked point $p_i$, one can thus associate a point $(x_i, g_i)$ in $I\mathcal{X}$ where $x_i = f(p_i)$, and $g_i \in \operatorname{Aut}_{\mathcal{X}}(x_i)$ is the image of $\exp(2 \pi \sqrt{-1}/r_i)$ under the induced homomorphism.
Given a family $\mathcal{C} \to S$ of marked orbifold curves, there may be nontrivial gerbe structure above the locus defined by the $i$-th marked point. For this reason there is generally not a well defined map \[
ev_i: \overline{\mathscr{M}}_{g,n}(\mathcal{X}, d) \to I\mathcal{X}. \] However, as explained in \cite{AGV} and \cite{CCLT} Section~2.2.2, it is still possible to define maps \[
ev_i^*: H^*_{CR}(\mathcal{X}) \to H^*(\overline{\mathscr{M}}_{g,n}(\mathcal{X}, d)) \] which behave \emph{as if the evaluation maps $ev_i$ are well defined}.
Let $X$ denote the coarse underlying space of the stack $\mathcal{X}$. There is a \emph{reification map} \[
\overline{\mathscr{M}}_{g,n}(\mathcal{X}, d) \to \overline{\mathscr{M}}_{g,n}(X, d), \] which forgets the orbifold structure of each map. For each marked point there is an associated line bundle, the $i^{th}$ universal cotangent line bundle, \[
\begin{array}{c} L_i \\ \downarrow \\ \overline{\mathscr{M}}_{g,n}(X, d) \end{array} \] with fiber $T^*_{p_i} C$ over $\{f: (C, p_1, \ldots, p_n) \to X\}$. Define the $i$-th $\psi$-class
by $\psi_i = r^*(c_1(L_i))$.
As in the non-orbifold setting, there exists a virtual fundamental class $[\overline{\mathscr{M}}_{g,n}(\mathcal{X}, d)]^{vir}$. \emph{Orbifold Gromov-Witten invariants} for $\mathcal{X}$ are defined as integrals \begin{equation*} \big\langle \alpha_1 \psi^{k_1}, \ldots , \alpha_n \psi^{k_n}\big\rangle_{g,n,d}^{\mathcal{X}} = \int_{[\overline{\mathscr{M}}_{g,n}(\mathcal{X}, d)]^{vir}} \prod_{i=1}^n ev_i^*(\alpha_i) \psi_i^{k_i}, \end{equation*} where $\alpha_i \in H^*_{CR}(\mathcal{X})$.
Let $\overline{\mathscr{M}}_{g,(g_1, \ldots, g_n)}(\mathcal{X}, d)$ denote the open and closed substack of $\overline{\mathscr{M}}_{g,n}(\mathcal{X}, d)$ such that $ev_i$ maps to a component $\mathcal{X}_{g_i}$ of $I\mathcal{X}$. The space $\overline{\mathscr{M}}_{g,(g_1, \ldots, g_n)}(\mathcal{X}, d)$ has (complex) virtual dimension \[
n + (g - 1)(\dim \mathcal{X} - 3) + \langle c_1(T\mathcal{X}),d \rangle
- \sum_{i=0}^n \operatorname{age}(\mathcal{X}_{g_i}). \] In other words, for homogeneous classes $\alpha_i \in H^*(\mathcal{X}_{g_i})$ the Gromov-Witten invariant \\ $\big\langle \alpha_1, \ldots , \alpha_n\big\rangle_{g,n,d}^{\mathcal{X}}$ will vanish unless \[
\sum_{i = 1}^n \deg_{CR}(\alpha_i) = 2\left(n + (g - 1)(\dim \mathcal{X} - 3)
+ \langle c_1(T\mathcal{X}),d \rangle\right). \]
\subsubsection{Quantum cohomology and the Dubrovin connection} \label{s:1.2.5}
Let $\{T_i\}_{i \in I}$ be a basis for $H^*_{CR}(\mathcal{X})$ and $\{ T^i \}_{i \in I}$ its dual basis. We can represent a general point in coordinates by \[
\mathbf{t} = \sum_i t^i T_i \in H^*_{CR}(\mathcal{X}). \] Gromov-Witten invariants allow us to define a family of product structures parameterized by $\mathbf{t}$ in a formal neighborhood of $0$ in $H^*_{CR}(\mathcal{X})$. The \emph{(big) quantum product} $*_\mathbf{t}$ is defined as \begin{equation}\label{e:product}
\alpha_1 *_\mathbf{t} \alpha_2 := \sum_d \sum_{n \geq 0} \sum_i
\frac{q^d}{n!} \langle \alpha_1, \alpha_2, T_i,
\mathbf{t}, \ldots, \mathbf{t} \rangle^\mathcal{X}_{0, 3+n, d} T^i , \end{equation} where the first sum is over the Mori cone of effective curve classes and the variables $q^d$ are in an appropriate Novikov ring $\Lambda$ used to guarantee formal convergence of the sum. The \emph{WDVV equations} (\cite{CK}, Section~8.2.3) imply the associativity of the product. The \emph{small quantum product} is defined by restricting the parameter of the quantum product to divisors $\mathbf{t} \in H^2(\mathcal{X})$ supported on the \emph{non-twisted sector}.
One can interpret $*_\mathbf{t}$ as defining a product structure on the tangent bundle $T H^*_{CR}(\mathcal{X}; \Lambda)$, such that for a fixed $\mathbf{t}$ the quantum product defines a (Frobenius) algebra structure on $T_{\mathbf{t}} H^*_{CR}(\mathcal{X}; \Lambda)$. This can be rephrased in terms of the \emph{Dubrovin connection}: \[
\nabla^z_{\frac{\partial}{\partial t^i}} \left( \sum_{j} a_j T_j\right) =
\sum_{j} \frac{\partial a_j} {\partial t^i} T_j
- \frac{1}{z} \sum_{j} a_j T_i *_\mathbf{t} T_j . \]
This defines a $z$-family of connections on $T H^*_{CR}(\mathcal{X}; \Lambda)$.
\begin{remark} Note that when $\mathbf{t}$, $T_i$ and $T_j$ are in $H^{even}_{CR}(\mathcal{X})$, then for dimension reasons $T_i *_\mathbf{t} T_j$ will be also be supported in even degree. Thus $\nabla^z$ restricts to a connection on $TH^{even}_{CR}(\mathcal{X}; \Lambda)$. When restricted to $TH^{even}_{CR}(\mathcal{X}; \Lambda)$, the quantum product is commutative. \end{remark}
\begin{remark} \label{r:1.3} For the purpose of this paper, we clarify here what we mean by ``$A$ model of $\mathcal{X}$''. Let $H := H^{even}_{CR}(\mathcal{X}; \Lambda)$. The (genus zero part of) \emph{$A$ model of $\mathcal{X}$} is the tangent bundle $TH$ with its natural (flat) fiberwise pairing and the Dubrovin connection restricted to $H^{1,1}_{CR}(\mathcal{X})$. \end{remark}
The commutativity and associativity of the quantum product implies that the Dubrovin connection is flat. The \emph{topological recursion relations} allow us to explicitly describe solutions to $\nabla^z$. Define \begin{equation}\label{e:bsol}
s_i(\mathbf{t},z) = T_i + \sum_d \sum_{n \geq 0} \sum_j \frac{q^d}{n!}
\bigg\langle \frac{T_i}{z - \psi_1}, T^j,
\mathbf{t}, \ldots, \mathbf{t} \bigg\rangle^\mathcal{X}_{0, 2+n, d} T_j \end{equation} where ${1}/{(z - \psi_1)}$ should be viewed as a power series in $1/z$. The sections $s_i$ form a basis for the $\nabla^z$-flat sections; see e.g.~\cite{CK}, Proposition 10.2.1. Thus we obtain a fundamental solution matrix $S = S(\mathbf{t}, z) = (s_{ij})$ given by \begin{equation}\label{e:sol}
s_{ij}(\mathbf{t},z)= ( T^i, s_j )_{CR}^\mathcal{X}. \end{equation}
If one restricts the base to divisors $\mathbf{t} \in H^2(\mathcal{X})$, the \emph{divisor equation} (\cite{AGV} Theorem~8.3.1) allows a substantial simplification of the formula for $s_i$ \[
s_i(\mathbf{t},z)|_{\mathbf{t} \in H^2(\mathcal{X})} =
e^{\mathbf{t} / z}\left(T_i + \sum_{d > 0} \sum_j
q^d e^{d\mathbf{t}} \bigg\langle \frac{ T_i}{z - \psi_1},
T^j \bigg\rangle^\mathcal{X}_{0, 2,d} T_j\right). \]
\subsection{Generating functions} \label{s:1.3}
Given an orbifold $\mathcal{X}$, Givental's (big) $J$-function is the first row vector of the fundamental solution matrix, obtained by pairing the solution vectors of the Dubrovin connection with $1$. \begin{equation*}
\begin{split}
J^\mathcal{X}_{big}(\mathbf{t}, z) & := \sum_{i} \left( s_i(\mathbf{t}), 1 \right)_{CR}^\mathcal{X} T^i \\
& = 1 + \sum_d \sum_{n \geq 0} \sum_i \frac{q^d}{n!} \bigg\langle \frac{T_i}{z - \psi_1}, 1, \mathbf{t}, \ldots, \mathbf{t} \bigg\rangle^\mathcal{X}_{0, 2+n, d} T^i \\
& = 1 + \frac{\mathbf{t}}{z} + \sum_{d}
\sum_{n \geq 0} \sum_i \frac{q^d}{n!} \bigg\langle
\frac{T_i}{z(z - \psi_1)}, \mathbf{t}, \ldots, \mathbf{t} \bigg\rangle^\mathcal{X}_{0,1+n, d} T^i ,
\end{split} \end{equation*} The last equality follows from the \emph{string equation}. It is also easy to see that the fundamental solution matrix $S(\mathbf{t}, z)$ of \eqref{e:sol} is equal to $z \nabla J_{big}$. As such, $J_{big}$ encodes all information about quantum cohomology.
However, the big $J$-function is often impossible to calculate directly. In the non-orbifold Gromov--Witten theory, when the cohomology is generated by divisors, the \emph{small $J$-function} proves much more computable, while powerful enough to solve many problems; see e.g.~\cite{aG1, aG2}. The small $J$-function for a nonsingular \emph{variety} $X$ is a function on $\mathbf{t} \in H^2(X)$: \[ \begin{split}
J^X_{small} (\mathbf{t},z) &:= J^X_{big} (\mathbf{t},z) |_{\mathbf{t} \in H^2(X)} \\
& = e^{\mathbf{t} / z}\left(1 + \sum_{d > 0} \sum_{i} q^d e^{d\mathbf{t}} \bigg\langle \frac{ T_i}{z - \psi_1}, 1 \bigg\rangle^X_{0, 2,d} T^i\right).
\end{split} \]
In orbifold theory, however, the Chen--Ruan cohomology is never generated by divisors except for trivial cases, due to the presence of the twisted sectors. Therefore, the knowledge of the small $J$-function alone is often not enough to reconstruct significant information about the orbifold quantum cohomology. (Note however that in Section~5 of \cite{CCLT} one way was found to circumvent this obstacle for weighted projective spaces.)
We propose the following definition of \emph{small $J$-matrix for orbifolds}.
\begin{definition} \label{d:sJ} For $\mathbf{t} \in H^2(\mathcal{X})$, define $J_g^\mathcal{X}$ as the cohomology-valued function \begin{equation}\label{e:sJ}
\begin{split}
J^{\mathcal{X}}_{g} (\mathbf{t}, z)|_{\mathbf{t} \in H^2(\mathcal{X})}
& := \sum_i \left(s_i(\mathbf{t})|_{\mathbf{t} \in H^2(X)}, \mathbb{1}_g\right)_{CR}^\mathcal{X} T^i \\
&= e^{\mathbf{t}/z}\left(\mathbb{1}_g + \sum_{d > 0} \sum_i q^d e^{d\mathbf{t}} \bigg\langle
\frac{T_i}{z - \psi_1}, \mathbb{1}_g \bigg\rangle^{\mathcal{X}}_{0,2,d} T^i\right),
\end{split} \end{equation} where $\mathbb{1}_g$ is the fundamental class on the component $\mathcal{X}_g$ of $I\mathcal{X}$.
The \emph{small $J$-matrix} is the matrix-valued function \[
J^{\mathcal{X}}_{small} (\mathbf{t}, z) =
\left[ J^{\mathcal{X}}_{g,i} (\mathbf{t}, z) \right]_{g \in G, i \in I}
= \left[ (J^{\mathcal{X}}_{g} (\mathbf{t}, z), T_i)^{\mathcal{X}}_{CR} \right]_{g \in G, i \in I}\; , \] where $G$ is the index set of the components of $I\mathcal{X}$, $I$ the index for the basis $\{T_i\}_{i \in I}$ of $H^*_{CR}(\mathcal{X})$ and $J^\mathcal{X}_{g,i}(\mathbf{t}, z)$ the coefficient of $T^i$ in $J^\mathcal{X}_g(\mathbf{t},z)$. \end{definition}
\begin{remark} \label{r:1.5} We believe that the small $J$-matrix is the right replacement of the small $J$-function in the orbifold theory, for its computability and structural relevance.
Structurally equation~\eqref{e:sol} shows that one needs to specify ``two-points'' (i.e.~a matrix) in the generating function in order to form
the fundamental solutions of the Dubrovin connection. Ideally, one would like to get the full $|I| \times |I|$ fundamental solution matrix $S = z \nabla J_{big}$ restricted to $\mathbf{t} \in H^2(\mathcal{X})$. This would give all information about the \emph{small} quantum cohomology. Unfortunately, a direct computation of $S(\mathbf{t})|_{\mathbf{t} \in H^2(\mathcal{X})}$ is mostly out of reach in the orbifold theory.
In the (non-orbifold) case when $H^*(X)$ is generated by divisors, as shown by A.~Givental, the small $J$-function is often enough to determine
the essential information for small quantum cohomology. One can think of the small $J$-function as a a submatrix of size $1 \times |I|$, indeed the first row vector, of $S$.
However, in the orbifold theory, the above matrix is not enough to determine useful information about small quantum cohomology except in the trivial cases. We believe that the smallest useful submatrix of $S$
is the small $J$-matrix (of size $|G| \times |I|$) defined above. We will show that it is both computable and relevant to the structure of orbifold quantum cohomology. In this paper we are able to calculate the small $J$-matrix of the
toric orbifold $\mathcal{Y} = [\mathbb{P}^4/\bar{G}]$, and we use a sub-matrix of the small $J$-matrix $J^{\mathcal{W}}_{small}$ to fully describe the solution matrix $S(\mathbf{t})|_{\mathbf{t} \in H^2(\mathcal{X})}$ of the mirror quintic $\mathcal{W}$. \end{remark}
\section{$J$-function of $[\mathbb{P}^4/\bar{G}]$} \label{s:2}
\subsection{Inertia orbifold of $[\mathbb{P}^4/\bar{G}]$} \label{s:2.1} Let $[x_0, x_1, x_2, x_3, x_4]$ be the homogeneous coordinates of $\mathbb{P}^4$. Denote \[
\zeta = \zeta_5 := e^{2 \pi \sqrt{-1}/5}. \] Let the group $\bar{G} \cong (\mathbb{Z}/5\mathbb{Z})^3$ be a (finite abelian) subgroup of the big torus of $\mathbb{P}^4$ acting via generators $e_1, e_2, e_3$: \begin{equation} \label{e:G}
\begin{split}
e_1[x_0, x_1, x_2, x_3, x_4] &=
[\zeta x_0, x_1, x_2, x_3, \zeta^{-1} x_4] \\
e_2[x_0, x_1, x_2, x_3, x_4] &=
[ x_0, \zeta x_1, x_2, x_3, \zeta^{-1} x_4] \\
e_3[x_0, x_1, x_2, x_3, x_4] &=
[ x_0, x_1, \zeta x_2, x_3, \zeta^{-1} x_4] .
\end{split} \end{equation} Let $\mathcal{Y} =[\mathbb{P}^4/ \bar{G}]$. As explained in the Introduction this orbifold plays an instrumental role in what follows so we give here a detailed presentation of its corresponding inertia orbifold.
The group $\bar{G}$ can be described alternatively as follows. Let \[
{G} := \{(\zeta^{r_0}, \ldots, \zeta^{r_4}) \, | \,
\sum_{i=0}^4 r_i \equiv 0 \, (\operatorname{mod} 5) \} \] and \[
\bar{G} \cong {G}/ \big\langle (\zeta, \ldots, \zeta) \big \rangle. \] The $\bar{G}$-action on $\mathbb{P}^4$ comes from coordinate-wise multiplication. By a slight abuse of notation, we will represent a group element $g \in {G}$ by the power of $\zeta$ in each coordinate: \[
G = \{ (r_0, \ldots , r_4) \, | \, \sum_{i=0}^4 r_i \equiv 0 \, (\operatorname{mod} 5),
0 \leq r_i \leq 4 \,\forall i \}. \] For an element $g \in G$, denote $[g]$ the corresponding element in $\bar{G}$.
Fix an element $\bar{g} \in \bar{G}$. Let $g = (r_0, \ldots , r_4) \in {G}$ be such that $[g] =\bar{g}$. Define \[
I(g) := \left\{ j \in \{0, 1, 2, 3, 4\} \, | \, r_j =0 \right\}, \] then \[
\mathbb{P}^4_g := \left\{x_j = 0\right\}_{j \notin I(g)} \subset \mathbb{P}^4 \] is a component of $(\mathbb{P}^4)^{\bar{g}}$. From this we see that each element $g \in {G}$ such that $[g] =\bar{g}$ corresponds to a connected component $\mathcal{Y}_{g}$ of $I\mathcal{Y}$ associated with $\mathbb{P}^4_g \subset (\mathbb{P}^4)^{\bar{g}}$. Note that if $g$ has no coordinates equal to zero then $\mathbb{P}^4_g$ is empty, and so is $\mathcal{Y}_g$. This gives us a convenient way of indexing components of $I\mathcal{Y}$.
We summarize the above discussions in the following lemma. \begin{lemma} \label{l:2.1} \begin{equation*}
I\mathcal{Y} = \coprod_{g \in S} \mathcal{Y}_{g}\:, \end{equation*} where \[
\mathcal{Y}_g = \{ (x, [g]) \in I\mathcal{Y} \, | \, x \in [\mathbb{P}^4_g / \bar{G} ]\} \] is a connected component and $S$ denotes the set of all $g = (r_0, \ldots, r_4)$ such that at least one coordinate $r_i$ is equal to $0$.
Consequently, a convenient basis $\{T_i\}$ for $H^*_{CR}(\mathcal{Y})$ is \[
\bigcup_{g \in S} \{\mathbb{1}_g, \mathbb{1}_g \tilde{H}, \ldots ,
\mathbb{1}_g\tilde{H}^{\dim(\mathcal{Y}_g)}\}. \] \end{lemma}
\subsection{$J$-functions} \label{s:2.2}
Recalling a basic fact about global quotient orbifolds, a map of orbifolds $f: \mathcal{C} \to [\mathbb{P}^4/\bar{G}]$ can be identified with a principal $\bar{G}$-bundle $C$, and a $\bar{G}$-equivariant map $\tilde{f}: C \to \mathbb{P}^4$ such that the following diagram commutes: \footnote{Technically f is identified with an equivalence class of such objects.} \begin{equation}\label{e:cover}
\xymatrix{C \ar[d]_{\pi_{C}}\ar[r]^{\tilde{f}} & \mathbb{P}^4 \ar[d]_{\pi_{\mathbb{P}^4}}\\
\mathcal{C}\ar[r]^{f} & [\mathbb{P}^4/\bar{G}].} \end{equation}
\begin{lemma} \label{l:2.2} (i) The map $\tilde{f}$ is representable if and only if $C$ is a nodal curve with each irreducible component a smooth variety.
(ii) There do not exist representable orbifold morphisms $f: \mathcal{C} \to \mathcal{Y}$ from a genus $0$ orbifold curve $\mathcal{C}$ with only one orbifold marked point. \end{lemma}
\begin{proof} (i) follows immediately from the definition of representability.
(ii) follows from (i): If $\mathcal{C}$ is irreducible, this is because there do not exist smooth covers of genus $0$ orbifold curves with only one point with nontrivial isotropy. An induction argument then shows that the same is true of reducible curves with only one orbifold marked point (we assume always that our nodes be balanced). \end{proof}
A line bundle on $[\mathbb{P}^4/\bar{G}]$ can be identified with a $\bar{G}$-equivariant line bundle on $\mathbb{P}^4$. Therefore, the Picard group on $[\mathbb{P}^4/\bar{G}]$ is a $\bar{G}$-extension of $\mathbb{Z}$. Let $L$ be \emph{any} line bundle on $[\mathbb{P}^4/\bar{G}]$ such that $\pi_{\mathbb{P}^4}^*L = H$, where $H$ is the hyperplane class on $\mathbb{P}^4$. By \eqref{e:cover}, we have the following equality \[
\int_{\mathcal{C}} f^*(L) = \frac{1}{125} \int_{C} \tilde{f}^*(H). \] We define the degree of a map $f:\mathcal{C} \to \mathcal{Y}$ by \[
d := \frac{1}{125} \int_{C} \tilde{f}^*(H). \]
This also allows us to determine necessary conditions on the triple $d$, $h = (r_0(h), \ldots, r_4(h))$ and $g = (r_0(g), \ldots, r_4(g))$ for \[
\overline{\mathscr{M}}_{0,h,g}(\mathcal{Y}, d) :=
\overline{\mathscr{M}}_{0,2}(\mathcal{Y}, d) \cap ev_1^{-1}(\mathbb{1}_h) \cap ev_2^{-1}(\mathbb{1}_g) \] to be nonempty.
\begin{proposition}\label{p:mapsallowed} The space $\overline{\mathscr{M}}_{0,h,g}(\mathcal{Y}, d) $ is nonempty only if \begin{enumerate} \item[(i)] $[h] = [g]^{-1}$ in $\bar{G}$; \item[(ii)] $r_i(h) + r_i(g) \equiv 5d \,(\operatorname{mod} 5)$ or equivalently ${\displaystyle \langle d \rangle = \langle (r_i(h) + r_i(g))/5 \rangle }$ for $0 \leq i \leq 4$. \end{enumerate} \end{proposition}
\begin{proof} We will first consider the case where the source curve is irreducible. Assume that there exists a map $\{f: \mathcal{C} \to \mathcal{Y}\}$ in $\overline{\mathscr{M}}_{0,h,g}(\mathcal{Y}, d)$ such that $\mathcal{C}$ is non-nodal. Consider the principal $\bar{G}$-bundle $\pi_C: C \to \mathcal{C}$. After choosing a generic base point $x \in \mathcal{C}$ and a point $\tilde{x}$ in $\pi_C^{-1}(x)$, we get a homomorphism $\phi: \pi_1(\mathcal{C}, x) \to \bar{G}$. We can specify generators $\rho_1$, and $\rho_2$ of $\pi_1(\mathcal{C}, x)$ such that $\rho_i$ is the class of loops wrapping once around $p_i$ in the counterclockwise direction. Then $\phi(\rho_1) = [h]$ and $\phi(\rho_2) = [g]$. Because $\rho_1\cdot \rho_2 = 1$ in $\pi_1(\mathcal{C},x)$, it must be the case that $[h]\cdot [g] = 1$ in $\bar{G}$. This proves (i) for $\mathcal{C}$ non-nodal.
Next we will show (ii) in the case where $\mathcal{C}$ is non-nodal. To see this, note that the only smooth connected cover of $\mathcal{C}$ is isomorphic
to $\mathbb{P}^1$. This cover is degree $r := |[h]|$,
so $C$ must consist of $|\bar{G}|/r$ components, each isomorphic to $\mathbb{P}^1$. In the case $h = (0,0,0,0,0)$, this implies that $C$ has 125 components, and so $d$ is an integer. Thus Condition (ii) holds trivially.
If $h \neq (0,0,0,0,0)$, then $r=5$. First note that (i) implies that $r_i(h)+r_i(g) \, (\operatorname{mod} 5)$ is the same for any $i$. Thus, we only need to prove the statement for one $i$. Let $\mu_5$ be the group generated by $[h]$. Let $C' \cong \mathbb{P}^1$ be one component of $C$ and let \[
f' := \tilde{f} \big|_{C'} : C' \to \mathbb{P}^4 \] be the $\mu_5$-equivariant morphism induced from the $\bar{G}$-equivariant morphism $\tilde{f}: C \to \mathbb{P}^4$. $(f')^*(\mathscr{O}(1))$ is a degree $5d$ line bundle on $C' =\mathbb{P}^1$. Therefore, any lifting of the torus action on $\mathbb{P}^1$ will have \emph{weights} $(w, w+5d)$ at the fibers of the 2 fixed points. Call these two fixed points $p'_1$ and $p'_2$. Since $\mu_5 = \langle[h]\rangle$ is a subgroup of the torus, the \emph{characters} of the $[h]$-action at the fibers of the 2 fixed points must be $(\zeta^w, \zeta^{w+5d})$, for some $w$ in $\{0, \ldots , 4\}$.
Let $q_1:= f'(p'_1)$ and $q_2 := f'(p'_2)$. By assumption, $q_1 \in \mathbb{P}^4_h$, $q_2 \in \mathbb{P}^4_g$. Choose an $i \in I(h)$ and $j \in I(g)$ such that $i \neq j$, $x_i(q_1) \neq 0$ and $x_j(q_2) \neq 0$. The action of $[h]$ on the fiber over $q_1$ and $q_2$ can be chosen to be $(\zeta^{r_i(h)}, \zeta^{- r_j(h)})$. By the above weight/character arguments, \[
r_i(h) - (- r_j(h)) \equiv 5d \, (\operatorname{mod} 5). \] Since $j \in I(g)$ and $i \in I(h)$, \[
r_j(h) = r_j(h) - r_i(h) = r_i(g) - r_j(g) = r_i(g), \] so we can rewrite the above as $r_i(h) + r_i(g) \equiv 5d \, (\operatorname{mod} 5)$.
The nodal case follows similarly. Consider a nodal curve $f: \mathcal{C} \to \mathcal{Y}$. Let $\mathcal{C}_1, \ldots, \mathcal{C}_n$ be the irreducible components connecting $p_1$ to $p_2$. It follows from Lemma~\ref{l:2.2}, each of these components will have 2 orbifold points (at either nodes or marked points) and these will be the only points in $\mathcal{C}$ with nontrivial orbifold structure. The above calculation for irreducible components plus the condition that all nodes be balanced in this situation then implies the claim. \end{proof}
Once condition (i) is satisfied, the degree of maps allowed is thus determined by the quantity \[ d(h,g) := \langle
(r_i(h) + r_i(g))/5 \rangle .\] Note that this number remains constant as $i$ varies.
We will define generating functions related to the $J$-functions $J^\mathcal{Y}_g$ which isolate the $2$-point invariants of $ \overline{\mathscr{M}}_{0,h,g}(\mathcal{Y}, d)$. Let \[
S(d,h) :=\{(b,k)\,\,|\,\,\, 0 < b \leq d, \, \,\,\, 0 \leq k \leq 4, \,\,\,\,
\langle b \rangle = r_k(h)/5 \}, \] and let \[
c(d, h) := \big|S(d,h)\big|. \]
Given $h, g \in G$ such that $[h] = [g]^{-1}$, define \[
Z_{h,g} := \sum_{d} Q^{c(d,h)} \sum_i \bigg\langle\frac{T^h_i}{z - \psi_1},
\mathbb{1}_g\bigg\rangle^\mathcal{Y}_{0,2,d} T^i_h, \] where $\{T_i^h\}$ is a basis for $H^*(\mathcal{Y}_h)$, and $\{T^i_h\}$ is the dual basis under the Chen-Ruan orbifold pairing. (The motivation behind this choice of exponent for $Q$ will become clear in what follows: it is chosen to simplify the recursion satisfied by our generating function). Notice that by the above lemma, the only degrees which contribute to $Z_{h,g}$ are $d$ such that $\langle d \rangle = d(h,g)$. Finally, let \[
Z_{g} := \mathbb{1}_g + \sum \limits_{\{h | \,[h] = [g]^{-1}\}} Z_{h,g}. \]
Let $T = (\mathbb{C}^*)^5$ (or $\mathbb{C}^*$) act on $\mathbb{C}^5$ with (generic) weights $-\lambda_0, \ldots , -\lambda_4$. This induces an action on $\mathbb{P}^4$ and $\mathcal{Y}$. Furthermore there is an induced $T$-action on the inertia orbifold $I\mathcal{Y}$ and on $\overline{\mathscr{M}}_{0,2}(\mathcal{Y}, d)$. We will consider an equivariant analogue $Z^T_g$ of $Z_g$ defined by replacing the coefficients of $Z_g$ with their equivariant counterparts: \[
Z^T_{h,g} := \sum_{d, i} Q^{c(d,h)} \bigg\langle\frac{T^h_i}{z - \psi_1},
\mathbb{1}_g\bigg\rangle^{\mathcal{Y}, T}_{0,2,d} T^i_h, \quad Z^T_g := \mathbb{1}_g + \sum \limits_{\{h | \,[h] = [g]^{-1}\}} Z^T_{h,g}. \] where $\{T^h_i\}$ is now a basis of the equivariant cohomology $H^*_T(\mathcal{Y}_h)$.
Consider the cohomology valued functions \begin{equation} \label{e:2.2.2}
Y^T_{h,g} := \sum_{\{d \,\vline \langle d \rangle = d(h,g)\}} Q^{c(d,h)}
\frac{\mathbb{1}_{h^{-1}}}{\prod \limits_{(b,k)\in S(d,h)} (bz + H - \lambda_k)}, \end{equation} where \[
h^{-1} := (-r_0(h), \ldots, -r_4(h)) \,(\operatorname{mod}5). \] As with $Z$, let \begin{equation} \label{e:2.2.3}
Y^T_g := \mathbb{1}_g + \sum \limits_{\{h | \,[h] = [g]^{-1}\}} Y^T_{h,g}. \end{equation}
\begin{theorem}\label{t:Zformula} We have the equality in equivariant cohomology: \[
Z_g^T = Y^T_{g}. \] In particular, taking the nonequivariant limit, we conclude that $Z_g = Y_g$, (where $Y_g$ is the obvious non-equivariant limit of $Y^T_g$.) \end{theorem}
\begin{remark} For those who are familiar with the computation of the small $J$-function for toric manifolds \cite{aG2}, the generating functions $Z$, as indicated above, play the role of the $J$-function. The hypergeometric-type functions $Y$ then take the place of the $I$-function. Recall that one way of formulating the computation of genus zero GW invariants is to say that the $J$-function is equal to the $I$-function after a change of variables, called the \emph{mirror map}. In the present case, the mirror map is trivial. \end{remark}
\subsection{Proof of Theorem~\ref{t:Zformula}} \label{s:2.3}
The proof follows from a localization argument similar in spirit to that in \cite{aG2}. The strategy is to apply the Localization Theorem (after inverting the equivariant characters $\lambda_0, \ldots, \lambda_4$ in the ring $H^*_{CR,T}(\mathcal{Y})$) on the equivariant generating functions to determine a recursion satisfied by $Z^T_g$. This recursion relation in fact determines $Z^T_g$ up to the constant term in the Novikov variables. We then show that $Y^T_g$ satisfies the same recursion. Since $Z^T_g$ and $Y^T_g$ have the same initial term and the same recursion relation, $Z^T_g = Y^T_g$.
\subsubsection{a lemma on $c(d,h)$}
We will first explain the seemingly strange appearance of the exponents $c(d,h)$ in the definition of $Z_{h,g}$.
\begin{lemma}\label{l:dime} Let \[
m_d = \dim(\overline{\mathscr{M}}_{0,h,g}(\mathcal{Y}, d)), \] then if $[h] = [g]^{-1}$ and $\langle d \rangle = d(h,g)$, we have \[
c(d,h) = m_d - \dim(\mathcal{Y}_h) + 1. \] \end{lemma}
\begin{proof} The standard formula for virtual dimension gives \[
m_d = 5d + 3 - \operatorname{age}(h) - \operatorname{age}(g). \] Note that for any presentation $g = (r_0(g), \ldots, r_4(g))$, $\operatorname{age}(g) = \sum_{i=0}^4 r_i(g)/5$. Because $[h] = [g]^{-1}$, we have that \[
r_i(g) - r_j(g) \equiv r_j(h) - r_i(h) \, (\operatorname{mod} 5). \] This allows us to write \[
\frac{r_k(g)}{5} = \left\{
\begin{array}{cc} - r_k(h)/5 + d(h,g) & d(h,g) \geq r_k(h)/5 \\
1 - r_k(h)/5+ d(h,g) & d(h,g)< r_k(h)/5
\end{array} \right., \] which gives \[
\begin{array}{ll}
m_d &= 5d + 3 - 5d(h,g) - |\{k\, | d(h,g) < r_k(h)/5\}| \\
& = 5\lfloor d \rfloor + |\{k\, | d(h,g) \geq r_k(h)/5\}| - 2.
\end{array} \] Now, for a fixed $k$, \[
|\{b\, | 0 \leq b \leq d, \,\,\, \langle b \rangle = r_k(h)/5 \}|
= \left\{ \begin{array}{cc} \lfloor d \rfloor & d(h,g) < r_k(h)/5 \\
1 + \lfloor d \rfloor & d(h,g) \geq r_k(h)/5
\end{array} \right\}. \] Summing over all $k$, we get that \[
m_d = |\{(b,k)\, | 0 \leq b \leq d, \quad 0 \leq k \leq 4,\quad
\langle b \rangle = r_k(h)/5 \}| - 2. \] Finally, \[
\dim(\mathcal{Y}_g) = |\{k\, |\, 0 = r_k(h)/5\}| - 1, \] which gives the desired equality. \end{proof}
\subsubsection{Setting up the localization} The action of $T$ on $\overline{\mathscr{M}}_{0,h,g}(\mathcal{Y}, d))$ allows us to reduce integrals on the moduli space to sums of integrals on the fixed point loci with respect to the torus action. As usual, this reduces us to considering integrals of certain graph sums (see \cite{tGrP}). The generating function $Z^T_g$ consists of integrals where the first insertion is the pull back of a class on \[
\coprod \limits_{\{h| [h] = [g]^{-1}\}} \mathcal{Y}_h. \] We will now express $Z_g$ in terms of a new basis for this space which interacts nicely with the localization procedure. For each coordinate $0 \leq i \leq 4$, $i$ is in $I(h)$ for exactly one $h$
in $\{h| [h] = [g]^{-1}\}$.
(Recall that the presentations $h \in \{h| [h] = [g]^{-1}\}$ index the fixed point sets of $\mathbb{P}^4$ with respect to $[h]$). Then for $i \in I(h)$, let $q_i$ be the $T$-fixed point of $\mathcal{Y}_h$ obtained by setting all coordinates $\{j \,\vline\, j \neq i\}$ equal to zero. Then, for $i \in I(h)$, let \[
\phi_i = \mathbb{1}_h\cdot \prod \limits_{j \in I(h) - i} H - \lambda_j. \] If we pair $Z^T_g$ with $\phi_i$, we obtain the function \begin{equation*}
Z^T_{i,g} = \frac{\delta^{i, I(g)}}{125} + \sum_d Q^{c(d, h)} \bigg\langle
\frac{\phi_i}{z - \psi_1}, \mathbb{1}_g\bigg\rangle^{\mathcal{Y}, T}_{0,2,d}, \end{equation*} where $\delta^{i, I(g)}$ equals $1$ if $i \in I(g)$ and $0$ otherwise.
The fixed point set of $\mathcal{Y}_h$ consists of $\{q_j | j \in I(h)\}$. Note that under the inclusion $i_j: \{ q_j \} \to \mathcal{Y}_h$, $H$ pulls back to $\lambda_j$. Therefore $i_j^* (\phi_i) = 0$ unless $i = j$. From this we see that the coefficients of $Z^T_{i,g}$ consist of integrals over graphs such that the first marked point is mapped to $q_i$.
We divide the remaining graphs into \emph{two types}: those in which the first marked point is on a contracted component, and those in which the first marked point is on a noncontracted component.
\begin{claim} There is no contribution from graphs of the first type. \end{claim}
\begin{proof} The proof is a dimension count. We will show that the contributions from graphs of the first type must contain as a multiplicative factor integrals of the form $\int_{M} \Psi$ such that $\deg_\mathbb{C} (\Psi) > \dim (M)$, and hence the vanishing claim.
The complex degree of $\phi_i$ is $\dim(\mathcal{Y}_h)$, so the invariant $\langle\phi_i \psi_1^k, \mathbb{1}_g\rangle^{\mathcal{Y},T}_{0,2,d}$ vanishes unless $k \geq m_d - dim(\mathcal{Y}_h)$. Thus we can simplify our expression for $Z^T_{i,g}$: \begin{equation*}
\begin{split}
Z^T_{i,g}&=\frac{\delta^{i, I(g)}}{125} + \sum_d Q^{c(d,h)}
\bigg\langle\frac{\phi_i}{z - \psi_1}, \mathbb{1}_g \bigg\rangle^{\mathcal{Y},T}_{0,2,d} \\
&= \frac{\delta^{i, I(g)}}{125} + \sum_d Q^{c(d,h)}\frac{1}{z} \sum_{k = 0}^\infty
\big\langle \phi_i (\psi_1/z)^k, \mathbb{1}_g\big\rangle^{\mathcal{Y},T}_{0,2,d} \\
&=\frac{\delta^{i, I(g)}}{125} +\sum_d Q^{c(d,h)}\frac{1}{z} \sum_{k = c(d,h)-1}^\infty
\big\langle\phi_i (\psi_1/z)^k, \mathbb{1}_g\big\rangle^{\mathcal{Y},T}_{0,2,d} \\
&=\frac{\delta^{i, I(g)}}{125} + \sum_d \Big(\frac{Q}{z} \Big)^{c(d,h)}
\bigg\langle
\frac{\phi_i \psi_1^{c(d,h)-1}}{1 - (\psi_1/z)}, \mathbb{1}_g\bigg\rangle^{\mathcal{Y},T}_{0,2,d}.
\end{split} \end{equation*} Here the third equality follows from Lemma \ref{l:dime}.
Now consider a fixed point graph $M_\Gamma$ such that $p_1$ is on a contracted component. At the level of virtual classes, we can write \begin{equation} \label{e:2.3.1}
\left[M_\Gamma\right] = F(\Gamma) \cdot \prod_k \left[M_{v_k}\right], \end{equation} where each $M_{v_k}$ represents a contracted component of the graph isomorphic to a component of $\overline{M}_{0, n}(B\mathbb{Z}_r, 0)$, and $F(\Gamma)$ is a factor determined by $\Gamma$. Let $M_{v_0}$ be the component containing $p_1$. $M_{v_0}$ contains at most $2$ orbifold marked points, and the number of non-orbifold marked points is restricted by $d$. In particular, each non-orbifold marked point corresponds to a (non-orbifold) edge of the dual graph. Each of these edges must have degree at least $1$, so if the total degree of the map is $d$, then there can be at most $\lfloor d \rfloor $ nontwisted marked points. Thus the dimension of $M_{v_0}$ is at most $\lfloor d\rfloor - 1$. Now, the proof of Lemma \ref{l:dime} shows that \[
c(d, h) - 1 = 5\lfloor d \rfloor + |\{k\,|\, r_k(h)/5 \leq d(h,g)\}|
- 2 - \dim(\mathcal{Y}_h). \]
But $\dim(\mathcal{Y}_h)$ is exactly $|\{k\,|\, r_k(h) =0 \}| - 1$, which implies that $$c(d, h) - 1 \geq 5 \lfloor d \rfloor - 1.$$ If $d \geq 1$, the above quantity is strictly greater than $\lfloor d \rfloor - 1$. Because there do not exist graphs such that $p_1$ is on a non-contracted component for $d<1$, we have that for $M_\Gamma$, $c(d, h) - 1 \gneq \dim(M_{v_0})$. But $\psi_1^{c(d, I) - 1}$ must therefore vanish on these graphs, proving the claim. \end{proof}
\subsubsection{Contributions from a graph of the second type}\label{s:2.3.3}
Now let us consider the contribution to $\langle\frac{\phi_i}{z - \psi_1}, \mathbb{1}_g\rangle^{\mathcal{Y},T}_{0,2,d}$ from a particular graph $\Gamma$ of the second type. In particular, we know that $p_1$ is on a noncontracted component. Call this component $\mathcal{C}_0$, and denote the rest of the graph $\Gamma '$. $\Gamma'$ and $\mathcal{C}_0$ connect at a node $p'$, which maps to some $q_k \in \mathcal{Y}$. Let $d'$ be the degree of one connected component of the principal $\bar{G}$-bundle above $\mathcal{C}_0$. We know from Proposition~\ref{p:mapsallowed} that $\langle d' \rangle = r_k(h)/5$. By identifying $p' \in \Gamma '$ as a marked point (replacing $p_1$ on $\mathcal{C}_0$), we can view $M_{\Gamma '}$ as a fixed point locus in $\overline{\mathscr{M}}_{0, h', g}(\mathcal{Y}, d - d')$, where $[h] = [h']$, but $r_k(h') = 0$. Our plan will be to express integrals on $M_\Gamma$ in terms of integrals on $M_{\Gamma '}$, thus reducing the calculation to one involving maps of strictly smaller degree. This will give us a recursion.
The factor $F(\Gamma)$ in Equation~\ref{e:2.3.1} is composed of three contributions: the automorphisms of the graph $\Gamma$ itself, a contribution from each edge of $\Gamma$ (the non-contracted components of curves in $M_\Gamma$), and a contribution from certain flags of $\Gamma$ (the nodes of curves in $M_\Gamma$). The edge corresponding to $\mathcal{C}_0$ maps to the line $q_{ik} \cong \mathbb{P}^1/\bar{G}$ connecting $q_i$ and $q_k$. (Note that the $\bar{G}$-action is a subgroup of the big torus $(\mathbb{C}^*)^4$ of $\mathbb{P}^4$, $\bar{G}$ naturally acts on $(\mathbb{C}^*)^4$ orbits.) The degree of the map upstairs is $5 d'$. Thus there is a contribution of $1/(5 d')$ to $F(\Gamma)$ from the automorphism of $M_\Gamma$ coming from rotating the underlying curve. The edge also contributes a factor of $1/25$ due to the fact that $q_{ik}$ is a $(\mathbb{Z}/5\mathbb{Z})^2$-gerbe. So the total contribution to $F(\Gamma)$ from the edge containing $p_1$ is $1/(125d')$. The contribution from the node $p'$ is $125/r$.
(Recall $r = |[h]|$, which is equal to the order of the isotropy at $p'$). There will be an additional factor of $r$ appearing when we examine deformations of $M_\Gamma$, thus canceling the $r$ in the denominator. We finally arrive at the relation \[
\left[M_\Gamma\right] = F(\Gamma) \cdot \prod_{\text{vertices }v \in \Gamma}
\left[M_{v}\right] \, = \, \frac{F(\Gamma ')}{d'} \cdot
\prod_{\text{vertices }v \in \Gamma '} \left[M_{v}\right]
\, = \, \frac{1}{d'}\left[M_{\Gamma '}\right]. \] By examining the localization exact sequence (see \cite{tGrP}), we have the following identity: \begin{equation}\label{e:norm}
e(N_\Gamma)
= \frac{e(H^0(\mathcal{C}_0, f^*T\mathcal{Y})^m)(\text{node smoothing at $p'$})}
{e(H^0(p', f^*T\mathcal{Y})^m)e(H^1(\mathcal{C}_0, f^*T\mathcal{Y})^m) e((H^0(\mathcal{C}_0, T\mathcal{C}_0)^m)
}e(N_{\Gamma '})
\end{equation} where $e$ denotes the equivariant Euler class, and as is standard we identify certain vector bundles with their fibers. Here the superscript $m$ denotes the moving part of the vector bundle with respect to the torus action. Let us calculate the factors in \eqref{e:norm}.
$\bullet \, (\text{node smoothing at $p'$})$: The node smoothing contributes a factor of \[
\left( \frac{\lambda_k - \lambda_i}{rd'} - \frac{\psi_1 '}{r} \right) =
\frac{1}{r}\left( \frac{\lambda_k - \lambda_i}{d'} - \psi_1 ' \right), \] where $\psi_1 '$ is the $\psi$-class corresponding to $p_1 '$ on $M_{\Gamma} '$. This factor of $r$ is what cancels with the previous factor mentioned above.
$\bullet \, e(H^0(\mathcal{C}_0, T\mathcal{C}_0)^m)$: Let $C$ be the principal $\bar{G}$-bundle over $\mathcal{C}_0$ induced from
$f|_{\mathcal{C}_0}: \mathcal{C}_0 \to [\mathbb{P}^4/\bar{G}]$. As was argued in Proposition~\ref{p:mapsallowed},
$C$ consists of $(|\bar{G}|/r)$ copies of $\mathbb{P}^1$. Let $C_0$ be one of these copies. Then $C_0$ is a principal $\langle [h]\rangle$-bundle over $\mathcal{C}_0$ and \[
H^0(\mathcal{C}_0, T\mathcal{C}_0) = H^0(C_0, TC_0 )^{\langle [h]\rangle}. \] The $\langle [h] \rangle$-invariant part of $H^0(C_0, TC_0 )$ is one dimensional. It is fixed by the torus action, thus the moving part of $H^0(\mathcal{C}_0, T\mathcal{C}_0)$ is trivial and $e(H^0(\mathcal{C}_0, T\mathcal{C}_0)^m) = 1$.
$\bullet \, e(H^1(\mathcal{C}_0, f^*T\mathcal{Y})^m)$: Let $C_0$ be as in the previous bullet, then \[
H^1(\mathcal{C}_0, f^*T\mathcal{Y}) = H^1(C_0, \tilde{f}^*T\mathbb{P}^4 )^{\langle [h]\rangle} = 0. \] Therefore $e(H^1(\mathcal{C}_0, f^*T\mathcal{Y})^m)=1$.
$\bullet \, e(H^0(\mathcal{C}_0, f^*T\mathcal{Y})^m)$: To calculate this term, note that \[
H^0(\mathcal{C}_0, f^*T\mathcal{Y})^m \cong
\left( H^0(C_0, \tilde{f}^*T\mathbb{P}^4)^{\langle [h] \rangle}\right)^m. \] We will look at the $\langle [h] \rangle$ invariant part of the short exact sequence \[
0 \to \mathbb{C} \to H^0(\mathscr{O}_{C_0}(r d' )) \otimes V \to H^0(\tilde{f}^*T\mathbb{P}^4)
\to 0, \] where $\mathbb{P}^4 = \mathbb{P}(V)$ and $V \cong \mathbb{C}^5$. The exact sequence comes from the pullback of the Euler sequence for $\mathbb{P}^4$ to $C_0$. (Note that the degree of $\tilde{f}: C_0 \to \mathbb{P}^4$ is $rd'$). The action of $[h]$ on the first term in the sequence is trivial.
Recall that $\mathbb{P}(V)$ has coordinates $[x_0, \ldots, x_4]$. Let $[s,t]$ be homogeneous coordinates on $C_0 \cong \mathbb{P}^1$, such that the preimage of $p_1$ in $C_0$ is $[0,1]$ and the preimage of $p'$ in $C_0$ is $[1,0]$.
Then the middle term of the sequence is spanned by elements of the form $s^at^b \frac{\partial}{\partial x_l}$ where $0 \leq l \leq 4$ and $a + b = rd'$. The action is given by \[
[h] . (s^at^b \frac{\partial}{\partial x_l})
= e^{2 \pi \sqrt{-1} (-a + r_l(h))/r}s^at^b \frac{\partial}{\partial x_l}, \] and so this summand is invariant under the $\langle [h] \rangle$-action if and only if $r_l(h)/r = \langle a/r \rangle$. The $\mathbb{C}^*$-action on this term has weight \[ \left(a/r d'\right)\lambda_k + \left(b/r d'\right)\lambda_i - \lambda_l, \] so we finally arrive at \[
\begin{split} &e(H^0(\mathcal{C}_0, f^*T\mathcal{Y})^m) \\
= & \prod_{\substack{\{(a,l)| 0 \leq a \leq rd'\,\, 0 \leq l \leq 4\,\, r_l(h)/r = \langle a/r \rangle \}\\ \setminus \{(0,i),\,\,(r d',k)\}}}
\left(\frac{a}{rd'}\lambda_k + \frac{rd'-a}{rd'}\lambda_i - \lambda_l\right) \\
=&\prod_{\substack{\{(a,l)| 0 \leq a \leq rd'\,\, 0 \leq l \leq 4\,\, r_l(h)/r = \langle a/r \rangle \}\\ \setminus \{(0,i),\,\,(r d',k)\}}}
\left(a\left(\frac{\lambda_k -\lambda_i}{rd'}\right)
+ \lambda_i - \lambda_l\right).
\end{split} \]
$\bullet \, e(H^0(p', f^*T\mathcal{Y})^m )$: Similarly, the node $p'$ is isomorphic to $B\mathbb{Z}_r$, and each of the
$|\bar{G}|/r$ points lying in the principal $\bar{G}$-bundle over $p'$ is a principal $\langle [h] \rangle$-bundle over $p'$. Thus $H^0(p', f^*T\mathcal{Y})^m \cong \left( (T_{q_k}\mathbb{P}^n)^{\langle [h] \rangle}\right)^m$ and \[
e(H^0(p', f^*T\mathcal{Y})^m ) = \prod_{l \in I(h') \setminus\{ k \}}
\left(\lambda_k - \lambda_l\right). \]
Finally note that $ev_1^*(\phi_i) = \prod_{l \in I(h) - i} (\lambda_i - \lambda_l)$. We can do one further simplification. On the graphs which we consider, namely those where $p_1$ is on a noncontracted component, $\psi_1$ restricts to $\frac{\lambda_k - \lambda_i}{d'}$. (In fact $e(T^*_{p_1} \mathcal{C}) \cong \frac{\lambda_k - \lambda_i}{rd'}$, but because we are following the convention that $\psi$-classes are pulled back from the reification, we must multiply this by a factor of $r$).
These calculations plus \eqref{e:norm} then give us the contribution to $\big\langle\frac{\phi_i \psi_1^{c(d,h) - 1}}{1 - \psi_1/z}, \mathbb{1}_{g} \big\rangle^{\mathcal{Y},T}_{0,2,d}$ from the graph $M_\Gamma$: \[ \begin{split}
&\int_{[M_\Gamma]} \frac{ev_1^*(\phi_i)\psi_1^{c(d,I) - 1}}
{e(N_\Gamma)\left(1 - \psi_1/z \right)} \\
= &\frac{\frac{\lambda_k - \lambda_i}{d'}^{c(d,I) - 1}
\prod_{l \in I(h) \setminus \{ i\}} (\lambda_i - \lambda_l)e(H^1(\mathcal{C}_0, f^*T\mathcal{Y})^m)}
{e(H^0(\mathcal{C}_0, f^*T\mathcal{Y})^m) (1 - \frac{\lambda_k - \lambda_i}{d'z})} \\
&\cdot\frac{1}{d '}\int_{[M_\Gamma ']} \frac{e(H^0(p', f^*T\mathcal{Y})^m)}
{(\text{node smoothing at $p'$})e(N_{\Gamma'})} \\
=&\frac{\frac{\lambda_k - \lambda_i}{d'}^{c(d,h) - 1}\prod_{l \in I(h)\setminus \{ i\}}
(\lambda_i - \lambda_l)}{(d' - \frac{\lambda_k - \lambda_i}{z})
\prod \limits_{\substack{\{(a,l)| 0 \leq a \leq rd'\,\, 0 \leq l \leq 4\,\, r_l(h)/r = \langle a/r \rangle \}\\ \setminus \{(0,i),\,\,(r d',k)\}}}
\left(a\left(\frac{\lambda_k -\lambda_i}{rd'}\right) + \lambda_i - \lambda_l\right)} \\
&\cdot \int_{[M_{\Gamma'}]} \frac{\prod_{l \in I(h')\setminus \{ k \}}
\left(\lambda_k - \lambda_l\right)}{(\frac{\lambda_k - \lambda_i}{d'} - \psi_1)
e(N_{\Gamma'})}. \end{split} \]
\subsubsection{Recursion relations} \label{s:2.3.4} We will formulate the above computations into a recursion relation. To do that, the following regularity lemma is needed.
\begin{lemma}[Regularity Lemma] $Z^T_{i,g}$ is an element of $\mathbb{Q}(\lambda_i,z)[[Q]]$. The coefficient of each $Q^D$ is a rational function of $\lambda_i$ and $z$ which is regular at $z = (\lambda_i -\lambda_j)/k$ for all $j\neq i$ and $k \geq 1$. \end{lemma}
\begin{proof} This follows from a standard localization argument, see e.g.\ Lemma 11.2.8 in \cite{CK}. \end{proof}
Using the Regularity Lemma, the above computation simplifies to \[\begin{split} &\left( \bigg\langle\frac{\phi_i \psi_1^{c(d, h) - 1}}{1 - \psi_1/z}, \mathbb{1}_{g}\bigg\rangle^{\mathcal{Y},T}_{0,2,d} \right)_{M_{\Gamma}} \\
&= C^{i,k}_{d'} \cdot \left(\frac{\lambda_k - \lambda_i}{d'}\right)^{c(d,h) - 1 - (c(d', h) - 1)} \cdot \left( \bigg\langle\frac{\phi_k}{z - \psi_1}, \mathbb{1}_{g}\bigg\rangle^{\mathcal{Y},T}_{0,2,d - d'} \right)_{M_{\Gamma'}} \bigg|_{z \mapsto \frac{\lambda_k - \lambda_i}{d'}}, \end{split} \] where \[
C^{i,k}_{d'} = \frac{1}{(d' - \frac{\lambda_k - \lambda_i}{z})
\prod \limits_{\{(a,l)\in S(d', h) \setminus \{ (d',k) \}\}} \left(a + d'\left(\frac{\lambda_i - \lambda_l}{\lambda_k -\lambda_i}\right)\right)} \] and $\left(-
\right)_{M_{\Gamma}}$ means the contribution of the fixed component $M_{{\Gamma}}$ to the expression in parentheses.
Due to the fact that $r_k(h)/5 = \langle d' \rangle$, one can check that \[
c(d, h) - c(d', h) = c(d - d', h') \] (see \eqref{e:recur}). We arrive at the expression \[
C^{i,k}_{d'} \cdot \left( Q^{c(d-d', k)} \bigg\langle\frac{\phi_k}{z - \psi_1},
\mathbb{1}_{g}\bigg\rangle^{\mathcal{Y},T}_{0,2,d - d'} \right)_{M_{\Gamma'}}
\bigg|_{z \mapsto \frac{\lambda_k - \lambda_i}{d'}, Q \mapsto \frac{\lambda_k - \lambda_i}{d'} }. \] After summing over all possible graphs, we obtain the recursion: \begin{equation}\label{e:recursion}
Z^T_{i,g} = \frac{\delta^{i, I(g)}}{125} +
\sum_{\{(d',k) | \frac{r_k(h)}{5} = \langle d'\rangle , k \neq i, d' \neq 0 \}}
\left(\frac{Q}{z}\right)^{c(d', h)}C^{i,k}_{d'}\cdot Z^T_{k, g}
\bigg|_{z \mapsto \frac{\lambda_k - \lambda_i}{d'}, Q \mapsto \frac{Q}{z}\frac{\lambda_k - \lambda_i}{d'} }. \end{equation} Although we have suppressed this in the notation, recall that in the above summand, $h$ is the presentation such that $\phi_i \in \mathcal{Y}_h$ ($i \in I(h)$).
We will now turn our attention to $Y^T_g$. Let us define the function $Y^T_{i,g}$ analogously to that of $Z^T_{i,g}$ , \[
Y^T_{i,g} := ( \phi_i, Y^T_g )^\mathcal{Y}_{CR}. \] For $i \in I(h)$, \[
Y^T_{i,g} = \frac{1}{125}\left(\delta^{i, I(g)}
+ \sum_{\langle d \rangle = d(h,g)} Q^{c(d,h)}
\frac{1}{\prod \limits_{(b,k) \in S(d, h)} (bz + \lambda_i - \lambda_k)}\right). \] \begin{claim} $Y^T_{i,g}$ satisfy the same recursion as $Z^T_{i,g}$ in \eqref{e:recursion}. \end{claim} \begin{proof} Consider the summand of $Y^T_{i,g}$ of degree $c(d,h)$ in $Q$, which we will denote $(Y^T_{i,g})^{c(d,h)}$. \[ \begin{split}
(Y^T_{i,g})^{c(d,h)}
= &\frac{1}{125} \left(\frac{Q}{z} \right)^{c(d,h)}
\frac{1}{\prod_{(b,k) \in S(d,h)}
\left(b + (\lambda_i - \lambda_k)/z\right)} \\
=&\frac{1}{125} \left(\frac{Q}{z} \right)^{c(d,h)}
\sum_{\{(b,k) | r_k(h)/5 = \langle b \rangle , k \neq i, b\neq 0 \}}
\frac{1}{\left(b + (\lambda_i - \lambda_k)/z\right)} \\
&\qquad \cdot \frac{1}{\prod \limits_{(m,l) \in S(d,h) \setminus \{ (b,k) \} }
\left( b (\lambda_i - \lambda_l)/(\lambda_k - \lambda_i) + m\right)}\\
=&\frac{1}{125} \left(\frac{Q}{z} \right)^{c(d,h)}
\sum_{\{(b,k) | r_k(h)/5 = \langle b \rangle , k \neq i, b \neq 0 \}} \\
&\left( \frac{ 1/\left(b + (\lambda_i - \lambda_k)/z\right)}
{\prod \limits_{\{(m,l) \in S(d,h) \setminus \{(b,k)\}| m \leq b\}}
\left( b (\lambda_i - \lambda_l)/(\lambda_k - \lambda_i) + m\right)} \right. \\
&\qquad \cdot \left.
\frac{1}{\prod \limits_{\{(m,l) \in S(d,h) \setminus \{(b,k)\}| m > b\}}
\left( b (\lambda_i - \lambda_l)/(\lambda_k - \lambda_i) + m\right)} \right). \end{split} \]
The last product from above can be rewritten as \[
\prod_{(n,l) \in S(d - b, h')} \left(n +
b\frac{\lambda_k - \lambda_l}{\lambda_k - \lambda_i}\right), \] where $h'$ is chosen such that $[h] = [h']$ and $k \in I(h')$. To see this note that if $(b, k)$ and $(m,l)$ are both in $S(d, h)$, then by definition $r_k(h)/5 = \langle b \rangle$ and $r_l(h)/5 = \langle m \rangle$. If $k \in I(h')$, then \[
\begin{split}
&\frac{r_l(h')}{5} = \frac{r_l(h')}{5} - \frac{r_k(h')}{5} \\
\equiv &\frac{r_l(h)}{5} - \frac{r_k(h)}{5} \equiv \langle m \rangle -
\langle b \rangle \equiv \langle m - b \rangle \, (\operatorname{mod} 1).
\end{split} \] In other words $r_l(h')/5 = \langle m - b \rangle$. This proves that if $(b, k)\in S(d, h)$, and $h'$ is chosen as above, then for pairs $(m, l)$ with $b < m \leq d$, \begin{equation}\label{e:recur} (m,l) \in S(d, h)\text{ if and only if }(m - b, l) \in S(d - b, h'). \end{equation} We arrive at the relation \begin{equation*} \begin{split}
&\left(Y^T_{i,g}\right)^{c(d,h)} \\
\quad = &\sum_{\{(b,k) | r_k(h)/5 = \langle b\rangle , k \neq i, b \neq 0 \}}
\left(\frac{Q}{z}\right)^{c(b, h)}C^{i,k}_{b}\left(Y^T_{k, g}\right)^{c(d-b, h')}
\bigg|_{z \mapsto \frac{\lambda_k - \lambda_i}{b}, Q \mapsto \frac{Q}{z}\frac{\lambda_k - \lambda_i}{b}}. \end{split} \end{equation*} We conclude that $Y^T_{i,g}$ satisfy the same recursion as $Z^T_{i,g}$. \end{proof}
The recursion relation and initial conditions imply $Y^T_{i,g} = Z^T_{i,g}$. The proof of Theorem~\ref{t:Zformula} is now complete. \begin{remark} As a corollary one may easily obtain an explicit formula for the small $J$-matrix $J^\mathcal{Y}_{small}(t, z)$ by isolating coefficients of the various $Z^\mathcal{Y}_g$. We give an explicit expression for certain specified rows of $J^\mathcal{Y}_{small}(t, z)$ in Corollary~\ref{c:ambientJformula}. \end{remark}
\section{$A$ model of the mirror quintic $\mathcal{W}$} \label{s:3}
\subsection{Fermat quintic and its mirror} \label{s:3.1}
Let $M \subset \mathbb{P}^4$ be the Fermat quintic defined by the equation $Q_0(x) = x_0^5 + x_1^5 + x_3^5 + x_4^5 + x_5^5$ \[
M := \{ Q_0(x) = 0 \} \subset \mathbb{P}^4. \] The Greene--Plesser's \emph{mirror construction} \cite{bGrP} gives the \emph{mirror orbifold} as the quotient stack \[
\mathcal{W} := [M / \bar{G}]. \] Note that the $\bar{G}$-action on $\mathbb{P}^4$ \eqref{e:G} preserves the quintic equation $Q_0(x)$ and therefore induces an action on $M$. Equivalently, \begin{equation} \label{e:3.1.1}
\mathcal{W} = \{ Q_0 =0 \} \subset \mathcal{Y} = [ \mathbb{P}^4/\bar{G}]. \end{equation}
\begin{remark} \label{r:3.1} Since in this section we will only be interested in the Gromov--Witten theory ($A$ model), which is deformation invariant, we will only speak of the mirror orbifold instead of the mirror family. \end{remark}
Recall in Lemma~\ref{l:2.1} the inertia orbifold of $\mathcal{Y} = [\mathbb{P}^4/\bar{G}]$ is indexed by $g \in G$. For a particular $g$, the dimension of $\mathcal{Y}_g$ is equal to
$\big|\{j | r_j = 0\}\big| - 1$, and can be identified with a linear subspace of $\mathcal{Y}$. The age shift of $\mathcal{Y}_g$ is $\operatorname{age}(g) = \sum_{i = 0}^4 r_i/5$.
The inertia orbifold of the mirror quintic $\mathcal{W}$ can be described by that of $\mathcal{Y}$. $\mathcal{W}$ intersects nontrivially with $\mathcal{Y}_g$ exactly when
$\big|\{j | r_j = 0 \}\big| \geq 2$. (that is, $\dim \mathcal{Y}_g \geq 1$.) Let \[
\bar{S} := \left\{g= (r_0, \ldots, r_4) \in G \big|\, 2 \leq \big|\{j | r_j = 0 \}\big| \, \right\}. \] (Note that $\bar{S}$ contains $e=(0, \ldots, 0)$.) Then \[
I\mathcal{W} = \coprod_{g \in \bar{S}} \mathcal{W}_g \,, \qquad
\mathcal{W}_g := \mathcal{W} \cap \mathcal{Y}_g. \] All nontrivial intersections are transverse, so \[
\dim(\mathcal{W}_g) = \dim(\mathcal{Y}_g) - 1 = \big|\{j | r_j = 0\}\big| - 2. \] It follows that the age shift of $\mathcal{W}_g$ is equal to the age shift of $\mathcal{Y}_g$. The cohomology of $\mathcal{W}$ is given by \[
H^*_{CR}(\mathcal{W}) = \bigoplus_{g \in \bar{S}} H^{* - 2\operatorname{age}(g)}(\mathcal{W}_g). \]
In the sequel, we will only be interested in the subring of $H^*_{CR}(\mathcal{W})$ consisting of classes of even (real) degree. We will denote this ring as $H^{even}_{CR}(\mathcal{W})$. It can be checked via a direct calculation that if
$i: \mathcal{W} \hookrightarrow \mathcal{Y}$ is the inclusion, \[
H_{CR}^{even}(\mathcal{W}) = i^* H^*_{CR} (\mathcal{Y}). \]
\begin{conventions} \label{conv:2} Let $H$ be the hyperplane class on $\mathbb{P}^4$. By an abuse of notation, we will denote $H$ any fixed choice of $L$ on $\mathcal{Y}$ such that $\pi_{\mathbb{P}^4}^*(L) = H$,
where $\pi_{\mathbb{P}^4}$ was defined in \eqref{e:cover}. We will also denote $H$ the induced class on $\mathcal{W}$. Even though there are as many as $|\bar{G}|$ choices of $L$, they are topologically equivalent and will serve the same purpose in our discussion. \end{conventions}
A convenient basis $\{T_i\}$ for $H_{CR}^{even}(\mathcal{W})$ is \begin{equation} \label{e:Wbasis}
\bigcup_{g \in \bar{S}} \{\mathbb{1}_g, \mathbb{1}_g {H}, \ldots ,
\mathbb{1}_g {H}^{\dim(\mathcal{W}_g)}\}. \end{equation}
We also note that $H_{CR}^{even}(\mathcal{W}) \subset H^*_{CR}(\mathcal{W})$ is a self-dual subring with respect to the Poincar\'e pairing of $H^*_{CR}(\mathcal{W})$. Furthermore, this basis is self-dual (up to a constant factor). Given $g = (r_0, \ldots, r_4) \in S$, let \[
g^{-1}:= (-r_1, \ldots, -r_4) \, (\operatorname{mod} 5). \] Then the Poincar\'e dual elements can be easily calculated: \[
\left( \mathbb{1}_g {H}^k \right)^{\vee}
= 25 \left( \mathbb{1}_{g^{-1}} {H}^{\dim(\mathcal{W}_g) - k} \right). \]
\subsection{$J$-functions of $\mathcal{W}$} \label{s:3.2}
\begin{conventions} \label{conv:3} By the matrix $J$-function of $\mathcal{W}$, we will mean the matrix consisting of the collection of $H^{even}_{CR}(\mathcal{W})$-valued functions with variable $\mathbf{t} = t H$. \begin{equation}\label{e:sJW}
J_g^{\mathcal{W}}(t,z) := e^{t{H}/z}\left(\mathbb{1}_g + \sum_{d , i} q^d e^{dt}
\bigg\langle \frac{T_i}{z - \psi_1}, \mathbb{1}_g \bigg\rangle^{\mathcal{W}}_{0,2,d}
T^i\right), \end{equation} where the basis $\{T_i\}$ is for $H^{even}_{CR}(\mathcal{W})$, as in \eqref{e:Wbasis}. Here as in Section~\ref{s:2}, by the degree $d$ of a map $f: \mathcal{C} \to \mathcal{W}$ we mean \[
d:= \int_{\mathcal{C}} f^*({H}). \] Note that if we extend the basis $\{T_i\}$ to full basis of $H^*_{CR}(\mathcal{C})$, the classes of odd (real) degree will not contribute to $J^{\mathcal{W}}_g(t, z)$, and thus \eqref{e:sJW} is equal to the $J_g$-function of \eqref{e:sJ}. \end{conventions}
As has been shown in Proposition~\ref{p:mapsallowed}, for an orbi-curve $\mathcal{C}$ with two marked points, the degree must be a multiple of $1/5$. Recall also from Proposition~\ref{p:mapsallowed} that the only nonzero contribution to the terms in $J_g^{\mathcal{W}}$ comes from elements $T_i$ supported on some $\mathcal{W}_h$ such that $[h] = [g^{-1}]$. From the definition of $\bar{S}$, it is required that \begin{equation} \label{e:3.1.2}
\big|\{j | r_j = 0 \}\big| \geq 2, \quad
\sum r_j \equiv 0 \, (\operatorname{mod} 5). \end{equation} We will enumerate all possible cases.
It follows from the conditions \eqref{e:3.1.2} that
$\big|\{j | r_j = 0 \}\big|$ must be equal to $2$, $3$ or $5$. That is, $\dim(\mathcal{W}_g)$ is equal to $0$, $1$ or $3$.
If $\dim(\mathcal{W}_g) = 3$, $g = e = (0,0,0,0,0)$ and $\mathbb{1}_e = 1$. The only basis elements which contribute to $J_e^{\mathcal{W}}$ come from the nontwisted sector. We have \begin{equation} \label{e:dim3}
J_e^{\mathcal{W}}(t,z) = e^{t{H}/z}\left(1 + \sum_{d > 0} q^d e^{dt} \bigg\langle
\frac{{H}^i}{z - \psi_1}, 1 \bigg\rangle^{\mathcal{W}}_{0,2,d}
(25{H}^{3-i})\right). \end{equation}
If $\dim(\mathcal{W}_g) = 1$, then up to a permutation of the entries, $g = (0,0,0, r_1, r_2)$ with $r_1 \neq r_2$. By definition of $\bar{S}$, other than $g$ there is no $h \in \bar{S}$ such that $[h] = [g]$. Therefore, the two basis elements which contribute nontrivially to $J_g^{\mathcal{W}}$ are $\mathbb{1}_{g^{-1}}$ and $\mathbb{1}_{g^{-1}} {H}$. We arrive at \begin{equation} \label{e:dim1}
\begin{split}
&J_g^{\mathcal{W}}(t,z) = e^{t {H}/z} \Bigg( \mathbb{1}_g + \\
&\sum_{d > 0} q^d e^{dt} \left(\bigg\langle \frac{\mathbb{1}_{g^{-1}}}{z - \psi_1}, \mathbb{1}_g
\bigg\rangle^{\mathcal{W}}_{0,2,d} (25\mathbb{1}_g{H}) + \bigg\langle
\frac{\mathbb{1}_{g^{-1}}{H}}{z - \psi_1}, \mathbb{1}_g \bigg\rangle^{\mathcal{W}}_{0,2,d}
(25\mathbb{1}_{g})\right) \Bigg).
\end{split} \end{equation}
If $\dim(\mathcal{W}_g) = 0$, then up to a permutation of the entries, $g = (0,0,r_1, r_1, r_2)$, with $r_1 \neq r_2$. There is only one other $g_1 \in \bar{S}$ such that $[g_1] =[g]$, namely, $g_1 = (-r_1, -r_1, 0,0, r_2 - r_1) \, (\operatorname{mod} 5)$. The two basis elements which contribute nontrivially to the invariants of $J_g^{\mathcal{W}}$ are $\mathbb{1}_{g^{-1}}$ and $\mathbb{1}_{(g_1)^{-1}}$. Thus we can express $J_g^{\mathcal{W}}(t,z)$ as \begin{equation} \label{e:dim0}
\begin{split}
&J_g^{\mathcal{W}}(t,z) = e^{t {H}/z} \Bigg( \mathbb{1}_g + \\
&\sum_{d > 0} q^d e^{dt}
\left(\bigg\langle \frac{\mathbb{1}_{g^{-1}}}{z - \psi_1}, \mathbb{1}_g
\bigg\rangle^{\mathcal{W}}_{0,2,d} (25\mathbb{1}_g) + \bigg\langle
\frac{\mathbb{1}_{(g_1)^{-1}}}{z - \psi_1}, \mathbb{1}_g \bigg\rangle^{\mathcal{W}}_{0,2,d}
(25\mathbb{1}_{g_1})\right)\Bigg).
\end{split} \end{equation} Thus for each twisted component $\mathcal{W}_g$, the $J$-function $J_g^{\mathcal{W}}$ has two components.
We will relate the functions $J_g^{\mathcal{W}}$ to certain hypergeometric functions, called $I$-functions. To start with, let us introduce ``bundled-twisted'' Gromov--Witten invariants. Let $E \to \mathcal{X}$ be a line bundle over the orbifold $\mathcal{X}$. We have the following diagram \[
\begin{CD}
@. E\\
@. @VVV \\
\mathcal{C} @>f>> \mathcal{X} \\
@VV{\pi}V @. \\
\overline{\mathscr{M}}_{0,n}(\mathcal{X}, d).
\end{CD} \]
The $E$-twisted Gromov--Witten invariants are defined to be \[
\big\langle \alpha_1 \psi^{k_1}, \ldots , \alpha_n \psi^{k_n}
\big\rangle_{0,n,d}^{\mathcal{X}, \operatorname{tw}} = \int_{[\overline{\mathscr{M}}_{0,n}(\mathcal{X}, d)]^{vir}}
\prod_{i=1}^n ev_i^*(\alpha_i) \psi_i^{k_i} \cup e(E_{0,n,d}), \] where \[
E_{o,n,d} := \pi_* f^*(E) \] and $e(E_{0,n,d})$ is the Euler class of the $K$-class. We can define a twisted pairing on $H^*_{CR}(\mathcal{X}; \Lambda)$ by \[
( \alpha_1, \alpha_2 )_{CR}^{\mathcal{X}, \operatorname{tw}} =
\int_\mathcal{X} \alpha_1 \cup I^*(\alpha_2) \cup e(E). \] With this, we can define a twisted $J$-function \[
J^{\mathcal{X}, \operatorname{tw}}(\mathbf{t},z) = 1 + \mathbf{t}/z + \sum_{d}
\sum_{n \geq 0} \sum_i \frac{q^d}{n!} \bigg\langle \frac{T_i}{z - \psi_1}, 1, \mathbf{t},
\ldots, \mathbf{t} \bigg\rangle_{0,2+k, d}^{\mathcal{X}, \operatorname{tw}} T^i. \] Here $T_i$ is a basis for $H^*_{CR}(\mathcal{X}; \Lambda)$ and $T^i$ is the dual basis with respect to the twisted pairing.
The twisted invariants are related to invariants on the hypersurface. In our case, $\mathcal{X} = \mathcal{Y} = [\mathbb{P}^4/\bar{G}]$, and $E = \mathscr{O} (5) \to \mathcal{Y}$. It is easy to see that $E_{0,n,d} = R^0 \pi_* f^*(\mathscr{O} (5))$ is a vector bundle. The embedding $i: \mathcal{W} \hookrightarrow \mathcal{Y}$ induces a morphism $\iota: \overline{\mathscr{M}}_{0,n}(\mathcal{W}, d)\hookrightarrow \overline{\mathscr{M}}_{0,n}(\mathcal{Y}, d)$. It is well-known that \begin{equation} \label{3.2.6}
\iota_* [\overline{\mathscr{M}}_{0,n}(\mathcal{W}, d)]^{\operatorname{vir}} =
e(E_{0,n,d}) \cap [\overline{\mathscr{M}}_{0,n}(\mathcal{Y}, d)]^{\operatorname{vir}}. \end{equation} A proof can be found in e.g.~\cite{CKL}. (That proof, given in the nonorbifold setting there, can be readily modified to the orbifold setting.) This relates the twisted invariants on $\mathcal{Y}$ to the invariants on $\mathcal{W}$. Assume that $\mathbf{t}$ is restricted to $H^{even}_{CR}(\mathcal{Y})$, then \[ J^\mathcal{W}(\mathbf{t}, z) = i^* J^{\mathcal{Y}, tw}(\mathbf{t}, z). \] Let us now further restrict $\mathbf{t}$ to $H^{2}_{CR}(\mathcal{Y})$. In our setting we may write an element of $H^{2}_{CR}(\mathcal{Y})$ as \begin{equation} \label{e:3.2.7}
\mathbf{t} = tH + \sum\limits_{\{g| \operatorname{age}(g) = 1\}} t^g \mathbb{1}_g. \end{equation} Write the $J$-function of $\mathcal{Y}$ as \[
J^{\mathcal{Y}} (\mathbf{t}) = \sum_d q^d J_d^{\mathcal{Y}}(\mathbf{t}). \] For each $d$, define the modification factor \[
M^{E/\mathcal{W}}_d := \prod_{m=1}^{5d} (5H + m z). \] (Note that we have taken the $\lambda =0$ limit in \cite{CCIT}.)
\begin{definition} \label{d:Ifunction} Define the \emph{twisted $I$-function} by \[
I^{E} (\mathbf{t}) := \sum_d q^d M^{E/\mathcal{W}}_d J_d^{\mathcal{Y}}(\mathbf{t}) \] Write \begin{equation} \label{e:3.2.8}
\begin{split}
I^{E} (\mathbf{t}, z) = &I^E_e (t,z) +
\frac{1}{z}\left(\sum\limits_{\{g| \operatorname{age}(g) = 1\}} t^g I^E_g(t,z) \right) \\
&+\frac{1}{z}\left( \sum \limits_{\{g_1, g_2| \operatorname{age}(g_i) = 1\}} t^{g_1}t^{g_2}
I^E_{g_1, g_2}(t,z) + \ldots \right).
\end{split} \end{equation} For $g$ such that $\operatorname{age}(g) \leq 1$ (including $g = e$), define the $A$ model hypergeometric functions \begin{equation}\label{e:IA}
I^A_g(t,z) = i^*\left(I^E_g(t, z)\right). \end{equation}
\end{definition}
\begin{theorem}\label{t:A-model} Given $g = (r_0, \ldots, r_4)$ such that the age shift of $\mathcal{W}_g$ is at most $1$, there exist functions $F_0(t)$, $G_0(t)$, and $H_g(t)$, determined explicitly by $I^E_g(t, z)$ such that $F_0$ and $H_g$ ($g \neq 0$) are invertible, and \begin{equation}\label{e:A-model}
J_g^{\mathcal{W}}(\tau(t),z) =
\frac{I^A_g(t,z)}{H_g(t)} \qquad \text{where }
\tau(t) = \frac{G_0(t)}{F_0(t)}. \end{equation} \end{theorem} \begin{remark} In the statement of the theorem, $F_0(t)$ and $G_0(t)$ do not depend on $g$, so the \emph{mirror map} $t \mapsto \tau(t) = G_0(t)/F_0(t)$ is well defined. \end{remark}
\subsection{Proof of Theorem~\ref{t:A-model}} \label{s:3.3}
There are two key ingredients in the proof. The first one is the version of \emph{quantum Lefschetz hyperplane theorem} (QLHT) for orbifolds proved in \cite{CCIT}. By Equation~\eqref{e:3.1.1}, $\mathcal{W}$ is a hyperplane section of $\mathcal{Y}$ and hence $J^{\mathcal{W}}$ can be calculated by QLHT. Corollary~5.1 in \cite{CCIT} in particular implies the following:
\begin{theorem}[\cite{CCIT}]\label{t:QLHT} Let the setting be as above, with $E= \mathscr{O}(5) \to \mathcal{Y}$. Then \begin{equation} \label{e:3.3.1}
I^{E}(\mathbf{t}, z) = F(\mathbf{t}) + \frac{G(\mathbf{t})}{z} + O(z^{-2}) \end{equation} for some $F$ and $G$ with $F$ scalar valued and invertible, and \begin{equation}\label{e:QLHT}
J^{\mathcal{Y}, \operatorname{tw}}(\tau(\mathbf{t}), z) = \frac{I^{E}(\mathbf{t}, z)}
{F(\mathbf{t})} \qquad \text{where } \tau(\mathbf{t}) = \frac{G(\mathbf{t})}{F(\mathbf{t})}. \end{equation} \end{theorem}
The second ingredient is the explicit formula of $J_g^{\mathcal{Y}}$ from Section~\ref{s:2}. Note that we are only concerned with those $g$ such that $i^* \mathbb{1}_g \neq 0$ and $\operatorname{age}(\mathbb{1}_g) \leq 1$. Therefore only those $J_g^{\mathcal{Y}}$ are listed. The following is a straightforward corollary of Theorem~\ref{t:Zformula}, \eqref{e:2.2.2} and \eqref{e:2.2.3} by equating the terms $Q^{c(d,h)} \mathbb{1}_{h^{-1}} H^k$ of $Z_g$ with the terms $q^d e^{dt} \mathbb{1}_{h^{-1}} H^k$ of $J^\mathcal{Y}_g$.
\begin{corollary} \label{c:ambientJformula} The functions $J^{\mathcal{Y}}_g(t,z)$ are given by the following formulas. \begin{enumerate} \item[(i)] If $g = e = (0,0,0,0,0)$, \begin{equation}\label{e:jdim4}
J_e^\mathcal{Y} = e^{t H/z}\left(1 + \sum_{\langle d\rangle = 0}
q^d e^{dt} \frac{1}{\prod \limits_{\substack{0 < b \leq d\\ \langle b \rangle = 0 }}
(bz - H)^5 }\right). \end{equation} \item[(ii)] If $g = (0,0,0, r_1, r_2)$, let $g_1 = (-r_1,-r_1,-r_1,0, r_2-r_1) \, (\operatorname{mod} 5)$ and let ${g_2 = (-r_2,-r_2,-r_2,r_1-r_2,0) \, (\operatorname{mod} 5)}$. Then \begin{align}
J^Y_g =
&e^{t H/z}\mathbb{1}_g \left(1 + \sum_{\langle d \rangle = 0} \frac{q^d e^{dt}}
{\prod \limits_{\substack{0 < b \leq d \\ \langle b \rangle = 0}} (H+bz)^3 \prod
\limits_{\substack{0 < b \leq d \\ \langle b \rangle = \left\langle \frac{r_2}{5} \right\rangle}}
(H+bz)\prod
\limits_{\substack{0 < b \leq d \\ \langle b \rangle = \left\langle \frac{r_1}{5}\right\rangle}}
(H+bz)} \right) \label{e:jdim1} \\ + &e^{t H/z}\mathbb{1}_{g_1}\left(
\sum_{\langle d \rangle = \left\langle \frac{r_1}{5}\right\rangle} \frac{q^d e^{dt}}{\prod
\limits_{\substack{0 < b \leq d \\ \langle b \rangle = \left\langle \frac{r_1}{5} \right\rangle}}
(H+bz)^3\prod
\limits_{\substack{0 < b \leq d \\ \langle b \rangle = 0}} (H+bz)\prod
\limits_{\substack{0 < b \leq d \\ \langle b \rangle = \left\langle \frac{2r_1}{5} \right\rangle}}
(H+bz)}\right) \nonumber \\ + &e^{t H/z}\mathbb{1}_{g_2} \left(
\sum_{\langle d \rangle = \left\langle \frac{r_2}{5}\right\rangle} \frac{q^d e^{dt}}{\prod
\limits_{\substack{0 < b \leq d \\ \langle b \rangle = \left\langle \frac{r_2}{5} \right\rangle}}
(H+bz)^3\prod
\limits_{\substack{0 < b \leq d \\ \langle b \rangle = \left\langle \frac{2r_2}{5} \right\rangle}}
(H+bz)\prod
\limits_{\substack{0 < b \leq d \\ \langle b \rangle = 0}} (H+bz)}\right). \nonumber \end{align} \item[(iii)] If $g = (0,0, r_1, r_1, r_2)$, let $g_1 = (-r_1, -r_1, 0,0,r_2 - r_1) \, (\operatorname{mod} 5)$ and let $g_2 = (-r_2, -r_2, r_1 - r_2, r_1 - r_2, 0 ) \, (\operatorname{mod} 5)$. Then \begin{align}
J^Y_g =
&e^{t H/z}\mathbb{1}_g \left( 1 + \sum_{\langle d \rangle = 0} \frac{q^d e^{dt}}
{\prod \limits_{\substack{0 < b \leq d \\ \langle b \rangle = 0}} (H+bz)^2\prod
\limits_{\substack{0 < b \leq d \\ \langle b \rangle = \left\langle \frac{3r_2}{5} \right\rangle}}
(H+bz)^2\prod
\limits_{\substack{0 < b \leq d \\ \langle b \rangle = \left\langle \frac{2r_1}{5}\right\rangle}}
(H+bz)} \right) \label{e:jdim0} \\
+ &e^{t H/z}\mathbb{1}_{g_1} \left(
\sum_{\langle d \rangle = \left\langle \frac{r_1}{5}\right\rangle}
\frac{q^d e^{dt}}{\prod
\limits_{\substack{0 < b \leq d \\ \langle b \rangle = \left\langle \frac{r_1}{5} \right\rangle}}
(H+bz)^2\prod \limits_{\substack{0 < b \leq d \\ \langle b \rangle = 0}}
(H+bz)^2\prod
\limits_{\substack{0 < b \leq d \\ \langle b \rangle = \left\langle \frac{r_2}{5} \right\rangle}}
(H+bz)}\right) \nonumber \\
+ &e^{t H/z}\mathbb{1}_{g_2} \left(
\sum_{\langle d \rangle = \left\langle \frac{r_2}{5}\right\rangle}
\frac{q^d e^{dt}}{\prod
\limits_{\substack{0 < b \leq d \\ \langle b \rangle = \left\langle \frac{r_2}{5} \right\rangle}}
(H+bz)^2\prod
\limits_{\substack{0 < b \leq d \\ \langle b \rangle = \left\langle \frac{2r_1}{5} \right\rangle}}
(H+bz)^2\prod \limits_{\substack{0 < b \leq d \\ \langle b \rangle = 0}} (H+bz)}\right).
\nonumber
\end{align} \end{enumerate}
In fact, due to the age requirement, there are only two choices in case (ii) up to permutation: $(r_1, r_2)=(2,3)$ or $(1,4)$. In case (iii), only $(r_1, r_2)=(1,3)$ or $(2,1)$ are possible. \end{corollary}
\begin{lemma} \label{l:3.8} There are scalar valued functions $F_0(t), G_0(t)$ and $G_g(t)$ for each $g$ with $\operatorname{age}(g) =1$, such that \begin{equation*}
i^* \left( I^E(\mathbf{t}, z) \right) = F_0(t) + \frac{G_0(t) H}{z} +
\sum_{\operatorname{age}(g)=1} \frac{t^g G_g(t) \mathbb{1}_g}{z} + R, \end{equation*} where $R$ denotes the \emph{remainder}, consisting of terms with either the degrees in $t^g$'s greater or equal to $2$ or the degree in $z^{-1}$ greater or equal to $2$. In other words, if we write $G(\mathbf{t})$ from \eqref{e:3.3.1} as \[
G(\mathbf{t}) = \overline{G_0}(\mathbf{t})H + \sum_g \overline{G_g}(\mathbf{t}) \mathbb{1}_g \] and denote $O(2)$ the terms with the degrees in $t^g$'s greater or equal to $2$, then \[
F(\mathbf{t}) = F_0(t) + O(2), \quad \overline{G_0}(\mathbf{t}) = G_0(t) + O(2),
\quad \overline{G_g}(\mathbf{t}) = t^g G_g(t) + O(2). \] \end{lemma}
\begin{proof} The proof of this lemma follows from Corollary~\ref{c:ambientJformula} together with the following observations. First, in case (ii) $i^*(\mathbb{1}_{g_1})= i^*(\mathbb{1}_{g_2}) =0$ due to dimensional reasons. Similarly with $i^*(\mathbb{1}_{g_2})=0$ in case (iii). Secondly, in case (iii) the $\mathbb{1}_{g_1}$ term has higher $z^{-1}$ power: The modification factor contributes terms of $z^{5d}$ plus lower order (in $z$) terms. $i^* J^{\mathcal{Y}}_g$ contributes $z^{-(5d+1)}$ plus higher order (in $z^{-1}$) terms. The combined contribution goes to the remainder $R$. \end{proof}
With all this preparation, it is easy to prove Theorem~\ref{t:A-model}.
\begin{proof}[Proof of Theorem~\ref{t:A-model}] Start by pulling back the equation \eqref{e:QLHT} to $\mathcal{W}$. Setting all $t^g =0$ we get \eqref{e:A-model} for the case $g=e$ if we let $H_e = F_0$: \[
I^A_e(t) = i^*I^E_e (t) = i^* I^E(\mathbf{t}) |_{\mathbf{t} =tH}. \] Here by $\mathbf{t}=tH$ we mean that setting all $t^g=0$ in \eqref{e:3.2.7}. In the case $g \neq e$, take the partial derivative of \eqref{e:QLHT} with respect to $t^g$ and then set all $t^g=0$. Note that from \eqref{e:3.2.8}, we have \[
I^A_g(t) = i^*I^E_g (t)
= z \frac{\partial}{\partial t^g} i^*I^E(\mathbf{t}) |_{\mathbf{t} =tH}.
\] By Lemma~\ref{l:3.8} all the ``extra terms'' vanish and \eqref{e:A-model} follows for $g \neq e$ after letting $H_g(t) = G_g(t)$. The proof is now complete. \end{proof}
\section{Periods and Picard--Fuchs equations} \label{s:4}
The theory of variation of Hodge structures (VHS) is closely related to the \emph{$B$ model} of a Calabi--Yau variety $X$, which encodes information about the deformations of complex structures on $X$. By the local Torelli theorem for Calabi--Yau's, the Kodaira--Spencer spaces inject to the tangent spaces of period domains and one can investigate the deformations of $X$ via VHS, which can be described by a system of flat connections on cohomology vector bundles.
For the benefit of the readers who come from the GWT side of mirror symmetry, we give a brief and self-contained summary of the parts of VHS theory which are related to our work: the Gauss--Manin connection and the associated notions of the period matrix and Picard--Fuchs equations. For a more detailed introduction the reader may consult \cite{pG2}, \cite{pG1}.
\subsection{Gauss--Manin connections, periods, and Picard--Fuchs equations}\label{s:4.1} Over a smooth family of projective varieties $\pi: \mathscr{X} \to S$ of relative dimension $n$, we can consider the higher direct image sheaf (tensored with $\mathscr{O}_S$) on $S$: \[
R^n \pi_*\mathbb{C} \otimes \mathscr{O}_S. \] The fiber over a point $t \in S$ of this sheaf is $H^n(X_t)$. This sheaf is locally free, and is naturally endowed with a \emph{flat} connection $\nabla^{GM}$, the \emph{Gauss--Manin} connection. It can be defined in terms of the flat sections given by the lattice $R^n \pi_* \mathbb{Z}$ in $R^n\pi_*\mathbb{C} \to S$, a \emph{local system}. The Hodge filtration can be described fiberwise by \[
\left(\mathscr{F}^p\right)_t \cong \oplus_{a \geq p} H^{a, n-a}(X_t). \]
We will be particularly interested in the case when the base $S$ is one dimensional. Suppose now $S$ is an open curve and the family $\pi$ extends to a flat family over a proper curve $\bar{S}$. The vector bundle $R^n\pi_*\mathbb{C} \otimes \mathscr{O}_{S}$ extends to a vector bundle $\mathscr{H} \to \bar{S}$ whose fiber over $t$ in $ S$ consists of the middle cohomology group $H^n(X_t)$. While it is not true that $\nabla^{GM}$ extends to a connection on all of $\mathscr{H}$, the singularities which arise are at worst a regular singularities \cite{pD}.
This means that after choosing local coordinates, the connection matrix
acquires at worst a logarithmic pole at $t= 0$. Nevertheless we may still speak of flat (multi-valued) sections of $\nabla^{GM}$, controlled by the monodromy.
Let $\{ \gamma_i \}$ be a basis of $H_n(X_{t_0})$. Since $\pi: \mathscr{X} \to S$ is smooth, it is a locally trivial fibration and $n$-cycles $\gamma_i$ can be extended to \emph{locally constant} cycles $\gamma_i(t)$. Let $\omega_t$ be a (local) section of $\mathscr{H}$. The functions $\int_{\gamma(t)} \omega_t$ are called the \emph{periods} and by the local constancy of $\gamma(t)$ \[
\frac{d}{d t}\left( \int_{\gamma(t)} \omega_t \right) =
\int_{\gamma(t)} \nabla^{GM}_t s(t). \]
The periods satisfy the \emph{Picard--Fuchs equations}, defined as follows. Taking successive derivatives of $\omega_t$ with respect to the connection gives a sequence of sections \[
\omega_t, \nabla^{GM}_t\omega_t, \ldots, \left(\nabla^{GM}_t\right)^k\omega_t,
\ldots . \] Because the rank of $\mathscr{H}$ is finite, for some $k$ there will exist a relation between these sections of the form \[
\left(\nabla_t^{GM}\right)^k \omega_t + \sum_{i = 0}^{k-1} f_{i}(t)\left(\nabla_t^{GW}\right)^i \omega_t = 0. \] The corresponding differential equation \begin{equation}\label{e:PF}
\left( \left(\frac{d}{d t}\right)^k +
\sum_{i = 0}^{k-1} f_{i}(t)\left(\frac{d}{d t}\right)^i \right) \left( \int_{\gamma(t)} \omega_t \right)=0 \end{equation} is the Picard--Fuchs equation for $\omega_t$. The situation when the dimension of $S$ is greater than one is essentially the same, but \eqref{e:PF} is replaced by a PDE.
Let $\{\phi_i\}_{i \in I}$ be a basis of sections of $\mathscr{H}$. Then if $\{\gamma_i\}_{i \in I}$ is a basis of locally constant $n$-cycles, we can write the fundamental solution matrix of the Gauss-Manin connection in coordinates as \[S = \left( s_{ij} \right) \text{ with } s_{ij} = \int_{\gamma_j} \phi_i.\] With this choice of basis, we see that the $i^{th}$ row of $S$ gives the periods for the section $\phi_i$.
\begin{remark} In the literature, often (but not always) the term \emph{periods} are reserved for the case when $\phi(t)$ is a (holomorphic) $n$-form, i.e.\ a section of $\mathscr{F}^n$, and Picard--Fuchs equations only for periods in this restricted sense. Here, we choose to use these terms in a more general sense defined above. Note, however, by the results in \cite{BG}, for Calabi--Yau threefolds the general Picard--Fuchs equations can be determined from the restricted ones. \end{remark}
\begin{remark} \label{r:4.2} Let $U$ denote the Kuranishi space of the Calabi-Yau $n$-fold $X$. For the purpose of this paper, we use the term (genus zero part of) \emph{$B$ model of $X$} to denote the vector bundle $\mathscr{H} \to U$ with the natural (flat) fiberwise pairing and the Gauss--Manin connection. \end{remark}
\subsection{Griffiths--Dwork method} \label{s:4.2}
Let us assume now that the family $X_t$ is a family of hypersurfaces defined by homogeneous polynomials $Q_t$ of degree $d$ in $\mathbb{P}^{n+1}$. In this case the \emph{Griffiths--Dwork method} can be employed to explicitly calculate the Picard--Fuchs equations. We summarize the relevant results of \cite{pG1} here.
The method relies on Griffiths' work in \cite{pG1} showing that one can calculate the period integrals on $X_t$ as one of \emph{rational} forms on $\mathbb{P}^{n+1}$. \emph{For the time being, let us fix $t$ and suppress it in the notation.} Griffiths first shows that in fact any class $\Omega$ in $H^{n+1}(\mathbb{P}^{n+1} \setminus X)$ can be represented in cohomology by a \emph{rational} $n+1$ form. In particular, let $\Omega_0$ be the canonical $n+1$-form on $\mathbb{P}^{n+1}$: $\Omega_0= \sum_{i = 0}^{n+1} (-1)^{i} x_i dx_0 \cdots \hat{dx_i} \cdots dx_{n+1}$. We can represent $\Omega$ by a rational form with poles in $X$, \[
\Omega = \frac{P(x)}{Q(x)^{k}} \Omega_0 \] where $P(x)$ is a homogeneous polynomial with degree $k d -(n+2)$.
The rational $n+1$ forms are then related to regular $n$ forms on $X$ via the residue map. More precisely, let $A^n_k(X)$ denote the space of rational $(n+1)$-forms on $\mathbb{P}^{n+1}$ with poles of order at most $k$ on $X$, and let \[
\mathcal{H}_k(X) := {A^{n+1}_k(X)}/{dA^{n}_{k-1}(X)}. \]
This gives an obvious filtration \[
\mathcal{H}_1(X) \subset \mathcal{H}_2(X) \subset \cdots \subset \mathcal{H}_{n+1}(X) =: \mathcal{H}(X). \] This description of rational forms interacts nicely with the Hodge filtration $F^p$ of the \emph{primitive classes}. Griffiths proves that the following diagram \begin{equation}\label{e:commdiagram}
\begin{array}{ccccccc}
\mathcal{H}_1(X) &\subset & \mathcal{H}_2(X) &
\subset & \cdots &\subset & \mathcal{H}_{n+1}(X) \\ \quad \downarrow \Res & & \quad \downarrow \Res & & & & \quad \downarrow \Res \\
F^{n}& \subset & F^{n-1}&
\subset & \cdots & \subset & F^{0}
\end{array} \end{equation} is commutative, and that each vertical arrow is surjective. In particular, $\mathcal{H}_{k+1}(X)/\mathcal{H}_{k}(X) \cong F^{n-k}/ F^{n-k+1}.$
Now, for each $n$-cycle $\gamma$ in $H_n(X)$, let \[
T: H_n(X) \to H_{n+1}(\mathbb{P}^{n+1} \setminus X) \] be the \emph{tube map} such that $T(\gamma)$ is a sufficiently small $S^1$-bundle around $\gamma$ in $\mathbb{P}^{n+1} \setminus X$. Griffiths then shows that the tube map is \emph{surjective} in general and also \emph{injective when $n$ is odd}. \begin{theorem} \label{t:4.2} All primitive classes on $X$ can be represented as residues of rational forms on $\mathbb{P}^{n+1}$ with poles on $X$. This representation is unique when $n$ is odd. \end{theorem} This follows from the surjectivity/injectivity of $\Res$ and $T$, as well as the residue formula \[
\frac{1}{2 \pi i} \int_{T(\gamma)} \Omega = \int_\gamma \Res(\Omega). \]
Next Griffiths relates the rational forms to the Jacobian ring. Let $J(Q) = \langle \partial Q/ \partial x_0, \ldots, \partial Q / \partial x_{n+1} \rangle$ be the Jacobian ideal of $Q$.
\begin{theorem} \label{t:4.3} \begin{equation} \label{e:filtration}
\mathbb{C}[ x_0, \ldots, x_{n+1}]_{dk - n - 1}/J(Q) \cong F^{n - k}/ F^{n + 1 - k} \cong PH^{n - k, k}(V). \end{equation} \end{theorem} The key relationship between rational forms is given by the following formula ((4.5) in \cite{pG1}) \begin{equation}\label{e:relate} \frac{\Omega_0}{Q(x)^{k}} \sum_{j = 0}^{n+1} B_j(x) \frac{\partial Q(x)}{\partial x_j} = \frac{1}{k-1}\frac{\Omega_0}{Q(x)^{k-1}} \sum_{j=0}^{n+1} \frac{\partial B_i(x)}{\partial x_j} + d \phi, \end{equation} where $\phi \in A^n_{k-1}$. Thus, the order of the pole of a form $\frac{P(x)}{{Q(x)}^{k}} \Omega_0$ can be lowered if and only if $P(x)$ is contained in $J(Q)$. Thus by identifying the form $\Res \left(\frac{P(x)}{Q(x)^{k}} \Omega_0\right)$ with the homogeneous polynomial $P$, one obtains the isomorphism.
The above results allow one to explicitly calculate the Picard--Fuchs equations for certain families of forms $\omega_t$ on $X_t$. As before, $X_t$ is a family of hypersurfaces defined by degree $d$ homogeneous polynomials $Q_t$. Then we can represent a family of forms as $\omega_t = \Res \left(\frac{P_t(x)}{Q_t(x)^{k}} \Omega_0\right)$. Let $\gamma_t$ be a locally constant $n$ cycle as before, then \[
\begin{split}
\frac{\partial}{\partial t} \int_{\gamma_t} \omega_t
=&\frac{\partial}{\partial t} \int_{\gamma_t}
\Res\left(\frac{P_t(x)}{Q_t(x)^{k}} \Omega_0\right)
= \frac{\partial}{\partial t} \int_{T(\gamma_t)}
\frac{P_t(x)}{Q_t(x)^{k}} \Omega_0 \\
= & \int_{T(\gamma_t)} \frac{\partial}{\partial t}
\left(\frac{P_t(x)}{Q_t(x)^{k}}\Omega_0\right)
= \int_{\gamma_t} \Res\left(\frac{\partial}{\partial t}
\left(\frac{P_t(x)}{Q_t(x)^{k}}\Omega_0\right)\right).
\end{split} \] The third equality follows because a small change in $T(\gamma(t))$ will not change its homology class. In other words, letting $\nabla^{GM}$ denote the Gauss--Manin connection, \[
\nabla^{GM}_t \Res\left(\frac{P_t(x)}{Q_t(x)^{k}}\Omega_0\right) =
\Res\left(\frac{\partial}{\partial t}
\left(\frac{P_t(x)}{Q_t(x)^{k}}\Omega_0\right)\right), \] allowing one to obtain the Picard--Fuchs equations of $\omega_t$ via explicit calculations of the polynomials (in the Jacobian rings). An explicit example is given in the next section.
\section{$B$ model of the Fermat quintic $M$} \label{s:5} We now turn to the specific case of the Fermat quintic threefold $M$ in $\mathbb{P}^4$. It has been shown that the Hodge diamonds of $M$ and $\mathcal{W}$ are mirror symmetric \[
h^{p,q}(M) = h^{3-p, q}(\mathcal{W}). \] In particular, the deformation family of $\mathcal{W}$ is one-dimensional while for $M$ the deformation is $101$ dimensional.
Recall in our study of the $A$ model of $\mathcal{W}$, we restrict the Dubrovin connection (i.e.~Frobenius structure) to to the ``small'' parameter $t$ corresponding to the hyperplane class $H$.
In the following discussions of the complex moduli of $M$, we will also study the full period matrix for the Gauss--Manin connection, but restricted to a particular deformation parameter.
Let \begin{equation} \label{e:5.1}
Q_\psi(x) = x_0^5 + x_1^5 + x_2^5 + x_3^5 + x_4^5 - \psi x_0x_1x_2x_3x_4, \end{equation} and define the family $M_\psi = \{Q_\psi(x) = 0\} \subset \mathbb{P}^4$. When writing the Picard-Fuchs equations it will later become convenient to the coordinate change $t = -5\log(\psi)$.
\subsection{Picard--Fuchs equations for $M_{\psi}$} \label{s:5.1}
In the specific case of the family $M_\psi$, there is a ``diagrammatic technique'', pioneered in \cite{CDR} and refined in \cite{DGJ}, which utilizes the symmetry of $Q_\psi$ and $P$ to simplify the bookkeeping.
The starting point is the equation \eqref{e:relate}. Consider the rational form \[
\omega_\psi = \frac{P(x)}{Q_\psi(x)^{k}} \Omega_0, \quad
P(x) = x_0^{r_0} \cdots x_4^{r_4}, \quad \text{with $\sum_{i=0}^4 r_i = 5(k - 1)$.} \] Fix $i$ between $0$ and $4$, and set $B_j = \delta_{ij} x_i P(x)$ for $0 \leq j \leq 4$. Noting that \[
\frac{\partial }{ \partial x_i}Q_\psi(x) = 5 x_i^4 -
\psi x_0 \cdots \hat{x_j} \cdots x_4, \] and applying \eqref{e:relate} with these choices of $B_j$ (and $k$ replaced by $k+1$), we arrive at \begin{equation} \label{e:5.1.1}
5\int_{T(\gamma)}\frac{\left(x_i^5\right)P }{Q_\psi^{k+1}}\Omega_0-
\psi\int_{T(\gamma)}\frac{\left( x_0 \ldots x_4\right)P }{Q_\psi^{k+1}}
\Omega_0
= \frac{1 + r_i}{k}\int_{T(\gamma)} \frac{P}{Q_\psi^k} \Omega_0 \end{equation} for any choice of cycle $\gamma \in H_n(X)$. Note, however, that there is a degenerate case in the above setting: in the case when $P(x)$ is independent of $x_i$, let $B_j = \delta_{ij} P(x)$. Then in \eqref{e:relate} we get \begin{equation} \label{e:5.1.2}
5\int_{T(\gamma)}\frac{\left(x_i^4 \right) P}{Q_\psi^{k+1}}\Omega_0-
\psi\int_{T(\gamma)}
\frac{\left( x_0 \ldots \hat{x_i} \ldots x_4\right)P }{Q_\psi^{k+1}}
\Omega_0 = 0. \end{equation} \emph{We can interpret this equation as allowing $r_i=-1$ in \eqref{e:5.1.1}}.
Furthermore, $\frac{\partial }{ \partial \psi}Q_\psi = -x_0 \cdots x_4$, and so we have the relationship \begin{equation} \label{e:der}
\frac{\partial}{\partial \psi} \int_{T(\gamma)}
\frac{P}{Q_\psi^{k}} \Omega_0 = k\int_{T(\gamma)}
\frac{\left( x_0 \cdots x_4\right)P }{Q_\psi^{k+1}}\Omega_0. \end{equation}
The authors in \cite{CDR, DGJ} apply \eqref{e:5.1.1} \eqref{e:5.1.2} and \eqref{e:der} recursively to get relations of the periods, hence the Picard--Fuchs equations. For convenience of bookkeeping, one can keep track of the polynomial $P(x)$ by its exponents $(r_0, \ldots, r_4)$. \eqref{e:5.1.1} can be understood symbolically as a relation between $(r_0, \ldots, r_4)$, $(r_0,\ldots, r_i+5, \ldots, r_4)$ and $(r_0+1, \ldots, r_4 + 1)$.
Consider for example the case $P=1$ corresponding to $(0,\ldots,0)$. Applying \eqref{e:der} four times, one may write the fourth derivative of $(0, \ldots, 0)$ as a multiple of $(4,\ldots,4)$. This may then be related to $(5, 5, 5, 5, 0)$ by \eqref{e:5.1.2}. Applying \eqref{e:5.1.1} to relate $(r_0, \ldots, r_4)$ to a linear combination of $(r_0,\ldots, r_i-5, \ldots, r_4)$ and $(r_0+1, \ldots, r_i-4, \ldots, r_4 + 1)$ repeatedly, one can reduce to terms with $r_i \leq 4$ for all $i$. In fact, eventually all terms will be of the form $\{ (r, r,\ldots, r) \}$ for $r =0, \ldots, 4$. This can be seen by noting that \emph{none of \eqref{e:5.1.1}, \eqref{e:5.1.1} or \eqref{e:der} changes $r_i -r_j \, (\operatorname{mod} 5)$.} Hence, we have found a relation between the fourth derivative of $(0,\ldots,0)$ and $\{ (r,\ldots, r) \}$ for $r =0, \ldots, 4$. By \eqref{e:der}, the various $(r,\ldots, r)$ are $r$-th derivatives of $(0,\ldots,0)$, and we obtain a fourth order ODE in $\psi$ for the period corresponding to $P=1$. (See Table~1 below for the equation.) Other cases can be computed similarly. These arguments can be illuminated by diagrams in \cite{CDR, DGJ},
hence the name \emph{diagrammatic technique}.
Now we apply this method to calculate the Picard--Fuchs equations for the period integrals we are interested in. For every $g = (r_0, \ldots, r_4) \in G$ (defined in Section~\ref{s:2.1}), define \[
P_g(x) = x_0^{r_0} \cdots x_4^{r_4} \] and \[
k = \left( \sum_{i = 0}^4 \frac{r_i}{5} \right) + 1 = \operatorname{age}(g) +1. \] We will consider specific families of the form \begin{equation} \label{e:5.1.4}
\omega_g (\psi)= \Res \left( \frac{\psi P_g(x)}{Q_\psi(x)^k} \Omega_0\right) \end{equation} For our purposes, \emph{it will be sufficient to consider families $\omega_g$ such that $P_g$ satisfies $\operatorname{age}(g) \leq 1$ (i.e.\ $\sum_{i = 0}^4 r_i \leq 5$) and at least two of the $r_i$'s equal $0$.} We observe that other $\omega_g$ can be obtained from differentiations \eqref{e:der} or relations \eqref{e:5.1.1} and \eqref{e:5.1.2} from the listed $\omega_g$. For example, $(1,1, 1,1,1)$ is the derivative of $(0,0,0,0,0)$; $(1,1,1,2,0)$ is related to $(0,0, 0, 1, 4)$ via \[
0 \equiv x_3 \partial_{x_4} Q_{\psi} = x_3 x_4^4 - \psi x_0 x_1 x_2 X_3^2 . \] We remark that these conditions on $g$ match the conditions on $A$ model computation in Section~\ref{s:3} perfectly. In Claim~\ref{claim:6.6} it is shown that the derivatives of these families generate all of $\mathscr{H}$.
Table~1 below gives the Picard--Fuchs equation satisfied by each of the above-mentioned forms. We label the forms by the corresponding 5-tuple $g = (r_0, \ldots, r_4)$. Note that permuting the $r_i$'s does not effect the differential equation, so we do not distinguish between permutations. Here \[t = -5 \log( \psi).\]
\begin{table}[htdp] \begin{center} \[
\begin{array}{|c|c|} \hline \text{type}&\text{Picard--Fuchs equation}\\ \hline \, &\, \\ (0,0,0,0,0)& (\frac{d}{dt})^4 - 5^5e^t(\frac{d}{dt} + \frac{1}{5}) (\frac{d}{dt} + \frac{2}{5})(\frac{d}{dt} + \frac{3}{5})(\frac{d}{dt} + \frac{4}{5})\\ \, &\, \\ \hline \, &\, \\ (0,0,0,1,4)& (\frac{d}{dt})^2 - 5^5e^t(\frac{d}{dt} + 2/5)(\frac{d}{dt} + 3/5)\\ \, &\, \\ \hline \, &\, \\ (0,0,0,2,3)& (\frac{d}{dt})^2 - 5^5e^t(\frac{d}{dt} + 1/5)(\frac{d}{dt} + 4/5)\\ \, &\, \\
\hline \, &\, \\ (0,0,1,1,3)& (\frac{d}{dt})(\frac{d}{dt}- 1/5) - 5^5e^t(\frac{d}{dt} + 1/5)(\frac{d}{dt} + 3/5)\\ \, &\, \\ \hline \, &\, \\ (0,0,2,2,1)& (\frac{d}{dt})(\frac{d}{dt}- 2/5) - 5^5e^t(\frac{d}{dt} + 1/5)(\frac{d}{dt} + 2/5)\\ \, &\, \\ \hline \end{array} \]
\caption{The Picard--Fuchs equations for forms $\omega_g$.}\label{table} \end{center} \end{table} The same computation was done in \cite{CDR, DGJ}. We note however that \emph{there are several differences} between the period integrals we consider, and those of \cite{DGJ}. First, our family $M_\psi$ differs from that in \cite{DGJ} by a factor of 5 in the first term. Second, the forms we consider \eqref{e:5.1.4} differ slightly from those considered in \cite{DGJ} by an extra factor of $\psi$ in the numerator (see remark~\ref{r:5.1}). Finally, our final equations use different coordinates than in \cite{DGJ}. However the same methods used in their paper can easily be modified to obtain the formulas we present here.
\begin{remark}\label{r:5.1} The factor of $\psi$ in the numerator of \eqref{e:5.1.4} might appear unnatural at the first glance, but it can be considered as a way to change the form of the Picard-Fuchs equation, as \[
\frac{d}{d t} e^{-t/5} f(t) =
e^{-t/5} \left( - \frac{1}{5} +\frac{d}{d t} \right) f(t). \] In the comparison of $A$ model and $B$ model this modification will ensure that the $I$ functions from both sides coincide. It is also used in the Mirror Theorem for the Fermat quintic. \end{remark}
\subsection{$I^B$-functions} \label{s:5.2} We can solve the above Picard-Fuchs equations with hypergeometric series. As in Section~\ref{s:2}, we will organize these solutions in the form of an $I$-function. For each of the above forms $\omega_g$, $I^B_g$ will be a function taking values in $H^*_{CR}(\mathcal{W}) \cong H^*(I\mathcal{W})$, whose components give solutions to the corresponding Picard--Fuchs equation. \begin{proposition}\label{p:5.1} For the $g$ listed in table~\ref{table}, the components of $I^B_g(t,1)$ give a basis of solutions to the Picard--Fuchs equations for $\omega_g$, where $I^B_g(t,z)$ is given below. \begin{enumerate} \item[(i)]If $g = e = (0,0,0,0,0)$, \begin{equation}\label{e:idim3} I^B_e(t,z) = e^{t H/z}\left(1 + \sum_{\langle d \rangle = 0} e^{dt} \frac{\prod \limits_{1 \leq m\leq5d}(5H + mz)}{\prod \limits_{\substack{0 < b \leq d \\ \langle b \rangle = 0}} (H + bz)^5}\right)
\end{equation} \item[(ii)] If $g = (0,0,0,r_1, r_2)$,
\begin{equation}\label{e:idim1}
\begin{split}
&I^B_g(t,z) = e^{t H/z}\mathbb{1}_g \\
&\Bigg(1 + \sum_{\langle d \rangle = 0} e^{dt}
\frac{\prod \limits_{1 \leq m\leq5d}(5H + mz)}{\prod
\limits_{\substack{0 < b \leq d \\ \langle b \rangle = 0}} (H + bz)^3\prod
\limits_{\substack{0 < b \leq d \\ \langle b \rangle = \left\langle \frac{r_2}{5}\right\rangle}}
(H + bz)\prod
\limits_{\substack{0 < b \leq d \\ \langle b \rangle = \left\langle \frac{r_1}{5}\right\rangle}}
(H + bz)}\Bigg)
\end{split} \end{equation} \item[(iii)] If $g = (0,0,r_1, r_1, r_2)$, let
$g_1 = (-r_1, -r_1, 0,0, r_2 - r_1) (\operatorname{mod} 5)$. Then
\begin{equation} \label{e:idim0}
\begin{split}
&I^B_g(t,z) = \\
e^{t H/z}\mathbb{1}_g &\left(1 + \sum_{\langle d \rangle = 0} e^{dt} \frac{\prod
\limits_{1 \leq m\leq5d}(5H+ mz)}{\prod
\limits_{ \substack{0 < b \leq d \\ \langle b \rangle = 0}} (H+bz)^2\prod
\limits_{\substack{0 < b \leq d \\ \left\langle b \right\rangle = \left\langle \frac{3r_2}{5} \right\rangle}}
(H+bz)^2\prod
\limits_{\substack{0 < b \leq d \\ \langle b \rangle = \left\langle \frac{2r_1}{5}\right\rangle}}
(H+bz)} \right) \\
+\:e^{t H/z}\mathbb{1}_{g_1} &\left(
\sum_{\langle d \rangle = \left\langle \frac{r_1}{5}\right\rangle} e^{dt}
\frac{\prod \limits_{1 \leq m\leq5d}(5H+ mz)}{\prod
\limits_{\substack{0 < b \leq d \\ \langle b \rangle = \left\langle \frac{r_1}{5} \right\rangle}}
(H+bz)^2\prod
\limits_{\substack{0 < b \leq d \\ \langle b \rangle = 0}} (H+bz)^2\prod
\limits_{\substack{0 < b \leq d \\ \langle b \rangle = \left\langle \frac{r_2}{5}\right\rangle}}
(H+bz)}\right)
\end{split} \end{equation} \end{enumerate} \end{proposition}
\begin{remark} \label{r:5.3} Note that the functions $I^B_g(t,z)$ in equations \eqref{e:idim3}, \eqref{e:idim1}, and \eqref{e:idim0}, are supported on spaces of dimension 3, 1, and 0 respectively. So for each $g$, the number of components of $I^B_g(t,z)$ equals the order of the corresponding Picard--Fuchs equation as desired. \end{remark}
\section{Mirror Theorem for the mirror quintic: $A(\mathcal{W}) \equiv B(M)$} In this section, we will show the ``mirror dual'' version of (the mathematical version of) the \emph{mirror conjecture} by Candelas--de la Ossa--Greene--Parkes \cite{CDGP}.
More specifically, we will show that the $A$ model of $\mathcal{W}$ is equivalent to the $B$ model of $M$, up to a mirror map.
We start in~\ref{s:6.1} by stating a ``classical'' mirror theorem relating the GWT of $\mathcal{W}$ with the periods of $M_\psi$ on the level of generating functions. This is exactly analogous to Givental's original formulation in \cite{aG1}. In~\ref{s:6.2} we give a brief explanation of how Givental's original mirror theorem implies a full correspondence between the $A$ model of $M$ and the $B$ model of $\mathcal{W}$. Finally in~\ref{s:6.3} we use similar methods as in~\ref{s:6.2} to prove a mirror theorem equating the $A$ model of $\mathcal{W}$ to the $B$ model of $M$.
\subsection{A correspondence of generating functions} \label{s:6.1} We will first show that the $I$-functions $I^A_g$ of the $A$ model of $\mathcal{W}$ (Definition~\ref{d:Ifunction}) are identical to the $I$-functions $I^B_g$ of the $B$ model of $M_\psi$ defined in Section~\ref{s:5.2}.
\begin{remark} Note that in the formula $I^A_g$, the Novikov variable $q$ always appears next to $e^t$. There is therefore no harm in setting $q = 1$. We apply this specialization in what follows. \end{remark}
\begin{proposition}\label{p:6.1} Let $g = (r_0, \ldots, r_4) \in G$ satisfies the conditions $\operatorname{age}(g) \leq 1$ and that at least two of $r_i$'s are equal to zero. We have an $A$-interpretation of $g$ as parameterizing a component of $\mathcal{W}_g$ in $I\mathcal{W}$. We have also a $B$-interpretation of $g$ in $\omega_g$ \eqref{e:5.1.4} where $P_g$ denote the polynomial $x_0^{r_0}\cdots x_4^{r_4}$. Then \[
I^A_g(t,z) = I^B_g(t,z). \]
\end{proposition}
\begin{proof} This follows from a direct comparison of formulas \eqref{e:jdim4}, \eqref{e:jdim1}, and \eqref{e:jdim0} from Corollary~\ref{c:ambientJformula} with formulas \eqref{e:idim3}, \eqref{e:idim1}, and \eqref{e:idim0} respectively. \end{proof}
Combining Proposition~\ref{p:6.1} with Theorem~\ref{t:A-model}, we conclude that some periods from VHS of $M$ correspond to the Gromov--Witten invariants of $\mathcal{W}$.
\begin{corollary} \label{c:6.2} For $g =(r_0, \ldots, r_4) \in G$ such that $\operatorname{age}(g) \leq 1$ and $\mathcal{W}_g$ is nonempty (i.e. at least two $r_i$'s vanish), we have \[
J_g^{\mathcal{W}}(\tau(t),z) =
\frac{I^B_g(t,z)}{H_g(t)} \qquad \text{where }
\tau(t) = \frac{G_0(t)}{F_0(t)}. \] In other words, under the mirror map \[
t \mapsto \tau = \frac{G_0(t)}{F_0(t)}, \] the periods of $\frac{\omega_g}{H_g(t)}$ are equal to the coefficients of $J^\mathcal{W}_g(\tau, 1)$. \end{corollary} This theorem should be viewed as an analogue of Givental's original mirror theorem~\ref{t:msquintic} stated below.
\subsection{Mirror Theorem for the Fermat quintic revisited} \label{s:6.2}
To get some insight of the full correspondence, we return to the ``classical'' mirror theorem for the Fermat quintic threefold. While this is not strictly necessary for the logical flow of the proof, we feel that it illuminates our approach in a simpler setting. We also strive to clarify certain points which are not entirely clear in the literature.
Let $J^M(t, z)$ denote the small $J$-function for $M$ where $t$ is the coordinate of $H^2(M)$ dual to the hyperplane class $H$. Let $\mathcal{W}_\psi$ denote the one dimensional deformation family defined by the vanishing of $Q_\psi$ (see \eqref{e:5.1}) in $\mathcal{Y}$. \begin{equation} \label{e:Qt}
\mathcal{W}_\psi := \{ Q_\psi(x) =0 \}
\subset \mathcal{Y}. \end{equation} Let \[ \omega = \Res\left(\frac{\psi \Omega_0}{Q_\psi(x)}\right). \] As in section~\ref{s:5} there exists an $H^*(M)$-valued $I$-function, $I^B_{\mathcal{W}_\psi}(t, z)$, such that the components of $I^B_{\mathcal{W}_\psi}(t,1)$ give a basis of solutions for the Picard--Fuchs equations for $\omega_\psi$, where $t= -5 \log \psi$.
\begin{theorem}[Mirror Theorem \cite{aG1}\cite{LLY}] \label{t:msquintic} There exist explicitly determined functions $F(t)$ and $G(t)$,
such that $F$ is invertible, and \[
J^{M}(\tau(t),z) = \frac{I^B_{\mathcal{W}_\psi}(t,z)}{F(t)} \qquad
\text{where } \tau(t) = \frac{G(t)}{F(t)}. \]
\end{theorem}
We will show how Theorem~\ref{t:msquintic} implies a correspondence between the fundamental solution matrix of the Dubrovin connection for $M$ and that of the Gauss--Manin connection for $\mathcal{W}_\psi$. In order to emphasize the symmetry between the $A$ model and $B$ model, we will denote the respective pairings as $(-,-)^A$ and $(-,-)^B$.
Let \[
s = e^t = \psi^{-5}, \] and consider the flat family $\mathcal{W}_s$ over $S = \operatorname{Spec}(\mathbb{C}[s])$. In the Calabi--Yau case, the $H$ expansion of $I^B$ always occurs in the form of a function of $H/z$,
in particular $I^B_{\mathcal{W}_s}$ is homogeneous of degree zero if one sets $\deg(z)=2$. The same is true of $J^M$. Thus, one may set $z=1$ without loss of information. $I^B_{\mathcal{W}_s}(t,1)$ gives a basis of solutions for the Picard--Fuchs equations of $\omega$. In other words after an appropriate choice of basis $\{s^B_0(t), \ldots, s^B_3(t)\}$ of solutions of $\nabla^{GM}$, \[
( s^B_i(t), \omega)^B = I^B_i(t,1), \] where $I^B_i(t,z)$ is the $H^i$ coefficient of $I^B_{\mathcal{W}_s}(t,z)$.
By the same argument, if we choose an appropriate basis $\{s^A_0(\tau), \ldots, s^A_3(\tau)\}$ of solutions for $\nabla^{z}$, Section~\ref{s:1} shows that the coefficients $J^M_i(\tau,1)$ of the function $J^M(\tau,1)$ give us the functions \[
( s^A_i(\tau), 1 )^A = J^M_i(\tau,1) . \]
Thus we can interpret Theorem~\ref{t:msquintic} as saying that after choosing correct bases of flat sections and applying the mirror map \[
t \mapsto \tau = \frac{G(t)}{F(t)}, \] we have the equality \[
( s^B_i(t), \omega/F(t) )^B = \frac{I_i(t,1)}{F(t)}
= J_i(\tau, 1) = ( s^A_i(\tau), 1 )^A . \]
To show the full correspondence between the solution matrix for the Dubrovin connection for $M$ and that of the Gauss--Manin connection on $S$, we must find a basis $\phi_0, \ldots , \phi_3$ of sections of $\mathscr{H}$ and a basis $T_0, \ldots, T_3$ of sections of $H^{even}(M)$ such that for all $i$ and $j$, \begin{equation}\label{e:mx}
( s^B_i, \phi_j )^B = ( s^A_i, T_j )^A \end{equation} As expected, we set $\phi_0 = \omega/F(t)$ and $T_0 = 1$.
\begin{claim} \label{claim:6.4} \[
\phi_j = \left( \nabla^{GM}_{t} \right)^j\phi_0 \:\text{ for } \:
0 \leq j \leq 3 \] gives a basis of sections for $\mathscr{H}$. \end{claim}
\begin{proof} This follows from standard Hodge theory for Calabi--Yau threefolds, but in this case can be explicitly calculated. \begin{align}
\nabla^{GM}_{t} \phi_0 =& \frac{d}{d t}\left(\frac{1}{F(t)}\right)\omega + \frac{1}{F(t)} \nabla^{GM}_{t} \omega \nonumber\\
=& -\frac{F'(t)}{F(t)}\phi_0 + \frac{1}{F(t)}\Res \left(\frac{d}{d t} \frac{\psi \Omega_0}{Q_\psi}\right) \nonumber\\
=& -\frac{F'(t)}{F(t)}\phi_0 +\frac{1}{F(t)}\Res\left( s \frac{d}{d s} \frac{\psi \Omega_0}{Q_\psi}\right)\nonumber \\
= & -\frac{F'(t)}{F(t)}\phi_0 +\frac{1}{F(t)}\Res\left( \frac{-\psi}{5} \frac{d}{d \psi} \frac{\psi\Omega_0}{Q_\psi}\right)\nonumber \\
=& -\frac{F'(t)}{F(t)}\phi_0 + \frac{-\psi}{5 F(t)}\Res \left( \frac{\Omega_0}{Q_\psi} + \frac{x_0 \cdots x_4}{Q_\psi^2}\Omega_0 \right).\label{e:basis} \end{align} Because of the last term in the above sum, the image of $\left(\nabla^{GM}_{t}\right) \phi_0$ in $\mathscr{F}^2 /\mathscr{F}^3$ is nonzero by \eqref{e:filtration}. Similarly, the image of $\left(\nabla^{GM}_{t}\right)^j \phi_0$ in $\mathscr{F}^{3-j}/\mathscr{F}^{3+1-j}$ for
$1 \leq j \leq 3$ is nonzero, thus the sections $\phi_0, \ldots, \phi_3$ must be linearly independent. \end{proof}
Note that \begin{align}\label{e:Abasis}
&( s^B_i, \phi_1 )^B = ( s^B_i, \nabla^{GM}_{t} \phi_0 )^B =\frac{\partial}{\partial t} ( s^B_i, \phi_0 )^B =\\ & \frac{\partial}{\partial t} ( s^A_i, T_0 )^A
= \left(\frac{\partial \tau}{\partial t}\right)
\frac{\partial}{\partial \tau} ( s^A_i, T_0 )^A
= \left( s^A_i, \left(\frac{\partial \tau}{\partial t} \right) \nabla^{z}_\tau T_0 \right)^A .
\nonumber \end{align} Therefore, if we set \[
T_1 = \frac{\partial (G/F)}{\partial t} \nabla^{z}_\tau T_0, \] we have the desired relationship \[
( s^B_i, \phi_1 )^B = ( s^A_i, T_1)^A. \] If we similarly set \[
T_k = \frac{\partial (G/F)}{\partial t} \nabla^{z}_\tau T_{k-1}, \] \eqref{e:mx} follows.
This shows that the mirror map lifts to an isomorphism of vector bundles, and the connection is preserved. Indeed, the fundamental solution of the Gauss--Manin connection is a $4$ by $4$ matrix, where $4$ is the rank of $H^3(\mathcal{W})$. On the other hand, the fundamental solution of the Dubrovin connection is also a $4$ by $4$ matrix, where $4$ is the rank of $H^{even}(M)$. We recall that the $J$-function can be thought of as the first row vectors of the fundamental solution matrix, as discussed in Section~\ref{s:1}. The above discussion shows that we can extend the correspondence between the first row of the fundamental solution to the full fundamental solution.
We summarize the above in the following theorem.
\begin{theorem} \label{t:MTfull} The fundamental solutions of the Gauss--Manin connection for $\mathcal{W}_s$ are equivalent, up to a mirror map, to the fundamental solutions of the Dubrovin connection for $M$, when restricted to $H^2(M)$. \end{theorem}
\subsection{Mirror Theorem for the mirror quintic}\label{s:6.3}
In this subsection, we will extend the partial correspondence in Section~\ref{s:6.1} between the periods of $M_\psi$ and the $A$ model of $\mathcal{W}$ to the full correspondence, generalizing the ideas in Section~\ref{s:6.2}.
Similar to the above,
consider the flat family $M_s$ over $S = \operatorname{Spec}(\mathbb{C}[s])$ defined by \eqref{e:5.1}, where $s = e^t = \psi^{-5}$.
Corollary~\ref{c:6.2} states that some periods of $M_s$ correspond to Gromov--Witten invariants on $\mathcal{W}$. We would like to extend this result to all periods.
First, we must choose a basis of sections of $\mathscr{H} \to S$. Let $\omega_e$ denote the holomorphic family of (3,0)-forms corresponding to $g= e=(0,\ldots,0)$ in~\eqref{e:5.1.4}. It is no longer true that derivatives of $\omega_e/F_0(t)$ with respect to the Gauss--Manin connection generate a basis of sections of $\mathscr{H}$, thus it becomes necessary to consider the other forms $\omega_g$ satisfying the conditions formulated in Corollary~\ref{c:6.2}. Namely, let $\phi_e = \omega/F_0(t)$ and let $\phi_g = \omega_g/H_g(t)$ where $g$ satisfies $\operatorname{age}(g) = 1$. Consider the set of sections \[
\{\phi_0, \nabla^{GM}_t \phi_0, (\nabla^{GM}_t)^2 \phi_0,
(\nabla^{GM}_t)^3 \phi_0\} \cup \{ \phi_g, \nabla^{GM}_t \phi_g\}. \]
\begin{claim} \label{claim:6.6} These forms comprise a basis of the Hodge bundle $\mathscr{H}$. \end{claim}
\begin{proof}
The proof is similar to Claim~\ref{claim:6.4}. We note that in the last four rows in Table~1, corresponding to age one type, the dimensions are $20, 20, 30,$ and $30$. Thus $|\{ \phi_g\}|=100$, and there are exactly 204 forms in the above set. One can check via \eqref{e:filtration} and another argument like in \eqref{e:basis} that these sections are in fact linearly independent.
\end{proof}
Then, as in~\eqref{e:Abasis} the periods of $(\nabla^{GM}_t)^k \phi_0$ correspond to the derivatives $\left(\frac{d}{d t}\right)^k J^{\mathcal{W}}_e(\tau, 1)$, and the periods of $\nabla^{GM}_t \phi_g$ correspond to $\left(\frac{d}{d t}\right) J^{\mathcal{W}}_g(\tau, 1)$.
Let $T_0 = 1$, and $T_k = \frac{\partial (G_0/F_0)}{\partial t} \nabla^{z}_\tau T_{k-1}$ for $0 \leq k \leq 3$. Let $T_g = \mathbb{1}_g$ and $T_g ' = \frac{\partial (G_0/F_0)}{\partial t} \nabla^{z}_\tau \mathbb{1}_g$. Then if we choose the correct basis of flat sections $\{s^B_i\}$ and $\{s^A_i\}$, we have that \[\begin{split}
( s^B_i, (\nabla^{GM}_t)^k \phi_0 )^B &= ( s^A_i, T_k )^A,\\
( s^B_i, \phi_g )^B &= ( s^A_i, T_g )^A\: \operatorname{and}\\
( s^B_i, \nabla^{GM}_t \phi_g )^B &= ( s^A_i, T_g ' )^A.
\end{split} \] This implies that the set \[
\{T_0, T_1, T_2, T_3\} \cup \{ T_g, T_g '\}, \] is a basis of $TH^{even}_{CR}(\mathcal{W})$, and that with these choices of bases the solution matrices for the two respective connections are identical after the mirror transformation. Thus we obtain the full correspondence.
In terms of the language of Theorem~\ref{t:MTfull}, we can formulate our final result in the following form. On the side of the $A$ model of $\mathcal{W}$, let $t$ be the dual coordinate of $H$; on the side of $B$ model of $M_s$, let $t = \log(s)$.
Then we have
\begin{theorem} \label{t:MMTfull} The fundamental solutions of the Gauss--Manin connection $\nabla^{GM}_t$ for $M_s$ is equivalent, up to a mirror map, to the fundamental solutions of the Dubrovin connection $\nabla^z_t$ for $\mathcal{W}$ restricted to $tH \in H^2(\mathcal{W})$. \end{theorem}
\begin{remark} Even though the base direction is constrained to one dimension instead of the full $101$-dimension deformation space, our fundamental solutions are full $204$ by $204$ matrices, as both ranks of $H^3(M)$ and $H^{even}(\mathcal{W})$ are $204$. \end{remark}
\end{document} | arXiv |
If $X$ and $Y$ are topological vector spaces over $\mathbb R$, then a map $f:X\to Y$ is called uniformly continuous if for each neighborhood $V\subseteq Y$ of $0\in Y$, there exists a neighborhood $W\subseteq X$ of $0\in X$ such that for any $x,y\in X$ satisfying $x-y\in W$, it follows that $f(x)-f(y)\in V$.
It is not difficult to show that addition $+:X\times X\to X$ is a uniformly continuous function when $X\times X$ is endowed with the product topology. Also, for a given $\alpha\in\mathbb R$, the map $x\mapsto \alpha x$ is a uniformly continuous function from $X$ to $X$.
However, I suspect that scalar multiplication $\cdot:\mathbb R\times X\to X$ is not necessarily uniformly continuous when $\mathbb R$ is endowed with the usual topology and $\mathbb R\times X$ is endowed with the product tvs structure. Indeed, if $X=\mathbb R$ under the usual topology, then the function $(\alpha,x)\mapsto \alpha x$ from $\mathbb R^2$ to $\mathbb R$ is clearly not uniformly continuous.
Can anybody confirm this? I'm asking because several textbook references on topological vector spaces claim that both addition and multiplication are uniformly continuous functions, but I suspect uniform continuity of scalar multiplication can only be established in a restricted sense, i.e., when it is interpreted for a given scalar and not when $\mathbb R$ is considered as a tvs in its own right to be multiplied with $X$ to form a product tvs. In particular, I'm confused as to which of these two notions of "scalar multiplication" is appropriate when one is talking about the uniform continuity of such an operation.
You're right, as can be seen in the example you mention, multiplication $\mathbb R\times\mathbb R \to\mathbb R$. And you say this is "clearly" not uniformly continuous, so I don't know if you have a question about proving it. You can take as $V$ the interval $(-1,1)$. Each open $W\subset \mathbb R\times \mathbb R$ containing $(0,0)$ also contains $(\delta,0)$ for some $\delta>0$, but $\left(\frac1\delta+\delta\right)\cdot\frac1\delta - \frac1\delta\cdot\frac1\delta = 1\not\in (-1,1)$ even though $\left(\frac1\delta+\delta,\frac1\delta\right)-\left(\frac1\delta,\frac1\delta\right)\in W$.
You might want to consider for which subsets $A\subset \mathbb R$ the restriction $A\times X\to X$ is uniformly continuous. It is necessary and sufficient that $A$ be bounded.
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Completability of a uniform space, metric space and topological vector space? | CommonCrawl |
Final 2015 Practice Exam 2 Solutions
Stony Brook Physics phy131studiof15:lectures
Trace: • Final 2015 Practice Exam 2 Solutions
PHY132 Studio Fall 2015
To get around the world quickly Santa relies on high speed circular orbits around the Earth, in which the centripetal force is provided by gravity. (The initial acceleration to get him to the required velocity for this orbit is provided by Rudolf).
a)(15 points) If he wants to complete a full orbit in 2 hrs at what height above the surface of the earth must he fly?
$\frac{mv^{2}}{r}=\frac{Gmm}{r^{2}}$
$\frac{2\pi r}{v}=2\times60\times60=7200\mathrm{s}$
$v=\frac{2\pi r}{7200}$
$\frac{(2\pi)^2r^{2}}{7200^{2}r}=\frac{GM}{r^{2}}$
$r^{3}=\frac{7200^{2}GM}{(2\pi)^{2}}$
$r^{3}=5.24\times10^{20}\mathrm{m^{3}}$
$r=8061\mathrm{km}$
$h=r-r_{e}=1681\mathrm{km}$
b) (5 points) At what speed is Santa moving while he is in this orbit?
$v=\frac{2\pi r}{7200}=\frac{2\pi 8.061\times10^{6}}{7200}=7034\mathrm{m/s}$
c) (5 points) Santa has 1000 kg of presents in his sleigh. What is the difference in the gravitational potential energy for those presents when they are in orbit compared to when they are on the ground?
$-\frac{GMm}{8.061\times10^6}+\frac{GMm}{6.380\times10^6}=1.3\times10^{10}\mathrm{J}$
An elf in Santa's workshop raises a $4\,\mathrm{kg}$ present from the workshop conveyer belt to Santa's loading bay by pulling on a rope that passes over a pulley which has mass $2\,\mathrm{kg}$ and radius $20\,\mathrm{cm}$. The bearing of the pulley is frictionless. For a solid disk the moment of inertia is $\frac{1}{2}MR^{2}$. The 4 kg present is initially at rest and the elf would like to lift it at a constant speed of $0.5\,\mathrm{ms^{-1}}$.
a) (5 points ) In order to have the weight be moving with this velocity $1\,\mathrm{s}$ after the elf starts pulling on the rope, with what force $F$ should the elf pull on the rope?(Assume uniform acceleration of the object.)
$a=0.5\mathrm{ms^{-2}}$
$Fr-Tr=\frac{1}{2}m_{pulley}r^{2}\alpha=\frac{1}{2}m_{pulley}r^{2}\frac{a}{r}$
$F-T=\frac{1}{2}m_{pulley}a$
$m_{gift}a=T-m_{gift}g\to T=m_{gift}(a+g)$
$F=m_{gift}(a+g)+\frac{1}{2}m_{pulley}a$
$F=4(10.3)+\frac{1}{2}\times2\times0.5=41.7\mathrm{N}$
b) (5 points) What is the angular acceleration of the pulley while the elf is doing this?
$\alpha=\frac{a}{r}=\frac{0.5}{0.2}=25\mathrm{s^{-2}}$
c) (5 points) What is the magnitude of the normal force exerted by the ground on the elf, who weighs 20kg, during parts (a) and (b)?
$F_{N}=m_{elf}g-F=20\times9.8-41.7=154.3\mathrm{N}$
d) (5 points) Once the object has reached a constant speed of $0.5\,\mathrm{ms^{-1}}$ what force should the elf now apply to maintain that speed?
$F=m_{gift}g=4\times9.8=39.2\mathrm{N}$
e) (5 points) What is the angular velocity (in rpm) of the pulley while the elf is raising the object with a constant speed of $0.5\,\mathrm{ms^{-1}}$?
$\omega=\frac{v}{r}=\frac{0.5}{0.2}=2.5\mathrm{s^{-1}}=23.8\,\mathrm{rpm}$
After numerous unfortunate incidents involving polar bears and tourists who have seen too many Coke$^{\mathrm{TM}}$ ads, Santa has decided to install a new sign at the North Pole. To do this he uses a uniform metal rod of length 0.8 m and a cable which he attaches 20 cm from the end of the metal rod. The new sign has mass 3 kg and hangs from the end of the rod, while the rod itself has mass 1 kg. The cable is attached the top of the pole and the rod is held by a hinge 30 cm down from the top of the pole.
a) (5 points) What angle $\theta$ does the cable make with the rod?
$\theta=\tan^{-1}\frac{30}{60}=26.57^{o}$
b) (10 points) What is the tension in the cable?
Vertical forces, up is positive $T\sin\theta+Fh_{y}=m_{sign}g+m_{rod}g$
Horizontal forces, right is positive $-T\cos\theta+Fh_{x}=0$
Torques $0.6T\sin\theta=0.8m_{sign}g+0.4m_{rod}g$
$T=\frac{0.8\times3\times9.8+0.4\times1\times9.8}{0.6\times\sin26.57^{o}}=102.24\mathrm{N}$
c) (5 points) What is the vertical component of the force exerted by the hinge on the rod? Does this force component point up or down?
$Fh_{y}=4.981-102.24\sin26.57^{o}=-6.53\mathrm{N}$
So force points down.
d) (5 points) What is the horizontal component of the force exerted by the hinge on the rod? Does this force component point to the left or to the right?
$Fh_{x}=102.24\cos26.57^{o}=91.4\mathrm{N}$
Force points to the right
Santa wants to give a child a ball that will float in water with exactly half of the ball above the surface and the other half below when the temperature is 20$\mathrm{^{o}C}$. The ball will be 20 cm in diameter and made from rubber which is 1 cm thick and has density 1800 $\mathrm{kg\, m^{-3}}$. The density of air at 20$\mathrm{^{o}C}$ under normal conditions, such as those that exist above the surface of the water, is 1.2 $\mathrm{kg\, m^{-3}}$. The density of water is 1000 $\mathrm{kg\, m^{-3}}$. The inside of the ball is to be filled with pressurized air, which will have a higher mass density than that outside the ball.
A. (10 points) What should the mass density of air inside the ball be so that ball floats as desired?
Weight of displaced fluid
$(\frac{1}{2}1000+\frac{1}{2}1.2)\frac{4}{3}\pi0.1^{3}$
Weight of ball including dense air
$1800\frac{4}{3}\pi(0.1^{3}-0.09^{3})+\rho_{air}\frac{4}{3}\pi0.09^{3}$
At equilibrium these two are equal, so
$\rho_{air}=\frac{(\frac{1}{2}1000+\frac{1}{2}1.2)\frac{4}{3}\pi0.1^{3}-1800\frac{4}{3}\pi(0.1^{3}-0.09^{3})}{\frac{4}{3}\pi0.09^{3}}$
$\rho_{air}=17.55\mathrm{kg\,m^{-3}}$
b) (5 points) What is the mass of the air inside the ball?
$m_{air}=17.55\times\frac{4}{3}\pi0.09^{3}=.054\mathrm{kg}$
c) (5 points) The molar mass of air is 29 g/mol. How many moles of air are there inside the ball?
$n=\frac{54}{29}=1.85 \mathrm{moles}$
d) (5 points) If we treat the air as an ideal gas, to what pressure should the ball be inflated to at a temperature of 20$\mathrm{^{o}C}$? Give your final answer in atm.
$PV=nRT$
$P=\frac{1.85\times8.314\times293}{\frac{4\pi}{3}0.09^{3}}=1471\mathrm{kPa}=14.6\mathrm{atm}$
A spring of spring constant $k=10 \,\mathrm{N/m}$ is hanging from the ceiling. A 500 g wooden Santa is attached to the spring changing it's equilibrium position by $x_{0}$. In all of the following questions you may neglect the mass of the spring.
a) (5 points) Find $x_{0}$.
$kx_{0}=mg$
$x_{0}=\frac{0.5\times9.8}{10}=0.49\mathrm{m}$
b) (5 points) The Santa is pushed back up so the spring has the same length as it had before the Santa was attached. How much potential energy has been added to the system?
$\Delta PE=\int_{x_{0}}^0 F_{Net}\,dx=\int_{x_{0}}^0(kx-mg)\,dx=-\frac{1}{2}kx_{0}^{2}+mgx_{0}=-\frac{1}{2}\times10\times0.49^{2}+0.5\times9.8\times0.49=1.2\mathrm{J}$
If we took the spring in the other direction then
$\Delta PE=\int_{x_{0}}^{2x_{0}} F_{Net}\,dx=\int_{x_{0}}^{2x_{0}}(kx-mg)\,dx=\frac{3}{2}kx_{0}^{2}-mgx_{0}=\frac{3}{2}\times10\times0.49^{2}-0.5\times9.8\times0.49=1.2\mathrm{J}$
So you can see that there is the same amount of energy stored in either direction, which is important for simple harmonic motion to occur.
c) (5 points) The Santa is then released and executes simple harmonic motion.What is the frequency of the simple harmonic motion?
$f=\frac{1}{2\pi}\sqrt{\frac{k}{m}}=\frac{1}{2\pi}\sqrt{\frac{10}{0.5}}=0.71\mathrm{Hz}$
d) (5 points) What is the magnitude of the maximum velocity during simple harmonic motion? Give the position of the point or points at which this occurs in terms of displacement from $x_{0}$.
$v_{max}=\omega A= 2\pi\times0.71\times0.49=2.19\mathrm{m/s}$ occurs at displacement $x=0\mathrm{m}$ from $x_{0}$
e) (5 points) What is the magnitude of the maximum acceleration during simple harmonic motion? Give the position of the point or points at which this occurs in terms of displacement from $x_{0}$.
$a_{max}=\omega^{2} A=(2\pi\times0.71)^{2}\times0.49=9.8\mathrm{m/s^{2}}$ occurs at displacements $x=-0.49\mathrm{m},x=0.49\mathrm{m}$ from $x_{0}$
Santa slides down a chimney which has a volume of 10\,m$^{3}$. The chimney is sealed at the bottom by a floor, and sealed at the top by Santa's belly. We can treat the air in the chimney as an ideal diatomic gas. (Note: Santa's suit has very good thermal insulation!)
a) (5 points) When Santa slides down the chimney from top to bottom he reduces the volume of the air by a factor of 5 (ie. $V_{final}=\frac{V_{initial}}{5}=2\,\mathrm{m^{3}}$). If the initial pressure is standard atmospheric pressure what is the final pressure. (Hint: No heat is added to the air in this process and we can approximate it is quasistatic.)
For an adiabatic process
$P_{i}V_{i}^{\gamma}=P_{f}V_{f}^{\gamma}$ where in this case $\gamma=\frac{7}{5}$
$P_{f}=P_{i}(\frac{V_{i}}{V_{f}})^{\gamma}=101.3\mathrm{kPa}\times{5}^{7/5}=964.2\mathrm{kPa}$
b) (5 points) If the initial temperature of the air in the chimney was $2\mathrm{^{o}C}$ what is the final temperature of the air in the chimney?
As the gas is ideal
$P_{i}V_{i}=nRT_{i}$
$P_{f}V_{f}=nRT_{f}$
$\frac{T_{f}}{T_{i}}=\frac{P_{f}V_{f}}{P_{i}V_{i}}$
$T_{f}=\frac{964.2}{101.3}\times\frac{1}{5}\times275=523.5\mathrm{K}$
c) (5 points) To get Santa out of the chimney a fire is lit under it which adds heat to the air, so that it expands isothermally. How much work is done by the gas as it expands isothermally back to it's original volume.
For an isothermal process
$W=nRT\ln{\frac{V_{f}}{V_{i}}}$
$W=964.2\times10^{3}\times 2\ln(5)=3103\mathrm{kJ}$
d) (5 points) What is the pressure in the chimney when Santa has returned to his original position?
$P_{i}V_{i}=P_{f}V_{f}$
$P_{f}=\frac{V_{f}}{V_{i}}P_{i}=\frac{964.2}{5}=192.8\mathrm{kPa}$
e) (5 points) Sketch a pressure-volume diagram which shows both processes, making sure to label each process and the direction in which they occur.
Red is first process, green is second
phy131studiof15/lectures/finalp2sol.txt · Last modified: 2015/12/02 09:49 by mdawber | CommonCrawl |
Quaternions as a solution to determining the angular kinematics of human movement
John H. Challis1
BMC Biomedical Engineering volume 2, Article number: 5 (2020) Cite this article
The three-dimensional description of rigid body kinematics is a key step in many studies in biomechanics. There are several options for describing rigid body orientation including Cardan angles, Euler angles, and quaternions; the utility of quaternions will be reviewed and elaborated.
The orientation of a rigid body or a joint between rigid bodies can be described by a quaternion which consists of four variables compared with Cardan or Euler angles (which require three variables). A quaternion, q = (q0, q1, q2, q3), can be considered a rotation (Ω = 2 cos−1(q0)), about an axis defined by a unit direction vector \( \left({q}_1/\sin \left(\frac{\Omega}{2}\right),{q}_2/\sin \left(\frac{\Omega}{2}\right),{q}_3/\sin \left(\frac{\Omega}{2}\right)\right) \). The quaternion, compared with Cardan and Euler angles, does not suffer from singularities or Codman's paradox. Three-dimensional angular kinematics are defined on the surface of a unit hypersphere which means numerical procedures for orientation averaging and interpolation must take account of the shape of this surface rather than assuming that Euclidean geometry based procedures are appropriate. Numerical simulations demonstrate the utility of quaternions for averaging three-dimensional orientations. In addition the use of quaternions for the interpolation of three-dimensional orientations, and for determining three-dimensional orientation derivatives is reviewed.
The unambiguous nature of defining rigid body orientation in three-dimensions using a quaternion, and its simple averaging and interpolation gives it great utility for the kinematic analysis of human movement.
The discovery of quaternions is generally attributed to William Rowan Hamilton (1805–1865) who had a sudden insight when walking with his wife on October 16, 1843. He was so excited by this insight, generalizing complex numbers into three-dimensions, that he carved a key formula and the date into the Broome Bridge in Dublin. He wrote,
$$ {i}^2={j}^2={k}^2= ijk=-1 $$
The quaternion was therefore,
$$ q={q}_0+{q}_1i+{q}_2j+{q}_3k $$
Where q0, q1, q2, and q3 are all real, and the imaginary components (i, j, k) are the fundamental quaternion units having the rules for multiplication inscribed on Broome Bridge. The name quaternion comes from the Latin quaternio, meaning a group of four. The term had been previously used to refer to a group of four soldiers by Milton in Paradise Lost (1663), and by Scott in The Waverly Novels (1832) to refer to a word with four syllables.
Although others had envisaged quaternions before Hamilton, for example Olinde Rodrigues [16] and Leonhard Euler [12], it was Hamilton who first started to formalize their algebra. William Thomson (1824–1907) an Irish mathematical physicist claimed that,
"Quaternions came from Hamilton after his really good work had been done, and though beautifully ingenious, have been an unmixed evil to those who have touched them in any way." [23]
Not all scientists of the time were scornful of quaternions (e.g., [6]). In the last century the development of computers, the introduction of computer graphics, and the automation of the capture of rigid body motion have revealed the full utility of the quaternions in modern biomechanics.
The three-dimensional description of rigid body kinematics is a key step in many studies in biomechanics. Once measured, kinematic data may require numerical procedures such as interpolation, averaging, and differentiations. For three-dimensional linear kinematic data these procedures are relatively straightforward as linear kinematics are defined in three-dimensional Euclidean space. In contrast, three-dimensional angular kinematics are defined on the surface of a unit hypersphere [14], as a consequence different numerical procedures are required for operations such as interpolation and averaging. The use of quaternions helps simplify some of these numerical procedures, and provide some advantages over other methods of describing three-dimensional angular kinematics such as Cardan and Euler angles. Here the utility of quaternions will be presented for: representations of rigid body orientation, determining three-dimensional orientation, avoiding singularities, averaging three-dimensional orientation, interpolating three-dimensional orientations, and for determining three-dimensional orientations derivatives. Where appropriate the performance of quaternions will be juxtaposed with that of Cardan and Euler angles. This review commences with a presentation of the general properties of quaternions.
There are three common ways of presenting quaternions. The first is as a complex number with three imaginary parts,
Where q0, q1, q2, and q3 are all real, and i, j, k are the imaginary components. The second is 7as a vector with four components,
$$ q=\left({q}_0,{q}_1,{q}_2,{q}_3\right) $$
This representation is the four-tuple form of the quaternion. Finally, the quaternion can be represented as a scalar (q0) and a three element vector (\( \underset{\_}{q}=\left(\ {q}_1,{q}_2,{q}_3\right) \)),
$$ q=\left(\ {q}_0,\underset{\_}{q}\right) $$
The products of the imaginary numbers can be described using the following Figure (Fig. 1).
Products of imaginary numbers comprising a quaternion. Given any starting point, moving in a counter-clockwise direction (with the arrows) gives the results of the products of the imaginary numbers (e.g., k.i = j). If the motion is clockwise then the product is negative (e.g., j.i = −kj)
There are a number of types of quaternions, given that the norm of a quaternion is,
$$ \mid q\mid =\sqrt{q_0^2+{q}_1^2+{q}_2^2+{q}_3^2} $$
the primary ones are,
Pure quaternion q = (0, q1, q2,q3)
Identity quaternion q = (1, 0, 0, 0)
Conjugate quaternion \( \overline{q}=\left({q}_0,-{q}_1,-{q}_{2,}-{q}_3\right) \)
Quaternion Inverse \( {q}^{-1}=\frac{\overline{q}}{{\left|q\right|}^2} \)
Unit Quaternion q = (q0, q1, q2,q3)
(where) ∣q ∣ = 1
The norm for the unit quaternion is equal to one, its inverse is therefore simply its conjugate. For describing rotations in three-dimensions unit quaternions are used, their intrinsic properties confering a number of advantages.
Representations of rigid body orientation
If a rigid body in three-dimensions undergoes translation and rotation then the new pose (position and orientation) of any point on that body can be described by,
$$ y= Rx+\underset{\_}{v} $$
Where y are points measured in pose 2, R is a 3 × 3 attitude matrix, x are points measured in pose 1, and \( \underset{\_}{v} \) is a 3 × 1 vector describing the translation from one pose to the other. The attitude matrix belongs to the special-orthogonal group of order three, R ∈ SO(3). As a consequence of being in this group the inverse of the attitude matrix also belongs to the special-orthogonal group, as does the product of any matrices in this group,
$$ {R}^T={R}^{-1}\kern0.5em R{R}^T={R}^TR=I\kern0.5em \det (R)=1 $$
The attitude matrix consists of nine direction cosines, but these elements do not convey the nature of three-dimensional rotations.
Derived from the work of Cayley [3] there is a relationship between the attitude matrix and a skew-symmetric matrix P,
$$ R=\left(I-P\right){\left(I+P\right)}^{-1} $$
Where I is the identity matrix, and P is a skew-symmetric matrix which has the following format,
$$ P\left\{p\right\}=\left[\begin{array}{ccc}0& -{p}_3& {p}_2\\ {}{p}_3& 0& -{p}_1\\ {}-{p}_2& {p}_1& 0\end{array}\right] $$
This analysis suggests that as P only has three unique elements; in theory the attitude matrix can be described by three elements only. The most common of these are the Cardan and Euler angles (e.g., [26]). Both of these angle conventions can be described as an ordered sequence of rotations about three coordinate axes. For the Cardan angles a sequence might be rotations about the X, Y, and Z axes respectively,
$$ {R}_{XYZ}={R}_Z\left(\gamma \right){R}_Y\left(\beta \right){R}_X\left(\alpha \right) $$
Where α, β, and γ are angles of rotation about the X Y, and Z axes respectively. For the Euler angles a sequence might be rotations about the Z, Y, and Z axes respectively,
$$ {R}_{ZXZ}={R}_Z\left(\gamma \right){R}_X\left(\beta \right){R}_Z\left(\alpha \right) $$
For this convention, the terminal rotations use the same axis, but in theory that axis has already been rotated by the middle rotation in the sequence so is in a different orientation for the second rotation about the axis. For both Cardan and Euler angles each can use six different permutations of axes.
For the Z, X, Z Euler sequence the attitude matrix can be expressed in terms of the three Euler angles (γ, β, α),
$$ {R}_{ZXZ}=\left[\begin{array}{ccc}c\left(\alpha \right)c\left(\beta \right)c\left(\gamma \right)-s\left(\alpha \right)s\left(\gamma \right)& -c\left(\alpha \right)c\left(\beta \right)s\left(\gamma \right)-s\left(\alpha \right)\mathit{\cos}\left(\gamma \right)& c\left(\alpha \right)s\left(\beta \right)\\ {}s\left(\alpha \right)c\left(\beta \right)c\left(\gamma \right)+c\left(\alpha \right)s\left(\gamma \right)& -s\left(\alpha \right)c\left(\beta \right)s\left(\gamma \right)+c\left(\alpha \right)c\left(\gamma \right)& s\left(\alpha \right)s\left(\beta \right)\\ {}-s\left(\beta \right)\mathrm{c}\left(\gamma \right)& s\left(\beta \right)s\left(\gamma \right)& c\left(\beta \right)\end{array}\right] $$
Note that cos(α) is represented by represented by c(α), and sin(α) by s(α) and similarly for the other angles. Inspection of the matrix reveals how the individual angles can be extracted from the matrix,
$$ \mathit{\cos}\left(\beta \right)={r}_{3,3} $$
$$ \mathit{\sin}\left(\alpha \right)=\frac{r_{2,3}}{\mathit{\sin}\left(\beta \right)} $$
$$ \mathit{\sin}\left(\gamma \right)=\frac{r_{1,3}}{\mathit{\sin}\left(\beta \right)} $$
If there is only rotation about one axis then it is relatively easy to visualize the change in orientation described by a set of Euler or Cardan angles, but it is harder when there is motion about two or three of the axes.
The change of rigid body orientation described by quaternions adds one more variable compared with Cardan or Euler angles (from three to four). A quaternion, q = (q0, q1, q2, q3), can be considered a rotation of angle Ω, about an axis defined by the unit direction vector e, where,
$$ {q}_0=\pm \cos \frac{\varOmega }{2} $$
$$ \left[\begin{array}{c}{q}_1\\ {}{q}_2\\ {}{q}_3\end{array}\right]=\pm \underset{\_}{e}\mathit{\sin}\frac{\varOmega }{2} $$
Where 0 ≤ Ω ≤ π. Therefore a quaternion can be directly visualized as a directed line in space about which there is a rotation. For example, see Fig. . 2, if a point r0 is transformed by a rotation matrix to point r1, then then this transformation can be visualized as a rotation (Ω) about a line (e).
The transformation of a point r0 by a rotation of Ω about a line e, to point r1. The left image shows the general representation of the transformation, and the right image shows a view in a plane normal to the axis of rotation
Therefore the change in the orientation of a rigid body can be visualized from its quaternion. The problem with characterizing a rotation using Cardan or Euler angles is that the user must define the axis sequence with each sequence corresponding to a different set angles for describing the same rigid body attitude, adding to the problems with this visualization (see Table 1). There is no such ambiguity in quaternions.
Table 1 The influence of different angle sequences on the resulting amounts of rotations about each axis for six different Cardanic angle sequences
From Eqs. 8 and 9 it can be seen that angles as defined by Cardan or Euler angles can be combined by taking the product of the matrices describing the rotations to be combined. In a similar fashion if the rotations described by two quaternions (q and r) are to be combined the quaternion product must be computed,
$$ qr=\left[\begin{array}{cccc}{q}_0& -{q}_1& -{q}_2& -{q}_3\\ {}{q}_1& {q}_0& -{q}_3& {q}_2\\ {}{q}_2& {q}_3& {q}_0& -{q}_1\\ {}{q}_3& -{q}_2& {q}_1& {q}_0\end{array}\right]\left[\begin{array}{c}{r}_0\\ {}{r}_1\\ {}{r}_2\\ {}{r}_3\end{array}\right] $$
Quaternion multiplication is not commutative, therefore,
$$ \mathrm{qr}\ne \mathrm{rq} $$
Codman's paradox was identified by Codman in 1934 when examining the function of the shoulder [7]. With the arm in an initial position it is hanging by the side with the thumb towards the front and the fingers pointing down, first rotate the arm to the horizontal (wing position), then rotate arm to the front (fingers are now pointing straight ahead), then bring the arm back down to the side. In this final position the arm has undergone an axial rotation, therefore the thumb is now pointing to the side (inward). In a Cardanic angle sequence this is explained because rotations about the terminal axes (first and third) produce motion about the middle axis. With a quaternion representation the sequence of rotations can be considered as a rotation about each axis in sequence (represented by i then j then k), which results in a rotation at the end of the sequence because ijk = − 1, one of the basic properties of quaternions identified by Hamilton.
Chasles theorem states that the motion of a rigid body can be considered to be a translation along, and a rotation about a suitable axis in space [5]. This theorem means the description of the motion of a rigid body, or motion of one rigid body relative to another rigid body, can be the motion along and around a helical axis. The helical axes have been useful to describe joint behavior (e.g., [1]). The finite helical axis describes the motion of a rigid body from one position to another, and is frequently used as an approximation to the instantaneous helical axis (e.g., [2]). The finite helical axis is defined by: the angle of rotation (Ω) about the axis, the unit direction vector (\( \underset{\_}{\mathrm{e}} \)) of the axis, the amount of translation (u) along the axis, and the location of a point (s) on the helical axis. From Eqs. 3, 14, and 15 the angle is computed from,
$$ \Omega =2\ {\mathit{\cos}}^{-1}(q) $$
and the unit direction vector (\( \underset{\_}{\mathrm{e}} \)),
$$ \underset{\_}{\mathrm{e}}=\frac{\boldsymbol{q}}{\left|\boldsymbol{q}\right|} $$
The amount of translation (u) along the axis comes from,
$$ u={\underset{\_}{e}}^T\underset{\_}{v} $$
Finally the location of a point (s) on the axis from,
$$ s=\frac{1+\mathit{\cos}\left(\Omega \right)}{2{\mathit{\sin}}^2\left(\Omega \right)}\left(I-{R}^T\right)\underset{\_}{v} $$
Eqs. 18 and 19 show the intrinsic relationship between quaternions and finite helical axes.
Determining the three-dimensional orientation
Determining three-dimensional orientation of a rigid body requires the computation of the attitude matrix (R), this occurs in two scenarios. One is the change in pose measured in one reference frame so the attitude matrix would represent the change in orientation. The other is to map from one reference frame to another, typically inertial and body fixed, so the attitude matrix would represent the orientation of one reference frame relative to another. Given the basic rigid body transformation equation, Eq. 5, a least-squares approach to the problem of determining R and \( \underset{\_}{v} \) would require the minimizing of,
$$ \frac{1}{n}\sum \limits_{i=1}^n{\left(R{x}_i+\underset{\_}{v}-{y}_i\right)}^T\left(R{x}_i+\underset{\_}{v}-{y}_i\right) $$
Where n is the number of non-coplanar points measured in both reference frames (n ≥ 3), yi is the ith point measured in pose 2, and xi is the ith point measured in pose 1. If the data are accurate (and noiseless) the result of this equation would be zero, but in reality this does not occur so R and \( \underset{\_}{v} \) are selected to make the result as close to zero as possible. The identification of R and \( \underset{\_}{v} \) was presented by Grace Wahba as a numerical problem to solved [24]. Since Wahba presented the challenge solutions have emerged in many domains including photogrammetry (e.g., [9]), mechanical engineering (e.g., [19]), space craft kinematics (e.g., [21]), computer vision (e.g., [10]), and biomechanics (e.g., [4]).
The singular value decomposition [8], can be used to compute R and \( \underset{\_}{v} \) given measurements of at least three no-coplanar points [4]. It revolves around the decomposition of the cross-dispersion matrix C which can be computed from,
$$ C=\frac{1}{n}\sum \limits_{i=1}^n{\left({y}_i-\overline{y}\right)}^T\left({x}_i-\overline{x}\right) $$
Where \( \overline{x} \) and \( \overline{y} \) are the mean vectors (\( \overline{x}=\frac{1}{n}\sum \limits_{i=1}^n{x}_i \), \( \overline{y}=\frac{1}{n}\sum \limits_{i=1}^n{y}_i \)). The singular value decomposition of C is computed,
$$ C= UD{V}^T $$
Where U is a 3 × 3 orthogonal matrix, consisting of vectors u1, u2,u3, D is a 3 × 3 diagonal matrix, whose elements are non-negative real values (the singular values), and V is a 3 × 3 orthogonal matrix, consisting of vectors, consisting of vectors v1, v2, v3. Then the attitude matrix, R, is computed from,
$$ R=U\left[\begin{array}{ccc}1& 0& 0\\ {}0& 1& 0\\ {}0& 0& \det \left(U{V}^T\right)\end{array}\right]{V}^T $$
The vector \( \underset{\_}{v} \) can be computed using the mean vectors,
$$ \underset{\_}{v}=\overline{y}-R\overline{x} $$
Given a unit quaternion the attitude matrix (R) can be computed from,
$$ R(q)=\left({q}_0^2-{\underset{\_}{q}}^T\underset{\_}{q}\right)I+2\underset{\_}{q}\ {\underset{\_}{q}}^T-2{q}_0S\left\{\underset{\_}{q}\right\} $$
Where \( S\left\{\underset{\_}{q}\right\} \) generates a skew-symmetric matrix from a vector, therefore for vector \( \underset{\_}{q}=\left({q}_1,{q}_2,{q}_3\right) \),
$$ S\left\{\underset{\_}{q}\right\}=\left[\begin{array}{ccc}0& -{q}_3& {q}_2\\ {}{q}_3& 0& -{q}_1\\ {}-{q}_2& {q}_1& 0\end{array}\right] $$
Which when Eq. 27 is expanded gives,
$$ R(q)=\left[\begin{array}{ccc}{q}_0^2+{q}_1^2-{q}_2^2-{q}_3^2& 2\left({q}_1{q}_2-{q}_3{q}_0\right)& 2\left({q}_1{q}_3-{q}_2{q}_0\right)\\ {}2\left({q}_1{q}_2+{q}_3{q}_0\right)& {q}_0^2-{q}_1^2+{q}_2^2-{q}_3^2& 2\left({q}_2{q}_3-{q}_1{q}_0\right)\\ {}2\left({q}_1{q}_3-{q}_2{q}_0\right)& 2\left({q}_2{q}_3+{q}_1{q}_0\right)& {q}_0^2-{q}_1^2-{q}_2^2+{q}_3^2\end{array}\right] $$
Note that the matrix is quadratic relative to the quaternions, and unlike other parameters extracted from the matrix does not contain transcendental functions, for example, Eq. 10. This can be an advantage, for example, when fast computations are required.
Inspection of Eq. 29 gives the following equations for the extraction of the quaternions from the attitude matrix,
$$ {q}_0=\pm \sqrt{r_{1,1}+{r}_{2,2}+{r}_{3,3}+1} $$
$$ {q}_1=\frac{r_{2,3}-{r}_{3,2}}{4\ {q}_0} $$
If the quaternion describes a rotation of π radians then then q0 = 0, therefore the remainder of the components of the quaternion are not defined using Eqs. 31, 32, and 33. If the data used to determine the attitude matrix are noisy then this problem can occur as the rotation approaches π radians. Shepperd [18] presented a numerically more robust method of extracting the quaternion from the attitude matrix. The first step is the estimate each of the components of the quaternion from,
$$ {q_0}^2=\frac{1}{4}\left(1+{r}_{1,1}+{r}_{2,2}+{r}_{3,3}\right) $$
$$ {q_1}^2=\frac{1}{4}\left(1+2{r}_{1,1}-\left({r}_{1,1}+{r}_{2,2}+{r}_{3,3}\right)\right) $$
Whichever of these equations provides the largest square root is used as the basis for computing the remainder of the quaternion components using the appropriate equations from the following,
$$ {q}_0{q}_1=\frac{1}{4}\left({r}_{2,3}-{r}_{3,2}\right) $$
$$ {q}_2{q}_3=\frac{1}{4}\left({r}_{2,3}+{r}_{3,2}\right) $$
For example, if Eq.35 gives the highest value for a quaternion component, then q0 is estimated from Eq. 38, q2 is estimated from Eq. 43, and q2 is estimated from Eq. 42.
There are numerical methods for determining the quaternion directly from common points measures in two poses (e.g., [9, 22]), these are still based around minimizing Eq. 22 and therefore give equivalent results.
Avoiding singularities
The attitude matrix consists of nine elements (3 × 3). As this matrix is orthogonal, this property imposes six constraints on its nine elements, a characteristic of matrices belonging to the special-orthogonal group of order three. The constraints suggest that it is feasible to described rigid body orientation using three parameters. However, the three-parameter representations of SO(3) for certain rigid body attitudes are singular.
To illustrate the problem with these singularities consider the Cardanic sequence X-Y-Z sequence (angles α, β, and γ),
$$ {R}_{XYZ}=\left[\begin{array}{ccc}c\left(\gamma \right)c\left(\beta \right)& c\left(\gamma \right)s\left(\beta \right)s\left(\alpha \right)-s\left(\gamma \right)c\left(\alpha \right)& c\left(\gamma \right)s\left(\beta \right)c\left(\alpha \right)+s\left(\gamma \right)s\left(\alpha \right)\\ {}s\left(\gamma \right)c\left(\beta \right)& s\left(\gamma \right)s\left(\beta \right)s\left(\alpha \right)+c\left(\gamma \right)c\left(\alpha \right)& s\left(\gamma \right)s\left(\beta \right)c\left(\alpha \right)-c\left(\gamma \right)s\left(\alpha \right)\\ {}-s\left(\beta \right)& c\left(\beta \right)s\left(\alpha \right)& c\left(\beta \right)c\left(\alpha \right)\end{array}\right] $$
If the middle rotation β = π/2, then then the matrix in Eq. 44 can be expressed in terms of the two terminal angles,
$$ {R}_{XYZ}=\left[\begin{array}{ccc}0& \mathit{\cos}\left(\gamma \right)\mathit{\sin}\left(\alpha \right)-\mathit{\sin}\left(\gamma \right)\mathit{\cos}\left(\alpha \right)& \mathit{\cos}\left(\gamma \right)\mathit{\cos}\left(\alpha \right)+\mathit{\sin}\left(\gamma \right)\mathit{\sin}\left(\alpha \right)\\ {}0& \mathit{\sin}\left(\gamma \right)\mathit{\sin}\left(\alpha \right)+\mathit{\cos}\left(\gamma \right)\mathit{\cos}\left(\alpha \right)& \mathit{\sin}\left(\gamma \right)\mathit{\cos}\left(\alpha \right)-\mathit{\cos}\left(\gamma \right)\mathit{\sin}\left(\alpha \right)\\ {}-1& 0& 0\end{array}\right] $$
This can be simplified to,
$$ {R}_{XYZ}=\left[\begin{array}{ccc}0& \mathit{\sin}\left(\alpha -\gamma \right)&\ \mathit{\cos}\left(\alpha -\gamma \right)\\ {}0& \mathit{\cos}\left(\alpha -\gamma \right)& -\mathit{\sin}\left(\alpha -\gamma \right)\\ {}-1& 0& 0\end{array}\right] $$
The matrix illustrates that the rotation depends only on the difference between the two angles (α − γ), and therefore only has one degree of freedom instead of two. The rotation of β = π/2 means motions of angles α and γ results in rotations about the same axis.
Inspection of Eq. 11, 12, and 13 show for an Euler angle sequence of α, β, and γ the terminal angles (α, γ) are undefined for β angles of ±nπ (n = 0, 1, 2, …), because to compute these two angles require division by sin(β), which is division by zero for β values of ±nπ. Similarly for Cardanic angles the terminal angles are undefined for β angles of ±(2n + 1)\( \frac{\pi }{2} \) (n = 0, 1, 2, …), see Eq. 44.
These singularities can be visualized by considering gimbal mechanism. A gimbal consists of three concentric rings, with axes through each ring representing a rotation. The two inner rings can be rotated so they completely overlap with one another, consequently the rotation about one axis cannot be separated from rotation about the other, thus there is a gimbal lock. In biomechanics to avoid gimbal lock the sequence of rotations for a Cardan or Euler sequence are selected so the system, for example a joint, never approaches a singularity (e.g., [28]).
Quaternions can be represented by positions on the surface of hypersphere, where its radius is equal to the quaternions norm (Fig. . 3). The quaternion representation means that a rigid bodies orientation can be visualized using two quaternions, (q0, q1, q2, q3) and (-q0, -q1, -q2, -q3). To avoid this ambiguity quaternions can be constrained to either the top or bottom hemisphere of the hypersphere. Given this constraint there are no singularities or ambiguities with the quaternion definition of rigid body attitude.
Quaternions represented on a hypersphere, where q and r are quaternions, and qr is the quaternion resulting from their product. Here the quaternions have been constrained to the upper-hemisphere
Averaging three-dimensional orientations
Three-dimensional angular kinematics are not defined in three-dimensional Euclidean space, unlike linear vectors, but exist on the surface of a non-linear manifold and as a consequence the average orientation is not simply a case of, for example, averaging a set of Cardan angles. Consider the Y, Z, X Cardan sequence with corresponding angles of \( \left(\frac{\pi }{2},\frac{\pi }{2},\frac{\pi }{2}\ \right) \), the attitude matrix is,
$$ R={R}_X\left(\frac{\pi }{2}\right){R}_Z\left(\frac{\pi }{2}\right){R}_Y\left(\frac{\pi }{2}\right)=\left[\begin{array}{ccc}0& -1& 0\\ {}1&\ 0& 0\\ {}0&\ 0& 1\end{array}\right] $$
If the other Cardan angles to average are \( \left(0,0,\frac{\pi }{2}\right) \) the "averaged" set of angles would be \( \left(\frac{\pi }{4},\frac{\pi }{4},\frac{\pi }{2}\ \right) \). The error in this analysis is illustrated if the attitude matrix is examined for a Y, Z, X Cardan sequence with corresponding angles of \( \left(0,0,\frac{\pi }{2}\right) \),
$$ R={R}_X(0){R}_Z(0){R}_Y\left(\frac{\pi }{2}\right)=\left[\begin{array}{ccc}0& -1& 0\\ {}1&\ 0& 0\\ {}0&\ 0& 1\end{array}\right] $$
As three-dimensional rigid body attitude is defined as positions on the surface of hypersphere, the simple averaging of Cardan or Euler angles can produce errors in the average attitude. These errors occur because the averaging of angles is equivalent to taking chords of a circle, but appropriate averaging should take into account the contour of the surface described by the hypersphere. Moakher [15] has demonstrated that the error in averaging the Cardan or Euler angles, for example to average orientations described by R1 and R2 is,
$$ {d}_E=2\sqrt{2}\left|\mathit{\sin}\frac{\theta }{2}\right| $$
Where \( \theta ={\cos}^{-1}\left(\frac{1}{2}\left( tr\left({R}_1^T{R}_2\right)-1\right)\right) \). The equation indicates that if the angular distance (θ) across the surface of the hypersphere is too great then the error in the average determined from the Cardan or Euler angles will also be large, quaternions offer a solution to this problem.
The average rigid body attitude can be computed from a sequence of quaternions (qi, i = 1, m), then the average quaternion can be computed (\( \overline{q} \)),
$$ \overline{q}=\frac{1}{m}\sum \limits_{i=1}^m{q}_i $$
With this approach after the averaging the quaternion is normalized to ensure the average is a unit quaternion. While such averaging is not statistically optimal it does provide superior results to the averaging of Cardan or Euler angles. An improved approach was presented by Markley et al. [13]. Once again given a sequence of m quaternions the matrix M is computed,
$$ M=\frac{1}{m}\sum \limits_{i=1}^m{q}_i{q}_i^T $$
The average quaternion is the eigenvector of matrix M corresponding to the maximum eigenvalue.
Rigid body averaging in biomechanics can occur in a number of circumstance, for example making multiple measurements so that averaging improves accuracy, or averaging repeat trials of the same task to produce a representative time series signal(s). In the former case the distribution of attitudes will be relatively small, and in the latter case potentially much larger. To illustrate averaging of rigid body attitudes, 1000 criterion attitude matrices were generated via exploiting a random number generator. For each criterion attitude matrix 10 noisy versions of the matrix were generated. The noisy matrices were generated based on the error model of Woltring et al. [27] where errors (Δφ) are multiplicative with an isotropic distribution. The noisy attitude matrix (\( \hat{R} \)) is,
$$ \hat{R}=\left(I+A\left(\Delta \upvarphi \right)\right)\ R $$
Where I is the identity matrix, and A(Δφ) is a skew-symmetric matrix,
$$ A\left(\Delta \varphi \right)=\left[\begin{array}{ccc}0& -\Delta {\varphi}_z& \Delta {\varphi}_y\\ {}\Delta {\varphi}_z& 0& -\Delta {\varphi}_x\\ {}-\Delta {\varphi}_y& {\Delta \varphi}_x& 0\end{array}\right] $$
The error vector Δφ refers to small rotational errors about the reference frame affixed to the body of interest. Three noisy conditions were examined, one with a noise standard deviation of 0.035° (e.g., [11]) to reflect errors which might occur in Roentgen stereo-photogrammetry, one with 2° to reflect the spread of performances which may occur if a subject performs the same task multiple times (e.g., [25]), and an extreme condition of 10°. For each of the 10 noisy attitude matrices the average rigid body attitude was determined by 1) computing the average of the Cardan angles determined from the noisy matrices, 2) computing the average of the quaternions determined from the noisy matrices, and 3) computing an average quaternion as the eigenvector of matrix M corresponding to the maximum eigenvalue. To assess the error in computing the average attitude the product of the attitude matrix estimate of the average and transpose of the criterion were computed,
$$ {R}_{err}={R}_{Est}{R}_{Criterion}^T $$
where RCriterion is the criterion attitude matrix, and REst the estimated average attitude matrix. The error matrix (Rerr) can be quantified as the error angle (θ) which can be computed from,
$$ \theta ={\mathit{\cos}}^{-1}\left(\frac{trace\left({R}_{err}\right)-1}{2}\right) $$
If the estimated average attitude matrix exactly equals the criterion matrix, then the error matrix would be the identity matrix, giving an error angle of zero. This error angle reflects the angle through which the rigid body attitude defined by the estimated average attitude matrix must be rotated so that it corresponds with the attitude defined by the criterion attitude matrix (Chasles Theorem).
When the noise level is low all three methods produce the same performance (Table 2), this is to be expected as taking the average of a set of angles takes the chord to the surface of a sphere which for small noise levels is a reasonable approximation to the surface. With increasing noise level, the taking the average of a set of angles introduces larger errors than the other two approaches. The method of Markley et al. [13] is superior to the simple averaging of quaternions, and subsequent normalization, but the latter approach gives a reasonable approximation if speed of processing is important.
Table 2 The error angle corresponding to the estimation of rigid body attitude from multiple rigid body attitude measurements. Three methods are compared: the average of Cardan angles, the average and subsequent normalization of quaternions, and using the method of Markley et al. [13] for processing a set of quaternions
Interpolating three-dimensional orientations
The three-dimensional attitude of a rigid body can be determined using a variety of methods, including image-based motion analysis or the use of an inertial measurement unit. Given these data there are a number of reasons why it might be interpolated. For example, when trying to temporally align signals collected at different rates, or to increase the temporal density of collected data. The increase of the temporal density of sampled data is appropriate if data collection has observed Shannon's sampling theorem [17].
Vectors are relatively easily interpolated as vectors exist in linear space. In contrast quaternions exist in a curved space as each quaternion corresponds to a point on a unit hypersphere, therefore appropriate interpolation between pairs of quaternions must allow for the shape of the hypersphere surface. An appropriate approach to interpolating quaternions will ensure a consistent angular velocity between a pair of quaternions. The procedure typically used for quaternion interpolation is called Slerp, a name which derived from Spherical linear interpolation [20]. The Slerp formula for interpolating between two quaternions q1 and q2 is,
$$ q=\frac{\sin \left(\left(1-f\right)\theta \right)}{\sin \theta }{q}_1+\frac{\sin \left( f\theta \right)}{\sin \theta }{q}_2 $$
Where θ is the angle between the two quaternions (which can be computed from their dot product), and f is the fraction of interval between the two quaternions for which a quaternion is to be estimated (0 < f < 1).
The errors which arise if rigid body attitude data is not appropriately interpolated parallel those which occur if these data are not appropriately averaged, with the errors arising being larger the greater the time interval over which interpolation is to be performed, as greater time is likely associated with greater motion. If linear interpolation is used the surface of the hypersphere is approximated by a chord (Fig. 4). Slerp ensures that interpolated points lie on the surface of the hypersphere.
Spherical interpolation between the quaternions, q1 and q2. If simple linear interpolation was used then the interpolated point would be based on a line passing through the hypershere (the red colored chord). Using spherical interpolation, the predicted point is on the surface of the hypersphere
Three-dimensional orientations derivatives
Angular velocity is the rate of change of the orientation of one reference frame with respect to another, therefore the angular velocities cannot simply be computed from the differentiation of the orientation angles. Angular velocities can be computed from Poisson's equation [26], for example given the attitude matrix (R) at a given time instant,
$$ A\left\{\omega \right\}=\dot{R}{R}^T $$
(Where) \( A\left\{\omega \right\}=\left[\begin{array}{ccc}0& -{\omega}_Z& {\omega}_Y\\ {}{\omega}_Z& 0& -{\omega}_x\\ {}-{\omega}_Y& {\omega}_X& 0\end{array}\right] \)
The computation of the angular velocities from quaternions is straightforward,
$$ \left[\begin{array}{c}{\omega}_X\\ {}{\omega}_Y\\ {}{\omega}_Z\end{array}\right]=\left[\begin{array}{c}-{q}_1\\ {}-{q}_2\\ {}-{q}_3\end{array}\begin{array}{c}-{q}_0\\ {}{q}_3\\ {}-{q}_2\end{array}\begin{array}{c}-{q}_3\\ {}{q}_0\\ {}{q}_1\end{array}\begin{array}{c}{q}_2\\ {}-{q}_1\\ {}{q}_0\end{array}\right]\left[\begin{array}{c}{q}_0.\\ {}{q}_1.\\ {}{q}_2.\\ {}{q}_3.\end{array}\right] $$
There is an efficiency to using quaternions. Compared with other approaches (e.g., Euler angles, Cardan angles), the quaternion does not suffer from singularities when defining rigid body orientation, and therefore avoids the gimbal lock. The quaternion represents the direction cosine matrix as a homogenous quadratic function of the components of the quaternion, unlike other approaches it does not require trigonometric or other transcendental function evaluations. It is also efficient for combining rotations, averaging rotations, interpolating rigid body orientations, and the computation of derivatives.
The rotation axis defined by a quaternion
i, j, k :
Imaginary components of a quaternion where i2 = j2 = k2 = ijk = − 1,
u :
The amount of translation along a helical axis
Ω:
The angle of the rotation caused by a quaternion
α, β ,γ:
Set of Euler or Cardan angles
R :
3 x 3 attitude matrix
SO(3):
The special-orthogonal group of order three
U, D, V :
Components of the singular value decomposition of a matrix, where U is square orthogonal matrix, D is a diagonal matrix whose elements are non-negative real values (the singular values), and V is a square orthogonal matrix
v :
is a 3 x 1 vector describing the translation from one pose to the another
q :
A quaternion where q = q0 + q1i + q2j + q3k; q0, q1, q2, and q3 - real, components of a quaternion
s :
The location of a point on a helical axis
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Biomechanics Laboratory, Pennsylvania State University, University Park, PA, 16802, USA
John H. Challis
This is the work of one author (JHC). The author(s) read and approved the final manuscript.
Correspondence to John H. Challis.
The author has no competing interests.
Challis, J.H. Quaternions as a solution to determining the angular kinematics of human movement. BMC biomed eng 2, 5 (2020). https://doi.org/10.1186/s42490-020-00039-z
DOI: https://doi.org/10.1186/s42490-020-00039-z
Averaging | CommonCrawl |
\begin{document}
\begin{abstract} We prove that Schur classes of nef vector bundles are limits of classes that have a property analogous to the Hodge-Riemann bilinear relations. We give a number of applications, including (1) new log-concavity statements about characteristic classes of nef vector bundles (2) log-concavity statements about Schur and related polynomials (3) another proof that normalized Schur polynomials are Lorentzian. \end{abstract} \title{On Hodge-Riemann Cohomology Classes}
\section{Introduction} Since the dawn of time, human beings have asked some fundamental questions: who are we? why are we here? is there life after death? Unable to answer any of these, in this paper we will consider cohomology classes on a compact projective manifold that have a property analogous to the Hard-Lefschetz Theorem and Hodge-Riemann bilinear relations.
To state our results let $X$ be a projective manifold of dimension $d\ge 2$. We say that a cohomology class $\Omega\in H^{d-2,d-2}(X;\mathbb R)$ has the \emph{Hodge-Riemann property} if the intersection form $$ Q_{\Omega}(\alpha,\alpha') := \int_X \alpha \Omega \alpha' \text{ for } \alpha,\alpha'\in H^{1,1}(X;\mathbb R)$$ has signature $(+,-,-,\ldots,-)$. We write $$\operatorname{HR}(X)= \{ \Omega \text{ with the Hodge Riemann property}\}$$ and $\overline{\operatorname{HR}}(X)$ for its closure.
This definition is made in light of the fact that the classical Hodge-Riemann bilinear relations say precisely that if $L$ is an ample line bundle on $X$, then $c_1(L)^{d-2}$ is in $\operatorname{HR}(X)$. A natural question, initiated by Gromov \cite{Gromov90}, is if there are other cohomology classes that have this property, and our first result answers this in terms of certain characteristic classes of vector bundles.
\begin{theorem*}[$\subseteq$ Theorem \ref{thm:derivedschurclassesareinHR}] Let $E$ be a nef vector bundle on $X$ and $\lambda$ be a partition of $d-2$. Then the Schur class $s_{\lambda}(E)$ lies in $\overline{\operatorname{HR}}(X)$. \end{theorem*}
In fact we can do better; for each $i$ define the \emph{derived Schur polynomials} $s_{\lambda}^{(i)}$ by requiring that
$$ s_{\lambda}(x_1+t,\ldots,x_e+t) = \sum_{i=0}^{|\lambda|} s_{\lambda}^{(i)}(x_1,\ldots,x_e) t^i.$$ \begin{theorem*}[$\subseteq$ Theorem \ref{thm:derivedschurclassesareinHR}] Let $E$ be a nef vector bundle on $X$ and $\lambda$ be a partition of $d-2+i$. Then the derived Schur class $s_{\lambda}^{(i)}(E)$ lies in $\overline{\operatorname{HR}}(X)$. \end{theorem*}
We prove moreover: \begin{itemize} \item Analogous statements hold for monomials of derived Schur classes of possibly different nef vector bundles (Theorem \ref{thm:monomialsderivedschurclassesareinHR}). \item If $E$ is perturbed by adding a sufficiently small ample class, then $s_{\lambda}(E)$ lies in $\operatorname{HR}(X)$ (rather than in just the closure) (Remark \ref{rmk:comparisonwitholdpaper}). \item The above holds even in the setting of compact K\"ahler manifolds, where nefness of $E$ is taken in the metric sense following Demailly-Peternell-Schneider (Theorem \ref{thm:derivedschurclassesareinHR:Kahler}). \end{itemize}
\begin{center} * \end{center}
Our above result is interesting even in the case that $E=\oplus_{i=1}^e L_i$ is a direct sum of ample line bundles, from which we deduce that the Schur polynomial $s_{\lambda}(c_1(L_1),\ldots,c_1(L_e))$ lies in $\overline{\operatorname{HR}}(X)$. As a concrete example, $s_{(1,1)}(x_1,x_2) = x_1^2 + x_1 x_2 + x_2^2$, so if $L_1$ and $L_2$ are ample line bundles on a fourfold the class \begin{equation}c_1(L_1)^2 + c_1(L_1)c_1(L_2) + c_1(L_2)^2 \in \overline{\operatorname{HR}}(X).\label{eq:simplexample}\end{equation} As already noted, the classical Hodge-Riemann bilinear relations tell us that the classes $c_1(L_1)^2$ and $c_1(L_2)^2$ both lie in $\operatorname{HR}(X)$, and it was proved by Gromov \cite{Gromov90} that the mixed term $c_1(L_1)c_1(L_2)$ also lies in $\operatorname{HR}(X)$. However in general having the Hodge-Riemann property is not preserved under taking convex combinations, and thus \eqref{eq:simplexample} is new. \\
From these considerations it is natural to ask which universal combinations of characteristic classes of ample (resp.\ nef) vector bundles lie in $\operatorname{HR}(X)$ (resp. $\overline{\operatorname{HR}}(X)$). Although we do not know the full answer to this, the following is a contribution in this direction.
\begin{theorem*}[$\subseteq$ Theorem \ref{thm:poyla}] Let $E$ be a nef vector bundle on a projective manifold of dimension $d$, and $\lambda$ be a partition of $d-2$. Suppose $\mu_0,\ldots,\mu_{d-2}$ is a P\'olya frequency sequence of non-negative real numbers. Then the combination $$ \sum_{i=0}^{d-2} \mu_i s_{\lambda}^{(i)}(E) c_1(E)^{i}$$ lies in $\overline{\operatorname{HR}}(X)$. \end{theorem*}
\begin{center} * \end{center}
As an application of these results we are able to give various new inequalities between characteristic classes of nef vector bundles. Continuing to assume $X$ is projective of dimension $d$, let $\lambda$ and $\mu$ be partitions of length $|\lambda|$ and $|\mu|$ respectively and assume $|\lambda| + |\mu|\ge d$.
\begin{theorem*}[= Theorem \ref{thm:generalizedKT}] Assume $E,F$ are nef vector bundles on $X$. Then the sequence
\begin{equation}\label{eq:generalizedKT}i\mapsto \int_X s_{\lambda}^{(|\lambda|+|\mu|-d-i)}(E)s_{\mu}^{(i)}(F) \end{equation} is log-concave \end{theorem*}
As a particular case, we get that if $E$ is a nef vector bundle and $\lambda$ a partition of $d$, then
$$j\mapsto \int_X s_{\lambda}^{(j)}(E)c_1(E)^{j} $$ is log-concave, which as a special case says the map $$i\mapsto \int_X c_i(E)c_1(E)^{d-i} $$ is also log-concave. One should think of these statements as higher-rank analogs of the Khovanskii-Tessier inequalities. We even get combinatorial applications of this, such as the following:
\begin{corollary*}[= Corollary \ref{cor:combinatoric}]
Let $\lambda$ and $\mu$ be partitions, and let $d$ be an integer with $d\le |\lambda| + |\mu|$. Assume $x_1,\ldots,x_e,y_1,\ldots,y_f\in \mathbb R_{\ge 0}$. Then the sequence
$$i\mapsto s^{(|\lambda| + |\mu| -d +i)}_{\lambda}(x_1,\ldots,x_e) s_{\mu}^{(i)}(y_1,\ldots,y_f)$$ is log concave. \end{corollary*}
\begin{corollary*}[= Corollary \ref{cor:combinatoric2}] Let $\lambda$ be a partition and $x_1,\ldots,x_e\in \mathbb R_{\ge 0}$. Then the sequence $$i\mapsto s_{\lambda}^{(i)}(x_1,\ldots,x_e)$$ is log-concave. \end{corollary*}
This last statement has been known for a long time for the partition $\lambda=(e)$, for then the derived Schur polynomials become the elementary symmetric polynomials $c_i$ (see Example \ref{ex:Chernclasses}). Then more is true namely, $i\mapsto c_i(x_1,\ldots,x_e)$ is ultra-log concave - a result which is due to Newton \cite{Newton} (see, for example, \cite[Chap.\ 11]{Cvetkovski} for a modern treatment).
As a final application we show how knowing that Schur classes of nef bundles lie in $\overline{\operatorname{HR}}(X)$ gives another proof of a result of Huh-Matherne-M\'esz\'aros-Dizier \cite{Huhetal} that the normalized Schur polynomials are Lorentzian.
\subsection{Comparison with previous work: } There is some overlap between Theorem \ref{thm:derivedschurclassesareinHR} and our original work on the subject in \cite{RossTomaHR}. A principal difference is that in \cite{RossTomaHR} we show that derived Schur classes of ample bundles have the Hodge-Riemann property, whereas here we settle in merely showing these classes are limits of classes with this property. So even though logically many of our results follow from \cite{RossTomaHR}, the proofs we give here are simpler and substantially shorter. In fact, our account here does not depend on any of the parts of \cite{RossTomaHR} and is self-contained relying only on a few standard techniques in the field (as contained say in \cite{Lazbook2}). The main tools we use are the Bloch-Gieseker theorem, and the cone classes of Fulton-Lazarsfeld that express Schur classes as pushforwards of certain Chern classes (which builds on the determinantal formula of Kempf-Laksov \cite{Kempf}) . The material on the non-projective case in \S\ref{sec:Kahler}, on convex combinations in \S\ref{sec:combinations} and on inequalities in \S\ref{sec:inequalities} is all new.
We refer the reader to \cite{RossTomaHR} for a survey of other works concerning Hodge-Riemann classes. Although there are many places in which log-convexity and Schur polynomials meet (e.g. \cite{Chen,Okounkov-logconcave,Huhetal,Lam,Gao,Richards}) we are not aware of any previous inequalities that cover precisely those studied here.
\subsection{Organization of the paper: } \S\ref{sec:notationandconvention}, \S\ref{sec:derivedschurclasses} and \S\ref{sec:coneclasses} contain preliminary material on Schur polynomials, derived Schur polynomials and cone classes. We also include in \S\ref{sec:fulton-lazarsfeld} a self-contained proof of a theorem of Fulton-Lazarsfeld concerning positivity of (derived) Schur polynomials. The main theorems about derived Schur classes having the Hodge-Riemann property is proved in \S\ref{sec:schurclassesareinHR}, and in \S\ref{sec:Kahler} we explain how this extends to the non-projective case. In \S\ref{sec:combinations} we consider convex combinations of Hodge-Riemann classes, and in \S\ref{sec:inequalities} we give our application to inequalities and our proof that normalized Schur polynomials are Lorentzian.
\subsection{Acknowledgments: } The first author thanks Ivan Cheltsov and Jinhun Park for their hospitality in Pohang and for the opportunity to present this work at the Moscow-Shanghai-Pohang Birational Geometry, K\"ahler-Einstein Metrics and Degenerations conference. The first author is supported by NSF grants DMS-1707661 and DMS1749447.
\section{Notation and convention}\label{sec:notationandconvention}
We work throughout over the complex numbers. For the majority of the paper we will take $X$ to be a projective manifold (which we always assume is connected), and $E$ a vector bundle (which we always assume to be algebraic). Given such a vector bundle $E$ we denote by $\pi: \mathbb P(E)\to X$ the space of one-dimensional quotients of $E$, and by $\pi:\mathbb P_{sub}(E)\to X$ the space of one-dimensional subspaces of $E$. We say that a vector bundle $E$ is ample (resp.\ nef) if the hyperplane bundle $\mathcal O_{\mathbb P(E)}(1)$ on $\mathbb P(E)$ is ample (resp.\ nef).
We will make use of the formalism of $\mathbb Q$-twisted bundles (see \cite[Section 6.2, 8.1.A]{Lazbook2}, \cite[p457]{Miyoka}). Given a vector bundle $E$ on $X$ of rank $e$ and an element $\delta \in N^1(X)_{\mathbb Q}$ the $\mathbb Q$-twisted bundle denoted $E\langle \delta \rangle$ is a formal object understood to have Chern classes defined by the rule \begin{equation}c_p(E\langle \delta \rangle) := \sum_{k=0}^p\binom{e-k}{p-k} c_k(E) \delta^{p-k} \text{ for } 0\le p\le e.\label{eq:defcherntwisted}\end{equation} Here and henceforth we abuse notation and write $\delta$ also for its image under $N^1(X)_{\mathbb Q} \to H^2(X;\mathbb Q)$, so the above intersection is taking place in the cohomology ring $H^*(X)$.
By the rank of $E\langle \delta \rangle$ we mean the rank of $E$. The above definition is made so if $\delta= c_1(L)$ for a line bundle $L$ on $X$ then $$c_p(E\langle c_1(L)\rangle) = c_p(E\otimes L).$$ If $E$ has Chern roots given by $x_1,\ldots,x_e$ then $E\langle \delta\rangle$ is understood to have Chern roots $x_1+\delta,\ldots,x_e+\delta$. The twist of an $\mathbb Q$-twisted bundle is given by the rule $E\langle \delta \rangle \langle \delta' \rangle = E\langle \delta + \delta' \rangle$. That \eqref{eq:defcherntwisted} continues to hold when $E$ is an $\mathbb Q$-twisted bundle is an elementary calculation - for convenience of the reader, we omit the proof.
We say that $E\langle \delta \rangle$ is ample (resp.\ nef) if the class $c_1(\mathcal O_{\mathbb P(E)}(1)) + \pi^* \delta$ is ample (resp.\ nef) on $\mathbb P(E)$.
Suppose $p(x_1,\ldots,x_e)$ is a homogeneous symmetric polynomial of degree $d'$ and $E$ is a $\mathbb Q$-twisted vector bundle of rank $E$ on $X$ with Chern roots $\tau_1,\ldots,\tau_e$. Then we have the well-defined characteristic class $$p(E): = p(\tau_1,\ldots,\tau_e) \in H^{d',d'}(X;\mathbb R).$$
By abuse of notation we let $c_i$ denote the $i$th elementary symmetric polynomial, so $c_i(E)\in H^{i,i}(X;\mathbb R)$ is unambiguously defined as the $i$th-Chern class of $E$.
\section{Derived Schur Classes}\label{sec:derivedschurclasses}
By a partition $\lambda$ of an integer $b\ge 1$ we mean a sequence $0\le \lambda_N\le \cdots \le \lambda_1$ such that $|\lambda|:=\sum_{i} \lambda_i =b$. For such a partition, the Schur polynomial $s_{\lambda}$ is the symmetric polynomial of degree $|\lambda|$ in $e\ge 1$ variables given by $$s_{\lambda} = \det \left(\begin{array}{ccccc} c_{\lambda_1} & c_{\lambda_1+1} &\cdots &c_{\lambda_1 +N-1}\\ c_{\lambda_2-1} & c_{\lambda_2} &\cdots &c_{\lambda_2 +N-2}\\ \vdots & \vdots & \vdots & \vdots\\ c_{\lambda_N-N+1} & c_{\lambda_N -N +2}&\cdots &c_{\lambda_N}\\ \end{array}\right)$$ where $c_i$ denotes the $i$-th elementary symmetric polynomial.
We will have use for the following symmetric polynomials associated to Schur polynomials.
\begin{definition}\label{def:derivedschur}[Derived Schur polynomials]
Let $\lambda$ be a partition. For any $e\ge 1$ we define $s_{\lambda}^{(i)}(x_1,\ldots,x_e)$ for $i=0,\ldots,|\lambda|$ by requiring that
$$ s_{\lambda}(x_1+t,\ldots,x_e+t) = \sum_{i=0}^{|\lambda|} s_{\lambda}^{(i)}(x_1,\ldots,x_e) t^i \text{ for all } t\in \mathbb R.$$
\end{definition}
In fact $s_{\lambda}^{(i)}$ depends also on $e$ but we drop that from the notation. By convention we set $s_{\lambda}^{(i)}=0$ for $i\notin \{0,\ldots,|\lambda|\}$. For $0\le i\le |\lambda|$, clearly $s_{\lambda}^{(i)}$ is a homogeneous symmetric polynomial of degree $|\lambda|-i$ and $s_{\lambda}^{(0)} = s_{\lambda}$.
Thus for any $\mathbb Q$-twisted vector bundle $E$ of rank $e$ we have classes
$$s_{\lambda}^{(i)}(E)\in H^{|\lambda|-i,|\lambda|-i}(X;\mathbb R),$$ and by construction if $\delta \in N^1(X)_{\mathbb Q}$ then
$$ s_{\lambda}(E\langle \delta\rangle) = \sum_{i=0}^{|\lambda|} s_{\lambda}^{(i)}(E) \delta^i.$$
\begin{example}[Chern classes]\label{ex:Chernclasses} Consider the partition of $\lambda = (p)$ consisting of just one integer. Then $s_{\lambda}=c_p$, and from standard properties of Chern classes of a tensor product if $\operatorname{rk} E=e\ge p$ then $$s_{\lambda}^{(i)}(E) = \binom{e-p+i}{i} c_{p-i}(E)\text{ for all } 0\le i\le p.$$ \end{example}
\begin{example}[Derived Schur polynomials of Low degree] We list some of the derived Schur classes of low degree for a bundle $E$ of rank $e$. First $$s_{(1)} = c_1, \quad s_{(1)}^{(1)} = e$$ and for $e\ge 2$, \begin{align*} s_{(2,0)} &= c_2 & s_{(2,0)}^{(1)} &= (e-1) c_1 & s_{(2,0)}^{(2)}&=\binom{e}{2}\\ s_{(1,1)} &= c_1^2 -c_2,& s_{(1,1)}^{(1)}&=(e+1)c_1 & s_{(1,1)}^{(2)}&= \binom{e+1}{2} \end{align*}
and for $e\ge 3$, \begin{align*} s_{(3,0,0)} &= c_3 & s_{(3,0,0)}^{(1)} &= (e-2)c_2 & s_{(3,0,0)}^{(2)}&= \binom{e-1}{2} c_1\\ &&&& s_{(3,0,0)}^{(3)}&=\binom{e}{3} \\ s_{(2,1,0)}&= c_1c_2 -c_3& s_{(2,1,0)}^{(1)}&= 2c_2 + (e-1)c_1^2 & s_{(2,1,0)}^{(2)}&=(e^2-1) c_1 \\ &&&& s_{(2,1,0)}^{(3)}&=2\binom{e+1}{3}\\ s_{(1,1,1)}&= c_1^3 -2c_1c_2 + c_3 & s_{(1,1,1)}^{(1)}&=(e+2) (c_1^2-c_2) &s_{(1,1,1)}^{(2)}&= \binom{e+2}{2} c_1 \\ &&&&s_{(1,1,1)}^{(3)}&= \binom{e+2}{3} \end{align*} \end{example}
\begin{example}[Lowest Degree Derived Schur Classes]\label{ex:lowestdegree} Suppose $e\ge \lambda_1$. Then we can write the Schur polynomial as a sum of monomials
$$s_{\lambda} (x_1,\ldots,x_e)= \sum_{|\alpha| = |\lambda|} c_\alpha x_1^{\alpha_1} \cdots x_e^{\alpha_e}$$ where $c_{\alpha}\ge 0$ for all $\alpha$ (in fact the $c_{\alpha}$ count the number of semistandard Young tableaux of weight $\alpha$ whose shape is conjugate to $\lambda$). Since $e\ge \lambda_1$, $s_{\lambda}$ is not identically zero, so at least one of the $c_{\alpha}$ is strictly positive. Thus in the expansion
$$s_{\lambda}(x_1+t,\ldots, x_e+t)=\sum_{i=0}^{|\lambda|} s_{\lambda}^{(i)}(x_1,\ldots,x_e) t^i $$
the coefficient in front of $t^{|\lambda|}$ is strictly positive, i.e. $s_{\lambda}^{(|\lambda|)}>0$.
So, in terms of characteristic classes, if $E$ has rank at least $\lambda_1$ then $$s_{\lambda}^{(|\lambda|)}(E) \in H^{0}(X;\mathbb R)=\mathbb R$$ is strictly positive. \end{example}
\section{Cone Classes}\label{sec:coneclasses}
We will rely on a construction exploited by Fulton-Lazarsfeld that express Schur classes as the pushforward of Chern classes, and we include a brief description here. Let $E$ be a vector bundle of rank $e$ on $X$ of dimension $d$ and suppose $0\le \lambda_N\le \lambda_{N-1} \le \cdots \le \lambda_1$ is a partition of length $|\lambda| = b\ge 1$ and $\lambda_1\ge e$.
Set $a_i: = e + i -\lambda_i$ and fix a vector space $V$ of dimension $e+N$. Then it is possible to find a nested sequence of subspaces $0\subsetneq A_1\subsetneq A_{2} \subsetneq\cdots \subsetneq A_N\subset V$ with $\dim(A_i) = a_i$.
We set $F: = V^*\otimes E=\operatorname{Hom}(V,E)$ and let $f + 1 = \operatorname{rk}(F) = e(e+N)$. Then inside $F$ define $$\hat{C} : = \{ \sigma\in \operatorname{Hom}(V,E): \dim \ker(\sigma(x))\cap A_i \ge i \text{ for all } i=1,\ldots, N \text{ and } x\in X\}$$ which is a cone in $F$. Finally set $$C = [\hat{C}] \subset \mathbb P_{\operatorname{sub}}(F).$$
\begin{proposition}\label{prop:pushforwardschur} $C$ has codimension $b$ and dimension $d+f-b$, has irreducible fibers over $X$ and is flat over $X$ (in fact it is locally a product). Moreover if \begin{equation}0 \to \mathcal O_{\mathbb P_{\operatorname{sub}}(F)}(-1) \to \pi^* F \to U \to 0\label{eq:tautologicalsequence}\end{equation} is the tautological sequence then
\begin{equation}s_{\lambda}(E) = \pi_* c_{f}(U|_C).\label{eq:pushforwasschur}\end{equation} \end{proposition}
\begin{proof} This is described by Fulton-Lazarsfeld in \cite{FultonLazarsfeld}. An account (that is written for the the case $|\lambda| = d$) can be found in \cite[(8.12)]{Lazbook2} and an account for general $|\lambda|$ is given in \cite[Proposition 5.1]{RossTomaHR} that is based on \cite{FultonIT}. We remark that in \cite[Proposition 5.1]{RossTomaHR} we made the additional assumption that $N\ge b$ and $e\ge 2$, but have since realized these are not necessary (we used this to ensure that $f\ge b$, but this actually follows immediately from $e\ge \lambda_1$). \end{proof}
This extends to $\mathbb Q$-twisted bundles $E' = E\langle \delta\rangle$. Here we identify $$P':=\mathbb P_{\operatorname{sub}}(F\langle \delta \rangle)\stackrel{\pi}{\to} X$$ with $\mathbb P_{\operatorname{sub}}(F)\stackrel{\pi}{\to} X$ but the quotient bundle $U$ on $P'$ is replaced by $U': =U \langle \pi^* \delta\rangle$. We consider the same cone $[C]\subset P'$. Then \eqref{eq:pushforwasschur} still holds in the sense that
\begin{equation}s_{\lambda}(E') = \pi_* c_{f}(U'|_C).\label{eq:pushforwasschur:twisted}\end{equation} To see this, observe that as $\delta\in N^1(X)_{\mathbb Q}$ we have $\delta =\frac{1}{m} c_1(L)$ for some $m\in \mathbb Z$ and line bundle $L$. Then for $t$ divisible by $m$ \begin{equation}\pi_* c_{f}(U\langle t\pi^* \delta\rangle) = \pi_* c_{f}(U \otimes \pi^* L^{\frac{t}{m}}) = s_{\lambda} (E\otimes L^{\frac{t}{m}}) = s_{\lambda}(E\langle t\delta\rangle)\label{eq:pushforwasschurt}\end{equation} where the second equality uses \eqref{eq:pushforwasschur}. But both sides of \eqref{eq:pushforwasschurt} are polynomials in $t$, so since this equality holds for infinitely many $t$ it must hold for all $t\in \mathbb Q$, in particular when $t=1$ which gives \eqref{eq:pushforwasschur:twisted}.
A key feature we will rely on is that if $E'$ is assumed to be nef then so is $U'$. For if $E'$ is nef then so is $F':=F\langle \delta\rangle$ and the formal surjection $F'\to U'$ coming from \eqref{eq:tautologicalsequence} implies that $U'$ is also nef (see \cite[Lemma 6.2.8]{Lazbook2} for these properties of nef $\mathbb Q$-twisted bundles).
Another extension is to the product of Schur classes of possibly different vector bundles $E_1,\ldots,E_p$ on $X$. Let $\lambda^{1},\ldots,\lambda^{p}$ be partitions and assume $\operatorname{rk}(E_j)\ge\lambda_1^j$ for $j=1,\ldots,p$. We consider again the corresponding cones $C_i$ that sit inside $F_i:=\operatorname{Hom}(V_i,E_i)$ for some vector space $V_i$. We may consider the fiber product $C:=C_1\times_X C_2 \times_X \cdots \times_X C_p$ inside $\oplus_j \operatorname{Hom}(V_i,E_i)=:F$ and its projectivization $[C]\subset \mathbb P_{\operatorname{sub}}(F)$. Then, using that each $C_i$ is flat over $X$, if $U$ is the tautological vector bundle on $ \mathbb P_{\operatorname{sub}}(F)$ of rank $f$ we have
\begin{equation}\pi_* c_{f}(U|_C) = \prod_j s_{\lambda^{j}}(E_j)\label{eq:productschur}\end{equation} (see \cite[8.1.19]{Lazbook2}, \cite[Sec 3c]{FultonLazarsfeld}).
\section{Fulton-Lazarsfeld Positivity}\label{sec:fulton-lazarsfeld} Using the cone construction we quickly get the following positivity statement, which is essentially a weak version of a result of Fulton-Lazarsfeld \cite{FultonLazarsfeld}. For the reader's convenience we include the short proof here.
\begin{proposition}\label{prop:fultonlazderivedshur}Let $X$ be smooth and projective of dimension $d$, $\lambda$ be a partition of length $d+i$ for some $i\ge 0$ and $E$ be an $\mathbb Q$-twisted nef vector bundle. Then $\int_X s_{\lambda}^{(i)}(E)\ge 0$. \end{proposition}
\begin{proof}
We first claim that if $E$ is a nef $\mathbb Q$-twisted bundle of rank $d$ on an irreducible projective variety $X$ of dimension $d$ then $\int_X c_d(E)\ge 0$. By taking a resolution we may assume $X$ is smooth. Let $h$ be an ample class on $X$. By the Bloch-Gieseker Theorem \cite{BlochGieseker} we have $\int_X c_{d}(E\langle t h\rangle) \neq 0$ for all $t>0$ since $E\langle t h\rangle$ is ample (here we allow $t$ to be irrational extending the notation in the obvious way, and observe that although the original Bloch Gieseker result is not stated for twisted bundles the same proof works in this setting, see \cite[p113]{Lazbook2} or \S\ref{sec:Kahler}). Expanding this as a polynomial in $t$ this gives
$$0\neq \int_X c_{d}(E) + t c_{d-1} (E) h + \cdots + t^d h^d \text{ for all } t\in \mathbb R_{>0}.$$ Clearly this polynomial is strictly positive for $t\gg 0$, and hence since it is nowhere-vanishing, is strictly positive for all $t>0$. In particular $\int_{X} c_d(E)\ge 0$ as claimed.
To prove the Proposition, we may assume $e:=\operatorname{rk}(E)\ge \lambda_1$ else $s_{\lambda}(E)=0$ and the statement is trivial. When $|\lambda|=d$, \eqref{eq:pushforwasschur:twisted} gives a map $\pi:C\to X$ from an irreducible variety $C$ of dimension $n$ and a nef $\mathbb Q$-twisted bundle $U$ of rank $n$ so that $\pi_* c_{n}(U) = s_{\lambda}(E)$. So by the previous paragraph $\int_X s_{\lambda}(E) = \int_C c_{n}(U)\ge 0.$
Finally suppose $i\ge 0$ and $|\lambda|=d+i$. Set $\hat{X} = X\times \mathbb P^i$ and $\tau = c_1(\mathcal O_{\mathbb P^1}(1))$. Since $|\lambda| = \dim(\hat{X})$ we have
$$0\le \int_{\hat{X}} s_{\lambda} (E\langle \tau\rangle) = \int_{\hat{X}} \sum_{j=0}^{|\lambda|+i} s_{\lambda}^{(j)}(E)\tau^j =\int_X s_{\lambda}^{(i)}(E) \int_{\mathbb P^i} \tau^i = \int_X s_{\lambda}^{(i)}(E).$$ \end{proof}
\begin{corollary}\label{cor:Fultonlazdoubleintersection:weak} Let $X$ be smooth and projective of dimension $d$, $\lambda$ be a partition of length $d+i-2$, let $E$ be a nef $\mathbb Q$-twisted bundle of rank $e\ge \lambda_1$ and $h$ be an ample class on $X$. Then $\int_{X} s_{\lambda}^{(i)}(E) h^2 \ge0$. \end{corollary} \begin{proof} Rescale so $h$ is very ample, and apply the previous theorem to the restriction of $E$ to the intersection of two general elements in the linear series defined by $h$.
\end{proof}
\begin{remark} [Derived Schur Polynomials are Numerically Positive] \label{rem:derivedschurarelinercombinationsofschur} If $|\lambda| = d+i$ then by taking a resolution of singularities, we have $\int_X s_{\lambda}^{(i)}(E) \ge 0$ for all nef vector bundles $E$ on any irreducible projective variety $X$ of dimension $d$. That is, $s_{\lambda}^{(i)}$ is a numerically positive polynomial in the sense of Fulton-Lazarsfeld, and hence by their main result \cite[Theorem I]{FultonLazarsfeld} we deduce $s_{\lambda}^{(i)}$ can be written as a non-negative linear combination of the Schur polynomials $\{s_{\mu} : |\mu|=d\}$. This answers a question of Xiao \cite[p10]{XiaoHighDegree}. \end{remark}
\begin{remark}[Monomials of Derived Schur Classes]\label{rem:monomialsofderivedschurclasses} It is easy to extend this to monomials of derived Schur polynomials. That is, if $E_1,\ldots,E_p$ are nef bundles on $X$ and $\lambda^{1},\ldots,\lambda^{p}$ are partitions such that $\sum_j |\lambda^{j}| = d$ then \begin{equation}\label{eq:monomialschupositive} \int_X \prod_j s_{\lambda^{j}}(E_j)\ge 0.\end{equation} We simply repeat the proof of Proposition \ref{prop:fultonlazderivedshur}
using \eqref{eq:productschur} in place of \eqref{eq:pushforwasschur:twisted}). For the derived case suppose we also have integers $i_1,\ldots,i_p$ and that our partitions are such that $\sum_j |\lambda^{(j)}| - i_j = d$. Then \begin{equation}\int_X \prod_j s_{\lambda^{j}}^{(i_j)}(E_j) \ge 0.\label{eq:monomialsofderivedarenonnegative}\end{equation} To see this consider the product $\hat{X}:=X\times \prod_j \mathbb P^{i_j}$ and let $\tau_j$ be the pullback of the hyperplane class in $\mathbb P^{i_j}$ to $\hat{X}$. Then \eqref{eq:monomialschupositive} applies to the class $\prod_j s_{\lambda^{j}}(E_j(\tau_j))$. Expanding this as a symmetric polynomial in the $\tau_j$ the coefficient of $\prod_j \tau_j^{i_j}$ is precisely $\prod_j s_{\lambda^{j}}^{(i_j)}(E_j)$ so \eqref{eq:monomialsofderivedarenonnegative} follows. The analog of Corollary \ref{cor:Fultonlazdoubleintersection:weak} also holds for monomials of derived Schur polynomials. \end{remark}
\section{Hodge-Riemann classes}\label{sec:HRclasses:definitions}
Let $X$ be a projective smooth variety dimension $d$ and let $\Omega\in H^{d-2,d-2}(X;\mathbb R)$. This defines an intersection form $$ Q_{\Omega}(\alpha,\alpha')= \int_{X} \alpha \Omega \alpha' \text{ for } \alpha,\alpha'\in H^{1,1}(X;\mathbb R).$$
\begin{definition}[Hodge-Riemann Property] We say that a bilinear form $Q$ on a finite dimensional vector space has the \emph{Hodge-Riemann property} if $Q$ is non-degenerate and has precisely one positive eigenvalue. We say that $\Omega\in H^{d-2,d-2}(X;\mathbb R)$ has the Hodge-Riemann property if $Q_{\Omega}$ does, and denote by $\operatorname{HR}(X)$ denote the set of all $\Omega$ with this property. \end{definition}
\begin{definition}[Weak Hodge-Riemann Property] A bilinear form $Q$ on a finite dimensional vector space is said to have the \emph{weak Hodge-Riemann property} if it is a limit of bilinear forms that have the Hodge-Riemann property. We say that $\Omega$ has the weak Hodge-Riemann property if $Q_{\Omega}$ does, and denotes by $\operatorname{HR_w}(X)$ the set of $\Omega$ with this property. \end{definition}
So $Q$ has the weak Hodge-Riemann property if and only if if has one eigenvalue that is non-negative, and all the others are non-positive. Clearly $$\overline{\operatorname{HR}}(X)\subset \operatorname{HR_w}(X)$$ but we do not claim these are equal (the issue being that in principle $Q_{\Omega}$ could be the limit of bilinear forms with the Hodge-Riemann property that do not come from classes in $H^{d-2,d-2}(X;\mathbb R)$). If $h$ is ample then by the classical Hodge-Riemann bilinear relations $h^{d-2}\in \operatorname{HR}(X)$, and so $\operatorname{HR_w}(X)$ is a non-empty closed cone inside $H^{d-2,d-2}(X;\mathbb R)$.
It is convenient to work with $\operatorname{HR_w}(X)$ as it behaves well with respect to pullbacks and pushforwards. This is captured by the following simple piece of linear algebra.
\begin{lemma}\label{lem:simplelinearalgebra} Let $f:V\to W$ be a linear map of vector spaces and $Q_V$ and $Q_W$ be bilinear forms on $V$ and $W$ respectively such that $$Q_W(f(v),f(v')) = Q_V(v,v') \text{ for all } v,v'\in V.$$ Suppose that $Q_W$ has the weak Hodge-Riemann property and there is a $v_0\in V\setminus\{0\}$ with $Q_V(v_0,v_0)\ge 0$. Then $Q_V$ has the weak Hodge-Riemann property. \end{lemma} \begin{proof} Let $N = \operatorname{ker}(f)$. Then $N$ is orthogonal to all of $V$ with respect to $Q_V$. The signature on a complementary subspace to $N$ is induced by $Q_W$. Thus $Q_{V}$ can only be negative semi-definite, or have the weak Hodge-Riemann property, and the assumption that $Q_V(v_0,v_0)\ge 0$ means it is the latter case that occurs. \end{proof}
\begin{lemma}[Pullbacks]\label{lem:pullbacksgiveHR}
Let $\pi:X'\to X$ be a surjective map between smooth varieties of dimension $d$. Let $\Omega\in H^{d-2,d-2}(X,\mathbb R)$ and suppose there is an $h\in H^{1,1}(X;\mathbb R)\setminus\{0\}$ with $\int_X \Omega h^2\ge 0$ and that $\pi^* \Omega \in {\operatorname{HR_w}}(X')$. Then $\Omega \in {\operatorname{HR_w}}(X)$. \end{lemma} \begin{proof}
This follows from Lemma \ref{lem:simplelinearalgebra} applied to $\pi^*: H^{1,1}(X;\mathbb R)\to H^{1,1}(X';\mathbb R)$ since
$Q_{\pi^* \Omega}(\pi^*\alpha,\pi^* \alpha') = \int_{X'} \pi^*( \Omega \alpha \alpha') =\deg(\pi) \int_X \Omega \alpha\alpha' = \deg(\pi) Q_{\Omega}(\alpha,\alpha')$. \end{proof}
\begin{lemma}[Pushforwards]\label{lem:HRpushforward} Let $\pi:X'\to X$ be a surjective map between smooth varieties. Let $\Omega'\in {\operatorname{HR_w}}(X')$ and suppose there is an $h\in H^{1,1}(X;\mathbb R)\setminus\{0\}$ with $\int_X (\pi_*\Omega') h^2\ge 0$. Then $\pi_* \Omega' \in {\operatorname{HR_w}}(X)$. \end{lemma} \begin{proof} This follows from Lemma \ref{lem:simplelinearalgebra} applied to $\pi^*: H^{1,1}(X;\mathbb R)\to H^{1,1}(X';\mathbb R )$ since from the projection formula, $$Q_{\Omega'}(\pi^*\alpha,\pi^*\alpha') = \int_{X'} \Omega' (\pi^* \alpha) (\pi^*\alpha') = \int_X \pi_*\Omega' \alpha \alpha' = Q_{\pi_*\Omega}(\alpha,\alpha').$$ \end{proof}
We will need the following variant that allows for an intermediate space that might not be smooth.
\begin{lemma}\label{lem:intermediatesingular} Let $X,Y,Z$ be irreducible projective varieties with morphisms $Z\stackrel{\sigma}{\to} Y \stackrel{\pi}{\to} X$ and assume that $Z$ and $X$ are smooth. Let $d=\dim X$ and assume $Z$ and $Y$ are of the same dimension $n$ and that $\sigma$ is surjective. Let $\Omega\in H^{2n-4}(Y;\mathbb R)$ be such that $\Omega': = \pi_* \Omega\in H^{d-2,d-2}(X;\mathbb R)$. Assume \begin{enumerate} \item $\sigma^*\Omega \in \operatorname{HR_w}(Z)$.
\item There exists an $h\in H^{1,1}(X;\mathbb R)\setminus\{0\}$ such that $\int_X (\pi_*\Omega) h^2 \ge 0$. \end{enumerate} Then $\pi_*\Omega\in \operatorname{HR_w}(X)$. \end{lemma} \begin{proof} Let $p= \pi\circ \sigma:Z\to X$. By the projection formula \begin{align*} Q_{\sigma^* \Omega}(p^*\alpha,p^*\alpha') &= \int_Z \sigma^* \Omega p^*\alpha p^*\alpha' = \int_Z \sigma^* \Omega \sigma^* \pi^* \alpha \sigma^* \pi^*\alpha' \\ &= \deg(\sigma) \int_Y \Omega \pi^*\alpha \pi^*\alpha' = \deg(\sigma) \int_Z (\pi_* \Omega) \alpha \alpha' = \deg(\sigma) Q_{\pi_* \Omega}( \alpha,\alpha').\end{align*} Thus the result follows from Lemma \ref{lem:simplelinearalgebra} applied to $p^*:H^{1,1}(X;\mathbb R)\to H^{1,1}(Z;\mathbb R)$. \end{proof}
\section{Schur classes are in $\overline{HR}$}\label{sec:schurclassesareinHR}
\begin{lemma}\label{lemma:cn2I} Let $X$ be a smooth projective manifold of dimension $d\ge 4$, and $E$ be a nef $\mathbb Q$-twisted bundle of rank $d-2$. Then $c_{d-2}(E)\in {\operatorname{HR_w}}(X)$. \end{lemma} \begin{proof} This is exactly as in \cite[Proposition 3.1]{RossTomaHR}. First assume that $E$ is ample and $X$ is smooth. By a consequence of the Bloch-Gieseker Theorem for all $t\in \mathbb R_{\ge 0}$ the intersection form $$Q_{t}(\alpha):= \int_X \alpha c_{d-2} (E\langle th \rangle) \alpha \text{ for } \alpha\in H^{1,1}(X;\mathbb R)$$ is non-degenerate (we remark that we are allowing possibly irrational $t$ here, and then $c_{d-2}(E\langle th \rangle)$ is to be understood as being defined as in \eqref{eq:defcherntwisted}). Now for small $t$ we have $$ c_{d-2}(E\langle th\rangle) = t^{d-2} h^{d-2} + O(t^{d-3}).$$ Observe that for an intersection form $Q$, having signature $(+,-\ldots,-)$ is invariant under multiplying $Q$ by a positive multiple, and is an open condition as $Q$ varies continuously. Thus since we know that $h^{d-2}$ has the Hodge-Riemann property, the intersection form $(\alpha,\beta)\mapsto \int_X \alpha h^{d-2} \beta$ has signature $(+,-\ldots,-)$, and hence so does $Q_t$ for $t$ sufficiently large. But $Q_t$ is non-degenerate for all $t\ge 0$, and hence $Q_t$ must have this same signature for all $t\ge 0$. Thus $c_{d-2}(E)\in \operatorname{HR}(X)$.
Since any $\mathbb Q$-twisted nef bundle $E$ can be approximated by an $\mathbb Q$-twisted ample vector bundle we deduce that $c_{d-2}(E)\in \overline{\operatorname{HR}}(X)\subset \operatorname{HR_w}(X)$. \end{proof}
\begin{theorem}[Derived Schur Classes are in $\overline{\operatorname{HR}}$]\label{thm:derivedschurclassesareinHR} Let $X$ be smooth and projective of dimension $d\ge 2$, let $\lambda$ be a partition of length $d+i-2$ and let $E$ be a $\mathbb Q$-twisted nef vector bundle on $X$. Then $$s_{\lambda}^{(i)}(E) \in \overline{\operatorname{HR}}(X).$$ \end{theorem} \begin{proof} The statement is trivial unless $e:=\operatorname{rk}(E)\ge \lambda_1$ and $d\ge 2$ which we assume is the case. When $d=3$, $s_{\lambda}^{(i)}$ is a positive multiple of $c_1$ and then the result we want follows from the classical Hodge-Riemann bilinear relations. So we can assume from now on that $d\ge 4$.
Fix an ample class $h$ on $X$. We first prove that $s_{\lambda}(E)\in \operatorname{HR_w}(X)$. Consider the case $i=0$ so $|\lambda| = d-2$. By Corollary \ref{cor:Fultonlazdoubleintersection:weak} $\int_X s_{\lambda}(E) h^2\ge 0$. Also, the cone construction described in \S\ref{sec:coneclasses} (particularly \eqref{eq:pushforwasschur:twisted}) gives an irreducible variety $\pi:C\to X$ of dimension $n$ and a nef $\mathbb Q$-twisted vector bundle $U$ of rank $n-2$ such that $$\pi_* c_{n-2}(U) = s_{\lambda}(E).$$
Since $C$ is irreducible we can take a resolution of singularities $\sigma:C'\to C$. Then $\sigma^*U$ is also nef, and Lemma \ref{lemma:cn2I} gives $c_{n-2}(\sigma^*U)\in \operatorname{HR_w}(C')$. Thus Lemma \ref{lem:intermediatesingular} implies $s_{\lambda}(E)\in {\operatorname{HR_w}}(X)$.
Consider next the case $i\ge 1$, so $|\lambda| = d+i-2$. Again by Corollary \ref{cor:Fultonlazdoubleintersection:weak}, $\int_X s_{\lambda}^{(i)}(E) h^2\ge 0$. Consider the product $\hat{X} = X\times \mathbb P^i$ and set $\tau = c_1(\mathcal O_{\mathbb P^i}(1))$. Suppressing pullback notation, the $\mathbb Q$-twisted bundle $E\langle \tau \rangle$ on $\hat{X}$ is nef, so by the previous paragraph $s_{\lambda}(E\langle \tau\rangle) \in \operatorname{HR_w}(\hat{X})$. Now
$$s_{\lambda}(E\langle \tau\rangle) = \sum_{j=0}^{|\lambda|} s_{\lambda}^{(j)}(E) \tau^j$$ so if $\pi:\hat{X}\to X$ is the projection $$\pi_* s_{\lambda}(E\langle \tau\rangle) = s_{\lambda}^{(i)}(E).$$ Thus by Lemma \ref{lem:HRpushforward} we get also $s_{\lambda}^{(i)}(E)\in \operatorname{HR_w}(X)$.
To complete the proof define $$\Omega_t = s_{\lambda}^{(i)}(E\langle th \rangle) \text{ for } t\in \mathbb Q_{\ge 0}$$ and $$ f(t) = \det(Q_{\Omega_t}).$$ Note that the leading term of $\Omega_t$ is a positive multiple of $h^{d-2}$ (this is Example \ref{ex:lowestdegree} and it is here we use that $e\ge\lambda_1$). In particular, for $t$ sufficiently large $Q_{\Omega_t}$ is non-degenerate (in fact it has the Hodge-Riemann property). Thus $f$ is not identically zero, and since it is a polynomial in $t$ this implies $f(t)\neq 0$ for all but finitely many $t$. Thus there is an $\epsilon>0$ so that $f(t)\neq 0$ for rational $0<t<\epsilon$ and we henceforth consider only $t$ in this range. Then $Q_{\Omega_t}$ is non-degenerate, and as $Q_{\Omega_t}(h,h)\ge 0$ it cannot be negative definite. The previous paragraph gives $\Omega_t\in {\operatorname{HR_w}}$, so we must actually have $\Omega_t \in \operatorname{HR}(X)$ for small $t\in \mathbb Q_{>0}$. Thus $\Omega_0 = s_{\lambda}^{(i)}(E)\in \overline{\operatorname{HR}}(X)$ as claimed. \end{proof}
\begin{remark}\label{rmk:comparisonwitholdpaper} Note the above proof gives more, namely that if $h$ is an ample class and $E$ is nef and $\lambda_1\le \operatorname{rk}(E)$ we have
$$s_{\lambda}^{(i)}(E\langle th\rangle)\in \operatorname{HR}(X) \text{ for all but possibly finitely many } t\in \mathbb Q_{>0}.$$ As mentioned in the introduction, the main result of \cite{RossTomaHR} says more namely that if $E$ is ample of rank at least $\lambda_1$ then $s_{\lambda}^{(i)}(E)\in \operatorname{HR}(X)$, but the proof of that statement is significantly harder. \end{remark}
\begin{theorem}[Monomials of Schur Classes are in $\overline{\operatorname{HR}}$]\label{thm:monomialsderivedschurclassesareinHR} Let $X$ be smooth and projective of dimension $d$ and $E_1,\ldots,E_p$ be nef vector bundles on $X$.
Let $\lambda^1,\ldots,\lambda^p$ be partitions such that $$\sum_i |\lambda^i| = d-2.$$ Then the monomial of Schur polynomials $$\prod_i s_{\lambda^i}(E_i)$$ lies in $\overline{\operatorname{HR}}(X)$. \end{theorem} \begin{proof} The proof is similar to what has already been said, so we merely sketch the details. Set $\Omega=\prod_i s_{\lambda^i}(E_i)$. Then \eqref{eq:productschur} gives a map $\pi:C\to X$ from an irreducible variety of dimension $n$ and nef bundle bundle $U$ on $C$ so $\pi_* c_{n-2}(U) = \Omega$. A small modification of the proof of Proposition \ref{prop:fultonlazderivedshur} and Corollary \ref{cor:Fultonlazdoubleintersection:weak} means that if $h$ is ample $\int_X \Omega h^2\ge 0$.
Consider $$\Omega_t : = \pi_* c_{n-2}(U\langle t \pi^*h \rangle)$$ and take a resolution $\sigma:C'\to C$. Then $\sigma^* U\langle \pi^*h\rangle$ remains nef, so Lemma \ref{lem:intermediatesingular} implies $\Omega_t \in \operatorname{HR_w}(X)$.
Now we can equally apply this construction replacing each $E_i$ with $E_i\otimes \mathcal O(th)$ for $t\in \mathbb N$ (which one can check does not change $\pi:C\to X$) giving $$\pi_* c_{n-2} ( U \langle th\rangle) = \prod_i s_{\lambda^i}(E_i\langle th\rangle) \text{ for } t\in \mathbb N.$$ In particular applying Example \ref{ex:lowestdegree} to each factor on the right hand side, the highest power of $t$ is a positive multiple of $h^{d-2}$. Thus for almost all $t\in \mathbb Q_{>0}$ we have $Q_{\Omega_t}$ is non-degenerate, and so in fact $Q_{\Omega_t}\in \operatorname{HR}(X)$. Taking the limit as $t\to 0$ gives the result we want. \end{proof}
\section{The K\"ahler case} \label{sec:Kahler}
The main place in which projectivity has been used so far is in the application of the Bloch-Gieseker Theorem, and here we explain how this projectivity assumption can be relaxed. Following Demailly-Peternell-Schneider \cite{DemaillyPeternellSchneider} we say a line bundle $L$ on a compact K\"ahler manifold $X$ is \emph{nef} if for all $\epsilon>0$ and all K\"ahler forms $\omega$ on $X$ there exists a hermitian metric $h$ on $L$ with curvature $dd^c \log h \ge -\epsilon \omega$. We say that a vector bundle $E$ on $X$ is nef if the hyperplane bundle $\mathcal O_{\mathbb P(E)}(1)$ is nef.
For the rest of this section let $(X,\omega)$ be a compact K\"ahler manifold of dimension $d$. Given a vector bundle $E$ and $\delta\in H^{1,1}(X;\mathbb R)$ we can consider the $\mathbb R$-twisted bundle $E\langle \delta \rangle$ whose Chern classes are defined just as in the case of $\mathbb Q$-twists in the projective case. We identify $\mathbb P(E\langle \delta\rangle)$ with $\mathbb P(E)$, and say that $E\langle \delta\rangle$ is nef if for any K\"ahler metric $\omega'$ on $\mathbb P(E)$, any $\epsilon>0$, and any closed $(1,1)$ form $\delta'$ on $X$ such that $[\delta']=\delta$, there exists a hermitian metric $h$ on $\mathcal O_{\mathbb P(E)}(1)$ such that $$dd^c \log h + \pi^* \delta' \ge -\epsilon \omega'.$$ We refer the reader to \cite{DemaillyPeternellSchneider} for the fundamental properties of nef bundles on compact K\"ahler manifolds, in particular to the statement that a quotient of a nef bundle is again nef, and the direct sum of two nef bundles is again nef (and each of these statements extend to the case of $\mathbb R$-twisted nef bundles with minor modifications of the proofs involved).
\begin{theorem}[Bloch-Gieseker for K\"ahler Manifolds] Let $E$ be a nef $\mathbb R$-twisted vector bundle of rank $e\le d$ and $t>0$. Let $e+j\le d$ and consider $$\Omega := c_{e}(E\langle t\omega\rangle) \wedge \omega^j.$$ Then then map $$H^{d-e-j}(X) \stackrel{\wedge \Omega}{\longrightarrow} H^{d+e+j}(X)$$ is an isomorphism. \end{theorem} \begin{proof} Write $E = E'\langle \delta\rangle$ where $E'$ is a genuine vector bundle. Fix $t>0$ and set $E_t: = E\langle t\omega\rangle = E'\langle \delta + t\omega\rangle$. Set $\pi:\mathbb P(E')\to X$ and define $\zeta' = c_1(\mathcal O_{\mathbb P(E')}(1))$ and $\zeta: = \zeta' + \pi^*(\delta+ t [\omega])$. Then $\zeta^e - c_1(E_t) \zeta^{e-1} + \cdots + (-1)^e c_e(E_t)=0$ where we supress pullback notation for convenience.
Suppose $a\in H^{d-e-j}(X)$ has $a c_e(E_t)\omega^j=0$, and we will show that $a=0$. To this end define $$ b = a.(\zeta^{e-1}- c_1(E_t) \zeta^{e-2} + \cdots +(-1)^{e-1} c_{e-1}(E_t))$$ so by construction $$ \zeta b \omega^j = \pm a c_e(E_t) \omega^j=0$$ We claim that $\zeta$ is a K\"ahler class. Given this for now, the Hard-Lefschetz property for $\zeta$ then gives $b\omega^j=0$ and hence $a\omega^j = \pi_* (b\omega^j)=0$ and hence $a=0$ by the Hard-Lefschetz property of $\omega^j$
It remains to show that $\zeta$ is K\"ahler, and the following is essentially what is described in \cite[proof of Theorem 1.12]{DemaillyPeternellSchneider}. Fix $\omega'$ a K\"ahler metric on $\mathbb P(E')$, and fix a hermitian metric on $E'$ which induces a hermitian metric $\hat{h}$ on $\mathcal O_{\mathbb P(E')}(1)$. Then $dd^c\log \hat{h}$ is strictly positive in the fiber directions, so there is a constant $C>0$ with $$ dd^c\log\hat{h} + C \pi^* \omega \ge C^{-1}\omega'.$$ Let $\delta'$ be a closed $(1,1)$-form on $X$ with $[\delta']=\delta$, and choose $\epsilon>0$ sufficiently small that $(t-C^2 \epsilon)\omega +C\epsilon \delta'>0$. Then as $E$ is assumed to be nef there is a hermitian metric $h$ on $\mathcal O_{\mathbb P(E')}(1)$ such that $dd^c \log h + \pi^* \delta'\ge -\epsilon \omega'$.
Then the class $\zeta = c_1(\mathcal O_{\mathbb P(E')}(1)) + \pi^*[\delta+t\omega]$ is represented by the form $$ (1-C\epsilon) dd^c \log h + C\epsilon dd^c \log \hat{h} + \pi^*(\delta' + t\omega) $$ which is bounded from below by \begin{align*} (1-C\epsilon) (-\epsilon \omega' - \pi^*\delta') &+ C\epsilon(C^{-1}\omega' - C\pi^*\omega) + \pi^*(t\omega + \delta')\\ &= C\epsilon^2 \omega' + (t-C^2\epsilon) \pi^*\omega + C\epsilon \pi^*\delta'\\ & \ge C\epsilon^2 \omega' >0. \end{align*} Thus $\zeta$ is a K\"ahler class as claimed. \end{proof}
\begin{corollary} Let $E$ be a nef $\mathbb R$-twisted vector bundle of rank $e\le d$ and $j=d-e$. Then $$\int_X c_e(E)\omega^j \ge 0$$
\end{corollary} \begin{proof} Let $f(t) = \int_X c_e(E\langle t\omega\rangle)\omega^j$. The Bloch-Gieseker theorem implies $f(t)\neq 0$ for all $t>0$, and since it is clearly positive for $t\gg 0$ $f$ is not identically zero. Since $f$ is polynomial in $t$ we get $f(t)>0$ for $t>0$ sufficiently small, which proves the statement. \end{proof}
From here almost all the results in this paper extend to the K\"ahler case, and the proofs have only trivial modifications. We state only one and leave the rest to the reader.
\begin{theorem}[Derived Schur classes of nef vector bundles on K\"ahler manifolds are in $\overline{\operatorname{HR}}$]\label{thm:derivedschurclassesareinHR:Kahler} Let $X$ be a compact K\"ahler manifold of dimension $d\ge 2$, let $\lambda$ be a partition of length $d+i-2$ and let $E$ be an $\mathbb R$-twisted nef vector bundle on $X$. Then $$s_{\lambda}^{(i)}(E) \in \overline{\operatorname{HR}}(X).$$ \end{theorem}
\section{Combinations of Derived Schur Classes}\label{sec:combinations}
An interesting feature of the Hodge-Riemann property for bilinear forms is that it generally is not preserved by taking convex combinations, and so there is no reason to expect that a convex combination of classes with the Hodge-Riemann property again has the Hodge-Riemann property. In fact this is true even for combinations of Schur classes of an ample vector bundle as the following example shows
\begin{example}[\!\!\protect{\cite[Section 9.2]{RossTomaHR}}]\label{example:convexcombinations} Let $X=\mathbb P^2\times \mathbb P^3$ Then $N^1(X)$ is two-dimensional, with generators $a,b$ that satisfy $a^3=0$, $a^2b^3=1$. Set $\mathcal O_X(a,b) = \mathcal O_{\mathbb P_2}(a) \boxtimes \mathcal O_{\mathbb P^3}(b)$ and consider the nef vector bundle $$ E = \mathcal O(1,0) \oplus \mathcal O(1,0) \oplus \mathcal O(0,1).$$ One computes that the form $$ (1-t) c_3(E) + t s_{(1,1,1)}(E)$$ gives an intersection form on $N^1(X)$ with matrix $$Q_t:= \left(\begin{array}{cc} t&2t \\ 2t&1+2t \end{array}\right).$$ For $t\in (0,1/2)$ the matrix $Q_t$ has two strictly positive eigenvalues. Thus fixing $t\in (0,1/2)$, any small pertubation of $E$ by an ample class gives an ample $\mathbb Q$-twisted bundle $E'$ so that $(1-t)c_3(E') + t s_{(1,1,1)}(E')$ does not have the Hodge-Riemann property.
\end{example}
Given this it is interesting to ask if there are particular convex combinations of (derived) Schur classes that do retain the Hodge-Riemann property. To state one such result we need the following definition, for which we recall a matrix is said to be \emph{totally positive} if all its minors have non-negative determinant,.
\begin{definition}[P\'olya Frequency Sequence] Let $\mu_0,\ldots,\mu_{N}$ be non-negative numbers, and set $\mu_i=0$ for $i<0$. We say $\mu_0,\ldots,\mu_N$ is a \emph{P\'olya frequency sequence} if the matrix $$ \mu:=(\mu_{i-j})_{i,j=0}^N$$ is totally positive.\end{definition}
\begin{theorem}\label{thm:poyla}
Suppose that $X$ has dimension $d\ge 4$ that $h$ is an nef class on $X$ and $E$ is a nef vector bundle. Let $|\lambda|=d-2$ and $\mu_0,\ldots,\mu_{d-2}$ be a P\'olya frequency sequence. Then the class \begin{equation}\sum_{i=0}^{d-2} \mu_i s_{\lambda}^{(i)} (E) h^{i}\label{eq:Poylaclass}\end{equation} lies in $\overline{\operatorname{HR}}(X)$. \end{theorem}
Theorem \ref{thm:poyla} follows quickly from the following statement, for which we recall $c_i$ denotes the $i$-th elementary symmetric polynomial.
\begin{proposition}\label{prop:poyla:divisor} Suppose that $X$ has dimension $d\ge 4$ and $E$ is a nef vector bundle. Let $\lambda$ be a partition of $d-2$. Let $D_1,\ldots,D_q$ be ample $\mathbb Q$-divisors on $X$ for some $q\ge 1$. Then for any $t_1,\ldots,t_q\in \mathbb Q_{>0}$ the class $$\sum_{i=0}^{d-2} s_{\lambda}^{(i)}(E) c_i(t_1 D_1,\ldots,t_q D_q)$$ lies in $\overline{\operatorname{HR}}(X)$ \end{proposition}
\begin{proof}[Proof of Theorem \ref{thm:poyla}] If all the $\mu_i$ vanish the statement is trivial, so we assume this is not the case. From the Aissen-Schoenberg-Whitney Theorem \cite{AissenSchoenbergWhitney}, the assumption that $\mu_i$ is a P\'olya frequency sequence implies that the generating function $$\sum_{i=0}^{d-2} \mu_i z^i$$ has only real roots, and since each $\mu_i$ is non-negative these roots are then necessarily non-positive. Writing these roots as $\{-t_j\}$ for $t_j\in \mathbb R_{\ge 0}$ means $$\sum_{i=0}^{d-2} \mu_i z^i = \kappa\prod_{j=0}^N (z+t_j) \text{ where }\kappa>0$$ which implies $$ \mu_i = \kappa c_i(t_1,\ldots,t_{ N}) \text{ for all }i.$$
Now for each $j$ let $t_j^{(n)}\in \mathbb Q_{>0}$ tend to $t_j$ as $n\to \infty$. Fix an ample divisor $h''$ and consider the class $h': = h + \frac{1}{n} h''$. Proposition \ref{prop:poyla:divisor} (applied with $q= N$ and $D_1=\cdots = D_q = h'$) implies $$\sum_{i=0}^{d-2} s_{\lambda}^{(i)}(E) c_i(t^{(n)}_1,\ldots,t^{(n)}_{ N}) (h')^{i}$$ lies in $\overline{\operatorname{HR}}(X)$. Taking the limit as $n\to \infty$ gives the statement we want. \end{proof}
\begin{proof}[Proof of Proposition \ref{prop:poyla:divisor}] Set $$ \Omega : = \Omega(D_1,\ldots,D_p): = \sum_{i=0}^{d-2} s_{\lambda}^{(i)}(E) c_i(D_1,\ldots,D_p).$$ Without loss of generality we may assume all the $D_i$ are integral and very ample. Write $t_j = r_j/s$ for some positive integers $r_j$ and $s$. By an iterated application of the Bloch-Gieseker covering construction, we find a finite $u:Y\to X$ and line bundles $\eta_j$ on $X'$ such that that $\eta_j^{\otimes s} = u^* \mathcal O(D_j)$. Thus $$ r_j c_1(\eta_j) = t_j u^* D_j.$$
Set $E' = u^* E$. Consider the cone construction for $E'$ as described in \S\ref{sec:coneclasses}. That is, there is a surjective $\pi: C\to Y$ from an irreducible variety $C$ of dimension $n$, and a nef vector bundle $U$ on $C'$ of rank $n-2$ such that $\pi_* c_{n-2}(U)= s_{\lambda}(E')$. In fact more is true namely;
\begin{lemma}\label{lemma:pushforwardderivedschur:latex}
\begin{equation}\pi_* c_{n-2-i}(U|_C) = s_{\lambda}^{(i)}(E') \text{ for } 0\le i\le |\lambda|.\label{eq:pushforwardderivedschur:removed}\end{equation} \end{lemma} \begin{sketchproof} Formally this is clear: for if $\delta'\in H^{1,1}(X;\mathbb R)$ then $c_{n-2}(U\langle\pi^* \delta'\rangle)= \sum c_{n-2-i}(U) (\pi^*\delta')^i$ and pushing this forward to $X$ gives a polynomial in $\delta'$ of classes on $X$ whose coefficients are the derived Schur classes $s_{\lambda}^{(i)}(E')$. For a full proof we refer the reader to \cite[Proposition 5.2]{RossTomaHR}. \end{sketchproof}
Continuing with the proof of the Proposition, set $$ F = \bigoplus_{i=1}^p \eta_i^{\otimes r_i}$$ so $$ c_j(F) = c_j ( r_1 c_1(\eta_1), \cdots, r_p c_1(\eta_p)) = u^* c_j( t_1D_1,\ldots, t_p D_p).$$ Then on $C'$ the bundle $$ \tilde{U} := U \oplus \pi^* F$$ is nef. Take a resolution $\sigma:C\to C'$, the vector bundle $\sigma^*U$ remains nef and so using Theorem \ref{thm:derivedschurclassesareinHR} and Lemma \ref{lem:intermediatesingular} $$\pi_* c_{n-2}(\tilde{U}) \in \operatorname{HR_w}(Y)$$ But \begin{align*} \pi_* c_{n-2}(\tilde{U})&= \pi_*( c_{n-2}(U) + c_{n-3}(U) \pi^* c_1(F) + \cdots + c_{n-2-d}(U) \pi^* c_d(F))\\ &= s_{\lambda}(E') + s_{\lambda}^{(1)}(E')c_1(F) + \cdots + s_{\lambda}^{(d-2)}(E') c_{d-2}(F) \\ &= u^*\Omega. \end{align*} So by Lemma \ref{lem:pullbacksgiveHR} applied to $u:Y\to X$ we conclude that $\Omega\in \operatorname{HR_w}(X)$.
To show that in fact $\Omega\in \overline{\operatorname{HR}}(X)$ we consider the effect of replacing each $D_i$ with $D_i +th$. Let $\Omega_t: = \Omega(D_1+th,\ldots,D_p+th)$ which is a polynomial in $t$ whose $t^{d-2}$ term is some positive multiple of $h^{d-2}$. Setting $f(t) = \det(Q_{\Omega_t})$ we conclude exactly as in the end of the proof of Theorem \ref{thm:derivedschurclassesareinHR} that $\Omega_t\in \operatorname{HR}(X)$ for $t\in \mathbb Q_{+}$ sufficiently small, and thus $\Omega\in \overline{\operatorname{HR}}(X)$ as required. \end{proof}
\begin{question} Suppose that $\mu_1,\ldots,\mu_{d-2}$ is a P\'olya frequency sequence with each $\mu_i$ strictly positive, and that $h$ and $E$ are ample. Is it then the case that the class in \eqref{eq:Poylaclass} is actually in $\operatorname{HR}(X)$? The difficulty here is that to follow the proof we have given above one needs to address the possibility that some of the $t_j$ are irrational. \end{question}
\begin{comment} \begin{remark}\label{rem:rmkspoyla}
Putting $h=c_1(E)$ into the theorem implies that setting $$p:= \sum_{i=0}^{d-2} \mu_i s_{\lambda}^{(i)} c_1^{i}$$ then $p(E)$ lies in $\overline{\operatorname{HR}}(X)$ for any nef $\mathbb Q$-twisted vector bundle $E$.
\item **is this even true?** This generalises to other symmetric homogeneous polynomials $p$ of degree $d-2$ in $e$ variables. For such a $p$ define the associated derived polynomials $p^{(i)}$ by requiring that $$ p(x_1+t,\ldots,x_e+t) = \sum_{i=0}^d p^{(i)}(x_1,\ldots,x_e) t^i.$$ We let $\mathcal P$ be the set of $p$ such that \begin{enumerate} \item $p^{(d)}>0$ \item $p(E)\in \overline{\operatorname{HR}}(X)$ for all $\mathbb Q$-twisted nef vector bundles $E$ of rank $e$ on a smooth projective variety $X$ of dimension $d$. \end{enumerate} Then the same proof as above shows if $p\in \mathcal P$ then $\sum_{i=0}^d \mu_i p^{(i)}(E) h^{d-i}\in \overline{\operatorname{HR}}(X)$ for any nef bundle $E$, any nef class $h$ and any P\'oyla frequency sequence $\mu_i$. \end{enumerate}
\end{remark} \end{comment}
\section{Inequalities}\label{sec:inequalities}
\subsection{Hodge-Index Type inequalities} The simplest and most fundamental inequality obtained from the Hodge-Riemann property is the Hodge-index inequality.
\begin{theorem}[Hodge-Index Theorem] Let $X$ be a manifold of dimension $d$ and $\Omega\in \operatorname{HR_w}(X)$. If $\beta \in H^{1,1}(X)$ is such that $\int_X \beta^2 \Omega\ge0$ then for any $\alpha \in H^{1,1}(X)$ it holds that \begin{equation} \int_X \alpha^2 \Omega \int_X \beta^2 \Omega \le \left( \int_X \alpha \beta \Omega\right)^2\label{eq:HI}\end{equation} Moreover if $\Omega\in \operatorname{HR}(X)$ and $\int_X \beta^2 \Omega>0$ then equality holds in \eqref{eq:HI} if and only if $\alpha$ and $\beta$ are proportional. \end{theorem} \begin{proof} The statement is about symmetric bilinear forms with the given signature and its proof is standard. Indeed, the case when $\int_X \beta^2 \Omega=0$ is trivial and the case when the intersection form is nondegenerate and $\int_X \beta^2 \Omega>0$ is classical. Finally, the case when the intersection form is degenerate and $\int_X \beta^2 \Omega>0$ reduces itself to the previous one by modding out the kernel of the intersection form.
\end{proof}
In particular (namely Theorem \ref{thm:derivedschurclassesareinHR}) the inequality \eqref{eq:HI} applies when $\Omega=s_{\lambda}(E)$ whenever $\lambda$ is a partition of $d-2$, $E$ is a nef $\mathbb Q$-twisted bundle on $X$ and $\beta$ is nef. We now prove a variant of this that gives additional information.
\begin{theorem}\label{thm:thmschurhodgeimproved}
Let $X$ be a projective manifold of dimension $d\ge 4$ and let $E$ be a $\mathbb Q$-twisted nef vector bundle and $h\in H^{1,1}(X;\mathbb R)$ be nef. Also let $\lambda$ be a partition of length $|\lambda|= d-1$. Then for all $\alpha\in H^{1,1}(X;\mathbb R)$, \begin{equation}\int_X \alpha^2 s_{\lambda}^{(1)}(E) \int_X h s_{\lambda}(E) \le 2 \int_X \alpha h s_{\lambda}^{(1)}(E) \int_X \alpha s_{\lambda}(E)\label{eq:schurhodgeimproved}\end{equation} \end{theorem}
\begin{remarks} \begin{enumerate} \item In the case that $\lambda = (d-1)$ and $\operatorname{rk}(E)=d-1$ the inequality \eqref{eq:schurhodgeimproved} becomes \begin{equation} \int_X \alpha^2c_{d-2}(E) \int_X h c_{d-1}(E) \le 2\int_X \alpha h c_{d-2}(E) \int_X \alpha c_{d-1}(E).\label{eq:schurhodgeimproved:chern} \end{equation} This was previously proved in \cite[Theorem 8.2]{RossTomaHR}. In fact \eqref{eq:schurhodgeimproved:chern} was shown to hold for all nef vector bundles of rank at least $d-1$ and if $E,h$ are assumed ample then equality holds in \eqref{eq:schurhodgeimproved:chern} if and only if $\alpha=0$. We imagine a similar statement holds in the context of Theorem \ref{thm:thmschurhodgeimproved}. \item Assume in the setting of Theorem \ref{thm:thmschurhodgeimproved} that $\int_X s_{\lambda}(E)h >0$ and let $W$ be the kernel of the map $H^{1,1}(X) \to \mathbb R$ given by $\alpha\mapsto \int_X \alpha s_{\lambda}(E)$. Then $W$ has codimension 1, and \eqref{thm:thmschurhodgeimproved} says that the intersection form $Q_{s_{\lambda}(E)}$ is negative semidefinite on the codimension one subspace $$ \{ \alpha \in H^{1,1}(X) : \int_X \alpha s_{\lambda}^{(1)}(E) =0\}.$$ This is different information to the Hodge-Index inequality which is essentially a reformulation of the fact that this intersection form is negative semidefinite on the orthogonal complement of $h$. \item The inequality \eqref{eq:schurhodgeimproved} generalizes to any homogeneous symmetric polynomial $p$ in $e$ variables with the property that $p(E)\in \overline{\operatorname{HR}}(X)$ for all $\mathbb Q$-twisted nef vector bundles $E$ of rank $e$ (with the obvious definition for the derived polynomials $p^{(i)})$. \end{enumerate} \end{remarks}
\begin{proof}[Proof of Theorem \ref{thm:thmschurhodgeimproved}]
If $e:=\operatorname{rk}(E)<\lambda_1$ the statement is trivial, so we assume $e\ge \lambda_1$. We start with some reductions. By continuity, it is sufficient to prove this under the additional assumption that $h$ is ample. Also replacing $E$ with $E\langle th\rangle$ for $t\in \mathbb Q_{>0}$ sufficiently small we may assume that $\int_X s_{\lambda}(E) h>0$.
Now set $\hat{X} = X\times \mathbb P^1$ and $\hat{E}= E\boxtimes \mathcal O_{\mathbb P^1}(1)$. Observe $\hat{E}$ is nef on $\hat{X}$ and $|\lambda| = \dim(\hat{X})-2$. So Theorem \ref{thm:derivedschurclassesareinHR} implies $$s_{\lambda}(\hat{E})\in \overline{\operatorname{HR}}(\hat{X}).$$
Let $\alpha\in H^{1,1}(X;\mathbb R)$ and denote by $\tau$ the hyperplane class on $\mathbb P^1$. Also to ease notation define $$\Omega: = s_{\lambda}(E)\in H^{d-1,d-1}(X;\mathbb R)\text{ and } \Omega':= s_{\lambda}^{(1)}(E)\in H^{d-2,d-2}(X;\mathbb R)$$ so $s_{\lambda}(\hat{E}) = \Omega + \Omega'\tau$.
Now define $$ \hat{\alpha} := \alpha - \kappa \tau \text{ where }\kappa: = \frac{\int_X \alpha\Omega' h}{\int_X \Omega h}$$ so $$\hat{\alpha} s_{\lambda}(\hat{E}) h = \hat{\alpha} (\Omega + \tau \Omega')h =0.$$ Also observe $$\int_{\hat{X}} s_{\lambda}(\hat{E}) h^2 = \int_X \Omega' h^2>0$$ so the Hodge-Index inequality applied to $s_{\lambda}(\hat{E})$ yields $$0\ge \int_{\hat{X}} \hat{\alpha}^2 s_{\lambda}(\hat{E}) = \int_{\hat{X}} (\alpha^2 - 2\kappa \alpha\tau)(\Omega + \tau\Omega') = \int_X \alpha^2 \Omega' - 2\kappa\int_X \alpha\Omega.$$ Rearranging this gives \eqref{eq:schurhodgeimproved}. \end{proof}
\subsection{Khovanskii-Tessier-type inequalities}
Let $X$ be smooth and projective of dimension $d$. Suppose that $E,F$ are vector bundles on $X$, and let $\lambda$ and $\mu$ be partitions of length $|\lambda|$ and $|\mu|$ respectively, and to avoid trivialities we assume $|\lambda| + |\mu| \ge d$.
\begin{definition} We say a sequence $(a_i)_{i\in \mathbb Z}$ of non-negative real numbers is \emph{log concave} if \begin{equation}\label{eq:logconcave}a_{i-1} a_{i+1}\le a_i^2 \text{ for all } i\end{equation} \end{definition} We note that for a finite sequence, say $a_i=0$ for $i<0$ and for $i>n$, log-concavity is equivalent to \eqref{eq:logconcave} holding in the range $i=1,\ldots,n-1$.
\begin{theorem}\label{thm:generalizedKT} Assume $E,F$ are nef. Then the sequence
\begin{equation}\label{eq:generalizedKT}i\mapsto \int_X s_{\lambda}^{(|\lambda|+|\mu|-d-i)}(E)s_{\mu}^{(i)}(F) \end{equation} is log-concave \end{theorem}
Before giving the proof, some special cases are worth emphasising.
\begin{corollary}\label{cor:lambdaandmuard}
Suppose that $|\lambda| = |\mu|=d$. Then the sequence $$i\mapsto \int_X s_{\mu}^{(d-i)}(E) s_{\lambda}^{(i)}(F)$$ is log-concave \end{corollary}
\begin{corollary}\label{cor:Enefandhnef} Suppose that $|\lambda| =d$ and let $h$ be a nef class on $X$. Then the sequence \begin{equation}i\mapsto \int_X s_{\lambda}^{(d-i)}(E) h^{d-i} \label{eq:logconcavewithh} \end{equation} is log-concave. In particular the map \begin{equation}i\mapsto \int_X c_i(E)h^{d-i} \label{eq:logconcavewithh:chern} \end{equation} is log-concave. \end{corollary} \begin{proof}[Proof of Corollary \ref{cor:Enefandhnef}] By continuity we may assume that $h$ is ample. Let $L$ be a line bundle with $c_1(L)=h$. By rescaling $h$ we may, without loss of generality, assume $L$ is globally generated giving a surjection $$\mathcal O^{\oplus f+1} \to L\to 0$$ for some integer $f$. Let $F^*$ be the kernel of this surjection. Then $F$ is a vector bundle of rank $f$ that is globally generated and hence nef. Now set $\mu = (f)$, so $s_{\mu}^{(j)}(F) = c_{f-j}(F)=h^{f-j}$. We now replace $i$ with $f-d+i$ in \eqref{eq:generalizedKT} (which is an affine linear transformation so does not affect log-concavity). Note that
$$|\lambda| + |\mu| -d - (f-d+i) = |\lambda|-i,$$ so Theorem \ref{thm:generalizedKT} gives \eqref{eq:logconcavewithh}
Finally \eqref{eq:logconcavewithh:chern} follows upon letting $e:=\operatorname{rk}(E)$ and putting $\lambda = (e)$ so $s_{\lambda}^{(j)}(E) = c_{e-j}(E)$ so $s_{\lambda}^{(|\lambda|-i)}(E)= c_i(E)$. \end{proof}
\begin{proof}[Proof of Theorem \ref{thm:generalizedKT}] The first thing to note is that all the quantities in \eqref{eq:generalizedKT} are non-negative (see Remark \ref{rem:monomialsofderivedschurclasses}). Also, we may as well assume $\operatorname{rk}(E)\ge \lambda_1$ and $\operatorname{rk}(F)\ge \mu_1$ else the statement is trivial.
Set
$$j = |\lambda| + |\mu| -d-i$$ and define $$ a_i:=\int_X s_{\lambda}^{(j)}(E)s_{\mu}^{(i)}(F)$$
so the task is to show that $(a_i)$ is log-concave. We observe that $a_i=0$ if either $i$ or $j$ are negative, or $i> |\mu|$ or $j>|\lambda|$. Thus the range of interest is
$$ \underline{i}: = \max\{ 0, |\mu|-d\} \le i \le \min \{ |\mu|, |\lambda| + |\mu|-d\} =: \overline{i}.$$ Fix such an $i$ in this range and consider $$ \hat{X} = X\times \mathbb P^{j+1} \times \mathbb P^{i+1}.$$
Let $\tau_1$ be the pullback of the hyperplane class on $\mathbb P^{j+1}$ and $\tau_2$ the pullback of the hyperplane class on $\mathbb P^{i+1}$ and consider $$\Omega = s_{\lambda} (E(\tau_1)) \cdot s_{\mu} (F(\tau_2)).$$
Observe that by construction $|\lambda| + |\mu| = d + i +j = \dim{\hat{X}} -2=: \hat{d}-2$. Expanding $\Omega$ as a polynomial in $\tau_1,\tau_2$ one sees that the coefficient of $\tau_1^j\tau_2^i$ is precisely $s_{\lambda}^{(j)} s_{\mu}^{(i)}$. Thus $$\int_{\hat{X}} \Omega \tau_1 \tau_2 = \int_X s_{\lambda}^{(j)} s_{\mu}^{(i)} \int_{\mathbb P^{j+1}} \tau_1^{j+1} \int_{\mathbb P^{i+1}} \tau_2^{i+1} = \int_X s_{\lambda}^{(j)} s_{\mu}^{(i)} = a_i.$$ Similarly $\int_{\hat{X}} \Omega \tau_1^2 = a_{i-1}$ and $\int_{\hat{X}} \Omega \tau_2^2= a_{i+1}$.
Now, since $E(\tau_1)$ and $F(\tau_2)$ are nef on $\hat{X}$ we know from Theorem \ref{thm:monomialsderivedschurclassesareinHR} that $\Omega\in \overline{\operatorname{HR}}(\hat{X})$. Thus the Hodge-Index inequality \eqref{eq:HI} applies with respect to the classes $\tau_1,\tau_2$ which is \begin{equation}\int_{\hat{X}} \Omega \tau_1^2 \int_{\hat{X}} \Omega \tau_2^2 \le \left(\int_{\hat{X}} \Omega \tau_1 \tau_2 \right)^2\label{eq:EandFnefHI}\end{equation} giving the log-concavity we wanted. \end{proof}
\begin{remark}
In \cite{RossTomaHR} we gave a slightly different proof of \eqref{eq:logconcavewithh} which gave more, namely that if $X$ is smooth and $E$ and $h$ are ample then the map in question is strictly log-concave. We expect that an analogous improvement can be made to Theorem \ref{eq:generalizedKT}, but it is not clear how this can be proved using the methods we have given here, since the bundle $F$ constructed in the above proof is only nef.
\end{remark}
\begin{question} Is there a natural statement along the lines of Theorem \ref{thm:generalizedKT} that applies to three or more nef vector bundles? For instance perhaps it is possible to package characteristic numbers into a homogeneous polynomial that can be shown to be Lorentzian in the sense of Br\"{a}nd\'{e}n-Huh \cite{BrandenHuh}. \end{question}
\begin{corollary}\label{cor:combinatoric}
Let $\lambda$ and $\mu$ be partitions, and let $d$ be an integer with $d\le |\lambda| + |\mu|$. Assume $x_1,\ldots,x_e,y_1,\ldots,y_f\in \mathbb R_{\ge 0}$. Then the sequence
$$i\mapsto s^{(|\lambda| + |\mu| -d +i)}_{\lambda}(x_1,\ldots,x_e) s_{\mu}^{(i)}(y_1,\ldots,y_f)$$ is log concave. \end{corollary} \begin{proof} By continuity we may assume the $x_i$ and $y_i$ are rational. Furthermore, by clearing denominators, we may suppose they all lie in $\mathbb N$. Then take $X=\mathbb P^d$ and $E = \bigoplus_{i=1}^e \mathcal O_{\mathbb P^d}(x_i)$ and $F = \bigoplus_{i=1}^f \mathcal O_{\mathbb P^d}(y_i)$. Then for any symmetric polynomial $p$ of degree $\delta$ we have $p(E) = p(x_1,\ldots,x_e)\tau^{\delta}$ and similarly for $F$. Thus what we want follows from Theorem \ref{thm:generalizedKT}. \end{proof}
Putting $e=f$ we can consider
$$ u_i : = s^{(|\lambda| + |\mu| -d +i)}_{\lambda}s_{\mu}^{(i)}$$
as a polynomial in $x_1,\ldots,x_e$. Still assuming $d\le |\lambda| + |\mu|$, Corollary \ref{cor:combinatoric} says that $$ (u_i^2 - u_{i+1} u_{i-1} )(x_1,\ldots,x_e) \ge 0 \text{ for any } x_1,\ldots,x_e\in \mathbb R_{\ge 0}.$$
\begin{question} Is $u_i^2 - u_{i+1} u_{i-1}$ monomial-positive (i.e.\ a sum of monomials with all non-negative coefficients)? \end{question}
\begin{corollary}\label{cor:combinatoric2} Let $\lambda$ be a partition and $x_1,\ldots,x_e\in \mathbb R_{\ge 0}$. Then the sequence $$i\mapsto s_{\lambda}^{(i)}(x_1,\ldots,x_e)$$ is log-concave. \end{corollary} \begin{proof}
By continuity we may assume $x_i\in \mathbb Q_{>0}$, and then by clearing denominators that they are all in $\mathbb N$. Set $d = |\lambda|$ and $X=\mathbb P^d$ and $E = \bigoplus_{j=1}^e \mathcal O_{\mathbb P^d}(x_i)$ and $h= c_1(E)$ which are both ample. Then for any symmetric polynomial $p$ of degree $d$ in $e$ variables we have $\int_X p(E) = p(x_1,\ldots,x_e)$. Thus Corollary \ref{cor:Enefandhnef} tells us that the map $$i\mapsto s_{\lambda}^{(d-i)}(x_1,\ldots,x_e) (x_1 + \cdots x_e)^{d-i}=:a_i$$ is log-concave That is $a_{i-1}a_{i+1}\le a_i^2$, and dividing both sides of this inequality by $(x_1+\ldots+x_e)^{2d-2i}$ gives that $i\mapsto s_{\lambda}^{(d-i)}(x_1,\ldots,x_e)$ is log-concave. Replacing $d-i$ with $i$ does not change the log-concavity, so we are done. \end{proof}
\begin{question} Do Corollary \ref{cor:combinatoric} or Corollary \ref{cor:combinatoric2} have a purely combinatorial proof? \end{question}
\subsection{Lorentzian Property of Schur polynomials}
We end with a discussion on how our results relate to those of Huh-Matherne-M\'esz\'aros-Dizier \cite{Huhetal}. To do so we need some definitions that come from \cite{BrandenHuh}. A symmetric homogeneous polynomial $p(x_1,\ldots,x_e)$ of degree $d$ is said to be \emph{strictly Lorentzian} if all the coefficients of $p$ are positive and for any $\alpha\in \mathbb N^e$ with $\sum_j \alpha_j=d-2$ we have $$\frac{\partial^{\alpha}p}{\partial x^{\alpha}} \text{ has signature } (+,-,\ldots,-).$$ We say $p$ is \emph{Lorentzian} if it is the limit of strictly Lorentzian polynomials.
Any homogeneous polynomial $p$ of degree $d$ can be written as $p = \sum_{\mu} a_{\mu} x^{\mu}$ where the sum is over $\mu\in \mathbb Z_{\ge 0}^e$ with $\sum \mu_j= d$. We write $[p]_{\mu}:= a_\mu$ for the coefficient of $x^{\mu}$. The \emph{normalization} of $p$ is defined by $$N(p) : = \sum_{\mu} \frac{a_{\mu}}{\mu!} x^{\mu}.$$
\begin{theorem}[Huh-Matherne-M\'esz\'aros-Dizier \protect{\cite[Theorem 3]{Huhetal}}]\label{thm:schurlorentzian} The normalized Schur polynomials $ N(s_{\lambda})$ are Lorentzian. \end{theorem}
Our proof needs a preparatory statement. For this we set $$t_j(x_1,\ldots,x_e) = x_j \text{ for each } j=1,\ldots, e.$$ \begin{lemma}\label{lem:sillypolynomial} Let $p(x_1,\ldots,x_e)$ be a homogeneous polynomial of degree $d$, let $e'$ be any integer satisfying $e'\ge\max_{1\le j\le e}\deg_{x_j}(p)$, where $\deg_{x_j}(p)$ is the degree of $p$ with respect to the indeterminate $x_j$,
and set $$ q(x_1,\ldots,x_e) := x_1^{e'} \cdots x_e^{e'} p(x_1^{-1},\ldots, x_e^{-1}).$$ Let $\alpha\in \mathbb Z_{\ge 0}^e$ with $\sum_j \alpha_j=d-2$ and set $\beta_j: = e'-\alpha_j$. Then $$ \frac{\partial^{\alpha}}{\partial x^{\alpha}} N(p) =\frac{1}{2} \sum_{1\le i,j\le e} [ q t_i t_j]_{\beta} x_i x_j.$$ \end{lemma} \begin{proof} For $1\le i\le e$ set $\delta_{i}=(0,\ldots,0,1,0,\ldots,0)\in \mathbb Z^e$ with $1$ at the $i$-th position. Then if $p$ is written as $p = \sum_{\mu} a_{\mu} x^{\mu}$, we get $$ \frac{\partial^{\alpha}}{\partial x^{\alpha}} N(p) =\frac{1}{2}\sum_{1\le i,j\le e}a_{\alpha+\delta_{i}+\delta_{j}}x_{i}x_{j}= \frac{1}{2}\sum_{1\le i,j\le e} [ q t_i t_j]_{\beta} x_i x_j,$$ as one can check by expanding $p$ in monomials. \end{proof}
\begin{proof}[Proof of Theorem \ref{thm:schurlorentzian}]
Take a partition $\lambda=(\lambda_{1},\ldots,\lambda_{N})$ of $d:=|\lambda|$ with $0\le \lambda_N\le \cdots \le \lambda_1$ and assume $\lambda_1\le e$ else the statement is trivial. Then $d$ is the degree of $s_{\lambda}(x_1,\ldots,x_e)$. Note that by adding zero members to the partition $\lambda$ we may increase $N$ without changing the value of $s_{\lambda}$. We may therefore suppose that in our case $N\ge e$. The dual partition to $\lambda$ is defined by $$\overline{\lambda}_i: = e-\lambda_{N-i} \text{ for } i=1,\ldots,N$$
so $|\overline{\lambda}| = Ne - |\lambda| = Ne-d$.
Applying the definition $$s_{\lambda} = \det \left(\begin{array}{ccccc} c_{\lambda_1} & c_{\lambda_1+1} &\cdots &c_{\lambda_1 +N-1}\\ c_{\lambda_2-1} & c_{\lambda_2} &\cdots &c_{\lambda_2 +N-2}\\ \vdots & \vdots & \vdots & \vdots\\ c_{\lambda_N-N+1} & c_{\lambda_N -N +2}&\cdots &c_{\lambda_N}\\ \end{array}\right)$$ to $$x_1^N \cdots x_e^N s_{\lambda}(x_1^{-1} ,\ldots,x_e^{-1})$$ and multiplying each row of the matrix defining $$ s_{\lambda}(x_1^{-1} ,\ldots,x_e^{-1})$$ with $x_1\cdots x_e$, we get $$x_1^N \cdots x_e^N s_{\lambda}(x_1^{-1} ,\ldots,x_e^{-1})=$$ $$ \det \left(\begin{array}{ccccc} c_{e-\lambda_1} & c_{e-\lambda_1-1} &\cdots &c_{e-\lambda_1 -N+1}\\ c_{e-\lambda_2+1} & c_{e-\lambda_2} &\cdots &c_{e-\lambda_2 -N+2}\\ \vdots & \vdots & \vdots & \vdots\\ c_{e-\lambda_N+N-1} & c_{e-\lambda_N +N -2}&\cdots &c_{e-\lambda_N}\\ \end{array}\right)= s_{\bar\lambda}(x_1,\ldots,x_e).$$
Thus $$s_{\overline{\lambda}}(x_1,\ldots,x_e) = x_1^N \cdots x_e^N s_{\lambda}(x_1^{-1} ,\ldots,x_e^{-1})$$ and, equivalently, $$s_{\lambda}(x_1,\ldots,x_e) = x_1^N \cdots x_e^N s_{\overline{\lambda}}(x_1^{-1} ,\ldots,x_e^{-1}).$$
It is tempting to now apply Lemma \ref{lem:sillypolynomial}, but before doing that we introduce a small perturbation. For $\epsilon>0$ set $\tilde{x}_j : = x_j + \epsilon \sum_p x_p$ and let $$q_\epsilon(x_1,\ldots,x_e) := s_{\overline{\lambda}}(\tilde{x}_1,\ldots,\tilde{x}_e)$$ and $$p_{\epsilon}(x_1,\ldots,x_e) := x_1^N \cdots x_e^N q_{\epsilon}(x_1^{-1},\cdots,x_e^{-1}),$$ so \begin{equation}q_{\epsilon}(x_1,\ldots,x_e) = x_1^N \cdots x_e^N p_{\epsilon}(x_1^{-1},\cdots,x_e^{-1}).\label{eq:defqepsilon}\end{equation} We will show that $N(p_{\epsilon})$ is strictly Lorentzian for small $\epsilon>0$, which completes the proof since $p_\epsilon$ tends to $s_{\lambda}$ as $\epsilon$ tends to zero.
To this end, let $\alpha\in \mathbb Z_{\ge 0}^e$ with $\sum_j \alpha_j=d-2$ and set $\beta_j : =N- \alpha_j$ and $$ X := \prod_{j=1}^e \mathbb P^{\beta_j}. $$ Let $\tau_j$ denote the pulback of the hyperplane class on $\mathbb P^{\beta_j}$ to $X$, and set $h:= \sum_j \tau_j$ which is ample. Next set $$E := \bigoplus_{j=1}^e \pi_j^* \mathcal O_{\mathbb P^{\beta_j}}(1) \text{ and } E' := E\langle \epsilon h\rangle.$$
Then $E$ is a nef vector bundle on $X$ and by construction $\dim X = Ne-d+2 = |\overline{\lambda}|+2$. So from Theorem \ref{thm:derivedschurclassesareinHR} we know $s_{\overline{\lambda}}(E) \in \overline{\operatorname{HR}}(X)$. In fact by Remark \ref{rmk:comparisonwitholdpaper} we actually have $s_{\overline{\lambda}}(E')\in \operatorname{HR}(X)$ for sufficiently small $\epsilon>0$ and we assume henceforth this is the case.
Now by \eqref{eq:defqepsilon} and Lemma \ref{lem:sillypolynomial}, \begin{equation}\frac{\partial^{\alpha}}{\partial x^{\alpha}} N(p_{\epsilon}) = \frac{1}{2}\sum_{1\le i,j\le e} [ q_{\epsilon} t_i t_j]_{\beta} x_i x_j\label{eq:proofschulorenzian1}\end{equation} and our goal is to show that this has the desired signature. But this is precisely what we already know, since thinking of $s_{\bar\lambda}(E') \tau_i \tau_j$ as a homogeneous polynomial in $\tau_1,\ldots,\tau_e$, integrating over $X$ picks out precisely the coefficient of $\tau^{\beta}$, and as $E'$ has Chern roots $\tau_1 + \epsilon h, \cdots ,\tau_e+\epsilon h$ this becomes $$\int_{X} s_{\bar\lambda}(E') \tau_i \tau_j = [ q_{\epsilon} t_i t_j]_{\beta}.$$ Hence the quadratic form in \eqref{eq:proofschulorenzian1} is precisely the intersection form $\frac{1}{2}Q_{s_{\bar\lambda}(E')}$ on $H^{1,1}(X)$, which has signature $(+,-,\ldots,-)$ and we are done. \end{proof}
\begin{remark} There is a lot of overlap between what we have here and the original proof in \cite{Huhetal}. For instance we rely here on our Theorem that Schur classes of (certain) ample vector bundles have the Hodge-Riemann property, which in turn relies on the Bloch-Gieseker theorem and thus on the classical Hard-Lefschetz Theorem. On the other hand, \cite{Huhetal} relies on the fact that the volume function on a projective variety is Lorentzian, which is a facet of the Hodge-index inequalities (that are a consequence of the Hodge-Riemann bilinear relations).
Also, instead of our cone classes discussed in \S\ref{sec:coneclasses}, the authors in \cite{Huhetal} use a different aspect of Schur classes that is also a degeneracy locus. Finally we remark the use of the dual partition $\overline{\lambda}$ also appears crucially in \cite{Huhetal}. Nevertheless there is a slightly different feel to the two proofs, and we leave it to the readers to decide if they consider them ``essentially the same" \cite{gowers_proofssame}. \end{remark}
\end{document} | arXiv |
Bifurcation values for a family of planar vector fields of degree five
Classical operators on the Hörmander algebras
February 2015, 35(2): 653-668. doi: 10.3934/dcds.2015.35.653
Chaos for the Hyperbolic Bioheat Equation
J. Alberto Conejero 1, , Francisco Rodenas 1, and Macarena Trujillo 1,
Dept. Matemàtica Aplicada and IUMPA, Universitat Politècnica de València, València, 46022, Spain, Spain, Spain
Received July 2013 Revised January 2014 Published September 2014
The Hyperbolic Heat Transfer Equation describes heat processes in which extremely short periods of time or extreme temperature gradients are involved. It is already known that there are solutions of this equation which exhibit a chaotic behaviour, in the sense of Devaney, on certain spaces of analytic functions with certain growth control. We show that this chaotic behaviour still appears when we add a source term to this equation, i.e. in the Hyperbolic Bioheat Equation. These results can also be applied for the Wave Equation and for a higher order version of the Hyperbolic Bioheat Equation.
Keywords: Hypercyclic $C_0$-semigroups, hyperbolic heat equation., dynamics of $C_0$-semigroups, chaotic $C_0$-semigroups.
Mathematics Subject Classification: Primary: 47A16; Secondary: 47D0.
Citation: J. Alberto Conejero, Francisco Rodenas, Macarena Trujillo. Chaos for the Hyperbolic Bioheat Equation. Discrete & Continuous Dynamical Systems, 2015, 35 (2) : 653-668. doi: 10.3934/dcds.2015.35.653
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J. Alberto Conejero Francisco Rodenas Macarena Trujillo | CommonCrawl |
\begin{definition}[Definition:Irreducible Polynomial/Definition 1]
Let $R$ be an integral domain.
An '''irreducible polynomial''' over $R$ is an irreducible element of the polynomial ring $R \left[{X}\right]$.
\end{definition} | ProofWiki |
\begin{document}
\setcounter{page}{1} \thispagestyle{empty}
\begin{abstract} We use localization method to understand the rational equivariant cohomology rings of real Grassmannians and oriented Grassmannians, then relate this to the Leray-Borel description which says the ring generators are equivariant Pontryagin classes, Euler classes in even dimension, and one more new type of classes in odd dimension, as stated by Casian and Kodama. We give additive basis in terms of equivariant characteristic polynomials and equivariant Schubert/canonical classes. We also calculate Poincar\'e series, equivariant Littlewood-Richardson coefficients and equivariant characteristic numbers. Since all these Grassmannians with torus actions are equivariantly formal, many results for equivariant cohomology have similar statements for ordinary cohomology. \end{abstract}
\title{Localization of equivariant cohomology rings of real Grassmannians}
\section{Introduction} \vskip 15pt
The study of homology and cohomology of real and complex Grassmannian was initially based on Ehresmann's work \cite{Eh37} in terms of Schubert cells. The relations between Schubert classes and characteristic classes were given by Chern \cite{Ch48} for real Grassmannians in $\mathbb{Z}/2$ coefficients. Using the techniques of spectral sequences of fibre bundles, Leray \cite{Le46,Le49} and Borel \cite{Bo53A,Bo53B} gave a unified way to describe the cohomology rings of certain homogeneous spaces in terms of characteristic classes.
These pioneering work applies not only to ordinary cohomology but also to equivariant cohomology. For compact connected Lie group $G$ and its connected closed subgroup $H$ of the same rank, the maximal torus $T$ acts from the left of the homogeneous space $G/H$ with the Leray-Borel description of its equivariant cohomology in $\mathbb{Q}$ coefficients: \[ H^*_T(G/H)=\mathbb{S}\mathfrak{t}^*\otimes_{(\mathbb{S}\mathfrak{t}^*)^{W_G}} (\mathbb{S}\mathfrak{t}^*)^{W_H} \] where $\mathbb{S}\mathfrak{t}^*$ is the symmetric algebra of the dual Lie algebra $\mathfrak{t}^*$, and $W_G,W_H$ are the Weyl groups of $G$ and $H$.
For example, if we consider complex flag varieties as quotients of unitary groups $U(n)$, then the Weyl groups are products of symmetric groups and the Weyl group invariants $(\mathbb{S}\mathfrak{t}^*)^{W_G}, (\mathbb{S}\mathfrak{t}^*)^{W_H}$ consist of symmetric polynomials in appropriate variables. In topological terminology, these invariants are polynomials of equivariant Chern classes of canonical bundles, with relations from Whitney product formula.
Similarly, for even dimensional oriented Grassmannians viewed as quotients of $SO(n)$, whose Weyl groups act by permutations and sign changes on polynomials in appropriate variables, the Leray-Borel description means theirs equivariant cohomology rings are generated by equivariant Pontryagin classes and equivariant Euler classes of canonical bundles and complementary bundles, with relations from Whitney product formula and square of Euler classes as top Pontryagin classes. Notice that oriented Grassmannians are natural $2$-fold covers of real Grassmannians, then we can identify the equivariant cohomology of real Grassmannians as $\mathbb{Z}/2$-invariant of the equivariant cohomology of oriented Grassmannians. Due to the lack of preferred orientations for subspaces in $\mathbb{R}^n$, there is no Euler class of canonical bundle or complementary bundle over real Grassmannians. These facts give even dimensional real Grassmannians their Leray-Borel description of equivariant cohomology rings generated by equivariant Pontryagin classes, with relations from Whitney product formula. This special case of Leray-Borel description for real Grassmannians are stated in Casian\&Kodama \cite{CK}. For odd dimensional oriented Grassmannians $SO(2n+2)/SO(2k+1)\times SO(2n-2k+1)$ in which $SO(2n+2)$ and $SO(2k+1)\times SO(2n-2k+1)$ have different ranks of maximal tori, similar Leray-Borel description was obtained by Takeuchi \cite{Ta62}.
In this paper, we will use localization methods to understand the rational equivariant cohomology rings of real and oriented Grassmannians and re-derive the Leray-Borel description. We will give additive basis both in characteristic polynomials and in canonical classes, compute Poincar\'e series and characteristic numbers, and try to relate these results with Schubert calculus on real Grassmannians.
Alternatively, Sadykov \cite{Sa17}, Carlson \cite{Carl} and He \cite{HeB} have given short derivations of the Leray-Borel-Takeuchi descriptions of the rational cohomology rings of real and oriented Grassmannians.
\textbf{Acknowledgement} This work was part of the author's PhD thesis. The author would like to thank Victor Guillemin and Jonathan Weitsman for guidance. The author would also like to thank Jeffrey Carlson for many useful discussions and for the references of Leray and Takeuchi.
\vskip 20pt \section{Torus actions, equivariant cohomology} \vskip 15pt In this section, we recall some basics of equivariant cohomology for torus actions.
\subsection{Torus actions and isotropy weights at fixed points}\label{subsec:T-action} Throughout the paper, a manifold $M$ is always assumed to be smooth, compact, but not necessarily oriented nor connected. Let torus $T$ act on a manifold $M$, we will denote $M^T$ as the fixed-point set. For any point $p$ in a connected component $C$ of $M^T$, there is the \textbf{isotropy representation} of $T$ on the tangent space $T_p M$, which splits into weighted spaces $T_p M = V_0 \oplus V_{[\alpha_1]} \oplus \cdots \oplus V_{[\alpha_r]}$ where the non-zero distinct weights $[\alpha_1],\ldots,[\alpha_r] \in \mathfrak{t}^*_\mathbb{Z}/{\pm 1}$ are determined only up to signs (If $M$ has a $T$-invariant stable almost complex structure, then those weights are determined without ambiguity of signs). Comparing with the tangent-normal splitting $T_p M = T_p C \oplus N_p C$, we get that $T_p C = V_0$ and $N_p C = V_{[\alpha_1]} \oplus \cdots \oplus V_{[\alpha_r]}$. Since $N_p C = V_{[\alpha_1]} \oplus \cdots \oplus V_{[\alpha_r]}$ is of even dimension, the dimensions of $M$ and of the components of $M^T$ will be of the same parity. If $\mathrm{dim}\, M$ is even, the smallest possible components of $M^T$ could be isolated points. If $\mathrm{dim}\, M$ is odd, the smallest possible components of $M^T$ could be isolated circles. Moreover, since $T$ acts on the normal space $N_p C$ by rotation, this gives the normal space $N_p C$ an orientation.
For any subtorus $K$ of $T$, we get two more actions automatically: the \textbf{sub-action} of $K$ on $M$ and the \textbf{residual action} of $T/K$ on $M^K$.
\subsection{Equivariant cohomology} Let torus $T=(S^1)^n$ act on a manifold $M$. The $T$-equivariant cohomology of $M$ is defined using the Borel construction $H^*_{T} (M) = H^*((ET\times M) / T)$, where $ET=(S^\infty)^{n}$ with $BT=ET/T=(\mathbb{C} P^\infty)^{n}$ and the coefficient ring will usually be $\mathbb{Q}$ throughout the paper, unless otherwise mentioned to be $\mathbb{Z}$. By this definition, if we denote $\mathfrak{t}^*$ as the dual Lie algebra of $T$, then $H^*_T (pt)=H^*(ET / T)=H^*((\mathbb{C} P^\infty)^{n})=\mathbb{S}\mathfrak{t}^*$ is a polynomial ring $\mathbb{Q}[\alpha_1,\dots,\alpha_n]$ or $\mathbb{Z}[\alpha_1,\ldots,\alpha_n]$ under the identification that $c_1(\gamma_i\rightarrow BT)=\alpha_i$, where $\gamma_i\rightarrow BT$ is the canonical complex line bundle on the $i$-th $\mathbb{C} P^\infty$-component of $BT$ and $\alpha_i \in \mathfrak{t}^*_\mathbb{Z}$ is the integral weight dual to the $i$-th $S^1$-component of $T$. The trivial map $\iota: M \rightarrow pt$ induces a homomorphism $\iota^*: H^*_T(pt)\rightarrow H^*_T(M)$ and hence makes $H^*_T(M)$ a $H^*_T (pt)$-module.
For a $T$-equivariant complex, oriented or real vector bundle $V$ over $M$, we can define the equivariant Chern, Euler or Pontryagin classes respectively as $c^T(V)=c((ET\times V) / T)\,,e^T(V)=e((ET\times V) / T)\,,p^T(V)=p((ET\times V) / T)\in H^*_T(M,\mathbb{Z})$.
The famous Atiyah-Bott-Berline-Vergne(ABBV) localization formula says:
\begin{thm}[ABBV Localization Formula, \cite{BV83,AB84}]\label{ABBV}
On an oriented $T$-manifold $M$, an equivariant cohomology class $\omega \in H^*_T (M)$ can be integrated as
\[
\int_M \omega = \sum_{C \subseteq M^T} \int_C \frac{\omega|_C}{e^T(NC)}
\]
where the summation is taken for every component $C \subseteq M^T$ with normal bundle $NC$ and equivariant Euler class $e^T(NC)$. \end{thm}
Inspired by this localization theorem, one can hope for more connections between the manifold $M$ and its fixed-point set $M^T$, if $H^*_T(M)$ is actually a free $H^*_T (pt)$-module.
\begin{dfn}
An action of $T$ on $M$ is \textbf{equivariantly formal} if $H^*_T (M)$ is a free $H^*_T (pt)$-module. \end{dfn}
Using the techniques of spectral sequences, equivariant formality has various equivalent expressions.
\begin{thm}[Equivalences of equivariant formality, \cite{AP93} pp.\,210 Thm\,3.10.4, \cite{GGK02} pp.\,206-207] \label{thm:formal} Let torus $T$ act on a manifold $M$, the following conditions about equivariant cohomology are equivalent: \begin{enumerate}
\item The $T$-action is equivariantly formal, i.e. $H^*_T (M)$ is a free $H^*_T (pt)$-module
\item The Leray-Serre sequence of the fibration $M \hookrightarrow (M \times ET) /T \rightarrow BT$ collapses with $E_\infty = E_2 = H^*(BT)\otimes H^*(M)$
\item $H^*_T(M)\cong H^*_T (pt)\otimes H^*(M)$ as $H^*_T (pt)$-module
\item $H^*_T(M)\rightarrow H^*(M)$, defined as the restriction to the fibre $M$, is surjective
\item Any additive basis of $H^*(M)$ can be lifted to $H^*_T(M)$, hence give an additive $H^*_T (pt)$-basis for $H^*_T(M)$
\item $\sum \mathrm{dim}\,H^*(M^T) = \sum \mathrm{dim}\,H^*(M)$. \end{enumerate} \end{thm} \begin{rmk}
The equivalences among (2)(3)(4)(5) are direct applications of the Leray-Hirsch theorem (which works not only for $\mathbb{Q}$ coefficients, but also for $\mathbb{Z}$ coefficients if $H^*(M,\mathbb{Z})$ is a free $\mathbb{Z}$-module). For the equivalence to the remaining conditions (1)(6) in $\mathbb{Q}$ coefficients, see the cited references. \end{rmk}
\begin{rmk}
When the Betti numbers of $M$ and $M^T$ are known, the equality $\sum \mathrm{dim}\,H^*(M^T) = \sum \mathrm{dim}\,H^*(M)$ is a handy way to verify the equivariant formality. \end{rmk}
\begin{rmk}
The fibre inclusion $M \hookrightarrow (M \times ET) /T$ induces a homomorphism $H^*_T(M) \rightarrow H^*(M)$ factoring through $\mathbb{Q}\otimes_{H^*_T (pt)} H^*_T(M)$, where $\mathbb{Q}$ has a $H^*_T (pt)=\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-algebra structure from the constant-term morphism $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]\rightarrow \mathbb{Q}: f(\alpha_1,\ldots,\alpha_n)\mapsto f(0)$. When the $T$-action is equivariantly formal, i.e. $H^*_T(M)\cong H^*_T (pt)\otimes_\mathbb{Q} H^*(M)$, then we can recover $H^*(M)$ as $\mathbb{Q}\otimes_{H^*_T (pt)} H^*_T(M)$. \end{rmk}
\vskip 20pt \section{GKM-type theorems} \vskip 15pt In this section, we recall the Chang-Skjelbred lemma, the GKM theorem and the author's recent work on GKM-type theorems for possibly non-orientable manifolds in odd dimensions.
\subsection{Chang-Skjelbred lemma and GKM condition}
Given an action $T\curvearrowright M$, for every point $p\in M$, its stabilizer is defined as $T_p=\{t \in T \mid t\cdot p = p\}$, and its orbit is $\mathcal{O}_p\cong T/T_p$. Let's set the $1$-skeleton $M_1=\{p \mid \textup{dim}\,\mathcal{O}_p\leq 1\}$, the union of $1$-dimensional orbits and fixed points. If $H^*_T (M)$ is a free $H^*_T (pt)$-module, i.e. the action is equivariantly formal, Chang and Skjelbred \cite{CS74} proved that $H^*_T (M)$ only depends on the fixed-point set $M^T$ and the $1$-skeleton $M_1$.
\begin{thm}[Chang-Skjelbred Lemma, \cite{CS74}]\label{Chang}
If an action $T\curvearrowright M$ is equivariantly formal, then
\[
H^*_T(M) \cong H^*_T(M_1)\hookrightarrow H^*_T(M^T).
\] \end{thm}
This Lemma enables one to describe the equivariant cohomology $H^*_T(M)$ as a sub-ring of $H^*_T(M^T)$, subject to certain algebraic relations determined by the $1$-skeleton $M_1$. To apply the Chang-Skjelbred Lemma, we will follow Goresky, Kottwitz and MacPherson's idea to start with the smallest fixed-point set $M^T$ and $1$-skeleton $M_1$.
\begin{dfn}[GKM condition in either even or odd dimensions]\label{GKMCond}
An action $T\curvearrowright M$ is \textbf{GKM} if
\begin{itemize}
\item[(1)] The fixed-point set $M^T$ consists of isolated points or isolated circles.
\item[(2)] At each fixed point $p\in M^T$, the non-zero weights $[\alpha_1],\ldots,[\alpha_n] \in \mathfrak{t}^*_\mathbb{Z}/{\pm 1}$ of the isotropy $T$-representation $T\curvearrowright T_pM$ are pair-wise independent.
\end{itemize} \end{dfn}
\begin{rmk}
As mentioned in the Subsection\,\ref{subsec:T-action}, the dimensions of $M$ and of $M^T$ have the same parity. Condition\,(1) in Definition\,\ref{GKMCond} means that $M^T$ consists of isolated points when $M$ is even dimensional or $M^T$ consists of isolated circles when $M$ is odd dimensional. \end{rmk}
\begin{rmk}
The GKM condition is equivalent to requiring the $1$-skeleton $M_1$ to be 2-dimensional when $M$ is even dimensional or 3-dimensional when $M$ is odd dimensional. \end{rmk}
\subsection{GKM theorem: the even dimensional case}
Goresky, Kottwitz and MacPherson \cite{GKM98} considered torus actions on algebraic varieties where the fixed-point set $M^T$ is finite and the $1$-skeleton $M_1$ is a union of spheres $S^2$. They proved that the cohomology $H^*_T(M)$ can be described in terms of congruence relations on a graph determined by the $1$-skeleton $M_1$. Goertsches and Mare \cite{GM14} observed that GKM theory also works in non-orientable case by adding $\mathbb{R} P_{[\alpha]}^2$ components to the $1$-skeleton $M_1$.
\begin{dfn}[GKM graph in even dimension]
If an action $T\curvearrowright M^{2m}$ is GKM, then its \textbf{GKM graph} consists of
\begin{description}
\item[Vertices] A $\bullet$ for each fixed point in $M^T$
\item[Edges$\,\&\,$Weights] A solid edge with weight $[\alpha]$ for each $S^1_{[\alpha]}$ joining two $\bullet$'s representing its two fixed points, and a dotted edge with weight $[\beta]$ for each $\mathbb{R} P^2_{[\beta]}$ joining a $\bullet$ to an empty vertex.
\end{description} \end{dfn}
\begin{thm}[GKM theorem in even dimension, \cite{GKM98} pp.\,26 Thm\,1.2.2, \cite{GM14} pp.\,7 Thm\,3.6]\label{thm:EvenGKM}
If the action of a torus $T$ on a (possibly non-orientable) manifold $M^{2m}$ is equivariantly formal and GKM, then we can construct its GKM graph $\Gamma$, with vertex set $V=M^T$ and weighted edge set $E$, moreover the equivariant cohomology has a graphic description
\[
H^*_T(M) = \big\{ f: V\rightarrow \mathbb{S}\mathfrak{t}^* \mid f_p \equiv f_q \mod{\alpha} \quad \mbox{for each solid edge $\overline{pq}$ with weight $\alpha$
in $E$}\big\}.
\] \end{thm}
\begin{rmk}
Since $H^*_T(\mathbb{R} P_{[\beta]}^2)=H^*_T(pt)$ does not contribute to congruence relations, we can erase all the dotted edges of $\mathbb{R} P_{[\beta]}^2$ and only keep the solid edges of $S^2_{[\alpha]}$ to construct an \textbf{effective} GKM graph, which does not necessarily have the same number of edges for every vertex. \end{rmk}
\subsection{GKM-type theorem: the odd dimensional case}
In odd dimensions, we can also consider smallest-dimensional $1$-skeleton, i.e. $M_1$ is $3$-dimensional. Then according to Chang-Skjelbred lemma, the localization boils down to understanding $S^1$-actions on $3$-manifolds, studied by the author \cite{He17}.
Similar to the even dimensional case, we can construct a graph for each odd dimensional $T$-manifold under the GKM condition\,\ref{GKMCond}.
\begin{dfn}[GKM graph in odd dimension]
If an action $T\curvearrowright M^{2m+1}$ (possibly non-orientable) is GKM, then its \textbf{GKM graph} consists of
\begin{description}
\item[Vertices] There will be two types of vertices.
\begin{itemize}
\item [$\circ$] for each fixed circle $C \subset M^T$.
\item [$\Box$] for each 3d connected component $N^3_{[\alpha]}$ in $M^{T_{[\alpha]}}$ of some codimension-$1$ subtorus $T_{[\alpha]}$ which has Lie algebra $\mathfrak{t_\alpha}\subset \mathfrak{t}$ annihilated by $\alpha$. The $\Box$ is then weighted with $[\alpha]$.
\end{itemize}
\item[Edges] An edge joins a $(\Box,N)$ to a $(\circ,C)$, if the 3d manifold $N$ contains the fixed circle $C$ and hence is a connected component of $M^{T_{[\alpha]}}$ for an isotropy weight $[\alpha]$ of $C$. There are no edges directly joining $\circ$ to $\circ$, nor $\Box$ to $\Box$.
\end{description} \end{dfn}
In order to derive a GKM-type graphic description of $H_{T}^*(M^{2m+1})$, we need to fix in advance an orientation $\theta_i$ for each fixed circle $C_i \subseteq M^T$, and also fix an orientation for each orientable $M^{T_{[\alpha]}} \subseteq M_1$.
\begin{thm}[A GKM-type theorem in odd dimension, \cite{HeA}]\label{thm:OddGKM}
If the action of a torus $T$ on (possibly non-orientable) manifold $M^{2n+1}$ is equivariantly formal and GKM, then we can construct its GKM graph $\Gamma$, with two types of vertex sets $V_\circ$ and $V_\Box$ and edge set $E$. An element of the equivariant cohomology $H^*_T(M)$ can be written as:
\[
(P,Q\theta): V_\circ \longrightarrow \mathbb{S}\mathfrak{t}^* \oplus \mathbb{S}\mathfrak{t}^* \theta
\]
where $\theta$ is the generator of $H^1(S^1)$,
under the relations that for each $\Box$ representing a 3d component $N$ of some $M^{T_{[\alpha]}}$ and the neighbour $\circ$'s representing the fixed circles $C_1,\ldots,C_k$ on this component,
\begin{itemize}
\item if $N$ is non-orientable,
\begin{equation*}
P_{C_1}\equiv P_{C_2}\equiv \cdots\equiv P_{C_k} \mod{\alpha}
\end{equation*}
\item if $N$ is orientable,
\begin{equation*}
P_{C_1}\equiv P_{C_2}\equiv \cdots\equiv P_{C_k} \mbox{ and } \sum_{i=1}^k \pm Q_{C_i}\equiv 0 \mod{\alpha}
\end{equation*}
where the sign for each $Q_{C_i}$ is specified by comparing the prechosen orientation $\theta_i$ with the induced orientation of $N$ on $C_i$.
\end{itemize} \end{thm}
\begin{rmk}
As discussed in \cite{HeA}, different choices of orientations from $C_i \subseteq M^T$ and from orientable $M^{T_{[\alpha]}} \subseteq M_1$ give the isomorphic equivariant cohomology. When $M$ has a $T$-invariant stable almost complex structure, then the isotropy weights $\alpha$ can be determined without ambiguity of signs. Moreover, $M^T$ and $M^{T_{\alpha}} \subseteq M_1$ are equipped with induced stable almost complex structures, hence are oriented canonically. \end{rmk}
\begin{rmk}
To describe the $\mathbb{S}\mathfrak{t}^*$-algebra structure of $H^*_T(M^{2m+1})$, it is convenient to write an element $(P,Q\theta)$ as $(P_C+Q_C\theta)_{C\subset M^T}$. Note $\theta^2=0$, then $(P_C+Q_C\theta)_{C\subset M^T} + (\bar{P}_C+\bar{Q}_C\theta)_{C\subset M^T}=([P_C+\bar{P}_C]+[Q_C+\bar{Q}_C]\theta)_{C\subset M^T}$, and $(P_C+Q_C\theta)_{C\subset M^T} \Cdot (\bar{P}_C+\bar{Q}_C\theta)_{C\subset M^T}=([P_C \bar{P}_C]+[P_C\bar{Q}_C+\bar{P}_C Q_C]\theta)_{C\subset M^T}$. For any polynomial $R \in \mathbb{S}\mathfrak{t}^*$, we have $R \Cdot (P_C+Q_C\theta)_{C\subset M^T} = (R P_C+R Q_C\theta)_{C\subset M^T}$. \end{rmk}
\vskip 20pt \section{Equivariant cohomology rings of complex Grassmannians} \vskip 15pt In this section, we recall the GKM description and Leray-Borel description of equivariant cohomology rings of complex Grassmannians, together with the characteristic basis and canonical basis of the additive structure. We use the notation $G_k(\mathbb{C}^n)$ for the Grassmannian of $k$-dimensional complex subspaces in $\mathbb{C}^n$.
\subsection{GKM description of complex Grassmannians} As shown by Guillemin, Holm and Zara \cite{GHZ06}, for compact connected group $G$ and its closed connected subgroup $H$ of the same rank as $G$, the homogeneous space $G/H$ is GKM and equivariantly formal under the left action of maximal torus $T$, hence has a graphic description for its equivariant cohomology. For example, the GKM graph of $T^n \curvearrowright U(n)/(U(k)\times U(n-k))$ is the Johnson graph $J(n,k)$, of which each vertex is a $k$-element subset $S \subseteq \{1,\ldots,n\}$ and two vertices $S,S'$ are joined by an edge if they differ by one element. For later use in the case of real Grassmannians, we will give an explicit description for the $1$-skeleta of complex Grassmannians.
\begin{prop}[$1$-skeleta of complex Grassmannians]
The $T^n$-action on $G_k(\mathbb{C}^n)$ has $\binom{n}{k}$ fixed points of the form $\oplus_{i\in S} \mathbb{C}_i$, where $S$ is a $k$-element subset of $\{1,2,\ldots,n\}$ and $\mathbb{C}_i$ is the $i$-th component of $\mathbb{C}^n$. The isotropy weights at $\oplus_{i\in S} \mathbb{C}_i$ are $\{\alpha_j-\alpha_i \mid i\in S,j\not \in S\}$, and join $\oplus_{i'\in S} \mathbb{C}_{i'}$ to $\oplus_{i'\in (S\smallsetminus\{i\})\cup \{j\}} \mathbb{C}_{i'}$ via $\{(\oplus_{i'\in S\smallsetminus\{i\}} \mathbb{C}_{i'})\oplus L \mid L \in \mathbb{P}(\mathbb{C}_i\oplus\mathbb{C}_j)\}\cong \mathbb{C} P^1$ in the $1$-skeleton. \end{prop} \begin{proof}
$T^n$ acts on $\mathbb{C}^n$ linearly by $(t_1,\ldots,t_n)\cdot (z_1,\ldots,z_n)=(t_1 z_1,\ldots,t_1 z_n)$ and hence induces an action on $G_k(\mathbb{C}^n)$ by mapping every $k$-dimensional subspace $V$ to $t\cdot V$ for each $t\in T^n$. A fixed point $V$ of the $T$-action on $G_k(\mathbb{C}^n)$ is exactly a $k$-dimensional sub-representation of $\mathbb{C}^n= \oplus_{i=1}^n \mathbb{C}_i$. Since $\oplus_{i=1}^n \mathbb{C}_i$ has distinct weights $\alpha_1,\ldots,\alpha_n$, a $k$-dimensional sub-representation is of the form $\oplus_{i\in S} \mathbb{C}_i$ for $S$, a $k$-element subset of $\{1,2,\ldots,n\}$.
To understand the isotropy weights at $\oplus_{i \in S} \mathbb{C}_i \in G_k(\mathbb{C}^n)$, notice that the tangent space of $G_k(\mathbb{C}^n)$ at $\oplus_{i \in S} \mathbb{C}_i$ is $Hom_\mathbb{C}(\oplus_{i \in S} \mathbb{C}_i, \oplus_{j \not \in S} \mathbb{C}_j)\cong (\oplus_{i \in S} \mathbb{C}_i)^* \otimes_\mathbb{C} (\oplus_{j \not \in S} \mathbb{C}_j)\cong\oplus_{i \in S} \oplus_{j \not \in S} (\mathbb{C}_i^* \otimes_\mathbb{C} \mathbb{C}_j)$ with pair-wise independent weights $\{\alpha_j-\alpha_i \mid i \in S \text{ and } j \not \in S\}$.
The finiteness of fixed points and pair-wise independence of isotropy weights at every fixed point verifies that the $T^n$ action on $G_k(\mathbb{C}^n)$ is GKM. Moreover, for a $k$-element subset $S\subseteq \{1,2,\ldots,n\}$ and a weight $\alpha_j-\alpha_i$ with $j\not \in S,\, i \in S$, the $2$-sphere
\[
\big\{(\underset{i'\in S\smallsetminus \{i\}}{\oplus} \mathbb{C}_{i'}) \oplus L \mid L \in \mathbb{P}(\mathbb{C}_i\oplus \mathbb{C}_j) \big\} \cong \mathbb{C} P^1
\]
connects the $T$-fixed point $\oplus_{i'\in S} \mathbb{C}_{i'}$ with the $T$-fixed point $\oplus_{i'\in (S\smallsetminus\{i\})\cup \{j\}} \mathbb{C}_{i'}$, and is fixed by the corank-$1$ subtorus torus $T_{\alpha_j-\alpha_i}$ whose Lie algebra is $\textup{Ker}(\alpha_j-\alpha_i)$. \end{proof}
Since $G_k(\mathbb{C}^n)$ doesn't have odd-degree cells, the canonical $T^n$ action on $G_k(\mathbb{C}^n)$ is equivariantly formal in $\mathbb{Z}$ coefficients and of course in $\mathbb{Q}$ coefficients. Then we can apply the even dimensional GKM theorem to $G_k(\mathbb{C}^n)$ using congruence relations on the Johnson graph $J(n,k)$.
\begin{thm}[GKM description of complex Grassmannians, \cite{GZ01}]\label{thm:GKMcplxGrass}
Let $\mathcal{S}$ be the collection of $k$-element subsets of $\{1,2,\ldots,n\}$, then the equivariant cohomology of the $T^n$ action on $G_k(\mathbb{C}^n)$ is
\[
H^*_{T}(G_k(\mathbb{C}^n),\mathbb{Q})=\big\{f:\mathcal{S}\rightarrow \mathbb{Q}[\alpha_1,\ldots,\alpha_n] \mid f_{S} \equiv f_{S'} \mod \alpha_j-\alpha_i \quad \text{for $S,S' \in \mathcal{S}$ with $S\cup\{j\}=S'\cup\{i\}$}\big\}.
\] \end{thm}
Using Morse theory on graphs, Guillemin and Zara analysed additive basis for equivariant cohomology of GKM manifolds.
\begin{thm}[Canonical basis of complex Grassmannians, \cite{GZ03}]\label{thm:CplxSchub}
There is a self-indexing Morse function on $\mathcal{S}$
\[
\phi: \mathcal{S} \longrightarrow \mathbb{R} : S \longmapsto 2(\sum_{i\in S} i) -k(k+1)
\]
and a canonical class $\tau_S \in H^{\phi(S)}_{T}(G_k(\mathbb{C}^n),\mathbb{Q})$ for each $S\in \mathcal{S}$ such that
\begin{enumerate}
\item $\tau_S$ is supported upward, i.e. $\tau_S(S')=0$ if $\phi(S')\leq \phi(S)$
\item $\tau_S(S)=\prod' (\alpha_j - \alpha_i)$ where the product is taken over the weights at $S$ connecting to $S'$ with $\phi(S')<\phi(S)$
\end{enumerate}
Moreover, $\{\tau_S, S\in \mathcal{S}\}$ give a $H^*_{T}(pt,\mathbb{Q})$-additive basis of $H^*_{T}(G_k(\mathbb{C}^n),\mathbb{Q})$. \end{thm}
\begin{rmk}
The canonical classes $\tau_S$ are exactly the equivariant Schubert classes via the relation that if $S$ consists of the elements $i_1<i_2<\cdots<i_k$, then the corresponding Schubert symbol is $(i_1-1,i_2-2,\ldots,i_k-k)$. \end{rmk}
\begin{rmk}
In this paper, we only need the existence of canonical classes $\tau_S$. The general formula of $\tau_S$ restricted at each fixed point was given by Guillemin\&Zara \cite{GZ03} and simplified by Goldin\&Tolman \cite{GT09}. \end{rmk}
\subsection{Leray-Borel description of complex Grassmannians} Besides the equivariant Schubert basis, equivariant characteristic classes and characteristic polynomials on complex Grassmannians will give ring generators and additive basis for their equivariant cohomology rings.
\begin{thm}[Leray-Borel description of complex Grassmannians, see \cite{BT82} pp.\,293 Prop\,23.2]
For the complex Grassmannians $G_k(\mathbb{C}^n)$, let $c_1,c_2,\ldots,c_k$ and $\bar{c}_1,\bar{c}_2,\ldots,\bar{c}_{n-k}$ be the Chern classes of the canonical bundle on $G_k(\mathbb{C}^n)$ and its complementary bundle respectively, then
\[
H^*(G_k(\mathbb{C}^n),\mathbb{Z})=\frac{\mathbb{Z}[c_1,c_2,\ldots,c_k;\bar{c}_1,\bar{c}_2,\ldots,\bar{c}_{n-k}]}{(1+c_1+c_2+\ldots+c_k)(1+\bar{c}_1+\bar{c}_2+\ldots+\bar{c}_{n-k})=1}.
\] \end{thm}
The relation $(1+c_1+c_2+\ldots+c_k)(1+\bar{c}_1+\bar{c}_2+\ldots+\bar{c}_{n-k})=1$ makes either $c_1,c_2,\ldots,c_k$ or $\bar{c}_1,\bar{c}_2,\ldots,\bar{c}_{n-k}$ as sets of ring generators of the cohomology $H^*(G_k(\mathbb{C}^n),\mathbb{Z})$. Certain monomials of $c_1,c_2,\ldots,c_k$ actually give an additive basis of the cohomology $H^*(G_k(\mathbb{C}^n),\mathbb{Z})$, stated by Carrell \cite{Ca78} for complex Grassmannians in $\mathbb{C}$ coefficients as a result of ``standard combinatorial reasoning''. Later, details of proof were supplied by Jaworowski \cite{Ja89} for real Grassmannians in $\mathbb{Z}/2$ coefficients which can be adapted to complex Grassmannians in $\mathbb{Z}$ coefficients.
\begin{thm}[Characteristic basis of complex Grassmannians, \cite{Ca78, Ja89}]
The set of monomials $c_1^{r_1}c_2^{r_2}\cdots c_k^{r_k}$ of cohomological degree $2d=\sum_{i=1}^{k} 2ir_i$ satisfying the condition $\sum_{i=1}^{k} r_i \leq n-k$ forms an additive basis for $H^{2d}(G_k(\mathbb{C}^n),\mathbb{Z}),0\leq d\leq k(n-k)$. \end{thm}
Notice that the cohomology $H^*(G_k(\mathbb{C}^n),\mathbb{Z})$ does not have odd-degree elements, the Leray-Serre sequence of $G_k(\mathbb{C}^n)\hookrightarrow ET\times_T G_k(\mathbb{C}^n) \rightarrow BT$ collapses at $E_2=H^*(BT,\mathbb{Z})\otimes_\mathbb{Z} H^*(G_k(\mathbb{C}^n),\mathbb{Z})$ with $H^*_T(G_k(\mathbb{C}^n),\mathbb{Z})\cong H^*(BT,\mathbb{Z})\otimes_\mathbb{Z} H^*(G_k(\mathbb{C}^n),\mathbb{Z})$ as $H^*(BT,\mathbb{Z})$-modules. Therefore, the action $T^n \curvearrowright G_k(\mathbb{C}^n)$ is equivariantly formal in $\mathbb{Z}$ coefficients, and of course in $\mathbb{Q}$ coefficients. However, $H^*_T(G_k(\mathbb{C}^n),\mathbb{Z})\cong H^*(BT,\mathbb{Z})\otimes_\mathbb{Z} H^*(G_k(\mathbb{C}^n),\mathbb{Z})$ is only a $H^*(BT,\mathbb{Z})$-module isomorphism which \begin{enumerate}
\item neither gives the $H^*(BT,\mathbb{Z})$-algebra structure of $H^*_T(G_k(\mathbb{C}^n),\mathbb{Z})$.
\item nor specifies a map $H^*_T(G_k(\mathbb{C}^n),\mathbb{Z}) \rightarrow H^*(BT,\mathbb{Z})\otimes_\mathbb{Z} H^*(G_k(\mathbb{C}^n),\mathbb{Z})$. \end{enumerate}
The above two problems can be resolved using the equivariant version of the Leray-Borel description which is usually given for a compact connected Lie group $G$ with maximal torus $T$ and a closed connected subgroup $H$ containing $T$ as \[ H^*_T(G/H)=\mathbb{S}\mathfrak{t}^*\otimes_{(\mathbb{S}\mathfrak{t}^*)^{W_G}} (\mathbb{S}\mathfrak{t}^*)^{W_H} \] where $W_G$ and $W_H$ are the Weyl groups of $G$ and $H$. For the complex Grassmannian $G_k(\mathbb{C}^n)=U(n)/(U(k)\times U(n-k))$, we have the Weyl group $W_G = S_n$, the symmetric group of $n$ elements, and the Weyl group $W_H = S_k \times S_{n-k}$. Under these Weyl group actions, it is well known that the invariant elements in $\mathbb{S}\mathfrak{t}^*$ are symmetric polynomials, or topologically the equivariant Chern classes:
\begin{thm}[Equivariant Leray-Borel description of complex Grassmannians, \cite{Tu10} pp.\,21]
For the complex Grassmannian $G_k(\mathbb{C}^n)=U(n)/(U(k)\times U(n-k))$, let $T^n$ be the maximal torus of $U(n)$ which acts on the left of $G_k(\mathbb{C}^n)$, and $\alpha_1,\alpha_2,\ldots,\alpha_n$ be the integral basis for its Lie dual algebra $\mathfrak{t}^*$, also let $c^T_1,c^T_2,\ldots,c^T_k$ and $\bar{c}^T_1,\bar{c}^T_2,\ldots,\bar{c}^T_{n-k}$ be the equivariant Chern classes of the canonical bundle on $G_k(\mathbb{C}^n)$ and its complementary bundle respectively, then
\[
H^*_T(G_k(\mathbb{C}^n),\mathbb{Z})=\frac{\mathbb{Z}[\alpha_1,\alpha_2,\ldots,\alpha_n][c^T_1,c^T_2,\ldots,c^T_k;\bar{c}^T_1,\bar{c}^T_2,\ldots,\bar{c}^T_{n-k}]}{c^T\bar{c}^T = \prod_{i=1}^{n}(1+\alpha_i)}.
\] \end{thm}
Since the equivariant Chern classes $c^T_1,c^T_2,\ldots,c^T_k$ and $\bar{c}^T_1,\bar{c}^T_2,\ldots,\bar{c}^T_{n-k}$ lift the ordinary Chern classes $c_1,c_2,\ldots,c_k$ and $\bar{c}_1,\bar{c}_2,\ldots,\bar{c}_{n-k}$, we get
\begin{thm}[Equivariant characteristic basis of complex Grassmannians]
The set of monomials $(c^T_1)^{r_1}(c^T_2)^{r_2}\cdots (c^T_k)^{r_k}$ satisfying the condition $\sum_{i=1}^{k} r_i \leq n-k$ form an additive $H^*_T(pt)$-basis for $H_T^*(G_k(\mathbb{C}^n),\mathbb{Z})$. \end{thm}
\begin{proof}
Combine the ordinary characteristic basis with the equivalence (5) of Theorem \ref{thm:formal}. \end{proof}
\subsection{Relations between the Leray-Borel and GKM descriptions}\label{subsec:BGKM} Since the characteristic monomials $(c^T)^I=(c_1^T)^{i_1}\cdots(c_k^T)^{i_k},\,\sum_{j=1}^{k} i_j \leq n-k$ in Leray-Borel description and the canonical classes $\tau_{S}$ in GKM description are both basis for the free $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-module $H^*_{T}(G_k(\mathbb{C}^n),\mathbb{Q})$, there will be transformations $K,\bar{K}$ between them such that \begin{align*} (c^T)^I &= \sum_S K^I_S \tau_S\\ \tau_S &= \sum_I \bar{K}_I^S (c^T)^I \end{align*} where $K^I_S,\bar{K}_I^S \in \mathbb{Q}[\alpha_1,\ldots,\alpha_n]$.
\begin{rmk}
The idea of considering the transformations between Schubert classes and characteristic classes dates back to Bernstein, Gelfand and Gelfand \cite{BGG73}. In this paper, we only need the existence of the transformations $K,\bar{K}$. For the complete flag manifold $Fl(\mathbb{C}^n)$, Kaji \cite{Ka} gave explicit algorithms on how to decide the polynomials $K^I_S,\bar{K}_I^S$. It would also be interesting to know what the $K^I_S,\bar{K}_I^S$ explicitly are for complex Grassmannians. \end{rmk}
The Littlewood-Richardson rule for equivariant Schubert classes is that there are polynomials $N_{S,S'}^{S''}\in \mathbb{Q}[\alpha_1,\ldots,\alpha_n]$ (see Knutson\&Tao \cite{KT03}) for the multiplication of Schubert classes \[ \tau_{S}\tau_{S'} =\sum_{S''}N_{S,S'}^{S''}\tau_{S''}. \]
On the other hand, the multiplication of the equivariant characteristic classes can be localized. We can express the total equivariant Chern classes $c^T,\bar{c}^T$ of the canonical bundle $\gamma$ and its complementary bundle $\bar{\gamma}$ in GKM description at each fixed point $\oplus_{i\in S} \mathbb{C}_i \in G_k(\mathbb{C}^n)$. Note that the canonical bundle $\gamma$, complementary bundle $\bar{\gamma}$ and tangent bundle $TG_k(\mathbb{C}^n)$ restricted at $\oplus_{i\in S} \mathbb{C}_i \in G_k(\mathbb{C}^n)$ for a $k$-element subset $S \subset \{1,\ldots,n\}$ are the vector spaces $\oplus_{i\in S} \mathbb{C}_i$, $\oplus_{j\not\in S} \mathbb{C}_j $ and $\oplus_{i \in S} \oplus_{j \not \in S} (\mathbb{C}_i^* \otimes_\mathbb{C} \mathbb{C}_j)$ respectively, we get \begin{align*}
c^T|_S &= c^T(\gamma|_S)=\prod_{i\in S} (1+\alpha_i)\\
\bar{c}^T|_S &= c^T(\bar{\gamma}|_S)=\prod_{j\not\in S} (1+\alpha_j)\\
e^T|_S &=e^T(T_SG_k(\mathbb{C}^n))=\prod_{i\in S}\prod_{j\not\in S} (\alpha_j - \alpha_i). \end{align*} Since $\gamma\oplus \bar{\gamma} = \sum_{i=1}^{n}\mathbb{C}_i$, this also shows why there is the relation $c^T\bar{c}^T = \prod_{i=1}^{n}(1+\alpha_i)$.
If we denote $e_l(x_1,\ldots,x_m)$ as the $l$-th elementary symmetric polynomial in variables $x_1,\ldots,x_m$, then $c^T_l|_S = e_l(\alpha_{i\in S}), \bar{c}^T_l|_S = e_l(\alpha_{j\not\in S})$. \begin{thm}[Equivariant Chern numbers of complex Grassmannians, \cite{Tu10} pp.\,21 Prop\,23]\label{thm:EquivChernNum}
Using the ABBV localization formula \ref{ABBV}, equivariant Chern numbers of complex Grassmannians can be given as
\begin{align*}
\int_{G_k(\mathbb{C}^n)} (c^T)^I
& = \sum_{S} \frac{((c_1^T)^{i_1}\cdots(c_k^T)^{i_k})|_S}{e^T_S} \\
& = \sum_{S} \frac{e^{i_1}_1(\alpha_{i\in S})\cdots e^{i_k}_k(\alpha_{i\in S})}{\prod_{i\in S}\prod_{j\not\in S} (\alpha_j - \alpha_i)}\in \mathbb{Q}[\alpha_1,\ldots,\alpha_n].
\end{align*}
where the sum is taken for all $k$-element subsets $S \subset \{1,\ldots,n\}$. When the cohomological degree of a characteristic polynomial matches with the dimension of a Grassmannian, then its equivariant integration results in a constant, i.e. an ordinary Chern number. \end{thm}
\begin{rmk}
By substituting any $\alpha_i=a_i \in \mathbb{R}$ such that $a_i\not =0, a_i \not = a_j$ into the above fraction, we can evaluate the equivariant Chern numbers. For example, a good choice will be $\alpha_i=i,\forall i$. If the cohomological degree of a characteristic polynomial does not match with the dimension of a Grassmannian, then such evaluation will be zero. The interesting case is when the degree matches the dimension. \end{rmk}
\begin{cor}\label{thm:OrdChernNum}
When the cohomological degree of a characteristic polynomial matches with the dimension of a Grassmannian, we then get a formula for the ordinary Chern numbers :
\[
\int_{G_k(\mathbb{C}^n)} c^I = \sum_{S} \frac{e^{i_1}_1(S)\cdots e^{i_k}_k(S)}{\prod_{i\in S}\prod_{j\not\in S} (j - i)}\in \mathbb{Q}
\]
where the sum is taken for all $k$-element subsets $S \subset \{1,\ldots,n\}$. \end{cor}
\begin{rmk}
The above characteristic numbers are with respect to the characteristic classes of the canonical bundle and complementary bundle, not the tangent bundle. However,
\[
c^T(T(G_k(\mathbb{C}^n)))|_S=c^T(\oplus_{i\in S} \oplus_{j\not\in S} (\mathbb{C}_i^* \otimes \mathbb{C}_j))=\prod_{i\in S}\prod_{j\not\in S} (1+\alpha_j - \alpha_i).
\]
We can also use the ABBV formula to calculate the equivariant (and ordinary) characteristic numbers of the tangent bundle. \end{rmk}
\begin{rmk}
The equivariant Chern classes $(c^T)^I$ are integral. Moreover, the canonical classes $\tau_S$ are actually the equivariant Schubert classes, hence also integral. Therefore, the coefficients $N,K,\bar{K}$ and characteristic numbers $\int (c^T)^I,\int c^I$ are all integral. \end{rmk}
\vskip 20pt \section{Equivariant cohomology rings of real Grassmannians} \vskip 15pt In this section, we give the GKM description and Leray-Borel description of equivariant cohomology rings of real Grassmannians, together with the canonical basis and characteristic basis of the additive structure. We use the notation $G_k(\mathbb{R}^n)$ for the Grassmannian of $k$-dimensional real subspaces in $\mathbb{R}^n$.
The dimension of $G_k(\mathbb{R}^n)$ is $k(n-k)$. Therefore, $G_{2k}(\mathbb{R}^{2n}), G_{2k}(\mathbb{R}^{2n+1}), G_{2k+1}(\mathbb{R}^{2n+1})$ are even dimensional, but $G_{2k+1}(\mathbb{R}^{2n+2})$ is odd dimensional. Moreover, the real Grassmannians $G_k(\mathbb{R}^n)$ differ from each other on Poincar\'e series and orientability according to the parities of $k$ and $n$, as shown by Casian and Kodama:
\begin{thm}[Poincar\'e series of real Grassmannians, \cite{CK} pp.\,11, Thm\,5.1]\label{thm:Poinc}
The relations between Poincar\'e series of real Grassmannians and complex Grassmannians are given as:
\begin{align*}
P_{G_{2k}(\mathbb{R}^{2n})}(t)=P_{G_{2k}(\mathbb{R}^{2n+1})}(t)&=P_{G_{2k+1}(\mathbb{R}^{2n+1})}(t)=P_{G_{k}(\mathbb{C}^{n})}(t^2)\\
P_{G_{2k+1}(\mathbb{R}^{2n+2})}(t)&=(1+t^{2n+1})P_{G_{k}(\mathbb{C}^{n})}(t^2).
\end{align*} \end{thm}
\begin{rmk}
The Poincar\'e series of complex Grassmannian is (see \cite{BT82} pp.\,292 Prop\,23.1)
\[
P_{G_{k}(\mathbb{C}^{n})}(t)=\frac{(1-t^2)\cdots(1-t^{2n})}{(1-t^2)\cdots(1-t^{2k})(1-t^2)\cdots(1-t^{2(n-k)})}.
\]
Using the relations between Poincar\'e series, we see that $G_{2k}(\mathbb{R}^{2n})$ and $G_{2k+1}(\mathbb{R}^{2n+2})$ have non-zero top Betti numbers, hence orientable; however, $G_{2k}(\mathbb{R}^{2n+1})$ and $G_{2k+1}(\mathbb{R}^{2n+1})$ have zero top Betti numbers, hence non-orientable. \end{rmk}
\subsection{GKM description of real Grassmannians} Similar to the case of complex Grassmannians, we will show the real Grassmannians also have appropriate torus actions that are equivariantly formal and GKM.
First, we specify the torus actions on real Grassmannians. Write the coordinates on $\mathbb{R}^{2n}$ as $(x_1,y_1,\ldots,x_n,y_n)$. Let $T^n$ act on $\mathbb{R}^{2n},\mathbb{R}^{2n+1},\mathbb{R}^{2n+2}$ so that the $i$-th $S^1$-component of $T^n$ exactly rotates the $i$-th pair of real coordinates $(x_{i},y_{i})$ and leaves the remaining coordinates free, hence we can write $\mathbb{R}^{2n}=\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]}$, $\mathbb{R}^{2n+1}=(\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}_0$ and $\mathbb{R}^{2n+2}=(\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}^2_0$ for their decompositions into weighted subspaces, where $[\alpha_i] \in \mathfrak{t}_\mathbb{Z}^*/\pm 1$. These actions induce $T^n$ actions on $G_{2k}(\mathbb{R}^{2n})$, $G_{2k}(\mathbb{R}^{2n+1})$, $G_{2k+1}(\mathbb{R}^{2n+1})$ and $G_{2k+1}(\mathbb{R}^{2n+2})$.
Since there are natural $T^n$-diffeomorphisms $G_{2k}(\mathbb{R}^{2n+1})\cong G_{2n-2k+1}(\mathbb{R}^{2n+1})$ identifying the second and the third types of real Grassmannians, in many discussions we will only consider the three cases of $G_{2k}(\mathbb{R}^{2n})$, $G_{2k}(\mathbb{R}^{2n+1})$ and $G_{2k+1}(\mathbb{R}^{2n+2})$.
\subsubsection{Fixed points} Similar to the observation in the case of complex Grassmannians, the $T^n$-fixed points of real Grassmannians are exactly some appropriate dimensional sub-representations of the ambient representations. The verification of sub-representations of $\mathbb{R}^{2n}=\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]}$, $\mathbb{R}^{2n+1}=(\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}_0$ for the real Grassmannians $G_{2k}(\mathbb{R}^{2n})$ and $G_{2k}(\mathbb{R}^{2n+1})$ are straightforward. Let's focus on the $2k+1$ dimensional sub-representations of $\mathbb{R}^{2n+2}=(\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}^2_0$. Notice that the sub-representation is odd dimensional, hence must have exactly one dimension in the part of trivial representation, therefore has the form $(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus L_0$ where $S$ is a $k$-element subset of $\{1,2,\ldots,n\}$ and $L_0\in \mathbb{P}(\mathbb{R}^2_0)$. For each $k$-element subset $S$, the connected component $C_S=\{(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus L_0 \mid L_0\in \mathbb{P}(\mathbb{R}^2_0)\}\cong \mathbb{R} P^1$ gives a fixed circle isolated from the other fixed circles. This gives all the fixed points of the $T^n$ action on $G_{2k+1}(\mathbb{R}^{2n+2})$.
\begin{rmk}
Because of the one-to-one correspondence between a $k$-element subset $S \subset \{1,\ldots,n\}$ with a fixed point or circle, sometimes we will use $S$ directly to mean a fixed point or circle. \end{rmk}
\subsubsection{Isotropy weights}\label{subsubsec:IsoWeights} Fixing a $k$-element subset $S$, let's describe the tangent spaces at the fixed points in the three cases of real Grassmannians. \begin{enumerate}
\item The tangent space at $\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]} \in G_{2k}(\mathbb{R}^{2n})$ is
\[
Hom_\mathbb{R}\Big(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]}, \oplus_{j \not\in S} \mathbb{R}^2_{[\alpha_j]}\Big) \cong (\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})^* \otimes_\mathbb{R} (\oplus_{j \not\in S} \mathbb{R}^2_{[\alpha_j]})\cong \oplus_{i \in S} \oplus_{j \not \in S} \big((\mathbb{R}^2_{[\alpha_i]})^* \otimes_\mathbb{R} \mathbb{R}^2_{[\alpha_j]}\big).
\]
\item The tangent space at $\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]} \in G_{2k}(\mathbb{R}^{2n+1})$ is
\[
Hom_\mathbb{R}\Big(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]}, (\oplus_{j \not\in S} \mathbb{R}^2_{[\alpha_j]})\oplus \mathbb{R}_0\Big) \cong \Big(\oplus_{i \in S} \oplus_{j \not \in S} \big((\mathbb{R}^2_{[\alpha_i]})^* \otimes_\mathbb{R} \mathbb{R}^2_{[\alpha_j]}\big)\Big)\oplus \oplus_{i \in S} (\mathbb{R}^2_{[\alpha_i]})^*.
\]
\item The tangent space at $(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus L_0 \in G_{2k+1}(\mathbb{R}^{2n+2})$, where $L_0 \in \mathbb{P}(\mathbb{R}^2_0)$ has a $L_0^\bot \in \mathbb{P}(\mathbb{R}^2_0)$ such that $L_0 \oplus L_0^\bot \cong \mathbb{R}^2_0$, is
\begin{align*}
&Hom_\mathbb{R}\Big((\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus L_0, (\oplus_{j \not\in S} \mathbb{R}^2_{[\alpha_j]})\oplus L_0^\bot\Big)\\
\cong& \Big(\oplus_{i \in S} \oplus_{j \not \in S} \big((\mathbb{R}^2_{[\alpha_i]})^* \otimes_\mathbb{R} \mathbb{R}^2_{[\alpha_j]}\big)\Big) \oplus \Big( \oplus_{i \in S} (\mathbb{R}^2_{[\alpha_i]})^*\otimes_\mathbb{R} L_0^\bot \Big) \oplus \Big( \oplus_{j \not\in S} L_0^*\otimes_\mathbb{R} \mathbb{R}^2_{[\alpha_j]} \Big) \\
&\oplus\Big( L_0^* \otimes_\mathbb{R} L_0^\bot \Big)
\end{align*}
among which the first three terms and the fourth term give respectively the normal space and tangent space of the fixed circle $C_S=\{(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus L_0 \mid L_0\in \mathbb{P}(\mathbb{R}^2_0)\}$. \end{enumerate} The isotropy weights are then determined by the following simple lemma: \begin{lem}
The weights of the tensor product $(\mathbb{R}^2_{[\alpha_i]})^* \otimes_\mathbb{R} \mathbb{R}^2_{[\alpha_j]}$ are $[\alpha_j-\alpha_i],[\alpha_j+\alpha_i] \in \mathfrak{t}_\mathbb{Z}^*/\pm 1$. \end{lem} \begin{proof}
The $T$-action on the dual space $(\mathbb{R}^2_{[\alpha_i]})^*$ is defined in an invariant way so that for $t \in T$, $l \in (\mathbb{R}^2_{[\alpha_i]})^*$, $v \in \mathbb{R}^2_{[\alpha_i]}$, and if we denote $\langle l,v\rangle$ as the natural pairing, we should have
$\langle t\cdot l,t\cdot v\rangle=\langle l,v\rangle$ or equivalently, $(t\cdot l)(v) = l(t^{-1}\cdot v)$. Notice that only the $i$-th and $j$-th $S^1$-component of $T^n$ have non-trivial actions on $(\mathbb{R}^2_{[\alpha_i]})^*$ or $\mathbb{R}^2_{[\alpha_j]}$, let $e^{\sqrt{-1}\theta_i} \in S^1_i$, $e^{\sqrt{-1}\theta_j} \in S^1_j$, and write elements of $(\mathbb{R}^2_{[\alpha_i]})^* \otimes_\mathbb{R} \mathbb{R}^2_{[\alpha_j]}$ as $2 \times 2$ matrices, then the $S^1_i \times S^1_j$ action on $(\mathbb{R}^2_{[\alpha_i]})^* \otimes_\mathbb{R} \mathbb{R}^2_{[\alpha_j]}$ can be given as:
\[
(e^{\sqrt{-1}\theta_i},e^{\sqrt{-1}\theta_j})\cdot
\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}=
\begin{pmatrix}
\cos \theta_j & -\sin \theta_j\\
\sin \theta_j & \cos \theta_j
\end{pmatrix}
\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}
\begin{pmatrix}
\cos \theta_i & \sin \theta_i\\
-\sin \theta_i & \cos \theta_i
\end{pmatrix}.
\]
Consider the following new basis of $(\mathbb{R}^2_{[\alpha_i]})^* \otimes_\mathbb{R} \mathbb{R}^2_{[\alpha_j]}$
\[
M_1 =
\begin{pmatrix}
1 & 0\\
0 & 1
\end{pmatrix} \quad
M_2 =
\begin{pmatrix}
0 & -1\\
1 & 0
\end{pmatrix} \quad
M_3 =
\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix} \quad
M_4 =
\begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix}.
\]
We get
\begin{align*}
(e^{\sqrt{-1}\theta_j}\cdot M_1,e^{\sqrt{-1}\theta_j}\cdot M_2) &= (M_1,M_2)
\begin{pmatrix}
\cos \theta_j & -\sin \theta_j\\
\sin \theta_j & \cos \theta_j
\end{pmatrix}\\
(e^{\sqrt{-1}\theta_i}\cdot M_1,e^{\sqrt{-1}\theta_i}\cdot M_2) &= (M_1,M_2)
\begin{pmatrix}
\cos \theta_i & \sin \theta_i\\
-\sin \theta_i & \cos \theta_i
\end{pmatrix}.
\end{align*}
In other words, $S^1_i \times S^1_j$ acts on $\mathbb{R} M_1 \oplus \mathbb{R} M_2$ with weight $[\alpha_j-\alpha_i]$.
Similarly, $S^1_i \times S^1_j$ acts on $\mathbb{R} M_3 \oplus \mathbb{R} M_4$ with weight $[\alpha_j+\alpha_i]$. \end{proof}
\subsubsection{$1$-skeleta} To begin with, let's work out the $1$-skeleton of the $T^2$-action on $G_2(\mathbb{R}^4)$. From the previous discussions, we know that there are two fixed points $\mathbb{R}^2_{[\alpha_1]},\mathbb{R}^2_{[\alpha_2]} \in G_2(\mathbb{R}^4)$, both have the same isotropy weights $[\alpha_2-\alpha_1]$ and $[\alpha_2+\alpha_1]$. Let $T_{\alpha_2-\alpha_1}$ be the subtorus of $T^2$ with Lie algebra annihilated by $\alpha_2-\alpha_1$, i.e. $T_{\alpha_2-\alpha_1}$ is the diagonal $\{(t,t) \in T^2\}$. Similarly, $T_{\alpha_2+\alpha_1}$, the subtorus with Lie algebra annihilated by $\alpha_2+\alpha_1$, is the anti-diagonal $\{(t,t^{-1}) \in T^2\}$.
Note that there is a natural diffeomorphism $\mathcal{F}:\mathbb{C}^2 \rightarrow \mathbb{R}^4$ by forgetting the complex structure. This induces an embedding $\mathbb{C} P^1 \hookrightarrow G_2(\mathbb{R}^4):L \mapsto \mathcal{F}(L)$ where $L$ is a complex line in $\mathbb{C}^2$ and $\mathcal{F}(L)$ its two dimensional real image in $\mathbb{R}^4$. Let $J:\mathbb{C}^2 \rightarrow \mathbb{C}^2: (z_1,z_2) \mapsto (z_1,\bar{z}_2)$ be the diffeomorphism with conjugation on the second variable. This also induces an embedding $\mathbb{C} P^1 \hookrightarrow G_2(\mathbb{R}^4):L \mapsto \mathcal{F}(J(L))$. We will denote the images of the two embeddings as $\mathbb{C} P^1$ and $\overline{\mathbb{C} P^1}$.
\begin{lem}
The fixed-point sets of $T_{\alpha_2-\alpha_1}$ and $T_{\alpha_2+\alpha_1}$ in $G_2(\mathbb{R}^4)$ are $\mathbb{C} P^1$ and $\overline{\mathbb{C} P^1}$ respectively, i.e. the $1$-skeleton of the $T^2$-action on $G_2(\mathbb{R}^4)$ is $\mathbb{C} P^1 \cup \overline{\mathbb{C} P^1}$ glued at the two $T^2$-fixed points $\mathbb{R}^2_{[\alpha_1]},\mathbb{R}^2_{[\alpha_2]} \in G_2(\mathbb{R}^4)$. \end{lem} \begin{proof}
Let $L_0=\mathbb{C} \oplus 0$ and $L_\infty = 0 \oplus \mathbb{C}$ be the two complex lines in $\mathbb{C}^2$, they are the two poles of both $\mathbb{C} P^1$ and $\overline{\mathbb{C} P^1}$, and are exactly the two $T^2$-fixed points $\mathbb{R}^2_{[\alpha_1]},\mathbb{R}^2_{[\alpha_2]} \in G_2(\mathbb{R}^4)$. The diagonal circle $T_{\alpha_2-\alpha_1}=\{(t,t) \in T^2\}$ fixes $\mathbb{C} P^1$ because $(t,t)\cdot [z_1,z_2]=[tz_1,tz_2]=[z_1,z_2]$ trivially, hence $\mathbb{C} P^1$ joins $\mathbb{R}^2_{[\alpha_1]}$ to $\mathbb{R}^2_{[\alpha_2]}$ with weight $[\alpha_2-\alpha_1]$. Similarly, $\overline{\mathbb{C} P^1}$ joins $\mathbb{R}^2_{[\alpha_1]}$ to $\mathbb{R}^2_{[\alpha_2]}$ with weight $[\alpha_2+\alpha_1]$. The $2$-spheres $\mathbb{C} P^1$ and $\overline{\mathbb{C} P^1}$ exhaust all the $T^2$-fixed points and the isotropy weights, therefore give the $1$-skeleton of the $T^2$-action on $G_2(\mathbb{R}^4)$. \end{proof}
Generally, let $T_{\alpha_j-\alpha_i}$ and $T_{\alpha_j+\alpha_i}$ be the subtori of $T^n$ with Lie algebras annihilated by $\alpha_j-\alpha_i$ and $\alpha_j+\alpha_i$ respectively. For the $T^n$-action on $\mathbb{R}^{2n}=\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]}$, the fixed-point sets of $T_{\alpha_j-\alpha_i}$ and $T_{\alpha_j+\alpha_i}$ on $G_2(\mathbb{R}^2_{[\alpha_i]}\oplus \mathbb{R}^2_{[\alpha_j]})$ are two $2$-spheres sharing the poles which are exactly the two $T^n$-fixed points $\mathbb{R}^2_{[\alpha_i]},\mathbb{R}^2_{[\alpha_j]} \in G_2(\mathbb{R}^2_{[\alpha_i]}\oplus \mathbb{R}^2_{[\alpha_j]})$. We will denote the $2$-spheres as $S^2_{[\alpha_j-\alpha_i]}$ and $S^2_{[\alpha_j+\alpha_i]}$ and keep in mind that every element $V$ in $S^2_{[\alpha_j-\alpha_i]}$ or $S^2_{[\alpha_j+\alpha_i]}$ is a $2$-plane in $\mathbb{R}^2_{[\alpha_i]}\oplus \mathbb{R}^2_{[\alpha_j]}$.
Now we are ready to describe the $1$-skeleta of the $T^n$ actions on the three types of real Grassmannians. Let $S$ be a $k$-element subset of $\{1,2,\ldots,n\}$, and $i\in S$, $j\not\in S$. \begin{enumerate}
\item For $G_{2k}(\mathbb{R}^{2n})$, the $T^n$-fixed point $V_S=\oplus_{i'\in S} \mathbb{R}^2_{[\alpha_{i'}]}$ is joined to $V_{(S\smallsetminus\{i\})\cup \{j\}}$ via $\{(\oplus_{i'\in S\smallsetminus\{i\}} \mathbb{R}^2_{[\alpha_{i'}]})\oplus V\mid V\in S^2_{[\alpha_j-\alpha_i]}\}\cong S^2$ of weight $[\alpha_j-\alpha_i]$ and also via $\{(\oplus_{i'\in S\smallsetminus\{i\}} \mathbb{R}^2_{[\alpha_{i'}]})\oplus V\mid V\in S^2_{[\alpha_j+\alpha_i]}\}\cong S^2$ of weight $[\alpha_j+\alpha_i]$.
\item For $G_{2k}(\mathbb{R}^{2n+1})$, the $T^n$-fixed point $V_S=\oplus_{i'\in S} \mathbb{R}^2_{[\alpha_{i'}]}$ is joined to $V_{(S\smallsetminus\{i\})\cup \{j\}}$ via $\{(\oplus_{i'\in S\smallsetminus\{i\}} \mathbb{R}^2_{[\alpha_{i'}]})\oplus V\mid V\in S^2_{[\alpha_j-\alpha_i]}\}\cong S^2$ of weight $[\alpha_j-\alpha_i]$ and also via $\{(\oplus_{i'\in S\smallsetminus\{i\}} \mathbb{R}^2_{[\alpha_{i'}]})\oplus V\mid V\in S^2_{[\alpha_j+\alpha_i]}\}\cong S^2$ of weight $[\alpha_j+\alpha_i]$. Moreover, $V_S$ is contained in $\{(\oplus_{i'\in S\smallsetminus\{i\}} \mathbb{R}^2_{[\alpha_{i'}]})\oplus V\mid V\in G_2(\mathbb{R}^2_{[\alpha_i]}\oplus \mathbb{R}_0)\}\cong \mathbb{R} P^2$ of weight $[\alpha_i]$ without other fixed points.
\item For $G_{2k+1}(\mathbb{R}^{2n+2})$, the $T^n$-fixed circle $C_S=\{(\oplus_{i'\in S} \mathbb{R}^2_{[\alpha_{i'}]})\oplus L_0 \mid L_0\in \mathbb{P}(\mathbb{R}^2_0)\}\cong \mathbb{R} P^1$ is joined to $C_{(S\smallsetminus\{i\})\cup \{j\}}$ via $\{(\oplus_{i'\in S\smallsetminus\{i\}} \mathbb{R}^2_{[\alpha_{i'}]})\oplus V \oplus L_0 \mid V\in S^2_{[\alpha_j-\alpha_i]}, L_0\in \mathbb{P}(\mathbb{R}^2_0)\}\cong S^2\times \mathbb{R} P^1$ with weight $[\alpha_j-\alpha_i]$ and also via $\{(\oplus_{i'\in S\smallsetminus\{i\}} \mathbb{R}^2_{[\alpha_{i'}]})\oplus V \oplus L_0 \mid V\in S^2_{[\alpha_j+\alpha_i]}, L_0\in \mathbb{P}(\mathbb{R}^2_0)\}\cong S^2\times \mathbb{R} P^1$ with weight $[\alpha_j+\alpha_i]$. Moreover, $C_S$ is contained in $\{(\oplus_{i'\in S\smallsetminus\{i\}} \mathbb{R}^2_{[\alpha_{i'}]})\oplus W \mid W\in G_3(\mathbb{R}^2_{[\alpha_{i}]}\oplus \mathbb{R}^2_0)\}\cong \mathbb{R} P^3$ and $\{(\oplus_{i'\in S} \mathbb{R}^2_{[\alpha_{i'}]})\oplus L \mid L\in \mathbb{P}(\mathbb{R}^2_{[\alpha_j]}\oplus\mathbb{R}^2_0)\}\cong \mathbb{R} P^3$ of weights $[\alpha_i]$ and $[\alpha_j]$ respectively without other fixed points. \end{enumerate}
\subsubsection{GKM graphs of real Grassmannians} Since $G_{2k}(\mathbb{R}^{2n}), G_{2k}(\mathbb{R}^{2n+1}), G_{2k+1}(\mathbb{R}^{2n+1})$ are even dimensional, but $G_{2k+1}(\mathbb{R}^{2n+2})$ is odd dimensional, we will construct GKM graphs according to the parity of dimensions.
\begin{exm}
We give some examples of GKM graphs for $G_k(\mathbb{R}^n)$ when $k$ or $n$ is small.
\begin{enumerate}
\item $\mathbb{R} P^{2n}$ as $G_{1}(\mathbb{R}^{2n+1})$ or $G_{2n}(\mathbb{R}^{2n+1})$
\begin{figure}
\caption{Complete GKM graph for $\mathbb{R} P^{2n}$}
\caption{Effective GKM graph for $\mathbb{R} P^{2n}$}
\caption{GKM graphs for $\mathbb{R} P^{2n}$}
\end{figure}
\item $\mathbb{R} P^{2n+1}$ as $G_{1}(\mathbb{R}^{2n+2})$ or $G_{2n+1}(\mathbb{R}^{2n+2})$
\begin{figure}
\caption{GKM graph for $\mathbb{R} P^{2n+1}$}
\caption{Condensed GKM graph for $\mathbb{R} P^{2n+1}$}
\caption{GKM graphs for $\mathbb{R} P^{2n+1}$}
\end{figure}
\item $G_{2}(\mathbb{R}^{4}),G_{2}(\mathbb{R}^{5}),G_{3}(\mathbb{R}^{5})$ as $G_{2k}(\mathbb{R}^{2n}), G_{2k}(\mathbb{R}^{2n+1}), G_{2k+1}(\mathbb{R}^{2n+1})$ when $k=1,n=2$.
\begin{figure}
\caption{GKM graph for $G_{2}(\mathbb{R}^{4})$}
\caption{Condensed GKM graph for $G_{2}(\mathbb{R}^{4})$}
\caption{GKM graphs for $G_{2}(\mathbb{R}^{4})$}
\end{figure}
\begin{figure}
\caption{Complete GKM graph}
\caption{Effective GKM graph}
\caption{Condensed GKM graph}
\caption{GKM graphs for $G_{2}(\mathbb{R}^{5})$}
\end{figure}
\begin{figure}
\caption{Complete GKM graph}
\caption{Effective GKM graph}
\caption{Condensed GKM graph}
\caption{GKM graphs for $G_{3}(\mathbb{R}^{5})$}
\end{figure}
\item $G_{3}(\mathbb{R}^{6})$ as $G_{2k+1}(\mathbb{R}^{2n+2})$ when $k=1,n=2$.
\begin{figure}
\caption{GKM graph for $G_{3}(\mathbb{R}^{6})$}
\caption{Condensed GKM graph for $G_{3}(\mathbb{R}^{6})$}
\caption{GKM graphs for $G_{3}(\mathbb{R}^{6})$}
\end{figure}
\end{enumerate} \end{exm}
\begin{rmk}
The graphs of $\mathbb{R} P^{2n}$ and $G_2(\mathbb{R}^5)$ have appeared in Goertsches\&Mare \cite{GM14}. \end{rmk}
\subsubsection{Formality, cohomology and canonical basis of real Grassmannians} We have given the $1$-skeleta and GKM graphs for real Grassmannians $G_k(\mathbb{R}^n)$ under appropriate torus actions. To apply the GKM-type theorems in even and odd dimensions, we still need to verify that those torus actions on $G_k(\mathbb{R}^n)$ are equivariantly formal.
\begin{prop}[Equivariant formality of torus actions on real Grassmannians]
The total Betti numbers of $G_k(\mathbb{R}^n)$ and of its fixed-point set are equal:
\begin{enumerate}
\item For the $T^n$-actions on $G_{2k}(\mathbb{R}^{2n}), G_{2k}(\mathbb{R}^{2n+1}), G_{2k+1}(\mathbb{R}^{2n+1})$, the isolated fixed points $V_S$ are all parametrized by $\mathcal{S}=\{S\subseteq\{1,2,\ldots,n\} \mid \#S=k\}$, and we have
\[
\sum \mathrm{dim}\,H^*(G_{2k}(\mathbb{R}^{2n})) = \sum \mathrm{dim}\,H^*(G_{2k}(\mathbb{R}^{2n+1})) = \sum \mathrm{dim}\,H^*(G_{2k+1}(\mathbb{R}^{2n+1})) = \# \mathcal{S} = \binom{n}{k}.
\]
\item For the $T^n$-action on $G_{2k+1}(\mathbb{R}^{2n+2})$, the isolated fixed circles $C_S$ are also indexed on $\mathcal{S}=\{S\subseteq\{1,2,\ldots,n\} \mid \#S=k\}$, and we have
\[
\sum\mathrm{dim}\,H^*(G_{2k+1}(\mathbb{R}^{2n+1})) = \# \mathcal{S} \cdot \sum\mathrm{dim}\,H^*(S^1) = 2\binom{n}{k}.
\]
\end{enumerate}
Therefore, the torus actions on $G_k(\mathbb{R}^n)$ are equivariantly formal. \end{prop} \begin{proof}
The verification is based on the equivalence (6) of Theorem \ref{thm:formal}. The total Betti numbers of $G_k(\mathbb{R}^n)$ can be calculated from the Casian-Kodama formula in Theorem \ref{thm:Poinc} by substituting $t=1$ in the Poincar\'e series.
\begin{align*}
\sum \mathrm{dim}\,H^*(G_{2k}(\mathbb{R}^{2n})) = \sum \mathrm{dim}\,H^*(G_{2k}(\mathbb{R}^{2n+1})) &= \sum \mathrm{dim}\,H^*(G_{2k+1}(\mathbb{R}^{2n+1})) = \sum \mathrm{dim}\,H^*(G_{k}(\mathbb{C}^{n}))\\
\sum \mathrm{dim}\,H^*(G_{2k+1}(\mathbb{R}^{2n+2})) &= 2\sum \mathrm{dim}\,H^*(G_{k}(\mathbb{C}^{n})).
\end{align*}
On the other hand, by the formality of the $T^n$-action on $G_{k}(\mathbb{C}^{n})$, which also has isolated points parametrized by $\mathcal{S}$, we have
\[
\sum \mathrm{dim}\,H^*(G_{k}(\mathbb{C}^{n})) = \# \mathcal{S} = \binom{n}{k}.
\]
Therefore, total Betti numbers of $G_k(\mathbb{R}^n)$ and of its fixed-point set are equal, and the torus actions on $G_k(\mathbb{R}^n)$ are equivariantly formal. \end{proof}
With the verifications of GKM conditions and equivariant formality, we can give the GKM description of the torus actions on $G_k(\mathbb{R}^n)$ by applying the generalized GKM-type Theorems \ref{thm:EvenGKM} and \ref{thm:OddGKM} in even and odd dimensions.
\begin{thm}[GKM description of equivariant cohomology of real Grassmannians]\label{thm:GKMrealGrass}
Let $\mathcal{S}$ be the collection of $k$-element subsets of $\{1,2,\ldots,n\}$.
\begin{enumerate}
\item For even dimensional Grassmannians $G_{2k}(\mathbb{R}^{2n}), G_{2k}(\mathbb{R}^{2n+1}), G_{2k+1}(\mathbb{R}^{2n+1})$ with $T^n$-actions, they have the same equivariant cohomology
\[
\big\{f:\mathcal{S}\rightarrow \mathbb{Q}[\alpha_1,\ldots,\alpha_n] \mid f_{S} \equiv f_{S'} \mod \alpha^2_j-\alpha^2_i \quad \text{for $S,S' \in \mathcal{S}$ with $S\cup\{j\}=S'\cup\{i\}$}\big\}.
\]
\item For odd dimensional Grassmannian $G_{2k+1}(\mathbb{R}^{2n+2})$ with $T^n$-action, an element of the equivariant cohomology is a set of polynomial pairs $(f_S, g_S \theta)$ to each $\circ$-vertex $S$ where $\theta$ is the unit volume form of $S^1$ such that
\begin{enumerate}
\item $g_{S} \equiv 0 \mod \prod_{i=1}^{n}\alpha_i$\quad for every $S$
\item $f_{S} \equiv f_{S'} , \quad g_{S} \equiv g_{S'}\mod \alpha^2_j-\alpha^2_i$\quad for $S,S' \in \mathcal{S}$ with $S\cup\{j\}=S'\cup\{i\}$.
\end{enumerate}
\end{enumerate} \end{thm}
\begin{rmk}
For convenience, we will write an element $f \in H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}))$ as $(f_S)_{S \in \mathcal{S}}$ and an element $(f,g\theta) \in H^*_{T^n}(G_{2k+1}(\mathbb{R}^{2n+2}))$ as $(f_S+g_S\theta)_{S \in \mathcal{S}}$, which are understood as tuples indexed with respect to $S \in \mathcal{S}$. \end{rmk}
\begin{rmk}
In the $1$-skeleton of the odd dimensional Grassmannian $G_{2k+1}(\mathbb{R}^{2n+2})$, every $\mathbb{R} P^3_{[\alpha_i]}$ containing a unique fixed circle $C_S$ contributes a relation $g_{S} \equiv 0 \mod \alpha_i$; every $S^2\times \mathbb{R} P^1$ with weight $\alpha_j\pm \alpha_i$ and two fixed circles $C_S,C_{S'}$ contributes two relations $f_{S} \equiv f_{S'} , g_{S} \equiv g_{S'}\mod \alpha_j\pm\alpha_i$. These simple components in $1$-skeleton resolve the sign issues in odd dimensional GKM-type Theorem\,\ref{thm:OddGKM}. \end{rmk}
\begin{rmk}
Note that in the above description, we have condensed some congruence relations because $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$ is a unique-factorization domain.
\begin{align*}
\begin{cases}
f_{S} \equiv f_{S'} \mod \alpha_j-\alpha_i\\
f_{S} \equiv f_{S'} \mod \alpha_j+\alpha_i
\end{cases}
&\Longleftrightarrow \quad f_{S} \equiv f_{S'} \mod \alpha^2_j-\alpha^2_i\\
\begin{cases}
g_{S} \equiv 0 \mod \alpha_1\\
\vdots \quad \qquad \vdots \quad \qquad \vdots\\
g_{S} \equiv 0 \mod \alpha_n
\end{cases}
&\Longleftrightarrow \quad g_{S} \equiv 0 \mod \prod_{i=1}^{n}\alpha_i.
\end{align*} \end{rmk}
Notice the similarity among the GKM descriptions of the even and odd dimensional real Grassmannians and the complex Grassmannians, we have \begin{thm}[Relations among equivariant cohomology of real and complex Grassmannians]\label{thm:AllGrass}
The relations between the equivariant cohomology of even, odd dimensional real Grassmannians and complex Grassmannians are
\begin{enumerate}
\item There are a series of $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-algebra isomorphisms:
\[
H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}))\cong H^*_{T^n}(G_{2k}(\mathbb{R}^{2n+1}))\cong H^*_{T^n}(G_{2k+1}(\mathbb{R}^{2n+1})).
\]
\item There is an element $r^T \in H^{2n+1}_{T^n}(G_{2k+1}(\mathbb{R}^{2n+2}))$ such that $(r^T)^2=0$, and there is a $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-algebra isomorphism
\[
H^*_{T^n}(G_{2k+1}(\mathbb{R}^{2n+2})) \cong H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}))[r^T]/(r^T)^2.
\]
\item There is a $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-algebra monomorphism:
\[
H^*_{T^n}(G_{2k}(\mathbb{R}^{2n})) \hookrightarrow H^*_{T^n}(G_{k}(\mathbb{C}^{n})).
\]
\end{enumerate} \end{thm} \begin{proof} All the Grassmannians with $T^n$-action are modelled on the same Johnson graph $J(n,k)$ with slightly different congruence relations. \begin{enumerate}
\item This is the part\,(1) of Theorem\,\ref{thm:GKMrealGrass}.
\item From Theorem\,\ref{thm:GKMrealGrass}, the GKM descriptions of even and odd dimensional real Grassmannians have the same congruence relations on the $f_S$ polynomials:
\[
f_{S} \equiv f_{S'}\mod \alpha^2_j-\alpha^2_i \quad \text{for $S,S' \in \mathcal{S}$ with $S\cup\{j\}=S'\cup\{i\}$}.
\]
But the odd dimensional real Grassmannian has extra part of $g_S \theta$ with congruence relations:
\begin{enumerate}
\item $g_{S} \equiv 0 \mod \prod_{i=1}^{n}\alpha_i$\quad for every $S$
\item $g_{S} \equiv g_{S'}\mod \alpha^2_j-\alpha^2_i$\quad for $S,S' \in \mathcal{S}$ with $S\cup\{j\}=S'\cup\{i\}$.
\end{enumerate}
The first set of congruence relations means that
\[g_{S}=\big(\prod_{i=1}^{n}\alpha_i\big) \cdot h_{S}\]
for a polynomial $h_{S}\in \mathbb{Q}[\alpha_1,\ldots,\alpha_n]$ and for every $S$. Substitute into the second set of congruence relations, and note that $\prod_{i=1}^{n}\alpha_i$ is coprime with $\alpha^2_j-\alpha^2_i$, then we get
\[
h_{S} \equiv h_{S'}\mod \alpha^2_j-\alpha^2_i \quad \text{ for $S,S' \in \mathcal{S}$ with $S\cup\{j\}=S'\cup\{i\}$}
\]
exactly the same as the congruence relations on the $f_S$ polynomials. Denote
\[
r^T=\big((\prod_{i=1}^{n}\alpha_i) \theta\big)_{S\in \mathcal{S}}
\]
which has $(r^T)^2=0$ because $\theta$ is the unit volume form of $S^1$, and has degree $2n+1$ because each $\alpha_i$ is of degree $2$ in cohomology. Then we can write
\[
(f_S+g_S\theta)_{S\in \mathcal{S}} = (f_S)_{S\in \mathcal{S}} + r^T\cdot (h_S)_{S\in \mathcal{S}}.
\]
This establishes the bijection
\[
H^*_{T^n}(G_{2k+1}(\mathbb{R}^{2n+2})) \cong H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}))[r^T]/(r^T)^2
\]
which can be easily verified to be a $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-algebra isomorphism.
\item From Theorem \ref{thm:GKMrealGrass}, the GKM description of even dimensional real Grassmannians has the congruence relations on the $f_S$ polynomials:
\[
f_{S} \equiv f_{S'}\mod \alpha^2_j-\alpha^2_i \quad \text{for $S,S' \in \mathcal{S}$ with $S\cup\{j\}=S'\cup\{i\}$}
\]
which automatically satisfy the congruence relations on the $f_S$ polynomials for the complex Grassmannians in Theorem \ref{thm:GKMcplxGrass}:
\[
f_{S} \equiv f_{S'}\mod \alpha_j-\alpha_i \quad \text{for $S,S' \in \mathcal{S}$ with $S\cup\{j\}=S'\cup\{i\}$}.
\]
This establishes the injection
\[
H^*_{T^n}(G_{2k}(\mathbb{R}^{2n})) \hookrightarrow H^*_{T^n}(G_{k}(\mathbb{C}^{n}))
\]
which is also easy to verify as a $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-algebra monomorphism. \end{enumerate} \end{proof}
\begin{rmk}
Those $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-algebra isomorphisms in Theorem\,\ref{thm:AllGrass} give ring isomorphisms among ordinary cohomology of real Grassmannians. But the ordinary version $H^*(G_{2k}(\mathbb{R}^{2n})) \rightarrow H^*(G_{k}(\mathbb{C}^{n}))$ is not injective simply due to fact that $G_{2k}(\mathbb{R}^{2n})$ is of dimension $4k(n-k)$, twice the real dimension of $G_{k}(\mathbb{C}^{n})$. \end{rmk}
\begin{thm}[Canonical basis of even dimensional real Grassmannians]\label{thm:RealSchub}
There is a self-indexing Morse function on $\mathcal{S}$
\[
\psi: \mathcal{S} \longrightarrow \mathbb{R} : S \longmapsto 4(\sum_{i\in S} i) -2k(k+1)
\]
and a canonical class $\sigma_S \in H^{\psi(S)}_{T^n}(G_{2k}(\mathbb{R}^{2n}),\mathbb{Q})$ for each $S\in \mathcal{S}$ such that
\begin{enumerate}
\item $\sigma_S$ is supported upward, i.e. $\sigma_S(S')=0$ if $\psi(S')\leq \psi(S)$
\item $\sigma_S(S)=\prod' (\alpha^2_j - \alpha^2_i)$ where the product is taken over the weights at $S$ connecting to $S'$ with $\psi(S')<\psi(S)$
\end{enumerate}
Moreover, $\{\sigma_S, S\in \mathcal{S}\}$ give an additive $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-basis of $H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}),\mathbb{Q})$. \end{thm} \begin{proof}
By Theorem \ref{thm:AllGrass}, we can identify $H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}))$ with its embedded image in $H^*_{T^n}(G_{k}(\mathbb{C}^{n}))$. Recall from Theorem \ref{thm:CplxSchub} on the canonical classes of complex Grassmannian $G_{k}(\mathbb{C}^{n})$, we used the function $\phi=\frac{\psi}{2}$, hence both $\psi$ and $\phi$ define the same partial order on $\mathcal{S}$. Moreover, there is a basis $\tau_S$ of $H^*_{T^n}(G_{k}(\mathbb{C}^{n}))$, such that
\begin{enumerate}
\item $\tau_S$ is supported upward, i.e. $\tau_S(S')=0$ if $\phi(S')\leq \phi(S)$
\item $\tau_S(S)=\prod' (\alpha_j - \alpha_i)$ where the product is taken over the weights at $S$ connecting to $S'$ with $\phi(S')<\phi(S)$
\end{enumerate}
Let's introduce the ring homomorphism:
\[
Sq: \mathbb{Q}[\alpha_1,\ldots,\alpha_n] \rightarrow \mathbb{Q}[\alpha_1,\ldots,\alpha_n] : f(\alpha_1,\ldots,\alpha_n) \mapsto f(\alpha^2_1,\ldots,\alpha^2_n).
\]
If $f_S \equiv f_{S'} \mod \alpha_j-\alpha_i$, i.e. $f_S - f_{S'}$ is a multiple of $\alpha_j-\alpha_i$, then $Sq(f_S) - Sq(f_{S'})=Sq(f_S - f_{S'})$ is a multiple of $Sq(\alpha_j-\alpha_i)=\alpha^2_j-\alpha^2_i$, i.e. $Sq(f_S) \equiv Sq(f_{S'}) \mod \alpha^2_j-\alpha^2_i$. The homomorphism $Sq$ not only refines the congruence relations of $H^*_{T^n}(G_{k}(\mathbb{C}^{n}))$, but also has image in $H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}))$, i.e. $Sq(H^*_{T^n}(G_{k}(\mathbb{C}^{n}))) \subseteq H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}))$.
Now we can define $\sigma_S = Sq(\tau_S) \in H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}))$, and we see that this collection of classes satisfies the required properties of being supported upward and $\sigma_S(S)=\prod' (\alpha^2_j - \alpha^2_i)$ over weights at $S$ connecting to $S'$ with $\psi(S')<\psi(S)$.
According to Guillemin\&Zara (\cite{GZ03} pp.\,125, Remark of Thm\,2.1), $\{\sigma_S\}$ give an additive basis of $H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}))$. \end{proof}
Since we have proved $H^*_{T^n}(G_{2k+1}(\mathbb{R}^{2n+2})) \cong H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}))[r^T]/(r^T)^2$ in Theorem \ref{thm:AllGrass}, then \begin{cor}[Canonical basis of odd dimensional real Grassmannians]
$\sigma_S$ and $r^T\sigma_S$ give an additive $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-basis of $H^*_{T^n}(G_{2k+1}(\mathbb{R}^{2n+2}))$. \end{cor}
\begin{rmk}
In the case of complex Grassmannian $G_{k}(\mathbb{C}^{n})$, a subset $S\subseteq \{1,2,\ldots,n\}$ with elements $ i_1<i_2<\cdots<i_k$ corresponds to Schubert symbol $(i_1-1,i_2-2,\ldots,i_k-k)$; there could be correspondences for real Grassmannians
\begin{enumerate}
\item For even dimensional Grassmannians $G_{2k}(\mathbb{R}^{2n}),G_{2k}(\mathbb{R}^{2n+1})$, let $S$ consist of $ i_1<i_2<\cdots<i_k$, then the $T^n$-fixed point $\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]}$, with pivot positions $(2i_1-1,2i_1, 2i_2-1,2i_2,\ldots,2i_k-1,2i_k)$ in its reduced echelon form, will correspond to Schubert symbol $(2i_1-2,2i_1-2, 2i_2-4,2i_2-4,\ldots,2i_k-2k,2i_k-2k)$.
\item For even dimensional Grassmannian $G_{2k+1}(\mathbb{R}^{2n+1})$, the $T^n$-fixed point $\mathbb{R}_0 \oplus (\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})$, with pivot positions $(1,2i_1,2i_1+1, 2i_2,2i_2+1,\ldots,2i_k,2i_k+1)$ in its reduced echelon form, will also correspond to Schubert symbol $(2i_1-2,2i_1-2, 2i_2-4,2i_2-4,\ldots,2i_k-2k,2i_k-2k)$.
\item For odd dimensional Grassmannian $G_{2k+1}(\mathbb{R}^{2n+2})$, besides the above Schubert symbols $(2i_1-2,2i_1-2, 2i_2-4,2i_2-4,\ldots,2i_k-2k,2i_k-2k)$, there is the class $r^T$, which is conjectured by Casian\&Kodama \cite{CK} to be the Schubert class with the hook Young diagram $1^{2k}\times (2(n-k)+1)$ of symbol $(1,\ldots,1,2(n-k)+1)$ where there are $2k$ copies of $1$. Following this conjecture, we can guess that a class $r^T\sigma_S$ with $S$ given by $i_1<i_2<\cdots<i_k$, corresponds to the Schubert symbol $(2i_1-1,2i_1-1, 2i_2-3,2i_2-3,\ldots,2i_k-2k+1,2i_k-2k+1,2(n-k)+1)$.
\end{enumerate} \end{rmk}
Recall from Subsection \ref{subsec:BGKM} of the equivariant Littlewood-Richardson rule for complex Grassmannian $G_k(\mathbb{C}^n)$ \[ \tau_{S}\tau_{S'} =\sum_{S''}N_{S,S'}^{S''}\tau_{S''} \] where $N_{S,S'}^{S''}\in \mathbb{Q}[\alpha_1,\ldots,\alpha_n]$. If we apply the ring homomorphism $Sq$ on both sides, then we get: \begin{thm}[Equivariant Littlewood-Richardson coefficients for real Grassmannians]
The equivariant Littlewood-Richardson coefficients for real Grassmannian $G_{2k}(\mathbb{R}^{2n})$ satisfy
\[
\sigma_{S}\sigma_{S'} =\sum_{S''}Sq(N_{S,S'}^{S''})\sigma_{S''}
\]
where $Sq(N_{S,S'}^{S''}) \in \mathbb{Q}[\alpha^2_1,\ldots,\alpha^2_n]$ is obtained from $N_{S,S'}^{S''}$ by replacing $\alpha_i$ to be $\alpha_i^2$. \end{thm}
\begin{rmk} Since $Sq$ keeps constant term unchanged, the Littlewood-Richardson rules for ordinary cohomology of complex Grassmannian $G_k(\mathbb{C}^n)$ and real Grassmannian $G_{2k}(\mathbb{R}^{2n})$ are the same. \end{rmk}
\subsection{Leray-Borel description of real Grassmannians} Similar to Leray-Borel description of equivariant (ordinary) cohomology of complex Grassmannians using equivariant (ordinary) Chern classes, we will show there is a Leray-Borel description of equivariant (ordinary) cohomology of real Grassmannians using equivariant (ordinary) Pontryagin classes.
\subsubsection{Equivariant Pontryagin classes} The $T^n$ actions on $\mathbb{R}^{2n}=\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]}$, $\mathbb{R}^{2n+1}=(\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}_0$ and $\mathbb{R}^{2n+2}=(\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}^2_0$ induce actions on $G_{2k}(\mathbb{R}^{2n})$, $G_{2k}(\mathbb{R}^{2n+1})$, $G_{2k+1}(\mathbb{R}^{2n+1})$ and $G_{2k+1}(\mathbb{R}^{2n+2})$. These actions induce further actions on the canonical bundles $\gamma$ and complementary bundles $\bar{\gamma}$ over those Grassmannians. Then we can consider their equivariant Pontryagin classes $p^T=p^T(\gamma)$ and $\bar{p}^T=p^T(\bar{\gamma})$.
First, let's compute a warm-up example of equivariant Pontryagin classes.
\begin{lem}\label{lem:Pont}
The total equivariant Pontryagin class of the vector space $\mathbb{R}^2_{[\alpha]}$ with weight $[\alpha] \in \mathfrak{t}^*_\mathbb{Z}/\pm 1$ over a point is $1+\alpha^2$. \end{lem} \begin{proof}
Think of the elements of $\mathbb{R}^2_{[\alpha]}$ as $2\times 1$ column vectors. For a Lie algebra element $\xi \in \mathfrak{t}$, the action of its group element $\exp(\xi) \in T$ on $\mathbb{R}^2_{[\alpha]}$ is given as a $2\times 2$ real matrix
\[
\begin{pmatrix}
\cos(\alpha(\xi)) & -\sin(\alpha(\xi))\\
\sin(\alpha(\xi)) & \cos(\alpha(\xi))
\end{pmatrix}
\qquad
\textup{or}
\qquad
\begin{pmatrix}
\cos(\alpha(\xi)) & \sin(\alpha(\xi))\\
-\sin(\alpha(\xi)) & \cos(\alpha(\xi))
\end{pmatrix}.
\]
Tensoring $\mathbb{R}^2_{[\alpha]}$ over $\mathbb{R}$-coefficients with $\mathbb{C}$ means that we can treat the above real matrices as complex matrices. Since both of them have the same characteristic function $\lambda^2-2\cos(\alpha(\xi))\lambda+1=(\lambda-e^{\sqrt{-1}\alpha(\xi)})(\lambda-e^{-\sqrt{-1}\alpha(\xi)})$, the two real matrices have the same diagonalization over $\mathbb{C}$-coefficients:
\[
\begin{pmatrix}
e^{\sqrt{-1}\alpha(\xi)} & 0\\
0 & e^{-\sqrt{-1}\alpha(\xi)}
\end{pmatrix}
\]
i.e. the $T$-action on the complex vector space $\mathbb{R}^2_{[\alpha]}\otimes_\mathbb{R} \mathbb{C}$ has weights $\alpha$ and $-\alpha$. Therefore, $c^T(\mathbb{R}^2_{[\alpha]}\otimes_\mathbb{R} \mathbb{C})=(1-\alpha)(1+\alpha)=1-\alpha^2$. Following Milnor-Stasheff's convention of signs, we get $p^T(\mathbb{R}^2_{[\alpha]})=1+\alpha^2$. \end{proof}
Second, let's specify the equivariant Pontryagin classes of canonical bundles, complementary bundles and tangent bundles of real Grassmannians in GKM description at each fixed point or circle of the real Grassmannians.
\begin{prop}\label{prop:Pont}
For all the four real Grassmannians $G_{2k}(\mathbb{R}^{2n})$, $G_{2k}(\mathbb{R}^{2n+1})$, $G_{2k+1}(\mathbb{R}^{2n+1})$ and $G_{2k+1}(\mathbb{R}^{2n+2})$ with $T^n$-actions, the equivariant Pontryagin classes $p^T=p^T(\gamma)$ and $\bar{p}^T=p^T(\bar{\gamma})$ of the canonical bundle and complementary bundle localized at each fixed point or circle indexed as a $k$-element subset $S \in \{1,\ldots,n\}$ are
\begin{align*}
p^T|_S &= p^T(\gamma|_S)=\prod_{i\in S} (1+\alpha^2_i)\\
\bar{p}^T|_S &= p^T(\bar{\gamma}|_S)=\prod_{j\not\in S} (1+\alpha^2_j)
\end{align*}
with the relation $p^T\bar{p}^T = \prod_{i=1}^{n}(1+\alpha^2_i)$. The equivariant Pontryagin classes of the tangent bundles are given at each fixed point or circle $S$ as
\begin{align*}
p^T(TG_{2k}(\mathbb{R}^{2n}))|_S&=\prod_{i \in S}\prod_{j \not \in S} \big[(1+(\alpha_j-\alpha_i)^2)(1+(\alpha_j+\alpha_i)^2)\big]\\
p^T(TG_{2k}(\mathbb{R}^{2n+1}))|_S&=\prod_{i \in S}\prod_{j \not \in S} \big[(1+(\alpha_j-\alpha_i)^2)(1+(\alpha_j+\alpha_i)^2)\big] \prod_{i \in S} (1+\alpha_i^2)\\
p^T(TG_{2k+1}(\mathbb{R}^{2n+1}))|_S&=\prod_{i \in S}\prod_{j \not \in S} \big[(1+(\alpha_j-\alpha_i)^2)(1+(\alpha_j+\alpha_i)^2)\big] \prod_{j \not \in S} (1+\alpha_j^2)\\
p^T(TG_{2k+1}(\mathbb{R}^{2n+2}))|_S&=\prod_{i \in S}\prod_{j \not \in S} \big[(1+(\alpha_j-\alpha_i)^2)(1+(\alpha_j+\alpha_i)^2)\big] \prod_{i \in S} (1+\alpha_i^2) \prod_{j \not \in S} (1+\alpha_j^2).
\end{align*} \end{prop} \begin{proof}
For $G_{2k}(\mathbb{R}^{2n})$, at each fixed point $S$, we have $\gamma|_S=\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]}$ and $\bar{\gamma}|_S=\oplus_{j \not\in S} \mathbb{R}^2_{[\alpha_j]}$, and furthermore $\gamma|_S\oplus\bar{\gamma}|_S=\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]}$. The claimed expressions of the localized Pontryagin classes then follow from the Lemma \ref{lem:Pont}. The cases of $G_{2k}(\mathbb{R}^{2n+1})$, $G_{2k+1}(\mathbb{R}^{2n+1})$ and $G_{2k+1}(\mathbb{R}^{2n+2})$ are similar. For the Pontryagin classes of the tangent bundles localized at each fixed point or circle, we can apply Lemma \ref{lem:Pont} to the weight decompositions (see Subsubsection \ref{subsubsec:IsoWeights}) of tangent bundles at each fixed point or circle. \end{proof}
\subsubsection{Characteristic basis of real Grassmannians} Think of $p^T$ and $\bar{p}^T$ as elements of the embedded image of $H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}))$ in $H^*_{T^n}(G_{k}(\mathbb{C}^{n}))$ using GKM description on the Johnson graph $J(n,k)$. If we compare the localized expressions of $p^T$ and $\bar{p}^T$ with $c^T$ and $\bar{c}^T$ in Subsection \ref{subsec:BGKM}, we get the formula \[ p^T=Sq(c^T) \qquad \bar{p}^T=Sq(\bar{c}^T) \] where the homomorphism $Sq$ is defined in Theorem \ref{thm:RealSchub}.
Recall in Subsection \ref{subsec:BGKM}, we discussed the transformations $K,\bar{K}$ between the characteristic monomials $(c^T)^I=(c_1^T)^{i_1}\cdots(c_k^T)^{i_k}$ in Leray-Borel description and the canonical classes $\tau_S$ in GKM description: \begin{align*} (c^T)^I &= \sum_S K^I_S \tau_S\\ \tau_S &= \sum_I \bar{K}_I^S (c^T)^I \end{align*} where $K^I_S,\bar{K}_I^S \in \mathbb{Q}[\alpha_1,\ldots,\alpha_n]$. Apply the homomorphism $Sq$ and recall $\sigma_S=Sq(\tau_S)$ from Theorem \ref{thm:RealSchub}, then \begin{align*} (p^T)^I &= \sum_S Sq(K^I_S) \sigma_S\\ \sigma_S &= \sum_I Sq(\bar{K}_I^S) (p^T)^I \end{align*} where $Sq(K^I_S),Sq(\bar{K}_I^S) \in \mathbb{Q}[\alpha_1,\ldots,\alpha_n]$.
Since $\{\sigma_S\}$ give a basis of $H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}))$, the above transformations imply:
\begin{thm}[Equivariant characteristic basis of real Grassmannians]\label{thm:EquivCharReal}
The set of monomials $(p^T_1)^{r_1}(p^T_2)^{r_2}\cdots (p^T_k)^{r_k}$ satisfying the condition $\sum_{i=1}^{k} r_i \leq n-k$ forms an additive $H^*_T(pt)$-basis for $H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}))\cong H^*_{T^n}(G_{2k}(\mathbb{R}^{2n+1}))\cong H^*_{T^n}(G_{2k+1}(\mathbb{R}^{2n+1}))$. Together with the set of monomials $r^T\cdot(p^T_1)^{r_1}(p^T_2)^{r_2}\cdots (p^T_k)^{r_k}$, they form an additive $H^*_T(pt)$-basis for $H^*_{T^n}(G_{2k+1}(\mathbb{R}^{2n+2}))$. \end{thm}
Now we can give the Leray-Borel description for equivariant cohomology of real Grassmannians. \begin{thm}[Equivariant Leray-Borel description of even dimensional real Grassmannians]\label{thm:EquivBReal}
For the even dimensional real Grassmannians $G_{2k}(\mathbb{R}^{2n})$, $G_{2k}(\mathbb{R}^{2n+1})$ and $G_{2k+1}(\mathbb{R}^{2n+1})$ with $T^n$-actions, their equivariant cohomology is the same:
\[
H^*_T(G_{2k}(\mathbb{R}^{2n}),\mathbb{Q})\cong\frac{\mathbb{Q}[\alpha_1,\alpha_2,\ldots,\alpha_n][p^T_1,p^T_2,\ldots,p^T_k;\bar{p}^T_1,\bar{p}^T_2,\ldots,\bar{p}^T_{n-k}]}{p^T\bar{p}^T = \prod_{i=1}^{n}(1+\alpha^2_i)}.
\] \end{thm} \begin{proof}
Apply the $Sq$ to the generators and relations in the Leray-Borel description of $H^*_T(G_k(\mathbb{C}^n))$. \end{proof}
\begin{thm}[Equivariant Leray-Borel description of odd dimensional real Grassmannians]\label{thm:EquivBReal2}
For the odd dimensional real Grassmannian $G_{2k+1}(\mathbb{R}^{2n+2})$ with $T^n$-actions, the equivariant cohomology is:
\[
H^*_T(G_{2k+1}(\mathbb{R}^{2n+2}),\mathbb{Q})\cong\frac{\mathbb{Q}[\alpha_1,\alpha_2,\ldots,\alpha_n][p^T_1,p^T_2,\ldots,p^T_k;\bar{p}^T_1,\bar{p}^T_2,\ldots,\bar{p}^T_{n-k};r^T]}{p^T\bar{p}^T = \prod_{i=1}^{n}(1+\alpha^2_i),\,(r^T)^2=0}.
\] \end{thm}
For a $T^n$-equivariantly formal space $M$, we can recover the ordinary cohomology from the equivariant cohomology by $H^*(M,\mathbb{Q})=H_T^*(M,\mathbb{Q})\otimes_{\mathbb{Q}[\alpha_1,\ldots,\alpha_n]} \mathbb{Q}$ where $\mathbb{Q}$ has a $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-algebra structure from the constant-term morphism $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]\rightarrow \mathbb{Q}: f(\alpha_1,\ldots,\alpha_n)\mapsto f(0)$. Therefore, the above Theorem \ref{thm:EquivCharReal}, \ref{thm:EquivBReal}, \ref{thm:EquivBReal2} have ordinary versions by ignoring the $\alpha_i$.
Let $r\in H^{2n+1}(G_{2k+1}(\mathbb{R}^{2n+2}),\mathbb{Q})$ be the ordinary image of the $r^T\in H^{2n+1}_T(G_{2k+1}(\mathbb{R}^{2n+2}),\mathbb{Q})$.
\begin{cor}[Ordinary characteristic basis of real Grassmannians]
The set of monomials $(p_1)^{r_1}(p_2)^{r_2}\cdots (p_k)^{r_k}$ satisfying the condition $\sum_{i=1}^{k} r_i \leq n-k$ forms an additive basis for $H^*(G_{2k}(\mathbb{R}^{2n}))\cong H^*(G_{2k}(\mathbb{R}^{2n+1}))\cong H^*(G_{2k+1}(\mathbb{R}^{2n+1}))$. Together with the set of monomials $r\cdot(p_1)^{r_1}(p_2)^{r_2}\cdots (p_k)^{r_k}$, they form an additive basis for $H^*(G_{2k+1}(\mathbb{R}^{2n+2}))$. \end{cor}
\begin{cor}[Ordinary Leray-Borel description of real Grassmannians]
For the even dimensional real Grassmannians $G_{2k}(\mathbb{R}^{2n})$, $G_{2k}(\mathbb{R}^{2n+1})$ and $G_{2k+1}(\mathbb{R}^{2n+1})$, their cohomology is the same:
\[
\frac{\mathbb{Q}[p_1,p_2,\ldots,p_k;\bar{p}_1,\bar{p}_2,\ldots,\bar{p}_{n-k}]}{p\bar{p} = 1}.
\]
For the odd dimensional real Grassmannian $G_{2k+1}(\mathbb{R}^{2n+2})$, the cohomology is:
\[
\frac{\mathbb{Q}[p_1,p_2,\ldots,p_k;\bar{p}_1,\bar{p}_2,\ldots,\bar{p}_{n-k};r]}{p\bar{p} = 1,\, r^2=0}
\]
where $r\in H^{2n+1}(G_{2k+2}(\mathbb{R}^{2n+2}),\mathbb{Q})$ is the ordinary image of the $r^T\in H^{2n+1}_T(G_{2k+2}(\mathbb{R}^{2n+2}),\mathbb{Q})$. \end{cor}
\begin{rmk}
This explicit Leray-Borel description for the real Grassmannians is stated in Casian\&Kodama \cite{CK}. The even dimensional case is a special case of the Leray-Borel description, and the odd dimensional case is due to Takeuchi \cite{Ta62}. \end{rmk}
\begin{rmk}
For $n\leq 7$, the ordinary cohomology groups of $G_k(\mathbb{R}^n)$ in $\mathbb{Z}$ coefficients were computed by Jungkind \cite{Ju79}. \end{rmk}
\vskip 20pt \section{Equivariant cohomology rings of oriented Grassmannians} \vskip 15pt In this section, we give the GKM description and Leray-Borel description of equivariant cohomology rings of oriented Grassmannians, together with the characteristic basis of the additive structure. We use the notation $\tilde{G}_k(\mathbb{R}^n)$ for the Grassmannian of $k$-dimensional oriented subspaces in $\mathbb{R}^n$.
The Pl\"{u}cker embedding of an oriented Grassmannian can be given as follows: for $V\in \tilde{G}_k(\mathbb{R}^n)$, we can choose an ordered orthonormal basis $v_1,\ldots,v_k$ of $V$, then the well-defined wedge product $v_1 \wedge \cdots \wedge v_k \in \tilde{G}_1(\wedge^k \mathbb{R}^n)=S(\wedge^k \mathbb{R}^n)$ in the unit sphere of $\wedge^k \mathbb{R}^n$ gives the embedding $\tilde{G}_k(\mathbb{R}^n) \hookrightarrow S(\wedge^k \mathbb{R}^n)$.
Similar to the case of real Grassmannians, we can consider the $T^n$-action on $\mathbb{R}^{2n}=\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]}$, $\mathbb{R}^{2n+1}=(\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}_0$ and $\mathbb{R}^{2n+2}=(\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}^2_0$ for their decompositions into weighted subspaces, where $\alpha_1,\ldots,\alpha_n$ are the standard basis of $\mathfrak{t}_\mathbb{Z}^*$. These actions induce $T^n$ actions on $\tilde{G}_{2k}(\mathbb{R}^{2n})$, $\tilde{G}_{2k}(\mathbb{R}^{2n+1})$, $\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$ and $\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$. More specifically, each $t \in T$ maps $v_1 \wedge \cdots \wedge v_l$ where $l=2k,2k+1$, to $t\cdot v_1 \wedge \cdots \wedge t \cdot v_l$, and it is easy to check the map is independent from the choice of a positive orthonormal basis $v_1,\ldots,v_k$.
Also since there are natural $T^n$-diffeomorphisms $\tilde{G}_{2k}(\mathbb{R}^{2n+1})\cong \tilde{G}_{2n-2k+1}(\mathbb{R}^{2n+1})$ identifying the second and the third types of real Grassmannians, in some discussions we will only consider the three cases of $\tilde{G}_{2k}(\mathbb{R}^{2n})$, $\tilde{G}_{2k}(\mathbb{R}^{2n+1})$ and $\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$.
\subsection{Oriented Grassmannians as $2$-covers over real Grassmannians} There are natural $2$-coverings of oriented Grassmannians over real Grassmannians $\pi:\tilde{G}_k(\mathbb{R}^n)\rightarrow G_k(\mathbb{R}^n): v_1 \wedge \cdots \wedge v_k \mapsto \mathrm{Span}_\mathbb{R}(v_1,\ldots,v_k)$ which induces a pull-back morphism $\pi^*: H^*(G_k(\mathbb{R}^n))\rightarrow H^*(\tilde{G}_k(\mathbb{R}^n))$ between their cohomology. The non-trivial deck transformation is defined by reversing orientations $\rho: \tilde{G}_k(\mathbb{R}^n)\rightarrow\tilde{G}_k(\mathbb{R}^n):v_1 \wedge \cdots \wedge v_k \mapsto -(v_1 \wedge \cdots \wedge v_k)$ which induces an isomorphism $\rho^*: H^*(\tilde{G}_k(\mathbb{R}^n)) \rightarrow H^*(\tilde{G}_k(\mathbb{R}^n))$. Both $\pi$ and $\rho$ commute with the $T$-actions that we introduced on the oriented Grassmannians and real Grassmannians.
For covering maps between compact spaces, or equivalently for free actions of finite groups, there is a well-known fact relating their cohomology in rational coefficients: \begin{lem}
Let $\pi:X\rightarrow Y$ be a covering between compact topological spaces with a finite deck transformation group $G$ which also acts on the cohomology $H^*(X,\mathbb{Q})$. Then $\pi^*: H^*(Y,\mathbb{Q}) \rightarrow H^*(X,\mathbb{Q})$ is injective with image $H^*(X,\mathbb{Q})^G$. This conclusion is also true for equivariant cohomology if a torus $T$ acts on $X$ and commutes with the action of $G$. \end{lem} \begin{proof}
For a cocycle $c$ of $X$, the averaged cocycle $\frac{1}{|G|}\sum_{g\in G} gc$ is invariant under $G$-action, hence comes from a cocycle of $Y$. Consider the averaging map $\pi_*: H^*(X,\mathbb{Q}) \rightarrow H^*(Y,\mathbb{Q}):[c]\mapsto \frac{1}{|G|}[\sum_{g\in G} gc]$, then the composition $\pi_* \pi^*$ is the identity map on $H^*(Y,\mathbb{Q})$, hence $\pi^*$ is injective. Note that every cohomology class in $H^*(X,\mathbb{Q})^G$ can be represented by a $G$-invariant cocycle using the averaging method. This proves the image of $\pi^*$ is exactly $H^*(X,\mathbb{Q})^G$.
For the $T^n$-equivariant version, though the Borel construction $X\times_{T^n} (S^\infty)^n,Y\times_{T^n} (S^\infty)^n$ is not compact, we can apply the ordinary version of current Lemma to the compact approximations $X\times_{T^n} (S^N)^n, Y\times_{T^n} (S^N)^n$ for $N\rightarrow \infty$. \end{proof}
\begin{rmk}
For the averaging method to work, we can relax the $\mathbb{Q}$ coefficients to be any coefficient ring that contains $\frac{1}{|G|}$. In $\mathbb{R}$ coefficients, the ordinary and equivariant de Rham theory together with the averaging method give a proof without using compact approximations. \end{rmk}
Applying this Lemma to the oriented Grassmannians as $T$-equivariant $2$-covers over real Grassmannians, we get \begin{prop}\label{prop:OrientVSReal}
The pull-back morphisms of ordinary and equivariant cohomology
\[
\pi^*: H^*(G_k(\mathbb{R}^n)) \hookrightarrow H^*(\tilde{G}_k(\mathbb{R}^n)) \qquad \text{and} \qquad H^*_T(G_k(\mathbb{R}^n)) \hookrightarrow H^*_T(\tilde{G}_k(\mathbb{R}^n))
\]
are both injective. Moreover,
\[
\pi^*(H^*(G_k(\mathbb{R}^n))) = H^*(\tilde{G}_k(\mathbb{R}^n))^{\mathbb{Z}/2} \qquad \text{and} \qquad \pi^*(H^*_T(G_k(\mathbb{R}^n))) = H^*_T(\tilde{G}_k(\mathbb{R}^n))^{\mathbb{Z}/2}
\]
identifies cohomology of real Grassmannians as the ${\mathbb{Z}/2}$-invariant subrings of cohomology of oriented Grassmannians, or equivalently as the $+1$-eigenspaces of $\rho^*$ on cohomology of oriented Grassmannians. \end{prop}
For odd dimensional oriented Grassmannian $\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$, the deck transformation $\rho:\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}) \rightarrow \tilde{G}_{2k+1}(\mathbb{R}^{2n+2}): v_1 \wedge \cdots \wedge v_{2k+1} \mapsto -(v_1 \wedge \cdots \wedge v_{2k+1})=(-v_1) \wedge \cdots \wedge (-v_{2k+1})$ is induced from the antipodal map $A:\mathbb{R}^{2n+2}\rightarrow \mathbb{R}^{2n+2}: v\mapsto -v$ which is homotopic to the identity map on $\mathbb{R}^{2n+2}$ via \[ \begin{pmatrix} \begin{matrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{matrix} & &\\ & \ddots &\\ & & \begin{matrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{matrix} \end{pmatrix} \] which is actually $T^n$-equivariant and further induces $T^n$-equivariant homotopy between $\rho$ and $id$ on $\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$.
\begin{thm}[Relations between odd dimensional oriented and real Grassmannians]\label{thm:OddGrass}
Since $\rho^*=id$ on ordinary and equivariant cohomology of odd dimensional oriented Grassmannians, we have
\begin{align*}
H^*(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})) &= H^*(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}))^{\mathbb{Z}/2} \cong H^*(G_{2k+1}(\mathbb{R}^{2n+2}))\\ H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})) &= H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}))^{\mathbb{Z}/2} \cong H^*_T(G_{2k+1}(\mathbb{R}^{2n+2})).
\end{align*} \end{thm}
\begin{cor}
The Poincar\'e series of $\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$ are
\[
P_{\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})}(t)=P_{G_{2k+1}(\mathbb{R}^{2n+2})}(t)=(1+t^{2n+1})P_{G_{k}(\mathbb{C}^{n})}(t^2).
\] \end{cor}
The relations between cohomology of oriented and real Grassmannians in even dimensions are more delicate. In next two subsections, we will try to understand the $\rho^*$-action on finer structures of the cohomology rings of oriented Grassmannians.
\subsection{GKM description of oriented Grassmannians} The GKM description of oriented Grassmannians is very similar to that of real Grassmannians in previous section. Hence most of the details will be omitted but referred to those of real Grassmannians.
\subsubsection{Orientations and Euler classes of canonical bundle and complementary bundle}\label{subsub:Euler} The preferred orientation on every oriented $k$-dimensional subspace in $\mathbb{R}^n$ brings new invariants.
For example, the $T^n$-fixed points of $\tilde{G}_{2k}(\mathbb{R}^{2n})$ as $2k$-dimensional $T^n$-subrepresentation of $\mathbb{R}^{2n}=\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]}$ are of the form $\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]}$ where $S$ is a $k$-element subset of $\{1,2,\ldots,n\}$. Though an orientation on $\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]}$ can not specify the signs of the individual weights $\alpha_i,i\in S$, it does specify the sign of the product of weights as either $\prod_{i\in S}\alpha_i$ or $-\prod_{i\in S}\alpha_i$, which is exactly the equivariant Euler class of an oriented $T$-representation over a point. We will denote $V_{S_+},V_{S_-}$ for the $T$-subrepresentation $\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]}$ with equivariant Euler classes $e^T$ as $\prod_{i\in S}\alpha_i,-\prod_{i\in S}\alpha_i$ respectively. Similarly, for $\tilde{G}_{2k}((\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}_0)$ we can introduce the same notations with the fixed points $V_{S_\pm}=(\oplus_{i\in S}\mathbb{R}^2_{[\alpha_i]},\pm \prod_{i\in S}\alpha_i)$.
For $\tilde{G}_{2k+1}((\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}_0)$, the fixed points are of the form $(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}_0$ which has equivariant Euler class $0$ because of the $0$-weight space $\mathbb{R}_0$. But an orientation on $(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}_0$ gives the complementary $T^n$-subrepresentation $\oplus_{j \not\in S} \mathbb{R}^2_{[\alpha_j]}$ an orientation hence an equivariant Euler class $\bar{e}^T$ either $\prod_{j \not\in S}\alpha_j$ or $-\prod_{j \not\in S}\alpha_j$. We will denote these fixed points as $V_{S_\pm}=((\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}_0,\pm \prod_{j \not\in S}\alpha_j)$.
For $\tilde{G}_{2k+1}((\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}_0^2)$, a fixed component is of the form $\tilde{C}_S=\{(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus L_0 \mid L_0\in \tilde{G}_1(\mathbb{R}^2_0)\}\cong S^1$. Since both $(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus L_0$ and its complement $(\oplus_{j \not\in S} \mathbb{R}^2_{[\alpha_j]})\oplus L_0^\perp$ have a $0$-weight part, the equivariant Euler classes of both $T^n$-subrepresentations are $0$.
\subsubsection{$1$-skeleta} We can describe the $1$-skeleta of oriented Grassmannians: \begin{prop}[$1$-skeleta of oriented Grassmannians]\label{prop:OrientSkeleton}
The fixed points, isotropy weights and $1$-skeletons of oriented Grassmannians can be given as
\begin{enumerate}
\item For $\tilde{G}_{2k}(\mathbb{R}^{2n})$, there are $2\binom{n}{k}$ fixed points of the form $V_{S_\pm}=(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]},\pm \prod_{i\in S}\alpha_i)$, where $S$ is a $k$-element subset of $\{1,2,\ldots,n\}$. The isotropy weights at both $V_{S_\pm}$ are $\{[\alpha_j\pm\alpha_i] \mid i\in S,j\not \in S\}$, among which $[\alpha_j-\alpha_i]$ joins $V_{S_\pm}$ via a $2$-sphere to $V_{((S\smallsetminus\{i\})\cup \{j\})_\pm}$, and $[\alpha_j+\alpha_i]$ joins $V_{S_\pm}$ via a $2$-sphere to $V_{((S\smallsetminus\{i\})\cup \{j\})_\mp}$.
\item For $\tilde{G}_{2k}(\mathbb{R}^{2n+1})$, there are $2\binom{n}{k}$ fixed points of the form $V_{S_\pm}=(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]},\pm \prod_{i\in S}\alpha_i)$, where $S$ is a $k$-element subset of $\{1,2,\ldots,n\}$. The isotropy weights at both $V_{S_\pm}$ are $\{[\alpha_j\pm\alpha_i] \mid i\in S,j\not \in S\}\cup \{[\alpha_i]\mid i \in S\}$, among which $[\alpha_j-\alpha_i]$ joins $V_{S_\pm}$ via a $2$-sphere to $V_{((S\smallsetminus\{i\})\cup \{j\})_\pm}$, and $[\alpha_j+\alpha_i]$ joins $V_{S_\pm}$ via a $2$-sphere to $V_{((S\smallsetminus\{i\})\cup \{j\})_\mp}$, and $[\alpha_i]$ joins $V_{S_+}$ via a $2$-sphere to $V_{S_-}$.
\item For $\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$, there are $2\binom{n}{k}$ fixed points of the form $V_{S_\pm}=((\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}_0,\pm \prod_{j \not\in S}\alpha_j)$, where $S$ is a $k$-element subset of $\{1,2,\ldots,n\}$. The isotropy weights at both $V_{S_\pm}$ are $\{[\alpha_j\pm\alpha_i] \mid i\in S,j\not \in S\} \cup \{[\alpha_j]\mid j \not \in S\}$, among which $[\alpha_j-\alpha_i]$ joins $V_{S_\pm}$ via a $2$-sphere to $V_{((S\smallsetminus\{i\})\cup \{j\})_\pm}$ and $[\alpha_j+\alpha_i]$ joins $V_{S_\pm}$ via a $2$-sphere to $V_{((S\smallsetminus\{i\})\cup \{j\})_\mp}$, and $[\alpha_j]$ joins $V_{S_+}$ via a $2$-sphere to $V_{S_-}$.
\item For $\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$, there are $\binom{n}{k}$ fixed circles of the form $\tilde{C}_S=\{(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus L_0 \mid L_0\in \tilde{G}_1(\mathbb{R}^2_0)\}\cong S^1$, where $S$ is a $k$-element subset of $\{1,2,\ldots,n\}$. The isotropy weights at $\tilde{C}_S$ are $\{[\alpha_j\pm\alpha_i] \mid i\in S,j\not \in S\} \cup \{[\alpha_i]\mid i\in S\}\cup \{[\alpha_j]\mid j \not \in S\}$, among which both $[\alpha_j+\alpha_i]$ and $[\alpha_j-\alpha_i]$ join $\tilde{C}_S$ via a $S^2\times S^1$ to $\tilde{C}_{ (S\smallsetminus\{i\})\cup \{j\}}$, and $[\alpha_i],[\alpha_j]$ join $\tilde{C}_S$ via a $S^3$ to no other fixed circles.
\end{enumerate} \end{prop} \begin{proof}
Similar to the case of real Grassmannians. \end{proof}
\subsubsection{GKM graphs of oriented Grassmannians} Using the $1$-skeleta of oriented Grassmannians, we can construct their GKM graphs:
\begin{exm}
We will give some examples of GKM graphs for $\tilde{G}_k(\mathbb{R}^n)$ when $k$ or $n$ is small.
\begin{enumerate}
\item $S^{2n}$ as $\tilde{G}_{1}(\mathbb{R}^{2n+1})$ or $\tilde{G}_{2n}(\mathbb{R}^{2n+1})$
\begin{figure}
\caption{GKM graph for $S^{2n}$}
\caption{Condensed GKM graph for $S^{2n}$}
\caption{GKM graphs for $S^{2n}$}
\end{figure}
\item $S^{2n+1}$ as $\tilde{G}_{1}(\mathbb{R}^{2n+2})$ or $\tilde{G}_{2n+1}(\mathbb{R}^{2n+2})$
\begin{figure}
\caption{GKM graph for $S^{2n+1}$}
\caption{Condensed GKM graph for $S^{2n+1}$}
\caption{GKM graphs for $S^{2n+1}$}
\end{figure}
\item $\tilde{G}_{2}(\mathbb{R}^{4}),\tilde{G}_{2}(\mathbb{R}^{5}),\tilde{G}_{3}(\mathbb{R}^{5})$ as $\tilde{G}_{2k}(\mathbb{R}^{2n}), \tilde{G}_{2k}(\mathbb{R}^{2n+1}), \tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$ when $k=1,n=2$.
\begin{figure}
\caption{GKM graph for $\tilde{G}_{2}(\mathbb{R}^{4})$}
\caption{GKM graph for $\tilde{G}_{2}(\mathbb{R}^{5})$}
\caption{GKM graph for $\tilde{G}_{3}(\mathbb{R}^{5})$}
\caption{GKM graphs for $\tilde{G}_{2}(\mathbb{R}^{4}),\tilde{G}_{2}(\mathbb{R}^{5}),\tilde{G}_{3}(\mathbb{R}^{5})$}
\end{figure}
\item $\tilde{G}_{3}(\mathbb{R}^{6})$ as $\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$ when $k=1,n=2$.
\begin{figure}
\caption{GKM graph for $\tilde{G}_{3}(\mathbb{R}^{6})$}
\caption{Condensed GKM graph for $\tilde{G}_{3}(\mathbb{R}^{6})$}
\caption{GKM graphs for $\tilde{G}_{3}(\mathbb{R}^{6})$}
\end{figure}
\end{enumerate} \end{exm}
\subsubsection{Formality, cohomology and canonical basis of oriented Grassmannians} \begin{prop}[Equivariant formality of torus actions on oriented Grassmannians]
The $T^n$-actions on all the four types of oriented Grassmannians $\tilde{G}_{2k}(\mathbb{R}^{2n}), \tilde{G}_{2k}(\mathbb{R}^{2n+1}), \tilde{G}_{2k+1}(\mathbb{R}^{2n+1}), \tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$ are equivariantly formal. All the four oriented Grassmannians have the same total Betti number $2\binom{n}{k}$. \end{prop} \begin{proof}
The even dimensional oriented Grassmannians $\tilde{G}_{2k}(\mathbb{R}^{2n}), \tilde{G}_{2k}(\mathbb{R}^{2n+1}), \tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$ can be viewed as homogeneous spaces of the form $G/H$ with $H$ a compact connected Lie subgroup and of the same rank as the connected compact Lie group $G$. As shown in \cite{GHZ06}, the actions of maximal tori on these homogeneous spaces are equivariantly formal. Their total Betti numbers are the same as their Euler characteristic numbers $|W_G/W_H|$ where $W_G,W_H$ are the Weyl groups of $G$ and $H$. Alternatively, we can compute the total Betti number as the number of fixed points given in Theorem \ref{prop:OrientSkeleton}, namely $2\binom{n}{k}$. The odd dimensional oriented Grassmannian $\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$ has the same equivariant cohomology as the real Grassmannian $G_{2k+1}(\mathbb{R}^{2n+2})$, hence is also equivariantly formal with total Betti number $2\binom{n}{k}$. \end{proof}
After verifying GKM conditions and equivariant formality, we can give the GKM description of the torus actions on $\tilde{G}_k(\mathbb{R}^n)$ by applying the even dimensional GKM Theorem as in Guillemin, Holm and Zara \cite{GHZ06} and the odd dimensional GKM-type Theorem \ref{thm:OddGKM}.
\begin{thm}[GKM description of equivariant cohomology of oriented Grassmannians]\label{thm:GKMorientGrass}
The following congruence relations are given for any two $k$-element subsets $S,S'\subset \{1,\ldots,n\}$ differed by one element with $S\cup\{j\}=S'\cup\{i\}$.
\begin{enumerate}
\item For the oriented Grassmannian $\tilde{G}_{2k}(\mathbb{R}^{2n})$, an element of equivariant cohomology is a set of polynomials $f_{S_\pm} \in \mathbb{Q}[\alpha_1,\ldots,\alpha_n]$ to each vertex $S_\pm$ such that
\begin{enumerate}
\item $f_{S_+} \equiv f_{S'_+}, \quad f_{S_-} \equiv f_{S'_-} \mod \alpha_j-\alpha_i$
\item $f_{S_+} \equiv f_{S'_-}, \quad f_{S_-} \equiv f_{S'_+} \mod \alpha_j+\alpha_i$.
\end{enumerate}
\item For the oriented Grassmannian $\tilde{G}_{2k}(\mathbb{R}^{2n+1})$, an element of equivariant cohomology is a set of polynomials $f_{S_\pm} \in \mathbb{Q}[\alpha_1,\ldots,\alpha_n]$ to each vertex $S_\pm$ such that
\begin{enumerate}
\item $f_{S_+} \equiv f_{S'_+} , \quad f_{S_-} \equiv f_{S'_-}\mod \alpha_j-\alpha_i$
\item $f_{S_+} \equiv f_{S'_-} , \quad f_{S_-} \equiv f_{S'_+}\mod \alpha_j+\alpha_i$
\item $f_{S_+} \equiv f_{S_-} \mod \prod_{i'\in S}\alpha_{i'}$.
\end{enumerate}
\item For the oriented Grassmannian $\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$, an element of equivariant cohomology is a set of polynomials $f_{S_\pm} \in \mathbb{Q}[\alpha_1,\ldots,\alpha_n]$ to each vertex $S_\pm$ such that
\begin{enumerate}
\item $f_{S_+} \equiv f_{S'_+} , \quad f_{S_-} \equiv f_{S'_-}\mod \alpha_j-\alpha_i$
\item $f_{S_+} \equiv f_{S'_-} , \quad f_{S_-} \equiv f_{S'_+}\mod \alpha_j+\alpha_i$
\item $f_{S_+} \equiv f_{S_-} \mod \prod_{j'\not\in S}\alpha_{j'}$.
\end{enumerate}
\item For the odd dimensional oriented Grassmannian $\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$ with $T^n$-action, an element of equivariant cohomology is a set of polynomial pairs $(f_S, g_S \theta)$ to each $\circ$-vertex $S$ where $\theta$ is the unit volume form of $S^1$ such that
\begin{enumerate}
\item $g_{S} \equiv 0 \mod \prod_{i=1}^{n}\alpha_i$
\item $f_{S} \equiv f_{S'} , \quad g_{S} \equiv g_{S'}\mod \alpha^2_j-\alpha^2_i$.
\end{enumerate}
\end{enumerate} \end{thm}
For odd dimensional oriented Grassmannians, the induced deck transformation $\rho^*: H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}))\rightarrow H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}))$ is the identity map hence acts trivially on the GKM description. Solving the same set of congruence equations as in Theorem \ref{thm:AllGrass} of $H^*_T({G}_{2k+1}(\mathbb{R}^{2n+2}))$, we will also get an element $\tilde{r}^T \in H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}))$ in GKM description localized at a fixed circle $\tilde{C}_S=\{(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus L_0 \mid L_0\in \tilde{G}_1(\mathbb{R}^2_0)\}\cong S^1$ to be $\tilde{r}^T_S = (\prod_{i=1}^{n}\alpha_i) \theta_{S^1}$ similar to the ${r}^T \in H^*_T({G}_{2k+1}(\mathbb{R}^{2n+2}))$ localized at ${C}_S=\{(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus L_0 \mid L_0\in {G}_1(\mathbb{R}^2_0)\}\cong \mathbb{R} P^1$ to be ${r}^T_S = (\prod_{i=1}^{n}\alpha_i) \theta_{\mathbb{R} P^1}$, where $\theta_{S^1}$ and $\theta_{\mathbb{R} P^1}$ are the unit volume forms of $S^1$ and $\mathbb{R} P^1$ respectively.
\begin{prop}[Canonical basis of $H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}))$]\label{prop:CanoBaseIII}
Let $\sigma_{S\in \mathcal{S}}$ be the canonical basis of $H^*_T({G}_{2k}(\mathbb{R}^{2n}))$ from Theorem \ref{thm:RealSchub} and $\tilde{r}^T_S = \prod_{i=1}^{n}\alpha_i \theta_{S^1}$ be the odd-degree generator. Then $\sigma_{S}, \tilde{r}^T \cdot \sigma_{S}$ give additive $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-basis of $ H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}))$. \end{prop}
However, there is a subtlety for the pullback $\pi^*:H^*_T({G}_{2k+1}(\mathbb{R}^{2n+2}))\rightarrow H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}))$ though this is an isomorphism. The $2$-fold covering $\pi: \tilde{G}_{2k+1}(\mathbb{R}^{2n+2}) \rightarrow {G}_{2k+1}(\mathbb{R}^{2n+2})$ restricts to a $2$-fold covering of fixed circles $\pi: (\tilde{C}_S\cong S^1) \rightarrow ({C}_S\cong \mathbb{R} P^1)$ which will give the localized pullback $\pi^*(\theta_{\mathbb{R} P^1})=2\theta_{S^1}$. Hence we get $\pi^*({r}^T)=2\tilde{r}^T$.
\begin{prop}[The explicit pullback of cohomology between odd dimensional Grassmannians]\label{prop:Pullr}
In the canonical basis, the pullback of cohomology of odd dimensional Grassmannian is
\begin{align*}
\pi^*:H^*_T({G}_{2k+1}(\mathbb{R}^{2n+2}))&\longrightarrow H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}))\\
\sigma_{S} &\longmapsto \sigma_{S}\\
{r}^T \Cdot \sigma_{S} &\longmapsto 2\tilde{r}^T \Cdot \sigma_{S}.
\end{align*} \end{prop}
For the even dimensional oriented Grassmannians, the deck transformation $\rho: \tilde{G}_k(\mathbb{R}^n)\rightarrow \tilde{G}_k(\mathbb{R}^n)$ switches any fixed point ${S_+}$ with its twin fixed point ${S_-}$ by reversing orientations. Then the induced deck transformation $\rho^*:H^*_T(\tilde{G}_k(\mathbb{R}^n))\rightarrow H^*_T(\tilde{G}_k(\mathbb{R}^n))$ in GKM description will switch any polynomial $f_{S_+}$ with $f_{S_-}$. Notice the symmetry in the GKM descriptions, we see that the switch of polynomials preserves the congruence relations.
Since $(\rho^*)^2=id$, both cohomology $H^*(\tilde{G}_k(\mathbb{R}^n)),H^*_T(\tilde{G}_k(\mathbb{R}^n))$ decompose
into $\pm 1$-eigenspaces of $\rho^*$.
\begin{prop}\label{prop:RhoEigen}
For the even dimensional oriented Grassmannians $\tilde{G}_{2k}(\mathbb{R}^{2n}),\tilde{G}_{2k}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$, the elements of $+1$-eigenspace of $\rho^*$ on the equivariant cohomology can be identified as those sets of polynomials $\{f_{S_\pm}, S \in \mathcal{S}\}$ where $\mathcal{S}$ is the collection of $k$-element subsets of $\{1,\ldots,n\}$ such that
\begin{align*}
f_{S_+}=f_{S_-}
\end{align*}
and the elements of $-1$-eigenspace of $\rho^*$ are those with
\[
f_{S_+}=-f_{S_-}.
\] \end{prop}
\begin{rmk}
As we have proved before, the $+1$-eigenspaces of $\rho^*$ on the equivariant cohomology of oriented Grassmannians are exactly the equivariant cohomology of real Grassmannians. \end{rmk}
Recall that we defined equivariant Euler classes at each fixed point $S$ for $\tilde{G}_{2k}(\mathbb{R}^{2n}),\tilde{G}_{2k}(\mathbb{R}^{2n+1})$ to be $e^T_{S_{\pm}}=\pm \prod_{i\in S} \alpha_i$ and for $\tilde{G}_{2k}(\mathbb{R}^{2n}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$ to be $\bar{e}^T_{S_{\pm}}=\pm \prod_{j\not\in S} \alpha_j$. It is easy to check that $\{e^T_{S_\pm}, S \in \mathcal{S}\}$ and $\{\bar{e}^T_{S_\pm}, S \in \mathcal{S}\}$ are elements of the GKM description of the corresponding equivariant cohomology. Since $\rho$ changes the signs of orientations, $\rho^*$ changes the signs of the equivariant Euler classes. Therefore, $\{e^T_{S_\pm}, S \in \mathcal{S}\}$ and $\{\bar{e}^T_{S_\pm}, S \in \mathcal{S}\}$ are in the $-1$-eigenspaces of $\rho^*$. Topologically, the localized classes $e^T,\bar{e}^T$ in GKM description are exactly the equivariant Euler classes of the canonical oriented bundles and complementary oriented bundles over the oriented Grassmannians.
\begin{prop}[Equivariant Euler class and top equivariant Pontryagin class]\label{prop:EulerPont}
Similar to the relations between ordinary Euler class and top ordinary Pontryagin class,
\begin{enumerate}
\item For $\tilde{G}_{2k}(\mathbb{R}^{2n})$ and $\tilde{G}_{2k}(\mathbb{R}^{2n+1})$, we have $(e^T)^2 = p^T_k $
\item For $\tilde{G}_{2k}(\mathbb{R}^{2n})$ and $\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$, we have $(\bar{e}^T)^2 = \bar{p}^T_{n-k}$
\item For $\tilde{G}_{2k}(\mathbb{R}^{2n})$, we have $e^T\bar{e}^T= \prod_{i=1}^{n}\alpha_i$
\end{enumerate} \end{prop} \begin{proof}
Let's prove this for $\tilde{G}_{2k}(\mathbb{R}^{2n})$ which covers the remaining cases of $\tilde{G}_{2k}(\mathbb{R}^{2n+1}), \tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$. In Proposition \ref{prop:Pont}, we have given the localized top equivariant Pontryagin classes of real Grassmannians as
\[
p^T_k|_S=\prod_{i \in S} \alpha_i^2 \qquad \qquad \bar{p}^T_{n-k}|_S=\prod_{j \not \in S} \alpha_j^2.
\]
Via the pullback $\pi^*: H^*_T({G}_{2k}(\mathbb{R}^{2n})) \rightarrow H^*_T(\tilde{G}_{2k}(\mathbb{R}^{2n}))$, the equivariant Pontryagin classes of ${G}_{2k}(\mathbb{R}^{2n})$ are identified as those of $\tilde{G}_{2k}(\mathbb{R}^{2n})$, and are in the $+1$-eigenspaces of $\rho^*$. Therefore
\[
p^T_k|_{S_\pm}=\prod_{i \in S} \alpha_i^2 \qquad \qquad \bar{p}^T_{n-k}|_{S_\pm}=\prod_{j \not \in S} \alpha_j^2.
\]
Comparing them with
\[
e^T_{S_{\pm}}=\pm \prod_{i\in S} \alpha_i \qquad \qquad \bar{e}^T_{S_{\pm}}=\pm \prod_{j\not\in S} \alpha_j
\]
we get the stated relations. \end{proof}
The induced deck transformation $\rho^*$ is a ring homomorphism, therefore the multiplication of an element in the $-1$-eigenspace with an element in the $+1$-eigenspace results in the $-1$-eigenspace.
\begin{prop}\label{prop:EulerMult}
Multiplication with the equivariant Euler classes $e^T,\bar{e}^T$ maps $+1$-eigenspaces of $\rho^*$ to $-1$-eigenspaces.
\begin{enumerate}
\item For $\tilde{G}_{2k}(\mathbb{R}^{2n+1})$, the multiplication with $e^T$ is an isomorphism between $+1$-eigenspace of $\rho^*$ to its $-1$-eigenspace.
\item For $\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$, the multiplication with $\bar{e}^T$ is an isomorphism between $+1$-eigenspace of $\rho^*$ to its $-1$-eigenspace.
\end{enumerate} \end{prop} \begin{proof}
For $\tilde{G}_{2k}(\mathbb{R}^{2n+1})$, denote $V_{+1}$ and $V_{-1}$ be the $+1$ and $-1$-eigenspaces of $\rho^*$ on $H^*_T(\tilde{G}_{2k}(\mathbb{R}^{2n+1}))$. The fact that $e^T$ is in the $-1$-eigenspace gives the multiplication $\times e^T: V_{+1}\rightarrow V_{-1}$. On the other hand, every element $\{f_{S_\pm}, S \in \mathcal{S}\}$ of $V_{-1}$ has the form $f_{S_+}=-f_{S_-}$ by Prop \ref{prop:RhoEigen}. Plug this into the congruence relation between $S_+$ and $S_-$ in Theorem \ref{thm:GKMorientGrass}, we get
\[
f_{S_+} \equiv f_{S_-}=-f_{S_+} \mod \prod_{i\in S}\alpha_{i}
\]
or equivalently, both $f_{S_+}$ and $f_{S_-}$ are multiples of $e^T_{S_{\pm}}=\pm \prod_{i\in S} \alpha_i$. Therefore, the localized quotients $f_{S_+}/e^T_{S_+}, f_{S_-}/e^T_{S_-}\in \mathbb{Q}[\alpha_1,\ldots,\alpha_n]$ are polynomials, and this defines a unique element $f/e^T \in V_{+1}$.
The case of $\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$ is similar. \end{proof}
\begin{rmk}
For $\tilde{G}_{2k}(\mathbb{R}^{2n})$, neither the multiplication by $e^T$ nor by $\bar{e}^T$ are isomorphisms between the $+1$ and $-1$-eigenspaces of $\rho^*$. We will try to understand the equivariant cohomology of $\tilde{G}_{2k}(\mathbb{R}^{2n})$ in next subsection. \end{rmk}
The above isomorphism between eigenspaces of $\rho^*$, together with the canonical basis $\sigma_{S}$ of $H^*_T({G}_{2k}(\mathbb{R}^{2n+1}))$ and $H^*_T({G}_{2k+1}(\mathbb{R}^{2n+1}))$, give \begin{prop}[Canonical basis of $H^*_T(\tilde{G}_{2k}(\mathbb{R}^{2n+1})),H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+1}))$]\label{prop:CanoBaseII}
Let $\sigma_{S\in \mathcal{S}}$ be the canonical basis of $H^*_T({G}_{2k}(\mathbb{R}^{2n+1}))$ and $H^*_T({G}_{2k+1}(\mathbb{R}^{2n+1}))$ from Theorem \ref{thm:RealSchub}. Then $\sigma_{S}, e^T \cdot \sigma_{S}$ and $\sigma_{S}, \bar{e}^T \cdot \sigma_{S}$ give additive $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-basis of $ H^*_T(\tilde{G}_{2k}(\mathbb{R}^{2n+1}))$ and $H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+1}))$ respectively. \end{prop}
\begin{cor}
The Poincar\'e series of $\tilde{G}_{2k}(\mathbb{R}^{2n+1})$ and $\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$ are
\begin{align*}
P_{\tilde{G}_{2k}(\mathbb{R}^{2n+1})}(t)&=(1+t^{2k})P_{{G}_{2k}(\mathbb{R}^{2n+1})}(t)=(1+t^{2k})P_{G_{k}(\mathbb{C}^{n})}(t^2)\\
P_{\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})}(t)&=(1+t^{2n-2k})P_{{G}_{2k+1}(\mathbb{R}^{2n+1})}(t)=(1+t^{2n-2k})P_{G_{k}(\mathbb{C}^{n})}(t^2).
\end{align*} \end{cor}
\begin{cor}[Relations between some oriented and real Grassmannians]\label{thm:Type2Grass}
The equivariant cohomology of $\tilde{G}_{2k}(\mathbb{R}^{2n+1})$ and $\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$ are $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-algebra extensions by $e^T,\bar{e}^T$ of the equivariant cohomology of ${G}_{2k}(\mathbb{R}^{2n+1})$ and ${G}_{2k+1}(\mathbb{R}^{2n+1})$, i.e.
\begin{align*}
H^*_T(\tilde{G}_{2k}(\mathbb{R}^{2n+1}))&\cong \frac{H^*_T({G}_{2k}(\mathbb{R}^{2n+1}))[e^T]}{(e^T)^2 = p^T_k} \\
H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+1}))&\cong \frac{H^*_T({G}_{2k+1}(\mathbb{R}^{2n+1}))[\bar{e}^T]}{(\bar{e}^T)^2 = \bar{p}^T_{n-k}}.
\end{align*} \end{cor} \begin{proof}
Using Prop \ref{prop:EulerPont}, \ref{prop:CanoBaseII} and dimension counting. \end{proof}
\subsection{Leray-Borel description of oriented Grassmannians} In this subsection, we will confirm the ring generators of equivariant cohomology of oriented Grassmannians to be characteristic classes, then determine the complete relations among them, and also give additive basis.
\subsubsection{Leray-Borel description of $\tilde{G}_{2k}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$} From Theorem \ref{thm:OddGrass} and Theorem \ref{thm:Type2Grass}, we have seen that equivariant cohomology rings of $\tilde{G}_{2k}(\mathbb{R}^{2n+1})$, $\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$ and $\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$ are ring extensions of the equivariant cohomology of their real counterparts. Hence the equivariant Leray-Borel descriptions and equivariant characteristic basis of those oriented Grassmannians can be extended from the related real Grassmannians.
\begin{thm}[Equivariant Leray-Borel description of $\tilde{G}_{2k}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$]
The equivariant cohomology rings of $\tilde{G}_{2k}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$ are generated by equivariant Pontryagin and Euler classes, and an odd-degree class $\tilde{r}^T$:
\begin{align*}
H^*_T(\tilde{G}_{2k}(\mathbb{R}^{2n+1})) &\cong \frac{\mathbb{Q}[\alpha_1,\alpha_2,\ldots,\alpha_n][p^T_1,p^T_2,\ldots,p^T_k;\bar{p}^T_1,\bar{p}^T_2,\ldots,\bar{p}^T_{n-k};e^T]}{p^T\bar{p}^T = \prod_{i=1}^{n}(1+\alpha^2_i),\, (e^T)^2=p^T_k}\\
H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})) &\cong \frac{\mathbb{Q}[\alpha_1,\alpha_2,\ldots,\alpha_n][p^T_1,p^T_2,\ldots,p^T_k;\bar{p}^T_1,\bar{p}^T_2,\ldots,\bar{p}^T_{n-k};\bar{e}^T]}{p^T\bar{p}^T = \prod_{i=1}^{n}(1+\alpha^2_i),\, (\bar{e}^T)^2=\bar{p}^T_{n-k}}\\
H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}))&\cong\frac{\mathbb{Q}[\alpha_1,\alpha_2,\ldots,\alpha_n][p^T_1,p^T_2,\ldots,p^T_k;\bar{p}^T_1,\bar{p}^T_2,\ldots,\bar{p}^T_{n-k};\tilde{r}^T]}{p^T\bar{p}^T = \prod_{i=1}^{n}(1+\alpha^2_i),\,(\tilde{r}^T)^2=0}.
\end{align*} \end{thm}
\begin{thm}[Equivariant characteristic basis of $\tilde{G}_{2k}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$]\label{thm:EquivCharOrient}
The sets of monomials $\{(p^T_1)^{r_1}(p^T_2)^{r_2}\cdots (p^T_k)^{r_k},\,e^T\cdot(p^T_1)^{r_1}(p^T_2)^{r_2}\cdots (p^T_k)^{r_k}\}$, $\{(p^T_1)^{r_1}(p^T_2)^{r_2}\cdots (p^T_k)^{r_k},\,\bar{e}^T\cdot(p^T_1)^{r_1}(p^T_2)^{r_2}\cdots (p^T_k)^{r_k}\}$ and $\{(p^T_1)^{r_1}(p^T_2)^{r_2}\cdots (p^T_k)^{r_k},\,\tilde{r}^T\cdot(p^T_1)^{r_1}(p^T_2)^{r_2}\cdots (p^T_k)^{r_k}\}$ satisfying the condition $\sum_{i=1}^{k} r_i \leq n-k$ form additive $H^*_T(pt)$-basis for $H^*_T(\tilde{G}_{2k}(\mathbb{R}^{2n+1})),H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})),H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}))$ respectively. \end{thm}
The above two theorems of equivariant ring generators and equivariant additive basis both have their ordinary versions by replacing $\alpha_i$ with $0$.
\begin{cor}[Ordinary Leray-Borel description of $\tilde{G}_{2k}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$]
The ordinary cohomology rings of $\tilde{G}_{2k}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$ are generated by Pontryagin and Euler classes, and an odd-degree class $\tilde{r}$:
\begin{align*}
H^*(\tilde{G}_{2k}(\mathbb{R}^{2n+1})) &\cong \frac{\mathbb{Q}[p_1,p_2,\ldots,p_k;\bar{p}_1,\bar{p}_2,\ldots,\bar{p}_{n-k};e]}{p\bar{p} = 1,\, e^2=p_k}\\
H^*(\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})) &\cong \frac{\mathbb{Q}[p_1,p_2,\ldots,p_k;\bar{p}_1,\bar{p}_2,\ldots,\bar{p}_{n-k};\bar{e}]}{p\bar{p} = 1,\, \bar{e}^2=\bar{p}_{n-k}}\\
H^*(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}))&\cong\frac{\mathbb{Q}[p_1,p_2,\ldots,p_k;\bar{p}_1,\bar{p}_2,\ldots,\bar{p}_{n-k};\tilde{r}]}{p\bar{p} = 1,\, \tilde{r}^2=0}.
\end{align*} \end{cor}
\begin{cor}[Ordinary characteristic basis of $\tilde{G}_{2k}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$]
The sets of monomials $\{p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k},\,e\cdot p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\}$, $\{p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k},\,\bar{e}\cdot p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\}$ and $\{p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k},\,\tilde{r}\cdot p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\}$ satisfying the condition $\sum_{i=1}^{k} r_i \leq n-k$ form additive basis for $H^*(\tilde{G}_{2k}(\mathbb{R}^{2n+1})),H^*(\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})),H^*(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}))$ respectively. \end{cor}
\subsubsection{Leray-Borel description of $\tilde{G}_{2k}(\mathbb{R}^{2n})$} Now let's turn to the remaining type of oriented Grassmannian $\tilde{G}_{2k}(\mathbb{R}^{2n})$. As we remarked in previous subsection, neither the multiplication by $e^T$ nor by $\bar{e}^T$ are isomorphisms between eigenspaces of $\rho^*$. However, we will show the multiplications by $e^T$ and $\bar{e}^T$, restricted on certain carefully chosen subspaces, do give isomorphism between $+1$ and $-1$-eigenspaces of $\rho^*$.
Notice the equivariant diffeomorphism ${G}_{2k}(\mathbb{R}^{2n}) \cong {G}_{2n-2k}(\mathbb{R}^{2n})$ by mapping an oriented $2k$-dimensional subspace to its perpendicular oriented $(2n-2k)$-dimensional subspace. Then the complementary characteristic monomials $(\bar{p}^T_1)^{r_1}(\bar{p}^T_2)^{r_2}\cdots (\bar{p}^T_{n-k})^{r_{n-k}}$ and $\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k}^{r_{n-k}}$, satisfying the condition $\sum_{i=1}^{n-k} r_i \leq k$, give additive basis for the equivariant and respectively ordinary cohomology of ${G}_{2k}(\mathbb{R}^{2n})$. Also recall from Prop \ref{prop:EulerPont} on the relations among top Pontryagin classes and Euler classes of the oriented Grassmannian $\tilde{G}_{2k}(\mathbb{R}^{2n})$ that $(e^T)^2=p^T_k,(\bar{e}^T)^2=\bar{p}^T_{n-k},e^T\bar{e}^T=\prod_{i=1}^{n}\alpha_i$ and $e^2=p_k,\bar{e}^2=\bar{p}_{n-k},e\bar{e}=0$.
\begin{prop}[Eigenspaces of $H^*(\tilde{G}_{2k}(\mathbb{R}^{2n}))$]\label{prop:OrientEigen}
Let $\rho$ be the non-trivial deck transformation of the covering $\pi: \tilde{G}_{2k}(\mathbb{R}^{2n})\rightarrow {G}_{2k}(\mathbb{R}^{2n})$ and identify $H^*({G}_{2k}(\mathbb{R}^{2n}))$ as the $+1$-eigenspace of $\rho^*$ on $H^*(\tilde{G}_{2k}(\mathbb{R}^{2n}))$.
\begin{enumerate}
\item The multiplications by $e^T$ and $\bar{e}^T$ are isomorphisms restricted on the following subspaces
\begin{align*}
e \times: \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1) & \overset{\cong}{\longrightarrow}
e\cdot \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1)\\
\bar{e} \times: \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k}^{r_{n-k}}\mid \sum_{i=1}^{n-k} r_i \leq k-1) & \overset{\cong}{\longrightarrow}
\bar{e}\cdot \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k}^{r_{n-k}}\mid \sum_{i=1}^{n-k} r_i \leq k-1).
\end{align*}
\item $e\cdot \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1) \oplus \bar{e}\cdot \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k}^{r_{n-k}}\mid \sum_{i=1}^{n-k} r_i \leq k-1)$ is the $(-1)$-eigenspace of $\rho^*$
\item $e\cdot H^*({G}_{2k}(\mathbb{R}^{2n})) \cap \bar{e} \cdot H^*({G}_{2k}(\mathbb{R}^{2n}))=0$ and $e\cdot H^*({G}_{2k}(\mathbb{R}^{2n})) \oplus \bar{e} \cdot H^*({G}_{2k}(\mathbb{R}^{2n}))$ is the $(-1)$-eigenspace of $\rho^*$
\item The kernels of $e \times$ and $\bar{e} \times$ on $H^*({G}_{2k}(\mathbb{R}^{2n}))$ are $\bar{p}_{n-k}\cdot \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k}^{r_{n-k}}\mid \sum_{i=1}^{n-k} r_i \leq k-1)$ and $p_k \cdot \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1)$ respectively
\item The following spaces are identical
\begin{align*}
e\cdot \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1) =
e\cdot \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k-1}^{r_{n-k-1}}\mid \sum_{i=1}^{n-k-1} r_i \leq k) = e\cdot H^*({G}_{2k}(\mathbb{R}^{2n}))\\
\bar{e}\cdot \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k}^{r_{n-k}}\mid \sum_{i=1}^{n-k} r_i \leq k-1) =
\bar{e}\cdot \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_{k-1}^{r_{k-1}}\mid \sum_{i=1}^{k-1} r_i \leq n-k) =
\bar{e} \cdot H^*({G}_{2k}(\mathbb{R}^{2n})).
\end{align*}
\end{enumerate} \end{prop} \begin{proof}
Note that the total Betti numbers of $H^*({G}_{2k}(\mathbb{R}^{2n}))$ and $H^*(\tilde{G}_{2k}(\mathbb{R}^{2n}))$ are $\binom{n}{k}$ and $2\binom{n}{k}$ respectively, hence the dimension of the $-1$-eigenspace of $\rho^*$ is $\binom{n}{k}$.
\begin{enumerate}
\item The composition of the surjective linear maps
\begin{align*}
&e \times: \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1) \longrightarrow
e\cdot \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1)\\
&e \times: e\cdot \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1) \longrightarrow e^2\cdot \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1)
\end{align*}
is
\[
p_k\times: \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1) \longrightarrow
p_k\cdot \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1)
\]
where we have used the relation $e^2=p_k$. The composition maps a sub-basis of $H^*({G}_{2k}(\mathbb{R}^{2n}))$ onto another sub-basis without common vectors, hence is a bijection. Therefore, each individual surjection is a bijection. Similarly, we get the bijection for the restricted $\bar{e} \times$.
\item
We have seen from the above that
\[
e \times: e\cdot \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1) \longrightarrow p_k\cdot \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1)
\]
is a bijection. However, $e \times$ takes $\bar{e} \cdot H^*({G}_{2k}(\mathbb{R}^{2n}))$ to zero, because $e\bar{e}=0$. Hence
\[
e\cdot \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1) \cap \bar{e} \cdot H^*({G}_{2k}(\mathbb{R}^{2n})) = 0.
\]
Similarly,
\[
\bar{e}\cdot \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k}^{r_{n-k}}\mid \sum_{i=1}^{n-k} r_i \leq k-1) \cap e\cdot H^*({G}_{2k}(\mathbb{R}^{2n})) = 0.
\]
Combine these two, we get
\[
e\cdot \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1) \cap \bar{e}\cdot \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k}^{r_{n-k}}\mid \sum_{i=1}^{n-k} r_i \leq k-1) = 0.
\]
However, as a subspace in $-1$-eigenspace of $\rho^*$, the sum $e\cdot\mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1) \oplus \bar{e}\cdot \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k}^{r_{n-k}}\mid \sum_{i=1}^{n-k} r_i \leq k-1)$ has dimension $\binom{n-1}{k}+\binom{n-1}{n-k}=\binom{n}{k}$ the same as dimension of the entire $-1$-eigenspace of $\rho^*$, hence is exactly the $-1$-eigenspace of $\rho^*$.
\item
The above series of zero intersections force $e\cdot\mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1)=e\cdot H^*({G}_{2k}(\mathbb{R}^{2n}))$ and $\bar{e}\cdot \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k}^{r_{n-k}}\mid \sum_{i=1}^{n-k} r_i \leq k-1)=\bar{e}\cdot H^*({G}_{2k}(\mathbb{R}^{2n}))$. Hence we get $e\cdot H^*({G}_{2k}(\mathbb{R}^{2n})) \cap \bar{e} \cdot H^*({G}_{2k}(\mathbb{R}^{2n}))=0$ and $e\cdot H^*({G}_{2k}(\mathbb{R}^{2n})) \oplus \bar{e} \cdot H^*({G}_{2k}(\mathbb{R}^{2n}))$ is the $(-1)$-eigenspace of $\rho^*$.
\item
We have proved $e\cdot H^*({G}_{2k}(\mathbb{R}^{2n}))=e\cdot\mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1)$ and they are of dimension $\binom{n-1}{k}$. Since $H^*({G}_{2k}(\mathbb{R}^{2n}))$ is of dimension $\binom{n}{k}$, the kernel of $e\times$ on $H^*({G}_{2k}(\mathbb{R}^{2n}))$ is then of dimension $\binom{n}{k}-\binom{n-1}{k}=\binom{n-1}{n-k}$. Because $e\cdot \bar{p}_{n-k}=e\cdot \bar{e}^2=0$, the subspace $\bar{p}_{n-k}\cdot \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k}^{r_{n-k}}\mid \sum_{i=1}^{n-k} r_i \leq k-1)$ of dimension $\binom{n-1}{n-k}$ is clearly in the kernel of $e\times$ on $H^*({G}_{2k}(\mathbb{R}^{2n}))$, hence is exactly the kernel. Similarly, we obtain the kernel of $\bar{e}\times$ on $H^*({G}_{2k}(\mathbb{R}^{2n}))$.
\item
The $e\times$-kernel subspace $\bar{p}_{n-k}\cdot \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k}^{r_{n-k}}\mid \sum_{i=1}^{n-k} r_i \leq k-1)$ of $H^*({G}_{2k}(\mathbb{R}^{2n}))$ has complementary subspace $\mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k-1}^{r_{n-k-1}}\mid \sum_{i=1}^{n-k-1} r_i \leq k)$. Hence the restriction
\[
e\times: \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k-1}^{r_{n-k-1}}\mid \sum_{i=1}^{n-k-1} r_i \leq k) \longrightarrow e\cdot H^*({G}_{2k}(\mathbb{R}^{2n}))
\]
is bijection, therefore $e\cdot \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k-1}^{r_{n-k-1}}\mid \sum_{i=1}^{n-k-1} r_i \leq k) = e\cdot H^*({G}_{2k}(\mathbb{R}^{2n}))$. The identification $e\cdot\mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1)=e\cdot H^*({G}_{2k}(\mathbb{R}^{2n}))$ is proved in (3). Similarly, we get the identifications for $\bar{e}\cdot H^*({G}_{2k}(\mathbb{R}^{2n}))$.
\end{enumerate} \end{proof}
The detailed discussion of $e\times$ and $\bar{e}\times$ between the eigenspaces of $\rho^*$ gives: \begin{cor}[Ordinary characteristic basis of $\tilde{G}_{2k}(\mathbb{R}^{2n})$]
The ordinary cohomology of $\tilde{G}_{2k}(\mathbb{R}^{2n})$ is generated by Pontryagin classes and Euler classes with an additive basis $\{p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k\}$ for the $+1$-eigenspace of $\rho^*$ and $\{e\cdot\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k-1}^{r_{n-k-1}}\mid \sum_{i=1}^{n-k-1} r_i \leq k\}\cup \{ \bar{e}\cdot p_1^{r_1}p_2^{r_2}\cdots p_{k-1}^{r_{k-1}}\mid \sum_{i=1}^{k-1} r_i \leq n-k\}$ for the $-1$-eigenspace. \end{cor}
\begin{rmk}
Using the various identifications of $e\cdot H^*({G}_{2k}(\mathbb{R}^{2n}))$ and $\bar{e}\cdot H^*({G}_{2k}(\mathbb{R}^{2n}))$ in Theorem \ref{prop:OrientEigen}, we can also give the additive basis of $\tilde{G}_{2k}(\mathbb{R}^{2n})$ in other forms. \end{rmk}
\begin{cor}
The Poincar\'e series of $\tilde{G}_{2k}(\mathbb{R}^{2n})$ are
\begin{align*}
P_{\tilde{G}_{2k}(\mathbb{R}^{2n})}(t)
&=P_{{G}_{2k}(\mathbb{R}^{2n})}(t)+t^{2k}P_{{G}_{2k}(\mathbb{R}^{2n-2})}(t)+t^{2n-2k}P_{{G}_{2k-2}(\mathbb{R}^{2n-2})}(t)\\
&=P_{G_{k}(\mathbb{C}^{n})}(t^2)+t^{2k}P_{G_{k}(\mathbb{C}^{n-1})}(t^2)+t^{2n-2k}P_{G_{k-1}(\mathbb{C}^{n-1})}(t^2).
\end{align*} \end{cor} \begin{proof}
Notice that the $-1$-eigenbasis $\{e\cdot\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k-1}^{r_{n-k-1}}\mid \sum_{i=1}^{n-k-1} r_i \leq k\}\cup \{ \bar{e}\cdot p_1^{r_1}p_2^{r_2}\cdots p_{k-1}^{r_{k-1}}\mid \sum_{i=1}^{k-1} r_i \leq n-k\}$ has factors $\{\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k-1}^{r_{n-k-1}}\mid \sum_{i=1}^{n-k-1} r_i \leq k\}$ and $ \{p_1^{r_1}p_2^{r_2}\cdots p_{k-1}^{r_{k-1}}\mid \sum_{i=1}^{k-1} r_i \leq n-k\}$ which also appear as the additive basis of $H^*({G}_{2k}(\mathbb{R}^{2n-2}))$ and $H^*({G}_{2k-2}(\mathbb{R}^{2n-2}))$ respectively. \end{proof}
\begin{rmk}
The Poincar\'e series of even dimensional oriented Grassmannians $\tilde{G}_{2k}(\mathbb{R}^{2n}), \tilde{G}_{2k}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$ were already computed by H. Cartan \cite{Car50}. \end{rmk}
\begin{thm}[Ordinary Leray-Borel description of $\tilde{G}_{2k}(\mathbb{R}^{2n})$]
The ordinary cohomology of $\tilde{G}_{2k}(\mathbb{R}^{2n})$ is a ring extension of the ordinary cohomology of ${G}_{2k}(\mathbb{R}^{2n})$:
\[
H^*(\tilde{G}_{2k}(\mathbb{R}^{2n}))\cong \frac{H^*({G}_{2k}(\mathbb{R}^{2n}))[e,\bar{e}]}{e^2=p_k,\,\bar{e}^2=\bar{p}_{n-k},\,e\bar{e}=0} \cong \frac{\mathbb{Q}[p_1,p_2,\ldots,p_k;\bar{p}_1,\bar{p}_2,\ldots,\bar{p}_{n-k};e,\bar{e}]}{p\bar{p} = 1,\, e^2=p_k,\,\bar{e}^2=\bar{p}_{n-k},\,e\bar{e}=0}.
\] \end{thm} \begin{proof}
Consider the ring homomorphism
\[
\frac{H^*({G}_{2k}(\mathbb{R}^{2n}))[e,\bar{e}]}{e^2=p_k,\,\bar{e}^2=\bar{p}_{n-k},\,e\bar{e}=0} \longrightarrow H^*(\tilde{G}_{2k}(\mathbb{R}^{2n}))
\]
which sends Pontryagin classes of ${G}_{2k}(\mathbb{R}^{2n})$ to the corresponding Pontryagin classes of $\tilde{G}_{2k}(\mathbb{R}^{2n})$ and sends the abstract symbols $e,\bar{e}$ to the actual Euler classes of the oriented canonical bundle and complementary bundle over $\tilde{G}_{2k}(\mathbb{R}^{2n})$. Since we have proved that $H^*(\tilde{G}_{2k}(\mathbb{R}^{2n}))$ is generated by Pontryagin classes and Euler classes, the above morphism is surjective. It is easy check that $H^*({G}_{2k}(\mathbb{R}^{2n}))[e,\bar{e}]/\{e^2=p_k,\,\bar{e}^2=\bar{p}_{n-k},\,e\bar{e}=0\}$ also has the same additive basis $\{p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k\}\cup\{e\cdot\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k-1}^{r_{n-k-1}}\mid \sum_{i=1}^{n-k-1} r_i \leq k\}\cup \{ \bar{e}\cdot p_1^{r_1}p_2^{r_2}\cdots p_{k-1}^{r_{k-1}}\mid \sum_{i=1}^{k-1} r_i \leq n-k\}$ as $H^*(\tilde{G}_{2k}(\mathbb{R}^{2n}))$. Hence, we get a ring isomorphism. \end{proof}
Since $T^n$ acts on $\tilde{G}_{2k}(\mathbb{R}^{2n})$ equivariantly formal, i.e. $H_T^*(\tilde{G}_{2k}(\mathbb{R}^{2n}))\cong\mathbb{Q}[\alpha_1,\dots,\alpha_n]\otimes_\mathbb{Q} H^*(\tilde{G}_{2k}(\mathbb{R}^{2n}))$ as $\mathbb{Q}[\alpha_1,\dots,\alpha_n]$-modules, we can lift the ordinary basis, characteristic classes and relations to be equivariant, then obtain the equivariant versions of characteristic basis and Leray-Borel description:
\begin{cor}[Equivariant Leray-Borel description and characteristic basis of $\tilde{G}_{2k}(\mathbb{R}^{2n})$]
The equivariant cohomology of $\tilde{G}_{2k}(\mathbb{R}^{2n})$ is a ring extension of the equivariant cohomology of ${G}_{2k}(\mathbb{R}^{2n})$:
\begin{align*}
H_T^*(\tilde{G}_{2k}(\mathbb{R}^{2n}))
&\cong \frac{H_T^*({G}_{2k}(\mathbb{R}^{2n}))[e^T,\bar{e}^T]}{(e^T)^2=p^T_k,\,(\bar{e}^T)^2=\bar{p}^T_{n-k},\,e^T\bar{e}^T=\prod_{i=1}^{n}\alpha_i} \\
&\cong \frac{\mathbb{Q}[\alpha_1,\alpha_2,\ldots,\alpha_n][p^T_1,p^T_2,\ldots,p^T_k;\bar{p}^T_1,\bar{p}^T_2,\ldots,\bar{p}^T_{n-k};e^T,\bar{e}^T]}{p^T\bar{p}^T = \prod_{i=1}^{n}(1+\alpha^2_i),\,(e^T)^2=p^T_k,\,(\bar{e}^T)^2=\bar{p}^T_{n-k},\,e^T\bar{e}^T=\prod_{i=1}^{n}\alpha_i}
\end{align*}
with additive $\mathbb{Q}[\alpha_1,\alpha_2,\ldots,\alpha_n]$-basis $\{(p^T_1)^{r_1}(p^T_2)^{r_2}\cdots (p^T_k)^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k\}$ for the $+1$-eigenspace of $\rho^*$ and $\{e^T\cdot(\bar{p}^T)_1^{r_1}(\bar{p}^T)_2^{r_2}\cdots (\bar{p}^T)_{n-k-1}^{r_{n-k-1}}\mid \sum_{i=1}^{n-k-1} r_i \leq k\}\cup \{ \bar{e}^T\cdot (p_1^T)^{r_1}(p_2^T)^{r_2}\cdots (p_{k-1}^T)^{r_{k-1}}\mid \sum_{i=1}^{k-1} r_i \leq n-k\}$ for the $-1$-eigenspace. \end{cor}
\begin{rmk}
The ordinary cohomology groups of $\tilde{G}_k(\mathbb{R}^n)$ in $\mathbb{Z}$ coefficients for $n\leq 8$ were computed by Jungkind \cite{Ju79}. The ordinary cohomology rings of $\tilde{G}_k(\mathbb{R}^n)$ in $\mathbb{R}$ coefficients for $k=2$ were computed by Shi\&Zhou \cite{SZ14}. \end{rmk}
\subsubsection{Characteristic numbers of orientable Grassmannians} All the oriented Grassmannians are canonically oriented. Among the real Grassmannians, only $G_{2k}(\mathbb{R}^{2n})$ and $G_{2k+1}(\mathbb{R}^{2n+2})$ have nonzero top Betti numbers and hence are orientable. We can integrate equivariant cohomology classes on these Grassmannians using the Atiyah-Bott-Berline-Vergne(ABBV) localization formula \ref{ABBV}. According to the additive $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-module structures of equivariant cohomology of theses Grassmannians, we shall need to understand the integration of equivariant characteristic classes in various cases for any multi-index $I=(i_1,\ldots,i_k)$: $(p^T)^I,\,e^T\cdot(p^T)^I,\,\bar{e}^T\cdot(p^T)^I,\,r^T\cdot(p^T)^I,\,\tilde{r}^T\cdot(p^T)^I$.
The equivariant Pontryagin classes of canonical bundles, complementary bundles and tangent bundles are given in Prop \ref{prop:Pont}. The equivariant Euler classes of canonical bundles and complementary bundles are given in Subsubsection \ref{subsub:Euler}. The $r^T,\tilde{r}^T$ are given in Theorem \ref{thm:AllGrass} and Prop \ref{prop:CanoBaseIII}. In order to apply the ABBV formula, we need a localized expression for the equivariant Euler class of normal bundle at each fixed point or fixed circle.
\begin{prop}
Let $S$ be a $k$-element subset of $\{1,\ldots,n\}$, the equivariant Euler class of normal bundle at a fixed point or fixed circle associated to $S,S_\pm$ is
\begin{enumerate}
\item For $G_{2k}(\mathbb{R}^{2n}),\tilde{G}_{2k}(\mathbb{R}^{2n})$,
\[
e^N_{S_\pm} = e^N_S = \prod_{i\in S}\prod_{j \not \in S}(\alpha^2_j-\alpha^2_i).
\]
\item For $\tilde{G}_{2k}(\mathbb{R}^{2n+1})$,
\[
e^N_{S_\pm} = \pm \prod_{l\in S}\alpha_l \prod_{i\in S}\prod_{j \not \in S}(\alpha^2_j-\alpha^2_i).
\]
\item For $\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$,
\[
e^N_{S_\pm} = \pm \prod_{l \not\in S}\alpha_l \prod_{i\in S}\prod_{j \not \in S}(\alpha^2_j-\alpha^2_i).
\]
\item For ${G}_{2k+1}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$,
\[
e^N_S = \prod_{l=1}^n\alpha_l \prod_{i\in S}\prod_{j \not \in S}(\alpha^2_j-\alpha^2_i).
\]
\end{enumerate} \end{prop} \begin{proof}
The tangent spaces with weight decomposition at each fixed point or fixed circle of real Grassmannians (hence also oriented Grassmannians) are given in Subsubsection \ref{subsubsec:IsoWeights}, therefore we get the equivariant Euler classes of normal bundles up to signs as the expressions claimed in current Proposition. To resolve the sign ambiguity, we just need to note that the claimed expressions are invariant under the Weyl groups of the oriented Grassmannians as homogeneous spaces $G/H$, and also invariant under the deck transformation $\rho^*$. \end{proof}
Next, we will compute and relate equivariant characteristic numbers of different Grassmannians.
\begin{thm}[Equivariant characteristic numbers of real\&oriented Grassmannians]
Let $I=(i_1,\ldots,i_k)$ be a multi-index and $\mathcal{S}$ be the collection of all $k$-element subsets of $\{1,\ldots,n\}$, then
\begin{align*}
&\int_{\tilde{G}_{2k}(\mathbb{R}^{2n})}(p^T)^I=\int_{\tilde{G}_{2k}(\mathbb{R}^{2n+1})}e^T \cdot (p^T)^I=\int_{\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})}\bar{e}^T \cdot (p^T)^I\\
=&2\int_{{G}_{2k}(\mathbb{R}^{2n})}(p^T)^I=2\int_{{G}_{2k+1}(\mathbb{R}^{2n+2})}r^T\cdot (p^T)^I=2\int_{\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})}\tilde{r}^T\cdot (p^T)^I\\
=& 2 \sum_{S \in \mathcal{S}}\frac{\big((p_1^T)^{i_1}\cdots(p_k^T)^{i_k}\big)|_S}{\prod_{i\in S}\prod_{j \not \in S}(\alpha^2_j-\alpha^2_i)}.
\end{align*} \end{thm} \begin{proof}
When applying the ABBV localization formula \ref{ABBV}, besides the localized Pontryagin classes, we just need to observe that for ${G}_{2k}(\mathbb{R}^{2n})$, $\tilde{G}_{2k}(\mathbb{R}^{2n})$, $\tilde{G}_{2k}(\mathbb{R}^{2n+1})$ and $\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$, they respectively have
\begin{align*}
\frac{e^T_{S}}{e^N_{S}}&=\frac{1}{\prod_{i\in S}\prod_{j \not \in S}(\alpha^2_j-\alpha^2_i)} &
\frac{e^T_{S_\pm}}{e^N_{S_\pm}}&=\frac{1}{\prod_{i\in S}\prod_{j \not \in S}(\alpha^2_j-\alpha^2_i)}\\
\frac{e^T_{S_\pm}}{e^N_{S_\pm}}&=\frac{\pm \prod_{l\in S}\alpha_l}{\pm \prod_{l\in S}\alpha_l \prod_{i\in S}\prod_{j \not \in S}(\alpha^2_j-\alpha^2_i)} &
\frac{\bar{e}^T_{S_\pm}}{e^N_{S_\pm}}&=\frac{\pm \prod_{l\not \in S}\alpha_l}{\pm \prod_{l\not\in S}\alpha_l \prod_{i\in S}\prod_{j \not \in S}(\alpha^2_j-\alpha^2_i)}
\end{align*}
for ${G}_{2k+1}(\mathbb{R}^{2n+2}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$, they respectively have
\begin{align*}
\frac{\int r^T_S}{e^N_S} &=\frac{\prod_{l=1}^n\alpha_l \int_{S^1}\theta_{S^1}}{\prod_{l=1}^n\alpha_l \prod_{i\in S}\prod_{j \not \in S}(\alpha^2_j-\alpha^2_i)} &
\frac{\int \tilde{r}^T_S}{e^N_S} &=\frac{\prod_{l=1}^n\alpha_l \int_{\mathbb{R} P^1}\theta_{\mathbb{R} P^1}}{\prod_{l=1}^n\alpha_l \prod_{i\in S}\prod_{j \not \in S}(\alpha^2_j-\alpha^2_i)}.
\end{align*}
All these fractions are equal to $\frac{1}{\prod_{i\in S}\prod_{j \not \in S}(\alpha^2_j-\alpha^2_i)}$. The difference by factor of $2$ comes from the fact that $\tilde{G}_{2k}(\mathbb{R}^{2n})$, $\tilde{G}_{2k}(\mathbb{R}^{2n+1})$ and $\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$ have fixed points indexed by $S_\pm$, while ${G}_{2k}(\mathbb{R}^{2n})$, ${G}_{2k+1}(\mathbb{R}^{2n+2})$ and $\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$ have fixed points or circles indexed by $S$. \end{proof}
\begin{rmk}
When the cohomological degree of a characteristic polynomial matches with the dimension of a Grassmannian, or equivalently $\sum_{j=1}^{k}j\cdot i_j=k(n-k)$, then the equivariant characteristic number will be a constant, i.e. an ordinary characteristic number. Moreover, we then get a formula of the ordinary characteristic numbers by substituting any $\alpha_i=a_i \in \mathbb{R}$ such that $a_i\not =0, a_i \not = \pm a_j$ into the localized expression of ABBV formula. For instance, we can choose $\alpha_i=i,\forall i$, or $\alpha_i=\sqrt{i},\forall i$. Moreover, we have the relations between ordinary Pontryagin characteristic numbers:
\begin{align*}
&\int_{\tilde{G}_{2k}(\mathbb{R}^{2n})}p^I=\int_{\tilde{G}_{2k}(\mathbb{R}^{2n+1})}e \cdot p^I=\int_{\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})}\bar{e} \cdot p^I\\
=&2\int_{{G}_{2k}(\mathbb{R}^{2n})}p^I=2\int_{{G}_{2k+1}(\mathbb{R}^{2n+2})}r\cdot p^I=2\int_{\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})}\tilde{r}\cdot p^I.
\end{align*} \end{rmk}
\begin{rmk}
Recall that localized equivariant Pontryagin class can be obtained from localized equivariant Chern class by replacing $\alpha_i$ with $\alpha^2_i$. Let $Sq:\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$ be the ring homomorphism by sending $\alpha_i$ to $\alpha^2_i$ for every $i$. Then we have
\[
\int_{{G}_{2k}(\mathbb{R}^{2n})}(p^T)^I = Sq\big(\int_{{G}_{k}(\mathbb{C}^{n})}(c^T)^I\big).
\]
Since the ring homomorphism $Sq$ keeps rational numbers unchanged, we have the relation of ordinary Pontryagin numbers and Chern numbers
\[
\int_{{G}_{2k}(\mathbb{R}^{2n})}p^I = \int_{{G}_{k}(\mathbb{C}^{n})}c^I = \sum_{S\in \mathcal{S}} \frac{e^{i_1}_1(S)\cdots e^{i_k}_k(S)}{\prod_{i\in S}\prod_{j\not\in S} (j - i)}
\]
where the second identity is given in Cor \ref{thm:OrdChernNum}. \end{rmk}
\begin{rmk}
Consider the $2$-covers of orientable Grassmannians $\pi: \tilde{G}_{2k}(\mathbb{R}^{2n}) \rightarrow {G}_{2k}(\mathbb{R}^{2n})$ and $\pi: \tilde{G}_{2k+1}(\mathbb{R}^{2n+2}) \rightarrow {G}_{2k+1}(\mathbb{R}^{2n+2})$. By the naturality of equivariant Pontryagin classes and the above relations among equivariant characteristic numbers, we see
\[
\int_{\tilde{G}_{2k}(\mathbb{R}^{2n})}\pi^*\big((p^T)^I\big)=\int_{\tilde{G}_{2k}(\mathbb{R}^{2n})}(p^T)^I = 2\int_{{G}_{2k}(\mathbb{R}^{2n})}(p^T)^I.
\]
From Prop \ref{prop:Pullr}, we have $ \pi^*{r}^T=2\tilde{r}^T$ , then
\[
\int_{\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})}\pi^*\big({r}^T\cdot (p^T)^I\big) = 2\int_{\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})}\tilde{r}^T\cdot (p^T)^I = 2\int_{{G}_{2k+1}(\mathbb{R}^{2n+2})}r^T\cdot (p^T)^I.
\] \end{rmk}
\vskip 20pt
\end{document} | arXiv |
What is the largest $n$ such that $a = 2^{306} \cdot 3^{340}$ is a perfect $n$th power?
We claim that $a$ is a perfect $n$th power if and only if $n$ divides both $306$ and $340$. To see this, suppose that $n \mid 306$ and $n \mid 340$. Then $2^{\frac{306}{n}} 3^{\frac{340}{n}}$ is an integer whose $n$th power is $a$. Conversely, suppose $b^n = a$. Then the only primes which divide $b$ are $2$ and $3$. Choose $c$ and $d$ so that $b=2^{c} 3^{d}$. Then $b^n = 2^{cn} 3^{dn} = 2^{306} 3^{340}$, which implies $n \mid 306$ and $n \mid 340$. This concludes our proof of the claim that $a$ is an $n$th power if and only if $n$ divides both $306$ and $340$.
The largest number which simultaneously divides two numbers is their GCD. Using the Euclidean algorithm, the GCD of $306$ and $340$ is the same as the GCD of $340$ and $340-306 = 34$. Since $34$ divides $340$, the GCD of these two is $34$, so the largest possible $n$ is $\boxed{34}$. | Math Dataset |
\begin{document}
\begin{frontmatter} \title{A zero-inflated Bayesian nonparametric approach for identifying differentially abundant taxa in multigroup microbiome data with covariates}
\runtitle{ZIBNP: Bayesian differential analysis for microbiome data with covariates}
\begin{aug}
\author[A]{\fnms{Archie}~\snm{Sachdeva}\ead[label=e1]{[email protected]}}, \author[A]{\fnms{Somnath}~\snm{Datta}\ead[label=e2]{[email protected]}\orcid{0000-0000-0000-0000}} \and \author[A]{\fnms{Subharup}~\snm{Guha}\ead[label=e3]{[email protected]}}
\address[A]{Department of Biostatistics, University of Florida\printead[presep={,\ }]{e1,e2,e3}}
\end{aug}
\begin{abstract} Scientific studies in the last two decades have established the central role of the microbiome in disease and health. Differential abundance analysis aims to identify microbial taxa associated with two or more sample groups defined by attributes such as disease subtype, geography, or environmental condition. The results, in turn, help clinical practitioners and researchers diagnose disease and develop new treatments more effectively. However, detecting differential abundance is uniquely challenging due to the high dimensionality, collinearity, sparsity, and compositionality of microbiome data. Further, there is a critical need for unified statistical approaches that can directly compare more than two groups and appropriately adjust for covariates. We develop a zero-inflated Bayesian nonparametric (ZIBNP) methodology that meets the multipronged challenges posed by microbiome data and identifies differentially abundant taxa in two or more groups, while also accounting for sample-specific covariates. The proposed hierarchical model flexibly adapts to unique data characteristics, casts the typically high proportion of zeros in a missing data framework, and mitigates high dimensionality and collinearity issues by utilizing the dimension reducing property of the semiparametric Chinese restaurant process. The approach relates the microbiome sampling depths to inferential precision and conforms with the compositional nature of microbiome data. In simulation studies and in the analyses of the CAnine Microbiome during Parasitism (CAMP) dataset on infected and uninfected dogs and a Global Gut Microbiome dataset
on human subjects belonging to three geographical regions, we compare ZIBNP with established statistical methods for differential abundance analysis in the presence of covariates. \end{abstract}
\begin{keyword} \kwd{Missing data} \kwd{Chinese restaurant process} \kwd{Compositional data} \kwd{Stochastic imputation} \kwd{MCMC} \kwd{ZIBNP} \end{keyword}
\end{frontmatter}
\section{Introduction}\label{S:introduction}
During the last two decades, numerous scientific studies have investigated the role of microbiome in disease diagnostics and therapeutics to establish the influence of
microbiome variations on health and disease \citep{WADE2013137,Shreiner2015,rautava_2016,Tang2019}. Giant strides in next-generation sequencing (NGS) technologies have enabled the simultaneous identification and quantification of relative abundances of all microbial taxa in study samples, offering a glimpse into the microbiome in our bodies \citep{Tang2019}.
Most study samples are sequenced using either amplicon or random shotgun sequencing. The former sequencing technique amplifies the highly variable regions of the 16S rRNA gene \citep{Kembel_16S2012}. Shotgun metagenomic sequencing covers all genetic variation in the DNA compared to the specific regions targeted by amplicon sequencing. Consequently, shotgun sequencing achieves better species-level resolution and ability to detect novel viruses \citep{BIBBY2013275,Relman2013}, but requires much larger volumes \citep{Sharpton_2014}. We refer the reader to \cite{Bharti2019} and \cite{Xia2018_ch1} for a comprehensive overview, along with the pros and cons, of different microbiome sequencing technologies. In general, raw microbiome sequence data are analyzed using a bioinformatics pipeline followed by taxonomic profile generation \citep{Samuel_2021,Bharti2019}, which maps sequences to known reference databases to obtain a high-dimensional vector of counts for each sample \citep{Schloss2011}. Each vector element corresponds to a \textit{taxon}, a generic term referring to any profiled set of microbiome features, e.g., operational taxonomic units (OTUs), amplicon sequence variants (ASVs), and metagenomic species. The data are arranged in an \textit{abundance table}, or matrix of microbiome counts, with each cell consisting of the abundance of a taxon (column) in a sample (row).
High-throughput sequencing technologies do not measure the actual taxa counts in a sample, but instead, measure the counts relative to the total number of reads \citep{ Aitchison1982}, referred to as the \textit{library size} or \textit{sampling depth}. These \textit{compositional} observations can be regarded as random points on a simplex \citep{aitchison_2008}. The sampling depths vary from sample to sample due to technical variations of the sequencing process, and are often normalized in different ways to enable meaningful group comparisons \citep{McKnight2019,Weiss2017}.
Microbiome data are also highly \textit{sparse} and are typically comprised of between $10\%$ and $70\%$ zeros. Some zeros may be caused by individual traits; for instance, low-fiber diets deplete certain gut bacteria \citep{diet2020}. However, most zeros occur due to sequencing process errors and highly variable sampling depths. These three types of zeros are referred to in the literature as biological, sampling, and technical zeros, respectively \citep{SILVERMAN20202789, Kaul2017,Weiss2017}.
The primary goal of the proposed approach is \textit{differential abundance analysis} or the identification of microbial taxa highly associated with multiple subpopulations of individuals or \textit{groups} of samples. Identifying differentially abundant (DA) taxa may help in developing novel treatments and support clinical practitioners in the effective diagnoses of diseases. The problem is especially challenging because of the distinctive characteristics of microbiome data, such as high dimensionality, collinearity, sparsity, and compositionality, that necessitate sophisticated statistical methods. Furthermore, it is critically important to appropriately adjust for individual-specific attributes in differential analysis \citep{Vujkovic-Cvijin2020}; for example, the diet and lifestyle of individuals significantly affect their microbial compositions and are potentially associated with disease subtypes as the groups.
An overview of state-of-the-art statistical methods for analyzing microbiome data can be found in \cite{Weiss2017}, \cite{Wallen2021}, and \cite{dattaGuha2021}. Statistical approaches for differential abundance analysis can be broadly classified as \textit{compositional} and \textit{count-based} \citep{Liu2021,Nearing2022}. Prominent among the compositional approaches are ALDEx2 \citep{Fernandes2014} and ANCOM-II \citep{Kaul2017}. ALDEx2 relies on Dirichlet-multinomial model to infer abundance from counts and performs centered log-ratio (clr) transformation, and ANCOM-II uses additive log-ratio (alr) transformations. Both methods use Wilcoxon rank-sum tests to detect the differential taxa among the groups.
On the other hand, count-based approaches recognize that the sampling depths are informative about measurement precision and are able to effectively capture zero-inflation and overdispersion of the abundances. Zero-inflated count approaches include metagenomeseq \citep{Paulson2013}, which uses cumulative sum scaling to compute an appropriate quantile, log-transforms the data after adding a pseudo count of 1, and models data sparsity by fitting a zero-inflated log-normal model. ANCOMBC \citep{Lin2020} uses a linear regression framework on the log scale with a bias-corrected random intercept. For two-group comparisons, MaAsLin2 \citep{maaslin2_2021} uses the GLM framework to analyze the data with a variety of options for normalization and probability distributions for the counts. The methods DESeq2 \citep{Love2014} and corncob \citep{Martin2020} model the counts using negative binomial and beta-binomial distributions, respectively, and assume a marginal distribution framework for the per-feature counts for each sample. Zero-inflated negative binomial approaches \citep{Risso2018, Xia2018_bookCH_ZI} are motivated by sparsity and overdispersion issues. However, few statistical methods perform reliably for different microbiome datasets~\citep{Weiss2017}.
This paper proposes a novel unified zero-inflated Bayesian nonparametric (ZIBNP) approach that effectively meets the multipronged challenges of high-dimensionality, collinearity, sparsity, and covariate confounding to accurately detect the DA taxa across multiple groups of study samples. As its name suggests, the hierarchical model comprises a two-component mixture model; one mixture component is a point mass at zero, and the other component is a flexible Bayesian nonparametric model whose posterior adapts to the specific characteristics of the microbiome dataset. Technical and sampling zeros are cast in a missing data framework and stochastically inferred as a by-product of the posterior inferences. High dimensionality and collinearity (specifically, small $n$, large $p$ issues) are mitigated by allocating the large number of taxa to fewer latent clusters defined by shared relative abundance patterns and induced by the Chinese restaurant process \citep{MullerMitra2013,Lijoi_Prunster_2010}. The approach accounts for the available sample-specific covariates in a manner that relates the varying sampling depths to inferential precision while also conforming with the compositional interpretation of microbiome data.
We apply the ZIBNP technique to analyze two microbiome datasets and infer the set of DA taxa while adjusting for covariates: \textit{(i)} The CAnine Microbiome during Parasitism (CAMP) study \citep{mdb} explores the impact of natural parasitic infection on the gut microbiome of domesticated dogs. In addition to microbiome abundances, the data comprises animal-specific covariates and two groups of dogs determined by infection status (infected or uninfected); and \textit{(ii)} Global Gut Microbiome data \citep{Yatsunenko2012} on human subjects belonging to three geographical regions (groups).
The paper is organized as follows. Section \ref{S:model} provides an overview of our proposed ZIBNP model. Section \ref{S:BNP} develops the nonparametric aspects responsible for achieving dimension reduction, incorporating covariate effects, and detecting the DA taxa. Section~\ref{S:missing data} introduces the missing data framework for stochastically imputing technical zeros. Section~\ref{S:inference} outlines the posterior inference and MCMC procedure for detecting DA taxa. The simulation study of Section \ref{S:simulation2}, demonstrates the high accuracy achieved by the proposed method relative to some well-established analytical approaches capable of adjusting for covariates. Section \ref{sec:dataAnalysis} analyzes the motivating microbiome datasets using the ZIBNP method. Section \ref{S:discussion} concludes with a brief discussion.
\section{A Novel Bayesian Hierarchical Approach} \label{S:model}
For $n$ samples and $p$ taxa, the microbiome data are arranged in an $n \times p$ matrix, with each cell representing the observed count or abundance of a taxon (column) in a particular study sample or subject (row). Without loss of generality, we assume that taxa corresponding to entire columns of $n$ zeros were previously eliminated during preprocessing to give $p$ columns of data with at least non-zero entry each. Denote the taxa abundance matrix by ${\boldsymbol{Z}} = (({Z}_{ij}))$, where $Z_{ij}$ is the number of reads of taxon $j$ in sample $i$, and $\boldsymbol{Z}_i=(Z_{i1},\ldots,Z_{ip})'$ is the vector of taxa abundances for subject $i$.
For $K\ge 2$, there are $K$ groups determined by a categorical attribute partitioning the samples, e.g., disease status, disease subtype, or geographical region. The group membership of sample~$i$ is denoted by $k_i$ and $n_k$ samples belong to the $k$th group. In addition to the attribute defining the groups, there are $T$ covariates, arranged in a matrix $\boldsymbol{X}=((X_{it}))$.
With ${L_i}$ representing sequencing depth or library size, the \textit{relative abundance} vector for the $i$th subject, denoted by $\hat{\boldsymbol{q}}_i=(\hat{q}_{i1},\ldots,\hat{q}_{ip})'$, is computed as $\hat{\boldsymbol{q}}_i=\boldsymbol{Z}_i/L_i$.
We wish to identify the set of DA microbial {taxa}. To accomplish this end-goal, we foster a statistical framework capable of quantifying the complex relationships between microbial communities and the grouping factor of interest. In addition to the challenges posed by high dimensionality and sparsity, the association is confounded by sample-specific characteristics or covariates that produce spurious microbial associations \citep{Vujkovic-Cvijin2020}.
We define a \textit{differential status variable} $\tilde{h}_j$, with $\tilde{h}_j=2$ ($\tilde{h}_j=1$) signifying that taxon $j$ is (not) DA. The set of DA taxa is then \begin{equation*}
\mathscr{D} = \left\{j: \tilde{h}_{j}=2, \,\, j=1,\ldots,p\right\}, \end{equation*}
and posterior inferences about unknown parameters $\tilde{h}_1,\ldots,\tilde{h}_p$ are of interest.
As discussed in Section \ref{S:introduction}, \textit{biological zeros} occur due to the actual absence of a taxon for a group, and manifest as blocks of taxa with zeros for an entire group of subjects. Therefore, these taxa are labeled as DA without further analysis. \textit{Technical zeros} are related to issues such as batch effects \citep{batchEffects2019} often encountered during the processing and sequencing of samples. \textit{Sampling zeros} randomly occur due to a taxon not being measured in a sample, e.g., because of small sampling depths. Analyzing the non-biological sources of sparsity is less straightforward and requires carefully designed statistical methods.
Consequently, our statistical framework primarily focuses on technical and sampling zeros. The proposed ZIBNP model relies on a two-component mixture for each element of $\boldsymbol{Z}$, namely, a point mass at zero (representing technical zeros) and a Bayesian nonparametric (BNP) model for the counts, denoted by $\mathscr{F}$, under which sampling zeros may stochastically occur. More formally, with $\mathscr{F}_{ij}$ denoting the marginal distribution of $Z_{ij}$ under BNP model $\mathscr{F}$, and $I_{\{0\}}$ representing a point mass at~0, the ZIBNP model assumes that \begin{equation}
Z_{ij} \overset{\text{indep}}{\sim} r_{ij} I_{\{0\}} + (1-r_{ij}) \mathscr{F}_{ij},\label{eq:mixture}
\end{equation} where $r_{ij}$ is the probability of a technical zero in sample $i$ and taxon $j$ and $\boldsymbol{R}=((r_{ij}))$ denotes the $n \times p$ matrix of probabilities of technical zeros. The second mixture component accounts for sampling zeros. Section \ref{S:BNP} develops BNP model $\mathscr{F}$. Section \ref{S:technical zeros} details the stochastic process for technical zeros.
\subsection{Bayesian nonparametric (BNP) model, $\mathscr{F}$}\label{S:BNP}
The taxa abundances are modeled as \begin{equation}
{\boldsymbol{Z}}_i \mid \boldsymbol{q}_i \overset{\text{indep}}{\sim} \text{Multinomial}({L_i}, \boldsymbol{q}_i), \quad i=1,\ldots, n, \label{eq:Zi} \end{equation}
where $\boldsymbol{q}_i=(q_{i1},\ldots,q_{ip})'$ is a probability $p$-tuple, i.e., $\sum_{j=1}^{p}q_{ij}=1$. The columns of the unknown row-stochastic matrix $\mathbf{Q}=((q_{ij}))$ are denoted by $\tilde{\boldsymbol{q}}_j$, $j=1,\ldots,p$. The marginal probability of a sampling zero under $\mathscr{F}$ is then \begin{equation}P\bigl[Z_{ij} =0\mid \mathscr{F}\bigr] = (1-q_{ij})^{L_i}.\label{eq:sampling0} \end{equation}
\paragraph*{Dimension reduction} Microbiome data usually involves far fewer samples than taxa, i.e., $n << p$, resulting in severe collinearity in the abundance matrix $\boldsymbol{Z}$. Model $\mathscr{F}$ mitigates collinearity by detecting lower-dimensional structure in the matrix columns and reducing them to $C$ unique motifs, where $C$ is unknown and far smaller than $p$.
Specifically, the model performs unsupervised clustering of the taxa into $C$ latent clusters.
For an allocation variable $c_j$, let $\{c_j=u\}$ symbolize the random event that the $j${th} taxon is allocated to the $u${th} latent cluster.
Let $m_u = \sum_{j=1}^{p}I(c_j =u)$ be the number of taxa in cluster $u$, so that $\sum_{u=1}^{C}m_u=p$. We associate the $u$th latent cluster with a common cluster-specific latent vector, $\boldsymbol{q}_u^*=(q_{1u}^*,\ldots,q_{nu}^*)'$. The columns of matrix $\mathbf{Q}$ corresponding to the $m_u$ taxa are assumed to be identically equal to $\boldsymbol{q}_u^*$ for clusters $u=1,\ldots,C$. That is,
\begin{equation}
\tilde{\boldsymbol{q}}_j=\boldsymbol{q}_{c_j}^*, \quad\text{for all taxa $j=1,\ldots,p$}. \label{eq:qtilde} \end{equation} This formulation eliminates redundancy in the large number of taxa by detecting patterns in the matrix $\mathbf{Q}$ columns, with the latent vectors $\boldsymbol{q}_{1}^*,\ldots,\boldsymbol{q}_{C}^*$ representing the cluster-specific motifs or common across-sample patterns shared by all taxa belonging to a cluster. Then, conditional on the vector $\boldsymbol{c}=(c_1, \ldots, c_p)$,
matrix $\mathbf{Q}$ is fully determined by $\mathbf{Q}^*=((q_{iu}^{*}))$ of dimension $n$ by $C$.
In conjunction with model~(\ref{eq:Zi}), this implies that all taxa $j$ belonging to a cluster have similar, but not necessarily identical, relative abundances $\hat{q}_{1j},\ldots,\hat{q}_{nj}$. Since $\mathbf{Q}$ is row-stochastic, each row of matrix $\mathbf{Q}^*$ satisfies
\begin{equation}
\sum_{u=1}^{C} m_u q_{iu}^* =1, \quad i=1,\ldots,n, \label{eq:constraint}
\end{equation}
implying that cluster motifs $\boldsymbol{q}_{1}^*,\ldots,\boldsymbol{q}_{C}^*$ are dependent.
Allocation vector $\boldsymbol{c}$ is given the semiparametric Chinese restaurant process (CRP) prior with precision or mass parameter $\alpha_c$ \citep{MullerMitra2013} because of its ability to achieve dimension reduction \citep[e.g., ][]{muller2013bayesian, Medvedovic_etal_2004,Kim_etal_2006,Dunson_Park_2008,guha2016nonparametric,guha2022predicting}. With $[\cdot]$ representing densities, we have \begin{equation}
[\boldsymbol{c}] \sim \frac{\Gamma(\alpha_c) \alpha_c^C}{\Gamma(\alpha_c + p)} \prod_{u=1}^{C} \Gamma(m_u), \quad \boldsymbol{c} \in \mathscr{P}_p, \label{eq:CRP} \end{equation}
where $\mathscr{P}_p$ is the set of all partitions of $p$ taxa into $C$ \textit{non-empty} clusters, and $1 \le C\le p$ is random.
In general, CRPs achieve dimension reduction because the random number of clusters, $C$, is asymptotically equivalent to $\alpha_c \log(p)$ as $p \to \infty$ \citep{Lijoi_Prunster_2010}.
Since matrix $\mathbf{Q}$ is a ``less noisy'' version of the relative abundance matrix, a consequence of model assumption (\ref{eq:qtilde}) is that differential statuses are shared by all taxa belonging to a latent cluster, which is itself collectively DA or non-DA. Analogously to the taxon differential status, we envision \textit{cluster differential status variables} $h_1,\ldots,h_C$, with the DA taxa equivalently given by \begin{equation}
\mathscr{D} = \left\{j: h_{c_j}=2, \,\, j=1,\ldots,p\right\}. \label{eq:D} \end{equation}
\paragraph*{Incorporating covariates}
We model the dependencies between the latent vectors $\boldsymbol{q}_{1}^*,\ldots,\boldsymbol{q}_{C}^*$ and the sample-specific covariates. First, we preselect and hold fixed a singleton cluster consisting only of a \textit{reference taxon}; strategies for choosing this special taxon are discussed below. Without loss of generality, the reference cluster and reference taxon are labeled $c=1$ and $j=1$, respectively. Next, since microbiome abundance data are compositional, we model the log ratios
\begin{align}
\eta_{iu} =\log(\frac{q_{iu}^*}{q_{i1}^*}) \stackrel{\text{indep}}\sim N\biggl(\beta_{0k_iu} + \sum_{l=1}^Tx_{il} \beta_{lk_iu}, \,\, \sigma_e^2\biggr), \quad\text{$i=1,\ldots,n$, and $u>1$,} \label{eq:eta} \end{align} where the latent elements are identifiable due to constraint (\ref{eq:constraint}). We assume that $\sigma_e^2$ is small enough that most of the variability in the $\eta_{iu}$'s is explained by the covariates. In the (unrealistic) situation in which the covariate effects are negligible, the latent elements $q_{iu}^*$ of all samples $i$ of a group are approximately equal.
The cluster- and group-specific regression parameters, $\boldsymbol{\beta}_{ku}=(\beta_{0ku},\ldots,\beta_{Tku})' \in \mathcal{R}^{T+1}$, quantify the covariate relationships for samples belonging to group $k$ and taxa belonging to non-reference clusters~$u=2,\ldots,K$. Since the regression vectors $\boldsymbol{\beta}_{1u},\ldots, \boldsymbol{\beta}_{Ku}$ adjust for sample characteristics, the shared differential status of each non-reference cluster $u$ and its member taxa are immediately available upon inspection of the regression vectors, as we discuss in the sequel.
Let $\boldsymbol{X}^{\dag}=[\mathbf{1}_n: \boldsymbol{X}]$ be the matrix of dimension $n$ by $(T+1)$, and let column vector $\nu\mathbf{1}_M=(\nu,\ldots,\nu)'$. With $\delta_{\boldsymbol{\mu}}$ denoting a point mass at $\boldsymbol{\mu}$ and $\text{Dir}_{M}(\boldsymbol{\gamma})$ denoting a Dirichlet distribution in $\mathcal{R}^M$ for a parameter vector $\boldsymbol{\gamma}$ with positive elements, the $(T+1)$-variate regression parameters
are assigned a $M$-component finite mixture hierarchical prior: \begin{align}
&\boldsymbol{\beta}_{ku} \stackrel{\text{i.i.d.}}\sim \sum_{m=1}^{M} \pi_m \delta_{\boldsymbol{\mu}_m}, \quad \text{$k=1,\ldots,K$, and $u> 1$, where}\label{eq:beta}\\
&\boldsymbol{\pi} =(\pi_1,\ldots,\pi_M)' \sim \text{Dir}_{M}\bigl(\frac{\alpha_0}{M}\mathbf{1}_M\bigr),\notag\\
&\boldsymbol{\mu}_m \stackrel{\text{i.i.d.}}\sim N_{T+1}\bigl(\mathbf{0},\tau^2 ({\boldsymbol{X}^{\dag}}^T\boldsymbol{X}^{\dag})^{-1}\bigr), \quad m=1,\ldots,M,\label{eq:mum_prior} \end{align} and $\tau^2$ follows an inverse-gamma hyperprior with parameters $a_{\tau}$ and $b_{\tau}$. The finite mixture prior is key because it allows the borrowing of strength between the latent clusters and groups through shared regression parameter vectors $\boldsymbol{\beta}_{ku}$. More specifically, \textit{membership variable} $v_{ku}$ records the $M$-mixture component from which regression vector $\boldsymbol{\beta}_{ku}$ is drawn. That is, \begin{equation}
\boldsymbol{\beta}_{ku}= \boldsymbol{\mu}_{v_{ku}}, \quad k=1,\ldots,K; u=1,\ldots,C.\label{eq:v_ku} \end{equation}
\noindent \textit{Choosing the reference taxon} \quad
Since they adjust for covariates, the group-specific regression coefficients in equation (\ref{eq:eta}) are informative about the unknown cluster differential statuses. However, because of assumption (\ref{eq:eta}), the differential status of the reference cluster is inextricably related to any criterion for calling the DA and non-DA clusters. Hence, by convention, we require that the reference taxon be non-DA. These are some options for choosing a reference taxon: \begin{itemize}
\item \textit{Minimum variance taxon} \quad Following \cite{Nearing2022}, select the taxon $j$ with the smallest variance of the relative abundances $\hat{q}_{1j},\ldots,\hat{q}_{nj}$. Then reorder the taxa so that the reference taxon and its singleton cluster have the label 1.
\item \textit{Artificial reference taxon} \quad Augment abundance matrix $\boldsymbol{Z}$ with an artificial taxon with label 1 and unit abundance counts for all samples. Increase $p$ by 1. Since the sampling depths are large, this additional ``taxon'' is likely to be non-DA and will not significantly change the analyses. \end{itemize}
We have used the second option in the simulation study and data analyses.
\paragraph*{Detecting differential statuses of taxa} The $K(C-1)$ regression coefficient vectors in expressions~(\ref{eq:eta}) and (\ref{eq:beta}) adjust for covariate effects to reveal the DA clusters. Furthermore, as discussed, the finite mixture prior in (\ref{eq:beta}) causes the regression vectors to be tied with positive probability. The common differential statuses of the non-reference clusters and their allocated taxa are then inferred in a straightforward member. Specifically, for non-reference clusters $u=2,\ldots,C$:
\begin{align} h_u= \begin{cases} 1 \qquad\text{if } \boldsymbol{\beta}_{1u}=\dots=\boldsymbol{\beta}_{Ku}, \\ 2 \qquad \text{otherwise}. \end{cases} \label{eq:DA} \end{align} In other words, a non-reference cluster is non-DA if and only if its $K$ regression vectors are identical. By design, the reference cluster (taxon) is always non-DA. Further, the differential statuses of the taxa are identical as their parent clusters by (\ref{eq:D}).
These ideas are demonstrated by a toy example in Figure \ref{fig:toyDA}, where we have $n=7$ subjects belonging to $K=2$ groups (adults and children), $p=50$ taxa, and $T$ subject-specific covariates. The taxa are allocated to $C=8$ CRP clusters. Vector $\boldsymbol{x}^{\dag}_i \in \mathcal{R}^{T+1}$ denotes the $i$th row of the $n$ by $(T+1)$ matrix $\boldsymbol{X}^{\dag}$, and $\epsilon_{iu} \stackrel{\text{i.i.d.}}\sim N(0, \sigma_e^2)$. Cluster 1, containing only reference taxon 1, is not represented in the figure. For the seven non-reference clusters, $K(C-1)=14$ vectors $\boldsymbol{\beta}_{ku}$, each consisting of $(T+1)$ regression coefficients including the intercept, arise from a finite mixture model (\ref{eq:beta}) with $M=5$ multivariate components. The colors in Figure \ref{fig:toyDA} represent distinct components of the finite mixture model. For example, the regression vectors $\boldsymbol{\beta}_{14}$, $\boldsymbol{\beta}_{24}$, and $\boldsymbol{\beta}_{28}$ are equal because they are drawn from the same mixture component (shown in blue). Applying expression (\ref{eq:DA}), cluster $4$ is non-DA, whereas clusters 2, 3, and 8 are DA. All taxa share the same differential attributes as their parent clusters, e.g., taxa 4, 5, 7, and 9 belonging to cluster 4 are all non-DA, whereas taxa 8 and 23 belonging to cluster 3 are~DA.
\begin{figure}
\caption{Cartoon illustration of the procedure for calling the taxa differential statuses. For $n=7$ subjects, there are $K=2$ groups (3 adults and 4 children), $p=50$ taxa, $T$ covariates, and $C=8$ latent clusters. Cluster~1, containing only reference taxon 1, is non-DA and not shown here. The colors represent the components of a finite mixture model (FMM) from which the group-cluster regression coefficients, $\boldsymbol{\beta}_{ku} \in \mathcal{R}^{T+1}$, for group $k=1,2$ and non-reference cluster $u> 1$ are drawn. The taxa allocated to clusters 2, 3, and 8 (i.e., taxa 2, 3, 6, 8, 10, 12, 23, and 50) are DA because, for all these clusters, the regression vectors of adults and children correspond to different FMM components or colors. Taxa 4, 5, 7, and 9, allocated to cluster 4, are non-DA because the cluster's regression vectors for adults and children, $\boldsymbol{\beta}_{14}$ and $\boldsymbol{\beta}_{24}$ respectively, are identical (blue). See the text for further details.}
\label{fig:toyDA}
\end{figure}
\subsection{Modeling technical zeros}\label{S:technical zeros}
Returning to the first mixture component in expression (\ref{eq:mixture}), we model $r_{ij}$, the probability of technical zeros in sample $i=1,\ldots,n$, and taxon $j=1,\ldots,p$. The log-sampling depth and covariates are important predictors of the proportion of technical zeros in a sample \citep{technicalZero2020}. For random effects vector, $\boldsymbol{\lambda}_{iu}=(\lambda_{0iu},\ldots,\lambda_{T+1,iu})'$, accounting for cluster-specific effects shared by the taxa, we assume that
\begin{equation}
\text{logit}(r_{ij}) = \lambda_{0ic_j} + \sum_{t=1}^{T} \lambda_{tic_j} x_{it} + \lambda_{T+1,ic_j} \log(L_i), \quad i=1,\ldots,n, \label{eq:zgp} \end{equation} where $\lambda_{tiu} \overset{\text{i.i.d.}}{\sim} N(0,\tau_{\lambda}^2)$ for all $t$, $i$, and $u$. In addition to the covariates, expression~(\ref{eq:zgp}) includes log-sampling depth as a known predictor of the logit-likelihood of technical zeros \citep{Jiang2021}. Denote by $\boldsymbol{\Lambda}$ the collection of cluster-specific random matrices $\Lambda_1, \ldots, \Lambda_C$, where $\Lambda_u=((\lambda_{tiu}))$ is the $n \times (T+2)$ matrix of regression coefficients for the $u$th cluster in equation (\ref{eq:zgp}). Because all $m_u$ taxa belonging to the $u$th cluster have the same probability of technical zeros for a given sample, we write \[r_{ic_j}^* = r_{ij}, \quad i=1,\ldots,n; \,j=1,\ldots,p.\]
From expressions (\ref{eq:mixture}) and (\ref{eq:sampling0}), and applying the CRP taxon-to-cluster allocations, the probability of a non-biological (i.e., either technical or sampling) zero in taxon $j$ of sample $i$ is \begin{equation}
P(Z_{ij}=0) = r_{ic_j}^* + (1-r_{ic_j}^*) (1-q_{ic_j}^*)^{L_i}.
\end{equation}
Due to the lower-dimensional structure imposed by submodel $\mathscr{F}$ and equation (\ref{eq:zgp}), the abundance matrix columns of all taxa in a latent cluster display similar zero patterns. This feature of the ZIBNP model helps overcome the challenges of data sparsity in accurately detecting the latent clusters.
\subsection{A missing data framework for technical zeros}\label{S:missing data}
BNP submodel $\mathscr{F}$ accounts for sampling zeros that are relevant to detecting DA taxa. By contrast, since technical zeros are related to sequencing process errors, they obfuscate the submodel $\mathscr{F}$ parameters in expression (\ref{eq:mixture}) and are not informative about the differential statuses of the taxa. Consequently, the ZIBNP model is cast in a missing data framework to make accurate inferences about the differential patterns of the groups.
Specifically, we interpret expression (\ref{eq:mixture}) as
an independent mechanism that, with probability $r_{ic_j}^*$, converts the abundance count, $\Tilde{Z}_{ij}$, arising from submodel $\mathscr{F}$ to a technical zero. We denote the true abundance matrix corresponding to $\mathscr{F}$ by $\Tilde{\boldsymbol{Z}}=((\Tilde{Z}_{ij}))$. Inferences about parameters directly relevant to the DA taxa rely on the partially latent matrix $\Tilde{\boldsymbol{Z}}$ rather than the observed matrix $\boldsymbol{Z}$.
Let $\delta_{ij}$ represent the missingness indicator that $\Tilde{Z}_{ij}$ is missing, so that $\delta_{ij} \stackrel{\text{indep}}\sim \text{Bernoulli}(1-r_{ic_j}^*)$. The observed abundance counts are then related to the true counts as \begin{equation*}
Z_{ij} =
\begin{cases}
\Tilde{Z}_{ij} & \text{ if } \delta_{ij}=1 ,\\
0 & \text{ if } \delta_{ij}=0.
\end{cases} \end{equation*} If $Z_{ij}>0$, we clearly have $\delta_{ij}=1$. On the other hand,
$Z_{ij} = 0$ may correspond to a sampling zero ($\delta_{ij}=1$) or technical zero ($\delta_{ij}=0$). Unlike standard missing data approaches, indicator $\delta_{ij}$ is unknown for zero abundances, and the probability of a technical zero is \begin{equation}P\bigl[\delta_{ij}=0 \mid Z_{ij} = 0\bigr]=\frac{r_{ic_j}^*}{r_{ic_j}^* + (1-r_{ic_j}^*) (1-q_{ic_j}^*)^{L_i}}, \label{eq:technical?} \end{equation} with the sampling zero probability given by the complementary event. Since the sampling depths are usually large, a technical zero is much more likely than a sampling zero, unless $q_{ic_j}^*$ is very small. In any case, equation (\ref{eq:technical?}) can be applied to iteratively call the technical and sampling zeros (i.e., a posteriori generate the unknown missingness indicators) in the MCMC~procedure outlined in Section \ref{S:MCMC}.
Let $J_i = \{j: \delta_{ij}=0, j=1,\ldots,p \}$ be the taxa with missing counts in the $i$th sample. Using latent matrix $\Tilde{\boldsymbol{Z}}$, the observed sampling depths are then $L_i = \sum_{j \notin J_i} \Tilde{Z}_{ij}$.
The \textit{latent sampling depth} relies on the missing taxa and is defined as $\tilde{S_i}= \sum_{j \in J_i} \Tilde{Z}_{ij}$. The \textit{true sampling depth} is defined as $\tilde{L_i} = \sum_{j =1}^p \Tilde{Z}_{ij}$ and equals $(L_i + \tilde{S_i})$.
With these definitions, the following result, whose proof appears in Supplementary Material, allows us to stochastically reconstruct the true abundance matrix $\Tilde{\boldsymbol{Z}}$ in the posterior:
\begin{theorem} \label{prop_2_1} Suppose the missingness indicators $\delta_{ij}$ corresponding to the zero abundances are given. For sample $i=1,\ldots,n$, let $\tilde{q}_i = \sum_{j \in J_i} q_{ic_j}^*$. Then
\begin{enumerate}
\item \label{prop2_1_1} Latent sampling depth $\tilde{S_i}$ has a negative binomial distribution:
\[\tilde{S_i} \sim \text{NegBin}\bigl(L_i,\, \tilde{q}_i\bigr).
\]
\item \label{prop2_1_2} Let the vector of $|J_i|$ missing taxa abundances be $\Tilde{\boldsymbol{Z}}_i^{(0)}=\bigl(\Tilde{Z}_{ij}:\delta_{ij}=0, \, j=1,\ldots,p\bigr)$. Define $w_{ij}=q_{ic_j}^*/\tilde{q}_i$ for $j \in J_i$, and probability vector $\boldsymbol{w}_i= (w_{ij}: j \in J_i)$ of length $|J_i|$. Then
\[\Tilde{\boldsymbol{Z}}_i^{(0)} \mid \tilde{S_i} \sim \text{Multinomial}\bigl(\tilde{S}_i, \,\boldsymbol{w}_i\bigr).
\]
\end{enumerate} \end{theorem}
Finally, we assign standard conjugate priors to the remaining hyperparameters. Figure \ref{fig:dag_zibnp} displays the directed acyclic graph (DAG) representation of the ZIBNP model.
\begin{figure}
\caption{DAG representation of the ZIBNP model. Circles represent stochastic model parameters, solid rectangles represent data and deterministic variables, and open rectangles represent prespecified constants. }
\label{fig:dag_zibnp}
\end{figure}
\section{Posterior Inferences}\label{S:inference}
\subsection{MCMC algorithm}\label{S:MCMC} The model parameters are initialized using the naive estimation techniques described in Supplementary Material.
Subsequently, all parameters are iteratively updated by MCMC procedures. All model parameters except $\eta_{iu}$ are updated by Gibbs samplers. Although the full conditional of $\eta_{iu}$ does not have a closed form, it is log-concave, and the $\eta_{iu}$'s can therefore be generated by adaptive rejection sampling \citep{GilksWild1992}; see Section 2 of Supplementary Material for an outline of the MCMC steps. Following burn-in, the MCMC sample is stored and post-processed for posterior inferences.
\subsection{Taxa differential statuses}\label{S:posterior DA taxa}
First, we derive expressions for conditional posterior probabilities of the differential statuses of the latent clusters and their member taxa under the ZIBNP model. The following results follow from law of total probability and are omitted for brevity.
\begin{theorem}\label{thm 2} For the ZIBNP model, let the symbol $\Theta^-$ generically denote all model parameters not deterministically related to the quantities discussed in each the following statements. Thus, the set of parameters indicated by $\Theta^-$ depends on the context. Given $\Theta^-$ and $\mathbf{Z}$, let $P^*[\cdot]$ denote the conditional posterior probability, $P\bigl[\cdot \mid \Theta^-, \mathbf{Z}\bigr]$. Then \begin{enumerate}
\item\label{thm 2:part 1} For clusters $u=1,\ldots,C$, the conditional posterior probability that the $u$th cluster is non-DA is $P^*[h_u=1]= \sum_{m=1}^{M} \prod_{k=1}^{K} P^*\bigl[v_{ku}=m\bigr]$, where membership variable $v_{ku}$ is defined in (\ref{eq:v_ku}).
\item\label{thm 2:part 2} For the non-reference taxa $j=2,\ldots,p$, the probability that the $j$th taxon is non-DA is $P^*[\tilde{h}_j=1]= \sum_{u=1}^{C} P^*[h_u=1] P^*[c_j=u]$, where the expression for $P^*[h_u=1]$ on the right hand side is given in Part \ref{thm 2:part 1} of the theorem. \end{enumerate} \end{theorem}
The above theoretical expressions suggest MCMC-based empirical average estimators of the (unrestricted) posterior probabilities of the taxa differential statuses. For example, let $\hat{P}^{(l)}[\cdot]$ denote conditional posterior probability $P\bigl[\cdot \mid {\Theta^-}^{(l)}, \mathbf{Z}\bigr]$, evaluated using the $l$th MCMC sample model parameters, and $L$ be the MCMC sample size. Then, applying Part \ref{thm 2:part 2} of Theorem \ref{thm 2}, an MCMC-based empirical average estimator of posterior probability $P\bigl[\tilde{h}_j=1 \mid \mathbf{Z}\bigr]$ is
\[ \widehat{P}\bigl[\tilde{h}_j=1 \mid \mathbf{Z}\bigr]= \frac{1}{L}\sum_{l=1}^{L} \hat{P}^{(l)}\bigl[\tilde{h}_j=1 \bigr]. \] The procedure is applied to infer the posterior probabilities of the differential statuses of the $p$ taxa along with uncertainty estimates. For a nominal FDR (say, 5\%), we apply the Bayesian FDR or \textit{direct posterior probability} approach \citep{pmid15054023} to select an appropriate posterior probability threshold, $\kappa$, for calling the DA taxa. An estimate of differential status variable $\tilde{h}_j$ is
\[ \widehat{\tilde{h}}_j = 1+ \mathcal{I}\biggl(\widehat{P}\bigl[\tilde{h}_j=1 \mid \mathbf{Z}\bigr]\ge \kappa\biggr), \] where $\mathcal{I}(\cdot)$ represents the indicator function. The set of inferred DA taxa is then $\hat{\mathscr{D}}=\left\{j: \widehat{\tilde{h}}_{j}=2, \,\, j=1,\ldots,p\right\}$.
\section{Simulation Studies}\label{S:simulation2} Using simulated microbiome datasets with different sparsity levels and effect sizes, we investigated the effectiveness of the proposed ZIBNP method, making comparisons with some existing statistical methods with available R packages and capable of detecting DA taxa in the presence of covariates.
\paragraph*{Data generation} For $ n=100$ subjects belonging to $K=2$ groups, we generated abundance matrix $\boldsymbol{Z}$ for $p=1,002$ taxa, with taxon 1 as the artificial reference taxon whose abundance counts, $Z_{i1},\ldots,Z_{100,1}$, were all equal to $1$. The following generation steps, which differ in several important aspects from the ZIBNP model, were implemented to generate the artificial datasets; for example, we relied on the SparseDOSSA \citep{sparseDOSSA2_2021} model, which assumes a log-normal distribution for the marginal feature abundances and a multinomial distribution for the counts.
Thirty datasets were generated in each of nine scenarios corresponding to the combinations of (i) three sparsity levels ($15\%, 25\%$, and $58\%$) matching typical microbiome data and regulated by simulation parameter~$\lambda_0$ in Step \ref{lambda_0} below, and (ii) three effect sizes quantified by the fold change, $\omega$, equal to $2,3$ and $4$ in the SparseDOSSA model of Step \ref{pt:sim2_omega} below. The specific details are as follows:
\begin{enumerate}
\item \label{pt:sim1_cov} \textbf{Covariates} \quad We used the covariates from a publicly available oral microbiome dataset \citep{Burcham2020} consisting of adults and children as the subject groups. We selected $T=4$ covariates, namely, three binary covariates (antibiotics taken during the last 6 months, brush teeth daily, and sex) and one continuous covariate (BMI).
Randomly sampling the covariates of $50$ subjects from this real dataset, we repeated the \textit{same} set of covariates in adults and children to obtain the covariate matrix, $\boldsymbol{X}$, of dimension $100$ by $4$. More formally, if $\boldsymbol{X}^{(0)}$ denotes the $50 \times 4$ covariate matrix sampled from the real oral microbiome dataset, then
\begin{equation}
\boldsymbol{X} = \left[
\begin{matrix}
\boldsymbol{X}^{(0)}\\
\boldsymbol{X}^{(0)}\\
\end{matrix}
\right], \label{eq:sim_X}
\end{equation}
where the first 50 subjects represent adults, and the remaining 50 subjects represent children.
\item \label{pt:sim1_cj} \textbf{Allocation vector}\quad We generated the true number of clusters, $C$, from a discrete uniform distribution on $\{8,9,\ldots,20\}$. Applying the \texttt{rpartitions} package in R, we generated a $C$-partition of $p$ objects using the function \texttt{rand\_partitions}. The true taxon-to cluster allocation variables, $c_1,\ldots,c_p$, were set equal to this randomly generated partition with $C$ clusters, and therefore relied on an entirely different stochastic mechanism than the CRP of the ZIBNP model.
\item\label{pt:sim2_hu} \textbf{Cluster differential statuses}\quad With $h_u=2$ signifying a DA $u$th cluster, the differential statuses were generated as $h_u \stackrel{i.i.d.}\sim 1+\text{Bernoulli}(\pi_0)$, $ u=1, \ldots,C$, setting $\pi_0=0.3$.
\item\label{pt:sim2_omega} \textbf{Regression coefficients}\quad Using the $T=4$ covariates of Step \ref{pt:sim1_cov}, the cluster-specific regression coefficients excluding the intercepts were generated as
\begin{equation*}
\boldsymbol{\beta}_u ^-\sim N_T(\boldsymbol{0}, (\boldsymbol{X}^T\boldsymbol{X})^{-1}), \quad u=1,\ldots,C.
\end{equation*}
The regression intercepts, $\psi_{ku}$, of the $2C$ group-cluster combinations determined the effect sizes, and were evaluated as follows:
\begin{equation}
\psi_{ku}=6 + \log(\omega) \mathcal{I}\bigl(k=2, h_u=2\bigr), \quad k=1,2, \label{psi_ku}
\end{equation} to obtain the regression vectors of the group-cluster combinations as $\boldsymbol{\beta}_{ku}= (\psi_{ku},\boldsymbol{\beta}_u^- )^T$. Construct (\ref{psi_ku}) ensures that the observed counts generated in the subsequent steps tend to be systematically different in the two groups for the DA clusters.
\item\textbf{True taxa abundances}\quad Using the sparseDOSSA model, the non-missing absolute abundances were generated using log-normal distribution:
\begin{align*}
A_{ij} &\stackrel{\text{indep.}}\sim \text{log} N(\mu_{ic_j},\sigma^2), \quad\text{where}\\
\mu_{iu} &={\boldsymbol{x}_i^{\dag}}^T \boldsymbol{\beta}_{k_{i}u}, \quad i=1,\ldots,n, j=1,\ldots,p,
\end{align*}
with $\sigma^2$ chosen so that $R^2=0.97$ in the linear regression analysis of $\{\log A_{ij}\}$.
To mimic the range of sampling depths observed in actual microbiome datasets, we generated the observed sampling depths, $\tilde{L}_i \stackrel{\text{i.i.d.}}\sim \text{Poisson}(10,000)\times \text{Poisson(100)} $, for subjects $i=1,\ldots,100$.
Then, following the sparseDOSSA approach, we calculated the relative abundances, $R_{ij} = \frac{A_{ij}}{\sum_{j=1}^{p} A_{ij}}$, and generated the true taxa abundances as
\begin{equation*}
\boldsymbol{\tilde{Z}}_i \sim \text{Multinomial}(\tilde{L}_i, \boldsymbol{R}_i).
\end{equation*}
\begin{table}
\centering \renewcommand{1.2}{1.2}
\begin{tabular}{@{}ll@{}}
\hline
$\lambda_0$ & \% zeros \\
\hline
-0.100 & 15\% \\
-0.059 & 25\% \\
0.023 & 58\% \\
\hline
\end{tabular}
\caption{Averaging over the 30 datasets in the simulation study, observed sparsity in taxa abundance matrix $\boldsymbol{Z}$ as a function of $\lambda_0$. } \label{tab:zeroperc} \end{table}
\item \label{lambda_0} \textbf{Observed taxa abundances} Setting $\boldsymbol{\lambda}_{iu}= \lambda_0 \boldsymbol{1}$, where $\lambda_0 \in \mathcal{R}$, and using the observed sampling depths $\tilde{L}_i$ (rather than $L_i$), we applied equation (\ref{eq:zgp}) to generate the probabilities $r_{ij}$.
The observed counts were generated using a zero-inflated log-normal model:
\begin{equation*}
Z_{ij} \stackrel{\text{indep.}}\sim \text{ZIlogN}(r_{ij}, \mu_{ic_{j}},\sigma^2), \quad\text{so that}
\end{equation*}
\begin{align*}
Z_{ij} =\begin{cases}
\Tilde{Z}_{ij}, \quad \text{w.p.} \quad 1-r_{ij},\\
0, \qquad \text{w.p. } \quad r_{ij}.
\end{cases}
\end{align*}
Varying simulation parameter $\lambda_0$ in the set $\{-0.1, -.059, .023\}$ produced the required range of sparsity levels in the simulation scenarios; see Table \ref{tab:zeroperc}. \end{enumerate} \begin{figure}
\caption{In the simulation study, comparative benchmarking of the competing statistical methods in the nine scenarios corresponding to the combinations of sparsity level $15\%, 25\%$ and $58\%$, and fold change 2, 3, and 4.}
\label{fig:sim2_overall}
\end{figure}
For the $30$ artificial datasets in each of the nine scenarios, and disregarding knowledge of the generation mechanism and its true parameter values, the Section~\ref{S:inference} procedure was applied to analyze each artificial dataset and make posterior inferences for the proposed ZIBNP model. Applying the Section \ref{S:posterior DA taxa} strategy, we post-processed the MCMC sample to
estimate the DA posterior probabilities, $P\bigl[\tilde{h}_j=2\mid \boldsymbol{Z}\bigr]$. As described in Section \ref{S:posterior DA taxa}, assuming a nominal FDR of 5\%, we applied the Bayesian FDR procedure to call the DA taxa, and thereby, evaluated the achieved false discovery rate (FDR), Matthews correlation coefficient (MCC), sensitivity, and specificity of ZIBNP for each simulated dataset. Unlike other performance metrics, MCC delivers a holistic evaluation of a method's accuracy by producing a high score near 1 only if a method's predictions are accurate with respect to all four confusion matrix metrics; namely, true positives, false negatives, true negatives, and false positives \citep{chicco2020advantages}.
We compared our technique with some well-established methods for differential abundance analysis with covariates, focusing only on methods implemented in publicly available R packages:
\begin{enumerate}
\item ANCOMBC \citep{Lin2020}: We used the \texttt{ancombc} function of R package ANCOMBC (1.0.5) with default settings. The function fits compositional data by making a log transformation and using a sample-specific offset term for bias correction.
\item DESeq2 \citep{Love2014}: We used the \texttt{DESeq2} function of package DESeq2 (1.30.1) with argument {\tt sizefactor} set to {``poscounts''}. The R function performs a likelihood ratio test for a reduced model containing all covariates except the grouping factor.
\item Metagenomeseq \citep{Paulson2013}: R function \texttt{fitZIG} of the metagenomeSeq (1.32.0) package was applied to fit a zero-inflated Gaussian version of the technique. The workflow involved normalizing the data using cumulative sum scaling \citep{Paulson2013} with default settings.
\item Maaslin2 \citep{maaslin2_2021}: The \texttt{Maaslin2} function of the Maaslin2 (1.4.0) R package was used with total sum scaling normalization and covariate fixed effects.
This technique fits a generalized linear model to each feature abundance with respect to the covariates, checking for significance using the Wald test.
\end{enumerate}
\begin{comment}
\begin{figure}
\caption{For the simulation study, ROC plots comparing the methods ZIBNP (proposed), ANCOMBC, fitZIG, and MaAsLin2. The panels represent different sparsity levels from 13\% to 60\% zeros.}
\label{fig:sim_roc_all}
\end{figure} \end{comment}
The competing statistical methods were corrected for multiple hypothesis testing using the Benjamini-Hochberg procedure \citep{BenjaminiHochberg_1995} to obtain taxa-specific adjusted p-values \citep{storey2003positive}.
Using the adjusted p-values, for a target FDR of 5\%, the DA taxa were called using the Benjamini-Hochberg procedure to evaluate each method's achieved FDR, MCC, sensitivity, and specificity in each dataset.
Summarizing over the $30$ datasets of each simulation scenario, Figure \ref{fig:sim2_overall} displays boxplots of different performance metrics of the competing methods in detecting the DA taxa. We find that fitZIG displays high sensitivity, but has highly inflated FDR, low MCC, and low specificity in all the scenarios. MaasLin2 exhibits inflated FDR and low MCC. It is interesting that the FDR of MaasLin2 \textit{increases} with increase in fold change, an anomaly also reported by other studies \citep[e.g., ][]{sparseDOSSA2_2021}. ANCOMBC successfully controls FDR at lower sparsity levels, but is unable to reliably detect DA taxa, as evidenced by its low sensitivity in all nine scenarios.
Although DESeq2 performs reliably for higher levels of fold change, its performance significantly deteriorates with smaller effect sizes.
From Figure \ref{fig:sim2_overall}, we observe that ZIBNP is generally reliable with impressive sensitivity, MCC, and specificity. Its FDR is well controlled for lower sparsity levels. Although FDR is somewhat affected for the highest sparsity level ($58\%$ zeros), ZIBNP nonetheless performs well even in these scenarios, exhibiting high MCC, sensitivity, and specificity. In summary, ZIBNP appreciably outperforms the methods DESeq2, ANCOMBC, fitZIG, and MaAsLin2 in inferential accuracy and is reliable for analyzing even sparse microbiome datasets.
\section{Data Analysis}\label{sec:dataAnalysis}
We applied the ZIBNP technique to analyze publicly available microbiome data from the CAMP and Global Gut studies, consisting of $K=2$ and $K=3$ groups, respectively. The results were caffected with other differential analysis techniques.
\subsection{CAMP study canine data} \quad The dataset was downloaded from the MicrobiomeDB resource \citep{mdb}. To investigate the association between eukaryotic parasite infection on the composition of the gut microbiome, the data was obtained by sequencing the V4 region of the 16S rRNA gene of the fecal samples of 155 infected (case) and 115 uninfected (control) dogs. Animal-specific attributes such as sterilization, pet ownership, age, and sex are available, allowing statistical methods for differential analysis to adjust for these covariates while detecting DA taxa.
Table \ref{tab:CAMP_dataxa} presents the DA taxa detected by ZIBNP between the case and control groups . Some of the detected taxa have been reported by previous studies. For example, bacteria of the genus \textit{Bacteroides} were detected as DA with low relative abundance in infected dogs, and bacteria of the genera \textit{Megamonas} and \textit{Prevotella} were significantly associated with the infection status of dogs in studies of the effect of parasite-induced infections on gut microbiome composition \citep{Berry2020,dogs_prevotella}. The genus \textit{Blautia} has been previously reported to have a strong dysbiosis in dogs with acute diarrhea \citep{dogs_prevotella}. In addition, microbes from the genera \textit{Streptococcus} \citep{dogs_prevotella}, \textit{Ruminococcus gnavus} \citep{Ruminococcus_ibd,dog_giardia}, and \textit{Alloprevotella} \citep{dog_giardia} have known associations with inflammatory bowel disease in humans and dogs.
The methods ANCOMBC, fitZIG, Maaslin2, DESeq2, and ZIBNP detected 9, 72, 7, 42, and 10 DA taxa, respectively. Table \ref{tab:jaccard_camp} displays the pairwise similarity measures using the Jaccard index \citep{Jaccard1901} between the DA taxa detected by the different statistical methods, all of which attempt to adjust for covariates. The Jaccard index (J) is a measure of similarity between two sets that ranges between $0\%$ to $100\%$. It is defined as the size of the intersection divided by the size of the union between two sets; $J = 0\%$ implies that the two sets have no overlap, whereas $J=100\%$ implies the sets completely overlap. Although all the methods in Table \ref{tab:jaccard_camp} display low overlap, there is moderate overlap between ANCOMBC and Maaslin2, possibly because both methods fit generalized linear models to the data. The proposed ZIBNP method has some shared DA taxa with ANCOMBC and Maaslin2. However, ZIBNP has a low overlap with fitZIG and DESeq2, respectively relying on parametric zero-inflated Gaussian and negative binomial models. This reveals that the findings of ZIBNP are substantially different from existing statistical methods. The genera \textit{Blautia} and \textit{Lachnoclostridium} were the common findings of the methods ZIBNP, ANCOMBC, and Maaslin2. Additionally, the genus \textit{Ruminococcus gnavus} was detected by ZIBNP as well as ANCOMBC.
\begin{figure}
\caption{Venn diagram of the number of DA taxa detected by the methods ANCOMBC, MaAsLin2, and ZIBNP in the CAMP canine dataset.}
\label{fig:my_label}
\end{figure}
\begin{table}[h!]
\centering
\resizebox{\textwidth}{!}{\begin{tabular}{lllllll} \hline Kingdom & Phylum & Class & Order & Family & Genus & Species \\ \hline Bacteria & Bacteroidota & Bacteroidia & Bacteroidales & Bacteroidaceae & Bacteroides & - \\ Bacteria & Bacteroidota & Bacteroidia & Bacteroidales & Prevotellaceae & Alloprevotella & -\\ Bacteria & Bacteroidota & Bacteroidia & Bacteroidales & Prevotellaceae & Prevotella & -\\ Bacteria & Firmicutes & Bacilli & Lactobacillales & Streptococcaceae & Streptococcus & - \\ Bacteria & Firmicutes & Clostridia & Clostridiales & Clostridiaceae & Clostridium sensu stricto 1 & - \\ Bacteria & Firmicutes & Clostridia & Lachnospirales & Lachnospiraceae & [Ruminococcus] gnavus group & - \\ Bacteria & Firmicutes & Clostridia & Lachnospirales & Lachnospiraceae & Blautia & -\\ Bacteria & Firmicutes & Clostridia & Lachnospirales & Lachnospiraceae & Lachnoclostridium & - \\ Bacteria & Firmicutes & Clostridia & Peptostreptococcales-Tissierellales & Peptostreptococcaceae & Peptoclostridium & - \\ Bacteria & Firmicutes & Negativicutes & Veillonellales-Selenomonadales & Selenomonadaceae & Megamonas & Megamonas funiformis \\ \hline
\end{tabular}}
\caption{In the CAMP study, DA taxa detected by ZIBNP between infected and uninfected dogs.}
\label{tab:CAMP_dataxa}
\end{table}
\begin{table}[] \centering \begin{tabular}{llllll} \hline & ANCOMBC & fitZIG & Maaslin2 & DESeq2 & ZIBNP \\ \hline ANCOMBC &1 & 0.03 & 0.6 & 0.11 & 0.19 \\ fitZIG & 0.03 & 1 & 0.03 & 0.28 & 0 \\ Maaslin2 & 0.6 & 0.03 & 1 & 0.11 & 0.13 \\ DESeq2 & 0.11 & 0.28 & 0.11 & 1 & 0.02 \\ ZIBNP & 0.19 & 0 & 0.13 & 0.02 & 1 \\ \hline \end{tabular} \caption{For the CAMP study data, pairwise Jaccard index of the DA taxa for different statistical methods.} \label{tab:my_label} \label{tab:jaccard_camp} \end{table}
\subsection{Global Gut Microbiome study}
We applied the proposed ZIBNP method to analyze the motivating Global Gut Microbiome data \citep{Yatsunenko2012}. The study examines the differences in the microbial abundance between samples collected from individuals residing in Malawia, Venezuela, and the USA. More specifically, the data consists of 100 US individuals, and 83 individuals each from Malawi and Venezuela, in addition to age and sex as covariates. The abundance matrix of $266$ subjects $\times$ $1,270$ taxa was comprised of $30\%$ zeros.
Along with the proposed ZIBNP approach, we analyzed the data using the methods ANCOMBC, fitZIG and DESeq2, all of which perform differential analysis with covariates. Since MaAsLin2 in its current form does not have a global significance test for more than two groups, we did not use this method to analyze the Global Gut data.
The results are shown in Table \ref{tab:globalgut_Jaccard} and Figure \ref{fig:venn_globalgut}. The four methods detected a common set of $37$ DA taxa, as seen in Figure \ref{fig:venn_globalgut}. The low overlap of detected DA taxa between ZIBNP and other competing methods reveals the benefits of the proposed nonparametric Bayesian approach in finding otherwise undetected taxa. ANCOMBC and fitZIG shared a large proportion of DA taxa and had a Jaccard index of $82\%$. FitZIG (MetagenomeSeq) detected $1,035$ (out of $1,270$) taxa as DA, and in fact, detected the largest number of DA taxa in both motivating datasets. However, as noted by several other studies \citep{Thorsen2016,Hawinkel2017} and Section \ref{S:simulation2}, the methods fitZIG and ANCOMBC often exhibit inflated FDR, and one should proceed with caution while interpreting their results.
Table S1 of Supplementary Material provides the list of $201$ DA taxa detected by ZIBNP. Among all DA taxa detected by ZIBNP, $5.5\%$ and $5\%$ taxa belonged to the genus \textit{Prevotella} and \textit{Bacteroides}, respectively. The original article \citep{Yatsunenko2012} reporting the Global Gut Microbiome results focused on pairwise comparisons, such US versus non-US individuals and Venezuela versus Malawi individuals. They are not, therefore, directly comparable to the multigroup comparisons of ZIBNP. However, \cite{Yatsunenko2012} reported several taxa belonging to the genus \textit{Prevotella} as differential for the (pairwise) geographical regions. Furthermore, they stated that taxa belonging to the genus \textit{Bacteroides} were significantly more abundant in US individuals compared to non-US individuals.
\begin{figure}
\caption{Venn diagram of the number of DA taxa detected by the methods ANCOMBC, DESeq2, fitZIG (MetagenomeSeq), and ZIBNP in the Global Gut dataset.}
\label{fig:venn_globalgut}
\end{figure}
\begin{table}[]
\centering
\begin{tabular}{|c|c|c|c|c|}
\hline
& ANCOMBC & fitZIG & DESeq2 & ZIBNP \\
\hline
ANCOMBC & 1 & 0.82 & 0.48 & 0.17 \\
fitZIG & 0.82 & 1 & 0.51 & 0.16 \\
DESeq2 & 0.48 & 0.51 & 1 & 0.06 \\
ZIBNP & 0.17 & 0.16 & 0.06 & 1 \\
\hline
\end{tabular}
\caption{Jaccard index calculated between the pair of methods used to detect DA taxa for the Global Gut Microbiome study.}
\label{tab:globalgut_Jaccard}
\end{table}
\section{Discussion}\label{S:discussion}
Differential abundance analyses between multiple groups of study samples help identify novel therapeutic targets for disease treatment. However, very few existing methods for differential analysis with covariates perform consistently well for different datasets \citep{Weiss2017}. Motivated by the challenges of high dimensionality, sparsity, and compositionality inherent to microbiome data, we propose a unified methodology based on a two-component mixture model that adjusts to distinctive data characteristics. A key contributor to our strategy's success is a model-based missing data framework that relies on the stochastic relationship between
latent sampling depths and the large number of zero counts to improve the accuracy of differential abundance analysis.
Latent clusters for the taxa induced by the semiparametric Chinese restaurant process alleviate collinearity issues caused by the small $n$, large $p$ problem. The compositional aspects of microbiome data are modeled by a regression framework for the log-ratios of the dependent cluster-specific probabilities. The taxa differential statuses are immediately available from the group-cluster regression parameters, whose common multivariate prior has a finite number of atoms. Through simulation experiments and data analyses, we demonstrate the potential of the proposed ZIBNP method as a reliable technique for detecting DA taxa in a wide variety of microbiome datasets.
As suggested by several studies \citep{Sankaran2014,Xiao2017}, the accuracy of statistical methods for differential analysis can be further improved by incorporating the phylogenetic distances between the taxa. Going forward, our research will focus on utilizing this potentially valuable information. We are also developing an R package for ZIBNP that will be made available on GitHub for differential analysis of multiple groups with covariates.
\begin{comment}
\begin{appendix} \section*{Title}\label{appn}
Appendices should be provided in \verb|{appendix}| environment, before Acknowledgements.
If there is only one appendix, then please refer to it in text as \ldots\ in the \hyperref[appn]{Appendix}. \end{appendix} \end{comment}
\begin{comment}
\begin{appendix} \section{Title of the first appendix}\label{appA} If there are more than one appendix, then please refer to it as \ldots\ in Appendix \ref{appA}, Appendix \ref{appB}, etc.
\section{Title of the second appendix}\label{appB} \subsection{First subsection of Appendix \protect\ref{appB}}
Use the standard \LaTeX\ commands for headings in \verb|{appendix}|. Headings and other objects will be numbered automatically. \begin{equation} \mathcal{P}=(j_{k,1},j_{k,2},\dots,j_{k,m(k)}). \label{path} \end{equation}
Sample of cross-reference to the formula (\ref{path}) in Appendix \ref{appB}. \end{appendix}
\begin{acks}[Acknowledgments] The authors would like to thank the anonymous referees, an Associate Editor and the Editor for their constructive comments that improved the quality of this paper. \end{acks}
\end{comment}
\begin{funding}
The third author (SG) was supported by the National Science Foundation under Award DMS-1854003. \end{funding}
\begin{supplement} \stitle{MCMC implementation } \sdescription{Section 1.1 describes the MCMC initialization for the ZIBNP model parameters. Section 1.2 outlines the MCMC updates.} \end{supplement}
\begin{supplement} \stitle{Proof of Theorem \ref{prop_2_1}} \sdescription{Section 2 of the supplementary material describes the proofs of Parts \ref{prop2_1_1} and \ref{prop2_1_2} of Theorem \ref{prop_2_1}.} \end{supplement}
\begin{supplement} \stitle{Table S1} \sdescription{Excel file listing the DA taxa detected by ZIBNP for the Global Gut Microbiome study.} \end{supplement}
\sg{Are Supp Material sections part of main paper in AOAS template?}
\subsection{MCMC initialization} \label{sec:mcmc_init}
The reference cluster ($u=1$) and its singleton taxon (without loss of generality, $j=1$) are selected as described in Section \ref{S:BNP} and held fixed throughout the inference procedure. For the initialization procedure only, we add $1$ to all the abundance counts in $\boldsymbol{Z}$. The model parameters are then initialized as follows:
\begin{enumerate}
\item \underline{Matrix $\mathbf{Q}$} \quad Let $q_{ij}= Z_{ij}/\sum_{j=1}^{p}Z_{ij} $.
\item \underline{Allocation vector, cluster motifs and cluster sizes} \quad For the reference cluster, set $m_1=1$ and $q_{i1}^{*}= \hat{q}_{i1}$ for $i =1,\ldots,n$. The number of clusters are initialized as $\text{log}(p)$,
and columns $2,\ldots,p$ of matrix $\mathbf{Q}$ are clustered using the k-means algorithm with $(C-1)=\text{log}(p)$ prespecified number of clusters. Thereby, evaluate taxa-to-cluster allocation vector $\boldsymbol{c} =(c_1, \ldots, c_p)$ along with the cluster sizes, $m_u$, of the non-reference clusters. Cluster motifs, $\boldsymbol{q}^*_{u}$, for $u>1$, are initialized as the vectorized cluster centers estimated by k-means.
\item \underline{Vector $\boldsymbol{\lambda}_{iu}$}\quad
Since the library sizes are large, the marginal probabilities of sampling zeros from equation (\ref{eq:sampling0}) are nearly equal to zero. So, for initialization purposes, we assume that the zero counts in matrix $\boldsymbol{Z}$ are all technical zeros and consider matrix $\boldsymbol{\Delta} = ((\delta_{ij}))$ such that \[\delta_{ij} = \begin{cases}
0 \qquad \text{if } Z_{ij}=0,\\
1 \qquad \text{if } Z_{ij}>0.
\end{cases}\]
Then we fit a random intercept - random slope logistic regression model given the covariates, cluster allocations and library sizes for each sample:
\begin{equation*}
\log\left(\frac{P(\delta_{ij}=0)}{P(\delta_{ij}=1)}\right) = \lambda_{0ic_j} + \sum_{t=1}^{T} \lambda_{tic_j} x_{it} + \lambda_{T+1,ic_j} \log(L_i), \quad i=1,\ldots,n,
\end{equation*}
Furthermore, $r_{ic_j}^*$, for $i=1,\ldots,n$ and $j=2,\ldots,p$, can be estimated using the $\boldsymbol{\lambda}_{iu}$ values.
\item \underline{Log ratio $\eta_{iu}$} \quad For $u>1$, set $\eta_{iu} = \log(q_{iu}^*/q_{i1}^*)$.
\item \underline{Variance $\sigma^2_e$} \quad Regarding $\boldsymbol{\Xi}=((\eta_{iu}: i=1,\ldots,n, \, u>1))$ as an outcome matrix with $(C-1)$ multivariate responses per sample, fit a linear regression model with $\boldsymbol{X}^{\dag}$ as the covariate matrix. Set $\sigma^2_e$ equal to the estimated residual variance.
\item \underline{Finite mixture model parameters} \quad
\begin{enumerate}
\item \label{beta_init}
For $u>1$ and $k=1,\ldots,K$, define vector $\boldsymbol{\eta}^{(k)}_{u}=(\eta_{iu}: k_i=k, i=1,\ldots,n)$, and
let ${\boldsymbol{X}^{\dag}}^{(k)}$ be the $n_k \times (T+1)$ covariate matrix of the $n_k$ samples in group~$k$. For every group-cluster combination, compute the least squares estimates $\hat{\boldsymbol{\beta}}_{ku}$ in the linear regression model:
$$\boldsymbol{\eta}_u^{(k)} = {\boldsymbol{X}^{\dag}}^{(k)} \boldsymbol{\beta}_{ku} + \boldsymbol{\epsilon}_{ku}. $$
\item Cluster the $K (C-1)$ vectors, $\hat{\boldsymbol{\beta}}_{ku}$, using k-means clustering with $M=\text{log}(K (C-1))$ to obtain $\boldsymbol{\mu}_m$, where $m=1,\ldots,M$, as the estimated cluster mean vectors, and {membership variables} $\{v_{ku}\}$.
\item Initialize ${\boldsymbol{\beta}}_{ku}$ using expression (\ref{eq:v_ku}).
\item In vector $\boldsymbol{\pi}$, set $\pi_m= \frac{\sum_{k=1}^K \sum_{u=2}^C I(v_{ku}=m)}{K(C-1)} $, $m=1,\ldots,M$.
\item The cluster differential statuses are available from (\ref{eq:DA}). The differential status of the $j$th taxa is $\tilde{h}_j=h_{c_j}$, $j=1,\ldots,p$.
\end{enumerate}
\item \underline{Variance $\tau^2$} \quad Let
$ \tau^2 = \frac{1}{M(T+1)} \sum_{m=1}^{M} \boldsymbol{\mu}_m^{T} {\boldsymbol{X}^{\dag}}^{T} \boldsymbol{X}^{\dag } \boldsymbol{\mu}_m$.
\end{enumerate}
\subsection{MCMC updates} \label{sec:mcmc_posterior}
Most of the parameters, except the log-ratios $\eta_{iu}$, have known full conditionals and can be updated using Gibbs sampling. The full conditional posterior for each parameter is described below. As before, $[\cdot]$ represents a density function with respective to a suitable dominating measure. For any set of parameters $\theta$, the symbol $[\theta \mid \boldsymbol{Z},\cdots]$ generically denotes the full conditional of $\theta$ in the following description:
\begin{itemize}
\item For variance $\sigma_e^2$ in equation (\ref{eq:eta}), we assume an inverse-gamma prior, Inv-gamma$(a_e,b_e)$, truncated to the interval $[\frac{(1-R^2_{ul})}{Var(\boldsymbol{\eta})^{-1}}, \frac{(1-R^2_{ll})}{Var(\boldsymbol{\eta})^{-1}}\big]$, where $R_{ll}=0.9$ and $R_{ul}=0.999$ . The full conditional of $\sigma_e^2$ is a truncated inverse-gamma distribution with parameters $(a_e+n(C-1))$ and $\bigl(b_e+ \sum_{i=1}^{n}\sum_{u>1} (\eta_{iu}- \beta_{0k_iu} + \sum_{l=1}^Tx_{il} \beta_{lk_iu})^2 /2\bigr)$.
\item Using the $M$-component finite mixture hierarchical prior defined in equation (\ref{eq:beta}), the regression parameter vector $\boldsymbol{\beta}_{ku}$ is updated using the membership variable $v_{ku}$ defined in (\ref{eq:v_ku}). The event $[v_{ku} =m]$ is same as $[\beta_{ku}=\mu_m]$, with the conditional posterior probability of $[\boldsymbol{\beta}_{ku}=\mu_m]$ defined as follows: $$P_{kum}=P^{*}(\boldsymbol{\beta}_{ku}=\mu_m)=P^{*}(v_{ku} =m) \propto \pi_m L(v_{ku}),$$ where $\pi_m$ is the prior probability of regression parameter vector being equal to the $m$th component of the finite mixture model and $L(v_{ku})$ is the likelihood term with $\boldsymbol{\beta}_{ku} = \boldsymbol{\mu}_m$ As a result, each $\boldsymbol{\beta}_{ku}$ can be updated from the corresponding multinomial distribution with posterior probability vector $\mathbf{P}_{ku}=(P_{ku1},\ldots,P_{kuM})$ of length $M$, that is
$$v_{ku} \sim \text{Multinomial}(1,\mathbf{P}_{ku}).$$
\item The mixture components $\boldsymbol{\mu_m}$ are updated as follows
\begin{align*}
[\boldsymbol{\mu}_m \bigm| \boldsymbol{\eta},\sigma^2_e,\tau^2, v_{ku}] \propto N(\boldsymbol{\mu}_m \bigm| \boldsymbol{0},\tau^2 ({\boldsymbol{X}^{\dag}}^T {\boldsymbol{X}^{\dag}})^{-1} ) \prod_{(i,u)\in \Lambda_m} N(\eta_{iu} \bigm| \boldsymbol{x}_i^{\dag}\boldsymbol{\mu}_m^{T},\sigma^2_e) , \end{align*} where $\Lambda_m = \{(i,u): v_{ku}=m, k_i=k,i=1,\ldots,n\}$ and $\boldsymbol{x}_i^{\dag}{\boldsymbol{\mu}}_m^{T} = \mu_{0m} + \sum_{l=1}^Tx_{il} \mu_{lm}$.
\noindent The posterior distribution of $\boldsymbol{\mu}_m $ is normal with mean vector $\boldsymbol{A}^*$ and variance matrix $\boldsymbol{B}^*$ where
\begin{align*}
\boldsymbol{A}^* &= W\hat{\boldsymbol{\mu}}_m + (\textbf{I}_{T+1}-W) \boldsymbol{0}=W\hat{\boldsymbol{\mu}}_m, \\
\boldsymbol{B}^* &= \sigma^2_e W ({\boldsymbol{X}^\dag}^T_m \boldsymbol{X}^{\dag}_m)^{-1},\\
W &= ({\boldsymbol{X}^\dag}^T_m \boldsymbol{X}^{\dag}_m + ({\boldsymbol{X}^{\dag}}^T \boldsymbol{X}^{\dag}) \frac{\sigma^2_e}{\tau^2})^{-1} ({\boldsymbol{X}^\dag}^T_m \boldsymbol{X}^{\dag}_m),
\end{align*}
and $\boldsymbol{X}^{\dag}_m$ and $\boldsymbol{\eta}_m$ are defined as follows.
For example, if $\boldsymbol{\beta}_{11}=\boldsymbol{\beta}_{12}=\boldsymbol{\beta}_{22}=\boldsymbol{\beta}_{25}=\boldsymbol{\beta}_{17}=\boldsymbol{\mu}_1$, then $\boldsymbol{\eta}_m$ is a vector of length $N^{'}=(3n_1+2n_2)$ with $\boldsymbol{\eta}_m=(\boldsymbol{\eta}_1^{(1)},\boldsymbol{\eta}_2^{(1)},\boldsymbol{\eta}_2^{(2)},\boldsymbol{\eta}_5^{(2)},\boldsymbol{\eta}_7^{(1)})$. In this case $\boldsymbol{X}^{\dag}_m=({\boldsymbol{X}^\dag}^{(1)},{\boldsymbol{X}^\dag}_1^{(1)},{\boldsymbol{X}^\dag}^{(2)},{\boldsymbol{X}^\dag}^{(2)},{\boldsymbol{X}^\dag}^{(1)})$ with dimension $N^{'}\times (T+1)$.
\item We assign a Dirichlet distribution prior $\boldsymbol{\pi} \sim Dir(\frac{\alpha_0}{M}, \ldots, \frac{\alpha_0}{M})$ for mixture probabilities $\boldsymbol{\pi}=(\pi_1, \ldots,\pi_M)$. The mixture probabilities are updated as \begin{align*}
\boldsymbol{\pi} &\sim Dir(\frac{\alpha_0}{M} + \textbf{d}), \end{align*} where $\textbf{d} =(d_1,\ldots,d_M)$ and $d_m =\sum_k \sum_u I[v_{ku}=m]$.
\item To update taxon-to cluster allocation variable $c_j$, we use the Chinese restaurant process prior defined in equation (\ref{eq:CRP}). The prior probability that $j^{th}$ taxon is assigned to $u^{th}$ cluster is then \begin{equation*}
[c_j =u \mid \boldsymbol{c}_{-j}] \propto
\begin{cases}
m_u^{(j-1)} \quad \text{if } u=1,\ldots,C^{(j-1)}\\
\alpha_c \quad \qquad \text{if } u = C^{(j-1)} +1
\end{cases} \end{equation*} where the probability of opening a new cluster is proportional to $\alpha_c$. The likelihood term is \begin{align*}
[\boldsymbol{Z} \mid c_j=u, \boldsymbol{c}_{-j}] = \prod_{i=1}^{n} \left( \frac{L_i!}{\prod_{j=1}^{p} Z_{ij}!} \prod_{u=1}^{C} (q_{iu}^*)^{\xi_{iu}} \right), u=2, \ldots,C, \end{align*} where $\xi_{iu} = \sum_{j \in S_u} Z_{ij} $.
The log-posterior probability $[c_j = u \mid \boldsymbol{Z}, \boldsymbol{c}_{-j}, \alpha_c ]$ is as follows:
\begin{multline*}
l_{ju} = \log(Pr[c_j = u \mid \boldsymbol{Z}, \boldsymbol{c}_{-j}, \alpha_c ] ) \propto \\
\begin{cases}
\log(m_u) + \sum_{i=1}^{n} \sum_{u=1}^{G} \xi_{iu} \log(q_{iu}^*),\\
\log(\alpha_c) + \sum_{i=1}^{n} \sum_{u=1}^{G} \xi_{iu} \log(q_{iu}^*)
\end{cases}
\end{multline*}
\sg{[Add "ifs'' in cases above: should it be $m_u^{(j-1)}$ in line 1? No summation over u in line 1? Line 2: integral of some sort? Please check formulas-- let's discuss if you're not sure.]} for $u=2, \ldots,C$. A taxa $j \in \{ 2,\ldots, p\}$ can belong to any of the $C-1$ clusters, where we consider $j=1$ as the reference taxa ($u=1$).
\item The posterior density for $\eta_{iu}$ is as follows \begin{align*}
[\eta_{iu}\mid \boldsymbol{\beta}_{ku}, \boldsymbol{Z}, C] &\propto \prod_{j=1}^{p} (q_{ij})^{Z_{ij}} N\biggl(\eta_{iu} \bigm| \beta_{0k_iu} + \sum_{l=1}^Tx_{il} \beta_{lk_iu}, \,\, \sigma_e^2\biggr) \\
&\propto \prod_{u=1}^{C} (q^*_{iu})^{\xi_{ij}} N\biggl(\eta_{iu} \bigm| \beta_{0k_iu} + \sum_{l=1}^Tx_{il} \beta_{lk_iu}, \,\, \sigma_e^2\biggr), \end{align*}
where $q_{iu}^* = \frac{\exp(\eta_{iu})}{\sum_{u=1}^{G}m_u \exp(\eta_{iu})}$, $m_1=1$, and $\eta_{i1} =0$ for $u=1, \ldots,C$.
Substituting $q_{iu}^*$, the right hand side can be written as \begin{equation}
\prod_{u=1}^{C} \frac{\exp(\eta_{iu} \xi_{iu})}{ (\sum_{u=1}^{C}m_u exp(\eta_{iu}))^{\xi_{iu}}} N\biggl(\eta_{iu} \bigm| \beta_{0k_iu} + \sum_{l=1}^Tx_{il} \beta_{lk_iu}, \,\, \sigma_e^2\biggr) .\label{eq:eta_poserior} \end{equation}
The posterior distribution of $\eta_{iu}$ in equation (\ref{eq:eta_poserior}) does not represent any standard distribution. Therefore, we use an alternative procedure to sample $\eta_{iu}$. Let the log of full conditional of $\eta_{iu}$ be denoted by $\mathscr{L}_{iu}$, so that \begin{align*}
\mathscr{L}_{iu}= \sum_{u=1}^{C} \eta_{iu} \xi_{iu} - \sum_{u=1}^{G}\xi_{iu} \log\left(\sum_{u=1}^{G}m_u exp(\eta_{iu})\right) -\frac{(\eta_{iu}-\beta_{0k_iu}-\sum_{l=1}^Tx_{il} \beta_{lk_iu})^2}{2 \sigma^2_e} + \mathscr{C}, \end{align*} where $\mathscr{C}$ includes all remaining terms independent of $\eta_{iu}$. Furthermore, \begin{align*}
\frac{\partial\mathscr{L}_{iu}}{\partial \eta_{iu}} &= \xi_{iu} - L_i \frac{m_u exp(\eta_{iu})}{\sum_{u=1}^{G} m_u exp(\eta_{iu})} -\frac{1}{\sigma^2_e} (\eta_{iu}-\beta_{0k_iu} -\sum_{l=1}^Tx_{il} \beta_{lk_iu}) \\
&= \xi_{iu} - L_i m_u q_{iu}^{*} -\frac{1}{\sigma^2_e} (\eta_{iu}-\beta_{0k_iu} -\sum_{l=1}^Tx_{il} \beta_{lk_iu}), \end{align*} and
\begin{align*}
\frac{\partial^2\mathscr{L}_{iu}}{\partial \eta_{iu}^2} &= - L_i \frac{m_u exp(\eta_{iu}) (\sum_{u=1}^{G} m_u exp(\eta_{iu}) - m_u exp(\eta_{iu}) ) }{(\sum_{u=1}^{G} m_u exp(\eta_{iu}))^2} -\frac{1}{\sigma^2_e} \\
&= - L_i m_u q_{iu}^{*} (1- m_u q_{iu}^{*}) -\frac{1}{\sigma^2_e} < 0, \end{align*} where $0 < q_{iu}^{*} < 1$. Since the second derivative of log of posterior is negative, the posterior distribution of $\eta_{iu}$ is log-concave. Consequently, we can use adaptive rejection sampling to sample from the full conditional posterior of $\eta_{iu}$. This is achieved by using the \texttt{ars} package in R.
\item For the hyperparameter $\tau^2$ in equation (\ref{eq:mum_prior}), we assume an inverse-gamma prior, Inv-gamma$(a_{\tau},b_{\tau})$. The full conditional of $\tau^2$ is an inverse-gamma distribution with parameters $a_{\tau} + {(T+1)M}/{2}$ and $b_{\tau} + {\sum_{m=1}^{M}\boldsymbol{\mu}_m^T ({\boldsymbol{X}^{\dag}}^T {\boldsymbol{X}^{\dag}}) \boldsymbol{\mu}_m}/{2}$.
\end{itemize}
\section{Proof of Theorem \ref{prop_2_1}}
We use the fusing and partitioning properties of the multinomial distribution to prove the result. Given $\boldsymbol{Y}=(Y_1,\ldots,Y_G) \sim \text{Multinomial}(N,(\pi_1, \ldots,\pi_G))$, these two properties are illustrated by examples:
\begin{enumerate}
\item \textbf{Fusing} \label{re:fuse} \qquad $\boldsymbol{Y}^*=(Y_1+Y_2,\ldots,Y_G) \sim \text{Multinomial}(N,(\pi_1+\pi_2, \ldots,\pi_G))$
\item \textbf{Partitioning} \label{re:part} \qquad Subvectors ($Y_1,Y_2$) and ($Y_3, \ldots, Y_G$) are conditionally independent and multinomial:
\begin{align*}
(Y_1, Y_2) &\sim \text{Multinomial}\bigl(z,(\frac{\pi_1}{\pi_1+\pi_2} ,\frac{\pi_2}{\pi_1+\pi_2} )\bigr), \quad\text{and}\\
(Y_3,\ldots,Y_G) &\sim \text{Multinomial}\bigl(N-z,(\frac{\pi_3}{\pi_3+\ldots+\pi_G} ,\ldots,\frac{\pi_G}{\pi_3+\ldots+\pi_G} )\bigr).
\end{align*}
\end{enumerate}
\textbf{{Theorem \ref{prop_2_1}, Part \ref{prop2_1_1}:}} As mentioned, the true sampling depth $\tilde{L}_i$ is equal to the sum of the observed sampling depth $L_i$ (known) and the unknown latent sampling depth $\tilde{S}_i$. We know that $\tilde{\boldsymbol{Z}}_i$ follows Multinomial($\tilde{L}_i,\boldsymbol{q}_i$), where $\boldsymbol{q}_i=(q_{i1}, \ldots,q_{ip})$. Using the fusing property, the probability of success of the sum of the censored counts $\tilde{S}_i$, is equal to $\tilde{q}_i = \sum_{j \in J_i} q_{ij} = \sum_{j \in J_i} q_{ic_j}^*$. The goal is to sample $\tilde{S}_i= \sum_{j \in J_i}Z_{ij}$, and so, the problem can be formulated as sampling the number of success before a fixed number ($L_i$) of failures occur in a series of Bernoulli trials. Hence, the random variable $\tilde{S}_i$ follows a negative binomial distribution with parameters $L_i$ and $\tilde{q}_i$.
\textbf{{Theorem \ref{prop_2_1}, Part \ref{prop2_1_2}}:} Given the latent sampling depth $\tilde{S}_i$, we can obtain the true sampling depth $\tilde{L}_i$. The $i^{th}$ row of the latent matrix $\tilde{\boldsymbol{Z}}$, denoted by $\tilde{\boldsymbol{Z}}_i$, consists of the observed abundances and missing taxa abundances, and follows Multinomial($\tilde{L}_i,\boldsymbol{q}_i$), where $\boldsymbol{q}_i=(q_{i1}, \ldots,q_{ip})$ and $\tilde{L}_i=\tilde{S}_i+ L_i$.
Using the partitioning property, the conditional distribution of the missing taxa abundances, denoted by $\Tilde{\boldsymbol{Z}}_i^{(0)}=\bigl(\Tilde{Z}_{ij}:\delta_{ij}=0, \, j=1,\ldots,p\bigr)$, is \[\Tilde{\boldsymbol{Z}}_i^{(0)} \mid \tilde{S_i} \sim \text{Multinomial}\bigl(\tilde{S}_i, \,\boldsymbol{w}_i\bigr). \]
where $w_{ij}=q_{ic_j}^*/\tilde{q}_i$ for $j \in J_i$ and $\tilde{q}_i = \sum_{j \in J_i} q_{ic_j}^*$.
\section{Table S1}
\nocite{neal2000markov}
\end{document} | arXiv |
Snake-to-isola transition and moving solitons via symmetry-breaking in discrete optical cavities
Exact solutions for periodic and solitary matter waves in nonlinear lattices
October 2011, 4(5): 1327-1340. doi: 10.3934/dcdss.2011.4.1327
Dark solitary waves in nonlocal nonlinear Schrödinger systems
David Usero 1,
Dpto. de Matemática Aplicada, Fac. CC. Químicas, Universidad Complutense de Madrid 28040, Spain
Received September 2009 Revised January 2010 Published December 2010
Dark soliton-like solutions are analyzed in the context of a certain nonlocal nonlinear Schrödinger Equation with nonlocal dispersive term of Kac-Baker type. Main purpose is to investigate such solutions with negative nonlinear term and the presence of general integral dispersive terms. First the model is presented and the properties of the fundamental solution, the continuous wave, is studied. Dark solitary waves are perturbations of this plane wave. The study of dark type of solutions is divided in two different cases black and dark solitary waves. The range of existence of such solutions is studied analytically, and also their physical quantities like norm, momentum and energy. Usual behavior of nonlinear systems under nonlocal dispersive terms is found.
Keywords: Kac-Baker interaction., Nonlinear Schrödinger equation, Dark soliton, nonlocal dispersion.
Mathematics Subject Classification: Primary: 35Q55, 35Q51; Secondary: 45K0.
Citation: David Usero. Dark solitary waves in nonlocal nonlinear Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1327-1340. doi: 10.3934/dcdss.2011.4.1327
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David Usero | CommonCrawl |
Tag: Mathieu
sporadic simple games
Published July 13, 2008 by lievenlb
About a year ago I did a series of posts on games associated to the Mathieu sporadic group $M_{12} $, starting with a post on Conway's puzzle M(13), and, continuing with a discussion of mathematical blackjack. The idea at the time was to write a book for a general audience, as discussed at the start of the M(13)-post, ending with a series of new challenging mathematical games. I asked : "What kind of puzzles should we promote for mathematical thinking to have a fighting chance to survive in the near future?"
Now, Scientific American has (no doubt independently) taken up this lead. Their July 2008 issue features the article Rubik's Cube Inspired Puzzles Demonstrate Math's "Simple Groups" written by Igor Kriz and Paul Siegel.
By far the nicest thing about this article is that it comes with three online games based on the sporadic simple groups, the Mathieu groups $M_{12} $, $M_{24} $ and the Conway group $.0 $.
the M(12) game
Scrambles to an arbitrary permutation in $M_{12} $ and need to use the two generators $INVERT=(1,12)(2,11)(3,10)(4,9)(5,8)(6,7) $ and $MERGE=(2,12,7,4,11,6,10,8,9,5,3) $ to return to starting position.
Here is the help-screen :
They promise the solution by july 27th, but a few-line GAP-program cracks the puzzle instantly.
Similar in nature, again using two generators of $M_{24} $. GAP-solution as before.
This time, they offer this help-screen :
the .0 game
Their most original game is based on Conway's $.0 $ (dotto) group. Unfortunately, they offer only a Windows-executable version, so I had to install Bootcamp and struggle a bit with taking screenshots on a MacBook to show you the game's starting position :
Dotto:
Dotto, our final puzzle, represents the Conway group Co0, published in 1968 by mathematician John H. Conway of Princeton University. Co0 contains the sporadic simple group Co1 and has exactly twice as many members as Co1. Conway is too modest to name Co0 after himself, so he denotes the group ".0" (hence the pronunciation "dotto").
In Dotto, there are four moves. This puzzle includes the M24 puzzle. Look at the yellow/blue row in the bottom. This is, in fact, M24, but the numbers are arranged in a row instead of a circle. The R move is the "circle rotation to the right": the column above the number 0 stays put, but the column above the number 1 moves to the column over the number 2 etc. up to the column over the number 23, which moves to the column over the number 1. You may also click on a column number and then on another column number in the bottom row, and the "circle rotation" moving the first column to the second occurs. The M move is the switch, in each group of 4 columns separated by vertical lines (called tetrads) the "yellow" columns switch and the "blue" columns switch. The sign change move (S) changes signs of the first 8 columns (first two tetrads). The tetrad move (T) is the most complicated: Subtract in each row from each tetrad 1/2 times the sum of the numbers in that tetrad. Then in addition to that, reverse the signs of the columns in the first tetrad.
Strategy hints: Notice that the sum of squares of the numbers in each row doesn't change. (This sum of squares is 64 in the first row, 32 in every other row.) If you manage to get an "8"in the first row, you have almost reduced the game to M24 except those signs. To have the original position, signs of all numbers on the diagonal must be +. Hint on signs: if the only thing wrong are signs on the diagonal, and only 8 signs are wrong, those 8 columns can be moved to the first 8 columns by using only the M24 moves (M,R).
Arnold's trinities version 2.0
Published June 20, 2008 by lievenlb
Arnold has written a follow-up to the paper mentioned last time called "Polymathematics : is mathematics a single science or a set of arts?" (or here for a (huge) PDF-conversion).
On page 8 of that paper is a nice summary of his 25 trinities :
I learned of this newer paper from a comment by Frederic Chapoton who maintains a nice webpage dedicated to trinities.
In his list there is one trinity on sporadic groups :
where $F_{24} $ is the Fischer simple group of order $2^{21}.3^{16}.5^2.7^3.11.13.17.23.29 = 1255205709190661721292800 $, which is the third largest sporadic group (the two larger ones being the Baby Monster and the Monster itself).
I don't know what the rationale is behind this trinity. But I'd like to recall the (Baby)Monster history as a warning against the trinity-reflex. Sometimes, there is just no way to extend a would be trinity.
The story comes from Mark Ronan's book Symmetry and the Monster on page 178.
Let's remind ourselves how we got here. A few years earlier, Fischer has created his 'transposition' groups Fi22, Fi23, and Fi24. He had called them M(22), M(23), and M(24), because they were related to Mathieu's groups M22,M23, and M24, and since he used Fi22 to create his new group of mirror symmetries, he tentatively called it $M^{22} $.
It seemed to appear as a cross-section in something even bigger, and as this larger group was clearly associated with Fi24, he labeled it $M^{24} $. Was there something in between that could be called $M^{23} $?
Fischer visited Cambridge to talk on his new work, and Conway named these three potential groups the Baby Monster, the Middle Monster, and the Super Monster. When it became clear that the Middle Monster didn't exist, Conway settled on the names Baby Monster and Monster, and this became the standard terminology.
Marcus du Sautoy's account in Finding Moonshine is slightly different. He tells on page 322 that the Super Monster didn't exist. Anyone knowing the factual story?
Some mathematical trickery later revealed that the Super Monster was going to be impossible to build: there were certain features that contradicted each other. It was just a mirage, which vanished under closer scrutiny. But the other two were still looking robust. The Middle Monster was rechristened simply the Monster.
And, the inclusion diagram of the sporadic simples tells yet another story.
Anyhow, this inclusion diagram is helpful in seeing the three generations of the Happy Family (as well as the Pariahs) of the sporadic groups, terminology invented by Robert Griess in his 100+p Inventiones paper on the construction of the Monster (which he liked to call, for obvious reasons, the Friendly Giant denoted by FG).
The happy family appears in Table 1.1. of the introduction.
It was this picture that made me propose the trinity on the left below in the previous post. I now like to add another trinity on the right, and, the connection between the two is clear.
Here $Golay $ denotes the extended binary Golay code of which the Mathieu group $M_{24} $ is the automorphism group. $Leech $ is of course the 24-dimensional Leech lattice of which the automorphism group is a double cover of the Conway group $Co_1 $. $Griess $ is the Griess algebra which is a nonassociative 196884-dimensional algebra of which the automorphism group is the Monster.
I am aware of a construction of the Leech lattice involving the quaternions (the icosian construction of chapter 8, section 2.2 of SPLAG). Does anyone know of a construction of the Griess algebra involving octonions???
Arnold's trinities
Referring to the triple of exceptional Galois groups $L_2(5),L_2(7),L_2(11) $ and its connection to the Platonic solids I wrote : "It sure seems that surprises often come in triples…". Briefly I considered replacing triples by trinities, but then, I didnt want to sound too mystic…
David Corfield of the n-category cafe and a dialogue on infinity (and perhaps other blogs I'm unaware of) pointed me to the paper Symplectization, complexification and mathematical trinities by Vladimir I. Arnold. (Update : here is a PDF-conversion of the paper)
The paper is a write-up of the second in a series of three lectures Arnold gave in june 1997 at the meeting in the Fields Institute dedicated to his 60th birthday. The goal of that lecture was to explain some mathematical dreams he had.
The next dream I want to present is an even more fantastic set of theorems and conjectures. Here I also have no theory and actually the ideas form a kind of religion rather than mathematics.
The key observation is that in mathematics one encounters many trinities. I shall present a list of examples. The main dream (or conjecture) is that all these trinities are united by some rectangular "commutative diagrams".
I mean the existence of some "functorial" constructions connecting different trinities. The knowledge of the existence of these diagrams provides some new conjectures which might turn to be true theorems.
Follows a list of 12 trinities, many taken from Arnold's field of expertise being differential geometry. I'll restrict to the more algebraically inclined ones.
1 : "The first trinity everyone knows is"
where $\mathbb{H} $ are the Hamiltonian quaternions. The trinity on the left may be natural to differential geometers who see real and complex and hyper-Kaehler manifolds as distinct but related beasts, but I'm willing to bet that most algebraists would settle for the trinity on the right where $\mathbb{O} $ are the octonions.
2 : The next trinity is that of the exceptional Lie algebras E6, E7 and E8.
with corresponding Dynkin-Coxeter diagrams
Arnold has this to say about the apparent ubiquity of Dynkin diagrams in mathematics.
Manin told me once that the reason why we always encounter this list in many different mathematical classifications is its presence in the hardware of our brain (which is thus unable to discover a more complicated scheme).
I still hope there exists a better reason that once should be discovered.
Amen to that. I'm quite hopeful human evolution will overcome the limitations of Manin's brain…
3 : Next comes the Platonic trinity of the tetrahedron, cube and dodecahedron
Clearly one can argue against this trinity as follows : a tetrahedron is a bunch of triangles such that there are exactly 3 of them meeting in each vertex, a cube is a bunch of squares, again 3 meeting in every vertex, a dodecahedron is a bunch of pentagons 3 meeting in every vertex… and we can continue the pattern. What should be a bunch a hexagons such that in each vertex exactly 3 of them meet? Well, only one possibility : it must be the hexagonal tiling (on the left below). And in normal Euclidian space we cannot have a bunch of septagons such that three of them meet in every vertex, but in hyperbolic geometry this is still possible and leads to the Klein quartic (on the right). Check out this wonderful post by John Baez for more on this.
4 : The trinity of the rotation symmetry groups of the three Platonics
where $A_n $ is the alternating group on n letters and $S_n $ is the symmetric group.
Clearly, any rotation of a Platonic solid takes vertices to vertices, edges to edges and faces to faces. For the tetrahedron we can easily see the 4 of the group $A_4 $, say the 4 vertices. But what is the 4 of $S_4 $ in the case of a cube? Well, a cube has 4 body-diagonals and they are permuted under the rotational symmetries. The most difficult case is to see the $5 $ of $A_5 $ in the dodecahedron. Well, here's the solution to this riddle
there are exactly 5 inscribed cubes in a dodecahedron and they are permuted by the rotations in the same way as $A_5 $.
7 : The seventh trinity involves complex polynomials in one variable
the Laurant polynomials and the modular polynomials (that is, rational functions with three poles at 0,1 and $\infty $.
8 : The eight one is another beauty
Here 'numbers' are the ordinary complex numbers $\mathbb{C} $, the 'trigonometric numbers' are the quantum version of those (aka q-numbers) which is a one-parameter deformation and finally, the 'elliptic numbers' are a two-dimensional deformation. If you ever encountered a Sklyanin algebra this will sound familiar.
This trinity is based on a paper of Turaev and Frenkel and I must come back to it some time…
The paper has some other nice trinities (such as those among Whitney, Chern and Pontryagin classes) but as I cannot add anything sensible to it, let us include a few more algebraic trinities. The first one attributed by Arnold to John McKay
13 : A trinity parallel to the exceptional Lie algebra one is
between the 27 straight lines on a cubic surface, the 28 bitangents on a quartic plane curve and the 120 tritangent planes of a canonic sextic curve of genus 4.
14 : The exceptional Galois groups
explained last time.
15 : The associated curves with these groups as symmetry groups (as in the previous post)
where the ? refers to the mysterious genus 70 curve. I'll check with one of the authors whether there is still an embargo on the content of this paper and if not come back to it in full detail.
16 : The three generations of sporadic groups
Do you have other trinities you'd like to worship?
bloomsday 2 : BistroMath
Exactly one year ago this blog was briefly renamed MoonshineMath. The concept being that it would focus on the mathematics surrounding the monster group & moonshine. Well, I got as far as the Mathieu groups…
After a couple of months, I changed the name back to neverendingbooks because I needed the freedom to post on any topic I wanted. I know some people preferred the name MoonshineMath, but so be it, anyone's free to borrow that name for his/her own blog.
Today it's bloomsday again, and, as I'm a cyclical guy, I have another idea for a conceptual blog : the bistromath chronicles (or something along this line).
Here's the relevant section from the Hitchhikers guide
Bistromathics itself is simply a revolutionary new way of understanding the behavior of numbers. …
Numbers written on restaurant checks within the confines of restaurants do not follow the same mathematical laws as numbers written on any other pieces of paper in any other parts of the Universe.
This single statement took the scientific world by storm. It completely revolutionized it.So many mathematical conferences got hold in such good restaurants that many of the finest minds of a generation died of obesity and heart failure and the science of math was put back by years.
Right, so what's the idea? Well, on numerous occasions Ive stated that any math-blog can only survive as a group-blog. I did approach a lot of people directly, but, as you have noticed, without too much success… Most of them couldnt see themselves contributing to a blog for one of these reasons : it costs too much energy and/or it's way too inefficient. They say : career-wise there are far cleverer ways to spend my energy than to write a blog. And… there's no way I can argue against this.
Whence plan B : set up a group-blog for a fixed amount of time (say one year), expect contributors to write one or two series of about 4 posts on their chosen topic, re-edit the better series afterwards and turn them into a book.
But, in order to make a coherent book proposal out of blog-post-series, they'd better center around a common theme, whence the BistroMath ploy. Imagine that some of these forgotten "restaurant-check-notes" are discovered, decoded and explained. Apart from the mathematics, one is free to invent new recepies or add descriptions of restaurants with some mathematical history, etc. etc.
One possible scenario (but I'm sure you will have much better ideas) : part of the knotation is found on a restaurant-check of some Italian restaurant. This allow to explain Conway's theory of rational tangles, give the perfect way to cook spaghetti to experiment with tangles and tell the history of Manin's Italian restaurant in Bonn where (it is rumoured) the 1998 Fields medals were decided…
But then, there is no limit to your imagination as long as it somewhat fits within the framework. For example, I'd love to read the transcripts of a chat-session in SecondLife between Dedekind and Conway on the construction of real numbers… I hope you get the drift.
I'm not going to rename neverendingbooks again, but am willing to set up the BistroMath blog provided
Five to ten people are interested to participate
At least one book-editor shows an interest
update : (16/06) contacted by first publisher
You can leave a comment or, if you prefer, contact me via email (if you're human you will have no problem getting my address…).
Clearly, people already blogging are invited and are allowed to cross-post (in fact, that's what I will do if it ever gets so far). Finally, if you are not willing to contribute blog-posts but like the idea and are willing to contribute to it in any other way, we are still auditioning for chanting monks
The small group of monks who had taken up hanging around the major research institutes singing strange chants to the effect that the Universe was only a figment of its own imagination were eventually given a street theater grant and went away.
And, if you do not like this idea, there will be another bloomsday-idea next year…
Farey symbols of sporadic groups
Published March 20, 2008 by lievenlb
John Conway once wrote :
There are almost as many different constructions of $M_{24} $ as there have been mathematicians interested in that most remarkable of all finite groups.
In the inguanodon post Ive added yet another construction of the Mathieu groups $M_{12} $ and $M_{24} $ starting from (half of) the Farey sequences and the associated cuboid tree diagram obtained by demanding that all edges are odd. In this way the Mathieu groups turned out to be part of a (conjecturally) infinite sequence of simple groups, starting as follows :
$L_2(7),M_{12},A_{16},M_{24},A_{28},A_{40},A_{48},A_{60},A_{68},A_{88},A_{96},A_{120},A_{132},A_{148},A_{164},A_{196},\ldots $
It is quite easy to show that none of the other sporadics will appear in this sequence via their known permutation representations. Still, several of the sporadic simple groups are generated by an element of order two and one of order three, so they are determined by a finite dimensional permutation representation of the modular group $PSL_2(\mathbb{Z}) $ and hence are hiding in a special polygonal region of the Dedekind's tessellation
Let us try to figure out where the sporadic with the next simplest permutation representation is hiding : the second Janko group $J_2 $, via its 100-dimensional permutation representation. The Atlas tells us that the order two and three generators act as
e:= (1,84)(2,20)(3,48)(4,56)(5,82)(6,67)(7,55)(8,41)(9,35)(10,40)(11,78)(12, 100)(13,49)(14,37)(15,94)(16,76)(17,19)(18,44)(21,34)(22,85)(23,92)(24, 57)(25,75)(26,28)(27,64)(29,90)(30,97)(31,38)(32,68)(33,69)(36,53)(39,61) (42,73)(43,91)(45,86)(46,81)(47,89)(50,93)(51,96)(52,72)(54,74)(58,99) (59,95)(60,63)(62,83)(65,70)(66,88)(71,87)(77,98)(79,80);
v:= (1,80,22)(2,9,11)(3,53,87)(4,23,78)(5,51,18)(6,37,24)(8,27,60)(10,62,47) (12,65,31)(13,64,19)(14,61,52)(15,98,25)(16,73,32)(17,39,33)(20,97,58) (21,96,67)(26,93,99)(28,57,35)(29,71,55)(30,69,45)(34,86,82)(38,59,94) (40,43,91)(42,68,44)(46,85,89)(48,76,90)(49,92,77)(50,66,88)(54,95,56) (63,74,72)(70,81,75)(79,100,83);
But as the kfarey.sage package written by Chris Kurth calculates the Farey symbol using the L-R generators, we use GAP to find those
L = e*v^-1 and R=e*v^-2 so
L=(1,84,22,46,70,12,79)(2,58,93,88,50,26,35)(3,90,55,7,71,53,36)(4,95,38,65,75,98,92)(5,86,69,39,14,6,96)(8,41,60,72,61,17, 64)(9,57,37,52,74,56,78)(10,91,40,47,85,80,83)(11,23,49,19,33,30,20)(13,77,15,59,54,63,27)(16,48,87,29,76,32,42)(18,68, 73,44,51,21,82)(24,28,99,97,45,34,67)(25,81,89,62,100,31,94)
R=(1,84,80,100,65,81,85)(2,97,69,17,13,92,78)(3,76,73,68,16,90,71)(4,54,72,14,24,35,11)(5,34,96,18,42,32,44)(6,21,86,30,58, 26,57)(7,29,48,53,36,87,55)(8,41,27,19,39,52,63)(9,28,93,66,50,99,20)(10,43,40,62,79,22,89)(12,83,47,46,75,15,38)(23,77, 25,70,31,59,56)(33,45,82,51,67,37,61)(49,64,60,74,95,94,98)
Defining these permutations in sage and using kfarey, this gives us the Farey-symbol of the associated permutation representation
L=SymmetricGroup(Integer(100))("(1,84,22,46,70,12,79)(2,58,93,88,50,26,35)(3,90,55,7,71,53,36)(4,95,38,65,75,98,92)(5,86,69,39,14,6,96)(8,41,60,72,61,17, 64)(9,57,37,52,74,56,78)(10,91,40,47,85,80,83)(11,23,49,19,33,30,20)(13,77,15,59,54,63,27)(16,48,87,29,76,32,42)(18,68, 73,44,51,21,82)(24,28,99,97,45,34,67)(25,81,89,62,100,31,94)")
R=SymmetricGroup(Integer(100))("(1,84,80,100,65,81,85)(2,97,69,17,13,92,78)(3,76,73,68,16,90,71)(4,54,72,14,24,35,11)(5,34,96,18,42,32,44)(6,21,86,30,58, 26,57)(7,29,48,53,36,87,55)(8,41,27,19,39,52,63)(9,28,93,66,50,99,20)(10,43,40,62,79,22,89)(12,83,47,46,75,15,38)(23,77, 25,70,31,59,56)(33,45,82,51,67,37,61)(49,64,60,74,95,94,98)")
sage: FareySymbol("Perm",[L,R])
[[0, 1, 4, 3, 2, 5, 18, 13, 21, 71, 121, 413, 292, 463, 171, 50, 29, 8, 27, 46, 65, 19, 30, 11, 3, 10, 37, 64, 27, 17, 7, 4, 5], [1, 1, 3, 2, 1, 2, 7, 5, 8, 27, 46, 157, 111, 176, 65, 19, 11, 3, 10, 17, 24, 7, 11, 4, 1, 3, 11, 19, 8, 5, 2, 1, 1], [-3, 1, 4, 4, 2, 3, 6, -3, 7, 13, 14, 15, -3, -3, 15, 14, 11, 8, 8, 10, 12, 12, 10, 9, 5, 5, 9, 11, 13, 7, 6, 3, 2, 1]]
Here, the first string gives the numerators of the cusps, the second the denominators and the third gives the pairing information (where [tex[-2 $ denotes an even edge and $-3 $ an odd edge. Fortunately, kfarey also allows us to draw the special polygonal region determined by a Farey-symbol. So, here it is (without the pairing data) :
the hiding place of $J_2 $…
It would be nice to have (a) other Farey-symbols associated to the second Janko group, hopefully showing a pattern that one can extend into an infinite family as in the inguanodon series and (b) to determine Farey-symbols of more sporadic groups. | CommonCrawl |
Greatest element and least element
In mathematics, especially in order theory, the greatest element of a subset $S$ of a partially ordered set (poset) is an element of $S$ that is greater than every other element of $S$. The term least element is defined dually, that is, it is an element of $S$ that is smaller than every other element of $S.$
Definitions
Let $(P,\leq )$ be a preordered set and let $S\subseteq P.$ An element $g\in P$ is said to be a greatest element of $S$ if $g\in S$ and if it also satisfies:
$s\leq g$ for all $s\in S.$
By switching the side of the relation that $s$ is on in the above definition, the definition of a least element of $S$ is obtained. Explicitly, an element $l\in P$ is said to be a least element of $S$ if $l\in S$ and if it also satisfies:
$l\leq s$ for all $s\in S.$
If $(P,\leq )$ is also a partially ordered set then $S$ can have at most one greatest element and it can have at most one least element. Whenever a greatest element of $S$ exists and is unique then this element is called the greatest element of $S$. The terminology the least element of $S$ is defined similarly.
If $(P,\leq )$ has a greatest element (resp. a least element) then this element is also called a top (resp. a bottom) of $(P,\leq ).$
Relationship to upper/lower bounds
Greatest elements are closely related to upper bounds.
Let $(P,\leq )$ be a preordered set and let $S\subseteq P.$ An upper bound of $S$ in $(P,\leq )$ is an element $u$ such that $u\in P$ and $s\leq u$ for all $s\in S.$ Importantly, an upper bound of $S$ in $P$ is not required to be an element of $S.$
If $g\in P$ then $g$ is a greatest element of $S$ if and only if $g$ is an upper bound of $S$ in $(P,\leq )$ and $g\in S.$ In particular, any greatest element of $S$ is also an upper bound of $S$ (in $P$) but an upper bound of $S$ in $P$ is a greatest element of $S$ if and only if it belongs to $S.$ In the particular case where $P=S,$ the definition of "$u$ is an upper bound of $S$ in $S$" becomes: $u$ is an element such that $u\in S$ and $s\leq u$ for all $s\in S,$ which is completely identical to the definition of a greatest element given before. Thus $g$ is a greatest element of $S$ if and only if $g$ is an upper bound of $S$ in $S$.
If $u$ is an upper bound of $S$ in $P$ that is not an upper bound of $S$ in $S$ (which can happen if and only if $u\not \in S$) then $u$ can not be a greatest element of $S$ (however, it may be possible that some other element is a greatest element of $S$). In particular, it is possible for $S$ to simultaneously not have a greatest element and for there to exist some upper bound of $S$ in $P$.
Even if a set has some upper bounds, it need not have a greatest element, as shown by the example of the negative real numbers. This example also demonstrates that the existence of a least upper bound (the number 0 in this case) does not imply the existence of a greatest element either.
Contrast to maximal elements and local/absolute maximums
A greatest element of a subset of a preordered set should not be confused with a maximal element of the set, which are elements that are not strictly smaller than any other element in the set.
Let $(P,\leq )$ be a preordered set and let $S\subseteq P.$ An element $m\in S$ is said to be a maximal element of $S$ if the following condition is satisfied:
whenever $s\in S$ satisfies $m\leq s,$ then necessarily $s\leq m.$
If $(P,\leq )$ is a partially ordered set then $m\in S$ is a maximal element of $S$ if and only if there does not exist any $s\in S$ such that $m\leq s$ and $s\neq m.$ A maximal element of $(P,\leq )$ is defined to mean a maximal element of the subset $S:=P.$
A set can have several maximal elements without having a greatest element. Like upper bounds and maximal elements, greatest elements may fail to exist.
In a totally ordered set the maximal element and the greatest element coincide; and it is also called maximum; in the case of function values it is also called the absolute maximum, to avoid confusion with a local maximum.[1] The dual terms are minimum and absolute minimum. Together they are called the absolute extrema. Similar conclusions hold for least elements.
Role of (in)comparability in distinguishing greatest vs. maximal elements
One of the most important differences between a greatest element $g$ and a maximal element $m$ of a preordered set $(P,\leq )$ has to do with what elements they are comparable to. Two elements $x,y\in P$ are said to be comparable if $x\leq y$ or $y\leq x$; they are called incomparable if they are not comparable. Because preorders are reflexive (which means that $x\leq x$ is true for all elements $x$), every element $x$ is always comparable to itself. Consequently, the only pairs of elements that could possibly be incomparable are distinct pairs. In general, however, preordered sets (and even directed partially ordered sets) may have elements that are incomparable.
By definition, an element $g\in P$ is a greatest element of $(P,\leq )$ if $s\leq g,$ for every $s\in P$; so by its very definition, a greatest element of $(P,\leq )$ must, in particular, be comparable to every element in $P.$ This is not required of maximal elements. Maximal elements of $(P,\leq )$ are not required to be comparable to every element in $P.$ This is because unlike the definition of "greatest element", the definition of "maximal element" includes an important if statement. The defining condition for $m\in P$ to be a maximal element of $(P,\leq )$ can be reworded as:
For all $s\in P,$ IF $m\leq s$ (so elements that are incomparable to $m$ are ignored) then $s\leq m.$
Example where all elements are maximal but none are greatest
Suppose that $S$ is a set containing at least two (distinct) elements and define a partial order $\,\leq \,$ on $S$ by declaring that $i\leq j$ if and only if $i=j.$ If $i\neq j$ belong to $S$ then neither $i\leq j$ nor $j\leq i$ holds, which shows that all pairs of distinct (i.e. non-equal) elements in $S$ are incomparable. Consequently, $(S,\leq )$ can not possibly have a greatest element (because a greatest element of $S$ would, in particular, have to be comparable to every element of $S$ but $S$ has no such element). However, every element $m\in S$ is a maximal element of $(S,\leq )$ because there is exactly one element in $S$ that is both comparable to $m$ and $\geq m,$ that element being $m$ itself (which of course, is $\leq m$).[note 1]
In contrast, if a preordered set $(P,\leq )$ does happen to have a greatest element $g$ then $g$ will necessarily be a maximal element of $(P,\leq )$ and moreover, as a consequence of the greatest element $g$ being comparable to every element of $P,$ if $(P,\leq )$ is also partially ordered then it is possible to conclude that $g$ is the only maximal element of $(P,\leq ).$ However, the uniqueness conclusion is no longer guaranteed if the preordered set $(P,\leq )$ is not also partially ordered. For example, suppose that $R$ is a non-empty set and define a preorder $\,\leq \,$ on $R$ by declaring that $i\leq j$ always holds for all $i,j\in R.$ The directed preordered set $(R,\leq )$ is partially ordered if and only if $R$ has exactly one element. All pairs of elements from $R$ are comparable and every element of $R$ is a greatest element (and thus also a maximal element) of $(R,\leq ).$ So in particular, if $R$ has at least two elements then $(R,\leq )$ has multiple distinct greatest elements.
Properties
Throughout, let $(P,\leq )$ be a partially ordered set and let $S\subseteq P.$
• A set $S$ can have at most one greatest element.[note 2] Thus if a set has a greatest element then it is necessarily unique.
• If it exists, then the greatest element of $S$ is an upper bound of $S$ that is also contained in $S.$
• If $g$ is the greatest element of $S$ then $g$ is also a maximal element of $S$[note 3] and moreover, any other maximal element of $S$ will necessarily be equal to $g.$[note 4]
• Thus if a set $S$ has several maximal elements then it cannot have a greatest element.
• If $P$ satisfies the ascending chain condition, a subset $S$ of $P$ has a greatest element if, and only if, it has one maximal element.[note 5]
• When the restriction of $\,\leq \,$ to $S$ is a total order ($S=\{1,2,4\}$ in the topmost picture is an example), then the notions of maximal element and greatest element coincide.[note 6]
• However, this is not a necessary condition for whenever $S$ has a greatest element, the notions coincide, too, as stated above.
• If the notions of maximal element and greatest element coincide on every two-element subset $S$ of $P,$ then $\,\leq \,$ is a total order on $P.$[note 7]
Sufficient conditions
• A finite chain always has a greatest and a least element.
Top and bottom
The least and greatest element of the whole partially ordered set play a special role and are also called bottom (⊥) and top (⊤), or zero (0) and unit (1), respectively. If both exist, the poset is called a bounded poset. The notation of 0 and 1 is used preferably when the poset is a complemented lattice, and when no confusion is likely, i.e. when one is not talking about partial orders of numbers that already contain elements 0 and 1 different from bottom and top. The existence of least and greatest elements is a special completeness property of a partial order.
Further introductory information is found in the article on order theory.
Examples
• The subset of integers has no upper bound in the set $\mathbb {R} $ of real numbers.
• Let the relation $\,\leq \,$ on $\{a,b,c,d\}$ be given by $a\leq c,$ $a\leq d,$ $b\leq c,$ $b\leq d.$ The set $\{a,b\}$ has upper bounds $c$ and $d,$ but no least upper bound, and no greatest element (cf. picture).
• In the rational numbers, the set of numbers with their square less than 2 has upper bounds but no greatest element and no least upper bound.
• In $\mathbb {R} ,$ the set of numbers less than 1 has a least upper bound, viz. 1, but no greatest element.
• In $\mathbb {R} ,$ the set of numbers less than or equal to 1 has a greatest element, viz. 1, which is also its least upper bound.
• In $\mathbb {R} ^{2}$ with the product order, the set of pairs $(x,y)$ with $0<x<1$ has no upper bound.
• In $\mathbb {R} ^{2}$ with the lexicographical order, this set has upper bounds, e.g. $(1,0).$ It has no least upper bound.
See also
• Essential supremum and essential infimum
• Initial and terminal objects
• Maximal and minimal elements
• Limit superior and limit inferior (infimum limit)
• Upper and lower bounds
Notes
1. Of course, in this particular example, there exists only one element in $S$ that is comparable to $m,$ which is necessarily $m$ itself, so the second condition "and $\geq m,$" was redundant.
2. If $g_{1}$ and $g_{2}$ are both greatest, then $g_{1}\leq g_{2}$ and $g_{2}\leq g_{1},$ and hence $g_{1}=g_{2}$ by antisymmetry.
3. If $g$ is the greatest element of $S$ and $s\in S,$ then $s\leq g.$ By antisymmetry, this renders ($g\leq s$ and $g\neq s$) impossible.
4. If $M$ is a maximal element, then $M\leq g$ since $g$ is greatest, hence $M=g$ since $M$ is maximal.
5. Only if: see above. — If: Assume for contradiction that $S$ has just one maximal element, $m,$ but no greatest element. Since $m$ is not greatest, some $s_{1}\in S$ must exist that is incomparable to $m.$ Hence $s_{1}\in S$ cannot be maximal, that is, $s_{1}<s_{2}$ must hold for some $s_{2}\in S.$ The latter must be incomparable to $m,$ too, since $m<s_{2}$ contradicts $m$'s maximality while $s_{2}\leq m$ contradicts the incomparability of $m$ and $s_{1}.$ Repeating this argument, an infinite ascending chain $s_{1}<s_{2}<\cdots <s_{n}<\cdots $ can be found (such that each $s_{i}$ is incomparable to $m$ and not maximal). This contradicts the ascending chain condition.
6. Let $m\in S$ be a maximal element, for any $s\in S$ either $s\leq m$ or $m\leq s.$ In the second case, the definition of maximal element requires that $m=s,$ so it follows that $s\leq m.$ In other words, $m$ is a greatest element.
7. If $a,b\in P$ were incomparable, then $S=\{a,b\}$ would have two maximal, but no greatest element, contradicting the coincidence.
References
1. The notion of locality requires the function's domain to be at least a topological space.
• Davey, B. A.; Priestley, H. A. (2002). Introduction to Lattices and Order (2nd ed.). Cambridge University Press. ISBN 978-0-521-78451-1.
| Wikipedia |
Socioeconomic and demographic characterization of an endemic malaria region in Brazil by multiple correspondence analysis
Raquel M. Lana ORCID: orcid.org/0000-0002-7573-13641,2,
Thais I. S. Riback2,
Tiago F. M. Lima3,
Mônica da Silva-Nunes4,
Oswaldo G. Cruz2,
Francisco G. S. Oliveira5,
Gilberto G. Moresco6,
Nildimar A. Honório7,8 &
Cláudia T. Codeço2
Malaria Journal volume 16, Article number: 397 (2017) Cite this article
This article has been updated
The Correction to this article has been published in Malaria Journal 2017 16:408
In the process of geographical retraction of malaria, some important endemicity pockets remain. Here, we report results from a study developed to obtain detailed community data from an important malaria hotspot in Latin America (Alto Juruá, Acre, Brazil), to investigate the association of malaria with socioeconomic, demographic and living conditions.
A household survey was conducted in 40 localities (n = 520) of Mâncio Lima and Rodrigues Alves municipalities, Acre state. Information on previous malaria, schooling, age, gender, income, occupation, household structure, habits and behaviors related to malaria exposure was collected. Multiple correspondence analysis (MCA) was applied to characterize similarities between households and identify gradients. The association of these gradients with malaria was assessed using regression.
The first three dimensions of MCA accounted for almost 50% of the variability between households. The first dimension defined an urban/rurality gradient, where urbanization was associated with the presence of roads, basic services as garbage collection, water treatment, power grid energy, and less contact with the forest. There is a significant association between this axis and the probability of malaria at the household level, OR = 1.92 (1.23–3.02). The second dimension described a gradient from rural settlements in agricultural areas to those in forested areas. Access via dirt road or river, access to electricity power-grid services and aquaculture were important variables. Malaria was at lower risk at the forested area, OR = 0.55 (1.23–1.12). The third axis detected intraurban differences and did not correlate with malaria.
Living conditions in the study area are strongly geographically structured. Although malaria is found throughout all the landscapes, household traits can explain part of the variation found in the odds of having malaria. It is expected these results stimulate further discussions on modelling approaches targeting a more systemic and multi-level view of malaria dynamics.
The American continent witnessed a large reduction in malaria incidence in the last century, with 18 out of 21 previously malaria-endemic countries now close to elimination [1]. Malaria activity is currently concentrated in the Amazonian region (including Brazil, Colombia, Peru, and Venezuela) where Plasmodium falciparum and Plasmodium vivax are the two main parasites. In Brazil, morbidity and mortality by falciparum malaria decreased since the 1980's [2] and, in 2010, it was mostly restricted to the Northwest Amazon, while more than 80% of malaria burden in the Brazilian and Peruvian Amazon [2, 3], and 67% in Colômbia [4] is attributed to P. vivax. Controlling vivax malaria is more challenging for several reasons: the large proportion of asymptomatic and sub-microscopic infections are more difficult to detect and cure; and its latent stage, lasting weeks to months, is refractory to standard treatments [5]. Relapses from latent infections account for 14–40% of malaria episodes in the Amazon [6].
Despite low mortality, vivax malaria imposes a high disease burden in malaria transmission hotspots, where individuals can experience several debilitating episodes per year, impairing their capacity to work and study. Although not frequent, P. vivax infection is associated with unusual complications [7]. During pregnancy, it is a cause of stillbirth and low birth weight, and during infancy, it may affect child development [5].
Subsequent waves of colonization and development in the Amazon have created strong spatial heterogeneities, with the coexistence of small and scattered communities along rivers, agricultural settlements along "fishbone" road networks, and towns with different levels of development [8]. These communities are connected by the flow of individuals seeking health, banking services, purchasing goods, attending school and working [9]. The population along rivers are composed mostly of indigenous and descendants of immigrants from the nineteenth and mid-twentieth centuries entering the rubber industry. In the 1940's, a large wave of colonization driven by rubber exploitation populated many of the larger localities found today. More recently, immigrants were attracted to the settlements created by the National Institute for Land Reform (INCRA) along new roads crossing the Amazonian region. These parallel secondary roads form "a fishbone land occupation scheme" that can easily be seen from satellite, creating an extended deforestation fringe. Within the Brazilian Amazon, three municipalities (Mâncio Lima, Rodrigues Alves and Cruzeiro do Sul) constitute a persistent malaria hotspot (defined as a site with significantly higher disease risk than average), for both P. falciparum and P. vivax parasites [10]. Here, this area will be referred to as the "Juruá hotspot" in reference to the administrative region where it is located. Many studies have assessed large-scale drivers of malaria in the Amazon, and more specifically, in the Juruá hotspot, associating malaria risk with forest cover [10], deforestation [11], and fish farming [12]. Malaria transmission in this area is found in both rural and urban zones, and the Juruá hotspot constitutes one of the seven sites for urban malaria studies of the International Centers of Excellence for Malaria Research (ICEMR) [13].
Micro-epidemiology is the study of fine-scale variations in risk? between households or other sub-village groupings within villages, or between neighbouring villages or other similar socio-spatial aggregations such as urban neighbourhoods, agricultural settlements and health centre catchment areas? [14]. The household survey described here is a micro-epidemiology study conducted in the Juruá hotspot to investigate the distribution of malaria and its interaction with socio-economic, behavioural and demographic parameters along gradients of development. Using multivariate analysis, living conditions associated with exposure to malaria were identified, providing a characterization of this hard-to-reach population at a geographical scale not studied before. As this population is under strong environmental and development change, this study provides a baseline for subsequent monitoring activities and guide the application of interventions.
The study area is located within two municipalities in the Alto Juruá region, Acre state, Brazil. Mâncio Lima (ML) (7.5468\({^\circ }\)S, 73.3709\(^{\circ }\)W) has 15,206 inhabitants (2.79 inhabitants/km\(^{2}\)), and Rodrigues Alves (RA) (7.8819\(^{\circ }\)S, 73.3709\(^{\circ }\)W) has 14,389 inhabitants (4.68 inhabitants/km\(^{2}\)) [15]. Their administrative centers are connected to each other (40 km) and to Cruzeiro do Sul (CS) by a paved road (CS-RA: 12 km, CS-ML: 43 km). CS is Acre's second-largest city (78,507 inhabitants), has an airport with daily flights to the state capital, Rio Branco. A paved 700 km road connects CS to Rio Branco. These modern modes of transportation coexist with the traditional and heavily used waterway transport (the Juruá River is the main commodity transportation route between the neighbour municipalities of the Alto Juruá region and Manaus, in the Amazon as state). Two-thirds of ML is occupied by protected areas, including the Serra do Divisor National Park and indigenous reserves. About half (57.3%) of the Mâncio Lima population lives in the administrative center (a town divided into 9 localities or neighbourhoods), the remainder distributed in 57 small localities scattered along dirt roads and rivers. In Rodrigues Alves (RA), 30% lives in a small town (5 localities or neighbourhoods) and 70% in 69 rural settlements.
Map of the municipalities of Mâncio Lima and Rodrigues Alves, Acre, Brazil, with the 40 localities included in the survey. The point labels are abbreviations of the localities' names (full names in Additional file 1). Points are colored according to the different zones: ML-r (red), ML-u (green), RA-r (blue) and RA-u (violet). Source of image: Google MAPS API (https://developers.google.com/maps/terms), 2017. Software QGis Version 2.18
The survey, conducted in 2015, included 40 localities, being 5 in the town of Mâncio Lima (ML-u), 9 in the town of Rodrigues Alves (RA-u), 13 in the rural area of Mâncio Lima (ML-r) and 13 in the rural area of Rodrigues Alves (RA-r). ML-r was surveyed in February when river level was high, the two towns (ML-u and RA-u) were surveyed in May, and the road accessible rural localities in RA-r were surveyed in July, during the dry season when roads are open (Fig. 1).
Riverine localities (ML-r)
ML-r localities are located along the Moa river and its tributary, the Azul river. They are accessible only by boat, taking up to 2 days to reach the farthest point during the rainy season. The landscape is mostly covered by forest; the population lives mainly by manioc production, fishing and, social welfare. From the total of 26 localities, 20 non-indigenous, 13 were surveyed (66%). Outdated official data indicated 399 households in the 13 localities but 182 households were found inquiring the local population, varying from 3 to 40 households per locality. For logistic reasons, the sampling effort was concentrated at the core of each locality, excluding isolated single households and not returning to those that were closed. A total of 107 households were interviewed in this area (sample effort = 58.7%), during the 6 days of the survey, corresponding to 505 dwellers.
Localities in the two towns (ML-u and RA-u)
All 15 within-town localities were included in the survey. The number of households in ML-u was 2589, varying from 32 to 730 per locality. RA-u had 1497 households, varying from 160 to 389 households per locality. A systematic sampling scheme was adapted to the geography of each town. In ML, every 8th to 12th house along a predefined route was surveyed, in a protocol similar to the Census, totaling 190 households (sample size = 7.3%) with 732 dwellers. In RA, 102 households (with 459 dwellers) were sampled, by choosing a side from each of the 102 lots and randomly sampling 1 household (sample size = 6.8%).
Rural localities accessible by road (RA-r)
RA-r localities are agricultural settlements distributed along 3 dirt roads radiating from the ML-RA main road, an example of a fishbone colonization pattern. The landscape is dominated by pastures but also includes areas of recent deforestation. During the monsoons, road access is mostly limited to 4 × 4 vehicles and motorcycles and may be completely blocked for weeks. Alternative access to some localities is possible via the Juruá river and its tributaries. The 13 surveyed localities varied in size from 18 to 133 households, totaling 2728 households. At each locality, the survey team sketched a map and selected houses the most evenly possible. The final sample size was 121 households (4.4%), with 578 dwellers.
The complete list of surveyed localities is found in Additional file 1, and representative images are found in Additional file 2.
The questionnaire was adapted from [16] to include questions that are specific to rural settings, such as the ownership of farm animals, as well as blocks covering mobility, access to health care, and behaviors associated with malaria exposure. A pilot study was conducted before the survey to test the questionnaire for clarity. The questionnaire is divided into 17 blocks of which, 14 are included in the present study (Table 1). Blocks A and B collect demographic information from the interviewee and the house mates, blocks C, D and E inquiry about their history of malaria infections, block F records the ownership of objects such as appliances and vehicles, block G records all the occupations and sources of income of the householders. Five blocks collect data on house construction material (H), accessibility (I), source of water (J), garbage disposal (L), and electricity (M). Questions on activities and habits associated with malaria exposure and usage of bed nets as well as other mosquito deterrents are in the last blocks (N, O). The questionnaire was answered by a householder, 18 years of age or more, who could respond for the other householders and agreed with the study, signing the informed consent. The research protocol was approved by Comitê de Ética da Escola Nacional de Saúde Pública Sérgio Arouca (Ethics Committee of the National School of Public Health Sérgio Arouca) (ENSP-Fiocruz), in 04/11/2014, Number 861.871.
Table 1 Questionnaire description
Analysis was carried out at the household level. Individual level variables (age, gender, schooling) were aggregated as following: children (presence of dwellers < 14 years old), elderly (presence of dwellers ≥ 60 years old), adult gender (only women, only men or both), adult gender 1 (uni or pluri gender), maximum schooling (with 9 categories), maximum schooling 1 (with 5 categories), illiteracy in dwellers aged ≥ 18 years 1 (0, ≥ 1), illiteracy in dwellers aged ≥ 18 years 2 (0, 1 or 2), illiteracy in dwellers aged ≥ 18 and ≤ 59 (0, 1 or 2), illiteracy in dwellers aged ≥ 60 years (0, 1 or 2). Information on previous malaria episodes was used to construct the variable household with malaria in the last 12 months (POS/NEG). Before proceeding, a preliminary analysis was carried out to identify and correct ill-formed variables. For example, variables with categories with fewer than 5 elements were recategorized; if two variables belonging to the same block were strongly correlated, one of them was chosen or a combination of both was created.
Multiple correspondence analysis (MCA)
Flowchart describing the steps for the construction of the MCA factor map
MCA is a multivariate exploratory analysis for visualizing large datasets of categorical variables [17, 18]. Its graphical visualization provides a structural organization for the variables and categories in a dimensional space that is useful for identifying patterns in the data and associations between the investigated parameters [18]. Here, MCA was used to elicit gradients of living conditions in the Juruá malaria hotspot. To carried out the analysis, variables were divided into 8 groups (Table 2). Malaria in the last 12 months (yes/no) was included as a supplementary variable. Supplementary variables do not contribute to the MCA but can be plotted together to provide useful visualization, in this case, of the distribution of malaria along the development gradients [19]. The MCA map was constructed step wisely (Fig. 2). For each of the 8 groups of variables, the MCA was applied and variables were selected using the squared cosine test. All variables with \(cos^{2}\) > 0.2 in at least one of the 3 first MCA dimensions were maintained in this first round. Some variables, such as 'presence of swamps', did not meet this criterion but were maintained due to their known association with malaria. In some instances, before excluding a variable, different categorizations were attempted and a variable was discarded only if no alternative version attended the selection criterion. Variables with missing data can strongly distort MCA results, so after comparing MCA with and without these variables, those with strong leverage effect were removed. The next step was to merge all selected variables into a single group and apply the MCA. Three rounds of MCA were performed in order to remove non-significant variables. In the first round, confidence ellipses were used for identifying nonsignificant categories that were collapsed into single categories [20]. This procedure helped to reduce the number of categories per variable. Second, the association between each variable and the response variable Household with or without malaria in the last 12 months was tested using chi-squared contingency test [20]. When confidence ellipses of all categories of a variable included the origin of the MCA plot and the chi-squared test was not significant (at p \(\le\) 0.2), the variable was excluded. However, some variables considered important from the theoretical perspective, such as those associated with malaria exposure (e.g., owning a boat) or a well-established indicator of income (owning a fridge) were maintained independently of these selection criteria. The same procedure was repeated in the second and third rounds, restricting to the application of the \(cos^{2}\) rule (Fig. 2). The number of dimensions maintained in the final model was determined based on their inertia, that is, the percentage of variance explained.
Table 2 Summary of the variables and its categories, organized by block, and the result of the chi-squared test for differences in their distribution between study zones (ML-u, ML-r, RA-u, RA-r) and between households with and without malaria in the last 12 months
Mixed logistic regression
To map the probability distribution of malaria along the MCA gradients, a mixed logistic regression model was fitted. The model has locality as a random intercept to control for clustering of households within localities and a fixed effect for each MCA dimension. Let \(Y_{ij}\) = report of malaria in the last 12 months in household i, at locality j. The model is:
$$\begin{aligned} logit(E(Y_{ij})) = b_{0} + b_{1} * dim1 + b_{2} * dim2 + b_{3} * dim3 + a_{j} \end{aligned}$$
where \(b_{0}\) is the intercept, \(b_{1}\), \(b_{2}\) and \(b_{3}\) are the fixed effects of the first, second and third MCA dimensions, and \(a_{j}\) is the random intercept with distribution \(a_{j} \sim N(0, \sigma ^{2})\). Preliminary analysis using additive models confirmed the linearity of the effect of the covariates on the response variable. All analysis were conducted in R 3.4.1 [21] using the FactoMineR package [20] and the lme4 package [22].
Characteristics of the surveyed households
The survey included 520 households totaling 2274 dwellers. Of these, 1112 (48.9%) were female and 1162 (51.1%) male. The mean age was 25.5 years old. A total of 442 (19.9%) persons were reported to have malaria in the last 12 months, 104 (23.53%) were interviewees, and 338 their housemates. At household level, 233 (44.8%) reported at least one episode of malaria in the last 12 months, distributed as: 56 households (10.77%) in ML-r, 80 (15.38%) in ML-u, 66 (12.69%) in RA-r, and 31 (5.96%) in RA-u (p < 0.01). Table 2 shows the Chi-squared association between surveyed variables and the report of malaria by the household. The variables most associated were adult gender, adult illiteracy, ownership of microwave, horse and chicken, working in agriculture, access to the household (river, road), piped water, dishwashing at the river bank, garbage collection, frequency of activities close to rivers and forest (Table 2).
Gradients of development
Plot of the MCA eigenvalues (black circles) and cumulative percent of inertia (white circles)
Distribution of households in a 1 × 2 dimension MCA factor map, b 1 × 3 dimension MCA factor map. Colors and circles refer to the location of the households in the four zones (ML-u and RA-u are the two urban areas, ML-r is the riverine area and RA-r is the road accessible rural area)
The first three MCA dimensions explained almost 50% of the variability among households. Dimension 1 contributed with 32% of the inertia, while dimension 2, contributed 9%, and dimension 3 with 7.04% (Fig. 3). These three dimensions were retained for the analysis. Figure 4a shows the V-shaped distribution of households along the first two MCA axes, with greater dispersion along the horizontal axis. In the MCA plot, the origin represents the average household and the dispersion around it indicates how they differ in relation to this average. Households from the two towns are clustered together in the second quadrant. The rural households aggregate in two clusters. One cluster, farthest to the right, is formed by households from the riverine zone (ML-r). The second cluster includes most households from the rural localities accessible by road (RA-r) and shows an overlap with the cluster formed by the two towns. Households at the intersection of these two clusters are located at Iracema (sigla.loc_IRA) and Pé-da-Terra (sigla.loc_PET), both at the periphery of ML. Figure 4b shows the distribution of households along the second and third axes. In this graph, an overlap between riverine and road accessible rural areas is observed. Variables defining the first dimension were associated with the availability of infrastructure and services, such as household accessibility via road or river, garbage disposal, source of water for domestic usage, and power grid electricity (Tables 2, 3; Additional file 3). Some variables directly associated with malaria exposure also contributed to the definition of this gradient, including the frequency at which householders entered the forest, worked in agriculture, had a boat, had crevices in their house walls. Income, measured in terms of ownership of non-essential appliances, such as washing machine, iron, blender, also contributed to this dimension. Overall, households at the high end of dimension 1 were accessible only by river, consumed water directly from the river, did not have bathrooms, had wood walls with crevices, electricity was often dependent on fuel-powered electric generator, relied on social benefits, and worked in family agriculture. In summary, they represent the households along rivers. Houses at the low end of dimension 1 were made of bricks without crevices, had piped water, bathroom with toilet, non-essential appliances, and received power grid electricity. They were accessed by paved roads, and had garbage collection. Most householders did not work in agriculture, nor required social benefits. From these features, dimension 1 can be interpreted as a gradient from the most developed areas and richer households found in the urban localities, to the most underdeveloped areas and poorer households located at the riverine localities (an urbanization/rurality gradient) (Additional file 3).
Table 3 Relative contributions of the variables to the 3 first dimensions of the MCA
Households did not spread as much along the second dimension as they did along the first one, differed mainly in their access, source of energy, and working activity. At the high end of the second axis, households were only accessible by river and were not power-grid-energy served (Additional file 3). At the low end of the second axis are rural households accessible by roads, with access to electricity and people working on aquaculture and agriculture activities. This second dimension discriminated between the riverine and road accessible rural households (Table 3; Additional file 3). While the riverine population works most in agriculture, the road rural population works in fish farming (a gradient of rural differentiation or riverine-road rural gradient).
The third dimension explained 7.04% of the variance. Despite the low variance, this dimension is important to discriminate households within the urban and road-accessible rural localities. The variables that most contributed to this axis are related to the physical characteristics of the house (materials) as well as their source of water. At the positive side of this axis, there are households made of brick, fully covered with roof lining, no crevices in their walls, piped water, bathroom with toilet, and fish ponds. At the negative side of the 3rd axis, households are made of wood with crevices, no roof lining, no bathroom, piped water only outside the house, no fishponds. From these features, the 3rd axis is interpreted as a housing construction gradient, ranging from primitive wood houses without plumbing to fully equipped brick houses. This gradient exists both in the towns and the road accessible zones, but not in the riverine zone (Fig. 4b).
Malaria distribution along the development gradients
Distribution of households in the a 1 × 2 dimension MCA factor map, and b the 1 × 3 dimension MCA factor map. Colors indicate the status of the household according to the presence of at least one member with malaria in the last 12 months
Figure 5 shows the distribution of households with and without malaria in the last 12 months in the 1 × 2 and 1 × 3 MCA factor maps. Although malaria does not cluster on any specific region, there is a slight trend towards the fourth quadrant of the 1 × 2 factor map (Fig. 5a) and the fourth and first quadrants of the 1 × 3 factor map (Fig. 5b). According to the mixed logistic regression model, the odds of observing a household with malaria increased significantly along the first MCA dimension (Table 4). The marginal probability of malaria along this axis ranged from ca. 30% in the most developed urban household to ca. 65% in the most forested primitive household (Additional file 4). Along the second dimension, there was a weak negative effect (p = 0.10), with the odds of having malaria increasing in households at the road accessible localities. Variation in malaria probability was lower in comparison with the first gradient. The third MCA dimension was not significantly associated with malaria. Figure 6 shows the predicted probability of malaria along the three MCA gradients.
Predicted probability of malaria along the three development gradients produced by the multivariate analysis
Table 4 Odds ratio of having a household with malaria cases along the three development gradients derived from the MCA
This study describes the micro-epidemiology of malaria within one of the most important malaria endemicity pockets in the Americas, where 3 municipalities contributed to the top 20 cities with the highest rates of malaria between 2012 and 2014 [2, 10, 23]: Mâncio Lima with 341.9 Annual Parasitic Index (API), Rodrigues Alves with 263.6 API and Cruzeiro do Sul with 195.2 API, in 2014 [23].
Three dimensions of development emerged from the household data, characterizing large independent axes of variation in living conditions in the region. The first dimension describes an index of rurality, with households at the low end of the scale is relatively urban, and the ones at the high end, the most rural. A continuous gradient of rurality has advantages over the traditional urban–rural dichotomy avoiding 'threshold traps' where households at the border are at risk of misclassification. As an objective concept, it can be used to trek the trajectories of rurality change over time [24]. At the locality level, greater rurality was associated with lower population size, lower population density, lower extent of built-in area and greater remoteness in relation to administrative centers, agreeing with the main dimensions considered by Waldorf [24].
The second dimension found in the study describes the different household conditions in riverine and road-accessible rural localities. It emphasizes differences in productive assets held by the household (having a fish pond), as well as communal assets to which they have access (roads and power grid electricity) that facilitate access to industrialized materials for house construction. In poor rural communities, these assets make their physical capital, which together with the social, human, natural and financial capitals, compose the household wealth [25]. Households along the rivers Moa and Azul had the lowest physical capital, in comparison to those accessible by roads. Lack of physical capital can be partially explained by the landscape. First, the transportation of goods in small boats along these rivers is costly and difficult. Second, available industrialized materials for house construction are not suitable for the flooded forest. These regions require specific malaria-proof housing solutions.
The third axis of development is an index of housing construction quality. Variation in this index is observed within the road accessible localities, but not within the river accessible households, due to the constraints above described. Housing construction quality is a proxy for wealth in several studies of malaria risk factors, as reviewed by Bannister-Tyrrell et al. [14].
All three development dimensions were required for the malaria probability model. The probability of malaria ranged from a minimum of 30% in the least rural (more urban) households, peaked at the households with intermediate rurality (60%) and decreased to ca. 50% in the most rural households. There is no evidence of a sharp transition in malaria risk between urban and rural localities that would suggest distinct levels of exposure. A stable spatial gradient encompassing all levels of rurality is of interest for management. This spatial pattern is consistent with malaria maintained in a source-sink metapopulation structure where localities with high and low transmission conditions are connected by human commutation [26]. The role of each locality as source or sink is not clear, however. Despite the low probability of malaria in the urban localities, it is known that localities in ML-u can maintain a large population of Anopheles darlingi and Anopheles albitarsis by their fish farming activities [12]. If this is sufficient to characterize the urban localities as sources, it is not known. The households with the highest probability of malaria were located at the end of the secondary roads, close to the deforestation borders. A similar pattern was observed by Silva-Nunes et al. [27] in the Acrelândia (Acre state) malaria hotspot, where a higher risk of malaria was associated with new plots in the periphery of more structured areas, generally occupied by new settlers from non-malaria regions. This is a pattern that seems to persist in the Amazon Basin and call for rethinking the current national colonization programme [28]. These populations are not isolated, but linked to each other by mobility, which enhances the importation of Plasmodium between hotspots and other regions [29], especially in regions of unstable populations that commute [10] in search of basic services such as health, banking and education, or for work [9] as the case of ML urban zone, the main attraction place of the study area. Fish farming is also found throughout the localities at intermediate rurality. This activity can contribute to maintaining transmission away from the forest frontier. Critical community size (CCS) is another factor that affects the stability of malaria. Localities with very small household density, as those found in the riverine region, may have environmental conditions for transmission, but not the minimum amount of individuals to sustain it. However, accurate information about CCS is somewhat challenging to achieve, as it is influenced by immunity, migrations, and other variables [30].
Characterizing the microgeography of the Juruá malaria hotspot is essential for better understanding the factors that maintain transmission and to design adequate control strategies. The place where one lives is a risk factor for malaria [27, 31]. In this study, it was detected the risk of malaria is higher for poorer individuals living in rural areas than for poorer individuals living in urban areas. Overall, the region presented low rates of development and poor infrastructure indicators such as water supply and adequate garbage disposal [15]. The control of malaria and other diseases is further challenged by the fact that most people live within or very close to environmentally protected areas and require specifically tailored strategies.
The study relied on self-reported malaria and third-party information. This may be a source of bias. However, the population is familiar with malaria symptoms which are debilitating enough to be remembered. Another potential source of bias is related to the sampling scheme. Designing a sampling strategy for this heterogeneous population presented several challenges, as discussed in Kondo et al. [32]. The population is highly mobile and scattered; available population counts were imprecise. Whole localities disappear while new ones are created in a few years, householders spend months away from their primary address either working or studying or can move their wood houses from one place to the other, as necessary. Despite these limitations, this study has provided demographic results that can inform future, more detailed, complex and expensive studies using serological techniques and genetic markers.
This work contributed to the characterization of one of the most important endemicity pockets in the Americas and provides a wealth of information that can effectively be used for the overall goal of malaria elimination. The methodological approach using MCA proved useful for mapping the distribution of living conditions associated with malaria. There is no sharp transition in the risk of malaria between rural and urban localities, and interventions should not deal separately with the urban and rural zones. Finally, the characterization of this population is of great relevance for Brazil, since it is a hard to reach population, living in extreme poverty in a situation of invisibility. Understanding local habits, identifying populations at greater social vulnerability is essential for combating poverty and improving the living conditions of this population.
After publication of the article [1], it has been brought to our attention that the y-axis of Fig. 6 has been labeled incorrectly. It should read "linear predictor". This has now been corrected in the original article.
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RML, CTC and NAH conceived and developed the questionnaire and the study design. RML, CTC, TISR, FGSO, MSN, GGM participated in the household survey. TL and RML developed the database. RML typed and organized the database. RML, CTC and OGC analyzed the data. RML and CTC wrote the manuscript. All authors read and approved the final manuscript.
Prefeitura Municipal de Mâncio Lima, Prefeitura Municipal de Rodrigues Alves, Endemias de Mâncio Lima and Rodrigues Alves. Mâncio Lima and Rodrigues Alves residents. SESACRE. Household survey and typing team. Leonardo S. Bastos and Marcelo F.C. Gomes for LaTeXsupport and suggestions.
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.
All participants signed an informed consent for publication.
Ethical considerations. The study protocol was approved by the Ethical Review Board of the National Public Health School, Brazil (Number 861.871), and written informed consent was obtained from each adult participant.
This work was funded by Grant E-26/102.287/2013 from Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro—FAPERJ (http://www.faperj.br), Conselho Nacional de Desenvolvimento Científico e Tecnológico—CNPq (http://www.cnpq.br) 305553/2014-3, 454665/2014-8. RML was supported by PhD fellowship from Brasil Sem Miséria Program of Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (http://www.capes.gov.br). The funding agencies had no role in the design, collection, analysis or interpretation of the data.
Programa de Pós-Graduação em Epidemilogia em Saúde Pública, Escola Nacional de Saúde Pública Sérgio Arouca, Fundação Oswaldo Cruz, Rua Leopoldo Bulhões, 1480, Manguinhos, Rio de Janeiro, RJ, 21041-210, Brazil
Raquel M. Lana
Programa de Computação Científica, Fundação Oswaldo Cruz, Residência Oficial, Avenida Brasil, 4365, Manguinhos, Rio de Janeiro, RJ, 21040-360, Brazil
, Thais I. S. Riback
, Oswaldo G. Cruz
& Cláudia T. Codeço
Laboratório de Engenharia e Desenvolvimento de Sistemas, Departamento de Computação e Sistemas, Instituto de Ciências Exatas e Aplicadas, Universidade Federal de Ouro Preto., Rua 36, n. 115, Loanda, João Monlevade, MG, 35931-008, Brazil
Tiago F. M. Lima
Centro de Ciências da Saúde, Universidade Federal do Acre, Campus Universitário-BR 364, km 4-Distrito Industrial, Rio Branco, AC, 69920-900, Brazil
Mônica da Silva-Nunes
Campus Cruzeiro do Sul, Universidade Federal do Acre, Estrada do Canela Fina, s/n, Cruzeiro do Sul, AC, 69980-000, Brazil
Francisco G. S. Oliveira
Coordenação Geral dos Programas Nacionais de Controle e Prevenção da Malária e das Doenças transmitidas pelo Aedes, Departamento de Vigilância das Doenças Transmissíveis, Secretaria de Vigilância em Saúde-Ministério da Saúde, SRTV 702, Via W 5 Norte, Ed. PO700-6 andar, Brasília, DF, 70723-040, Brazil
Gilberto G. Moresco
Laboratório de Mosquitos Transmissores de Hematozoários-Lathema, Instituto Oswaldo Cruz, FIOCRUZ, Avenida Brasil, 4365, Manguinhos, Rio de Janeiro, RJ, 21040-360, Brazil
Nildimar A. Honório
Núcleo Operacional Sentinela de Mosquitos Vetores-Nosmove/FIOCRUZ, Avenida Brasil, 4365, Manguinhos, Rio de Janeiro, RJ, 21040-360, Brazil
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Correspondence to Raquel M. Lana.
The original version of the article was revised to correct an error in figure 6.
A correction to this article is available online at https://doi.org/10.1186/s12936-017-2058-7.
12936_2017_2045_MOESM1_ESM.pdf
Additional file 1. Information of the study localities.
Additional file 2. Images of exemplification of the types of localities in the study area.
Additional file 3. Complementary MCA plots.
Additional file 4. Details of mixed logistic regression models analyzes.
Lana, R.M., Riback, T.I.S., Lima, T.F.M. et al. Socioeconomic and demographic characterization of an endemic malaria region in Brazil by multiple correspondence analysis. Malar J 16, 397 (2017) doi:10.1186/s12936-017-2045-z
Urban malaria
Rurality
Multiple correspondence analysis
Micro-epidemiology | CommonCrawl |
\begin{definition}[Definition:Right Derived Functor]
Let $\mathbf A$ be an abelian category with enough injectives.
Let $\mathbf B$ be an abelian category.
Let $F: \mathbf A \to \mathbf B$ be a left exact functor.
Let $X$ and $Y$ be objects of $\mathbf A$.
Let $f: X \to Y$ be a morphism of $\mathbf A$.
Let $I$ be an arbitrary injective resolution of $X$.
Let $J$ be an arbitrary injective resolution of $Y$.
Let $\tilde f : I \to J$ be a morphism of cochain complexes induced by $f$.
Let $\map F I$ denote the cochain complex defined by applying the functor on cochains induced by $F$ to $I$.
Let $i \in \Z_{\ge 0}$ be a non-negative integer.
Let $\map {H^i} {\map F I}$ denote the $i$-th cohomology of $\map F I$.
The '''$i$-th right derived functor''' $\mathrm R^i F : \mathbf A \to \mathbf B$ of $F$ is defined on objects as:
:$\mathrm R^i \map F X := \map {H^i} {\map F I}$
{{explain|If $\mathrm R^i \map F X$ is just defined the same as $\map {H^i} {\map F I}$, then why define it at all?
This article defines a sequence of functors $\mathrm R^i F$ attached to $F$. The definition of the right derived functors of a functor is a central definition in homological algebra and should not be omitted. --Wandynsky (talk) 11:00, 28 July 2021 (UTC)}}
{{explain|It is not clear what exactly is being defined here. Do the following lines contribute to the definition? Can't figure out exactly what is what. <br/> As has been done here in the above rewrite, the best approach to defining something (and standard {{ProofWiki}} style) is: a) Write at the start all the objects that contribute to the definition: "Let... let... let..." b) State the definition in terms of all those objects. Do not use the word "any", it is ambiguous and loose.
Tried to fix it. Does it look better now? It's a bit tricky in this case. --Wandynsky (talk) 08:22, 28 July 2021 (UTC)
Definite improvement, but some way to go. Further explain templates have been added. Once I understand what this page says, I will be able to try to put it into a form that others on my level (I failed my CSE mathematics) can get to grips with.
How straightforward would it be to go to a source work and present the material as presented there?
In extremis I may reconcile it with my copy of Freyd, but "derived functor" is in an exercise right at the end, and I'd need to work through the book to understand it, and I've barely cracked it open.}}
{{explain|Are there in fact two different definitions being set up here? If that is the case, we need two different pages for them. Perhaps transclude one inside the other.}}
The '''$i$-th right derived functor''' $\mathrm R^i F$ of $F$ is defined on morphisms as follows:
Define $\mathrm R^i \map F f: \mathrm R^i \map F X \to \mathrm R^i \map F Y$ by the induced map $\map {H^i} {\map F {\tilde f} } : \map {H^i} {\map F I} \to \map {H^i} {\map F J}$.
\end{definition} | ProofWiki |
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February 2011, 5(1): 1-17. doi: 10.3934/ipi.2011.5.1
Modified iterated Tikhonov methods for solving systems of nonlinear ill-posed equations
Adriano De Cezaro 1, , Johann Baumeister 2, and Antonio Leitão 3,
Institute of Mathematics Statistics and Physics, Federal University of Rio Grande, Av. Italia km 8, 96201-900 Rio Grande, Brazil
Fachbereich Mathematik, Johann Wolfgang Goethe Universität, Robert–Mayer–Str. 6–10, 60054 Frankfurt am Main
Department of Mathematics, Federal University of St. Catarina, P.O. Box 476, 88040-900 Florianópolis
Received October 2009 Revised September 2010 Published February 2011
We investigate iterated Tikhonov methods coupled with a Kaczmarz strategy for obtaining stable solutions of nonlinear systems of ill-posed operator equations. We show that the proposed method is a convergent regularization method. In the case of noisy data we propose a modification, the so called loping iterated Tikhonov-Kaczmarz method, where a sequence of relaxation parameters is introduced and a different stopping rule is used. Convergence analysis for this method is also provided.
Keywords: Ill-posed equations, iterated Tikhonov method, Nonlinear systems, Regularization, Kaczmarz method..
Mathematics Subject Classification: Primary: 65J20; Secondary: 47J0.
Citation: Adriano De Cezaro, Johann Baumeister, Antonio Leitão. Modified iterated Tikhonov methods for solving systems of nonlinear ill-posed equations. Inverse Problems & Imaging, 2011, 5 (1) : 1-17. doi: 10.3934/ipi.2011.5.1
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Adriano De Cezaro Johann Baumeister Antonio Leitão | CommonCrawl |
\begin{document}
\title{Experimental Quantum Error Rejection for Quantum Communication} \author{Yu-Ao Chen} \affiliation{Physikalisches Institut, Universit\"{a}t Heidelberg, Philosophenweg 12, D-69120 Heidelberg, Germany} \author{An-Ning Zhang} \affiliation{Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230027, People's Republic of China} \author{Zhi Zhao} \affiliation{Physikalisches Institut, Universit\"{a}t Heidelberg, Philosophenweg 12, D-69120 Heidelberg, Germany} \affiliation{Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230027, People's Republic of China} \author{Xiao-Qi Zhou} \affiliation{Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230027, People's Republic of China} \author{Jian-Wei Pan} \affiliation{Physikalisches Institut, Universit\"{a}t Heidelberg, Philosophenweg 12, D-69120 Heidelberg, Germany}\affiliation{Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230027, People's Republic of China}
\pacs{03.67.Mn, 03.67.Pp, 42.50.Dv} \date{\today }
\begin{abstract} We report an experimental demonstration of a bit-flip error rejection protocol for error-reduced transfer of quantum information through a noisy quantum channel. In the experiment, an unknown state to be transmitted is encoded into a two-photon entangled state, which is then sent through an engineered noisy quantum channel. At the final stage, the unknown state is decoded by a parity measurement, successfully rejecting the erroneous transmission over the noisy quantum channel.
\end{abstract} \maketitle
A crucial step in the full realization of long-distance quantum communication is to overcome the problems caused by decoherence and exponential photon loss in the noisy quantum channel \cite{gisinrmp}. As a general solution, two distant parties could first share highly entangled photon pairs, the transmission of quantum states for various applications in quantum communication can then be achieved by using ancilla entanglement. As the quantum repeater \cite{Briegel98}, combining entanglement purification \cite{bennett96} and entanglement swapping \cite{zuk93}, could provide an efficient way to generate highly entangled states between distant locations, many experimental efforts have been made to achieve entanglement swapping, entanglement purification and quantum memory \cite{pan98a,pan03,kuzmich04,polzik04}, and even the demonstration of a prototype of quantum relay \cite{zhao03,gisin04}. However, one still has a long way to go before the above techniques can be realistically applied to long-distance quantum communication.
Meanwhile, in the context of quantum error correction (QEC) the way to protect a fragile unknown quantum state is to encode the state into a multi-particle entangled state \cite{shor95,steane96,laflamme96}. Then, the subsequent measurements, i.e. the so-called decoding processes, can find out and correct the error during the quantum operations. Several QEC protocols have been experimentally demonstrated in the NMR \cite{cory,knill} and ion-trap \cite{ion} systems. Although the QEC are primarily designed for large scale quantum computing, the similar idea was also inspired to implement error-free transfer of quantum information through a noisy quantum channel \cite{dik01}.
The main idea in the original scheme is to encode an unknown quantum state of single particle into a two-particle entangled state \cite{dik01}. After the encoded state is transmitted over the noisy quantum channel, a parity check measurement \cite{pan98b} is sufficient to reject the transmission with bit-flip error. Such a scheme has the advantage of avoiding the difficult photon-photon controlled-NOT gates necessary for the usual QEC. Moreover, the error rejection scheme proposed promises additional benefit of high efficiency, compared to the QEC based on linear optics quantum logic operations \cite{klm01}. This is because the crucial feed-forward operations in linear optics QEC will lead to very low efficiency. Although the original scheme is within the reach of the current technology as developed in the recent five-photon experiments \cite{zhao04,zhao05}, it is not optimal in its use of ancilla entangled state because the encoding process is implemented via a Bell-state measurement.
Remarkably, it is found recently \cite{wang04} that one pair of ancilla entangled state is sufficient to implement the two-photon coding through two quantum parity measurements. Thus, an elegant modification of the previous experiment on four-photon entanglement \cite{pan01-1} would allow a full experimental realization of the error rejection code.
In this letter, we report an experimental realization of bit-flip error rejection for fault-tolerant transfer of quantum states through a noisy quantum channel. An unknown state to be transmitted is first encoded into a two-photon entangled state, which is then sent through an engineered noisy quantum channel. At the final stage, the unknown state is decoded by quantum parity measurement, successfully rejecting the erroneous transmission over the noisy quantum channel.
\begin{figure}
\caption{Scheme for one bit-flip quantum error-rejection.}
\label{figure1}
\end{figure}
Let us first consider the scenario that Alice wants to send a single photon in an unknown polarization state $\alpha\left\vert H\right\rangle_1 +\beta \left\vert V\right\rangle_1$ to Bob through a noisy quantum channel. As shown in Fig. 1, instead of directly sending it to Bob, Alice can encode her unknown state onto a two-photon entangled state with an ancilla pair of entangled photons: \begin{equation} \left\vert \phi^{+}\right\rangle _{23}=\tfrac{1}{\sqrt{2}}\left( \left\vert HH\right\rangle _{23}+\left\vert VV\right\rangle _{23}\right) . \end{equation} The photon in the unknown polarization state and one photon out of the ancilla entangled photon\ are superposed at a polarization beam splitting (PBS$_{1}$). Behind the PBS$_{1}$, with a probability of 50\% we obtain the renormalized state corresponding to the three-fold coincidence among modes $1^\prime$, $2^\prime$ and 3 \begin{equation} \left\vert \psi\right\rangle _{1^{\prime}2^{\prime}3}=\alpha\left\vert HHH\right\rangle _{1^{\prime}2^{\prime}3}+\beta\left\vert VVV\right\rangle _{1^{\prime}2^{\prime}3}. \end{equation} Conditional on detecting photon $1^{\prime}$ in the $\left\vert +\right\rangle $ polarization state with a probability of 50\%, where $\left\vert \pm\right\rangle =\frac{1}{\sqrt{2}}\left( \left\vert H\right\rangle \pm\left\vert V\right\rangle \right)$, the remaining two photons will then be projected onto the following entangled state: \begin{equation} \left\vert \psi\right\rangle _{2^{\prime}3}=\alpha\left\vert HH\right\rangle _{2^{\prime}3}+\beta\left\vert VV\right\rangle _{2^{\prime}3}. \end{equation} Thus, through a quantum parity measurement between modes $1^\prime$ and $2^\prime$, a two-photon encoding operation can be realized.
After finishing the encoding process, Alice sends photons $2^{\prime}$ and 3 to Bob through a noisy quantum channel and Bob will recombine the two photons at the PBS$_{2}$ in order to identify and reject the erroneous transmission. If there is no error in the quantum channel, Bob will obtain the same quantum state as in (3) after PBS$_{2}$. Projecting photon $2^{\prime\prime}$ into the $\left\vert +\right\rangle $ state with a success probability of $50\%$, photon $3^{\prime}$ will be left in the unknown state $\alpha\left\vert H\right\rangle +\beta\left\vert V\right\rangle $. Through the decoding process, i.e. conditional on detecting in mode $2^{\prime\prime}$ one and only one $\left\vert +\right\rangle $-polarized photon, Bob can recover the state originally sent by Alice.
If a bit-flip error occurred for one of the two transmitted photons, the two photons will have different polarizations and exit the PBS$_{2}$ in the same output arm. Therefore, no coincidence will be observed between modes $2^{\prime\prime}$ and $3^\prime$. That is to say, the bit-flip error during the transmission of quantum states over the noisy channel has been simply rejected by the final quantum parity measurement. However, if both bit-flip errors occurred simultaneously for the two transmitted photons, Bob would finally obtain the polarization state of $\alpha\left\vert V\right\rangle +\beta\left\vert H\right\rangle $ via the same quantum parity measurement for decoding operation and the error can not be effectively rejected.
Moreover, the detection of photon $1'$ in the $\left\vert-\right\rangle$ state also leads to encoding of the initial quantum state in a two photon state, provided the associated phase flip is taken into account. Obviously, the same holds for the decoding at Bob's: projection onto the $\left\vert-\right\rangle$ state is associated with a phase flip that can be compensated for. The coding and decoding efficiency can thus be increased by a factor of two each.
Specifically, suppose that Alice would send photons to Bob in one of the three complementary bases of $\left\vert H\right\rangle/\left\vert V\right\rangle $, $\left\vert +\right\rangle/\left\vert -\right\rangle $ and $\left\vert R\right\rangle/\left\vert L\right\rangle $ (where $\left\vert R\right\rangle =\tfrac{1}{\sqrt{2}}\left( \left\vert H\right\rangle + i\left\vert V\right\rangle \right)$, and $\left\vert L\right\rangle =\tfrac{1}{\sqrt{2}}\left( \left\vert H\right\rangle - i\left\vert V\right\rangle \right)$), and each qubit is directly sent through the noisy quantum channel with bit-flip error rate of $E_{0}=p$. The quantum bit error rate (QBER) after the decoding process \cite{wang04} will be: \begin{equation} E_{1}=\frac{p^{2}}{\left(1-p\right)^{2}+p^{2}}, \end{equation} for the polarization states of $\left\vert H\right\rangle/\left\vert V\right\rangle $ and $\left\vert R\right\rangle/\left\vert L\right\rangle $, and no error occurs for the $\left\vert +\right\rangle/\left\vert-\right\rangle $. Therefore, the QBER of $E_{1}$ will be lower, compared to the QBER of $E_{0}$ for any $p<1/2$. For small $p$, $E_{1}$ is on the order of $p^{2}$. The transmission fidelity can thus be greatly improved by using the quantum error rejection code.
Note that, conditional detection of photons in mode $1^{\prime}$ implies that there is either zero or one photon in the mode $2^{\prime}$. But, as any further practical application of such a coding involves a final verification step, detecting a threefold coincidence makes sure that there will be exactly one photon in each of the modes $2^{\prime} $ and $3$. This feature allows us to perform various operations like, for example, the recombination of two photons at PBS$_{2}$ before the final detection. This makes our encoding scheme significantly different from a previous two-photon encoding experiment \cite{pittman04}, where there are certain probabilities of containing two photons in one of two encoding modes. Thus, the previous two-photon encoding experiment cannot be applied to the error-rejection code.
Moreover, we would like to emphasize that, compared to the two recent experiments on fault-tolerant quantum information transmission \cite{Ricci04,Massar05} our protocol has two essential advantages. On the one hand, the work in \cite{Ricci04} can only encode and send a known state instead of encoding and sending arbitrary unknown states required by many quantum communication protocols. On the other hand, the experiment in \cite{Massar05} can only filtrate half of the single phase-shift error. Thus, if the error rate of the channel is $p$, after applying the error filtration method the remaining QBER is still larger than $p/2$ even in the ideal case. Note that, the error filtration probability in \cite{Massar05} can be increased by coding a qubit in a larger number of time-bins, however, this would need much more resources. While our method can in principle reject any one bit-flip error with certainty as analyzed before. In fact, the ability to suppress the first order error ($p$) to the second order ($p^2$) is essential to overcome the channel noise in scalable quantum communication.
\begin{figure}
\caption{Experimental set-up for quantum error-rejection.}
\label{figure2}
\end{figure}
A schematic drawing of the experimental realization of the error rejection is shown in Fig. 2. A UV pulse (with a duration of 200fs, a repetition rate of 76MHz and a central wavelength of 394nm) passes through a BBO crystal twice to generate two entangled photon pairs $1 $, $4$ and $2$, $3$ in the state $\left\vert \phi^{+}\right\rangle $ \cite{kwiat}. The high quality of two-photon entanglement is confirmed by observing a visibility of $(94\pm1)\%$ in the $\left\vert+\right\rangle/\left\vert-\right\rangle$ basis. One quarter wave plate (QWP) and one polarizer (Pol.) in front of detector $D_4$ are used to perform the polarization projection measurement such that the input photon in mode 1 is prepared in the unknown state.
The two photons in modes 1 and 2 are steered to the PBS$_{1}$ where the path length of photon $1$\ have been adjusted by moving the delay mirror Delay 1 such that they arrive simultaneously. Conditional on detecting photon $1^{\prime}$ in the $\left\vert +\right\rangle $ polarization, the unknown polarization state was encoded into the modes in $2^{\prime}$ and $3$. The encoded two-photon state is transmitted through the engineered noisy quantum channel and then recombined at the PBS$_{2}$. Furthermore, the path length of photon $2^{\prime}$ has been adjusted by moving the Delay 2 such that photons in modes $2^{\prime}$ and $3$ arrive at the PBS$_{2}$ simultaneously. Through the whole experiment, spectral filtering (with a FWHM 3nm, F in Fig. 2) and fiber-coupled single-photon detectors have been used to ensure good spatial and temporal overlap between photons in modes $1$ and $2$, and photons in modes $2^{\prime}$ and $3$ \cite{zuk01}.
To characterize the quality of the encoding and decoding process, we first measure the interference visibility at the PBS$_1$. Since photon pairs 1-4 and 2-3 are in the state $\left\vert\phi^{+}\right\rangle$, it is easy to see that the four-fold coincidence in $1^{\prime}$, $2^{\prime}$, $3$ and $4$ corresponds to a four-photon GHZ state $\tfrac{1}{\sqrt{2}}(\left\vert HHHH\right\rangle _{1^{\prime}2^{\prime}34}+\left\vert VVVV\right\rangle _{1^{\prime}2^{\prime}34})$ \cite{pan01-1}. The four-photon entanglement visibility in the $\left\vert+\right\rangle/\left\vert-\right\rangle$ basis was observed to be $(83\pm3)\%$. Similarly, the four-photon entanglement visibility in modes $1^{\prime}$, $2^{\prime\prime}$, $3^{\prime}$ and $4$ is observed to be $(80\pm3)\%$, before introducing artificial channel noise. Note that, the visibility is obtained after compensating the birefringence effect of the PBSes \cite{pan03-1}.
In the experiment, the noisy quantum channels are simulated by one half wave plate (HWP) sandwiched with two QWPs. Each of two QWP is set at 90$^{0}$ such that the horizontal and vertical polarization will experience 90$^{0}$ phase shift after passing through the QWPs. By randomly setting the HWP axis to be oriented at $\pm\theta$ with respect to the horizontal direction, the noisy quantum channel can be engineered with a bit-flip error probability of $p=\sin^{2}\left( 2\theta\right)$.
In order to show that our experiment has successfully achieved the error rejection code, the quantum states to be transmitted in mode $1$ are prepared along one of the three complementary bases of $\left\vert H\right\rangle/\left\vert V\right\rangle$, $\left\vert +\right\rangle/\left\vert-\right\rangle$, and $\left\vert R\right\rangle/\left\vert L\right\rangle$. The error rates in the engineered quantum channel can be varied by simultaneously changing the axis of each half-wave plate. Specifically, we vary the angle $\theta$ to achieve various error rates from 0 to $0.40$ with an increment $0.05$ in the quantum channel.
\begin{figure}
\caption{Experimental results for three different initial states $\left\vert V\right\rangle$ (a), $\left\vert -\right\rangle$ (b) and $\left\vert L\right\rangle$ (c), and (d) shows the average QBER (calculated over all the six states). The quadrangle and the triangle dots are corresponding to the cases that error-rejection and no error-rejection was made, and the solid curves and the dot curves show the theoretically prediction of QBER for the cases without and with error rejection respectively.}
\end{figure}
The experimental results of three different input states, after passing through the noisy quantum channel, are shown in Fig. 3, The triangle dots in Fig. 3, corresponding to the bit-flip error rates of single photons, were measured by directly sending the quantum state of photon 1 (after passing through a PBS and some wave-plates for state preparing) through the engineered quantum channel while with both PBS$_1$ and PBS$_2$ removed. These dots also shows the quality of the simulated error of the quantum noisy channel. The quadrangle dots show the final bit-flip error rates after performing error rejection operation with the help of PBS$_1$ and PBS$_2$. Fig. 3a, 3b and 3c shows the experimental results for the input states $\left\vert V\right\rangle$, $\left\vert -\right\rangle$, and $\left\vert L\right\rangle$, respectively. The other three input states have the similar results as the one with the same basis respectively. And Fig. 3d shows the average QBER calculated over all six input states.
In Fig. 3, one can clearly see that our error-rejection operation itself also introduces significant error rates, even with $E_0=0$. Therefore, if the original $E_0$ is comparable with the error rate caused by the experimental imperfection, no improvement will be gained after error-rejection. In the $\left\vert H\right\rangle$/$\left\vert V\right\rangle$ experiment, the experimental error rate is about $5\%$. In both $\left\vert +\right\rangle$/$\left\vert -\right\rangle$ and $\left\vert R\right\rangle$/$\left\vert L\right\rangle$ experiments an experimental error rate of $10\%$ is observed.
We notice that, whereas both the $\left\vert +\right\rangle$/$\left\vert -\right\rangle$ and $\left\vert R\right\rangle$/$\left\vert L\right\rangle$ experiments have roughly the same visibility, a better visibility is obtained in the $\left\vert H\right\rangle$/$\left\vert V\right\rangle$ experiment. This is mainly due to our two-photon entanglement source, which has a better visibility in the $\left\vert H\right\rangle$/$\left\vert V\right\rangle$ basis ($97\%$) than in the $\left\vert +\right\rangle$/$\left\vert -\right\rangle$ or $\left\vert R\right\rangle$/$\left\vert L\right\rangle$ basis ($94\%$). Moreover, it is partly due to the imperfect birefringent compensation at the PBS$_1$ and PBS$_2$ \cite{pan03-1}, which leads a reduction of interference visibility, hence imperfect encoding and decoding process. Moreover, the imperfect encoded state passing through the noisy channel also leads that in the $\left\vert +\right\rangle$/$\left\vert -\right\rangle$ basis the result becomes deteriorate as increasing of artificial noise.
From Fig. 3a and 3c, it is obvious that our error-rejection method can significantly reduce the bit-flip error as long as $E_0$ is larger than the experimental error rates. However, although ideally in the $\left\vert +\right\rangle $/$\left\vert -\right\rangle$ experiment no error should occur after the error-rejection operation, an error rate no less than $10\%$ is observed, which is in accordance with the limited visibility of $80\%$.
Although our experimental results are imperfect, they are sufficient to show a proof of principle of a bit-flip error rejection protocol for error-reduced transfer of quantum information through a noisy quantum channel. Moreover, Fig. 3d shows that for a substantial region our experimental method does provide an improved QBER over the standard scheme in a six-state quantum key distribution (QKD). This implies, with further improvement, the error-rejection protocol may be used to improve the threshold of tolerable error rate over the quantum noisy channel in QKD \cite{wang01}.
Our experimental realization of bit-flip error rejection deserves some further comments. First, the same method can be applied to reject the phase-shift error because phase errors can be transformed into bit-flip errors by a $45^0$ polarization rotation. In this way we can reject all the 1 bit phase-shift error instead of bit-flip error. Second, by encoding unknown states into higher multi-photon ($N$-photon) entanglement and performing multi-particle parity check measurement \cite{pan98b} either the higher order (up to $N-1$) bit-flip error or phase-shift error can be rejected for more delicate quantum communication.
In summary, our experiment shows a proof of principle of a bit-flip error rejection protocol for error-reduced transfer of quantum information through a noisy quantum channel. Moreover, by further improvement of the quality of the resource for multi-photon entanglement, the method may also be used to enhance the bit error rate tolerance \cite{Chau,lo} over the noisy quantum channel and offer a novel way to achieve long-distance transmission of the fragile quantum states in the future QKD.
\begin{acknowledgments} We are grateful to X.-B. Wang for valuable discussion. This work was supported by the Alexander von Humboldt Foundation, the Marie Curie Excellence Grant of the EU and the Deutsche Telekom Stiftung. This work was also supported by the NNSFC and the CAS. \end{acknowledgments}
\end{document} | arXiv |
\begin{document}
\title{Subwavelength atom localization via amplitude and phase control of the absorption spectrum-II} \author{Kishore T. Kapale} \email{[email protected]} \affiliation{Jet Propulsion Laboratory, California Institute of Technology, Mail Stop 126-347, 4800 Oak Grove Drive, Pasadena, California 91109-8099}
\author{M. Suhail Zubairy} \email{[email protected]} \affiliation{Institute for Quantum Studies and Department of Physics, Texas A\&M University, College Station, TX 77843-4242}
\begin{abstract} Interaction of the internal states of an atom with spatially dependent standing-wave cavity field can impart position information of the atom passing through it leading to subwavelength atom localization. We recently demonstrated a new regime of atom localization [Sahrai {\it et al.}, Phys. Rev. A {\bf 72}, 013820 (2005)], namely sub-half-wavelength localization through phase control of electromagnetically induced transparency. This regime corresponds to extreme localization of atoms within a chosen half-wavelength region of the standing-wave cavity field. Here we present further investigation of the simplified model considered earlier and show interesting features of the proposal. We show how the model can be used to simulate variety of energy level schemes. Furthermore, the dressed-state analysis is employed to explain the emergence and suppression of the localization peaks, and the peak positions and widths. The range of parameters for obtaining clean sub-half-wavelength localization is identified. \end{abstract} \pacs{42.50.Ct, 42.50.Pq, 42.50.Gy, 32.80.Lg}
\maketitle \section{Introduction} Precise localization of atoms has attracted considerable attention in recent years. Optical manipulations allow probing the center-of-mass degrees of freedom of atoms with subwavelength precision. The interest in subwavelength atom localization is largely due to its applications to many areas requiring manipulations of atomic center-of-mass degrees of freedom, such as laser cooling~\cite{ChuMetcalf}, Bose-Einstein condensation~\cite{Collins96}, and atom lithography~\cite{lithography} alongwith fundamentally important issues such as measurement of the center-of-mass wavefunction of moving atoms~\cite{KapaleWavefunction}.
Optical techniques for position measurements of the atom are of considerable interest from both theoretical and experimental point of view. Several scheme have been proposed for the localization of an atom using optical methods~\cite{welch9193}. It is well known that optical methods provide better spatial resolution in position measurement of the atom. For example, in the optical virtual slits scheme the atom interacts with a standing-wave field and imparts a phase shift to the field. Measurement of this phase shift then gives the position information of the atom~\cite{WallsZollerWilkens}. In another related idea based on phase quadrature measurement is considered in Ref.~\cite{Walls95}. Kunze {\it et al.}~\cite{Kunze97} demonstrated how the entanglement between the atomic position and its internal state allows one to localize the atom without directly affecting its spatial wave function. It is shown that, by using Ramsey interferometry, the use of a coherent-state cavity field is better than the classical field to get a higher resolution in position information of the atom~\cite{Kien97}. Resonance imaging methods have also been employed in experimental studies of the precision position measurement of the moving atoms~\cite{Thomas90,Bigelow97}.
More recently, atom-localization methods based on the detection of the spontaneously emitted photon during the interaction of an atom with the classical standing-wave field, are considered~\cite{Zoller96,Herkomer97,QamarPRA2000,QamarOC2000}. It is, however, important to note that from an experimental point of view, observation of spontaneous emission spectrum is very tricky and difficult. In this context, another scheme based on a three-level $\Lambda$-type system interacting with two fields, a probe laser field and a classical standing wave coupling field, is used for atom localization by Paspalakis and Knight~\cite{Knight2001}. They observe that in the case of a weak probe field, measurement of the population in the upper level leads to sub-wavelength localization of the atom during its motion in the standing wave. Thus, in essence, this scheme uses absorption of a probe field for atom localization. Atomic coherence effects, such as coherent population trapping, have also been shown to be useful for subwavelength localization of atoms by Agarwal and Kapale~\cite{AgarwalKapaleAL}, where monitoring the coherence of the trapping state gives rise to subwavelength localization of an atom to a precision prechosen through the ratio of the square of Rabi frequencies of the strong standing-wave drive field and a weak probe field.
The authors (with collaborators) recently proposed a subwavelength atom localization scheme through phase control of the absorption of a weak probe field by the atom. A modified $\Lambda$-type level scheme with an extra level and the drive fields forming a complete loop was shown to introduce a phase dependence in the response of the atomic medium to a weak probe field. This phase controllable atomic response was shown to give rise to tunable group velocity from subluminal to superluminal in a single system~\cite{Sahrai:2004}. By considering one of the drive fields to be a standing-wave field of a cavity it was shown that the same scheme can be used to localize an atom flying through the standing wave field to subwavelength domain~\cite{Sahrai:2005}.
This article is a sequel to the earlier article~\cite{Sahrai:2005}, henceforth referred to as $\mathscr{I}$. In $\mathscr{I}$ a restricted parameter range of the model was considered to show the possibility of sub-half-wavelength localization. In this article further investigations of the analytical results are carried out to show how the model can be used to simulate variety of atomic systems with varying energy level spacings, different atomic dipole matrix elements and decay properties. Appropriate parameters required to obtain different regimes of localization are studied in detail. A dressed-states approach is also considered to give insight into the results obtained.
The article is organized as follows: For completeness, a brief description of the procedure to determine the susceptibility of the atom to a weak probe field is given in Sec.~\ref{Sec:Model}. Then, in Sec.~\ref{Sec:Results} A, the susceptibility expression is studied in detail to arrive at the conditions for observing atom localization. Various parameter ranges for the drive field Rabi frequencies and decay properties are considered in order to simulate variety of atomic species and to clarify experimentally controllable features and properties of the model and numerical results and their explanation through the analytical probe susceptibility expression is presented in Sec.~\ref{Sec:Results} B. A simple dressed states treatment is presented in Sec.~\ref{Sec:Results}C in order to explain the results obtained in the earlier sections. Finally the conclusion is presented.
\section{The Model and Equations} \label{Sec:Model} The schematics of the proposed scheme are shown in Fig.~\ref{Fig:Scheme}. We consider an atom, moving in the $z$ direction, as it passes through a classical standing-wave field of a cavity. The cavity is taken to be aligned along the $x$ axis. The internal energy level structure of the atom is shown in Fig.~\ref{Fig:Scheme}(b).
\begin{figure}
\caption{The Model: $(a)$ The cavity supports the standing wave field (1) corresponding to Rabi frequency $\Omega_1$. Two other fields (2, 3) are applied at an angle as shown. The atom enters the cavity along the $z$ axis and interacts with the three drive fields. The whole process takes place in the $x$-$z$ plane. $(b)$ The energy level structure of the atom. Probe field, denoted by $\mathcal{E}_p$, is detuned by an amount $\Delta$ from the $\ket{a_1}-\ket{c}$ transition. The fields (2, 3) shown in $(a)$ part of the figure correspond to the fields with Rabi frequencies $\Omega_2$ and $\Omega_3$ respectively. The decay rates from the upper levels $\ket{a_1}$ and $\ket{a_2}$ are taken to be $\gamma_1$ and $\gamma_2$ respectively.}
\label{Fig:Scheme}
\end{figure}
The radiative decay rates from the level $\ket{a_1}$ and
$\ket{a_2}$ to level $\ket{c}$ are taken to be $\gamma_1$ and $\gamma_2$. The upper level $\ket{a_1}$ is coupled to the level $\ket{a_2}$ and further the level $\ket{a_2}$ is coupled to level $\ket{b}$ via classical fields with Rabi frequencies $\Omega_3$ and $\Omega_2$, respectively. In addition, the upper level $\ket{a_1}$ is coupled to level $\ket{b}$ via a classical standing-wave field having Rabi frequency $\Omega_1$. It should be noted that the Rabi frequency of the standing wave is position dependent and is taken to be $\Omega_1(x)=\Omega_1 \sin \kappa x$ . Here $\Omega_1(x)$ is defined to include the position dependence and $\kappa$ is the wave vector of the standing wave field, defined as $\kappa =2 \pi/{\lambda}$, where $\lambda$ is the wavelength of the standing-wave field of the cavity. We assume that the atom is initially in the state $\ket{c}$ and interacts with a weak probe field that is near resonant with $\ket{c}\rightarrow\ket{a_1}$ transition. The detuning of the probe field on this transition is taken to be $\Delta$. We assume that the center-of-mass position distribution of the atom is nearly uniform along the direction of the standing wave. Therefore, we apply the Raman-Nath approximation and neglect the kinetic part of the atom from the Hamiltonian~\cite{Meystre:1999}. Under these circumstances, the Hamiltonian of the system in the rotating wave approximation can be written as \begin{equation} \label{eq:hamiltonian1} \mathcal{H} = \mathcal{H}_0 + \mathcal{H}_I \end{equation} where \begin{equation} \label{eq:hamiltonian2}\mathcal{H}_0 = \hbar \omega_{a_1}\ket{a_1}\bra{a_1} + \hbar \omega_{a_2}\ket{a_2}\bra{a_2} + \hbar\omega_b\ket{b}\bra{b} + \hbar\omega_c\ket{c}\bra{c}, \end{equation} and \begin{multline} \label{eq:hamiltonian3}\mathcal{H}_I=-\frac{\hbar}{2} \left[ \Omega_1 {\rm e}^{- {\rm i} \nu_1 t} \sin{\kappa x}\, \ket{a_1}\bra{b}\right. \\ + \Omega_2 {\rm e}^{{\rm i} k x \cos \theta_2} {\rm e}^{- {\rm i} \nu_2 t}\ket{a_2}\bra{b} \\ + \left.\Omega_3 {\rm e}^{{\rm i} k x \cos{\theta_3}} {\rm e}^{- {\rm i} \nu_3 t}\ket{a_1}\bra{a_2} +\frac{\mathcal{E}_p\wp_{a_1 c}}{\hbar} {\rm e}^{- {\rm i} \nu_p t}\ket{a_1}\bra{c}\right]+ \mbox{H.c.} \end{multline} Here $\omega_i$ are the frequencies of the states $\ket{i}$ and $\nu_i$ are the frequencies of the optical fields, and $\theta_{2}$, $\theta_{3}$ are the angles made by the propagation direction of the fields $\Omega_{2}$ and $\Omega_{3}$ with respect the $x$ axis respectively. The subscript $p$ stands for the quantities corresponding to the probe field\mdash i.e., ${\mathcal{E}_p}$ and $\nu_p$ are the amplitude and frequency of the probe field. Also $\wp_{a_1 c}$ is the dipole matrix element of the $\ket{c}\rightarrow \ket{a_1}$ transition. For simplicity, we assume that the Rabi frequencies $\Omega_1$ and $\Omega_2$ are real and $\Omega_3$ is complex\mdash i.e., $\Omega_3=\Omega_3 {\rm e}^{- {\rm i} \varphi }$. This choice of imparting a carrying phase to field 3, is only for the convenience of calculations. As will become clear later, only the relative phase of the three fields is important and absolute phases do not matter. The dynamics of the system is
described using density matrix approach as:
\begin{equation} \dot{\rho} = - \frac{{\rm i}}{\hbar} [H,\rho] - \frac{1}{2} \{\Gamma,\rho\}, \end{equation}
where $\{\Gamma,\rho\} = \Gamma\rho + \rho \Gamma$. Here the decay rate is incorporated into the equation by a relaxation matrix
$\Gamma$, which is defined, by the equation $\langle n| \Gamma | m \rangle = \gamma_n \delta_{nm}$. The detailed calculations of these equations are given in the Appendix of $\mathscr{I}$.
Our goal is to obtain information about the atomic position from the susceptibility of the system at the probe frequency. Therefore, we need to determine the steady state value of the off-diagonal the density matrix element, $\rho_{a_1 c}$. After necessary algebraic calculation and moving to appropriate rotating frames, we obtain a set of density matrix equations. To determine $\rho_{a_1 c}$ we only need following equations \begin{align} \dot{\tilde{\rho}}_{a_1 c} &= - [{\rm i} (\omega_{a_1 c}-\nu_p) + \half\gamma_1]\tilde{\rho}_{a_1 c} + \frac{{\rm i}}{2} \Omega_3 {\rm e}^{- {\rm i} \varphi } {\rm e}^{ {\rm i} k x \cos \theta_3 } \tilde{\rho}_{a_2 c} \nonumber \\ &\qquad+ \frac{{\rm i}}{2} \Omega_1 \sin \kappa x \tilde{\rho}_{b c} - {\rm i} \frac{\mathcal{E}_p \wp_{a_1 c} }{2 \hbar} (\tilde{\rho}_{a_1 a_1} - \tilde{\rho}_{cc}), \nonumber \\ \dot{\tilde{\rho}}_{a_2 c} &= -[{\rm i} (\omega_{a_2 c} - (\nu_p- \nu_3)) + \half\gamma_2] \tilde{\rho}_{a_2 c} \nonumber \\ &\qquad+
\frac{{\rm i}}{2} \Omega_2 {\rm e}^{ {\rm i} k x \cos \theta_2 }\tilde{\rho}_{b c} + \frac{{\rm i}}{2} \Omega_3 {\rm e}^{ {\rm i} \varphi } {\rm e}^{- {\rm i} k x \cos \theta_3 } \tilde{\rho}_{a_1 c} \nonumber \\ &\qquad \qquad- {\rm i} \frac{\mathcal{E}_p \wp_{a_1 c}}{2 \hbar} \tilde{\rho}_{a_2 a_1}, \nonumber \\ \dot{\tilde{\rho}}_{b c} &= - [{\rm i} ( \omega_{b c} + \nu_1 -\nu_p ) + \gamma_{b c}]\tilde{\rho}_{b c} + \frac{{\rm i}}{2} \Omega_1 \sin \kappa x \tilde{\rho}_{a_1 c} \nonumber \\ &\qquad+ \frac{{\rm i}}{2}\Omega_2 {\rm e}^{- {\rm i} k x \cos \theta_2 } \tilde{\rho}_{a_2 c} - {\rm i} \frac{\mathcal{E}_p \wp_{a_1 c}}{2\hbar} \tilde{\rho}_{b a_1}. \label{eq:tilderhodot} \end{align}
As we know, the dispersion and absorption are related to the susceptibility of the system and is determined by $\rho_{a_1 c}$. We take the probe field to be weak, and calculate the polarization of the system to lowest order in $\mathcal{E}_p$. We keep all the terms of the driving fields but keep only linear terms in the probe field. The atom is initially in the ground state $\ket{c}$, therefore we use \begin{equation} \tilde{\rho}_{cc}^{(0)} =1, \quad \tilde\rho_{b a_1}^{(0)}=0, \quad \tilde\rho_{a_2 a_1 }^{(0)} = 0, \quad \tilde\rho_{a_1 a_1}^{(0)} = 0. \label{Eq:initial} \end{equation} Equation~(\ref{eq:tilderhodot}) can then be simplified considerably to obtain \begin{align} \dot{\tilde{\rho}}_{a_1 c} &= - ({\rm i} \Delta + \half\gamma_1)\tilde{\rho}_{a_1 c} + \frac{{\rm i}}{2} \Omega_3\, {\rm e}^{- {\rm i} \varphi } {\rm e}^{ {\rm i} k x \cos \theta_3 } \tilde{\rho}_{a_2 c} \nonumber \\ &\qquad+ \frac{{\rm i}}{2} \Omega_1 \sin{\kappa x} \,\tilde{\rho}_{b c} + {\rm i} \frac{\mathcal{E}_p \wp_{a_1 c} }{2 \hbar} , \nonumber \\ \dot{\tilde{\rho}}_{a_2 c} &= - ({\rm i} \Delta + \half\gamma_2)\tilde{\rho}_{a_2 c} + \frac{{\rm i}}{2} \Omega_3\, {\rm e}^{{\rm i} \varphi } {\rm e}^{- {\rm i} k x \cos \theta_3 } \tilde{\rho}_{a_1 c} \nonumber \\ &\qquad+ \frac{{\rm i}}{2} \Omega_2 {\rm e}^{{\rm i} k x \cos \theta_2 }\tilde{\rho}_{b c}, \nonumber \\ \dot{\tilde{\rho}}_{b c} &= - {\rm i}\, \Delta\, \tilde{\rho}_{b c} + \frac{{\rm i}}{2} \Omega_1 \sin{\kappa x} \,\tilde{\rho}_{a_1 c} \nonumber \\ &\qquad + \frac{{\rm i}}{2} \Omega_2 {\rm e}^{- {\rm i} k x \cos \theta_2 } \tilde{\rho}_{a_2 c}. \label{eq:tilderhodot2} \end{align} Here we have introduced the detuning of the probe field and the frequency difference between levels $\ket{a_1}$ and $\ket{c}$, \begin{equation} \Delta = \omega_{a_1c} - \nu_p = \omega_{a_2c}+ \nu_3 - \nu_p = \omega_{bc} + \nu_1 - \nu_p. \label{Eq:Delta} \end{equation} Here we have also assumed that $\gamma_{b c}=0$. It can be easily seen that these set of equations can also be used to simulate a variety of level schemes as shown in Fig.~\ref{Fig:levelschemes}, after redefining the decay rates accordingly as discussed in the caption. The schme in Fig.~\ref{Fig:levelschemes}$(b)$ requires special attention as the positions of the states $\ket{a_1}$ and $\ket{a_2}$ are reversed in the order of increasing energy compared to the other levelschemes. This entails small change in the rotating frame that is chosen to arrive at the simplified density matrix equations. The transformation required can be accomplished by replacing the complex Rabi frequency $\Omega_3$ by its complex conjugate $\Omega_3^*$ and changing its frequency $\nu_3$ to $-\nu_3$. The density matrix equations so obtained are identical to the set~\eqref{eq:tilderhodot2} given above except for the redifinition of the phase from $\varphi\rightarrow -\varphi$. However, as will be seen later, the phase enters through the term $\cos\varphi$ in the response of the atoms to a weak probe field, thus the final results are identical for all the models discussed in Fig.~\ref{Fig:levelschemes} \begin{figure*}
\caption{ Several level schemes that can be studied using our model in Fig. 1$(b)$, within the weak probe limit. The decay rates are defined as follows: $(a)$ $\gamma_1 = \gamma_{a_1b} + \gamma_{a_1c}$ and $\gamma_2=\gamma_{a_2b}+\gamma_{a_2c}$; $(b)$ $\gamma_1 = \gamma_{a_1b} + \gamma_{a_1c}$ and $\gamma_2 = \gamma_{a_2a_1}+\gamma_{a_2b} + \gamma_{a_2c}$; $(c)$ $\gamma_1 =\gamma_{a_1a_2}+ \gamma_{a_1b} + \gamma_{a_1c}$ and $\gamma_2=0$; $(d)$ $\gamma_1 =\gamma_{a_1a_2}+ \gamma_{a_1b} + \gamma_{a_1c}$ and $\gamma_2=\gamma_{a_2b}$. Here $\gamma_{ij}$ corresponds to spontaneous decay rate from level $\ket{i}$ to level $\ket{j}$. It can be noted that the positions of levels $\ket{a_1}$ and $\ket{a_2}$ in $(b)$ are reversed compared to the other schemes. Slight modifications in the equations are needed to simulate level scheme in $(b)$ with the equations given in the text. The transformations required is: the complex Rabi frequency $\Omega_3 \rightarrow \Omega_{3}^{*}$---i.e., $\nu_3 \rightarrow - \nu_3$. It can be easily shown that the results remain unchanged under these transformations, as discussed in the text.}
\label{Fig:levelschemes}
\end{figure*}
This set of equations can be solved analytically; the detailed discussion can be found in the appendix of $\mathscr{I}$. Thus, the off-diagonal density-matrix element corresponding to the probe transition is obtained as \begin{equation} {\rho}_{a_1 c} = \tilde{\rho}_{a_1 c} {\rm e}^{- {\rm i} \nu_p t} = \frac{1}{Y \hbar} (\Omega_2^2 - 4 \Delta^2 + 2 {\rm i} \gamma_2 \Delta) \mathcal{E}_p \wp_{a_1 c} {\rm e}^{- {\rm i} \nu_p t}, \label{eq:tilderhoa1c} \end{equation} where we have chosen, without loss of generality, $\theta_3=\pi/ 4$, $\theta_2=\pi/2+\pi/ 4$, moreover, $Y$ is defined to be \begin{align} Y = A + {\rm i} B, \end{align} with \begin{align} A &= - 8 \Delta^3 + 2 \Delta (\Omega_1^2 \sin^2 \kappa x + \Omega_2^2 + \Omega_3^2 ) \nonumber \\ &\qquad + 2 \gamma_1 \gamma_2 \Delta + \Omega_1 \Omega_2 \Omega_3 ({\rm e}^{{\rm i} \varphi} + {\rm e}^{-{\rm i} \varphi}) \sin \kappa x , \nonumber \\ B &= 4 \Delta^2 (\gamma_1 +\gamma_2) - \gamma_1 \Omega_2^2 - \gamma_2 \Omega_1^2 \sin^2 \kappa x. \end{align}
The susceptibility at the probe frequency can be written as \begin{equation} \chi =\frac{2 N \wp_{a_1c} \rho_{a_1c} }{\epsilon_0 \mathcal{E}_p } {\rm e}^{{\rm i} \nu_p t}
=\frac{2 N |\wp_{a_1c}|^2}{\epsilon_0 } \frac{ (\Omega_2^2 - 4 \Delta^2 + 2 {\rm i} \gamma_2 \Delta) }{Y \hbar}, \end{equation} where $N$ is the atom number density in the medium. The real and imaginary parts of susceptibility are given as \begin{align}
\chi' & = \frac{2 N |\wp_{a_1 c}|^2 }{ \epsilon_0 \hbar Z} \{ (\Omega_2^2 - 4 \Delta^2) A + 2 \gamma_2 \Delta B\},\\ \chi'' & =
\frac{2 N |\wp_{a_1 c}|^2 }{ \epsilon_0 \hbar Z} \{ 2 \gamma_2 \Delta A - (\Omega_2^2 - 4 \Delta^2) B)\}, \label{Eq:Chiprime} \end{align} where $Z = Y Y^*$ and $\chi=\chi'+ {\rm i} \chi''$. It is imperative to point out that the phase enters the susceptibility expression only through the quantities $A$ and $Y$. Even the phase dependence of $Y$ is only through the quantity $A$. Moreover, we observe that the phase dependent term in $A$ is $\Omega_1 \Omega_2 \Omega_3 ({\rm e}^{{\rm i} \varphi} + {\rm e}^{-{\rm i} \varphi}) \sin \kappa x $. Thus the phase factor could very well have come from either of the three driving fields. As pointed out earlier, if all the fields had phase dependence,
only the collective phase would be important and no individual phase-dependent terms would occur. This is because the Rabi frequencies $\Omega_{i}$ in all the other terms appear through $\Omega_{i}^{2}$, which is $|\Omega_{i}|^{2}$ for a complex Rabi
frequency $\Omega_{i}=|\Omega_{i}| {\rm e}^{{\rm i} \phi_{i}}$. The collective phase can be easily determined to be $\varphi=\phi_{2}+\phi_{3}-\phi_{1}$, by repeating the susceptibility calculation. Here $\phi_{i}$ is the phase of the complex Rabi frequency $\Omega_{i}$ of the $i$th driving field.
In the next section we consider the imaginary part of the susceptibility $\chi''$ in detail and obtain various conditions for subwavelength localization of the atom.
\section{Results and Discussions} \label{Sec:Results} We study the expression~(\ref{Eq:Chiprime}) for the imaginary part of the susceptibility on the probe transition in greater detail in the following discussion. It is clear that $\chi''$\mdash i.e., probe absorption\mdash depends on the controllable parameters of the system like probe field detuning, amplitudes and phases of the driving fields. First we present analytical considerations of the probe absorption maxima and its relation to the atom localization. Then we present the results of the numerical study for a variety of different sets of values of the parameters. In the end we present the dressed-state analysis to shed some light on the numerical results.
\subsection{The probe absorption maxima} Noting the dependence of $\chi''$ on $\sin{\kappa x}$, it is, in principle, possible to obtain information about the $x$ position of the atom as it passes through the cavity by measuring the probe absorption. Nevertheless, for precise localization of the atom the susceptibility should show maxima or peaks at certain $x$ positions. We obtain the conditions for the presence of peaks in $\chi''$ in the discussion to follow. Eq.~(\ref{Eq:Chiprime}) can be rewritten as follows, using
$\mathscr{N}={2 N |\wp_{a_1c}|^2}/{\hbar \epsilon_0}$, \begin{widetext} \begin{align} \frac{\chi''}{\mathscr{N}} &= \frac
{A+ B(\kappa x)} {
\gamma_1\,[A + 2\, B(\kappa x)]
+ \gamma_2^2\,(\Omega_1^2\,\sin^2\kappa x -4\Delta^2)^2
+ [8 \Delta^3 - 2\,\Delta \,(\Omega_1^2\,\sin^2{\kappa x} + \Omega_2^2 + \Omega_3^2) - 2\,\Omega_1 \Omega_2 \Omega_3\, \cos\varphi\, \sin{\kappa x}]^2 } \nonumber \\ &=\frac
{
A+B(\kappa x)}{ \gamma_1\,[A + 2\, B(\kappa x) ]
+ \gamma_2^2 \Omega_1^4(\sin \kappa x - R_1)^2(\sin \kappa x - R_2)^2 + 4\Delta^2 \Omega_1^4(\sin \kappa x-R_3)^2(\sin\kappa x - R_4)^2} \label{Eq:Rep2} \end{align} where \begin{align} A&= \gamma_1\,( 4\,\Delta^2\,\gamma_2^2 +
(\Omega_2^2-4\,\Delta^2)^2)\,, \nonumber \\ B(\kappa x)&= \gamma_2
( \Omega_1^2\,\Omega_2^2\sin^2{\kappa x}\, +
4\,\Delta\, \Omega_1\,\Omega_2\,\Omega_3 \,\cos\varphi\,\sin{\kappa x}\,
+ 4\,\Delta^2\,\Omega_3^2)=\gamma_2 \, \Omega_1^2\,\Omega_2^2\,(\sin\kappa x-L_1)(\sin\kappa x - L_2)\,, \nonumber \\
L_{1,2}&= \frac{2\,\Delta\,\Omega_3}{\Omega_1\,\Omega_2}
{\left( - \cos\varphi \pm
{\sqrt{\cos^2\varphi -1}} \right) }\,, \\
R_{1,2}&=\mp\frac{2\,\Delta }{{{\Omega }_1}}\,,\nonumber \\
R_{3,4}&=\frac{1}{2\,\Delta \,\Omega_1}\Biggl[-\Omega_2\,\Omega_3 \cos\varphi \pm
\sqrt{\Omega_2^2\,\Omega_3^2 \cos^2\varphi - 4 \Delta^2(\Omega_2^2 +
\Omega_3^2- 4 \Delta^2)}\Biggr]\,.\nonumber \end{align} \end{widetext} It can be seen that the probe field absorption would peak at positions satisfying \begin{equation} \sin{\kappa x}=R_{1,2,3,4}\,. \end{equation} The roots $L_{1,2}$ do not contribute to the probe absorption maxima as they are the roots of the numerator as well, and these contributions mutually cancel. Moreover, for a given set of parameters not all four roots contribute to the probe absorption maxima, as they have different weighting factors given by $\gamma_2 \Omega_1^4$ and $4\Delta^2\Omega_1$. The dominant weighting factor, being independent of position, governs which set of roots $\{R_{1,2}\}$ or $\{R_{3,4}\}$ will be important for dictating the atom-localization positions.
It can be clearly seen that for $\gamma_2=0$ the maxima positions gverned by $R_{1,2}$ do not occur; whereas, for $\Delta=0$ the roots $R_{1,2}$ are more important compared to $R_{3,4}$. In the regime where both $\Delta$ and $\gamma_2$ are zero interesting consequences follow. This competition of the roots gives rise to various interesting regimes of parameters and possibilities in the atom localization. In the following we will throw light on the novel propreties arising due to this freedom. It can be noted that in $\mathscr{I}$ the roots $L_{1,2}$ and $R_{1,2}$ did not appear as $\gamma_2$ was taken to be zero, which leads to $B(\kappa x)=0$. For completeness we give the expression of $\chi''$ as used in $\mathscr{I}$: \begin{widetext} \begin{align}
\chi'' &= \frac{2 N |\wp_{a_1c}|^2}{\hbar \epsilon_0} \frac{\gamma_1 (\Omega_2^2 - 4 \Delta^2)^2} {\gamma_1^2(\Omega_2^2 -4\Delta^2 )^2 + ( 8 \Delta^3 - 2\,\Delta \,(\Omega_1^2\,\sin^2{\kappa x} + \Omega_2^2 + \Omega_3^2) - 2\,\Omega_1 \Omega_2 \Omega_3\, \cos\varphi\, \sin{\kappa x})^2} \nonumber \\
&=\frac{2 N |\wp_{a_1c}|^2}{\hbar \epsilon_0} \frac{\gamma_1 (\Omega_2^2 - 4 \Delta^2)^2} {\gamma_1^2(\Omega_2^2-4\Delta^2 )^2 + 4 \Delta^2 \Omega_1^4\, (\sin{\kappa x}-R_3)^2(\sin{\kappa x}-R_4)^2}\,. \label{Eq:chirootOld} \end{align} \end{widetext} \forget{To illustrate the behavior of the probe absorption in detail for different parameter regimes, we performed extensive numerical study of the above expression; the results are summarized in the discussion to follow. To understand the numerical results we have also looked at the dressed-states approach and provided qualitative explanations of the findings; they will be presented in the subsection to follow.}
A direct calculation of $\chi''$ from the equation~\eqref{Eq:Chiprime} shows that the positions of maxima of do not strongly depend on the decay parameters, and are function of only the drive field Rabi frequencies and phases. However, for a chosen value of the detuning the widths of the peaks observed in the plots of $\chi''$ vs $\kappa x$ depend on the values of the decay parameters. To make connection with the positions of maxima predicted by the roots of the denominator in Eq.~\eqref{Eq:Rep2}, i.e, the roots $R_{1,2,3,4}$, we study these roots in detail in the following discussion.
The probe field detunings required to obtain probe field absorption peaks as a function of the $x$ coordinate along the cavity field axis can be obtained by solving equations $\sin{\kappa x} = R_{1,2,3,4}$ for $\Delta$. We denote the solutions for $\sin \kappa x = R_{1,2}$ as $\delta_{1,2}$ and the solutions for $\sin \kappa x = R_{3, 4}$ as $\delta_{3,4,5}$. It can be easily shown that \begin{align} \delta_{1,2} = \mp\frac{\Omega_1}{2}\sin\kappa x \end{align} and $\delta_{3,4,5}$ are the solutions of $\sin \kappa x = R_{3,4}$\mdash i.e., the cubic equation, \begin{equation} 4 \delta^3 - \delta(\Omega_1^2 \sin^2 \kappa x + \Omega_2^2 + \Omega_3^2) - \Omega_1 \Omega_2 \Omega_3 \sin \kappa x \cos \varphi = 0. \label{Eq:delta345} \end{equation} When the relative phase $\varphi=\pi/2$, the above equation can be readily solved to give \begin{align} \delta_{3}&=0,\quad\delta_{4,5}= \pm \frac{1}{2}\, \sqrt{\Omega_1^2 \sin^2 \kappa x + \Omega_2^2 + \Omega_3^2}\,. \end{align} Thus, for $\varphi=\pi/2$ the above equations give the values of the probe detuning for observing probe absorption maxima as a function of the spatial position along the standing-wave field. It is clear that for $\delta=\Delta=0$ there is no atom localization possible as the probe absorption would be the same at all spatial positions. We do not give expressions for $\delta_{3,4,5}$ for the case of $\varphi=0i$ as they are quite complicated; however, they can be readily evaluated numerically to verify the predictions.
It can also be noted that for the simplified case of $\Omega_2=\Omega_3=\Omega$ the $\delta_{3,4,5}$ expressions are considerably simplified and are given by \begin{align} \delta_{3}&= -\frac{1}{2}\,{\Omega_1 \sin \kappa x}\,, \nonumber \\ \delta_{4,5}&=\frac{1}{4}\,\[\Omega_1 \sin \kappa x \pm \sqrt{\Omega_1^2 \sin^2 \kappa x + 8 \Omega^2}\right] \quad\text{ for } \varphi = 0. \end{align} \forget{and \begin{align} \delta_{3}&=0 \,, \nonumber \\ \delta_{4,5}&= \pm \frac{1}{2}\, \sqrt{\Omega_1^2 \sin^2 \kappa x + 2 \Omega^2}\quad\text{ for } \varphi = \pi/2. \end{align}} This means that $\delta_3=\delta_1$ for $\varphi = 0$; however, we also have \begin{align} L_{1,2} &= -\frac{2 \Delta}{\Omega_1}\quad &\text{for } \varphi=0\,, \nonumber \\ L_{1,2} &= \frac{2 \Delta}{\Omega_1}(-1\pm {\rm i})\quad &\text{for } \varphi=\pi/2 \,. \label{Eq:L12R} \end{align} Thus, $L_{1,2} = R_1$, therefore, the peaks arising from $\sin \kappa x = R_1$---i.e., $\delta_{1}$---will be completely suppressed for $\varphi=0$. Morever, $\delta_2$ will only appear if $\gamma_2$ is considerably larger compared to all other parameters of the system. Numerical study presented in the next subsection suggests that $\gamma_2>10\, \Delta_{\text{max}}$ for the roots $\delta_{1,2}$ to start showing up. It can also be seen that $\delta_{1}$ starts showing up for $\varphi=0$ if $\Omega_1 > 10\, \Omega_2$. These features can be understood by observing Eq.~\eqref{Eq:Rep2} and comparing the weighting coefficients of various roots of the numerator and denominator. This features are confirmed by the numerical study presented in the next subsection.
\forget{The results are as follows: \begin{widetext} \begin{align} \sin \kappa x &= R_{1,2} \Rightarrow \Delta=\pm \frac{\Omega_1}{2}\sin \kappa x= \delta_{1,2}(\kappa x) \nonumber \\ \sin \kappa x &= R_{3,4} \Rightarrow \Delta=\delta_{3,4,5}(\kappa x) \nonumber \\ \text{where} \nonumber \\ \delta_{3} &= \frac{\sqrt[3]{3}\,({{{\Omega }_1}}^2\,{\sin \kappa x}^2 + {{{\Omega }_2}}^2 +
{{{\Omega }_3}}^2) + \sqrt[3]{D^2}
}
{2\,\sqrt[3]{9D}}\nonumber \\
\delta_{4}&=\frac{}{4\,\sqrt[3]{9D}} \text{with}\nonumber \\ D&={9\, \Omega_1\,\Omega_2\,
\Omega_3\,\sin \kappa x\,\cos \varphi
+ {\sqrt{3}}\,
{\sqrt{27\,\Omega_1^2\,\Omega_2^2\,
\Omega_3^2\,{\sin^2 \kappa x}\,{\cos^2 \varphi}
- {(\Omega_1^2\,\sin^2 \kappa x +
\Omega_2^2 + \Omega_3^2)}^3}}} \end{align} \end{widetext} }
\subsection{Numerical considerations} In the discussion to follow we plots the roots $\delta_{1,2,3,4,5}$ as a function of $\kappa x$ and show their connection with the behavior of $\chi''$ vs the probe detuning along the cavity field.
To make contact with our earlier work, $\mathscr{I}$, we first consider the parameter range with $\gamma_2=0$ and $\Omega_2=\Omega_3$ and study the effect of increasing $\gamma_2$ on that result. The findings are summarized in Fig.~\ref{Fig:effectofgamma}. In the first column we plot the roots $\delta_{1,2,3,4,5}$ so that their relation to the probe absorption maxima can be established. Then the contour-density plot of $\chi''$ vs the probe-field detuning $\Delta$ and the $\kappa x$ and $\chi''$ vs $\kappa x$ for chosen value of $\Delta$ are plotted for different values of $\gamma_2$ starting with $\gamma_2=0$. The color of the line plots corresponds to the horizontal lines in first-column plots for the respective phase value. This correspondence helps to determine the positions and number of the peaks in the line plots from the places at which the horizontal line intersects the roots $\delta_{1,2,3,4,5}(\kappa x)$. It is clear that the roots $\delta_{3,4,5}$, denoted by solid lines, are dominant most of the times as opposed to $\delta_{1,2}$. The same conclusion can be drawn from the dressed-states approach but for different reasons as discussed in the next subsection. \begin{figure*}
\caption{ The effect of $\gamma_2$ on the localization. The parameters are $\Omega_1=30, \Omega_2=\Omega_3=20$ $\gamma_1=1$. Top row: $\varphi=0$ and Bottom row: $\varphi=\pi/2$. The first column shows the plots of the roots $\delta_{1,2}$ using dashed lines and that of $\delta_{3,4,5}$ by solid lines. The colored horizontal lines correspond to values of detuning $\Delta$ chosen to plot $\chi''$ vs $\kappa x$ in the line-plots shown later in each row for different values of $\gamma_2$ shown on top. The line plots are preceded by contour plots of $\chi''$ to give an idea of its dependence on $\Delta$ as well as position, $\kappa x$. The brightness of a given location in the contour plot is proportional to its height in a 3D plot of $\chi''$ vs $\Delta$ and $\kappa x$. It can observed that for $\Omega_2=\Omega_3=\Omega$ the roots $\delta_1$ and $\delta_3$ coincide. The roots $\delta_{4,5}$ loose their significance as $\gamma_2$ increases. This result can be explained through the dressed states approach as described in the text. For $\Delta=0$ and $\gamma_2=10\gamma$ the root $\delta_{1}=\delta_{3}$ dominates as opposed to $\delta_{4,5}$ as expected. Despite expectation the root $\delta_2$ never appears for $\varphi$ different from $\pi/2$. It can be noticed that for $\Delta=0$ and non-zero $\gamma_2$ one might expect both the roots $\delta_{1,2}$ to dominate, however, this is case only when $\varphi=\pi/2$.}
\label{Fig:effectofgamma}
\end{figure*}
We label the dominant roots, $\delta_{3,4,5}$, such that the root crossing the $\Delta=0$ line (for $\varphi=0$) or the $\Delta=0$ line itself (for $\varphi=\pi/2$) as $\delta_{3}$, the root above the $\Delta=0$ line as $\delta_{4}$, and the one below the $\Delta=0$ line as $\delta_{5}$ as seen in the first-column plots in Fig.~\ref{Fig:effectofgamma}. Among the non-dominant roots---denoted by dashed lines in the first-column plots in Fig.~\ref{Fig:effectofgamma}---the root that coincides with $\delta_{3}$ is denoted as $\delta_1$ and the other one is $\delta_2$. It can be noted that the $\delta_{1,2}$ are independent of the relative phase of the drive fields $\varphi$. This labeling of the roots will be used for the rest of the discussion.
In Fig.~\ref{Fig:effectofgamma}, we further observe that with increasing $\gamma_2$, the roots $\delta_{4,5}$ start diminishing. It is, however, to be noted that this behavior can only be seen from the density plots of $\chi''$ and the plots of $\delta_{3,4,5}$ themselves do not give this information. Another way to explain the peak widths and their dominance is through the decay rates of the dressed-states. We discuss the implications in the next subsection where we evaluate the dressed-states.
We consider the results depicted in the line-plots in Fig.~\ref{Fig:effectofgamma} in further detail. The effect of increasing $\gamma_2$ can be easily seen from the peaks arising due to the roots $\delta_{4,5}$, as seen in the blue plots in Fig.~\ref{Fig:effectofgamma} for both the values of $\varphi=0,\pi/2$. We first consider the results for $\varphi=0$---For $\gamma_2=0$, out of the four blue peaks (corresponding to $\Delta=13\gamma$) occurring the first half-wavelength region, the outer ones arise from $\delta_4$ and the inner ones from $\delta_{3}$. Thus, the expectation--from the density plots--would be that the inner roots would remain sharp and dominant while the outer ones will loose their height and sharpness; this expectation is confirmed by the line-plots for $\gamma_2=\gamma$ and $\gamma_2=10\gamma$. The green ($\Delta=5\gamma$) peaks which arise solely through $\delta_{3}$ are unaffected by increasing $\gamma_2$. The same is true for the red line-plots which correspond to the probe detuning of $\Delta=0$ giving rise to peaks at the nodes of the cavity standing-wave field. Now we consider the case of $\varphi=\pi/2$---Here $\delta_{3}$ coincides with the zero line, hence for $\gamma_2=0$ the red line-plot ($\Delta=0$) gives equal absorption at all spacial points but starts showing spacial dependence as $\gamma_2$ increases which can also be clearly seen from the density plots in the $\Delta=0$ region. The blue ($\Delta=16\gamma$) peaks, in this case, arise from $\delta_{4}$ and therefore diminish in height as $\gamma_2$ increases. The green ($\Delta=12\gamma$) plots show that $\delta_{1,2}$ do not contribute the this particular choice of parameters and show zero absorption for all values of $\gamma_2$. The results depicted in the line plots coincide very well with the corresponding density plots.
The red curves in Fig.~\ref{Fig:effectofgamma}, corresponding to the detuning $\Delta=0$, show different behavior for different phase values. For $\varphi=0$, the height and width of the peaks observed at the zero detuning of the probe field are immaterial of the lifetime of level $\ket{2}$, as they arise from the zero eigenvalue of the dressed state which does not contain any $\ket{2}$ component. This corresponds to a regime of localization that is very common in several other localization proposals, namely, observance of localization peaks at the nodes of the standing-wave cavity field. For $\varphi=\pi/2$, the $\Delta=0$ value is special as it does no show any localization for $\gamma_2=0$ (observe the red plots in the lower row of the Fig.~\ref{Fig:effectofgamma}). It can be seen that these peaks become sharper with increasing $\gamma_2$, whereas the green and blue plots still show dimishing height and increasing width of the peaks. This can be explained as follows: at $\Delta=0$ and $\gamma_2\neq0$ roots $\delta_{1,2}$ dominate as opposed to $\delta_{3,4,5}$ for all other values of the detunings. Thus, for $\varphi=\pi/2$, the root $\delta=0$ starts losing its significance as $\gamma_2$ increases and only the nodal points show peaks which arise from $\delta_{1,2}$ shown by dashed-line plots in first column. This feature is absent for $\varphi=0$ as $\delta_{1}$ does not occur for the parameters of Fig.~\ref{Fig:effectofgamma}---as explained in the context of Eq.~\eqref{Eq:L12R}---and $\gamma_2$ is not large enough for $\delta_{2}$ to show up compared to the dominant root $\delta_{3}$. However, with a different parameter range of values this competition of roots can be seen for $\varphi=0$ as depicted in Fig.~\ref{Fig:interplay}. It is also interesting to note that for $\varphi=0$, in the current figure, the localization peaks at the nodes of the cavity standing-wave field are much sharper than the ones observed for $\varphi=\pi/2$.
Thus, the general conclusion that can be drawn from Fig.~\ref{Fig:effectofgamma} is that as $\gamma_2$ increases only the maxima due to the root $\delta_3$ (for $\varphi=0, \pi$) and root $\delta_{1,2}$ (for $\varphi=\pi/2$) show sharp peaks and the other peaks diminish in magnitude and sharpness. The drive field parameters chosen in Fig.~\ref{Fig:effectofgamma} were as that of the earlier work in $\mathscr{I}$ except for the non-zero $\gamma_2$.
Now we study the effect of varying the amplitudes of the drive fields and go beyond the condition $\Omega_2=\Omega_3$ on the probe field absorption. The results are summarized in Fig.~\ref{Fig:interplay}. \begin{figure*}
\caption{ Localization characteristics for non-identical drive field intensities and interplay of different roots. The parameters are $\Omega_1=30, \Omega_2=20, \Omega_3=10$ $\gamma_1=1$. Top row: $\varphi=0$ and bottom row: $\varphi=\pi/2$. The structure of the figure is the same as Fig.~\ref{Fig:effectofgamma}. The roots $\delta_{4,5}$ loose their significance as $\gamma_2$ increases. This result can be explained through the dressed states approach as described in the text. For $\Delta=0$ and $\gamma_2=10\gamma$ the roots $\delta_{1,2}$ dominate as opposed to $\delta_{3,4,5}$ as expected for $\varphi=\pi/2$. This can be clearly seen as $\delta_3=0$ line becomes insignificant and only the nodal points (arising from $\delta_{1,2}$) remain with increasing $\gamma_2$.}
\label{Fig:interplay}
\end{figure*}
The results for $\varphi=\pi/2$ in Fig.~\ref{Fig:interplay} are very similar to the one in Fig.~\ref{Fig:effectofgamma}, except for the green ($\Delta=12\gamma$) plots. Here the detuning values are the same in both the figures, however, the roots $\delta_{4,5}$ have a larger range than before due to their dependence on the drive-field Rabi frequencies. Thus, new roots appear for the green plots in Fig.~\ref{Fig:interplay} as opposed to no roots in Fig.~\ref{Fig:effectofgamma}. These roots however diminish as $\gamma_2$ is increased making them useless for atom localization for larger $\gamma_2$. For the case of $\varphi=0$ the roots $\delta_{3,4,5}$ have completely different profiles compared to their counterparts in Fig.~\ref{Fig:effectofgamma}. The red plots have the same behavior as in Fig.~\ref{Fig:effectofgamma} being a very commonly observed localization regime for the probe detuning $\Delta=0$. Moreover, the disappearance of the roots $\delta_{4,5}$ with increasing $\gamma_2$ exists in this parameter range as well. Due to this the blue and green peaks loose their height and sharpness with increasing $\gamma_2$.
We observe that an interesting regime arises when $\gamma_2\neq 0$ and the detuning of the probe field $\Delta=0$. In this regime the roots $\delta_{1,2}$ dominate compared to $\delta{3}=0$ for $\varphi=\pi/2$; it can be clearly seen from the last column of plots in Fig.~\ref{Fig:interplay}. The significance of this is clearly apparent for $\varphi=\pi/2$, where the probe absorption is uniform over all spatial points for $\delta_{3}=0$ when $\gamma_2=0$; however as $\delta_{1,2}$ become dominant due to increasing $\gamma_2$ absorption peaks start emerging at positions corresponding to the nodes of the standing-wave field. The behavior for $\varphi=0$ is a bit different and it needs complete dressed-states analysis to explain that has been found to be quite complicated.
Another interesting feature observable from Fig.~\ref{Fig:interplay} is that the root given by $\delta_{3}$ is not dominant at all spacial positions as it is in Fig.~\ref{Fig:effectofgamma}. With increasing $\gamma_2$ the relatively flat regions in the plot of $\delta_{3}$ vs $\kappa x$ start loosing their significance as $\gamma_2$ is increased. This can be ascribed to the broadening of the resonances owing to increased $\gamma_2$. When $\delta_{3}$ remains close close to the line $(\Omega_1/2) \sin \kappa x$, the state is very close to the first eigenstate discussed in Eq.~\ref{Eq:eigenphi0} and these parts remain sharp $\gamma_2$ does not affect the sharpness. Whereas, departure of $\delta_{3}$ from $(\Omega_1/2) \sin \kappa x$ lines can be ascribed to increasing components of state $\ket{a_2}$, which decays with $\gamma_2$, in the dressed state. Thus, the flat regions loose their significance for localization with increasing $\gamma_2$.
\forget{We observe that an interesting regime arises when $\gamma_2\neq 0$ and the detuning of the probe field $\Delta=0$. In this regime the root $\delta_{1}$ dominates for $\varphi=0$ and both roots $\delta_{1,2}$ dominate for $\varphi=\pi/2$; it can be clearly seen in last column of plots in Fig.~\ref{Fig:interplay}. The significance of this is clearly apparent for $\varphi=\pi/2$, where the probe absorption is uniform over all spatial points for $\delta_{3}=0$ when $\gamma_2=0$; however as $\delta_{1,2}$ become dominant due to increasing $\gamma_2$ absorption peaks start emerging at positions corresponding to the nodes of the standing-wave field. The behavior for $\varphi=0$ is a bit different as only $\delta_1$ remains significant and the root $\delta_2$ which arises from $R_2=-2\Delta/\Omega_1$ vanishes as it is very close to $L_1$, the root of the numerator and looses its significance. Further numerical study shows that if $\Omega_1\gg \Omega_2$, and $\varphi=0$, the relative weight of $R_{1,2}$ is much larger than $L_{1,2}$ even if $R_{2}$ is close in value to $L_{1,2}$ and both $\delta_{1,2}$ appear immaterial of the phase value.
Another interesting feature that can be observed from Fig.~\ref{Fig:interplay} is that the root given by $\delta_{3}$ is not dominant at all spacial positions as it is in Fig.~\ref{Fig:effectofgamma}. With increasing $\gamma_2$ the relatively flat regions in the plot of $\delta_{3}$ vs $\kappa x$ start loosing their significance as $\gamma_2$ is increased. This is not due to the broadening of the resonances owing to increased $\gamma_2$, as the dressed-states analysis suggests linewidth of the $\delta_{3}$ resonance is independent of $\gamma_2$. This feature can be ascribed to the dominance of the roots shifting from $\delta_{1,2}$ to $\delta_{3,4,5}$ as $\Delta$ is increased for a given value of $\gamma_2$. Near $\Delta=0$ for large $\gamma_2$ the roots $\delta_{1,2}$ are supposed to be dominant. $\delta_{1,2}$ remain dominant upto a region where $\Delta < \gamma_2/2$. The flat regions of $\delta_{3}$ lie in this region for large enough $\gamma_2$ and hence they loose their significance. For larger $\Delta$ once again the roots $\delta_{3,4,5}$ should become dominant however, $\delta_{4,5}$ become less sharper with increasing $\gamma_2$ and $\delta_3$ does not extend far enough for larger $\Omega$.}
Noting that for $\gamma_2\gg\Delta$ we can expect the behavior of the probe absorption to be completely dominated by $\delta_{1,2}$ as opposed to $\delta_{3,4,5}$ we choose appropriate values for the parameters and consider the density plots of $\chi''$ in Fig.~\ref{Fig:Unrealistic}. \begin{figure}
\caption{ Dominance of the roots $R_{1,2}$. The parameter values are same as in Fig.~\ref{Fig:effectofgamma} except for $\gamma_2$. $(a)$ $\gamma_2=10\gamma$. $(b)$ $\gamma_2= 10^3\gamma$ $(c)$ $\gamma_2=10^4\gamma$. It can be seen that the less dominant maxima slowly vanish as $\gamma_2$ is increased. Thus for large $\gamma_2$ compared to $\gamma_1$ only $\Delta=0$ shows probe peak absorption. As $\gamma_2$ increases $R_1$ starts dominating which is the same as $R_3$, however for somewhat unrealistically larger $\gamma_2$, $R_{3,4}$ roots completely dominate. The apparance of $R_4$ can not be explained by the dressed state approach.}
\label{Fig:Unrealistic}
\end{figure} However, this parameter range is unrealistic and also not very useful as there will be four peaks observed for the detuning lying in the interesting regime as both the $\delta_{1,2}$ roots exist giving rise to four intersection points in one wavelength for a chosen value of the probe detuning. Nevertheless, the observations made in the context of Eq.~\eqref{Eq:L12R} can be confirmed from the results in Fig.~\ref{Fig:Unrealistic}. Both the curves appearing for $\varphi=\pi/2$ have equal characteristics as they both are due to $\delta_{1,2}$, which arise from the same term in the denominator. However, for $\varphi=0$, as discussed earlier, $\delta_{1}$ is cancelled by the roots of the numerator and instead of $\delta_{1}$, the sharper root $\delta_{3}$ appears. Only when $\gamma_2$ is sufficiently large $\delta_{3}$ and $\delta_{2}$ acquire same sharpness as seen in the last column plots of Fig.~\ref{Fig:Unrealistic}.
Deeper understanding of the interplay of different roots can be achieved through the dressed states calculation. We determine the dressed states in the next subsection and explain the above obtained results from a different point of view.
\subsection{Dressed-states approach} To understand the emergence of several roots for the maxima of the probe absorption we consider the dressed states approach. The effective Hamiltonian, taking into account only the strong drive fields, can be expressed as \begin{widetext} \begin{equation} \mathscr{H}_{\rm eff}= \frac{{\rm i}}{2} \( \begin{matrix} 0 & \Omega_3\,{\rm e}^{-{\rm i} \varphi}\,{\rm e}^{{\rm i} k x \cos \theta_3}& \Omega_1\,\sin \kappa x\\ \Omega_3\,{\rm e}^{{\rm i} \varphi}\,{\rm e}^{-{\rm i} k x \cos \theta_3} & 0 & \Omega_2\,{\rm e}^{{\rm i} k x \cos \theta_2}\\
\Omega_1\,\sin \kappa x & \Omega_2\,{\rm e}^{-{\rm i} k x \cos \theta_2} & 0 \end{matrix} \) \label{Eq:EffHam} \end{equation} \end{widetext} in the basis
$\{ \ket{a_1}, \ket{a_2}, \ket{b}\} $. Choosing $\theta_2=\pi/4$ and $\theta_3=\pi/2 + \pi/4$, in the above Eq.~\eqref{Eq:EffHam} we arrive at the secular equation \begin{equation} 4 \lambda^3 - \lambda(\Omega_1^2 \sin^2 \kappa x + \Omega_2^2 + \Omega_3^2) - \Omega_1 \Omega_2 \Omega_3 \sin \kappa x \cos \varphi = 0, \label{Eq:secular} \end{equation} where $\lambda$ are the eigenenergies of the Hamiltonian. It can be noted that Eq.~\eqref{Eq:secular} is identical to Eq.~\eqref{Eq:delta345}. Thus, there is a direct connection between the detuning values for the probe field at which it experiences maximum absorption, $\delta_{3,4,5}$, and the dressed state eigenvalues. The dressed state eigenvalues $\lambda$ give the Stark-shifts in the energy of the state $\ket{a}$. When this Stark-shifted transition $\ket{a}$--$\ket{c}$ is probed by the weak probe field, the resonances will occur at the points where the probe frequency matches the energy level difference between the Stark-shifted levels $\ket{a}$ and level $\ket{c}$. If the probe field frequency, or the detuning $\Delta$, is chosen such that it is in resonance with one of the dressed states then it experiences absorption maxima. This can be expressed by a condition $\lambda=\delta$. It can, however, be noted that only the detuning solutions $\delta_{3,4,5}$ can be explained through the dressed-states approach and not the solutions $\delta_{1,2}$, as explained later.
Actual form of the dressed states for general parameters is quite complicated and in not required as we are only interested in locating the positions of the resonances in the frequency space. In general the probe absorption peaks are quite sharp except at the stationary points along the $x$ axis and when $\gamma_2$ increases. This calculation can be extended further to obtain the spontaneous decay rates of the dressed states. These decay rates could then give more information about the widths of the probe absorption maxima and the loss of sharpness with increasing $\gamma_2$.
For a completely general set of parameters evaluating the actual form of the dressed states and their spontaneous decay rates is sufficiently involved compared to the information that can be gained by such an exercise. Therefore, we consider a restricted regime of parameters to extract information about the sharpness of the probe absorption peaks through the dressed states approach. Assuming $\Omega_2=\Omega_3=\Omega$ and $\varphi=0$ we obtain the eigenvalues to be \begin{align} \left\{ -\frac{1}{2}{\Omega_1\,\sin\kappa x }\,, \frac{1}{4}\({\Omega_1\sin\kappa x\, \pm
{\sqrt{8\,{\Omega }^2 + \Omega_1^2
\sin^2 \kappa x}}}\)\right\} \end{align} with the corresponding eigenstates \begin{align} \label{Eq:eigenphi0} \frac{1}{\sqrt{2}}\( \begin{matrix} -\ket{a_1} \\ 0 \\ \ket{b} \end{matrix} \),\quad \mathscr{N}^{(\pm,0)}\( \begin{matrix} \ket{a_1} \\ c_{2}^{(\pm,0)}\ket{a_2}\\ \ket{b} \end{matrix} \) \end{align} where \begin{equation} c_{2}^{(\pm,0)}= \frac{{\rm e}^{\frac{-{\rm i} \,k\,x}{{\sqrt{2}}}}\Omega \,\left( 3\Omega_1\sin\kappa x\,
\pm
{\sqrt{8\,{\Omega }^2 + \Omega_1^2
\sin^2 \kappa x}} \right) }
{2\,{\Omega }^2 + \Omega_1^2
\sin^2 \kappa x \pm
\Omega_1\sin\kappa x\,
{\sqrt{8\,{\Omega }^2 + \Omega_1^2
\sin^2 \kappa x}}}\, \end{equation} and $\mathscr{N}^{(\pm,0)}$ is the appropriate normalization constant. Whereas, for $\varphi=\pi/2$ we obtain the eigenvalues \begin{align} \{0, \pm\frac{1}{2}\sqrt{2 \Omega^2 + \Omega_1^2 \sin\kappa x}\} \end{align} with the corresponding eigenstates \begin{align} \mathscr{N}_1^{(\pi/2)}\( \begin{matrix} {\rm i} \ket{a_1} \\ -{\rm i}\frac{1}{\Omega} {\rm e}^{\frac{-{\rm i} \,k\,x}{\sqrt{2}}}\Omega_1 \sin\kappa x \ket{a_2} \\ \ket{b} \end{matrix} \),\nonumber \\ \mathscr{N}_2^{(\pm,\pi/2)}\( \begin{matrix} c_1^{(\pm, \pi/2)}\ket{a_1} \\ c_2^{(\pm, \pi/2)}\ket{a_2} \\ \ket{b} \end{matrix} \) \end{align} where \begin{align} c_1^{(\pm, \pi/2)} = \frac{-\Omega_1 \sin \kappa x \pm {\rm i} \sqrt{2 \Omega^2 + \Omega_1^2 \sin^2 \kappa x}}{{\rm i}\, \Omega_1 \sin \kappa x \mp \sqrt{2 \Omega^2 + \Omega_1^2 \sin^2 \kappa x}} \nonumber \\ c_2^{(\pm, \pi/2)} =\frac{- 2\, \Omega\, {\rm e}^{\frac{-{\rm i} \,k\,x}{\sqrt{2}}}}{{\rm i}\, \Omega_1 \sin \kappa x \mp \sqrt{2\, \Omega^2 + \Omega_1^2 \sin^2 \kappa x}} \end{align} with $\mathscr{N}_1^{(\pi/2)}$ and $\mathscr{N}_2^{(\pm,\pi/2)}$ being the appropriate normalization constants.
The message to be taken from the dressed states representation $\mathscr{N} (c_{a_1} \ket{a_1} + c_{a_2} \ket{a_2} + c_b \ket{b})$ in the bare atomic levels is that the decay rate of the corresponding dressed-state is given by $\gamma= |c_{a_1}|^2 \gamma_1 + |c_{a_2}|^2 \gamma_2$, as the level $\ket{b}$ is the ground state. Therefore, it is clear the for $\varphi=0$ the first dressed-state has the deay rate $\gamma_1/4$ whereas the other states decay at the rate proportional to $|\mathscr{N}^{(\pm,0)}|^2 (\gamma_1 + |c_{2}^{(\pm,0)}|^2 \gamma_2)$. Resulting in sharp localization peaks when the probe field is in resonance with the first dressed state and not so sharp localization peaks when the probe field is in resonance with the other two dressed states. In fact with increasing $\gamma_2$, as seen already in the numerical solutions, the latter two states contribute wider and wider resonances which are increasingly useless for atom localization. Similar conclusions can be drawn for the case of $\varphi=\pi/2$; all three roots are equally sharp when $\gamma_2=0$ and the latter two roots increasingly loose their sharpness and decrease in amplitude for larger $\gamma_2$. This observation can be confirmed through the plots in Figs.~\ref{Fig:effectofgamma} and~\ref{Fig:interplay}.
Another important conclusion that can be drawn from the dressed-state eigenvalues is that for the case of $\varphi=0$ the eigenvalues can be made to be well separated by choosing $\Omega_1$ to be little smaller than $\Omega_2=\Omega_3=\Omega$. In such a case the three roots do not overlap and the detuning can be chosen in the range $\{0, \Omega_1/2\}$ to obtain sub-half-wavelength localization. We illustrate this regime in Fig.~\ref{Fig:GSHWL}.
\begin{figure}
\caption{ Illustrating the appropriate conditions to obtain good sub-half-wavelength probe absorption peaks along the cavity field\mdash i.e., sub-half-wavelength localization (observe the green plots in last column.). The parameters are $\Omega_1=20\gamma$, $\Omega_2=22\gamma$, $\Omega_3=25\gamma$. $\gamma_1=\gamma_2=\gamma$. In the first column we show the plots of the roots $\delta_{1,2}$ in dashed lines and that of the roots $\delta_{3,4,5}$ in solid lines alongwith horizontal lines for chosen values of probe detuning $\Delta$ that will be considered for more study in column 3. In the second column we show the density plots of $\chi''$ and in the third column we choose $\Delta=5\gamma$ (green) and show the probe absorption peaks to illustrate the regime of sub-half-wavlength localization and its dependence on the phase $\varphi=0$. We choose other $\Delta$ values as well shown in blue and red to contrast the sub-half-wavelength regime. Note $\Omega_2, \Omega_3 > \Omega_1$ seperates the central root, $\delta_{3}$, from the other ones $\delta_{4,5}$ and provides a phase dependent localization for $\varphi=0$ and no localization for $\varphi=\pi/2$. The localization peaks would appear in the second sub-half-wavelength region if $\varphi=\pi$.}
\label{Fig:GSHWL}
\end{figure}
Moreover, this result holds true even when the drive fields $\Omega_2$ and $\Omega_3$ do not have the same value, $\Omega_2\neq\Omega_3$, and when $\Omega_1<\Omega_{2},\Omega_3$. Another message to be taken from Fig.~\ref{Fig:GSHWL} is that the results for $\varphi=\pi$ are mirror image of that of $\varphi=0$ taken around the vertical line $\kappa x = 0$. This holds true for all parameter ranges, hence we have plotted only the non-trivial cases $\varphi=0$ and $\varphi=\pi/2$ in all the other plots. The range of detuning $\Delta$ spanned by the root $\delta_{3}$ (for $\varphi=0,\pi$) gives an ideal range where sub-half-wavelength localization can be observed, which can always be calculated by solving the Eqs.~\eqref{Eq:secular} or~\eqref{Eq:delta345}, when it is very well separated from the other roots $\delta_{4,5}$. This happens as discussed above when $\Omega_1 < \Omega_2,\Omega_3$.
In spite of the use of the restricted parameters for the evaluation of the dressed-states the results are valid in general as our numerical studies show. Neverthelsess, it can be noted that the parameter range where $\gamma_2$ has a role to play on the dominance of the roots $\delta_{1,2}$ or when $\gamma_2\gg\Omega_{1,2,3}$ can not be explained through the dressed-states approach (See Fig.~\ref{Fig:Unrealistic}). This breakdown of the dressed-states approach for large $\gamma_2$ is easy to understand. Dressed-state calculation is usually done with the assumption that the Drive field Rabi frequencies are large compared to all the other parameters of the system, which breaks down in the large $\gamma_2\gg\Omega$ limit, giving rise to roots which are not predictable by the dressed-states.
\section{Conclusions} We have studied of a variant of a $\Lambda$-type EIT, where a phase dependence is introduced through the application of three driving fields in a loop-configuration. The advantage of the phase dependence is in the tunability that becomes available to manipulate the response of the atomic medium to a weak probe field. By choosing one of the drive fields to be a standing-wave field of the cavity the phase dependence can be extended to obtain atom localization. We have given equations that could be used to simulate several, apparently quite different, energy level schemes. Effect of different parameters are studied with analytical as well as numerical techniques. A dressed-states approach is developed and it is used to explain the peak probe absorption and the peak widths. Also a region of parameters is identified which gives clean sub-half-wavelength localization for a wide range of probe detunings; thus, increasing the applicability of the model. In this range of parameters we show how the choice of phase governs whether localization would be observed or not.
\acknowledgments Part of this work was carried out (by K.T.K.) at the Jet Propulsion Laboratory under a contract with the National Aeronautics and Space Administration (NASA). K.T.K. acknowledges support from the National Research Council and NASA, Codes Y and S. M.S.Z. acknowledges support of the Air Force Office of Scientific Research, DARPA-QuIST, TAMU Telecommunication and Informatics Task Force (TITF) Initiative, and the Office of Naval Research.
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Evaluating an integrated care pathway for frail elderly patients in Norway using multi-criteria decision analysis
M. Kamrul Islam ORCID: orcid.org/0000-0003-4751-15131,2,
Sabine Ruths3,4,
Kristian Jansen3,5,
Runa Falck6,
Maureen Rutten-van Mölken7 &
Jan Erik Askildsen1
BMC Health Services Research volume 21, Article number: 884 (2021) Cite this article
To provide value-based care for patients with multi-morbidity, innovative integrated care programmes and comprehensive evaluations of such programmes are required. In Norway, a new programme called "Holistic Continuity of Patient Care" (HCPC) addresses the issue of multi-morbidity by providing integrated care within learning networks for frail elderly patients who receive municipal home care services or a short-term stay in a nursing home. This study conducts a multi-criteria decision analysis (MCDA) to evaluate whether the HCPC programme performs better on a large set of outcomes corresponding to the 'triple aim' compared to usual care.
Prospective longitudinal survey data were collected at baseline and follow-up after 6-months. The assessment of HCPC was implemented by a novel MCDA framework. The relative weights of importance of the outcomes used in the MCDA were obtained from a discrete choice experiment among five different groups of stakeholders. The performance score was estimated using a quasi-experimental design and linear mixed methods. Performance scores were standardized and multiplied by their weights of importance to obtain the overall MCDA value by stakeholder group.
At baseline in the HCPC and usual care groups, respectively, 120 and 89 patients responded, of whom 87 and 41 responded at follow-up. The average age at baseline was 80.0 years for HCPC and 83.6 for usual care. Matching reduced the standardized differences between the groups for patient background characteristics and outcome variables. The MCDA results indicated that HCPC was preferred to usual care irrespective of stakeholders. The better performance of HCPC was mostly driven by improvements in enjoyment of life, psychological well-being, and social relationships and participation. Results were consistent with sensitivity analyses using Monte Carlo simulation.
Frail elderly with multi-morbidity represent complex health problems at large costs for society in terms of health- and social care. This study is a novel contribution to assessing and understanding HCPC programme performance respecting the multi-dimensionality of desired outcomes. Integrated care programmes like HCPC may improve well-being of patients, be cost-saving, and contribute to the pursuit of evidence based gradual reforms in the care of frail elderly.
The global phenomenon of increasing life expectancy and proportion of elderly [1, 2] is also seen in Norway. The 2020 national population projections in Norway predict more elderly people in the population than previous projections. The share of persons aged 65 years or over will increase from 18 to 30% by 2070 [3]. Even though general health status of the elderly population in Norway has improved (for example, better self-reported health, being more physically active, and having less anxiety and depression), the demand for health care has not decreased during the last twenty years. There is little change in the number of people living with chronic disease or requiring assistance in activities of daily living [4] and resources spent in elderly care have increased [5]. The Organisation for Economic Co-operation and Development (OECD) projections predict that budgetary pressures in the coming decades are likely to come mainly from rising long-term care expenditure due to population ageing [6].
In Norway, hospitals are state-owned enterprises governed by appointed administrative boards. Municipalities are responsible for providing primary care and social services. Well-coordinated healthcare programmes can help the elderly with chronic conditions smoothly navigate the health and social care system. Within this endeavour, in 2012 the Norwegian Care Coordination Reform [7] was launched. An important intention of the reform was to develop coordinated clinical pathways across primary care and specialist care. The three primary objectives of the Coordination Reform are: (i) a more cohesive and coordinated approach to health and care services; (ii) a greater proportion of health and care services to be provided in the local communities; (iii) greater focus on preventative measures and improving public health.
Through the Norwegian Care Coordination Reform, the municipalities were given increased responsibility for community-based treatment, care and rehabilitation, to substitute specialized care by more affordable primary care [8]. A key challenge which the Care Coordination Reform intends to address, is that the common disease-specific approach in specialist care suffers from fragmentation and lack of person-centeredness when applied to people with multi-morbidity [9, 10]. As a result, generic clinical pathways for patients with multiple chronic diseases and frail elderly were developed, based on the involved municipalities' experiences, as well as empirical research. Based on a previous model in the municipality of Trondheim by Røsstad and Grimsmo [11], in 2013 the Norwegian Ministry of Health and Care Services, the Norwegian Directorate of Health, the Norwegian Institute of Public Health and the Norwegian Association of Local and Regional Authorities developed generic clinical pathways for patients with multiple chronic diseases and frail elderly through the programme Learning network of "Holistic Continuity of Patient Care" (HCPC) [12]. Learning networks as a method is based on Breakthrough Series, developed in the USA in 1995 [13], in which teams from different health services gather in collaboratives for learning and improvement.Footnote 1 The HCPC programme aims at improving patient pathways for the frail elderly by:
Moving attention from "what is the matter with you" to "what is important to you" so as to strengthen the service receiver's role and contribute to equity and empowerment of the patient.
More systematic collaboration between municipalities and hospitals.
Follow-up of patient pathways using standardised measures.
The programme is targeted at frail elderly people living at home that have recent functional deterioration requiring additional municipal care services. The program involves early assessment, patient-centered follow-up, early involvement of the patient's general practitioner (GP), and assigning a designated primary care contact.
There is a strong need for integrated care models [15, 16] for elderly people with multi-morbidity, as well as evaluations of such interventions and measures, in particular regarding the meso organisational and macro system-level care integration strategies [17,18,19,20]. Integrated care models for people with multi-morbidity are complex multi-faceted interventions that aim to improve a wide range of outcomes, often referred to as the Triple Aim (i.e. improving health/wellbeing, experience with care, costs [21]. In order to evaluate such models, a broad evaluation framework is required, in which a wide range of outcome measures are included. The current evidence of integrated care for frail elderly is inconsistent; the quality of studies is weak and they are inconclusive [22,23,24]. In particular, much of the previous studies evaluated the effectiveness and cost-effectiveness of these interventions for frail elderly by considering health outcomes such as functional limitations and health-related quality of life [24, 25]. These outcomes might not be appropriate for frail elderly people whose health is deteriorating [26, 27]. Hence, traditional cost-effectiveness studies seem unable to capture outcomes relevant to frail older people. Multi-Criteria Decision Analysis (MCDA) can provide a framework for evaluating integrated care on a broader set of outcomes [28, 29]. In MCDA the outcomes (i.e. criteria) can be reviewed separately but also integrated into one score by applying a relative weighting of importance to the outcomes. In addition, MCDA can consider multiple perspectives by using weights reflecting the preferences of different stakeholders [28, 30].
To our knowledge, no thorough scientific evaluation of patient outcomes related to introducing HCPC has so far been completed. Within the HCPC context, a qualitative exploration was conducted by Nilsen et al. [31]. They studied home care nurses' experiences with implementation and systematic use of functional and wellbeing checklists developed for improving continuity of care and quality of care in the pathways provided to old and chronically ill patients in the communities. The checklists aimed to be person-centred and function-based as suggested by the findings from the Patient Trajectory for Home-dwelling elders (PaTH) project [32]. Using MCDA, this study aims to investigate the impact of introducing a specific model of integrated care for frail elderly patients, HCPC. We investigate specifically whether the HCPC programme contributes to improved health and well-being, experience of care and resource utilisation. The programme is evaluated as part of the European Union -financed SELFIE project (Sustainable integrated care models for multi-morbidity, delivery, financing and performance).Footnote 2
The three core differences between the HCPC programme (i.e. integrated care programme) and usual care (control group) are first the initial and follow-up (6 weeks) assessment of the patient's level of functioning by validated tools; second the "everyday-rehabilitation" informed by the patient's own goals for activities of daily living; and third the early involvement of the patient's GP. As part of the programme a new professional role has been developed; a designated primary contact person (coordinator), notably a nurse or a social worker, responsible for individual patient follow up. The designated primary care contact works in the municipal care service. Focus is on functional ability rather than on disease and impairment. A patient's GP is involved within 2 weeks after enrolment, i.e. through consultation at GP surgery or a home visit. The primary care teams comprise at least the patient's coordinator and the GP. Other primary care professionals (e.g. physiotherapist, occupational therapist, social worker) are involved when appropriate. The HCPC and patient care involved are financed through the participating municipalities' general budgets. There are no direct financial incentives towards municipalities.
Population and data collection
The study population comprised frail elderly patients with multi-morbidity starting or extending their use of municipal home care services, or having a short-term stay in a nursing home, because of functional deterioration. While many of these patients were discharged from hospital recently, hospitalization was not mandatory for inclusion. The intervention group was recruited from municipalities participating in HCPC, and the usual care from municipalities not (yet) enrolled to this programme.
The HCPC group was derived from 12 municipalities in South- and Mid-Norway. Two of these municipalities were very small (< 2000 inhabitants), eight were small (5000 to 26,000 inhabitants) and two were mid-sized (44,000 to 46,000 inhabitants). The programme owner, The Norwegian Association of Local and Regional Authorities, introduced and recommended the study to the municipalities enrolled in HCPC through their contact network and during network meetings. Members of the research team attended several meetings, provided written and oral information to the municipalities and signed collaboration contracts. Primary contact persons in the enrolled municipalities recruited participants consecutively, i.e. they conducted home visits to eligible patients, provided study information, and collected informed consent. Within the eligible study population, there were no exclusion criteria. For patients with dementia or other conditions interfering with competence to give consent or answer survey questions, next of kin was invited instead. Three questionnaires at baseline and three at follow-up (for the same patients) were, and one questionnaire only at follow-up was completed by proxy. The enrolled patients' primary contact person completed the "SELFIE-questionnaire for frail elderly" (we will refer to this questionnaire as the "SELFIE-Questionnaire" hereafter) based on a face-to-face patient interview in the patient's home, at baseline (enrolment) and after 6 months. Data collection was conducted from September 2017 to June 2019.
The usual care group was derived from four municipalities in West-Norway not yet enrolled in the learning network programme of HCPC. Two of the municipalities were small (8000 to 16,000 inhabitants), one was mid-sized (30,000 inhabitants) and one was large (284,000 inhabitants). In the three small and mid-sized municipalities, research assistants (nurses/assistant nurses) conducted home visits to eligible patients, provided study information, and collected informed consent. In the large municipality, none of several attempts to recruite eligible home-dwelling people were successful. Instead, research assistants (employees/students at the University of Bergen, Norway) recruited eligible patients during short-term rehabilitation stays in nursing homes, before discharge to their own home. As with the HCPC group, there were no exclusion criteria in the usual care group. Next of kin was invited to fill out the questionnaire in cases where patients were unable to answer the questions by themselves. However, none of them filled out the questionnaire to a sufficient extent to be included in the study. For the enrolled patients, the research assistants completed the "SELFIE-Questionnaire" based on a face-to-face patient interview in the nursing home at baseline (enrolment) and in the patients' home (after 6 months). Data collection was conducted from September 2018 to October 2019.
Multi-criteria decision analysis (MCDA)
The SELFIE MCDA framework was developed based on established guidelines and follows the seven recommended steps: i) establish the decision-context, ii) identify and structure criteria, iii) determine the performance on criteria, iv) determine the weights of the criteria, v) create an overall value score, vi) perform sensitivity analyses, vii) interpret results [28, 29]. In the first step the aim is to establish what the likely decisions are that need to be made and thus how the MCDA will be used [28, 29]. Earlier qualitative study including document analyses and interviews with programme-initiators, managers, representatives of payer organisations and care providers of the HCPC programme has shown that the programme needs to provide evidence on the effectiveness of the intervention to help establish the long-term sustainability and potentially wider implementation of the programme throughout Norway [30, 33]. The aim of the MCDA was to inform these decisions quantitively by comparing the HCPC programme to usual care on the improvement in health and well-being, experience of care and resource utilisation. Moreover, it was important to identify the relevant stakeholders in this decision-making process. The stakeholders that were considered pertinent were five groups: Patients, Partners and other informal caregivers, Professionals, Payers, and Policy makers [33].
Outcomes/criteria (the second step)
The decision criteria that were used in the MCDA were the outcome variables that were measured with the "SELFIE-Questionnaire" specially developed for the integrated care programmes targeting elderly populations (see Appendix 1). Table 1 illustrates the validated items from well-established and widely used "instruments" for defining the outcomes included in the "SELFIE-Questionnaire". All outcomes were related to one of the three domains of the Triple aim, i.e. health/well-being, experience of care and resource utilisation/costs (Berwick et al., 2008). A core set of outcomes included physical functioning, psychological well-being, enjoyment of life, social relationships and participation, resilience, person-centeredness, continuity of care and total health and social care costs.Footnote 3 In addition, for the elderly patients, programme-specific outcomes included autonomy, informal care costs, long term institutional admission, and falls leading to hospital admissions. The distinction between a core set of outcomes and programme-specific outcomes was made within the SELFIE project, where the core set was measured in all evaluation studies as part of that project.
Table 1 Outcome measures included in the SELFIE-Questionnaire and their scale range
To estimate and compare total health and care cost between HCPC and usual care, the cost components were identified and quantified from a societal perspective. The health and social care costs included the costs arising from the consequences of treatment (i.e. costs incurred with primary care services, e.g. GP and nurse, drug, and hospital costs) and formal social care costs. Costs related to informal care were estimated separately. Data on health, social care and informal care use were collected with an adapted version of the iMTA Medical Consumption Questionnaire and included within the "SELFIE- Questionnaire" [34].Footnote 4 The questionnaire includes questions about contacts with GPs, primary care nurses, GP assistants, physiotherapists, dieticians, psychologists, dentists, social workers, welfare workers, and medical specialists, as well as hospital inpatient- and outpatient admissions, home care services, residential care and nursing homes, and informal care services during the past 3 months.
To quantify the total health and social care costs in monetary terms (Norwegian kroner - NOK) and to get the unit price of the relevant components, we used several different published documents and articles. In particular, national tariffs for GP services come from the Normal tariff for General Practitioners and Out-Of-Hours Emergency medical services 2018–19 document [36]. We converted all costs into 2019 prices using Consumer Price Index (changes in the price level of a weighted average market basket of consumer goods and services purchased by households) provided by Statatistics Norway (see [37]). The consultation fee for a GP was calculated as the weighted average of the fee for a non-specialist GP (NOK 165 (16.7 EURFootnote 5)) and an approved GP specialist (NOK 257 (26 EUR)). In estimating actual health care provider cost, out-of-pocket costs (co-payments) were assumed to be 30% of actual provider cost [39]. Per-diem in-patient hospital cost information was gathered from a recent published study from Norway [40]. Using their estimation, inflated by the Consumer Price Index, we estimated the average costs of a general ward bed day in Norwegian hospitals at NOK 8400 (853 Euros in 2019 price).
To estimate the medication costs we have collected the Defined Daily Dose, i.e. the assumed average maintenance dose per day for a drug used for its main indication in adults, by using the Defined Daily Dose Index 2020 [41]. The medication's price was obtained using the joint directory for drugs [42]. The unit cost for social care and home care inpatient and outpatient components were gathered from reports published by the Norwegian Health Directorate [43, 44]. To estimate the informal care, we used relevant unit cost for home care services. The details on unit costs for social care and home care services by different municipalities are provided in Table A1 in Appendix 2.
Statistical analysis of performance scores (the third step)
The performance scores on the decision criteria of the MCDA were defined as the scores on the outcome measures described above. When estimating causal effects of using observational data, it is crucial to reduce systematic differences in the empirical distribution of the baseline (pre-intervention) confounders [45]. We performed a quasi-experimental approach called inverse probability of treatment weighting (IPTW) using the propensity score, to minimise the impact of any potential selection bias between HCPC and usual care at baseline. Logit regression was used to estimate the propensity score in which treatment status was regressed on selected observed baseline socio-demographic characteristics: age, gender, living condition, smoking status, multi-morbidity status (number of health problems≥2), and the baseline values of two core-set outcome variables, namely physical health, and social relationships and participation. The IPTW assigns a weight to each UC patient based on the similarity of the patient to the HCPC patients. In the IPTW the weights of the HCPC patients were set to 1 and the weights of the usual care patients were calculated with the formula Weight = propensity score/(1–propensity score).
The baseline differences between the HCPC and usual care patients were assessed before and after the IPTW. Overall matching results were assessed by examining and reporting three test statistics. First, the mean (median) absolute standardised bias (i.e. the mean/median of the ratios of the difference of the sample means in the HCPC and usual care groups over the square root of the average of the variances in both groups); second, Rubin's B, defined as the standardized difference of the means of the linear index of the propensity score in HCPC and usual care group, and third, Rubin's R, defined as the ratio of HCPC and usual care variances of the linear index of the propensity score [46]. Statistical significance of the standardised differences have also been presented for all covariates before and after matching.
We estimated the "average treatment effect on the treated" for all outcome measures on the IPTW data. To analyse the outcomes we used repeated measurement models with individual level random intercept effects (i.e we assumed that the coefficients were fixed but the intercept varied randomly). We used models assuming continuous outcomes for ease of interpretation. Formally, we estimated the following equation:
$$ {y}_{jt}={\beta}_0+{\beta}_1{I}_j+{\beta}_2{T}_t+{\beta}_3{I}_j\times {T}_t+\chi {X}_{jt}+{\psi}_j+{\varepsilon}_{jt} $$
where yji is the outcome for individual j at time t; Ij is a dummy variable categorising intervention group (variable equals to 1 if the observation is from the HCPC cohort) and Tt is a dummy variable for time (equals to 1 if the observation is from T1 period), respectively. The coefficient for \( {\hat{\beta}}_3 \) describes the treatment effect. Xjt includes individual jth age at time period t. ψj is the random error term for the jth individual and εjt is the remaining error term for jth individual observed in the tth period.
To calculate performance scores we predicted the mean score of the HCPC group at 6 months based on the regressions results. In addition we calculated the mean score of the usual care group assuming they had the same baseline score as the HCPC group. In this way the calculated performance scores could be directly compared between the HCPC and usual care group. This was done separately for each outcome.
Weghting the criteria (the fourth step)
Relative weights for the different criteria (i.e., outcomes) among stakeholders were elicited in an online weight elicitation study among Norwegian patients, informal caregivers, professional care providers, payers and policy makers. In the weight elicitation study, a discrete choice experiment (DCE) was used to obtain weights for the core set of outcomes (for full details of how these preference weight were obtained see [30, 47]. Table 2 gives the relative weights of the outcomes included in the MCDA, and as shown, all five stakeholder groups put relatively high weights on enjoyment of life and the lowest weight on cost.
Table 2 Relative DCE weights of the core set of outcomes used in the Multi-Criteria Decision Analysis by type of stakeholder
Overall value calculation (the fifth step)
In the MCDA, the mean predicted outcome scores at T1 were first standardised on a 0–1 scale. This was done using relative standardisation with the equation:
$$ {S}_{aj}=\frac{x_{aj}}{{\left({x}_{aj}^2+{x}_{bj}^2\right)}^{1/2}} $$
Where xajis the raw performance score in terms of mean predicted values for outcome j (on the natural scale) for the HCPC group and; \( {x}_{aj}^2 and\ {x}_{bj}^2 \) are the square of mean predicted values for outcome j for the HCPC group and the usual care group respectively. For all outcomes, the standardised score was set so that a higher score indicates better performance. To achieve this, in the above-mentioned equation, x was replaced by 1/x for reversely coded outcome measures (i.e outcomes where a higher score on the natural scale indicates a worse performance). We implemented an additive MCDA model where the standardised outcomes were weighted and subsequently summed to obtain a single overall value score for the HCPC group and a single overall value score for the usual care group.
Sensitivity analysis (the sixth step)
Probabilistic sensitivity analysis using Monte Carlo simulation was performed to evaluate the joint uncertainity of preference weights and performance scores. Cholesky decomposition was conducted for 10,000 replications to obtain a single overall value score. We calculated confidence (uncertainty) intervals around the overall value scores for each stakeholder group. The difference between the overall value score of the HCPC and the usual care is statistically significant if the confidence intervals do not overlap.
In the deterministic sensitivity analyses, we additionally conducted a swing weighting method for eliciting preferences [30, 48]. Within this analysis a wider range of outcomes (along with the eight outcomes in the core set), namely the programme-specific outcomes for frail elderly care programmes, were also included.Footnote 6
All statistical analyses were performed with STATA 16.
Sample characteristics
The study comprised information from 120 and 89 patients at baseline from the HCPC and usual care group respectively, and 86 patients in the HCPC group and 41 patients from usual care group at follow-up after 6 months. After scrutinizing the data, we found that there were some missing observations (varied from minimum 1 to highest 10) for a few outcome variables. To be pragmatic and to consider all available patients responding at baseline, we imputed these missing data on the outcome variables by their mean values and by the HCPC and usual care groups. However, we omitted four patients who responded at follow-up only but not at baseline.
Table 3 provides descriptive statistics on baseline characteristics of patients for two groups, and for pre-and post-matching data. In particular, before matching, the average age of the elderly patients at baseline was around 80.0 years for the HCPC group, while the corresponding average age was higher for the usual care group, at 83.6 years. Overall, as illustrated in the matching statistics given in the lower panel of Table 3, matching led to an improvement in the comparability of the two groups. The statistics satisfied the required criteria with recommended ranges (Rubin [46] suggested that the value of B < 25; 0.5 < R < 2 for sufficient balance). In particular, mean (median) standardised bias was 27.9 (28.0) in the unmatched sample, and after implementing the IPTW using the propensity score approach in the matched sample it was reduced to 6.3 (3.5); Rubin's B was 88.6 in unmatched sample and reduced to 22.8 for the matched sample, and Rubin's R showed 0.79 in unmatched sample and changed to 0.91 in matched sample.
Table 3 Baseline characteristics before and after propensity score matching
Figure 1 shows the distribution of the core outcome variables between the HCPC and usual care groups before and after matching. As is visualized, after matching the distributions are quite overlapping for all core-set outcomes provided in the figure.
The distribution of the core outcome variables between HCPC (treated) and usual care groups (untreated) before and after matching
Effects on outcomes
Table 4 illustrates the results of the statistical analyses of all core set outcomes included in the MCDA. The estimated "treatment effects on the treated", \( {\hat{\beta}}_3 \) (coefficients of the interaction term between time and intervention), is provided in the last column of Table 4. The five health and wellbeing outcomes (the first 5 items/rows) consistently favour HCPC, and the experience outcomes favour UC, but there is no statistically significant effect (at the 5% level) of HCPC for any of these core-set outcome variables. A weak positive effect of the intervention is observed for the outcome variable resilience. A significant effect is observed for the programme-specific outcome autonomy ("remaining in charge and making own decisions on how one lives his/her own life"). The estimated coefficient indicates that HCPC negatively affects patients' autonomy as measured by the Pearlin Mastery Scale (see Table 1), which is reduced by 2.9 points for the patients belonging to HCPC compared to usual care. A weak (at the 10% level) effect is observed for the informal care cost outcome in favour of HCPC.
Table 4 Health/Wellbeing and Experience Outcomesd: within and between group differences at 6 months
Table A2 (in Appendix 2) describes the health and social care costs differences for the last 3 months prior to interview date) by HCPC versus usual care at follow-up. The average total health and social care costs and informal care costs were lower for HCPC than usual care at follow-up. However, when categorising the total costs, we found that the costs of the care provided by the the GP, medical specialist and psychologist were higher for HCPC than usual care. Costs related to nurses, physiotherapists, hospital emergency care, hospital stay, inpatient and outpatient social care were substantially lower for HCPC than usual care.
As shown in Table 4, the treatment effect is also negative for health and social care costs (Euro − 2999), indicating that HCPC is less costly than usual care, as is the case also for costs of informal care. The estimate indicates a reduction in informal care costs of HCPC by Euro 2097 compared to usual care.
MCDA overall scores
Table 5 reports the overall value score in the Multi-Criteria Decision Analysis (MCDA) for five Norwegian stakeholder groups (5Ps): Patients, Partners, Professionals, Payers and Policy makers. For All stakeholder groups, HCPC performed better than usual care, although the difference was small. Regarding the individual outcomes, the standardized performance score was higher for HCPC than usual care, except for continuity of care.
Table 5 Stakeholders' value scores in the Multi-Criteria Decision Analysis
The robustness of the results was investigated by considering new sets of coefficients for treatment effect and importance weights using the Cholesky decomposition approach. The lower panel of Table 5 shows the uncertainty around the value score calculated with Monte Carlo simulation. For all five stakeholders, the MCDA results were very similar to our base analyses, indicating that the overall value scores were higher for integrated care than usual care. The proportion of the 10,000 simulations showing a preference for integrated care was between 94 and 99% for all stakeholders. The 95% confidence intervals did not overlap.
Table A4 in Appendix 3 also presents the value scores in the MCDA with the swing weights. The weights differed slightly, and the estimated score for the programme-specific outcome autonomy also here favoured usual care rather than HCPC. Overall, the MCDA results were highly consistent with our base analyses. The weighted scores were of similar magnitudes, and higher for integrated care than usual care for all five stakeholder groups.
Summary of findings (the seventh step)
HCPC is an integrated care programme targeting frail elderly patients with multi-morbidity that have just started or extended their use of municipal home care services or have a short-term stay in a nursing home. The results from an MCDA, where a large set of outcomes corresponding to the 'Triple aim' (i.e. health/well-being, experiences with care, and costs) was taken into consideration, indicated that elderly patients in the municipalities participating in learning networks of HCPC fare better than elderly patients in the municipalities offering usual care. The result held for the perspectives of patients, partners, professionals, policy makers and payers. The differences in total MCDA scores between HCPC and usual care were constant over the five stakeholder groups. The main driver of this result seemed to be the difference in performance of the HCPC along the dimension enjoyment of life. Here HCPC improved patients' experienceconsiderably more than usual care, although the effect was not statistically significant. There was little difference of opinion between the stakeholders with respect to enjoyment of life being the highest valued outcome measure. Furthermore, it appeared that the HCPC programme increased cost incurred for primary health care, but reduced comparatively costly components attributed to hospital stay and nursing home care, and cost associated with informal care. The municipalities cover the establishment costs for the HCPC programme, however, we did not have data on this intervention cost component (e.g. costs associated with intensified collaboration between sectors, costs of planning individual care pathways) at our disposal and it was not included in this analysis. Moreover, it is difficult to make any sensible assumptions on the costs related with collaboration between sectors and cost related with planning individual care pathways. We have been in contact with some of the municipalities to have them indicate costs associated without them being able to give us precise estimates. But they are not likely to be very high, since also individual pathways have to planned both in the HCPC and the usual care groups. Besides, when it comes to costs borne by the municipalities, who run the programmes, there is no strong indication of favourable or adverse financial incentives.
An interesting difference was observed when weighting outcomes with swing weights instead of DCE weights. Enjoyment of life was attached a lower weight, while autonomy was weighted quite highly. Furthermore, usual care patients fared better than HCPC patients when considering how they feel about mastering their own life. This may reflect that involving more health personnel in HCPC may provide better care but at the cost of more decision making left with professional care providers, which may negatively affect the patients' feeling of autonomy.
Strengths and limitations
The study is a novel contribution to investigating and understanding how an integrated health care programme performs when taking into consideration the multi-dimensionality of relevant outcomes. The SELFIE MCDA provides a structured and explicit evaluation framework for assesing an integrated programme with multiple health and wellbeing outcomes relevant to frail elderly patients. Even though we are aware of the methodological challenges discussed in the literature, particularly regarding which outcomes to include and how to assess uncertainty [49], the study provides some evidence that pure cost-effective analyses may miss important aspects of relevant outcome space. An international valuation study has shown that different stakeholder groups - patients, partners, professionals and policy makers - both within and between countries appreciated the enjoyment of life, social relationships and participation, and personal autonomy in addition to traditional health outcomes [47]. This study showed that on certain outcomes HCPC performed better than usual care and on others it did not. MCDA allowed us to use an explicit framework to aggregate the various outcomes and to determine an overall value score. Furthermore, we showed overall value scores from the perspectives of different stakeholders. The latter is important because aligning the preferences of different stakeholders is likely to contribute to the success of an integrated care approach. There is good reason to have confidence in the weights attached to the outcomes, obtained in a discrete choice experiment among a total of 776 stakeholders (~ 150 per stakeholder group) (Rutten-van Mölken et al., 2020).
Another strength is that the performance scores were estimated using a quasi-experimental framework (i.e. IPTW) by including a control group which provided better estimates of effectiveness than many previous studies in this field that are before-after studies. Nevertheless, the uncertainty around the overall value scores was formally incorporated using probabilistic sensitivity Monte-Carlo simulations on both weights and scores.
Methodologically, we have shown benefits from using an MCDA framework in evaluating care programmes aimed at patients that are in a situation with several, and often conflicting, demands. Hoverver, there are some methodological limitations with the MCDA framework discussed in the literature [49]. In particular, the creation of a composite measure of benefit without knowing the threshold value for the willingness to pay or the opportunity costs of one additional unit of that benefit (as researchers do know for QALYs); and whether cost should be consider as one of the criteria/outcomes in the MCDA. Questions could also be raised on the utility of the value scores from the different perspectives. Is there a perspective that should prevail? Or, should analysts average the perspectives? In our study we assumed that the performance scales and weight scales were linearly associated. Simplicity, transparency and implementational convenience favour the additive MCDA models. To avoid inappropriate double counting of value, the outcomes in an additive model should not overlap and the outcomes should be preferentially independent [49]. The potential violations of these key assumptions may not be ruled out in our additive MCDA framework. To study this may involve, for example, mapping between the discrete choice experiment levels and the performance scale. However, changing our assumption of linearity is unlikely to affect our MCDA results because the performance scores of the HCPC and usual care do not differ very much. Although the SELFIE MCDA framework has made efforts to overcome several challenges facing the approach (see [30]), future work could be attained on the actual use of the MCDA tool by decision makers to clarify how stakeholders want to engage with MCDA, and precisely when and how it contributes the most.
Further limitalions are connected with the patient recruitment and the impact evaluation. Patient recruitment in both HCPC and usual care municipalities was difficult for several reasons. It was particularly difficult to establish the usual care municipalities. The Norwegian Association of Local and Regional Authorities made efforts to recruit municipalities that were planning to join the HCPC programme, expecting these might have an interest in a comparative study. However, several municipalities declined, often due to a lack of resources to assist in collecting data, or general overload of work also related to planned merging of municipalities. Furthermore, although the HCPC group was easier to establish, patient recruitment here proved difficult since so many of the municipalities were small with few eligible patients. Another issue to consider was that frail elderly patients in general may have problems answering the lengthy questionnaire that was used for data collection on health and wellbeing, experience of care, and individual costs. For these patients, the follow-up interview was particulary problematic, since their health condition might worsen. Recall bias associated with health and social care, and informal care resource use could be a potential concern for the study using a retrospective self-reported questionnaire. However, such a bias would not be systematically different between the HCPC and usual care groups. Attrition rate as a result of losses to follow-up was relatively higher for the usual care group. At the 6-month follow-up, 41 patients from the usual care group (more than 46% sample at the baseline) responded, whereas 86 patients (around 72% of baseline sample) responded from the HCPC group. Ifloss to follow-up was systematically related to the patients' underlying characteristics such as age, multimorbidity status etc., differences in the attrition rates for usual care and HCPC group could result in selection biases. High attrition rates are common in studies on elderly patients [50]. The HCPC group comprised exclusively home-dwelling frail elderly people, while the UC group was mainly recruited from short-term nursing home rehabilitation stays (93%) and mostly from one large municipality. The difficulty to recruit home-dwelling patients in this control municipality was attributable to the lack of dedicated primary care contact persons. This strategy may have introduced bias which we attempted to overcome through a successful implementation of the IPTW approach at baseline by controlling for the living condition attribute in estimating the propensity score. Moreover, different persons were involved in recruiting and interviewing patients in the different municipalities. Even though they were given instructions on patient selection and how to assist patients in answering, it is difficult to rule out reporting bias.
Generalizability
The study population was derived from 16 municipalities of various sizes in different parts of the country, supporting transferability of the results to similar municipalities in other regions. The generic clinical pathways for patients with multi-morbidity in the HCPC programme ensured that the included patients have many different chronic conditions and thus comprised a representative population with multi-morbidity in need of municipal care services. Transferability of results to other countries depends on organisation and cooperation of health and social services, and how different outcomes would be weighted by different stakeholders.
Comparison with existing literature
It is difficult to directly compare the MCDA results of the HCPC programme to other integrated care programmes. Many integrated primary care approaches targeting frail older persons have emerged over the years. As mentioned in the onset, however, the quality of previous studies was not good and evidence for their effectiveness and cost-effectiveness remained mixed [22,23,24, 51]. The integrated care programmes were highly multifaceted interventions and as such comparing them to each other introduced difficulties due to differences in interventions, outcomes and populations [52]. Nevertheless, in a recent comprehensive review, Baxter and colleagues [52] concluded that integrated care programmes/models might enhanced patient satisfaction, increased perceived quality of care, and enabled access to services, although the evidence for other outcomes including health and wellbeing outcomes and service costs remained uncertain.
Implications for research and/or practice
The numbers of patients with multiple chronic diseases are increasing, which is a challenge to the public purse. To establish relevant and cost-efficient care it is vital to investigate the main needs and care demands of these patients. This project highlights the importance of providing care along different dimensions. The Norwegian HCPC programme for frail elderly patients is an integrated programme that has received much attention from the municipalities in Norway. The results of this study are important contributions to better understand the effects of integrated care programmes and may be useful for decision-makers at the municipal and central level for prioritisation of often-costly initiatives. Future research should focus on developing stronger links between outcomes and weights in the MCDA. Our results support furthering resources for a randomised controlled trial that may provide stronger conclusions on causality than possible in a quasi-experimental design.
This study shows the importance of evaluating reforms and new initiatives for chronically ill and frail elderly patients, often with multi-morbidities, in a broad framework. Cost-effectiveness analyses may give a first-hand insight into the acceptability of spending resources on pure health and cost dimensions. However, the pure health state dimension does not capture all relevant features of what is relevant to these patients. The results clearly show the importance of a broad perspective such as the MCDA framework when considering care delivery in a transparent way. A national programme like HCPC, with moderate changes, may improve well-being of the patients, in the long term be cost saving, and overall be worthwhile pursuing in gradual care reforms for the frail elderly patients.
The stakeholder data used for weighing is anonymous (provided in Table 2), and can be available on request from the corresponding author (MKI). The data that support the findings provided in Tables 3, 4 and 5 are not publicly available due to them containing information that could compromise research participant privacy/consent.
Learning networks as a method has been internationally used and evaluated for a range of issues, evidence summary can be accessed [14]. In Norway, three national programmes have the short name "Learning network" (for children, for drug abuse/psychiatric care and for elderly).
Among other aims, the SELFIE project evaluates impact of 17 different integrated chronic care (ICC) programmes for patients with chronic conditions and multi-morbidities in eight European countries [30].
In the SELFIE project, several different integrated programmes were investigated. Some outcomes were common for all programmes, termed 'core outcomes', whereas other outcomes were specific for a smaller groups of treatment programs, like those involving frail elderly. The latter are termed 'program specific outcomes'. The selection of the outcomes was based on a literature review, workshops with representatives from the five stakeholder groups, and focus groups with individuals with multi-morbidity [33].
The link to where people can order the questionnaires at [35].
Annual exchange rate for 1 euro = NOK 9.8527 in 2019 [38].
Table A3 in Appendix 3 shows the swing weights for five Norwegian stakeholders. We found some difference in the ranking of the weights over attributes and by stakeholders. For example, in swing weight, patients put relatively highest weight on physical functioning and then autonomy, while other stakeholders put relatively higher weights on enjoyment of life and autonomy.
MCDA:
Multi Criterion Decision Analyses
OECD:
The Organisation for Economic Co-operation and Development
HCPC:
Holistic Continuity of Patient Care
DCE:
Discrete Choice Experiment
PSM:
Propensity Score Matching
IPTW:
Inverse Probability Treatment Weighting
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The authors are grateful to two anonymous referees for helpful comments and suggestions. We are also thankful to many different organisations and individuals for their valuable support during study initiation and data collection process. Our sincere thanks go to Sigrid Askum and Torun Risnes from Norwegian association of local and regional authorities; Anders Vege from Norwegian Institute of Public Health; Anders Grimsmo from Norwegian Health Network; research assistants, Tord Lauvland Bjornevik and Nina Lunde; to all participating municipalities.
This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 634288. The funder had no role in the design of the study; in the collection, analysis, and interpretation of data; and in writing the manuscript. The content of this report reflects only the SELFIE groups' views and the European Commission is not liable for any use that may be made of the information contained herein.
Department of Economics, University of Bergen, Postboks 7802, 5020, Bergen, Norway
M. Kamrul Islam & Jan Erik Askildsen
Department of Social Sciences, NORCE Norwegian Research Centre, Bergen, Norway
M. Kamrul Islam
Research Unit for General Practice, NORCE Norwegian Research Centre, Bergen, Norway
Sabine Ruths & Kristian Jansen
Department of Global Public Health and Primary Care, University of Bergen, Bergen, Norway
Sabine Ruths
Department of Nursing homes, Municipality of Bergen, Bergen, Norway
Kristian Jansen
Department of Comparative Politics, University of Bergen, Bergen, Norway
Runa Falck
School of Health Policy and Management, Erasmus University Rotterdam, Rotterdam, the Netherlands
Maureen Rutten-van Mölken
Jan Erik Askildsen
MKI, SR, MRvM and JEA contributed to the study conception and design. MKI wrote the first draft of the manuscript. All authors made substantial contributions to the revision of the manuscript for important intellectual content. SR obtained approvals. SR, RF, KJ and JEA acquired data. MKI conducted the statistical analyses. MKI and JEA contributed to the interpretation of the results. All authors reviewed, edited, read and approved the final version of the manuscript.
Correspondence to M. Kamrul Islam.
The Regional Committee for Medical and Health Research Ethics has approved this study (2017/632–3). The study was conducted in accordance with the provisions of the Declaration of Helsinki (1996) and Good Clinical Practice guidelines. All patients provided written informed consent before any study procedure was performed. The collected data have been stored on a separate SAFE server ("Clinton") at the University of Bergen, with quality assured firewall and restricted, password-protected access. Only researchers associated with the project have access to these data. No one has access to the data in identifiable form.
Islam, M.K., Ruths, S., Jansen, K. et al. Evaluating an integrated care pathway for frail elderly patients in Norway using multi-criteria decision analysis. BMC Health Serv Res 21, 884 (2021). https://doi.org/10.1186/s12913-021-06805-6
Frail elderly
Continuity of care
Multi-criteria decision analysis
Multi-morbidity
Quasi-experimental design
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\begin{document}
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\maketitle
\begin{abstract} We establish the twisted functoriality in nonabelian Hodge theory in positive characteristic. As an application, we obtain a purely algebraic proof of the fact that the pullback of a semistable Higgs bundle with vanishing Chern classes is again semistable. \end{abstract} \section{Introduction} In the classical nonabelian Hodge theory \cite{Sim92}, one has the following Simpson correspondence: Let $X$ be a compact K\"{a}hler manifold. There is an equivalence of categories $$ C^{-1}_{X}: \mathrm{HIG}(X)\to \mathrm{MIC}(X), $$ where $\mathrm{HIG}(X)$ is the category of polystable Higgs bundles over $X$ with vanishing first two Chern classes and $\mathrm{MIC}(X)$ is the category of semisimple flat bundles over $X$. The equivalence is independent of the choice of a background K\"{a}hler metric, and the following functoriality holds: Let $f: Y\to X$ be a morphism of compact K\"{a}hler manifolds. Then for any $(E,\theta)\in \mathrm{HIG}(X)$, one has a natural isomorphism in $\mathrm{MIC}(Y)$ \begin{equation}\label{functoriality over C} C_{Y}^{-1}f^*(E,\theta)\cong f^*C^{-1}_{X}(E,\theta). \end{equation} In the nonabelian Hodge theory in positive characteristic \cite{OV}, Ogus-Vologodsky established an analogue of \ref{functoriality over C} for derived categories, with the $W_2$-lifting assumption on $f$ (see Theorem 3.22 \cite{OV}). In a recent preprint \cite{La2}, A. Langer proved the equality \ref{functoriality over C} under the assumption that the $W_2(k)$-lifting of $f$ is good (see Definition 5.1 and Theorem 5.3 in loc. cit.). However, such an assumption on the lifting of $f$ is quite restrictive.
Let $k$ be a perfect field of characteristic $p>0$. Let $X$ be a smooth variety over $k$ and $D$ a reduced normal crossing divisor in $X$. One forms the log smooth variety $X_{\log}$ whose log structure is the one determined by $D$. Equip $k$ and $W_2(k)$ with the trivial log structure. Assume that the log morphism $X_{\log}\to k$ is liftable to $W_2(k)$. Choose and then fix such a lifting $\tilde X_{\log}$. Then one has the inverse Cartier transform \footnote{Theorem 6.1 \cite{LSYZ19} deals with only the case of SNCD. However, a simple \'{e}tale descent argument extends the construction to the reduced NCD case.} (which is in general not an equivalence of categories without further condition on the singularities of modules along $D$) $$ C^{-1}_{X_{\log}\subset \tilde X_{\log}}: \mathrm{HIG}_{\leq p-1}(X_{\log}/k)\to \mathrm{MIC}_{\leq p-1}(X_{\log}/k). $$ Let $Y_{\log}=(Y,B)$ be a log smooth variety like above, together with a $W_2(k)$-lifting $\tilde Y_{\log}$. Our main result is the following analogue of \ref{functoriality over C} in positive characteristic: \begin{theorem}\label{main result} Notion as above. Then for any object $(E,\theta)\in \mathrm{HIG}_{\leq p-1}(X_{\log}/k)$, one has a natural isomorphism $$ C_{Y_{\log}\subset \tilde Y_{\log}}^{-1}f^{\circ}(E,\theta)\cong f^*C^{-1}_{X_{\log}\subset \tilde X_{\log}}(E,\theta), $$ where $f^{\circ}(E,\theta)$ is the twisted pullback of $(E,\theta)$. \end{theorem} The twisted pullback of $(E,\theta)$ refers to a certain deformation of $f^*(E,\theta)$ along the obstruction class of lifting $f$ over $W_2(k)$. When the obstruction class vanishes, the twisted pullback is just the usual pullback. See \S2 for details. Hence, one has the following immediate consequence. \begin{corollary}\label{main cor} Let $f: Y_{\log}\to X_{\log}$ be a morphism of log smooth varieties over $k$. Assume $f$ is liftable to $W_2(k)$. Then for any object $(E,\theta)\in \mathrm{HIG}_{\leq p-1}(X_{\log}/k)$, one has a natural isomorphism in $\mathrm{MIC}_{\leq p-1}(Y'_{\log}/k)$ $$ C_{Y_{\log}\subset \tilde Y_{\log}}^{-1}f^{*}(E,\theta)\cong f^*C^{-1}_{X_{\log}\subset \tilde X_{\log}}(E,\theta). $$ \end{corollary} The notion of \emph{twisted pullback} and the corresponding \emph{twisted functoriality} as exhibited in Theorem \ref{main result} was inspired by the work of Faltings in the $p$-adic Simpson correspondence \cite{Fa05}. It is a remarkable fact that char $p$ and $p$-adic Simpson correspondences have many features in common. As an application, we obtain the following result. \begin{theorem}\label{semistability} Let $k$ be an algebraically closed field and $f: (Y,B)\to (X,D)$ a morphism between smooth projective varieties equipped with normal crossing divisors over $k$. Let $(E,\theta)$ be a semistable logarithmic Higgs bundles with vanishing Chern classes over $(X,D)$. If either $\textrm{char}(k)=0$ or $\textrm{char}(k)=p>0$, $f$ is $W_2(k)$-liftable and ${\rm rank}(E)\leq p$, then the logarithmic Higgs bundle $f^*(E,\theta)$ over $(Y,B)$ is also semistable with vanishing Chern classes.
\end{theorem} For $\textrm{char}(k)=0$ and $D=\emptyset$, the result is due to C. Simpson by transcendental means \cite{Sim92}. Our approach is to deduce it from the char $p$ statement by mod $p$ reduction and hence is purely algebraic.
\section{Twisted pullback} We assume our schemes are all noetherian. Let $(R,M)$ be an affine log scheme. Let $f: Y\to X$ be a morphism of log smooth schemes over $R$. Fix an $r\in {\mathbb N}$. Choose and then fix an element $\tau\in \mathrm{Ext}^1(f^*\Omega_{X/R},{\mathcal O}_Y)$. The aim of this section is to define the twisted pullback along $\tau$ as a functor $$ \mathrm{TP}_{\tau}: \mathrm{HIG}_{\leq r}(X/R)\to \mathrm{HIG}_{\leq r}(Y/R), $$ under the following assumption on $r$ \begin{assumption}\label{basic assumption on r}
$r!$ is invertible in $R$. \end{assumption}
Let $\Omega_{X/R}$ be the sheaf of relative logarithmic K\"{a}hler differentials and $T_{X/R}$ be its ${\mathcal O}_X$-dual. They are locally free of rank $\dim X-\dim R$ by log smoothness. The symmetric algebra $\mathrm{Sym}^{\bullet}T_{X/R}=\bigoplus_{k\geq 0} \mathrm{Sym}^kT_{X/R}$ on $T_{X/R}$ is ${\mathcal O}_X$-algebra, and one has the following morphisms of ${\mathcal O}_X$-algebras whose composite is the identity: $$ {\mathcal O}_X\to \mathrm{Sym}^{\bullet}T_{X/R}\to {\mathcal O}_X. $$ It defines the zero section of the natural projection $\Omega_{X/R}\to X$, where we view $\Omega_{X/R}$ as a vector bundle over $X$ (see Ex 5.18, Ch. II \cite{Ha77}). Set $${\mathcal A}_r:=\mathrm{Sym}^{\bullet}(T_{X/R})/\mathrm{Sym}^{\geq r+1}(T_{X/R}),$$ which is nothing but the structure sheaf of the closed subscheme $(r+1)X$ of $\Omega_{X/R}$ supported along the zero section. In below, we shall use the notations ${\mathcal A}_r$ and ${\mathcal O}_{(r+1)X}$ interchangeably. Note as ${\mathcal O}_X$-module, ${\mathcal A}_{r}={\mathcal O}_X\oplus T_{X/R}\oplus \cdots \oplus \mathrm{Sym}^{r}T_{X/R}$. The following lemma is well-known. \begin{lemma}\label{correspondence} The category of nilpotent (quasi-)coherent Higgs modules over $X/R$ of exponent $\leq r$ is equivalent to the category of (quasi-)coherent ${\mathcal O}_{(r+1)X}$-modules. \end{lemma} \begin{proof} The natural inclusion $\iota: X\to \Omega_{X/R}$ of zero section induces an equivalence of categories between the category of sheaves of abelian groups over $X$ and the category of sheaves of abelian groups over $\Omega_{X/R}$ whose support is contained in the zero section. Let $E$ be a sheaf of abelian groups over $X$. It has a Higgs module structure if it has \begin{itemize} \item [(i)] a ring homomorphism $\theta^0:{\mathcal O}_X\to {\rm End}(E)$; \item [(ii)] an ${\mathcal O}_X$-linear homomorphism $\theta^1: T_{X/R}\to {\rm End}_{{\mathcal O}_X}(E)$. \end{itemize} Since $\mathrm{Sym}^{\bullet}T_{X/R}$ is generated by $T_{X/R}$ as ${\mathcal O}_X$-algebra, $\theta^0$ and $\theta^1$ together extend to a ring homomorphism $$ \theta^{\bullet}: \mathrm{Sym}^{\bullet}T_{X/R}\to {\rm End}_{{\mathcal O}_X}(E)\subset {\rm End}(E). $$ If $\theta^1$ is nilpotent of exponent $\leq r$, then $\mathrm{Sym}^{\geq r+1}(T_{X/R})\subset \mathrm{Ann}(E)$. Therefore, we obtain an ${\mathcal A}_r$-module structure on $E$. So we obtain a sheaf of ${\mathcal O}_{(r+1)X}$-module. As $E$ is (quasi-)coherent as ${\mathcal O}_X$-module, it is (quasi-)coherent as ${\mathcal O}_{(r+1)X}$-module. Conversely, for a quasi-coherent ${\mathcal O}_{{\mathcal A}_r}$-module $E$, one obtains a ring homomorphism $$ {\mathcal A}_r\to {\rm End}(E). $$ Restricting it to the degree zero part, one obtains the ${\mathcal O}_X$-module structure on $E$. While restricting to the degree one component, one obtains a morphism of sheaf of abelian groups $$ \theta: T_{X/R}\to {\rm End}(E), v\mapsto \theta_v:=\textrm{the multiplication by}\ v. $$ Since for any $v\in T_{X/R}$, $v^{r+1}=0$ in ${\mathcal A}_r$, it follows $\theta^{r+1}=0$, that is the exponent of $\theta\leq r$. For any $f\in{\mathcal O}_X, v\in T_{X/R}$ and any $e\in E$, one verifies that $$ \theta_v(fe)=\theta_{fv}(e)=f\theta_v(e), $$ which means that the image of $\theta$ is contained in ${\rm End}_{{\mathcal O}_X}(E)$. The obtained ${\mathcal O}_X$-module is nothing but the pushforward of $E$ along the composite $(r+1)X\nrightarrow \Omega_{X/R}\to X$ which is finite. Therefore, $E$ is (quasi-)coherent as ${\mathcal O}_X$-module if it is (quasi-)coherent as ${\mathcal O}_{(r+1)X}$-module. \end{proof} \begin{remark}\label{equivalence between A-module and Higgs module} An $f^*\Omega_{X/R}$-Higgs module is a pair $(E,\theta)$ where $E$ is an ${\mathcal O}_Y$-module and $\theta: E\to E\otimes f^*\Omega_{X/R}$ is an ${\mathcal O}_Y$-linear morphism satisfying $\theta\wedge \theta=0$. A modification of the above argument shows that the category of nilpotent (quasi-)coherent $f^*\Omega_{X/R}$-Higgs modules is equivalent to the category of (quasi-)coherent $f^*{\mathcal A}_r$-modules. \end{remark}
{\bf $1^{st}$ construction:} For an $r$ satisfying Assumption \ref{basic assumption on r}, we have a natural morphism: $$ \exp: H^1(Y,f^*T_{X/R})\to H^1(Y, (f^*{\mathcal A}_r)^*), \tau\mapsto \exp(\tau)=1+\tau+\cdots+\frac{\tau^r}{r!}, $$ where $(f^*{\mathcal A}_r)^*$ is the unit group of $f^*{\mathcal A}_r$. An element of $f^*{\mathcal A}_r$ is invertible iff its image under $f^*{\mathcal A}_r\to {\mathcal O}_{Y}$ is invertible. So we obtain an $f^*{\mathcal A}_r$-module ${\mathcal F}^{r}_{\tau}$ of rank one. We introduce an intermediate category $\mathrm{HIG}_{\leq r}(f^*\Omega_{X/R})$, which is the category of nilpotent quasi-coherent $f^*\Omega_{X/R}$-Higgs modules of exponent $\leq r$. We define the functor $$ \mathrm{TP}^{{\mathcal F}}_{\tau}: \mathrm{HIG}_{\leq r}(X/R)\to \mathrm{HIG}_{\leq r}(f^*\Omega_{X/R}) $$ as follows: For an $E\in \mathrm{HIG}_{\leq r}(X/R)$, define $$ \mathrm{TP}^{{\mathcal F}}_{\tau}(E):={\mathcal F}^r_{\tau}\otimes_{f^*{\mathcal A}_r}f^{*}E $$ as $f^*{\mathcal A}_r$-module. Next, for a morphism $\phi: E_1\to E_2$ in $\mathrm{HIG}_{\leq r}(X/R)$, $$\mathrm{TP}^{{\mathcal F}}_{\tau}(\phi):=id\otimes f^*\phi: \mathrm{TP}^{{\mathcal F}}_{\tau}(E_1)\to \mathrm{TP}^{{\mathcal F}}_{\tau}(E_2)$$ is a morphism of $f^*{\mathcal A}_r$-modules. One has the natural functor from $\mathrm{HIG}_{\leq r}(f^*\Omega_{X/R})$ to $\mathrm{HIG}_{\leq r}(\Omega_{Y/R})$ induced by the differential morphism $f^*\Omega_{X/R}\to \Omega_{Y/R}$. We define the functor $\mathrm{TP}^1_{\tau}$ to be composite of functors $$ \mathrm{HIG}_{\leq r}(X/R)\stackrel{\mathrm{TP}^{{\mathcal F}}_{\tau}}{\longrightarrow}\mathrm{HIG}_{\leq r}(f^*\Omega_{X/R})\to \mathrm{HIG}_{\leq r}(Y/R). $$ This is how Faltings \cite{Fa05} defines twisted pullback in the $p$-adic setting, at least for those small $\tau$s. \\
{\bf $2^{nd}$ construction:} This is based on the method of \emph{exponential twisting} \cite{LSZ15}, whose basic construction is given as follows:
{\itshape Step 0}: Take an open affine covering $\{U_{\alpha}\}_{\alpha\in \Lambda}$ of $X$ as well as an open affine covering $\{V_{\alpha}\}_{\alpha\in \Lambda}$ of $Y$ such that $f: V_{\alpha}\to U_{\alpha}$. Let $\{\tau_{\alpha\beta}\}$ be a Cech representative of $\tau$. That is, $\tau_{\alpha\beta}\in \Gamma(V_{\alpha\beta}, f^*T_{X/R})$ satisfying the cocycle relation $$ \tau_{\alpha\gamma}=\tau_{\alpha\beta}+\tau_{\beta\gamma}. $$
{\itshape Step 1}: Let $(E,\theta)$ be a nilpotent Higgs module over $X$, whose exponent of nilpotency satisfies Assumption \ref{basic assumption on r}. For any $\alpha$, set $(E_{\alpha},\theta_{\alpha})=(E,\theta)|_{U_{\alpha}}$. Then one forms the various local Higgs modules $\{(f^*E_{\alpha},f^*\theta_{\alpha})\}$ via the usual pullback. \\
{\itshape Step 2}: Define $$ G_{\alpha\beta}=\exp(\tau_{\alpha\beta}\cdot f^*\theta)=\sum_{i\geq 0}\frac{(\tau_{\alpha\beta}\cdot f^*\theta)^n}{n!}. $$
The expression makes sense since each term $\frac{(\tau_{\alpha\beta}\cdot f^*\theta)^n}{n!}$ is well defined by assumption. Obviously, $G_{\alpha\beta}\in \mathrm{Aut}_{{\mathcal O}_{Y}}(f^*E|_{V_{\alpha\beta}})$. Because of the cocycle relation, $\{G_{\alpha\beta}\}$ satisfies the cocycle relation $$ G_{\alpha\gamma}=G_{\beta\gamma}G_{\alpha\beta}. $$ Then we use the set of local isomorphism $\{G_{\alpha\beta}\}$ to glue the local $\Omega_{Y/R}$-Higgs modules $\{(f^*E_{\alpha},f^*\theta_{\alpha})\}$, to obtain a new Higgs module over $Y$. The verification details are analogous to \S2.2 \cite{LSZ15}. It is tedious and routine to verify the glued Higgs module, up to natural isomorphism, is independent of the choice of affine coverings and Cech representatives of $\tau$. We denote it by $\mathrm{TP}^2_{\tau}(E)$. For a morphism $\phi: E_1\to E_2$ of Higgs modules, it is not difficult to see that $f^*\phi$ induces a morphism $\mathrm{TP}^2_{\tau}(\phi):\mathrm{TP}^2_{\tau}(E_1)\to \mathrm{TP}^2_{\tau}(E_2)$. \begin{proposition}\label{Faltings twisted pullback is exponential twisting} The two functors $\mathrm{TP}^1_{\tau}$ and $\mathrm{TP}^2_{\tau}$ are naturally isomorphic. \end{proposition} \begin{proof}
One uses the equivalence in Remark \ref{equivalence between A-module and Higgs module}. It suffices to notice that the element $\exp(\tau_{\alpha\beta})\in f^*{\mathcal A}_r$ has its image $G_{\alpha\beta}$ in $\mathrm{Aut}_{{\mathcal O}_{Y}}(f^*E|_{V_{\alpha\beta}})$. \end{proof} By the above proposition, we set $\mathrm{TP}_{\tau}$ to be either of $\mathrm{TP}^i_{\tau}, i=1.2$. \begin{proposition} The functor $\mathrm{TP}_{\tau}$ has the following properties: \begin{itemize}
\item [(i)] it preserves rank;
\item [(ii)] it preserves direct sum;
\item [(iii)] Let $E_i, i=1,2$ be two nilpotent Higgs modules over $X/R$ whose exponents of nilpotency satisfies $(r_1+r_2)!$ being invertible in $R$. Then there is a canonical isomorphism of Higgs modules over $Y$:
$$
\mathrm{TP}_{\tau}(E_1\otimes E_2) \cong \mathrm{TP}_{\tau}(E_1)\otimes\mathrm{TP}_{\tau}(E_2).
$$ \end{itemize} \end{proposition} \begin{proof} The first two properties are obvious. To approach (iii), one uses the second construction. Note that when $\tau=0$, it is nothing but the fact $f^*(E_1\otimes E_2)=f^*E_1\otimes f^*E_2$. When the exponents $r_i, i=1,2$ satisfies the condition, one computes that
$$
\exp(\tau\cdot (f^*\theta_1\otimes id+id\otimes f^*\theta_2))=\exp(\tau\cdot f^*\theta_1\otimes id)\exp(id\otimes\tau\cdot f^*\theta_2),
$$ using the equality
$$
(f^*\theta_1\otimes id)(id\otimes f^*\theta_2)=(id\otimes f^*\theta_2)(f^*\theta_1\otimes id)=f^*\theta_1\otimes f^*\theta_2.
$$ \end{proof}
To conclude this section, we shall point out that there is one closely related construction that works for all $r\in {\mathbb N}$. \\
{\bf $3^{rd}$ construction:} Note that the element $\tau\in \mathrm{Ext}^1(f^*\Omega_X,\Omega_Y)\cong \mathrm{Ext}^1({\mathcal O}_Y,f^*T_{X/R})$ corresponds to an extension of ${\mathcal O}_Y$-modules $$ 0\to f^*T_{X/R}\to {\mathcal E}_{\tau}\stackrel{pr}{\to} {\mathcal O}_{Y} \to 0. $$ Notice that ${\mathcal E}_{\tau}$ admits a natural $f^*\mathrm{Sym}^{\bullet}T_{X/R}$-module structure: In degree zero, this is ${\mathcal O}_{Y}$-structure; in degree one, $$ f^*T_{X/R}\otimes_{{\mathcal O}_{Y}}{\mathcal E}_{\tau}\stackrel{id\otimes pr}{\longrightarrow} f^*T_{X/R}\otimes_{{\mathcal O}_{Y}}{\mathcal O}_{Y}=f^*T_{X/R}\subset {\mathcal E}_{\tau}, $$ and therefore $\theta: f^*T_{X/R}\to {\rm End}_{{\mathcal O}_{Y}}({\mathcal E}_{\tau})$. By construction, $\theta\neq 0$ but $\theta^2=0$. For any $r\in {\mathbb N}$, set $$ {\mathcal E}^r_{\tau}:=\mathrm{Sym}^r{\mathcal E}_{\tau}. $$ The proof of the next lemma is straightforward. \begin{lemma}\label{basic property of E^r}
For any $r\in {\mathbb N}$, ${\mathcal E}^r_{\tau}$ is a nilpotent $f^*\Omega_{X/R}$-Higgs bundle of exponent $r$. It admits a filtration $F^{\bullet}$ of $f^*\Omega_{X/R}$-Higgs subbundles:
$$
{\mathcal E}^r_{\tau}=F^0\supset F^1\supset\cdots \supset F^r\supset 0,
$$
whose associated graded $Gr_{F^{\bullet}}E^r_{\tau}$ is naturally isomorphic to $f^*{\mathcal A}_r$. When $\tau=0$, ${\mathcal E}^r_{\tau}=f^*{\mathcal A}_r$ as $f^*\Omega_{X/R}$-Higgs bundle. \end{lemma} By the lemma, ${\mathcal E}^r_{\tau}$ is an $f^*{\mathcal A}_r$-module of rank one. Therefore, one may replace the tensor module in the definition of $\mathrm{TP}^{{\mathcal F}}_{\tau}$ with ${\mathcal E}^{r}_{\tau}$. This defines a new functor $\mathrm{TP}^{{\mathcal E}}_{\tau}$ and hence the third twisted pullback functor $\mathrm{TP}^3_{\tau}$.
\begin{remark}
When one is interested only in coherent objects, one may drop the nilpotent condition in the construction. This is because by Cayley-Hamilton, there is an element of form
$v^r-a_1v^{r-1}+\cdots+(-1)^ra_r\in \mathrm{Sym}^{\bullet}T_{X/R}$ annihilating $E$, so that $\mathrm{Sym}^{\bullet}T_{X/R}$-module structure on $E$ factors though $\mathrm{Sym}^{\bullet}T_{X/R}\to {\mathcal A}_r$. \end{remark} We record the following statement for further study. \begin{proposition} Assume $r\in {\mathbb N}$ satisfy Assumption \ref{basic assumption on r}. Then as $f^*{\mathcal A}_r$-modules, \begin{itemize}
\item [(i)] ${\mathcal F}^r_{\tau}\cong {\mathcal E}^r_{\tau}$ for $r\leq 1$;
\item [(ii)] ${\mathcal F}^r_{\tau}\ncong {\mathcal E}^r_{\tau}$ for $r>1$. \end{itemize} \end{proposition} \begin{proof} Obviously, ${\mathcal F}^0_{\tau}\cong {\mathcal E}^0_{\tau}\cong {\mathcal O}_Y$. Assume $r\geq 1$. We illustrate our proof by looking at the case of $X/R$ being a relative curve. We describe ${\mathcal E}^r_{\tau}$ in terms of local data: Take $r=1$ first. Let $U_{\alpha}$ be an open subset of $X$ with $\partial_{\alpha}$ a local basis of $\Gamma(U_{\alpha}, T_{X/R})$. Assume that $V_{\alpha}$ to be an open subset of $Y$ such that $f: V_{\alpha}\to U_{\alpha}$. We may assume the gluing functions between two different local basis are identity. Let $\{\tau_{\alpha\beta}\}$ be a Cech representative of $\tau$. Write $\tau_{\alpha\beta}=a_{\alpha\beta}f^*\partial_{\alpha\beta}$. Then ${\mathcal E}_{\tau}$ is the ${\mathcal O}_{Y}$-module obtained by gluing $\{{\mathcal O}_{V_{\alpha}}\oplus f^*T_{U_{\alpha}/R}\}$ via the following gluing matrix: $$
\left(
\begin{array}{c}
1 \\
f^*\partial_{\alpha} \\
\end{array}
\right)=\left(
\begin{array}{cc}
1 & a_{\alpha\beta} \\
0 & 1 \\
\end{array}
\right)\left(
\begin{array}{c}
1 \\
f^*\partial_{\beta} \\
\end{array}
\right). $$
Under the assumption for $r$, ${\mathcal E}^r_{\tau}$ is obtained by gluing $$\{f^*{\mathcal A}_{r}|_{U_{\alpha}}={\mathcal O}_{V_{\alpha}}\oplus f^*T_{U_{\alpha}/R}\oplus\cdots \oplus f^*T^{\otimes r}_{U_{\alpha}/R}\}$$ via the gluing matrix: $$ \left(
\begin{array}{c}
\frac{1}{r!} \\
\frac{f^*\partial_{\alpha}}{(r-1)!} \\
\vdots\\
f^*\partial^{r-1}_{\alpha} \\
f^*\partial^r_{\alpha} \\
\end{array} \right)=\left(
\begin{array}{ccccc}
1& a_{\alpha\beta} & \frac{a^2_{\alpha\beta}}{2!} & \ldots & \frac{a^r_{\alpha\beta}}{r!} \\
0& 1 & a_{\alpha\beta} & \ldots & \frac{a^{r-1}_{\alpha\beta}}{(r-1)!} \\
\vdots & \vdots & \ddots & \ddots & \vdots \\
0 & 0 & \ldots & 1 & a_{\alpha\beta} \\
0 & 0 & \ldots & \ldots & 1 \\
\end{array}
\right)\left(
\begin{array}{c}
\frac{1}{r!} \\
\frac{f^*\partial_{\beta}}{(r-1)!} \\
\vdots\\
f^*\partial^{r-1}_{\beta} \\
f^*\partial^r_{\beta} \\
\end{array} \right). $$
As comparison, ${\mathcal F}^r_{\tau}$ is obtained by gluing $\{f^*{\mathcal A}_{r}|_{U_{\alpha}}\}$ via the following transition functions $$ \left( \begin{array}{c} 1 \\ f^*\partial_{\alpha} \\ \vdots\\ f^*\partial^{r-1}_{\alpha} \\ f^*\partial^r_{\alpha} \\ \end{array} \right)=\left( \begin{array}{ccccc} 1& a_{\alpha\beta} & \frac{a^2_{\alpha\beta}}{2!} & \ldots & \frac{a^r_{\alpha\beta}}{r!} \\ 0& 1 & a_{\alpha\beta} & \ldots & \frac{a^{r-1}_{\alpha\beta}}{(r-1)!} \\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & 0 & \ldots & 1 & a_{\alpha\beta} \\ 0 & 0 & \ldots & \ldots & 1 \\ \end{array} \right)\left( \begin{array}{c} 1 \\ f^*\partial_{\beta} \\ \vdots\\ f^*\partial^{r-1}_{\beta} \\ f^*\partial^r_{\beta} \\ \end{array} \right). $$ Therefore, ${\mathcal F}^r_{\tau}$ and ${\mathcal E}^{r}_{\tau}$ are isomorphic as ${\mathcal O}_Y$-modules. However, when $r\geq 2$, the Higgs structures of these two bundles differ: For ${\mathcal F}^r_{\tau}$, the Higgs field along $\partial_{\alpha}$ is given by $$ \theta_{\partial_\alpha}\left( \begin{array}{c} 1 \\ f^*\partial_{\alpha} \\ \vdots\\ f^*\partial^{r-1}_{\alpha} \\ f^*\partial^r_{\alpha} \\ \end{array} \right)=\left( \begin{array}{ccccc} 0& 1 & 0 & \ldots & 0 \\ 0& 0 & 1 & \ldots & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & 0 & \ldots & 0 & 1 \\ 0 & 0 & \ldots & \ldots & 0 \\ \end{array} \right)\left( \begin{array}{c} 1 \\ f^*\partial_{\alpha} \\ \vdots\\ f^*\partial^{r-1}_{\alpha} \\ f^*\partial^r_{\alpha} \\ \end{array} \right), $$ while for Higgs field action for ${\mathcal E}^r_{\tau}$ is given by $$ \theta_{\partial_\alpha}\left( \begin{array}{c} \frac{1}{r!} \\ \frac{f^*\partial_{\alpha}}{(r-1)!} \\ \vdots\\ f^*\partial^{r-1}_{\alpha} \\ f^*\partial^r_{\alpha} \\ \end{array} \right)=\left( \begin{array}{ccccc} 0& \frac{1}{r} & 0 & \ldots & 0 \\ 0& 0 & \frac{1}{r-1} & \ldots & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & 0 & \ldots & 0& 1 \\ 0 & 0 & \ldots & \ldots & 0 \\ \end{array} \right)\left( \begin{array}{c} \frac{1}{r!} \\ \frac{f^*\partial_{\alpha}}{(r-1)!} \\ \vdots\\ f^*\partial^{r-1}_{\alpha} \\ f^*\partial^r_{\alpha} \\ \end{array} \right). $$
\end{proof}
\section{Twisted functoriality} Now we come back to the setting in \S1. First we make the following \begin{definition}\label{twisted pullback} Let $k$, $f: Y_{\log}\to X_{\log}$ and $\tilde X_{\log}, \tilde Y_{\log}$ be as in \S1. For a Higgs module $(E,\theta)\in \mathrm{HIG}_{\leq p-1}(X_{\log}/k)$. Then the twisted pullback $f^\circ(E,\theta)$ is defined to be $\mathrm{TP}_{ob(f)}(E,\theta)$, where $ob(f)$ is the obstruction class of lifting $f$ to a morphism $\tilde Y_{\log}\to \tilde X_{\log}$ over $W_2(k)$. \end{definition} Assume that $f$ admits a $W_2(k)$-lifting $\tilde f$. In Langer's proof of functoriality Theorem 5.3 \cite{La19}, the existence of local logarithmic Frobenius liftings $F_{\tilde X_{\log}}$ and $F_{\tilde Y_{\log}}$ such that $F_{\tilde X_{\log}}\circ \tilde f=\tilde f\circ F_{\tilde Y_{\log}}$ is crucial-this is where the condition of $\tilde f$ being good enters. However, one notices that any local logarithmic Frobenius liftings on $\tilde X$ and $\tilde Y$ commute with $\tilde f$ \emph{up to homotopy}. A heuristic reasoning shows that this homotopy should be intertwined with the homotopies caused by local logarithmic Frobenius liftings of both $\tilde X$ and $\tilde Y$, as well as the one caused by local liftings of the morphism (no $W_2$-lifting on $f$ is assumed any more). Turning this soft homotopy argument into exact differential calculus in positive characteristic yields the proof for the claimed twisted functoriality.
To start with proof of Theorem \ref{main result}, we take an \'{e}tale covering $\mathcal X=\coprod_i X_i\to X$ with $X_i$ affine and the pullback of $D$ along each $X_i\to X$ simple normal crossing. Then we take an \'{e}tale covering $\pi: \mathcal Y=\coprod_i Y_i\to Y$ with similar properties and $f$ restricts to a local morphism $f_i: Y_{i,\log}\to X_{i,\log}$ for each $i$. For each $i$, we choose logarithmic Frobenius lifting over $W_2(k)$ $$ F_{\tilde X_{i,\log}}: \tilde X_{i,\log}\to \tilde X_{i,\log},\quad F_{\tilde Y_{i,\log}}: \tilde Y_{i,\log}\to \tilde Y_{i,\log}, $$ and also a $W_2(k)$-lift $\tilde f_i: \tilde Y_{i,\log}\to \tilde X_{i,\log}$. Such local lifts exist. Set $$ (V_1,\nabla_1)=C_{Y_{\log}\subset \tilde Y_{\log}}^{-1}f^{\circ}(E,\theta),\quad (V_2,\nabla_2)=f^*C^{-1}_{X_{\log}\subset \tilde X_{\log}}(E,\theta). $$ In below, we exhibit an isomorphism between $(V_i,\nabla_i),i=1,2$ after pulling back to the \'{e}tale covering ${\mathcal Y}$ which satisfies the descent condition. The whole proof is therefore divided into two steps.\\
{\itshape Step 1: Isomorphism over ${\mathcal Y}$}\\ As ${\mathcal Y}$ is a disjoint union of open affine log schemes $\{Y_{i,\log}\}$s, it suffices to construct an isomorphism for each open affine. In the foregoing argument, we drop out the subscript $i$ everywhere. Notice first that the two morphisms $\tilde f^*\circ F_{\tilde X_{\log}}^*$ and $F_{\tilde Y_{\log}}^*\circ \tilde f^*$ coincide after reduction modulo $p$. Thus, it defines an element $$ \nu_{f}\in {\rm Hom}_{{\mathcal O}_{X}}(\Omega_{X_{\log}/k},{\mathcal O}_Y) $$ such that $$ \nu_f\circ d=\frac{1}{p}(F_{\tilde Y_{\log}}^*\circ \tilde f^*-\tilde f^*\circ F_{\tilde X_{\log}}^*). $$ So we get $\nu_f\cdot \theta\in \Gamma(Y,{\rm End}_{{\mathcal O}_Y}(g^*E))$, where $g=F_X\circ f=f\circ F_Y$. \begin{lemma} $\exp(\nu_f\cdot \theta)$ defines an isomorphism $(V_1,\nabla_1)\to (V_2,\nabla_2)$. That is, there is a commutative diagram: $$\CD
V_1 @> \exp(\nu_f\cdot \theta) >> V_2 \\
@V \nabla_1 VV @VV \nabla_2V \\
V_1\otimes \Omega_{Y_{\log}/k} @>\exp(\nu_f\cdot \theta)\otimes id>> V_2\otimes \Omega_{Y_{\log}/k}. \endCD$$ \end{lemma} \begin{proof} Recall that over $Y$, $V_1=V_2=g^*E$. So $\exp(\nu_f\cdot E)$ defines an isomorphism from $V_1$ to $V_2$. Moreover, the connections are given by $$ \nabla_1=\nabla_{can}+(id\otimes \frac{dF_{\tilde Y_{\log}}}{p})(F_Y^*f^*\theta), $$ and respectively by $$ \nabla_2=f^*(\nabla_{can}+(id\otimes \frac{dF_{\tilde X_{\log}}}{p})(F_X^*\theta)). $$ Now we are going to check the commutativity of the above diagram. Take a local section $e\in E$. Then $$ \exp(\nu_f\cdot \theta)\otimes id\circ \nabla_1(g^*e)= \exp(\nu_f\cdot \theta)(id\otimes \frac{dF_{\tilde Y_{\log}}}{p})(F_Y^*f^*\theta(e)). $$ On the other hand, $\nabla_2\circ \exp(\nu_f\cdot \theta)(e)$ equals $$ \exp(\nu_f\cdot \theta)d(\nu_f\cdot \theta)(g^*e)+\exp(\nu_f\cdot \theta)(f^*(id\otimes \frac{dF_{\tilde X_{\log}}}{p})(F_X^*\theta(e))). $$ We take a system of local coordinates $\{x_i\}$ for $\tilde X$ and use the same notion for its reduction modulo $p$. Write $\theta=\sum_i\theta_idx_i$, and $\nu_f=\sum_iu_i\partial_{x_i}$ with $u_i\in {\mathcal O}_Y$. Thus $$ d(\nu_f\cdot \theta)=d(\sum_ig^*\theta_i\cdot u_i). $$ As $d$ is ${\mathcal O}_X$-linear, it equals $$ \sum_ig^*\theta_i\cdot du_i=\sum_ig^*\theta_i\cdot d(\frac{(F_{\tilde Y_{\log}}^*\circ \tilde f^*-\tilde f^*\circ F_{\tilde X_{\log}}^*)(x_i)}{p}). $$ So $d(\nu_f\cdot \theta)(g^*e)=\sum_ig^*\theta_i(e)\cdot \frac{(F_{\tilde Y_{\log}}^*\circ \tilde f^*-\tilde f^*\circ F_{\tilde X_{\log}}^*)(x_i)}{p}$. On the other hand, \begin{eqnarray*} (id\otimes \frac{dF_{\tilde Y_{\log}}}{p})(F_Y^*f^*\theta(e))&=& \sum_i g^*\theta_i(e)\cdot (\frac{id\otimes dF_{\tilde Y_{\log}}}{p})(F_Y^*f^*(dx_i))\\ &=&\sum_ig^*\theta_i(e)\cdot \frac{d(F_{\tilde Y_{\log}}^*\tilde f^*(x_i))}{p}, \end{eqnarray*} and similarly, \begin{eqnarray*} f^*(id\otimes \frac{dF_{\tilde X_{\log}}}{p})(F_X^*\theta(e))&=& \sum_i g^*\theta_i(e)\cdot \frac{f^*d(F^*_{\tilde X_{\log}}(x_i))}{p}\\ &=&\sum_i g^*\theta_i(e)\cdot \frac{d(\tilde f^*F^*_{\tilde X_{\log}}(x_i))}{p}. \end{eqnarray*} This completes the proof.
\end{proof} {\itshape Step 2: Descent condition}\\
In Step 1, we have constructed an isomorphism $\exp(\nu_f\cdot \theta): \pi^*(V_1,\nabla_1)\to \pi^*(V_2,\nabla_2)$ whose restriction to $Y_{i,\log}$ is given by $\exp(\nu_{f_i}\cdot \theta)$. Let $p_i: {\mathcal Y}\times_Y {\mathcal Y}\to {\mathcal Y}, i=1,2$ be two projections. In below, we show that $$ p_1^*(\exp(\nu_f\cdot \theta))=p_2^*(\exp(\nu_f\cdot \theta)). $$ The obstruction class $ob(F_X)$ (resp. $ob(F_Y)$ and $ob(f)$) of lifting $F_X$ (resp. $F_Y$ and $f$) over $W_2$ has its Cech representative landing in $\Gamma(X_{ij},F^*_{X}T_{X_{\log}/k})$ (resp. $\Gamma(Y_{ij},F^*_{Y}T_{Y_{\log}/k})$ and $\Gamma(Y_{ij},f^*T_{X_{\log}/k})$). We have the following natural maps: $$ f^*: H^1(X,F_X^*T_{X_{\log}/k})\to H^1(Y,f^*F_X^*T_{X_{\log}/k})=H^1(Y,g^*T_{X_{\log}/k}), $$ $F_Y^*: H^1(Y,f^*T_{X_{\log}/k})\to H^1(Y,g^*T_{X_{\log}/k})$, and $$ f_*: H^1(Y,F_Y^*T_{Y_{\log}/k})\to H^1(Y,F_Y^*f^*T_{X_{\log}/k})=H^1(Y,g^*T_{X_{\log}/k}), $$ which is induced by $f_*: T_{Y_{\log}/k}\to f^*T_{X_{\log}/k}$. \begin{lemma}\label{cech relation} One has an equality in $\Gamma(Y_{ij},g^*T_{X_{\log}/k})$, where $Y_{ij}=Y_i\times_YY_j$: $$ \nu_{f_i}-\nu_{f_j}=ob(F_Y)_{ij}+ob(f)_{ij}-ob(F_X)_{ij} $$ where we understand the obstruction classes as their images via the natural morphisms. Consequently, there is an equality in $H^1(Y,g^*T_{X_{\log}/k})$: $$ [\nu_{f_i}-\nu_{f_j}]=ob(F_Y)+ob(f)-ob(F_X). $$ \end{lemma} \begin{proof} First, we observe the following identity \begin{eqnarray*} \frac{1}{p}(F^*_{\tilde Y_{i,\log}}\circ \tilde f_i^*-F^*_{\tilde Y_{j,\log}}\circ \tilde f_j^*)&=& \frac{1}{p}[(F^*_{\tilde Y_{i,\log}}-F^*_{\tilde Y_{j,\log}})\circ \tilde f_i^*]+\frac{1}{p}[F^*_{\tilde Y_{j,\log}}\circ (\tilde f_i^*-\tilde f_j^*)]\\ &=& \frac{F^*_{\tilde Y_{i,\log}}-F^*_{\tilde Y_{j,\log}}}{p}\circ f_i^*+F_{Y_{j}}^*\circ \frac{\tilde f_i^*-\tilde f_j^*}{p}. \end{eqnarray*} It follows that \begin{eqnarray*} (\nu_{f_i}-\nu_{f_j})\circ d&=&\frac{1}{p}(F^*_{\tilde Y_{i,\log}}\circ \tilde f_i^*-F^*_{\tilde Y_{j,\log}}\circ \tilde f_j^*)-\frac{1}{p}(\tilde f_i^*\circ F^*_{\tilde X_{i,\log}} -\tilde f_j^*\circ F^*_{\tilde X_{j,\log}})\\ &=& \frac{F^*_{\tilde Y_{i,\log}}-F^*_{\tilde Y_{j,\log}}}{p}\circ f_i^*+F_{Y_{j}}^*\circ \frac{\tilde f_i^*-\tilde f_j^*}{p}-f_j^*\circ\frac{F^*_{\tilde X_{i,\log}}-F^*_{\tilde X_{j,\log}}}{p}-\frac{\tilde f_i^*-\tilde f_j^*}{p}\circ F_{X_i}^*\\ &=&ob(F_Y)_{ij}\circ f_i^*\circ d+F_{Y_{j}}^*\circ ob(f)_{ij}\circ d-f_j^*\circ ob(F_X)_{ij}\circ d. \end{eqnarray*} In the second equality, the last term vanishes because $$ \frac{\tilde f_i^*-\tilde f_j^*}{p}\circ F_{X_i}^*=ob(f)\circ (d F_{X_i}^*)=0. $$ \end{proof} Now we turn the above equality into an equality required in the descent condition. The transition function of $V_1$ is given by $$ a_{ij}:=\exp(ob(F_Y)_{ij}\cdot f_i^*\theta)\cdot\exp (F_{Y_j}^*(ob(f)_{ij}\cdot \theta)), $$ while the transition function for $V_2$ is given by $$ b_{ij}:=f_j^*\exp((ob(F_X)_{ij}\cdot \theta)). $$ Then Lemma \ref{cech relation} implies the commutativity of the following diagram over $Y_{ij}$: $$ \CD
V_1|_{Y_i} @>\exp(\nu_{f_i}\cdot \theta)>> V_2|_{Y_i} \\
@V a_{ij} VV @VVb_{ij}V \\
V_1|_{Y_j} @>>\exp(\nu_{f_j}\cdot \theta)> V_2|_{Y_j}. \endCD $$ The commutativity is nothing but the descent condition for the isomorphism $\exp(\nu_f\cdot\theta)$. So we are done.
\section{Semistability under pullback} Semistablity is not always preserved under pullback. After all, semistability refers to some given ample line bundle and an ample line bundle does not necessarily pulls back to an ample line bundle. Even worse, in the postive characteristic case, there are well-known examples of semistable vector bundles over curves which pull back to unstable bundles under Frobenius morphism.
For a polystable Higgs bundle with vanishing Chern classes in characteristic zero, this is handled by the existence of Higgs-Yang-Mills metric-it is a harmonic bundle by this case and harmonic bundles pulls back to harmonic bundles. Consequently, the pullback of a polystable Higgs bundle with vanishing Chern classes is again polystable with vanishing Chern classes. For a semistable Higgs bundle with vanishing Chern classes, one takes a Jordan-H\"older filtration of the Higgs bundle and the semistability of the pullback follows from that of the polystable case. In the following, we provide a purely algebraic approach to the semistable case. We proceed to the proof of Theorem \ref{semistability}. \begin{proof} We resume the notations of Theorem \ref{semistability}. Since taking Chern class commutes with pullback, the statement about vanishing Chern classes of the pullback is trivial. We focus on the semistability below. In the following discussion, we choose and then fix an arbitrary ample line bundle $L$ (resp. $M$) over $X$ (resp. $Y$). We consider first the char $p$ setting. Fix a $W_2(k)$-lifting $\tilde f: \tilde Y_{\log}=(\tilde Y, \tilde B)\to \tilde X_{\log}=(\tilde X, \tilde D)$. First, we observe that the proof of Theorem A.4 \cite{LSZ} works verbatim for a semistable logarithmic Higgs bundle, so that there exists a filtration $Fil_{-1}$ on $E$ such that $\mathrm{Gr}_{Fil_{-1}}(E,\theta)$ is semistable. Now applying \cite[Theorem A.1]{LSZ}, \cite[Theorem 5.12]{La1} to the \emph{nilpotent} semistable Higgs bundle $\mathrm{Gr}_{Fil_{-1}}(E,\theta)$, we obtain a flow of the following form: $$
\xymatrix{
(E,\theta)\ar[dr]^{Gr_{Fil_{-1}}} && (H_0,\nabla_0)\ar[dr]^{Gr_{Fil_0}} && (H_1,\nabla_1)\ar[dr]^{Gr_{Fil_1}} \\
&(E_0,\theta_0) \ar[ur]^{C_{X_{\log}\subset \tilde X_{\log}}^{-1}} & & (E_1,\theta_1) \ar[ur]^{C_{X_{\log}\subset \tilde X_{\log}}^{-1}}&&
\cdots,
}
$$ in which each Higgs term in bottom is semistable. Next, because of Corollary \ref{main cor}, we may obtain the pullback flow as follows: $$
\xymatrix{
f^*(E,\theta)\ar[dr]^{Gr_{f^*Fil_{-1}}} && f^*(H_0,\nabla_0)\ar[dr]^{Gr_{f^*Fil_0}} && f^*(H_1,\nabla_1)\ar[dr]^{Gr_{f^*Fil_1}} \\
&f^*(E_0,\theta_0) \ar[ur]^{C_{Y_{\log}\subset \tilde Y_{\log}}^{-1}} & & f^*(E_1,\theta_1) \ar[ur]^{C_{Y_{\log}\subset \tilde Y_{\log}}^{-1}}&&
\cdots
}
$$ Now as the Higgs terms $(E_i,\theta_i)$s in the first flow are semistable of the same rank and of vanishing Chern classes, the set $\{(E_i,\theta_i)\}_{i\geq 0}$ form a bounded family. So the set $\{f^*(E_i,\theta_i)\}_{i\geq 0}$ also forms a bounded family. In particular, the degrees of subsheaves in $\{f^*E_i\}_{i\geq 0}$ have an upper bound $N$. Suppose $f^*(E,\theta)$ is unstable, that is, there exists a saturated Higgs subsheaf $(F,\eta)$ of positive degree $d$ in $f^*(E,\theta)$. Then $\mathrm{Gr}_{f^*Fil_{-1}}(F,\eta)\subset f^*(E_0,\theta_0)$ is a Higgs subsheaf of degree $d$. It implies that $\mathrm{Gr}_{f^*Fil_0}\circ C^{-1}_{Y_{\log}\subset \tilde Y_{\log}}(F,\eta)\subset f^*(E_1,\theta_1)$ is of degree $pd$. Iterating this process, one obtains a subsheaf in $f^*(E_i,\theta_i)$ whose degree exceeds $N$. Contradiction. Therefore, $f^*(E,\theta)$ is semistable.
Now we turn to the char zero case. By the standard spread-out technique, there is a regular scheme $S$ of finite type over ${\mathbb Z}$, and an $S$-morphism $\mathfrak{f}: ({\mathcal Y},{\mathcal B})\to ({\mathcal X},{\mathcal D})$ and an $S$-relative logarithmic Higgs bundle $({\mathcal E},\Theta)$ over $({\mathcal X},{\mathcal D})$, together with a $k$-rational point in $S$ such that $\{\mathfrak f: ({\mathcal Y},{\mathcal B})\to ({\mathcal X},{\mathcal D}), ({\mathcal E},\Theta)\}$ pull back to $\{f: (Y,B)\to (X,D), (E,\theta)\}$. In above, we may assume that ${\mathcal X}$ (resp. ${\mathcal Y}$) is smooth projective over $S$ and ${\mathcal D}$ (resp. ${\mathcal B}$) is an $S$-relative normal crossing divisor in ${\mathcal X}$ (resp. ${\mathcal Y}$). For a geometrically closed point $s\in S$ and a $W_2(k(s))$-lifting $\tilde s\to S$, we obtain a family $\mathfrak{f}_s: ({\mathcal Y},{\mathcal B})_s\to ({\mathcal X},{\mathcal D})_s$ over $k(s)$ which is $W_2$-liftable. Once taking an $s\in S$ such that $\textrm{char}(k(s))\geq {\rm rank}({\mathcal E}_s)={\rm rank}(E)$, we are in the previous char $p$ setting. Hence it follows that $\mathfrak{f}_s^*({\mathcal E},\Theta)_s$ is semistable. From this, it follows immediately that $f^*(E,\theta)$ is also semistable.
\end{proof}
\end{document} | arXiv |
Functional Analytic Square Root of Hamiltonian Alternative to Dirac
I was thinking about the history of the Dirac equation and asked myself, what happens if one simply considers the Schrödinger equation $i\hbar\frac{\partial\phi}{\partial t}=\sqrt{-c^2\hbar^2\Delta+m^2c^4}\phi$? The literature seems to suggest that the square root is troublesome. However, in spectral theory it is well known how to take the square root of positive self-adjoint operators. So, what goes wrong with this?
quantum-mechanics schroedinger-equation hamiltonian dirac-equation non-locality
Qmechanic♦
141k1919 gold badges325325 silver badges16731673 bronze badges
Iván Mauricio BurbanoIván Mauricio Burbano
$\begingroup$ It is not clear what benefit we have from using $\sqrt{…}$? We must consider two roots (particles, antiparticles) as $\pm \sqrt{…}$ and consider $\phi$ as a spinor. In Dirac's formulation, these are 4 linear equations. In this formulation, these are 4 equations with a fractional degree of operator. $\endgroup$ – Alex Trounev Apr 4 '20 at 19:58
$\begingroup$ Just a comment. The Klein-Gordon-Fock-Schrödinger equation (today known only the first two names) was around by the end of 1926. The probabilistic interpretation came around in 1927 and in 1928 Dirac ignores the square root possibility and comes up with his equations. The mathematics allowing us to speak about a sqrt of a linear oporator on a Hilbert space was just under creation by Von Neumann in Germany and M.Stone in the US by that time. So by the time it was ready, the success of Dirac equations (reinterpreted in terms of QFT) made returning to the sqrt option uninteresting. $\endgroup$ – DanielC Apr 5 '20 at 0:11
$\begingroup$ Related: What's wrong with the square root version of the Klein-Gordon equation? and Delocalization in the square root version of Klein-Gordon equation $\endgroup$ – Chiral Anomaly Apr 5 '20 at 15:40
I think that the problem is that the square root of the Laplacian is a non-local opertor and non-locality is usully regarded as a bad thing in physics. The long range nature shows up in the general expression
$$ (-\nabla^2)^s f(x)\equiv \frac{4^s}{\pi^{n/2}}\frac{\Gamma(s+n/2)}{\Gamma(-s)} \int_{{\mathbb R}^n} d^ny \frac{\left\{f(x)-f(y)\right\}}{|x-y|^{2s+n}}, \quad 0<s<1, $$ for fractional powers of the Laplacian as a convolution integral. (The limit on the range of $s$ is to ensure that no further subtractions are required to define $(-\nabla^2)^s$ as a distribution.)
mike stonemike stone
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The Distance Formula
In analytic geometry, the distance between two points of the xy-plane can be found using the distance formula. Distance Formula is used to calculate the distance between two points. The distance between (x1, y1) and (x2, y2) is given by:
\[\large d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\]
Example For The Distance Formula
Question: Given the points (-1, -2) and (-3, 5), find the distance between them.
Label the points as follows
$\left(x_{1},y_{1}\right)=\left(-1, -2\right) and \left(x_{2},y_{2}\right)=\left(-3, 5\right)$
Therefore: $x_{1}=-1,\: y_{1}=-2,\: x_{2}=-3, and\: y_{2}=5$
To find the distance (d) between the points, use the distance formula:
$d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$
\(=\sqrt{\left(-3-\left(-1\right)\right)^{2}+\left(5-\left(-2\right)\right)^{2}}\)
$=\sqrt{\left(-3+1\right)^{2}+\left(5+2\right)^{2}}$
$=\sqrt{\left(-2\right)^{2}+\left(7\right)^{2}}$
$=\sqrt{4+49}$
$=\sqrt{53}$
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\begin{document}
\title{Study of Novel Sparse Array Design Based on the Maximum Inter-Element Spacing Criterion \\
\author{Wanlu Shi, and Yingsong Li, \IEEEmembership{Senior Member, IEEE},~Rodrigo C. de Lamare,~\IEEEmembership{Senior Member,~IEEE}
} \thanks{} \thanks{Wanlu Shi is with the College of Information and Communication Engineering, Harbin Engineering University, Harbin 150001, China.} \thanks{Yingsong Li is with the Key Laboratory of Intelligent Computing and Signal Processing Ministry of Education, Anhui University, Hefei, Anhui, China (e-mail: [email protected]){\it (Corresponding author: Yingsong Li)}} \thanks{R. C. de Lamare is with the Centre for Telecommunications Research (CETUC), Pontifical Catholic University of Rio de Janeiro (PUC-Rio), G¨¢vea, Rio de Janeiro - Brazil and the University of York, UK.} }
\maketitle \begin{abstract} A novel sparse array (SA) structure is proposed based on the maximum inter-element spacing (IES) constraint (MISC) criterion. Compared with the traditional MISC array, the proposed SA configurations, termed as improved MISC (IMISC) has significantly increased uniform degrees of freedom (uDOF) and reduced mutual coupling. In particular, the IMISC arrays are composed of six uniform linear arrays (ULAs), which can be determined by an IES set. The IES set is constrained by two parameters, namely the maximum IES and the number of sensors. The uDOF of the IMISC arrays is derived and the weight function of the IMISC arrays is analyzed as well. The proposed IMISC arrays have a great advantage in terms of uDOF against the existing SAs, while their mutual coupling remains at a low level. Simulations are carried out to demonstrate the advantages of the IMISC arrays. \end{abstract} \begin{IEEEkeywords} Uniform degrees of freedom, sparse array (SA), mutual coupling, maximum inter-element spacing constraint, direction-of-arrival (DOA) estimation. \end{IEEEkeywords}
\section{Introduction} There is a growing body of literature that emphasizes the vital role of sensor array techniques in diverse applications, such as seismology~\cite{optimum_array_processing}, radar~\cite{radar}, sonar~\cite{sonar}, and others. {An important problem in sensor array technique is direction finding~\cite{music}, especially the case when there are more targets than the number of sensors~\cite{optimum_array_processing,fourth_order}.} In this context, the difference coarray (DCA) principle is one of the feasible choices to produce plenty uniform degrees of freedom (uDOF), {which makes possible to use fewer sensors estimate more targets~\cite{coarray,MRA,MHA}.} Based on the DCA concept, many classical sparse arrays (SAs) have been introduced, among which the nested arrays (NAs)~\cite{NESTED} and coprime arrays (CAs) are very popular~\cite{coprimeC,COPRIME,Extended_coprime,stap_coprime,listomp}. The NAs can achieve $O(N^2/2)$ uDOF using $N$ sensors but are sensitive to the mutual coupling (MC) among sensors. Unlike NAs, CAs are robust to MC, but their uDOF are less than those of NAs~\cite{coprimeC,COPRIME}. \par Many existing SAs have been developed via the prototype NAs and CAs ~\cite{improved_nested,CADIS,ANA,TSONA,Extended_coprime}. Early SAs have focused on increasing the achievable uDOF. To this end, the improved NA (INA)~\cite{improved_nested}, the CA with displaced subarrays (CADiS)~\cite{CADIS} and the two side-extended nested array (TSENA)~\cite{TSONA} are proposed. These SAs show considerable improvement on uDOF. However, with the growing of electromagnetic equipments, MC is inevitable and needs to be considered when designing SAs. In this regard, several SAs have been proposed based on NAs and CAs. For example, the super nested arrays (SNAs)~\cite{SNAC1,SNAC2,SNA1,SNA2}, the extended padded coprime arrays (ePCAs)~\cite{padded_coprime} and the improved coprime nested arrays (ICNA)~\cite{ICNA} are examples of SAs designed based on NAs and CAs. It is worth noting that the ICNA realizes $O( 4Q^2/7)$ uDOFs with $Q$ sensor elements and MC among sensors is very low. Another general SA design method is the ULA fitting scheme, based on which the ULA fitting with 4 base layer (UF-4BL) has been proposed with low MC~\cite{UF,UF_ICASSP}. Beyond the aforementioned SA design approaches, the maximum inter-element spacing (IES) constraint (MISC) criterion is also promising~\cite{MISC}. The MISC criterion aims to design SAs with an inter-element spacing set, which is constrained by the maximum element spacing and the number of sensors. The MISC arrays provide a good balance between MC and uDOF.
In this work, we develop an improved MISC (IMISC) array based on the MISC approach with an increased uDOF and reduced MC as compared to the traditional MISC array. In particular, we carry out an analysis that shows that IMISC arrays reach $O( 2Q^2/3)$ uDOF, which is significantly higher than those uDOF achieved by existing sparse arrays. Moreover, the MC of the proposed IMISC arrays remains at a reduced level. Simulations illustrate the excellent performance of the proposed IMISC arrays.
\section{Coarray Concept and Mutual Coupling}\label{fundamental} \subsection{Difference Coarray Model}
Let us assume an $Q$-sensor linear array with location set \begin{equation} \mathbb{L} = \left\{ {{p_0},p_1, p_2,\dots,p_{Q-1}} \right\}, \label{position_set} \end{equation} then the associated steering vector can be obtained as $\mathbf{v}_{\phi}=\left[e^{j \frac {2\pi}{\lambda} p_0/ \sin(\phi)},\cdots,e^{j \frac {2\pi}{\lambda} p_{Q-1}/ \sin(\phi)}\right]^T$, where $\phi$ is the signal impinging direction. Then imagine $R$ uncorrelated far-field narrowband signals impinging from directions $\left\{\phi_i,i=1,\cdots,R\right\}$ with powers $\left\{\sigma^2_i,i=1,\cdots,R\right\}$. In this regard, the observed signal can be expressed as \begin{equation} {\mathbf{x}} = {\mathbf{V}}{\mathbf{s}} + {\mathbf{n}}, \label{received_data_one_snapshot} \end{equation} where $\mathbf{V}\triangleq\left[\mathbf{v}_{\phi_1},\cdots,\mathbf{v}_{\phi_R}\right]$ is the array manifold matrix, $\mathbf{s} \triangleq \left[\mathbf{s}_1,\cdots,\mathbf{s}_R\right]^T$ denotes the signal vector, ${\mathbf{n}}$ is the white Gaussian noise with zero mean. Then, the covariance matrix of (\ref{received_data_one_snapshot}) can be calculated as \begin{equation} {\mathbf{R}_{\mathbf{x}}} \triangleq E[\mathbf{x}{\mathbf{x}^H}] = \mathbf{V}\mathbf{R}_{\mathbf{s}}{\mathbf{V}^H} + \sigma_n^2 \mathbf{I}, \label{covariance_matrix} \end{equation} where $\sigma ^2_n$ is the power of noise and $\mathbf{R}_{\mathbf{s}}$ represents the signal covariance matrix. In this case, the vectorized version of~(\ref{covariance_matrix}) can be obtained as \begin{equation} {\mathbf{w}}\triangleq{\rm{vec}}({\mathbf{R}_{\mathbf{x}}}) = ({\mathbf{V}}^*\odot{\mathbf{V}}){\mathbf{q}}+ \sigma_n^2 {\mathbf{1}}_n, \label{vectorized_covariance_matrix} \end{equation} where ${\mathbf{1}}_n={\rm{vec}}({\mathbf{I}}_N)$, ${\mathbf{q}} = \left[\sigma ^2_1,\cdots,\sigma ^2_R\right]^T$, and $\odot$ means the Khatri-Rao product. The DCA concept originates from~(\ref{vectorized_covariance_matrix}), where ${\mathbf{q}}$ serves as the observed signal of the DCA. Based on~(\ref{vectorized_covariance_matrix}), the DCA of array $\mathbb{L}$ has the following position set~\cite{NESTED} \begin{equation} \mathbb{D}= \left\{ {{p_a-p_b},a,b = 0,1, \cdots Q-1} \right\}. \label{DCA} \end{equation} \subsection{Coupling Leakage} In practice, the MC among sensors should be considered. To this end, the MC matrix $\mathbf{A}$ can be interpolated into~(\ref{received_data_one_snapshot}), resulting in \begin{equation} {\mathbf{x}} = {\mathbf{A}} {\mathbf{V}}{\mathbf{s}} + {\mathbf{n}}, \label{received_data_one_snapshot_with_coupling} \end{equation} Based on~\cite{MISC,coupling1,coupling3}, the MC matrix for linear arrays can be formulated as {a $D$-banded matrix given by}
\begin{equation} {\mathbf{A}}_{b,c}=\left\{{\begin{array}{*{20}{l}}
a_{|p_b-p_c|},&|p_b-p_c|\le D,\\ 0,&{\rm{elsewhere}}, \end{array}} \right. \label{c_approximate} \end{equation} where $p_b,p_c\in \mathbb{L}$ and $a_d,d\in [0,D]$ represent elements in $\mathbf{A}$ that satisfy
\begin{equation} \left\{{\begin{array}{*{20}{l}}
a_0=1\textgreater |a_1|\textgreater |a_2|\textgreater\cdots\textgreater|a_D|,\\
|a_i/a_j|=j/i,\quad i,j\in [{\color{red}1},D]. \end{array}} \right. \label{c_coefficients} \end{equation} Coupling leakage is a parameter to evaluate the MC effect, which is expressed as~\cite{SNA1,ANA,MISC}
\begin{equation}
E=\frac {||{\mathbf{A}}-{\rm{diag}}({\mathbf{A}})||_F} {||{\mathbf{A}}||_F}. \label{coupling_leakage} \end{equation} Typically, higher coupling leakage means larger MC. Moreover, equations~(\ref{c_approximate}),~(\ref{c_coefficients}) and~(\ref{coupling_leakage}) reveal the relationship between MC and the weight function $w(n)$. For linear arrays, the {weight function $w(n)$ indicates} the number of element pairs {whose} IES {is} $n$. {For SA $\mathbb{L}$, $w(n)$ can be obtained as~\cite{MISC}
\begin{equation}\nonumber w(n)=|\{ (p_i,p_j)|p_i-p_j = n;p_i,p_j \in \mathbb{L}\}|. \end{equation} Based on the analysis above,} it is noted that $w(1)$, $w(2)$ and $w(3)$ dominate the MC.
\section{Proposed IMISC array} In this section, the IMISC array is developed, which is devised based on the MISC criterion{~\cite{MISC}}. {Compared with the traditional MISC array, the proposed IMISC SA has higher uDOF and lower coupling leakage.} The IMISC SA has key merits. First, IMISC has a closed-form expression for sensor location and uDOF. Moreover, compared with the existing SAs, IMISC has higher uDOF and provides better {\color{red}a} balance between uDOF and MC.
The IMISC array can be identified using an IES set for a given sensor amount $Q$. Particularly, the IES set is expressed as $\mathbb{S}$ and the maximum IES is denoted as $M$. In this regard, $\mathbb{S}$ for IMISC can be expressed as \begin{equation} \mathbb{S}_{\text{IMISC}}=\left\{ \begin{array}{l} \underbrace {2,...,2}_{\frac{M}{4} - 1},1,1,\frac{M}{2} - 2,\\ \underbrace {\frac{M}{2} - 1,...,\frac{M}{2} - 1}_{\frac{M}{4} - 2},\underbrace {M,...,M}_{Q - M},\\ \frac{M}{2} + 1,\underbrace {\frac{M}{2} + 1,...,\frac{M}{2} + 1}_{\frac{M}{4} - 2},2,\underbrace {2,...,2}_{\frac{M}{4} - 1}, \end{array} \right\} \label{spacing} \end{equation} where \begin{equation} M = 4\lfloor \frac{Q+2}{6}\rfloor,Q\ge10,\\ \label{M} \end{equation} where $\lfloor \cdot \rfloor$ is the floor operator. The location set associated to~(\ref{spacing}) is given as \begin{equation} \begin{aligned} &\mathbb{L}_{\text{IMISC}} =\\ \small&\left\{ \begin{array}{l} \underbrace {0,...,\frac{M}{2} - 2}_{{\text{ULA 1, IES=2}}},\underbrace {\frac{M}{2} - 1,\frac{M}{2}}_{{\text{ULA 2, IES=1}}},\underbrace {M - 2,...,\frac{{{M^2}}}{8} - \frac{M}{4}}_{{\text{ULA 3, IES=}}\frac{M}{2} - 1},\\ \underbrace {\frac{{{M^2}}}{8} + \frac{{3M}}{4},...,MQ - \frac{{7{M^2}}}{8} - \frac{M}{4}}_{{\text{ULA 4, IES=}}M},\\ \underbrace {MQ - \frac{{7{M^2}}}{8} + \frac{M}{4} + 1,...,MQ - \frac{{3{M^2}}}{4} - \frac{M}{2} - 1}_{{\text{ULA 5, IES=}}\frac{M}{2}+1},\\ \underbrace {MQ - \frac{{3{M^2}}}{4} - \frac{M}{2} + 1,...,MQ - \frac{{3{M^2}}}{4} - 1}_{{\text{ULA 6, IES=2}}}. \end{array} \right\} \end{aligned} \label{Structure} \end{equation} One can tell from~(\ref{Structure}) that IMISC is composed of 6 sub-ULAs and a specific array configuration of IMISC is provided in~Fig.~\ref{IMISC_STRUCTURE}, where ${\text{s-}}i$, $i=1,...,6$ represent sub-ULA~1,...,sub-ULA~6, respectively. \begin{figure}
\caption{IMISC array structure with~$Q=10$ and $M=8$.}
\label{IMISC_STRUCTURE}
\end{figure}
\section{Analysis of IMISC} In this section, we carry out an analysis of IMISC arrays to obtain the achievable uDOF and their weight functions. \subsection{Achievable uDOF} Defining $\mathbb{D}_{\text{IMISC}}$ as the DCA of the IMISC, then the consecutive part of $\mathbb{D}_{\text{IMISC}}$ can be expressed as \begin{equation} \mathbb{C}_{\text{IMISC}}=[-MQ + \frac{{3{M^2}}}{4} + \frac{M}{2} - 1,MQ - \frac{{3{M^2}}}{4} - \frac{M}{2} + 1]. \label{consecutive_part1} \end{equation} The proof of~(\ref{consecutive_part1}) can be found in the Appendix. Based on~(\ref{consecutive_part1}), the final uDOF of IMISC is \begin{equation} {\text{uDOF}}_{\text{IMISC}}=2MQ - \frac{{3{M^2}}}{2} - M + 3. \label{uDOF1} \end{equation} Substituting~(\ref{M}) to~(\ref{uDOF1}) and omitting the floor operator, the uDOF of IMISC can be approximately obtained as \begin{equation} {\text{uDOF}}_{\text{IMISC}}=\frac{{{\rm{2}}{Q^2}}}{3} - \frac{{2Q}}{3} - 1. \label{uDOF2} \end{equation} Considering the fact that $M$ and $Q$ are integers, the accurate expression for ${\text{uDOF}}_{\text{IMISC}}$ is \begin{equation} \begin{aligned} {\text{uDOF}}_{\text{IMISC}}=\left\{ {\begin{array}{*{20}{l}} \frac{{{\rm{2}}{Q^2}}}{3} - \frac{{2Q}}{3} - 1,&Q\% 6 = 4,3,\\ \frac{{{\rm{2}}{Q^2}}}{3} - \frac{{2Q}}{3} + \frac{5}{3},&Q\% 6 = 5,2,\\ \frac{{{\rm{2}}{Q^2}}}{3} - \frac{{2Q}}{3} + 3,&Q\% 6 = 0,1, \end{array}} \right. \label{uDOF3} \end{aligned} \end{equation} where $\%$ is the remainder operator. Based on~(\ref{uDOF2}) and~(\ref{uDOF3}), one can find that IMISC can use $O(Q)$ sensors {to} produce $O( 2Q^2/3)$ uDOF, which is higher than most existing SAs. For comparison, the uDOF for the traditional MISC array is provided here {\begin{equation} \begin{aligned} {\text{uDOF}}_{\text{MISC}}=\left\{ {\begin{array}{*{20}{l}} \frac{{{Q^2}}}{2} + 3Q - 8.5,&Q\% 4 = 1,\\ \frac{{{Q^2}}}{2} + 3Q - 9,&Q\% 2 = 0,\\ \frac{{{Q^2}}}{2} + 3Q - 10.5,&Q\% 4 = 3. \end{array}} \right. \end{aligned} \label{uDOF_MISC} \end{equation}} From~(\ref{uDOF3}) and~(\ref{uDOF_MISC}), the IMISC has a higher uDOF than traditional MISC. \subsection{Weight Function} Based on~(\ref{spacing}), the first three values in the weight function can be obtained as \begin{equation} \begin{aligned} &w(1)=2, \quad w(2)=\left\{ {\begin{array}{*{20}{l}} 2\lfloor\frac{Q+2}{6}\rfloor,&Q\ge 16,\\ 5,&16>Q\ge10, \end{array}} \right.\\ &\quad \quad \quad \quad w(3)=\left\{ {\begin{array}{*{20}{l}} 1,&Q\ge 16,\\ 2,&16>Q\ge10. \end{array}} \right. \label{w1} \end{aligned} \end{equation} For comparison, the first three values in the weight function for MISC are given by \begin{equation} \begin{aligned} w(1)=1,w(2)=2\lfloor\frac{Q}{4}\rfloor-3, w(3)=\left\{ {\begin{array}{*{20}{l}} 1,&Q\neq 9,\\ 2,&Q=9. \end{array}} \right. \label{w2} \end{aligned} \end{equation} Based on~(\ref{w1}) and~(\ref{w2}), it is obvious that the IMISC has higher $w(1)$ but lower $w(2)$ than MISC. Therefore, when $Q$ is sufficiently large, the coupling leakage of the IMISC will lower than the traditional MISC. \section{Numerical Simulations}\label{NE}
In this section, the proposed IMISC array is compared with the Co-prime ~\cite{Extended_coprime}, MISC~\cite{MISC}, SNA~\cite{SNA1,SNA2}, TSONA~\cite{TSONA}, ePCA~\cite{padded_coprime}, ICNA~\cite{ICNA} and UF-4BL~\cite{UF} arrays. DOA estimations are carried out utilizing the spatial smoothing MUSIC algorithm, and the number of snapshots is set to 1000~\cite{NESTED}. In all examples, coupling parameters are chosen according to~$a_i=a_1e^{-j(i-1)\pi/8}/i, \, i=2,\dots,100$. In this regard, the larger $|a_1|$ indicates higher MC effect. Other parameter estimation algorithms that exploit sparsity can also be considered \cite{intadap,jidf,sjidf,rrdoa}.
\subsection{uDOF and Coupling Leakage}
In the first example, the uDOF and coupling leakage versus number of sensors are presented, respectively, where the number of sensors vary from 20~to~100. Fig.~\ref{uDOF-CL}(a) shows the variation of the uDOF when the number of sensors is altered. It is clear that the proposed IMISC array has the highest uDOF among the relevant SA configurations. Besides, with the increasing of the number of sensors, the advantage of IMISC on uDOF is increased as well. This is due to the fact that the IMISC reaches $O(2Q^2/3)$ uDOF, while the maximum uDOF for other SAs is $O(4Q^2/7)$. \par Fig.~\ref{uDOF-CL}(b) shows the coupling leakage versus the number of sensors. One can see that the couple leakage for IMISC is lower than that of the traditional MISC array but higher than ICNA and UF-4BL. Therefore, based on the results shown in~Fig.~\ref{uDOF-CL}, we claim that the proposed IMISC array provides better balance between uDOF and coupling leakage than existing SAs.
\begin{figure}
\caption{uDOF and coupling leakage versus number of sensors.}
\label{uDOF-CL}
\end{figure}
\subsection{Root-Mean-Square Error (RMSE) Performance with various SNR, $|a_1|$, and snapshots}
In the second example, the RMSE performance in different conditions is considered. All the RMSE results are obtained through~500 trials.
\begin{figure}
\caption{RMSE performance in various conditions.}
\label{SNR-C1}
\end{figure}
Firstly, the RMSE performance versus SNR in the presence of coupling is presented. All the SAs have 34 sensors and there are 39 sources in $[-60^\circ,60^\circ]$, and $a_1=0.3e^{j\pi/3}$. From~Fig.~\ref{SNR-C1}(a), it is observed that IMISC has the best performance, due to the longest uDOF and low MC.
Next, the RMSE versus $|a_1|$ is investigated. Here, all the SAs consist of $35$ sensors, the number of sources is set to $50$ within azimuth $[-60^\circ,60^\circ]$, SNR=~0~dB, and $|a_1|$ varies from 0 to 0.5. As shown in~Fig.~\ref{SNR-C1}(b), when $|a_1|<0.4$, IMISC gives the best performance, while when $|a_1|>0.4$, the ICNA and UF-4BL perform better than IMISC.
{Finally, the RMSE performance versus the number of snapshots is presented in~Fig.~\ref{SNR-C1}(c), where all the SAs have~37 sensors, the number of sources is~45 within azimuth $[-60^\circ,60^\circ]$, SNR=0~dB, and $|a_1|=0.3$. We can see that the proposed IMISC provides the best performance over snapshots.}
\section{Conclusion}\label{conclusion} In this letter, based on the MISC principle, a novel SA structure, termed as IMISC has been proposed. Compared with the MISC array the IMISC array possesses higher uDOF and lower coupling among sensors. The IMISC SA can be uniquely obtained by an inter-element spacing set, which is constrained by the maximum inter-element spacing and the number of sensors. Moreover, the uDOF and the weight function of the IMISC SAs are analyzed in detail. Simulations verify that IMISC can provide a great balance between uDOF and coupling leakage, and hence has better performance in DOA estimation than existing SAs. Future work will focus on how to further reduce the MC among sensors.
\begin{appendices} \section{Consecutive part in the DCA of IMISC}\label{proof_proposition_SDCA_symmetric}
Based on~~(\ref{consecutive_part1}) and due to the symmetric property of DCA, we only need to prove $\mathbb{D}_{\text{IMISC}}^{+} \supset \mathbb{C}_{\text{IMISC}}^{+}$, where $\mathbb{D}_{\text{IMISC}}^{+}$ is the positive set of $\mathbb{D}_{\text{IMISC}}$ and $\mathbb{C}_{\text{IMISC}}^{+}=[1,MQ - \frac{{3{M^2}}}{4} - \frac{M}{2} + 1]$. \par In IMISC, assume that ULA~$i$ has the following position set \begin{equation} \begin{aligned} \mathbb{P}_{\text{ULA }i}=\{p_{i}(x),x=1,\dots,Q_i\}, \end{aligned} \label{ULAi} \end{equation} where $Q_i$ is the number of sensors in ULA~$i$. Similarly, setting the position set of ULA~$j$ as \begin{equation} \begin{aligned} \mathbb{P}_{\text{ULA }j}=\{p_{j}(y),y=1,\dots,Q_j\}, \end{aligned} \label{ULAj} \end{equation} where $Q_j$ is the number of sensors in ULA~$j$. Then, defining $\mathbb{D}_{i,j}$ as the DCA between ULA~$i$ and ULA~$j$, which has the following expression \begin{equation} \begin{aligned} \mathbb{D}_{i,j}=\{p_{j}(y)-p_{i}(x),x=1,\dots,Q_i,y=1,\dots,Q_j\}. \end{aligned} \label{D_i_j} \end{equation} In particular, we have \begin{equation}\nonumber \begin{aligned} \mathbb{D}_{i,i}=\{p_{i}(x),x=1,\dots,Q_i\}-p_{i}(1). \end{aligned} \label{D_i_i} \end{equation} In IMISC, the number of ULAs is~6, in this regard, $\mathbb{D}_{\text{IMISC}}^{+}$ can be expressed as \begin{equation} \begin{aligned} \mathbb{D}_{\text{IMISC}}^{+}=\bigcup_{i=1}^{5} \mathbb{D}_{i,j},j=i,i+1,\dots,6, \end{aligned} \label{D_MISC+} \end{equation} where $\cup$ is the union operator. Based on~(\ref{Structure}) and~(\ref{D_i_j}), some DCA expressions in~(\ref{D_MISC+}) are provided in~(\ref{expressions_D}). The proof of $\mathbb{D}_{\text{IMISC}}^{+} \supset \mathbb{C}_{\text{IMISC}}^{+}$ can be completed by finding consecutive lags within the range $[1,MQ - \frac{{3{M^2}}}{4} - \frac{M}{2} + 1]$ based~(\ref{D_MISC+}) and~(\ref{expressions_D}) (at the bottom of this page). \par Based on~(\ref{expressions_D}), one can tell that $\mathbb{D}_{1,2}\cup\mathbb{D}_{1,3}\cup\mathbb{D}_{3,4}\cup\mathbb{D}_{4,5}\cup\mathbb{D}_{5,6}$ can generate consecutive range \begin{equation} [1,\frac{M^2}{8}-\frac{M}{4}], \label{cons1} \end{equation} and $\mathbb{D}_{1,4}\cup\mathbb{D}_{2,4}\cup\mathbb{D}_{3,4}\cup\mathbb{D}_{4,5}\cup\mathbb{D}_{4,6}\cup\mathbb{D}_{4,4}$ can generate consecutive range \begin{equation} [\frac{M^2}{8}-\frac{M}{4}+1,MQ-\frac{7M^2}{8}-\frac{5M}{4}+3], \label{cons2} \end{equation} and $\mathbb{D}_{1,5}\cup\mathbb{D}_{1,6}\cup\mathbb{D}_{2,5}\cup\mathbb{D}_{2,6}\cup\mathbb{D}_{3,5}\cup\mathbb{D}_{3,6}\cup\mathbb{D}_{1,4}\cup\mathbb{D}_{2,4}\cup\mathbb{D}_{4,6}$ can generate consecutive range \begin{equation} [MQ-\frac{7M^2}{8}-\frac{5M}{4}+3,MQ - \frac{{3{M^2}}}{4} - \frac{M}{2} + 1]. \label{cons3} \end{equation} Equations~(\ref{cons1}),~(\ref{cons2}) and~(\ref{cons3}) complete the proof. \small
\hrule \begin{align}\nonumber &\mathbb{D}_{\text{1,2}}=\Big\{ 1,2,\dots,\frac{M}{2} \Big\}, \quad\mathbb{D}_{\text{2,6}}=\Big\{ MN-\frac{3M^2}{4}-M+1+i+j,i=0,1,j=0,2,\dots,\frac{M}{2}-2 \Big\}, \\\nonumber &\mathbb{D}_{\text{1,3}}=\Big\{ \frac{M}{2}+i+j,i=0,2,\dots,\frac{M}{2}-2,j=(0,1,\dots,\frac{M}{4}-2)(\frac{M}{2}-1) \Big\},\\\nonumber &\mathbb{D}_{\text{1,4}}=\Big\{ \frac{M^2}{8} + \frac{M}{4} + 2+i+j,i=0,2,\dots,\frac{M}{2}-2,j=(0,1,\dots,N-M-1)(M) \Big\},\\\nonumber &\mathbb{D}_{\text{2,4}}=\Big\{ \frac{M^2}{8} + \frac{M}{4}+i+j,i=0,1,j=(0,1,\dots,N-M-1)(M) \Big\},\\\nonumber &\mathbb{D}_{\text{3,4}}=\Big\{ M+i+j,i=(0,1,\dots,\frac{M}{4}-2)(\frac{M}{2}-1),j=(0,1,\dots,N-M-1)(M) \Big\},\\\nonumber &\mathbb{D}_{\text{4,5}}=\Big\{ \frac{M}{2}+1+i+j,i=(0,1,\dots,\frac{M}{4}-2)(\frac{M}{2}+1),j=(0,1,\dots,N-M-1)(M) \Big\},\\\label{expressions_D} &\mathbb{D}_{\text{4,6}}=\Big\{ \frac{M^2}{8} - \frac{M}{4} + 1+i+j,i=0,2,\dots,\frac{M}{2}-2,j=(0,1,\dots,N-M-1)(M) \Big\},\\\nonumber &\mathbb{D}_{\text{5,6}}=\Big\{ 2+i+j,i=0,2,\dots,\frac{M}{2}-2,j=(0,1,\dots,\frac{M}{4}-2)(\frac{M}{2}+1) \Big\},\\\nonumber &\mathbb{D}_{\text{1,5}}=\Big\{ MN-\frac{7M^2}{8}-\frac{M}{4}+3+i+j,i=0,2,\dots,\frac{M}{2}-2,j=(0,1,\dots,\frac{M}{4}-2)(\frac{M}{2}+1) \Big\},\\\nonumber &\mathbb{D}_{\text{1,6}}=\Big\{ MN-\frac{3M^2}{4}-M+3+i+j,i=0,2,\dots,\frac{M}{2}-2,j=0,2,\dots,\frac{M}{2}-2 \Big\},\\\nonumber &\mathbb{D}_{\text{2,5}}=\Big\{ MN-\frac{7M^2}{8}-\frac{M}{4}+1+i+j,i=0,1,j=(0,1,\dots,\frac{M}{4}-2)(\frac{M}{2}+1) \Big\},\\\nonumber &\mathbb{D}_{\text{3,5}}=\Big \{MN-M^2+\frac{M}{2}+1+i+j,i=(0,1,\dots,\frac{M}{4}-2)(\frac{M}{2}-1),j=(0,1,\dots,\frac{M}{4}-2)(\frac{M}{2}+1)\Big\},\\\nonumber &\mathbb{D}_{\text{3,6}}=\Big \{MN-\frac{7M^2}{8}-\frac{M}{4}+1+i+j,i=(0,1,\dots,\frac{M}{4}-2)(\frac{M}{2}-1),j=0,2,\dots,\frac{M}{2}-2 \Big\}.\\\nonumber \end{align}\nonumber
\end{appendices}
\end{document} | arXiv |
Find the least positive integer $x$ that satisfies $x+4609 \equiv 2104 \pmod{12}$.
Subtract 4609 from both sides of the congruence to obtain $x\equiv -2505\pmod{12}$. By dividing 2505 by 12, we find that the least integer $k$ for which $-2505+12k>0$ is $k=209$. Adding $12\cdot 209$ to $-2505$, we find that $x\equiv 3\pmod{12}$. Thus $\boxed{3}$ is the least integer satisfying the given congruence. | Math Dataset |
How to prove that a function is $C^{\infty}$?
I would like to show that the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined as
$$ h(t) = \left\{ \begin{array}{ll} 0 & \mbox{if } t \leq 0 \\ \exp{(-\frac{1}{t})} & \mbox{if } t > 0 \end{array} \right. $$
is $C^{\infty}$. It seems to me that the problem is at $t = 0$ and I assume I have to use the definition of the derivative and induction, but I don't know how to generalize this result to $\infty$.
real-analysis
Tim BrenenTim Brenen
By induction show that for $t>0$ we have that $h^{(n)}(t)$ is a polynomial in $\frac1t$ times $\exp(-1/t)$ (and of course $h^{(n)}(t)=0$ for $t<0$). Conclude from this fact (for $n-1$) that $h^{(n)}(0)=0$ and $h^{(n)}$ is continuous. In other words, show that there exists polynomials $f_n(X)\in\mathbb R[X]$, $n\in\mathbb N_0$ such that $$ h^{(n)}(t)=\begin{cases}0&\text{if }t\le 0\\f_n(\tfrac 1t)\exp(-\tfrac 1t)&\text{if }t>0.\end{cases}$$
Hagen von EitzenHagen von Eitzen
$\begingroup$ I was thinking on something like that, but is it enough to show that $h \in C^{\infty}$. $\endgroup$ – Tim Brenen Nov 13 '13 at 12:27
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