uuid int64 541B 3,299B | dataset stringclasses 1
value | text stringlengths 1 4.29M |
|---|---|---|
3,212,635,537,906 | arxiv | \section{Observables in Probabilistic Theories}
This paper is based on the stimulating article \cite{bhs12} by Busch, Heinosaari and Schultz. The authors should be congratulated for introducing a useful new tool for measuring the compatibility of a probabilistic theory (PT). In this paper, we present a simpler, but coarser, measure of compatibility that we believe will also be useful.
A \textit{probabilistic theory} is a $\sigma$-convex subset ${\mathcal K}$ of a real Banach space ${\mathcal V}$. That is, if
$0\le\lambda _i\le 1$ with $\sum\lambda _i=1$ and $v_i\in{\mathcal K}$, $i=1,2,\ldots$, then $\sum\lambda _iv_i$ converges in norm to an element of ${\mathcal K}$. We call the elements of ${\mathcal K}$ \textit{states}. There is no loss of generality in assuming that ${\mathcal K}$ generates ${\mathcal V}$ in the sense that the closed linear hull of ${\mathcal K}$ equals ${\mathcal V}$. Denote the collection of Borel subsets of $\mathbb R ^n$ by ${\mathcal B} (\mathbb R ^n)$ and the set of probability measures on ${\mathcal B} (\mathbb R ^n)$ by
${\mathcal M} (\mathbb R ^n)$. If ${\mathcal K}$ is a PT, an $n$-\textit{dimensional observable} on ${\mathcal K}$ is a $\sigma$-affine map
$M\colon{\mathcal K}\to{\mathcal M} (\mathbb R ^n)$. We denote the set of $n$-dimensional observables by ${\mathcal O} _n({\mathcal K} )$ and write
${\mathcal O} ({\mathcal K} )={\mathcal O} _1({\mathcal K} )$. We call the elements of ${\mathcal O} ({\mathcal K} )$ \textit{observables}. For
$M\in{\mathcal O} ({\mathcal K} )$, $s\in{\mathcal K}$, $A\in{\mathcal B} (\mathbb R )$, we interpret $M(s)(A)$ as the probability that $M$ has a value in $A$ when the system is in state $s$.
A set of observables $\brac{M_1,\ldots ,M_n}\subseteq{\mathcal O} ({\mathcal K} )$ is \textit{compatible} or \textit{jointly measurable} if there exists an $M\in{\mathcal O} _n({\mathcal K} )$ such that for every $A\in{\mathcal B} (\mathbb R )$ and every $s\in{\mathcal K}$ we have
\begin{align*}
M&(s)(A\times\mathbb R\times\cdots\times\mathbb R )=M_1(s)(A)\\
M&(s)(\mathbb R\times A\times\mathbb R\times\cdots\times\mathbb R)=M_2(s)(A)\\
\vdots&\\
M&(s)(\mathbb R\times\mathbb R\times\cdots\times\mathbb R\times A)=M_n(s)(A)
\end{align*}
In this case, we call $M$ a \textit{joint observable} for $\brac{M_1,\ldots ,M_n}$ and we call $\brac{M_1,\ldots ,M_n}$ the
\textit{marginals} for $M$. It is clear that if $\brac{M_1,\ldots ,M_n}$ is compatible, then any proper subset is compatible. However, we suspect that the converse is not true. If a set of observables is not compatible we say it is
\textit{incompatible}.
It is clear that convex combinations of observables give an observable so ${\mathcal O} ({\mathcal K} )$ forms a convex set. In the same way, ${\mathcal O} _n({\mathcal K} )$ is a convex set. Another way of forming new observables is by taking functions of an observable. If $f\colon\mathbb R\to\mathbb R$ is a Borel function and $M\in{\mathcal O} ({\mathcal K} )$, the observable $f(M)\colon{\mathcal K}\to{\mathcal M} (\mathbb R )$ is defined by $f(M)(s)(A)=M(s)\paren{f^{-1}(A)}$ for all $s\in{\mathcal K}$, $A\in{\mathcal B} (\mathbb R )$.
\begin{thm}
\label{thm11}
If $M_1,M_2\in{\mathcal O} ({\mathcal K} )$ are functions of a single observable $M$, then $M_1$, $M_2$ are compatible.
\end{thm}
\begin{proof}
Suppose $M_1=f(M)$, $M_2=g(M)$ where $f$ and $g$ are Borel functions. For $A,B\in{\mathcal B} (\mathbb R )$, $s\in{\mathcal K}$ define
$\widetilde{M} (s)$ on $A\times B$ by
\begin{equation*}
\widetilde{M} (s)(A\times B)=M(s)\sqbrac{f^{-1}(A)\cap g^{-1}(B)}
\end{equation*}
By the Hahn extension theorem, $\widetilde{M} (s)$ extends to a measure in ${\mathcal M} (\mathbb R ^2)$. Hence,
$\widetilde{M}\in{\mathcal O} _2({\mathcal K} )$ and the marginals of $\widetilde{M}$ are $f(M)$ and $g(M)$. We conclude that $M_1=f(M)$ and $M_2=g(M)$ are compatible
\end{proof}
It follows from Theorem~\ref{thm11} that an observable is compatible with any Borel function of itself and in particular with itself. In a similar way we obtain the next result.
\begin{thm}
\label{thm12}
If $M_1,M_2\in{\mathcal O} ({\mathcal K} )$ are compatible and $f$, $g$ are Borel functions, then $f(M_1)$ and $g(M_2)$ are compatible.
\end{thm}
\begin{proof}
Since $M_1$, $M_2$ are compatible, they have a joint observable $M\in{\mathcal O} _2({\mathcal K} )$. For $A,B\in{\mathcal B} (\mathbb R )$,
$s\in{\mathcal K}$ define $\widetilde{M} (s)$ on $A\times B$ by
\begin{equation*}
\widetilde{M} (s)(A\times B)=M(s)\sqbrac{f^{-1}(A)\times g^{-1}(B)}
\end{equation*}
As in the proof of Theorem~\ref{thm11}, $\widetilde{M} (s)$ extends to a measure in ${\mathcal M} (\mathbb R ^2)$. Hence, $\widetilde{M}\in{\mathcal O} ({\mathcal K} )$ and the marginals of $\widetilde{M}$ are
\begin{align*}
\widetilde{M} (s)(A\times\mathbb R )&=M(s)\sqbrac{f^{-1}(A)\times\mathbb R}=M_1(s)\sqbrac{f^{-1}(A)}=f(M_1)(s)(A)\\
\widetilde{M} (s)(\mathbb R\times A)&=M(s)\sqbrac{\mathbb R\times g^{-1}(A)}=M_2(s)\sqbrac{g^{-1}(A)}=g(M_2)(s)(A)
\end{align*}
We conclude that $f(M_1)$ and $g(M_2)$ are compatible.
\end{proof}
The next result is quite useful and somewhat surprising.
\begin{thm}
\label{thm13}
Let $M_i^j\in{\mathcal O} ({\mathcal K} )$ for $i=1,\ldots ,n$, $j=1,\ldots ,m$ and suppose $\brac{M_i^1,\ldots ,M_i^m}$ is compatible, $i=1,\ldots ,n$. If $\lambda _i\in\sqbrac{0,1}$ with $\sum\lambda _i=1$, $i=1,\ldots ,n$, then
\begin{equation*}
\brac{\sum _{i=1}^n\lambda _iM_i^1,\sum _{i=1}^n\lambda _iM_i^2,\ldots ,\sum _{i=1}^n\lambda _iM_i^m}
\end{equation*}
is compatible.
\end{thm}
\begin{proof}
Let $\widetilde{M} _i\in{\mathcal O} _m({\mathcal K} )$ be the joint observable for $\brac{M_i^1,\ldots ,M_i^m}$, $i=1,\ldots ,n$. Then
$\widetilde{M}\!=\!\sum _{i=1}^n\lambda _i\widetilde{M} _i$ is an $m$-dimensional observable with marginals
\begin{align*}
\widetilde{M}&(s)(A\times\mathbb R\times\cdots\times\mathbb R)=\sum _{i=1}^n\lambda _i\widetilde{M} _i(s)(A\times\mathbb R\times\cdots\times\mathbb R )
=\sum _{i=1}^n\lambda _iM_i^1(s)(A)\\
\widetilde{M}&(s)(\mathbb R\times A\times\mathbb R\times\cdots\times\mathbb R )
=\sum _{i=1}^n\lambda _i\widetilde{M} _i(s)(\mathbb R\times A\times\mathbb R\times\cdots\times\mathbb R )\\
&\hskip 11pc =\sum _{i=1}^n\lambda _iM_i^2(s)(A)\\
\vdots&\\
\widetilde{M}&(s)(\mathbb R\times\mathbb R\times\cdots\times\mathbb R\times A)
=\sum _{i=1}^n\lambda _i\widetilde{M} _i(s)(\mathbb R\times\mathbb R\times\cdots\times\mathbb R\times A)\\
&\hskip 11pc =\sum _{I=1}^n\lambda _iM_i^m(s)(A)
\end{align*}
The result now follows
\end{proof}
\begin{cor}
\label{cor14}
Let $M,N,P\in{\mathcal O} ({\mathcal K} )$ and $\lambda\in\sqbrac{0,1}$. If $M$ is compatible with $N$ and $P$, then $M$ is compatible with $\lambda N+(1-\lambda )P$.
\end{cor}
\begin{proof}
Since $\brac{M,N}$ and $\brac{M,P}$ are compatible sets, by Theorem~\ref{thm13}, we have that
$M=\lambda M+(1-\lambda )M$ is compatible with $\lambda N+(1-\lambda )P$.
\end{proof}
\section{Noisy Observables}
If $p\in{\mathcal M} (\mathbb R )$, we define the \textit{trivial observable} $T_p\in{\mathcal O} ({\mathcal K} )$ by $T_p(s)=p$ for every $s\in{\mathcal K}$. A trivial observable represents noise in the system. We denote the set of trivial observables on ${\mathcal K}$ by ${\mathcal T} ({\mathcal K} )$. The set ${\mathcal T} ({\mathcal K} )$ is convex with
\begin{equation*}
\lambda T_p+(1-\lambda )T_q=T_{\lambda p+(1-\lambda )q}
\end{equation*}
for every $\lambda\in\sqbrac{0,1}$ and $p,q\in{\mathcal M} (\mathbb R )$. An observable $M\in{\mathcal O} ({\mathcal K} )$ is compatible with any $T_p\in{\mathcal T} ({\mathcal K} )$ and a joint observable $\widetilde{M}\in{\mathcal O} _2({\mathcal K} )$ is given by
\begin{equation*}
\widetilde{M} (s)(A\times B)=p(A)M(s)(B)
\end{equation*}
If $M\in{\mathcal O} ({\mathcal K} )$, $T\in{\mathcal T} ({\mathcal K} )$ and $\lambda\in\sqbrac{0,1}$ we consider $\lambda M+(1-\lambda )T$ as the observable $M$ together with noise. Stated differently, we consider $\lambda M+(1-\lambda )T$ to be a noisy version of $M$. The parameter $1-\lambda$ gives a measure of the proportion of noise and is called the \textit{noise index}. Smaller $\lambda$ gives a larger proportion of noise. As we shall see, incompatible observables may have compatible noisy versions.
The next lemma follows directly from Corollary~\ref{cor14}. It shows that if $M$ is compatible with $N$, then $M$ is compatible with any noisy version of $N$.
\begin{lem}
\label{lem21}
If $M\in{\mathcal O} ({\mathcal K} )$ is compatible with $N\in{\mathcal O} ({\mathcal K} )$, then $M$ is compatible with $\lambda N+(1-\lambda )T$ for any $\lambda\in\sqbrac{0,1}$ and $T\in{\mathcal T} ({\mathcal K} )$.
\end{lem}
The following lemma shows that for any $M,N\in{\mathcal O} ({\mathcal K} )$ a noisy version of $N$ with noise index $\lambda$ is compatible with any noisy version of $M$ with noise index $1-\lambda$. The lemma also shows that if $M$ is compatible with a noisy version of $N$, then $M$ is compatible with a still noisier version of $N$.
\begin{lem}
\label{lem22}
Let $M,N\in{\mathcal O} ({\mathcal K} )$ and $S,T\in{\mathcal T} ({\mathcal K} )$.
{\rm (a)}\enspace If $\lambda\in\sqbrac{0,1}$, then $\lambda M+(1-\lambda )T$ and $(1-\lambda )N+\lambda S$ are compatible.
{\rm (b)}\enspace If $M$ is compatible with $\lambda N+(1-\lambda )T$, then $M$ is compatible with $\mu N+(1-\mu )T$ where
$0\le\mu\le\lambda\le 1$.
\end{lem}
\begin{proof}
(a)\enspace Since $\brac{M,S}$ and $\brac{T,N}$ are compatible sets, by Theorem~\ref{thm13}
$\lambda M+(1-\lambda )T$ is compatible with $\lambda S+(1-\lambda )N$.
(b)\enspace We can assume that $\lambda >0$ and we let $\alpha =\mu /\lambda$ so $0\le\alpha\le 1$. Since
$\brac{M,\lambda N+(1-\lambda )T}$ and $\brac{M,T}$ are compatible sets, by Theorem~\ref{thm13},
$M=\alpha M+(1-\alpha )M$ is compatible with
\begin{align*}
\alpha\sqbrac{\lambda N+(1-\lambda )T}+(1-\alpha )T&=\alpha\lambda N+\sqbrac{\alpha (1-\lambda )+(1-\alpha )}T\\
&=\mu N+(1-\mu )T\qedhere
\end{align*}
\end{proof}
The \textit{compatibility region} $J(M_1,M_2,\ldots ,M_n)$ of observables $M_i\in{\mathcal O} ({\mathcal K} )$, $i=1,\ldots ,n$, is the set of points $(\lambda _1,\lambda _2,\ldots ,\lambda _n)\in\sqbrac{0,1}^n$ for which there exist $T_i\in{\mathcal T} ({\mathcal K} )$, $i=1,2,\ldots ,n$, such that
\begin{equation*}
\brac{\lambda _iM_i+(1-\lambda _i)T_i}_{i=1}^n
\end{equation*}
form a compatible set. Thus, $J(M_1,M_2,\ldots ,M_n)$ gives parameters for which there exist compatible noisy versions of $M_1,M_2,\ldots ,M_n$. It is clear that $0=(0,\ldots ,0)\in J(M_1,M_2,\ldots ,M_n)$ and we shall show that $J(M_1,M_2,\ldots ,M_n)$ contains many points. We do not know whether $J(M_1,M_2,\ldots ,M_n)$ is\newline
symmetric under permutations of the $M_i$. For example, is $J(M_1,M_2)=J(M_2,M_1)$?
\begin{thm}
\label{thm23}
$J(M_1,M_2,\ldots M_n)$ is a convex subset of $\sqbrac{0,1}^n$.
\end{thm}
\begin{proof}
Suppose $(\lambda _1,\ldots ,\lambda _n),(\mu _1,\ldots ,\mu _n)\in J(M_1,\ldots ,M_n)$. We must show that
\begin{align*}
\lambda (\lambda _1,\ldots ,\lambda _n)+&(1-\lambda )(\mu _1,\ldots ,\mu _n)\\
&=(\lambda\lambda _1+(1-\lambda )\mu _1,\ldots ,\lambda\lambda _n+(1-\lambda )\mu _n)\in J(M_1,\ldots ,M_n)
\end{align*}
for all $\lambda\in\sqbrac{0,1}$. Now there exist $S_1,\ldots ,S_n,T_1,\ldots ,T_n\in{\mathcal T} ({\mathcal K} )$ such that
$\brac{\lambda _iM_i+(1-\lambda _i)S_i}_{i=1}^n$ and $\brac{\mu _iM_i+(1-\mu _i)T_i}_{i=1}^n$ are compatible. By
Theorem~\ref{thm13} the set of observables
\begin{align*}
&\brac{\lambda\sqbrac{\lambda _iM_i+(1-\lambda _i)S_i}+(1-\lambda )\sqbrac{\mu _iM_i+(1-\mu _i)T_i}}\\
&\quad =\brac{(\lambda\lambda _i+(1-\lambda )\mu _i)M_i+\lambda (1-\lambda _i)S_i+(1-\lambda )(1-\mu _i)T_i}
\end{align*}
is compatible. Since
\begin{align*}
\lambda (1-\lambda _i)+(1-\lambda )(1-\mu _i)&=1-\lambda\lambda _i-\mu _i+\lambda\mu _i\\
&=1-\sqbrac{\lambda\lambda _i+(1-\lambda )\mu _i}
\end{align*}
letting $\alpha _i=\lambda\lambda _i+(1-\lambda )\mu _i$ we have that
\begin{equation*}
U_i=\frac{1}{1-\alpha _i}\sqbrac{\lambda (1-\lambda _i)S_i+(1-\lambda )(1-\mu _i)T_i}\in{\mathcal T} ({\mathcal K} )
\end{equation*}
Since $\brac{\alpha _iM_i+(1-\alpha _i)U_i}_{i=1}^n$ forms a compatible set, we conclude that\newline
$(\alpha _1,\ldots ,\alpha _n)\in J(M_1,\ldots ,M_n)$.
\end{proof}
Let $\Delta _n=\brac{(\lambda _1,\ldots ,\lambda _n)\in\sqbrac{0,1}^n\colon\sum\lambda _i\le 1}$. To show that
$\Delta _n$ forms a convex subset of $\sqbrac{0,1}^n\subseteq\mathbb R ^n$, let
$(\lambda _1,\ldots ,\lambda _n),(\mu _1,\ldots ,\mu _n)\in\Delta _n$ and $\lambda\in\sqbrac{0,1}$. Then
$\lambda (\lambda _1,\ldots ,\lambda _n+(1-\lambda )(\mu _1,\ldots ,\mu _n)\in\sqbrac{0,1}^n$ and
\begin{equation*}
\sum _{i=1}^n\sqbrac{\lambda\lambda _i+(1-\lambda )\mu _i}
=\lambda\sum\lambda _i+(1-\lambda )\sum\mu _i\le\lambda +(1-\lambda )=1
\end{equation*}
\begin{thm}
\label{thm24}
If $\brac{M_1,\ldots ,M_n}\subseteq{\mathcal O} ({\mathcal K} )$, then $\Delta _n\subseteq J(M_1,\ldots ,M_n)$.
\end{thm}
\begin{proof}
Let $\delta _0=(0,0,\ldots ,0)\in\mathbb R ^n$, $\delta _i=(0,\ldots ,0,1,0,\ldots ,0)\in\mathbb R ^n$, $i=1,\ldots ,n$ where 1 is in the $i$th coordinate. It is clear that
\begin{equation*}
\delta _i\in J(M_1,\ldots ,M_n)\cap\Delta _n,\quad i=0,1,\ldots ,n
\end{equation*}
If $\lambda =(\lambda _1,\ldots ,\lambda _n)\in\Delta _n$, letting $\mu =\sum\lambda _i$ we have that $0\le\mu\le 1$,
$\sum\lambda _i+(1-\mu )=1$ and
\begin{equation*}
\lambda =\sum _{i=1}^n\lambda _i\delta _i+(1-\mu )\delta _0
\end{equation*}
It follows that $\Delta _n$ is the convex hull of $\brac{\delta _0,\delta _1,\ldots ,\delta _n}$. Since
\begin{equation*}
\brac{\delta _0,\delta _1,\ldots ,\delta _n}\subseteq J(M_1,\ldots ,M_n)
\end{equation*}
and $J(M_1,\ldots ,M_n)$ is convex, it follows that $\Delta _n\in J(M_1,\ldots ,M_n)$.
\end{proof}
The $n$-\textit{dimensional compatibility region} for PT ${\mathcal K}$ is defined by
\begin{equation*}
J_n({\mathcal K} )=\cap\brac{J(M_1,\ldots ,M_n)\colon M_i\in{\mathcal O} ({\mathcal K} ),i=1,\ldots ,n}
\end{equation*}
We have that $\Delta _n\subseteq J_n({\mathcal K} )\subseteq\sqbrac{0,1}^n$ and $J_n({\mathcal K} )$ is a convex set that gives a measure of the incompatibility of observables on ${\mathcal K}$. As $J_n({\mathcal K} )$ gets smaller, ${\mathcal K}$ gets more incompatible and the maximal incompatibility is when $J_n({\mathcal K} )=\Delta _n$. For the case of quantum states ${\mathcal K}$, the set $J_2({\mathcal K} )$ has been considered in detail in \cite{bhs12}.
We now introduce a measure of compatibility that we believe is simpler and easier to investigate than $J_2(M,N)$ For
$M,N\in{\mathcal O} ({\mathcal K} )$, the \textit{compatibility interval} $I(M,N)$ is the set of $\lambda\in\sqbrac{0,1}$ for which there exists a $T\in{\mathcal T} ({\mathcal K} )$ such that $M$ is compatible with $\lambda N+(1-\lambda )T$. Of course, $0\in T(M,N)$ and $M$ and $N$ are compatible if and only if $1\in I(M,N)$. We do not know whether $I(M,N)=I(N,M)$. It follows from Lemma~\ref{lem22}(b) that if $\lambda\in T(M,N)$ and $0\le\mu\le\lambda$, then $\mu\in I(M,N)$. Thus, $I(M,N)$ is an interval with left endpoint 0. The \textit{index of compatibility} of $M$ and $N$ is $\lambda (M,N)=\sup\brac{\lambda\colon\lambda\in I(M,N)}$. We do not know whether $\lambda (M,N)\in I(M,N)$ but in any case $I(M,N)=\sqbrac{0,\lambda (M,N)}$ or $I(M,N)=\sqparen{0,\lambda (M,N)}$. For a PT ${\mathcal K}$, we define the \textit{interval of compatibility} for ${\mathcal K}$ to be
\begin{equation*}
I({\mathcal K} )=\cap\brac{I(M,N)\colon M,N\in{\mathcal O} ({\mathcal K} )}
\end{equation*}
The \textit{index of compatibility} of ${\mathcal K}$ is
\begin{equation*}
\lambda ({\mathcal K} )=\inf\brac{\lambda (M,N)\colon M,N\in{\mathcal O} ({\mathcal K} )}
\end{equation*}
and $I({\mathcal K} )=\sqbrac{0,\lambda ({\mathcal K} )}$ or $I({\mathcal K} )=\sqparen{0,\lambda ({\mathcal K} )}$. Again, $\lambda ({\mathcal K} )=0$ gives a measure of incompatibility of the observables in ${\mathcal O} ({\mathcal K} )$.
\bigskip
\noindent\textbf{Example 1.}\ (Classical Probability Theory)\enspace Let $(\Omega ,{\mathcal A} )$ be a measurable space and let ${\mathcal V}$ be the Banach space of real-valued measures on ${\mathcal A}$ with the total variation norm. If ${\mathcal K}$ is the
$\sigma$-convex set of probability measures on ${\mathcal A}$, then ${\mathcal K}$ generates ${\mathcal V}$. There are two types of observables on ${\mathcal K}$, the \textit{sharp} and \textit{fuzzy} observables. The sharp observables have the form $M_f$ where $f$ is a measurable function $f\colon\Omega\to\mathbb R$ and $M_f(s)(A)=s\sqbrac{f^{-1}(A)}$. If $M_f$, $M_g$ are sharp observables, form the unique 2-dimensional observable $\widetilde{M}$ satisfying
\begin{equation*}
\widetilde{M} (s)(A\times B)=s\sqbrac{f^{-1}(A)\cap g^{-1}(B)}
\end{equation*}
Then $\widetilde{M}$ is a joint observable for $M_f$, $M_g$ so $M_f$ and $M_g$ are compatible. The unsharp observables are obtained as follows. Let ${\mathcal F} (\Omega )$ be the set of measurable functions $f\colon\Omega\to\sqbrac{0,1}$. Let
$\widehat{M}\colon{\mathcal B} (\mathbb R )\to{\mathcal F} (\Omega )$ satisfy $\widehat{M} (\mathbb R )=1$, $\widehat{M} (\dot{\cup} A_i)=\sum\widehat{M} (A_i)$. An unsharp observable has the form
\begin{equation*}
M(s)(A)=\int\widehat{M} (A)ds
\end{equation*}
Two unsharp observables $M,N$ are also compatible because we can form the joint observable $\widetilde{M}$ given by
\begin{equation*}
\widehat{M} (S)(A\times B)=\int\widehat{M} (A)\widehat{N} (B)ds
\end{equation*}
We conclude that $J({\mathcal K} )=\sqbrac{0,1}\times\sqbrac{0,1}$ and $I({\mathcal K} )=\sqbrac{0,1}$ so ${\mathcal K}$ has the maximal amount of compatibility.
\bigskip
\noindent\textbf{Example 2.}\ (Quantum Theory)\enspace Let $H$ be a separable complex Hilbert space and let ${\mathcal K}$ be the $\sigma$-convex set of all trace 1 positive operators on $H$. Then ${\mathcal K}$ generates the Banach space of self-adjoint trace-class operators with the trace norm. It is well known that $M\in{\mathcal O} ({\mathcal K} )$ if and only if there exists a positive operator-valued measure (POVM) $P$ such that $M(s)(A)=\mathrm{tr}\sqbrac{sP(A)}$ for every $s\in{\mathcal K}$, $A\in{\mathcal B} (\mathbb R )$. It is shown in \cite{bhs12} that if $\dim H=\infty$, then there exist $M_1,M_2\in{\mathcal O} ({\mathcal K} )$ such that
$J_2(M_1,M_2)=\Delta _2$ and hence $J({\mathcal K} )=\Delta _2$. If $\dim H<\infty$, then $J({\mathcal K} )$ is not known, although partial results have been obtained and it is known that $J({\mathcal K} )\to\Delta _2$ as $\dim H\to\infty$
\bigskip
Now let $H$ be an arbitrary complex Hilbert space with $\dim H\ge 2$. Although the Pauli matrices $\sigma _x$, $\sigma _y$ are 2-dimensional, we can extend them from a 2-dimensional subspace $H_0$ of $H$ to all of $H$ by defining
$\sigma _x\psi =0$ for all $\psi\in H_0^\perp$. Define the POVMs $M_x$, $M_y$ on $H$ by
$M_x(\pm 1)=\tfrac{1}{2}(I\pm\sigma _x)$, $M_y(\pm 1)=\tfrac{1}{2}(I\pm\sigma _y)$. It is shown in \cite{bhs12} that
\begin{equation*}
J(M_x,M_y)=\brac{(\lambda ,\mu )\in\sqbrac{0,1}\times\sqbrac{0,1}\colon\lambda ^2+\mu ^2\le 1}
\end{equation*}
Thus, $J(M_x,M_y)$ is a quadrant of the unit disk. We conclude that $M_x$ is compatible with $\mu M_y+(1-\mu )T$ for $T\in{\mathcal T} ({\mathcal K} )$ if and only if $1+\mu ^2\le 1$. Therefore, $\mu =0$, so $I(M_x,M_y)=\brac{0}$ and $\lambda (M_x,M_y)=0$. Thus, $I({\mathcal K} )=\brac{0}$ and $\lambda ({\mathcal K} )=0$. We conclude that quantum mechanics has the smallest index of compatibility possible for a PT. The index of compatibility for a classical system is 1, so we have the two extremes. It would be interesting to find $\lambda ({\mathcal K} )$ for other PTs.
\section{Concrete Quantum Logics}
We now consider a PT that seems to be between the classical and quantum PTs of Examples~1 and~2. A collection of subsets ${\mathcal A}$ of a set $\Omega$ is a $\sigma$-\textit{class} if $\emptyset\in{\mathcal A}$, $A^c\in{\mathcal A}$ whenever $A\in{\mathcal A}$ and if $A_i$ are mutually disjoint, $i=1,2,\ldots$, then $\cup A_i\in{\mathcal A}$. If ${\mathcal A}$ is a $\sigma$-class on $\Omega$, we call
$(\Omega ,{\mathcal A} )$ a \textit{concrete quantum logic}. A $\sigma$-\textit{state} on ${\mathcal A}$ is a map
$s\colon{\mathcal A}\to\sqbrac{0,1}$ such that $s(\Omega )=1$ and if $A_i\in{\mathcal A}$ are mutually disjoint, then
$s(\cup A_i)=\sum s(A_i)$. If ${\mathcal K}$ is the set of $\sigma$-states on $(\Omega ,{\mathcal A} )$, we call ${\mathcal K}$ a
\textit{concrete quantum logic} PT. Let ${\mathcal A} _\sigma$ be the $\sigma$-algebra generated by ${\mathcal A}$. A $\sigma$-state
$s$ is \textit{classical} if there exists a probability measure $\mu$ on ${\mathcal A} _\sigma$ such that $s=\mu\mid{\mathcal A}$. As in the classical case, an observable is \textit{sharp} if it has the form $M_f(s)(A)=s\sqbrac{f^{-1}(A)}$ for an ${\mathcal A}$-measurable function $f\colon\Omega\to\mathbb R$. If $f$ and $g$ are ${\mathcal A}$-measurable functions satisfying
$f^{-1}(A)\cap g^{-1}(B)\in{\mathcal A}$ for all $A,B\in{\mathcal B} (\mathbb R )$, then $M_f$ and $M_g$ are compatible because they have a joint observable $M$ satisfying $M(s)(A\times B)=s\sqbrac{f^{-1}(A)\cap g^{-1}(B)}$ for all $s\in{\mathcal K}$, $A,B\in{\mathcal B} (\mathbb R )$. We do not know whether $M_f$ and $M_g$ compatible implies that $f^{-1}(A)\cap g^{-1}(B)\in{\mathcal A}$ holds for every
$A,B\in{\mathcal B} (\mathbb R )$, although we suspect it does not.
\bigskip
\noindent\textbf{Example 3.}This is a simple example of a concrete quantum logic. Let $\Omega =\brac{1,2,3,4}$ and let
${\mathcal A}$ be the collection of subsets of $\Omega$ with even cardinality. Then
\begin{equation*}
{\mathcal A} =\brac{\emptyset ,\Omega ,\brac{1,2},\brac{3,4},\brac{1,3},\brac{2,4},\brac{1,4},\brac{2,3}}
\end{equation*}
Let ${\mathcal K}$ be the sets of all states on ${\mathcal A}$. Letting $a=\brac{1,2}$, $a'=\brac{3,4}$, $b=\brac{1,3}$, $b'=\brac{3,4}$,
$c=\brac{1,4}$, $c'=\brac{2,3}$ we can represent an $s\in{\mathcal K}$ by
\begin{align*}
\widehat{s}&=\paren{s(a),s(a'),s(b),s(b'),s(c),s(c')}\\
&=\paren{s(a),1-s(a),s(b),1-s(b),s(c),1-s(c)}
\end{align*}
Thus, every $s\in{\mathcal K}$ has the form
\begin{equation*}
s=\paren{\lambda _1,1-\lambda _1,\lambda _2,1-\lambda _2,\lambda _3,1-\lambda _3}
\end{equation*}
for $0\le\lambda _i\le 1$, $i=1,2,3$. The pure (extremal) classical states are the 0-1 states:
$\delta _1=(1,0,1,0,1,0)$, $\delta _3=(1,0,0,1,0,1)$, $\delta _3=(0,1,1,0,0,1)$, $\delta _4=(0,1,0,1,1,0)$. The pure nonclassical states are the 0-1 states: $\gamma _1=1-\delta _1$, $\gamma _2=1-\delta _2$, $\gamma _3=1-\delta _3$,
$\gamma _4=1-\delta _4$ where $1=(1,1,1,1,1,1)$. For example, to see that $\gamma _1$ is not classical, we have that
$\gamma _1=(0,1,0,1,0,1)$. Hence,
$\gamma _1\paren{\brac{3,4}}=\gamma _1\paren{\brac{2,4}}=\gamma _1\paren{\brac{2,3}}=1$. If there exists a probability measure $\mu$ such that $\gamma _1=\mu\mid{\mathcal A}$ we would have
$\mu\paren{\brac{1}}=\mu\paren{\brac{2}}=\mu\paren{\brac{3}}=\mu\paren{\brac{4}}=0$ which is a contradiction. The collection of sharp observable is very limited because a measurable function $f\colon\Omega\to\mathbb R$ can have at most two values. Thus, if $M_f$ is a sharp observable there exists $a,b\in\mathbb R$ such that $M_f(s)\paren{\brac{a,b}}=1$ for every $s\in{\mathcal K}$. There are many observables with more than two values (non-binary observables) and these are not sharp. Even for this simple example, it appears to be challenging to investigate the region and interval of compatibility.
\section{Vector-Valued Measures}
Let ${\mathcal K}$ be a PT with generated Banach space ${\mathcal V}$ and ${\mathcal V} ^*$ be the Banach space dual of ${\mathcal V}$.
A \textit{normalized vector-valued measure} (NVM) for ${\mathcal K}$ is a map $\Gamma\colon{\mathcal B} (\mathbb R )\to{\mathcal V} ^*$ such that $A\mapsto\Gamma(A)(s)\in{\mathcal M} (\mathbb R )$ for every $s\in{\mathcal K}$. Thus, $\Gamma$ satisfies the conditions:
\begin{list} {(\arabic{cond})}{\usecounter{cond}%
\setlength\itemindent{-7pt}}
\item $\Gamma (\mathbb R )(s)=1$ for every $s\in{\mathcal K}$,
\item $0\le\Gamma (A)(s)\le 1$ for every $s\in{\mathcal K}$, $A\in{\mathcal B} (\mathbb R )$,
\item If $A_i\in{\mathcal B} (\mathbb R )$ are mutually disjoint, $i=1,2,\ldots$, then
\begin{equation*}
\Gamma (\cup A_i)(s)=\sum\Gamma (A_i)(s)
\end{equation*}
for every $s\in{\mathcal K}$.
\end{list}
This section shows that there is a close connection between observables on ${\mathcal K}$ and NVMs for ${\mathcal K}$.
\begin{thm}
\label{thm41}
If $\Gamma$ is a NVM for ${\mathcal K}$, then $M\colon{\mathcal K}\to{\mathcal M} (\mathbb R )$ given by\newline
$M(s)(A)=\Gamma (A)(s)$, $s\in{\mathcal K}$, $A\in{\mathcal B} (\mathbb R )$, is an observable on ${\mathcal K}$.
\end{thm}
\begin{proof}
Since $A\mapsto\Gamma (A)(s)\in{\mathcal M} (\mathbb R )$ we have that $A\mapsto M(s)(A)\in{\mathcal M} (\mathbb R )$. Let
$\lambda _i\in\sqbrac{0,1}$ with $\sum\lambda _i=1$, $s_i\in{\mathcal K}$, $i=1,2,\ldots$, and suppose that $s=\sum\lambda _is_i$. Then $\lim\limits _{n\to\infty}\sum\limits _{i=1}^n\lambda _is_i=s$ in norm and since $s\mapsto\Gamma (A)(s)\in{\mathcal V} ^*$, for every $A\in{\mathcal B} (\mathbb R )$ we have
\begin{align*}
M(s)(A)&=M\paren{\sum\lambda _is_i}(A)=\Gamma (A)\paren{\sum\lambda _is_i}
=\Gamma (A)\paren{\lim _{n\to\infty}\sum _{i=1}^n\lambda _is_i}\\
&=\lim _{n\to\infty}\Gamma (A)\paren{\sum _{i=1}^n\lambda _is_i}=\lim _{n\to\infty}\sum _{i=1}^n\lambda _i\Gamma (A)(s_i)\\
&=\lim _{n\to\infty}\sum _{i=1}^n\lambda _iM(s_i)(A)=\sum _{i=1}^\infty\lambda _iM(s_i)(A)
\end{align*}
It follows that $M\paren{\sum\lambda _is_i}=\sum\lambda _iM(s_i)$ so $M\in{\mathcal O} ({\mathcal K} )$.
\end{proof}
The converse of Theorem~\ref{thm41} holds if some mild conditions are satisfied. To avoid some topological and measure-theoretic technicalities, we consider the special case where ${\mathcal V}$ is finite-dimensional. Assuming that ${\mathcal K}$ is the base of a generating positive cone ${\mathcal V} ^+$, we have that every element $v\in{\mathcal V} ^+$ has a unique form $v=\alpha s$,
$\alpha \ge 0$, $s\in{\mathcal K}$ and that ${\mathcal V} ={\mathcal V} ^+\oplus{\mathcal V} ^-$ where ${\mathcal V} ^-=-{\mathcal V} ^+$ and
${\mathcal V} ^+\cap{\mathcal V} ^-=\brac{0}$. If $M\in{\mathcal O} ({\mathcal K} )$, then for every $A\in{\mathcal B} (\mathbb R )$, $s\mapsto M(s)(A)$ is a convex, real-valued function on ${\mathcal K}$. A standard argument shows that this function has a unique linear extension
$\widehat{M} (A)={\mathcal V} ^*$ for every $A\in{\mathcal B} (\mathbb R )$. Hence
\begin{equation}
\label{eq41}
\widehat{M} (A)(s)=M(s)(A)
\end{equation}
for every $s\in{\mathcal K}$, $A\in{\mathcal B} (\mathbb R )$. Since $A\mapsto\widehat{M} (A)(s)=M(s)(A)\in{\mathcal M} (\mathbb R )$ we conclude that
$A\mapsto\widehat{M} (A)$ is a NVM and $\widehat{M}$ is the unique NVM satisfying \eqref{eq41}. It follows that the converse of
Theorem~\ref{thm41} holds in this case.
\bigskip
\noindent\textbf{Example $1'$.}\ (Classical Probability Theory)\enspace In this example ${\mathcal V} ^*$ is the Banach space of bounded measurable functions $f\colon\Omega\to\mathbb R$ with norm $\doubleab{f}=\sup\ab{f(\omega )}<\infty$ and duality given by
\begin{equation*}
\elbows{\mu ,f}=f(\mu )=\int fd\mu
\end{equation*}
The function $1(\omega )=1$ for every $\omega\in\Omega$ is the natural unit satisfying $1(\mu )=1$ for every $\mu\in{\mathcal K}$. In this case, ${\mathcal K}$ is a base for the generating positive cone ${\mathcal V} ^+$ of bounded measures and the converse of Theorem~\ref{thm41} holds. Then a NVM $\Gamma$ has the form $0\le\Gamma (A)(\omega )\le 1$ for every
$A\in{\mathcal B} (\mathbb R )$, $\omega\in\Omega$ and $\Gamma (\mathbb R )=1$. Thus $\Gamma (A)\in{\mathcal F} (\Omega )$ and if $M$ is the corresponding observable, then
\begin{equation*}
M(\mu )(A)=\Gamma (A)(\mu )=\int\Gamma (A)d\mu
\end{equation*}
In particular, if $T_p\in{\mathcal T} ({\mathcal K} )$ then the corresponding NVM $\Gamma _p$ has the form
\begin{equation*}
\Gamma _p(A)(\mu )=T_p(\mu )(A)=p(A)
\end{equation*}
so $\Gamma _p(A)$ is the constant function $p(A)$. Moreover, if $M_p\in{\mathcal O} ({\mathcal K} )$ is sharp, then the corresponding NVM $\Gamma _f$ satisfies
\begin{equation*}
\int\Gamma _f(A)d\mu=\Gamma _f(A)(\mu )=M_f(\mu )(A)=\mu\sqbrac{f^{-1}(A)}=\int\chi _{f^{-1}(A)}d\mu
\end{equation*}
Hence, $\Gamma _f(A)=\chi _{f^{-1}(A)}$ for every $A\in{\mathcal B} (\mathbb R )$.
\bigskip
\noindent\textbf{Example $2'$.}\ (Quantum Theory)\enspace In this example ${\mathcal V} ^*$ is the Banach space ${\mathcal B} (H)$ of bounded linear operators on $H$ with norm
\begin{equation*}
\doubleab{L}=\sup\brac{\doubleab{L\psi}\colon\doubleab{\psi}=1}
\end{equation*}
and duality given by
\begin{equation*}
\elbows{s,L}=L(a)=\mathrm{tr}(sL)
\end{equation*}
The identity operator $I$ is the natural unit satisfying $I(s)=1$ for all $s\in{\mathcal K}$. In this case, ${\mathcal K}$ is a base for the generating cone ${\mathcal V} ^+$ of positive trace class operators and the converse of Theorem~\ref{thm41} holds, If $\Gamma$ is a NVM, then $\Gamma (A)$ is a positive operator satisfying $0\le\Gamma (A)\le I$ called an \textit{effect} and
$\Gamma (\mathbb R )=I$. According to the converse of Theorem~\ref{thm41}, if $M$ is an observable, then there exists a POVM
$\Gamma$ such that
\begin{equation*}
M(s)(A)=\mathrm{tr}\sqbrac{s\Gamma (A)}
\end{equation*}
for every $s\in{\mathcal K}$ and $A\in{\mathcal B} (\mathbb R )$. In particular, if $T_p\in{\mathcal T} ({\mathcal K} )$, then the corresponding NVM
$\Gamma _p$ has the form
\begin{equation*}
\mathrm{tr}\sqbrac{s\Gamma _p(A)}=\Gamma _p(A)(s)=T_p(s)(A)=p(A)=\mathrm{tr}\sqbrac{sp(A)I}
\end{equation*}
so $\Gamma _p(A)=p(A)I$ for all $A\in{\mathcal B} (\mathbb R )$.
\bigskip
Similar to a NVM, we define an $n$-\textit{dimensional} NVM to be a map\newline
$\Gamma\colon{\mathcal B} (\mathbb R ^n)\to{\mathcal V} ^*$ such that $A\mapsto\Gamma (A)(s)\in{\mathcal M} (\mathbb R ^b)$ for every $s\in{\mathcal K}$. Moreover, a set $\brac{\Gamma _1,\ldots ,\Gamma _n}$ of NVMs for ${\mathcal K}$ is compatible if there exists an $n$-dimensional NVM $\Gamma$ such that
\begin{align*}
\Gamma (A&\times\mathbb R\times\cdots\times\mathbb R )=\Gamma _1(A)\\
&\vdots\\
\Gamma (\mathbb R&\times\mathbb R\times\cdots\times\mathbb R\times A)=\Gamma _n(A)
\end{align*}
for every $A\in{\mathcal B} (\mathbb R )$. The proof of the following theorem is straightforward.
\begin{thm}
\label{thm42}
If $\brac{M_1,\ldots ,M_n}\subseteq{\mathcal O} ({\mathcal K} )$ and $\brac{\Gamma _1,\ldots\Gamma _n}$ are the corresponding NVM for ${\mathcal K}$, then $\brac{M_1,\ldots ,M_n}$ are compatible if and only if $\brac{\Gamma _1,\ldots ,\Gamma _n}$ are compatible.
\end{thm}
|
3,212,635,537,907 | arxiv | \section{#2
\newcommand{\Sec}[1]{{Sec.~\ref{sec:#1}}}
\renewcommand{\subsection}{\@startsection{subsection}{2}{0pt}%
{-3.25ex plus -1ex minus -.2ex}%
{1.5ex plus .2ex}%
{\centering\normalsize\itshape}}
\newcommand{\startappendices}{%
\setcounter{equation}{0}%
\setcounter{section}{1}%
\setcounter{subsection}{1}%
\renewcommand{\thesection}{\Alph{section}}}
\newcommand\fakesection{\@startsection {section}{1}{\z@}%
{-3.5ex \@plus -1ex \@minus -.2ex}%
{2.3ex \@plus.2ex}%
{\centering\normalsize\bfseries}}
\newcounter{appendixcount}%
\setcounter{appendixcount}{0}%
\renewcommand{ |
3,212,635,537,908 | arxiv | \section{Introduction}
\subsection{The results}
\label{ov}
The goal of this paper is to discuss rationality of smooth Fano threefolds over algebraically non-closed fields of characteristic~$0$.
In~\cite{KP19} we considered the case of geometrically rational Fano threefolds with geometric Picard number~$\uprho(X_{\bar{\mathsf{k}}}) = 1$
and here we switch the focus to the case of geometrically rational Fano threefolds~$X$ with Picard numbers
\begin{equation}
\label{eq:rho-assumptions}
\uprho(X) = 1
\qquad\text{and}\qquad
\uprho(X_{\bar{\mathsf{k}}}) > 1.
\end{equation}
In fact, Fano threefolds satisfying~\eqref{eq:rho-assumptions} have been classified in~\cite{Prokhorov-GFano-2},
and~\cite{Alzati-Bertolini-1992a} explains which of these are geometrically rational.
A combination of these results gives the following
\begin{theorem}[{\cite[Theorem~1.2]{Prokhorov-GFano-2}}, {{\cite{Alzati-Bertolini-1992a}}}]
\label{thm:classif}
There are exactly six families of geometrically rational Fano threefolds satisfying~\eqref{eq:rho-assumptions}
as listed in Table~\xref{table:fanos}.
\begin{table}[h]
\begin{tabular}
{lcccccp{0.5\textwidth}}
\label{ta:res1}
\\\hline
\\[-2ex]
&$\upiota(X_{{\bar{\mathsf{k}}}})$&$\uprho(X_{{\bar{\mathsf{k}}}})$
&$-K_{X}^3$
&${\mathrm{g}}(X)$
&$\mathrm{h}^{1,2}(X_{{\bar{\mathsf{k}}}})$\raisebox{-1.5ex}{\ }
&\multicolumn{1}{c}{$X_{{\bar{\mathsf{k}}}}$}\\
\hline
\\[-2ex]
\type{3,3}
& $1$&$2$& $20$ & $11$ & $3$ &
an intersection of three divisors of bidegree $(1,1)$ in~$\mathbb{P}_{{\bar{\mathsf{k}}}}^3\times \mathbb{P}_{{\bar{\mathsf{k}}}}^3$;
\\ [5pt]
\type{1,1,1,1}
& $1$
& $4$& $24$& $13$ & $1$ &
a divisor of multidegree $(1,1,1,1)$ in~$(\mathbb{P}_{{\bar{\mathsf{k}}}}^1)^4$;
\\ [5pt]
\type{4,4}
& $1$& $2$& $28$& $15$ & $0$ &
the blow-up of a
smooth quadric $Q_{{\bar{\mathsf{k}}}} \subset \mathbb{P}_{{\bar{\mathsf{k}}}}^4$ along a linearly normal smooth rational quartic curve;
\\ [5pt]
\type{2,2,2}
& $1$
& $3$& $30$& $16$ & $0$ &
an intersection of three divisors
of multidegrees~$(0,1,1)$, $(1,0,1)$, $(1,1,0)$ in~$\mathbb{P}_{{\bar{\mathsf{k}}}}^2\times \mathbb{P}_{{\bar{\mathsf{k}}}}^2\times \mathbb{P}_{{\bar{\mathsf{k}}}}^2$;
\\
\type{2,2}
& $2$
&$2$& $48$& $25$ & $0$ &
a divisor of bidegree $(1,1)$ in~$\mathbb{P}_{{\bar{\mathsf{k}}}}^2\times \mathbb{P}_{{\bar{\mathsf{k}}}}^2$;
\\ [5pt]
\type{1,1,1}
& $2$
& $3$& $48$ & $25$ & $0$ &
$\mathbb{P}_{{\bar{\mathsf{k}}}}^1\times \mathbb{P}_{{\bar{\mathsf{k}}}}^1\times \mathbb{P}_{{\bar{\mathsf{k}}}}^1$.\raisebox{-1.5ex}{\ }
\\\hline
\end{tabular}
\smallskip
\caption{Geometrically rational Fano threefolds~$X$
satisfying~\eqref{eq:rho-assumptions}}
\label{table:fanos}
\end{table}
\end{theorem}
The first column of Table~\ref{table:fanos} contains the name for the family we use in this paper,
the next columns contain the index~$\upiota(X_{\bar{\mathsf{k}}})$, defined as
\begin{equation*}
\upiota(X_{\bar{\mathsf{k}}}) = \max\left\{ i \ \left|\ \tfrac1i K_{X_{\bar{\mathsf{k}}}} \in \Pic(X_{\bar{\mathsf{k}}}) \right.\right\},
\end{equation*}
the geometric Picard number~$\uprho(X_{\bar{\mathsf{k}}})$, the anticanonical
degree~$(-K_X)^3$, the genus~${\mathrm{g}}(X)$, defined by
\begin{equation*}
(-K_X)^3= 2{\mathrm{g}}(X) - 2
\end{equation*}
and the Hodge number $\mathrm{h}^{1,2}(X_{\bar{\mathsf{k}}})$ of the threefold,
while the last column provides a geometric description of these varieties
over an algebraic closure~${\bar{\mathsf{k}}}$ of the base field.
We discuss some geometric properties of threefolds from Table~\ref{table:fanos} in~\S\ref{sec:contractions}.
In particular, we describe their extremal contractions over~${\bar{\mathsf{k}}}$ and identify their Hilbert schemes of lines and conics,
as well as the subschemes of the Hilbert schemes of twisted cubic curves passing through a general point.
However, our main interest is in rationality criteria, and the next theorem is our main result.
\begin{theorem}
\label{thm:unirat}
Let $X$ be a Fano threefold from Table~\xref{table:fanos};
in particular we assume~$\uprho(X) = 1$.
\begin{enumerate}
\item
\label{thm:unirat:unirat}
$X$ is unirational if and only if~$X({\mathsf{k}}) \ne \varnothing$.
\item
\label{thm:unirat:rat}
If $X$ has type~\type{4,4}, \type{2,2,2}, \type{2,2}, or~\type{1,1,1}
then~$X$ is ${\mathsf{k}}$-rational if and only if $X({\mathsf{k}}) \ne \varnothing$.
\item
\label{thm:unirat:nonrat}
If $X$ has type~\type{3,3} then~$X$ is never
${\mathsf{k}}$-rational.
\end{enumerate}
\end{theorem}
Note that over an algebraically closed field threefolds of types~\type{4,4}, \type{2,2,2}, \type{2,2},
and~\type{1,1,1} have~$\mathrm{h}^{1,2} = 0$, hence trivial intermediate Jacobians, while the intermediate Jacobians of threefolds
of types~\type{3,3} and~\type{1,1,1,1} over~${\bar{\mathsf{k}}}$ are Jacobians of curves of genus~$3$ and~$1$, respectively
(and ${\mathsf{k}}$-forms of these over~${\mathsf{k}}$);
this explains the difference in the behavior.
It is a classical fact that the existence of a ${\mathsf{k}}$-point is necessary for rationality or unirationality,
so the major part of the proof of the theorem consists of proving rationality or unirationality under this assumption.
We use for this a case-by-case analysis
(see~\S\ref{subsec:sketch} for a description of our approach).
The theorem is thus a combination of the following results (we assume everywhere~$X({\mathsf{k}}) \ne \varnothing$):
\begin{itemize}
\item
rationality for threefolds of type~\type{1,1,1} is proved in Corollary~\ref{cor:x111};
\item
rationality for threefolds of type~\type{2,2} is proved in Proposition~\ref{prop:x22};
\item
rationality for threefolds of type~\type{2,2,2} is proved in Proposition~\ref{prop:x222};
\item
rationality for threefolds of type~\type{4,4} is proved in Proposition~\ref{prop:x44};
\item
unirationality for threefolds of type~\type{1,1,1,1} is proved in Proposition~\ref{prop:x1111};
\item
unirationality for threefolds of type~\type{3,3} is proved in Proposition~\ref{prop:x33};
\item
non-rationality for threefolds of type~\type{3,3} is proved in Corollary~\ref{cor:x33}.
\end{itemize}
Theorem~\ref{thm:unirat} provides nice criteria for rationality of the five out of six types of Fano threefolds listed in Table~\ref{table:fanos}.
For the remaining type~\type{1,1,1,1} we have a conjecture and a partial result.
\begin{conjecture}
\label{conj:x1111}
If $X$ has type~\type{1,1,1,1} and~$\uprho(X) = 1$ then~$X$ is never ${\mathsf{k}}$-rational.
\end{conjecture}
To explain the partial result we need to introduce some notation.
Let~$X$ be a Fano threefold of type~\type{1,1,1,1}.
As we show in Lemma~\ref{lemma:picard}, the action of the Galois group~${\mathrm{G}}({\bar{\mathsf{k}}}/{\mathsf{k}})$ on~$\Pic(X_{\bar{\mathsf{k}}})$
factors through the group~$\mathfrak{S}_4$ that acts by permutations of the pullbacks of the point classes of the factors of the ambient~$(\mathbb{P}_{{\bar{\mathsf{k}}}}^1)^4$,
and the assumption~$\uprho(X) = 1$ means that the subgroup
\begin{equation*}
{\mathrm{G}}_X := \Ima({\mathrm{G}}({\bar{\mathsf{k}}}/{\mathsf{k}}) \longrightarrow \mathfrak{S}_4) \subset \mathfrak{S}_4
\end{equation*}
is \emph{transitive},
hence belongs to the following list of (conjugacy classes) of transitive subgroups of~$\mathfrak{S}_4$:
\begin{equation*}
\label{eq:transitive-subgroups}
{\mathrm{G}}_X \in \{ \mathfrak{S}_4, \mathfrak{A}_4, \mathrm{D}_4, \mathrm{V}_4, \mathrm{C}_4 \},
\end{equation*}
where~$\mathfrak{A}_4$ is the alternating subgroup,
$\mathrm{D}_4$ is the dihedral group of order~8 (a Sylow 2-subgroup in~$\mathfrak{S}_4$),
$\mathrm{V}_4$ is the Klein group of order~$4$,
and~$\mathrm{C}_4$ is the cyclic group of order~$4$.
Note that all of these groups contain~$\mathrm{V}_4$ except for~$\mathrm{C}_4$.
\begin{theorem}
\label{thm:x1111-non-st-rat}
Let~${\mathrm{G}} \subset \mathfrak{S}_4$ be a subgroup containing the Klein group~$\mathrm{V}_4 \subset \mathfrak{S}_4$.
Let~${\mathsf{k}}$ be a field such that there is an epimorphism~${\mathrm{G}}({\bar{\mathsf{k}}}/{\mathsf{k}}) \twoheadrightarrow {\mathrm{G}}$.
Then for the field of rational functions~$K = {\mathsf{k}}(t)$ there exists a variety~$X$ over~$K$ of type~\type{1,1,1,1}
such that~${\mathrm{G}}_X = {\mathrm{G}}$, $\uprho(X) = 1$, and~$X(K) \ne \varnothing$,
but~$X$ is not stably rational over~$K$.
\end{theorem}
\subsection{The proofs}
\label{subsec:sketch}
For (uni)rationality constructions it is natural to use ${\mathsf{k}}$-Sarkisov links:
\begin{equation}
\label{eq:sl-general}
\vcenter{
\xymatrix{
&
\tilde X \ar[dl]_{\sigma}\ar@{-->}^{\psi}[rr] \ar[dr]^{\phi}
&&
\tilde X^+ \ar[dr]^{\sigma_+} \ar[dl] \ar[dl]_{\phi_+}
\\
X &&
\bar X &&
X^+,
} }
\end{equation}
where $\sigma$ is the blowup of a ${\mathsf{k}}$-irreducible subvariety,
$\phi$ and $\phi_+$ are small crepant birational contractions,
$\psi$ is a flop, and
$\sigma_+$ is a Mori extremal contraction.
Note that such a link is completely determined by the center of the blowup~$\sigma$ ---
the contractions and the flop are obtained by the ${\mathsf{k}}$-Minimal Model Program applied to~${\tilde{X}}$
(note that~$\uprho({\tilde{X}}) = 2$, so the output of the MMP is unambiguous);
in particular the link is defined over~${\mathsf{k}}$.
For our purpose it is enough to consider two types of Sarkisov links:
\begin{itemize}
\item
Sarkisov links where~$\sigma$ is the blowup of a ${\mathsf{k}}$-point;
\item
Sarkisov links where~$\sigma$ is the blowup of a reduced ${\mathsf{k}}$-irreducible singular conic.
\end{itemize}
We construct the corresponding links accurately for threefolds of type~\type{4,4} in~\S\ref{sec:x44} (see Theorem~\ref{theorem:x44-links})
by using standard MMP arguments.
Of course, a similar construction could be given for other types of Fano threefolds from Table~\ref{table:fanos},
but to make the argument less tedious we use the fact that all other among these threefolds are ${\mathsf{k}}$-forms
of complete intersections in products of projective spaces and deduce the required (uni)rationality constructions
from an appropriate birational transformation for a product of projective spaces.
With this goal in mind we construct in~\S\ref{sec:toric}
a toric birational transformation between the product~$(\mathbb{P}^n)^r$ of projective spaces
and a $\mathbb{P}^r$-bundle over the product~$(\mathbb{P}^{n-1})^r$ of smaller projective spaces,
see Theorem~\ref{proposition:toric-link}
(in fact, we construct a birational transformation in a slightly more general situation,
but the setup described above is the only one that we need for applications in the paper).
This theorem has a consequence of independent interest, Corollary~\ref{corollary:product-rational},
saying that a ${\mathsf{k}}$-form of a product of projective spaces is ${\mathsf{k}}$-rational if and only if it has a ${\mathsf{k}}$-point.
This corollary immediately gives the required rationality construction for Fano threefolds of type~\type{1,1,1} (Corollary~\ref{cor:x111}),
and with a bit of more work provides rationality constructions for threefolds of types~\type{2,2} (Proposition~\ref{prop:x22})
and~\type{2,2,2} (Proposition~\ref{prop:x222})
as well as a unirationality construction for threefolds of type~\type{1,1,1,1} (Proposition~\ref{prop:x1111}).
In the case of a variety~$X$ of type~\type{3,3} with a ${\mathsf{k}}$-point~$x$
we again use the toric transformation of Theorem~\ref{proposition:toric-link}
to construct a birational equivalence of~$X$ with a divisor~$X^+$ of bidegree~$(2,2)$ in a ${\mathsf{k}}$-form of~$\mathbb{P}^2 \times \mathbb{P}^2$.
If~$x$ lies on a ${\bar{\mathsf{k}}}$-line in~$X$, we check that~$X^+$ contains a ${\mathsf{k}}$-form of the quadric surface~$\mathbb{P}^1 \times \mathbb{P}^1$
and use this to deduce unirationality of~$X$ (Proposition~\ref{prop:x33}).
If~$x$ does not lie on a line, we check in Proposition~\ref{prop:conic-bundle-x33}
that~$X^+$ described above is, in fact, the mid-point of a Sarkisov link,
that ends with a conic bundle over~$\mathbb{P}^2$ which has a smooth quartic curve~$\Gamma \subset \mathbb{P}^2$ as discriminant.
We also check that the discriminant double covering~$\tilde\Gamma \to \Gamma$
associated to
this conic bundle
is trivial over a quadratic extension~${\mathsf{k}}'$ of the base field~${\mathsf{k}}$ but nontrivial over~${\mathsf{k}}$,
and that the conic bundle has a rational section over~${\mathsf{k}}'$.
We check in Theorem~\ref{prop:non-rationality-conic-bundle} that these geometric properties
characterize the non-rational conic bundles constructed by Benoist and Wittenberg in~\cite{BW}
and deduce in Corollary~\ref{cor:x33} non-rationality of~$X$ from~\cite[Proposition~3.4]{BW}.
In the last part of the paper, \S\ref{sec:x1111}, we discuss Fano threefolds of type~\type{1,1,1,1}.
To prove Theorem~\ref{thm:x1111-non-st-rat} we use a degeneration technique.
Namely we construct a family of Fano threefolds of type~\type{1,1,1,1} over~$\mathbb{P}^1_{\mathsf{k}}$
with special fiber a singular toric threefold (with ordinary double points)
which is well-known not to be stably rational.
Since stable rationality is specialization-closed by a result of Nicaise and Shinder~\cite{NS},
we conclude that the general fiber of the constructed family is also not stably rational.
\subsection*{Acknowledgements.}
We would like to thank Sergey Gorchinskiy, Zhenya Shinder
and Costya Shramov for useful discussions. We are also grateful to the anonymous
referee for correcting a mistake in the original statement of Theorem~\ref{thm:x1111-non-st-rat} 1.4 and for useful
comments. This work was performed at the Steklov International Mathematical Center
and supported by the Ministry of Science and Higher Education of the Russian Federation
(agreement no. 075-15-2022-265). The paper was also partially supported by the HSE
University Basic Research Program.
\section{Extremal contractions and Hilbert schemes of curves}
\label{sec:contractions}
In this section we describe the geometry of Fano threefolds of index~$1$ from Table~\ref{table:fanos}.
In particular, we describe their extremal contractions over~${\bar{\mathsf{k}}}$
as well as their Hilbert schemes of lines and conics, and Hilbert schemes of twisted cubic curves passing through a fixed point.
To start with, recall that for most Fano threefolds the anticanonical linear system is very ample and
the anticanonical image is an intersection of quadrics;
in fact Fano threefolds which do not enjoy these nice properties (hyperelliptic and trigonal ones)
have been classified and listed in~\cite{Iskovskikh-1980-Anticanonical}.
It is easy to check that Fano threefolds from Table~\ref{table:fanos} are not in this list;
therefore we obtain
\begin{theorem}
[{\cite[Chapter~2, Theorems~2.2 and~3.4]{Iskovskikh-1980-Anticanonical}}]
\label{th:bht}
Let $X$ be a Fano threefold from Table~\textup{\ref{table:fanos}}.
The anticanonical class $-K_X$ is very ample and the anticanonical image
\begin{equation*}
X = X_{2g-2} \subset \mathbb{P}^{g+1}
\end{equation*}
is an intersection of quadrics \textup(as a scheme\textup), where $g =
{\mathrm{g}}(X)$.
\end{theorem}
\subsection{Contractions over~${\bar{\mathsf{k}}}$}
\label{subsec:contractions}
Assume~$X$ is a Fano threefold of index~$1$ from Table~\ref{table:fanos}, i.e.,
a threefold of either of types~\type{2,2,2}, \type{4,4}, \type{3,3},
\type{1,1,1,1}.
Then there is an embedding
\begin{equation}
\label{eq:x-y}
X_{\bar{\mathsf{k}}} \subset Y \cong (\mathbb{P}^n)^r,
\end{equation}
(we will see in Lemma~\ref{lemma:picard} that~$r = \uprho(X_{\bar{\mathsf{k}}})$, hence the notation), where
\begin{equation*}
(n,r) = (2,3),\ (4,2),\ (3,2),\ \text{or}\ (1,4).
\end{equation*}
Indeed, for types~\type{2,2,2}, \type{3,3}, \type{1,1,1,1} this holds by definition
and for type~\type{4,4} this follows from the following
\begin{lemma}
\label{lemma:X44}
Let $\Gamma_1 \subset Q_1 \subset \mathbb{P}^4$ be a linearly normal smooth rational quartic curve in a smooth quadric threefold.
If~$H_1$ is the hyperplane class of~$Q_1$ then the linear system~$|2H_1 - \Gamma_1|$ of quadrics through~$\Gamma_1$
defines a birational morphism $\pi_2 \colon \Bl_{\Gamma_1}Q_1 \to Q_2 \subset \mathbb{P}^4$ onto
another smooth quadric threefold~$Q_2$
and this morphism is itself the blowup of a linearly normal smooth rational
quartic curve~$\Gamma_2 \subset Q_2$,
so that
\begin{equation*}
\Bl_{\Gamma_1}(Q_1) \cong \Bl_{\Gamma_2}(Q_2).
\end{equation*}
Moreover, if~$X$ is a Fano threefold of type~\type{4,4} there is a natural embedding
\begin{equation*}
X_{\bar{\mathsf{k}}} \xhookrightarrow{\hspace{1.1em}} Q_1 \times Q_2 \subset \mathbb{P}^4 \times \mathbb{P}^4
\end{equation*}
such that~$-K_{X_{\bar{\mathsf{k}}}}$ is the sum of the pullbacks of the hyperplane classes of
the factors.
\end{lemma}
\begin{proof}
The curve~$\Gamma_1$ is an intersection of six quadrics in~$\mathbb{P}^4$; therefore it is an intersection of five quadrics in~$Q_1$.
Hence, if~$E_1$ is the exceptional divisor of the blowup~$\pi_1 \colon X_{{\bar{\mathsf{k}}}} \to Q_1$
and~$H_1$ is the pullback of the hyperplane class of~$Q_1$,
the linear system~$|2H_1-E_1|$ on~$X_{\bar{\mathsf{k}}}$ is 4-dimensional and base point free.
Therefore, this linear system defines a morphism~$\pi_2 \colon \Bl_{\Gamma_1}(Q_1)\to \mathbb{P}^4$;
moreover, standard intersection theory gives~$(2H_1 - E_1)^3 = 2$.
Hence the image of~$\pi_2$ (which is not contained in a hyperplane by definition)
is a quadric~\mbox{$Q_2 \subset \mathbb{P}^4$} and~$\pi_2$ is birational.
Since $-K_{X_{\bar{\mathsf{k}}}}$ is ample on the fibers of~$\pi_2$ and $\uprho(X_{\bar{\mathsf{k}}}) = 2$,
we see that~$\pi_2$ is an extremal Mori contraction.
By \cite{Mori-1982} the quadric~$Q_2$ is smooth and~$\pi_2$ is the blowup of a curve which must be a linearly normal smooth rational quartic curve.
For the last statement just note that~$H_1 + (2H_1 - E_1) = 3H_1 - E_1$ is the anticanonical class of~$X_{\bar{\mathsf{k}}}$.
\end{proof}
We denote by~$H_i$, $1 \le i \le r$, the pullbacks to~$Y = (\mathbb{P}^n)^r$ of the hyperplane classes of the factors and,
abusing the notation, also their restrictions to~$X_{\bar{\mathsf{k}}}$ via the embedding~\eqref{eq:x-y}.
\begin{lemma}
\label{lemma:picard-simple}
If~$X$ is a threefold of either of
types~\type{2,2,2}, \type{4,4}, \type{3,3}, \type{1,1,1,1} then
the Picard group~$\Pic(X_{\bar{\mathsf{k}}})$ is freely generated by the classes~$H_i$:
\begin{equation*}
\Pic(X_{\bar{\mathsf{k}}}) = \bigoplus_{i=1}^r \mathbb{Z} H_i.
\end{equation*}
Moreover,
\begin{equation}
\label{eq:kx}
-K_{X_{\bar{\mathsf{k}}}} = H := H_1 + \dots + H_r;
\end{equation}
\end{lemma}
\begin{proof}
For type~\type{4,4} this follows from Lemma~\ref{lemma:X44},
and for the other types the first statement follows from the Lefschetz Hyperplane Theorem
and the second from adjunction and the description of Table~\ref{table:fanos}.
\end{proof}
For each subset $I \subset \{1,\dots,r\}$ we consider the projection
\begin{equation}
\label{eq:pi-i}
\pi_I \colon X_{\bar{\mathsf{k}}} \xhookrightarrow{\hspace{1.1em}}
Y \longrightarrow \prod_{i \in I} \mathbb{P}^n \cong (\mathbb{P}^n)^{|I|}.
\end{equation}
Especially useful are the morphisms~$\pi_I$ for $I$ of cardinality~$r - 1$,
so we introduce the notation
\begin{equation*}
{\widehat{\imath}} := \{1,\dots,r\} \setminus \{i\}
\end{equation*}
and write
\begin{equation}
\label{eq:pi-hi}
\pi_{\widehat{\imath}} \colon X_{\bar{\mathsf{k}}} \longrightarrow (\mathbb{P}^n)^{r-1}
\end{equation}
for the corresponding morphisms.
Note that in the case~$r = 2$ we have~${\widehat{\imath}} = \{ 3 - i \}$, so these morphisms are the same as morphisms~$\pi_{3-i}$.
The next lemma describes~$X_{\bar{\mathsf{k}}}$ in terms of the~$\pi_{\widehat{\imath}}$.
\begin{lemma}
\label{lemma:pi-i}
The morphism~$\pi_{\widehat{\imath}}$ is birational onto its image and the exceptional divisor~$E_{\widehat{\imath}}$ of~$\pi_{\widehat{\imath}}$ is irreducible.
More precisely, the morphism~$\pi_{\widehat{\imath}}$ identifies~$X_{\bar{\mathsf{k}}}$ as follows:
\begin{enumerate}
\item\label{lemma:pi-i:222}
if $X$ has type~\type{2,2,2} the map~$\pi_{\widehat{\imath}}$ is the blowup of a smooth divisor~$W_{\widehat{\imath}} \subset \mathbb{P}^2 \times \mathbb{P}^2$ of bidegree~$(1,1)$
along a smooth rational curve~$\Gamma_{\widehat{\imath}} \subset W_{\widehat{\imath}}$ of bidegree~$(2,2)$ whose projections to the factors~$\mathbb{P}^2$ are closed embeddings;
the divisor class~$H_i$ is equal to~$\sum_{j\ne i}H_j - E_{\widehat{\imath}}$;
\item\label{lemma:pi-i:44}
if $X$ has type~\type{4,4} the map~$\pi_i$ is the blowup of a $3$-dimensional
quadric~$Q_{i}$ along a smooth linearly normal rational curve~$\Gamma_{i} \subset Q_{i}$ of degree~$4$;
the divisor class~$H_{{\widehat{\imath}}}$ is equal to~$2H_{i} - E_i$;
\item\label{lemma:pi-i:33}
if $X$ has type~\type{3,3} the map~$\pi_i$ is the blowup of~$\mathbb{P}^3$ along a
smooth curve~$\Gamma_{i} \subset \mathbb{P}^3$ of genus~$3$ and degree~$6$;
the divisor class~$H_{{\widehat{\imath}}}$ is equal to~$3H_{i} - E_i$;
\item
\label{lemma:pi-i:x1111}
if $X$ has type~\type{1,1,1,1} the map~$\pi_{\widehat{\imath}}$ is the blowup of~$\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$
along a smooth elliptic curve~$\Gamma_{\widehat{\imath}} \subset (\mathbb{P}^1)^3$ of multidegree~$(2,2,2)$;
the divisor class~$H_i$ is equal to~$\sum_{j\ne i}H_j - E_{\widehat{\imath}}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Part~\ref{lemma:pi-i:44} is proved in Lemma~\ref{lemma:X44}.
So, assume~$X$ is a variety of either of types~\type{2,2,2}, \type{3,3}, or~\type{1,1,1,1}.
Birationality of the projection~$\pi_{\widehat{\imath}}$ is clear from the descriptions of Table~\ref{table:fanos};
it also follows that all fibers of~$\pi_{\widehat{\imath}}$ are linear subspaces in~$\mathbb{P}^n$
and~$-K_{X_{\bar{\mathsf{k}}}}$ restricts to each of them as the hyperplane class by~\eqref{eq:kx}.
Also, it is easy to see that the image of~$\pi_{\widehat{\imath}}$ is smooth in all cases
(for type~\type{2,2,2} if~$W_{\widehat{\imath}} \subset \mathbb{P}^2 \times \mathbb{P}^2$ is singular
then its preimage in~$\mathbb{P}^2 \times \mathbb{P}^2 \times \mathbb{P}^2$ is singular along a plane,
hence~$X_{\bar{\mathsf{k}}}$, which is the intersection of this preimage with two other divisors, must be singular;
and for types~\type{3,3} and~\type{1,1,1,1} the image is just~$\mathbb{P}^3$ or~$\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$, respectively).
By Lemma~\ref{lemma:picard-simple} the relative Picard number of~$\pi_{\widehat{\imath}}$
is~$1$ and~$-K_{X_{\bar{\mathsf{k}}}}$ is ample, hence~$\pi_{\widehat{\imath}}$ is an extremal Mori contraction.
Since both the source and target of~$\pi_{\widehat{\imath}}$ are smooth,
it follows from~\cite{Mori-1982} that the morphism~$\pi_{\widehat{\imath}}$ is either the blowup of a smooth curve or the blowup of a smooth point.
In the latter case the restriction of~$-K_{X_{\bar{\mathsf{k}}}}$ to the nontrivial fiber~$\mathbb{P}^2$ of~$\pi_{\widehat{\imath}}$ would be isomorphic to~${\mathscr{O}}_{\mathbb{P}^2}(2)$,
contradicting to the above observation, hence~$\pi_{\widehat{\imath}}$ is the blowup of a smooth curve.
The remaining assertions
are easy and left to the reader (see also \cite{Mori1981-82}).
\end{proof}
\begin{lemma}
\label{lemma:picard}
The classes~$H_i$ are semiample and generate the nef cone of~$X_{\bar{\mathsf{k}}}$.
The Galois group~${\mathrm{G}}({\bar{\mathsf{k}}}/{\mathsf{k}})$ permutes these classes in a transitive way.
In other words, the natural homomorphism~$\varpi_X \colon {\mathrm{G}}({\bar{\mathsf{k}}}/{\mathsf{k}}) \to \Aut(\Pic(X_{\bar{\mathsf{k}}}))$
factors through the subgroup~$\mathfrak{S}_r \subset \Aut(\Pic(X_{\bar{\mathsf{k}}}))$
and its image
\begin{equation}
\label{eq:gx}
{\mathrm{G}}_X := \Ima({\mathrm{G}}({\bar{\mathsf{k}}}/{\mathsf{k}}) \xrightarrow{\ \varpi_X\ } \mathfrak{S}_r)
\end{equation}
is a transitive subgroup of~$\mathfrak{S}_r$.
\end{lemma}
\begin{proof}
The classes~$H_i$ are pullbacks of ample classes on~$\mathbb{P}^n$, hence semiample,
and they generate~$\Pic(X_{\bar{\mathsf{k}}})$ by Lemma~\ref{lemma:picard-simple}.
If~$\Lambda_i$ is the class of a non-trivial fiber of~$\pi_{\widehat{\imath}}$, we have
\begin{equation*}
H_j \cdot \Lambda_i= \delta_{ij},
\end{equation*}
therefore~$H_j$ generate the rays of the nef cone.
It follows that the Galois group permutes the~$H_i$, hence its action on~$\Pic(X_{\bar{\mathsf{k}}})$ factors through the permutation group.
Transitivity of the subgroup~${\mathrm{G}}_X \subset \mathfrak{S}_r$ follows from the equality~$\uprho(X) = 1$.
\end{proof}
We say that a surface~$\Pi \subset X_{\bar{\mathsf{k}}}$ is an {\sf $H$-plane} if~$\Pi \cong \mathbb{P}^2_{\bar{\mathsf{k}}}$ and~$H\vert_\Pi$ is the line class.
\begin{corollary}
\label{cor:planes}
Fano threefolds of index~$1$ from Table~\textup{\ref{table:fanos}} contain no $H$-planes over~${\bar{\mathsf{k}}}$.
\end{corollary}
\begin{proof}
If~$\Pi \subset X_{\bar{\mathsf{k}}}$ is an $H$-plane, the
restriction~$(H_1 + \dots + H_r)\vert_\Pi$ is the line class.
Since~all the~$H_i$
are nef, it follows
that~$H_j\vert_\Pi \sim 0$ for all $j \ne i$ and some~$i$,
hence~$\Pi$ is contracted to a point by the projection~$\pi_{\widehat{\imath}}$.
It remains to note that the fibers of~$\pi_{\widehat{\imath}}$ are at most 1-dimensional by Lemma~\ref{lemma:pi-i}.
\end{proof}
\subsection{Lines}
\label{subsec:lines}
By a \textsf{line on~$X$} we understand a curve (defined over~${\bar{\mathsf{k}}}$) of anticanonical degree~$1$.
We denote by~$\mathrm{F}_1(X)$ the Hilbert scheme of lines on~$X$.
Note that $\mathrm{F}_1(X)_{\bar{\mathsf{k}}} \cong \mathrm{F}_1(X_{\bar{\mathsf{k}}})$.
\begin{lemma}
\label{lemma:lines-1}
Let $X$ be a Fano threefold of types~\type{2,2,2}, \type{4,4}, \type{3,3}, or~\type{1,1,1,1}.
A line on~$X$ is a fiber of the exceptional divisor of one of the projections~\eqref{eq:pi-hi}.
In particular
\begin{equation*}
\mathrm{F}_1(X_{\bar{\mathsf{k}}}) \cong \bigsqcup_{i=1}^r\ \Gamma_{\widehat{\imath}},
\end{equation*}
where the smooth curves~$\Gamma_{\widehat{\imath}}$ have been described in Lemma~\xref{lemma:pi-i}.
The normal bundle of each line is
\begin{equation}
\label{eq:normal-lines}
{\mathscr{N}}_{L/X_{\bar{\mathsf{k}}}} \cong {\mathscr{O}}_L \oplus {\mathscr{O}}_L(-1).
\end{equation}
Finally,
the action of the Galois group~${\mathrm{G}}({\bar{\mathsf{k}}}/{\mathsf{k}})$ on the set of connected components of the Hilbert scheme of lines
factors through the group~${\mathrm{G}}_X$ and is transitive.
\end{lemma}
\begin{proof}
Since the classes $H_i$ are semiample, it follows from~\eqref{eq:kx} that for
each ${\bar{\mathsf{k}}}$-line~$L$ on~$X$
there is a unique~$i$ such that $L \cdot H_i = 1$ and $L \cdot H_j = 0$ for $j \ne i$
(i.e., $[L] = \Lambda_i$ in the notation of Lemma~\ref{lemma:picard}).
Thus, $L$ is contracted by the projection~$\pi_{\widehat{\imath}}$, hence it is equal to a fiber of the exceptional divisor of this projection.
Taking into account the description of the projections~$\pi_{\widehat{\imath}}$ from Lemma~\ref{lemma:pi-i}, we obtain the description of~$\mathrm{F}_1(X_{\bar{\mathsf{k}}})$.
Further, the description of the normal bundle of~$L$ follows from the exact sequence
\begin{equation*}
0 \longrightarrow {\mathscr{N}}_{L/E_{\widehat{\imath}}} \longrightarrow {\mathscr{N}}_{L/X_{\bar{\mathsf{k}}}} \longrightarrow {\mathscr{N}}_{E_{\widehat{\imath}}/X_{\bar{\mathsf{k}}}}\vert_L \longrightarrow 0,
\end{equation*}
because the first term is trivial and the last is~${\mathscr{O}}_L(-1)$.
Finally, factorization of the Galois action on the set of connected components
of~$\mathrm{F}_1(X_{\bar{\mathsf{k}}})$ and its transitivity follow from Lemma~\ref{lemma:picard}.
\end{proof}
For a ${\bar{\mathsf{k}}}$-point~$x \in X$ we denote
by~$\mathrm{F}_1(X_{\bar{\mathsf{k}}},x) \subset \mathrm{F}_1(X_{\bar{\mathsf{k}}})$ the
subscheme parameterizing lines passing through~$x$.
We will need the following observation.
\begin{lemma}
\label{lemma:lines}
Let $X$ be a Fano threefold of
types~\type{2,2,2}, \type{4,4}, \type{3,3}, or~\type{1,1,1,1}.
If~$x \in X({\bar{\mathsf{k}}})$, the scheme~$\mathrm{F}_1(X_{\bar{\mathsf{k}}},x)$ is a finite reduced scheme of length at most~$r = \uprho(X_{\bar{\mathsf{k}}})$.
If, moreover, $x \in X({\mathsf{k}})$ then either~$\mathrm{F}_1(X_{\bar{\mathsf{k}}},x) = \varnothing$,
or~$\mathrm{F}_1(X_{\bar{\mathsf{k}}},x)$ is a reduced scheme of length~$r$
and the Galois group~${\mathrm{G}}({\bar{\mathsf{k}}}/{\mathsf{k}})$ action on~$\mathrm{F}_1(X_{\bar{\mathsf{k}}},x)$
factors through the group~${\mathrm{G}}_X$ and is transitive.
\end{lemma}
\begin{proof}
By Lemma~\ref{lemma:lines-1} for each ${\bar{\mathsf{k}}}$-point~$x$ of~$X$ there is at
most one line
from each of the connected components of the Hilbert scheme $\mathrm{F}_1(X_{\bar{\mathsf{k}}})$
passing through~$x$.
This proves that~$\mathrm{F}_1(X_{\bar{\mathsf{k}}},x)$ is finite and reduced and gives the bound for
its length.
Now assume $x$ is a point defined over~${\mathsf{k}}$ and let $L$ be a ${\bar{\mathsf{k}}}$-line
through~$x$.
Then for any~$g \in {\mathrm{G}}({\bar{\mathsf{k}}}/{\mathsf{k}})$ the line $g(L)$ also passes through~$x$.
Transitivity of the Galois action on the set of components of~$\mathrm{F}_1(X_{\bar{\mathsf{k}}})$
then implies that there is a unique line of each type through~$x$, hence the
length of~$\mathrm{F}_1(X_{{\bar{\mathsf{k}}}},x)$ is~$r$,
and the ${\mathrm{G}}({\bar{\mathsf{k}}}/{\mathsf{k}})$-action on~$\mathrm{F}_1(X_{\bar{\mathsf{k}}},x)$ factors
through~${\mathrm{G}}_X$ and is transitive.
\end{proof}
\subsection{Conics}
\label{subsec:conics}
By a \textsf{conic on~$X$} we understand a connected curve (defined over~${\bar{\mathsf{k}}}$) of anticanonical degree~$2$.
We denote by~$\mathrm{F}_2(X)$ the Hilbert scheme of conics on~$X$.
Note that $\mathrm{F}_2(X)_{\bar{\mathsf{k}}} \cong \mathrm{F}_2(X_{\bar{\mathsf{k}}})$.
\begin{lemma}
\label{lemma:conics}
Let $X$ be a Fano threefold of types~\type{2,2,2}, \type{4,4}, \type{3,3},
or~\type{1,1,1,1}.
We have the following descriptions of the Hilbert schemes of
conics~$\mathrm{F}_2(X_{{\bar{\mathsf{k}}}})$:
\begin{eqnarray*}
\mathrm{F}_2((\mtype{2,2,2})_{\bar{\mathsf{k}}}) &\cong& \mathbb{P}^2_{\bar{\mathsf{k}}} \sqcup \mathbb{P}^2_{\bar{\mathsf{k}}} \sqcup \mathbb{P}^2_{\bar{\mathsf{k}}},\\
\mathrm{F}_2((\mtype{4,4})_{\bar{\mathsf{k}}}) &\cong& \Gamma_1 \times \Gamma_2,\\
\mathrm{F}_2((\mtype{3,3})_{\bar{\mathsf{k}}}) &\cong& \Sym^2\Gamma_1 \cong \Sym^2\Gamma_2,\\
\mathrm{F}_2((\mtype{1,1,1,1})_{\bar{\mathsf{k}}}) &\cong& \bigsqcup_6 \ (\mathbb{P}^1_{\bar{\mathsf{k}}} \times \mathbb{P}^1_{\bar{\mathsf{k}}}) ,
\end{eqnarray*}
where $\Gamma_i$ are the curves described in Lemma~\xref{lemma:pi-i}.
Moreover, the morphism from each component of the universal conic to~$X_{\bar{\mathsf{k}}}$ is dominant.
\end{lemma}
\begin{proof}
First, note that no conic on~$X$ is contracted by the projections~$\pi_{\widehat{\imath}}$,
since by~Lemma~\xref{lemma:pi-i} any reduced connected curve contracted by~$\pi_{\widehat{\imath}}$ is a line
and lines do not support nonreduced conics by~\eqref{eq:normal-lines} and~\cite[Remark~2.1.7]{KPS}.
Therefore, we deduce from~\eqref{eq:kx} that for each ${\bar{\mathsf{k}}}$-conic~$C \subset X_{{\bar{\mathsf{k}}}}$
there is a pair of indices~$1 \le i_1 < i_2 \le r$ such that
\begin{equation}
\label{eq:intersections-conic}
H_{i_1} \cdot C = H_{i_2} \cdot C = 1
\qquad\text{and}\qquad
H_j \cdot C = 0\quad\text{for $j \not\in \{i_1,\,i_2\}$}.
\end{equation}
If~$r \ge 3$, i.e., if~$X$ is of type~\type{2,2,2} or~\type{1,1,1,1}, such~$C$ is contracted by one of the projections
\begin{equation}
\label{eq:conic-bundles}
\pi_i \colon X_{\bar{\mathsf{k}}} \longrightarrow \mathbb{P}^2_{\bar{\mathsf{k}}}
\qquad\text{or}\qquad
\pi_{i_1,i_2} \colon X_{\bar{\mathsf{k}}} \longrightarrow \mathbb{P}^1_{\bar{\mathsf{k}}} \times \mathbb{P}^1_{\bar{\mathsf{k}}},
\end{equation}
respectively.
It is easy to see that the maps~\eqref{eq:conic-bundles} are flat conic bundles, hence~$C$ is a fiber of one of them,
and therefore~$\mathrm{F}_2(X_{\bar{\mathsf{k}}})$ is the disjoint union of~$\mathbb{P}^2_{\bar{\mathsf{k}}}$, or of~$\mathbb{P}^1_{\bar{\mathsf{k}}} \times \mathbb{P}^1_{\bar{\mathsf{k}}}$, respectively.
Assume~$X$ is of type~\type{4,4}.
Applying Corollary~\ref{cor:f2x-gamma1} twice we obtain a morphism
\begin{equation*}
\varphi = (\varphi_1, \varphi_2) \colon \mathrm{F}_2(X_{\bar{\mathsf{k}}}) \longrightarrow \Gamma_1 \times \Gamma_2
\end{equation*}
that takes a smooth conic~$C \subset X_{\bar{\mathsf{k}}}$ to the unique pair of lines~$(L_2,L_1)$ of different types such that~$C \cap L_i \ne \varnothing$.
We will show that~$\varphi$ is an isomorphism.
First, note that by~\eqref{eq:intersections-conic} if~$C \subset X$ is a conic then~$\pi_1(C) \subset Q_1$ and~$\pi_2(C) \subset Q_2$ are lines,
and by Lemma~\ref{lemma:pi-i}\ref{lemma:pi-i:44} they intersect the curves~$\Gamma_1$ and~$\Gamma_2$, respectively.
Thus, by Corollary~\ref{cor:f2x-gamma1} for~$x_1 \in \Gamma_1$ if~$[C] \in \varphi_1^{-1}(x_1)$ the line~$\pi_1(C) \subset Q_1$ passes through~$x_1$.
Since any line on~$Q_1$ through~$x_1$ lies in the embedded tangent space to~$Q_1$ at~$x_1$,
and the intersection of this tangent space with~$Q_1$ is a 2-dimensional quadratic cone with vertex at~$x_1$,
it follows that
\begin{equation*}
\varphi_1^{-1}(x_1) \cong \mathbb{P}^1
\end{equation*}
for any~$x_1 \in \Gamma_1$.
Since also~$\Gamma_2 \cong \mathbb{P}^1$, the morphism~$\varphi$ is a morphism of~$\mathbb{P}^1$-bundles over~$\Gamma_1$,
and to show that it is an isomorphism, it is enough to check that it is birational.
So, consider a general pair~$(L_2,L_1)$ of lines on~$X$ of different types.
It follows from~\eqref{eq:intersections-conic} and Lemma~\ref{lemma:pi-i}\ref{lemma:pi-i:44}
that~$\bar{L}_1 := \pi_1(L_1)$ is a line on~$Q_1$ bisecant to~$\Gamma_1$, $x_1 := \pi_1(L_2)$ is a point on~$\Gamma_1$,
and by Corollary~\ref{cor:f2x-gamma1} the preimage~$\varphi^{-1}(L_2,L_1)$
is the Hilbert scheme of lines~$L \subset Q_1$ passing through~$x_1$ and intersecting~$\bar{L}_1$.
By genericity we may assume~$x_1 \not\in \bar{L}_1$ (i.e., that the lines~$L_1$ and~$L_2$ do not intersect).
Then any line~$L$ as above is contained in the intersection of the plane spanned by~$\bar{L}_1$ and~$x_1$ with~$Q_1$,
which is equal to the union of the line~$\bar{L}_1$ with a residual line.
Therefore, $L$ must be equal to the residual line, hence the scheme~$\varphi^{-1}(L_2,L_1)$ consists of a single point,
so~$\varphi$ is birational, and hence it is an isomorphism.
Since the embedded tangent space to~$Q_1$ at a general point~$x \in Q_1$ intersects the quartic curve~$\Gamma_1$ at~$4$ points,
there are~$4$ lines on~$Q_1$ through~$x$ intersecting~$\Gamma_1$,
hence the universal conic is dominant of degree~$4$ over~$X_{\bar{\mathsf{k}}}$.
Finally, assume $X$ is of type~\type{3,3}.
By~\eqref{eq:intersections-conic} and Lemma~\ref{lemma:pi-i}\ref{lemma:pi-i:33}
the image of~$C$ with respect to the blowup~$\pi_i \colon X_{\bar{\mathsf{k}}} \to \mathbb{P}^3$ is a line
intersecting the curve~$\Gamma_{i} \subset \mathbb{P}^3$ at two points.
This defines a morphism
\begin{equation*}
\mathrm{F}_2(X_{\bar{\mathsf{k}}}) \longrightarrow \Sym^2\Gamma_{i},
\end{equation*}
and it is easy to see that it is an isomorphism.
It is also easy to see that for a general point~$x \in \mathbb{P}^3$ there are seven lines passing through~$x$ and bisecant to~$\Gamma_1$;
therefore the universal conic on~$X_{\bar{\mathsf{k}}}$ is dominant of degree~$7$ over~$X_{\bar{\mathsf{k}}}$.
\end{proof}
\begin{remark}
\label{rem:f2x-theta}
Let~$X$ be a threefold of type~\type{4,4}.
Clearly a general line on the quadric~$Q_1$ passing through a point~$x \in \Gamma_1$ is not bisecant to~$\Gamma_1$
and its strict transform in~$X$ intersects the line~$L_2 = \pi_1^{-1}(x)$ transversally.
This means that a general conic intersecting~$L_2$ is smooth and intersects~$L_2$ transversally.
\end{remark}
For a given curve~$\Theta \subset X$ we denote by~$\mathrm{F}_2(X,\Theta)$
the subscheme of the Hilbert scheme~$\mathrm{F}_2(X)$
that parameterizes conics intersecting the curve~$\Theta$
and by~$\mathscr{C}_2(X,\Theta) \subset \mathrm{F}_2(X,\Theta) \times X$ the restriction of the universal family of conics.
\begin{lemma}
\label{lemma:conics-c0}
If $X$ is of type~\type{4,4} and $\Theta$ is a singular conic then~$\mathrm{F}_2(X_{\bar{\mathsf{k}}},\Theta) \cong \Gamma_1 \cup \Gamma_2$
is the union of the two rulings of the surface~$\mathrm{F}_2(X_{\bar{\mathsf{k}}}) \cong \Gamma_1 \times \Gamma_2$.
Moreover, the natural projection~$\mathscr{C}_2(X,\Theta) \to X$ is birational onto
an anticanonical divisor~$R_{\Theta} \subset X$ passing through each component of the curve~$\Theta$ with multiplicity~$3$.
\end{lemma}
\begin{proof}
Let $L_1$ and $L_2$ be the irreducible components (over~${\bar{\mathsf{k}}}$) of the conic~$\Theta$.
The argument of Lemma~\ref{lemma:conics} shows that~$L_i$ are lines of two different types and
\begin{equation*}
\mathrm{F}_2(X_{\bar{\mathsf{k}}},\Theta) = \mathrm{F}_2(X_{\bar{\mathsf{k}}},L_1) \cup \mathrm{F}_2(X_{\bar{\mathsf{k}}},L_2).
\end{equation*}
Recall that by Lemma~\ref{lemma:lines-1} the curves~$\Gamma_1$
and~$\Gamma_2$
can be identified with the two connected components of~$\mathrm{F}_1(X_{\bar{\mathsf{k}}})$
and the isomorphism $\mathrm{F}_2(X_{\bar{\mathsf{k}}}) \cong \Gamma_1 \times \Gamma_2$ of
Lemma~\ref{lemma:conics}
is defined by taking a conic~$C$ to the unique pair of lines of different
types intersecting~$C$.
This means that
\begin{equation*}
\mathrm{F}_2(X_{\bar{\mathsf{k}}},\Theta) \cong
(\Gamma_1 \times [L_1]) \cup ([L_2] \times \Gamma_2)
\subset \Gamma_1 \times \Gamma_2;
\end{equation*}
thus~$\mathrm{F}_2(X_{\bar{\mathsf{k}}},\Theta)$ is the union of two rulings of~$\mathrm{F}_2(X)$
and~$\mathscr{C}_2(X_{\bar{\mathsf{k}}},\Theta) = \mathscr{C}_2(X_{\bar{\mathsf{k}}},L_1) \cup \mathscr{C}_2(X_{\bar{\mathsf{k}}},L_2)$.
Furthermore, it follows from the description of Lemma~\ref{lemma:conics} that
the morphism~$\mathscr{C}_2(X_{\bar{\mathsf{k}}},L_2) \to X_{\bar{\mathsf{k}}}$ is birational onto
the hyperplane section tangent to~$Q_1$ at the point~$\pi_1(L_2)$;
it contains the line~$\pi_1(L_1)$ with multiplicity~$1$ and
has multiplicity~$2$ at the point~$\pi_1(L_2)$.
Similarly, the morphism~$\mathscr{C}_2(X_{\bar{\mathsf{k}}},L_1) \to X_{\bar{\mathsf{k}}}$ is birational onto
the hyperplane section containing the line~$\pi_2(L_2)$ with multiplicity~$1$
and having multiplicity~$2$ at the point~$\pi_2(L_1)$.
Thus, the morphism~\mbox{$\mathscr{C}_2(X,\Theta) \to X$} is birational onto a divisor of
class $(H_1 - L_1 - 2L_2) + (H_2 - L_2 - 2L_1) = H - 3\,\Theta$.
\end{proof}
\subsection{Twisted cubic curves}
Finally, we describe the Hilbert scheme~$\mathrm{F}_3(X,x)$ of subschemes of~$X$
with Hilbert polynomial~$3t + 1$ with respect to~$H$ that pass through a point~$x$;
since~$X$ is an intersection of quadrics (Theorem~\ref{th:bht}) and contains no planes (Corollary~\ref{cor:planes})
every such subscheme is a union of rational curves (see~\cite[Lemma~2.9]{KP19}),
so we will use the name {\sf rational normal cubic curves} for subschemes parameterized by~$\mathrm{F}_3(X,x)$.
We denote by~$\mathscr{C}_3(X,x) \subset \mathrm{F}_3(X,x) \times X$ the restriction of the universal family of curves.
Recall the curves~$\Gamma_{\widehat{\imath}}$ described in Lemma~\xref{lemma:pi-i}.
\begin{lemma}
\label{lemma:cubics-x}
Let $X$ be a Fano threefold of types~\type{2,2,2}, \type{4,4}, \type{3,3},
or~\type{1,1,1,1}.
If~$x$ is a ${\mathsf{k}}$-point on~$X$ not lying on a ${\bar{\mathsf{k}}}$-line,
one has the following descriptions of the schemes~$\mathrm{F}_3(X_{\bar{\mathsf{k}}},x)$:
\begin{eqnarray*}
\mathrm{F}_3((\mtype{2,2,2})_{\bar{\mathsf{k}}},x) &\cong& \Gamma_{1,2} \cong \Gamma_{1,3} \cong \Gamma_{2,3},\\
\mathrm{F}_3((\mtype{4,4})_{\bar{\mathsf{k}}},x) &\cong& \mathbb{P}^1_{\bar{\mathsf{k}}} \sqcup \mathbb{P}^1_{\bar{\mathsf{k}}},\\
\mathrm{F}_3((\mtype{3,3})_{\bar{\mathsf{k}}},x) &\cong& \Gamma_1 \sqcup \Gamma_2,\\
\mathrm{F}_3((\mtype{1,1,1,1})_{\bar{\mathsf{k}}},x) &\cong& \bigsqcup_8 \mathbb{P}^1_{\bar{\mathsf{k}}}.
\end{eqnarray*}
Moreover, for threefolds of type~\type{4,4}
the natural projection~$\mathscr{C}_3(X,x) \to X$ is birational onto
an anticanonical divisor~$R_x \subset X$ passing through the point~$x$ with multiplicity~$4$.
\end{lemma}
\begin{proof}
First, consider a threefold~$X$ of type~\type{2,2,2}.
If~$C$ is a rational normal cubic curve and~\mbox{$H_i \cdot C = 0$} for some~$i$
then $C$ is contracted by one of the conic bundles~\eqref{eq:conic-bundles},
hence~$C$ is supported on a fiber of~\eqref{eq:conic-bundles}.
But the conormal bundle of any such fiber is trivial, hence it cannot support a nonreduced
curve of arithmetic genus~$0$ and degree more than~$2$.
This means that we have~$H_i \cdot C = 1$ for each~$i$, and we conclude from this and Lemma~\xref{lemma:pi-i}
that the image of~$C$ under the map~\mbox{$\pi_{1,2} \colon X_{\bar{\mathsf{k}}} \to W_{1,2}$}
is a rational curve of bidegree~$(1,1)$ intersecting the curve~$\Gamma_{1,2}$ and passing through~$x$.
The argument analogous to that of Corollary~\ref{cor:f2x-gamma1} shows that there is a morphism
\begin{equation*}
\varphi_{1,2} \colon \mathrm{F}_3(X,x) \longrightarrow \Gamma_{1,2}
\end{equation*}
that takes a twisted cubic curve~$C$ to the unique point~$x_{1,2} \in \Gamma_{1,2}$ such that~$C \cap \pi_{1,2}^{-1}(x_{1,2}) \ne \varnothing$.
This morphism is an isomorphism, because
on~$W_{1,2}$ there is a unique curve of bidegree~$(1,1)$ through a given pair of points
(unless they lie on a fiber of either of the projections~$W_{1,2} \to \mathbb{P}^2_{\bar{\mathsf{k}}}$, in which case~$x$ lies on a line in~$X$).
The same argument proves isomorphisms of~$\mathrm{F}_3(X,x)$ with the curves~$\Gamma_{1,3}$ and~$\Gamma_{2,3}$.
Next, consider a threefold of type~\type{4,4}.
If $H_i \cdot C = 0$ for some~$i$ then $C$ is contracted by~$\pi_i$, hence is supported on a line.
But the conormal bundle of a line is globally generated by~\eqref{eq:normal-lines},
hence a line cannot support a nonreduced
curve of arithmetic genus~$0$ and degree more than~$1$.
This means that~$C$ has bidegree~$(1,2)$ or~$(2,1)$.
In the first case the image of~$C$ under~$\pi_1$ is a line on the quadric~$Q_1$ passing through~$x$;
hence the corresponding component of~$\mathrm{F}_3(X,x)_{\bar{\mathsf{k}}}$ is isomorphic to~$\mathbb{P}^1_{\bar{\mathsf{k}}}$.
It also follows that the corresponding component of~$\mathscr{C}_3(X,x)$ is a Hirzebruch surface that maps birationally onto
the hyperplane section of~$Q_1$ tangent at~$x$, i.e., a
divisor of class~$H_1$ passing through~$x$ with multiplicity~$2$.
The second component is described analogously.
The total divisor class of the image~$R_x$ of~$\mathscr{C}_3(X,x) \to X$
is~$H_1 + H_2 - 4x$, i.e., it is the anticanonical class passing through~$x$ with multiplicity~$4$.
Next, consider a threefold of type~\type{3,3}.
The same argument as above shows that~$C$ has bidegree~$(1,2)$ or~$(2,1)$.
In the first case the image of~$C$ under~$\pi_1$ is a line on~$\mathbb{P}^3_{\bar{\mathsf{k}}}$ passing through~$x$ and intersecting the curve~$\Gamma_1$.
Since for any point of~$\Gamma_1$ there is a unique line through it and~$x$,
the corresponding component of~$\mathrm{F}_3(X_{\bar{\mathsf{k}}},x)$ is isomorphic to~$\Gamma_1$.
Analogously, the second component is isomorphic to~$\Gamma_2$.
Finally, consider a threefold of type~\type{1,1,1,1}.
Then, of course, $H_i \cdot C = 0$ for some~$i$.
The argument used for threefolds of type~\type{2,2,2} shows this cannot hold for two distinct~$i$.
So, assume this holds for~$i = 1$.
By Lemma~\ref{lemma:pi-i}\ref{lemma:pi-i:x1111} the image of~$C$ under the map~$\pi_{1,2,3}$
is a curve of multidegree~$(0,1,1)$ on~$\mathbb{P}^1_{\bar{\mathsf{k}}} \times \mathbb{P}^1_{\bar{\mathsf{k}}} \times \mathbb{P}^1_{\bar{\mathsf{k}}}$
intersecting the curve~$\Gamma_{1,2,3}$ and passing through~$x$.
In other words, it is a curve of bidegree~$(1,1)$ on the surface~$\mathbb{P}^1_{\bar{\mathsf{k}}} \times \mathbb{P}^1_{\bar{\mathsf{k}}}$ passing through~$x$
and either of the two points of intersection of~$\Gamma_{1,2,3}$ with this surface
(note that these points cannot collide because otherwise~$x$ would lie on a line in~$X$).
Therefore, there are two pencils of such curves.
Using the same argument for other~$i$ we see that altogether there are~8
pencils of twisted cubic curves on~$X$ passing through~$x$.
\end{proof}
\begin{remark}
\label{rem:f3x-x}
Let~$X$ be a threefold of type~\type{4,4}.
Since a general line on~$Q_1$ passing through a point~$x \not\in \Gamma_1$ does
not intersect~$\Gamma_1$,
it follows that a general twisted cubic curve on~$X$ passing through~$x$ is
smooth.
\end{remark}
\section{A birational transformation for a product of projective spaces}
\label{sec:toric}
In this section we construct a birational transformation for a product of projective spaces
and deduce a consequence for the rationality of its ${\mathsf{k}}$-forms;
in particular we prove the rationality criterion for threefolds~\type{1,1,1}.
\subsection{Product of projective spaces}
\label{subsec:ppp}
Consider the product
\begin{equation*}
Y = \mathbb{P}^{n_1} \times \mathbb{P}^{n_2} \times \dots \times \mathbb{P}^{n_r} = \mathbb{P}(V_1) \times
\mathbb{P}(V_2) \times \dots \times \mathbb{P}(V_r)
\end{equation*}
of projective spaces.
Assume that $r = p + q$ and
\begin{equation}
\label{eq:n-i}
n_1 \ge n_2 \ge \dots \ge n_p \ge 2,
\qquad
n_{p+1} = \dots = n_{p+q} = 1.
\end{equation}
Let $y \in Y$ be a point, and let $(v_1,v_2,\dots,v_r)$, $0 \ne v_i \in V_i$, be the corresponding collection of vectors.
Consider the blowup
\begin{equation*}
{\tilde{Y}} = \Bl_y(Y)
\end{equation*}
and let $E \subset {\tilde{Y}}$ be its exceptional divisor.
Let~$\PGL(V_i)_{v_i} \subset \PGL(V_i)$ be the stabilizer of the point~$[v_i] \in \mathbb{P}(V_i)$ in the projective linear group $ \PGL(V_i)$.
The group
\begin{equation*}
G = \prod_{i=1}^r \PGL(V_i)_{v_i}
\end{equation*}
acts naturally on~${\tilde{Y}}$ and has
finitely many orbits, which can be described as follows.
First, for each $1 \le i \le r$ let
\begin{equation}
\label{eq:divisor-ei}
{\tilde{Y}}_i := \Bl_y\Big( \mathbb{P}(V_1) \times \dots \times \mathbb{P}(V_{i-1}) \times [v_i] \times \mathbb{P}(V_{i+1}) \times \dots \times \mathbb{P}(V_r)\Big) \subset {\tilde{Y}}.
\end{equation}
Furthermore, for any subset $I \subsetneq\{1,\dots,r\}$ denote
\begin{equation}
\label{eq:eI}
{\tilde{Y}}_I := \bigcap_{i \in I} {\tilde{Y}}_i
\qquad\text{and}\qquad
E_I := E \cap {\tilde{Y}}_I.
\end{equation}
Finally, set
\begin{equation}
\label{eq:eI-circ}
{\tilde{Y}}_I^\circ := {\tilde{Y}}_I \setminus \left( E_I \cup \bigcup_{I \subsetneq J}
{\tilde{Y}}_J \right)
\qquad\text{and}\qquad
E_I^\circ := E_I \setminus \left( \bigcup_{I \subsetneq J} E_J \right).
\end{equation}
Then ${\tilde{Y}}_\varnothing^\circ$ is the open orbit, $E_\varnothing^\circ$
and~${\tilde{Y}}_i^\circ$, $p+1 \le i \le q$, are the orbits of codimension~$1$,
and all other orbits have higher codimension.
To describe the other side of the transformation, denote
\begin{equation*}
\bar{V}_i := V_i / {\mathsf{k}} v_i
\end{equation*}
and choose splittings $V_i = {\mathsf{k}} v_i \oplus \bar{V}_i$.
They induce a direct sum decomposition
\begin{equation*}
V_1 \otimes \dots \otimes V_{r} =
\bigoplus_{I \subset \{1,\dots,r\}} \bar{V}_I,
\qquad\text{where}\quad
\bar{V}_I := \bigotimes_{i \in I} \bar{V}_i.
\end{equation*}
Note that the point $y$ corresponds to the summand~$\bar{V}_\varnothing = {\mathsf{k}}$,
and the tangent space to~$Y$ at~$y$ corresponds to the sum of the summands~$\bar{V}_I$ with~$|I| = 1$.
Note also that for $i \ge p + 1$ one has $\mathbb{P}(\bar{V}_i) \cong \Spec({\mathsf{k}})$.
Let
\begin{equation*}
Y^+ := \prod_{i=1}^r \mathbb{P}(\bar{V}_i) =
\mathbb{P}(\bar{V}_1) \times \mathbb{P}(\bar{V}_2) \times \dots \times \mathbb{P}(\bar{V}_p) \cong
\mathbb{P}^{n_1-1} \times \mathbb{P}^{n_2-1} \times \dots \times \mathbb{P}^{n_p-1}.
\end{equation*}
Denote by~$h_i$ the pullback to~$Y^+$ of the hyperplane class of the $i$-th factor
(note that~$h_i = 0$ for~$i \ge p + 1$ because, as we noticed above, $\mathbb{P}(\bar{V}_i)$ is just a point)
and for $I \subset \{1,\dots,r\}$ set
\begin{equation*}
h_I := \sum_{i \in I} h_i.
\end{equation*}
Consider the vector bundle~${\mathscr{E}}$ of rank~$r+1$ on~$Y^+$ defined by
\begin{equation}
\label{eq:ce}
{\mathscr{E}} := \bigoplus_{|I| \ge r - 1} {\mathscr{O}}\left( - h_I \right).
\end{equation}
Denote by
\begin{equation*}
s_i \colon Y^+ \longrightarrow \mathbb{P}_{Y^+}({\mathscr{E}})
\end{equation*}
the section~of~\eqref{eq:ce} corresponding to the summand with $I = \{1,\dots,i-1,i+1,\dots,r\}$.
Set
\begin{equation}
\label{def:hyp-typ}
{\hat{Y}}^+ := \mathbb{P}_{Y^+}({\mathscr{E}}),
\qquad
{\tilde{Y}}^+ := \Bl_{s_{p+1}(Y^+) \sqcup \dots \sqcup s_{p+q}(Y^+)}({\hat{Y}}^+).
\end{equation}
and let~$E_i \subset {\tilde{Y}}^+$, $p + 1 \le i \le p + q$, be the exceptional divisors.
The group~$G$ acts transitively on~$Y^+$, the vector bundle~${\mathscr{E}}$ is
$G$-equivariant, and its summands ${\mathscr{O}}(-h_I)$ with $|I| = r - 1$
are~$G$-invariant.
Therefore, the action of~$G$ lifts naturally to~${\hat{Y}}^+$ and~${\tilde{Y}}^+$.
Moreover, the action of~$G$ on~${\tilde{Y}}^+$ still has a finite number of orbits,
which can be described as follows.
For a subset $J \subsetneq \{1,\dots,r\}$ denote
\begin{equation}
\label{eq:yplus-i}
\bar{{\mathscr{E}}}_J = \bigoplus_{J \subset I,\ |I| = r - 1} {\mathscr{O}}\left( - h_I \right);
\end{equation}
this is a subbundle in~${\mathscr{E}}$ of corank~$1 + |J|$.
Let~${\tilde{Y}}^+_J \subset {\tilde{Y}}^+$ denote the strict transform of~$\mathbb{P}_{Y^+}(\bar{{\mathscr{E}}}_J)$.
Then the $G$-orbits are
\begin{align*}
({\tilde{Y}}^+)^\circ &= {\tilde{Y}}^+ \setminus \left( {\tilde{Y}}^+_\varnothing \cup \bigcup_{i = p +
1}^q E_i \right),
&
({\tilde{Y}}^+_J)^\circ &= {\tilde{Y}}^+_J \setminus \left(\bigcup_{i = p + 1}^q E_i \right),
\\
E_i^\circ &= E_i \setminus {\tilde{Y}}^+_\varnothing,
&
E_{i,J}^\circ &= (E_i \cap {\tilde{Y}}^+_J) \setminus \left( \bigcup_{J \subsetneq K} E_i \cap {\tilde{Y}}^+_K \right),
\end{align*}
where in the last formula we assume~$i \not\in J$.
Note that~$({\tilde{Y}}^+)^\circ$ is the open orbit,
$({\tilde{Y}}^+_\varnothing)^\circ$ and~$E_i^\circ$ are the orbits of codimension~$1$,
and all other orbits have higher codimension.
The linear projection out of~$[v_i]$ defines a~$\PGL(V_i)_{v_i}$-equivariant rational map $\mathbb{P}(V_i)
\dashrightarrow \mathbb{P}(\bar{V}_i)$ which is regular if $i \ge p + 1$.
The product of these maps is a $G$-equivariant rational map, which we denote by~$\psi_0 \colon Y \dashrightarrow Y^+$.
It gives rise to the following birational transformation.
\begin{theorem}
\label{proposition:toric-link}
There is a small birational $G$-equivariant isomorphism~$\psi \colon {\tilde{Y}}
\dashrightarrow {\tilde{Y}}^+$ that fits into the commutative diagram
\begin{equation}
\label{eq:sl-ppp}
\vcenter{
\xymatrix@C=5em{
{\tilde{Y}} \ar[dd]_{\sigma} \ar@{-->}^{\psi}[r] \ar@{..>}[dr]_{\hat\psi} &
{\tilde{Y}}^+ \ar^{\tilde\sigma_+}[d] \ar@/^3em/[dd]^{\sigma_+}
\\
&
{\hat{Y}}^+ \ar[d]^{\hat\sigma_+}
\\
Y \ar@{-->}[r]^{\psi_0} \ar@{..>}[ur]^{\hat\psi_0} &
Y^+,
} }
\end{equation}
where~$\hat\sigma_+ \colon {\hat{Y}}^+ = \mathbb{P}_{Y^+}({\mathscr{E}}) \to Y^+$ and
\mbox{$\tilde\sigma_+ \colon {\tilde{Y}}^+ = \Bl_{s_{p+1}(Y^+) \sqcup \dots \sqcup s_{p+q}(Y^+)}({\hat{Y}}^+) \to {\hat{Y}}^+$} is the projection and the blowup, respectively,
$\sigma_+ := \hat\sigma_+ \circ \tilde\sigma_+$
and, such that~$\psi$ induces isomorphisms of~$G$-orbits
\begin{equation*}
{\tilde{Y}}_\varnothing^\circ \cong ({\tilde{Y}}^+)^\circ,
\qquad
E_\varnothing^\circ \cong ({\tilde{Y}}^+_\varnothing)^\circ,
\qquad\text{and}\qquad
{\tilde{Y}}_i^\circ \cong E_i^\circ
\end{equation*}
of codimension~$0$ and~$1$.
Moreover, if
\begin{itemize}
\item
$H_i$, $1 \le i \le r$, are the hyperplane classes of $\mathbb{P}(V_i)$
and
$H = H_1 + \dots + H_r$,
\item
$E$ is the exceptional divisor of $\sigma$,
\item
$h$ is the relative hyperplane class of the projective bundle~$\hat\sigma_+$,
\item
$h_i$, $1 \le i \le p$, are the hyperplane classes of~$\mathbb{P}(\bar{V}_i)$, and
\item
$e_i$, $p+1 \le i \le p+q$, are the exceptional divisor classes of the blowup~$\tilde\sigma_+$,
\end{itemize}
then in the Picard group $\Pic({\tilde{Y}}) = \Pic({\tilde{Y}}^+)$ there are the following
equalities
\begin{equation}
\label{eq:toric-pic-1}
\begin{array}{lclll}
h_i &=& H_i - E,\quad && 1 \le i \le p,\\
e_i &=& H_i - E,\quad && p+1 \le i \le p+q,\\
h &=& H - (r - 1)E.\qquad&&
\end{array}
\end{equation}
Conversely, one has
\begin{equation}
\label{eq:toric-pic-e}
E = h - \sum_{i=1}^p h_i - \sum_{j=p+1}^{p+q} e_j,
\qquad
H_i =
\begin{cases}
h_i + E, & \text{for $1 \le i \le p$},\\
e_i + E, & \text{for $p + 1 \le i \le p + q$}.
\end{cases}
\end{equation}
\end{theorem}
The maps~$\hat\psi_0$ and~$\hat\psi$ in~\eqref{eq:sl-ppp} will be defined in the course of proof.
\begin{proof}
For each $u_i \in V_i$ denote by~$\bar{u}_i \in \bar{V}_i$ the image of~$u_i$ under the linear projection from the fixed vector~$v_i \in V_i$.
Then the rational map~$\psi_0 \colon Y \dashrightarrow Y^+$ is given by the
formula
\begin{equation*}
(u_1,\dots,u_r) \longmapsto (\bar{u}_1,\dots,\bar{u}_r).
\end{equation*}
This map is regular on the open orbit~$Y^\circ \subset Y$ (given by the
conditions~$\bar{u}_i \ne 0$ for all~\mbox{$1 \le i \le r$})
and it extends regularly to the orbits~$Y_i^\circ \subset Y$ of codimension~$1$
(given by the condition~$\bar{u}_i = 0$ for some~$p + 1 \le i \le p+ q$
and~$\bar{u}_j \ne 0$ for all $j \ne i$).
Now consider the rational $G$-equivariant map
\begin{equation}
\label{eq:tpsi-1}
\hat\psi_0 \colon Y \dashrightarrow {\hat{Y}}^+,
\qquad
(u_1,\dots,u_r) \longmapsto \left((\bar{u}_1,\dots,\bar{u}_r), \sum_{|I| \ge r - 1}
\bigotimes_{i \in I} \bar{u}_i\right).
\end{equation}
Here we consider the summand~$\otimes_{i \in I} \bar{u}_i$
as a point in the fiber of the line bundle~${\mathscr{O}}(-h_I)$ and their sum for $|I|
\ge r - 1$
as a point in the fiber (of the projectivization) of~${\mathscr{E}}$.
The map~$\hat\psi_0$ induces an isomorphism of the open orbit~$Y^\circ \subset
Y$
onto the open orbit~$\mathbb{P}_{Y^+}({\mathscr{E}}) \setminus \mathbb{P}_{Y^+}(\bar{{\mathscr{E}}}_{\varnothing})$ in~${\hat{Y}}^+$
and contracts each orbit~$Y_i^\circ$ of codimension~$1$ to the section~$s_i(Y^+) \subset {\hat{Y}}^+$, $p + 1 \le i \le p + q$.
Now consider the composition~$\hat\psi = \hat\psi_0 \circ \sigma \colon {\tilde{Y}} \dashrightarrow {\hat{Y}}^+$.
The restriction of~$\hat\psi$ to the exceptional divisor~$E$ is given by
\begin{equation}
\label{eq:tpsi-2}
E = \mathbb{P}(\bar{V}_1 \oplus \dots \oplus \bar{V}_r) \dashrightarrow {\hat{Y}}^+,
\qquad
(\bar{u}_1 + \dots + \bar{u}_r) \longmapsto \left((\bar{u}_1,\dots,\bar{u}_r),
\sum_{|I| = r - 1} \bigotimes_{i \in I} \bar{u}_i\right).
\end{equation}
It maps the orbit~$E_\varnothing^\circ \subset {\tilde{Y}}$ isomorphically
onto the $G$-orbit~$({\tilde{Y}}^+_{\varnothing})^\circ = \mathbb{P}_{Y^+}(\bar{{\mathscr{E}}}) \setminus \left( \bigcup_{i=1}^r
\mathbb{P}_{Y^+}(\bar{{\mathscr{E}}}_i) \right)$ of codimension~$1$.
By the above arguments it also gives an isomorphism of open $G$-orbits
and contracts the orbits~${\tilde{Y}}_i^\circ \cong Y_i^\circ$, $p + 1 \le i \le p + q$,
to the sections~$s_i(Y^+) \subset {\hat{Y}}^+$.
Therefore, $\hat\psi$ induces a birational isomorphism
\begin{equation*}
\psi \colon {\tilde{Y}} \dashrightarrow
\Bl_{s_{p+1}(Y^+) \sqcup \dots \sqcup s_{p+q}(Y^+)}(\mathbb{P}_{Y^+}({\mathscr{E}})) = {\tilde{Y}}^+.
\end{equation*}
Finally, it is easy to see that the induced map~${\tilde{Y}}_i^\circ \to E_i^\circ$ is an isomorphism for all~$p + 1 \le i \le p + q$.
This gives the commutative diagram~\eqref{eq:sl-ppp} and proves that~$\psi$ is
small.
The first two lines in~\eqref{eq:toric-pic-1} follow easily from the
formulas~\eqref{eq:tpsi-1}, \eqref{eq:tpsi-2}, and~\eqref{eq:divisor-ei}.
The last line follows from the equality of the canonical classes of~${\tilde{Y}}$
and~${\tilde{Y}}^+$
expressed in terms of~$H_i$ and~$E$ on the one hand,
and~$h_i$, $h$, and~$e_i$ on the other hand.
Finally,~\eqref{eq:toric-pic-e} follows from~\eqref{eq:toric-pic-1}.
\end{proof}
\begin{remark}
\label{rem:flips}
Alternatively, one can use the fact that the varieties~${\tilde{Y}}$ and~${\tilde{Y}}^+$, as
well as the birational isomorphism~$\psi$ are toric.
Thus, to check that~$\psi$ is small, it is enough to identify the generators of
rays of the corresponding fans.
Moreover, comparing the other cones in the fans one can check that the
map~$\psi$ factors as the composition
\begin{equation*}
\xymatrix@1@C=4em{
{\tilde{Y}}\ \ar@{-->}[r]^{\psi_1} &
\ {\tilde{Y}}'\ \ar@{-->}[r]^{\psi_2} &
\ \dots\ \ar@{-->}^{\psi_{r-2}}[r] &
\ {\tilde{Y}}^{(r-2)}\ \ar@{-->}^{\psi_{r-1}}[r] &
\ {\tilde{Y}}^+
}
\end{equation*}
of standard (anti)flips~$\psi_l$ in the strict transforms of~${\tilde{Y}}_I$
for~\mbox{$|I| = l$}, $1 \le l \le r - 2$, and
for~\mbox{$|I| = r - 1$} with~$\{p+1,\dots,p+q\} \subset I$, respectively.
\end{remark}
\subsection{Rationality of forms of products of projective spaces}
Here we apply the birational transformation of the previous subsection to
deduce the following
corollary (see~\cite{Zak07} for a different proof).
\begin{corollary}
\label{corollary:product-rational}
Let $Y$ be a ${\mathsf{k}}$-form of $\mathbb{P}^{n_1} \times \mathbb{P}^{n_2} \times \dots \times \mathbb{P}^{n_r}$.
For any~$y \in Y({\mathsf{k}})$ the diagram~\eqref{eq:sl-ppp} is defined over~${\mathsf{k}}$
and if~$Y({\mathsf{k}}) \ne \varnothing$ then~$Y$ is~${\mathsf{k}}$-rational.
\end{corollary}
\begin{proof}
First, we prove that for any~$y \in Y({\mathsf{k}})$ the diagram~\eqref{eq:sl-ppp} is defined over~${\mathsf{k}}$.
The divisor classes~$H = \sum_{i=1}^r H_i$ and~\mbox{$H' := \sum_{i=1}^p H_i$}
on~$Y_{\bar{\mathsf{k}}}$ are Galois-invariant,
and since we have~\mbox{$Y({\mathsf{k}}) \ne \varnothing$} by assumption, we conclude
that they are defined over~${\mathsf{k}}$.
Also~${\tilde{Y}}$ and~$E$ are defined over~${\mathsf{k}}$ as~$y$ is a ${\mathsf{k}}$-point.
Therefore, the divisor classes
\begin{equation*}
\sum_{i=1}^p h_i = H' - pE,
\qquad
h = H - (r - 1)E,
\qquad\text{and}\qquad
-\sum_{i=p+1}^{p+q} e_i = H - H' - qE
\end{equation*}
(which by Theorem~\ref{proposition:toric-link} are equal to the strict transforms of the classes that are
ample on~$Y^+$, relatively ample for~${\hat{Y}}^+ \to Y^+$ and for~${\tilde{Y}}^+ \to {\hat{Y}}^+$, respectively)
are defined over~${\mathsf{k}}$, hence the varieties~$Y^+$, ${\hat{Y}}^+$, and~${\tilde{Y}}^+$, equal
to the images of~${\tilde{Y}}$ under the maps
given by their appropriate linear combinations, are defined over~${\mathsf{k}}$, as well
as the remaining arrows in the diagram.
Now to prove~${\mathsf{k}}$-rationality of~$Y$ we argue by induction in $\dim(Y) = \sum n_i$.
If the dimension is zero, there is nothing to prove.
So, assume the dimension is positive and consider the diagram~\eqref{eq:sl-ppp}.
By Theorem~\ref{proposition:toric-link} the variety~$Y^+$
is a ${\mathsf{k}}$-form of $Y_{\bar{\mathsf{k}}}^+ = \mathbb{P}^{n_1-1}_{\bar{\mathsf{k}}} \times \mathbb{P}^{n_2-1}_{\bar{\mathsf{k}}} \times \dots \times \mathbb{P}^{n_r-1}_{\bar{\mathsf{k}}}$.
By the Nishimura lemma (see \cite{nishimura-55})
we have~$Y^+({\mathsf{k}}) \ne \varnothing$, hence $Y^+$ is ${\mathsf{k}}$-rational by the induction assumption.
Furthermore, the morphism~${\hat{Y}}^+ \to Y^+$ is a ${\mathsf{k}}$-form of a projective bundle,
and by~\eqref{eq:toric-pic-e} the strict transform of the exceptional divisor~$E$ of~${\tilde{Y}}$
provides for it a relative hyperplane section.
But~$E$ is defined over~${\mathsf{k}}$, therefore ${\tilde{Y}}^+$ is rational over~$Y^+$, hence it is ${\mathsf{k}}$-rational.
It remains to note that the morphisms~$\sigma$, $\psi$, and~$\tilde\sigma_+$ in~\eqref{eq:sl-ppp} are birational,
hence $Y$ is ${\mathsf{k}}$-rational as well.
\end{proof}
Applying this to the case of a~${\mathsf{k}}$-form of~$(\mathbb{P}^1)^3$ we obtain
\begin{corollary}
\label{cor:x111}
If $X$ is a Fano threefold of type~\type{1,1,1} with~$X({\mathsf{k}}) \ne \varnothing$
then~$X$ is ${\mathsf{k}}$-rational.
\end{corollary}
For other applications of the theorem we will often use the following
observation.
Recall the definitions~\eqref{eq:divisor-ei} and~\eqref{eq:yplus-i} of the
subvarieties~$Y_i \subset Y$ of codimension~$n_i$
and subbundles~$\bar{{\mathscr{E}}}_i \subset {\mathscr{E}}$ of corank~$2$.
\begin{proposition}
\label{prop:x-transform}
Let~$Y$ be a~${\mathsf{k}}$-form of~$(\mathbb{P}^n)^r$ where~$n \ge 2$ and assume~$Y$ has a
${\mathsf{k}}$-point~\mbox{$y \in Y({\mathsf{k}})$}.
Let~$X \subset Y$ be a closed ${\mathsf{k}}$-subvariety containing the point~$y$ such
that
\begin{equation*}
X_{\bar{\mathsf{k}}} = \bigcap_{\alpha = 1}^c D_\alpha \subset (\mathbb{P}^n_{{\bar{\mathsf{k}}}})^r
\end{equation*}
is a complete intersection of divisors~$D_\alpha \subset (\mathbb{P}^n_{{\bar{\mathsf{k}}}})^r$, $1 \le
\alpha \le c$.
Let~${\tilde{D}}_\alpha \subset {\tilde{Y}}_{\bar{\mathsf{k}}}$ and~${\tilde{D}}_\alpha^+ \subset {\tilde{Y}}^+_{\bar{\mathsf{k}}} =
{\hat{Y}}^+_{\bar{\mathsf{k}}}$ be the strict transforms of~$D_\alpha$ and set
\begin{equation*}
{\tilde{X}}^+_{\bar{\mathsf{k}}} := \bigcap_{\alpha = 1}^c {\tilde{D}}_\alpha^+ \subset
{\tilde{Y}}^+_{\bar{\mathsf{k}}} = {\hat{Y}}^+_{\bar{\mathsf{k}}} =
\mathbb{P}_{(\mathbb{P}^{n-1}_{{\bar{\mathsf{k}}}})^r}({\mathscr{E}}).
\end{equation*}
If~$X$ is smooth at~$y$ and for each~$1 \le i \le r$ one has
\begin{equation}
\label{eq:dimension-conditions}
\dim ( X_{\bar{\mathsf{k}}} \cap (Y_i)_{\bar{\mathsf{k}}} ) < \dim(X_{\bar{\mathsf{k}}})
\qquad\text{and}\qquad
\dim ( {\tilde{X}}^+_{\bar{\mathsf{k}}} \cap \mathbb{P}_{Y^+_{\bar{\mathsf{k}}}}(\bar{{\mathscr{E}}_i}) ) < \dim(X_{\bar{\mathsf{k}}})
\end{equation}
then the strict transform~${\tilde{X}}^+ = \psi_*(\Bl_y(X))$ of~$X$ in~${\tilde{Y}}^+$ is a
${\mathsf{k}}$-form of the complete intersection~${\tilde{X}}^+_{\bar{\mathsf{k}}}$
and~$X$ is birational to~${\tilde{X}}^+$ over~${\mathsf{k}}$.
\end{proposition}
\begin{proof}
By Corollary~\ref{corollary:product-rational} the diagram~\eqref{eq:sl-ppp} is defined over~${\mathsf{k}}$,
so it is enough to check that the complete intersection~${\tilde{X}}^+_{\bar{\mathsf{k}}}$
is equal to the strict transform of~$\Bl_y(X_{\bar{\mathsf{k}}})$.
First, note that the assumption that~$y$ is a smooth point of~$X$ implies that
the strict transform~$\Bl_y(X_{\bar{\mathsf{k}}})$ of~$X_{\bar{\mathsf{k}}}$ in~${\tilde{Y}}$
is the complete intersection of the divisors~${\tilde{D}}_\alpha$.
Furthermore, the first part of the assumptions~\eqref{eq:dimension-conditions}
implies
that the intersection of~$\Bl_y(X_{\bar{\mathsf{k}}})$ with the open~$G$-orbit in~${\tilde{Y}}_{\bar{\mathsf{k}}}$
is dense in~$\Bl_y(X_{\bar{\mathsf{k}}})$.
Therefore, the strict transform of~$\Bl_y(X_{\bar{\mathsf{k}}})$ in~${\tilde{Y}}^+_{\bar{\mathsf{k}}}$ is contained in~${\tilde{X}}^+_{\bar{\mathsf{k}}}$.
So, it remains to check that~${\tilde{X}}^+_{\bar{\mathsf{k}}}$ is irreducible of dimension~$\dim(X_{\bar{\mathsf{k}}})$.
This is definitely true for the intersection of~${\tilde{X}}^+_{\bar{\mathsf{k}}}$ with the
complement of the union of projective subbundles~$\mathbb{P}_{Y^+_{\bar{\mathsf{k}}}}(\bar{{\mathscr{E}}}_i)$,
because the map~$\psi$ defines an isomorphism of this complement with an open subset of~${\tilde{Y}}_{\bar{\mathsf{k}}}$.
On the other hand, the second part of the assumptions~\eqref{eq:dimension-conditions}
gives a bound for the dimension of the intersections with these projective subbundles, which implies the irreducibility.
\end{proof}
\section{Rationality and unirationality of types \type{2,2}, \type{2,2,2},
and~\type{1,1,1,1}}
\label{sec:constructions}
In this section we prove rationality of Fano threefolds of types~\type{2,2}
and~\type{2,2,2}
as well as unirationality of threefolds of type~\type{1,1,1,1}
under the assumption~$X({\mathsf{k}}) \ne \varnothing$.
\subsection{Rationality of \type{2,2}}
To start with we deal with threefolds of type~\type{2,2}.
\begin{proposition}
\label{prop:x22}
Let $X$ be a Fano threefold of type~\type{2,2}.
If $X({\mathsf{k}}) \ne \varnothing$ then~$X$ is ${\mathsf{k}}$-rational.
\end{proposition}
\begin{proof}
Let~$x$ be a ${\mathsf{k}}$-point of~$X$.
By definition~$X$ is a smooth divisor of bidegree~$(1,1)$ in a ${\mathsf{k}}$-form~$Y$ of~$\mathbb{P}^2 \times \mathbb{P}^2$.
Since the birational isomorphism~$\psi \colon \Bl_x(Y) = {\tilde{Y}} \dashrightarrow {\tilde{Y}}^+ = \mathbb{P}_{Y^+}({\mathscr{E}})$ is small by Theorem~\ref{proposition:toric-link},
it follows that~$X$ is birational to a ${\mathsf{k}}$-form of a divisor
\begin{equation*}
{\tilde{X}}^+_{\bar{\mathsf{k}}} \subset \mathbb{P}_{\mathbb{P}^1 \times \mathbb{P}^1}({\mathscr{E}}) = \mathbb{P}_{\mathbb{P}^1 \times
\mathbb{P}^1}({\mathscr{O}}(-1,-1) \oplus {\mathscr{O}}(-1,0) \oplus {\mathscr{O}}(0,-1))
\end{equation*}
which by~\eqref{eq:toric-pic-e} has type~$H_1 + H_2 - E = h$.
Any such divisor corresponds to a morphism
\begin{equation*}
\xi \colon {\mathscr{O}}(-1,-1) \oplus {\mathscr{O}}(-1,0) \oplus {\mathscr{O}}(0,-1) \longrightarrow {\mathscr{O}}.
\end{equation*}
Furthermore, the divisor~${\tilde{X}}^+ \subset {\tilde{Y}}^+$ comes with a morphism~$\sigma_+ \colon {\tilde{X}}^+ \to Y^+$ defined over~${\mathsf{k}}$.
By the Nishimura lemma we have~${\tilde{X}}^+({\mathsf{k}}) \ne \varnothing$, hence~$Y^+({\mathsf{k}}) \ne \varnothing$,
and since~$Y^+$ is a ${\mathsf{k}}$-form of~$\mathbb{P}^1 \times \mathbb{P}^1$, it is ${\mathsf{k}}$-rational by Corollary~\ref{corollary:product-rational}.
Finally, the general fiber of the morphism~$\sigma_+ \colon {\tilde{X}}^+ \to Y^+$ is a $1$-dimensional linear section of a form of a projective plane,
hence it is isomorphic to~$\mathbb{P}^1$, hence~${\tilde{X}}^+$ is rational over~$Y^+$, hence is~${\mathsf{k}}$-rational, hence so is~$X$.
\end{proof}
\begin{remark}
One can check that the birational isomorphism~$\psi \colon {\tilde{X}} \dashrightarrow
{\tilde{X}}^+$
is a flop in the union of the strict transforms of two ${\bar{\mathsf{k}}}$-lines passing
through the point~$x \in X$
and that~$\sigma_+ \colon {\tilde{X}}^+ \to Y^+$ is the projectivization of the
vector bundle~$\Ker(\xi)$ of rank~$2$ over~$Y^+$,
and that these maps provide a Sarkisov link~\eqref{eq:sl-general}.
This is an example of a pseudoisomorphism between almost del Pezzo varieties of degree~$5$,
see~\cite[Lemma~5.4 and proof of Theorem~1.2]{KP22}.
\end{remark}
\subsection{Rationality of~\type{2,2,2}}
A similar argument works for threefolds of type~\type{2,2,2}.
\begin{proposition}
\label{prop:x222}
Let $X$ be a Fano threefold of type~\type{2,2,2}.
If $X({\mathsf{k}}) \ne \varnothing$ then~$X$ is ${\mathsf{k}}$-rational.
\end{proposition}
\begin{proof}
Let~$x$ be a ${\mathsf{k}}$-point of~$X$.
By definition~$X_{\bar{\mathsf{k}}}$ is a complete intersection of three divisors in~$Y_{\bar{\mathsf{k}}}
= \mathbb{P}(V_1) \times \mathbb{P}(V_2) \times \mathbb{P}(V_3) \cong \mathbb{P}^2 \times \mathbb{P}^2 \times
\mathbb{P}^2$
of multidegree~$(1,1,0)$, $(1,0,1)$, and~$(0,1,1)$, respectively.
Denote by
\begin{equation*}
F_{12} \in V_1^\vee \otimes V_2^\vee,
\qquad
F_{13} \in V_1^\vee \otimes V_3^\vee,
\qquad
F_{23} \in V_2^\vee \otimes V_3^\vee,
\end{equation*}
their equations.
We apply Proposition~\ref{prop:x-transform}; for this we consider the intersection
\begin{equation*}
{\tilde{X}}^+_{\bar{\mathsf{k}}} \subset \mathbb{P}_{\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1}({\mathscr{E}}) =
\mathbb{P}_{\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1}({\mathscr{O}}(-1,-1,-1) \oplus {\mathscr{O}}(-1,-1,0) \oplus
{\mathscr{O}}(-1,0,-1) \oplus {\mathscr{O}}(0,-1,-1) )
\end{equation*}
of the three strict transforms of the above divisors, which by~\eqref{eq:toric-pic-e} have types
\begin{equation*}
H_1 + H_2 - E = h - h_3,
\qquad
H_1 + H_3 - E = h - h_2,
\qquad
H_2 + H_3 - E = h - h_1,
\end{equation*}
hence correspond to a morphism of vector bundles
\begin{multline*}
\xi \colon {\mathscr{O}}(-1,-1,-1) \oplus {\mathscr{O}}(-1,-1,0) \oplus {\mathscr{O}}(-1,0,-1) \oplus
{\mathscr{O}}(0,-1,-1)
\longrightarrow \\ \longrightarrow
{\mathscr{O}}(0,0,-1) \oplus {\mathscr{O}}(0,-1,0) \oplus {\mathscr{O}}(-1,0,0).
\end{multline*}
It is easy to see that~$\xi$ is given by the matrix
\begin{equation}
\label{eq:xi-x222}
\xi =
\begin{pmatrix}
\bar{F}_{12} & 0 & F_{12}(-,v_2) & F_{12}(v_1,-) \\
\bar{F}_{13} & F_{13}(-,v_3) & 0 & F_{13}(v_1,-) \\
\bar{F}_{23} & F_{23}(-,v_3) & F_{23}(v_2,-) & 0
\end{pmatrix},
\end{equation}
where we write~$x = (v_1,v_2,v_3)$, choose splittings~$V_i = {\bar{\mathsf{k}}} v_i \oplus \bar{V}_i$,
write $\bar{F}_{ij}$ for the
restriction of~$F_{ij}$ to~$\bar{V}_i \otimes \bar{V}_j$,
and consider~$F_{ij}(v_i,-)$ and~$F_{ij}(-,v_j)$ as linear functions
on~$\bar{V}_j$ and~$\bar{V}_i$ by restriction.
Let us check the dimension conditions~\eqref{eq:dimension-conditions}.
Since~$(Y_i)_{\bar{\mathsf{k}}}$ is a fiber of the projection
\begin{equation*}
\pi_{i} \colon Y_{\bar{\mathsf{k}}} = \mathbb{P}^2 \times \mathbb{P}^2 \times \mathbb{P}^2 \longrightarrow \mathbb{P}^2,
\end{equation*}
it follows from Lemma~\ref{lemma:conics} that~$X_{\bar{\mathsf{k}}} \cap (Y_i)_{\bar{\mathsf{k}}}$ is a
conic, hence the first part of the dimension conditions is satisfied.
To check the second part, we need to show that the restriction
\begin{equation*}
\xi_{2,3} \colon {\mathscr{O}}(-1,0,-1) \oplus {\mathscr{O}}(0,-1,-1) \longrightarrow {\mathscr{O}}(0,0,-1)
\oplus {\mathscr{O}}(0,-1,0) \oplus {\mathscr{O}}(-1,0,0)
\end{equation*}
of~$\xi$ to the last two summands of~${\mathscr{E}}$ (given by the last two columns
of~\eqref{eq:xi-x222}) cannot be everywhere degenerate,
and similarly for the restrictions~$\xi_{1,3}$ and~$\xi_{1,2}$.
Assuming that~$\xi_{2,3}$ is everywhere degenerate we conclude
from~\eqref{eq:xi-x222} that
\begin{equation*}
F_{12}(v_1,-) = F_{13}(v_1,-) = 0
\quad\text{or}\quad
F_{12}(-,v_2) = F_{23}(v_2,-) = 0
\quad\text{or}\quad
F_{13}(v_1,-) = F_{23}(v_2,-) = 0.
\end{equation*}
In any case it would follow that
at least two of the bilinear forms~$F_{i,j}$ are degenerate, hence
at least two of the divisors~$W_{i,j} \subset \mathbb{P}(V_i) \times \mathbb{P}(V_j)$
(defined by the equation~$F_{i,j}$) are singular, which contradicts Lemma~\ref{lemma:pi-i}\ref{lemma:pi-i:222}.
Thus, the conditions~\eqref{eq:dimension-conditions} are satisfied, and we
conclude from Proposition~\ref{prop:x-transform}
that~$X$ is ${\mathsf{k}}$-birational to a ${\mathsf{k}}$-form of the complete
intersection~${\tilde{X}}^+_{\bar{\mathsf{k}}}$.
Finally, the subvariety~${\tilde{X}}^+ \subset {\tilde{Y}}^+$ comes with a morphism~$\sigma_+
\colon {\tilde{X}}^+ \to Y^+$ defined over~${\mathsf{k}}$.
By the Nishimura lemma we have~${\tilde{X}}^+({\mathsf{k}}) \ne \varnothing$, hence~$Y^+({\mathsf{k}}) \ne \varnothing$,
and since~$Y^+$ is a ${\mathsf{k}}$-form of~$\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$, it is
${\mathsf{k}}$-rational by Corollary~\ref{corollary:product-rational}.
Moreover, the general fiber of the morphism~$\sigma_+ \colon {\tilde{X}}^+ \to Y^+$ is
a 0-dimensional linear section of a form of a projective space,
hence this morphism is birational, hence~${\tilde{X}}^+$ is~${\mathsf{k}}$-rational, hence so is~$X$.
\end{proof}
\begin{remark}
If the point~$x$ does not lie on a ${\bar{\mathsf{k}}}$-line, i.e., $\mathrm{F}_1(X,x) = \varnothing$,
one can check that the birational isomorphism~$\psi \colon {\tilde{X}} \dashrightarrow {\tilde{X}}^+$
is a flop in the union of the strict transforms of three smooth ${\bar{\mathsf{k}}}$-conics
passing through the point~$x \in X$,
that~$\sigma_+ \colon {\tilde{X}}^+ \to Y^+$ is the blowup of a smooth
geometrically rational curve of multidegree~$(2,2,2)$
and that these maps provide a Sarkisov link~\eqref{eq:sl-general}.
\end{remark}
\subsection{Unirationality of~\type{1,1,1,1}}
Finally, we deal with threefolds of type~\type{1,1,1,1}.
\begin{proposition}
\label{prop:x1111}
Let $X$ be a Fano threefold of type~\type{1,1,1,1}.
If $X({\mathsf{k}}) \ne \varnothing$ then~$X$ is ${\mathsf{k}}$-unirational.
\end{proposition}
\begin{proof}
Let~$x$ be a ${\mathsf{k}}$-point of~$X$.
By definition~$X$ is a smooth divisor of multidegree~$(1,1,1,1)$ in
a~${\mathsf{k}}$-form~$Y$ of~$\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$.
The birational isomorphism~$\psi \colon \Bl_x(Y) = {\tilde{Y}} \dashrightarrow {\tilde{Y}}^+ = \Bl_{s_1,s_2,s_3,s_4}(\mathbb{P}^4)$
is small by Theorem~\ref{proposition:toric-link},
so it follows that~$X$ is birational to a ${\mathsf{k}}$-form of a divisor
\begin{equation*}
{\tilde{X}}^+_{\bar{\mathsf{k}}} \subset \Bl_{s_1,s_2,s_3,s_4}(\mathbb{P}^4)
\end{equation*}
which by~\eqref{eq:toric-pic-e} has type~$H_1 + H_2 + H_3 + H_4 - E = 3h - 2
\sum_{i=1}^4 e_i$, in particular, ${\tilde{X}}^+$ is a cubic hypersurface.
Moreover, the exceptional divisor~$E \cap \Bl_x(X) \subset {\tilde{Y}} = \Bl_x(Y)$ of
the blowup~$\Bl_x(X) \to X$
is an irreducible ${\mathsf{k}}$-rational surface birational to a ${\mathsf{k}}$-form of the
complete intersection of~${\tilde{X}}^+_{\bar{\mathsf{k}}}$
with the linear span~$\mathbb{P}^3 \subset \mathbb{P}^4$ of the points~$s_i$.
Let us prove that~${\tilde{X}}^+_{\bar{\mathsf{k}}}$ is not a cone.
Indeed, ${\tilde{X}}^+_{\bar{\mathsf{k}}}$ is smooth away from the linear span~$\mathbb{P}^3$ of the~$s_i$,
because the map~$\psi$ from Theorem~\ref{proposition:toric-link} is an
isomorphism over it and~$X_{\bar{\mathsf{k}}}$ is smooth,
so if~${\tilde{X}}^+_{\bar{\mathsf{k}}}$ is a cone, its vertex belongs to the~$\mathbb{P}^3$.
But then its intersection with the~$\mathbb{P}^3$ (which has been shown to be an
irreducible~${\mathsf{k}}$-rational surface)
is itself a cone and has a singular point at each of the~$s_i$.
But it is easy to see that any such cone is reducible;
this contradiction proves the claim.
Now we conclude that~${\tilde{X}}^+$ is~${\mathsf{k}}$-unirational by~\cite[Theorem~1.2]{Kollar:cubic}.
\end{proof}
\section{Rationality of type \type{4,4}}
\label{sec:x44}
In this section we prove rationality of Fano threefolds of type \type{4,4}.
\subsection{Sarkisov links}
\label{subsec:sl-x44-existence}
We start with a construction of two Sarkisov links.
Recall that~$\mathrm{F}_1(X,x)$ denotes the Hilbert scheme of lines on~$X$ passing
through~$x$, see~\S\ref{subsec:lines};
and that by Lemma~\ref{lemma:lines} if~$x \in X({\mathsf{k}})$ and~$\mathrm{F}_1(X,x)$
is not empty,
$\mathrm{F}_1(X,x)$ is the union of two reduced~${\bar{\mathsf{k}}}$-points swapped by the Galois action.
If~$L_1$ and~$L_2$ are the corresponding~${\bar{\mathsf{k}}}$-lines on~$X$ (passing
through~$x$), then
\begin{equation*}
\Theta(x) := L_1 \cup L_2
\end{equation*}
is a singular ${\mathsf{k}}$-conic on~$X$ irreducible over~${\mathsf{k}}$ and
with~$\Sing(\Theta(x)) = \{x\}$.
Recall that a quintic del Pezzo threefold is a Fano threefold of index~$2$ and half-anticanonical degree~$5$.
Over an algebraically closed field it can be realized as a complete intersection
of the Grassmannian~$\Gr(2,5)$
with a linear subspace of codimension~$3$,
see~\cite[Chapter~2, Theorem~1.1]{Iskovskikh-1980-Anticanonical}.
\begin{theorem}
\label{theorem:x44-links}
Let $X$ be a Fano threefold of type~\type{4,4} and let~$x \in X({\mathsf{k}})$ be a
${\mathsf{k}}$-point.
\begin{enumerate}
\item
\label{prop:sl:point-2-blowup-Q}
If\/ $\mathrm{F}_1(X,x) = \varnothing$ there exists a Sarkisov link~\eqref{eq:sl-general} defined over~${\mathsf{k}}$, where
\begin{itemize}
\item
$\sigma$ is the blowup of the point~$x$,
\item
$X^{+}$ is a smooth quintic del Pezzo threefold, and
\item
$\sigma_+$ is the blowup of a smooth ${\mathsf{k}}$-irreducible curve~$B^+ \subset X^+$ of degree~$4$ with two geometrically rational ${\bar{\mathsf{k}}}$-components.
\end{itemize}
\item
\label{prop:sl-conic}
If\/ $\mathrm{F}_1(X,x) \ne \varnothing$ there exists a Sarkisov link~\eqref{eq:sl-general} defined over~${\mathsf{k}}$, where
\begin{itemize}
\item
$\sigma$ is the blowup of the singular ${\mathsf{k}}$-irreducible
conic~$\Theta(x)$,
\item
$X^{+}$ is a smooth Fano threefold of type~\type{2,2}, and
\item
$\sigma_+$ is the blowup of a singular ${\mathsf{k}}$-irreducible curve~$B^+ \subset X^+$ of degree~$6$ with two geometrically rational ${\bar{\mathsf{k}}}$-components.
\end{itemize}
\end{enumerate}
\end{theorem}
The proof of the theorem takes~\S\ref{subsec:sl-x44-existence} and~\S\ref{subsec:sl-x44-details}:
in the rest of~\S\ref{subsec:sl-x44-existence} we prove the existence of the links,
and in~\S\ref{subsec:sl-x44-details} we describe them in detail.
The proofs of cases~\ref{prop:sl:point-2-blowup-Q} and~\ref{prop:sl-conic} are completely analogous,
so to carry them on simultaneously we introduce the following convenient notation
\begin{equation}
\label{eq:m-x}
m = m(x) :=
\begin{cases}
2, & \text{if $\mathrm{F}_1(X,x) = \varnothing$},\\
1, & \text{if $\mathrm{F}_1(X,x) \ne \varnothing$}.
\end{cases}
\end{equation}
The proof of the existence of the links is analogous to the first parts of~\cite[Theorem~5.17 and Theorem~5.9]{KP19},
so we use below some results from~\cite[\S5.1]{KP19}.
Let
\begin{equation*}
\sigma \colon {\tilde{X}} \longrightarrow X
\end{equation*}
be the blowup of~$X$ at~$x$ or at~$\Theta(x)$, respectively.
We denote by~$H$ (the pullback to~${\tilde{X}}$ of) the anticanonical class of~$X$ and
by~$E$ the exceptional divisor of~$\sigma$.
First, note that for $m = m(x)$ the anticanonical linear system
\begin{equation*}
|-K_{\tilde{X}}| = |H - mE|
\end{equation*}
is base-point free by Theorem~\ref{th:bht} and~\cite[Lemma~5.7 and~5.5]{KP19}.
Moreover, combining~\cite[(5.1.9) and~(5.1.7)]{KP19}, we can uniformly write
\begin{equation}
\label{eq:hhe}
H^3 = 2g - 2,
\qquad
H^2 \cdot E = 0,
\qquad
H \cdot E^2 = 2(m - 2),
\qquad
E^3 = m - 1,
\end{equation}
where we recall from Table~\ref{table:fanos} that~$g = {\mathrm{g}}(X) = 15$.
We will also need the following observation.
\begin{lemma}
\label{lemma:m}
The linear system ${\mathscr{M}} := |H - (m+1)E|$
on the blowup~${\tilde{X}}$ of~$X$ has positive dimension
\begin{equation}
\label{eq:dim-m}
\dim {\mathscr{M}} \ge g - m - 7 \ge 6,
\end{equation}
and has no fixed components.
\end{lemma}
\begin{proof}
The dimension is estimated in~\cite[Lemma~5.4(iii) and~(i)]{KP19}.
To prove that ${\mathscr{M}}$ has no fixed components, note that the linear system~$|kE|$ is 0-dimensional for any~$k \ge 0$
(since~$E$ is the exceptional divisor of a blowup),
hence the only possibility for a fixed component of~${\mathscr{M}}$ is provided by the divisor~$E$ with some multiplicity.
So, assume
\begin{equation*}
|H - (m+1)E|=(a - m - 1)E + |H - aE|,
\end{equation*}
where $a\ge m + 2$ and $E$ is not a fixed component of the linear system~$|H - aE|$.
Since the linear system~$|H - mE|$ is base-point free and $|H - aE|$ has no fixed components, using~\eqref{eq:hhe} we obtain
\begin{equation*}
0 \le (H - aE)^2\cdot (H - mE)= 2g-2-a^2(m^2 - 3m + 4) + 4am(m-2).
\end{equation*}
When $m = 2$ this gives $a^2 \le 14$, hence $a \le 3$, and when $m = 1$ this
gives $(a+1)^2 \le 15$, hence~\mbox{$a \le 2$}.
In both cases this contradicts the assumption~$a \ge m + 2$.
\end{proof}
Now we can deduce the existence of the Sarkisov links.
\begin{proposition}
\label{prop:sl-x44-existence}
Let $X$ be a Fano threefold of type~\type{4,4} with a ${\mathsf{k}}$-point~$x$.
\begin{enumerate}
\item
If~$\mathrm{F}_1(X,x) = \varnothing$, there exists a Sarkisov
link~\eqref{eq:sl-general},
where~$\sigma$ is the blowup of~$x$.
\item
If~$\mathrm{F}_1(X,x) \ne \varnothing$, there exists a Sarkisov
link~\eqref{eq:sl-general},
where $\sigma$ is the blowup of~$\Theta(x)$.
\end{enumerate}
In both cases the link is defined over~${\mathsf{k}}$.
\end{proposition}
\begin{proof}
We use notation~\eqref{eq:m-x}.
Recall that the anticanonical class $H = -K_X$ is very ample
and the image of the anticanonical embedding $X \subset \mathbb{P}^{g + 1}$ is an
intersection of quadrics (see Theorem~\ref{th:bht}).
The anticanonical morphism $\phi \colon {\tilde{X}} \to \mathbb{P}^{g-m-1}$ cannot contract a divisor~$D$,
because by~\cite[Lemma~5.7 and~5.5]{KP19} this divisor is then a fixed component of~${\mathscr{M}}$,
but by Lemma~\ref{lemma:m} this linear system has no fixed components.
Therefore, the required link exists and is defined over~${\mathsf{k}}$ by~\cite[Lemma~5.7 and~5.5]{KP19}.
\end{proof}
\subsection{The second contraction}
\label{subsec:sl-x44-details}
By Proposition~\ref{prop:sl-x44-existence} we have the
diagram~\eqref{eq:sl-general},
so to finish the proof of Theorem~\ref{theorem:x44-links} it remains to describe the extremal contraction~$\sigma_+$.
During this step we systematically use the classification of extremal contractions from~\cite{Mori-1982} and~\cite{Cutkosky-1988}.
We denote by $H^+,E^+ \in \Pic({\tilde{X}}^+)$ the strict transforms of the
classes~$H,E \in \Pic({\tilde{X}})$.
Note that
\begin{equation*}
- K_{{\tilde{X}}^+} = H^+ - mE^+
\end{equation*}
because $-K_{\tilde{X}} = H - mE$ by definition of~${\tilde{X}}$ and the map~$\psi$ is an isomorphism in codimension one.
Consider also the strict transform
\begin{equation*}
{\mathscr{M}}^+ := |H^+ - (m+1)E^+|
\end{equation*}
of the linear system~${\mathscr{M}}$.
We denote by $\Upsilon_i \subset {\tilde{X}}$ the flopping curves and by~$\Upsilon_i^+
\subset {\tilde{X}}^+$ the corresponding flopped curves.
Finally, when $\mathrm{F}_1(X,x) = \varnothing$ we denote by~$C$ a general twisted
cubic curve on~$X$ passing through~$x$
and otherwise we denote by~$C$ a general conic meeting~$\Theta(x)$
(recall Lemma~\ref{lemma:cubics-x} and Lemma~\ref{lemma:conics-c0} for the
description of the corresponding Hilbert schemes).
Note that
\begin{equation}
\label{eq:h-dot-c}
H \cdot C = m + 1.
\end{equation}
We denote by~${\tilde{C}}$ the strict transform of~$C$ in~${\tilde{X}}$
and by~${\tilde{C}}^+$ the strict transform of~${\tilde{C}}$ in~${\tilde{X}}^+$.
Note that by Remark~\ref{rem:f2x-theta} and Remark~\ref{rem:f3x-x} the curve~${\tilde{C}}$ is smooth and
\begin{equation}
\label{eq:e-dot-c}
E \cdot {\tilde{C}} = 1;
\end{equation}
in particular, $(H - mE)\cdot {\tilde{C}} = 1$ and~${\tilde{C}}$ does not contain the curves~$\Upsilon_i$.
\begin{lemma}
\label{lemma:sigma+}
The nef cone of~${\tilde{X}}^+$ is generated by the anticanonical class $-K_{{\tilde{X}}^+}$ and~$M^+ \in {\mathscr{M}}^+$,
and the Mori cone of~${\tilde{X}}^+$ is generated by the class of the
curves~$\Upsilon_i^+$ and the class of\/~${\tilde{C}}^+$.
In particular, the extremal contraction~$\sigma_+$ is given by a multiple of the linear
system~${\mathscr{M}}^+$ and contracts the extremal ray generated by~${\tilde{C}}^+$.
\end{lemma}
\begin{proof}
Since~$\phi$ is crepant and~$\psi$ is a flop, the morphism~$\phi_+$ is crepant as well.
Moreover, the anticanonical linear system~$|-K_{\tilde{X}}|$ is base-point free
by~\cite[Lemma~5.7]{KP19}, hence~$|-K_{{\tilde{X}}^+}|$ is base-point free as well.
On the other hand, we have~$(H - mE) \cdot \Upsilon_i = -K_{{\tilde{X}}} \cdot \Upsilon_i = 0$,
and since $H \cdot \Upsilon_i > 0$, we conclude that $E \cdot \Upsilon_i > 0$.
Therefore, for~$M \in {\mathscr{M}}$ we have
\begin{equation*}
M \cdot \Upsilon_i =
(H - (m+1)E)\cdot \Upsilon_i =
-E \cdot \Upsilon_i < 0.
\end{equation*}
If $M^+ \in {\mathscr{M}}^+$ is the strict transform of~$M$,
this implies that $M^+ \cdot \Upsilon_i^+ > 0$ by one of the definitions of a flop, see, e.g.,~\cite[Definition~6.10]{Kollar-Mori:88}.
Now if~$M^+$ is not nef, it is negative on the extremal ray~$\mathrm{R}$
corresponding to the contraction~$\sigma_+ \colon {\tilde{X}}^+ \to X^+$.
Since the canonical class is also negative on~$\mathrm{R}$, the contraction~$\sigma_+$ cannot be small
(see \cite[Theorem~0]{Benveniste} or \cite[Corollary~6.3.4]{MP:1pt}),
hence curves in~$\mathrm{R}$ sweep a subvariety of~${\tilde{X}}^+$ of dimension~$\ge 2$,
hence the base locus of~${\mathscr{M}}^+$ is at least 2-dimensional, which contradicts Lemma~\ref{lemma:m}.
This proves that~$M^+$ is nef.
Now we have $-K_{\tilde{X}} \cdot {\tilde{C}} = (H - mE) \cdot {\tilde{C}} = 1$ by~\eqref{eq:h-dot-c} and~\eqref{eq:e-dot-c},
hence $-K_{{\tilde{X}}^+} \cdot {\tilde{C}}^+ = 1$.
On the other hand, since a general divisor~$H$ meets~$C$ away from the
indeterminacy locus of the map~$X\dashrightarrow \tilde X^+$,
we have~$H^+\cdot {\tilde{C}}^+ \ge H\cdot C=m+1$ and so~$E^+\cdot {\tilde{C}}^+ \ge 1$.
Thus,
\begin{equation*}
M^+ \cdot {\tilde{C}}^+ = (-K_{{\tilde{X}}^+} - E^+) \cdot {\tilde{C}}^+ = 1 - E^+ \cdot {\tilde{C}}^+ \le 0.
\end{equation*}
Since~$M^{+}$ is nef, $M^+ \cdot {\tilde{C}}^+= 0$.
Combining the above computations we conclude that the nef cone of~${\tilde{X}}^+$ is
generated by~$-K_{{\tilde{X}}^+}$ and~$M^+$
and the Mori cone is generated by~$\Upsilon_i^+$ and~${\tilde{C}}^+$.
The rest of the lemma follows from the Mori contraction
theorem~\cite[Theorems~3.1 and~3.2]{Mori-1982}.
\end{proof}
Now we can finally prove Theorem~\ref{theorem:x44-links}.
\begin{proof}[Proof of Theorem~\xref{theorem:x44-links}]
Let $X$ be a Fano threefold of type~\type{4,4}.
By Proposition~\ref{prop:sl-x44-existence}
there exists a Sarkisov link~\eqref{eq:sl-general}
and it remains to describe the contraction~$\sigma_+$.
Since~$\sigma_+$ is an extremal contraction, we have $\uprho(X^+) =
\uprho({\tilde{X}}^+) - 1 = \uprho({\tilde{X}}) - 1 = \uprho(X)$, hence~$\uprho(X^+) = 1$.
Similarly, we have~$\uprho(X^+_{\bar{\mathsf{k}}}) \le \uprho({\tilde{X}}^+_{\bar{\mathsf{k}}}) - 1 =
\uprho({\tilde{X}}_{\bar{\mathsf{k}}}) - 1 = \uprho(X_{\bar{\mathsf{k}}}) = 2$, hence
\begin{equation}
\label{eq:rho-xplus-bkk}
\uprho(X^+_{\bar{\mathsf{k}}}) \le 2.
\end{equation}
On the other hand, in the case $\mathrm{F}_1(X,x) \ne \varnothing$, the
varieties~${\tilde{X}}$ and~${\tilde{X}}^+$ are not smooth and
arguing as in the proof of~\cite[Theorem~5.9]{KP19} we obtain
\begin{equation}
\label{eq:rk-cl-txp}
\rk \Cl({\tilde{X}}^+) = 2,
\qquad
\rk \Cl({\tilde{X}}^+_{\bar{\mathsf{k}}}) = 5 - m.
\end{equation}
Since~$\phi$ and~$\phi_+$ are crepant morphisms, the projection formula implies
that any triple intersection product of divisor classes on~${\tilde{X}}^+$ which
includes~$K_{{\tilde{X}}^+}$
is equal to the analogous triple product on~${\tilde{X}}$, so
using~\eqref{eq:hhe}
we compute
(recall that~$g = {\mathrm{g}}(X) = 15$)
\begin{equation}
\label{eq:new1}
(-K_{\tilde X^+})^3 = 2(g - m - 3) {} = 24 - 2m,\quad
(-K_{\tilde X^+})^2\cdot E^+ = 4,\quad
(-K_{\tilde X^+})\cdot (E^+)^2 = -2.
\end{equation}
On the other hand, by Lemma~\ref{lemma:sigma+} and primitivity of~$H^+ -
(m + 1)E^+$ we have
\begin{equation}
\label{eq:new2}
H^+ - (m + 1)E^+ = \sigma_+^*A^+,
\end{equation}
where $A^+$ is the ample generator of the Picard group of~$X^+$.
We have
\begin{equation}
\label{eq:intersections}
(\sigma_+^*A^+)^2 \cdot(-K_{{\tilde{X}}^+}) = (H^+ - (m + 1)E^+)^2\cdot (-K_{X^+}) =
2(g - m - 8) = 14 - 2m > 0,
\end{equation}
therefore~$\sigma_+$ is not a del Pezzo fibration.
Similarly, if~$\sigma_+$ is a conic bundle, it follows that
\begin{equation*}
(A^+)^2 = 7 - m,
\end{equation*}
hence~$X^+$ is a smooth quintic or sextic del Pezzo surface,
which of course contradicts the inequality~\eqref{eq:rho-xplus-bkk}.
Therefore, the morphism~$\sigma_+$ is birational.
By Lemma~\ref{lemma:sigma+} the morphism~$\sigma_+$ contracts the strict
transform~$R^+$ of the divisor swept by curves~$C$,
i.e., the strict transform of the divisor~$R_x \subset X$ if~$\mathrm{F}_1(X,x) =
\varnothing$, or of the divisor~$R_{\Theta(x)}$ otherwise.
In both cases Lemma~\ref{lemma:cubics-x} and Lemma~\ref{lemma:conics-c0} show
that~$R^+$
has over~${\bar{\mathsf{k}}}$ two irreducible components swapped by the Galois group.
Therefore, it follows from~\eqref{eq:rk-cl-txp} that
\begin{equation*}
\rk \Cl(X^+) = 1,
\qquad
\rk \Cl(X^+_{\bar{\mathsf{k}}}) = 3 - m,
\end{equation*}
and~$\sigma_+$ is the blowup of two ${\bar{\mathsf{k}}}$-curves or two ${\bar{\mathsf{k}}}$-points.
Furthermore, by Lemmas~\ref{lemma:cubics-x} and~\ref{lemma:conics-c0} we have
\begin{equation*}
R^+ \sim H^+ - (m + 2)E^+.
\end{equation*}
Denoting by~$i_+$ the index of~$X^+$ and by~$a_+$ the discrepancy of the exceptional divisor~$R^+$ of~$\sigma_+$,
and computing the anticanonical class of~${\tilde{X}}^+$ in two ways we obtain the equality
\begin{equation*}
H^+ - mE^+ = i_+(H^+ - (m+1)E^+) - a_+(H^+ - (m+2)E^+).
\end{equation*}
Solving this equation, we obtain~$i_+ = 2$ and~$a_+ = 1$.
Thus, $X^+$ is a Fano threefold of index~$2$
and~$\sigma_+$ is either the blowup of a ${\mathsf{k}}$-irreducible curve~$B^+$ with two~${\bar{\mathsf{k}}}$-components,
or of two rational double ${\bar{\mathsf{k}}}$-points on~$X^+$ swapped by the Galois action.
Moreover, using the equality from~\eqref{eq:intersections} we obtain
\begin{equation*}
14 - 2m= (\sigma_+^*A^+)^2\cdot (-K_{{\tilde{X}}^+}) = (\sigma_+^*A^+)^2 \cdot
(2\sigma_+^*A^+ - R^+) = 2(\sigma_+^*A^+)^3,
\end{equation*}
hence $X^+$ is a quintic or sextic del Pezzo threefold, respectively.
Finally, if~$X^+$ is singular, its class group~$\Cl(X^+)$ has rank greater than~$1$ (see~\cite[Theorem~1.7]{Prokhorov-GFano-1}),
which contradicts to the equality~$\rk\Cl(X^+) = 1$ obtained above.
Thus, $X^+$ is smooth and~$\sigma_+$ is the blowup of a curve~$B^+$
such that~$B^+_{{\bar{\mathsf{k}}}}$ has two irreducible~${\bar{\mathsf{k}}}$-components swapped by the Galois group.
To finally compute the degree of~$B^+$, recall that~$R^+$ is the exceptional divisor of~$\sigma_+$.
Note that on the one hand, equalities~\eqref{eq:new1} and~\eqref{eq:new2} imply that
\begin{equation*}
(\sigma_+^*A^+) \cdot (-K_{{\tilde{X}}^+})^2 = 2g - 2m - 10 = 20 - 2m,
\end{equation*}
and on the other hand, this expression is equal to
\begin{equation*}
(\sigma_+^*A^+) \cdot (2\sigma_+^*A^+ - R^+)^2 =
4(\sigma_+^*A^+)^3 + (\sigma_+^*A^+) \cdot (R^+)^2 = 4(7 - m)- \deg(B^+)
\end{equation*}
(where the degree is computed with respect to~$A^+$).
Thus,
\begin{equation*}
\deg(B^+) = 8 - 2m,
\end{equation*}
hence~$B^+$ is a quartic or sextic curve with two connected ${\bar{\mathsf{k}}}$-components (swapped by the Galois action),
i.e., a union of two conics or two cubic curves.
\end{proof}
\subsection{Rationality}
Now we use the constructed links to prove rationality of threefolds of
type~\type{4,4}.
\begin{proposition}
\label{prop:x44}
Let $X$ be a Fano threefold of type~\type{4,4}.
If $X({\mathsf{k}}) \ne \varnothing$ then~$X$ is ${\mathsf{k}}$-rational.
\end{proposition}
\begin{proof}
Let~$x \in X({\mathsf{k}})$ be a~${\mathsf{k}}$-point.
First, assume that~$\mathrm{F}_1(X,x) = \varnothing$.
Then by Theorem~\ref{theorem:x44-links}\ref{prop:sl:point-2-blowup-Q}
the variety~$X$ is birational to a smooth quintic del Pezzo threefold~$X^+$.
But~$X^+$ is ${\mathsf{k}}$-rational by~\cite[Theorem~3.3]{KP19}, hence so is~$X$.
Now assume that $\mathrm{F}_1(X,x) \ne \varnothing$.
Then by Theorem~\ref{theorem:x44-links}\ref{prop:sl-conic}
the variety $X$ is birational to a smooth Fano threefold $X^+$ of
type~\type{2,2}.
Moreover, by the Nishimura lemma we have~$X^+({\mathsf{k}}) \ne \varnothing$.
Therefore, $X^+$ is ${\mathsf{k}}$-rational by~Proposition~\ref{prop:x22}, hence so
is~$X$.
\end{proof}
\section{Fano threefolds of type~\type{3,3}}
In this section we prove that a Fano threefold~$X$ of type~\type{3,3} is
${\mathsf{k}}$-unirational if~$X({\mathsf{k}}) \ne \varnothing$,
but not ${\mathsf{k}}$-rational if~$\uprho(X) = 1$.
\subsection{The discriminant curve}
\label{subsec:discriminant}
Let~$X$ be a Fano threefold of type~\type{3,3} with~$X({\mathsf{k}}) \ne \varnothing$.
Recall from Lemma~\ref{lemma:picard} that the image~${\mathrm{G}}_X$ of the Galois group~${\mathrm{G}}({\bar{\mathsf{k}}}/{\mathsf{k}})$ in~$\Aut(\Pic(X_{\bar{\mathsf{k}}}))$
is a group of order~$2$ swapping the generators~$H_1$ and~$H_2$ of~$\Pic(X_{\bar{\mathsf{k}}})$.
The homomorphism ${\mathrm{G}}({\bar{\mathsf{k}}}/{\mathsf{k}})\to {\mathrm{G}}_X$ therefore defines a quadratic extension~${\mathsf{k}}'/{\mathsf{k}}$
such that $H_1$ and~$H_2$ are defined on~$X_{{\mathsf{k}}'}$, hence
\begin{equation*}
X_{{\mathsf{k}}'} \cong \Big(\mathbb{P}(V_1) \times \mathbb{P}(V_2)\Big) \cap \mathbb{P}(A^\perp),
\end{equation*}
where~$V_i$ are ${\mathsf{k}}'$-vector spaces of dimension~$4$ and~$A \subset V_1^\vee \otimes V_2^\vee$
is the $3$-dimensional subspace of linear equations of~$X_{{\mathsf{k}}'}$.
Note that the~${\mathsf{k}}'$-spaces $V_1 \otimes V_2$ and~$A$ are defined over~${\mathsf{k}}$,
as well as the inclusion~$A \subset V_1^\vee \otimes V_2^\vee$.
We think of vectors~$a \in A$ as of bilinear forms on~$V_1 \otimes V_2$ and denote by
\begin{equation*}
\Gamma \xhookrightarrow{\ \ } \mathbb{P}(A)
\end{equation*}
the \textsf{discriminant curve} parameterizing degenerate bilinear forms; it is also
defined over~${\mathsf{k}}$.
\begin{lemma}
\label{lemma:gamma-x33}
The curve~$\Gamma$ is a smooth plane quartic curve;
in particular it is a non-hyperelliptic curve of genus~$3$.
\end{lemma}
\begin{proof}
The discriminant divisor in~$\mathbb{P}(V_1^\vee \otimes V_2^\vee)$, i.e., the divisor parameterizing degenerate bilinear forms,
is a quartic hypersurface, hence~$\Gamma$ is a quartic curve or the entire plane.
To prove that~$\Gamma$ is a smooth curve we can work over~${\mathsf{k}}'$,
and it is enough to show that the tangent space to~$\Gamma$ at any point is 1-dimensional.
Assume to the contrary that the tangent space at a point~$[a] \in \mathbb{P}(A)$ is 2-dimensional; then
\begin{enumerate}
\item
\label{item:rank-a}
either the bilinear form~$a(-,-) \in V_1^\vee \otimes V_2^\vee$ has corank at least~$2$,
\item
\label{item:tangency-a}
or~$a$ has corank~$1$ and if the vectors~$v_1 \in V_1$ and~$v_2 \in V_2$ generate its left and right kernels, respectively,
the point $([v_1],[v_2]) \in \mathbb{P}(V_1) \times \mathbb{P}(V_2)$ belongs to~$X_{{\mathsf{k}}'}$.
\end{enumerate}
In case~\ref{item:rank-a}, if~$K_1 \subset V_1$ and~$K_2 \subset V_2$ are the left and right kernels of~$a$
(they have dimension~$\ge 2$ by assumption), the form~$a$ vanishes on~$\mathbb{P}(K_1) \times \mathbb{P}(K_2)$,
hence the intersection~$\big(\mathbb{P}(K_1) \times \mathbb{P}(K_2)\big) \cap X_{{\mathsf{k}}'}$
is a codimension-2 linear section of~$\mathbb{P}(K_1) \times \mathbb{P}(K_2)$, hence it is non-empty.
Therefore, in case~\ref{item:rank-a}, similarly to the
case~\ref{item:tangency-a},
there is a point $([v_1],[v_2]) \in X_{{\mathsf{k}}'}$ such that~$v_1$ and~$v_2$ belong to the left and right kernels of some~$a$.
Then the hyperplane section of~$\mathbb{P}(V_1) \times \mathbb{P}(V_2)$ by the hyperplane corresponding to~$a$ is singular at~$([v_1],[v_2])$,
hence~$X_{{\mathsf{k}}'}$ is also singular at this point.
\end{proof}
As explained in~Lemma~\ref{lemma:pi-i}\ref{lemma:pi-i:33}
the projections~$\pi_1 \colon X_{{\mathsf{k}}'} \to \mathbb{P}(V_1)$ and~$\pi_2 \colon X_{{\mathsf{k}}'} \to \mathbb{P}(V_2)$ defined over~${\mathsf{k}}'$, but not over~${\mathsf{k}}$,
are the blowups of curves $\Gamma_i\subset \mathbb{P}(V_i)$ of genus~$3$ and degree~$6$ also defined over~${\mathsf{k}}'$.
The next lemma relates the ${\mathsf{k}}'$-curves~$\Gamma_i$ to the discriminant curve~$\Gamma$ defined over~${\mathsf{k}}$.
\begin{lemma}
\label{lemma:gamma-gamma-i}
There is a natural isomorphism $\Gamma_i \cong \Gamma_{{\mathsf{k}}'}$ of curves
over~${\mathsf{k}}'$.
\end{lemma}
\begin{proof}
The fiber of the projection~$\pi_1$ over a point $[v_1] \in \mathbb{P}(V_1)$
is the intersection of the projectivizations of the orthogonals of~$v_1$ with respect to all bilinear forms~$a \in A$.
Therefore, it has positive dimension if and only if~$v_1$ belongs to the left kernel of one of the forms.
Furthermore, if~$v_1$ belongs to the left kernel of two distinct forms in~$A$,
the fiber of~$\pi_1$ over~$[v_1]$ contains a plane, which contradicts Corollary~\ref{cor:planes}.
This means that the morphism
\begin{equation*}
\gamma_i \colon \Gamma_{{\mathsf{k}}'} \longrightarrow \mathbb{P}(V_i),
\qquad
a \longmapsto \Ker_i(a),
\end{equation*}
where $\Ker_1$ and $\Ker_2$ denote the left and right kernels of the bilinear form~$a$, respectively,
is an isomorphism $\Gamma_{{\mathsf{k}}'} \to \Gamma_i$.
\end{proof}
\begin{remark}
It is also easy to check that if $H_i\vert_\Gamma$
are the pullbacks of the hyperplane classes of~$\Gamma_i\subset \mathbb{P}(V_i)$ to~$\Gamma_{{\mathsf{k}}'}$
under the isomorphism of Lemma~\ref{lemma:gamma-gamma-i} then~$H_1\vert_\Gamma + H_2\vert_\Gamma = 3K_\Gamma$
and that the divisor classes~$H_i\vert_\Gamma - K_\Gamma$ are non-effective and swapped by the~${\mathrm{G}}({\mathsf{k}}'/{\mathsf{k}})$-action.
Conversely, given two such classes on a curve~$\Gamma_{{\mathsf{k}}'}$ one can reconstruct the variety~$X$.
\end{remark}
\subsection{The double projection from a point}
\label{subsec:x33-double-projection}
Recall the quadratic extension~${\mathsf{k}}'/{\mathsf{k}}$ defined in~\S\ref{subsec:discriminant}.
Recall also the canonical embedding~$X \subset Y$, where~$Y$ is a ${\mathsf{k}}$-form of~$\mathbb{P}^3 \times \mathbb{P}^3$.
We consider the birational transformation of Theorem~\ref{proposition:toric-link}
for the variety~$Y_{{\mathsf{k}}'} = \mathbb{P}(V_1) \times \mathbb{P}(V_2)$ associated with a~${\mathsf{k}}$-point
\begin{equation*}
x_0 = ([v_1],[v_2]) \in X \subset Y.
\end{equation*}
As in~\S\ref{subsec:ppp} we denote ${\bar{V}}_i := V_i / {\mathsf{k}}' v_i$ and choose a splitting~$V_i = {\mathsf{k}}' v_i \oplus \bar{V}_i$.
The transformation of Theorem~\ref{proposition:toric-link} in this case looks as follows
\begin{equation}
\label{eq:link-y33}
\vcenter{\xymatrix@C=4em{
{\tilde{Y}} \ar[d]_\sigma
\ar@{-->}[rr]^-\psi &&
{\tilde{Y}}^+ \ar[d]^{\sigma_+}
\\
Y &
&
Y^+
}}
\end{equation}
where~$\sigma$ is the blowup of~$x_0$, $Y^+_{{\mathsf{k}}'} \cong \mathbb{P}(\bar{V}_1) \times
\mathbb{P}(\bar{V}_2)$,
$\sigma_+$ is the projectivization of the vector bundle
\begin{equation}
\label{eq:ce-x33}
{\mathscr{E}} = {\mathscr{O}}(-h_1-h_2) \oplus {\mathscr{O}}(-h_1) \oplus {\mathscr{O}}(-h_2)
\end{equation}
(here $h_i$ stand for the hyperplane classes of~$\mathbb{P}({\bar{V}}_i)$) over~$Y^+_{{\mathsf{k}}'}$,
and the map~$\psi$ is a small birational isomorphism.
Note that all varieties and maps in~\eqref{eq:link-y33} are defined over~${\mathsf{k}}$.
Recall also the relations~\eqref{eq:toric-pic-1} in~$\Pic({\tilde{Y}}_{{\mathsf{k}}'}) =
\Pic({\tilde{Y}}^+_{{\mathsf{k}}'})$
between the hyperplane classes $H_i$ of the factors~$\mathbb{P}(V_i)$ of~$Y_{{\mathsf{k}}'}$,
the class~$E$ of the exceptional divisor of~$\sigma$,
the hyperplane classes~$h_i$, and the relative hyperplane class~$h$
of~${\tilde{Y}}^+_{{\mathsf{k}}'} = \mathbb{P}_{Y^+_{{\mathsf{k}}'}}({\mathscr{E}})$:
\begin{equation}
\label{eq:pic-relations-x33}
\begin{cases}
h_1 = H_1 - E,\\
h_2 = H_2 - E,\\
h_{\phantom{2}} = H_1 + H_2 - E
\end{cases}
\qquad
\begin{cases}
\hbox to 1.2em{$H_1$} = h - h_2,\\
\hbox to 1.2em{$H_2$} = h - h_1,\\
\hbox to 1.2em{$E$} = h - h_1 - h_2.
\end{cases}
\end{equation}
Since~$X$ is a smooth linear section of~$Y$, containing the point~$x_0$,
it is a complete intersection of three divisors $D_\alpha$, $1 \le \alpha \le 3$, in the linear system~$|H_1 + H_2|$
whose strict transforms on~${\tilde{Y}}$ belong to the linear system~$|H_1 + H_2 - E|$.
Now it follows from~\eqref{eq:pic-relations-x33} that their strict
transforms~${\tilde{D}}^+_\alpha$ on~${\tilde{Y}}^+$ belong to the linear system~$|h|$.
As in Proposition~\ref{prop:x-transform} we consider the complete intersection
\begin{equation*}
{\tilde{X}}^+_{{\mathsf{k}}'} := {\tilde{D}}^+_1 \cap {\tilde{D}}^+_2 \cap {\tilde{D}}^+_3 \subset \mathbb{P}_{Y^+_{{\mathsf{k}}'}}({\mathscr{E}}).
\end{equation*}
It follows that~${\tilde{X}}^+_{{\mathsf{k}}'}$ is determined by a morphism of vector bundles
\begin{equation*}
\xi \colon {\mathscr{E}} \longrightarrow A^\vee \otimes {\mathscr{O}},
\end{equation*}
and if we choose a basis~$a_1,a_2,a_3$ in~$A$, it is easy to see that~$\xi$ is
given by the matrix
\begin{equation}
\label{eq:xi-x33}
\xi =
\begin{pmatrix}
\bar{a}_1(-,-) & a_1(-,v_2) & a_1(v_1,-) \\
\bar{a}_2(-,-) & a_2(-,v_2) & a_2(v_1,-) \\
\bar{a}_3(-,-) & a_3(-,v_2) & a_3(v_1,-)
\end{pmatrix},
\end{equation}
where~$\bar{a}_i \in \bar{V}_1^\vee \otimes \bar{V}_2^\vee$ denotes the restriction of the bilinear form~$a_i$ to~$\bar{V}_1 \otimes \bar{V}_2$,
while~$a_i(-,v_2)\in V_1^\vee$ and~$a_i(v_1,-)\in V_2^\vee$ are considered as linear functions
on~$\bar{V}_1$ and~$\bar{V}_2$, respectively.
\begin{proposition}
\label{prop:x33-plus}
The threefold~$X$ is ${\mathsf{k}}$-birational to the ${\mathsf{k}}$-form~${\tilde{X}}^+$ of the threefold~${\tilde{X}}^+_{{\mathsf{k}}'}$ defined by~\eqref{eq:xi-x33}
and to a ${\mathsf{k}}$-form~$X^+$ of its image in~$Y^+$
\begin{equation*}
X^+_{{\mathsf{k}}'} = \sigma_+({\tilde{X}}^+_{{\mathsf{k}}'}) = \{ \det(\xi) = 0 \} \subset \mathbb{P}(\bar{V}_1) \times \mathbb{P}(\bar{V}_2),
\end{equation*}
which is a geometrically irreducible and normal divisor of bidegree~$(2,2)$.
Moreover,
\begin{itemize}
\item
if~$\mathrm{F}_1(X,x_0) = \varnothing$ then ${\tilde{X}}^+ \cong {\tilde{X}} = \Bl_x(X)$ is smooth,
the morphism~$\sigma_+ \colon {\tilde{X}}^+ \to X^+$ is induced by the double projection from~$x_0$,
and it is a small resolution of singularities;
\item
if~$\mathrm{F}_1(X,x_0) \ne \varnothing$ then~$X^+$ contains a ${\mathsf{k}}$-form
of a quadric surface~$\mathbb{P}^1 \times \mathbb{P}^1 \subset \mathbb{P}(\bar{V}_1) \times \mathbb{P}(\bar{V}_2)$ rational over~${\mathsf{k}}$.
\end{itemize}
\end{proposition}
\begin{proof}
To prove birationality of~${\tilde{X}}$ and~${\tilde{X}}^+ = \psi_*({\tilde{X}})$ we
apply Proposition~\ref{prop:x-transform},
so we need to verify the dimension conditions~\eqref{eq:dimension-conditions}.
We have~$(Y_1)_{{\mathsf{k}}'} = [v_1] \times \mathbb{P}(V_2)$, hence
\begin{equation*}
X_{{\mathsf{k}}'} \cap (Y_1)_{{\mathsf{k}}'} = ([v_1] \times \mathbb{P}(V_2)) \cap \mathbb{P}(A^\perp)
\end{equation*}
is a fiber of the projection~$\pi_1 \colon X_{{\mathsf{k}}'} \to \mathbb{P}(V_1)$.
By Lemma~\ref{lemma:pi-i} it is a point or a line.
A similar argument for~$X_{{\mathsf{k}}'} \cap (Y_2)_{{\mathsf{k}}'}$ shows that the first part of~\eqref{eq:dimension-conditions} holds.
Moreover, this argument also shows that in the case~$\mathrm{F}_1(X,x_0) = \varnothing$, the blowup~${\tilde{X}}_{{\mathsf{k}}'}$ of~$X_{{\mathsf{k}}'}$
has empty intersection with the indeterminacy locus~$({\tilde{Y}}_1)_{{\mathsf{k}}'} \sqcup ({\tilde{Y}}_2)_{{\mathsf{k}}'}$ of the map~$\psi$.
On the other hand, the subbundle~$\bar{{\mathscr{E}}}_1 \subset {\mathscr{E}}$ is just the
summand~${\mathscr{O}}(-h_2)$ in~\eqref{eq:ce-x33},
hence the corresponding intersection~${\tilde{X}}^+_{{\mathsf{k}}'} \cap
\mathbb{P}_{Y^+_{{\mathsf{k}}'}}(\bar{{\mathscr{E}}}_1)$ is the zero locus of the morphism
\begin{equation*}
\xi_2 \colon {\mathscr{O}}(-h_2) \longmapsto A^\vee \otimes {\mathscr{O}}
\end{equation*}
given by the last column of~\eqref{eq:xi-x33}.
It is easy to see that this is empty, if~$\mathrm{F}_1(X,x_0) = \varnothing$, or isomorphic to a line otherwise.
A similar argument works for~${\tilde{X}}^+_{{\mathsf{k}}'} \cap \mathbb{P}_{Y^+_{{\mathsf{k}}'}}(\bar{{\mathscr{E}}}_2)$;
therefore, the second part of~\eqref{eq:dimension-conditions} also holds.
This proves that~${\tilde{X}}^+ = \psi_*({\tilde{X}})$ is a ${\mathsf{k}}$-form of~${\tilde{X}}^+_{{\mathsf{k}}'}$, which is ${\mathsf{k}}$-birational to~$X$,
and if~$\mathrm{F}_1(X,x_0) = \varnothing$, it is isomorphic to~${\tilde{X}}$,
and in particular in this case it is smooth.
Now we describe the image of~${\tilde{X}}^+$ in~$Y^+$.
By definition, ${\tilde{X}}^+_{{\mathsf{k}}'}$ parameterizes points in the projectivizations of kernel spaces of~$\xi$;
therefore, its image in~$Y^+_{{\mathsf{k}}'} = \mathbb{P}(\bar{V}_1) \times \mathbb{P}(\bar{V}_2)$
is the degeneracy locus~$X^+_{{\mathsf{k}}'}$ of~$\xi$
which is, of course, given by the equation~\mbox{$\det(\xi) = 0$}.
Since~$\det({\mathscr{E}}) \cong {\mathscr{O}}(-2h_1 - 2h_2)$ by~\eqref{eq:ce-x33}, this is a divisor of bidegree~$(2,2)$, which is
geometrically irreducible because~${\tilde{X}}^+_{{\mathsf{k}}'}$ is.
Moreover, fibers of the morphism~\mbox{$\sigma_+ \colon {\tilde{X}}^+_{{\mathsf{k}}'} \to X^+_{{\mathsf{k}}'}$} are linear spaces,
so since both the source and the target are 3-dimensional, the morphism is birational.
To prove that~$X^+_{\bar{\mathsf{k}}}$ is normal we consider the Koszul resolution
\begin{equation*}
0 \longrightarrow
{\mathscr{O}}_{\mathbb{P}_{Y^+_{\bar{\mathsf{k}}}}(-{\mathscr{E}})}(-3h) \longrightarrow
{\mathscr{O}}_{\mathbb{P}_{Y^+_{\bar{\mathsf{k}}}}(-{\mathscr{E}})}(-2h)^{\oplus 3} \longrightarrow
{\mathscr{O}}_{\mathbb{P}_{Y^+_{\bar{\mathsf{k}}}}(-{\mathscr{E}})}(-h)^{\oplus 3} \longrightarrow
{\mathscr{O}}_{\mathbb{P}_{Y^+_{\bar{\mathsf{k}}}}(-{\mathscr{E}})} \longrightarrow
{\mathscr{O}}_{{\tilde{X}}^+_{\bar{\mathsf{k}}}} \longrightarrow 0.
\end{equation*}
Pushing it forward to~$Y^+_{\bar{\mathsf{k}}}$, we obtain the following exact sequence
\begin{equation*}
0 \longrightarrow {\mathscr{O}}_{Y^+_{\bar{\mathsf{k}}}}(-2h_1-2h_2) \longrightarrow {\mathscr{O}}_{Y^+_{\bar{\mathsf{k}}}} \longrightarrow \sigma_{+*}{\mathscr{O}}_{{\tilde{X}}^+_{\bar{\mathsf{k}}}} \longrightarrow 0.
\end{equation*}
It follows that~$\sigma_{+*}{\mathscr{O}}_{{\tilde{X}}^+_{\bar{\mathsf{k}}}} \cong {\mathscr{O}}_{X^+_{\bar{\mathsf{k}}}}$, and since~${\tilde{X}}^+_{\bar{\mathsf{k}}}$ is normal, so is~$X^+_{\bar{\mathsf{k}}}$.
Now assume~$\mathrm{F}_1(X,x_0) = \varnothing$.
In this case the pullback along~$\sigma_+$ of the ample class~$h_1 + h_2$
on~$\mathbb{P}(\bar{V}_1) \times \mathbb{P}(\bar{V}_2)$
by~\eqref{eq:pic-relations-x33} equals~$H_1 + H_2 - 2E$, the anticanonical
class of~${\tilde{X}}^+ \cong {\tilde{X}}$,
hence the morphism~$\sigma_+$ is the double projection from the point~$x_0$.
Consequently, it is small by the argument of~\cite[Theorem~5.17]{KP19}.
Indeed, by~\cite[Lemma~5.4(iii)]{KP19} we have~$\dim|H_1 + H_2 - 3E| \ge g - 9 = 2$
(recall that~$g = {\mathrm{g}}(X) = 11$, see Table~\ref{table:fanos}),
hence by~\cite[Lemma~5.7(ii)]{KP19} any divisor~$D$ contracted by~$\sigma_+$
must be a fixed component of~$|H_1 + H_2 - 3E|$,
and at the same time by~\cite[(5.1.8)]{KP19} its class should be a multiple
of~$H_1 + H_2 - 5E$,
and these two conclusions are incompatible.
Finally, assume that~$\mathrm{F}_1(X,x_0) \ne \varnothing$.
As it was explained in Lemma~\ref{lemma:gamma-gamma-i} this means that (for appropriate ${\mathsf{k}}'$-basis in~$A$) we have
\begin{equation*}
a_1(v_1,-) = 0
\qquad\text{and}\qquad
a_2(-,v_2) = 0
\end{equation*}
as linear functions on~$\bar{V_2}$ and~$\bar{V}_1$, respectively; moreover,
$[a_1],[a_2] \in \mathbb{P}(A)$ as above are unique and swapped by the Galois action.
Consider the surface
\begin{equation*}
\{a_1(-,v_2) = 0,\ a_2(v_1,-) = 0\} \subset \mathbb{P}(\bar{V}_1) \times \mathbb{P}(\bar{V}_2)
= Y^+_{{\mathsf{k}}'}.
\end{equation*}
(isomorphic to~$\mathbb{P}^1_{{\mathsf{k}}'} \times \mathbb{P}^1_{{\mathsf{k}}'}$).
The equations, defining it are Galois-conjugate, hence it comes from a~${\mathsf{k}}$-surface in~$Y^+$.
This surface is the image of the exceptional divisor~$E$ of~$\sigma$, hence it is~${\mathsf{k}}$-rational.
It is clear from~\eqref{eq:xi-x33} that this surface is contained in the degeneracy locus~$X^+$ of~$\xi$.
\end{proof}
\subsection{A conic bundle structure}
\label{subsec:x33-conic-bundle}
In this section we work under the assumption~$\mathrm{F}_1(X,x_0) = \varnothing$
and show that in this case~$X$ admits a nice conic bundle structure.
We will need a general result about what we call \textsf{Springer resolutions}.
Let~$M$ be a variety, let~$\xi \colon {\mathscr{E}}_1 \to {\mathscr{E}}_2^\vee$ be a morphism of vector bundles on~$M$ of the same rank
and let~$\xi^\vee \colon {\mathscr{E}}_2 \to {\mathscr{E}}_1^\vee$ be its dual morphism.
Assume the degeneracy locus $Z \subset M$ of~$\xi$ is a geometrically integral
divisor.
Let~$Z_1 \subset \mathbb{P}_M({\mathscr{E}}_1)$ and~$Z_2 \subset \mathbb{P}_M({\mathscr{E}}_2)$ be the zero loci
of the morphisms
\begin{equation*}
{\mathscr{O}}(-h_{{\mathscr{E}}_1}) \hookrightarrow p_1^*{\mathscr{E}}_1 \xrightarrow{\quad p_1^*\xi \quad} p_1^*{\mathscr{E}}_2^\vee
\qquad\text{and}\qquad
{\mathscr{O}}(-h_{{\mathscr{E}}_2}) \hookrightarrow p_2^*{\mathscr{E}}_2 \xrightarrow{\quad p_2^*\xi^\vee \quad} p_2^*{\mathscr{E}}_1^\vee,
\end{equation*}
where $p_i \colon \mathbb{P}_M({\mathscr{E}}_i) \to M$ are the projections,
$h_{{\mathscr{E}}_i}$ are their relative hyperplane classes, and the first arrows are the tautological embeddings.
\begin{lemma}
\label{lemma:springer}
If one of the morphisms
\begin{equation*}
p_1\vert_{Z_1} \colon Z_1 \longrightarrow Z
\qquad\text{or}\qquad
p_2\vert_{Z_2} \colon Z_2 \longrightarrow Z
\end{equation*}
is birational then so is the other.
Moreover, if one of them is small then so is the other, and there is an equality
\begin{equation}
\label{eq:springer-pic}
(h_{{\mathscr{E}}_1} + {\mathrm{c}}_1(p_1^*{\mathscr{E}}_1)) + (h_{{\mathscr{E}}_2} + {\mathrm{c}}_1(p_2^*{\mathscr{E}}_2)) = 0
\end{equation}
in the group~$\Cl(Z_1) \cong \Cl(Z) \cong \Cl(Z_2)$,
where the isomorphisms of the class groups are induced by the small
birational morphisms~$p_i$.
\end{lemma}
\begin{proof}
Let~$Z^{\ge c} \subset Z$ be the locus of points where the corank of~$\xi$ is at
least~$c$ (so that~$Z = Z^{\ge 1}$).
Then both morphisms~$p_i\vert_{Z_i}$ are~$\mathbb{P}^{c-1}$-fibrations
over~$Z^{\ge c} \setminus Z^{\ge c + 1}$.
In particular if one of the morphisms is birational then $\dim Z^{\ge c} \le \dim Z - c$ for~$c \ge 2$
and then the other morphism is also birational.
Similarly, if one of the morphisms is small then $\dim Z^{\ge c} \le \dim Z - c - 1$ for~$c \ge 2$
and then the other morphism is also small.
Finally, assuming that the morphisms are small, we have
\begin{equation*}
\Cl(Z_1) = \Cl(Z_1 \setminus p_1^{-1}(Z^{\ge 2})) = \Cl(Z \setminus Z^{\ge 2}) =
\Cl(Z_2 \setminus p_2^{-1}(Z^{\ge 2})) = \Cl(Z_2),
\end{equation*}
and when restricted to~$Z \setminus Z^{\ge 2}$ the morphism~$\xi$ has constant corank~$1$,
the summands in~\eqref{eq:springer-pic} are equal to~${\mathrm{c}}_1(\Ima(\xi))$ and~${\mathrm{c}}_1(\Ima(\xi^\vee))$, respectively,
and~\eqref{eq:springer-pic} follows from the natural duality isomorphism~$\Ima(\xi^\vee) \cong \Ima(\xi)^{\vee}$.
\end{proof}
Now, coming back to the threefold~$X$ of type~\type{3,3} and assuming~$\mathrm{F}_1(X,x_0) = \varnothing$
we recall
that~$X^+ \subset Y^+$ is the degeneracy locus of~$\xi \colon {\mathscr{E}} \to A^\vee \otimes {\mathscr{O}}$
and note that~${\tilde{X}}^+ \subset \mathbb{P}_{Y^+}({\mathscr{E}})$ is one of its Springer resolutions.
Consider the other Springer resolution
\begin{equation*}
{\tilde{X}}^{+\!+} \subset \mathbb{P}_{Y^+}(A \otimes {\mathscr{O}}) \cong Y^+ \times \mathbb{P}(A),
\end{equation*}
which by definition is the zero locus of the morphism
\begin{equation*}
{\mathscr{O}}(-h_A) \hookrightarrow A \otimes {\mathscr{O}} \xrightarrow{\quad \xi^\vee \quad}
{\mathscr{E}}^\vee,
\end{equation*}
where $h_A$ is the hyperplane class of~$\mathbb{P}(A)$ and we suppress the pullbacks in the notation.
In view of~\eqref{eq:ce-x33} the scheme~$X_{{\mathsf{k}}'}^{+\!+}$ is just a complete intersection of divisors
of types $h_1 + h_2 + h_A$, $h_1 + h_A$, and~$h_2 + h_A$ in~$\mathbb{P}(\bar{V}_1) \times \mathbb{P}(\bar{V}_2) \times \mathbb{P}(A)$
that correspond to the columns of~\eqref{eq:xi-x33}.
\begin{proposition}
\label{prop:conic-bundle-x33}
If~$\mathrm{F}_1(X,x_0) = \varnothing$ there is a commutative diagram defined
over~${\mathsf{k}}$
\begin{equation}
\label{eq:link-x33}
\vcenter{\xymatrix@C=4em{
{\tilde{X}} \ar[d]_\sigma
\ar@{=}[rr]^-\psi &&
{\tilde{X}}^+ \ar[dr]^{\sigma_+}
\ar@{-->}[rr]^-{\psi_+} &&
{\tilde{X}}^{+\!+} \ar[dl]_{\sigma_{+\!+}} \ar[d]^f
\\
X &
&&
X^+ &
\mathbb{P}(A),
}}
\end{equation}
where~${\tilde{X}}^+$ and~${\tilde{X}}^{+\!+}$ are the Springer resolutions of the degeneracy
locus~$X^+ \subset Y^+$ of~$\xi$,
the morphisms~$\sigma_+$ and~$\sigma_{+\!+}$ are small birational contractions,
and $\psi_+ = \sigma_{+\!+}^{-1} \circ \sigma_+$ is a flop.
Moreover, ${\tilde{X}}^{+\!+}$ is smooth, $f$ is a flat conic bundle whose discriminant curve is the curve~$\Gamma$ defined in~\textup\S\xref{subsec:discriminant},
the map $f \circ \psi_+ \circ \psi \colon {\tilde{X}} \dashrightarrow \mathbb{P}(A)$ is given by the linear system $|H_1 + H_2 - 3E|$,
and the exceptional divisor~$E \subset {\tilde{X}}$ of~$\sigma$ dominates~$\mathbb{P}(A)$.
\end{proposition}
\begin{proof}
The morphism~$\sigma_+$ is small by Proposition~\ref{prop:x33-plus},
hence~$\sigma_{+\!+}$ is small by Lemma~\ref{lemma:springer};
moreover, it follows that both morphisms are crepant.
Now, the relation~\eqref{eq:springer-pic} implies that the $\sigma_+$-antiample class~$-h$
is $\sigma_{+\!+}$-ample, hence~$\psi_+ := \sigma_{+\!+}^{-1} \circ \sigma_+$ is a flop.
Since~${\tilde{X}}^+ \cong {\tilde{X}}$ is smooth and~$\psi_+$ is a flop, ${\tilde{X}}^{+\!+}$ is smooth as well, see~\cite[Theorem~2.4]{Kollar1989}.
Next, we show that~$f$ is a conic bundle and identify its discriminant.
For this note that by definition the fiber of~$f$ over a point~$[a] \in \mathbb{P}(A)$ is
given in~$\mathbb{P}(\bar{V}_1) \times \mathbb{P}(\bar{V}_2)$ by the equations
\begin{equation*}
a(-,v_2) = a(v_1,-) = \bar{a}(-,-) = 0.
\end{equation*}
The first is a linear function on~$\bar{V}_1$, the second is a linear function
on~$\bar{V}_2$,
and both are non-zero because~$\mathrm{F}_1(X,x_0) = \varnothing$, so their common zero
locus is~$\mathbb{P}^1_{{\mathsf{k}}'} \times \mathbb{P}^1_{{\mathsf{k}}'} \subset \mathbb{P}(\bar{V}_1) \times \mathbb{P}(\bar{V}_2)$.
The last equation~$\bar{a}(-,-) = 0$ cuts a divisor of bidegree~$(1,1)$ on
this~$\mathbb{P}^1_{{\mathsf{k}}'} \times \mathbb{P}^1_{{\mathsf{k}}'}$, i.e., a conic;
and if~$\bar{a}(-,-)$ vanishes identically, then the corresponding bilinear
form~$a$ vanishes on~$\mathbb{P}^2_{{\mathsf{k}}'} \times \mathbb{P}^2_{{\mathsf{k}}'} \subset \mathbb{P}(V_1) \times \mathbb{P}(V_2)$,
hence has corank~$2$, which is impossible by the argument of
Lemma~\ref{lemma:gamma-x33} as~$X$ is smooth.
This shows that~$f$ is a flat conic bundle.
Finally, note that if~$\bar{v}'_i$, $\bar{v}''_i$, $\bar{v}'''_i$ are bases of vector spaces~$\bar{V}_i$
such that~$a(v_1,\bar{v}'_2) = a(v_1,\bar{v}''_2) = a(\bar{v}'_1,v_2) = a(\bar{v}''_1,v_2) = 0$ then the matrix of~$a$ has the form
\begin{equation*}
\begin{pmatrix}
0 & 0 & 0 & a(v_1,\bar{v}'''_2) \\
0 & \bar{a}(\bar{v}'_1,\bar{v}'_2) & \bar{a}(\bar{v}'_1,\bar{v}''_2) & * \\
0 & \bar{a}(\bar{v}''_1,\bar{v}'_2) & \bar{a}(\bar{v}''_1,\bar{v}''_2) & * \\
a(\bar{v}'''_1,v_2) & * & * & *
\end{pmatrix}
\end{equation*}
with non-zero entries~$a(v_1,\bar{v}'''_2)$ and~$a(\bar{v}'''_1,v_2)$, and with the 2-by-2 matrix in the middle
giving the equation of the conic~$f^{-1}(a)$ in~$\mathbb{P}^1_{\bar{\mathsf{k}}} \times \mathbb{P}^1_{\bar{\mathsf{k}}}$.
Therefore, the conic is singular if and only if~$\det(a) = 0$, i.e., if and only if~$[a] \in \Gamma$.
Thus, the discriminant curve of~$f$ equals~$\Gamma$.
Finally, using~\eqref{eq:springer-pic}, \eqref{eq:ce-x33}, and~\eqref{eq:pic-relations-x33}
we deduce that the map~$f \circ \psi_+ \circ \psi$ is given by the linear system
\begin{equation*}
h_A = 2(h_1 + h_2) - h = H_1 + H_2 - 3E.
\end{equation*}
Since the canonical class of~${\tilde{X}}$ is equal to~$H_1 + H_2 - 2E$ and~$\psi_+$ is a flop,
it follows that~$\psi_*(E)$ is a relative anticanonical divisor for~$f$, hence it dominates~$\mathbb{P}(A)$.
\end{proof}
If $f \colon {\mathscr{X}} \to S$ is a flat conic bundle over a surface~$S$ (not necessarily proper) with a smooth
discriminant curve~$\Delta \subset S$,
consider the preimage~${\mathscr{X}}_\Delta := f^{-1}(\Delta)$, its
normalization~${\mathscr{X}}_\Delta^\nu \to {\mathscr{X}}_\Delta$,
and the Stein factorization
\begin{equation*}
{\mathscr{X}}_\Delta^\nu \longrightarrow \tilde\Delta \longrightarrow \Delta.
\end{equation*}
Then the first arrow is a~$\mathbb{P}^1$-bundle and the second arrow is an \'etale
double covering (because~$\Delta$ was assumed to be smooth).
We will say that the \'etale covering~$\tilde\Delta \to \Delta$ is \textsf{the
discriminant double covering} of the conic bundle~$f$.
\begin{lemma}
\label{lemma:tilde-gamma}
The discriminant double covering of the conic bundle~$f \colon {\tilde{X}}^{+\!+} \to
\mathbb{P}(A)$ has the form
\begin{equation*}
\tilde\Gamma \cong \Gamma \times_{\mathsf{k}} {\mathsf{k}}' \longrightarrow \Gamma.
\end{equation*}
\end{lemma}
\begin{proof}
If the conic~$f^{-1}([a])$ is singular, it is a union of two components that
correspond to the two factors~$\mathbb{P}(\bar{V}_i)$ in~$Y^+_{{\mathsf{k}}'}$
and each of them is contracted by appropriate projection~$Y^+_{{\mathsf{k}}'} \to
\mathbb{P}(\bar{V}_i)$.
Therefore, the discriminant double covering~$\tilde\Gamma \to \Gamma$ becomes trivial after the extension of scalars to~${\mathsf{k}}'$,
while it is non-trivial over~${\mathsf{k}}$, hence the claim.
\end{proof}
\subsection{Unirationality}
\label{subsec:unirat-x33}
In this section we prove unirationality of~$X$ assuming that~$X({\mathsf{k}}) \ne
\varnothing$.
We start with the following observation which might be useful in other
situations.
\begin{lemma}
\label{lemma:unirationality-22}
Let~$Y$ be a ${\mathsf{k}}$-form of~$\mathbb{P}^2 \times \mathbb{P}^2$ and let~$W \subset Y$
be a ${\mathsf{k}}$-rational ${\mathsf{k}}$-form of a quadric surface~$\mathbb{P}^1 \times \mathbb{P}^1 \subset \mathbb{P}^2 \times \mathbb{P}^2$.
Any geometrically irreducible normal divisor $Z \subset Y$ of bidegree~$(2,2)$
such that $W \subset Z$ is~${\mathsf{k}}$-unirational.
\end{lemma}
\begin{proof}
Consider the toric birational isomorphism
\begin{equation*}
\xymatrix{
& \Bl_{\mathbb{P}^1 \times \mathbb{P}^1}(\mathbb{P}^2 \times \mathbb{P}^2) \ar[dl] \ar@{=}[r] &
\Bl_{\mathbb{P}^1 \sqcup \mathbb{P}^1}(\mathbb{P}^4) \ar[dr]
\\
\mathbb{P}^2 \times \mathbb{P}^2 \ar@{-->}[rrr]^\chi &&&
\mathbb{P}^4.
}
\end{equation*}
analogous to the birational transformation of Theorem~\ref{proposition:toric-link} (see also~\cite[Proposition~3]{Zak07}).
Here the map~$\chi$ is the projection from the linear span of~$W$ under the Segre embedding~$\mathbb{P}^2 \times \mathbb{P}^2 \subset \mathbb{P}^8$,
and the right arrow is the blowup of two skew lines in~$\mathbb{P}^4$.
Denoting by~$\hat{e}$ the class of the exceptional divisor of the left blowup,
by~$\hat{h}$ the hyperplane class of~$\mathbb{P}^4$, and by~$e_1$ and~$e_2$ the classes
of the exceptional divisors of the right blowup,
it is easy to check that we have the relations
\begin{equation*}
\begin{cases}
\hbox to .85em{$\hat{h}$} = h_1 + h_2 - \hat{e},\\
\hbox to .85em{$e_1$} = h_1 - \hat{e},\\
\hbox to .85em{$e_2$} = h_2 - \hat{e},
\end{cases}
\qquad \text{and} \qquad
\begin{cases}
\hbox to .95em{$h_1$} = \hat{h} - e_2,\\
\hbox to .95em{$h_2$} = \hat{h} - e_1,\\
\hbox to .95em{$\hat{e}$} = \hat{h} - e_1 - e_2.
\end{cases}
\end{equation*}
In particular,
the map~$\chi$ is given by the linear system~$|h_1 + h_2 - \hat{e}|$, hence it is defined over~${\mathsf{k}}$.
Furthermore, we have $2h_1 + 2h_2 - \hat{e} = 3\hat{h} - e_1 - e_2$, and since~$Z$ is normal,
the strict transform of~$Z$ under the map~$\chi$ is a cubic
threefold~$\hat{Z} \subset \mathbb{P}^4$
passing through the pair of skew lines~$\mathbb{P}^1 \sqcup \mathbb{P}^1$.
Moreover, we have $\hat{e} = \hat{h} - e_1 - e_2$, hence the image of~$\hat{E}$
is the hyperplane section of this cubic threefold (by the linear span of these lines).
On the other hand, $\hat{E}$ is birational to the ${\mathsf{k}}$-rational surface~$W$, hence it is~${\mathsf{k}}$-rational.
In particular, $\hat{Z}({\mathsf{k}}) \ne \varnothing$.
Now, if~$\hat{Z}$ is not a cone, it is ${\mathsf{k}}$-unirational by Koll\'ar's theorem~\cite[Theorem~1.2]{Kollar:cubic}.
Otherwise, if~$\hat{Z}$ is a cone and its vertex lies away from the hyperplane spanned by the two skew lines,
then the base of the cone is the ${\mathsf{k}}$-rational surface~$\hat{E}$, hence
the cone~$\hat{Z}$ is also~${\mathsf{k}}$-rational.
Finally, if the vertex of the cone lies on~$\hat{E}$, then~$\hat{E}$ itself must be a cubic cone in~$\mathbb{P}^3$,
and since it also contains two skew lines, it is not geometrically irreducible, which is absurd.
\end{proof}
Now we can deduce unirationality of~$X$.
\begin{proposition}
\label{prop:x33}
If $X$ is a Fano threefold of type~\type{3,3} with~$X({\mathsf{k}}) \ne \varnothing$
then~$X$ is ${\mathsf{k}}$-unirational.
\end{proposition}
\begin{proof}
Let $x_0$ be a ${\mathsf{k}}$-point on~$X$.
First, assume~$\mathrm{F}_1(X,x_0) = \varnothing$.
By Proposition~\ref{prop:conic-bundle-x33} we have a ${\mathsf{k}}$-birational
map~\mbox{$X\dashrightarrow {\tilde{X}}^{+\!+}$},
where~$f \colon {\tilde{X}}^{+\!+} \to \mathbb{P}(A)$ is a conic bundle.
Moreover, the ${\mathsf{k}}$-rational surface~\mbox{$E \cong \mathbb{P}(T_{x_0}X) \subset {\tilde{X}}$} dominates the base of this conic bundle.
Therefore, ${\tilde{X}}$ is ${\mathsf{k}}$-unirational
(see, e.g. \cite[Lemma~4.14(i)]{KP19}); and hence so is~$X$.
Now assume that $\mathrm{F}_1(X,x_0) \ne \varnothing$.
By Proposition~\ref{prop:x33-plus} we have a birational
map~$X\dashrightarrow X^+$
where~$X^+$ is a geometrically irreducible normal divisor of bidegree~$(2,2)$ in a ${\mathsf{k}}$-form of~$\mathbb{P}^2 \times \mathbb{P}^2$
that contains a ${\mathsf{k}}$-form of a ${\mathsf{k}}$-rational quadric surface~$\mathbb{P}^1 \times \mathbb{P}^1$.
Therefore, $X^+$ is ${\mathsf{k}}$-unirational by Lemma~\ref{lemma:unirationality-22},
hence so is~$X$.
\end{proof}
\subsection{Non-rationality}
In this section we prove non-rationality of Fano threefolds of type~\type{3,3}.
We will use the following reformulation of a result of Benoist--Wittenberg
from~\cite{BW}.
\begin{theorem}
\label{prop:non-rationality-conic-bundle}
Let ${\mathscr{X}} \to S$ be a flat conic bundle over a smooth ${\mathsf{k}}$-rational surface~$S$
with smooth connected discriminant curve~$\Delta \subset S$.
Assume the discriminant double covering takes the form
\begin{equation*}
\tilde\Delta \cong \Delta \times_{\mathsf{k}} {\mathsf{k}}' \longrightarrow \Delta
\end{equation*}
where ${\mathsf{k}}'/{\mathsf{k}}$ is a quadratic extension of the base field.
If the conic bundle~${\mathscr{X}}_{{\mathsf{k}}'} \to S_{{\mathsf{k}}'}$ admits a rational section
and the curve~$\Delta$ is not hyperelliptic then~${\mathscr{X}}$ is not~${\mathsf{k}}$-rational.
\end{theorem}
Note that we require neither the surface~$S$ nor the curve~$\Delta$ to be proper;
moreover, during the proof we will further shrink~$S$ but keep (the generic point of) the curve~$\Delta$ in~$S$.
\begin{proof}
Since $S$ is normal and $f$ is proper any rational section of~$f$ extends to codimension~$1$ points,
hence defines a regular section over the complement of a finite subscheme of~$S$.
Moreover, over the complement of this finite subscheme
the section does not pass through singular points of fibers of~$f$,
hence it defines a section of the morphism ${\mathscr{X}}_\Delta^\nu \to \Delta$,
where recall that~${\mathscr{X}}_\Delta^\nu$ is the normalization of~${\mathscr{X}}_\Delta =
f^{-1}(\Delta)$.
Therefore it also gives a section of the discriminant double covering~$\tilde\Delta \to \Delta$.
If the original section is defined over~${\mathsf{k}}$, we obtain a contradiction with the isomorphism~$\tilde\Delta \cong \Delta \times_{\mathsf{k}} {\mathsf{k}}'$;
this means that the morphism~$f$ has no rational sections defined over~${\mathsf{k}}$.
Now consider a rational section of $f \colon {\mathscr{X}}_{{\mathsf{k}}'} \to S_{{\mathsf{k}}'}$.
Removing if necessary a finite subscheme from~$S$ we may assume that this section is regular.
Its intersection with the conjugate section (with respect to the~${\mathrm{G}}({\mathsf{k}}'/{\mathsf{k}})$-action)
projects to a curve in~$S$ which is disjoint from~$\Delta$
(because a regular section does not pass through singular points of fibers).
So, shrinking~$S$ further we may assume that the section and its conjugate do not intersect.
Then the union
\begin{equation*}
Z \subset {\mathscr{X}}
\end{equation*}
of the section and its conjugate is a $2$-section of~$f$ defined over~${\mathsf{k}}$;
moreover, $Z \cong S \times_{\mathsf{k}} {\mathsf{k}}'$ and in particular~$Z$ is \'etale over~$S$.
Consider the bundles ${\mathscr{V}} := (f_*\omega_{\mathscr{X}}^{-1})^\vee$ of rank~$3$
and~${\mathscr{V}}_Z := (f_*\omega_{\mathscr{X}}^{-1}\vert_Z)^\vee$ of rank~$2$ on~$S$.
The restriction morphism~$\omega_{\mathscr{X}}^{-1} \to \omega_{\mathscr{X}}^{-1}\vert_Z$ induces an embedding of vector bundles~${\mathscr{V}}_Z \hookrightarrow {\mathscr{V}}$
and a Cartesian square
\begin{equation*}
\xymatrix{
Z \ar[r] \ar[d] &
\mathbb{P}_S({\mathscr{V}}_Z) \ar[d]
\\
{\mathscr{X}} \ar[r] &
\mathbb{P}_S({\mathscr{V}}),
}
\end{equation*}
where all arrows are the natural embeddings.
Shrinking the surface~$S$ again but keeping an open part of the curve~$\Delta$
in it
we may assume that the bundles~${\mathscr{V}}$ and~${\mathscr{V}}_Z$ are trivial and that
the subvarieties~\mbox{${\mathscr{X}} \subset \mathbb{P}_S({\mathscr{V}})$} and~$Z \subset \mathbb{P}_{S}({\mathscr{V}}_Z)$
are given by a quadratic form~$q \in \Sym^2{\mathscr{V}}^\vee$ and its restriction $q_Z
\in \Sym^2{\mathscr{V}}_Z^\vee$ to~${\mathscr{V}}_Z$, respectively.
Since $Z$ is \'etale over~$S$, the form~$q_Z$ is everywhere non-degenerate and
can be written as follows
\begin{equation*}
q_Z = x^2 - \alpha y^2,
\end{equation*}
where $(x,y)$ are homogeneous coordinates in the fiber of~$\mathbb{P}_S({\mathscr{V}}_Z)
\cong S \times \mathbb{P}^1$
and $\alpha \in {\mathsf{k}}^\times$ is such that~${\mathsf{k}}' = {\mathsf{k}}(\sqrt{\alpha})$.
Now, considering the orthogonal complement to~${\mathscr{V}}_Z$ in~${\mathscr{V}}$ we see that~$q$
takes the form
\begin{equation*}
q = x^2 - \alpha y^2 - Fz^2,
\end{equation*}
where $F$ is an equation of~$\Delta$ on~$S$.
Thus, the conic bundle ${\mathscr{X}} \to S$ is birational to conic bundles considered in~\cite[\S3.3.1]{BW},
hence~${\mathscr{X}}$ is not~${\mathsf{k}}$-rational by~\cite[Proposition~3.4]{BW}.
\end{proof}
Now we apply this to prove non-rationality of threefolds of type~\type{3,3}.
\begin{corollary}
\label{cor:x33}
If $X$ is a Fano threefold of type~\type{3,3} then~$X$ is not~${\mathsf{k}}$-rational.
\end{corollary}
\begin{proof}
If $X$ is not~${\mathsf{k}}$-unirational, there is nothing to prove.
So, assume~$X$ is ${\mathsf{k}}$-unirational.
Then there exists a ${\mathsf{k}}$-point $x_0 \in X$ such that~$\mathrm{F}_1(X,x_0) = \varnothing$.
Consider the conic bundle ${\tilde{X}}^{+\!+} \to \mathbb{P}^2$ constructed in Proposition~\ref{prop:conic-bundle-x33}.
By Lemma~\ref{lemma:gamma-x33} the discriminant curve of~$f$ is the smooth non-hyperelliptic curve~$\Gamma$ defined in~\S\ref{subsec:discriminant}
and by Lemma~\ref{lemma:tilde-gamma} the discriminant double covering has the form~$\tilde\Gamma \cong \Gamma\times_{\mathsf{k}} {\mathsf{k}}'$.
Finally, the description of Proposition~\ref{prop:conic-bundle-x33} shows that~$f$ admits a rational section after base change to~${\mathsf{k}}'$.
Therefore, Theorem~\ref{prop:non-rationality-conic-bundle} applies and proves that~${\tilde{X}}^{+\!+}$ is not~${\mathsf{k}}$-rational,
hence~$X$ is not ${\mathsf{k}}$-rational as well.
\end{proof}
\section{Fano threefolds of type \type{1,1,1,1}}
\label{sec:x1111}
In this section we apply the degeneration technique of~\cite{NS} to prove Theorem~\ref{thm:x1111-non-st-rat}.
\subsection{Toric degeneration}
To start with we consider ${\mathrm{Y}}_0 = (\mathbb{P}^1)^4$, denote by $(u_i:v_i)$ the homogeneous coordinates on the $i$-th factor,
and consider the point
\begin{equation*}
{\mathrm{y}}_0 := (1,1,1,1) \in {\mathrm{Y}}_0.
\end{equation*}
Clearly, ${\mathrm{Y}}_0$ is a toric variety with respect to the action of the split torus $\mathbb{G}_{\mathrm{m}}^4$ that rescales the~$v_i$.
We also consider the action of~$\mathfrak{S}_4$ on~${\mathrm{Y}}_0$ that permutes the factors.
It normalizes the torus action, and together they generate an action of
the group~$\mathbb{G}_{\mathrm{m}}^4 \rtimes \mathfrak{S}_4$.
Finally, consider the subtorus
\begin{equation}
\label{eq:t0}
{\mathrm{T}}_0 := \big\{ (t_1,t_2,t_3,t_4) \in \mathbb{G}_{\mathrm{m}}^4 \mid t_1t_2t_3t_4 = 1 \big\}
\end{equation}
and the collection of three 1-parametric subgroups
\begin{equation}
\label{eq:rt-4}
{\mathrm{T}}_0^{i_1,i_2;i_3,i_4} := \big\{ (t_1,t_2,t_2,t_4) \mid t_{i_1} = t_{i_2} = t_{i_3}^{-1} = t_{i_4}^{-1} \big\} \subset {\mathrm{T}}_0,
\end{equation}
where~$(i_1,i_2)(i_3,i_4) \in \mathrm{V}_4 \setminus \{1\} \subset \mathfrak{S}_4$ is a nontrivial element of the Klein subgroup.
\begin{lemma}
\label{lemma:x1111-toric}
The subvariety
\begin{equation*}
{\mathrm{X}}_0^{\mathrm{toric}} := \{ u_1u_2u_3u_4 - v_1v_2v_3v_4 = 0 \} \subset {\mathrm{Y}}_0
\end{equation*}
is the unique ${\mathrm{T}}_0$-invariant divisor of multidegree~$(1,1,1,1)$ in~${\mathrm{Y}}_0$ which contains the point~${\mathrm{y}}_0$.
It is a toric variety with~$6$ ordinary double points
\begin{equation*}
{\mathrm{x}}_{p,q} = \big\{ ((u_1:v_1),(u_2:v_2),(u_3:v_3),(u_4:v_4)) \mid
\text{$u_i = 0$ if~$i \in \{p,q\}$, and $v_i = 0$ if~$i \not\in \{p,q\}$} \big\}.
\end{equation*}
For each permutation~$(i_1,i_2)(i_3,i_4) \in \mathrm{V}_4 \setminus \{1\} \subset \mathfrak{S}_4$ the curve
\begin{equation}
\label{eq:c-iiii}
\mathrm{C}_{i_1,i_2;i_3,i_4} := \overline{{\mathrm{T}}_0^{i_1,i_2;i_3,i_4}\cdot {\mathrm{y}}_0} \subset {\mathrm{X}}_0^{\mathrm{toric}}
\end{equation}
is a smooth rational curve and~$\mathrm{C}_{i_1,i_2;i_3,i_4} \cap \Sing({\mathrm{X}}_0^{\mathrm{toric}}) = \{ {\mathrm{x}}_{i_1,i_2}, {\mathrm{x}}_{i_3,i_4} \}$.
\end{lemma}
\begin{proof}
The monomial basis in the space of homogeneous polynomials of multidegree~$(1,1,1,1)$
is a weight basis for the action of~${\mathrm{T}}_0$,
and all weights are different except for the weight~$0$ which has multiplicity~2
and the corresponding weight space is spanned by the monomials~$u_1u_2u_3u_4$ and~$v_1v_2v_3v_4$.
Therefore, every ${\mathrm{T}}_0$-invariant divisor is either given by a monomial equation
(but then it does not contain the point~${\mathrm{y}}_0$),
or it is given by a linear combination of~$u_1u_2u_3u_4$ and~$v_1v_2v_3v_4$,
and if it contains the point~${\mathrm{y}}_0$, it is equal to~${\mathrm{X}}_0^{\mathrm{toric}}$.
The latter is obviously a toric variety with respect to the natural action of~${\mathrm{T}}_0$.
In the affine chart $v_1 \ne 0$, $v_2 \ne 0$, $u_3 \ne 0$, $u_4 \ne 0$ we can set $v_1 = v_2 = u_3 = u_4 = 1$
and use~$u_1$, $u_2$, $v_3$, $v_4$ as coordinates.
Then the equation of~${\mathrm{X}}_0^{\mathrm{toric}}$ takes the form
\begin{equation*}
u_1u_2 - v_3v_4 = 0,
\end{equation*}
which means that the origin of the chart, i.e., the point ${\mathrm{x}}_{3,4} = (0,0,\infty,\infty) \in {\mathrm{Y}}_0$
is an ordinary double point of~${\mathrm{X}}_0^{\mathrm{toric}}$.
Considering similarly the other charts, we see that the singular locus of~${\mathrm{X}}_0^{\mathrm{toric}}$
is the $\mathfrak{S}_4$-orbit of the point~${\mathrm{x}}_{3,4}$;
in particular each singular point of~${\mathrm{X}}_0^{\mathrm{toric}}$ is an ordinary double point.
Moreover, we see that the hypersurface~${\mathrm{X}}_0^{\mathrm{toric}}$ is normal.
The orbits of the point~${\mathrm{y}}_0$ under the 1-parametric subgroups~\eqref{eq:rt-4} are
\begin{equation*}
\{ (t,t,t^{-1},t^{-1}) \mid t \in \mathbb{G}_{\mathrm{m}} \},
\qquad
\{ (t,t^{-1},t,t^{-1}) \mid t \in \mathbb{G}_{\mathrm{m}} \},
\qquad
\{ (t,t^{-1},t^{-1},t) \mid t \in \mathbb{G}_{\mathrm{m}} \}.
\end{equation*}
It is easy to see that the closure of the first orbit is smooth and contains the point~${\mathrm{x}}_{1,2}$ and~${\mathrm{x}}_{3,4}$,
and similarly for the other two orbits.
\end{proof}
For a field extension~${\mathsf{k}}'/{\mathsf{k}}$ we denote by~$\Res_{{\mathsf{k}}'/{\mathsf{k}}} \colon \Sch_{{\mathsf{k}}'} \to \Sch_{\mathsf{k}}$
the Weil restriction of scalars functor from the category of~${\mathsf{k}}'$-schemes to the category of ${\mathsf{k}}$-schemes,
the right adjoint to the extension of scalars~\mbox{$- \otimes_{\mathsf{k}} {\mathsf{k}}' \colon \Sch_{\mathsf{k}} \to \Sch_{{\mathsf{k}}'}$}.
Consider the projective line~$\mathbb{P}^1_{{\mathsf{k}}'}$,
the torus~$\mathbb{G}_{\mathrm{m}}$ acting faithfully on~$\mathbb{P}^1_{{\mathsf{k}}'}$ and denote by~$0,\infty \in \mathbb{P}^1_{{\mathsf{k}}'}$ its fixed points.
\begin{proposition}
\label{prop:forms-x44}
For a field extension~${\mathsf{k}}'/{\mathsf{k}}$ of degree~$4$ consider the ${\mathsf{k}}$-forms
\begin{equation}
\label{eq:y-t-res}
Y := \Res_{{\mathsf{k}}'/{\mathsf{k}}}(\mathbb{P}^1_{{\mathsf{k}}'}),
\qquad
T := \Ker\left(\Res_{{\mathsf{k}}'/{\mathsf{k}}}\mathbb{G}_{\mathrm{m}} \longrightarrow \mathbb{G}_{\mathrm{m}}\right)
\end{equation}
of~${\mathrm{Y}}_0 = (\mathbb{P}^1)^4$ and of the torus~${\mathrm{T}}_0$, and the natural faithful $T$-action on~$Y$.
Let
\begin{equation*}
y \in Y
\end{equation*}
be the ${\mathsf{k}}$-point that corresponds to the point~$1 \in \mathbb{P}^1_{{\mathsf{k}}'}$.
Then
\begin{enumerate}
\item
\label{item:x-varpi}
The half-anticanonical linear system of~$Y$ is defined over~${\mathsf{k}}$
and it contains a unique $T$-invariant divisor~$X^{\mathrm{toric}} \subset Y$ passing through~$y$.
\item
\label{item:x-sing}
The divisor~$X^{\mathrm{toric}}$ is integral and has ordinary double points in the sense of~\cite[Definition~4.2.1]{NS}
with the singular locus of length~$6$.
\end{enumerate}
\end{proposition}
\begin{proof}
The fact that~$Y$ and~$T$ are ${\mathsf{k}}$-forms of~${\mathrm{Y}}_0$ and~${\mathrm{T}}_0$ is obvious from the definition of Weil restriction of scalars,
and the ${\mathsf{k}}$-point~$y$ is obtained from the extension--restriction adjunction.
Note that upon extension of scalars to~${\bar{\mathsf{k}}}$,
the triple~$(Y,T,y)$ becomes isomorphic to the triple~$({\mathrm{Y}}_0,{\mathrm{T}}_0,{\mathrm{y}}_0)$.
\ref{item:x-varpi}
Let~$H$ denote the Segre class of~$Y$ (the half of the anticanonical class);
it is obviously Galois-invariant, and since~$Y$ has a ${\mathsf{k}}$-point, $H$ is defined over~${\mathsf{k}}$.
Therefore, the linear system
\begin{equation}
\label{eq:fp}
\mathfrak{P} = |H - y| \cong \mathbb{P}^{14}
\end{equation}
of divisors in~$|H|$ containing~$y$ is defined over~${\mathsf{k}}$.
We define~$X^{\mathrm{toric}} \subset Y$ as the closure of the~$T$-orbit of the point~$y$;
the uniqueness of~$X^{\mathrm{toric}}$ follows from Lemma~\ref{lemma:x1111-toric}.
\ref{item:x-sing}
The extension of scalars of~$X^{\mathrm{toric}} \subset Y$ to~${\bar{\mathsf{k}}}$ coincides with~\mbox{${\mathrm{X}}_0^{\mathrm{toric}} \subset {\mathrm{Y}}_0$},
hence its singular locus~$Z := \Sing(X^{\mathrm{toric}})$ has length~$6$
and if~$E$ is the exceptional divisor of the blowup
\begin{equation*}
{\tilde{X}} := \Bl_{Z}(X^{\mathrm{toric}})
\end{equation*}
then~$E \to Z$ is a smooth quadric bundle.
According to~\cite[Definition~4.2.1]{NS} it remains to check that this bundle has a section.
For this note that the union of the 1-parametric subgroups~${\mathrm{T}}_0^{i_1,i_2;i_3,i_4} \subset {\mathrm{T}}_0$ defined in~\eqref{eq:rt-4}
is Galois-invariant, hence it comes from a ${\mathsf{k}}$-subset in the torus~$T$,
and hence the closure of the image of the point~$y$ under the action of this subset
is a curve~$C \subset X$ defined over~${\mathsf{k}}$.
Furthermore, the extension of scalars of~$C$ to~${\bar{\mathsf{k}}}$ is the union of the curves~\eqref{eq:c-iiii}.
In particular, the curve~$C$ contains the singular locus~$Z$,
and the intersection of its strict transform to the blowup~${\tilde{X}}$ with the exceptional divisor~$E$
provides a section for~\mbox{$E \to Z$}.
\end{proof}
Now let~$X$ be a smooth Fano threefold of type~\type{1,1,1,1}.
Recall the definition~\eqref{eq:gx} of the Galois group~${\mathrm{G}}_X \subset \mathfrak{S}_4$ of~$X$.
In the next lemma we use notation introduced in Proposition~\ref{prop:forms-x44}.
\begin{lemma}
\label{lem:x1111-models}
If~${\mathsf{k}}'/{\mathsf{k}}$ is the field extension of degree~$4$ associated with an epimorphisma
${\mathrm{G}}({\bar{\mathsf{k}}}/{\mathsf{k}}) \twoheadrightarrow{\mathrm{G}}$ onto a transitive subgroup ${\mathrm{G}}\subset \mathfrak{S}_4$,
then a general divisor~$X \subset Y$ from the linear system~\eqref{eq:fp}
is a smooth Fano threefold of type~\type{1,1,1,1} with~${\mathrm{G}}_X = {\mathrm{G}}$, $\uprho(X) = 1$, and~$X({\mathsf{k}}) \ne \varnothing$.
\end{lemma}
\begin{proof}
The smoothness of a general divisor~$X$ in the linear system~$\mathfrak{P}$ follows from the Bertini theorem,
the property~$X({\mathsf{k}}) \ne \varnothing$ is obvious because~$X$ contains the ${\mathsf{k}}$-point~$y$,
the equality~${\mathrm{G}}_X = {\mathrm{G}}$ follows from the construction, and~$\uprho(X) = 1$ follows from transitivity of~${\mathrm{G}} \subset \mathfrak{S}_4$.
\end{proof}
\begin{remark}
One can also prove the converse statement: any Fano threefold~$X$ of type~\type{1,1,1,1}
with~$X({\mathsf{k}}) \ne \varnothing$ and~${\mathrm{G}}_X = {\mathrm{G}}$ is isomorphic to a divisor in the linear system~$\mathfrak{P}$,
see~\cite[Proposition~7.16]{K22}.
\end{remark}
Now we are ready to prove Theorem~\ref{thm:x1111-non-st-rat}.
\begin{proof}[Proof of Theorem~\xref{thm:x1111-non-st-rat}]
We consider the field extension~${\mathsf{k}}'/{\mathsf{k}}$ as in Lemma~\ref{lem:x1111-models}
and use the construction and notation of Proposition~\ref{prop:forms-x44};
in particular the linear system~$\mathfrak{P} \cong \mathbb{P}^{14}_{\mathsf{k}}$ of half-anticanonical divisors in~$Y$.
Let~$\mathfrak{p}_0 \in \mathfrak{P}$ be the point that corresponds to the toric divisor~$X^{\mathrm{toric}} \subset Y$.
Note that the space of lines in~$\mathfrak{P}$ through the point~$\mathfrak{p}_0$ is the projective space~$\mathbb{P}^{13}_{\mathsf{k}}$,
in particular ${\mathsf{k}}$-points are Zariski dense in it.
Therefore, there is a line~$L \subset \mathfrak{P}$ through~$\mathfrak{p}_0$ defined over~${\mathsf{k}}$.
We denote by
\begin{equation*}
{\mathscr{X}} \to L
\end{equation*}
the corresponding family of half-anticanonical divisors in~$Y$.
Then the general point of~$L$ corresponds to a smooth variety~${\mathscr{X}}_L$ of type~\type{1,1,1,1}
over the field~${\mathsf{k}}(L) \cong {\mathsf{k}}(t)$.
Since for a general ${\mathsf{k}}$-point $\mathfrak{p} \in L$
the fiber of~${\mathscr{X}}_{\mathfrak{p}}$ has ${\mathrm{G}}_{X_{\mathfrak{p}}} = {\mathrm{G}}$ by Lemma~\ref{lem:x1111-models},
and since the natural restriction morphism of Galois groups~${\mathrm{G}}(\overline{{\mathsf{k}}(L)}/{\mathsf{k}}(L)) \to {\mathrm{G}}({\bar{\mathsf{k}}}/{\mathsf{k}})$ is surjective,
we have~${\mathrm{G}}_{{\mathscr{X}}_L} = {\mathrm{G}}$.
Since~${\mathrm{G}} \subset \mathfrak{S}_4$ is transitive, this implies~$\uprho({\mathscr{X}}_L) = 1$.
Finally, ${\mathscr{X}}_L$ by construction contains the ${\mathsf{k}}(L)$-point~$y \times_{\mathsf{k}} {{\mathsf{k}}(L)}$, hence~${\mathscr{X}}_L({\mathsf{k}}(L)) \ne \varnothing$.
Assume~${\mathscr{X}}_L$ is stably rational.
Consider the point~$\mathfrak{p}_0 \in L$ as a special point of the family~${\mathscr{X}}/L$.
By Proposition~\ref{prop:forms-x44} the corresponding variety~${\mathscr{X}}_{\mathfrak{p}_0} = X^{\mathrm{toric}}$
is integral with ordinary double points,
hence by~\cite[Proposition~4.2.9]{NS} the family~${\mathscr{X}}/L$ is~$\mathbb{L}$-faithful in the sense of~\cite[Definition~4.2.7]{NS}.
Therefore, by~\cite[Proposition~4.2.10]{NS} the special fiber~$X^{\mathrm{toric}}$ is stably rational.
On the other hand, by definition the Galois group of the extension~${\mathsf{k}}'/{\mathsf{k}}$ coincides with~${\mathrm{G}}_X$
and contains the Klein group~$\mathrm{V}_4$.
By~\cite[\S~2.4.8]{Vos} for any smooth compactification~$V\supset T$
one has
\begin{equation*}
H^1({\mathrm{G}}({\bar{\mathsf{k}}}/{\mathsf{k}}),\, \Pic(V_{{\bar{\mathsf{k}}}})) = H^1({\mathrm{G}}_X,\, \Pic(V_{{\bar{\mathsf{k}}}}))\neq 0.
\end{equation*}
Since this
group is a stable birational invariant
(see, e.g., \cite[\S4.4]{Vos}, or~\cite[\S2.A]{Colliot-Thelene-Sansuc-1987}),
the torus~$T$ and the corresponding toric variety~$X^{\mathrm{toric}}$ are not stably rational.
This contradiction shows that~${\mathscr{X}}_L$ is not stably rational and completes the proof of the theorem.
\end{proof}
|
3,212,635,537,909 | arxiv | \section{Introduction}
According to Einstein's theory of gravitation,
space-time is locally flat, and the Einstein form of the
equivalence principle states that the
outcome of any non-gravitational
experiment should be independent of where
and when in the Universe it is performed.
Among the most accurately measured
quantities in physics, we find transition frequencies
in simple atomic systems and $g$ factor experiments,
both for free and bound leptons (electrons and muons).
Leptons are described, in curved
space-time, by the gravitationally and
electromagnetically coupled Dirac equation.
Here, we derive a generalized Dirac Hamiltonian
which describes both mentioned couplings for light fermions,
in electromagnetic and (weak) gravitational fields,
and establish its properties under a
particle-antiparticle transformation.
We also find its nonrelativistic form by
a Foldy--Wouthuysen transformation.
As convincingly demonstrated by the
Shapiro delay, measured to
excellent accuracy by the Cassini spacecraft
in superior conjunction~\cite{BeIeTo2003}, we must assign a
coordinate dependence to the vacuum permittivity
and vacuum permeability, in global coordinates.
Based on these assumptions, we investigate the position-dependence
of atomic transitions and
find (in agreement with Ref.~\cite{Wi1974prd})
that the position-dependence of their frequencies
is largely compatible with the equivalence principle.
For free and bound $g$ factor experiments,
gravitational frequency shifts of spin-flip transitions
have been the subject of rather intense
discussions~\cite{MoFuSh2018ptep,MoFuSh2018remark1,MoFuSh2018remark2,Vi2018,Ni2018,Gu2018}.
Our paper addresses part of these questions but otherwise has
a much broader scope. Furthermore, it has long been conjectured that
subtle limitations to the Einstein equivalence
principle should occur within a full quantum theory.
We find such limitations, both due to the Fokker
precession as well as due to the noncommutativity of the electron's
momentum operator with the global space-time coordinates.
It is our goal to present a comprehensive and
relatively
easily digestible account of related matters,
despite the length of the current article.
For clarification, we should point out that
throughout this paper, we consider gravitational effects for an
atom at rest with respect to a center of gravity, in contrast to
Refs.~\cite{Pa1980prd,PaPi1982,AdChVa2012},
where the authors refer to an atom in a
freely falling reference frame.
Note that in Ref.~\cite{AuMa1994},
the results of Refs.~\cite{Pa1980prd,PaPi1982,AdChVa2012} are
generalized to accelerated and rotating
reference frames; such frames are not of interest for the
current study.
Furthermore, we assume, throughout the
paper, that local Lorentz invariance
is conserved. Conceivable correction terms beyond
this approximation are considered
in Ref.~\cite{GaHa1990}.
In detail, the paper is organized as follows.
In Sec.~\ref{sec2}, we consider the
gravitationally and electromagnetically coupled
Dirac equation, and the scaling of atomic transition
frequencies, and bound-state $g$ factors,
induced by the gravitational coupling.
The interrelation of quantum mechanics and
Einstein's equivalence principle is
being studied in Sec.~\ref{sec3}.
Roughly speaking, the question is whether a
non-deterministic theory (namely, quantum mechanics)
can in principle be fully compatible with a
fully deterministic theory (namely, general relativity),
given the fact that position and momentum operators
in quantum mechanics behave differently from their
classical counterparts.
We shall find tiny, but important corrections
to the so-called $\sqrt{T}$ scaling which otherwise
ensures the compatibility of the gravitationally
corrected frequencies with the equivalence principle.
The measurability of the higher-order gravitational
corrections is discussed in Sec.~\ref{sec4}.
Conclusions are reserved for Sec.~\ref{sec5}.
\section{Gravity and Scaling}
\label{sec2}
\subsection{Coupled Dirac Hamiltonian}
\label{sec2A}
We use units with $\hbar = c = \epsilon_0 = 1$.
The relativistic (gravitationally coupled)
Dirac--Schwarzschild Hamiltonian
is~\cite{JeNo2013pra}
\begin{equation}
\label{HDS}
H_{\rm DS} =
\frac12 \, \left\{ \vec\alpha \cdot \vec p,
\left( 1 - \frac{r_s}{r} \right) \right\} +
\beta m \left( 1 - \frac{r_s}{2 r} \right) \,,
\end{equation}
where
$\vec \alpha =
\left( \begin{array}{cc}
0 & \vec \sigma \\
\vec \sigma & 0
\end{array} \right)$
and
$\beta =
\left( \begin{array}{cc}
\mathbbm{1}_{2 \times 2} & 0 \\
0 & \mathbbm{1}_{2 \times 2}
\end{array} \right)$ are Dirac matrices,
and $m$ denotes the fermion (electron) mass.
The $(2 \times 2)$ Pauli
spin matrices are denoted as $\vec \sigma$.
Hermiticity properties of
this Hamiltonian are discussed in App.~\ref{appA1},
while a comparison of this result to
other literature references is the subject
of App.~\ref{appA2}.
The Foldy--Wouthuysen (FW) transformed
Dirac--Schwarzschild Hamiltonian
has been found in Ref.~\cite{JeNo2013pra},
\begin{subequations}
\label{HDSFW}
\begin{align}
H_{\rm FW} =& \;
\beta \, \left( m + \frac{\vec p^{\,2}}{2 m} -
\frac{\vec p^{\,4}}{8 m^3}
- \beta \, \frac{m \, r_s}{2 \, r}
- \frac{3 r_s}{8 m} \,
\left\{ \vec p^{\,2}, \frac{1}{r} \right\} \right.
\nonumber\\
& \; + \left.
\frac{3 \pi r_s}{4 m} \delta^{(3)}(\vec r) \,
+ \frac{3 r_s}{8 m} \, \frac{\vec\Sigma \cdot \vec L}{r^3} \right) \,,
\end{align}
which can be reformulated as
\begin{multline}
H_{\rm FW} = \beta \,
\left( m + \frac{\vec p^{\,2}}{2 m} \right)
\\
- \beta \, \left( \frac{\vec p^{\,4}}{8 m^3} - \frac{m \, r_s}{2 \, r}
- \frac{3}{16 m} \left\{ \vec \Sigma \cdot \vec p,
\left\{ \vec \Sigma \cdot \vec p, \frac{r_s}{r} \right\} \right\}
\right),
\end{multline}
\end{subequations}
where $r_s$ is the Schwarzschild radius, and
$r$ is the radial variable in Eddington coordinates~\cite{Ed1924}.
The latter form is obtained from the first
by applying the operator identity
$\{ A, \{ A, B \} \} =
2 \{ A^2, B \} - [ A, [A, B]] $,
for $A = \vec \Sigma \cdot \vec p$ and $B = 1/r$,
where the $(4 \times 4)$ spin matrices are
$\vec \Sigma = \left( \begin{array}{cc}
\vec\sigma & 0 \\
0 & \vec\sigma \end{array} \right)$.
The generalization of the Dirac--Schwarzschild
Hamiltonian~\eqref{HDS} to the case of an additional
external electromagnetic fields
[denoted here as the Dirac--Schwarzschild--Coulomb
(DSC) Hamiltonian]
involves the replacement of the
kinetic momentum operators $\vec p$ by the
canonical momentum operators $\vec \pi = \vec p - e \, \vec A$,
and the addition of the scalar potential term $e \, A^0$.
Here, $e = -|e|$ is the
physical electron charge.
It reads as follows,
\begin{equation}
\label{HDSC}
H_{\rm DSC} =
\frac12 \, \left\{ \left( 1 - \frac{r_s}{r} \right),
\vec\alpha \cdot \vec \pi \right\} +
e \, A^0 \mathbbm{1}_{4 \times 4} +
\beta m \left( 1 - \frac{r_s}{2 r} \right).
\end{equation}
After a Foldy--Wouthuysen
transformation, one obtains the Hamiltonian $H_{\rm EM}$
which describes the coupling to
external electromagnetic fields,
\begin{multline}
\label{HEM}
H_{\rm EM} =
\beta \, \left( m +
\frac{(\vec\Sigma \cdot \vec\pi)^2}{2 m} -
\frac{(\vec\Sigma \cdot \vec\pi)^4}{8 m^3} \right)
+ e \, A^0 \mathbbm{1}_{2 \times 2}
\\[1.0ex]
- \beta \, \left( \frac{m \, r_s}{2 \, r}
+ \frac{3}{16 m}
\left\{ \vec \Sigma \cdot \vec \pi,
\left\{ \vec \Sigma \cdot \vec \pi,
\frac{r_s}{r} \right\} \right\} \right)
\\[1.0ex]
+ \left\{ 1 + \frac{r_s}{r}, \frac{e}{16 m^2} \,
\left( \vec\nabla \cdot \vec E +
\vec \Sigma \cdot (\vec E \times \vec \pi -
\vec \pi \times \vec E)
\right) \right\} \,.
\end{multline}
This Hamiltonian is a $(4 \times 4)$-matrix,
diagonal in the space of $(2 \times 2)$-submatrices.
The $(2 \times 2)$-particle Hamiltonian
$ H^+_{\rm EM}$ is obtained by
replacing $\beta \to 1$:
\begin{multline}
\label{HEMplus}
H^+_{\rm EM} = m +
\frac{(\vec\sigma \cdot \vec\pi)^2}{2 m} -
\frac{(\vec \sigma \cdot \vec\pi)^{4}}{8 m^3}
+ e \, A^0
\\[0.1133ex]
- \frac{m \, r_s}{2 \, r}
- \frac{3}{16 m}
\left\{ \vec \sigma \cdot \vec \pi,
\left\{ \vec \sigma \cdot \vec \pi,
\frac{r_s}{r} \right\} \right\}
\\[0.1133ex]
+ \left\{ 1 + \frac{r_s}{r}, \frac{e}{16 m^2} \,
\left( \vec\nabla \cdot \vec E +
\vec \sigma \cdot (\vec E \times \vec \pi -
\vec \pi \times \vec E)
\right) \right\} \,.
\end{multline}
The antiparticle Hamiltonian $H^-_{\rm EM}$
is obtained from $H_{\rm EM}$ by
replacing $\beta \to -1$, taking into account an overall factor $-1$
due to the reinterpretation principle,
replacing $\vec \Sigma \to -\vec\sigma$,
and $\vec p \to - \vec p$, again due to
reinterpretation for antiparticles.
One can convince oneself that the antiparticle Hamiltonian
$H^-_{\rm EM}$ can be obtained from the
particle Hamiltonian $H^+_{\rm EM}$ by the replacement $e \to -e$
(charge conjugation, hence $\vec \pi \to \vec \pi' = \vec p + e \, \vec A$)),
while all the gravitational terms
are invariant under the particle-antiparticle transformations~\cite{JeNo2013pra},
establishing the equivalence principle for anti-particles.
We now continue to work with the particle Hamiltonian~\eqref{HEMplus},
which can be simplified based on the identity
$(\vec\sigma \cdot \vec \pi)^2
= \vec \pi^2 - e \, \vec \sigma \cdot \vec B$, which implies that
\begin{multline}
\label{HFW}
H^+_{\rm EM} = m + \frac{\vec \pi^{\,2}}{2 m}
- \frac{\vec \pi^{\,4}}{8 m^3}
- \frac{e}{2 m} \, \vec \sigma \cdot \vec B
\\[0.1133ex]
+ e \, A^0
+ \frac{e}{8 m^3} \, \{ \vec \sigma \cdot \vec B, \vec\pi^{\,2} \}
- \frac{m \, r_s}{2 \, r}
+ \frac{3 \pi r_s}{4 m} \delta^{(3)}(\vec r)
\\[0.1133ex]
- \frac{3}{8 m}
\left\{ \vec \pi^{\,2} - e \, \vec \sigma \cdot \vec B,
\frac{r_s}{r} \right\}
+ \frac{3 r_s}{8 m r^3}
\vec \sigma \cdot \vec r \times \vec \pi
\\[0.1133ex]
+ \left\{ 1 + \frac{r_s}{r}, \frac{e}{16 m^2} \,
\left( \vec\nabla \cdot \vec E +
\vec \sigma \cdot \vec E \times \vec \pi -
\vec \sigma \cdot \vec \pi \times \vec E
\right) \right\} \,.
\end{multline}
We should note that related calculations have
recently been considered in
other contexts~\cite{ObSiTe2014,ObSiTe2016,ObSiTe2017},
with an important clarifying remark given in the text following
Eq.~(7.33) of Ref.~\cite{ObSiTe2017}
(see also Ref.~\cite{JeNo2014jpa}).
We now discuss a general metric for weak gravitational fields and
gravitational red shifts. For inspiration,
we start with the Schwarzschild metric~\cite{Sc1916}
in isotropic form (Sec.~43 of Chap.~3 of Ref.~\cite{Ed1924}),
\begin{subequations}
\begin{equation}
\label{eddington}
\mathrm{d} s^2 =
\left( \frac{1 - r_s/(4 r)}{1 + r_s/(4 r)} \right)^2 \, \mathrm{d} t^2
-\left( 1 + \frac{r_s}{4 r} \right)^4 \; \mathrm{d} \vec r^{\,2} \,.
\end{equation}
This metric can be expanded to first order in the
potential $\Phi(\vec r) = - G M/r$ ,
where $M$ is the mass of the central gravitational object,
and generalized
to arbitrary (weak) gravitational potentials $\Phi$,
\begin{align}
\label{ds2}
\mathrm{d} s^2
= & \; \left( 1 - \frac{r_s}{r} \right) \, \mathrm{d} t^2
- \left( 1 + \frac{r_s}{r} \right) \, \mathrm{d} \vec r^{\,2}
\nonumber\\[0.1133ex]
=& \; \left( 1 + 2 \, \Phi \right) \, \mathrm{d} t^2
- \left( 1 - 2 \, \Phi \right) \, \mathrm{d} \vec r^{\,2}
\nonumber\\[0.1133ex]
=& \; T \, \mathrm{d} t^2 - H \, \mathrm{d} \vec r^{\,2}
= \overline g_{\mu\nu} \, \mathrm{d} x^\mu \, \mathrm{d} x^\nu \,.
\end{align}
\end{subequations}
Here, $\overline g_{\mu\nu} = {\rm diag}(T, -H, -H,-H)$ is the
curved-space metric, while we reserve the symbol
$\widetilde g_{\mu\nu} $ for the metric of free space~\cite{JeNo2013pra}.
In the following,
we use the symbols $T$ and $H$ for the case of a general
gravitational potential $\Phi$.
In a metric of the form~\eqref{ds2}
(see Refs.~\cite{OhRu1994,As2002,Pa2010})
one has for light, which travels on a zero geodesic
with $\mathrm{d} s^2 = 0$,
\begin{equation}
\label{shapiro}
\left| \frac{\mathrm{d} \vec r }{ \mathrm{d} t } \right|^2 =
\frac{ 1 + 2 \, \Phi }{ 1 - 2 \, \Phi } =
\frac{T}{H} \approx 1 + 4 \, \Phi \,.
\end{equation}
We thus generalize~\eqref{shapiro}
to general gravitational fields.
The Shapiro time delay~\cite{Sh1964,ShEtAl1968,Sh1999,Lo1988,KrTr1988}
is consistent
with an effective speed of light, of the form
$c_{\rm eff} = 1 + 2 \, \Phi = \sqrt{ T/H } $,
to first order in the gravitational potential.
This implies that in electrodynamics,
we must assign a slight gravitational dependence
to the vacuum permittivity $\epsilon$ and vacuum permeability $\mu$,
so that
\begin{equation}
\label{ceff}
c_{\rm eff}^2 = \frac{1}{\epsilon \, \mu} = \frac{T}{H} \,,
\qquad
\epsilon = \mu = \sqrt{ \frac{H}{T} } \,,
\end{equation}
consistent with Eq.~(4) of Ref.~\cite{Wi1974prd}.
\subsection{Gravity and atomic transitions}
\label{sec2B}
The generalization of the Hamiltonian~\eqref{HDSC}
to a general gravitational potential $\Phi$
can be found by realizing that the derivation,
outlined in Ref.~\cite{JeNo2013pra}, goes through
for a general metric of the form given in Eq.~\eqref{ds2}.
The Hamiltonian reads as
\begin{align}
H_{\rm DSC} =& \;
\frac12 \, \left\{ 1 + 2 \Phi,
\vec\alpha \cdot \vec \pi \right\} +
e \, A^0 + \beta m \left( 1 + \Phi \right)
\\[0.1133ex]
=& \; \frac12 \, \left\{ \sqrt{\frac{T}{H}},
\vec\alpha \cdot \vec \pi \right\} +
e \, A^0 + \beta m \sqrt{T} \,.
\end{align}
If we ignore commutators of the gravitational fields
and the momentum operators, then
we may approximate
\begin{equation}
\label{HDSCapprox}
H_{\rm DSC} \approx
\sqrt{\frac{T}{H}} \, \vec \alpha \cdot \vec \pi
+ \sqrt{T} \, \beta \, m
+ e \, A^0 \,.
\end{equation}
We here confirm the result given in Eq.~(14) of Ref.~\cite{Wi1974prd},
and show that anticommutators are needed in
order to turn the Hamiltonian into a manifestly
Hermitian entity.
The approximation~\eqref{HDSCapprox} is valid
if we assume that $T$ and $H$ remain constant to very
good approximation, over the distance scales
relevant to the described quantum mechanical phenomena.
We consider the Hamiltonian~\eqref{HDSCapprox}
for the case $\vec A = \vec 0$, and
$e \, A^0 = - Z e^2/(4 \pi \epsilon |\vec \rho|)$,
where $|\vec \rho|$ is the distance to the atomic nucleus.
In this case, the Hamiltonian becomes
\begin{equation}
\label{HDSCproblem}
H_{\rm DSC} = \sqrt{\frac{T}{H}} \, \vec \alpha \cdot \vec p
+ \sqrt{T} \, \beta \, m
- \frac{Z e^2}{4 \pi \epsilon |\vec \rho|} \,,
\end{equation}
where the subscripts refer to Dirac, Schwarzschild and Coulomb (DSC).
The energy eigenvalue equation is
\begin{equation}
H_{\rm DSC} \, \psi = E \, \psi \,.
\end{equation}
With Ref.~\cite{Wi1974prd}, we now perform the following
scaling,
\begin{equation}
\label{scaling}
m = \bar m \, \frac{1}{\sqrt{H}} \,, \qquad
e^2 = \bar e^2 \sqrt{\frac{T}{H}} \, \epsilon \,, \qquad
E = \bar E \, \sqrt{\frac{T}{H}} \,,
\end{equation}
which turns the eigenvalue problem~\eqref{HDSCproblem} into
\begin{equation}
\label{saye1}
\left( \vec \alpha \cdot \vec p
+ \beta \, \bar m
- \frac{Z \bar e^2}{4 \pi |\vec \rho|} \right) \, \psi =
\bar E \, \psi \,.
\end{equation}
The energy can be given in terms of the scaled function
$f(n, J, Z\alpha)$ which has been introduced by
Sapirstein and Yennie in Ref.~\cite{SaYe1990},
\begin{subequations}
\label{escaling}
\begin{align}
\label{saye2}
\bar E =& \; \bar m \, f(n,J, Z \bar \alpha) \,,
\\[0.1133ex]
f(n,J, Z \bar \alpha) =& \; \left( 1 + \frac{ (Z \bar \alpha)^2 }%
{ [n_r + \sqrt{ (J + 1/2)^2 - (Z\bar\alpha)^2 } ]^2 } \right)^{-\tfrac12} \,,
\end{align}
where $n_r = n - J - 1/2$ is the
so-called reduced principal quantum number.
The electron's orbital angular momentum quantum number is
$\ell$, while its total angular momentum is $J$.
Finally, the gravitationally ``modified'' (as it turns out, invariant)
fine-structure constant is
\begin{equation}
\label{alphaconst}
\bar \alpha
= \frac{\bar e^2}{4 \pi}
= \sqrt{\frac{H}{T}} \, \frac{e^2}{4 \pi \epsilon}
= \sqrt{\frac{H}{T}} \, \frac{e^2}{4 \pi} \, \sqrt{\frac{T}{H}}
= \frac{e^2}{4 \pi} = \alpha \,.
\end{equation}
\end{subequations}
The position-independence of the fine-structure
constant has been verified experimentally,
in a dedicated experiment described in
Ref.~\cite{TuEtAl1983}.
We should notice that experimental possibilities
to search for a temporal as well as
spatial variation of the fine-structure
constant have since dramatically improved
in accuracy~\cite{FiEtAl2004,PeEtAl2004,GoEtAl2014,HuEtAl2014}.
The scaling of the bound-state energy
is found as
\begin{equation}
\label{yes}
E = \sqrt{\frac{T}{H}} \, \bar E
= \sqrt{\frac{T}{H}} \, \bar m \, f(n,J, Z \alpha)
= \sqrt{T} \, m \, f(n,J, Z \alpha) \,,
\end{equation}
valid for both main-structure (change in the principal
quantum number) as well as fine-structure transitions.
\subsection{Gravity and $\maybebm{g}$ factor}
\label{sec2C}
We start from Eq.~\eqref{HDSCapprox},
but this time we include the static vector potential
$\vec A = \frac12 \, (\vec B \times \vec r)$,
which describes a constant $\vec B$ field. Hence,
$H_{\rm DSC}$ attains the form
\begin{equation}
\label{HDSCapprox_full}
H_{\rm DSC} = \sqrt{\frac{T}{H}} \, \vec \alpha \cdot
(\vec p - e \, \vec A)
+ \sqrt{T} \, \beta \, m + e \, A^0 \,.
\end{equation}
Taking into account that $\vec A = \tfrac12 \, (\vec B \times \vec r)$,
one can write the Hamiltonian $H_M$ which describes the
magnetic coupling of the electron to the
external field as
\begin{equation}
\label{HM}
H_M = - \sqrt{\frac{T}{H}} \, e \,
\, \vec \alpha \cdot \vec A
= - \sqrt{\frac{T}{H}} \, \frac{e}{2} \,
\, \vec \alpha \cdot (\vec B \times \vec r) \,.
\end{equation}
Canonically, one assumes that $\vec B$ is directed
along the $z$ axis~\cite{GlSh2002}.
The Land\'{e} $g$ factor
(written as $g_J$) and the expectation value for a hydrogenic
state in a homogeneous $\vec B$ field can be
expressed as
\begin{equation}
\label{HMexpec}
\left< H_M \right> =
- g_J \, \frac{e}{2 m} \, | \vec B | \, \mu_J \,,
\end{equation}
where $\mu_J$ is the projection of the
electron's total angular momentum
(angular$+$spin) onto the axis of the $\vec B$ field.
The expectation value in Eq.~\eqref{HMexpec}
is to be taken in an eigenstate of the unperturbed
problem, i.e., in a (gravitationally modified)
Dirac--Coulomb eigenstate of the Hamiltonian~\eqref{HDSCproblem}.
Let us therefore consider the Hamiltonian~\eqref{HDSCapprox_full}
under the same scaling as the one used in Eq.~\eqref{scaling}.
The eigenvalue problem transforms into
\begin{equation}
\left( \vec \alpha \cdot (\vec p - \bar e\, \vec A)
+ \beta \, \bar m
- \frac{Z \bar e^2}{4 \pi |\vec \rho|} \right) \, \psi =
\bar E \, \psi \,.
\end{equation}
For the magnetic-field coupling Hamiltonian,
written in terms of the scaled variables,
\begin{equation}
\label{HMbar}
\bar H_M = - \frac{\bar e}{2} \,
\, \vec \alpha \cdot (\vec B \times \vec r) \,,
\end{equation}
we therefore have the following relation,
which holds in view of the analogy with the
unperturbed Dirac problem (see Ref.~\cite{GlSh2002}):
\begin{align}
\left< \bar H_M \right> =& \;
- \bar g_J \, \frac{\bar e}{2 \bar m} \, | \vec B | \, \mu_J \,,
\\[0.1133ex]
\bar g_J =& \; \frac{\varkappa}{J (J+1)} \left( \varkappa \,
\frac{\bar E}{\bar m} - \frac12 \right)
\nonumber\\[0.1133ex]
=& \; \frac{\varkappa}{J (J+1)} \left( \varkappa \,
f(n,J, Z \alpha) - \frac12 \right) \,.
\end{align}
Here, $\varkappa$ is the Dirac angular quantum
number, which is given as
$\varkappa = (-1)^{J+ \ell +1/2} \, \left(J + \frac12 \right) $.
A comparison of the Hamiltonian $H_M$ given in Eq.~\eqref{HM}
to the Hamiltonian $\bar H_M$ given in Eq.~\eqref{HMbar}
reveals that $\left< H_M \right> =
-(T/H)^{1/2} \, \left< \bar H_M \right> $
so that, in view of Eq.~\eqref{HMexpec},
\begin{equation}
\label{gJTH}
g_J = \frac{\sqrt{T}}{H} \, \bar g_J \,
= \frac{\sqrt{T}}{H} \,
\frac{\varkappa}{J (J+1)} \left( \varkappa \,
f(n,J, Z \alpha) - \frac12 \right) \,.
\end{equation}
For the ground state, one has with $\varkappa = -1$
and $J = 1/2$,
\begin{equation}
\label{bargJTH}
g_J = \frac{\sqrt{T}}{H} \,
\frac43 \, \left( \sqrt{1 - (Z\alpha)^2} + \frac12 \right) \,.
\end{equation}
The free-electron $g$ factor is obtained from this
expression, in the limit $Z\alpha \to 0$, and is equal to
$g_S = 2 \sqrt{T}/H$. One can convince oneself that this result is
compatible with the terms proportional to $\vec \sigma \cdot \vec B$
in Eq.~\eqref{HFW}; these determine the g factor.
At this stage, we have clarified the gravitational corrections
to the {\em non-anomalous} part of the electron's magnetic moment.
For the anomalous part, we need to consider the
generalized Dirac equation, which necessitates
the introduction of form factors.
We recall that in flat space,
the electromagnetically coupled
Dirac equation reads as
$\left[ \widetilde\gamma^\mu (p_\mu - e \, A_\mu) - m \right] \, \psi = 0$,
where the $\widetilde\gamma^\mu$ are Dirac $\gamma$
matrices which fulfill the anti-commutator relations
$\{ \widetilde\gamma^\mu, \widetilde\gamma^\nu \}
= 2 \widetilde g^{\mu\nu}
= {\rm diag}(1, -1, -1, -1)$. In order to describe the
anomalous magnetic moment,
one replaces the Dirac $\gamma$ matrices by
a form-factor expression (see Chap.~7 of Ref.~\cite{ItZu1980}),
\begin{equation}
\label{formfactor}
\widetilde\gamma^\mu \to \widetilde\gamma^\mu \, F_1(q^2) +
\frac{\mathrm{i} \widetilde\sigma^{\mu\nu} \, q_\nu}{2 m} F_2(q^2) \,,
\end{equation}
where the spin matrices are given as $\widetilde\sigma^{\mu\nu} =
\tfrac{\mathrm{i}}{2} [ \widetilde\gamma^\mu, \widetilde\gamma^\nu ]$.
The replacement leads to the modified Dirac
(MD) Hamiltonian~\cite{JePa2002},
\begin{align}
\label{HDDM}
H_{\rm MD} =& \; \vec{\alpha} \cdot
\left[\vec{p} - e \, F_1(\vec\nabla^2 ) \, \vec{A}\right]
+ \beta\,m + {\mathrm e} \, F_1(\vec\nabla^2 ) \, A^0
\nonumber\\[0.1133ex]
& \; + F_2(\vec\nabla^2 ) \, \frac{e}{2\,m} \, \left({\mathrm i}\,
\vec{\gamma} \cdot \vec{E} - \beta \, \vec{\sigma} \cdot \vec{B} \right) \,.
\end{align}
In the following, we shall approximate
\begin{equation}
\label{approx}
F_1(q^2) \approx F_1(0) = 1 \,,
\qquad
F_2(q^2) \approx F_2(0) = \kappa \approx \frac{\alpha}{2 \pi} \,,
\end{equation}
and set the external electric field equal to zero, $\vec E = \vec 0$.
[We remember that, if we set $\vec E$ equal to the
Coulomb electric field, the corresponding term in Eq.~\eqref{HDDM}
describes the anomalous magnetic-moment correction to the Lamb shift.]
Here, $\kappa$ describes the anomalous magnetic moment
correction to the electron's spin $g$ factor,
and is approximated by the Schwinger term $\alpha/(2 \pi)$.
With the approximations outlined in Eq.~\eqref{approx},
the Hamiltonian~\eqref{HDDM} becomes
\begin{equation}
\label{HDm}
H_{\rm MD} = \vec{\alpha} \cdot
\left(\vec{p} - e \, \vec{A}\right)
+ \beta\,m + e \, A^0 -
\kappa \, \frac{e}{2\,m} \,
\beta \, \vec{\sigma} \cdot \vec{B} \,.
\end{equation}
We carry out a replacement analogous to Eq.~\eqref{formfactor} in curved space,
\begin{equation}
\overline\gamma^\mu \to \overline\gamma^\mu \, F_1(q^2) +
\frac{\mathrm{i} \overline\sigma^{\mu\nu} \, q_\nu}{2 m} F_2(q^2) \,,
\qquad
\{ \overline\gamma^\mu, \overline\gamma^\nu \} =
2\overline g^{\mu\nu} \,,
\end{equation}
where
$\overline g^{\mu\nu} = {\rm diag}(1/T, -1/H, -1/H, -1/H)$
is the inverse of the metric $\overline g_{\mu\nu}$ given
in Eq.~\eqref{ds2}. The curved-space
Dirac spin matrices $\overline\sigma^{\mu\nu} = \frac12 \,
[ \overline\gamma^\mu, \overline\gamma^\nu ]$ fulfill
$\overline\sigma^{\mu\nu} =
\widetilde\sigma^{\mu\nu} / H$,
and the gravitational modification of
Eq.~\eqref{HDm} reads as
\begin{equation}
\label{HX}
H = \sqrt{\frac{T}{H}} \, \vec \alpha \cdot
(\vec p - e \, \vec A)
+ \sqrt{T} \, \beta \, m
+ e \, A^0
- \frac{\sqrt{T}}{H} \, \kappa \, \frac{e}{2\,m} \,
\beta \, \vec{\sigma} \cdot \vec{B} \,.
\end{equation}
The gravitationally modified
electron $g_J$ factor (for the $1S$ state) thus is, in view of
Eq.~\eqref{bargJTH} and Eq.~\eqref{HX},
\begin{equation}
\label{gJ1S}
g_J = \frac{\sqrt{T}}{H} \,
\frac43 \, \left( \sqrt{1 - (Z\alpha)^2} + \frac12 +
\frac32 \, \kappa \right) \,,
\end{equation}
where the free-electron term is obtained in the limit
$Z\alpha \to 0$. The scaling with $\sqrt{T}/H$ is thus established as a
universal scaling of the free-electron and bound-electron $g$
factors, including the anomalous-magnetic-moment
correction.
\subsection{Equivalence principle and $\maybebm{g}$ factor}
\label{sec2D}
According to Eq.~\eqref{yes}, atomic transition frequencies receive a
gravitational correction proportional to $\sqrt{T}$,
while according to Eq.~\eqref{gJ1S},
the $1S$ electron $g$ factor
receives a correction proportional to
$\sqrt{T}/H$.
The prefactor $\sqrt{T}$ in Eq.~\eqref{yes} describes the
transition from coordinate time to laboratory time,
This is evident from the metric~\eqref{ds2},
$\mathrm{d} s^2 = T \, \mathrm{d} t^2 - H \, \mathrm{d} \vec r^{\,2} = \mathrm{d} \tau^2$,
where $ \mathrm{d} \tau^2 $ measures the (square of the) time
interval in the local Lorentz frame.
We can convert the time derivative operator from
coordinate time to the time elapsed in the local Lorentz frame,
\begin{equation}
\sqrt{T} \, \mathrm{d} t = \mathrm{d} \tau \,,
\qquad
\mathrm{i} \frac{\partial}{\partial \tau} =
\frac{\mathrm{i}}{\sqrt{T}} \, \frac{\partial}{\partial t} \,.
\end{equation}
The energy~\eqref{yes} is formulated with respect to the
coordinate time, and so, the energy in the laboratory
can be obtained by dividing the energy $E$ given in Eq.~\eqref{yes}
by a factor $1/\sqrt{T}$, and
one obtains the laboratory atomic energy levels as being
given by the expression $ m \, f(n,J, Z \alpha) $.
Let us put this statement into the context
of the weak and strong forms of the equivalence principle.
The ``weak equivalence principle'' (WEP) asserts the
proportionality of ``mass'' (``inertial mass'') and ``weight'' (which enters
the gravitational force law).
The Einstein equivalence principle (EEP) states that {\em (i)} WEP is valid,
{\em (ii)} the outcome of any local non-gravitational experiment is independent
of the velocity of the freely-falling reference frame in which it is performed
(local Lorentz invariance, LLI), and
{\em (iii)} the outcome of any local non-gravitational experiment is
independent of where and when in the universe it is performed
(local position invariance, LPI).
The scaling with $\sqrt{T}/H$ of the electron's $g$ factor,
in coordinate time, taken at face value, would
imply a scaling with $1/H$ in the local Lorentz frame
of each laboratory, after dividing out the factor $\sqrt{T}$.
This would make the outcome of a non-gravitational experiment
(the measurement of the electron's $g$ factor)
dependent on the position,
limit the validity of the principle of local position invariance, and, hence, the EEP.
In order to resolve the problem,
we note that we have assumed, in our derivation, that the
$\vec A$ field is given in terms of
the components of the covariant basis,
\begin{equation}
\vec A = A^i \, \vec e_i \,,
\qquad
\vec e_i \cdot \vec e_j = H \, \delta_{ij} \,.
\end{equation}
Latin indices indicate spatial components ($i,j,k,\ldots = 1,2,3$).
However, the ``Cartesian'' unit vectors (index $c$)
which span the local Lorentz frame are
\begin{equation}
\hat e_i = \frac{1}{\sqrt{H}} \, \vec e_i \,,
\qquad
\hat e_i \cdot \hat e_j = \delta_{ij} \,.
\end{equation}
Let $x^i$ denote the components of the
position vector $\vec r$ in the basis spanned by the
$\vec e_i $, while the components $x_c^j $ are relevant to the
basis spanned by the $\hat e_i$. Then,
\begin{equation}
x_c^j = \sqrt{H} \, x^j \,,
\qquad
A_c^i = \sqrt{H} \, A^j \,.
\end{equation}
We denote by $\epsilon^{ijk}$ the totally antisymmetric
Levi-Civit\`{a} tensor (under the normalization
$\epsilon^{123} =1 $). Then, we have
\begin{equation}
\vec A = \frac12 \, \vec B_c \times \vec r_c
= \frac12 \, \hat e_i \, \epsilon^{ijk} \, B^j \, x_c^k \,,
\end{equation}
which is the appropriate vector potential for a
magnetic field with ``Cartesian'' components $B_c^i$,
measured in the local Lorentz frame.
The curl of $\vec A$ enters into Eq.~\eqref{bargJTH};
it is calculated with
momentum operators $\vec p = -\mathrm{i} \vec\nabla$ where
$\nabla^k = -\mathrm{i} \partial/\partial x^k$, and hence,
\begin{equation}
B^i = \epsilon^{ijk} \frac{\partial}{\partial x^j} A^k
= \frac12 \, \epsilon^{ijk} \frac{\partial}{\partial x^j}
\epsilon^{k \ell m} \, B_c^\ell \, \sqrt{H} \, x^m
= \sqrt{H} \, B_c^i \,.
\end{equation}
For the vector $\vec B$, this means that
\begin{equation}
\label{ottifant}
\vec B = B^i \, \vec e_i
= (\sqrt{H}\, B_c^i) \, (\sqrt{H} \hat e_i )
= H\, B_c^i \, \hat e_i \,.
\end{equation}
Thus, the $i$th component of $\vec B$, written in our
basis, is equal to $H$ times the $B$ field measured by a local
observer, in his or her own Lorentz frame.
This implies that, when normalized to the local $B$ field,
spin-flip frequencies transform with a factor
$\sqrt{T}$, not $\sqrt{T}/H$, respecting the equivalence principle.
We note that the same factor $H$ is obtained in Ref.~\cite{Wi1974prd}
for the transformation of the hyperfine-structure generating
$B$ field of a nucleus, from global coordinates to the local
Lorentz frame; however, the derivation proceeds in a completely
different way [see Eqs.~(32)---(34) of Ref.~\cite{Wi1974prd}].
One notes that the restoration of the $\sqrt{T}$ scaling
actually is absolutely crucial for the validity of the current
adjustment of the fundamental constants~\cite{MoNeTa2016}.
\section{Quantum Mechanics and Equivalence Principle}
\label{sec3}
\subsection{Leading order
and $\maybebm{\sqrt{T}}$ scaling}
\label{sec3A}
We recall, from Sec.~\ref{sec2A},
that relatively weak gravitational fields give rise
to a metric
\begin{align}
\mathrm{d} s^2 =& \; T \, \mathrm{d} t^2 - H \, \mathrm{d} \vec r^{\,2} \,,
\\[0.1133ex]
T =& \; 1 + 2 \, \Phi \,, \qquad H = 1 - 2 \, \Phi \,,
\end{align}
where $\Phi$ is the gravitational potential.
Hence, if, in global coordinates, an energy goes as
\begin{equation}
\label{common_prefactor}
E = \sqrt{T} \, E_c \,,
\end{equation}
where $E_c$ is the energy measured in a
local, Cartesian Lorentz frame, then this effect is
physically unobservable if the experiment
is carried out locally, because
the time derivative operator $\mathrm{d}/\mathrm{d} \tau$
with respect to the proper time has the
eigenvalue
\begin{equation}
\mathrm{i} \frac{\partial}{\partial \tau} \psi =
\frac{\mathrm{i}}{\sqrt{T}} \, \frac{\partial}{\partial t} \psi
= \frac{E}{\sqrt{T}} \, \psi = E_c \, \psi \,.
\end{equation}
All factors that go with $\sqrt{T}$ are unobservable since
they can be absorbed in going to local, Cartesian
coordinates.
It is highly instructive (and non-obvious)
to convince oneself that the
leading kinetic terms in the Dirac--Schwarzschild
Hamiltonian~\eqref{HDSFW}
follow the $\sqrt{T}$ scaling.
This observation, in particular, implies that the
gravitational Breit term
\begin{equation}
- \frac{3 r_s}{8 m} \,
\left\{ \vec p^{\,2}, \frac{1}{r} \right\} \,,
\end{equation}
does not lead to an observable gravitational
shift. At face value, one could otherwise assume
that it induces a numerically large,
$(1/n^2)$-dependent shift
on hydrogen energy levels
(where $n$ is the principle quantum number), because the
operator $1/r$, where $r$ is the radial variable
with respect to the gravitational center (e.g., the Earth),
commutes, to an excellent approximation, with the
momentum operator of the electron,
and in fact, the difference of the
operator $(1/r) \, \vec p^{\,2}$ and the
anti-commutator $(1/2) \, \{ r^{-1}, \vec p^{\,2} \}$
can be ignored altogether on the level
of first-order perturbation theory.
This is because one has
$\langle \psi^{\mbox{{\bf{\tiny +}}}} | r^{-1} \, \vec p^{\,2} | \psi \rangle =
\frac12 \, \langle \psi^{\mbox{{\bf{\tiny +}}}} | \{ r^{-1}, \, \vec p^{\,2} \} | \psi \rangle$
for any reference state $\psi$.
(The Hermitian adjoint, as opposed to the Dirac
adjoint $\overline \psi$, is denoted as $\psi^{\mbox{{\bf{\tiny +}}}}$.)
The kinetic terms from Eq.~\eqref{HDSFW}
read as follows,
\begin{align}
\label{Hkin_spec}
H_{\rm kin} = & \;
m - \frac{r_s}{2 \, r} \, m
+ \frac{\vec p^{\,2}}{2 m}
- \frac{3 r_s}{8 m} \, \left\{ \vec p^{\,2}, \frac{1}{r} \right\}
\nonumber\\[0.1133ex]
\to & \;
m \left(1 - \frac{r_s}{2 \, r} \right) +
\left( 1 - \frac{3}{2} \, \frac{r_s}{r} \right) \,
\frac{\vec p^{\,2}}{2 m} \,,
\end{align}
where we ignore the commutator and specialize the
Hamiltonian to particles as opposed to
anti-particles (i.e., we replace the
Dirac $\beta$ matrix by the unit matrix).
For a central gravitational field, one has
\begin{equation}
T = 1 + 2 \, \Phi = 1 - \frac{r_s}{r} \,,
\qquad
H = 1 - 2 \, \Phi = 1 + \frac{r_s}{r} \,,
\end{equation}
where $r_s = 2 G M$.
To first order in $r_s$, we can thus reformulate the
gravitational dependence as follows,
\begin{equation}
\label{Hkin}
H_{\rm kin} \sim
\sqrt{T} \, m + \frac{\sqrt{T}}{H} \,
\frac{\vec p^{\,2}}{2 m} =
\sqrt{T} \, \left( m +
\frac{\vec p_c^{\,2}}{2 m} \right) \,.
\end{equation}
Here, we have transformed the momentum operator
to local Cartesian coordinates, as follows,
\begin{equation}
p_c^j
= -\mathrm{i} \frac{\partial}{\partial x_c^j}
= -\mathrm{i} \frac{1}{\sqrt{H}} \frac{\partial}{\partial x^j}
= \frac{1}{\sqrt{H}} p^j \,.
\end{equation}
This implies that the gravitational Breit
term does not contribute to an observable
gravitational energy difference among atomic
energy levels.
The Schr\"{o}dinger Hamiltonian is completed
by adding the Coulomb term
\begin{equation}
\label{Hcoul}
H_{\rm coul} = - \frac{Z e^2}{4 \pi \epsilon \, \rho}
= - \sqrt{\frac{T}{H}} \frac{Z e^2}{4 \pi \, \rho}
= - \sqrt{T} \, \frac{Z e^2}{4 \pi \, \rho_c} \,,
\end{equation}
where $\epsilon = \sqrt{H/T}$ is the gravitationally
modified vacuum permittivity,
$\rho = | \vec \rho |$ is the distance from the
atomic nucleus, and $\rho_c = \sqrt{H} \, \rho$.
It is instructive to compare the scaling
outlined above to the relativistic formalism
used in Sec.~\ref{sec2B}.
Adding the kinetic term from Eq.~\eqref{Hkin}
and the Coulomb term given in Eq.~\eqref{Hcoul},
and subtracting the rest mass term,
which is irrelevant for atomic transitions,
one obtains the gravitationally
modified Schr\"{o}dinger Hamiltonian
\begin{equation}
\label{HST}
H_S = \sqrt{T} \, \left(
\frac{\vec p_c^{\,2}}{2 m}
- \frac{Z \alpha }{\rho_c} \right) \,,
\end{equation}
where $\alpha = e^2/(4 \pi)$ is the fine-structure
constant.
Interestingly, one could hypothesize about the
physical consequences of the gravitational Breit
term for high-precision atomic clocks~\cite{HiEtAl2013,HuEtAl2016},
which currently operate on a precision level of $10^{-18}$
or better, if the Breit term were to contribute
to an observable energy difference and the
Penrose conjecture were to hold in the renormalized
form~\eqref{penrose_ren}.
In this case, the renormalized gravitational
energy difference~\eqref{penrose_ren}
among the different atomic levels
involved in the atomic clock transition,
in view of $r_s/r \sim 10^{-9}$ for the Earth,
would result in gravitational collapse of the
atomic state on a relative frequency level of
$10^{-9}$, which is the ratio of the
gravitational Breit term to the
leading nonrelativistic kinetic term in the atomic
Hamiltonian [$\vec p^{\,2}/(2m)$].
This would prevent continuous
interrogation of the atomic clock and thus make
the experiments~\cite{HiEtAl2013,HuEtAl2016}
(and also the clock comparison,
see Refs.~\cite{GoEtAl2014,HuEtAl2014}) infeasible,
because the hypothetical gravitational effect
would limit the clock precision to a
level of $10^{-9}$. Indeed, the Yb$^+$ clock
transitions described in Refs.~\cite{HiEtAl2013,HuEtAl2016}
involve atomic transitions with a change in the
principal quantum number, and thus, the expectation value
of the $\vec p^{\,2}$ operator becomes state dependent.
This hypothetical consideration is included here
in order to illustrate that care is required in the
treatment of the gravitational terms;
one can easily be fooled into obtaining
excessively large effects if one does not
carry through the analysis correctly
(see also
Refs.~\cite{MoFuSh2018ptep,MoFuSh2018remark1,MoFuSh2018remark2,Vi2018,Ni2018,Gu2018}).
\begin{table*}[ht!]
\begin{center}
\begin{minipage}{0.9\linewidth}
\begin{center}
\caption{\label{table1} Order-of-magnitude estimate
(rows 1, 2, 4, 5) and numerical values (remaining rows)
of the quantum limitations of the EEP, for the effects
$\delta E^{(i)}$, $\delta E^{(ii)}$, $\delta E^{(iii)}$
$\delta E^{(iv)}$ as described in the text
[see Eqs.~\eqref{dEi},~\eqref{dEii}, and~\eqref{dEiii}].
The shifts are
evaluated for astrophysical objects of interest.
The column labeled ``Earth due to Sun'' is
included because, despite the large distance
from the Earth to the Sun (about $146 \times 10^9 \, {\rm m}$),
the large solar mass of about $M_\odot = 1.989 \times 10^{30} \, {\rm kg}$
could be assumed to lead to large
gravitational shifts. However, because of the
suppression of the gravitational effects by
$R^{-n}$, with $n \geq 2$, the effects due to the Sun
are numerically suppressed.}
\begin{tabular}{l@{\hspace*{0.3cm}}S[table-format=-1.2e-2]@{\hspace*{0.3cm}}%
S[table-format=-1.2e-2]@{\hspace*{0.3cm}}S[table-format=-1.2e-2]%
@{\hspace*{0.3cm}}S[table-format=-1.2e-2]}
\hline
\hline
\multicolumn{1}{c}{Effect} &
\multicolumn{1}{c}{Earth} &
\multicolumn{1}{c}{Earth} &
\multicolumn{1}{c}{White} &
\multicolumn{1}{c}{Neutron} \\
& & \multicolumn{1}{c}{Due to Sun} &
\multicolumn{1}{c}{Dwarf} &
\multicolumn{1}{c}{Star} \\
\hline
\rule[-1mm]{0mm}{4mm}
$\delta E^{(i)}/E_h$ [Eq.~\eqref{dEi}, estimate]
& 8.99e-40 & 2.50e-47 & 4.19e-34 & 2.72e-26 \\
\hline
\rule[-1mm]{0mm}{4mm}
$\delta E^{(ii)}/E_h$ [Eq.~\eqref{dEii}, estimate] &
1.17e-44 & 4.74e-51 & 2.55e-33 & 1.06e-22 \\
\rule[-1mm]{0mm}{4mm}
$\delta E^{(ii)}/E_h$ [Eq.~\eqref{dEii_calc}, hydrogen~$2S$] &
1.77e-37 & 7.13e-44 & 3.84e-26 & 1.59e-15 \\
\hline
\rule[-1mm]{0mm}{4mm}
$\delta E^{(iii)}/E_h$ [Eq.~\eqref{dEiii}, estimate] &
4.79e-44 & 1.33e-51 & 2.23e-38 & 1.45e-30 \\
\hline
\rule[-1mm]{0mm}{4mm}
$\delta E^{(iv)}/E_h$ [Eq.~\eqref{Eiv_estim}, estimate] &
1.99e-19 & 1.26e-22 & 9.28e-14 & 1.89e-8 \\
\rule[-1mm]{0mm}{4mm}
$\delta E^{(iv)}/E_h$ [Eq.~\eqref{Eiv_calc}, HF] &
3.17e-20 & 2.02e-23 & 1.48e-14 & 3.01e-9 \\
\rule[-1mm]{0mm}{4mm}
$\delta E^{(iv)}/E_h$ [Eq.~\eqref{Eiv_calc}, N$_2$] &
8.89e-19 & 5.65e-22 & 4.14e-13 & 8.43e-8 \\
\rule[-1mm]{0mm}{4mm}
$\delta E^{(iv)}/E_h$ [Eq.~\eqref{Eiv_calc}, Cl$_2$] &
-1.32e-18 & -8.41e-22 & -6.17e-13 & -1.25e-7 \\
\hline
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{center}
\end{table*}
\subsection{Higher orders
and broken $\maybebm{\sqrt{T}}$ scaling}
\label{sec3B}
\subsubsection{Overview}
\label{sec3B1}
From this consideration,
it becomes obvious that only gravitational
effects on atomic transitions which go beyond the
``common prefactor'' $\sqrt{T}$
[see Eq.~\eqref{common_prefactor}], could lead to
observable consequences (competing effects
are discussed in Apps.~\ref{appB} and~\ref{appC}).
We therefore attempt, for a central gravitational field,
to analyze the leading effects which could
contribute to quantum limitations of the EEP,
in view of a breaking of the $\sqrt{T}$ scaling.
There are three competing effects
to compare and to analyze,
{\em (i)} a first-order plain gravitational shift,
obtained by expanding the Newtonian gravitational
potential over the size of the atom,
{\em (ii)} a second-order gravitational shift,
again obtained on the basis of
the Newtonian gravitational potential,
{\em (iii)}
commutator-induced shifts due to higher-order operators
in the Dirac--Schwarzschild--Coulomb Hamiltonian.
A fourth effect, quite surprisingly,
exists for diatomic molecules.
\subsubsection{Gravitation and size of the atom}
\label{sec3B2}
We denote by $\vec R$ the coordinate of
the atomic nucleus with respect to the
gravitational center, and by $\vec \rho$ the
distance of the electron from the atomic nucleus.
Then, if $\vec \rho$ denotes the vector
from the gravitational center to the atomic electron,
one has
\begin{align}
V =& \; - \frac{m \, r_s}{2 \, r}
= m \, \Phi = -\frac{G m M}{r} \,,
\nonumber\\[0.1133ex]
\frac{1}{r} = & \;
\frac{1}{| \vec R + \vec \rho | } =
\frac{1}{R} - \frac{\vec R \cdot \vec \rho}{R^3}
+ \frac{3 \, (\vec R \cdot \vec \rho)^2 - R^2 \, \rho^2}{2 \, R^5}
\nonumber\\[0.1133ex]
& \; - \vec \rho^{\,2} \, 4 \pi \delta^{(3)}(\vec R) +
{\mathcal{O}}(\rho^3) \,,
\end{align}
where the Dirac-$\delta$ term can be ignored
if the atom is sufficiently
displaced from the point $\vec R = \vec 0$,
which can be safely assumed to be
the case for practically important applications.
One writes
\begin{subequations}
\begin{align}
V =& \; V^{[0]} + V^{[1]} + V^{[2]} \,,
\\[0.1133ex]
V^{[0]} =& \; -\frac{G m M}{R} \,,
\\[0.1133ex]
V^{[1]} =& \; G m M \frac{\vec R \cdot \vec \rho}{R^3}
\propto \Phi^2 \,,
\\[0.1133ex]
V^{[2]} =& \; - \frac{G m M}{2}
\frac{3 \, (\hat R \cdot \vec \rho)^2 - \vec \rho^2}{R^3}
\propto \Phi^3 \,.
\end{align}
\end{subequations}
The term $V^{[0]}$ is absorbed in the scaling factor $\sqrt{T}$
which multiplies the mass term in $H_{\rm kin}$,
as given in Eqs.~\eqref{Hkin_spec} and~\eqref{Hkin}.
The expectation value of the leading correction $V^{[1]}$
vanishes on any atomic energy eigenstate, due to parity.
However, as shown in the following,
nontrivial effects can be expected for diatomic molecules.
The effect scales as $R^{-2}$ and thus is
proportional to $\Phi^2$, where $\Phi = -G M/R$ is the gravitational
potential.
The first nonvanishing correction is due to the
quadrupole term $V^{[2]}$, which scales with $\Phi^3$.
For an atom, $|\vec \rho| \sim a_0$ where $a_0$ is the Bohr radius.
The induced shift is of order
\begin{equation}
\label{dEi}
\delta E^{(i)} = \langle V^{[2]} \rangle \sim
\frac{G \, m \, M \, a_0^2}{R^3}
= 8.99 \times 10^{-40} \, E_h \,,
\end{equation}
where $E_h = \alpha^2 m c^2 \approx 27.2 \, {\rm eV}$ is the Hartree energy
and the shift has been evaluated for the Earth
($M \to M_\oplus$, $R \to R_\oplus$).
For other systems, see Table~\ref{table1}.
The effect is of first order in the
gravitational potential and addresses
point {\em (i)} listed above.
For completeness, we should point out that
we use the following parameters: the Earth mass
$M_{\mbox{{\bf{\tiny +}}}} = 5.974 \times 10^{24} \, {\rm kg}$,
the Sun's mass $M_\odot = 1.989 \times 10^{30} \, {\rm kg}$,
a typical white dwarf mass of $M_{\rm wd} = 1.4 \, M_\odot$,
with a radius of $R_{\rm wd}$ being equal to the
radius of the Earth, $R_{\rm wd} = R_\oplus = 6.378 \times 10^6 \, {\rm m}$,
as well as a neutron star of mass $M_{\rm ns} = 2.8 \, M_\odot$,
and a radius of $R_{\rm ns} = 20 \, {\rm km}$.
The second-order perturbation due to $V^{[1]}$, on an atomic state,
can be expressed as
\begin{align}
\label{dEii}
\delta E^{(ii)} =& \; \left< V^{[1]} \, \frac{1}{(E - H)'} \, V^{[1]} \right>
\nonumber\\[0.1133ex]
\sim & \;
\frac{G^2 \, m \, M^2 \,a_0^2}{\alpha^2 \, c^2 \, R^4}
\sim 1.17 \times 10^{-44} \, E_h \,,
\end{align}
where $[1/(E- H)']$ is the atomic reduced Green function,
and the numerical value is obtained for a point on the
surface of the Earth.
We here assume that there are no quasi-degenerate levels
which are displaced from the atomic reference state
by an energy shift which is far less than a typical
atomic energy level difference of
$E - E_n \sim E_h \equiv \alpha^2 \, m \, c^2$,
where $E$ is the reference-state energy, and
$E_n$ is the virtual-state energy.
The Hartree energy is denoted as $E_h \approx 27.2 \, {\rm eV}$.
In the absence of such quasi-degenerate levels,
the order-of-magnitude estimate
$[1/(E- H)'] \sim 1/E_h$ is valid.
One may consult Table~\ref{table1}
for numerical estimates of $\delta E^{(ii)}$ for other astrophysical
systems. We have thus addressed point {\em (ii)} listed above.
A remark is in order.
The estimate given above in Eq.~\eqref{dEii}
should be taken with a grain of
salt, in part, because quasi-degenerate levels
can otherwise alter the predictions quite drastically.
E.g., for the hydrogen $1S$--$2S$
transition~\cite{FiEtAl2004,PaEtAl2011},
the $2P_{1/2}$ levels are displaced from the
$2S$ state only by the Lamb shift, while
the $2P_{3/2}$ levels are separated by the fine structure.
With the following data
[see Eq.~(42) of~\cite{AdEtAl2017vdWi}]
for the $2S$--$2P_{1/2}$ Lamb shift energy
interval ${\mathcal{L}}$ and the
$2P_{3/2}$--$2P_{1/2}$ fine-structure
interval ${\mathcal{F}}$,
\begin{align}
{\mathcal{L}} =& \; 1.61 \times 10^{-7} \, E_h \,,
\\[0.1133ex]
{\mathcal{F}} =& \; 1.67 \times 10^{-6} \, E_h \,,
\end{align}
we have
[see Eq.~(17) of~\cite{AdEtAl2017vdWi}],
\begin{equation}
\left< z\, \frac{1}{(E - H)'} \, z \right>
= 3 \, a_0^2 \, \left( \frac{1}{{\cal L}} -
\frac{2}{{\cal F}} \right)\,.
\end{equation}
The estimates in the second row of
Table~\ref{table1} should thus be
multiplied by a factor
\begin{equation}
\label{dEii_calc}
3 \left( \frac{1}{{\mathcal{L}}} - \frac{2}{{\mathcal{F}}} \right) =
1.504 \times 10^7 \,,
\end{equation}
to obtain numbers for the hydrogen $2S$ state.
The modified estimates,
adjusted for the hydrogen $2S$ state,
are given in the third row of Table~\ref{table1}.
\begin{table*}[th!]
\begin{center}
\begin{minipage}{0.9\linewidth}
\begin{center}
\caption{\label{table2} Order-of-magnitude estimate
(rows 1, 2, 4, 5) and numerical values (remaining rows) for the
$C_n(M)$ coefficients for those gravitational shifts
of atomic transitions
which break the $\sqrt{T}$ scaling,
as defined in Eq.~\eqref{defCM}.}
\begin{tabular}{l@{\hspace*{0.3cm}}S[table-format=-1.2e-2]@{\hspace*{0.3cm}}%
S[table-format=-1.2e-2]@{\hspace*{0.3cm}}S[table-format=-1.2e-2]%
@{\hspace*{0.3cm}}S[table-format=-1.2e-2]}
\hline
\hline
\multicolumn{1}{c}{Effect} &
\multicolumn{1}{c}{Earth} &
\multicolumn{1}{c}{Earth} &
\multicolumn{1}{c}{White} &
\multicolumn{1}{c}{Neutron} \\
& & \multicolumn{1}{c}{Due to Sun} &
\multicolumn{1}{c}{Dwarf} &
\multicolumn{1}{c}{Star} \\
\hline
\rule[-1mm]{0mm}{4mm}
$C_3(M)$ for $\delta E^{(i)}$ [$\hbar \omega_0 = E_h$]
& 8.99e-40 & 8.11e-51 & 4.14e-51 & 1.03e-51 \\
\hline
\rule[-1mm]{0mm}{4mm}
$C_2(M)$ for $\delta E^{(ii)}$ [$\hbar \omega_0 = E_h$]
& 1.17e-44 & 2.24e-53 & 1.17e-44 & 1.19e-39 \\
\rule[-1mm]{0mm}{4mm}
$C_2(M)$ for $\delta E^{(ii)}$ [hydrogen~$2S$, $\omega_0 = \omega_{1S2S}$] &
4.71e-37 & 8.99e-46 & 4.71e-37 & 4.79e-32 \\
\hline
\rule[-1mm]{0mm}{4mm}
$C_3(M)$ for $\delta E^{(iii)}$ [$\hbar \omega_0 = E_h$]
& 4.79e-44 & 4.32e-55 & 2.20e-55 & 5.51e-56 \\
\hline
\rule[-1mm]{0mm}{4mm}
$C_2(M)$ for $\delta E^{(iv)}$ [$\hbar \omega_0 = E_h$]
& 1.99e-19 & 5.98e-25 & 4.27e-25 & 2.13e-25 \\
\rule[-1mm]{0mm}{4mm}
$C_2(M)$ for $\delta E^{(iv)}$ [HF, $\omega_0 = \omega_{\rm ioni}$]
& 1.41e-19 & 4.24e-25 & 3.03e-25 & 1.51e-25 \\
\rule[-1mm]{0mm}{4mm}
$C_2(M)$ for $\delta E^{(iv)}$ [N$_2$, $\omega_0 = \omega_{\rm ioni}$]
& 1.55e-18 & 4.66e-24 & 3.33e-24 & 1.66e-24 \\
\rule[-1mm]{0mm}{4mm}
$C_2(M)$ for $\delta E^{(iv)}$ [Cl$_2$, $\omega_0 = \omega_{\rm ioni}$]
& -3.14e-18 & -9.42e-24 & -6.73e-24 & -3.37e-24 \\
\hline
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{center}
\end{table*}
\subsubsection{Fokker precession term}
\label{sec3B3}
The Fokker precession term
\begin{equation}
H_{\rm FP} =
\frac{3 r_s}{8 m} \, \frac{\vec\sigma \cdot \vec L}{R^3}
\end{equation}
in the Dirac--Schwarzschild--Coulomb Hamiltonian~\eqref{HDSFW}
is proportional to $|\Phi|^3$,
where $\Phi = -G M /r$ is the gravitational potential.
This term is generated by the difference of the
exact Foldy--Wouthuysen Hamiltonian, given in Eq.~\eqref{HDSFW},
and the approximate form~\eqref{HDSCproblem},
due to commutators of the momentum operators and the
gravitational potential, while
we had obtained the approximate form~\eqref{HDSCproblem}
by {\em ignoring} the commutators.
It thus goes beyond the terms considered in Ref.~\cite{Wi1974prd},
which exhibit the universal scaling with
$\sqrt{T} = \sqrt{1 + 2 \Phi}$,
and leads to an energy shift of the order of
\begin{equation}
\label{dEiii}
\delta E^{(iii)} = \langle H_{\rm FP} \rangle \sim
\frac{\hbar^2 G M}{m R^3 c^2} =
4.79 \times 10^{-44} \, E_h \,.
\end{equation}
The numerical estimate is obtained for the Earth
($M = M_\oplus$ and $R = R_\oplus$).
Again, one may consult Table~\ref{table1}
for numerical estimates of $\delta E^{(iii)}$ for other astrophysical
systems. We have addressed point {\em (iii)} listed above.
\subsubsection{Atoms and limit of vanishing Bohr radius}
\label{sec3B4}
One might argue that the variation
of the gravitational potential around the
atomic center does not constitute a quantum limitation
of the EEP, because it is simply given
as the expectation value of
a gravitational effect, evaluated on the
atomic wave function.
However, it leads to an observable frequency shift
and to a deviation from the universal $\sqrt{T}$
scaling of the atomic transition frequencies.
The effect would vanish
if the electron could be perfectly localized, which
however is incompatible with fundamental
postulates of quantum mechanics. In particular,
perfect localization of the
electron's wave packet
would be incompatible with Heisenberg's uncertainty principle.
It is instructive to observe that the
energy shifts $\delta E^{(i)}$ and $\delta E^{(ii)}$
vanish in the limit $a_0 \to 0$, which
would correspond to the classical limit of a
perfectly localizable electron.
The energy shifts $\delta E^{(iii)}$,
by contrast, is nonvanishing even in the
limit $a_0$ and constitutes a genuine
quantum correction to the EEP, due to
the Fokker precession acting on the bound
atomic electron.
\subsubsection{Diatomic molecules}
\label{sec3B5}
For a diatomic molecule, the situation is essentially more
interesting because the expectation value
\begin{equation}
\label{Eiv}
\delta E^{(iv)} =
\langle V^{[1]} \rangle =
\left< G m M \frac{\vec R \cdot \vec \rho}{R^3} \right>
\end{equation}
can be nonvanishing.
It is known that diatomic molecules typically have
nonvanishing electric dipole moments~\cite{HuHe1979}.
Indeed, it is known~\cite{TaSi1977} that, e.g.,
hydrogen fluoride (HF) has a dipole moment of
$1.82 \, {\rm D}$, where $\rm D$ denotes the Debye,
which is a canonical unit of an atomic dipole moment,
equal to $0.20819434 \, |e| \, \mbox{\AA}$,
where $|e|$ is the elementary charge.
A calculation using GAUSSIAN 2.0~\cite{Kl2018Priv}
reveals that the hydrogen fluoride ion (HF$^+$)
has a dipole moment of $2.36 \, {\rm D}$
(in units of Debye), which is
measured with respect to the center-of-mass
of the hydrogen fluoride ion
(by convention). However, the {\em electric} dipole moment is of
no significance when it comes to the
evaluation of gravitational corrections.
Namely, for the evaluation of gravitational corrections,
one should consider the fact that the
mass of the atom is concentrated in the
atomic nuclei. The two nuclei in a diatomic
molecule are separated by the bond length.
If the energetically highest molecular orbital
is bonding, then the bond length will increase upon
excitation into energetically higher states,
with the maximum change reached for excitations
close to the ionization threshold.
An example is HF, which has a bond length of
\begin{equation}
\label{ell_start}
\ell_{\rm HF} = 0.917\, \mbox{\AA},
\end{equation}
to be contrasted with HF$^+$, which has a
bond length of
\begin{equation}
\ell_{\rm HF^+} = 1.001\,\mbox{\AA} \,,
\end{equation}
according to Refs.~\cite{TaSi1977,HuHe1979}.
By contrast, if the energetically highest molecular orbital
is anti-bonding, then the bond length will decrease upon
excitation into energetically higher states,
An example is CL$_2$, whose bond length decreases
from
\begin{equation}
\ell_{\rm CL_2} = 1.99\,\mbox{\AA} \qquad \to \qquad
\ell_{\rm CL_2^+} = 1.89\,\mbox{\AA}
\end{equation}
upon ionization into CL$_2^+$
(see Ref.~\cite{HuHe1979}).
For N$_2$, the bond length changes
according to
\begin{equation}
\label{ell_end}
\ell_{\rm N_2} = 1.12\,\mbox{\AA} \qquad \to \qquad
\ell_{\rm N_2^+} = 1.29\,\mbox{\AA} \,.
\end{equation}
We can thus conclude that, in a diatomic molecule,
if we hold the position of one of the nuclei
(mass $m_1$) fixed to the origin, then
there will be an energy correction of the form
\begin{align}
\label{Eiv_estim}
\delta E^{(iv)} =& \;
\langle V^{[1]} \rangle =
\left< G M \frac{m_2 \vec R \cdot \vec L}{R^3} \right>
\nonumber\\[0.1133ex]
\sim & \;
\frac{G \, m_p \, M \, a_0}{R^2} \,.
\end{align}
Here, $\vec L$ is the bond length vector,
$m_2$ is the mass of the respective other nucleus,
while $m_p$ is the proton mass.
In formulating the order-of-magnitude estimate,
we use the proton mass (mass of the nucleus of the hydrogen
atom) as a measure for $m_2$; of course, this
assumption has to be adjusted according to
the molecule under consideration.
The ionization energy of a diatomic molecule
in a gravitational field thus changes according to
\begin{equation}
\label{Eiv_calc}
\delta E^{(iv)} = \frac{G m_2 M \, \Delta \ell}{R^2}
\end{equation}
upon ionization, if the axis of the diatomic molecule
is aligned along the $\vec R$ vector.
Here, $\Delta \ell$ is the change in the bond
length upon ionization.
This is because directly under the ionization
threshold, the bond length will
asymptotically approach that of the ion.
According to the above considerations,
given in Eqs.~\eqref{ell_start}---\eqref{ell_end}, one has
\begin{subequations}
\begin{align}
\Delta \ell_{\rm HF} =& \; 0.084 \, \mbox{\AA}\,,
\\[0.1133ex]
\Delta \ell_{{\rm N}_2} =& \; 0.17 \, \mbox{\AA}\,,
\\[0.1133ex]
\Delta \ell_{{\rm Cl}_2} =& \; -0.10 \, \mbox{\AA}\,.
\end{align}
\end{subequations}
Numerical estimates of the gravitational
effects can be quite large for typical diatomic
For absolute clarity, we should
point out that, for a successful
measurement of the gravitation frequency shift,
the diatomic molecules need to
be aligned with reference to the gravitational field;
of course, the effect vanishes when averaged over an
ensemble of unaligned molecules.
\section{Measurement of the Higher--Order Shifts}
\label{sec4}
According to Table~\ref{table1},
the dominant effects for either the hydrogen
$1S$--$2S$ transition or molecular
transitions are given by the shifts
$\delta E^{(ii)}$ and $\delta E^{(iv)}$.
It is instructive to study their
dependence on the gravitational potential,
and the measurability of the effects.
As evident from Eq.~\eqref{dEii}, the
shift $\delta E^{(ii)}$ can be written as
\begin{equation}
\label{before}
\delta E^{(ii)} =
\left( \frac{\Phi}{\Phi_0} \right)^4 \,
C_4(M) \, (\hbar \omega_{1S2S}) \,,
\end{equation}
where $\omega_{1S2S}$ is the unperturbed $1S$--$2S$
frequency, and $C_4(M)$ is a coefficient whose value
depends on the mass of the gravitational center.
The gravitational potential is $\Phi = -G M/R$,
and we have normalized the potential with respect to
$\Phi_0 = G M_\oplus/R_\oplus$,
where $M_\oplus$ is the Earth mass,
and $R_\oplus$ is the Earth's radius.
Also, $\delta E^{(iv)}$ given in Eq.~\eqref{Eiv_estim}
can be written as
\begin{equation}
\delta E^{(iv)} = \left( \frac{\Phi}{\Phi_0} \right)^2 \,
C_2(M) \, (\hbar \omega_{\rm ioni}) \,,
\end{equation}
where $\omega_{\rm ioni}$ is the
angular unperturbed ionization frequency,
and $C_2(M)$ [which may be different from the coefficient
used in Eq.~\eqref{before}]
is a mass-dependent coefficient.
Let us assume that a general higher-order
gravitational frequency shift,
which limit the validity of the universal $\sqrt{T} $ scaling,
can be written in the functional form
\begin{equation}
\label{defCM}
\delta E =
\left( \frac{|\Phi|}{\Phi_0} \right)^n \,
C_n(M) \, \hbar \omega_0 \,,
\end{equation}
where, for the cases studied above,
one would have either $n = 2,3,4$.
The coefficient $C_n(M)$ depends on the
mass of the gravitational center,
while $\omega_0$ is the unperturbed frequency.
We can thus write a gravitationally corrected
transition energy $E$ as
\begin{align}
E =& \; \sqrt{T} \, \omega_0 + \delta E
\nonumber\\[0.1133ex]
=& \; \left( \sqrt{1 + 2 \Phi} +
| \Phi |^n \, C(M) \right) \, \hbar \omega_0 \,.
\end{align}
In units with $\hbar = 1$, we have
\begin{equation}
E = \frac{\mathrm{d} \theta}{\mathrm{d} t} \,,
\qquad
\omega_0 = \frac{\mathrm{d} \theta}{\mathrm{d} \tau} \,,
\end{equation}
where $t$ is the global coordinate time,
and $\tau$ is the proper time measured by the local
observer, while $\theta$ is the rotation angle
of the oscillation. Then,
\begin{equation}
\frac{\mathrm{d} \tau}{\mathrm{d} t} = \sqrt{1 + 2 \Phi} +
| \Phi |^n \, C_n(M) \,.
\end{equation}
Comparing two atomic clocks at different
altitudes (points labeled $1$ and $2$ in the
gravitational field), which is the essence of relativistic
geodesy~\cite{MaMu2013}, one arrives at the result
\begin{equation}
\label{corr_term}
\frac{\mathrm{d} \tau_1}{\mathrm{d} \tau_2} =
\frac{\sqrt{1 + 2 \Phi_1} + | \Phi_1 |^n \, C_n(M)}%
{\sqrt{1 + 2 \Phi_2} + | \Phi_2 |^n \, C_n(M)} \,.
\end{equation}
We reemphasize that the numerical value of the
coefficient $C_n(M)$, as well as the value of $n$,
are not universal, but depend on the
atomic system and the transition under study
(see Table~\ref{table2}).
The prediction thus is that,
if one expands this result in $\Phi_1$ and $\Phi_2$ to order $n$, then
the coefficients of order less than $n$ will
agree with the expansion of the leading term
$\sqrt{1 + 2 \Phi_1}/\sqrt{1 + 2 \Phi_2}$,
while at order $n$, there will be an additional correction
\begin{align}
\label{expansion}
\frac{\mathrm{d} \tau_1}{\mathrm{d} \tau_2} \sim & \;
\left(
\sum_{k=0}^n \left( \begin{array}{c}
\sfrac12 \\ k \end{array} \right) (2 \Phi_1)^k
\right)
\left(
\sum_{k=0}^n \left( \begin{array}{c}
\sfrac12 \\ k \end{array} \right) (2 \Phi_2)^k
\right)^{-1}
\nonumber\\[0.1133ex]
& \; + C_n(M) \, \left( | \Phi_1 |^n \, - | \Phi_2 |^n \right) \,,
\end{align}
which describes the deviation from Einstein's
equivalence principle.
Here,
\begin{equation}
\left( \begin{array}{c}
n \\ k \end{array} \right) =
\frac{\Gamma(n+1)}{\Gamma(k+1) \, \Gamma(n - k + 1)}
\end{equation}
is the binomial coefficient.
Terms of order $n+1$ and higher in the
gravitational potentials have
been neglected in writing Eq.~\eqref{expansion}.
Numerical results for the coefficient $C_n(M)$
are given in Table~\ref{table2}.
We use the ionization energies
$6.12\,{\rm eV}$ for HF,
$15.58 \, {\rm eV}$ for N$_2$,
and $11.48 \, {\rm eV}$ for Cl$_2$,
as well as the known $1S$--$2S$ frequency
for hydrogen (see Ref.~\cite{PaEtAl2011}).
\section{Conclusions}
\label{sec5}
Let us summarize the main results of the
current, lengthy, paper.
We shall proceed section by section.
In Sec.~\ref{sec2},
we derive generally applicable Hamiltonians
for the combined gravitational-electromagnetic interaction
in a central gravitational field,
which add relativistic corrections to the
leading-order (nonrelativistic) result
[see Eqs.~\eqref{HEM} and~\eqref{HEMplus}].
Furthermore, we show that the interplay of the
gravitationally modified Dirac equation,
and the gravitationally modified vacuum permittivity
and permeability, leads to a value of the fine-structure
constant independent of gravity [see Eq.~\eqref{alphaconst}].
As a result, we confirm (see Ref.~\cite{Wi1974prd}) that
atomic transition energies are (to an excellent approximation)
compatible with the equivalence principle
[see Eq.~\eqref{yes}].
We also derive a universal gravitational scaling
for the electron's $g$ factor, including the
bound-state corrections, and the anomalous magnetic moment
term [see Eq.~\eqref{gJ1S}].
Only a careful consideration of the transformation of the
magnetic-field components from global coordinates to a local
Lorentz frame, restores the validity of the EEP
[see Eq.~\eqref{ottifant}].
In Sec.~\ref{sec3}, we first discuss
gravitational energy shifts which scale with the
universal prefactor $\sqrt{T} = \sqrt{1 + 2 \, \Phi}$.
Our discussion culminates in Eq.~\eqref{HST},
where we derive the gravitationally
corrected Schr\"{o}dinger--Coulomb Hamiltonian,
to complement Eq.~\eqref{yes}.
Furthermore, we treat
four effects which go beyond the
universal prefactor $\sqrt{T}$,
and which, therefore, in the
language of Ref.~\cite{Wi1974prd},
limit the compliance of transition frequencies
with the Einstein equivalence principle.
These effects are mainly caused by the
non-deterministic nature of quantum mechanics,
which prevents us from perfectly localizing an
electron at a given point in time,
as described by Heisenberg's uncertainty principle.
Specifically, we have an energy correction $\delta E^{(i)}$,
due to a quadrupole term in the gravitational field,
given in Eq.~\eqref{dEi}, which leads to
a nontrivial effect due to
the nonvanishing extent of the quantum mechanical
wave function. A second correction
$\delta E^{(ii)}$ is due to a second-order effect
involving the dipole expansion about the
gravitational center of the atom,
and $\delta E^{(iii)}$ is described by
the Fokker precession term.
One notices that the energy shift $\delta E^{(iii)}$ does not vanish
in the limit $a_0 \to 0$.
The effect thus does not require the
gravitational field to change significantly over the
dimension of the atom, at variance with a remark
issued in the text following Eq.~(12.13) of Ref.~\cite{Pa1980prd}.
Then, for diatomic molecules, quite remarkably,
the dipole term $\delta E^{(iv)}$ due to first-order
perturbation theory
involves the dipole expansion about the
gravitational center of the atom;
its expectation value does not vanish and leads to a direction-dependent
energy shift. Numerical values
for the energy shifts $\delta E^{(i)}$,
$\delta E^{(ii)}$, $\delta E^{(iii)}$, and $\delta E^{(iv)}$,
are given in Table~\ref{table1}.
In Sec.~\ref{sec4}, we discuss the
measurability of the gravitational shifts
in atomic-clock comparisons.
One first observes that the energy shifts $\delta E^{(i)}$,
$\delta E^{(ii)}$, $\delta E^{(iii)}$, and $\delta E^{(iv)}$,
which limit the validity of the $\sqrt{T}$ scaling,
have a functional $| \Phi |^n$ dependence, where $\Phi$ is the
gravitational potential.
They thus lead to a correction term
in the atomic-clock comparison, as given in
in Eq.~\eqref{corr_term},
which could in principle be measured in an
accurate comparison of atomic clocks
running at places with different gravitational potentials.
Equation~\eqref{corr_term} is one of the main
results of the current paper.
Data for the $C_n(M)$ coefficients,
which enter Eq.~\eqref{defCM},
are given in Table~\ref{table2}.
One should remember that the conclusions of
Ref.~\cite{Wi1974prd} crucially depend on the approximation that
commutator terms between the gravitational couplings and the kinetic operators
in the Hamiltonian can be neglected. Only under this assumption can
the fundamental $\sqrt{T}$ scaling of the atomic energy levels be
derived. Here,
we go beyond this approximation and quantify those effects which do not
follow the universal $\sqrt{T}$ scaling.
We reemphasize that the Fokker precession term
does not vanish in the limit of a pointlike atom (vanishing Bohr radius),
and leads to a manifest deviation
of the gravitational modification of
atomic transition frequencies from the fundamental $\sqrt{T}$ scaling,
which is otherwise crucial in establishing the compatibility
of high-precision spectroscopy experiments
with the equivalence principle~\cite{Wi1974prd}.
The tiny gravitational corrections beyond the
$\sqrt{T}$ scaling should be compared to
effects due to space-time noncommutativity~\cite{SeWi1999,ChSJTu2001,DeEtAl2017}
(see App.~\ref{appB}),
and a conceivable limitation of the
achievable accuracy due to a gravitationally
induced collapse of the wave function
(Penrose conjecture, see Refs.~\cite{Pe1996penrose,Pe1998penrose,Pe2014penrose},
see App.~\ref{appC}).
The conclusion is that under reasonable assumptions,
they do not preclude the measurability of the
quantum corrections outlined in Eqs.~\eqref{corr_term} and~\eqref{defCM},
as explained in detail in App.~\ref{appB3} and App.~\ref{appC3}.
In view of seemingly unstoppable progress in high-precision
spectroscopy~\cite{PrEtAl2013}, the effects could be
of phenomenological relevance sooner than otherwise expected.
\centerline{\bf Acknowledgements}
The author acknowledges helpful conversations with
Professor Sir Roger~Penrose, as well as
Professors Clifford M.~Will, Ulrich Bonse and Gregory S.~Adkins.
The formalism outlined in
Ref.~\cite{Wi1974prd} has been an utmost important
inspiration for our studies. This research was
supported by the National Science Foundation (grant PHY--1710856)
and by the Missouri Research Board.
|
3,212,635,537,910 | arxiv | \section{Introduction}
\label{sec:intro}
Nonlinear dynamical systems with non-integrable differential constraints, the so-called nonholonomic systems, have been attracting many researchers and engineers for the last three decades. A theorem in \cite{brockett_dgct83} gave a challenging and negative fact that there does not exist any smooth time-invariant feedback control law to be able to stabilize nonholonomic systems. The applications include various types of robotic vehicles and manipulation. Some of them have been often used as a kind of benchmark platform to demonstrate the performance of a proposed controller for not only a control problem of a single robotic system and also a distributed control problem of multiagent robotic systems.
A V/STOL aircraft without gravity~(\cite{hauser_autom92}), an underactuated manipulator~(\cite{arai_ieee-tra98}), and an underactuated hovercraft~(\cite{he_robotica16}) belong to a class of dynamic nonholonomic systems which are subject to acceleration constraints. The mathematical representation of these systems can be transformed to the second-order chained form by a coordinate and input transformation. The second-order chained form is a canonical form for dynamic nonholonomic systems.
Several control approaches to the second-order chained-form system have been developed so far. Most of them focuses on avoiding the theorem of \cite{brockett_dgct83}. \cite{ge_ijc01} and \cite{he_robotica16} exploit discontinuity in their stabilizing controllers; \cite{deluca_ijrr02} and \cite{aneke_ijrnc03} reduce the control problem into a trajectory tracking problem. Other than those, \cite{yoshikawa_icra00} and \cite{ito_electron19} consider a motion planning problem (in other words, a feedforward control problem).
For motion planning of the second-order chained form system, this paper presents a novel control approach based on switching a state.
The second-order chained form system is divided into three subsystems.
Two of them are the so-called double integrators; the other subsystem is a nonlinear system depending on one of the double integrators.
In other words, the input matrix of the latter subsystem depends on a single state of the double integrators.
The double integrator is linearly controllable, which enables to switch the value of the position state in order to modify the nature of the nonlinear subsystem. Steering the value into one corresponds to modifying the nonlinear subsystem into the double integrator; steering the value into zero corresponds to modifying the nonlinear subsystem into a linear autonomous system. This nature is the basis of the proposed control approach.
The proposed approach is composed of such state-switching and also sinusoidal control inputs. Its effectiveness is validated by a simulation result.
\section{Subsystem Decomposition of the Second-Order Chained Form System}
\label{sec:sub-syst-decomp_socf-sys}
Consider the following second-order chained form system:
\begin{equation}
\label{eq:2nd-cf}
\od{{}^2}{t^2}\bm{\xi} =
\begin{bmatrix}
1 & 0\\
0 & 1\\
\xi_2 & 0
\end{bmatrix}
\bm{u},
\end{equation}
where $\bm{\xi} := \tp{[\, \xi_1,\, \xi_2,\, \xi_3 \,]}$ and
$\bm{u} = \tp{[\, u_1,\, u_2 \,]}$.
By defining a state vector as
$\bm{z} = \tp{[\, z_1,\, \dotsc, z_6 \,]} :=
\tp{[\, \tp{\bm{\xi}},\, \tp{\dot{\bm{\xi}}} \,]}$,
the system~\eqref{eq:2nd-cf} is described as
\begin{equation}
\label{eq:affine-sys}
\od{}{t} \bm{z} =
\begin{bmatrix}
z_4\\ z_5\\ z_6\\ 0\\ 0\\ 0
\end{bmatrix}
+ \begin{bmatrix}
0\\ 0\\ 0\\ 1\\ 0\\ z_2
\end{bmatrix} u_1
+ \begin{bmatrix}
0\\ 0\\ 0\\ 0\\ 1\\ 0
\end{bmatrix} u_2.
\end{equation}
This affine nonlinear system~\eqref{eq:affine-sys} has equilibrium points
at $\bm{z}_e := \tp{[\, z_1^\star, \, z_2^\star, \, z_3^\star, \:
0, \, 0, \, 0 \,]}$,\ $z_1^\star, z_2^\star, z_3^\star \in \R$
with $u_1 = u_2 = 0$.
By using the theorem of \cite{sussmann_siam-jco87},
we can easily confirm that the system~\eqref{eq:affine-sys}
(or \eqref{eq:2nd-cf}) is small-time local controllable at $\bm{z}_e$.
From the viewpoint to separate the control inputs,
the system~\eqref{eq:2nd-cf} is divided into the following two subsystems:
\begin{subequations}
\label{eq:subsysts}
\begin{align}
\label{eq:1st-subsyst}
\od{}{t}
\begin{bmatrix}
z_2 \\ z_5
\end{bmatrix}
&=
\begin{bmatrix}
0 & 1\\
0 & 0
\end{bmatrix}
\begin{bmatrix}
z_2 \\ z_5
\end{bmatrix}
+
\begin{bmatrix}
0 \\ 1
\end{bmatrix}
u_2,
\\
\label{eq:2nd-subsyst}
\od{}{t}
\begin{bmatrix}
z_1 \\ z_3 \\ z_4 \\ z_6
\end{bmatrix}
&=
\begin{bmatrix}
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{bmatrix}
\begin{bmatrix}
z_1 \\ z_3 \\ z_4 \\ z_6
\end{bmatrix}
+
\begin{bmatrix}
0 \\ 0 \\ 1 \\ z_2
\end{bmatrix}
u_1.
\end{align}
\end{subequations}
The subsystem~\eqref{eq:1st-subsyst} is
a second-order linear system---the so-called double integrator;
the subsystem~\eqref{eq:2nd-subsyst} looks a four-order linear system
but its input matrix~$\bm{b}_2$ depends on a state~$z_2$.
The subsystem~\eqref{eq:2nd-subsyst} is furthermore decomposed to the following subsystems:
\begin{subequations}
\label{eq:sub2systs}
\begin{align}
\label{eq:subsyst11}
\od{}{t}
\begin{bmatrix}
z_1 \\ z_4
\end{bmatrix}
&=
\begin{bmatrix}
0 & 1\\
0 & 0
\end{bmatrix}
\begin{bmatrix}
z_1 \\ z_4
\end{bmatrix}
+
\begin{bmatrix}
0 \\ 1
\end{bmatrix}
u_1,
\\
\label{eq:subsyst31}
\od{}{t}
\begin{bmatrix}
z_3 \\ z_6
\end{bmatrix}
&=
\begin{bmatrix}
0 & 1\\
0 & 0
\end{bmatrix}
\begin{bmatrix}
z_3 \\ z_6
\end{bmatrix}
+
\begin{bmatrix}
0 \\ z_2
\end{bmatrix}
u_1.
\end{align}
\end{subequations}
The next section introduces the proposed control approach based on
the above-mentioned subsystem decomposition.
\section{State-Switching Control Approach}
\label{sec:stsw-contr-approach}
The section addresses a motion planning problem between equilibrium points of the second-order chained form system~\eqref{eq:2nd-cf}.
The author presents a control approach based on the subsystem decomposition that divides the second-order chained form system~\eqref{eq:2nd-cf} into the three subsystems~\eqref{eq:1st-subsyst}, \eqref{eq:subsyst11}, and \eqref{eq:subsyst31}.
The input matrix of the subsystem~\eqref{eq:subsyst31} depends on a state of the subsystem~\eqref{eq:1st-subsyst}, $z_2$.
The state $z_2$ can be constant by a control input $u_2$ because the subsystem~\eqref{eq:1st-subsyst} is linear controllable.
Now let us consider that $z_2$ between $0$ and $1$ is switched.
When $z_2$ is equivalent to $1$, the subsystems~\eqref{eq:subsyst11} and \eqref{eq:subsyst31} are completely same under a shared control input~$u_1$.
The shared control input brings that the state differences between those controlled subsystems then depend on their values at the moment when $z_2$ just becomes $1$
In meanwhile, when $z_2$ is equivalent to $0$, the subsystem~\eqref{eq:subsyst11} can be controlled independently of another subsystem~\eqref{eq:subsyst31}.
From this perspective, the author conceived the following basic maneuver for the above-mentioned motion planning problem:
\begin{enumerate}[
label=\texttt{Step \arabic{enumi}}),%
leftmargin=5em,labelsep=.5em,itemsep=0em%
]
\item
Steer $z_2$ from any initial value to $1$ by using $\func{u_1}{t} = 0,\ \func{u_2}{t} = \func{q_1}{t}$;
\item
Steer $z_3$ from any initial value to any desired value (in conjunction with it, $z_1$ is also driven) by using $\func{u_1}{t} = \func{q_2}{t},\ \func{u_2}{t} = 0$;
\item
Steer $z_2$ from $1$ to $0$ by using $\func{u_1}{t} = 0,\ \func{u_2}{t} = \func{q_3}{t}$;
\item
Steer $z_1$ from a value to any desired value by using $\func{u_1}{t} = \func{q_4}{t},\ \func{u_2}{t} = 0$;
\item
Steer $z_2$ from $0$ to any desired value by using $u_1 = 0,\ u_2 = q_5(t)$,
\end{enumerate}
where $q_i(t)$ is an appropriate sinusoidal function in the $i$-th phase ($i = 1, 2, \dotsc, 5$).
Note that sinusoidal steering is inspired by the result of \cite{ito_electron19}.
\begin{rem}
Some conventional approaches such as in \cite{aneke_ijrnc03} and \cite{hably_cdc14} also exploit similar (but different) subsystem decomposition. Note that those approaches focus on not motion planning but stabilizing (via trajectory tracking).
\end{rem}
\section{Numerical Example}
\label{sec:num-ex}
A numerical example is shown to demonstrate the effectiveness of the proposed control approach.
Consider a motion planning problem between $\func{\bm{z}}{0} = \tp{[\, 3,\, 0.5,\, 1,\; 0,\, 0,\, 0 \,]}$ and $\func{\bm{z}}{4T} = \tp{[\, 0,\, 0,\, 0,\; 0,\, 0,\, 0 \,]}$.
The basic maneuver is adjusted as follows:
\begin{enumerate}[
label=\texttt{Step \arabic{enumi}}),%
leftmargin=5em,labelsep=.5em,itemsep=0em%
]
\item
Steer $z_2$ using from $0.5$ to $1$;
\item
Steer $z_3$ from $1$ to $0$;
\item
Steer $z_2$ from $1$ to $0$;
\item
Steer $z_1$ from a resultant value in \texttt{Step 2} to $0$.
\end{enumerate}
To execute this procedure, the following control inputs were adopted:
\begin{equation}
\label{eq:contr-input4case2b}
\left\{%
\begin{array}{@{\,}ll@{\,}}
\func{u_1}{t} = 0 \;\;\mbox{and} &\\[.25em]
\multicolumn{1}{r}{%
\qquad\func{u_2}{t} = \phantom{-}a_1 \omega^2 \sin{\omega t}},
\quad &\mbox{for}\;\; t \in [0, T],
\\[.75em]
\func{u_1}{t} = -a_2 \omega^2 \sin{\omega t} &\\[.25em]
\multicolumn{1}{r}{%
\mbox{and}\quad \func{u_2}{t} = 0},
\quad &\mbox{for}\;\; t \in (T, 2T],
\\[.75em]
\func{u_1}{t} = 0 \;\;\mbox{and} &\\[.25em]
\multicolumn{1}{r}{%
\func{u_2}{t} = -a_3 \omega^2 \sin{\omega t}},
\quad &\mbox{for}\;\; t \in (2T, 3T],
\\[.75em]
\func{u_1}{t} = -a_4 \omega^2 \sin{\omega t} &\\[.25em]
\multicolumn{1}{r}{%
\mbox{and}\quad \func{u_2}{t} = 0},
\quad &\mbox{for}\;\; t \in (3T, 4T].
\end{array}
\right.
\end{equation}
\refig[Figure]{case2b_time-plots} depicts
a simulation result with $\omega = 2\pi$, $T = 1$, $a_1 = 0.5/(2\pi)$, $a_2 = a_3 = 1/(2\pi)$, and $a_4 = 2/(2\pi)$.
It is obvious that the desired motion is successfully planned. The proposed control approach, therefore, was confirmed to be useful for motion planning.
The results of the conventional approaches such as in \cite{aneke_ijrnc03} and \cite{hably_cdc14} imply that the proposed approach can be (partially) combined with stabilization and trajectory tracking.
\begin{figure}[b]
\centering
\includegraphics[width=.5\textwidth]{./figs/case2b_t-all}
\caption{Time plots.}
\label{fig:case2b_time-plots}
\end{figure}
\section{Conclusion}
\label{sec:conc}
For a motion planning problem of the second-order chained form system, this paper has proposed a state-switching control approach based on subsystem decomposition.
The subsystem decomposition divides the second-order chained form system to three subsystems.
One of the subsystems has the input matrix that depends on a state of the other subsystem.
Switching the state between one and zero modifies the nature of the associated subsystem.
This is the key point of the proposed control approach.
The effectiveness of the proposed approach was shown by the simulation result.
Future directions of this study are:
\begin{itemize}
\item
to compare the proposed approach with the other related ones;
\item
to investigate further properties of the proposed approach; and
\item
to extend the second-order chained form system into
the higher-order one.
\end{itemize}
|
3,212,635,537,911 | arxiv |
\section{Introduction}
The non-Gaussian nature of cosmic large-scale structure (LSS) implies that statistics beyond the power spectrum carry significant information about the universe's origin, evolution, and constituents. The volume and precision of the data from upcoming galaxy surveys, such as, PFS\footnote{\url{https://pfs.ipmu.jp/}} \cite{PFSTeam:2012fqu}, DESI\footnote{\url{https://www.desi.lbl.gov}} \cite{DESI:2016fyo}, SPHEREx\footnote{\url{https://www.jpl.nasa.gov/missions/spherex}} \cite{Dore:2014cca}, Euclid\footnote{\url{https://www.euclid-ec.org}} \cite{Amendola:2016saw}, and Roman Space Telescope\footnote{\url{https://roman.gsfc.nasa.gov/}} \cite{Wang:2021oec} will allow for higher signal-to-noise measurement of higher-order statistics such as the bispectrum. However, in addition to challenges in accounting for observational effects, modeling the signal and the covariance estimation of the galaxy bispectrum is rather challenging. Therefore, in addition to a growing number of recent works on extracting information from the LSS bispectrum\footnote{See \cite{Philcox:2021hbm,Gualdi:2022kwz,Hou:2022wfj,Philcox:2022hkh} for recent works on galaxy 4-point statistics, \cite{Valogiannis:2021chp,Eickenberg:2022qvy, Valogiannis:2022xwu} for statistics beyond $n$-point correlation functions using wavelets, and \cite{White:2016yhs,Massara:2020pli,Massara:2022zrf,Bonnaire:2021sie,Paillas:2022wob} for summary statistics capturing the environmental dependence of clustering.} (e.g. \cite{Gil-Marin:2016wya,Hahn:2019zob, Hahn:2020lou,Gualdi:2021yvq,MoradinezhadDizgah:2020whw,Eggemeier:2021cam,Philcox:2021kcw,Ivanov:2021kcd,Cabass:2022wjy,Cabass:2022ymb,Philcox:2022frc,DAmico:2022gki,DAmico:2022osl,Coulton:2022rir}), several alternative and more efficient estimators that compress the information in the bispectrum via some weighted averaging, have been proposed in the literature \cite{Fergusson:2010ia,Schmittfull:2012hq,Schmittfull:2014tca,Chiang:2015pwa,Eggemeier:2015ifa,Eggemeier:2016asq,Wolstenhulme:2014cla,Byun:2017fkz}.
Among the proposed statistics, the weighted galaxy skew spectra \cite{Schmittfull:2014tca,MoradinezhadDizgah:2019xun,Dai:2020adm,Schmittfull:2020hoi,Chakraborty:2022aok} are simple and computationally efficient proxy statistics for the galaxy bispectrum, constructed from the cross-correlation of the observed galaxy density field with an appropriately weighted square of it\footnote{Optimally weighted skew spectra in harmonic space, which saturate the Cramer-Rao bound of the bispectrum in the weakly non-Gaussian limit, were first introduced in the context of constraining primordial non-Gaussianity from the CMB data \cite{Komatsu:2003iq,Munshi:2009ik} (see \cite{Munshi:2020ofi,Munshi:2021uwn,Chakraborty:2022aok} for some recent works on the harmonic-space skew and kurto spectra).}. Since the forms of the skew spectra are derived from the maximum likelihood estimator of the amplitude of the bispectrum \cite{Schmittfull:2014tca}, in the limit of weak non-Gaussianity, they carry the same Fisher information as the full bispectrum for parameters that appear as overall amplitudes of the separable contributions of tree-level bispectrum. Therefore they represent a lossless compression of the bispectrum for constraining the galaxy biases, the logarithmic growth rate, and the amplitude of the primordial power spectrum and bispectrum\footnote{Analogous estimators for trispectrum, often referred to as kurto spectra, involve correlations of two quadratic fields or that of a cubic and linear field \cite{Abidi:2018eyd,Munshi:2021uwn}.} \cite{MoradinezhadDizgah:2019xun}. In real space, and considering Gaussian initial conditions (GICs), the list of quadratic fields include a squared galaxy density, a tidal term, and a shift term \cite{Schmittfull:2014tca}. While the first two depend on nonlinear biases $b_2$ and $b_{\mathcal G_2}$, the third only depends on linear bias $b_1$. Therefore, the corresponding three skew spectra optimally capture the bispectrum information on the three galaxy biases. In the presence of primordial non-Gaussianity, additional skew spectra should be constructed \cite{Schmittfull:2014tca}. For instance, the correlation of galaxy density filtered by the inverse of the matter transfer function with a weighted square of it captures the imprint of primordial bispectrum on matter bispectrum. The weights, in this case, are determined by the shape of the primordial bispectrum \cite{,MoradinezhadDizgah:2019xun}. In redshift space (and for GICs), to capture the full information of the bispectrum on amplitude-like parameters, one needs a total of fourteen skew spectra, the forms of which are determined by the tree-level bispectrum in redshift-space \cite{Schmittfull:2020hoi}. In this case, each of the spectra has a different dependence on the logarithmic growth rate, the amplitude of fluctuations, and galaxy biases.
There are several advantages to using the skew spectra as opposed to the bispectrum; first, they are simple to interpret since they are functions of a single wavenumber and not triangle shapes. Second, the computational cost of measuring them from the data is comparable to that of the power spectrum. While accounting for all bispectrum triangles requires ${\mathcal O}(N^2)$ operations\footnote{See \cite{Scoccimarro:2015bla,Slepian:2015qwa} for fast estimators of Fourier- and configuration-space 3-point statistics.}, capturing the full information of the bispectrum using the skew spectra requires ${\mathcal O}(N \log N)$ operations, where $N = (k_{\rm max}/\Delta k)^3$ is the number of 3D Fourier-space grid points at which the fields are evaluated given a small-scale cutoff of $k_{\rm max}$. Third, estimating the covariance matrices of the skew spectra from mocks requires a significantly smaller number of mocks\footnote{For a given data vector estimating the covariance from the mocks requires a significantly larger number of realizations than the length of the data vector. Several proposals are made in the literature to speed up and decrease the number of required simulations for the estimation of bispectrum covariance \cite{Joachimi:2016xhk,Hall:2018umb,Chartier:2021frd,Philcox:2020zyp}.}. Lastly, in comparison to the bispectrum \cite{Pardede:2022udo}, accounting for the survey window function is expected to be considerably simpler for the skew spectra \cite{Hou:2022aaa}.
Previously, the perturbative models of the weighted skew spectra in real and redshift space have been tested against N-body simulations \cite{Schmittfull:2014tca, Schmittfull:2020hoi}, and their constraining power for amplitude-like model parameters in real space has been investigated \cite{Schmittfull:2014tca,MoradinezhadDizgah:2019xun}. However, the full information content of the galaxy skew spectra for constraining cosmological parameters has not been explored before\footnote{See also Ref. \cite{Dai:2020adm} for forecasts for a small subset of skew spectra, corresponding to the $\langle \delta^2 \delta \rangle$ correlation.}. In this paper, we investigate how much the complete set of redshift-space galaxy skew spectra, introduced in Ref. \cite{Schmittfull:2020hoi}, can improve the cosmological information from the power spectrum multipoles and how the improvement compares with that of the bispectrum. For this purpose, we use simulated datasets that are constructed to perform numerical Fisher forecasts. Since the forecasts do not rely on utilizing theoretical models of the observables (the derivatives and covariance matrices are directly measured from simulations), we can explore the information encoded on the smaller-scale fluctuations, which normally are excluded since the perturbative models of the observables fail as one approaches the non-linear scales.
We first focus on the Fisher forecasts using the halo catalogs from the Quijote suite of simulations \cite{Villaescusa-Navarro:2019bje} \href{https://github.com/franciscovillaescusa/Quijote-simulations}{\faGithub}\footnote{\url{ https://github.com/franciscovillaescusa/Quijote-simulations}}. This analysis can be considered an idealized case, where the halo biases are perfectly known, and only cosmological parameters are varied. To make the analysis more realistic, similar to some of the previous forecasts with this dataset~\cite{Naidoo:2021dxz,Coulton:2022rir}, we vary a single nuisance parameter, the minimum halo mass cut $M_{\rm min}$ as a proxy for unknown halo bias. The halo mass cut captures the difference in the halo number densities and is marginalized over. Next, we go beyond a simple single-parameter proxy of the halo bias
and investigate the impact of marginalization over parameters of a more complex tracer-to-matter biasing relation. For this analysis, we use the Molino galaxy mock catalogs \cite{Hahn:2020lou} \href{https://changhoonhahn.github.io/molino}{\faGlobe}\footnote{\url{ https://changhoonhahn.github.io/molino}}, which are built on Quijote halo catalogs by applying a five-parameter Halo Occupation Distribution (HOD) prescription. Therefore, they allow us to simultaneously vary cosmological and HOD parameters and study the extent to which the information content of the skew spectra is affected by marginalization. For both datasets, we test the robustness of the numerical forecasts in computing the derivatives~\cite{Coulton:2022rir} as well as the noise due to the estimation of the covariance matrices from a finite number of simulations. We find that on both datasets, the skew spectra provide significant improvements over the power spectrum results. We, thus, advocate the skew spectra as powerful, efficient, and easy-to-interpret summary statistics for extracting non-Gaussian information of the LSS.
The rest of the paper is organized as follows. In \S \ref{sec:th_skewspecs}, we review the theoretical construction of the redshift-space skew spectra of biased tracers and describe the measurement pipeline that we apply to the simulated data. We then briefly describe the synthetic datasets from the Quijote and Molino simulations in \S \ref{sec:sims}, and outline the numerical Fisher forecast methodology in \S \ref{sec:fisher}. We present our results in \S \ref{sec:res}, together with the measurements of the skew spectra, their signal-to-noise ratio, and the forecasted parameter constraints. We draw our conclusions in \S \ref{sec:conc}. We present additional results in a series of appendices pertaining to the information in individual skew spectra \S\ref{app:indSkew}, tests of numerical stability of Fisher forecasts \S\ref{app:convergence}, the effect of shot noise subtraction \S\ref{app:shot}, and lastly, the results for Molino galaxies in light of the peculiar characteristics of the sample \S\ref{app:Molino}.
\section{Weighted Skew Spectra in Redshift Space}\label{sec:th_skewspecs}
This section reviews the basic ingredients for constructing the galaxy skew spectra and the perturbative model of its leading contributions, focusing on redshift-space estimators. We also describe the numerical pipeline we use to measure the skew spectra from simulated halo/galaxy catalogs. We refer the interested reader to Ref.~\cite{Schmittfull:2020hoi} for more details on modeling and the comparison of the theory predictions against the simulated dark matter overdensity field.
\subsection{Skew Spectra as Maximum Likelihood Estimators}
For a theoretical model of the galaxy bispectrum, $A_b^{\rm th} B_g^{\rm th}({\bf k}_1,{\bf k}_2,{\bf k}_3)$, the maximum likelihood estimator of its amplitude, $A_b^{\rm th}$, in the weak non-Gaussian limit is given by the projection of the inverse-variance weighted cubic estimator onto the theoretical template \cite{Fergusson:2010ia},
\begin{align}\label{eq:maxlike}
{\hat A}_b^{\rm th} &= \iint\frac{d^3 k\, d^3q}{N_{\rm th}(2\pi)^3} \ \frac{B_g^{\rm th}({\bf q},-{\bf k},{\bf k}-{\bf q})}{P_g^{\rm obs}({\bf k})P_g^{\rm obs}({\bf k}-{\bf q})P_g^{\rm obs}({\bf k})} \notag \\
& \hspace{1in} \times \left[\delta_g^{\rm obs}({\bf q})\delta_g^{\rm obs}({\bf k}-{\bf q})\delta_g^{\rm obs}(-{\bf q}) - 3 \av{\delta_g^{\rm obs}({\bf q})\delta_g^{\rm obs}({\bf k}-{\bf q})} \delta_g^{\rm obs}(-{\bf k})\right].
\end{align}
Here, $N_{\rm th}$ is the normalization which depends on the assumed theory bispectrum, $\delta_g^{\rm obs}$ represents a noisy measurement of the galaxy and $P_g^{\rm obs}$ is the measured power spectrum including the shot noise. The term linear in $\delta_g^{\rm obs}$ is only relevant for the statistically inhomogeneous field and is dropped in the rest of our discussions.
For a theoretical bispectrum that consists of a sum of product-separable terms,
$B_g^{\rm th}({\bf k}_1,{\bf k}_2,{\bf k}_3)\\ =\sum_i f_i({\bf k}_1) g_i({\bf k}_2)h_i({\bf k}_3)$ (as is the case for the leading-order contributions due to gravitational evolution and most types of primordial bispectrum), the above estimator for each separable contribution can be recast in terms of cross-correlations between appropriately filtered quadratic and linear fields,
\begin{align}\label{eq:skew_opt}
{\hat A}_b^{\rm th} &= \int \frac{d^3 k}{N_{\rm th}} \left[\frac{f\delta_g^{\rm obs}}{P_g^{\rm obs}} \star \frac{g\delta_g^{\rm obs}}{P_g^{\rm obs}}\right]({\bf k}) \left[\frac{h \delta_g^{\rm obs}}{P_g^{\rm obs}}\right](-{\bf k}) \notag \\
&= \frac{1}{N_{\rm th}}\int k^2 dk \ {\hat {\mathcal P}}_{\frac{f\delta_g}{P_g}\star \frac{g\delta_g}{P_g}, \frac{h \delta_g}{P_g}}(k).
\end{align}
Here, $\star$ denotes the convolution of two filtered galaxy fields, and we defined $\hat {\mathcal P}$ as the angle-averaged skew spectrum between the filtered quadratic and linear fields. Instead of summing over all Fourier modes to obtain the overall amplitude of the bispectrum, one can retain the shape information and consider a set of skew spectra to extract the information on various contributions to the bispectrum optimally. The estimator in Eq.~\eqref{eq:maxlike} includes the information from all triangles and is unbiased. Therefore, the weighted skew spectra defined in Eq.~\eqref{eq:skew_opt} provide a lossless compression of the (tree-level) bispectrum information for parameters appearing as overall amplitudes of separable contributions to the model bispectrum. In the rest of this paper, as in Ref.~\cite{Schmittfull:2020hoi}, we drop the inverse-variance weighting of the skew spectra estimators and denote the redshift-space skew spectra as $\hat {\mathcal P}_{{\mathcal S}_n {\tilde \delta_g}}(k)$, with ${\mathcal S}_n$ and $\tilde \delta_g$ being the filtered quadratic and linear fields, respectively. We note that when analyzing real data with both shot noise and observational effects, the inverse-variance weighting of the observed density field should be accounted for.
Since the skew spectra involve integration over a wide range of Fourier modes, some of the information from small scales are imprinted on larger scales. The input field should be smoothed to remove the information from small scales (in particular if using perturbative theoretical models of the skew spectra based on tree-level bispectrum). For a tophat filter, smoothing the fields is equivalent to imposing a cutoff $k_{\rm max}$ to ensure that wavevectors above some threshold are not included in the analysis. As in Ref. \cite{Schmittfull:2014tca, Schmittfull:2020hoi}, to avoid edge effects in our measurements, we apply a Gaussian smoothing $W_R(k) = \exp({- k^2R^2/2})$ to the observed density field when computing the quadratic fields.
\subsection{Clustering Component}
To define the redshift-space skew spectra, we consider the tree-level galaxy bispectrum. Neglecting the derivative bias operators and stochastic contributions, up to second order in perturbation theory, the over-density of a biased tracer (such as halos or galaxies denoted generically as $\delta_g$) is given in terms of the matter overdensity and tidal field as \cite{Desjacques:2016bnm},
\begin{equation}
\delta_g({\bf x}) = b_1 \delta({\bf x}) + \frac{1}{2} b_2 \delta^2({\bf x}) + b_{{\mathcal G}_2} {\mathcal G}_2({\bf x}).
\label{eqn:deltag_b1b2bg2}
\end{equation}
Here, $\delta$ refers to matter overdensity and ${\mathcal G}_2$ is the Galileon operator given in terms of the Newtonian gravitational potential $\Phi$ as
\begin{equation}
{\mathcal G}_2(\Phi) \equiv \left(\partial_i \partial_j \Phi \right)^2 - \left(\partial^2 \Phi\right)^2.
\label{eq:Galileon_G2}
\end{equation}
Neglecting the effective field theory counterterms (including fingers-of-god contributions), the tree-level gravitationally induced halo/galaxy bispectrum in redshift space is given by
\begin{equation}\label{eq:rsd_bis}
B_g({\bf k}_1,{\bf k}_2,{\bf k}_3) = 2 Z_1({\bf k}_1) Z_1({\bf k}_2) Z_2({\bf k}_1,{\bf k}_2) P_{\rm lin}(k_1) P_{\rm lin}(k_2) + 2 \ {\rm perms.},
\end{equation}
where $P_{\rm lin}$ is the linear matter power spectrum. The functions $Z_1$ and $Z_2$ are the redshift-space perturbation theory kernels and are given by
\begin{align}
Z_1({\bf k}_1) &= b_1 + f \frac{k^2_{1\parallel}}{k_1^2}, \notag \\
Z_2({\bf k}_1,{\bf k}_2) &= b_1F_2({\bf k}_1,{\bf k}_2) + \frac{b_2}{2} + b_{{\mathcal G}_2} S^2({\bf k}_1,{\bf k}_2)
+ f \frac{k_{\parallel}^2}{k^2} G_2({\bf k}_1,{\bf k}_2)
\nonumber \\
&\quad
+ b_1 f \frac{k_{\parallel}}{2} \left(\frac{k_{1\parallel}}{k_1^2}+\frac{k_{2\parallel}}{k_2^2}\right)
+f^2 \frac{k_{\parallel}}{2}
\left(
\frac{k_{1\parallel}}{k_1^2}\frac{k_{2\parallel}^2}{k_2^2} + \frac{k_{1\parallel}^2}{k_1^2}\frac{k_{2\parallel}}{k_2^2}
\right).
\end{align}
Here, $f$ is the logarithmic growth rate, and $k_{\parallel}$ is parallel to the line-of-sight (LoS) component of wavevector $\bf k$. The $F_2$ and $G_2$ functions are the standard perturbation theory kernels for matter density and velocity, while $S^2$ is the Fourier transform of the Galileon operator,
\begin{equation}
S^2({\bf k}_1,{\bf k}_2) \equiv \left(\frac{{\bf k}_1\cdot{\bf k}_2}{k_1 k_2}\right)^2 - 1.
\end{equation}
The tree-level bispectrum in Eq.~\eqref{eq:rsd_bis} can be cast as a sum of separable contributions of the general form $B_n({\bf k}_1,{\bf k}_2,{\bf k}_3) = P_{\rm lin}({\bf k}_1)P_{\rm lin}({\bf k}_2) D_n({\bf k}_1,{\bf k}_2) h({\bf k}_3)$. Therefore, the full information of the bispectrum on galaxy biases, growth rate, and the amplitude of the primordial power spectrum and bispectrum can be fully captured by a set of 14 skew spectra ${\mathcal P}_{{\mathcal S}_n \tilde \delta}$, where ${\mathcal S}_n$ are smoothed fields quadratic in galaxy overdensity (see the definition below). The general form of the expectation value of the skew spectra estimators can be written as
\begin{equation}
{\mathcal P}_{{\mathcal S}_n \tilde \delta}(k) = \int \frac{d\Omega_k}{4\pi} \av {{\mathcal S}_n({\bf k}) \tilde \delta_g(-{\bf k})},
\label{eqn:quad_field}
\end{equation}
where the quadratic fields ${\mathcal S}_n$ of two smoothed fields $\delta_g^R({\bf k}) = W_R(k) \delta_g({\bf k})$ are defined as
\begin{equation}
{\mathcal S}_n({\bf k}) = \int \frac{d^3 q}{(2\pi)^3} D_n({\bf q},{\bf k}-{\bf q}) W_R({\bf q}) W_R({\bf k}-{\bf q}) \delta_g({\bf q})\delta_g({\bf k}-{\bf q}),
\label{eq:ss_definition}
\end{equation}
and the filtered field is given by $\tilde \delta_g({\bf k}) = h({\bf k}) \delta_g({\bf k})$. In redshift space, the kernels $D_n$ can have a dependence on the direction with respect to the LoS. To make the notation simpler, in the rest of the paper, we drop the tilde of the filtered density field.
Each quadratic operator $\mathcal S_n$ picks up a different combination of bias parameters and growth rate $f$. The explicit forms of the skew spectra were obtained in Ref. \cite{Schmittfull:2020hoi}. Here, we duplicate them for completeness,
\begin{align}\label{eq:skew_list}
b_1^3: & \qquad \mathcal S_1 = F_2[\delta,\delta],\\
b_1^2b_2: & \qquad \mathcal S_2 = \delta^2, \\
b_1^2 b_{\mathcal G_2}: & \qquad \mathcal S_{3} = S^2[\delta,\delta],\\
b_1^3f: & \qquad \mathcal S_4 = \hat z_i\hat z_j\,\partial_i\left(\delta\frac{\partial_j}{\nabla^2}\delta\right), \label{eq:S4}\\
b_1^2f: & \qquad \mathcal S_5 = 2F_2[\delta^\parallel,\delta] + G_2^\parallel[\delta,\delta], \\
b_1b_2 f: & \qquad \mathcal S_6 = \delta\delta^\parallel, \\
b_1 b_{\mathcal G_2} f: & \qquad \mathcal S_7 = S^2[\delta,\delta^{\parallel}], \\
b_1^2f^2: & \qquad \mathcal S_8 = \hat z_i\hat z_j\partial_i\left(\delta\frac{\partial_j}{\nabla^2}\delta^\parallel
+2\delta^\parallel \frac{\partial_j}{\nabla^2}\delta
\right),
\\
b_1f^2: & \qquad \mathcal S_9 = F_2[\delta^\parallel,\delta^\parallel] + 2 G_2^\parallel[\delta^\parallel,\delta],\\
b_2f^2: & \qquad \mathcal S_{10} = \big(\delta^\parallel\big)^2,\\
b_{\mathcal G_2} f^2: & \qquad \mathcal S_{11} = S^2(\delta^{\parallel}, \delta^{\parallel}),\\
b_1f^3: & \qquad \mathcal S_{12} = \hat z_i\hat z_j\partial_i\left(\delta^{\parallel\parallel}\frac{\partial_j}{\nabla^2}\delta
+2\delta^\parallel \frac{\partial_j}{\nabla^2}\delta^\parallel
\right),\\
f^3: & \qquad \mathcal S_{13} = G_2^\parallel[\delta^\parallel,\delta^\parallel],\\
f^4: & \qquad \mathcal S_{14} = \hat z_i\hat z_j\partial_i\left(\delta^{\parallel\parallel}\frac{\partial_j}{\nabla^2}\delta^\parallel \right).
\end{align}
In the above expressions, all products are in pixel space. We defined the redshift-space operators,
\begin{align}
\mathcal O^\parallel &=
\hat z_i\hat z_j\frac{\partial_i\partial_j}{\nabla^2}\mathcal O, \\
\mathcal O^{\parallel\parallel} &=
\hat z_i\hat z_j\hat z_m\hat z_n\frac{\partial_i\partial_j\partial_m\partial_n}{\nabla^4}\mathcal O,
\end{align}
and the operators $\mathcal O [a,b]$ that act on arbitrary fields $a$ and $b$,
\begin{equation}
\label{eq:F2Def}
{\mathcal O}[a,b]({\bf k}) \equiv \int\frac{d^3 q}{(2\pi)^3} \,\frac{1}{2}\Big[a({\bf q})b({\bf k}-{\bf q})+b({\bf q})a({\bf k}-{\bf q})\Big] \, {\mathcal O}({\bf q},{\bf k}-{\bf q}).
\end{equation}
\vspace{0.1in}
\subsection{Shot Noise Component}\label{subsec:shot}
In addition to the clustering component described above, the skew spectra receive a shot noise contribution due to the discrete nature of halos and galaxies. Assuming the Poisson shot noise of the bispectrum, it is straightforward to derive the expressions for the skew spectra shot noise. Extending the calculation of real-space shot noise of the skew spectra \cite{Schmittfull:2014tca} to redshift space, for each of the 14 skew spectra, the (Poisson) shot noise is given by
\begin{equation}
{\mathcal P}^{\rm shot}_{{\mathcal S_n}\delta}(k) = \frac{1}{2} \int d\mu_k \left\{ \left[\frac{1}{\bar{n}^2} + \frac{P_g(k,\mu_k)}{\bar{n}}\right] J_{D_n}(k,\mu_k) + \frac{2}{\bar{n}}\tilde{J}_{D_n}(k,\mu_k)\right\},
\label{eq:shot_th}
\end{equation}
where $\mu_k = k_{\parallel}/k$ and
\begin{align}
J_{D_n}(k,\mu_k) &= \int \frac{d^3q}{(2\pi)^3}W_R({\bf q})W_R({\bf k}-{\bf q}) D_n({\bf q}, {\bf k}-{\bf q}), \notag \\
\tilde{J}_{D_n}(k,\mu_k) &= \int \frac{d^3q}{(2\pi)^3} W_R({\bf q})W_R({\bf k}-{\bf q}) D_n({\bf q}, {\bf k}-{\bf q}) P_g(q,\mu_q).
\end{align}
Deviations from the Poisson shot noise can be captured by including additional nuisance parameters in the $1/{\bar n}$ and $1/{\bar n}^2$ terms, in analogy with the bispectrum \cite{Schmittfull:2014tca,MoradinezhadDizgah:2019xun}.
For the theoretical prediction of the shot noise, which we compare with the measured shot noise of the skew spectrum estimator, we assume the shot noise to be Poissonian, and use the linear Kaiser model of the redshift-space halo power spectrum,
\begin{equation}
P_g(k,\mu_k) = (b_1 + f\mu_k^2)^2 P_{\rm lin}(k).
\end{equation}
\subsection{Measurement Pipeline}\label{subsec:pipeline}
To measure the clustering components of the skew-spectra, we use the Python package \textsc{skewspec} \cite{Schmittfull:2020hoi} \href{https://github.com/mschmittfull/skewspec}{\faGithub}\footnote{\url{https://github.com/mschmittfull/skewspec}}, which employs \textsc{nbodykit} \cite{Hand:2017pqn} \href{https://github.com/bccp/nbodykit}{\faGithub}\footnote{\url{https://github.com/bccp/nbodykit}} to compute the quadratic fields ${\mathcal S}_n$ and their cross-spectrum with the input halo/galaxy density fields. While the quadratic fields can be computed efficiently as the product of fields in configuration space at the same location ${\bf x}$, applying the filters in Fourier space is more straightforward than taking derivatives of the real-space data. Therefore, to compute the quadratic fields, we Fourier transform two copies of the input density, apply the filters corresponding to each ${\mathcal S}_n$ by multiplying each copy by appropriate factors in $k$, Fourier transform back to real space, and multiply the two fields there. In order to reduce the statistical noise on the measurements of the skew spectra, we extend the \textsc{skewspec} package to include the three LoS directions and average over the measurements along the three directions.
Furthermore, we implement routines for measuring the shot noise of biased tracers. In performing the numerical Fisher forecasts, while for numerical derivatives, the shot noise cancels at the leading order, it can non-trivially enter the covariance and lead to artificial response when mocks have different number densities (for instance, when applying different halo mass-cut as discussed in figure~\ref{fig:2dcontour_ss_smooth10_shotonxoff_nuisMmin}). Therefore, we account for the contribution of the shot noise of the estimator to the covariance by subtracting the shot noise components from the skew spectra estimators. Below we describe in more detail how the shot noise estimator is constructed.
We define a shot-noise estimator based on the theoretical prediction in Eq.~\eqref{eq:shot_th}.
To refrain from performing convolution in Fourier space, we define a unity field in Fourier space, $\mathbb{I}({\bf k})$, on which the relevant filters are applied. With this, the estimator of the shot noise of the skew spectra, which we apply to simulated halo/galaxy catalogs, is given by
\begin{align}\label{eqn:shot_pipeline}
\av{{\hat {\mathcal P}}^{\rm shot}_{{\mathcal S_n}\delta} (k)}&= \frac{1}{L^3} \int \frac{d\Omega_k}{4\pi} {\tilde {\mathcal O}}^{\bf m''}_{n''}(-{\bf k}) \Biggl\{
\left(\frac{1}{{\bar n}^2}+\frac{P_g({\bf k})}{\bar{n}}\right)\ft{{\mathcal O}^{\bf m m'}_{nn'}({\bf x})} +\ft{\frac{2}{\bar n}{\mathcal Q}^{\bf mm'}_{nn'}({\bf x})} \Biggr\},
\end{align}
where $\mathcal{FT}$ refers to Fourier transform, we use the tilde symbol to denote quantities in Fourier space. Here, $\tilde{{\mathcal O}}^{\bf m}_n({\bf k})$ is the filtered unity field in Fourier space, ${\mathcal O}^{\bf m m'}_{nn'}({\bf x})$ is the filtered quadratic unity field in real space, and ${\mathcal Q}^{\bf mm'}_{nn'}({\bf x})$ is a filtered cubic field built from quadratic density field of the biased tracer and a unity field in real space. Here indices ${\bf m, m',m''}$ are 3-vectors, e.g. ${\bf m}=(m_x,m_y,m_z)$, and $n',n'$ and $n''$ are real numbers. These parameters specify the filtering of the unity and halo density fields by
\begin{align}
{\tilde {\mathcal O}}_n^{\bf m}({\bf k}) &\equiv k^n {\bf k}^{\bf m} \mathbb{I}({\bf k}), \notag \\
{\tilde {\mathcal Q}}_n^{\bf m}({\bf k}) &\equiv k^n {\bf k}^{\bf m} {\mathcal Q}({\bf k}),
\end{align}
where the filtered 3D power spectrum ${\mathcal Q}({\bf k})\equiv W_R({\bf k})P_g({\bf k})$. We define the element-wise exponentiation of a 3D vector as
\begin{equation}
{\bf k}^{\bf m} \equiv (k_x)^{m_x} (k_y)^{m_y} (k_z)^{m_z},
\end{equation}
where e.g. $(k_x)^{m_x}$ denotes the $x$-component of ${\bf k}$ raised to the power $m_x$. The powers $k^n$ are the square of the magnitude of the wavevectors. The quadratic fields are defined as
\begin{align}
{\tilde {\mathcal O}}_{nn'}^{\bf m m'}({\bf k}) &\equiv \int \frac{d^3q}{2\pi^3} {\tilde {\mathcal O}}_n^{\bf m}({\bf q}) {\tilde {\mathcal O}}_{n'}^{\bf m'}({\bf k}-{\bf q}) \notag \\
\tilde{{\mathcal Q}}_{nn'}^{\bf m m'}({\bf k}) &\equiv \int \frac{d^3q}{2\pi^3} {\tilde {\mathcal Q}}_n^{\bf m}({\bf q}) {\tilde {\mathcal O}}_{n'}^{\bf m'}({\bf k}-{\bf q}).
\end{align}
These convolutions in Fourier space can be evaluated by taking the Fourier transform of the real-space products of the fields,
\begin{align}
{\mathcal O}_{nn'}^{\bf m m'}({\bf x}) &= {\mathcal O}_{n}^{\bf m}({\bf x}) {\mathcal O}_{n'}^{\bf m'}({\bf x}) \notag \\
{\mathcal Q}_{nn'}^{\bf m m'}({\bf x}) &= {\mathcal Q}_{n}^{\bf m}({\bf x}) {\mathcal O}_{n'}^{\bf m'}({\bf x})
\end{align}
In the current implementation, measuring all 14 skew-spectra on a single CPU core takes about four minutes; the most time-consuming calculations are those involving the tidal Galileon operators, including $\mathcal{P}_{{\mathcal S}_3\delta}$, $\mathcal{P}_{{\mathcal S}_7\delta}$, and $\mathcal{P}_{{\mathcal S}_{11}\delta}$ since they involve a nested loop over the three LoS directions. The shot noise measurements require almost the same amount of time. Although the quadratic unity fields only need to be computed once for the first two contributions in Eq.~\eqref{eqn:shot_pipeline}, the last term requires the operator to be applied to a 3D power spectrum-like field and a unity field, whereas the power spectrum varies from mocks to mocks.
\section{Synthetic Datasets}\label{sec:sims}
In this section, we review the characteristics of the two sets of simulations that we use in our Forecasts, the Quijote halo catalogs~\cite{Villaescusa-Navarro:2019bje}, and the Molino galaxy catalogs~\cite{Hahn:2020lou}. We refer to the corresponding references for more details.
\subsection{N-body Quijote Halo Catalogs}
The Quijote simulation suite \cite{Villaescusa-Navarro:2019bje} consists of 44,000 N-body simulations run with several different cosmological models to perform numerical Fisher forecasts and as training data for machine learning applications. The simulations have a box size of $L_{\rm box}=1 \ {h^{-1}\rm{Gpc}}$ and are run with Gadget-III TreePM+SPH code~\cite{Springel:2005mi}, evolving the particles from initial condition at redshift $z=127$ to $z=0$. In this work, we use $15,000$ realizations of Quijote Suite with fiducial cosmology at ${\bm \theta}=\{\Omega_{\rm m}, \Omega_{\rm b}, h, n_s, \sigma_8\} = \{0.3175, 0.049, 0.6711, 0.9624, 0.834\}$ with total neutrino mass being zero. For each parameter we use a pair of 500 realizations at a step lower $\Delta\theta_i^-=\theta_i^--\theta_i$ and higher $\Delta\theta_i^+=\theta_i^+-\theta_i$ compared to the fiducial set. Additionally, we use a sets of simulations including massive neutrinos $\{M_{\nu}^+, M_{\nu}^{++}, M_{\nu}^{+++}\}$. The simulations with massless neutrinos have $512^3$ dark matter particles, and they were generated using initial conditions with second-order perturbation theory (2LPT) to compute particle displacements and peculiar velocities for a given input matter power spectrum. The simulations with massive neutrinos use the Zel'dovich approximation for generating the initial condition, where the $512^3$ neutrino particles and $512^3$ dark matter particles are both treated as collisionless and pressureless fluids. The Friend-of-Friend (FoF) halo finding algorithm \cite{Davis:1985rj} is used to identify halos, assuming a linking length of $b=0.2$. The step size for cosmological parameters are given by $\{\Delta\Omega_{\rm m}, \Delta\Omega_{\rm b}, \Delta h, \Delta n_s, \Delta \sigma_8\}=\{0.02, 0.004, 0.04, 0.04, 0.03\}$, with $\Delta\theta_i \equiv \Delta\theta_i^+-\Delta\theta_i^-$.
We apply a halo mass cut of $M_{\rm min}>3.2 \times 10^{13} M_{\odot}$, which corresponds to a mean halo density of $\bar{n}=1.55 \times 10^{-4}\, h^3 {\rm Mpc}^{-3}$. As in Ref. \cite{Coulton:2022rir}, for computing the derivatives, we take the average of the measurement along the three LoS directions, while for the covariance matrices, we only use the projection along a single LoS to avoid biasing the covariance estimation by the correlations between different LoS directions.
\subsection{HOD-based Molino Galaxy Catalogs}
The Molino suite of galaxy catalogs~\cite{Hahn:2020lou} consists of 75,000 mocks that are constructed upon the application of a standard 5-parameter HOD model to Quijote halo catalogs. In general, a HOD model provides a prescription for building the relationship between the underlying dark matter and the biased tracers; the halos are populated by galaxies such that the mean number of galaxies is given by the sum of central and satellite galaxies,
\begin{equation}
\langle N_{\rm gal} \rangle = \langle N_{\rm cent} \rangle + \langle N_{\rm sat} \rangle.
\end{equation}
In the standard five-parameter HOD model \cite{Zheng:2007zg}, used in Molino catalogs, the mean occupation function of the central galaxies can be described by a step-like function with a soft cutoff profile to account for the scatter between galaxy luminosity and host halo mass. The mean satellite galaxy occupation is often modeled as a power law form at high halo masses (with the slope close to unity) and drops steeper than the power law at lower masses;
\begin{equation}
\langle N_{\rm cent} \rangle = \frac{1}{2}\left[1+{\rm erf}\left(\frac{\log M_h - \log M_{\rm min}}{\sigma_{\log M}} \right)\right], \qquad \langle N_{\rm sat} \rangle = \langle N_{\rm cent} \rangle\left(\frac{M_h-M_0}{M_1}\right)^\alpha.
\end{equation}
The two free parameters of the central distribution are the minimum mass scale of halos that can host central galaxies above the luminosity threshold $M_{\rm min}$, and the width of the cutoff profile $\sigma_{\log M}$. The three free parameters of the satellite galaxies are the mass scale $M_0$ at which the mean occupation of satellites drops faster than a power law, $M_1$ characterizes the amplitude of the satellite mean occupation function, and $\alpha$ is the asymptotic slope at high halo mass. Central galaxies are placed at the center of the halo, while the spatial distribution of satellite galaxies inside halos is assumed to follow the NFW profile~\cite{Navarro:1996gj}.
The fiducial cosmology of Molino catalogs assumes the following values of the HOD parameters, $\left\{ \log M_{\rm min}, \sigma_{\log M}, \log M_0, \alpha, \log M_1 \right\} = \left\{13.65, 0.2, 14.0, 1.1, 14.0 \right\}$. To compute the numerical derivatives with respect to the HOD parameters, the Molino suite additionally includes mocks with a variation of a single parameter above and below the fiducial values of HOD parameters applied to 500 Quijote boxes in the fiducial cosmology. The step size of these variations is $\left\{ \Delta \log M_{\rm min}, \Delta \sigma_{\log M}, \Delta \log M_0, \Delta \alpha, \Delta \log M_1 \right\} = \left\{ 0.05, 0.02, 0.2, 0.2, 0.2 \right\}$. To obtain more accurate measurements of the derivatives, each simulation box is populated with galaxies with five different initial seeds. The redshift-space measurements are performed along the three LoS directions. Therefore, the derivatives used in the Fisher forecasts are the average over 15 measurements. As in Quijote measurements, the covariance matrices are obtained from the simulations with the fiducial cosmology along a single LoS.
\section{Forecasting Methodology}\label{sec:fisher}
The accuracy with which a given dataset can constrain cosmological and nuisance parameters are commonly estimated using the Fisher information matrix formalism. In general the Fisher matrix is defined as \cite{Tegmark:1996bz,Tegmark:1997rp}
\begin{equation}
F_{\alpha \beta} = -\left\langle \frac{\partial^2 \mathcal{L({\bf d}|{\bm \theta})}}{\partial\theta_\alpha\partial\theta_\beta}\right\rangle,
\end{equation}
where ${\mathcal L}$ is the likelihood of the data ${{\bf d}} $ given the parameters ${\bm \theta}$. Assuming that the likelihood ${\mathcal L}$ is Gaussian, the forecasted marginalized uncertainty on the $\alpha$-th parameter is given by $\sigma^2(\theta_\alpha) = (F^{-1})_{\alpha \alpha}$. As a consequence of the Cramér-Rao inequality, for an unbiased estimator, this uncertainty is always larger than the unmarginalized uncertainty $1/\sqrt{F_{\alpha \alpha}}$. For multivariate Gaussian probability distribution with nonzero mean and covariance that is independent of parameters of interest, the Fisher matrix is further simplified and can be written in terms of the change of the mean of the data with respect to the model parameters and the data covariance.
In our analysis, we consider the data vector consisting of the 14 redshift-space skew spectra in the selected $k$-bins, ${\bf d} \equiv \left\{ {\bm{\mathcal P}}_{{\mathcal S}_1 \delta}, ..., {\bm{\mathcal P}}_{{\mathcal S}_{14} \delta} \right\}$. When exploring the constraining power of power spectrum and skew spectra combined, the data vector additionally consists of power spectrum multipoles, ${\bf d} \equiv \left\{ {\bm{\mathcal P}}_{{\mathcal S}_1 \delta}, ..., {\bm{\mathcal P}}_{{\mathcal S}_{14}\delta}, {\bf P}_0, {\bf P}_2, {\bf P}_4 \right\}$, where ${\bf P}_n $ are the data vectors power spectrum multipoles of different $k$-bins, respectively. The shot noise is subtracted in the estimators for both skew spectra and power spectrum multipoles\footnote{As will be described later, in our analysis of the Molino galaxies, we do not subtract the shot noise contributions of the skew spectra.}. The Fisher information matrix for the above data vectors is then given by
\begin{equation}\label{eq:fisher}
F_{\alpha\beta} = \sum_{i,j = 1}^{N_{\rm d}} \frac{\partial d_i}{\partial \theta_\alpha} \mathbb{C}^{-1}_{ij} \frac{\partial d_j}{\partial\theta_\beta}\, ,
\end{equation}
where $N_{\rm d}$ is the total size of the data vector in consideration. Binning the skew spectra and the power spectrum multipoles in $N_{\rm b}$ bins of width $\Delta k$, we have $N_{\rm d} = 14 N_{\rm b}$ for skew spectra only and $N_{\rm d} = 14 N_{\rm b} + 3 N_{\rm b}$ for the joint data vector of skew spectra and power spectrum multipoles. The covariance matrix in Eq.~\eqref{eq:fisher} is the full $N_{\rm d} \times N_{\rm d}$ covariance matrix for the data vector $\bf d$ defined by
\begin{equation}
\mathbb{C}_{ij} \equiv
\frac{1}{N_{\rm s}-1}\sum_n^{N_{\rm s}} \left(d_i^{(n)}-\bar{d}_i\right)\left(d_j^{(n)}-\bar{d}_j\right),
\end{equation}
with $N_{\rm s}$ being the number of mocks. We correct the inverse of the covariance by the Hartlap factor~\cite{Hartlap:2006kj} given that the covariance is estimated from a limited number of mocks. We use the Quijote and the Molino suites to compute the derivatives and the covariance matrix numerically.
For cosmological parameters, $\{\Omega_{\rm m}, \Omega_{\rm b}, h, n_s, \sigma_{8}\}$, and the HOD parameters, we evaluate the derivatives from the averaged measurements of the skew spectra and power spectrum over realization at $\theta^+$ and $\theta^-$,
\begin{equation}
\frac{\partial d_i}{\partial \theta_i} = \frac{d_i(\theta^+) - d_i(\theta^-)}{\theta^+ - \theta^-}.
\end{equation}
For $M_\nu$, since the fiducial value is zero and negative masses do not have any physical meaning, we use simulations with $\{M_{\nu}^{+}, M_{\nu}^{++}, M_{\nu}^{+++}\}=\{0.1\, {\rm eV}, 0.2\, {\rm eV}, 0.4\,{\rm eV}\}$, and estimate the derivatives using
\begin{equation}
\frac{\partial d_i}{\partial M_{\nu}} = \frac{-21 d_i(\theta_{\rm fid}^{\rm ZA})+32 d_i(M_{\nu}^+)-12 d_i(M_{\nu}^{++})+d_i(M_{\nu}^{+++})}{12\,\delta M_{\nu}},
\end{equation}
which is accurate up to the second order in $\mathcal{O}(\delta M_{\nu}^2)$\footnote{See Ref.~\cite{Massara:2022zrf} for more detailed discussion of the different estimators for neutrino derivatives.}. For the Quijote halos, we add the minimum halo mass-cut $M_{\rm min}$ as a nuisance parameter to account for an unknown effective bias parameter. We take $M^+_{\rm min} = 3.3 \times 10^{13}\, h^{-1}M_\odot$ and $M^-_{\rm min} = 3.1 \times 10^{13}\, h^{-1}M_\odot$. In Appendix~\ref{app:convergence} we study the stability of the numerical derivatives and the covariance matrix estimation from a finite number of mocks.
We set the $k$-bin width to be the fundamental frequency of the simulation box with side length $L$, {\it i.e.}, $k_f=2\pi/L$ and the minimum $k$-bin edge to be half of the fundamental mode.
We chose the maximum $k$-value for our base analysis to be $k_{\rm max} = 0.25\, {{\rm Mpc}^{-1} h}$. Therefore we have $N_{\rm b} = 39$ for each of the observables in the data vector. Our base analysis assumes the smoothing scale of $R=10\, {h^{-1}\rm{Mpc}}$, which is motivated by half of the average inter-particle separation distance $\av{\Delta r}\sim 18\,{h^{-1}\rm{Mpc}}$ for the Quijote halo catalogs. To test the dependence of the constraints on the smallest scales included in the forecasts, we also present the results for the smoothing scales of $R= 20\, {h^{-1}\rm{Mpc}}$, and a range of other values of $k_{\rm max}$.
\section{Results}\label{sec:res}
In this section, we first present the measurements of the skew spectra and their covariance matrix. Next, we investigate the impact of the non-Gaussian off-diagonal elements of the covariance matrix due to mode coupling on the signal-to-noise ratio (SNR) of individual skew spectra. Finally, we present the forecasted parameter constraints for both Quijote halo and Molino galaxy samples from skew spectra alone and in combination with power spectrum multipoles. For the covariance estimation, we measure 15,000 skew spectra at fiducial cosmology. This is the same for both Quijote and Molino. For the derivative estimation, we measure 500 pairs of simulations left- and right-ward of the reference cosmology for 6 cosmological parameters. In addition, for Quijote, we also carry out measurement using two different halo mass cuts; while for Molino, instead of the halo mass cuts we measure the skew spectra with the 5 HOD parameters. In summary, at each smoothing scale, we measure a total of $24,000 + 15,000 = 39,000$ skew spectra for Quijote and $180,000 + 15,000 = 195,000$ for Molino.
\subsection{Measured Skew Spectra and Their Covariance Matrix}
In figure \ref{fig:quijote_ss_smooth10x20_shoton_fid} we show the measurements of the 14 halo skew spectra at fiducial cosmology of Quijote simulations, applying smoothing scales of $R=10\, {h^{-1}\rm{Mpc}}$ (red) and $R=20\, {h^{-1}\rm{Mpc}}$ (blue). The error bars correspond to the variance of the spectra, averaged over 15,000 simulations at reference cosmology. As expected, for the larger smoothing scale, all skew spectra drop faster towards higher-$k$ modes, and almost all vanish when approaching $k_{\rm max}\sim 0.3\, {{\rm Mpc}^{-1} h}$. When using a smaller smoothing scale, the overall amplitudes of the skew spectra increase and the peak positions shift towards smaller scales. The shape of skew spectra can be classified into three categories: (i) Constant type with density square $\delta^2$: $\mathcal{P}_{\mathcal{S}_2\delta}$, $\mathcal{P}_{\mathcal{S}_6\delta}$, $\mathcal{P}_{\mathcal{S}_{10}\delta}$, which are characterized by a peak on very large scales and a relatively sharp fall off towards smaller scales.
(ii) Displacement type with operator $\partial_i\partial_j/\nabla^2$: $\mathcal{P}_{\mathcal{S}_1\delta}$, $\mathcal{P}_{\mathcal{S}_{4-5}\delta}$, $\mathcal{P}_{\mathcal{S}_{8-9}\delta}$, $\mathcal{P}_{\mathcal{S}_{12-14}\delta}$, which are characterized by a positive ``bump". For the operators that have dependence on the LoS, the resulting skew spectra have a ``zero-crossing" feature on large scales.
(iii) Tidal type with operator $S^2$: $\mathcal{P}_{\mathcal{S}_3\delta}$, $\mathcal{P}_{\mathcal{S}_7\delta}$, $\mathcal{P}_{\mathcal{S}_{11}\delta}$, these skew spectra all have a negative ``bump" feature and are anti-correlated with the rest of skew spectra on small scales.
The three categories are intrinsically related to the real-space skew spectra (see figure 1 in~\cite{Schmittfull:2014tca}). Despite the similarity within the three groups, there are high correlations among these 14 skew spectra, and each of them contributes almost equally to the total information content, and we can not simply truncate the data vector by selecting only a subset of them. However, the data vector of skew spectra can be potentially optimally compressed by finding combinations of different spectra that maximize the Fisher information. We defer further investigation of this compression to future works.
\begin{figure}[t]
\centering
\includegraphics[width=0.86\textwidth]{figs/quijote_ss_smooth10x20_shotoff_fid.pdf}
\caption{Measured skew spectra with fiducial cosmology on Quijote halo catalogs setting smoothing scales of $R=10\, {h^{-1}\rm{Mpc}}$ (red) and $R=20\, {h^{-1}\rm{Mpc}}$ (blue). The error bars correspond to the variance of the spectra, averaged over 15,000 simulations at reference cosmology.}
\label{fig:quijote_ss_smooth10x20_shoton_fid}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=.47\textwidth]{figs/quijote_covariance_ssxpk_smooth10_shotoff_fid.pdf}
\includegraphics[width=.47\textwidth]{figs/quijote_covariance_ssxpk_smooth20_shotoff_fid.pdf}
\caption{Correlation matrix of the combined data vector $\mathbf{d}\equiv \{\mathcal{P}_{{\mathcal S}_1\delta}, \ldots \mathcal{P}_{{\mathcal S}_{14}\delta}, P_0, P_2\}$ for halo skew spectra and power spectrum multipoles setting $k_{\rm max}=0.5\, {{\rm Mpc}^{-1} h}$ measured from Quijote simulations. On the left, we show the combined data vector assuming a smoothing scale of $R=10\, {h^{-1}\rm{Mpc}}$ for the skew spectra, while on the right, we used $R=20\, {h^{-1}\rm{Mpc}}$. Each block denotes the covariance between the $i$-th component in the combined data vector with the $j$-th component. Within each block, the structure shows the covariance for different wave numbers $k$.}
\label{fig:quijote_covariance_smooth10x20_shoton_fid}
\end{figure}
Having a large number of mocks of the Quijote simulation suite, we estimate the full covariance matrix of the redshift-space skew spectra and the power spectrum multipoles for the first time. Figure \ref{fig:quijote_covariance_smooth10x20_shoton_fid} displays the joint correlation matrix of the lowest two power spectrum multipoles ($\ell=0,2$) and the skew spectra with $R=10\,{h^{-1}\rm{Mpc}}$ (left panel) and $R=20\,{h^{-1}\rm{Mpc}}$ (right panel). Each block denotes the (cross-)covariance structure between two skew spectra $\mathcal{P}_{{\mathcal S}_i\delta}$ with $\mathcal{P}_{{\mathcal S}_j\delta}$ for $i, j=\{1,\ldots,14\}$, between power spectrum multipoles $P_\ell$ for $\ell =\{ 0,2\}$, and between $P_\ell$ and $\mathcal{P}_{{\mathcal S}_i\delta}$. Within each block, the structure shows the covariance for different wave numbers, i.e., the mode coupling contributions. The white bands correspond to scales where the covariance vanishes since smoothing the field washes out small-scale fluctuations. The covariance for the larger smoothing scale goes to zero at $k \sim\! 0.3\, {{\rm Mpc}^{-1} h}$, the scale at which the skew spectra themselves approach zero as shown in figure \ref{fig:quijote_ss_smooth10x20_shoton_fid}. The dark blobs appearing in the white bands on the right plot are numerical artifacts from the fact that to compute the correlation matrix, we divided the covariance matrix elements by diagonal elements, which on those scales approach zero. Overall, the correlations of the skew spectra and power spectrum multipoles are smaller than the cross-correlations between different skew spectra. Unlike the power spectrum, the skew spectra can have positive or negative signs, as shown in figure \ref{fig:quijote_ss_smooth10x20_shoton_fid}. Therefore, different spectra can be correlated (red blocks) or anti-correlated (blue blocks), depending on the sign of the skew spectra. We note that for Quijote halos, we only include the power spectrum monopole and quadrupole for this paper. This is because the hexadecapole for Quijote halos has negligible signal-to-noise. As we will discuss later in this section, this is not the case for Molino galaxies. Therefore, in Molino analysis, we include the hexadecapole.
\begin{figure}[t]
\centering
\includegraphics[width=0.55\textwidth]{figs/quijote_diag_cov_smooth10x20_shotoff_fid.pdf}
\caption{{\it Upper panel}: Diagonal of the 14 skew spectra covariance matrix for smoothing scale $R=10\, {h^{-1}\rm{Mpc}}$ (red) and $=20\, {h^{-1}\rm{Mpc}}$ (blue) with $k_{\rm max}=0.25\, {{\rm Mpc}^{-1} h}$. {\it Lower panel}: square root ratio of the diagonal elements of 14 skew spectra covariance matrix for the two smoothing scales.}
\label{fig:quijote_diag_cov_smooth10x20_shoton_fid}
\vspace{.1in}
\end{figure}
To better compare the relative size of the covariance matrices for the two smoothing scales, in the top panel of figure \ref{fig:quijote_diag_cov_smooth10x20_shoton_fid}, we show the diagonal elements of the covariance matrix of the skew spectra for $R=10\, {h^{-1}\rm{Mpc}}$ in red and $R=20\, {h^{-1}\rm{Mpc}}$ in blue. The square root of the ratio of the two is shown in the bottom panel. The points lying between each of the two vertical dotted lines correspond to a given skew spectrum with wavenumbers increasing from left to right. Overall, the skew spectra with a smaller smoothing scale have a larger variance, with a slower drop towards the smaller scales. This is due to the fact that applying a large smoothing scale washes away the fluctuations on small scales. While this trend is seen for all skew spectra, the scale dependencies of the variances of different spectra differ from one another, reflecting the difference in their shapes shown in figure \ref{fig:quijote_ss_smooth10x20_shoton_fid}.
\subsection{Impact of the Non-Gaussian Covariance}
Before presenting the parameter constraints, we investigate the importance of the off-diagonal contributions\footnote{The non-Gaussian contributions to diagonal elements are accounted for in the measured covariances.} to the covariance matrix, focusing on the signal-to-noise ratio (SNR) of individual skew spectra. While the impact of off-diagonal elements for the halo/galaxy power spectrum and bispectrum has been previously studied ({\it e.g.}~\cite{Sato:2013mq,Kayo:2012nm,Chan:2016ehg,Sugiyama:2018yzo}), here we present it for the first time for the skew spectra. It is worth noting that the off-diagonal elements of the skew spectra covariance matrix receive both Gaussian and non-Gaussian contributions. This was explicitly shown in the theoretical prediction of the covariance matrix of real-space skew spectra using tree-level perturbation theory \cite{Schmittfull:2014tca, MoradinezhadDizgah:2019xun}. Even if retaining only the diagonal Gaussian contributions of the bispectrum covariance, the skew spectra (even on large scales) have non-vanishing off-diagonal elements.
In figure \ref{fig:quijote_snr_smooth10x20_shoton_Mnu}, we show the SNR for each of the skew spectra as a function of small-scale cutoff using $R=10\, {h^{-1}\rm{Mpc}}$ (red) and $R=20\, {h^{-1}\rm{Mpc}}$ (blue). The dashed lines show the SNR assuming the covariance matrix to be diagonal, while the solid lines include the off-diagonal elements. As expected, the diagonal covariance approximation artificially enhances the SNR. When accounting for the mode coupling in the covariance, the SNR grows slower\footnote{This saturated trend of the SNR of the skew spectra resembles one of the power spectra in figure 19 of Ref.~\cite{Chan:2016ehg}.} and nearly levels off at $k_{\rm max} \sim 0.4\, {{\rm Mpc}^{-1} h}$. Going from $k_{\rm max}=0.3 \, {{\rm Mpc}^{-1} h}$ to $k_{\rm max}=0.4 \, {{\rm Mpc}^{-1} h}$, there is an increase by 25\% in signal-to-noise ratio when using the full covariance, while an increase by 54\% under the diagonal covariance approximation.
This trend is the same for both values of $R$.
\begin{figure}
\centering
\includegraphics[width=0.85\textwidth]{figs/quijote_snr_smooth10x20_shotoff_fid.pdf}
\caption{The signal-to-noise ratio of individual skew spectra given by ${\rm SNR}=(\mathcal{P}_{\mathcal{S}_i\delta} C^{-1}_{ij} \mathcal{P}_{\mathcal{S}_j\delta})^{1/2}$ for $R=10\, {h^{-1}\rm{Mpc}}$ (red) $R=20\, {h^{-1}\rm{Mpc}}$ (blue). The solid lines are computed using the full covariance matrix, while only the diagonal elements are included for the dashed lines.}
\label{fig:quijote_snr_smooth10x20_shoton_Mnu}
\end{figure}
The impact of the off-diagonal covariance element on the SNR can be clearly understood with an analytic approximation, as was done for the power spectrum covariance in Ref.~\cite{Carron:2014hja}. We assume that the covariance matrix $\mathbb{C}_{[i]}$ of the $i$-th skew spectrum can be decomposed into a diagonal and an off-diagonal components, $\mathbb{C}_{[i]}\equiv\mathbb{D}+ \sigma_{\rm min}^2 {\mathcal{P}_{\mathcal{S}_{i}\delta}}{\mathcal{P}_{\mathcal{S}_{i}\delta}}^T$, with $\mathbb{D}$ being the diagonal part for which $\sigma_{\rm min}^2$ is zero. In the case of power spectrum, $\sigma^2_{\rm min}$ has the interpretation of minimum achievable variance \cite{Carron:2014hja}, while for the skew spectra, the definition can be extended to the cumulants of $\av{\delta^3(x_1)\delta^3(x_2)}_c \propto \sigma^8\xi(x_1-x_2)$ \cite{Bernardeau:1995ty}. Using the Sherman-Morrison-Woodbury formula~\cite{Sherman:1950, Woodbury:1950}, the inverse covariance can be expanded in terms of the inverse of the diagonal part plus a correction,
\begin{eqnarray}
\mathbb{C}_{[i]}^{-1} \equiv \left(\mathbb{D}+ \sigma_{\rm min}^2{\mathcal{P}_{\mathcal{S}_{i}\delta}}{\mathcal{P}_{\mathcal{S}_{i}\delta}}^T\right)^{-1} = \mathbb{D}^{-1} - \sigma_{\rm min}^2\frac{\mathbb{D}^{-1} {\mathcal{P}_{\mathcal{S}_{i}\delta}}{\mathcal{P}_{\mathcal{S}_{i}\delta}}^T\mathbb{D}^{-1}}{1+ \sigma_{\rm min}^2{\mathcal{P}_{\mathcal{S}_{i}\delta}}^T \mathbb{D}^{-1} {\mathcal{P}_{\mathcal{S}_{i}\delta}}}.
\label{eqn:SM_expansion}
\end{eqnarray}
Therefore, the SNR for the full covariance, $({\rm S}/{\rm N})^2$, is related to SNR in the limit of diagonal covariance, $({\rm S}/{\rm N})^2_G$, as \cite{Carron:2014hja}
\begin{eqnarray}
({\rm S}/{\rm N})^2 = \frac{({\rm S}/{\rm N})^2_G}{1+\sigma_{\rm min}^2({\rm S}/{\rm N})^2_G}.
\end{eqnarray}
It is thus clear that including the non-diagonal part of the covariance denoted by the term associated with $\sigma_{\rm min}^2$ always decreases the SNR.
Our results for the SNR align with the Fisher forecasts presented in Ref.~\cite{MoradinezhadDizgah:2019xun}, which showed that neglecting the off-diagonal elements significantly affects the parameter constraints, underestimating the expected uncertainties. Therefore, our forecasts presented below use the full covariance matrix shown in figure \ref{fig:quijote_covariance_smooth10x20_shoton_fid}.
\subsection{Forecasted Parameter Constraints}
\label{subsec:forecast_param}
\vspace{0.05in}
\underline{Quijote Halo Catalogs}
\vspace{0.05in}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\textwidth]{figs/quijote_2dcontour_pk_ss_smooth10x20_shotoff_nuisMmin.pdf}
\caption{Marginalized 1- and 2-$\sigma$ parameter constraints from Quijote halo power spectrum multipoles (grey), and skew spectra with two smoothing scales of $R=10\,{h^{-1}\rm{Mpc}}$ (blue) and $R=20\, {h^{-1}\rm{Mpc}}$ (red). The small-scale cutoff is set to $k_{\rm max} = 0.25 \ {{\rm Mpc}^{-1} h}$ in all cases.}
\label{fig:2dcontour_pk_ss_smooth10x20}\vspace{0.2in}
\end{figure}
\noindent To compare the information content of the skew spectra with that of the power spectrum multipoles ($\ell = 0,2$), in figure \ref{fig:2dcontour_pk_ss_smooth10x20}, we show the 2D marginalized constraints on cosmological parameters and minimum halo mass (as a proxy for the unknown halo bias) from the full skew spectra data vector. The gray contours correspond to the power spectrum multipoles, while the blue and red ones are from the skew spectra with $R=20\, {h^{-1}\rm{Mpc}}$ and $R=10\, {h^{-1}\rm{Mpc}}$, respectively. The small-scale cutoff is set to $k_{\rm max} = 0.25\, {{\rm Mpc}^{-1} h}$ in all three cases. Compared to the power spectrum multipoles, the skew spectra improve the constraints on all parameters by $\sim (23-62)\%$, with the improvements of $M_\nu$ and $\sigma_8$ being the two most prominent. The choice of the smoothing scale has nearly no impact on the constraint on the total mass of neutrinos. On the contrary, the constraint on $\sigma_8$ improves by $\sim 35\%$ when reducing the smoothing scale from $R=20\ {h^{-1}\rm{Mpc}}$ to $R=10\ {h^{-1}\rm{Mpc}}$. For other parameters, the improvement is at most $\sim\!20\%$. The insensitivity of the neutrino constraints to the choice of smoothing scale can be better understood by inspecting the shape of the normalized derivatives of skew and power spectra as a function of wavenumber. As shown in figure~\ref{fig:quijote_response_smooth10x20_Mnu} of Appendix~\ref{app:responses}, the responses (which are the inverse-variance normalized derivatives) for both smoothing scales are of similar amplitudes. For individual skew spectra, the off-diagonal mode-coupling contributions of the covariance counteract the small difference between the response functions for the two choices of smoothing scale such that the parameter constraints, shown in figure \ref{fig:1sigma_kmax_ind_smoothing_mnu}, are nearly unaffected by the value of smoothing scale. When combining all skew spectra, the constraints become even more comparable, presumably due to additional covariance between different skew spectra.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\textwidth]{figs/quijote_1sigma_ss_smooth10x20_shotoff_pk02_nuisMmin.pdf}
\caption{Marginalized 1-$\sigma$ parameter constraint from power spectrum multipoles (grey), skew spectra with smoothing scales of $R=10\,{h^{-1}\rm{Mpc}}$ (blue) and $R=20\, {h^{-1}\rm{Mpc}}$ (red), as a function of small-scale cutoff $k_{\rm max}$.}
\label{fig:quijote_1sigma_ss_smooth10x20_shotoff_pk02_nuisMmin}
\end{figure}
In the results above, we have applied a conservative small-scale cutoff, regardless of the smoothing scale. We illustrate the dependence of the constraints on the choice of $k_{\rm max}$ in figure~\ref{fig:quijote_1sigma_ss_smooth10x20_shotoff_pk02_nuisMmin}. The magenta and blue lines show the results from skew spectra with $R=10\, {h^{-1}\rm{Mpc}}$ and $R=20\, {h^{-1}\rm{Mpc}}$, while the grey lines are from the power spectrum multipoles. For $R=20\, {{\rm Mpc}^{-1} h}$, we only plot the errors up to $k_{\rm max} = 0.3\, {{\rm Mpc}^{-1} h}$ since beyond this scale the covariance is nearly vanishing and the Fisher matrix is not reliable. On the largest scale (the lowest values of $k_{\rm max}$), the skew spectra provide remarkably better constraints on all cosmological parameters compared to the power spectrum multipoles. This is not unexpected since being a correlation between composite quadratic fields and the original halo field, the non-Gaussian information from small-scale fluctuations gets imprinted on the large-scale skew spectra. Therefore, even when imposing the small-scale cutoff of $k_{\rm max}$, modes smaller than this cutoff still contribute to the observed skew spectra due to convolution in the quadratic field. For $M_\nu$, the skew spectra consistently provide tighter constraints than the power spectrum for all choices of $k_{\rm max}$ (ranging from a factor of 2.5 to 3 better). For $\Omega_b$ and $h$, the ratio between the parameter constraints from skew and power spectra stays nearly constant for $k_{\rm max} \gtrsim 0.2 \, {{\rm Mpc}^{-1} h}$, while for $\Omega_m, n_s$ and $\sigma_8$ the skew spectra and power spectrum constraints approach one another increasing the $k_{\rm max}$. These conclusions may differ when considering a more complex bias model. Breaking degeneracies between cosmological and nuisance bias parameters can lead to even more superior constraints from the skew spectra. This is because they capture the information in higher-order statistics not imprinted on the power spectrum. We investigate this by considering the Molino galaxy sample and varying the HOD parameters. The discussion will follow later in this section.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\textwidth]{figs/quijote_2dcontour_pkxss10_shotoff_nuisMmin.pdf}
\caption{Marginalized 1- and 2-$\sigma$ constraints from Quijote halo power spectrum multipoles (grey), skew spectra with smoothing scales $R=10\, {h^{-1}\rm{Mpc}}$ (red), and the two combined (green). The smoothing scale is set to $k_{\rm max} = 0.25 \ {{\rm Mpc}^{-1} h}$ in all cases.}
\label{fig:2dcontour_pk_ss_joint_smooth10}
\end{figure}
Next, we demonstrate how much the joint analysis of the skew spectra and power spectrum multipoles improves upon their individual constraints. In figure \ref{fig:2dcontour_pk_ss_joint_smooth10}, we compare the 2D marginalized 1- and 2-$\sigma$ constraints from the power spectrum multipoles (grey), the skew spectra with $R= 10\, {h^{-1}\rm{Mpc}}$ (red), and the two combined (green). We set $k_{\rm max}=0.25\, {{\rm Mpc}^{-1} h}$ in all three cases. The constraints on cosmological parameters from the joint analysis are dominated by the skew spectra, and the combination with the power spectrum multipoles only improves the constraints slightly, ranging from $(9-18)\%$, the smallest and largest gains are on $M_\nu$ and $\Omega_{\rm b}$, respectively. Most parameters exhibit identical degeneracy directions in power and skew spectra. Therefore, the improvement in the constraints is not driven by the breaking of degeneracies when the two statistics are combined. As we will show later in this section, for Molino galaxies, the conclusions differ as a result of varying a larger number of nuisance parameters characterizing a more complex matter-tracer biasing relation. In table~\ref{tab:quijote}, we report the marginalized 1-$\sigma$ uncertainties on cosmological parameters from skew and power spectra individually and combined, marginalizing over the minimum halo mass cut $M_{\rm min}$.
In the previous analysis of the halo power spectrum (monopole and quadrupole) and bispectrum (monopole) of the Quijote dataset presented in Ref.~\cite{Hahn:2019zob}, in addition to $M_{\rm min}$, an additional nuisance parameter characterizing the overall amplitude of the two summary statistics was varied. To compare our results with theirs, we perform an additional forecast (setting $k_{\rm max}=0.2\ {{\rm Mpc}^{-1} h}$ as in their forecasts), allowing the overall amplitudes of the power spectrum multipoles and the skew spectra to vary. In that case, our power spectrum constraints are comparable to theirs\footnote{We use a slightly different k binning than Ref.~\cite{Hahn:2019zob}, but the overall effect in terms of parameter constraints is no more than 10\%}, and on average, the constraints from skew spectra are very competitive to the one from bispectrum.
We note, however, that the comparison between constraints from skew spectra and bispectrum monopole has scale dependence. The constraints from skew spectra reach a plateau, resulting from the imposed smoothing scale. On the contrary, the bispectrum constraints continue improving with increased $k_{\rm max}$. For example, at $k_{\rm max}=0.3\,{{\rm Mpc}^{-1} h}$, we find the constraints from skew spectra on most parameters are weaker.
\begin{table}[t]
\centering
\begin{tabular}{c|c|cccccc}
\toprule
Quijote & \begin{tabular}[c]{@{}c@{}}$k_{\rm max}$\\$[{\rm Mpc}^{-1} h]$\end{tabular} & $\Omega_{\rm m}$ & $\Omega_{\rm b}$ & $h$ & $n_s$ & $\sigma_8$ & \begin{tabular}[c]{@{}c@{}}$M_{\nu}$\\ $[{\rm eV}]$\end{tabular} \\
\hline
\multirow{3}{*}{$\mathcal{P}^{R_{20}}_{\mathcal{S}\delta}$} & $0.15$ & $0.035$ & $0.016$ & \begin{tabular}[c]{@{}c@{}}$0.171$\\\end{tabular} & $0.185$ & $0.058$ & $0.315$ \\
& $0.20$ & $0.027$ & $0.014$ & $0.141$ & $0.144$ & $0.036$ & $0.258$ \\
& $0.25$ & $0.025$ & $0.012$ & $0.123$ & $0.122$ & $0.027$ & $0.212$ \\
\hline
\multirow{4}{*}{$\mathcal{P}^{R_{10}}_{\mathcal{S}\delta}$} & $0.15$ & $0.027$ & $0.014$ & $0.151$ & $0.156$ & $0.038$ & $0.299$ \\
& $0.20$ & $0.021$ & $0.011$ & $0.115$ & $0.113$ & $0.025$ & $0.251$ \\
& $0.25$ & $0.020$ & $0.010$ & $0.106$ & $0.100$ & $0.020$ & $0.212$ \\
& $0.50$ & $0.014$ & $0.007$ & $0.066$ & $0.049$ & $0.013$ & $0.118$ \\
\hline
\multirow{4}{*}{$P_{\ell=0,2}$} & $0.15$ & $0.058$ & $0.029$ & $0.399$ & $0.487$ & $0.198$ & $0.845$ \\
& $0.20$ & $0.032$ & $0.015$ & $0.171$ & $0.195$ & $0.081$ & $0.673$ \\
& $0.25$ & $0.026$ & $0.014$ & $0.153$ & $0.157$ & $0.049$ & $0.568$ \\
& $0.50$ & $0.017$ & $0.011$ & $0.110$ & $0.065$ & $0.019$ & $0.322$ \\
\hline
$\mathcal{P}^{R_{10}}_{\mathcal{S}\delta}+P_{\ell=0,2}$ & $0.25$ & $0.017$ & $0.008$ & $0.088$ & $0.086$ & $0.018$ & $0.196$ \\
\bottomrule
\end{tabular} \vspace{0.2in}
\caption{Marginalized 1-$\sigma$ constraints on cosmological parameters from halo skew and power spectra measured on Quijote simulations. We show the results from skew spectra with $R=20\,{h^{-1}\rm{Mpc}}$ and $R=10\,{h^{-1}\rm{Mpc}}$, contrasted with those from the power spectrum monopole and quadrupole. The last row shows the constraints from the combination of the skew spectra with $R=10\ {h^{-1}\rm{Mpc}}$ and the power spectrum multipoles. }
\label{tab:quijote}\vspace{-0.2in}
\end{table}
An important question to address in order to establish the skew spectra as (nearly) optimal proxy statistics for the bispectrum is how much of the information of the bispectrum is captured by the skew spectra. While at $k_{\rm max} = 0.5\, {{\rm Mpc}^{-1} h}$, the comparison with bispectrum results of Ref.~\cite{Hahn:2019zob} clearly indicates information loss (even compared to just bispectrum monopole), we note that, as pointed out in Ref.~\cite{Coulton:2022rir}, the results of numerical Fisher forecasts of the bispectrum should be interpreted with some caution. This is primarily due to the fact that a lack of convergence of the numerical derivatives with respect to the number of mocks used in the measurements can result in artificially tight parameter constraints~\cite{Coulton:2022rir}. Therefore, a more conclusive statement of the information content of the skew spectra in comparison with the bispectrum is not possible at the moment. We perform extensive stability tests for convergence of numerical derivatives of skew spectra in Appendix~\ref{app:convergence}, quantifying the amount by which the parameter uncertainties may be underestimated due to lack of full convergence and estimating the required number of simulations to achieve full convergence. We conclude that the convergence rate of the skew spectra is comparable with that of the power spectrum multipoles. Therefore, the relative gain in constraints from the skew spectra compared to the power spectrum should not be affected by the convergence of the numerically estimated derivatives and covariance.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{figs/molino_2dcontour_pk4shot0xssshot1_k0d25.pdf}
\caption{Marginalized 1- and 2-$\sigma$ constraints for Molino galaxy mocks. The plots show constraints obtained from power spectrum multipoles (tan), skew spectra with smoothing scale of $R=10\, {h^{-1}\rm{Mpc}}$ (red), and the two combined (teal).}
\label{fig:molino_2dcontour_pk4shot1xssshot1_k0d25}
\end{figure}
\vspace{0.2in}
\noindent \underline{Molino Galaxy catalogs}
\vspace{0.05in}
\noindent In order to understand the impact of marginalization over parameters of a more complex tracer-matter relation, we next study the galaxy catalog from the HOD-based Molino suite. Before presenting the results, let us make two remarks on the differences between our analysis of Quijote and Molino datasets. First, in contrast to the Quijote analysis, we include the three lowest multipoles of the power spectrum. This is because the Molino galaxy sample has a large hexadecapole and a fair comparison with skew spectra (which capture the anisotropic clustering information) should include the power spectrum hexadecapole. In Appendix~\ref{app:Molino}, we test the information content of the hexadecapole for this sample and provide further discussion of how the adopted calibration of the HOD model is at the root of the observed large hexadecapole. Second, we compare the results of the skew spectra without subtraction of the shot noise to the shot-noise-subtracted power spectrum. This choice was made solely for computational convenience in measuring the skew spectra. We found that for the power spectrum of Molino galaxies, the shot noise plays a marginal role in terms of constraints on cosmological parameters. Given that we found that the cosmological constraints are largely insensitive to subtraction of the shot noise for Quijote catalogs (presented in Appendix \ref{app:shot}), we expect that our results for Molino should also not be affected by the shot noise subtraction.
Figure~\ref{fig:molino_2dcontour_pk4shot1xssshot1_k0d25} shows marginalized 1- and 2-$\sigma$ constraints for Molino galaxy mocks obtained from power spectrum multipoles (orange), skew spectra with smoothing scale of $R=10\, {h^{-1}\rm{Mpc}}$ (red), and the two combined (teal) at $k_{\rm max}=0.25\,{{\rm Mpc}^{-1} h}$. We find that marginalization over the HOD parameters, which capture a complex biasing relation between dark matter and tracers, does not affect the degeneracy directions among the cosmological parameters (in comparison with Quijote results in figure~\ref{fig:2dcontour_pk_ss_joint_smooth10}). Although there is no guarantee that parameter degeneracies should always remain the same when opening up parameter space, we do not observe a strong (anti)correlation between the HOD parameters and any of the cosmological parameters in Molino. This is likely to be the reason why in Molino analysis, the degeneracies among cosmological parameters are still determined by the underlying physical processes. Nonetheless, as we discuss in Appendix~\ref{app:Molino}, further investigation of the consequence of the galaxy assignment scheme using a HOD parameterization that is tuned to realistic samples remains interesting.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\textwidth]{figs/molino_1sigma_ss_smooth10x20_shot1_pk024_shot0.pdf}
\caption{Marginalized 1-$\sigma$ constraints as a function of $k_{\rm max}$ from Molino galaxies. We show the constraints from the lowest two (three) multipoles in gray (yellow), from skew spectra with $R=20\, {h^{-1}\rm{Mpc}}$ in blue, and from skew spectra with $R=10\, {h^{-1}\rm{Mpc}}$ in red.}
\label{fig:molino_1sigma_ss_smooth10x20_shot1_pk024_shot0}
\end{figure}
Compared to the power spectrum up to hexadecapole, skew spectra improve the constraints by $(32-71)\%$. Combining the power spectrum and skew spectra improves the constraints from skew spectra alone by an additional $(10-34)\%$, with the least improvement again in $M_\nu$ and the largest in $\sigma_8$. Overall, the relative gain between skew spectra and power spectrum in cosmological parameters is more significant on Molino than Quijote, by at most $50\%$ in $n_s$ (for $M_\nu$ the constraints are slightly degraded). In addition, we also see that skew spectra, alone and combined with power spectra, tightly constrain the HOD parameters. In particular, the degeneracy in the ${\rm log}M_{\rm min}-{\rm log}M_{0}$ plane and $\sigma_{\rm log M}-{\rm log}M_{0}$ is broken by the skew spectra.
We found that the skew spectra can achieve very competitive results compared to the bispectrum monopole at linear to mildly non-linear scales ($k<0.3 \ {{\rm Mpc}^{-1} h}$). In particular, we found that for Molino the skew spectra not only outperform the bispectrum monopole but also have a larger relative gain compared to the power spectrum than using the bispectrum monopole. In Table \ref{tab:molino}, we summarize the 1-$\sigma$ constraints from skew spectra, power spectrum multipoles ($\ell=0,2,4$), and the combination of the two measured on Molino galaxies. Going from Quijote to Molino, the gain in these higher-order statistics estimators to power spectrum is more obvious for skew spectra than bispectrum monopole at linear to mildly non-linear scales. There could be various reasons driving this difference in the relative gain. For instance, the power excess (rising slope of satellite power spectrum towards high-$k$) in the satellite galaxies implies that the satellite velocities can induce large anisotropy in redshift space. This information can be captured by the LoS-dependent skew spectra while is averaged over in bispectrum monopole. As we discuss in more detail in Appendix~\ref{app:Molino}, the conclusions can be different for a realistic galaxy sample.
We demonstrate the dependence of the constraints on the small-scale cutoff in figure~\ref{fig:molino_1sigma_ss_smooth10x20_shot1_pk024_shot0}, which shows the marginalized 1-$\sigma$ constraints as a function of $k_{\rm max}$. To highlight the information content of hexadecapole of Molino galaxies, we show the constraints for the lowest two and three multipoles of the power spectrum in grey and in yellow, respectively. Additionally, we show the constraints from the skew spectra with smoothing scales of $R=20\,{h^{-1}\rm{Mpc}}$ in blue and $R=10\,{h^{-1}\rm{Mpc}}$ in red. Again, for $R=20\,{h^{-1}\rm{Mpc}}$, we show only up to $k_{\rm max}=0.3\,{{\rm Mpc}^{-1} h}$ because this smoothing washes out the small-scale information. The 2D marginalized constraints are shown in figure~\ref{fig:molino_2dcontour_pk_shot0_ss_shot1_k0d25} of Appendix \ref{app:Molino}. For the Molino sample, we also find that the skew spectra provide more significant improvement in cosmological constraints at large scales in comparison to the power spectrum. The improvements gradually plateau when approaching smaller scales. Furthermore, the hexadecapole adds non-negligible information to the constraints (due to the large satellite contribution in the Molino sample).
The Molino galaxy mocks provide valuable insight into the impact of marginalization over (nuisance) parameters characterizing a more complex biasing relation. However, for realistic spectroscopic galaxy samples, the conclusions about the relative gain in parameter constraints from skew spectra could be different. This is due to the fact that apart from the underestimation of the parameter uncertainties intrinsic to all Fisher forecasts, as described above and in Appendix D, the characteristics of the Molino sample do not resemble the known populations of galaxies probed by spectroscopic galaxy surveys. Therefore, establishing the impact of marginalization over HOD nuisance parameters requires performing a similar study as ours applied to a HOD parameterization calibrated to match realistic clustering signals.
\begin{table}[t]
\centering
\begin{tabular}{c|c|cccccc}
\toprule
Molino & \begin{tabular}[c]{@{}c@{}}$k_{\rm max}$\\$[{\rm Mpc}^{-1} h]$\end{tabular} & $\Omega_{\rm m}$ & $\Omega_{\rm b}$ & $h$ & $n_s$ & $\sigma_8$ & \begin{tabular}[c]{@{}c@{}}$M_{\nu}$\\ $[{\rm eV}]$\end{tabular} \\
\hline
\multirow{3}{*}{$\mathcal{P}^{R_{20}}_{\mathcal{S}\delta}$} & $0.15$ & $0.044$ & $0.018$ & $0.201$ & $0.201$ & $0.127$ & $0.384$ \\
& $0.20$ & $0.036$ & $0.015$ & $0.162$ & $0.161$ & $0.078$ & $0.301$ \\
& $0.25$ & $0.032$ & $0.013$ & $0.139$ & $0.132$ & $0.058$ & $0.244$ \\
\hline
\multirow{4}{*}{$\mathcal{P}^{R_{10}}_{\mathcal{S}\delta}$} & $0.15$ & $0.037$ & $0.017$ & $0.192$ & $0.181$ & $0.068$ & $0.339$ \\
& $0.20$ & $0.031$ & $0.012$ & $0.143$ & $0.142$ & $0.053$ & $0.285$ \\
& $0.25$ & $0.028$ & $0.011$ & $0.126$ & $0.126$ & $0.041$ & $0.234$ \\
& $0.50$ & $0.019$ & $0.008$ & $0.077$ & $0.074$ & $0.022$ & $0.127$ \\
\hline
\multirow{3}{*}{$P_\ell=0,2,4$} & $0.15$ & $0.056$ & $0.027$ & $0.339$ & $0.392$ & $0.247$ & $0.868$ \\
& $0.20$ & $0.051$ & $0.022$ & $0.283$ & $0.347$ & $0.179$ & $0.708$ \\
& $0.25$ & $0.042$ & $0.019$ & $0.227$ & $0.261$ & $0.140$ & $0.565$ \\
& $0.50$ & $0.028$ & $0.012$ & $0.133$ & $0.148$ & $0.053$ & $0.275$ \\
\hline
$\mathcal{P}^{R_{10}}_{\mathcal{S}\delta}+P_\ell=0,2,4$ & $0.25$ & $0.021$ & $0.009$ & $0.106$ & $0.109$ & $0.027$ & $0.216$ \\
\bottomrule
\end{tabular}\vspace{0.2in}
\caption{Same as table~\ref{tab:quijote}, but for Molino galaxy catalogs.}
\label{tab:molino}
\vspace{-0.2in}
\end{table}
\section{Conclusion}\label{sec:conc}
In this paper, we have focused on quantifying the cosmological information of skew spectra of biased tracers in redshift space. The skew spectra are cross-correlations between the observed galaxy field and several appropriately weighted squares of it. They have been proposed as efficient proxy statistics to extract the non-Gaussian information of the LSS encoded in the bispectrum of biased tracers. By construction, the skew spectra correspond to the maximum-likelihood estimators for the parameters that appear as overall amplitudes in the tree-level bispectrum model. Therefore, they optimally constrain (perturbative) galaxy bias parameters, growth rate, and amplitudes of the primordial power spectrum and bispectrum. Their information content for other cosmological parameters, which is the focus of this paper, has not been explored before.
The main advantages of the skew spectra compared to the bispectrum lies in their efficiency and interpretability, which result from their low dimensionality; In contrast to the redshift-space bispectrum, which is a function of five variables, the skew spectra are just a function of a single variable, similar to the power spectrum multipoles. Furthermore, being pseudo-power spectra, it is expected that for the skew spectra, accounting for observational effects (in particular the survey window function) is simpler than the bispectrum. Lastly, the extensive previous works on the analysis of galaxy clustering power spectrum, which now has reached considerable maturity, should be largely applicable to the analysis of the skew spectra. Motivated by these potential advantages, we set out to investigate cosmological constraints from redshift-space skew spectra.
We used two sets of synthetic data, the Quijote halo and the Molino galaxy catalogs, to perform numerical Fisher forecasts for six-parameter $\nu\Lambda$CDM model, varying $\{\Omega_{\rm m},\Omega_{\rm b}, h, n_s, \sigma_8, M_\nu\}$. After presenting the measured skew spectra and their covariance matrix, we illustrated the impact of off-diagonal elements of the covariance on the SNR from individual skew spectra. We then showed the forecasted parameter constraints from the redshift-space skew spectra, contrasted with those from the power spectrum multipoles. We investigated several ingredients and assumptions of the Fisher forecasts, including the impact of subtraction of the shot noise and the choice of the smoothing scale, the information content of individual skew spectra, and the stability of the forecasts w.r.t. a number of mocks used in the measurements.
For Quijote halos and using scales up to $k_{\rm max}=0.25\,{{\rm Mpc}^{-1} h}$, we found that the skew spectra (with $R=10\ {h^{-1}\rm{Mpc}}$) provide constraints on cosmological parameters that are tighter than those from the power spectrum multipoles $(\ell=0,2)$ by $(23-62)\%$. Combining the skew and power spectra, the constraints are further improved (by up to 18\%), compared to skew spectra alone. We did not find the shot noise to have a major impact on the forecasted constraints of cosmological parameters. The constraints from skew spectra are competitive with those from the bispectrum monopole at $k_{\rm max}=0.2\,{{\rm Mpc}^{-1} h}$ and become less constraining at $k\sim\!0.3\,{{\rm Mpc}^{-1} h}$ due to the smoothing of the density field. Using Molino galaxy catalogs, we investigated the effect of marginalization over parameters of a more complex biasing relation between tracers and DM. For this dataset, upon marginalization over HOD parameters, we found an improvement of $(32-71)\%$ in cosmological constraints from skew spectra compared to power spectrum multipoles ($\ell = 0, 2, 4$). Given the excessive power of satellite galaxies at small scales, in this analysis, we included the power spectrum hexadecapole. Interestingly enough, despite the strong random-motion-associated redshift space distortion on the small scales, the shapes of the 14 skew spectra and degeneracy directions between cosmological parameters on the Molino sample are unaffected.
We examined how the choice of smoothing scale affects the constraints and found that using a smoothing scale of $R=10\,{h^{-1}\rm{Mpc}}$ tightens the constraints on cosmological parameters, on average by $\sim 20\%$, compared to the analysis with $R=20\,{h^{-1}\rm{Mpc}}$. While the perturbative theoretical model of the skew spectra (based on tree-level bispectrum prediction) was shown to be more reliable with $R=20\,{h^{-1}\rm{Mpc}}$ \cite{Schmittfull:2020hoi}, in this paper, we presented both scales but considered the results for the smaller smoothing scale as our baseline analysis. On the one hand, the theoretical model can be improved by going beyond the tree-level bispectrum, and on the other hand, non-standard analysis methods, which do not rely on the availability of theoretical predictions, such as likelihood-free inference~\cite{Hahn2022simbig1, Hahn2022simbig2}, can potentially enable us to extract the information from these smaller scales.
Apart from the simplifying assumptions intrinsic to Fisher forecasts, in interpreting simulation-based numerical forecasts, testing the stability of the results with respect to variation of the number of mocks used in the estimation of derivatives and covariance matrix is essential. We estimated the additional variance that enters the sampling covariance, which is estimated from a limited number of realizations, and found that for 15,000 mocks, the noise in the covariance leads to an additional variance of 3.8\% for the skew spectra, compared to a sub-percent level for the power spectrum. Testing the convergence with regards to the noise in the measurement of the derivatives, we showed that the derivatives estimated from the pairs of mocks and averaged over the 3 LoS, using 500 of such pairs, could lead to an underestimation of the constraints by $(20-30)\%$ on neutrino masses. This is the case for skew spectra or power spectra and both on Quijote or Molino samples. For this reason, instead of quoting the absolute number, we largely focused on the relative improvement between the two statistics.
While the results presented in this work are certainly very encouraging, the information content of the skew spectra is far from established for realistic observational data, and this work is just the first step in this regard. There are several directions in which this work can be extended, some of which we will pursue in upcoming works. First, the fact that the 14 skew spectra can be classified into three categories implies that there may be more efficient ways to group the information. Currently, sorting the skew spectra according to their power in growth rate and bias is motivated by being a maximum-likelihood estimator for the ``amplitude-like" parameters. It remains an interesting question to compress the data vector in a more efficient way for generalized applications. Second, In order to perform theory-based likelihood analysis of the observational data, more accurate perturbative theoretical models of the skew spectra, in particular when assuming a smaller smoothing scale, are necessary. Third, in order to be applied to survey data, the construction of survey estimators, including the observational effects, and accounting for the survey window in theoretical predictions, is required.
\section*{Acknowledgments}
We are grateful to William Coulton, Yangyang Li, and Zvonomir Vlah for very helpful discussions. We thank Paco Francisco Villaescusa-Navarro for guidance on the use of Quijote data, and Oliver Philcox, Marcel Schmittfull, and Zack Slepian for their helpful feedback on the draft of this manuscript. JH has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Sk\l{}odowska-Curie grant agreement No 101025187. AMD acknowledges funding from Tomalla Foundation for Research in Gravity and Boninchi Foundation. The authors acknowledge the University of Florida Research Computing for providing computational resources and support that have contributed to the research results reported in this publication.
|
3,212,635,537,912 | arxiv | \section{Introduction}
The chiral $SU(N_f)_L\times SU(N_f)_R$ symmetry of the QCD Lagrangian with $N_f$ (massless) quark flavors is known to be broken down spontaneously to $SU(N_f)_V$ in the
nonperturbative QCD vacuum. With inclusion of explicit chiral symmetry breaking by the nonvanishing quark masses, this symmetry-breaking pattern is visibly manifest in the spectrum of the low-mass pseudoscalar meson octet for $N_f = 3$. This octet figures as the massless Goldstone bosons of spontaneously broken chiral $SU(3)_L\times SU(3)_R$ symmetry in the limit of massless $u$, $d$, and $s$ quarks. An equivalent but not directly observable manifestation of spontaneous chiral symmetry breaking (S$\chi$SB) is the appearance of a nonzero quark condensate. Its observable counterpart is the pion decay constant, $f_\pi$, which acts as an order parameter for S$\chi$SB and defines a chiral-symmetry-breaking scale, $4\pi f_\pi \sim 1$ GeV, briefly referred to as the chiral scale. Further evidence for S$\chi$SB is the absence of parity doublets in the lower mass area of the hadron spectrum, an important example being the large mass gap between the $\rho$ $(J^P = 1^-)$ and $a_1$ $(J^P = 1^+)$ mesons. Thus it is well established that at zero temperature, QCD is in the Nambu-Goldstone realization of chiral symmetry. Were it in the Wigner-Weyl realization with a trivial vacuum and parity doublets, the $\rho$ and $a_1$ would be degenerate. One expects this latter situation to be reached as the temperature is raised toward a critical value, $T_c \sim 0.2$ GeV, for the chiral transition. Around this point and beyond, the $\rho$ and $a_1$ are expected to melt into the quark-antiquark continuum.
Lattice QCD results~\cite{Cheng:2006qk, Aoki:2006br} do indicate that the spontaneously broken chiral
symmetry is restored at high temperature. The spectral functions of chiral partners such as the (isovector) vector and and axial-vector mesons are then supposed to become degenerate. In this context, concerning the in-medium behavior of vector mesons, measurements of dileptons produced
in relativistic heavy-ion collisions have attracted great interest over the past decades (see, e.g., Ref. \cite{NA60}). Dileptons as well as photons are excellent probes carrying information about vector meson spectral functions at the high temperatures and densities reached in the collision zone. In particular, the $\rho$ meson, as the lowest dipole excitation of the QCD vacuum, has been in the focus
of such investigations.
The issue of in-medium hadronic properties persists as a basic theme ever since the suggestion of Brown and
Rho (BR)~\cite{Brown:1991kk} that hadron masses should decrease systematically when a hadronic medium undergoes the transition toward chiral symmetry restoration. The BR scaling hypothesis states that the dropping of hadron masses, with the exception of the pion mass, should go in parallel with the dropping of the pion decay constant $f_\pi$ (here the one related to the time component of the axial current) in hot and dense hadronic matter. In general, a sufficient condition for chiral symmetry restoration is that the in-medium mass difference between the $\rho$ and $a_1$ mesons tends to zero, although it is not mandatory that the masses vanish individually. In practice, however, these mass-shift scenarios are overshadowed by strong spectral-broadening effects. The $\rho$ and $a_1$ resonances have large widths already in the vacuum. Their interactions with hadrons in the medium increase their decay widths significantly through collision broadening. Distinguishing ``mass shift'' from ``broadening'' scenarios is obviously not a meaningful issue once the spectral functions become very broad and do not show clear resonance structures.
In a situation like this, it nevertheless still makes sense to perform a sum-rule analysis of such broad spectral distributions, focusing on their lowest moments and defining a mean mass through
the first moment of the in-medium spectral function. Such an analysis is made possible by identifying the characteristic chiral-symmetry-breaking scale, $\Lambda_{\mathrm{CSB}}\approx4\pi f_\pi$, with the continuum threshold separating the low-energy-resonance region from the high-energy continuum. Consider the example of the vector spectral function, to be specified in detail later:
\begin{equation}
R_V(s)=R_V^{res}(s)\,\Theta(s_V-s)+R_V^{cont}\,\Theta(s-s_V)~,
\end{equation}
where $R_V^{res}(s)$ denotes the resonance part. This low-energy part is separated by the scale $s_V$ from the high-energy continuum part $R_V^{cont}$ determined by perturbative QCD. In vacuum, the identification $\sqrt{s_V}=4\pi f_\pi$ is supported by current algebra relations and spectral sum rules, as we point out later. The extension to the case of finite baryon density at zero temperature has been studied in our previous work~\cite{Kwon:2008vq} where it is demonstrated that the aforementioned scale analysis works with $\sqrt{s_V(\rho)} = 4\pi f_\pi^*(\rho)$ now interpreted in terms of the in-medium change of the chiral order parameter.
Before turning to finite temperatures, it is useful to recall Weinberg's sum rules ~\cite{Weinberg:1967kj}. These sum rules are entirely based on current algebra and establish rigorous relations for the difference between vector and axial-vector spectral functions:
\begin{eqnarray}
\int^{\infty}_0\mathrm{d} s\,\big[R_V(s)-R_A(s)\big]&=&0~,
\label{eq:Wsr1}\\
\int^{\infty}_0\mathrm{d} s\,s\big[R_V(s)-R_A(s)\big]&=&0~.
\label{eq:Wsr2}
\end{eqnarray}
Assume first the following schematic forms for the vector and axial-vector spectral functions:
\begin{equation}
\begin{split}
R_{V}(s)&=R_{V}^{res}(s)\,\Theta(s_{V}-s)+c_0\,\Theta(s-s_{V})~,\\
R_{A}(s)&=R_{A}^{res}(s)\,\Theta(s_{A}-s)+c_0\,\Theta(s-s_{A})~,\\
\end{split}
\label{eq:specftn0}
\end{equation}
where $c_0$ multiplying the continuum parts of the spectral functions is determined by perturbative QCD $(c_0 = 3/2$ in leading order for the $\rho$ and $a_1$ channels). The equality of these vector and axial-vector continuum pieces is indicative of chiral symmetry restoration
at high energy beyond the scales $s_{V,A}$. For the resonant parts, choose for simplicity a zero-width ansatz (as realized
in the large-$N_c$ limit):
\begin{equation}
\begin{split}
R_V^{res}(s)&=12\pi^2\,f_V^2 m_V^2\,\delta(s-m_V^2)~,\\
R_A^{res}(s)&=12\pi^2\,\big[f_\pi^2\,\delta(s-m_\pi^2)+f_A^2m_A^2\,\delta(s-m_A^2)\big]~,
\end{split}
\label{eq:specftn0prime}
\end{equation}
where $f_{V/A}$ are dimensionless vector and axial-vector couplings respectively. The $a_1$ spectral function includes a pion pole term with its residue determined by the pion decay constant $f_\pi$. In the chiral limit ($m_\pi\rightarrow 0$), and assuming equal continuum thresholds for the vector and axial-vector channels, $s_V = s_A$, the Weinberg sum rules (\ref{eq:Wsr1},\ref{eq:Wsr2}) read as follows:
\begin{equation}
\begin{split}
f_V^2m_V^2-f_A^2m_A^2&=f_\pi^2~,\\
f_V^2m_V^4-f_A^2m_A^4&=0~.
\end{split}
\label{eq:largeNWSR}
\end{equation}
These equations, together with the KSRF relation~\cite{Kawarabayashi:1966kd,Riazuddin:1966sw},
\begin{equation}
f_V^2m_V^2=2f_\pi^2~,
\label{eq:lKSFR}
\end{equation}
imply that the vacuum $a_1$ and $\rho$ masses satisfy the well-known relation
\begin{equation}
m_A^2=2m_V^2~.
\label{eq:mass}
\end{equation}
The Weinberg sum rules are based on the observation that chiral symmetry is restored in its Wigner-Weyl realization asymptotically, at high-energy scales, where QCD is perturbative. The duality between the resonant and asymptotic parts of the spectral functions derived from current correlators is one of the basic ideas of QCD sum rules, to which we now proceed. The present study constructs finite energy sum rules (FESR) for $\rho$ and $a_1$ meson at nonzero temperature and vanishing baryon chemical potential. We show how finite temperature FESRs can provide constraints for the pattern and trend toward chiral symmetry restoration. Realistic vacuum spectral functions are employed as input in comparison with schematic $\delta$ function spectra, and the $\rho$-$a_1$ parity-mixing scenario is used to describe temperature-dependent effects.
\section{Finite energy sum rules}
Following these preparations, the starting point is now the time-ordered current correlation function
\begin{equation}
\Pi^{\mu\nu}(q)=i\int\mathrm{d}^4x\,e^{iq\cdot x}\langle\mathcal{T}
j^\mu(x)j^\nu(0)\rangle_T,
\label{CCcorelator}
\end{equation}
with the vector current $j_V^\mu(x)=\frac{1}{2}\left(\bar{u}\gamma^\mu u-\bar{d}\gamma^\mu d\right)$ and the axial-vector
current $j_A^\mu=\frac{1}{2}\left(\bar{u}\gamma^\mu\gamma_5u-\bar{d}\gamma^\mu\gamma_5d\right)$ carrying the quantum
numbers of $\rho$ and $a_1$ meson, respectively. The bracket $\langle\mathcal{O}\rangle_T$ indicates the thermal expectation value of an operator $\mathcal{O}$,
\begin{equation}
\langle\mathcal{O}\rangle_T=\frac{\mathrm{tr}\,\mathcal{O}\,\exp\left(-H/T\right)}{\mathrm{tr}\,\exp\left(-H/T\right)},
\label{eq:TherAV}
\end{equation}
where $H$ is the Hamiltonian.
In vacuum, the tensor correlation function (\ref{CCcorelator}) can be related to a single invariant correlator, $\Pi(q^2)=\frac{1}{3}g_{\mu\nu}\Pi^{\mu\nu}$. In a medium, the longitudinal and transverse parts of the correlator differ as a consequence of broken Lorentz invariance. In the rest frame of the medium and for the case in which the mesons have vanishing three-momentum $\mathbf{q}=0$, however, longitudinal and transverse correlation functions coincide and again are given as a single function $\Pi(\omega,\mathbf{q}=0)$. This correlator is written in the form of a twice-subtracted dispersion relation:
\begin{equation}
\Pi(q^2)=\Pi(0)+\Pi^\prime(0)\,q^2+\frac{q^4}{\pi}\int{\mathrm{d} s}\frac{\mathrm{Im}\Pi(s)}{s^2(s-q^2-i\epsilon)}.
\label{dispersion}
\end{equation}
On the other hand, the operator product expansion (OPE) is used to represent the correlator at large spacelike momentum, $q^2=-Q^2<0$:
\begin{equation}
12\pi^2\Pi(q^2 = -Q^2)=-c_0Q^2\ln\left(\frac{Q^2}{\mu^2}\right)+c_1+\frac{c_2}{Q^2}+\frac{c_3}{Q^4}+\cdots~,
\label{ope}
\end{equation}
with the coefficients
\begin{equation}
\begin{split}
c_0&=\frac{3}{2}\left(1+\delta_N\right)~,\\
c_1&=-\frac{9}{2}(m^2_u+m^2_d)~,\\
c_2&=\frac{\pi^2}{2}\left\bra\frac{\alpha_s}{\pi}G^2\right\ket_T\pm6\pi^2\left(m_u\langle\bar{u}u\rangle_T+m_d\langle\bar{d}d\rangle_T\right)~.
\end{split}
\label{c_n}
\end{equation}
Here, $\delta^{}_N$ in $c_0$ denotes the radiative corrections in perturbative QCD. The explicit
form of $\delta^{}_N$ up to order $\alpha_s^3(s)$ can be found in Ref.~\cite{Kwon:2008vq}. The difference between vector and axial-vector channels results to this order from the sign of the quark condensate term in $c_2$. In the chiral limit ($m_{u,d}\rightarrow 0$), $c_1$ and the second term of $c_2$ vanish. The first appearance of a difference between $\rho$ and $a_1$ spectral functions is then in the term proportional to $c_3$ in the expansion (\ref{ope}). This term involves four-quark condensates,
\begin{equation}
\begin{split}
c_3&=-6\pi^3\alpha_s\big[\langle(\bar{u}\gamma_\mu\gamma_5\lambda^au\mp\bar{d}\gamma_\mu\gamma_5\lambda^ad)^2\rangle_T\\
&\quad+\frac{2}{9}\langle(\bar{u}\gamma_\mu\lambda^au+\bar{d}\gamma_\mu\lambda^ad)\sum_{q=u,d,s}\bar{q}\gamma^\mu\lambda^aq\rangle_T\big]~,
\end{split}
\label{c_3}
\end{equation}
that are subject to large uncertainties. To evaluate these condensates, a factorization approximation is frequently used, assuming that intermediate states are saturated by the QCD ground state:
\begin{equation}
\langle(\bar{q}\gamma_\mu\gamma_5\lambda^a{q})^2\rangle=-\langle(\bar{q}\gamma_\mu\lambda^a{q})^2\rangle
=\frac{16}{9}\kappa\,\langle\bar{q}q\rangle^2~,
\label{4q}
\end{equation}
with $\kappa$ introduced to parametrize the deviation from exact factorization ($\kappa=1$).
However, the factorization approximation for the four-quark condensates is not sufficiently accurate for any quantitative considerations~\cite{Kwon:2008vq}. This basic uncertainty can be avoided by arranging sum rules in terms of moments of the spectral function and restricting the analysis to the lowest two moments:
\begin{eqnarray}
\int^{s_0(T)}_0\mathrm{d} s\,R(s,T)&=& \nonumber
\\s_0(T)\,c_0 &+& c_1-12\pi^2\,\Pi(0)~,
\label{0momsr}\\
\int^{s_0(T)}_0\mathrm{d} s\,s\,R(s,T)&=&\frac{s^2_0(T)}{2}\,c_0-c_2(T)~.
\label{1momsr}
\end{eqnarray}
The dimensionless, temperature-dependent spectral function $R(s,T)$ stands generically for the vector or axial channel and is defined as
\begin{equation}
R(s,T)=-\frac{12\pi}{s}\mathrm{Im}\Pi(s,T)~. \nonumber
\end{equation}
Its high-energy continuum is separated from the low-energy part by the scale $s_0(T)$ that is expected to shift downward with increasing temperature. Again, this $s_0$ stands for the continuum
threshold scale in either vector or axial-vector channel. These scales, denoted in the following by $s_V(T)$ and $s_A(T)$, are generally different and to be determined by the detailed sum-rule analysis. The last term on the right-hand side of Eq.~(\ref{0momsr}) vanishes in the vector channel and represents the pion pole contribution in the axial-vector channel, with $\Pi(0)=f_\pi^2$ at $T=0$.
In the asymptotic region ($Q^2\to\infty$) where the OPE is valid, all nonperturbative scales appear in the form of power corrections to the perturbative calculations. These nonperturbative contributions are separated into the respective condensates. At low temperatures, it can be assumed that, apart from the scales $s_{V,A}(T)$, the $T$ dependence is only in the condensates.
\section{Evaluation of $\langle\mathcal{O}\rangle_T$}
The evaluation of the $T$-dependent condensates follows the method employed previously in Ref.~\cite{Hatsuda:1992bv}. In Eq.~(\ref{eq:TherAV}), the vacuum state and the lowest excitations of the hadron gas, namely pions, are taken into account as eigenstates of the Hamiltonian to calculate the thermal expectation values at low temperatures:
\begin{equation}
\langle\mathcal{O}\rangle_T=\langle\mathcal{O}\rangle_0+\sum^3_{a=1}\int\frac{\mathrm{d}^3p}{2E(2\pi)^3}\,\langle\pi^a(p)|\mathcal{O}|\pi^a(p)\rangle\, n^{}_B~,
\label{eq:matrixT}
\end{equation}
where $n^{}_B=\left(e^{E/T}-1\right)^{-1}$ denotes Bose-Einstein distributions of thermal pions with $E^2=m_\pi^2+p^2$.
The pion matrix element in Eq.~(\ref{eq:matrixT}) can be evaluated in the soft pion limit:
\begin{equation}
\langle\pi^a(p)|\mathcal{O}|\pi^a(p)\rangle=-\frac{1}{f_\pi^2}\bra0|\left[\mathcal{Q}^a_5,\,\left[\mathcal{Q}^a_5,\,\mathcal{O}\right]\right]|0\rangle+\cdots,
\label{eq:softpion}
\end{equation}
where $\mathcal{Q}_5^a$ is the axial charge operator. Equations (\ref{eq:matrixT}) and (\ref{eq:softpion}), when applied to the scalar quark operator, $\bar{q}q$, give the leading order $T$ dependence of the chiral condensate,
\begin{equation}
\bra\bar{q}q\ket_T=\bra\bar{q}q\ket_0\left[1-\frac{T^2}{8f_\pi^2}B_1\left(\frac{m_\pi}{T}\right)\right]~,
\label{eq:LOqucondT}
\end{equation}
with
\begin{equation}
B_1(x)=\frac{6}{\pi^2}\int^\infty_x \mathrm{d}
y\,\frac{\sqrt{y^2-x^2}}{e^y-1}~.
\end{equation}
Equation (\ref{eq:LOqucondT}) is the well-known result derived from chiral effective field theory~\cite{Gasser:1986vb,Gerber:1988tt,Kaiser:1999mt}.
The application of Eq.~(\ref{eq:matrixT}) to the gluon condensate makes use of the QCD trace anomaly,
\begin{equation}
\Theta^\mu_\mu=-\frac{1}{8}\left(11-\frac{2}{3}N_f\right)\frac{\alpha_s}{\pi}G^2+\sum_q m_q\bar{q}q~,
\end{equation}
to calculate the pionic matrix element of the relevant gluon operator
\cite{Hatsuda:1990uw}. For $N_f=3$, the resulting temperature dependent gluon condensate becomes
\begin{equation}
\left\bra\frac{\alpha_s}{\pi}G^2\right\ket_T=\left\bra\frac{\alpha_s}{\pi}G^2\right\ket_0-\frac{1}{9}m_\pi^2T^2B_1\left(\frac{m_\pi}{T}\right)~,
\label{eq:glcond_T}
\end{equation}
in which the second term of the right-hand side gives a numerically minor contribution to the sum rule. In the actual calculation, the Gell-Mann$-$Oakes$-$Renner (GOR) relation is used for evaluating the vacuum quark condensate,
$m_q\langle\bar{q}q\rangle=-(0.11\,\mathrm{GeV})^4$, while the charmonium sum rules~\cite{Ioffe:2005ym} constrain the vacuum gluon condensate as $\left\bra\frac{\alpha_s}{\pi}G^2\right\ket=0.005\pm0.004\,\mathrm{GeV}^4$, with large uncertainty. In practice, this uncertainty is not prohibitive since the overall correction induced by $c_2$ in Eq. (\ref{1momsr}) is small compared to the dominant term proportional to $s_0^2(T)$.
Apart from the in-medium modifications of the scalar condensates, new spin-dependent operators appear in the OPE because of broken Lorentz invariance in the heat bath. Such operators are classified by their canonical dimension and twist ($\tau=\text{dimension}-\text{spin}$). The $c_2(T)$ in the sum rule for the first moment now includes the contribution from the twist-2 operator $\mathcal{ST}\,\bar{q}\gamma_\mu D_\nu q$, where the symbol $\mathcal{ST}$ makes the operator symmetric and traceless with respect to its Lorentz indices. The pion matrix element of this operator in Eq. (\ref{eq:softpion}) is evaluated using its relation to the quark distribution function in the pion~\cite{Hatsuda:1992bv}. In the actual calculation, this contribution to $c_2(T)$ is about three times smaller than the leading $T$-dependent correction to the quark condensate in Eq. (\ref{eq:LOqucondT}).
\section{Phenomenology at low $T$}
The mixing of vector and axial-vector correlation functions at finite temperature is introduced following Ref. \cite{Dey:1990ba} and translates correspondingly to the spectral functions:
\begin{eqnarray}
R_V(s,T)&=&\left[1-\epsilon(T)\right]\,R_V(s,0) + \epsilon(T)\,R_A(s,0)~,\nonumber\\
R_A(s,T)&=& \epsilon(T)\,R_V(s,0) + \left[1-\epsilon(T)\right]\,R_A(s,0).
\label{eq:V-Amixing}
\end{eqnarray}
The mixing parameter $\epsilon$ is given by the thermal pion loop integral
\begin{equation}
\epsilon(T)=\frac{2}{f_\pi^2}\int\frac{\mathrm{d}^3p}{(2\pi)^3}\frac{1}{E(e^{E/T}-1)}~.
\end{equation}
In the chiral limit ($m_\pi\to0$) it reduces to $\epsilon=T^2/(6f_\pi^2)$. At the temperature $T_d$ where $\epsilon\simeq0.5$, the vector and axial-vector spectral functions are maximally mixed and become degenerate, $R_V(s,T)=R_A(s,T)$. The temperature at which this degeneracy takes place,
$T_d = \sqrt{3}\,f_\pi\simeq 151$ MeV (with $f_\pi = 87$ MeV in the chiral limit), is not far from the characteristic temperature commonly associated with the chiral crossover transition ($T_c = 170 - 190$ MeV) \cite{Cheng:2006qk, Aoki:2006br}. Of course, as one approaches this temperature range,
terms of higher order in $T^2$ become important and need to be taken into account.
It was pointed out in Ref.~\cite{Dey:1990ba} that the vector meson mass changes weakly at low
temperature (i.e., at order $T^2$). We demonstrate this feature next by performing the FESRs introduced in the previous sections.
\subsection{Spectral functions with zero width}
It is instructive to test the FESR for the vector and axial-vector spectral functions with zero width,
the limiting situation realized in the large $N_c$ limit of QCD:
\begin{equation}
\begin{split}
R_V(s,T=0)&=12\pi^2\,f_V^2\,m_V^2\,\delta(s-m_V^2)~,\\
R_A(s,T=0)&=12\pi^2\,f_A^2\,m_A^2\,\delta(s-m_A^2)~.
\label{eq:spftn0}
\end{split}
\end{equation}
The vector and axial-vector couplings, $f_{V/A}$, are determined by
\begin{equation}
f_V^2\,m_V^2=2f_\pi^2,\qquad
f_A^2\,m_A^2\simeq f_\pi^2+\frac{s_A-s_V}{8\pi^2}~.
\label{eq:fVA}
\end{equation}
The first part is just the KSRF relation; the second one is derived by applying the KSRF relation
to the first Weinberg sum rule in Eq.~(\ref{eq:largeNWSR}) and adding the correction coming from the difference between $s_A$ and $s_V$. Then the lowest two moments of the vector sum rule are:
\begin{eqnarray}
12\pi^2\,f_V^2\,m_V^2\left[1-\epsilon(T)\right]&=&\frac{3}{2}\,s_V(T)-12\pi^2f_\pi^2\,\epsilon(T)~,\nonumber\\
\label{eq:chiLVsr0}\\
12\pi^2\,f_V^2\,m_V^4\left[1-\epsilon(T)\right]&=&\frac{3}{4}\,s_V^2(T)-c_2(T)~,
\label{eq:chiLVsr1}
\end{eqnarray}
where the $a_1$ meson contribution has been omitted because in the actual calculation, $\sqrt{s_V}$ is found to be located at $s_V\leq m_A^2$ so that the $a_1$ pole does not contribute to the integrals of the spectral moments.
With this setup, the continuum threshold in the vector channel is evaluated to be $\sqrt{s_V}\simeq1.1~\mathrm{GeV}$ at $T=0$ from the sum rules, using $f_\pi\simeq87~\mathrm{MeV}$ in the chiral limit and $m_V=770~\mathrm{MeV}$. Note that $\sqrt{s_V}$ turns out to be perfectly close to the chiral scale, $4\pi\,f_\pi$.
The finite temperature behavior of $\sqrt{s_V(T)}$ extracted from the sum rules to leading order in $\epsilon$ is shown in Fig.~\ref{fig:chirho}. The sum rules for the lowest two moments give consistent results at low temperature. However, although the identification $\sqrt{s_V}=4\pi f_\pi$ emerges naturally from Eq.~(\ref{eq:chiLVsr0}) at $T=0$, the temperature dependence of $\sqrt{s_V}$ evolves
more slowly than the canonical $f_\pi(T)=\left[1-\epsilon(T)/2\right]f_\pi(T=0)$ found in chiral perturbation
theory~\cite{Gasser:1986vb}.
\begin{figure}[ht]
\centering
\includegraphics[width=8cm]{rhoTdsth}
\caption{(Color online) Continuum threshold $\sqrt{s_V}$ in the vector meson channel as a function of $T$ obtained from the sum rules for the 0th (black solid line) and 1st (red dashed line) moment of the spectral function. The $T$ dependence of chiral scale $4\pi f_\pi(T)$ (blue dotted line) is also displayed.}
\label{fig:chirho}
\end{figure}
Defining an average mass $\bar{m}$ by the normalized first moment of the spectral function,
\begin{equation}
\bar{m}^2\equiv\frac{\int\mathrm{d}{s}\,sR}{\int\mathrm{d}{s}\,R}~,
\label{eq:avmass}
\end{equation}
we confirm that the $\rho$ meson mass remains unchanged for $T\neq0$ and stays at the vacuum pole position, $\bar{m}_V=770~\mathrm{MeV}$, to leading order in $\epsilon$. This is consistent with the statement \cite{Dey:1990ba,Eletsky:1994rp} that the vector meson pole position remains unchanged at order $T^2$.
\begin{figure}[ht]
\centering
\includegraphics[width=8cm]{a1Tdsth}
\caption{(Color online) Axial-vector continuum threshold $\sqrt{s_A}$ as a function of $T$ obtained from the sum rules for the 0th (black solid line) and 1st (red dashed line) moment of the $a_1$ spectral function. In contrast to the $\rho$ meson, the $a_1$-meson mass (black dotted line) decreases with rising temperature.}
\label{fig:chia1}
\end{figure}
In contrast to the $\rho$ meson case in which the $a_1$ pole does not contribute to the integrals of the spectral moments (since $s_V\leq m_A^2$), the left-hand side of the sum rules in the axial-vector channel receives contributions from both $\rho$ and $a_1$ poles. By using Eq.~(\ref{eq:fVA}) the sum rule for the
first moment of the $a_1$ spectral function becomes
\begin{equation}
12\pi^2\left[f_A^2\,m_A^4(1-\epsilon)+f_V^2\,m_V^4\,\epsilon\right]=\frac{3}{4}s_A^2-c_2~.
\label{eq:chiLAsr1}
\end{equation}
With $m_A=1.26~\mathrm{GeV}$, this gives $\sqrt{s_A}\simeq1.54~\mathrm{GeV}$ at $T=0$ and thus $f_A^2\simeq0.014$, which agrees with the empirical values of $f_A^2$ in Ref.~\cite{Sakurai:1980js}. As seen in Fig.~\ref{fig:chia1}, the average axial-vector mass $\bar{m}_A$ defined by Eq.~(\ref{eq:avmass}) decreases in parallel with the continuum threshold $\sqrt{s_A}$ as the temperature rises.
One observes that the $a_1$ mass $\bar{m}_A\simeq1.09~\mathrm{GeV}$ does not yet approach the $\rho$ mass $\bar{m}_V=770~\mathrm{MeV}$ at temperatures around $T_d\simeq151~\mathrm{MeV}$, where $\epsilon=0.5$, suggesting that higher order effects in $\epsilon$ become important as one gets closer to the critical region for chiral restoration.
\subsection{Realistic spectral functions with finite width}
In this section we study realistic empirical spectral functions with finite widths as displayed in Fig.~\ref{fig:specftn}. The continuum is still described by a step function. In previous
work~\cite{Kwon:2008vq}, we have shown that, both in the vacuum and at finite baryon density, the detailed choice for modeling the continuum threshold is not decisive for the outcome of the sum-rule analysis. In practice, the continuum threshold can be given an improved description using a
\begin{figure}[ht]
\centering
\includegraphics[width=8cm]{spectral}
\caption{(Color online) Vector (black curve) and axial-vector (red curve) spectral distributions in vacuum, compared to $e^+e^-\to{n}\pi$ data with $n$ even~\cite{Aloisio:2004bu,Dolinsky:1991vq} and data from hadronic $\tau$ decays~\cite{Barate:1998uf,Ackerstaff:1998yj}. Here $s_V$ and $s_A$ stand for the continuum thresholds in vector and axial-vector channels, respectively. The ramping function (blue dashed line) shows an example of smooth threshold modeling (see text and appendix).}
\label{fig:specftn}
\end{figure}
ramping function, resulting in the blue solid curve in Fig.~\ref{fig:specftn}, which reproduces the experimental data very well. We show that for the finite-temperature sum rules considered here, the results are also consistently stable, independent of the detailed threshold modeling, as long as the slope of the ramping into the continuum is chosen sufficiently large (see Appendix~\ref{appx2}).
\begin{figure}[ht]
\centering
\includegraphics[width=8cm]{mixspectral}
\caption{(Color online) The $\rho$ and $a_1$ meson spectra at finite temperature.}
\label{fig:specFT}
\end{figure}
Figure \ref{fig:specFT} exhibits the temperature-dependent $\rho$ and $a_1$ spectra generated by Eq.~(\ref{eq:V-Amixing}) with physical values of the pion mass and decay constant, $m_\pi=139.6~\mathrm{MeV}$ and $f_\pi=92.4~\mathrm{MeV}$. The pole position of the vector meson resonance stays at its vacuum mass. However, when examining the average mass defined by Eq.~(\ref{eq:avmass}),
\begin{equation}
\bar{m}_V^2(T) = \frac{\frac{c_0}{2}s^2_V(T)-c_2(T)}{c_0\,s_V(T) +c_1-12 \pi^2 f_\pi^2\, \epsilon(T)}~,
\label{eq:avVmass}
\end{equation}
the broad widths of the $\rho$ and $a_1$ spectra affect the $\rho$ mass distribution.
\begin{table}[ht]
\centering
\begin{tabular}{|c|c|c|}
\hline
$T$ (MeV) & $\bar{m}_V$ (MeV) & $\sqrt{s_V}$ (GeV) \\
\hline
$0$ & ~$791\pm2$\,\, & ~$1.139\pm0.007$\,\, \\
$50$ & ~$792\pm2$\,\, & ~$1.138\pm0.007$\,\, \\
$100$ & ~$800\pm4$\,\, & ~$1.133\pm0.008$\,\, \\
$120$ & ~$807\pm5$\,\, & ~$1.128\pm0.009$\,\, \\
\hline
\end{tabular}
\caption{Average vector mass and continuum threshold at various $T$ from Eqs.~(\ref{eq:avVmass}) and (\ref{1momsr}). The errors include uncertainties in the value of $\alpha_s$ (entering the NLO perturbative QCD corrections) and of the gluon condensate. }
\label{tab:avVmass}
\end{table}
From Table~{\ref{tab:avVmass}}, it is apparent that the average vector mass tends to slightly move upward with temperature, picking up weight from the mixing with the $a_1$, while $\sqrt{s_V}$ moves downward. When using the tightly constrained vacuum spectral functions as input at $T=0$, it turns out that there can be a small (<5\%) mismatch between the left-hand and the right-hand sides of the sum rules. The error assignments in the values of $\sqrt{s_V}$ from the sum rule for the first spectral moment take this small uncertainty into account in Table~{\ref{tab:avVmass}}. In practice, the errors in
Table~\ref{tab:avVmass} come primarily from the uncertainties of the gluon vacuum condensate and the running strong coupling $\alpha_s(s)$.
As the temperature increases the continuum onset scale $s_V(T)$ shows evidentally a tendency to decrease. At temperatures $T > 140$ MeV, however, solutions for $\sqrt{s_V}$ cease to exist,
and hence the average mass at that high $T$ cannot be determined via Eq.~(\ref{eq:avVmass}). This is partly a consequence of the restrictions imposed by the treatment to leading order in $\epsilon(T)$. Even more so, it is related to the fact that, with realistic spectral functions, the broad $a_1$ distribution does not permit a separation of scales between low- and high-energy parts of the spectrum. The increased mixing of the $a_1$ into the $\rho$ spectrum at high temperature enhances this effect.
Thus, with realistic spectral distributions, the FESR approach has its applicability limited to
the range well below the ``melting'' temperature of the resonances.
As mentioned in the introduction, the difference between vector and axial-vector correlators serves as an order parameter of spontaneous chiral symmetry breaking and restoration. Direct substraction of the FESRs in the axial-vector and vector channels gives
\begin{equation}
\begin{split}
\int^{s_V(T)}_0\mathrm{d} s\,R_V&(s,T)-\int^{s_A(T)}_0\mathrm{d} s\,R_A(s,T)\\
=c_0&\left[s_V(T)-s_A(T)\right]+12\pi^2f_\pi^2\left[1-2\epsilon(T)\right]~,\\
\int^{s_V(T)}_0\mathrm{d} s\,s\,R_V&(s,T)-\int^{s_A(T)}_0\mathrm{d} s\,s\,R_A(s,T)\\
&=\frac{c_0}{2}\left[s_V^2(T)-s_A^2(T)\right]~.
\label{eq:V-Asr}
\end{split}
\end{equation}
Keeping the leading order mixing ansatz (\ref{eq:V-Amixing}) for the spectral functions, one finds $\sqrt{s_V}\simeq1.12$ GeV and
$\sqrt{s_A}\simeq1.49$ GeV at $T=0$ from Eqs. (\ref{eq:V-Asr}). Note that $\sqrt{s_V}$ is again close to the chiral scale $4\pi f_\pi$. It is important to realize that $s_A$, as it results from the consistency conditions of the sum-rule analysis, is different from (i.e., significantly larger than) $s_V$. In fact, the $a_1$ mass itself is comparable to $4\pi f_\pi$ so that the continuum threshold in the axial-vector channel must necessarily be located at a higher scale than $s_V$ once the large $a_1$ width is taken into account. Assuming $s_A=s_V \equiv s_0$ as in the schematic Weinberg sum rules (\ref{eq:largeNWSR}), one finds that the leading order temperature dependence, $1-2\epsilon$, just drops out in Eq.~(\ref{eq:V-Asr}) and yields a trivial result,
\begin{equation}
\begin{split}
\int^{s_0}_0\mathrm{d} s\,\left[R_V(s,0)-R_A(s,0)\right]&=12\pi^2f_\pi^2~,\\
\int^{s_0}_0\mathrm{d} s\,s\left[R_V(s,0)-R_A(s,0)\right]&=0~,
\end{split}
\end{equation}
independent of temperature.
\section{Higher order corrections}
\subsection{$T^4$ corrections}
Effects at order $T^4$ have been treated systematically in Ref.~\cite{Eletsky:1994rp}. It is shown there that these corrections amount to replacing the mixing parameter $\epsilon$ by
$\epsilon\,\to\,\epsilon(1-\epsilon/2)$. The temperature $T_d$ at which the $\rho$ and $a_1$ spectral distributions become degenerate is now shifted upward to $T_d = \sqrt{6} f_\pi$.
The $T^4$ correction incorporates interacting pions with nonvanishing momentum in the heat bath, which contribute to the Lorentz noninvariant part of the current correlation
function. According to the Ref.~\cite{Eletsky:1994rp}, this contribution can be expressed in the frame with
$\mathbf{q}=0$ as follows:
\begin{equation}
\begin{split}
\Pi_V(q^2 = -Q^2,T)~&=~ \Pi_V(-Q^2,T=0)\\
\quad-~\epsilon\left(1-\frac{\epsilon}{2}\right)&\left[\Pi_V(-Q^2,T=0)-\Pi_A(-Q^2,T=0)\right]\\
+&\frac{4\pi^2\,M_2}{15 \,Q^2}\,T^4~,
\label{eq:corT^4}
\end{split}
\end{equation}
where $M_2$ is the first moment of quark distributions in the pion:
\begin{equation}
M_2=\frac{1}{2}\int^1_0\mathrm{d} x\,x\left[v(x)+2s(x)\right]~,
\end{equation}
with valence and sea quark distributions, $v(x)$ and $s(x)$, respectively. We use the value $M_2 \simeq 0.12$ as discussed in \cite{Eletsky:1994rp}.
By transferring Eq.~(\ref{eq:corT^4}) to the FESR, it turns out that Eq.~(\ref{1momsr})
is modified by the new term of order $T^4$ as follows:
\begin{equation}
\int^{s_V(T)}_0\mathrm{d} s\,s\,R_V(s,T) -\frac{16\pi^4M_2}{5}\,T^4=\frac{c_0}{2} s^2_V(T)-c_2(T)~,
\label{1momsrc}
\end{equation}
while the denominator of Eq.~(\ref{eq:avVmass}) receives a small correction from the replacement
$\epsilon\,\to\,\epsilon(1-\epsilon/2)$.
On the OPE side, the $T$ dependence of the quark condensates is also improved up to order $T^4$.
Chiral perturbation theory gives~\cite{Gerber:1988tt}
\begin{equation}
\bra\bar{q}q\ket_T=\bra\bar{q}q\ket_0\left(1-\frac{3}{4}\epsilon-\frac{3}{32}\epsilon^2\right).
\end{equation}
However, this correction is numerically small in the actual calculation.
The substitution of Eq.~(\ref{1momsrc}) in the numerator of Eq.~(\ref{eq:avVmass}) introduces a relatively small negative mass shift,
\begin{equation}
\delta \bar{m}_V^2 = -\frac{16\pi^4M_2\,T^4}{5 \int^{s_V(T)}_0\mathrm{d} s\,R_V(s,T)}~,
\end{equation}
consistent with the findings of Ref.~\cite{Eletsky:1994rp},
which compensates for the increase of the vector average mass by the finite width effect. For instance the average mass at $T=100$ MeV is now obtained as $\bar{m}_V\simeq 798\pm4$ MeV, slightly less than in Table~\ref{tab:avVmass} but only marginally different.
\subsection{Massive states}
For higher temperatures the contributions from massive excitations such as $K$ and $\eta$ in Eq.~(\ref{eq:TherAV}) need to be considered. In the $T$ dependence of the quark and gluon condensates, these contributions are included as in Ref.~\cite{Hatsuda:1992bv}:
\begin{equation}
\begin{split}
\frac{\bra\bar{q}q\ket_T}{\bra\bar{q}q\ket_0}&=1-\frac{T^2}{8f_\pi^2}\left[B_1\left(\frac{m_\pi}{T}\right)+\frac{7}{9}B_1\left(\frac{m_K}{T}\right)\right],\\
\left\bra\frac{\alpha_s}{\pi}G^2\right\ket_T&=\left\bra\frac{\alpha_s}{\pi}G^2\right\ket_0-\frac{T^2}{9}\left[m_\pi^2B_1\left(\frac{m_\pi}{T}\right)\right.\\
&\qquad\qquad\qquad\quad+\left.\frac{5}{3}m_K^2B_1\left(\frac{m_K}{T}\right)\right].
\end{split}
\end{equation}
In practice, the $T$ dependence of the gluon condensate is just a few percent of its vacuum value and negligible. These numerically minor corrections in the OPE are of only little significance to the sum-rule analysis.
\section{Spectral functions from an effective field theory}
So far we have used empirical input for the spectral distributions at zero temperature.
In this section, we illustrate how the sum rule works for the example of a
chiral effective field theory in which the vector meson spectral distribution can be
explicitly calculated and tested with respect to its sum-rule consistency.
The example chosen is the chiral Lagrangian based on generalized hidden local
symmetry (GHLS)~\cite{ghls}, which explicitly includes the axial-vector meson in addition to
pion and vector meson. Details of the formalism at one loop are found in Ref.~\cite{our}.
The parity mixing in the $\rho$ meson spectrum at finite temprature is generated from
a process in which $\pi$ and $a_1$ mesons circulate in a loop attached to the $\rho$ meson,
with the pion coming from the heat bath. In this approach one does not have to rely on
equations such as Eqs.~(\ref{eq:V-Amixing}) or (\ref{eq:corT^4}).
\begin{figure}[ht]
\centering
\includegraphics[width=8cm]{rhoHLS.eps}
\caption{The $\rho$ meson resonance at finite temperature calculated in the GHLS model.}
\label{fig:ghls}
\end{figure}
Figure~\ref{fig:ghls} shows the vector spectral function of the GHLS approach at
several temperatures.
The spectrum exhibits the usual resonant peak, the width of which is governed by the
$\rho \rightarrow 2\pi$ decay. The $\rho$-$a_1$ mixing involves mechanisms in which the
energy of the timelike
$\rho$ meson splits into two branches corresponding to the processes
$\rho + \pi \to a_1$ and $\rho \to a_1 + \pi$, with thresholds
$\sqrt{s} = m_{a_1} - m_\pi$ and $\sqrt{s} = m_{a_1} + m_\pi$.
This produces the threshold effects seen as a shoulder at
$\sqrt{s}=m_{a_1}- m_\pi$ (and a bump above $\sqrt{s}=m_{a_1} + m_\pi$, not
seen in the figure). The height of the $\rho$ spectrum gets reduced with increasing temperature,
whereas the $a_1$-meson contribution is enhanced via the mixing effect.
The average vector mass defined in Eq.~(\ref{eq:avmass}) and the continuum
threshold in the vector channel as a function of temperature are summarized
in Table~\ref{tab:sthHLS}.
\begin{table}[ht]
\centering
\begin{tabular}{|c|c|c|}
\hline
$T$ (MeV) & $\bar{m}_V$ (MeV) & $\sqrt{s_V}$ (GeV) \\
\hline
$0$ & ~$787\pm2$\,\, & ~$1.126\pm0.007$\,\, \\
$40$ & ~$787\pm2$\,\, & ~$1.126\pm0.007$\,\, \\
$80$ & ~$787\pm2$\,\, & ~$1.122\pm0.007$\,\, \\
$120$ & ~$786\pm3$\,\, & ~$1.111\pm0.008$\,\, \\
$140$ & ~$786\pm3$\,\, & ~$1.102\pm0.008$\,\, \\
\hline
\end{tabular}
\caption{The average $\rho$-meson mass and the continuum
threshold $\sqrt{s_V}$ at various temperatures, following from the FESR consistency test of the
spectral function calculated in the GHLS approach \cite{our}.}
\label{tab:sthHLS}
\end{table}
One observes again that $\bar{m}_V$ stays unchanged from its vacuum value over the whole
temperature range $T < 140$ MeV. The continuum threshold scale $\sqrt{s_V}$ shows a systematic decrease toward higher temperature, but at a rate considerably smaller than the expected
behavior of the chiral order parameter, $f_\pi(T)$.
\section{Conclusions}
In this work we have constructed FESRs in order to study the behavior of $\rho$ and $a_1$ mesons as well as their mixing at finite temperature, with the aim of exploring the pattern of chiral symmetry restoration. The sum rules for the lowest two spectral moments of vector and axial-vector spectral functions involve only the leading QCD condensates as corrections. With inclusion of perturbative QCD terms up to order $\alpha_s^3$, these sum rules permit a reliable quantitative analysis, unaffected by the large uncertainties from condensates of higher dimension such as the four-quark condensates.
The leading temperature corrections involve thermal pion loops. To order $T^2$, the temperature dependence is generated entirely by the $\rho - a_1$ mixing effect caused by the nonvanishing $\rho\,\pi \,a_1$ coupling in the presence of a pionic heat bath, as pointed out previously by Dey, Eletsky, and Ioffe \cite{Dey:1990ba}. To this order, there is no shift of the pole mass in the thermal vector meson propagator. Next-to-leading effects of order $T^4$ involve $\pi\pi$ intermediate states and can lead to moderate mass shifts.
In the large-$N_c$ limit the $\rho$ and $a_1$ widths vanish. In this limit, schematic distributions with $\delta$ function resonances and a step-function parametrization of the high-energy continuum, when inserted in the FESR analysis, reproduce the well-known current algebra and chiral sum rules.
As an interesting feature, one finds that at zero temperature the continuum threshold $s_V$ in the vector channel is identified with the scale characteristic of spontaneous chiral symmetry breaking:
$\sqrt{s_V} = 4\pi\,f_\pi$. Above this scale, chiral symmetry is restored in its Wigner-Weyl realization. In the resonance region below this scale, the symmetry is in the spontaneously broken Nambu-Goldstone realization.
When realistic spectral functions with large widths are implemented, this ``clean'' separation of scales
is no longer rigorously maintained, but the FESR analysis is still useful, with the pole mass in the vector correlator now replaced by the normalized first moment of the spectral distribution. Within its range of applicability (up to temperatures of about 140 MeV), the sum-rule analysis consistently shows an almost constant behavior of this average vector meson spectral mass.
The primary temperature dependence of the spectral function comes from $\rho$-$a_1$ mixing in the thermal pionic medium. The $a_1$ mass, again identified with the normalized first moment of the
axial-vector spectral distribution, decreases with rising temperature. This indicates the expected tendency of the $\rho$ and $a_1$ spectra becoming identical (degenerate) when chiral symmetry is restored. However, the present analysis does not support the BR scaling hypothesis of a dropping $\rho$ meson mass, at least not up to temperatures $T\sim140$ MeV, which are not far from the chiral crossover transition temperature, $170-190$ MeV.
The continuum threshold scale $s_V$ in the vector channel (even when smoothed by a ramping function) systematically moves downward in energy as the temperature increases. This feature is
observed in all cases studied. However, while the identification of $\sqrt{s_V}$ with the chiral scale
$4\pi\,f_\pi$ emerges naturally at $T = 0$, the downward evolution with temperature of $\sqrt{s_V}$ is
significantly slower than that of $f_\pi(T)$ deduced from chiral perturbation theory.
\section*{ACKNOWLEDGEMENTS}
This work has been supported in part by BMBF, GSI, and the DFG Cluster of Excellence Origin and Structure of the Universe. Two of us (WW and CS) thank the organizers of the program `New Frontiers in QCD' for their kind hospitality at the Yukawa Institute of Theoretical Physics, Kyoto, where this article was finalized. YK is grateful to S. H. Lee and S. i. Nam for useful discussions.
CS acknowledges parital support by the Hessian
LOEWE initiative through the Helmholtz International
Center for FAIR (HIC for FAIR).
\begin{appendix}
\section{Continuum threshold modeling}
\label{appx2}
Here we test the reliability of the continuum threshold parametrization by a schematic step function.
Such a test can be performed by replacing the step function with a ramp function to yield a smooth transition between resonance and continuum regions, as follows:
\begin{equation}
R(s)=R_{res}(s)\,\Theta(s_2-s)+R_c(s)\,W(s)~,
\end{equation}
where the weight function, $W(s)$, is defined as
\begin{equation}
W(x)=\left\{\begin{array}{cl}
\vspace{2mm}
0 & \text{ for } x\leq s_1\\
\vspace{1.5mm}
\displaystyle\frac{x-s_1}{s_2-s_1} & \text{ for }s_1\leq x\leq s_2\\
1 & \text{ for } x\geq s_2~.
\end{array}\right.
\label{eq:Wofs}
\end{equation}
The step function behavior is recovered for $W(x)$ in the limit $s_1\rightarrow s_2$.
\begin{figure}[ht]
\includegraphics[width=8.5cm]{ramp100MeV}
\caption{Dependence of $\sqrt{s_V}$ [determined from Eqs.~(\ref{app:0thsr})-(\ref{app:s_v})] on the slope
$(s_2-s_1)^{-1}$ of the ramp function $W(s)$ describing the onset of the continuum for the example of $T=100~\mathrm{MeV}$.
The grey band indicates the uncertainty range of the result obtained with step-function parametrization of the continuum.}
\label{app:ramp}
\end{figure}
By using the function $W(s)$, the modified sum rules for the lowest two moments of the spectrum $R(s)$ become
\begin{equation}
\begin{split}
\int^{s_2}_0\mathrm{d} s\,R_{res}(s)&=s_2\left(c_0+\frac{3}{2}\varepsilon_0\right)+c_1-12\pi^2\Pi(0)\\
&-\left[c_0-R_{res}(s_2)\right]\int^{s_2}_{s_1}\mathrm{d} s\,W(s)~,
\label{app:0thsr}
\end{split}
\end{equation}
\begin{equation}
\begin{split}
\int^{s_2}_0\mathrm{d} s\,sR_{res}(s)&=\frac{s^2_2}{2}\left(c_0+\frac{3}{2}\varepsilon_1\right)-c_2\\
&-\left[c_0-R_{res}(s_2)\right]\int^{s_2}_{s_1}\mathrm{d} s\,sW(s)~.
\label{app:1stsr}
\end{split}
\end{equation}
Sets of intervals $[s_1, s_2]$ are then determined so as to satisfy both sum rules (\ref{app:0thsr}, \ref{app:1stsr}),
and the scale $s_V$, defined by
\begin{equation}
s_V=\frac{s_1+s_2}{2}~,
\label{app:s_v}
\end{equation}
is now introduced to characterize the continuum threshold.
In any temperature region where the sum rules are valid the present analysis does not depend sensitively on details of the threshold modeling. For an example of the $\rho$ meson channel, it is demonstrated in Fig.~\ref{app:ramp} that the resulting $\sqrt{s_V}$ at finite temperature is
stable with respect to variations in the slope $(s_2-s_1)^{-1}$ of the ramp function $W(s)$. In this test the
uncertainties of $\alpha_s(Q^2)$ and of the gluon condensate have been disregarded for simplicity.
\vspace{0.5cm}
\end{appendix}
|
3,212,635,537,913 | arxiv | \section{Introduction}
The classical Grushin plane $\mathbb{G}$ is defined as the space $\mathbb{R}^2$ equipped with the sub-Riemannian (Carnot-Carath\'eodory) metric $d_\mathbb{G}$ generated by the vector fields
\[X_1 = \partial_{x_1} \quad \text{ and } \quad X_2 = x_1\partial_{x_2}.\]
This means more precisely that the distance between points $p,q\in\mathbb{G}$ is
\[ d_{\mathbb{G}}(p,q) = \inf_{\gamma} \int_0^1 \sqrt{x_1'(t)^2 + \frac{x_2'(t)^2}{x_1(t)^2}} dt, \]
where the infimum is taken over all paths $\gamma = (x_1(t),x_2(t))\colon [0,1] \to \mathbb{G}$, with $\gamma(0) = p$ and $\gamma(1) = q$, that are absolutely continuous in the Euclidean metric. The Grushin plane is one of the simplest examples of a sub-Riemannian manifold, as well as a basic example of the \textit{almost Riemannian} manifolds studied by Agrachev, Boscain, Charlot, Ghezzi, and Sigalotti \cite{ABCGS}, \cite{ABS}. For additional background on the Grushin plane and sub-Riemannian spaces in general, see Bella\"iche \cite{Bellaiche}.
Recently, Seo \cite{Seo} proved a general characterization of spaces admitting a bi-Lipschitz embedding into some Euclidean space $\mathbb{R}^n$, from which it follows that $\mathbb{G}$ admits such an embedding. In contrast, the Heisenberg group can not be embedded bi-Lipschitz in any Euclidean space \cite{Semmes}. While Seo's result does not give the optimal target dimension, Wu \cite{Wu} constructed an explicit bi-Lipschitz embedding of $\mathbb{G}$ into $\mathbb{R}^3$, where the dimension 3 is the smallest possible.
In the present article, Wu's result is extended to the generalized Grushin plane $\mathbb{G}_\alpha$, $\alpha\geq 0$, studied first by Franchi and Lanconelli \cite{FrLa}. Similarly to $d_\mathbb{G}$, the metric $d_{\mathbb{G}_{\alpha}}$ is generated by the vector fields
\[X_1 = \partial_{x_1} \quad \text{ and } \quad X_2 = |x_1|^{\alpha}\partial_{x_2}.\]
For integer values of $\alpha$, $|x_1|^{\alpha}$ can be replaced by $x_1^{\alpha}$ and the space $\mathbb{G}_\alpha$ is a sub-Riemannian manifold of step $\alpha+1$. For noninteger values of $\alpha$, this space is technically not sub-Riemannian, but this distinction does not matter for the purposes of this paper. Meyerson \cite{Meyerson} and Ackermann \cite{Ack} have shown that $\mathbb{G}_\alpha $ is quasisymmetric to the Euclidean space $\mathbb{R}^2$ for any $\alpha \geq 0$. Moreover, it can be deduced by Seo's theorem \cite[Theorem 4.4]{Seo} that $\mathbb{G_\alpha}$ is bi-Lipschitz embeddable into some Euclidean space when $\alpha>0$, though without identifying the smallest target dimension.
In this paper, we construct for each $\alpha\geq 0$ a bi-Lipschitz embedding of $\mathbb{G}_\alpha $ into $\mathbb{R}^{[\alpha ]+2}$ where $[\alpha]$ is the greatest integer that is less or equal to $\alpha$. A point of interest in both Wu's and our construction is that the image of $\mathbb{G}_\alpha$ is a quasiplane in $\mathbb{R}^{[\alpha]+2}$.
\begin{thm}\label{thm:main}
For all integers $N\geq 0$ and $n \geq 1$, there exists $L>1$ depending only on $N,n$ such that for any $\alpha\in [N,N+\frac{n-1}n]$, there exists an $L$-bi-Lipschitz homeomorphism of $\mathbb{G}_{\alpha}$ onto a $2$-dimensional quasiplane $\mathcal{P}_{\alpha}$ in $\mathbb{R}^{N+2}$.
\end{thm}
A \emph{$k$-dimensional quasiplane} $\mathcal{P}$ in $\mathbb{R}^n$, with $k<n$, is the image of a $k$-dimensional hyperplane in $\mathbb{R}^n$ under a quasiconformal self-map of $\mathbb{R}^n$. Complete characterizations of these spaces in terms of their geometric structure exist only for $n=2$, $k=1$ by Ahlfors \cite{Ah}. While such intrinsic characterizations have been elusive for $n\geq 3$, several intriguing examples of quasiplanes and quasispheres have been constructed \cite{Bishop, DToro, Lewis, Meyer, PW, VW2, Wu}.
A couple of remarks are in order. The target dimension $N+2 = [\alpha ]+2$ in Theorem \ref{thm:main} is minimal. Indeed, by (\ref{eq:grushinQuasimetric}), the \emph{singular line} $\{x_1=0\}$ of $\mathbb{G}_{\alpha}$ is bi-Lipschitz homeomorphic to the ``snowflaked'' space $(\mathbb{R},|\cdot| ^{1/(1+\alpha)})$ which, by a well-known theorem of Assouad \cite[Proposition 4.12]{Assouad}, embeds bi-Lipschitz into $\mathbb{R}^{[\alpha]+2}$ with the target dimension $[\alpha]+2$ being the smallest possible when $\alpha>0$. It is noteworthy that, for $\alpha>0$, $\mathbb{G}_\alpha$ embeds in the same Euclidean space that its singular line embeds in.
The same result of Assouad also justifies the dependence of the constant $L$ on $n$. For if there was a uniform $L$ such that $\mathbb{G}_\alpha$ was $L$-bi-Lipschitz embeddable in $\mathbb{R}^{N+2}$ for all $\alpha \in [N,N+1)$, then by a simple Arzel\`a-Ascoli limiting argument (see Lemma \ref{lem:BLembed}), it would follow that $\mathbb{G}_{N+1}$, thus the singular line in $\mathbb{G}_{N+1}$, is also embeddable in $\mathbb{R}^{N+2}$ which is false.
The following corollary is an immediate consequence of Theorem \ref{thm:main}.
\begin{cor}\label{cor:BLcor}
If $\alpha\in [0,1)$ then $\mathbb{G}_{\alpha}$ is bi-Lipschitz homeomorphic to $\mathbb{R}^2$.
\end{cor}
Therefore, $\mathbb{G}_{\alpha}$ is bi-Lipschitz homeomorphic to $\mathbb{G}_{\beta}$ whenever $\alpha,\beta \in [0,1)$. In contrast, if $\alpha \geq 1$ then $\mathbb{G}_\alpha$ has Hausdorff dimension $\alpha+1$, and is bi-Lipschitz homeomorphic to $\mathbb{G}_\beta$ only when $\alpha = \beta$. Combined with the Beurling-Ahlfors quasiconformal extension \cite{BerAhl}, Corollary \ref{cor:BLcor} yields the following result.
\begin{cor}\label{cor:ext}
If $\alpha\in[0,1)$, then any bi-Lipschitz embedding of the singular line of $\mathbb{G}_{\alpha}$ into $\mathbb{R}^2$ extends to a bi-Lipschitz homeomorphism of $\mathbb{G}_{\alpha}$ onto $\mathbb{R}^2$.
\end{cor}
An alternative proof of Corollary \ref{cor:BLcor} along with new results on questions of quasisymmetric parametrizability and bi-Lipschitz embeddability of high-dimensional Grushin spaces can be found in a recent paper of Wu \cite{Wu2}.
\subsection{Outline of the proof of Theorem \ref{thm:main}.}\label{sec:outline}
The proof of Theorem \ref{thm:main} comprises two parts. In Section \ref{sec:rationalproof} we show Theorem \ref{thm:main} for rationals $\alpha\geq 0$ and in Section \ref{sec:irrationalproof} we use an Arzel\`a-Ascoli limiting argument to prove Theorem \ref{thm:main} for all real values $\alpha\geq 0$. The proof of Corollary \ref{cor:ext} is also given in Section \ref{sec:proofofmainthm}.
Much of the proof of Theorem \ref{thm:main} for rational $\alpha\geq 0$ follows the method of Wu in \cite{Wu}. The crux of the proof is the construction, for each rational $\alpha\in[N,N+\frac{n-1}n]$, of a quasisymmetric mapping $F_{\alpha}:\mathbb{R}^{N+2}\to \mathbb{R}^{N+2}$, such that in each ball $B(x,\frac{1}{2}\dist(x,\{0\}\times\mathbb{R}))$, $F_{\alpha}$ is the product of $\dist(x,\{0\}\times\mathbb{R})^{-\frac{1}{1+\alpha}}$ and a $\lambda$-bi-Lipschitz mapping with $\lambda$ depending only on $N,n$. Such a mapping is $\frac{1}{1+\alpha}$-\emph{snowflaking} on $\{0\}\times\mathbb{R} \subset \mathbb{R}^{N+1}\times\mathbb{R}$ (i.e. $|F_{\alpha}(x) - F_{\alpha}(y)| \simeq |x-y|^{\frac{1}{1+\alpha}}$ for all $x,y \in \{0\}\times\mathbb{R}$) and maps the $2$-dimensional plane $\mathbb{R}\times\{0\}\times\mathbb{R}$ onto a quasiplane $\mathcal{P}_{\alpha}$.
Composed with a quasisymmetric homeomorphism of $\mathbb{G}_\alpha$ onto $\mathbb{R}^2$, we obtain a bi-Lipschitz homeomorphism $f_{\alpha}$ of $\mathbb{G}_\alpha$ into $\mathcal{P}_{\alpha}$.
The quasisymmetric mappings $F_\alpha$ are constructed in Section \ref{sec:proprational} by iterating a finite number of bi-Lipschitz mappings $\Theta$ which are defined in Section \ref{sec:blocks} as in \cite{Wu}. However, a straightforward generalization of Wu's method, without additional care, would give no control on the local bi-Lipschitz constant $\lambda$ (thus on the bi-Lipschitz constant of $f_{\alpha}$), and the proof of Theorem \ref{thm:main} for irrational values of $\alpha$ would not be possible. To overcome this issue, we construct in Section \ref{sec:blocks} two sets of bi-Lipschitz mappings $\Theta_z$, corresponding to $z = 0$ and $z = N+\frac{n-1}n$, and then periodically alternate between these when constructing the quasisymmetric mapping $F_\alpha$.
Our inability to define $F_{\alpha}$ for irrational $\alpha>0$ is the reason for considering the irrational case separately; see Remark \ref{rem:distinction}.
\medskip
\textbf{Acknowledgements.} The authors are grateful to Jang-Mei Wu and Jeremy Tyson for suggesting this problem and for valuable discussions.
\section{Preliminaries}
A homeomorphism $f\colon D\to D'$ between two domains in $ \mathbb{R}^n$ is called $K$-\emph{quasiconformal} if it is orientation-preserving, belongs to $ W_{\text{loc}}^{1,n}(D)$, and satisfies the distortion inequality
\[|Df(x)|^n \le K J_f(x) \quad \text{a. e.} \,\,\, x \in D,\]
where $Df$ is the formal differential matrix and $J_f$ is the Jacobian.
An embedding $f$ of a metric space $(X,d_X)$ into a metric space $(Y,d_Y)$ is said to be $\eta$-\emph{quasisymmetric} if there exists a homeomorphism $\eta \colon [0,\infty) \to [0,\infty)$ such that for all $x,a,b \in X$ and $t>0$ with $d_X(x,a) \leq t d_X(x,b)$,
\[d_Y(f(x),f(a)) \leq \eta(t)d_Y(f(x),f(b)). \]
A quasisymmetric mapping between two domains in $\mathbb{R}^n$ is quasiconformal. On the other hand, a quasiconformal mapping defined on a domain $D\subset \mathbb{R}^n$ is quasisymmetric on each compact set $E \subset D$. In $\mathbb{R}^n$ the two notions coincide: if $f:\mathbb{R}^n \to \mathbb{R}^n$ is $K$-quasiconformal then it is $\eta$-quasisymmetric for some $\eta$ depending only on $K,n$. For a systematic treatment of quasiconformal mappings see \cite{Vais1}.
A mapping $f\colon X \to Y$ between metric spaces is \emph{$L$-bi-Lipschitz} if there exists a constant $L \geq 1$ such that $L^{-1}d_X(x,y) \leq d_Y(f(x),f(y)) \leq Ld_X(x,y)$ for all $x,y \in X$.
In the following, we write $u\lesssim v$ (resp. $u \simeq v$) when the ratio $u/v$ is bounded above (resp. bounded above and below) by positive constants. These constants may vary, but are described in each occurrence.
\section{Basic geometric constructions}\label{sec:blocks}
This section extends the construction by Wu \cite{Wu} to higher-dimensional targets; the notational conventions follow those of Wu as much as possible. Our goal is to build certain annular tubes and bi-Lipschitz maps between these tubes which are used in Section \ref{sec:proprational} to define quasiconformal homeomorphisms of $\mathbb{R}^{N+2}$.
These constructions are based on examples of Bonk and Heinonen \cite{BH} and Assouad \cite{Assouad}.
\subsection{Definitions and notation}
An \emph{$N$-cube} $\mathcal{C}$ is the product $\Delta_1\times\cdots\times\Delta_N$ of bounded closed intervals $\Delta_i\subset\mathbb{R}$ of equal length. A \emph{$j$-face} of $\mathcal{C}$ is a product $\Delta_1'\times\cdots\times\Delta_N'$ where, for $j$ indices, $\Delta_i' = \Delta_i$ and for the other $N-j$ indices $\Delta_i'$ is an endpoint of $\Delta_i$. The $0$-faces of a cube $\mathcal{C}$ are its \emph{vertices}.
For an $N$-cube $\mathcal{C}$ and integer $0 \leq k \leq N$, we define a \emph{$k$-flag} of $\mathcal{C}$ to be a sequence $\{\mathcal{C}^j\}_{j=0}^k$ where $\mathcal{C}^j$ is a $j$-face of $\mathcal{C}$ and $\mathcal{C}^{j-1} \subset \mathcal{C}^j$ for all $1 \leq j \leq k$. Observe that for $N$-cubes $\mathcal{C}$ and $\widetilde{\mathcal{C}}$ and $(N-2)$-flags $\{\mathcal{C}^j\}$ and $\{\widetilde{\mathcal{C}}^j\}$, there exists a unique orientation-preserving similarity $\psi: \mathbb{R}^N \to \mathbb{R}^N$ such that $\psi(\mathcal{C})= \widetilde{\mathcal{C}}$ and $\psi(\mathcal{C}^j) = \widetilde{\mathcal{C}}^j$ for each $0 \leq j \leq N-2$.
For a point $x = (x_1,\dots,x_{N}) \in \mathbb{R}^{N}$ and a number $r>0$, define the cube
\[ \mathcal{C}^{N}(x,r) = [x_1-r/2, x_1 +r/2]\times \dots \times [x_{N}-r/2,x_{N}+r/2] \]
and denote $\mathfrak{C}^{N} = \mathcal{C}^{N}(0,1)$ where $0$ here denotes the origin in $\mathbb{R}^{N}$.
Slightly abusing the notation, we define for two numbers $0<r<R<\infty$ the \emph{cubic annulus}
\[ \mathcal{A}^{N}(r,R) = \overline{(R\mathfrak{C}^{N}) \setminus (r\mathfrak{C}^{N})} = [-R/2,R/2]^N \setminus (-r/2,r/2)^N.\]
Here and for the rest, for $X\subset \mathbb{R}^N$ and $c > 0$, we write $c X = \{c x : x\in X\}$.
Finally, for a polygonal arc $\ell \subset \mathbb{R}^{N}$ and some $\epsilon>0$, define the \emph{cubic thickening} of $\ell$
\[ \mathcal{T}^{N}(\ell,\epsilon) = \overline{\bigcup \mathcal{C}^{N}(x,\epsilon)}\]
where the union is taken over all $x\in \ell$ such that their distances from the endpoints of $\ell$ are at least $\epsilon/2$.
For the rest of Section \ref{sec:blocks} we fix integers $N\geq 0$, $n\geq 1$ and set
\[ p = p_{N,n} = N+\frac{n-1}{n}\text{ and }M = M_{N,n} = 9^{n(N+2)}.\]
The dependence of quantities and sets on $N,n$ is omitted whenever possible.
\subsection{Blocks}\label{sec:tubes}
Let $I\subset\mathfrak{C}^{N+1}\times[0,1]$ be the straight-line path from $(0,\dots,0)$ to $(0,\dots,0,1)$ and $L\subset\mathfrak{C}^{N+1}\times[0,1]$ be the straight-line path from $(0,\dots,0)$ to $(0,\dots,0,\frac{1}2)$ concatenated with the straight-line path from $(0,\dots,0,\frac{1}2)$ to $(\frac{1}2,0,\dots,0,\frac{1}2)$.
We define three types of blocks that are used throughout the paper:
\begin{enumerate}
\item the \emph{$I$-block} $Q_I = \mathcal{T}^{N+2}(I,\frac{M-2}M) = (\frac{M-2}{M}\mathfrak{C}^{N+1})\times [0,1]$;
\item the \emph{$L$-block} $Q_L = \mathcal{T}^{N+2}(L,\frac{M-2}M) $
\[= (\frac{M-2}{M}\mathfrak{C}^{N+1}\times[0,\frac{M-1}{M}])\cup ([\frac{1}{2},1]\times \frac{M-2}{M}\mathfrak{C}^{N+1}) \]
\item the \emph{regular block} $\mathsf{Q} = \mathfrak{C}^{N+1}\times[0,1]$.
\end{enumerate}
On each of these blocks, the entrance, exit and side are defined as follows.
\begin{enumerate}
\item The \emph{entrance} of $Q_{I}$ is $\text{en}(Q_{I}) = Q_{I} \cap \{x_{N+2} = 0\}$,
\item the \emph{exit} of $Q_I$ is $\text{ex}(Q_I) = Q_I \cap \{x_{N+2} = 1\}$,
\item the \emph{side} of $Q_{I}$ is $\text{s}(Q_{I}) = \overline{\partial Q_{I} \setminus(\text{en}(Q_{I}) \cup \text{ex}(Q_{I}))}$.
\end{enumerate}
Analogous definitions can be made for $Q_L$ and $\mathsf{Q}$ with the difference that the \emph{exit} of $Q_L$ is $\text{ex}(Q_L) = Q_L \cap \{x_{1} = \frac12\}$. These definitions are applied to images of the respective objects under similarity maps. For a similarity map $h$ and $\ell \in \{h(I), h(L)\}$, we write $Q_\ell$ in place of $h(Q_I)$ or $h(Q_L)$. We call $Q_\ell$ the \emph{block associated with the segment $\ell$}; note that $Q_\ell$ naturally inherits a direction from $\ell$.
\subsection{Cores}\label{sec:cores}
From each block $Q_I$, $Q_L$ and $\mathsf{Q}$ we remove a \emph{core} from its interior, which we describe in this section.
In Section \ref{sec:paths} we construct a simple polygonal path $J_I = J_I(N,n) \subset Q_I$ from $(0,\ldots, 0, 0)$ to $(0, \ldots, 0, 1)$ consisting of $M^{1+p}$ many $I$- and $L$-segments $\ell_1, \ldots, \ell_{M^{1+p}}$ of length $1/M$ labelled according to their order in $J_I$ with the following properties.
\begin{enumerate}
\item The segments $\ell_1$, $\ell_{M^{1+p}}$, and $\ell_{(M^{1+p}+1)/2}$ are $I$-segments.
\item For all $1 \leq m < M^{1+p}$, $Q_{\ell_m} \cap Q_{\ell_{m+1}}$ is the exit of $Q_{\ell_m}$ and the entrance of $Q_{\ell_{m+1}}$. If $1 \leq l, m \leq M^{1+p}$ and $|m-l|>1$, then $Q_{\ell_m} \cap Q_{\ell_l} = \emptyset$.
\item $\text{en}(Q_{\ell_1}) = Q_{\ell_1} \cap \partial Q_I \subset \text{en}(Q_{I})$ and $\text{ex}(Q_{\ell_{M^{1+p}}}) = Q_{\ell_{M^{1+p}}} \cap \partial Q_I \subset \text{ex}(Q_{I})$. For $2 \leq m \leq M^{1+p}-1$, $Q_{\ell_m} \cap \partial Q_I = \emptyset$.
\item $J_I$ is symmetric with respect to the plane $x_{N+2} = \frac12$.
\item $J_I$ is unknotted in $Q_I$, in the sense that there every bi-Lipschitz homeomorphism $\theta: (\partial Q_I,J_I) \to (\partial\mathsf{Q},I)$ extends to a bi-Lipschitz homeomorphism $\Theta : Q_I \to \mathsf{Q}$.
\end{enumerate}
Similarly, in Section \ref{sec:paths} we construct a simple polygonal path $J_L = J_L(N,n) \subset Q_L$ satisfying the same properties, except that $\ell_{(M^{1+p}+1)/2}$ is an $L$-segment and $J_L$ is symmetric with respect to the plane $x_1 + x_{N+2} = \frac12$.
Given $J_I = \bigcup_{m=1}^{M^{1+p}} \ell_m$ as above, define the \emph{core}
\[\kappa_p(Q_I) = \bigcup_{m=1}^{M^{1+p}} Q_{\ell_m} = \mathcal{T}^{N+2}(J_I,\frac{M-2}{M^2}).\]
We similarly define the core $\kappa_p(Q_L)$. The entrance, the exit and the side of $\kappa_p(Q_I),\kappa_p(Q_L)$ are canonically defined. A second set of cores $\kappa_0(Q_I),\kappa_0(Q_L)$ in $Q_I,Q_L$, respectively, is defined as follows. Write $I = \bigcup_{m=1}^{M}\ell_m$ with $\ell_m = \{0\}\times[m-1,m] \subset \mathbb{R}^{N+1}\times I$ and set
\[ \kappa_{0}(Q_I) = \bigcup_{m=1}^M Q_{\ell_m} = \mathcal{T}^{N+2}(I,\frac{M-2}{M^2}).\]
Similarly write $L = \bigcup_{m=1}^{M}\ell_m'$ where $\ell_{m}'$ is an $L$-segment if $m=\frac{M+1}{2}$ and an $I$-segment otherwise; and each $\ell_m'$ has length $1/M$. Set $\kappa_{0}(Q_L) = \bigcup_{m=1}^M Q_{\ell'_m} = \mathcal{T}^{N+2}(L,\frac{M-2}{M^2})$.
To simplify the notation, in what follows we write $Q_m$ instead of $Q_{\ell_m}$.
Two types of cores are similarly defined for the regular block $\mathsf{Q}$. For each $z \in \{0,p\}$ let
\[ \mathsf{k}_z(\mathsf{Q}) = (M^{-1-z}\mathfrak{C}^{N+1}) \times [0,1]\]
which is composed of $M^{1+z}$ consecutive blocks
\[\mathsf{Q}_m = M^{-1-z}\left ( \mathfrak{C}^{N+1} \times [m-1, m]\right ), \quad m = 1, \ldots, M^{1+z}.\]
\subsection{Flag-edges}\label{sec:edges}
We introduce in this section \emph{flag-edges} and \emph{flag-paths}, which generalize the \emph{edges} and \emph{edge paths} used by Wu \cite[Section 2.3]{Wu} to blocks of arbitrary dimensions. These play an important bookkeeping role later when defining bi-Lipschitz maps between annular tubes.
For the rest fix an $(N-1)$-flag $\mathcal{F}_0 = \{\mathcal{C}^j\}_{j=1}^{N-1}$ of $\mathfrak{C}^{N+1}$. We call the collection of faces $e_{\mathcal{F}_0} = \{(\frac{M-2}{M}\mathcal{C}^j)\times [0,1]\}_{j=0}^{N-1}$ a \emph{flag-edge} on $Q_I$.
Before defining flag-edges on $Q_L$, we first define faces on the side $\text{s}(Q_L)$ inductively. If $\mathcal{C}^0 = (x_1,\dots,x_{N+1})$, $x_j \in \{\pm \frac{M-2}{2M}\}$, is a $0$-face of $\text{en}(Q_L)$ then define the $L$-type path
\[ P(\mathcal{C}^0) = (\{(x_1,\dots,x_{N+1})\}\times [0,\frac12-x_{1}]) \cup ([x_1,\frac12]\times\{(x_{2},\dots,x_{N+1},\frac12-x_{1})\}.\]
Suppose that for every $j$-face $\mathcal{C}^j$ of $\text{en}(Q_L)$, the set $P(\mathcal{C}^j)$ has been defined. Let $\mathcal{C}^{j+1}$ be a $(j+1)$-face of $\text{en}(Q_L)$ and let $\mathcal{C}^{j}_1, \dots, \mathcal{C}^{j}_{2(j+1)}$ be the $j$-faces of $\mathcal{C}^{j+1}$. Then define $P(\mathcal{C}^{j+1})$ to be the union of all line segments with endpoints on $\bigcup_{i=1}^{2(j+1)} P(\mathcal{C}^{j}_{i})$ that lie entirely on $\text{s}(Q_L)$. We call $P(C^{j+1})$ a $(j+2)$-face on $\partial Q_L$.
Let now $\mathcal{F} = \{\mathcal{C}^j\}_{j=0}^{N-1}$ be an $(N-1)$-flag of $\mathfrak{C}^{N+1}$. We call the collection $e_{\mathcal{F}} = \{P(\frac{M-2}{M}\mathcal{C}^j)\}_{j=0}^{N-1}$ a flag-edge on $Q_L$.
We now define \emph{flag-paths} along the cores $\kappa_z(Q_I)$, $\kappa_z(Q_L)$ for each value $z \in \{0, p\}$. We start with the $Q_L$ case. Rescaling an $(N-1)$-flag $\mathcal{F}$ of $\mathfrak{C}^{N+1}$, we obtain an $(N-1)$-flag $\mathcal{F}_1$ on the entrance of the first block $Q_1$ of $\kappa_z(Q_L)$. For a $j$-face $\mathcal{C}^j_1 \in \mathcal{F}_1$ define $P(\mathcal{C}^j_1) \subset \text{s}(Q_1)$ as above and note that $P(\mathcal{C}^j_1)$ defines uniquely a $j$-face $\mathcal{C}^j_2$ on the entrance of the block $Q_2$. Continuing inductively we obtain $j$-faces $\mathcal{C}^j_m$ on $\text{en}(Q_m)$ and $(j+1)$-faces $P(\mathcal{C}^j_m)$ on $\text{s}(Q_m)$. Define the \emph{flag-path} $w_{\mathcal{F}} = \{ \bigcup_{m=1}^{M^{1+z}}P(\mathcal{C}^j_m)\}_{j=0}^{N-1}$. For each block $Q_m$ in $\kappa_z(Q_L)$, $m \in \{1, \ldots, M^{1+z}\}$, we call $w_{\mathcal{F}} \cap Q_m$ the \emph{marked flag-edge of $Q_m$ derived from the data $(Q_L, e_{\mathcal{F}})$}.
A corresponding flag-path $w_{\mathcal{F}_0}$ is defined similarly for $\kappa_z(Q_I)$. For this we use the flag $\mathcal{F}_0$ instead of an arbitrary $(N-1)$-flag $\mathcal{F}$ of $\mathfrak{C}^{N+1}$.
In addition, let $\mathsf{e} = \{\mathcal{C}^j \times [0,1] : \mathcal{C}_j \in \mathcal{F}_0\}$ be a flag-edge of $\mathsf{Q}$ and $\mathsf{w} = \{(M^{-1-z}\mathcal{C}^j) \times [0,1] : \mathcal{C}_j \in \mathcal{F}_0\}$ be a flag-path along $\mathsf{k}_z(\mathsf{Q})$. As before we omit the dependency on $z$ in the notation for $\mathsf{w}$.
\subsection{Annular tubes}\label{sec:annulartubes}
For each $z \in \{0,p\}$, define the \emph{annular tubes}
\[ \tau_{z}(Q_I) = \overline{Q_I \setminus \kappa_{z}(Q_I)} \text{, } \tau_{z}(Q_L) = \overline{Q_{L} \setminus \kappa_{z}(Q_L)} \text{ and }\mathsf{t}_z(\mathsf{Q}) = \overline{\mathsf{Q}\setminus\mathsf{k}_z(\mathsf{Q})}.\]
For $Q \in \{Q_I,Q_L\}$, we define the \emph{entrance} and \emph{exit} of each $\tau_z(Q)$ as $\text{en}(Q) \cap \tau_z(Q)$ and $\text{ex}(Q) \cap \tau_z(Q)$, respectively. These are isometric to the cubic annulus $A = \frac{M-2}{M}\mathcal{A}^{N+1}(\frac{1}{M},1)$. The remaining part of $\partial \tau_{z}(Q_I)$ is composed of the side $\text{s}(Q_I)$ of block $Q_I$ and the side $\text{s}(\kappa_{z}(Q_I))$ of the core $\kappa_{z}(Q_I)$. The boundary of $\tau_{z}(Q_L)$ can be similarly partitioned.
Define similarly the entrance and exit of $\mathsf{t}_z(\mathsf{Q})$. These are isometric to the cubic annulus $\mathsf{A}_z = \mathcal{A}^{N+1}(M^{-1-z},1)$. Note that $\mathsf{A}_z$ depends on $z$ while $A$ does not.\
If $\sigma$ is a similarity mapping of $Q_{I}$ onto some block $\sigma(Q_I)$, we denote by $\kappa_{z}(\sigma(Q_I))$ the image $\sigma(\kappa_{z}(Q_I))$ with $z\in\{0,p\}$. The sets $\kappa_{z}(\sigma(Q_L))$, $\mathsf{k}_z(\sigma(\mathsf{Q}))$, $\tau_z(\sigma(Q_I))$, $\tau_z(\sigma(Q_L))$ and $\mathsf{t}_z(\sigma(\mathsf{Q}))$ are defined similarly when $\sigma$ is a similarity mapping.
\subsection{Bi-Lipschitz maps between annular tubes}\label{sec:BLmaps}
For each $z \in \{0, p\}$, each $Q\in \{Q_I, Q_L\}$, and every $(N-1)$-flag $\mathcal{F}$ of $\mathfrak{C}^{N+1}$, we define in this section bi-Lipschitz homeomorphisms $\Theta_z^{\mathcal{F}}: (\mathsf{t}_z(\mathsf{Q}), \mathsf{e}, \mathsf{w}) \to (\tau_z(Q), e_{\mathcal{F}}, w_{\mathcal{F}})$ where $\mathcal{F}=\mathcal{F}_0$ if $Q=Q_I$.
The construction of these maps is performed in 4 steps. In Step 1 we define the mappings on $\text{s}(\mathsf{Q})$, in Step 2 we define them on $\text{s}(\mathsf{k}_z(\mathsf{Q}))$ and in Step 3 we define them on the entrance and exit of $\mathsf{t}_z(\mathsf{Q})$. Combining the first three steps we obtain bi-Lipschitz mappings $\theta_z^{\mathcal{F}}: (\partial \mathsf{t}_z, \mathsf{e}, \mathsf{w}) \to (\partial \tau_z(Q), e_{\mathcal{F}}, w_{\mathcal{F}})$. Finally, in Lemma \ref{lem:BLext} we extend the mappings on the whole $\mathsf{t}_z(\mathsf{Q})$.
For each $(N-1)$-flag $\mathcal{F}$ of $\mathfrak{C}^{N+1}$ let $\psi_{\mathcal{F}}$ be the unique rotation on $\mathbb{R}^{N+2}$ that maps $\mathfrak{C}^{N+1}$ onto itself and $\mathcal{F}$ onto $\mathcal{F}_0$.
\emph{Step 1}. Define $\theta_z^{\mathcal{F}_0}: (\text{s}(\mathsf{Q}), \mathsf{e}) \to (\text{s}(Q_I), e_{\mathcal{F}_0})$ by $\theta_z^{\mathcal{F}_0}(x,t) = (\frac{M-2}{M}x,t)$, where $x \in \partial \mathfrak{C}^{N+1}$ and $t \in [0,1]$. To define $\theta_z^{\mathcal{F}}$ onto $(\text{s}(Q_L), e_{\mathcal{F}})$ first observe that $\text{s}(Q_L)$ is the union of $L$-type $1$-fibers
\begin{align*}
L_x = &(\{\frac{M-2}{M}(x_1,\dots,x_{N+1})\}\times [0,1/2-\frac{M-2}{M}x_1]) \\
&\cup ([\frac{M-2}{M}x_1,1/2]\times\{(0,\dots,0,1/2)+\frac{M-2}{M}(x_2,\dots,x_{N+1},-x_1)\}
\end{align*}
where $x =(x_1,\dots,x_{N+1})\in\partial \mathfrak{C}^{N+1}$. Similarly, $\text{s}(\mathsf{Q})$ is the union of $1$-fibers $I_{x} = \{x\}\times [0,1]$ where $x\in \partial \mathfrak{C}^{N+1}$. Define $\theta_z^{\mathcal{F}}$ on $\text{s}(\mathsf{Q})$ by mapping each $I_{x}$ to $L_{\psi_{\mathcal{F}}(x)}$ by arc-length parametrization. Note that for this step, the mappings $\theta_z^{\mathcal{F}}$ do not actually depend on $z$.
\emph{Step 2}. We extend each $\theta_z^{\mathcal{F}}$ to the inner side $\text{s}(\mathsf{k}_z(\mathsf{Q}))$ of $\partial\mathsf{t}_z(\mathsf{Q})$. Given a block $\mathsf{Q}_m$ of $\mathsf{k}_z(\mathsf{Q})$ let $\zeta_z^m$ be the similarity map in $\mathbb{R}^{N+2}$ that maps $(\mathsf{Q}, \mathsf{e})$ onto $(\mathsf{Q}_m, \mathsf{w} \cap \mathsf{Q}_m)$. Similarly, given a block $Q_m$ in the core $\kappa_z(Q)$, let $\varepsilon(Q_m)$ be the marked flag-edge $w_{\mathcal{F}} \cap Q_m$ derived from $(Q, e_{\mathcal{F}})$ and let $\sigma_z^m: (Q(Q_m), e_{\mathcal{F}(Q_m)}) \to (Q_m, \varepsilon(Q_m))$ be a similarity map for some unique $Q(Q_m) \in \{Q_I, Q_J\}$ and $(N-1)$ flag $\mathcal{F}(Q_m)$ of $\mathfrak{C}^{N+1}$. (The dependence of $\sigma_z^m$ on $\mathcal{F}$ is omitted to simplify the notation.) Define now $\theta_z^{\mathcal{F}}$ on the inner side $\text{s}(\mathsf{k}_z(\mathsf{Q}))$ by taking $\theta_z^{\mathcal{F}}|\text{s}(\mathsf{Q}_m) = \sigma_z^m \circ \theta_z^{\mathcal{F}(Q_m)} \circ (\zeta_z^m)^{-1}$ for all $1 \leq m \leq M^{1+z}$. Since the union of marked flag-edges of the $Q_m$ is the flag-path $w_{\mathcal{F}}$, the map $\theta_z^{\mathcal{F}}$ is defined consistently on the intersection of consecutive blocks and thus is well-defined.
\emph{Step 3}. For each $(N-1)$-flag $\mathcal{F}$ of $\mathfrak{C}^{N+1}$, let $\phi_z^{\mathcal{F}} : \mathsf{A}_z \to A$ with
\[ \phi_z^{\mathcal{F}}(xt) = \frac{M-2}{M}\left ( \frac{M-1}{M-M^{-z}}(t-1)+1 \right ) \psi_{\mathcal{F}}(x)\]
where $t\in[M^{-1-z},1]$ and $x\in \partial\mathfrak{C}^{N+1}$. Define $\theta_z^{\mathcal{F}}$ on the entrance and exit of $\mathsf{t}_z(\mathsf{Q})$ by $\phi_z^{\mathcal{F}}$ modulo an isometry chosen in such a way that the mappings $\theta_z^{\mathcal{F}}: (\partial \mathsf{t}_z(\mathsf{Q}), \mathsf{e}, \mathsf{w}) \to (\partial \tau_z(Q), e_{\mathcal{F}}, w_{\mathcal{F}})$ are homeomorphisms. Then $\theta_z^{\mathcal{F}}$ are in fact bi-Lipschitz.
The final step in the construction of the mappings $\Theta_z^{\mathcal{F}}$ is given in the next lemma.
\begin{lem}\label{lem:BLext}
Every bi-Lipschitz map $\theta_z^{\mathcal{F}} $ extends to a bi-Lipschitz map
\[ \Theta_z^{\mathcal{F}} : (\mathsf{t}_z(\mathsf{Q}), \mathsf{e}, \mathsf{w}) \to (\tau_z(Q), e_{\mathcal{F}}, w_{\mathcal{F}}).\]
\end{lem}
We verify the lemma first for $z=0$. If $Q=Q_I$ then define $\Theta_0^{\mathcal{F}_0} : \mathsf{t}_0(\mathsf{Q}) \to \tau_0(Q_I)$ with $\Theta_0^{\mathcal{F}_0}(xt,t') = (\phi_{0,\mathcal{F}_0}(xt),t')$. If $Q=Q_L$ then note that $\tau_0(Q_I)$ is the union of $1$-fibers $I_x$ and $\tau_0(Q_L)$ is the union of $1$-fibers $L_x$ where $I_x,L_x$ are as in Step 1 and $x\in \frac{M-2}{M}\mathcal{A}^{N+1}(M^{-1},1)$. Let $\theta_{\mathcal{F}} : \tau_0(Q_I) \to \tau_0(Q_L)$ be the bi-Lipschitz mapping that maps each $I_{x}$ to $L_{\psi_{\mathcal{F}}^{-1}(x)}$ by arc-length parametrization. Set $\Theta_0^{\mathcal{F}} = \theta_{\mathcal{F}}\circ\Theta_0^{\mathcal{F}_0}$.
The proof of Lemma \ref{lem:BLext} when $z=p$ relies on the structure of the paths $J_I$, $J_L$ and is deferred until Section \ref{sec:exten_proof}.
\section{Quasisymmetric snowflaking homeomorphisms in $\mathbb{R}^{N+2}$}\label{sec:proprational}
The key part of the proof of Theorem \ref{thm:main} for a rational $\alpha\in[N,N+\frac{n-1}{n}]$ is the construction of a quasisymmetric mapping $F_{\alpha} \colon \mathbb{R}^{N+2} \to \mathbb{R}^{N+2}$ that maps Whitney squares of a $2$-dimensional plane of $\mathbb{R}^{N+2}$ into sets which are bi-Lipschitz homeomorphic to the Whitney squares of $\mathbb{G}_{\alpha}$.
\begin{prop}\label{prop:qsimilarity}
For all integers $N\geq 0$ and $n\geq 1$, there exists $\lambda >1$ depending only on $N,n$ satisfying the following. For each rational $\alpha\in[N,N+\frac{n-1}{n}]$, there exists an $\eta$-quasisymmetric map $F_{\alpha} : \mathbb{R}^{N+2} \to \mathbb{R}^{N+2}$ with $\eta$ depending only on $N,n$ such that,
\begin{equation}\label{eq:quasisim2}
|x'|^{\frac{\alpha}{1+\alpha}}F_{\alpha}|_{B(x,\frac{1}{2}|x'|)} \quad \text{ is }\lambda\text{-bi-Lipschitz}
\end{equation}
for all $x = (x',x'')\in\mathbb{R}^{N+1}\times\mathbb{R}$ with $|x'| \neq 0$.
\end{prop}
The mapping $F_{\alpha}$ is $\frac{1}{1+\alpha}$-snowflaking on $\{0\}\times\mathbb{R} \subset \mathbb{R}^{N+1}\times\mathbb{R}$.
\begin{cor}\label{cor:qsimilarity}
Let $F_{\alpha} \colon \mathbb{R}^{N+2} \to \mathbb{R}^{N+2}$ be the mapping of Proposition \ref{prop:qsimilarity}. Then, there exists $\lambda'>1$ depending only on $N,n$ such that
\[ (\lambda')^{-1}|x-y|^{\frac{1}{1+\alpha}} \leq |F_{\alpha}(x) - F_{\alpha}(y)| \leq \lambda'|x-y|^{\frac{1}{1+\alpha}}\]
for all $x = (x',x''),y=(y',y'') \in \mathbb{R}^{N+1}\times\mathbb{R}$ with $|x''-y''| \geq \frac{1}{2}\max\{|x'|,|y'|\}$.
\end{cor}
\begin{proof}
Let $\hat{x} = (w,x'')$ and $\hat{y} = (w,y'')$ where $w\in\mathbb{R}^{N+1}$ satisfies $|w|=3|x-y|$. Note that $|x-y| \simeq |\hat{x}-x| \simeq |\hat{x}-\hat{y}|$. By (\ref{eq:quasisim2}) we have that $|F_{\alpha}(\hat{x}) - F_{\alpha}(\hat{y})| \simeq |w|^{-\frac{\alpha}{1+\alpha}}|x-y|$ and applying the fact that $F_{\alpha}$ is quasisymmetric twice (to the points $x,y,\hat{x}$ and to the points $\hat{x},x,\hat{y}$), $|F_{\alpha}(x) - F_{\alpha}(y)| \simeq |F_{\alpha}(\hat{x}) - F_{\alpha}(\hat{y})| \simeq |x-y|^{\frac{1}{1+\alpha}}$.
\end{proof}
The rest of this section is devoted to the proof of Proposition \ref{prop:qsimilarity}. As mentioned in Section \ref{sec:outline}, the construction in \cite{Wu} can be used to deduce Proposition \ref{prop:qsimilarity} for all rational $\alpha\in [N,N+1)$ but with no control on $\lambda$ and $\eta$. For this reason, while in \cite{Wu} the mapping $F_{\alpha}$, for $\alpha=1$, is obtained by iterating one family of bi-Lipschitz mappings $\Theta^{\mathcal{F}}$, here $F_{\alpha}$ is obtained by a periodic iteration of $2$ families of bi-Lipschitz mappings $\Theta^{\mathcal{F}}_{z}$ (for the two values $z\in\{0,N+\frac{n-1}{n}\}$) in the alternating fashion of Section \ref{sec:prelimarr}.
For the rest we fix integers $N\geq 0$, $n\geq 1$ and a rational number $\alpha\in[N,p_{N,n}]$ where, as before, $p_{N,n} = N+\frac{n-1}{n}$. In Section \ref{sec:qcinQ} we define the map $F_{\alpha}$ on the block $\mathsf{Q}$ and in Section \ref{sec:QCext} we extend it in all $\mathbb{R}^{N+2}$.
\subsection{A preliminary arrangement}\label{sec:prelimarr}
Suppose that $\frac{\alpha}{p_{N,n}} = \frac{a}{a+b}$ for some $a,b\in\mathbb{N}$. Using the next lemma we create a periodic sequence $(z_k)_{k\geq 1}$ that takes only the two values $0,p_{N,n}$ and $|z_1+\dots +z_k- k\alpha| \leq p_{N,n}$ for all $k \geq 1$.
\begin{lem}\label{lem:sum}
Suppose that $y<z<x$ are such that $(a+b)z = ax +by$ for some nonnegative integers $a,b$. Then, there exists a finite sequence $(z_k)_{1}^{a+b}$ which has $a$ terms $x$ and $b$ terms $y$ such that, for all $k=1,\dots,a+b$,
\begin{equation}\label{eq:sum}
|z_1+\dots +z_k - kz| \leq x-y.
\end{equation}
\end{lem}
\begin{proof}
We may assume that $a,b\neq 0$; otherwise the claim is immediate.
Define $z_k$ inductively as follows. Set $z_1 = x$. Suppose that the terms $z_1,\dots,z_k$ have been defined; if $z_1+\dots + z_k \geq kz$, set $z_{k+1} = y$ and if $z_1+\dots z_k < kz$, set $z_{k+1} = x$.
Suppose that for some $k_0 < a+b$, the sequence $\{z_1,\dots,z_{k_0}\}$ contains exactly $b$ terms $y$. Then, $z_1+\dots + z_{k_0} - k_0z = (a+b-k_0)(z-x) <0$ and thus, $z_{k_0+1} = x$. Similarly, $z_k = x$ for all $k=k_0+1,\dots,a+b$ and $(z_k)^{a+b}_1$ has exactly $b$ terms $y$ and $a$ terms $x$. The same arguments apply if, for some $k_0 < a+b$, the sequence $\{z_1,\dots,z_{k_0}\}$ contains exactly $a$ terms $x$.
To show (\ref{eq:sum}) we apply induction on $k$. If $k=1$ then $|z_1-z| = |x-z| < x-y$. Suppose that (\ref{eq:sum}) is true for some $k<a+b$. Without loss of generality, assume that $z_1+\dots+z_{k} - k z \geq 0$. Then, $z_{k+1} = y$ and
\[y-x < y-z \leq z_1+\dots+z_{k+1} - (k+1)z \leq (x-y) + (y-z) \leq x-y.\qedhere\]
\end{proof}
By Lemma \ref{lem:sum}, there exists a sequence $(z_k)_{1}^{a+b}$ having $a$ terms $p_{N,n}$ and $b$ terms $0$ such that, for each $k = 1,\dots,a+b$,
\[ |z_1 + \dots + z_k - k\alpha| \leq p_{N,n}.\]
Extend $z_k$ to all $k\in\mathbb{N}$ with $z_k = z_{k'}$ if $k \equiv k' \mod (a+b)$.
\subsection{A quasiconformal map on $\mathsf{Q}$}\label{sec:qcinQ}
We define a quasiconformal mapping $f: (\mathsf{Q},\mathsf{e}) \to (Q_I, e_{\mathcal{F}_0})$, iterating the mappings $\Theta^{\mathcal{F}}_z$ as in \cite[Section 2.6]{Wu}.
Set $\mathsf{K}_0 = \mathsf{t}(\mathsf{Q})$ and for $k\geq 1$,
\[ \mathsf{K}_{-k} = (M^{-k-z_1-\dots-z_{k}}\mathfrak{C}^{N+1})\times [0,1].\]
Moreover, define $\mathsf{T}_{-k} = \overline{\mathsf{K}_{-k} \setminus \mathsf{K}_{-k-1}}$ with $k \geq 0$. Then
\[ \mathsf{Q} = (\{0\}\times[0,1]) \cup \bigcup_{k \geq 0} \mathsf{T}_{-k}.\]
Set also $K_0 = Q_I$, $K_{-1} = \kappa_{z_1}(Q_I)$, and $T_0 = \overline{K_0 \setminus K_{-1}} = \tau_{z_1}(Q)$. Let $f|\mathsf{T}_0 = \Theta_{z_1}^{\mathcal{F}_0}: \mathsf{T}_0 \to T_0$ noticing that $\mathsf{T}_0 = \mathsf{t}_{z_1}(\mathsf{Q})$.
For every $l \in [1, M^{1+z_1}]$, let $\varepsilon_l$ be the marked flag-edge on $Q_l$ derived from $(Q_I, e_{\mathcal{F}_0})$, and let $\sigma^l_{z_1}: (Q(l), e_{\mathcal{F}(l)}) \to (Q_l, \varepsilon_l)$ be the similarity mapping defined in Section \ref{sec:BLmaps} where $Q(l)\in\{Q_I,Q_L\}$ and $\mathcal{F}(l)$ is a $(N-1)$-flag of $\mathfrak{C}^{N+1}$. The similarity $\sigma^l_{z_1}$ induces naturally a core $\kappa_{z_2}(Q_l)$, consequently a tube $\tau_{z_2}(Q_l) = \overline{Q_l \setminus \kappa_{z_2}(Q_l)}$ to each block $Q_l$ in $K_{-1}$.
Set $K_{-2} = \bigcup_l \kappa_{z_2}(Q_l)$ and $T_{-1} = \overline{K_{-1} \setminus K_{-2}} = \bigcup_l \tau_{z_2}(Q_l)$. Since $\mathsf{T}_{-1} = \bigcup_l \mathsf{t}_l$, the mapping $f|\mathsf{T}_{-1}: \mathsf{T}_{-1} \to T_{-1}$ is defined by gluing together homeomorphisms
\[f|\mathsf{t}_l = \sigma_{z_1}^l \circ \Theta^{\mathcal{F}(l)}_{z_2} \circ (\zeta_{z_1}^l)^{-1}: \mathsf{t}_l \to \tau_{z_2}(Q_l)\]
where $\zeta^l_{z_1} : (\mathsf{Q},\mathsf{e}) \to (\mathsf{Q}_l,\mathsf{w}\cap \mathsf{Q}_l)$ is the similarity defined in Section \ref{sec:BLmaps}.
The union $W_{-1}$ of marked flag-edges $\varepsilon_l$ is a flag-path along $\kappa_{z_2}(Q_I)$ going from $\{(\frac{M-2}{M^2}\mathcal{C}^j)\times\{0\} : \mathcal{C}^j \in \mathcal{F}_0\}$ to $\{(\frac{M-2}{M^2}\mathcal{C}^j)\times\{1\} : \mathcal{C}^j \in \mathcal{F}_0\}$, and the restrictions of $f|\mathsf{t}_l$ to the entrance and to the exit of $\mathsf{t}_l$ are identical modulo isometries for all $l$. Hence, we conclude that the gluing, therefore the homeomorphism $f|\mathsf{T}_{-1}$, is well-defined. We now have the extension $f: \mathsf{T}_0 \cup \mathsf{T}_{-1} \to T_0 \cup T_{-1}$.
For the next step, the index $l$ in the previous step is replaced by $l_1$.
Fix $l_1 \in \{1,\dots, M^{1+z_1}\}$. Associated to each of the $M^{1+z_2}$ blocks $Q_{l_1, l_2}$ ($1 \leq l_2\leq M^{1+z_2}$) in the core $\kappa_{l_1} = \kappa_{z_2}(Q_{l_1})$, the process of defining $f|\mathsf{t}_{l_1}$ has uniquely defined a core $\kappa_{l_1, l_2}=\kappa_{z_3}(Q_{l_1,l_2})$, a tube $\tau_{l_1, l_2} = \tau_{z_3}(Q_{l_1,l_2})$, a marked flag-edge $\varepsilon_{l_1,l_2}$, a block $Q(l_1, l_2) \in \{Q_I, Q_J\}$, an $(N-1)$-flag $\mathcal{F}(l_1, l_2)$ of $\mathfrak{C}^{N+1}$, and a similarity mapping
\[ \sigma_{z_1,z_2}^{l_1,l_2} : (Q(l_1,l_2), e_{\mathcal{F}(l_1,l_2)}) \to (Q_{l_1,l_2}, \varepsilon_{l_1,l_2}).\]
Similarly, we define for each $l_2=1,\dots,M^{1+z_2}$ a similarity mapping
\[ \zeta_{z_1,z_2}^{l_1,l_2} : (\mathsf{Q}, \mathsf{e}) \to (\mathsf{Q}_{l_1,l_2}, \mathsf{w}\cap \mathsf{Q}_{l_1,l_2}).\]
The union $W_{-2}$ of these $M^{2 + z_1 + z_2}$ marked flag-edges is a flag-path along $K_{-2}$ from $\{(\frac{M-2}{M^3}\mathcal{C}^j)\times\{0\} : \mathcal{C}^j \in \mathcal{F}_0\}$ to $\{(\frac{M-2}{M^3}\mathcal{C}^j)\times\{1\} : \mathcal{C}^j \in \mathcal{F}_0\}$, and the union $K_{-3}$ of the cores of these $M^{2+z_1+z_2}$ new blocks is a topological $(N+2)$-cube. Set $T_{-2} = \overline{K_{-2} \setminus K_{-3}}$. We now extend $f: \mathsf{T}_0 \cup \mathsf{T}_{-1} \cup \mathsf{T}_{-2} \to T_0 \cup T_{-1} \cup T_{-2}$ by gluing together the homeomorphisms
\[f|\mathsf{t}_{l_1,l_2} = \sigma_{z_1,z_2}^{l_1,l_2} \circ \Theta_{z_3}^{\mathcal{F}(l_1,l_2)} \circ (\zeta_{z_1,z_2}^{l_1,l_2})^{-1}|\mathsf{t}_{l_1,l_2} \to \tau_{l_1, l_2}.\]
Continuing this process inductively in a self-similar manner, we obtain a homeomorphism $f : \mathsf{Q}\setminus (\{0\}\times [0,1]) \to Q_I \setminus \gamma$, where $\gamma$ is the snowflake open curve $\gamma = \bigcap_{k=1}^\infty K_{-k}$.
\begin{lem}\label{lem:qsimil}
There exists $C > 1$ depending only on $N,n$ such that $M^{- \alpha k}f$ is $C$-bi-Lipschitz on each of the $M^{k+z_1+\dots+z_k}$ tubes in $\mathsf{T}_{-k}$.
\end{lem}
\begin{proof} The scaling factor of each $\zeta_{z_1,\dots,z_k}^{l_1,\dots,l_k}$ is $M^{-k-z_1-\dots-z_k}$ and the scaling factor of each $\sigma_{z_1,\dots,z_k}^{l_1,\dots,l_{k}}$ is $\frac{M-2}{M}M^{-k}$. Moreover, only a finite number of different bi-Lipschitz mappings $\Theta_{z}^{\mathcal{F}}$ have been used in the definition of $f$. Therefore, by Lemma \ref{lem:sum}, $M^{-\alpha k}f$ is $C$-bi-Lipschitz on each of the $M^{k+z_1+\dots+z_k}$ tubes in $\mathsf{T}_{-k}$, for some constant $C > 1$ depending on $M$, $p_{N,n}$, and the bi-Lipschitz constants of the maps $\Theta^{\mathcal{F}}_{0},\Theta^{\mathcal{F}}_{p_{N,n}}$; thus $C$ depends only on $N,n$.
\end{proof}
Hence, the mapping $f: \mathsf{Q}\setminus(\{0\}\times [0,1]) \to Q_I \setminus \gamma$ is $K$-quasiconformal for some $K$ depending only on $N,n$. By a theorem of V\"ais\"al\"a for removable singularities \cite[Theorem 35.1]{Vais1}, $f$ can be extended to a $K$-quasiconformal mapping from $\mathsf{Q}$ onto $Q_I$.
\begin{rem}
Note the following self-similar property on $I$-blocks: whenever $Q_{l_1,\dots,l_{a+b}}$ is an $I$-block of $K_{-a-b}$ then
\begin{equation}\label{eq:selfsimil}
f|\mathsf{t}_{l_1,\dots,l_{a+b}} = \sigma_{z_1,\dots,z_{a+b}}^{l_1,\dots,l_{a+b}} \circ f|\mathsf{t} \circ (\zeta_{z_1,\dots z_{a+b}}^{l_1,\dots,l_{a+b}})^{-1}|\mathsf{t}_{l_1,\dots,l_{a+b}}.
\end{equation}
\end{rem}
In particular, the periodicity of $\{z_{k}\}$ with period $a+b$ implies the periodicity of $f$ (up to similarities) to tubes $\mathsf{t}_{l_1,\dots,l_{k}}$ and $\mathsf{t}_{l_1,\dots,l_{k+a+b}}$ when $Q_{l_1,\dots,l_{k}}$, $Q_{l_1,\dots,l_{k+a+b}}$ are $I$-blocks.
Finally, note that the snowflake curve $\gamma = \bigcup_{k=1}^{\infty} K_{-k}$ is the image of the line segment $\{0\}\times[0,1]$ under $f$.
\subsection{Quasiconformal extension to $\mathbb{R}^{N+2}$}\label{sec:QCext}
We now extend the mapping $f: \mathsf{Q} \to Q_I$ to a quasiconformal homeomorphism of $\mathbb{R}^{N+2}$ by backward iteration.
Fix an $I$-block $Q_{l_1,\dots,l_{a+b}}$ in some core $\kappa_{l_1,\dots,l_{a+b}}$ of $Q_I$ with $l_i \neq 1, M^{1+z_i}$. Such a block exists by the first property of the path $J_I$ in Section \ref{sec:cores}.
Let $\zeta = \zeta_{z_{1},\dots,z_{a+b}}^{l_1,\dots,l_{a+b}}$ be the similarity in $\mathbb{R}^{N+2}$ that maps $(\mathsf{Q}, \mathsf{e})$ to $(\mathsf{Q}_{l_1,\dots,l_{a+b}}, \mathsf{w} \cap \mathsf{Q}_{l_1,\dots,l_{a+b}})$, and $\sigma = \sigma_{z_1,\dots,z_{a+b}}^{l_1,\dots,l_{a+b}}$ be the similarity in $\mathbb{R}^{N+2}$ that maps $(Q_I, e_{\mathcal{F}_0})$ to $(Q_{l_1,\dots,l_{a+b}}, w_{\mathcal{F}_0} \cap Q_{l_1,\dots,l_{a+b}})$ as in Section \ref{sec:qcinQ}. Note that $\zeta$ has a scaling factor $M^{-(a+b)(1+\alpha)}$ and $\sigma$ has a scaling factor $\frac{M-2}{M}M^{-(a+b)}$.
Because $l_i \neq 1, M^{1+z_i}$, the space $\mathbb{R}^{N+2}$ is the union of an increasing sequence of $I$-blocks and regular blocks
\[ \mathbb{R}^{N+2} = \bigcup_{k \geq 0} \sigma^{-k} Q_I = \bigcup_{k \geq 0} \zeta^{-k} \mathsf{Q}.\]
If $l_i = 1$ for all $i=1,\dots,a+b$ or $l_i = M^{1+z_i}$ for all $i=1,\dots,a+b$ then these unions would be proper subsets of $\mathbb{R}^{N+2}$.
Define homeomorphisms $F^{(k)}: \zeta^{-k}\mathsf{Q} \to \sigma^{-k}Q_I$, $k \geq 0$, by
\begin{equation}\label{eq:selfsimil2}
F^{(k)} = \sigma^{-k} \circ f \circ \zeta^k.
\end{equation}
The self similar property \eqref{eq:selfsimil} implies that $f \circ \zeta|\mathsf{Q} = \sigma \circ f$. Therefore, $F^{(k)}|\mathsf{Q} = \sigma^{-k} \circ f \circ \zeta^k|\mathsf{Q} = f$ for all $k \geq 0$, and $F^{(k')}|\zeta^{-k} \mathsf{Q} = F^{(k)}$ for all $k' \geq k \geq 0$. Thus, the mapping $F_{\alpha} = \lim_{k \to \infty} F^{(k)}: \mathbb{R}^{N+2} \to \mathbb{R}^{N+2}$ is well-defined. Moreover, since all mappings $F^{(k)}$ are $K$-quasiconformal, $F_{\alpha}$ is $K$-quasiconformal and therefore, $F_{\alpha}$ is $\eta$-quasisymmetric for some $\eta$ depending only on $N,n$.
\begin{rem}\label{rem:distinction}
The backward iteration depends on the fact that $\alpha$ is rational. In fact, for any real number $\alpha \in [N, p_{N,n}]$, the arguments of Lemma \ref{lem:sum} can be used to find a sequence $(z_k)_{k\geq 0}$ having terms in $\{0,p_{N,n}\}$ such that $|z_1+\cdots+z_k - k\alpha| \leq p_{N,n}$ for all $k\geq 0$. Therefore, a quasiconformal map $f: \mathsf{Q} \to Q_I$ can be constructed as in Section \ref{sec:qcinQ}. However, if $\alpha$ is irrational the sequence $(z_k)$ is not periodic and the backward iteration cannot be used to extend this map in all $\mathbb{R}^{N+2}$.
\end{rem}
We show now that the quasisymmetric mapping $F_{\alpha}$ satisfies (\ref{eq:quasisim2}).
\begin{proof}[{Proof of Proposition \ref{prop:qsimilarity}}]
By the self similar property (\ref{eq:selfsimil2}) and the scaling factors of $\zeta,\sigma$, it is enough to show (\ref{eq:quasisim2}) only for $x=(x',x'') \in \mathsf{Q}$ with $|x'|\neq 0$. Suppose that $x \in t_{l_1\cdots l_k}$. Then, $B(x,\frac{1}{2}|x'|)$ intersects at most $m$ annulus tubes $t_{l_1\cdots l_{k'}}$ for some $m$ depending only on $M$, thus on $N,n$. Since $|x'| \simeq M^{-k-z_1-\cdots-z_k} \simeq M^{-k(\alpha+1)}$, we deduce (\ref{eq:quasisim2}) by Lemma \ref{lem:qsimil}.
\end{proof}
\section{Proof of Theorem \ref{thm:main}}\label{sec:proofofmainthm}
In this section we give the proof of Theorem \ref{thm:main}. Using Proposition \ref{prop:qsimilarity}, we first show in Section \ref{sec:rationalproof} the theorem when $\alpha\geq 0$ is a rational number and then, in Section \ref{sec:irrationalproof}, we prove the theorem for all real numbers $\alpha\geq 0$ applying an Arzel\`a-Ascoli limiting argument.
Assuming Theorem \ref{thm:main}, the proof of Corollary \ref{cor:ext} is as follows.
\begin{proof}[{Proof of Corollary \ref{cor:ext}}]
Suppose that $\alpha\in[0,1)$ and $g$ is a bi-Lipschitz embedding of the singular line $\Gamma = \{x_1=0\}\subset \mathbb{G}_{\alpha}$ into $\mathbb{R}^2$. We show that $g$ extends to a bi-Lipschitz embedding of $\mathbb{G}_{\alpha}$ onto $\mathbb{R}^2$.
Let $f \colon \mathbb{G}_{\alpha} \to \mathbb{R}^2$ be the bi-Lipschitz mapping of Theorem \ref{thm:main}. Then, $g(\Gamma)$ and $f(\Gamma)$ are quasilines in $\mathbb{R}^2$ and $g\circ f^{-1}$ is a bi-Lipschitz homeomorphism between these quasilines. Consider an $\eta$-quasisymmetric mapping $h : \mathbb{R} \to f(\Gamma)$. By the Beurling-Ahlfors quasiconformal extension \cite{BerAhl}, there exists a $K$-quasiconformal extension $F : \mathbb{R}^2 \to \mathbb{R}^2$ of $h$, with $K$ depending only on $\eta$ that satisfies
\[ \diam{F(I)} \simeq |DF(x)| \diam{I}\]
for every arc $I\subset \mathbb{R}\times\{0\}$ and every point $x\in\mathbb{R}^2$ for which $\dist(x,I)\simeq |I|$. Here the ratio constants depend only on $\eta$. Similarly, there exists a quasiconformal mapping $G : \mathbb{R}^2 \to \mathbb{R}^2$ extending $g\circ f^{-1} \circ h$ satisfying the properties of $F$.
We claim that $F = G \circ F^{-1}\circ f$ is bi-Lipschitz extension of $g$. Indeed, for any point $x\in \mathbb{R}^2$ we have $|DF(x)|/|DG(x)| \simeq \diam{F(I)}/\diam{G(I)}$ for some suitable $I\subset \mathbb{R}\times\{0\}$. Since $g\circ f^{-1}$ is bi-Lipschitz, the last ratio is comparable to 1. Therefore, $|DF(x)|\simeq|DG(x)|$ and $G \circ F^{-1}$ is bi-Lipschitz.
\end{proof}
\subsection{Proof of Theorem \ref{thm:main} when $\alpha$ is rational}\label{sec:rationalproof}
We first recall two basic properties of the generalized Grushin metric.
The dilation property states that for any $\alpha \geq 0$ and any $\delta >0$,
\[d_{\mathbb{G}_{\alpha}}((\delta x_1,\delta^{1+\alpha}x_2),(\delta y_1,\delta^{1+\alpha}y_2)) = \delta d_{\mathbb{G}_{\alpha}}((x_1,x_2),(y_1,y_2)).\]
This can be found in \cite{Bellaiche} for the case $\alpha = 1$, but it applies equally to the case of arbitrary $\alpha \geq 0$.
Given $x = (x_1,x_2), y = (y_1,y_2) \in \mathbb{G}_{\alpha}$ we define the \emph{Grushin quasidistance}
\begin{equation*} \label{eq:grushinQuasimetric}
d_{\alpha}(x,y) = |x_1 - y_1| + \min\left\{ |x_2-y_2|^{\frac{1}{1+\alpha}}, \frac{|x_2-y_2|}{|x_1|^{\alpha}}\right\}.
\end{equation*}
It is well-known that the quasidistance $d_{\alpha}(x,y)$ is comparable to $d_{\mathbb{G}_{\alpha}}(x,y)$; see e.g. \cite[Theorem 2.6]{FrLa}. In fact, the following result holds true.
\begin{lem}\label{lem:grushinQuasimetric}
For each $m\geq 0$ there exists $C(m)>1$ such that for all $\alpha \in [0,m]$ and all $x,y \in \mathbb{G}_{\alpha}$
\[ C(m)^{-1}d_{\alpha}(x,y) \leq d_{\mathbb{G}_{\alpha}}(x,y) \leq C(m)d_{\alpha}(x,y)\]
\end{lem}
The proof of Lemma \ref{lem:grushinQuasimetric} is identical to that of Lemma 2 in \cite{Ack}.
The next lemma is a simple application of the Mean Value Theorem and its proof is left to the reader.
\begin{lem}\label{lem:MVT}
For all $m\geq 0$ there exists $c(m)>1$ such that for all $\alpha\in [0,m]$ and $x, y\in\mathbb{R}$ with $|x|\geq|y|$,
\[ c(m)^{-1} |x|^{\alpha}|x-y| \leq |x|x|^{\alpha} - y|y|^{\alpha}| \leq c(m)|x|^{\alpha}|x-y|.\]
\end{lem}
For each number $\alpha\in [N,N+\frac{n-1}{n}]$ define $H_{\alpha}: \mathbb{G}_{\alpha} \to \mathbb{R}\times\{0\}\times\mathbb{R} \subset \mathbb{R}^{N+2}$ to be the mapping
\[ H_{\alpha}(x_1, x_2) = (x_1|x_1|^{\alpha},0,\dots,0, x_2).\]
It is known that $H_{\alpha}$ is an $\eta'$-quasisymmetric mapping with $\eta'$ depending only on $N,n$; see e.g. \cite[Theorem 2]{Ack}.
We are now ready to prove Theorem \ref{thm:main} for rational $\alpha\geq 0$. The argument in this case is analogous to those of \cite[Theorem 1.1]{Wu} and \cite[Theorem 5.1]{Wu2}.
\begin{prop}\label{prop:rational}
For all integers $N\geq 0$ and $n\geq 1$, there exists $L>1$ depending only on $N,n$ such that, for each rational $\alpha \in [N,N+\frac{n-1}{n}]$, there exists an $L$-bi-Lipschitz homeomorphism of $\mathbb{G}_{\alpha}$ onto a $2$-dimensional quasiplane $\mathcal{P}_{\alpha} \subset \mathbb{R}^{N+2}$.
\end{prop}
\begin{proof}
Fix a rational $\alpha\in [N,N+\frac{n-1}n]$ and let $\lambda$ and $F_{\alpha}: \mathbb{R}^{N+2} \to \mathbb{R}^{N+2}$ be the constant and $\eta$-quasisymmetric map, respectively, of Proposition \ref{prop:qsimilarity} with $\lambda$ and $\eta$ depending only on $N,n$. The composition $F_{\alpha} \circ H_{\alpha}$ is a homeomorphism from $\mathbb{G}_{\alpha}$ onto the quasiplane $\mathcal{P}_{\alpha} = F_{\alpha}(\mathbb{R}\times\{0\}\times\mathbb{R})$. We show that $F_{\alpha} \circ H_{\alpha}$ is $L$-bi-Lipschitz with $L$ depending only on $\lambda$, the quasisymmetric data $\eta,\eta'$ of $F_{\alpha},H_{\alpha}$, respectively, the constant $C(N+\frac{n-1}{n})$ of Lemma \ref{lem:grushinQuasimetric} and the constant $c(N+\frac{n-1}{n})$ of Lemma \ref{lem:MVT}; thus $L$ depends only on $N,n$. The comparison constants below depend at most on $N,n$.
Let $x = (x_1, x_2)$, $y = (y_1, y_2)$ be points in $\mathbb{G}_{\alpha}$ and assume that $|x_1| \geq |y_1|$.
The proof splits into four cases.
\emph{Case I.} $|x_1| > 0$, $|x_1 - y_1| \leq |x_1|/4$, and $|x_2 - y_2| \leq |x_1|^{1+\alpha}/2$. Then, $|x_1| \simeq |y_1|$ and the Grushin distance satisfies $d_{\mathbb{G}_{\alpha}}(x,y) \simeq |x_1 - y_1| + |x_1|^{-\alpha}|x_2-y_2|$. Moreover, by Lemma \ref{lem:MVT}, $|H_{\alpha}(x)-H_{\alpha}(y)| \simeq |x_1|^{\alpha}|x_1-y_1| + |x_2-y_2|$.
If $H_{\alpha}(y) \in B(H_{\alpha}(x),\frac{1}{2}|x_1|^{1+\alpha})$ then Proposition \ref{prop:qsimilarity} yields $|F_{\alpha} \circ H_{\alpha}(x) - F_{\alpha} \circ H_{\alpha}(y)| \simeq |x_1-y_1| + |x_1|^{-\alpha}|x_2 - y_2| \simeq d_{\mathbb{G}_{\alpha}}(x,y)$.
Otherwise, $|H_{\alpha}(x)-H_{\alpha}(y)| \simeq |x_1|^{1+\alpha}$. Let $z\in\mathbb{G}_{\alpha}$ such that $|H_{\alpha}(x)-H_{\alpha}(z)| = |x_1|^{1+\alpha}/2$. Then the quasisymmetry of $F_{\alpha}$ and $H_{\alpha}$ implies $|F_{\alpha} \circ H_{\alpha}(x) - F_{\alpha} \circ H_{\alpha}(y)| \simeq |F_{\alpha} \circ H_{\alpha}(x) - F_{\alpha} \circ H_{\alpha}(z)| \simeq d_{\mathbb{G}_{\alpha}}(x,z) \simeq d_{\mathbb{G}_{\alpha}}(x,y)$.
\emph{Case II.} $|x_1| > 0$, $|x_1 - y_1| \geq |x_1|/4$, and $|x_2 - y_2| \leq |x_1|^{1+\alpha}/2$. Then, $d_{\mathbb{G}_{\alpha}}(x,y) \simeq |x_1 - y_1| \simeq |x_1|$ and, by Lemma \ref{lem:MVT}, $|H_{\alpha}(x)-H_{\alpha}(y)| \simeq |x_1|^{1+\alpha}$. Similar to the second part of Case I, $|F_{\alpha} \circ H_{\alpha}(x) - F_{\alpha} \circ H_{\alpha}(y)| \simeq d_{\mathbb{G}_{\alpha}}(x,y)$.
\emph{Case III.} $|x_1| > 0$ and $|x_2 - y_2| \geq |x_1|^{1+\alpha}/2$. Then, $d_{\mathbb{G}_{\alpha}}(x,y) \simeq |x_1 - y_1| + |x_2 - y_2|^{1/(1+\alpha)} \simeq |x_2 - y_2|^{1/(1+\alpha)}$.
By Corollary \ref{cor:qsimilarity}, $|F_{\alpha} \circ H_{\alpha}(x) - F_{\alpha} \circ H_{\alpha}(y)| \simeq |H_{\alpha}(x) - H_{\alpha}(y)|^{\frac{1}{1+\alpha}} \simeq |x_2-y_2|^{\frac{1}{1+\alpha}} \simeq d_{\mathbb{G}_{\alpha}}(x,y)$.
\emph{Case IV.} $x_1 = 0$. Then, $|F_{\alpha} \circ H_{\alpha}(x) - F_{\alpha} \circ H_{\alpha}(y)| \simeq d_{\mathbb{G}_{\alpha}}(x,y)$ by taking limits in Case III.
\end{proof}
\subsection{Proof of Theorem \ref{thm:main} when $\alpha$ is irrational}\label{sec:irrationalproof}
The following lemma deals with the bi-Lipschitz embeddability of $\mathbb{G}_{\alpha}$ into $\mathbb{R}^{[\alpha]+2}$ for all real $\alpha\geq 0$.
\begin{lem}\label{lem:BLembed}
For all integers $N\geq 0$ and $n\geq 1$, there exists $L>1$ depending only on $N,n$ such that for all $\alpha \in [N,N+\frac{n-1}{n}]$ there exists an $L$-bi-Lipschitz embedding $f_{\alpha} : \mathbb{G}_{\alpha} \to \mathbb{R}^{N+2}$.
\end{lem}
\begin{proof}
Fix a number $\alpha \in [N,N+\frac{n-1}{n}]$ and let $(q_k)_{k\in\mathbb{N}}$ be a sequence of rational numbers in $[N,N+\frac{n-1}{n}]$ converging to $\alpha$. Note that $\lim_{k\to\infty}d_{\mathbb{G}_{q_k}}(x,y) = d_{\mathbb{G}_{\alpha}}(x,y)$ for each $x,y\in\mathbb{R}^{2}$. By Proposition \ref{prop:rational}, there exists $L>1$ depending only on $N,n$ such that, for each $q_k$, there is an $L$-bi-Lipschitz map $f_{q_k} : \mathbb{G}_{q_k} \to \mathbb{R}^{N+2}$. It is clear by their construction that each $f_{q_k}$ maps $(0,0)$ to $(0,\dots,0) \in\mathbb{R}^{N+2}$.
Let $\mathscr{A}= \{a_1,a_2,\dots\}$ be a countable dense set in $(\mathbb{G}_{\alpha},d_{\mathbb{G}_\alpha})$. Note that, for each $i\in\mathbb{N}$, $|f_{q_k}(a_i)| \leq L d_{\mathbb{G}_{q_k}}(a_i,(0,0))$. Hence, for each $i\in\mathbb{N}$, the sequence $(f_{q_k}(a_i))_{k\in\mathbb{N}}$ is bounded. Define, for each $i\in\mathbb{N}$, a subsequence of $(f_{q_k})_{k\in\mathbb{N}}$ as follows. Set $(f^0_k)_{k\in\mathbb{N}} = (f_{q_k})_{k\in\mathbb{N}}$ and for each $i \in\mathbb{N}$ let $(f^i_k)_{k\in\mathbb{N}}$ be a subsequence of $(f^{i-1}_k)_{k\in\mathbb{N}}$ so that $(f^i_k(a_i))_{k\in\mathbb{N}}$ converges. Then, for each $a_i\in \mathscr{A}$, the sequence $(f^k_k(a_i))_{k\in\mathbb{N}}$ converges. Set $f(a_i) = \lim_{k\to\infty} f^k_k(a_i)$.
We claim that $f: (\mathscr{A},d_{\mathbb{G}_{\alpha}}) \to \mathbb{R}^{N+2}$ is $L$-bi-Lipschitz. Let $z_1,z_2 \in \mathscr{A}$ and $\epsilon>0$. Choose $k\in\mathbb{N}$ big enough so that
\begin{equation}\label{eq:BL1}
|f^k_k(z_i)-f(z_i)| \leq \frac{\epsilon}{3} \quad \text{ for each } i=1,2
\end{equation}
and if $f^k_k = f_{q(k)}$ for some $q(k)\in\{q_1,q_2,\dots\}$ then
\begin{equation}\label{eq:BL2}
|d_{\mathbb{G}_{q(k)}}(z_1,z_2) - d_{\mathbb{G}_{\alpha}}(z_1,z_2)| \leq \frac{\epsilon}{3L}.
\end{equation}
Combining (\ref{eq:BL1}) and (\ref{eq:BL2}) we have that
\begin{equation*}
|f(z_1)-f(z_2)|\leq Ld_{\mathbb{G}_{\alpha}}(z_1,z_2) + \epsilon.
\end{equation*}
Similarly, $|f(z_1)-f(z_2)| \geq \frac{1}{L}d_{\mathbb{G}_{\alpha}}(z_1,z_2) - \epsilon$. Since $\epsilon$ is arbitrary, the claim follows.
Using the density of $\mathscr{A}$ in $\mathbb{G}_{\alpha}$, the mapping $f$ can be extended to all $\mathbb{G}_{\alpha}$ uniquely. It remains to show that $f: \mathbb{G}_{\alpha} \to \mathbb{R}^{N+2}$ is $L$-bi-Lipschitz. Let $x_1,x_2 \in \mathbb{G}_{\alpha}$ and $\epsilon>0$. Find $z_1,z_2\in \mathscr{A}$ such that for each $i=1,2$, $d_{\mathbb{G}_{\alpha}}(x_i,z_i) < \frac{\epsilon}{4L}$ and $|f(x_i)-f(z_i)| < \frac{\epsilon}{4}$. Then,
\begin{align*}
|f(x_1)-f(x_2)| \leq L d_{\mathbb{G}_{\alpha}}(z_1,z_2) + \frac{\epsilon}{2}
\leq L d_{\mathbb{G}_{\alpha}}(x_1,x_2) + \epsilon.
\end{align*}
Similarly, $|f(x_1)-f(x_2)| \geq \frac{1}{L}d_{\mathbb{G}_{\alpha}}(x_1,x_2) - \epsilon$. Since $\epsilon$ is arbitrary, $f$ is $L$-bi-Lipschitz.
\end{proof}
We now prove Theorem \ref{thm:main}.
\begin{proof}[{Proof of Theorem \ref{thm:main}}]
Let $\alpha\in[N,N+\frac{n-1}{n}]$ and $(q_k)$ be a sequence of rationals in $[N,N+\frac{n-1}{n}]$ converging to $\alpha$ such that the $L$-bi-Lipschitz maps $f_{q_k} = F_{q_k} \circ H_{q_k}$ converge to an $L$-bi-Lipschitz map $f_{\alpha}: \mathbb{G}_{\alpha} \to \mathbb{R}^{N+2}$ as in the proof of Lemma \ref{lem:BLembed}. Here $F_{q_k}:\mathbb{R}^{N+2} \to \mathbb{R}^{N+2}$ is the quasisymmetric mapping of Proposition \ref{prop:qsimilarity}, $H_{q_k}(x,y) = (x|x|^{q_k},0,\dots,0,y)$ is the quasisymmetric mapping of $\mathbb{G}_{q_k}$ onto $\mathbb{R}^2$ and $L$ depends only on $N,n$. Note that the mappings $H_{q_k}$ converge pointwise to the mapping $H_{\alpha} = (x|x|^{\alpha},0,\dots,0,y)$ and that the mappings $F_{q_k}$ fix the origin of $\mathbb{R}^{N+2}$ and the vector $(0,\dots,0,1)$. By \cite[Corollary 10.30]{Heinonen}, passing to a subsequence, we may assume that $F_{q_k}$ converges to a quasisymmetric mapping $F_{\alpha}$. Then, $f_{\alpha} = H_{\alpha}\circ F_{\alpha}$, and the image of $f_{\alpha}$ is $F_{\alpha}(\mathbb{R}\times\{0\}\times\mathbb{R})$ which is a $2$-dimensional quasiplane in $\mathbb{R}^{N+2}$.
\end{proof}
\section{Appendix} \label{sec:paths}
This section gives the construction of the paths $J_I(N,n),J_L(N,n)$ used in Section \ref{sec:cores} and the proof of Lemma \ref{lem:BLext}. In Section \ref{sec:firstpaths}, we construct for each integer $N\geq 0$ and each integer $M = 4k+5\geq 9$ paths $\mathcal{J}^{N}_I(M)$, $\mathcal{J}^{N}_I(M)$ which serve as a base for the construction of paths $J_I(N,n),J_L(N,n)$ in Section \ref{sec:pathsN}. Then, in Section \ref{sec:exten_proof} we show Lemma \ref{lem:BLext}.
\subsection{Auxiliary paths}\label{sec:firstpaths}
Let $N\geq 0$ and $M= 4k+5 \geq 9$ be integers. The paths $\mathcal{J}^{N}_I(M), \mathcal{J}^{N}_I(M)$ are defined by induction on $N$.
For an integer $M= 4k+5 \geq 9$ let $\mathcal{J}^{0}_I(M)$ be the segment $I\subset\mathbb{R}^2$ which we divide into $M$ disjoint $I$-segments $\ell_m$ of length $1/M$. Similarly, let $\mathcal{J}^{0}_L(M)$ be the segment $L\subset\mathbb{R}^2$ which we divide into $M$ disjoint $I$- and $L$-segments $\ell_m$ of length $1/M$ where $\ell_{\frac{M-1}{2}}$ is an $L$-segment and the rest are $I$-segments.
To obtain $\mathcal{J}^{1}_I(M)$, replace each pair of $I$-segments $\ell_{m}\cup\ell_{m+1}$, where $m \in \{2, 4, \dots, \frac{M-5}{2}, \frac{M+5}{2},\dots, M-4, M-2\}$, by a swath containing $\frac{M-1}{2}$ $I$- and $L$-segments of length $1/M$ running in the negative $x_1$-direction; see Figure \ref{figure:IPath} for a schematic representation. Precisely, $\mathcal{J}^{1}_I(M)$ contains $\frac{M-5}{2}$ swaths and each swath contains $4$ $L$-segments and $\frac{M-5}{2}$ pairs of consecutive $I$-segments. Here we make use of the fact that $M=4k+5$.
To obtain $\mathcal{J}^{2}_I(M)$ replace each of the $(M-5)^2/4$ pairs of consecutive $I$-segments in $\mathcal{J}^{2}_I(M)$ by a swath containing $\frac{M-1}{2}$ many $I$- and $L$-segments of length $1/M$ running in the positive $x_3$-direction; see Figure \ref{figure:IPath}. Note that $\mathcal{J}^{2}_I(M)$ contains $(M-5)^2/4$ new swaths, each containing $\frac{M-5}{2}$ pairs of consecutive $I$-segments.
Proceeding inductively, we obtain for each integer $N\geq 0$ and each integer $M = 4k+5\geq 9$ a path $\mathcal{J}^{N}_I(M)$. Denote by $(\#\mathcal{J}^{N}_I(M))$ the total number of $I$- and $L$-segments in $\mathcal{J}^{N}_I(M)$, and by $(\#\mathcal{J}^{N}_I(M))^*$ the total number of pairs of consecutive $I$-segments. Then, $(\#\mathcal{J}^{0}_I(M)) = M$, $(\#\mathcal{J}^{0}_I(M))^* = \frac{M-5}{2}$ and for $N\geq 1$
\begin{eqnarray*}
&(\#\mathcal{J}^{N}_I(M)) &= (\#\mathcal{J}^{N-1}_I(M)) + (\#\mathcal{J}^{N-1}_I(M))^*(M-3)\\
&(\#\mathcal{J}^{N}_I(M))^* &= (\#\mathcal{J}^{N-1}_I(M))^* \frac{M-5}{2}.
\end{eqnarray*}
Therefore,
\[ (\#\mathcal{J}^{N}_I(M)) = M + (M-3)(M-5)\frac{(M-5)^{N}-2^N}{2^{N+1}(M-7)}\]
and
\[ (\#\mathcal{J}^{N}_I(M))^* = \frac{(M-5)^{N+1}}{2^{N+1}}.\]
The paths $\mathcal{J}^{N}_L(M)$ are constructed similarly; see Figure \ref{figure:IPath}.
\begin{figure}[h]
\centering
{
\includegraphics[width=1.3in]{Full_IPath}
}
\hspace{.1in}
{
\includegraphics[width=1.3in]{Full_LPath}
}
\hspace{.1in}
{
\includegraphics[width=1.3in]{IPath3}
}
\caption{The paths $\mathcal{J}^0_I(M)$, $\mathcal{J}^0_L(M)$ and a swath in the extra dimension.} \label{figure:IPath}
\end{figure}
\subsection{Construction of the paths $J_I,J_L$} \label{sec:pathsN}
Fix integers $N\geq 0$ and $n\geq 1$ and set $M = M_{N,n}=9^{n(N+2)}$. We first construct paths $\tilde{J}_I(N,n)$ and $\tilde{J}_L(N,n)$ as an extension of $\mathcal{J}^N_{I}$ and $\mathcal{J}^N_{L}$, respectively, in an extra dimension. The required paths $J_I(N,n)$ and $J_L(N,n)$ are obtained after applying a suitable rotation to $\tilde{J}_I(N,n)$ and $\tilde{J}_L(N,n)$ respectively.
We work first for $\tilde{J}_{I}(N,n)$. To construct $\tilde{J}_I(N,n)$ we use the path $\mathcal{J}^{N}_I(M)$ which contains $M' = (\#\mathcal{J}^{N}_I(M))^* = 2^{-N-1}(M-5)^{N+1}$ pairs of consecutive $I$-segments. Replace each pair of $I$-segments $\ell_m\cup \ell_{m+1}$ in $\mathcal{J}^{N}_I(M)$, $m=1,\dots,M'$, by a swath consisting of $2k_m+2$ many $I$- and $L$-segments, running in the positive $x_{N+2}$-direction. Here, $0 \leq k_m \leq \frac{M-3}2$ (if $k_m = 0$ then the swath contains only $\ell_m,\ell_{m+1}$ and if $k_m\geq 1$ then it contains $4$ $L$-segments and $2(k_m-1)$ $I$-segments). The resulting path is denoted by $\tilde{J}_I(N,n)$. Moreover we require that the swaths are chosen in such a way that $\tilde{J}_I(N,n)$ is symmetric with respect to the plane $x_{N+1} = 1/2$. Hence, for each $m\in \{1,\dotsm, M'\}$ there is $m' \in \{1,\dots,M'\}$, $m\neq m'$ such that $k_m = k_{m'}$.
The path $\tilde{J}_I(N,n)$ must consist of $M^{1+p_{N,n}} = M^{N+2-1/n}$ many $I$- and $L$-segments of length $1/M$. Thus, we require that
\[ 2(k_1 + k_2 +\cdots + k_{M'}) + 2M' + ((\#\mathcal{J}^{N}_I(M)) - 2M') = M^{N+2-1/n}\]
or equivalently
\begin{equation}\label{eq:paths}
k_1 + k_2 +\cdots + k_{M'} = \frac{M^{N+2-1/n} - (\#\mathcal{J}^{N}_I(M))}{2}.
\end{equation}
The symmetry of $\tilde{J}_I(N,n)$ implies that the left hand side of (\ref{eq:paths}) is even. Moreover, since $M$ is a multiple of $9$, the right hand side of (\ref{eq:paths}) is also even. Since each $k_m$ can take any integer value in $[0,M-3]$, the left hand side of (\ref{eq:paths}) can take any even integer value in $[0,2(M-3)M']$ and it is enough to show that
\[ 2(M-3)M' \geq \frac{M^{N+2-1/n} - (\#\mathcal{J}^{N}_I(M))}{2}.\]
Indeed, since $M = 9^{n(N+2)}$,
\[ 2(M-3)M' = \frac{(M-3)(M-5)^{N+1}}{2^N} \geq \left ( \frac{M-5}{2} \right )^{N+2} \geq M^{N+2-\frac{1}{n}}.\]
Properties (1)--(4) of Section \ref{sec:cores} are immediate. The proof of property (5) is almost identical to the proof of Lemma \ref{lem:BLext} in the following section. The path $\tilde{J}_{L}(N,n)$ is obtained similarly. In this case we require symmetry with respect to the plane $x_1+x_{N+1}= \frac{1}{2}$.
\subsection{Proof of Lemma \ref{lem:BLext}} \label{sec:exten_proof}
We show Lemma \ref{lem:BLext} for $z=p$ and $Q=Q_I$. Similar arguments apply when $Q=Q_L$. For the rest, $\mathcal{F} = \mathcal{F}_0$.
By Section \ref{sec:pathsN}, each $J_I(N,n)$ is constructed as a sequence of paths $I = J_1,J_2,\dots,J_{N+2} = J_I(N,n)$ where each $J_k$ lies in a $k$-dimensional subspace of $\mathbb{R}^{N+2}$ and $J_{k+1}$ is constructed by replacing pairs of $I$-segments $\ell_m \cup \ell_{m+1}$ of $J_k$ by swaths $\mathscr{S} = I_m\cup\mathscr{S}_m\cup I_{m+1}'$. Here, $I_m\subset\ell_m$, $I_{m+1}'\subset\ell_{m+1}$ are line segments and $\mathscr{S}_m$ is a polygonal arc perpendicular to the $k$-plane containing $J_k$. Associated to each $J_k$ we consider a core $\kappa_k = \mathcal{T}^{N+2}(J_k,\frac{M-2}{M^2})$.
Each core $\kappa_k$ consists of $M_k$ many $I$- and $L$-blocks $Q_{k,m}$, $m=1,\dots,M_k$. Here $M_k = (\#\mathcal{J}^{k}_I(M))$, if $k=0,\dots,N+1$, and $M_{N+2}=M^{1+p_{N,n}}$ with $M=9^{n(N+2)}$. Similar to the path $J_k$, each core $\kappa_k$ is constructed by removing certain pairs of $I$-blocks from $\kappa_{k-1}$ and replacing these pairs by solid swaths $\mathcal{S} = \mathcal{T}^{N+2}(\mathscr{S},\frac{M-2}{M^2})$. Note that $\kappa_1 = \kappa_0(Q)$.
For each $k,m$ the side $\text{s}(Q_{k,m})$ has a fibration into $I$-segments (if $Q_{k,m}$ is an $I$-block) or $L$-segments (if $Q_{k,m}$ is an $L$-block) similar to the fibrations $\{I_x\},\{L_x\}$ of Section \ref{sec:BLmaps}. The fibrations of the sides of $Q_{k,m}$ induce a fibration of the side $\text{s}(\kappa_k) = \bigcup_{u} \Gamma_{k,u}$ where $\Gamma_{k,u}$ is a polygonal arc, $u\in \partial\mathfrak{C}^{N+1}$ and $\Gamma_{k,u}\cap \text{s}(Q_{k,m})$ is a fiber of $\text{s}(Q_{k,m})$. As with the paths $J_k$, each $\Gamma_{k+1,u}$ is constructed by replacing certain line segments of $\Gamma_{k,u}$ by fibers which lie on the new solid swaths of $\kappa_{k+1}$. Note that if $u$ is a vertex of $\mathfrak{C}^{N+1}$ then $\Gamma_{k,u}$ is an edge of $\kappa_k$ and $\Gamma_{N+2,u}$ is an element of the flag-path $w_{\mathcal{F}_0}$ of $ \kappa_p(Q_I)$.
For the construction of $\Theta_p^{\mathcal{F}_0}$ we first map $\tau_p(Q)$ onto $\tau_{0}(Q)$ and then we compose with $\Theta_0^{\mathcal{F}_0}$.
\emph{Step 1: We map $(Q,\kappa_{N+2})$ onto $(Q,\kappa_{1})$}. We construct a bi-Lipschitz map in $Q$ which fixes $\partial Q$ and maps $\kappa_{N+2}$ onto $\kappa_{N+1}$ by compressing each solid swath onto the two $I$-blocks of $\kappa_{N+1}$ which it replaced. The map is defined in a neighbourhood of each solid swath.
For each solid swath $\mathcal{S} \subset \kappa_{N+2}$, consider a $(N+2)$-box $\widetilde{Q}(\mathcal{S}) \subset Q$ which contains $\mathcal{S}$ and satisfies the following properties,
\begin{enumerate}
\item each face of $\widetilde{Q}(\mathcal{S})$ is parallel to a coordinate $(N+1)$-hyperplane;
\item $\widetilde{Q}(\mathcal{S}) \cap \kappa_{N+2} = \mathcal{S}$;
\item $\widetilde{Q}(\mathcal{S})$ and $\widetilde{Q}(\mathcal{S}')$ have disjoint interiors if $\mathcal{S} \neq \mathcal{S}'$.
\end{enumerate}
For each solid swath $\mathcal{S} \subset \kappa_{N+2}$ we construct a bi-Lipschitz isotopy $\Phi_{\mathcal{S}}: \widetilde{Q}(\mathcal{S}) \times I \to \widetilde{Q}(\mathcal{S})$ such that $\Phi_{\mathcal{S}}(\cdot, t)|\partial \widetilde{Q}(\mathcal{S}) = \text{id}$ for all $t \in [0,1]$, $\Phi_{\mathcal{S}}(\cdot, 0) = \text{id}$, and $\Phi_{\mathcal{S}}(\cdot, 1)|\mathcal{S}$ is a PL bi-Lipschitz map of $\mathcal{S}$ onto the two $I$-blocks of $\kappa_{N+1}$ that $\mathcal{S}$ replaced. By PL bi-Lipschitz isotopy, we mean that the induced mapping $g_{t_1t_2} = \Phi_{\mathcal{S}}(\Phi_{\mathcal{S}}^{-1}(\cdot , t_1),t_2)$ is piecewise linear and $(1+C|t_2-t_1|)$-bi-Lipschitz for some constant $C>0$ and all $t_1, t_2 \in [0,1]$. Note that $g_{t_1t_2}^{-1} = g_{t_2t_1}$.
Fix a solid swath $\mathcal{S} \subset \kappa_{N+2}$ and write $\widetilde{Q}(\mathcal{S}) = \widetilde{Q}$ and $\Phi_{\mathcal{S}} = \Phi$. Suppose that $\mathcal{S} = Q_1'\cup\cdots\cup Q_{2m}'$ where $Q_{i}'$ are blocks of $\kappa_{N+2}$ and that $\mathcal{S}$ has replaced two $I$-blocks $Q_1\cup Q_2$ of $\kappa_{N+1}$.
If $m=1$ then $Q_1=Q_1'$, $Q_2=Q_2'$ and $\Phi$ is the identity in $\widetilde{Q}$.
Suppose now that $m\geq 2$. We write $Q_i = \mathcal{T}^{N+2}(\ell_i,\frac{M-2}{M^2})$ and $Q_j' = \mathcal{T}^{N+2}(\ell_j',\frac{M-2}{M^2})$ for $i=1,2$ and $j=1,\dots,2m$ where $\ell_1,\ell_2$ are $I$-segments, $\ell_i'$ is an $L$-segment when $i=1,m,m+1,2m$ and an $I$-segment otherwise. Let $\widehat{\ell}$ be an $I$-segment of length $\frac{1}{10M}$ intersecting both $\ell_1$ and $\ell_2$. Define $\widehat{\ell}_1 = \ell_1\setminus \widehat{\ell}$, $\widehat{\ell}_{2m} = \ell_2\setminus \widehat{\ell}$ and $\{\widehat{\ell}_j\}_{j=1}^{2m-1}$ be a partition of $\widehat{\ell}$ into $I$-segments of length $\frac{1}{(2m-2)10M}$.
Let $\Phi: \partial(\widetilde{Q}\setminus \mathcal{S}) \times I \to \widetilde{Q}$ be a PL bi-Lipschitz isotopy on $\partial(\widetilde{Q}\setminus \mathcal{S})$ such that $\Phi(\cdot, t)|\partial \widetilde{Q} = \text{id}$ for all $t \in [0,1]$, $\Phi(\cdot, 0) = \text{id}$, and $\Phi(\cdot, 1)|\partial\mathcal{S}$ maps each $Q_{j}'$ onto $\widehat{Q}_j = \mathcal{T}(\widehat{\ell}_j, \frac{M-2}{M^2})$, while each $\Gamma_{N+2,u}\cap Q_j'$ is mapped onto $\Gamma_{N+1,u}\cap \widehat{Q}_j$ by arc-length parametrization. Let $\Sigma_t = \Phi(\partial(\widetilde{Q}\setminus \mathcal{S}),t)$.
We use the following theorem of V\"ais\"al\"a on bi-Lipschitz extensions.
\begin{thm}[{\cite[Corollary 5.20]{Vais:86}}]\label{thm:vais_ext}
Let $n\geq 2$ and $\Sigma \subset \mathbb{R}^n$ be a compact piecewise linear manifold of dimension $n$ or $n-1$ with or without boundary. Then, there exist $L',L>1$ depending on $\Sigma$, such that every $L$-bi-Lipschitz embedding $f: \Sigma \to \mathbb{R}^n$ extends to an $L'$-bi-Lipschitz map $F: \mathbb{R}^n \to \mathbb{R}^n$.
\end{thm}
By Theorem \ref{thm:vais_ext}, for each $t\in[0,1]$, there are constants $L_t, L_t' >1$ such that any $L_t$-bi-Lipschitz map $f: \Sigma_t \to \mathbb{R}^{N+2}$ has an $L_t'$-bi-Lipschitz extension $F: \mathbb{R}^{N+2} \to \mathbb{R}^{N+2}$. For all $t \in [0,1]$, there is an open interval $\Delta_t$ such that $1+C|s-t|<L_t$ for all $s \in \Delta_t$. Cover $[0,1]$ with finitely many intervals $\{\Delta_{t_j}\}_{j=1}^l$, where $0 = t_0 < t_1 < \cdots < t_l = 1$ and $\Delta_{t_{j-1}}\cap \Delta_{t_j} \neq \emptyset$. For each $j=1,\dots,l$ set $a_{2j} = t_j$ and $a_{2j-1} \in \Delta_{t_{j-1}}\cap\Delta_{t_j}$. Then, each $g_{a_{j}a_{j+1}}$ extends to a bi-Lipschitz map $G_{a_{j}a_{j+1}}: \mathbb{R}^{N+2} \to \mathbb{R}^{N+2}$. Hence, $G_{a_{2l-1}a_{2l}}\circ \cdots \circ G_{a_0a_1}$ is a bi-Lipschitz self-map of $\widetilde{Q}$. The respective bi-Lipschitz maps on each $\widetilde{Q}$ can be pasted together and the resulting map is still bi-Lipschitz.
Similarly we use a bi-Lipschitz map in $Q$ that maps $\kappa_{N+1}$ onto $\kappa_{N}$ satisfying all the properties of the previous bi-Lipschitz map. Inductively, we obtain a bi-Lipschitz map
\[ \Theta' : (Q,\kappa_{N+2}) \to (Q,\kappa_{1})\]
such that $\Theta'$ is identity on $\partial Q$, maps each $\Gamma_{N+2,u}$ on $\Gamma_{1,u}$ and every block $Q_{N+2,m}$ in the core $\kappa_{N+2}$ is mapped to a block $\mathcal{T}^{N+2}(\ell,\frac{M-2}{M^2})$ where $\ell = \ell(m)$ is a straight line segment lying on $J_{1}$. Note that $J_1$ and all fibers $\Gamma_{1,u}$ of $\kappa_1$ are straight line segments isometric to each other.
\emph{Step 2: We straighten the images of $\Gamma_{N+2,u}$.} Consider the line segments
\[ \Gamma'_{N+2,u}(m) = \Theta' (\Gamma_{N+2,u}\cap Q_{N+2,m})\]
and let $\Gamma'_{N+2,u} = \bigcup_{m=1}^{M_{N+2}}\Gamma'_{N+2,u}(m)$. The family $\{\Gamma'_{N+2,u}\}_{u\in \mathfrak{C}^{N+1} }$ is a fibration of $\kappa_1 = \kappa_0(Q)$ and if $u$ is a vertex of $\mathfrak{C}^{N+1}$ then $\Gamma'_{N+2,u}$ is an edge of $\kappa_1$. Let $\Theta'' : Q \to Q$ be a bi-Lipschitz mapping which is identity on $\partial Q$ and linear on each $\Gamma'_{N+2,u}(m)$. Moreover, for all $u,m$, $\Theta''(\Gamma'_{N+2,u}(m))$ lies on $\Gamma'_{N+2,u}$ and its length is $1/M$. Define now $\Theta_z^{\mathcal{F}_0} = (\Theta')^{-1}\circ(\Theta'')^{-1}\circ\Theta_0^{\mathcal{F}_0}$ and the proof is complete.
\bibliographystyle{abbrv}
|
3,212,635,537,914 | arxiv | \section{Introduction}
AC optimal power flow (OPF) is a canonical power systems operation problem that is at the heart of optimizing large-scale power networks. Newton-based methods are at the core of many AC OPF solvers used in grids today. However, traditional Newton-Raphson can suffer from slow and numerically unstable Jacobian matrix inversions. To reduce the computational burden associated with calculating the full Jacobian and its inverse, many Quasi-Newton methods attempt to find a solution to the optimality conditions by leveraging an approximate Jacobian matrix \cite{QN1,QN2}. For example, the full Jacobian can be replaced with a function of the gradient of the objective, or the chord method can be used, which fixes the Jacobian to a constant value from the first iteration. Various ways to construct the Jacobian using data-driven techniques have been explored for power flow in \cite{JacobianData16,JacobianNN} with promising results. Here, we avoid calculating a Jacobian by replacing the Newton-Raphson step with a purely data-driven machine learning (ML) model that learns subsequent iterations. The ML model is trained on Newton-Raphson iterations and learns how to imitate the Newton-Raphson algorithm without having to construct Jacobian matrices or calculate matrix inverses.
Learning for OPF is a rapidly growing area due to the immense power that ML and deep learning in particular can provide for representing extremely complex variable relationships and performing inference (making predictions) extremely quickly. For example, \cite{chatzos2020highfidelity, OPFGNN} attempt to learn AC OPF solutions; \cite{Zamzam_learn_19} develops a method to learn \emph{feasible} AC OPF solutions; ML was used to reduce AC OPF complexity in \cite{hasan2020hybrid,Baker_jointCC}, and warm-start points for AC OPF were obtained using ML in \cite{Baker_learning,Diehl2019WarmStartingAO}, to name a few applications (a more in-depth overview can be found in \cite{Amin20}). Instead of learning the AC OPF solution directly, this paper aims to develop an iterative model that learns Newton-Raphson descent directions depending on the current candidate solution. This technique aims to learn a representation such as ``Given my current step, what direction should I move to decrease the objective function?" rather than ``Given an initial guess, what is the exact optimal solution of the nonconvex AC OPF?" A similar technique was explored in \cite{li2016learning} where a reinforcement learning agent was trained to solve general unconstrained optimization problems with great success.
The model is comprised of a fully connected three-layer neural network $F_R$ with feedback, where input ${\bf x}^{k}$ is the candidate optimal solution vector at iteration $k$. Reminiscent of a simple recurrent neural network, the model iteratively uses feedback from the output layer as inputs until convergence ($||{\bf x}^{k+1}-{\bf x}^{k}|| \leq \epsilon$). The model thus bypasses any construction of a Jacobian matrix or associated inverse. Simulation results demonstrate that although the solution is approximate, a solution can be obtained orders of magnitudes faster than traditional Newton-Raphson, and numerical issues due to nearly-singular Jacobians are avoided.
\section{Learning-boosted Quasi-Newton} \label{sec:LBQN}
A general nonconvex optimization problem with $n$-dimensional optimization variable vector ${\bf x}$, cost function $f(\cdot): \mathbb{R}^{n} \rightarrow \mathbb{R}$, $M$ equality constraints $g_i(x) = 0$, $g_i(\cdot): \mathbb{R}^{n} \rightarrow \mathbb{R}$, and $P$ inequality constraints $h_j(x) \leq 0$, $h_j(\cdot): \mathbb{R}^{n} \rightarrow \mathbb{R}$ can be written as
\begin{subequations} \label{eqn:gen_opt}
\begin{align}
\min_{\substack{{\bf x}}} ~~ f({\bf x})& \\
\mathrm{s.t:~} g_i({\bf x}) &= 0, ~~i=1, ... M\\
h_j({\bf x}) &\leq 0, ~~j=1, ..., P
\end{align}
\end{subequations}
Newton-Raphson (sometimes just called Newton's Method) utilizes first and second-order derivatives of the Karush Kuhn Tucker (KKT) optimality conditions of \eqref{eqn:gen_opt} to find a stationary point of the KKT conditions. For AC OPF, this results in finding a local (or maybe global) minimum of the nonconvex problem \eqref{eqn:gen_opt}. The Newton-step, that is, the iterative update of candidate optimal solution ${\bf x}^{k+1}$ at iteration $k$ can be written as
\begin{equation}\label{eqn:OGNR}
{\bf x}^{k+1} = {\bf x}^{k} - \alpha J^{-1}({\bf x}^{k})d({\bf x}^{k}),
\end{equation}
\noindent where $d({\bf x}^{k})$ is a vector of KKT conditions evaluated at the current candidate solution ${\bf x}^{k}$, $J^{-1}({\bf x}^{k})$ is the Jacobian matrix of the KKT conditions evaluated at ${\bf x}^{k}$, and $\alpha$ is an optional step-size parameter where $0 < \alpha \leq 1$. Equation \eqref{eqn:OGNR} is repeated until convergence, which is typically when iterations cease to change significantly (e.g., $||{\bf x}^{k+1}-{\bf x}^{k}|| \leq \epsilon$) or when the KKT conditions are nearly satisfied (e.g. $d({\bf x}^{k}) \leq \epsilon$) for some small $\epsilon$, for example.
\subsection{Quasi-Newton Methods}
The term ``Quasi-Newton" simply refers to methods where the Jacobian matrix in \eqref{eqn:OGNR} or its inverse is approximated. These methods are typically used for two reasons: First, calculating the Jacobian and its inverse is expensive and time-consuming, which may not be appropriate for large problems or fast-timescale optimization. Second, the Jacobian can be singular or close to singular at the optimal solution \cite{singularJ05, BakerJacobian13}, making the inverse challenging. Thus, Quasi-Newton methods replace $J^{-1}$ with an approximate Jacobian $\tilde{J}^{-1}$ or sometimes with an approximate Jacobian inverse.
The learning-boosted method replaces $\alpha^{k}J^{-1}({\bf x}^{k})d({\bf x}^{k})$ in \eqref{eqn:OGNR} with a fully connected neural network (NN) model $F_R(\cdot): \mathbb{R}^n \rightarrow \mathbb{R}^n$ that takes in ${\bf x}^{k}$ as an input and provides ${\bf x}^{k+1}$ as an output; e.g.
\begin{equation}\label{eqn:LBNR}
{\bf x}^{k+1} = F_R({\bf x}^{k}).
\end{equation}
A fully connected three-layer NN is used here. The variable vector ${\bf x}^{k} = [{\bf v}^{k},~ {\utwi{\theta}}^k,~ {\bf P}^k_g,~ {\bf Q}^k_g]^T$, where ${\bf v}^{k}$ contains the complex voltage magnitudes at each bus, ${\utwi{\theta}}^k$ contains the complex voltage angles, and ${\bf P}_g$ and ${\bf Q}_g$ are the real and reactive power outputs at each generator, respectively.
\subsection{Network architecture}
Choosing the number of nodes in the hidden layer can be performed by using a popular heuristic \cite{masters93}: $N_h \approx \sqrt{N_i \cdot N_o} = \sqrt{(2N_L+n) \cdot n}$, where $N_i$ is the number of nodes in the input layer and $N_o$ is the number of nodes in the output layer. For the considered problem, the number of inputs is the number of optimization variables $n$ and the number of loads $2N_L$ ($N_L$ real demands and $N_L$ reactive demands). Note that even though the loads are inputs to the NN, they are fixed and do not change as $k$ changes. These numbers were later refined through trial-and-error. A rectified linear unit (ReLU) was used as the activation function on the input layer; a hyperbolic tangent (tanh) activation function was used in the hidden layer, and a thresholded linear function was used on the output.
\section{Convergence Analysis} \label{sec:convergence}
Quasi-Newton iterations are fixed-point iterations. That is, they can be written in the form ${\bf x}^{k+1} = F({\bf x}^{k})$, where $F({\bf x}^{k})$ is the right-hand side of \eqref{eqn:OGNR}. Unfortunately, when solving AC OPF it is generally difficult to show ${\bf x}^{k+1} = {\bf x}^{k}$ for any $k$. However, we can design our NN to guarantee convergence.
In \cite{RNNcont}, conditions on the activation functions and bounds on the weights are given to ensure convergence of a recurrent neural network (RNN). The proposed NN architecture is a basic version of an RNN, as the output of the last layer becomes an input of the input layer. Consider the mapping $F_R$ in \eqref{eqn:LBNR}. In order to show that the proposed NN converges to a unique fixed point for any initial $x^0$ and that $F_R$ is a contraction, it must first be true that each activation function $f_i$ must be bounded, continuous, differentiable, real-valued, and have bounded derivatives. In the proposed NN, thresholds on both the input (ReLU) and output (Linear) layers were placed to bound their output. The ReLU function on input $x$, $\max\{0,x\}$, is techncially not differentiable at $x=0$. Thus, a more formal analysis would have to be performed to determine if exactly the same convergence results apply in every situation with a ReLU, although they were achieved in simulation. Similar but slightly worse results were achieved with a tanh function on the input layer, which does satisfy these requirements.
From Theorem 1 and 2 in \cite{RNNcont} we have a bound on each weight $w_{ij}$ connecting nodes $i$ and $j$ that must hold for the network to converge to a unique fixed point for any given $x^0$:
\begin{equation}\label{eqn:weightbound}
|w_{ij}| < c^* < \frac{1}{N_n \cdot f'_{max}},
\end{equation}
\noindent where $'$ denotes a first-order derivative, $N_n$ is the total number of network nodes, $f_l$ is activation function $l = 1,...,N_n$, and
\begin{equation}
f_{max} = \max_{\substack{y \in \mathbb{R}}, l=1,...,N_n}(|f'_l(y)|).
\end{equation}
\noindent Then $F_R$ is a contraction with $0 < c < 1$, and $c = N_n c^* |f'_{max}|$, and $w_{ij}$ can be designed by \eqref{eqn:weightbound}.
\section{Network Architecture} \label{sec:net_params}
Four networks were considered: The IEEE 30 bus, IEEE 300 bus, PG-lib 500-bus \cite{PGlib-OPF}, and 1,354-bus PEGASE networks \cite{pegase}. The number of loads, lines, generators, and total (real) generation capacity for each network is shown in Table \ref{tab:networks}. Line flow constraints were neglected (although some networks did not have them to begin with). The data generation, training and testing of the network, and simulations were performed on a 2017 MacBook Pro laptop with 16 GB of memory. Keras with the Tensorflow backend was used to train the neural network using the Adam optimizer. The chosen number of nodes in the hidden layer and training dataset sizes are shown in Table \ref{tab:hyper}.
\begin{table}[]\centering
\small \caption{Considered network parameters}
\begin{tabular}{|l|l|l|l|l|}
\hline
\textbf{Case} & \textbf{\begin{tabular}[c]{@{}l@{}}\# of \\ Loads\end{tabular}} & \textbf{\begin{tabular}[c]{@{}l@{}}\# of \\ Lines\end{tabular}} & \textbf{\begin{tabular}[c]{@{}l@{}}\# of \\ Gens\end{tabular}} & \textbf{\begin{tabular}[c]{@{}l@{}}Total Real \\ Gen. Capacity\end{tabular}} \\ \hline
\textbf{30-bus} & 20 & 41 & 6 & 335 MW \\ \hline
\textbf{300-bus} & 191 & 411 & 69 & 32.68 GW \\ \hline
\textbf{500-bus} & 200 & 597 & 56 & 12.19 GW \\ \hline
\textbf{1,354-bus} & 621 & 1991 & 260 & 128.74 GW \\ \hline
\end{tabular}\label{tab:networks}
\end{table}
\begin{table}[]\centering\small \caption{Number of nodes and training samples}
\begin{tabular}{|l|l|l|l|l|}
\hline
\textbf{Case} & \textbf{\begin{tabular}[c]{@{}l@{}}Input\\ Nodes\end{tabular}} & \textbf{\begin{tabular}[c]{@{}l@{}}Output\\ Nodes\end{tabular}} & \textbf{\begin{tabular}[c]{@{}l@{}}Hidden\\ Nodes\end{tabular}} & \textbf{\begin{tabular}[c]{@{}l@{}}Training\\ Samples\end{tabular}} \\ \hline
\textbf{30-bus} & 112 & 72 & 100 & 72,111 \\ \hline
\textbf{300-bus} & 1,120 & 738 & 800 & 91,432 \\ \hline
\textbf{500-bus} & 1,512 & 1,112 & 2,300 & 111,674 \\ \hline
\textbf{1,354-bus} & 4,470 & 3,228 & 6,000 & 126,724 \\ \hline
\end{tabular}\label{tab:hyper}
\end{table}
\subsection{Dataset generation}\label{sec:datagen}
The MATPOWER Interior Point Solver (MIPS) \cite{MATPOWER} was used to generate the data and was used as the baseline for comparison with the NN model. The termination tolerance of the MIPS solver was set to $10^{-9}$ for data generation and $10^{-4}$ for testing. The tolerance of the learning-boosted solver was set to $10^{-4}$, where convergence is reached when $||{\bf x}^{k+1}-{\bf x}^k|| \leq \epsilon$. A smaller tolerance was used for data generation to promote smoother convergence and ``basins of attraction" within the ML model. For a fair comparison, the same convergence criteria was used for the NN model and MIPS during testing. 1,000 different loading scenarios were randomly generated at each load bus from a uniform distribution of $\pm40\%$ around the given base loading scenario in MATPOWER. In some cases, the generated load profile resulted in an infeasible solution; these samples were excluded.
Table \ref{tab:hyper} shows the number of training samples $({\bf x}^{k}, {\bf x}^{k+1})$ generated for each scenario. Note that a different number of iterations to convergence is encountered for each scenario and network. It is recognized that generating a diverse and representative dataset is an important and essential thrust of research within learning-based OPF methods. This is an important direction of future work.
\section{Simulation Results} \label{sec:sim_result}
\subsection{Prediction results}
1,000 testing scenarios were generated for each network. Each simulation was initialized with a flat start (${\bf v}^0 = {\bf P}^0_g = {\bf Q}^0_g = 1$ and ${\utwi{\theta}}^0 = 0$). 1,000 test scenarios were generated using the same methodology in \ref{sec:datagen}. The mean absolute error (MAE) for voltage magnitudes (in pu) and real power generation (in MW) is shown in Table \ref{tab:error}. The fourth column in the table shows the mean absolute percentage error (MAPE) for the OPF objective function across all scenarios. Interestingly, the 300-bus case had a much worse MAE than other cases. This could be because many solutions in the training dataset had nearly-singular Jacobians near the optimal solution, and thus iterations in some areas were less well-defined and harder for the NN to learn.
\begin{table}[]\centering \small \caption{Mean absolute error across testing dataset}
\begin{tabular}{|l|l|l|l|}
\hline
\textbf{Case} & \textbf{\begin{tabular}[c]{@{}l@{}}MAE: Voltage\\ Magnitude (pu)\end{tabular}} & \textbf{\begin{tabular}[c]{@{}l@{}}MAE: Active\\ Power (MW)\end{tabular}} & \textbf{\begin{tabular}[c]{@{}l@{}}MAPE: \\Cost (\%)\end{tabular}} \\ \hline
\textbf{30-bus} & 0.004 pu & 0.64 MW & 0.29\% \\ \hline
\textbf{300-bus} & 0.009 pu & 10.47 MW & 0.65\% \\ \hline
\textbf{500-bus} & 0.099 pu & 0.62 MW & 0.66\% \\ \hline
\textbf{1,354-bus} & 0.019 pu & 7.55 MW & 1.16\% \\ \hline
\end{tabular}\label{tab:error}
\end{table}
Fig. \ref{fig:gen30} shows 200 test scenarios for the 30-bus system. The NN model does a good job of approximating the actual optimal generation values, given as black dashed lines, for each generator.
\begin{figure}[h!]
\centering
\includegraphics[width=0.7\textwidth]{gen30.png}
\caption{Predicted generation values for 200 test scenarios (colors) and actual optimal values (black dashed line) for the IEEE 30-bus system.}
\label{fig:gen30}
\end{figure}
\subsection{Computation time}
The mean, worst-case (max), and variance of time to convergence for the MIPS solver and the proposed NN is shown for all networks across all test scenarios in Table \ref{tab:time}.
\begin{table}[h!]\centering \small \caption{Time to convergence for each network.}
\begin{tabular}{|l|l|l|l|l|}
\hline
\textbf{Case} & \textbf{\begin{tabular}[c]{@{}l@{}}Mean\\ Time (s)\end{tabular}} & \textbf{\begin{tabular}[c]{@{}l@{}}Max\\ Time (s)\end{tabular}} & \textbf{\begin{tabular}[c]{@{}l@{}}Variance \\ in Time (s)\end{tabular}} & \textbf{\begin{tabular}[c]{@{}l@{}}Mean \\ Speedup\end{tabular}} \\ \hline
\textbf{30-bus MIPS} & 0.04 & 0.41 & 4.52e-04 & \cellcolor[HTML]{656565} \\ \hline
\textbf{30-bus NN} & 0.06 & 0.42 & 3.53e-04 & \textbf{-0.66x} \\ \hline
\textbf{300-bus MIPS} & 1.09 & 10.87 & 2.09 & \cellcolor[HTML]{656565}{\color[HTML]{656565} } \\ \hline
\textbf{300-bus NN} & 0.03 & 0.42 & 0.001 & \textbf{36.3x} \\ \hline
\textbf{500-bus MIPS} & 1.46 & 2.96 & 0.49 & \cellcolor[HTML]{656565} \\ \hline
\textbf{500-bus NN} & 0.08 & 0.45 & 3.25e-4 & \textbf{18.3x} \\ \hline
\textbf{1,354-bus MIPS} & 7.64 & 19.89 & 15.55 & \cellcolor[HTML]{656565}{\color[HTML]{656565} } \\ \hline
\textbf{1,354-bus NN} & 0.34 & 0.69 & 0.0016 & \textbf{22.5x} \\ \hline
\end{tabular}\label{tab:time}
\end{table}
There is almost no benefit for using a data-driven approach for smaller networks, as these can already be solved in real-time using a solver. However, for larger networks, the learning-boosted Quasi-Newton method obtains solutions extremely quickly compared to the MIPS solver. In addition, the learning-boosted approach has lower variance in computational time, making it more reliable for providing solutions on regular intervals. Interestingly, the 300-bus system takes MIPS longer to solve for some OPF scenarios than the 500-bus. In these cases, the Jacobian was close to singular.
\begin{figure}[h!]
\centering
\hspace*{-3mm}\includegraphics[width=0.7\textwidth]{norm_iter.png}
\caption{Norm of ${\bf x}^{k+1}-{\bf x}^{k}$ for two scenarios in the 500-bus (left) and 1,354-bus (right) systems.}
\label{fig:norm_iter}
\end{figure}
Fig. \ref{fig:norm_iter} shows the norm between two iterations, $||{\bf x}^{k+1}-{\bf x}^{k}||_2$, for two typical test scenarios from the 500 and 1,354-bus systems. While MIPS almost always results in fewer iterations to convergence (because it is using the exact Jacobian), each iteration takes longer to perform. Thus, although each iteration in the learning-boosted method is inexact, each iteration is very fast. In addition, convergence of the NN is guaranteed if the conditions are satisfied in Section \ref{sec:convergence}.
\subsection{Assessing feasibility}
Variable bounds are ensured by thresholding the output of the output layer on the NN. However, while convergence of the NN is guaranteed, there is no guarantee of convergence to an AC-feasible solution (just as with DC OPF). In some cases, like with the 30-bus system, the NN actually outputs solutions with smaller mean constraint violations than MIPS. This is likely due to the fact that the training data was generated for a convergence tolerance of $\epsilon = 10^{-9}$ but during testing, the algorithm is terminated at $\epsilon = 10^{-4}$. Despite the prediction error for the 1,354-bus being relatively low (with a mean optimality gap of $1.16\%$ across the testing set), high errors in the satisfaction of the power flow equations indicate a possible need for an increased training dataset size. Table \ref{tab:feasibility} shows the mean constraint violation for the AC power flow constraints across all networks and test scenarios.
\begin{table}[h!]\centering \small \caption{Mean constraint violation across all networks}
\begin{tabular}{|l|l|l|l|l|}
\hline
\textbf{Case} & \textbf{30-bus} & \textbf{300-bus} & \textbf{500-bus} & \textbf{1,354-bus} \\ \hline
\textbf{\begin{tabular}[c]{@{}l@{}}Mean Constraint \\ Violation (pu)\end{tabular}} & 0.05 & 0.43 & 0.32 & 9.95 \\ \hline
\end{tabular}\label{tab:feasibility}
\end{table}
\subsection{Tracking optimal solutions}
Assuming measurements of load at each bus are made available on a one-second basis (which is a reasonable assumption; in fact, SCADA systems should provide measurements with latency of less than one second \cite{SCADAdelay}), the benefit of a real-time optimization approach can be further assessed. Typically, real-time adjustments to generators are done via automatic generation control (AGC); a rule-based, suboptimal affine control policy. Here, a simple experiment was performed where both MATPOWER and the learning-boosted algorithm received load updates every one second. Figure \ref{fig:500track} shows the ability of the learning-boosted approach to track optimal generation setpoints with higher accuracy than MATPOWER, despite being an approximation, due to the extremely fast time to obtain new setpoints. This figure illustrates thirty-seconds of the slack bus output for the 500-bus system, which is the generator whose output varies the most in this test case.
\begin{figure}[t!]
\centering
\includegraphics[width=0.65\textwidth]{500track.png}
\caption{Slack bus generation in the 500-bus system. The learning-boosted solution, although an approximation, is able to track faster changes in demand more accurately than MIPS.}
\label{fig:500track}
\end{figure}
As MATPOWER took nearly three seconds to obtain each solution, setpoints were not updated on a one-second scale. Alternatively, the learning-boosted approach provides reasonably accurate solutions in less than a second and keeps up with one-second fluctuations in demand. In reality, generators would adjust their real-time outputs in accordance with the aforementioned AGC heuristics; for illustrative purposes, we show only OPF solutions here. With computationally light methodologies for calculating approximate OPF solutions, optimal solutions could potentially be provided in real-time.
\section{Conclusion} \label{sec:conclusion}
A data-driven model for approximating Newton-Raphson for AC OPF was implemented here. Results show that for smaller networks, not much benefit, other than avoiding singular Jacobian matrices and taking inverses of ill-conditioned matrices, can be found. However, for larger networks, the NN model can provide approximate solutions extremely quickly and at very regular intervals compared to a state-of-the-art OPF solver. The optimality gap was small for all considered cases; however, the feasibility gap for the 1,354-bus system was large. This suggests a need for more training data or an increase in model complexity, as the proposed model only contained one hidden layer.
\bibliographystyle{IEEEtran}
|
3,212,635,537,915 | arxiv | \section{Introduction}
\label{s:intro}
Fragmentation processes offer a random model for particles which break apart
as time passes. Informally, we imagine a single particle, characterised by its mass,
which after some random time splits into two or more daughter particles, distributing its
mass between them according to some law. The new particles act independently
of one another and evolve in the same way.
Variants of such processes have been studied over many years, with applications
across the natural sciences \cite{BCP-frag,
BA-cell-growth,Cx-size-dist}.
One large class of fragmentation models,
encompassing the so-called homogeneous fragmentation processes,
has been particularly successful, and a comprehensive discussion can be found
in the book of \citet{Ber-fc}.
Compensated fragmentation processes were defined by \citet{Ber-cfrag}
as a generalisation of
homogeneous fragmentations,
and permit high-intensity fragmentation and Gaussian fluctuations of the sizes of fragments.
The processes arise as the limits of homogeneous fragmentations
under dilation \cite[Theorem 2]{Ber-cfrag},
and may also be thought of as being related to
a type of branching Lévy process, for which the branching occurs at the jump times of the process.
From this viewpoint, they may be regarded as the simplest example in the class of
so-called Markovian growth-fragmentation processes \cite{BeMGF},
and for this reason they are sometimes called \emph{homogeneous} growth-fragmentation processes.
Other examples in the
class of Markovian growth-fragmentations
can be obtained by slicing planar random maps with boundary,
as discovered by \citet{BBCK-maps}, or by considering
the destruction of an infinite recursive tree, as in \citet{BB-ou}.
\skippar
The main purpose of this work is to give a complete spine decomposition
for compensated fragmentation processes.
This is motivated by the many applications that such decompositions have
found in proving powerful results
across the spectrum of branching process models.
Since the foundational work of \citet{LPP-LlogL}
on `conceptual' proofs of the $L \log L$ criterion for Galton--Watson processes,
a large literature has emerged, of which we offer here only a selection,
focusing on the applications we have in mind.
In the context of branching random walks, the spine decomposition has
been used to prove martingale convergence theorems and to study
the asymptotics, fluctuations and genealogy of the largest particle;
see \cite{Shi-BRW} for a detailed monograph with historical references.
For branching Brownian motion, spine techniques were
used by \citet{CR-kpp} to describe asymptotic presence
probabilities, and by
\citet{Kyp-FKPP} and \citet{RY-dm}
to study solutions of reaction-diffusion equations of
Fisher--Kolmogorov--Petrowski--Piscounov (FKPP) type.
In the context of superprocesses, we mention the study of strong laws of large numbers
by \citet{EKW-slln}, which also contains a thorough review of the literature.
Spine techniques have lent themselves well to the study of
homogeneous (pure) fragmentation processes.
Convergence theorems were proved by \citet{BR-disc}, and
the decomposition was used by \citet{Haa-loss} to study the fragmentation equation,
\citet{HKK-lln} for the proof of strong laws of large numbers,
and \citet{BHK-FKPP} to look at solutions of FKPP equations.
Returning to the topic of growth-fragmentation processes,
\citet[\S 4]{BBCK-maps} used a spine decomposition in order to study
certain random planar maps, and the results presented in this
article overlap with theirs under certain parameter choices (see \autoref{r:mto}.)
\citet[\S 3.2]{BerSte} gave an explicit decomposition
for compensated fragmentation processes in
the case of finite fragmentation rate and applied it to
the phenomenon of local explosion, and
\citet[\S 6]{BW-gfe} made implicit use of a
spine decomposition in studying the growth-fragmentation equation.
\medskip\noindent
Our object of study is the compensated fragmentation process
$\mathbf{Z} = (\mathbf{Z}(t), t\ge 0)$,
where $\mathbf{Z}(t) = (Z_1(t),Z_2(t),\dotsc)$ is an element of
$\ell^{2\downarrow} = \{ \mathbf{z} = (z_1,z_2,\dotsc) \in \ell^2 : z_1 \ge z_2 \ge \dotsb \ge 0 \}$.
The values $Z_1(t),Z_2(t),\dotsc$
are regarded as the ranked sizes of \emph{fragments}
as seen at time $t$.
Unless otherwise specified, we will assume that $\mathbf{Z}(0) = (1,0,\dotsc)$.
The law of $\mathbf{Z}$ is characterised by a triple $(a,\sigma,\nu)$
of \emph{characteristics},
where $a \in \RR$, $\sigma \ge 0$ and $\nu$ is a measure on the space
\[
\Pp =\mbigl\{ \mathbf{p} = (p_1,p_2,\dotsc) : p_1\ge p_2 \ge \dotsb\ge 0, \sum_{i=1}^\infty p_i \le 1 \mr\},
\]
satisfying the moment condition
\begin{equation}\label{e:nu}
\int_{\Pp} (1-p_1)^2 \,\nu(\dd\pp) < \infty .
\end{equation}
Loosely speaking, $a$ describes deterministic growth or decay of the fragments and
$\sigma$ describes the magnitude of Gaussian fluctuations
in their sizes. The measure $\nu$ is called the
\emph{dislocation measure}, and $\nu(\dd\pp)$ represents the rate at which
a fragment of size $x$ splits into a cloud of particles of sizes $xp_1,xp_2,\dotsc$.
The connection between $\mathbf{Z}$ and the triple is given by
the \emph{cumulant} $\kappa$,
which is defined by the equation
$e^{t \kappa(q)} = \lE\mbigl[\sum_{i\ge 1} Z_i(t)^q\mr]$.
It is given by the following expression, akin to the L\'evy--Khintchine formula for
L\'evy processes:
\begin{equation}\label{e:kappa}
\kappa(q)
=
\frac{1}{2}\sigma^2q^2 + aq
+ \int_{\Pp} \mBigl[ \sum_{i\ge 1} p_i^q - 1 + (1-p_1)q \mr] \, \nu(\dd \pp), \qquad q\in \RR. \end{equation}
The function $\kappa$ takes values in $\RR\cup\{\infty\}$.
We regard
$\dom \kappa := \{ q\in \RR : \kappa(q) < \infty \}$ as the function's domain.
Condition \eqref{e:nu} entails that
\begin{equation}\label{eq:dom}
q \in \dom \kappa \quad \text{if and only if}\quad \int_{\Pp} \sum_{i\ge 2} p_i^q \, \nu(\dd \pp) <\infty ,
\end{equation}
and that $[2,\infty) \subset \dom \kappa$.
One notable property of $\kappa$ is that it is strictly convex and smooth on the interior
of $\dom\kappa$.
If the measure $\nu$ satisfies the stronger moment condition
$\int_{\Pp} (1-p_1) \, \nu(\dd\pp)<\infty$,
and $\sigma = 0$,
then $\kappa$ is the cumulant of a homogeneous fragmentation process
$\mathbf{Z}$ in the sense of \cite{Ber-fc}, with additional deterministic exponential
growth or decay.
\skippar
We shall prove a spine decomposition for $\mathbf{Z}$
under a change of measure.
In particular,
for $\omega \in \dom \kappa$, we define the \emph{(exponential) additive martingale} $\mg{\omega}{\cdot}$
as follows:
\begin{equation*}
\mg{\omega}{t} = e^{-t\kappa(\omega)} \sum_{i\ge 1} Z_i(t)^q, \wh t \ge 0.
\end{equation*}
Since this is a unit-mean martingale
(see the forthcoming \autoref{l:Zbar-bp}),
we may define a new, `tilted' probability measure
$\lQ$, as follows. Fix $t \ge 0$, and let $A$ be an event depending only on the
path of $\mathbf{Z}$ up to time $t$.
Then, define
\begin{equation*} \lQ(A) = \lE[ \Ind_A \mg{\omega}{t} ] . \end{equation*}
Our first main result is \autoref{t:mto}, in which we show
that under $\lQ$, the process $\mathbf{Z}$
may be regarded as
the exponential of a single spectrally negative Lévy process (the \emph{spine})
with Laplace exponent
$\kappa(\cdot + \omega) - \kappa(\omega)$, onto whose jumps are grafted independent
copies of $\mathbf{Z}$ (under the original measure $\lP$).
This is the \emph{spine decomposition}, also known as a
\emph{full many-to-one theorem}.
In order to illustrate the power of this spine decomposition,
we study the \emph{derivative martingale} associated with $\mathbf{Z}$. For $\omega$ in the interior of $\dom \kappa$, this is defined by
\begin{equation}\label{}
\MD(\omega, t)
= \frac{\partial }{\partial \omega}\mg{\omega}{t}
= e^{-t \kappa(\omega)}
\sum_{i\ge 1} \mbigl( -t \kappa'(\omega) + \log Z_i(t) \mr)
Z_i(t)^\omega, \qquad t\geq 0.
\end{equation}
Since this martingale can take both positive and negative values,
it is not immediately obvious whether its limit as $t\to\infty$
exists.
Using our decomposition, we prove our second main result,
\autoref{t:DerMart}, which states that the derivative martingale converges
to a strictly negative limit under certain conditions.
This limit is closely related to the process representing the largest
fragment of the compensated fragmentation.
Our theorem is the counterpart of results on the asymptotics of the derivative martingale
which have been found in the context
of homogeneous (pure) fragmentation processes \cite{BR-disc},
branching random walks \cite{BK-mc,Shi-BRW}
and branching Brownian motion \cite{Kyp-FKPP}.
In the case of compensated fragmentation processes,
\citet{Dadoun:agf} studied the discrete-time skeletons of the
derivative martingale via a branching random walk, and used their convergence to obtain
asymptotics for the largest fragment.
Our work complements and extends this by showing the almost sure
convergence of the martingale in continuous time and
showing that the expectation of the terminal value is infinite;
we also obtain somewhat weaker conditions.
\skippar
This work lays the foundations for future research in two principal directions.
The first concerns more general Markovian growth-fragmentations, and in particular
we anticipate that it should be possible to extend
the spine decomposition to
growth-fragmentations based on generalised Ornstein--Uhlenbeck processes,
as studied in \cite{Shi-ougf,BB-ou}.
The second concerns applications for the homogeneous processes studied here.
Our asymptotic
results for the derivative martingale
may be used to study the size of the largest fragment and
the existence and uniqueness of travelling wave solutions to
FKPP equations, much as in \cite{BHK-FKPP}.
\medskip\noindent
The organisation of this paper is as follows. In \autoref{s:cf}, we give a rigorous
definition of the branching L\'evy process, outlining the truncation
argument of \cite{Ber-cfrag} and simultaneously define a new labelling
scheme for its particles.
In \autoref{s:backward}, we consider the measure $\lQ$
just presented, additionally distinguishing a single particle
by picking from those particles alive at time $t$ in a size-biased way.
In \autoref{s:forward}, we present a complete construction of a Markov process
with a single distinguished particle, which we claim gives the law of the
process $\mathbf{Z}$ with distinguished particle under $\lQ$; this claim is then
proven in \autoref{s:fb}.
Finally, we discuss the asymptotic properties of the derivative martingale in
\autoref{s:dm}.
\section{The branching L\'evy process}
\label{s:cf}
Our goal in this section is to establish a genealogical structure for the
compensated fragmentation
process $\mathbf{Z}$, that is to represent it as a random infinite marked tree.
This is what allows us to study the spine decomposition.
To be specific, we will define a family of L\'evy processes, $(\Zz_u, u\in \tree)$,
labelled by the nodes
of a tree $\tree$. For $t\ge 0$, let $\tree_t$ be the set of
individuals present at time $t$. We will be able to list the elements of
$\tree_t$ by $u_1,u_2,\dotsc$ such that $\Zz_{u_1}(t) \ge \Zz_{u_2}(t) \ge \dotsb$.
The compensated fragmentation process at time $t$ is then given by
\[ \mathbf{Z}(t) = (\exp(\Zz_{u_1}(t)),\exp(\Zz_{u_2}(t)), \dotsc). \]
We also define a related point measure-valued process, called the \emph{branching L\'evy process}:
\[\Zz(t) = \sum_{u\in \tree_t} \delta_{\Zz_u(t)}. \]
One can easily recover the compensated fragmentation process $\mathbf{Z}$ from $\Zz$.
Therefore, for convenience, we shall always work with $\Zz$ from now on and state all our results in terms of $\Zz$.
\subsection{L\'evy processes}
\label{s:LP}
Since our main object of study is a branching L\'evy process, it is unsurprising that
L\'evy processes play a key role. We give a short summary of the relevant definitions
and properties.
A stochastic process $\xi = (\xi(t), t\ge 0)$
under a probability measure $\sP$ is called a \emph{L\'evy process} if it has
stationary, independent increments and c\`adl\`ag paths,
and satisfies $\xi(0) = 0$ almost surely. The process
$\xi$ is said to be \emph{spectrally
negative} if the only points of discontinuity of its paths are negative jumps.
The usual way to characterise the law of such a process is through its Laplace
exponent; this is a function $\Psi \from \RR \to \RR\cup\{\infty\}$, such that
for every $t\ge 0$, $\sE[ e^{q\xi(t)} ] = e^{t\Psi(q)}$. It is well-known that
$\Psi$ satisfies the so-called \emph{L\'evy--Khintchine formula}, as follows:
\begin{equation}
\label{e:lk}
\Psi(q) = \frac{1}{2}\gamma^2 q^2 + \mathtt{a}q + \int_{(-\infty,0)} \mbigl[ e^{qx} - 1 - qx\Indic{x>-1} \mr] \, \Pi(\dd x), \qquad q\in\RR,
\end{equation}
and $\Psi(q) < \infty$ if $q\ge 0$.
Here, $\mathtt{a} \in \RR$ is called the \emph{centre} of $\xi$, $\gamma \ge 0$ is the
\emph{Gaussian coefficient}, and $\Pi$ is a measure, called the \emph{L\'evy measure},
on $(-\infty,0)$, which satisfies the moment condition
$\int_{(-\infty,0)} \min \{ 1, x^2\} \, \Pi(\dd x) <\infty$.
The classification of L\'evy processes is made
more precise by the \emph{L\'evy--It\^o decomposition}, which we now describe.
Let $\mathtt{M}$ be a Poisson random measure on $[0,\infty)\times (-\infty,0)$
with intensity measure $\text{Leb}\times\Pi$.
Let $B = (B(t) , t \ge 0)$ be a standard Brownian motion
independent of $\mathtt{M}$.
Then, a L\'evy process $\xi$ with Laplace exponent $\Psi$ can
be constructed as:
\[
\xi(t) = \gamma B(t) + \mathtt{a}t
+ \int_{[0,t]\times (-\infty,-1]} x \, \mathtt{M}(\dd s, \dd x)
+ \lim_{\epsilon\downto 0} \int_{[0,t]\times (-1,-\epsilon)} x\, \bigl[ \mathtt{M}(\dd s, \dd x) - \dd s \Pi(\dd x)\bigr]
,
\]
and the limit of compensated small jumps which appears as the last term
is guaranteed to exist in the $L^2$ sense.
We refer to the measure $\mathtt{M}$ as the \emph{jump measure} of $\xi$.
Standard works on this class of processes are the books \cite{Ber-Levy,Kyp2,Sato}.
We mention here only one additional feature which will be useful in our
study of the spine decomposition.
If $\Psi(\omega)<\infty$, then the process
$M_\omega(t) = e^{\omega \xi(t) - t\Psi(\omega)}$ is a unit-mean martingale
under $\sP$ for the natural filtration of $\xi$,
and if we define a new measure $\sP_{\omega}$ via
\[
\sP_{\omega}(A) \coloneqq \sE[ \Ind_A M_\omega(t) ] ,
\qquad A \in \sigma(\xi(s),\, s\le t), \; t \ge 0,
\]
then $\xi$ under $\sP_{\omega}$ is a spectrally negative
L\'evy process \cite[\S 8.1]{Kyp2}. Its Laplace exponent is the function $\Ess_\omega \Psi$
defined by
\[ \Ess_{\omega}\Psi(q) \coloneqq \Psi(q+\omega) - \Psi(\omega), \qquad q\ge 0. \]
This new process has
centre
$\mathtt{a}_\omega \coloneqq \mathtt{a} + \gamma^2 \omega + \int_{(-1,0)} x(e^{\omega x}-1)\, \Pi(\dd x)$,
Gaussian coefficient $\gamma$,
and L\'evy measure $\Pi_\omega(\dd x) \coloneqq e^{\omega x} \Pi(\dd x)$.
The function $\Ess_\omega \Psi$ is referred to as the
\emph{Esscher transform} of $\Psi$.
\subsection{Construction and truncation of the branching L\'evy process}
\label{s:construct-truncate}\label{s:labels}
In this section, we give a rigorous definition of the branching Lévy process.
Our presentation is inspired by \citet{Ber-cfrag}, and the main idea is first to define,
given a sequence of numbers $b_n \ge 0$,
a collection of \emph{truncated} processes $\bar{\Zz}^{(b_n)}$
representing the positions, and attached labels, of particles which do not
land `too far' (i.e., at a distance greater than $b_n$)
from their parent.
This is necessary since the rate of fragmentation is, in general, infinite.
These processes will be constructed such that they are consistent with one another,
in a sense
which will shortly be made precise, and such that taking $n\to\infty$ reveals all of the
particles. The main innovation compared to \cite{Ber-cfrag} is the
inclusion of labels for the particles,
and this is what allows us to study the spine decomposition.
Readers who are already familiar with the construction
of \cite{Ber-cfrag} may wish to skip this
section on first reading, and simply assume the existence of
a set of particle labels which is consistent under truncation.
\skippar
Let us introduce some notation.
The set of labels will be given by
$\tree = \cup_{j\ge 0} (\NN^3)^j$, where we use the convention
$(\NN^3)^0 = \{\varnothing\}$,
and we will denote elements of this set in the following way: if $u_i \in \NN^{3}$
for $i=1,\dotsc,I$, then we will write $(u_1,u_2,\dotsc,u_I)$ as $u_1 u_2 \dotsb u_I$.
The label $\varnothing$ represents the progenitor particle which is alive at time $0$,
sometimes called the `Eve' particle; and each offspring of the particle with label $u \in \tree$
receives a label $u(m,k,i)$, for some choice of $m,k,i$ which will be explained shortly.
Note that we use a Crump-Mode-Jagers type labelling scheme, in which
the closest of the `offspring' of a particle at each branching event retains the parent's identity;
see \cite{Jag-gbp} for a discussion of this so-called `general branching process' framework.
Our system is reminiscent of the
one adopted in \cite{BeMGF}, which also uses immortal particles
with labels based on the size of the jumps, but for which the labels are purely
generational.
We mention here also an alternative approach to the genealogy
by \citet{BM-bLp}, based upon a restriction to dyadic rational times,
which is of quite a different style.
Let $(a,\sigma,\nu)$ be a triple of characteristics
satisfying the conditions outlined in the introduction,
and let $\kappa$ be the cumulant given by \eqref{e:kappa}.
We assume throughout
that $\nu(\{\mathbf{0}\}) = 0$, where $\mathbf{0} \coloneqq (0,\dotsc)$
Our results will still hold without this condition,
but it simplifies notation and proofs by allowing us to ignore the possibility that particles are killed outright.
Let $(b_n)_{n\ge 0} \subset [0,\infty)$ be a strictly increasing sequence such
that $b_0 = 0$ and $b_n \to \infty$; this will be a fixed sequence of truncation
levels, which will be assumed given throughout this work.
For $b\ge 0$, we
let $k_b\from \Pp\to\Pp$ be given by
\begin{equation}\label{e:kb}
k_b(p_1,p_2,p_3,\dotsc) = (p_1,p_2\Indic{p_2 > e^{-b}},p_3\Indic{p_3> e^{-b}},\dotsc) ,
\end{equation}
and define the \emph{truncated dislocation measure}
via the pushforward $\nu^{(b)} = \nu \circ k_b^{-1}$.
\skippar
\label{s:intuitive}%
We now consider $n\ge 0$ to be fixed; we are going to define the \emph{branching Lévy process
truncated at level $b_n$}.
Since the labelling is a little more complex than usual, let us first give an intuitive
description of this process.
The process begins at time zero with a single particle having label $\varnothing$, and positioned
at the origin.
The spatial position of the particle
follows a spectrally negative Lévy process $\xi_{\varnothing}$ with
Laplace exponent $\Psi^{(b_n)}$ defined by
\[ \Psi^{(b_n)}(q) = \frac{1}{2}\sigma^2q^2 + \mBigl( a + \int_{\Pp\setminus\Pp_1} (1-p_1)\nu^{(b_n)}(\dd \pp) \mr)q
+ \int_{\Pp_1} \mbigl[ p_1^q-1+(1-p_1)q \mr] \, \nu^{(b_n)}(\dd \pp),
\]
where
$\Pp_1$ is the set of sequences with at most one non-zero element,
\[ \Pp_1 = \{ \pp \in \Pp: p_2 = 0 \}. \]
Crucially, the moment condition \eqref{e:nu}
implies that the pushforward
$\nu^{(b_n)}\rvert_{\Pp_1} \circ \log^{-1}$
is indeed a L\'evy measure, so $\Psi^{(b_n)}$ is the Laplace exponent
of a L\'evy process.
Moreover, $\nu^{(b_n)}$ restricted to $\Pp\setminus \Pp_1$
is finite.
At time $T_{\varnothing,1}$, having an exponential distribution
with parameter
$\lambda_{b_n} \coloneqq \nu^{(b_n)}(\Pp\setminus\Pp_1)<\infty$, the particle
$\varnothing$ branches. Take $\pp$ to be a random variable
with distribution $\nu^{(b_n)}\rvert_{\Pp\setminus\Pp_1}/ \lambda_{b_n}$,
and scatter particles in locations $\xi_{\varnothing}(T_{\varnothing,1}-) + \log p_i$,
for $i\ge 1$.
The particle in location $\xi_{\varnothing}(T_{\varnothing,1}-) + \log p_1$ retains the
label $\varnothing$. The particles in the other locations receive labels
$\varnothing(m,1,j) = (m,1,j)$, where $m\le n$ is the unique natural number
such that $e^{-b_{m-1}} \ge p_i > e^{-b_m}$, and $j$ is the minimal
natural number such the initial location of $(m,1,j)$
in $\RR$ is less than or equal to that of $(m,1,j-1)$
(recall that particles are scattered downwards.)
After this first branching event, the particle $\varnothing$ continues
to perform a Lévy process, and then at time $T_{\varnothing,1}+T_{\varnothing,2}$,
with $T_{\varnothing,2}$ independent of and
equal in distribution to $T_{\varnothing,1}$, it branches
again. Particles are scattered according to the same rule, this time receiving
labels $(m,2,j)$, with the $2$ indicating that this is the second branching event
for $\varnothing$. The particle then proceeds in this manner.
Meanwhile, each particle $u$ which was already born has the same evolution.
It performs a Lévy process $\xi_{u}$ with the same law as $\xi_{\varnothing}$,
and after waiting a period $T_{u,1}$, independent of and
equal in distribution to $T_{\varnothing,1}$, it branches.
Its children are scattered in the same way as before, but they receive labels $u(m,1,j)$;
and, subsequently, at the $k$-th branching event of $u$, the children receive labels $u(m,k,j)$.
A sketch illustrating the labelling scheme appears in \autoref{f:Z}.
\skippar Having established the main idea, we now give a rigorous definition of the
branching Lévy process truncated at level $b_n$.
Strictly speaking, all the symbols we define in the next few paragraphs
should have an annotation of the sort $\cdot^{(b_n)}$, but this would be rather cumbersome.
The notations $a_\cdot$, $\xi_\cdot$, $N_\cdot$, $T_\cdot$, $\Delta^{(\cdot)}$ and $\mathcal{Q}_{\cdot}$,
shortly to be defined,
will not appear again in the sequel, so we warn that they depend implicitly on $n$;
and all other notations will either receive an annotation or will turn out not to depend on $n$ after all.
\newcommand\ML{\mathsf{ML}}
\newcommand\Lu{\mathsf{L}}
Emulating \cite{Ber-cfrag}, we define the following random elements.
\begin{itemize}
\item
$(\xi_u)_{u\in \tree}$, a family of i.i.d.\ Lévy processes with Laplace exponent $\Psi^{(b_n)}$.
\item
$(N_u)_{u\in\tree}$, a family of i.i.d.\ Poisson processes $(N_u(t), t\ge 0)$ with rate
$\lambda_{b_n} = \nu^{(b_n)}(\Pp\setminus\Pp_1)$.
Let us denote by $(T_{u,p})_{p\ge 0}$ the inter-arrival times of $N_u$, so that
$T_{u,0} = 0$ and $T_{u,p} = \inf\{ t\ge T_{u,p-1} : N_u(t) = N_u(T_{u,p-1})+1 \} - T_{u,p-1}$, for $p\ge 1$.
\item
$(\Delta^{(u,p)})_{u\in\tree, p\ge 1}$, a family of i.i.d.\ elements of $\Pp\setminus\Pp_1$ with distribution
$\nu^{(b_n)}\rvert_{\Pp\setminus\Pp_1}/ \lambda_{b_n}$.
\end{itemize}
In the above list, $\xi_u$ represents the motion of the particle with label $u$,
ignoring the times at which it branches; $N_u$ jumps at the branching times of $u$; and
the mass-partition
$\Delta^{(u,p)} = (\Delta^{(u,p)}_i)_{i\ge 1}$
encodes the relative locations of $u$ and its children at the $p$-th time that $u$ branches. Moreover, these three families are independent one of the
others.
Our first step is to divide the $\Delta^{(u,p)}_\cdot$ into (disjoint) classes, which correspond to the
truncation level of the children they represent.
Define
\begin{equation}\label{e:L}
\Lu(y) = \min\{ m \ge 1 : e^{-b_{m-1}} \ge y > e^{-b_m} \} .
\end{equation}
For $l\ge 1$, let $\Delta^{(u,p,l)} = \mbigl(\Delta^{(u,p)}_j : j\ge 2 \text{ such that } \Lu(\Delta^{(u,p)}_j) = l\mr)^{\downarrow}$,
where $\cdot^{\downarrow}$ indicates decreasing rearrangement of the sequence.
For every $l\ge 1$, we regard the finite sequence $\Delta^{(u,p,l)}$ as being an element of $\Pp$,
by filling the tail with zeroes. Note that $\Delta^{(u,p,l)} = \mathbf{0}$ for all $l>n$.
Next, for each label $u$, we will give
definitions for certain random elements. These are:
$a_u \in[0,\infty)$, the birth time of $u$;
$\Zz_u = (\Zz_u(t), t\ge 0)$, with $\Zz_u(t) \in \RR$ representing
the position of $u$ at time $t$;
and $K_u^{(b_n)} = (K_u^{(b_n)}(t), t\ge 0)$, with $K_u^{(b_n)}(t) = (K_u^{(b_n)}(t,l) : l\ge 1) \in (\NN\cup \{0\})^{\NN}$.
The latter sequence has the interpretation that
$K_u^{(b_n)}(t,l)$ is the number of branching events which particle $u$ has had up to time
$t$ in which at least one child with label of the form $u(l,k,i)$, for any $k,i \in \NN$, was born.
For the particle $\varnothing$, let
\begin{eqnarr*}
a_\varnothing = 0;
\qquad \Zz_\varnothing(t) &=& \xi_\varnothing(t) + \sum_{p=1}^{N_\varnothing(t)} \log\Delta^{(\varnothing,p)}_1, \quad t \ge 0; \\
K_\varnothing^{(b_n)}(t, l) &=& \sum_{p=1}^{N_\varnothing(t)} \Indic{\Delta^{(\varnothing,p,l)} \ne \mathbf{0}}, \quad t\ge 0,\,l\ge 1.
\end{eqnarr*}
For the remaining particles, we first need a bit of notation:
let
\[
\mathcal{Q}_{u,m}(k) = \inf \mBigl\{ P \in \NN : \sum_{p=1}^{P} \Indic{\Delta^{(u,p,m)} \ne \mathbf{0}} = k \mr\},
\qquad k \in \NN,
\]
with the convention that $\inf\emptyset = \infty$. Thus, $\mathcal{Q}_{u,m}(k)$ is the number of
birth events of $u$ which take place until the $k$-th event at which the sequence $\Delta^{(u,p)}$ contains
at least one element $y$ with $\Lu(y) = m$.
Fix $u\in\tree$ and $(m,k,i) \in \NN^{3}$ arbitrary, and write $u' = u(m,k,i)$. Then let:
\begin{eqnarr*}
a_{u'} &=& a_u + \sum_{p=1}^{\mathcal{Q}_{u,m}(k)} T_{u,p}; \\
\Zz_{u'}(t) &=& \Zz_{u}(a_{u'}-) + \log \Delta^{(u,\mathcal{Q}_{u,m}(k),m)}_i
+ \xi_{u'}(t-a_{u'})
+ \sum_{p=1}^{N_{u'}(t-a_{u'})} \log \Delta^{(u',p)}_1 , \qquad t\ge a_{u'};
\\
K_{u'}^{(b_n)}(t, l) &=& \sum_{p=1}^{N_{u'}(t-a_{u'})} \Indic{\Delta^{(u',p,l)} \ne \mathbf{0}}, \qquad t\ge a_{u'},\, l\ge 1.
\end{eqnarr*}
We define $a_{u'} = \infty$ if either $\mathcal{Q}_{u,m}(k) = \infty$ or $\Delta^{(u,\mathcal{Q}_{u,m}(k),m)}_i = 0$.
In particular, $a_{u'} = \infty$ for every $u' = u(m,k,i)$ with $m>n$.
We are now in a position to define the following elements:
\begin{defn}
Let $n\ge 0$, then
\[ \Zz^{(b_n)}(t) = \sum_{u\in\tree} \delta_{\Zz_u(t)}\Indic{a_u \le t}, \qquad t\ge 0, \]
is the \emph{branching Lévy process truncated at level $b_n$},
and
\[ \bar\Zz^{(b_n)}(t) = \sum_{u\in\tree} \delta_{(u,K_u^{(b_n)}(t),\Zz_u(t))}\Indic{a_u\le t}, \qquad t\ge 0, \]
is the \emph{labelled branching Lévy process truncated at level $b_n$}.
\end{defn}
From the latter, let us also define
\[ \tree_t^{(b_n)} = \{ u \in \tree : \exists k,z \text{ such that } \bar\Zz^{(b_n)}(t)\{(u,k,z)\} = 1 \} , \qquad t\ge 0, \]
which is the set of labels of particles present at time $t$.
We introduce now the following function, which will be required to
understand the un-truncated process.
For $u\in\tree$, define
\[ \ML(u)
= \max\{ r \ge 1: \text{there exist } u', k, i, u'' \text{ such that }
u = u'(r,k,i)u'' \}.
\]
Thus, $\ML(u)$ can be seen as the maximum value of $r$ for which a particle
with label $u$ could appear in
the construction of $\bar{\Zz}^{(b_r)}$,
and indeed, if $u \in \tree_t^{(b_n)}$, then $\ML(u) \le n$.
Of these processes, $\Zz^{(b_n)}$ is a branching
L\'evy process with characteristics
$(a,\sigma,\nu^{(b_n)})$ in the sense of \citet[Definition 1]{Ber-cfrag},
and the others are
our extensions.
In particular, we have by \cite[Theorem 1]{Ber-cfrag} that
$\lE\mbigl[ \sum_{u\in\tree_t^{(b_n)}} e^{q\Zz_u(t)}\mr] = e^{t\kappa^{(b_n)}(q)}$, for all $q\in \RR$,
where
\begin{eqnarr*}
\kappa^{(b_n)}(q) &\coloneqq& \frac{1}{2}\sigma^2q^2 + aq + \int_{\Pp} \mBigl[ \sum_{i\ge 1} p_i^q - 1 + (1-p_1)q \mr] \, \nu^{(b_n)}(\dd \pp)
\\
&=& \frac{1}{2}\sigma^2q^2 + aq + \int_{\Pp} \mBigl[ p_1^q-1+(1-p_1)q + \sum_{i\ge 2} p_i^q\Indic{p_i>e^{-b_n}} \mr] \, \nu(\dd \pp), \qquad q \in \RR.
\end{eqnarr*}
This function represents the cumulant of the truncated branching Lévy process.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\textwidth]{fig-truncation3-Zb3}
\caption{A sketch of the construction and labels of a (truncated)
branching L\'evy process,
with truncation levels marked at certain birth events.
The path in solid black represents the process $\Zz^{(b_1)}$, which in this
particular instance includes only the Eve particle $\varnothing$. The paths
in dashed red represent the particles in the process
$\Zz^{(b_2)}\setminus\Zz^{(b_1)}$; note that these are precisely the particles
$u$ for which $\ML(u) = 2$.
The paths in dotted blue represent the particles in the process
$\Zz^{(b_3)}\setminus\Zz^{(b_2)}$.
}
\label{f:Z}
\end{figure}
\begin{rem}
In the construction above, the role of the the component $K^{(b_n)}_u$,
which records some information about the children of $u$,
is simply
to ensure that the process $\bar{\Zz}^{(b_n)}$ is Markov
(see the forthcoming \autoref{l:Zbar-bp}.)
Without the inclusion of this mark,
if a particle $u$ branches at time $t$, it is not possible to determine
the labels of its children solely from $\bar{\Zz}^{(b_n)}(t)$.
We emphasise that the unlabelled process, $\Zz^{(b_n)}$, is always Markov
\cite[p.~1272]{Ber-cfrag}.
\end{rem}
\skippar
Having defined the truncated branching Lévy process $\bar{\Zz}^{(b_n)}$,
we introduce the idea of further truncating it at level $b_m\le b_n$. That is,
we consider keeping, at each branching event, the child particle which is the closest to
the parent, and suppressing the other children if and only if their distance to the position of the parent
prior to branching is larger than or equal to $b_m$, together with their descendants.
Mathematically, for $m\le n$, we let
\[ (\tree_t^{(b_n)})^{(b_m)} = \mbigl\{u\in \tree_t^{(b_n)}~:~ \ML(u) \le m\mr\} , \quad t\ge 0.\]
We then define
\begin{equation}
(\Zz^{(b_n)})^{(b_m)}(t) = \sum_{u \in(\tree_t^{(b_n)})^{(b_m)}} \delta_{\Zz_u(t)} , \quad t\ge 0,
\label{e:bnbm}
\end{equation}
which is the \emph{truncation of $\Zz^{(b_n)}$ to level $b_m$},
and similarly
\begin{eqnarr*}
(K_u^{(b_n)})^{(b_m)}(t,l) &=& K_u^{(b_n)}(t,l)\Indic{l\le m}, \quad t\ge 0,\, l\ge 1;
\\
(\bar\Zz^{(b_n)})^{(b_m)}(t) &=& \sum_{u\in(\tree_t^{(b_n)})^{(b_m)}}\delta_{(u,(K_u^{(b_n)})^{(b_m)}(t),\Zz_u(t))}, \quad t\ge 0.
\end{eqnarr*}
With this definition, we get the following lemma.
\begin{lem}\relax\label{l:consistency}
Let $m\le n$. Then $(\Zz^{(b_n)})^{(b_m)}$ is equal in law to $\Zz^{(b_m)}$
and $(\bar\Zz^{(b_n)})^{(b_m)}$ is equal in law to $\bar\Zz^{(b_m)}$.
\end{lem}
\begin{proof}
The first statement is \cite[Lemma 3]{Ber-cfrag}, and the second follows by considering
the intuitive description of the labels beginning on page~\pageref{s:intuitive}:
if all $u$ with $\ML(u) > m$ are removed,
then those elements do not appear in $\bar\Zz$, and the sequence $(K_{u'}^{(b_n)}(t))^{(b_m)}$
for the remaining $u'$ simply erases the record of birth events that would have given rise to those
erased $u$.
\end{proof}
We therefore see that both the labels and the positions of the particles are consistent
under truncation, as are the marks $K_u^{(b_\cdot)}$. By the Kolmogorov extension theorem,
we can construct, simultaneously on the same probability space,
a collection of processes $(\Zz^{(b_n)})_{n\ge 0}$ and $(\bar\Zz^{(b_n)})_{n\ge 0}$
with the property that the equality in law of \autoref{l:consistency}
is replaced by almost sure equality.
\skippar
Thus, we are able to define the following (un-truncated) processes:
\begin{defn}
The \emph{branching Lévy process with characteristics $(a,\sigma,\nu)$} is
\[ \qquad \Zz(t) = \bigcup_{n\ge 0} \Zz^{(b_n)}(t), \qquad t\ge 0.\]
\end{defn}
For the (un-truncated) process $\Zz$, the set of labels of particles present up to time $t$ is
\[ \tree_t = \bigcup_{n\ge 0} \tree_t^{(b_n)}, \qquad t\ge 0. \]
Furthermore, let $K_u(t,l) = K_u^{(b_l)}(t,l)$ for each $t\ge 0$, $u\in\tree_t$ and $l\ge 1$,
and $K_u(t) = (K_u(t,l))_{l\ge 1}$.
\begin{defn}
The \emph{labelled branching Lévy process with characteristics $(a,\sigma,\nu)$} is
\[ \bar\Zz(t) = \sum_{u\in\tree_t} \delta_{(u,K_u(t),\Zz_u(t))}, \qquad t\ge 0. \]
\end{defn}
In particular, since $\kappa^{(b_n)}(q) \upto \kappa(q)$ whenever $q\in\dom\kappa$,
we have that \[ \lE\mBigl[ \sum_{u\in\tree_t} e^{q\Zz_u(t)}\mr] = e^{t\kappa(q)}, \qquad q\in\dom\kappa, \] which
is an important property of the process.
\begin{rem}\label{r:misc}
\begin{enumerate}
\item
In \cite{BM-bLp,BBCK-maps}, growth-fragmentations
are studied in which
upward jumps of the particle locations (with or without associated branching)
are permitted. This can be accommodated in our construction
as well, simply by removing
the restriction that the processes $\xi_\cdot$ be spectrally negative
(and, if necessary, incorporating branching at upward jumps)
thereby giving versions of these processes with labels and genealogies.
\item
The processes $\Zz^{(b_n)}$ and $\bar\Zz^{(b_n)}$ are (labelled) branching
L\'evy processes in their own right, having characteristics
$(a,\sigma,\nu^{(b_n)})$.
\item\label{i:misc:3}
We wish to emphasise that, despite the technical appearance of our label
definitions, they can be found deterministically once the unlabelled
branching L\'evy process
is known. In particular, if we have all $\Zz^{(b_n)}$ defined on the same
probability space, and we are given a single sample from this space,
then a sample of the process $\bar{\Zz}^{(b_n)}$ can be constructed, without
extra randomness, using the intuitive
definition of the labels on page \pageref{s:intuitive}.
This will be important in \autoref{s:forward}.
\end{enumerate}
\end{rem}
\subsection{Regularity and the branching property}\label{s:branching}
One of the key results of \cite{Ber-cfrag} was the branching property of the compensated
fragmentation $\Zb$. This result extends naturally to $\Zz$, and we shall shortly
give an explicit statement of it for $\bar\Zz$. However, we first elaborate
a little on the state space of $\bar\Zz$, and consider the regularity of the process.
We first expand on the space $\tree$. Some of the definitions here will not be
needed until the next section, but we give them here for ease of reference.
We define relations
$\preceq$ and $\prec$ on $\tree$ to denote ancestry,
so $u\preceq v$ if there exists some $u' \in \tree$ such that $v = uu'$,
and $u\prec v$ if $u\preceq v$ and $u\ne v$.
Using this, we define ancestors and descendants
as follows, which is a little subtle due to immortality of particles.
If $s<t$ and $v\in\tree_t$, we define $u = \Anc(s;v)$ to be the
largest (with respect to $\preceq$) element of $\tree_s$ such that $u\preceq v$.
Conversely, for $u\in\tree_s$, we define $\Desc(s,u;t) = \{ v \in \tree_t : u = \Anc(s;v)\}$.
We also define $\abs{u}$ to be the unique $n \in \NN\cup\{0\}$
such that $u \in (\NN^3)^n$,
that is, the generation of $u$;
and $(u_i)_{1\le i\le n}$ to be those elements of $\NN^3$ such that $u = u_1\dotsb u_n$.
We extend this so that $u_i = (0,0,0)$ if $\abs{u} < i$.
Finally, we consider $\tree$ be endowed with the metric
$\rho(u,v) = \sum_{i\ge 1} \norm{u_i - v_i}$, where here $\norm{\,\cdot\,}$ is the
usual Euclidean norm on $\RR^3$.
Define the space $\mathcal{L}$ to consist of those sequences $K = (K(l))_{l\ge 1}$ in the set $(\NN\cup\{0\})^{\NN}$
for which the function $\norm{K}_{\mathcal{L}} = \sum_{l\ge 1} \exp(-l-e^{4b_l}) \abs{K(l)}$
is finite; then $(\mathcal{L},\norm{\,\cdot\,}_{\mathcal{L}})$ is a normed vector
space which is isometrically isomorphic to $\ell^1$.
Now let $\mathbf{X} = \tree\times \mathcal{L} \times \RR$.
This is a complete, separable metric space when given the usual product metric.
It will prove useful to define
$\mathcal{M}_p(\mathbf{X})$ to be the set of point measures on $\mathbf{X}$
which are finite on bounded subsets of $\mathbf{X}$. We give this a metric as follows
(see \cite[\S A2.6]{DV-pp1}). Let $q \in (\dom \kappa)^\circ$ be chosen arbitrarily,
and let $x_0 = (\varnothing,\mathbf{0},0) \in \mathbf{X}$.
If $\mu,\mu'$ are point measures on $\mathbf{X}$, let
\[ d_q(\mu,\mu') = \int_0^\infty q e^{-qr} \frac{d^{(r)}(\mu_r,\mu_r')}{1+d^{(r)}(\mu_r,\mu_r')} \, \dd r , \]
where
$\mu_r = \mu\rvert_{B_r(x_0)}$ is the measure $\mu$ restricted to
the open ball $B_r(x_0)$ of radius $r \ge 0$ around $x_0$, and
$d^{(r)}$ is the L\'evy--Prokhorov metric
on $B_r(x_0)$; this is defined as:
\begin{multline*} d^{(r)}(\mu,\mu') = \inf\{ \epsilon \ge 0 : \text{for all } F \subset B_r(x_0) \text{ closed, } \\
\mu_r(F) \le \mu_r'(F^{\epsilon})+\epsilon \text{ and } \mu_r'(F) \le \mu_r(F^{\epsilon})+\epsilon \}, \end{multline*}
where $F^\epsilon \coloneqq \{x \in B_r(x_0) : \text{there exists } y\in F \text{ such that } d(x,y) < \epsilon\}$.
For a labelled branching L\'evy process $\bar\Zz(t) = \sum_{u\in\tree_t} \delta_{(u,K_u(t),\Zz_u(t))}$, one may show that for any $u\in \tree$ and $t\ge 0$, $\norm{K_u(t)}_{\mathcal{L}}<\infty$
almost surely.
Therefore, we may regard $\bar\Zz$ as taking values in
the complete separable metric space $\mathcal{M}_p(\mathbf{X})$
with metric $d_q$. Furthermore,
we have the following pair of results:
\begin{lem}\label{l:cv-Zbn}
For $q \in (\dom\kappa)^\circ$ and $t\ge 0$, $\sup_{s\le t} d_q( \bar\Zz(s),\bar\Zz^{(b_n)}(s)) \to 0$ in probability.
\begin{proof}
Fix $q \in (\dom \kappa)^\circ$ and $t\ge 0$.
To begin with,
\begin{multline}\label{e:cadlag-cv-inter}
d_q(\bar\Zz(s),\bar\Zz^{(b_n)}(s))
\le
d_q\mBigl( \sum_{u \in \tree_s} \delta_{(u,K_u(s),\Zz_u(s))},
\sum_{u\in\tree_s^{(b_n)}} \delta_{(u,K_u(s),\Zz_u(s))} \mr)
\\
{} +
d_q\mBigl( \sum_{u \in \tree_s^{(b_n)}} \delta_{(u,K_u(s),\Zz_u(s))},
\sum_{u\in\tree_s^{(b_n)}} \delta_{(u,K_u^{(b_n)}(s),\Zz_u(s))} \mr).
\end{multline}
We study the two terms on the right-hand side separately.
We first look at the second term.
Using the definition of the L\'evy--Prokhorov
metric and the fact that $\Zz^{(b_n)} \subset \Zz$,
we find that for every $r \ge 0$,
\[
d^{(r)}\mBigl( \sum_{u \in \tree_s^{(b_n)}} \delta_{(u,K_u(s),\Zz_u(s))},
\sum_{u\in\tree_s^{(b_n)}} \delta_{(u,K_u^{(b_n)}(s),\Zz_u(s))} \mr)
\le \max\{ \norm{K_u^{(b_n)}(s) - K_u(s)}_{\mathcal{L}} : u \in \tree_s^{(b_n)} \}.
\]
The crude estimate $\sup_{s\le t} K_u(s,l) = K_u(t,l) \le \card\Zz^{(b_l)}(t)$,
the latter being the number of particles in $\Zz^{(b_l)}(t)$, then yields
\begin{eqnarr*}
\lE\mbigl[\sup_{s\le t}\max\{ \norm{K_u^{(b_n)}(s) - K_u(s)}_{\mathcal{L}} : u \in \tree_s^{(b_n)} \}\mr]
&\le& \sum_{l\ge n+1} \exp(-(l+e^{4b_l})) \lE[\card \Zz^{(b_l)}(t)]
\\
&=& \sum_{l\ge n+1} \exp\mbigl(\kappa^{(b_l)}(0) t -l-e^{4b_l}\mr) .
\end{eqnarr*}
Noticing that
\[e^{-2 b_l}\kappa^{(b_l)}(0) = \int_{\Pp} \sum_{i\ge 2}e^{-2 b_l} \Indic{p_i> e^{-b_l}}\, \nu (\dd \pp) \le \int_{\Pp} \sum_{i\ge 2}p_i^2\, \nu (\dd \pp) =: C<\infty, \]
we have
\[\sum_{l\ge n+1} \exp\mbigl(t\kappa^{(b_l)}(0) -l-e^{4b_l}\mr) \le \sum_{l\ge n+1} e^{-l} \exp\mbigl(e^{2b_l}(Ct - e^{2b_l})\mr). \]
For fixed $t\ge 0$,
the right-hand side tends to zero as $n\to\infty$.
This ensures that the second term of \eqref{e:cadlag-cv-inter} converges to
zero in probability.
Turning to the first term in \eqref{e:cadlag-cv-inter}, we have
\[
d^{(r)}\mBigl( \sum_{u \in \tree_s} \delta_{(u,K_u(s),\Zz_u(s))},
\sum_{u\in\tree_s^{(b_n)}} \delta_{(u,K_u(s),\Zz_u(s))} \mr)
\le \sum_{u \in \tree_s\setminus \tree_s^{(b_n)}} \Indic{\Zz_u(s) \in (-r,r)} . \]
We now integrate in order to study the $d_q$-distance, and use the
bound $\Indic{\Zz_u(s) \in (-r,r)} \le e^{q'(r+\Zz_u(s))}$, where
$q' \in \dom \kappa$ is chosen arbitrarily such that $q' < q$ holds:
\begin{eqnarr*}
d_q\mBigl( \sum_{u \in \tree_s} \delta_{(u,K_u(s),\Zz_u(s))},
\sum_{u\in\tree_s^{(b_n)}} \delta_{(u,K_u(s),\Zz_u(s))} \mr)
&\le& \int_0^\infty qe^{-qr}
\sum_{u \in \tree_s\setminus \tree_s^{(b_n)}} \Indic{\Zz_u(s) \in (-r,r)} \, \dd r
\\
&\le& \int_0^\infty qe^{(q'-q)r}
\sum_{u \in \tree_s\setminus \tree_s^{(b_n)}} e^{q' \Zz_u(s)} \, \dd r \\
&=& \frac{q}{q-q'} \mBigl(
\sum_{u \in \tree_s} e^{q' \Zz_u(s)}
- \sum_{u \in \tree_s^{(b_n)}} e^{q' \Zz_u(s)}
\mr).\IEEEyesnumber \label{e:trick}
\end{eqnarr*}
Now, the proof is completed using Doob's maximal inequality exactly
as in \cite[Proof of Lemma 4]{Ber-cfrag}.
\end{proof}
\end{lem}
\begin{cor}[regularity of $\bar\Zz$]\label{c:cadlag}
The process $\bar\Zz$ possesses a c\`adl\`ag version in $\mathcal{M}_p(\mathbf{X})$.
\begin{proof}
This follows from the above lemma exactly as in \cite[Proposition 2]{Ber-cfrag}.
\end{proof}
\end{cor}
Thanks to this result, we can consider $\lP$ to be defined on the space
$\Omega = D([0,\infty),\mathcal{M}_p(\mathbf{X}))$ of
càdlàg functions from $[0,\infty)$ to $\mathcal{M}_p(\mathbf{X})$,
endowed with the Skorokhod topology; we refer the reader to
\cite{Bil-conv} for more details on this space.
The process $\bar\Zz$ has the Markov property, which in this context is usually called
the \emph{branching property} and which we now explain.
We first define translation operators for $u \in \tree$ and $t\ge 0$, as follows.
Let $\theta_{u,t}\from\Omega \to \Omega$ be such that,
if
\[ \bar\Zz(s+t) = \sum_{uu'\in\tree_{s+t}} \delta_{(uu', K_{uu'}(s+t),\Zz_{uu'}(s+t))}
+ \sum_{\substack{u''\in\tree_{s+t}, \\u \not\preceq u''}} \delta_{(u'',K_{u''}(s+t),\Zz_{u''}(s+t))}, \qquad s \ge 0 , \]
then
\[ \bar\Zz(s)\circ\theta_{u,t} = \sum_{uu'\in \tree_{s+t}} \delta_{(u',K_{uu'}(s+t)-K_{uu'}(t),\Zz_{uu'}(s+t)-\Zz_{uu'}(t))}, \qquad s \ge 0. \]
That is, $\theta_{u,t}$ shifts the particle process such that one only observes the particle with label $u$
and its descendants which are born strictly after $t$; and the particle represented by $u$ is shifted to start
at the origin, at time $0$, with label $\varnothing$ and no recollection of its genealogical history.
Let $(\FF_t)_{t\ge 0}$ be the natural filtration of
$\bar\Zz$, namely $\FF_t = \sigma(\bar\Zz(s), s\le t)$,
and define $\FF_\infty = \sigma\bigl(\cup_{t\ge 0} \FF_t\bigr)$.
We then have the following simple result.
\begin{lem}[branching property]\relax\label{l:Zbar-bp}
For each $u \in \tree$, let $F_u$ be a bounded, measurable functional. Then
\[
\lE\mBigl[ \prod_{u\in \tree_t} F_u( \bar\Zz(s) \circ \theta_{u,t}, s \ge 0) \mm\vert \FF_t \mr]
= \prod_{u \in \tree_t} \lE[ F_u(\bar \Zz(s), s \ge 0) ] .
\]
\end{lem}
\begin{proof}
This follows directly from the branching property of $\Zz$ in \cite[p.~1272]{Ber-cfrag}
and the construction of the labels.
\end{proof}
We remark that, as a consequence of \autoref{c:cadlag}, the constant time $t$
in the above lemma may be replaced by any $(\FF_t)$-stopping time,
or indeed by a stopping line in the sense of \cite[\S 4]{BHK-FKPP}.
\section{Change of measure and backward selection of the spine}
\label{s:backward}
For $\omega \in \dom \kappa$, we define the \emph{exponential additive martingale} $\mg{\omega}{\cdot}$
just as we did in the introduction:
\begin{equation*}
\mg{\omega}{t} = e^{-t\kappa(\omega)} \sum_{u\in \tree_t} e^{\omega \Zz_u(t)} , \wh t \ge 0.
\end{equation*}
It has been proved in \cite[Corollary 3]{Ber-cfrag} that this is a
martingale with unit mean.
As such, we may make a martingale change of measure, as follows. We define a measure
$\lQ$ on $\FF_\infty$ by setting, for $A\in \FF_t$,
\begin{equation}\label{e:lQ-W} \lQ(A) = \lE[ \Ind_A W(\omega,t) ] . \end{equation}
The martingale property of $W(\omega,\cdot)$ ensures
that this change of measure is consistent across different choices of $t$, and
also implies that the process $\bar\Zz$ under $\lQ$ remains a Markov process.
$\lQ$ is often referred to as an `exponential tilting' of the probability measure $\lP$.
\medskip\noindent
Under this tilted measure, we isolate a single particle as the `spine'.
We first expand the basic probability space $\Omega$ to
produce $\hat\Omega = \Omega\times\tree^{[0,\infty)}$,
and introduce for each $t\ge 0$ a random variable $U_t$ such that,
for $A\subset\Omega$ measurable,
$A\times\{U_t = u\} = A \times \{ g \in \tree^{[0,\infty)} : g(t) = u\}$.
Let $\hat\FF_t = \sigma(\FF_t; U_s, s\le t)$ and $\hat\FF_\infty = \sigma\bigl(\cup_{t\ge 0} \hat\FF_t\bigr)$.
We may then extend the definition of $\lQ$ to sets in $\hat\FF_\infty$.
For $A \in \FF_t$ and $u\in \tree$, let
\begin{equation}\label{e:lQ}
\lQ(A; U_t=u) = e^{-t\kappa(\omega)} \lE[ \Ind_A e^{\omega \Zz_u(t)}].
\end{equation}
It is well-known
(see, for instance, \cite[Theorem 4.2]{HH-spine})
that events $\hat A \in \hat \FF_t$ may be written as
$\hat A = \bigcup_{u\in\tree} (A_u \cap \{ U_t = u \})$, with $A_u \in \FF_t$, and so \eqref{e:lQ}
is equivalent to defining
\begin{equation}
\lQ(\hat A) = e^{-t\kappa(\omega)} \lE \mBigl[\sum_{u\in \tree_t} \Ind_{A_u} e^{\omega \Zz_u(t)} \mr] , \qquad \hat A \in \hat{\FF}_t.
\label{e:lQ'}
\end{equation}
The measure $\lQ$ is well-defined,
in that, if $\hat{A}\in\FF_t$, then the right-hand side of \eqref{e:lQ'}
reduces simply to \eqref{e:lQ-W}.
However, in terms of the definition on $\hat{\FF}_\infty$,
$\lQ$ distinguishes the label $U_t$
at time $t$, and we call this the \emph{spine label}.
For each fixed $t\ge 0$, if we define $U_t$ via \eqref{e:lQ},
we can project it backward by setting $U_s = \Anc(s;U_t)$
for $s\le t$.
Due to the branching property of $\Zz$, this is
consistent with evaluating $\lQ$ on $\hat\FF_s$,
as is made precise in the following lemma.
\begin{lem}[consistency of $\lQ$]\relax\label{l:lQ-consistent}
Let $s< t$ and $u \in \tree$. Let $\lQ^t$ indicate the measure $\lQ$
defined on $\hat\FF_t$ by means of \eqref{e:lQ} and back-projection of $U_t$, and $\lQ^s$ similarly
for $\lQ$ defined on $\hat\FF_s$.
If $A \in \FF_s$, then
\[ \lQ^s(A; U_s = u) = \lQ^t(A; U_s = u) . \]
\end{lem}
\begin{proof}
Firstly, we have
\[
e^{-t\kappa(\omega)}\lE\mBigl[ \sum_{v \in \Desc(s,u;t)} e^{\omega \Zz_v(t)} \mgiven \FF_s \mr]
= e^{-s\kappa(\omega)} e^{\omega\Zz_u(s)},
\]
due to the branching property. Then,
\begin{eqnarr*}
\lQ^s( A; U_s=u)
&=& e^{-s\kappa(\omega)}\lE[ \Ind_A e^{\omega\Zz_u(s)}]
\\
&=& e^{-t\kappa(\omega)}
\lE\mbiggl\{
\Ind_A
\lE\mBigl[ \sum_{v \in \Desc(s,u;t)} e^{\omega\Zz_v(t)} \mgiven \FF_s \mr] \mr\} \\
&=& \lE\mBigl[ \Ind_A \sum_{v \in \Desc(s,u;t)} e^{\omega\Zz_v(t)} \mr]
= \mathbb{E}_{\lQ^t} \mBigl[\Ind_A \sum_{v \in \Desc(s,u;t)} \Indic{U_t=v} \mr]
= \lQ^t( A; U_s=u). \qedhere
\end{eqnarr*}
\end{proof}
\skippar
We refer to the process $(\bar\Zz,U) = ( (\bar\Zz(t),U_t), t\ge 0)$ as the
\emph{branching L\'evy process with spine}.
In order for it to be useful, it is important that
$(\bar\Zz,U)$ retain the branching property.
For the sake of clarity, we keep the time-annotation
$\lQ^t$ which was introduced in the last lemma.
\begin{lem}[branching property of $(\bar \Zz,U)$]\relax\label{l:Zbar-U-bp}
Fix $t\ge s\ge 0$. Let $F_v$ be an $\FF_{t-s}$-measurable functional for each $v\in \tree$, and
let $G$ be $\sigma(U_{t-s})$-measurable. Then,
\[ \lQ^t \mbiggl[ G\circ \theta_{s,U_s} \cdot \prod_{v\in\tree_s} F_v\circ\theta_{s,v} \mm\vert \hat\FF_s \mr]
= \mbigl.\prod_{\substack{v\in \tree_s \\ v\ne U_s}} \lP[F_v] \lQ^{t-s}[G\cdot F_u]\mr\rvert_{u=U_s} .
\]
\end{lem}
\begin{proof}
By Kolmogorov's definition of conditional expectation and the definition of $\hat\FF_s$,
it is sufficient to prove that, for $K$ an $\FF_s$-measurable functional and $u\in\tree$,
\begin{equation}\label{e:Zbar-U-bp-inter}
\lQ^t\mbiggl[ K \Indic{U_s = u} G\circ \theta_{s,u} \cdot \prod_{v\in\tree_s} F_v \circ \theta_{s,v} \mr]
= \lQ^t\mbiggl[ K \Indic{U_s=u} \prod_{\substack{v\in\tree_s \\v\ne u}} \lP[F_v] \lQ^{t-s}[G\cdot F_u] \mr].
\end{equation}
Fixing $G = \Indic{U_{t-s}=u'}$, for some $u'\in\tree$, the left-hand side is equal to
\begin{eqnarr*}
\eqnarrLHS{
e^{-t\kappa(\omega)}
\lP\mbiggl[ K \Indic{u=\Anc(s;uu')} e^{\omega \Zz_{uu'}(t)} \prod_{v\in \tree_s} F_v\circ\theta_{s,v} \mr]
}
\qquad \qquad &=&
e^{-t\kappa(\omega)} \lP\mbiggl[ K e^{\omega \Zz_u(s)} \prod_{\substack{v\in\tree_s \\ v \ne u}} \lP[F_v]
\cdot \lP[ F_u e^{\omega \Zz_{u'}(t-s)} \Indic{u'\in \tree_{t-s}} ] \mr]
\\
&=&
e^{-s\kappa(\omega)}\lP\mbiggl[ K e^{\omega\Zz_u(s)} \prod_{\substack{v \in \tree_s\\ v\ne u}} \lP[F_v]
\lQ^{t-s}[ F_u \Indic{U_{t-s} = u'}] \mr]
\\
&=&
\lQ^s \mbiggl[ K \Indic{U_s=u} \prod_{\substack{v \in \tree_s\\ v\ne u}} \lP[F_v]
\lQ^{t-s}[ G\cdot F_u] \mr]
\end{eqnarr*}
where in the second line we have used \autoref{l:Zbar-bp} and the fact that the event $u=\Anc(s;uu')$ is equivalent
to the event that $uu'$ is born after time $s$ (or $u'=\varnothing$);
and in the third and fourth lines we have used the definition of $\lQ^\cdot$. An
appeal to \autoref{l:lQ-consistent} yields \eqref{e:Zbar-U-bp-inter}, which completes the proof.
\end{proof}
\skippar
From now on we will drop the time-annotations $\lQ^t$ and simply use the notation $\lQ$.
Our primary goal in the remainder of the article is to characterise the law
of the process $(\bar\Zz,U)$ in terms of well-understood objects.
\section{Forward construction of the process with spine}
\label{s:forward}
In this section, we give a construction of a Markov process with values
in the set of point measures and with a certain distinguished line of
descent. The process, which we will write as $(\bar\Yy,V)$,
is regarded as being defined under $\lQ$,
and we call it the
\emph{decorated spine process} with parameters $(a,\sigma,\nu,\omega)$.
In the next section, we will
show that it coincides in law with the process $(\bar{\Zz},U)$
described in \autoref{s:backward}.
We start with a candidate for the motion of the spine particle
itself.
Let $\xi$ be a spectrally negative L\'evy process
whose
Laplace exponent has the L\'evy--Khintchine representation
\[ \Ess_\omega\kappa(q) \coloneqq \kappa(q+\omega)-\kappa(\omega) =
\frac{1}{2}\sigma^2q^2 + a_{\omega}q + \int_{(0,1)} (y^q-1-ql(y)) \, \pi(\dd y) , \qquad q \ge 0, \]
where
\begin{eqnarr*}
\pi(\dd y) &=& \sum_{i\ge 1} y^\omega \nu(p_i \in \dd y)
\text{, that is, } \int_{(0,1)} f(y) \, \pi(\dd y) = \int_{\Pp} \sum_{i\ge 1} p_i^\omega f(p_i) \, \nu(\dd \pp), \\
l(y) &=& \Indic{\abs{\log y} \le 1}\log y, \\
a_\omega &=& a+\omega\sigma^2 + \int_{\Pp} \mBigl[ 1 - p_1 + \sum_{i\ge 1} p_i^\omega l(p_i) \mr] \,\nu(\dd \pp) .
\end{eqnarr*}
Note that
in particular, the Lévy measure of $\xi$ is given by
the pushforward $\Pi \coloneqq \pi \circ \log^{-1}$.
The motivation for this definition of $\xi$ is that, if
$\nu(\Pp\setminus\Pp_1) <\infty$,
then by \cite[Proposition 3.4]{BerSte}
the process $(\Zz_{U_t}(t), t \ge 0)$ under $\lQ$
is known to be equal in law to the
process $\xi$; this is not difficult to prove
even in the absence of said finiteness condition,
but it will be a corollary of the main theorem in the next
section, so we do not pursue this here.
\skippar
We regard $\xi$ as representing the position of the spine particle,
and our goal is now to construct the rest of the branching L\'evy process
around it. There will be
three steps to this: firstly, we take the Poisson random measure
giving the jump times and sizes
of $\xi$. We then add decorations to this which indicate the additional offspring
which should be present due to the branching structure; and in the final step,
we graft independent branching L\'evy processes (under $\lP$) onto this structure.
\skippar
Next we require a short lemma establishing the existence of a conditional measure.
\begin{lem}\relax\label{l:cond}
For each $i \in \NN$,
and $y>0$,
there exists a probability measure $\nu_i(\dd\pp \mid y)$ on $\Pp$ such that
\[ \int_{\Pp} f(\pp) \, \nu(\dd\pp) = \int_{\Pp\times (0,1)} f(p_1,\dotsc,p_{i-1},y,p_{i+1},\dotsc) \, \nu_i(\dd \pp \mid y) \nu(p_i \in \dd y). \]
\begin{proof}
{ \let\rho\varrho
Let $\rho_i \from \Pp \to (0,1)$ be given by $\rho_i(p_1,p_2,\dotsc) = p_i$, and let $\nu_i = \nu \circ \rho_i^{-1}$.
We seek measures $\nu_i(\cdot \given y)$, such that each $\nu_i(\cdot \given y)$ is a probability measure,
$\nu_i(\Pp\setminus\rho_i^{-1}(y)\given y) = 0$, and $\nu(\dd\pp) = \int_{(0,1)} \nu_i(\dd\pp \given y) \nu_i(\dd y)$.
We define first
\[ h_i(y) = \begin{cases}
(1-y)^2 , & i = 1, \\
y^2, & i \ge 2.
\end{cases}
\]
Thus, $h_i(p_i) \le (1-p_1)^2$ for all $i$ and $\pp$, and in particular
$\int_{\Pp} h_i(p_i)\, \nu(\dd\pp) = \int_{(0,1)} h_i(y) \, \nu_i(\dd y) < \infty$.
Define $\bar{\lambda}_i(\dd\pp) = h_i(p_i)\nu(\dd\pp)/\int h_i\, \dd \nu_i$, a probability measure.
Then by standard results on disintegration of measures (see \cite{Sim}, for instance)
there exist measures $\nu_i(\cdot \given y)$ for each $y$
such that $\nu_i(\Pp\setminus\rho_i^{-1}(y)\given y) = 0$ and
\begin{eqnarr*} \int_{\Pp} \frac{f(\pp)h_i(p_i)}{\int h_i\,\dd\nu_i} \, \nu(\dd\pp)
= \int_{\Pp} f(\pp) \, \bar{\lambda}_i(\dd\pp)
&=& \int_{(0,1)} \int_{\Pp} f(\pp) \, \nu_i(\dd\pp \given y) \, \bar{\lambda}_i\circ\rho_i^{-1}(\dd y)
\\
&=& \int_{(0,1)} \int_{\Pp} \frac{f(\pp) h_i(p_i)}{\int h_i\, \dd\nu_i} \, \nu_i(\dd\pp \given y) \, \nu_i(\dd y).
\end{eqnarr*}
This completes the proof.
}
\end{proof}
\end{lem}
We are now in a position to define all the relevant quantities, and assemble
them into the decorated process.
\begin{defn}\label{d:eta}\relax
\begin{enumerate}
\item
Let $\mathtt{M}(\dd s, \dd z)$ be the
jump measure of $\xi$, that is,
a Poisson random measure with intensity $\dd s \,\Pi(\dd z)$.
Define $M$ to be the pushforward of $\mathtt{M}$ under
the function $(s,z) \mapsto (s,e^z)$. Thus, $M(\dd s,\dd y)$ is a Poisson
random measure with intensity $\dd s\, \pi(\dd y)$.
\item
Now let $\lambda_i(\dd y) = y^\omega \nu(p_i\in \dd y)$.
Observe that $\lambda_i$ is absolutely continuous
with respect to $\pi$, and define $g_i = \dd \lambda_i / \dd \pi$, the Radon-Nikodym derivative.
In particular, $\sum_{i\ge 1} g_i \equiv 1$ $\pi$-a.e..
\item
We define a probability kernel from $[0,\infty)\times (0,1)$ to $\NN\times \Pp$ by
\[ q(s,y,\dd i,\dd \pp) = g_i(y) \nu_i(\dd\pp \mid y) \zeta(\dd i) , \]
where $\zeta$ is counting measure on $\NN$,
and a measure $\eta$ on $[0,\infty)\times (0,1)\times\NN\times \Pp$ by
\[ \eta(\dd s, \dd y, \dd i, \dd \pp) \coloneqq q(s,y,\dd i,\dd\pp) \pi(\dd y) \dd s
= y^\omega \nu(p_i\in \dd y) \nu_i(\dd \pp\mid y) \dd s \zeta(\dd i) . \]
This has the following consistency properties:
\[
\int_{\NN\times \Pp} \eta(\dd s, \dd y, \dd i, \dd \pp)
= \pi(\dd y) \dd s \quad \text{and} \quad
\int_{(0,1)\times \NN} \eta(\dd s, \dd y, \dd i, \dd \pp)
=
\sum_{i\ge 1} p_i^{\omega}\nu(\dd \pp)\dd s .
\]
\item
Let $N(\dd s, \dd y, \dd i, \dd \pp)$ be the $q$-randomisation of $M$, in the sense
of \cite[Ch.~12, p.~226]{Kal-found}. It is readily checked (via \cite[Lemma 12.2]{Kal-found}, say)
that $N$ is a Poisson random measure with
intensity $\eta(\dd s, \dd y, \dd i, \dd \pp)$.
\end{enumerate}
\end{defn}
This completes the definition of the decorations, and we will now
define a process $\Yy = (\Yy(t), t\ge 0)$.
We regard the definition as being given under the probability measure $\lQt$,
and we assume that the underlying probability space has been enlarged as required to
accommodate it.
\begin{defn}\label{d:decorated-spine}
Let $(\Zz^{[s,j]})_{s\in \RR, j\in \NN}$ denote a collection of independent
branching L\'evy processes with triple $(a,\sigma,\nu)$. Under the probability measure
$\lQt$, the \emph{decorated Lévy process} $\Yy$, with parameters $(a,\sigma,\nu,\omega)$,
is defined
as follows:
\[ \Yy(t) = \delta_{\xi(t)} + \int_{[0,t]\times(0,1)\times\NN\times\Pp}
\sum_{j \ne i} \mbigl[\Zz^{[s,j]}(t-s) + \xi(s-) + \log p_j \mr] \, N(\dd s,\dd y, \dd i, \dd \pp) , \qquad t\ge 0,
\]
where the sum appearing on the right-hand side is over only those $j$ for which
$p_j > 0$. The summand has the following interpretation:
if $\mu = \sum_{i \in I} \delta_{\mu_i}$ is a point measure and $z \in \RR$, then
$\mu + z \coloneqq \sum_{i\in I} \delta_{\mu_i +z} $.
\end{defn}
\medskip\noindent
Let us consider the process $\Yy$ under truncation. Formally, this is required
to give the particles labels; however, the truncated processes
will also be a vital component in showing the equivalence of the two spine constructions.
Let $b>0$, and recall that $k_b$ is given by \eqref{e:kb}.
We define a random measure $N_b$
by the mapping
\begin{equation}
\int N_b(\dd s,\dd y, \dd i, \dd\pp) f(s,y,i,\pp) = \int N(\dd s,\dd y, \dd i, \dd \pp) f(s,y,i, k_b(\pp)).
\label{e:Nb}
\end{equation}
Let
\[ A_b = \{ (y,i,\pp) : i \ge 2 \text{ and } y \le e^{-b} \} \]
and define the first entry time
\[ \tau_b \coloneqq \inf\{ t\ge 0: N(\{t\}\times A_b) > 0 \} ,\]
which is a stopping time in the natural filtration of $N$.
Then $\tau_{b}$ is the time at which the spine is killed under truncation at level $b$, and it has an exponential distribution with parameter
\[ \theta_b \coloneqq \int_{\Pp} \sum_{i\ge 2}p_i^\omega \Indic{p_i\le e^{-b}} \, \nu(\dd\pp) < \infty . \]
We define the process $\Yy^{(b)}$ by the expression
\begin{multline} \Yy^{(b)}(t) = \delta_{\xi(t)} \Indic{t<\tau_b} \\
{} + \int_{[0,t]\times (0,1)\times \NN\times\Pp} N_b(\dd s,\dd y, \dd i,\dd\pp)
\sum_{j\ne i} \mbigl[ (\Zz^{[s,j]})^{(b)}(t-s) + \xi(s-)+\log p_j\mr] \Indic{s\le \tau_b},
\label{e:Yb}
\end{multline}
where $(\Zz^{[s,j]})^{(b)}$ indicates that the immigrated copy of $\Zz$ is truncated at level $b$.
With this definition, we have
all the processes $\Yy^{(b_n)}$, for $n\ge 0$, defined on the same probability space as $\Yy$.
Moreover, following \autorefref{r:misc}{i:misc:3}, we also have the processes
$\bar\Yy^{(b_n)}$ all defined on the same space.
Now suppose that $m<n$, and denote by $(\Yy^{(b_n)})^{(b_m)}$ the result of applying
the truncation method of \eqref{e:bnbm} to the process $\Yy^{(b_n)}$.
It follows that $(\Yy^{(b_n)})^{(b_m)} = \Yy^{(b_m)}$ almost surely;
this can be verified by comparing the particles present at the first braching time $T_{b_n} \coloneqq \inf\{t\ge 0: \card\Yy^{(b_n)} \ge 2 \}$,
and then proceeding iteratively.
Thus, we have that, for every $t\ge 0$, $\Yy(t) = \cup_{n\ge 0} \Yy^{(b_n)}(t)$ almost surely,
with $\bar\Yy(t)$ being defined similarly.
\skippar
We now specify a \emph{distinguished line of descent} in $\bar\Yy$, which we denote by
$V = (V_t, t\ge 0)$ with $V_t \in \tree$. We want it to track the particle whose
position is given by $\xi$, and it may be found explicitly as follows.
Fix $t\ge 0$. Observe that
\[ \int_{[0,t]\times(0,1)\times\NN\times\Pp} \Indic{i \ne 1} \eta(\dd s,\dd y,\dd i,\dd \pp) = t\int \sum_{i\ge 2} p_i^\omega \,\nu(\dd \pp) < \infty , \]
and so we may enumerate the atoms of $N\rvert_{[0,t]\times(0,1)\times(\NN\setminus\{1\})\times\Pp}$
as $(s_j,y_j,i_j,\pp^{(j)})_{1\le j\le J(t)}$, with $0\le J(t)<\infty$ and $s_j< s_{j+1}$;
moreover let $s_{J(t)+1}=t$.
If $J(t)=0$, that is, $N$ restricted to ${[0,t]\times(0,1)\times(\NN\setminus\{1\})\times\Pp}$ has no atoms,
we set $V_t = \varnothing$.
Otherwise, we proceed by recursion as follows.
Let $V_0 = \varnothing$.
For $j\ge 1$, we consider the children of particle $V_{s_{j-1}}$ which are born at time $s_j$.
Among these offspring, we pick $V_{s_j}$ such that $\Yy_{V_{s_j}}(s_j) = \xi(s_j-)+\log y_j$.
To be entirely explicit, recall the definition of $\Lu$ from \eqref{e:L},
define
$k_j = i_j - \card\{ k \ge 1: \Lu(p_k) < \Lu(p_{i_j})\}$,
and let
$V_{s_j} = V_{s_{j-1}}(\Lu(y_j),K_{V_{s_{j-1}}}(s_j,\Lu(y_j)),k_j)$.
For $s \in [s_{j-1},s_j)$ and $j\ge 1$, we let $V_s = V_{s_{j-1}}$.
Thus, in particular, $V_t = V_{s_{J(t)}}$.
\begin{defn}\label{d:deco2}
The \emph{decorated spine process} with parameters $(a,\sigma,\nu,\omega)$
is the process $(\bar\Yy(t),V_t)_{t\ge 0}$ under the measure $\lQt$.
\end{defn}
We remark that, by its construction,
$(\bar{\Yy},V) = ((\bar\Yy(t),V_t), {t\ge 0})$ is a Markov process,
and in particular it possesses a branching property exactly analogous to \autoref{l:Zbar-U-bp}.
Moreover, it has similar regularity properties, as we now show.
We need the following
lemma, whose proof is quite technical but requires nothing more than
the definition of $\Yy$ and an understanding of the additive martingale
$W(\omega,\cdot)$
of $\Zz$.
\begin{lem}\label{l:cv-Ybn}
For $q\in(\dom \kappa)^{\circ}\cap (1, \infty)$ and $t\ge 0$,
$\sup_{s\le t} d_q(\bar{\Yy}(s),\bar{\Yy}^{(b_n)}(s)) \to 0$ in probability.
\begin{proof}
In the proof, we will use similar notation ($\tree_s$, $K_u(s)$, etc.)
for the atoms of $\bar{\Yy}$ to that which we used for the atoms of $\bar{\Zz}$.
The proof follows very similar lines to the proof of \autoref{l:cv-Zbn},
and we again begin by using the triangle inequality to obtain
\begin{multline}\label{e:cadlag-Y-inter}
d_q(\bar\Yy(s),\bar\Yy^{(b_n)}(s))
\le
d_q\mBigl( \sum_{u \in \tree_s} \delta_{(u,K_u(s),\Yy_u(s))},
\sum_{u\in\tree_s^{(b_n)}} \delta_{(u,K_u(s),\Yy_u(s))} \mr)
\\
{} +
d_q\mBigl( \sum_{u \in \tree_s^{(b_n)}} \delta_{(u,K_u(s),\Yy_u(s))},
\sum_{u\in\tree_s^{(b_n)}} \delta_{(u,K_u^{(b_n)}(s),\Yy_u(s))} \mr).
\end{multline}
To show that the second term vanishes as $n\to\infty$, we can use the same method as in \autoref{l:cv-Zbn},
so long as we can adequately bound $\lEQ[\# \Yy^{(b_n)}(t)]$. We do this as follows,
beginning with:
\begin{eqnarr*}
\lEQ[ \# \Yy^{(b_n)}(t)]
&=&
\lEQ \biggl[ \Indic{t<\tau_{b_n}}
+ \int_{[0,t]\times (0,1)\times \NN \times \Pp}
N_{b_n}(\dd s, \dd y, \dd i, \dd \pp)
\sum_{j\ne i} \# \bigl(\Zz^{[s,j]}\bigr)^{(b_n)}(t-s) \Indic{s\le \tau_{b_n}}
\biggr] \\
&=& \lQ(t<\tau_{b_n})
+ \lEQ\biggl[ \int_{[0,\tau_{b_n}]\times A_{b_n}^c}
\eta_{b_n}(\dd s, \dd y, \dd i, \dd \pp)
\sum_{j \ne i} \lE[\# \Zz^{(b_n)}(t-s) ]\biggr],
\end{eqnarr*}
where in the final equality we use the fact that $N_{b_n}$ restricted to
$[0,\infty)\times A_{b_n}^c$ is independent of $\tau_{b_n}$, together with the
compensation formula for the Poisson random measure $N_{b_n}$ with
intensity measure $\eta_{b_n}$,
which is $\eta$ restricted to $[0,\infty)\times A_{b_n}^c$.
Recall that $\tau_b$ is an exponentially distributed
random variable with rate $\theta_b$.
Moreover,
$\eta_b\rvert_{[0,\infty)\times A_b^c} = \eta^{(b)}$, where
$\eta^{(b)}$ is
the measure $\eta$ constructed as in \autoref{d:eta} for the parameters
$(a,\sigma,\nu^{(b)},\omega)$, that is,
\begin{equation}\relax\label{e:eta-n}
\eta^{(b)}(\dd s, \dd y, \dd i, \dd \pp)
= y^\omega \nu^{(b)}(p_i\in \dd y) \nu^{(b)}_i(\dd \pp\mid y) \dd s \zeta(\dd i).
\end{equation}
Thus we can rewrite the previous expression to obtain that
\begin{eqnarr*}
\lEQ[\# \Yy^{(b_n)}(t)]
&=&
e^{-t \theta_{b_n}}
+ \int_{[0,t] \times (0,1)\times \NN\times \Pp}
\eta^{(b_n)}(\dd s, \dd y, \dd i, \dd \pp)
(\#\pp - 1) \lE[\# \Zz^{(b_n)}(t-s)] e^{-s \theta_{b_n}},
\end{eqnarr*}
where $\# \pp$ is the number of non-zero elements in $\pp$.
Continuing to evaluate the components of this expression, we
obtain
\begin{eqnarr*}
\lEQ[\# \Yy^{(b_n)}(t)]
&=& e^{-t\theta_{b_n}}
+ \int_0^t e^{(t-s)\kappa^{(b_n)}(0)} e^{-s\theta_{b_n}}
\cdot \int_{\Pp} (\# \pp - 1) \sum_{i\ge 1} p_i^\omega \, \nu^{(b_n)}(\dd \pp).
\end{eqnarr*}
We observe that
\[
\int_{\Pp} (\# \pp - 1) \sum_{i\ge 1} p_i^\omega \, \nu^{(b_n)}(\dd \pp)
= \int_{\Pp \setminus \Pp_1}\Indic{p_1 > e^{-b_n}} \mbigl( \sum_{i\ge 2} \Indic{p_i > e^{-b_n}}\mr) \mbigl(p_1^{\omega}+\sum_{i\ge 2} p_i^\omega \mr)\, \nu^{(b_n)}(\dd \pp).
\]
If $\omega \ge 0$, then $p_1^\omega \le 1$,
whereas if $\omega<0$, then
$\Indic{p_1>e^{-b_n}}p_1^\omega \le e^{-\omega b_n}$.
In either case, we have
\begin{eqnarr*}
\int_{\Pp \setminus \Pp_1}\Indic{p_1 > e^{-b_n}} \mbigl( \sum_{i\ge 2} \Indic{p_i > e^{-b_n}}\mr) p_1^{\omega}\, \nu^{(b_n)}(\dd \pp)
& \le& \max(1, e^{-\omega b_n} ) \int_{\Pp} \mbigl( \sum_{i\ge 2} \Indic{p_i > e^{-b_n}}\mr) \,\nu^{(b_n)}(\dd\pp) \\
&=&\max(1, e^{-\omega b_n} ) \kappa^{(b_n)}(0).
\end{eqnarr*}
Using
$\int_{\Pp} \sum_{j\ge 2} p_j^\omega \nu^{(b_n)}(\dd\pp) \le \int_{\Pp} \sum_{j\ge 2} p_j^\omega \nu(\dd\pp) <\infty$,
and
$\sum_{i\ge 2} \Indic{p_i> e^{-b_n}} \le e^{b_n}$, we obtain
\[\int_{\Pp \setminus \Pp_1}\Indic{p_1 > e^{-b_n}} \mbigl( \sum_{i\ge 2} \Indic{p_i > e^{-b_n}}\mr) \mbigl(\sum_{i\ge 2} p_i^\omega \mr)\, \nu^{(b_n)}(\dd \pp)
\le e^{b_n} \int_{\Pp} \sum_{i\ge 2} p_i^{\omega} \nu(\dd\pp).\]
It follows that
\[
\lEQ[\# \Yy^{(b_n)}(t)]
\le
e^{-t\theta_{b_n}}
+ \frac{1-e^{-t(\kappa^{(b_n)}(0) + \theta_{b_n})}}{\kappa^{(b_n)}(0) + \theta_{b_n}}
e^{t\kappa^{(b_n)}(0)}
\cdot
\mBigl(
\max(1, e^{-\omega b_n} )\kappa^{(b_n)}(0)
+ e^{b_n} \int_{\Pp} \sum_{i\ge 2} p_i^{\omega} \nu(\dd\pp)
\mr).
\]
Recall from the proof of \autoref{l:cv-Zbn} that $\kappa^{(b_n)}(0)<C e^{2 b_n}$
for some $C>0$ depending only on $\nu$; thus, for some $C'>0$, we have
\[
\lEQ[\# \Yy^{(b_n)}(t)]\le e^{C e^{2 b_n}+ C' b_n}.
\]
This is sufficient for our method of bounding the second term in \eqref{e:cadlag-Y-inter}
to work.
We turn now to the first term of \eqref{e:cadlag-Y-inter}.
Using the same trick as in \eqref{e:trick}, we select
$q'$ arbitrarily such that $q>q'$ and $q' \in (\dom \kappa)^\circ$,
and obtain
\begin{eqnarr*}
d_q\mBigl( \sum_{u \in \tree_s} \delta_{(u,K_u(s),\Yy_u(s))},
\sum_{u\in\tree_s^{(b_n)}} \delta_{(u,K_u(s),\Yy_u(s))} \mr)
&\le&
\frac{q}{q-q'}
\sum_{u \in \tree_s\setminus \tree_s^{(b_n)}} e^{q' \Yy_u(s)}.
\end{eqnarr*}
We now use the definition of $\Yy$ and $\Yy^{(b_n)}$ to write
$
\sum_{u \in \tree_s\setminus \tree_s^{(b_n)}} e^{q' \Yy_u(s)}
= I_1(s) + I_2(s) + I_3(s),
$
where for reasons of brevity the terms $I_i$ will be defined as we proceed.
The first of these is
\begin{eqnarr*}
I_1(s)
&=&
\int_{[0,s]\times (0,1)\times \NN\times \Pp} N(\dd v, \dd y,\dd i, \dd \pp)
\sum_{j\ne i} \sum_{u \in \tree_{s-v}^{[v,j]}}
e^{q' \bigl[ \Zz_u^{[v,j]}(s-v) + \log p_j + \xi(v-) \bigr]}
\\
&&
{} -
\int_{[0,s]\times (0,1)\times \NN\times \Pp} N(\dd v, \dd y,\dd i, \dd \pp)
\sum_{j\ne i} \sum_{u \in \tree_{s-v}^{[v,j]}}
e^{q' \bigl[ (\Zz_u^{[v,j]})^{(b_n)}(s-v) + \log p_j + \xi(v-) \bigr]}.
\end{eqnarr*}
If we define
$M_{v,j}^{(n)}(t)
= \sum_{u\in \tree_t} e^{q' \Zz^{[v,j]}_u(t)} - \sum_{u\in \tree_t^{(b_n)}} e^{q' \Zz^{[v,j]}_u(t)}$
for arbitrary $v\ge 0$ and $j \ge 1$,
and observe that this is a non-negative martingale in its own filtration,
it then follows that
\begin{eqnarr*}
\eqnarrLHS{\lEQ\mbigl[ \min\mbigl\{ \sup_{s\le t}
I_1(s) , 1\mr\} \mr]}
&\le&
\lEQ\mbiggl[
\int_{[0,t]\times (0,1)\times \NN\times \Pp} N(\dd v, \dd y,\dd i, \dd \pp)
e^{q'\xi(v-)}
\sum_{j\ne i}
p_j^{q'}
\min\bigl\{
\sup_{s\le t}
M_{v,j}^{(n)}(s-v)
,1\bigr\}
\mr]
\\
&\le&
\lEQ\mbiggl[
\int_{[0,t]\times (0,1)\times \NN\times \Pp} N(\dd v, \dd y,\dd i, \dd \pp)
e^{q'\xi(v-)}
\sum_{j\ne i}
p_j^{q'}
\min
\bigl\{\sup_{w\le t} M_{v,j}^{(n)}(w) , 1 \bigr\}
\mr]
\\
&=&
\int_0^t e^{v \Ess_\omega \kappa(q')} \, \dd v
\cdot
\int_{\Pp} \sum_{i\ge 1} \sum_{j\ne i} p_i^\omega p_j^{q'} \, \nu(\dd \pp)
\cdot
\lE\biggl[
\min \bigl\{ \sup_{w\le t} M_{0,1}^{(n)}(w) ,1\bigr\}
\biggr].\IEEEyesnumber \label{e:cadlag-Y-I1}
\end{eqnarr*}
We first claim that if $q' \ge 1$ and $q'\in \dom \kappa$, then
\begin{equation}\label{e:bd-int-N}
\int_{\Pp} \sum_{i\ge 1} \sum_{j\ne i} p_i^\omega p_j^{q'} \, \nu(\dd \pp)
=
\int_{\Pp\setminus \Pp_1} \sum_{i\ge 1} \sum_{j\ne i} p_i^\omega p_j^{q'} \, \nu(\dd \pp)
< \infty.
\end{equation}
We begin with the estimate
\[
\sum_{i\ge 1} \sum_{j\ne i} p_i^\omega p_j^{q'}
\le
2p_1^\omega \sum_{j\ge 2} p_j^{q'}
+ \Bigl( \sum_{i\ge 2} p_i^\omega\Bigr)\Bigl( \sum_{i\ge 2} p_i^{q'}\Bigr) .
\]
If $\omega \ge 0$, then
\eqref{e:bd-int-N} follows from
the fact that $q'\in \dom \kappa$ and $q' \ge 1$.
If $\omega < 0$, then since $\pp\in\Pp\setminus\Pp_1$,
we have $p_1^\omega \le p_2^\omega \le \sum_{i\ge 2} p_i^\omega$
and \eqref{e:bd-int-N} again follows.
Finally, using
Doob's maximal inequality just as in \autoref{l:cv-Zbn},
we see that $\sup_{w\le t} M_{0,1}^{(n)}(w)$ converges to $0$ in probability as $n\to\infty$.
Thus, the right-hand side of \eqref{e:cadlag-Y-I1} approaches $0$ also,
and so $\sup_{s\le t} I_1(s)$ tends to $0$ in probability.
This deals with the term $I_1$, which is the main difficulty. The term
$I_2$ is defined as
\begin{eqnarr*}
I_2(s)
&=&
\int_{[0,s]\times (0,1)\times \NN\times \Pp} N(\dd v, \dd y,\dd i, \dd \pp)
\sum_{j\ne i} \sum_{u \in \tree_{s-v}^{[v,j]}}
e^{q' \bigl[ (\Zz_u^{[v,j]})^{(b_n)}(s-v) + \log p_j + \xi(v-) \bigr]}
\\
&&
{} -
\int_{[0,s]\times (0,1)\times \NN\times \Pp} N_{b_n}(\dd v, \dd y,\dd i, \dd \pp)
\sum_{j\ne i}
\sum_{u \in \tree_{s-v}^{[v,j]}}
e^{q' \bigl[ (\Zz_u^{[v,j]})^{(b_n)}(s-v) + \log p_j + \xi(v-) \bigr]}
\\
&=&
\int_{[0,s]\times (0,1)\times \NN\times \Pp} N(\dd v, \dd y,\dd i, \dd \pp)
\sum_{j\ne i} p_j^{q'} \Indic{j\ne 1 \text{ and } p_j<e^{-b_n}} \sum_{u \in \tree_{s-v}^{[v,j]}}
e^{q' \bigl[ (\Zz_u^{[v,j]})^{(b_n)}(s-v) + \xi(v-) \bigr]}.
\end{eqnarr*}
Using a similar technique to the one for the term $I_1$, we obtain
\begin{multline}
\lEQ\bigl[ \min\bigl\{ \sup_{s\le t} I_2(s) ,1 \bigr\}\bigr]
\\ {} \le
\int_0^t e^{v\Ess_\omega \kappa(q')} \, \dd v
\cdot
\int_{\Pp} \sum_{i\ge 1} \sum_{j\ne i} p_i^\omega p_j^{q'}
\Indic{j\ne 1 \text{ and } p_j<e^{-b_n}}
\, \nu(\dd \pp)
\cdot
\lE\mbigl[ \min\bigl\{\sup_{r\le t} W(q',r),1\bigr\} \mr].
\label{e:cadlag-Y-I2}
\end{multline}
We can then make the estimate
\begin{equation}\label{e:bd-supW}
\lE\mbigl[ \min\bigl\{\sup_{r\le t} W(q',r),1\bigr\} \mr]
\le \lE\mbigl[ \min\bigl\{\sup_{r\le t} W(q',r),1\bigr\}^2 \mr]^{1/2}
\le \lE[W(q',t)]^{1/2} < \infty,
\end{equation}
where in the second inequality,
we use a variation on Doob's $L^2$-%
inequality (see the proof of Corollary II.1.6 in \cite{RY-cmbm}.)
Moreover, take $\epsilon>0$ such that $q'-\epsilon>1$ and $q'-\epsilon \in \dom \kappa$, then
\[
\sum_{i\ge 1} \sum_{j\ne i} p_i^\omega p_j^{q'}
\Indic{j\ne 1 \text{ and } p_j<e^{-b_n}}
\le
e^{-\epsilon b_n} \sum_{i\ge1}\sum_{j\ne i} p_i^\omega p_j^{q'-\epsilon },
\]
and just as in the $I_1$ case, we know that
$\int_{\Pp} \sum_{i\ge 1}\sum_{j\ne i} p_i^\omega p_j^{q'-\epsilon } \,\nu(\dd\pp) <\infty$.
It follows that the right hand side of \eqref{e:cadlag-Y-I2} tends to zero,
and thus $\sup_{s\le t} I_2(s) \to 0$ in probability.
Lastly, we turn to $I_3$. This term is defined as
\begin{eqnarr*}
I_3(s)
&=&
\int_{[0,s]\times (0,1)\times \NN\times \Pp} N_{b_n}(\dd v, \dd y,\dd i, \dd \pp)
\sum_{j\ne i}
\sum_{u \in \tree_{s-v}^{[v,j]}}
e^{q' \bigl[ (\Zz_u^{[v,j]})^{(b_n)}(s-v) + \log p_j + \xi(v-) \bigr]}
\\
&&
{} -
\int_{[0,s]\times (0,1)\times \NN\times \Pp} N_{b_n}(\dd v, \dd y,\dd i, \dd \pp)
\sum_{j\ne i}
\sum_{u \in \tree_{s-v}^{[v,j]}}
e^{q' \bigl[ (\Zz_u^{[v,j]})^{(b_n)}(s-v) + \log p_j + \xi(v-) \bigr]}
\Indic{v\le \tau_{b_n}}
\\
&=&
\int_{(\tau_{b_n},s]\times(0,1) \times \NN\times \Pp}
N_{b_n}(\dd v, \dd y,\dd i, \dd \pp)
e^{q'\xi(v-)} \sum_{j\ne i} p_j^{q'}
\sum_{u \in \tree_{s-v}^{[v,j]}} e^{q'(\Zz_u^{[v,j]})^{(b_n)}(s-v)}.
\end{eqnarr*}
In particular,
\begin{eqnarr*}
\sup_{s\le t} I_3(s)
&\le&
\int_{(\tau_{b_n},t]\times(0,1) \times \NN\times \Pp}
N_{b_n}(\dd v, \dd y,\dd i, \dd \pp)
e^{q'\xi(v-)} \sum_{j\ne i} p_j^{q'}
\sup_{w\le t}
\sum_{u \in \tree_{w}^{[v,j]}} e^{q'(\Zz_u^{[v,j]})^{(b_n)}(w)} .
\end{eqnarr*}
Making a change of variable in the integral, and using the independence
properties of the Poisson point process $N_{b_n}$, we obtain
\begin{IEEEeqnarray*}{rCll}
\IEEEeqnarraymulticol{4}{l}{
\lEQ\mbigl[\min\bigl\{\sup_{s\le t} I_3(s),1\bigr\}\mr]
} \\
&\le&
\lEQ\biggl[
&
\Indic{\tau_{b_n} \le t}
e^{q' \xi(\tau_{b_n})}
\\
&&& {} \cdot
\lEQ\biggl[
\int_{(r,t]\times (0,1) \times \NN\times \Pp}
N_{b_n}(\dd v, \dd y,\dd i, \dd \pp)
e^{q' \xi(v-)}
\sum_{j\ne i}
p_j^{q'}
\min\bigl\{ \sup_{w \le t} e^{q'(\Zz_u^{[v,j]})^{(b_n)}(w)} ,1 \bigr\}
\biggr]_{r = \tau_{b_n}}
\biggr] \\
&\le&
\IEEEeqnarraymulticol{2}{l}{
\lEQ[ e^{q' \xi(\tau_{b_n})} \Indic{\tau_{b_n} \le t}]
\cdot
\int_0^t e^{v\Ess_\omega \kappa(q')} \, \dd v
\cdot
\int_{\Pp} \sum_{i\ge 1} \sum_{j\ne i} p_i^\omega p_j^{q'}
\, \nu(\dd \pp)
\cdot
\lE\mbigl[ \min\bigl\{\sup_{r\le t} W(q',r),1\bigr\} \mr]
}.
\end{IEEEeqnarray*}
By \eqref{e:bd-int-N} and \eqref{e:bd-supW}, we are left with just the first expectation, for which we have:
\begin{eqnarr*}
\lEQ[\Indic{\tau_{b_n} \le t} e^{q'\xi(\tau_{b_n})} ]
&\le&
\lEQ[\Indic{\tau_{b_n} \le t} \sup_{s\le t} e^{q'\xi(s)} ]
\le
\lQ(\tau_{b_n} \le t)^{1/2}
\lE\bigl[ \bigl(\sup_{s\le t} e^{q'\xi(s)} \bigr)^2\bigr]^{1/2}.
\end{eqnarr*}
The second term on the right-hand side may be bounded using Doob's $L^2$-%
inequality for the exponential martingale of the L\'evy process $\xi$;
and the first term approaches zero as $n\to\infty$ since
$\tau_{b_n}$ has an exponential distribution whose parameter approaches zero.
It follows that $\sup_{s\le t} I_3(s) \to 0$ in probability.
Having shown the necessary convergence for each term $I_i$, we have now proved
that the first term in \eqref{e:cadlag-Y-inter} converges to zero in probability,
and this
completes the proof.
\end{proof}
\end{lem}
An immediate consequence of the lemma is the regularity of $\bar{\Yy}$.
\begin{cor}
Let $q\in (\dom \kappa)^{\circ}\cap (1,\infty)$. Then the process $\bar{\Yy}$ possesses a c\`adl\`ag version in $\mbigl(\Mmp(\mathbf{X}), d_q\mr)$.
\end{cor}
We will fix from now on a metric $d_q$ with $q\in (\dom \kappa)^{\circ}\cap (1, \infty)$, and assume that the process $\bar{\Yy}$ is c\`adl\`ag.
\section{The spine decomposition theorem}
\label{s:fb}
We now show that the forward and backward constructions of the process
with distinguished spine under $\lQ$, i.e.\ $(\bar \Zz(t),U_t)_{t\ge 0}$
and $(\bar \Yy(t),V_t)_{t\ge 0}$,
in fact have the same law.
We use a truncation technique,
recalling the definitions of $\Zz^{(b)}$, $\nu^{(b)}$
and the sequence $(b_n)$ from \autoref{s:construct-truncate}.
In order to simplify notation in the proof, we define the measure $\lPn$ such that
the law of $(\bar \Zz,\tree_\cdot)$ under $\lPn$ is that of $(\bar \Zz^{(b_n)},\tree^{(b_n)}_\cdot)$ under $\lP$.
For $n\ge 1$, we consider on the one hand the
measure $\lQn$ constructed from $\lPn$ as follows:
\begin{equation}\label{e:Qn}
\lEQn[ F(\bar \Zz(s), s \le t) \Indic{U_t=u}]
= e^{-t\kappa^{(b_n)}(\omega)} \lE^{(b_n)}[ F(\bar \Zz(s), s \le t) e^{\omega \Zz_u(t)}],
\end{equation}
where $F$ is a continuous bounded functional on $D([0,\infty),\mathcal{M}_p(\mathbf{X}))$,
and we use the convention $e^{\Zz_u(t)} = 0$ if $u\notin \tree_t$.
On the other hand, we
regard the process $(\bar \Yy,V)$ under $\lQtn$ as being the decorated spine process
with parameters $(a,\sigma,\nu^{(b_n)},\omega)$.
\begin{lem}\relax\label{l:fb-n}
Under $\lQn$, $(\bar \Zz(t),U_t)_{t\ge 0}$ is equal in law to $(\bar \Yy(t),V_t)_{t\ge 0}$.
\end{lem}
\begin{proof}
We verify that the two processes have the same decomposition at the first branching time;
since both are Markov, this is sufficient.
We start with $(\bar \Zz,U)$ under $\lQn$. Let $T$ denote the time of the first branching event, that is,
\[ T = \inf\{ t\ge 0 : \card \bar \Zz(t) \ge 2 \}, \]
where $\card\bar\Zz(t) = \bar\Zz(t)(\mathbf{X})$ is the number of atoms in $\bar\Zz(t)$.
From the construction of the
truncated processes, we know that
under $\lPn$, $T$ has an exponential distribution with rate $\lambda_{b_n} = \nu^{(b_n)}(\Pp\setminus\Pp_1)$.
The point measure $\bar \Zz(T)$ has a countable number of atoms; let $(u^{(j)})_{j\ge 1}$
be their labels, such that $u^{(1)} = \varnothing$ and $(u^{(j)})_{j\ge 2}$ is lexicographically ordered;
in particular, this implies that $\Zz_{u^{(j)}}(T) \eqd \Zz_{\varnothing}(T-) + \log p_j$, where $\pp$
is sampled from $\nu^{(b_n)}\rvert_{\Pp\setminus\Pp_1}/\lambda_{b_n}$.
Furthermore, the translates $\Zz \circ \theta_{u^{(j)},T}$ are independent of each other and
of $\FFs_T$, where
we recall that $\FFs_t = \sigma( \bar\Zz(s), U_s; s\le t)$, for $t \ge 0$.
Additionally, $(\Zz_\varnothing(s),s< T)$ is independent
of $T$ and $\pp$, and has the law of a L\'evy process with Laplace exponent
$\Psi^{(b_n)}$ killed at an independent exponential time of rate $\lambda_{b_n}$.
All of these facts add up to the following computation, in which $F_j$ is a $\FFs_{t}$-measurable
functional, $G_j$ is a measurable function of $\RR$ and $J$ is a measurable functional on path space;
and $u \in \tree$.
Let $i$ be such that $u = u^{(i)}v$, with $i=1$ only if $u^{(j)}\not\prec u$ for all $j\ge 2$,
and as a shorthand denote $\Delta\Zz_{u^{(j)}}(T) = \Zz_{u^{(j)}}(T) - \Zz_{\varnothing}(T-)$.
\begin{eqnarr*}
\eqnarrLHS{
\lQn[ J(\Zz_\varnothing(s),s<T) \prod_{j\ge 1} F_j \circ \theta_{u^{(j)},T}
\, G_j(\Delta \Zz_{u^{(j)}}(T)) ; U_{T+t} = u ]
}
&=& \lEn\mbiggl[
e^{-T\kappa^{(b_n)}(\omega)} e^{\omega \Zz_{\varnothing}(T-)} J(\Zz_{\varnothing}(s),s<T)
\prod_{j\ge 1} G_j(\Delta \Zz_{u^{(j)}}(T)) e^{\omega \Delta \Zz_{u^{(i)}}(T)} \\
&& \quad {} \cdot \lEn\mBigl[
e^{-t\kappa^{(b_n)}(\omega)}\prod_{j\ge 1} F_j\circ \theta_{u^{(j)},T}
\, e^{\omega(\Zz_u(T+t)-\Zz_{u^{(i)}}(T)}
\mm\vert \FFs_T
\mr]
\mr]
\\
&=&
\int_0^\infty \dd r \, \exp(-\mu_{b_n}r) \sE[ J(\chi^{(b_n)}_{\omega}(s),s< r)] \\
&& \quad {} \cdot
\int_{\Pp\setminus\Pp_1} \nu^{(b_n)}(\dd \pp)\, p_i^\omega \prod_{j\ge 1} G_j(\log p_j) \lEQn[F_i; U_t = v] \prod_{j\ne i} \lEn[F_j],
\end{eqnarr*}
where in the last line we define
\[ \mu_{b_n} \coloneqq \lambda_{b_n} + \kappa^{(b_n)}(\omega) - \Psi^{(b_n)}(\omega)
= \int_{\Pp\setminus\Pp_1} \sum_{i\ge 1} p_i^\omega \, \nu^{(b_n)}(\dd \pp) ,
\]
and $\chi^{(b_n)}_{\omega}$, defined under a probability measure $\sP$, is a Lévy process whose
Laplace exponent is an Esscher transform of
$\Psi^{(b_n)}$, namely $\Ess_{\omega} \Psi^{(b_n)} \coloneqq \Psi^{(b_n)}(\omega + \cdot) - \Psi^{(b_n)}(\omega)$.
We now turn to the process $(\bar\Yy,V)$, again under $\lQn$.
We again define the branching time,
\[ T = \inf\{ t\ge 0 : \card \bar \Yy(t) \ge 2 \}
= \inf\{t \ge 0: N(\{t\}\times \mathcal{A}) > 0 \}
,
\]
where $\mathcal{A} = (0,1) \times \NN \times (\Pp\setminus\Pp_1)$;
that is, $T$ is the first time that a jump of $\xi$ is accompanied by immigration.
We consider the quantity
\begin{equation}\label{e:bn-functional}
\lEQtn \mbigl[ J(\Yy_{\varnothing}(s), s < T)
\prod_{j\ge 1} F_j \circ \theta_{u^{(j)},T}
\, G_j(\Delta\Yy_{u^{(j)}}(T)); V_{T+t} = u \mr],
\end{equation}
where $F_j, G_j, J$ are measurable functionals as above.
Observe that, under $\lQn$, $N$ is a Poisson random measure with intensity
$\eta^{(b_n)}$ as defined in \eqref{e:eta-n}.
Now, by the definition of $T$ and standard properties of Poisson random
measures \cite[\S O.5]{Ber-Levy}, we know that
$T$ has an exponential distribution with rate
\[ \int_{[0,1]\times\mathcal{A}} \eta^{(b_n)}(\dd s,\dd y, \dd i, \dd \pp)
= \int_{\Pp\setminus\Pp_1} \sum_{i\ge 1} p_i^\omega \, \nu^{(b_n)}(\dd \pp)
\eqqcolon \mu_{b_n} .
\]
In fact, we can say more: the restriction $N\rvert_{[0,T)\times (0,1)\times\NN\times \Pp}$
has same law as the restriction
$N\rvert_{[0,\tau)\times\mathcal{A}^c}$,
where $\tau$ is an exponentially-distributed random variable with rate $\mu_{b_n}$
which is independent of $N$,
and $\mathcal{A}^c = (0,1)\times\NN\times\Pp_1$.
This has implications for the process $(\Yy_{\varnothing}(s),s<T)$
which, importantly, is the same as the spine process $\xi$ on the
time interval in question;
it remains a L\'evy process with Gaussian coefficient $\sigma$,
but has two changes: first, it is killed independently at rate $\mu_{b_n}$.
Second, the law of its jump measure, which we recall is the pushforward of
$N(\dd s, \dd y, \NN,\Pp)$ by the map $(s,y)\mapsto (s,\log y)$,
is altered because the law of $N$ is altered.
Working with the L\'evy--It\^o decomposition, we see that $(\Yy_{\varnothing}(s),s<T)$
has Laplace exponent given by
\begin{eqnarr*}
\eqnarrLHS{\frac{1}{2}\sigma^2 q^2 + c_{b_n,\omega}q + \int_{[0,1]\times(0,1)\times\{1\}\times\Pp_1} \mbigl( y^q-1-ql(y) \mr) \, \eta^{(b_n)}(\dd s,\dd y, \dd i, \dd \pp) - \mu_{b_n}}
&=& \frac{1}{2}\sigma^2 q^2 + c_{b_n,\omega}q + \int_{\Pp_1} \mbigl( p_1^q-1-ql(p_1) \mr) p_1^\omega\, \nu^{(b_n)}(\dd\pp) - \mu_{b_n}
\\
&=& \Ess_\omega\Psi^{(b_n)}(q) - \mu_{b_n}, \qquad q \ge 0,
\IEEEyesnumber\label{e:fb-n:le}
\end{eqnarr*}
where
\begin{eqnarr*}
c_{b_n,\omega}
&=&
a + \omega\sigma^2 + \int_{\Pp} (1-p_1) \, \nu^{(b_n)}(\dd\pp)
+ \int_{\Pp_1} p_1^\omega l(p_1)\, \nu^{(b_n)}(\dd\pp) .
\end{eqnarr*}
Note that the centre $c_{b_n,\omega}$ differs from the centre of $\xi$ due to
the change in compensation of the small jumps.
It follows that $(\Yy_{\varnothing}(s),s<T)$
has the law of $\chi^{(b_n)}_{\omega}$ killed at an independent
exponential time with rate $\mu_{b_n}$.
Considering the particles born at time $T$, define the children $(u^{(j)})_{j\ge 1}$
of $\Yy_\varnothing$ as for the previous part of the proof,
and assume that again $u = u^{(i)}v$, with the convention that $i = 1$ only if $u^{(j)}\not\prec u$
for all $j\ge 2$.
Using again the properties of Poisson random measures, the atom $(T,y,k,\pp)$
of $N$ appearing at time $T$ is such
that $(y,k,\pp)$ has distribution $\frac{\eta^{(b_n)}([0,1],\cdot)\rvert_{\mathcal{A}}}{\mu_{b_n}}$,
and we are further restricted in \eqref{e:bn-functional}
to the event $V_{T+t}=u$, which implies that here
we are restricted to the event $\{k=i\}$.
Finally, from the construction of the decorated spine process,
we know that each child $u^{(j)}$ is initially positioned at $\Yy_\varnothing(T-)+\log p_j$,
and
that the translate $\Yy \circ \theta_{u^{(i)},T}$ has the law of $\Yy$ under $\lQtn$,
while the translates $\Yy \circ \theta_{u^{(j)},T}$ are independent of one another
and have the law of $\Zz$ under $\lP^{(b_n)}$.
The discussion above essentially proves the required decomposition, but for clarity we
provide the following calculation, in which $J, F_j, G_j$ are measurable functionals as above.
\begin{eqnarr*}
\eqnarrLHS{
\lEQtn \mbigl[ J(\Yy_{\varnothing}(s), s < T)
\prod_{j\ge 1} F_j \circ \theta_{u^{(j)},T} G_j(\Delta\Yy_{u^{(j)}}(T)); V_{T+t} = u \mr]
}
&=&
\int_0^{\infty} \dd r \,
\mu_{b_n} \exp(-\mu_{b_n}r) \sE[ J(\chi^{(b_n)}_{\omega}(s), s < r) ] \\
&& {} \cdot \int_{(0,1)\times(\Pp\setminus \Pp_1)}
\frac{1}{\mu_{b_n}}\eta^{(b_n)}([0,1],\dd y, \{i\}, \dd \pp) \,
G_i(\log y) \lEQtn[F_i; V_t = v] \prod_{j\ne i} \lEn[F_j] G_j(\log p_j)
\\
&=& \int_0^{\infty} \dd r \, \exp(-\mu_{b_n}r) \sE[ J(\chi^{(b_n)}_{\omega}(s), s < r) ]
\\
&& {} \cdot
\int_{\Pp\setminus\Pp_1} \nu^{(b_n)}(\dd\pp) \, p_i^\omega \prod_{j\ge 1} G_j(\log p_j)
\lEQtn[F_i; V_t = v] \prod_{j\ne i} \lEn[F_j].
\end{eqnarr*}
This completes the proof.
\end{proof}
Having established the result for these truncated processes, we need to
remove the truncation, and this proves the following
theorem on the spine decomposition,
which is our main result.
\begin{thrm}[Spine decomposition]\label{t:mto}
Under $\lQ$, $(\bar \Zz(t),U_t)_{t\ge 0}$
is equal in law to $(\bar \Yy(t),V_t)_{t\ge 0}$.
\end{thrm}
\begin{proof}
Since the processes $(\bar{\Zz},U)$ and $(\bar{\Yy},V)$ are both Markov,
it is sufficient to fix $t\ge 0$ and prove that $(\bar{\Zz}(t),U_t)$
has the same distribution as $(\bar{\Yy}(t),V_t)$.
For the measure
$\lQn$, \eqref{e:Qn} implies
\[
\lEQn[ F(\bar \Zz(t)) \Indic{U_t=u}]
= e^{-t\kappa^{(b_n)}(\omega)} \lE[ F(\bar \Zz^{(b_n)}(t)) e^{\omega \Zz^{(b_n)}_u(t)}]
\]
for continuous bounded $F$.
Under $\lP$, $\bar{\Zz}^{(b_n)}(t)\to \bar{\Zz}(t)$ weakly on $\Mmp(\mathbf{X})$.
Furthermore, for every $\omega\in\dom\kappa$,
$\kappa^{(b_n)}(\omega)\upto\kappa(\omega)$.
Hence, certainly
the distribution of $(\bar{\Zz}(t),U_t)$
under
$\lQn$ converges weakly to the distribution of $(\bar{\Zz}(t),U_t)$
under $\lQ$.
We now address the convergence
of the law of $(\bar\Yy(t),V_t)$.
Consider first the process $\bar \Yy^{(b_n)}$, which was defined in \autoref{s:forward},
using the notation $A_{b_n}$ and $\tau_{b_n}$.
We may consider
the joint process $(\bar\Yy^{(b_n)},V^{(b_n)})$, by adjoining a `cemetery' element $\partial$
to the collection of labels, and defining
$V_t^{(b_n)} = V_t\Indic{t < \tau_{b_n}} + \partial \Indic{t \ge \tau_{b_n}}$; thus,
$V_t^{(b_n)} = \partial$ indicates that the distinguished line of descent has been
killed before time $t$ in the process $\bar\Yy^{(b_n)}$.
We will start by showing that, for $F$ a continuous bounded functional on $\Mmp(\mathbf{X})$
and $u\in\tree$,
\begin{eqnarr*}
\lEQt[ F(\bar \Yy^{(b_n)}(t)) \Indic{V_t^{(b_n)}=u} \mid V_t^{(b_n)} \ne \partial]
&=& \lEQtn[ F(\bar \Yy(t)) \Indic{V_t = u}] \IEEEyesnumber \label{e:fb-inter1} \\
&=& \lEQn[ F(\bar\Zz(t)) \Indic{U_t = u}].
\end{eqnarr*}
The second equality is an immediate corollary of
\autoref{l:fb-n}, so we have only to prove the first equality.
The conditioning on the left-hand side of \eqref{e:fb-inter1} is the same as conditioning
on the event $\{ t < \tau_{b_n}\}$, where $\tau_{b_n}$ is the hitting time of the set $A_{b_n}$ for the
Poisson random measure $N$.
We notice that, given $\{t < \tau_{b_n}\}$, we have the equality
\begin{equation*}
\Yy^{(b_n)}(t) = \delta_{\xi(t)}
+ \int_{[0,t]\times (0,1)\times\NN\times\Pp} {N}_{b_n}(\dd s,\dd y, \dd i, \dd \pp)
\sum_{j\ne i} \mbigl[ (\Zz^{[s,j]})^{(b_n)} + \xi(s-) + \log p_j \mr],
\end{equation*}
where
$N_{b_n}$ is defined in \eqref{e:Nb}.
Using standard properties of Poisson random measures
\cite[\S O.5]{Ber-Levy},
we see that, under $\lQt(\cdot \mid t < \tau_{b_n})$,
$N$ has the same law as
its restriction
$N\rvert_{[0,\infty)\times A_{b_n}^c}$.
Thus, under $\lQt(\cdot\mid t<\tau_{b_n})$,
$N_{b_n}$ is a Poisson random measure whose intensity
is the measure $\eta^{(b_n)}$ given in \eqref{e:eta-n}.
The change in the law of the measure $N$ which is induced by this
conditioning causes a corresponding change to the jump measure of $\xi$,
which we again recall is the pushforward of $N(\dd s, \dd y, \NN, \Pp)$ under
$(s,y) \mapsto (s,\log y)$.
Using the L\'evy--It\^o decomposition much as in the proof of \eqref{e:fb-n:le},
we may show
that under $\lQt(\cdot \mid t < \tau_{b_n})$,
$(\xi,N_{b_n})$ has the same law as $(\xi,N)$ does under $\lQtn$.
Finally, $\bar{\Yy}^{(b_n)}$ is measurable with respect to
$N_{b_n}$ and $\xi$, and the same is true of $V_t^{(b_n)}$
on the event $\{t<\tau_{b_n}\}$.
This completes the proof of \eqref{e:fb-inter1}.
We now need to take $n\to\infty$. The right-hand side of \eqref{e:fb-inter1}
converges to $\lEQ[F(\bar\Zz(t)) \Indic{U_t = u}]$, as discussed
at the beginning of the proof.
The left-hand side of \eqref{e:fb-inter1} is equal to
\begin{equation}\relax\label{e:fb-inter2}
\frac{\lEQt\mbigl[ F(\bar{\Yy}^{(b_n)}(t)) \Indic{V^{(b_n)}_t = u}\mr]}%
{\lQt(V_t^{(b_n)}\ne \partial)}.
\end{equation}
For every $t\ge 0$ and every realisation of the process,
$\{V_t^{(b_n)}\ne \partial\}$ holds for
large enough $n$; moreover,
by \autoref{l:cv-Ybn} we have $\bar{\Yy}^{(b_n)}(t)\to \bar{\Yy}(t)$ in probability,
and hence (extracting a subsequence if necessary) also almost surely.
It follows from the dominated convergence
theorem that \eqref{e:fb-inter2} converges to
$\lQt[F(\bar \Yy(t)) \Indic{V_t = u}]$. This completes the proof.
\end{proof}
\begin{rem}\label{r:mto}
The theorem above establishes a `full many-to-one theorem' in
the language of \cite{HH-spine}.
We stress that a version of this theorem has been proved, by \citet{BBCK-maps},
for the case of binary branching ($\nu(\dd\pp)$ being supported by those $\pp$
such that $p_3 = 0$) under the condition that $\kappa(\omega) = 0$,
though their description of the decomposition differs somewhat from ours due to their
view of the genealogy.
\end{rem}
An immediate corollary is the following useful expression for certain functionals of $\bar\Zz(t)$:
\begin{cor}[Many-to-one formula]\label{c:mto}
For a non-negative Borel function $f: \RR \to [0,\infty)$,
\[ \lE\mBigl[\sum_{u\in \tree_t} f(\Zz_u(t)) \mr] = e^{t\kappa(\omega)} \sE[ f(\xi(t)) e^{-\omega\xi(t)} ] ,\qquad t\ge 0, \]
where $\xi$ under the measure $\sP$ is a L\'evy process with Laplace exponent $\Ess_\omega\kappa$.
\end{cor}
We also point out the following consequence for the process $\bar\Yy$. Recall
that $(\bar{\Yy},V)$ is Markov; this result says the same is true even if
we forget $V$.
\begin{cor}
$\bar \Yy$ is a Markov process under $\lQ$.
\end{cor}
\begin{proof}
$\bar\Zz$ is defined (without the distinguished
particle $U$)
by a change of measure of a Markov process with respect to the martingale
$\mg{\omega}{\cdot}$, and is therefore a Markov process in its own right.
$\bar\Yy$ is equal in distribution to $\bar\Zz$ under $\lQ$, and this
completes the proof.
\end{proof}
\section{The derivative martingale}
\label{s:dm}
For every $\omega\in (\dom \kappa)^{\circ}$, the interior of $\dom \kappa$, let $\partial W(\omega,\cdot)$ denote the derivative martingale, given by
\[ \partial W(\omega,t) = e^{-t\kappa(\omega)}
\sum_{u\in \tree_t} \big(\Zz_u(t) - t\kappa'(\omega)\big) e^{\omega \Zz_u(t)}, \qquad t \ge 0 . \]
Our purpose is to study the asymptotic properties of this martingale.
Before stating our main result, Theorem~\ref{t:DerMart}, let us first distinguish two regimes of $\omega$.
By the convexity of $\kappa$, we observe that the function $q\mapsto q \kappa'(q) -\kappa(q)$ is increasing on $(\dom \kappa)^{\circ}$ and has at most one sign change. From now on, we assume that
\begin{equation}\label{H}
\text{ there exists (a unique) } \ws>0, \text{ such that } \ws \in (\dom \kappa)^{\circ} ~\text{and}~\ws\kappa'(\ws) -\kappa(\ws)=0. \tag*{(H)}
\end{equation}
The value $\ws$ has proved to be critical for the study of the uniform integrability of the exponential additive martingale $\MA(\omega, \cdot)$; see \cite{Dadoun:agf, BM-mcbLp}.
We point out that the assumption \ref{H} entails
that $\kappa(0)\in (0,+\infty]$, so the \emph{non-extinction}
event has strictly positive probability:
\[ \lP\mbigl(\#\Zz(t) >0 \text{ for all }t\ge 0 \mr)\in (0,1],\]
where $\#\Zz(t) \coloneqq \Zz(t)(\RR)$ denotes the number of atoms at time $t$.
Write $\lP^*$ for the probability measure $\lP$ conditional on non-extinction.
We recall our standing assumption $\nu(\{\mathbf 0\}) = 0$,
implying that particles are never killed;
this in fact implies that non-extinction occurs $\lP$-almost surely,
and so in fact $\lP^* = \lP$ for us. However, we retain the notation $\lP^*$
in order to make clear how our results would look without our assumption.
\skippar
We now state the main result of this section.
\begin{thrm}\label{t:DerMart} Suppose that \ref{H} holds.
\begin{enumerate}
\item Let $\omega \geq \ws$, then the derivative martingale $\MD(\omega, t)$ converges $\lP$-almost surely to a finite non-positive limit $\MD(\omega,\infty)$ as $t\to \infty$.
\item Let $\omega > \ws$, then $\MD(\omega,\infty) = 0$ holds $\lP$-almost surely,
\item For $\omega=\ws$, there is $\lE[\MD(\ws,\infty)] = -\infty$, and $\MD(\ws,\infty) <0$ holds $\lP^*$-almost surely.
\end{enumerate}
\end{thrm}
In \cite[Corollary 2.10(b)]{Dadoun:agf}, Dadoun has shown
the $\lP^*$-almost sure negativity of the random variable $\partial W(\bar{\omega},\infty)$,
identified there as the almost sure limit of the discrete martingale
$(\partial W(\bar{\omega},n), n = 0,1,,\dotsc)$ Our theorem improves upon
\cite{Dadoun:agf} by proving convergence of the continuous-time martingale
and finding the expected value of the limit random variable.
Furthermore, we do not require condition (2.7) of \cite{Dadoun:agf}.
The limit $\partial W(\bar{\omega},\infty)$ has an intimate connection with the asymptotic behaviour of the largest fragment and Seneta--Heyde norming for $\MA(\ws, \cdot)$; see \cite[Corollary~2.10 and Remark~2.11]{Dadoun:agf}.
Analogues of \autoref{t:DerMart} were proved for multitype branching random walks by \citet{BK-mc}, for branching Brownian motion by \citet{Kyp-FKPP} and for pure fragmentation processes by \citet{BR-disc}. A thorough exposition of the theory for branching random walks is given in the monograph of \citet{Shi-BRW}.
The common approach of the works described above is
a technique based upon stopping particles moving at a certain speed,
and we stress that the spine decomposition plays a central role in these arguments.
Our proof, which is primarily modelled
on that of \citet{BR-disc}, is postponed to \autoref{s:proofD}; in the coming \autoref{s:sm}, we prepare for it by investigating a related family of martingales.
\begin{rem}
For generic branching L\'evy processes \cite{BM-bLp} in which upward jumps of the particle locations are permitted, the same arguments apply to prove (i) and (ii) of \autoref{t:DerMart}, but not (iii).
For $\omega=\bar\omega$, we expect that an additional assumption in terms of the dislocation measure $\nu$ is needed to make the limit non-trivial.
In the case of branching random walks \cite{Chen-dm} and branching Brownian motion \cite{RY-dm}, optimal moment conditions have been found, and the martingale limits are proven to be zero when these conditions do not hold.
\end{rem}
\subsection{The stopped martingales}\label{s:sm}
In this subsection we fix $a>0$ and $\omega \in (\dom \kappa)^{\circ}$, and define a process
\[
\MD_a(\omega,t):= \sum_{u\in \tree_t} \mbigl( a +t \kappa'(\omega) -\Zz_u(t) \mr) e^{-t \kappa(\omega) + \omega\Zz_u(t)}\Indic{ a+r \kappa'(\omega) - \Zz_{ \Anc(r,u)}(r)>0 \text{ for } r\le t},\quad t\ge 0,
\]
where $\Anc(r,u)$ denotes the ancestor of $u$ at time $r$ as in \autoref{s:branching}. It is clear that $\MD_a(\omega,t)$ is always
non-negative.
We use this to define a new measure on $\FF_\infty$ by
\[ \lQD (A) := \frac{1}{a}\lE[\partial W_a(\omega,t) \Ind_A ] , \qquad t \ge 0, A \in \FF_t, \]
and extend it to $\hat\FF_\infty$ by
\begin{multline}\label{e:lQD}
\lQD (A; U_t = u) \\
\coloneqq \frac{1}{a}\lE\mbigl[\mbigl(a + t\kappa'(\omega) - \Zz_u(t)\mr) e^{-t \kappa(\omega)+\omega \Zz_u(t)}
\Indic{a+s\kappa'(\omega)-\Zz_{\Anc(r;u)}(r) > 0 \text{ for }r\le t} \Ind_A \mr]. \end{multline}
To justify that the measure $\lQD$ is well-defined and does not depend on the choice of $t$, we consider the interpretation of $\lQD$ as having a density with respect to $\lQ$ on $\hat\FF_\infty$.
Recall that under $\lQ$, we have a process $\bar\Zz$ together with a spine label $U$, and the spine $(\Zz_{U_t}(t), t\geq 0)$ is a L\'evy process with Laplace exponent $\Ess_\omega\kappa$. Write
\begin{equation}\label{e:lambda}
\lambda(t) := a + t\kappa'(\omega) - \Zz_{U_t}(t), \qquad t\ge 0,
\end{equation}
then it follows that $\lambda$ under $\lQ$ is a L\'evy process
with respect
to the filtration $(\hat \FF_t)_{t\ge 0}$, started at $a$. The process $\lambda$
is spectrally positive, in the sense that it has only positive jumps, and it
has Laplace exponent $\kappa'(\omega)q - \Ess_\omega\kappa(q)$,
meaning that $\lEQ[e^{-q(\lambda(t)-a)}] = e^{-t(\kappa'(\omega)q-\Ess_\omega\kappa(q))}$
for $q\ge 0$. (This is a slight change in notation compared to
\eqref{e:lk}, but it follows the usual convention for the Laplace exponent of a
spectrally positive process.)
In particular, $\lEQ [ \lambda(t)] = a$ for every $t\geq 0$, which implies that $\lambda$ is a $\lQ$-martingale with respect to $(\hat\FF_t)_{t\ge 0}$.
Let
\[ \zeta = \inf\{ t\ge 0: \lambda(t) < 0 \}, \]
then it follows from \autoref{c:mto} that
\begin{equation}\label{e:cm-lambda}
\lQD (A) =a^{-1}\lEQ\mBigl[ \lambda(t) \Indic{t < \zeta} \Ind_{A} \mr] , \qquad t \ge 0, A \in \hat\FF_t. \end{equation}
Using the fact that the stopped martingale $(\lambda(t\wedge \zeta)= \lambda(t) \Indic{t < \zeta}, t\geq 0) $ remains a $\lQ$-martingale (see \cite[Corollary II.3.6]{RY-cmbm}), we justify the previous definition of $\lQD$ as a consistent change of measure.
As a consequence, $\partial W_a (\omega, \cdot)$ is a non-negative $\lP$-martingale, and therefore converges $\lP$-almost surely to a limit $\MD_a(\omega, \infty)$ as $t\to \infty$.
\skippar
The main object of this subsection is to establish the following result, which will be crucially used in the proof of \autoref{t:DerMart}.
\begin{prop}\label{l:MDa}Suppose that \ref{H} holds.
\begin{enumerate}
\item For $\omega> \ws$, we have $\lP$-almost surely $\MD_a(\omega,\infty)= 0$.
\item For $\omega= \ws$, the martingale $\MD_a(\ws,t)$ converges to $\MD_a(\ws,\infty)$ in $L^1(\lP)$.
\end{enumerate}
\end{prop}
To prove \autoref{l:MDa}, the key idea is to use the `forward' construction (\autoref{d:deco2} and \autoref{t:mto}) of $(\bar\Zz,U)$ under $\lQ$, as a L\'evy process $\xi$ with Laplace exponent $\Ess_\omega\kappa$ whose jumps are decorated with independent branching L\'evy processes with law $\lP$, each
positioned according to the atoms of a random measure $N$.
By a slight abuse of notation, the measure $N$ under $\lQ$
can be seen as an
integer-valued random measure on $[0,\infty)\times E$, with $E = (0,1)\times\NN\times\Pp$,
and its support is a random set having the form
$\{ (s,(e^{-\Delta \xi(s)},i_s,\pp_s)) : \Delta\xi(s)\ne 0\}$.
Further, $N$ is Poisson with the (non-random) intensity measure $\eta$.
Since $\lQD$ is absolutely continuous with respect to $\lQ$
on every $\hat\FF_t$, the process under $\lQD$ has the same structure; however, the laws of the process $\xi$ and the random measure $N$ may be different.
The following pair of lemmas provides
more detail on the discussion above;
we refer to \citet[\S II.1]{JS-limit} for a thorough discussion of
random measures, and in particular the
notion of the predictable compensator of a random measure.
Note that hereafter, when we say \emph{predictable},
we will always mean predictable with respect to the filtration
$(\hat\FF_t)_{t\ge 0}$.
\begin{lem}\label{l:clambda}
Under $\lQD$, the process $\lambda$ defined as in \eqref{e:lambda} is a spectrally positive
L\'evy process starting from $a$ with
Laplace exponent $q \mapsto \kappa'(\omega)q - \Ess_\omega\kappa(q)$, conditioned to
be positive in the sense of \cite{Cha-cond,CD-cond}.
In particular, we have that $\inf_{t\ge 0} \lambda(t) > 0$, $\lQD$-almost surely.
\begin{proof}
Recall that $\lambda$ is a (unconditioned) L\'evy process with the given Laplace exponent under $\lQ$.
In the work of \citet{CD-cond}, it is shown that conditioning the L\'evy process $\lambda$
to remain
positive is equivalent to performing a martingale change of measure with respect to the
martingale $U_-(\lambda(t))\Indic{t<\zeta}$, where $U_-$ is the potential function
of the downward ladder height subordinator.
Since $\lambda$ has no negative jumps and
has constant mean $a$, it follows that
$U_-(x) = x \Indic{x>0}$ (see \cite[\S 6.5.2]{Kyp2} for
the analogous case of processes with no positive jumps.)
Therefore, conditioning $\lambda$ to remain positive gives rise to
$\lQD$ as the conditioned measure.
This completes the characterisation of $\lambda$ under $\lQD$.
Finally, since $\lambda$ under $\lQ$ is a centred L\'evy process with only positive jumps, the fact that the overall infimum of $\lambda$ under $\lQD$ is positive is implied by \cite[Theorem 1(a)]{CD-cond}.
\end{proof}
\end{lem}
\begin{lem}\label{l:cN}
The predictable compensator of the random measure $N$ under $\lQD$ is given by
\[ \eta'(\dd s, \dd y, \dd i, \dd \pp) \coloneqq
\frac{\lambda(s-) - \log y}{\lambda(s-)}
\, \eta(\dd s, \dd y, \dd i, \dd \pp) .
\]
\begin{proof}
We first point out that $\zeta$ is predictable:
since $\lambda$ is a spectrally positive L\'evy process under $\lQ$, it can only pass
below $0$ continuously. Thus, defining
$T_n = \inf\{ t \ge 0: \lambda(t) < 1/n\} < \zeta$,
we have that $\zeta = \sup_{n} T_n$, which implies in particular that $\zeta$ is predictable (by \cite[Theorem I.2.15(a)]{JS-limit}.)
Now, since $N$ is Poisson under $\lQ$, its compensator under $\lQ$ is the (non-random) intensity measure $\eta$, and moreover
the density process for the change of measure is
\[
\frac{\dd \lQD}{\dd \lQ}\bigg|_{\hat\FF_s}
= a^{-1}\lambda(s) \Indic{s<\zeta}.
\]
For any predictable random function $(s,(y,i,\pp)) \mapsto U_s(y,i,\pp)$,
we have that
\begin{multline*}
\lEQ \mbiggl[
\int_{[0,\infty)\times E} \frac{\lambda(s)}{\lambda(s-)} \Indic{s<\zeta}
U_s(y,i,\pp) \, N(\dd s, \dd y,\dd i,\dd \pp) \mr] \\
=
\lEQ \mbiggl[
\int_{[0,\infty)\times E} \frac{\lambda(s-)-\log y}{\lambda(s-)} \Indic{s<\zeta}
U_t(y,i,\pp) \, N(\dd s, \dd y,\dd i,\dd \pp) \mr],
\end{multline*}
and the random function
$(s,(y,i,\pp)) \mapsto \frac{\lambda(s-)-\log y}{\lambda(s-)} \Indic{s<\zeta}$
is predictable. Having made these observations, the result
follows by the Girsanov theorem for random measures
\cite[Theorem III.3.17(b)]{JS-limit}.
Note that under $\lQD$, $\zeta = \infty$ by \autoref{l:clambda}.
\end{proof}
\end{lem}
We now need one final technical result to prepare for the
main proposition in this section.
\begin{lem}\label{l:lambda-int}
For every $p>0$,
\[ \lQD \mbiggl[ \int_{0}^\infty \mBigl(\lambda(r)+1+\frac{1}{\lambda(r)}\mr) e^{-p \lambda(r)} \, \dd r \mr] < \infty . \]
\begin{proof}
Let $V(a, \dd y) = \lQD \mbigl[ \int_0^\infty \Indic{\lambda(r)\in \dd y} \, \dd r \mr]$,
and $U^\dag(a,\dd y) = \lQ \mbigl[ \int_0^{\zeta} \Indic{\lambda(r)\in \dd y}\, \dd r \mr]$.
The former is the potential of a Lévy process conditioned to stay positive, and the latter
is that of a Lévy process killed upon going below the level $0$. Due to the $h$-transform
connecting their semigroups, they are related by the formula
\[ V(a,\dd y) = \frac{y}{a}U^\dag(a,\dd y). \]
Moreover, by \cite[Corollary 8.8]{Kyp2}, we have
\[ U^\dag(a,\dd y) = (\WW(y)-\WW(y-a)) \, \dd y , \qquad y \ge 0, \]
where $\WW$ is the scale function of the spectrally negative Lévy process $-\lambda$,
with the convention that $\WW(x)=0$ for $x< 0$,
and $k >0$ is a constant.
Thus, we have that
\begin{eqnarr*}
\lQD \mbiggl[ \int_{0}^\infty \mBigl(\lambda(r)+1+\frac{1}{\lambda(r)}\mr) e^{-p \lambda(r)} \, \dd r \mr]
&=& \int_{[0,\infty)} \mBigl(y+1+\frac{1}{y}\mr) e^{-py} \, V(a,\dd y) \\
&=& k \int_0^\infty \frac{y}{a}\mBigl(y+1+\frac{1}{y}\mr) e^{-py} (\WW(y)-\WW(y-a)) \, \dd y.
\end{eqnarr*}
Finally, by \cite[equation (VII.4)]{Ber-Levy}
and the renewal theorem \cite[Theorem III.21]{Ber-Levy}, we know that
$\WW(y)-\WW(y-a) \to ac/{m_+}$ as $y \to \infty$, where $m_+$ is the mean of the ascending
ladder height process of $\lambda$ and $c$ is a meaningless constant.
This implies that the integral above converges at $\infty$.
We then note that the integrand is equivalent to $ k a^{-1} (y^2+y+1) \WW(0)$ as $y\to 0$,
and this completes the proof.
\end{proof}
\end{lem}
\begin{rem}
In \cite{BR-disc}, the result
\[
\inf_{t\ge 0} \lambda(t) > 0
\qquad \text{and}\qquad
\lim_{t\to \infty}
\frac{\log \lambda(t)}{\log t}
= \frac{1}{2} \qquad\quad \text{$\lQD$-almost surely},
\]
stated in \cite[equation (21)]{BR-disc},
is used. This would suffice for our purposes also.
However, since the proof of \autoref{l:lambda-int} is not very long,
we offer it for the sake of completeness.
\end{rem}
\skippar
We are now in a position to prove \autoref{l:MDa}.
\begin{proof} [Proof of \autoref{l:MDa}]
(i)~By a fundamental result in measure theory (see e.g. \cite[Corollary~1]{Ath-change}), it suffices to prove that
\[
\MD_a(\omega,\infty) = \infty, \quad \lQD\text{-a.s.}
\]
For $\lambda$ defined by \eqref{e:lambda} and every $t\geq 0$, it is clear that
\[\MD_a(\omega,t) \geq \lambda(t) \exp \mbigl(a\omega+\omega \kappa'(\omega) t- \kappa(\omega)t + \omega\lambda(t)\mr) .\]
As $\omega > \ws$, there is $\kappa'(\omega)\omega > \kappa(\omega)$. Moreover, we know from \autoref{l:clambda} that $\inf_{t\ge 0} \lambda(t) > 0$. The claim follows as a consequence.
\bigskip
(ii)~Let us start with a useful estimate. Under assumption \ref{H}, we can choose $\epsilon>0$ small enough such that $\ws-\epsilon\in (\dom \kappa)^{\circ}$ and that $\ws-\epsilon>0$.
Then there is
\begin{equation}\label{e:h-dm}
\int_{\Pp} \mBigl( \sum_{i\ge 2} p_i^{\omega} \mr) \mBigl(\log_+\mbigl(\sum_{i\ge 2} p_i^{\omega}\mr) \mr)^{\rho} \, \nu(\dd \pp) <\infty, \qquad \text{for all}~ \omega\in [\ws-\epsilon, \ws], \rho\in[1,2].
\end{equation}
To prove \eqref{e:h-dm}, for any $\omega\in [\ws-\epsilon, \ws]$ and $\rho\in[1,2]$, we next choose $\gamma >0$ small enough, such that
$\omega + \rho \gamma (\omega -1) \in \dom \kappa$.
Using the inequality $\log_+ y \le \gamma^{-1} y^{\gamma}$ for all $y\ge 0$ and Jensen's inequality, we have
\[
\mBigl( \sum_{i\ge 2} p_i^{\omega} \mr) \mBigl(\log_+\mbigl(\sum_{i\ge 2} p_i^{\omega}\mr) \mr)^{\rho} \le \gamma^{-2} \Big( \sum_{i\ge 2} p_i p_i^{\omega-1} \Big)^{1+\rho \gamma}
\le \gamma^{-2} \sum_{i\ge 2} p_i^{\omega+\rho \gamma(\omega-1)}.
\]
Since $\omega + \rho \gamma (\omega -1) \in \dom \kappa$, by \eqref{eq:dom} we deduce \eqref{e:h-dm}.
\skippar
We now come back to the proof of the proposition.
By \cite[Lemma 4.2]{Shi-BRW} it suffices to show that
\begin{equation}\label{e:liminf}
\liminf_{t\to \infty} \lQDc \mbigl[ \MD_a(\ws,t) \mm\vert \GG_{\infty} \mr]<\infty, \quad \lQDc \text{-a.s.},
\end{equation}
where $\GG_{\infty}:= \sigma(\Zz_{U(t)}(t),U(t), t\geq 0) \subset \hat{\FF}_{\infty}$.
Recall that $\lQDc$ is related to $\lQc$ via the change of measure \eqref{e:cm-lambda}, and that
$\Zz$ under $\lQc$ can be described as a decorated spine process as in \autoref{d:decorated-spine}. With notation therein, we claim that
\begin{equation}\label{e:Q=P}
\lQDc\mbigl[\MD_a(\ws,t)\mm\vert \GG_{\infty} \mr]=S_t, \qquad \lQDc-a.s. ,
\end{equation}
where, with $\lambda(t)= a - \xi(t) + t\kappa'(\ws)$ and $\zeta = \inf\{ t\ge 0: \lambda(t) < 0 \}$,
\begin{multline}
S_t:= \lambda(t) e^{a\ws - \ws\lambda(t)}\Indic{t<\zeta} \\
+ \int_{[0,t]\times(0,1)\times\NN\times\Pp}
e^{a\ws - \ws\lambda(r-)} \Indic{r<\zeta} \sum_{j \ne i} (\lambda(r-)- \log p_j) p_j^{\bar{\omega}} \, N(\dd r,\dd y, \dd i, \dd \pp). \label{e:MD-g}
\end{multline}
We postpone for a moment the proof of \eqref{e:Q=P} and turn our attention to $S_t$.
Let
\[
X:= \sum_{i\ge 2} p_i^{\ws}, \quad \text{and}\quad \tilde{X} := \sum_{i\ge 2} p_i^{\ws-\epsilon}.
\]
Fix $\theta\in (0,\ws-\epsilon)$, let
\begin{eqnarr*}
A_t&:=& \lambda(t) e^{a\ws - \ws\lambda(t)}\Indic{t<\zeta},\\
B_t&:=& \int_{[0,t]\times(0,1)\times\NN\times\Pp}
e^{a\ws - \ws\lambda(r-)} \Indic{e^{\theta\lambda(r-)}>X} \Indic{e^{\theta\lambda(r-)}> \tilde{X}}
\sum_{j \ne i} (\lambda(r-)- \log p_j) p_j^{\bar{\omega}} \, N(\dd r,\dd y, \dd i, \dd \pp),\\
D_t&:=&\int_{[0,t]\times(0,1)\times\NN\times\Pp}
e^{a\ws - \ws\lambda(r-)} \Indic{e^{\theta\lambda(r-)}\le X} \Indic{i\ne 1}
\sum_{j \ne i} (\lambda(r-)- \log p_j) p_j^{\bar{\omega}} \, N(\dd r,\dd y, \dd i, \dd \pp),\\
D'_t&:=&\int_{[0,t]\times(0,1)\times\NN\times\Pp}
e^{a\ws - \ws\lambda(r-)} \Indic{e^{\theta\lambda(r-)}\le \tilde{X}} \Indic{i\ne 1}
\sum_{j \ne i} (\lambda(r-)- \log p_j) p_j^{\bar{\omega}} \, N(\dd r,\dd y, \dd i, \dd \pp).\\
E_t&:=&\int_{[0,t]\times(0,1)\times\NN\times\Pp}
e^{a\ws - \ws\lambda(r-)} \Indic{i= 1}
\sum_{j \ne i} (\lambda(r-)- \log p_j) p_j^{\bar{\omega}} \, N(\dd r,\dd y, \dd i, \dd \pp).
\end{eqnarr*}
Then it is clear that
\[
S_t \le A_t+B_t+D_t + D'_t + E_t.
\]
We shall study the asymptotics of the five terms separately.
\skippar
Let us start with $B_t$.
Using the compensator of $N$ under $\lQDc$ given in \autoref{l:cN} and \autoref{d:eta}, we deduce that
\[
\lQDc \mbigl[ B_t \mr]
= \int_{[0,t]} \lQDc \mBigl[e^{a\ws - \ws\lambda(r)}
\mbigl(C_1 \lambda(r)+ C_0 +C_{-1}\lambda(r)^{-1}\mr)\mr]\, \dd r,
\]
where $C_1$, $C_0$ and $C_{-1}$ are given by
\[
C_1 := \int_{\Pp\setminus \Pp_1} \Indic{e^{\theta\lambda(r)}>X} \Indic{e^{\theta\lambda(r)}> \tilde{X}} \mbigl( \sum_{i\ge 1}\sum_{j \neq i}p_i^{\ws}p_j^{\ws}\mr)\nu(\dd \pp),
\]
\[
C_0 := -\int_{\Pp\setminus \Pp_1} \Indic{e^{\theta\lambda(r)}>X} \Indic{e^{\theta\lambda(r)}> \tilde{X}} \mbigl( \sum_{i\ge 1}\sum_{j \neq i} p_i^{\ws}p_j^{\ws}(\log p_i + \log p_j) \mr)\nu(\dd \pp),
\]
\[
C_{-1} := \int_{\Pp\setminus \Pp_1} \Indic{e^{\theta\lambda(r)}>X} \Indic{e^{\theta\lambda(r)}> \tilde{X}} \mbigl( \sum_{i\ge 1}\sum_{j \neq i} p_i^{\ws}p_j^{\ws}\log p_i\log p_j\mr)\nu(\dd \pp).
\]
By the inequality $|\log y| \le \epsilon^{-1} y^{-\epsilon}$ for $y\in [0,1]$, there is
\[ \Indic{e^{\theta\lambda(r)}>\tilde{X}} \sum_{i\geq 1}\sum_{j\neq i}p_i^{\ws}p_j^{\ws}\log p_i\log p_j
\le \epsilon^{-2}\Indic{e^{\theta\lambda(r)}>\tilde{X}} \sum_{i\geq 1}\sum_{j\neq i}p_i^{\ws-\epsilon}p_j^{\ws-\epsilon}
\le
\epsilon^{-2} \Indic{e^{\theta\lambda(r)}>\tilde{X}} (\tilde{X}+2 p_1^{\ws-\epsilon}) \tilde{X}.
\]
We also note that $p_1^{\ws-\epsilon}\le 1 $.
It follows that
\[C_{-1}
\le
\epsilon^{-2} \mbigl(2+e^{\theta\lambda(r)} \mr) \int_{\Pp\setminus \Pp_1} \tilde{X} \, \nu(\dd \pp) .
\]
As $\ws-\epsilon\in \dom \kappa$, by \eqref{eq:dom} we have
$C_{-1}\le c_{-1} (e^{\theta\lambda(r)} + 1)$, with $c_{-1}>0$ a finite constant.
Similarly, we can prove that $C_0\le c_0 (e^{\theta\lambda(r)}+1)$ and $C_{1}\le c_{1} (e^{\theta\lambda(r)}+1)$, with $c_0,c_{1}>0$ finite constants.
Take $\bar{c}:= \max(c_1, c_0, c_{-1})>0$, then
\[
\lQDc \mbigl[ B_t \mr]
\leq \bar{c}e^{a\ws} \int_{[0,t]} \dd r \lQDc\mBigl[( e^{- (\ws-\theta)\lambda(r)} +e^{- \ws\lambda(r)})\Indic{r< \zeta } \mbigl(\lambda(r)+ 1 +\lambda(r)^{-1} \mr) \mr].
\]
By \autoref{l:lambda-int}, we deduce that
\[
\liminf_{t\to \infty} \lQDc[ B_t ]<\infty.
\]
Using similar arguments, that we omit for conciseness, we can deduce that $ \liminf_{t\to \infty} \lQDc [ E_t ]<\infty$.
By \autoref{l:lambda-int}, we also see that
$ \liminf_{t\to \infty}\lQDc [ A_t ]<\infty. $
Then Fatou's lemma yields
\[
\liminf_{t\to \infty} (A_t + B_t +E_t) <\infty, \qquad \qquad \lQDc-a.s.
\]
\skippar
We next estimate $D_t$. To this end, let us consider
\[
H_{\infty}:=\int_{[0,\infty)\times(0,1)\times\NN\times\Pp}
\Indic{e^{\theta\lambda(r-)}\le X}\Indic{i\ne 1} \, N(\dd r,\dd y, \dd i, \dd \pp).
\]
Using again \autoref{l:cN}, \autoref{d:eta} and the inequality $|\log y| \le \epsilon^{-1} y^{-\epsilon}$ for $y\in [0,1]$, we deduce that
\[
\lQDc \mbigl[ H_{\infty} \mr]
= \int_{\Pp} \mbiggl(
\int_{[0,\infty)} \lQDc \mBigl[
\Indic{e^{\theta\lambda(r)}\le X}
\mbigl( X + \lambda(r)^{-1} \epsilon^{-1}\tilde{X} \mr)
\mr] \, \dd r
\mr) \nu(\dd \pp).
\]
By similar arguments as in the proof of \autoref{l:lambda-int}, with notations therein, we obtain that
\[
\int_{[0,\infty)} \lQDc \mBigl[
\Indic{e^{\theta\lambda(r)}\le X} \mbigl( X + \lambda(r)^{-1} \epsilon^{-1}\tilde{X} \mr)
\mr] \, \dd r
= k \int_0^{\theta^{-1}\log_+ X} \frac{y}{a}\mbigl( X + y^{-1} \epsilon^{-1} \tilde{X} \mr) (\WW(y)-\WW(y-a)) \, \dd y.
\]
we know that
$\WW(y)-\WW(y-a) \to ac/{m_+}$ as $y \to \infty$, where $m_+$ is the mean of the ascending
ladder height process of $\lambda$ and $c$ is a meaningless constant.
This implies that, there exists a constant $C_3$ large enough, such that
\[
\lQDc \mbigl[ H_{\infty} \mr] \le
C_3 \int_{\Pp} \mBigl(
\theta^{-2} X ( \log_+ X )^2 + \theta^{-1} \tilde{X} \log_+ X
\mr) \, \nu(\dd \pp).
\]
Since $\tilde{X} \log_+ X \le (X \log_+ X + \tilde{X} \log_+ \tilde{X})$, by \eqref{e:h-dm} the right-hand-side of the above expression is finite.
Hence $H_{\infty}$ is $\lQDc$-a.s. finite, which yields that
$\sup_{t\ge 0} D_t<\infty$ holds $\lQDc$-a.s.
Indeed,
\[
\sup_{t\ge 0} D_t \le \int_{[0,\infty) \times(0,1)\times\NN\times\Pp}
e^{a\ws - \ws\lambda(r-)}
\Big| \sum_{j \ne i} (\lambda(r-)- \log p_j) p_j^{\bar{\omega}} \Big| \, \Indic{e^{\theta\lambda(r-)}\le X}\Indic{i\ne 1} N(\dd r,\dd y, \dd i, \dd \pp).\\
\]
The right-hand-side is an integral over a random point measure, whose total mass is $H_{\infty}$.
So the fact that $H_{\infty}$ is $\lQDc$-a.s. finite yields that the integral is $\lQDc$-a.s. a finite sum.
In the same manner, we can also deduce that $\sup_{t\ge 0} D'_t<\infty$ holds $\lQDc$-a.s.
This would require that $\int_{\Pp} X (\log_+ \tilde{X})^2 \, \nu(\dd \pp)<\infty$, which is again a consequence of \eqref{e:h-dm}.
Having assumed \eqref{e:Q=P}, this completes the proof of \eqref{e:liminf}.
\skippar
It remains to justify \eqref{e:Q=P}.
We first consider $\lEQc \mbigl[ \MD_a(\ws,t) \mm\vert \GG_{\infty} \mr]$.
Recall that $\Zz$ under $\lQc$ can be described as a decorated spine process as in \autoref{d:decorated-spine}. With notation therein, we have $\lambda(t)= a - \xi(t) + t\kappa'(\ws)$. Notice that each $\Zz_u(t)$ with $u\in \tree_{t}$ corresponds bijectively to a $\Zz^{[r,j]}_v(t-r)$ with $v\in \tree^{[r,j]}_{t-r}$, such that
\[
\Zz_u(t) = \Zz^{[r,j]}_v(t-r) + \xi(r-)+\log p_j = \Zz^{[r,j]}_v(t-r) - \lambda(r-)+ r\kappa'(\bar{\omega}) + a +\log p_j.
\]
Replacing $\Zz_u(t)$ in $\MD_a(\ws,t)$ by the right-hand-side of the identity above, conditioning to $\GG_{\infty}$ and using the fact that $\ws \kappa'(\ws) =\kappa(\ws)$, we have that
\begin{eqnarr*}
\eqnarrLHS{ \lEQc\mbigl[\MD_a(\ws,t)\mm\vert \GG_{\infty} \mr] -\lambda(t) e^{a\ws - \ws\lambda(t)}\Indic{t<\zeta}}
&=& \int_{[0,t]\times(0,1)\times\NN\times\Pp}
e^{a\ws - \ws\lambda(r-)} \Indic{r<\zeta} \sum_{j \ne i} \lEQc\mbigl[\MD^{[r,j]}(a^{[r,j]}, \ws,t-r)\mm\vert \GG_{\infty} \mr] p_j^{\bar{\omega}} \, N(\dd r,\dd y, \dd i, \dd \pp),
\end{eqnarr*}
where $\zeta := \inf\{s\geq 0: \lambda(s)\leq 0\}$, $a^{[r,j]}:=\lambda(r-)- \log p_j$, and $\MD^{[r,j]}(a^{[r,j]}, \ws,t-r)$ denotes the stopped derivative martingale of the branching L\'evy process $\Zz^{[r,j]}$, i.e.
\[\sum_{v\in \tree^{[r,j]}_{t-r}} \mbigl( a^{[r,j]} +(t-r) \kappa'(\bar\omega) -\Zz^{[r,j]}_v(t-r) \mr) e^{-(t-r) \kappa(\bar\omega) + \bar\omega\Zz^{[r,j]}_v(t-r)}\Indic{ -s \kappa'(\bar\omega) + \Zz^{[r,j]}_{ \Anc(s,v)}(s)<a^{[r,j]}, \forall s\in[0,t-r]}.
\]
By the independence of $\Zz^{[r,j]}$ and $\GG_{\infty}$, we have the identity \[\lEQc\mbigl[\MD^{[r,j]}(a^{[r,j]}, \ws,t-r)\mm\vert \GG_{\infty} \mr] = a^{[r,j]}=\lambda(r-)- \log p_j. \]
Summarizing, we have that $\lEQc \mbigl[\MD_a(\ws,t)\mm\vert \GG_{\infty} \mr]$ is equal to $S_t$ as in \eqref{e:MD-g}.
We can now prove \eqref{e:Q=P}.
For every $s>t$, let $\GG_{s}:= \sigma(\Zz_{U(r)}(r),U(r), r\in [0,s]) \subset \hat{\FF}_{s}$.
By the change of measure \eqref{e:cm-lambda}, for every $A\in \GG_{s}$ we have that
\begin{eqnarr*}
\lQDc \mbigl[\MD_a(\ws,t) \Ind_{A} \mr]
& =&a^{-1} \lEQc\mbigl[ \MD_a(\ws,t) \lambda(s)\Indic{s<\zeta}\Ind_{A} \mr]\\
&=&a^{-1}\lEQc\mBigl[ \lEQc\mbigl[\MD_a(\ws,t)\mm\vert \GG_{\infty} \mr]\lambda(s)\Indic{s<\zeta}\Ind_{A} \mr], \\
&=& a^{-1}\lEQc\mBigl[ S_t\lambda(s)\Indic{s<\zeta}\Ind_{A} \mr],
\end{eqnarr*}
where the second equality is due to the fact that $\lambda(s)\Indic{s<\zeta}$ is also $\GG_{\infty}$-measurable. Since \eqref{e:MD-g} shows that $S_t$ is $\hat{\FF}_t$-measurable, it is also $\hat{\FF}_s$-measurable for $s>t$. Using again the change of measure \eqref{e:cm-lambda}, we have
\[\lQDc\mbigl[\MD_a(\ws,t)\mm\vert \GG_{s} \mr]=S_t, \qquad \lQDc-a.s.\]
Letting $s\to \infty$, L\'evy's zero-one law leads to \eqref{e:Q=P}.
\end{proof}
\subsection{Proof of \autoref{t:DerMart}}\label{s:proofD}
We are now ready to prove \autoref{t:DerMart}. We tackle each part separately.
\bigskip
(i)
For every $a>0$, it is clear that $\MD(\omega,t)$ is equal to $a \MA(\omega, t)-\MD_a(\omega,t)$ for all $t\ge 0$ in the event
\[ B_{\omega, a}:= \mbigl\{ \sup_{t\geq 0}\mbigl(\sup_{u\in\tree_t} \Zz_u(t)- \kappa'(\omega) t \mr)<a \mr\}.\]
We know from \citet[Theorem 2.3]{Dadoun:agf} that for the additive martingale the following convergence holds $\lP$-almost surely:
\begin{equation}\label{e:am}
\lim_{t\to \infty} \mg{\omega}{t} = 0, \qquad \text{for any}~\omega\ge \ws.
\end{equation}
Then $\MD(\omega,t)$ converges to a finite non-positive limit
\begin{equation}\label{e:Ba}
\MD(\omega,\infty) := -\MD_a(\omega,\infty), \text{ in the event } B_{\omega,a}.
\end{equation}
On the other hand, since
\[
\sup_{u\in\tree_t} \Zz_u(t)- \omega^{-1}\kappa(\omega) t \leq\omega^{-1} \log \mg{\omega}{t}, \qquad \text{for every } t>0,
\]
and $\kappa'(\omega)\ge \omega^{-1}\kappa(\omega)$, letting $t\to \infty$, we deduce that
\begin{equation}\label{e:Z1}
\lim_{t\to \infty} \mbigl( \sup_{u\in\tree_t} \Zz_u(t)- \kappa'(\omega) t\mr)= -\infty,\qquad \lP-\text{almost surely}.
\end{equation}
Then $\lP \mbigl(\lim_{a\uparrow \infty} B_{\omega,a} \mr)= 1$ as a consequence. We hence conclude that $\MD(\omega,t)$ converges $\lP$-almost surely to a finite non-positive limit.
\bigskip
(ii)
As $\omega>\ws$, it follows from \eqref{e:Ba} and \autoref{l:MDa} that $\MD(\omega,\infty)=0$ in $B_{\omega,a}$ for every $a>0$. Since $\lP \mbigl(\lim_{a\uparrow \infty} B_{\omega,a} \mr)= 1$, we deduce that $\MD(\omega,\infty) = 0$ holds $\lP$-almost surely.
\bigskip
(iii)
For every $a>0$, we observe from \eqref{e:Z1} that $\lP$-almost surely
\[
\liminf_{t\to \infty} \inf_{u\in \tree_t} \mBigl[\mbigl( a +t \kappa'(\ws) -\Zz_u(t) \mr) e^{-t \kappa(\ws) + \ws\Zz_u(t)} \mr]\geq 0,\]
which entails that $\lP$-almost surely
\[\lim_{t\to \infty} (a\MA(\ws,t) - \MD(\ws,t)) \geq\lim_{t\to \infty} \MD_a(\ws,t).\]
We hence deduce from \eqref{e:am} and \autoref{l:MDa} that
\[ \lE[\MD(\ws,\infty)] \leq \lE[-\MD_a(\ws,\infty)] = -a. \]
Since $a>0$ is arbitrary, we readily have $ \lE[\MD(\ws,\infty)] = -\infty$.
It remains to prove that $\MD(\ws,\infty) <0$, $\lP^*$-almost surely. The following arguments are modified from the proof of Proposition 8~(iii) in \cite{BR-disc}. For every $v \in \tree$ and $t\geq 0$, denote
\[\MD^{(v)}(\ws,t):= e^{-t\kappa(\ws)}
\sum_{u\in \tree_{t+1}, v \preceq u, b_u > 1} \mbigl(\Zz_u(t+1) -\Zz_v(1) - t\kappa'(\ws)\mr) e^{\ws (\Zz_u(t+1) - \Zz_v(1))}
\]
and
\[\MA^{(v)}(\ws,t):= e^{-t\kappa(\ws)}\sum_{u\in \tree_{t+1}, v \preceq u, b_u > 1} e^{\ws (\Zz_u(t+1)-\Zz_v(1))},\]
with convention $\MD^{(v)}(\ws,t)= \MA^{(v)}(\ws,t) = 0$ whenever $v\not\in \tree_1$.
Then we have the following decomposition:
\begin{equation}\label{e:t+1}
\MD(\ws,t+1)= e^{-\kappa(\ws)}\mbiggl( \sum_{v\in \tree_1} e^{\ws \Zz_v(1) }\MD^{(v)}(\ws,t)
+ \sum_{v\in \tree_1} e^{\ws \Zz_v(1) }(\Zz_v(1) -\kappa'(\ws) ) \MA^{(v)}(\ws,t) \mr).
\end{equation}
Let us start with proving the following technical result:
\begin{equation}\label{e:cvto0}
\sum_{v\in \tree_1} e^{\ws \Zz_v(1) }(\Zz_v(1) -\kappa'(\ws) ) \MA^{(v)}(\ws,t) \underset{t\to \infty}{\longrightarrow} 0 \quad \text{ in probability with respect to }\lP.
\end{equation}
Let $c>0$ be small enough such that $\ws-c\in \dom \kappa$. One observes that $|\log y| \leq c^{-1} (y^{-c} + y^{c})$ for every $y>0$, then
\begin{multline*}
\lE \mbiggl[ \sum_{u\in \tree_1} |\Zz_u(1)-\kappa'(\ws)| e^{\ws \Zz_u(1)} \mr]
\le c^{-1}\lE \mbiggl[ \mBigl(\sum_{u\in \tree_1} \mbigl( e^{(\ws+c) \Zz_u(1)} + e^{(\ws-c) \Zz_u(1)}+ |\kappa'(\ws)| e^{\ws \Zz_u(1)} \mr) \mr) \mr].
\end{multline*}
The second expectation is finite, so is the first one.
Fix an enumeration of $\tree$ and denote for every $u\in \tree$ its index by $I_u\in \NN$. Then for every $\epsilon, \delta>0$, there exists $n_0\in \NN$, depending on $\epsilon$ and $\delta$, such that
\[ \lE \mbiggl[ \sum_{v\in \tree_1, I_v>n_0} |\Zz_v(1)-\kappa'(\ws)| e^{\ws \Zz_v(1)} \mr] \leq \delta \epsilon.
\]
Furthermore, by conditioning on $\FFs_1$ and using the branching property, \autoref{l:Zbar-bp}, we deduce the identity
\[ \lE \mbiggl[ \sum_{v\in \tree_1, I_v>n_0} |\Zz_v(1)-\kappa'(\ws)| e^{\ws \Zz_v(1)} \MA^{(v)}(\ws,t) \mr]= \lE \mbiggl[ \sum_{v\in \tree_1, I_v>n_0} |\Zz_v(1)-\kappa'(\ws)| e^{\ws \Zz_v(1)} \mr].
\]
Then an application of Markov inequality yields
\[ \lP \mbiggl[ \sum_{v\in \tree_1, I_v>n_0} |\Zz_v(1)-\kappa'(\ws)| e^{\ws \Zz_v(1)} \MA^{(v)}(\ws,t) > \delta \mr] \leq \epsilon.
\]
Moreover, we see from \eqref{e:am} that each $\MA^{(v)}(\ws,t)$ converges $\lP$-almost surely to $0$. It follows that
\[ \sum_{v\in \tree_1, I_v\leq n_0} |\Zz_v(1)-\kappa'(\ws)| e^{\ws \Zz_v(1)}\MA^{(v)}(\ws,t) \underset{t\to\infty}{\longrightarrow} 0, \qquad \lP\text{-a.s. }
\]
Hence we have proved \eqref{e:cvto0}.
We now go back to \eqref{e:t+1}. By the branching property, $(\MD^{(v)}(\ws,\cdot), v\in \tree_1)$ are independent copies of $\MD(\ws, \cdot)$, also independent of $\FFs_1$. Then we see from part~(i) that each $\MD^{(v)}(\ws,t)$ converges $\lP$-almost surely to a non-positive limit $\MD^{(v)}(\ws,\infty)$, which has the same law as $\MD(\ws,\infty)$. Letting $t\to \infty$ in \eqref{e:t+1} and using \eqref{e:cvto0}, we deduce that, for every $u\in \tree$,
\[
\MD(\ws,\infty)= \Indic{u\in \tree_1} e^{-\kappa(\ws)}e^{\ws Z_{u}(1) }\MD^{(u)}(\ws,\infty)+R,\qquad \lP\text{-a.s.},
\]
where \[R := \lim_{t\to \infty}e^{-\kappa(\ws)}\sum_{v\in \tree_1, v\ne u} e^{\ws \Zz_v(1) } \MD^{(v)}(\ws,t), \quad \text{ in probability with respect to }\lP.
\]
Since $\MD^{(u)}(\ws,\infty)$ is independent of $R$, the above identity entails that
\[
\lP \mbigl( \MD(\ws,\infty) > 0 \mr)
\geq \rho \cdot \lP \mbigl( R >0\mr) ,
\]
where $\rho:= \lP \mbigl( \MD(\ws,\infty) = 0 \mr) = \lP \mbigl( \MD^{(u)}(\ws,\infty) = 0 \mr)$.
Recall from part~(i) that $\MD(\ws,\infty)$ is non-positive, i.e. $\lP \mbigl( \MD(\ws,\infty) > 0 \mr) =0$. Suppose now that
$\rho >0$ (otherwise there is nothing to prove), then $\lP \mbigl( R > 0 \mr) = 0$. It follows that
\[
\MD(\ws,\infty)\leq \Indic{u\in \tree_1} e^{-\kappa(\ws)}e^{\ws Z_{u}(1) }\MD^{(u)}(\ws,\infty) ,\qquad \lP\text{-a.s.}
\]
This inequality holds for every $u\in \tree$, we hence deduce by the independence of the family $\big( \MD^{(u)}(\ws,\infty), u\in \tree\big)$ that
\begin{equation}\label{e:extinct}
\rho \leq \lE\mbigl[ \rho^{\# \tree_1}\mr] , \end{equation}
with $\# \tree_1$ the number of particles at time $1$. It has been proved in (ii) that
$\lE \mbigl[ \MD(\ws,\infty) \mr] = -\infty$, so we also have $\rho <1$.
Hence \eqref{e:extinct} entails that $\rho =\lP \mbigl( \MD(\ws,\infty) = 0 \mr)$ is smaller or equal to the extinction probability.
On the other hand, it is clear that the extinction event is included in $\{\MD(\ws,\infty) = 0\}$.
We conclude that $\MD(\ws,\infty) < 0$, $\lP^*$-almost surely.
This completes the proof.
\section*{Acknowledgements}
We would like to thank Jean Bertoin and Andreas E. Kyprianou for their helpful comments
on earlier drafts of this work.
Q.S.\ was supported by math--STIC of Paris University 13 and the SNSF fellowship P2ZHP2\_171955.
|
3,212,635,537,916 | arxiv | \section{Introduction}
In this paper, we consider the one-dimensional discrete Schr\"odinger operator:
$$
(Hu)(n) = u(n+1)+u(n-1) + V(n)u(n)\quad\mbox{on $l^2(\mathbb{Z})$},
$$
where $V(n)$ is a complex-valued periodic potential with period $L$
containing only one nonzero value $v$ within a period.
As the period $L$ increases, this potential becomes sparse, which we call the sparse potential.
If $v$ is real, $H$ is a bounded self-adjoint operator.
For real $v$, the spectrum $\sigma(H)$ of $H$ is purely absolutely continuous and consists of at most $L$ closed intervals on the real axis.
These results are proved in general dimension using the method of direct integral decomposition (see, e.g., Reed-Simon~\cite{RS4}) following Gel'fand~\cite{Gelfand}.
It describes the behavior of electrons or holes in a one-dimensional crystal (e.g., Kittel~\cite{Kittel}).
Each closed interval of the spectrum is called a {\it spectral band}.
In condensed matter physics, it is essential to determine the location of the bands, especially the band-to-band gap ({\it band gap}).
Estimating the band gap is applied to study the electrical properties of semiconductors.
Even if $v$ is not real, $H$ is still a bounded operator.
Thus, while its spectrum $\sigma(H)$ is a compact set on the complex plane, its shape often becomes more complicated.
In the case of self-adjoint operators, a nice theory, including the spectral decomposition theorem,
can be applied.
On the other hand, for non-self-adjoint cases, no such nice general theory exists and
must be analyzed on a problem-by-problem basis.
Because of this inconvenience, the spectral theory for complex-valued potential literature has been relatively few.
Non-self-adjoint Schr\"odinger-type operators (non-Hermitian Hamiltonian in physics)
have emerged naturally in $\mathcal{PT}$-symmetric (parity-time symmetric) quantum theory (see, e.g., Bender~\cite{Bender}),
providing a strong incentive for their study.
For differential operators with complex-valued periodic potentials, Valiev~\cite{OV}
is an excellent guide for researchers in this field.
In the discrete (Jacobi matrix) case, a theory similar to that of differential operators can be constructed.
Moreover, the direct and inverse spectral theory for more general Jacobi matrices with complex periodic coefficients has also been obtained (Hochstadt~\cite{Hochstadt}, Papanicolaou~\cite{PapaJOP}).
An example of the significant difference between complex-valued periodic potentials and real-valued periodic potentials is given by Gasimov~\cite{Gasimov}(see also Guillemin and Uribe~\cite{GU}).
In continuous case, if $V(x)\in L^2_{\text{\rm loc}}(\mathbb{R})$ is real-valued, a famous theorem of Borg~\cite{Borg}, that is, $\sigma(-\frac{d^2}{dx^2}+V)=[0, \infty)$ if and only if $V(x)=0$ a.e..
In the case of complex-valued periodic potentials,
the results are very different: Gasimov~\cite{Gasimov} showed that if
$$
V(x) = \sum_{k=1}^{\infty}c_ke^{ikx},\quad \sum_{k=1}^{\infty}|c_k|<\infty,
$$
then $\sigma(-\frac{d^2}{dx^2}+V)=[0, \infty)$.
Papanicolaou~\cite{PapaJOP} showed a discrete version of this Gasimov's theorem.
The discriminant (defined in Section \ref{sec:discriminant}) determines the spectrum of the one-dimensional Schr\"odinger operator with periodic potential.
The discriminant is a polynomial in $E$ with the period of the potential as its degree.
Papanicolaou~\cite{PapaJOP} proved there exists at most $L!$ different $L$-periodic potentials whose discriminants are the same. As a result, the spectra of these operators coincide.
Papanicolaou~\cite{PapaJOP} used the fact that the coefficients of the discriminant are represented by the elementary symmetric polynomials of $V(1), V(2), \ldots, V(L)$ to show this result.
This representation is suitable for inverse problems but not for the perturbation theory of discriminant because of the difficulty in obtaining information about the value of the discriminant.
The most fundamental problem in the study of operators is determining the operator's spectrum.
As described in Section \ref{sec:discriminant}, the problem of determining the spectrum for a periodic potential with period $L$ is equivalent to the problem of finding the intersection of two algebraic curves of degree $L$. It becomes more difficult as $L$ increases; thus, a few examples have been studied in detail. The sparse potentials treated in this paper can be analyzed even for large $L$.
This paper presents a Chebyshev polynomial~(defined in Section \ref{sec:Chebyshev}) representation of the discriminant, which is then applied to the spectrum analysis.
This paper is organized as follows.
Section 2 defines the discriminant, states that the discriminant describes the spectrum of $H$, and introduces the Floquet spectrum.
Section 3 presents the Chebyshev polynomial representation of the discriminant, which is the main theorem of this paper, after introducing Chebyshev polynomials and listing their properties. Furthermore, we apply this theorem to derive two properties concerning integrals in $[-2, 2]$ of the discriminant.
Section 4 derives the first-order Taylor polynomials of the discriminant using the main theorem.
Section 5 shows that exactly $L$ spectral bands appear for nonzero real $v$ and that the band outside $[-2, 2]$ converges to a point as $L$ is large, applying the result of Section 4.
Section 6 studies the Floquet spectrum by perturbation method for $v\in\mathbb{C}$ with small and large $|v|$.
\section{Discriminant and Spectrum of $H$}\label{sec:discriminant}
The spectrum of the one-dimensional discrete Schr\"odinger operator with
periodic potential is represented by the {\it Hill discriminant}.
For every $E\in\mathbb{C}$, the equation $Hu = Eu$ can be uniquely solved by giving initial values $u(0)$ and $u(1)$.
Precisely, the solution $u(n)$ can be represented as
$$
\left(\begin{array}{c}
u(n+1) \\
u(n)
\end{array}
\right) = \Phi_n(E)\left(\begin{array}{c}
u(1) \\
u(0)
\end{array}
\right),
$$
where
$$
\Phi_n(E) =
\left\{\begin{array}{ll}
A_n(E)\cdots A_1(E) & (n\geq 1) \\
I & (n=0) \\
A_{n+1}(E)^{-1}\cdots A_0(E)^{-1} & (n\leq -1)
\end{array}\right.
$$
and
$$
A_n(E) = \left(\begin{array}{cc}
E - V(n) & - 1 \\
1 & 0
\end{array}\right).
$$
Note that $\det\Phi_n(E) = 1$ since $\det A_n(E) = 1$.
The spectrum $\sigma(H)$ of $H$ can be characterized as
$$
\sigma(H) = \{E\in\mathbb{C}: \Delta_L(E) \in [-2, 2] \},
$$
where $\Delta_L(E) = \text{\rm tr}\Phi_L(E)$ is the Hill discriminant (``discriminant'' for short) of $H$.
Note that $\Delta_L(E)$ is a monic polynomial.
This result was proved by Rofe-Beketov~\cite{Rofe} for continuous Schr\"odinger operators with complex coefficients, but it can also be established for the discrete case.
We note that in the case $v=0$, i.e., free Laplacian $H_0$, $\Delta_L(E) = 2T_L(E/2)$ for the first kind Chebyshev polynomial $T_L$ of degree $L$~(described in Section \ref{sec:Chebyshev}).
In this case, $\Delta_L(E) = 2\cos (L\cos^{-1}(E/2))$ by the definition of the Chebyshev polynomial, which shows that $\sigma(H_0) = [-2, 2]$.
We introduce the {\it Floquet spectrum}
$$
\sigma_{\kappa}(H) = \{E\in\mathbb{C}: \Delta_L(E) = 2\cos\kappa \}
$$
for $\kappa\in [0, \pi]$, then $\sigma(H)$ can be written as the disjoint union
\begin{equation}\label{eq:union}
\sigma(H) = \bigcup_{\kappa\in [0, \pi]}\sigma_{\kappa}(H).
\end{equation}
$\Delta_L(E) = 2\cos\kappa$ is equivalent to $\Phi_L(E)$ having eigenvalues $e^{i\kappa}, e^{-i\kappa}$.
Since the discriminant $\Delta_L(E)$ is a polynomial of degree $L$,
$\Delta_L(E)-2\cos\kappa$ has $L$ roots $E_1(\kappa), \ldots, E_L(\kappa)$ with multiplicity(these are called $\kappa$-{\it Floquet eigenvalues}),
each $E_j(\kappa)$ of which is continuous with respect to $\kappa$.
Therefore, (\ref{eq:union}) leads that $\sigma(H)$ consists of $L$ closed bounded analytic arcs lying in the complex plane at most.
\begin{figure}[htbp]
\centering
\includegraphics[width=100mm]{L3a2explain.png}
\caption{The spectrum for $L=3$ and $v=2i$}
\label{fig:L3a2explain}
\end{figure}
\begin{example}\label{ex:L3}
By setting $E=x+iy~(x, y)\in\mathbb{R}^2$,
the Floquet spectrum $\sigma_{\kappa}(H)$ can be represented as the intersection of the curves
$\Re\Delta_L(E) = \Re\Delta_L(x+iy) = 2\cos\kappa$ and $\Im\Delta_L(E) = \Im\Delta_L(x+iy) = 0$.
These two curves are real zeros of polynomials of total degree $L$ in $x$ and $y$.
The two curves represented by the real and imaginary parts of a holomorphic function on the complex plane intersect perpendicularly.
Therefore, $\Re\Delta_L(x+iy) = 2\cos\kappa$ and $\Im\Delta_L(x+iy)=0$ have perpendicular intersections.
For example, consider the case $L=3$ and $v=2i$. In this case,
$\Re\Delta_3(x+iy) = x^3 - 3xy^2 - 3x + 4xy$ and
$\Im\Delta_3(x+iy) = 3x^2y-y^3-3y - 2(x^2-y^2-1)$.
In Figure \ref{fig:L3a2explain}, the dashed line represents $\Re\Delta_3(x+iy) = \pm 2$
and the dotted line represents $\Im\Delta_3(x+iy) = 0$.
The part of the dotted line that is solid is the arc of the spectrum,
and the three curves that intersect these curves are $\Re\Delta_3(x+iy) = 0.8~(\kappa=\cos^{-1}0.4)$.
These intersections (three filled circles in Figure \ref{fig:L3a2explain}) correspond to the Floquet spectrum $\sigma_{\cos^{-1}0.4}(H)$.
\end{example}
\begin{rem}
It is easy to verify that the spectrum is symmetric about the imaginary axis for pure imaginary $v$.
\end{rem}
\section{Representation of the discriminant using Chebyshev polynomials}\label{sec:Chebyshev}
In this section, we list several formulas for the Chebyshev polynomials (see, e.g., Mason-Handscomb~\cite{Chabyshev}) to be used later,
and represent the discriminant $\Delta_L(E)$ using Chebyshev polynomials.
\begin{defn}\label{defn:Chebyshev}
The Chebyshev polynomials of the first kind $T_n(x)$ are defined by
$T_{n}(\cos\theta)=\cos(n\theta)~(n\geq 0)$.
Similarly, define the Chebyshev polynomials of the second kind $U_{n}(x)$ are defined by
$U_{n}(\cos\theta)\sin\theta=\sin(n+1)\theta~(n\geq 0)$.
\end{defn}
It is easy to verify that $T_n(-x) = (-1)^nT_n(x)$ and $U_n(-x) = (-1)^nU_n(x)$.
The first few Chebyshev polynomials of the first and second kind are $T_0(x)=1$, $T_1(x)=x$, $T_2(x)=2x^2-1$, $T_3(x)=4x^3-3x$, $U_0(x)=1$, $U_1(x)=2x$, $U_2(x)=4x^2-1$, $U_3(x)=8x^3-4x$.
It is also easily seen that
\begin{eqnarray*}
T_n(\cosh\xi) &=& \cosh n\xi, \\
U_n(\cosh\xi) &=& \frac{\sinh(n+1)\xi}{\sinh\xi}.
\end{eqnarray*}
\begin{prop}\label{prop:identity}
The Chebyshev polynomials satisfy the following recursive relations:
\begin{eqnarray*}
T_{n+1}(x) &=& 2xT_n(x) - T_{n-1}(x) ~(n\geq 1), \\
U_{n+1}(x) &=& 2xU_n(x) - U_{n-1}(x) ~(n\geq 1), \\
T_n(x) &=& \frac{1}{2}(U_n(x)-U_{n-2}(x)) ~(n\geq 2).
\end{eqnarray*}
\end{prop}
\begin{prop}\label{prop:identity-diff}
The following formulas hold for the derivative of Chebyshev polynomials.
\begin{eqnarray*}
T'_n(x) &=& nU_{n-1}(x) ~(n\geq 1), \\
U'_n(x) &=& \frac{(n+1)T_{n+1}(x) - xU_n(x)}{x^2-1} ~(n\geq 0)
\end{eqnarray*}
\end{prop}
\begin{prop}\label{prop:orthogonality}
Both $\{T_n(x)\}_{n=0}^{\infty}$ and $\{U_n(x)\}_{n=0}^{\infty}$ form a sequence of orthogonal polynomials in the following sense:
\begin{eqnarray*}
\frac{2}{\pi}\int_{-1}^1T_n(x)T_m(x)\frac{1}{\sqrt{1-x^2}}dx &=& \delta_{nm} \\
\frac{2}{\pi}\int_{-1}^1U_n(x)U_m(x)\sqrt{1-x^2}dx &=& \delta_{nm},
\end{eqnarray*}
where $\delta_{nm}$ is the Kronecker Delta.
\end{prop}
\begin{prop}\label{prop:Chebyshevzeros}
\begin{eqnarray*}
T_n(x) &=& 2^{n-1}\prod_{k=1}^n\left(x-\cos\frac{(2k-1)\pi}{2n}\right), \\
U_n(x) &=& 2^{n}\prod_{k=1}^n\left(x-\cos\frac{k\pi}{n+1}\right)
\end{eqnarray*}
\end{prop}
\begin{lem}\label{lem:poermatrix}
Let $\Phi_0(E)$ be the matrix
$$
\Phi_0(E) =\left(\begin{array}{cc}
E & -1 \\
1 & 0
\end{array}
\right),
$$
then, $\text{\rm tr}(\Phi_0(E)^n) = 2T_n(E/2)$.
\end{lem}
\begin{proof}
We first consider the case of $\Phi_0(E)$ has eigenvalues $\lambda, \lambda^{-1}$ satisfying $\lambda^2\ne 1$.
By direct computation, we learn
\begin{equation}\label{eq:power}
\Phi_0(E)^n = \frac{\lambda^n-\lambda^{-n}}{\lambda-\lambda^{-1}}\Phi_0(E) - \frac{\lambda^{n-1}-\lambda^{-(n-1)}}{\lambda-\lambda^{-1}}I.
\end{equation}
Let $\lambda=e^{i\theta}$, i.e., $E=2\cos\theta$~($\theta$ need not be a real number).
Taking trace of both sides of (\ref{eq:power}), combining the second and the third identities in Proposition \ref{prop:identity} yields
\begin{eqnarray*}
\text{\rm tr}(\Phi_0(E)^n) &=& \frac{e^{in\theta}-e^{-in\theta}}{e^{i\theta}-e^{-i\theta}}E - 2\frac{e^{i(n-1)\theta}-e^{-i(n-1)\theta}}{e^{i\theta}-e^{-i\theta}} \\
&=& EU_{n-1}(E/2) - 2U_{n-2}(E/2) \\
&=& U_n(E/2) - U_{n-2}(E/2) = 2T_n(E/2).
\end{eqnarray*}
We consider the case $\lambda^2=1$, i.e., $E=\pm 2$.
In this case, $\lambda$ is a double root of the eigenpolynomial of $\Phi_0(E)$, then we have
\begin{equation}\label{eq:power2}
\Phi_0(E)^n = n\lambda^{n-1}\Phi_0(E) - (n-1)\lambda^nI.
\end{equation}
Taking trace of both sides of (\ref{eq:power2}), we have
$$
\text{\rm tr}(\Phi_0(E)^n) = n(\pm 1)^{n-1}(\pm 2) - 2(n-1)(\pm 1)^n.
$$
Here is the double sign in the same order.
Then, $\text{\rm tr}(\Phi_0(E)^n)=2$ for $E=2$, and $=2(-1)^n$ for $E=-2$, these are consistent with $2T_n(E/2)$.
\end{proof}
\noindent
Then, our main theorem is stated as follows:
\begin{thm}\label{thm:represent}
The discriminant $\Delta_L(E)$ can be represented as
\begin{equation}\label{eq:mainrepresentation}
\Delta_L(E) = 2T_L(E/2) - vU_{L-1}(E/2)
\end{equation}
for $L\geq 1$.
\end{thm}
\begin{proof}
From the cyclic invariance of the matrix trace, the discriminant $\Delta_L(E)$
equals the trace of the matrix given by
\begin{equation}\label{eq:basic}
\Phi_0(E)^{L-1}
\left(\begin{array}{cc}
E - v& -1 \\
1 & 0
\end{array}
\right)=\Phi_0(E)^L
-v\Phi_0(E)^{L-1}
\left(\begin{array}{cc}
1& 0 \\
0 & 0
\end{array}
\right).
\end{equation}
\noindent
For $L=1, 2$, by direct computation, we have
\begin{eqnarray*}
\Delta_1(E) &=& E - v = 2T_1(E/2) - vU_0(E/2), \\
\Delta_2(E) &=& E^2 - 2 - vE = 2T_2(E/2) - vU_1(E/2).
\end{eqnarray*}
Next, assume $L\geq 3$,
using (\ref{eq:power}), we have
$$
\Phi_0(E)^{L-1}
\left(\begin{array}{cc}
1& 0 \\
0 & 0
\end{array}
\right) =
\frac{\lambda^{L-1}-\lambda^{-(L-1)}}{\lambda-\lambda^{-1}}\left(\begin{array}{cc}
E & 0 \\
0 & 0
\end{array}
\right) - \frac{\lambda^{L-2}-\lambda^{-(L-2)}}{\lambda-\lambda^{-1}}\left(\begin{array}{cc}
1& 0 \\
0 & 0
\end{array}
\right).
$$
Taking trace in both sides shows that
\begin{eqnarray}
&& \text{\rm tr}\left\{\Phi_0(E)^{L-1}
\left(\begin{array}{cc}
1& 0 \\
0 & 0
\end{array}
\right)\right\} \nonumber \\
&=& \frac{\lambda^{L-1}-\lambda^{-(L-1)}}{\lambda-\lambda^{-1}}E-\frac{\lambda^{L-2}-\lambda^{-(L-2)}}{\lambda-\lambda^{-1}} \nonumber \\
&=& EU_{L-2}(E/2) - U_{L-3}(E/2) = U_{L-1}(E/2). \label{eq:U}
\end{eqnarray}
From Lemma \ref{lem:poermatrix}, we learn that $\text{\rm tr}(\Phi_0(E)^L) = 2T_L(E/2)$.
Substituting $\text{\rm tr}(\Phi_0(E)^L) = 2T_L(E/2)$ and (\ref{eq:U}) into (\ref{eq:basic}),
we have
\begin{eqnarray*}
\Delta_L(E) &=& \text{\rm tr}(\Phi_L(E))-v\text{\rm tr}\left\{\Phi_0(E)^{L-1}
\left(\begin{array}{cc}
1& 0 \\
0 & 0
\end{array}
\right)\right\} \\
&=& 2T_L(E/2) - vU_{L-1}(E/2).
\end{eqnarray*}
\end{proof}
\begin{cor}\label{cor:zeros}
Let $\beta_{k}$ be the $k$-th root of $U_{L-1}(E/2)$ is given by
$$
\beta_{k} = 2\cos\frac{k\pi}{L}\quad (k=1, 2, \ldots, L-1).
$$
Then, $\{\beta_1, \beta_2, \ldots, \beta_{L-1}\}\subset\sigma(H)$. Moreover,
$2\in\sigma(H)$ if and only if $v\in [0, 4/L]$, also, $-2\in\sigma(H)$ if and only if $v\in[-4/L, 0]$.
In particular, $\pm 2\not\in\sigma(H)$ if $v\not\in\mathbb{R}$.
\end{cor}
\begin{proof}
Substituting $\beta_{k}$ into (\ref{eq:mainrepresentation}) yields
$\Delta_L(\beta_{k}) = 2\cdot (-1)^{k}\in [-2, 2]$.
Furthermore, substituting $E = \pm 2$ into (\ref{eq:mainrepresentation}) yields
$\Delta_L(2) = 2 - Lv$ and $\Delta_L(-2) = (-1)^L(2 + Lv)$.
$2\in\sigma(H)$ if and only if $\Delta_L(2) = 2 - Lv\in[-2, 2]$, which means that $v\in [0, 4/L]$.
Similarly, it can be shown that $-2\in\sigma(H)$ and that $v\in[-4/L, 0]$ is equivalent.
\end{proof}
Corollary \ref{cor:zeros} shows that the roots $\beta_1, \beta_2, \ldots, \beta_{L-1}$ of $\Delta_L(E)$
that corresponding to one of the endpoints of each spectral arc is concentrated at endpoints $E = \pm 2$
of the spectrum of the free Laplacian as $L$ is large,
and their density is approximately $\displaystyle\frac{1}{\pi\sqrt{4-E^2}}$ on the real axis.
\begin{cor}\label{rem:osc}
\begin{equation}\label{eq: integral}
\int_{-2}^2\Delta_L(E)dE = \left\{\begin{array}{cl}
-\frac{4}{L^2-1} & \mbox{$L$: even} \\
-\frac{2v}{L} & \mbox{$L$: odd}.
\end{array}
\right.
\end{equation}
\end{cor}
\begin{proof}
Using (\ref{eq:mainrepresentation}) and the definition of the Chebyshev polynomial, and
substituting $E=2\cos\theta$, the integral then becomes
\begin{eqnarray*}
\int_{-2}^2\Delta_L(E)dE &=& \int_{-2}^2(2T_L(E/2) - vU_{L-1}(E/2))dE \\
&=& \int_0^{\pi}(2\cos L\theta\sin\theta - v\sin L\theta)d\theta.
\end{eqnarray*}
This integral can be easily computed to obtain the desired result.
\end{proof}
This integral (\ref{eq: integral}) approaches $0$ as $L$ is large, which means that $\Delta_L(E)$ oscillates intensively within
$[-2, 2]$ as large $L$.
\begin{thm}\label{thm:Parseval}
\begin{eqnarray*}
\frac{1}{2\pi}\int_{-2}^2|\Delta_L(E)|^2\sqrt{4-E^2}dE &=& 2 + |v|^2 \quad(L\geq 2) \\
\frac{1}{2\pi}\int_{-2}^2|\Delta_1(E)|^2\sqrt{4-E^2}dE &=& 1 + |v|^2
\end{eqnarray*}
\end{thm}
\begin{proof}
For $L=1$, direct computation of the integral yields
\begin{eqnarray*}
&& \frac{1}{2\pi}\int_{-2}^2|\Delta_1(E)|^2\sqrt{4-E^2}dE \\
&=& \frac{1}{2\pi}\int_{-2}^2|E-v|^2\sqrt{4-E^2}dE \\
&=& \frac{1}{2\pi}\int_{-2}^2(E^2-(v+\overline{v})E + |v|^2)\sqrt{4-E^2}dE \\
&=& \frac{1}{2\pi}\int_{-2}^2(E^2 + |v|^2)\sqrt{4-E^2}dE = 1 + |v|^2.
\end{eqnarray*}
We consider the case $L\geq 2$.
Theorem \ref{thm:represent} and the third identity in Proposition \ref{prop:identity} together shows that
\begin{eqnarray*}
\Delta_L(E) &=& 2T_L(E/2) - vU_{L-1}(E/2) \\
&=& U_L(E/2)-U_{L-2}(E/2) - vU_{L-1}(E/2).
\end{eqnarray*}
Using Proposition \ref{prop:orthogonality} and Parseval's identity, one can show
\begin{eqnarray*}
&& \frac{1}{2\pi}\int_{-2}^2|\Delta_L(E)|^2\sqrt{4-E^2}dE \\
&=& \frac{1}{2\pi}\int_{-2}^2|U_L(E/2)|^2\sqrt{4-E^2}dE + \frac{1}{2\pi}\int_{-2}^2|U_{L-2}(E/2)|^2\sqrt{4-E^2}dE \\
&& + |v|^2\frac{1}{2\pi}\int_{-2}^2|U_{L-1}(E/2)|^2\sqrt{4-E^2}dE= 2 + |v|^2.
\end{eqnarray*}
\end{proof}
The integral in Theorem \ref{thm:Parseval} does not depend on $L(\geq 2)$, but only on $v$.
Theorem \ref{thm:Parseval} that the value of the weight $\sqrt{4-E^2}$
becomes smaller near $\pm 2$ indicates that as $L$ becomes larger,
$|\Delta_L(E)|$ becomes larger near $\pm 2$.
From this fact and Corollary \ref{rem:osc}, we learn that $\Delta_L(E)$ oscillates intensively near $\pm 2$ for large $L$.
\section{The first-order Taylor approximation of the discriminant}
In this section, we give the first-order Taylor polynomials at $\alpha_j~(j=1, 2, \ldots, L)$,
$\beta_{k}~(k=1, 2, \ldots, L-1)$, and $\pm 2$ on the real axis of the discriminant for application to the analysis of the spectrum of $H$ near $[-2, 2]$.
\begin{lem}\label{lem:Taylor}
The first-order Taylor polynomials at $\alpha_j~(j=1, 2, \ldots, L)$,
$\beta_{k}~(k=1, 2, \ldots, L-1)$, and $\pm 2$ are given by:
{\small
\begin{eqnarray}
\Delta_L(E) &=& \frac{v(-1)^j}{\sin\theta_j} + \frac{2L\sin^2\theta_j-v\cos\theta_j}{2\sin^3\theta_j}(E-\alpha_j) + O((E-\alpha_j)^2) \label{eq:alpha}\\
\Delta_L(E) &=& 2(-1)^{k} + \frac{v(-1)^{k}}{2\sin^3\phi_{k}}(E-\beta_{k}) + O((E-\beta_{k})^2), \label{eq:beta} \\
\Delta_L(E) &=& 2-vL + \left\{L^2-\frac{v}{4}L(L^2-1)\right\}(E-2) + O((E-2)^2) \label{eq:2}\\
\Delta_L(E) &=& (-1)^L(2+vL) + (-1)^{L}\left\{L^2+\frac{v}{4}L(L^2-1)\right\}(E+2) \nonumber \\
&& + O((E+2)^2),\label{eq:-2}
\end{eqnarray}
}
\noindent
where $\alpha_j = 2\cos\theta_j = 2\cos\frac{(2j-1)\pi}{2L}$ and $\beta_{k} = 2\cos\phi_{k}=2\cos\frac{k\pi}{L}$.
\end{lem}
\begin{proof} Taking derivative of (\ref{eq:mainrepresentation}) and Proposition \ref{prop:identity-diff} together shows that
\begin{eqnarray}
\Delta_L'(E) &=& T_L'(E/2) - \frac{v}{2}U_{L-1}'(E/2) \nonumber \\
&=& LU_{L-1}(E/2) - \frac{v}{2}\cdot\frac{LT_L(E/2) - \frac{E}{2}U_{L-1}(E/2)}{(E/2)^2-1} \nonumber \\
&=& LU_{L-1}(E/2) - v\frac{2LT_L(E/2)-EU_{L-1}(E/2)}{E^2-4}.\label{eq: ChevDeri2}
\end{eqnarray}
Since $\alpha_j$ satisfies $T_L(\alpha_j/2)=0$ and $\beta_{k}$ satisfies $U_{L-1}(\beta_{k}/2)=0$,
substituting $\alpha_j$ and $\beta_{k}$ into (\ref{eq: ChevDeri2}) yields (\ref{eq:alpha}) and (\ref{eq:beta}).
Substituting $E = 2\cos\theta~(0<\theta<\pi)$ into (\ref{eq: ChevDeri2}), we have
\begin{eqnarray}
\Delta_L'(2\cos\theta) &=& LU_{L-1}(\cos\theta) - v\frac{2LT_L(\cos\theta)-2\cos\theta U_{L-1}(\cos\theta)}{4\cos^2\theta-4} \nonumber \\
&=& L\frac{\sin L\theta}{\sin\theta} + v\frac{2L\cos L\theta - 2\cos\theta\frac{\sin L\theta}{\sin\theta}}{4\sin^2\theta} \nonumber \\
&=& L\frac{\sin L\theta}{\sin\theta} + v\frac{L\sin\theta\cos L\theta - \cos\theta\sin L\theta}{2\sin^3\theta}. \label{eq:limit2}
\end{eqnarray}
Taking limit $\theta\to 0$ of both sides of (\ref{eq:limit2}), we have
\begin{equation}\label{eq:der2}
\Delta_L'(2) = L^2 -\frac{v}{4}L(L^2-1).
\end{equation}
Combining (\ref{eq:der2}) and $\Delta_L(2)=2-vL$ yields (\ref{eq:2}).
Using
\begin{eqnarray*}
\Delta_L(-E) &=& 2T_L(-E/2)-vU_{L-1}(-E/2) \\
&=& 2(-1)^LT_L(E/2) - v(-1)^{L-1}U_{L-1}(E/2),
\end{eqnarray*}
we have
$$
\Delta_L'(-2\cos\theta) = L(-1)^L\frac{\sin L\theta}{\sin\theta} + v(-1)^{L-1}\frac{L\sin\theta\cos L\theta - \cos\theta\sin L\theta}{2\sin^3\theta}
$$
and then, taking limit $\theta\to 0$, we have
\begin{equation}\label{eq:der-2}
\Delta_L'(-2) = (-1)^LL^2 - (-1)^{L-1}\frac{v}{4}L(L^2-1).
\end{equation}
Combining (\ref{eq:der-2}) and $\Delta_L(-2)=(-1)^L(2+vL)$ yields (\ref{eq:-2}).
\end{proof}
\begin{rem}
Since (\ref{eq: ChevDeri2}) is the derivative of polynomial $\Delta_L(E)$, (\ref{eq: ChevDeri2}) is also a polynomial, therefore, $2LT_L(E/2)-EU_{L-1}(E/2)$ becomes a divisor of $E^2-4$.
For example, in the case of $L=3$, it is equals to $2\cdot 3T_3(E/2)-EU_{2}(E/2) = 2E^3-8E=2E(E^2-4)$.
\end{rem}
\begin{rem}
From (\ref{eq:alpha}) in Lemma \ref{lem:Taylor}, we have
$$
\Delta_L(\alpha_j) = \frac{v(-1)^j}{\sin\frac{(2j-1)\pi}{2L}}.
$$
This indicates that the discriminant's amplitude increases almost in proportion to $Lv$ near both ends of $[-2, 2]$. It is consistent with Corollary \ref{rem:osc} and Theorem \ref{thm:Parseval}.
\end{rem}
\section{Spectrum for real $v$}
In this section, we consider the case of real $v$ compared to where $v$ is not real.
$H$ is a self-adjoint operator if $v$ is real, therefore, the spectrum $\sigma(H)\subset\mathbb{R}$.
Throughout this section, interval means an interval on the real axis.
\begin{thm}(A part of Theorem 4.10 in Kato~\cite{Kato}) \label{thm:Kato}
Let $T$ be self-adjoint, and $A$ bounded symmetric. Then $S=T+A$ is self-adjoint and
$$
\sup_{\zeta\in\sigma(S)}\text{\rm dist}(\zeta, \sigma(T))\leq\|A\|,
$$
where $\text{\rm dist}(\zeta, D) = \inf\{|\zeta-x| : x\in D\}$.
\end{thm}
Theorem \ref{thm:Kato} implies the roughest estimate of the location of the spectrum as follows.
\begin{prop}
$$
\sup_{\zeta\in\sigma(H)}\text{\rm dist}(\zeta, [-2, 2])\leq |v|,
$$
\noindent
that is, $\sigma(H)\subset [-2-|v|, 2 + |v|]$.
\end{prop}
Since $\Delta_L(E)$ is a polynomial of degree $L$, the number of extreme values of the discriminant is at most $L-1$; therefore, $\sigma(H)=\{E\in\mathbb{R}:\Delta_L(E)\in[-2, 2]\}$ consists of at most $L$ closed intervals.
The structure of the spectrum can be further detailed, as in the following Theorem \ref{thm:realstructure}.
\begin{thm} \label{thm:realstructure}
If $v\in\mathbb{R}$ is nonzero, $\sigma(H)$ consists of exactly $L$ closed intervals.
\end{thm}
\begin{proof}
Let $\beta_{k}~(k=1, 2, \ldots, L-1)$ be defined in Lemma \ref{lem:Taylor}, then
$\Delta_L(\beta_{k}) = 2(-1)^{k}$.
Since the first-order coefficient of (\ref{eq:beta}), i.e.,
$$
\eta_{k} = \frac{v(-1)^{k}}{2\sin^3\frac{k\pi}{L}}
$$
is never zero, therefore,
$\Delta_L(E)$ intersects $2(-1)^{k}$ transversely at $E=\beta_{k}~(k=1, 2, \ldots, L-1)$.
This result means that $\beta_{k}$ is an endpoint of a spectral band.
Since it can be shown the same way for $v<-2$ or $L$ is even,
we will only prove the $v>2$ and odd $L$ case here.
Since $\Delta_L(-2) = -(2+Lv)<-2$ and $\Delta_L(\beta_{L-1})=2$, there exists $\gamma_{L-1}\in (-2, \beta_{L-1})$ such that $\Delta_L(\gamma_{L-1})=-2$.
Therefore, $[\gamma_{L-1}, \beta_{L-1}]$ contains at least one spectral band $I_{L-1}$ of $\sigma(H)$.
Next, since $\Delta_L'(\beta_{L-1})=\eta_{L-1}>0$ and $\Delta_L(\beta_{L-2}) = -2$,
there exists $\gamma_{L-2}\in (\beta_{L-1}, \beta_{L-2})$ and $[\gamma_{L-2}, \beta_{L-2}]$
contains at least one spectral band $I_{L-2}$.
In the same way, we show that there exists $\gamma_{L-3}\in (\beta_{L-2}, \beta_{L-3}), \cdots, \gamma_{1}\in (\beta_{2}, \beta_1)$ such that all $[\gamma_{k}, \beta_{k}]$ each contain a spectral band $I_{k}$ for every $k = 1, 2, \ldots, L-1$.
Since $\Delta_L(E)$ is a monic polynomial, there exists $\beta_0>\beta_1$ such that $\Delta_L(\beta_0) = 2$.
Therefore, since $\Delta_L(\beta_1) = -2$, $\Delta_L'(\beta_1) = \eta_1<0$, there exists $\gamma_0\in (\beta_1, \beta_0)$ such that $\Delta_L(\gamma_0) = -2$, that is, $[\gamma_0, \beta_0]$ cotains at least a spectral band $I_0$.
From the above, there exist at least $L$ spectral bands.
Since there exist at most $L$ spectral bands, this indicates that $I_{k}=[\gamma_{k}, \beta_{k}]~(k = 0, 1, \ldots, L-1)$.
Therefore, we conclude
$$
\sigma(H) = \bigcup_{k=0}^{L-1}[\gamma_{k}, \beta_{k}].
$$
Thus, we complete the proof.
\end{proof}
By carefully observing the proof of Theorem \ref{thm:realstructure}
and the value at $E = \pm 2$ of the discriminant $\Delta_L(E)$,
the following result is easily derived.
\begin{cor}\label{cor:bandlocation}
Suppose that $\sigma(H) = \bigcup_{k = 0}^{L-1}I_{k}$, where $I_{k}=[\gamma_{k}, \beta_{k}]$ with $\gamma_{L-1}<\beta_{L-1}<\cdots<\gamma_0<\beta_0$. Then, $[-2, 2]\cap[\gamma_0, \beta_0]\ne\emptyset$ if and only if $0\leq v\leq 4/L$, and $[-2, 2]\cap[\gamma_{L-1}, \beta_{L-1}]\ne\emptyset$ if and only if $-4/L\leq v\leq 0$.
\end{cor}
We remark that Corollary \ref{cor:bandlocation} implies that for $|v|>4/L$, $[-2, 2]$ contains exactly $L-1$ spectral bands, and there exists only one band outside $[-2, 2]$.
If $L$ is large enough, $|v|>4/L$ is satisfied; moreover, we have a good approximation of the location of the spectral band outside $[-2, 2]$ as follows.
\begin{cor}
The spectral band $I_{*}$ outside $[-2, 2]$ closes a point
either $\sqrt{4+v^2}$ for $v>0$, or $-\sqrt{4+v^2}$ as $L\to\infty$.
That is,
$$
\sup_{E\in I_{*}}\text{\rm dist}(E, \{\text{\rm sgn}(v)\sqrt{4+v^2}\})\to 0\quad (L\to\infty),
$$
where $\text{\rm sgn}$ is the signum function.
\end{cor}
\begin{proof}
It is sufficient to show that all Floquet eigenvalues in the spectral band outside $[-2, 2]$
converge to either $\sqrt{4+v^2}$ for $v>0$, or $-\sqrt{4+v^2}$ for $v<0$ as $L\to\infty$.
By letting $E(\kappa)=2\cosh\xi(\kappa)~(\xi(\kappa)>0)$ be a Floquet eigenvalue larger than $2$, we obtain
\begin{eqnarray*}
\Delta_L(2\cosh\xi(\kappa)) &=& 2T_L(\cosh\xi(\kappa)) - vU_{L-1}(\cosh\xi(\kappa)) \\
&=& 2\cosh L\xi(\kappa) - v\frac{\sinh L\xi(\kappa)}{\sinh\xi(\kappa)} = 2\cos\kappa.
\end{eqnarray*}
Thus, we have
\begin{equation}\label{eq:limit}
2\coth L\xi(\kappa) - \frac{v}{\sinh\xi(\kappa)} = \frac{2\cos\kappa}{\sinh L\xi(\kappa)}.
\end{equation}
Taking limit $L\to\infty$ of both sides of (\ref{eq:limit}), we learn that $\sinh\xi(\kappa)\to v/2$ as $L\to\infty$, therefore,
$E=2\cos\xi(\kappa)\to\sqrt{4+v^2}$ as $L\to\infty$.
In the case $v<0$, $E\to -\sqrt{4+v^2}$ can be found similarly.
\end{proof}
\section{Floquet spectrum for small and large $v$}
In this section, we study perturbations of the Floquet spectrum for small and large $v$.
We first note that from Proposition \ref{prop:Chebyshevzeros}, all roots of $T_n(x)$ and $U_n(x)$ are simple.
\begin{thm}(Lagrange inversion formula, 3.6.6. in Abramowitz and Stegun\cite{AS})\label{thm:Lagrange}
Suppose $w=f(z)$ is analytic at a point $z_0$ and $f'(z_0)\ne 0$.
Then $z=g(w)$ given by a power series has a nonzero radius of convergence:
$$
g(w) = z_0 + \sum_{n=1}^{\infty}\frac{g_n}{n!}(w-f(z_0))^n,
$$
where
$$
g_n = \lim_{z\to a}\frac{d^{n-1}}{dz^{n-1}}\left\{\left(\frac{z-z_0}{f(z)-f(z_0)}\right)^n\right\}.
$$
\end{thm}
\begin{lem}\label{lem:Taylorapprox}
Let $f(z)$ be a polynomial with $\alpha$ as a simple root, and let $g(z)$ be a polynomial satisfying $g(\alpha)\ne 0$.
Then, for $t \in\mathbb{C}$ with sufficiently small $|t|$, $f(z) - sg(z) - t$ has a simple root
$\alpha(v, t)$ near $\alpha$ such that
$$
\alpha(v, t) = \alpha + \frac{g(\alpha)}{f'(\alpha)}s + \frac{t}{f'(\alpha)} + O((|s|+|t|)^2)
$$
\end{lem}
\noindent
{\it Sketch of the proof of Lemma \ref{lem:Taylorapprox} }
\noindent
First, we note that $f'(\alpha)\ne 0$ since $\alpha$ is a simple root of $f(z)$.
We consider the polynomial $h(z) = f(z)-sg(z)-t$.
If $|s|$ and $|t|$ are small enough, $h(z)$ has a simple root $\alpha(s, t)$ near $\alpha$.
Theorem \ref{thm:Lagrange} shows
\begin{equation}\label{eq:alphavt}
\alpha(s, t) = \alpha(s) + \frac{t}{f'(\alpha(s))-sg'(\alpha(s))} + O(|t|^2),
\end{equation}
where $\alpha(s)$ is a root of $f(z)-sg(z)=0$.
$\alpha(s)$ is close enough to $\alpha$ for every sufficiently small $|s|$.
Since $f(z)/g(z)$ is analytic at $\alpha$ the assumption, we can use Theorem \ref{thm:Lagrange} again, it shows
\begin{eqnarray}
\alpha(s) &=& \alpha + \frac{s}{\left.\left(\frac{f(z)}{g(z)}\right)'\right|_{z=\alpha}} + O(|s|^2) \nonumber \\
&=& \alpha + \frac{g(\alpha)}{f'(\alpha)}s + O(|s|^2) \label{eq:alphav}
\end{eqnarray}
Substituting (\ref{eq:alphav}) into (\ref{eq:alphavt}) and expanding the first-order term for $t$ using the geometric series for small $|s|$, we have
\begin{eqnarray*}
\alpha(s, t) &=& \alpha + \frac{g(\alpha)}{f'(\alpha)}s + \frac{t}{f'(\alpha(s))-sg'(\alpha(s))} + O(|s|^2) + O(|t|^2) \\
&=& \alpha + \frac{g(\alpha)}{f'(\alpha)}s + \frac{t}{f'(\alpha(s))}\sum_{n=0}^{\infty}\left(\frac{g'(\alpha(s))}{f'(\alpha(s))}\right)^ns^n + O(|s|^2) + O(|t|^2) \\
&=& \alpha + \frac{g(\alpha)}{f'(\alpha)}s + \frac{t}{f'(\alpha(s))} + O(|st|) + O(|s|^2) + O(|t|^2)
\end{eqnarray*}
\noindent
Since it follows from (\ref{eq:alphav}) that $f'(\alpha(s)) =f'(\alpha) + O(|s|)$, we have
$$
\frac{1}{f'(\alpha(s))} = \frac{1}{f'(\alpha)} + O(|s|).
$$
Summarize the above to reach the desired result.
\hfill $\Box$
\begin{thm}\label{thm:localstructure}
Let $\alpha_j$ be defined in Lemma \ref{lem:Taylor}, and $\alpha_j(v, \kappa)$ be the element nearest $\alpha_j$ of the Floquet spectrum $\sigma_{\kappa}(H)$. For $v\in\mathbb{C}$ with sufficiently small $|v|$ and $\kappa\in [0, \pi]$ with sufficiently small $|\cos\kappa|$,
$$
\alpha_j(v, \kappa) = \alpha_j + \frac{v}{L} + 2(-1)^{j-1}\frac{\sin\frac{(2j-1)\pi}{2L}}{L}\cos\kappa
+ O((|v|+|\cos\kappa|)^2).
$$
\end{thm}
\begin{proof}
Consider the equation $\Delta_L(E) - 2\cos\kappa = 2T_L(E/2) - vU_{L-1}(E/2) - 2\cos\kappa = 0$.
By applying Lemma \ref{lem:Taylorapprox} to this equation, we have
\begin{equation}\label{eq:alphaperturb}
\alpha_j(v, \kappa) = \alpha_j + \frac{U_{L-1}(\alpha_j/2)}{T_L'(\alpha_j/2)}v + \frac{2\cos\kappa}{T_L'(\alpha_j/2)} + O(|v|^2+|\cos\kappa|^2).
\end{equation}
Proposition \ref{prop:identity-diff} leads
$$
T_L'(\alpha_j/2)=LU_{L-1}(\alpha_j/2)=\frac{(-1)^{j-1}L}{\sin\frac{(2j-1)\pi}{2L}}.
$$
Substituting the above into (\ref{eq:alphaperturb}) completes the proof.
\end{proof}
Theorem \ref{thm:localstructure} implies that $\sigma(H)$ has $L$ arcs approximately parallel to the real axis near $\alpha_j+v/L$ for $j = 1, 2, \ldots, L$.
Moreover, as $L$ increases, the potential becomes sparse, therefore,
the spectrum close to the set $[-2, 2]$ on the real line.
\begin{figure}[htbp]
\centering
\includegraphics[width=100mm]{L5a05.png}
\caption{The spectrum for $L=5$ and $v=i/2$}
\label{fig:L5a05}
\end{figure}
\begin{example}\label{ex:L5a05}
Figure \ref{fig:L5a05} shows the spectrum for $L=5$, $v=i/2$.
In Figure \ref{fig:L5a05}, the dashed lines are the curves represented by $\Re\Delta_5(E) = \Re(2T_5(E/2)) + \frac{1}{2}\Im U_4(E/2)=\pm 2$ and the dotted lines are the curves represented by $\Im\Delta_5(E) = \Im(2T_5(E/2)) - \frac{1}{2}\Re U_4(E/2)=0$, where
\begin{eqnarray*}
\Re(2T_5(E/2)) &=& x^5 - 10x^3y^2 - 5x^3 + 5xy^4 + 15xy^2 + 5x \\
\Im(2T_5(E/2)) &=& 5x^4y - 10x^2y^3 - 15x^2y + y^5 + 5y^3 + 5y \\
\Re U_4(E/2) &=& x^4 - 6x^2y^2 - 3x^2 + y^4 + 3y^2 + 1 \\
\Im U_4(E/2) &=& 4x^3y - 4xy^3 - 6xy
\end{eqnarray*}
for $E=x+iy~((x, y)\in\mathbb{R}^2)$.
The solid lines are the spectrum (spectral arcs), and the filled circles on the real axis are $\alpha_1, \alpha_2, \ldots,\alpha_5$ in order from right to left.
The figure shows that the spectral arcs are almost parallel to the real axis in the neighborhood of each $\alpha_j+v/L=\alpha_j + i/10(j=1, 2, \ldots, 5)$.
\end{example}
\medskip
We next consider the case of large $|v|$.
\begin{thm}\label{thm:localstructure2}
Let $\beta_k$ be defined in Lemma \ref{lem:Taylor}, and $\beta_k(v, \kappa)$ be the element nearest $\beta_k$ of the Floquet spectrum $\sigma_{\kappa}(H)$.
For $v$ with sufficiently large $|v|$ and $\kappa\in [0, \pi]$ with sufficiently small $|\cos\kappa/v|$,
$$
\beta_{k}(v, t) = \beta_k - \frac{2\sin^2\frac{k\pi}{L}}{L}\frac{1}{v} -(-1)^k\frac{2\sin^2\frac{k\pi}{L}}{L}\frac{\cos\kappa}{v} + O((|1/v|+|\cos\kappa/v|)^2),
$$
where $\beta_k$ is defined in Lemma \ref{lem:Taylor}.
\end{thm}
\begin{proof} The equation $\Delta_L(E) - 2\cos\kappa = 0$ can be rewritten as
\begin{equation}
U_{L-1}(E/2) - \frac{2}{v}T_L(E/2) + \frac{2\cos\kappa}{v} = 0.
\end{equation}
We recall
$$
T_L(\beta_k/2)=(-1)^k, \quad U_{L-1}(\beta_k/2)=-\frac{L(-1)^k}{\sin^2\frac{k\pi}{L}},
$$
and Lemma \ref{lem:Taylorapprox}, hence we have
\begin{eqnarray*}
\beta_k(v, \kappa) &=& \beta_k + \frac{2T_{L}(\beta_k/2)}{U_{L-1}'(\beta_k/2)}\frac{1}{v} - \frac{2\cos\kappa}{U_{L-1}'(\beta_k/2)}\frac{1}{v} + O((|1/v|+|\cos\kappa/v|)^2) \\
&=& \beta_k - \frac{2\sin^2\frac{k\pi}{L}}{L}\frac{1}{v} -(-1)^k\frac{2\sin^2\frac{k\pi}{L}}{L}\frac{\cos\kappa}{v} + O((|1/v|+|\cos\kappa/v|)^2).
\end{eqnarray*}
\end{proof}
\begin{figure}[htbp]
\centering
\includegraphics[width=100mm]{L5a5.png}
\caption{The spectrum for $L=5$ and $v=5i$}
\label{fig:L5a5}
\end{figure}
\begin{example}
Figure \ref{fig:L5a5} shows the spectrum for $L=5$, $v=5i$.
The discriminant is the same as in Example \ref{ex:L5a05} except for the value of $v$.
In Figure \ref{fig:L5a5}, the dashed lines are the curves represented by $\Re\Delta_5(E) = \Re(2T_5(E/2)) + 5\Im U_4(E/2)=\pm 2$
and the dotted lines are the curves represented by $\Im\Delta_5(E) = \Im(2T_5(E/2)) - 5\Re U_4(E/2)=0$ for $E=x+iy~((x, y)\in\mathbb{R}^2)$.
The solid lines are the spectral arcs, and the filled circles on the real axis are $\beta_1, \beta_2, \beta_3$, and $\beta_4$ in order from right to left.
Four curves extend from $\beta_1$, $\beta_2$, $\beta_3$, and $\beta_4$ almost in the direction of the imaginary axis, i.e., in the direction of $-1/v$.
In the case of small $v$, five connected components of the spectrum exist, as in Figure \ref{fig:L5a05}, but as $v$ increases, there are four.
\end{example}
\medskip
\noindent
{\small\bf Acknowledgements} \\
The author would like to thank Dr. Yu Morishima for helpful suggestions on drawing figures in Python.
\small
|
3,212,635,537,917 | arxiv |
\section{Method}
\label{sec:method}
Our approach for generating automated IoT device traffic has two main components: 1) Configuring a robotic arm to rigorously test
user interface interactions with IoT devices, and 2) Automating network traffic collection during robotic arm interactions via existing IoT traffic analysis tools~\cite{iot_inspector}.
Implementing this approach involves four primary challenges: 1) Efficiently obtaining correct input parameters for the robotic arm such that it interacts with desired user interface elements on the IoT device, 2) Ensuring interaction accuracy between the robotic arm and the IoT device, 3) Designing interaction sequences that thoroughly explore the space of possible device behaviors, and 4) Automating network traffic collection during robot/device interaction.
This section describes our implementation, including how we addressed these challenges for the Amazon Echo Show 5 and Sensi Wi-Fi Smart Thermostat devices. Our source code is publicly available for use in future research\footnote{\href{https://github.com/Chasexj/Automated\_IoT\_Traffic\_Generation}{https://github.com/Chasexj/Automated\_IoT\_Traffic\_Generation}}.
\subsection{Experiment Setup}
\subsubsection{Robotic Arm} The main robotic arm used in this study is the Arduino Braccio Robotic Arm~\cite{braccio} with 6 degrees of freedom (DoF) and motors/servos connected to the Braccio shield. The assembled arm is fixed to the working station as illustrated in Figure~\ref{fig:set_up} and is controlled via a Arduino UNO REV3 board~\cite{r3}. We choose this particular model of robotic arm for this study for three main reasons: 1) Unlike industrial robotic arms which often sell for tens of thousands of dollars, the Arduino Braccio Robotic Arm retails for less than \$250. This keeps our method accessible for IoT researchers with limited budgets, 2) Compared to other robotic arms in the same price range, this model has relatively high precision ($\pm2$mm), which is desirable for interacting with devices with small buttons, and 3) The Arduino Braccio Robotic Arm is compatible with the Arduino UNO REV3 board, which is widely used and comes with comprehensive usage documentation as well as many useful third-party libraries.
We configure the robotic arm using the open-source Arduino Software (IDE)~\cite{arduino_ide} which allows for controlling sequential movements of the arm given parameters such as arm rotations and movement delays. Given the correct arm rotations, the robotic arm can press and interact with the buttons on the IoT device. The IoT device is also fixed to the working station, and all buttons on the device are within the maximum reach of the arm.
\subsubsection{IoT Devices}The two devices tested in this study are the Amazon Echo Show 5 (Echo-Show5)~\cite{echo} and the Emerson Sensi Wi-Fi Smart Thermostat (Sensi-Thermostat)~\cite{sensi}.
Echo-Show5 is a Amazon smart display capable of performing actions such as video/voice calls, video streaming, online shopping, weather inquiry, music playing, etc. The device is equipped with the Amazon's Alexa~\cite{alexa} voice assistant, a touch screen, as well as four buttons on the top of the device. The four buttons include a mechanical switch for the camera shutter, a mic mute button, and two buttons responsible for volume up/down. We chose Echo-Show5 as one of the two testing devices in this study for two reasons. First, it is representative of the IoT smart speaker/display market (Amazon is a major IoT manufacturer with over 51\% U.S. smart speaker market share as of 2020~\cite{amazon_data}). Second, compared to simple IoT devices such as smart light bulbs, the Echo-Show5 offers a significantly wider variety of functionalities, such as video/music streaming, video calls, and web-browsing. This allows us to evaluate our approach for consumer IoT products with similar multipurpose functionalities.
Sensi-Thermostat is a smart Wi-Fi thermostat that is compatible with software such as Amazon Alexa and Google Assistant~\cite{google}. It is designed to control conventional household heating/cooling. While it can be connected to a user's smartphone for remote control, there are also six physical buttons on the device that can be used to raise/lower temperature, turn on/off the fan, and change the heating/cooling mode and schedule. Similar to Echo-Show5, Sensi-Thermostat is a commonly used IoT device. Unlike Echo-Show5, the Sensi-Thermostat is a single-purpose device that is dedicated to monitoring and controlling household heating and cooling temperature settings. This makes the Sensi-Thermostat ideal for validating our approach for automating interactions with simpler IoT products.
\subsubsection{Network Traffic Collection} To capture network traffic to and from the devices, we set up a Raspberry Pi Wi-Fi access point using
existing ``IoT Inspector'' software~\cite{iot_inspector}. This configures the Raspberry Pi such that all traffic through the Wi-Fi network is captured and stored locally as PCAP files. Upon setting up the robotic arm, IoT device, and Raspberry Pi access point, we instruct the arm to interact with the device to generate and capture the network traffic for analysis.
Since the recorded PCAP files contain the source and destination addresses of captured network packets, we can use tools such as Wireshark~\cite{wireshark} to separate packets generated specifically by the IoT device from background traffic.
For testing this approach with Echo-Show5 and Sensi-Thermostat, we specifically filter the captured traffic to include only
Transmission Control Protocol (TCP) packets sent to or from these devices' Ethernet MAC addresses.
\begin{figure}
\centering
\includegraphics[width=8cm]{figures/set_up.png}
\caption{Sample hardware setup with robotic arm \textit{(left)} and IoT device \textit{(right)} fixed to the table.}
\label{fig:set_up}
\end{figure}
\subsection{Inverse Kinematics for Robotic Arm Input Parameters}
\label{sec:inverse-kinematics}
Once the robotic arm and IoT devices are fixed in place, we must efficiently acquire the input parameters, specifically arm joint rotations, needed for the arm to reach the user interface elements on the device. A na\"{i}ve approach for getting these rotation parameters is through trial and error, i.e.~repeatedly testing and adjusting parameters until the arm reaches the desired UI element. However, as IoT devices often differ from each other in terms of the numbers and locations of UI elements, it is excessively time-consuming to repeat such this trial-and-error process for every UI element on every new device.
Rather than relying on trial and error to obtain the arm rotations needed for each button on each device, we improve the efficiency of the process by utilizing inverse kinematics. We use the inverse kinematics scripts from~\cite{invkin}, which take the lengths of each section of the arm and the button's 2D coordinates as inputs and output the needed arm rotations to reach the button. The inverse kinematics scripts from \cite{invkin} apply only to robotic arms with 3 degrees of freedom (DoF), since applying inverse kinematics with higher DoF increases the computation time. As a result, instead of considering all 6 DoF of our Arduino Braccio robotic arm, we treat the arm as having 4 DoF: one base joint that can rotate horizontally and three arm joints that can rotate vertically. For each desired button, we first manually measure the arm base rotation needed to align the arm with with button. After confirming the base rotation, we then measure the button's vertical 2-dimensional coordinates with respect to the origin/base of the robotic arm and apply inverse kinematics using the robotic arm lengths to obtain the three vertical rotations needed for the arm to reach the button. This eliminates the need to re-test the arm rotations for each new button, significantly reducing the amount of time and manual effort needed to acquire the robotic arm input parameters for each IoT device.
\subsection{Ensuring Interaction Accuracy}
The buttons on most IoT devices are designed for precise interactions with human fingers and are often relatively small in size (usually less than 1 $cm^2$). This can be problematic, as the robotic arm can experience small deviations ($\pm2$mm) during every movement, resulting in accumulating misalignment with the intended button over the course of long experiments. If not corrected, this can cause missed presses or unintended presses of adjacent buttons.
We overcome this problem by physically enlarging the buttons' surface areas to overcome the accumulating misalignment from each arm rotation. This is achieved by attaching additional hard surfaces on top of the original buttons (Figure~\ref{fig:buttons}). Given the larger surface areas, accumulated rotation biases become insignificant and missed presses are avoided.
\begin{figure}
\centering
\includegraphics[width=8cm]{figures/buttons.png}
\caption{IoT device (Sensi-Thermostat) with physically enlarged buttons to eliminate robotic arm misalignment errors.}
\label{fig:buttons}
\end{figure}
While we address the accumulating movement bias, it is worth noting that the robotic arm also experiences delays in executing movement commands: when a movement is requested via the IDE command execution, a very small delay occurs before the robotic arm motors actually move to adjust the arm to the desired position. However, this delay in execution time is on the order of milliseconds and, unlike the positional bias, it is fixed and constant for each movement. This means that the time delay between button press command timestamps and actual button presses is predictable, does not accumulate over multiple commands, and is insignificant compared to the time scale of the actual UI interaction sequences.
\subsection{Designing Interaction Sequences}
Our implementation instructs the robotic arm to move to specific locations by providing a set of arm rotations (acquired using inverse kinematics) as the input parameters (Section~\ref{sec:inverse-kinematics}). Given the limited number of unique physical buttons on each IoT device, we can configure the robotic arm to iterate through buttons in a specific order, or ``interaction sequence,'' by feeding in consecutive sets of arm rotations.
Given the difficulty of collecting real user-generated IoT traffic for research purposes, one of the main research goals of this study is to design robot interaction sequences that
can consider all possible real user interactions. This is especially difficult as most user interactions with IoT devices are random and do not necessarily follow a strict pattern. Meanwhile, different users may have different preferences or habits when interacting with the same devices. As a result, we want to supply the robotic arm with interaction sequences
that provide thorough coverage of possible user interactions.
Our implementation generates comprehensive IoT device traffic by using permutation-based interaction sequences. First, we assign each button on the testing device a unique number and create a set containing these numbers/buttons. For example, given a device with three buttons, we create the set \{1,2,3\} where each number represents a specific button on the device. We then construct a test suite for the experiment that permutes this set of buttons, generating all possible linear orderings. We configure the robotic arm to press the buttons according to each of the permutations, which causes the arm to attempt all possible sequences of interactions with the buttons on the device. This permutation-based approach
provides thorough coverage of the space of possible interaction sequences. Our test suite also allows for randomization within permutation-based sequences to further mimic real user activities (Section~\ref{sec:future}).
For example, we can choose to use permutation with random repetition to interact with individual user interface elements multiple times per interaction sequence or to randomize the time between interactions to simulate unpredictable user behavior.
\subsection{Demonstrating Effectiveness with Machine Learning}
\label{sec:mlmethod}
To evaluate the effectiveness of our approach, we want to verify that the button presses performed by the robotic arm actually lead to the collection of network traffic useful for IoT network research. More specifically, we want to showcase high correlations between the button presses and the captured IoT device traffic such that
we can predict which buttons were pressed on the device from the network traffic alone.
This allows us to infer that many of the captured packets are indeed triggered by the automated button presses.
This also allows us to conclude that the captured traffic provides substantial information about user interactions with the IoT device and is therefore relevant for followup network, security, or privacy analysis.
We use machine learning to test the correlations between the button presses and the collected traffic as follows: 1) labeling IoT device traffic with the respective buttons pressed immediately prior to collection, and 2) training and testing a supervised machine learning algorithm using the button-labeled traffic data to determine if, given a packet in the collected traffic, we can correctly predict the button press that caused the packet.
\subsubsection{Training Data}
We collect training data by instructing the robotic arm to press each button on the device repeatedly. For example, the arm might perform 15 presses of button \#1 followed by 15 presses of button \#2 and 15 presses of button \#3. The delay between each button press is fixed and constant (10 seconds).
At the same time, we record the timestamp for each button press using Coordinated Universal Time (UTC).
The network traffic captured as PCAP files during these interaction sequences is converted to CSV format prior to labeling using nprint~\cite{holland2020nprint}. Each individual packet in the PCAP file is encoded by nprint as a row in the resulting CSV file containing the packet's source and destination IP addresses, payloads, IPv4 and TCP headers, and relative timestamps.
We then label the packets with the button pressed immediately prior by matching packet timestamps and button press timestamps. Because we can assume that a button must be pressed before a corresponding packet is sent (if the packet is sent before the button press, then the button press did not cause the sending of the packet), we expect to observe slight timestamp differences between button presses and corresponding packets. This means we can confidently match the buttons and packets even if their timestamps do not exactly align.
A single button press can cause network activity consisting of multiple packets,
so for each button press with timestamp (\emph{t\textsubscript{button}}), we treat this button as the label for all packets with timestamps (\emph{t\textsubscript{packet}}) within the following range: \[\emph{t\textsubscript{button}}\leq\emph{t\textsubscript{packet}}<(\emph{t\textsubscript{button}+10s)}\]
This labeling is reasonable because all associated packets are transmitted a few seconds after a button is pressed, and there is a 10 second gap between consecutive button presses.
\subsubsection{Random Forest Classifier}
We train a random forest classifier from the scikit-learn library\cite{scikit-learn} to predict the button press corresponding to collected packets. We use a randomized 0.75/0.25 train/test split on the collected packet data and conduct a grid search
to tune the hyperparameters of the classifier, including the criterion (function to measure the quality of a split), max\_depth (maximum depth of the tree), min\_sample\_split (minimum number of samples required to split an internal node), and n\_estimators (number of trees in the forest). The grid search utilizes a 10 fold cross-validation over the button presses in the training set and uses multi-metric scoring (accuracy, precision, and recall) to assess the performance of the cross-validated model. Specific hyperparameter ranges searched when conducting the grid search are shown in Table~\ref{tab:grid_parm}, and the optimal values found are given in Section~\ref{sec:evaluation}.
\begin{table}[t]
\centering\footnotesize
\begin{tabular}{llll}
\toprule
Hyperparameter & Value 1 & Value 2 & Value 3 \\ \midrule
criterion & ``gini'' & ``entropy'' & \\
max\_depth & 20 & 40 & 80 \\
min\_sample\_split & 2 & 5 & 10 \\
n\_estimators & 200 & 400 & 800 \\ \toprule
\end{tabular}
\caption{Grid search values for random forest hyperparameters.}
\label{tab:grid_parm}
\end{table}
Using the best hyperparameters from the grid search, we test the classifier on the test set and record the test scores (F\textsubscript{1}, precision, recall). To obtain more reliable results, we repeat the above procedures (randomized train/test split, grid search, and classifier testing) 50 times and record all results to compute the means and variances of the test scores. Higher scores indicate that we can confidently associate captured packets with the button presses that generated those packets.
\section{Limitations}
\label{sec:limit}
Using a robotic arm to automate physical IoT device interactions is an effective way to scale the collection of IoT traffic accross device behaviors. However, the nature of collecting network data from physical IoT devices via robotics has certain constraints and drawbacks:
\subsection{Non-Physical User Interface Elements}
While a robotic arm can perform many types of interactions with IoT devices, such as pressing buttons or sliding switches, it cannot perform non-physical interactions such as voice commands. Even certain phyiscal user interface elements, such as touch screens, can prove challenging for off-the-shelf robotic arms. Unlike physical buttons, there are an infinite amount of possible interactions to test for comprehensive coverage of touch screen presses, and the location of touch screen input elements may change as the result of prior interactions. Considering the fact that some IoT devices, including the Echo-Show5, have touch screens and voice commands as part of their major functions, it is important to explore additional techniques (such as software emulators) that can automate these types of interactions.
Despite this limitation, we believe that the method described in this paper will be useful to IoT researchers. Physical device interfaces (e.g.~buttons and switches) have proved the most challenging to automate thus far, and are almost ubiquitous in inexpensive consumer IoT products. Combining a robotic arm with a microphone and an instrumented smartphone running devices' associated mobile applications would be sufficient to explore the all possible user inputs for many common devices without requiring
tedious manual button pressing or dedicated development of device-specific emulators.
\subsection{IoT Device Size}
Configuring the robotic arm to interact with the Echo-Show5 and the Sensi-Thermostat was feasible as both of these devices are small enough for the robotic arm to reach any location on the devices. In general, the target IoT device's physical size is limited by the size of the robotic arm. Significantly larger devices (such as a smart refrigerator) exceed the maximum reach or range of motion of most hobbyist robotic arms. While larger and more precise robotic arms are available on the market, these robotic devices are substantially more expensive. One possible solution is to have multiple smaller robotic arms operate on a single large device simultaneously, but this would require precise coordination of the arms to inter-operate with each other. However, such a limitation does not raise significant concern since most popular consumer IoT devices are not large appliances and would be suitable for our approach.
\subsection{Environmental Sensor Data}
The approach described in this paper is intended to automate user interactions with IoT devices to collect network traffic from a comprehensive set of device behaviors. However, many IoT devices also include environmental sensors, such as thermometers, light sensors, accelerometers, and gyroscopes that also determine their behaviors and network communications~\cite{apthorpe2019keeping}.
The scope of possible readings from these sensors is not explored by our robotic arm approach, and would require a software emulator or a laboratory setting with local environment controls. While it would be possible to place both a robotic arm and an IoT device into a chamber with controllable temperature or lighting and conduct permutation-based testing with each of these variables, this is outside the scope of this paper.
\section{Results}
\label{sec:evaluation}
Our study demonstrates that a robotic arm can be used to automate interactions with IoT devices in order to collect network traffic for research. Testing this approach with the Echo-Show5 and Sensi-Thermostat shows that it can collect IoT traffic that provides rigorous coverage of device behaviors with high correlations between button presses and captured packets.
\subsection{Visual Support for Automated IoT Traffic Collection}
\label{sec:results-visual}
To show that our approach is effective in generating IoT traffic, we first visualize traffic collected from the Echo-Show5 (Figure~\ref{fig:echo_io_shade}) and the Sensi-Thermostat (Figure~\ref{fig:therm_io}) when tested with interaction sequences of all four physical buttons on each device.
\subsubsection{Echo Show 5} Traffic collection from the Echo-Show5 started at the relative timestamp of 0s and ended at the relative timestamp of 1533s. The robotic arm started pressing buttons on the device according to our permutation-based interaction sequences starting at 50s and concluded all button presses at 1000s.
\begin{figure}[t]
\centering
\includegraphics[width=7.5cm]{figures/echo_io_shade.jpg}
\caption{Echo-Show5 sample TCP traffic with significantly more packets during automated robotic arm interactions.}
\label{fig:echo_io_shade}
\end{figure}
It is clear that there are spikes of packets sent by the device during the time period when the automated button presses occurred: The period with button presses produced a total of 8391 packets with an average of 8.83 packets per second. In contrast, the rest of the packet capture only had a total of 1202 packets and an average of 2.25 packets per second. Note that we excluded packets prior to the beginning of the button presses when calculating these average rates, as these early packets are the result of device initialization at startup
and are not caused by button presses.
The significantly higher amount of network traffic during the button presses suggests that our robotic arm's interactions with the Echo-Show5 indeed generates activity-related traffic compared to periods with no robot interaction. This is further supported by the substantial decrease in the average number of packets and the frequency of packets after the robot interactions stopped at 1000s.
\subsubsection{Sensi-Thermostat}
\begin{figure}[t]
\centering
\includegraphics[width=7.5cm]{figures/therm_io.png}
\caption{Sensi-Thermostat sample TCP traffic with significantly more packets during automatic robotic arm interactions.}
\label{fig:therm_io}
\end{figure}
We started collecting traffic from the Sensi-Thermostat at the relative timestamp of 0s and ended at the relative timestamp of 950s. The robotic arm button presses occurred from 0s to 950s. Unlike the Echo-Show5, the Sensi-Thermostat does not generate any traffic when the robotic arm is inactive. The observed packet spikes at a rate of 2 packets per second per button press are directly a result of the robotic arm interactions.
The direct visual correlation between these spikes and the button presses validates the effectiveness of our automated interaction and data collection approach.
Compared to the Echo-Show5, there are significantly fewer captured packets for the Sensi-Thermostat; however, this is expected, as the Sensi-Thermostat is a simpler device.
\subsection{ML Support for Automated IoT Traffic Collection}
We use machine learning along with the IoT traffic captured from the Echo-Show5 to verify the correlation between button presses and captured packets (Section~\ref{sec:mlmethod}).
Confidently associating captured packets with particular button presses strongly indicates that the collected packets were triggered by the automated button presses.
This further supports the ability of our automated interaction method to produce traffic containing information about device behavior that would be useful for network, security, or privacy research.
We train and test a machine learning model (random forest classifier) on the Echo-Show5 data instead of the traffic from the Sensi-Thermostat, because the Sensi-Thermostat produced such a small amount of traffic (41 packets) that 1) it is easy to visually verify correlations between traffic spikes and the button presses (Section~\ref{sec:results-visual}) and 2) there is not enough data to train a ML model.
In comparison, the Echo-Show5 produced significantly more captured traffic with our permutation-based interaction sequences (8391 packets).
We labeled each packet in the collected Echo-Show5 traffic with the number of the button most likely responsible for it being sent (Section~\ref{sec:mlmethod}) as shown in Figure~\ref{fig:alexa_ml_io}. We randomly split the button-labeled Echo-Show5 traffic into train/test sets with the ratio of 0.75/0.25 and conducted a grid search using the training data to obtain the optimal hyperparameters for random forest classifier. We then tested the classifier with the optimal hyperparameters on the test set and recorded the test scores with ``micro,'' ``macro,'' and ``weighted'' averaging options\footnote{Averaging options: (1) micro: all samples are weighted equally; (2) macro: all classes are weighted equally; (3) weighted: each individual class's contribution to the averaged score is weighted by its relative cardinality.} (Table~\ref{tab:scores}).
\begin{figure}[t]
\centering
\includegraphics[width=7.5cm]{figures/repeated_alexa.jpg}
\caption{Echo-Show5 TCP traffic with labeling of repeated button presses.}
\label{fig:alexa_ml_io}
\end{figure}
We repeated this training and testing process 50 times using the captured traffic as described in Section~\ref{sec:mlmethod}. All average F\textsubscript{1}, precision, and recall scores are approximately 0.96 with variances less than 1.0e-4, illustrating strong correlation between robotic arm button presses and the captured traffic.
We expect that these scores would not be a perfect 1.0, because there will be some background traffic sent by the device that is not correlated with the button presses.
Nevertheless, these high scores allow us to conclude that the captured traffic does provide substantial information about the robot arm interactions and would be useful for followup research about the network, security, or privacy implications of user interactions with the IoT device.
\begin{table}[t]
\centering\footnotesize
\begin{tabular}{llll}
\toprule
Averaging Option & F\textsubscript{1} & Precision & Recall \\ \midrule
"micro" & \begin{tabular}[c]{@{}l@{}}0.961\\ ($4.888\mathrm{e}{-5}$)\end{tabular} & \begin{tabular}[c]{@{}l@{}}0.961\\ ($4.888\mathrm{e}{-5}$)\end{tabular} & \begin{tabular}[c]{@{}l@{}}0.961\\ ($4.888\mathrm{e}{-5}$)\end{tabular} \\
"macro" & \begin{tabular}[c]{@{}l@{}}0.960\\ ($5.079\mathrm{e}{-5}$)\end{tabular} & \begin{tabular}[c]{@{}l@{}}0.961\\ ($4.680\mathrm{e}{-5}$)\end{tabular} & \begin{tabular}[c]{@{}l@{}}0.959\\ ($6.295\mathrm{e}{-5}$)\end{tabular} \\
"weighted" & \begin{tabular}[c]{@{}l@{}}0.961\\ ($4.835\mathrm{e}{-5}$)\end{tabular} & \begin{tabular}[c]{@{}l@{}}0.961\\ ($4.572\mathrm{e}{-5}$)\end{tabular} & \begin{tabular}[c]{@{}l@{}}0.961\\ ($4.888\mathrm{e}{-5}$)\end{tabular} \\ \toprule
\end{tabular}
\caption{Average test scores over 50 repetitions. Variances are specified in parentheses.}
\label{tab:scores}
\end{table}
\section{Conclusion}
\label{sec:conclusion}
In this study, we design a general method for automating the collection of IoT traffic across a variety of user interactions and device behaviors: configuring a robotic arm to interact with physical user interface elements of IoT devices according to formalized permutation-based interaction sequences.
We describe the steps required to implement this method, including
applying inverse kinematics to efficiently obtain robotic arm movement parameters and physically enlarging the IoT device buttons to improve interaction accuracy.
We also describe the creation of permutation-based interaction sequences to provide comprehensive coverage of possible user interface interactions.
We tested the effectiveness of this approach by applying it to two representative IoT devices, an Amazon Echo Show~5 and an Emerson Sensi Wi-Fi Smart Thermostat.
We were able to generating IoT traffic from sequences of automated button presses while using machine learning to demonstrate that the collected traffic was highly correlated with the robot interactions, suggesting that the traffic contains information about device behaviors that could be useful for network, security, or privacy analyses.
Compared to prior methods of IoT network data collection, our automated approach provides high interaction coverage and does not require tedious manual effort. Our method readily adaptable to different consumer IoT devices and for testing multi-device interactions.
We have made our source code and other reference materials needed for this approach publicly available for future research.\footnote{\href{https://github.com/Chasexj/Automated\_IoT\_Traffic\_Generation}{https://github.com/Chasexj/Automated\_IoT\_Traffic\_Generation}}
\section{Future Work}
\label{sec:future}
Apart from the limitations stated in Section~\ref{sec:limit}, our automated approach to IoT device interactions is amenable to many additions and improvements that could be the topic of future work. We hope that others will adopt and adapt this approach to automate and scale IoT network research.
\subsection{Randomness in Interaction Sequences}
One straightforward extension of our approach is to create permutation-based interaction sequences with increased randomness to better replicate the variety of real user interactions. Although our current implementation explores all possible unique interaction permutations, additional randomness could be added to the testing suite by 1) introducing randomized delays between interactions, 2) randomly repeating button presses within interaction sequences, 3) repeating interaction sequences for a random number of iterations.
Each of these methods would result in longer testing times due to longer delays between interactions and/or more robotic arm movements required per test. Ideally, the scope of the randomness would be tuned to the type of the IoT device, introducing more variety in interaction sequences to devices typically involving less predictable user inputs.
A follow-up study could also test whether increases in interaction randomness actually increase the variety of collected network traffic. If network traffic is predominantly linked to the most recent interaction, introducing random delays in interaction frequency may not actually produce traffic of greater interest for later analysis.
\subsection{Additional IoT Devices}
While we chose two popular consumer IoT devices to test our automated interaction approach, we expect that this method is amenable to a wide variety of IoT products in consumer, medical, and other contexts.
We are aware of several universities and consumer advocacy groups with access to many IoT devices and we recommend their use of our technique for automating network security and privacy analyses at scale.
We would especially like to see this approach used to evaluate medical devices, as they often have many physical buttons and may be under-audited from network security perspective~\cite{burns2016brief}.
\subsection{Multi-Device Interactions}
Although this study explores the performance of our automated interaction approach with individual devices, future work could apply the approach to simultaneous interactions with multiple devices using multiple robotic arms.
Some IoT devices are designed to communicate with other devices on the local network, and the traffic from these communications would not be visible from robotic arm interactions with a single device. A setup with multiple devices and multiple robotic arms could test interleaved interaction sequences across devices to record traffic from device-device communications.
As discussed in~\cite{apthorpe2019keeping}, patterns of network traffic from multiple IoT devices within a household may allow privacy-violating inferences not achievable with traffic from a single device alone.
Applying our approach to multiple devices could reveal the possibility of such inferences and potentially reveal other privacy or security vulnerabilities related to local device communications as well.
As more manufacturers promote ``ecosystems'' of IoT devices, the ability to automate network research of multi-device interactions becomes increasingly important.
\subsection{Computer Vision UI Identification}
Our approach provides substantial automation over IoT network data collection with manual device interactions. The only remaining manual process is to configure the robotic arm with the positions of each of the user interface elements on the device prior to automated interactions (Section~\ref{sec:inverse-kinematics}). This initialization step could also be automated in future work by adding a camera to the setup and using computer vision to identify the type and location of user interface elements on the device. This would provide an extremely low barrier to entry for network data collection from any IoT device at the expense of the additional camera hardware and increased potential for misidentification of UI elements.
\section{Introduction}
\label{sec:introduction}
The Internet of things (IoT) refers to the wide variety of physical objects increasingly connected to the Internet. IoT devices range from common household items, such as thermostats, light bulbs, and door locks, to medical products, wearables, and industrial sensors.
For example, a ``smart'' (IoT) thermostat may be able to receive location data from a smart car and automatically adjust home temperature when a user leaves work.
Although IoT devices provide unprecedented convenience and efficiency, they also raise concerns about security and privacy. Users report privacy fears associated with constant connectivity and always-on environmental sensors~\cite{naeini2017privacy, zheng2018user, apthorpe2020you, huang2020amazon}, and researchers have identified insecurities in IoT device software and network communications~\cite{loi2017systematically, chu2018security, wang2019looking}. These issues, combined with the increasing popularity of consumer IoT devices, are motivating many studies analyzing IoT network traffic to detect and prevent privacy and security vulnerabilities.
Collecting IoT network traffic for these studies typically involves researchers manually interacting with devices in a laboratory setting. Researchers press buttons, touchscreens, or other user interface elements on the devices in attempt to mimic real-world user behaviors or collect traffic from as many device states as possible. For example, a researcher may attempt to record network traffic while pressing every button on a device in sequence or by pressing patterns of buttons (or other user interface elements) in order to examine the full scope of device behavior~\cite{apthorpe2019keeping}. However, these types of manual device interactions are time consuming and are unlikely comprehensive, posing a significant challenge to IoT network research (Section~\ref{sec:related}).
IoT studies have also utilized crowdsourcing to acquire consumer IoT network traffic~\cite{huang2020iot, mazhar2020characterizing}. This can provide large quantities of realistic data; however, crowdsourcing typically requires extensive data collection platform development, and user reports of device behavior during traffic collection are prone to inaccuracy.
In this paper, we present an alternative method for automating IoT network traffic collection: configuring a robotic arm to physically interact with IoT devices according to formalized interaction sequences (Section~\ref{sec:method}).
This approach can simulate real user behaviors, provide comprehensive coverage of possible user/device interactions, and eliminate tedious manual button pressing.
Recording Internet traffic to and from the device during robotic arm interactions provides a scalable source of network data that can be used for security audits, privacy evaluation, or other IoT research.
We demonstrate the effectiveness of this approach by configuring a robotic arm to press physical buttons on two devices: an Amazon Echo Show 5~\cite{echo} (a smart speaker with multiple user interfaces) and an Emerson Sensi Wi-Fi Smart Thermostat~\cite{sensi} (a household thermostat with only physical buttons) (Section~\ref{sec:evaluation}).
We verify that this approach produces network behavior correlated with robot/device interactions by testing a variety of permutation-based button press sequences while collecting Internet traffic.
For example, we observe 3.9 times more TCP traffic to and from the Echo Show when the robotic arm is actively pressing button sequences on its user interface. We also
train a machine learning model (random forest classifier) that can accurately infer ($F_1 > 0.95$) when specific buttons are pressed on the device from network traffic alone.
This suggests that the captured traffic provides substantial information about interactions with the device and would be useful for network, security, or privacy analyses.
Employing robotics for collecting network data from physical IoT devices eliminates many drawbacks of laboratory data collection or crowdsourcing. Our approach provides rigorous interaction coverage and high scalability, allowing for easier collection of IoT traffic for research purposes. We hope that this automated approach and our provided source code\footnote{\href{https://github.com/Chasexj/Automated\_IoT\_Traffic\_Generation}{https://github.com/Chasexj/Automated\_IoT\_Traffic\_Generation}} will facilitate continued research in IoT security, privacy, and networking (Sections~\mbox{\ref{sec:limit}--\ref{sec:future}}).
\section{Background \& Related Work}
\label{sec:related}
Collecting network data from consumer IoT devices has posed a consistent challenge for IoT research. Studies typically use either laboratory data collection or crowdsourcing to collect IoT network traffic; however, both approaches have substantial drawbacks that limit rigor and scalability.
\subsection{Laboratory IoT Traffic Collection}
Many studies of consumer IoT network traffic involve data collection in a controlled ``laboratory'' environment. Researchers acquire devices relevant to their research question then instrument the devices directly or connect the devices to an instrumented network that allows for the collection of network traffic.
Laboratory data collection has several benefits. First, researchers can precisely control the network environment, ensuring that recorded network traffic actually corresponds to specific devices or user interactions. This greatly simplifies ground-truth labeling of collected network traffic for input into supervised machine learning algorithms or for other follow-up analyses.
Second, researchers can collect data while subjecting the device or network to active attacks that would be unethical outside of a controlled environment. For example, researchers can flood a device with denial-of-service traffic or attempt to install bogus TLS certificates.
Unfortunately, laboratory data collection also has several serious drawbacks that have limited consumer IoT research. First, collecting network traffic from all possible user interactions and device behaviors is usually infeasible. Most consumer IoT devices do not have emulator support, so user interactions must be tested on a physical device. Testing repeated sequences of specific user interactions (e.g. button presses) on a device is quite tedious, limiting data collection to a small set of interactions and a correspondingly small volume of network traffic.
Second, researcher interactions with consumer IoT devices are unlikely representative of real user behavior over varying timescales. This means that network traffic generated via laboratory interactions may not be representative of traffic generated by devices in consumer households.
Laboratory data collection has been extensively employed to audit consumer IoT devices for a variety of security and privacy concerns. Studies focused on privacy-violating inferences from IoT network traffic~\cite{apthorpe2019keeping, acar2020peek, edu2020smart} and network vulnerabilities in specific classes of devices (e.g. children's toys~\cite{chu2018security, shasha2019playing}) have sparked increased consumer awareness, regulatory action, and manufacturer attention to consumer IoT security and privacy.
\subsection{Crowdsourcing}
A more recent approach to acquiring consumer IoT network traffic involves crowdsourcing data collection to real users who have adopted IoT devices~\cite{huang2020iot, mazhar2020characterizing}. This typically involves the creation of custom hardware or software that allows users to instrument their own devices or home networks.
Crowdsourcing IoT traffic has several benefits. Crowdsourced data is more externally valid than laboratory data, because it comes from real users interacting with IoT devices in the wild.
With substantial recruitment effort, crowdsourcing can also produce more data from a wider variety of devices and interaction patterns than laboratory collection.
However, crowdsourced data has other drawbacks. It may require extensive development efforts to create a platform for data collection that participants feel comfortable incorporating into their homes. Recruiting participants is also challenging. Crowdsourcing platforms such as Amazon Mechanical Turk are not well suited to IoT data collection, which does not fit well into the Human Intelligence Task (HIT) framework.
Participant compensation for large-scale crowdsourcing campaigns can also be quite expensive, especially if participants are required to install hardware in their homes. Some studies avoid this expense by asking interested individuals to volunteer data without monetary compensation, sometimes by providing details about local IoT device behavior that may be of interest to privacy or security-conscious users~\cite{huang2020iot}. However, this approach usually limits study participation to technically-savvy participants or those with existing privacy or security concerns, potentially introducing results bias.
Crowdsourced data also suffers from limited reliability in terms of data labels. Participants may not accurately report what devices they own or what interactions they perform with the devices. \textit{Post hoc} identification of device types and user interactions from network data is a research problem in its own right~\cite{huang2020iot}.
\subsection{Our Automated Approach}
The automated data collection approach described in this paper combines the scalability and reduced tediousness of crowdsourcing with the control and verifiability of laboratory data collection. By performing device interactions with a robotic arm, we are able to examine network traffic from a breadth of device interactions with minimal manual effort. We are also able to rigorously test permutations of device interactions to collect data for all possible (or all reasonable) interactions with a device. These permutations can include different interface elements (buttons, etc.) as well as different timings between interactions. |
3,212,635,537,918 | arxiv | \section{1. Symmetry classification of pairing channels}
Electrons in 3D Dirac semi-metal are described by a four component bi-spinor
creation operator, $\psi _{\alpha }^{\dagger }= \psi _{L\uparrow }^{\dagger
},\psi _{L\downarrow }^{\dagger },\psi _{R\uparrow }^{\dagger }$,\\
$\psi_{R\downarrow }^{\dagger }$, whose index $\gamma $ takes four values. Here
we classify the possible local superconducting order parameters, written
generally as
\begin{equation}
\widehat{M}=\int_{\mathbf{r}}\psi _{\alpha }^{+}\left( \mathbf{r}\right)
M_{\alpha \beta }\psi _{\beta }^{+}\left( \mathbf{r}\right) \text{,}
\label{SA1a}
\end{equation}%
with constant antisymmetric matrix $M$ according to representations of the
3D rotation group. The representations of the rotation group therefore
characterize various possible superconducting phases. Generator of rotations
consists of the orbital momentum operator $\mathbf{L}$ and the spin operator%
\begin{equation}
S^{i}=\int_{\mathbf{r}}\psi _{\gamma }^{+}\left( r\right) \mathbf{\Sigma }%
_{\gamma \delta }^{i}\psi _{\delta }\left( r\right) , \label{SA1b}
\end{equation}%
Due to the rotation symmetry they transform covariantly under the action of $%
\mathbf{J}=\mathbf{L}+\mathbf{S}$. The global quantity in Eq.(\ref{SA1a})
transforms as
\begin{eqnarray}
\left[ \widehat{M},J^{i}\right] &=&\int_{\mathbf{r,r}^{\prime }}\left[ \psi
_{\alpha }^{+}\left( r\right) M_{\alpha \beta }\psi _{\beta }^{+}\left(
r\right) ,\psi _{\gamma }^{+}\left( r^{\prime }\right) \mathbf{\Sigma }%
_{\gamma \delta }^{i}\psi _{\delta }\left( r^{\prime }\right) \right]
\label{transformation} \\
&=&-2\int_{r}\psi _{\gamma }^{+}\left( r\right) \mathbf{\Sigma }_{\gamma
\delta }^{i}M_{\delta \kappa }\psi _{\kappa }^{+}\left( r\right) \text{.}
\notag
\end{eqnarray}%
Out of 16 possible matrices $M$ six are antisymmetric. They transform into
each other forming the following irreducible representations.
(i) a triplet of matrices $\left\{ T_{x},T_{y},T_{z}\right\} =\left\{ \beta
\alpha _{z},-\gamma _{x}\gamma _{y}\gamma _{z},\beta \alpha _{x}\right\} $
transforms as a vector
\begin{equation}
\left[ \widehat{M_{k}^{T}},J^{l}\right] =i\varepsilon _{klm}\widehat{%
M_{m}^{T}} \label{vector}
\end{equation}
\ \ \ \ \ \ \ \ (ii) three scalar multiplets: $S_{1}=i\alpha _{y};$ \ \ $%
S_{2}=i\Sigma _{y};$ \ \ $S_{3}=-i\beta \alpha _{y}\gamma _{5}$. \
Which one of the condensates is realized at zero temperature is determined
by the Hamiltonian.
\section{2. Microscopic equations for local pairing}
\subsection{Gor'kov equations}
To treat the pairing the general gaussian approximation can be employed.
Using the standard formalism, the Matsubara Green's functions ($\tau $ is
the Matsubara time),
\begin{eqnarray}
G_{\alpha \beta }\left( \mathbf{r},\tau ;\mathbf{r}^{\prime },\tau ^{\prime
}\right) &=&-\left\langle T_{\tau }\psi _{\alpha }\left( \mathbf{r},\tau
\right) \psi _{\beta }^{\dagger }\left( \mathbf{r}^{\prime },\tau ^{\prime
}\right) \right\rangle \text{;} \label{GFdef} \\
F_{\alpha \beta }^{\dagger }\left( \mathbf{r},\tau ;\mathbf{r}^{\prime
},\tau ^{\prime }\right) &=&\left\langle T_{\tau }\psi _{\alpha }^{\dagger
}\left( \mathbf{r},\tau \right) \psi _{\beta }^{\dagger }\left( \mathbf{r}%
^{\prime },\tau ^{\prime }\right) \right\rangle \text{,} \notag
\end{eqnarray}%
obey the Gor'kov equations:%
\begin{gather}
-\frac{\partial G_{\gamma \kappa }\left( \mathbf{r},\tau ;\mathbf{r}^{\prime
},\tau ^{\prime }\right) }{\partial \tau }-\int_{\mathbf{r}^{\prime \prime
}}\left\langle \mathbf{r}\left\vert \widehat{K}_{\gamma \beta }\right\vert
\mathbf{r}^{\prime \prime }\right\rangle G_{\beta \kappa }\left( \mathbf{r}%
^{\prime \prime },\tau ;\mathbf{r}^{\prime },\tau ^{\prime }\right)
\label{Gorkov} \\
-gF_{\beta \gamma }\left( \mathbf{r},\tau ;\mathbf{r},\tau \right) F_{\beta
\kappa }^{\dagger }\left( \mathbf{r},\tau ,\mathbf{r}^{\prime },\tau
^{\prime }\right) =\delta ^{\gamma \kappa }\delta \left( \mathbf{r-r}%
^{\prime }\right) \delta \left( \tau -\tau ^{\prime }\right) ; \notag \\
\frac{\partial F_{\gamma \kappa }^{\dagger }\left( \mathbf{r},\tau ;\mathbf{r%
}^{\prime },\tau ^{\prime }\right) }{\partial \tau }-\int_{\mathbf{r}%
^{\prime \prime }}\left\langle \mathbf{r}\left\vert \widehat{K}_{\gamma
\beta }^{t}\right\vert \mathbf{r}^{\prime \prime }\right\rangle F_{\beta
\kappa }^{\dagger }\left( \mathbf{r}^{\prime \prime },\tau ;\mathbf{r}%
^{\prime },\tau ^{\prime }\right) \notag \\
-gF_{\gamma \beta }^{\dagger }\left( \mathbf{r},\tau ;\mathbf{r},\tau
\right) G_{\beta \kappa }\left( \mathbf{r},\tau ,\mathbf{r}^{\prime },\tau
^{\prime }\right) =0\text{.} \notag
\end{gather}%
In the homogeneous case the Gor'kov equations for Fourier components of the
Greens functions simplify considerably,
\begin{eqnarray}
D_{\gamma \beta }^{-1}G_{\beta \kappa }\left( \omega ,p\right) -\Delta
_{\gamma \beta }F_{\beta \kappa }^{\dagger }\left( \omega ,p\right)
&=&\delta ^{\gamma \kappa }\text{;} \label{Gorkov_uniform} \\
D_{\beta \gamma }^{-1}F_{\beta \kappa }^{\dagger }\left( \omega ,p\right)
+\Delta _{\gamma \beta }^{\ast }G_{\beta \kappa }\left( \omega ,p\right) &=&0%
\text{,} \notag
\end{eqnarray}%
where $\omega =\pi T\left( 2n+1\right) $ is the Matsubara frequency and$\
D_{\gamma \beta }^{-1}=\left( i\omega -\mu \right) \delta _{\gamma \beta
}+v_{F}p^{j}\alpha _{\alpha \beta }^{j}$.
\bigskip The matrix gap function can be chosen as ($\Delta $ real)
\begin{equation}
\widehat{\Delta }_{\beta \gamma }=gF_{\gamma \beta }\left( 0\right) =\Delta
M_{\gamma \beta }\text{.} \label{delta}
\end{equation}%
These equations are conveniently presented in matrix form (superscript $t$
denotes transposed and $I$ - the identity matrix):
\begin{eqnarray}
D^{-1}G-\Delta F^{\dagger } &=&I\text{;} \label{matrixeq} \\
D^{t-1}F^{\dagger }+\Delta ^{\ast }G &=&0\text{.} \notag
\end{eqnarray}%
Solving these equations, one obtains
\begin{eqnarray}
G^{-1} &=&D^{-1}+\Delta D^{t}\Delta ^{\ast }\text{;} \label{solution} \\
F^{\dagger } &=&-D^{t}\Delta ^{\ast }G\text{,} \notag
\end{eqnarray}%
with the gap function, Eq.(\ref{gap}), found from the consistency condition.
Now we find solutions of this equation for each of the possible
superconducting phases.
\subsection{Triplet solution of the gap equation}
In this phase rotational symmetry is spontaneously broken simultaneously
with the electric charge $U\left( 1\right) $ (global gauge invariance)
symmetry. Assuming $z$ direction of the $p$ - wave condensate the order
parameter matrix takes a form: $\Delta =\Delta _{T}M_{z}^{T}=\Delta
_{T}\beta \alpha _{x}$. In this Section we use the units of $v_{F}=1,\hbar
=1 $ and the energy scale will be set by the Debye cutoff, $T_{D}=1$, of the
electron - phonon interactions, see below. The off-diagonal matrix element
of the matrix gap equation, for real $\Delta _{T}>0$ is:
\begin{equation}
\frac{1}{g}=\sum\limits_{\omega q}\frac{\Delta _{T}^{2}+p_{\perp
}^{2}-p_{z}^{2}+\mu ^{2}+\omega ^{2}}{\left( \Delta _{T}^{2}+\omega
^{2}\right) ^{2}+\left( p^{2}-\mu ^{2}\right) ^{2}+2\left( p^{2}+\mu
^{2}\right) \omega ^{2}+2\Delta _{T}^{2}\left( p_{\perp }^{2}-p_{z}^{2}+\mu
^{2}\right) }\text{,} \label{gap}
\end{equation}%
where $p_{\perp }^{2}=p_{x}^{2}+p_{y}^{2}$. The spectrum of elementary
excitations obtained from the four poles of the Greens function, see
Fig.SM1, is\ (in physical units)
\begin{equation}
E_{\pm }^{2}=\Delta _{T}^{2}+v_{F}^{2}p^{2}+\mu ^{2}\pm 2v_{F}\sqrt{\Delta
_{T}^{2}p_{z}^{2}+p^{2}\mu ^{2}}\text{.} \label{spectrum}
\end{equation}%
There are two nodes at $p_{x}=p_{y}=0,v_{F}p_{z}=\pm \sqrt{\Delta
_{T}^{2}+\mu ^{2}}$, when the branches $+\left\vert E_{-}\right\vert $ and $%
-\left\vert E_{-}\right\vert $ cross, see Fig.SM1a and a section $p_{\perp
}=0$ in Fig.SM1b. There is also a saddle points with energy gap, $2\Delta
_{T}$ on the circle $p_{x}^{2}+p_{y}^{2}=\mu ^{2},p_{z}=0$ see the section
in the $p_{z}=0$ direction in Fig. SM1c. The higher energy band $E_{+}$
touches the lower band at $p=0$, so that there is a Dirac point for
quasiparticles, see Fig. SM1d.
\begin{figure}[tbp]
\centering
\subfigure[]{\includegraphics[width=6cm]{SM1a.pdf}}
\subfigure[]{\includegraphics[width=6cm]{SM1b.pdf}}
\subfigure[]{\includegraphics[width=6cm]{SM1c.pdf}}
\subfigure[]{\includegraphics[width=6cm]{SM1d.pdf}}
\caption{Spectrum of triplet excitations. a. section $p_{\perp }=0$ in b.
There is also a saddle points with energy gap, c. $2\Delta _{T}$ on the
circle $p_{x}^{2}+p_{y}^{2}=\protect\mu ^{2},p_{z}=0$ see the section in the
$p_{z}=0$ direction in . d. The higher energy band $E_{+}$ touches the lower
band at $p=0$, so that there is a Dirac point for quasiparticles.}
\end{figure}
Integration over $\omega $ gives using polar coordinates for $p$ and $x=\cos
\theta ,$ $\zeta =\sqrt{\Delta _{T}^{2}x^{2}+\mu ^{2}}$,%
\begin{equation}
\frac{1}{g}=\frac{1}{8\pi ^{2}}\int_{p=\max \left[ \mu -1,0\right] }^{\mu
+1}\int_{x=0}^{1}\frac{p^{2}}{\zeta }\left\{ \frac{\zeta +px^{2}}{\sqrt{%
\Delta _{T}^{2}+p^{2}+\mu ^{2}+2p\zeta }}+\frac{\zeta -px^{2}}{\sqrt{\Delta
_{T}^{2}+p^{2}+\mu ^{2}-2p\zeta }}\right\} \text{.} \label{gapeqT}
\end{equation}%
The lower bound on the momentum integration is nonzero when chemical
potential $\mu $ exceeds $T_{D}$, see Fig. SM2. The integral over $x$ was
performed analytically, while the last integral was done numerically.
\begin{figure}[tbp]
\centering
\subfigure[]{\includegraphics[width=6cm]{SM2a.pdf}} \subfigure[]{%
\includegraphics[width=6cm]{SM2b.pdf}}
\caption{Chemical potential in Dirac semi - metals and the phonon mediated
pairing. (a) Chemical potential relative to Dirac point is smaller that
typical energy of phonons, the Debye energy $T_{D}$. (b) The BCS
approximation limit: the chemical potential is much larger than the Debye
energy $T_{D}$.}
\end{figure}
\subsection{Singlet representations}
It turns out that the second singlet in Eq.(\ref{gapsinglet}) gives results
identical to that of the first one, while the third singlet does not have a
solution in the physically interesting range of parameters. Therefore we
assume the order parameter in the matrix form $\ \Delta =\Delta
_{S}M_{1}^{S}=i\Delta _{S}\alpha ^{y}$. The relevant matrix element of the
matrix gap equation, is for real $\Delta _{S}$:
\begin{equation}
\frac{1}{g}=\sum\limits_{\omega p}\frac{\Delta _{S}^{2}+p^{2}+\mu
^{2}+\omega ^{2}}{\left( \Delta _{S}^{2}+p^{2}\right) ^{2}+\left( \mu
^{2}+\omega ^{2}+2\Delta _{S}^{2}\right) \left( \mu ^{2}+\omega ^{2}\right)
+2p^{2}\left( \omega ^{2}-\mu ^{2}\right) }\text{.} \label{gapsinglet}
\end{equation}%
Spectrum (in physical units) now is isotropic,%
\begin{equation}
E_{\pm }^{2}=\Delta _{S}^{2}+\left( v_{F}\left\vert p\right\vert \pm \mu
\right) ^{2}\text{.} \label{spectrum_S}
\end{equation}%
Integration over $\omega $ gives%
\begin{equation}
\frac{1}{g}=\mu \sum\limits_{\mu -T_{D}<\varepsilon _{p}<\mu +T_{D}}\frac{p}{%
r_{+}r_{-}\left( r_{+}-r_{-}\right) }\text{,} \label{gap1_S}
\end{equation}%
where $r_{\pm }=\sqrt{\Delta _{S}^{2}+\left( \left\vert p\right\vert \pm \mu
\right) ^{2}}$, while the $p$ integration results in:%
\begin{equation}
\frac{16\pi ^{2}}{g}=\Phi \left( \mu +1,\mu \right) -\Phi \left( \max \left[
\mu -1,0\right] ,\mu \right) \label{gapeq2_S}
\end{equation}%
with%
\begin{equation}
\Phi \left( p,\mu \right) =r_{-}\left( p+3\mu \right) +r_{+}\left( p-3\mu
\right) -\left( \Delta _{S}^{2}-2\mu ^{2}\right) \log \left[ \left(
p+r_{-}-\mu \right) \left( p+r_{+}+\mu \right) \right] \label{Fidef}
\end{equation}
The solution is presented in Fig. 2 of the paper as lines of constant
chemical potential. Having found the order parameter, one has to determine
what symmetry breaking is realized by comparing energies of the solutions
as explained in the text.
\section{3. The BCS and the strong coupling limits}
\subsection{Triplet}
In several limiting cases the integrals can be performed analytically. At
zero chemical potential the results are presented in Section IV, while here
we list the BCS limit of $\mu >>T_{D}$ and the strong coupling case of $g\mu
^{2}>>1$, $\Delta _{T}\propto g$.
(i) In the BCS limit one has%
\begin{equation}
\frac{1}{g}=\frac{a_{T}\mu ^{2}}{4\pi ^{2}}\sinh ^{-1}\frac{T_{D}}{\Delta
_{T}}\text{,} \label{gapBCS_T}
\end{equation}%
with $a_{T}=0.69$, leading to exponential gap dependence on $\lambda $ when
it is small:%
\begin{equation}
\Delta _{T}=T_{D}/\sinh \left( 1/2a_{T}\lambda \right) \simeq
2T_{D}e^{-1/2a_{T}\lambda }\text{.} \label{dT_BCS}
\end{equation}
(ii) In the strong coupling one obtains with solution
\begin{equation}
\Delta _{T}=\frac{g}{12\pi ^{2}}\left\{
\begin{array}{c}
6\mu ^{2}+2\text{ \ for }\mu <1 \\
\left( \mu +1\right) ^{3}\text{\ for }\mu >1%
\end{array}%
\right. \text{.} \label{dT_sc}
\end{equation}%
Usually the local coupling does not prefer the triplet pairing and the
singlet channels of coupling are realized. We therefore turn to them.
\subsection{\protect\bigskip Singlet}
For singlet one has
(i) BCS, $\mu >>T_{D}$
\begin{equation}
\Delta _{S}=T_{D}/\sinh \left( 1/2\lambda \right) \simeq
2T_{D}e^{-1/2\lambda }\text{.} \label{BCS_S}
\end{equation}
(ii) Strong coupling
\begin{equation}
\Delta _{S}=\frac{2\lambda \left( T_{D}+\mu \right) ^{3}}{3\mu ^{2}}\text{.}
\label{S_sc}
\end{equation}
\subsection{Energies}
In limiting cases, one obtains expressions in closed form.
(i) BCS, $\mu >T_{D}$, using Eq.(\ref{gapBCS_T}) and Eq.(\ref{dT_BCS}) for
the triplet and Eq.(\ref{BCS_S}) for the singlet, one has the energy density:%
\begin{equation}
F_{T,S}=-\frac{a_{T,S}\mu ^{2}T_{D}}{2\pi ^{2}v_{F}^{3}\hbar ^{3}}\left(
\sqrt{\Delta _{T}^{2}+T_{D}^{2}}-T_{D}\right) \simeq -\frac{a_{T,S}}{\pi ^{2}%
}\frac{\mu ^{2}T_{D}^{2}}{v_{F}^{3}\hbar ^{3}}\exp \left( -\frac{1}{%
a_{T,S}\lambda }\right) , \label{F_BCS}
\end{equation}%
with $a_{T}=0.69$, while $a_{S}=1$ and assuming $\lambda <<1$. The ratio of
the two phases gives
\begin{equation}
\frac{F_{T}}{F_{S}}=0.69e^{-0.45/\lambda }\text{.} \label{ratio}
\end{equation}
:
\bigskip (ii) Strong coupling limit, using Eq.(\ref{dT_sc}) for triplet and
Eq.(\ref{S_sc}) for the singlet,
\begin{equation}
F_{T}=F_{S}=-\frac{1}{72\pi ^{4}v_{F}^{3}\hbar ^{3}}\left\{
\begin{array}{c}
4\left( 3\mu ^{2}+T_{D}^{2}\right) ^{2}\text{ \ for }\mu <T_{D} \\
T_{D}^{-2}\left( \mu +T_{D}\right) ^{6}\text{\ \ for }\mu >T_{D}%
\end{array}%
\right. \text{.} \label{energy_sc}
\end{equation}%
The difference appears at order $1/g$. To summarize, in most of the
parameter range shown triplet is a bit higher than that of the singlet, but
the two condensates are nearly degenerate.
\section{ 4. Magnetic impurities}
After averaging over impurities, the Gor'kov equations, Eq.(\ref%
{Gorkov}) acquires an additional term In components (no Nambu notations)%
\begin{eqnarray}
I &=&D^{-1}G-NG-LF^{+} \label{S4a} \\
0 &=&\left( D^{t-1}-N^{+}\right) F^{+}-\left( L^{+}-\Delta ^{\ast }\right) G
\notag
\end{eqnarray}%
where the normal disorder average
\begin{eqnarray}
N^{\alpha \beta ^{\prime }}\left( r-r^{\prime },\tau -\tau ^{\prime }\right)
&=&J^{2}\left\langle \sum\nolimits_{a,b}S_{a}^{i}S_{b}^{i}\delta \left(
r-r_{a}\right) \delta \left( r^{\prime }-r_{b}\right) \right\rangle
_{dis}\times \label{N} \\
\Sigma _{\alpha \beta }^{i}\Sigma _{\alpha ^{\prime }\beta ^{\prime
}}^{i^{\prime }}\left\langle T\psi _{\beta }\left( r,\tau \right) \psi
_{\alpha ^{\prime }}^{+}\left( r^{\prime },\tau ^{\prime }\right)
\right\rangle &=&-C\delta \left( r-r^{\prime }\right) \Sigma _{\alpha \beta
}^{i}G_{\beta \alpha ^{\prime }}\left( 0,\tau -\tau ^{\prime }\right) \Sigma
_{\alpha ^{\prime }\beta ^{\prime }}^{i} \notag
\end{eqnarray}%
lead to the renormalization of the chemical potential and relaxation time
that can be safely neglected for our purposes. The second, anomalous
disorder average%
\begin{eqnarray}
L_{imp}^{+\beta \beta ^{\prime }}\left( r-r^{\prime },\tau -\tau ^{\prime
}\right) &=&-J^{2}\left\langle \sum\nolimits_{a,b}S_{a}^{i}S_{b}^{i}\delta
\left( r-r_{a}\right) \delta \left( r^{\prime }-r_{b}\right) \right\rangle
_{dis}\times \label{L} \\
\Sigma _{\alpha \beta }^{i}\Sigma _{\alpha ^{\prime }\beta ^{\prime
}}^{i}\left\langle T\psi _{\alpha }^{+}\left( r,\tau \right) \psi _{\alpha
^{\prime }}^{+}\left( r^{\prime },\tau ^{\prime }\right) \right\rangle
&=&-C\delta \left( r-r^{\prime }\right) \Sigma ^{it}F^{+}\left( 0,\tau -\tau
^{\prime }\right) \Sigma ^{i}\text{,} \notag
\end{eqnarray}%
determines the influence of the disorder on the condensate. For singlet one
has in Fourier space%
\begin{equation}
\sum\nolimits_{q}F_{\beta \gamma }^{+}\left( q,\omega \right) =\frac{i}{g}%
\Delta ^{S}\left( \omega \right) \alpha _{\beta \gamma }^{y}; \label{Fomega}
\end{equation}%
leading via
\begin{equation}
i\Sigma ^{it}\alpha ^{y}\Sigma ^{i}=i\left(
\begin{array}{cc}
\sigma _{i}^{t} & \\
& \sigma _{i}^{t}%
\end{array}%
\right) \left(
\begin{array}{cc}
\sigma _{y} & \\
& -\sigma _{y}%
\end{array}%
\right) \left(
\begin{array}{cc}
\sigma _{i} & \\
& \sigma _{i}%
\end{array}%
\right) =-3i\alpha _{y} \label{algebra}
\end{equation}%
to Eq.(L) in the main text from which the bifurcation point is found.
Similarly for triplet
\begin{equation}
L_{imp}^{+}\left( p,\omega \right) =-C\Sigma
^{it}\sum\nolimits_{q}F^{+}\left( q,\omega \right) \Sigma ^{i}=-\frac{C}{g}%
\Delta ^{T}\left( \omega \right) \gamma ^{x}\text{.} \label{tripletL}
\end{equation}%
since now%
\begin{equation}
\sum\nolimits_{q}F_{\beta \gamma }^{+}\left( q,\omega \right) =\frac{1}{g}%
\Delta ^{T}\left( \omega \right) \gamma _{\beta \gamma }^{x}
\label{Ftriplet}
\end{equation}%
and%
\begin{equation}
\Sigma ^{it}\gamma ^{x}\Sigma ^{i}=\left(
\begin{array}{cc}
\sigma _{i}^{t} & 0 \\
0 & \sigma _{i}^{t}%
\end{array}%
\right) \left(
\begin{array}{cc}
0 & -\sigma _{x} \\
\sigma _{x} & 0%
\end{array}%
\right) \left(
\begin{array}{cc}
\sigma _{i} & 0 \\
0 & \sigma _{i}%
\end{array}%
\right) =\gamma ^{x} \label{algtriplet}
\end{equation}%
Note opposite signs of the singlet and triplet. At bifurcation point
(destruction of the condensate) the function $f$ will be now%
\begin{equation}
f_{T}\left( \omega \right) =\frac{1}{4}\text{tr}\sum\nolimits_{\mathbf{q}%
}D^{t}\gamma ^{x}D\gamma ^{x}>0\text{.} \label{fT}
\end{equation}%
To solve the equations for the critical disorder strength $C^{c}$, one
integrates over $\omega $,
\begin{equation}
\frac{1}{g}=\sum\nolimits_{\omega }\frac{f_{T}\left( \omega \right) }{%
1-C^{c}f_{T}\left( \omega \right) } \label{ft}
\end{equation}%
that has no solution.
\end{document}
|
3,212,635,537,919 | arxiv | \section{A cordial sync} \label{sec:method}
To address the aforementioned two challenges we develop: (a) a novel action sampling procedure named \textbf{S}ynchronize \textbf{Y}our actio\textbf{N}s \textbf{C}oherently (\mbox{\sc{SYNC}}\xspace) and (b) an intuitive \& effective multi-agent training loss named the \textbf{C}o\textbf{ordi}n\textbf{a}tion \textbf{L}oss (\mbox{\sc{CORDIAL}}\xspace).
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth,trim={0 0 2cm 0},clip]{arxiv_figs/model_decentral_v5.pdf}
\caption{Model overview for 2 communicative agents in the decentralized setting. \textit{Left}: all decentralized methods in this paper have the same TBONE~\cite{jain2019CVPRTBONE} backbone architecture. \textit{Right}: marginal vs \mbox{\sc{SYNC}}\xspace-policies. With marginal policies, the standard in prior work, each agent constructs its own policy and independently samples from this policy. With \mbox{\sc{SYNC}}\xspace-policies, agents communicate to construct a distribution $\alpha$ over multiple ``strategies'' which they then sample from using a shared random seed}
\label{fig:model}
\end{figure}
\noindent\textbf{Addressing challenge 1: \mbox{\sc{SYNC}}\xspace-policies.} %
For readability, we consider
$N=2$ agents and illustrates an overview in \figref{fig:model}. The joint probability tensor $\Pi_t$ is hence a matrix of size $|\cA|\times|\cA|$.
Recall our goal: %
using little communication, multiple agents should sample their actions from a high-rank joint policy. This is difficult as (i) little communication means that, except in degenerate cases, no agent can form the full joint policy and (ii) even if all agents had access to the joint policy it is not obvious how to ensure that the decentralized agents will sample a valid coordinated action. %
To achieve this note that, for any rank $m\leq |\cA|$ matrix $L\in\bR^{|\cA|\times |\cA|}$, there are vectors $v_1,w_1,\ldots,v_m,w_m\in\bR^{|\cA|}$ such that
$L = \sum_{j=1}^m v_j\otimes w_j$. Here, $\otimes$ denotes the outer product.
Also, the \emph{non-negative rank} of a matrix $L\in\bR^{|\cA|\times |\cA|}_{\geq 0}$ equals the smallest integer $s$ such that $L$ can be written as the sum of $s$ non-negative rank-one matrices.
Furthermore, a non-negative matrix $L\in\bR^{|\cA|\times |\cA|}_{\geq 0}$ has non-negative rank bounded above by $|\cA|$. Since $\Pi_t$ is a $|\cA|\times |\cA|$ joint probability matrix, \ie, $\Pi_t$ is non-negative and its entries sum to one, it has non-negative rank $m\leq |\cA|$, \ie, there exist non-negative vectors $\alpha\in\bR_{\geq 0}^m$ and
$p_1, q_1,\ldots,p_m,q_m\in\bR_{\geq 0}^{|\cA|}$ whose entries sum to one such that
$\Pi_t=\sum_{j=1}^m\alpha_j \cdot p_j\otimes q_j$.
We call a sum of the form
$\sum_{j=1}^m\alpha_j \cdot p_j\otimes q_j$ a \emph{mixture-of-marginals}. With this decomposition at hand, randomly sampling action pairs $(a^1,a^2)$ from
$\sum_{j=1}^m\alpha_j \cdot p_j\otimes q_j$ can be interpreted as a
two step process: first sample an index
$j\sim \text{Multinomial}(\alpha)$ and then sample
$a^1\sim \text{Multinomial}(p_j)$ and
$a^2\sim \text{Multinomial}(q_j)$.
This stage-wise procedure suggests a strategy for sampling actions in a multi-agent setting, which we refer to as
\emph{\mbox{\sc{SYNC}}\xspace-policies}. Generalizing to an $N$ agent setup, suppose
that agents $(A^i)_{i=1}^N$ have access to a shared random stream of numbers.
This can be accomplished if all agents
share a random seed or if all agents initially communicate their individual random
seeds and sum them to obtain a shared
seed. Furthermore, suppose that all agents locally store a shared function %
$f_\theta: \mathbb{R}^K\to \Delta_{m-1}$ where $\theta$ are learnable
parameters, $K$ is the dimensionality of all communication
between the agents in a timestep, and $\Delta_{m-1}$ is the
standard $(m-1)$-probability simplex. Finally, at time $t$ suppose
that each agent $A^i$ produces not a single policy $\pi^i_t$ but instead a
collection of policies $\pi^i_{t,1}, \ldots, \pi^i_{t,m}$. Let
$C_t\in\bR^K$ be all communication sent between agents at time $t$. Each agent $A^i$ then samples its action as follows:
(i) compute
the shared probabilities $\alpha_t = f_\theta(C_t)$, (ii) sample an
index $j\sim \text{Multinomial}(\alpha_t)$ using the shared random
number stream, (iii) sample, independently, an action $a^i$ from the
policy $\pi^i_{t, j}$. Since both
$f_\theta$ and the random number stream are shared,
the quantities in (i) and (ii) are
equal across all agents despite being computed individually.
This sampling procedure is equivalent to sampling
from the tensor
$\sum_{j=1}^m \alpha_j \cdot \pi^1_{t, j} \otimes \ldots \otimes \pi^N_{t, j}$ which,
as discussed above, may have rank up to $m$. Intuitively, \mbox{\sc{SYNC}}\xspace\ enables decentralized agents to have a more expressive joint policy by allowing them to agree upon a strategy by sampling from $\alpha_t$.
\noindent\textbf{Addressing challenge 2: \mbox{\sc{CORDIAL}}\xspace. } %
We encourage agents to rapidly learn to choose coordinated actions via a new loss. In particular,
letting $\Pi_t$ be the joint policy of our agents, we propose the
\emph{coordination loss} (\mbox{\sc{CORDIAL}}\xspace)
\begin{align}
\text{CL}_\beta(S_t,\Pi_t) = -\beta \cdot \langle S_t, \log(\Pi_t) \rangle \ / \sum_{1\leq i,j\leq |\cA|} (S_t)_{ij},
\label{eq:loss}
\end{align}
where $\log$ is applied element-wise, $\langle * , * \rangle$ is the usual Frobenius inner product, and $S_t$ is defined in \secref{sec:task}. Notice that \mbox{\sc{CORDIAL}}\xspace encourages agents to have a near
uniform policy over the actions which are coordinated.
We use this loss to replace the standard entropy encouraging loss in policy gradient algorithms (\eg, the A3C algorithm~\cite{MnihEtAlPMLR2016}). Similarly to the
parameter for the entropy loss in A3C, $\beta$ is chosen to be
a small positive constant so as to not overly discourage
learning.
Note that the coordination loss is less
meaningful when $\Pi_t = \pi^1\otimes \cdots \otimes \pi^N$, \ie, when $\Pi_t$ is rank-one.
For instance, suppose that $S_t$ has
ones along the diagonal, and zeros elsewhere, so that we wish to
encourage the agents to all take the same action. In this case it is
straightforward to show that
$\text{CL}_\beta(S_t,\Pi_t) = -\beta \sum_{i=1}^N \sum_{j=1}^M (1/M)
\log\pi^i_{t}(a^j)$ so that $\text{CL}_\beta(S_t,\Pi_t)$ simply
encourages each agent to have a uniform distribution over its actions and thus actually encourages the agents to place a large amount of probability mass on uncoordinated actions. Indeed, \tabref{tab:cl_study} shows that using \mbox{\sc{CORDIAL}}\xspace without \mbox{\sc{SYNC}}\xspace leads to poor results.
\section{Supplementary Material
} \label{sec:supp}
This supplementary material provides:
\begin{itemize}
\item[\ref{sec:action-restrictions}] The conditions for a collection of actions to be considered \textit{coordinated}.
\item[\ref{sec:rank-one-challenge-example}] An example showing that standard independent multi-agent action sampling makes it impossible to, even in principle, obtain an optimal joint policy
\item[\ref{sec:extra-training-details}] Training details including hyperparameter choices, hardware configurations, and reward structure. We also discuss our upgrades to AI2-THOR\xspace.
\item[\ref{sec:quant-eval-extra-details}] Additional discussion, tables, and plots regarding our quantitative results.
\item[\ref{sec:qualitative-extra-details}] Additional discussion, tables, and plots of our qualitative results including a description of our supplementary video as well as an in-depth quantitative evaluation of communication learned by our agents.
\end{itemize}
\subsection{Action restrictions}
\label{sec:action-restrictions}
We now comprehensively describe the restrictions defining when actions taken by agents are globally consistent with one another. In the following we will, for readability, focus on the two agent setting. All conditions defined here easily generalize to any number of agents. Recall the sets $\mathcal{A}^{\text{NAV}}, \mathcal{A}^{\textsc{MWO}}, \mathcal{A}^{\textsc{MO}}$, and $\mathcal{A}^{\textsc{RO}}$ defined in \secref{sec:task}. We call these sets the \emph{modalities of action}. Two actions $a^1,a^2\in\cA$ are said to be of the same modality if they both are an element of the same modality of action. Let $a^1$ and $a^2$ be the actions chosen by the
two agents. Below we describe the conditions when $a^1$ and $a^2$ are considered \emph{coordinated}. If the agents' actions are uncoordinated, both actions fail and
no action is taken for time $t$. These conditions are summarized in \figref{fig:coordination_matrix_furnmove}.
\noindent\textbf{Same action modality.} A first necessary, but not
sufficient, condition for successful coordination is that the agents
agree on the modality of action to perform. Namely that both $a^1$ and
$a^2$ are of the same action modality. Notice the block diagonal structure in \figref{fig:coordination_matrix_furnmove}.
\noindent\textbf{No independent movement.} Our second condition models
the intuitive expectation that if one agent wishes to reposition
itself by performing a single-agent navigational action, the other
agent must remain stationary. %
Thus, if $a^1,a^2\in\mathcal{A}^{\text{NAV}}$, then
$(a^1,a^2)$ are coordinated if and only if one of $a^1$ or $a^2$ is a \textsc{Pass} action. The $\{1,2,3,4\}^2$ entries of the matrix in \figref{fig:coordination_matrix_furnmove} show coordinated pairs of single-agent navigational actions.
\noindent\textbf{Orientation synchronized object movement.} Suppose that
both agents wish to move (with) the object in a direction so that
$a^1,a^2\in\mathcal{A}^{\textsc{MWO}}$ or
$a^1,a^2\in\mathcal{A}^{\textsc{MO}}$. Note that, as actions are
taken from an egocentric perspective, it is possible, for example,
that moving ahead from one agent's perspective is the same as moving
left from the other's. This condition requires that the direction specified
by both of the agents is consistent globally. Hence $a^1,a^2$ are
coordinated if and only if the direction specified by both
actions is the same in a global reference frame. For example, if both agents
are facing the same direction this condition requires that
$a^1=a^2$ while if the second agent is rotated 90 degrees
clockwise from the first agent then $a^1=\textsc{MoveObjectAhead}$ will
be coordinated if and only if $a^2=\textsc{MoveObjectLeft}$. See the multicolored 4$\times$4 blocks in \figref{fig:coordination_matrix_furnmove}.
\noindent\textbf{Simultaneous object rotation.} For the lifted
object to be rotated, both agents must rotate it in the same direction in a global reference frame. As we only allow the
agents to rotate the object in a single direction (clockwise) this
means that %
$a^1=\textsc{RotateObjectRight}$ %
requires
$a^2=a^1$. See the (9, 9) entry of the matrix in \figref{fig:coordination_matrix_furnmove}.
While a pair of uncoordinated actions are always unsuccessful, it need not be true that a pair of coordinated actions is successful.
A pair of coordinated actions will be unsuccessful in two cases: performing the action pair would result in (a) an agent, or the
lifted object, colliding with one another or another object in the
scene; or (b) an agent moving to a position more than 0.76m from the lifted object. Here (a) enforces the physical constraints of the environment while (b) makes the
intuitive requirement that an agent has a finite reach and cannot lift an object when being far away.
\subsection{Challenge 1 (rank-one joint policies) example}
\label{sec:rank-one-challenge-example}
We now illustrate how requiring two agents to independently sample actions from marginal policies can result in failing to capture an optimal, high-rank, joint policy.
Consider two agents $A^1$ and $A^2$ who must work together to play
rock-paper-scissors (RPS) against some adversary $E$. In particular, our game takes place in a single timestep where each agent $A^i$, after perhaps communicating with the other agent, must choose some action $a^i\in\cA=\{R, P, S\}$. During this time the adversary also chooses some action $a^E \in \cA$. Now, in our game, the pair of agents $A^1,A^2$ lose if they choose different actions (\ie, $a^1\not=a^2$), tie with the adversary if all players choose the same action (\ie, $a^1=a^2=a^E$), and finally win or lose if they jointly choose an action that beats or losses against the adversary's choice following the normal rules of RPS (\ie, win if $(a^1,a^2,a^E)\in\{(R,R,S),$ $(P,P,R),$ $(S,S,P)\}$, lose if $(a^1,a^2,a^E)\in\{(S,S,R),$ $(R,R,P),$ $(P,P,S)\}$).
Moreover, we consider the challenging setting where $A^1,A^2$
communicate in the open so that the adversary can view their joint policy $\Pi$
before choosing the action it wishes to take. Notice that we've dropped the $t$ subscript on $\Pi$ as there is only a single timestep. Finally, we treat this game as zero-sum so that our agents obtain a reward of 1 for victory, 0 for a tie, and -1 for a loss. We refer to the optimal joint policy as $\Pi^*$. If the agents operate in a decentralized manner using their own (single) marginal policies, their effective rank-one joint policy equals $\Pi = \pi^1\otimes\pi^2$.\\
\noindent\textbf{Optimal joint policy:} It is well known, and easy to show, that
the optimal joint policy equals $\Pi^* = I_{3} / 3$, where $I_{3}$ is the identity matrix of size $3$. Hence, the agents take
multi-action ($R, R$), ($P,P$), or ($S,S$)
with equal probability obtaining an expected reward of zero.\\
\noindent\textbf{Optimal rank-one joint policy:} $\Pi^*$ (the optimal joint policy) is of rank three and thus cannot be captured by $\Pi$ (an outer product of marginals). Instead, brute-force symbolic minimization, using Mathematica \cite{Mathematica}, shows that an optimal strategy for $A^1$ and $A^2$ is to let $\pi^1=\pi^2$ with
\begin{align}
\pi^1(R) &= 2-\sqrt{2} \approx 0.586, \\
\pi^1(P) &= 0, \text{ and }\\
\pi^1(S) &= 1 - \pi^1(R) \approx 0.414.
\end{align}
The expected reward from this strategy is $5 - 4\sqrt{2}\approx- .657$, far
less than the optimal expected reward of $0$.
\subsection{Training details}
\label{sec:extra-training-details}
\subsubsection{Centralized agent.}
\figref{fig:central_model} provides an overview of the architecture of the centralized agent. The final joint policy is constructed using a single linear layer applied to a hidden state. As this architecture varies slightly when changing the number of agents and the environment (\ie, AI2-THOR\xspace or our gridworld variant of AI2-THOR\xspace) we direct anyone interested in exact replication to our codebase.
\begin{figure}[t]
\centering
\includegraphics[trim={0 0 2cm 0},clip,width=\linewidth]{arxiv_figs/model_central_v1.pdf}
\caption{\textbf{Central model architecture.} The central backbone observes the aggregate of all agents' observations. Moreover, the actor in the central model explicitly captures the joint policy distribution.}
\label{fig:central_model}
\end{figure}
\subsubsection{AI2-THOR upgrades.}
As we described in \secref{sec:experiments} we have made several upgrades to AI2-THOR\xspace in order to make it possible to run our \mbox{\sc{FurnMove}}\xspace task. These upgrades are described below. \\
\noindent\textbf{Implementing \mbox{\sc{FurnMove}}\xspace methods in AI2-THOR\xspace's Unity codebase.} The AI2-THOR\xspace simulator has been built using C\# in Unity. While multi-agent support exists in AI2-THOR\xspace, our \mbox{\sc{FurnMove}}\xspace task required implementing a collection of new methods to support randomly initializing our task and moving agents in tandem with the lifted object. Initialization is accomplished by a randomized search procedure that first finds locations in which the lifted television can be placed and then determines if the agents can be situated around the lifted object so that they are sufficiently close to the lifted object and looking at it. Implementing the joint movement actions (recall $\mathcal{A}^{\textsc{MWO}}$) required checking that all agents and objects can be moved along straight-line paths without encountering collisions. \\
\noindent\textbf{Top-down Gridworld Mirroring AI2-THOR\xspace.} To enable fast prototyping and comparisons between differing input modalities, we built an efficient gridworld mirroring AI2-THOR\xspace. See \figref{fig:grid_vision_thor_comparison} for a side-by-side comparison of AI2-THOR\xspace and our gridworld. This gridworld was implemented primarily in Python with careful caching of data returned from AI2-THOR\xspace.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{./arxiv_figs/vision-grid-comparison.pdf}
\caption{\textbf{Directly comparing visual AI2-THOR\xspace with our gridworld.} The same scene with identical agent, TV, and TV-stand, positions in AI2-THOR\xspace and our gridworld mirroring AI2-THOR\xspace. Gridworld agents receive clean, task-relevant, information directly from the environment while, in AI2-THOR\xspace, agents must infer this information from complex images.}
\label{fig:grid_vision_thor_comparison}
\end{figure}
\subsubsection{Reward structure.}
Rewards are provided to each agent individually at every step. These rewards include: (a) $+1$ whenever the lifted object is moved closer, in Euclidean distance, to the goal object than it had been previously in the episode, (b) a constant $-0.01$ step penalty to encourage short trajectories, and (c) a penalty of $-0.02$ whenever the agents action fails. The minimum total reward achievable for a single agent is $-7.5$
corresponding to making only failed actions, while the maximum total reward equals $0.99 \cdot d$ where $d$ is the total number of steps it would take to move the lifted furniture directly to the goal avoiding all obstructions. Our models are trained to maximize the expected discounted cumulative gain with discounting factor $\gamma=0.99$.
\subsubsection{Optimization and learning hyperparameters.}
For all tasks, we train our agents using reinforcement learning, particularly the popular A3C algorithm~\cite{MnihEtAlPMLR2016}. For {\mbox{\sc{FurnLift}}\xspace}, we follow~\cite{jain2019CVPRTBONE} and additionally use a warm start via imitation learning (DAgger~\cite{RossAISTATS2011}). When we deploy the coordination loss ({\mbox{\sc{CORDIAL}}\xspace}), we modify the A3C algorithm by replacing the entropy loss with the coordination loss {\mbox{\sc{CORDIAL}}\xspace} defined in~\equref{eq:loss}.
In our experiments we anneal the $\beta$ parameter from a starting value of $\beta = 1$ to a final value of $\beta=0.01$ over the first $5000$ episodes of training. We use an ADAM optimizer with a learning rate of $10^{-4}$, momentum parameters of $0.9$ and $0.999$, with optimizer statistics shared across processes. Gradient updates are performed in an unsynchronized fashion using a HogWild!~style approach \cite{RechtNIPS2011}. Each episode has a maximum length of $250$ total steps per agent. Task-wise details follow:\\
\begin{itemize}
\item {\mbox{\sc{FurnMove}}\xspace}: Visual agents for {\mbox{\sc{FurnMove}}\xspace} are trained for $500,000$ episodes, across $8$ TITAN V or TITAN X GPUs with $45$ workers and take approximately three days to train.
\item {\mbox{Gridworld-\sc{FurnMove}}\xspace}: Agents for {\mbox{Gridworld-\sc{FurnMove}}\xspace} are trained for 1,000,000 episodes using $45$ workers. Apart from parsing and caching the scene once, gridworld agents do not need to render images. Hence, we train the agents with only $1$ G4 GPU, particularly the \texttt{g4dn.16xlarge} virtual machine on AWS. Agents (\ie, two) for {\mbox{Gridworld-\sc{FurnMove}}\xspace} take approximately 1 day to train.
\item {\mbox{Gridworld-\sc{FurnMove}}\xspace}-3Agents: Same implementation as above, except that agents (\ie, three) for {\mbox{Gridworld-\sc{FurnMove}}\xspace}-3Agents take approximately 3 days to train. This is due to an increase in the number of forward and backward passes and a CPU bottleneck. Due to the action space blowing up to $|\mathcal{A}|\times|\mathcal{A}|\times|\mathcal{A}| = 2197$ (\vs 169 for two agents), positive rewards become increasingly sparse. This leads to grave inefficiency in training, with no learning for $\sim$500k episodes. To overcome this, we double the positive rewards for the RL formulation for all methods within the three agent setup.
\item {\mbox{\sc{FurnLift}}\xspace}: We adhere to the exact training procedure laid out by Jain~\etal~\cite{jain2019CVPRTBONE}. Visual agents for {\mbox{\sc{FurnLift}}\xspace} are trained for 100,000 episodes with the first 10,000 being warm started with a DAgger-styled imitation learning. Reinforcement learning (A3C) takes over after the warm-start period.
\end{itemize}
\subsubsection{Integration with other MARL methods.} As mentioned in~\secref{sec:related_work}, our contributions are orthogonal to the RL method deployed. Here we give some pointers for integration with a deep Q-Learning and a policy gradient method.\\
\noindent\textbf{QMIX.} While we focus on policy-gradients and QMIX~\cite{rashid2018qmix} uses Q-learning, we can formulate a SYNC for Q-Learning (and QMIX). Analogous to an actor with multiple policies, consider a value head where each agent's Q-function $Q_i$ is replaced by a collection of Q-functions $Q^a_i$ for $a\in A$. Action sampling is done stage-wise, i.e. agents jointly pick a strategy as $\arg\max_a Q_{SYNC}($communications$, a)$, and then individually choose action $\arg\max_{u^i} Q^a_i(\tau^i, u^i)$. These $Q^a_i$ in turn can incorporated into the QMIX mixing network.\\
\noindent\textbf{COMA/MADDPG.} Both these policy gradient algorithms utilize a centralized critic. Since our contributions focus on the actor head, we can directly replace their per-agent policy with our SYNC policies and thus benefit directly from the counterfactual baseline in COMA~\cite{FoersterAAAI2018} or the centralized critic in MADDPG~\cite{LoweNIPS2017}.
\subsection{Quantitative evaluation details}
\label{sec:quant-eval-extra-details}
\subsubsection{Confidence intervals for metrics reported.} In the main paper, we mentioned that we mark the best performing decentralized method in \textbf{bold} and \hl{highlight it in green} if it has non-overlapping $95$\% confidence intervals. In this {
\section{The furniture moving task (\mbox{\sc{FurnMove}}\xspace)} \label{sec:task}
We describe our new multi-agent task \mbox{\sc{FurnMove}}\xspace, grounded in the real-world experience of moving furniture. We begin by introducing notation.
\noindent\textbf{RL background and notation.}
Consider $N\geq 1$ collaborative agents $A^1$, $\ldots,$ $A^N$. At every timestep $t\in\mathbb{N}=\{0,1,\ldots\}$ the agents, and environment, are
in some state $s_t\in\mathcal{S}$ and each agent $A^i$ obtains an
observation $o^i_t$ recording some partial information about
$s_t$. For instance, $o^i_t$ might be the egocentric visual view of an
agent $A^i$ embedded in some simulated environment. %
From observation $o^i_t$ and history $h^i_{t-1}$, which records prior observations and decisions made by the agent, each agent $A^i$ forms a policy
$\pi^i_t:\cA\to[0,1]$ where $\pi^i_t(a)$ is the probability that agent $A^i$ chooses to take action $a\in\cA$ from a finite set of options $\cA$ at time $t$.
After the agents execute their respective
actions $(a^1_t,\ldots,a^N_t)$, which we call a \emph{multi-action}, they enter
a new state $s_{t+1}$ and receive individual rewards
$r^1_{t},\ldots,r^N_{t}\in\mathbb{R}$. For more on RL
see~\cite{SuttonMIT1998,MnihNature2015,MnihEtAlPMLR2016}.
\noindent\textbf{Task definition.} \mbox{\sc{FurnMove}}\xspace is set in the near-photorealistic and physics enabled simulated environment AI2-THOR\xspace~\cite{ai2thor}. In \mbox{\sc{FurnMove}}\xspace, $N$ agents collaborate to move a lifted object through an indoor environment with the goal of placing this object above a visually distinct target as illustrated in \figref{fig:teaser}. Akin to humans moving large items, agents must navigate around other furniture and frequently walk in-between obstacles on the floor. %
In \mbox{\sc{FurnMove}}\xspace, each agent at every timestep receives an egocentric observation (a $3\times 84 \times 84$ RGB image) from AI2-THOR\xspace. In addition, agents are allowed to communicate with other agents at each timestep via a low bandwidth communication channel. Based on their local observation and communication, each agent must take an action from the set $\mathcal{A}$.
The space of actions $\mathcal{A} = \mathcal{A}^{\text{NAV}}\cup \mathcal{A}^{\textsc{MWO}}\cup \mathcal{A}^{\textsc{MO}}\cup \mathcal{A}^{\textsc{RO}}$ available to an agent is comprised
of the
four single-agent navigational actions $\mathcal{A}^{\text{NAV}} = \{$\textsc{MoveAhead}, \textsc{RotateLeft}, \textsc{RotateRight}, \textsc{Pass}$\}$ used to move the agent independently,
four actions $\mathcal{A}^{\textsc{MWO}}=\{$\textsc{MoveWithObjectX} $\mid X\in \{$\textsc{Ahead, Right, Left, Back}$\}\}$ used to move the lifted object and the agents simultaneously in the same direction,
four actions $\mathcal{A}^{\textsc{MO}}=\{$\textsc{MoveObjectX}$ \mid X\in \{$\textsc{Ahead, Right, Left, Back}$\}\}$ used to move the lifted object while the agents stay in place,
and a single action used to rotate the lifted object clockwise $\mathcal{A}^{\textsc{RO}}= \{\textsc{RotateObjectRight}\}$.
We assume that all movement actions for agents and the lifted object result in a displacement of 0.25 meters (similar to~\cite{jain2019CVPRTBONE,habitat19iccv}) and all rotation actions result in a rotation of 90 degrees (counter-)clockwise when viewing the agents from above. %
\begin{figure}[t]
\centering
{
\phantomsubcaption\label{fig:coordination_matrix_furnmove}
\phantomsubcaption\label{fig:coordination_matrix_furnlift}
}
\begin{tabular}{@{\hskip0pt}c@{\hskip15pt}c@{\hskip0pt}}
\includegraphics[height=5.5cm]{arxiv_figs/coordination_matrix_move_v3.pdf} &
\includegraphics[height=5.5cm]{arxiv_figs/coordination_matrix_lift_v3.pdf}
\\
(a) {\mbox{\sc{FurnMove}}\xspace} & (b) {\mbox{\sc{FurnLift}}\xspace}
\end{tabular}
\caption{\textbf{Coordination matrix for tasks.} The matrix $S_t$ records the validity of multi-action $(a^1,a^2)$ for different relative orientations of agents \textcolor{red}{$A^1$} \& \textcolor{cyan}{$A^2$}. (a) Overlay of $S_t$ for all four relative orientation of two agents, for {\mbox{\sc{FurnMove}}\xspace}. Notice that only $16/169=9.5\%$ multi-actions are coordinated at any given relative orientation, (b) {\mbox{\sc{FurnLift}}\xspace} where single agent actions are always valid and coordination is needed only for \textsc{PickUp} action, i.e. at least $16/25=64\%$ actions are always valid.}
\label{fig:coordination_matrix}
\end{figure}
Close and on-going collaboration is required in \mbox{\sc{FurnMove}}\xspace\ due to restrictions on the set of actions which can be successfully completed jointly by all the agents. These restrictions reflect physical constraints: for instance, if two people attempt to move in opposite directions while carrying a heavy object they will either fail to move or drop the object. For two agents, we summarize these restrictions using the \emph{coordination matrix} shown in Fig.~\ref{fig:coordination_matrix_furnmove}. For comparison, we include a similar matrix in \figref{fig:coordination_matrix_furnlift} corresponding to the \mbox{\sc{FurnLift}}\xspace task from~\cite{jain2019CVPRTBONE}. We defer a more detailed discussion of these restrictions to \secref{sec:action-restrictions} of the
\section{Conclusion} \label{sec:conclusion}
We introduce \mbox{\sc{FurnMove}}\xspace, a collaborative, visual, multi-agent task requiring close coordination between agents and develop novel methods that allow for moving beyond existing marginal action sampling procedures, these methods lead to large gains across a diverse suite of metrics.
\section{Models}
\label{sec:baselines}
We study four distinct policy types: \emph{central}, \emph{marginal}, \emph{marginal w/o comm}, and {\emph{SYNC}\xspace}. \emph{Central} samples actions from a joint policy generated by a central agent with access to observations from all agents.
While often unrealistic in practice due to communication bottlenecks, \emph{central} serves as an informative baseline.
\emph{Marginal} follows prior work, \eg, \cite{jain2019CVPRTBONE}: each agent independently samples its actions from its individual policy after communication. \emph{Marginal w/o comm} is identical to \emph{marginal} but does not permit agents to communicate explicitly (agents may still see each other). %
Finally, {\emph{SYNC}\xspace} is our newly proposed policy described in \secref{sec:method}. %
For a fair comparison,
all decentralized agents (\ie, \emph{SYNC}\xspace, \emph{marginal}, and \emph{marginal w/o comm}), use the same TBONE backbone architecture from \cite{jain2019CVPRTBONE}, see \figref{fig:model}. We have ensured that parameters are fairly balanced so that our proposed \emph{SYNC}\xspace has close to (and never more) parameters than the \emph{marginal} and \emph{marginal w/o comm} nets.
Note, we train \emph{central} and {\emph{SYNC}\xspace} with {\mbox{\sc{CORDIAL}}\xspace}, and the \emph{marginal} and \emph{marginal w/o comm} without it. This choice is mathematically explained in~\secref{sec:method} and empirically validated in~\secref{sec:quantitative}.
\noindent \textbf{Architecture details:} %
For readability we describe the policy and value net for the 2 agent setup while noting that it can be trivially extended to any number of agents.
As noted above, decentralized agents use the TBONE backbone from~\cite{jain2019CVPRTBONE}. Our primary architectural novelty extends TBONE to \mbox{\sc{SYNC}}\xspace-policies. An overview of the TBONE backbone and differences between sampling with \emph{marginal} and \emph{SYNC}\xspace policies is shown in \figref{fig:model}.
As a brief summary of TBONE, agent $i$ observes at time $t$ inputs $o_t^i$, \ie, %
a $3\times 84 \times 84$ RGB image returned from AI2-THOR\xspace which represents the $i$-th agent's egocentric view. For each agent, the observation is encoded by a four layer CNN and combined with an agent specific learned embedding (that encodes the ID of that agent) along with the history embedding $h^i_{t-1}$. The resulting vector is fed into a long-short-term-memory (LSTM) \cite{HochreiterNC1997} unit to produce a $512$-dimensional embedding $\tilde{h}^i_t$ corresponding to the $i^\text{th}$ agent.
The agents then undergo two rounds of communication resulting in two final hidden states $h^1_t, h^2_t$ and messages $c^i_{t,j}\in\bR^{16}$, $1\leq i,j\leq 2$ with message $c^i_{t,j}$ being produced by agent $i$ in round $j$ and then sent to the other agent in that round. In \cite{jain2019CVPRTBONE}, the value of the agents' state as well as logits corresponding to the policy of the agents are formed by applying linear functions to $h^1_t, h^2_t$.
We now show how \mbox{\sc{SYNC}}\xspace can be integrated into TBONE to allow our agents to represent high rank joint distributions over multi-actions (see Fig.~\ref{fig:model}). First each agent computes the logits corresponding to $\alpha_t$. This is done using a $2$-layer MLP applied to the messages sent between the agents, at the second stage. In particular,
$\alpha_t = {\bf W}_3\ \text{ReLU}({\bf W}_2\ \text{ReLU}({\bf W}_1\ [c^1_{t,2}; c^2_{t,2}] + {\bf b}_1) + {\bf b}_2) + {\bf b}_3$ where ${\bf W}_1\in\bR^{64\times 32}, {\bf W}_2\in\bR^{64\times 64}$,
${\bf W}_3\in\bR^{m\times 64}$,
${\bf b}_1\in\bR^{32},{\bf b}_2\in\bR^{64}$, and ${\bf b}_3\in\bR^m$
are a learnable collection of weight matrices and biases. After computing $\alpha_t$ we compute a collection of policies $\pi^i_{t, 1}, \dots, \pi^i_{t, m}$ for $i\in\{1,2\}$. Each of these policies is computed following the TBONE architecture but using $m-1$ additional, and learnable, linear layers per agent. %
\section{Introduction}
\label{sec:introduction}
Collaboration is the defining principle of our society. %
Humans have refined strategies to efficiently collaborate, developing verbal, deictic, and kinesthetic means. In contrast, progress towards enabling artificial embodied agents to learn collaborative strategies is still in its infancy. Prior work mostly studies collaborative agents in grid-world like environments. Visual, multi-agent, collaborative tasks have not been studied until very recently~\cite{das2018tarmac,jain2019CVPRTBONE}. While existing tasks are well designed to study some aspects of collaboration, they often don't require agents to closely collaborate \emph{throughout} the task. Instead such tasks either require initial coordination (distributing tasks) followed by almost independent execution, or collaboration at a task's end (\eg, verifying completion). Few tasks require frequent coordination, and we are aware of none within a visual setting.
To study our algorithmic ability to address tasks which require close and frequent collaboration,
we introduce the furniture moving (\mbox{\sc{FurnMove}}\xspace) task (see Fig.~\ref{fig:teaser}), set in the %
AI2-THOR\xspace environment. Given only their egocentric visual observations, agents jointly hold a lifted piece of furniture in a living room scene and must collaborate to move it to a visually distinct goal location. %
As a piece of furniture cannot be moved without both agents agreeing on the direction, agents must explicitly \emph{coordinate at every timestep}. Beyond coordinating actions, high performance in our task requires agents to visually anticipate possible collisions, handle occlusion due to obstacles and other agents, and estimate free space. %
Akin to the challenges faced by a group of roommates relocating a widescreen television, this task necessitates extensive and ongoing coordination amongst all agents at every time step.
In prior work, collaboration between multiple agents has been enabled primarily by (i) sharing observations or (ii) learning low-bandwidth communication. (i) is often implemented using a \emph{centralized} agent, \ie, a single agent with access to all observations from all agents~\cite{boutilier1999sequential,peng2017multiagent,usunier2016episodic}. While effective it is also unrealistic: the real world poses restrictions on communication bandwidth, latency, and modality. %
We are interested in the more realistic \emph{decentralized} setting enabled via option (ii). %
This is often
implemented by one or more rounds of message passing between agents before they choose their actions~\cite{FoersterNIPS2016,LoweNIPS2017,jain2019CVPRTBONE}. %
Training decentralized agents when faced with \mbox{\sc{FurnMove}}\xspace's requirement of coordination at each timestep leads to two technical challenges. Challenge 1: as each agent independently samples an action from its policy at every timestep, the joint probability tensor of all agents' actions at any given time is rank-one. This severely limits which multi-agent policies are representable. %
Challenge 2: the number of possible mis-steps or failed actions increases dramatically when requiring that agents closely coordinate with each other, complicating training. %
Addressing challenge 1, we introduce \mbox{\sc{SYNC}}\xspace{} (\textbf{S}ynchronize \textbf{Y}our actio\textbf{N}s \textbf{C}oherently) policies
which permit expressive (\ie, beyond rank-one) joint policies for decentralized agents while using interpretable communication. %
To ameliorate challenge 2 we introduce the \textbf{C}o\textbf{ordi}n\textbf{a}tion \textbf{L}oss (\mbox{\sc{CORDIAL}}\xspace) that replaces the standard entropy loss in actor-critic algorithms and guides agents away from actions that are mutually incompatible. %
A 2-agent system using \mbox{\sc{SYNC}}\xspace and \mbox{\sc{CORDIAL}}\xspace obtains a 58\% success rate on test scenes in \mbox{\sc{FurnMove}}\xspace, an impressive absolute gain of 25 %
percentage points %
over the baseline from \cite{jain2019CVPRTBONE} (76\% relative gain). In a 3-agent setting, this difference is even more extreme.
In summary, our contributions are: (i) \mbox{\sc{FurnMove}}\xspace, a new multi-agent embodied task that demands ongoing coordination, (ii) \mbox{\sc{SYNC}}\xspace, a collaborative mechanism that permits expressive joint action policies for decentralized agents, (iii) \mbox{\sc{CORDIAL}}\xspace, a training loss for multi-agent setups which, when combined with \mbox{\sc{SYNC}}\xspace, leads to large gains, and (iv) improvements to the open-source AI2-THOR environment including a $16\times$ faster gridworld equivalent enabling fast prototyping.
\section{Related work}
\label{sec:related_work}
We start by reviewing single agent embodied AI tasks followed by non-visual Multi-Agent RL (MARL) and end with visual MARL.
\noindent\textbf{Single-agent embodied systems:} Single-agent embodied systems have been considered extensively in the literature. For instance,
literature on visual navigation, \ie, locating an object of interest given only visual input, %
spans geometric and learning based methods.
Geometric approaches have been proposed separately for mapping and planning phases of navigation. Methods entailing structure-from-motion and SLAM~\cite{tomasi1992shape,frahm2016structurefrommotion,thorpe2000structure,cadena2016past,smith1986on,smith1986estimating} were used to build maps. Planning algorithms on existing maps~\cite{CannyMIT1988,KavrakiRA1996,Lavalle2000} and combined mapping \& planning~\cite{elfes1989using,kuipers1991byun,konolige2010viewbased,fraundorfer2012visionbased,aydemir2013active} are other related research directions.
While these works propose geometric approaches, the task of navigation can be cast as a reinforcement learning (RL) problem, mapping pixels to policies in an end-to-end manner. RL approaches~\cite{oh2016control,abel2016exploratory,daftry2016learning,giusti2016he,kahn2017plato,toussaint2003learning,mirowski2017learning,tamar2016value} have been proposed to address navigation in synthetic layouts like mazes, arcade games and other visual environments~\cite{wymann2013torcs,bellemare2013the,kempka2016vizdoom,lerer2016learning,johnson2016the,SukhbaatarARXIV2015}. %
Navigation within photo-realistic environments%
~\cite{BrodeurARXIV2017,SavvaARXIV2017Minos,Chang3DV2017Matterport,ai2thor,xia2018gibson,stanford2d3d,GuptaCVPR2018,xia2019interactive,habitat19iccv} led development of \textit{embodied} AI agents. The early work~\cite{ZhuARXIV2016} addressed object navigation (find an object given an image) in AI2-THOR\xspace. Soon after,~\cite{GuptaCVPR2018} showed how imitation learning permits agents to learn to build a map from which they navigate. Methods also investigate the utility of topological and latent memory maps~\cite{GuptaCVPR2018,savinov2018semiparametric,henriques2018mapnet,wu2019bayesian}, graph-based learning~\cite{wu2019bayesian,yang2018visual}, meta-learning~\cite{wortsman2019learning}, unimodal baselines~\cite{thomason2019shifting}, 3D point clouds~\cite{Wijmans2019EQAPhoto}, and effective exploration~\cite{wang2019reinforced,savinov2018semiparametric,Chaplot2020Explore,ramakrishnan2020exploration} to improve embodied navigational agents. Embodied navigation also aids AI agents to develop behavior such as instruction following~\cite{HillARXIV2017,anderson2018vision,Suhr2019CerealBar,wang2019reinforced,anderson2019NeuripsChasing}, city navigation~\cite{chen2019touchdown,mirowski2018learningcity,mirowski2019streetlearn,de2018talkthewalk}, question answering~\cite{DasCVPR2018,DasECCV2018,GordonCVPR2018,Wijmans2019EQAPhoto,das2020probing}, and active visual recognition~\cite{yang2018visualsemantic,yang2019embodied}. Recently, with visual and acoustic rendering, agents have been trained for audio-visual embodied navigation~\cite{chen2019audio,gao2020visualechoes}.
In contrast to the above single-agent embodied tasks and approaches, we focus on collaboration between multiple embodied agents. Porting the above single-agent architectural novelties (or a combination of them) to multi-agent systems such as ours is an interesting direction for future work.
\noindent\textbf{Non-visual MARL:}
Multi-agent reinforcement learning (MARL) is challenging due to non-stationarity when learning. Multiple methods have been proposed to address resulting issues~\cite{TanICML1993,TesauroNIPS2004,TampuuPLOS2017,FoersterARXIV2017}. For instance, permutation invariant critics have been developed recently~\cite{LiuCORL2019}. In addition, for MARL, cooperation and competition between agents has been studied~\cite{LauerICML2000,Panait2005,MatignonIROS2007,Busoniu2008,OmidshafieiARXIV2017,GuptaAAMAS2017,LoweNIPS2017,FoersterAAAI2018,LiuCORL2019}.
Similarly, communication and language in the multi-agent setting has been investigated~\cite{GilesICABS2002,KasaiSCIA2008,BratmanCogMod2010,MeloMAS2011,LazaridouARXIV2016,FoersterNIPS2016,SukhbaatarNIPS2016,MordatchAAAI2018,Baker2019EmergentTU} in maze-based setups, tabular tasks, or Markov games. These algorithms mostly operate on low-dimensional observations such as kinematic measurements (position, velocity, \etc) and top-down occupancy grids. For a survey of centralized and decentralized MARL methods, kindly refer to~\cite{zhang2019multi}.
Our work differs from the aforementioned MARL works in that we consider complex visual environments. Our contribution of SYNC-Policies is largely orthogonal to RL loss function or method. For a fair comparison to~\cite{jain2019CVPRTBONE}, we used the same RL algorithm (A3C) but it is straightforward to integrate SYNC into other MARL methods~\cite{rashid2018qmix,FoersterAAAI2018,LoweNIPS2017} (for details, see~\secref{sec:extra-training-details} of the supplement).
\noindent\textbf{Visual MARL:}
Recently, Jain \etal~\cite{jain2019CVPRTBONE} introduced a collaborative task for two embodied visual agents, which we refer as \mbox{\sc{FurnLift}}\xspace. In this task, two agents are randomly initialized in an AI2-THOR\xspace living room scene, must visually navigate to a TV, and, in a singe coordinated \textsc{PickUp} action, work to lift that TV up. Note that \mbox{\sc{FurnLift}}\xspace doesn't demand that agents coordinate their actions at each timestep. Instead, such coordination only occurs at the last timestep of an episode. Moreover, as success of an action executed by an agent is independent (with the exception of the \textsc{PickUp} action), a high performance joint policy need not be complex, \ie, it may be near low-rank. More details on this analysis and the complexity of our proposed \mbox{\sc{FurnMove}}\xspace task are provided in \secref{sec:task}.
Similarly, a recent preprint~\cite{chen2019visual} proposes a visual hide-and-seek task, where agents can move independently. Das~\etal~\cite{das2018tarmac} enable agents to learn who to communicate with, on predominantly 2D tasks. In visual environments they study the task where multiple agents parallely navigate to the same object. Jaderberg~\etal~\cite{jaderberg2019human} recently studied the game of Quake III and Weihs~\etal~\cite{weihs2019artificial} develop agents to play an adversarial hiding game in AI2-THOR\xspace. Collaborative perception for semantic segmentation and recognition classification have also been investigated recently~\cite{Liu_2020_CVPR,liu2020who2com}.
To the best of our knowledge, all previous visual or non-visual MARL in the decentralized setting operate with a single marginal probability distribution per agent, \ie, a rank-one joint distribution.
Moreover, \mbox{\sc{FurnMove}}\xspace is the first multi-agent collaborative task in a visually rich domain requiring close coordination between agents at every timestep.
\section{Experiments}
\label{sec:experiments}
\subsection{Experimental setup}
\label{sec:environment}
\noindent\textbf{Simulator.}
We evaluate our models using the AI2-THOR\xspace environment~\cite{ai2thor} with several novel upgrades. First, we introduce new methods which permit to (a) randomly initialize the lifted object and agent locations close to each other and looking towards the lifted object, and (b) simultaneously move agents and the lifted object in a given direction with collision detection.
Secondly, we build a top-down gridworld version of AI2-THOR\xspace for faster prototyping, that is $16\times$ faster than~\cite{jain2019CVPRTBONE}. For details about framework upgrades, see \secref{sec:extra-training-details} of the
|
3,212,635,537,920 | arxiv | \section{introduction}
Composite fermions \cite{Jain89}, the topological bound states of electrons and an even number ($2s$) of quantized vortices, lead to an explanation of the fractional quantum Hall effect \cite{Tsui82,Prange87} at fractions $\nu=n/(2sn\pm 1)$ as the integer quantum Hall effect of composite fermions, and they allow a calculation of the topological and non-topological features of these fractional quantum Hall states\cite{DasSarma07,Heinonen98,Jain07,Halperin20}.
Greiter and Wilczek \cite{Greiter90,Greiter92b,Greiter21} proposed an adiabatic approach, wherein the fractional quantum Hall effect is connected to the integer quantum Hall effect by continuously tuning the strength of the vortex attached to electrons from zero to $2s$, while at the same time varying the external magnetic field in such a manner that the effective magnetic field remains unchanged. The two limiting cases are familiar and well-studied. When the number of attached vortices is zero, we of course have the integer quantum Hall effect of non-interacting electrons. When the number of attached vortices is an even integer, the base particles are composite fermions (CFs), producing the fractional quantum Hall effect of electrons at $\nu=n/(2sn\pm 1)$. This article is concerned with the intermediate states, when the number of vortices attached to each particle is a rational fraction and the base particles are anyons obeying fractional braiding statistics \cite{Leinaas77,Wilczek82}. Recently, a Chern-Simons field theory with fluctuating dynamical gauge field has also been used to forge a bridge connecting the integer and fractional quantum Hall states~\cite{Hansson21}.
We note that the excitations of the fractional quantum Hall states have been predicted to obey fractional braiding statistics~\cite{Halperin84,Arovas84,Jain07}.
In contrast, we are dealing in our study with fictitious anyons designed to interpolate between the integer and fractional quantum Hall states.
The theoretical studies on anyons have been attempted through various methods, including field theories \cite{Iengo92,Chen89,Iengo91,Iengo90,Hansson96}, exact diagonalization \cite{Sporre91,Sporre93,Canright89,Canright89a,Hanna89,Xie90,Hatsugai91,Kudo20,Ouvry19}, density functional theory \cite{Hu21}, wave functions \cite{Wu84,Thouless85,Girvin90,Chin92,Lundholm17,Lundholm13,Fayyazuddin93,Fayyazuddin93b}, and other methods \cite{Laughlin88,Fetter89,Wen90c,Lee91,Chitra92,Li92,Correggi17}. The wave function approach gives an explicit description of the many-particle states and allows direct calculations of both topological and non-topological physical quantities. Earlier, the wave function studies were mainly based on disk geometry \cite{Wu84,Girvin90,Chin92,Lundholm17,Lundholm13}.
In this work, we revisit the problem of constructing an anyon wave function on a torus for general filling factors. The torus geometry offers certain special advantages. One of them is that the torus is compact, which avoids complications from edge states hosted by open boundaries. The shape of the torus and the boundary conditions are tunable, which makes it an ideal geometry to study topological bulk quantities such as Chern number and Hall viscosity. The wave functions for CFs carrying $2s$ vortices have been constructed on a torus in Ref.~\cite{Pu17}. However, unlike in the disk geometry, a generalization from CFs to anyons cannot be accomplished by simply replacing the integer number of attached vortices by a fractional number. The interplay of the periodic boundary conditions and fractional statistics imposes a nontrivial braiding group for anyons on a torus \cite{Einarsson90}, which requires the wave functions for anyons to be multi-component. An alternative way to understand the origin of the multi-component structure has been discussed by Fayyazuddin \cite{Fayyazuddin93}, as arising from the coupling of the gauge field and the particle degrees of freedom. This multi-component structure is consistent with the exact diagonalization results on a lattice Hamiltonian on a torus \cite{Wen92c,Hatsugai91,Kudo20} and also with the Chern-Simons theory \cite{Iengo91,Iengo92,Hosotani92}. Ref.~\cite{Fayyazuddin93b} has studied anyon ground-state wave functions on a torus using the Chern-Simons gauge transformation.
We achieve a construction of trial wave functions for degenerate ground states as well as excited states of anyons for general statistical parameters and filling factors such that the effective filling factor is an integer. These wave functions are more general and have a simpler form than those constructed previously in the literature, reducing to the Jain CF wave functions \cite{Pu17} when the number of attached fluxes to each particle is an even integer. We believe, from experience with the CF theory, that the lowest Landau level (LLL) projections of these wave functions should provide a good account of anyons interacting by a repulsive interaction, such as the Coulomb interaction. However, we have not investigated the quantitative validity of these wave functions. We calculate below variational excitation gaps, as well as several topological properties of the incompressible states of anyons, which are expected to be insensitive to the details of the wave function.
The remainder of the paper is organized as follows. In Sec.~\ref{wf}, we briefly review the braiding group of anyons on a torus. Then we construct a complete set of multi-component anyon wave functions that provide a representation of this braiding group. We also show that our wave functions have the expected ground state degeneracies. In Sec.~\ref{energy}, we calculate the variational values for the charge gaps for anyons interacting via the Coulomb interaction, and we find that the gap is preserved as we tune the number of attached vortices; this supports the view that the process is adiabatic and is also consistent with exact diagonalization findings by Kudo and Hatsugai\cite{Kudo20}. In Secs.~\ref{Chern number} and ~\ref{Hall viscosities}, we calculate the Chern number and the Hall viscosity analytically and numerically. We find that the total Hall viscosity can be viewed as the sum of the Hall viscosities of different factors in the wave function, and it encodes information on the number of filled effective Landau levels and the anyon statistics. We summarize our results in Sec.~\ref{summary}. Our results are consistent with the work by Kudo and Hatsugai \cite{Kudo20}, who have diagonalized a lattice model Hamiltonian in the torus geometry and numerically calculated the ground-state degeneracies, gaps, and Chern numbers.
\section{multi-component anyon wave functions and ground state degeneracies}
\label{wf}
We consider a two-dimensional many-particle system on the surface of a torus with a perpendicular external magnetic field applied. We assume that there are $N$ anyons and $N_\phi$ external magnetic flux quanta. The filling factor of anyons is $\nu=N/N_\phi$. The adiabatic transport of an anyon around another along a closed loop results in a statistical phase $2\theta$ (which defines our anyon).
In what follows, we define
\begin{equation}
\theta=\pi\left(1+{p\over q}\right)=\pi{p'\over q}
\end{equation}
with $p'=p+q$.
When ${\theta\over \pi}$ is an odd (even) integer, the particles are fermions (bosons).
The anyons can be mapped into fermions or bosons
with gauge fluxes attached. Assuming that the base particles are fermions,
the number of effective magnetic flux quanta felt by them (counting both the external magnetic field and the statistical field) is
\begin{equation}
N_\phi^{f}=N_\phi-\left({\theta\over \pi}-1\right)N=N_\phi-{p\over q}N,
\end{equation}
and the effective filling factor for fermions, $\nu_f$, is given by
\begin{equation}
{1\over \nu_f}={1\over \nu}-{\theta\over \pi}+1={1\over \nu}-{p\over q}.
\end{equation}
It reduces to the standard CF theory when $p/q$ is an even integer.
If the base particles are chosen to be bosons instead, the number of effective magnetic flux quanta felt by them is
\begin{equation} N_\phi^{b}=N_\phi-{\theta\over \pi}N=N_\phi-{p'\over q}N,
\end{equation}
and the inverse of the bosonic filling factor is
\begin{equation}
{1\over \nu_b}={1\over \nu}-{\theta\over \pi}={1\over \nu}-{p'\over q}.
\end{equation}
(We note that we use a different convention for the definition of $\theta$ compared with that in Ref.~\cite{Kudo20}. Our $\theta$ corresponds to $2\pi-\theta$ of that paper. We choose our convention because it is more natural for CFs, as $\theta$ is simply $m\pi$ at filling $1/m$ for CFs. Our convention is
consistent with that of Refs.~\cite{Greiter90,Greiter92b}.)
A torus can be mapped into a parallelogram on a complex plane with quasi-periodic boundary conditions imposed. We define the real axis along one edge of the parallelogram and name the length of that edge as $L_1$. The other edge of the parallelogram is defined as $L_2=L_1\tau$, where $\tau=\tau_1+i\tau_2$ is a complex number called the modular parameter \cite{Gunning62} of the torus. In this work, we assume the external magnetic field is $\vec{B}=-B\hat{z}$. Then, in the symmetric gauge,
the LLL wave function is a holomorphic function of the particle coordinates $z_i=x_i+iy_i$ times a Gaussian factor $e^{-{|z|^2\over 4\ell^2}}$, where $\ell=\sqrt{\hbar c/eB}$ is the magnetic length. Later we will also use the reduced particle coordinates $\theta_{1,i}, \theta_{2,i}\in [0,1)$ which are defined through $z_i=L_1\theta_{1,i}+L_2\theta_{2,i}$. The total area of the torus is $V=L_1^2\tau_2=2\pi N_\phi \ell^2$. The quasi-periodic boundary conditions under external magnetic field are defined through magnetic translation operators \cite{Zak64,Brown64}. For Hall viscosities, it is convenient to choose the $\tau$ gauge $(A_x,A_y)=B(y,-{\tau_1\over \tau_2}y)$ \cite{Fremling14,Pu20}. In this gauge, the magnetic translation operators acting on a single particle $z=L_1\theta_1+L_2\theta_2$ are defined as:
\begin{equation}
\label{magnetic translation operator}
t\left(\alpha L_1+\beta L_2\right)=e^{\alpha\partial_1+\beta\partial_2+i2\pi\beta N_{\phi}\theta_1},
\end{equation}
where $\partial_1\equiv{\partial \over {\partial \theta_1}}$ and $\partial_2\equiv{\partial \over {\partial \theta_2}}$ .
Fermions or bosons satisfy the quasi-periodic boundary conditions on a torus:
\begin{equation}
\label{pbc}
t(L_i)\psi(z,\bar{z})=e^{i\phi_i}\psi(z,\bar{z}) \quad i=1,2.
\end{equation}
However, this is not the case for anyons since this equation is inconsistent with the fractional statistics. As shown in Ref.~\cite{Birman69}, the braiding can be accomplished on a torus by wrapping two particles along the two periodic loops. If the periodic boundary conditions are just represented by phases, the braiding statistics can only be integer multiples of $\pi$. Ref.~\cite{Einarsson90,Hatsugai91} have shown that the boundary conditions for anyons on a torus are given by:
\begin{equation}
\label{bg1}
t_j(L_1)\Psi=e^{i\phi_1}e^{-i2j\theta}
\begin{pmatrix}
1&\cdots&\\
&c&\cdots\\
\vdots&\vdots&\vdots\\
\cdots&&c^{q-1}\\
\end{pmatrix}
\Psi
\end{equation}
\begin{equation}
\label{bg2}
t_j(L_2)\Psi=e^{i\phi_2}e^{i2j\theta}
\begin{pmatrix}
0&1&0&\cdots&0\\
0&0&1&\cdots&0\\
\vdots&\vdots&\vdots&\vdots\\
1&0&0&\cdots&0\\
\end{pmatrix}
\Psi,
\end{equation}
with $\theta=\pi{p'\over q}$, $c=e^{i2\pi{p'\over q}}$. The twisted boundary conditions are defined by phases $\phi_1$ and $\phi_2$, which represent $2\pi$ times the number of magnetic flux quanta through the two holes of the torus \cite{Hatsugai91}.
Here $t_j$ is the magnetic translation operator acting on the $j$th particle. $\Psi$ is a $q$-component vector.
For the special case of CFs, $q=1$, and the above boundary conditions are satisfied by the Jain CF wave functions constructed in Ref.~\cite{Pu17}.
In the remaining part of this section, we first find a solution for Eq.~\ref{bg1} and Eq.~\ref{bg2} for ${1\over \nu_f}=1$ by ansatz, just as was done in Ref.~\cite{Haldane85} for Laughlin wave functions. Then we show how to generalize the solution to other fillings. Finally, we show how to obtain wave functions for all degenerate ground states.
\subsection{$\nu_f=1$}
The effective filling factor $\nu_f=1$ is obtained when anyons with statistics $\theta=\pi{p'\over q}$ have a filling factor $\nu={q\over p'}$.
Here, the fermions fill the lowest Landau level, and the ground state
has $q$ components. In what follows, we obtain trial wave functions for the ground and excited states.
We make the following ansatz for the wave function (we use superscript $\alpha$ to label the degeneracy and subscript $k$ to label the component):
\begin{equation}
\label{anyon wf}
\Psi^{(\alpha)}=(\Psi^{(\alpha)}_0,\Psi^{(\alpha)}_1,\Psi^{(\alpha)}_2,\cdots,\Psi^{(\alpha)}_{q-1})^T
\end{equation}
\begin{equation}
\label{wf2}
\Psi^{(\alpha)}_k[z_i]=e^{i\pi\tau N_\phi\sum_i \theta_{2,i}^2}\prod_{i=1}^NJ_iF^{(\alpha)}_k(Z),
\end{equation}
\begin{equation}
\label{J1}
J_i=\prod_{j>i}e^{{p'\over q} \ln \elliptic{1/2}{1/2}{{z_i-z_j\over L_1}}{\tau}}.
\end{equation}
Here $Z=\sum_{i=1}^Nz_i$ is the center-of-mass coordinate, and we use the Jacobi theta function with rational characteristics\cite{Mumford07}; its definition and some of its properties are listed in Appendix \ref{theta-function}. The ansatz wave function has three parts. The first part $e^{i\pi\tau N_\phi\sum_i \theta_{2,i}^2}$
appears in the $\tau$ gauge. The second part $\prod_{i=1}^NJ_i$ is purely made up of the relative coordinates $z_i-z_j$; it is analogous to the Jastrow factor $\prod_{i<j}\left(z_i-z_j\right)$ in the disk geometry. The
coefficient ${p'\over q}$ in the exponential of Eq.~\ref{J1} is fixed by the braiding statistics and defines the number of attached vortices (it is replaced by an even integer for CFs). The last part $F^{(\alpha)}_k(Z)$ is the center-of-mass part, which carries both the degeneracy index and component index. In making this ansatz, we assume that the wave function can be written as a product of the relative part and the center-or-mass part,
which is known to be true for $q=1$, \ie, for the Laughlin states.
Note that Eq.~\ref{J1} has a branch cut. We adopt the convention that when $z\rightarrow z\pm L_1$, $\ln \elliptic{1/2}{1/2}{{z\over L_1}}{\tau}\rightarrow \ln \elliptic{1/2}{1/2}{{z\over L_1}}{\tau}\pm i\pi$, \ie, when the particle coordinate moves to the right (left) across the boundary, the $\ln \vartheta$ function goes up (down) on the Riemann surface.
Now we need to solve for the center-of-mass part $F^{(\alpha)}_k(Z)$. According to Eq.~\ref{bg1} and Eq.~\ref{bg2}, it satisfies
\begin{equation}
\label{Fbg1}
F^{(\alpha)}_k(Z+L_1)=e^{i\left[\phi_1+2\pi{p'\over q}\left(k-{N+1\over 2}\right)\right]}F^{(\alpha)}_k(Z)
\end{equation}
\begin{equation}
\label{Fbg2}
F^{(\alpha)}_k(Z+L_2)=e^{-i\left({2\pi p'Z\over qL_1}-\phi_2-{p'\over q}\pi(N+1)+{\pi p' \tau\over q}\right)}F^{(\alpha)}_{k+1}(Z).
\end{equation}
with $k=0,1,\cdots q-1$ and $F^{(\alpha)}_q(Z)=F^{(\alpha)}_0(Z)$.
We can do a Fourier expansion of $F^{(\alpha)}_k(Z)$ according to Eq.~\ref{Fbg1}
\begin{equation}
F^{(\alpha)}_k(Z)=\sum_n d_{k,n}^{(\alpha)}e^{i\left(2\pi n+\phi_1+2\pi{p'\over q}\left(k-{N+1\over 2}\right)\right){Z\over L_1}}.
\end{equation}
Through Eq.~\ref{Fbg2} and $F^{(\alpha)}_q(Z)=F^{(\alpha)}_0(Z)$, the coefficients are fixed as
\begin{equation}
d_{k+1,n}^{(\alpha)}=e^{i\left(2\pi n+\phi_1+2\pi{p'\over q}\left(k-{N\over 2}\right)\right)\tau-i\left(\phi_2+{p'\over q}\pi(N+1)\right)}d_{k,n}^{(\alpha)}
\end{equation}
\begin{equation}
d_{0,n+p'}^{(\alpha)}=e^{i\left(2\pi nq+\pi p'q+q\phi_1-p'\pi(N+1)\right)\tau-i(q\phi_2+p'\pi (N+1))}d_{0,n}^{(\alpha)}
\end{equation}
Because there are $p'$ independent coefficients, evidently, this tells us that there are $p'$ independent solutions, depending on our choice of the coefficients $d_{0,0}^{(\alpha)}$, $d_{0,1}^{(\alpha)}$, $\cdots$ $d_{0,p'-1}^{(\alpha)}$. These solutions can be written in an elegant form using theta functions:
\begin{equation}
\label{FK}
F_k^{(\alpha)}(Z)=\elliptic{a_k}{b_\alpha}{Z\over L_1}{{q\over p'}\tau}
\end{equation}
\begin{equation}
a_k={1\over 2\pi}\left(\phi_1-{p'\over q}\pi (N+1)+2\pi k{p'\over q}\right)
\end{equation}
\begin{equation}
b_\alpha=-{1\over 2\pi}\left({q\over p'}(\phi_2+2\pi\alpha)-2\pi(N-1){q\over p'}+\pi(N+1)\right)
\end{equation}
with $k=0,1,\cdots q-1$ and $\alpha=0,1,\cdots p'-1$. Eq.~\ref{anyon wf}, Eq.~\ref{wf2}, Eq.~\ref{J1}, and Eq.~\ref{FK} together give the $p'$-fold degenerate $q$-component ground state wave functions.
If we apply the center-of-mass magnetic translation $t_{\rm CM}\left(L_2/N_\phi\right)=\prod_{i=1}^Nt_i\left(L_2/N_\phi\right)$ on $\Psi^{(\alpha)}$, we get
\begin{eqnarray}
t_{\rm CM}\left(L_2/N_\phi\right)\Psi^{(\alpha)}&=&e^{i{q\over p'}\left(\phi_2+2\pi(\alpha-N+1)+{p'\over q}\pi (N+1)\right)}\Psi^{(\alpha)} \nonumber\\
\end{eqnarray}
The degenerate states have different eigenvalues, and hence are orthogonal. On the other hand, they can be transformed into one another by applying $t_{\rm CM}\left(L_1/N_\phi\right)$:
\begin{eqnarray}
t_{\rm CM}\left(L_1/N_\phi\right)\Psi^{(\alpha)}&=&\Psi^{(\alpha-1)}
\end{eqnarray}
One may notice that for the special case of $\nu=1/m$, which corresponds to the Laughlin state for fermions or bosons, Eq.~\ref{anyon wf} does not have the familiar form given in general literature. (For instance, one can compare Eq.~\ref{anyon wf} to Eq. 6 in Ref.~\cite{Pu20b}.) Actually, they are related by an $m$-dimensional unitary transformation. While Eq.~\ref{anyon wf} is an eigenstate of $t_{\rm CM}\left(L_2/N_\phi\right)$, the more familiar Laughlin wave function (e.g., see Ref.~\cite{Pu20b}) is chosen to be the eigenstates of $t_{\rm CM}\left(L_1/N_\phi\right)$. If one defines the periodic properties Eq.~\ref{bg1} and Eq.~\ref{bg2} such that $t_n(L_2)$ is diagonal and $t_n(L_1)$ is non-diagonal, the more familiar form will be recovered.
\subsection{$\nu_f=n$}
Now let us consider the case of more general $\nu_f=n$. This corresponds to the case of $N$ anyons with statistical parameter $\theta=\pi(1+{p\over q})$, in a magnetic field with flux number $N_\phi=({1\over n}+{p\over q})N$, at $\nu={nq \over q+np}$.
When we model the anyons as fermions with ${p\over q}$ vortices attached to them, the fermions fill $n$ Landau levels in the effective magnetic field, \ie $\nu_f=n$.
In this case, we construct below the wave function as a product of a $q$-component anyon wave function and the fermionic or bosonic scalar wave function in the effective magnetic field. We show that the ground state degeneracy is given by $q+np$. (For $n=1$, which corresponds to $\nu_f=1$, this gives a ground state degeneracy of $q+p=p'$, consistent with the previous subsection.)
Following the standard CF construction, we first write the wave function as a product state:
\begin{widetext}
\begin{equation}
\label{anyon wf2}
\Psi_{n;{p\over q}}=\Psi_n\Psi_{p\over q}^{(N-1)}
\end{equation}
Here, the $q$-component $\Psi_{p\over q}^{(N-1)}$ is given by Eq.~\ref{anyon wf} with $N_\phi\rightarrow {p\over q}N$, $p'\rightarrow p$, $\phi_1\rightarrow \phi_1^a$, $\phi_2 \rightarrow \phi_2^a$. We choose $\alpha=N-1$ just to simplify the phase factor under $t_{\rm CM}\left({qL_2\over pN}\right)$. The other part $\Psi_n$ is the (single-component) wave function of $n$-filled Landau levels \cite{Pu20}:
\begin{equation}
\label{eq:Psi_n}
\Psi_n[z_i,\bar{z}_i]=e^{i\pi\tau N_\phi^* \sum_{j=1}^{N}\theta_{2,j}^2}{1\over \sqrt{N}}\chi_n[f_{i}(z_j,\bar{z}_j)].
\end{equation}
\begin{equation}
\label{upro chi}
\chi_m[f_{i}(z_j,\bar{z}_j)]=
\begin{vmatrix}
f_0^{(0)}(z_1)&f_0^{(0)}(z_2)&\ldots&f_0^{(0)}(z_N) \\
f_0^{(1)}(z_1)&f_0^{(1)}(z_2)&\ldots &f_0^{(1)}(z_N)\\
\vdots&\vdots&\vdots \\
f_0^{(N_\phi^*-1)}(z_1)&f_0^{(N_\phi^*-1)}(z_2)&\ldots&f_0^{(N_\phi^*-1)}(z_N) \\
f_1^{(0)}(z_1,\bar{z}_1)&f_1^{(0)}(z_2,\bar{z}_2)&\ldots&f_1^{(0)}(z_N,\bar{z}_N) \\
f_1^{(1)}(z_1,\bar{z}_1)&f_1^{(1)}(z_2,\bar{z}_2)&\ldots &f_1^{(1)}(z_N,\bar{z}_N)\\
\vdots&\vdots&\vdots \\
f_{n-1}^{(N_\phi^*-1)}(z_1,\bar{z}_1)&f_{n-1}^{(N_\phi^*-1)}(z_2,\bar{z}_2)&\ldots&f_{n-1}^{(N_\phi^*-1)}(z_N,\bar{z}_N)\\
\end{vmatrix}.
\end{equation}
\begin{equation}
\label{eq:psi_n_basis}
f_n^{(k)}(z,\bar{z})=
\sum_{t\in\mathbb{Z}+\frac{k}{N_{\phi}^*}+{\phi_1^f\over 2\pi N_\phi^*}}e^{i\pi N_{\phi}^*\tau t^2}e^{i2\pi N_{\phi}^*t\left(\frac{z}{L_1}-{\phi_2^f\over 2\pi N_\phi}\right)}
H_{n}\left(\frac{\tau_{2}L_1}{\ell_{B}}\left(\theta_2+t\right)\right),
\end{equation}
where $N_\phi^*=N/n$ and $H_{n}(x)$ are the Hermite polynomials. (We omit the normalization factors here.)
\end{widetext}
Since the phases generated by magnetic translation operators simply add, the above wave function satisfies Eq.~\ref{bg1} and Eq.~\ref{bg2} on the condition that $\phi_1=\phi_1^f+\phi_1^a$ and $\phi_2=\phi_2^f+\phi_2^a$.
We find it natural to make the following choice for the phases:
\begin{equation}
\phi_i^a={np\over np+q}\phi_i, \quad i=1,2
\end{equation}
\begin{equation}
\phi_i^f={q\over np+q}\phi_i, \quad i=1,2.
\end{equation}
In this choice, the magnetic fields through the two holes of the torus felt by the fermions and anyons are in the same proportion as the magnetic fields perpendicular to the torus felt by the fermions and anyons. We conjecture that for this choice, there exists at least one momentum sector in which the wave function is well defined for all $\phi_1$ and $\phi_2$. We prove this conjecture in Appendix~\ref{appen a} for $\nu_f=1$, and we have found it to be valid for all cases below.
One might at first think that the wave function in Eq.~\ref{anyon wf2} has a $p$-fold degeneracy, in contrast to the expected ($q+np$)-fold degeneracy~\cite{Kudo20}. Below we show how to reproduce the $q+np$ degenerate wave functions from Eq.~\ref{anyon wf2}. First, we note that $\Psi_n$ satisfies:
\begin{equation}
t_{\rm CM}\left({nL_2\over N}\right)\Psi_n=e^{i(\phi_2^f+\pi(N-1))n}\Psi_n
\end{equation}
while $\Psi_{p\over q}$ satisfies:
\begin{equation}
\label{pq}
t_{\rm CM}\left({qL_2\over pN}\right)\Psi_{p\over q}^{(N-1)}=e^{i\left({q\over p}\phi_2^a+\pi(N+1)\right)}\Psi_{p\over q}^{(N-1)}.
\end{equation}
Therefore, Eq.~\ref{anyon wf2} satisfies:
\begin{equation}
t_{\rm CM}\left({qnL_2\over N}\right)\Psi_{n;{p\over q}}=e^{i\left(qn\phi_2+n\pi(N+1)(p+q)\right)}\Psi_{n;{p\over q}}.
\end{equation}
We can define a momentum projection operator:
\begin{equation}
\label{P}
P_\alpha={1\over \sqrt{np+q}}\sum_{j=0}^{np+q-1}\left[e^{-i{qn\phi_2+n\pi(N+1)(p+q)+2\pi\alpha\over np+q}}t_{\rm CM}\left({L_2\over N_\phi}\right)\right]^j.
\end{equation}
This generates the degenerate eigenstates:
\begin{equation}
\label{degeneracy}
t_{\rm CM}\left(L_2/N_\phi\right)P_\alpha \Psi_{n;{p\over q}}=e^{i{qn\phi_2+n\pi(N+1)(p+q)+2\pi\alpha\over np+q}}P_\alpha \Psi_{n;{p\over q}}
\end{equation}
where $\alpha=0,1,2,\cdots np+q-1$ corresponds to the $np+q$-fold ground state degeneracy. The $np+q$-fold degenerate states are related to each other by $t_{\rm CM}\left(L_1/N_\phi\right)$. With some possible gauge transformation, the degenerate states have such relation
\begin{equation}
t_{\rm CM}\left(L_1/N_\phi\right)P_\alpha \Psi_{n;{p\over q}}=e^{i{nq\phi_1+\pi q(N-n)\over np+q}}P_{\alpha-nq} \Psi_{n;{p\over q}}
\end{equation}
To ensure the wave function after momentum projection does not vanish, we have to choose $\phi_i^f={q\over q+pn}\phi_i$ and $\phi_i^a={np\over q+np}\phi_i$ for $i=1,2$, as explained in Appendix~\ref{appen a}. Note that the degeneracy is equal to $q+np$, which is equal to the denominator of the filling factor $\nu={nq\over q+np}$ only when $nq$ and $q+np$ are mutually coprime. For example, for $n=2$ and $p/q=1/4$, we have $\nu=4/3$ while the degeneracy is $6$. This result is consistent with the exact diagonalization results shown in Ref.~\cite{Kudo20}.
For $n\ge 2$, $\Psi_{n;{p\over q}}$ given by Eq.~\ref{anyon wf2} is not fully in the LLL. In general, one can apply a direct LLL projection following Ref.~\cite{Girvin84b}. However, the direct-projected wave functions cannot be used to calculate systems typically with more than ten particles. An alternative Jain-Kamilla projection can be applied to evaluate large systems \cite{Jain97,Jain97b}. If ${p\over q}\ge 2$, the modified Jain-Kamilla projection \cite{Pu17,Pu20} can be implemented as:
\begin{equation}
\label{anyon wf3}
\Psi_{n;{p\over q}}=(\Psi_0,\Psi_1,\Psi_2,\cdots,\Psi_{q-1})^T
\end{equation}
\begin{equation}
\label{projected wf}
\Psi_k=
e^{\i\pi\tau N_\phi\sum_i \theta_{2,i}^2}F_k(Z)\prod_{i=1}^N\bar{J}_i
\chi_n[\hat{g}_{i}(z_j)\tilde{J}_j],
\end{equation}
\begin{equation}
\label{chi-det}
{\chi_n}[\hat{g}_{i}(z_j)\tilde{J}_j]=
\begin{vmatrix}
\hat{g}_0^{(0)}(z_1)\tilde{J}_1&\ldots&\hat{g}_0^{(0)}(z_N)\tilde{J}_N \\
\vdots&\vdots&\vdots \\
\hat{g}_1^{(0)}(z_1)\tilde{J}_1^p&\ldots&\hat{g}_1^{(0)}(z_N)\tilde{J}_N \\
\vdots&\vdots&\vdots \\
\end{vmatrix},
\end{equation}
\begin{equation}
\bar{J}_i=\prod_{j\neq i}e^{\left({p\over 2q}-1\right) \ln \elliptic{1/2}{1/2}{{z_i-z_j\over L_1}}{\tau}}.
\end{equation}
\begin{equation}
\tilde{J}_i=\prod_{j\neq i}e^{\ln \elliptic{1/2}{1/2}{{z_i-z_j\over L_1}}{\tau}}.
\end{equation}
where $F_k(Z)$ is given by Eq.~\ref{FK} with $p'\rightarrow p$, $\phi_1\rightarrow \phi_1^a$, $\phi_2 \rightarrow \phi_2^a$. The general form of $\hat{g}_n^{(k)}(z)$ was derived in detail in Refs.~\cite{Pu17,Pu20}. Here we give the form for the lowest three Landau levels (without including any normalization factors):
\begin{widetext}
\begin{equation}
\hat{g}_0^{(k)}(z)=f_0^{(k)}(z)=\elliptic{{k\over N_\phi^*}+{\phi_1^f\over 2\pi N_\phi^*}}{-{\phi_2^f\over 2\pi}}{N_\phi^* z\over L_1}{N_\phi^* \tau},
\end{equation}
\begin{equation}
\label{2nd LL matrix element}
\hat{g}_1^{(k)}(z)=(N_\phi^*-N_\phi)\frac{\partial f_0^{(k)}(z)}{\partial z}+N_\phi^*f_0^{(k)}(z)2\frac{\partial}{\partial z},
\end{equation}
\begin{equation}
\hat{g}_2^{(k)}(z)= (N_\phi-N_\phi^*)^2{\partial^2 f_0^{(k)}(z)\over\partial z^2}
-2N_\phi^*(N_\phi-N_\phi^*){\partial f_0^{(k)}(z)\over\partial z}2{\partial\over \partial z}
+N_\phi^{*2}f_0^{(k)}\left(2{\partial\over \partial z}\right)^2,
\end{equation}
\end{widetext}
We mention a caveat for the projected wave function Eq.~\ref{projected wf}. Compared to the Jastrow factor in the unprojected wave function Eq.~\ref{J1}, we changed $\prod_{i<j}{\rm exp}\left({{p\over q} \ln \elliptic{1/2}{1/2}{{z_i-z_j\over L_1}}{\tau}}\right)$ to $\prod_{i\neq j}{\rm exp}\left({{p\over 2q} \ln \elliptic{1/2}{1/2}{{z_i-z_j\over L_1}}{\tau}}\right)$. For composite fermions, i.e. when $p\over q$ is an even integer, this process only generates a factor of $(-1)^{pN(N-1)\over 4q}$, which is of no significance. However, the Jastrow factors of anyons have branch cuts, and the definition of how the Jastrow factors change across the branch cuts is very subtle. If we use the definition for the multi-valued Jastrow factors mentioned right after Eq.~\ref{branch}, we find the projected wave function no longer satisfies Eq.~\ref{bg1} and Eq.~\ref{bg2}. On the other hand, if we confine the particles to the principal region of the torus (\ie the parallelogram spanned by $L_1$ and $L_2$), the projected wave function captures the lowest Landau level part of the unprojected wave function, which satisfies the imposed braiding group. Therefore, the projected wave function is still sufficient for calculating local physical quantities such as energies, Berry curvatures, and Hall viscosities.
An important property that is required for wave functions on a torus is modular covariance. We discuss this issue in Appendix~\ref{appen b} and show that the anyon wave function constructed above is modular covariant.
\section{energy gap and the adiabatic principle}
\label{energy}
The key point of the adiabatic principle is that the ground states remain gapped as we tune the strength of the attached vortex, or in other words the statistical phase $\theta$, in such a manner that the effective filling factor remains constant. In this section, we numerically confirm this statement by calculating the transport gaps for $\nu_f=1,2$ using our ansatz wave functions, assuming Coulomb interaction between the anyons.
We calculate the transport gaps by creating a quasiparticle state and a quasihole state separately. A quasiparticle can be obtained from Eq.~\ref{anyon wf3} by occupying an extra orbital in the lowest unoccupied effective Landau level in the Slater determinant part. Similarly, a quasihole can be obtained by leaving an unoccupied orbital in the highest occupied effective Landau level in the Slater determinant part. For $\nu_f=1$, the transport gap is calculated as:
\begin{eqnarray}
\label{delta}
\Delta_{n=1;{p\over q}}(N)&=&E^{\rm qp}\left(N,N_\phi=\left(1+{p\over q}\right)N+1\right)\nonumber \\
&&+E^{\rm qh}\left(N,N_\phi=\left(1+{p\over q}\right)N-1\right)\nonumber \\
&&-2E^{\rm 0}\left(N,N_\phi=\left(1+{p\over q}\right)N\right).
\end{eqnarray}
where $E^{\rm qp}$, $E^{\rm qh}$ and $E^{\rm 0}$ are the energies of the quasiparticle, the quasihole and the ground states. For $\nu_f=2$, the transport gap is calculated as:
\begin{eqnarray}
\label{delta2}
\Delta_{n=2;{p\over q}}(N)&=&E^{\rm qp}\left(N-1,N_\phi=\left({1\over 2}+{p\over q}\right)N-{p\over q}\right)\nonumber \\
&&+E^{\rm qh}\left(N+1,N_\phi=\left({1\over 2}+{p\over q}\right)N+{p\over q}\right) \nonumber \\
&&-2E^{\rm 0}\left(N,N_\phi=\left({1\over 2}+{p\over q}\right)N\right)
\end{eqnarray}
We assume Coulomb interaction between particles.
We use variational Monte Carlo and the anyon wave function Eq.~\ref{anyon wf3} to calculate the transport gaps. The results are shown in Fig.~\ref{gap}. Because the energy only depends on the relative part of the wave function, we only use the first component in Eq.~\ref{anyon wf3} to calculate the energy and multiply the values by the number of components to save the computation time. For $\nu_f=1$, we calculate the transport gaps for many anyon states between two Laughlin states $\nu=1/3$ and $\nu=1/5$. For $\nu_f=2$, we calculate the transport gaps for many anyon states between two Jain states $\nu=2/5$ and $\nu=2/9$. As Fig.~\ref{gap} shows, the gaps vary smoothly with the change of $\theta$ and remain nonzero. This is a justification of the adiabatic heuristic principle proposed by Greiter and Wilczek \cite{Greiter90,Greiter92b}. A similar result is obtained by diagonalizing lattice Hamiltonian of smaller systems in Ref.~\cite{Kudo20}.
We note that we do not connect $\nu=1/3$ to $\nu=1$ or $\nu=2/5$ to $\nu=2$. The reason is technical: we are not able to perform the Jain-Kamilla projection for anyons in this filling factor region. However, in light of the above results, there is no reason to doubt that analogous adiabatic continuity in that filling factor range also holds.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{./gap.pdf}
\caption{
The transport gaps for $\nu_f=1$ and $\nu_f$=2 as a function of $1/\nu$ for $N=20$, obtained from lowest-Landau-level projected variational wave functions. Coulomb interaction is assumed between the anyons, and the energies are quoted in units of $e^2/\epsilon \ell$.
}
\label{gap}
\end{figure}
\section{Chern numbers and Hall conductivity}
\label{Chern number}
In this section, we calculate the Chern number for the anyon wave function in Eq.~\ref{anyon wf} following the approach used by Niu, Thouless, and Wu~\cite{Niu85} and Tao and Haldane~\cite{Tao86}. The Chern number is defined as:
\begin{equation}
\label{chern}
C=-i2\pi\sum_\alpha\langle{\partial J_2^{(\alpha)}\over \partial \phi_1}-{\partial J_1^{(\alpha)}\over \partial \phi_2}\rangle.
\end{equation}
Here $\langle\rangle$ refers to the average in $(\phi_1,\phi_2)$ space, and the summation is over all degenerate ground states. $J_i^{(\alpha)}$ is defined as:
\begin{equation}
J_i^{(\alpha)}=\sum_k\langle\Psi_k^{(\alpha)}|{\partial\over \partial \phi_i}|\Psi_k^{(\alpha)}\rangle
\end{equation}
where the wave function is normalized, \ie $\sum_k\langle\Psi_k^{(\alpha)}|\Psi_k^{(\alpha)}\rangle=1$.
To see the periodicity of our wave function in the $(\phi_1,\phi_2)$ space, we need the identities Eq.~\ref{thetaa1} and Eq.~\ref{thetab1}.
Given these identities and the assumption that the overall normalization factor does not depend on $\phi_1$ and $\phi_2$, it is straightforward to see:
\begin{widetext}
\begin{equation}
\label{C1}
\left(P_\alpha \Psi_{n;{p\over q}}\right)_k\left(\phi_1+2\pi(q+pn),\phi_2\right)=e^{i\pi (N-n)q}\left(P_\alpha \Psi_{n;{p\over q}}\right)_k\left(\phi_1,\phi_2\right)
\end{equation}
\begin{equation}
\label{C2}
\left(P_\alpha \Psi_{n;{p\over q}}\right)_k\left(\phi_1,\phi_2+2\pi(q+pn)\right)=e^{-i2\pi nq\left[{\phi_1\over 2\pi}-{p\over 2q}(N+1)+{pk\over q}+{N-n\over 2n}\right]}\left(P_\alpha \Psi_{n;{p\over q}}\right)_k\left(\phi_1,\phi_2\right)
\end{equation}
Therefore, the average can be taken in the space $(0,2(q+pn)\pi)\otimes (0,2(q+pn)\pi)$. With the above identities, we can now prove
\begin{eqnarray}
C&=&-{i2\pi\over(2\pi(q+pn))^2}\sum_\alpha\int_0^{2(q+pn)\pi}d\phi_1\int_0^{2(q+pn)\pi}d\phi_2\left[{\partial J_2^{(\alpha)}\over \partial\phi_1}-{\partial J_1^{(\alpha)}\over \partial \phi_2}\right]\\ \nonumber
&=&-{i2\pi\over(2\pi(q+pn))^2}\sum_\alpha\int_0^{2\pi (q+pn)}d\phi_2\left(J_2^{(\alpha)}(2(q+pn)\pi,\phi_2)-J_2^{(\alpha)}(0,\phi_2)\right)\\ \nonumber
&&+{i2\pi\over(2\pi(q+pn))^2}\sum_\alpha\int_0^{2\pi(q+pn)}d\phi_1\left(J_1^{(\alpha)}(\phi_1,2\pi (q+pn))-J_1^{(\alpha)}(\phi_1,0)\right)\\ \nonumber
&=&{qn}
\end{eqnarray}
\end{widetext}
The Chern number $qn$ depends not only on the fermionic filling factor $n$ but also on the statistical phase $\theta=\pi(1+p/q)$. Only when $\theta$ is an integer multiple of $\pi$ does the Chern number equal to $n$. Hence, it is not the Chern number but rather $C/q$ that remains invariant under the adiabatic evolution. This is also consistent with the finding of Ref.~\cite{Kudo20}. (We note that the Chern number in Ref.~\cite{Kudo20} is actually equal to our $C/q$.) As shown in Ref.~\cite{Niu85}, the Hall conductivity in units of $e^2/h$ is the Chern number per degenerate ground state. Thereby, it is ${nq\over q+np}{e^2\over h}$ for our anyon states.
As shown in Ref.~\cite{Kudo19} for fractional quantum Hall states, the integration or average over the twist angles in Eq.~\ref{chern} is not necessary when the system size is large enough, since the Berry curvature is already uniform. To see whether this is also true for anyon wave function, we calculate the Berry curvature at different points in the $(\phi_1,\phi_2)$ plane for $\nu_f=1$, $\nu=2/3$, $N=12$. As shown in Fig.~\ref{chern2D}, the Berry curvature is uniform to an extremely high degree, at the value derived above. We then calculate the $C/q=-i{2\pi\over q}\sum_\alpha \left({\partial J_2^{(\alpha)}\over \partial \phi_1}-{\partial J_1^{(\alpha)}\over \partial \phi_2}\right)$ without integration for different anyon wave functions with $\nu_f=1,2$, $N=12$. The results are shown in Fig.~\ref{chernno}. The numerical results are quantized at $\nu_f$, which agree with our analytical derivations above.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{./chern2D.pdf}
\caption{
The Berry curvature divided by $q$, \ie $-i{2\pi\over q}\sum_\alpha \left({\partial J_2^{(\alpha)}\over \partial \phi_1}-{\partial J_1^{(\alpha)}\over \partial \phi_2}\right)$, at different points in the $(\phi_1,\phi_2)$ plane for our anyon wave function with $\nu_f=1$, $\nu=2/3$, $N=12$. The value is expected to be $1$.) We note the Berry curvature is uniform to a high degree, varying in a narrow range from $0.994$ to $1.002$. The sudden change in the color is an artifact, arising because the Berry curvature has been evaluated only for a discrete set of $(\phi_1,\phi_2)$ points.
}
\label{chern2D}
\end{figure}
\begin{figure}[t]
\includegraphics[width=\columnwidth]{./chern.pdf}
\caption{
This figure shows $C/q=-i{2\pi\over q}\sum_\alpha \left({\partial J_2^{(\alpha)}\over \partial \phi_1}-{\partial J_1^{(\alpha)}\over \partial \phi_2}\right)$ evaluated at randomly chosen $(\phi_1,\phi_2)$ points for anyon ground states with $\nu_f=1,2$, $N=12$, where $C$ is the total Chern number. The values of $C/q$ are well quantized at $\nu_f$ as the statistics is varied, which agrees with the analytical result. The unprojected wave functions have been used for the calculation.
}
\label{chernno}
\end{figure}
\section{Hall viscosities for anyons}
\label{Hall viscosities}
In addition to the Chern number, another topological quantity that can be easily calculated in torus geometry is the Hall viscosity. Avron, Seiler, and Zograf \cite{Avron95} showed that the Hall viscosity can be computed as Berry curvature through adiabatic deformation of the geometry of the torus:
\begin{equation}
\label{berry-curv}
\eta^A=-{\hbar \tau_2^2 \over V}\mathcal{F}_{\tau_1,\tau_2},
\end{equation}
where
\begin{equation}
\label{BC}
\mathcal{F}_{\tau_1,\tau_2}=-2{\rm Im}\bigg\langle {\partial \Psi \over \partial \tau_1}\bigg|{\partial \Psi \over \partial \tau_2}\bigg\rangle.
\end{equation}
Based on Eq.~\ref{berry-curv}, Read proposed \cite{Read09,Read10} that for fermionic and bosonic gapped states $\eta^A$ is given by
\begin{equation}
\label{hall visc}
\eta^A=\mathcal{S}{\hbar \rho \over 4}.
\end{equation}
where $\rho=N/V$ and the ``shift" $\mathcal{S}$ is a topological quantum number defined in the spherical geometry, given by $\mathcal{S}={N\over \nu}-N_\phi$.
This relation has been derived or numerically confirmed for Laughlin states, Pfaffian states, and Jain states by various approaches \cite{Read09,Read10,Tokatly09,Cho14,Fremling14,Lapa18,Lapa18b,Pu20}. In particular, Ref.~\cite{Pu20} developed an analytical derivation for microscopic wave functions. The main result of that work is that if a wave function is a product of several components, then the Hall viscosity is the sum of the Hall viscosities for different components provided that the normalization factor satisfies certain behavior in the thermodynamic limit. This statement holds for the unprojected as well as the projected Jain wave functions.
Clearly, Eq.~\ref{anyon wf2} is in a product form, and we can apply the theorem stated above. The fermionic part $\Psi_n$ contributes ${nN\hbar\over 4V}$ to the Hall viscosity. The remaining question is: how much does the anyonic part $\Psi_{p\over q}$ contribute? As it is shown in Refs.~\cite{Tokatly09,Milovanovic10}, if the wave function (disregarding the normalization factor $\mathcal{N}$) is a holomorphic function of $\tau$, which is the case for $\Psi_{p\over q}$, then its contribution to Hall viscosity is given by
\begin{equation}
{\hbar \tau_2^2\over 2V}\left[\left({\partial\over \partial \tau_1}\right)^2+\left({\partial\over \partial \tau_2}\right)^2\right]\ln \mathcal{N}.
\end{equation}
We further note that $\Psi_{p\over q}$ is very similar to the Laughlin wave function. They can both be separated into a center-of-mass part and a relative part [which is written in terms of $(z_i-z_j)$]. Furthermore, the relative part of $\Psi_{p\over q}$ has the same form as the relative part of the Laughlin wave function at $\nu=1/m$ with $m=p/q$.
We now argue that the contribution of the center-of-mass to the Hall viscosity vanishes in the thermodynamic limit.
The contribution of the center-of-mass part to $\ln \mathcal{N}$ is on the order of $\ln N$ ($N$ is the particle number), implying that its contribution to the Hall viscosity vanishes as $\ln N\over N$ in the thermodynamic limit. In fact, when deriving the Hall viscosity for Laughlin states, Tokatly and Vignale~\cite{Tokatly09} used the cylindrical geometry in which the center-of-mass part is absent, which also is valid only if the contribution of the center-of-mass part is unimportant. Hence, the total contribution of $\Psi_{p\over q}$ is ${p\rho\hbar\over 4q}$, and the Hall viscosity for the anyon wave function in Eq.~\ref{anyon wf2} is $\left(n+{p\over q}\right){\rho\hbar \over 4}$, or in terms of $\nu$ and $\nu_f$:
\begin{equation}
\eta^A=\left({1\over \nu}-{1\over \nu_f}+\nu_f\right){\hbar \rho\over 4}.
\end{equation}
We also calculate the Hall viscosity of the anyon wave function Eq.~\ref{anyon wf2} directly through Eq.~\ref{berry-curv}. As mentioned above, the Hall viscosity is dominated by the relative part, so we only use the first component of the wave function and multiply the result by the number of components, just as we have done for energy. [We mention a slight subtlety in the calculation. Because of the presence of branch cuts in the Jastrow factors, we have to manually correct the jumps between different Riemann sheets. For instance, when we vary the geometry of the torus by a tiny amount, the imaginary part of $\ln \elliptic{1/2}{1/2}{{z_i-z_j\over L_1}}{\tau}$ might change by $\delta\pm2\pi$, where $\delta$ is a tiny number. In that case, we manually correct the change to $\delta$.]
The results are shown in Fig.~\ref{hallvisc}. We consider a system of 20 particles on a square torus $\tau=i$. We calculate the Hall viscosity at $\nu_f=1,2$ for different filling factors by varying the statistical phase $\theta$. According to the analysis above, the Hall viscosity for $\nu_f=1$ is ${1\over \nu}{\rho\hbar \over 4}$ and for $\nu_f=2$ is $\left({3\over 2}+{1\over \nu}\right){\rho\hbar \over 4}$. The numerical results agree with these values. We also note that the unprojected wave functions and projected wave functions have the same Hall viscosity, as is also the case with the Jain states \cite{Pu20}.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{./Hall_visc.pdf}
\caption{
The Hall viscosity in units of $\hbar \rho\over 4$, for $\nu_f=1,2$. The straight lines indicate the theoretically predicted values, which are given by ${1\over \nu}$ and ${3\over 2}+{1\over \nu}$, respectively. The system contains 20 particles on a square torus. The $\nu_f=2$ results are for the unprojected wave functions (stars), but wherever possible, we have also evaluated the Hall viscosity for the LLL projected wave functions (circles). The $\nu_f=1$ wave functions are automatically in the LLL.
}
\label{hallvisc}
\end{figure}
\section{Summary}
\label{summary}
In summary, we have achieved a construction of multi-component anyon wave functions in the torus geometry. The wave functions are representations of the braiding group and have the expected ground state degeneracy. In the special cases in which $\theta$ is an integer multiples of $\pi$, the anyon wave functions return to the Jain CF wave functions \cite{Pu17}. When ${p\over q}\ge 2$, we can project the wave function to the LLL with the efficient modified Jain-Kamilla method. We calculate the transport gaps by evaluating the ground state, quasiparticle, and quasihole energies. The transport gap varies smoothly as we vary the statistical parameter $\theta$. We also calculate the Chern number, and we find that $C\over q$ is an adiabatic invariant, \ie it is invariant with the change of $\theta$. This is consistent with the exact diagonalization results of Kudo and Hatsugai\cite{Kudo20}. We also evaluate the Hall viscosity and find it to be $\left(n+{p\over q}\right){\rho\hbar \over 4}$ for $\nu_f=n,\theta=(1+{p\over q})\pi$. The results are summarized in Table.~\ref{table1}.
\begin{table*}
\makebox[\textwidth]{\begin{tabular}{|c|c|}
\hline
$\nu_f$&$n$\\ \hline
$\nu$&${nq\over np+q}$\\ \hline
$\theta$&$\pi\left(1+{p\over q}\right)$\\ \hline
number of components&$q$\\ \hline
ground state degeneracy&$q+np$\\ \hline
$C$&$nq$\\ \hline
Hall conductivity&${nq\over q+np}{e^2\over h}$\\ \hline
$\eta^A$&${\hbar \rho\over 4}\left({p\over q}+n\right)$\\ \hline
\end{tabular}}
\caption{\label{table1} Summary of the number of components, ground state degeneracy, Chern number ($C$), and Hall viscosity ($\eta^A$) of the anyon wave function at filling factor $\nu$ with statistical phase $\theta$ and effective filling $\nu_f$. The integer quantum Hall states correspond to $q=1$ and $p=0$, whereas the composite fermion states to $p=2s$ and $q=1$.}
\end{table*}
\begin{acknowledgments}
We are grateful to Yayun Hu, Koji Kudo and Bin Wang for helpful discussions. This work was supported by the U. S. Department of Energy, Office of Basic Energy Sciences, under Grant No. DE-SC0005042. The numerical part of this research was conducted with Advanced CyberInfrastructure computational resources provided by the Institute for CyberScience at the Pennsylvania State University.
\end{acknowledgments}
\begin{appendix}
\section{Jacobi theta function with rational characteristics}
\label{theta-function}
The Jacobi theta function with rational characteristics\cite{Mumford07} is defined as
\begin{equation}
\elliptic[\displaystyle]abz\tau=\sum_{n=-\infty}^{\infty}e^{i\pi \left(n+a\right)^2\tau}e^{i2\pi \left(n+a\right)\left(z+b\right)}.
\end{equation}
The zeros of $\elliptic[\displaystyle]abz\tau$ lie at $z={1\over 2}-b+m+\left({1\over 2}-a+n\right)\tau$, where $m$ and $n$ are integers. We list here several periodic properties of Jacobi theta functions that have been used in our derivations in the main text:
\begin{equation}
\elliptic[\displaystyle]ab{z+1}\tau=e^{i2\pi a}\elliptic[\displaystyle]abz\tau ,
\end{equation}
\begin{equation}
\elliptic[\displaystyle]ab{z+\tau}\tau=e^{-i\pi [\tau+2(z+b)]}\elliptic[\displaystyle]abz\tau ,
\end{equation}
\begin{equation}
\elliptic[\displaystyle]ab{z+w}{w\tau}=\elliptic[\displaystyle]a{b+w}z{w\tau} ,
\end{equation}
\begin{equation}
\label{branch}
\elliptic[\displaystyle]ab{z+\tau}{w\tau}=e^{-i{2\pi \over w}\left(z+b+{\tau\over 2}\right)}\elliptic[\displaystyle]{a+{1\over w}}bz{w\tau} ,
\end{equation}
\begin{equation}
\label{thetaa1}
\elliptic{a+1}{b}{z}{\tau}=\elliptic{a}{b}{z}{\tau},
\end{equation}
\begin{equation}
\label{thetab1}
\elliptic{a}{b+1}{z}{\tau}=e^{i2\pi a}\elliptic{a}{b}{z}{\tau},
\end{equation}
where $w$ is a real number.
\section{Distribution of the twisted boundary phases}
\label{appen a}
In this appendix, we discuss the constrain on the distribution of the twisted boundary phases $\phi_i^a$ and $\phi_i^f$ ($i=1,2$) for composite anyon wave functions. As mentioned in the main text and Ref.~\cite{Pu17}, the preservation of periodic boundary conditions only require $\phi_i=\phi_i^a+\phi_i^f$. This implies, in general:
\begin{equation}
\phi_i^a=\gamma\phi_i,\quad i=1,2,
\end{equation}
\begin{equation}
\phi_i^f=(1-\gamma)\phi_i,\quad i=1,2,
\end{equation}
where $\gamma$ is a real number. We fix $\gamma$ as follows.
The phases correspond to the effective magnetic field through the holes of the torus felt by the attached vortices and fermions, which are in the proportion $np/q$. Requiring the same proportion for the phases yields
$\gamma={np\over np+q}$. In this context, we note that the
wave function in Eq.~\ref{C1} and Eq.~\ref{C2} produces a different Chern number for $\gamma\neq {np\over np+q}$;
the reason is that then the wave function $P_\alpha\Psi_{n;{p\over q}}$ vanishes for some values of $\phi_1$ and $\phi_2$. Hence, the calculation of Chern only makes sense for $\gamma={np\over np+q}$. We next demonstrate this statement
for $n=1$, leaving
the generalization to other values of $n$ as an open question.
The anyon wave function Eq.~\ref{anyon wf2} can be written as a superposition of different momentum eigenstates.
\begin{equation}
\label{decom}
\Psi_{1;{p\over q}}=\Psi_1\Psi_{p\over q}^{(N-1)}=\sum_{\alpha=0}^{p+q-1}c_\alpha \Psi_{1+{p\over q}}^{(\alpha)}
\end{equation}
Here $\Psi_{1+{p\over q}}^{(\alpha)}$ are the momentum eigenstates. They can be obtained by applying $P_\alpha$ defined in Eq.~\ref{P}. For the special case $n=1$, there is an easier way: they are simply given by Eq.~\ref{anyon wf} with the replacement $p'\rightarrow p+q$. Our task is to show all $c_\alpha$ are nonzero for arbitrary $\phi_1$ and $\phi_2$ only when $\gamma={p\over p+q}$.
The Jastrow factors on the left-hand-side and right-hand-side of Eq.~\ref{decom} are obviously identical. The coefficients are
hence determined by the center-of-mass part. In other words, Eq.~\ref{decom} can be rewritten as:
\begin{widetext}
\begin{equation}
\label{decom2}
\elliptic{a^f}{b^f}{Z\over L_1}{\tau}\elliptic{a^a_{k}}{b^a}{Z\over L_1}{q\tau\over p}\sim\sum_{\alpha=0}^{p+q-1} c_\alpha \elliptic{a_k}{b_\alpha}{Z\over L_1}{q\tau\over p+q}
\end{equation}
\end{widetext}
where the parameters are given by:
\begin{equation}
a^f={N-1\over 2}+{\phi_1^f\over 2\pi}
\end{equation}
\begin{equation}
b^f={N-1\over 2}-{\phi_2^f\over 2\pi}
\end{equation}
\begin{equation}
a^a_k={1\over 2\pi}\left(\phi_1^a-{p\pi\over q}(N+1)+2\pi {pk\over q}\right)
\end{equation}
\begin{equation}
b^a=-{1\over 2\pi}\left({q\over p}\phi_2^a+\pi(N+1)\right)
\end{equation}
\begin{equation}
a_k={1\over 2\pi}\left(\phi_1-{p+q\over q}\pi(N+1)+2\pi{p+q\over q}k\right)
\end{equation}
\begin{equation}
b_\alpha=-{1\over 2\pi}\left({q\over p+q}\left(\phi_2+2\pi \alpha\right)-2\pi (N-1){q\over p}+\pi (N+1)\right)
\end{equation}
In Eq.~\ref{decom2} we use $\sim$ instead of $=$ since, for simplicity, we
have omitted normalization factors in this equation and below.
This does not influence our judgment whether $c_\alpha$ is zero.
To solve for $c_\alpha$, we expand the theta function according to its definition and compare the coefficients of $e^{i2\pi(n+a_3){Z\over L_1}}$ on left-hand-side and right-hand-side:
\begin{widetext}
\begin{equation}
c_\alpha e^{i\pi(n+a_k)^2{q\tau\over p+q}+i2\pi(n+a_k)b_\alpha}\sim \sum_{m_1,m_2}\delta_{n+a_k,m_1+m_2+a^f+a^a_k}e^{i\pi(m_1+a^f)^2\tau+i\pi(m_2+a^a_k)^2{q\tau\over p}+i2\pi(m_1+a^f)b^f+i2\pi(m_2+a^a_k)b^a}.
\end{equation}
\end{widetext}
Since $c_\alpha$ cannot depend on $n$ or $k$, we
choose $n=0,k=0$. With some algebra, we get
\begin{equation}
c_\alpha\sim \elliptic{a^f-a_0{q\over p+q}}{b^f-b^a}{0}{{q+p\over p}\tau}.
\end{equation}
The condition for $c_\alpha=0$ is
\begin{equation}
a^f-a_0{q\over p+q}={\phi_1\over 2\pi}\left({p\over p+q}-\gamma\right)+N={1\over 2}+l_1
\end{equation}
\begin{equation}
b^f-b^a={\phi_2\over 2\pi}\left({q+p\over p}\gamma-1\right)+N={1\over 2}+l_2
\end{equation}
Here $l_1$ and $l_2$ are two arbitrary integers and $N$ is the particle number.
When $\gamma={p\over p+q}$
it is not possible to satisfy the above two equations for any $\phi_1$ and $\phi_2$.
For $\gamma\neq {p\over p+q}$, one can always find values of $\phi_1$ and $\phi_2$ to make $c_\alpha$ zero. This proves our statement that we have to choose $\gamma={p\over p+q}$ to ensure that the wave function remains non-zero in the entire $(\phi_1,\phi_2)$ space.
\section{Modular covariance of the anyon wave functions}
\label{appen b}
As mentioned in the main text, the geometry of a torus is parameterized by $L_1$ and $L_2$. However, the parametrization is not unique. The geometry is unchanged under a modular transformation of $L_1$ and $L_2$,
\begin{equation}
\bigl(
\begin{smallmatrix}
L_2'\\
L_1'
\end{smallmatrix}
\bigr)
=
\bigl(
\begin{smallmatrix}
a &b\\
c &d
\end{smallmatrix}
\bigr)
\bigl(
\begin{smallmatrix}
L_2\\
L_1
\end{smallmatrix}
\bigr),
\end{equation}
where $a,b,c,d\in\mathbb{Z}$ with $ad-bc=1$. These transformations form the modular group, which is spanned by two elements $\mathcal{T}$: $\bigl(\begin{smallmatrix}
1 &1\\
0 &1
\end{smallmatrix}\bigr)$ and $\mathcal{S}$: $\bigl(\begin{smallmatrix}
0 &-1\\
1 &0
\end{smallmatrix}\bigr)$. The geometry itself is unchanged by these transformations. If we redefine the twisted periodic boundary phases $\phi_1$ and $\phi_2$ consistently with the modular transformations, all the physical quantities should be invariant under modular transformations. To guarantee this, the wave functions must be covariant under modular transformations. To be more specific, the Hilbert space of the degenerate ground states is invariant under modular transformations, and the transformations of degenerate ground states are described by a unitary matrix. In this appendix, we show that our anyon wave functions do possess theses properties.
First let us consider the case in which there is no degeneracy, \ie $p=1$. In this case, the ground state is nondegenerate and thus should be invariant under modular transformation. This is true for fermions and bosons, whose wave functions are single-component. However, for anyons, the transformation is more subtle. Under a $\mathcal{S}$ transformation, the new $L_1$ direction is the original $L_2$ direction. Thereby, $t_i(L_1)$ acting on the original wave function is now represented by a nondiagonal matrix instead of a diagonal matrix. One can, however, recover the forms of Eq.~\ref{bg1} and Eq.~\ref{bg2} by performing a unitary transformation on the original ground state wave function $\Psi_k$:
\begin{equation}
\tilde{\Psi}_k=\sum_{k'}U_{kk'}\Psi_{k'}\quad k,k'=0,1,2\cdots q-1
\end{equation}
In other words, the components are mixed and reordered. The matrix can be
obtained by comparing the periodic properties of $\tilde{\Psi}_k$ and $\Psi_k$. The matrices for $\mathcal{T}$ and $\mathcal{S}$ are
\begin{equation}
U_{kk'}\left(\mathcal{T}\right)={1\over \sqrt{q}}\delta_{kk'}c^{k(k-1)\over 2}
\end{equation}
\begin{equation}
U_{kk'}\left(\mathcal{S}\right)={1\over \sqrt{q}}c^{-kk'}
\end{equation}
with $c=e^{i2\pi{p'\over q}}$ as defined in the main text.
When the ground state degeneracy is present, there is another set of matrices $V$ that describes the mixing of degenerate ground states $\Psi^{(\alpha)}=P_\alpha \Psi_{n;{p\over q}}$ under modular transformation. The $V$ matrices can be derived
by comparing the properties of $\Psi^{(\alpha)}$ under $t_{\rm CM}\left(L_1\over N_\phi\right)$ and $t_{\rm CM}\left(L_2\over N_\phi\right)$ before and after the modular transformations. The matrices for $\mathcal{T}$ and $\mathcal{S}$ are
\begin{equation}
V_{\alpha \alpha'}\left(\mathcal{T}\right)={1\over\sqrt{np+q}}e^{i\left({2\pi(\alpha-\alpha')\over np+q}+\theta_0\right){\alpha'-\alpha\over nq}}
\end{equation}
\begin{equation}
V_{\alpha \alpha'}\left(\mathcal{S}\right)={1\over\sqrt{np+q}}e^{i\left({\theta_1\alpha-\theta_2\alpha'\over nq}-{2\pi\alpha\alpha'\over nq(np+q)}\right)}
\end{equation}
where $\theta_0={\pi q(nN_\phi-N)\over np+q}$, $\theta_1={\pi q(N-n)-n\pi (N+1)(p+q)\over np+q}$, and $\theta_2={\pi q(N-n)+n\pi (N+1)(p+q)\over np+q}$. The final modular transformation is described by the direct products of $U$ and $V$:
\begin{equation}
\tilde{\varphi}_k^{(\alpha)}=U_{kk'}V_{\alpha \alpha'}\varphi_{k'}^{(\alpha')}
\end{equation}
Since the direct product $U\otimes V$ is unitary, the physical quantities are guaranteed to be invariant under modular transformations.
As shown in Ref.~\cite{Fremling19}, the modified Jain-Kamilla projection preserves the modular covariance of the wave functions.
To confirm the modular covariance of the wave functions numerically, we calculate the Hall viscosities for wave functions that are related by modular transformations. The result is shown in Fig.~\ref{modcov}. We choose $\nu_f=1,\theta=3\pi/2,N=4$. As a result of the covariance under $T$ transformation, blue circles and red stars are supposed to be coincident, and the data are expected to be symmetric with respect to the $y$-axis because of the covariance under the $S$ transformation.
The numerical results in Fig.~\ref{modcov}
are explicitly consistent with these expectations,
thus demonstrating that the wave functions are modular covariant.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{./modcov.pdf}
\caption{
The Hall viscosity for different shapes of torus in units of $\hbar \rho\over 4$, for $\nu_f=1,\theta=3\pi/2,N=4$. The red and blue symbols are related by the $T$ transformation, and the left-hand side and right-hand side of the $y$-axis are related by the $S$ transformation.
}
\label{modcov}
\end{figure}
\end{appendix}
|
3,212,635,537,921 | arxiv | \section{Introduction}
William James \cite{Jam90} developed a theory of two selves. The first one is the "Me" self, and the second one is the "I" self. The "me" self refers to the aspects of someone that come from that person's experiences. The "me" self can break down into three sections, The Material Self, The Social Self, and The Spiritual Self. The "I" self is categorized into the thinking self. The Pure Ego was the name given to the "I" self. James also thinks our body is the innermost part of the material self. Gallagher's \cite{gal00} studies between philosophy of mind and the other cognitive science that are focused on two aspects of self. They are the minimal self and the narrative self. The minimal self is the procedure of human brain processes and ecologically embedded body. He considered sense of ownership is an integration of information (i. e. visual, tactile, somatosensory etc.) processing in human brains. According to these various studies, we know the illusory ownership can be evoked in some conditions that the other objects can be recognized as the subjects' own body parts by themselves. Recently, due to the fast development of both computer graphic technologies and computer hardware performances, VR has gained more attention now. It can replicate an environment to allow the users achieving an immersive experience. We have studied approaches and methods that can evoke the illusory ownership in the VR environment. Especially, we use the Head Mount Display (HMD) to evoke illusory ownerships with various conditions.
\section{Previous researches}
Botvinick and Cohen reported a phenomenon that called Rubber Hand Illusion (RHI) \cite{BC98}, which a rubber hand placed in front of a healthy subject can misunderstand it as his or her own hand under certain circumstances. Many researches on RHI have been conducted recently. Some researches replaced the rubber hand with other objects such as: a table \cite{AR03}, a rake, a curtain, and another stick-shape objects \cite{HP10}. They all made it clear that the subjects felt if these objects were the part of their own bodies, but the other studies (using stick-shape object \cite{TH05}) also pointed out that the non-hand-shape objects cannot evoke the RHI.
Full Body Illusion (FBI) is based on the research of human brain disease that called Autoscopic Phenomena. The patients have a strong illusion of the ownership of other human bodies. However, the FBI can also occur in healthy humans. It possesses a similar arousal method to the RHI. The research by Petkova \& Ehrsson \cite{PE08} and Slater et al. \cite{SPES09} used both a first-person visual perceptive of the mannequin and tactile stimulus to the test subjects. The report evaluates the subjects' illusion using verbal questionnaires. Lenggenhager et al. \cite{LMB09} and Aspell et al. \cite{ALB09} tries some tests using a third person visual perceptive input, but the subjects still felt the illusion by using the same evaluation methods.
In most of previous researches, the illusory ownership was evaluated by the investigations about how strong the subjects can feel the illusory ownership. However, we think oral investigations are still unconvincing. With the other methods, electromyography (EMG) was used to measure the difference of muscle activities when a strong blow to the fake hand \cite{THA*12}. They also verified that the EMG difference and strength of illusory ownership are positively related. Furthermore, in \cite{AR03}, the human skin resistivity was also measured if an illusory ownership is evoked. He/she is easy to sweat when he/she is nervous or feels fear. So they made an assumption that skin resistance can vary when illusory ownership was evoked.
\section{RHIs and FBIs measured by body temperature}
In this paper, we evaluated the strength of illusory ownership by measuring the body temperature, which based on the research of Moseley et al. \cite{MOV*08}. When the RHI was evoked, the subjects' hand temperatures were dropped.
\subsection{RHI}
First, we tried to evoke the RHI by using a virtual hand displayed in a HMD. There were 2 group of participants. Here, 8 participants (2 females and 6 males, $mean \pm $SD age = $24.5 \pm $1.59 years) were tested in RHI trial (Trial 1) and RHI control trial (Trial 2) in actual world base on the experiment of Moseley et al. \cite{MOV*08}. The other 8 participants (3 females and 5 males, $mean \pm $SD age=$25 \pm $2.27 years) were tested in RHI in VR space trial (Trial 3). None of them have any cognition diseases and the knowledge about RHI. We were keeping the indoor temperature under $25^\text{o}$C. The rubber hand and virtual hand displayed in HMD are the right hand.
\textbf{Trial 1:} Participants sat with their right hand resting on a table, and we set a block to prevent the participants seeing their own hand. A rubber right hand was place in front of the participants and a towel was placed over the participants' shoulder, hand, and the end of the rubber hand as if the rubber hand was connected to the participants' shoulder. Then we use two paintbrushes to stroke the rubber hand and the participants' subject hand synchronously.
\textbf{Trial 2:} The rubber hand was removed. The participants were instructed to look at the position where the rubber hand was placed in Trial 1. The other conditions remain unchanged.
\textbf{Trial 3:} Participants sat and wore a HMD, where a virtual hand resting on a virtual table are displayed as they are in front of the participant. A paintbrush stroking animation is added on the virtual hand with the same stroking pattern to the previous trials. The other conditions remain unchanged. Paintbrush stroking in actual world keep synchronized to the animation in VR space.
\textbf{Method:} For each trial, the test lasts 5 minutes and the hand's temperature was taken from a single spot on each participant's hand by a non-contact thermometer in every 30 seconds, which gave a total of 11 temperature readings per participant.
\begin{figure}[htb]
\centering
\includegraphics[width=.8\linewidth]{fig2_1.jpg}
\caption{\label{fig:2}
The average hand temperatures of all subjects in Trial 1, 2, and 3.}
\end{figure}
As we expected, in Trial 1 and Trial 3, all participants' hand temperatures decreased about 0.6 degrees as illustrated in Figure ~\ref{fig:2}. The T-test of the average temperature between Trial 2 (control) and Trial 3 (VR RHI) is [t (10, 0.005) = 3.16, P = 0.0002 < 1\%]. It has a significant difference. However, the T-test of Trial 1 (RHI) and Trial 3 (VR RHI) is [t (10, 0.025) = 2.228, P = 0.4592 > 5\%]. There is no significant difference between Trail 1 and Trial 3. Thus, we can safely conclude that RHI in VR space (Trial 3) and the RHI in actual world are related. The RHI can be also evoked in VR space and the skin temperature decreases can be observed.
\subsection{FBI}
According to \cite{LMB09} and \cite{PE08}, they evaluated the strength of illusory ownership in both First Person Perceptive (1pp) and Third Person Perceptive (3pp). In their tests, the subjects were asked if they felt the body they are observing were their own body or the mannequin stand in front of them were themselves. We considered, if there was a contradiction between 1pp and 3pp which can affect the physical body temperature decreases and it may not affect the result of investigation.
Here, 5 participants (1 female and 4 males, $mean \pm $SD age = $24.5 \pm $1.59 years) were participant in both 1pp and 3pp experiment. None of them have any cognition diseases and the knowledge about RHI and FBI. We keep the indoor temperature under $25^\text{o}$C. For each participant, we provided both a sleeveless shirt with two symmetrical holes on its back (the same level as shoulder blade) and a shorts to the knee with two symmetrical holes, in order to measure temperatures in 6 spots (four on the limbs and two on the back). Then, the participants sat and wore the HMD.
\textbf{1pp experiment:} Participants sat and could see a seated virtual body as if the participants see their own body through the HMD screen. The participants' chest was stroked while the virtual body's chest was stroked either synchronously or asynchronously.
\textbf{3pp experiment:} Participants sat and could see a virtual mannequin as if the mannequin was sitting in front of them through the HMD. The participant's chest and the mannequin's chest was stroked either synchronously or asynchronously.
\textbf{Method:} Two experiments were performed in order. Each experiment has two trials, a synchronous and an asynchronous trial. For each trial, stokes lasted 5 minutes and the body temperatures were taken from the 6 spots in order by a non-contact thermometer in every 30 seconds, which gave 11 body temperatures per spot, 66 body temperature measurements at one trial per participant.
\begin{figure}[htb]
\centering
\includegraphics[width=.9\linewidth]{fig3_1.jpg}
\caption{\label{fig:3}(a) The temperature comparison between synchronous and asynchronous stroke in 6 spot. (b) The average hand temperature changing between 1pp and 3pp synchronous trial.}
\end{figure}
In 1pp experiment, the body temperatures variations are showing in Figure ~\ref{fig:3}. (a). We set the synchrony and the temperatures in different spots to be two factors. The average body temperatures at each symmetrical spots on the left and right were evaluated by ANOVA test. The significant temperature differences in the symmetrical left and right spots such as: the left hand and the right hand [F (2,4) = 0.001, P = 0.982 > 5\%], the left calf and the right calf [F (2,4) = 0.001, P = 0.687 > 5\%], and the left back and the right back [F (2,4) = 0.001, P = 0.778 > 5\%] are all rejected. On the other hand, when we assume the synchronicities are the significant factors, the all significant average temperature differences on the hand [F (2,4) = 0.019 < 5\%], the calf [F (2,4) = 0.005 < 5\%], and the back [F (2,4) = 0.015 < 5\%] cannot be rejected. Therefore, we can safely conclude that a similar mechanism of human brain functions can work on both RHI and FBI. Furthermore, when the illusory ownership is evoked, a consistent temperature dropping can be observed.
Next, we illustrated the average hand temperature variations of both 1pp and 3pp synchronous trials by using T-test as indicated in Figure ~\ref{fig:3}. (b). There is a significant difference [t (10, 0.005) = 3.169, P = 0.0008<1\%]. We only utilized the temperature on the hand, because we have already found that the significant temperature differences over the 6 spots are rejected. For 1pp experiment, there was a significant difference in hand temperature decreases [F (2,4) = 0.001, P = 0.019 < 5\%]. But for 3pp experiment [F (2,4) = 1.861, P = 0.198 > 5\%], the significant temperature decreases was rejected. But, the previous researches by \cite{LMB09} and \cite{ALB09} claims that they did evoked the FBI in 3pp and they did not show any physical evidence whether the FBI is evoked or not. We observed these physical evidences only in 1pp experiment. Therefore, we considered the actual systemic illusion occurred on both physical body and mental spirit only in the first person perceptive.
\section{FBI for different size virtual bodies}
As some other researches suggest FBI can be affected by the authenticity of virtual mannequin. Banakou et al. \cite{BGS13} tested with bigger and smaller virtual mannequins. In their tests, the subjects gave positive oral answers on the illusory ownership for the both conditions. Here, we intended to accept more explicit responses from the subjects varying the body sizes.
There were 10 participants (all male, $mean \pm $SD age = $25.5 \pm $1.25 years) with the average height 175 cm. None of them have any cognition diseases and the knowledge about RHI and FBI. We kept the indoor temperature under $25^\text{o}$C. And prepared 4 different size virtual bodies with the height of 140cm, 175cm, 250cm and 500cm. Using each different height body, the participants wore the HMD, and were provided 5 minutes to adapt to their new bodies.
\textbf{140cm experiment:} A 140cm height mannequin was displayed sitting in front of a mirror as participants could see themselves in the mirror. There are two trials, each participant's chest was stroked while the mannequin's chest was stroked either synchronously or asynchronously. In the end of each trial, participants were asked if they felt the ownership to the mannequin.
\textbf{175cm experiment:} A 175cm height mannequin was displayed. The other procedures were identical to 140cm experiment.
\textbf{250cm experiment:} A 250cm height mannequin was displayed. The other procedures were identical to 140cm experiment.
\textbf{500cm experiment:} A 500cm height mannequin was displayed. The other procedures were identical to 140cm experiment.
\textbf{Method:} The method of these four experiments are the same as the previous FBI tests except that we only measure the hand temperatures.
As it shows in Figure ~\ref{fig:4}. (a) and (b), we observed the average temperature decreases only in the synchronous trials. There is a significant temperature difference between Figure ~\ref{fig:4}. (a) and (b) [F (1,36) = 38.399, P = 0.0001 < 0.1\%]. As the results of four different height mannequins in Figure ~\ref{fig:4}. (a), all the temperature measurements of four heights in the synchronous trials were decreased, and we set the heights as the main effect can be rejected [F (3,36) = 0.08, P = 0.971 > 5\%]. The consequence of these four experiments we conducted is consistent with the previous research \cite{BGS13}.
However, although the factor of mannequin heights had no significant effect on the body temperature, we found that the decrease rate of the body temperature is significantly different varying the different height mannequins. In Figure ~\ref{fig:4}. (c), on the synchronous trials, the 175cm and the 250cm trials have a large variation from time 30s to 120s. Thus, we classified the 175cm and the 250cm trials to be group 1 and the other two heights to be group 2. Then we found there is a significant difference between two groups [F (1,18) = 4.89, P = 0.042 < 5\%]. Furthermore, as it shows in Figure ~\ref{fig:4}. (d), on the asynchronous trials, the temperature did not change significantly and it can be tolerated as normal skin temperature changes. On the synchronous trials, the body temperature decreases for each height are, respectively, $0.42^\text{o}$C, $0.57^\text{o}$C, $0.58^\text{o}$C and $0.39^\text{o}$C. Also, the temperature decrease amplitude of group 1 is larger than the group 2 can be observed in Figure ~\ref{fig:4}. (d). And the difference in body temperature decreases in these two groups with a significant difference level [F (1,38) = 8.10, P = 0.035 < 5\%].As the result shows the temperature changing rate and the temperature decreases amplitude between these two group are different. Thus, we can say the group 1 has a stronger feeling of illusory ownership. Also the 175cm height is the average height of the participants, and one of strongest feeling of illusory ownership was evoked in this height. Therefore, the authenticity of virtual mannequin heights can affect the strength of illusory ownership.
\begin{figure}[htb]
\centering
\includegraphics[width=0.9\linewidth]{fig4_1.jpg}
\caption{\label{fig:4}The average temperature decreases between the (a) synchronous and (b) asynchronous trials. (c) The temperature changing rate of 4 heights under synchronous trials. (d) the amplitude of body temperature decreases between synchronous and asynchronous trials.}
\end{figure}
\section{Conclusions}
Illusory ownership is evoked in VR space by using HMD. The evoked FBI can be measured by using the body temperature decreases. The FBI can only be evoked in 1pp tests. We consider the evocation conditions of RHI and FBI are identical, and they can be summarized into three conditions as follows:
1. Human body must receive both the synchronized visual and somatosensory signals in temporal coincidence;
2. The visual signal must be the first person perceptive (1pp);
3. The subject and the virtual body needs to be the same height as much as possible.
The FBI can or may strongly induce or enhance the strong sense of unity with the virtual characters or avatars in VR spaces. The FBI can be used for teleoperation of avatar robots and can enhance the immersive feeling in VR spaces such as those in 4D cinemas. Manipulation of robots with different sizes is one of the unresolved problems, and FBI may help to resolve this problem. However, further researches are necessary to answer these questions. Also we will conduct some case studies to help the VR to be more comfortable for the sensitive users and collaborate with other fields (neuroscience e.g.) by using openBCI device to find out the impact on long-term VR users in the future work.
\bibliographystyle{eg-alpha-doi}
|
3,212,635,537,922 | arxiv | \section{Introduction}
\label{sec:int}
The presence of an electric quadrupole moment in the ground state of
the deuteron \cite{kel39,nor40} can be explained by including static
tensor terms in the microscopic nucleon--nucleon force as first
suggested by Rarita {\it et al.}~\cite{rar37,rar41a,rar41b}. This
procedure is today commonly adopted by all the modern microscopic
nucleon--nucleon interactions \cite{mac87,sto93,sto94,pud97,wir95}.
In a description of the nucleon--nucleon interaction based on a
meson--exchange picture \cite{mac87}, the strongest of the tensor
components, the tensor--isospin term, is dominated by the exchange
of a single pion. Since the pion is the lightest meson, this means
that the interaction range of the tensor--isospin term is the longest
one inside the nucleon--nucleon interaction.
For many years, tensor terms have not been considered in effective
interactions used in mean--field theories such as Hartree--Fock (HF) and
random--phase approximation (RPA), commonly used to
describe medium and heavy nuclei. An exception to this is represented
by the semirealistic M3Y--P interactions~\cite{nak03,nak08}. These
interactions are based on the effective M3Y-Paris
interaction~\cite{ana83} which has been constructed to describe
inelastic nucleon--nucleus processes. The new M3Y--P interactions
are obtained by including a density--dependent zero--range term and by
modifying some of the force parameters. The tensor
and tensor--isospin terms of the M3Y--P3 and of the M3Y--P5 interactions
are the same as those used in the original M3Y--Paris interaction.
In almost all the existing parameterizations of the most used
effective interaction in HF and RPA self--consistent calculations, the
Skyrme interaction, the tensor term is neglected, even though a
zero--range tensor term was proposed in the original formulation of the force
\cite{Sky56,Sky58}. In the last years, tensor terms have been
included either on top of existing Skyrme parametrizations like the
SIII~\cite{bei75} and the SLy5~\cite{cha97,cha98a,cha98b} forces (see
Refs.~\cite{sta77,col07}) or by inserting them in the global fit
procedure producing new parametrizations of Skyrme
interactions~\cite{bro06,les07,ben09}. In the following we shall
indicate as SLy5$_{\rm\,T}$\, the parametrization introduced in
Ref.~\cite{col07} where tensor terms have been added on top
of the SLy5 interaction.
Finite--range effective Gogny--like interactions are less used in HF
and RPA calculations than the Skyrme ones. As far as the tensor terms
are concerned, we find in the literature only few works where they
have been introduced in this type of forces. A first effort in this
direction was done by Onishi and Negele~\cite{oni78} who added a
tensor term to an effective force of finite range which was taken of
Gaussian form. After the introduction of the Gogny force in
1980~\cite{dec80}, some further attempts of including a finite--range
tensor term have been done. Otsuka {\it et al.}~\cite{ots05} proposed
the GT2 force obtained by adding to the standard central channels of
the original Gogny force~\cite{dec80} a finite--range tensor--isospin
term of Gaussian form, and by refitting all the parameters. An
alternative procedure was proposed in Ref.~\cite{co11} where the D1ST
and the D1MT forces were constructed by adding on top
of the D1S~\cite{ber91} and D1M~\cite{gor09} parameterizations a
finite--range tensor--isospin term chosen to reproduce the energy of
the first 0$^-$ state in $^{16}$O in self--consistent HF plus RPA
calculations.
The inclusion of tensor terms in effective interactions allows us to have
the same operator structure of the microscopic nucleon--nucleon interactions, and, moreover,
it is necessary to describe observables related to both single particle (s.p.)
\cite{col07,col08,mor10} and collective properties of medium and heavy
nuclei~\cite{cao09,don09,nes10,co11,ang11,co12b}.
The s.p. proton (neutron) energy gaps at $Z$ ($N$) = 8, 20, and 28 have been
investigated in Refs.~\cite{mor10,wan11} by using both non--relativistic and
relativistic HF techniques. These studies showed that nuclear systems corresponding to
$Z$ or $N =$ 8 and 20 are particularly suitable to study the neutron--proton tensor component
of the effective interaction.
On the other hand, the s.p. proton (neutron) energy gaps along
isotonic (isotopic) chains may be sensitive to the like--nucleon
component of the tensor interaction. Experimental observations
indicate that the $N=14$ neutron gap in oxygen isotopes increases when
going from $^{16}$O to $^{22}$O. A similar behavior is found in
calcium isotopes for the $N=28$ neutron s.p energy gap which increases
from $^{40}$Ca to $^{48}$Ca. The shell--model calculations of
Ref.~\cite{sar10} describe this behavior and predict an analogous
increase of the $N=90$ neutron gap from $^{132}$Sn to $^{140}$Sn. In
that work, the previous effect has been attributed to the three--body
terms of the interaction. Here we show that, in the framework of
mean--field HF theory, the evolution of these three neutron gaps
strongly depends on the presence of the like--nucleon component of the
tensor term.
For the Skyrme interaction, several works exist in the literature
where both the neutron--proton and the like--nucleon tensor
contributions have been analyzed. An extensive study of the effects
generated by these two contributions has been carried out in Ref.~\cite{les07}. In the Gogny
case, a detailed analysis where the two components are studied separately is still missing. As
already mentioned, in some recent works a finite--range tensor term
has been introduced only in the isospin dependent channel with a
single parameter to be chosen \cite{ots05,co11}. As it will be
discussed in the next section, this implies that the neutron--proton
and the like--nucleon contributions are proportional and have the same
sign, that is, they are both attractive or repulsive. Considering
that realistic and semirealistic nucleon--nucleon forces include both
types of tensor terms (pure tensor and tensor isospin), we propose
here to take into account both terms also for the effective Gogny
interaction. This implies the introduction of a second parameter which
allows us to separately tune the neutron--proton and the like--nucleon
tensor contributions of the effective interaction.
The work is organized as follows. In Sec.~\ref{sec:compSk} we present
the physics case of the $N=28$ neutron gap in calcium isotopes. We
describe the experimental energies by using the Gogny D1ST and the
Skyrme SLy5$_{\rm\,T}$\, interactions, and we show the need of including both
tensor and tensor--isospin terms in the Gogny interaction. In
Sec.~\ref{sec:neutron} we discuss
the implementation of the two finite--range tensor terms in the Gogny interaction, and we
propose two possibilities for the choice of the parameters. In
Sec.~\ref{sec:res} we apply these two parameterizations of the
tensor terms in the Gogny interaction to describe the evolution of the neutron gaps in
oxygen, calcium and tin isotopes. We compare our results with
experimental data (where available) and with the results of the HF
calculations carried on with the Skyrme interaction. Finally, we draw
in Sec.~\ref{sec:con} our conclusions and we discuss the perspectives
of future applications of the Gogny plus tensor interaction.
\section{Neutron $N=28$ energy gap in calcium isotopes}
\label{sec:compSk}
It has been experimentally established that the $N=28$ neutron energy gap,
that is, the difference between the s.p. energies of the 2$p_{3/2}$ and
1$f_{7/2}$ neutron levels, increases when going from $^{40}$Ca to
$^{48}$Ca. The experimental situation is summarized in Fig.~7 of
Ref.~\cite{sor08} and indicates a change from a value of about
2.2 MeV in $^{40}$Ca to 4.8 MeV in $^{48}$Ca.
We have calculated the evolution of this energy gap in the HF
framework by using Skyrme and Gogny interactions. Our results are
presented in Fig.~\ref{fig:skd1}. In the panel (b) we show with
solid and dotted lines, respectively, the results obtained with the
Skyrme SLy5 and SLy5$_{\rm\,T}$\, interactions. The values of the energy gaps we have obtained are,
in general, larger than the experimental ones. Despite this deficiency, we observe that the
interaction without tensor terms, the SLy5, does not describe the
trend of the energy gap, which is slightly decreasing in this calculation.
On the other hand, the result obtained with the SLy5$_{\rm\,T}$\,
force, which includes tensor terms, shows an increasing behavior of
the energy gap.
The behavior of the energy gap is controlled by the like--nucleon term of
the tensor force. In the Skyrme interaction the contribution of tensor
components to the energy density of the system can be written as
\cite{col07,mor10}
\begin{equation}
\Delta E_{\rm T}(r) = \frac{1}{2} \alpha_{\rm T}
\left[J^2_p(r) + J^2_n(r)\right] + \beta_{\rm T} J_p(r)
J_n(r)
\,\,\,,
\label{eq:skyten}
\end{equation}
where $J_p(r)$ and $J_n(r)$ are the proton and neutron spin--orbit densities.
The parameters $\alpha_{\rm T}$ and $\beta_{\rm T}$ rule, respectively, the
like--nucleon and the proton-neutron terms of the tensor interaction. In the
SLy5$_{\rm\,T}$\, force \cite{col07} these parameters assume the values -170~MeV~fm$^{5}$
and 100~MeV~fm$^{5}$, respectively.
It is easy to show that the effect of the tensor interaction is
almost zero in spin-saturated nuclear systems, since the effect on the
$j=l+1/2$ s.p. level is canceled by that on the $j=l-1/2$ one. The
global effect would be exactly zero if the radial wave functions of
the two levels were the same. Since calcium isotopes are
spin-saturated in protons, the like--nucleon tensor term
does not act on protons, and the neutron--proton
contribution is not active in the evolution of the neutron gap. The consequences of this
in the excitation of magnetic states in calcium isotopes have been widely
discussed in \cite{co12b}.
The sensitivity of our results to the like--nucleon tensor term is
shown in the panel (b) of Fig.~\ref{fig:skd1} by the dashed line,
obtained by changing the sign of the parameter $\alpha_{\rm T}$. This
modification leads to a decreasing energy gap going from $^{40}$Ca to
$^{48}$Ca.
In the panel (a) of Fig.~\ref{fig:skd1} we show the results of HF calculations carried out
with the Gogny interaction. The black full line indicates the result obtained with the D1S
force \cite{dec80} which does not contain tensor terms. The behavior
of the energy gap is analogous to that obtained with the Skyrme
interaction without tensor term. The dotted line shows the result
obtained with the D1ST interaction. In this case the behavior of the
energy gap is opposite with respect to the experimental one, and also
with respect to that obtained with the SLy5$_{\rm\,T}$\, interaction. If the sign
of the parameter that determines the strength of the tensor term in
the D1ST interaction is changed, the results indicated by the blue dashed line in panel (a)
of Fig.~\ref{fig:skd1} are obtained. We remark that this operation on the D1ST force acts only
on the tensor--isospin dependent term and, therefore, changes both the
like--nucleon and unlike components of the tensor force. In this way,
nuclear properties depending on the neutron--proton tensor interaction
that are well described by the D1ST force are not any more
reproduced. For example, the energy of the first $0^-$ state in
$^{16}$O whose experimental value of 10.94 MeV was used to tune the
tensor force term in the D1ST interaction, appears at 14.48 MeV when
the sign of the total strength is changed. Evidently, a unique
tensor-isospin term in the D1S force is not able to reproduce
simultaneously both nuclear properties.
\section{Tensor terms and the Gogny interaction}
\label{sec:neutron}
The D1ST and D1MT interactions have been constructed
by adding a tensor-isospin term to the Gogny D1S and D1M interactions,
respectively~\cite{co11}. The radial part of this term was based on
the analogous one in the microscopic Argonne V18
interaction~\cite{wir95}. Specifically, we have considered
\begin{equation}
v_{\rm Tt}(r) \,= \, v_{{\rm Tt}, {\rm AV18}} (r) \, \left[ 1\, -\,
\exp \left(-\,b \, r^2 \right) \right] \, ,
\label{eq:vtensor}
\end{equation}
where the radial part of the Argonne V18 tensor
isospin term ~\cite{wir95}, $v_{\rm Tt, AV18}(r)$,
has been multiplied by a function that simulates the
effect of the short-range correlations \cite{ari07}. Here
$b$ is a free parameter.
The inclusion of this tensor-isospin term was done without changing
the values of the other force parameters but the strength of the
spin-orbit term. The values of the two free parameters, one for the
tensor and the other one for the spin-orbit term, have been chosen to
reproduce, in an iterative HF plus RPA calculation chain, the energy
of the first $0^-$ state and the s.p. energy gap between the
$1p_{3/2}$ and $1p_{1/2}$ neutron states, in $^{16}$O \cite{ang11}.
In the present work we use an expression for the tensor interaction
similar to that proposed by Onishi and Negele~\cite{oni78}
\begin{eqnarray}
\nonumber
V_{\rm tensor} (r_1,r_2) &=& \left( V_{{\rm T}1} + V_{{\rm T}2} \,
P^{\tau}_{12} \right)\, S_{12} \,
\exp \left[ -(r_1-r_2)^2/\mu_{\rm T}^2 \right] \\
&=& \left[ \left(V_{{\rm T}1} + \frac{1}{2} V_{{\rm T}2} \right) \,
+ \frac{1}{2} V_{{\rm T}2} \, \mbox{\boldmath $\tau$}(1) \cdot \mbox{\boldmath $\tau$}(2) \right]\, S_{12} \,
\exp \left[ -(r_1-r_2)^2/\mu_{\rm T}^2 \right]
\label{eq:1fit}
\end{eqnarray}
where we have indicated with $P^{\tau}$ the usual isospin exchange
where $S_{12}$ and $\mbox{\boldmath $\tau$}$ represent the usual tensor and
isospin Pauli operators. In the second line we have
separated the pure tensor and tensor-isospin terms. In this approach
the radial part of the two independent tensor terms is identical, and
it has been chosen of Gaussian form. In our calculations we used
$\mu_{\rm T}=1.2$ fm, corresponding to the longest range of the D1S
interaction.
In this approach, the strength of the full tensor force is ruled by the two parameters
$V_{{\rm T}1}$ and $V_{{\rm T}2}$. A calculation of the isospin matrix elements for the
interaction (\ref{eq:1fit}) indicates that the strength of the force acting in
like--nucleon pairs is given by $V_{{\rm T}1}+V_{{\rm T}2}$,
while that between proton-neutron pairs is $V_{{\rm T}2}$. These
combinations of the parameters are, respectively, analogous to the
$\alpha_{\rm T}$ and $\beta_{\rm T}$ parameters of the Skyrme
interaction given in Eq. (\ref{eq:skyten}).
The two tensor terms in Eq. (\ref{eq:1fit}) have been added to the D1S
force without changing any other parameter value, including the
strength of the spin-orbit. In this way we are able to analyze
exclusively the effect of the tensor force.
In order to choose the values of the two free
parameters, $V_{{\rm T}1}$ and $V_{{\rm T}2}$, we have used two
observables. The first one is the energy difference between the
$1f_{5/2}$ and $1f_{7/2}$ s.p. neutron states in $^{48}$Ca. As already
discussed in Sect. \ref{sec:compSk}, this observable depends only on
the like--nucleon tensor contribution and therefore is ruled by
$V_{{\rm T}1}+V_{{\rm T}2}$. We show in Table~\ref{tab:tuno} the
energy difference between these two s.p. states obtained for
various values of $V_{{\rm T}1}+V_{{\rm T}2}$. We have verified that
by changing $V_{{\rm T}1}$ and $V_{{\rm T}2}$ the result is the same
if the sum does not change. The experimental value of the energy
difference is 8.8 MeV~\cite{cot08}, therefore we have chosen $V_{{\rm
T}1}+V_{{\rm T}2}=-20$~MeV.
The second observable we have considered is the energy of the first $0^-$ state in
the $^{16}$O nucleus. In Ref.~\cite{co11} a large sensitivity of the energies of the
$0^-$ states in doubly magic nuclei to the tensor-isospin term of the interaction was
observed. We show in Fig.~\ref{fig:e0} the excitation energy of this
state calculated in the HF plus RPA approach for different values of
$V_{{\rm T}2}$. All the calculations shown by the black solid line have been carried out by
using $V_{{\rm T}1}+V_{{\rm T}2}=-20$~MeV. For $V_{{\rm T}2}=115$~MeV we obtain for the energy
of the $0^-$ the value of 10.72 MeV, close to the experimental value
of 10.96 MeV \cite{led78}. This choice of $V_{{\rm T}2}$, together
with $V_{{\rm T}1}+V_{{\rm T}2}=-20$~MeV, implies $V_{{\rm T}1}=-135$
MeV. We label this parameterization D1ST2a.
In order to identify the general features of our results we have
implemented another parameterization of the tensor terms, which we call D1ST2b. In this case,
we selected the like-nucleon part of the tensor force to reproduce the $N=28$ neutron gap
increase from $^{40}$Ca to $^{48}$Ca as obtained in the HF calculation with the SLy5$_{\rm T}$
force. We obtained this results with the value of $V_{{\rm T}1}+V_{{\rm T}2}=-80$ MeV. As in the
previous case, the other observable we have chosen to select the value of $V_{{\rm T}2}$ is
the excitation energy of the first $0^-$ state in $^{16}$O. The blue
dotted line in Fig.~\ref{fig:e0} indicates the value $V_{{\rm T}2}$ = $102$ MeV.
In Fig.~\ref{fig:compare} we compare the two terms of the D1ST2a and D1ST2b tensor force with
the analogous ones of the effective M3YP5~\cite{nak08} and
microscopic AV18~\cite{wir95} interactions. The M3YP5 tensor isospin term is of the same order
than that of our interactions. In the case of the $v_T$ term, M3YP5 presents an attractive part for small
$q$, that becomes repulsive for $q>1$ fm$^{-1}$. It is interesting to notice that all the
effective interactions have a repulsive $v_T$ term and an attractive $v_{T \tau} $ term. On the contrary, in the
microscopic AV18 interaction both terms are attractive. This is an indication of the important role
played by both short and long range correlations in modifying the interaction.
\section{Neutron gaps}
\label{sec:res}
The results we discuss in this section have been obtained in the HF
framework. Pairing correlations are not included in our calculations
since the nuclei we have considered, $^{16}$O, $^{22}$O, $^{40}$Ca,
$^{48}$Ca, $^{132}$Sn and $^{140}$Sn, have a well defined closed-shell
character. The study has been conducted by comparing results obtained
by using interactions with (D1ST2a, D1ST2b and SLy5$_{\rm\,T}$\,) and
without (D1S and Sly5) tensor terms.
\subsection{N=28 and N=90}
\label{su:gaps}
The gap evolution in calcium and tin isotopes, are rather similar. The
case $N=28$ involves the 2$p_{3/2}$ and 1$f_{7/2}$ neutron s.p. levels
in $^{40}$Ca and $^{48}$Ca. The s.p. energies of these states are
shown in Fig.~\ref{fig:gap28}, for the Gogny (panel (a)) and Skyrme
(panel (b)) interactions. The corresponding gap values are shown in
panels (c) and (d) of the same figure.
The effects of the tensor on the energies of the 2$p_{3/2}$ state are
rather small, while those on the 1$f_{7/2}$ state are more evident,
producing a lowering of the energy value in $^{48}$Ca, much pronounced
in the case of the Skyrme and D1ST2b interactions. In both type of calculations
(Skyrme and Gogny) only the presence of the tensor terms produces an increase of the gap,
in agreement with the experimental evidence \cite{sor08}. In the
shell--model calculations of Ref.~\cite{sar10} the energy of the 1$f_{7/2}$ level is lowered
and that of the 2$p_{3/2}$ level is increased. This last effect is not present in our
calculations.
The case $N=90$ involves similar s.p. states which differ from the
$N=28$ case only for the principal quantum numbers. The results
obtained are presented in Fig.~\ref{fig:gap90} and show behaviours
similar to those shown in Fig.~\ref{fig:gap28}. In this case, we found
an increase of the energy gap already in the D1S calculation. This
effect is enhanced by the inclusion of the tensor term and is more evident for the D1ST2b force.
No experimental data are available for the $N=90$ gap, however shell model calculations carried
out with microscopic interactions indicate an increase of the $N=90$ gap \cite{sor08}.
\subsection{N=14}
\label{su:gap14}
We discuss another case, in a different region of the nuclear chart,
where the experimental values of the s.p energies are
known. We consider the energy gap $N=14$ between the 2$s_{1/2}$ and
the 1$d_{5/2}$ neutron states in oxygen isotopes. The experimental
value of this gap in $^{16}$O is 0.87 MeV \cite{vau72}. From the
study of the excited states in $^{21-23}$O nuclei through their
$\gamma$ decay, Stanoiu {\it et al.} \cite{sta04} deduced a value of
the energy gap of 4.11 MeV in $^{22}$O. This value is relatively large
and, for this reason, $^{22}$O can be considered a doubly magic
nucleus. This is also supported by the observation that the value of
the excitation energy of the first $2^+$ state in $^{22}$O is almost
twice that observed in the neighboring even--even nuclei.
The results of our calculations are shown in Fig.~\ref{fig:gap14}.
Also in this case the behavior found for the two types of
interactions, Gogny and Skyrme, are rather similar. The major effects
of the tensor terms of the force are present on the neutron 1$d_{5/2}$
s.p. energies which, in $^{22}$O are remarkably lower than those
obtained without tensor, mainly for the Skyrme
interaction and in the case of the D1ST2b force. This effect produces an
increase of the energy gap, even though the energies of the 2$s_{1/2}$
states remain unchanged. The results obtained with the D1S and SLy5
force show a decreasing gap.
\section{Conclusions}
\label{sec:con}
In this work, we have first pointed out the need of including in the
effective Gogny interaction two independent tensor terms acting
separately on like--nucleon and proton-neutron pairs. We have included
these two independent terms under the form of tensor and
tensor--isospin components to be added on top of effective Gogny
forces. To the best of our knowledge, only tensor--isospin terms have
been considered up to now for these finite--range interactions.
We have proposed two different parameterizations of the tensor force. In a first one,
the strength of the like-nucleon part of the tensor force has been chosen to reproduce the
experimental value of the splitting between the $1f_{7/2}$ and $1f_{5/2}$ neutron s.p. energies
in $^{48}$Ca. In the second parameterization, the strength of this part of the interaction has
been chosen to reproduce the neutron gap increase in $N=28$ going from $^{40}$Ca to $^{48}$Ca as
obtained with the SLy5$_{\rm T}$ interaction. In both parameterizations the remaining term ruling
the proton-neutron term has been selected to the energy of the first $0^-$ excited state in
$^{16}$O
Using these parameterizations, we have calculated the neutron energy gap for $N=$
14, 28, and 90, in oxygen, calcium and tin isotopes respectively. Our
results show that both parameterizations reproduce the trend for the neutron gaps obtained with
the Skyrme SLy5$_{\rm\,T}$\, interaction, better in the case of the D1ST2b fit. This trend is
in agreement with the experimental behavior in oxygen and calcium
isotopes, and with the results of shell--model calculations in tin isotopes.
The inclusion of two tensor terms allows us to reproduce the
experimental trends of the neutron energy gaps in the isotope chains we have
investigated. This is our main result. From the quantitative point of
view it is evident that the two observables related to the
like-nucleon term of the tensor interaction are not compatible in HF
calculations. The parameterization D1ST2a built to reproduce the
s.p. splitting of the $f$ states in $^{48}$Ca produces the correct
behaviour of the neutron energy gap, but its value is quantitative too
small. Probably, a good quantitative description of these two
quantities requires to go beyond mean-field calculations, and to
consider explicitly the effects of the coupling between s.p. and
collective degrees of freedom.
We consider the present work as a step forward in the direction of
constructing a new parameterization of the Gogny interaction which
include tensor terms. We have the perspective of validating these new
tensor terms by using them in the description of observables where the
particle--like contribution of the tensor force is expected to play a
role, for example, in the excitation of unnatural parity states in
nuclei with neutron excess and with closed proton shells \cite{ang11}.
Of course, a more accurate fit would require to simultaneously modify
all the parameters of the Gogny force, especially the spin--orbit strength
which has a strong interplay with the tensor force, in both the
s.p. energies and the excitation of magnetic states.
\acknowledgments
This work has been partially supported by the PRIN (Italy) {\sl
Struttura e dinamica dei nuclei fuori dalla valle di stabilit\`a}, by
the Spanish Ministerio de Ciencia e Innovaci\'on (Contract
Nos. FPA2009-14091-C02-02 and ACI2009-1007) and by the Junta de
Andaluc\'{\i}a (FQM0220).
\newpage
|
3,212,635,537,923 | arxiv | \section{GENERAL ASPECTS OF RESOURCE DESTROYING MAPS}
Here we provide detailed discussions of general properties of the
theory of resource destruction. Specifically, we show that the
convexity of the set of free states is a necessary condition for
an associated resource destroying map to be linear, and analyse
the robustness of the resource-free conditions.
\subsection{Linearity of resource destroying map}
We prove the following result that
relates the convexity of the theory and the linearity of a resource destroying map:
\begin{thm}
Let $S$ be a set of states. Suppose $S$ is not convex, then there does not exist a linear map that stabilizes the states in $S$ only, i.e., no linear map $\lambda$ satisfies $\lambda(\rho)=\rho$ for all $\rho\in S$, and $\lambda(\sigma)\neq\sigma$ for all $\sigma\not\in S$.
\label{convex}
\end{thm}
\begin{proof}
By the nonconvexity of $S$, one can always find $\rho_1,\rho_2\in S$ such that $p\rho_1+(1-p)\rho_2\not\in S$ for some probability $p$. Suppose there is such a linear $\lambda$. Then $\lambda(p\rho_1+(1-p)\rho_2)=p\lambda(\rho_1)+(1-p)\lambda(\rho_2)=p\rho_1+(1-p)\rho_2$ by plugging in the property that $\lambda(\rho)=\rho$ when $\rho\in S$. This contradicts the other defining property of $\lambda$ that $\Lambda(\sigma)\neq\sigma$ when $\sigma\not\in S$.
\end{proof}
Recall that only free states are stabilized by a resource destroying map.
Therefore, for a theory with nonconvex $F$, no linear map can satisfy both requirements on all inputs. This implies that nonconvex theories do not admit resource destroying channels.
\subsection{Robustness of resource-free conditions}
When the set of free states $F$ is not a singleton, i.e., contains more than one element,
the definition of resource destroying map $\lambda$ is not unique.
This leads to the question of whether different choices of $\lambda$
for a given $F$ define different resource nongenerating,
nonactivating and commuting conditions (and their selective versions).
As mentioned in the main text, the resource nongenerating condition is robust:
since $\lambda$ is always surjective onto $F$ by the nonresource-fixing
requirement, this condition exclusively selects out the operations
under which $F$ is closed. Here we show that, in contrast, resource
nonactivating operations and thus commuting operations can depend on
$\lambda$.
For example, consider a peculiar $\lambda$ that maps all
$\rho\not\in F$ to a particular $\rho_0\in F$. In other words,
$\rho_0$ and all states outside $F$ form a family and all other
free states are ``orphans'' without any parent states. If an
operation does not stabilize all free states, then it has to
map all states to the same image (a free state) to satisfy the
nonactivating condition. Such a requirement is clearly stronger
than the general case.
An explicit example in the context of coherence is as follows. We define
an extreme coherence-destroying map that takes all coherent states
to one incoherent state $\rho_0$, while all other incoherent states are orphans.
Under this map, a partial depolarizing channel fails the
nonactivating condition
since it
maps $\rho_0$ to $\tau\rho_0+(1-\tau)I/d$, which is still incoherent,
while any coherent state remains coherent, thus always mapped to
$\rho_0$. However, the partial depolarizing channel
obviously satisfies the (selective) nonactivating condition with $\Pi$.
In addition, if we restrict the input to be free states, the
resource nongenerating condition allows the same set
of operations, but the nonactivating condition trivially holds for all
channels. Combining all these observations, we see that only
the nongenerating condition is robust under the choice of
resource destroying map.
Note that the above robustness results hold for the selective version of each condition.
It would be interesting to study resource non-activation
under appropriate restrictions on the choices for $\lambda$.
For example, one could require that no orphans exist,
or consider a class of resource destroying maps instead of a
single one. As usual, we are primarily interested in physically
motivated definitions. Despite the fragility of the
resource nonactivating and commuting conditions under
variations of resource destroying maps and input states, these
conditions are still meaningful and nontrivial for physical
definitions of $\lambda$.
\subsection{Selective monotonicity}
We say a resource measure $f(\rho)$ exhibits selective monotonicity
if it is monotone nonincreasing under selective measurements on average.
That is, $f(\rho)\geq \sum_\mu p_\mu f(\mathcal{E}_\mu(\rho))$
where $p_\mu = {\rm tr}\,\mathcal{E}_\mu(\rho)$.
Now consider a valid distance measure $D(\cdot,\cdot)$ that
satisfies
$D(\rho,\sigma)\geq \sum_\mu p_\mu
D(\mathcal{E}_\mu(\rho),\mathcal{E}_\mu(\sigma))$,
such as quantum relative entropy $S(\rho||\sigma)$ \cite{vedralplenio}.
For selective commuting operations, i.e., $[\mathcal{E}_\mu,\lambda]=0$ for all $\mu$:
\begin{eqnarray}
\tilde{\mathfrak{D}}(\rho) &:=& D(\rho,\lambda(\rho)) \nonumber\\
&\geq& \sum_\mu p_\mu D(\mathcal{E}_\mu(\rho),\mathcal{E}_\mu(\lambda(\rho))) \nonumber\\
&=& \sum_\mu p_\mu D(\mathcal{E}_\mu(\rho),\lambda(\mathcal{E}_\mu(\rho)))\nonumber\\
&\equiv& \sum_\mu p_\mu \tilde{\mathfrak{D}}(\mathcal{E}_\mu(\rho)),
\end{eqnarray}
where the second line follows from the given property of $D$, and the third line follows from the selective commuting condition.
Thus, strong monotonicity holds for the simple measure $\tilde{\mathfrak{D}}$ introduced in Eq.\ (4) with proper $D$, e.g., the relative entropy between a state and its resource-destroyed counterpart, under selective commuting operations.
\section{COHERENCE}\label{appa}
Here we present a detailed analysis of the application of the theory of
resource destroying maps to the resource theory of coherence.
We first analyze the comparative power of coherence-free classes defined by the theory of $\Pi$, namely $\bar{X}(\Pi)$, $\bar{X}^*(\Pi)$ and $X(\Pi)$ and their selective counterparts $\bar{X}_s(\Pi)$, $\bar{X}_s^*(\Pi)$ and $X_s(\Pi)$. We give some new examples of operations that exhibit characteristic behaviors in this theory.
Note that $\bar{X}_s(\Pi)$ and $X_s(\Pi)$ are respectively equivalent to Incoherent Operations (IO) \cite{baumgratz} and Strictly Incoherent Operations (SIO) \cite{sio} in literature. $\bar{X}(\Pi)$, namely coherence nongenerating operations, is recently studied by one of the authors \cite{hung}. In the preparation of this work, we became aware that $X(\Pi)$ is independently studied as Dephasing-covariant Incoherent Operations (DIO) in Refs.\ \cite{speakable,examination}.
Notably, Ref.\ \cite{examination} argued that this class is a maximal extension of the ``physically consistent'' class, which is another interesting interpretation of $X(\Pi)$.
The structure of this theory is illustrated in Fig.\ \ref{cohfig}.
For comparison, we also briefly discuss about some coherence-free definitions arising from other scenarios, including Translationally Invariant Operations (TIO) \cite{time,speed} and Genuinely Incoherent Operations (GIO) \cite{genuine,powerg}. In this section, $\{|i\rangle\langle i|\}$ is the incoherent basis.
\begin{figure}[H]
\centering
\includegraphics[width=0.36\columnwidth]{coh_rd.png}
\caption{\label{cohfig}A Venn diagram of coherence-free operations arising from the theory of resource destruction.}
\end{figure}
\subsection{$\bar{X}_s(\Pi)$ and $X(\Pi)$ are incomparable}
We first show that neither $\bar{X}_s(\Pi)$ and $X(\Pi)$ contains the other by constructing quantum operations belonging to $\bar{X}_s(\Pi)\backslash X(\Pi)$ and $X(\Pi)\backslash\bar{X}_s(\Pi)$. To achieve this, we derive the conditions on the entries of Kraus operators for operations in $\bar{X}_s(\Pi)$ and $X(\Pi)$. For an operation in $\bar{X}_s(\Pi)$, there is a Kraus decomposition such that
\begin{equation}
K^{(\mu)}_{ki}K^{(\mu)*}_{li}=0,\forall k\neq l
\end{equation}
for all Kraus operators $K^{(\mu)}$, where $K^{(\mu)}_{ki}=\langle{k}|K^{(\mu)}|{i}\rangle$. On the other hand, an operation in $X(\Pi)$ requires the summation \begin{equation}
\label{Eq:Pi_Kraus}\sum_\mu K^{(\mu)}_{ki}K^{(\mu)*}_{li}=0,\forall k\neq l.
\end{equation}
Consider a qubit operation $\mathcal E_1(\rho)=K_1(\rho)K_1^\dagger+K_2(\rho)K_2^\dagger$ with $K_1=|0\rangle\langle +|$ and $K_2=|1\rangle\langle-|$. $K_1$ and $K_2$ are both incoherent so $\mathcal E_1\in\bar{X}_s(\Pi)$. However, it can be checked that $\mathcal E_1(\Pi(|+\rangle\langle+|))=I/2$ but $\Pi(\mathcal E_1(|+\rangle\langle+|))=|0\rangle\langle0|$, thus $\mathcal E_1\in \bar{X}_s(\Pi)\backslash X(\Pi)$. Since $\bar{X}_s(\Pi)\subset \bar{X}(\Pi)$, this operation is also an example of $\bar{X}(\Pi)\backslash X(\Pi)$.
Next consider a qutrit operation $\mathcal E_2$ with Kraus operators
\begin{equation}
\begin{array}{ccc}
K_1=\left(\begin{array}{ccc}x_1&0&0\\
0&a&0\\
0&-b&0\end{array}\right), &
K_2=\left(\begin{array}{ccc}0&0&x_2\\
0&b^*&0\\
c^*&a^*&0\end{array}\right), &
K_3=\left(\begin{array}{ccc}0&0&0\\
0&0&x_3\\
a&-c&0\end{array}\right),
\end{array}
\end{equation}
where the parameters satisfy $|x_1|^2+|c|^2+|a|^2=2|a|^2+2|b|^2+|c|^2=|x_2|^2+|x_3|^2=1$. It can be checked that any linear combination of the three Kraus operators is not incoherent. Since any other Kraus decomposition $\{M_i\}_{i=1}^d$ of $\mathcal{E}_2$ is related to $\{K_1,K_2,K_3\}$ by a $d$-dimensional unitary transformation $[u_{ij}]_{i,j=1}^d$ as $M_i=\sum_{j=1}^3u_{ij}K_j$, (and hence $M_i$ are not incoherent), we conclude that $\mathcal E_2\not\in\bar{X}_s(\Pi)$. Meanwhile, $\{K_1,K_2,K_3\}$ satisfy Eq.\ (\ref{Eq:Pi_Kraus}), thus $\mathcal E_2\in X({\Pi})\backslash\bar{X}_s(\Pi)$.
We also note that $\bar{X}_s(\Pi)\cap X(\Pi)$ is not empty, because $X_s(\Pi)$ (studied in \cite{sio}) is a subset of both. So $\bar{X}_s(\Pi)$ and $X(\Pi)$ are incomparable but not disjoint.
\subsection{Incoherent-measure-and-prepare operations are in $\bar{X}_s^\ast(\Pi)$}
By previous results and references cited in the introduction,
we already have a full characterization of the coherence
nongenerating and $\Pi$-commuting classes. The question
of non-activation has not been studied before, however.
Accordingly, we exhibit here a class of operations belonging
to $\bar{X}_s^\ast(\Pi)$ (and thus $\bar{X}^\ast(\Pi)$). Some of them are able to generate coherence, so we also have examples of $\bar{X}_s^\ast(\Pi)\backslash X(\Pi)$.
Consider the following type of operations such that the
Kraus operators take the form $K_i=|f_i\rangle\langle i|$.
Such operations represent the following measure-and-prepare procedure:
one first performs a projective measurement in the incoherent basis,
and then prepares the system the corresponding state $|f_i\rangle$
upon measuring $i$. Such operations are entanglement
breaking when acting locally \cite{entbreak}.
Here we show that they belong to $\bar{X}_s^\ast(\Pi)$. Let $\mathcal{E}^{\rm MP}$ be such an operation.
Notice that the measuring step of $\mathcal{E}^{\rm MP}$ destroys coherence, so $\mathcal{E}^{\rm MP}\circ\Pi=\mathcal{E}^{\rm MP}$ automatically holds. More explicitly, for any $\rho$,
\begin{eqnarray}
\mathcal{E}^{\rm MP}(\Pi(\rho))&=&\sum_{ij} |f_i\rangle\langle i|j\rangle\langle j|\rho|j\rangle\langle j|i\rangle\langle f_i| \nonumber\\
&=& \sum_i |f_i\rangle\langle i|\rho|i\rangle\langle f_i| = \mathcal{E}^{\rm MP}(\rho),
\end{eqnarray}
where we used $\langle i|j \rangle=\delta_{ij}$ for the second equality.
The same condition also holds for each Kraus arm.
So $\mathcal{E}^{\rm MP}(\rho)\in\bar{X}_s^\ast(\Pi)$.
Now suppose there exists some $i$ such that $|f_i\rangle$ is coherent: then the operation ceases to be coherence nongenerating (simply take $|i\rangle$ as the input). Therefore, such operations reside in $\bar{X}_s^\ast(\Pi)\backslash X(\Pi)$.
\subsection{Other definitions of coherence-free operations}
Besides those derived from coherence destruction $\Pi$ as shown above, there are also some other proposals of coherence-free operations arising from different contexts.
Two notable ones are Translationally Invariant Operations (TIO) \cite{time,speed} and Genuinely Incoherent Operations (GIO) \cite{genuine,powerg}. However, we argue that both of them are not theories of coherence with respect to a specified observable in a precise sense.
TIO naturally arises from the asymmetry theory \cite{refrmp,asymgour}, since coherence can be viewed as asymmetry relative to time translations generated by some preferred Hamiltonian \cite{time,speed}.
An operation $\mathcal E^{\rm TI}$ is said to be translationally-invariant with respect to a Hamiltonian $H$ if it satisfies
\begin{equation}
\mathcal E^{\rm TI}(e^{-iHt}\rho e^{iHt})=e^{-iHt}\mathcal E^{\rm TI}(\rho) e^{iHt},\forall t,
\end{equation}
for any state $\rho$. It turns out that the power of TIO depends on whether $H$ exhibits degeneracy or not. For general $H$, it is possible to generate coherence within the decoherence-free subspaces using TIO, so this class is technically not even contained in the maximal class $\bar{X}(\Pi)$. However, when $H$ has a nondegenerate spectrum, the resulting class ${\rm TIO}^\ast$) defines more precisely a theory of coherence with respect to the eigenbasis of $H$. It can be shown that ${\rm TIO}^\ast\subset X_s(\Pi)$ \cite{sio} (earlier Ref.\ \cite{speed} showed that ${\rm TIO}^\ast\subset \bar{X}_s(\Pi)$).
The concept of GIO is proposed in order to remove the dependence of incoherence on specific experimental realizations. In other words, the Kraus arms are unable to create coherence for \emph{all} Kraus decompositions of a GIO, or equivalently, all Kraus operators are diagonal in the incoherent basis \cite{genuine}.
A consequence is that all incoherent states are invariant under GIO.
Therefore, in some sense, GIO represents a theory with more constraints in addition to those imposed by incoherence since it cannot even achieve transformations among incoherent (free) states.
Indeed, it is known that $\mathrm{GIO}\subset \mathrm{TIO}^\ast$ \cite{speed} and $\mathrm{TIO}^\ast\subset X_s(\Pi)$ \cite{sio}: $\mathrm{GIO}$ is strictly weaker than the weakest class given by the theory of $\Pi$. We include a more intuitive and straightforward proof of $\mathrm{GIO}\subset X_s(\Pi)$ here.
Let $\mathcal{E}^{\rm GI}$ be a GIO.
Suppose a Kraus decomposition of $\mathcal{E}^{\rm GI}$ reads $\mathcal{E}^{\rm GI}(\rho)=\sum_\mu K_\mu \rho K^\dagger_\mu$ and $\mathcal{E}^{\rm GI}_\mu(\rho):=K_\mu \rho K^\dagger_\mu$. By Theorem 1 of Ref.\ \cite{genuine}, $K_\mu$ is diagonal in the incoherent basis. In other words, $K_\mu|i\rangle=\alpha_{i\mu}|i\rangle$ for all $i,\mu$, where $\alpha_{il}$ is some constant.
Notice that any state $\rho$ can be written in the form of
\begin{equation}
\rho=\Pi(\rho)+\sum_{j \neq i}|i\rangle\langle i|\rho|j\rangle\langle j|,
\end{equation}
which separates the diagonal and off-diagonal parts. By linearity of $\mathcal{E}^{\rm GI}_\mu$,
\begin{eqnarray}
\mathcal{E}^{\rm GI}_\mu(\rho)&=&\mathcal{E}^{\rm GI}_\mu(\Pi(\rho))+\sum_{j \neq i} K_\mu|i\rangle\langle i|\rho|j\rangle\langle j|K^\dagger_\mu\nonumber\\
&=&\mathcal{E}^{\rm GI}_\mu(\Pi(\rho))+\sum_{j \neq i} \alpha_{i\mu}\alpha_{j\mu}^\ast|i\rangle\langle i|\rho|j\rangle\langle j|.\label{lin}
\end{eqnarray}
Notice that the off-diagonal parts remain off-diagonal, which are erased by a following $\Pi$. That is, $\Pi(\mathcal{E}^{\rm GI}_\mu(\rho))=\Pi(\rho)=\mathcal{E}^{\rm GI}_\mu(\Pi(\rho))$ for all $\rho$, i.e., $\mathcal{E}^{\rm GI}_\mu\circ\Pi=\Pi\circ\mathcal{E}^{\rm GI}_\mu$.
So $\mathrm{GIO}\subseteq X_s(\Pi)$.
To see that the containment is proper, consider an erasure channel that maps everything to $|0\rangle\langle 0|$, where $|0\rangle$ is an incoherent basis state. This channel obviously belongs to $X_s(\Pi)$. But it is not a GIO since it does not leave incoherent states invariant except $|0\rangle\langle 0|$.
\section{DISCORD}\label{appb}
It is much more difficult to study discord-free conditions since the discord destroying map $\pi$ (which acts locally) is dependent on the input and not uniquely defined within degenerate subspaces.
To obtain some preliminary understandings of this $\pi$ theory, we examine the power of some of the most typical quantum operations acting locally on the same subsystem as $\pi$ ( without loss of generality, subsystem $A$).
We show that local unitary-isotropic channels (mixture of some unitary channel and depolarization, or unitary with white noise) exhibit the strongest classicality: they belong to both $X_A(\pi_A)$ and $X_{s,A}(\pi_A)$. Nevertheless, rank-one projective measurements fail to be nonactivating: they reside in $\bar{X}_{s,A}(\pi_A)\backslash X_A(\pi_A)$. In addition, a peculiar feature that distinguishes nonlinear resource destroying maps from linear ones is that selective classes are not necessarily contained in their original counterparts. We confirm this in the $\pi$ theory by showing that $\bar{X}_{s,A}(\pi_A)\backslash \bar{X}_A(\pi_A)$ contain certain qudit ($d>2$) mixed-unitary channels.
We also provide a measure-and-prepare protocol that is able to generate but not activate discord. However, this protocol does not represent a channel since it depends on the eigenbasis of the input.
In this section, we follow the notations used in the main text: $\rho_{AB}$ is an arbitrary bipartite state, $\rho_A=\sum_i p_i|i\rangle\langle i|$ ($\{|i\rangle\langle i|\}$ diagonalizes the reduced density operator of $A$).
\subsection{Unitary-isotropic channels are in $X_A(\pi_A)$ and $X_{s,A}(\pi_A)$}
\begin{comment}
First let's consider the simplest case of unitary channels. Let $u$ be a unitary channel: $u(\rho):=U\rho U^\dagger$, where $U$ is an unitary operator.
We have
\begin{eqnarray}
(u_A \otimes I_B)(\pi_A(\rho_{AB}))&=&U_A\left(\sum_{i} |i\rangle_A\langle i| \rho_{AB} |i\rangle_A\langle i|\right) U^\dagger_A \nonumber\\
&=& \sum_i U|i\rangle_A\langle i|U^\dagger\otimes\langle i|\rho_{AB}|i\rangle,
\end{eqnarray}
where ${\rm tr}\langle i|\rho_{AB}|i\rangle=p_i$.
On the other hand, one can write $\rho_{AB}=\sum_i |i\rangle_A\langle i|\otimes\langle i|\rho_{AB}|i\rangle+\sum_{j\neq k} |j\rangle_A\langle k|\otimes\langle j|\rho_{AB}|k\rangle$ where ${\rm tr}\langle j|\rho_{AB}|k\rangle=0$. So
\begin{equation}
(u_A \otimes I_B)(\rho_{AB})=\sum_i U|i\rangle_A\langle i|U^\dagger\otimes\langle i|\rho_{AB}|i\rangle +\sum_{j\neq k} U|j\rangle_A\langle k|U^\dagger\otimes\langle j|\rho_{AB}|k\rangle. \label{ui}
\end{equation}
Notice that ${\rm tr}_B(u_A \otimes I_B)(\rho_{AB})=\sum_i p_i U|i\rangle\langle i|U^\dagger $, so $\{U|i\rangle\langle i| U^\dagger\}$ is a new eigenbasis. Therefore,
\begin{eqnarray}
\pi_A((u_A \otimes I_B)(\rho_{AB}))&=&\sum_{i} U|i\rangle_A\langle i| U^\dagger (u_A \otimes I_B)(\rho_{AB}) U^\dagger|i\rangle_A\langle i| U \nonumber\\
&=&
\sum_i U|i\rangle_A\langle i|U^\dagger\otimes\langle i|\rho_{AB}|i\rangle\nonumber\\
&=&(u_A \otimes I_B)(\pi_A(\rho_{AB})),
\end{eqnarray}
where the second equality is obtained by plugging in Eq.\ (\ref{ui}): the latter part vanishes. That is, $(u_A \otimes I_B)\circ\pi_A=\pi_A\circ(u_A \otimes I_B)$: $u\in X_A(\pi_A)$.
Since $u$ itself represents the only Kraus arm, it trivially belongs to $X_{s,A}(\pi_A)$.
\end{comment}
Unitary-isotropic channels take the form $\tilde{u}^\gamma(\rho)=(1-\gamma)U\rho U^\dagger+\gamma I/d$, where $U$ is unitary, $\gamma$ characterizes the degree of depolarization ($\gamma\in[0,d^2/(d^2-1)]$ so that $\tilde{u}^\gamma$ is completely positive \cite{2016arXiv161007504B}), and $d$ is the dimension of the Hilbert space.
Unitary channels ($\gamma=0$) and depolarizing channels ($U=I$) are special cases of unitary-isotropic channels.
On the one hand, $\pi_A$ is a local measurement in the $\{|i\rangle\langle i|\}$ basis, so
\begin{eqnarray}
\pi_A(\rho_{AB}) &=& \sum_{i} |i\rangle_A\langle i| \rho_{AB} |i\rangle_A\langle i|,\\
(\tilde{u}^\gamma_A \otimes I_B)(\pi_A(\rho_{AB}))&=&(1-\gamma)U_A\left(\sum_{i} |i\rangle_A\langle i| \rho_{AB} |i\rangle_A\langle i|\right) U^\dagger_A + \frac{\gamma}{d_A}I_A\otimes\rho_B\nonumber\\
&=& (1-\gamma)\sum_i U_A|i\rangle_A\langle i|U_A^\dagger\otimes\langle i|\rho_{AB}|i\rangle+\frac{\gamma}{d_A}I_A\otimes\rho_B,
\end{eqnarray}
where ${\rm tr}\langle i|\rho_{AB}|i\rangle=p_i$.
On the other hand,
\begin{equation}
(\tilde{u}^\gamma_A\otimes I_B)(\rho_{AB})=(1-\gamma)(U_A\otimes I_B)\rho_{AB}(U_A^\dagger\otimes I_B)+\frac{\gamma}{d_A} I_A\otimes\rho_B.
\end{equation}
Notice that ${\rm tr}_B(\tilde{u}^\gamma_A \otimes I_B)(\rho_{AB})=\sum_i ((1-\gamma)p_i+\gamma/d) U_A|i\rangle_A\langle i|U_A^\dagger $, so $\{U|i\rangle\langle i| U^\dagger\}$ is a new eigenbasis, which implies that
\begin{eqnarray}
\pi_A((\tilde{u}^\gamma_A\otimes I_B)\rho_{AB})&=&(1-\gamma)\sum_i U_A|i\rangle_A\langle i|U_A^\dagger U_A\rho_{AB}U_A^\dagger U_A|i\rangle_A\langle i|U_A^\dagger+\frac{\gamma}{d_A}\sum_i U_A|i\rangle_A\langle i|U_A^\dagger I_A U_A|i\rangle_A\langle i|U_A^\dagger\otimes\rho_B\nonumber\\
&=& (1-\gamma)\sum_i U_A|i\rangle_A\langle i|U_A^\dagger\otimes\langle i|\rho_{AB}|i\rangle+\frac{\gamma}{d_A}I_A\otimes\rho_B\nonumber\\
&=&(\tilde{u}^\gamma_A\otimes I_B)(\pi_A(\rho_{AB})).
\end{eqnarray}
That is, $(\tilde{u}^\gamma_A\otimes I_B)\circ \pi_A =\pi_A \circ (\tilde{u}^\gamma_A\otimes I_B)$: $\tilde{u}^\gamma\in X_A(\pi_A)$.
Now notice that applying Heisenberg-Weyl operators uniformly at random (Heisenberg-Weyl twirling) on any qudit gives the maximally mixed state \cite{wilde}. This indicates that unitary-isotropic channels admit a Kraus decomposition by unitaries (belong to mixed-unitary channels). More explicitly,
\begin{equation}
\tilde{u}^\gamma(\rho)=(1-\gamma)U\rho U^\dagger +\frac{\gamma}{d^2}\sum_{i,j=0}^{d-1}\mathsf{X}^i\mathsf{Z}^j\rho{\mathsf{Z}^\dagger}^j{\mathsf{X}^\dagger}^i \equiv \left(1-\gamma\frac{d^2-1}{d^2}\right)U\rho U^\dagger + \frac{\gamma}{d^2}\sum_{\substack{i,j=0\\i+j\neq 0}}^{d-1}U\mathsf{X}^i\mathsf{Z}^j\rho{\mathsf{Z}^\dagger}^j{\mathsf{X}^\dagger}^i U^\dagger,
\end{equation}
where $\mathsf{X}$ and $\mathsf{Z}$ are generalized Pauli operators acting unitarily as $\mathsf{X}|j\rangle=|{j+1}\mod d\rangle$ (cyclic shift) and $\mathsf{Z}|j\rangle=e^{i2\pi j/d}|j\rangle$ (phase). $\mathsf{X}^i\mathsf{Z}^j$ for $i,j=0,\cdots, d-1$ are Heisenberg-Weyl operators. Since unitaries belong to $X_A(\pi_A)$, we conclude that $\tilde{u}^\gamma\in X_{s,A}(\pi_A)$.
\subsection{Rank-one projective measurements are in $\bar{X}_{s,A}(\pi_A)\backslash X_A(\pi_A)$}
Now consider a local projective measurement in the basis $\{|\psi_j\rangle\langle\psi_j|\}$, denoted by $\Psi$. This operation is obviously commutativity-preserving (thus in $\bar{X}({\pi})$) since its output is always diagonal in the specified basis. In fact, each projection is trivially commutativity-preserving, so $\Psi\in\bar{X}_{s,A}(\pi_A)$.
Then we consider if $\Psi_A$ always commutes with $\pi_A$. On the one hand,
\begin{equation}
(\Psi_A\otimes I_B)(\pi_A(\rho_{AB}))=
\sum_{i,j}|\psi_j\rangle_A\langle\psi_j|\otimes\langle\psi_j|i\rangle_A\langle{i}|\rho_{AB}|i\rangle_A\langle{i}|\psi_j\rangle.
\label{d1}
\end{equation}
On the other hand,
\begin{equation}
(\Psi_A\otimes I_B)(\rho_{AB})= \sum_{j}|\psi_j\rangle_A\langle\psi_j|\otimes\langle\psi_j|\rho_{AB}|\psi_j\rangle,\label{d2}
\end{equation}
which is already classical-quantum, so $\pi_A((\Psi_A\otimes I_B)(\rho_{AB}))=(\Psi_A\otimes I_B)(\rho_{AB})$.
Since $\rho_{AB}$ is arbitrary, the right hand sides of Eqs.\ (\ref{d1}) and (\ref{d2}) always coincide if and only if $\{|i\rangle\}=\{|\psi_i\rangle\}$. (The equality always holds if we restrict to classical-quantum inputs so that $|i\rangle_A\langle{i}|\otimes I_B$ and $\rho_{AB}$ commute.)
This indicates that $\Psi_A$ do not commute with $\pi_A$ when $\Psi_A$ is not an eigenbasis of $\rho_A$. Therefore, $\Psi\in\bar{X}_{s,A}(\pi_A)\backslash X_A(\pi_A)$.
\subsection{Some mixed-unitary channels are in $\bar{X}_{s,A}(\pi_A)\backslash \bar{X}_A(\pi_A)$}
By previous results, mixed-unitary channels belong to all selective classes. Here we argue that certain mixed-unitary channels live outside $\bar{X}_A(\pi_A)$, thus confirming that selective conditions are not necessarily stronger than their original versions in theories with nonlinear resource destroying maps.
When $A$ is a qubit, $\bar{X}_A(\pi_A)$ (commutativity-preserving channels) is composed of unital channels, which is known to be equivalent to mixed-unitary channels (qubit quantum Birkhoff theorem) \cite{watrous}, and semiclassical (SC) channels (the outputs of a semiclassical channel are diagonal in the same basis) \cite{local,hucpc}. Therefore, the following classes collapse:
\begin{equation}
\bar{X}_A(\pi_A)\backslash{\rm SC} = \text{Mixed-unitary} = \text{Unital}.
\end{equation}
When $A$ is a qudit with dimension $d>2$, however, the above classes form a strict hierarchy. In this case, $\bar{X}_A(\pi_A)\backslash{\rm SC}$ are composed of isotropic channels (which takes the form $(1-\gamma)\Gamma({\rho})+\gamma I/d$, where $\Gamma$ is either unitary or antiunitary) \cite{hucpc,guohou}, which belong to mixed-unitary channels (unitary-isotropic case: by previous results; antiunitary-isotropic case: Ref.\ \cite{2016arXiv161007504B}). However, since $\Gamma$ preserves eigenvalues, $\mu$ must isotropically ``shrink'' the spectrum towards uniformity by degree $\lambda$ (which does not distinguish between reference frames, as its name suggests). For example, given any two pure states as inputs, the respective outputs of an isotropic channel must have the same spectrum. Therefore, the property of isotropy places a strong uniformity requirement on the mixtures of unitaries.
An example of anisotropic mixed-unitary channels is given as follows. Let $\mu(\rho)=\mathsf{X}\rho\mathsf{X^\dagger}/2+\mathsf{X^2}\rho\mathsf{{X^\dagger}^2}/2$ be a qutrit mixed-unitary channel, where $\mathsf{X}$ is the cyclic shift generalized Pauli operator as defined earlier.
Define $|\chi\rangle:=(|0\rangle+|1\rangle+|2\rangle)/\sqrt{3}$, which is an eigenstate of both $\mathsf{X}$ and $\mathsf{X}^2$. Then
\begin{equation}
\mu(|\chi\rangle\langle\chi|)=|\chi\rangle\langle\chi|,
\end{equation}
which remains a pure state, but
\begin{equation}
\mu(|0\rangle\langle 0|)=\frac{1}{2}|1\rangle\langle 1|+\frac{1}{2}|2\rangle\langle 2|,
\end{equation}
which is mixed. So $\mu$ is not an isotropic channel. Moreover, it is well known that there exist qudit unital channels that is not mixed-unitary (no quantum Birkhoff theorem) \cite{watrous,shorppt}. In conclusion, for $d>2$,
\begin{equation}
\bar{X}_A(\pi_A)\backslash{\rm SC} \subsetneq \text{Mixed-unitary} \subsetneq \text{Unital}.
\end{equation}
Since $\mu(|\chi\rangle\langle\chi|)$ and $\mu(|0\rangle\langle 0|)$ clearly do not commute, $\mu$ is also not semiclassical. So $\mu\not\in\bar{X}_A(\pi_A)$.
\subsection{A measure-and-prepare map}
Lastly, we define the following measure-and-prepare protocol that generates discord, but cannot activate it. Given an input $\rho_{AB}$, consider the measure-and-prepare operation with Kraus operators $K_i=|g_i\rangle\langle i|$ (recall that $\{|i\rangle\}$ diagonalizes $\rho_A$), but there exists $j\neq k$ such that $\langle g_j|g_k\rangle\neq0$. Let $\xi$ be such a measure-and-prepare map. By definition, it is not commutativity-preserving, thus able to create discord. However, for any $\rho_{AB}$,
\begin{eqnarray}
(\xi_A\otimes I_B)(\pi_A(\rho_{AB}))&=&\sum_{ij} |g_i\rangle_A\langle i|j\rangle_A\langle j|\rho_{AB}|j\rangle_A\langle j|i\rangle_A\langle g_i| \nonumber\\
&=& \sum_i |g_i\rangle_A\langle i|\rho_{AB}|i\rangle_A\langle g_i| = (\xi_A\otimes I_B)(\rho_{AB}),
\end{eqnarray}
where we used $\langle i|j\rangle=\delta_{ij}$ for the second equality.
That is, $\xi_A\otimes I_B=(\xi_A\otimes I_B)\circ\pi_A$. So $\xi$ is nonactivating.
Note that the above protocol is not linear.
It remains an open question as to whether there are quantum channels in $\bar{X}^\ast_A(\pi_A)\backslash X_A(\pi_A)$.
\end{document} |
3,212,635,537,924 | arxiv | \section{INTRODUCTION}
\label{sec:intro}
It is generally believed that supermassive black holes (SMBHs) reside
in many galaxies, but only a fraction of them exhibit active galactic
nuclei (AGN), while others are dormant \citep{kori1995}. Tidal
disruption events can provide a unique way to find and study the
dormant SMBHs \citep{re1988}. Such an event occurs when a star
approaches a SMBH and is tidally disrupted and subsequently accreted
\citep{lioz1979,re1988}. The mass of the SMBH should be
$\lesssim$$10^{8}$ \msun\ for such events to occur outside the event
horizon for solar-type stars. Tidal disruption events are expected to
be transient and rare, with the average occurrence rate of
$\sim10^{-4}$ yr$^{-1}$ per galaxy \citep{re1990}. They are predicted
to have a fast rise, with a timescale of half a year, and the decay
can last on the order of months to years, with the luminosity decaying
as $L\propto t^{-5/3}$ \citep{lioz1979,re1988,re1990}. The peak of the
flare is expected to reach the Eddington luminosity and be dominated
by thermal UV or X-ray emission.
A few tidal disruption event candidates were found from the {\it
ROSAT} All-Sky Survey, such as \object{RX J1624.9+7554}, \object{RX
J1242.6-1119} and \object{NGC 5905}
\citep{grthle1999,kogr1999,koba1999}. Their host galaxies were
confirmed to be inactive or only weakly active (\object{NGC 5905})
using {\it Hubble Space Telescope} spectroscopy
\citep{gehako2003}. They had peak soft X-ray luminosities up to
$\sim$10$^{44}$ erg s$^{-1}$ and showed ultrasoft X-ray spectra with
blackbody temperatures of $\sim$0.04--0.1 keV
\citep{ko2002,ko2008}. \object{NGC 5905} is the best observed
candidate and is the first one found to follow approximately the
$L\propto t^{-5/3}$ evolution. Some candidates were detected recently
from the {\it XMM-Newton}\ Slew Survey \citep{essafr2007,essako2008} in the X-rays
and from the {\it Galaxy Evolution Explorer} Deep Imaging Survey in
the UV \citep{gemami2006,gebama2008,gehece2009}. Very recently, the
transient source \object{Swift J164449.3+573451} was suspected to be
due to a tidal disruption event \citep{blgime2011, bukegh2011}. In
contrast with the candidates above and the theoretical prediction,
this source is hard in X-rays, with a photon index around 1.8
\citep{bukegh2011}. Its tidal disruption event explanation still needs
to be confirmed by future long-term monitoring.
We are carrying out a project of classifying a sample of sources in
the Second {\it XMM-Newton}\ Serendipitous Source (2XMM) Catalog
\citep{wascfy2009}. Here we report on the discovery of an ultrasoft
X-ray transient source, \object{2XMMi~J184725.1-631724}, whose
position is RA=18:47:25.16, Dec=-63:17:24.96 (J2000) from the 2XMM
catalog, with a 1-$\sigma$ error of 0\farcs26. It is in the direction
of the center of the galaxy IC 4765-f01-1504 \citep{camein2006}. This
source has negligible emission above 2 keV. We describe the
multi-wavelength observations of the source and the data reduction in
Section~\ref{sec:reduction}. In Section~\ref{sec:results}, we first
give the multi-wavelength detections of the source, followed by
presentations of its detailed X-ray spectral and timing properties. We
discuss its possible nature in Section~\ref{sec:discussion} and draw
our conclusions in Section~\ref{sec:conclusion}.
\section{DATA ANALYSIS}
\label{sec:reduction}
\subsection{{\it XMM-Newton}\ Observations}
\tabletypesize{\scriptsize}
\setlength{\tabcolsep}{0.03in}
\begin{deluxetable*}{lcccccccc}
\tablecaption{{\it XMM-Newton}\ Observation Log\label{tbl:obslog}}
\tablewidth{0pt}
\tablehead{ \colhead{Observation ID} &\colhead{Date}&\colhead{off-axis angles (arcmin)} &\colhead{Duration} &\colhead{Exposure(ks)} &\colhead{Filter} \\
& &\colhead{pn/MOS1/MOS2}&\colhead{(ks)}&\colhead{pn/MOS1/MOS2} &
}
\startdata
0405550401(XMM1) & 2006-09-06.98 &3.9/3.0/3.7 &28.0&19.5/27.6/27.6 &medium \\
0405380501(XMM2) & 2007-04-16.31 &9.0/8.5/9.4 &34.7&20.5/32.2/31.9 &thin1
\enddata
\end{deluxetable*}
\object{2XMMi~J184725.1-631724} was observed twice by {\it XMM-Newton}\
(Table~\ref{tbl:obslog}), on 2006 September 7 and 2007 April 16. These
two observations of this source will be referred to hereafter as XMM1
and XMM2, respectively. The source was detected in all the three
European Photon Imaging Cameras in the imaging mode, i.e., pn, MOS1,
and MOS2 \citep{jalual2001,stbrde2001,tuabar2001}, in both
observations. The source was also detected by the Optical Monitor
\citep[OM;][]{mabrmu2001} in XMM1, but it was not in the FOV of the OM
in XMM2. In XMM1, the two UV filters UVW1 and UVM2 were used, and we
obtained the source detection information directly from the pipeline
products.
We used SAS 10.0.0 and the calibration files of 2010 November for
reprocessing the X-ray event files and follow-up analysis. The data in
strong background flare intervals, mostly at the end of the XMM2
observation in the pn camera, are excluded following the SAS thread
for the filtering against high backgrounds. The final exposures used
are given in Table~\ref{tbl:obslog}. We extracted the source spectra
of the pn, MOS1, and MOS2 cameras from a circular region centered on
the source using 15$''$ and 35$''$ radii for XMM1 and XMM2,
respectively. A smaller radius was used for XMM1 because the source
was fainter and near the CCD gap. The background spectrum was
extracted from a large circular region with a radius of 100$''$ near
the source in each camera. The event selection criteria followed the
default values in the pipeline (see Table~5 in \citet{wascfy2009}). We
rebinned the spectra to have at least 20 counts in each bin so as to
adopt the $\chi^2$ statistic for the spectral fits.
We also extracted light curves from the pn camera, which has a larger
effective area and a higher timing resolution than the MOS cameras,
using the same apertures as those for spectral extraction. We first
extracted background-subtracted light curves with a bin size of 250~s,
using the SAS task {\it epiclccorr} to apply relative corrections. To
create the power density spectra (PDS), we also extracted light curves
from the source region using the frame time as the bin size, which is
199.1 ms for XMM1 (using the extended-full-frame mode) and 73.4 ms for
XMM2 (using the full-frame mode). Considering that the source is very
soft and the background dominates above 2 keV, all light curves were
extracted in the energy range 0.2--2.0 keV. We calculated the PDS
using a similar procedure as, e.g., \citet{gorore2006}. The XMM1 199.1
ms and XMM2 73.4 ms pn light curves were split into segments each with
32768 and 65536 data bins, respectively, resulting in four segments
for XMM1 and five for XMM2. The PDS was calculated for each segment,
and all PDS for each light curve were merged and averaged by binning
in frequency using a logarithmic factor of 1.1, under the condition that
each bin contains at least 20 individual PDS measurements. The errors
were calculated from the sample standard deviation of PDS measurements
in each bin.
\subsection{{\it ROSAT} and {\it Swift} Observations}
Our source was not detected in the {\it ROSAT} All-Sky Survey in 1990,
which had a detection limit of 0.1--2.4 keV flux 5$\times$10$^{-13}$
erg s$^{-1}$ cm$^{-2}$ \citep{voasbo1999}. Our source was in the FOV
of one {\it ROSAT} PSPC pointed observation (the sequence number
800256, 1992 October, $\sim$11 ks), at an off-axis angle of
$\sim$2.6$'$. It was not detected either and was (thus) not listed in
the WGA catalog of the {\it ROSAT} point sources
\citep{whgian1994}. We calculated the confidence interval of the
source detection using Bayesian statistics as described in
\citet{krbuno1991}. Circular source and background regions with radii
of 40$\arcsec$ and 2$\arcmin$ respectively were used. The
corresponding (ancillary plus photon redistribution) response matrix
was generated and used to convert the count rates to the fluxes.
At our request, the {\it Swift} Gamma Ray Burst Explorer mission
\citep{gechgi2004} observed the field of
\object{2XMMi~J184725.1-631724} on 2011 February 23 for a total of 5
ks (observation ID 00031930001). The X-ray telescope
\citep[XRT;][]{buhino2005} was operated in Photon Counting mode
\citep{hibuno2004}. X-ray data were reduced with the task {\it
xrtpipeline} version 0.12.1. We found an enhanced count rate at the
position of our source, but it is very weak. We also calculated the
confidence interval of the detection. Radii of 23\farcs5 and
235$\arcsec$ were used for the circular source and background regions,
respectively. The corresponding response matrix was generated using
the calibration files of 2011 February. The UV-Optical Telescope
\citep[UVOT;][]{rokema2005} was operated using the UVW1 filter for 5
ks. The magnitude and flux were measured with the task {\it
uvotsource} version 3 based on the most recent UVOT calibration as
described in \citet{pobrpa2008} and \citet{becuho2010}. Circular
source and background regions with radii of 5$\arcsec$ and
20$\arcsec$, respectively, were used.
\subsection{Optical Observations}
Our source is in the direction of the center of the galaxy IC
4765-f01-1504 \citep{camein2006}. This galaxy is located in the
background of the rich group of galaxies IC 4765 (also known as Abell
S0805, $z$=0.01497). It was imaged with the 1.3 m Warsaw telescope at
Las Campanas Observatory in Chile through the standard Johnson V and
Cousins I filters in 1998. We used the V- and I-filter images from
\citet{camein2006} to derive the main photometric parameters of the
galaxy with a S\'{e}rsic model in GALFIT \citep{pehoim2010}. The
images have a FWHM of the PSF of about 1\farcs2.
\citet{camein2006} also obtained an optical spectrum of the galaxy on
1999 June 19 with the Wide Field CCD camera mounted on the 2.5 m Du
Pont Telescope at the Las Campanas Observatory in Chile, but it has
poor quality. We obtained a new longslit spectrum of this galaxy with
the Gemini Multi-Object Spectrograph \citep[GMOS,][]{hojoal2004} at
the Gemini South Telescope in the queue mode. The observation was made
on the night of 2011 March 19 (UT) during bright time (illumination
fraction 0.99), under photometric conditions and $\sim$1\arcsec\
seeing. The 400 lines/mm ruling density grating (R400) centered at
5500\AA\ was used, to minimize the effect of moon illumination. The
slit width was set to 1$\arcsec$. A total of four exposures of 900 s
each were obtained. Small offsets in the spectral direction (50\AA)
towards the blue and the red were applied between exposures to allow
for the gaps between CCDs and to avoid any loss of important lines
present in the spectra. Spectroscopic flats and comparison lamp
(CuAr) spectra were taken after each science exposure. In addition,
the spectrophotometric standard star LTT 7379 was observed at the end
of the night to flux calibrate the science spectrum.
The observations were processed with the Gemini IRAF package version
1.9 in IRAF. All science exposures, comparison lamps and spectroscopic
flats were bias subtracted and trimmed. Spectroscopic flats were
processed by removing the calibration unit plus GMOS spectral response
and the calibration unit uneven illumination, normalizing and leaving
only the pixel-to-pixel variations and the fringing. The resulting
two-dimensional spectra were then wavelength calibrated, corrected by
S-shape distortions, sky-subtracted, extracted to a one-dimensional
format using a fixed aperture of 7\farcs8 in diameter, and then
average combined. The final spectrum has a resolution of $\sim$8.8
\AA\ (FWHM) and a dispersion of $\sim$1.36 \AA\ pixel$^{-1}$, covering a
wavelength interval of $\sim$4000--7600~\AA. The signal-to-noise ratio
is about 40 at 5500 \AA.
We measured the redshift of the galaxy with two methods. In the first
method, we cross-correlated the spectrum with a high signal-to-noise
template using the {\it fxcor} routine in the IRAF RV package. The
error was estimated using the R statistic of \citet{toda1979}:
$\sigma_{v}=(3/8)(w/(1+R))$, where $w$ is the FWHM of the correlation
peak and $R$ is the ratio of the correlation peak height to the
amplitude of the antisymmetric noise. In the second method, we
identified the most prominent absorption lines (as no clear emission
lines were detected) in the spectrum and derived the redshift by
employing a line-by-line Gaussian fit using the {\it rvidline} routine
in the IRAF RV package.
\section{RESULTS}
\label{sec:results}
\subsection{The Source and the Multi-wavelength Observations}
\label{sec:srcandobs}
\tabletypesize{\scriptsize}
\setlength{\tabcolsep}{0.03in}
\begin{deluxetable*}{lcccccccccccc}
\tablecaption{The counterpart candidates in UV, optical, and IR\label{tbl:counterpart}}
\tablewidth{0pt}
\tablehead{\multicolumn{3}{c}{UV (XMM1 OM)} &&\multicolumn{4}{c}{Optical (USNO B1.0)}&&\multicolumn{4}{c}{IR (2MASS PSC)}\\
\cline{1-3} \cline{5-8} \cline{10-13}
\colhead{$r$} & \colhead{UVW1} & \colhead{UVM2} &&\colhead{$r$} & \colhead{B2}&\colhead{R2}&\colhead{I} &&\colhead{$r$} &\colhead{J}&\colhead{H}&\colhead{K}\\
\colhead{(arcsec)} &\multicolumn{2}{c}{(AB mag/flux($10^{-16}$ erg s$^{-1}$ cm$^{-2}$ \AA$^{-1}$))} &&\colhead{(arcsec)} &\multicolumn{3}{c}{(mag)} &&\colhead{(arcsec)} &\multicolumn{3}{c}{(mag)}
}
\startdata
0.52 & $19.65$$\pm$0.10/1.78$\pm$0.17 & $19.83$$\pm$$0.19$/2.38$\pm$0.41 &&0.16& 16.1 &15.33 &15.40 && 0.31& 15.29$\pm$0.07 & 14.59$\pm$0.09 &14.30$\pm$0.08
\enddata
\tablecomments{The $r$ columns are the offsets of the counterparts from \object{2XMMi~J184725.1-631724}. The magnitudes/fluxes are not corrected for the Galactic reddening.}
\end{deluxetable*}
\begin{figure}
\centering
\includegraphics{optimg_lin.eps}
\caption{The optical image of the galaxy in the V band. The pixel size
is 0\farcs414. The green plus marks the central position of the
galaxy obtained from the fits to its V- and I-band profiles using a
S\'{e}rsic model, and it is at RA=18:47:25.14,
Dec=$-$63:17:24.77 (J2000). The red circle is centered at the X-ray
position, with the radius corresponding to the 3-$\sigma$
error. \label{fig:optimg}}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=0.99\textwidth]{optical_spec.eps}
\caption{The smoothed spectrum of the galaxy IC 4765-f01-1504 from the Gemini South Telescope, with the most important absorption lines identified. The drop at 6870 \AA\ is due to the atmosphere OH absorption. \label{fig:optspec}}
\end{figure*}
\object{2XMMi~J184725.1-631724} was detected in X-rays in both XMM1
and XMM2. We see no clear X-ray emission of our source from the {\it
ROSAT} observation in 1992 October and the {\it Swift} observation in
2011 February, indicating variability factors of $>$64 and $>$12,
respectively, compared with XMM2 (the 0.2--10 keV absorbed flux; see
Section~\ref{sec:rossw}). Here we concentrate on observations in other
wavelengths.
The source position from the 2XMM catalog has been astrometrically
corrected by matching with the optical catalog USNO B1.0
\citep{moleca2003}. For XMM1 and XMM2, a very small correction was
invoked (a fraction of an arcsec). We compared the corrected positions
of ten of the brightest X-ray sources in each observation with the
positions of optical counterparts from the USNO B1.0 catalog and found
that most of the offsets between the matches are less than 0.5$''$,
indicating successful astrometric corrections.
Table~\ref{tbl:counterpart} gives the detection of a UV source in UVW1
and UVW2 from the XMM1 OM at a position within the 2-$\sigma$
positional error from \object{2XMMi~J184725.1-631724} and is probably
its UV counterpart. There is also a UV source detected near our source
in UVW1 from the {\it Swift} UVOT. After applying the astrometric
correction using the USNO-B1.0 catalog with the {\it XMM-Newton}\ SAS task {\it
eposcorr}, we obtain its position of RA=18:47:25.14 and
Dec=-63:17:25.04 (J2000), within the 1-$\sigma$ error from the X-ray
position. Its magnitude and flux are 18.67$\pm$0.04 and
(1.34$\pm$0.05)$\times$10$^{-16}$ erg~s$^{-1}$~cm$^{-2}$~\AA$^{-1}$,
respectively. We note that the UV filter set in the UVOT is different
from that of the {\it XMM-Newton}\ OM, and the above values should not be directly
compared with the OM measurements in XMM1 above. \citet{grkoga2008}
measured an offset between the magnitudes from the two instruments of
$W1_{\rm OM} - W1_{\rm UVOT} = 0.78$ by comparing several field stars
in the images of the AGN Mkn 335. With this taken into account, there
seems to be little variability in the UV between the two epochs.
The optical and IR counterpart candidates of the source from the USNO
B1.0 and 2MASS Point Source Catalogs are given in
Table~\ref{tbl:counterpart}. The optical counterpart, the galaxy IC
4765-f01-1504, is shown in Figure~\ref{fig:optimg}
\citep{camein2006}. Our fits of the V- and I-band images using a
S\'{e}rsic model give integrated magnitudes of
16.99$\pm$0.01 and 15.62$\pm$0.02, effective radii of
2\farcs52$\pm$0\farcs04 and 2\farcs94$\pm$0\farcs09, S\'{e}rsic
indices of 3.54$\pm$0.06 and 4.14$\pm$0.13, and apparent axis ratios
of 0.238$\pm$0.003 and 0.271$\pm$0.003, respectively. This galaxy may
be an elliptical galaxy, which typically has a S\'{e}rsic index of
4. The axis ratios above would imply a high inclination of this galaxy
if its intrinsic ellipticity is low.
Figure~\ref{fig:optspec} shows the spectrum of IC 4765-f01-1504 from
the Gemini South Telescope. No clear emission lines were detected,
supporting the identification as an elliptical galaxy. A redshift of
$z$=0.0353$\pm$0.0001 was obtained from both the cross-correlation
method, with $R=12.80$, and the absorption line fit method, indicating
a perfect agreement between them. This redshift disagrees with the
value of 0.0869 obtained by \citet{camein2006}, which used the
cross-correlation method (as there were no significant emission lines
detected). Considering that our new spectrum has much better quality,
we adopt this new redshift. We measured a 3-$\sigma$ upper limit of
[OIII] 5007 \AA\ of 0.9$\times$$10^{-15}$ erg s$^{-1}$
cm$^{-2}$. Assuming a flat universe with the Hubble constant $H_0$=73
km s$^{-1}$ Mpc$^{-1}$ and the matter density $\Omega_{\rm M}$=0.27,
this redshift corresponds to a comoving radial distance of 143.9 Mpc
and a luminosity distance of 149.0 Mpc, which will be used in this
paper. The absolute V and K magnitudes of this galaxy are -19.2 and
-21.6, respectively, after the Galactic extinction correction
\citep{scfida1998}. Based on the BH mass vs. bulge luminosity
relations from \citet{gr2007}, \citet{lafari2007} and
\citet{mahu2003}, the above magnitudes imply the mass of the SMBH in
IC 4765-f01-1504 to be about 10$^7$ and 10$^6$ \msun\ if the
bulge/total luminosity ratio is 1 or 0.1, respectively. Because the
sample of galaxies in the above studies were generally brighter than
IC 4765-f01-1504 and these relations have large intrinsic scattering,
these mass estimates might have an uncertainty as large as one order
of magnitude.
\subsection{X-ray Spectral Modeling}
\label{sec:spmod}
\tabletypesize{\scriptsize}
\setlength{\tabcolsep}{0.03in}
\begin{deluxetable*}{lccccccccccc}
\tablecaption{Spectral modeling results\label{tbl:mcd+pl}}
\tablewidth{0pt}
\tablehead{\colhead{Model} &\colhead{Obs} &\colhead{$N_{\rm H}$} &
\colhead{$kT_{\rm MCD/BB}$} &\colhead{$N_{\rm MCD/BB}$} &
\colhead{$\Gamma_{\rm PL/SIMPL}$} & \colhead{$N_{\rm PL}/f_{\rm SC}$} &
\colhead{$\chi^2_\nu(\nu)$} & \colhead{$f_{\rm MCD/BB}$} &
\colhead{$F_{\rm abs}$} & \colhead{$F_{\rm unabs}$} & \colhead{$L$}\\
& & \colhead{($10^{20}$~${\rm cm}^{-2}$)} &
\colhead{(eV)} & \colhead{($10^{4}$)} &
& \colhead{($10^{-5}/\%$)} &
&(\%) &\multicolumn{2}{c}{($10^{-12}$ erg s$^{-1}$ cm$^{-2}$)} & ($10^{43}$ erg s$^{-1})$
}
\startdata
\multirow{2}{*}{MCD+PL} & XMM1 & \multirow{2}{*}{$8.6$$\pm$$ 0.5$} & $65.8$$\pm$$ 5.0$ & $1.56^{+1.21}_{-0.63}$ & $3.72$$\pm$$ 0.62$ & $1.79$$\pm$$ 0.32$ &0.92(115) & $84.5^{+7.3}_{-14.8}$ & $0.22$$\pm$$ 0.01$ & $1.76$$\pm$$ 0.19$ &$ 1.70$$\pm$$ 0.44$\\
&XMM2 & & $93.1$$\pm$$ 2.2$ & $1.45^{+0.34}_{-0.23}$ & $3.27$$\pm$$ 0.70$ & $3.75$$\pm$$ 1.39$ &1.20(284) & $96.4^{+1.9}_{-6.1}$ & $1.93$$\pm$$ 0.03$ & $10.18^{+1.00}_{-0.57}$ &$ 6.38$$\pm$$ 0.66$\\
\hline
\multirow{2}{*}{SIMPL(MCD)} &XMM1 & \multirow{2}{*}{$8.5$$\pm$$ 0.5$} & $65.1$$\pm$$ 5.5$ & $1.73^{+1.65}_{-0.73}$ & $3.71$$\pm$$ 0.63$ & $3.33^{+3.98}_{-1.69}$ &0.92(115) & $90.0^{+3.7}_{-7.1}$ & $0.22$$\pm$$ 0.01$ & $1.75^{+0.35}_{-0.24}$ &$ 1.81$$\pm$$ 0.57$\\
& XMM2 & & $92.9$$\pm$$ 2.4$ & $1.49$$\pm$$ 0.32$ & $3.33$$\pm$$ 0.78$ & $0.83^{+1.32}_{-0.49}$ &1.20(284) & $97.7^{+0.9}_{-2.2}$ & $1.92$$\pm$$ 0.03$ & $10.21$$\pm$$ 0.89$ &$ 6.40$$\pm$$ 0.72$\\
\hline
\multirow{2}{*}{BB+PL} &XMM1 & \multirow{2}{*}{$7.4$$\pm$$ 0.7$} & $57.8$$\pm$$ 3.9$ & $1.63^{+1.13}_{-0.56}$ & $3.71$$\pm$$ 0.59$ & $1.76$$\pm$$ 0.28$ &0.92(115) & $78.2^{+9.5}_{-17.2}$ & $0.22$$\pm$$ 0.01$ & $1.23$$\pm$$ 0.11$ &$ 0.59$$\pm$$ 0.14$\\
& XMM2 & & $78.2$$\pm$$ 1.6$ & $2.37$$\pm$$ 0.41$ & $3.89$$\pm$$ 0.61$ & $5.54$$\pm$$ 1.46$ &1.18(284) & $86.0^{+7.6}_{-14.4}$ & $1.92$$\pm$$ 0.03$ & $7.59^{+1.15}_{-0.51}$ &$2.79$$\pm$$ 0.39$\\
\hline
\multirow{2}{*}{SIMPL(BB)} &XMM1 & \multirow{2}{*}{$7.2$$\pm$$ 0.5$} & $57.7$$\pm$$ 4.1$ & $1.81^{+1.3}_{-0.65}$ & $3.71$$\pm$$ 0.59$ & $5.48^{+4.81}_{-2.31}$ &0.92(115) & $87.8^{+3.9}_{-7.1}$ & $0.22$$\pm$$ 0.01$ & $1.18$$\pm$$ 0.18$ &$ 0.59$$\pm$$ 0.15$\\
& XMM2 & & $77.5$$\pm$$ 2.2$ & $2.58^{+0.71}_{-0.45}$ & $4.25$$\pm$$ 0.83$ & $3.35^{+3.56}_{-1.63}$ &1.18(284) & $94.2^{+2.4}_{-4.6}$ & $1.92$$\pm$$ 0.03$ & $7.2$$\pm$$ 0.68$ &$ 2.70$$\pm$$ 0.31$
\enddata
\tablecomments{The column $f_{\rm MCD/BB}$ refers to the unabsorbed flux fraction of the MCD/BB component in the 0.2--10 keV energy band (for the models SIMPL(MCD) and SIMPL(BB), it refers to the unscattered part). $F_{\rm abs}$ and $F_{\rm unabs}$ are the total absorbed and unabsorbed fluxes in the 0.2--10 keV energy band, respectively. The luminosity $L$ was calculated using the unabsorbed bolometric flux of each spectral component (the PL component was integrated down to 0.2 keV). All errors are at a 90\%-confidence level.}
\end{deluxetable*}
\begin{figure*}
\centering
\includegraphics[width=0.99\textwidth]{specfitplot_raw.eps}
\caption{The unfolded spectra and the fit residuals using the model MCD+PL. For clarity, only the pn spectra are shown for the unfolded spectra. The dotted, dashed, and solid lines are for the MCD and PL components and the total model, respectively. The residuals are shown for all three cameras (black/red/green for pn/MOS1/MOS2, respectively).\label{fig:spfits}}
\end{figure*}
\begin{figure}
\centering
\includegraphics{Flux_Tbb.eps}
\caption{The MCD flux versus $kT_{\rm MCD}$ using the models MCD+PL and SIMPL(MCD). The solid, dotted, and dashed lines plot the $L\propto T^4$, $L\propto T^3$, and $L\propto T^5$ relations, as a guidance. \label{fig:spfits2}}
\end{figure}
We fitted the spectra of \object{2XMMi~J184725.1-631724} from both XMM1
and XMM2 using various spectral models. We jointly fitted the spectra
from all three cameras, i.e., pn, MOS1, and MOS2, and their relative
normalizations were left free. We only report the normalization results
corresponding to the pn camera. MOS1 and MOS2 differ by about 10$\%$
(the largest one $\sim$20\%), less than the error bars. We fitted the
spectra in the 0.2--10 keV energy band.
We were unsure the X-ray emission mechanism of our source. Thus we
first tested the common single-component models to see whether any of
them can describe our X-ray spectra well: a single temperature
blackbody (BB), a multi-color disk (MCD), a PL, a broken PL, a cut-off
PL, an APEC thermal plasma model, and a thermal bremsstrahlung
spectrum. They are models bbodyrad, diskbb, powerlaw, bknpower,
cutoffpl, APEC, and bremss in XSPEC, respectively. All models include
the absorption described by the WABS model in XSPEC; our results
change little with alternative absorption models such as PHABS or
TBABS in XSPEC. All these simple models fail to describe one or both
of the {\it XMM-Newton}\ spectra, with residuals above 1 keV typically seen. For
indication, we report the PL index $\Gamma_{\rm PL}$ of the fits using
the PL model. We obtained $\Gamma_{\rm PL}=5.86$$\pm$0.27 for XMM1
($\chi^2_\nu$($\nu$)=1.13(117)) and 6.88$\pm$$0.09$ for XMM2
(($\chi^2_\nu$($\nu$)=2.25(286)). The lower $\chi^2$ value for XMM1 to
some degree is due to poorer data.
We next attempted to fit the spectra with the double-component models
MCD+PL and BB+PL, finding that they describe both the XMM1 and XMM2
spectra much better than the above common single-component models,
with the $\chi^2$ values decreased by more than 140 for the total
degrees of freedom of about 400 of both the XMM1 and XMM2 spectra. As
a way to model the hard component self-consistently, we also fitted
the spectra with SIMPL(MCD) and SIMPL(BB). SIMPL \citep[in
XSPEC12;][]{stnamc2009} is an empirical convolution model of
Comptonization in which a fraction ($f_{\rm SC}$) of the input seed
photons are converted into a power law parametrized by an index
($\Gamma_{\rm SIMPL}$). We assume that all the scattered photons are
up-scattered in energy in this model.
The best-fitting values of the column density are consistent between
XMM1 and XMM2, with $N_{\rm H}$=$(7.6^{+1.5}_{-2.6})$ and
$(8.6$$\pm$$0.6)$$\times$$10^{20}$ cm$^{-2}$, respectively, using the
model MCD+PL. Thus, we chose to fit both spectra with a common value
of $N_{\rm H}$. The final results are given in
Table~\ref{tbl:mcd+pl}. The best-fitting values of $N_{\rm H}$ are
slightly higher than the Galactic value of 6.1$\times$$10^{20}$~${\rm
cm}^{-2}$ from the Leiden/Argentine/Bonn Survey of Galactic HI
\citep{kabuha2005}, probably indicating a small intrinsic absorption.
For the model MCD+PL, we show the unfolded spectra and residuals in
Figure~\ref{tbl:mcd+pl}. In this model, the spectra are dominated by
the MCD component at energies below 1 keV. The fraction of the MCD
component is about 84.5\% and 96.4\% for XMM1 and XMM2, respectively
(the 0.2--10 keV unabsorbed flux; Table~\ref{tbl:mcd+pl}). The 0.2--10
keV flux increases from XMM1 to XMM2 by a factor of 8.8 (absorbed) or
5.8 (unabsorbed). We also estimate the luminosity, using the
bolometric flux of each spectral component. The disk inclination is
uncertain, and we assume it to be 60$\degr$. The PL component diverges
at low energies, and we integrate its flux above 0.2 keV. We obtain
luminosities of 1.70 and 6.38$\times$10$^{43}$ erg s$^{-1}$ for XMM1
and XMM2, respectively (Table~\ref{tbl:mcd+pl}). For comparison, the
corresponding 0.2--10 keV luminosities are 0.47 and
2.70$\times$10$^{43}$ erg s$^{-1}$, respectively.
We plot the MCD bolometric flux versus its temperature at the inner
disk radius $kT_{\rm MCD}$ in Figure~\ref{fig:spfits2} (the upper
panel). We can see that the evolution of the MCD luminosity is
consistent with the $L\propto T^4$ track (the solid line), which
implies a constant inner disk radius with the change in luminosity. We
note that this is based on the only two observations available. The
disk temperature is relatively low, only $kT_{\rm MCD}=65.8$ and
$93.1$ eV for XMM1 and XMM2, respectively. The PL component is weak,
and its parameter values have relatively large uncertainties. Its
index is consistent between XMM1 and XMM2 and is relatively steep,
with $\Gamma_{\rm PL}$ about 3.5. Forcing XMM1 and XMM2 to have the
same value of $\Gamma_{\rm PL}$ in the fit, we see a change of the PL
normalization $N_{\rm PL}$ by a factor of 2.6 (4.5 $\sigma$).
The model SIMPL(MCD) gives results very similar to the model MCD+PL
(Table~\ref{tbl:mcd+pl} and Figure~\ref{fig:spfits2}), in terms of the
MCD temperature, the thermal fraction, etc. It infers that only about
$f_{\rm SC}$$=$$3\%$ and $1\%$ of the thermal disk emission is
Comptonized to the hard emission in XMM1 and XMM2, respectively. This
model, with a natural cutoff at low energies for the hard component,
infers luminosities similar to those of the model MCD+PL obtained by
integrating the PL flux down to 0.2 keV
(Table~\ref{tbl:mcd+pl}).
The spectra can also be fitted almost equally well using the models
BB+PL and SIMPL(BB) (Table~\ref{tbl:mcd+pl}). The BB component
dominates in both XMM1 and XMM2, contributing $\gtrsim$80\% of the
0.2--10 keV flux, similar to the MCD component in the models MCD+PL
and SIMPL(MCD). Its effective temperature $kT_{\rm BB}$ is also low,
about 58 and 78 eV for XMM1 and XMM2, respectively. The MCD and BB
models have very similar spectral shapes at high energies
\citep{mamami1986}, but their differences become large at low
energies. We estimate their differences in the UV. We have
measurements from two UV filters, i.e., UVW1 and UVM2, from XMM1. The
flux densities of the MCD component in the model MCD+PL from XMM1 are
($2.15$$\pm$$0.69$) and ($3.68$$\pm$$1.18$)$\times$10$^{-17}$ erg
s$^{-1}$ cm$^{-2}$ \AA$^{-1}$ at the effective wavelengths of UVW1
(2910 \AA) and UVM2 (2310 \AA), respectively. The corresponding values
for the BB component in the model BB+PL from XMM1 are
($0.40$$\pm$$0.16$) and ($1.00$$\pm$$0.40$)$\times$10$^{-19}$ erg
s$^{-1}$ cm$^{-2}$ \AA$^{-1}$, respectively. The corresponding flux
densities measured with UVW1 and UVW2 are ($3.00$$\pm$$0.47$) and
($5.08$$\pm$$1.45$)$\times$10$^{-16}$ erg s$^{-1}$ cm$^{-2}$
\AA$^{-1}$, respectively, after the Galactic dust extinction
correction using a reddening value of $E_{\rm (B-V)}=0.098$
\citep{scfida1998} and assuming a spectral shape of a MCD model at low
frequencies (i.e., a power law with a photon index of $2/3$). We see
that the UV flux from the OM detection is much higher than the BB flux
in the UV. It is closer to the MCD flux in the UV, but still there is
about an order of magnitude difference, which will be discussed in
Section~\ref{sec:discussion}. The flux of the PL component in the UV
is hard to assess as this model is too steep and diverges at low
energies (more than two orders of magnitude higher than that measured
by the OM) and must be cut off below some energy. The models
SIMPL(MCD) and SIMPL(BB) show no such problem, and their fluxes in the
UV, from the whole model or only from the thermal components, are very
close to the MCD and BB fluxes in the UV obtained above.
\subsection{Fast X-ray Variability}
\label{sec:pds}
\tabletypesize{\scriptsize}
\setlength{\tabcolsep}{0.03in}
\begin{deluxetable}{lcccccccc}
\tablecaption{The fit results of the PDS using a PL plus a constant. \label{tbl:timing}}
\tablewidth{0pt}
\tablehead{\colhead{Obs} &
\colhead{$\Gamma_{\rm PL}$} & \colhead{$N_{\rm PL}$ ($10^{-5}$)} &
\colhead{$C_{\rm P}$} &
\colhead{$\chi^2_\nu(\nu)$} &
\colhead{rms(\%)}
}
\startdata
XMM1 & $1.74^{+1.13}_{-0.57}$ & $3.49^{+27.70}_{-3.48}$ & 25.59$\pm$0.17 & 1.07(75) & 21.2$\pm$5.3 \\
XMM2 &1.65$\pm$0.19 &$8.17^{+20.42}_{-6.75}$ & 2.77$\pm$0.01 & 1.06(82) &21.1$\pm$1.9
\enddata
\tablecomments{The column rms refers to 0.0001--0.01 Hz fractional rms after subtracting the Poisson level. $N_{\rm PL}$ is the PL normalization at 1 Hz. All errors are at a 90\%-confidence level, except for the rms, whose 1-$\sigma$ errors are given.}
\end{deluxetable}
\begin{figure*}
\centering
\includegraphics{lcsppower_s217743.eps}
\caption{Left panels: The pn 250~s 0.2--2.0~keV background-subtracted light curves. Right panels: The PDS of the pn 0.2--2.0 keV light curves binned at 199.1 ms for XMM1 and 73.4 ms for XMM2. The black solid constant line is the average PDS above 0.1 Hz, representing the Poisson level, and the dotted line is the best-fitting model of a PL plus a constant. \label{fig:lcfft}}
\end{figure*}
The left panels of Figure~\ref{fig:lcfft} show the pn 250 s
background-subtracted light curves. The background is at a level of
about 3$\%$ and 2$\%$ for XMM1 and XMM2, respectively. The variations
of the source count rate can be clearly seen for both
observations. For XMM2, which has higher count rates, we see that the
source varies by a factor of $\sim$4 within 5 ks.
The right panels of Figure~\ref{fig:lcfft} show the pn PDS of XMM1 and
XMM2. Both PDS are flat at frequencies above $0.1$ Hz, representing
the Poisson level. Their averages above $0.1$ Hz weighted by the
errors differ from the expected Poisson noise values by
$<$$0.3\%$. Below 0.01 Hz, both PDS show a clear deviation from the
Poisson level (the black solid line). We successfully fitted both PDS
with a power-law (PL) plus a constant $C_{\rm P}$, accounting for the
Poisson level, and the results are given in
Table~\ref{tbl:timing}. The PL index $\Gamma_{\rm PL}$ is about 1.7
for both observations. We evaluate the fractional rms within 0.0001 to
0.01 Hz after subtracting the Poisson level and obtain a value of
(21.2$\pm$5.3)\% and (21.1$\pm$1.9)\% for XMM1 and XMM2,
respectively. The above results indicate that the source shows similar
variability in both XMM1 and XMM2.
The large fast variability seen above should be due to the thermal
component from the models in Table~\ref{tbl:mcd+pl}, as the hard
component is very weak (e.g., $<$4\% in XMM2, in terms of the 0.2--2.0
keV pn count rates). To see the cause of the variability, we extracted
the high- and low-state spectra from the bright observation XMM2,
corresponding to intervals with the pn count rate higher or lower than
0.8 counts~s$^{-1}$, respectively (Figure~\ref{fig:lcfft}). We fitted
these two spectra simultaneously using the model SIMPL(MCD) with a
common value of $N_{\rm H}$. We find that the MCD temperature in the
low state is smaller than in the high state by 10 eV at a 4.4-$\sigma$
confidence level. Their MCD normalizations are consistent with being
the same within the error bars. The parameters of the hard component
have large uncertainties, making it hard to constrain any trend. Based
on the MCD model, the above results provide one explanation of our
large fast variability: it is caused by the fast variations in the
mass accretion rate, with the disk truncated at a constant radius.
\subsection{Comparison with ROSAT and Swift Observations}
\label{sec:rossw}
We obtain count rates of $<$1.9$\times$10$^{-3}$ and
1.3$^{+2.4}$$\times$10$^{-3}$ counts s$^{-1}$ for the {\it ROSAT} PSPC
observation in 1992 October and the {\it Swift} XRT observation in
2011 February, respectively. The 3-$\sigma$ upper bounds are given
above, and the lower bounds are zero. Using the response matrices
corresponding to the source extraction regions used and the fit
results of the model SIMPL(MCD) for XMM1, these count rates correspond
to the 0.2--10 keV absorbed fluxes of $<$0.3 and
0.6$^{+1.0}$$\times$$10^{-13}$ erg s$^{-1}$ cm$^{-2}$, respectively,
and the 0.2--10 keV unabsorbed fluxes of $<$2.4 and
4.5$^{+8.3}$$\times$$10^{-13}$ erg s$^{-1}$ cm$^{-2}$,
respectively. Thus, the source varied by a factor of $>$64 and $>$43
between the {\it ROSAT} pointed observation in 1992 and XMM2, using
the 0.2--10 keV absorbed and unabsorbed fluxes, respectively. The
corresponding variation factors between XMM2 and the {\it Swift}
observation in 2011 are $>$12 and $>$8, respectively.
\section{DISCUSSION}
\label{sec:discussion}
\subsection{The Nature and Implication of the Soft Component}
The remarkable features of \object{2XMMi~J184725.1-631724} are the
extreme softness of its X-ray spectra and the large
variability. Understanding the nature and implication of the soft
component which dominates the X-ray spectra will help to pin down the
nature of the source. As it is in the direction coincident with the
center of the galaxy IC 4765-f01-1504, we first assume that its X-ray
emission is associated with the SMBH in this galaxy. This can be due
to either a tidal disruption event or an AGN.
The fits with the models MCD+PL and SIMPL(MCD) above assume that there
was a thermal disk emission, whose luminosity fraction was inferred to
be very high, $\sim$90\%. The evolution of the MCD luminosity is
consistent with $L\propto T^4$, though only two observations are
available. These properties are very similar to the thermal state of
BH X-ray binaries \citep{remc2006}, in which the accretion disk is
believed to be truncated at the innermost stable circular orbit
(ISCO). Thus we assume that the disk is also truncated at the ISCO
during these two observations and roughly estimate the BH mass from
the MCD normalization $N_{\rm MCD}$. Using a distance of 143.9 Mpc and
assuming a disk inclination of 60$\degr$, we infer the BH mass to be
$\sim$3$\times$$10^5$ \msun, neglecting factors such as the spin and
the hardening effect. If we replace the MCD model with the more
realistic accretion disk model around a Kerr black hole kerrbb
\citep[in XSPEC;][]{lizina2005} and explore the parameter spaces of
the disk inclination 0--75$\degr$, the spin parameter $a^*$ 0--1, and
the hardening factor 1--1.7, we obtain a range of the BH mass of
(0.06--3.81)$\times$10$^6$ \msun. Assuming the BH mass to be
5$\times$$10^5$ \msun, the source would be at about 0.3 and 1.0
Eddington luminosity in XMM1 and XMM2, respectively, common values
seen in the thermal state of BH X-ray binaries \citep{dogiku2007}. We
note that the light crossing time of the inner accretion disk around a
BH with this mass is $\sim$50 s, about the timescale on which the
source begins to show strong variability (Figure~\ref{fig:lcfft}). The
above mass estimate is consistent with that using the BH mass
vs. bulge luminosity relations (Section~\ref{sec:srcandobs}),
considering the large uncertainties of both methods.
Some AGN can be very soft, showing strong soft excesses
\citep{pumaco1992}. The soft excess refers to the excess of emission
below $\sim$2 keV with respect to the extrapolation of the power-law
fit of the continuum above 2 keV and is commonly observed in type-I
AGN \citep{tupo1989}. The strongest soft X-ray excesses and
variability are found in Narrow Line Seyfert 1 galaxies \citep[NLS1s;
e.g.,][]{bobrfi1996,le1999a,le1999b,grkole2010}. The nature of the
soft excess is still unclear. The above thermal disk model is one of
the several competing models invoked \citep[e.g.,][]{wafi1993}. The
problem with this explanation is that the soft excesses from a sample
of AGN with a large range of mass and luminosity have characteristic
temperatures spanning a narrow range ($\sim$0.1--0.2 keV), which is
hard to explain \citep{gido2004,crfaga2006}. The narrow range of the
characteristic temperatures of the soft excesses finds a natural
explanation if they are due to atomic processes. There are two main
scenarios, i.e., absorption and reflection. \citet{gido2004} proposed
the soft excess as an artifact of strong, relativistically smeared,
partially ionized absorption. We test this model using the swind1
model from XSPEC. We use the model swind1(PL), with $\Gamma_{\rm PL}$
required to be $<$3.5 to make sure that it is not the steep PL
describing the soft excess. We obtain the values of $\chi^2_\nu(\nu)$
to be 1.32(114) and 1.31(283) for XMM1 and XMM2, respectively. Both
observations require strong velocities ($>$0.5 and 0.29$\pm$0.02 speed
of light for XMM1 and XMM2, respectively) to smear the
absorption/emission lines, which is hard to achieve from a radiatively
driven accretion disk wind \citep{scdo2007,scdopr2009}. The difference
between XMM1 and XMM2 is mainly due to different absorption column
densities and smearing velocities, requiring dramatic changes in the
absorber.
In the reflection model, a series of soft X-ray emission lines below 2
keV, if strongly relativistically blurred, can produce the smooth soft
excess feature. We follow \citet{crfaga2006} to use the model
kdblur(PL+reflionx), where kdblur is the relativistic convolution and
reflionx is a table model of the ionized reflection (see their
references therein). In the fits we force XMM1 and XMM2 to have common
values of $N_{\rm H}$, the inclination, and the Fe abundance. We
obtain $\chi^2_\nu(\nu)$=0.95(111) and 1.34(281) and reflection flux
fractions $\sim$0.94 and 1 (0.2--10 keV, unabsorbed) for XMM1 and
XMM2, respectively. The fits require a low inclination
(17.1$\pm$7.6$\degr$), a high Fe abundance ($8.37_{-2.65}^{+0.79}$
solar value), and a steep illuminating power-law spectrum (reaching
the upper index limit of 3.3 allowed in the model). The disk
emissivity index is different between these two observations
($5.37_{-0.27}^{+0.87}$ and 8.99$\pm$0.89 for XMM1 and XMM2
respectively), implying a very different disk structure. This model
infers a highly spinning BH, with the inner disk radius at
2.38$\pm$0.38 and $1.87_{-0.19}^{+0.53}$ gravitational radii for XMM1
and XMM2 respectively. Compared with the results from
\citet{crfaga2006}, the above values are extreme but not unique.
We see that both the absorption and reflection models invoke extreme
environments to explain our source. The former requires absorbers at
very high velocities and varying dramatically. The latter requires
very steep illuminating spectra and very different disk emissivities
between XMM1 and XMM2. In comparison, the thermal disk
emission explanation for the soft component in our source is more
reasonable. In this model, the difference between XMM1 and XMM2 is
simply due to the change in the accretion rate. We note that the
inferred inner disk temperatures are lower than the typical
characteristic temperatures of the soft excesses in AGN, which might
indicate that the soft component of our source has a different nature
from soft excesses in AGN.
Our spectral fits did not combine both the UV and X-ray spectra. There
is about one order of magnitude difference (about 9$\sigma$) between
the UV fluxes of the thermal disk inferred from the soft X-ray
spectral fits and the OM measurements
(Section~\ref{sec:results}). This can be due to several
factors. First, we have only used the simple MCD model, while a more
realistic disk spectral model is probably needed to fit broadband data
\citep[about three decades in frequency
here;][]{lone1989,rofami1992,vafa2009}. Secondly, the starlight or hot
gas emission might be significant in the UV. This is supported by the
little variability of the UV emission between XMM1 and the {\it Swift}
observation in 2011 February. Finally, some UV emission might come
from the reprocessing of the X-ray emission in the outer disk or
surrounding gas.
\subsection{The Tidal Disruption Event Explanation}
\begin{figure}
\includegraphics{longtermlc.eps}
\caption{The long-term luminosity curve inferred from X-ray spectral
fits. Arrows represent 3-$\sigma$ upper bounds. Note that the {\it
ROSAT} observation was made in 1992 October. The solid curve is ${\rm
Luminosity}=3.45\times 10^{43}[({\rm Time}-2006.60 {\rm yr})/(1 {\rm
yr})]^{-5/3}$ erg s$^{-1}$, and the dotted curve is ${\rm
Luminosity}=7.57\times 10^{43}[({\rm Time}-2006.18 {\rm yr})/(1 {\rm
yr})]^{-5/3}$ erg s$^{-1}$. \label{fig:longtermlc}}
\end{figure}
We show above that the X-ray emission in XMM1 and XMM2 can be best
explained as coming from a thermal disk around a SMBH with a mass of
$\sim$$10^5$--$10^6$ \msun. The transient nature of the source makes
it a great tidal disruption event candidate. This is further supported
by the extreme softness of its X-ray spectra and the inactivity of the
nucleus of the candidate host galaxy IC 4765-f01-1504. The inactivity
of IC 4765-f01-1504, i.e., not an AGN, is indicated by its lack of
significant optical emission lines (Figure~\ref{fig:optspec}). As our
new optical spectrum was made four years after the flare, this also
implies no detection of optical emission lines due to the flare at
this stage. The 2MASS IR colors (Table~\ref{tbl:counterpart}) also put
this galaxy in the region occupied by inactive galaxies in the
color-color diagram \citep[e.g.,][]{hyal1982}. The estimated BH mass
allows the tidal disruption of a solar-type star to be observable
\citep{lioz1979, re1988}. The luminosity reached 6.4$\times$10$^{43}$
erg~s$^{-1}$, which is about the average seen in other candidates
\citep{ko2002, essafr2007, gehece2009}. However, no previous
candidates had soft X-ray spectra with such high quality during the
peak of the flare to allow for the detailed spectral studies here.
One main feature of tidal disruption events is the temporal
evolution. Such events are predicted to rise on timescales of several
months and decay on timescales of months/years \citep{lioz1979,
re1988, re1990}. The decay approximately follows $L\propto (t-t_{\rm
D})^{-5/3}$, where $t_{\rm D}$ is the tidal disruption time, for most
candidates \citep{mauler2010,essako2008,hageko2004,koba1999},
consistent with the theory. XMM1 is fainter than XMM2, and they are
separated by 211 days. If our source is due to tidal disruption, XMM1
should probably be in the rising phase, and XMM2 in the decay
phase. We plot the luminosity curve in
Figure~\ref{fig:longtermlc}. Two decay curves following $L\propto
t^{-5/3}$ are also plotted, with XMM2 assumed to be in the decay. The
dotted and solid curves assume $t_{\rm D}$ to be at half a year and
one month before XMM1, respectively. These assumptions are reasonable,
as the minimum period for the material to return to the SMBH (so as to
accrete) after disruption is on the order of a month
\citep{re1988,evko1989}. These curves predict that the source
luminosity in the {\it Swift} observation in 2011 February should be a
factor of about 20 less than XMM2. Our detection limit is consistent
with this. In the above, we concentrate only on the decay, as the
connection from the rise to the decay is uncertain. The rise could
probably take months, and there could be a short period of several
months when the luminosity is maintained near the Eddington limit
\citep{re1988,re1990}. Future sensitive X-ray observations in the
decay are needed to constrain the long-term evolution.
We see that our source showed large fast X-ray variability, with an
rms of $\sim$21\% (Section~\ref{sec:pds}) in both {\it XMM-Newton}\
observations. If they are really dominated by the thermal disk
emission, the above rms value is high, compared with the values of
$\lesssim$5\% typically seen in the thermal state of neutron-star (NS)
or BH low-mass X-ray binaries \citep{lireho2007,remc2006,mcre2006}. We
note that our rms was integrated over two decades in frequency, the
same as the above studies. Besides, in low-mass X-ray binaries, the
power of the thermal disk typically scales with the frequency as
$\nu^{-1}$ \citep{mcre2006}, while the power for our source is
steeper, scaling approximately as $\nu^{-1.7}$. Large fast variability
and steep power can be seen sometimes in the thermal state of X-ray
binaries, such as the flaring branch of bright NSs \citep{hovajo2002},
which was ascribed to some local instability in the inner disk by
\citet{lireho2009}. For the case of our source, we have shown in
Section~\ref{sec:pds} that its fast variability can be explained as
due to fast variations in the mass accretion rate, which, in the
context of a tidal disruption event, could reasonably be ascribed to
shocks during drastic compression and distortion of the stellar
material \citep{re1988}. This does not occur in low-mass X-ray
binaries, in which the mass is transferred through the Roche
lobe. Thus we speculate that the large fast variability and the steep
power in the thermal state is intrinsic to tidal disruption
events. Among the previous tidal disruption event candidates with
thermal X-ray peaks, the peak of \object{NGC 5905} was the best
observed. It showed an increase of a factor of $\sim$3 during the
peak, but it was over four days \citep{bakoda1996}, and the data
quality was not high enough to investigate the variability on much
shorter timescales.
\subsection{Comparison with AGN}
\object{2XMMi~J184725.1-631724} is unlikely to be an AGN from the lack
of bright optical emission lines. In the following we briefly compare
the X-ray properties of our source with ultrasoft AGN to show their
similarities and differences. NLS1 galaxies typically show the
steepest soft-X-ray spectra among AGN and have a typical photon index
of around 3 \citep{bobrfi1996,grkole2010}. The simple fits of the XMM1
and XMM2 spectra using the PL model indicate an extreme softness of
our source that is not seen in NLS1s.
We estimate the optical-to-X-ray spectral slope $\alpha_{\rm ox}$
using the flux densities in the rest frame of 2500 \AA~and 2 keV
\citep{taavbr1979}. We obtain $\alpha_{\rm ox}$=1.76$\pm$0.03. The
luminosity density in the rest frame of 2500 \AA~is
$l_{2500}$=(1.48$\pm$0.24)$\times$10$^{28}$ erg s$^{-1}$
Hz$^{-1}$. With this value of $l_{2500}$, the value of $\alpha_{\rm
ox}$ should be $\lesssim$1.3, based on the sample of 92 Seyfert 1
galaxies in \citet[][their Figure 15]{grkole2010}. In fact
\object{2XMMi~J184725.1-631724} has about the highest value of
$\alpha_{\rm ox}$ and the lowest value of $l_{2500}$ and is an
outlier, compared with their sample. We note, however, the possible
large uncertainty of $\alpha_{\rm ox}$ for our source, whose star
light contamination might be large, as discussed above.
There have been several ultrasoft AGN claimed in the literature. The
NLS1 galaxy WPVS 007 has the softest X-ray spectrum among AGN detected
during the {\it ROSAT} All-Sky Survey, with $\Gamma_{\rm PL}$$\sim$8
if fitted with a PL or $kT$$\sim$$20$ eV with a BB
\citep{grbema1995}. It can be explained as emission from the inner
disk \citep{grbema1995}, but the quality of the data and the lack of
simultaneous observations above 2.4 keV could make its soft excess due
to the presence of a warm absorber \citep{grleko2008}. The narrow-line
quasar PHL 1092 has {\it XMM-Newton}\ observations, and the X-ray spectra are
steep ($\Gamma_{\rm PL}$$\sim$4--5), but the PL fit is not good when
it is bright \citep{gabobr2004, mifabr2009}. Its bright spectrum in
2003 was fitted with a model of MCD+PL plus an absorption line and a
reflection component by \citet{gabobr2004}; its soft excess was
ascribed to the MCD component ($kT$=114$\pm$4 eV), as in our
study. The NLS1 galaxy 1H 0707-495 also shows an intense soft
excess. Its X-ray spectra from {\it XMM-Newton}\ show $\Gamma_{\rm PL}$$\sim$3.8
\citep{bofasu2002}. They were fitted with a model of BB+PL plus a
reflection component by \citet{fazoro2009}. From this model, both the
thermal disk emission and the reflection contribute to the soft
excess, but the latter dominates. We note that the above ultrasoft AGN
show large short-term variability factors of a few and/or long-term
variability factors of a few hundred, similar to our source. Thus the
variability of our source is not extreme compared with NLS1s, but the
softness of its X-ray spectra is hardly challenged. We note that the
soft component in our source cools in low states, which is not
generally seen in NLS1s \citep[e.g.,][]{mifabr2009}.
\subsection{Alternative Explanations}
We explore the possibility that \object{2XMMi~J184725.1-631724} is not
associated with the SMBH in IC 4765-f01-1504. If it is an
ultraluminous X-ray source (ULX) in this galaxy, it would have a
luminosity about one order of magnitude brighter than the brightest
ULX reported thus far, i.e., HLX-1 \citep{faweba2009}. However, as
derived above, both the spectral fits and the variability argument
imply that this source most probably has a mass of
$\sim$10$^5$--10$^6$ \msun, and it should be within 0.5 kpc (3-$\sigma$
error) of the galaxy center. Being so massive and so close to the
galaxy center, it seems unlikely to be a source other than the central
SMBH. Thus we deem that our source is not a ULX in IC 4765-f01-1504.
The other possibility is that \object{2XMMi~J184725.1-631724} is a
foreground Galactic source. Following \citet{haru2009} and using the
0.2--10 keV absorbed flux of 1.92$\times$10$^{-12}$ erg s$^{-1}$
cm$^{-2}$ (Table~\ref{tbl:mcd+pl}), we have the X-ray to IR flux ratio
$f_{\rm X}/f_{\rm J}\sim 5$ for our source, making it unlikely to be a
coronally active star. These stars generally have $f_{\rm X}/f_{\rm
J}<0.03$ \citep{haru2009}.
The softness of the source with characteristic temperatures of a few
tens of eV makes it similar to the super-soft X-ray sources
\citep[SSS;][]{kava2006,gr2000}. This class of objects have BB
temperatures in the range 20--100 eV, which are about two orders of
magnitude lower than X-ray binaries containing an accreting NS or
BH. There are various types of SSS. One main class is cooling white
dwarfs \citep[WDs;][]{kava2006}. \object{2XMMi~J184725.1-631724}
brightened by a factor of $\gtrsim$64, thus ruling out this
hypothesis. A large fraction of SSS can be interpreted as nuclear
burning of the hydrogen-rich matter on the surface of a white dwarf
that accretes matter from the companion, such as in the so-called
close binary super-soft sources and super-soft novae
\citep{kava2006,gr2000}. It seems unlikely that our source is such
based on the following considerations. In our Galaxy, there are only
about a dozen such sources detected since the {\it Einstein}
Observatory observations \citep{gr2000}, indicating a very low density
of such objects in the sky or a very low life duty cycle. The
probability for any of them lying in the direction of the center of a
galaxy is simply negligible. Besides, these objects are typically
observed at luminosities of $\sim$10$^{36}$--10$^{38}$ ergs~s$^{-1}$,
while our source has much lower luminosities, $<$10$^{34}$
ergs~s$^{-1}$, if it is 5 kpc away. These objects are mostly found at
distances of $<$5 kpc \citep{gr2000}.
\section{CONCLUSION}
\label{sec:conclusion}
\object{2XMMi~J184725.1-631724} is an ultrasoft X-ray transient source
with characteristic temperatures of a few tens of eV. It was bright in
two {\it XMM-Newton}\ observations in 2006--2007, but was not detected in a {\it
ROSAT} pointed observation in 1992, implying a variation factor of
$\gtrsim$64 in the 0.2--10 keV absorbed flux. It was undetected again
in a {\it Swift} observation in 2011 February, implying a flux
decrease by a factor of $\gtrsim$12. It lies toward the center of the
galaxy IC 4765-f01-1504 at a redshift of 0.0353. No bright optical
emission lines were detected from this galaxy, making this source a
good tidal disruption event candidate. The fits to the two {\it XMM-Newton}\
spectra using a thermal disk plus a weak hard component indicate that
the accretion disk luminosity appears to follow the $L\propto T^4$
relation and that the BH mass is around 10$^5$--10$^6$ \msun. The
source showed large fast variability in both {\it XMM-Newton}\ observations, which
can be explained as due to fast variations in the mass accretion
rate. To further check whether this is a tidal disruption event,
future long-term X-ray monitoring is necessary to see whether it
follows the decay expected for a tidal disruption event.
\acknowledgments
Acknowledgments: We thank the anonymous referee for the helpful
comments. We acknowledge the use of public data from the {\it ROSAT},
{\it Swift} and {\it XMM-Newton}\ data archives, and the 2XMM Serendipitous Source
Catalog, constructed by the XMM-Newton Survey Science Center on
behalf of ESA. We want to thank the {\it Swift} PI Neil Gehrels for
approving our ToO request to observe the field of
\object{2XMMi~J184725.1-631724}. {\it Swift} is supported at PSU by
NASA contract NAS5-00136. The optical spectroscopy is based on
observations obtained at the Gemini Observatory which is operated by
the Association of Universities for Research in Astronomy, Inc., under
a cooperative agreement with the NSF on behalf of the Gemini
partnership: the National Science Foundation (United States), the
Science and Technology Facilities Council (United Kingdom), the
National Research Council (Canada), CONICYT (Chile), the Australian
Research Council (Australia), Minist\' erio da Ci\^ encia e Tecnologia
(Brazil) and Ministerio de Ciencia, Tecnolog\'{\i}a e Innovaci\' on
Productiva (Argentina). The observations were carried out as part of
program GS-2011A-Q-90. SAF acknowledges funding from the Australian
Research Council.
|
3,212,635,537,925 | arxiv | \section{Introduction}
Verification of probabilistic weather forecasts is an area of intensive research and growing interest as ensemble forecasting is becoming a standard approach in numerical weather prediction. Ensemble prediction systems (EPS) issue a sample of possible future states of the atmosphere \citep{lewis05,leut08}. The forecasts can be interpreted in the form of a predictive distribution and probabilistic products can be derived in order to support and optimize forecast-based decision-making \citep{krz83}. Appropriate tools for the assessment of probabilistic products from this perspective are therefore essential.
Such tools already exist for probabilistic products expressed in the form a probability forecast. The relative operating characteristic (ROC) curve is a common verification tool for the assessment of probability forecasts \citep{mason82}. The ROC curve is related to decision-making analysis and the corresponding fundamental property of the forecast is called \textit{discrimination}.
Forecast discrimination assesses whether the forecast can be used to successfully discriminate between the observations \citep{murphy91} or, said differently, whether appropriate decisions can be taken based on a forecast.
Discrimination is translated in terms of \textit{economic value} using a simple cost-loss model that allows the specificity of a user to be taken into account through the definition of a \textit{cost-loss ratio}. The derived quantitative measure is called \textit{value score} or \textit{relative value} and is usually represented in the form of a probability value plot showing the forecast value as a function of the user's cost-loss ratio \citep{richardson2000,wilks2001,zhu02}. The value of a forecast is defined as the benefit to a user as a result of making decisions based on a forecast and has to be distinguished from forecast quality, the overall agreement between forecast and observation \citep{murphy93}. In a verification process, value and quality can be seen as being from the point of view of the forecast user and from the point of view of the forecast provider, respectively. The distinction between the two \textit{types of goodness}, value and quality, is crucial since a non-linear relationship between them can lead to situations where a large improvement in the forecast quality does not imply an increase in the forecast value, or conversely, a small improvement in forecast quality can bring a notable benefit in terms of forecast value \citep{chen87,buizza2000,pinson2013}.
Probabilistic products can be expressed in terms of a probability when the focus is on a particular event of interest, but also in terms of a quantile when the focus is on a particular probability level of interest. While a probability forecast first requires the definition of an event, i.e. the categorization of the original information, a quantile forecast is a 'single-valued' forecast expressed in the unit of the variable being forecast. Considering here probabilistic products derived from EPS simulations for continuous variables, such as temperature, wind speed or global radiation, quantile forecasts allow one to work with a continuous forecast as the original one by defining a nominal probability level. The choice of a probability level is directly related to the user's loss function: a quantile forecast at a given probability level is the optimal forecast for users with a specific asymmetry in their loss function \citep{koenker99,fh07,gneiting2011b}.
Based on the relationship between user's loss function and quantile forecast level, the quantile score (QS) is the natural scoring rule for assessing the quality of quantile forecasts \citep{koenker99,fh07,gneiting2011b}. More recently, the verification of quantile forecasts has benefited from the tradition and concepts stemming from the probability forecast verification framework. It has been shown that QS is a \textit{proper} scoring rule and a decomposition of the score has been proposed \citep{Bentzien2014}. The QS decomposition provides information about \textit{reliability} and \textit{resolution}, two other fundamental attributes of a probabilistic forecast \citep{toth2003}.
The aim of the paper at hand is to extend the range of verification methods dedicated to the assessment of quantile forecasts. In particular, the assessment of quantile forecasts from the user's perspective, in a decision-making framework, is explored here. Based on a simple cost-loss model, the concepts of forecast discrimination and forecast value are revisited focusing on a specific user rather than on an specific event. First, a new tool is proposed for the analysis of user-based discrimination. The so-called relative user characteristic (RUC) curve and the associated summary measure are shown to be adequate for the assessment of quantile forecast discrimination ability. Secondly, quantile forecast value is discussed as an application of the value score to quantile forecasts. The quantile value plot, showing the economic value of a forecast as a function of a range of events of interest, is proposed as a new tool for the visualization of quantile forecast performance. Finally, the relationship between quantile forecast value and quantile skill score is discussed in the same vein as the relationship between probability forecast value and Brier skill score \citep{murphy69}. The concepts developed are first illustrated with the help of synthetic datasets and in a second step applied to probabilistic forecasts derived from an EPS.
The manuscript is organized as follows: Section \ref{sec:data} describes the datasets that are used to illustrate the discussion.
Section \ref{sec:def} introduces definitions and notations and describes the relationship between quantile forecast and forecast user within a cost-loss model framework. Section \ref{sec:discr} discusses the concept of discrimination and Section \ref{sec:value} the application of the economic value score to quantile forecasts. Section \ref{sec:conc} presents the conclusions.
\section{ Data}
\label{sec:data}
\subsection{Synthetic datasets}
\label{subsec:toy}
In order to illustrate the concepts discussed hereafter, we make use of synthetic and real datasets. The synthetic data are derived from a toy-model based on normal distributions often used to illustrate verification discussions \citep[e.g.][]{hamill2001,weigel2011}. The toy-model is kept simple in order to facilitate the interpretation of the results.
We consider a signal \(s\), normally distributed, written \(s \sim \mathcal{N}(0,1)\). We assume that the
observations are randomly drawn from a distribution $\mathcal{N} (s,1)$ and the associated predictive distribution described by $\mathcal{N} (s+\beta,\sigma)$
where \(\beta\) is the unconditional bias parameter and \(\sigma\) the dispersion parameter. We define the following test-cases:
\begin{enumerate}
\item[\emph{\(A_0\)}]: \(\beta=0\), \(\sigma=1\) (a perfect probabilistic forecast) ,
\item[\emph{\(A_1\)}]: \(\beta=-0.75\), \(\sigma=1\) (a biased forecast),
\item[\emph{\(A_2\)}]: \(\beta=0\), \(\sigma=1/3\) (an underdispersive forecast),
\item[\emph{\(B\)}]: \(\beta=\epsilon_B\), \(\sigma=1\) (a forecast with white noise),
\end{enumerate}
where \(\epsilon_B\) is derived from a uniform distribution defined on \(]-5,5[\). The first three datasets \(A_0\), \(A_1\) and \(A_2\) differ only in terms of biases while the fourth dataset \(B\) corresponds to a forecast with a dynamically disturbed signal.
\subsection{COSMO-DE-EPS}
\label{subsec:cdeps}
Real datasets are provided by COSMO-DE-EPS, a regional ensemble prediction system run operationally at Deutscher Wetterdienst, Offenbach, Germany. The ensemble system is based on a 2.8 km grid resolution version of the COSMO model \citep{stepp03,bald11}
with a model domain that covers Germany and parts of the neighbouring countries. The ensemble comprises 20 members including variations in initial conditions, physics parameterisations and boundary conditions \citep{gtpb10,pbtg12}.
COSMO-DE-EPS has been first developed focusing on high-impact weather events \citep{bbtg2013,zbb2013} and is planned to be used for energy-applications. The focus in this paper is on global radiation which is the main weather variable affecting solar energy forecasts.
Verification is applied to the 0300UTC run with a forecast horizon ranging between 5 and 15 hours. Two periods of 3 months are compared: winter (December, January, February) 2012/2013 and summer (June, July, August) 2013. The observation dataset consists of pyranometer measurements from 32 stations distributed over Germany and quality controlled \citep{becker2012}.
Global radiation forecasts and observations are transformed into clearness index before verification. The clearness index is defined as the ratio between global radiation at ground and global radiation at the top of the atmosphere \citep{badescu}. This pre-processing of the data allows climatological effects and misinterpretation of the verification results to be avoided \citep{hamilljuras2006}.
\section{Definitions and framework}
\label{sec:def}
\subsection{Quantile forecast, quantile score, and quantile skill score}
We first consider the quantity to be forecast (or \textit{observation}) \(\Omega \in \Re\) that we assume to be a continuous random variable driven by a stochastic process. An observed event \(E\) is defined by a threshold \(\omega\) as \(E :\Omega \ge \omega\). The base rate \(\pi\) of an event \(E\) (or climatological frequency) corresponds to:
\begin{equation}
\pi = Pr( \Omega \ge \omega).
\label{equ:pi}
\end{equation}
Consider now a predictive cumulative distribution \(F(x)\).
The probability forecast \(p_\omega\) of event \(E\) is defined as:
\begin{equation}
p_\omega= 1- F(\omega).
\label{equ:prob}
\end{equation}
The quantile forecast \(q_\tau\) at probability level \(\tau\) ($0 \le \tau \le 1$) is defined as:
\begin{equation}
q_\tau := F^{-1}(\tau)=\text{inf}\{y:F(x)\ge \tau \}
\label{equ:cdf}
\end{equation}
such the relationship between a probability forecast and a quantile forecast is expressed as:
\begin{equation}
p_{q_\tau}=1-\tau.
\label{equ:pq}
\end{equation}
Figure \ref{fig:cdf} shows an example of a cumulative distribution function \(F(x)\). A threshold \(\omega\) and the associated probability forecast \(1-p_\omega\) as well as a probability level \(\tau\) and the associated quantile forecast \(q_\tau\) are shown on the plot.
The quantile score (QS) is the scoring rule applied in order to assess the quality of a quantile forecast. QS is based on an asymmetric piecewise linear function \(\rho_\tau\) called the check function. The check function was first defined in the context of quantile regression \citep{koenker78}:
\begin{equation}
\rho_{\tau}(u) = u[\tau - I(u<0)] =
\left\lbrace
\begin{array}{ccc}
\tau u & \mbox{if} & u \ge 0\\
(\tau-1)u & \mbox{if} & u < 0\\
\end{array}\right.
\label{equ:check}
\end{equation}
where $I(.)$ is an indicator function having value 1 if the condition in parenthesis is true and zero otherwise.
QS results from the mean of the check function applied to the pairs \(i=1,...,N\) of observation \(\Omega_i\) and quantile forecast \(q_{\tau,i}\) following
\begin{equation}
QS = \dfrac{1}{N} \sum_{i=1}^{N} {\rho_\tau(\Omega_i-q_{\tau,i})},
\label{equ:QS}
\end{equation}
where \(N\) is the size of the verification sample. Developing Eq. \eqref{equ:QS} we can write
\begin{equation}
QS = \dfrac{1-\tau}{N} \sum_{i:\Omega_i < q_{\tau,i}} ( q_{\tau,i} - \Omega_i)
+ \dfrac{\tau}{N} \sum_{i:\Omega_i \geq q_{\tau,i}} (\Omega_i - q_{\tau,i} )
\label{equ:QSf}
\end{equation}
The scoring rule consists of penalties per unit \(1-\tau\) and \(\tau\) associated with under-forecasting and over-forecasting, respectively.
Skill scores are computed in order to measure the relative benefit of using a forecast compared to a reference forecast \citep{wilks}. The quantile skill score (QSS) measures the skill of a quantile forecast compared to a reference quantile forecast. Considering the climatology as reference, QSS corresponds to:
\begin{equation}
QSS= \dfrac{QS_\text{forecast} -QS_\text{climate}}{QS_\text{perfect}-QS_\text{climate}} = 1-\dfrac{QS_\text{forecast}}{QS_\text{climate}}
\label{equ:QSS}
\end{equation}
where \( QS_\text{forecast}\), \( QS_\text{perfect}\) and \(QS_\text{climate}\) represent the quantile scores of the forecast under assessment, of a perfect deterministic forecast and of a climatological \(\tau\)-quantile forecast, respectively. \( QS_\text{perfect}\), by definition, equals 0 and a climatological \(\tau\)-quantile forecast, noted \(\Omega_\tau\), is here defined as the \(\tau\)-quantile of the observation distribution over the verification sample.
\begin{figure}
\centering
\includegraphics[width=6cm]{./fig_cumudistr.eps}
\caption{
Example of a predictive cumulative distribution function $F(x)$. Probabilistic products are derived either fixing a threshold \(\omega\) and deriving the associated probability forecast \(p_\omega\), or fixing a probability level \(\tau\) and deriving the associated quantile forecast \(q_\tau\).
}
\label{fig:cdf}
\end{figure}
\subsection{Cost-loss model and optimal decision-making}
\label{sec:costmodel}
The framework used to discuss the concept of \textit{use}r and \textit{decision-making} is based on a static cost-loss model \citep{thompson62,katz1997}. The cost-loss model describes situations of dichotomous decisions: a user has to decide whether or not to take protective action against potential occurrence of an event \(E\). The decision is made based on a decision variable (or forecast) \(\Lambda\). A decision criterion \(\lambda\) applied to the decision variable defines an action \(A: \Lambda \ge \lambda\). Taking action implies a cost \(C\). In the case of occurrence of the event \(E\) without preventive action, a loss \(L\) is encountered. The cost-loss ratio is denoted \(\alpha\):
\begin{equation}
\alpha=\dfrac{C}{L}.
\label{equ:alpha}
\end{equation}
A user with cost-loss ratio \(\alpha\) is called hereafter an \(\alpha\)-user. Based on this simple model the optimal decision strategy of an \(\alpha\)-user can be discussed \citep[e.g.][]{Richardson2011}. The problem consists of finding, for a decision variable \(\Lambda\), the \textit{critical} decision criterion \(\lambda_\alpha\) that minimizes the \(\alpha\)-user mean expense if actions are taken when \(\Lambda \ge \lambda_\alpha\).
Consider first the case of a probability forecast \(p_\omega\) as a decision variable. Based on \(p_\omega\), does the user have to take action or not? In order to answer this question, the average expenses in the cases of positive and negative answers are compared. If the answer is yes, the user encounters a cost \(C\) on every occasion, so the average expense \(\bar{E}_\text{yes}\) is simply
\begin{equation}
\bar{E}_\text{yes}=C.
\label{equ:yes}
\end{equation}
If the answer is no, the user has no cost but a loss \(L\) on each occasion where the event occurs, so on average
the user's expense \(\bar{E}_\text{no}\) is
\begin{equation}
\bar{E}_\text{no}=L Pr(\Omega \geq \omega \mid p_\omega),
\label{equ:no}
\end{equation}
where \(Pr(\Omega \geq \omega \mid p_\omega)\) is the probability that the event occurs when the probability forecast \(p_\omega\) is issued.
So, users with a cost-loss ratio \(\alpha < Pr(\Omega \geq \omega \mid p_\omega)\) should take preventive action, while users with a greater cost-loss ratio should not.
The critical decision criterion \(p_\omega^\star\) associated with the decision variable \(p_\omega\) is thus defined as
\begin{equation}
p_\omega^\star = \{p_\omega \ | \Pr(\Omega \geq \omega \mid p_\omega) = \alpha \}.
\label{equ:optdec}
\end{equation}
Thus, the action based on the probability forecast \(A: p_\omega \geq p_\omega^\star\) optimizes the user's mean expense in the long term.
If the forecast is reliable, we have by definition \( \Pr(\Omega \geq \omega \mid p_\omega) = p_\omega\): the event actually happens with an observed relative frequency consistent with the forecast probability \citep{brocker2009}. The optimal decision is then to take action if
\begin{equation}
p_\omega \geq \alpha.
\label{equ:optp}
\end{equation}
When the probability forecast is compared to the cost-loss ratio in order to decide whether or not to take action (without additional information about forecast reliability), we say that the probability forecast is taken at face value. For example, consider users who have to decide whether or not to take preventive action against precipitation occurrence. If the forecast probability of precipitation is 10\%, users with cost-loss ratio lower than 10\% take action. If the forecast is not reliable, the critical decision criterion is no longer \(\alpha\) but has to be adjusted following Eq. \eqref{equ:optdec}. Statistical adjustments of the forecast based on past data is usually referred as \textit{forecast calibration} \citep[e.g.][]{gneit2007}.
Consider now a quantile forecast \(q_{\tau}\) as a decision variable. We apply the same reasoning as for a probability forecast. The critical decision criterion \(q_\tau^\star\) associated with \(q_{\tau}\) is defined as
\begin{equation}
q_\tau^\star = \{ q_\tau \ | \ Pr( \Omega \geq \omega \mid q_\tau ) = \alpha \}
\label{equ:cdcq}
\end{equation}
such that taking action when \(q_\tau \geq q_\tau^\star \) minimizes the user mean expense. By definition, a quantile forecast is reliable if it satisfies
\begin{equation}
Pr( \Omega \geq \omega \mid q_\tau = \omega ) = 1-\tau ,
\label{equ:qrelia}
\end{equation}
i.e.\ the observed relative frequency of the event defined by the quantile forecast is consistent with the quantile forecast probability level. Eq. \eqref{equ:cdcq} has a straightforward solution
\begin{equation}
q^\star_\tau = \omega
\label{equ:optq}
\end{equation}
when the decision variable is the quantile forecast at probability level \(\tau\) defined as
\begin{equation}
\tau = 1-\alpha.
\label{equ:taualpha}
\end{equation}
Taking action when \(q_\tau \geq \omega\) with \(\tau= 1-\alpha\) is equivalent to taking action when \(p_\omega \geq \alpha\) since the cumulative probability distribution function \(F(x)\) is by definition monotonically increasing (see e.g. Figure \ref{fig:cdf}). Hence, a quantile forecast is taken at face value when the user's decision is made based on the comparison of the forecast with the event threshold \(\omega\). In our example, if the 90\%-quantile forecast of precipitation is greater than zero, a user with cost-loss ratio \(\alpha=1-0.9=0.1\) takes preventive action.
In a general form, the critical decision criterion \(\lambda_\alpha\) for an \(\alpha\)-user is defined by
\begin{equation}
\lambda_\alpha = \{ \lambda \ | \ Pr( \Omega \geq \omega \mid \lambda ) = \alpha \}
\label{equ:cdc}
\end{equation}
where the decision variable could equally be the probability forecast \(p_\omega\) or the quantile forecast \(q_\tau\) with \(\tau=1-\alpha\). Provided that the forecasts are reliable, the critical decision criteria are known and have a simple expression (Eqs (\ref{equ:optp},\ref{equ:optq})). In the following, we say that the decision variable is taken at \textit{face value} when the user applies the decision criterion valid for a reliable forecast, irrespective of whether the forecast is actually reliable or not.
\subsection{Quantile forecast user}
\label{subsec:user}
\begin{figure}
\centering
\includegraphics[width=9cm]{./fig_cl_functions.eps}
\caption{
(a) Cost (dashed line) as a function of the level of protection \(x\) and loss (full line) as a function of the observation \(\Omega\). An observation \(\Omega_i\) is represented by a vertical line.
(b) Expense as a function of the difference between the observation \(\Omega_i\) and the level of protection \(x\). The horizontal line indicates the expense for a perfect level of protection.
}
\label{fig:CLEx}
\end{figure}
The dichotomous decision problem is extended to a continuous decision problem considering the cost \(C\) and the loss \(L\) as unitary cost and unitary loss, respectively \citep{epstein69,roulston2003}. The cost of taking protection is a linear function of the level of protection \(x\) and the loss without protection is a linear function of the observation \(\Omega\), as illustrated in Figure \ref{fig:CLEx}. The optimization problem consists of finding the level of protection that minimizes the expected user expense.
Considering a variable defined on \(\Re+\) (the generalization to variables defined on \(\Re\) is straightforward), the expense associated with a level of protection \(x\) corresponds to \(C x\). If the observation is \(\Omega\), then protection is perfect if $x = \Omega$. But if $x>\Omega$, then there is an unnecessary expense due to a larger level of protection than is actually needed. If the observation \(\Omega\) is greater than the level of protection, then additionally a loss \(L (\Omega- x)\) is encountered. Formally, we can write the expense function $E$ as
\begin{equation}
E =
\ \begin{cases}
C (x -\Omega) & \text{if } \Omega < x \\
(L -C) (\Omega- x) & \text{if } \Omega \geq x.
\end{cases}
\label{equ:expi}
\end{equation}
The expense function is represented in Figure \ref{fig:CLEx}.
If divided by $L$, the expense function is an asymmetric loss function equivalent to the check function defined in Eq. \eqref{equ:check}, where the asymmetry is given by \(\tau = \frac{L-C}{L}\). Thus the optimal level of protection \(x^\star\) which minimizes the user's mean expense corresponds to the $1-\alpha$ quantile of the true predictive distribution of $\Omega$.
This result is not new: quantile forecasts arise as an optimal solution for users with an asymmetric linear loss function \citep{koenker78,chris97}. More recently, it has been shown that quantile forecasts are optimal forecasts in a stochastic optimization framework for a more general class of loss functions \citep{,gneiting2011}.
Asymmetric loss functions find a number of applications, in particular for operational decision-making problems related to the integration of renewable energies into the electricity grid. For example, asymmetric loss functions can be associated with market participants who want to optimize their bids or system operators who have to optimize their reserves. The user's optimal forecast corresponds then to a specific quantile of the predictive distribution where the probability is defined by the user's cost-loss ratio \citep{pinson2007,pinson2013}.
\section{Discrimination}
\label{sec:discr}
Based on the discussion developed in the previous Section, continuous decision making is seen in the following as a \textit{continuum} of dichotomous decisions. For each threshold \(\omega\) of the event spectrum, the question is whether to take action for the next unit of the variable. The adequate decision for a user in order to minimize the expected expense is a function of his (her) cost-loss ratio as defined in Eq. \eqref{equ:cdc}. Moreover, the relationship between cost-loss ratio and quantile probability level, \(\tau=1-\alpha\), makes implicit the cost-loss ratio \(\alpha\) of a user as soon as the level \(\tau\) of the quantile forecast used as decision variable is selected.
\subsection{General verification framework}
A general framework for forecast verification is based on the joint distribution of forecasts and observations \citep{murphy87}. The overall agreement between forecasts and observations is called quality and is measured by scoring rules, like QS for quantile forecasts. In order to access more information about the forecast performance, two factorizations of the joint distribution, into conditional and marginal distributions, can be applied: the \textit{calibration-refinement} (CR) factorization when conditioning on the forecasts and the \textit{likelihood-base rate} (LBR) factorization when conditioning on the observations. Summary measures based on these two factorizations are associated with attributes, fundamental characteristics of the forecast. Reliability and resolution are derived from the CR factorization while discrimination is derived from the LBR factorization \citep{murphy92}.
Here the focus is on discrimination, the key forecast attribute for decision-making processes. A general definition of discrimination is "the ability of a forecasting system to produce different forecasts for those occasions having different realized outcomes" \citep{wilks}. Discrimination assessment is discussed in terms of \textit{event} and \textit{action} within the dichotomous decision framework. Regarding the LBR factorization, it is common practice to analyse discrimination in terms of hit rate \(H\) and false alarm rate \(F\) defined as
\begin{equation}
H = Pr( \Lambda \ge \lambda \mid \Omega \ge \omega )
\label{equ:H}
\end{equation}
and
\begin{equation}
F = Pr( \Lambda \ge \lambda \mid \Omega < \omega),
\label{equ:F}
\end{equation}
respectively. Actions \(A: \Lambda \ge \lambda \) and events \(E:\Omega \ge \omega\) are dichotomous, each presenting two alternatives, so H and F can be easily derived from the construction of a \(2\times2\) contingency table. No discrimination corresponds to the case where:
\begin{equation}
H = F
\label{equ:HFlambda}
\end{equation}
for all \(\lambda \in \Lambda\) and \(\omega \in \Omega\), meaning that actions and event occurrence are independent \citep{brocker2014}.
\subsection{Event-based discrimination}
We first focus on one particular event defined by a threshold \(\omega\), with event-specific hit rate \(H_\lambda\) and false alarm rate \(F_\lambda\). A popular way to assess discrimination (Eq. \eqref{equ:HFlambda}) is to plot the set of points (\(F_\lambda,H_\lambda\)) for a range of actions with \(\lambda \in \Lambda\). The resulting curve is known as the relative operating characteristic (ROC) curve. When action and event occurrence are independent, the ROC curve is a diagonal line. Concavity of the curve indicates a discrimination ability in the forecast and the area under the curve (AUC) becomes a quantitative measure of forecast discrimination \citep{mason82}. Figure \ref{fig:roccas} (a) shows an example of a ROC curve for the synthetic dataset \(A_0\). The event of interest is \(E: \Omega \geq 0\) with a base rate $\pi = Pr( \Omega \geq 0)$ of 0.5. The respective forecast probability \(p_0 =1-F(0)\) is used as decision variable.
The interpretation of the ROC curve can be related to the dichotomous decision model described in Section \ref{sec:costmodel} as discussed for example in \cite{Richardson2011}. In order to describe this relationship, we consider the slope of the ROC curve, defining first the gradient of a line joining two successive ROC points
(\(F_{\lambda},H_{\lambda}\)) and (\(F_{\lambda+\Delta \lambda},H_{\lambda+\Delta \lambda}\)):
\begin{equation}
\dfrac{H_{\lambda}-H_{\lambda+\Delta \lambda}}{F_{\lambda}-F_{\lambda+\Delta \lambda}} =
\dfrac
{ Pr( \Lambda \ge \lambda \mid \Omega \ge \omega ) - Pr( \Lambda \ge \lambda+\Delta \lambda \mid \Omega \ge \omega )}
{ Pr( \Lambda \ge \lambda \mid \Omega < \omega ) - Pr( \Lambda \ge \lambda+\Delta \lambda \mid \Omega < \omega )}.
\end{equation}
The slope of the curve \(\gamma\) is obtained when \(\Delta \lambda\) tends to \(0\):
\begin{equation}
\gamma(\lambda,\omega) = \dfrac{Pr( \Lambda = \lambda \mid \Omega \ge \omega) } {Pr( \Lambda = \lambda \mid \Omega < \omega) }
\label{equ:LR1}
\end{equation}
where the ratio is also know as the \textit{likelihood ratio} \citep{brock2011}. Using the Bayes rule and the definition of the critical decision criterion of an \(\alpha\)-user in Eq. \eqref{equ:cdc}, we can write
\begin{equation}
\gamma(\lambda_\alpha,\omega) = \dfrac{1-\pi}{\pi} \dfrac{\alpha}{1-\alpha}
\label{equ:LR2}
\end{equation}
where \(\pi = Pr(\Omega \ge \omega)\) is the base rate of an event \(E:\Omega \ge \omega\) and \(\lambda_\alpha\) the corresponding critical decision criterion of an \(\alpha\)-user.
The range of decision criterion \(\lambda\) used to derive the ROC curve (\(F_{\lambda},H_{\lambda}\)) corresponds to a range of critical decision criteria associated with users with different cost-loss ratios. Each point of the ROC curve is associated with a specific \(\alpha\)-user that is identified by the slope of the curve at that point. The slope possibly ranges between \(0\) and \(+\infty\) at the right-top and the bottom-left corners of the ROC plot respectively. Moving along the curve from the top to the bottom consists in varying the cost-loss ratio \(\alpha\) between \(0\) and \(1\).
For example, consider a user with a cost-loss ratio \(\alpha= 50\%\). In Figure \ref{fig:roccas}, the point of the ROC curve with slope \(\gamma= 1 \) is highlighted (\(\alpha =0.5,\pi=0.5\) in Eq. \eqref{equ:LR2}). This point indicates the performance of the forecast in terms of \(H\) and \(F\) for this particular user. Conversely, the decision criterion applied to obtain this point corresponds to the critical decision criterion for the 50\%-user.
The ROC curve applied to a decision variable, then, corresponds to testing whether actions and event occurrence are independent for one event and a range of users with different cost-loss ratios. The ROC curve is an \textit{event specific} but \textit{user unspecific} discrimination tool and is therefore well-adapted to probability forecast discrimination assessment.
\begin{figure}[htb]
\centering
\includegraphics[width=5cm]{./fig_roc_roci_tang.eps}
\caption{
Discrimination curves for decision variables from the synthetic dataset \(A_0\). The diagonal lines are the no discrimination lines. The points correspond to the (\(F\),\(H\)) pair for the event \(\Omega \geq 0\) and the action associated with the $50\%$-users.
(a) ROC curve of the probability forecast \(p_{0}\) for the event \(E: \Omega \geq 0\), with base rate \(\pi=0.5\), and equi-cost lines (in grey) of slope \(\gamma =1\).
(b) RUC curve of the quantile forecast \(q_{0.5}\) for the user with cost-loss ratio \(\alpha=0.5\).
}
\label{fig:roccas}
\end{figure}
\subsection{User-based discrimination}
We focus now on a user with cost-loss ratio \(\alpha\). The critical decision criterion \(\lambda_\alpha\) defines the action of this specific user with respect to an event. We define then the user-specific hit rate \(H_\omega\) and false alarm rate \(F_\omega\) as in Eqs \eqref{equ:H} and \eqref{equ:F} for a fixed \(\alpha\). In order to test Eq. \eqref{equ:HFlambda}, the set of points (\(F_\omega,H_\omega\)) are plotted for a range of events. We call the resulting curve a \textit{relative user characteristic} (RUC) curve because it is a comparison of two user characteristics (\(F_\omega\) and \(H_\omega\)) as the event definition varies. As for the ROC curve, the no discrimination line corresponds to the diagonal line and concavity of the curve indicates forecast discrimination ability.
Figure \ref{fig:roccas} (b) shows an example of a RUC curve valid for a user with cost-loss ratio \(\alpha=50\%\). In this example, the decision variable is the 50\%-quantile forecast from the synthetic dataset \(A_0\). Moving along the RUC curve from the bottom left corner to the top right corner involves varying the event under focus, the event's base rate varying from \(0\) to \(1\), respectively. The point with slope \(\gamma=1\) corresponds to the event \(E:\Omega \geq 0\) with base rate \(\pi=0.5\). This point is obviously the same as in Figure \ref{fig:roccas} (a).
In order to produce a RUC curve, critical decision criteria have to be known for a range of events. They can be estimated resolving Eq. \eqref{equ:cdcq} numerically. In practice, critical decision criteria can also be estimated by means of a reliability diagram. For example, a reliability diagram for quantile forecasts plots the conditional observed quantile as a function of quantile forecast categories \citep{Bentzien2014}. With regard to Eq. \eqref{equ:qrelia}, we can deduce that the mean forecast in each forecast category (horizontal axis of the diagram) is an estimation of the critical decision criteria associated with the events defined by the corresponding conditional observed quantile (vertical axis of the diagram).
The RUC curve is user specific (and event unspecific) and therefore well-adapted to quantile forecast discrimination. A summary measure of quantile discrimination ability is obtained mimicking the ROC framework: the area under the RUC curve, noted here \(AUC^\prime\), is proposed as a quantitative measure of discrimination for quantile forecasts. Considering \(n_E\) events \(E_i: \Omega \geq \omega_i\), \(i=1,...,n_E\) with increasing base rate, \(AUC^\prime\) is estimated by a trapezoidal approximation as
\begin{equation}
AUC^\prime = \sum_{i=0}^{n_E} 0.5 ( H_{\omega_{i+1}} + H_{\omega_{i}}) ( F_{\omega_{i+1}} - F_{\omega_{i}})
\label{equ:aucprime}
\end{equation}
with the trivial points \(H_{\omega_{0}}=F_{\omega_{0}}=0\) (for an event of base rate 0) and \(H_{\omega_{n_E+1}}=F_{\omega_{n_E+1}}=1\) (for an event of base rate 1). In order to reduce the biases introduced by the limited number of RUC points, the RUC curve can be fitted under a bi-normal assumption. The procedure involves considering \(F_\omega\) and \(H_\omega\) as both expressed as integrations of the standard normal distribution \citep{mason82}. The bi-normal model has been shown to be valid in most cases when applied in the ROC framework \citep{mason2002,atger2004}.
The properties of the RUC curve and \(AUC^\prime\) are discussed with the help of illustrative examples based on 4 simple simulation test cases (see Section \ref{subsec:toy}). In Figure \ref{fig:ABCDE}, the forecast attributes reliability, resolution and discrimination are shown as a function of the probability level \(\tau\) of the \(\tau\)-quantile forecast under assessment. RUC curves for the 50\%-quantile forecasts are also shown. Quantile forecast reliability and resolution are estimated using the decomposition of the quantile score \citep{Bentzien2014} while discrimination curves and summary measures are estimated based on the bi-normal assumption.
Figures \ref{fig:ABCDE} (a) shows the lack of reliability, which occurs by construction in the simulations \(A_1\), \(A_2\) and \(B\). In Figures \ref{fig:ABCDE} (b) and \ref{fig:ABCDE} (c), resolution and discrimination measures deliver a similar message comparing the different simulations which illustrates the idea that "resolution and discrimination are the two faces of the same coin" \citep{brocker2014}. Resolution and discrimination exhibit however different behaviours as a function of the probability level reflecting the fact that the first takes the forecaster's perspective and the second the user's perspective. Moreover, discrimination ability is identical for the simulations \(A_0\), \(A_1\) and \(A_2\): they are unaffected by biases and dispersion errors. Indeed, \(AUC^\prime\) is by construction insensitive to conditional and unconditional biases. In contrast, the forecast derived from simulation \(B\) with a perturbed signal presents less discrimination ability than forecasts from the other simulations, in particular for the 50\%-quantile forecast. Focusing on users with cost-loss ratio \(\alpha=0.5\) ($\tau=0.5$) , RUC curves for the 50\%-quantile forecasts of simulations \(A_0\), \(A_1\), \(A_2\), and \(B\) are shown in Figure \ref{fig:ABCDE} (d). The largest discrepancies between simulations $A$ and $B$ are visible at the centre of the RUC curves, so for events with intermediate base rates, while for events with small or large base rates the RUC curves tend to overlap.
\begin{figure*}[thb]
\centering
\includegraphics[width=17cm]{./fig_scores_simuABCDE.eps}
\caption{
(a) Reliability, (b) resolution and (c) discrimination as a function of the probability level \(\tau\) of the \(\tau\)-quantile forecasts and (d) RUC curves for the 50\%-quantile forecasts ($\tau=0.5$). The results are shown for the simulation test cases \emph{\(A_0\)} (full lines), \emph{\(A_1\)} (dashed lines), \emph{\(A_2\)} (dotted lines) and \emph{\(B\)} (full grey line).
}
\label{fig:ABCDE}
\end{figure*}
\section{ Value of quantile forecasts}
\label{sec:value}
\subsection{Economic value}
The cost-loss model described in Section \ref{sec:costmodel} has been used to develop the concept of economic value of a probabilistic forecast. The forecast value is assessed considering decision-making made by an \(\alpha\)-user about the occurrence of an event. The value of a forecast (also called value score or relative value) is defined as
\begin{equation}
V = \dfrac{\bar{E}_\text{climate}- \bar{E}_\text{forecast}} {\bar{E}_\text{climate}- \bar{E}_\text{perfect}},
\label{equ:Vdef}
\end{equation}
where the mean expense \(\bar{E}\) of an \(\alpha\)-user is estimated when decisions are based on a forecast (\(\bar{E}_\text{forecast}\)), on a perfect deterministic forecast (\(\bar{E}_\text{perfect}\)), or on climatological information (\(\bar{E}_\text{climate}\)) \citep{richardson2000,wilks2001,zhu02}. \(V\) is a measure of the economic gain (or reduction of mean expense) when using a forecast relative to the gain when using a perfect deterministic forecast.
Following \textit{e.g.} \cite{Richardson2011}, the mean expense of a forecast user can be written as
\begin{equation}
\bar{E}_\text{forecast} = F(1-\pi)C -H\pi(L-C)+ \pi L,
\label{equ:Efct}
\end{equation}
where \(H\) and \(F\) are the hit rate and false alarm rate as defined in Eqs \eqref{equ:H} and \eqref{equ:F}, respectively, and \(\pi\) the base rate of the event of interest. A user with a perfect deterministic forecast at hand has to face costs only. The user mean expense corresponds in this case to:
\begin{equation}
\bar{E}_\text{perfect} = \pi C.
\label{equ:Eperf}
\end{equation}
For a user who bases his (her) decision on climatological information, the optimal mean expense is expressed as
\begin{equation}
\bar{E}_\text{climate} = \left\lbrace
\begin{array}{ccc}
C & \mbox{if} & \alpha < \pi\\
\pi L & \mbox{if} & \alpha \ge \pi,\\
\end{array}\right.
\label{equ:Eclim}
\end{equation}
depending on the relationship between cost-loss ratio and base rate.
Combining Eqs \eqref{equ:Efct}-\eqref{equ:Eclim},
the value of a forecast can finally be written as:
\begin{equation}
V=\left\lbrace
\begin{array}{ccc}
(1-F) -\left(\dfrac{\pi}{1-\pi} \right)\left(\dfrac{1-\alpha}{\alpha} \right) (1-H)& \mbox{if} & \alpha < \pi \\
H -\left(\dfrac{1-\pi}{\pi} \right)\left(\dfrac{\alpha}{1-\alpha} \right) F & \mbox{if} & \alpha \geq \pi. \\
\end{array}\right.
\label{equ:Vdef2}
\end{equation}
So, the economic value \(V\) is defined for an event with base rate \(\pi\) and a user with cost-loss ratio \(\alpha\). \(V\) depends on the forecast performance in terms of H and F.
\begin{figure}[htb]
\centering
\includegraphics[width=6cm]{./maxvcriteria_valq_v2.eps}
\caption{
(a) Value \(V\) of the probability forecast from simulation $A_0$ for the event defined as \(E:\omega \geq 0\) with base rate \(\pi=0.5\). The dashed lines represent the forecast value when the probability levels $0.1,0.2,...,0.9$ are chosen as decision criterion. The full line represents the envelope of the dashed lines.
(b) Value \(V\) for users with cost-loss ratio \(\alpha=0.7\) of the \(30\%\)-quantile forecasts taken at face value from the 4 synthetic datasets: \(A_0\) (full black line, square), \(A_1\) (dashed line, triangle), \(A_2\) (dotted line, circle) and \(B\) (full grey line, cross). The black point is the common point of the two plots: value of the simulation \(A_0\) for the event with base rate $\pi=0.5$ and a user with cost loss ratio $\alpha=0.7$.
}
\label{fig:maxval}
\end{figure}
Applied to a probability forecast, the event's base rate is fixed and the value of a probability forecast is generally represented in the form of a probability value plot showing \(V\) as a function of \(\alpha\). An example is provided in Fig. \ref{fig:maxval} (a), applied to simulation $A_0$ considering the event \(E:\omega \geq 0\). The forecast value curves are plotted for a range of probabilities as decision criterion, then the optimal values for each \(\alpha\)-user (the upper envelope of the relative value curves) is selected to represent the value of the probabilistic forecast system \citep[e.g.][]{richardson2000,wilks2001}. The probability value plot is related to the ROC framework since the pairs $(F,H)$ of Eq. \eqref{equ:Vdef2} are the ones used to draw the ROC curve. It has also been shown that the overall value of a probability forecast, considering all potential users, corresponds to the Brier skill score of the forecast if the distribution of cost-loss ratio is uniform over all users \citep{murphy69,Richardson2011}.
\subsection{Quantile value plot }
Applied to a quantile forecast, so focusing on a \(\alpha\)-user, the value score is evaluated for a range of events of interest defined for example by their base rate \(\pi\). A new tool is therefore proposed for the assessment of quantile forecast performance:
the quantile value plot which represents how \(V\) varies as a function of \(\pi\). This is illustrated in Figure \ref{fig:maxval} (b). The value of the \(30\%\)-quantile forecasts is plotted when the quantile forecasts derived from simulations \emph{\(A_0\)}, \emph{\(A_1\)}, \emph{\(A_2\)}, and \emph{\(B\)} are taken at face value.
Taking a quantile at face value means using it as it is, so for each event it implies considering the event threshold as decision criterion (see Section \ref{sec:costmodel}). An alternative is to apply the critical decision criteria, i.e. to use the $(F,H)$ pairs from the RUC curve to estimate the value in Eq. \eqref{equ:Vdef2}. We talk then about \textit{potential} value since it corresponds to the maximum value of the forecast, i.e. the maximum that could be potentially reached if an adequate calibration is applied to the forecast. Indeed, value and \textit{potential} value are by definition identical if the forecast is reliable.
A parallel between probability value plot and quantile value plot can be draw. In a probability value plot, the decision variable is a probability forecast, the base rate \(\pi\) of the event under focus is fixed and the forecast value \(V\) is then plotted for a range of cost-loss ratios. The role of \(\alpha \) and \(\pi\) are inverted in order to produce a quantile value plot rather than a probability value plot. The cost-loss ratio is defined by the quantile probability level and a range of events of interest are scanned. It results that
the cost-loss ratio of the end-user does not appear explicitly in a quantile value plot as is the case for the value plot for probability forecasts.
The fundamental properties of \(V\) are however the same when focusing on one event or on one user. These properties \cite[demonstrations can be found e.g. in][]{Richardson2011} are recalled here. First, the forecast value reaches its maximum when \(\pi=\alpha\) (or noted differently when \(\pi=1-\tau\)). For instance, a forecast user with a cost-loss ratio of \(\alpha = 0.1\) draws a maximum benefit from a forecast if his (her) event of interest has a climatological probability of occurrence of \(10\%\). Secondly, the value of a reliable forecasts (full line in Figure \ref{fig:maxval} (b)) is always greater than the value of the same forecast with biases (dashed and dotted lines in Figure \ref{fig:maxval} (b)). The value of the reliable forecast corresponds to the potential value of the two other datasets. Finally, the potential value is by definition always positive.
\subsection{A real example}
\label{subsec:cdeps}
The tools introduced for the assessment of quantile forecast discrimination and value are here applied to a real dataset. Quantile forecasts of global radiation are derived from COSMO-DE-EPS and assessed for two periods of the year 2013. Results for the winter period are shown in Figure \ref{fig:cdeps1} and results for the summer period in Figure \ref{fig:cdeps2}. Quantile discrimination is estimated with the area under the RUC curve (\(AUC^\prime\)) for probability levels \(\tau=0.1,0.2,...,0.9\). A deeper analysis is performed for the 10\%-, 50\%- and 90\%-quantile forecasts with the help of quantile value plots.
The discrimination ability of the EPS quantile forecasts varies as a function of the probability level but is greater than 0.80 which can be interpreted as good performance. For the winter season, discrimination is higher for high and low probability levels than intermediate ones whereas for the summer season, discrimination is approximately constant over the probability levels with a tendency to decrease for high levels.
Inspection of the quantile value plot allows a deeper insight into the forecast potential performance. This could be relevant for quantile users with a specific interest in only one part of the event spectrum. The potential value of the quantile forecasts is plotted as a function of event in terms of the clearness index in \% to simplify the reading of the plots. However, an event has a different base rate for each season which complicates a direct comparison of the quantile value plot in Figures \ref{fig:cdeps1} and \ref{fig:cdeps2}.
\subsection{Overall value and Quantile Skill Score}
As a final step in drawing a parallel between probability forecast verification and quantile forecast verification, the relationship between value and skill score with climatology as a reference is explored. It has been shown that the overall value of a probability forecast is equivalent to its Brier Skill Score (BSS) when the users have a uniform distribution of cost-loss ratio \citep{murphy69,Richardson2011}. Similarly, we now investigate the relationship between the overall value of a quantile forecast
and its QSS.
For this purpose, we extend the cost-loss model to more than two observation categories assuming that the cost \(C\) and the loss \(L\) of the cost-loss model are the unitary increment of cost and loss per unit of variable, respectively, as discussed in Section \ref{subsec:user}.
Following \cite{Richardson2011}, the overall value is defined as the ratio
\begin{equation}
V_{all}= \dfrac{T_{C}- T_{F}} {T_{C}- T_{P}}
\label{equ:Vall}
\end{equation}
where the total mean expense $T$ of a user is estimated when decisions are based on a climatological forecast ($T_C$), on a perfect deterministic forecast ($T_P$) or on a given forecast ($T_F$) so that Eq. \eqref{equ:Vall} is the extension of Eq. \eqref{equ:Vdef} to all possible events.
The total expense for a perfect deterministic forecast corresponds to the sum of the costs \(C\) associated with each observation. The total mean expense \(T_P\) can then be expressed as
\begin{equation}
T_P= \dfrac{1}{N} \sum_{i=1}^{N} C \Omega_i .
\label{equ:TP}
\end{equation}
For a climatological quantile forecast \(\Omega_\tau\), the total expense corresponds to the sum of the costs associated with \(\Omega_\tau\) and the losses encountered when the observations are greater than the climatological forecast (\(\Omega_i\geq \Omega_\tau\)). The total mean expense for a climatological forecast \(T_C\) is written as
\begin{equation}
T_C= \dfrac{1}{N} \sum_{i=1}^{N} C \Omega_\tau +
\dfrac{1}{N} \sum_{i:\Omega_i\geq \Omega_\tau} L (\Omega_i- \Omega_\tau ) .
\label{equ:TC}
\end{equation}
Considering now a sample of quantile forecasts \(q_{\tau,i}\) and the corresponding observations \(\Omega_i\), the total expense of a forecasts user corresponds in that case to the sum of the costs associated with each forecast \(q_{\tau,i}\) and the losses encountered when \(\Omega_i \geq q_{\tau,i}\), given by
\begin{equation}
T_F = \dfrac{1}{N} \sum_{i=1}^N C q_{\tau,i} +
\dfrac{1}{N} \sum_{i:\Omega_i \geq q_{\tau,i}} L (\Omega_i - q_{\tau,i}) .
\label{equ:TF}
\end{equation}
Combining Eqs \eqref{equ:TP}-\eqref{equ:TF}, it is shown in the Appendix that the overall value \(V_{all}\) corresponds to QSS (Eq. \eqref{equ:QSS}) with the climatology as a reference based on the assumption of constant cost-loss ratio for all outcomes.
In other words, extending the dichotomous event-action framework to a continuous framework allows one to turn back to the `classical` or `natural` measure of performance for quantile forecast. Conversely, using the dichotomous framework provides the keys to making a deeper analysis of the quantile performance at the event level.
\begin{figure*}[htb]
\centering
\includegraphics[width=17cm]{./fig_scores_simuCDEPS1.eps}
\caption{ Verification results for COSMO-DE-EPS global radiation forecasts during winter 2012/2013:
quantile discrimination ability (\(AUC^\prime\)) as a function of the probability level (a),
potential value of the 10\%-quantile forecast (b), 50\%-quantile forecast (c) and 50\%-quantile forecast (d) as a function of the event of interest defined by thresholds of the clearness index in \%.
}
\label{fig:cdeps1}
\end{figure*}
\begin{figure*}[htb]
\centering
\includegraphics[width=17cm]{./fig_scores_simuCDEPS2.eps}
\caption{
Same as Figure \ref{fig:cdeps1} but for summer 2013.
}
\label{fig:cdeps2}
\end{figure*}
\section{Conclusion}
\label{sec:conc}
Verification measures and tools related to users' decision-making are provided here for quantile forecasts as decision variables. Drawing a parallel with the verification of probability forecasts, the new verification tools allow the scuite of verification methods for quantile forecasts to be completed. In particular, the concepts of forecast discrimination and forecast value are discussed based on a simple cost-loss model.
First, the RUC curve is shown to be the counterpart of the ROC curve when the focus is on a given user rather than on a given event. The areas under the RUC and ROC curves are summary measures of discrimination adapted to quantile and probability forecasts, respectively. Both measures share the same properties, such as non-sensitivity to calibration.
Second, the translation of discrimination ability into value is explored with the help of the value score. The definition of the forecast value is directly adopted from the probability forecast verification framework. Forecast value and forecast potential value are estimated when the decision variable is a quantile forecast, so focusing on a user with a specific cost-loss ratio. The first is obtained when the forecast is taken at face value and the second when critical decision criteria are applied. The value of a quantile forecast can then be plotted as a function of a range of events of interest, defined for example in terms of base rates. The derived plot is called a quantile value plot and provides a valuable insight into the performance of a quantile forecast. As a real example, the discrimination ability and value of global radiation forecasts from COSMO-DE-EPS are demonstrated over a summer and a winter period.
Finally, it is shown that the overall value of a quantile forecast corresponds to the quantile skill score with climatology as reference when a constant cost-loss ratio for all outcomes is assumed. In the same spirit as the weighted version of the continuous ranked probability score proposed by \cite{ranjan2011}, a weighted version of the quantile skill score could be envisaged in order to take into account specific use of quantile forecasts.
\section*{Acknowledgement}
This work has been initiated in the framework of the EWeLiNE project (\textit{Erstellung innovativer Wetter- und Leistungsprognosemodelle f\"ur die Netzintegration wetterabh\"angiger Energietr\"ager}) funded by the German Federal Ministry for Economic Affairs and Energy.
\section*{Appendix}
\label{app:qss}
\subsection*{Overall value and Quantile Skill Score}
From Eqs \eqref{equ:TP} and \eqref{equ:TC}, the difference in expense between climatological and perfect deterministic forecasts can be written as
\begin{equation}
\begin{split}
T_C - T_P= \dfrac{1}{N} \sum_{i=1}^{N} C (\Omega_\tau - \Omega_i) +
\dfrac{1}{N} \sum_{i:\Omega_i \geq \Omega_\tau} L (\Omega_i- \Omega_\tau)
\label{equ:num}
\end{split}
\end{equation}
Considering the relationship $\tau= 1-\dfrac{C}{L}$ and setting \(L\) equal to 1 in the following demonstration without loss of generality, we obtain
\begin{equation}
T_C - T_P= \dfrac{(1-\tau)}{N} \sum_{i=1}^{N} (\Omega_\tau - \Omega_i) +
\dfrac{1}{N} \sum_{i:\Omega_i \geq \Omega_\tau} (\Omega_i-\Omega_\tau)
\end{equation}
and with some algebra
\begin{equation}
\begin{split}
T_C - T_P= \dfrac{(1-\tau)}{N} \sum_{i:\Omega_i\leq \Omega_\tau} (\Omega_\tau-\Omega_i) +
\dfrac{\tau}{N} \sum_{i:\Omega_i \geq \Omega_\tau} (\Omega_i- \Omega_\tau)
\end{split}
\label{equ:tctp}
\end{equation}
This mean expense difference, $T_C - T_P$, corresponds to the definition of the quantile score for a climatological forecast (\(QS_\text{climate}\)).
In the same manner, from Eqs \eqref{equ:TF} and \eqref{equ:TC}, the difference between climatological forecast expense and the quantile forecast expense is written as
\begin{equation}
\begin{split}
T_C - T_F = \dfrac{1}{N} \sum_{i=1}^{N} C \Omega_\tau +
\dfrac{1}{N} \sum_{i:\Omega_i \geq \Omega_\tau} L (\Omega_i -\Omega_\tau) \\
- \dfrac{1}{N} \sum_{i=1}^N C q_{\tau,i}
- \dfrac{1}{N} \sum_{i:\Omega_i \geq q_{\tau,i}} L (\Omega_i- q_{\tau,i})
\end{split}
\end{equation}
which becomes after some algebra
\begin{equation}
\begin{split}
T_C - T_F =
\dfrac{(1-\tau)}{N} \sum_{i:\Omega_i\leq \Omega_\tau} (\Omega_\tau - \Omega_i) +
\dfrac{\tau}{N} \sum_{i:\Omega_i \geq \Omega_\tau} (\Omega_i -\Omega_\tau) \\
- \bigg( \dfrac{(1-\tau)}{N} \sum_{i:\Omega_i\leq q_{\tau,i}} (q_{\tau,i} - \Omega_i) +
\dfrac{\tau}{N} \sum_{i:\Omega_i \geq q_{\tau,i}} (\Omega_i- q_{\tau,i}) \bigg)
\end{split}
\label{equ:tctf}
\end{equation}
where the first term corresponds to the definition of the quantile score for a climatological forecast (\(QS_\text{climate}\), Eq. \eqref{equ:tctp}), and the second term to the quantile score (\(QS_\text{forecast}\), Eq. \eqref{equ:QSf}). With regard to the definition of the quantile skill score and of the overall value (Eqs \eqref{equ:QSS} and \eqref{equ:Vall}, respectively), we end up with:
\begin{equation}
V_{all}= QSS
\label{equ:vall}
\end{equation}
|
3,212,635,537,926 | arxiv | \section{Introduction}
When propagation in a coupled-waveguide system is described in terms
of a system matrix, exceptional points of degeneracy are points in
the parameter space of such a system at which simultaneous eigenvalue
and eigenvector degeneracies occur \cite{Heiss}. Interest in EPDs
has recently risen due to Parity-Time (PT) symmetric systems, wherein
non-Hermitian Hamiltonians can nevertheless exhibit real spectra,
representing physical observables. PT-symmetry has led to a range
of interesting phenomena in quantum mechanics and photonic systems
\cite{Bender1,Bender2,Ruter,Hodaei,Othman1,Hassan}, and in metamaterials
research \cite{Sounas,Monticone1,Monticone2,Chen1}, with applications
to cloaking, negative refraction, imaging, field transformation, and
sensing, among others. In a system whose evolution is described with
a system matrix, EPDs are associated with a Jordan block, corresponding
to a deficient (incomplete) set of eigenfunctions, and algebraically
growing solutions of generalized (associated) eigenvectors at the
EPD. Moreover, in the vicinity of EPDs, by virtue of small detuning,
eigenvalues exhibit unconventional perturbations following a fractional
power-law expansion in the perturbation parameters \cite{kato}.
It is important to point out that EPDs are manifest in the parameter
space of a system's eigenstates' temporal evolution (e.g., such as
certain coupled resonators with loss and gain), or of a system's eigenstates'
spatial evolution. This latter case represents the evolution of eigenwaves
in a given spatial direction, such as in a multimode waveguide with
prescribed loss and gain, which is investigated in this paper, where
the multimode waveguide is a pair of uniform coupled transmission
lines. Some of the earliest examples of EPDs have been also observed
in structures with spatial periodicity which are explored, for instance,
in \cite{FV1,FV2,OC,VS}, such as those exhibiting degenerate band
edges or stationary inflection points. Although EPDs are usually viewed
from a linear algebra standpoint, and are associated with systems
described by matrices with Jordan blocks \cite{FV1,Heiss}, it has
been observed that they also represent points in configuration space
where multiple branches of spectra connect, and are linked to branch
points in the space of control variables \cite{Hern,Hern2}.
In this work, we consider a coupled uniform transmission-line system,
recently examined in \cite{EPD}, and demonstrate several new aspects
of EPDs in these systems. Specifically, we stress that for a coupled
uniform transmission line, eigenvalue degeneracies always result in
eigenvector degeneracies, such that all eigenvalue degeneracies represent
EPDs. We derive closed-form expressions for the branch-point singularities/EPDs
using bifurcation theory. We show and discuss in detail the connection
of EPDs with previous work on fold-point and branch-point singularities
in waveguiding systems \cite{A1,A2,A3,A4,A6,A7,A11,A12,A14} associated
with mode degeneracies and mode interactions, which provides a complementary
viewpoint for understanding EPDs.
\section{Coupled Transmission-Line Formulation\label{sec:Coupled-Transmission-Line}}
We consider two uniform coupled transmission lines (CTLs) as depicted
in Fig. 1.
\begin{figure}[!th]
\noindent \begin{centering}
\includegraphics[width=3in]{Geometry}
\par\end{centering}
\caption{Two coupled transmission lines with mutual capacitive and inductive
coupling, invariant along $z$. They exhibit EPDs under certain conditions
described in the paper.}
\label{fig1}
\end{figure}
We refer to the formulation given in \cite{EPD} for the analysis
of eigenwaves propagating along the $z$-direction in a CTL (the $e^{i\omega t}$
time-harmonic evolution is implicitly assumed). Here we summarize
the mathematical steps carried out to obtain the eigenwaves supported
by such a guiding system. The CTL equations for a two-line network
consisting of uniform transmission lines are given by the telegraphers
equations \cite{Paul,MM}
\begin{equation}
\frac{d\mathbf{V}\left(z\right)}{dz}=-\mathbf{\underline{\underline{\mathbf{Z}}}\ I}\left(z\right),\ \ \frac{d\mathbf{I}\left(z\right)}{dz}=-\mathbf{\underline{\underline{\mathbf{Y}}}\ V}\left(z\right)\label{GE}
\end{equation}
where the voltage and current are 2-dimensional vectors, $\mathbf{V}(z)=\left[V_{1}\left(z\right)\ \ V_{2}\left(z\right)\right]^{\text{T}}$
and $\mathbf{I}(z)=\left[I_{1}\left(z\right)\ \ I_{2}\left(z\right)\right]^{\text{T}}$,
whereas $\underline{\underline{\mathbf{Z}}}$ and $\underline{\underline{\mathbf{Y}}}$
are $2\times2$ matrices,
\begin{equation}
\underline{\underline{\mathbf{Z}}}\left(\omega\right)=\left[\begin{array}{cc}
Z_{11} & Z_{12}\\
Z_{21} & Z_{22}
\end{array}\right],\ \ \underline{\underline{\mathbf{Y}}}\left(\omega\right)=\left[\begin{array}{cc}
Y_{11} & Y_{12}\\
Y_{21} & Y_{22}
\end{array}\right],
\end{equation}
where the off-diagonal elements represent coupling between the two
transmission lines. Furthermore, the per-unit-length series impedance
and shunt admittance matrices are given by $\underline{\underline{\mathbf{Z}}}=i\omega\underline{\underline{\mathbf{L}}}+\underline{\underline{\mathbf{R}}}$
and $\underline{\underline{\mathbf{Y}}}=i\omega\mathbf{\underline{\underline{C}}}+\mathbf{\underline{\underline{G}}}$,
where $\mathbf{\underline{\underline{\mathbf{R}}},\underline{\underline{G}},\underline{\underline{\mathbf{L}}},\mathbf{}}$and
$\underline{\underline{\mathbf{C}}}$ are matrices of the per-unit-length
distributed CTL parameters, assumed nondispersive for simplicity.
The matrices $\underline{\underline{\mathbf{L}}}$ and $\mathbf{\mathbf{\underline{\underline{C}}}}$
are positive definite and symmetric \cite{Paul,MM}, and the off-diagonal
entries of $\mathbf{\mathbf{\underline{\underline{C}}}}$ and $\underline{\mathbf{\underline{G}}}$
are negative. In general $\underline{\underline{\mathbf{R}}}$ and
$\underline{\mathbf{\underline{G}}}$ are positive definite if they
represent losses (no gain), and in the following they are assumed
to be diagonal for simplicity. In addition, note also that the per-unit-length
impedance and admittance matrices may possess cutoff capacitance and
inductance terms, respectively, as done in Ch. 7 in \cite{H1961},
and also in \cite{OTper1} to model waveguide cutoff. Since we do
not investigate cutoff related degeneracies, we simply ignore these
terms in the CTL formulations above.
\subsection{EPD from a Linear Algebra Perspective}
Decoupling (\ref{GE}), we obtain two second-order wave equations
for the voltage and current vectors
\begin{equation}
\frac{d^{2}\mathbf{V}\left(z\right)}{dz^{2}}=\mathbf{\underline{\underline{\mathbf{Z}}}\,\underline{\underline{\mathbf{Y}}}\ V}\left(z\right),\ \ \frac{d^{2}\mathbf{I}\left(z\right)}{dz^{2}}=\mathbf{\underline{\underline{\mathbf{Y}}}\,\underline{\underline{\mathbf{Z}}}\ I}\left(z\right).\label{2d}
\end{equation}
The two systems lead to the same wavenumber solutions though in general,
$\underline{\underline{\mathbf{Z}}}$ and $\underline{\underline{\mathbf{Y}}}$
do not necessarily commute; one common exception is for lossless lines
in a homogeneous environment characterized by $\mu,\varepsilon$,
in which case $\underline{\underline{\mathbf{Z}}}\,\underline{\underline{\mathbf{Y}}}=-\omega^{2}\mu\varepsilon\mathbf{\underline{\underline{1}}}$,
where $\underline{\underline{\mathbf{1}}}$ is the $2\times2$ identity
matrix. Alternatively, one may form a four-dimensional state vector
$\mathbf{\Psi}\left(z\right)=\left[V_{1}\left(z\right)\ \ V_{2}\left(z\right)\ \ I_{1}\left(z\right)\ \ I_{2}\left(z\right)\right]^{\text{T}}$,
leading to
\begin{equation}
\frac{d}{dz}\mathbf{\Psi}\left(z\right)=-i\mathbf{\underline{M}}\left(\omega\right)\mathbf{\Psi}\left(z\right)\label{evolEq}
\end{equation}
where
\begin{equation}
\mathbf{\underline{M}}\left(\omega\right)=\left[\begin{array}{cc}
\mathbf{\underline{\underline{0}}} & -i\underline{\underline{\mathbf{Z}}}\\
-i\underline{\underline{\mathbf{Y}}} & \underline{\underline{\mathbf{0}}}
\end{array}\right].\label{M}
\end{equation}
Assuming that the transmission line is invariant along $z$, the homogeneous
solutions to (\ref{2d}) and (\ref{evolEq}) are found to be in the
form $\mathbf{\Psi}\left(z\right)\propto e^{-ikz}$ with $k$ being
the wavenumber. As such, (\ref{2d}) and (\ref{evolEq}) become
\begin{align}
-\left(\mathbf{\underline{\underline{\mathbf{Z}}}\,\underline{\underline{\mathbf{Y}}}}\right)\left(\omega\right)\mathbf{\ V}\left(z\right) & =k^{2}\mathbf{V}\left(z\right),\label{evp}\\
-\left(\underline{\underline{\mathbf{Y}}}\,\underline{\underline{\mathbf{Z}}}\right)\left(\omega\right)\mathbf{\ I}\left(z\right) & =k^{2}\mathbf{I}\left(z\right),\nonumber \\
\ \ \mathbf{\underline{M}}\left(\omega\right)\mathbf{\Psi}\left(z\right) & =k\mathbf{\Psi}\left(z\right).\nonumber
\end{align}
Note that the first two equations in (\ref{evp}) have two eigenvalues
$k^{2}$ (and both signs of $k$ are possible), whereas the third
equation in (\ref{evp}) has four eigenvalues $k$. All three eigenvalue
problems lead to the same four eigenvalues, and encompass the same
physics, which is thoroughly explained in \cite{EPD}. Here, we wish
to make several new observations about these eigenproblems from two
different but complementary perspectives, which opens up new ways
for utilizing such EPDs and conceiving new operational principles
for a variety of microwave devices. For simplicity, we assume reciprocity,
i.e., $Y_{21}=Y_{12}$ and $Z_{21}=Z_{12}$.
We denote the algebraic multiplicity for eigenvalues $\lambda$ (i.e.,
the order of the eigenvalue degeneracy) as $m(\lambda)$. The geometric
multiplicity of the eigenvalue (the span of the eigenvector space
associated with the eigenvalue) is denoted as $l(\lambda)$. We make
the following observations related to EPDs:
\begin{enumerate}
\item For the systems of CTLs considered above, when an EPD occurs one has
$m\left(\lambda\right)>l\left(\lambda\right)$, i.e., all degenerate
eigenvalues have a deficient eigenspace, and the matrices $\mathbf{\underline{M},\underline{\underline{\mathbf{Z}}}\,\underline{\underline{\mathbf{Y}}}},\underline{\underline{\mathbf{Y}}}\,\underline{\underline{\mathbf{Z}}}$
\textit{cannot} be diagonalized (except for the trivial degeneracy
at $k$=0 and in uncoupled lines). In particular, for the two uniform
CTLs considered here, EPDs are associated with $l\left(\lambda\right)=2$,
and $m\left(\lambda\right)=1$.
\item EPDs imply the presence of square-root branch points in the complex-frequency
plane. As such, these complex-frequency plane singularities are generally
unavailable for monochromatic problems, but may be accessed in certain
pulse shaping scenarios \cite{AA,CB,CB2}.
\item The analysis of EPD from a linear algebra perspective can analogously
be studied as fold singularities of mappings in bifurcation theory.
\item PT-symmetric conditions lead to EPDs on the real-frequency axis, and,
thus, to physically observable phenomena in monochromatic problems.
\end{enumerate}
\bigskip{}
In the following, we examine the aforementioned statements and provide
analytical expressions for the eigenvalues and eigenvectors to reveal
the origin of EPDs and their relation to eigenvalue and eigenvector
degeneracies and branch points. In Section \ref{subsec:EPD-from-a}
we examine EPDs from a different prospective, that of bifurcation
theory.
We first consider the $2\,\textrm{\ensuremath{\times}}\,2$ eigenvalue
problem in (\ref{evp}); $-\mathbf{\underline{\underline{\mathbf{Z}}}\,\underline{\underline{\mathbf{Y}}}}$
having eigenvalues $k_{1,2}^{2}$ and $regular$ voltage eigenvectors
$\mathbf{V}_{1,2}$, obtained analytically as
\begin{equation}
k_{n}^{2}=\frac{1}{2}\left(-T+(-1)^{n}D\right),\ \ \mathbf{V}_{n}=\left[\begin{array}{c}
-\frac{1}{2N_{1}}\left(N_{2}+(-1)^{n}D\right)\\
1
\end{array}\right]\label{eq:k1,2_V1,2}
\end{equation}
where $n=1,2$ , then $N_{1}=Y_{11}Z_{12}+Y_{12}Z_{22}$, $N_{2}=-Y_{11}Z_{11}+Y_{22}Z_{22}$
and
\begin{equation}
D=\sqrt{T^{2}-4\text{det}\left(\mathbf{\underline{\underline{\mathbf{Z}}}\,\underline{\underline{\mathbf{Y}}}}\right)}.\label{D}
\end{equation}
The trace $T$ and determinant of $\underline{\underline{\mathbf{Z}}}\,\underline{\underline{\mathbf{Y}}}$
are given by
\begin{align}
T & =\text{Tr}\left(\mathbf{\underline{\underline{\mathbf{Z}}}\,\underline{\underline{\mathbf{Y}}}}\right)=2Y_{12}Z_{12}+Y_{22}Z_{22}+Y_{11}Z_{11},\label{T}\\
\text{det}\left(\mathbf{\underline{\underline{\mathbf{Z}}}\,\underline{\underline{\mathbf{Y}}}}\right) & =\left(Y_{11}Y_{22}-Y_{12}^{2}\right)\left(Z_{11}Z_{22}-Z_{12}^{2}\right).\label{Trace}
\end{align}
For the $-\underline{\underline{\mathbf{Y}}}\,\underline{\underline{\mathbf{Z}}}$
formulation in (\ref{evp}), everything is analogous; the same eigenvalues
are obtained, and the regular current $\mathbf{I}_{1,2}$ eigenvectors
are retrieved using (\ref{eq:k1,2_V1,2}) by replacing $N_{1}\rightarrow Y_{22}Z_{12}+Y_{12}Z_{11}$.
It is obvious that, without considering the trivial eigenvalue degeneracy
at $k=0$, eigenvalue degeneracies occur when $D=0$, and, moreover,
from (\ref{eq:k1,2_V1,2}) it is clear that at this point eigenvectors
are also degenerate; $m\left(k^{2}\right)=2$ and $l\left(k^{2}\right)=1$
since $\mathbf{V}_{1}=\mathbf{V}_{2}$.
For the formulation in (\ref{evp}) involving the $4\,\textrm{\ensuremath{\times}}\,4$
matrix $\mathbf{\underline{M}}$, one finds the four eigenvalues and
$regular$ eigenvectors as
\begin{align}
k_{n} & =\left(\pm\right)\frac{1}{\sqrt{2}}\sqrt{-T+\nu_{n}D},\ \ \label{eq:1psi}\\
\mathbf{\Psi}_{n} & \mathbf{=}\left[\begin{array}{c}
\left(\pm\right)i\frac{\sqrt{-T+\nu_{n}D}}{\sqrt{2}}\frac{-N_{2}-\nu_{n}D}{N_{3}-\nu_{n}YD}\\
\left(\pm\right)i\frac{2\sqrt{-T+\nu_{n}D}}{\sqrt{2}}\frac{N_{1}}{N_{3}-\nu_{n}Y_{12}D}\\
\frac{\left(-N_{2}-\nu_{n}D\right)Y_{11}+2Y_{12}N_{1}}{N_{3}-\nu_{n}Y_{12}D}\\
1
\end{array}\right],\label{kPsi}
\end{align}
where the $+$ sign in front is for $n=1,2$ , the $-$ sign in front
is for $n=3,4$ , $\nu_{n}=(-1)^{n}$, $N_{3}=Y_{11}\left(Y_{12}Z_{11}+2Y_{22}Z_{12}\right)+Y_{12}Y_{22}Z_{22}$,
and again both eigenvalues and eigenvectors become simultaneously
degenerate when $D=0$, and $m_{\mathbf{}}\left(\pm k\right)=2>l\left(\pm k\right)=1$.
Therefore, excepting the case of uncoupled identical lines \cite{FN}
and $k=0$, for all system descriptions in (\ref{evp}) eigenvector
degeneracies are simultaneous with eigenvalue degeneracies. Thus,
these simultaneous eigenvalue and eigenvector degeneracies are, by
definition, an EPD, where $k=\pm k_{e}$ with $k\equiv\sqrt{-T}/\sqrt{2}$.
Indeed at such points the matrices in (\ref{evp}) are deficient and
cannot be diagonalized because there are not enough eigenvectors to
form a complete basis. This proves Item 1 above. From the above analysis,
Item 2 is also demonstrated, since $D=D\left(\omega\right)$ clearly
represents a square-root type branch point in the complex-$\omega$
plane.
Conditions for EPDs were also presented in \cite{EPD}; here we briefly
comment on those and the connection with the condition $D=0$. In
\cite{EPD}, it was shown that the conditions
\begin{align}
& T=\text{Tr}\left(\mathbf{\underline{\underline{\mathbf{Z}}}\,\underline{\underline{\mathbf{Y}}}}\right)=-2k^{2},\label{tr}\\
& \text{det}\left(\mathbf{\underline{\underline{\mathbf{Z}}}\,\underline{\underline{\mathbf{Y}}}}\right)=k^{4},\label{det3}
\end{align}
are necessary for an eigenvalue degeneracy (and so, in fact, are necessary
and sufficient for an EPD as described previously, excepting $k=0$
and uncoupled lines). These two conditions combined yield det$\left(\mathbf{\underline{\underline{\mathbf{Z}}}\,\underline{\underline{\mathbf{Y}}}}\right)=T^{2}/4$,
which is the condition under which $D=0$.
Furthermore, when, e.g., $\mathbf{\underline{M}}$ is similar to a
diagonal matrix (away from the EPD) it can be written in the form
\begin{equation}
\mathbf{\underline{M}}=\mathbf{\underline{U}}\,\mathbf{\underline{\Lambda}}\,\mathbf{\underline{U}}^{-1}\label{SimM}
\end{equation}
\noindent where $\mathbf{\underline{U}}\,$ is a $4\,\textrm{\ensuremath{\times}}\,4$
matrix representing the similarity transformation of $\mathbf{\underline{M}}$
that brings it to a diagonal form, and $\mathbf{\underline{\Lambda}}$
is a diagonal matrix whose diagonal entries are the eigenvalues $k_{n}$
in (\ref{eq:1psi}). It was shown in \cite{EPD} that the condition
$\mathrm{det}\left(\mathbf{\underline{U}}\right)=0$ provides necessary
and sufficient conditions for an eigenvector degeneracy (at which
point the regular eigenvectors must be augmented with associated eigenvectors,
and, rather than a diagonal form, the simplest matrix representation
is given by the Jordan canonical form \cite{LT}). Forming\textit{\ }
\begin{equation}
\det\left(\mathbf{\underline{U}}\right)=-16\frac{Y_{11}}{N_{2}^{3}}D^{2}\left(Y_{12}Z_{22}+Y_{11}Z_{12}\right)\sqrt{T^{2}-D^{2}}=0\label{U}
\end{equation}
it is observed that $\mathrm{det}\left(\mathbf{\underline{U}}\right)=0$
occurs when $D=0$ (or when $Y_{12}Z_{22}+Y_{11}Z_{12}=0$, which
seems to not be of practical interest, and note that $D=T$ cannot
be true since, using (\ref{D}), it would hold only if det$\left(\mathbf{\underline{\underline{\mathbf{Z}}}\,\underline{\underline{\mathbf{Y}}}}\right)=0$,
which is not true). Alternatively, assuming a similarity transformation
analogous to that in (\ref{SimM}) but that diagonalizes the 2 $\times$
2 matrix $-\mathbf{\underline{\underline{\mathbf{Z}}}\,\underline{\underline{\mathbf{Y}}}}$,
\begin{equation}
\det\left(\mathbf{\underline{\underline{\mathbf{U}}}}\right)=-\frac{D}{Y_{11}Z_{12}+Y_{12}Z_{22}}
\end{equation}
which again occurs at $D=0$. Therefore, the previously stated conditions
in \cite{EPD} are, for uniform CTLs modeled by nondispersive $\mathbf{\underline{\underline{\mathbf{R}}},\underline{\underline{G}},\underline{\underline{\mathbf{L}}},\mathbf{}}$and
$\underline{\underline{\mathbf{C}}}$ parameters, alternative ways
of stating the $D=0$ EPD condition.
\textbf{\textcolor{black}{Puiseux series}}\textcolor{black}{. In what
follows, it will be useful to cast the eigenvalue problems (\ref{evp})
in the form
\begin{equation}
H\left(k,\omega\right)=\mathrm{det}\left(\mathbf{A}\left(\omega,\bm{\xi}\right)-k\mathbf{1}\right)=0\label{eq:H}
\end{equation}
where $\bm{\xi}$ is the vector of geometrical and material parameters
of the system, and $\mathbf{1}$ is the identity matrix. In particular,
in the following, all the partial derivatives in $\omega$ could be
substituted with partial derivatives in $\bm{\xi}$ and analogous
conclusions would be reached relative to the dispersion diagram $\left(k,\bm{\xi}\right)$
and associated BPs. In (\ref{eq:H}), the matrix $\mathbf{A}$ represents
either the 2$\times$2 system for which $\mathbf{\underline{\underline{\mathbf{A}}}=-\underline{\underline{\mathbf{Z}}}\,\underline{\underline{\mathbf{Y}}}}$,
or $\underline{\underline{\mathbf{A}}}=-\mathbf{\underline{\underline{\mathbf{Y}}}\,\underline{\underline{\mathbf{Z}}}}$
(in which case the eigenvalue is $k^{2}$ rather than $k$) or the
4$\times$4 system $\mathbf{\underline{\mathbf{A}}=\underline{M}}$.
In the following we suppress the dependence on $\bm{\xi}$. The condition
(\ref{eq:H}) leads to }
\begin{equation}
k^{4}+k^{2}\text{Tr}\left(\underline{\underline{\mathbf{A}}}\right)+\text{det}\left(\underline{\underline{\mathbf{A}}}\right)=0,\label{HE0}
\end{equation}
which is also given in \cite{EPD}. \textcolor{black}{Denoting derivatives
as
\begin{equation}
H_{\varsigma}^{(m)}\left(k_{e},\omega_{e}\right)=\left.\frac{\partial^{(m)}H(k,\omega)}{\partial\varsigma^{m}}\right|_{(k_{e},\omega_{e})},
\end{equation}
for $\varsigma=k,\omega$, an }\textit{\textcolor{black}{m}}\textcolor{black}{th-order
eigenvalue degeneracy (i.e., an }\textit{\textcolor{black}{m}}\textcolor{black}{th-order
root of $H\left(k,\omega\right)$) will satisfy
\begin{equation}
H(k_{e},\omega_{e})=H_{k}^{\prime}(k_{e},\omega_{e})=...=H_{k}^{(m-1)}(k_{e},\omega_{e})=0,\label{eq:H1}
\end{equation}
\begin{equation}
H_{k}^{(m)}\left(k_{e},\omega_{e}\right)\neq0,\label{H2}
\end{equation}
}
\noindent \textcolor{black}{where $k_{e}$ is the degenerate wavenumber
and $\omega_{e}$ is the frequency at which the wavenumbers become
degenerate. }For a second-order EPD, the condition $H_{k}^{\prime}(k,\omega)=0$
is
\begin{equation}
k\left(T+2k^{2}\right)=0,
\end{equation}
which is equivalent to the trace condition (\ref{tr}) for $k\neq0$,
and leads to $k=\pm\sqrt{-T}/\sqrt{2}$, consistent with the general
eigenvalue at the EPD.\textcolor{black}{{} As described briefly in \cite{EPD}
but of more direct importance here, the eigenvalues of the CTL at
such a degeneracy can be written as a convergent Puiseux series \cite{AM,kato}
\begin{equation}
k_{n}(\omega)=k_{e}+\alpha_{1}\zeta^{n}(\omega-\omega_{e})^{\frac{1}{m}}+{\displaystyle \sum_{p=2}^{\infty}\alpha_{p}}(\zeta^{n}(\omega-\omega_{e})^{\frac{1}{m}})^{p}\label{eq:PS}
\end{equation}
for $n=0,1,2,...,m-1$, where $\zeta=e^{i\frac{2\pi}{m}}$. The first-order
coefficient is given by
\begin{equation}
\alpha_{1}=\left(-\frac{H'_{\omega}\left(k_{e},\omega_{e}\right)}{\frac{1}{m!}H{}_{k}^{(m)}\left(k_{e},\omega_{e}\right)}\right)^{\frac{1}{m}}.\label{eq:alpha_1}
\end{equation}
The Puiseux series is a direct consequence of the Jordan Block form
(see for example page 65 in \cite{kato}) hence it is always relevant
in systems that exhibit an EPD to describe the eigenvalue perturbation
away from the EPD. Applying the fractional power expansion (\ref{eq:PS})
to the 2nd order EPD in the uniform CTL above, and ignoring expansion
terms with order equal or higher than $\omega-\omega_{e}$, one arrives
at}
\noindent \textcolor{black}{
\begin{equation}
k(\omega)\simeq k_{e}\pm\alpha_{1}\sqrt{(\omega-\omega_{e})}+O(\omega-\omega_{e}).\label{eq:EPD_first_terms}
\end{equation}
The first two terms in (\ref{eq:EPD_first_terms}) show the occurrence
of the branch-point singularity in the complex-frequency plane, resulting
from the square-root function. Associated with this series is the
condition \cite{AM}
\begin{equation}
H'_{\omega}\left(k_{e},\omega_{e}\right)\neq0,\label{H3}
\end{equation}
and so the first-order coefficient $\alpha_{1}$ is nonzero. An important
aspect of the Puiseux series is that it provides the characteristic
form of the solution in the vicinity of the EPD, as shown later in
relation to Fig. 2. Regarding Statement 3, the conditions (\ref{eq:H1})-(\ref{H2}),
and (\ref{H3}) will be reconsidered in Section \ref{subsec:EPD-from-a}
from the viewpoint of singularity and bifurcation theory.}
\textbf{Jordan Block and Generalized Eigenvectors. }At an EPD in a
uniform 2-CTL, the eigenvalue degeneracy corresponds to an eigenvector
degeneracy as we have previously discussed. This can be also shown
by noticing that when the eigenvalues of a $2\times2$ system matrix,
as in the first two systems in (\ref{evp}), are identical then it
is either proportional to an identity matrix (hence with two independent
eigenvectors) or otherwise it must be proportional to a $2\times2$
Jordan block (that exhibit the eigenvector degeneracy). For the $4\times4$
system matrix $\mathbf{\underline{M}}$ as in the third system in
(\ref{evp}) the situation is more involved. At an EPD the system
matrix $\mathbf{\underline{M}}$ is similar to a matrix containing
two Jordan blocks as
\begin{equation}
\mathbf{\underline{M}}=\mathbf{\underline{U}}\left[\begin{array}{cc}
\mathbf{\underline{\underline{\mathbf{J}}}_{+}} & \mathbf{\underline{\underline{0}}}\\
\mathbf{\underline{\underline{0}}} & \mathbf{\underline{\underline{\mathbf{J}}}_{-}}
\end{array}\right]\mathbf{\mathbf{\underline{U}}}^{-1},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathbf{\underline{\underline{\mathbf{J}}}_{\pm}}=\left[\begin{array}{cc}
\pm k_{e} & 1\\
0 & \pm k_{e}
\end{array}\right]
\end{equation}
where $\mathbf{\underline{U}}$ is a 4$\times$4 matrix constituting
a similarity transformation and containing the generalized eigenvectors
of $\mathbf{\underline{M}}$ namely $\mathbf{\underline{S}}=\left[\Psi_{1}\,\,|\,\,\Psi_{1}^{g}\,\,|\,\,\Psi_{3}\,\,|\,\,\Psi_{3}^{g}\right]$
that are constructed through the Jordan chain procedure (\cite{EPD},
\cite{Meyer}, see also \cite{HN} for the differential operator case)
\begin{align}
\left(\mathbf{\underline{M}}-k_{e}\mathbf{\underline{1}}\right)\Psi_{1}\, & =0,\,\,\,\,\,\,\,\,\,\,\,\left(\mathbf{\underline{M}}-k_{e}\mathbf{\mathbf{\underline{1}}}\right)\Psi_{1}^{g}\,=\Psi_{1}\\
\left(\mathbf{\underline{M}}+k_{e}\mathbf{\mathbf{\underline{1}}}\right)\Psi_{3}\, & =0,\,\,\,\,\,\,\,\,\,\,\,\left(\mathbf{\underline{M}}+k_{e}\mathbf{\mathbf{\underline{1}}}\right)\Psi_{3}^{g}\,=\Psi_{3}\label{GeneralizedPsi}
\end{align}
with $\Psi_{1}$ and $\Psi_{1}^{g}$ being the regular and generalized
eigenvectors associated with the wavenumber $k_{e}$ at the second-order
EPD, and similarly $\Psi_{3}$ and $\Psi_{3}^{g}$ are the regular
and generalized eigenvectors associated with the wavenumber $-k_{e}$.
We consider the general solution of (\ref{evolEq}) subject to an
initial condition at an arbitrary $z=z_{0}$ given by $\Psi(z_{0})=\Psi_{0}.$
Its general and unique solution is given by
\begin{align*}
\Psi(z) & =\exp(-i\mathbf{\underline{M}}z)\Psi_{0}\\
& =\mathbf{\underline{U}}\left[\begin{array}{cc}
\exp(-i\mathbf{\underline{\underline{\mathbf{J}}}_{+}}z) & \underline{\mathbf{\underline{0}}}\\
\underline{\mathbf{\underline{0}}} & \exp(-i\mathbf{\underline{\underline{\mathbf{J}}}_{+}}z)
\end{array}\right]\mathbf{\underline{U}}^{-1}\Psi_{0},\\
& =\mathbf{\mathbf{\underline{U}}}\left(\left[\begin{array}{cccc}
e^{-ik_{e}z} & -ize^{-ik_{e}z} & 0 & 0\\
0 & e^{-ik_{e}z} & 0 & 0\\
0 & 0 & e^{+ik_{e}z} & ize^{ik_{e}z}\\
0 & 0 & 0 & e^{+ik_{e}z}
\end{array}\right]z\right)\mathbf{\mathbf{\underline{U}}}^{-1}\Psi_{0}
\end{align*}
which provides growing solutions along $z$ as $\Psi(z)\propto ze^{-ik_{e}z}$
discussed in \cite{EPD}.
\subsection{EPD from a Theory of Singular and Bifurcation Points Perspective\label{subsec:EPD-from-a}}
Here, we address Statement 3, and connect the previous analysis with
an entirely different method based on singularity and bifurcation
theory \cite{A9,A10}. We consider the implicit dispersion equation
(\ref{eq:H}), $H(k,\omega)=\mathrm{det}(\mathbf{A}(\omega)-k\mathbf{1})=0.$
Here, $H(k,\omega)$ is more generally understood as a mapping $\mathrm{\mathbb{C^{\mathrm{2}}\rightarrow C}}$,
$H(k,\omega)=z$. Obviously, the modal solutions of interest occur
for $z=0$, although viewing $H$ more generally as a mapping facilitates
the analysis below. For many waveguiding structures one must solve
$H(k,\omega)=0$ numerically, via a complex-plane root search, but
for the CTLs of interest here an explicit solution can be obtained,
$k_{n}(\omega)=\left(\pm\right)\frac{1}{\sqrt{2}}\sqrt{-T+\nu_{n}D},$
as given in (\ref{eq:1psi}).
The mapping $H(k,\omega)=z$ defines a surface in $\mathbb{C\mathrm{^{2}}}$,
and for the simple case of $(k,\omega,z)\in\mathbb{R}$, this is depicted
in Fig. \ref{fig1-1}. The particular case of $H(k,\omega)=0$ defines
a curve (solid line in Fig. \ref{fig1-1}), which is the dispersion
curve of interest, and the smoothness of that curve at a given point
determines important modal properties. In particular, one can define
regular and singular points of the curve associated with certain modal
behavior \cite{A1,A2,A3,A4}. In the following we consider $k$ as
the unknown and $\omega$ as a distinguished parameter, although the
roles can also be reversed.
\begin{figure}[!th]
\noindent \begin{centering}
\includegraphics[width=3in]{RP}
\par\end{centering}
\caption{Depiction of the surface defined by $H(k,\omega)=z$, for $(k,\omega,z)\in\mathbb{R}$.
The surface $H(k,\omega)=z$ may intersect the $(k,\omega)-$plane,
at $H(k,\omega)=0$, resulting in the curved line of intersection
shown that represents a standard dispersion diagram. If $H(k,\omega)=0$
does not have solutions for $(k,\omega)\in\mathbb{R}$, then solutions
can be be found in complex space. }
\label{fig1-1}
\end{figure}
We first define a regular point on the curve $H(k,\omega)=0$ as a
point where $\partial H/\partial k\neq0$. At a regular point the
implicit function theorem \cite{A5} can be used to show that a unique
smooth curve $k=k(\omega)$ exists in the neighborhood of the point.
Except for a finite number of non-regular points (a set of measure
zero), all points of modal dispersion are regular points, wherein
the dispersion curve is smooth and single-valued. It is also worthwhile
to note that differentiation $d/d\omega$ of $H(k,\omega)=0$ leads
to, via the chain rule,
\begin{equation}
\frac{dk}{d\omega}=-\frac{\partial H/\partial\omega}{\partial H/\partial k},\label{eq:T}
\end{equation}
and, therefore, at a regular point the tangent of $k(\omega)$ (related
to the group velocity) is well-defined. However, of particular interest
are the singular points \cite{A4} of the mapping $H$, which ultimately
lead to branch points in the complex-frequency plane \cite{A2,A6}.
The point $(k_{s},\omega_{s})$ is said to be a singular point of
the mapping $H$ if \cite[p. 2]{A9}
\begin{equation}
H(k_{s},\omega_{s})=H_{k}^{\prime}(k_{s},\omega_{s})=0.\label{Eq4-1}
\end{equation}
\noindent Obviously, in this case the tangent (\ref{eq:T}) is undefined.
In \cite[p. 45]{A9} it is shown that $H_{k}^{\prime}(k_{s},\omega_{s})=0$
is a necessary condition for the solution of $H(k_{s},\omega_{s})=0$
to be a bifurcation point (a point where the number of solutions changes).
For the two coupled transmission lines described above, Fig. \ref{fig1-2}
shows a plot of $H(k,\omega)$ in the vicinity of the EPD $(k_{s},\omega_{s})=(k_{e},\omega_{e})$
(the green, curved surface; numerical values of the CTL parameters
are the same as given in Section \ref{subsec:EDPs-as-branch}). The
intersection with the zero plane (solid blue) is clearly visible,
which forms the dispersion curve; the 2D dispersion is shown as the
black solid line (see also Fig. 4a in \cite{EPD}). A plot of the
function $H_{k}^{\prime}(k,\omega)$ is also shown in Fig. \ref{fig1-2}
(the slanted orange plane); units of $H(k,\omega)$ (green) and $H'_{k}(k,\omega)$
(pink) are in $m^{-4}$ and $m^{-3}$ , respectively. The intersection
of $H_{k}^{\prime}(k,\omega)$ with the $z=0$ plane forms the line
$H_{k}^{\prime}(k,\omega)=0$ shown in the figure with a black dashed
curve. The intersection of $H(k,\omega)$ and $H_{k}^{\prime}(k,\omega)$
on the $z=0$ plane is at the singular point (EPD) denoted by a black
solid circle (note that for $\omega<\omega_{e}$ the solid and dashed
curves seem to overlap. This is merely due to the scale of the plot;
the two lines actually only intersect at the EPD). For both $H$ and
$H_{k}^{\prime}$ the real part of the function is shown, as the imaginary
parts are negligible.
\begin{figure}[!th]
\noindent \begin{centering}
\includegraphics[scale=0.5]{plot_3D_png}
\par\end{centering}
\caption{The functions $H(k,\omega)$ (green), $H'_{k}(k,\omega)$ (pink),
and the zero plane (blue) vs. $k,\omega$ in the vicinity of the EPD
(solid dot). The 2D dispersion $H(k,\omega)=0$ is also shown (solid
black line). The black dashed line is $H_{k}^{\prime}(k,\omega)=0$.
Units of $H(k,\omega)$ (green) and $H'_{k}(k,\omega)$ (pink) are
in $m^{-4}$ and $m^{-3}$ , respectively.}
\label{fig1-2}
\end{figure}
In addition to the conditions (\ref{Eq4-1}), we defined a fold bifurcation
point (also know as a turning point, or limit point) when $H$ satisfies
(\ref{Eq4-1}) together with
\begin{equation}
H_{kk}^{\prime\prime}(k_{s},\omega_{s})\neq0,\,\,\,H_{\omega}^{\prime}(k_{s},\omega_{s})\neq0.\label{Eq4-1a}
\end{equation}
The zero conditions (\ref{Eq4-1}) together with the nonzero condition
$H_{kk}^{\prime\prime}(k_{s},\omega_{s})\neq0$ indicates that the
degeneracy is of second-order, i.e., where two modal eigenvalues coalesce,
as given in (\ref{eq:H1})-(\ref{H2}). The\textcolor{black}{{} nonzero
condition $H_{\omega}^{\prime}(k_{s},\omega_{s})\neq0$ serves as
a sufficient condition for $\omega_{s}$ to be a BP in the complex
$\omega-$plane, as proved in \cite{A2} using the Weierstrass preparation
theorem. }In \cite{A1,A2,A3,A4,A7,A8} the importance of fold singular
points in modal interaction phenomena on guided-wave structures has
been addressed in connection with the fold bifurcation from bifurcation
theory \cite{A9,A10}.
\textcolor{black}{Notably, the zero and non-zero conditions }(\ref{Eq4-1})-(\ref{Eq4-1a})\textcolor{black}{{}
are the same as (\ref{eq:H1})-(\ref{H2}), and (\ref{H3}) that arise
from linear algebra analysis. Thus, it can be concluded that the fold
singular point considered in, e.g., \cite{A1,A2,A3,A4,A7,A8} is in
fac}t an EPD which may reside generally in the complex plane $(k,\omega)\in\mathbb{C}^{2}$.
Therefore, in the following we denote $(k_{s},\omega_{s})$ as $(k_{e},\omega_{e})$.\textcolor{red}{{}
}An analogous treatment of EPDs using the conventional coupled-mode
theory \cite{CMT} is briefly outlined in the appendix.
\textbf{\textcolor{black}{Characteristic form. }}In the local neighborhood
of the fold point (FP)/EPD $(k_{e},\omega_{e})$ the qualitative behavior
of the mapping $H$ can be represented by the normal form \cite[p. 308-309]{A10}
, \cite[p. 196-198]{A9},
\begin{align}
(k-k_{e})^{2}+(\omega-\omega_{e})=0,\,\,\,\Delta & >0,\label{Eq7}\\
(k-k_{e})^{2}-(\omega-\omega_{e})=0,\,\,\,\Delta & <0\nonumber
\end{align}
where $\Delta=H_{kk}^{\prime\prime}(k_{e},\omega_{e})H_{\omega}^{\prime}(k_{e},\omega_{e})$,
leading to the dispersion function
\begin{align}
k(\omega) & =k_{e}\pm i\sqrt{\omega-\omega_{e}},\,\,\,\Delta>0,\label{Eq8}\\
k(\omega) & =k_{e}\pm\sqrt{\omega-\omega_{e}},\,\,\,\Delta<0.\nonumber
\end{align}
For the case of $\Delta>0$ with $\omega<\omega_{e}$ two branching
solutions $\Re{(k(\omega))}$ of $(k-k_{e})^{2}+(\omega-\omega_{e})$
generate a parabola, and for $\omega>\omega_{e}$ two equal solutions
$\Re{(k(\omega))}$ exist as a straight line $k(\omega)=k_{e}$. This
corresponds to the characteristic intersection of a parabola and a
straight line that occurs at a point of fold bifurcation \cite{A9,A10},
as shown in Fig. \ref{fig1-2} (see also \cite{EPD}). When $\omega=\omega_{e}$
there is only one solution $(k_{e},\omega_{e})$ corresponding to
the fold point. Also, $\Im{(k(\omega))}$ for $\omega<\omega_{e}$
yields the solution $k(\omega)=0$, and for $\omega>\omega_{e}$ two
branching solutions form a parabola in the imaginary plane of $k(\omega)$.
A similar analysis can be applied to the case of $\Delta<0$.
It should be noted that the conditions (\ref{Eq4-1}) and (\ref{Eq4-1a})
define both real and complex FPs/EPDs, however, the normal form (\ref{Eq7})
is applicable for real valued FPs, where $\Delta$ is real-valued.
Otherwise, the quantitative behavior of the local structure of the
function $H(k,\omega)$ in the vicinity of FP/EPD can be obtained
with a Taylor series expansion. Explicitly, the Taylor series in the
vicinity of the EPD can be written as
\begin{align}
H\left(k,\omega\right)= & H\left(k_{e},\omega_{e}\right)+H_{k}^{\prime}\left(k-k_{e}\right)+H_{\omega}^{\prime}\left(\omega-\omega_{e}\right)\\
& +\frac{1}{2}H_{kk}^{\prime\prime}\left(k-k_{e}\right)^{2}+H_{k\omega}^{\prime\prime}\left(k-k_{e}\right)\left(\omega-\omega_{e}\right)\nonumber \\
& +\frac{1}{2}H_{\omega\omega}^{\prime\prime}\left(\omega-\omega_{e}\right)^{2}+...=0.\nonumber
\end{align}
Since $H(k_{e},\omega_{e})=H_{k}^{\prime}(k_{e},\omega_{e})=0$, and
discarding the higher-order terms,
\begin{equation}
k-k_{e}\simeq\pm\alpha_{1}\left(\omega-\omega_{e}\right)^{1/2}+\alpha_{2}\left(\omega-\omega_{e}\right)\pm\alpha_{3}\left(\omega-\omega_{e}\right)^{3/2}+O((\omega-\omega_{e})^{2})
\end{equation}
where
\begin{equation}
\alpha_{1}=\sqrt{-2\frac{H_{\omega}^{\prime}}{H_{kk}^{\prime\prime}}},\ \ \alpha_{2}=-\frac{H_{\omega k}^{\prime\prime}}{H_{kk}^{\prime\prime}},\ \ \alpha_{3}=\frac{\alpha_{1}}{2}\frac{(H_{\omega k}^{\prime\prime})^{2}-H_{\omega\omega}^{\prime\prime}H_{kk}^{\prime\prime}}{-2H_{\omega}^{\prime}H_{kk}^{\prime\prime}}.
\end{equation}
The coefficient $\alpha_{1}$ is the same as (\ref{eq:alpha_1}),
and the higher-order coefficients are the same as given in \cite{AM}
retaining the same order of terms.
\subsection{EDPs leading to branch points in the complex-frequency plane\label{subsec:EDPs-as-branch}}
Regarding Statements 2 and 3, it is clear from several points of view
that $D=0$ defines a degeneracy in the eigenvalue plane, and a square-root-type
BP in the complex-frequency plane (since $\mathbf{A}=\mathbf{A}\left(\omega\right)$
for $\mathbf{A=-\underline{\underline{\mathbf{Z}}}\,\underline{\underline{\mathbf{Y}}}}$,
$-\mathbf{\underline{\underline{\mathbf{Y}}}\,\underline{\underline{\mathbf{Z}}}}$,
or $\mathbf{\underline{\mathbf{M}}}$). Solving $D=0$ leads to the
frequency where the BP/EPD occurs (this also can be obtained by substituting
$k^{2}=-T/2$ from (\ref{tr}) into (\ref{det3})). Assuming for simplicity
that $\mathbf{\underline{\underline{\mathbf{G}}}}=\mathbf{\underline{\underline{\mathbf{0}}}}$,
this leads to
\begin{equation}
\omega_{e}^{2}a+\omega_{e}b+c=0,
\end{equation}
where,
for $L_{11}=L_{22}=L$ and $C_{11}=C_{22}=C$ ($C_{nm}$ is the $nm$th
element of the capacitance matrix),
\begin{align}
a & =4\left(C_{12}L+CL_{12}\right)^{2},\\
b & =-4iC_{12}\left(CL_{12}+LC_{12}\right)\left(R_{11}+R_{22}\right),\nonumber \\
c & =-2R_{22}R_{11}\left(2C_{12}^{2}-C^{2}\right)-C^{2}\left(R_{11}^{2}+R_{22}^{2}\right).\nonumber
\end{align}
If $\left(R_{11}+R_{22}\right)\neq0$, then $\omega_{e}$ will not
be on the real-$\omega$ axis, assuming $\left(CL_{12}+LC_{12}\right)\neq0$.
For the PT-symmetric case, $R_{11}=-R=-R_{22}$,
\begin{equation}
\omega_{e}=\sqrt{\frac{-c}{a}}=R\frac{\sqrt{C^{2}-C_{12}^{2}}}{C_{12}L+CL_{12}}.\label{wBP}
\end{equation}
This will occur on the real-$\omega\,$\ axis, since one expects
$C^{2}>C_{12}^{2}$, proving Statement 4. Note that from a design
point of view, expression (\ref{wBP}) leads to the needed value of
$R$ for a desired value of $\omega_{e}$.
If we assume that $\mathbf{\underline{\underline{\mathbf{G}}}}\ne\mathbf{\underline{\underline{\mathbf{0}}}}$,
for $L_{11}=L_{22}=L$, $C_{11}=C_{22}=C$, and the PT-symmetric case,
$R_{11}=-R=-R_{22}$ and $G_{11}=-G=-G_{22}$,
\[
\omega_{e}^{2}=-\frac{G^{2}L_{12}^{2}+R^{2}C_{12}^{2}-X-\left(G^{2}L^{2}+C^{2}R^{2}\right)}{\left(CL_{12}+LC_{12}\right)^{2}}
\]
where $X=2GR\left(LC+C_{12}L_{12}\right)$. If $R=0$,
\begin{equation}
\omega_{e}=\frac{G\sqrt{L^{2}-L_{12}^{2}}}{C_{12}L+CL_{12}}
\end{equation}
which is the dual of (\ref{wBP}).
\begin{figure}[htp!]
{[}a{]}\includegraphics[width=0.4\textwidth]{figa}
{[}b{]}\includegraphics[width=0.4\textwidth]{figc}
{[}c{]}\includegraphics[width=0.4\textwidth]{figb}
{[}d{]}\includegraphics[width=0.4\textwidth]{figd}
\caption{Dispersion behavior near an EPD for coupled transmission lines with
(a) $R_{11}=-R_{22}=-73.172$ ohms (PT-symmetric case), as $\omega$
varies from $0.5\omega_{e}$ to $1.5\omega_{e}$ along the real-$\omega$
axis. (b) Same as (a) but for $R_{11}=-1.2R_{22}$, where the EPD
lies above the real-frequency axis. (c) Same as (a) but for $R_{11}=-0.8R_{22}$,
such that the EPD is below the real-frequency axis. (d) Same as (b)
but for $\Re(\omega)$ varying from $0.5\omega_{e}$ to $1.5\omega_{e}$
at a constant value $\Im(\omega)=\Im(\omega_{s})=(0.022\omega_{e})$.
In all cases the pair $(k_{e},\omega_{e})$ are the values at PT-symmetry,
$(k_{e},\omega_{e})=(28.649\,\mathrm{m^{-1}},2\pi10^{9}\,\mathrm{s^{-1}})$,
$R_{22}=73.172$ ohms, and the star indicates the BP/EPD. }
\label{fig2}
\end{figure}
As an example, Fig. \ref{fig2} shows $\Im(k/k_{e})$ versus $\Re(k/k_{e})$
in the vicinity of the fold point for numerical parameters taken from
\cite{EPD} corresponding to two coupled microstrip lines (strip width
3 mm, gap between strips 0.1 mm; substrate height 0.75 mm, and dielectric
constant of 2.2); $C_{11}=C_{22}=C=0.12\,\,\textrm{nF/m},L_{11}=L_{22}=L=0.18\,\,\mu\textrm{H/m},L_{12}=L_{21}=49.24\,\,\textrm{nH/m},C_{12}=C_{21}=-25.83\,\,\textrm{pF/m},\textrm{ and }\mathbf{\underline{\underline{\mathbf{G}}}}=\mathbf{\underline{\underline{\mathbf{0}}}}$.
Setting a target frequency of $\omega_{e}=2\pi10^{9}\,\mathrm{s^{-1}}$,
from (\ref{wBP}), to place the EPD on the real frequency axis at
$\omega_{e}$ requires $R_{11}=-R_{22}=-73.172$ ohms. The corresponding
value of wavenumber at the EPD is $k_{e}=28.649$ ${\rm m}^{-1}$.
A two-dimensional root search of (\ref{Eq4-1})-(\ref{Eq4-1a}) yields
$(k_{s}/k_{e},\omega_{s}/\omega_{e})=(1,1)$ as expected. Dispersion
behavior in the vicinity of the fold point is shown in Fig. \ref{fig2}a.
For other values of $R_{11}=-R_{22}$ (i.e., maintaining PT-symmetry)
the fold point remains on the $\Re(\omega)$ axis, but moves to lower
or higher frequencies as indicated in (\ref{wBP}). Upon breaking
PT-symmetry by using $R_{11}\neq-R_{22}$, the BP/EPD does not occur
on the real-frequency axis, as shown in Figs. \ref{fig2}b,c,d, where
in all cases $R_{22}=73.172$ ohms. For $R_{11}=-1.2R_{22}$ the 2D
root search of (\ref{Eq4-1})-(\ref{Eq4-1a}) yields $(k_{s}/k_{e},\omega_{s}/\omega_{e})=(1.1+i0.025,1.1+i0.022)$,
where $(k_{e},\omega_{e})$ are the values given above under the PT-symmetry
conditions, $(k_{e},\omega_{e})=(28.649\,\mathrm{m^{-1}},2\pi10^{9}\,\mathrm{s^{-1}})$.
As such, the EPD lies above the real-frequency axis, and Fig. \ref{fig2}b
shows the corresponding dispersion behavior. Since a scanning of an
operating frequency (assumed real) does not pass through the branch
point, the eigenvalues do not become degenerate. Alternatively, Fig.
\ref{fig2}c shows the dispersion behavior when $R_{11}=-0.8R_{22}$,
such that the EPD is below the real-frequency axis and the modes have
interchanged with their counterparts in Fig. \ref{fig2}b. Fig. \ref{fig2}d
shows the dispersion behavior for the case $R_{11}=-1.2R_{22}$, when
the real part of frequency is varied while keeping a constant $\Im(\omega)=\Im(\omega_{s})=(0.022\omega_{e})$,
and so passing through the singular point (EPD), at which point the
modal degeneracy is recovered at a complex-valued $k$ . In this complex
frequency case a BP is clearly visible and occurs at a complex value
wavenumber. Regarding Figs. \ref{fig2}b,c, note that to interchange
the modal solutions it is not necessarily to encircle the EPD/BP (as
done in, for example \cite{Hassan}, \cite{R1}). It is shown in Figs.
\ref{fig2}b,c that the interchange of solutions is due to varying
the frequency path above or below the BP \cite{A6,A14}.
\section{Conclusions}
We have examined several aspects of EPDs on two coupled transmission
lines, demonstrating that in the framework of the eigenvalue problem
the eigenvalue degeneracies are always coincident with eigenvector
degeneracies, such that all eigenvalue degeneracies correspond to
EPDs. We also discussed the fact that EPDs are related to branch-point
singularities in the complex-frequency plane, as can be ascertained
from both linear algebra concepts and from the theory of singular
points of complex mappings and bifurcation theory. Moreover, we have
provided a connection between the linear algebra approach and an approach
based on singularity and bifurcation theories, previously used to
study modal interactions on guided-wave structures. We have presented
simple closed-form expressions for the complex-frequency plane EPDs,
and showed that under PT-symmetry these branch points reside on the
real-frequency axis and generalized the branch point discussion to
complex frequency and wavenumbers. \bigskip{}
\section*{Acknowledgments}
This material is based upon M.O and F.C work supported by the Air
Force Office of Scientific Research under award number FA9550-15-1-0280.
\bigskip{}
\section*{Appendix: Coupled-Mode Theory}
In addition to the transmission-line treatment of EPDs, here we briefly
comment on the matrix that arises from conventional so called ``coupled-mode
theory'' \cite{CMT}. For simplicity, we consider the PT-symmetric
case for otherwise identical individual transmission lines (e.g.,
one will have loss and one will have gain). Then, the individual (uncoupled)
lines have propagation constants $\beta$ and $\beta^{\ast}$, which,
when brought into proximity, become $\beta+\delta$ and $\beta^{\ast}+\delta^{\ast}$
under the coupling constant $\kappa$. The coupled system modes obey
the evolution equation \cite{PRL}
\begin{equation}
i\frac{d}{dz}\left[\begin{array}{c}
a_{1}\\
a_{2}
\end{array}\right]=\left[\begin{array}{cc}
\beta+\delta & \kappa\\
\kappa^{\ast} & \left(\beta+\delta\right)^{\ast}
\end{array}\right]\left[\begin{array}{c}
a_{1}\\
a_{2}
\end{array}\right]=\mathbf{\underline{\underline{\beta}}}\left[\begin{array}{c}
a_{1}\\
a_{2}
\end{array}\right]\label{ee}
\end{equation}
where $a_{1}$ and $a_{2}$ are the wave amplitudes in transmission
lines 1 and 2, respectively. One can proceed with examination of the
eigenvectors and eigenvalues, but it suffices to consider, analogous
to (\ref{eq:H}), the dispersion relation
\begin{eqnarray}
H\left(k,\omega\right) & = & \left\vert \left[\begin{array}{cc}
\beta\left(\omega\right)+\delta\left(\omega\right) & \kappa\left(\omega\right)\\
\kappa^{\ast}\left(\omega\right) & \left(\beta\left(\omega\right)+\delta\left(\omega\right)\right)^{\ast}
\end{array}\right]-k\left[\begin{array}{cc}
1 & 0\\
0 & 1
\end{array}\right]\right\vert =0\nonumber \\
& = & k^{2}-k\mathrm{Tr}\left(\mathbf{\underline{\underline{\beta}}}\right)+\det\left(\mathbf{\mathbf{\underline{\underline{\beta}}}}\right)=0\label{ee1}
\end{eqnarray}
where $\mathbf{\mathbf{\underline{\underline{\beta}}}}$ is the 2$\times$2
matrix in (\ref{ee}). Obviously, (\ref{ee1}) is analogous to (\ref{HE0}).
Furthermore,
\begin{equation}
H'_{k}\left(k,\omega\right)=2k-\left(\beta^{\ast}+\delta^{\ast}+\beta+\delta\right)=0
\end{equation}
leads to
\begin{equation}
k=\frac{1}{2}\left(\beta^{\ast}+\delta^{\ast}+\beta+\delta\right)={\textstyle \Re}\left(\beta+\delta\right)=\frac{1}{2}\mathrm{Tr}\left(\mathbf{\mathbf{\mathbf{\underline{\underline{\beta}}}}}\right)
\end{equation}
and using (\ref{ee1}) one obtains
\begin{equation}
\mathrm{Tr}^{2}\left(\underline{\underline{\beta}}\left(\omega\right)\right)-4\det\left(\underline{\underline{\beta}}\left(\omega\right)\right)=0
\end{equation}
which is the condition $D=0$ in (\ref{D}), and which leads to the
value of the EPD frequency $\omega=\omega_{\text{e}}$. The nonzero
condition $H_{\omega}^{\prime}(k,\omega)\neq0$ can be evaluated if
all matrix entries are known as a function of frequency. Thus, coupled-mode
theory leads to the same analysis of EPDs as the CTL formulation presented
in Section \ref{sec:Coupled-Transmission-Line}, and, therefore, can
also be analyzed using bifurcation theory.
\bigskip{}
|
3,212,635,537,927 | arxiv | \section{Preliminaries}
\label{sec:prelim}
Let $H$ be a Hilbert space, $(G,\cdot)$ a topological group, and $\{T(g)\mid g\in G\}$ be an isometric representation of $G$ on $H$. In other words, $T(g): H \rightarrow H$ is an isometry satisfying
\begin{enumerate}[label = (\roman*)]
\item $T(e) = \text{Identity}$
\item $T(g_1\cdot g_2) = T(g_1) \circ T(g_2)$ for all $g_1,g_2\in G$
\item $G\times H \ni (g,u) \mapsto T(g)u \in H$ is a continuous map.
\end{enumerate}
The $\textit{orbit}$ of an element $u\in H$ is the set $\{T(g)u\mid g\in G\}$. We say that a subset $A\subset H$ is $\textit{invariant}$ (under $G$) if $T(g)A \subset A$ for all $g\in G$. A functional $\varphi: H \rightarrow \mathbb{R}$ is said to be $\textit{invariant}$ if $\varphi\circ T(g) = \varphi$ for all $g\in G$. A mapping $R: A_1 \rightarrow A_2$ between subsets of $H$ is called $\textit{equivariant}$ if $R\circ T(g) = T(g) \circ R$ for all $g\in G$.
The set of \textit{fixed points} of $H$ under the representation $\{T(g)\}$, which we refer to as the set of \textit{(most) symmetric elements} of $H$ (under $G$), is the closed subspace of $H$ defined by
\begin{equation*}
\text{Fix}(G) = \{u\in H\mid T(g)u = u, \ \forall g\in G\}.
\end{equation*}
If $u_0\in H$ is an isolated critical point of $\varphi \in C^2(H, \mathbb{R})$ which is {\it nondegenerate}, i.e. the bilinear form $\varphi''(u_0): H\times H \rightarrow \mathbb{R}$ is nondegenerate, the \textit{Morse Index} of $\varphi$ at $u_0$, which we denote by
\begin{equation*}
i_M(\varphi,u_0),
\end{equation*}
is the supremum of $k\in \mathbb{N}$ such that $\varphi''(u_0)$ is \textit{negative} {\it definite} on a $k$-dimensional subspace of $H$.
Moreover, recall that the eigenvalues of the problem
\begin{gather}\label{eigprob}
\begin{cases}
-h''(t) = \lambda h(t)\\
h(0) = h(2\pi)\\
h'(0) = h'(2\pi)
\end{cases}
\end{gather}
are $\lambda_j = j^2$ where $j\in \mathbb{Z}^{0+} = \{z\in \mathbb{Z}\mid z \geq 0\}$, with corresponding eigenfunctions $h_j(t) = C_j\cos(jt) + D_j\sin(jt)$ (and so $h_0(t) = C_0)$, where $C_j,D_j$ are arbitrary constants. Note that $\lambda_0 = 0$ is simple while $\lambda_j, \ j\geq 1$ are double. Furthermore, for $j \geq 1$, the eigenfunctions are periodic with minimal period $\frac{2\pi}{j}$.
Consider the following spaces
\begin{align*}
H^1_{per}[0,2\pi] &= \{u\in H^1[0,2\pi]\mid u(0) = u(2\pi), \ u'(0) = u'(2\pi)\};\\
V_j &= \{u\in H^1_{per}[0,2\pi]\mid \text{u is $2\pi$/j periodic}\};\\
E_j &= \{C_1\cos(jt) + C_2\sin(jt)\mid C_1,C_2\in \mathbb{R}\}, \ j\in \mathbb{Z}^+;\\
E_0 &= \mathbb{R}.
\end{align*}
Note that $E_j$ is the $j$th eigenspace of \eqref{eigprob}. We endow $H^1_{per}[0,2\pi]$ with the norm and inner product, respectively,
\begin{gather*}
||u||=\left(\int_0^{2\pi}(|u'|^2 + |u|^2)\ dt\right)^{1/2} \ \ \text{and} \ \ \langle u,v \rangle = \int_0^{2\pi} (u'v' + uv)\ dt.
\end{gather*}
\section{Introduction}
\label{sec:intro}
We are interested in problems of the form
\begin{equation}\label{eq:problem}
\begin{cases}
-u''(t) + g(u(t)) = f(t)\\
u(0) = u(2\pi)\\
u'(0) = u'(2\pi)
\end{cases}
\end{equation}
where $g:\mathbb{R} \rightarrow \mathbb{R}$ is a given $C^1$ function and $f:[0,2\pi]\rightarrow \mathbb{R}$ is continuous. Let $G$ be a finite group such that $G = \mathbb{Z}_m$ for some $m$. Suppose $u \in H_{per}^1[0,2\pi]$, then let $\{T(g): g\in G\}$ be an isometric topological representation of $G$ on $H_{per}^1[0, 2\pi]$ such that the action of $T(g)$ on $H_{per}^1[0, 2\pi]$ is given by $T(g)u = \hat{u}(t+ \frac{2\pi g}{m})$ where $\hat{u}$ is the periodic extension of $u$ on the real line. Here
\begin{align*}
\text{Fix}(G) = \{u\in H_{per}^1[0,2\pi]\mid u\text{ is $2\pi/m$-periodic}\}.
\end{align*}
For this group action, we call elements of Fix($g$) \textit{translationally symmetric}, since the action preserves symmetry under translation. For such a problem as \eqref{eq:problem}, three questions are of use to us.
The first question concerns that of \textit{preservation of symmetry}. That is, if $f(t)$ has a certain symmetry, say periodicity on the real line with period $P$, then if all solutions to \eqref{eq:problem} also have period $P$, we say that the symmetry is preserved.
Conversely, the second question concerns the \textit{breaking of symmetry}. That is, if $f(t)$ is periodic on the real line with period $P$, then a solution $v(t)$ to $\eqref{eq:problem}$ breaks symmetry if $v(t)$ is not periodic with period $P$.
The third question concerns \textit{existence and multiplicity of solutions using symmetry}. In other words, given an isometric representation of a topological group $G$ on $H$, there exist either one or multiple solutions that are geometrically distinct, i.e. solutions that are not within the orbit of the other.
Results on preservation and breaking of symmetry have appeared for both ordinary and partial differential equations such as in Willem \cite{Willem1989}, Dancer \cite{Dancer1983}, Lazer-McKenna \cite{Lazer1988}, just to mention a few pioneering references on the subject. Costa-Fang \cite{CostaFang2019} proved a result for breaking of symmetry that concerns translational symmetry using $\mathbb{Z}_p$ ($p$ prime) group actions, i.e. they showed that there exists a function that is $2\pi/\hat{p}$ ($\hat{p} \neq p$ and $\hat{p}$ being prime) periodic such that (\ref{eq:problem}) has a solution that does not have the same periodicity. The theorem relies on the hypothesis that the derivative on $g$ must be bounded above and below by certain eigenvalues. In this way, this is a type of "symmetry" imposed on $g$. Our breaking of symmetry theorem extends the hypothesis to cases when the order of the group action and the order of the eigenvalues are relatively prime. On the other hand, our preservation of symmetry result shows that all solutions of (\ref{eq:problem}) must satisfy a certain symmetry if the nonlinear term satisfies a symmetry based on the eigenvalues.
There have also been results that exhibit existence of solutions for Hamiltonian systems that preserve a type of symmetry. Major approaches in this field include Lusternik-Schnirelmann Theory and Index Theory as in Costa-Willem \cite{Costa1986}. In particular, existence of distinct subharmonic solutions (i.e. solutions with period $kT$ where $k \in \mathbb{N}$) for Hamiltonian systems with period $T$ have been explored in Rabinowitz \cite{Rabinowitz1980}, Tarentello \cite{Tarentello1988}, and Liu-Wang \cite{Liu1993}.
\section{Preservation of Symmetry}
\label{sec:preserv}
For the proof of preservation of symmetry, we first state a result from Mawhin, the proof of which is found in \cite{Mawhin1976}.
\begin{thm}\label{thm:MMT}
Given a real Hilbert space $H$, with inner product $\langle, \rangle$ and norm $|\cdot|$, let $L: \text{dom}(L)\subset H \rightarrow H$ be a linear, self-adjoint operator and $N: H \rightarrow H$ be a mapping with a bounded, linear G{\^ a}teaux derivative $N'$ on H such that, for each $x\in H$, $N'(x)$ is a symmetric operator. Denote by $\rho(A)$, $\sigma(A)$, and $r_\sigma(A)$ the resolvent set, the spectrum, and the spectral radius, respectively, of any linear operator $A$ in $H$.
Suppose there exist real numbers $\lambda < \mu$ such that
\begin{equation*}
(\lambda,\mu) \subset \rho(L), \quad \lambda,\mu \in \sigma(L)
\end{equation*}
and real numbers $p,q$ with
\begin{equation*}
\lambda < q \leq p < \mu
\end{equation*}
such that, for each $x\in H$,
\begin{equation*}
qI \leq N'(x) \leq pI.
\end{equation*}
Then, $L - N$ is one-to-one,
\begin{equation*}
(L-N)(\text{dom}(L)) = H,
\end{equation*}
and $(L-N)^{-1}$ is globally Lipschitzian.
\end{thm}
We now present our result on preservation of symmetry, which is inspired by \cite{Willem1989}.
\begin{thm}
Let $g:\mathbb{R} \rightarrow \mathbb{R}$ be a given $C^1$ function. Let $s$ be any positive integer. Suppose there exists $i\geq 1$ such that
\begin{equation}\label{eq:condition}
\lambda_i < \min_{t\in \mathbb{R}} g'(u(t)) < \max_{t\in \mathbb{R}} g'(u(t)) < \lambda_{i+1}
\end{equation}
then, for every $f\in L^2[0,2\pi]$ such that $f$ is $2\pi/s$-periodic, all solutions of $\eqref{eq:problem}$ are $2\pi/ks$-periodic for some $k\in \mathbb{N}$ (and so is also $2\pi/s$ periodic).
\end{thm}
\textbf{Remark 1.}
Note that the last part of the statement of Theorem 2 suggests that if the hypotheses are satisfied, then all solutions will retain the same periodicity. However, this does not mean that the minimal period will be the same -- in fact, it could be this period divided by a positive integer.
\textbf{Remark 2.}
In light of using Theorem 1 to prove Theorem 2, it turns out that there exists a unique solution. Hence, Theorem 2 can be rephrased as saying that there exists a unique solution to (\ref{eq:problem}) that preserves the symmetry for every periodic function $f \in L^2[0,2\pi]$.
\begin{proof}
Let $P$ be the orthogonal projector on the space of $2\pi/s$-periodic functions. Define the self-adjoint operator $L:D(L) \subset L^2 \rightarrow L^2$ by
\begin{align*}
D(L) &= H^1_{per}[0,2\pi] \cap H^2[0,2\pi]\\
Lu &= -u''.
\end{align*}
For $u\in L^2$, we write $u = v+w$ for $v = Pu$ and $w = (I-P)u$. Then, \eqref{eq:problem} is equivalent to
\begin{align*}
Lv &= -Pg(v+w) + f\\
Lw &= -(I-P)g(v+w).
\end{align*}
Since $v$ is periodic with minimal period $2\pi/ks$ for some $k\in \mathbb{N}$, then $g(v)$ is also periodic with minimal period $2\pi/ks$. Note that if $w\equiv 0$, then
\begin{equation*}
L(0) = -(I-P)g(v+0) = -(I-P)g(v) = 0.
\end{equation*}
Hence, $w = 0$ is a solution to $\eqref{eq:problem}$. In view of \eqref{eq:condition}, applying Theorem $\ref{thm:MMT}$, with $\lambda = \lambda_i, \mu = \lambda_{i+1}$, $q = \min_{t\in \mathbb{R}} g'(u(t))$, and $p = \max_{t\in \mathbb{R}} g'(u(t))$, we see that for every $v\in \text{Range}(P)$, there is exactly one solution $w\in \text{Range}(I-P)$. In particular, $w = 0$ for every $v$. Hence, $u = v$. \qed
\end{proof}
\section{Breaking of Symmetry}
\label{sec:break}
For breaking of symmetry, we present a result inspired by \cite{CostaFang2019}. In fact, the following is a generalization of \cite{CostaFang2019} to arbitrary coprime positive integers.
\begin{thm}
Let $g:\mathbb{R} \rightarrow \mathbb{R}$ be a given $C^1$ function satisfying
\begin{gather}
|g(t)| \leq C_g, \ \text{for all $t\in \mathbb{R}$}
\end{gather}
and
\begin{gather}
G(t) \rightarrow -\infty \ \text{as} \ |t|\rightarrow\infty,
\end{gather}
where $G(t) = \int_0^t g(s)\ ds$. Suppose there exists $t_0,t_1\in \mathbb{R}$ such that
\begin{gather}\label{eigeninequality2}
\lambda_{r-1} < g'(t_0) < \lambda_r < g'(t_1) < \lambda_{r+1}
\end{gather}
for some positive integer $r$ and let $s$ be another positive integer, $\gcd(r,s)=1$. Then there exists $\hat{f}:\mathbb{R}\rightarrow\mathbb{R}$, $\hat{f}\in V_s$ such that $\eqref{eq:problem}$ has at least one solution which is not in $V_s$.
\end{thm}
\textbf{Remark 3.}
Let us understand the difference between Theorem 2 and Theorem 3. In the case of Theorem 2, if the derivative of the non-linearity $g$ is bounded between consecutive eigenvalues, each of which correspond to the periodic eigenfunctions, then the period of the solution(s) (if they exist) must retain at least the same period as the forcing term $f$. In contrast, if the derivative "crosses" an eigenvalue, then Theorem 3 states that one can find an appropriate $f$ where the periodicity of the solution(s) is \emph{not} preserved.
\begin{proof}
Define quadratic forms on $V_s^\perp$ by
\begin{gather*}
Q_i(h) = \int_0^{2\pi} \left(|h'|^2 - g'(t_i)|h|^2\right)\ dt, \ i=0,1.
\end{gather*}
Note that since $r$ and $s$ are coprime, then $\sin(rt),\cos(rt)\in V_s^\perp$. Additionally, from \eqref{eigeninequality2}, $Q_i, i = 0,1,$ are non-degenerate.
Since $g'$ is continuous, then there exists $\delta_0 > 0$ such that $\lambda_{r-1} < g'(t) < \lambda_r$ for all $t\in [t_0-\delta_0,t_0+\delta_0]$. Consequently,
\begin{gather*}
(r-1)^2 < g'(\delta_0\sin(st) + t_0) < r^2, \ \forall t\in \real.
\end{gather*}
Letting $u^*_0(t) \coloneqq \delta_0\sin(st) + t_0$, we define the quadratic form
\begin{gather*}
\Tilde{Q}_0(h) = \int_0^{2\pi} (|h'|^2 - g'(u_0^*)|h|^2)\ dt, \ h\in V_s^\perp,
\end{gather*}
which is non-degenerate and has Morse index $m_0$. Similarly, define $\Tilde{Q}_1(h)$, which has Morse index $m_1$, using $u^*_1(t) \coloneqq \delta_1\sin(st) + t_1$ where $\delta_1 > 0$ is such that $\lambda_r < g'(t) < \lambda_{r+1}$ for all $t\in [t_1-\delta_1, t_1+\delta_1]$.
The rest of the proof follows as in \cite{CostaFang2019} with $s$ taking the role of $\hat{p}$ and $r$ taking the role of $p$. \qed
\end{proof}
\section{Acknowledgement}
We would like to thank Professor David Costa for his valuable suggestions for our manuscript and his inspiring mentorship.
|
3,212,635,537,928 | arxiv | \subsubsection*{\bibname}}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{url,booktabs,nicefrac,microtype}
\usepackage{xcolor}
\definecolor{linkblue}{rgb}{0.1,0.4,0.7}
\usepackage[colorlinks=true,citecolor=linkblue]{hyperref}
\usepackage{multicol}
\usepackage{float}
\usepackage{cancel}
\usepackage{physics}
\usepackage{amsfonts,amsmath,amsthm,amssymb}
\usepackage{mathtools}
\usepackage{wrapfig}
\usepackage{pifont}
\usepackage{tikz}
\usetikzlibrary{bayesnet}
\usetikzlibrary{arrows}
\newcommand{ \textcolor{green!60!black}{\ding{51}} }{ \textcolor{green!60!black}{\ding{51}} }
\newcommand{ \textcolor{red!60!black}{\ding{55}} }{ \textcolor{red!60!black}{\ding{55}} }
\usepackage{wrapfig}
\usepackage{colortbl}
\usepackage{tabularx}
\usepackage{graphbox}
\usepackage{comment}
\usepackage{multirow}
\usepackage{booktabs}
\usepackage{algorithm, algpseudocode, algcompatible}
\usepackage{enumitem}
\usepackage{caption}
\usepackage{subcaption}
\input{math.tex}
\begin{document}
\twocolumn[
\arxivtitle{Variational multiple shooting for Bayesian ODEs with Gaussian processes}
\arxivauthor{Pashupati Hegde \And \c{C}a\u{g}atay Y{\i}ld{\i}z \And Harri L{\"a}hdesm{\"a}ki \And Samuel Kaski \And Markus Heinonen}
\arxivaddress{Department of Computer Science\\
Aalto University, Finland\\
\texttt{first.last@aalto.fi} \\} ]
\begin{abstract}
Recent machine learning advances have proposed black-box estimation of \textit{unknown continuous-time system dynamics} directly from data. However, earlier works are based on approximative solutions or point estimates. We propose a novel Bayesian nonparametric model that uses Gaussian processes to infer posteriors of unknown ODE systems directly from data. We derive sparse variational inference with decoupled functional sampling to represent vector field posteriors. We also introduce a probabilistic shooting augmentation to enable efficient inference from arbitrarily long trajectories. The method demonstrates the benefit of computing vector field posteriors, with predictive uncertainty scores outperforming alternative methods on multiple ODE learning tasks.
\end{abstract}
\section{Introduction}
\label{section:intro}
Ordinary differential equations (ODEs) are powerful models for continuous-time non-stochastic systems, which are ubiquitous from physical and life sciences to engineering \citep{hirsch2012differential}. In this work, we consider non-linear ODE systems
\begin{align}
\mathbf{x}(t) &= \mathbf{x}_0 + \int_0^t \mathbf{f}( \mathbf{x}(\tau)) d\tau \label{eq:odeproblemtraj} \\
\dot{\mathbf{x}}(t) &:= \frac{d\mathbf{x}(t)}{dt} = \mathbf{f}(\mathbf{x}(t)), \label{eq:odeproblem}
\end{align}
where the state vector $\mathbf{x}(t) \in \mathbb{R}^D$ evolves over time $t \in \mathbb{R}_+$ from an initial state $\mathbf{x}_0$ following its time derivative $\dot{\mathbf{x}}(t)$, and $\tau$ is an auxiliary time variable. Our goal is to learn the differential function $\mathbf{f} : \mathbb{R}^D \mapsto \mathbb{R}^D$ from state observations, when the functional form of $\mathbf{f}$ is unknown.
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{plots/ils1_gpode_model.pdf}
\caption{Illustration of GPODE: The model learns a GP posterior (a) of a vector field. Valid ODE trajectories are sampled from the posterior process as shown in (b) and (c).}
\label{fig:gpode_illustration_1}
\end{figure*}
The conventional mechanistic approach involves manually defining the equations of dynamics and optimizing their parameters \citep{butcher2008numerical}, or inferring their posteriors \citep{girolami2008bayesian} from data. However, the equations are unknown or ambiguous for many systems, such as human motion \citep{4359316}. Early seminal works explored fitting unknown ODEs with splines \citep{henderson2014network}, Gaussian processes \citep{aijo2009learning} or kernel methods \citep{heinonen2014learning} by resorting to less accurate gradient matching approximations \citep{varah1982spline}. Recently, accurate estimation of free-form non-linear dynamics was proposed using Gaussian processes with sensitivity equations \citep{heinonen2018learning} and neural networks with adjoints \citep{chen2018neural}. However, both approaches are restricted to learning point estimates of the dynamics, limiting their uncertainty characterization and generalization. Furthermore, the gradient descent training in both approaches is ill-suited for complex or long-horizon ODEs with typically highly non-linear integration maps \citep{diehl2017}.
In this work, we introduce Bayesian learning of unknown, non-linear ODEs. Our contributions are:
\begin{itemize}
\item We introduce a way of learning posteriors of vectorfields using Gaussian processes as flexible priors over differentials $\mathbf{f}$. We adapt decoupled functional sampling to simulate ODEs in linear time from vector field posteriors.
\item For the difficult problem of gradient optimizations of ODEs, we introduce a novel probabilistic shooting method. It is motivated by the canonical shooting methods from optimal control and makes inference stable and efficient on long trajectories.
\item We empirically show the effectiveness of the proposed method even while learning from a limited number of observations. We demonstrate the ability to infer arbitrarily long trajectories efficiently with the shooting extension.
\end{itemize}
\section{Related Works}
\begin{table*}[!t]
\caption{Related work: for each method, we indicate if it can learn posterior over vector fields, does it assume an unknown system dynamics, and whether the inference is performed directly over the observations or on the empirical gradients}
\label{tab:related_works}
\centering
\resizebox{1.00\textwidth}{!}{
\begin{tabular}{lcccl}
\toprule
& ODE & Freeform & Exact & \\
Method & posterior & dynamics & gradients & Reference \\
\midrule
NeuralODE & \textcolor{red!60!black}{\ding{55}} & \textcolor{green!60!black}{\ding{51}} & \textcolor{green!60!black}{\ding{51}} & \citet{chen2018neural} \\
npODE & \textcolor{red!60!black}{\ding{55}} & \textcolor{green!60!black}{\ding{51}} & \textcolor{green!60!black}{\ding{51}} & \citet{heinonen2018learning} \\
Gradient matching & \textcolor{red!60!black}{\ding{55}} & \textcolor{green!60!black}{\ding{51}} & \textcolor{red!60!black}{\ding{55}} & \citet{ramsay2007parameter,heinonen2014learning} \\
Mechanistic GM & \textcolor{green!60!black}{\ding{51}} & \textcolor{red!60!black}{\ding{55}} & \textcolor{red!60!black}{\ding{55}} & \citet{dondelinger2013ode,wenk2020odin} \\
GPODE & \textcolor{green!60!black}{\ding{51}} & \textcolor{green!60!black}{\ding{51}} & \textcolor{green!60!black}{\ding{51}} & this work \\
\bottomrule
\end{tabular}
}
\end{table*}
\paragraph{Mechanistic ODE models.} In mechanistic modelling the equation $\mathbf{f}_{\boldsymbol{\theta}}$ is predefined with a set of coefficients ${\boldsymbol{\theta}}$ to be fitted \citep{butcher2008numerical}.
Several works have proposed embedding mechanistic models within Bayesian or Gaussian process models \citep{calderhead2008,dondelinger2013ode,macdonald2015}. Recently both Julia and Stan have introduced support for Bayesian analysis of parametric ODEs \citep{rackauckas2017differentialequations,stan}.
\paragraph{Free-form ODE models.} Multiple works have proposed fitting unknown, non-linear and free-form ODE differentials with gradient matching \citep{ramsay2007parameter} using splines \citep{henderson2014network}, Gaussian processes \citep{aijo2009learning,aijo2013,ridderbusch2020learning,wenk2020odin} or kernel methods \citep{heinonen2014learning}. Recently, \citet{heinonen2018learning} proposed accurate \textit{maximum a posteriori} optimisation of vector fields with sensitivity equation gradients \citep{kokotovic1967direct}. Neural ODEs \citep{chen2018neural} introduced adjoint gradients \citep{pontryagin1962mathematical} along with flexible black-box neural network vector fields. Several extensions to learning latent ODEs have been proposed \citep{yildiz2019ode2vae,rubanova2019}. \citet{bhouri2021} proposed a hybrid model combining neural networks and Gaussian processes for sparse ODE system discovery \citep{brunton2016discovering}.
\paragraph{Discrete-time state-space models.} There is a large literature on Markovian state-space models that operate over discrete time increments \citep{wang2005,turner2010state, frigola2014variational}. Typically nonlinear state transition functions are modeled with Gaussian processes and applied to latent state estimation or system identification problems with dynamical systems \citep{eleftheriadis2017identification, doerr2018probabilistic, ialongo2019overcoming}. In this paper, we focus strictly on continuous-time models and leave the study of discrete vs. continuous formulations for future work.
\section{Methods}
\label{section:methods}
We consider the problem of learning ODEs \eqref{eq:odeproblem} and propose a Bayesian model to infer posteriors over the differential $\mathbf{f}(\cdot)$.
\subsection{Bayesian modeling of ODEs using GPs}
We assume a sequence of $N$ observations $\mathbf{Y} = (\mathbf{y}_1, \mathbf{y}_2, \ldots \mathbf{y}_N)^T \in \mathbb{R}^{N \cross D}$ along a trajectory, with $\mathbf{y}_i \in \mathbb{R}^D$ representing the noisy observation of the unknown state $\mathbf{x}(t_i) \in \mathbb{R}^D$ at time $t_i$. We assume a zero mean vector-valued Gaussian process prior over $\mathbf{f}$,
\begin{align}
\mathbf{f}(\mathbf{x}) &\sim \mathcal{GP}(\mathbf{0}, K(\mathbf{x}, \mathbf{x}')),
\end{align}
which defines a distribution of differentials $\mathbf{f}(\mathbf{x})$ with covariance $\cov[\mathbf{f}(\mathbf{x}), \mathbf{f}(\mathbf{x}')] = K(\mathbf{x}, \mathbf{x}')$, where $K(\mathbf{x}, \mathbf{x}') \in \mathbb{R}^{D \cross D}$ is a stationary matrix-valued kernel \citep{alvarez2012}. We follow the commonly used sparse inference framework for GPs using inducing variables \citep{titsias2009variational}, and augment the full model with inducing values $\mathbf{U} = (\mathbf{u}_1, \ldots ,\mathbf{u}_M)^T \in \mathbb{R}^{M \times D}$ and inducing locations $\mathbf{Z} = (\mathbf{z}_1, \ldots, \mathbf{z}_M )^T \in \mathbb{R}^{M \times D}$ such that $\mathbf{u}_m = \mathbf{f}(\mathbf{z}_m)$. The inducing variables are trainable `landmark' state-differential pairs, from which the rest of the differential field is interpolated (See Figure \ref{fig:gpode_illustration_1}, where arrow locations are the $\mathbf{z}_m$ and arrow end-points are the $\mathbf{u}_m$). The inducing augmentation leads to the following prior and conditionals \citep{hensman2013gaussian}:
\begin{align}
p(\mathbf{U}) &= \mathcal{N}(\mathbf{U} | \mathbf{0}, \mathbf{K}_{\mathbf{Z}\Z}), \\
p(\mathbf{f} | \mathbf{U}; \mathbf{Z}) &= \mathcal{N}(\mathbf{f} | \mathbf{A} \mathrm{vec}(\mathbf{U}), \mathbf{K}_{\mathbf{X} \mathbf{X}} - \mathbf{A} \mathbf{K}_{\mathbf{Z} \mathbf{Z}} \mathbf{A}^T),
\end{align}
where $\mathbf{X} = (\mathbf{x}_1, \mathbf{x}_2, \ldots \mathbf{x}_{N'})^T \in \mathbb{R}^{N' \times D}$ collects all the intermediate state evaluations $\mathbf{x}(t_i)$ encountered along a numerical approximation of the true continuous ODE integral \eqref{eq:odeproblemtraj}, $\mathbf{f} = (\mathbf{f}(\mathbf{x}_1)^T, \ldots, \mathbf{f}(\mathbf{x}_N')^T)^T \in \mathbb{R}^{N'D \times 1}$, $\mathbf{K}_{\mathbf{X}\X}$ is a block-partitioned matrix of size $N'D \times N'D$ with $D \times D$ blocks, so that block $(\mathbf{K}_{\mathbf{X}\X})_{i,j} = K(\mathbf{x}_i, \mathbf{x}_j)$, and $\mathbf{A} = \mathbf{K}_{\mathbf{X}\mathbf{Z}} \mathbf{K}^{-1}_{\mathbf{Z}\Z}$. For notational simplicity, we assume that the measurement time points are among the time points of the intermediate state evaluations.
The joint probability distribution follows
\begin{align}
p(\mathbf{Y}, \mathbf{f}, \mathbf{U}, \mathbf{x}_0) &= p(\mathbf{Y}|\mathbf{f},\mathbf{x}_0)p(\mathbf{f},\mathbf{U})p(\mathbf{x}_0) \\
&\hspace{-10mm} = \prod\limits_{i=1}^{N}p(\mathbf{y}_i|\mathbf{f}, \mathbf{x}_0) p(\mathbf{f}|\mathbf{U}) p(\mathbf{U}) p(\mathbf{x}_0),
\end{align}
where the conditional distribution $p(\mathbf{y}_i| \mathbf{f}, \mathbf{x}_0) = p(\mathbf{y}_i|\mathbf{x}_i)$ computes the likelihood over ODE state solutions $\mathbf{x}_i = \mathbf{x}_0 + \int_0^{t_i} \mathbf{f}( \mathbf{x}(\tau)) d\tau$.
\subsection{Variational inference for GP-ODEs}
In contrast to earlier approaches that estimate MAP solutions \citep{heinonen2018learning,ridderbusch2020learning}, our goal is to infer the posterior distribution $p(\mathbf{f},\mathbf{x}_0 | \mathbf{Y})$ of the vector field $\mathbf{f}$ and initial value $\mathbf{x}_0$ from observations $\mathbf{Y}$. The posterior is intractable due to the non-linear integration map $\mathbf{x}_0 \overset{\mathbf{f}}{\mapsto} \mathbf{x}(t)$.
We use the stochastic variational inference (SVI) formulation for sparse GPs \citep{hensman2013gaussian} in this work. We introduce a factorized Gaussian posterior approximation for the inducing variables across state dimensions $q(\mathbf{U}) = \prod_{d=1}^{D}\mathcal{N}(\mathbf{u}_d|\mathbf{m}_d, \mathbf{Q}_d), \mathbf{u}_d \in \mathbb{R}^M$ where $\mathbf{m}_d \in \mathbb{R}^M, \mathbf{Q}_d \in \mathbb{R}^{M\times M}$ are the mean and covariance parameters of the variational Gaussian posterior approximation for the inducing variables. We treat the inducing locations $\mathbf{Z}$ as optimized hyperparameters.
The posterior distribution for the variational approximation can be written as
\begin{align}
q(\mathbf{f}) &= \int p(\mathbf{f}|\mathbf{U}) q(\mathbf{U}) d\mathbf{U} \\
&\hspace{-8mm}= \int \mathcal{N}\left(\mathbf{f} | \mathbf{A} \mathrm{vec}(\mathbf{U}), \mathbf{K}_{\mathbf{X}\X} - \mathbf{A} \mathbf{K}_{\mathbf{Z}\Z}\mathbf{A}^T \right) q(\mathbf{U}) d\mathbf{U}. \label{eq:inducing_posterior_gp}
\raisetag{2\normalbaselineskip}
\end{align}
The posterior inference goal then translates to estimating the posterior $p(\mathbf{f}, \mathbf{U}, \mathbf{x}_0 | \mathbf{Y})$ of the inducing points $\mathbf{U}$ and initial state $\mathbf{x}_0$. Under variational inference this learning objective
\begin{align}
\argmin_{q} \: \KL\big[ \, q(\mathbf{f},\mathbf{U},\mathbf{x}_0) \, || \, p(\mathbf{f},\mathbf{U},\mathbf{x}_0|\mathbf{Y}) \, \big]
\end{align}
translates into maximizing the evidence lowerbound (ELBO) \citep{blei2017variational},
\begin{align}
\log p(\mathbf{Y}) &\ge \sum_{i=1}^N \overbrace{\mathbb{E}_{q(\mathbf{f}, \mathbf{x}_0)} \log p(\mathbf{y}_i | \mathbf{f}, \mathbf{x}_0)}^{\text{variational likelihood}} - \overbrace{\KL[ q(\mathbf{U}) || p(\mathbf{U})]}^\text{inducing KL} \notag \\
&\quad - \underbrace{\KL[ q(\mathbf{x}_0) || p(\mathbf{x}_0)]}_\text{initial state KL},
\end{align}
where we also assume variational approximation $q(\mathbf{x}_0) = \mathcal{N}(\mathbf{a}_0, {\Sigma}_0)$ for the initial state $\mathbf{x}_0$. See supplementary section 1.1 for detailed derivations of the above equations.
\begin{figure*}[!h]
\centering
\begin{subfigure}[b]{0.9\columnwidth}
\centering
\includegraphics[width=1.0\textwidth]{plots/ils2_nonshooting_illustration.pdf}
\caption{The full model formulation}
\end{subfigure}
\qquad
\begin{subfigure}[b]{0.9\columnwidth}
\centering
\includegraphics[width=1.0\textwidth]{plots/ils3_shooting_illustration.pdf}
\caption{Shooting augmentation}
\end{subfigure}
\caption{Illustrations of GPODE formulations: the full model formulation (a) follows the long trajectory integration, whereas the shooting version (b) splits the long trajectory into multiple short subintervals.}
\label{fig:shooting_illustration}
\end{figure*}
\subsection{Sampling ODEs from Gaussian processes}
The Picard-Lindel\"{o}f theorem \citep{lindelof} ensures valid ODE systems define unique solutions to the initial value problem (IVP) \eqref{eq:odeproblemtraj}. In order to sample valid state trajectories for the IVP, we need to efficiently sample GP functions $\mathbf{f}(\cdot) \sim q(\mathbf{f})$ \eqref{eq:inducing_posterior_gp}. This way, we can evaluate the sample function $\mathbf{f}(\mathbf{x}(t))$ at arbitrary states $\mathbf{x}(t)$ encountered during ODE forward integration, while accounting for both the inducing and interpolation distributions of Equation \eqref{eq:inducing_posterior_gp}. Unfortunately, function-space sampling of such GPs has prohibitive cubic complexity \citep{rasmussen2006gaussian,pmlr-v108-ustyuzhaninov20a}, while the more efficient weight-space sampling with Fouriers cannot accurately express the posterior \eqref{eq:inducing_posterior_gp} \citep{wilson2020efficiently}.
We use the decoupled sampling that decomposes the posterior into two parts \citep{wilson2020efficiently},
\begin{align}
\label{eq:decoupled_conditional}
\overbrace{\mathbf{f}(\mathbf{x})|\mathbf{U}}^\text{posterior} &= \overbrace{\mathbf{f}(\mathbf{x})}^\text{prior} + \overbrace{K(\mathbf{x},\mathbf{Z})K(\mathbf{Z},\mathbf{Z})^{-1}(\mathbf{U} - \mathbf{f}_{\mathbf{Z}}))}^\text{update}. \\
&\approx \sum_{i=1}^{S} \mathbf{w}_i \boldsymbol{\phi}_i(\mathbf{x}) + \sum_{j=1}^{M} \boldsymbol\nu_j K(\mathbf{x}, \mathbf{z}_j),
\end{align}
where we use $S$ Fourier bases $\boldsymbol{\phi}_i(\cdot)$ with $\mathbf{w}_i \sim \mathcal{N}(\mathbf{0},I)$ \citep{rahimi2007random,brault2016random} to represent the stationary prior, and function basis $K(\cdot,\mathbf{z}_j)$ for the posterior update with $\boldsymbol\nu = K(\mathbf{Z},\mathbf{Z})^{-1}(\mathbf{U} - \boldsymbol{\Phi} \mathbf{W})$, $\boldsymbol{\Phi} = \boldsymbol{\phi}(\mathbf{Z}) \in \mathbb{R}^{M \times S}, \mathbf{W} \in \mathbb{R}^{S \times D}$. By combining these two steps, we can accurately evaluate functions from the posterior \eqref{eq:inducing_posterior_gp} in linear time at arbitrary locations. We refer the reader to the supplementary section 1.2 for more details. We note that concurrent works by \citet{mikheeva2021aligned} and \citet{ensinger2021symplectic} also utilize the decoupled-sampling to infer ODE posteriors with GPs.
\begin{figure*}[t]
\centering
\includegraphics[width=0.8\textwidth]{plots/exp1_vdp_illustration_without_gm.pdf}
\caption{Learning the 2D Van der Pol dynamics \textbf{(a)} with alternative methods \textbf{(b-d)}. Column 1 shows the vector fields while columns 2 and 3 show the state trajectories $x_1(t)$ and $x_2(t)$. GPODE learns the posterior accurately.}
\label{fig:vdp_illustration}
\end{figure*}
\subsection{Augmenting the ODE model with shooting system}
A key bottleneck in ODE modeling is the poor gradient descent performance over long integration times $\mathbf{x}_{0:T}$, which can exhibit vanishing or exploding gradients \citep{haber2017,choromanski2020,kim2021stiff}. Earlier approaches tackled this issue mainly with more accurate numerical solvers \citep{zhuang2020,zhuang2021mali}. The nonlinearity of the integration map $\mathbf{x}_0 \overset{\mathbf{f}}{\mapsto} \mathbf{x}_t$ motivates us to instead propose to segment the full integration $\mathbf{x}_{0:T}$ into short segments, which are easier to optimize and can be trivially parallelized \citep{aydogmus2021modified}. This is called the \emph{multiple shooting} method in optimal control literature \citep{osborne1969,bock1984}, in the context of parameter estimation of ODEs \citep{vandomselaar1975nonlinear,bock1983recent}; see \citet{diehl2017} and \citet{peifer2005parameter} for reviews. We introduce probabilistic shooting for the Gaussian process posterior inference of ODEs.
We begin by introducing shooting state variables $\mathbf{S} = (\mathbf{s}_0,\mathbf{s}_1,\ldots,\mathbf{s}_{N-1})$, $\mathbf{s}_i \in \mathbb{R}^D$, and segment the continuous state function $\mathbf{x}(t;\mathbf{x}_0)$ \eqref{eq:odeproblemtraj} into $N$ segments $\{(\mathbf{s}_{i-1},\mathbf{x}(t_i; \mathbf{s}_{i-1}))\}_{i=1}^{N}$ that branch from the shooting variables $\mathbf{s}_{i-1}$ (See Figure \ref{fig:shooting_illustration});
\begin{align}
\mathbf{x}(t_i; \mathbf{s}_{i-1}) &= \mathbf{s}_{i-1} + \int_{t_{i-1}}^{t_i} \mathbf{f}(\mathbf{x}(\tau)) d\tau \label{eq:shooting_ivps}.
\end{align}
In addition, every shooting variable is approximately matched with the ODE state evolution from the previous shooting state,
\begin{align}
\mathbf{s}_i &= \mathbf{x}(t_i; \mathbf{s}_{i-1}) + \boldsymbol{\xi} \label{eq:shooting_constraints},
\end{align}
where $\boldsymbol{\xi} \in \mathbb{R}^D$ represents the tolerance parameter controlling the shooting approximation. The augmented system is equivalent to the original ODE system in case the constraints $\mathbf{s}_i = \mathbf{x}(t_i; \mathbf{s}_{i-1})$ are satisfied exactly at the limit $\boldsymbol{\xi} \to \mathbf{0}$. We place a Gaussian prior over the tolerance parameter $\boldsymbol{\xi} \sim \mathcal{N}(\mathbf{0}, \sigma^2_\xi\textbf{\textrm{I}})$, which translates into the following prior over shooting variables
\begin{align}
p(\mathbf{s}_i|\mathbf{s}_{i-1}) &= \mathcal{N}(\mathbf{s}_i|\mathbf{x}(t_i; \mathbf{s}_{i-1}), \sigma^2_\xi \textbf{\textrm{I}}). \label{eq:shooting_prior}
\end{align}
Further, the joint probability of the augmented model after placing a GP prior over the vectorfield $\mathbf{f}$ can be written as
\begin{align}
p(\mathbf{Y},\mathbf{S},\mathbf{f}) &= \prod\limits_{i=1}^{N} p(\mathbf{y}_{i}|\mathbf{s}_{i-1}, \mathbf{f}) \prod\limits_{i=1}^{N-1}p(\mathbf{s}_{i}|\mathbf{s}_{i-1}, \mathbf{f})p(\mathbf{s}_0) p(\mathbf{f}).
\end{align}
\subsection{Variational inference for the augmented model}
To infer the augmented posterior $p(\mathbf{f},\mathbf{U}, \mathbf{S}| \mathbf{Y})$ we introduce variational approximation for the shooting variables $q(\mathbf{S}) = q(\mathbf{s}_0) \cdots q(\mathbf{s}_{N-1})$, where each distribution $q(\mathbf{s}_i) = \mathcal{N}(\mathbf{s}_i|\mathbf{a}_i,{\Sigma}_i)$ is a Gaussian. This results in the joint variational approximation
\begin{align}
q(\mathbf{S},\mathbf{f},\mathbf{U}) &= \prod_{i=0}^{N-1} q(\mathbf{s}_i) p(\mathbf{f}|\mathbf{U}) q(\mathbf{U}),
\end{align}
and the following evidence lower bound for the shooting model,
\begin{align}
\L_{\mathrm{shooting}} &= \sum_{i=1}^N \mathbb{E}_{q(\mathbf{s}_{i-1},\mathbf{f})} \Big[\log p(\mathbf{y}_i|\mathbf{s}_{i-1}, \mathbf{f})\Big] \nonumber\\
&\hspace{-15mm} + \sum_{i=1}^{N-1} \mathbb{E}_{q(\mathbf{s}_{i},\mathbf{s}_{i-1},\mathbf{f})} \Big[ \log p\left(\mathbf{s}_i | \mathbf{s}_{i-1}, \mathbf{f} \right) \Big] - \mathbb{E}_{q(\mathbf{s}_i)} \Big[ \log q(\mathbf{s}_{i}) \Big] \nonumber\\
&\hspace{-15mm} - \KL[q(\mathbf{s}_0) \, || \, p(\mathbf{s}_0)] - \KL[ q(\mathbf{U}) \, || \, p(\mathbf{U})].
\end{align}
The ELBO consists of an expected log-likelihood term, which matches the state evolution \eqref{eq:shooting_ivps} from every shooting variable to the corresponding observation. In addition, the posterior approximation for every shooting variable is also matched with the ODE evolution of the approximated posterior of the previous shooting state, leading to corresponding cross-entropy and entropy terms.
The ELBO for the augmented shooting model requires solving only the short segments $\eqref{eq:shooting_ivps}$ with simpler integration maps, thus great at mitigating problems with vanishing/exploring gradients. Since the involved numerical ODE integrations can be done in parallel, the shooting model is also computationally faster than the full model in practice. See supplementary section 1.3 for a plate diagram and detailed derivation of the approach.
\begin{table*}[t]
\centering
\caption{VDP system learning performance on extrapolation task with observations on regular (task 1) and irregular time intervals (task 2). We report mean $\pm$ standard error over 5 runs from different random initialization, the best values bolded. ($\uparrow$): higher is better, ($\downarrow$) lower is better}
\resizebox{0.7\textwidth}{!}{
\begin{tabular}{l c c c c}
\toprule
& \multicolumn{2}{c}{Task 1: Regular time-grid} & \multicolumn{2}{c}{Task 2: Irregular time-grid}\\
\cmidrule(lr){2-3}
\cmidrule(lr){4-5}
& MNLL ($\downarrow$)& MSE ($\downarrow$)& MNLL ($\downarrow$) & MSE ($\downarrow$) \\
\midrule
GP gradient matching & $1.47 \pm 0.02$ & $1.27 \pm 0.01$ & - & -\\
NeuralODE & - & $2.45 \pm 0.18$ & - & $2.26 \pm 0.09$\\
npODE & $7.42 \pm 1.88$ & $0.29 \pm 0.06$ & $11.82 \pm 2.78$ & $1.24 \pm 0.20$ \\
GPODE & $\mathbf{0.60 \pm 0.03}$ & $\mathbf{0.13 \pm 0.01}$ & $\mathbf{0.41 \pm 0.18}$ & $\mathbf{0.21 \pm 0.07}$ \\
\bottomrule
\end{tabular}
}
\label{table:vdp_illustration}
\end{table*}
\begin{figure*}[t]
\centering
\includegraphics[width=0.9\textwidth]{plots/exp2_fhn_interpolation.pdf}
\caption{Learning FHN with gaps in data (gray region). GPODE results in a useful posterior even during missing data, while both NeuralODE and npODE result in some biases.}
\label{fig:fhn_interpolation}
\end{figure*}
\section{Experiments}
\label{section:experiments}
We validate the proposed method on Van der Pol (VDP) and FitzHugh–Nagumo (FHN) systems and also on the task of learning human motion dynamics (MoCap). We use 16 inducing points in VDP and FHN experiments and 100 inducing points for the MoCap experiments. Except for the NeuralODE model, we assume Gaussian observation likelihood, and infer the unknown noise scale parameter from the training data. All the experiments use squared exponential kernel with ARD. Along with the variational parameters, kernel lengthscales, signal variance, noise scale, and inducing locations are optimized as hyperparameters. In addition, for the shooting model, we fix the constraint tolerance parameter to a small value $\sigma^2_\xi = 1e^{-6}$ consistently across all the experiments.
\begin{table}
\caption{Missing data results on the FHN system.}
\centering
\resizebox{0.8\columnwidth}{!}{
\begin{tabular}{lcc}
\toprule
& MNLL ($\downarrow$) & MSE ($\downarrow$) \\
\midrule
NeuralODE & - & $0.18 \pm 0.00$ \\
npODE & $6.49 \pm 1.49$ & $0.08 \pm 0.01$ \\
GPODE & $\mathbf{0.09 \pm 0.05}$ & $\mathbf{0.07 \pm 0.02}$ \\
\bottomrule
\end{tabular}
}
\label{table:fhn_interpolation}
\end{table}
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth]{plots/exp3_vdp_seqlen.pdf}
\caption{Varying sequence length and observation noise: shooting formulation makes GPODE feasible for long sequences, outperforming the non-shooting version and competing methods.}
\label{fig:seqlen_illustration}
\end{figure*}
We use the implicit \texttt{dopri5} solver with tolerance parameters \texttt{rtol}$=1e^{-5}$ and \texttt{atol}$=1e^{-5}$, and use the adjoint method for computing loss gradients with \texttt{torchdiffeq}\footnote{\url{https://github.com/rtqichen/torchdiffeq}} package \citep{chen2018neural}. All the experiments are repeated 5 times with random initialization, and means and standard errors are reported over multiple runs. We measure the predictive performance of different models using mean squared error (MSE) and mean negative log likelihood (MNLL) metrics.
\subsection{Learning Van der Pol dynamics}
We first illustrate the effectiveness of the proposed method by inferring the vector field posterior on a two-dimensional VDP (see Figure \ref{fig:vdp_illustration}),
\begin{align}
\dot{x}_1 = x_2, \quad \dot{x}_2 = -x_1 + 0.5 x_2 (1-x_1^2).
\end{align}
We simulate a trajectory of 50 states following the true system dynamics from the initial state $\left(x_1(0), x_2(0)\right) = \left(-1.5, 2.5 \right)$, and add Gaussian noise with $\sigma^2=0.05$ to generate the training data. We explore two scenarios with training time interval $t \in [0,7]$ and forecasting interval $t \in [7,14]$: (1) over a regularly sampled time grid, (2) over an irregular grid using uniform random sampling of time points. Task (2) demonstrates one of the key advantages of continuous-time models with the ability to handle irregular data. We compare our GPODE model with npODE \citep{heinonen2018learning}, NeuralODE \citep{chen2018neural}, and gradient matched GPs \citep{ridderbusch2020learning}.
Figure \ref{fig:vdp_illustration}(b) shows that GPODE learns a vector field posterior whose posterior mean closely matches the ground truth, with low variance (\textcolor{blue}{blue regions}) near the observed data. The posterior reduces back to the prior away from data (\textcolor{orange}{orange regions}), indicating a good uncertainty characterization. The npODE seems to overfit, while the gradient matching GP cannot fit the model adequately. NeuralODE learns an appropriate vector field, but its long-term forecasting is biased. The NeuralODE and npODE models do not represent uncertainty, while the gradient matching GP posterior is poorly fit. A quantitative evaluation of the model fits in Table \ref{table:vdp_illustration} indicates superior performance of GPODE. In addition, we also verify that GPODE performs better than the competing methods in terms of quantitative evaluation of the inferred velocity field compared against the ground truth (see supplementary for details).
\begin{figure}[t]
\centering
\includegraphics[width=0.8\columnwidth]{plots/exp3_shooting_runtime_multiple.pdf}
\caption{Optimization efficiency with GPODE models.}
\label{fig:gpode_shooting_efficiency}
\end{figure}
\subsection{Learning with missing observations}
We illustrate the usefulness of learning Bayesian ODE posteriors under missing data with the FHN oscillator
\begin{align}
\dot{x}_1 &= 3(x_1 - x_1^3/3 + x_2), \\ \nonumber
\dot{x}_2 &= (0.2 - 3x_1 - 0.2 x_2)/3.
\end{align}
We generate a training sequence by simulating 25 regularly-sampled time points from $t \in [0, 5.0]$ with added Gaussian noise with $\sigma^2 = 0.025$. We remove all observations at the quadrant $x_1>0, x_2<0$ and evaluate model accuracy in this region. Figure \ref{fig:fhn_interpolation} shows that all models adequately learn smooth forecasts of missing states. The point estimates of npODE and NeuralODE have biases, while the GPODE posterior captures the uncertainty well (See Table \ref{table:fhn_interpolation}).
\subsection{Learning long trajectories with the shooting formulation}
We demonstrate the necessity of the shooting formulation for working with long training trajectories. We use the VDP system with four observations per unit of time for $T = (25,40,55)$ corresponding to $N = (100,160,220)$ observed states. We also vary the observation variance as $\sigma^2 = (0.01,0.05,0.1)$ and test the model for forecasting additional 50 time points.
Figure \ref{fig:seqlen_illustration} demonstrates that vanilla-GPODE and NeuralODE fail to fit the data with long sequences on all noise levels. In contrast, inference for the shooting model is successful in all settings. The npODE is remarkably robust to long trajectories. All methods use high-performance gradient matching initialization for a realistic study (see supplementary)
Figure \ref{fig:gpode_shooting_efficiency} shows a runtime trace comparison between vanilla GPODE and the shooting variant in wall-clock time for a fixed budget of 2000 optimization steps on the VDP system with $N=100$, $T=25$ and $\sigma^2=0.01$. The shooting model converges approximately 10 times faster. The speedup stems from the parallelization of the shooting ODE solver, since the shooting method splits the full IVP problem into numerous short and less non-linear IVPs. In addition, the shooting method relaxes the inference problem with its auxiliary augmentation. This experiment was conducted on a system with AMD Ryzen 5 3600 processor and Nvidia GeForce GTX 1660S GPUs.
\begin{figure*}[t]
\centering
\includegraphics[width=0.8\textwidth]{plots/exp4_mocap_39_long.pdf}
\caption{Learning the walking dynamics of subject \texttt{39}: The true dynamics and predicted dynamics (mean) for the first three components in PCA space are shown in (a). Corresponding trajectories in the observation space for 6 different sensors are shown in (b) (We do not plot the observation noise variance)}
\label{fig:mocap_illustration}
\end{figure*}
\subsection{Learning human motion dynamics}
We learn the dynamics of human motion from noisy experimental data from CMU MoCap database for three subjects, \texttt{09}, \texttt{35} and \texttt{39}. The dataset consists of 50 sensor readings from different parts of the body while walking or running. We follow the preprocessing of \citet{4359316} and center the data. The dataset was further split into train, test, and validation sequences. We observed that both the NeuralODE and npODE models suffer from over-fitting, and we remedy this by applying early stopping by monitoring the validation loss during optimization.
We project the original 50-dimensional data into a 5-dimensional latent space using PCA and learn the dynamics in the latent space \citep{heinonen2018learning}. To compute the data likelihood, we project the latent dynamics back to the original data space by inverting the PCA. We divide the experiment into sub-tasks MoCap-short and MoCap-long, based on the length of the sequence considered for model training (see the supplementary section for more details on the dataset and experimental setup). We measure the predictive performance on unseen test sequences in both tasks.
Table \ref{table:mocap} indicates that GPODE outperforms the competing npODE and NeuralODE methods. Figure \ref{fig:mocap_illustration} visualizes the predicted dynamics for a test sequence. The GPODE variants have reasonable posterior uncertainties, while NeuralODE and npODE tend to be overconfident and make more mistakes (see Figure \ref{fig:mocap_illustration} (b), sensors \texttt{05}, \texttt{41} and \texttt{47}) . We note that some variations in the data space cannot be accurately estimated due to the low-dimensional PCA projection.
\begin{table*}[ht]
\centering
\caption{Test MNLL and MSE metrics for dynamics prediction task on CMU MoCap dataset.}
\resizebox{0.9\textwidth}{!}{
\begin{tabular}{c l c c c c c c}
\toprule
\multirow{2}*{Metric} & \multirow{2}*{Method} & \multicolumn{2}{c}{Subject 09}
& \multicolumn{2}{c}{Subject 35}
& \multicolumn{2}{c}{Subject 39} \\
\cmidrule(lr){3-4}
\cmidrule(lr){5-6}
\cmidrule(lr){7-8}
& & short & long
& short & long
& short & long \\
\midrule
\multirow{3}*{MNLL($\downarrow$)}
& npODE & $2.09 \pm 0.01$ & $1.97 \pm 0.10$ & $1.78 \pm 0.09$ & $1.71 \pm 0.06$ & $2.06 \pm 0.04$ & $1.89 \pm 0.06$ \\
& GPODE-vanilla & $1.30 \pm 0.02$ & $1.26 \pm 0.02$ & $1.27 \pm 0.04$ & $1.39 \pm 0.04$ & $1.29 \pm 0.01$ & $\mathbf{1.13 \pm 0.01}$ \\
& GPODE-shooting & $\mathbf{1.19 \pm 0.02}$ & $\mathbf{1.14 \pm 0.02}$ & $\mathbf{1.25 \pm 0.06}$ & $\mathbf{1.08 \pm 0.02}$ & $\mathbf{1.25 \pm 0.01}$ & $1.36 \pm 0.02$ \\
\midrule
\multirow{3}*{MSE($\downarrow$)} & NeuralODE & $27.3 \pm 2.8$ & $21.0 \pm 2.8$ & $26.4 \pm 0.6$ & $21.4 \pm 0.1$ & $111.7 \pm 15.2$ & $41.3 \pm 3.8$ \\
& npODE & $53.5 \pm 18.5$ & $17.7 \pm 1.6$ & $29.1 \pm 3.4$ & $16.1 \pm 3.7$ & $90.5 \pm 21.6$ & $41.0 \pm 5.3$ \\
& GPODE-vanilla & $15.78 \pm 0.6$ & $12.6 \pm 1.1$ & $16.1 \pm 0.9$ & $15.5 \pm 0.7$ & $\mathbf{20.7 \pm 1.2}$ & $23.6 \pm 1.8$ \\
& GPODE-shooting & $\mathbf{9.1 \pm 0.3}$ & $\mathbf{8.3 \pm 1.2}$ & $\mathbf{10.1 \pm 0.8}$ & $\mathbf{11.7 \pm 0.7}$ & $26.7 \pm 0.6$ & $\mathbf{21.1 \pm 2.8}$ \\
\bottomrule
\end{tabular}
}
\label{table:mocap}
\end{table*}
\section{Conclusion and Discussion}
We proposed a novel model for Bayesian inference of ODEs using Gaussian processes. With this approach, one can model unknown ODE systems directly from the observational data and learn posteriors of the continuous-time vector fields. In contrast, earlier works produce point estimate solutions. We believe this to be a significant addition to the data-descriptive ODE modeling methods, especially for applications where uncertainty quantification is critical. Many conventional machine learning algorithms have been interpreted and modeled as continuous-time dynamical systems, with applications to generative modeling \citep{grathwohl2019ffjord} and probabilistic alignment \citep{pmlr-v108-ustyuzhaninov20a}, among others. However, scaling GPs to high-dimensional datasets (such as images) can be a bottleneck. The applicability of the proposed model as a plug-in extension for these applications can be studied as part of future work.
We also highlighted a problem of learning black-box ODE models on long trajectories and proposed a probabilistic shooting framework enabling efficient inference on such tasks. This framework can be applied to other existing approaches, such as NeuralODEs. However, the proposed shooting augmentation introduces model approximation and involves approximating inference over auxiliary shooting variables. Hence the benefits of the shooting augmentation can be task specific, especially on short sequences. More rigorous assessment of the shooting formulation can be considered as part of future work. One could also consider improving the variational approximations analogously to the works \citep{ialongo2019overcoming} on discrete domains.
\newpage
\bibliographystyle{plainnat}
|
3,212,635,537,929 | arxiv | \section{Introduction}
The notion of interpretability between theories of mathematics has been studied using the framework of modal logic.
Visser~\cite{Vis88,Vis90} introduced the logic $\IL$ which is an extension of the logic $\GL$ of provability in the language of $\GL$ augmented by the binary modal operator $\rhd$.
In this framework, the formula $A \rhd B$ is intended as ``$T + A$ is interpretable in $T + B$'' for some suitable theory $T$.
Visser also introduced extensions $\ILM$ and $\ILP$ of $\IL$, and it has been shown that these logics are complete with respect to such arithmetical interpretations.
In particular, Visser~\cite{Vis90} proved that the logic $\ILP$ which is obtained from $\IL$ by adding the persistence principle $\PP$: $A \rhd B \to \Box(A \rhd B)$ as an axiom is arithmetically complete for finitely axiomatizable appropriate theories $T$.
De Jongh and Sambin~\cite{Sam76} independently proved that $\GL$ has the fixed point property (FPP), that is, for any modal formula $A(p)$ in the language of $\GL$, if each occurrence of $p$ in $A(p)$ is in some scope of $\Box$, then there exists a modal formula $F$ such that $\var(F) \subseteq \var(A(p)) \setminus \{p\}$ and $F \leftrightarrow A(F)$ is provable in $\GL$, where $\var(F)$ is the set of all propositional variables contained in $F$.
This is a modal counterpart of the Fixed Point Lemma for theories of arithmetic used in a standard proof of G\"odel's incompleteness theorems.
De Jongh and Visser~\cite{DeJVis91} proved that the de Jongh--Sambin fixed point theorem can be extended to the language of $\IL$, and that $\IL$ has FPP.
The authors have analyzed the question of which sublogics of $\IL$ are sufficient to enjoy FPP.
First,~\cite[Definition 2.1]{KO21} introduced the logic $\IL^-$ as a basis for semantical investigations of sublogics of $\IL$, and proved the completeness and the incompleteness of several logics between $\IL^-$ and $\IL$ with respect to some relational semantics.
Then,~\cite{IKO20} studied FPP for these sublogics.
In particular, it was proved that the fixed point theorem holds for the sublogic $\IL^-(\J{2}_+, \J{5})$, and that it does not hold for several sublogics of $\IL$.
Moreover,~\cite{IKO20} introduced a weaker version $\ell$FPP of the fixed point property, and proved that the sublogic $\IL^-(\J{4}, \J{5})$ has $\ell$FPP.
Furthermore,~\cite{Okawa} proved that every element of the infinite descending sequence $\langle \IL^-(\J{2}_+, \J{5}^n) \rangle_{n \geq 1}$ (resp.~$\langle \IL^-(\J{4}, \J{5}^n) \rangle_{n \geq 1}$) has FPP (resp.~$\ell$FPP).
These logics are not the only extensions of $\IL^-$ having FPP or $\ell$FPP.
De Jongh and Visser~\cite{DeJVis91} gave a simple proof that a sublogic $\mathbf{SR_1}$ of $\ILP$ has FPP.
Then, in our context, it immediately follows that $\IL^-(\J{4}_+, \PP)$ also has FPP, although $\IL^-(\J{4}_+)$ does not have FPP as shown in~\cite{IKO20}.
This observation suggests that the persistence principle $\PP$ may be logically well-suited to sublogics of $\IL$.
Our motivation for the present paper is to analyze the behavior and effects of the persistence principle $\PP$ over the logic $\IL^-$.
Our main research focus is the logic $\IL^-(\PP)$ obtained by adding $\PP$ into $\IL^-$.
In fact, although it was proved in~\cite{IKO20} that $\IL^-$ does not have $\ell$FPP, in Section~\ref{Sec_Pre}, we show that $\IL^-(\PP)$ enjoys $\ell$FPP.
Thus, it can be seen that the principle $\PP$ has a beneficial effect on $\IL^-$ from the viewpoint of the fixed point theorem.
We study the logic $\IL^-(\PP)$ from three aspects: proof theoretic, relational semantic, and arithmetical semantic aspects in Sections~\ref{Sec_CE},~\ref{Sec_SV}, and~\ref{Sec_AC}, respectively.
In Section~\ref{Sec_CE}, we investigate proof theoretic aspects of $\IL^-(\PP)$.
The systems of sequent calculus for the logics $\IL$ and $\ILP$ have been introduced and studied by Sasaki \cite{Sas02_1,Sas02_2,Sas03}.
In the present paper, we introduce the systems $(\IL^-)^G$ and $(\IL^-(\PP))^G$ of sequent calculus for the logics $\IL^-$ and $\IL^-(\PP)$, respectively.
We prove that the cut-elimination theorem holds for $(\IL^-(\PP))^G$.
Then, we prove that Maehara's method can be applied to the systems $(\IL^-)^G$ and $(\IL^-(\PP))^G$, that is, every sequent having a cut-free proof has a Craig interpolant.
Therefore, we conclude that $\IL^-(\PP)$ has the Craig interpolation property.
On the other hand, it was shown in~\cite{IKO20} that $\IL^-$ does not have the Craig interpolation property, and hence we obtain that the cut-elimination theorem does not hold for $(\IL^-)^G$.
From these results, we can see that $\PP$ is a principle that behaves proof theoretically well for $\IL^-$.
In Section~\ref{Sec_SV}, we investigate relational semantics of $\IL^-(\PP)$.
The logics $\IL$ and $\ILP$ have relational semantics which are extensions of Kripke semantics.
A triple $(W, R, \{S_w\}_{w \in W})$ is called a \textit{Veltman frame} if $(W, R)$ is a $\GL$-frame and for each $w \in W$, $S_w$ is a transitive and reflexive binary relation on $R[w] = \{x \in W \mid w R x\}$ satisfying $(\forall x, y \in W) (w R x \ \&\ x R y \Rightarrow x S_w y)$.
De Jongh and Veltman~\cite{DeJVel90} proved the completeness theorems of $\IL$ and $\ILP$ with respect to Veltman semantics.
The treatment of the family $\{S_w\}_{w \in W}$ in Veltman semantics is somewhat complicated, and then a simplified semantics was introduced by Visser~\cite{Vis88}.
A triple $(W, R, S)$ is called a \textit{simplified Veltman frame} or a \textit{Visser frame} if $(W, R)$ is a $\GL$-frame and $S$ is a transitive and reflexive binary relation on $W$ satisfying $(\forall x, y \in W) (x R y \Rightarrow x S y)$.
Then, Visser proved that $\IL$ and $\ILP$ are also complete with respect to simplified Veltman semantics.
The notion of Veltman frames can be generalized.
We say that a triple $(W, R, \{S_w\}_{w \in W})$ is an $\IL^-\!$-frame if $(W, R)$ is a $\GL$-frame and for each $w \in W$, $S_w \subseteq R[w] \times W$.
Then,~\cite{KO21} proved that $\IL^-$ is characterized by the class of all finite $\IL^-\!$-frames.
In this context, it seems natural to generalize the notion of simplified Veltman semantics as well.
Let us consider triples $(W, R, S)$ such that $(W, R)$ is a $\GL$-frame and $S$ is simply a binary relation on $W$.
Then, unlike the case of Veltman semantics, in the present paper, we prove that the logic $\IL^-(\PP)$ is valid in all such frames.
Moreover, we prove that $\IL^-(\PP)$ is characterized by the class of all finite such frames.
Therefore, $\IL^-(\PP)$ is the natural basis of a generalization of simplified Veltman semantics.
This result also shows that simplified Veltman semantics is a useful device for analyzing logics containing the principle $\PP$.
Also, as an application of that result, we show that $\IL^-(\PP)$ is faithfully embeddable into several extensions of the fusion $\GLK$ of $\GL$ and $\K$.
Finally, in Section~\ref{Sec_AC}, we investigate an arithmetical semantics of $\IL^-(\PP)$.
Inspired from our embedding result of $\IL^-(\PP)$ into bimodal logics, we introduce appropriate arithmetical semantics for $\IL^-(\PP)$ based on Fefermanian provability predicates satisfying L\"ob's derivability conditions (See Visser~\cite{Vis21}).
Then, by tracing the proof of the main result in~\cite{Kur18_2}, we prove that $\IL^-(\PP)$ is sound and complete with respect to such arithmetical semantics.
\section{Sublogics of $\IL$ and fixed point properties}\label{Sec_Pre}
In this section, we introduce the logics $\IL$ and $\IL^-$.
For several logics related to them, we summarize known results on the fixed point theorem and the Craig interpolation theorem.
We also show that the logic $\IL^-(\PP)$ has $\ell$FPP.
The language $\mathcal{L}(\rhd)$ consists of countably many propositional variables $p, q, r, \ldots$, the logical constant $\bot$, Boolean connectives $\neg, \land, \lor$, and $\to$, the unary modal operator $\Box$, and the binary modal operator $\rhd$.
The logical constant $\top$ and the modal operator $\Diamond$ are introduced as abbreviations for $\neg \bot$ and $\neg \Box \neg$, respectively.
The axioms of the logic $\IL$ in the language $\mathcal{L}(\rhd)$ are as follows:
\begin{description}
\item [L1] All tautologies in $\mathcal{L}(\rhd)$;
\item [L2] $\Box(A \to B) \to (\Box A \to \Box B)$;
\item [L3] $\Box(\Box A \to A) \to \Box A$;
\item [J1] $\Box(A \to B) \to A \rhd B$;
\item [J2] $(A \rhd B) \land (B \rhd C) \to A \rhd C$;
\item [J3] $(A \rhd C) \land (B \rhd C) \to (A \lor B) \rhd C$;
\item [J4] $A \rhd B \to (\Diamond A \to \Diamond B)$;
\item [J5] $\Diamond A \rhd A$.
\end{description}
The inference rules of $\IL$ are Modus Ponens $\dfrac{A \to B \quad A}{B}$ and Necessitation $\dfrac{A}{\Box A}$.
Several sublogics were introduced in~\cite{KO21}, and the basis for them is the logic $\IL^-$.
The axioms of $\IL^-$ are $\G{1}$, $\G{2}$, $\G{3}$, $\J{3}$, and $\J{6}$: $\Box A \leftrightarrow (\neg A) \rhd \bot$.
The inference rules of $\IL^-$ are Modus Ponens, Necessitation, $\R{1}$ $\dfrac{A \to B}{C \rhd A \to C \rhd B}$, and $\R{2}$ $\dfrac{A \to B}{B \rhd C \to A \rhd C}$.
Let $\IL^-(\Sigma_1, \ldots, \Sigma_n)$ be the logic obtained from $\IL^-$ by adding the schemes $\Sigma_1, \ldots, \Sigma_n$ as axioms.
The following schemes have been analyzed in previous studies. (See~\cite{DeJVis91,KO21,Okawa,Vis88}):
\begin{description}
\item [J2$_+$] $(A \rhd (B \lor C)) \land (B \rhd C) \to A \rhd C$;
\item [J4$_+$] $\Box (A \to B) \to (C \rhd A \to C \rhd B)$;
\item [J5$^n$] $\Diamond^n A \rhd A$ ($n \geq 1$);
\item [E1] $\Box(A \leftrightarrow B) \to (A \rhd C \leftrightarrow B \rhd C)$;
\item [E2] $\Box(A \leftrightarrow B) \to (C \rhd A \leftrightarrow C \rhd B)$;
\item [P] $A \rhd B \to \Box(A \rhd B)$.
\end{description}
For these axiom schemes, it has been shown that the following holds.
\begin{fact}[cf.~\cite{KO21,Okawa}]\label{IL-fact}\leavevmode
\begin{enumerate}
\item $\IL^- \vdash \Box(A \to B) \to (B \rhd C \to A \rhd C)$.
Hence, $\IL^- \vdash \mathbf{E1}$.
\item $\IL^-(\J{4}_+) \vdash \J{4} \land \mathbf{E2}$.
\item $\IL^-(\J{2}_+) \vdash \J{2} \land \J{4}_+$.
\item $\IL^-(\J{2}) \vdash \J{4}$.
\item $\IL^-(\J{5}^n) \vdash \J{5}^{n+1}$ for each $n \geq 1$.
\item The logics $\IL$, $\IL^-(\J{1}, \J{2}, \J{5})$, and $\IL^-(\J{1}, \J{2}_+, \J{5})$ are deductively equivalent.
\end{enumerate}
\end{fact}
For any $\mathcal{L}(\rhd)$-formula $A$, let $\var(A)$ denote the set of all propositional variables appearing in $A$.
\begin{defn}[FPP]\leavevmode
\begin{enumerate}
\item We say that a propositional variable $p$ is \textit{modalized} in an $\mathcal{L}(\rhd)$-formula $A$ if all occurrences of $p$ in $A$ are in scope of some modal operator $\Box$ or $\rhd$.
\item The logic $L$ is said to have the \textit{fixed point property (FPP)} if for any propositional variable $p$ and any $\mathcal{L}(\rhd)$-formula $A(p)$ in which $p$ is modalized, there exists an $\mathcal{L}(\rhd)$-formula $F$ such that $\var(F) \subseteq \var(A(p)) \setminus \{p\}$ and $L \vdash F \leftrightarrow A(F)$.
Such a formula $F$ is called a \textit{fixed point} of $A(p)$.
\end{enumerate}
\end{defn}
De Jongh and Visser~\cite[p.~47]{DeJVis91} proved that the logic $\IL$ has FPP.
This result was slightly improved by showing that the fixed point theorem holds without the axiom scheme $\J{1}$, namely, the logic $\IL^-(\J{2}_+, \J{5})$ also has FPP (\cite[Corollary 5.1 and Theorem 5.2]{IKO20}).
Moreover, it was proved that for each $n \geq 2$, the logic $\IL^-(\J{2}_+, \J{5}^n)$ has FPP (\cite[Theorem 3.21.1]{Okawa}).
Hence, each element of the infinite decreasing sequence $\langle \IL^-(\J{2}_+, \J{5}^n) \rangle_{n \geq 1}$ of sublogics of $\IL^-(\J{2}_+, \J{5})$ has FPP, and the intersection of all the logics of the sequence is exactly $\IL^-(\J{2}_+)$ (\cite[Proposition 3.11]{Okawa}).
It was also shown that the logic $\IL^-(\J{2}_+)$ does not have FPP (\cite[Corollary 6.2]{IKO20}).
De Jongh and Visser also proved the fixed point theorem in another way.
They proved that the logic containing $\G{1}$, $\G{2}$, $\G{3}$, $\PP$, $\mathbf{E1}$, and $\mathbf{E2}$, and is closed under Modus Ponens and Necessitation has FPP (\cite[Corollary 2.5.(a)]{DeJVis91}).
From their result, it immediately follows that the logic $\IL^-(\J{4}_+, \PP)$ has FPP.
For some technical reasons, the following weaker version of FPP was introduced in~\cite[Definition 3.8]{IKO20}:
\begin{defn}[$\ell$FPP]\leavevmode
\begin{enumerate}
\item We say that a propositional variable $p$ is \textit{left-modalized} in an $\mathcal{L}(\rhd)$-formula $A$ if $p$ is modalized in $A$ and for any subformula $B \rhd C$ of $A$, $p \notin \var(C)$.
\item The logic $L$ is said to have \textit{$\ell$FPP} if for any propositional variable $p$ and any $\mathcal{L}(\rhd)$-formula $A(p)$ in which $p$ is left-modalized, there exists an $\mathcal{L}(\rhd)$-formula $F$ such that $\var(F) \subseteq \var(A(p)) \setminus \{p\}$ and $L \vdash F \leftrightarrow A(F)$.
\end{enumerate}
\end{defn}
Then, it was proved that the logic $\IL^-(\J{4}, \J{5})$ has $\ell$FPP (\cite[Theorem 5.9]{IKO20}).
Moreover, as in the case of FPP, it was proved that for each $n \geq 2$, the logic $\IL^-(\J{4}, \J{5}^n)$ also has $\ell$FPP (\cite[Theorem 3.21.2]{Okawa}).
The fact that the logic $\IL^-(\J{4}_+, \PP)$ has FPP suggests that some weaker logics are expected to have $\ell$FPP.
Indeed, we prove:
\begin{thm}\label{lFPP}
The logic $\IL^-(\PP)$ has $\ell$FPP.
\end{thm}
In the proof, we use the following fact.
\begin{fact}[{\cite[Proposition~3.6]{IKO20}}]\label{Sbsti}
Let $A, B$, and $C(p)$ be $\mathcal{L}(\rhd)$-formulas.
\begin{enumerate}
\item If $p \notin \var(E)$ for every subformula $D \rhd E$ of $C(p)$, then
\[
\IL^- \vdash (A \leftrightarrow B) \land \Box (A \leftrightarrow B) \to \bigl(C(A) \leftrightarrow C(B)\bigr).
\]
\item If $p$ is left-modalized in $C(p)$, then
\[
\IL^- \vdash \Box(A \leftrightarrow B) \to \bigl(C(A) \leftrightarrow C(B)\bigr).
\]
\end{enumerate}
\end{fact}
\begin{proof}[Proof of Theorem~\ref{lFPP}]
As in a usual proof of the fixed point theorem, it suffices to find a fixed point of $A(p) \rhd B$ in which $p$ is left-modalized.
Let $F$ be $A(\top) \rhd B$, and we prove that $F$ is a fixed point of $A(p) \rhd B$ in $\IL^-(\PP)$.
Obviously, $\var(F) \subseteq \var\bigl(A(p) \rhd B\bigr) \setminus \{p\}$.
We show $\IL^-(\PP) \vdash F \leftrightarrow A(F) \rhd B$.
Since $\IL^- \vdash F \to (\top \leftrightarrow F)$, we have $\IL^- \vdash \Box F \to \Box(\top \leftrightarrow F)$.
By Fact~\ref{Sbsti}.2, we have $\IL^- \vdash \Box F \to \bigl(A(\top) \rhd B \leftrightarrow A(F) \rhd B\bigr)$.
Hence,
\begin{eqnarray}\label{eq2}
\IL^- \vdash \Box F \to (F \leftrightarrow A(F) \rhd B).
\end{eqnarray}
Therefore, $\IL^-(\PP) \vdash F \to A(F) \rhd B$.
On the other hand, $\IL^- \vdash A(F) \rhd B \to (\Box F \to F)$, and then $\IL^- \vdash \Box(A(F) \rhd B) \to \Box (\Box F \to F)$.
Hence, $\IL^- \vdash \Box(A(F) \rhd B) \to \Box F$.
We get $\IL^-(\PP) \vdash A(F) \rhd B \to \Box F$.
By (\ref{eq2}) again, $\IL^-(\PP) \vdash A(F) \rhd B \to F$.
We conclude $\IL^-(\PP) \vdash F \leftrightarrow A(F) \rhd B$.
\end{proof}
Notice that our proof of Theorem~\ref{lFPP} is essentially the same as de Jongh and Visser's proof of FPP for $\IL^-(\J{4}_+, \PP)$.
Then, our fixed points obtained in our proof are the same as ones given by de Jongh and Visser.
The properties FPP and $\ell$FPP are closely related to the Craig interpolation property.
\begin{defn}[CIP]
We say that a logic $L$ has the \textit{Craig interpolation property (CIP)} if for any $\mathcal{L}(\rhd)$-formulas $A$ and $B$, if $L \vdash A \to B$, then there exists an $\mathcal{L}(\rhd)$-formula $C$ such that $\var(C) \subseteq \var(A) \cap \var(B)$, $L \vdash A \to C$, and $L \vdash C \to B$.
\end{defn}
Areces, Hoogland, and de Jongh~\cite[Theorem 1]{AHD01} proved that $\IL$ has CIP.
Moreover, it was proved that $\IL^-(\J{2}_+, \J{5})$ has CIP, but some sublogics including $\IL^-$ do not have CIP (See \cite{IKO20}).
The failure of CIP for some logics was derived by showing the failure of FPP for these logics through the following fact.
\begin{fact}[{\cite[Lemmas 3.10 and 3.11]{IKO20}}]\label{CIPFPP}
Let $L$ be any extension of $\IL^-$ that is closed under substituting a formula for a propositional variable.
\begin{enumerate}
\item If $L$ has CIP, then $L$ has $\ell$FPP.
\item If $L$ is an extension of $\IL^-(\J{4}_+)$ and has CIP, then $L$ has FPP.
\end{enumerate}
\end{fact}
In the next section, we prove that $\IL^-(\PP)$ has CIP.
Then, we also obtain an alternative proof of Theorem~\ref{lFPP}.
\section{Proof theoretic aspects of $\IL^-(\PP)$}\label{Sec_CE}
In this section, we investigate proof theoretic aspects of $\IL^-(\PP)$.
This section consists of three subsections.
In the first subsection, we introduce the systems $(\IL^-)^G$ and $(\IL^-(\PP))^G$ of sequent calculi and prove that they exactly correspond to $\IL^-$ and $\IL^-(\PP)$, respectively.
The second subsection is devoted to proving the cut-elimination theorem for $(\IL^-(\PP))^G$.
In the last subsection, we show that Maehara's method for proving the existence of interpolants for sequents with a cut-free proof can be applied to $(\IL^-)^G$ and $(\IL^-(\PP))^G$.
As a consequence, we obtain that $\IL^-(\PP)$ enjoys CIP, but the cut-elimination theorem does not hold for $(\IL^-)^G$.
\subsection{The systems $(\IL^-)^G$ and $(\IL^-(\PP))^G$}
As an exception in the present paper, the language of our systems of sequent calculi does not contain the modal symbol $\Box$, and we assume that $\Box A$ is an abbreviation of $(\neg A )\rhd \bot$.
This corresponds to the scheme $\J{6}$.
Throughout this section, capital Greek letters $\Gamma, \Delta, \Sigma, \ldots$ always denote finite \textit{sets} of $\mathcal{L}(\rhd)$-formulas.
As usual, for each $\mathcal{L}(\rhd)$-formula $A$, the expressions $\Gamma, \Delta$ and $\Gamma, A$ denote $\Gamma \cup \Delta$ and $\Gamma \cup \{A\}$, respectively.
Let $\Box \Gamma$ be the set $\{\Box A \mid A \in \Gamma\}$.
A \textit{sequent} is an expression of the form $\Gamma \Rightarrow \Delta$.
\begin{defn}[Sequent calculus $(\IL^-)^G$]
The system $(\IL^-)^G$ is defined by the following initial sequents and inference rules:
\begin{description}
\item[Initial sequents:]
\begin{align*}
A \Rightarrow A & & \bot \Rightarrow
\end{align*}
\item[Logical rules:]
\begin{align*}
\infer[(wl)]{\Gamma, A \Rightarrow \Delta}{\Gamma \Rightarrow \Delta} & &
\infer[(wr)]{\Gamma \Rightarrow A, \Delta}{\Gamma \Rightarrow \Delta} \\
\infer[(\neg l)]{\Gamma, \neg A \Rightarrow \Delta}{\Gamma \Rightarrow A, \Delta} & &
\infer[(\neg r)]{\Gamma \Rightarrow \neg A, \Delta}{\Gamma, A \Rightarrow \Delta} \\
\infer[(\land l)]{\Gamma, A_1 \land A_2 \Rightarrow \Delta}{\Gamma, A_i \Rightarrow \Delta \ (i =1,2)} & &
\infer[(\land r)]{\Gamma \Rightarrow A \land B, \Delta}
{\Gamma \Rightarrow A, \Delta & \Gamma \Rightarrow B, \Delta} \\
\infer[(\lor l)]{\Gamma, A \lor B \Rightarrow \Delta}
{\Gamma, A \Rightarrow \Delta & \Gamma, B \Rightarrow \Delta} & &
\infer[(\lor r)]{\Gamma \Rightarrow A_1 \lor A_2, \Delta}{\Gamma \Rightarrow A_i, \Delta \ (i=1,2)} \\
\infer[(\to l)]{\Gamma, A \to B \Rightarrow \Delta}
{\Gamma \Rightarrow A, \Delta & \Gamma, B \Rightarrow \Delta} & &
\infer[(\to r)]{\Gamma \Rightarrow A \to B, \Delta}{\Gamma, A \Rightarrow B, \Delta}
\end{align*}
\item[Cut:]
\[
\infer[(cut)]{\Gamma, \Sigma \Rightarrow \Delta, \Pi}{\Gamma \Rightarrow \Delta, A & A, \Sigma \Rightarrow \Pi}
\]
\item[Modal rules:]
\[
\infer[(\Box)]{\Box \Gamma \Rightarrow \Box A}{\Box \Gamma, \Gamma, \Box A \Rightarrow A}
\quad
\infer[(\rhd)]{\{ X_i \rhd Y_i \mid i < n \} \Rightarrow A \rhd B}{ A \Rightarrow \{ X_i \mid i < n\} & \langle Y_i \Rightarrow B \rangle_{i < n} }
\]
\end{description}
In the rule $(cut)$, the formula $A$ is said to be the \textit{cut-formula}.
In the rule $(\rhd)$, $n$ is some natural number, which is possibly $0$.
Also, in the rule $(\rhd)$, $\langle Y_i \Rightarrow B \rangle_{i < n}$ indicates that all $n$ elements of this sequence are upper sequents of the rule, allowing for duplication.
\end{defn}
Note that our system of sequent calculus is set-based, and hence the structural rules are only $(wl)$ and $(wr)$.
If there is no room for misreading, repeated trivial applications of propositional rules are sometimes abbreviated by a double line.
Notice that the rule $(\Box)$ is known as the rule for $\GL$, that is, it corresponds to $\G{1}$, $\G{2}$, $\G{3}$, and Necessitation.
Also, the following derivation shows that the rule $\dfrac{\Gamma \Rightarrow A}{\Box \Gamma \Rightarrow \Box A}$ for the smallest normal modal logic $\K$ is admissible only from the rule $(\rhd)$: For $\Gamma = \{X_0, \ldots, X_{n-1}\}$,
\begin{equation*}
\infer[(\rhd)]{\{(\neg X_i) \rhd \bot \mid i < n\} \Rightarrow (\neg A) \rhd \bot}
{\infer={\neg A \Rightarrow \{\neg X_i \mid i < n\}}
{\{X_i \mid i < n\} \Rightarrow A}
&
\bot \Rightarrow \bot & \cdots & \bot \Rightarrow \bot
}.
\end{equation*}
For the sake of simplicity, we sometimes abbreviate the set $\{X_i \mid i < n\}$ of formulas as $\{X_i\}$, but we believe that this will not cause any confusions.
We prove that the system $(\IL^-)^G$ is equivalent to $\IL^-$.
\begin{prop}\label{IL^-equiv}
Let $A$ be any $\mathcal{L}(\rhd)$-formula.
\begin{enumerate}
\item If $\IL^- \vdash A$, then $(\IL^-)^G \vdash (\Rightarrow A)$.
\item If $(\IL^-)^G \vdash \Gamma \Rightarrow \Delta$, then $\IL^- \vdash \bigwedge \Gamma \to \bigvee \Delta$.
\end{enumerate}
\end{prop}
\begin{proof}
1. We prove the proposition by induction on the length of proofs in $\IL^-$.
Notice that the axiom scheme $\J{6}$ is built in $(\IL^-)^G$.
The following derivation shows that $(\IL^-)^G$ proves the sequents corresponding to the axiom scheme $\J{3}$.
\[
\infer[(\rhd): n = 2]{A \rhd C, B \rhd C \Rightarrow (A \lor B) \rhd C}
{\deduce{A \lor B \Rightarrow A, B}{\vdots}
&
C \Rightarrow C
&
C \Rightarrow C
}.
\]
Then, it suffices to show that the rules $\R{1}$ and $\R{2}$ are admissible in $(\IL^-)^G$.
\begin{description}
\item [R1]
\[
\infer[(\rhd): n = 1]{C \rhd A \Rightarrow C \rhd B}
{C \Rightarrow C
&
A \Rightarrow B
}.
\]
\item [R2]
\[
\infer[(\rhd): n = 1]{B \rhd C \Rightarrow A \rhd C}
{A \Rightarrow B
&
C \Rightarrow C
}.
\]
\end{description}
2. It is sufficient to show that the rule $(\rhd)$ is admissible in $\IL^-$.
Suppose that $\IL^- \vdash A \to \bigvee \{X_i\}$ and $\IL^- \vdash Y_j \to B$ for each $j < n$.
Since $\IL^- \vdash Y_i \to \bigvee \{Y_j\}$ for each $i < n$, we have $\IL^- \vdash X_i \rhd Y_i \to X_i \rhd \bigvee \{Y_j\}$ by the rule $\R{1}$.
Then, $\IL^- \vdash \bigwedge \{X_i \rhd Y_i \} \to \bigwedge \{X_i \rhd \bigvee \{Y_j\} \}$.
By the $n-1$ times applications of $\J{3}$, we obtain
\begin{equation}\label{seq1}
\IL^- \vdash \bigwedge \{X_i \rhd Y_i\} \to \left( \bigvee \{X_i\} \right) \rhd \bigvee \{Y_j\}.
\end{equation}
Since $\IL^- \vdash A \to \bigvee \{X_i\}$, by the rule $\R{2}$, we have
\begin{equation}\label{seq2}
\IL^- \vdash \left( \bigvee \{X_i\} \right) \rhd \bigvee\{Y_j\} \to A \rhd \bigvee \{Y_j\}.
\end{equation}
Since $\IL^- \vdash \bigvee \{Y_j\} \to B$, we obtain
\[
\IL^- \vdash A \rhd \bigvee \{Y_j\} \to A \rhd B
\]
by $\R{1}$.
By combining this with (\ref{seq1}) and (\ref{seq2}), we conclude
\[
\IL^- \vdash \bigwedge \{X_i \rhd Y_i\} \to A \rhd B.
\]
\end{proof}
We will prove the failure of the cut-elimination theorem for $(\IL^-)^G$ at the end of this section.
In contrast, when we add the persistence principle $\PP$ to $(\IL^-)^G$, the cut-elimination method works well.
We introduce $(\IL^-(\PP))^G$ the system of sequent calculus for $\IL^-(\PP)$.
For the sake of simplicity, throughout this section, $\Omega$ and $\Theta$, with subscripts in some cases, always denote finite sets of $\mathcal{L}(\rhd)$-formulas of the forms $C \rhd D$.
Then, $\IL^-(\PP) \vdash \bigwedge \Omega \to \Box \bigwedge \Omega$.
\begin{defn}[Sequent calculus $(\IL^-(\PP))^G$]
The system $(\IL^-(\PP))^G$ is obtained from the system $(\IL^-)^G$ by replacing the two modal rules $(\Box)$ and $(\rhd)$ with the following single modal rule $(\rhd_P)$:
\begin{center}
$\infer[(\rhd_P)]{\Omega, \{ X_i \rhd Y_i \mid i < n\} \Rightarrow A \rhd B}
{\Omega, \{ X_i \rhd Y_i \mid i < n\}, A \rhd B, A \Rightarrow \{ X_i \mid i < n\}
&
\langle Y_i \Rightarrow B \rangle_{i < n}
}.$
\end{center}
In the rule, the set $\Omega$ is possibly empty.
The elements of $\{X_i \rhd Y_i \mid i < n\}$ are said to be the \textit{principal formulas} of the rule.
Also, the formula $A \rhd B$ is said to be the \textit{diagonal formula} of the rule.
\end{defn}
The following proposition states the equivalence between $\IL^-(\PP)$ and $(\IL^-(\PP))^G$.
\begin{prop}
Let $A$ be any $\mathcal{L}(\rhd)$-formula.
\begin{enumerate}
\item If $\IL^-(\PP) \vdash A$, then $(\IL^-(\PP))^G \vdash (\Rightarrow A)$.
\item If $(\IL^-(\PP))^G \vdash \Gamma \Rightarrow \Delta$, then $\IL^-(\PP) \vdash \bigwedge \Gamma \to \bigvee \Delta$.
\end{enumerate}
\end{prop}
\begin{proof}
1. First, the following derivations show that rules $(\Box)$ and $(\rhd)$ in $(\IL^-)^G$ are admissible in $(\IL^-(\PP))^G$.
\begin{description}
\item [$(\Box)$] For $\Gamma = \{X_0, \ldots, X_{n-1}\}$,
\begin{equation*}
\footnotesize
\infer[(\rhd_P): \Omega = \varnothing]{\{(\neg X_i) \rhd \bot \mid i < n\} \Rightarrow (\neg A) \rhd \bot}
{\infer={\{(\neg X_i) \rhd \bot \mid i < n\}, (\neg A) \rhd \bot, \neg A \Rightarrow \{\neg X_i \mid i < n\}}
{\{(\neg X_i) \rhd \bot \mid i < n\}, \{X_i \mid i < n\}, (\neg A) \rhd \bot \Rightarrow A}
&
\bot \Rightarrow \bot & \cdots & \bot \Rightarrow \bot
}.
\normalsize
\end{equation*}
\item [$(\rhd)$]
\begin{equation*}
\infer[(\rhd_P): \Omega = \varnothing]{\{X_i \rhd Y_i \mid i < n\} \Rightarrow A \rhd B}
{\infer={\{X_i \rhd Y_i \mid i < n\}, A \rhd B, A \Rightarrow \{X_i \mid i < n\}}
{A\Rightarrow \{X_i \mid i < n\}}
&
\langle Y_i \Rightarrow B \rangle_{i < n}
}.
\end{equation*}
\end{description}
Then, Proposition~\ref{IL^-equiv}.1 shows that the sequents corresponding to theorems of $\IL^-$ are provable in $(\IL^-(\PP))^G$.
Then, it suffices to show that the sequents corresponding to $\PP$ are provable in $(\IL^-(\PP))^G$.
\begin{align*}
\infer[(\rhd_P): n = 0\ \text{and}\ \Omega = \{A \rhd B\}]{A \rhd B \Rightarrow (\neg(A \rhd B)) \rhd \bot}
{\infer[(\neg l)]{A \rhd B, (\neg(A \rhd B)) \rhd \bot, \neg(A \rhd B) \Rightarrow}
{\infer[(wl)]{A \rhd B, (\neg(A \rhd B)) \rhd \bot \Rightarrow A \rhd B}
{A \rhd B \Rightarrow A \rhd B}
}
}.
\end{align*}
2. It suffices to show that the rule $(\rhd_P)$ is admissible in $\IL^-(\PP)$.
Assume that $\IL^-(\PP)$ proves
\begin{equation*}
\bigwedge \Omega \land \bigwedge \{ X_i \rhd Y_i \} \land (A \rhd B) \land A \to \bigvee \{ X_i \}
\end{equation*}
and $Y_j \to B$ for each $j < n$.
Then,
\begin{align}\label{seq3}
\IL^-(\PP) & \vdash \bigwedge \Omega \land \bigwedge \{ X_i \rhd Y_i \} \land (A \rhd B) \to \left(A \to \bigvee \{ X_i \} \right), \nonumber\\
& \vdash \bigwedge \Box \Omega \land \bigwedge \Box \{ X_i \rhd Y_i \} \land \Box (A \rhd B) \to \Box \left(A \to \bigvee \{ X_i \}\right), \nonumber\\
& \vdash \bigwedge \Omega \land \bigwedge \{ X_i \rhd Y_i \} \land \Box (A \rhd B) \to \Box \left(A \to \bigvee \{ X_i \}\right).
\end{align}
Also, as in the proof of Proposition~\ref{IL^-equiv}.2, we have $\IL^- \vdash \bigwedge \{ X_i \rhd Y_i \} \to \left( \bigvee \{ X_i \} \right) \rhd \bigvee \{ Y_i \}$.
By Fact \ref{IL-fact}.1,
\begin{equation*}
\IL^- \vdash \Box \left(A \to \bigvee \{ X_i \} \right) \to \left( \left( \bigvee \{ X_i \} \right) \rhd \bigvee \{ Y_i \} \to A \rhd \bigvee \{ Y_i \} \right).
\end{equation*}
Thus, we obtain
\begin{equation*}
\IL^- \vdash \bigwedge \{ X_i \rhd Y_i \} \to \left( \Box \left(A \to \bigvee \{ X_i \} \right) \to A \rhd \bigvee \{ Y_i \} \right).
\end{equation*}
From this with (\ref{seq3}),
\begin{equation*}
\IL^-(\PP) \vdash \bigwedge \Omega \land \bigwedge \{ X_i \rhd Y_i \} \land \Box (A \rhd B) \to A \rhd \bigvee \{ Y_i \}.
\end{equation*}
Since $\IL^-(\PP) \vdash \bigvee \{ Y_i \} \to B$, by the rule $\R{1}$, $\IL^-(\PP) \vdash A \rhd \bigvee \{ Y_i \} \to A \rhd B$.
Hence, we obtain
\begin{equation*}
\IL^-(\PP) \vdash \bigwedge \Omega \land \bigwedge \{ X_i \rhd Y_i \} \land \Box (A \rhd B) \to A \rhd B.
\end{equation*}
Finally,
\begin{align}\label{seq4}
\IL^-(\PP) & \vdash \bigwedge \Omega \land \bigwedge \{ X_i \rhd Y_i \} \to ( \Box (A \rhd B) \to A \rhd B), \\
& \vdash \bigwedge \Box \Omega \land \bigwedge \Box \{ X_i \rhd Y_i \} \to \Box ( \Box (A \rhd B) \to A \rhd B), \nonumber \\
& \vdash \bigwedge \Omega \land \bigwedge \{ X_i \rhd Y_i \} \to \Box (A \rhd B), \nonumber \\
& \vdash \bigwedge \Omega \land \bigwedge \{ X_i \rhd Y_i \} \to A \rhd B. \tag{By (\ref{seq4})} \nonumber
\end{align}
\end{proof}
\subsection{Proof of the cut-elimination theorem}
In this subsection, we prove the cut-elimination theorem for $(\IL^-(\PP))^G$.
That is, if a sequent $\Gamma \Rightarrow \Delta$ is derivable in $(\IL^-(\PP))^G$, then it is also derivable without using $(cut)$s.
We prove the cut-elimination theorem by double induction on degree and height.
For an instance of $(cut)$ of the form
$$\infer[(cut)]{\Gamma, \Sigma \Rightarrow \Delta, \Pi}{\deduce{\Gamma \Rightarrow \Delta, A}{\pi} & \deduce{A, \Sigma \Rightarrow \Pi}{\sigma}}$$
where $\pi$ and $\sigma$ are cut-free subproofs, we define the degree and height of this application of the rule $(cut)$ as follows:
\begin{itemize}
\item The \textit{degree} of the $(cut)$ is the number of occurrences of connectives and operators in the cut-formula $A$.
\item The \textit{height} of the $(cut)$ is the sum of the maximum lengths of the subproofs $\pi$ and $\sigma$.
\end{itemize}
The usual cut-elimination procedure searches a topmost instance of the $(cut)$ rule and converts it into a cut-free proof of the same conclusion.
The conversion method is given by primary induction on the degree and secondary induction on the height.
If the cut-formula of such a topmost instance is one of the propositional variables, $\bot$, $\neg A$, $A \lor B$, $A \land B$, or $A \to B$, then it is verified in the standard way.
Thus, we focus only on the rule $(cut)$ where the cut-formula is of the form $A \rhd B$.
Even in this case, if at least one of the last applications of $\pi$ and $\sigma$ is a logical rule, then the secondary induction hypothesis goes well.
Therefore, the only essential case is that the last applications of $\pi$ and $\sigma$ are both $(\rhd_P)$.
Let us consider the following derivation:
\begin{equation*}
\infer[(cut)]
{\Omega_1, \Omega_2, \{ X_i \rhd Y_i \}, \{ Z_j \rhd W_j \} \Rightarrow C \rhd D}
{\deduce{\Omega_1, \{ X_i \rhd Y_i \} \Rightarrow A \rhd B}{\pi}
&
\deduce{\Omega_2, \{ Z_j \rhd W_j \}, A \rhd B \Rightarrow C \rhd D}{\sigma}
},
\end{equation*}
where $\pi$ and $\sigma$ are cut-free subproofs whose last applications are both $(\rhd_P)$.
If the cut-formula $A \rhd B$ is non-principal in the last application of $\sigma$, then the following argument shows that a cut-free proof of the lower sequent is easily obtained from the induction hypothesis.
Suppose that $\sigma$ is of the form:
\[
\infer[(\rhd_P)]{A \rhd B, \Omega_2, \{ Z_j \rhd W_j \} \Rightarrow C \rhd D}
{\deduce{A \rhd B, \Omega_2, \{ Z_j \rhd W_j \}, C \rhd D, C \Rightarrow \{ Z_j \}}{\sigma_L}
&
\langle \deduce{W_j \Rightarrow D}{\sigma_j} \rangle
}.
\]
From $\pi$ and $\sigma_L$, applying $(cut)$ with the cut-formula $A \rhd B$, we obtain a proof of
\[
\left(
\begin{array}{ccc}
\Omega_1, & \{ X_i \rhd Y_i \}, & C \rhd D, \\
\Omega_2, & \{ Z_j \rhd W_j \}, & C
\end{array}
\Rightarrow \{ Z_j \}
\right).
\]
By the induction hypothesis, this application of $(cut)$ is eliminable because its height is smaller than that of the original $(cut)$.
Therefore, we obtain a cut-free proof of the above sequent.
The required cut-free proof is obtained as follows:
\[
\infer[(\rhd_P)]{\Omega_1, \Omega_2, \{ X_i \rhd Y_i \}, \{ Z_j \rhd W_j \} \Rightarrow C \rhd D}
{
\deduce{\begin{array}{ccc}
\Omega_1, & \{ X_i \rhd Y_i \}, & C \rhd D, \\
\Omega_2, & \{ Z_j \rhd W_j \}, & C
\end{array}
\Rightarrow
\{ Z_j \}}{\vdots}
&
\deduce{\langle W_j \Rightarrow D \rangle}{\sigma_j}
}.
\]
Notice that the formulas in $\{ X_i \rhd Y_i \}$ are non-principal in this application of $(\rhd_P)$.
So, it suffices to prove the following main proposition.
\begin{prop}\label{mainP}
Suppose that the following proof is given:
\begin{equation*}
\infer[(cut)]
{\Omega_1, \Omega_2, \{ X_i \rhd Y_i \}, \{ Z_j \rhd W_j \} \Rightarrow C \rhd D}
{\deduce{\Omega_1, \{ X_i \rhd Y_i \} \Rightarrow A \rhd B}{\pi}
&
\deduce{\Omega_2, \{ Z_j \rhd W_j \}, A \rhd B \Rightarrow C \rhd D}{\sigma}
},
\end{equation*}
where $\pi$ and $\sigma$ are cut-free subproofs of the following forms respectively:
\begin{itemize}
\item $\pi: \infer[(\rhd_P)]
{\Omega_1, \{ X_i \rhd Y_i \} \Rightarrow A \rhd B}
{\deduce{\Omega_1, \{ X_i \rhd Y_i \}, A \rhd B, A \Rightarrow \{ X_i \}}{\pi_L}
&
\langle \deduce{Y_i \Rightarrow B}{\pi_i} \rangle
}$,
\item \footnotesize $\sigma: \infer[(\rhd_P)]
{\Omega_2, \{ Z_j \rhd W_j \}, A \rhd B \Rightarrow C \rhd D}
{\deduce{\Omega_2, \{ Z_j \rhd W_j \}, A \rhd B, C \rhd D, C \Rightarrow \{ Z_j \}, A}{\sigma_L}
&
\langle \deduce{W_j \Rightarrow D}{\sigma_j} \rangle
&
\deduce{B \Rightarrow D}{\sigma_B}}.$
\normalsize
\end{itemize}
Then, there exists a cut-free proof of $(\Omega_1, \Omega_2, \{ X_i \rhd Y_i \}, \{ Z_j \rhd W_j \} \Rightarrow C \rhd D)$.
\end{prop}
The remainder of this subsection is devoted to proving Proposition \ref{mainP}.
For this purpose, we prove some lemmas along the lines of a modification of Borga's argument \cite{Bor83}.
We assume that we already have the fixed cut-free proofs $\pi$, $\sigma$, $\pi_L$, $\pi_i (i<n)$, $\sigma_L$, $\sigma_j (j < m)$, and $\sigma_B$ as indicated in the statement of Proposition~\ref{mainP}.
We say that a proof is an \textit{$(A, B)^\star$-proof} if every cut-formula of the rule $(cut)$ occurring in the proof is either $A$ or $B$.
For the subproof $\pi_L$, we inductively define \textit{$(A \rhd B)$-explicit} sequents of $\pi_L$ as follows:
\begin{enumerate}
\item The root sequent $(\Omega_1, \{ X_i \rhd Y_i \}, A \rhd B, A \Rightarrow \{X_i\})$ is $(A \rhd B)$-explicit;
\item If $\dfrac{S_1 \quad \cdots \quad S_n}{S_0}$ is an application of some rule in $\pi_L$, $S_0$ is $(A \rhd B)$-explicit, and $A \rhd B$ occurs in the antecedent of the sequent $S_i$, then $S_i$ is also $(A \rhd B)$-explicit.
\end{enumerate}
Clearly, the set of all $(A\rhd B)$-explicit sequents of $\pi_L$ form a subtree of $\pi_L$.
\begin{lem}\label{LemA}
Suppose that there exists an $(A, B)^\star$-proof of $(\Omega_1, \{ X_i \rhd Y_i \}, \Theta, A \Rightarrow \{ X_i \})$.
Then, for any $(A \rhd B)$-explicit sequent $S = (A \rhd B, \Gamma \Rightarrow \Delta)$ of $\pi_L$, there exists an $(A, B)^\star$-proof of $(\Omega_1, \{ X_i \rhd Y_i \}, \Theta, \Gamma \Rightarrow \Delta)$.
\end{lem}
\begin{proof}
We prove the lemma by top-down induction on the position of $S$ in the subtree of $(A \rhd B)$-explicit sequents of $\pi_L$.
\textbf{Base case:} The sequent $S$ in question must be (1): an initial sequent $(A \rhd B \Rightarrow A \rhd B)$ or (2): a lower sequent of $\dfrac{\Gamma \Rightarrow \Delta}{A \rhd B, \Gamma \Rightarrow \Delta} (wl)$ where $\Gamma$ does not contain $A \rhd B$.
For Case (1), applying $(wl)$ to $\pi$ several times, we get a cut-free proof of $(\Omega_1, \{ X_i \rhd Y_i \}, \Theta \Rightarrow A \rhd B)$.
For Case (2), a cut-free proof of $(\Omega_1, \{ X_i \rhd Y_i \}, \Theta, \Gamma \Rightarrow \Delta)$ is easily obtained from $(\Gamma \Rightarrow \Delta)$ by applying suitable $(wl)$s.
\textbf{Inductive step:}
(i) Suppose that $S$ is a lower sequent of an application of $(\rhd_P)$.
We distinguish the following two cases:
\begin{enumerate}
\item If $A \rhd B$ is non-principal, then the application is of the form:
\begin{equation*}
\infer[(\rhd_P)]{\Omega', A \rhd B, \{ X_k' \rhd Y_k' \} \Rightarrow C' \rhd D'}
{\Omega', A \rhd B, \{ X_k' \rhd Y_k' \}, C' \rhd D', C' \Rightarrow \{ X_k' \}
&
\langle Y_k' \Rightarrow D' \rangle
}.
\end{equation*}
By the induction hypothesis, there exists an $(A,B)^\star$-proof of
\begin{equation*}
\left(
\begin{array}{cccc}
\Omega_1, & \{ X_i \rhd Y_i \}, & \Theta, & C' \rhd D', \\
\Omega', & \{ X_k' \rhd Y_k' \}, & & C'
\end{array}
\Rightarrow \{ X_k' \}\right).
\end{equation*}
For this and $\langle Y_k' \Rightarrow D' \rangle$, we apply ($\rhd_P$).
Notice that each sequent in $\langle Y_k' \Rightarrow D' \rangle$ has a cut-free proof because it is in $\pi_L$.
Then, we obtain an $(A, B)^\star$-proof of
$( \Omega_1, \Omega', \{ X_i \rhd Y_i \}, \{ X_k' \rhd Y_k' \}, \Theta \Rightarrow C' \rhd D')$.
Recall that $\Theta$ contains only $\rhd$-formulas, and here the principal formulas in this application of $(\rhd_P)$ are $\{ X_k' \rhd Y_k' \}$.
\item If $A \rhd B$ is principal, then the application is of the form:
\small
\begin{equation*}
\infer[(\rhd_P)]{\Omega', \{ X_k' \rhd Y_k' \}, A \rhd B \Rightarrow C' \rhd D'}
{\Omega', \{ X_k' \rhd Y_k' \}, A \rhd B, C' \rhd D', C' \Rightarrow \{ X_k' \}, A
&
\langle Y_k' \Rightarrow D' \rangle
&
B \Rightarrow D'
}.
\end{equation*}
\normalsize
By the induction hypothesis, there exists an $(A,B)^\star$-proof of
\begin{equation*}
\left(\begin{array}{cccc}
\Omega_1, & \{ X_i \rhd Y_i \}, & \Theta, & C' \rhd D', \\
\Omega', & \{ X_k' \rhd Y_k' \}, & & C'
\end{array}
\Rightarrow \{ X_k' \}, A \right).
\end{equation*}
For this and an $(A,B)^\star$-proof of $(\Omega_1, \{ X_i \rhd Y_i \}, \Theta, A \Rightarrow \{ X_i \})$ in the supposition of the lemma, we apply the rule $(cut)$ with the cut-formula $A$.
Then, we obtain an $(A, B)^\star$-proof of
\begin{equation*}
\left(
\begin{array}{cccc}
\Omega_1, & \{ X_i \rhd Y_i \}, & \Theta, & C' \rhd D', \\
\Omega', & \{ X_k' \rhd Y_k' \}, & & C'
\end{array}
\Rightarrow
\begin{array}{c}
\{ X_i \}, \\
\{ X_k' \}
\end{array}
\right).
\end{equation*}
For each $i$, $\pi_i$ is a cut free proof of $(Y_i \Rightarrow B)$.
Also, the elements of $\langle Y_k' \Rightarrow D' \rangle$ and the sequent $(B \Rightarrow D')$ have cut-frees proof because they are in $\pi_L$.
Then, by applying $(cut)$, each $(Y_i \Rightarrow D')$ has an $(A, B)^\star$-proof.
Apply $(\rhd_P)$ as follows:
\footnotesize
\begin{equation*}
\infer[(\rhd_P)]{
\begin{array}{cc}
\Omega_1, & \{ X_i \rhd Y_i \}, \\
\Omega', & \{ X_k' \rhd Y_k' \},
\end{array}
\Theta \Rightarrow C' \rhd D'}
{\begin{array}{cccc}
\Omega_1, & \{ X_i \rhd Y_i \}, & \Theta, & C' \rhd D', \\
\Omega', & \{ X_k' \rhd Y_k' \}, & & C'
\end{array}
\Rightarrow
\begin{array}{c}
\{ X_i \}, \\
\{ X_k' \}
\end{array}
&
\langle Y_i \Rightarrow D' \rangle
&
\langle Y_k' \Rightarrow D' \rangle.
}
\end{equation*}
\normalsize
Therefore, $(\Omega_1, \Omega', \{X_i \rhd Y_i\}, \{X_k' \rhd Y_k'\}, \Theta \Rightarrow C' \rhd D')$ has an $(A, B)^\star$-proof.
\end{enumerate}
(ii) The case that $S$ is a lower sequent of a logical rule is clear by the induction hypothesis.
For instance, if $\dfrac{S_1 \quad S_2}{A \rhd B, \Gamma \Rightarrow \Delta}$ is some application of the logical rule $(\ast)$, then both $S_1$ and $S_2$ are also $(A \rhd B)$-explicit.
Then, they are of the forms $(A \rhd B, \Gamma_1 \Rightarrow \Delta_1)$ and $(A \rhd B, \Gamma_2 \Rightarrow \Delta_2)$, respectively.
By the induction hypothesis, there exist $(A, B)^\star$-proofs of $(\Omega_1, \{X_i \rhd Y_i\}, \Theta, \Gamma_1 \Rightarrow \Delta_1)$ and $(\Omega_1, \{X_i \rhd Y_i\}, \Theta, \Gamma_2 \Rightarrow \Delta_2)$.
Then, by the rule $(\ast)$, we obtain a required $(A, B)^\star$-proof of $(\Omega_1, \{X_i \rhd Y_i\}, \Theta, \Gamma \Rightarrow \Delta)$.
\end{proof}
In the subtree consisting of $(A \rhd B)$-explicit sequents in $\pi_L$, let $\overline{\Theta} := \{ E \rhd F \mid E \rhd F$ is a diagonal formula of the lowermost application of $(\rhd_P)$ in which $A \rhd B$ is principal $\}$.
\begin{lem}\label{LemB}
There exists a cut-free proof of $(\Omega_1, \{ X_i \rhd Y_i \}, \overline{\Theta}, A \Rightarrow \{ X_i \})$.
\end{lem}
\begin{proof}
By top-down induction on the position of $S$ in the subtree of $(A \rhd B)$-explicit sequents of $\pi_L$, we prove that for any $(A \rhd B)$-explicit sequent $S: (A \rhd B, \Gamma \Rightarrow \Delta)$ of $\pi_L$, there exists a cut-free proof of $(\Omega_1, \{X_i \rhd Y_i\}, \overline{\Theta}, \Gamma \Rightarrow \Delta)$.
The proof is done by tracing that of Lemma~\ref{LemA} while replacing the phrase `$(A, B)^\star$-proofs' with `cut-free proofs'.
The only case that should be considered is that $S$ is a lower sequent of an application of $(\rhd_P)$ and $A \rhd B$ is a principal formula of the rule.
In this case, $S$ of the form $(A \rhd B, \Gamma \Rightarrow E \rhd F)$ for some $E \rhd F \in \overline{\Theta}$.
Then, we obtain a cut-free proof
$$
\infer={\Omega_1, \{ X_i \rhd Y_i \}, \overline{\Theta}, \Gamma \Rightarrow E \rhd F}
{E \rhd F \Rightarrow E \rhd F}.
$$
We have completed the proof by induction.
Finally, for the root sequent $(\Omega_1, \{ X_i \rhd Y_i \}, A \rhd B, A \Rightarrow \{X_i \} )$ of $\pi_L$, we obtain a cut-free proof of $(\Omega_1, \{ X_i \rhd Y_i \}, \overline{\Theta}, A \Rightarrow \{X_i \})$.
\end{proof}
\begin{lem}\label{LemC}
For any $\Theta \subseteq \overline{\Theta}$, there exists an $(A, B)^\star$-proof of $(\Omega_1, \{ X_i \rhd Y_i \}, \Theta, A \Rightarrow \{ X_i \})$.
\end{lem}
\begin{proof}
We prove the lemma by downward induction on the cardinality $|\Theta|$ of the set $\Theta \subseteq \overline{\Theta}$.
For $\overline{\Theta}$, there exists a cut-free proof $(\Omega_1, \{ X_i \rhd Y_i \}, \overline{\Theta}, A \Rightarrow \{X_i \})$ by Lemma~\ref{LemB}.
Then, suppose that the lemma holds for subsets of $\overline{\Theta}$ with the cardinality $k+1$.
Let $\Theta \subseteq \overline{\Theta}$ be such that $|\Theta| = k$.
By top-down induction on the position of $S$ in the subtree of $(A \rhd B)$-explicit sequents of $\pi_L$, we prove that for any $(A \rhd B)$-explicit sequent $S: (A \rhd B, \Gamma \Rightarrow \Delta)$ of $\pi_L$, there exists an $(A, B)^\star$-proof of $(\Omega_1, \{X_i \rhd Y_i\}, \Theta, \Gamma \Rightarrow \Delta)$.
This is also done by tracing the proof of Lemma~\ref{LemA}, and the only case that should be considered is that $S$ is a lower sequent of an application of $(\rhd_P)$, $A \rhd B$ is a principal formula of the rule, and the diagonal formula $E \rhd F$ of the rule is not in $\Theta$.
In this case, the application of the rule is of the form:
\begin{equation*}
\small
\infer[(\rhd_P)]{\Omega', \{ X_k' \rhd Y_k' \}, A \rhd B \Rightarrow E \rhd F}
{\deduce{\Omega', \{ X_k' \rhd Y_k' \}, A \rhd B, E \rhd F, E \Rightarrow \{ X_k' \}, A}{\rho}
&
\langle Y_k' \Rightarrow E \rangle
&
\deduce{B \Rightarrow E}{\rho_B}}.
\normalsize
\end{equation*}
Since $|\Theta \cup \{ E \rhd F \}| = k + 1$, there is an $(A, B)^\star$-proof $\pi''$ of
\[
(\Omega_1, \{ X_i \rhd Y_i \}, \Theta, E \rhd F, A \Rightarrow \{ X_i \})
\]
by the induction hypothesis.
Applying Lemma~\ref{LemA} to $\pi''$ and $\rho$, we get a $(A,B)^\star$-proof of
\begin{equation*}
\left(
\begin{array}{cc}
\Omega_1, & \{ X_i \rhd Y_i \}, \\
\Omega', & \{ X_k' \rhd Y_k' \},
\end{array}
\Theta, E \rhd F, E \Rightarrow \{ X_k' \}, A \right).
\end{equation*}
Together with $\pi''$, we obtain an $(A, B)^\star$-proof of
\begin{equation*}
\left(
\begin{array}{cc}
\Omega_1, & \{ X_i \rhd Y_i \} \\
\Omega', & \{ X_k' \rhd Y_k' \},
\end{array}
\Theta, E \rhd F, E \Rightarrow
\begin{array}{c}
\{ X_i \}, \\
\{ X_k' \}
\end{array}
\right)
\end{equation*}
by applying $(cut)$ with the cut-formula $A$.
Notice that each of $(Y_i \Rightarrow B)$, $(Y_k' \Rightarrow E)$, and $(B \Rightarrow E)$ has a cut-free proof.
Then, $(Y_i \Rightarrow E)$ has an $(A, B)^\star$-proof by applying the rule $(cut)$ with the cut-formula $B$.
Finally, we apply $(\rhd_P)$ as follows:
\footnotesize
\begin{equation*}
\infer[(\rhd_P)]{
\begin{array}{cc}
\Omega_1, & \{ X_i \rhd Y_i \}, \\
\Omega', & \{ X_k' \rhd Y_k' \},
\end{array}
\Theta \Rightarrow E \rhd F}
{\begin{array}{cc}
\Omega_1, & \{ X_i \rhd Y_i \}, \\
\Omega', & \{ X_k' \rhd Y_k' \},
\end{array} \Theta, E \rhd F, E \Rightarrow
\begin{array}{c}
\{ X_i \}, \\
\{ X_k' \}
\end{array}
&
\langle Y_i \Rightarrow E \rangle
&
\langle Y_k' \Rightarrow E \rangle
}.
\end{equation*}
\normalsize
Therefore, we have an $(A, B)^\star$-proof of $(\Omega_1, \Omega', \{X_i \rhd Y_i\}, \{X_k' \rhd Y_k'\}, \Theta \Rightarrow E \rhd F)$.
Consequently, for the root sequent $(\Omega_1, \{X_i \rhd Y_i\}, A \rhd B, A \Rightarrow \{X_i\})$ of $\pi_L$, we get an $(A, B)^\star$-proof of $(\Omega_1, \{ X_i \rhd Y_i \}, \Theta, A \Rightarrow \{ X_i \})$.
\end{proof}
By Lemma~\ref{LemC}, we get an $(A, B)^\star$-proof $\pi'$ of $(\Omega_1, \{ X_i \rhd Y_i \}, A \Rightarrow \{ X_i \})$.
\begin{proof}[Proof of Proposition~\ref{mainP}.]
By applying the rule $(cut)$ for $\pi$ and $\sigma_L$ with the cut-formula $A \rhd B$, we obtain a proof of
\[
\left(
\begin{array}{ccc}
\Omega_1, & \{ X_i \rhd Y_i \}, & C \rhd D, \\
\Omega_2, & \{ Z_j \rhd W_j \}, & C
\end{array}
\Rightarrow \{ Z_j \}, A
\right).
\]
This application of $(cut)$ is eliminable by the induction hypothesis on the height.
So we have a cut-free proof of the above sequent.
From this and $\pi'$, we apply $(cut)$ with the cut-formula $A$.
Then, we get an $(A, B)^\star$-proof of
\begin{equation*}
\left(
\begin{array}{ccc}
\Omega_1, & \{ X_i \rhd Y_i \}, & C \rhd D, \\
\Omega_2, & \{ Z_j \rhd W_j \}, & C
\end{array}
\Rightarrow
\begin{array}{c}
\{ X_i \}, \\
\{ Z_j \}
\end{array}
\right).
\end{equation*}
Apply $(\rhd_P)$ as follows:
\begin{equation*}
\infer[(\rhd_P)]{
\begin{array}{cc}
\Omega_1, & \{ X_i \rhd Y_i \}, \\
\Omega_2, & \{ Z_j \rhd W_j \}
\end{array}
\Rightarrow C \rhd D
}
{\left(
\begin{array}{ccc}
\Omega_1, & \{ X_i \rhd Y_i \}, & C \rhd D, \\
\Omega_2, & \{ Z_j \rhd W_j \}, & C
\end{array}
\Rightarrow
\begin{array}{c}
\{ X_i \}, \\
\{ Z_j \}
\end{array}
\right)
&
\deduce{\langle Y_i \Rightarrow D \rangle}{\pi_i'}
&
\deduce{\langle W_j \Rightarrow D \rangle}{\sigma_j}
}.
\end{equation*}
Here $\pi_i'$ is obtained from $\pi_i$ and $\sigma_B$ by applying $(cut)$ with the cut-formula $B$.
This is an $(A, B)^\star$-proof, and therefore the sequent $(\Omega_1, \Omega_2, \{ X_i \rhd Y_i \}, \{ Z_j \rhd W_j \} \Rightarrow C \rhd D)$ also has an $(A, B)^\star$-proof.
The degree of every $(cut)$ in this proof is smaller than that of the original $(cut)$.
Then, by the induction hypothesis, we obtain a cut-free proof of $(\Omega_1, \Omega_2, \{ X_i \rhd Y_i \}, \{ Z_j \rhd W_j \} \Rightarrow C \rhd D)$.
\end{proof}
\begin{thm}[The cut-elimination theorem for $(\IL^-(\PP))^G$]\label{CE}
For any sequent $S$, if $S$ is provable in $(\IL^-(\PP))^G$, then there exists a cut-free proof of $S$ in $(\IL^-(\PP))^G$.
\end{thm}
\subsection{Craig interpolation property}
In this section, we show that Maehara's method of obtaining Craig interpolants of sequents having cut-free proofs can be applied to our systems $(\IL^-)^G$ and $(\IL^-(\PP))^G$.
Therefore, we conclude that $\IL^-(\PP)$ enjoys CIP.
Moreover, it follows that the cut-elimination theorem does not hold for our system $(\IL^-)^G$.
For a sequent $(\Gamma \Rightarrow \Delta)$, we say that a pair $[(\Gamma_1; \Delta_1), (\Gamma_2; \Delta_2)]$ of pairs of finite sets of formulas is a \textit{separation of $(\Gamma \Rightarrow \Delta)$} if $\Gamma_1 \cap \Gamma_2 = \Delta_1 \cap \Delta_2 = \varnothing$, $\Gamma = \Gamma_1 \cup \Gamma_2$, and $\Delta = \Delta_1 \cup \Delta_2$.
For a finite set $\Gamma$ of formulas, let $\var(\Gamma) : = \bigcup\{\var(A) \mid A \in \Gamma\}$.
\begin{thm}\label{MM}
Let $L$ be either $(\IL^-)^G$ or $(\IL^-(\PP))^G$.
Suppose that $(\Gamma \Rightarrow \Delta)$ has a cut-free proof in $L$.
Then, for any separation $[(\Gamma_1; \Delta_1),(\Gamma_2; \Delta_2)]$ of $(\Gamma \Rightarrow \Delta)$, there exists a formula $C$ which satisfies the following conditions:
\begin{enumerate}
\item $L \vdash (\Gamma_1 \Rightarrow \Delta_1, C)$ and $L \vdash (\Gamma_2, C \Rightarrow \Delta_2)$;
\item $\var(C) \subseteq \var(\Gamma) \cap \var(\Delta)$.
\end{enumerate}
\end{thm}
\begin{proof}
(i) First, we prove the theorem for $L = (\IL^-(\PP))^G$ by top-down induction on the position of the sequent $(\Gamma \Rightarrow \Delta)$ in a cut-free proof $\pi$ of the sequent in $(\IL^-(\PP))^G$.
The cases for an initial sequent and an instance of logical rules are straightforward.
We only give a proof of the case that the last application of $\pi$ is $(\rhd_P)$.
Such an application of $(\rhd_P)$ is of the form:
\begin{equation*}
\infer[(\rhd_P)]{
\begin{array}{cc}
\Omega_1, & \{ X_i \rhd Y_i \}, \\
\Omega_2, & \{ Z_j \rhd W_j \}
\end{array}
\Rightarrow A \rhd B}
{\begin{array}{cc}
\Omega_1, & \{ X_i \rhd Y_i \}, \\
\Omega_2, & \{ Z_j \rhd W_j \},
\end{array}
A \rhd B, A \Rightarrow
\begin{array}{c}
\{ X_i \} , \\
\{ Z_j\}
\end{array}
&
\langle Y_i \Rightarrow B \rangle
&
\langle W_j \Rightarrow B \rangle
}.
\end{equation*}
Here we consider a separation $[(\Gamma_1; \Delta_1),(\Gamma_2; \Delta_2)]$ of the lower sequent $(\Omega_1, \Omega_2, \{X_i \rhd Y_i\}, \{Z_j \rhd W_j\} \Rightarrow A \rhd B)$ with $\Gamma_1 = \Omega_1 \cup \{ X_i \rhd Y_i \}$ and $\Gamma_2 = \Omega_2 \cup \{ Z_j \rhd W_j \}$.
We distinguish the following two cases.
\begin{enumerate}
\item Suppose that $\Delta_1 = \{A \rhd B\}$ and $\Delta_2 = \varnothing$.
Take the following separations of the upper sequents:
\begin{align*}
& [(\Omega_1, \{ X_i \rhd Y_i \}, A \rhd B, A; \{ X_i \}),(\Omega_2, \{ Z_j \rhd W_j \}; \{ Z_j \})], \\
& [(\varnothing ; B), (W_j; \varnothing)], \text{ for each }j.
\end{align*}
By the induction hypothesis, there exist $D$ and $E_j$ for each $j$ such that $(\IL^-(\PP))^G$ proves
\begin{itemize}
\item $(\Omega_1, \{ X_i \rhd Y_i \}, A \rhd B, A \Rightarrow \{ X_i \}, D)$,
\item $(D, \Omega_2, \{ Z_j \rhd W_j \} \Rightarrow \{ Z_j \})$,
\item $(\Rightarrow B, E_j)$, and
\item $(E_j, W_j \Rightarrow)$.
\end{itemize}
Let $E'$ be the formula $\bigvee \neg E_j$.
Then, we have $(\IL^-(\PP))^G \vdash E' \Rightarrow B$ because $(\IL^-(\PP))^G \vdash \neg E_j \Rightarrow B$ for each $j$.
Notice that if $\{ Z_j \rhd W_j \}$ is an empty set, then $E'$ is $\bot$, and hence $(\IL^-(\PP))^G \vdash E' \Rightarrow B$ also holds.
Similarly, we also have $(\IL^-(\PP))^G \vdash W_j \Rightarrow E'$ for each $j$.
Consider the following derivations:
\begin{itemize}
\item \footnotesize
\[
\infer[(\neg r)]{\Omega_1, \{ X_i \rhd Y_i \} \Rightarrow A \rhd B, \neg (D \rhd E')}
{\infer[(\rhd_P)]{\Omega_1, \{ X_i \rhd Y_i \}, D \rhd E' \Rightarrow A \rhd B}
{\infer[(wl)]{\Omega_1, \{ X_i \rhd Y_i \}, D \rhd E', A \rhd B, A \Rightarrow \{ X_i \}, D}
{\Omega_1, \{ X_i \rhd Y_i \}, A \rhd B, A \Rightarrow \{ X_i \}, D}
&
\langle Y_i \Rightarrow B \rangle
&
E' \Rightarrow B
}},
\] \normalsize
\item
\[
\infer[(\neg l)]{\Omega_2, \{ Z_j \rhd W_j \}, \neg (D \rhd E') \Rightarrow }
{\infer[(\rhd_P)]{\Omega_2, \{ Z_j \rhd W_j \} \Rightarrow D \rhd E'}
{\infer[(wl)]{\Omega_2, \{ Z_j \rhd W_j \}, D \rhd E' , D \Rightarrow \{ Z_j \}}
{\Omega_2, \{ Z_j \rhd W_j \}, D \Rightarrow \{ Z_j \}}
&
\langle W_j \Rightarrow E' \rangle
}}.
\]
\end{itemize}
Then, $C : \equiv \neg (D \rhd E')$ fulfills Conditions 1 and 2 in the statement of the theorem.
\item Suppose that $\Delta_1 = \varnothing$ and $\Delta_2 = \{A \rhd B\}$.
Take the following separations of the upper sequents:
\begin{align*}
& [(\Omega_1, \{ X_i \rhd Y_i \} ; \{ X_i \}), (\Omega_2, \{ Z_j \rhd W_j \}, A \rhd B, A ; \{ Z_j \} )], \\
& [(Y_i ; \varnothing ), (\varnothing ; B )], \text{ for each } i.
\end{align*}
By the induction hypothesis, there exist interpolants $D$ and $E_i$ for each $i$ such that $(\IL^-(\PP))^G$ proves
\begin{itemize}
\item $(\Omega_1, \{ X_i \rhd Y_i \} \Rightarrow \{ X_i \}, D)$,
\item $(D, \Omega_2, \{ Z_j \rhd W_j \}, A\rhd B, A \Rightarrow \{ Z_j \})$,
\item $(Y_i \Rightarrow E_i)$, and
\item $(E_i \Rightarrow B)$.
\end{itemize}
Let $E'$ be $\bigvee E_i$.
Then, we have $(\IL^-(\PP))^G \vdash (E' \Rightarrow B)$ and $(\IL^-(\PP))^G \vdash (Y_i \Rightarrow E')$ for each $i$.
Consider the following derivations:
\begin{itemize}
\item
\[
\infer[(\rhd_P)]{\Omega_1, \{ X_i \rhd Y_i \} \Rightarrow (\neg D) \rhd E'}
{\infer[(wl)]{\Omega_1, \{ X_i \rhd Y_i \}, (\neg D) \rhd E', \neg D \Rightarrow \{ X_i \}}
{\infer[(\neg l)]{\Omega_1, \{ X_i \rhd Y_i \}, \neg D \Rightarrow \{ X_i \}}
{\Omega_1, \{ X_i \rhd Y_i \} \Rightarrow \{ X_i \}, D}
}
&
\langle Y_i \Rightarrow E' \rangle
},
\]
\item
\[
\scriptsize
\infer[(\rhd_P)]{\Omega_2, \{ Z_j \rhd W_j \}, (\neg D) \rhd E' \Rightarrow A \rhd B}
{\infer[(wl)]{\Omega_2, \{ Z_j \rhd W_j \}, (\neg D) \rhd E', A \rhd B , A \Rightarrow \{ Z_j \}, \neg D}
{\infer[(\neg r)]{\Omega_2, \{ Z_j \rhd W_j \}, A \rhd B , A \Rightarrow \{ Z_j \}, \neg D}
{D, \Omega_2, \{ Z_j \rhd W_j \}, A \rhd B, A \Rightarrow \{ Z_j \}}}
&
\langle W_j \Rightarrow B \rangle
&
E' \Rightarrow B
}.
\normalsize
\]
\end{itemize}
Then, $C : \equiv (\neg D) \rhd E'$ fulfills Conditions 1 and 2 of the theorem.
\end{enumerate}
This finishes the proof of the theorem for $L = (\IL^-(\PP))^G$.
(ii) Secondly, we give a proof for $L = (\IL^-)^G$.
A proof is similar as in the case of $L = (\IL^-(\PP))^G$, and so we only sketch a proof of the case of the rule $(\rhd)$.
An instance of the rule is
\begin{equation*}
\infer[(\rhd)]{\{ X_i \rhd Y_i \}, \{ Z_j \rhd W_j \} \Rightarrow A \rhd B}
{A \Rightarrow \{ X_i \} ,\{ Z_j\}
&
\langle Y_i \Rightarrow B \rangle
&
\langle W_j \Rightarrow B \rangle
},
\end{equation*}
with $\Gamma_1 = \{ X_i \rhd Y_i \}$ and $\Gamma_2 = \{ Z_j \rhd W_j \}$.
We distinguish the following two cases.
\begin{enumerate}
\item Suppose that $\Delta_1 = \{ A \rhd B \}$ and $\Delta_2 = \varnothing$. Take the following separations of the upper sequents:
\begin{align*}
& [(A ; \{X_i \} ) , (\varnothing ; \{ Z_j\})], \\
& [(\varnothing ; B) , (W_j ; \varnothing)] \text{ for each } j.
\end{align*}
Let $D$ and $E_j$ be corresponding interpolants obtained from the induction hypothesis, and put $E' :\equiv \bigvee \neg E_j$.
The following derivations show that the formula $C \equiv \neg (D \rhd E')$ is indeed an interpolant of the separation in question:
\begin{itemize}
\item
\[ \infer[(\neg r)]{\{ X_i \rhd Y_i \} \Rightarrow A \rhd B, \neg (D \rhd E')}
{\infer[(\rhd)]{\{X_i \rhd Y_i \}, D \rhd E' \Rightarrow A \rhd B}
{A \Rightarrow \{ X_i \}, D
&
\langle Y_i \Rightarrow B \rangle
&
E' \Rightarrow B
}
}, \]
\item
\[ \infer[(\neg l)]{\neg D \rhd E', \{ Z_j \rhd W_j \} \Rightarrow }
{\infer[(\rhd)]{\{Z_j \rhd W_j \} \Rightarrow D \rhd E'}
{D \Rightarrow \{ Z_j \}
&
\langle W_j \Rightarrow E' \rangle
}
}.\]
\end{itemize}
\item Suppose that $\Delta_1 = \varnothing$ and $\Delta_2 = \{ A \rhd B \}$.
Take the following separations of the upper sequents:
\begin{align*}
& [(\varnothing ; \{ X_i \}), (A ; \{ Z_j \} )], \\
& [(Y_i ; \varnothing ), (\varnothing ; B )], \text{ for each } i.
\end{align*}
Let $D$ and $E_i$ be corresponding interpolants, and put $E' :\equiv \bigvee E_i$.
The following derivations show that the formula $C \equiv (\neg D) \rhd E'$ is indeed an interpolant of the separation.
\begin{itemize}
\item \[ \infer[(\rhd)]{\{ X_i \rhd Y_i \} \Rightarrow (\neg D) \rhd E'}
{\infer[(\neg l)]{\neg D \Rightarrow \{ X_i \}}{ \Rightarrow \{X_i \}, D}
&
\langle Y_i \Rightarrow E' \rangle
}, \]
\item \[ \infer[(\rhd)]{\{ Z_j \rhd W_j \}, (\neg D) \rhd E' \Rightarrow A \rhd B}
{\infer[(\neg r)]{A \Rightarrow \{Z_j \}, \neg D}{D, A \Rightarrow \{Z_j\}}
&
\langle W_j \Rightarrow B \rangle
&
E' \Rightarrow B
}.\]
\end{itemize}
\end{enumerate}
\end{proof}
\begin{thm}[The Craig interpolation theorem for $\IL^-(\PP)$]\label{CIP}
The logic $\IL^-(\PP)$ enjoys CIP.
\end{thm}
From Fact~\ref{CIPFPP} and this theorem, we obtain an alternative proof of Theorem~\ref{lFPP} stating that $\IL^-(\PP)$ has $\ell$FPP.
According to \cite[Section 6]{IKO20}, $\IL^-$ does not have CIP.
Therefore, we conclude the failure of the cut-elimination theorem for $(\IL^-)^G$.
\begin{cor}\label{FCE}
There exists a sequent $(\Gamma \Rightarrow \Delta)$ that is provable in $(\IL^-)^G$ but cannot be proved without the rule $(cut)$ in $(\IL^-)^G$.
\end{cor}
\section{Relational semantic aspects of $\IL^-(\PP)$}\label{Sec_SV}
In this section, we investigate relational semantics and the modal completeness for $\IL^-(\PP)$.
This section consists of three subsections.
In the first subsection, we introduce two kinds of relational semantics of $\IL^-(\PP)$, namely, $\IL^-\!$-frames and their simplifications.
In particular, we show that $\IL^-(\PP)$ is valid in all such simplified frames.
Thus, $\IL^-(\PP)$ is the natural basis for the investigation of a certain relational semantics.
In the second subsection, we prove the modal completeness theorems of $\IL^-(\PP)$ with respect to these relational semantics.
Finally, in the last subsection, we show that $\IL^-(\PP)$ is faithfully embeddable into several extensions of the fusion $\GLK$ of $\GL$ and $\K$.
\subsection{Two relational semantics}
First, we introduce \textit{$\IL^-\!$-frames} which were originally introduced by Visser~\cite{Vis88} as Veltman prestructures (See also~\cite[Definition 2.2]{KO21}).
\begin{defn}
We say that a triple $(W, R, \{S_{w}\}_{w \in W})$ is an \textit{$\IL^-\!$-frame} if the following conditions hold:
\begin{enumerate}
\item $W$ is a non-empty set;
\item $R$ is a transitive and conversely well-founded binary relation on $W$;
\item For each $w \in W$, $S_{w} \subseteq R[w] \times W$ where $R[w] := \{x \in W \mid w {R} x\}$.
\end{enumerate}
A quadruple $(W, R, \{S_{w}\}_{w \in W}, \Vdash)$ is said to be an \textit{$\IL^-\!$-model} if $(W, R, \{S_{w}\}_{w \in W})$ is an $\IL^- \!$-frame and $\Vdash$ is a binary relation between $W$ and the set of all $\mathcal{L}(\rhd)$-formulas satisfying the usual conditions for satisfaction relation and the following conditions:
\begin{itemize}
\item $w \Vdash \Box A :\iff (\forall x \in W)(w {R} x \Rightarrow x \Vdash A)$;
\item $w \Vdash A \rhd B :\iff (\forall x \in W)\bigl(w {R} x \ \&\ x \Vdash A \Rightarrow (\exists y \in W)(x {S_{w}} y\ \&\ y \Vdash B)\bigr)$.
\end{itemize}
A modal formula $A$ is said to be \textit{valid} in an $\IL^- \!$-frame $(W, R, \{S_{w}\}_{w \in W})$ if for all $\IL^- \!$-models $(W, R, \{S_{w}\}_{w \in W}, \Vdash)$ based on the frame and all $w \in W$, $w \Vdash A$.
\end{defn}
It is easily shown that every theorem of $\IL^-$ is valid in all $\IL^-\!$-frames.
Moreover, the modal completeness of $\IL^-$ with respect to $\IL^- \!$-frames has been proved.
\begin{fact}[{\cite[Theorem 5.1]{KO21}}]
For any $\mathcal{L}(\rhd)$-formula $A$, the following are equivalent:
\begin{enumerate}
\item $\IL^- \vdash A$.
\item $A$ is valid in all $\IL^-\!$-frames.
\item $A$ is valid in all finite $\IL^- \!$-frames.
\end{enumerate}
\end{fact}
The validity of an $\mathcal{L}(\rhd)$-formula in an $\IL^-\!$-frame is sometimes characterized by some condition on the binary relations on the frame (cf.~\cite{KO21,Vis88}).
Regarding the principle $\PP$, it is easily shown that the following known condition on Veltman frames works also for $\IL^-\!$-frames (See Visser~\cite[p.~15]{Vis88}).
\begin{prop}\label{FCP}
Let $\mathcal{F} = (W, R, \{S_{w}\}_{w \in W})$ be any $\IL^- \!$-frame.
Then, the following are equivalent.
\begin{enumerate}
\item $\PP$ is valid in $\mathcal{F}$.
\item $(\forall w, x, y, z \in W) (w {R} x {R} y {S_{w}} z \Rightarrow y {S_{x}} z)$.
\end{enumerate}
\end{prop}
The treatment of the family $\{S_w\}_{w \in W}$ in $\IL^-\!$-frames is somewhat complicated.
The idea of simplifying them by converting them into a single relation $S$ was first done by Visser~\cite[Section 16]{Vis88} for $\IL^-\!$-frames where $\IL$ is valid.
Our second semantics is obtained by implementing the idea on $\IL^-\!$-frames.
\begin{defn}\label{Def:SILPF}
We say that a triple $(W, R, S)$ is a \textit{simplified $\IL^-(\PP)$-frame} if the following conditions hold:
\begin{enumerate}
\item $W$ is a non-empty set;
\item $R$ is a transitive and conversely well-founded binary relation on $W$;
\item $S$ is a binary relation on $W$.
\end{enumerate}
A \textit{simplified $\IL^-(\PP)$-model} is a quadruple $(W, R, S, \Vdash)$ where $(W, R, S)$ is a simplified $\IL^-(\PP)$-frame and $\Vdash$ is a satisfaction relation satisfying the following condition:
\begin{itemize}
\item $w \Vdash A \rhd B :\iff (\forall x \in W)\bigl(w {R} x\ \&\ x \Vdash A \Rightarrow (\exists y \in W)(x {S} y\ \&\ y \Vdash B)\bigr)$.
\end{itemize}
\end{defn}
In the literature such as~\cite{Vis88}, a satisfaction relation $\Vdash$ on simplified frames is usually defined so that $w \Vdash A \rhd B$ is equivalent to
\[
(\forall x \in W)\bigl(w {R} x\ \&\ x \Vdash A \Rightarrow (\exists y \in W)(w {R} y\ \&\ x {S} y\ \&\ y \Vdash B)\bigr).
\]
In contexts that adopt the latter definition of $\Vdash$, the logics under consideration contain the axiom scheme $\J{4}_+$, and models based on the latter definition always validate $\J{4}_+$.
On the other hand, we adopt the former definition of $\Vdash$ because we are dealing with logics that do not necessarily contain $\J{4}_+$ as an axiom scheme.
However, in contrast to the case $\IL^-\!$-frames, our simplified $\IL^-(\PP)$-frames always validate the persistence principle $\PP$.
This is the reason why we adopted the terminology `simplified $\IL^-(\PP)$-frames' in Definition~\ref{Def:SILPF}.
\begin{prop}\label{Psou}
The principle $\PP$ is valid in all simplified $\IL^-(\PP)$-frames.
\end{prop}
\begin{proof}
Let $(W, R, S, \Vdash)$ be any $\IL^-(\PP)$-model and let $w \in W$.
Suppose $w \Vdash A \rhd B$.
We show $w \Vdash \Box(A \rhd B)$.
Let $x, y \in W$ be such that $w {R} x$, $x {R} y$, and $y \Vdash A$.
Since $w {R} y$ and $w \Vdash A \rhd B$, there exists a $z \in W$ such that $y {S} z$ and $z \Vdash B$.
Therefore, we conclude $x \Vdash A \rhd B$ and we obtain $w \Vdash \Box(A \rhd B)$.
\end{proof}
In the next subsection, we prove that the logic $\IL^-(\PP)$ is actually characterized by the class of all simplified $\IL^-(\PP)$-frames.
\subsection{Modal completeness for $\IL^-(\PP)$}
In this subsection, we prove the modal completeness theorems for $\IL^-(\PP)$.
Before proving our theorems, we prepare some definitions.
For any set $\Phi$ of $\mathcal{L}(\rhd)$-formulas, let
\[
\Phi_{\rhd} := \{B \mid \text{there exists a} \, \, C \, \, \text{such that} \, \, B \rhd C \in \Phi \, \, \text{or} \, \, C \rhd B \in \Phi\}.
\]
For any $\mathcal{L}(\rhd)$-formula $A$, let
\begin{eqnarray*}
{\sim} A :\equiv \left\{
\begin{array}{ll}
B & \text{if}\ A\ \text{is of the form}\ \neg B\ \text{for some}\ B, \\
\neg A & \text{otherwise.}
\end{array}
\right.
\end{eqnarray*}
A finite set $\Gamma$ of $\mathcal{L}(\rhd)$-formulas is said to be \textit{consistent} if $\IL^{-}(\PP) \nvdash \bigwedge \Gamma \to \bot$.
Let $\Phi$ be any finite set of $\mathcal{L}(\rhd)$-formulas.
A subset $\Gamma$ of $\Phi$ is said to be \textit{$\Phi$-maximally consistent} if $\Gamma$ is consistent and for any $A \in \Phi$, either $A \in \Gamma$ or ${\sim}A \in \Gamma$.
Notice that if $X \subseteq \Phi$ is consistent, then there exists a $\Phi$-maximally consistent set $\Gamma$ including $X$.
Moreover, if $\Gamma$ is a $\Phi$-maximally consistent set and $\IL^-(\PP) \vdash \bigwedge \Gamma \to A$ for $A \in \Phi$, then $A \in \Gamma$.
\begin{defn}\label{Def:ade}
We say that a set $\Phi$ of $\mathcal{L}(\rhd)$-formulas is \textit{adequate} if it satisfies the following conditions:
\begin{enumerate}
\item $\Phi$ is closed under taking subformulas and applying $\sim$;
\item $\bot \in \Phi_{\rhd}$;
\item If $B, C \in \Phi_{\rhd}$, then $B \rhd C \in \Phi$;
\item If $B \rhd C \in \Phi$, then $\Box(B \rhd C) \in \Phi$;
\item If $B \in \Phi_{\rhd}$, then $\Box{\sim}B \in \Phi$;
\item If $B, C_{1}, \ldots, C_{n} \in \Phi_{\rhd}$, then $\Box(B \to \bigvee_{1 \leq i \leq n} C_{i}) \in \Phi$.
\end{enumerate}
\end{defn}
The following proposition is easily proved.
\begin{prop}\label{finade}
Let $X$ be any finite set of $\mathcal{L}(\rhd)$formulas.
Then, there exists a finite adequate set $\Phi$ including $X$.
\end{prop}
Notice that the finiteness of $\Phi$ in Proposition~\ref{finade} is guaranteed by our setting that $\Box$ is in our language $\mathcal{L}(\rhd)$ as a primitive symbol, and $\Box A$ is not an abbreviation for $(\neg A) \rhd \bot$.
We are ready to prove our first modal completeness theorem.
\begin{thm}[The modal completeness theorem for $\IL^-(\PP)$ with respect to $\IL^-\!$-frames]
\label{MCILP}
For any $\mathcal{L}(\rhd)$-formula $A$, the following are equivalent:
\begin{enumerate}
\item $\IL^-(\PP) \vdash A$.
\item $A$ is valid in all $\IL^- \!$-frames in which $\PP$ is valid.
\item $A$ is valid in all finite $\IL^- \!$-frames in which $\PP$ is valid.
\end{enumerate}
\end{thm}
\begin{proof}
The implications $(1 \Rightarrow 2)$ and $(2 \Rightarrow 3)$ are straightforward.
We prove $(3 \Rightarrow 1)$.
Suppose $\IL^-(\PP) \nvdash A$.
We show that there exist a finite $\IL^- \!$-model $(W, R, \{S_{w}\}_{w \in W}, \Vdash)$ and a $w \in W$ such that $w \nVdash A$.
There exists a finite adequate set $\Phi$ including $\{{\sim}A\}$ by Proposition~\ref{finade}.
Let
\[
K_{\Phi} := \{\Gamma \subseteq \Phi \mid \Gamma \ \text{is a} \ \Phi\text{-maximal consistent set}\}.
\]
Then, $K_{\Phi}$ is also a finite set.
We define the binary relations $\prec$ and $\prec_C$ for each $C \in \Phi_\rhd$ on $W$ as follows:
For $\Gamma, \Delta \in K_{\Phi}$,
\begin{enumerate}
\item $\Gamma \prec \Delta :\Longleftrightarrow$ $B, \Box B \in \Delta$ for any $\Box B \in \Gamma$, and there exists a $\Box C \in \Delta \setminus \Gamma$.
\item $\Gamma \prec_{C} \Delta :\Longleftrightarrow$ $\Gamma \prec \Delta$ and ${\sim}B \in \Delta$ for any $B \rhd C \in \Gamma$.
\end{enumerate}
Notice that $\prec$ is transitive and irreflexive.
Also, if $\Gamma \prec \Delta$, then $\Gamma \prec_{\bot} \Delta$ by Conditions 2 and 5 in Definition~\ref{Def:ade}.
\begin{lem}\label{MClem1}
Let $\Gamma, \Delta, \Theta \in K_{\Phi}$ and $C \in \Phi_{\rhd}$.
If $\Gamma \prec \Delta \prec_{C} \Theta$, then $\Gamma \prec_{C} \Theta$.
\end{lem}
\begin{proof}
Suppose $\Gamma \prec \Delta \prec_{C} \Theta$.
We have $\Gamma \prec \Theta$ by the transitivity of $\prec$.
Let $B \rhd C \in \Gamma$.
Since $\Box(B \rhd C) \in \Phi$ and $\IL^-(\PP) \vdash \bigwedge \Gamma \to \Box(B \rhd C)$, we have $\Box(B \rhd C) \in \Gamma$.
Hence, $B \rhd C \in \Delta$ by $\Gamma \prec \Delta$.
Therefore, ${\sim}B \in \Theta$ by $\Delta \prec_{C} \Theta$, and we conclude $\Gamma \prec_{C} \Theta$.
\end{proof}
The following two facts were proved in~\cite{KO21}.
\begin{fact}[{\cite[Lemma 4.6]{KO21}}]\label{MClem2}
Let $\Gamma \in K_{\Phi}$ and $C, D \in \Phi_{\rhd}$.
If $C \rhd D \notin \Gamma$, then there exists a $\Delta \in K_{\Phi}$ such that $C \in \Delta$ and $\Gamma \prec_{D} \Delta$.
\end{fact}
Notice that Condition 6 in Definition~\ref{Def:ade} is used in proving Fact~\ref{MClem2}.
\begin{fact}[{\cite[Lemma 4.7]{KO21}}]\label{MClem3}
Let $\Gamma, \Delta \in K_{\Phi}$ and $C, D, E \in \Phi_{\rhd}$.
If $C \rhd D \in \Gamma$, $\Gamma \prec_{E} \Delta$ and $C \in \Delta$, then there exists a $\Theta \in K_{\Phi}$ such that $D, {\sim}E \in \Theta$.
\end{fact}
Let $(W, R, \{S_{w}\}_{w \in W}, \Vdash)$ be a quadruple defined as follows:
\begin{enumerate}
\item $W :=\{(\Gamma, B) \mid \Gamma \in K_{\Phi}$ and $B \in \Phi_{\rhd} \}$;
\item $(\Gamma, B) \mathrel{R} (\Delta, C) : \iff \Gamma \prec \Delta$;
\item $(\Delta, C) \mathrel{S_{(\Gamma, B)}} (\Theta, D) : \iff (\Gamma, B) \mathrel{R} (\Delta, C)$ and if $\Gamma \prec_{C} \Delta$, then ${\sim} C \in \Theta$;
\item $(\Gamma, B) \Vdash p : \iff p \in \Gamma$.
\end{enumerate}
Let $\Gamma_{0}$ be a $\Phi$-maximally consistent set with ${\sim}A \in \Gamma_{0}$.
Then, $W$ is a non-empty set because $(\Gamma_{0}, \bot) \in W$.
Also, since $\Phi$ is finite, so is $W$.
Obviously, $R$ is a transitive and irreflexive binary relation on $W$.
Since $W$ is finite, $R$ is conversely well-founded.
Trivially, $S_{(\Gamma, B)} \subseteq R[(\Gamma, B)] \times W$ for each $(\Gamma, B) \in W$.
Therefore, $(W, R, \{S_{w}\}_{w \in W}, \Vdash)$ is a finite $\IL^-\!$-model.
We show that $\PP$ is valid in $(W, R, \{S_{w}\}_{w \in W})$.
Suppose $(\Gamma, B) \mathrel{R} (\Delta, C) \mathrel{R} (\Theta, D) \mathrel{S_{(\Gamma, B)}} (\Lambda, E)$.
Suppose $\Delta \prec_{D} \Theta$.
Since $\Gamma \prec \Delta \prec_{D} \Theta$, by Lemma~\ref{MClem1}, $\Gamma \prec_{D} \Theta$.
By $(\Theta, D) \mathrel{S_{(\Gamma, B)}} (\Lambda, E)$, we have ${\sim} D \in \Lambda$.
Therefore, we conclude $(\Theta, D) \mathrel{S_{(\Delta, C)}} (\Lambda, E)$.
Then, $\PP$ is valid in $(W, R, \{S_{w}\}_{w \in W})$ by Proposition~\ref{FCP}.
\begin{cl}
Let $A' \in \Phi$ and $(\Gamma, B) \in W$.
Then,
\[
(\Gamma, B) \Vdash A' \iff A' \in \Gamma.
\]
\end{cl}
\begin{proof}
We prove the claim by induction on the construction of $A'$.
We only prove the case that $A'$ is of the form $C \rhd D$.
($\Rightarrow$):
Suppose $C \rhd D \notin \Gamma$.
By Fact~\ref{MClem2}, there exists a $\Delta \in K_{\Phi}$ such that $C \in \Delta$ and $\Gamma \prec_{D} \Delta$.
Then, $(\Delta, D) \in W$ and $(\Gamma, B) \mathrel{R} (\Delta, D)$.
Also, $(\Delta, D) \Vdash C$ by the induction hypothesis.
Let $(\Theta, E) \in W$ be such that $(\Delta, D) \mathrel{S_{(\Gamma, B)}} (\Theta, E)$.
Then, ${\sim}D \in \Theta$.
By the induction hypothesis, we have $(\Theta, E) \nVdash D$.
Therefore, we conclude $(\Gamma, B) \nVdash C \rhd D$.
($\Leftarrow$):
Suppose $C \rhd D \in \Gamma$ and let $(\Delta, E) \in W$ be such that $(\Gamma, B) \mathrel{R} (\Delta, E)$ and $(\Delta, E) \Vdash C$.
By the induction hypothesis, $C \in \Delta$.
We distinguish the following two cases.
\begin{enumerate}
\item Suppose $\Gamma \prec_{E} \Delta$.
By Fact~\ref{MClem3}, there exists a $\Theta \in K_{\Phi}$ such that $D, {\sim}E \in \Theta$.
Then, $(\Theta, E) \in W$ and $(\Delta, E) \mathrel{S_{(\Gamma, B)}} (\Theta, E)$.
Also, $(\Theta, E) \Vdash D$ by the induction hypothesis.
Therefore, we conclude $(\Gamma, B) \Vdash C \rhd D$.
\item Suppose $\Gamma \nprec_{E} \Delta$.
Since $\Gamma \prec_{\bot} \Delta$, there exists a $\Theta \in K_{\Phi}$ such that $D \in \Theta$ by Fact~\ref{MClem3}.
The rest of the proof is completely the same as in the first case. \qedhere
\end{enumerate}
\end{proof}
Since $A \notin \Gamma_0$, we conclude $(\Gamma_{0}, \bot) \nVdash A$ by the claim.
This finishes our proof of Theorem~\ref{MCILP}.
\end{proof}
Next, we prove the modal completeness for $\IL^-(\PP)$ with respect to simplified $\IL^-(\PP)$-frames.
Moreover, we consider the following condition $(\dagger)$ on simplified $\IL^-(\PP)$-frames $(W, R, S)$:
\[
\text{There exist no}\ x, y, z \in W\ \text{such that}\ x S y S z. \hspace{1in} (\dagger)
\]
\begin{thm}\label{SVC}
For any $\mathcal{L}(\rhd)$-formula $A$, the following are equivalent:
\begin{enumerate}
\item $\IL^-(\PP) \vdash A$.
\item $A$ is valid in all simplified $\IL^-(\PP)$-frames.
\item $A$ is valid in all finite simplified $\IL^-(\PP)$-frames satisfying the condition \textup{($\dagger$)}.
\end{enumerate}
\end{thm}
\begin{proof}
$(1 \Rightarrow 2)$: This is immediate from Proposition~\ref{Psou}.
$(2 \Rightarrow 3)$: Trivial.
$(3 \Rightarrow 1)$:
Suppose $\IL^-(\PP) \nvdash A$.
Then, by Theorem~\ref{MCILP}, there exist a finite $\IL^- \!$-model $(W, R, \{S_{w}\}_{w \in W}, \Vdash)$ and $w_0 \in W$ such that $\PP$ is valid in $(W, R, \{S_{w}\})$ and $w_0 \nVdash A$.
We would like to find some finite simplified $\IL^-(\PP)$-model $(W', R', S', \Vdash')$ and $w' \in W'$ such that $w' \nVdash' A$.
Let $(W', R', S', \Vdash')$ be a quadruple satisfying the following conditions:
\begin{enumerate}
\item $W' :=\{\seq{w_1, \ldots, w_n} \mid n \geq 1, (\forall i \leq n)(w_{i} \in W), \, \, \text{and} \, \, (\forall i < n)(w_{i} {R} w_{i+1}) \}$;
\item $\seq{x_1, \ldots, x_n} \mathrel{R'} \seq{y_1, \ldots, y_m} :\iff n < m$ and $(\forall i \leq n)(x_i = y_i)$;
\item $\seq{x_1, \ldots, x_n} \mathrel{S'} \seq{y_1, \ldots, y_m} :\iff n > 1$, $m=1$, and $x_{n} {S_{x_{n-1}}} y_{m}$;
\item $\seq{w_1, \ldots, w_n} \Vdash' p :\iff w_n \Vdash p$.
\end{enumerate}
$W'$ is a non-empty set because $\seq{w_0} \in W'$.
Furthermore, since $W$ is finite and $R$ is a conversely well-founded binary relation, $W'$ is also finite.
Therefore, $(W', R', S', \Vdash')$ is a finite simplified $\IL^-(\PP)$-model.
For any elements $\seq{x_1, \ldots, x_n}$, $\seq{y_1, \ldots, y_m}$, $\seq{z_1, \ldots, z_k}$ of $W'$, if $\seq{x_1, \ldots, x_n} \mathrel{S'} \seq{y_1, \ldots, y_m}$, then $m = 1$, and hence $\seq{y_1, \ldots, y_m}\mathrel{S'} \seq{z_1, \ldots, z_k}$ does not hold.
This means that the frame $(W', R', S')$ satisfies the condition $(\dagger)$.
\begin{cl}
Let $A'$ be any $\mathcal{L}(\rhd)$-formula and let $\seq{w_1, \ldots, w_n} \in W'$.
Then,
\[
\seq{w_1, \ldots, w_n} \Vdash' A' \iff w_n \Vdash A'.
\]
\end{cl}
\begin{proof}
We prove the claim by induction on the construction of $A'$.
We only give a proof of the case that $A'$ is of the form $B \rhd C$.
($\Rightarrow$):
Suppose $\seq{w_1, \ldots, w_n} \Vdash' B \rhd C$.
Let $x \in W$ be such that $w_{n} {R} x$ and $x \Vdash B$.
Then, we have $\seq{w_1, \ldots, w_n, x} \in W'$ and $\seq{w_1, \ldots, w_n} \mathrel{R'} \seq{w_1, \ldots, w_n, x}$.
By the induction hypothesis, $\seq{w_1, \ldots, w_n, x} \Vdash B$.
By our supposition, there exists a $\seq{y} \in W'$ such that $\seq{w_1, \ldots, w_n, x} \mathrel{S'} \seq{y} \Vdash' C$.
By the definition of $S'$ and the induction hypothesis, we have $x {S_{w_{n}}} y$ and $y \Vdash C$.
Hence, $w_n \Vdash B \rhd C$.
($\Leftarrow$): Suppose $w_n \Vdash B \rhd C$.
Let $\seq{x_1,\ldots, x_{m}} \in W'$ be such that $\seq{w_1, \ldots, w_n} \mathrel{R'} \seq{x_1,\ldots, x_{m}}$ and $\seq{x_1,\ldots, x_{m}} \Vdash' B$.
By the definition of $R'$ and the induction hypothesis, we have $n < m$, $w_{n} {R} x_{m}$ and $x_m \Vdash B$.
By our supposition, there exists a $y \in W$ such that $x_{m} {S_{w_{n}}} y$ and $y \Vdash C$.
Since $n < m$, either $n = m-1$ or $n < m-1$.
Therefore, we have either $w_{n} = x_{m-1}$ or $w_{n} {R} x_{m-1}$ by the definition of $R'$.
If $w_{n} = x_{m-1}$, then $x_{m} {S_{x_{m-1}}} y$ is obvious.
If $w_{n} {R} x_{m-1}$, then $x_{m} {S_{x_{m-1}}} y$ holds by Proposition~\ref{FCP} because $\PP$ is valid in $(W, R, \{S_{w}\}_{w \in W})$.
In either case, we obtain $x_{m} {S_{x_{m-1}}} y$.
Then, $\seq{x_1,\ldots, x_{m}} \mathrel{S'} \seq{y}$ because $m > n \geq 1$.
Also, we have $\seq{y} \Vdash' C$ by the induction hypothesis.
Therefore, we conclude $\seq{w_1, \ldots, w_n} \Vdash' B \rhd C$.
\qedhere
\end{proof}
Since $w_{0} \nVdash A$, we obtain $\seq{w_{0}} \nVdash' A$ by the claim.
\end{proof}
\subsection{An embedding of $\IL^-(\PP)$ into bimodal logics}
In this subsection, as an application of Theorem~\ref{SVC}, we show that $\IL^-(\PP)$ is faithfully embedded into several bimodal logics.
The language $\mathcal{L}_2$ of bimodal propositional logic is the language of propositional logic equipped with two unary modal operators $[0]$ and $[1]$.
The logic $\GLK$ in the language $\mathcal{L}_2$ has the following axioms and rules:
\begin{itemize}
\item All tautologies in the language $\mathcal{L}_2$;
\item $[0] (A \to B) \to ([0] A \to [0] B)$;
\item $[0] ([0] A \to A) \to [0] A$;
\item $[1] (A \to B) \to ([1] A \to [1] B)$;
\item $\dfrac{A \to B \quad A}{B}$;
\item $\dfrac{A}{[k] A}$ for $k \in \{0, 1\}$.
\end{itemize}
The logic $\GLK$ is called the \textit{fusion} of $\GL$ and $\K$.
Let $(\GLK) \oplus [1][1]\bot$ be the logic obtained from $\GLK$ by adding $[1][1]\bot$ as an axiom.
We say that a triple $(W, R_0, R_1)$ is a \textit{$\GLK$-frame} if $W$ is a nonempty set, $R_0$ and $R_1$ are binary relations on $W$, and $R_0$ is transitive and conversely well-founded.
Here for each $k \in \{0, 1\}$, the binary relation $R_k$ corresponds to the modal operator $[k]$.
From a general result about fusions of modal logics, it is known that $\GLK$ is characterized by the class of all finite $\GLK$-frames (cf.~Kurucz~\cite[Theorem 3]{Kuru07}).
We introduce a translation $\chi$ from $\mathcal{L}(\rhd)$-formulas into $\mathcal{L}_2$-formulas defined as follows:
\begin{enumerate}
\item $\chi(\bot)$ is $\bot$;
\item $\chi(p)$ is $p$ for each propositional variable $p$;
\item $\chi(\neg A)$ is $\neg \chi(A)$;
\item $\chi(A \circ B)$ is $\chi(A) \circ \chi(B)$ for $\circ \in \{\land, \lor, \to\}$;
\item $\chi(\Box A)$ is $[0] \chi(A)$;
\item $\chi(A \rhd B)$ is $[0]\bigl(\chi(A) \to \langle 1 \rangle \chi(B)\bigr)$.
\end{enumerate}
Here $\langle 1 \rangle$ is the abbreviation for $\neg [1] \neg$.
Then, we prove the following embedding result.
\begin{prop}\label{Embed}
For any modal formula $A$, the following are equivalent:
\begin{enumerate}
\item $\IL^-(\PP) \vdash A$.
\item $\GLK \vdash \chi(A)$.
\item $(\GLK) \oplus [1][1]\bot \vdash \chi(A)$.
\end{enumerate}
\end{prop}
\begin{proof}
$(1 \Rightarrow 2)$:
We prove the implication by induction on the length of proofs in $\IL^-(\PP)$.
\begin{itemize}
\item If $A$ is one of $\G{1}$, $\G{2}$, and $\G{3}$, then $\chi(A)$ is also an axiom of $\GLK$.
\item ($\J{3}$): Since
\begin{align*}
\bigl(\chi(A) \to \langle 1 \rangle \chi(C)\bigr) \land \bigl(\chi(B) \to \langle 1 \rangle \chi(C)\bigr) \to \bigl(\chi(A \lor B) \to \langle 1 \rangle \chi(C)\bigr)
\end{align*}
is a tautology, we have
\[
\GLK \vdash \chi(A \rhd C) \land \chi(B \rhd C) \to \chi((A \lor B) \rhd C).
\]
\item ($\J{6}$): Notice that $\chi((\neg A) \rhd \bot)$ is $[0] (\neg \chi(A) \to \langle 1 \rangle \bot)$.
Since $\GLK \vdash \neg \langle 1 \rangle \bot$, $\GLK \vdash \chi((\neg A) \rhd \bot) \leftrightarrow \chi(\Box A)$.
\item ($\PP$): Since $\chi(A \rhd B)$ is of the form $[0] C$ for some $C$, $\GLK \vdash \chi(A \rhd B) \to \chi(\Box (A \rhd B))$.
\item If $A$ is derived by using Modus Ponens or Necessitation, then $\GLK \vdash \chi(A)$ is obvious by using the induction hypothesis.
\item If $A$ is derived by using the rule $\R{1}$ from $B \to C$, then $A$ is of the form $D \rhd B \to D \rhd C$ for some $D$.
By the induction hypothesis, $\GLK \vdash \chi(B) \to \chi(C)$.
Then, $\GLK \vdash \langle 1 \rangle \chi(B) \to \langle 1 \rangle \chi(C)$, and hence
\[
\GLK \vdash \bigl(\chi(D) \to \langle 1 \rangle \chi(B)\bigr) \to \bigl(\chi(D) \to \langle 1 \rangle \chi(C)\bigr).
\]
Thus, $\GLK \vdash \chi(D \rhd B) \to \chi(D \rhd C)$.
\item If $A$ is derived from $B \to C$ by using the rule $\R{2}$, then $A$ is of the form $C \rhd D \to B \rhd D$.
By the induction hypothesis, $\GLK \vdash \chi(B) \to \chi(C)$, and then
\[
\GLK \vdash \bigl(\chi(C) \to \langle 1 \rangle \chi(D)\bigr) \to \bigl(\chi(B) \to \langle 1 \rangle \chi(D)\bigr).
\]
We obtain $\GLK \vdash \chi(C \rhd D) \to \chi(B \rhd D)$.
\end{itemize}
$(2 \Rightarrow 3)$: Obvious.
$(3 \Rightarrow 1)$: Suppose $\IL^-(\PP) \nvdash A$.
By Theorem~\ref{SVC}, there exists a finite simplified $\IL^-(\PP)$-model $(W, R, S, \Vdash)$ satisfying the condition ($\dagger$) and an element $w \in W$ such that $w \nVdash A$.
Then, $(W, R, S)$ is also a $\GLK$-frame.
Let $\Vdash^*$ be a satisfaction relation on the $\GLK$-frame $(W, R, S)$ satisfying $x \Vdash^* p \iff x \Vdash p$ for any $x \in W$ and propositional variable $p$.
Then, it is shown by induction on the construction of $B$ that for any $\mathcal{L}(\rhd)$-formula $B$ and $x \in W$, $x \Vdash^* \chi(B)$ if and only if $x \Vdash B$.
Hence, we obtain $w \nVdash^* \chi(A)$.
Also, by the condition $(\dagger)$, $[1][1]\bot$ is valid in the $\GLK$-frame $(W, R, S)$.
Therefore, every theorem of $(\GLK) \oplus [1][1]\bot$ is valid in the frame.
Hence, we conclude that $(\GLK) \oplus [1][1] \bot \nvdash \chi(A)$.
\end{proof}
From this embedding result, $\IL^-(\PP)$ is faithfully embeddable into bimodal logics $L$ between the logics $\GLK$ and $(\GLK) \oplus [1][1]\bot$.
Prominent examples of such logics $L$ are $\GL \otimes \mathbf{K4} = (\GLK) \oplus ([1]A \to [1][1]A)$ and $\FGL = (\GLK) \oplus ([1]([1]A \to A) \to [1]A)$.
Finally, we prove the failure of FPP for the logic $(\GLK) \oplus [1][1] \bot$.
\begin{prop}\label{FailureofFPP}
For any $\mathcal{L}_2$-formula $A$,
\[
(\GLK) \oplus [1][1] \bot \nvdash A \leftrightarrow [0] \neg [1] A.
\]
Consequently, the logic $(\GLK) \oplus [1][1] \bot$ does not enjoy FPP.
\end{prop}
\begin{proof}
Let $(W, R_0, R_1)$ be a triple with $W : = \{x, y\}$, $R_0 := \{(x, y)\}$, and $R_1 := \{(y, x)\}$.
It is easily shown that $(W, R_0, R_1)$ is a $\GLK$-frame in which $[1][1] \bot$ is valid.
Let $A$ be any $\mathcal{L}_2$-formula and $\Vdash$ be any satisfaction relation on the frame.
Since $x \Vdash A$ if and only if $y \Vdash [1]A$, and $y \Vdash [1]A$ if and only if $x \Vdash \neg [0] \neg [1] A$, we obtain that $x \Vdash A \leftrightarrow \neg [0] \neg [1] A$.
Therefore, $x \nVdash A \leftrightarrow [0] \neg [1] A$.
We conclude $(\GLK) \oplus [1][1]\bot \nvdash A \leftrightarrow [0] \neg [1] A$.
\end{proof}
As a benefit of our embedding result, we get the following corollary in contrast to Theorem~\ref{lFPP}.
\begin{cor}\label{FailureofFPP2}
For any $\mathcal{L}(\rhd)$-formula $A$,
\[
\IL^-(\PP) \nvdash A \leftrightarrow \top \rhd \lnot A.
\]
Hence, $\IL^-(\PP)$ does not enjoy FPP.
\end{cor}
\begin{proof}
Suppose, towards a contradiction, that $\IL^-(\PP) \vdash A \leftrightarrow \top \rhd \lnot A$ for some $\mathcal{L}(\rhd)$-formula $A$.
Then, by Proposition~\ref{Embed}, $(\GLK) \oplus [1][1]\bot \vdash \chi(A) \leftrightarrow \chi(\top \rhd \neg A)$.
It follows that $(\GLK) \oplus [1][1]\bot$ proves $\chi(A) \leftrightarrow [0] \neg [1] \chi(A)$.
This contradicts Proposition~\ref{FailureofFPP}.
\end{proof}
We say that the \textit{uniqueness of fixed points (UFP)} holds for a logic $L$ if for any $\mathcal{L}(\rhd)$-formula $A(p)$ such that $p$ is modalized in $A(p)$ and $A(p)$ does not contain $q$,
\[
L \vdash \boxdot(p \leftrightarrow A(p) ) \land \boxdot (q \leftrightarrow A(q) ) \to ( p \leftrightarrow q),
\]
where $\boxdot F$ is an abbreviation for $\Box F \land F$.
It was essentially proved in de Jongh and Visser~\cite[Theorem 2.1]{DeJVis91} that UFP holds for $\IL^-(\J{4}_+)$.
As shown in~\cite[Lemma 3.10]{IKO20}, for any extension $L$ of $\IL^-$, if $L$ has CIP and UFP holds for $L$, then $L$ has FPP.
This explains why Fact~\ref{CIPFPP}.2 holds.
From Theorem~\ref{CIP} and Corollary~\ref{FailureofFPP2}, we obtain that UFP does not hold for $\IL^-(\PP)$.
\section{Arithmetical semantic aspects of $\IL^-(\PP)$}\label{Sec_AC}
In this section, we investigate arithmetical semantics for $\IL^-(\PP)$.
First, we introduce appropriate arithmetical interpretations for $\IL^-(\PP)$ inspired from our embedding $\chi$.
Secondly, we strengthen Theorem~\ref{SVC}.
Then, by using such a strengthening, we prove the arithmetical completeness theorem for $\IL^-(\PP)$.
Throughout this section, we assume that $T$ always denotes a consistent recursively enumerable extension of $\PA$ in the language $\mathcal{L}_A$ of arithmetic.
We say that a formula $\tau(u)$ is a \textit{numeration} of $T$ if for any natural number $n$, $n$ is the G\"odel number of an element of $T$ if and only if $\PA \vdash \tau(\overline{n})$.
For each numeration $\tau(u)$ of $T$, we can construct a formula $\Prf_{\tau}(x, y)$ saying that $y$ is the G\"odel number of a proof of a sentence whose G\"odel number is $x$ from the theory axiomatized by the set of sentences defined by $\tau$.
Let $\PR_\tau(x)$ be the formula $\exists y \Prf_\tau(x, y)$.
Then, the following fact holds:
\begin{fact}[Feferman~\cite{Fef60}]\label{DC}
Let $\tau(u)$ be any numeration of $T$ and let $\varphi$ and $\psi$ be any $\mathcal{L}_A$-formulas.
\begin{enumerate}
\item If $T \vdash \varphi$, then $\PA \vdash \PR_\tau(\gn{\varphi})$.
\item $\PA \vdash \PR_\tau(\gn{\varphi \to \psi}) \to (\PR_\tau(\gn{\varphi}) \to \PR_\tau(\gn{\psi}))$.
\item If $\varphi$ is a $\Sigma_1$ sentence, then $\PA \vdash \varphi \to \PR_\tau(\gn{\varphi})$.
\end{enumerate}
\end{fact}
If $\tau(u)$ is a $\Sigma_1$ numeration of $T$, then it is known that $\PR_\tau(x)$ is also $\Sigma_1$.
Hence, $\PA \vdash \PR_\tau(\gn{\varphi}) \to \PR_\tau(\gn{\PR_\tau(\gn{\varphi})})$ holds for any $\mathcal{L}_A$-formula $\varphi$.
On the other hand, there exist $\Sigma_2$ numerations $\tau(u)$ of $T$ such that $T \nvdash \PR_\tau(\gn{\varphi}) \to \PR_\tau(\gn{\PR_\tau(\gn{\varphi})})$ (see~\cite{Kur18_1,Kur18_2,Vis21}).
We say that a numeration $\tau(u)$ of $T$ satisfies the \textit{L\"ob condition} if $\PA \vdash \PR_\tau(\gn{\varphi}) \to \PR_\tau(\gn{\PR_\tau(\gn{\varphi})})$ for any $\mathcal{L}_A$-formula $\varphi$.
A mapping $f$ from the set of all propositional variables to a set of $\mathcal{L}_A$-sentences is called an \textit{arithmetical interpretation}.
For any pair $(\tau_0, \tau_1)$ of numerations of $T$, every arithmetical interpretation $f$ is extended to a mapping $f'$ whose domain is the set of all $\mathcal{L}_2$-formulas by the following clauses:
\begin{enumerate}
\item $f'(\bot)$ is $0 = 1$;
\item $f'(\neg A)$ is $\neg f'(A)$;
\item $f'(A \circ B)$ is $f'(A) \circ f'(B)$ for $\circ \in \{\land, \lor, \leftrightarrow\}$;
\item $f'([k] A)$ is $\PR_{\tau_k}(\gn{f'(A)})$ for $k \in \{0, 1\}$.
\end{enumerate}
In particular, it was proved that $\mathbf{GLK} = (\GLK) \oplus ([0]A \to [1]A)$ is arithmetically sound and complete with respect to the classes of all pairs $(\tau_0, \tau_1)$ of numerations such that $\tau_0(u)$ is $\Sigma_1$ and $\PA \vdash \forall x(\PR_{\tau_0}(x) \to \PR_{\tau_1}(x))$ (\cite[Corollary 4.11]{Kur18_1}).
Also, Beklemishev~\cite[Theorem 1]{Bek92} proved that $\mathbf{CS}_1 = (\FGL) \oplus ([0]A \to [1][0]A) \oplus ([1]A \to [0][1]A)$ and its extensions are arithmetically sound and complete with respect to some appropriate classes of pairs of $\Sigma_1$ numerations.
Here we extend arithmetical interpretations to mappings of $\mathcal{L}(\rhd)$-formulas inspired by our embedding result of $\IL^-(\PP)$ into bimodal logics.
Since the translation $\chi$ introduced in the last section is such an embedding, the mapping $f' \circ \chi$ seems to be appropriate.
In fact, by Proposition~\ref{Embed}, for any theorem $A$ of $\IL^-(\PP)$, we obtain $\GLK \vdash \chi(A)$, and if $\tau_0$ satisfies the L\"ob condition, then it is easily shown that $\PA \vdash (f' \circ \chi)(A)$.
From this observation, we directly extend every arithmetical interpretation $f$ to a mapping $f_{\tau_0, \tau_1}$ from the set of all $\mathcal{L}(\rhd)$-formulas to a set of $\mathcal{L}_A$-sentences as follows:
\begin{enumerate}
\item $\FTT{\bot}$ is $0 = 1$;
\item $\FTT{\neg A}$ is $\neg \FTT{A}$;
\item $\FTT{A \circ B}$ is $\FTT{A} \circ \FTT{B}$ for $\circ \in \{\land, \lor, \leftrightarrow\}$;
\item $\FTT{\Box A}$ is $\PR_{\tau_0}(\gn{\FTT{A}})$;
\item $\FTT{A \rhd B}$ is $\PR_{\tau_0}(\gn{\FTT{A} \to \Con_{\tau_1 + \FTT{B}}})$.
\end{enumerate}
Here for each $\mathcal{L}_A$-sentence $\psi$, $(\tau_1 + \psi)(u)$ is the numeration $\tau_1(u) \lor u = \gn{\psi}$ of the theory $T + \psi$.
The following arithmetical soundness of $\IL^-(\PP)$ follows from the above argument.
\begin{prop}[Arithmetical soundness for $\IL^-(\PP)$]\label{AS}
Let $A$ be any $\mathcal{L}(\rhd)$-formula $A$ and $(\tau_0, \tau_1)$ be any pair of numerations of $T$.
If $\IL^-(\PP) \vdash A$ and $\tau_0$ satisfies the L\"ob condition, then for any arithmetical interpretation $f$, $\PA \vdash \FTT{A}$.
\end{prop}
The remainder of this section is devoted to proving the converse of Proposition~\ref{AS}.
Furthermore, we prove the following uniform version of the arithmetical completeness theorem.
\begin{thm}[Uniform arithmetical completeness theorem for $\IL^-(\PP)$]\label{AC}
There exist a pair $(\tau_0, \tau_1)$ of $\Sigma_2$ numerations of $T$ such that both $\tau_0$ and $\tau_1$ satisfy the L\"ob condition and an arithmetical interpretation $f$ such that for any $\mathcal{L}(\rhd)$-formula $A$, the following are equivalent:
\begin{enumerate}
\item $\IL^-(\PP) \vdash A$.
\item $\PA \vdash \FTT{A}$.
\item $T \vdash \FTT{A}$.
\end{enumerate}
\end{thm}
From Proposition~\ref{AS} and Theorem~\ref{AC}, we obtain the following corollary.
\begin{cor}
For any $\mathcal{L}(\rhd)$-formula $A$, the following are equivalent:
\begin{enumerate}
\item $\IL^-(\PP) \vdash A$.
\item For any pair $(\tau_0, \tau_1)$ of numerations of $T$ in which $\tau_0$ satisfies the L\"ob condition and any arithmetical interpretation $f$, $\PA \vdash \FTT{\chi(A)}$.
\item For any pair $(\tau_0, \tau_1)$ of $\Sigma_2$ numerations of $T$ such that both $\tau_0$ and $\tau_1$ satisfy the L\"ob condition and any arithmetical interpretation $f$, $T \vdash \FTT{\chi(A)}$.
\end{enumerate}
\end{cor}
Before proving Theorem~\ref{AC}, we strengthen Theorem~\ref{SVC}.
\begin{defn}
For each $\mathcal{L}(\rhd)$-formula $A$, we define the natural number $d(A)$ recursively as follows:
\begin{enumerate}
\item $d(p) = d(\bot) = 0$;
\item $d(A \circ B) = \max\{d(A), d(B)\}$ for $\circ \in \{\land, \lor, \to\}$;
\item $d(\neg A) = d(\Box A) = d(A)$;
\item $d(A \rhd B) = \max\{d(A), d(B)+1\}$.
\end{enumerate}
\end{defn}
\begin{thm}\label{SVC2}
The logic $\IL^-(\PP)$ is sound and complete with respect to the class of all finite simplified $\IL^-(\PP)$-frames $(W, R, S)$ such that $S$ is transitive and $R \cup S$ is conversely well-founded.
\end{thm}
\begin{proof}
Suppose $\IL^-(\PP) \nvdash A$, then by Theorem~\ref{SVC}, there exists a finite simplified $\IL^-(\PP)$-model $\mathcal{M} = (W, R, S, \Vdash)$ satisfying the condition $(\dagger)$ and an element $w \in W$ such that $w \nVdash A$.
We define a new simplified $\IL^-(\PP)$-model $\mathcal{M}^* = (W^*, R^*, S^*, \Vdash^*)$ as follows:
\begin{itemize}
\item $W^* : = \{(x, n) \mid x \in W$ and $0 \leq n \leq d(A)\}$;
\item $(x, n) R^* (y, m) : \iff x R y$ and $n = m$;
\item $(x, n) S^* (y, m) : \iff x S y$ and $n = m+1$;
\item $(x, n) \Vdash^* p : \iff x \Vdash p$.
\end{itemize}
The frame $(W^*, R^*, S^*)$ consists of the $d(A)+1$ copies of the original frame $(W, R, S)$ with different levels as indicated in the second components of elements of $W$.
The $R^*$-transition does not change the level, and the $S^*$-transition reduces the level by one.
By the condition $(\dagger)$ for $(W, R, S)$ and the definition of our $S^*$, the frame $(W^*, R^*, S^*)$ also satisfies the condition $(\dagger)$.
In particular, we have that $S^*$ is transitive.
The value of the second component of each element of $W^*$ is not changed by the $R^*$-transition, but it is decreased by the $S^*$-transition.
So every $R^* \cup S^*$-chain of elements of $W^*$ contains only a finite number of $S^*$-transitions.
Obviously $R^*$ is conversely well-founded because so is $R$.
Hence, the relation $R^* \cup S^*$ is also conversely well-founded.
It suffices to show that $A$ is not valid in the model $\mathcal{M}^*$.
\begin{cl}
For any $\mathcal{L}(\rhd)$-formula $B$ and any $(x, n) \in W^*$, if $d(B) \leq n$, then
\[
(x, n) \Vdash^* B \iff x \Vdash B.
\]
\end{cl}
\begin{proof}
We prove the claim by induction on the construction of $B$.
We give only a proof of the case that $B$ is of the form $C \rhd D$.
Assume $d(C \rhd D) \leq n$.
$(\Rightarrow)$:
Suppose $(x, n) \Vdash^* C \rhd D$.
Let $y \in W$ be any element such that $x R y$ and $y \Vdash C$.
Then, $(x, n) R^* (y, n)$.
Since $d(C) \leq d(C \rhd D) \leq n$, by the induction hypothesis, $(y, n) \Vdash^* C$.
Then, there exists a $(z, m) \in W^*$ such that $(y, n) S^* (z, m)$ and $(z, m) \Vdash^* D$.
In this case, $y S z$ and $n = m + 1$.
Since $d(D) + 1 \leq d(C \rhd D) \leq n = m + 1$, we have $d(D) \leq m$.
Then, by the induction hypothesis, $z \Vdash D$.
Therefore, $x \Vdash C \rhd D$.
$(\Leftarrow)$:
Suppose $x \Vdash C \rhd D$.
Let $(y, m) \in W^*$ be such that $(x, n) R^* (y, m)$ and $(y, m) \Vdash^* C$.
Then, $x R y$ and $n = m$.
Since $d(C) \leq d(C \rhd D) \leq n = m$, by the induction hypothesis, $y \Vdash C$.
Then, there exists a $z \in W$ such that $y S z$ and $z \Vdash D$.
Since $d(D) + 1 \leq d(C \rhd D) \leq n$, we have $d(D) \leq n -1$.
By the induction hypothesis, $(z, n - 1) \Vdash^* D$.
Also, $(y, m) S^* (z, n - 1)$ because $m = (n - 1) + 1$.
We conclude $(x, n) \Vdash^* C \rhd D$.
\end{proof}
By the claim, we obtain $(w, d(A)) \nVdash^* A$, and hence $A$ is not valid in $\mathcal{M}^*$.
\end{proof}
Our proof of Theorem~\ref{AC} is merely a tracing of the proof of~\cite[Theorem 4.1]{Kur18_2}.
Therefore, we only give an outline of a proof and we leave the details to that paper.
From our proofs of Theorems~\ref{SVC} and~\ref{SVC2}, we obtain a primitive recursively represented simplified $\IL^-(\PP)$-model $\mathcal{M} = (W, R, S, \Vdash)$ satisfying the following conditions:
\begin{itemize}
\item $W = \omega$;
\item For any $x \in W \setminus \{0\}$, $0 R x$;
\item $S$ is transitive and $R \cup S$ is conversely well-founded;
\item The restriction of $\mathcal{M}$ to the set $W \setminus \{0\}$ is a disjoint union of infinitely many finite simplified $\IL^-(\PP)$-models;
\item For any $\mathcal{L}(\rhd)$-formula $A$, if $\IL^-(\PP) \nvdash A$, then there exists an $i \in W \setminus \{0\}$ such that $i \nVdash A$.
\end{itemize}
Let $R^*$ be the transitive closure of $R \cup S$.
Also, let $\sim$ be an equivalence relation on $W \setminus \{0\}$ defined by
\[
i \sim j : \iff i\ \text{and}\ j\ \text{belong to the same model in the disjoint union}.
\]
Let $x R y$, $x S y$, $x R^* y$, and $x \sim y$ be $\Delta_1(\PA)$ formulas naturally representing the corresponding binary relations in $\PA$.
We may assume that $\PA$ proves several basic facts about these relations.
For example, for each $i \neq 0$, $\PA \vdash \forall x (\overline{i} R x \leftrightarrow \bigvee_{i R j} x = \overline{j})$.
Let $\sigma(u)$ be a $\Sigma_1$ numeration of $T$ such that $T \nvdash \neg \Con_T^n$ for all $n \in \omega$, where $\Con_T^n$ is the sentence obtained by applying $\PR_T(\cdot)$ to $0=1$ $n$ times.
The existence of such a numeration is proved in~\cite[Lemma 7]{Bek90}.
Notice that the assumption that $T$ is an extension of $\PA$ is used to guarantee the existence of $\sigma(u)$.
Since $R^*$ is transitive and conversely well-founded, as in the usual proof of the uniform arithmetical completeness theorem of $\GL$, we obtain a $\Sigma_2$ Solovay formula $\alpha(x)$ satisfying the following conditions (cf.~\cite[Chapter 9]{Boo93}):
\begin{enumerate}
\item $\PA \vdash \exists x \alpha(x)$;
\item $\PA \vdash \forall x \forall y(\alpha(x) \land \alpha(y) \to x = y)$;
\item $\PA \vdash \forall x \forall y(\alpha(x) \land x R^* y \to \neg \PR_{\sigma}(\gn{\neg \alpha(\dot{y})}))$;
\item $\PA \vdash \forall x(\alpha(x) \land x \neq 0 \to \PR_{\sigma}(\gn{\exists z(\dot{x} R^* z \land \alpha(z))}))$;
\item $\N \models \alpha(0)$.
\end{enumerate}
Here $\N$ is the standard model of arithmetic.
In the following, let $Q \in \{R, S\}$.
Define $\delta_Q(x, u)$ to be the $\Sigma_1$ formula
\[
\exists z(x \sim z \land u = \gn{\neg \alpha(\dot{z})} \land \neg x Q z).
\]
Also, define $\gamma_Q(u)$ to be the $\Sigma_2$ formula
\[
\exists x(\alpha(x) \land x \neq 0 \land \delta_Q(x, u)).
\]
Let $\tau_Q(u)$ be the $\Sigma_2$ formula $\sigma(u) \lor \gamma_Q(u)$.
Since $\PA \vdash \alpha(0) \to \forall u \neg \gamma_Q(u)$, we have $\PA \vdash \alpha(0) \to \forall u \bigl(\tau_Q(u) \leftrightarrow \sigma(u) \bigr)$.
Then, $\tau_Q(u)$ is a $\Sigma_2$ numeration of $T$ (cf.~\cite[Lemma 4.13]{Kur18_2}).
As in the proof of~\cite[Theorem 4.1]{Kur18_2}, we can prove the following lemmas.
\begin{lem}[{cf.~\cite[Lemma 4.6]{Kur18_2}}]\label{Lem1}\leavevmode
\begin{enumerate}
\item $\PA \vdash \forall x \forall y \bigl(\alpha(x) \land x \neq 0 \land x \sim y \land \neg x Q y \to \PR_{\tau_Q}(\gn{\neg \alpha(\dot{y})}) \bigr)$.
\item $\PA \vdash \forall x \bigl(\alpha(x) \land x \neq 0 \to \PR_{\tau_Q}(\gn{\exists z (\dot{x} Q z \land \alpha(z))}) \bigr)$.
\end{enumerate}
\end{lem}
\begin{lem}[{cf.~\cite[Lemma 4.10]{Kur18_2}}]\label{Lem2}
If $i Q j$, then $\PA \vdash \alpha(\overline{i}) \to \neg \PR_{\tau_Q}(\gn{\neg \alpha(\overline{j})})$.
\end{lem}
\begin{lem}[{cf.~\cite[Lemmas 4.3, 4.7 and 4.8]{Kur18_2}}]\label{Lem3}\leavevmode
\begin{enumerate}
\item $\PA \vdash \forall x \forall y \bigl(x \neq 0 \land x Q y \to (\delta_Q(x, u) \to \delta_Q(y, u)) \bigr)$.
\item $\PA \vdash \exists x \exists z \bigl(x \neq 0 \land x Q z \land \alpha(z) \land \PR_{\sigma \lor \delta_Q(x)}(\gn{\varphi}) \bigr) \to \PR_{\tau_Q}(\gn{\varphi})$ for any $\mathcal{L}_A$-formula $\varphi$.
\item $\PA \vdash \PR_{\tau_Q}(\gn{\varphi}) \to \PR_{\tau_Q}(\gn{\PR_{\tau_Q}(\gn{\varphi})})$ for any $\mathcal{L}_A$-formula $\varphi$.
\end{enumerate}
\end{lem}
Lemma~\ref{Lem3}.3 means that $\tau_Q(u)$ satisfies the L\"ob condition.
Notice that the transitivity of $Q$ is used in the proof of Lemma~\ref{Lem3}.1.
Let $f$ be the arithmetical interpretation defined by $f(p) \equiv \exists x (\alpha(x) \land x \Vdash p)$.
Finally, we prove the following lemma.
\begin{lem}\label{Lem4}
Let $B$ be any $\mathcal{L}(\rhd$)-formula and $i \neq 0$.
\begin{enumerate}
\item If $i \Vdash B$, then $\PA \vdash \alpha(\overline{i}) \to \FRS{B}$;
\item If $i \nVdash B$, then $\PA \vdash \alpha(\overline{i}) \to \neg \FRS{B}$.
\end{enumerate}
\end{lem}
\begin{proof}
We prove Clauses 1 and 2 simultaneously by induction on the construction of $B$.
We give only a proof of the case that $B$ is of the form $C \rhd D$.
1. Suppose $i \Vdash C \rhd D$.
Let $j \in W$ be such that $i R j$.
If $j \nVdash C$, then by the induction hypothesis, $\PA \vdash \alpha(\overline{j}) \to \neg \FRS{C}$.
If $j \Vdash C$, then there exists a $k \in W$ such that $j S k$ and $k \Vdash D$.
By the induction hypothesis, $\PA \vdash \alpha(\overline{k}) \to \FRS{D}$.
Then, $\PA \vdash \neg \PR_{\tau_S}(\gn{\neg \alpha(\overline{k})}) \to \Con_{\tau_S + \FRS{D}}$.
By Lemma~\ref{Lem2}, we get $\PA \vdash \alpha(\overline{j}) \to \neg \PR_{\tau_S}(\gn{\neg \alpha(\overline{k})})$, and hence $\PA \vdash \alpha(\overline{j}) \to \Con_{\tau_S + \FRS{D}}$.
We have proved that
\[
\PA \vdash \forall x \bigl(\overline{i} R x \land \alpha(x) \to (\FRS{C} \to \Con_{\tau_S + \FRS{D}}) \bigr).
\]
Then,
\[
\PA \vdash \PR_{\tau_R}(\gn{\exists x(\overline{i} R x \land \alpha(x))}) \to \PR_{\tau_R}(\gn{\FRS{C} \to \Con_{\tau_S + \FRS{D}}}).
\]
Since $\PA \vdash \alpha(\overline{i}) \to \PR_{\tau_R}(\gn{\exists x(\overline{i} R x \land \alpha(x))})$ by Lemma~\ref{Lem1}.2, we obtain
\[
\PA \vdash \alpha(\overline{i}) \to \PR_{\tau_R}(\gn{\FRS{C} \to \Con_{\tau_S + \FRS{D}}}).
\]
This means $\PA \vdash \alpha(\overline{i}) \to \FRS{C \rhd D}$.
2. Suppose $i \nVdash C \rhd D$.
Then, there exists a $j \in W$ such that $i R j$, $j \Vdash C$ and for all $k \in W$ with $j S k$, $k \nVdash D$.
Then, by the induction hypothesis, $\PA$ proves $\alpha(\overline{j}) \to \FRS{C}$ and $\forall x \bigl(\overline{j} S x \land \alpha(x) \to \neg \FRS{D} \bigr)$.
Then, $\PA \vdash \PR_{\tau_S}(\gn{\exists x(\overline{j} S x \land \alpha(x))}) \to \PR_{\tau_S}(\gn{\neg \FRS{D}})$.
Since $\PA \vdash \alpha(\overline{j}) \to \PR_{\tau_S}(\gn{\exists x(\overline{j} S x \land \alpha(x))})$ by Lemma~\ref{Lem1}.2, we have $\PA \vdash \alpha(\overline{j}) \to \PR_{\tau_S}(\gn{\neg \FRS{D}})$.
We have $\PA \vdash \alpha(\overline{j}) \to \FRS{C} \land \PR_{\tau_S}(\gn{\neg \FRS{D}})$.
Then, we obtain
\[
\PA \vdash \neg \PR_{\tau_R}(\gn{\neg \alpha(\overline{j})}) \to \neg \PR_{\tau_R}(\gn{\FRS{C} \to \Con_{\tau_S + \FRS{D}}}).
\]
Since $\PA \vdash \alpha(\overline{i}) \to \neg \PR_{\tau_R}(\gn{\neg \alpha(\overline{j})})$ by Lemma~\ref{Lem2}, we get
\[
\PA \vdash \alpha(\overline{i}) \to \neg \PR_{\tau_R}(\gn{\FRS{C} \to \Con_{\tau_S + \FRS{D}}}).
\]
We conclude $\PA \vdash \alpha(\overline{i}) \to \neg \FRS{C \rhd D}$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{AC}]
Suppose $\IL^-(\PP) \nvdash A$.
Then, there exists an $i \neq 0$ such that $i \nVdash A$.
By Lemma~\ref{Lem4}, $\PA \vdash \alpha(\overline{i}) \to \neg \FRS{A}$.
Then, $\PA \vdash \neg \PR_{\sigma}(\gn{\neg \alpha(\overline{i})}) \to \neg \PR_{\sigma}(\gn{\FRS{A}})$.
Since $0 R^* i$, $\PA \vdash \alpha(0) \to \neg \PR_{\sigma}(\gn{\neg \alpha(\overline{i})})$, and hence $\PA \vdash \alpha(0) \to \neg \PR_{\sigma}(\gn{\FRS{A}})$.
Since $\N \models \alpha(0)$, we have $\N \models \neg \PR_{\sigma}(\gn{\FRS{A}})$.
Therefore, we conclude $T \nvdash \FRS{A}$.
\end{proof}
\section{Concluding remarks}
In the present paper, we have analyzed properties of the logic $\IL^-(\PP)$ from various perspectives.
In particular, we found that $\IL^-(\PP)$ satisfies some logically good properties that $\IL^-$ does not.
For example, $\IL^-$ does not have CIP and $\ell$FPP, whereas $\IL^-(\PP)$ does have these properties (Theorems~\ref{CIP} and \ref{lFPP}).
Also, for the systems of sequent calculi $(\IL^-)^G$ and $(\IL^-(\PP))^G$ that we introduced, the cut-elimination theorem does not hold for the former, but it does for the latter (Theorem \ref{CE} and Corollary \ref{FCE}).
We also proved that $\IL^-(\PP)$ is a base logic for some relational semantics, that is, $\IL^-(\PP)$ is characterized by the class of all simplified $\IL^-(\PP)$-frames (Theorem \ref{SVC}).
Furthermore, we also proved that $\IL^-(\PP)$ has a connection with arithmetic, that is, it is complete with respect to some suitable arithmetical semantics (Theorem \ref{AC}).
Therefore, we can say that the persistence principle $\PP$ is a well behaved principle for $\IL^-$.
On the other hand, $\IL^-(\PP)$ does not have FPP, and UFP does not hold for $\IL^-(\PP)$, so it cannot be said that the logic can fully express the properties of arithmetic (Corollary \ref{FailureofFPP2}).
In this sense, it may be expected in the future to study meaningful extensions of $\IL^-(\PP)$.
One candidate is $\IL^-(\J{4}_+, \PP)$, which is a sublogic of $\ILP$ and enjoys FPP.
Furthermore, we would like to mention as a candidate another logic that seems to make sense arithmetically.
In our arithmetical semantics for $\IL^-(\PP)$, $\Box$ is interpreted by a provability predicate $\PR_{\tau_0}(\cdot)$, which is not necessarily $\Sigma_1$.
If we restrict our argument to interpreting $\Box$ by a $\Sigma_1$ provability predicate, then $\mathcal{L}(\rhd)$-formulas that are not contained in $\IL^-(\PP)$ would become arithmetically valid.
For example, the following Ignatiev's axiom $\mathbf{Sa}$ is such a formula:
\begin{description}
\item [Sa] $A \rhd B \to \bigl(A \land (C \rhd D) \bigr) \rhd \bigl(B \land (C \rhd D) \bigr)$.
\end{description}
We propose to study the logic $\IL^-(\PP, \mathbf{Sa})$ following the work of the present paper.
In Section~\ref{Sec_SV}, we proved that $\IL^-(\PP)$ is faithfully embeddable into bimodal logics $\GLK$ and $\FGL$ (Proposition \ref{Embed}).
Inspired from this result, we introduced arithmetical semantics for $\IL^-(\PP)$ and directly proved that $\IL^-(\PP)$ is arithmetically complete.
From this situation, we conjecture that the bimodal logics $\GLK$ and $\FGL$ are complete for the corresponding arithmetic semantics.
\begin{prob}\leavevmode
\begin{enumerate}
\item Is the logic $\GLK$ complete with respect to the arithmetical semantics based on pairs $(\tau_0, \tau_1)$ of numerals such that $\tau_0$ satisfies the L\"ob condition?
\item Is the logic $\FGL$ complete with respect to the arithmetical semantics based on pairs $(\tau_0, \tau_1)$ of numerals such that both $\tau_0$ and $\tau_1$ satisfy the L\"ob condition?
\end{enumerate}
\end{prob}
From the previous studies on extensions of $\IL^-$, we know that two different logics $\IL^-(\J{2}_+, \J{5})$ and $\IL^-(\J{4}_+, \PP)$ have FPP.
Then, we propose the following problem.
\begin{prob}
Does the intersection of $\IL^-(\J{2}_+, \J{5})$ and $\IL^-(\J{4}_+, \PP)$ enjoy FPP?
\end{prob}
\bibliographystyle{plain}
|
3,212,635,537,930 | arxiv | \section{Introduction}\label{sec:intro}
For more than a decade, the system of the $B\to\pi K$ decays is an outstanding
topic in heavy-flavour physics (for a review, see \cite{RF-rev}).
Thanks to the $B$ factories, we could obtain valuable insights into
these decays, raising the possibility of having a modified electroweak (EW) penguin
sector through the impact of new physics (NP). The following discussion follows
closely the strategy developed in \cite{BFRS}, and explores the
picture after the experimental updates that were reported in the summer of 2006
\cite{BFRS-07}. The corresponding working assumptions for the treatment of
the hadronic sector can be summarised as follows:
\begin{itemize}
\item[i)] {$SU(3)$ flavour symmetry:} however, $SU(3)$-breaking
corrections are included through ratios of decay constants and form factors whenever they arise, and the sensitivity of the numerical results on non-factorizable
$SU(3)$-breaking effects is explored.
\item[ii)] {Neglect of the penguin annihilation and exchange topologies:} these
contributions can be probed and controlled through the $B_d\to K^+K^-$,
$B_s\to\pi^+\pi^-$ system, which can be fully exploited at LHCb.
\end{itemize}
All consistency checks which can be performed with the current data support these
working assumptions and do not indicate any anomalous behaviour. Concerning
the treatment of NP, we assume -- although we are basically performing
a Standard-Model (SM) analysis -- that it manifests itself only in the electroweak (EW)
penguin sector. Such a kind of physics beyond the SM can be accommodated,
e.g., in SUSY, and models with extra $Z'$ bosons and extra dimension
scenarios. The topic of having NP in the EW penguin sector of $B\to\pi K$ decays
has received a lot of attention in the literature (see, e.g., \cite{BpiK-papers}).
In the following discussion \cite{BFRS,BFRS-07}, we use the notation
\begin{eqnarray
\lefteqn{\frac{\Gamma(B^0_d(t)\to f)-
\Gamma(\bar B^0_d(t)\to \bar f)}{\Gamma(B^0_d(t)\to f)+
\Gamma(\bar B^0_d(t)\to \bar f)}}\nonumber\\
&&={\cal A}_{\rm CP}^{\rm dir}\,\cos(\Delta M_d t)+
{\cal A}_{\rm CP}^{\rm mix}\,\sin(\Delta M_d t),\label{ACP-timedep}
\end{eqnarray}
where ${\cal A}_{\rm CP}^{\rm dir}$ and ${\cal A}_{\rm CP}^{\rm mix}$ denote the
``direct" and ``mixing-induced" CP-violating observables, respectively \cite{RF-rev};
a sign convention similar to that of (\ref{ACP-timedep}) will also be used for
self-tagging $B_d$ and charged $B$ decays.
\boldmath
\section{The Starting Point: $B\to\pi\pi$}\label{sec:Bpipi}
\unboldmath
We have seen interesting progress in the exploration of CP violation in
$B^0_d\to\pi^+\pi^-$. In the SM, the decay amplitude of this decay can be
written as follows \cite{RF-Bpipi}:
\begin{equation
A(B^0_d\to\pi^+\pi^-)=-|\tilde T| e^{i\delta_{\tilde T}}
\left[e^{i\gamma}-de^{i\theta}\right],
\end{equation}
where the $\tilde T$ amplitude is governed by the colour-allowed tree topologies,
and the CP-conserving hadronic parameter $de^{i\theta}$ describes, sloppily
speaking, the ratio of penguin to tree contributions. There is now -- for the
first time -- a nice agreement between the BaBar and Belle measurements of the
mixing-induced CP asymmetry:
\begin{equation}
{\cal A}_{\rm CP}^{\rm mix}(B_d\to\pi^+\pi^-)=
\left\{\begin{array}{ll}
0.53\pm0.14\pm0.02 & \mbox{(BaBar)}\\
0.61\pm0.10\pm0.04 & \mbox{(Belle),}
\end{array}\right.
\end{equation}
which yields the average of ${\cal A}_{\rm CP}^{\rm mix}(B_d\to\pi^+\pi^-)=
0.59\pm0.09$ \cite{HFAG}.
On the other hand, the picture of direct CP violation is still {\it not} settled:
\begin{equation}\label{ACP-dir-pipi-ex}
{\cal A}_{\rm CP}^{\rm dir}(B_d\to\pi^+\pi^-)=\left\{
\begin{array}{cc}
-0.16\pm0.11\pm0.03 & \mbox{(BaBar)}\\
-0.55\pm0.08\pm0.05 & \mbox{(Belle).}
\end{array}\right.
\end{equation}
This unsatisfactory situation can be resolved with the help of the $B^0_d\to\pi^-K^+$
mode, which is governed by QCD penguin contributions (this feature holds for all
$B\to\pi K$ decays). Direct CP violation in this channel is now experimentally well
established, with a nice agreement between the BaBar, Belle and CDF results,
yielding an average of ${\cal A}_{\rm CP}^{\rm dir}(B_d\to\pi^\mp K^\pm)=
0.095\pm0.013$ \cite{punzi}. In the SM, the $B^0_d\to\pi^-K^+$ decay amplitude
can be written as follows:
\begin{equation
A(B^0_d\to \pi^- K^+)=P'\left[1-re^{i\delta}e^{i\gamma}\right].
\end{equation}
Using the $SU(3)$ flavour symmetry and the dynamical assumptions specified in
Section~\ref{sec:intro}, we obtain
\begin{equation
re^{i\delta}=\frac{\epsilon}{d}e^{i(\pi-\theta)}
\end{equation}
with $\epsilon\equiv\lambda^2/(1-\lambda^2)=0.05$, implying the relation \cite{RF-Bpipi}:
\begin{eqnarray
\lefteqn{H_{\rm BR}\equiv
\frac{1}{\epsilon}\left(\frac{f_K}{f_\pi}\right)^2\left[\frac{\mbox{BR}
(B_d\to\pi^+\pi^-)}{\mbox{BR}(B_d\to\pi^\mp K^\pm)}\right]}\nonumber\\
&&=-\frac{1}{\epsilon}\left[\frac{{\cal A}_{\rm CP}^{\rm dir}(B_d\to\pi^\mp
K^\pm)}{{\cal A}_{\rm CP}^{\rm dir}(B_d\to\pi^+\pi^-)}\right].
\end{eqnarray}
Since the CP-averaged branching ratios and the direct CP asymmetry
${\cal A}_{\rm CP}^{\rm dir}(B_d\to\pi^\mp K^\pm)$ are well measured, we may use
this relation to {\it predict} the following value:
\begin{equation
{\cal A}_{\rm CP}^{\rm dir}(B_d\to\pi^+\pi^-)=-0.24\pm0.04,
\end{equation}
which favours the BaBar result in (\ref{ACP-dir-pipi-ex}). Since we can express
$H_{\rm BR}$, ${\cal A}_{\rm CP}^{\rm dir}(B_d\to\pi^\mp K^\pm)$ and
${\cal A}_{\rm CP}^{\rm mix}(B_d\to\pi^+\pi^-)$ in terms of $\gamma$ and
$d$, $\theta$, these parameters can be extracted from the data:
\begin{equation}\label{gam-d-theta}
\gamma=\left(70.0^{+3.8}_{-4.3}\right)^\circ,\quad
d=0.46\pm0.02,\quad \theta=(155\pm4)^\circ.
\end{equation}
The value of $\gamma$ is in agreement with the SM fits of the unitarity
triangle, and will be used for the remainder of this analysis.
Applying the isospin symmetry, we may write
\begin{equation}
\begin{array}{rcl}
\sqrt{2}A(B^0_d\to\pi^0\pi^0)&=&|P|e^{i\delta_P}
\left[1+(x/d)e^{i\gamma}e^{i(\Delta-\theta)}\right]\\
\sqrt{2}A(B^+\to\pi^+\pi^0)&=&-|\tilde T|e^{i\delta_{\tilde T}}e^{i\gamma}
\left[1+xe^{i\Delta}\right],
\end{array}
\end{equation}
where the hadronic parameter $xe^{i\Delta}$ denotes the ratio of
``colour-suppressed" to ``colour-allowed tree'' amplitudes. The experimental
values of the ratios of the CP-averaged $B\to\pi\pi$ branching ratios allow
an extraction of this quantity, with the following result:
\begin{equation
x=0.92_{-0.09}^{+0.08},\quad \Delta=-(50_{-14}^{+11})^\circ.
\end{equation}
Complementing these numbers with those in (\ref{gam-d-theta}), the
following {\it predictions} can be made in the SM:
\begin{eqnarray}
{\cal A}_{\rm CP}^{\rm dir}(B_d\to\pi^0\pi^0)
&=&-(0.40^{+0.14}_{-0.21})\label{ACPdir00}\\
{\cal A}_{\rm CP}^{\rm mix}(B_d\to\pi^0\pi^0)&=&
-(0.71^{+0.16}_{-0.17}),
\end{eqnarray}
which offer the exciting perspective of observing {\it large} CP violation in the
$B^0_d\to\pi^0\pi^0$ channel. So far, only data for the direct CP asymmetry
are available from the BaBar and Belle collaborations, with the average of
${\cal A}_{\rm CP}^{\rm dir}(B_d\to\pi^0\pi^0)=-(0.36^{+0.33}_{-0.31})$, which is --
note the signs -- in remarkable agreement with (\ref{ACPdir00}), giving us
further confidence in our analysis.
\boldmath
\section{The Main Target: $B\to\pi K$}\label{sec:BpiK}
\unboldmath
The $B\to\pi K$ decays are dominated by QCD penguin topologies, and can
be divided into two classes, depending on the impact of EW penguins:
\begin{itemize}
\item The EW penguins are colour-suppressed, leading to tiny contributions:
$B^0_d\to\pi^-K^+$, $B^+\to\pi^+K^0$.
\item The EW penguins are colour-allowed, leading to sizeable effects:
$B^0_d\to\pi^0K^0$, $B^+\to\pi^0K^+$.
\end{itemize}
\subsection{Observables with Tiny EW Penguin Effects}
Let us first have a closer look at the $B\to\pi K$ observables with a tiny
impact of the EW penguins. For the determination of $\gamma$
discussed above, we have already used the CP-averaged branching ratio
and the direct CP asymmetry of $B^0_d\to\pi^- K^+$, yielding a value of $\gamma$
in excellent agreement with the SM fits of the unitarity triangle.
Another decay with colour-suppressed EW penguins is at our disposal, with
the following amplitude:
\begin{equation}
A(B^+\to\pi^+K^0)=-P'\left[1+\rho_{\rm c}e^{i\theta_{\rm c}}
e^{i\gamma}\right],
\end{equation}
where the doubly Cabibbo-suppressed parameter $\rho_{\rm c}e^{i\theta_{\rm c}}$ is
usually neglected, implying vanishing direct CP violation. This feature is nicely
supported by the experimental average
${\cal A}_{\rm CP}^{\rm dir}(B^\pm\to\pi^\pm K)=-0.009\pm0.025$ \cite{HFAG}.
Finally, the working assumptions specified in Section~\ref{sec:intro}
allow us to predict the following ratio:
\begin{eqnarray}
\lefteqn{R\equiv \left[\frac{\mbox{BR}(B_d\to\pi^\mp K^\pm)}{\mbox{BR}(B^\pm
\to\pi^\pm K)}\right]\frac{\tau_{B^+}}{\tau_{B^0_d}}}\nonumber\\
&&\stackrel{\rm SM}{=}0.942\pm0.012
\stackrel{\rm exp}{=}0.93\pm0.05.
\end{eqnarray}
Consequently, we obtain an excellent agreement with the SM, and no anomalous
value of $\rho_{\rm c}$ is indicated,\footnote{This picture of $\rho_{\rm c}$ follows
also from $B^\pm\to K^\pm K$ decays \cite{FR}.} thereby ruling out toy models of
final-state interaction effects that were discussed several years ago.
The strategy developed in \cite{BFRS} allows also the prediction of the observables
of the $B_s\to K^+K^-$ decay, where the impact of EW penguins is tiny
(colour-suppressed) as well. In the SM, the corresponding CP asymmetries are
predicted as follows:
\begin{eqnarray}
{\cal A}_{\rm CP}^{\rm dir}(B_s\to K^+K^-)&=&
0.093\pm0.015\\
{\cal A}_{\rm CP}^{\rm mix}(B_s\to K^+K^-)&=&-0.234_{-0.014}^{+0.017}.
\end{eqnarray}
In contrast to the CP asymmetries, an $SU(3)$-breaking form-factor
ratio enters the prediction of the CP-averaged branching ratio. Using
the result of a recent QCD sum-rule calculation \cite{Khod} yields
\begin{equation}\label{BsKK-BR}
\mbox{BR}(B_s\to K^+K^-)=\left\{
\begin{array}{ll}
(27.9_{-5.1}^{+7.1})\times 10^{-6} & \mbox{[$B\to\pi\pi$]}\\
(28.1_{-5.1}^{+7.0})\times 10^{-6} & \mbox{[$B\to\pi K$].}
\end{array}
\right.
\end{equation}
As indicated, there are two options for the prediction of this branching ratio, using
either $B\to\pi\pi$ or $B\to\pi K$ data, which are in remarkable agreement with each
other. The $B_s\to K^+K^-$ channel has recently been observed at CDF,
with the following branching ratio \cite{punzi}:
\begin{equation}\label{BsKK-exp}
\mbox{BR}(B_s\to K^+K^-)=(24.4\pm1.4\pm4.6)\times10^{-6}.
\end{equation}
Within the uncertainties, (\ref{BsKK-BR}) is in nice agreement with (\ref{BsKK-exp}),
which is another support of the assumptions listed in Section~\ref{sec:intro}.
The $B_s\to K^+K^-$, $B_d\to\pi^+\pi^-$ system offers a powerful $U$-spin
strategy for the extraction of $\gamma$ at LHCb \cite{RF-Bpipi,nardulli}; the
predictions and hadronic parameters given above are useful for further
experimental studies to prepare the real data taking at the LHC.
\subsection{Observables with Sizeable EW Penguin Effects}
The following ratios are key quantities for an analysis of the $B\to\pi K$ system:
\begin{eqnarray}
R_{\rm c}&\equiv&2\left[
\frac{\mbox{BR}(B^\pm\to\pi^0K^\pm)}{\mbox{BR}(B^\pm\to\pi^\pm K^0)}\right]
\stackrel{\rm exp}{=}1.11\pm0.07\\
R_{\rm n}&\equiv&
\frac{1}{2}\left[
\frac{\mbox{BR}(B_d\to\pi^\mp K^\pm)}{\mbox{BR}(B_d\to\pi^0K^0)}\right]
\stackrel{\rm exp}{=}0.99\pm0.07.
\end{eqnarray}
The EW penguins, which provide an interesting avenue for NP to
manifest itself \cite{EWP-NP}, enter here in colour-allowed form through
the modes involving neutral pions, and are theoretically described by two
parameters: $q$, which measures the ``strength" of the EW penguin with
respect to the tree contributions, and a CP-violating phase $\phi$. In the SM,
the $SU(3)$ flavour symmetry allows a prediction of $q=0.60$ \cite{NR}, and $\phi$
{\it vanishes.}
\begin{figure}
\includegraphics[width=0.42\textwidth]{RnRc-timeline}
\caption{\label{fig:timeline} The time evolution of the experimental values of
$R_{\rm c,n}$.}
\end{figure}
If we look at Fig.~\ref{fig:timeline} showing the time evolution of the experimental
values of $R_{\rm c}$ and $R_{\rm n}$, we observe that the central values have significantly moved up (partly due to radiative corrections affecting final states with
charged particles \cite{BarIsi}), while the errors were only marginally reduced.
In Fig.~\ref{fig:RnRc}, we show the situation in the plane of the observables
$R_{\rm n}$ and $R_{\rm c}$: the contours correspond to different values of
$q$, and are parametrized through the phase $\phi$. We see that the SM
prediction (on the right-hand side) is very stable in time, having now significantly
reduced errors. On the other hand, the $B$-factory data have moved quite
a bit towards the SM. Converting the experimental values of
$R_{\rm n}$ and $R_{\rm c}$ into $q$ and $\phi$ yields
\begin{equation
q = 0.65_{-0.35}^{+0.39},\quad
\phi = -(52^{+21}_{-50})^\circ.
\end{equation}
A similar trend -- see, in particular, the time evolution of
$(\sin2\beta)_{\phi K_{\rm S}}$ -- is also present in the analysis of
CP violation in $b\to s$ penguin-dominated decays \cite{HFAG}.
\begin{figure}[t]
\includegraphics[width=0.42\textwidth]{RnRc06}
\caption{\label{fig:RnRc} The situation in the $R_{\rm n}$--$R_{\rm c}$ plane.}
\end{figure}
\begin{figure}
\vspace*{0.5truecm}
\includegraphics[width=0.42\textwidth]{AmixAdirpi0Ks}
\caption{\label{fig:ACP}The
${\cal A}_{\rm CP}^{\rm mix}(B_d\to\pi^0K_{\rm S})$--${\cal A}_{\rm CP}^{\rm
dir}(B_d\to\pi^0K_{\rm S})$ plane.}
\end{figure}
Let us now have a closer look at the CP asymmetries of the
$B^0_d\to\pi^0 K_{\rm S}$ and $B^\pm\to\pi^0K^\pm$ channels, which have
received a lot of attention and can also be analysed in the strategy of \cite{BFRS}.
As can be seen in Fig.~\ref{fig:ACP}, SM predictions for the CP-violating observables
of $B^0_d\to\pi^0K_{\rm S}$ are obtained that are much sharper than the current
$B$-factory data. In particular ${\cal A}_{\rm CP}^{\rm mix}(B_d\to\pi^0K_{\rm S})$
offers a very interesting quantitiy. We also see that the experimental central
values can be reached for large {\it positive} values of $\phi$. Concerning
direct CP violation in $B^\pm\to\pi^0K^\pm$, the following SM prediction
arises:
\begin{equation
{\cal A}^{\rm dir}_{\rm CP}(B^\pm\to\pi^0K^\pm)=
-0.001^{+0.049}_{-0.041},
\end{equation}
which is in good agreement with the experimental result $-0.047\pm0.026$
within the errors. For the new input data, this feature turns interestingly out to
be almost independent of NP. Consequently, the non-vanishing experimental
value of
\begin{eqnarray
\Delta A&\equiv& {\cal A}_{\rm CP}^{\rm dir}(B^\pm\to\pi^0K^\pm)-
{\cal A}_{\rm CP}^{\rm dir}(B_d\to\pi^\mp K^\pm)\nonumber\\
&\stackrel{\rm exp}{=}&-0.140\pm0.030,
\end{eqnarray}
which differs from zero at the $4.7\,\sigma$ level, is likely to be generated through hadronic effects, i.e.\ not through the presence of NP.
Performing, finally, a fit to $R_{\rm n}$, $R_{\rm c}$ and the CP asymmetries
of $B^0_d\to\pi^0K_{\rm S}$ yields
\begin{equation
q=1.7_{-1.3}^{+0.5},\quad \phi=+\left(73_{-18}^{+6}\right)^\circ.
\end{equation}
Interestingly, these parameters -- in particular the large {\it positive} phase --
would also allow us to accommodate the experimental values of
$(\sin2\beta)_{\phi K_{\rm S}}$ and the CP asymmetries of other
$b\to s$ penguin modes with central values smaller than
$(\sin2\beta)_{\psi K_{\rm S}}$. The large value of $q$ would be excluded
by constraints from rare decays in simple scenarios where NP enters only
through $Z$ penguins \cite{BFRS}, but could still be accommodated in other
scenarios, e.g.\ in models with leptophobic $Z'$ bosons.
\section{Conclusions}\label{sec:concl}
The strategy developed in \cite{BFRS} continues to provide a powerful tool
for the theoretical interpretation of the $B\to\pi\pi,\pi K$ data \cite{BFRS-07}.
Thanks to the progress at the $B$ factories, now only data could be used
where the BaBar and Belle collaborations are in full agreement with each other.
However, the corresponding SM predictions are very stable, with almost
unchanged central values since the original analysis of 2003, and significantly
reduced errors.
Using the braching ratio and direct CP asymmetry of the
$B^0_d\to\pi^-K^+$ channel, the picture of direct CP violation
in $B^0_d\to\pi^+\pi^-$ could be clarified, with the prediction of
${\cal A}_{\rm CP}^{\rm dir}(B_d\to\pi^+\pi^-)=-0.24\pm0.04$, which
favours the BaBar result, and the extraction of
$\gamma=\left(70.0^{+3.8}_{-4.3}\right)^\circ$, which is in agreement with
the SM fits of the unitarity triangle.
The current status of the $B\to\pi K$ system can be summarized
as follows:
\begin{itemize}
\item All modes with colour-suppressed EW penguins are in excellent
agreement with the SM.
\item The data for the $R_{\rm n,c}$ have moved quite a bit towards the SM predictions,
which are almost unchanged, thereby reducing the ``$B\to\pi K$ puzzle" for the
CP-averaged branching ratios.
\vspace*{0.2truecm}
\item The non-zero experimental value of $\Delta A$ seems
to be caused by hadronic and not by NP effects.
\item On the other hand, the mixing-induced CP violation in $B^0_d\to\pi^0K_{\rm S}$
still looks puzzling, and can straightforwardly be accommodated through a modified
EW penguin sector with a large, positive value of the CP-violating NP phase $\phi$.
\end{itemize}
Unfortunately, we still cannot draw definite conclusions about the presence of
NP in the $B\to\pi K$ system (and other $b\to s$ penguin decays, such as
$B^0_d\to\phi K_{\rm S}$). It will be interesting to keep track of the picture
of these decays once the data improve further.
\begin{acknowledgments}
I would like to thank Giovanni Punzi for valueable discussions about the
most recent CDF results.
\end{acknowledgments}
|
3,212,635,537,931 | arxiv | \section{Introduction}
\defcitealias{gillon14}{G14}
\defcitealias{southworth15}{S15}
\defcitealias{southworth16}{S16}
\defcitealias{wollert2015}{W15}
\defcitealias{ngo16}{N16}
\defcitealias{cartier17}{C17}
Transiting extrasolar planets are key objects for the understanding of worlds beyond our Solar System, as they provide a wealth of information about their systems. Thanks to their special orbital configuration, we can not only measure their radius, true mass (by either radial velocity measurements or via transit timing variations in multi-planet systems), and orbital parameters, but we can also study their atmosphere and thus gain a complete picture of their chemical and physical properties (e.g. \citealt{demingseager}, \citealt{winn}, \citealt{burrowsreview}, \citealt{madhureviewbis}). These atmospheric studies are conducted using mainly the transmission and emission (spectro)photometry techniques. During a transit, some of the starlight passes through the planetary atmosphere and, depending on the atmospheric extent, temperature, and composition, wavelength-dependent variations are seen in the amount of absorbed flux. Thus, from multi-wavelength transit light curves, a transmission spectrum of the upper atmosphere at the day-night terminator region can be obtained (e.g. \citealt{seager2000}, \citealt{charbonneau2002}). At the opposite conjunction, when the planet passes behind the star during a secondary eclipse (occultation), one can measure the flux drop caused by the elimination of the flux component originating from the dayside of the planet (e.g. \citealt{deming2005}, \citealt{charbonneau2005}). Using this technique at different wavelengths allows to probe the emission spectrum of the planet's dayside, from which insights on its atmospheric composition and vertical pressure-temperature ($P-T$) profile can be gained.\\
\indent
A broad diversity of transmission spectra has been found across the population of close-in transiting gas giant exoplanets (see e.g. \citealt{sing2016}, \citealt{fu2017}, \citealt{tsiaras2017}). Some planets have sufficiently clear atmospheres to allow detections of atomic and molecular species, in particular Na (e.g. \citealt{nikolov14h1}), K (e.g. \citealt{sing2011pot}), and $\mathrm{H_{2}O}$ (e.g. \citealt{deming2013}), while others appear to contain high-altitude clouds or hazes that completely mute the spectral features of the atmospheric components (see e.g. \citealt{gibson2013}, \citealt{line2013h12}, \citealt{lendl2016}). Even when detected, atmospheric spectral signatures are often less pronounced than predicted by theoretical models of clear atmospheres with solar abundances (e.g. \citealt{deming2013}, \citealt{madhureview}), suggesting that an extra opacity source is still present, at some level, in otherwise predominantly cloud-free atmospheres. This picture is supported by evidences for high-altitude atmospheric hazes reported for several hot Jupiters (see e.g. \citealt{nikolov2015}, \citealt{sing2015}, \citealt{sing2016}), based on their Rayleigh or Mie scattering signature in the planets' transmission spectra.\\
\indent
On the emission side, most observations gathered so far seem to indicate atmospheric vertical pressure-temperature profiles without significant thermal inversions. Temperature inversions were predicted by early theoretical studies of highly irradiated giant planets, that suggested two classes of hot Jupiters based on their degree of irradiation (e.g. \citealt{hubeny2003}, \citealt{fortney2008}); the hotter class was predicted to host thermal inversions in their atmospheres due to strong absorption of incident UV/visible irradiation at high altitude by high-temperature absorbers, such as gaseous TiO and VO (commonly found in low-mass stars and brown dwarfs), while cooler atmospheres were expected to be devoid of thermal inversions due to the condensation of these absorbing compounds. Thermal inversions have been previously claimed for several hot Jupiters based on \textit{Spitzer} observations (e.g. \citealt{knutson2008}, \citealt{machalek2008}, \citealt{knutson2009}), but these detections have been seriously called into question since then (e.g. \citealt{hansen2014}, \citealt{diamond2014}, \citealt{schwarz2015}). It has been suggested that TiO and VO may not remain suspended in the upper atmospheres of hot Jupiters due to cold-trapping, that would occur either deeper in the dayside atmosphere or on the cooler nightside, and would cause their condensation and downward drag by gravitational settling (e.g. \citealt{spiegel2009}). Inversion-causing compounds may also be photodissociated by high chromospheric emission from the host star, so that the formation of inversions may be affected by stellar activity (e.g. \citealt{knutsonact}). Another important factor is the atmospheric chemistry; for example, the atmospheric carbon-to-oxygen ratio (C/O) can control the abundance of TiO/VO, with a C/O $\geq$ 1 causing substantial depletion of TiO/VO, most available oxygen being taken up by CO molecules in this case thus leaving no oxygen for gaseous TiO/VO (\citealt{madhu2012}). Alternatively, the apparently isothermal emission spectra observed for many hot Jupiters (e.g. \citealt{hansen2014}) may result from the presence of high-altitude cloud decks in their dayside atmospheres that could prevent some thermal inversions from being detected (e.g. \citealt{singW12}). Nevertheless, hottest planets are still the best candidates to look for thermal inversions. Indeed, the planets showing the strongest evidence to date for temperature inversions are WASP-33\,b (\citealt{haynes2015}, \citealt{Nugroho2017}) and WASP-121\,b (\citealt{evans2017}), which are among the most highly irradiated hot Jupiters currently known. In this work, we study the atmospheric properties of another ultra-hot gas giant, \hbox{WASP-103\,b} (\citealt{gillon14}, hereafter \citetalias{gillon14}).\\
\indent
This extreme hot Jupiter, discovered by the WASP Collaboration (\citealt{pollacco2006}, \citealt{andrew2007}, \citealt{hellier2011}), has a mass of $\sim$1.5 $M_{\mathrm{Jup}}$, an inflated radius of $\sim$1.6 $R_{\mathrm{Jup}}$, and is in an ultra-short-period orbit ($\sim$22.2hrs) around a relatively bright (\hbox{$V$ = 12.1}, \hbox{$K$ = 10.8)} F8V star (\citetalias{gillon14}). With an incident stellar flux of \hbox{$\sim$$9.1\times10^{9}$ erg $\mathrm{s}^{-1} \mathrm{cm}^{-2}$} (\hbox{$\sim$$9.1\times10^{6}$ W $\mathrm{m}^{-2}$}), it is one of the most highly irradiated hot Jupiters known to date. Assuming a null Bond albedo, it is heated to an equilibrium temperature close to \hbox{2500 K}. These properties make \hbox{WASP-103\,b} an exquisite target for atmospheric characterization. Another interesting fact about this planet is that its orbital semi-major axis is also only $\sim$1.16 times larger than its Roche limit, meaning that the planet might be close to tidal disruption. \hbox{WASP-103\,b} is thus also a favorable object for studying the atmospheric properties of hot Jupiters in the last stages of their evolution.\\
\indent
\citeauthor{southworth15} ({\color{blue}2015}, hereafter \citetalias{southworth15}) published high-precision follow-up transit photometry of WASP-103\,b in several broad-band optical filters, which they used to refine the physical and orbital parameters of the system. They also detected a slope in the resulting broad-band transmission spectrum, larger values of the effective planetary radius being obtained at bluer wavelengths, which they found to be too steep to be mainly caused by Rayleigh scattering in the planetary atmosphere. Subsequent to their study, a previously unresolved faint star was found via lucky imaging by \citeauthor{wollert2015} ({\color{blue}2015}, hereafter \citetalias{wollert2015}) at an angular separation of only 0.24$\arcsec$ from WASP-103. This object, which was also recently imaged by \citeauthor{ngo16} ({\color{blue}2016}, hereafter \citetalias{ngo16}), is significantly redder than WASP-103 and may be either gravitationally bound or simply aligned along the line of sight. Contamination from this redder star, if not accounted for, is expected to produce a blueward slope in the transmission spectrum of WASP-103\,b, the transit signal being more strongly diluted at longer wavelengths than at shorter ones. This led \citeauthor{southworth16} ({\color{blue}2016}, hereafter \citetalias{southworth16}) to publish a reanalysis of the data presented in \citetalias{southworth15}, accounting for the presence of the contaminating star. They found the inclusion of contaminating light from the faint star in their analysis to have no significant effect on the derived system physical properties. They also reported a corrected broad-band transmission spectrum showing, instead of a steep slope, a minimum effective planetary radius around 760 nm and increasing values towards both bluer and redder wavelengths. This ``V-shape" is not well reproduced by existing theoretical models of transmission spectra. Very recently, \cite{lendl2017} reported an optical transmission spectrum of WASP-103\,b obtained at medium spectral resolution between 550 and \hbox{960 nm} using Gemini/GMOS. While they found signs of strong Na and K absorption, they did not observe any evidence for the V-shape pattern reported by \citetalias{southworth16}.\\
\indent
Recently, \citeauthor{cartier17} ({\color{blue}2017}, hereafter \citetalias{cartier17}) presented near-infrared occultation spectrophotometry of WASP-103\,b obtained using the \textit{Hubble Space Telescope}/Wide Field Camera 3. After correction for flux contamination from the nearby star, they found the dayside emission spectrum of WASP-103\,b to be indistinguishable from isothermal from 1.1 to 1.7 $\mu$m. They noted that several atmospheric models, besides an isothermal one, can result in an apparently isothermal emission spectrum across this wavelength range, for example an atmosphere with a thermal inversion layer just above the layer probed by their observations, an atmosphere with a monotonically decreasing temperature-pressure profile and a C/O>1, or an atmosphere harboring clouds or hazes at high altitude. This highlights the need for additional eclipse observations at other wavelengths to help differentiate between these potential atmospheric scenarios.\\
\indent
To improve the atmospheric characterization of \hbox{WASP-103\,b}, we carried out an intense ground-based photometric monitoring of its occultations, with the aim of probing its dayside emission spectrum in the $z'$ (0.9 $\mu$m) and $K_{\mathrm{S}}$ (2.1 $\mu$m) bands. We complemented the data acquired in the frame of this program with some additional transit photometry and radial velocity (RV) measurements, combined all these new observations with the data previously published in \citetalias{gillon14} and \citetalias{southworth15}, and performed a global analysis of the resulting extensive dataset, taking into account the contamination from the faint star. The Gemini/GMOS transmission and HST/WFC3 emission data were published during the final stages of the preparation of this manuscript, so we did not include them in our global analysis but we discuss these measurements along with our results in the scientific discussion. The paper is organized as follows. The new observations and their reduction are described in Section \ref{obs}, as well as the archival data used in our global analysis. In Section \ref{analysis}, we present our detailed data analysis and results. We discuss these results in Section \ref{discuss}, before concluding in Section \ref{concl}.
\vspace{-0.2cm}
\section{Observations and Data Reduction}
\label{obs}
\subsection{New data}
Between May 2014 and July 2015, we gathered a total of nineteen eclipse light curves of WASP-103\,b. Sixteen of these light curves were acquired during occultations of the planet and three during transits. This follow-up photometry was obtained using three different instruments: the 0.6m TRAPPIST robotic telescope and the EulerCam CCD camera on the 1.2m \textit{Euler}-Swiss telescope, both located at ESO La Silla Observatory (Chile), as well as the WIRCam near-infrared imager on the 3.6m Canada-France-Hawaii Telescope (CFHT) at Mauna Kea Observatory (Hawaii). We complemented this dataset with five new RV measurements obtained between Sept. 2013 and Sept. 2014 with the CORALIE spectrograph mounted on the \textit{Euler} telescope. The follow-up light curves are summarized in the upper part of Table \ref{tablelog}, while the RVs are presented in Table \ref{tablervs}. We describe these new data in the sections below.
\begin{table*}
\centering
\caption{Summary of follow-up photometry obtained for WASP-103. For each light curve, this table shows the night of acquisition (UT), the used instrument, the eclipse type, the filter ($BB$=blue-blocking) and exposure time, the number of data points, the selected baseline function, the standard deviation of the best-fit residuals (unbinned and binned per intervals of 2 min), and the deduced values for $\beta_{w}$, $\beta_{r}$ and $CF=\beta_{w} \times \beta_{r}$ (see Section \ref{analysis_method} for details). For the baseline function, p($\epsilon^{N}$) denotes, respectively, a $N$-order polynomial function of time ($\epsilon=t$), airmass ($\epsilon=a$), PSF full-width at half maximum ($\epsilon=f$) or radius for the WIRCam data ($\epsilon=r$), background ($\epsilon=b$), and $x$ and $y$ positions ($\epsilon=xy$). $o$ denotes an offset fixed at the time of the meridian flip.}
\vspace{-0.1cm}
\begin{tabular}{cccccccccccc}
\hline
\hline
Date&Instrument&Eclipse&Filter&$T_{\mathrm{exp}}$&$N_{\mathrm{p}}$&Baseline&$\sigma$&$\sigma_{120\mathrm{s}}$& $\beta_{w}$&$\beta_{r}$&$CF$\\
(UT) & & type & & (s) & &function&(\%)&(\%)& & &\\
\hline
New data & & & & & & & & & & &\\
\hline
2014 May 08-09&TRAPPIST&Occultation&$z'$&55&338&p($t^{1}$) + $o$&0.25&0.19&1.10&1.08&1.18\\
2014 May 20-20&WIRCam&Occultation&$K_{\mathrm{S}}$&5&1083&p($t^{2}$+$r^{1}$+$b^{1}$)&0.32&0.12&1.56&1.00&1.57\\
2014 June 10-11&EulerCam&Transit&$r'$&80&140&p($t^{1}$+$b^{1}$)&0.18&0.18&1.89&1.00&1.89\\
2014 June 16-17&TRAPPIST&Occultation&$z'$&50&341&p($t^{1}$) + $o$&0.25&0.18&1.08&1.07&1.16\\
2014 June 29-30&EulerCam&Occultation&$z'$&100&130&p($t^{1}$+$f^{1}$)&0.10&0.10&1.20&1.00&1.20\\
2014 July 05-06&EulerCam&Transit&$r'$&80&150&p($t^{1}$)&0.09&0.09&1.16&1.49&1.72\\
2014 July 11-12&EulerCam&Occultation&$z'$&100&65&p($t^{1}$+$b^{1}$)&0.09&0.09&1.18&1.10&1.30\\
2014 July 12-13&TRAPPIST&Occultation&$z'$&48&281&p($t^{1}$) + $o$&0.26&0.18&1.05&1.91&2.01\\
2014 July 25-26&TRAPPIST&Occultation&$z'$&48&308&p($t^{1}$) + $o$&0.23&0.16&1.00&1.10&1.10\\
2015 Apr. 02-03&TRAPPIST&Transit&$BB$&8&910&p($a^{1}$) + $o$&0.35&0.14&0.87&1.08&0.95\\
2015 May 17-18&TRAPPIST&Occultation&$z'$&36&328&p($t^{1}$) + $o$&0.32&0.23&1.28&1.22&1.56\\
2015 May 30-31&TRAPPIST&Occultation&$z'$&36&364&p($a^{1}$) + $o$&0.36&0.26&1.32&1.00&1.32\\
2015 June 11-12&EulerCam&Occultation&$z'$&100&139&p($t^{2}$+$f^{1}$+$b^{1}$)&0.14&0.14&1.56&2.10&3.27\\
2015 June 12-13&TRAPPIST&Occultation&$z'$&40&315&p($a^{1}$) + $o$&0.36&0.26&1.43&1.00&1.43\\
2015 July 07-08&TRAPPIST&Occultation&$z'$&46&365&p($t^{1}$) + $o$&0.30&0.22&1.29&1.51&1.94\\
2015 July 07-08&EulerCam&Occultation&$z'$&100&126&p($a^{2}$+$f^{1}$+$b^{1}$)&0.11&0.11&1.27&1.42&1.82\\
2015 July 19-20&TRAPPIST&Occultation&$z'$&46&327&p($t^{1}$) + $o$&0.35&0.24&1.37&1.17&1.60\\
2015 July 19-20&EulerCam&Occultation&$z'$&100&134&p($t^{1}$+$f^{1}$+$b^{1}$)&0.10&0.10&1.13&1.22&1.39\\
2015 July 20-21&EulerCam&Occultation&$z'$&60&210&p($t^{1}$+$f^{1}$)&0.10&0.08&1.10&1.37&1.51\\
\hline
Archival data & & & & & & & & & & &\\
\hline
2013 June 15-16&TRAPPIST&Transit&$BB$&7&802&p($t^{1}$+$f^{1}$) + $o$&0.32&0.13&0.69&1.48&1.02\\
2013 June 28-29&EulerCam&Transit&$r'$&120&103&p($t^{1}$+$f^{1}$+$b^{2}$+$xy^{1}$)&0.09&0.09&1.43&1.08&1.54\\
2013 July 11-12&EulerCam&Transit&$r'$&80&105&p($t^{1}$+$b^{1}$)&0.12&0.12&1.59&1.23&1.96\\
2013 July 23-24&TRAPPIST&Transit&$BB$&9&941&p($t^{2}$) + $o$&0.48&0.19&1.23&1.22&1.50\\
2013 Aug. 04-05&TRAPPIST&Transit&$BB$&10&935&p($t^{2}$) + $o$&0.29&0.13&0.80&1.51&1.21\\
2014 Apr. 19-20&DFOSC&Transit&$R$&100-105&134&p($t^{1}$)&0.07&0.07&1.05&2.53&2.65\\
2014 May 01-02&DFOSC&Transit&$I$&110-130&113&p($t^{1}$)&0.08&0.08&1.06&1.14&1.21\\
2014 June 08-09&DFOSC&Transit&$R$&100-130&130&p($t^{1}$)&0.10&0.10&1.03&1.41&1.45\\
2014 June 22-23&DFOSC&Transit&$R$&50-120&195&p($t^{1}$)&0.13&0.13&1.05&2.62&2.75\\
2014 June 23-24&DFOSC&Transit&$R$&100&112&p($t^{1}$)&0.06&0.06&1.01&1.44&1.45\\
2014 July 05-06&DFOSC&Transit&$R$&100&118&p($t^{1}$)&0.06&0.06&1.01&1.81&1.83\\
2014 July 05-06&GROND&Transit&$g'$&100-120&122&p($t^{2}$)&0.12&0.12&1.03&1.44&1.49\\
2014 July 05-06&GROND&Transit&$r'$&100-120&125&p($t^{2}$)&0.07&0.07&1.01&2.20&2.22\\
2014 July 05-06&GROND&Transit&$i'$&100-120&119&p($t^{2}$)&0.08&0.08&1.04&1.42&1.48\\
2014 July 05-06&GROND&Transit&$z'$&100-120&121&p($t^{2}$)&0.11&0.11&1.03&2.49&2.58\\
2014 July 17-18&DFOSC&Transit&$R$&90-110&139&p($t^{1}$)&0.07&0.07&1.00&1.21&1.22\\
2014 July 18-19&DFOSC&Transit&$R$&60-110&181&p($t^{2}$)&0.06&0.05&1.02&1.19&1.21\\
2014 July 18-19&GROND&Transit&$g'$&98-108&126&p($t^{2}$)&0.08&0.08&1.02&1.44&1.48\\
2014 July 18-19&GROND&Transit&$r'$&98-108&143&p($t^{2}$)&0.06&0.06&1.04&1.64&1.70\\
2014 July 18-19&GROND&Transit&$i'$&98-108&142&p($t^{2}$)&0.09&0.09&1.05&1.66&1.74\\
2014 July 18-19&GROND&Transit&$z'$&98-108&144&p($t^{2}$)&0.09&0.09&1.01&1.13&1.14\\
2014 Aug. 12-13&CASLEO&Transit&$R$&90-120&129&p($t^{2}$)&0.15&0.15&1.04&1.51&1.58\\
\hline
\hline
\end{tabular}
\label{tablelog}
\vspace{-0.3cm}
\end{table*}
\vspace{-0.2cm}
\subsubsection{TRAPPIST eclipse photometry}
We observed one transit and nine occultations of \hbox{WASP-103\,b} using the 0.6m TRAPPIST robotic telescope and its thermoelectrically-cooled 2K$\times$2K CCD (field of view of 22\arcmin$\times$22\arcmin, plate scale of 0.65\arcsec/pixel). For details of TRAPPIST, see \cite{gillontrap} and \cite{jehintrap}. The transit was observed in a blue-blocking ($BB$) filter that has a transmittance $>$90\% from 500 nm to beyond 1000 nm (effective wavelength = 696.8 nm), with each frame exposed for 8s. The occultations were acquired through a Sloan-$z'$ filter (effective wavelength = 895.5 nm), with exposure times between 36s and 55s. Throughout observations, the telescope was kept in focus and the positions of the stars on the chip were retained on the same few pixels, thanks to a ``software guiding" system that regularly derives an astrometric solution on the images and sends pointing corrections to the mount when needed.\\
\indent
After bias, dark, and flat-field corrections, stellar fluxes were extracted from the images using the \textsc{iraf/daophot}\footnote{\textsc{iraf} is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.} aperture photometry software (\citealt{stetson}). For each observation, a careful selection of both the photometric aperture size and of stable reference stars having a brightness similar to \hbox{WASP-103} was performed to obtain optimal differential photometry. The resulting light curves are shown in Figs. \ref{phot_occz} (raw occultation light curves), \ref{phoc_occz} (detrended occultation light curves, see Section \ref{analysis_method} for details about the modeling), and \ref{transits1} (raw transit light curve).
\vspace{-0.2cm}
\subsubsection{Euler/EulerCam eclipse photometry}
Two transit and six occultation light curves of WASP-103\,b were obtained with EulerCam, the imager of the 1.2m \textit{Euler}-Swiss telescope. EulerCam is a nitrogen-cooled 4K$\times$4K CCD camera with a field of view of 15.68\arcmin$\times$15.73\arcmin$\:$at a plate scale of 0.215\arcsec/pixel. The transits were observed in a Gunn-$r'$ filter (effective wavelength = 664.1 nm) with an exposure time of 80s. The occultations were acquired through a Gunn-$z'$ filter (effective wavelength = 912.3 nm), with exposure times between 60s and 100s. A slight defocus was applied to the telescope to optimize the observation efficiency and to minimize pixel-to-pixel effects. This resulted in stellar PSFs with a typical FWHM between 1.1\arcsec$\:$and 2.5\arcsec. Here too, the positions of the stars on the detector were kept within a box of a few pixels throughout the observations, thanks to the ``Absolute Tracking'' system of EulerCam that matches the point sources in each image with a catalog and adjusts the telescope pointing between exposures when needed. The reduction procedure used to extract the eclipse light curves was similar to that performed on TRAPPIST data. The resulting light curves are shown in Figs. \ref{phot_occz} (raw occultation light curves), \ref{phoc_occz} (detrended occultation light curves, see Section \ref{analysis_method} for details about the modeling), and \ref{transits1} (raw transit light curves). Further details of the EulerCam instrument and data reduction procedures can be found in \cite{lendleuler}.
\vspace{-0.2cm}
\subsubsection{CFHT/WIRCam occultation photometry}
We observed one occultation of WASP-103\,b with the Wide-field InfraRed Camera (WIRCam, \citealt{wircam}) on the 3.6m Canada-France-Hawaii Telescope. WIRCam consists of four 2K$\times$2K HgCdTe HAWAII-2RG arrays, arranged in a 2$\times$2 mosaic. The instrument has a total field of view of 20.5\arcmin$\times$20.5\arcmin$\:$(with gaps of 45\arcsec$\:$between adjacent chips) at a scale of 0.3\arcsec/pixel. We used the $K_{\mathrm{S}}$ broad-band filter, which has a bandwidth of 0.325 $\mu$m centered at \hbox{2.146 $\mu$m}. The observations took place on 2014 May 20 from 06:50 to 12:40 UT, covering the 2.6hrs long predicted occultation (assuming a circular orbit) together with 3.2hrs of out-of-eclipse observations. Conditions were photometric, with a median seeing of 0.6\arcsec, and airmass decreased from 2.4 to 1.09 during the run. The data were gathered in staring mode (\citealt{devost}), with the target and reference stars observed continuously for several hours on the same pixels without any dithering. This mode has been used by other authors for similar observations with WIRCam (see, e.g., \citealt{wang} and \citealt{croll2015}) and has proven to yield an optimal photometric precision. The pointing was carefully selected to ensure that the target and reference stars did not fall near bad pixels, as well as to maximize the number of suitable reference stars located on the same WIRCam chip as WASP-103 (see below).
The scientific sequence consisted of 1092 exposures, each 5s and read out with correlated double sampling\footnote{A CDS image is constructed by subtracting a first read of the array, done immediately after reset, from a second read of the array, performed at the end of the exposure.} (CDS). A defocus of 2mm was applied to the telescope, in order to reduce the impact of imperfect flat-fielding, inter- and intra-pixel variations on the photometry, as well as to keep the counts of the target and reference stars in the regime where detector non-linearity is minimized (linearity to within 1\% below $\sim$10 kADU)\footnote{\burl{www.cfht.hawaii.edu/Instruments/Imaging/WIRCam/WIRCamNonlinearity.html}\label{note}}. This resulted in a ring-shaped PSF with a radius of $\sim$4.5\arcsec ($\sim$15 pixels). A short set of dithered (ten offset positions) in-focus images was taken before and after the scientific sequence in order to construct a sky flat (see below).\\
\indent
The data were reduced independently of the traditional WIRCam \`{}I\`{}iwi pipeline\footnote{\burl{http://www.cfht.hawaii.edu/Instruments/Imaging/WIRCam/IiwiVersion2Doc.html}}, following the prescription of \cite{croll2015} for the reduction of WIRCam staring mode data. We refer the reader to that paper for a detailed description of the reduction procedure and give here an outline of the main steps. The frames from each detector were reduced separately. In each image, the pixels with CDS values above 36 kADU were flagged as saturated. The data were corrected for the small non-linearity\footref{note} present below \hbox{10 kADU}, following the iterative approach of \cite{vacca} for applying a non-linearity correction to CDS images. Each frame was dark subtracted and divided by a sky flat, which was created by taking the median stack of the dithered images acquired before and after the scientific sequence. A bad pixel map was constructed from this sky flat, where pixels flagged as bad were those that deviate by more than 2\% from the median of the array. Finally, in each image, bad and saturated pixels had their value replaced by the median value of the adjacent pixels (provided that they were not themselves bad or saturated).
\begin{figure}
\centering
\vspace{0.1cm}
\includegraphics[scale=0.43] {lcs_Ks_new.pdf}
\vspace{-0.5cm}
\caption{Occultation photometry obtained with CFHT/WIRCam in the $K_{\mathrm{S}}$-band. \textit{Top panel:} the raw, unbinned light curve, together with the best-fit full (photometric baseline $\times$ occultation in the $K_{\mathrm{S}}$-band) model (overplotted in red, see Section \ref{analysis_method} for details about the modeling). \textit{Second panel:} same as the top panel except that the data are binned in 7.2min bins. \textit{Third panel:} binned light curve divided by the best-fit photometric baseline model. The best-fit occultation model in the $K_{\mathrm{S}}$-band is overplotted in red. \textit{Bottom panel:} best-fit residuals. The RMS of the residuals is 524 ppm (7.2min bins). \textit{General note:} the data and models are not corrected for the dilution by the nearby star here.}
\label{lcs_Ks}
\vspace{-0.25cm}
\end{figure}
\indent
Aperture photometry was performed for the target and reference stars using \textsc{iraf/daophot}. Apertures were centered using intensity-weighted centroids. We tested a set of constant apertures, as well as apertures that varied from image to image as a function of the mean radius of the ring-shaped stellar PSFs. The best result was obtained with a variable aperture of 1.1 times the mean radius. Using a variable photometric aperture, rather than a fixed one, allowed us to account for the varying atmospheric conditions, as well as to find for each individual image a balance between choosing a small aperture to minimize the sky noise and a large aperture to encompass all the stellar light. For other examples of exoplanet near-infrared photometry extracted using a variable aperture size, see e.g. \cite{lendl2013} or \cite{zhouw19}. The sky annulus was kept constant for all images, with an inner radius of 30 pixels and an outer radius of 50 pixels. Differential photometry of WASP-103 was finally obtained. We tested all possible combinations of stable reference stars having a brightness similar to the target. We found the best photometry using eight reference stars located on the same WIRCam detector as WASP-103. The resulting light curve is shown in Fig. \ref{lcs_Ks}.
\vspace{-0.2cm}
\subsubsection{Euler/CORALIE radial velocities}
Five new spectroscopic measurements of WASP-103 were gathered with the CORALIE spectrograph mounted on the \textit{Euler} telescope (\citealt{queloz2000}). The spectra, all obtained with an exposure time of 30min, were processed with the CORALIE standard data reduction pipeline (\citealt{baranne96}). RVs were then computed from the spectra by weighted cross-correlation (\citealt{pepe2002}), using a numerical G2-spectral template that provides optimal precisions for late-F to early-K dwarfs. These RVs are presented in Table \ref{tablervs} and shown in Fig. \ref{rvs}. The cross-correlation function (CCF) FWHM and bisector span (BS, \citealt{queloz2001}) values are also given in Table \ref{tablervs}.
\begin{table}
\centering
\caption{New CORALIE RVs for WASP-103. The last two columns give the cross-correlation function FWHM and bisector span values, respectively.}
\label{tablervs}
\vspace{-0.1cm}
\begin{tabular}{ccccc}
\hline
BJD & RV & $\mathrm{\sigma}_{\mathrm{RV}}$ & FWHM & BS\\
- 2 450 000 & (km $\mathrm{s}^{-1}$) & (km $\mathrm{s}^{-1}$) & (km $\mathrm{s}^{-1}$) & (km $\mathrm{s}^{-1}$)\\
\hline
6537.506214 & -42.19467 & 0.12930 & 15.47857 & 0.04812\\
6837.661259 & -41.95497 & 0.05763 & 14.90643 & 0.07458\\
6852.650727 & -41.78295 & 0.06256 & 14.92542 & 0.26543\\
6880.562742 & -41.57979 & 0.07013 & 15.07407 & 0.45440\\
6920.487957 & -41.86693 & 0.09804 & 14.64620 & -0.40181\\
\hline
\end{tabular}
\end{table}
\begin{figure}
\centering
\includegraphics[scale=0.43] {rvs.pdf}
\vspace{-3.8cm}
\caption{\textit{Top:} Euler/CORALIE RV measurements period-folded on the best-fit transit ephemeris from our global MCMC analysis (see Section \ref{analysis_method}), with the best-fit Keplerian model overplotted in red. The data published in \citetalias{gillon14} are plotted in grey, while our new measurements are plotted in black. \textit{Bottom:} corresponding residuals.}
\label{rvs}
\end{figure}
\subsection{Archival data}
\vspace{0.1cm}
We also included in our global analysis the data previously published in \citetalias{gillon14} and \citetalias{southworth15}:
\begin{itemize}
\item three TRAPPIST transit light curves (blue-blocking \hbox{filter)};
\item two Euler/EulerCam transit light curves (Gunn-$r'$ filter);
\item eight transit light curves gathered with the DFOSC (Danish Faint Object Spectrograph and Camera) instrument on the 1.54m Danish telescope located at ESO La Silla Observatory (Bessel $R$ and $I$ filters);
\item eight transit light curves obtained using the GROND (Gamma-Ray Burst Optical/Near-Infrared Detector) instrument on the 2.2m MPG/ESO telescope (Sloan-$g'$, -$r'$, -$i'$, and -$z'$ filters);
\item one transit light curve acquired with the 2.15m telescope located at the CASLEO (Complejo Astronomico El Leoncito) Observatory (Johnson-Cousins $R$ filter);
\item eighteen CORALIE RVs.
\end{itemize}
\noindent
We refer the reader to \citetalias{gillon14} and \citetalias{southworth15} for more details about these data. The archival light curves are summarized in the lower part of Table \ref{tablelog} and shown in Figs. \ref{transits1} (TRAPPIST and Euler/EulerCam), \ref{transits2} (Danish/DFOSC), and \ref{transits3} (2.2m/GROND and CASLEO/2.15m), while the RVs are shown in Fig. \ref{rvs}.
\vspace{0.1cm}
\section{Data Analysis}
\label{analysis}
\vspace{0.2cm}
\subsection{Contamination from the nearby star}
\label{contam}
\vspace{0.2cm}
\citetalias{wollert2015} reported the detection of a stellar object located 0.242$\pm$0.016$\arcsec$ from WASP-103, fainter by $\Delta i'=3.11\pm0.46$ and $\Delta z'=2.59\pm0.35$. \citetalias{ngo16} presented further observations of this nearby star in the near-infrared and found magnitude differences with WASP-103 of $\Delta J=2.427\pm0.030$, $\Delta H=2.2165\pm0.0098$, and $\Delta K_{\mathrm{S}}=1.965\pm0.019$. The astrometric measurements from these two studies are inconclusive as to whether this star is gravitationally bound to the planetary system or not. Due to the very small angular separation of WASP-103 and the nearby object, both stars are contained in all photometric apertures used to extract the eclipse light curves that we included in our global analysis (new and archival data). Although a detailed characterization of the nearby star is beyond the scope of this work, we must estimate the dilution correction factor $(F_{\mathrm{W103}}+F_{\mathrm{cont}})/F_{\mathrm{W103}}=1+F_{\mathrm{cont}}/F_{\mathrm{W103}}$ (where $F_{\mathrm{cont}}/F_{\mathrm{W103}}$ is the contaminant-to-target flux ratio) for each of the observed passbands, in order to account for this contamination in our data analysis.
\begin{table}
\centering
\vspace{0.1cm}
\begin{tabular}{ccc}
\hline
Filter & WASP-103 & Contaminant\\
\hline
$J$ & 11.210 $\pm$ 0.029 & 13.637 $\pm$ 0.053\\
$H$ & 10.993 $\pm$ 0.032 & 13.209 $\pm$ 0.039\\
$K_{\mathrm{S}}$ & 10.932 $\pm$ 0.023 & 12.897 $\pm$ 0.037\\
\hline
\end{tabular}
\caption{Individual apparent magnitudes of WASP-103 and the nearby star in the $J$, $H$, and $K_{\mathrm{S}}$ bands.}
\label{magnitude}
\vspace{0.3cm}
\end{table}
\indent
To this end, we first derived the individual apparent magnitudes of WASP-103 and the nearby star in the $J$, $H$, and $K_{\mathrm{S}}$ bands based on their combined 2MASS magnitudes (\citealt{2mass}) and the magnitude differences reported by \citetalias{ngo16} in these bands. The resulting apparent magnitudes are given in Table \ref{magnitude}. While the $J-H$, $H-K_{\mathrm{S}}$, and $J-K_{\mathrm{S}}$ color indices of WASP-103 agree well with a F8V star, the colors of the nearby star suggest a spectral type comprised between K1 and M4 if it is on the main sequence (\citealt{colortype}). We computed the flux ratios $F_{\mathrm{cont}}/F_{\mathrm{W103}}$ in the passbands of interest assuming each of these two spectral types for the contaminant. We used for this purpose PHOENIX model spectra (\citealt{husser13}) of WASP-103 and the nearby star that we integrated over the passbands of interest. The flux ratio in a given passband can be expressed as:
\begin{equation}
\frac{F_{\mathrm{cont}}}{F_{\mathrm{W103}}} = f^{2} \frac{M_{\mathrm{cont}}}{M_{\mathrm{W103}}}
\end{equation}
where $M_{\mathrm{W103}}$ and $M_{\mathrm{cont}}$ are the band-integrated model fluxes of WASP-103 and the contaminant, respectively, and $f$ is a geometric factor defined as:
\begin{equation}
f=\frac{R_{\mathrm{cont}}}{R_{\mathrm{W103}}} \frac{d_{\mathrm{W103}}}{d_{\mathrm{cont}}}
\end{equation}
with $R_{\mathrm{W103}}$ (resp. $R_{\mathrm{cont}}$) and $d_{\mathrm{W103}}$ (resp. $d_{\mathrm{cont}}$) denoting, respectively, the radius and the distance of WASP-103 (resp. the contaminant). For WASP-103, we used a model spectrum interpolated to the effective temperature $T_{\mathrm{eff}}$, surface gravity log $g_{\star}$, and metallicity [Fe/H] reported in \citetalias{gillon14}. For the nearby star, two different model spectra, of typical K1V ($T_{\mathrm{eff}}$=5100K, log $g_{\star}$=4.5, [Fe/H]=0.0) and M4V ($T_{\mathrm{eff}}$=3200K, log $g_{\star}$=5.0, [Fe/H]=0.0) stars, were used. In each of these two cases, we calculated first the wavelength-independent factor $f$ by comparing the ratio of the model fluxes $M_{\mathrm{cont}}/M_{\mathrm{W103}}$ integrated over the $J$-band with the flux ratio $F_{\mathrm{cont}}/F_{\mathrm{W103}}$ measured in this band by \citetalias{ngo16}. The flux ratios in the passbands of interest were then computed using Equation (1). The final value for the flux ratio in each passband was taken as the average of the two values obtained using the K1V and M4V model spectra for the contaminating star, with an error bar large enough to encompass both values. The resulting flux ratios are given in Table \ref{dilution}. The values derived in the $i'$ and $z'$ bands are consistent with the measurements from \citetalias{wollert2015} in these two bands ($0.0622\pm0.0250$ and $0.0969\pm0.0302$, respectively), but more precise (in the $z'$-band). In the $K_{\mathrm{S}}$-band, we obtained a flux ratio of $0.1496\pm0.0178$, consistent with the value of $0.1637\pm0.0029$ measured by \citetalias{ngo16}, but significantly less precise. For this passband, we thus used the measurement from \citetalias{ngo16}.\\
\indent
In their reanalysis, \citetalias{southworth16} also estimated the contamination from the nearby star in the passbands observed in \citetalias{southworth15}, using the near-infrared magnitude differences reported by \citetalias{ngo16}.
They obtained values in good agreement with ours, albeit with much smaller error bars (see the last column of their Table 1). This is due to the fact that they only used the $\Delta J$ and $\Delta K_{\mathrm{S}}$ magnitude differences, thus discarding the $H$-band measurement, which results in a smaller uncertainty on the spectral type of the contaminating star. As we see no reason to discard any of the three measurements, we chose to adopt the safer procedure outlined above, giving equal weight to the three color indices.
\begin{table}
\centering
\vspace{-0.1cm}
\begin{tabular}{cc}
\hline
Filter & $F_{\mathrm{cont}}/F_{\mathrm{W103}}$\\
\hline
$g'$ & 0.0275 $\pm$ 0.0243 \\
$r'$ & 0.0419 $\pm$ 0.0322 \\
$R$ & 0.0447 $\pm$ 0.0318 \\
$BB$ & 0.0552 $\pm$ 0.0255 \\
$i'$ & 0.0586 $\pm$ 0.0250 \\
$I$ & 0.0633 $\pm$ 0.0236 \\
$z'$ & 0.0800 $\pm$ 0.0145 \\
$K_\mathrm{S}$ & 0.1637 $\pm$ 0.0029 \\
\hline
\end{tabular}
\vspace{-0.1cm}
\caption{Contaminant-to-target flux ratios in the observed passbands.}
\label{dilution}
\end{table}
\vspace{0.2cm}
\subsection{Global data analysis}
\vspace{0.2cm}
\label{analysis_method}
To obtain the strongest constraints on the system parameters, we performed a global Bayesian analysis of the whole dataset (41 eclipse light curves and 23 RVs). We used for this purpose the most recent version of the adaptive Markov Chain Monte-Carlo (MCMC) code described in \citet[and references therein]{gillonmcmc}, that derives the posterior probability distribution functions of the global model parameters, basing on stochastic simulations. Each UT time of mid-exposure was converted to the $\mathrm{BJD}_{\mathrm{TDB}}$ time-scale (\citealt{eastman}). To model the photometry, we used the eclipse model of \cite{mandelagol} multiplied by a different baseline model for each light curve (see below), while the RVs were modeled using a Keplerian orbit (e.g. \citealt{murraycorreia}) combined to a systemic velocity. A quadratic limb-darkening law was assumed for the transits.\\
\indent
The photometric baseline models, different for each light curve, allowed to account for photometric variations not related to the eclipses but rather to external astrophysical, instrumental, or environmental effects. They consisted of different polynomials with respect to, e.g., time, airmass, PSF full-width at half maximum, background, stellar position on the detector, or any combination of these parameters. For each light curve, the optimal baseline function (see Table \ref{tablelog}) was selected by way of minimizing the Bayesian Information Criterion (BIC, \citealt{schwarz}). For the TRAPPIST light curves, a normalization offset was also part of the baseline model to represent the effect of the meridian flip; that is, the $180^{\circ}$ rotation that the German equatorial mount telescope has to undergo when the meridian is reached. This movement results in different positions of the stellar images on the detector before and after the flip, and the normalization offset allows to account for a possible consecutive jump in the differential photometry at the time of the flip.
\indent
The jump parameters in our MCMC analysis (i.e. the parameters that are randomly perturbed at each step of the MCMC) were:
\begin{itemize}
\item the transit depth in the $R$-band \hbox{$\mathrm{d}F_{R}$ = $(R_{\mathrm{p,}R}/R_{\star})^{2}$}, where $R_{\mathrm{p,}R}$ is the planetary radius in the $R$-band and $R_{\mathrm{\star}}$ is the stellar radius;
\item the transit depth differences in the other wavelength bands $\mathrm{dd}F_{j}$ = $\mathrm{d}F_{j} - \mathrm{d}F_{R}$ (where \hbox{$j$ = $g'$,} $r'$, $BB$, $i'$, $I$, $z'$);
\item the occultation depths in the $z'$ and $K_{\mathrm{S}}$-bands, noted $\mathrm{d}F_{\mathrm{occ,}z'}$ and $\mathrm{d}F_{\mathrm{occ,}K_{\mathrm{S}}}$, respectively;
\item the transit impact parameter in the case of a circular orbit \hbox{$b'$ = $a$ $\mathrm{cos}$ $i_{\mathrm{p}}/R_{\star}$}, where $a$ is the orbital semi-major axis and $i_{\mathrm{p}}$ is the orbital inclination;
\item the transit width (from $1^{\mathrm{st}}$ to $4^{\mathrm{th}}$ contact) $W$;
\item the time of mid-transit $T_{0}$;
\item the orbital period $P$;
\item the stellar effective temperature $T_{\mathrm{eff}}$ and metallicity [Fe/H];
\item the parameter \hbox{$K_{2} = K \sqrt{1-e^{2}}\:\:P^{1/3}$}, where $K$ is the RV orbital semi-amplitude and $e$ is the orbital eccentricity;
\item the two parameters \hbox{$\sqrt{e}$ cos $\omega$} and \hbox{$\sqrt{e}$ sin $\omega$}, where $\omega$ is the argument of the periastron;
\item the linear combinations of the quadratic limb-darkening coefficients ($u_{1,j}$, $u_{2,j}$) in each wavelength band, \hbox{$c_{1,j} = 2 u_{1,j} + u_{2,j}$} and \hbox{$c_{2,j}=u_{1,j} - 2 u_{2,j}$} (where \hbox{$j$ = $g'$,} $r'$, $R$, $BB$, $i'$, $I$, $z'$).
\end{itemize}
The baseline model parameters were not jump parameters; they were determined by linear least-squares minimization from the residuals at each step of the MCMC, thanks to their linear nature
in the baseline functions. For this purpose, a Singular Value Decomposition (SVD) method was used (\citealt{Press1992}). This approach allows to increase significantly the efficiency of the MCMC implementation by reducing the number of jump parameters and thus the dimension of the space to probe.\\
\indent
Normal prior probability distribution functions were assumed for $T_{\mathrm{eff}}$, [Fe/H], $u_{1,j}$, and $u_{2,j}$. For $T_{\mathrm{eff}}$ and [Fe/H], the priors were based on the values reported in \citetalias{gillon14}, with expectations and standard deviations corresponding to the quoted measurements and errors, respectively. As for the normal priors imposed on $u_{1,j}$ and $u_{2,j}$, their parameters were interpolated from the theoretical tables of \cite{claret}. All these normal prior distributions are presented in Table \ref{priors}. Uniform prior distributions were assumed for the other jump parameters. At each MCMC step, a value for $F_{\mathrm{cont}}/F_{\mathrm{W103}}$ for each wavelength band was drawn from the normal distribution having as expectation and standard deviation the value and error given in Table \ref{dilution} for this band, respectively. The dilution correction factor for each band was computed as 1+$F_{\mathrm{cont}}/F_{\mathrm{W103}}$.
\begin{table}
\centering
\begin{tabular}{cc}
\hline
Parameter & Prior\\
\hline
$T_{\mathrm{eff}}$ & $\mathcal{N}(6110,160^{2})$ K \\
$[\mathrm{Fe/H}]$ & $\mathcal{N}(0.06, 0.13^{2})$ dex \\
$u_{\mathrm{1,}g'}$ & $\mathcal{N}(0.502, 0.032^{2})$ \\
$u_{\mathrm{2,}g'}$ & $\mathcal{N}(0.253, 0.020^{2})$ \\
$u_{\mathrm{1,}r'}$ & $\mathcal{N}(0.337, 0.024^{2})$ \\
$u_{\mathrm{2,}r'}$ & $\mathcal{N}(0.305, 0.010^{2})$ \\
$u_{\mathrm{1,}R}$ & $\mathcal{N}(0.316, 0.023^{2})$ \\
$u_{\mathrm{2,}R}$ & $\mathcal{N}(0.304, 0.010^{2})$ \\
$u_{\mathrm{1,}BB}$ & $\mathcal{N}(0.316, 0.046^{2})$ \\
$u_{\mathrm{2,}BB}$ & $\mathcal{N}(0.304, 0.020^{2})$ \\
$u_{\mathrm{1,}i'}$ & $\mathcal{N}(0.260, 0.020^{2})$ \\
$u_{\mathrm{2,}i'}$ & $\mathcal{N}(0.298, 0.008^{2})$ \\
$u_{\mathrm{1,}I}$ & $\mathcal{N}(0.242, 0.020^{2})$ \\
$u_{\mathrm{2,}I}$ & $\mathcal{N}(0.296, 0.008^{2})$ \\
$u_{\mathrm{1,}z'}$ & $\mathcal{N}(0.207, 0.018^{2})$ \\
$u_{\mathrm{2,}z'}$ & $\mathcal{N}(0.290, 0.007^{2})$ \\
\hline
\end{tabular}
\caption{Prior probability distribution functions assumed in our MCMC analysis. $\mathcal{N}(\mu,\sigma^{2})$ represents a normal distribution with an expectation $\mu$ and a variance $\sigma^{2}$.}
\label{priors}
\vspace{-0.2cm}
\end{table}
\indent
The physical parameters of the system were deduced from the jump parameters at each step of the MCMC, so that their posterior probability distribution functions could also be constructed. At each MCMC step, a value for the stellar mean density $\rho_{\star}$ was first derived from the Kepler's third law and the jump parameters $\mathrm{d}F_{R}$, $b'$, $W$, $P$, \hbox{$\sqrt{e}$ cos $\omega$} and \hbox{$\sqrt{e}$ sin $\omega$} (see e.g. \citealt{seager} and \citealt{winn}). This $\rho_{\star}$ and values for $T_{\mathrm{eff}}$ and $[$Fe/H$]$ drawn from their normal prior distributions were used to determine a value for the stellar mass $M_{\star}$ through an empirical law \hbox{$M_{\star}$($\rho_{\star}$, $T_{\mathrm{eff}}$, $[$Fe/H$]$)} (\citealt{enoch}, \citealt{gillon3}) that is calibrated using the set of well-constrained detached eclipsing binary (EB) systems presented by \cite{southworth}. Here, we chose to reduce this set to the 116 stars with a mass between 0.7 and 1.7 $M_{\odot}$, the goal of this selection being to benefit from our preliminary knowledge of the mass of WASP-103 (\citetalias{southworth15} give $M_{\star}$ = 1.204 $\pm$ 0.089 $M_{\odot}$) to improve the determination of the system parameters. In order to propagate correctly the error on the empirical law, the parameters of the selected subset of calibration stars were normally perturbed within their observational error bars and the coefficients of the law were redetermined at each MCMC step. We furthermore took into account the inability of the empirical law to perfectly reproduce the distribution of the stellar masses by determining at each step of the MCMC the quadratic difference between the RMS of the residuals of the modeling of the EB masses by the empirical law and the mean mass error for the EB sample. At each MCMC step, a new value for $M_{\star}$ was drawn from a normal distribution having as expectation and standard deviation the mass originally determined by the empirical law and the quadratic difference mentioned above, respectively. The stellar radius $R_{\star}$ was derived from $M_{\star}$ and $\rho_{\star}$, and the other physical parameters of the system were then deduced from the jump parameters and stellar mass and radius.
\begin{figure}
\centering
\includegraphics[scale=0.43] {occ_combz.pdf}
\vspace{-3.3cm}
\caption{\textit{Top:} combined occultation photometry obtained in the $z'$-band with TRAPPIST and Euler/EulerCam. The data are period-folded on the best-fit transit ephemeris from our global MCMC analysis (see Section \ref{analysis_method}), corrected for the photometric baseline, and binned in 5min bins (for visual convenience). The best-fit occultation model in the $z'$-band is overplotted in red. The data and model are not corrected for the dilution by the nearby star here. \textit{Bottom:} corresponding residuals. The RMS of the residuals in the shown interval is 305 ppm (5min bins).}
\label{occ_combz}
\end{figure}
\begin{figure*}
\centering
\includegraphics[scale=0.80] {transits_comb.pdf}
\vspace{-1.0cm}
\caption{\textit{Left:} combined transit light curves in each of the observed passbands. The data are period-folded on the best-fit transit ephemeris from our global MCMC analysis (see Section \ref{analysis_method}), corrected for the photometric baseline, and binned in 5min bins (for visual convenience). For each band, the overplotted, solid line is our best-fit transit model. The data and models are not corrected for the dilution by the nearby star here. The light curves are shifted along the \textit{y}-axis for clarity. \textit{Right:} corresponding residuals. The RMS of the residuals in the interval [-0.09,0.09] days are (from top to bottom) 477, 354, 249, 655, 392, 582, and 496 ppm (5min bins).}
\label{transits_comb}
\vspace{0.3cm}
\end{figure*}
\indent
Although the photometric errors were computed considering scintillation, sky, dark, readout and photon noises, they are known to be often moderately underestimated. A preliminary MCMC analysis, consisting of one chain of \hbox{50 000} steps, was performed to determine the correction factors $CF$ to be applied to the error bars of each photometric time series, as described in \cite{gillonmcmc}. For each light curve, $CF$ is the product of two contributions, $\beta_{w}$ and $\beta_{r}$. On one side, $\beta_{w}$ represents the under- or overestimation of the white noise of each measurement. It is computed as the ratio between the standard deviation of the residuals and the mean photometric error. On the other side, $\beta_{r}$ allows to account for possible correlated noise present in the light curve. It is obtained by comparing the standard deviations of the binned and unbinned residuals for different binning intervals ranging from 5 to 120min, with the largest value being kept as $\beta_{r}$. The values deduced for $\beta_{w}$, $\beta_{r}$, and \hbox{$CF$ = $\beta_{w} \times \beta_{r}$} for each light curve are given in Table \ref{tablelog}. Similarly, this preliminary analysis allowed us to assess the need to rescale the RV error bars, but it was unnecessary here (the best-fit RV model already gives a reduced $\chi^{2}$ = 1.0).\\
\indent
With the corrected photometric error bars, two analyses were then performed: one assuming a circular orbit \hbox{($e$ = 0)} and one with a free eccentricity. Each analysis consisted of three chains of 100 000 steps, whose convergence was checked using the statistical test of \cite{GR}. The first 20\% of each chain was considered as its burn-in phase and discarded. A model comparison based on the Bayes factor, as estimated from the BIC, strongly favored the circular model (Bayes factor of 4915 in its favor) over the eccentric one. We thus adopted the circular orbit as our nominal solution. The corresponding derived system parameters and 1-$\sigma$ error bars are presented in Table \ref{results}. The best-fit eclipse models are shown in Figs. \ref{lcs_Ks} (third panel, occultation model in the $K_{\mathrm{S}}$-band), \ref{occ_combz} (occultation model in the $z'$-band), and \ref{transits_comb} (transit models in each of the observed passbands), while the best-fit RV model is displayed in \hbox{Fig. \ref{rvs}}.
\begin{table*}
\centering
\caption{\textbf{System parameters:} median values and 1-$\sigma$ limits of the posterior probability distribution functions derived from our global MCMC analysis.}
\begin{tabular}{lcl}
\hline
\textbf{Parameters} & \textbf{Values} & \textbf{Units} \\
\hline
\textit{Stellar parameters} & & \\
\hline
Effective temperature $T_{\mathrm{eff}}$ & 6110 $\pm$ 160 & K \\
Metallicity [Fe/H] & 0.06 $\pm$ 0.13 & dex \\
Surface gravity log $g_{\star}$ & $4.219_{-0.014}^{+0.013}$ & cgs \\
Mean density $\rho_{\star}$ & $0.427_{-0.006}^{+0.004}$ & $\rho_{\odot}$ \\
Mass $M_{\star}$ & 1.21 $\pm$ 0.11 & $M_{\odot}$ \\
Radius $R_{\star}$ & 1.416 $\pm$ 0.043 & $R_{\odot}$ \\
\hline
\textit{Planet parameters} & & \\
\hline
Transit depth (in the $R$-band) $\mathrm{d}F_{R}$ = $(R_{\mathrm{p,}R}/R_{\star})^{2}$ & $1.323_{-0.031}^{+0.046}$ & $\%$ \\
Transit impact parameter $b' = a\:\mathrm{cos}\:i_{\mathrm{p}}/R_{\star}$ & $0.06_{-0.05}^{+0.06}$ & $R_{\star}$ \\
Transit width $W$ & 0.1090 $\pm$ 0.0003 & d \\
Time of inferior conjunction $T_{0}$ & 2 456 836.296427 $\pm$ 0.000063 & $\mathrm{BJD_{TDB}}$ \\
Orbital period $P$ & 0.92554517 $\pm$ 0.00000058 & d \\
RV semi-amplitude $K$ & 270 $\pm$ 14 & $\mathrm{m\:s^{-1}}$ \\
Scaled semi-major axis $a/R_{\star}$ & $3.010_{-0.013}^{+0.008}$ & -- \\
Orbital semi-major axis $a$ & $0.01979_{-0.00061}^{+0.00057}$ & AU \\
Orbital inclination $i_{\mathrm{p}}$ & $88.8_{-1.1}^{+0.8}$ & deg \\
Equilibrium temperature$^{a}$ $T_{\mathrm{eq}}$ & 2484 $\pm$ 67 & K \\
Surface gravity log $g_{\mathrm{p}}$ & $3.171_{-0.024}^{+0.027}$ & cgs \\
Mean density $\rho_{\mathrm{p}}$ & $0.353_{-0.024}^{+0.028}$ & $\rho_{\mathrm{Jup}}$ \\
Mass $M_{\mathrm{p}}$ & 1.51 $\pm$ 0.11 & $M_{\mathrm{Jup}}$ \\
Radius (in the $R$-band) $R_{\mathrm{p,}R}$ & $1.623_{-0.053}^{+0.051}$ & $R_{\mathrm{Jup}}$ \\
Roche limit$^{b}$ $a_{\mathrm{R}}$ & $0.01760_{-0.00082}^{+0.00079}$ & AU \\
$a/a_{\mathrm{R}}$ & $1.124_{-0.026}^{+0.029}$ & \\
\hline
\textit{Planet parameters corrected for asphericity (Section \ref{sphere})} & & \\
\hline
Radius (in the $R$-band) $R_{\mathrm{p,}R}$ & 1.681 $\pm$ 0.063 & $R_{\mathrm{Jup}}$ \\
Mean density $\rho_{\mathrm{p}}$ & 0.318 $\pm$ 0.035 & $\rho_{\mathrm{Jup}}$ \\
\hline
\textit{Planet/star radius ratio $R_{\mathrm{p}}/R_{\star}$ (transmission spectrum)} & & \\
\hline
$R_{\mathrm{p,}g'}/R_{\star}$ (0.48 $\mu$m) & $0.1180_{-0.0016}^{+0.0018}$ & -- \\
$R_{\mathrm{p,}r'}/R_{\star}$ (0.62 $\mu$m) & $0.1155_{-0.0018}^{+0.0016}$ & -- \\
$R_{\mathrm{p,}R}/R_{\star}$ (0.66 $\mu$m) & $0.1150_{-0.0014}^{+0.0020}$ & -- \\
$R_{\mathrm{p,}BB}/R_{\star}$ (0.70 $\mu$m)$^{c}$ & $0.1109_{-0.0018}^{+0.0024}$ & -- \\
$R_{\mathrm{p,}i'}/R_{\star}$ (0.76 $\mu$m) & $0.1116_{-0.0019}^{+0.0023}$ & -- \\
$R_{\mathrm{p,}I}/R_{\star}$ (0.82 $\mu$m) & 0.1133 $\pm$ 0.0013 & -- \\
$R_{\mathrm{p,}z'}/R_{\star}$ (0.90 $\mu$m) & $0.1143_{-0.0012}^{+0.0013}$ & -- \\
\hline
\textit{Occultation depths $\mathrm{d}F_{\mathrm{occ}}$ (emission spectrum)} & & \\
\hline
$\mathrm{d}F_{\mathrm{occ,}z'}$ (0.90 $\mu$m) & 699 $\pm$ 110 & ppm \\
$\mathrm{d}F_{\mathrm{occ,}K_{\mathrm{S}}}$ (2.15 $\mu$m) & $3567_{-350}^{+400}$ & ppm \\
\hline
\end{tabular}
\begin{tablenotes}
\item[]
\textbf{Notes.} $ ^{a}$Assuming a null Bond albedo.
\item[]
$ ^{b}$Using \hbox{$a_{\mathrm{R}}$ = 2.46 $R_{\mathrm{p}}(M_{\star}/M_{\mathrm{p}})^{1/3}$} (\citealt{chandr}).
\item[]
$ ^{c}$TRAPPIST blue-blocking filter.
\end{tablenotes}
\label{results}
\end{table*}
\section{Discussion}
\label{discuss}
\vspace{-0.2cm}
\subsection{System physical parameters and correction for asphericity}
\label{sphere}
As expected, we find a slightly larger radius ($1.623_{-0.053}^{+0.051}$ $R_{\mathrm{Jup}}$) and a slightly lower mean density ($0.353_{-0.024}^{+0.028}$ $\rho_{\mathrm{Jup}}$) for the planet than those reported by \citetalias{gillon14} ($1.528_{-0.047}^{+0.073}$ $R_{\mathrm{Jup}}$, $0.415_{-0.053}^{+0.046}$ $\rho_{\mathrm{Jup}}$) and \citetalias{southworth15} ($1.554\pm0.044$ $R_{\mathrm{Jup}}$, $0.367\pm0.027$ $\rho_{\mathrm{Jup}}$), who did not take into account the contamination from the nearby star, unknown at that time. Our values for these two parameters agree well with those found by \citetalias{southworth16} ($1.596_{-0.054}^{+0.044}$ $R_{\mathrm{Jup}}$, $0.339\pm0.023$ $\rho_{\mathrm{Jup}}$). The other physical parameters of the system are in very good agreement with those reported by \citetalias{gillon14}, \citetalias{southworth15}, and \citetalias{southworth16}.\\
\indent
With an orbital semi-major axis only $\sim$1.12 times larger than its Roche limit, WASP-103\,b is expected to be significantly deformed by tides (e.g. \citealt{budaj11}). We calculated values for the planetary radius and mean density corrected for asphericity using the same method as \citetalias{southworth15} (also applied to the case of WASP-121\,b by \citealt{delrez121}). In brief, we used the Roche model of \cite{budaj11} to compute the Roche shape of the planet which would have the same cross-section during transit as the one we inferred from our observations assuming a spherical planet (eclipse model of \citealt{mandelagol}, see Section \ref{analysis_method}). The main inputs of the model were the orbital semi-major axis ($a=4.26_{-0.13}^{+0.12}$ $R_{\odot}$), the star-to-planet mass ratio ($M_{\star}/M_{\mathrm{p}}=840\pm137$), and the planetary radius obtained assuming a spherical shape ($R_{\mathrm{p,}R}=1.623_{-0.053}^{+0.051}$ $R_{\mathrm{Jup}}$). We found a corrected value for the planetary radius of $1.681\pm0.063$ $R_{\mathrm{Jup}}$ (radius of the sphere that would have the same volume as the Roche surface of the planet) and a corresponding revised mean density of $0.318\pm0.035$ $\rho_{\mathrm{Jup}}$. These corrected values are also included in Table \ref{results}.
\subsection{Atmospheric properties of WASP-103\,b}
We use our data to constrain the atmospheric properties of WASP-103\,b. For this purpose, we modeled the atmospheric spectra of WASP-103\,b, both in thermal emission and in transmission, using an exoplanetary atmospheric modeling and retrieval method based on \cite{madhu2009} (also see \citealt{madhu2011}, \citealt{madhu2012}). Here, we briefly summarize the method. The model consists of a 1-D radiative transfer solver which computes the observed spectrum of an exoplanetary atmosphere for a given geometry, in transit or at secondary eclipse, assuming plane parallel geometry. The temperature profile and chemical composition of the atmosphere are free parameters of the model, with 6 parameters for the $P-T$ profile and \hbox{4-6} free parameters for the chemical species; one parameter for each relevant atom/molecule. We include the major opacity sources expected in hot hydrogen-dominated atmospheres, namely H$_2$O, CO, CH$_4$, CO$_2$, C$_2$H$_2$, HCN, TiO, VO, and collision-induced absorption (CIA) due to H$_{2}-$H$_2$, as described in \cite{madhu2012}. The model assumes hydrostatic equilibrium and local thermodynamic equilibrium (LTE) and while computing thermal emission spectra ensures global energy balance with the incident radiation. The parametric temperature structure and molecular abundances in the model allow exploration of a wide range of temperature profiles and chemical compositions in search of the best-fit models to fit the data.
\subsubsection{Emission spectrum}
We clearly detect the thermal emission from the dayside of the planet in both the $z'$ (0.9 $\mu$m) and $K_{\mathrm{S}}$ (2.1 $\mu$m) bands, the measured occultation depths being 699$\pm$110 ppm (6.4-$\sigma$ detection) and \hbox{$3567_{-350}^{+400}$ ppm} (10.2-$\sigma$ detection), respectively. These two measurements suggest a peculiar feature. The occultation depth in the $K_{\mathrm{S}}$-band corresponds to a brightness temperature ($T_{\mathrm{B}}$) of $3171_{-130}^{+144}$ K which is marginally higher ($\sim$1.7-$\sigma$) than the $T_{\mathrm{B}}$ obtained in the \hbox{$z'$-band} of $2914_{-87}^{+80}$ K. Generally, for hot Jupiters, the $T_{\mathrm{B}}$ in the $z'$-band is found to be consistent with or higher than that in the $K_{\mathrm{S}}$-band (see e.g. \citealt{anderson2013} and \citealt{lendl2013} for \hbox{WASP-19\,b}, \citealt{madhu2012} for \hbox{WASP-12\,b}, or \citealt{haynes2015} for WASP-33\,b). This is because the \hbox{$z'$-band} contains strong spectral features due to TiO. On the other hand, the $K_{\mathrm{S}}$-band is relatively devoid of strong spectral features due to TiO and other molecules relevant for hot Jupiters, with the exception of some weak features due to CO. Thus, when TiO is absent all the near-infrared ground-based photometric bands ($z'$, $J$, $H$, and $K_{\mathrm{S}}$) provide windows in atmospheric opacity and probe the temperatures in the deep atmosphere which tends to be isothermal for hot Jupiters. Therefore, the brightness temperatures in all these channels are expected to be similar, as observed for several hot Jupiters (e.g. \citealt{anderson2013}, \citealt{lendl2013}, \citealt{madhu2012}). On the other hand, in hot Jupiters where TiO is present, it can lead to a thermal inversion in the atmosphere (e.g. \citealt{fortney2008}) and cause spectral features in emission. In this case, the TiO features in the $z'$-band lead to a higher $T_{\mathrm{B}}$ in that band compared to that in the $K_{\mathrm{S}}$-band which still has limited opacity (e.g. \citealt{haynes2015}). Therefore, it is rarely the case that the $T_{\mathrm{B}}$ in the $z'$-band is lower than that in the $K_{\mathrm{S}}$-band, assuming a solar composition atmosphere.
\begin{figure}
\centering
\includegraphics[scale=0.55] {spectra_wasp103b_new_new.pdf}
\vspace{-0.5cm}
\caption{Observations and model spectra of dayside thermal emission from WASP-103\,b. The black circles with error bars show our $z'$ and $K_{\mathrm{S}}$-band measurements, while the HST/WFC3 data reported by \citetalias{cartier17} are plotted in blue. A zoom on the HST/WFC3 measurements is shown in the top inset. The colored curves show best-fit model spectra corresponding to three model scenarios: a blackbody model with a temperature of 2900 K (grey), a model with a thermal inversion and a solar composition but a low TiO abundance (green), and a 0.5$\times$solar model without a thermal inversion (red). The bottom inset shows the corresponding pressure-temperature profiles for the models. The green and red circles give the band-integrated fluxes of the corresponding models in the observed photometric bands (bottom black curves), for comparison to the data.}
\label{spectraem}
\end{figure}
\indent
Fig.~\ref{spectraem} shows our occultation measurements along with model emission spectra of WASP-103\,b. The HST/WFC3 measurements recently reported by \citetalias{cartier17} are also shown. We consider three model scenarios in order to explain the data. Firstly, we find that the $z'$-band and WFC3 data are very well explained by a featureless blackbody spectrum, with a temperature of $\sim$2900 K (grey model in Fig.~\ref{spectraem}). A blackbody spectrum is possible if either the atmosphere is isothermal, as shown here, or the metallicity is very low, i.e. providing very low molecular opacity irrespective of the temperature profile. However, the $K_{\mathrm{S}}$-band point is inconsistent with this blackbody model at $\sim$3-$\sigma$. Secondly, a model with a thermal inversion (green model in Fig.~\ref{spectraem}) can fit the current data at nearly the same level as the blackbody. Here, the inversion model has a solar composition atmosphere, but with 0.1$\times$solar TiO, and an inverted $P-T$ profile shown in the inset of Fig.~\ref{spectraem}. The inversion model is able to achieve such a fit because the continuum of this model is at the same temperature as the blackbody model, with only a few strong emission features in the 1.4 $\mu$m H$_2$O band due to a moderately steep $P-T$ profile. Thirdly, a 0.5$\times$solar non-inverted model (red model in Fig.~\ref{spectraem}) provides a slightly worse fit to the data. The non-inverted model has a higher continuum than the blackbody and strong H$_2$O absorption, neither of which is observed in the WFC3 spectrum. In summary, an isothermal temperature profile and/or a low H$_2$O abundance atmosphere provide the best fits to the $z'$-band and WFC3 data. A low H$_2$O abundance is achievable with either a low metallicity or a high C/O ratio (\citealt{madhu2012}, \citealt{moses2013}). Our model inferences are consistent with those of \citetalias{cartier17} who based their results on the WFC3 data alone.\\
\indent
On the other hand, as can be seen in Fig.~\ref{spectraem}, our $K_{\mathrm{S}}$-band data point is matched only at 2-3 $\sigma$ by our current models. To reproduce this high $K_{\mathrm{S}}$-band flux, a strong source of opacity in this wavelength band would be needed. Such a scenario would also require a strong thermal inversion, as well as low H$_2$O and TiO abundances. A thermal inversion would be needed to produce a $K_{\mathrm{S}}$-band emission feature from the above opacity source over the continuum blackbody which is required in the WFC3 band. And, the low H$_2$O and TiO abundances would be required to satisfy the lack of features in the WFC3 and $z'$-bands, respectively. However, as pointed out previously, the $K_{\mathrm{S}}$-band is not expected to contain strong spectral features due to any of the molecules relevant for a hot-Jupiter atmosphere, making the above scenario rather unlikely. Further observations in the $K_{\mathrm{S}}$-band are thus required to confirm our measurement, which is based on a single occultation light curve, before any adequate interpretation of the data can be given.\\
\indent
Additional data are necessary to better characterize the dayside atmosphere of WASP-103\,b. Most notably, observations with \textit{Spitzer} can distinguish between our three model scenarios. The \textit{Spitzer}/IRAC photometric bands at 3.6 and \hbox{4.5 $\mu$m} together should be able to provide strong constraints on the CO absorption versus emission features between the non-inverted and inverted models in the 4-5 micron region. The joint constraints on the CO and H$_2$O abundances could then help constrain the C/O ratio of the dayside atmosphere (\citealt{madhu2012}). As mentioned above, we also encourage further observations in the $K_{\mathrm{S}}$-band to confirm our high emission measurement in that band. On a longer time frame, the \textit{James Webb Space Telescope} (JWST) should be able to provide conclusive constraints on the $P-T$ profile and composition of this planetary atmosphere.
\subsubsection{Transmission spectrum}
The planet-to-star radius ratios $(R_{\mathrm{p}}/R_{\star})$ obtained for each of the observed passbands are given in Table \ref{results} and shown in Fig. \ref{spectratrans}. We find the same ``V-shape'' pattern as \citetalias{southworth16}, with a minimum effective planetary radius around \hbox{700 nm} and increasing values towards both shorter and longer wavelengths. This pattern is however less significant in our transmission spectrum, due to our more conservative error bars (cf. Section \ref{contam}).
\begin{figure}
\vspace{-0.2cm}
\centering
\includegraphics[scale=0.43] {transmission_wasp103_new.pdf}
\vspace{-2.8cm}
\caption{Broad-band measurements of the planet-to-star radius ratio $(R_{\mathrm{p}}/R_{\star})$ as a function of wavelength compared to model transmission spectra of WASP-103\,b. The various measurements are shown as black circles with error bars. The colored curves show three different plausible models: a fiducial model with solar abundance composition in thermochemical equilibrium (TE) without TiO (blue), a solar abundance model in TE with TiO (green), and a solar abundance model with enhanced scattering due to a possible haze with a scattering index of -3.5 (magenta). The colored circles give the band-integrated fluxes of the corresponding models in the observed photometric bands, for comparison to the data.}
\label{spectratrans}
\end{figure}
\indent
We compared our broad-band measurements to different model transmission spectra of \hbox{WASP-103\,b}, with the aim to derive constraints on the atmospheric properties of the planet at the day-night terminator region. The possible sources of opacity in the spectral range covered (0.4$-$1.0 $\mu$m) for a typical hot Jupiter in the temperature regime of WASP-103\,b, with an equilibrium temperature of $\sim$2500 K are: (\textit{a}) H$_2$O at the redder wavelengths, (\textit{b}) TiO and VO over the entire range, (\textit{c}) Na and K doublet line opacity peaking at 0.59 $\mu$m and \hbox{0.78 $\mu$m}, respectively, (\textit{d}) Rayleigh scattering due to H$_2$, and (\textit{e}) potential high-temperature refractory hazes, e.g. of Fe particles. Amongst all these sources of opacity, the most prominent sources are TiO and scattering. Given the broad-band photometric nature of the data, we are unable to resolve any particular spectral features. However, firstly, the prominent sources of opacity, e.g. TiO, are significantly broad to influence the data. Secondly, the full visible coverage of the data allows us to investigate a possible blueward slope in the data which, in turn, could constrain the sources of scattering in the atmosphere, including the presence of hazes.\\
\indent
We explored the model parameter space of the opacity sources discussed above in search of models that can explain the data. Fig. \ref{spectratrans} shows three models with different degrees of fit:
\begin{itemize}
\item a fiducial model with solar abundance composition in thermochemical equilibrium (TE), but with no TiO (blue);
\item a solar abundance model in TE with TiO (green);
\item a solar abundance model with enhanced scattering due to a possible haze with a scattering index of -3.5, instead of -4 for H$_{2}$ Rayleigh scattering (magenta).
\end{itemize}
The corresponding $\chi^{2}$ values are given in Table \ref{chi}. When considering the entire dataset, we find that the model with haze provides a better fit to the data than the two other models. However, we note cautiously that this result relies heavily on the high $R_{\mathrm{p}}/R_{\star}$ measured in the bluest wavelength band in our data, i.e. the $g'$-band. When this data point is not considered, the best fit is obtained by the fiducial solar-composition model without TiO (Table \ref{chi}, last column). It is clear however that none of these models match the data well.\\
\indent
During the refereeing process of this paper, a higher resolution optical transmission spectrum of WASP-103\,b obtained with Gemini/GMOS was published by \cite{lendl2017}. These data, covering the wavelength range between 550 and \hbox{960 nm}, do not show any signs of the V-shape pattern found in our measurements. Instead, they show increased absorption in the cores of the Na and K line features, without any other evident trend, thus pointing to a rather clear atmosphere for \hbox{WASP-103\,b} at the pressure levels probed (between 0.01 and 0.1 bar). We have no satisfactory explanation for this discrepancy at this point. Our transmission spectrum is mostly based on the dataset published by \citetalias{southworth15} and re-analysed by \citetalias{southworth16} (17 out of the 25 transit light curves we used). Our independent data analysis gives similar results to theirs, suggesting that the unusual profile of the measured transmission spectrum is intrinsic to the data and not related to the data analysis process.
\begin{table}
\centering
\begin{tabular}{ccc}
\hline
Model & All & Without the $g'$-band\\
& data & measurement\\
\hline
Solar without TiO & 9.6 & 4.0 \\
Solar with TiO & 10.3 & 6.0 \\
Solar with haze & 7.5 & 5.9 \\
\hline
\end{tabular}
\caption{$\chi^{2}$ values calculated from the data and the various models of the transmission spectrum of WASP-103\,b, considering all data (second column) and omitting the $g'$-band measurement (third column).}
\label{chi}
\end{table}
Extending WASP-103\,b's measured transmission spectrum towards near-infrared wavelengths, for example with HST/WFC3, would allow a more detailed characterization of its atmosphere at the day-night terminator region by constraining its H$_2$O abundance. Additional data at shorter wavelengths than the spectral range covered by the GMOS data (i.e. bluewards of 550 nm) would also be useful to definitely assess the presence of a scattering slope possibly related to hazes in \hbox{WASP-103\,b's} transmission spectrum. At the high temperature ($\gtrsim$2000 K) of WASP-103\,b, most of the usual condensate species are in gas phase (see e.g. \citealt{marley2013}), with the exception of few such as Fe and Al$_2$O$_3$. Characterizing the presence of hazes in such atmospheres is important to be able to make robust determinations of chemical abundances from future spectroscopic data using HST and JWST.
\vspace{-0.2cm}
\section{Conclusions}
\label{concl}
In this work, we presented a total of nineteen new eclipse light curves for the ultra-short-period hot Jupiter \hbox{WASP-103\,b}. Sixteen of these light curves were obtained during occultations and three during transits. We also obtained five new RV measurements. We combined these new observations with previously published data and performed a global MCMC analysis of the resulting extensive dataset (41 eclipse light curves and 23 RVs), taking into account the contamination from a faint nearby star.\\
\indent
Using the approach presented in \cite{lendl2013}, that involves combining a large number (here fifteen) of occultation light curves obtained with $\sim$1m-class telescopes to mitigate the effects of correlated noise and progressively extract the occultation signal from the noise, we detected the dayside emission of the planet in the $z'$-band at better than 6-$\sigma$ \hbox{(699$\pm$110 ppm)}. From a single occultation light curve acquired with the CFHT/WIRCam facility, we also detected the planet's dayside emission in the $K_{\mathrm{S}}$-band at better than 10-$\sigma$, the measured occultation depth being \hbox{$3567_{-350}^{+400}$ ppm}. We compared these two measurements, along with recently published HST/WFC3 spectrophotometric data, to model emission spectra of WASP-103\,b with different temperature profiles and chemical compositions. On one hand, we found that the $z'$-band and WFC3 data are best fit by an isothermal atmosphere at a temperature of $\sim$2900 K, or an atmosphere with a low metallicity or a high C/O ratio. On the other hand, we found an unexpectedly high flux in the $K_{\mathrm{S}}$-band when compared to these atmospheric models, which requires confirmation with additional observations before any interpretation can be given.\\
\indent
From our global data analysis, we also derived a broad-band optical transmission spectrum that shows a minimum around 700 nm and increasing values towards both shorter and longer wavelengths. This is in agreement with the results of the study by \citetalias{southworth16}, which was based on a large fraction of the archival transit light curves included in our analysis. The unusual profile of this transmission spectrum is poorly matched by theoretical spectra and is not confirmed by more recent observations at higher spectral resolution reported by \cite{lendl2017}.\\
\indent
Future observations with existing (HST, \textit{Spitzer}) or planned (JWST) facilities, both in emission and transmission, should be able to provide better constraints on the $P-T$ profile and chemical composition of WASP-103\,b's atmosphere. In particular, we encourage further observations in the $K_{\mathrm{S}}$-band to confirm our high emission measurement in that band. Improving our understanding of this planetary system also requires a more precise characterization of the faint nearby star, by obtaining additional adaptive-optics observations with a large-aperture telescope.
\section*{Acknowledgements}
The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2013-2016) under grant agreement number 312430 (OPTICON). The Canada-France-Hawaii Telescope (CFHT) is operated by the National Research Council of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii. The authors thank the CFHT staff, especially Pascal Fouqu\'e, for scheduling, helping prepare, and conducting the CFHT/WIRCam observations used in this work. TRAPPIST is a project funded by the Belgian Fund for Scientific Research (Fonds National de la Recherche Scientifique, F.R.S.-FNRS) under grant FRFC 2.5.594.09.F, with the participation of the Swiss National Science Fundation (SNF). The Swiss {\it Euler} Telescope is operated by the University of Geneva, and is funded by the Swiss National Science Foundation. L. Delrez acknowledges support from the Gruber Foundation Fellowship. M. Gillon and \hbox{E. Jehin} are F.R.S.-FNRS Research Associates. The authors thank the anonymous reviewers for their valuable suggestions.
\bibliographystyle{mnras}
|
3,212,635,537,932 | arxiv | \section{Introduction}
In this paper we consider only simple graphs. Let $G$ be a graph with vertex set $V(G)$ and
edge set $E(G)$.
Given $v\in V(G)$, let $N_G(v)$ denote the set of vertices adjacent to with $v$ in $G$ and $d_G(v)=|N_G(v)|$. The minimum vertex degree in graph $G$ is denoted by $\delta(G)$.
We write
$N_G[v]=N_G(v)\cup \{v\}$. Given $D\subseteq V(G)$, let $N_D(V)=N_G[v]\cap D$. For $X\subseteq V(G)$, the subgraph of $G$ whose vertex set is $X$ and whose edge set consists of the edges of $G$ joining vertices of $X$ is called the subgraph of $G$ induced by $X$ and is denoted by $G[X]$.
Let $g,f$ be two non-negative integer-valued function such that $g(v)\leq f(v)$ and $g(v)\equiv f(v)\pmod 2$ for all $v\in V(G)$.
A spanning subgraph $F$ of $G$ is called \emph{$(g,f)$-parity factor} if $d_F(v)\equiv f(v)\pmod 2$ and $g(v)\leq d_F(v)\leq f(v)$ for all $v\in V(G)$.
A $(g,f)$-parity factor is called \emph{$f$-factor} if $f(v)=g(v)$ for all $v\in V(G)$. If $f(v)=k$ for all $v\in V(G)$, then an $f$-factor is called a \emph{$k$-factor}. Let $a,b$ be two integers such that $a\leq b$ and $a\equiv b\pmod 2$. If $f(v)=b$ and $g(v)=a$ for all $v\in V(G)$, then a $(g,f)$-parity factor is called an \emph{$(a,b)$-parity factor}.
Lov\'asz \cite{Lov72} gave a characterization of graphs having $(g,f)$-parity factors. Amahashi \cite{Amha85} found a Tutte's type characterization for $(1,k)$-odd factors, which was generalized to $(1,f)$-odd factors by Cui and Kano \cite{CK88}.
\begin{theorem}[Lov\'asz, \cite{Lov72}]\label{lov72}
A graph $G$ has a $(g,f)$-parity factor if and only if for any two disjoint subsets $S,T$ of $V(G)$,
\[
\eta(S,T)=f(S)-g(T)+\sum_{x\in T}d_{G-S}(x)-q(S,T)\geq 0,
\]
where $q(S,T)$ denotes the number of components $C$ of $G-S-T$, called $g$-odd components, such that $g(V(C))+e_G(V(C),T)\equiv 1 \pmod 2$.
\end{theorem}
\begin{theorem}[Amahashi, \cite{Amha85}]\label{Am85}
Let $k\geq 1$ be an odd integer.
A graph $G$ contains an $(1,k)$-parity factor if and only if for any subset $S\subseteq V(G)$,
\[
c_0(G-S)\leq k|S|,
\]
where
$c_0(G-S)$ denotes the number of odd components of $G-S$.
\end{theorem}
Nishimura{\cite{Nishi}} gave a degree conditions for a graph to have a $k$-factor.
\begin{theorem}\label{Ni92}
Let $k$ be an integer such that $k\geq 3$, and let $G$ be a connected
graph of order $n$ with $n\geq 4k-3$, $kn$ even, and minimum degree at
least $k$. If $G$ satisfies
\begin{equation*}
\max\{d_G(u),d_G(v)\}\geq n/2
\end{equation*}
for each pair of nonadjacent vertices $u,v$ in $G$, then $G$ has a $k$-factor.
\end{theorem}
Li and Cai {\cite{Cai}} give a degree conditions for a graph to have an $(a,b)$-factor, which extended Nishimura's result.
\begin{theorem}
Let $G$ be a graph of order $n$, and let $a$ and $b$ be integers such that $1\leq a<b$.
Then $G$ has an [a,b]-factor if $\delta(G)\geq a, n\geq 2a+b+\frac{a^2-a}{b}$ and
\begin{align}\label{main-degree-condition}
\max\{d_G(u),d_G(v)\}\geq \frac{an}{a+b}
\end{align}
for any two nonadjacent vertices $u$ and $v$ in $G$.
\end{theorem}
In this paper we give a sufficient condition for a graph to have an $(a,b)$-parity factor in term of the minimum degree of graph $G$. Our main result generalizes Nishimura's result and improves Li and Cai's result in some sense.
\begin{theorem}\label{Main-Theorem}
Let $a,b,n$ be three integers such that $a\equiv b\pmod 2$, $na$ is even and $n\geq b(a+b)(a+b+2)/(2a)$.
Let $G$ be a graph of order $n$. If $\delta (G)\geq a+\frac{b-a}{a}$ and
\begin{equation}\label{degree_condition}
max\{d_G(u),d_G(v)\}\geq \frac{an}{a+b}
\end{equation}
for any two nonadjacent vertices, then $G$ has an $(a,b)$-parity factor.
\end{theorem}
\section{Proof of Theorem \ref{Main-Theorem}}
Firstly, we show that Theorem \ref{Main-Theorem} holds for $a=1$.
\begin{lemma}\label{a=1}
Let $k,n$ be two positive integers such that $k$ is odd, $n$ is even and $n\geq k+1$.
Let $G$ be a connected graph with order $n$. If $G$ satisfies
\begin{equation}\label{degree_condition2}
max\{d_G(u),d_G(v)\}\geq \frac{n}{1+k}
\end{equation}
for each pair of nonadjacent vertices, then $G$ has a $(1,k)$-odd factor.
\end{lemma}
\pf Suppose that $G$ contains no $(1,k)$-parity factors. By Theorem \ref{Am85}, there exists a subset $S\subset V(G)$ such that
\begin{align*}
c_o(G-S)> k|S|.
\end{align*}
Let $C_1,\ldots, C_q$ be these odd components of $G-S$ such that $|C_1|\leq \cdots\leq |C_q|$. Note that $n$ is even and $G$ is connected. By parity, one can see that
\begin{align}\label{q>=ks+2}
q=c_o(G-S)\geq k|S|+2\ \mbox{and}\ S\neq \emptyset.
\end{align}
Let $u\in V(C_1)$ and $v\in V(C_2)$.
By (\ref{q>=ks+2}), we infer that
\begin{align*}
\max\{d_G(u),d_G(v)\}&\leq \{|C_1|-1+|S|,|C_2|-1+|S|\}\\
&\leq |C_2|-1+|S|\\
&\leq \frac{n-|S|-1}{q-1}-1+|S|\\
&\leq \frac{n}{k|S|+1}+|S|-1-\frac{1}{k}\quad \mbox{(since $|S|\geq 1$ and $n\geq (k+1)|S|+2$)}\\
&< \frac{n}{k+1},
\end{align*}
contradicts with (\ref{degree_condition2}) since $uv\notin E(G)$. \qed
\noindent\textbf{Proof of Theorem \ref{Main-Theorem}.}
By Lemma \ref{a=1}, we may assume that $a\geq 2$. By Theorem \ref{Ni92}, we may assume that $a\leq b-2$.
Suppose that $G$ contains no $(a,b)$-parity factors. By Theorem \ref{lov72}, there exists two disjoint vertex sets $S$ and $T$ such that
\begin{align}\label{eq_lov1}
\eta(S,T)=b|S|-a|T|+\sum_{x\in T}d_{G-S}(x)-q(S,T)\leq -2,
\end{align}
where $q(S,T)$ denotes the number of components $C$ of $G-S-T$, called $a$-odd components, such that $g(V(C))+e_G(V(C),T)\equiv 1 \pmod 2$.
We write $s=|S|$, $t=|T|$ and $w=q(S,T)$. From (\ref{eq_lov1}), one can see that
\begin{align}\label{eq_lov2}
\eta(S,T)=bs-at+\sum_{x\in T}d_{G-S}(x)-w\leq -2.
\end{align}
If $S\cup T=\emptyset$, we have $w\geq 2$ by (\ref{lov72}), which implies that $G$ consists of
at least two components. However, this contradicts the connectedness of $G$.
So we may assume that
\begin{align}\label{ST-nonempty}
S\cup T\neq \emptyset.
\end{align}
If $w\geq 1$, let
$C_1,C_2,\dots C_w$ denote these $a$-odd components of $G-S-T$, and $m_i=|V(C_i)|$ for $1\leq i\leq w$. Put $U=\bigcup_{1\leq i\leq w}V(C_i)$.
We pick $S$ and $T$ such that $U$ is minimal and $V(G)-S-T-U$ is maximal.
\medskip
\textbf{ Claim 1.~} $d_{G-S}(u) \geq a+1$ and $ e_{G}(u,T) \leq b-1$ for every vertex $u\in V(U)$.
\medskip
Firstly, suppose that there exists $u\in U$ such that
\[
d_{G-S}(u) \leq a.
\]
Let $T'=T\cup \{u\}$. One can see that
\begin{align*}
\eta(S,T')&=bs-a|T'|+\sum_{x\in T'}d_{G-S}(x)-q(S,T')\\
&=bs-at-a+\sum_{x\in T}d_{G-S}(x)+d_{G-S}(u)-q(S,T')\\
&\leq bs-at+\sum_{x\in T}d_{G-S}(x)-(q(S,T)-1)\\
&\leq -1,
\end{align*}
which implies by parity
\begin{align*}
\eta(S,T')=bs-a|T'|+\sum_{x\in T'}d_{G-S}(x)-q(S,T')\leq -2,
\end{align*}
contradicting the minimality of $U$.
Secondly, suppose that there exists $u\in U$ such that
\[
e_G(u,T) \geq b.
\]
Let $S'=S\cup \{u\}$. One can see that
\begin{align*}
\eta(S',T)&=b|S'|-at+\sum_{x\in T}d_{G-S'}(x)-q(S',T)\\
&=bs+b-at+\sum_{x\in T}d_{G-S}(x)-e_G(u,T)-q(S',T)\\
&\leq bs-at+\sum_{x\in T}d_{G-S}(x)-(q(S,T)-1)\\
&\leq -1,
\end{align*}
which implies by parity
\begin{align*}
\eta(S',T)=b|S'|-at+\sum_{x\in T}d_{G-S'}(x)-q(S',T)\leq -2,
\end{align*}
contradicting the minimality of $U$ again. This completes Claim 1. \qed
\medskip
\textbf{ Claim 2.~} Let $C_{i_1},\ldots,C_{i_{\tau}}$ be any $\tau$ components of $G[U]$ and let $U'=\bigcup_{j=1}^{\tau}V(C_{i_j})$. $d_{G[T\cup U']}(u)\leq a-1+\tau$ for every vertex $u\in T$.
\medskip
Suppose that there exists $u\in T$ such that $d_{G[T\cup U']}(u)\geq a+\tau$. Let $T'=T-u$.
One may see that
\begin{align*}
\eta(S,T')&=bs-a|T'|+\sum_{x\in T'}d_{G-S}(x)-q(S,T')\\
&= bs-at+a+\sum_{x\in T}d_{G-S}(x)-d_{G-S}(u)-q(S,T')\\
&\leq bs-at+a+\sum_{x\in T}d_{G-S}(x)-(a+\tau)-(q(S,T)-\tau)\\
&=bs-at+\sum_{x\in T}d_{G-S}(x)-q(S,T)\leq -2,
\end{align*}
contradicting to the maximality of $V(G)-S-T-U$. This completes Claim 2. \qed
From the definition of $U$, we have
\begin{equation}\label{maxorder}
|U| \geq m_1+m_2(w-1).
\end{equation}
By Claim 1, one can see that for every $u\in C_j\ (1\leq j\leq w)$,
\begin{equation}\label{eq-m_j-s-r}
d_G(u)\leq (m_{j}-1)+s+r
\end{equation}
where $r=\min\{b,t\}$.
Let $u_1\in V(C_1)$ and $u_2\in V(C_2)$. It follow from (\ref{eq-m_j-s-r}) that
\begin{equation}\label{eqm1m2}
\max\{d_G(u_1),d_G(u_2)\}\leq (m_{2}-1)+s+r
\end{equation}
\medskip
\textbf{Claim 3.} $S\neq\emptyset$.
\medskip
Suppose that $S=\emptyset$.
By (\ref{ST-nonempty}), one may see that $t\geq 1$.
Note that $\delta(G)\geq a+\frac{b-a}{a}$. So we have
$d_G(x_1)\geq a+\frac{b-a}{a}$. From Theorem (\ref{lov72}), one can see that
\begin{align}\label{w-low-bound}
w=q(S,T)\geq \sum\limits_{v\in T}d_{G}(v)-at+2 \geq \frac{b-a}{a}t+2.
\end{align}
If $w>\frac{b}{a}+2$, since $b\geq a+2$, then it follows that
\begin{align}\label{w>=(b+2)/a+2}
w\geq \frac{b+2}{a}+2.
\end{align}
Combining (\ref{eqm1m2}) and (\ref{w>=(b+2)/a+2}), one can see that
\[
\max\{d_G(u_1),d_G(u_2)\}\leq m_2-1+s+r\leq \frac{n-t-1}{w-1}+b<\frac{an}{a+b+2}+b<\frac{an}{a+b},
\]
contradicting to (\ref{main-degree-condition}).
So we may assume that $w\leq \frac{b}{a}+2$. From (\ref{w-low-bound}), we infer that
\[
\frac{b-a}{a}t+2\leq \frac{b}{a}+2,
\]
i.e.,
\begin{align}\label{t-bound-cl3}
t\leq \frac{b}{b-a}.
\end{align}
From (\ref{w-low-bound}), we have
\begin{align*}\label{m1_cl1}
m_1\leq \frac{a(n-t)}{a+b}.
\end{align*}
Consider $H=G[V(C_1)\cup T]$. By Claim 2, for every $y\in T$, one can see that
\[
d_G(y)\leq a-1+w\leq \frac{b}{a}+1+a<\frac{an}{a+b}.
\]
By Claims 1 and 2, $d_{G-S}(u)=d_H(u)\geq a+1$ for every $u\in V(C_1)$ and $d_H(v)\leq a$ for every $v\in T$. Thus there exists two non-adjacent vertices $u\in V(C_1)$ and $v\in T$. If $m_1\leq \frac{an}{a+b}-\frac{b}{b-a}$, then one can see that
\[
\max\{d_G(u),d_G(v)\}\leq m_1-1+t<\frac{an}{a+b},
\]
a contradiction.
Thus we may assume that $ m_1\geq \frac{an}{a+b}-\frac{b}{b-a}$.
We claim that there exists $u\in V(C_1)$ such that $e_G(u,T)=0$, otherwise, by Claim 2 and (\ref{t-bound-cl3}), we have
\[
\frac{ab}{b-a}\geq at\geq \sum_{x\in T}d_{G[V(C_1)\cup T]}(x)\geq m_1\geq \frac{an}{a+b}-\frac{b}{b-a},
\]
i.e.,
\[
n\leq \frac{(a+1)b(a+b)}{a(b-a)},
\]
a contradiction. It follows
\[
\max\{d_G(u),d_G(v)\}\leq m)\leq m_1-1+s=m_1-1<\frac{an}{b+a},
\]
a contradiction. This completes Claim 3. \qed
\medskip
\textbf{Claim 4.} $T\neq\emptyset$.
\medskip
Suppose that $T=\emptyset$. By Theorem \ref{lov72}, then we have
\[
w\geq bs+2.
\]
By Claim 1, we have $|V(C_i)|\geq a+1$ for $1\leq i\leq w$. Thus we infer that
\[
n\geq (a+1)w+s\geq (a+1)(bs+2)+s>(a+1)bs,
\]
which implies that
\begin{equation*}
s\leq \frac{n}{(a+1)b}.
\end{equation*}
Hence we have
\[
m_2-1+s\leq \frac{n-s}{w-1}+s\leq \frac{n}{bs+1}+s<\frac{an}{a+b},
\]
a contradiction. This completes Claim 4. \qed
Put $h_1:=\min\{ d_{G- S}(v)\ |\ v\in T\}$,
and let $x_1\in T$ be a vertex satisfying $d_{G- S}(x_1)=h_1$. We write $p=|N_T[x_1]|$. Further, if $T- N_T[x_1]\neq\emptyset$,
let $ h_2:=\min\{ d_{G- S}(v)\ |\ v\in T-N_T[x_1]\}$ and let $x_2\in T-N_T[x_1]$ such that $d_{G- S}(x_2)=h_2$.
By the definition of $x_i$, we have
\begin{align}\label{dx1x2}
\max\{d_G(x_1),d_G(x_2)\}&\leq \max\{h_1+s,h_2+s\}\leq h_2+s.
\end{align}
Now we discuss four cases.
\medskip
\textbf{Case 1.} $h_1\geq a$.
\medskip
By Theorem \ref{lov72}, one can see that
\begin{align*}
w &\geq bs-at+\sum_{v\in T}d_{G- S}(v)+2\\
& \geq bs+(h_1-a)t+2\\
&\geq bs+2,
\end{align*}
i.e.,
\begin{align}\label{lov-case1-1}
w \geq bs+2.
\end{align}
Note that $n\geq w+s+t$. From (\ref{lov-case1-1}), we infer that
\begin{align*}
s<\frac{n}{b+1}.
\end{align*}
Hence we have
\begin{align*}
m_2-1+s+r&\leq \frac{n-t}{bs+1}+s+b\\
&\leq \frac{n}{bs+1}+s+b-1\\
&<\frac{an}{a+b},
\end{align*}
a contradiction.
So we may assume that $h_1<a$.
\medskip
\textbf{Case 2.} $T=N_T[x_1]$.
\medskip
We write $t=|N_T[x_1]|$. Since $h_1<a$, we have $t\leq a$. By Claim 1, one can see that for every $u\in V(C_1)$, $d_{G-S}(u)\geq a+1>h_1$. Thus we infer that $V(C_1)-N_G(x_1)\neq \emptyset$, i.e., there exists a vertex $v\in V(C_1)$ such that $x_1v\notin E(G)$.
By Theorem \ref{lov72},
\begin{align*}
w&\geq bs+(h_1-a)t+2\\
&\geq bs+(h_1-a)(h_1+1)+2\\
&\geq b(a+\frac{b}{a}-1-h_1)+(h_1-a)(h_1+1)+2\quad \mbox{(since $s+h_1\geq\delta(G)\geq a+\frac{b}{a}-1$)}\\
&=h_1^2-(a+b-1)h_1+ab+\frac{b^2}{a}-a-b+2\quad \mbox{(since $1\leq h_1\leq a$ and $a< b$)}\\
&\geq \frac{b^2}{a}-b+2>0,
\end{align*}
i.e.,
\begin{align}\label{Case21-w}
w &\geq bs+(h_1-a)(h_1+1)+2>2.
\end{align}
One can see that
\begin{align*}
\max\{d_G(v),d_G(x_1)\}&\leq m_1+s+t-1\\
&\leq \frac{n-s-t}{w}+s+t-1\\
&\leq \frac{n-s-t}{bs+(h_1-a)(h_1+1)+2}+s+t-1\\
& \leq \frac{n-s-h_1-1}{bs+(h_1-a)(h_1+1)+2}+s+h_1\\
&=\frac{n-h_1-1+\frac{1}{b}(h_1-a)(h_1+1)}{bs+(h_1-a)(h_1+1)+2}-\frac{1}{b}+s+h_1,
\end{align*}
i.e.,
\begin{align*}
\max\{d_G(v),d_G(x_1)\}&\leq \frac{n-h_1-1+\frac{1}{b}(h_1-a)(h_1+1)}{bs+(h_1-a)(h_1+1)+2}-\frac{1}{b}+s+h_1.
\end{align*}
We write
\begin{align}
f(s)= \frac{n-h_1-1+\frac{1}{b}(h_1-a)(h_1+1)}{bs+(h_1-a)(h_1+1)+2}-\frac{1}{b}+s+h_1.
\end{align}
So we have
\begin{align}\label{f'(s)}
f'(s)=-\frac{b(n-h_1-1)+(h_1-a)(h_1+1)}{(bs+(h_1-a)(h_1+1)+2)^2}+1.
\end{align}
Now we discuss two subcases.
\medskip
\textbf{Case 2.1.~} $s\leq \frac{an}{a+b}-h_1-1$.
\medskip
By (\ref{Case21-w}) and (\ref{f'(s)}), we infer that
\begin{align}\label{cov-f(s)}
f(s)\leq \max \{f(a+\frac{b}{a}-h_1-1), f(\frac{an}{a+b}-h_1-1)\}.
\end{align}
Hence one can see that
\begin{align*}
max\{d_G(x_1),d_G(x_2)\}&\leq m_1-1+s+t\\
&\leq \max\{f(a+\frac{b}{a}-h_1-1), f(\frac{an}{a+b}-h_1-1)\}\\
&< \frac{an}{a+b},
\end{align*}
contradicting to the degree condition.
\medskip
\textbf{Case 2.2.~} $s>\frac{an}{a+b}-h_1-1$.
\medskip
One can see that
\begin{align*}
n&\geq s+t+w\\
&\geq s+t+bs+(h_1-a)t+2\\
&\geq (b+1)\frac{an}{a+b}-(b+1)h_1-(b+1)+(h_1-a)(h_1+1)+2\\
&=(b+1)\frac{an}{a+b}+h_1^2-(a+b)h_1-a-b+1\quad \mbox{(since $0\leq h_1\leq a$)}\\
&\geq \frac{abn}{a+b}+\frac{an}{a+b}-ab-a-b+1>n \quad \mbox{(since $a\geq 2$)},
\end{align*}
a contradiction.
\medskip
\textbf{Case 3.~} $h_2\geq a$.
\medskip
By Lov\'{a}sz Theorem \ref{lov72},
\begin{align*}
w&\geq bs+\sum\limits_{v\in T}d_{G-S}(v)-at+2 \\
&\geq bs+(h_1-a)p+(h_2-p)(t-p)+2 \\
&\geq bs+(h_1-a)p+2
\end{align*}
Now we discuss two subcases.
\medskip
\textbf{Subcase 3.1.~} $h_2\leq \frac{1}{4}(a^2+6a+5)$.
\medskip
By Lov\'asz Theorem \ref{lov72}, we find
$$s\geq \frac{an}{a+b}-h_2\geq\frac{an}{a+b}-\frac{1}{4}(a^2+6a+5).$$
Hence one can see that
\begin{align*}
n&\geq w+s+t\\
&\geq (b+1)s+(h_1-a)p+2+t\\
&\geq (b+1)s+(h_1-a+1)p+2\\
&\geq (b+1)(\frac{an}{a+b}-\frac{1}{4}(a^2+6a+5)+1)+(h_1-a+1)(h_1+1)+2\\
&>n,
\end{align*}
a contradiction.
\medskip
\textbf{Subcase 3.2.~} $h_2\geq \frac{1}{4}(a^2+6a+5)$.
\medskip
By Lov\'{a}sz Theorem \ref{lov72},
\begin{align*}
w&\geq bs+\sum\limits_{v\in T}d_{G-S}(v)-at+2 \\
&\geq bs+(h_1-a)p+(h_2-a)(t-p)+2 \\
&\geq bs+(h_1-a)(h_1+1)+h_2-a+2\\
&\geq bs-\frac{1}{4}(a+1)^2+h_2-a+2\\
&\geq bs+3.
\end{align*}
We find
\begin{align*}
n\geq s+t+w\geq (b+1)s+2,
\end{align*}
which implies that
\begin{align*}
s\leq \frac{n-2}{b+1}.
\end{align*}
Thus by Claim 1, we infer that
\begin{align*}
m_2-1+s+b&\leq \frac{n-s-t}{bs+2}+s+b-1\\
&\leq \frac{n-s}{bs+2}+s+b-1\\
&<\frac{n-2}{bs+1}+s+b\\
&<\frac{an}{a+b},
\end{align*}
a contradiction.
\medskip
\textbf{Case 4.~} $0\leq h_1\leq h_2\leq a-1$.
\medskip
By (\ref{main-degree-condition}), we infer that
\begin{align}\label{case4s-bound}
s\geq \frac{an}{a+b}-h_2.
\end{align}
By Lovasz Theorem \ref{lov72},
\begin{align*}
w &\geq bs+\sum\limits_{v\in T}d_{G-S}(v)-at+2 \\
&\geq bs+(h_1-a)p+(h_2-a)(t-p)+2,
\end{align*}
where $p=|N_T[x_1]|$ and naturally there is $p\leq a$,
\begin{align}\label{case4w-bound}
w &\geq bs+(h_1-a)p+(h_2-a)(t-p)+2.
\end{align}
Thus we get
\begin{align*}
n&\geq s+t+w\\
&\geq (b+1)s+(h_1-a)p+(h_2-a)(t-p)+2+t\\
&= (b+1)s+(h_1-h_2)p+(h_2+1-a)t+2\\
&\geq (b+1)(\frac{an}{a+b}-h_2)+(h_1-h_2)(h_1+1)+(h_2-a+1)(\frac{bn}{a+b}+h_2)+2\\
&\geq (b+1)\frac{an}{a+b}+(-a+1)(\frac{bn}{a+b}+h_2)+2+h_2(\frac{bn}{a+b}+h_2-b-1)+(h_1-h_2)(h_1+1)\\
&\geq n+2,
\end{align*}
a contradiction.
This completes the proof. \qed
\noindent\textbf{Remark :}
These minimum degree conditions are sharp. Let $a,b,m$ be three integers, such that $m$ is sufficiently large. Consider graph $K_{ma,mb+1}$. Denote $K_{ma,mb+1}$ by $G$. Li and Cai \cite{Cai} show that $K_{ma,mb+1}$ contains no $[a,b]$-factors. One can see that
\[
\frac{an}{a+b}>\delta(G)\geq ma> \frac{a|V(G)|}{a+b}-1.
\]
Other hand. Let $C_1,\ldots, C_q$ be $q$ copies of $K_m$, where $q=a+\lceil\frac{b-a}{a}\rceil-1$. Let $G'$ be a graph obtained form $C_1,\ldots,C_q$ by adding a new vertex $v$ connecting one of vertices of each copy. Clearly, $G'$ is connected and $\delta(G')= a+\lceil\frac{b-a}{a}\rceil-1$. By taking $S=\emptyset$ and $T=\{v\}$, we infer that $G'$ contains no $(a,b)$-parity factor by Lovasz's Theorem \ref{lov72}.
|
3,212,635,537,933 | arxiv | \section{Introduction}
The immense amount of available bandwidth at the millimeter-wave (mmWave) band is an outstanding resource for supporting Gigabit-per-second (Gbps) data rates for backhaul and fronthaul applications in wireless networks. More importantly, the mmWave spectrum can alleviate the spectrum shortage at sub-$6$ GHz frequencies~\cite{rqppaport2013Itwillwork}~\cite{rappaport2015wideband}.
\subsection{Motivation}
In this paper, we focus our attention on developing channel models for spectrum in the $73$ GHz band. Our research mission is motivated by the shortage of channel measurement data in this band for airports and other environments. It is important to develop such channel models, since the available spectrum in the $73$ GHz band can support more than ten times the data rates of that of the sub-$6$ GHz spectrum. Moreover, to overcome the significant path loss in the $73$ GHz band, an accurate understanding of the channel models that govern this spectrum are needed to devise more appropriate beamforming and antenna structure~\cite{shad2019}~\cite{ almasi2019lens} for use in this band~\cite{rangan2014millimeter}.
The second focus of this paper is directed at characterizing the $73$ GHz channel models for airport indoor and outdoor settings. This research is motivated by the fact that in the near future, unmanned aerial vehicles (UAVs) will operate in conjunction with piloted airplanes in airport airspaces. More importantly, it is anticipated that automation through the use of machines and robots will affect and improve airport operations in every aspect. To make all of this possible, effective and reliable communication is a necessity. This, in turn will require the use of mmWave bands such as the $73$ GHz spectrum.
In this paper, for the first time, consider channel models at $73$ GHz in indoor and outdoor environments at an airport. Moreover, the resulting measurement data will go a long way in provide the communication research community with a richer data set on channel models in this band. This is important, since different channel measurements obtained by different teams in the mmWave band have resulted in a variety of path loss models for similar propagation environment~\cite{rqppaport2013Itwillwork, samimi20153}. This motivates the need for more channel measurement campaigns in the mmWave band, in this case the $73$ GHz band.
\subsection{Related Work}
Several companies and research groups have been carrying out channel measurements at mmWave frequencies~\cite{rqppaport2013Itwillwork, rappaport2015wideband, samimi20153}. However, as stated above, there is a lack of actual measurement results for the $73$ GHz frequency band. For instance, the authors in \cite{samimi20153} developed a $3$-D statistical model at both the $28$ GHz and $73$ GHz bands for outdoor line-of-sight (LOS) and non-line of sight (NLOS) environments. Subsequently, we computed the channel parameters at $28$ GHz and $73$ GHz for both LOS and NLOS scenarios in Boise~\cite{mahfuza2017} using the MATLAB statistical channel simulator~\cite{nyusim}. Other than the simulation works mentioned above, in ~\cite{mahfuza2018}, a channel measurement campaigns was conducted to obtain path loss models for $60$ GHz frequency band within airport environments.
\subsection{Contributions}
To the best of the authors' knowledge, this paper, for the first time, presents an extensive channel measurement campaign at $73$ GHz in an airport environment. The channel measurement campaigns were conducted both outside on the taxiways and the airport tarmac and inside the concourse and gate areas. To collect a set of comparative measurement data, we conducted a second set of measurement campaigns at Boise State University in both indoor and outdoor scenarios. The channel measurements were made with directional transmit and receive antennas with a $24$ dBi gain at different receive antenna heights. From the measured data, we obtained the parameters of two path loss models, i.e., the close-in reference distance model (CIM) and the floating-intercept model (FIM). Results show that the path loss exponents estimated from the CIM are very close to that of the free-space path loss model, while the FIM provides a better fit to the measurement data.
This paper is organized as follows: Section~\ref{sec:measurement_hardware} describes the channel measurement hardware setup, and measurement environments. Section ~\ref{sec:large_scale_fading} presents the large-scale fading channel models used in this paper. In Section ~\ref{sec:results}, we present and analyze the collected measurement data, while, Section ~\ref{sec:conclusion} concludes the paper.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.9\linewidth,height=5.5cm]{Figures/block_diagram.png}
\end{center}
\caption{Block diagram of the (a) Tx and (b) Rx for the mm-Wave propagation measurements at 73 GHz}
\label{Fig1}
\end{figure}
\section{measurement hardware and procedure}
\label{sec:measurement_hardware}
In this section the measurement hardware and the overall setup for both the indoor and outdoor channel measurement campaigns are described.
\subsection{Measurement Setup}
The channel measurement hardware setup is shown in Fig.~\ref{Fig1}. At the transmitter, the signal is digitally modulated at $4$ GHz inside the arbitrary waveform generator (AWG), with binary phase shift keying (BPSK) modulation with a $1$ GHz symbol rate. Following this, the RF output from the AWG is connected to intermediate frequency (IF) port of the up-converter mixer. This signal is then further up-converted to $73$ GHz. The mixer uses a local oscillator (LO) that operates at $38.5$ GHz. Following upconversion, a bandpass filter (BPF) is used to suppress the undesired out of band signal components. Finally, a power amplifier (PA) with a gain of $20$ dB is placed before the transmitter antenna. A directional horn antenna with a gain of $24$ dBi and a beamwidth of $7^{\circ}$ elevation and $11^{\circ}$ azimuth is used at both the transmitter and receiver. At the receiver, the antenna is connected a bandpass filter and then a low noise amplifier. A down-converter is used to shift the $73$ GHz signal to the baseband frequency of $4$ GHz. The received signal is then fed to the o-scope, where the received signal strength is estimated. All the hardware specifications are listed in Table~\ref{Tab_1}.
\begin{table}[t]
\caption{\bf{ Hardware Specification for the $73$ GHz Channel Measurement Campaign}}
\label{Tab_1}
\centering
\begin{tabular}{|c|c|}
\hline
\multicolumn{2}{|c|}{73 GHz Channel Measurement Campaign} \\ \hline
Carrier Frequency & 73 GHz \\ \hline
Tx antenna gain & 24 dBi \\ \hline
Rx antenna gain & 24 dBi \\ \hline
3dB beamwidth in V-plane & 7 $^{\circ}$ \\ \hline
3dB beamwidth in H-plane & 11 $^{\circ}$ \\ \hline
Modulation scheme & BPSK \\ \hline
Bandwidth & 1.3 GHz \\ \hline
Max Tx Power & 3 dBm \\ \hline
Max meas. path loss & 112 dB \\ \hline
\end{tabular}
\end{table}
\begin{figure}[t]
\begin{center}
\subfigure []
{ \includegraphics[width=0.47\linewidth,height=5cm]{Figures/hallway.png}\label{fig.hallway}}
\subfigure []
{\includegraphics[width=0.47\linewidth,height=5cm]{Figures/airport.jpg}\label{fig.gate}}
\end{center}
\caption{(a) Floor map of the Micron Engineering building. (b) The airport concourse C gate.}
\label{Fig_2}
\end{figure}
\subsection{Indoor Measurements}
During September of 2018, our first measurement campaign at $73$ GHz was conducted in the hallway of the Micron Engineering building (MEC) at Boise State University. The overall hallway layout is shown in Fig~\ref{fig.hallway}. The size of the MEC building hallway is about $ 32\times 2.2 \times 1.9$ $m^3$. The walls are made of sheetrock over metal studs, the ceiling tiles are made of a fiberboard material, and the ground is comprised of concrete. The transmitter and receiver were organized with two movable carts equipped with the instruments. The measurements were taken in thirteen different receiver locations while keeping the position of the transmit antenna fixed. The antennas were manually rotated to find the strongest received power for each unique Tx-Rx location.
The second set of measurements was completed in the indoor gate area of the Boise Airport. The gated area was specifically selected since it is a crowded environment with metallic objects and chairs. The overall layout of the airport gated area has been shown in Fig.~\ref{fig.gate}. The data were collected at three different receiver antenna heights. The transmitter antenna height was fixed at 1.6 meters relative to the ground.
\begin{figure}[t]
\begin{center}
\subfigure [ ]
{ \includegraphics[width=0.47\linewidth,height=5cm]{Figures/picture_tarmec}\label{fig.airport_picture}}
\subfigure []
{\includegraphics[width=0.47\linewidth,height=5cm]{Figures/tarmac}\label{fig.airport_map}}
\end{center}
\caption{(a) Photo of the outdoor setting in the airport. (b) Overhead image of the outdoor setting at the Boise Airport showing the transmitter and receiver at various locations.}
\label{Fig_3}
\end{figure}
\subsection{Outdoor Measurements}
In 2018 of September, two outdoor propagation measurement campaigns were conducted at $73$ GHz at the Boise Airport and at the Boise State University. The data capturing methodology is similar to that of the prior subsection. Fig.~\ref{fig.airport_picture} shows the outdoor setting at the airport just outside of the gate areas. The height of the transmitter antenna was fixed at $1.6$ m from the ground level, and ten receiver locations with three antenna heights, i.e., $1.6$ m, $1.4$ m and $1.3$ m, were selected. The airplanes shown in Fig.~\ref{fig.airport_map} was not present during the data collection.
In addition, another outdoor campaign was organized at Boise State University for LOS scenarios. The transmitter location was fixed at a height of $1.21 m$ from the ground level and various receiver locations (height of $1.18m$) were selected. The overall layout for this scenario is depicted in Fig.~\ref{Fig_4}. At each transceiver separation , the transmit antenna was manually tilted down toward the receiver antenna, and the receiver antenna was adjusted in such a way as to receive the highest signal to noise ratio. All antennas were placed in vertical-to-vertical (V-V) polarization in both indoor and outdoor scenarios.
\begin{figure}[t]
\begin{center}
\subfigure [ ]
{ \includegraphics[width=0.47\linewidth,height=5cm]{Figures/picture_outside.jpg}\label{fig.campus_picture}}
\subfigure []
{\includegraphics[width=0.47\linewidth,height=5cm]{Figures/out.jpg}\label{fig.campus_map}}
\end{center}
\caption{(a) Photo of the outdoor environment at the Boise State campus. (b) Overhead image of the outdoor environment at Boise State.}
\label{Fig_4}
\end{figure}
\begin{table*}[ht!]
\renewcommand{\arraystretch}{1.2}
\caption{\bfseries Comparison of key path loss parameters in CIM (1-meter reference distance) and that of FIM for the measurements at Boise Airport and at Boise State University}
\label{tab2}
\centering
\begin{tabular}{|c|l|l|l|l|l|l|l|l|}
\hline
\multicolumn{9}{|c|}{Directional Path Loss Models}
\\ \hline
\multirow{2}{*}{Environments} & \multirow{2}{*}{Scenario} & \multirow{2}{*}{$h_{Tx}$, m} & \multirow{2}{*}{$h_{Rx}$, m }& \multicolumn{2}{c|}{CIM} & \multicolumn{3}{c|}{FIM} \\ \cline{5-9}
& & & & n & $\sigma$, dB& $\alpha$, dB & $ \beta$ & $\sigma$, dB\\ \hline
Indoor (airport gate) & LOS & 1.21 & 1.18 & 2.1 & 2.6 & 75 & 1.8 & 2.4 \\ \hline
Indoor (MEC building) & LOS & 1.21 & 1.18 & 1.8 & 2.05 & 73 & 1.42 & 1.2 \\ \hline
outdoor (MEC building) & LOS & 1.2 & 1.8 & 1.8 & 2.2 & 73 & 1.5 & 1.8 \\ \hline
\end{tabular}
\end{table*}
\begin{table*}[ht!]
\renewcommand{\arraystretch}{1.5}
\caption{\bfseries The FIM parameters for Boise Airport in both indoor and outdoor scenarios at $73$ GHz for different receiver antenna heights. Transceiver separation ranges from $1$ m to $30$ m}
\label{tab3}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{Environments} & \multirow{2}{*}{$h_{Tx}$, m} & \multirow{2}{*}{$h_{Rx}$, m} & \multirow{2}{*}{Path Loss Scenarios} & \multicolumn{3}{c|}{Parameters for directional floating-intercept model} \\ \cline{5-7}
& & & & $\alpha$, dB & $\beta$ & $\sigma$, dB \\ \hline
\multirow{3}{*}{Indoor Airport} & \multirow{3}{*}{1.6} & 1.6 & \multirow{3}{*}{LOS} & 72 & 1.95 & 1.6 \\ \cline{3-3} \cline{5-7}
& & 1.47 & & 77 & 1.6 & 2.4 \\ \cline{3-3} \cline{5-7}
& & 1.3 & & 92 & 0.4 & 6.5 \\ \hline
\multirow{3}{*}{Outdoor Airport} & \multirow{3}{*}{1.6} & 1.6 & \multirow{3}{*}{LOS} & 73 & 1.7 & 1.24 \\ \cline{3-3} \cline{5-7}
& & 1.47 & & 76 & 1.6 & 2.8 \\ \cline{3-3} \cline{5-7}
& & 1.3 & & 89 & 0.62 & 6.11 \\ \hline
\end{tabular}
\end{table*}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.9\linewidth,keepaspectratio]{Figures/hall_los_co.pdf}
\end{center}
\caption{FIM and CIM along with the measurement data taken from the MEC building in LOS indoor scenario ($h_{tx} = 1.21$ m and $h_{rx}= 1.18$ m ). The blue square and solid black line represent the measurement LOS data and free-space path loss at $73$ GHz respectively. The dashed red line and dotted blue line show the FIM and CIM path loss models respectively}
\label{Fig5}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.9\linewidth,keepaspectratio]{Figures/gate_los_sameh.pdf}
\end{center}
\caption{FIM and CIM along with the measurement data taken from the airport gate in LOS indoor scenario ($h_{tx} = 1.21$ m and $h_{rx} = 1.18$ m ). The blue square and solid black line represent the measurement LOS data and free-space path loss at 73 GHz respectively. The dashed red line and dotted blue show the FIM and CIM path loss models respectively }
\label{Fig6}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.9\linewidth,keepaspectratio]{Figures/threeheights_indoor_air_update.pdf}
\end{center}
\caption{The FIM model at 73 GHz with different Rx antenna heights from the ground level when the Tx antenna height is kept at $1.6$ m. The blue star, purple square, and red circles represent the measurement LOS data for $ h_{Rx} = 1.6$, $1.47$ and $1.3$ m respectively and the dashed green, dotted blue, and dash dot orange lines show the FIM model corresponding those measurement data collected from the airport gate in LOS indoor scenario. The solid line represents the free-space path loss model at 73 GHz }
\label{Fig7}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.9\linewidth,keepaspectratio]{Figures/threeheights_out_air_update.pdf}
\end{center}
\caption{The FIM model at 73 GHz with different Rx antenna heights from the ground level when the Tx antenna height is kept at $1.6$ m from the ground level. The blue star, purple square, and red circles represent the measurement LOS data for $ h_{Rx} = 1.6$, $1.47$ and $1.3$ m respectively and the dashed green, dotted blue, and dash dot orange lines show the FIM model corresponding those measurement data collected from the airport surface area in LOS outdoor scenario. The solid line represents the free-space path loss model at 73 GHz}
\label{Fig8}
\end{figure}
\section{Large scale fading models}
\label{sec:large_scale_fading}
The directional path loss channel models are obtained for the indoor and outdoor scenarios using CIM and FIM methods.
\subsection{Close-in free space reference distance}
The Close-in free space reference distance path loss is given by~\ref{eq1} \cite{rappaport2014millimeter,maccartney2014omnidirectional,rappaport2015wideband}
\begin{align}
\label{eq1}
PL(d)[\text{dB}] =& PL(d_{0}) + 10 n \cdot \log_{10} ( \frac{d}{d_{0}} )
\nonumber
\\
&+ \chi_{\sigma}, \textrm{for} \quad d \geq d_{0}
\end{align}
where, $d_{0} $ is the close-in free space reference distance, $\chi_{\sigma}$ is a normal random variable with mean $0$ dB and standard deviation $\sigma$ \cite{rappaport2014millimeter}, $n$ is the PLE, and $ PL(d_{0})$ is the close-in free space path loss in dB that is given by.
\begin{equation}
\label{eq2}
PL(d_{0}) = 20\log_{10} \frac{4\pi d_{0}}{\lambda}.
\end{equation}
The parameters of this model are obtained by finding the best minimum mean-square error line fit to the measurement data. In this paper, $d_{0} = 1$ m is used for simplicity.
\subsection{Floating-intercept path loss model}
The floating-intercept (FIM) model is used in the WINNER II and 3GPP~\cite{GPP,kyosti2007winner} channel models, presented in~\eqref{PL_FREQ}.
\begin{align}
\label{PL_FREQ}
PL(\text{dB})= \alpha + 10 \beta \log_{10}(d)+ \chi_{\sigma}
\end{align}
where, $\alpha$ is the floating intercept in dB, and $\beta$ is the linear slope, and $\chi_{\sigma}$ is a normal random variable with standard deviation $\sigma$. The LS regression approach \cite{rappaport2014millimeter,mahfuza2017,maccartney2013path} creates a line-of-best fit to the empirical data.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.9\linewidth,keepaspectratio]{Figures/out_los_co.pdf}
\end{center}
\caption{FIM and CIM along with the measurement data taken from the MEC building in LOS outdoor scenario ($h_{tx} = 1.21$ m and $h_{rx}= 1.18$ m ). The blue square and solid black line represent the measurement LOS data and free-space path loss at 73 GHz respectively. The dashed red line and dotted blue show the FIM and CIM path loss models respectively }
\label{Fig9}
\end{figure}
\section{Propagation Path Loss Results}
\label{sec:results}
The empirical data and the corresponding regression line plots are shown in Figs.~\ref{Fig5} and~\ref{Fig6} for the university building and the airport gate area respectively in the LOS indoor scenario. The PLE (< 2) may be explained by the presence of a waveguide effect caused by the corridor walls. The shadowing factor is higher in the airport gated area than in the hallway corridor. This can be attributed to the many objects and metallic chairs in the gated area.
The larger path loss values are found at some locations in Fig.~\ref{Fig6}. This is most likely due to the movements of the passengers at the airport gate during taking the measurement. The path loss parameters extracted from FIM and CIM models are tabulated in Table~\ref{tab2}. Here, $\text{h}_{\text{Rx}}$ denotes the height of the receiver antenna, $\text{h}_{\text{Tx}}$ denotes the height of the transmitter antennas. Our results show that the CIM model generates the PLE of 1.8, 2.1, and 1.8 in the building hallway, indoor airport gate, and the outdoor campus areas, respectively which are very close to the free space PLE, $n = 2 $.
Figs.~\ref{Fig7} and~\ref{Fig8} show the path loss values for the LOS indoor and outdoor scenario in the airport gate areas respectively. The measurement data were captured when the $\text{h}_{\text{Rx}}$ was varied at three heights: $ 1.6$, $1.47$ and $1.3$m from the ground level keeping the $\text{h}_{\text{Tx}}$ at $1.6$ m. The results show that when the Rx antenna height decreases, the slope is decreased, but the shadowing factor increases. The slope values, $\beta$, are found to be $ 0.4$ and $0.62$ for airport indoor and outdoor environments, respectively when the transmit and receive antennas have the smallest height difference. The shadow factors determined for the path loss models developed from the
indoor and outdoor airport gate measurements using the
floating intercept are shown in Table~\ref{tab3}. The results show that the channel has high losses when both antennas are not at the same height. In addition to the airport outdoor campaign, Fig.~\ref{Fig9} shows the scattered measurement data along with the FIM and CIM for the outdoor campus environment. The CIM provides PLE and shadow factor 1.8 and 2.2 dB respectively in this scenario.
\section{Conclusion}
\label{sec:conclusion}
This paper presented the results of our extensive channel measurement campaign at 73 GHz at Boise Airport, and Boise State University. The FIM and CIM path loss models were developed for both indoor and outdoor airport scenarios. Our work shows that the PLEs from the CIM model are close to the PLE (=2) of the free-space model, whereas the FIM gives a better fit to the measured data. These results also show that the indoor airport environment is uniquely different from other indoor settings due to its large and open nature. Future work includes more measurements at another E-band frequency (81 GHz) for channel parameter estimation.
\section*{Acknowledgment}
This work was funded by NASA$'$s Aeronautics Research Mission Directorate. The authors would like to thank the Boise Airport Operation Team.
\bibliographystyle{IEEEtran}
|
3,212,635,537,934 | arxiv | \section{Introduction}
The spacetime geometry in the exterior of the asymptotic end state of a generic gravitational collapse for a rotating matter cloud is expected to be given by the Kerr solution of the Einstein equations \cite{Mars00}. The Kerr metric \cite{Kerr63} has conventionally been used to describe a rotating black hole or a naked singularity, depending on the angular momentum parameter $J$ and mass $M$ of the compact object. When the Kerr bound $J \leq M^2$ is not violated, the final spacetime contains a Kerr black hole, that is, formation of an event horizon takes place, and in the case otherwise, a Kerr naked singularity occurs.
Properties of both classes of Kerr spacetimes, namely the black hole and naked singularity configurations, have been explored in great detail in various contexts such as, the structure of ergoregions \cite{Visser09, Chakraborty+17a} and shadows \cite{EHT19, Hioki_Maeda09}, efficiency of energy transfer from accretion disks \cite{Gimon_Horava09}, highly energetic particle collisions \cite{Banados+09}, gravitomagnetic spin-precession \cite{Chakraborty+17b}, to name a few. The overall picture that emerges is that if both Kerr black holes and Kerr naked singularities exist in nature, it might be possible to distinguish them from astrophysical observations.
However, the question of whether these objects do, in fact, exist ubiquitously in nature falls within the ambit of a stability analysis of their corresponding exterior spacetimes. The quasi-normal mode frequencies (QNFs) of Kerr black holes were obtained after a vigorous study \cite{Teukolsky73, Teukolsky_Press74, Detweiler80, Leaver86}, and it was eventually established that Kerr black holes are indeed mode stable, in the seminal work of Whiting \cite{Whiting89}. On the other hand, the QNFs were also obtained recently for the Kerr naked singularity spacetimes \cite{Dotti+08}, and it is likely that the Kerr naked singularities may not be mode stable.
Now, given that the Kerr naked singularity spacetime is unstable against mode perturbations, does this mean that studies of its properties must be abandoned? Following \cite{Gimon_Horava09}, we adopt the following perspective. Singular metrics are solutions of the classical Einstein equations and the expectation is that a deeper theory of quantum gravity would smear out these singularities, irrespective of whether or not they are covered from asymptotic observers by event horizons. Therefore, it is possible that quantum gravity could introduce classes of legitimate compact objects such that their exterior geometries are described by metrics that were classically nakedly singular, but with their central singular regions excised and replaced by regions governed by Planckian physics. As argued by \cite{Gimon_Horava09}, string theory, a popular candidate for the quantum theory of gravity, has proven to be exceptionally good at resolving spacetime geometries with various timelike singularities, and such singularities in general relativity (GR) could in fact represent new classes of legitimate compact objects in the string-theoretic completion of GR (see for example the pair of papers, \cite{Breckenridge+97, Gimon_Horava04}.)
Therefore, following \cite{Gimon_Horava09}, we introduce the notion of a Kerr superspinar, a third exotic class of Kerr compact objects, whose exterior geometry is given by the overspinning Kerr metric ($J > M^2$). However, this hypothetical exotic object is assumed to have a finite size which, in the usual Boyer-Lindquist coordinates, is denoted by $r=r_0 > 0$. That is, the exterior spacetime of the Kerr superspinar is identical to that of the Kerr naked singularity, but its interior metric is treated as being unknown, and assumed to be provided by the full UV-complete theory of gravity.
An attempt to study the mode stability of these singularity-excised spacetimes was recently made \cite{Pani+10}. To better understand what this entails, it is useful to remember that the study of mode stability of any given spacetime requires us to solve the linearized Einstein equations with `appropriate boundary conditions.' Irrespective of the spacetime under consideration, one imposes the condition that there are no sources at asymptotic infinity, i.e. there is no incoming radiation at infinity. As for the boundary condition at the inner edge, in the case of black hole spacetimes, one naturally imposes perfectly absorbing boundary conditions at the horizon. However, for exotic objects like Kerr superspinars, there is no such `natural' boundary condition for incoming modes at its surface, and one must find quasinormal modes and their frequencies for each possible boundary condition. Pani \textit{et al} \cite{Pani+10} used a variety of boundary conditions that included all previous studies, and they found employing a numerical approach, that superspinars are typically likely to be unstable.
The equation governing the evolution of quasi-normal modes of the Kerr spacetime was discovered by Teukolsky \cite{Teukolsky73}. Recently, motivated by insight into the pole structure of the Teukolsky equation, we revisited the issue of mode stability of near-extremal ($J \gtrsim M^2$) superspinars in \cite{Nakao+18}. As a first step, we imposed boundary conditions identical to those that are imposed in the case of a black hole at its horizon, i.e. purely absorbing boundary conditions at the surface of the superspinar $r=r_0$. In this particular case, $r_0$ can be thought of as being the location of a `stringy horizon' of the Kerr superspinar. As an ansatz, we set the quasi-normal frequency spectrum of the near-extremal superpsinar to be identical to the near-extremal black hole and showed that for these boundary conditions, this spectrum is allowed. This ansatz was backed by the results of the numerical studies in \cite{Pani+10}. There it was discussed that near-extremal superspinars with stringy horizons are indeed mode stable. Therefore, the overspinning Kerr geometry could possibly admit legitimate interior solutions and must hence not be discarded without further exploration.
In the current work, our purpose is to maximally extend our results to include arbitrary boundary conditions, imposed at the surface of the superspinar. We find that barring a zero-measure set of boundary conditions, corresponding to `almost perfectly reflecting boundary conditions', near-extremal superspinars are in fact generically mode stable.
\section{The Teukolsky Equation}
The Kerr metric in Boyer-Lindquist (BL) coordinates $(t, r, \theta, \phi)$ is given as,
\begin{align} \label{eq:KerrMetricBL}
ds^2 =& -\left(1- \frac{2 M r}{\rho^2}\right)dt^2 - \frac{4Mar\sin^2\theta}{\rho^2}dtd\phi \\
& \quad\quad + \frac{A\sin^2\theta}{\rho^2} d\phi^2 + \frac{\rho^2}{\Delta}dr^2 + \rho^2 d\theta^2 \nonumber,
\end{align}
where we have employed geometrized units $G \! = \! c \! = \!1$. In the above, we have introduced the specific angular momentum $a = Ma_* = J/M$, with $\Delta=r^2-2Mr+a^2$, $\rho^2=r^2+a^2 \cos^2\theta$ and $A = (r^2 + a^2)^2 -a^2 \Delta\sin^2\theta$.
The discovery of a single master equation that determined the evolution of separable perturbations of various types of fields (scalar, electromagnetic and gravitational), with a spin-weight parameter $s$ characterizing the type of the field was reported in a seminal paper by Teukolsky \cite{Teukolsky73}. There, it was argued that with the introduction of the Kinnersley complex null tetrad \cite{Kinnersley69}, the electromagnetic field can be characterized by its Newman-Penrose \cite{Newman_Penrose62} components. Further, it was identified that the $\phi_0$ and $\phi_2$ components correspond to the ingoing and outgoing radiative parts of the field. Similarly, gravitational radiation is described by perturbations in the Weyl tensor $C_{\alpha\beta\gamma\delta}$, the traceless component of the Riemann tensor, and has ingoing and outgoing radiative parts given by $\psi_0$ and $\psi_4$. These quantities are well-behaved invariants under gauge transformations and infinitesimal tetrad rotations. Teukolsky unified the perturbation treatment of all types of perturbative fields (scalar, electromagnetic, gravitational) by introducing a master variable,
\begin{equation}
\psi=e^{-i\omega t+im\varphi}R_{lm}(r)S_{lm}(\theta),
\end{equation}
where $\omega$ will acquire the interpretation of being the characteristic quasi-normal frequency (QNF), when the appropriate boundary conditions are imposed and is, in general, complex. As mentioned already, the appropriate boundary conditions for the above perturbations to be treated as quasi-normal modes are that there be no incoming waves at spatial infinity. Imposing this condition, we investigate here the QNFs of outgoing gravitational field perturbations, corresponding to $\psi_4$. In this case, we can write $\psi_4=(r-ia\cos\theta)^{-4}~\psi$, as can be seen from Table 1 of \cite{Teukolsky73}, for $s=-2$.
The governing linearized Einstein equations of motion for this class of perturbations $\psi$ are called the Teukolsky equations, and in terms of the radial $R_{lm}$ and angular $S_{lm}$ functions, are given as,
\begin{align}
&\Delta^{-s}\frac{d}{dr}\left(\Delta^{s+1}\frac{dR_{lm}}{dr}\right) \label{eq:R-eq} \\
& \quad\quad +\left(\frac{K^2-2is(r-M)K}{\Delta}+4is\omega r-\lambda\right)R_{lm}=0, \nonumber \\
&\frac{1}{\sin\theta}\frac{d}{d\theta}\left(\sin\theta\frac{dS_{lm}}{d\theta}\right) + \biggl[\left(a\omega\cos\theta+s\right)^2 \label{eq:S-eq}\\
&\quad\quad -\left(\frac{m+s\cos\theta}{\sin\theta}\right)^2-s(s-1)+F\biggr]S_{lm}=0, \nonumber
\end{align}
where $|s|$ denotes the spin of the perturbing field, i.e., $|s| = 0, 1, 2$ for scalar, electromagnetic and gravitational ($|s|=2$) perturbations. $F={}_sF^l_{m,\omega}$ with the integer $l$ larger than or equal to ${\rm max}(|m|,|s|)$ is the separation constant equivalent to the eigenvalue of (\ref{eq:S-eq}) with the boundary conditions of regularity at $\theta=0$ and $\pi$, $K:=(r^2+a^2)\omega-am$ and $\lambda:=F+a^2\omega^2-2am\omega$.
In the case of $a^2<M^2$, $r=r_\pm:=M\pm\sqrt{M^2-a^2}$ are real roots of $\Delta=0$; $r=r_+$ corresponds to the event horizon and $r=r_-$ is the location of the Cauchy horizon. In the extremal case, $a^2=M^2$, $r_+$ and $r_-$ agree with each other, and there is only one degenerate event horizon. In the case of $a^2>M^2$, i.e., a naked singularity or a superspinar, there is no real root of $\Delta=0$, and correspondingly no event horizon exists.
Now, following \cite{Cardoso04}, we introduce,
\begin{equation}
R_{lm} = \Delta^{-s}\tilde{R} \exp\left(-i\int\frac{K}{\Delta}dr\right),
\end{equation}
so that Eq.~(\ref{eq:R-eq}) becomes
\begin{align}
&\Delta\frac{d^2\tilde{R}}{dr^2}-\left[2i\omega (r^2+a^2)-2(\tilde{s}+1)(r-M)-2iam\right] \nonumber\\
&\times\frac{d\tilde{R}}{dr} -\left[2(2\tilde{s}+1)i\omega r+\tilde{\lambda}\right]\tilde{R}=0, \label{eq:tR-eq}
\end{align}
where, using $F=E-s(s+1)$, we have introduced
$\tilde{s}=-s$ and $\tilde{\lambda}=\lambda+2s=E+a^2\omega^2-2am\omega-\tilde{s}(\tilde{s}+1)$.
\section{Quasi-Normal Stability of Near Extremal Superspinars}
We consider a near-extremal Kerr spacetime and hence we write the Kerr parameter in the form $$ a=M(1-\epsilon), $$ assuming $0<|\epsilon|\ll1$. The spacetime contains a superspinar in the case of $\epsilon<0$,
whereas there is a black hole in the case of $\epsilon> 0$.
In the case of black hole, it is known that the quasi-normal mode (QNM) frequency $\omega$ approaches $m/2M$ for $m=l$ in the limit of $\epsilon\rightarrow0_+$ \cite{Detweiler80}. The numerical study in Ref.~\cite{Pani+10} has revealed that even in the superspinar case, the QNM frequency $\omega$ approaches $m/2M$ for $m=l$ modes in the limit $\epsilon\rightarrow0_-$. Hence, hereafter we focus on the modes of $m=l$ and assume
\begin{equation}
M\omega-\frac{m}{2}={\cal O}\left(|\epsilon|^p\right), \label{assumption}
\end{equation}
where $p$ is a positive constant.
We rewrite Eq.~(\ref{eq:tR-eq}) in terms of the the dimensionless variables $y:=(r-M)/M$ and $\tilde{\omega}:=M\omega$ as,
\begin{align}
(y^2&-2\epsilon+\epsilon^2)\frac{d^2\tilde{R}_{lm}}{dy^2} -\left[2i\tilde{\omega} y^2+2(2i\tilde{\omega}-\tilde{s}-1)y \right. \nonumber\\
& \left. +2i(2\tilde{\omega}-m)(1-\epsilon) + 2i\tilde{\omega}\epsilon^2\right]\frac{d\tilde{R}_{lm}}{dy} \nonumber \\
& -\left[ 2(2\tilde{s}+1)i\tilde{\omega}(y+1)+\tilde{\lambda}\right]\tilde{R}_{lm}=0.
\label{Basic-eq}
\end{align}
We now divide this equation in two regions, the far zone defined as $y \gg max[\sqrt{|\epsilon|},|\epsilon|^p]$ and the near zone defined as $y \ll 1$. The solution to Eq.~(\ref{eq:tR-eq}) in the far zone is written in terms of confluent hypergeometric functions $_1F_1(\alpha;\gamma;z)$;
\begin{align}
\tilde{R}_{lm}^{~\text{far}}
&= A y^{-\tilde{s}-1/2+2i\tilde{\omega}+i\delta} \nonumber\\
&\times{}_1F_1\left(\frac{1}{2}+\tilde{s}+2i\tilde{\omega}+i\delta;1+2i\delta;2i\tilde{\omega} y\right)\nonumber \\
&+B y^{-\tilde{s}-1/2+2i\tilde{\omega}-i\delta}\nonumber\\
&\times{}_1F_1\left(\frac{1}{2}+\tilde{s}+2i\tilde{\omega}-i\delta;1-2i\delta;2i\tilde{\omega} y\right),
\label{solution-far}
\end{align}
where $A$ and $B$ are integration constants, and
$$
\delta^2:=4\tilde{\omega}^2-\frac{1}{4}-\tilde{\lambda}-\tilde{s}(\tilde{s}+1)\simeq \frac{1}{4}(7m^2-1)-E.
$$ is a constant. For the near-zone analysis, we keep terms only of leading order in $\epsilon$ and introduce a new radial variable, $x:=y-\sqrt{2\epsilon}$. The solution of Eq.~(\ref{eq:tR-eq}) in the near-zone is expressed by using
Gauss's hypergeometric function ${}_2F_1(\alpha,\beta;\gamma;z)$ in the form
\begin{align}
\tilde{R}_{lm}^{~\text{near}} =& C~x^{-\tilde{s}+4i\tau/\sigma} {}_2F_1(1/2-2i\tilde{\omega}+i\delta+4i\tau/\sigma, \nonumber \\
& 1/2-2i\tilde{\omega}-i\delta+4i\tau/\sigma; 1-\tilde{s}+4i\tau/\sigma;-x/\sigma) \nonumber \\
& +D~ {}_2F_1(1/2+\tilde{s}-2i\tilde{\omega}+i\delta,1/2+\tilde{s}-2i\tilde{\omega}-i\delta;\nonumber \\
&1+\tilde{s}-4i\tau/\sigma;-x/\sigma), \label{solution-near}
\end{align}
where
\begin{align}
\sigma :=& 2\sqrt{2\epsilon}~~~{\rm and}~~~
\tau :=& (1+\sqrt{2\epsilon})\tilde{\omega}-\frac{m}{2},
\end{align}
and $C$ and $D$ are integration constants. As shown in \cite{Cardoso04}, the boundary condition for purely ingoing waves at the black hole boundary is given by $R^{near}_{lm}=1$. If we compare our case with a black hole, then the term with $C$ blows up for points close to the horizon. Thus ($C=0$, $D=1$) is the boundary condition for purely ingoing waves and hence $C$ must be the reflection and $D$ the transmission coefficient.
Both solutions (\ref{solution-far}) and (\ref{solution-near}) are valid in the over-lapping region.
In the limit $y\rightarrow0$, the solution (\ref{solution-far}) behaves as
$
\tilde{R}_{lm}\rightarrow Ay^{-\tilde{s}-1/2+2i\tilde{\omega}+i\delta} +By^{-\tilde{s}-1/2+2i\tilde{\omega}-i\delta}.
$
In the limit $y\rightarrow \infty$, the solution (\ref{solution-near}) behaves as
$
\tilde{R}_{lm}\rightarrow {\cal A}y^{-\tilde{s}-1/2+2i\tilde{\omega}+i\delta}+{\cal B} y^{-\tilde{s}-1/2+2i\tilde{\omega}-i\delta},
$
where $\cal A$ and $\cal B$ are given by
\begin{align}
&{\cal A}=\sigma^{1/2-2i\tilde{\omega}-i\delta}\Gamma(2i\delta) \nonumber \\
&\times\Biggl(
\frac{C\sigma^{4i\tau/\sigma}\Gamma(1-\tilde{s}+4i\tau/\sigma)}
{\Gamma(1/2-\tilde{s}+2i\tilde{\omega}+i\delta)\Gamma(1/2-2i\tilde{\omega}+i\delta+4i\tau/\sigma)} \nonumber \\
&+\frac{D\sigma^{\tilde{s}}\Gamma(1+\tilde{s}-4i\tau/\sigma)}
{\Gamma(1/2+\tilde{s}-2i\tilde{\omega}+i\delta)\Gamma(1/2+2i\tilde{\omega}+i\delta-4i\tau/\sigma)}
\Biggr), \nonumber\\
\label{calA}\\
&{\cal B}={\cal A}|_{\delta\rightarrow-\delta}. \label{calB}
\end{align}
Thus, equating the two solutions in the overlapping region we get
\begin{equation}
A={\cal A}~~~~{\rm and}~~~~B={\cal B}. \label{matching}
\end{equation}
From the far-zone solution (\ref{solution-far}), for $y\rightarrow\infty$, we have
\begin{equation}
\tilde{R}_{lm}^{\ \rm far}\simeq Z_{\rm out}~ y^{-(1-4i\tilde{\omega})}e^{2i\tilde{\omega} y} +Z_{\rm in}~ y^{-(2\tilde{s}+1)},\nonumber
\end{equation}
where
\begin{align}
Z_{\rm in}&=A\frac{(-2i\tilde{\omega})^{-1/2-\tilde{s}-2i\tilde{\omega}-i\delta}\Gamma(1+2i\delta)}
{\Gamma(1/2-\tilde{s}-2i\tilde{\omega}+i\delta)} \cr
&+B\frac{(-2i\tilde{\omega})^{-1/2-\tilde{s}-2i\tilde{\omega}+i\delta}\Gamma(1-2i\delta)}{\Gamma(1/2-\tilde{s}-2i\tilde{\omega}-i\delta)}, \cr
Z_{\rm out}&=Z_{\rm in}|_{\tilde{s}\rightarrow-\tilde{s},\tilde{\omega}\rightarrow-\tilde{\omega}}. \nonumber
\end{align}
Thus, together with Eq.~(\ref{matching}),
the no-incoming wave condition, $Z_{\rm in}=0$, leads to
\begin{equation}
{\cal A}\frac{(-2i\tilde{\omega})^{-i\delta}\Gamma(1+2i\delta)}{\Gamma(1/2-\tilde{s}-2i\tilde{\omega}+i\delta)}
+\left(\delta\rightarrow-\delta\right)=0
\label{QNM-condition}
\end{equation}
Eq. (\ref{QNM-condition}) along with Eq. (\ref{calA}) and (\ref{calB}) gives us the expression of the QNM frequencies in term of the reflection and transmission coefficient at the Superspinar boundary. Substituting Eq. (\ref{calA}) and Eq. (\ref{calB}) into Eq. (\ref{QNM-condition}) we get
\begin{equation} \label{CD}
D\sigma^{\tilde{s}}(f_2+\sigma^{2i\delta}f_4)=-C\sigma^{4i\tau/\sigma}(f_1+\sigma^{2i\delta}f_3)
\end{equation}
where $f_1, f_2, f_3, f_4$ are given by
\begin{align}{
f_1=&\Bigg(\frac{\Gamma(2i\delta)\Gamma(1+2i\delta)(-2i\tilde{\omega})^{-i\delta}}{\Gamma(1/2-\tilde{s}+2i\tilde{\omega}+i\delta)\Gamma(1/2-\tilde{s}-2i\tilde{\omega}+i\delta)}\Bigg)\nonumber\\
&\times\frac{\Gamma(1-\tilde{s}+4i\tau/\sigma)}{\Gamma(1/2-2i\tilde{\omega}+i\delta+4i\tau/\sigma)}\\
f_2=&\Bigg(\frac{\Gamma(2i\delta)\Gamma(1+2i\delta)(-2i\tilde{\omega})^{-i\delta}}{\Gamma(1/2+\tilde{s}-2i\tilde{\omega}+i\delta)\Gamma(1/2-\tilde{s}-2i\tilde{\omega}+i\delta)}\Bigg)\nonumber\\
& \times\frac{\Gamma(1+\tilde{s}-4i\tau/\sigma)}{\Gamma(1/2+2i\tilde{\omega}+i\delta-4i\tau/\sigma)}\\
f_3=&\Bigg(\frac{\Gamma(-2i\delta)\Gamma(1-2i\delta)(-2i\tilde{\omega})^{i\delta}}{\Gamma(1/2-\tilde{s}+2i\tilde{\omega}-i\delta)\Gamma(1/2-\tilde{s}-2i\tilde{\omega}-i\delta)}\Bigg)\nonumber\\
& \times\frac{\Gamma(1-\tilde{s}+4i\tau/\sigma)}{\Gamma(1/2-2i\tilde{\omega}-i\delta+4i\tau/\sigma)}\\
f_4=&\Bigg(\frac{\Gamma(-2i\delta)\Gamma(1-2i\delta)(-2i\tilde{\omega})^{i\delta}}{\Gamma(1/2+\tilde{s}-2i\tilde{\omega}-i\delta)\Gamma(1/2-\tilde{s}-2i\tilde{\omega}-i\delta)}\Bigg)\nonumber\\
& \times\frac{\Gamma(1+\tilde{s}-4i\tau/\sigma)}{\Gamma(1/2+2i\tilde{\omega}-i\delta-4i\tau/\sigma)}}\end{align}
With $C$ being the reflection coefficient and $D$ the transmission coefficient, we have $D^2=1-C^2$. In the regime $\sigma \to 0$, we can apply Stirling's formula to the above equations and simplify to get $f_1$, $f_2$, $f_3$ and $f_4$ respectively as
\begin{eqnarray}
&\Bigg(&\frac{(4i\tau/\sigma)^{1/2-\tilde{s}+2i\tilde{\omega}-i\delta}\Gamma(2i\delta)\Gamma(1+2i\delta)(-2i\tilde{\omega})^{-i\delta}}{\Gamma(1/2-\tilde{s}+2i\tilde{\omega}+i\delta)\Gamma(1/2-\tilde{s}-2i\tilde{\omega}+i\delta)}\Bigg)\nonumber \\
&=&(4i\tau/\sigma)^{1/2-\tilde{s}+2i\tilde{\omega}-i\delta}g_1\\
&\Bigg(&\frac{(4i\tau/\sigma)^{1/2+\tilde{s}-2i\tilde{\omega}-i\delta}\Gamma(2i\delta)\Gamma(1+2i\delta)(-2i\tilde{\omega})^{-i\delta}}{\Gamma(1/2+\tilde{s}-2i\tilde{\omega}+i\delta)\Gamma(1/2-\tilde{s}-2i\tilde{\omega}+i\delta)}\Bigg)\nonumber \\
&=&(4i\tau/\sigma)^{1/2+\tilde{s}-2i\tilde{\omega}-i\delta}g_2
\end{eqnarray}
\begin{eqnarray}
&\Bigg(&\frac{(4i\tau/\sigma)^{1/2-\tilde{s}+2i\tilde{\omega}+i\delta}\Gamma(-2i\delta)\Gamma(1-2i\delta)(-2i\tilde{\omega})^{i\delta}}{\Gamma(1/2-\tilde{s}+2i\tilde{\omega}-i\delta)\Gamma(1/2-\tilde{s}-2i\tilde{\omega}-i\delta)}\Bigg)\nonumber \\
&=&(4i\tau/\sigma)^{1/2-\tilde{s}+2i\tilde{\omega}+i\delta}g_3\\
&\Bigg(&\frac{(4i\tau/\sigma)^{1/2+\tilde{s}-2i\tilde{\omega}+i\delta}\Gamma(-2i\delta)\Gamma(1-2i\delta)(-2i\tilde{\omega})^{i\delta}}{\Gamma(1/2+\tilde{s}-2i\tilde{\omega}-i\delta)\Gamma(1/2-\tilde{s}-2i\tilde{\omega}-i\delta)}\Bigg)\nonumber \\
&=&(4i\tau/\sigma)^{1/2+\tilde{s}-2i\tilde{\omega}+i\delta}g_4
\end{eqnarray}
Putting this back in Eq. (\ref{CD}),
\begin{equation}\label{25}
\frac{C}{\sqrt{1-C^2}}=A\Bigg(\frac{g_2+(4i\tau)^{2i\delta}g_4}{g_1+(4i\tau)^{2i\delta}g_3}\Bigg)
\end{equation}
where we have defined
\[A=-\sigma^{\tilde{s}-4i\tau/\sigma} (-1)^{1/2+\tilde{s}-2i\tilde{\omega}-i\delta} (4i\tau/\sigma)^{2(\tilde{s}-2i\tilde{\omega})}\].
Note that the particular use of casting the equation in this form is to shift all the dependence of $\sigma$ to the term $A$. Writing $\tilde{\omega}=a+ib$ and simplifying $A$ we get
\begin{align}
A&=(4\tau/\sigma)^{\tilde{s}+2b+\sqrt{2}\sin{(1/2-p)}|\epsilon| ^{p-1/2}}\nonumber \\
&\times [(-1)^{3/2+3\tilde{s}/2-2ia+4b-i\delta}(4i\tau)^{\tilde{s}-4i\tau/\sigma}\nonumber \\
&\times (4i\tau/\sigma)^{\sqrt{2}i\cos{(1/2-p)}\pi|\epsilon| ^{1/2-p}-2ia}]
\end{align}
Keeping consistent with the argument presented in \cite{Nakao+18}, we would take $p<1/2$ from here onwards. It should be noted that in the above equation, the term inside $[...]$ is finite, but the term outside $(4\tau/\sigma)^{\tilde{s}+2b+\sqrt{2}\sin{(1/2-p)}|\epsilon| ^{1/2-p}}\rightarrow \infty$ since for $p<1/2$, $(4\tau/\sigma)\rightarrow \infty$ and $|\epsilon| ^{p-1/2}\rightarrow \infty$ in the limit $\epsilon\rightarrow 0_\pm$ and $\sigma \rightarrow 0$. Simplifying Eq. (\ref{25}) by substituting $g_1, g_2, g_3, g_4$, we get
\begin{equation}\label{27}
\Bigg(\frac{(-8\tilde{\omega} \tau)^{2i\delta}-q_2e^{i\chi_2}}{(-8\tilde{\omega} \tau)^{2i\delta}-q_1e^{i\chi_1}}\Bigg)=A'\frac{\sqrt{1-C^2}}{C}
\end{equation}
where we have defined
\begin{align}-q_1e^{i\chi_1}=&\frac{\Gamma(1/2+\tilde{s}-2i\tilde{\omega}-i\delta)\Gamma(1/2-\tilde{s}-2i\tilde{\omega}-i\delta)}{\Gamma(1/2+\tilde{s}-2i\tilde{\omega}+i\delta)\Gamma(1/2-\tilde{s}-2i\tilde{\omega}+i\delta)}\nonumber \\
&\times \frac{\Gamma(2i\delta)\Gamma(1+2i\delta)}{\Gamma(-2i\delta)\Gamma(1-2i\delta)}\end{align}
\begin{align}-q_2e^{i\chi_2}=&\frac{\Gamma(1/2-\tilde{s}+2i\tilde{\omega}-i\delta)\Gamma(1/2-\tilde{s}-2i\tilde{\omega}-i\delta)}{\Gamma(1/2-\tilde{s}+2i\tilde{\omega}+i\delta)\Gamma(1/2-\tilde{s}-2i\tilde{\omega}+i\delta)}\nonumber \\
&\times \frac{\Gamma(2i\delta)\Gamma(1+2i\delta)}{\Gamma(-2i\delta)\Gamma(1-2i\delta)}\end{align}
and
\[A'=A\Bigg(\frac{\Gamma(1/2-\tilde{s}+2i\tilde{\omega}-i\delta)\Gamma(1/2-\tilde{s}-2i\tilde{\omega}-i\delta)}{\Gamma(1/2+\tilde{s}-2i\tilde{\omega}-i\delta)\Gamma(1/2-\tilde{s}-2i\tilde{\omega}-i\delta)}\Bigg)\]
Taking dividendo of Eq. (\ref{27}) and rearranging we get
\begin{equation}
(-8\tilde{\omega} \tau)^{2i\delta}-q_3q_1e^{i(\chi_1+\chi_3)}=0
\end{equation}
where
\begin{equation}\label{31}
q_3e^{i\chi_3}=1+\Bigg(\frac{C}{A'\sqrt{1-C^2}-C}\Bigg)\Bigg(1-\frac{q_2}{q_1}e^{i(\chi_2-\chi_1)}\Bigg)
\end{equation}
Following a similar analysis as \cite{Cardoso04}, we show that by taking $-8\omega \tau=\rho e^{i\zeta}$, the solution to the above equation is given by
\begin{equation}
\rho=e^{(\chi_1+\chi_3-2n\pi)/2\delta}
\end{equation}
\begin{equation}
\zeta=-\frac{1}{2\delta}\log{(q_3q_1)}
\end{equation}
Since we are working in the limit $\omega \sim m\omega_+$, we can rewrite $-8\tilde\omega \tau$ as $-8\tilde{\omega}\tau=-4m(\tilde{\omega}-m/2)$ , to obtain \cite{Sasaki_Nakamura90}
\[\tilde{\omega}=-\frac{\rho}{4m}\cos{\zeta}+m/2-i\frac{\rho}{4m}\sin{\zeta}\]
Therefore it follows that the stability of the superspinar depends on the sign of $\sin{\zeta}$. It is evident from Eq. (\ref{31}) that as $A\rightarrow \infty$, $q_3\to 1$. It has already been shown in \cite{Nakao+18} that for $\zeta=-\frac{1}{2\delta}\log{(q_1)}$, $\sin{\zeta}>0$. If $q_3\to 1$, $\log{(q_3q_1)}\to$ $\log{(q_1)}$ and $\zeta\to-\frac{1}{2\delta}\log{(q_1)}$, and thus $\sin{\zeta}>0$, irrespective of what $C$ is. Thus, we conclude that the imaginary part of the QNM frequencies is always negative, except for the case $C=1$. This is because our analysis stems from the limiting behaviour of $\sigma$ in Eq. (\ref{31}) and in the case of $C=1$, $(1-C)$ goes to $0$ and hence the limiting behaviour of $\sigma$ does not affect the analysis. This case must be solved exactly and our analysis is not suited for this case. It should also be noted here that for the case $C=0$, Eq. (\ref{31}) gives $q_3=1$ and hence stability is ensured as argued above, in agreement to the results of \cite{Nakao+18}. For the case $C=1$, \cite{Pani+10} has shown that the superspinar is unstable. Thus, in the entire parameter space of $C$, the extremal superspinar is stable for all boundary conditions except for the case $C=1$ and a zero-measure set very close to it.
\section{Conclusion}
In \cite{Nakao+18} the question raised was, are there boundary conditions for which the Superspinar is stable? In answer to this, it was suggested that we can take the frequency as the input parameter and set the imaginary part of it to be negative, and then find values of $C$ and $D$, the boundary parameters. Thus for a negative imaginary part of frequency there could exist an infinite set of $(C,D)$ for which the superspinar might be stable. In the present analysis, the new result that comes out is that, irrespective of what $C$ and $D$ are, a near extremal Kerr Superspinar would almost always be stable except for the $C=1$ case. The physical nature of the boundary corresponding to these boundary conditions is unclear at the moment and further study is required in this respect.
The present analysis is restricted to near extremal case for $l=m$ modes. But our result shows that for this particular case, the Kerr superspinar is almost always stable, in contrast to previous studies which concluded them to unstable. We do not exactly know yet what the boundary condition for the Superspinar at $r_0$ would be in physical reality. The interesting part of our result is that irrespective of what that boundary condition is, near extremal Kerr superspinars with $l=m$ modes would always be stable. With that being said, the statement in our earlier paper\cite{Nakao+18} is more strong since the argument could be extended to non-extremal case with $l \ne m$ modes. An interesting question then arises, namely, how strongly
the properties of a classical nakedly singular metric, such as e.g. mode stability, shadows, images etc, would depend on the
existence of a central naked singularity itself, rather than there being a superspinar as the central exotic object. These issues will be discussed separately in a different paper.
\iffalse
\section{References}
\fi
|
3,212,635,537,935 | arxiv | \section{Introduction}
It is known that exact {\it gravitational plane waves}
are very simple time dependent plane symmetric solutions of Einstein's
equations \cite{bel26}. Nevertheless, they show two main nontrivial global
features, namely:
i) the absence of a global Cauchy surface,
which is a consequence of the focusing effect that the waves exert on
null rays \cite{pen65},
ii) the presence of a Killing-Cauchy horizon which may be physically
understood as the caustic produced by the focusing of null rays \cite{pir89}.
The inverse of
the focusing time is a measure of the strength of the wave. For an
Einstein-Maxwell plane wave such inverse
time equals the electromagnetic energy per unit surface of the wave. This
makes exact plane waves very
different from their linearized counterparts, which have no focusing points
and admit a globally hyperbolic
space-time structure. One expects that exact plane waves may be relevant for
the study of the strong time
dependent gravitational fields that may be produced in the collision of black
holes \cite{dea79,fer80} or to
represent travelling waves on strongly gravitating cosmic strings
\cite{gar89-90}. In recent years these
waves have been used in classical general relativity to test some conjectures
on the stability of Cauchy
horizons
\cite{ori92,yur93}, and in string theory to test classical and quantum string
behaviour in strong
gravitational fields \cite{veg84-90,veg91,jof94}. Their interest also stems
from the fact that plane waves are a
subclass of exact classical solutions to string theory
\cite{ama84-88,hor90,tse93fes94rus95}.
In Einstein-Maxwell theory the particular class of plane symmetric
waves are seen to contain only a non-null component of the Ricci tensor and
only a non-null component of the Weyl tensor. In particular,
the single component of the Weyl tensor may be conveniently
interpreted as the {\em transverse wave component} in the direction of
propagation of the wave. In that sense,
the modulus term of the Weyl component can be identified with the
{\em amplitude} of the wave and the phase term with the {\em polarization}
of the wave. Furthermore, depending on whether the Ricci component or the Weyl
component is zero we will distinguish in between {\em pure gravitational
plane waves} or {\em pure electromagnetic plane waves} respectively.
When we consider a plane wave collision, we should analyze separately the
collision between pure gravitational waves,
between pure electromagnetic waves or between mixed waves. Namely:
i) when two pure gravitational
plane waves interact, the focusing effect of each
wave distorts the causal structure of the space-time
near the null horizons that these waves contain and either a spacelike
curvature singularity or a new regular Killing-Cauchy
horizon is created,
ii) when two pure electromagnetic plane waves interact, the situation
is more subtle. In fact,
in the full Einstein-Maxwell
theory, Maxwell's equations remain linear indicating non direct
electromagnetic interaction between the waves. However,
there is a non-linear interaction of the waves through the gravitational
field generated by their electromagnetic energy, which is
similar to the magnitude of the interaction between
pure gravitational waves. In that sense, the collision of two
electromagenic waves is seen to produce gravitational waves.
iii) in the case of mixed collisions, the pure electromagnetic wave is
partially reflected by the incident pure gravitational wave. The
gravitational wave, however, is not necessarily reflected.
Note that, the presence of a Killing-Cauchy horizon in a colliding
plane wave space-time implies a breakdown of predictability since the
geometry beyond the horizon is not uniquely determined by the initial
data posed by the incoming colliding waves. Also, the singularities
derived from plane wave collisions are not the
result of the collapse of matter but the result of the non-linear
effects of pure gravity, i.e. the {\em self-gravitation} of the
gravitation field.
When waves are coupled to quantum fields there is
neither vacuum polarization nor the spontaneous creation of particles.
In that sense they behave
very much as electromagnetic or
Yang-Mills plane waves in flat space-time \cite{des75,gib75}.
However, as a result of the non-linear interaction,
the creation of quantum particles is expected in a plane wave collision.
The interaction of quantum fields with colliding plane waves was first
considered by Yurtsever \cite{yur89} for the singular Khan-Penrose
solution \cite{kha71}, which describes the collision of two plane impulsive
gravitational waves. In that case,
an unambiguous ``out'' vacuum state was possible to define in a relatively
simple way. More recently, Dorca and Verdaguer \cite{dor93,dor94}
noticed that the presence
of a Killing-Cauchy horizon in a non singular colliding plane wave
space-time could be used to define an unambiguous ``out'' vacuum state
related to the preferred Hadamard state introduced by Kay and Wald in more
generic space-times with Killing-Cauchy horizons \cite{kay91}.
With this premise,
Dorca and Verdaguer studied the interaction of quantum fields in a
particular non-singular colliding plane wave space-time, the
interaction region of which was isometric to a region
inside the event horizon of
a Schwarzschild black hole \cite{fer87cha86,hay89}. Later on, the same
premise was applied by Feinstein and Sebasti\'an \cite{fei95}
to the Bel-Szekeres
solution \cite{bel74}, which represents the head
on collision of two electromagnetic
plane waves with an interaction region isometric to the Bertotti-Robinson
universe \cite{rob54ber59} filled with an uniform electric field.
In all these examples it was found that the initial state, defined
to be the vacuum state in the flat region before the arrival of the waves,
contained a spectrum of ``out'' particles consistent, in the
long wavelength limit, with a thermal spectrum with a temperature inversely
proportional to the focusing time of the waves.
A further step in the study of the interaction of quantum fields with
colliding plane waves is the computation of the expectation value of
the stress-energy tensor. Again, this problem was first considered by
Yurtsever \cite{yur89} for the Khan-Penrose solution \cite{kha71}.
In that case it was possible to determine the behavior of the stress-energy
tensor near the singularity of the interaction region.
It was shown that for the conformal coupling case (i.e. $\xi =1/6$) the
energy density and two of the principal pressures were
positive and unbounded towards the singularity.
This problem has been also considered by Dorca and Verdaguer in
the mentioned above non-singular colliding plane wave spacetime with an
interaction region isometric to an interior region of a Schwarzschild
black hole. As in the case of Yurtsever for
the Khan-Penrose solution, the expectation value of the stress-energy
tensor was calculated in the state representing the Minkowski vacuum
in the flat region before the arrival of the waves. This value
was first computed in a region close to both the Killing-Cauchy
horizon and the topological singularities, the {\em folding singularities},
that the colliding plane wave space-time contains \cite{gri91}.
In that particular region, the calculations were simplified due to
the blueshift effect on the energy of the initial quantum modes as
they reached the Killing-Cauchy horizon \cite{dor96}. In a recent work
\cite{dor97}, an approximation procedure was proposed by the author
in order to calculate such an expectation value throughout the causal
past region of the collision center. In both calculations, it was found that
the stress-energy diverged as the Killing-Cauchy horizon was approached.
The rest energy density was positive and unbounded towards the horizon.
Two of the principal pressures were negative and of the same order of
magnitude of the energy density. It was also pointed out that such a
behavior suggested that the non singular Killing-Cauchy horizon is
indeed unstable under quantum perturbations and a curvature singularity
would be the general outcome of a generic plane wave space-time
when backreaction is taking into account.
In the present paper, the approximation procedure introduced in
\cite{dor97} is applied to the non singular Bel-Szekeres space-time as
a first attempt to generalize such an approximation to more generic
colliding plane wave space-times.
The plan of the paper is the following. In section 2 the geometry of the
Bel-Szekeres solution is
briefly reviewed. In section 3 an adequate approximation in the
space-time geometry is introduced throughout the causal past of the
collision center. Then, the mode solutions of a massless scalar field
which represent the vacuum state before the arrival of the waves
are calculated all over this particular region. In section 4 the
Hadamard function, which is the key ingredient for
the computation of the stress-energy tensor,
is calculated and regularized by means of
the {\em point-splitting} technique. In section 5
the vacuum expectation value of
the stress-energy tensor is calculated. In section 6 a summary and
some consequences of that result are given. In order to help to
maintain the main body of the paper reasonably clear, some final
results are stored in the Appendices.
\section{Description of the geometry}
The Bel-Szekeres solution \cite{bel74} represents the collision of two
electromagnetic shock waves followed by trailing radiation. The interaction
region is isometric to the Bertotti-Robinson universe
\cite{rob54ber59}, which is the static conformaly flat solution of
Einstein-Maxwell equations with an uniform electric field.
Such a geometry is similar to the throat of the Reissner-Nordstrom
solution for the special case $M=Q$ \cite{whe73}.
The space-time
contains four space-time regions, given by
\begin{equation} \left.ds^2_{IV}\right.=4L_1L_2dudv-dx^2-dy^2, \label{eq:ibIV}\end{equation}
\begin{equation} \left.ds^2_{III}\right.=4L_1L_2dudv-
\cos ^2v\left( dx^2+ dy^2\right),\label{eq:ibIII}\end{equation}
\begin{equation} \left.ds^2_{II}\right.=4L_1L_2dudv-
\cos ^2u\left(dx^2+dy^2\right),\label{eq:ibII}\end{equation}
\begin{equation} \left.ds^2_{I}\right.=4L_1L_2dudv-
\cos^2 (u+v)\, dx^2-\cos ^2(u-v)\, dy^2,\label{eq:ibI}\end{equation}
where for convenience we have used $u$ and $v$ as dimensionless
null coordinates, and where $L_1$ and $L_2$, are length
parameters such that $L_1L_2$ is directly related to the focusing time
of the collision, i.e. to the inverse of the strength of the waves,
which is a measure of the amount of nonlinearity of the gravitational
waves \cite{gri91}.
This colliding wave space-time, as shown in Fig. 1, consists of two
approaching waves, regions II and III, in a flat background, region
IV, and an interaction region, region I. The two waves move in the
direction of two null coordinates $u$ and $v$, and since they have
translational symmetry along the transversal $x$-$y$
planes, the interaction region retains a
two-parameter symmetry group of motions generated by the Killing
vectors $\partial _x$ and $\partial _y$.
The four space-time regions
are separated by the two null wave fronts $u=0$ and $v=0$. Namely, the
boundary between regions I and II is $\{0\leq u<\pi /2,\; v=0\}$, the
boundary between regions I and III is $\{u=0,\; 0\leq v <\pi /2\}$, and
the boundary of regions II and III with region IV is
$\{u\leq 0,\;v=0\}\cup\{u=0,\;v\leq 0\}$. Region I meets region IV
only at the surface $u=v=0$. The Killing-Cauchy horizon in the region
I corresponds to the hypersurface $u+v=\pi /2$ and plane wave regions
II and III meet such a Killing-Cauchy horizon
only at ${\cal P}=\{u=\pi /2,\; v=0\}$ and
${\cal P}'=\{u=0,\; v=\pi /2\}$ respectively.
Observe that plane wave regions II
and III contain a singularity
at $u=\pi /2$, for region II, and $v=\pi /2$, for region III. These
singularities are not curvature singularities but a type of
topological singularity commonly referred to as a folding
singularity \cite{gri91}. This terminology arises from the fact that
the whole singularity $u=\pi /2$ in region II (or $v=\pi /2$ in region III)
must be identified (i.e. ``folded'') with $\cal P$ (or ${\cal P}'$)
(see \cite{dor93} for more details and for a 3-dimensional plot of a
space-time of this type).
\section{Mode propagation}
For simplicity we will consider in this section a massless scalar field,
which satisfies the usual Klein-Gordon equation,
\begin{equation} \Box\phi =0.\label{eq:kG}\end{equation}
Following the directions of the approximation procedure introduced
in the previous work \cite{dor97},
we will be interested in the value of the quantum field
$\phi$ all over the causal past region of the collision center.
The reason is essentially because the calculations can be greatly
simplified in this region. We will
start with the field solution in the flat region prior to the arrival
of the waves, which is chosen to be the usual vacuum in Minkowski
space-time. This vacuum solution will set a well posed initial value
problem on the null boundary $\Sigma =\{u=0,\; v\leq
0\}\cup\{u\leq 0,\; v=0\}$, by means of
which a unique solution for the field equation can be found throughout
the space-time, i.e., in the plane wave regions (regions II and III),
and in the interaction region (region I).
We will consider the line element,
\begin{equation} ds^2=2{\rm e}^{-M(u,v)}dudv-{\rm e}^{-U(u,v)}\left(
{\rm e}^{V(u,v)}dx^2+{\rm e}^{-V(u,v)}dy^2
\right), \label{eq:dsG} \end{equation}
which applies globally to the four space-time regions, and where the
functions $U$, $V$ and $M$, can be directly read off
(\ref{eq:ibIV})-(\ref{eq:ibI}). Then, the
field equation can
be separated in a plane-wave form solution for the transversal
coordinates $x$ and $y$, with $k_x$ and $k_y$, respectively, as
separation constants. This plane-wave separation is just a trivial
consequence of the translational symmetry of the space-time on the planes
spanned by the Killing vectors $\partial _x$ and $\partial _y$. The
field solution is thus,
\begin{equation} \phi (u,v,x,y)={\rm e}^{U(u,v)/2}\, f(u,v)\, {\rm e}^{ik_xx+ik_yy},
\label{eq:phiG}\end{equation}
where the function $f(u,v)$ satisfies the following second order
differential equation,
\begin{equation} f_{,uv}+\Omega (u,v)\, f=0;\;\;\; \Omega (u,v)=-
{\left({\rm e}^{-U/2}\right)_{,uv}\over {\rm e}^{-U/2}}+{1\over 2}
{{\rm e}^{-M+U}}\left( k_x^2{\rm e}^{-V}+k_y^2{\rm e}^V\right).
\label{eq:f(u,v)}\end{equation}
Equation (\ref{eq:f(u,v)}) can be straightforwardly solved in the
flat region (region IV). Then, this solution determines
on the null boundary $\Sigma$ a well posed set of
initial conditions for
the solutions of equation (\ref{eq:f(u,v)}) in plane wave
regions II and III. Finally, the field solution in regions II,
III and IV is,
\begin{equation} \phi (u,v,x,y)={1\over\sqrt{2k_-(2\pi)^3}}\,{\rm e}^{ik_xx+ik_yy}\,
\left\{\begin{array}{lll}
\displaystyle
{1\over\cos u}{\rm e}^{-i2{\hat k}_+\tan u-i2{\hat k}_-v};
& {\rm in\; region\; II}, & (a)\\
& &\\
\displaystyle
{1\over\cos v}{\rm e}^{-i2{\hat k}_+u-i2{\hat k}_-\tan v};
& {\rm in\; region\; III}, & (b)\\
& &\\
\displaystyle
{\rm e}^{-i2{\hat k}_+u-i2{\hat k}_-v};
& {\rm in\; region\; IV}, & (c)
\end{array}\right. \label{eq:phiII/III}\end{equation}
where we have used two new separation constants $k_\pm$, which
are related to the previous ones $k_x$ and $k_y$ by the relation
$4\, k_+k_-=k_x^2+k_y^2$. For convenience, we define also the
following set of dimensionless constants:
\begin{equation} {\hat k}_\pm = \sqrt{L_1L_2}\, k_\pm ,\;\;\;\;
k_1 = \sqrt{L_1L_2}\, k_x,\;\;\;\; k_2 = \sqrt{L_1L_2}\, k_y.
\label{eq:kiL}\end{equation}
Even though $\Sigma =\,\{(u=0,\,v<0\}\cup\{u<0,\,v=0\}$ is a
null hypersurface, a well defined scalar product is given by (see
\cite{dor93} for details),
\begin{equation} (\phi _1,\phi _2)=-i\int dx dy\left[
\int _{-\infty} ^0 \left.\left(\phi _1
{\buildrel\leftrightarrow\over\partial}_u \phi
_2^*\right)\right|_{v=0}\, du +
\int _{-\infty} ^0 \left.\left(\phi _1
{\buildrel\leftrightarrow\over\partial}_v \phi
_2^*\right)\right|_{u=0}\, dv\right]. \label{eq:scalarproduct}\end{equation}
Notice that the initial modes
(\ref{eq:phiII/III}) are well normalized on the boundary $\Sigma$
between the flat region and the plane wave regions, and this means,
from general grounds, that they remain well normalized on the boundary
$\Sigma _{\rm I}=\,\{u=0,\, 0\leq v<\pi /2\}\cup\{0\leq u<\pi /2,\, v=0\}$
between the plane waves and the interaction region. Thus,
the Cauchy problem for the interaction region is now well posed.
However, although it has been rather easy to find the solution of the
field equation in regions II and III which smoothly matches with the
Minkowski vacuum, it turns out to be a difficult problem for the
interaction region. Observe that the Cauchy data for the interaction
region is imposed by the field modes (\ref{eq:phiII/III}a,b) on the lines
$\Sigma _{\rm I}=\,\{u=0,\, 0\leq v<\pi /2\}\cup\{0\leq u<\pi /2,\, v=0\}$,
which are characteristic lines for the differential
equation (\ref{eq:f(u,v)}). Thus, the only independent Cauchy data are the
values of the function $f(u,v)$ on them. Furthermore, we are only
interested in finding the field solution on the causal past of the
collision center, which is determined by the simple condition
$u=v$. Then, the only relevant Cauchy data lie on the segments,
${\bar\Sigma}_{\rm I}=\,\{u=0,\,0\leq v<\pi /4\}\cup\{0\leq u<\pi /4,\, v=0\}$
(see \cite{gara64} for details).
In order to solve this partial problem, we start with the following
change of coordinates,
\begin{equation} {t} =u+v,\;\;\; {z}=v-u, \label{eq:xieta}\end{equation}
in equation (\ref{eq:f(u,v)}) and we obtain,
\begin{equation} f_{,{t}{t}}-f_{,{z}{z}}+\Omega ({t} ,{z})f=0, \label{eq:fxieta}\end{equation}
where the term $\Omega ({t} ,{z})$, using (\ref{eq:kiL}), is given by
\begin{equation} \Omega ({t} ,{z})={k_1^2+(\sin ^2{t})/4\over\cos ^2{t}}+
{k_2^2-(\sin ^2{z})/4\over\cos ^2{z}}. \label{eq:Vxieta}\end{equation}
From now on, we will denote the causal past region of the collision
center by ${\cal S}=\,\{0\leq {t}<\pi /2,\, -\pi /4<z<\pi /4\}$.
Observe that the behaviour of the variables ${t}$ and ${z}$ in equation
(\ref{eq:fxieta}) in region $\cal S$ is
very different. Since ${t}$ runs from $0$ to $\pi /2$ and ${z}$ runs
from $-\pi /4$ to $\pi /4$ in this region, the term
(\ref{eq:Vxieta}) blows up as coordinate ${t}$ goes to $\pi /2$, but
is perfectly smooth over the entire range of coordinate
${z}$. This fact suggests that in the whole region $\cal S$
the physical results that we may expect are directly
related to the coordinate ${t}$ and we may not expect any physically
remarkable change if we take ${z} =0$ in equation (\ref{eq:fxieta}).
However, if
we want to be consistent with such an approximation, we must also
modify the boundary conditions that lie on the line segments
${\bar\Sigma}_{\rm I}$. Since on the boundary ${\bar\Sigma}_{\rm I}$
we have that
${t} =\pm{z}$ and coordinate ${t}$ runs from $-\pi /4$ to $\pi /4$,
we must also take ${t} ={z} =0$. This means that the boundary
conditions on ${\bar\Sigma}_{\rm I}$, given in (\ref{eq:phiII/III}a,b), reduce
in such an approximation to the flat boundary conditions given in
(\ref{eq:phiII/III}c). Therefore we
change the mode propagation problem for the colliding wave space-time
into a rather simpler Schr\"odinger-type problem, which is
clear from Fig. 2, and which requires only that we find a solution
to equation (\ref{eq:fxieta}) with initial conditions given by the
Minkowski flat modes below the hypersurface
$\{{t} =0,\, -\pi /4 <{z}<\pi /4\}$.
Recall, however, that
none of the discussion above is
applicable when a solution of equation
(\ref{eq:fxieta}) in the neighborhood of the folding singularities
${\cal P}$ and ${\cal P}'$ is required. This is not
only because in that case both coordinates ${t} $
and ${z}$ take values near $\pi
/2$ and thus the potential term (\ref{eq:Vxieta}) is unbounded as
$z\rightarrow\pi /2$, but
also because the boundary conditions (\ref{eq:phiII/III}a,b) are also
unbounded as the folding singularities at ${t} =\pm{z} =\pi /2$ are approached.
In that case the mode propagation problem is much more complicated and
a more detailed discussion is required (see
\cite{dor93,dor94,dor96,fei95}) for details).
In fact, rather than relying on the discussed approximations in the
exact field equation (\ref{eq:fxieta}), it will be necessary to
rewrite a new field equation using an adequate approximation to the
space-time geometry throughout the causal past region of the collision
center. This is essentially because the process of renormalization involves
the subtraction of the infinite divergences that arise from the
formal definition of the stress-energy tensor, and these divergences
can be expressed as entirely geometric terms, which are independent
of any possible
approximations in the field equation.
Approximating the field equation (\ref{eq:fxieta}) in the
causal past of the collision center by taking ${z} =0$ is essentially
equivalent to changing
the line element, in the causal
past of the collision center, by a related line element
obtained from (\ref{eq:ibI}) by setting ${z} =0$, i.e.,
\begin{equation}
d{\hat s}^2_{\rm I}=L_1L_2\left(d{t} ^2-d{z} ^2\right)
-\cos ^2{t}\, dx^2-dy^2.
\label{eq:dshatIb}
\end{equation}
We will suppose that the line element (\ref{eq:dshatIb})
applies all over the causal past of the collision center, not
only in the interaction region but also through the plane wave regions
II and III in the sense of Fig. 2.
The plane wave collision starts at $t=0$ but
to avoid smoothness problems derived from such an
approximation, we will suppose that (\ref{eq:dshatIb}) applies
exactly on a range $\epsilon <{t}<\pi /2$, for a certain
$\epsilon >0$. In the range
$0\leq {t}\leq\epsilon$, as described
below, we will interpolate a line element
which smoothly matches with the flat space at $t=0$.
Nevertheless, the details of this matching will not affect the main physical
features.
The
exact field equation for this approximate space-time is,
\begin{equation}
\left(\Box +\xi R\right)\phi =0,
\label{eq:Appfieldeq}\end{equation}
where it is necessary to consider a coupling curvature term in the field
equation because, although the exact space-time is a vacuum
solution, we have a bounded nonzero value for
$R$ in the approximated space-time.
In order to solve this new field equation, we start rewriting the line
element (\ref{eq:dshatIb}) in the following general way,
\begin{equation}
ds^2=(f_1f_2f_3)\, d{{t} ^*}^2-\left(f_1f_2\over f_3\right)\, d{z} ^2
-\left(f_2f_3\over f_1\right)\, dx^2-
\left(f_1f_3\over f_2\right)\, dy^2,
\label{eq:dsgen}
\end{equation}
where the $f_i$ are functions of coordinate ${t}$ alone, which
for values of $0<\epsilon <{t}<\pi /2$, can be
straightforwardly determined by direct comparison with
(\ref{eq:dshatIb}) as $f_1({t})=\sqrt{L_1L_2}$,
$f_2({t})=\sqrt{L_1L_2}\, \cos{t}$, $f_3({t})=\cos{t}$.
For values ${t}\leq 0$ we take $f_1({t})=f_2({t})=\sqrt{L_1L_2}$,
$f_3({t})=1$, which correspond to their values in flat
space. Finally, in
the interval $0\leq{t}\leq\epsilon$, we smoothly interpolate each $f_i({t})$
($i=1,2,3$) between these values.
Also,
in order to prevent singularities in the field
equation, we conveniently reparametrize coordinate ${t}$, by ${t}
^*({t})$, as follows,
\begin{equation}
{d{t} ^*\over d{t}}={1\over f_3({t})}.
\label{eq:dxi*dxi}\end{equation}
Now, we use the following ansatz for the field solutions,
\begin{equation}
\phi _k=h({t} ^*)\,{\rm e}^{ik_xx+ik_yy+ik_{z}{z}},
\label{eq:ansatz}\end{equation}
where the plane wave factor in coordinates $x$, $y$ is related to the
translational symmetry of the space-time along the transversal directions
$x$, $y$, and the plane wave factor in
coordinate ${z}$ is just a consequence of our approximation.
Then equation (\ref{eq:Appfieldeq})
directly leads to the following Schr\"odinger-like differential
equation for the function $h({t} ^*)$,
\begin{equation}
h_{,{t} ^*{t} ^*}+\omega ^2({t} )\, h=0,\;\;\;\;
V({t})\equiv\omega ^2({t} )=f_0^2({t} )+f_1^2({t} )\,
k_x^2+f_2^2({t} )\, k_y^2+
f_3^2({t} )\, k_z^2,
\label{eq:heq}\end{equation}
where the function $f_0({t})$ stands for,
\begin{equation}
f_0^2({t} )=\left[f_1({t} )f_2({t} )f_3({t} )\right]\,\xi R.
\label{eq:omegafi}\end{equation}
Such differential equation can be WKB solved, essentially because the
short wavelength condition holds, i.e. $\omega
^{-1}d/d{t}^*\ln\omega\ll 1$. Observe that this condition reduces to
$(d{t}/d{t}^*)\, dV/d{t}\ll 2\,\omega ^3$, which becomes
particularly accurate when the Killing-Cauchy horizon is approached
since in that case $d{t}/d{t}^*=f_3({t})\rightarrow 0$. Therefore, the mode
solutions
$\phi _k$ which reduce to the flat mode solutions in the region prior
to the arrival of the waves, are
\begin{equation}
\phi _k={{\hat\omega}^{1/2}\over\sqrt{(2\pi)^32k_-W({t} )}}
{\rm e}^{ik_xx+ik_yy+ik_3{z}-i\int ^{{t} ^*}W(\zeta )d\zeta ^*},
\label{eq:phifi}\end{equation}
where we denote ${\hat\omega}^2=k_1^2+k_2^2+k_3^2$
with $k_1=\sqrt{L_1L_2}\, k_x$, $k_2=\sqrt{L_1L_2}\, k_y$,
$k_3=k_z$ and where $W({t} )$ stands for an
adiabatic series in powers of the time-dependent frequency
$\omega ({t} )$ of the
modes and its derivatives. Up to adiabatic order four (i.e. up
to terms involving four derivatives of $\omega ({t})$)
$W({t})$ it is given by,
\begin{equation} W({t} )=\omega +{A_2\over\omega ^3} +{B_2\over\omega ^5}
+{A_4\over\omega ^5}
+{B_4\over\omega ^7} +{C_4\over\omega ^9} +{D_4\over\omega
^{11}},\label{eq:Wxi}\end{equation}
where, using the notation ${\dot V}\equiv dV/d{t}^*$,
\begin{equation} A_2=-{{\ddot V}\over 8},\;\;\;\; B_2={5\over 32}\,{{\dot
V}^2},\label{eq:An}\end{equation}
$$ A_4={{\buildrel{....}\over V}\over 32},\;\;\;\; B_4=-
{28\,{\dot V}\,{\buildrel{...}\over V}+19\,{\ddot
V}^2\over 128},\;\;\;\; C_4={221\over 258}\,{{\dot V}^2\,{\ddot V}},\;\;\;\;
D_4=-{1105\over
2048}\,{{\dot V}^4},$$
and
$A_n$, $B_n$, ... denote the $n$ adiabatic terms in
$W({t})$.
Up to adiabatic order zero it is simply
$W({t} )=\omega({t} )$. Observe the two following facts:
\noindent
(i) Near the horizon ${t}=\pi /2$ we have $W({t})\simeq\omega({t})$.
This is because the higher adiabatic corrections vanish at the horizon.
\noindent
(ii) In the flat region prior to the arrival of the waves we have
$W({t} )={\hat\omega} =\left(k_1^2+k_2^2+k_3^2\right)^{1/2}$. In that case,
since $f_3=1$, we can use (\ref{eq:dxi*dxi}) to set
$ {t} ^*={t}$, where without loss of generality we choose the
value ${t}^*=0$ at ${t=0}$.
Therefore, the mode solutions (\ref{eq:phifi}) in the flat region reduce to,
$$
\phi _k^{{\rm IV}}={1\over\sqrt{(2\pi)^32k_-}}{\rm
e}^{ik_xx+ik_yy+ik_{z}{z} -i{\hat\omega}{t}},
$$
which indeed are the flat mode solutions defined in (\ref{eq:phiII/III}c),
recalling that the new separation constant $k_{z}=k_3$ is related to the
original $k_\pm$ by the ordinary null momentum relations, i.e.,
\begin{equation}
{\hat\omega}={\hat k}_++{\hat k}_-,\;\;{k_{z}}={\hat k}_+-{\hat k}_-.
\label{eq:nullmom}\end{equation}
It is important to understand that we are constructing a set of mode
solutions as an adiabatic series in terms of derivatives of the
frequency $\omega ({t})$ in the differential equation
(\ref{eq:heq}). This procedure is similar but not equivalent to the
construction of an {\em adiabatic vacuum state} where the field modes
are expanded as an adiabatic series in terms of the derivatives of the
metric coefficients (see for example \cite{bir82} for details). In
fact, observe for instance that the term $f^2_0({t})$ in (\ref{eq:heq})
involves two derivatives of the metric since it is directly related to
the curvature scalar. Thus, it would be a second order term for
an eventual adiabatic vacuum, but it is simply a zeroth order term in
our adiabatic series in derivatives of $\omega ({t})$.
\section{Hadamard function in the interaction region}
The key ingredient to calculate the vacuum expectation value of the
stress-energy tensor is the {\em Hadamard function} $G^{(1)}(x,x')$,
which is defined as the vacuum expectation value of the anticommutator
of the field, i.e.,
\begin{equation} G^{(1)}(x,x')=\langle\{\phi (x),\phi (x')\}\rangle =\sum _k
\left\{u_{k}(x)\, u^*_{k}(x')+u_{k}(x')\,
u^*_{k}(x)\right\}. \label{eq:defHadamard}\end{equation}
This Hadamard function contains non-physical divergence terms which
can be subtracted by the following point-splitting prescription,
\begin{equation} G_B^{(1)}(x,x')=G^{(1)}(x,x')-S(x,x'),
\label{eq:GB}\end{equation}
where $S(x,x')$ is the {\em midpoint expansion} of a locally
constructed quantity commonly referred as
a {\em Hadamard elementary solution}
(see for example \cite{wal94}) and given by
\begin{eqnarray}
S(x,x')&=&{1\over 8\pi ^2}\left\{
-{2\over\sigma}
-2{\Delta ^{(2)}}_{{\bar\mu}{\bar\nu}}\,{\sigma ^{\bar\mu}\sigma
^{\bar\nu}\over\sigma}
-2{\Delta ^{(4)}}_{{\bar\mu}{\bar\nu}{\bar\rho}{\bar\tau}}
\,{\sigma ^{\bar\mu}\sigma ^{\bar\nu}\sigma ^{\bar\rho}\sigma
^{\bar\tau}\over\sigma}
-a_1^{(0)}\,\ln (\mu ^{-2}\sigma )
\right.\nonumber
\\
& & \left.
-\left[
\left(
a_1^{(0)}{\Delta ^{(2)}}_{{\bar\mu}{\bar\nu}}
+{a_1 ^{(2)}}_{{\bar\mu}{\bar\nu}}
\right)\sigma ^{\bar\mu}\sigma ^{\bar\nu}
-{1\over 2}a_2^{(0)}\sigma
\right]\,\ln (\mu ^{-2}\sigma )
-{3\over 4}a_2^{(0)}\sigma
\right\},
\label{eq:Hada1}\end{eqnarray}
where the coefficients ${\Delta ^{(2)}}_{{\bar\mu}{\bar\nu}}$,
${\Delta ^{(4)}}_{{\bar\mu}{\bar\nu}{\bar\rho}{\bar\tau}}$,
$a_1^{(0)}\cdots$ are written
in Appendix A. We use the standard definition for the {\em geodetic
biscalar} $\sigma (x,x')=(1/2)s^2(x,x')$, being $s(x,x')$ the proper
distance between the points $x$ and $x'$ on a non-null geodesic
connecting them. Also,
$\sigma _{\bar\mu} (x,x') =(\partial /\partial x^{\bar\mu})\sigma (x,x')$
is a geodesic tangent vector at the point $\bar x$ with modulus $s(x,x')$,
being $\bar x$ the {\em midpoint} between $x$ and $x'$ on the geodesic.
The parameter $\mu$ in the
logarithmic term of (\ref{eq:Hada1}) is an arbitrary length parameter,
which is related to the two-parameter ambiguity of the point-splitting
regularization scheme \cite{wal94}.
Then we can compute $\langle T_{\mu\nu} \rangle$ by means of the
following differential operation,
\begin{equation} \langle T_{\mu\nu}(x)\rangle =\lim _{x\rightarrow x'}\, {\cal
D}_{\mu\nu}G^{(1)}(x,x'), \label{eq:limDT}\end{equation}
where ${\cal D}_{\mu\nu}$ is a nonlocal differential operator, which
in the conformal coupling case ($\xi =1/6$) is given by,
\begin{eqnarray}{\cal D}_{\mu\nu}&=&
{1\over 6}\,\left(\nabla _{\mu '}\nabla _{\nu}+\nabla
_{\nu'}\nabla_{\mu}\right)
-{1\over 24}\, g_{\mu\nu}\,\left(
\nabla _{\alpha '}\nabla ^{\alpha}+
\nabla _{\alpha }\nabla ^{\alpha '}\right)-\nonumber\\
& &{1\over 12}\,\left(\nabla _{\mu }\nabla _{\nu}+\nabla
_{\mu'}\nabla_{\nu '}\right)
+ {1\over 48}\, g_{\mu\nu}\,\left(
\nabla _{\alpha }\nabla ^{\alpha }+
\nabla _{\alpha '}\nabla ^{\alpha '}\right)-\nonumber\\
& &{1\over 12}\,\left(R_{\mu\nu}-{1\over 4}R\, g_{\mu\nu}\right).
\label{eq:Dopdif}\end{eqnarray}
However, the above differential operation and its limit have no immediate
covariant meaning because
$G^{(1)}(x,x')$ is not an ordinary function but a {\it biscalar} and the
differential operator ${\cal
D}_{\mu\nu}$ is {\it nonlocal}; thus we need to deal with the nonlocal
formalism of {\it bitensors} (see, for example \cite{dew60,chr76} or the
Appendix B of reference \cite{dor96} for a review on
this subject).
The regularization procedure (\ref{eq:GB}), however,
fails to give a covariantly conserved
stress-energy tensor essentially because the locally constructed
Hadamard function (\ref{eq:Hada1}) is not in
general symmetric on the endpoints $x$ and $x'$ (i.e. it satisfies the
field equation at the point $x$ but fails to satisfy it at $x'$)
(see \cite{wal78} for details).
Thus, to ensure covariant conservation, we must introduce an additional
prescription:
\begin{equation} \langle T_{\mu\nu}(x)\rangle =\langle T_{\mu\nu}^B (x)\rangle
- {a^{(0)}_2(x)\over 64\pi ^2}\, g_{\mu\nu}. \label{eq:GBT}\end{equation}
Note that this last term is responsible for the trace anomaly in the conformal
coupling case, because even though
$\langle T_{\mu\nu}^B (x)\rangle$ has null trace when $\xi =1/6$, the trace of
$\langle T_{\mu\nu}(x)\rangle$
is given by
$\langle T^{\mu}_{\mu}\rangle =
- {a^{(0)}_2(x)/(16\pi ^2)}$.
The regularization prescription just given in (\ref{eq:GBT})
satisfies the well known four
Wald's axioms
\cite{wal94,wal76,wal77-78b,chr75}, a set of properties that any physically
reasonable expectation value of the
stress-energy tensor of a quantum field should satisfy. There is still an
ambiguity in this prescription since
two independent conserved local curvature terms, which are quadratic in the
curvature, can be added to this
stress-energy tensor. In particular, the $\mu$-parameter ambiguity in
(\ref{eq:Hada1}) is a consequence of this (see \cite{wal94} for details).
Such a two-parameter ambiguity, however, cannot be
resolved within the limits of the
semiclassical theory, it may be resolved in a complete quantum theory
of gravity
\cite{wal94}. Note, however, that in some sense this ambiguity does not affect
the knowledge of the matter
distribution because a tensor of this kind belongs properly to the left hand
side of Einstein equations, i.e.
to the geometry rather than to the matter distribution.
After this preliminary introduction on the point-splitting regularization
technique, we may proceed to calculate the Hadamard function $G^{(1)}(x,x')$
in the interaction region for the initial vacuum state
defined by the modes $\phi _k$, (\ref{eq:phifi}). The
Hadamard function can be written as,
\begin{equation} G^{(1)}(x,x')=\sum _k\phi
_k(x)\,\phi ^*_k(x')\; +{c.c.}
\label{eq:G(1)}\end{equation}
Note that solutions $\phi _k$ contain the function $h({t} ^*)$, which
cannot be calculated analytically but may be approximated
up to any adiabatic order as described in (\ref{eq:Wxi})-(\ref{eq:An}).
Thus, we have the
inherent ambiguity of where to
cut the adiabatic series. In fact, this is an asymptotic expansion,
which has a well stablished ultraviolet limit but it may have convergence
problems in the low-energy limit.
However, observe from (\ref{eq:dxi*dxi}) and (\ref{eq:heq})
that since $dt/dt^*\rightarrow 0$ and
$V({t})=\omega ^2({t})\rightarrow k_1^2$
towards the horizon, the adiabatic series (\ref{eq:Wxi})
reduces to $W\simeq\omega$ near the horizon. This means that we could cut the
adiabatic series (\ref{eq:Wxi}) at order zero if we were
interested in a calculation near the horizon. However, this is only
partially true. In fact, it would be true if we were only interested in the
particle production problem \cite{fei95} but it is not
sufficient for the calculation of the vacuum expectation value of the
stress-energy tensor.
This is because
$G^{(1)}$ calculated with $h({t}^*)$
at order zero does not reproduce the short-distance singular
structure of a Hadamard elementary
solution (\ref{eq:Hada1}) in the coincidence limit $x\rightarrow x'$. The
smallest adiabatic
order for the function $h({t}^*)$
which we need to recover the singular structure of $G^{(1)}$ is order
four, basically because our adiabatic construction of the mode
solutions is similar (but not equivalent) to an {\em adiabatic vacuum
state} (see \cite{bir82} for details).
Although expanding the function $h({t}^*)$ in (\ref{eq:G(1)}) up to
adiabatic order four will give an accurate value for
the stress-energy tensor near the horizon, it will also give a suitable
approximate value for this tensor
all over the causal past of the collision center
(region $\cal S$ in Fig. 2). The reason is that even though the
short-wavelength condition, i.e.
$\omega ^{-1}d/d{t}^*\ln\omega\ll 1$, is particularly accurate near the
horizon it also holds throughout region $\cal S$.
In the mode sum (\ref{eq:G(1)}) we use the shortened notation
$\sum
_k\equiv
\int ^{\infty}_{0}{dk_-/ k_-}\,
\int ^{\infty}_{-\infty}dk_x\,
\int ^{\infty}_{-\infty}dk_y$ or equivalently
$\sum _k\equiv
(L_1L_2)^{-1}
\int ^{\infty}_{-\infty}dk_1\,
\int ^{\infty}_{-\infty}dk_2\,
\int ^{\infty}_{-\infty}{dk_3/ {\hat\omega}}$,
where the change of variables
(\ref{eq:nullmom}) and the usual notation (\ref{eq:kiL}) have been used.
Therefore we have,
\begin{eqnarray} G^{(1)}(x,x')&=& {1\over 2(2\pi)^3\, L_1L_2}\,
\int ^{\infty}_{-\infty}\int ^{\infty}_{-\infty}\int
^{\infty}_{-\infty}
{dk_1\, dk_2\, dk_3\over \sqrt{W({t})W({t} ')}}\,\times \nonumber\\
& &{\rm e}^{-i\int _{{t^*} '}^{{t^*}}W({\zeta})d{\zeta}^*+ik_x
(x-x')+ik_y (y-y')+
ik_{z} ({z} -{z} ')}\;\; +c.c.
\label{eq:G(1)2}\end{eqnarray}
We assume that the points
$x$ and $x'$ are connected by a
non-null geodesic in such a way that they are at the same proper distance
$\epsilon$ from a third midpoint
$\bar x$. We parametrize the geodesic by its proper distance $\tau$ and
with abuse of notation we denote
the end points by
$x$ and $x'$, which should not be confused with the third
component of $({t} ,\;{z} ,\; x,\; y)$.
Then we expand the integrand
function in powers of $\epsilon$ and we finally integrate term by term
to get an expression up to $\epsilon ^2$. The details of such a tedious
calculation can be found in ref. \cite{dor97}. The result is,
\begin{eqnarray} G^{(1)}(x,x')&=&{\bar A}+\sigma\,{\bar b}
+{C}_{{\bar \alpha}{\bar \beta}}\,{\sigma}^{\bar \alpha}{\sigma}^{\bar\beta}
+{D}_{{\bar \alpha}{\bar \beta}{\bar \gamma}{\bar \delta}}\,
{\sigma}^{\bar \alpha}{\sigma}^{\bar\beta}{\sigma}^{\bar
\gamma}{\sigma}^{\bar\delta}
+{1\over 8\pi ^2}\left\{
-{2\over\sigma}
-2{\Delta ^{(2)}}_{{\bar\mu}{\bar\nu}}\,{\sigma ^{\bar\mu}\sigma
^{\bar\nu}\over\sigma}
\right.
\label{eq:Hada2}\\
& & \left.
-2{\Delta ^{(4)}}_{{\bar\mu}{\bar\nu}{\bar\rho}{\bar\tau}}
\,{\sigma ^{\bar\mu}\sigma ^{\bar\nu}\sigma ^{\bar\rho}\sigma
^{\bar\tau}\over\sigma}
-a_1^{(0)}\, L
-\left[
\left(
a_1^{(0)}{\Delta ^{(2)}}_{{\bar\mu}{\bar\nu}}
+{a_1 ^{(2)}}_{{\bar\mu}{\bar\nu}}
\right)\sigma ^{\bar\mu}\sigma ^{\bar\nu}
-{1\over 2}a_2^{(0)}\sigma
\right]\, L
\right\}
\nonumber\end{eqnarray}
where $L$ is a logarithmic term defined as
$L=2\gamma+\ln (\sigma\,\xi R/2)$,
being $\gamma$ Euler's constant and where all the involved
coefficients
$\bar A$, $\bar b$, ${C}_{{\bar \alpha}{\bar \beta}}$... are given in
Appendix A.
According to (\ref{eq:GB}), the Hadamard function can be regularized using the
elementary
Hadamard solution (\ref{eq:Hada1}) and finally the regularized expression for
$G^{(1)}(x,x')$ up to order $\epsilon ^2$ is,
\begin{eqnarray} G^{(1)}_B(x,x')&=&{\bar A}+\sigma\,{\bar B}
+{C}_{{\bar \alpha}{\bar \beta}}\,{\sigma}^{\bar \alpha}{\sigma}^{\bar \beta}
+{D}_{{\bar \alpha}{\bar \beta}{\bar \gamma}{\bar \delta}}\,
{\sigma}^{\bar \alpha}{\sigma}^{\bar\beta}{\sigma}^{\bar
\gamma}{\sigma}^{\bar\delta}
\nonumber\\
& &+{1\over 8\pi ^2}\left\{
-a_1^{(0)}\, {\hat L}
-\left[
\left(
a_1^{(0)}{\Delta ^{(2)}}_{{\bar\mu}{\bar\nu}}
+{a_1 ^{(2)}}_{{\bar\mu}{\bar\nu}}
\right)\sigma ^{\bar\mu}\sigma ^{\bar\nu}
-{1\over 2}a_2^{(0)}\sigma
\right]\, {\hat L}
\right\},
\label{eq:G(1)fR}\end{eqnarray}
where ${\hat L}$ is a bounded logarithmic term given by,
${\hat L}=2\gamma+\ln (\mu ^{2}\,\xi R/2)$, $\mu$ being the arbitrary
length parameter introduced in (\ref{eq:Hada1}),
and where the coefficient $\bar B$ is
${\bar B}=b+3\, a_2^{(0)}/(32\pi ^2)$, which is given also in Appendix A.
From (\ref{eq:G(1)fR}) we can directly read off the regularized mean
square field in
the ``in" vacuum state as
$\langle\phi ^2\rangle ={\bar A}/2-a_1^{(0)}{\hat L}/(16\pi ^2)$.
It is important to remark, however, that the term
${D}_{{\bar \alpha}{\bar \beta}{\bar \gamma}{\bar \delta}}$ in
(\ref{eq:G(1)fR})
appears only as a consequence of our approximate procedure of
calculating the Hadamard function, i.e. using an adiabatic order four
expansion for the initial modes in powers of the
mode frequency $\omega ({t})$ and its derivatives.
Had we used an exact expression for the
initial modes (or an adiabatic vacuum state \cite{bir82}), such
a term would not appear.
\section{Expectation value of the stress-energy tensor}
To calculate the vacuum expectation value of the stress-energy
tensor we have to apply the differential operator
(\ref{eq:Dopdif}) to
(\ref{eq:G(1)fR}). As we have already pointed out, this is not straightforward
because we
work with nonlocal quantities. Note first that the operator (\ref{eq:Dopdif})
acts on bitensors which depend on the end points
$x$ and $x'$, but the expression (\ref{eq:G(1)fR}) for $G^{(1)}_B$ depends on
the midpoint $\bar x$. This means
that we need to covariantly expand (\ref{eq:G(1)fR}) in terms of
the endpoints $x$ and $x'$. Also, the presence of quartic $\sigma ^\mu$
terms in (\ref{eq:G(1)fR}) gives, after differentiation, path dependent
terms which must be conveniently averaged.
The details of such a calculation
may be found for instance in \cite{dor96,dor97}.
Then, in
the orthonormal basis
$\theta _1={g^{1/2}_{{t}{t}}}\,d{t}$,
$\theta _2={g^{1/2}_{{z}{z}}}\, d{z}$, $\theta _3={g^{1/2}_{xx}}\, dx$,
$\theta _4={g^{1/2}_{yy}}\, dy$, using the trace anomaly
prescription (\ref{eq:GBT}), we obtain
the following expectation values $\langle T_{\mu\nu}\rangle$ in
the conformal coupling case and for values
$0<\epsilon<{t}<\pi /2$ of coordinate ${t}$
\begin{eqnarray}\langle T_{\mu\nu}\rangle &=&
{2\gamma+\ln (\mu ^{2}\, R/12)\over 2880\, (L_1L_2)^2\,
\pi ^2}\,{\rm diag}(-1,\, -1,\, 1,\, -1)
+{1\over 4}\langle T^{\tau}_{\tau}\rangle g_{\mu\nu}+
\nonumber
\\
&&
{\rm
diag}\left(\rho _1({t}) ,\; -\rho _2({t}) ,\; \rho _1({t})+2\rho
_2({t}),\; -\rho _2({t})\right),
\label{eq:TMNconforme}\end{eqnarray}
where we have used for simplicity the
notation $z=\sin{t}$.
The trace anomaly term in that case is given
by (\ref{eq:GBT}) as,
$${1\over 4}\langle T^{\tau}_{\tau}\rangle g_{\mu\nu} =
-{a_2\over 64\,\pi ^2}\, g_{\mu\nu}=
-{{\rm diag}\left(1 ,\, -1 ,\, -1,\, -1\right)
\over 2880\, (L_1L_2)^2\, \pi ^2},
$$
and the functions $\rho _1({t})$ and $\rho _2({t})$
are given by,
\begin{equation}\rho _1({t}) ={-247247+1456234\, z^2+792789\, z^4\over
69189120\, (L_1L_2)^2\,\pi ^2\,(1-z)^2(1+z)^2},\label{eq:rho1}\end{equation}
\begin{equation}\rho _2({t}) ={502931+179686\, z^2+923887\, z^4\over
69189120\, (L_1L_2)^2\, \pi ^2\,(1-z)^2(1+z)^2}.\label{eq:rho2}\end{equation}
Recall that $\rho _1({t})$ is a positive definite function in an
interval $0<\epsilon <{t}<\pi /2$. Both functions are unbounded at the horizon
$({t}=\pi /2)$, and the expectation value of the stress-energy tensor
near the horizon is approximately given by,
\begin{equation}\left.\langle T_{\mu\nu}\rangle\right|_{{t}\simeq\pi /2} ={\rm
diag}\left(\rho ({t}) ,\; -\rho ({t}) ,\; 3\rho ({t})
,\; -\rho ({t})\right)+
{1\over 4}\langle T^{\tau}_{\tau}\rangle g_{\mu\nu},
\label{eq:TMapprox}\end{equation}
where $\rho ({t})=\Lambda (L_1L_2)^{-2}\cos{t}^{-4}$ and $\Lambda\simeq
0.0029$. The behavior of $\langle T_{\mu\nu}\rangle$ entirely agrees
with the previous result \cite{dor97}.
For values of $t\leq 0$, $\langle T_{\mu\nu}\rangle =0$,
and to be consistent with the approximation
we have used for the space-time geometry, we should
require that the value of $\langle T_{\mu\nu}\rangle$
(\ref{eq:TMNconforme}), which is valid for $0<\epsilon <{t}<\pi /2$, goes
smoothly to zero as ${t}\rightarrow 0$. In fact, this can be achieved
using an adequate matching of the line element (\ref{eq:dshatIb}) with the
flat line element through the interval $0\leq{t}\leq\epsilon$.
Observe that the logarithmic term in the stress-energy
tensor (\ref{eq:TMNconforme}) appears as
a consequence of a similar term in the Hadamard function
(\ref{eq:G(1)fR}). The argument of this
logarithm depends on the curvature scalar
and thus it will grow unbounded as the flat region is approached.
However, the coefficient that will appear in front of such a logarithm
in the Hadamard function (\ref{eq:G(1)fR}),
depends only on locally constructed curvature terms. Therefore,
with an adequate matching of the space-time
geometry, this coefficient will also
smoothly vanish towards the flat space region, below $t=0$, and it
will not give a contribution to the stress-energy tensor.
The details of such a matching, however, will not affect
the main features of the stress-energy tensor (\ref{eq:TMNconforme}),
particularly when the Killing-Cauchy horizon is approached.
We must recall, however, that
although the value (\ref{eq:TMNconforme}) for
$\langle T_{\mu\nu}\rangle$ satisfies asymptotically
the conservation equation near
the Killing-Cauchy horizon, it does not satisfy exactly the conservation
equation throughout region $\cal S$, essentially
because it is obtained by means of an
approximation in the field modes. Nevertheless, we could obtain a
truly conserved $\langle T_{\mu\nu}\rangle$, in the context of the
present approximation, by solving the conservation equation
considering a $\langle {T^{\mu}}_{\nu}\rangle$ given by
the following set of non-null components
$\{\langle T^t_t({t})\rangle ,\, \langle T^z_z({t})\rangle ,\,
\langle T^x_x({t})\rangle ,\, \langle T^y_y({t})\rangle\}$ with the
conditions:
i) $\langle T^y_y({t})\rangle =\langle T^z_z({t})\rangle$,
which is compatible with
(\ref{eq:TMNconforme}) and it is a physical consequence of the isotropy
of the metric (\ref{eq:dshatIb}) along the $y$-$z$ directions,
ii) trace anomaly condition, i.e.
$\langle T^x_x({t})\rangle =\langle T^{\mu}_{\mu}\rangle
-\langle T^t_t({t})\rangle -2\, \langle T^z_z({t})\rangle$,
iii) the ansatz $\langle T^t_t({t})\rangle =\rho _1({t})$,
which is the approximate value of $\langle T^t_t({t})\rangle $ obtained in
our calculation. Finally, the conservation equation gives
straightforwardly values for
$\langle T^z_z({t})\rangle$ and $\langle T^x_x({t})\rangle$
which are compatible with the values
$\langle T^z_z({t})\rangle =\rho _2({t})$ and
$\langle T^x_x({t})\rangle =-\rho _1({t})-2\rho _2({t})$ obtained in our
approximation. In particular, they have the same behaviour near the
Killing-Cauchy horizon.
Inspection of (\ref{eq:TMNconforme}) shows that not only is
the {\em weak energy
condition} satisfied \cite{wal84}, which means that the energy density is
nonnegative for any observer, but also
the {\em strong energy condition} is satisfied.
\section{Conclusions}
We have calculated the expectation value of the stress-energy tensor
of a massless scalar field in a space-time representing the head on
collision of two electromagnetic plane waves throughout the causal
past of the collision center and in the field state which
corresponds to the physical vacuum state before the collision takes
place. We have performed the calculations in this particular region
essentially because, following the directions of a previous work
\cite{dor97}, we could introduce a suitable approximation to the
space-time metric (see Fig. 2). This approximation not only has allowed us to
dramatically simplify the calculations but also to keep unchanged the
main physical features, in particular the behavior of the
stress-energy tensor near the Killing-Cauchy horizon of the
interaction region. In fact, such an approximation is also valid for more
generic plane wave space-times, and this will be the subject of a
forthcoming paper.
The results we have obtained are
entirely compatible with the previous result \cite{dor97}
and they may be briefly described as follows: before the collision
of the waves $\langle T_{\mu\nu}\rangle =0$, which correspond to the
lower edge Fig. 2. Then, after the collision the value of
$\langle T_{\mu\nu}\rangle$ starts to increase until it grows
unbounded towards the Killing-Cauchy horizon of the interaction
region. The
weak energy condition is satisfied, the rest energy density is positive and
diverges as $\cos ^{-4}{t}$. Two of
the principal pressures are negative and of the same order of magnitude
of the energy density. The
strong energy condition is also satisfied,
$\langle T^{\mu}_{\mu}\rangle$ is finite but $\langle
T^{\mu\nu}\rangle \langle T_{\mu\nu}\rangle$ diverges at the horizon and we
may
use ref \cite{hel93} on the stability of Cauchy horizons to argue that the
horizon will acquire by backreaction
a curvature singularity.
Thus, contrary to simple plane waves,
which do not polarize the vacuum \cite{des75,gib75}, the nonlinear collision
of these waves polarize the vacuum
and the focusing effect that the waves exert seems to produce,
in general, an unbounded positive energy density at the focusing points.
Therefore, when the colliding waves produce a Killing-Cauchy horizon, that
horizon may be, in general, unstable by vacuum
polarization.
In the more generic case when the wave collision produces a spacelike
singularity it seems clear that the
vacuum expectation value of the stress-energy tensor will also grow unbounded
near the singularity. In fact, in a forthcoming paper we will extend
the approximation introduced in the present work to a more generic
plane wave spacetime with the objective of more generally proving that the
negative pressures associated to the quantum fields could not prevent
the formation of the singularity.
\vskip 1.25 truecm
{\Large{\bf Acknowledgements}}
\vskip 0.5 truecm
\noindent
I am grateful to R. M. Wald, R. Geroch, E. Verdaguer, A. Campos,
E. Calzetta, A. Feinstein,
J. Iba{\~n}ez and A. Van Tonder for helpful
discussions. I am also grateful to the Physics Department of Brown
University for their hospitality and to the Grup de F\'{\i}sica
Te\`orica (IFAE) de l'Universitat Aut\`onoma de Barcelona.
This work has been partially supported by
NSF grant PHY 95-14726 to The University of Chicago and by the Direcci\'o
General de Recerca de la
Generalitat de Catalunya through the grant 1995BEAI300165.
|
3,212,635,537,936 | arxiv | \section{Introduction}
The {\it Kepler} mission (NASA) has detected thousands of planet candidates and has provided the opportunity to study planets statistically as a class of astrophysical objects.
{\it Kepler} has indeed provided clues about the occurrence rate and physical properties of small planets with orbital periods up to about 500 days (e.g., Marcy et al. 2014a,b).
\par
Several studies have been dedicated to the investigation of the physical properties of exoplanets (e.g., Lissauer et al. 2011; Lopez et al. 2012; Marcy et al. 2013; Rogers 2015) and of the relations among their physical and orbital properties (e.g., Fischer \& Valenti 2005; Mazeh \& Zucker 2003 ; Guillot et al. 2006; Burrows et al. 2007; Mayor et al. 2011; Miller \& Fortney 2011; Buchhave et al. 2012; Howard et al. 2012).
In this short Letter we investigate the
correlation between
planetary radius and orbital period for the {\it Kepler} candidates.
We
focus on planets with radii smaller than 4 $R_{\oplus}$, i.e., Neptune-size planets and smaller, planets that are now known to be most common around other stars.
\section{Statistical Analysis}
We use all the {\it Kepler} candidates with radii up to 4 $R_{\oplus}$ and with orbital periods between 0.5 and 500 days. The choice of 4 $R_{\oplus}$ is based on the standard division of small planets and giant planets (e.g., Weiss \& Marcy 2014; Marcy et al. 2014b). We choose this orbital period range in order to include most of the data.
Figure 1 shows the planetary radius ($R_p$) as a function of orbital
period ($P$) for our sample. The plot seems to suggest that there is a correlation between
the two parameters, and indeed, the correlation coefficient between $\log R_p$ and $\log P$ turns out to
be 0.5120. However, it is naturally harder to detect planets with
small radii at longer orbital periods,
which is a selection effect that already introduces some
correlation. Therefore, we must make sure that the correlation is not
simply caused by this selection effect. In order to quantify this influence, we use the completeness values for {\it Kepler} planet
detection, as derived by Silburt et al.~(2015, see their Figure 3). The
curve corresponding to a completeness value of
$80\%$ is also shown in the Figure. Applying this completeness criterion leaves us
with a sample of 2,955 planets (the blue dots above the sloped line in the Figure, henceforth the 'full sample').
In order to quantify the statistical significance of the correlation
we perform a bootstrap test (e.g., Efron \& Tibshirani 1993) in which we randomly draw a new sample
of $(P,R_p)$ pairs from the two separate samples of $R_p$ and $P$.
We exclude pairs which do not meet the completeness criterion
and calculate the correlation coefficient once this random sample
reaches the size of the original (full) sample.
This procedure leaves the marginal distributions of $R_p$ and $P$
essentially unchanged, it correctly accounts for the effect of the completeness criterion, and it ruins any residual
correlation that is not caused by the incompleteness.
We repeat this resampling procedure $10^6$ times and in this
way obtain the null distribution of the correlation coefficient, against which
we can test the hypothesis of the existence of correlation.
\par
Out of $10^6$ random resamplings, none of the cases yields a correlation coefficient higher than that obtained for the true data.
Figure 2 presents the distribution of the resampled correlation coefficients. This distribution is centred around a positive value of $\sim0.18$, and not zero, which reflects the effect of the completeness criterion.
The arrow in the plot represents the correlation coefficient obtained for the true data (0.5120). Since none of the random resamplings produced a value higher than the actual correlation, we can conclude that the correlation is statistically significance with a $P$ value smaller than $10^{-6}$.
As can be seen from Figure 1, the {\it Kepler} candidates are not distributed uniformly in the radius-period diagram, and most of them are concentrated in the region between $\sim3 $ and 100 days. To understand which region dominates the correlation we further divide the sample into three regions that exhibit somewhat different behaviors: (I) candidates with short periods of 0.5-3 days (II) candidates with intermediate periods, between 3 and 100 days, and (III) long-period candidates with periods between 100 and 500 days. The different regions are delineated in Figure 1. We repeat the analysis for each region separately including the bootstrap procedure.
Table~1 summarizes the derived correlation coefficients and corresponding $P$ values.
Region (II) seems to dominate the correlation, and indeed Figure 1 qualitatively suggests a slope of ~0.5--0.6 in the log-log plane. However, a close examination of Figure 1 suggests an additional contributing factor, which is the lack of large planets in short orbital periods in region (I).
We therefore divide region (I) into two sub-regions: (Ia) planets smaller than 2 $R_\oplus$; (Ib) planets larger than 2 $R_\oplus$.
There are 348 planets in region (Ia) and 62 in region (Ib). The areas of the two regions in the $\log P$-$\log R_p$ plane are 0.3847 (Ia) and 0.3010 (Ib).
Assuming, as our null hypothesis, a uniform distribution of the planets in this plane, the number of planets in (Ia) should follow a binomial distribution with $N=410$ and $p=0.5610$.
Under this null hypothesis the probability to obtain a number of 348 (or more) in region (Ia), is around $2\times10^{-36}$ -- undisputedly significant.
\section{Discussion}
Our analysis and its results suggest that there is a correlation
between the planetary radii and the orbital periods for
planets that are smaller than 4 $R_{\oplus}$. If indeed true, this correlation implies
that larger planets are more likely to exist at larger
radial distances.
The correlation follows from two contributing factors. The first is a correlation
between $R_p$ and $P$ for intermediate-size planets. Figure 1 suggests
a power-law relation of $R_p \sim P^{0.5-0.6}$. The second
contributing factor is a depletion of large planets with short periods.
This depletion has been noticed already by different authors (e.g., Ikoma \& Hori 2012; Owen \& Wu 2013; Lopez et al. 2013; Ciardi et al. 2013; Wu \& Lithwick 2013) and is most likely a signature of photo-evaporation of close-in
planets with atmospheres.
This process naturally results in naked cores and small radii for planets that orbit very close to their stars.
\par
It should be noted, that the analysis we present here does not consider the uncertainties of the radii and periods. While the errors of the periods are usually negligible, one might still wonder whether the relatively large errors of the radii could affect our findings. This could have been the case, if there were any correlation between those errors and the periods. We have calculated this correlation (not shown) and reassuringly found out it was negligible (less than 0.001). As a result, we can conclude that the correlation is robust and should not be affected by the measurement uncertainties. Future studies that account for a different sample of planets and perform a different statistical analysis could lead to a different value of the correlation coefficient but the correlation is expected to persist.
Theoretical investigations are required to fully understand
the correlation and how it can constrain planet formation theories and
the evolution history of planetary systems.
Future data from space
missions dedicated to detect transiting exoplanets such as {\it TESS,
CHEOPS} and {\it PLATO 2.0} and also ground-based observations,
can be used to confirm and sharpen our results.
\subsection*{Acknowledgments}
We thank the anonymous referee for valuable comments that helped to significantly improve this paper. R.~H.~acknowledges support from the Israel Space Agency under grant 3-11485.
\renewcommand{\baselinestretch}{.9}
|
3,212,635,537,937 | arxiv | \section{Introduction}
The concept of metric dimension was first studied in the context of navigation
system in various graphical networks \cite{HararyVertex}. There the robot
moves from one vertex of the network to another, and some of the vertices are
considered to be a landmark which helps a robot to establish its position in a
network. Then the problem of establishing the smallest set of landmarks in a
network becomes a problem of determining a smallest metric generator in a
graph \cite{KhullerVertex}.
Another interesting application is in chemistry where the structure of a
chemical compound is frequently viewed as a set of functional groups arrayed
on a substructure. This can be modeled as a labeled graph where the vertex and
edge labels specify the atom and bond types, respectively, and the functional
groups and substructure are simply subgraphs of the labeled graph
representation. Determining the pharmacological activities related to the
feature of compounds relies on the investigation of the same functional groups
for two different compounds at the same point \cite{ChartrandVertex}. Various
other aspects of the notion were studied \cite{BuczkowskiVertex, FehrVertex,
KleinVertex, MelterVertex} and a lot of research was dedicated to the
behaviour of metric dimension with respect to various graph operations
\cite{CaceresVertex, ChartrandVertex, SaputroVertex, YeroCoronaVertex}.
In this paper, we consider only simple and connected graphs. By $d(u,v)$ we
denote the distance between a pair of vertices $u$ and $v$ in a graph $G$. A
vertex $s$ from $G$ \emph{distinguishes} or \emph{resolves} a pair of vertices
$u$ and $v$ from $G$ if $d(s,u)\not =d(s,v).$ We say that a set of vertices
$S\subseteq V(G)$ is a \emph{vertex metric generator,} if every pair of
vertices in $G$ is distinguished by at least one vertex from $S.$ The
\emph{vertex metric dimension} of $G,$ denoted by $\mathrm{dim}(G),$ is the
cardinality of a smallest vertex generator in $G$. This variant of metric
dimension, as it was introduced first, is sometimes called only metric
dimension and the prefix "vertex" is omitted.
In \cite{TratnikEdge} it was noticed that there are graphs in which none of
the smallest metric generators distinguishes all pairs of edges, so this was
the motivation to introduce the notion of the edge metric generator and
dimension, particularly to study the relation between $\mathrm{dim}(G)$ and
$\mathrm{edim}(G).$
The distance $d(u,vw)$ between a vertex $u$ and an edge $vw$ in a graph $G$ is
defined by $d(u,vw)=\min\{d(u,v),d(u,w)\}.$ Recently, two more variants of
metric dimension were introduced, namely the edge metric dimension and the
mixed metric dimension of a graph $G.$ Similarly as above, a vertex $s\in
V(G)$ \emph{distinguishes} two edges $e,f\in E(G)$ if $d(s,e)\neq d(s,f).$ So,
a set $S\subseteq V(G)$ is an \emph{edge metric generator} if every pair of
vertices is distinguished by at least one vertex from $S,$ and the cardinality
of a smallest such set is called the \emph{edge metric dimension} and denoted
by $\mathrm{edim}(G).$ Finally, a set $S\subseteq V(G)$ is a \emph{mixed
metric generator} if it distinguishes all pairs from $V(G)\cup E(G),$ and the
\emph{mixed metric dimension}, denoted by $\mathrm{mdim}(G)$, is defined as
the cardinality of a smallest such set in $G$.
This new variant also attracted a lot of attention \cite{GenesonEdge,
HuangApproximationEdge, PeterinEdge, ZhangGaoEdge, ZhuEdge, ZubrilinaEdge},
with one particular direction of research being the study of unicyclic graphs
and the relation of the two dimensions on them \cite{Knor, SedSkreBounds,
SedSkreUnicyclic}. The mixed metric dimension is then a natural next step, as
it unifies these two concepts. It was introduced in \cite{KelencMixed} and
further studied in \cite{SedSkrekMixed, SedSkreTheta}. A wider and systematic
introduction to these three variants of metric dimension can be found in
\cite{KelPhD}.
In this paper we establish the vertex and the edge metric dimension of cactus
graph, using the approach from \cite{SedSkreUnicyclic} where the two
dimensions were established for unicyclic graphs. The extension is not
straightforward, as in cactus graphs a problem with indistinguishable pairs of
edges and vertices may arise from connecting two cycles, so additional
condition will have to be introduced.
\section{Preliminaries}
A \emph{cactus} graph is any graph in which all cycles are pairwise edge
disjoint. Let $G$ be a cactus graph with cycles $C_{1},\ldots,C_{c}$ and let
$g_{i}$ denote the length of a cycle $C_{i}$ in $G.$ For a vertex $v$ of a
cycle $C_{i},$ denote by $T_{v}(C_{i})$ the connected component of
$G-E(C_{i})$ which contains $v.$ If $G$ is a unicyclic graph, then
$T_{v}(C_{i})$ is a tree, otherwise $T_{v}(C_{i})$ may contain a cycle. When
no confusion arises from that, we will use the abbreviated notation $T_{v}.$ A
\emph{thread} hanging at a vertex $v\in V(G)$ of degree $\geq3$ is any path
$u_{1}u_{2}\cdots u_{k}$ such that $u_{1}$ is a leaf, $u_{2},\ldots,u_{k}$ are
of degree $2,$ and $u_{k}$ is connected to $v$ by an edge. The number of
threads hanging at $v$ is denoted by $\ell(v).$
We say that a vertex $v\in V(C_{i})$ is \emph{branch-active} if $\deg(v)\geq4$
or $T_{v}$ contains a vertex of degree $\geq3$ distinct from $v$. We denote
the number of branch-active vertices on $C_{i}$ by $b(C_{i}).$ If a vertex $v$
from a cycle $C_{i}$ is branch-active, then $T_{v}$ contains both a pair of
vertices and a pair of edges which are not distinguished by any vertex outside
$T_{v}$, see Figure \ref{Fig_branching}.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.9]{Figure01.pdf}
\end{center}
\caption{A cactus graph with two cycles. On the cycle $C$ vertices $v$ and $w$
are branch-active, and a pair of vertices is marked in $T_{v}$ and $T_{w}$
which is not distinguished by any vertex outside $T_{v}$ and $T_{w}$,
respectively.}%
\label{Fig_branching}%
\end{figure}
Now, we will introduce a property called "branch-resolving" which a set of
vertices $S\subseteq V(G)$ must possess in order to avoid this problem of non
distinguished vertices (resp. edges) due to branching. First, a thread hanging
at a vertex $v$ of degree $\geq3$ is $S$\emph{-free} if it does not contain a
vertex from $S.$ Now, a set of vertices $S\subseteq V(G)$ is
\emph{branch-resolving} if at most one $S$-free thread is hanging at every
vertex $v\in V(G)$ of degree $\geq3$. Therefore, for every branch-resolving
set $S$ it holds that $\left\vert S\right\vert \geq L(G)$ where
\[
L(G)=\sum_{v\in V(G),\ell(v)>1}(\ell(v)-1).
\]
It is known in literature \cite{TratnikEdge, KhullerVertex} that for a tree
$T$ it holds that $\mathrm{dim}(G)=\mathrm{edim}(G)=L(G).$
Also, given a set of vertices $S\subseteq V(G),$ we say that a vertex $v\in
V(C_{i})$ is $S$\emph{-active} if $T_{v}$ contains a vertex from $S.$ The
number of $S$-active vertices on a cycle $C_{i}$ is denoted by $a_{S}(C_{i}).$
If $a_{S}(C_{i})\geq2$ for every cycle $C_{i}$ in $G,$ then we say the set $S$
is \emph{biactive}. For a biactive branch-resolving set $S$ the following
holds: if a vertex $v$ from a cycle $C_{i}$ is branch-active, then $T_{v}$
contains a vertex with two threads hanging at it or $T_{v}$ contains a cycle,
either way $T_{v}$ contains a vertex from $S,$ so $v$ is $S$-active.
Therefore, for a biactive branch-resolving set $S$ we have $a_{S}(C_{i})\geq
b(C_{i})$ for every $i$.
\begin{lemma}
\label{Lemma_biactive_branchResolving}Let $G$ be a cactus graph and let
$S\subseteq V(G)$ be a set of vertices in $G.$ If $S$ is a vertex (resp. an
edge) metric generator, then $S$ is a biactive branch-resolving set.
\end{lemma}
\begin{proof}
Suppose to the contrary that a vertex (resp. an edge) metric generator $S$ is
not a biactive branch-resolving set. If $S$ is not branch-resolving, then
there exists a vertex $v$ of degree $\geq3$ and two threads hanging at $v$
which do not contain a vertex from $S.$ Let $v_{1}$ and $v_{2}$ be two
neighbors of $v,$ each belonging to one of these two threads. Then $v_{1}$ and
$v_{2}$ (resp. $v_{1}v$ and $v_{2}v$) are not distinguished by $S,$ a contradiction.
Assume now that $S$ is not biactive. We may assume that $G$ has at least one
cycle, otherwise $G$ is a tree and there is nothing to prove. Notice that an
empty set $S$ cannot be either a vertex or an edge metric generator in a
cactus graph unless $G=K_{2}$ but then it is a tree. Therefore, if $S$ is not
biactive, there must exist a cycle $C_{i}$ with precisely one $S$-active
vertex $v$ and let $v_{1}$ and $v_{2}$ be the two neighbors of $v$ on $C_{i}.$
Then $v_{1}$ and $v_{2}$ (resp. $v_{1}v$ and $v_{2}v$) are not distinguished
by $S,$ a contradiction.
\end{proof}
The above lemma gives us a necessary condition for $S$ to be a vertex (resp.
an edge) metric generator in a cactus graph. In \cite{SedSkreUnicyclic}, a
more elaborate condition for unicyclic graphs was established, which is both
necessary and sufficient. In this paper we will extend that condition to
cactus graphs, but to do so we first need to introduce the following
definitions from \cite{SedSkreUnicyclic}. Let $C_{i}$ be a cycle in a cactus
graph $G$ and let $v_{i},$ $v_{j}$ and $v_{k}$ be three vertices of $C_{i},$
we say that $v_{i},$ $v_{j}$ and $v_{k}$ are a \emph{geodesic triple} on
$C_{i}$ if
\[
d(v_{i},v_{j})+d(v_{j},v_{k})+d(v_{i},v_{k})=|V(C_{i})|.
\]
It was shown in \cite{SedSkreBounds} that a biactive branch-resolving set with
a geodesic triple of $S$-active vertices on every cycle is both a vertex and
an edge metric generator. This result is useful for bounding the dimensions
from above. Also, we need the definition of the five graph configurations from
\cite{SedSkreUnicyclic}.
\begin{figure}[ph]
\begin{center}
$%
\begin{array}
[c]{ll}%
\text{a) \raisebox{-1\height}{\includegraphics[scale=0.8]{Figure07.pdf}}} &
\text{b) \raisebox{-1\height}{\includegraphics[scale=0.8]{Figure08.pdf}}}\\
\text{c) \raisebox{-1\height}{\includegraphics[scale=0.8]{Figure09.pdf}}} &
\text{d) \raisebox{-1\height}{\includegraphics[scale=0.8]{Figure10.pdf}}}\\
\text{e) \raisebox{-1\height}{\includegraphics[scale=0.8]{Figure11.pdf}}} &
\text{f) \raisebox{-1\height}{\includegraphics[scale=0.8]{Figure12.pdf}}}%
\end{array}
$
\end{center}
\caption{All six graphs shown in this figure are unicyclic graphs with a
biactive branch-resolving set $S\ $comprised of vertices $s_{i}$. Vertices on
a cycle which are $S$-active are marked by a dashed circle and connected to
its antipodal vertices by a dashed line. Each graph in this figure contains at
least one of the five configurations, namely: a) $\mathcal{A}$, b)
$\mathcal{B}$ and also $\mathcal{D},$ c) $\mathcal{C}$, d) $\mathcal{D},$ e)
$\mathcal{E}$ on even cycle and also $\mathcal{C}$, f) $\mathcal{E}$ on odd
cycle. A pair $x$ and $x^{\prime}$ of undistinguished vertices and/or edges is
highlighted in each of the graphs. Notice that the graph in b) which contains
$\mathcal{B}$ consequently also contains $\mathcal{D},$ but graph in d) which
contains $\mathcal{D}$ does not contain $\mathcal{B}.$ Similarly, the graph in
e) which contains $\mathcal{E}$ also contains $\mathcal{C},$ but the graph
from c) which contains $\mathcal{C}$ does not contain $\mathcal{E}$.}%
\label{Fig_configurations}%
\end{figure}
\begin{definition}
Let $G$ be a cactus graph, $C$ a cycle in $G$ of the length $g$, and $S$ a
biactive branch-resolving set in $G$. We say that $C=v_{0}v_{1}\cdots v_{g-1}$
is \emph{canonically labeled} with respect to $S$ if $v_{0}$ is $S$-active and
$k=\max\{i:v_{i}$ is $S$-active$\}$ is as small as possible.
\end{definition}
Let us now introduce five configurations which a cactus graph can contain with
respect to a biactive branch-resolving set $S.$ All these configurations are
illustrated by Figure \ref{Fig_configurations}.\
\begin{definition}
Let $G$ be a cactus graph, $C$ a canonically labeled cycle in $G$ of the
length $g$, and $S$ a biactive branch-resolving set in $G$. We say that the
cycle $C$ \emph{with respect} to $S$ \emph{contains} configurations:
\begin{description}
\item {$\mathcal{A}$}. If $a_{S}(C)=2$, $g$ is even, and $k=g/2$;
\item {$\mathcal{B}$}. If $k\leq\left\lfloor g/2\right\rfloor -1$ and there is
an $S$-free thread hanging at a vertex $v_{i}$ for some $i\in\lbrack
k,\left\lfloor g/2\right\rfloor -1]\cup\lbrack\left\lceil g/2\right\rceil
+k+1,g-1]\cup\{0\}$;
\item {$\mathcal{C}$}. If $a_{S}(C)=2$, $g$ is even, $k\leq g/2$ and there is
an $S$-free thread of the length $\geq g/2-k$ hanging at a vertex $v_{i}$ for
some $i\in\lbrack0,k]$;
\item $\mathcal{D}$. If $k\leq\left\lceil g/2\right\rceil -1$ and there is an
$S$-free thread hanging at a vertex $v_{i}$ for some $i\in\lbrack
k,\left\lceil g/2\right\rceil -1]\cup\lbrack\left\lfloor g/2\right\rfloor
+k+1,g-1]\cup\{0\}$;
\item {$\mathcal{E}$}. If $a_{S}(C)=2$ and there is an $S$-free thread of the
length $\geq\left\lfloor g/2\right\rfloor -k+1$ hanging at a vertex $v_{i}$
with $i\in\lbrack0,k].$ Moreover, if $g$ is even, an $S$-free thread must be
hanging at the vertex $v_{j}$ with $j=g/2+k-i$.
\end{description}
\end{definition}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.9]{Figure14.pdf}
\end{center}
\caption{A cactus graph $G$ from Example \ref{Example_conf}. A smallest
biactive branch-resolving set $S=\{s_{1},s_{2},s_{3},s_{4},s_{5}\}$ is marked
in the figure by squares. Vertices on the cycles which are $S$-active are
marked by a dashed circle. Since every cycle contains a configuration with
respect to $S,$ for each cycle $C_{i}$ there is a pair of vertices and/or
edges $x_{i}$ and $x_{i}^{\prime}$ which are not distinguished by $S,$ these
pairs are also highlighted in the figure.}%
\label{Fig_configExample}%
\end{figure}
Notice that only an even cycle can contain configuration $\mathcal{A}$ or
$\mathcal{C}$. Also, configurations $\mathcal{B}$ and $\mathcal{D}$ are almost
the same, they differ only if $C$ is odd where the index $i$ can take two more
values in $\mathcal{D}$ than in $\mathcal{B}.$ Finally, for configurations
$\mathcal{A}$, $\mathcal{C}$, and $\mathcal{E}$ it holds that $a_{S}(C)=2,$ so
there are only two $S$-active vertices on the cycle $C$ and hence no geodesic
triple of $S$-active vertices. On the other hand, for configurations
$\mathcal{B}$ and $\mathcal{D}$ there might be more than two $S$-active
vertices on the cycle $C,$ but the bounds $k\leq\left\lfloor g/2\right\rfloor
-1$ and $k\leq\left\lceil g/2\right\rceil -1$ again imply there is no geodesic
triple of $S$-active vertices on $C.$ Therefore, we can state the following
observation which is useful for constructing metric generators.
\begin{remark}
\label{Obs_geodTriple}If there is a geodesic triple of $S$-active vertices on
a cycle $C$ of a cactus graph $G,$ then $C$ does not contain any of the
configurations $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$, $\mathcal{D}$, and
$\mathcal{E}$ with respect to $S.$
\end{remark}
The following result regarding configurations $\mathcal{A}$, $\mathcal{B}$,
$\mathcal{C}$, $\mathcal{D}$, and $\mathcal{E}$ was established for unicyclic
graphs (see Lemmas 6, 7, 13 and 14 from \cite{SedSkreUnicyclic}).
\begin{theorem}
\label{Lemma_configurations}Let $G$ be a unicyclic graph with the cycle $C$
and let $S$ be a biactive branch-resolving set in $G$. The set $S$ is a vertex
(resp. an edge) metric generator if and only if $C$ does not contain any of
configurations $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ (resp.
$\mathcal{A}$, $\mathcal{D}$, and $\mathcal{E}$) with respect to $S.$
\end{theorem}
In this paper we will extend this result to cactus graphs and then use it to
determine the exact value of the vertex and the edge metric dimensions of such
graphs. We first give an example how this approach with configurations can be
extended to cactus graphs.
\begin{example}
\label{Example_conf}Let $G$ be the cactus graph from Figure
\ref{Fig_configExample}. The graph $G$ contains six cycles and the set
$S=\{s_{1},s_{2},s_{3},s_{4},s_{5}\}$ is a smallest biactive branch-resolving
set in $G$. In the figure the set of $S$-active vertices on each cycle is
marked by a dashed circle. The cycle $C_{1}$ (resp. $C_{2}$, $C_{3}$, $C_{4}$,
$C_{5}$) with respect to $S$ contains configuration $\mathcal{A}$ (resp.
$\mathcal{B}$ and also $\mathcal{D}$, $\mathcal{C}$, $\mathcal{E}$ on odd
cycle, $\mathcal{E}$ on even cycle), so in each of these cycles there is a
pair of vertices and/or edges $x_{i}$ and $x_{i}^{\prime}$ which is not
distinguished by $S.$ The cycle $C_{6}$ does not contain any of the five
configurations as there is a geodesic triple of $S$-active vertices on
$C_{6},$ so all pairs of vertices and all pair of edges in $C_{6}$ are
distinguished by $S$.
\end{example}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.9]{Figure03.pdf}
\end{center}
\caption{A cactus graph $G$ with three cycles in which the unicyclic region
$G_{i}$ of the cycle $C_{i}$ is distinguished with $b_{1}$ and $b_{2}$ being
the boundary vertices of $G_{i}$. The set $S=\{s_{1},s_{2},s_{3}\}$ is a
smallest biactive branch-resolving set in $G$ for which a set of $S$-active
vertices is marked on each cycle. For the set $S,$ the regional set in $G_{i}$
is $S_{i}=\{s_{2},b_{1},b_{2}\}.$}%
\label{Fig_region}%
\end{figure}
Besides this configuration approach, in cactus graphs an additional condition
will have to be introduced for the situation when a pair of cycles share a vertex.
\section{Metric generators in cacti}
Let $G$ be a cactus graph with cycles $C_{1},\ldots,C_{c}.$ We say that a
vertex $v\in V(G)$ \emph{gravitates} to a cycle $C_{i}$ in $G$ if there is a
path from $v$ to a vertex from $C_{i}$ which does not share any edge nor any
internal vertex with any cycle of $G.$ A \emph{unicyclic region} of the cycle
$C_{i}$ from $G$ is the subgraph $G_{i}$ of $G$ induced by all vertices that
gravitate to $C_{i}$ in $G.$ The notion of unicyclic region of a cactus graph
is illustrated by Figure \ref{Fig_region}.
Notice that each unicyclic region $G_{i}$ is a unicyclic graph with its cycle
being $C_{i}.$ Also, considering the example from Figure \ref{Fig_region}, one
can easily notice that two distinct unicyclic regions may not be vertex
disjoint, as the path connecting vertex $b_{2}$ and the cycle $C_{i}$ belongs
both to $G_{i}$ and $G_{j}.$ But, it does hold that all unicyclic regions
cover the whole $G.$ We say that a subgraph $H$ of a graph $G$ is an
\emph{isometric} subgraph if $d_{H}(u,v)=d_{G}(u,v)$ for every pair of
vertices $u,v\in V(H).$ The following observation is obvious.
\begin{remark}
The unicyclic region $G_{i}$ of a cycle $C_{i}$ is an isometric subgraph of
$G.$
\end{remark}
Finally, we say that a vertex $v$ from a unicyclic region $G_{i}$ is a
\emph{boundary} vertex if $v\in V(C_{j})$ for $j\not =i.$ In the example from
Figure \ref{Fig_region}, the boundary vertices of the region $G_{i}$ are
$b_{1}$ and $b_{2}$.
Let $S$ be a biactive branch-resolving set in $G$ and let $G_{i}$ be a
unicyclic region in $G.$ For the set $S$ we define the \emph{regional set}
$S_{i}$ as the set obtained from $S\cap V(G_{i})$ by introducing all boundary
vertices from $G_{i}$ to $S$. For example, in Figure \ref{Fig_region} the set
$S_{i}=\{s_{2},b_{1},b_{2}\}$ is the regional set in the region of the cycle
$C_{i}$.
\begin{lemma}
\label{Lemma_regionGenerator}Let $G$ be a cactus graph with $c$ cycles
$C_{1},\ldots,C_{c}$ and let $S\subseteq V(G)$. If $S$ is a vertex (resp.
edge) metric generator in $G,$ then the regional set $S_{i}$ is a vertex
(resp. edge) metric generator in the unicyclic region $G_{i}$ for every
$i\in\{1,\ldots,c\}$.
\end{lemma}
\begin{proof}
Suppose first that there is a cycle $C_{i}$ in $G$ such that the regional set
$S_{i}$ is not a vertex (resp. an edge) metric generator in the unicyclic
region $G_{i}.$ This implies that there exists a pair of vertices (resp.
edges) $x$ and $x^{\prime}$ in $G_{i}$ which are not distinguished by $S_{i}.$
We will show that $x$ and $x^{\prime}$ are not distinguished by $S$ in $G$
either. Suppose the contrary, i.e. there is a vertex $s\in S$ which
distinguishes $x$ and $x^{\prime}$ in $G.$ If $s\in V(G_{i}),$ then $s\in
S_{i}.$ Since $G_{i}$ is an isometric subgraph of $G,$ then $x$ and
$x^{\prime}$ would be distinguished by $s\in S_{i}$ in $G_{i},$ a
contradiction. Assume therefore that $s\not \in V(G_{i}).$ Notice that the
shortest path from every vertex (resp. edge) in $G_{i}$ to $s$ leads through a
same boundary vertex $b$ in $G_{i}.$ The definition of $S_{i}$ implies $b\in
S_{i},$ so $x$ and $x^{\prime}$ are not distinguished by $b.$ Therefore, we
obtain%
\[
d(x,s)=d(x,b)+d(b,s)=d(x^{\prime},b)+d(b,s)=d(x^{\prime},s),
\]
so $x$ and $x^{\prime}$ are not distinguished by $s$ in $G,$ a contradiction.
\end{proof}
Notice that the condition from Lemma \ref{Lemma_regionGenerator} is necessary
for $S$ to be a metric generator, but it is not sufficient as is illustrated
by the graph shown in Figure \ref{Fig_incidence} in which every regional set
$S_{i}$ is a vertex (resp. an edge) metric generator in the corresponding
region $G_{i}$, but there still exists a pair of vertices (resp. edges) which
is not distinguished by $S,$ so $S$ is not a vertex (resp. an edge) metric
generator in $G$.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.9]{Figure04.pdf}
\end{center}
\caption{A cactus graph $G$ with three cycles and a smallest biactive
branch-resolving set $S=\{s_{1},s_{2}\}$ in $G.$ Every regional set $S_{i}$ is
a vertex (resp. an edge) metric generator in the corresponding region $G_{i}$,
but still a pair of vertices $v_{1}$ and $v_{2}$ is not distinguished by $S$.
The pair of edges $v_{1}v$ and $v_{2}v$ is also not distinguished by $S$ and
the same holds for the pair of edges $w_{1}w$ and $w_{2}w.$}%
\label{Fig_incidence}%
\end{figure}
Next, we will introduce notions which are necessary to state a condition which
will be both necessary and sufficient for a biactive branch-resolving set $S$
to be a vertex (resp. an edge) metric generator in a cactus graph $G.$ An
$S$\emph{-path} of the cycle $C_{i}$ is any subpath of $C_{i}$ which contains
all $S$-active vertices on $C_{i}$ and is of minimum possible length. We
denote an $S$\emph{-}path of the cycle $C_{i}$ by $P_{i}$. Notice that the
end-vertices of an $S$-path are always $S$-active, otherwise it would not be
shortest. For example, on the cycle $C_{2}$ in Figure \ref{Fig_incidence}
there are two different paths connecting $S$-active vertices $v$ and $w,$ one
is of the length $3$ and the other of length $5,$ so the shorter one is an
$S$-path. Also, an $S$\emph{-}path $P_{i}$ of a cycle $C_{i}$ may not be
unique, as there may exist several shortest subpaths of $C_{i}$ containing all
$S$-active vertices on $C_{i},$ but if the length of $P_{i}$ satisfies
$\left\vert P_{i}\right\vert \leq\left\lceil g_{i}/2\right\rceil -1$ then
$P_{i}$ is certainly unique and its end-vertices are $v_{0}$ and $v_{k}$ in
the canonical labelling of $C_{i}.$
\begin{definition}
Let $G$ be a cactus graph with cycles $C_{1},\ldots,C_{c}$ and let $S$ be a
biactive branch-resolving set in $G.$ We say that a vertex $v\in V(C_{i})$ is
\emph{vertex-critical} (resp. \emph{edge-critical}) on $C_{i}$ with respect to
$S$ if $v$ is an end-vertex of $P_{i}$ and $\left\vert P_{i}\right\vert
\leq\left\lfloor g_{i}/2\right\rfloor -1$ (resp. $\left\vert P_{i}\right\vert
\leq\left\lceil g_{i}/2\right\rceil -1$).
\end{definition}
Notice that the notion of a vertex-critical and an edge-critical vertex
differs only on odd cycles. We say that two distinct cycles $C_{i}$ and
$C_{j}$ of a cactus graph $G$ are \emph{vertex-critically incident} (resp.
\emph{edge-critically incident}) with respect to $S$ if $C_{i}$ and $C_{j}$
share a vertex $v$ which is vertex-critical (resp. edge-critical) with respect
to $S$ on both $C_{i}$ and $C_{j}$. Notice that on odd cycles the required
length of an $S$-path $P_{i}$ for $v$ to be vertex-critical differs from the
one required for $v$ to be edge-critical, while on even cycles the required
length is the same.
To illustrate this notion, let us consider the cycle $C_{2}$ in the graph from
Figure \ref{Fig_incidence}. Vertices $v$ and $w$ are both vertex-critical and
edge-critical on $C_{2}$ with respect to $S$ from the figure. Vertex $v$
belongs also to $C_{1}$ and it is also both vertex- and edge-critical on
$C_{1}.$ Therefore, cycles $C_{1}$ and $C_{2}$ are both vertex- and
edge-critically incident, the consequence of which is that a pair of vertices
$v_{1}$ and $v_{2}$ which are neighbors of $v$ and a pair of edges $v_{1}v$
and $v_{2}v$ which are incident to $v$ are not distinguished by $S.$ On the
other hand, vertex $w$ belongs also to $C_{3}$ on which it is edge-critical,
but it is not vertex-critical since $P_{3}$ is not short enough. So, $C_{2}$
and $C_{3}$ are edge-critically incident, but not vertex-critically incident.
Consequently, a pair of edges $w_{1}w$ and $w_{2}w$ is not distinguished by
$S,$ but a pair of vertices $w_{1}$ and $w_{2}$ is distinguished by $S.$ We
will show in the following lemma that this holds in general.
\begin{lemma}
\label{Lemma_incidence}Let $G$ be a cactus graph with $c$ cycles $C_{1}%
,\ldots,C_{c}$ and let $S$ be a biactive branch-resolving set in $G$. If $S$
is a vertex (resp. an edge) metric generator in $G,$ then there is no pair of
cycles in $G$ which are vertex-critically (resp. edge critically) incident
with respect to $S$.
\end{lemma}
\begin{proof}
Let $S$ be a vertex (resp. an edge) metric generator in $G.$ Suppose the
contrary, i.e. there are two distinct cycles $C_{i}$ and $C_{j}$ in $G$ which
are vertex-critically (resp. edge-critically) incident with respect to $S$.
This implies that $C_{i}$ and $C_{j}$ share a vertex $v$ which is
vertex-critical (resp. edge-critical) on both $C_{i}$ and $C_{j}$. Let $x$ and
$x^{\prime}$ be a pair of vertices (resp. edges) which are neighbors (resp.
incident) to $v$ on cycles $C_{i}$ and $C_{j}$ respectively, but which are not
contained on paths $P_{i}$ and $P_{j}.$ The length of paths $P_{i}$ and
$P_{j}$ which is required by the definition of a vertex-critical (resp.
edge-critical) vertex implies that a shortest path from both $x$ and
$x^{\prime}$ to all vertices from $P_{i}$ and $P_{j}$ leads through $v.$ Since
$P_{i}$ and $P_{j}$ contain all $S$-active vertices on $C_{i}$ and $C_{j},$
this further implies that a shortest path from both $x$ and $x^{\prime}$ to
all vertices from $S$ leads through $v.$ Since $d(x,v)=d(x^{\prime},v)$, it
follows that $x$ and $x^{\prime}$ are not distinguished by $S,$ a contradiction.
\end{proof}
Each of Lemmas \ref{Lemma_regionGenerator} and \ref{Lemma_incidence} gives a
necessary condition for a biactive branch-resolving set $S$ to be a vertex
(resp. an edge) metric generator in a cactus graph $G$. Let us now show that
these two necessary conditions taken together form a sufficient condition for
$S$ to be a vertex (resp. an edge) metric generator.
\begin{lemma}
\label{Lemma_sufficient}Let $G$ be a cactus graph with $c$ cycles
$C_{1},\ldots,C_{c}$ and let $S$ be a biactive branch-resolving set in $G$. If
a regional set $S_{i}$ is a vertex (resp. an edge) metric generator in the
unicyclic region $G_{i}$ for every $i=1,\ldots,c$ and there are no
vertex-critically (resp. edge-critically) incident cycles in $G,$ then $S$ is
a vertex (resp. an edge) metric generator in $G.$
\end{lemma}
\begin{proof}
Let $x$ and $x^{\prime}$ be a pair of vertices (resp. edges) from $G.$ We want
to show that $S$ distinguishes $x$ and $x^{\prime}.$ In order to do so, we
distinguish the following two cases.
\medskip\noindent\textbf{Case 1:}\emph{ }$x$\emph{ and }$x^{\prime}$\emph{
belong to a same unicyclic region }$G_{i}$\emph{ of }$G.$ Since the regional
set $S_{i}$ is a vertex (resp. an edge) metric generator in $G_{i},$ there is
a vertex $s\in S_{i}$ which distinguishes $x$ and $x^{\prime}$ in $G_{i}.$ If
$s\in S,$ then the fact that $G_{i}$ is an isometric subgraph of $G$ implies
that the pair $x$ and $x^{\prime}$ is distinguished by the same $s$ in $G$ as
well. Assume therefore that $s\not \in S,$ so the definition of the regional
set $S_{i}$ implies that $s$ is a boundary vertex of $G_{i}.$ Let $s^{\prime
}\in S$ be a vertex in $G$ such that the shortest path from $s^{\prime}$ to
both $x$ and $x^{\prime}$ leads through the boundary vertex $s.$ Recall that
such a vertex $s^{\prime}$ must exist since $S$ is biactive. The fact that $s$
distinguishes $x$ and $x^{\prime}$ in $G_{i}$, implies $d(x,s)\not =%
d(x^{\prime},s),$ which further implies%
\[
d(x,s^{\prime})=d(x,s)+d(s,s^{\prime})\not =d(x^{\prime},s)+d(s,s^{\prime
})=d(x^{\prime},s^{\prime}),
\]
so the pair $x$ and $x^{\prime}$ is distinguished by $S$ in $G$.
\medskip\noindent\textbf{Case 2:} $x$\emph{ and }$x^{\prime}$\emph{ do not
belong to a same unicyclic region of }$G.$ Let us assume that $x$ belongs to
$G_{i}$ and $x^{\prime}$ does not belong to $G_{i},$ and say it belongs to
$G_{j}$ for $j\not =i$. If $x$ and $x^{\prime}$ are distinguished by a vertex
$s\in S\cap V(G_{i}),$ then the claim is proven, so let us assume that $x$ and
$x^{\prime}$ are not distinguished by any $s\in S\cup V(G_{i}).$ Since $x$ and
$x^{\prime}$ do not belong to a same unicyclic region, there exists a boundary
vertex $b$ of the unicyclic region $G_{i}$ such that the shortest path from
$x$ to $x^{\prime}$ leads through $b.$ Let $s_{b}$ be a vertex from $S$ such
that the shortest path from $x$ to $s_{b}$ also leads through $b,$ which must
exist since $S$ is biactive. We want to prove that $x$ and $x^{\prime}$ are
distinguished by $s_{b}.$ Let us suppose the contrary, i.e. $d(x,s_{b}%
)=d(x^{\prime},s_{b}).$ Then we have the following%
\[
d(x,b)+d(b,s_{b})=d(x,s_{b})=d(x^{\prime},s_{b})\leq d(x^{\prime}%
,b)+d(b,s_{b}),
\]
from which we obtain
\begin{equation}
d(x,b)\leq d(x^{\prime},b). \label{For_riste}%
\end{equation}
Now, we distinguish the following two subcases.
\medskip\noindent\textbf{Subcase 2.a:} $b$\emph{ does not belong to }%
$V(C_{i}).$ Notice that by the definition of the unicyclic region, any acyclic
structure hanging at $b$ in $G$ is not included in $G_{i}$, as is illustrated
by $b_{2}$ from Figure \ref{Fig_region}, which implies $b$ is a leaf in
$G_{i}.$ Let $b^{\prime}$ be the only neighbor of $b$ in $G_{i}.$ The
inequality (\ref{For_riste}) further implies $d(x,b^{\prime})<d(x^{\prime
},b^{\prime})$ since $x$ belongs to $G_{i}$ and $x^{\prime}$ does not. Let
$v_{0}$ be the vertex from $C_{i}$ closest to $b,$ which implies $v_{0}$ is
$S$-active on $C_{i}.$ Let $v_{k}$ be an $S$-active vertex on $C_{i}$ distinct
from $v_{0},$ such a vertex $v_{k}$ must exist on $C_{i}$ because we assumed
$S$ is biactive. So, we have
\[
d(x,v_{k})\leq d(x,b^{\prime})+d(b^{\prime},v_{k})<d(x^{\prime},b^{\prime
})+d(b^{\prime},v_{k})=d(x^{\prime},v_{k}).
\]
Let $s_{k}$ be a vertex from $S$ which belongs to the connected component of
$G-E(C_{i})$ containing $v_{k}.$ Then we have%
\[
d(x,s_{k})\leq d(x,v_{k})+d(v_{k},s_{k})<d(x^{\prime},v_{k})+d(v_{k}%
,s_{k})=d(x^{\prime},s_{k}).
\]
Therefore, $S$ distinguishes $x$ and $x^{\prime},$ so $S$ is a vertex (resp.
an edge) metric generator in $G.$
\medskip\noindent\textbf{Subcase 2.b:} $b$\emph{ belongs to }$V(C_{i}).$ Since
$b$ is a boundary vertex of the unicyclic region $G_{i},$ this implies there
is a cycle $C_{l},$ for $l\not =i,$ such that $b\in V(C_{l})$. Therefore,
cycles $C_{i}$ and $C_{l}$ share the vertex $b.$ Notice that any acyclic
structure hanging at $b$ belongs to both $G_{i}$ and $G_{l}$, as is
illustrated by $b_{1}$ from Figure \ref{Fig_region}. If $x^{\prime}$ belongs
to $G_{l},$ then neither $x$ nor $x^{\prime}$ can belong to an acyclic
structure hanging at $b,$ as we assumed that $x$ and $x^{\prime}$ do not
belong to a same component. On the other hand, if $x^{\prime}$ does not belong
to $G_{l}$ and $x$ belongs to an acyclic structure hanging at $b,$ then we
switch $G_{i}$ by $G_{l}$ and assume that $x$ belongs to $G_{l}.$ This way we
assure that neither $x$ not $x^{\prime}$ belong to an acyclic structure
hanging at $b.$
If $d(x,b)<d(x^{\prime},b),$ let $v_{k}$ be an $S$-active vertex on $C_{i}$
distinct from $b,$ which must exist as $S$ is biactive. From (\ref{For_riste})
we obtain%
\[
d(x,v_{k})\leq d(x,b)+d(b,v_{k})<d(x^{\prime},b)+d(b,v_{k})=d(x^{\prime}%
,v_{k}),
\]
so similarly as in previous subcase $x$ and $x^{\prime}$ are distinguished by
a vertex $s_{k}\in S$ which belongs to the connected component of $G-E(C_{i})$
which contains $v_{k}.$
Therefore, assume that $d(x,b)=d(x^{\prime},b).$ If a shortest path from $x$
to all $S$-active vertices on $C_{i}$ and a shortest path from $x^{\prime}$ to
all $S$-active vertices on $C_{l}$ leads through $b,$ then $x^{\prime}$
belongs to $C_{l},$ i.e. $j=l$, and the pair of cycles $C_{i}$ and $C_{l}$ are
vertex-critically (resp. edge-critically) incident, a contradiction. So, we
may assume there is an $S$-active vertex $v_{k}$, say on $C_{i}$, such that a
shortest path from $x$ to $v_{k}$ does not lead through $b.$ Therefore,
\[
d(x,v_{k})<d(x,b)+d(b,v_{k})=d(x^{\prime},b)+d(b,v_{k})=d(x^{\prime},v_{k}).
\]
But now, similarly as in previous cases we have that $x$ and $x^{\prime}$ are
distinguished by a vertex $s_{k}\in S$ which is contained in the connected
component of $G-E(C_{i})$ containing $v_{k}.$
\end{proof}
Let us now relate these results with configurations $\mathcal{A},$
$\mathcal{B}$, $\mathcal{C}$, $\mathcal{D}$ and $\mathcal{E}$.
\begin{lemma}
\label{Obs_RegionConfiguration}Let $G$ be a cactus graph and let $S$ be a
biactive branch-resolving set in $G.$ A cycle $C_{i}$ of the graph $G$
contains configuration $\mathcal{A}$ (or $\mathcal{B}$ or $\mathcal{C}$ or
$\mathcal{D}$ or $\mathcal{E}$) with respect to $S$ in $G$ if and only if
$C_{i}$ contains the respective configuration with respect to $S_{i}$ in
$G_{i}.$
\end{lemma}
\begin{proof}
Let $G$ be a cactus graph with cycles $C_{1},\ldots,C_{c}$ and let $S$ be a
biactive branch-resolving set in $G.$ Since $S$ is a biactive set, for every
boundary vertex $b$ in a unicyclic region $G_{i},$ there is a vertex $s\in S$
such that the shortest path from $s$ to $C_{i}$ leads through $b,$ as it is
shown in Figure \ref{Fig_region}. This implies that the set of $S$-active
vertices on $C_{i}$ in $G$ is the same as the set of $S_{i}$-active vertices
on $C_{i}$ in $G_{i}.$ Since the presence of configurations $\mathcal{A}$,
$\mathcal{B}$, $\mathcal{C}$, $\mathcal{D}$ and $\mathcal{E}$ on a cycle
$C_{i}$, with respect to a set $S,$ by definition depends on the position of
$S$-active vertices on $C_{i},$ the claim follows.
\end{proof}
Notice that Lemmas \ref{Lemma_regionGenerator}, \ref{Lemma_incidence} and
\ref{Lemma_sufficient} give us a condition for $S$ to be a vertex (resp. an
edge) metric generator in a cactus graph, which is both necessary and
sufficient. In the light of Lemma \ref{Obs_RegionConfiguration}, we can
further apply Theorem \ref{Lemma_configurations} to obtain the following
result which unifies all our results.
\begin{theorem}
\label{Cor_generatorCharacterization}Let $G$ be a cactus graph with $c$ cycles
$C_{1},\ldots,C_{c}$ and let $S$ be a biactive branch-resolving set in $G$.
The set $S$ is a vertex (resp. an edge) metric generator if and only if each
cycle $C_{i}$ does not contain any of the configurations $\mathcal{A}$,
$\mathcal{B}$, and $\mathcal{C}$ (resp. $\mathcal{A}$, $\mathcal{D}$, and
$\mathcal{E}$) and there are no vertex-critically (resp. edge-critically)
incident cycles in $G$ with respect to $S$.
\end{theorem}
\begin{proof}
Let $S$ be a vertex (resp. an edge) metric generator in $G$. Then Lemma
\ref{Lemma_regionGenerator} implies that $S_{i}$ is a vertex (resp. edge)
metric generator in the unicyclic region $G_{i}$ and the Theorem
\ref{Lemma_configurations} further implies that every cycle does not contain
any of the configurations $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$
(resp. $\mathcal{A}$, $\mathcal{D}$, and $\mathcal{E}$) with respect to $S.$
Also, Lemma \ref{Lemma_incidence} implies that there are no vertex-critically
(resp. edge-critically) incident cycles in $G$ with respect to $S$.
The other direction is the consequence of Lemma \ref{Lemma_sufficient} and
Theorem \ref{Lemma_configurations}.
\end{proof}
\section{Metric dimensions in cacti}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.8]{Figure15.pdf}
\end{center}
\caption{A cactus graph with three cycles and two different smallest biactive
branch-resolving sets $S=\{s_{1},s_{2}\}$ and $S^{\prime}=\{s_{1}^{\prime
},s_{2}^{\prime}\}.$ With respect to $S$ the cycle $C_{2}$ contains
configuration $\mathcal{A}$ and $C_{3}$ configurations $\mathcal{B},$
$\mathcal{C}$ and $\mathcal{D}.$ With respect to $S^{\prime}$ cycles $C_{2}$
and $C_{3}$ contain none of the five configurations. The cycle $C_{1}$ has
$b(C_{1})=2,$ so the set of $S$-active vertices $\{u,v\}$ on $C_{1}$ is the
same for all smallest biactive branch-resolving sets and $C_{1}$ contains
configurations $\mathcal{B}$ and $\mathcal{D}$ with respect to all of them,
including $S$ and $S^{\prime}$ shown in the figure. Therefore, in this graph
cycle $C_{1}$ is both $\mathcal{ABC}$- and $\mathcal{ADE}$-positive, and
cycles $C_{2}$ and $C_{3}$ are both $\mathcal{ABC}$- and $\mathcal{ADE}%
$-negative.}%
\label{Fig_avoiding}%
\end{figure}Let $G$ be a cactus graph and $S$ a smallest biactive
branch-resolving set in $G.$ Then
\[
\left\vert S\right\vert =L(G)+B(G),
\]
where $B(G)=\sum_{i=1}^{c}\max\{0,2-b(C_{i})\}.$ If $b(C_{i})\geq2$, then the
set of $S$-active vertices on $C_{i}$ is the same for all smallest biactive
branch-resolving sets $S$. The set of $S$-active vertices may differ only on
cycles $C_{i}$ with $b(C_{i})<2.$ Therefore, such a cycle $C_{i}$ may contain
one of the configurations with respect to one smallest biactive
branch-resolving set, but not with respect to another. This is illustrated by
Figure \ref{Fig_avoiding}.
\begin{definition}
We say that a cycle $C_{i}$ from a cactus graph $G$ is $\mathcal{ABC}%
$\emph{-negative} (resp. $\mathcal{ADE}$\emph{-negative}), if there exists a
smallest biactive branch-resolving set $S$ in $G$ such that $C_{i}$ does not
contain any of the configurations $\mathcal{A},$ $\mathcal{B},$ and
$\mathcal{C}$ (resp. $\mathcal{A},$ $\mathcal{D},$ and $\mathcal{E}$) with
respect to $S.$ Otherwise, we say that $C_{i}$ is $\mathcal{ABC}%
$\emph{-positive} (resp. $\mathcal{ADE}$\emph{-positive}). The number of
$\mathcal{ABC}$-positive (resp. $\mathcal{ADE}$-positive) cycles in $G$ is
denoted by $c_{\mathcal{ABC}}(G)$ (resp. $c_{\mathcal{ADE}}(G)$).
\end{definition}
For two distinct smallest biactive branch-resolving sets $S$, the set of
$S$-active vertices may differ only on cycles with $b(C_{i})\leq1.$ Let
$C_{i}$ and $C_{j}$ be two such cycles in $G$ and notice that the choice of
the vertices included in $S$ from the region of $C_{i}$ is independent of the
choice from $C_{j}.$ Therefore, there exists at least one smallest biactive
branch-resolving set $S$ such that every $\mathcal{ABC}$-negative (resp.
$\mathcal{ADE}$-negative) avoids the three configurations with respect to $S.$
Notice that there may exist more than one such set $S,$ and in that case they
all have the same size, so among them we may choose the one with the smallest
number of vertex-critical (resp. edge-critical) incidencies. Therefore, we say
that a smallest biactive branch-resolving set $S$ is \emph{nice} if every
$\mathcal{ABC}$-negative (resp. $\mathcal{ADE}$-negative) cycle $C_{i}$ does
not contain the three configurations with respect to $S$ and the number of
vertex-critically (resp. edge-critically) incident pairs of cycles with
respect to $S$ is the smallest possible. The niceness of a smallest biactive
branch-resolving set is illustrated by Figure \ref{Fig_optimality}.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.8]{Figure16.pdf}
\end{center}
\caption{A cactus graph with three cycles and two different smallest biactive
branch-resolving sets $S=\{s_{1},s_{2}\}$ and $S^{\prime}=\{s_{1}^{\prime
},s_{2}^{\prime}\}.$ With respect to both $S$ and $S^{\prime}$ all three
cycles do not contain any of the five configurations. The difference is that
with respect to $S$ the cycle $C_{2}$ is both vertex- and edge-critically
incident with both $C_{1}$ and $C_{3}$, and with respect to $S^{\prime}$ the
cycle $C_{2}$ is both vertex- and edge-critically incident with $C_{3}$, but
only edge-critically incident with $C_{1}$. Therefore, the set $S$ is not
nice. Since the critical incidences of $S^{\prime}$ cannot be further avoided
the set $S^{\prime}$ is nice.}%
\label{Fig_optimality}%
\end{figure}
A set $S\subseteq V(G)$ is a \emph{vertex cover} if it contains a least one
end-vertex of every edge in $G.$ The cardinality of a smallest vertex cover in
$G$ is the \emph{vertex cover number} denoted by $\tau(G).$ Now, let $G$ be a
cactus graph and let $S$ be a nice smallest biactive branch-resolving set in
$G.$ We define the \emph{vertex-incident graph} $G_{vi}$ (resp.
\emph{edge-incident graph} $G_{ei}$) as a graph containing a vertex for every
cycle in $G,$ where two vertices are adjacent if the corresponding cycles in
$G$ are $\mathcal{ABC}$-negative and vertex-critically incident (resp.
$\mathcal{ADE}$-negative and edge-critically incident) with respect to $S$.
For example, if we consider the cactus graph $G$ from Figure \ref{Fig_example}%
, then $V(G_{vi})=V(G_{ei})=\{C_{i}:i=1,\ldots,7\},$ where $E(G_{vi}%
)=\{C_{3}C_{4},C_{4}C_{5}\}$ and $E(G_{ei})=\{C_{2}C_{3},C_{3}C_{4},C_{4}%
C_{5}\}.$
We are now in a position to establish the following theorem which gives us the
value of the vertex and the edge metric dimensions in a cactus graph.
\begin{theorem}
\label{Tm_dim}Let $G$ be a cactus graph. Then
\[
\mathrm{dim}(G)=L(G)+B(G)+c_{\mathcal{ABC}}(G)+\tau(G_{vi}),
\]
and
\[
\mathrm{edim}(G)=L(G)+B(G)+c_{\mathcal{ADE}}(G)+\tau(G_{ei}).
\]
\end{theorem}
\begin{proof}
If there is a cycle in $G$ with $b(C)=0,$ then $G$ is a unicyclic graph. For
unicyclic graphs with $b(C)=0$ we have $B(G)=2$ and if the three
configurations $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$ (resp.
$\mathcal{A}$, $\mathcal{D}$, $\mathcal{E}$) cannot be avoided by choosing two
vertices into $S,$ then $c_{\mathcal{ABC}}(G)=1$ (resp. $c_{\mathcal{ADE}%
}(G)=1$) and also the third vertex must be introduced to $S,$ so the claim
holds. In all other situations $B(G)$ equals the number of cycles in $G$ with
$b(C_{i})=1.$
Let $S$ be a smallest vertex (resp. edge) metric generator in $G$. Due to
Lemma \ref{Lemma_biactive_branchResolving} the set $S$ must be
branch-resolving. Let $S_{1}\subseteq S$ be a smallest branch-resolving set
contained in $S,$ so $\left\vert S_{1}\right\vert =L(G).$ Since according to
Lemma \ref{Lemma_biactive_branchResolving} the set $S$ must also be biactive,
let $S_{2}\subseteq S\backslash S_{1}$ be a smallest set such that $S_{1}\cup
S_{2}$ is biactive. Obviously, $S_{1}\cap S_{2}=\phi$ and $\left\vert
S_{2}\right\vert =B(G).$
Since $S_{1}\cup S_{2}$ is a smallest biactive branch-resolving set in $G,$ it
follows that every $\mathcal{ABC}$-positive (resp. $\mathcal{ADE}$-positive)
cycle in $G$ contains at least one of the three configurations with respect to
$S_{1}\cup S_{2},$ so according to Theorem \ref{Cor_generatorCharacterization}
the set $S_{1}\cup S_{2}$ is not a vertex (resp. an edge) metric generator in
$G.$ Therefore, each $\mathcal{ABC}$-positive (resp. $\mathcal{ADE}$-positive)
cycle $C_{i}$ must contain a vertex $s_{i}\in S\backslash(S_{1}\cup S_{2}),$
where we may assume that $s_{i}$ is chosen so that it forms a geodesic triple
on $C_{i}$ with vertices from $S_{1}\cup S_{2},$ so according to Observation
\ref{Obs_geodTriple} the cycle $C_{i}$ will not contain any of the
configurations with respect to $S.$ Denote by $S_{3}$ the set of vertices
$s_{i}$ from every $\mathcal{ABC}$-positive (resp. $\mathcal{ADE}$-positive)
cycle in $G.$ Obviously, $S_{3}\subseteq S,$ $S_{1}\cap S_{2}\cap S_{3}=\phi$
and $\left\vert S_{3}\right\vert =c_{\mathcal{ABC}}(G)$ (resp. $\left\vert
S_{3}\right\vert =c_{\mathcal{ADE}}(G)$).
Notice that $S_{1}\cup S_{2}\cup S_{3}$ is a biactive branch-resolving set in
$G$ such that every cycle $C_{i}$ in $G$ does not contain any of the
configurations $\mathcal{A},$ $\mathcal{B},$ $\mathcal{C}$ (resp.
$\mathcal{A},$ $\mathcal{D},$ $\mathcal{E}$) with respect to it. Notice that
$S_{1}\cup S_{2}\cup S_{3}$ still may not be a vertex (resp. an edge) metric
generator, as there may exist vertex-critically (resp. edge-critically)
incident cycles in $G$ with respect to $S_{1}\cup S_{2}\cup S_{3}.$ Since $S$
is a smallest vertex (resp. edge) metric generator, we may assume that $S_{2}$
is chosen so that a smallest biactive branch-resolving set $S_{1}\cup S_{2}$
is nice. Therefore, the graph $G_{vi}$ (resp. $G_{ei}$) contains an edge for
every pair of cycles in $G$ which are $\mathcal{ABC}$-negative and
vertex-critically incident (resp. $\mathcal{ADE}$-negative and edge-critically
incident) with respect to $S_{1}\cup S_{2}.$ Let us denote $S_{4}%
=S\backslash(S_{1}\cup S_{2}\cup S_{3}).$ For each edge $xy$ in $G_{vi}$
(resp. $G_{ei}$) the set $S_{4}$ must contain a vertex from $C_{x}$ or
$C_{y},$ chosen so that it forms a geodesic triple of $S$-active vertices on
$C_{x}$ or $C_{y}$ with other vertices from $S.$ Therefore, $S_{4}$ must
contain at least $\tau(G_{vi})$ (resp. $\tau(G_{ei})$) vertices in order for
$S$ to be a vertex (resp. an edge) metric generator. Since $S$ is a smallest
vertex (resp. edge) metric generator, it must hold $\left\vert S_{4}%
\right\vert =\tau(G_{vi})$ (resp. $\left\vert S_{4}\right\vert =\tau(G_{ei})$).
We have established that $S=S_{1}\cup S_{2}\cup S_{3}\cup S_{4},$ where
$S_{1}\cap S_{2}\cap S_{3}\cap S_{4},$ so $\left\vert S\right\vert =\left\vert
S_{1}\right\vert +\left\vert S_{2}\right\vert +\left\vert S_{3}\right\vert
+\left\vert S_{4}\right\vert .$ Since we also established $\left\vert
S_{1}\right\vert =L(G),$ $\left\vert S_{2}\right\vert =B(G),$ $\left\vert
S_{3}\right\vert =c_{\mathcal{ABC}}(G)$ (resp. $\left\vert S_{3}\right\vert
=c_{\mathcal{ADE}}(G)$) and $\left\vert S_{4}\right\vert =\tau(G_{vi})$ (resp.
$\left\vert S_{4}\right\vert =\tau(G_{ei})$), the proof is finished.
\end{proof}
The formulas for the calculation of metric dimensions from the above theorem
are illustrated by the following examples.
\begin{example}
Let us consider the cactus graph $G$ from Figure \ref{Fig_avoiding}. The set
$S^{\prime}=\{s_{1}^{\prime},s_{2}^{\prime}\}$ is an optimal smallest biactive
branch-resolving set in $G.$ But, since $C_{1}$ is both $\mathcal{ABC}$- and
$\mathcal{ADE}$-positive, the set $S^{\prime}$ is neither a vertex nor an edge
metric generator. Let $s_{3}^{\prime}$ be any vertex from $C_{1}$ which forms
a geodesic triple with two $S^{\prime}$-active vertices on $C_{1}.$ Then the
set $S=\{s_{1}^{\prime},s_{2}^{\prime},s_{3}^{\prime}\}$ is a smallest vertex
(resp. edge) metric generator, so we obtain
\[
\mathrm{dim}(G)=L(G)+B(G)+c_{\mathcal{ABC}}(G)+\tau(G_{vi})=0+2+1+0=3.
\]
and%
\[
\mathrm{edim}(G)=L(G)+B(G)+c_{\mathcal{ADE}}(G)+\tau(G_{ei})=0+2+1+0=3.
\]
\end{example}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.9]{Figure13.pdf}
\end{center}
\caption{A cactus graph $G$ from Example \ref{Example_calc}.}%
\label{Fig_example}%
\end{figure}
Let us now give an example of determining the vertex and the edge metric
dimensions on a cactus graph which is a bit bigger.
\begin{example}
\label{Example_calc}Let $G$ be the cactus graph from Figure \ref{Fig_example}.
The following table gives the choice and the number of vertices for every
expression in the formulas for metric dimensions from Theorem \ref{Tm_dim}
\[%
\begin{tabular}
[c]{|l||l|l|}\hline
& vertices & value\\\hline\hline
$L(G)$ & $s_{1},s_{2},s_{3},s_{4}$ & $4$\\\hline
$B(G)$ & $s_{5}$ & $1$\\\hline
$c_{\mathcal{ABC}}(G)$ & $s_{6}$ & $1$\\\hline
$\tau(G_{vi})$ & $s_{7}$ & $1$\\\hline
$c_{\mathcal{ADE}}(G)$ & $s_{8},s_{9}$ & $2$\\\hline
$\tau(G_{ei})$ & $s_{10},s_{11}$ & $2$\\\hline
\end{tabular}
\ \ \
\]
Therefore, the set $S=\{s_{1},s_{2},s_{3},s_{4},s_{5},s_{6},s_{7}\}$ is a
smallest vertex metric generator, so we obtain
\[
\mathrm{dim}(G)=L(G)+B(G)+c_{\mathcal{ABC}}(G)+\tau(G_{vi})=4+1+1+1=7.
\]
On the other hand, the set $S=\{s_{1},s_{2},s_{3},s_{4},s_{5},s_{8}%
,s_{9},s_{10},s_{11}\}$ is a smallest edge metric generator, so we have%
\[
\mathrm{edim}(G)=L(G)+B(G)+c_{\mathcal{ADE}}(G)+\tau(G_{ei})=4+1+2+2=9.
\]
\end{example}
Notice that $c_{\mathcal{ABC}}(G)\leq c.$ Also, if $\tau(G_{vi})\geq1$ then
$c_{\mathcal{ABC}}(G)+\tau(G_{vi})<c.$ The similar holds for $c_{\mathcal{ADE}%
}(G)$ and $\tau(G_{ei}).$ From this and Theorem \ref{Tm_dim} we immediately
obtain the following result.
\begin{corollary}
\label{Cor_boundB}Let $G$ be a cactus graph with $c$ cycles. Then
$\mathrm{dim}(G)\leq L(G)+B(G)+c$ and $\mathrm{edim}(G)\leq L(G)+B(G)+c.$
\end{corollary}
Further, notice that in a cactus graph with at least two cycles every cycle
has at least one branch-active vertex. Therefore, in such a cactus graph $G,$
we have $B(G)=\sum_{i=1}^{c}\max\{0,2-b(C_{i})\}\leq c$ with equality holding
only if $b(C_{i})=1$ for every cycle $C_{i}$ in $G$. Since $c_{\mathcal{ABC}%
}(G)+\tau(G_{vi})=c$ if and only if $c_{\mathcal{ABC}}(G)=c$ and $\tau
(G_{vi})=0,$ and similarly holds for the edge version of metric dimension,
Theorem \ref{Tm_dim} immediately implies the following simple upper bound on
the vertex and edge metric dimensions of a cactus graph $G$.
\begin{corollary}
Let $G$ be a cactus graph with $c\geq2$ cycles. Then
\[
\mathrm{dim}(G)\leq L(G)+2c
\]
with equality holding if and only if every cycle in $G$ is $\mathcal{ABC}%
$-positive and contains precisely one branch-active vertex.
\end{corollary}
\begin{corollary}
Let $G$ be a cactus graph with $c\geq2$ cycles. Then
\[
\mathrm{edim}(G)\leq L(G)+2c
\]
with equality holding if and only if every cycle in $G$ is $\mathcal{ADE}%
$-positive and contains precisely one branch-active vertex.
\end{corollary}
Notice that the upper bound from the above corollary may not hold for $c=1$,
i.e. for unicyclic graphs, as for the cycle $C$ of unicyclic graph it may hold
that $b(C)=0.$ As for the tightness of these bounds, we have the following proposition.
\begin{proposition}
For every pair of integers $b\geq0$ and $c\geq2,$ there is a cactus graph $G$
with $c$ cycles such that $L(G)=b$ and $\mathrm{dim}(G)=\mathrm{edim}%
(G)=L(G)+2c.$
\end{proposition}
\begin{proof}
For a given pair of integers $b\geq0$ and $c\geq2,$ we construct a cactus
graph $G$ in a following way. Let $G_{0}$ be a graph on $b+2$ vertices, with
one vertex $u$ of degree $b+1$ and all other vertices of degree $1$, i.e.
$G_{0}$ is a star graph. Let $H$ be a graph obtained from the $6$-cycle by
introducing a leaf to it and let $G_{1},\ldots,G_{c}$ be $c$ vertex disjoint
copies of $H$. Denote by $v_{i}$ the only vertex of degree $3$ in $G_{i}.$ Let
$G$ be a graph obtained from $G_{0},G_{1},\ldots,G_{c}$ by connecting them
with an edge $uv_{i}$ for $i=1,\ldots,c$. Obviously, $G$ is a cactus graph
with $c$ cycles and $L(G)=b$. On each of the cycles in $G$ the vertex $v_{i}$
is the only branch-active vertex. If $S\subseteq V(G)$ is a smallest
branch-resolving set in $G$ such that there is a cycle $C_{i}$ in $G$ with
only two $S$-active vertices, then because of the leaf pending on $v_{i}$ the
cycle $C_{i}$ contains either configuration $\mathcal{A}$ if the pair of
$S$-active vertices on $C_{i}$ is an antipodal pair or both configuration
$\mathcal{B}$ and $\mathcal{D}.$ Either way, $S$ is not a vertex nor an edge
metric generator.
On the other hand, the set $S$ consisting of $b$ leaves hanging at $u$ in
$G_{0}$ and a pair of vertices from each $6$-cycle which form a geodesic
triple with $v_{i}$ on the cycle is both a vertex and an edge metric generator
in $G.$ Since $\left\vert S\right\vert =b+2c=L(G)+2c,$ the claims hold.
\end{proof}
\section{An application to zero forcing number}
The results from previous section enable us to solve for cactus graphs a
conjecture posed in literature \cite{Eroh} which involves the vertex metric
dimension, the zero forcing number and the cyclomatic number $c(G)=\left\vert
E(G)\right\vert -\left\vert V(G)\right\vert +1$ (which is sometimes called the
cycle rank number and denoted by $r(G)$) of a graph $G$. Notice that in a
cactus graph $G$ the cyclomatic number $c(G)$ equals the number of cycles in
$G.$ Let us first define the zero forcing number of a graph.
Assuming that every vertex of a graph $G$ is assigned one of two colors, say
black and white, the set of vertices which are initially black is denoted by
$S.$ If there is a black vertex with only one white neighbor, then the
\emph{color-change rule} converts the only white neighbor also to black. This
is one iteration of color-change rule, it can be applied iteratively. A
\emph{zero forcing set} is any set $S\subseteq V(G)$ such that all vertices of
$G$ are colored black after applying the color-change rule finitely many
times. The cardinality of the smallest zero forcing set in a graph $G$ is
called the \emph{zero forcing number} of $G$ and it is denoted by $Z(G).$ In
\cite{Eroh} it was proven that for a unicyclic graph $G$ it holds that
$\mathrm{dim}(G)\leq Z(G)+1$, and it was further conjectured the following.
\begin{conjecture}
\label{Con_zero}For any graph $G$ it holds that $\mathrm{dim}(G)\leq
Z(G)+c(G).$
\end{conjecture}
Moreover, they proved for cacti with even cycles the bound $\mathrm{dim}%
(G)\leq Z(G)+c(G).$ We will use our results to prove that for cacti the
tighter bound from the above conjecture holds.
\begin{proposition}
Let $G$ be a cactus graph. Then $\mathrm{dim}(G)\leq Z(G)+c(G)$ and
$\mathrm{edim}(G)\leq Z(G)+c(G).$
\end{proposition}
\begin{proof}
Due to Corollary \ref{Cor_boundB} it is sufficient to prove that
$L(G)+B(G)\leq Z(G).$ Let $S\subseteq V(G)$ be a zero forcing set in $G.$ Let
us first show that $S$ must be a branch-resolving set. Assume the contrary,
i.e. that $S$ is not a branch-resolving set and let $v\in V(G)$ be a vertex of
degree $\geq3$ with at least two $S$-free threads hanging at $v.$ But then $v$
has at least two white neighbors, one on each of the $S$-free threads hanging
at it, which cannot be colored black by $S,$ so $S$ is not a zero forcing set,
a contradiction.
Let $S_{1}\subseteq S$ be a smallest branch-resolving set contained in $S$ and
let $S_{2}=S\backslash S_{1}.$ Obviously, $\left\vert S_{1}\right\vert =L(G)$
and $S_{1}\cap S_{2}=\phi.$ We now wish to prove that $\left\vert
S_{2}\right\vert \geq B(G).$ If $G$ is a tree, then the claim obviously holds,
so let us assume that $G$ contains at least one cycle. Let $C_{i}$ be a cycle
in $G$ such that $b(C_{i})\leq1.$ If $b(C_{i})=0,$ then $G$ is a unicyclic
graph and $C_{i}$ the only cycle in $G$. Since $b(C_{i})=0,$ we have
$S_{1}=\phi,$ so $S_{2}=S.$ Since a zero forcing set in unicyclic graph must
contain at least two vertices, we obtain $\left\vert S_{2}\right\vert
=\left\vert S\right\vert \geq2=B(G)$ and the claim is proven.
Assume now that for every cycle $C_{i}$ with $b(C_{i})\leq1$ it holds that
$b(C_{i})=1.$ Let $v$ be the branch-active vertex on such a cycle $C_{i}$ and
notice that $S_{1}$ can turn only $v$ black on $C_{i}.$ Therefore, in order
for $S$ to be a zero forcing set it follows that $S_{2}$ must contain a vertex
from every such cycle, i.e. $\left\vert S_{2}\right\vert \geq B(G).$
Therefore, $\left\vert S\right\vert =\left\vert S_{1}\right\vert +\left\vert
S_{2}\right\vert \geq L(G)+B(G).$
\end{proof}
The above proposition, besides proving for cacti the cycle rank conjecture
which was posed for $\mathrm{dim}(G),$ also gives a similar result for
$\mathrm{edim}(G).$ So, this motivates us to pose for $\mathrm{edim}(G)$ the
counterpart conjecture of Conjecture \ref{Con_zero}.
\section{Concluding remarks}
In \cite{SedSkreBounds} it was established that for a unicyclic graph $G$ both
vertex and edge metric dimensions are equal to $L(G)+\max\{2-b(G),0\}$ or
$L(G)+\max\{2-b(G),0\}+1.$ In \cite{SedSkreUnicyclic} a characterization under
which both of the dimensions take one of the two possible values was further
established. In this paper we extend the result to cactus graphs where a
similar characterization must hold for every cycle in a graph, and also the
additional characterization for the connection of two cycles must be
introduced. This result enabled us to prove the cycle rank conjecture for
cactus graphs.
Moreover, the results of this paper enabled us to establish a simple upper
bound on the value of the vertex and the edge metric dimension of a cactus
graph $G$ with $c$ cycles%
\[
\mathrm{dim}(G)\leq L(G)+2c\quad\hbox{ and }\quad\mathrm{edim}(G)\leq
L(G)+2c.
\]
Since the number of cycles can be generalized to all graphs as the cyclomatic
number $c(G)=\left\vert E(G)\right\vert -\left\vert V(G)\right\vert +1,$ we
conjecture that the analogous bounds hold in general.
\begin{conjecture}
Let $G$ be a connected graph. Then, $\mathrm{dim}(G)\leq L(G)+2c(G).$
\end{conjecture}
\begin{conjecture}
Let $G$ be a connected graph. Then, $\mathrm{edim}(G)\leq L(G)+2c(G).$
\end{conjecture}
In \cite{SedSkrekMixed} it was shown that the inequality $\mathrm{mdim}%
(G)<2c(G)$ holds for $3$-connected graphs. Since $\mathrm{dim}(G)\leq
\mathrm{mdim}(G)$ and $\mathrm{edim}(G)\leq\mathrm{mdim}(G),$ the previous two
conjectures obviously hold for $3$-connected graphs.
Also, motivated by the bound on edge metric dimension of cacti involving zero
forcing number, we state the following conjecture for general graphs, as a
counterpart of Conjecture \ref{Con_zero}.
\begin{conjecture}
Let $G$ be a connected graph. Then, $\mathrm{edim}(G)\leq Z(G)+c(G).$
\end{conjecture}
\bigskip
\bigskip\noindent\textbf{Acknowledgments.}~~Both authors acknowledge partial
support of the Slovenian research agency ARRS program\ P1-0383 and ARRS
projects J1-1692 and J1-8130. The first author also the support of Project
KK.01.1.1.02.0027, a project co-financed by the Croatian Government and the
European Union through the European Regional Development Fund - the
Competitiveness and Cohesion Operational Programme.
|
3,212,635,537,938 | arxiv |
\section{Preliminaries}
I skip the basic definitions.
\noindent \textbf{Allocation function, social welfare, VCG payments}.
\begin{itemize}
\item Vector of bids: $\boldsymbol{b} \in \mathbb{R}^N$
\item Allocation function: $f: \mathcal{N}\rightarrow \mathcal{N}$ from ads to slots
\item Social welfare: $$\text{SW}(f; \boldsymbol{b}) = \sum_{i\in\mathcal{N}} \Lambda_{f(i)}q_i b_i$$
\item Optimal allocation: $$f^*(\cdot; \boldsymbol{b}) = \arg\max_f \text{SW}(f; \boldsymbol{b})$$
\item Payments for the optimal allocation:
\begin{align}
p_i^W(\boldsymbol{b}) &= \text{SW}(f^*_{-i}(\cdot;\boldsymbol{b});\boldsymbol{b}) - \text{SW}_{-i}(f^*(\cdot;\boldsymbol{b});\boldsymbol{b}) \label{eq:pay.vcg.optimal.sw}\\
&=\sum_{j\neq i} \Lambda_{f^*_{-i}(j;\boldsymbol{b})} q_jb_j - \sum_{j\neq i}\Lambda_{f^*(j;\boldsymbol{b})}q_jb_j \label{eq:pay.vcg.optimal.ref}\\
&= \Lambda_{f^*(i;\boldsymbol{b})}q_ib_i - \int_{0}^{b_i} \Lambda_{f^*(i; \boldsymbol{b}_{-i},u)}q_i du.\label{eq:pay.vcg.optimal.tardos}
\end{align}
\item Contingent payments
\begin{align*}
\tilde{p}_i^W(\boldsymbol{b}) =
\begin{cases} \frac{p_i^W(\boldsymbol{b})}{q_i\Lambda_{f^*(i;\boldsymbol{b})}} &\mbox{if $click$ at $f^*(i;\boldsymbol{b})$} \\
0 & \mbox{otherwise}, \end{cases}
\end{align*}
so that $E_c[\tilde{p}_i^W(\boldsymbol{b})] = p_i^W(\boldsymbol{b})$.
\end{itemize}
\noindent \textbf{Known prominence, unknown quality}.
\begin{itemize}
\item Estimator $$ \widehat{q}_i = \frac{1}{S_{i1}} \sum_{s \in \mathcal{S}_{i1}} click^i_1(s)$$
\item Concentration bound $$\left| q_i - \widehat{q}_i \right| \leq \sqrt{\frac{N}{\tau} \log{\frac{2N}{\delta}}}:= \eta_{q}$$
\item Upper-confidence bound $$\hat{q}_i^+=\min\{\hat{q}_i+\eta_q, 1\}$$
\end{itemize}
\begin{itemize}
\item Empirical social welfare: $$\widehat{\text{SW}}(f; \boldsymbol{b}) = \sum_{i\in\mathcal{N}} \Lambda_{f(i)}\tilde{q}^+_i b_i$$
\item Empirical optimal allocation
$$ \hat{f}(\cdot; \boldsymbol{b}) = \arg\max_f \widehat{\text{SW}}(f; \boldsymbol{b}) = \arg\max_f \sum_{i\in\mathcal{N}} \frac{\tilde{q}^+_i}{q_i} \Lambda_{f(i)} q_ib_i = \arg\max_f \sum_{i\in\mathcal{N}} w_i v_i(f),$$
where $w_i = \frac{\tilde{q}^+_i}{q_i}$ is an allocation-independent weight and $v_i(f)$ is an allocation-dependent value.
\item Payments for the empirical optimal allocation:
\begin{align}
p_i^\Lambda(\boldsymbol{b}) &= \text{SW}(\hat{f}_{-i}(\cdot;\boldsymbol{b});\boldsymbol{b}) - \text{SW}_{-i}(\hat{f}(\cdot;\boldsymbol{b});\boldsymbol{b}) \label{eq:pay.vcg.emp.sw1}\\
&=\Lambda_{\hat{f}(i;\boldsymbol{b})}q_ib_i - \int_{0}^{b_i} \Lambda_{\hat{f}(i; \boldsymbol{b}_{-i},u)}q_i du \label{eq:pay.vcg.emp.tardos}\\
&= \frac{q_i}{\tilde{q}^+_i} \Big[\widehat{\text{SW}}(\hat{f}_{-i}(\cdot;\boldsymbol{b});\boldsymbol{b})) - \widehat{\text{SW}}_{-i}(\hat{f}(\cdot;\boldsymbol{b});\boldsymbol{b}))\Big] \label{eq:pay.vcg.emp.sw2}
\end{align}
\item Contingent payments
\begin{align*}
\tilde{p}_i^\Lambda(\boldsymbol{b}) =
\begin{cases} \frac{p_i^L(\boldsymbol{b})}{q_i\Lambda_{f^*(i;\boldsymbol{b})}} = \frac{1}{\tilde{q}^+_i\Lambda_{f^*(i;\boldsymbol{b})}} \Big[\widehat{\text{SW}}(\hat{f}_{-i}(\cdot;\boldsymbol{b});\boldsymbol{b})) - \widehat{\text{SW}}_{-i}(\hat{f}(\cdot;\boldsymbol{b});\boldsymbol{b}))\Big] &\mbox{if $click$ at $\hat{f}(i;\boldsymbol{b})$} \\
0 & \mbox{otherwise}, \end{cases}
\end{align*}
which is computable.
\end{itemize}
\noindent \textbf{Mechanism Babaioff}.
\noindent \textit{Remark:} if $\Lambda$ is unknown, the previous contingent payments cannot be computed and $\hat{f}$ cannot be written as the solution to an affine optimization problem, thus we cannot use the weighted VCG mechanism. We need a different mechanism.
\begin{itemize}
\item Given a parameter $\mu$, for each ad $i$ with original bid $b_i$, compute $(x_i, y_i)$ as
\begin{align*}
(x_i,y_i) &= \begin{cases} (b_i,b_i) & \mbox{w.p. } 1-\mu \\ (b''_i,b'_i) \mbox{ with } b_i'\sim\mathcal{U}([0,b_i]), b_i''=\text{rec}(b_i') & \mbox{otherwise }\end{cases}
\end{align*}
\item Compute the optimal allocation $f^*(\cdot; \mathbf{x})$.
\item Babaioff payments
\begin{align}\label{eq:pay.babaioff}
\bar{p}_i^B(\mathbf{x},\mathbf{y};\boldsymbol{b}) = \Lambda_{f^*(i;\mathbf{x})}q_ib_i - R_i^B(\mathbf{x},\mathbf{y};\boldsymbol{b}),
\end{align}
with rebate
\begin{align}\label{eq:pay.babaioff}
R_i^B(\mathbf{x},\mathbf{y};\boldsymbol{b}) =
\begin{cases} \frac{1}{\mu}\Lambda_{f^*(i;\mathbf{x})}q_ib_i &\mbox{if $y_i<b_i$} \\
0 & \mbox{otherwise}, \end{cases}
\end{align}
\item Expected Babaioff payments (equivalent to Tardos for the randomized allocation)
\begin{align}\label{eq:pay.babaioff.exp}
p_i^B(\boldsymbol{b}) = \mathbb{E}_{\mathbf{x},\mathbf{y}}[\bar{p}_i^B(\mathbf{x},\mathbf{y};\boldsymbol{b})] = \mathbb{E}_{\mathbf{x}}[\Lambda_{f^*(i;\mathbf{x})}|\boldsymbol{b}] q_ib_i - \int_{0}^{b_i} \mathbb{E}_{\mathbf{x}}[\Lambda_{f^*(i; \mathbf{x})}|(\boldsymbol{b}_{-i},u)]q_i du
\end{align}
The critical part of the proof is the $\mathbb{E}_{\mathbf{x},\mathbf{y}}[R_i^B(\mathbf{x},\mathbf{y};\boldsymbol{b})|\boldsymbol{b}]$
\begin{align*}
\mathbb{E}_{\mathbf{x},\mathbf{y}}&[R_i^B(\mathbf{x},\mathbf{y};\boldsymbol{b})|\boldsymbol{b}] = \mathbb{E}_{\mathbf{x},y_i}[R_i^B(\mathbf{x},y_i;\boldsymbol{b})|\boldsymbol{b}] = \mathbb{E}_{y_i} \big[\mathbb{E}_{\mathbf{x}|y_i}[R_i^B(\mathbf{x},y_i;\boldsymbol{b})|\boldsymbol{b}]\big]\\
&= \mathbb{P}[y_i=b_i]\big[\mathbb{E}_{\mathbf{x}|y_i}[R_i^B(\mathbf{x},b_i;\boldsymbol{b})|\boldsymbol{b}] + \mathbb{P}[y_i< b_i]\mathbb{E}_{y_i|y_i<b_i} \big[\mathbb{E}_{\mathbf{x}|y_i}[R_i^B(\mathbf{x},y_i;\boldsymbol{b})|\boldsymbol{b}]\big]\\
&= 0+ \mu\mathbb{E}_{y_i|y_i<b_i} \big[\mathbb{E}_{\mathbf{x}|y_i}[\frac{1}{\mu}\Lambda_{f^*(i;\mathbf{x})}q_ib_i|\boldsymbol{b}]\big]\\
&= \mathbb{E}_{y_i|y_i<b_i} \big[\mathbb{E}_{\mathbf{x}|y_i}[\Lambda_{f^*(i;\mathbf{x})}q_ib_i|\boldsymbol{b}]\big]\\
&= \int_{0}^{b_i} \mathbb{E}_{\mathbf{x}|y_i=u}[\Lambda_{f^*(i;\mathbf{x})}q_ib_i|\boldsymbol{b}]\frac{1}{b_i} du\\
&= \int_{0}^{b_i} \mathbb{E}_{\mathbf{x}|y_i=u}[\Lambda_{f^*(i;\mathbf{x})}|\boldsymbol{b}] q_idu\\
&= \int_{0}^{b_i} \mathbb{E}_{\mathbf{x}}[\Lambda_{f^*(i;\mathbf{x})}|(\boldsymbol{b}_{-i},u)] q_idu
\end{align*}
\item Babaioff contingent payments
\begin{align*}
\tilde{p}_i^B(\mathbf{x},\mathbf{y};\boldsymbol{b}) &= \begin{cases} \frac{\bar{p}_i^B(\mathbf{x},\mathbf{y};\boldsymbol{b})}{\Lambda_{f^*(i;\mathbf{x})}q_i} & \mbox{if $click$ at $f^*(i;\mathbf{x})$} \\ 0 & \mbox{otherwise} \end{cases}\\
&= \begin{cases} b_i -
\begin{cases} \frac{1}{\mu}b_i &\mbox{if $y_i<b_i$} \\
0 & \mbox{otherwise}, \end{cases}
& \mbox{if $click$ at $f^*(i;\mathbf{x})$} \\ 0 & \mbox{otherwise} \end{cases}
\end{align*}
thus $\mathbb{E}_{c}[\tilde{p}_i^B(\mathbf{x},\mathbf{y};\boldsymbol{b})] = \bar{p}_i^B(\mathbf{x},\mathbf{y};\boldsymbol{b})$ and $\mathbb{E}_{\mathbf{x},\mathbf{y}}[\mathbb{E}_{c}[\tilde{p}_i^B(\mathbf{x},\mathbf{y};\boldsymbol{b})]]=p_i^B(\boldsymbol{b})$. Notice that the click event $\{click \text{ at } f^*(i; \mathbf{x})\}$ displays a direct dependency on $f^*$ and the resampled bids $\mathbf{x}$. This means that the expectation wrt to the click event and the randomness of the mechanism cannot be swapped.
\end{itemize}
\noindent \textbf{Learning mechanism Babaioff}.
\begin{itemize}
\item During the exploitation phase, instead of $f^*$ we use the empirical optimal allocation $\hat{f}$.
\item Learning Babaioff payments
\begin{align*}
\bar{p}_i^L(\mathbf{x},\mathbf{y};\boldsymbol{b}) = \Lambda_{\hat{f}(i;\mathbf{x})}q_ib_i -
\begin{cases} \frac{1}{\mu}\Lambda_{\hat{f}(i;\mathbf{x})}q_ib_i &\mbox{if $y_i<b_i$} \\
0 & \mbox{otherwise}, \end{cases}
\end{align*}
\item Expected learning Babaioff payments
\begin{align}\label{eq:pay.learn.babaioff.exp}
p_i^L(\boldsymbol{b}) = \mathbb{E}_{\mathbf{x},\mathbf{y}}[p_i^L(\mathbf{x},\mathbf{y};\boldsymbol{b})] = \mathbb{E}_{\mathbf{x}}[\Lambda_{\hat{f}(i;\mathbf{x})}|\boldsymbol{b}] q_ib_i - \int_{0}^{b_i} \mathbb{E}_{\mathbf{x}}[\Lambda_{\hat{f}(i; \mathbf{x})}|(\boldsymbol{b}_{-i},u)]q_i du
\end{align}
\item Learning Babaioff contingent payments
\begin{align*}
\tilde{p}_i^L(\mathbf{x},\mathbf{y};\boldsymbol{b}) &= \begin{cases} \frac{\bar{p}_i^L(\mathbf{x},\mathbf{y};\boldsymbol{b})}{\Lambda_{\hat{f}(i;\mathbf{x})}q_i} & \mbox{if $click$ at $\hat{f}(i;\mathbf{x})$} \\ 0 & \mbox{otherwise} \end{cases}\\
&= \begin{cases} b_i -
\begin{cases} \frac{1}{\mu}b_i &\mbox{if $y_i<b_i$} \\
0 & \mbox{otherwise}, \end{cases}
& \mbox{if $click$ at $\hat{f}(i;\mathbf{x})$} \\ 0 & \mbox{otherwise} \end{cases}
\end{align*}
thus $\mathbb{E}_{c}[\tilde{p}_i^L(\mathbf{x},\mathbf{y};\boldsymbol{b})] = \bar{p}_i^L(\mathbf{x},\mathbf{y};\boldsymbol{b})$ and $\mathbb{E}_{\mathbf{x},\mathbf{y}}[\mathbb{E}_{c}[\tilde{p}_i^L(\mathbf{x},\mathbf{y};\boldsymbol{b})]]=p_i^L(\boldsymbol{b})$.
\item Modification vector $\mathbf{s}\in\{0,1\}^N$
$$ s_i = \begin{cases} 1 & \mbox{if } x_i=b_i \\ 0 & \mbox{otherwise (i.e., $x_i\in[0,b_i)$)}\end{cases}$$
Counters of the modification vector $S_1 = \sum_{i}s_i$, $S_0 = N-S_1$.\\
Conditional expectation of the modifications
$$\mathbb{E}_{\mathbf{x}|\mathbf{s}}[\cdot|\boldsymbol{b}]$$
refers to the expectation over the randomized mechanism given the initial bids $\boldsymbol{b}$ and the modification vector $\mathbf{s}$. Notice that for all the ads $i$ for which $s_i=1$ the corresponding $x_i$ has a Dirac distribution over $b_i$, while it has a specific distribution over $[0,b_i)$ if $s_i=0$.
\end{itemize}
\section{Regret Analysis}
\subsection{Per-ad regret definition in the exploitation phase}
\begin{align}
r_i(\boldsymbol{b}) = p_i^W(\boldsymbol{b}) - p_i^L(\boldsymbol{b}) = \underbrace{p_i^W(\boldsymbol{b}) - p_i^B(\boldsymbol{b})}_{\text{Babaioff regret}} + \underbrace{p_i^B(\boldsymbol{b}) - p_i^L(\boldsymbol{b})}_{\text{learning regret}} = r_i^B(\boldsymbol{b}) + r_i^L(\boldsymbol{b}).
\end{align}
\subsection{Babaioff Regret}
For any arbitrary vector $\mathbf{z}\in\Re^N$ we have
\begin{align*}
\mathbb{E}_{\mathbf{x}}[\Lambda_{f^*(i;\mathbf{x})}|\mathbf{z}] = \mathbb{P}[\mathbf{s}=\boldsymbol{1}] \Lambda_{f^*(i; \boldsymbol{b})} + \mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] \mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{f^*(i;\mathbf{x})}|\mathbf{z}]
\end{align*}
We rewrite the expected Babaioff payments (eq.~\ref{eq:pay.babaioff.exp}) as
\begin{align*}
p_i^B(\boldsymbol{b}) &= \bigg(\mathbb{P}[\mathbf{s}=\boldsymbol{1}] \Lambda_{f^*(i; \boldsymbol{b})} + \mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] \mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{f^*(i;\mathbf{x})}|\boldsymbol{b}]\bigg)q_ib_i \\
&\quad - \int_{0}^{b_i} \bigg(\mathbb{P}[\mathbf{s}=\boldsymbol{1}] \Lambda_{f^*(i; (\boldsymbol{b}_{-i},u))} + \mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] \mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{f^*(i;\mathbf{x})}|\boldsymbol{b}_{-i},u]\bigg)q_idu\\
&=\mathbb{P}[\mathbf{s}=\boldsymbol{1}] \bigg(\Lambda_{f^*(i; \boldsymbol{b})}q_ib_i - \int_{0}^{b_i}\Lambda_{f^*(i; (\boldsymbol{b}_{-i},u))} q_idu\bigg)\\
&\quad+\mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] \bigg(\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{f^*(i;\mathbf{x})}|\boldsymbol{b}]q_ib_i - \int_{0}^{b_i}\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{f^*(i;\mathbf{x})}|\boldsymbol{b}_{-i},u] q_idu\bigg)\\
&=\mathbb{P}[\mathbf{s}=\boldsymbol{1}] p_i^W(\boldsymbol{b}) + \mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] \bigg(\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{f^*(i;\mathbf{x})}|\boldsymbol{b}]q_ib_i - \int_{0}^{b_i}\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{f^*(i;\mathbf{x})}|\boldsymbol{b}_{-i},u] q_idu\bigg)
\end{align*}
The per-ad regret is
\begin{align*}
r_i^B&(\boldsymbol{b}) = p_i^W(\boldsymbol{b}) - p_i^B(\boldsymbol{b}) \\
&= p_i^W(\boldsymbol{b}) - \mathbb{P}[\mathbf{s}=\boldsymbol{1}] p_i^W(\boldsymbol{b}) - \mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] \bigg(\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{f^*(i;\mathbf{x})}|\boldsymbol{b}]q_ib_i - \int_{0}^{b_i}\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{f^*(i;\mathbf{x})}|\boldsymbol{b}_{-i},u] q_idu\bigg)\\
&= \mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] p_i^W(\boldsymbol{b}) - \mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] \bigg(\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{f^*(i;\mathbf{x})}|\boldsymbol{b}]q_ib_i - \int_{0}^{b_i}\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{f^*(i;\mathbf{x})}|\boldsymbol{b}_{-i},u] q_idu\bigg)\\
&\leq \mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] p_i^W(\boldsymbol{b}) + \mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] \bigg|\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{f^*(i;\mathbf{x})}|\boldsymbol{b}]q_ib_i - \int_{0}^{b_i}\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{f^*(i;\mathbf{x})}|\boldsymbol{b}_{-i},u] q_idu\bigg|\\
&\leq \mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] (p_i^W(\boldsymbol{b})+b_{\max})\\
&\leq 2\mathbb{P}\big[\exists j: s_j = 0\big] b_{\max}\\
&\leq 2\sum_{j\in\mathcal{N} }\mathbb{P}[s_j = 0] b_{\max} = 2N\mu b_{\max}
\end{align*}
\subsection{Learning Regret}
For any $\mathbf{x}$ we have
\begin{small}
\begin{align*}
d_{\text{SW}}&(f^*, \hat{f}; \mathbf{x}) = \text{SW}(f^*(\cdot;\mathbf{x}); \mathbf{x}) - \text{SW}(\hat{f}(\cdot;\mathbf{x}); \mathbf{x})\\
&= \text{SW}(f^*(\cdot;\mathbf{x}); \mathbf{x}) - \widehat{\text{SW}}(f^*(\cdot;\mathbf{x}); \mathbf{x}) + \widehat{\text{SW}}(f^*(\cdot;\mathbf{x}); \mathbf{x}) - \widehat{\text{SW}}(\hat{f}(\cdot;\mathbf{x}); \mathbf{x}) + \widehat{\text{SW}}(\hat{f}(\cdot;\mathbf{x}); \mathbf{x}) - \text{SW}(\hat{f}(\cdot;\mathbf{x}); \mathbf{x})\\
&\leq \text{SW}(f^*(\cdot;\mathbf{x}); \mathbf{x}) - \widehat{\text{SW}}(f^*(\cdot;\mathbf{x}); \mathbf{x}) + \widehat{\text{SW}}(\hat{f}(\cdot;\mathbf{x}); \mathbf{x}) - \text{SW}(\hat{f}(\cdot;\mathbf{x}); \mathbf{x})\\
&= \sum_{i\in\mathcal{N}} \Lambda_{f^*(i;\mathbf{x})}b_i (q_i-\tilde{q}^+_i) + \sum_{i\in\mathcal{N}} \Lambda_{\hat{f}(i;\mathbf{x})}b_i (\tilde{q}^+_i-q_i)\\
&\leq \sum_{i\in\mathcal{N}} \Lambda_{\hat{f}(i;\mathbf{x})}b_i 2\eta_q \leq 2Kb_{\max}\eta_q
\end{align*}
\end{small}
\noindent Similarly to the previous section, we write the learning Babaioff expected payments (eq.~\ref{eq:pay.learn.babaioff.exp}) as
\begin{align*}
p_i^L(\boldsymbol{b}) =\mathbb{P}[\mathbf{s}=\boldsymbol{1}] p_i^\Lambda(\boldsymbol{b}) + \mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] \bigg(\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\hat{f}(i;\mathbf{x})}|\boldsymbol{b}]q_ib_i - \int_{0}^{b_i}\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\hat{f}(i;\mathbf{x})}|\boldsymbol{b}_{-i},u] q_idu\bigg)
\end{align*}
\noindent Then the per-ad regret is
\begin{align*}
r_i^L &= p_i^B(\boldsymbol{b}) - p_i^L(\boldsymbol{b})\\
&= \mathbb{P}[\mathbf{s}=\boldsymbol{1}] (p_i^W(\boldsymbol{b})- p_i^\Lambda(\boldsymbol{b})) \\
&\quad + \mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] \bigg(\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{f^*(i;\mathbf{x})}|\boldsymbol{b}]q_ib_i - \int_{0}^{b_i}\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{f^*(i;\mathbf{x})}|\boldsymbol{b}_{-i},u] q_idu\\
&\quad\quad\quad\quad\quad\quad\quad -\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\hat{f}(i;\mathbf{x})}|\boldsymbol{b}]q_ib_i + \int_{0}^{b_i}\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\hat{f}(i;\mathbf{x})}|\boldsymbol{b}_{-i},u] q_idu\bigg)\\
&\leq (p_i^W(\boldsymbol{b})- p_i^\Lambda(\boldsymbol{b}))\\
&\quad + \mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] \bigg|\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{f^*(i;\mathbf{x})}|\boldsymbol{b}]q_ib_i - \int_{0}^{b_i}\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{f^*(i;\mathbf{x})}|\boldsymbol{b}_{-i},u] q_idu\\
&\quad\quad\quad\quad\quad\quad\quad -\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\hat{f}(i;\mathbf{x})}|\boldsymbol{b}]q_ib_i + \int_{0}^{b_i}\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\hat{f}(i;\mathbf{x})}|\boldsymbol{b}_{-i},u] q_idu\bigg|\\
&\leq (p_i^W(\boldsymbol{b})- p_i^\Lambda(\boldsymbol{b})) + N\mu b_{\max}\\
\end{align*}
\noindent Moving to the global regret, the first term is bound as in the previous paper, while the second is just summed up over $N$.
\section{Preliminaries}\label{s:notation}
\subsection{Economic Mechanisms}
\input{sec/021economicMechanisms}
\subsection{Multi--Armed Bandit}
\input{sec/022MultiArmedBandit}
\section{Monotonicity and Myerson's payments} \label{ap:monotonicity}
Consider a generic direct--revelation mechanism $M = (\mathcal{N}, \Theta, V, f, \{p_i\}_{i \in \mathcal{N}})$ as defined in Section~\ref{ssec:md}. In single--parameter linear environments, i.e., when
\begin{itemize}
\item the type of each agent $i$ is a scalar $v_i$ (single--parameter assumption),
\item utility function of agent~$i$ is $u_i(\hat{\mathbf{v}}) = z_i(f(\hat{\mathbf{v}})) v_i - p_i(\hat{\mathbf{v}})$ where $z_i$ is a function defined on the allocation (linear assumption),
\end{itemize}
monotonicity requires that $z_i(f(\hat{\mathbf{v}}_{-i}, v_i'')) \geq z(f(\hat{\mathbf{v}}_{-i}, v_i''))$ as $v_i''\geq v_i'$. In such environments, it is possible to design a DSIC mechanism imposing the following payments~\cite{tardos_sp}:
\begin{equation} \label{eq:p_tardos}
p_i(\hat{\mathbf{v}}) = h_i(\hat{\mathbf{v}}_{-i}) + z_i(f(\hat{\mathbf{v}})) \hat{v}_i - \int_{0}^{\hat{v}_i} z_i(f(\hat{\mathbf{v}}_{-i}, u)) du
\end{equation}
\noindent where $h_i(\hat{\mathbf{v}}_{-i})$ is a generic function not depending on the type of agent~$i$.
\section{Related Works}\label{s:related}
\section{Contextual Multi--Slot Auctions}\label{s:context}
In real--world SSAs, the characteristics of the auctions (e.g., the quality) highly depend on contextual information such as the content of the webpage, the query submitted by the user, and her profile. In this section, we further generalize the auction with externalities to the case of contextual auctions. More precisely, we denote by $\mathcal{X}$ a subset of the Euclidean space $\mathbb{R}^s$ and we assume that $x\in\mathcal{X}$ summarizes all the contextual information necessary to define the auction. In particular, for each ad $i$, the quality is a function $q_i: \mathcal{X} \rightarrow [0,1]$, while we assume that the values $v_i$ and the discount factors $\Gamma_k(f)$ are independent from $x$.\footnote{The generalization to $v_i(x)$ and $\Gamma_k(f;x)$ is straightforward.} When the functions $q_i$ are known in advance, for each $x$ we can still apply the VCG and charge adv $a_i$ with a payment
\begin{align*}
p_i(x) = \text{SW}(f^*_{-i}; x) - \text{SW}_{-i}(f^*; x),
\end{align*}
where $\text{SW}(f; x)$ is defined according to the qualities $q_i(x)$. The learning algorithm should now approximate each quality function over the whole domain $\mathcal{X}$. Although any regression algorithm could be employed to approximate $q_i$, here we consider a least squares approach for which performance bounds are available (see e.g.,~\cite{gyorfi2002distribution}). We denote by $\phi(\cdot)=\big(\varphi_1(\cdot),\ldots,\varphi_d(\cdot)\big)^\top$ a $d$-dimensional feature vector with features $\varphi_i : \mathcal X \rightarrow [0,1]$, and by $\mathcal{A}=\{\alpha_w(\cdot)=\phi(\cdot)^\top w\}$ the linear space of functions spanned by the basis functions in $\phi$. Similar to the previous settings, the algorithm first explores all the advs for $\tau$ rounds with an arbitrary exploration allocation $f_t$. At each round $t$, we assume a context $x_t$ to be independently drawn from a stationary distribution $\mu$ over the context space $\mathcal{X}$. At the end of the exploration phase, each ad $i$ has been impressed $S_i=K\tau/N$ times and we build the training set $\{x_{s}, c_{f_s(i)}^i(s)\}_{s=1}^{S_i}$ where $c_{f_s(i)}^i(s)$ is the click--event for ad $i$ when displayed at slot $f_s(i)$ in context $x_s$, and we compute the approximation
\begin{align*}
\tilde{q}_i = \alpha_{\hat w_i} = \arg\min_{\alpha\in\mathcal{A}} \sum_{s=1}^{S_i} \Big(\alpha(x_s) - \frac{c_{f_s(i)}^i(s)}{\Gamma_{f_s(i)}(f_s)} \Big)^2.
\end{align*}
Since $\mathcal{A}$ is linear, we can easily compute the coefficient vector $\hat w_i$ in closed form. Let $\Phi_i=[\phi(x_{s})^\top;\ldots;\phi(x_{S_i})^\top]$ be the feature matrix corresponding to the training set and $c_i = (\frac{c_{f_1(i)}^i(1)}{\Gamma_{f_1(i)}(f_1)}, \ldots, \frac{c_{f_1(S_i)}^{i}(S_i)}{\Gamma_{f_{S_i}(i)}(f_{S_i})})$ be the re-weighted vector of observations. Then we have
\begin{align}\label{eq:hat.w}
\hat w_i = (\Phi_i^\top \Phi_i)^{-1} \Phi_i^\top c_i.
\end{align}
During the exploitation phase, for any $x$, the A--VCG uses the quality $\tilde{q}_i(x)$ to compute the allocation and define the payments. In particular, the estimated social welfare in a context $x$ is defined as $\widehat{\text{SW}}(f; x) = \sum_{i=1}^n \Gamma_{f(i)}(f) \tilde{q}_i(x) v_i$ and the expected payments become
\begin{align*}
\hat{p}_i(x) = \big( \widehat{\text{SW}}(\hat{f}_{-i};x) - \widehat{\text{SW}}_{-i}(\hat{f};x) \big) \frac{q_i(x)}{\tilde{q}_i(x)}.
\end{align*}
Unlike the settings considered in the previous sections, we cannot expect to minimize the regret in each possible context $x\in\mathcal{X}$, thus we redefine the regret as the expectation w.r.t. the context distribution $\mu$
\begin{align*
R_{T,\mu} = T\mathbb{E}_{x\sim\mu} \Big[\sum_{i=1}^N p_i(x)\Big] - \sum_{t=1}^T\mathbb{E}_{x\sim\mu}\Big[ \sum_{i=1}^N p_{it}(x)\Big],
\end{align*}
where $p_{it}$ is equal to $0$ for the first $t\leq \tau$ explorative rounds and is equal to $\hat{p}_i$ during the exploitation phase.
In order to derive the regret bound, we need two technical assumptions (we further discuss them in the remarks).
\begin{assumption}\label{a:realizable}
The function space $\mathcal{A}$ contains all the quality functions $q_i$ (i.e., the approximation error of $\mathcal{A}$ is 0), that is for any $i$
\begin{align*}
\inf_{\alpha\in\mathcal{A}} || \alpha - q_i ||_{\mu} = 0.
\end{align*}
\end{assumption}
\begin{assumption}\label{a:lower-bound}
The function space $\mathcal{A}$ is such that for any $\alpha\in\mathcal{A}$, $\left\| 1/\alpha^2 \right\|_{\mu} \leq \xi.$
\end{assumption}
It is worth noting that here we no longer use an upper--confidence bound on $q_i$ as before. In fact, it is not possible to build an upper--confidence bound for each context $x$ since the accuracy of the approximated functions $\tilde{q}_i$ can only be bounded in expectation w.r.t. the context distribution $\mu$, as reported in the following lemma~\cite{gyorfi2002distribution}.\footnote{We recall that the $\mu$--weighted $L_2$--norm of a function $\alpha$ is defined as $||\alpha||_{\mu}^2 = \mathbb{E}_{x\sim\mu}[\alpha(x)^2]$.}
\begin{lemma}\label{lem:least.squares}
Let $\alpha_{\hat w_i}$ be computed as in (\ref{eq:hat.w}) with $S_i = K\tau/N$ samples in a $d$-dimensional linear space $\mathcal{A}$, then for any $i\in\mathcal{N}$
\begin{equation}\label{eq:least.squares}
||\alpha_{\hat w_i} - q_i||_{\mu} \leq \frac{64}{\Gamma_{\min}} \sqrt{\frac{(d+1)N}{K\tau} \log\big(\frac{324NTe^2}{\delta}\big) }:= \chi
\end{equation}
with probability $1-\delta$ (w.r.t. the random contexts and clicks), where $\Gamma_{\min} = \min_{f, k} \Gamma_k(f)$.
\end{lemma}
Given the two assumptions and Lemma~\ref{lem:least.squares}, we can now derive the following regret bound.
\begin{theorem}\label{thm:context}
Let us consider a contextual auction on the domain $\mathcal{X}$ with $n$ adv, $K$ slots, and $T$ rounds. Let $\mu$ be a distribution over $\mathcal{X}$. The auction have position/ad--dependent externalities and cumulative discount factors $\{\Gamma_k(f)\}_{k=1}^K$. For any parameter $\tau$ and $\delta$, the A--VCG is always truthful and it achieves a regret
\begin{align*}
R_{T,\mu} \leq v_{\max} K \left[ 6\xi(T - \tau) \chi + \tau + \delta T \right].
\end{align*}
By setting the parameters to
\begin{align*}
\delta &= n(TK)^{-1/3} \\
\tau &= 24^{1/3} T^{2/3} \Gamma_{\min}^{-2/3} K^{-1/3} n (d+1)^{1/3}(\log{(K^{1/3}T^{4/3}}))^{1/3},
\end{align*}
the corresponding regret is
\begin{align*
R_{T,\mu} \leq 24^{1/3} v_{\max}\xi T^{2/3} \Gamma_{\min}^{-2/3} K^{2/3} n (d+1)^{1/3} (\log{(KT}))^{1/3}.
\end{align*}
\end{theorem}
\myremark{1 (Bound).} As we can notice, the bound obtained in the contextual case has exactly the same dependency on $T$, $K$, and $N$ as in the regret in (\ref{eq:regret.extern}). The main difference is that two additional terms appear in the bound, the dimensionality of the space $d$ and the lower--bound $\xi$ on the functions in $\mathcal{A}$. It is interesting to notice that the regret grows as $\tilde O(d^{1/3})$ implying that the larger the number of features in $\mathcal{F}$, the worse the regret. This dependency is an immediate result of the fact that in (\ref{eq:hat.w}) we learn a $d$-dimensional vector $\hat w_i$ and as $d$ increases, the number of samples needed to have an accurate estimate of $q_i$ increases as well, thus lengthening the exploration phase. Finally, the term $\xi$ (see Assumption~\ref{a:lower-bound}) plays a similar role as $q_{\min}^{-1}$ since it bounds the norm of the inverse of the functions in $\mathcal{A}$.\\
\myremark{2 (Assumptions).} Assumption~\ref{a:realizable} is the so--called \textit{realizable} assumption, which implies that the functions to be approximated (the qualities $q_i$) belong to the function space $\mathcal{A}$. This assumption is reasonable whenever some prior knowledge about the ads is available and the features can be properly designed. Similarly, Assumption~\ref{a:lower-bound} is strictly related to the way the function space $\mathcal{A}$ is designed. In fact, it requires any function $\alpha\in\mathcal{A}$ to be lower--bounded away from $0$. This could be easily achieved by thresholding the prediction of each $\alpha$. It is also worth noting that the two assumptions together imply that for any $i$, $\min_{x\in X}q_i(x) \geq \xi$, i.e., the advs have a quality which is at least $\xi$ over the whole context space $\mathcal{X}$, thus suggesting that below a certain threshold the advs would not participate to the auction at all. \\
\myremark{3 (Regression algorithm).} Although in describing the algorithm we refer to least squares, any regression algorithm such as neural networks or logistic regression could be used. Nonetheless, in order to derive the regret bound, a specific replacement for Lemma~\ref{lem:least.squares} is needed.
\section{Conclusions and Future Work}\label{s:conclusions}
In this paper, we studied the problem of learning the click through rates of ads in sponsored search auctions with truthful mechanisms. This problem is highly challenging, combining online learning tools (i.e., regret minimization algorithms) together with economic tools (i.e., truthful mechanisms). While almost all the literature focused on single--slot scenarios, here we focused on multi--slot scenarios. With multiple slots it is necessary to adopt a user model to characterize the valuations of the users over the different slots. Here, we adopted the cascade model, that is the most common model used in the literature. In the paper, we studied a number of scenarios, each with a specific information setting of unknown parameters. For each scenario, we designed a truthful learning mechanism, studied its economic properties, derived an upper bound over the regret, and, for some mechanisms, also a lower bound. We considered both the regret over the auctioneer's revenue and the social welfare.
We showed that for the cascade model with only position--dependent externalities it is possible to design a truthful no--regret learning mechanism for the general case in which all the parameters are unknown. Our mechanism presents a regret $O(T^{2/3})$ and it is incentive compatible in expectation over the random component of the mechanism. However, it remains open whether or not it is possible to obtain a regret $O(T^{1/2})$. For specific sub cases, in which some parameters are known to the auctioneer, we obtained better results in terms of either incentive compatibility, obtaining dominant strategy truthfulness, or regret, obtaining a regret of zero. We showed that for the cascade model with the position-- and ad--dependent externalities it is possible to design a dominant strategy truthful mechanism with a regret $O(T^{2/3})$ when only the quality is unknown. Instead, even when the cascade model is only with ad--dependent externalities and no parameter is known it is not possible to obtain a no--regret dominant strategy truthful mechanism. The proof of this result would seem to suggest that the same result holds also when truthfulness is in expectation. However, we did not produce any proof for that, leaving it for future works. Finally, we empirically evaluated the bounds we provided, showing that the dependency of each bound from the parameters is empirically confirmed.
Two main questions deserve future investigation. The first question concerns the study of a lower bound for the case in which there are only position--dependent externalities for different notions of truthfulness in expectation, e.g., both in expectation over the click realizations and in expectation over the random component of the mechanism. Furthermore, it is open whether the separation of exploration and exploitation phases is necessary and, in the negative case, whether it is possible to obtain a regret $O(T^{1/2})$. The second question concerns a similar study related to the case with only ad--dependent externalities.
\section{Proofs of Section~4}
\noindent We start by reporting the straightforward proof of Proposition~\ref{p:hoeffding}.
\begin{pf}\textit{(Proposition~\ref{p:hoeffding})}
The derivation is a simple application of the Hoeffding's bound. We first notice that each of the terms in the empirical average $\tilde{q}_i$ (Eq.~\ref{eq:est.q}) is bounded in $[0; 1/\Lambda_{\pi(i;\theta_t)}]$. Thus we obtain
\begin{equation*}
\mathbb{P}\left( |q_i - \tilde{q}_i | \geq \epsilon \right) \leq 2 \exp\bigg(-\frac{2|B_i|^2\epsilon^2}{\sum_{t \in B_i}\big(\frac{1}{\Lambda_{\pi(i;\theta_t)}}-0\big)^2}\bigg) = \frac{\delta}{N}.
\end{equation*}
Reordering the terms in the previous expression leads to
\begin{align*}
\epsilon &= \sqrt{\left(\sum_{t \in B_i}\frac{1}{\Lambda^2_{\pi(i;\theta_t)}}\right) \frac{1}{2|B_i|^2} \log{\frac{2N}{\delta}}},
\end{align*}
which guarantees that all the empirical estimates $\tilde{q}_i$ are within $\epsilon$ of $q_i$ for all the ads.
\qed
\end{pf}
Before stating the main result of this section, we need the following technical lemma.
\begin{lemma}\label{lem:ratio}
For any slot $s_m$ with $m \in \mathcal{K}$
\begin{equation}\label{eq:ratio}
\frac{\max\limits_{i\in \mathcal{N}} (q_i \hat{v}_i; m)}{\max\limits_{i\in \mathcal{N}} (\tilde{q}^+_i \hat{v}_i; m)} \leq 1,
\end{equation}
where the operator $\max(\cdot;\cdot)$ is defined as in Section~\ref{s:constant} and probability $1-\delta$.
\end{lemma}
\begin{pf} \textit{(Lemma~\ref{lem:ratio})}
The proof is a straightforward application of Proposition~\ref{p:hoeffding}.
The lemma holds since it is true that $\max\limits_{i\in \mathcal{N}}(\tilde{q}^+_i \hat{v}_i; m) \geq \max\limits_{i\in \mathcal{N}}(q_i \hat{v}_i; m)$. In fact, consider the allocation $\theta^*$ in~(\ref{eq:efficient-alloc}) and $\tilde{\theta}$ in~(\ref{eq:optimalallocationestimatedq}) and consider $a_h$ s.t. $h = \alpha(m;\theta^*) = \arg \max\limits_{i \in \mathcal{N}}(q_i \hat{v}_i; m)$. There are two options:
\begin{itemize}
\item If $\pi(h;\tilde{\theta}) < m$ (the ad is displayed into a higher slot in the approximated allocation $\tilde{\theta}$), then
$\exists j \in \mathcal{N}$ s.t. $\pi(j;\theta^*) < m \wedge \pi(j;\tilde{\theta}) \geq m$. Thus
$$ \max\limits_{i \in \mathcal{N}}(\tilde{q}^+_i \hat{v}_i; m) \geq \tilde{q}^+_j \hat{v}_j \geq q_j \hat{v}_j \geq q_h \hat{v}_h = \max\limits_{i \in \mathcal{N}}(q_i \hat{v}_i; m)$$
where the second inequality holds with probability $1 - \delta$;
\item If $\pi(h;\tilde{\theta}) \geq m$ (the ad is displayed into a lower or equal slot in the approximated allocation $\tilde{\theta}$), then
$$ \max\limits_{i \in \mathcal{N}}(\tilde{q}^+_i \hat{v}_i; m) \geq \tilde{q}^+_h \hat{v}_h \geq q_h v_h = \max\limits_{i \in \mathcal{N}}(q_i \hat{v}_i; m)$$
where the second inequality holds with probability $1 - \delta$.
\end{itemize}
In both cases, the statement follows.\qed
\end{pf}
\begin{pf}\textit{(Theorem~\ref{thm:constant})}
The proof follows similar steps as in~\cite{devanur2009price}. Given the definition of A--VCG \ in Section~\ref{s:constant}, we notice that the expected payments are such that for all the slot indexes $m \in \mathcal{K}$ at t-th iteration of the auction:
\begin{align*}
\tilde{p}_{\alpha(m; \theta_t)}(\hat{\mathbf{v}}) = \left\{
\begin{array}{ll}
0 & \text{if } t \leq \tau \text{ (\textit{exploration})}\\
\tilde{p}_{\alpha(m; \tilde{\theta})}(\hat{\mathbf{v}}) & \text{if } t > \tau \text{ (\textit{exploitation})}
\end{array} \right.
\end{align*}
Thus, we can just focus on the average (w.r.t. clicks) per--round regret during the exploitation phase. According to the definition of payments in Section~\ref{s:constant}, defining $\Delta_l := \Lambda_l - \Lambda_{l+1}$, at each round of the exploitation phase we bound the per--round regret $r$ defined as the difference between the payments as
\begin{align*}
r &= \sum_{m = 1}^K (p^*_{\alpha(m;\theta^*)}(\hat{\mathbf{v}}) - \tilde{p}_{\alpha(m;\tilde{\theta})}(\hat{\mathbf{v}})) \\
&= \sum_{m = 1}^K \sum_{l = m}^{K} \Delta_l \left( \max\limits_{i\in \mathcal{N}}(q_i \hat{v}_i; l+1) - \frac{ \max\limits_{i\in \mathcal{N}}(\tilde{q}^+_i \hat{v}_i; l+1) }{\tilde{q}^+_{\alpha(m;\tilde{\theta})}} q_{\alpha(m;\tilde{\theta})} \right) \\
&= \sum_{m = 1}^K \sum_{l = m}^{K} \Delta_l \frac{ \max\limits_{i\in \mathcal{N}}(\tilde{q}^+_i \hat{v}_i; l+1)}{\tilde{q}^+_{\alpha(m;\tilde{\theta})}} \left( \frac{ \max\limits_{i\in \mathcal{N}}(q_i \hat{v}_i;l+1)}{\max\limits_{i\in \mathcal{N}}(\tilde{q}^+_{i} \hat{v}_i;l+1)} \tilde{q}^+_{\alpha(m;\tilde{\theta})} - q_{\alpha(m;\tilde{\theta})} \right) \\
&= \sum_{m = 1}^K \sum_{l = m}^{K} \Delta_l\frac{ \max\limits_{i\in \mathcal{N}}(\tilde{q}^+_{i} \hat{v}_i;l+1)}{\max\limits_{i\in \mathcal{N}}(\tilde{q}^+_{i} \hat{v}_i;m)} \hat{v}_{\alpha(m; \tilde{\theta})} \left( \frac{\max\limits_{i \in \mathcal{N}}(q_i \hat{v}_i;l+1)}{\max\limits_{i\in \mathcal{N}}(\tilde{q}^+_{i} \hat{v}_i;l+1)} \tilde{q}^+_{\alpha(m;\tilde{\theta})} - q_{\alpha(m;\tilde{\theta})} \right).
\end{align*}
By definition of the max operator, since $l+1 > m$, it follows that
\begin{align}\label{eq:step.loose}
\frac{\max\limits_{i \in \mathcal{N}} (\tilde{q}^+_{i} \hat{v}_i;l+1)}{\max\limits_{i \in \mathcal{N}} (\tilde{q}^+_{i} \hat{v}_i;m)} \leq 1.
\end{align}
Finally, from Lemma~\ref{lem:ratio} and $\hat{v}_{\alpha(m; \tilde{\theta})} \leq v_{\max}$, it follows that
\begin{align} \label{eq:boundVCG.exactvsest}
r \leq \sum_{m = 1}^K \sum_{l = m}^{K} v_{\max} \Delta_l(\tilde{q}^+_{\alpha(m;\tilde{\theta})} - q_{\alpha(m;\tilde{\theta})}) \leq v_{\max} \sum_{m = 1}^K \left[ (\tilde{q}^+_{\alpha(m;\tilde{\theta})} - q_{\alpha(m;\tilde{\theta})})\sum_{l = m}^{K} \Delta_l\right],
\end{align}
with probability at least $1-\delta$.
Notice that, by definition of $\Delta_l$, $\sum_{l=m}^{K} \Delta_l = \Lambda_{m} - \Lambda_{K+1} = \Lambda_{m}$. Furthermore, from the definition of $\tilde{q}^+_i$ and using Equation~(\ref{eq:eta}) we have that for any ad $a_i$, $\tilde{q}^+_{i} - q_i = \tilde{q}_i - q_i + \eta \leq 2\eta$, with probability at least $1 - \delta$. Thus, the difference between the payments becomes\footnote{Notice that in the logarithmic term the factor of 2 we have in Proposition~\ref{p:hoeffding} disappears since in this proof we only need the one-sided version of it.}
\begin{align}\label{eq:brs4}
r & \leq 2 v_{\max}\eta\sum_{m = 1}^K \Lambda_m \leq 2v_{\max} \left(\sum_{m = 1}^K \Lambda_m\right)\sqrt{\Bigg(\sum_{m=1}^{K} \frac{1}{\Lambda_{m}^2}\Bigg) \frac{N}{K^2 \tau} \log \frac{N}{\delta}}.
\end{align}
with probability $1-\delta$. The difference $r$ represents the regret at each round of the exploitation phase. In order to get the final regret bound we need to consider the whole time horizon $T$ and turn the bound into expectation. During the first $\tau$ rounds of exploration, the A--VCG\ sets all the payments to 0 and the per--round regret is
\begin{align*}
r = \sum_{m = 1}^K (p^*_{\alpha(m;\theta^*)}(\hat{\mathbf{v}}) - 0) = \sum_{m = 1}^K \sum_{l = m}^{K} \Delta_l \max_{i\in\mathcal{N}}(q_i\hat{v}_i; l+1) \leq v_{\max}\sum_{m = 1}^K \Lambda_m.
\end{align*}
On the other hand, in the remaining $T-\tau$ rounds the regret is bounded by~(\ref{eq:brs4}) with probability $1-\delta$. Thus we obtain
\begin{align*}
R_T &\leq v_{\max} \left(\sum_{m = 1}^K \Lambda_m \right)\Bigg( 2(T - \tau) \sqrt{\left(\sum_{m=1}^{K} \frac{1}{\Lambda_{m}^2}\right) \frac{N}{K^2 \tau} \log \frac{N}{\delta}} + \tau + \delta T \Bigg).
\end{align*}
We can simplify the bound by considering $\sum_{m = 1}^K \Lambda_m\leq K$:
\begin{equation*}
R_T \leq v_{\max} K\Bigg( 2(T - \tau) \sqrt{\left(\sum_{m=1}^{K} \frac{1}{\Lambda_{m}^2}\right) \frac{N}{K^2 \tau} \log \frac{N}{\delta}} + \tau + \delta T \Bigg) =: B(\tau).
\end{equation*}
A rough optimization of the bound $B_\tau$ is obtained in the following way:
\begin{equation*}
\frac{\partial B(\tau)}{\partial \tau} = v_{\max} K\bigg(- \frac{T+\tau}{\tau} \sqrt{\left(\sum_{m=1}^K\frac{1}{\Lambda_m^2}\right) \frac{N}{K^2 \tau} \log{\frac{N}{\delta}}} + 1\bigg) \geq 0\\
\end{equation*}
Using $T + \tau \leq 2 T$, we can make the previous inequality more restrictive and obtain
\begin{align*}
& \frac{2T}{\tau} \sqrt{ \left(\sum_{m=1}^K\frac{1}{\Lambda_m^2}\right) \frac{N}{K^2 \tau} \log{\frac{N}{\delta}}} \leq 1\\
& \tau^3 \geq 4 T^2 \left(\sum_{m=1}^K\frac{1}{\Lambda_m^2}\right) \frac{N}{K^2} \log{\frac{N}{\delta}} \\
& \tau \geq 2^\frac{2}{3} K^{-\frac{1}{3}} T^\frac{2}{3} N^\frac{1}{3} \left(\sum_{m=1}^K\frac{1}{\Lambda_m^2}\right) \left(\log{\frac{N}{\delta}}\right)^\frac{1}{3}.
\end{align*}
Finally we can use the inequality $\sum\limits_{m=1}^{K} 1/\Lambda_{m}^2\leq K/\Lambda_{\min}^2$, with $\Lambda_{\min} = \min_{m \in \mathcal{K}} \Lambda_m$, and obtain
\begin{equation*}
\tau = 2^{\frac{2}{3}} K^{-\frac{1}{3}} T^{\frac{2}{3}} N^{\frac{1}{3}} \Lambda_{\min}^{-\frac{2}{3}} \left( \log{\frac{N}{\delta}}\right)^{\frac{1}{3}},
\end{equation*}
which corresponds to a total regret of
\begin{align*}
R_T &\leq v_{\max} K \bigg(2T \tau^{-\frac{1}{2}} \sqrt{\frac{2N}{\Lambda_{\min}^2 K} \log \frac{N}{\delta}} - \underbrace{\tau^{\frac{1}{2}} \sqrt{\frac{2N}{\Lambda_{\min}^2 K} \log \frac{N}{\delta}}}_{\leq 0} + \tau + \delta T\bigg)\\
& = 2^\frac{5}{3} v_{\max} K^\frac{2}{3} T^\frac{2}{3} N^\frac{1}{3} \Lambda_{\min}^{-\frac{2}{3}} \left( \log{\frac{N}{\delta}}\right)^{\frac{1}{3}} + v_{\max} K \delta T
\end{align*}
Finally, we choose the parameter $\delta\in (0,1)$ in order to optimize the asymptotic rate (i.e., ignoring constants and logarithmic factors) and we set:
\[
\delta = K^{-\frac{1}{3}} T^{-\frac{1}{3}} N^{\frac{1}{3}}\\
\]
\noindent with the constraint that $T \geq \frac{N}{K}$ (given by $\delta \leq 1$), leading to:
\[
R_T \leq \big( 2^\frac{5}{3} \Lambda_{\min}^{-\frac{2}{3}} + 1\big) v_{\max} K^\frac{2}{3} T^\frac{2}{3} N^\frac{1}{3} \left[ \log \big( K^\frac{1}{3} T^\frac{1}{3} N^\frac{2}{3} \big) \right]^{\frac{1}{3}}
\]
\noindent which concludes the proof.\qed
\end{pf}
\begin{pf}\textit{(Theorem~\ref{thm:constant.l.baba})}
We begin the proof computing an upper bound on the per--ad regret $r_i = p_i^* - p_i$ for each round of the auction.
Before bounding the regret we introduce the following definitions used to study the randomization of the mechanism in~\cite{babaioff_impl_pay}:
\begin{itemize}
\item $\mathbf{s} \in \{0,1\}^N$ is a vector where each element $s_i$ represents whether the $i$-th bid coincide with the original bid or it has been modified by the self--resampling procedure, i.e., if $s_i=1$ then $x_i=\hat{v}_i$, otherwise $x_i < \hat{v}_i$. Notice that $\mathbf{s}$ does not provide information about the modified values $x_i$;
\item $\mathbb{E}_{\mathbf{x}|\mathbf{s}}[\Lambda_{\pi(i; f(\mathbf{x}))}|\hat{\mathbf{v}}]$ is the expected value of prominence associated to the slots allocated to ad $a_i$, when the declared bids $\hat{\mathbf{v}}$ are perturbed according to the vector $\mathbf{s}$, i.e., for which ads the bids have been preserved.
\end{itemize}
Define $S' = \{\mathbf{s}' | \pi(i;f^*(\hat{\mathbf{v}}))\leq K + 1 \Rightarrow s'_i = 1 \}$, i.e. all the random realization where the self--resampling procedure does not modify the bids of the first $K+1$ ads on the basis of the decreasing order of $q\hat{v}$, i.e. the $K$ ads displayed applying $f^*$ to the true bids $\hat{\mathbf{v}}$ and the first non allocated.
Given the previous definitions, we rewrite the expected payments $p_i(\hat{\mathbf{v}})$ as
\begin{align*}
p_i(\hat{\mathbf{v}}) &= \bigg(\mathbb{P}[\mathbf{s} \in S'] \Lambda_{\pi(i;f^*(\hat{\mathbf{v}}))} + \mathbb{P}[\mathbf{s} \not \in S'] \mathbb{E}_{\mathbf{x}|\mathbf{s} \not \in S'}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}]\bigg)q_i \hat{v}_i \\
&\quad - \int_{0}^{\hat{v}_i} \bigg(\mathbb{P}[\mathbf{s} \in S'] \Lambda_{\pi(i; f^*(\hat{\mathbf{v}}_{-i},u))} + \mathbb{P}[\mathbf{s} \not \in S'] \mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u]\bigg)q_i du\\
&=\mathbb{P}[\mathbf{s} \in S'] \bigg(\Lambda_{\pi(i;f^*(\hat{\mathbf{v}}))}q_i \hat{v}_i - \int_{0}^{\hat{v}_i}\Lambda_{\pi(i; f^*(\hat{\mathbf{v}}_{-i},u))} q_i du\bigg)\\
&\quad+\mathbb{P}[\mathbf{s} \not \in S'] \bigg(\mathbb{E}_{\mathbf{x}|\mathbf{s} \not \in S'}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}]q_i \hat{v}_i - \int_{0}^{\hat{v}_i}\mathbb{E}_{\mathbf{x}|\mathbf{s} \not \in S'}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u] q_i du\bigg)\\
&=\mathbb{P}[\mathbf{s} \in S'] p_i^*(\hat{\mathbf{v}}) \\
&\quad+ \mathbb{P}[\mathbf{s} \not \in S'] \bigg(\mathbb{E}_{\mathbf{x}|\mathbf{s} \not \in S'}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}]q_i \hat{v}_i - \int_{0}^{\hat{v}_i}\mathbb{E}_{\mathbf{x}|\mathbf{s} \not \in S'}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u] q_i du \bigg),
\end{align*}
where in the last expression we used the expression of the VCG payments according to~\cite{tardos_sp}. The per--ad regret is
\begin{align*}
r_i&(\hat{\mathbf{v}}) = p_i^*(\hat{\mathbf{v}}) - p_i(\hat{\mathbf{v}}) \\
&= p_i^*(\hat{\mathbf{v}}) - \mathbb{P}[\mathbf{s} \in S'] p_i^*(\hat{\mathbf{v}}) \\
&\quad- \mathbb{P}[\mathbf{s} \not \in S'] \bigg(\mathbb{E}_{\mathbf{x}|\mathbf{s} \not \in S'}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}]q_i \hat{v}_i - \int_{0}^{\hat{v}_i}\mathbb{E}_{\mathbf{x}|\mathbf{s} \not \in S'}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u] q_i du\bigg)\\
&= \mathbb{P}[\mathbf{s} \not \in S'] p_i^*(\hat{\mathbf{v}}) \\
&\quad- \mathbb{P}[\mathbf{s} \not \in S'] \underbrace{\bigg(\mathbb{E}_{\mathbf{x}|\mathbf{s} \not \in S'}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}]q_i \hat{v}_i - \int_{0}^{\hat{v}_i}\mathbb{E}_{\mathbf{x}|\mathbf{s} \not \in S'}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u] q_i du\bigg)}_{r_{i,1}^B}.
\end{align*}
Since in the integral we have that $u \leq \hat{v}_i$ and since the allocation function stated in~\cite{babaioff_impl_pay} is monotone, we have that
\begin{align*}
\mathbb{E}_{\mathbf{x} | \mathbf{s} \not \in S'}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u] \leq \mathbb{E}_{\mathbf{x} | \mathbf{s} \not \in S'}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}],
\end{align*}
which implies that $r_{i,1}^B$ is non--negative. Thus the regret $r_i^B$ can be bounded as
\begin{align}
r_i^B(\hat{\mathbf{v}}) &= \mathbb{P}[\mathbf{s} \not \in S'] p_i^*(\hat{\mathbf{v}}) \underbrace{ - \mathbb{P}[\mathbf{s} \not \in S'] r_{i,1}^B}_{\leq 0}\\
&\leq \mathbb{P}[\mathbf{s} \not \in S'] p_i^*(\hat{\mathbf{v}}) \leq \mathbb{P}\big[\exists j: s_j = 0 \land \pi(j;f^*(\hat{\mathbf{v}})) \leq K+1\big] v_{\max}\\
&\leq \sum_{j\in \mathcal{N}: \pi(j;f^*(\hat{\mathbf{v}})) \leq K+1 }\mathbb{P}[s_j = 0] v_{\max} = \left( K + 1 \right) \mu v_{\max} \leq 2 K \mu v_{\max}. \label{bnd:rSRP}
\end{align}
We can now compute the bound on the global regret $R_T$. We recall that this mechanism does not require an estimation phase, thus the regret is:
\begin{align*}
R_T & \leq 2 K^2 \mu v_{\max} T
\end{align*}
\end{pf}
\section{Proofs of Section~4.3}
\TODO{Alex: in tutta questa dimostrazione parliamo di VCG e non WVCG e usiamo i pagamenti Tardos della VCG, e' corretto? vedi anche il mio commento sul fatto che i pagamenti Tardos possano dover contenere dei pesi.}
\begin{pf}\textit{(Theorem~\ref{thm:constant.ql})}
Initially, we compute an upper bound on the per--ad regret $r_i = p_i^* - p_i$ for each round of the exploitation phase and we later use this result to compute the upper bound for the regret over the whole time interval ($R_T$).
We first recall a series of payment definitions which are relevant to the proof. When all the auction parameters (i.e., quality $\{q_i\}_{i \in \mathcal{N}}$ and prominence $\{\Lambda_m\}_{m \in \mathcal{K}}$) are known and are used to compute the exact allocation function $f^*$, we define two possible payments
\begin{itemize}
\item $p_i^*(\hat{\mathbf{v}})$ are the VCG payments defined in~(\ref{eq:payment});
\item $p_i^{B^*}(\hat{\mathbf{v}})$ are the expected payments obtained by the random mechanism based on the self--resampling procedure of cSRP is applied to the declared bids $\hat{\mathbf{v}}$. Formally, using the formulation of payments from~\cite{tardos_sp}, we define
\begin{align*}
p_i^{B^*}(\hat{\mathbf{v}}) &= \mathbb{E}_{\mathbf{x}}\big[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}\big]q_i \hat{v}_i - \int_{0}^{\hat{v}_i} \mathbb{E}_{\mathbf{x}|\mathbf{s}}\big[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u\big]q_i du.
\end{align*}
\end{itemize}
Furthermore, we use two payments when the approximated allocation function $\tilde{f}$ defined using the estimated qualities as
\begin{itemize}
\item $\tilde{p}_i(\hat{\mathbf{v}})$ are the payments defined in Eq.~(\ref{eq:pay.vcg.emp.tardos});
\item $p_i(\hat{\mathbf{v}})$ are the expected payments obtained by the random mechanism and defined in Eq.~(\ref{eq:pay.tardos.rand}).
\end{itemize}
We divide the per--ad regret in two different components:
\begin{align}
r_i(\hat{\mathbf{v}}) &= p_i^*(\hat{\mathbf{v}}) - p_i(\hat{\mathbf{v}}) \\
&= \underbrace{p_i^*(\hat{\mathbf{v}}) - p_i^{B^*}(\hat{\mathbf{v}})}_{\text{cSRP regret}} + \underbrace{p_i^{B^*}(\hat{\mathbf{v}}) - p_i(\hat{\mathbf{v}})}_{\text{learning regret}} = r_i^B(\hat{\mathbf{v}}) + r_i^L(\hat{\mathbf{v}}) \nonumber,
\end{align}
where
\begin{itemize}
\item $r_i^B(\hat{\mathbf{v}})$ is the regret due to the use of the approach proposed in~\cite{babaioff_impl_pay} instead of the VCG payments, when all the parameters are known;
\item $r_i^L(\hat{\mathbf{v}})$ is the regret due to the uncertainty on the parameters when the payments defined in~\cite{babaioff_impl_pay} are considered.
\end{itemize}
For the definitions of $\mathbf{s}$ and $\mathbb{E}_{\mathbf{x}|\mathbf{s}}[\Lambda_{\pi(i; f(\mathbf{x}))}|\hat{\mathbf{v}}]$ refer to the proof of Theorem~\ref{thm:constant.l.baba}.
\TODO{[ALTERNATIVA]}
Before bounding the two sources of regret we introduce the following definitions used to study the randomization of the mechanism in~\cite{babaioff_impl_pay}:
\begin{itemize}
\item $\mathbf{s} \in \{0,1\}^N$ is a vector where each element $s_i$ represents whether the $i$-th bid coincide with the original bid or it has been modified by the self--resampling procedure, i.e., if $s_i=1$ then $x_i=\hat{v}_i$, otherwise $x_i < \hat{v}_i$. Notice that $\mathbf{s}$ does not provide information about the modified values $x_i$;
\item $\mathbb{E}_{\mathbf{x}|\mathbf{s}}[\Lambda_{\pi(i; f(\mathbf{x}))}|\hat{\mathbf{v}}]$ is the expected value of prominence associated to the slots allocated to ad $a_i$, when the declared bids $\hat{\mathbf{v}}$ are perturbed according to the vector $\mathbf{s}$, i.e., for which ads the bids have been preserved.
\end{itemize}
\noindent\textbf{Step 1: the cSRP regret.}
We can reuse the result obtained in the proof of Theorem~\ref{thm:constant.l.baba}. In particular, we can use the bound~\ref{bnd:rSRP}, i.e. $r_i^B(\hat{\mathbf{v}}) \leq \left(K+1\right) \mu v_{\max}$. Given that we have assumed $N > K$, in the remaining parts of this proof we will use the following upper bound: $r_i^B(\hat{\mathbf{v}}) \leq \left(K+1\right) \mu v_{\max} \leq \leq N \mu v_{\max} $.
\TODO{[ALTERNATIVA]}
Given the previous definitions, we rewrite the expected payments $p_i^{B^*}(\hat{\mathbf{v}})$ as
\begin{align*}
p_i^{B^*}(\hat{\mathbf{v}}) &= \bigg(\mathbb{P}[\mathbf{s}=\boldsymbol{1}] \Lambda_{\pi(i;f^*(\hat{\mathbf{v}}))} + \mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] \mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}]\bigg)q_i \hat{v}_i \\
&\quad - \int_{0}^{\hat{v}_i} \bigg(\mathbb{P}[\mathbf{s}=\boldsymbol{1}] \Lambda_{\pi(i; f^*(\hat{\mathbf{v}}_{-i},u))} + \mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] \mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u]\bigg)q_i du\\
&=\mathbb{P}[\mathbf{s}=\boldsymbol{1}] \bigg(\Lambda_{\pi(i;f^*(\hat{\mathbf{v}}))}q_i \hat{v}_i - \int_{0}^{\hat{v}_i}\Lambda_{\pi(i; f^*(\hat{\mathbf{v}}_{-i},u))} q_i du\bigg)\\
&\quad+\mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] \bigg(\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}]q_i \hat{v}_i - \int_{0}^{\hat{v}_i}\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u] q_i du\bigg)\\
&=\mathbb{P}[\mathbf{s}=\boldsymbol{1}] p_i^*(\hat{\mathbf{v}}) \\
&\quad+ \mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] \bigg(\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}]q_i \hat{v}_i - \int_{0}^{\hat{v}_i}\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u] q_i du \bigg),
\end{align*}
where in the last expression we used the expression of the VCG payments according to~\cite{tardos_sp}. The per--ad regret is
\begin{align*}
r_i^B&(\hat{\mathbf{v}}) = p_i^*(\hat{\mathbf{v}}) - p_i^{B^*}(\hat{\mathbf{v}}) \\
&= p_i^*(\hat{\mathbf{v}}) - \mathbb{P}[\mathbf{s}=\boldsymbol{1}] p_i^*(\hat{\mathbf{v}}) \\
&\quad- \mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] \bigg(\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}]q_i \hat{v}_i - \int_{0}^{\hat{v}_i}\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u] q_i du\bigg)\\
&= \mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] p_i^*(\hat{\mathbf{v}}) \\
&\quad- \mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] \underbrace{\bigg(\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}]q_i \hat{v}_i - \int_{0}^{\hat{v}_i}\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u] q_i du\bigg)}_{r_{i,1}^B}.
\end{align*}
Since in the integral we have that $u \leq \hat{v}_i$ and since the allocation function stated in~\cite{babaioff_impl_pay} is monotone, we have that
\begin{align*}
\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u] \leq \mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}],
\end{align*}
which implies that $r_{i,1}^B$ is non--negative. Thus the regret $r_i^B$ can be bounded as
\begin{align*}
r_i^B(\hat{\mathbf{v}}) &= \mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] p_i^*(\hat{\mathbf{v}}) \underbrace{ - \mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] r_{i,1}^B}_{\leq 0}\\
&\leq \mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] p_i^*(\hat{\mathbf{v}}) \leq \mathbb{P}\big[\exists j: s_j = 0\big] v_{\max} \leq \sum_{j\in\mathcal{N} }\mathbb{P}[s_j = 0] v_{\max} = N \mu v_{\max}.
\end{align*}
\noindent\textbf{Step 2: the learning regret.}
Similar to the previous step, we write the learning expected payments based on the cSRP (Eq.~\ref{eq:pay.tardos.rand}) as
\begin{align*}
p_i(\hat{\mathbf{v}}) =\mathbb{P}[\mathbf{s}=\boldsymbol{1}] \tilde{p}_i(\hat{\mathbf{v}}) + \mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] \bigg(\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\pi(i; \tilde{f}(\mathbf{x}))}|\hat{\mathbf{v}}] q_i \hat{v}_i - \int_{0}^{\hat{v}_i}\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\pi(i; \tilde{f}(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u] q_i du\bigg).
\end{align*}
Then the per-ad regret is
\begin{align*}
r_i^L(\hat{\mathbf{v}}) &= p_i^{B^*}(\hat{\mathbf{v}}) - p_i(\hat{\mathbf{v}})\\
&= \mathbb{P}[\mathbf{s}=\boldsymbol{1}] (p_i^*(\hat{\mathbf{v}})- \tilde{p}_i(\hat{\mathbf{v}})) +\\
&\quad + \mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] \bigg(\underbrace{\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}]q_i \hat{v}_i - \int_{0}^{\hat{v}_i}\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u] q_i du}_{\leq v_{\max}} +\\
&\quad\quad\quad\quad\quad\quad\quad \underbrace{-\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\pi(i;\tilde{f}(\mathbf{x}))}|\hat{\mathbf{v}}]q_i \hat{v}_i + \int_{0}^{\hat{v}_i}\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\pi(i;\tilde{f}(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u] q_i du}_{= - r_{i,1}^B \leq 0}\bigg)\\
&\leq p_i^*(\hat{\mathbf{v}})- \tilde{p}_i(\hat{\mathbf{v}}) + N\mu v_{\max}.
\end{align*}
\footnote{\TODO{Alex: qui c'era un ``with probability $1-\delta$.'' che non ho capito e ho tolto, corretto? M: si' concorderei, piu' che altro bisogna magari rivedere dove metterlo e dove non metterlo}}Since the WVCG payments corresponding to the estimated allocation function $\tilde{f}$ can be written as
\begin{align*}
\tilde{p}_i(\hat{\mathbf{v}}) &=\Lambda_{\pi(i;\tilde{f}(\hat{\mathbf{v}}))}q_i\hat{v}_i - \int_{0}^{\hat{v}_i} \Lambda_{\pi(i; \tilde{f}(\mathbf{v}_{-i},u))}q_i du\\
&= \frac{q_i}{\tilde{q}^+_i} \Big[\widetilde{SW}\big(\tilde{f}_{-i}\left(\hat{\mathbf{v}}\right),\hat{\mathbf{v}}\big) - \widetilde{SW}_{-i}\big(\tilde{f}\left(\hat{\mathbf{v}}\right), \hat{\mathbf{v}}\big)\Big],
\end{align*}
we can use the results stated in proof of Theorem~\ref{thm:constant} and from Eq.~(\ref{eq:boundVCG.exactvsest}) we can conclude that
\begin{align*}
\sum_{i: \pi(i; f^*(\hat{\mathbf{v}})\leq K)} \left(p_i^*\left(\hat{\mathbf{v}}\right)- \tilde{p}_i\left(\hat{\mathbf{v}}\right)\right) \leq 2v_{\max}\eta \left( \sum_{m=1}^K \Lambda_m \right) \leq 2Kv_{\max}\eta.
\end{align*}
\noindent\textbf{Step 3: the global regret.} We now bring together the two instantaneous regrets and have that at each round of the the exploitation phase we have the regret $r = \sum_{i=1}^N r_i$.
We first notice that the expected instantaneous regret $r_i$ for each ad $a_i$ is defined as the difference between the VCG payment $p_i^*(\hat{\mathbf{v}})$ and the (expected) payments computed by the estimated randomized mechanism $p_i(\hat{\mathbf{v}})$. We notice that $p_i^*(\hat{\mathbf{v}})$ can be strictly positive only for the $K$ displayed ads, while $p_i(\hat{\mathbf{v}}) \geq 0 \ \forall i \in \mathcal{N}$, due to the mechanism randomization. Thus, $p_i^*(\hat{\mathbf{v}}) - p_i(\hat{\mathbf{v}}) > 0$ only for at most $K$ ads. Thus we obtain the per--round regret
\begin{align*}
r &\leq \sum_{i: \pi(i;f^*(\hat{\mathbf{v}})) \leq K} r_i = \sum_{i: \pi(i;f^*(\hat{\mathbf{v}})) \leq K} \left(r_i^B + r_i^L\right)\\
& \leq K N \mu v_{\max} + \sum_{i: \pi(i;f^*(\hat{\mathbf{v}})) \leq K} \left(p_i^*\left(\hat{\mathbf{v}}\right)- \tilde{p}_i\left(\hat{\mathbf{v}}\right) + N\mu v_{\max} \right)\\
&\leq K N \mu v_{\max} + 2 K v_{\max} \eta_q + K N \mu v_{\max} = 2 K v_{\max} \eta_q + 2 K N \mu v_{\max}
\end{align*}
\noindent We can now compute an upper bound for the global regret:
\begin{align*}
R_T & \leq v_{\max} K \left[\left( T-\tau \right) \left(2 \sqrt{\frac{N}{\tau} \log \frac{2N}{\delta}} + 2 \mu N \right) + \tau + \delta T \right] =: B_\tau.
\end{align*}
\noindent A rough optimization of the bound $B_\tau$ is obtained in the following way:
\begin{align*}
\frac{\partial B_\tau}{\partial \tau} &= v_{\max} K \left( - 2 \sqrt{\frac{N}{\tau} \log{\frac{2N}{\delta}}} - 2 \mu N - \frac{(T - \tau)}{ \tau} \sqrt{\frac{N}{\tau} \log{\frac{2N}{\delta}}} + 1
\right) \geq 0\\
& -\frac{T+\tau}{\tau} \sqrt{\frac{N}{\tau}\log{\frac{2N}{\delta}}} - 2 \mu N + 1 \geq 0\\
& \frac{T+\tau}{\tau} \sqrt{\frac{N}{\tau}\log{\frac{2N}{\delta}}} \leq 1 - 2 \mu N \leq 1\\
& \frac{(T + \tau)^2}{\tau^2} \frac{N}{\tau} \log{\frac{2N}{\delta}} \leq 1
\end{align*}
\noindent by approximating $T + \tau \approx 2 T$:
\begin{align*}
& \frac{4 T^2}{\tau^2} \frac{N}{\tau} \log{\frac{2N}{\delta}} \leq 1\\
& \tau^3 \geq 4 T^2 N \log{\frac{2N}{\delta}} \\
& \tau \geq 2^\frac{2}{3} T^\frac{2}{3} N^\frac{1}{3} \left( \log{\frac{2N}{\delta}} \right)^\frac{1}{3}
\end{align*}
\noindent Thus, by considering:
\[
\tau = 2^\frac{2}{3} T^\frac{2}{3} N^\frac{1}{3} \left( \log{\frac{2N}{\delta}} \right)^\frac{1}{3}
\]
\noindent leading to a total regret of:
\begin{align*}
R_T &\leq 2 K T v_{\max} \sqrt{\frac{N}{\tau} \log \frac{2N}{\delta}} + 2 \mu K N T v_{\max} + \tau K v_{\max} + \delta K v_{\max} T +\\
&\underbrace{- 2 K v_{\max} \tau \sqrt{\frac{N}{\tau} log \frac{2N}{\delta}} - 2 \mu \tau K N v_{\max}}_{\leq 0}\\
&\leq 2^\frac{2}{3} v_{\max} K T^\frac{2}{3} N^\frac{1}{3} \left( \log \frac{2N}{\delta} \right)^\frac{1}{3} + v_{\max} K 2 \mu N T + \\
&+ 2^\frac{2}{3} T^\frac{2}{3} N^\frac{1}{3} \left( \log \frac{2N}{\delta} \right)^\frac{1}{3} K v_{\max} + \delta v_{\max} K T\\
&= O\left( K v_{\max} T^\frac{2}{3} N^\frac{1}{3} \left( \log \frac{2N}{\delta} \right)^\frac{1}{3} \right) + O\left( \mu K v_{\max} N T \right)+ O\left( \delta K v_{\max} T \right)
\end{align*}
We choose $\delta$ and $\mu$ s.t. the asymptotic order is not increased (not considering logarithmic factors), we set:
[DA SCEGLIERE]
\noindent with the constraint that $?$ (given by $\delta \leq 1$), leading to:
[DA SCEGLIERE]
Possible constraints to impose to $\delta$ and $\mu$:
\begin{itemize}
\item
\begin{itemize}
\item To have the second and third terms with the same order of the first one
\item $\mu = N^{-\frac{2}{3}} T^{-\frac{1}{3}}$. $\mu$ is always $\leq 1$
\item $\delta = N^\frac{1}{3} T^{-\frac{1}{3}}$. $\delta \leq 1$, thus $T \geq N$
\item
\begin{align*}
R_T &\leq 2^\frac{2}{3} v_{\max} K T^\frac{2}{3} N^\frac{1}{3} \left( \log 2 N^\frac{2}{3} T^\frac{1}{3} \right)^\frac{1}{3} + 2 N^\frac{1}{3} v_{\max} K T^\frac{2}{3} + N^\frac{1}{3} T^{-\frac{1}{3}} K v_{\max} T\\
&= 2^\frac{2}{3} v_{\max} K T^\frac{2}{3} N^\frac{1}{3} \left( \log 2 N^\frac{2}{3} T^\frac{1}{3} \right)^\frac{1}{3} + 2 N^\frac{1}{3} v_{\max} K T^\frac{2}{3} + N^\frac{1}{3} K v_{\max} T^\frac{2}{3}\\
&\leq 4.6 N^\frac{1}{3} K v_{\max} T^\frac{2}{3} \max \left\lbrace 1, \left( \log 2 N^\frac{2}{3} T^\frac{1}{3} \right)^\frac{1}{3} \right\rbrace
\end{align*}
\end{itemize}
\item
\begin{itemize}
\item To remove the N dependency in $r_i^B$
\item $\mu = \frac{1}{N} T^{-\frac{1}{3}}$. $\mu$ is always $\leq 1$
\item $\delta = N^\frac{1}{3} T^{-\frac{1}{3}}$. $\delta \leq 1$, thus $T \geq N$
\item \begin{align*}
R_T &\leq 2^\frac{2}{3} v_{\max} K T^\frac{2}{3} N^\frac{1}{3} \left( \log 2 N^\frac{2}{3} T^\frac{1}{3} \right)^\frac{1}{3} + 2 \frac{1}{N} N v_{\max} K T^\frac{2}{3} + N^\frac{1}{3} T^{-\frac{1}{3}} K v_{\max} T\\
&= 2^\frac{2}{3} v_{\max} K T^\frac{2}{3} N^\frac{1}{3} \left( \log 2 N^\frac{2}{3} T^\frac{1}{3} \right)^\frac{1}{3} + 2 v_{\max} K T^\frac{2}{3} + N^\frac{1}{3} K v_{\max} T^\frac{2}{3}\\
&\leq 4.6 N^\frac{1}{3} K v_{\max} T^\frac{2}{3} \max \left\lbrace 1, \left( \log 2 N^\frac{2}{3} T^\frac{1}{3} \right)^\frac{1}{3} \right\rbrace
\end{align*}
\end{itemize}
\end{itemize}\qed
\end{pf}
\section{Proofs of Section~5.1}
\begin{pf}\textit{(Theorem~\ref{thm:extern})}
We first compute the instantaneous per--ad regret $r_i = p^*_i(\hat{\mathbf{v}}) - \tilde{p}_i(\hat{\mathbf{v}})$ at each round of the exploitation phase for each ad $a_i$. According to the definition of payments we have
\begin{align*}
r_i = \underbrace{\text{SW}(f^*_{-i}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \widetilde{\text{SW}}(\tilde{f}_{-i}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \frac{q_i}{\tilde{q}^+_{i}}}_{r^1_i} + \underbrace{\widetilde{\text{SW}}_{-i}(\tilde{f}(\hat{\mathbf{v}}),\hat{\mathbf{v}}) \frac{q_i}{\tilde{q}^+_{i}} - \text{SW}_{-i}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}})}_{r^2_i}.
\end{align*}
We bound the first term through the following inequalities
\begin{align*}
r^1_i &= \text{SW}(f^*_{-i}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \widetilde{\text{SW}}(f^*_{-i}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \frac{q_i}{\tilde{q}^+_{i}} + \widetilde{\text{SW}}(f^*_{-i}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \frac{q_i}{\tilde{q}^+_{i}} - \widetilde{\text{SW}}(\tilde{f}_{-i}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \frac{q_i}{\tilde{q}^+_{i}} \\
&\leq \max_{f \in \mathcal{F}_{-i}} \left( \text{SW}(f(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \widetilde{\text{SW}}(f(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \frac{q_i}{\tilde{q}^+_{i}} \right) + \underbrace{\left( \widetilde{\text{SW}}(f^*_{-i}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \max_{f \in \mathcal{F}_{-i}} \widetilde{\text{SW}}(f(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \right)}_{\leq 0} \frac{q_i}{\tilde{q}^+_{i}} \\
&\leq \max_{f \in \mathcal{F}_{-i}} \sum_{j: \pi(j; f(\hat{\mathbf{v}}))\leq K} \Gamma_{\pi(j; f(\hat{\mathbf{v}}))} \hat{v}_j \left( q_j - \tilde{q}^+_{j} \frac{q_i}{\tilde{q}^+_{i}} \right) \\
& \leq v_{\max} \max_{f \in \mathcal{F}_{-i}} \sum_{j: \pi(j; f(\hat{\mathbf{v}}))\leq K} \left( q_j - q_j \frac{q_i}{\tilde{q}^+_{i}} + q_j \frac{q_i}{\tilde{q}^+_{i}} - \tilde{q}^+_{j} \frac{q_i}{\tilde{q}^+_{i}} \right) \\
&= v_{\max} \max_{f \in \mathcal{F}_{-i}} \sum_{j: \pi(j; f(\hat{\mathbf{v}}))\leq K} \Bigg[ q_j \left( \frac{\tilde{q}^+_{i} - q_i}{\tilde{q}^+_{i}} \right) + \underbrace{(q_j - \tilde{q}^+_{j})}_{\leq 0} \frac{q_i}{\tilde{q}^+_{i}} \Bigg] \\
& \leq \frac{v_{\max}}{q_{\min}} \max_{f \in \mathcal{F}_{-i}}\sum_{j: \pi(j; f(\hat{\mathbf{v}}))\leq K} \left( \tilde{q}_i - q_i + \eta \right) \leq \frac{2K v_{\max}}{q_{\min}} \eta,
\end{align*}
\noindent with probability $1-\delta$. We rewrite $r^2_i$ as
\begin{align*}
r^2_i &= \left( \widetilde{\text{SW}}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \Gamma_{\pi(i; \tilde{f}(\hat{\mathbf{v}}))}(\tilde{f}(\hat{\mathbf{v}})) \tilde{q}^+_{i} \hat{v}_i \right)\frac{q_i}{\tilde{q}^+_{i}} - \text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) + \Gamma_{\pi(i; f^*(\hat{\mathbf{v}}))}(f^*(\hat{\mathbf{v}})) q_i \hat{v}_i \\
&= \underbrace{\widetilde{\text{SW}}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \frac{q_i}{\tilde{q}^+_{i}} - \text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}})}_{r^{3}_i} + \left(\Gamma_{\pi(i; f^*(\hat{\mathbf{v}}))}\left(f^*(\hat{\mathbf{v}})\right) - \Gamma_{\pi(i; \tilde{f}(\hat{\mathbf{v}}))}(\hat{f}(\hat{\mathbf{v}}))\right) q_i \hat{v}_i.
\end{align*}
\noindent We now focus on the term $r^3_i$
\begin{align*}
r^3_i &= \widetilde{\text{SW}}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \frac{q_i}{\tilde{q}^+_{i}} - \text{SW}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) + \underbrace{\text{SW}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \max_{f \in \mathcal{F}} \text{SW}(f(\hat{\mathbf{v}}), \hat{\mathbf{v}})}_{\leq 0} \\
&\leq \max_{f \in \mathcal{F}} \left(\widetilde{\text{SW}}(f(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \frac{q_i}{\tilde{q}^+_{i}} - \text{SW}(f(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \right) \\
&= \max_{f \in \mathcal{F}} \sum_{j: \pi(j; f(\hat{\mathbf{v}}))\leq K} \Gamma_{\pi(j; f(\hat{\mathbf{v}}))}(f(\hat{\mathbf{v}})) \hat{v}_j \left( \tilde{q}^+_{j} \frac{q_i}{\tilde{q}^+_{i}} - q_j \right) \\
& \leq v_{\max} \max_{f \in \mathcal{F}} \sum_{j: \pi(j; f(\hat{\mathbf{v}}))\leq K} \left( \tilde{q}^+_{j} \frac{q_i}{\tilde{q}^+_{i}} - q_j \right) \\
& \leq v_{\max} \max_{f \in \mathcal{F}} \sum_{j: \pi(j; f(\hat{\mathbf{v}}))\leq K} (\tilde{q}^+_j - q_j) \leq 2K v_{\max} \eta.
\end{align*}
\noindent We define $I = \{i |\pi(i; f^∗(\hat{\mathbf{v}})) \leq K \lor \pi(i; \tilde{f}(\hat{\mathbf{v}})) \leq K, i \in \mathcal{N}\}$, $|I| \leq 2K$. It is clear that only the ads $a_i$ s.t. $i \in I$ have a regret $r_i \not = 0$. The other ads, $i \not \in I$, have both $p^*_i(\hat{\mathbf{v}}) = 0$ and $\tilde{p}_i(\hat{\mathbf{v}})=0$. Thus, we can bound the regret $r$, at each exploitative round, in the following way
\begin{align*}
r &= \sum_{i \in I} (r^1_i + r^2_i) \\
& \leq \sum_{i \in I} \Big( \frac{2K v_{\max}}{q_{\min}} \eta + 2K v_{\max} \eta \Big) + \sum_{i \in I} \left(\Gamma_{\pi(i; f^*(\hat{\mathbf{v}}))}(f^*(\hat{\mathbf{v}})) - \Gamma_{\pi(i;\tilde{f}(\hat{\mathbf{v}}))}(\tilde{f}(\hat{\mathbf{v}})) \right)q_i \hat{v}_i \\
& = \sum_{i \in I} \Big( \frac{2K v_{\max}}{q_{\min}} \eta + 2K v_{\max} \eta \Big) + \sum_{i=1}^N \left(\Gamma_{\pi(i; f^*(\hat{\mathbf{v}}))}(f^*(\hat{\mathbf{v}})) - \Gamma_{\pi(i;\tilde{f}(\hat{\mathbf{v}}))}(\tilde{f}(\hat{\mathbf{v}})) \right)q_i \hat{v}_i \\
&\leq \frac{8K^2 v_{\max}}{q_{min}} \eta + \text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \text{SW}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \\
& = \frac{8K^2 v_{\max}}{q_{min}} \eta + \text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \widetilde{\text{SW}}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}})+\\
& + \underbrace{\widetilde{\text{SW}}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \max_{f \in \mathcal{F}} \widetilde{\text{SW}}(f)}_{\leq 0} + \widetilde{\text{SW}}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \text{SW}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \\
& \leq \frac{8K^2 v_{\max}}{q_{min}} \eta + \underbrace{\text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \widetilde{\text{SW}}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}})}_{r^1} + \underbrace{\widetilde{\text{SW}}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \text{SW}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}})}_{r^2}
\end{align*}
\noindent The two remaining terms $r^1$ and $r^2$ can be easily bounded as
\begin{align*}
r^1 &= \text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \widetilde{\text{SW}}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \leq \max_{f \in \mathcal{F}} \big( \text{SW}(f(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \widetilde{\text{SW}}(f(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \big) \\
& = \max_{f \in \mathcal{F}} \sum_{j: \pi(j; f(\hat{\mathbf{v}}))\leq K} \Gamma_{\pi(j; f(\hat{\mathbf{v}}))}(f(\hat{\mathbf{v}})) \hat{v}_j \left( q_j - \tilde{q}^+_j \right) \\
&\leq v_{\max} \max_{f \in \mathcal{F}} \sum_{j: \pi(j; f(\hat{\mathbf{v}}))\leq K} ( q_j - \tilde{q}^+_j ) \leq 0,
\end{align*}
\noindent and
\begin{align*}
r^2 &= \widetilde{\text{SW}}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \text{SW}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \leq \max_{f \in \mathcal{F}} \left( \widetilde{\text{SW}}(f(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \text{SW}(f(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \right) \\
& = \max_{f \in \mathcal{F}} \sum_{j: \pi(j;f(\hat{\mathbf{v}}))\leq K} \Gamma_{\pi(j;f(\hat{\mathbf{v}}))}(f(\hat{\mathbf{v}})) \hat{v}_j \left( \tilde{q}^+_{j} - q_j \right) \\
&\leq v_{\max} \max_{f \in \mathcal{F}} \sum_{j: \pi(j; f(\hat{\mathbf{v}}))\leq K} \big( \tilde{q}^+_j - q_j \big) \\
&= v_{\max} \max_{f \in \mathcal{F}} \sum_{j: \pi(j; f(\hat{\mathbf{v}}))\leq K} \left(\tilde{q}_j + \eta - q_j \right) \leq 2K v_{\max} \eta.
\end{align*}
\noindent Summing up all the terms we finally obtain $$r \leq \frac{10K^2 v_{\max}}{q_{\min}} \eta$$ with probability $1-\delta$. Now, considering the instantaneous regret of the exploration and exploitation phases, we obtain the final bound on the cumulative regret $R_T$ as follows
\begin{align*}
R_T \leq v_{\max} K \left[ (T - \tau) \left( \frac{10 K}{\Gamma_{\min}q_{\min}} \sqrt{\frac{N}{2K\tau} \log \frac{N}{\delta}} \right) + \tau + \delta T \right] =: B_\tau.
\end{align*}
Defining $c := \frac{10}{\Gamma_{\min} q_{\min}}$, a rough optimization of the bound $B_\tau$ is obtained as:
\begin{align*}
\frac{\partial B_\tau}{\partial \tau} &= v_{\max} K \left( 1 - \frac{cK(T - \tau)}{2 \tau} \sqrt{\frac{N}{2K\tau} \log{\frac{N}{\delta}}} - cK \sqrt{\frac{N}{2K\tau} \log{\frac{N}{\delta}}} \right) \geq 0\\
& \frac{c^2K^2(T + \tau)^2}{4 \tau^2} \frac{N}{2K\tau} \log{\frac{N}{\delta}} \leq 1\\
& \frac{c^2KN(T + \tau)^2}{8 \tau^3} \log{\frac{N}{\delta}} \leq 1
\end{align*}
\noindent and approximating $T + \tau \approx 2 T$:
\begin{align*}
& \tau^3 \geq \left(\frac{1}{2}\right)^3 c^2 K T^2 N \log{\frac{N}{\delta}}\\
& \tau \geq \frac{1}{2} c^\frac{2}{3} K^\frac{1}{3} T^\frac{2}{3} N^\frac{1}{3} \left( \log{\frac{N}{\delta}} \right)^\frac{1}{3}
\end{align*}
\noindent The bound over the regret is equal to:
\begin{align*}
R_T &\leq \left( \frac{1}{2} \right)^{\frac{1}{2}} c v_{\max} K^\frac{3}{2} T N^\frac{1}{2} \tau^{-\frac{1}{2}} \left(\log{\frac{N}{\delta}}\right)^\frac{1}{2} \underbrace{ - \left( \frac{1}{2} \right)^{\frac{1}{2}} c v_{\max} K^2 N^\frac{1}{2} \tau^{\frac{1}{2}} \left(\log{\frac{N}{\delta}}\right)^\frac{1}{2}}_{\leq 0} +\\
& + v_{\max} K \tau + \delta v_{\max} K T\\
& = \left( \frac{1}{2} \right)^{\frac{1}{2}} c v_{\max} K^\frac{3}{2} T N^\frac{1}{2} \left(\log{\frac{N}{\delta}}\right)^\frac{1}{2} \left( \frac{1}{2} \right)^{-\frac{1}{2}} c^{-\frac{1}{3}} K^{-\frac{1}{6}} T^{-\frac{1}{3}} N^{-\frac{1}{6}} \left(\log{\frac{N}{\delta}}\right)^{-\frac{1}{6}} +\\
& + v_{\max} K \frac{1}{2} c^\frac{2}{3} K^\frac{1}{3} T^\frac{2}{3} N^\frac{1}{3} \left(\log{\frac{N}{\delta}}\right)^\frac{1}{3} + \delta v_{\max} K T\\
& = c^\frac{2}{3} v_{\max} K^\frac{4}{3} T^\frac{2}{3} N^\frac{1}{3} \left(\log{\frac{N}{\delta}}\right)^\frac{1}{3} +\\
& + \frac{1}{2} c^\frac{2}{3} v_{\max} K^\frac{4}{3} T^\frac{2}{3} N^\frac{1}{3} \left(\log{\frac{N}{\delta}}\right)^\frac{1}{3} + \delta v_{\max} K T\\
& = O\left(c^\frac{2}{3} K^\frac{4}{3} T^\frac{2}{3} N^\frac{1}{3} \left( \log{\frac{N}{\delta}} \right)^\frac{1}{3} \right)
\end{align*}
We choose $\delta$ s.t. the asymptotic order is not increased (not considering logarithmic factors), we set:
[DA SCEGLIERE]
\noindent with the constraint that $?$ (given by $\delta \leq 1$), leading to:
[DA SCEGLIERE]
Bound with definition of $\delta$:
\begin{itemize}
\item $\delta = K^\frac{1}{3} N^\frac{1}{3} c^\frac{2}{3} T^{-\frac{1}{3}}$ In order to have $\delta \leq 1$ we have that $T \geq K N c^2$
\[
R_T \leq v_{\max} K^\frac{4}{3} T^\frac{2}{3} N^\frac{1}{3} \frac{5 \cdot 10^\frac{2}{3}}{2 \Gamma_{\min}^\frac{2}{3} q_{\min}^\frac{2}{3}} \left(\log{\frac{N^\frac{2}{3} \Gamma_{\min}^\frac{2}{3} q_{\min}^\frac{2}{3} T^\frac{1}{3}}{10^\frac{2}{3}}}\right)^\frac{1}{3}
\]
\item $\delta = T^{-\frac{1}{3}}$
\[
R_T \leq v_{\max} K^\frac{4}{3} T^\frac{2}{3} N^\frac{1}{3} \frac{5 \cdot 10^\frac{2}{3}}{2\Gamma_{\min}^\frac{2}{3} q_{\min}^\frac{2}{3}} \left(\log{N T^\frac{1}{3}}\right)^\frac{1}{3}
\]
\end{itemize}
This bound imposes constraints on the value of $T$, indeed, $T>\tau$, thus $T > \frac{1}{2} c^\frac{2}{3} K^\frac{1}{3} T^\frac{2}{3} N^\frac{1}{3} \left( \log{\frac{N}{\delta}} \right)^\frac{1}{3}$ leading to:
$$
T > \frac{1}{2} c^2 K N \log{\frac{N}{\delta}} > c^2 K N
$$
thus favoring the first choice of $\delta$ which reduces the logarithmic order without introducing new constraints. [???????????]
The problem of the previous bound is that $\tau$ and $\delta$ depends on $q_{\min}$, which is an unknown quantity. We rewrite the bounds removing that dependence.
We optimize in a suboptimal way choosing:
\[
\tau = \frac{1}{2} \left( \frac{10}{\Gamma_{\min}} \right)^{\frac{2}{3}} K^{\frac{1}{3}} T^{\frac{2}{3}} N^{\frac{1}{3}} \left( \log{\frac{N}{\delta}} \right)^{\frac{1}{3}}
\]
Defining $d = \frac{10}{\Gamma_{\min}}$, we have:
\begin{align*}
R_T & \leq \frac{d^\frac{2}{3}}{q_{\min}} v_{\max} K^\frac{4}{3} T^\frac{2}{3} N^\frac{1}{3} \left( \log{\frac{N}{\delta}} \right)^\frac{1}{3} + \frac{1}{2} \frac{d^\frac{2}{3}}{q_{\min}} v_{\max} K^\frac{4}{3} T^\frac{2}{3} N^\frac{1}{3} \left(\log{\frac{N}{\delta}}\right)^\frac{1}{3} + \delta v_{\max} K T\\
& = O\left( \frac{d^\frac{2}{3}}{q_{\min}} K^\frac{4}{3} T^\frac{2}{3} N^\frac{1}{3} \left( \log{\frac{N}{\delta}} \right)^\frac{1}{3} \right)
\end{align*}
We choose $\delta$ s.t. the asymptotic order is not increased (not considering logarithmic factors), we set: [DA SCEGLIERE]
\noindent with the constraint that $?$ (given by $\delta \leq 1$), leading to:
[DA SCEGLIERE]
\begin{itemize}
\item $\delta = K^\frac{1}{3} N^\frac{1}{3} d^\frac{2}{3} T^{-\frac{1}{3}}$ In order to have $\delta \leq 1$ we have that $T \geq K N d^2$
\[
R_T \leq v_{\max} K^\frac{4}{3} T^\frac{2}{3} N^\frac{1}{3} \frac{5 \cdot 10^\frac{2}{3}}{2 \Gamma_{\min}^\frac{2}{3} q_{\min}} \left(\log{\frac{N^\frac{2}{3} \Gamma_{\min}^\frac{2}{3} q_{\min} T^\frac{1}{3}}{10^\frac{2}{3}}}\right)^\frac{1}{3}
\]
\item $\delta = T^{-\frac{1}{3}}$
\[
R_T \leq v_{\max} K^\frac{4}{3} T^\frac{2}{3} N^\frac{1}{3} \frac{5 \cdot 10^\frac{2}{3}}{2 \Gamma_{\min}^\frac{2}{3} q_{\min}} \left(\log{N T^\frac{1}{3}}\right)^\frac{1}{3}
\]
\end{itemize}
Constraint imposed by $\tau$ on $T$:\\
We know that $T>\tau$, thus $T > \frac{1}{2} d^\frac{2}{3} K^\frac{1}{3} T^\frac{2}{3} N^\frac{1}{3} \left( \log{\frac{N}{\delta}} \right)^\frac{1}{3}$ leading to:
\[
T > \frac{1}{2} d^2 K N \log{\frac{N}{\delta}} > d^2 K N
\]
thus favoring the first choice of $\delta$ which reduces the logarithmic order without introducing new constraints.
\qed
\end{pf}
\section{Deviation Regret}\label{app:deviation.regret}
The definition of regret in (\ref{eq:regret}) measures the cumulative difference between the revenue of a VCG compared to the one obtained by A--VCG\ over $T$ rounds. Upper--bounds on this quantity guarantees that the loss in terms of revenue does not linearly increase with $T$. As illustrated in the previous sections, the key passage in the proofs is the upper--bounding of the regret at each round of the exploitation phase (i.e., $r = \sum_{i=1}^N (p_i - \hat{p}_i)$). Nonetheless, we notice that this quantity could be negative. Let us consider the following simple example. Let $N=3$, $K=1$, $v_i=1$ for all the ads, and $q_1=0.1$, $q_2=0.2$, and $q_3=0.3$. Let assume that after the exploration phase we have $\tilde{q}^+_1 = 0.1$, $\tilde{q}^+_2 = 0.29$, $\tilde{q}^+_3 = 0.3$. A standard VCG mechanism allocates adv $a_3$ and asks for a payment $p=0.2$. During the exploitation phase A--VCG\ also allocates $a_3$ but asks for an (expected) payment $\hat{p} = (\tilde{q}^+_2 / \tilde{q}^+_3) q_3 = 0.29$. Thus, the regret in each exploitation round is $r = p - \hat{p} = -0.09$. Although this result might seem surprising, it is due to the fact that while both A--VCG\ and VCG are truthful, in general A--VCG\ is not efficient. We recall that a mechanism is efficient if for any set of advs it always maximizes the social welfare. In the example, if for instance the true quality of adv $a_3$ is $q_3 = 0.28$, then the allocation induced by $\tilde{q}^+$s is not efficient anymore. By dropping the efficiency constraint, it is possible to design mechanisms with larger revenues than the VCG. For this reason, we believe that a more complete characterization of the behavior of A--VCG\ compared to the VCG should consider the \textit{deviation} between their payments and not only the loss in the revenue.
In particular, let us define the regret as the deviation between the VCG and the approximated VCG:
\begin{align}\label{eq:aregret}
\tilde{R}_T(\mathfrak{A}) = \sum_{t=1}^T \Big| \sum_{i=1}^N (p_i - p_{it}) \Big|,
\end{align}
We prove an upper--bound for the single--slot case (the extension of the multi--slot results is straightforward).
\begin{theorem}\label{thm:a-constant}
Let us consider an auction with $n$ advs and $T$ rounds. For any parameter $\tau$ and $\delta$, the A--VCG\ is always \textbf{truthful} and it achieves a regret
\begin{align}\label{eq:a-regret.const.exact}
\tilde{R}_T \leq v_{\max} (\tau + \frac{2\eta}{q_{\min}} + \delta T)
\end{align}
where $q_{\min} = \min_i q_i$. By setting the parameters to
\begin{align*}
\delta &= 1/T \\
\tau &= \tau = 2^{1/3}T^{2/3}n^{1/3} (\log NT)^{1/3},
\end{align*}
the regret is
\begin{align}\label{eq:a-regret.const}
\tilde{R}_T \leq v_{\max}\Bigg(\frac{2^{5/3}}{q_{\min}} T^{2/3}N^{1/3} (\log NT)^{1/3} + 1\Bigg).
\end{align}
\end{theorem}
\begin{pf}
We bound the difference between $p$ and $\hat{p}$ during the exploitation phase. We consider the two sides separately. Let $i^* = \arg\max_i \tilde{q}^+_i v_i$. We have
\begin{align*}
r_1 &= p - \hat{p} \\
&= \text{smax}_i(q_i v_i) - \frac{\text{smax}_i(\tilde{q}^+_i v_i)}{\tilde{q}^+_{i^*}}q_{i^*} \\
&= \frac{\text{smax}_i(\tilde{q}^+_i v_i)}{\tilde{q}^+_{i^*}} \Bigg( \frac{\text{smax}_i(q_i v_i)}{\text{smax}_i(\tilde{q}^+_i v_i)}\tilde{q}^+_{i^*} - q_{i^*}\Bigg)\\
&= v_{i^*}\frac{\text{smax}_i(\tilde{q}^+_i v_i)}{\tilde{q}^+_{i^*}v_{i^*}} \Bigg( \frac{\text{smax}_i(q_i v_i)}{\text{smax}_i(\tilde{q}^+_i v_i)}\tilde{q}^+_{i^*} - q_{i^*}\Bigg)\\
&\leq v_{\max} \Bigg( \frac{\text{smax}_i(q_i v_i)}{\text{smax}_i(q_i v_i)}\tilde{q}^+_{i^*} - q_{i^*}\Bigg)\\
&\leq v_{\max} \big( \tilde{q}^+_{i^*} - q_{i^*}\big) \leq 2v_{\max} \eta,
\end{align*}
with probability $1-\delta$.
Now we bound the other side.
\begin{align*}
r_2 &= \hat{p} - p \\
&= \frac{\text{smax}_i(\tilde{q}^+_i v_i)}{\tilde{q}^+_{i^*}}q_{i^*} - \text{smax}_i(q_i v_i) \\
&= \text{smax}_i(q_i v_i) \Bigg( \frac{\text{smax}_i(\tilde{q}^+_i v_i)}{\text{smax}_i(q_i v_i)}\frac{q_{i^*}}{\tilde{q}^+_{i^*}} - 1\Bigg)\\
&\leq \text{smax}_i(q_i v_i) \Bigg( \frac{\text{smax}_i((q_i + 2\eta) v_i)}{\text{smax}_i(q_i v_i)}\frac{q_{i^*}}{q_{i^*}} - 1\Bigg)\\
&\leq \text{smax}_i(q_i v_i) \Bigg( \frac{\text{smax}_i(q_i v_i) + 2\eta \text{smax}_i (v_i)}{\text{smax}_i(q_i v_i)} - 1\Bigg)\\
&\leq v_{\max} \Bigg( \frac{\text{smax}_i((q_i v_i) + 2\eta \text{smax}_i (v_i)}{\text{smax}_i(q_i v_i)} - 1\Bigg)\\
&\leq v_{\max} \Bigg( 1+ 2\eta\frac{\text{smax}_i (v_i)}{\text{smax}_i(q_i v_i)} - 1\Bigg)\\
&\leq v_{\max} 2\eta\frac{\frac{1}{q_{\min}}\text{smax}_i (q_i v_i)}{\text{smax}_i(q_i v_i)} = 2v_{\max} \frac{\eta}{q_{\min}},
\end{align*}
with probability $1-\delta$. As a result we have
\begin{align*}
|p - \hat{p}| \leq 2v_{\max} \frac{\eta}{q_{\min}},
\end{align*}
with probability $1-\delta$. The final bound on the expected regret is thus
\begin{align}
\tilde{R}_T \leq v_{\max} (\tau + \frac{2\eta}{q_{\min}} + \delta T)
\end{align}
By optimizing $\tau = 2^{1/3}T^{2/3}N^{1/3} (\log NT)^{1/3}$ and setting $\delta=1/T$ we have the final bound
\begin{align}
\tilde{R}_T \leq v_{\max}\Bigg(\frac{2^{5/3}}{q_{\min}} T^{2/3}N^{1/3} (\log NT)^{1/3} + 1\Bigg).
\end{align}
\qed
\end{pf}
\myremark{(the bound).} We notice that the bound is very similar to the bound for the regret $R_T$ but now an inverse dependency on $q_{\min}$ appears. This suggests that bounding the deviation between the two mechanisms is more difficult than bounding the revenue loss and that as the qualities become smaller, the A--VCG\ could be less and less efficient and, thus, have a larger and larger revenue. This result has two important implications. \textit{(i)} If social welfare maximization is an important requirement in the design of the learning mechanism, we should analyze the loss of A--VCG\ in terms of social welfare and provide (probabilistic) guarantees about the number of rounds the learning mechanism need in order to be efficient (see~\cite{gonen2007incentive-compatible} for a similar analysis). \textit{(ii)} If social welfare is not a priority, this result implies that a learning mechanism could be preferable w.r.t. to a standard VCG mechanism. We believe that further theoretical analysis and experimental validation are needed to understand better both aspects.
\section{Numerical Simulations}\label{s:experiments}
In this section we report numerical simulations to validate the theoretical bounds over the regret of the auctioneer's revenue presented in the previous sections.\footnote{The bounds over the regret of the social welfare present a structure similar to those over the auctioneer's revenue and their empirical analysis is omitted, providing similar results.} In particular, we analyze the accuracy with which our bounds predict the dependency of the regret on the main parameters of the auctions such as $T$, $N$, $K$, and $q_{\min}$. All the simulations share the way the ads are generated. The qualities $\{q_i\}_{\mathcal{N}}$ are drawn from a uniform distribution in $[0.01, 0.1]$, while the values $\{v_i\}_{\mathcal{N}}$ are randomly drawn from a uniform distribution on $[0, 1]$ ($v_{\max} = 1$). Since the main objective is to test the accuracy of the bounds, we report the \textit{relative regret}
$$\overline{R}_T = \frac{R_T}{B(T,K,N, q_{\min}, \Gamma_{\min})},$$
where $B(T,K,N, q_{\min}, \Gamma_{\min})$ is the value of the bound for the specific setting (i.e., (\ref{eq:regret.const}) and (\ref{eq:regret.posdep.qlu}) for position--dependent, and (\ref{eq:regret.extern}) for position/ad--dependent externalities). We analyze the accuracy of the bound w.r.t. each specific parameter, changing only its value and keeping the values of all the others fixed. We expect the relative regret to be always smaller than $1$, indeed we expect $B$ to be an actual upper--bound on the real regret $R_T$. All the results presented in the following sections have been obtained by setting $\tau$ and $\delta$ as suggested by the bounds derived in the previous sections and, where it is not differently specified, by averaging over 100 independent runs.
\subsection{Position--Dependent Externalities}\label{s:exp.constant}
\subsubsection{Unknown $\{q_i\}_{i \in \mathcal{N}}$}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.45\textwidth]{pics/const_T.eps}
\includegraphics[width=0.45\textwidth]{pics/const_N.eps}
\end{center}
\vspace{-0.4cm}
\caption{Position--dependent externalities with unknown $\{q_i\}_{i \in \mathcal{N}}$. Dependency of the relative regret on $T$, $N$.}\label{f:const}
\vspace{-0.4cm}
\end{figure*}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.45\textwidth]{pics/const_K.eps}
\includegraphics[width=0.45\textwidth]{pics/const_K_fixed.eps}
\end{center}
\vspace{-0.4cm}
\caption{Position--dependent externalities with unknown $\{q_i\}_{i \in \mathcal{N}}$. Dependency of the relative regret on $K$ for two different choice of the the qualities $q$.}\label{f:const2}
\vspace{-0.2cm}
\end{figure*}
First of all we analyze the accuracy of the bound provided in Section~\ref{ssec:uq}, where the model presents only position--dependent externalities and the qualities of the ads are unknown. We design the simulations such that $\lambda_m=\lambda$ for every $m$ with $\Lambda_1 = 1$ and $\Lambda_K = 0.8$ (i.e., $\lambda = \sqrt[K-1]{0.8}$). Thus, $\Lambda_{\min} = 0.8$ in all the experiments.
In Fig.~\ref{f:const} we analyze the accuracy of the bound w.r.t. the parameters $T$ and $N$. All the three curves in the left plot are completely flat (except for white noise) showing that the value of the relative regret $\overline{R}_T$ for different values of $K$ and $N$ not change as $T$ increases. This suggests that the bound in Theorem~\ref{thm:constant} effectively predicts the dependency of the regret $R_T$ w.r.t. the number of rounds $T$ of the auction as $\tilde O(T^{2/3})$. The right plot represents the dependency of the relative regret $\overline{R}_T$ on the number of ads $N$. In this case we notice that it is relatively accurate as $N$ increases but there is a transitory effect for smaller values of $N$ where the regret grows faster than predicted by the bound (although $B(T,K,N, q_{\min}, \Lambda_{\min})$ is still an upper--bound to $R_T$). Finally, the left plot of Fig.~\ref{f:const2} suggests that the dependency on $K$ in the bound of Theorem~\ref{thm:constant} is over--estimated, since the relative regret $\overline{R}_T$ decreases as $K$ increases. As discussed in the comment to the proof in Section~\ref{s:constant} this might be explained by the over--estimation of the term $\frac{\max_i(\tilde{q}^+_{i} \hat{v}_i;l)}{\max_i(\tilde{q}^+_{i} \hat{v}_i;k)}$ in the proof. In fact, this term is likely to decrease as $K$ increases. In order to validate this intuition, we have identified some instances for which the bound seems to accurately predict the dependency on $K$. For these instances $q_1 = 0.1$, $q_2=0.095$, and $q_i=0.09$ for every $2<i\leq K$. As a result, the ratio between the qualities $q_i$ is fixed (on average) and does not change with $K$. The right plot of Fig.~\ref{f:const2} shows that, with these values of $q_i$, the ratio $\overline{R}_T$ is constant for different values of $N$, implying that in this case the bound accurately predicts the behavior of $R_T$.
\subsubsection{Unknown $\{\Lambda_m\}_{m \in \mathcal{K}}$}
\TODO{Questa sezione è in fase di scrittura}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.45\textwidth]{pics/posdepli_T.eps}
\includegraphics[width=0.45\textwidth]{pics/posdepli_K.eps}
\end{center}
\vspace{-0.4cm}
\caption{Position--dependent externalities with unknown $\{\Lambda_m\}_{m \in \mathcal{K}}$. Dependency of the relative regret on $T$ and $K$.}\label{f:pd.TK}
\vspace{-0.4cm}
\end{figure*}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.45\textwidth]{pics/posdepli_mu.eps}
\includegraphics[width=0.45\textwidth]{pics/baba_var_mu.eps}
\end{center}
\vspace{-0.4cm}
\caption{Position--dependent externalities with unknown $\{\Lambda_m\}_{m \in \mathcal{K}}$. Dependency of the relative regret on $\mu$. Variance of "Babaioff" payments}\label{f:pd.mu}
\vspace{-0.2cm}
\end{figure*}
We now investigate the accuracy of the bound derived for algorithm A--VCG2$^\prime$ in Section~\ref{sssec:l.uc.m}.
The unknown $\{\lambda_m\}_{m \in \mathcal{K}}$, in the instances used for the simulation, is generated uniformly from the interval $ [0.98, 1] $. Thus the loss in prominence from a slot to another is very low. The simulations performed on instances randomly generated in this setting seem to confirm the bound derived in Section~\ref{sssec:l.uc.m}, while, with other instances the bound seems to overestimate the dependences over $K$ and $\mu$.
The left plot of Figure~\ref{f:pd.TK} shows the dependence of the ratio $\overline{R}_T$ w.r.t. to $T$ when $\mu = 0.01$. Despite the noise, the ratio seems to be constant unresponsive \TODO{trovare parola migliore} to the variation of T, confirming the bound provided. In the right plot the ratio follows the same behaviour when $K$ varies and $T = 100000$, $\mu = 0.01$. With small values of $K$ the bound seems to overestimate the dependence.
Finally, in the left plot of Figure~\ref{f:pd.mu} the ratio $\overline{R}_T$ seem to be constant w.r.t. the variation of parameter $\mu$ when $T = 500000$.
We conclude the analysis studying the variance of the payments as $\mu$ varies (Figure~\ref{f:pd.mu} right). Indeed, the bound suggests to choose a $\mu \rightarrow 0$ in order to reduce the regret, but this causes an undesired increase in the variance of the payments. Thus, the choice of $\mu$ should consider both these two dimensions of the problem. In the simulations for this last study, we repeat the auction for a finite number of times ($500000$) and compute the variance of the revenue of the auctioneer.
\subsubsection{Unknown $\{\Lambda_m\}_{m \in \mathcal{K}}$ and $\{q_i\}_{i \in \mathcal{N}}$}
In this section we analyze the bound provided in Section~\ref{ssec:uql} for position--dependent auctions where both the prominences and the qualities are unknown. For these simulations we generate $\{\lambda_m\}_{m \in \mathcal{K}}$ samples from a uniform distribution over $[0.5,1]$. In the simulations we adopted the values of $\tau$, $\delta$ and $\mu$ derived for the bound. In particular, in order to balance the increase of variance of the payments when $\mu$ decreases, the number of rounds is not constant, but it changes in function of $\mu$, i.e. $\frac{1000}{\mu}$. This means that, in expectation, the bid of a generic ad $a_i$ is modified $1000$ times over the number of the rounds.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.45\textwidth]{pics/pos_dep_T_primoa5000.eps}
\includegraphics[width=0.45\textwidth]{pics/pos_dep_N.eps}
\end{center}
\vspace{-0.4cm}
\caption{Position--dependent externalities with unknown $\{q_i\}_{i \in \mathcal{N}}$ and $\{\Lambda_m\}_{m \in \mathcal{K}}$. Dependency of the relative regret on $T$, $N$.}\label{f:ql}
\vspace{-0.4cm}
\end{figure*}
In the plots of Fig.~\ref{f:ql}, we show that the bound~(\ref{eq:regret.posdep.qlu}) accurately predicts the dependence of the regret w.r.t. the parameters $T$ and $N$. Indeed, except for the white noise due to the high variance of the payments based on the cSRP, the two plots shows that fixing the other parameters, the ratio $\overline{R}_T$ is constant as $T$ and $N$ increase, respectively.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.45\textwidth]{pics/pos_dep_K.eps}
\end{center}
\vspace{-0.4cm}
\caption{Position--dependent externalities with unknown $\{q_i\}_{i \in \mathcal{N}}$ and $\{\Lambda_m\}_{m \in \mathcal{K}}$. Dependency of the relative regret on $K$.}\label{f:ql_depK}
\vspace{-0.4cm}
\end{figure*}
The plot in Fig.~\ref{f:ql_depK} represents the dependency of the relative regret w.r.t. the parameter $K$. We can deduce that the bound $R_T$ over--estimate the dependency on $K$ for small values of the parameters, while, with larger values, the bound accurately predicts the behavior, the curves being flat.
\subsection{Position/Ad--Dependent Externalities}\label{s:exp.externalities}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.45\textwidth]{pics/extern_T.eps}
\includegraphics[width=0.45\textwidth]{pics/extern_q.eps}
\end{center}
\vspace{-0.4cm}
\caption{Dependency on $T$ and $q_{\min}$ in auctions with position/ad--dependent externalities.}\label{f:extern}
\vspace{-0.4cm}
\end{figure*}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.45\textwidth]{pics/extern_N.eps}
\end{center}
\vspace{-0.4cm}
\caption{Dependency of the relative regret $\overline{R}_T$ on $N$.}\label{f:add-extern}
\vspace{-0.2cm}
\end{figure}
In this section we analyze the bound provided in Section 5.1 for auctions with position--dependent and ad--dependent externalities where both only the qualities are unknown.
In the bound provided in Theorem~\ref{thm:extern} the regret $R_T$ presents a linear dependency on $N$ and an inverse dependency on the smallest quality $q_{\min}$. The relative regret $\overline{R}_T$ is now defined as $R_T/B$ where $B$ is bound (\ref{eq:regret.extern}). In the left plot of Fig.~\ref{f:extern} we report $\overline{R}_T$ as $T$ increases. As it can be noticed, the bound accurately predicts the behavior of the regret w.r.t. $T$ as in the case of position--dependent externalities. In the right plot of Fig.~\ref{f:extern} we report $\overline{R}_T$ as we change $q_{\min}$. According to the bound in (\ref{eq:regret.extern}) the regret should decrease as $q_{\min}$ increases (i.e., $R_T \leq \tilde O(q_{\min}^{-1}$)) but it is clear from the plot that $R_T$ has a much smaller dependency on $q_{\min}$, if any\footnote{From this experiment is not clear whether $\overline{R}_T = \tilde O(q_{\min}$), thus implying that $R_T$ does not depend on $q_{\min}$ at all, or $\overline{R}_T$ is sublinear in $q_{\min}$, which would correspond to a dependency $R_T = \tilde O(q_{\min}^{-f})$ with $f<1$.}. Finally, we study the dependency on $N$. The numerical analysis confirms the dependence derived in the bound: i.e. $\tilde O(N^\frac{1}{3})$.
We do not report results on $K$ since the complexity of finding the optimal allocation $f^*$ becomes intractable for values of $K$ larger than~8, as shown in~\cite{aamas2013}, making the empirical evaluation of the bound impossible.
\section{Learning with Position-- and Ad--Dependent Externalities}\label{s:externalities}
In this section we deal with the general model where both position-- and ad--dependent externalities are present, as formalized in~(\ref{eq:coeff2}), and we provide several partial results. In Section~\ref{sse:uqpad}, we analyze the problem of designing a DSIC mechanism when only the qualities of the ads are unknown. In Section~\ref{sse:pepad} we highlight some problems that rise when also other parameters are uncertain.
\subsection{Unknown quality} \label{sse:uqpad}
\begin{figure}[t]
\bookbox{
\begin{algorithmic}
\STATE \textbf{Input:} Length of exploration phase $\tau$, confidence $\delta$, position--dependent parameters $\{\Gamma_m\}_{m \in \mathcal{K}}$
\STATE
\STATE \textit{Exploration phase}
\FOR{$t = 1,\ldots,\tau$}
\STATE Allocate ads according to (\ref{eq:explorativeallocations})
\STATE Ask for no payment
\STATE Observe the clicks $\{click_{\pi(i;\theta_t)}^i(t)\}_{i=1}^{N}$
\ENDFOR
\STATE Compute the estimated quality $\tilde{q}_i = \frac{1}{|B_i|}\sum_{t \in B_i} \frac{click_{\pi(i; \theta_t)}^i(t)}{\Gamma_{\pi(i; \theta_t)(\theta_t)}}$
\STATE Compute $\tilde{q}^+_i = \tilde{q}_i + \eta$ where $\eta$ is given by (\ref{eq:eta2})
\STATE
\STATE \textit{Exploitation phase}
\FOR{$t = \tau+1,\ldots, T$}
\STATE Allocate ads according to $\tilde{f}$
\IF{Ad $a_i$ is clicked}
\STATE Ask for payment $\tilde p^c_i$ defined in (\ref{eq:hpay.extern})
\ENDIF
\ENDFOR
\end{algorithmic}}
\caption{Pseudo--code for the PAD--A--VCG\ mechanism.}\label{f:algpad}
\end{figure}
In this section we analyze the problem where the only unknown parameters are the qualities $\{q_i\}_{i \in \mathcal{N}}$ of the ads and the externality model includes position-- and ad--dependent externalities. As we do in Section~\ref{ssec:uq}, we focus on DSIC mechanisms and we leave open the question whether better bounds over the regret can be found by employing IC in expectation. Therefore we study MAB algorithms that separate the exploration and exploitation phases. The structure of the mechanism we propose, called PAD--A--VCG, is similar to the \avcg1 and is reported in Fig.~\ref{f:algpad}.
\paragraph{\indent Exploration phase.} During the exploration phase with length $\tau \leq T$ steps we collect $K$ samples of click or no--click events. Given a generic exploration policy $\{\theta_t\}_{0 \leq t \leq \tau}$, the estimate quality $\tilde{q}_i$ is computed as:
\begin{align*}
\tilde{q}_i = \frac{1}{|B_i|}\sum_{t \in B_i} \frac{click_{\pi(i; \theta_t)}^i(t)}{\Gamma_{\pi(i; \theta_t)}(\theta_t)},
\end{align*}
\noindent where we identify the set $B_i = \{t: \pi(i; \theta_t) \leq K, t \leq \tau\}$.
The explorative allocations $\theta_t$ have an impact on the discount $\Gamma_m(\theta_t)$ and thus a variation of Proposition~\ref{p:hoeffding} holds in which (\ref{eq:hoeffproposition1}) is substituted by:
\begin{align*
| q_i - \tilde{q}_i | \leq \sqrt{\Bigg(\sum_{t \in B_i} \frac{1}{\Gamma_{\pi(i; \theta_t)}(\theta_t)^2}\Bigg) \frac{1}{2 |B_i|^2} \log \frac{N}{\delta}}.
\end{align*}
For each exploration policy such that $|B_i| = \lfloor K\tau / N \rfloor$ $\forall i \in \mathcal{N}$, e.g. policy (\ref{eq:explorativeallocations}), we redefine $\eta$ as
\begin{align}\label{eq:eta2}
| q_i - \tilde{q}_i | \leq \frac{1}{\Gamma_{\min}}\sqrt{\frac{N}{2 K \tau} \log \frac{N}{\delta}} := \eta,
\end{align}
where $\Gamma_{\min} = \min\limits_{\theta \in \Theta, m \in \mathcal{K}} \{\Gamma_m(\theta)\}$. We define the upper--confidence bound $\tilde{q}^+_i = \tilde{q}_i + \eta$.
During the exploration phase, in order to preserve the DSIC property, the allocations $\{\theta_t\}_{0 \leq t \leq \tau}$ do not depend on the reported values of the advertisers and no payments are imposed to the advertisers.
\paragraph{\indent Exploitation phase} We define the estimated social welfare as
\begin{align*
\widetilde{\text{SW}}(\theta, \hat{\mathbf{v}}) = \sum_{i=1}^N \Gamma_{\pi(i; \theta)}(\theta) \tilde{q}^+_i \hat{v}_i = \sum_{m=1}^K \Gamma_m(\theta) \tilde{q}^+_{\alpha(m;\theta)} \hat{v}_{\alpha(m;\theta)} .
\end{align*}
We denote by $\tilde{\theta}$ the allocation maximizing $\widetilde{\text{SW}}(f(\hat{\mathbf{v}}), \hat{\mathbf{v}})$ and by $\tilde{f}$ the allocation function returning $\tilde{\theta}$:
\[
\tilde{\theta} = \tilde{f}(\hat{\mathbf{v}}) = \arg\max_{\theta\in\Theta}\widetilde{\text{SW}}(\theta, \hat{\mathbf{v}}).
\]
Once the exploration phase is over, the ads are allocated on the basis of $\tilde{f}$. Since $\tilde{f}$ is an affine maximizer, the mechanism can impose WVCG payments to the advertisers satisfying the DSIC property. In a \emph{pay--per--click} fashion, if ad $a_i$ is clicked, the advertiser is charged
\begin{align}\label{eq:hpay.extern}
\tilde p_i^c(\hat{\mathbf{v}}, click_{\pi(i;\tilde{\theta})}^i) = \frac{\widetilde{\text{SW}}(\tilde{\theta}_{-i}) - \widetilde{\text{SW}}_{-i}(\tilde{\theta})}{\Gamma_{\pi(i; \tilde{\theta})}(\tilde{\theta}) \tilde{q}^+_i}
\end{align}
which corresponds, in expectation, to the WVCG payment $\tilde p_i = \tilde p_i^c \Gamma_{\pi(i; \tilde{\theta})}(\tilde{\theta}) q_i$.
We are interested in bounding the regret of the auctioneer's revenue due to PAD--A--VCG\ compared to the auctioneer's revenue of the VCG mechanism when all the parameters are known.
\begin{theorem}\label{thm:extern}
Let us consider an auction with $N$ advs, $K$ slots, and $T$ rounds. The auction has position/ad--dependent externalities and cumulative discount factors $\{\Gamma_m(\theta)\}_{m=1}^K$ and $\eta$ defined as in (\ref{eq:eta2}). For any parameter $\tau \in \{0, \ldots, T\}$ and $\delta \in [0,1]$, the PAD--A--VCG\ achieves a regret:
\begin{align}\label{eq:regret.extern.exact}
R_T \leq v_{\max} K \left[ (T - \tau) \left( \frac{3\sqrt{2}n}{\Gamma_{\min}q_{\min}} \sqrt{\frac{N}{K\tau} \log \frac{N}{\delta}} \right) + \tau + \delta T \right],
\end{align}
\noindent where $q_{\min} = \min_{i \in \mathcal{N}} q_i$. By setting the parameters to
\begin{align*}
\delta &=K^\frac{1}{3} N^\frac{1}{3} \frac{6}{\Gamma_{\min}}^\frac{2}{3} T^{-\frac{1}{3}} \\
\tau &\geq \left( \frac{1}{2} \right)^\frac{1}{3} \frac{6}{\Gamma_{\min}}^\frac{2}{3} K^\frac{1}{3} T^\frac{2}{3} N^\frac{1}{3} \left[ \log\left(\frac{6}{\Gamma_{\min}}^{-\frac{2}{3}} N^\frac{2}{3} K^{-\frac{1}{3}} T^\frac{1}{3} \right)\right]^\frac{1}{3},
\end{align*}
the regret is
\begin{align}\label{eq:regret.extern}
R_T \leq v_{\max} K^\frac{4}{3} T^\frac{2}{3} N^\frac{1}{3} \frac{3 \cdot 2^\frac{1}{6} 6^\frac{2}{3}}{\Gamma_{\min}^\frac{2}{3} q_{\min}} \left[\log \left(\frac{N^\frac{2}{3} \Gamma_{\min}^\frac{2}{3} q_{\min} T^\frac{1}{3}}{6^\frac{2}{3}}\right)\right]^\frac{1}{3}.
\end{align}
\end{theorem}
\myremark{1 (Differences w.r.t. position--dependent externalities.)} Up to constants and logarithmic factors, the previous distribution--free bound is $R_T\leq \tilde O(T^\frac{2}{3} N^\frac{1}{3} K^\frac{4}{3})$.\footnote{We notice that in~\cite{glt} the authors provide a bound $O(T^\frac{2}{3} N K^\frac{2}{3})$ that does not match with their numerical simulations and thus they conjecture that the actual bound is $O(T^\frac{2}{3} N^\frac{1}{3} K^\frac{4}{3})$. Here we show that the conjecture is correct.} We first notice that moving from position-- to position/ad--dependent externalities does not change the dependency of the regret on both the number of rounds $T$ and the number of ads $N$. Moreover, the per--round regret still decreases to 0 as $T$ increases.
The main difference w.r.t. the bound in Theorem~\ref{thm:constant} is in the dependency on $K$ and on the smallest quality $q_{\min}$. We believe that the augmented dependence in $K$ is mostly due to an intrinsic difficulty of the position/ad--dependent externalities. The intuition is that now, in the computation of the payment for each ad $a_i$, the errors in the quality estimates cumulate through the slots (unlike the position--dependent case where they are scaled by $\Gamma_{k}-\Gamma_{k+1}$). This cumulated error should impact only on a portion of the ads (i.e., those which are actually impressed according to the optimal and the estimated optimal allocations) whose cardinality can be upper--bounded by $2K$. Thus we observe that the bound shows a super--linear dependency in the number of slots.
The other main difference is that now the regret has an inverse dependency on the smallest quality $q_{\min}$. Inspecting the proof, this dependency appears because the error of a quality estimation for an ad $a_i$ might be amplified by the inverse of the quality itself $\frac{1}{q_i}$. As discussed in Remark 2 of Theorem~\ref{thm:constant}, this dependency might follow from that fact the we have a distribution--free bound.
We investigate whether this dependency is an artifact of the proof or it is intrinsic in the algorithm in the numerical simulations reported in Section~\ref{s:experiments}. [\textbf{ANTICIPARE QUALCOSA?}]
\myremark{2 (Optimization of the parameter $\tau$).} We are considering an environment where $\{q_i\}_{i \in \mathcal{N}}$ are unknown, but if, at least, a guess about the value of $q_{\min}$ is available, it could be used to better tune $\tau$ by multiplying it by $(q_{\min})^{-\frac{2}{3}}$, thus reducing the regret from $\tilde O((q_{\min})^{-1})$ to $\tilde O((q_{\min})^{-\frac{2}{3}})$.
\myremark{3 (Externalities--dependent bound).} We notice that the above bound does not reduce to the bound (\ref{eq:regret.const}) in which only position--dependent externalities are present even disregarding the constant terms. Indeed, the dependency on $K$ is different in the two bounds: in (\ref{eq:regret.const}) we have $K^{\frac{2}{3}}$ while in (\ref{eq:regret.extern}) we have $K^{\frac{4}{3}}$. This means that bound (\ref{eq:regret.extern}) over--estimates the dependency on $K$ whenever the auction has position--dependent externalities. It is an interesting open question whether it is possible to derive an \textit{auction--dependent} bound where the specific values of the discount factors $\gamma_k(f)$ explicitly appear in the bound and that it reduces to (\ref{eq:regret.const}) for position--dependent externalities.
\textit{(Comment to the proof).} [NON L'HO TOCCATO, ASPETTO DI RIVEDERE E SISTEMARE LA PROOF] For the lack of space we do not report the proof, which can be found in the online appendix. While the proof of Thm.~\ref{thm:constant} could exploit the specific definition of the payments for position--dependent slots and it is a fairly standard extension of~\cite{devanur2009price}, in this case the proof is more complicated because of the dependency of the discount factors on the actual allocations and decomposes the regret of the exploitation phase in components due to the different allocations ($\hat{f}$ instead of $f^*$) and the different qualities as well ($\tilde{q}^+$ instead of $q$).
Using the mechanism described before, it is possible to derive an upper--bound over the global regret, when the regret, as in~\cite{babaioff_impl_pay}, is computed over the social welfare of the allocation. We obtain the same dependence over $T$, as for the regret on the payment. Thus $R^{SW}_T\leq\tilde{O}(T^\frac{2}{3})$. In particular notice that PAD--A--VCG\ is a zero--regret algorithm.
\begin{theorem} \label{th:pad_q_sw}
Let us consider an auction with $N$ advs, $K$ slots, and $T$ rounds. The auction has position/ad--dependent externalities and cumulative discount factors $\{\Gamma_m(\theta)\}_{m=1}^K$ and $\eta$ defined as in (\ref{eq:eta2}). For any parameter $\tau \in \{0, \ldots, T\}$ and $\delta \in [0,1]$, the PAD--A--VCG\ achieves a regret:
\begin{align}
R^{SW}_T \leq v_{\max} K \left[ (T - \tau) \frac{2}{\Gamma_{\min}} \sqrt{\frac{N}{2K\tau} \log \frac{N}{\delta}} + \tau + \delta T \right],
\end{align}
By setting the parameters to
\begin{align*}
\delta &= K^{-\frac{1}{3}} N^\frac{1}{3} T^{-\frac{1}{3}}\\
\tau & \geq 2^\frac{1}{3} \Gamma_{\min}^{-\frac{2}{3}} N^\frac{1}{3} K^{-\frac{1}{3}} T^\frac{2}{3} \left[\log{\left(N^\frac{2}{3} K^\frac{1}{3} T^\frac{1}{3}\right)}\right]^\frac{1}{3},
\end{align*}
the regret is
\begin{align}
R^{SW}_T \leq 3 \cdot 2^\frac{1}{3} \cdot v_{\max} \Gamma_{\min}^{-\frac{2}{3}} K^\frac{2}{3} N^\frac{1}{3} T^\frac{2}{3} \left[ \log{\left(N^\frac{2}{3} K^\frac{1}{3} T^\frac{1}{3}\right)} \right]^\frac{1}{3}.
\end{align}
\end{theorem}
Notice that using $\tau$ and $\delta$ defined in Theorem~\ref{thm:extern}, the bound for $R_T^{SW}$ is $\tilde{O}(T^\frac{2}{3})$, even if the parameters are not optimal for this second framework.
\subsection{Further extensions} \label{sse:pepad}
In this section we provide a negative, in terms of regret, result under DSIC truthfulness when the parameter $\gamma_{i,m}$ depends only on the ad $i$ (as in~\cite{Kempe2008}, we denote it by $c_i$) and this parameter is the only uncertain parameter.
We focus on the exploitation phase, supposing the exploration phase has produced the estimates $\{\tilde{c}^+_i\}_{i \in \mathcal{N}}$ for the continuation probabilities $\{c_i\}_{i \in \mathcal{N}}$. The allocation function $f$ presented in~\cite{Kempe2008} is able to compute the optimal allocation when $\{c_i\}_{i \in \mathcal{N}}$ values are known, but it is not an affine maximizer when applied to the estimated values $\{\tilde{c}^+_i\}_{i \in \mathcal{N}}$. Indeed, we call this allocation function $\tilde{f}$:
\begin{equation} \label{eq:tf}
\tilde{f}(\hat{\mathbf{v}}) = \arg\max_{\theta \in \Theta} \sum_{m=1}^K q_{\alpha(m;\theta)} \hat{v}_{\alpha(m;\theta)} \prod_{h=1}^{m-1} \tilde{c}^+_{\alpha(h; \theta)}.
\end{equation}
In this case, a weight depending only on a single ad cannot be isolated. Furthermore, we show also that this allocation function is not monotonic.
\begin{proposition}
The allocation function $\tilde{f}$ is not monotonic.
\end{proposition}
\begin{pf}
The proof is by counterexample.\TODO{Alex: questa devo ancora ricontrollarla bene.} Consider an environment with 3 ads and 2 slots such that:
\begin{table}[!h]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
ad & $v_i$ & $\tilde{c}^+_i$ & $c_i$\\
\hline
$a_1$ & $0.85$ & $1$ & $0.89$\\
\hline
$a_2$ & $1$ & $0.9$ & $0.9$\\
\hline
$a_3$ & $1.4$ & $0$ & $0$\\
\hline
\end{tabular}
\end{center}
\end{table}
\noindent and $q_i = 1$ $\forall i \in \mathcal{N}$. The optimal allocation $\tilde{\theta}$ found by $\tilde{f}$ when agents declare their true values $\mathbf{v}$ is: ad $a_2$ is allocated in the first slot and $a_3$ in the second one. We have $CTR_{a_3}(\tilde{\theta}) = 0.9$.
If advertiser $a_3$ reports a larger value: $\hat{v}_3 = 1.6$, in the allocation $\hat{\theta}$ found by $\tilde{f}(\hat{v}_3, \mathbf{v}_{-3})$, ad $a_1$ is displayed into the first slot and $a_3$ into the second one. In this case $CTR_{a_3}(\hat{\theta}) = 0.89 < CTR_{a_3}(\tilde{\theta})$, thus the allocation function $\tilde{f}$ is not monotonic. \qed
\end{pf}
On the basis of the above result, we can state the following theorem.
\begin{theorem}\label{thm:regretsocialwelfarec}
Let us consider an auction with $N$ advertisers, $K$ slots, and $T$ rounds, with ad--dependent cascade model with parameters $\{c_i\}_{i=1}^N$ whose value are unknown. Any online learning DSIC mechanism achieves an expected regret $R^{SW}_T=\Theta(T)$ over the social welfare.
\end{theorem}
\begin{pf}
Call $f(\hat{\mathbf{v}}| \mathbf{c})$ the allocation function maximizing the social welfare given parameters $\mathbf{c}$. As shown above, $f(\hat{\mathbf{v}}| \tilde{\mathbf{c}})$ cannot be adopted in the exploitation phase, the mechanism would not be DSIC otherwise. However, it can be easily observed that a necessary condition to have a no--regret algorithm is that the allocation function used in the exploitation phase, say $g(\hat{\mathbf{v}}| \tilde{\mathbf{c}})$, is such that $g(\hat{\mathbf{v}}| \mathbf{c}) = f(\hat{\mathbf{v}}| \mathbf{c})$ for every $\hat{\mathbf{v}}$ and $\mathbf{c}$ (that is, they always return the same allocation) given that $\tilde{\mathbf{c}}$ are consistent estimates and $\tilde{\mathbf{c}}\rightarrow \mathbf{c}$ as $T\rightarrow +\infty$. Otherwise, since allocations are finite and the difference between the values of the allocations is generically strictly positive, the algorithm would suffer from a strictly positive regret when $T\rightarrow +\infty$ and therefore it would not be a no--regret mechanism. However, any such a $g$ would not be monotonic and therefore it cannot be adopted in a DSIC mechanism. As a result, any online learning DSIC mechanism is not a no--regret mechanism.
To complete the proof, we need to provide a mechanism with regret $\Theta(T)$. Such a mechanism can be easily obtained by partitioning ads in groups such that in each group the ads compete only for a single slot. Therefore, each ad can appear in only one slot. \qed
\end{pf}
The above result shows that no approach similar to the approach described in~\cite{babaioff_impl_pay} can be adopted even for IC in expectation. Indeed, the approach described in~\cite{babaioff_impl_pay} requires in input a monotonic allocation function. This would suggest a negative result in terms of regret also when IC in expectation. However, in this paper we leave the study of this case open.
Finally, we provide a result on the regret over the auctioneer's revenue, whose proof is straightforward given that the (W)VCG cannot be adopted due to the above result and therefore the regret over the payments cannot go to zero as $T$ goes to $\infty$.
\begin{theorem}\label{thm:constant.l}
Let us consider an auction with $N$ advertisers, $K$ slots, and $T$ rounds, with ad--dependent cascade model with parameters $\{c_i\}_{i=1}^N$ whose value are unknown. Any online learning DSIC mechanism achieves an expected regret $R_T=\Theta(T)$.\TODO{M: regret su cosa?}
\end{theorem}
\section{Proofs of Section~4.1 and 5.1 (SW)}
\begin{proof}\textit{(Theorem~\ref{th:pd_q_sw} and \ref{th:pad_q_sw}) }
We now prove the bound on the social welfare, starting from the cumulative instantaneous regret during the exploitation phase.
\begin{align*}
r &= \text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \text{SW}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \\
& = \text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \widetilde{\text{SW}}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}})+\\
& + \underbrace{\widetilde{\text{SW}}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \max_{f \in \mathcal{F}} \widetilde{\text{SW}}(f)}_{\leq 0} + \widetilde{\text{SW}}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \text{SW}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \\
&\leq \underbrace{\text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \widetilde{\text{SW}}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}})}_{r^1} + \underbrace{\widetilde{\text{SW}}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \text{SW}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}})}_{r^2}
\end{align*}
\noindent The two remaining terms $r^1$ and $r^2$ can be easily bounded as
\begin{align*}
r^1 &= \text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \widetilde{\text{SW}}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \leq \max_{f \in \mathcal{F}} \big( \text{SW}(f(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \widetilde{\text{SW}}(f(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \big) \\
& = \max_{f \in \mathcal{F}} \sum_{j: \pi(j; f(\hat{\mathbf{v}}))\leq K} \Gamma_{\pi(j; f(\hat{\mathbf{v}}))}(f(\hat{\mathbf{v}})) \hat{v}_j \left( q_j - \tilde{q}^+_j \right) \\
&\leq v_{\max} \max_{f \in \mathcal{F}} \sum_{j: \pi(j; f(\hat{\mathbf{v}}))\leq K} ( q_j - \tilde{q}^+_j ) \leq 0,
\end{align*}
\noindent with probability $1-\delta$ and
\begin{align*}
r^2 &= \widetilde{\text{SW}}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \text{SW}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \leq \max_{f \in \mathcal{F}} \left( \widetilde{\text{SW}}(f(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \text{SW}(f(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \right) \\
& = \max_{f \in \mathcal{F}} \sum_{j: \pi(j;f(\hat{\mathbf{v}}))\leq K} \Gamma_{\pi(j;f(\hat{\mathbf{v}}))}(f(\hat{\mathbf{v}})) \hat{v}_j \left( \tilde{q}^+_{j} - q_j \right) \\
&\leq v_{\max} \max_{f \in \mathcal{F}} \sum_{j: \pi(j; f(\hat{\mathbf{v}}))\leq K} \big( \tilde{q}^+_j - q_j \big) \\
&= v_{\max} \max_{f \in \mathcal{F}} \sum_{j: \pi(j; f(\hat{\mathbf{v}}))\leq K} \left(\tilde{q}_j + \eta - q_j \right) \leq 2K v_{\max} \eta.
\end{align*}
\noindent with probability $1 - \delta$.
This computation is valid for both the framework of Sections~4.1 and 5.1.
We first notice that using the values of $\tau$ and $\delta$ identified in the proof of the bound for the payments, we obtain a bound $\tilde{O}(T^\frac{2}{3})$ for the global regret $R_T$. Anyway, in the following we use a rough optimization in order to find values of the two variables more appropriate for the regret on the SW.
$$
R_T \leq v_{\max} K \left[ 2 (T - \tau) \eta + \tau + \delta T \right].
$$
\paragraph*{Section 4.1}
$$
R_T \leq v_{\max} K \left[ (T - \tau) \frac{2}{K \Lambda_{\min}} \sqrt{\frac{N}{2\tau} \log \frac{N}{\delta}} + \tau + \delta T \right] =: B_\tau.
$$
A rough optimization of the bound $B_\tau$ is obtained as:
\begin{align*}
\frac{\partial B_\tau}{\partial \tau} &= v_{\max} K \left( 1 - \frac{(T - \tau)}{\tau} \frac{1}{K \Lambda_{\min}} \sqrt{\frac{N}{2\tau} \log{\frac{N}{\delta}}} - \frac{1}{K \Lambda_{\min}} \sqrt{\frac{N}{2\tau} \log{\frac{N}{\delta}}} \right) \geq 0\\
& \frac{T + \tau}{\tau} \frac{1}{K \Lambda_{\min}} \sqrt{\frac{N}{2\tau} \log{\frac{N}{\delta}}} \leq 1
\end{align*}
\noindent and approximating $T + \tau \approx 2 T$:
\begin{align*}
& \frac{2T^2}{\tau^3} \frac{N}{K^2 \Lambda_{\min}^2} \log{\frac{N}{\delta}} \leq 1\\
& \tau^3 \geq 2 \Lambda_{\min}^{-2} N K^{-2} T^2 \log{\frac{N}{\delta}}\\
& \tau \geq 2^\frac{1}{3} \Lambda_{\min}^{-\frac{2}{3}} N^\frac{1}{3} K^{-\frac{2}{3}} T^\frac{2}{3} \left(\log{\frac{N}{\delta}}\right)^\frac{1}{3}
\end{align*}
\noindent The bound over the regret is equal to:
\begin{align*}
R_T &\leq 2^\frac{1}{3} v_{\max} \Lambda_{\min}^{-\frac{2}{3}} K^\frac{1}{3} N^\frac{1}{3} T^\frac{2}{3} \left(\log{\frac{N}{\delta}}\right)^\frac{1}{3}+\\
& + 2^\frac{1}{3} v_{\max} \Lambda_{\min}^{-\frac{2}{3}} N^\frac{1}{3} K^\frac{1}{3} T^\frac{2}{3} \left(\log{\frac{N}{\delta}}\right)^\frac{1}{3} + \delta v_{\max} K T\\
& = O\left(\Lambda^{-\frac{2}{3}} K^\frac{1}{3} N^\frac{1}{3} T^\frac{2}{3} \left( \log{\frac{N}{\delta}} \right)^\frac{1}{3} \right) + \delta v_{\max} K T
\end{align*}
We choose $\delta$ s.t. the asymptotic order is not increased (not considering logarithmic factors), we set:
[DA SCEGLIERE]
\noindent with the constraint that $?$ (given by $\delta \leq 1$), leading to:
[DA SCEGLIERE]
\begin{align*}
\delta & = K^{-\frac{2}{3}} N^\frac{1}{3} T^{-\frac{1}{3}}\\
R_T & \leq 3 \cdot 2^\frac{1}{3} \cdot v_{\max} \Lambda_{\min}^{-\frac{2}{3}} K^\frac{1}{3} N^\frac{1}{3} T^\frac{2}{3} \left( \log{N^\frac{2}{3} K^\frac{2}{3} T^\frac{1}{3}} \right)^\frac{1}{3}
\end{align*}
\paragraph*{Section 5.1}
$$
R_T \leq v_{\max} K \left[ (T - \tau) \frac{2}{\Gamma_{\min}} \sqrt{\frac{N}{2K\tau} \log \frac{N}{\delta}} + \tau + \delta T \right] =: B_\tau.
$$
A rough optimization of the bound $B_\tau$ is obtained as:
\begin{align*}
\frac{\partial B_\tau}{\partial \tau} &= v_{\max} K \left( 1 - \frac{(T - \tau)}{\tau} \frac{1}{\Gamma_{\min}} \sqrt{\frac{N}{2K\tau} \log{\frac{N}{\delta}}} - \frac{1}{\Gamma_{\min}} \sqrt{\frac{N}{2K\tau} \log{\frac{N}{\delta}}} \right) \geq 0\\
& \frac{T + \tau}{\tau} \frac{1}{\Gamma_{\min}} \sqrt{\frac{N}{2K\tau} \log{\frac{N}{\delta}}} \leq 1
\end{align*}
\noindent and approximating $T + \tau \approx 2 T$:
\begin{align*}
& \frac{2T^2}{\tau^3} \frac{1}{\Gamma_{\min}^2} \frac{N}{K} \log{\frac{N}{\delta}} \leq 1\\
& \tau^3 \geq 2 \Gamma_{\min}^{-2} N K^{-1} T^2 \log{\frac{N}{\delta}}\\
& \tau \geq 2^\frac{1}{3} \Gamma_{\min}^{-\frac{2}{3}} N^\frac{1}{3} K^{-\frac{1}{3}} T^\frac{2}{3} \left(\log{\frac{N}{\delta}}\right)^\frac{1}{3}
\end{align*}
\noindent The bound over the regret is equal to:
\begin{align*}
R_T &\leq 2^\frac{1}{3} v_{\max} \Gamma_{\min}^{-\frac{2}{3}} K^\frac{2}{3} N^\frac{1}{3} T^\frac{2}{3} \left(\log{\frac{N}{\delta}}\right)^\frac{1}{3}+\\
& + 2^\frac{1}{3} v_{\max} \Gamma_{\min}^{-\frac{2}{3}} N^\frac{1}{3} K^\frac{2}{3} T^\frac{2}{3} \left(\log{\frac{N}{\delta}}\right)^\frac{1}{3} + \delta v_{\max} K T\\
& = O\left(\Gamma^{-\frac{2}{3}} K^\frac{2}{3} N^\frac{1}{3} T^\frac{2}{3} \left( \log{\frac{N}{\delta}} \right)^\frac{1}{3} \right) + \delta v_{\max} K T
\end{align*}
We choose $\delta$ s.t. the asymptotic order is not increased (not considering logarithmic factors), we set:
[DA SCEGLIERE]
\noindent with the constraint that $?$ (given by $\delta \leq 1$), leading to:
[DA SCEGLIERE]
\begin{align*}
\delta & = K^{-\frac{1}{3}} N^\frac{1}{3} T^{-\frac{1}{3}}\\
R_T & \leq 3 \cdot 2^\frac{1}{3} \cdot v_{\max} \Gamma_{\min}^{-\frac{2}{3}} K^\frac{2}{3} N^\frac{1}{3} T^\frac{2}{3} \left( \log{N^\frac{2}{3} K^\frac{1}{3} T^\frac{1}{3}} \right)^\frac{1}{3}
\end{align*}
\end{proof}
\section{Proof of Section~4.3 (SW)}
\begin{proof}\textit{(Theorem~\ref{th:pd_lq_sw})}
We now prove the bound on the social welfare, starting from the cumulative instantaneous regret during the exploitation phase.
\begin{align*}
r &= \text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \mathbb{E}_{\mathbf{x}}[\text{SW}(\tilde{f}(\mathbf{x})], \hat{\mathbf{v}})|\hat{\mathbf{v}}] \\
&= \underbrace{\mathbb{P}[\mathbf{s} = \boldsymbol{1}]}_{\leq 1} \left(\text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \text{SW}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}})\right) + \underbrace{\mathbb{P}[\mathbf{s} \not = \boldsymbol{1}]}_{\leq N\mu} \left(\text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \mathbb{E}_{\mathbf{x} | \mathbf{s} \not = \boldsymbol{1}}[\text{SW}(\tilde{f}(x), \mathbf{v})|\hat{\mathbf{v}}]\right)\\
& \leq \text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \text{SW}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) + N \mu \underbrace{\left(\text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \underbrace{\mathbb{E}_{\mathbf{x} | \mathbf{s} \not = \boldsymbol{1}}[\text{SW}(\tilde{f}(x), \mathbf{v})|\hat{\mathbf{v}}]}_{\geq 0}\right)}_{\leq K v_{\max}}\\
& \leq v_{\max} K 2 \eta + v_{\max} \mu N K
\end{align*}
\begin{align*}
R_T & \leq v_{\max} K \left[ (T - \tau) ( 2 \eta + N \mu) + \tau + \delta T \right]\\
& \leq v_{\max} K \left[ (T - \tau) \left( 2 \sqrt{\frac{N}{\tau} \log{\frac{2N}{\delta}}} + N \mu \right) + \tau + \delta T \right] =: B_{\tau}
\end{align*}
A rough optimization of the bound $B_\tau$ is obtained as:
\begin{align*}
\frac{\partial B_\tau}{\partial \tau} &= v_{\max} K \left( 1 - \frac{(T - \tau)}{\tau} \sqrt{\frac{N}{\tau} \log{\frac{2N}{\delta}}} - \sqrt{\frac{N}{\tau} \log{\frac{2N}{\delta}}} - N \mu \right) \geq 0\\
& \frac{T + \tau}{\tau} \sqrt{\frac{N}{\tau} \log{\frac{2N}{\delta}}} \leq 1 - N \mu
\end{align*}
\noindent and approximating $T + \tau \approx 2 T$:
\begin{align*}
& \frac{4T^2}{\tau^3} N \log{\frac{2N}{\delta}} \leq 1 - N \mu < 1\\
& \tau^3 \geq 2^2 N T^2 \log{\frac{2N}{\delta}}\\
& \tau \geq 2^\frac{2}{3} N^\frac{1}{3} T^\frac{2}{3} \left(\log{\frac{2N}{\delta}}\right)^\frac{1}{3}
\end{align*}
\noindent The bound over the regret is equal to:
\begin{align*}
R_T &\leq 2^\frac{2}{3} v_{\max} K N^\frac{1}{3} T^\frac{2}{3} \left(\log{\frac{2N}{\delta}}\right)^\frac{1}{3}+\\
& + 2^\frac{2}{3} v_{\max} K N^\frac{1}{3}T^\frac{2}{3} \left(\log{\frac{2N}{\delta}}\right)^\frac{1}{3} + N \mu K v_{\max} + \delta v_{\max} K T\\
& = O\left( K N^\frac{1}{3} T^\frac{2}{3} \left( \log{\frac{N}{\delta}} \right)^\frac{1}{3} \right)+ N \mu K v_{\max} + \delta v_{\max} K T
\end{align*}
We choose $\delta$ s.t. the asymptotic order is not increased (not considering logarithmic factors), we set:
[DA SCEGLIERE]
\noindent with the constraint that $?$ (given by $\delta \leq 1$), leading to:
[DA SCEGLIERE]
\begin{align*}
\delta & = N^\frac{1}{3} T^{-\frac{1}{3}}\\
\mu & = N^{-\frac{2}{3}} T^{-\frac{1}{3}}\\
R_T & \leq 2^\frac{8}{3} \cdot v_{\max} K N^\frac{1}{3} T^\frac{2}{3} \left( \log{N^\frac{2}{3} T^\frac{1}{3}} \right)^\frac{1}{3}
\end{align*}
[ALTERNATIVA]
Define $S' = \{\mathbf{s}' | if \pi(i;f*(\hat{\mathbf{v}}))\leq K, then s'_i = 1 \}$, i.e. all the random realization where the self--resampling procedure does not modify the bids of the ads displayed when the allocation function is $f^*$ is applied to the true bids $\hat{\mathbf{v}}$.
\begin{align*}
r &= \text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \mathbb{E}_{\mathbf{x}}[\text{SW}(\tilde{f}(\mathbf{x}), \hat{\mathbf{v}})|\hat{\mathbf{v}}] \\
&= \underbrace{\mathbb{P}[\mathbf{s} \in S']}_{\leq 1} \left(\text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \mathbb{E}_{\mathbf{x}|\mathbf{s} \in S'}[\text{SW}(\tilde{f}(x), \hat{\mathbf{v}})|\hat{\mathbf{v}}]\right) +\\
& + \underbrace{\mathbb{P}[\mathbf{s} \not \in S']}_{\leq K\mu} \left(\text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \mathbb{E}_{\mathbf{x} | \mathbf{s} \not \in S'}[\text{SW}(\tilde{f}(x), \mathbf{v})|\hat{\mathbf{v}}]\right)\\
& \leq \text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \mathbb{E}_{\mathbf{x}|\mathbf{s} \in S'}[\text{SW}(\tilde{f}(\mathbf{x}), \hat{\mathbf{v}})|\hat{\mathbf{v}}] + K \mu \underbrace{\left(\text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \underbrace{\mathbb{E}_{\mathbf{x} | \mathbf{s} \not \in S'}[\text{SW}(\tilde{f}(x), \mathbf{v})|\hat{\mathbf{v}}]}_{\geq 0}\right)}_{\leq K v_{\max}}\\
& \leq \underbrace{\text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \mathbb{E}_{\mathbf{x}|\mathbf{s} \in S'}[\widetilde{\text{SW}}(f^*(\mathbf{x}), \hat{\mathbf{v}})|\hat{\mathbf{v}}]}_{\leq 0} + \underbrace{\mathbb{E}_{\mathbf{x}|\mathbf{s} \in S'}[\widetilde{\text{SW}}(f^*(\mathbf{x}), \hat{\mathbf{v}})|\hat{\mathbf{v}}] - \mathbb{E}_{\mathbf{x}|\mathbf{s} \in S'}[\widetilde{\text{SW}}(\tilde{f}(\mathbf{x}), \hat{\mathbf{v}})|\hat{\mathbf{v}}]}_{\leq 0}+ \\
& + \mathbb{E}_{\mathbf{x}|\mathbf{s} \in S'}[\widetilde{\text{SW}}(\tilde{f}(\mathbf{x}), \hat{\mathbf{v}})|\hat{\mathbf{v}}] - \mathbb{E}_{\mathbf{x}|\mathbf{s} \in S'}[\text{SW}(\tilde{f}(\mathbf{x}), \hat{\mathbf{v}})|\hat{\mathbf{v}}] + v_{\max} \mu K^2\\
& \leq \mathbb{E}_{\mathbf{x}|\mathbf{s} \in S'}[\widetilde{\text{SW}}(\tilde{f}(\mathbf{x}), \hat{\mathbf{v}})|\hat{\mathbf{v}}] - \text{SW}(\tilde{f}(\mathbf{x}), \hat{\mathbf{v}})|\hat{\mathbf{v}}] + v_{\max} \mu K^2\\
& \leq \max_{f \in \mathcal{F}} \left( \mathbb{E}_{\mathbf{x}|\mathbf{s} \in S'}[\widetilde{\text{SW}}(f(\mathbf{x}), \hat{\mathbf{v}}) - \text{SW}(f(\mathbf{x}), \hat{\mathbf{v}})|\hat{\mathbf{v}}] \right) + v_{\max} \mu K^2\\
& \leq \max_{f \in \mathcal{F}} \left( \mathbb{E}_{\mathbf{x}|\mathbf{s} \in S'}[\widetilde{\text{SW}}(f(\mathbf{x}), \hat{\mathbf{v}}) - \text{SW}(f(\mathbf{x}), \hat{\mathbf{v}})|\hat{\mathbf{v}}] \right) + v_{\max} \mu K^2\\
& \leq \max_{f \in \mathcal{F}} \left( \sum_{j: \pi(j; f(x)) \leq K} \Lambda_{\pi(j; f(x))} v_j (\tilde{q}_j - q_j) \right) + v_{\max} \mu K^2\\
& \leq v_{\max} \max_{f \in \mathcal{F}} \left( \sum_{j: \pi(j; f(x)) \leq K} (\tilde{q}_j - q_j) \right) + v_{\max} \mu K^2\\
& \leq v_{\max} \max_{f \in \mathcal{F}} \left( K 2 \eta \right) + v_{\max} \mu K^2\\
& \leq 2 v_{\max} K \eta + v_{\max} \mu K^2
\end{align*}
To understand the bound $\text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \mathbb{E}_{\mathbf{x}|\mathbf{s} \in S'}[\widetilde{\text{SW}}(f^*(\mathbf{x}), \hat{\mathbf{v}})|\hat{\mathbf{v}}] \leq 0$, notice that, given that the bids of the ads displayed in $f^*(\hat{\mathbf{v}})$ are not modified we have that $\alpha(m; f^*(\hat{\mathbf{v}})) = \alpha(m; f^*(\mathbf{x}))$ where $m\leq K$ and $\mathbf{x}$ s.t. $\mathbf{s} \in S'$.
To understand the bound $\mathbb{E}_{\mathbf{x}|\mathbf{s} \in S'}[\widetilde{\text{SW}}(f^*(\mathbf{x}), \hat{\mathbf{v}})|\hat{\mathbf{v}}] - \mathbb{E}_{\mathbf{x}|\mathbf{s} \in S'}[\widetilde{\text{SW}}(\tilde{f}(\mathbf{x}), \hat{\mathbf{v}})|\hat{\mathbf{v}}] \leq 0$, notice that, given the bids of the ads s.t. $\pi(j; f^*(\mathbf{x})) \leq K$ are not modified and $x_i \leq \hat{v}_i \ \forall i \in \mathcal{N}$, $\widetilde{\text{SW}}(f^*(\mathbf{x}), \hat{\mathbf{v}}) = \widetilde{\text{SW}}(f^*(\mathbf{x}), \mathbf{x}) \leq \max_{\theta \in \Theta} \widetilde{\text{SW}}(\theta, \mathbf{x}) = \widetilde{\text{SW}}(\tilde{f}(\mathbf{x}), \mathbf{x}) \leq \widetilde{\text{SW}}(\tilde{f}(\mathbf{x}), \hat{\mathbf{v}})$.
\begin{align*}
R_T & \leq v_{\max} K \left[ (T - \tau) ( 2 \eta + K \mu) + \tau + \delta T \right]\\
& \leq v_{\max} K \left[ (T - \tau) \left( 2 \sqrt{\frac{N}{\tau} \log{\frac{2N}{\delta}}} + K \mu \right) + \tau + \delta T \right] =: B_{\tau}
\end{align*}
A rough optimization of the bound $B_\tau$ is obtained as:
\begin{align*}
\frac{\partial B_\tau}{\partial \tau} &= v_{\max} K \left( 1 - \frac{(T - \tau)}{\tau} \sqrt{\frac{N}{\tau} \log{\frac{2N}{\delta}}} - \sqrt{\frac{N}{\tau} \log{\frac{2N}{\delta}}} \right) \geq 0\\
& \frac{T + \tau}{\tau} \sqrt{\frac{N}{\tau} \log{\frac{2N}{\delta}}} \leq 1
\end{align*}
\noindent and approximating $T + \tau \approx 2 T$:
\begin{align*}
& \frac{4T^2}{\tau^3} N \log{\frac{2N}{\delta}} \leq 1\\
& \tau^3 \geq 2^2 N T^2 \log{\frac{2N}{\delta}}\\
& \tau \geq 2^\frac{2}{3} N^\frac{1}{3} T^\frac{2}{3} \left(\log{\frac{2N}{\delta}}\right)^\frac{1}{3}
\end{align*}
\noindent The bound over the regret is equal to:
\begin{align*}
R_T &\leq 2^\frac{2}{3} v_{\max} K N^\frac{1}{3} T^\frac{2}{3} \left(\log{\frac{2N}{\delta}}\right)^\frac{1}{3}+\\
& + 2^\frac{2}{3} v_{\max} N^\frac{1}{3} K T^\frac{2}{3} \left(\log{\frac{2N}{\delta}}\right)^\frac{1}{3} + N \mu K v_{\max} + \delta v_{\max} K T\\
& = O\left( K N^\frac{1}{3} T^\frac{2}{3} \left( \log{\frac{N}{\delta}} \right)^\frac{1}{3} \right)+ \mu K^2 v_{\max} + \delta v_{\max} K T
\end{align*}
We choose $\delta$ s.t. the asymptotic order is not increased (not considering logarithmic factors), we set:
[DA SCEGLIERE]
\noindent with the constraint that $?$ (given by $\delta \leq 1$), leading to:
[DA SCEGLIERE]
\begin{align*}
\delta & = N^\frac{1}{3} T^{-\frac{1}{3}}\\
\mu & = K^{-1} T^{-\frac{1}{3}} (facoltativo N^\frac{1}{3})\\
R_T & \leq 2^\frac{8}{3} \cdot v_{\max} K N^\frac{1}{3} T^\frac{2}{3} \left( \log{N^\frac{2}{3} T^\frac{1}{3}} \right)^\frac{1}{3}
\end{align*}
\end{proof}
\section{Relationship between Revenue and Welfare Regrets}
We refer to the general formulation of the payments in Section~\ref{sse:uqpad}, notably the expected payments of Eq.~\ref{eq:hpay.extern}. Let $r_i^W$ and $r_i^R$ be the per--ad welfare and revenue regrets, respectively. We have the following sequence of inequalities
\begin{align*}
r_i^W &= \text{SW}(\theta^*_{-i}) - \text{SW}_{-i}(\theta^*) - \big(\widetilde{\text{SW}}(\tilde{\theta}_{-i}) - \widetilde{\text{SW}}_{-i}(\tilde{\theta})\big) \\
&= \text{SW}(\theta^*_{-i}) - \text{SW}_{-i}(\theta^*) - \frac{\tilde{q}^+_i}{q_i}\big(\widetilde{\text{SW}}(\tilde{\theta}_{-i}) - \widetilde{\text{SW}}_{-i}(\tilde{\theta})\big)\frac{q_i}{\tilde{q}^+_i} \\
&=\text{SW}(\theta^*_{-i}) - \text{SW}_{-i}(\theta^*) - \Big(1+\frac{2\eta}{q_i}\Big)\big(\widetilde{\text{SW}}(\tilde{\theta}_{-i}) - \widetilde{\text{SW}}_{-i}(\tilde{\theta})\big)\frac{q_i}{\tilde{q}^+_i}\\
&=\text{SW}(\theta^*_{-i}) - \text{SW}_{-i}(\theta^*) - \big(\widetilde{\text{SW}}(\tilde{\theta}_{-i}) - \widetilde{\text{SW}}_{-i}(\tilde{\theta})\big)\frac{q_i}{\tilde{q}^+_i} - \frac{\eta}{q_i}\big(\widetilde{\text{SW}}(\tilde{\theta}_{-i}) - \widetilde{\text{SW}}_{-i}(\tilde{\theta})\big)\\
&=r_i^R - \frac{\eta}{q_i}\big(\widetilde{\text{SW}}(\tilde{\theta}_{-i}) - \widetilde{\text{SW}}_{-i}(\tilde{\theta})\big)\frac{q_i}{\tilde{q}^+_i} \leq r_i^R,
\end{align*}
which implies that any upper--bound on the revenue regret immediately translates into a bound for the welfare regret. Indeed, also the reverse is true through the following a similar sequence of inequalities which leads to
\begin{align*}
r_i^R &= p_i^* - \tilde p_i \\
&\leq r_i^W - \frac{2\eta}{q_i}\big(\text{SW}(\theta^*_{-i}) - \text{SW}_{-i}(\theta^*)\big) \leq r_i^W - 2v_{\max}\frac{\eta}{q_{\min}}.
\end{align*}
\section{Learning with Position--Dependent Externalities}\label{s:constant}
In this section we study the multi--slot auctions with only position--dependent cascade model. The CTRs depend only on the quality of the ads and on the position of the slots in which the ads are allocated. Formally, parameters $\gamma_{m,i}$ are such that they coincide with the prominence parameter (i.e., $\gamma_{m,i}=\lambda_m$ for every $m$ and $i$). As a result, the cumulative probability of observation, defined in~(\ref{eq:coeff2}), reduces to
\begin{align} \label{eq:coeff}
\Lambda_m = \Gamma_m(\theta) = \left\{
\begin{array}{ll}
1 & \text{if } m=1 \\
\prod\limits_{l=1}^{m-1} \lambda_{l} & \text{if } 2 \leq m\leq K\\
0 & \text{otherwise}
\end{array} \right.,
\end{align}
where we use $\Lambda_m$ instead of $\Gamma_m(\theta)$ for consistency with most of the literature on position--dependent externalities and to stress the difference with respect to the general case.
When all the parameters are known by the auctioneer, the efficient allocation $\theta^*$ prescribes that the ads are allocated to the slots in decreasing order w.r.t. their expected reported value $q_i \hat{v}_i$. More precisely, for any $k\in \mathcal{K}'$, let $\max\limits_{i \in \mathcal{N}} (q_i \hat{v}_i; k)$ be the operator returning the $k$--th largest value in the set $\{q_1 \hat{v}_1, \ldots, q_N \hat{v}_N\}$, then $\theta^*$ is such that, for every $m\in \mathcal{K}'$, the ad displayed at slot $m$ is
\begin{align}\label{eq:pos.dep.efficient.alloc}
\alpha(m; \theta^*) = \arg\max\limits_{i \in \mathcal{N}} (q_i \hat{v}_i; m).
\end{align}
This condition also simplifies the definition of the efficient allocation $\theta^*_{-i}$ when $a_i$ is removed from $\mathcal{N}$. In fact, for any $i,j\in \mathcal{N}$, if $\pi(j; \theta^*)<\pi(i; \theta^*)$ (i.e., ad $a_j$ is displayed before $a_i$) then $\pi(j; \theta^*_{-i}) = \pi(j;\theta^*)$, while if $\pi(j; \theta^*)>\pi(i; \theta^*)$ then $\pi(j; \theta^*_{-i}) = \pi(j; \theta^*)-1$ (i.e., ad $j$ is moved one slot upward), and $\pi(i; \theta^*_{-i}) = N$. By recalling the definition of VCG payments $p^*_i$ in (\ref{eq:payment}), in case of position--dependent externalities we obtain the simplified formulation:
\begin{align}
p^*_i(\hat v) = \begin{cases}\sum\limits_{l=\pi(i; \theta^*)+1}^{K+1} \left[(\Lambda_{l-1} - \Lambda_l) \max\limits_{j\in \mathcal{N}}(q_j \hat{v}_j; l)\right] & \text{if } \pi(i; \theta^*)\leq K\\
0 & \text{otherwise}\end{cases},
\end{align}
which can be easily written as a per--slot payment as:
\begin{align}
p^*_{\alpha(m; \theta^*)}(\hat v) = \begin{cases}\sum\limits_{l=m+1}^{K+1} \left[(\Lambda_{l-1} - \Lambda_l) \max\limits_{i\in \mathcal{N}}(q_i \hat{v}_i; l)\right] & \text{if } m\leq K\\
0 & \text{otherwise}\end{cases}.
\end{align}
In the following sections we study the problem of designing incentive compatible mechanisms under different conditions of lack of information over the parameters $\{q_i\}_{i \in \mathcal{N}}$ and $\{\Lambda_m\}_{m \in \mathcal{K}}$. In particular, in Section~\ref{ssec:uq}, we assume that the actual values of $\{q_i\}_{i \in \mathcal{N}}$ are unknown by the auctioneer, while those of $\{\Lambda_m\}_{m \in \mathcal{K}}$ are known. In Section~\ref{ssec:ul}, we assume that the actual values of $\{\Lambda_m\}_{m \in \mathcal{K}}$ are unknown by the auctioneer, while those of $q_i$s are known. Finally, in Section~\ref{ssec:uql}, we assume that the actual values of both $\{q_i\}_{i \in \mathcal{N}}$ and $\{\Lambda_m\}_{m \in \mathcal{K}}$ are unknown.
\subsection{Unknown qualities $\{q_i\}_{i \in \mathcal{N}}$} \label{ssec:uq}
In this section we assume that the qualities of the ads ($\{q_i\}_{i \in \mathcal{N}}$) are unknown, while $\{\Lambda_m\}_{m \in \mathcal{K}}$ are known. We initially focus on DSIC mechanisms and subsequently we discuss about mechanisms IC in expectation.
As in~\cite{devanur2009price,babaioff2008characterizing}, we formalize the problem as a multi--armed bandit problem and we study the properties of a learning mechanism where the exploration and exploitation phases are separated, such that during the exploration phase, we estimate the values of $\{q_i\}_{i \in \mathcal{N}}$ and during the exploitation phase we use the estimated qualities $\{\tilde{q}_i\}_{i \in \mathcal{N}}$ to implement an IC mechanism.
The pseudo code of the algorithm A--VCG1 (Adaptive VCG1) is given in Fig.~\ref{f:alg}. The details of the algorithm follow.
\begin{figure}[t]
\bookbox{
\begin{algorithmic}
\STATE \textbf{Input:} Length of exploration phase $\tau$, confidence $\delta$, position--dependent parameters $\{\Lambda_m\}_{m \in \mathcal{K}}$
\STATE
\STATE \textit{Exploration phase}
\FOR{$t = 1,\ldots,\tau$}
\STATE Allocate ads according to (\ref{eq:pos.dep.efficient.alloc})
\STATE Ask for no payment
\STATE Observe the clicks $\{click_{\pi(i;\theta_t)}^i(t)\}_{i=1}^{N}$
\ENDFOR
\STATE Compute the estimated quality $\tilde{q}_i = \frac{1}{|B_i|}\sum_{t \in B_i} \frac{click_{\pi(i; \theta_t)}^i(t)}{\Lambda_{\pi(i; \theta_t)}}$
\STATE Compute $\tilde{q}^+_i = \tilde{q}_i + \eta$ where $\eta$ is given by (\ref{eq:eta})
\STATE
\STATE \textit{Exploitation phase}
\FOR{$t = \tau+1,\ldots, T$}
\STATE Allocate ads according to $\tilde{f}$ defined in~(\ref{eq:optimalallocationestimatedq})
\IF{Ad $a_i$ is clicked}
\STATE Ask for payment $\tilde p^c_i$ defined in (\ref{eq:hpay.const.ppc})
\ENDIF
\ENDFOR
\end{algorithmic}}
\caption{Pseudo--code for the A--VCG1 mechanism.}\label{f:alg}
\end{figure}
\paragraph*{\indent Exploration phase} The exploration phase takes $\tau\geq N/K$ rounds.\footnote{Notice that we need $\tau > N/K$ in order to guarantee that all the ads have at least one sample to initialize the estimates $\tilde{q}_i$.} During this phase, the algorithm receives as input the parameters $\{\Lambda_m\}_{m \in \mathcal{K}}$ and collects data to estimate the quality of each ad. Unlike the single--slot case, where we collect only one sample of click or no--click events per round, here we can exploit the fact that each ad $a_i$ has a non--zero CTR whenever it is allocated to a slot $s_m$ with $m\leq K$. As a result, at each round of the exploration phase, we collect $K$ samples (click or no--click events), one from each slot. Let $\theta_t$ (for $t \leq \tau$) be a sequence of (potentially arbitrary) allocations independent from the advertisers' bids. The set $B_i = \{t: \pi(i; \theta_t) \leq K, t\leq\tau\}$ contains all the time instants when ad $a_i$ is allocated to a valid slot, so that $|B_i|$ corresponds to the total number of (click/no--click) samples available for ad $a_i$. We denote by $click_{\pi(i; \theta_t)}^i(t)\in\{0,1\}$ the click event at time $t$ for ad $a_i$ when displayed at slot $\pi(i; \theta_t)$. Depending on the slot in which the click event happens, the ad $a_i$ has different CTRs, thus we weigh each click sample by the probability of observation $\Lambda_m$ related to the slot in which the ad was allocated. The estimated quality $\tilde{q}_i$ is computed as
\begin{align}\label{eq:est.q}
\tilde{q}_i = \frac{1}{|B_i|}\sum_{t \in B_i} \frac{click_{\pi(i; \theta_t)}^i(t)}{\Lambda_{\pi(i; \theta_t)}},
\end{align}
which is an unbiased estimate of $q_i$ (i.e., $\mathbb{E}_{click} [\tilde{q}_i] = q_i$, where $\mathbb{E}_{click}$ is the expectation w.r.t. the realization of the clicks). By applying the Hoeffding's inequality~\cite{hoeffding1963probability}, we obtain a bound over the error of the estimated quality $\tilde{q}_i$ for each ad $i$.\footnote{\TODO{Alex: giusto per completezza metterei la dimostrazione in appendice. M: fatto, ma potrebbe mancare un 2 al numeratore nel log. DA CONTROLLARE}}
\begin{proposition}\label{p:hoeffding}
For any ad $i \in \mathcal{N}$
\begin{align}\label{eq:hoeffproposition1}
| q_i - \tilde{q}_i | \leq \sqrt{\Bigg(\sum_{t \in B_i} \frac{1}{\Lambda_{\pi(i; \theta_t)}^2}\Bigg) \frac{1}{2 |B_i|^2} \log \frac{N}{\delta}},
\end{align}
with probability $1-\delta$ (w.r.t. the click events).
\end{proposition}
During the exploration phase, at each round $t=1,\ldots,\tau$, we adopt the following sequence of allocations
\begin{equation}\label{eq:explorativeallocations}
\theta_t=\{\langle s_1, a_{(t \text{ mod } N) + 1} \rangle, \ldots, \langle s_{N}, a_{(t+N-1 \text{ mod } N) + 1} \rangle\},
\end{equation}
obtaining $|B_i| = \lfloor K \tau / N \rfloor$ for all the ads $a_i$. Thus, given that $\lfloor K \tau / N \rfloor \geq \frac{\tau K}{2N}$, Equation~(\ref{eq:hoeffproposition1}) becomes\footnote{For the sake of presentation, from now on we drop the rounding and we use $ \tau K / N$. \TODO{Alex: use $\lfloor K \tau / N \rfloor \geq \tau K/(2N)$ instead.} M: ok}
\begin{align}\label{eq:eta}
| q_i - \tilde{q}_i | \leq \sqrt{\Bigg(\sum_{m=1}^{K} \frac{1}{\Lambda_{m}^2}\Bigg) \frac{2 N}{K^2 \tau} \log \frac{N}{\delta}} := \eta.
\end{align}
During this phase, in order to guarantee DSIC, the advertisers cannot be charged with any payment, i.e. all the payments in rounds $t\leq \tau$ are set to 0. In fact, as shown in~\cite{babaioff2008characterizing}, any bid--dependent payment could be easily manipulated by bidders with better estimates of the CTRs, thus obtaining a non--truthful mechanism, whereas non--bid--dependent payments could make the mechanism not to be IR and thus bidders may prefer not to participate to the mechanism.
\paragraph*{\indent Exploitation phase} Once the exploration phase is concluded, an upper--confidence bound over each quality is computed as $\tilde{q}^+_i = \tilde{q}_i + \eta$ and the exploration phase is started and run for the remaining $T-\tau$ rounds. We define the \emph{estimated social welfare} as:
\begin{align*}
\widetilde{\text{SW}}(\theta,\hat{\mathbf{v}})= \sum_{i=1}^N \Lambda_{\pi(i;\theta)} \tilde{q}^+_i \hat{v}_i
\end{align*}
and we define $\tilde{f}$ as the allocation function returning the efficient allocation $\tilde{\theta}$ on the basis of the estimated qualities as:
\begin{align}\label{eq:optimalallocationestimatedq}
\tilde{\theta}=\tilde{f}(\hat{\mathbf{v}}) = \arg~\max\limits_{\theta \in \Theta}~ \{\widetilde{\text{SW}}(\theta,\hat{\mathbf{v}})\}
\end{align}
Our mechanism adopts $\tilde{f}$ during all the steps of the exploitation phase. Notice that $\tilde{f}$ is an affine maximizer, given that
\[
\tilde{f}(\hat{\mathbf{v}}) = \arg\max\limits_{\theta \in \Theta} \sum_{i=1}^N \Lambda_{\pi(i; \theta)} \tilde{q}^+_i \hat{v}_i = \arg\max\limits_{\theta \in \Theta} \sum_{i=1}^N \frac{\tilde{q}^+_i }{q_i} \Lambda_{\pi(i; \theta)} q_i \hat{v}_i = \arg\max\limits_{\theta \in \Theta} \sum_{i=1}^N w_i \Lambda_{\pi(i; \theta)} q_i \hat{v}_i
\]
where each weight $w_i= \frac{\tilde{q}^+_i }{q_i} $ is independent of the advertisers' types $v_i$. Hence, we can apply the WVCG (weighted--VCG) payments (here denoted by $\tilde{p}$ because based on estimated parameters) satisfying the DSIC property
\begin{align}\label{eq:hpay.const}
\tilde p_i (\hat{\mathbf{v}}) &= \frac{1}{w_i} \sum_{l=\pi(i; \tilde{\theta})+1}^{K+1} (\Lambda_{l-1} - \Lambda_l) \max\limits_{j \in \mathcal{N}}(\tilde{q}^+_j \hat{v}_j; l) \nonumber\\
&=\frac{q_i}{\tilde{q}^+_i} \sum_{l=\pi(i;\tilde{\theta})+1}^{K+1} (\Lambda_{l-1} - \Lambda_l) \max\limits_{j \in \mathcal{N}}(\tilde{q}^+_j \hat{v}_j; l).
\end{align}
These payments cannot be computed by the auctioneer, since the actual $\{q_i\}_{i \in \mathcal{N}}$ are unknown. However, it can apply \emph{pay--per--click} payments $\{\tilde{p}_i^c\}_{i\in \mathcal{N}}$ such as:
\begin{align}\label{eq:hpay.const.ppc}
\tilde p^c_i (\hat{\mathbf{v}},click^i_{\pi(i; \tilde{\theta})}) =
\begin{cases}
0, & \textnormal{if } click^i_{\pi(i; \tilde{\theta})}=0 \\
\frac{1}{\Lambda_{\pi(i; \tilde{\theta})}\tilde{q}^+_i}\bigg(\sum\limits_{l=\pi(i; \tilde{\theta})+1}^{K+1} (\Lambda_{l-1} - \Lambda_l) \max\limits_{j \in \mathcal{N}}(\tilde{q}^+_j \hat{v}_j; l)\bigg), & \textnormal{if } click^i_{\pi(i; \tilde{\theta})}=1 \\
\end{cases}
\end{align}
which in expectation are equal to the WVCG payments $\tilde{p}_i (\hat{\mathbf{v}}) = \tilde p^c_i(\hat{\mathbf{v}},click^i_{\pi(i; \tilde{\theta})}) \Lambda_{\pi(i; \tilde{\theta})}q_i$. Unlike the payments $\tilde{p}_i (\hat{\mathbf{v}})$, these payments can be computed simply relying on the estimates $\tilde{q}^+_i$ and on the knowledge of the probabilities $\Lambda_m$.
We can state the following.
\begin{proposition}
The A--VCG1 is DSIC, IR \emph{a posteriori}, and WBB \emph{a posteriori}.
\end{proposition}
\begin{pf}
It trivially follows from the fact that the mechanism is a WVCG mechanism and that the payments are pay--per--click.\qed
\end{pf}
We now move to the analysis of the performance of A--VCG1 in terms of regret the mechanism cumulates through $T$ rounds.
\begin{theorem}\label{thm:constant}
Let us consider a sequential auction with $N$ advertisers, $K$ slots, and $T$ rounds with position--dependent cascade model with parameters $\{\Lambda_m\}_{m=1}^K$ and accuracy $\eta$ as defined in~(\ref{eq:eta}). For any parameter $\tau \in \{0, \ldots, T\}$ and $\delta \in [0,1]$, the A--VCG1 achieves a regret:
\begin{align}\label{eq:regret.const.exact}
R_T &\leq v_{\max} \left( \sum_{m = 1}^K \Lambda_m \right)\Big( 2(T - \tau) \eta + \tau + \delta T \Big).
\end{align}
\noindent By setting the parameters to
\begin{align*}
\delta &= K^{-\frac{1}{3}} T^{-\frac{1}{3}} N^{\frac{1}{3}}\\
\tau &= 2^{\frac{1}{3}} K^{-\frac{1}{3}} T^{\frac{2}{3}} N^{\frac{1}{3}} \Lambda_{\min}^{-\frac{2}{3}} \left[ \log \left( K^\frac{1}{3} T^\frac{1}{3} N^\frac{2}{3} \right) \right]^{\frac{1}{3}},
\end{align*}
where $\displaystyle \Lambda_{\min} = \min_{m \in \mathcal{K}} \Lambda_m, \ \Lambda_{\min} > 0$, then the regret is
\begin{align}\label{eq:regret.const}
R_T \leq \left( 2^\frac{4}{3} \Lambda_{\min}^{-\frac{2}{3}} + 1\right) v_{\max} K^\frac{2}{3} T^\frac{2}{3} N^\frac{1}{3} \left[ \log \left( K^\frac{1}{3} T^\frac{1}{3} N^\frac{2}{3} \right) \right]^{\frac{1}{3}}
\end{align}
\end{theorem}
We initially introduce some remarks about the above results, and subsequently discuss the sketch of the proof of the theorem.
\myremark{1 (The bound).} Up to numerical constants and logarithmic factors, the previous bound~(\ref{eq:regret.const}) is $R_T \leq \tilde O(T^\frac{2}{3} K^\frac{2}{3} N^\frac{1}{3})$.
We first notice that A--VCG1 is a no--regret algorithm since its per--round regret ($R_T/T$) decreases to 0 as $T^{-\frac{1}{3}}$, thus implying that it asymptotically achieves the same performance as the VCG. Furthermore, we notice that for $K=1$ the bound reduces (up to constants) to the single--slot case analyzed in~\cite{devanur2009price}. Unlike the standard bound for multi--armed bandit algorithms, the regret scales as $\tilde O(T^\frac{2}{3})$ instead of $\tilde O(T^\frac{1}{2})$. As pointed out in~\cite{devanur2009price} and \cite{babaioff2008characterizing} this is the unavoidable price the bandit algorithm has to pay to be DSIC.
Finally, the dependence of the regret on $N$ is sub--linear ($N^\frac{1}{3}$) and therefore an increase of the number of advertisers does not significantly worsen the regret. The dependency on the number of slots $K$ is similar: according to the bound (\ref{eq:regret.const}) the regret has a sublinear dependency $\tilde O(K^\frac{2}{3})$, meaning that whenever one slot is added to the auction, the performance of the algorithm does not significantly worsen. By analyzing the difference between the payments of the VCG and A--VCG1, we notice that during the exploration phase the regret is $O(\tau K)$ (e.g., if all the ads allocated into the $K$ slots are clicked at each explorative round), while during the exploitation phase the error in estimating the qualities sum over all the $K$ slots, thus suggesting a linear dependency on $K$ for this phase as well. Nonetheless, as $K$ increases, the number of samples available per ad increases as $\tau K/N$, thus improving the accuracy of the quality estimates by $\tilde O(K^{-\frac{1}{2}})$ (see Proposition~\ref{p:hoeffding}). As a result, as $K$ increases, the exploration phase can be shortened (the optimal $\tau$ actually decreases as $K^{-\frac{1}{3}}$), thus reducing the regret during the exploration, and still have accurate enough estimations to control the regret of the exploitation phase.
\myremark{2 (Distribution--free bound).} The bound derived in Theorem~\ref{thm:constant} is a \textit{distribution--free} (or worst--case) bound, since it holds for any set of advertisers (i.e., for any $\{q_i\}_{i\in\mathcal{N}}$ and $\{v_i\}_{i\in\mathcal{N}}$). This generality comes at the price that, as illustrated in other remarks and in the numerical simulations (see Section~\ref{s:experiments}), the bound could be inaccurate for some specific sets of advertisers. On the other hand, distribution--dependent bounds (see e.g., the bounds of UCB~\cite{auer2002finite-time}), where $q$ and $v$ appear explicitly, would be more accurate in predicting the behavior of the algorithm. Nonetheless, they could not be used to optimize the parameters $\delta$ and $\tau$, since they would then depend on unknown quantities.
\myremark{3 (Parameters).} The choice of parameters $\tau$ and $\delta$ reported in Theorem~\ref{thm:constant} is obtained by rough minimizing the upper--bound (\ref{eq:regret.const.exact}). Each parameter can be computed by knowing the characteristics of the auction (number of rounds $T$, number of slots $K$, number of ads $N$, and $\Lambda_m$). Moreover, since the values are obtained optimizing an upper--bound of the regret and not directly the true global regret, these values can provide a good guess for the parametrization, but there could be other values that better optimize the regret. Thus, in practice, the regret could be optimized by searching the space of the parameters around the values suggested in Theorem~\ref{thm:constant}.
\myremark{4 (IC in expectation).} Two interesting problems we do not solve in this paper once IC in expectation (over the click realizations and/or realizations of the random component of the mechanism) is adopted are whether or not it is possible to avoid the separation of the exploration and exploitation phases and whether it is possible to obtain a regret of $O(T^{1/2})$ as it is possible in the case of $K=1$~\cite{babaioff_impl_pay}. Any attempt we tried to extend the result presented in~\cite{babaioff_impl_pay} to the multi--slot case conducted us to a non--IC mechanism. We briefly provide some examples of adaptation to our framework of the two MAB presented~\cite{babaioff_impl_pay}. None of these attempts provided a monotone allocation function. We have tried to extend the UCB1 in different ways, e.g. introducing $N \cdot K$ estimators, one for each ad for each slot, or maintaining $N$ estimators weighting in different ways click obtained in different slots. The second MAB algorithm, called NewCB, is based on the definition of a set of active ads, the ones that can be displayed. We have considered extensions with a single set for all the slots and with multiple sets, one for each slot, without identifying monotone allocation algorithms.
\TODO{Alex: report the full proof when ready}
\textit{(Comments to the proof).} The proof uses relatively standard arguments to bound the regret of the exploitation phase. As discussed in Remark 2, the bound is distribution--free and some steps in the proof are conservative upper--bounds on quantities that might be smaller for specific auctions. For instance, the inverse dependency on the smallest cumulative discount factor $\Lambda_{\min}$ in the final bound could be a quite inaccurate upper--bound on the quantity $\sum_{m=1}^{K} 1/ \Lambda_{m}^2$. In fact, the parameter $\tau$ itself could be optimized as a direct function of $\sum_{m=1}^{K} 1 /\Lambda_{m}^2$, thus obtaining a more accurate tuning of the length of the exploration phase and a slightly tighter bound (in terms of constant terms). Furthermore, we notice that the step $\max\limits_{i \in \mathcal{N}}(\tilde{q}^+_{i} v_i;h) / \max\limits_{i \in \mathcal{N}}(\tilde{q}^+_{i} v_i;m) \leq 1$ is likely to become less accurate as the difference between $h$ and $m$ increases (see Eq.~\ref{eq:step.loose} in the proof). For instance, if the qualities $q_i$ are drawn from a uniform distribution in $(0,1)$, as the number of slots increases this quantity reduces as well (on average) thus making the upper--bound by $1$ less and less accurate. The accuracy of the proof and the corresponding bound are further studied in the simulations in Section~\ref{s:experiments}.
In a similar way, adopting the same mechanism as before, it is also possible to derive an upper--bound over the global regret, when the regret, as in~\cite{babaioff_impl_pay} is computed over the social welfare of the allocation. In particular we obtain, that, even in this case, A--VCG1 is a no--regret algorithm and $R^{SW}_T\leq\tilde{O}(T^\frac{2}{3})$.
\begin{theorem} \label{th:pd_q_sw}
Let us consider a sequential auction with $N$ advertisers, $K$ slots, and $T$ rounds with position--dependent cascade model with parameters $\{\Lambda_m\}_{m=1}^K$ and $\eta$ as defined in~(\ref{eq:eta}). For any parameter $\tau \in \{0, \ldots, T\}$ and $\delta \in [0,1]$, the A--VCG1 achieves a regret:
\begin{align}
R^{SW}_T &\leq v_{\max} K \left( 2 \left(T - \tau \right) \eta + \tau + \delta T \right).
\end{align}
\noindent By setting the parameters to
\begin{align*}
\delta &= K^{-\frac{2}{3}} T^{-\frac{1}{3}} N^{\frac{1}{3}}\\
\tau &= 2^\frac{1}{3} \Lambda_{\min}^{-\frac{2}{3}} N^\frac{1}{3} K^{-\frac{2}{3}} T^\frac{2}{3} \left[ \log{\left(N^\frac{2}{3} K^\frac{2}{3} T^\frac{1}{3}\right)} \right]^\frac{1}{3},
\end{align*}
where $\displaystyle \Lambda_{\min} = \min_{m \in \mathcal{K}} \Lambda_m, \ \Lambda_{\min} > 0$, then the regret is
\begin{align}
R^{SW}_T \leq 3 \cdot 2^\frac{1}{3} v_{\max} \Lambda_{\min}^{-\frac{2}{3}} K^\frac{1}{3} N^\frac{1}{3} T^\frac{2}{3} \left[ \log{\left(N^\frac{2}{3} K^\frac{2}{3} T^\frac{1}{3}\right)} \right]^\frac{1}{3}
\end{align}
\end{theorem}
Notice that using $\tau$ and $\delta$ defined in Theorem~\ref{thm:constant}, the bound for $R_T^{SW}$ is $\tilde{O}(T^\frac{2}{3})$, even if the parameters are not optimal for this second framework.
\subsection{Unknown $\{\Lambda_m\}_{m \in \mathcal{K}}$} \label{ssec:ul}
We now focus on the situation when the auctioneer knows $\{q_i\}_{i \in \mathcal{N}}$, while $\{\Lambda_m\}_{m \in \mathcal{K}}$ are unknown. By definition of cascade model, $\{\Lambda_m\}_{m \in \mathcal{K}}$ are strictly non--increasing in $m$. This dramatically simplifies the allocation problem since the optimal allocation can be found without knowing the actual values of $\{\Lambda_m\}_{m \in \mathcal{K}}$. Indeed, allocation $\theta^*$ such that $\alpha(m; \theta^*) = \arg\max\limits_{i \in \mathcal{N}} (q_i \hat{v}_i; m)$ is optimal for all possible $\{\Lambda_m\}_{m \in \mathcal{K}}$. However, the lack of knowledge about $\{\Lambda_m\}_{m \in \mathcal{K}}$ makes the design of a truthful mechanism not straightforward because they appear in the calculation of the payments. Differently from what we presented in the previous section, here we initially focus on IC in expectation mechanisms, providing two mechanisms (the first is IC in expectation over the click realizations and the second is IC in expectation over the realizations of the random component of the mechanism), and subsequently we produce some considerations about DSIC mechanisms.
\subsubsection{IC in expectation over the click realizations mechanism}
\begin{figure}[t]
\bookbox{
\begin{algorithmic}
\STATE \textbf{Input:} Qualities parameters $\{q_i\}_{i \in \mathcal{N}}$
\STATE
\FOR{$t = 1,\ldots, T$}
\STATE Allocate ads according to $f^*$ as prescribed by~(\ref{eq:pos.dep.efficient.alloc})
\IF{Ad $a_i$ is clicked}
\STATE Ask for payment $p^c_i$ defined in (\ref{eq:olppc})
\ENDIF
\ENDFOR
\end{algorithmic}}
\caption{Pseudo--code for the A--VCG2 mechanism.}\label{f:alg2}
\end{figure}
In this case, we do not need any estimation of the parameters $\{\Lambda_m\}_{m \in \mathcal{K}}$ and therefore we do not resort to the multi--armed bandit framework and the mechanism does not present separate phases. The pseudo code of the algorithm A--VCG2 (Adaptive VCG2) is given in Fig.~\ref{f:alg2}. On the basis of the above considerations, we can adopt the allocatively efficient allocation function $f^*$ as prescribed by~(\ref{eq:pos.dep.efficient.alloc}) even if the mechanism does not know the actual values of the parameters $\{\Lambda_m\}_{m \in \mathcal{K}}$. The VCG payments in this case are:
\begin{equation} \label{eq:ol}
p_i (\hat{\mathbf{v}})= \sum_{j \in \mathcal{N}: \pi(j;\theta^*) > \pi(i;\theta^*)} \left(\Lambda_{\pi(j;\theta^*_{-i})} - \Lambda_{\pi(j;\theta^*)} \right) q_j \hat{v}_j.
\end{equation}
Obviously, these payments cannot be computed, $\{\Lambda_m\}_{m \in \mathcal{K}}$ not being known by the mechanism. However, by resorting to execution--contingent payments (generalizing the pay--per--click approach\footnote{In pay--per--click payments, an advertiser pays only once its ad is clicked; in our execution--contingent payments, an advertiser pays also once the ads of other advertisers are clicked.}), we can impose computable payments that, in expectation, are equal to the ones in~(\ref{eq:ol}). More precisely, these payments are:
\begin{align} \label{eq:olppc}
p_{i}^c&(\hat{\mathbf{v}},\{click_{\pi(j; \theta^*)}^j(t)\}) \\
&=\sum\limits_{\pi(i;\theta^*) \leq m \leq K } click_m^{\alpha(m;\theta^*)}(t) \cdot \frac{q_{\alpha(m;\theta^*_{-i})} \cdot \hat{v}_{\alpha(m;\theta^*_{-i})}}{q_{\alpha(m;\theta^*)}} \nonumber\\
&\quad\quad\quad- \sum\limits_{\pi(i;\theta^*) < m \leq K } click_m^{\alpha(m;\theta^*)}(t) \cdot \hat{v}_{\alpha(m;\theta^*)}\nonumber
\end{align}
Notice that the payment $p_{i}^c$ depends not only on the click of ad $a_i$, but also on the clicks of all the ads displayed in the slots below. In expectation, the two terms of $p_i^c$ are:
\begin{align*}
\mathbb{E}_{click}\left[\sum\limits_{\pi(i;\theta^*) \leq m \leq K } click_m^{\alpha(m;\theta^*)}(t) \cdot \frac{q_{\alpha(m;\theta^*_{-i})} \cdot \hat{v}_{\alpha(m;\theta^*_{-i})}}{q_{\alpha(m;\theta^*)}}\right] & = \sum_{\pi(j;\theta^*) \geq \pi(i;\theta^*)} \Lambda_{\pi(j;\theta^*_{-i})} q_j \hat{v}_j \\
\mathbb{E}_{click}\left[\sum\limits_{\pi(i;\theta^*) < m \leq K } click_m^{\alpha(m;\theta^*)}(t) \cdot \hat{v}_{\alpha(m;\theta^*)}\right] & = \sum_{\pi(j;\theta^*) > \pi(i;\theta^*)} \Lambda_{\pi(j;\theta^*)} q_j \hat{v}_j
\end{align*}
and therefore, in expectation, the payment equals $\text{SW}(\theta^*_{-i},\hat{\mathbf{v}}_{-i}) - \text{SW}_{-i}(\theta^*,\hat{\mathbf{v}})$. Thus, we can state the following.
\begin{proposition}
The A--VCG2 is IC, IR, WBB in expectation (over click realizations) and AE.
\end{proposition}
\begin{pf}
It trivially follows from the fact that the allocation function is AE and the payments in expectation equal the VCG payments. \qed
\end{pf}
We discuss further properties of the mechanism in what follows.
\begin{proposition}
The A--VCG2 is not DSIC \emph{a posteriori} (w.r.t. click realizations).
\end{proposition}
\begin{pf}
The proof is by counterexample. Consider an environment with 3 ads $\mathcal{N}=\{a_1, a_2, a_3\}$ and 2 slots $S=\{s_1,s_2\}$ s.t. $q_1=0.5$, $v_1=4$, $q_2=1$, $v_2=1$, $q_3=1$, $v_3=0.5$, which correspond to expected values of $2$, $1$, and $0.5$.
The optimal allocation $\theta^*$ consists in allocating $a_1$ in $s_1$ and $a_2$ in $s_2$. Consider a time $t$ when both ad $a_1$ and $a_2$ are clicked, from Eq.~\ref{eq:olppc} we have that the payment of $a_2$ is:
\[
p_{2}^c = \frac{1}{q_2}q_3v_3 = 0.5
\]
If ad $a_2$ reports a value $\hat{v}_2=3$, the optimal allocation is now $a_2$ in $s_1$ e $a_1$ in $s_2$. In the case both $a_1$ and $a_2$ are clicked, the payment of $a_2$ is:
\[
p_{2}^c = \frac{1}{q_2} q_1 v_1 + \frac{1}{q_1} q_3 v_3 - v_1 = 2 + 1 - 4 = -1
\]
Given that, in both cases, the utility is $u_2 = v_2 - p_{2}^c$, reporting a non--truthful value is optimal. Thus, we can conclude that the mechanism is not DSIC.
\end{pf}
\begin{proposition}
The A--VCG2 is IR \emph{a posteriori} (w.r.t. click realizations). \label{prop:AVGC2IRaposteriori}
\end{proposition}
\begin{pf}
Rename the ads $\{a_1, \ldots, a_N\}$ such that $q_1 v_1 \geq q_2 v_2 \geq \ldots \geq q_N v_N$. We can write payments~(\ref{eq:olppc}) as:
\[
p_{i}^c = \sum_{j=i}^K \frac{click_{j}^j(t)}{q_j} q_{j+1} v_{j+1} - \sum_{j=i+1}^K click_{j}^j(t) v_j
\]
Thus, the utility for advertiser $a_i$ is:
\begin{align*}
u_i &= click_{j}^j(t) v_i + \sum_{j=i+1}^K click_{j}^j(t) v_j - \sum_{j=i}^K \frac{click_{j}^j(t)}{q_j} q_{j+1} v_{j+1}\\
&= \sum_{j=i}^K click_{j}^j(t) v_j - \sum_{j=i}^K \frac{click_{j}^j(t)}{q_j} q_{j+1} v_{j+1}\\
&= \sum_{j=i}^K \left( click_{j}^j(t) v_j - \frac{click_{j}^j(t)}{q_j} q_{j+1} v_{j+1} \right) \\
&= \sum_{j=i}^K click_{j}^j(t) v_j - \frac{click_{j}^j(t)}{q_j} q_{j+1} v_{j+1} \\
&= \sum_{j=i}^K \frac{click_{j}^j(t)}{q_j} ( q_j v_j - q_{j+1} v_{j+1}).
\end{align*}
Since $\frac{click_{j}^j(t)}{q_j} \geq 0$ by definition and $q_j v_j - q_{j+1} v_{j+1} \geq 0$ because of the chosen ordering of the ads, then the utility is always positive and we can conclude the mechanism is IR \emph{a posteriori}. \qed
\end{pf}
\begin{proposition}
The A--VCG2 is not WBB \emph{a posteriori} (w.r.t. click realizations).
\end{proposition}
\begin{pf}
The proof is by counterexample. Consider an environment with 3 ads $\mathcal{N}=\{a_1, a_2, a_3\}$ and 2 slots $S=\{s_1,s_2\}$ s.t. $q_1=1$, $v_1=2$, $q_2=0.5$, $v_2=1$, $q_3=1$, $v_3=\epsilon$, where $\epsilon > 0$ is a small number.
The optimal allocation $\theta^*$ consists in allocating $a_1$ in $s_1$ e $a_2$ in $s_2$. Consider a time instant $t$ when both ad $a_1$ and $a_2$ are clicked, their payments are:
\[
p_{1}^c = \frac{1}{q_1}q_2v_2 + \frac{1}{q_2} q_3 v_3 - v_2 = 0.5 + 2 \epsilon - 1 = 2 \epsilon - 0.5 < 0
\]
\[
p_{2}^c = \frac{1}{q_2} q_3 v_3 = 2 \epsilon
\]
Thus, $\sum_{i=1}^3 p_{i}^c = 4 \epsilon - 0.5 < 0$, and we can conclude that the mechanism is not WBB \emph{a posteriori}.\qed
\end{pf}
Now we state the following theorem, whose proof is straightforward.
\begin{theorem}\label{thm:constant.l}
Let us consider an auction with $N$ advertisers, $K$ slots, and $T$ rounds, with position--dependent cascade model with parameters $\{\Lambda_m\}_{m=1}^K$. The A--VCG2 achieves an expected regret $R_T=0$.
\end{theorem}
An important property of this mechanism is that the expected payments are exactly the VCG payments for the optimal allocation when all the parameters are known. Moreover, the absence of an exploration phase allows us to obtain an instantaneous expected regret of zero and, thus, the cumulative regret over the $T$ rounds of auction $R_T=0$. Similar considerations can be applied to the study of the regret over the social welfare, obtaining the following.
\begin{corollary}
The A--VCG2 has an expected regret over the social welfare of zero.
\end{corollary}
\subsubsection{IC in expectation over random component realizations mechanism} \label{sssec:l.uc.m}
\begin{figure}[t]
\bookbox{
\begin{algorithmic}
\STATE \textbf{Input:} Length of exploration phase $\tau$, confidence $\delta$
\STATE
\STATE \textit{Exploitation phase}
\FOR{$t = \tau+1,\ldots, T$}
\STATE Allocate ads according to $f^{*'}$ as prescribed by Algorithm~1
\IF{Ad $a_i$ is clicked}
\STATE Ask for payment $p^B_i$ defined in (\ref{eq:pay.babaioffAAAAAXXXX})
\ENDIF
\ENDFOR
\end{algorithmic}}
\caption{Pseudo--code for the A--VCG2$^\prime$ mechanism.}\label{f:alg22}
\end{figure}
As for the previous mechanism, here we have only the exploitation phase. Differently from the previous mechanism, the mechanism has a random component as proposed in~\cite{babaioff_impl_pay}. The mechanism, called A--VCG2$^\prime$ is reported in Fig.~\ref{f:alg22}. It is obtained applying the approach described in~\cite{babaioff_impl_pay} to allocation function $f^*$.
Since $f^*$ is monotonic~(see~\ref{ap:monotonicity}) and the problem is with single parameter and linear utilities, payments assuring DSIC can be written as~\cite{tardos_sp}:
\begin{equation} \label{eq:pay.vcg.emp.tardos}
\tilde{p}_{i}(\hat{\mathbf{v}}) = \Lambda_{\pi(i;f^*(\hat{\mathbf{v}}))} q_i \hat{v}_i - \int_{0}^{\hat{v}_i} \Lambda_{\pi(i;f^*(\hat{\mathbf{v}}_{-i},u))} q_i du .
\end{equation}
However, these payments are not directly computable, because parameters $\{\Lambda_m\}_{m \in \mathcal{K}}$ in the integral are unknown (and, as in the case discussed in Section~\ref{ssec:ul}.1, we cannot replace them by empirical estimates). We could obtain these payments in expectation by using execution--contingent payments associated with non--optimal allocations where the report $\hat{v}_i$ is modified between 0 and the actual value. This can be obtained by resorting to the approach proposed in~\cite{babaioff_impl_pay}. More precisely, the approach proposed in~\cite{babaioff_impl_pay} takes in input a generic allocation function $f$ and introduces a randomized component into it, producing a new allocation function that we denote by $f'$. This technique, at the cost of reducing the efficiency of $f$, allows the computation of the allocation and the payments at the same time even when payments described in~\cite{tardos_sp} cannot be computed directly.
We apply the approach proposed in~\cite{babaioff_impl_pay} to our $f^*$ obtaining a new allocation function $f^{*'}$. With $f^{*'}$, the advertisers' reported values $\{\hat{v}_i\}_{i \in \mathcal{N}}$ are modified, each with a (small) probability $\mu$. The (potentially) modified values are then used to compute the allocation (using $f^*$) and the payments. More precisely, with a probability of $(1-\mu)^N$, $f^{*'}$ returns the same allocation $f^*$ would return, while it does not with a probability of $1 - (1-\mu)^N$. The reported values $\{\hat{v}_i\}_{i \in \mathcal{N}}$ are modified through the \emph{canonical self--resampling procedure} (cSRP) described in~\cite{babaioff_impl_pay} that generates two samples: $x_i(\hat{v}_i,\omega_i)$ and $y_i(\hat{v}_i,\omega_i)$, where $\omega_i$ is the random seed. We sketch the result of cSRP where the function `rec' is defined in~\cite{babaioff_impl_pay}:
\begin{align*}
(x_i,y_i) = cSRP(\hat{v}_i)=\begin{cases} (\hat{v}_i,\hat{v}_i) & \mbox{w.p. } 1-\mu \\ (\hat{v}''_i,\hat{v}'_i) & \mbox{otherwise }\end{cases},
\end{align*}
where $\hat{v}_i'\sim\mathcal{U}([0,\hat{v}_i])$ and $\hat{v}_i''=\text{rec}(\hat{v}_i')$.
\begin{algorithm}
\begin{algorithmic}[1]
\begin{scriptsize}
\FORALL {$a_i \in N$}
\STATE $(x_i, y_i)=cSRP(\hat{v}_i)$ \label{s:csrp}
\STATE $\mathbf{x}=(x_1,\ldots,x_N)$
\ENDFOR
\STATE $\theta = f^*(\mathbf{x})$ \label{s:alloc}
\end{scriptsize}
\end{algorithmic}
\caption{$f^{*'}(\hat{\mathbf{v}})$}
\label{alg:babalg}
\end{algorithm}
Algorithm~\ref{alg:babalg} shows how $f^{*'}$ works when the original allocation function is $f^*$. The reported values $\{\hat{v}_i\}_{i \in \mathcal{N}}$ are perturbed through the canonical self--resampling procedure~(Step~\ref{s:csrp}) and then it returns the allocation found by applying the original allocation function $f^*$ to the new values $\mathbf{x}$ (Step~\ref{s:alloc}).
Finally, the payments are computed as
\begin{multline} \label{eq:pay.babaioffAAAAAXXXX}
p^B_i(\mathbf{x}, t) = \begin{cases} \frac{\bar{p}_i^B(\mathbf{x},\mathbf{y};\hat{\mathbf{v}})}{\Lambda_{\pi(i; f^*(\mathbf{x}))} q_i} & \mbox{if } click_{\pi(i; f^*(\mathbf{x}))}^{i}(t)=1 \\ 0 & \mbox{otherwise} \end{cases} =\\
= \begin{cases} \hat{v}_i -
\begin{cases} \frac{1}{\mu} \hat{v}_i &\mbox{if $y_i<\hat{v}_i$} \\
0 & \mbox{otherwise}, \end{cases}
& \mbox{if } click_{\pi(i; f^*(\mathbf{x}))}^{i}(t)=1 \\ 0 & \mbox{otherwise} \end{cases}
\end{multline}
where $\mathbf{y}=(y_1,\ldots,y_N)$ and the expected value of payments~(\ref{eq:pay.babaioffAAAAAXXXX}) w.r.t. the randomization of the mechanism are the payments~\cite{tardos_sp} for the randomized allocation function $f^{*'}$. The result presented in~\cite{babaioff_impl_pay} assures that the resulting mechanism is IC in expectation over the realizations of the random component and \emph{a posteriori} w.r.t. the click realizations.
We state the following results on the properties of the above mechanism.
\begin{theorem}\label{thm:constant.l.baba}
Let us consider an auction with $N$ advertisers, $K$ slots, and $T$ rounds, with position--dependent cascade model with parameters $\{\Lambda_m\}_{m=1}^K$. The A--VCG2 $^\prime$ achieves an expected regret $R_T \leq 2 K^2 \mu v_{\max} T$.
\end{theorem}
Adopting $\mu = \frac{1}{T^\alpha}$ with $\alpha>1$ then $R_T \rightarrow 0$, but, as we will show in Section~\ref{s:experiments}, the smaller $\mu$ the larger the variance of the payments. We provide a similar result for the regret over the social welfare.
\begin{theorem}\label{thm:constant.l.sw.baba}
Let us consider an auction with $N$ advertisers, $K$ slots, and $T$ rounds, with position--dependent cascade model with parameters $\{\Lambda_m\}_{m=1}^K$. The A--VCG2 $^\prime$ achieves an expected regret $R^{SW}_T \leq 2 K \mu v_{\max} T$.
\end{theorem}
\subsubsection{Considerations about DSIC mechanisms}
At the cost of worsening the regret, one may wonder whether there exists some no--regret DSIC mechanism. In what follows, resorting to the same arguments used in~\cite{sarma2010multi-armed}, we show that the answer to such question is negative.
\begin{theorem}\label{thm:constant.l}
Let us consider an auction with $N$ advertisers, $K$ slots, and $T$ rounds, with position--dependent cascade model with parameters $\{\Lambda_m\}_{m=1}^K$ whose value are unknown. Any online learning DSIC \textit{a posteriori} (w.r.t. click realizations) mechanism achieves an expected regret $R_T=\Theta(T)$.
\end{theorem}
\begin{pf}\textbf{(sketch)} Basically, the A--VCG2 mechanism is only IC in expectation (and not DSIC) because it adopts execution--contingent payments in which the payment of advertiser $a_i$ depends also on the clicks over ads different from $a_i$. The above payment technique---i.e., payments reported in~(\ref{eq:olppc})---is necessary to obtain in expectation the values $\text{SW}(\theta^*_{-i},\hat{\mathbf{v}}_{-i})$ and $\text{SW}_{-i}(\theta^*,\hat{\mathbf{v}})$, since parameters $\{\Lambda_m\}_{m \in \mathcal{K}}$ are not known. In order to have DSIC \textit{a posteriori} (i.e., truthful for any realization of the clicks), we need payments $p_i$ that are deterministic w.r.t. the clicks over other ads different from $a_i$ (i.e., pay--per--click payments are needed).
We notice that even if $\Lambda_m$ have been estimated (e.g., in an exploitation phase), we cannot have payments leading to DSIC. Indeed, with estimates $\tilde{\Lambda}_m$, the allocation function maximizing $\widetilde{SW}$ (computed with $\tilde{\Lambda}_m$) is not an affine maximizer and therefore the adoption of WVCG mechanism would not guarantee DSIC. As a result, only mechanisms with payments defined as in~\cite{tardos_sp} can be used. However, these payments, if computed exactly (and not estimated in expectation), require the knowledge about the actual $\Lambda_m$ related to each slot $s_m$ in which an ad can be allocated for each report $\hat{v}\leq v$.
To prove the theorem, we provide a characterization of DSIC mechanisms. Exactly, we need a monotonic allocation function and the payments defined in~\cite{tardos_sp}. These payments, as said above, require the knowledge about the actual $\Lambda_m$ related to the slot $s_m$ in which an ad can be allocated for each report $\hat{v}\leq v$. Thus we have two possibilities:
\begin{itemize}
\item In the first case, an ad can be allocated only in one slot and its report determines only whether it is displayed or not. That is, the ads are partitioned and each partition is associated with a slot and the ad with the largest expected valuation is chosen at each slot independently. This case is equivalent to multiple separate--single slot auctions and therefore each auction is DSIC as shown in~\cite{devanur2009price}. However, as shown in~\cite{sarma2010multi-armed}, this mechanism would have a regret $\Theta(T)$.
\item In the second case, an ad can be allocated in more than one slot on the basis of its report. In this case, to compute the payments, it would be necessary to know the exact CTRs of the ad for each possible slot, but this is possible only in expectation either by using the above execution--contingent as we do in Section~4.2.1 or by generating non--optimal allocation as we do in Section~4.2.2.
\end{itemize}
Thus, in order to have DSIC, we need to adopt the class of mechanisms described in the first case, obtaining $R_T=\Theta(T)$.\qed
\end{pf}
\subsection{Unknown $\{\Lambda_{m}\}_{m \in \mathcal{K}}$ and $\{q_i\}_{i \in \mathcal{N}}$} \label{ssec:uql}
In this section we study the situation in which both $\{q_i\}_{i \in \mathcal{N}}$ and $\{\Lambda_{m}\}_{m \in \mathcal{K}}$ are unknown. From the results discussed in the previous section, we know that adopting DSIC as solution concept we would obtain $R_T=\Theta(T)$. Thus, we focus only on IC in expectation.
First of all, we remark that the mechanisms presented in Sections~\ref{ssec:uq} and~\ref{ssec:ul} cannot be adopted here, but the study of a new mechanism is required. The mechanism we design is given by the combination of A--VCG1 and A--VCG2$^\prime$. The pseudo code of the algorithm A--VCG3 (Adaptive VCG3) is given in Fig.~\ref{f:alg3}. As in the case in which only $\{q_i\}_{i \in \mathcal{N}}$ are unknown, we formalize the problem as a multi--armed bandit where the exploration and exploitation phases are separate and where, during the exploration phase, we estimate the values of $\{q_i\}_{i \in \mathcal{N}}$. Details of the algorithm follow.
\begin{figure}[t]
\bookbox{
\begin{algorithmic}
\STATE \textbf{Input:} Length of exploration phase $\tau$, confidence $\delta$
\STATE
\STATE \textit{Exploration phase}
\FOR{$t = 1,\ldots,\tau$}
\STATE Allocate ads according to (\ref{eq:explorativeallocations})
\STATE Ask for no payment
\STATE Observe the clicks $\{click_{1}^i(t)\}_{i=1}^{N}$
\ENDFOR
\STATE Compute the estimated quality $\tilde{q}_i = \frac{1}{|B_i|}\sum_{t \in B_i} click_{1}^i(t)$
\STATE Compute $\tilde{q}^+_i = \tilde{q}_i + \eta$ where $\eta$ is given by (\ref{eq:hoeff})
\STATE
\STATE \textit{Exploitation phase}
\FOR{$t = \tau+1,\ldots, T$}
\STATE Allocate ads according to $\tilde{f}'$ as prescribed by Algorithm~1
\IF{Ad $a_i$ is clicked}
\STATE Ask for payment $\tilde{p}^B_i$ defined in (\ref{eq:pay.babaioff.ppc})
\ENDIF
\ENDFOR
\end{algorithmic}}
\caption{Pseudo--code for the A--VCG3 mechanism.}\label{f:alg3}
\end{figure}
\paragraph*{\indent Exploration phase} During the first $\tau$ rounds of the auction, estimates of $\{q_i\}_{i \in \mathcal{N}}$ are computed. We use the same exploration policy of Section~\ref{ssec:uq}, but the estimations are computed just using samples from the first slot, since $\Lambda_m$ with $m>1$ are unknown.\footnote{In the following, we report some considerations about the case in which also the samples from the slots below the first are considered.} Define $B_i = \{t: \pi(i; \theta_t) = 1, t\leq\tau\}$ the set of rounds $t\leq \tau$ where $a_i$ is displayed in the first slot, the number of samples collected for $a_i$ is $|B_i| = \lfloor \frac{\tau}{N} \rfloor \geq \frac{\tau}{2N}$. The estimated value of $q_i$ is computed as:
\begin{align*}
\tilde{q}_i = \frac{1}{|B_i|}\sum_{t \in B_i} click_1^i(t).
\end{align*}
such that $\tilde{q}_i$ is an unbiased estimate of $q_i$ (i.e., $\mathbb{E}_{click} [\tilde{q}_i] = q_i$, where $\mathbb{E}_{click}$ is in expectation w.r.t. the realization of the clicks). By applying the Hoeffding's inequality we obtain an upper bound over the error of the estimated quality $\tilde{q}_i$ for each ad $a_i$.\footnote{\TODO{Alex: c'e' una ragione specifica per chiamare questo $\eta_q$ mentre nella sezione precedente usavamo $\eta$? M: no, vada per $\eta$, DA SISTEMARE ANCHE NELLE PROOF}}
\begin{proposition}\label{p:hoeffding.ql}
For any ad $\{a_i\}_{i \in \mathcal{N}}$
\begin{align}\label{eq:hoeff}
| q_i - \tilde{q}_i | \leq \sqrt{\frac{1}{2 |B_i|} \log \frac{2N}{\delta}} \leq \sqrt{\frac{N}{\tau} \log \frac{2N}{\delta}} =: \eta,
\end{align}
with probability $1-\delta$ (w.r.t. the click events).
\end{proposition}
After the exploration phase, an upper--confidence bound over each quality is computed as $\tilde{q}^+_i = \tilde{q}_i + \eta$.
\paragraph*{\indent Exploitation phase} We first focus on the allocation function. During the exploitation phase we want to use an allocation $\tilde{\theta}=\tilde{f}(\hat{\mathbf{v}})$ maximizing the estimated social welfare with estimated $\{\tilde{q}^+_i\}_{i \in \mathcal{N}}$ and the parameters $\{\Lambda_m\}_{m \in \mathcal{K}}$. Since the actual parameters $\{\Lambda_m\}_{m \in \mathcal{K}}$ are monotonically non--increasing we can use an allocation $\{\langle s_m, a_{\alpha(m; \tilde{\theta})} \rangle\}_{m \in \mathcal{K}'}$, where
\begin{align*}
\alpha(m; \tilde{\theta}) = \arg\max_{i \in \mathcal{N}} (\tilde{q}^+_i \hat{v}_i; m) = \arg\max_{i \in \mathcal{N}} (\tilde{q}^+_i \Lambda_m \hat{v}_i; m).
\end{align*}
We now focus on payments. Allocation function $\tilde{f}$ is an affine maximizer (due to weights depending on $\tilde{q}_i$ as in Section~\ref{ssec:uq}), but WVCG payments cannot be computed given that parameters $\{\Lambda_m\}_{m \in \mathcal{K}}$ are unknown. Neither the adoption of execution--contingent payments, like in~(\ref{eq:olppc}), is allowed, given that $q_i$ is unknown and only estimates $\tilde{q}_i$ are available.
Thus, we resort to implicit payments as in Section~4.2.2. More precisely, we use the same exploitation phase we used in Section~4.2.2 except that we adopt $\tilde{f}$ in place of $f^*$. In this case, we have that the per--click payments are:
\begin{multline} \label{eq:pay.babaioff.ppc}
p^B_i(\mathbf{x}, t) = \begin{cases} \frac{\bar{p}_i^B(\mathbf{x},\mathbf{y};\hat{\mathbf{v}})}{\Lambda_{\pi(i; \tilde{f}(\mathbf{x}))} q_i} & \mbox{if } click_{\pi(i; \tilde{f}(\mathbf{x}))}^{i}(t)=1 \\ 0 & \mbox{otherwise} \end{cases}
= \\ \begin{cases} \hat{v}_i -
\begin{cases} \frac{1}{\mu} \hat{v}_i &\mbox{if $y_i<\hat{v}_i$} \\
0 & \mbox{otherwise}, \end{cases}
& \mbox{if } click_{\pi(i; \tilde{f}(\mathbf{x}))}^{i}(t)=1 \\ 0 & \mbox{otherwise} \end{cases}
\end{multline}
We can state the following.
\begin{theorem}
The A--VCG3 is IC and WBB in expectation (over the realizations of the random component of the mechanism) and IR \emph{a posteriori} (w.r.t. the random component of the mechanism). These properties hold \emph{a posteriori} w.r.t. the click realizations.
\end{theorem}
\begin{pf}
The proof of IC in expectation and WBB in expectation easily follows from the definition of the adopted mechanism as discussed in~\cite{babaioff_impl_pay}. The proof of IR \emph{a posteriori} is similar to the proof of Proposition~\ref{prop:AVGC2IRaposteriori}. The fact that the properties hold \emph{a posteriori} w.r.t. the click realizations follows from~\cite{babaioff_impl_pay}.\qed
\end{pf}
Now we want to analyze the performance of the mechanism in terms of regret cumulated through $T$ rounds. Notice that in this case we have to focus on two different potential sources of regret: the adoption of a sub--optimal (randomized) allocation function and the estimation of the unknown parameters.
\begin{theorem}\label{thm:constant.ql}
Let us consider an auction with $N$ advertisers, $K$ slots, and $T$ rounds, with position--dependent cascade model with parameters $\{\Lambda_m\}_{m=1}^K$. For any parameter $\tau$ and $\delta$, the A--VCG3 achieves a regret
\begin{align*}
R_T & \leq v_{\max} \left[\left( T-\tau \right) \left(2K \eta_q + 2 \mu K N \right) + \tau K + \delta K T \right]\\
& = v_{\max} K \left[\left( T-\tau \right) \left(2 \eta_q + 2 \mu N \right) + \tau + \delta T \right]
\end{align*}
\noindent By setting the parameters to
\begin{itemize}
\item $\mu = N^{-\frac{2}{3}} T^{-\frac{1}{3}}$. $\mu$ is always $\leq 1$
\item $\delta = N^\frac{1}{3} T^{-\frac{1}{3}}$. $\delta \leq 1$, thus $T \geq N$
\item $\tau = 2^\frac{2}{3} T^\frac{2}{3} N^\frac{1}{3} \left( \log{\frac{2N}{\delta}} \right)^\frac{1}{3}$
\end{itemize}
then the regret is
\begin{align}
R_T &\leq 2^\frac{2}{3} b_{\max} K T^\frac{2}{3} N^\frac{1}{3} \left( \log 2 N^\frac{2}{3} T^\frac{1}{3} \right)^\frac{1}{3} + 2 N^\frac{1}{3} b_{\max} K T^\frac{2}{3} + N^\frac{1}{3} T^{-\frac{1}{3}} K b_{\max} T\\
&= 2^\frac{2}{3} b_{\max} K T^\frac{2}{3} N^\frac{1}{3} \left( \log 2 N^\frac{2}{3} T^\frac{1}{3} \right)^\frac{1}{3} + 2 N^\frac{1}{3} b_{\max} K T^\frac{2}{3} + N^\frac{1}{3} K b_{\max} T^\frac{2}{3}\\
&\leq 4.6 N^\frac{1}{3} K b_{\max} T^\frac{2}{3} \max \left\{1, \left( \log 2 N^\frac{2}{3} T^\frac{1}{3} \right)^\frac{1}{3} \right\} \label{eq:regret.posdep.qlu}
\end{align}
\end{theorem}
\myremark{1 (The bound).} Up to numerical constants and logarithmic factors, the previous bound is $R_T \leq \tilde O(T^\frac{2}{3} K N^\frac{1}{3})$.
We first notice we match the lowest possible complexity for the parameter $T$ when exploration and exploitation phases are separate. Moreover observe that the proposed mechanism is a no--regret algorithm, thus asymptotically it achieves the same performances of VGC (when all the parameter are known), since its per--round regret ($R_T/T$) decreases to 0 as $T^{-\frac{1}{3}}$.
We can observe that, with respect to the case of Section~\ref{ssec:uq}, the dependence of the cumulative regret in the parameter $K$ is augmented by a factor $K^\frac{1}{3}$. The reason resides in the exploration phase, indeed, in this last case, we cannot take advantage of all data we can collect, given that we estimate the qualities only on the basis of their visualization in the first slot. Instead, the dependency on $N$ is the same of the one in the case studied in Section~\ref{ssec:uq}.
\myremark{2 (Non--separate phases and $O(T^{1/2})$).} The questions whether or not it is possible to avoid the separation of the exploration and exploitation phases preserving IC in expectation (in some form) and whether or not it is possible to obtain a regret of $O(T^{1/2})$ are open. We conjecture that, if it is possible to have $R_T=O(T^{1/2})$ when only $\{q_i\}_{i \in \mathcal{N}}$ are unknown, then it is possible to have $R_T=O(T^{1/2})$ also when $\{q_i\}_{i \in \mathcal{N}}$ and $\{\Lambda_m\}_{m \in \mathcal{K}}$ are unknown. However, such a problem is still open.
\myremark{3 (Using samples from multiple slots).} The question whether it is possible to exploit the samples from the slots below the first one to improve the accuracy of the estimates and to reduce the length of the exploration phase is open. The critical issue here is that the samples from those slots are about the product of two random variables, i.e., $\Lambda_s$ and $q_i$, and it is not trivial to find a method to use these samples to improve the esteems. However, in the case it is possible to exploit these samples, we would obtain a reduction of the regret bound of at most $K^{1/3}$, given that the dependency from $K$ cannot be better than in the case discussed in Section~\ref{ssec:uq} (i.e., $O(K^{\frac{2}{3}})$).
A--VCG3 allows also the identification of an upper--bound over the regret on the social welfare. The derivation is not straightforward with respect to the bound over the regret on the payments, but, using the value of the parameters identified in Theorem~\ref{thm:constant.ql}, the bound is $\tilde{O}(T^\frac{2}{3})$. Optimising the parameters w.r.t. to the regret over the social welfare, we obtain the following.
\begin{theorem} \label{th:pd_lq_sw}
Let us consider an auction with $N$ advertisers, $K$ slots, and $T$ rounds, with position--dependent cascade model with parameters $\{\Lambda_m\}_{m=1}^K$. For any parameter $\tau$ and $\delta$, the A--VCG3 achieves a regret
\begin{align*}
R_T^{SW} & \leq v_{\max} K \left[ (T - \tau) ( 2 \eta + N \mu) + \tau + \delta T \right]\\
& \leq v_{\max} K \left[ (T - \tau) \left( 2 \sqrt{\frac{N}{\tau} \log{\frac{2N}{\delta}}} + N \mu \right) + \tau + \delta T \right]
\end{align*}
\noindent By setting the parameters to
\begin{itemize}
\item $\mu = N^{-\frac{2}{3}} T^{-\frac{1}{3}}$. $\mu$ is always $\leq 1$
\item $\delta = \text{depende da che punto guardi il mondo tutto dipende}$ \TODO{}
\item $\tau = 2^\frac{2}{3} T^\frac{2}{3} N^\frac{1}{3} \left( \log{\frac{2N}{\delta}} \right)^\frac{1}{3}$
\end{itemize}
then the regret is
\begin{align*}
R_T^{SW} &\leq 2^\frac{8}{3} \cdot v_{\max} K N^\frac{1}{3} T^\frac{2}{3} \left( \log{N^\frac{2}{3} T^\frac{1}{3}} \right)^\frac{1}{3}.
\end{align*}%
\end{theorem}
\section{Problem statement}\label{s:statement}
In this section we introduce all the notation used throughout the rest of the paper. In particular, we formalize the sponsored search auction model, we define the mechanism design problem, and we introduce the learning process.
\subsection{Sponsored search auction model}
We resort to the standard model of sponsored search auctions~\cite{Narahari2009}. We denote by $\mathcal{N}=\{1,\ldots,N\}$ the set of ads indexes and by $a_i$ with $i \in \mathcal{N}$ the $i$--th ad (we assume w.l.o.g. each advertiser has only one ad and therefore we can identify by $a_i$ the $i$--th ad and the $i$--th advertiser indifferently). Each ad~$a_i$ is characterized by a \emph{quality} $q_i$ corresponding to the probability that $a_i$ is clicked once observed by the user, and by a \emph{value} $v_i\in\mathcal V$, with $\mathcal{V}=[0,V]$ and $V\in \mathbb{R}^+$, which $a_i$ receives when clicked ($a_i$ receives a value of zero if not clicked). We denote by $\mathbf{v}$ the profile $(v_1,\ldots,v_N)$ and, as customary in game theory, by $\mathbf{v}_{-i}$ the profile obtained by removing $v_i$ from $\mathbf{v}$. While qualities $\{q_i\}_{i \in \mathcal{N}}$ are commonly known by the auctioneer, values $\{v_i\}_{i \in \mathcal{N}}$ are private information of the advertisers. We denote by $\mathcal{K}=\{1,\ldots,K\}$ with $K < N$,\footnote{Although $K<N$ is the most common case, the results could be smoothly extended to $K>N$.} the set of slot indexes and by $s_m$ with $m\in \mathcal{K}$ the $m$--th slot from top to bottom. For notational convenience, we also define the extended set of slots indexes $\mathcal{K}'=\mathcal{K}\cup\{K+1,\ldots,N\}$.
We denote by the ordered pair $\langle s_m, a_i\rangle$ that ad $a_i$ is allocated into slot $s_m$, by $\theta$ a generic \emph{allocation} and by $\Theta$ the set of all the possible allocations. Although in an auction only $K$ ads can be actually displayed, we define an allocation as $\theta=\{\langle m,i\rangle: m\in \mathcal{K}',i \in \mathcal{N}\}$ where both $m$ and $i$ occur exactly once and any ad assigned to a slot $m>K$ is not displayed. We define two maps $\pi:\mathcal{N}\times\Theta \rightarrow \mathcal{K}'$ and $\alpha:\mathcal{K}'\times\Theta \rightarrow \mathcal{N}$ such that $\pi(i;\theta)$ returns the slot in which $a_i$ is displayed in allocation $\theta$ and $\alpha(m;\theta)$ returns the ad allocated in slot~$s_m$ in allocation $\theta$. Given $\theta \in \Theta$, we have that $\pi(i;\theta)=m$ if and only if $\alpha(m;\theta)=i$.
With more than one slot, it is necessary to adopt a model of the user describing how the expected value of an advertiser varies over the slots. We assume that the user behaves according to the popular \emph{cascade model} defined by~\cite{Kempe2008,Aggarwal2008}. In particular, the user's behavior can be modeled as a Markov chain whose states correspond to the slots, which are observed sequentially from the top to the bottom, and the transition probability corresponds to the probability of observing the ad $a_i$ displayed in the next slot; with the remaining probability the user stops observing the ads. This probability may depend on the index of the slot (i.e., $\pi(i;\theta)$), in this case the externalities are said \emph{position--dependent}, and/or on the ad that precedes $a_i$ in the current allocation $\theta$ (i.e., $\alpha(\pi(i;\theta)-1;\theta)$), in this case the externalities are said \emph{ad--dependent}.
In the general case, the cascade model can be described by introducing parameters $\gamma_{m,i}$ defined as the probability that a user observing ad~$a_i$ in slot~$s_{m}$ observes the ad in the next slot $s_{m+1}$. It can be easily seen that there are $KN$ different parameters $\gamma_{m,i}$. The (cumulative) probability that a user observes the ad displayed at slot $s_m$ in allocation $\theta$ is denoted by $\Gamma_m(\theta)$ and it is defined as:
\begin{align} \label{eq:coeff2}
\Gamma_m(\theta) = \left\{
\begin{array}{ll}
1 & \text{if } m=1 \\
\prod\limits_{l=1}^{m-1} \gamma_{l,\alpha(l;\theta)} & \text{if } 2 \leq m\leq K\\
0 & \text{otherwise}
\end{array} \right.
\end{align}
Given an allocation $\theta$, the \emph{click through rate} (CTR) of ad $a_i$ is the probability to be clicked once allocated according to $\theta$ and it is equal to $\Gamma_{\pi(i;\theta)}(\theta) q_{i}$. Similarly, the CTR of the ad displayed at slot $m$ can be computed as $\Gamma_m(\theta) q_{\alpha(m;\theta)}$. We notice that, according to this model, the user might click multiple ads at each impression. Given an allocation $\theta$, the \emph{expected value} (w.r.t. the user's clicks) of advertiser $a_i$ from $\theta$ is $\Gamma_{\pi(i;\theta)}(\theta) q_{i} v_i$, that is, the product of the CTR $\Gamma_{\pi(i;\theta)}(\theta) q_{i}$ by the value of the advertiser $v_i$. The advertisers' cumulative expected value from allocation $\theta$, commonly referred to as \emph{social welfare}, is:
\begin{align*}
\text{SW}(\theta,\mathbf{v})= \sum_{i=1}^N \Gamma_{\pi(i;\theta)}(\theta) q_{i} v_i
\end{align*}
In~\cite{Kempe2008,Aggarwal2008}, the authors factorize the probability $\gamma_{m,i}$ as the product of two independent terms: the \emph{prominence} $\lambda_m$, which only depends on the slot $s_m$, and the \emph{continuation probability} $c_i$, which only depends on the ad $a_i$. This leads to a reduction of the number of the parameters from $KN$ to $K+N$.\footnote{The allocation problem when either all the prominence probabilities $\lambda_m$s or all the continuation probabilities $c_i$s are equal to one can be solved in polynomial time, while, although no formal proof is known, the allocation problem with $\lambda_m$s and $c_i$s different from one is commonly believed to be $\mathcal{NP}$--hard~\cite{Kempe2008}. However, the allocation problem can be solved exactly for concrete settings and for very large settings approximation algorithms can be adopted as shown in~\cite{aamas2013}. In this paper, we just focus on optimal allocation functions.}
Finally, we denote by $click^i_{m}\in \{0,1\}$ the click/no--click event for ad $a_i$ allocated in slot $m$.
\subsection{Mechanism design problem} \label{ssec:md}
A direct--revelation economic mechanism for sponsored search auctions is formally defined as a tuple $(\mathcal{N},\mathcal V,\Theta,f,\{p_i\}_{i\in \mathcal{N}})$ where $\mathcal{N}$ is the set of agents (i.e., the advertisers), $\mathcal V$ is the set of possible actions available to the agents (i.e., the possible reported values), $\Theta$ is the set of the outcomes (i.e., the allocations), $f$ is the allocation function $f:\mathcal V^{N}\rightarrow \Theta$, and $p_i$ is the payment function of advertiser $a_i$ defied as $p_i:\mathcal V^{N}\rightarrow \mathbb R$. We denote by $\hat{v}_i$ the value reported by advertiser $a_i$ to the mechanism, by $\hat{\mathbf{v}}$ the profile of reported values and $\hat{\mathbf{v}}_{-i}$ the profile obtained by removing $\hat{v}_i$ from $\hat{\mathbf{v}}$.
At the beginning of an auction, each advertiser $a_i$ reports its value $\hat{v}_i$. The mechanism chooses the allocation on the basis of the advertisers' reports as $f(\hat{\mathbf{v}})$ and subsequently computes the payment of each advertiser $a_i$ as $p_i(\hat{\mathbf{v}})$. The expected utility of advertiser $a_i$ is defined as $\Gamma_{\pi(i;f(\hat{\mathbf{v}}))}f(\hat{\mathbf{v}}) q_{i} v_i - p_i(\hat{\mathbf{v}})$. Since each advertiser is an expected utility maximizer, it will misreport its value (i.e., $\hat v_i \neq v_i$) whenever this may lead its utility to increase. Mechanism design aims at finding an allocation function $f$ and a vector of payments $\{p_i\}_{i \in \mathcal{N}}$ such that some desirable properties---discussed in Section~2.1---are satisfied~\cite{mas-colell1995microeconomic}.
When all the parameters $q_i$ and $\gamma_{m,i}$ are known, the VCG mechanism satisfies IC in expectation (over click realizations), IR in expectation (over click realizations), WBB \emph{a posteriori} (w.r.t. click realizations), and AE. In the VCG mechanism, the allocation function, denoted by $f^*$, maximizes the social welfare given the reported types as:
\begin{align}
\label{eq:efficient-alloc}
\theta^*=f^*(\hat{\mathbf{v}}) = \arg\max_{\theta \in \Theta}~ \{\text{SW}(\theta,\hat{\mathbf{v}})\}
\end{align}
\noindent and payments are defined as
\begin{align}\label{eq:payment}
p^*_i(\hat{\mathbf{v}}) = \text{SW}(\theta^*_{-i},\hat{\mathbf{v}}_{-i}) - \text{SW}_{-i}(\theta^*,\hat{\mathbf{v}}),
\end{align}
where:
\begin{itemize}
\item $\theta^*_{-i}=f^*(\hat{\mathbf{v}}_{-i})$, i.e., the optimal allocation when advertiser $a_i$ is not present,
\item $\text{SW}_{-i}(\theta^*,\hat{\mathbf{v}})=\sum\limits_{j=1,j\neq i}^N \Gamma_{\pi(j;\theta^*)}(\theta^*) q_{j} \hat{v}_j$, i.e., the cumulative expected value of the optimal allocation $\theta^*$ minus the expected value of advertiser $a_i$.
\end{itemize}
In words, the payment of advertiser $a_i$ is the difference between the social welfare that could be obtained from allocation $\theta_{-i}^*$ computed removing ad $a_i$ from the auction and the social welfare of the efficient allocation $\theta^*$ without the contribution of advertiser $a_i$. The extension of the VCG mechanism do weighted ads (the WVCG mechanism) is straightforward. The weighted social welfare is $\text{SW}^w(\theta,\mathbf{v})= \sum_{i=1}^N \Gamma_{\pi(i;\theta)}(\theta) q_{i} v_i w_i$ where $w_i$ is the weight of advertiser $i$. In the WVCG, the allocation maximizing the weighted social welfare is chosen, while the payment is defined as $ p^w_i(\hat{\mathbf{v}}) = \frac{1}{w_i}(\text{SW}^w(\theta^*_{-i},\hat{\mathbf{v}}_{-i}) - \text{SW}^w_{-i}(\theta^*,\hat{\mathbf{v}}))$.
The previous mechanism is IC and IR in expectation, but it is not DSIC and IR \emph{a posteriori} w.r.t. the clicks (an advertiser may have a positive payment even when its ad has not been clicked). Nonetheless, the mechanism can be easily modified to satisfy DSIC and IR \emph{a posteriori} w.r.t. the clicks by using \emph{pay--per--click} payments $p^{c}_i$ as follows:
\begin{align}\label{eq:payment-contingent}
p^{c}_i(\hat{\mathbf{v}},click^i_{\pi(i; \theta^*)}) =
\begin{cases}
0 & \textnormal{if } click^i_{\pi(i; \theta^*)}=0 \\
\dfrac{\text{SW}(\theta^*_{-i},\hat{\mathbf{v}}_{-i}) - \text{SW}_{-i}(\theta^*,\hat{\mathbf{v}})}{\Gamma_{\pi(i;\theta^*)}(\theta^*) q_i} & \textnormal{if } click^i_{\pi(i; \theta^*)}=1
\end{cases}
\end{align}
so that $\mathbb E[ p^c_i(\hat{\mathbf{v}},click^i_{\pi(i; \theta^*)})] = p^*_i(\hat{\mathbf{v}})$, where the expectation is taken w.r.t. the click event, which is distributed as a Bernoulli random variable with parameter coinciding with the CTR of ad $a_i$ in allocation $\theta^*$, i.e., $\Gamma_{\pi(i; \theta^*)}q_i$. Similar definitions hold for the WVCG.
\subsection{Online learning mechanism design problem}\label{ss:online.mechanism}
In many practical problems, the parameters (i.e., $q_i$ and $\gamma_{m,i}$) are not known in advance by the auctioneer and must be estimated at the same time as the auction is deployed. This introduces a tradeoff between \textit{exploring} different possible allocations so as to collect information about the parameters and \textit{exploiting} the estimated parameters so as to implement a truthful high--revenue auction (i.e., a VCG mechanism). This problem could be easily casted as a multi--arm bandit problem \cite{robbins1952some} and standard techniques could be used to solve it, e.g., \cite{auer2002finite-time}. Nonetheless, such an approach would completely overlook the strategic dimension of the problem: advertisers may choose their reported values at each round $t$ to influence the outcome of the auction at $t$ and/or in future rounds after $t$ in order to increase the cumulative utility over all the rounds of the horizon $T$. Thus, in this context, truthfulness requires that reporting the truthful valuation maximizes the cumulative utility over all the horizon $T$. The truthfulness can be: in dominant strategies if advertisers know everything (including, e.g., the ads that will be clicked at each round $t$ if displayed) or in expectation. As customary, we adopt three forms of truthfulness in expectation: IC in expectation over the click realizations and \emph{a posteriori} w.r.t. the realizations of the random component of the mechanism (if such a component is present), IC in expectation over the realizations of the random component of the mechanism and \emph{a posteriori} w.r.t. the click realizations, and, finally, IC in expectation over both randomizations. We consider IC in expectation over the click realizations weaker than IC in expectation over the realizations of the random mechanism since each advertiser could control the clicks by using software bots.
Thus, here we face the more challenging problem where the exploration--exploitation dilemma must be solved so as to maximize the revenue of the auction under the hard constraint of incentive compatibility. Let $\mathfrak{A}$ be an IC mechanism run over $T$ rounds. We assume, as it is common in practice, that the advertisers' reports can change during these $T$ rounds. At each round~$t$, $\mathfrak{A}$ defines an allocation $\theta_t$ and prescribes an expected payment $p_{i,t}(\hat{\mathbf{v}})$ for each ad $a_i$. The objective of $\mathfrak{A}$ is to obtain a revenue as close as possible to a VCG mechanism computed on the basis of the actual parameters.\footnote{We refer the reader to~\ref{app:deviation.regret} for a slightly different definition of regret measuring the deviation from the revenue of a VGC mechanism.} More precisely, we measure the performance of $\mathfrak{A}$ as its cumulative regret over $T$ rounds:
\begin{align*}
\mathcal R_T(\mathfrak{A}) = T \sum_{i=1}^n p_i^*(\hat{\mathbf{v}}) - \sum_{t=1}^T \sum_{i=1}^n p_{i,t}(\hat{\mathbf{v}}).
\end{align*}
We remark that the regret is not defined on the basis of the pay--per--click payments asked on a specific sequence of clicks but on the expected payments $p_{i,t}(\hat{\mathbf{v}})$. Furthermore, since the learning mechanism $\mathfrak{A}$ estimates the parameters from the observed (random) clicks, the expected payments $p_{i,t}(\hat{\mathbf{v}})$ are random as well. Thus, in the following we will study the expected regret:
\begin{align}\label{eq:regret}
R_T(\mathfrak{A}) = \mathbb E[\mathcal R_T(\mathfrak{A})],
\end{align}
where the expectation is taken w.r.t. random sequences of clicks and possibly the randomness of the mechanism.
The mechanism $\mathfrak{A}$ is a \textit{no--regret} mechanism if its per--round regret $R_T(\mathfrak{A})/T$ decreases to 0 as $T$ increases, i.e., $\lim\limits_{T\rightarrow \infty} R_T(\mathfrak{A}) / T = 0$. Another popular definition of performance \cite{gonen2007incentive-compatible,babaioff2008characterizing} is the social welfare regret, denoted by $R_T^{SW}$ and measured as the difference between the (expected) social welfare of the optimal allocation $\theta^*$ and the (expected) social welfare of the best allocation $\tilde{\theta}$ found with the estimated parameters (i.e., $\text{SW}(\theta^*,\hat{\mathbf{v}}) - \text{SW}(\tilde{\theta},\hat{\mathbf{v}})$). We notice that minimizing the social welfare regret does not coincide with minimizing $R_T$. In fact, once the quality estimates are accurate enough, such that $\theta_t$ is equal to $\theta^*$, the social welfare regret drops to zero. On the other hand, since $p_{i,t}(\hat{\mathbf{v}})$ is defined according to the estimated qualities, $R_T(\mathfrak{A})$ might still be positive even if $\theta_t = \theta^*$. In addition, we believe that in practical applications providing a theoretical bound over the regret of the auctioneer's revenue is more important rather than a bound on the regret of the social welfare.\footnote{However, we show that our bounds over the regret of auctioneer's revenue can be easily extended also to the regret of the social welfare.}
The study of the problem when $K=1$ is well established in the literature.
More precisely, the properties required to have a DSIC mechanism are studied in~\cite{devanur2009price} and it is shown that any learning algorithm must split the exploration and the exploitation in two separate phases in order to incentivize the advertisers to report their true values.
This condition has a strong impact on the regret $R_T(\mathfrak{A})$ of the mechanism. In fact, while in a standard bandit problem the distribution--free regret is of order $\Omega(T^{1/2})$, in single--slot auctions, DSIC mechanisms cannot achieve a regret smaller than $\Omega(T^{2/3})$. In~\cite{devanur2009price} a truthful learning mechanism is designed with a nearly optimal regret of order $\tilde O(T^{2/3})$.\footnote{The $\tilde O$ notation hides both constant and logarithmic factors, that is $R_T \leq \tilde O(T^{2/3})$ if there exist $a$ and $b$ such that $R_T \leq a T^{2/3} \log^b T$.} Similar structural properties for DSIC mechanisms are also studied in~\cite{babaioff2008characterizing} and similar lower--bounds are derived for the social welfare regret.
The authors show in~\cite{babaioff_impl_pay} that, by introducing a random component in the allocation function and resorting to truthfulness in expectation over the realizations of the random component of the mechanism, the separation of exploration and exploitation phases can be avoided. In this case, the upper bound over the regret over the social welfare is $O(T^{1/2})$ matching the best bound of standard distribution--free bandit problems. However, the payments of this mechanism suffer of potentially high variance. Although it is expected that with this mechanism also the regret over the auctioneer revenue is of the order of $O(T^{1/2})$, no formal proof is known.
On the other hand, the study of the problem when $K>1$ is still mostly open. In this case, a crucial role is played by the CTR model. While with only one slot, the advertisers' CTRs coincide to their qualities $q_i$, with multiple slots the CTRs may also depend on the slots and the allocation of the other ads. The only results on learning mechanisms for sponsored search auction with $K>1$ are described in~\cite{sarma2010multi-armed}, where the authors characterize DSIC mechanisms and provide theoretical bounds over the social welfare regret. More precisely, the authors assume a simple CTR model in which the CTR itself depends on the ad $i$ and the slot $m$. This model differs from the cascade model (see Section~2.1) where the CTR is a more complex function of the quality $q_i$ of an ad and the cumulative probability of observation $\Gamma_m(\theta)$ which, in general, depends on both the slot $m$ and the full allocation $\theta$ (i.e., the ads allocated before slot $s_m$). It can be easily shown that the model studied in~\cite{sarma2010multi-armed} does not include and, at the same time, is not included by the cascade model. However, the two models correspond when the CTRs are separable in two terms in which the first is the agents' quality and the second is a parameter in $[0,1]$ monotonically decreasing in the slots (i.e., only--position--dependent cascade model). Furthermore, while the cascade model is supported by an empirical activity which confirms its validity as a model of the user behavior~\cite{Craswell,Joachims}, the model considered in~\cite{sarma2010multi-armed} has not been empirically studied. In~\cite{sarma2010multi-armed}, the authors show that when the CTRs are unrestricted (e.g., they are not strictly monotonically decreasing in the slots), then the regret over the social welfare is $\Theta(T)$ and therefore at every round (of repetition of the auction) a non--zero regret is accumulated. In addition, the authors provide necessary and, in some situations, sufficient conditions to have DSIC in restricted environments (i.e., higher slot higher click probability, separable CTRs in which only ads qualities need to be estimated), without presenting any bound over the regret (except for reporting an experimental evidence that the regret is $\Omega(T^{2/3})$ when the CTRs are separable).
We summarize in Tab.~\ref{tab::results} the known results from the literature and, in bold font, the original results provided in this paper.
\begin{table}[h]
\begin{scriptsize}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
slots & CTR model & unknown & solution & regret over & regret over \\
& & parameters & concept & social welfare & auctioneer revenue \\ \hline \hline
1 & -- & $q_i$ & DSIC & $\Theta(T^{2/3})$ & $\Theta(T^{2/3})$ \\ \cline{4-6}
& & & IC in exp. & $O(T^{1/2})$ & unknown \\ \hline
$>1$ & (unconstrained) $CTR_{i,m}$ & $CTR_{i,m}$ & DISC & $\Theta(T)$ & unknown \\ \cline{2-6}
& (unfactorized) cascade & $q_i$ & DISC & $\mathbf{O(T^{2/3})}$ & $\mathbf{\Theta(T^{2/3})}$ \\ \cline{3-6}
& & $\gamma_{i,s}$ & DISC & $\mathbf{\Theta(T)}$ & $\mathbf{\Theta(T)}$ \\ \cline{2-6}
& position--dep. cascade / & $\lambda_m$ & DSIC & $\mathbf{\Theta(T)}$ & $\mathbf{\Theta(T)}$ \\ \cline{4-6}
& separable $CTR_{i,m}$ & & IC in exp. & $\mathbf{0}$ & $\mathbf{0}$ \\ \cline{3-6}
& & $q_i$, $\lambda_m$ & DSIC & $\mathbf{\Theta(T)}$ & $\mathbf{\Theta(T)}$ \\ \cline{4-6}
& & & IC in exp. & $\mathbf{O(T^{2/3})}$ & $\mathbf{O(T^{2/3})}$ \\ \cline{2-6}
& ad--dependent cascade & $c_i$ & DSIC & $\mathbf{\Theta(T)}$ & $\mathbf{\Theta(T)}$ \\ \cline{3-6}
& & $q_i$, $c_i$ & DSIC & $\mathbf{\Theta(T)}$ & $\mathbf{\Theta(T)}$ \\ \hline
\end{tabular}
\end{center}
\end{scriptsize}
\caption{Known results on regret bounds for sponsored search auction. We remark with bold font the results provided in this paper.}
\label{tab::results}
\end{table}
\section{Introduction}\label{s:introduction}
Sponsored search auctions (SSAs) constitute one of the most successful applications of \emph{microeconomic mechanisms}, producing a revenue of about \$6 billion dollars in the US alone in the first half of 2010~\cite{IABreport2010}. In a SSA, a number of \emph{advertisers} bid to have their \emph{sponsored links} (from here on \textit{ads}) displayed in some slot alongside the search results of a keyword. Sponsored search auctions currently adopt a \emph{pay--per--click} scheme, requiring positive payments to an advertiser only if its ad has been clicked. Given an allocation of ads over the slots, each ad is associated with a \emph{click--through--rate} (CTR) defined as the probability that such ad will be clicked by the user. CTRs are estimated by the auctioneer and play a crucial role in the auction, since they are used by the auctioneer to find the optimal allocation (in expectation) and to compute the payments for each ad.
There is a large number of works formalizing SSAs as a \emph{mechanism design} problem~\cite{Narahari2009}, where the objective is to design an auction mechanism that incentivizes advertisers to bid their \emph{truthful} valuations (needed for \emph{economic stability}) and that assures both the advertisers and the auctioneer to have a non--negative utility. The most common SSA mechanism is the \emph{generalized second price} (GSP) auction~\cite{Edelman2007,Varian2007}. This mechanism is proved not to be truthful and advertisers may implement bidding strategies that gain more than bidding their truthful valuations as shown in~\cite{Edelman2007}. While in complete information settings the worst Nash equilibrium in the GSP gives a revenue to the auctioneer equal to the revenue given by the Vickrey--Clarke--Groves (VCG) equilibrium~\cite{Edelman2007}, in Bayesian settings the worst Bayes--Nash equilibrium in the GSP can provide a much smaller revenue than the VCG---a lower bound of $\frac{1}{8}$ is provided in~\cite{paes}. The implementation of the VCG mechanism (assuring truthfulness) for SSAs has been investigated in~\cite{Narahari2009}. Although the VCG mechanism is not currently adopted by the search engines (but it is, e.g., by Facebook), a number of scientific theoretical results builds upon it.
In this paper, we focus on the problem of designing truthful mechanisms when the CTRs are not known and need to be estimated in SSAs with multiple slots. This problem is particularly relevant in practice because the assumption that all the CTRs are known beforehand is rarely realistic. Furthermore, it also poses interesting scientific challenges since it represents one of the first examples where learning theory is paired with mechanism design techniques to obtain effective methods to learn under equilibrium constraints (notably the truthfulness property).
The problem of estimating the CTRs and to identify the best allocation of ads is effectively formalized as a \textit{multi--arm bandit problem}~\cite{robbins1952some} where each ad is an arm and the objective is to minimize the cumulative regret (i.e., the revenue loss w.r.t. an optimal allocation defined according to the exact CTRs). The problem of budgeted advertisers (i.e., auctions where the total amount of money each advertiser is willing to pay is limited) with multiple queries is considered in~\cite{pandey2006handling}. This problem is formalized as a budgeted multi--bandit multi--arm problem, where each bandit corresponds to a query, and an algorithm is proposed with explicit bounds over the regret on the revenue. Nonetheless, the proposed method works in a non--strategic environment, where the advertisers do not try to influence the outcome of the auction and always bid their true values. The strategic dimension of SSAs is partially taken into consideration in~\cite{langford2010maintaining} where the advertisers are assumed to play a bidding strategy at the equilibrium w.r.t. a set of estimated CTRs which are available to both the auctioneer and the advertisers. The authors introduce a learning algorithm which explores different rankings on the ads so as to improve the CTR estimates and, at the same, not to introduce incentives for the advertisers to deviate from the previous equilibrium strategy. A more complete notion of truthfulness for bandit algorithms in multi--slot SSAs is studied in~\cite{gonen2007incentive-compatible}. In particular, they build on the action elimination algorithm in~\cite{even-dar2006action} and they report a probably approximately correct (PAC) analysis of its performance. Unfortunately, as pointed in~\cite{devanur2009price} and \cite{babaioff2008characterizing} the mechanism is not guaranteed to be truthful and thus it only works when the advertisers bid their true values. An extension to the action elimination algorithm is also proposed in~\cite{gonen2007an-adaptive} for the more general setting where budgeted advertisers are allowed to enter and exit the auction at different time instants that they declare along with their bid. The authors derive an algorithm that approximately achieves the best social welfare under the assumption that the gain of untruthful declarations is limited. Finally, single--slot online advertising is studied also in~\cite{nazerzadeh2008dynamic} where the notion of Bayesian incentive compatibility (BIC) is taken into consideration and an asymptotically BIC and \textit{ex ante} efficient mechanism is introduced. The most complete study of truthful bandit mechanisms so far is reported in~\cite{devanur2009price} and \cite{babaioff2008characterizing}. These works first provided a complete analysis on the constraints truthfulness forces on the multi--arm bandit algorithm with single--slot SSAs, showing that no \emph{dominant--strategy} truthful bandit mechanism can achieve a regret (over the social welfare and over the auctioneer's revenue) smaller than $\tilde\Omega(T^\frac{2}{3})$ and that the exploration and exploitation phases must be separate. Furthermore, they also suggest nearly--optimal algorithms. Instead, when the notion of truthfulness is relaxed, adopting truthfulness \emph{in expectation} w.r.t. click (and possibly mechanism) randomness, it is possible to obtain a regret $\tilde O(T^\frac{1}{2})$ (over the social welfare) without separating the exploration and exploitation phases in the case of single--slot SSAs~\cite{babaioff_impl_pay}.
When multiple slots are present, a user model is needed to describe how the valuations of the advertisers change over the slots. All the models available in the literature assume the separation of the CTR as the product of two terms, the first capturing the probability that an ad will be clicked once observed by the user, while the second capturing the probability that the user will observe such an ad given the displayed allocation. The basic model (commonly referred to as \emph{separability model}) prescribes that the probability of observing an ad depends only on its position~\cite{Narahari2009}. Recently, more accurate models have been proposed and the most famous model is the \emph{cascade model} according to which the user scans the slots from top to bottom and the probability with which the user moves from a slot to the next one depends on the ad and on the slot (this kind of user is commonly called \emph{Markovian user})~\cite{Kempe2008,Aggarwal2008}, while with the remaining probability the user stops to observe ads. As a result, the probability of observing an ad depends on position of the ad and on all the ads allocated above. The validity of the cascade model has been evaluated and supported by a wide range of experimental investigations~\cite{Craswell,Joachims}. The only results on learning mechanisms for SSAs with multiple slots are described in~\cite{sarma2010multi-armed}, where the authors characterize dominant--strategy truthful mechanisms and provide theoretical bounds over the social welfare regret for the separability model. However, these results are partial, e.g., they do not solve the common case in which the slot--dependent parameters are monotonically decreasing in the slots, and they cannot easily be extended to the more challenging case of the cascade model (see discussion in Section~\ref{ss:online.mechanism}).
In the present paper, we build on the results available in the literature and we provide a number of contributions when the separability model and the cascade model are adopted. More precisely, our results can be summarized as follow.
\begin{itemize}
\item \emph{Separability model with monotone parameters/only position--dependent cascade model}: in this case, there are two groups of parameters, one related to the ads (called \emph{quality}) and one to the slots (called \emph{prominence}). We studied all configurations of information incompleteness. When only qualities are unknown, we provide a regret analysis in dominant--strategy truthfulness obtaining a regret of $\tilde O(T^\frac{2}{3})$ (while it is open whether it is possible to obtain a better upper bound adopting truthfulness in expectation). When only prominences are unknown, we provide a regret analysis in truthfulness in expectation obtaining a regret of $0$, whereas we show that any dominant--strategy truthful learning mechanism would have a regret of $\tilde{\Theta}(T)$. When both groups of parameters are unknown, we provide a regret analysis in truthfulness in expectation obtaining a regret of $\tilde {O}(T^\frac{2}{3})$ (while it is open whether it is possible to obtain a better upper bound adopting truthfulness in expectation), whereas any dominant--strategy truthful learning mechanism would have a regret of $\tilde{\Theta}(T)$.
\item \emph{Cascade model}: in the non--factorized cascade model (i.e., when the observation probabilities can be any) we show that it is possible to obtain a regret of $\tilde {O}(T^\frac{2}{3})$ in dominant--strategy truthful learning mechanisms when only the qualities of the ads are unknown. We show also that in the factorized cascade model (i.e., when the observation probabilities are the products of terms depending only on the position or on the ads as used in~\cite{Kempe2008}), in the very special case in which the ad--dependent parameters are unknown we obtain a regret of $\tilde{\Theta}(T)$ in dominant--strategy truthful learning mechanisms (while it is open whether it is possible to obtain a better upper bound adopting truthfulness in expectation).
\item \emph{Learning parameters}: for each setting of uncertainty we study we provide functions, to be used in practice, to set the learning parameters in order to minimize the bound over the regret given the parameters in input.
\item \emph{Numerical simulations}: we investigate the accuracy of all the bounds we provide in the paper in predicting the dependency of the regret on the auction parameters by numerical simulations. We show that the theoretical dependency matches the actual dependency we observed by simulation.
\end{itemize}
The paper is organized as follows. In Section~\ref{s:notation} we briefly review the basics of mechanism design and multi--armed bandit learning. Section~\ref{s:statement} formalizes sponsored search auctions and introduces the corresponding online learning mechanism design problem. In Section~\ref{s:statement} we also provide a more formal overview of existing results in comparison with the findings of this paper. In Sections~\ref{s:constant} and~\ref{s:externalities} we report and discuss the main regret bounds in the case of position--dependent and position-- and ad--dependent externalities. In Section~\ref{s:experiments} we report numerical simulations aiming at testing the accuracy of the theoretical bounds. Section~\ref{s:conclusions} concludes the paper and proposes future directions of investigation. The detailed proofs of the theorems are reported in Appendix.
\section{Numerical Simulations}\label{s:experiments}
In this section we report numerical simulations to validate the theoretical bounds over the regret of the auctioneer's revenue presented in the previous sections.\footnote{The bounds over the regret of the social welfare present a structure similar to those over the auctioneer's revenue and their empirical analysis is omitted, providing similar results.} In particular, we analyze the accuracy with which our bounds predict the dependency of the regret on the main parameters of the auctions such as $T$, $N$, $K$, and $q_{\min}$. All the simulations share the way the ads are generated. The qualities $\{q_i\}_{\mathcal{N}}$ are drawn from a uniform distribution in $[0.01, 0.1]$, while the values $\{v_i\}_{\mathcal{N}}$ are randomly drawn from a uniform distribution on $[0, 1]$ ($v_{\max} = 1$). Since the main objective is to test the accuracy of the bounds, we report the \textit{relative regret}
$$\overline{R}_T = \frac{R_T}{B(T,K,N, q_{\min}, \Gamma_{\min})},$$
where $B(T,K,N, q_{\min}, \Gamma_{\min})$ is the value of the bound for the specific setting (i.e., (\ref{eq:regret.const}) and (\ref{eq:regret.posdep.qlu}) for position--dependent, and (\ref{eq:regret.extern}) for position/ad--dependent externalities). We analyze the accuracy of the bound w.r.t. each specific parameter, changing only its value and keeping the values of all the others fixed. We expect the relative regret to be always smaller than $1$, indeed we expect $B$ to be an actual upper--bound on the real regret $R_T$. All the results presented in the following sections have been obtained by setting $\tau$ and $\delta$ as suggested by the bounds derived in the previous sections and, where it is not differently specified, by averaging over 100 independent runs.
\subsection{Position--Dependent Externalities}\label{s:exp.constant}
\subsubsection{Unknown $\{q_i\}_{i \in \mathcal{N}}$}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.45\textwidth]{pics/const_T.eps}
\includegraphics[width=0.45\textwidth]{pics/const_N.eps}
\end{center}
\vspace{-0.4cm}
\caption{Position--dependent externalities with unknown $\{q_i\}_{i \in \mathcal{N}}$. Dependency of the relative regret on $T$, $N$.}\label{f:const}
\vspace{-0.4cm}
\end{figure*}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.45\textwidth]{pics/const_K.eps}
\includegraphics[width=0.45\textwidth]{pics/const_K_fixed.eps}
\end{center}
\vspace{-0.4cm}
\caption{Position--dependent externalities with unknown $\{q_i\}_{i \in \mathcal{N}}$. Dependency of the relative regret on $K$ for two different choice of the the qualities $q$.}\label{f:const2}
\vspace{-0.2cm}
\end{figure*}
First of all we analyze the accuracy of the bound provided in Section~\ref{ssec:uq}, where the model presents only position--dependent externalities and the qualities of the ads are unknown. We design the simulations such that $\lambda_m=\lambda$ for every $m$ with $\Lambda_1 = 1$ and $\Lambda_K = 0.8$ (i.e., $\lambda = \sqrt[K-1]{0.8}$). Thus, $\Lambda_{\min} = 0.8$ in all the experiments.
In Fig.~\ref{f:const} we analyze the accuracy of the bound w.r.t. the parameters $T$ and $N$. All the three curves in the left plot are completely flat (except for white noise) showing that the value of the relative regret $\overline{R}_T$ for different values of $K$ and $N$ not change as $T$ increases. This suggests that the bound in Theorem~\ref{thm:constant} effectively predicts the dependency of the regret $R_T$ w.r.t. the number of rounds $T$ of the auction as $\tilde O(T^{2/3})$. The right plot represents the dependency of the relative regret $\overline{R}_T$ on the number of ads $N$. In this case we notice that it is relatively accurate as $N$ increases but there is a transitory effect for smaller values of $N$ where the regret grows faster than predicted by the bound (although $B(T,K,N, q_{\min}, \Lambda_{\min})$ is still an upper--bound to $R_T$). Finally, the left plot of Fig.~\ref{f:const2} suggests that the dependency on $K$ in the bound of Theorem~\ref{thm:constant} is over--estimated, since the relative regret $\overline{R}_T$ decreases as $K$ increases. As discussed in the comment to the proof in Section~\ref{s:constant} this might be explained by the over--estimation of the term $\frac{\max_i(\tilde{q}^+_{i} \hat{v}_i;l)}{\max_i(\tilde{q}^+_{i} \hat{v}_i;k)}$ in the proof. In fact, this term is likely to decrease as $K$ increases. In order to validate this intuition, we have identified some instances for which the bound seems to accurately predict the dependency on $K$. For these instances $q_1 = 0.1$, $q_2=0.095$, and $q_i=0.09$ for every $2<i\leq K$. As a result, the ratio between the qualities $q_i$ is fixed (on average) and does not change with $K$. The right plot of Fig.~\ref{f:const2} shows that, with these values of $q_i$, the ratio $\overline{R}_T$ is constant for different values of $N$, implying that in this case the bound accurately predicts the behavior of $R_T$. In fact, as commented in Theoerm~\ref{thm:constant}, we derive distribution--independent bounds where the qualities $q_i$ do not appear in the bound. As a result, $R_T$ should be intended as a worst case w.r.t. all the possible configurations of qualities and the externalities.
\subsubsection{Unknown $\{\Lambda_m\}_{m \in \mathcal{K}}$}
\begin{figure*}[th]
\begin{center}
\includegraphics[width=0.45\textwidth]{pics/posdepli_T.eps}
\includegraphics[width=0.45\textwidth]{pics/posdepli_K5000.eps}
\end{center}
\vspace{-0.4cm}
\caption{Position--dependent externalities with unknown $\{\Lambda_m\}_{m \in \mathcal{K}}$. Dependency of the relative regret on $T$ and $K$.}\label{f:pd.TK}
\vspace{-0.4cm}
\end{figure*}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.45\textwidth]{pics/posdepli_mu5000.eps}
\includegraphics[width=0.45\textwidth]{pics/baba_var_mu.eps}
\end{center}
\vspace{-0.4cm}
\caption{Position--dependent externalities with unknown $\{\Lambda_m\}_{m \in \mathcal{K}}$. Dependency of the relative regret on $\mu$. Variance of the revenue of the auctioneer}\label{f:pd.mu}
\vspace{-0.2cm}
\end{figure*}
We now investigate the accuracy of the bound derived for algorithm A--VCG2$^\prime$ presented in Section~\ref{sssec:l.uc.m}. We used several probability distributions to generate the values of $\{\lambda_m\}_{m \in \mathcal{K}}$. We observed that, when they are drawn uniformly from the interval $[0.98, 1.00]$, the numerical simulations confirm our bound (as we show below), whereas the bound seems to overestimate the dependences over $K$ and $\mu$ when the support of the probability distribution is larger (i.e., $[<0.98, 1.00]$); we do not report any plot for this second case.
The left plot of Figure~\ref{f:pd.TK} shows the dependence of the ratio $\overline{R}_T$ w.r.t. $T$ when $\mu = 0.01$. Despite the noise, the ratio seems not to be affected by the variation of $T$, confirming our bound. In the right plot of Figure~\ref{f:pd.TK}, the ratio follows the same behaviour as $K$ varies when $T = 10^5$ and $\mu = 0.01$ except that the bound seems to overestimate the dependence when $K$ assumes small values (as it happens in practice).
In the left plot of Figure~\ref{f:pd.mu}, the ratio $\overline{R}_T$ seems to be constant as $\mu$ varies when $T = 10^5$.
We conclude our analysis studying the variance of the payments as $\mu$ varies. The bound over $R_T$, provided in Section~\ref{sssec:l.uc.m}, suggests to choose a $\mu \rightarrow 0$ in order to reduce the regret. Nonetheless, the regret bounds are obtained in expectation w.r.t. all the sources of randomization (including the mechanism) and do not consider the possible deviations. Thus in the right plot of Figure~\ref{f:pd.mu} we investigate the variance of the payments. In fact, The variance is excessively high for small values of $\mu$, making the adoption of these value inappropriate. Thus, the choice of $\mu$ should consider both these two dimensions of the problem: the regret and the variance of the payments.
\subsubsection{Unknown $\{\Lambda_m\}_{m \in \mathcal{K}}$ and $\{q_i\}_{i \in \mathcal{N}}$}
In this section we analyze the bound provided in Section~\ref{ssec:uql} for position--dependent auctions where both the prominences and the qualities are unknown. For these simulations we generate $\{\lambda_m\}_{m \in \mathcal{K}}$ samples from a uniform distribution over $[0.5,1]$. In the simulations we adopted the values of $\tau$, $\delta$ and $\mu$ derived for the bound. In particular, in order to balance the increase of variance of the payments when $\mu$ decreases, the number of rounds is not constant, but it changes as a function of $\mu$, i.e. $\frac{1000}{\mu}$. This means that, in expectation, the bid of a generic ad $a_i$ is modified $1000$ times over the number of the rounds.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.45\textwidth]{pics/pos_dep_T_primoa5000.eps}
\includegraphics[width=0.45\textwidth]{pics/pos_dep_N.eps}
\end{center}
\vspace{-0.4cm}
\caption{Position--dependent externalities with unknown $\{q_i\}_{i \in \mathcal{N}}$ and $\{\Lambda_m\}_{m \in \mathcal{K}}$. Dependency of the relative regret on $T$, $N$.}\label{f:ql}
\vspace{-0.4cm}
\end{figure*}
In the plots of Fig.~\ref{f:ql}, we show that the bound~(\ref{eq:regret.posdep.qlu}) accurately predicts the dependence of the regret w.r.t. the parameters $T$ and $N$. Indeed, except for the white noise due to the high variance of the payments based on the cSRP, the two plots shows that fixing the other parameters, the ratio $\overline{R}_T$ is constant as $T$ and $N$ increase, respectively.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.45\textwidth]{pics/pos_dep_K.eps}
\end{center}
\vspace{-0.4cm}
\caption{Position--dependent externalities with unknown $\{q_i\}_{i \in \mathcal{N}}$ and $\{\Lambda_m\}_{m \in \mathcal{K}}$. Dependency of the relative regret on $K$.}\label{f:ql_depK}
\vspace{-0.4cm}
\end{figure*}
The plot in Fig.~\ref{f:ql_depK} represents the dependency of the relative regret w.r.t. the parameter $K$. We can deduce that the bound $R_T$ over--estimate the dependency on $K$ for small values of the parameters, while, with larger values, the bound accurately predicts the behavior, the curves being flat.
\subsection{Position/Ad--Dependent Externalities}\label{s:exp.externalities}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.45\textwidth]{pics/extern_T.eps}
\includegraphics[width=0.45\textwidth]{pics/extern_q.eps}
\end{center}
\vspace{-0.4cm}
\caption{Dependency on $T$ and $q_{\min}$ in auctions with position/ad--dependent externalities.}\label{f:extern}
\vspace{-0.4cm}
\end{figure*}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.45\textwidth]{pics/extern_N.eps}
\end{center}
\vspace{-0.4cm}
\caption{Dependency of the relative regret $\overline{R}_T$ on $N$.}\label{f:add-extern}
\vspace{-0.2cm}
\end{figure}
In this section we analyze the bound provided in Section 5.1 for auctions with position--dependent and ad--dependent externalities where both only the qualities are unknown.
In the bound provided in Theorem~\ref{thm:extern} the regret $R_T$ presents a linear dependency on $N$ and an inverse dependency on the smallest quality $q_{\min}$. The relative regret $\overline{R}_T$ is now defined as $R_T/B$ where $B$ is bound (\ref{eq:regret.extern}). In the left plot of Fig.~\ref{f:extern} we report $\overline{R}_T$ as $T$ increases. As it can be noticed, the bound accurately predicts the behavior of the regret w.r.t. $T$ as in the case of position--dependent externalities. In the right plot of Fig.~\ref{f:extern} we report $\overline{R}_T$ as we change $q_{\min}$. According to the bound in (\ref{eq:regret.extern}) the regret should decrease as $q_{\min}$ increases (i.e., $R_T \leq \tilde O(q_{\min}^{-1}$)) but it is clear from the plot that $R_T$ has a much smaller dependency on $q_{\min}$, if any\footnote{From this experiment is not clear whether $\overline{R}_T = \tilde O(q_{\min}$), thus implying that $R_T$ does not depend on $q_{\min}$ at all, or $\overline{R}_T$ is sublinear in $q_{\min}$, which would correspond to a dependency $R_T = \tilde O(q_{\min}^{-f})$ with $f<1$.}. Finally, we study the dependency on $N$ (Figure~\ref{f:add-extern}). In this case $\overline{R}_T$ slightly increases and then it tends to flat as $N$ increases. This result suggests that the, theoretically derived, $N^{1/3}$ dependency of $R_T$ w.r.t. the number of ads might be correct.
We do not report results on $K$ since the complexity of finding the optimal allocation $f^*$ becomes intractable for values of $K$ larger than~8, as shown in~\cite{aamas2013}, making the empirical evaluation of the bound impossible.
\section{Proofs of Social-Welfare Regret in Theorems~\ref{th:pd_q_sw} and~\ref{th:pad_q_sw}}
Before stating the main result of this section, we need the following technical lemma.
\begin{lemma}\label{lem:SW.rexpl}
Let us consider an auction with N advertisers, K slots, and T rounds, and a mechanism that separates the exploration ($\tau$ rounds) and the exploitation phases ($T-\tau$ rounds).
Consider an arbitrary space of allocation functions $\mathcal{G}$, $\tilde{g} \in \arg \max_{g' \in \mathcal{G}} \widetilde{\text{SW}}\left(g'(\hat{\mathbf{v}}), \hat{\mathbf{v}} \right)$ and $| q_i - \tilde{q}^+_i |\leq \eta$ with probability $1 - \delta$. For any $g \in \mathcal{G}$, an upper bound of the global regret over the SW ($R_T^{SW}$) of the mechanism adopting $\tilde{g}$ instead of $g$ is:
$$
R_T^{SW} \leq v_{\max} K \left[ 2 (T - \tau) \eta + \tau + \delta T \right].
$$
\end{lemma}
\begin{pf}
We now prove the bound on the social welfare, starting from the cumulative instantaneous regret during the exploitation phase.
\begin{align*}
r &= \text{SW}(g(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \text{SW}(\tilde{g}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \\
& = \text{SW}(g(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \widetilde{\text{SW}}(g(\hat{\mathbf{v}}), \hat{\mathbf{v}})+\\
& + \underbrace{\widetilde{\text{SW}}(g(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \max_{g' \in \mathcal{G}} \widetilde{\text{SW}}(g'(\hat{\mathbf{v}}), \hat{\mathbf{v}})}_{\leq 0} + \widetilde{\text{SW}}(\tilde{g}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \text{SW}(\tilde{g}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \\
&\leq \underbrace{\text{SW}(g(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \widetilde{\text{SW}}(g(\hat{\mathbf{v}}), \hat{\mathbf{v}})}_{r^1} + \underbrace{\widetilde{\text{SW}}(\tilde{g}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \text{SW}(\tilde{g}(\hat{\mathbf{v}}), \hat{\mathbf{v}})}_{r^2}
\end{align*}
\noindent The two remaining terms $r^1$ and $r^2$ can be easily bounded by using Lemma~\ref{lem:welfare}
\begin{align*}
r &\leq r_1 + r_2 \leq 0 + 2Kv_{\max} \eta = 2 K v_{\max}\eta
\end{align*}
\noindent with probability $1 - \delta$.
Thus, we can conclude that:
$$
R^{SW}_T \leq v_{\max} K \left[ 2 (T - \tau) \eta + \tau + \delta T \right].
$$
\end{pf}
\begin{pf}\textit{(Theorem~\ref{th:pd_q_sw}) }
\noindent \textbf{Step 1: global regret.}
We apply Lemma~\ref{lem:SW.rexpl} to the position--dependent cascade model with $\{q_i\}_{i \in \mathcal{N}}$ unknowns, obtaining
\begin{align*}
R^{SW}_T &\leq v_{\max} K \left[ 2 (T - \tau) \eta + \tau + \delta T \right]\\
&\leq v_{\max} K \left[2 (T - \tau) \frac{\sqrt{2}}{\Lambda_{\min}} \sqrt{\frac{N}{K \tau} \log \frac{2N}{\delta}} + \tau + \delta T \right]
\end{align*}
\noindent \textbf{Step 2: parameter optimization.}
First we notice that adopting the value of the parameters identified in Theorem~\ref{thm:constant} we obtain an upper bound $\tilde{O}(T^\frac{2}{3})$ for the global regret $R_T^{SW}$.
In order to find values that better optimize the bound over $R_T^{SW}$, let $e := \frac{\sqrt{2}}{\Lambda_{\min}}$, then we first simplify the previous bound as
\begin{align*}
R^{SW}_T &\leq v_{\max} K \left[ 2 e \sqrt{\frac{N}{K \tau} \log \frac{2N}{\delta}} + \tau + \delta T \right]
\end{align*}
Taking the derivative of the previous bound w.r.t. $\tau$ leads to
$$ v_{\max} K \left( - \tau^{-\frac{3}{2}} e T \sqrt{\frac{N}{K} \log \frac{2N}{\delta}} + 1 \right) = 0,$$
which leads to
$$\tau = e^\frac{2}{3} T^\frac{2}{3} N^\frac{1}{3} K^{-\frac{1}{3}} \left( \log \frac{2N}{\delta} \right)^\frac{1}{3}$$
Once replaced in the bound, we obtain
$$ R_T^{SW} \leq v_{\max} K \left[ 3 e^\frac{2}{3} T^\frac{2}{3} N^\frac{1}{3} K^{-\frac{1}{3}} \left( \log \frac{2N}{\delta} \right)^\frac{1}{3} + \delta T \right]$$
Finally, we choose $\delta$ to optimize the asymptotic order by setting
$$ \delta = e^\frac{2}{3} K^{-\frac{1}{3}} N^\frac{1}{3} T^{-\frac{1}{3}} $$
given that $\delta < 1$ this imply that $T > e^2 K^{-1} N$.
The final bound is
$$ R_T^{SW} \leq 4 v_{\max} e^\frac{2}{3} K^\frac{2}{3} N^\frac{1}{3} T^\frac{2}{3} \left( \log 2 e^{-\frac{2}{3}} N^\frac{2}{3} K^\frac{1}{3} T^\frac{1}{3} \right)^\frac{1}{3}$$
\end{pf}
\begin{pf}\textit{(Theorem~\ref{th:pad_q_sw}) }
\noindent \textbf{Step 1: global regret.}
We apply Lemma~\ref{lem:SW.rexpl} to the model with position-- and ad--dependent externalities with $\{q_i\}_{i \in \mathcal{N}}$ unknowns, obtaining
\begin{align*}
R^{SW}_T &\leq v_{\max} K \left[ 2 (T - \tau) \eta + \tau + \delta T \right]\\
&\leq v_{\max} K \left[ 2 (T - \tau) \frac{\sqrt{2}}{\Gamma_{\min}} \sqrt{\frac{N}{K \tau} \log \frac{2N}{\delta}} + \tau + \delta T \right]
\end{align*}
\noindent \textbf{Step 2: parameter optimization.}
First we notice that adopting the value of the parameters identified in Theorem~\ref{thm:extern} we obtain an upper bound $\tilde{O}(T^\frac{2}{3})$ for the global regret $R_T^{SW}$.
In order to find values that better optimize the bound over $R_T^{SW}$, it is possible to use the procedure followed in the proof of Theorem~\ref{th:pd_q_sw} with $e := \frac{\sqrt{2}}{\Gamma_{\min}}$:
$$ R_T^{SW} \leq 4 v_{\max} e^\frac{2}{3} K^\frac{2}{3} N^\frac{1}{3} T^\frac{2}{3} \left( \log 2 e^{-\frac{2}{3}} N^\frac{2}{3} K^\frac{1}{3} T^\frac{1}{3} \right)^\frac{1}{3}$$
\end{pf}
\section{Proof of Social-Welfare Regret in Theorem~\ref{thm:constant.l.sw.baba}}
\begin{pf}\textit{(Theorem~\ref{thm:constant.l.sw.baba})}
The bound over the global regret on the social welfare ($R_T^{SW}$) can be easily derived considering that each bid is modified by the self--resampling procedure with a probability of $\mu$.
Thus we can define $S' = \{\mathbf{s}' | \mathbf{s}'\in\{0,1\}^N, \pi(i;f^*(\hat{\mathbf{v}}))\leq K \Rightarrow s'_i = 1 \}$, i.e. all the random realization where the self--resampling procedure does not modify the bids of the ads displayed when the allocation function is $f^*$ is applied to the true bids $\hat{\mathbf{v}}$.
Thus we have:
\begin{align*}
R_T^{SW} &\leq T \left( \mathbb{P}\left[\mathbf{s} \in S'\right] \cdot 0 + \underbrace{\mathbb{P}\left[\mathbf{s} \not \in S'\right]}_{\leq K \mu} K v_{\max} \right) \leq K^2 \mu v_{\max} T
\end{align*}
\end{pf}
\section{Proof of Social-Welfare Regret Theorem~\ref{th:pd_lq_sw}}
\begin{pf}\textit{(Theorem~\ref{th:pd_lq_sw})}
\noindent \textbf{Step 1: instantaneous regret.}
We start computing the instantaneous regret over the SW during the exploitation phase.
First of all we introduce the following definition: $S' = \{\mathbf{s}' |\mathbf{s}'\in\{0,1\}^N, \pi(i;f^*(\hat{\mathbf{v}}) )\leq K \Rightarrow s'_i = 1 \}$, i.e. all the random realization where the self--resampling procedure does not modify the bids of the ads displayed when the allocation function is $f^*$ is applied to the true bids $\hat{\mathbf{v}}$.
We now provide the bound over the regret.
\begin{align*}
r &= \text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \mathbb{E}_{\mathbf{x}}\left[\text{SW}(\tilde{f}(\mathbf{x}), \hat{\mathbf{v}})|\hat{\mathbf{v}}\right] \\
&= \underbrace{\mathbb{P}[\mathbf{s} \in S']}_{\leq 1} \left(\text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \mathbb{E}_{\mathbf{x}|\mathbf{s} \in S'}\left[\text{SW}(\tilde{f}(x), \hat{\mathbf{v}})|\hat{\mathbf{v}}\right]\right) +\\
& + \underbrace{\mathbb{P}[\mathbf{s} \not \in S']}_{\leq K\mu} \left(\text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \mathbb{E}_{\mathbf{x} | \mathbf{s} \not \in S'}\left[\text{SW}(\tilde{f}(x), \mathbf{v})|\hat{\mathbf{v}}\right]\right)\\
& \leq \text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \mathbb{E}_{\mathbf{x}|\mathbf{s} \in S'}\left[\text{SW}(\tilde{f}(\mathbf{x}), \hat{\mathbf{v}})|\hat{\mathbf{v}}\right] + \\ & + K \mu \underbrace{\left(\text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \underbrace{\mathbb{E}_{\mathbf{x} | \mathbf{s} \not \in S'}\left[\text{SW}(\tilde{f}(x), \mathbf{v})|\hat{\mathbf{v}}\right]}_{\geq 0}\right)}_{\leq K v_{\max}}\\
& \leq \underbrace{\text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \mathbb{E}_{\mathbf{x}|\mathbf{s} \in S'}\left[\widetilde{\text{SW}}(f^*(\mathbf{x}), \hat{\mathbf{v}})|\hat{\mathbf{v}}\right]}_{r_1 \leq 0}+ \\&+ \underbrace{\mathbb{E}_{\mathbf{x}|\mathbf{s} \in S'}\left[\widetilde{\text{SW}}(f^*(\mathbf{x}), \hat{\mathbf{v}})|\hat{\mathbf{v}}\right] - \mathbb{E}_{\mathbf{x}|\mathbf{s} \in S'}\left[\widetilde{\text{SW}}(\tilde{f}(\mathbf{x}), \hat{\mathbf{v}})|\hat{\mathbf{v}}\right]}_{r_2 \leq 0}+ \\
& + \mathbb{E}_{\mathbf{x}|\mathbf{s} \in S'}\left[\widetilde{\text{SW}}(\tilde{f}(\mathbf{x}), \hat{\mathbf{v}})|\hat{\mathbf{v}}\right] - \mathbb{E}_{\mathbf{x}|\mathbf{s} \in S'}\left[\text{SW}(\tilde{f}(\mathbf{x}), \hat{\mathbf{v}})|\hat{\mathbf{v}}\right] + v_{\max} \mu K^2\\
& \leq \max_{f \in \mathcal{F}} \left( \mathbb{E}_{\mathbf{x}|\mathbf{s} \in S'}\left[\widetilde{\text{SW}}(f(\mathbf{x}), \hat{\mathbf{v}}) - \text{SW}(f(\mathbf{x}), \hat{\mathbf{v}})|\hat{\mathbf{v}}\right] \right) + v_{\max} \mu K^2\\
& \leq \max_{f \in \mathcal{F}} \left( \sum_{j: \pi(j; f(x)) \leq K} \Lambda_{\pi(j; f(x))} v_j (\tilde{q}_j - q_j) \right) + v_{\max} \mu K^2\\
& \leq v_{\max} \max_{f \in \mathcal{F}} \left( \sum_{j: \pi(j; f(x)) \leq K} (\tilde{q}_j - q_j) \right) + v_{\max} \mu K^2\\
& \leq 2 v_{\max} K \eta + v_{\max} \mu K^2 = v_{\max} K \left( 2 \eta + K \mu \right)
\end{align*}
We provide a brief intuition of bounds $r_1$ and $r_2$. The bound $r_1$ can be explained noticing that when the bids of the ads displayed in $f^*(\hat{\mathbf{v}})$ are not modified we have that $\alpha(m; f^*(\hat{\mathbf{v}})) = \alpha(m; f^*(\mathbf{x}))$ where $m\leq K$ and $\mathbf{x}$ s.t. $\mathbf{s} \in S'$. The bound for $r_2$ can be understood noticing that when the bids of the ads s.t. $\pi(j; f^*(\mathbf{x})) \leq K$ are not modified and $x_i \leq \hat{v}_i \ \forall i \in \mathcal{N}$, we obtain $\widetilde{\text{SW}}(f^*(\mathbf{x}), \hat{\mathbf{v}}) = \widetilde{\text{SW}}(f^*(\mathbf{x}), \mathbf{x}) \leq \max_{\theta \in \Theta} \widetilde{\text{SW}}(\theta, \mathbf{x}) = \widetilde{\text{SW}}(\tilde{f}(\mathbf{x}), \mathbf{x}) \leq \widetilde{\text{SW}}(\tilde{f}(\mathbf{x}), \hat{\mathbf{v}})$.
\noindent \textbf{Step 2: global regret.}
We can now compute the upper bound for the global regret
\begin{align*}
R_T^{SW} & \leq v_{\max} K \left[ (T - \tau) ( 2 \eta + K \mu) + \tau + \delta T \right]\\
& \leq v_{\max} K \left[ (T - \tau) \left( 2 \sqrt{\frac{N}{\tau} \log{\frac{2N}{\delta}}} + K \mu \right) + \tau + \delta T \right]
\end{align*}
\noindent \textbf{Step 3: parameter optimization.}
We first simplify the previous bound as
\begin{align*}
R_T^{SW} & \leq v_{\max} K \left[ 2 T \sqrt{\frac{N}{\tau} \log{\frac{2N}{\delta}}} + K \mu T + \tau + \delta T \right]
\end{align*}
Taking the derivative of the previous bound w.r.t. $\tau$ leads to
$$ v_{\max} K \left( -\tau^{-\frac{3}{2}} T \sqrt{N\log{\frac{2N}{\delta}}} + 1 \right) = 0,$$
which leads to
$$\tau = N^\frac{1}{3} T^\frac{2}{3} \left(\log{\frac{2N}{\delta}}\right)^\frac{1}{3}$$
Once replaced in the bound, we obtain
\begin{align*}
R_T^{SW} & \leq 3 v_{\max} K N^\frac{1}{3} T^\frac{2}{3} \left(\log{\frac{2N}{\delta}}\right)^\frac{1}{3}+ \mu K^2 v_{\max} T + \delta v_{\max} K T
\end{align*}
Finally, we choose $\delta$ and $\mu$ to optimize the asymptotic order by setting
\begin{align*}
\delta & = N^\frac{1}{3} T^{-\frac{1}{3}}\\
\mu & = K^{-1} T^{-\frac{1}{3}} N^\frac{1}{3}
\end{align*}
given that $\delta < 1$ this imply that $T > N$ and, given that $\mu < 1$ we have that $T>\frac{N}{K^3}$.
The final bound is
$$ R_T^{SW} \leq 5 \cdot v_{\max} K N^\frac{1}{3} T^\frac{2}{3} \left( \log{2 N^\frac{2}{3} T^\frac{1}{3}} \right)^\frac{1}{3}$$
\end{pf}
\section{Proof of Revenue Regret in Theorem~\ref{thm:constant}}
\noindent We start by reporting the proof of Proposition~\ref{p:hoeffding}.
\begin{pf}\textit{(Proposition~\ref{p:hoeffding})}
The derivation is a simple application of the Hoeffding's bound. We first notice that each of the terms in the empirical average $\tilde{q}_i$ (\ref{eq:est.q}) is bounded in $[0; 1/\Lambda_{\pi(i;\theta_t)}]$. Thus we obtain
\begin{equation*}
\mathbb{P}\left( |q_i - \tilde{q}_i | \geq \epsilon \right) \leq 2 \exp\bigg(-\frac{2|B_i|^2\epsilon^2}{\sum_{t \in B_i}\big(\frac{1}{\Lambda_{\pi(i;\theta_t)}}-0\big)^2}\bigg) = \frac{\delta}{N}.
\end{equation*}
By reordering the terms in the previous expression we have
\begin{align*}
\epsilon &= \sqrt{\left(\sum_{t \in B_i}\frac{1}{\Lambda^2_{\pi(i;\theta_t)}}\right) \frac{1}{2|B_i|^2} \log{\frac{2N}{\delta}}},
\end{align*}
which guarantees that all the empirical estimates $\tilde{q}_i$ are within $\epsilon$ of $q_i$ for all the ads with probability, at least, $1 - \delta$.
\qed
\end{pf}
Before stating the main result of this section, we need the following technical lemma.
\begin{lemma}\label{lem:ratio}
For any slot $s_m$ with $m \in \mathcal{K}$, with probability $1-\delta$,
\begin{equation}\label{eq:ratio}
\frac{\max\limits_{i\in \mathcal{N}} (q_i \hat{v}_i; m)}{\max\limits_{i\in \mathcal{N}} (\tilde{q}^+_i \hat{v}_i; m)} \leq 1,
\end{equation}
where the operator $\max(\cdot;\cdot)$ is defined as in Section~\ref{s:constant}.
\end{lemma}
\begin{pf}
The proof is a straightforward application of Proposition~\ref{p:hoeffding}.
We consider the optimal allocation $\theta^*$ defined in (\ref{eq:efficient-alloc}) and the estimated allocation $\tilde{\theta}$ defined in (\ref{eq:optimalallocationestimatedq}). We denote $h = \alpha(m;\theta^*) = \arg \max\limits_{i \in \mathcal{N}}(q_i \hat{v}_i; m)$, i.e., the index of the ad allocated in a generic slot in position $m$. There are two possible scenarios:
\begin{itemize}
\item If $\pi(h;\tilde{\theta}) < m$ (the ad is displayed into a higher slot in the approximated allocation $\tilde{\theta}$), then
$\exists j \in \mathcal{N}$ s.t. $\pi(j;\theta^*) < m \wedge \pi(j;\tilde{\theta}) \geq m$. Thus
$$ \max\limits_{i \in \mathcal{N}}(\tilde{q}^+_i \hat{v}_i; m) \geq \tilde{q}^+_j \hat{v}_j \geq q_j \hat{v}_j \geq q_h \hat{v}_h = \max\limits_{i \in \mathcal{N}}(q_i \hat{v}_i; m)$$
where the second inequality holds with probability $1 - \delta$;
\item If $\pi(h;\tilde{\theta}) \geq m$ (the ad is displayed into a lower or equal slot in the approximated allocation $\tilde{\theta}$), then
$$ \max\limits_{i \in \mathcal{N}}(\tilde{q}^+_i \hat{v}_i; m) \geq \tilde{q}^+_h \hat{v}_h \geq q_h v_h = \max\limits_{i \in \mathcal{N}}(q_i \hat{v}_i; m)$$
where the second inequality holds with probability $1 - \delta$.
\end{itemize}
In both cases, the statement follows.\qed
\end{pf}
\begin{pf}\textit{(Theorem~\ref{thm:constant})}
\noindent\textbf{Step 1: expected payments.}
The proof follows steps similar to those in~\cite{devanur2009price}. We first recall that for any ad $a_i$ such that $\pi(i; \theta^*)\leq K$, the expected payments of the VCG mechanism in this case reduce to (\ref{eq:pay.opt.click.vcg.posdep.ad}):
\begin{align*}
p^*_i(\hat{\mathbf{v}}) = \sum_{l=\pi(i; \theta^*)+1}^{K+1} \left[(\Lambda_{l-1} - \Lambda_l) \max\limits_{j\in \mathcal{N}}(q_j \hat{v}_j; l)\right],
\end{align*}
while, given the definition of \avcg1 \ reported in Section~\ref{s:constant}.1, the expected payments for at $t$--th iteration of the auction are
\begin{align*}
\tilde{p}_i(\hat{\mathbf{v}}) = \left\{
\begin{array}{ll}
0 & \text{if } t \leq \tau \text{ (\textit{exploration})}\\
\tilde{p}_{i}(\hat{\mathbf{v}}) & \text{if } t > \tau \text{ (\textit{exploitation})}
\end{array} \right.
\end{align*}
where the payment for any ad $a_i$ such that $\pi(i; \tilde\theta)\leq K$ is defined in (\ref{eq:hpay.const}) as
\begin{align*}
\tilde p_i (\hat{\mathbf{v}}) &=\frac{q_i}{\tilde{q}^+_i} \sum_{l=\pi(i;\tilde{\theta})+1}^{K+1} (\Lambda_{l-1} - \Lambda_l) \max\limits_{j \in \mathcal{N}}(\tilde{q}^+_j \hat{v}_j; l).
\end{align*}
\noindent\textbf{Step 2: exploration regret.} Since for any $t\leq \tau$ A--VCG\ sets all the payments to 0 the per--round regret is
\begin{align}\label{eq:exploration.regret}
r_t = \sum_{m = 1}^K (p^*_{\alpha(m;\theta^*)}(\hat{\mathbf{v}}) - 0) = \sum_{m = 1}^K \sum_{l = m}^{K} \Delta_l \max_{i\in\mathcal{N}}(q_i\hat{v}_i; l+1) \leq v_{\max}\sum_{m = 1}^K \Lambda_m,
\end{align}
where $\Delta_l = \Lambda_l - \Lambda_{l+1}$. The exploration regret is obtained by summing up $r_t$ over $\tau$ steps.
\noindent\textbf{Step 3: exploitation regret.} Now we focus on the expected (w.r.t. clicks) per--round regret during the exploitation phase. According to the definition of payments, at each round $t \in \{\tau + 1, \ldots, T\}$ of the exploitation phase we bound the per--round regret $r_t$ as
\begin{align*}
r_t &= \sum_{m = 1}^K (p^*_{\alpha(m;\theta^*)}(\hat{\mathbf{v}}) - \tilde{p}_{\alpha(m;\tilde{\theta})}(\hat{\mathbf{v}})) \\
&= \sum_{m = 1}^K \sum_{l = m}^{K} \Delta_l \left( \max\limits_{i\in \mathcal{N}}(q_i \hat{v}_i; l+1) - \frac{ \max\limits_{i\in \mathcal{N}}(\tilde{q}^+_i \hat{v}_i; l+1) }{\tilde{q}^+_{\alpha(m;\tilde{\theta})}} q_{\alpha(m;\tilde{\theta})} \right) \\
&= \sum_{m = 1}^K \sum_{l = m}^{K} \Delta_l \frac{ \max\limits_{i\in \mathcal{N}}(\tilde{q}^+_i \hat{v}_i; l+1)}{\tilde{q}^+_{\alpha(m;\tilde{\theta})}} \left( \frac{ \max\limits_{i\in \mathcal{N}}(q_i \hat{v}_i;l+1)}{\max\limits_{i\in \mathcal{N}}(\tilde{q}^+_{i} \hat{v}_i;l+1)} \tilde{q}^+_{\alpha(m;\tilde{\theta})} - q_{\alpha(m;\tilde{\theta})} \right) \\
&= \sum_{m = 1}^K \sum_{l = m}^{K} \Delta_l\frac{ \max\limits_{i\in \mathcal{N}}(\tilde{q}^+_{i} \hat{v}_i;l+1)}{\max\limits_{i\in \mathcal{N}}(\tilde{q}^+_{i} \hat{v}_i;m)} \hat{v}_{\alpha(m; \tilde{\theta})} \left( \frac{\max\limits_{i \in \mathcal{N}}(q_i \hat{v}_i;l+1)}{\max\limits_{i\in \mathcal{N}}(\tilde{q}^+_{i} \hat{v}_i;l+1)} \tilde{q}^+_{\alpha(m;\tilde{\theta})} - q_{\alpha(m;\tilde{\theta})} \right).
\end{align*}
By definition of the max operator, since $l+1 > m$, it follows that
\begin{align}\label{eq:step.loose}
\frac{\max\limits_{i \in \mathcal{N}} (\tilde{q}^+_{i} \hat{v}_i;l+1)}{\max\limits_{i \in \mathcal{N}} (\tilde{q}^+_{i} \hat{v}_i;m)} \leq 1.
\end{align}
Finally, from Lemma~\ref{lem:ratio} and $\hat{v}_{\alpha(m; \tilde{\theta})} \leq v_{\max}$, it follows that
\begin{align} \label{eq:boundVCG.exactvsest}
r_t \leq \sum_{m = 1}^K \sum_{l = m}^{K} v_{\max} \Delta_l(\tilde{q}^+_{\alpha(m;\tilde{\theta})} - q_{\alpha(m;\tilde{\theta})}) \leq v_{\max} \sum_{m = 1}^K \Big[ (\tilde{q}^+_{\alpha(m;\tilde{\theta})} - q_{\alpha(m;\tilde{\theta})})\sum_{l = m}^{K} \Delta_l\Big],
\end{align}
with probability at least $1-\delta$.
Notice that, by definition of $\Delta_l$, $\sum_{l=m}^{K} \Delta_l = \Lambda_{m} - \Lambda_{K+1} = \Lambda_{m}$. Furthermore, from the definition of $\tilde{q}^+_i$ and using (\ref{eq:eta}) we have that for any ad $a_i$, $\tilde{q}^+_{i} - q_i = \tilde{q}_i - q_i + \eta \leq 2\eta$, with probability at least $1 - \delta$. Thus, the difference between the payments becomes\footnote{Notice that in the logarithmic term the factor of 2 we have in Proposition~\ref{p:hoeffding} disappears since in this proof we only need the one-sided version of it.}
\begin{align}\label{eq:brs4}
r_t & \leq 2 v_{\max}\eta\sum_{m = 1}^K \Lambda_m \leq 2v_{\max} \left(\sum_{m = 1}^K \Lambda_m\right)\sqrt{\Bigg(\sum_{m=1}^{K} \frac{1}{\Lambda_{m}^2}\Bigg) \frac{2N}{K^2 \tau} \log \frac{N}{\delta}}.
\end{align}
with probability $1-\delta$.
\noindent\textbf{Step 4: global regret.} By summing up the regrets reported in (\ref{eq:exploration.regret}) and (\ref{eq:brs4}), we obtain
\begin{align*}
R_T &\leq v_{\max} \left(\sum_{m = 1}^K \Lambda_m \right)\Bigg( 2(T - \tau) \sqrt{\left(\sum_{m=1}^{K} \frac{1}{\Lambda_{m}^2}\right) \frac{2N}{K^2 \tau} \log \frac{N}{\delta}} + \tau + \delta T \Bigg),
\end{align*}
that can be further simplified give that $\sum_{m = 1}^K \Lambda_m\leq K$ as
\begin{equation}\label{eq:bound.thm.constant}
R_T \leq v_{\max} K\Bigg( 2(T - \tau) \sqrt{\left(\sum_{m=1}^{K} \frac{1}{\Lambda_{m}^2}\right) \frac{2N}{K^2 \tau} \log \frac{N}{\delta}} + \tau + \delta T \Bigg).
\end{equation}
\noindent\textbf{Step 5: parameters optimization.}
Beside describing the performance of \avcg1, the previous bound also provides guidance for the optimization of the parameters $\tau$ and $\delta$. We first simplify the bound in (\ref{eq:bound.thm.constant}) as
\begin{align}\label{eq:bound.thm.constant.simple}
R_T &\leq v_{\max} K\bigg( 2T \sqrt{\left(\sum_{m=1}^{K} \frac{1}{\Lambda_{m}^2}\right) \frac{2N}{K^2 \tau} \log \frac{2N}{\delta}} + \tau + \delta T \bigg) \notag\\
&\leq v_{\max} K\bigg( \frac{2T}{\Lambda_{\min}} \sqrt{\frac{2N}{K \tau} \log \frac{N}{\delta}} + \tau + \delta T \bigg),
\end{align}
where we used $\tau \leq T$ and $\sum\limits_{m=1}^{K} 1/\Lambda_{m}^2\leq K/\Lambda_{\min}^2$, with $\Lambda_{\min} = \min_{m \in \mathcal{K}} \Lambda_m$. In order to find the optimal value of $\tau$, we take the derivative of the previous bound w.r.t. $\tau$ and set it to zero and obtain
\begin{align*}
v_{\max} K\Big(-\tau^{-\frac{3}{2}} \frac{T}{\Lambda_{\min}} \sqrt{\frac{2N}{K} \log \frac{N}{\delta}} + 1\Big) = 0,
\end{align*}
which leads to
\begin{equation*}
\tau = 2^\frac{1}{3} K^{-\frac{1}{3}} T^{\frac{2}{3}} N^{\frac{1}{3}} \Lambda_{\min}^{-\frac{2}{3}} \Big( \log{\frac{N}{\delta}}\Big)^{\frac{1}{3}}.
\end{equation*}
Substituting this value of $\tau$ into (\ref{eq:bound.thm.constant.simple}) leads to the optimized bound
\begin{align*}
R_T &\leq v_{\max} K \bigg(3 \cdot 2^\frac{1}{3} K^{-\frac{1}{3}} T^{\frac{2}{3}} N^{\frac{1}{3}} \Lambda_{\min}^{-\frac{2}{3}} \Big( \log{\frac{N}{\delta}}\Big)^{\frac{1}{3}} + \delta T\bigg).\end{align*}
We are now left with the choice of the confidence parameter $\delta\in (0,1)$, which can be easily set to optimize the asymptotic rate (i.e., ignoring constants and logarithmic factors) as
\[
\delta = K^{-\frac{1}{3}} T^{-\frac{1}{3}} N^{\frac{1}{3}}\\
\]
\noindent with the trivial constraint that $T > \frac{N}{K}$ (given by $\delta < 1$). We thus obtain the final bound
\[
R_T \leq 4 \cdot 2^\frac{1}{3} v_{\max} \Lambda_{\min}^{-\frac{2}{3}} K^\frac{2}{3} T^\frac{2}{3} N^\frac{1}{3} \left[ \log \big( K^\frac{1}{3} T^\frac{1}{3} N^\frac{2}{3} \big)\right]^{\frac{1}{3}},
\]
\noindent which concludes the proof.\qed
\end{pf}
\section{Proof of Revenue Regret in Theorem~\ref{thm:constant.l.baba}}
Unlike the setting considered in Theorem~\ref{thm:constant}, here the regret is only due to the use of a randomized mechanism, since no parameter estimation is actually needed.
\begin{pf}\textit{(Theorem~\ref{thm:constant.l.baba})}
\noindent\textbf{Step 1: payments and additional notation.}
We recall that according to~\cite{tardos_sp} and~\cite{greenLaffont} the expected VCG payments can be written as in (\ref{eq:pay.vcg.emp.tardos}) in the form
\begin{align*}
p^*_i(\hat{\mathbf{v}}) = \Lambda_{\pi(i;f^*(\hat{\mathbf{v}}))} q_i \hat{v}_i - \int_{0}^{\hat{v}_i} \Lambda_{\pi(i;f^*(\hat{\mathbf{v}}_{-i},u))} q_i du,
\end{align*}
while the A--VCG2$^\prime$ mechanism prescribes contingent payments as in (\ref{eq:pay.opt.babaioff.click}), which lead to expected payments
\begin{align}\label{eq:pay.babaioff.exp}
p_i^{B,*}(\hat{\mathbf{v}}) &= \mathbb{E}_{\mathbf{x}}\big[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}\big]q_i \hat{v}_i - \int_{0}^{\hat{v}_i} \mathbb{E}_{\mathbf{x}}\big[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u\big]q_i du.
\end{align}
Given the randomness of the allocation function of A--VCG2$^\prime$, we need to introduce the following additional notation:
\begin{itemize}
\item $\mathbf{s} \in \{0,1\}^N$ is a vector where each element $s_i$ denotes whether the $i$--th bid has been preserved or it has been modified by the self--resampling procedure, i.e., if $x_i=\hat{v}_i$ then $s_i=1$, otherwise if $x_i < \hat{v}_i$ then $s_i=0$. Notice that $\mathbf{s}$ does not provide information about the actual modified values $\mathbf{x}$;
\item $\mathbb{E}_{\mathbf{x}|\mathbf{s}}[\Lambda_{\pi(i; f(\mathbf{x}))}|\hat{\mathbf{v}}]$ is the expected value of prominence associated with the slots allocated to ad $a_i$ conditioned on the declared bids $\hat{\mathbf{v}}$ being perturbed as in $\mathbf{s}$.
\end{itemize}
Let $S = \{\mathbf{s} | \pi(i;f^*(\hat{\mathbf{v}}))\leq K + 1 \Rightarrow s_i = 1\ \forall i \in \mathcal{N} \}$ be all the realizations where the self--resampling procedure does not modify the bids of the first $K+1$ ads, i.e., the $K$ ads displayed applying $f^*$ to the true bids $\hat{\mathbf{v}}$ and the first non-allocated ad.
\noindent\textbf{Step 2: the regret.}
We proceed by studying the per--ad regret $r_i(\hat{\mathbf{v}}) = p_i^*(\hat{\mathbf{v}}) - p_i^{B,*}(\hat{\mathbf{v}})$. Given the previous definitions, we rewrite the expected payments $p_i^{B,*}(\hat{\mathbf{v}})$ as
\begin{align*}
p_i^{B,*}(\hat{\mathbf{v}}) &= \bigg(\mathbb{P}[\mathbf{s} \in S] \Lambda_{\pi(i;f^*(\hat{\mathbf{v}}))} + \mathbb{P}[\mathbf{s} \not \in S] \mathbb{E}_{\mathbf{x}|\mathbf{s} \not \in S}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}]\bigg)q_i \hat{v}_i \\
&\quad - \int_{0}^{\hat{v}_i} \bigg(\mathbb{P}[\mathbf{s} \in S] \Lambda_{\pi(i; f^*(\hat{\mathbf{v}}_{-i},u))} + \mathbb{P}[\mathbf{s} \not \in S] \mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u]\bigg)q_i du\\
&=\mathbb{P}[\mathbf{s} \in S] \bigg(\Lambda_{\pi(i;f^*(\hat{\mathbf{v}}))}q_i \hat{v}_i - \int_{0}^{\hat{v}_i}\Lambda_{\pi(i; f^*(\hat{\mathbf{v}}_{-i},u))} q_i du\bigg)\\
&\quad+\mathbb{P}[\mathbf{s} \not \in S] \bigg(\mathbb{E}_{\mathbf{x}|\mathbf{s} \not \in S}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}]q_i \hat{v}_i - \int_{0}^{\hat{v}_i}\mathbb{E}_{\mathbf{x}|\mathbf{s} \not \in S}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u] q_i du\bigg)\\
&=\mathbb{P}[\mathbf{s} \in S] p_i^*(\hat{\mathbf{v}}) \\
&\quad+ \mathbb{P}[\mathbf{s} \not \in S] \bigg(\mathbb{E}_{\mathbf{x}|\mathbf{s} \not \in S}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}]q_i \hat{v}_i - \int_{0}^{\hat{v}_i}\mathbb{E}_{\mathbf{x}|\mathbf{s} \not \in S}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u] q_i du \bigg),
\end{align*}
where in the last expression we used the expression of the VCG payments in (\ref{eq:pay.vcg.emp.tardos}) according to~\cite{tardos_sp} and~\cite{greenLaffont}.
The per--ad regret is
\begin{align*}
r_i&(\hat{\mathbf{v}}) = p_i^*(\hat{\mathbf{v}}) - p_i^{B,*}(\hat{\mathbf{v}}) \\
&= p_i^*(\hat{\mathbf{v}}) - \mathbb{P}[\mathbf{s} \in S] p_i^*(\hat{\mathbf{v}}) \\
&\quad- \mathbb{P}[\mathbf{s} \not \in S] \bigg(\mathbb{E}_{\mathbf{x}|\mathbf{s} \not \in S}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}]q_i \hat{v}_i - \int_{0}^{\hat{v}_i}\mathbb{E}_{\mathbf{x}|\mathbf{s} \not \in S}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u] q_i du\bigg)\\
&= \mathbb{P}[\mathbf{s} \not \in S] p_i^*(\hat{\mathbf{v}}) \\
&\quad- \mathbb{P}[\mathbf{s} \not \in S] \underbrace{\bigg(\mathbb{E}_{\mathbf{x}|\mathbf{s} \not \in S}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}]q_i \hat{v}_i - \int_{0}^{\hat{v}_i}\mathbb{E}_{\mathbf{x}|\mathbf{s} \not \in S}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u] q_i du\bigg)}_{r_{i,1}^B}.
\end{align*}
Since we have that $u \leq \hat{v}_i$ in the integral and since the allocation function defined in~\cite{babaioff_impl_pay} is monotone, we have that
\begin{align*}
\mathbb{E}_{\mathbf{x} | \mathbf{s} \not \in S}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u] \leq \mathbb{E}_{\mathbf{x} | \mathbf{s} \not \in S}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}],
\end{align*}
which implies that $r_{i,1}^B$ is non--negative. Thus the regret $r_i^B$ can be bounded as
\begin{align}\label{bnd:rSRP}
r_i^B(\hat{\mathbf{v}}) &= \mathbb{P}[\mathbf{s} \not \in S] p_i^*(\hat{\mathbf{v}}) \underbrace{ - \mathbb{P}[\mathbf{s} \not \in S] r_{i,1}^B}_{\leq 0} \nonumber \\
&\leq \mathbb{P}[\mathbf{s} \not \in S] p_i^*(\hat{\mathbf{v}}) \leq \mathbb{P}\big[\exists j: s_j = 0 \land \pi(j;f^*(\hat{\mathbf{v}})) \leq K+1\big] v_{\max} \nonumber\\
&\leq \sum_{j\in \mathcal{N}: \pi(j;f^*(\hat{\mathbf{v}})) \leq K+1 }\mathbb{P}[s_j = 0] v_{\max} = \left( K + 1 \right) \mu v_{\max} \leq 2 K \mu v_{\max}.
\end{align}
We can now compute the bound on the global regret $R_T$. Since this mechanism does not require any estimation phase, the regret is simply
\begin{align*}
R_T & \leq 2 K^2 \mu v_{\max} T.
\end{align*}
\noindent\textbf{Step 3: parameters optimization.} In this case, the bound would suggest to choose a $\mu \rightarrow 0$, but it is necessary to consider that with $\mu \rightarrow 0$ the variance of the payment goes to infinity.
\end{pf}
\section{Proof of Revenue Regret in Theorem~\ref{thm:constant.ql}}
The proof of Theorem~\ref{thm:constant.ql} needs to combine the result of Theorem~\ref{thm:constant.l.baba} and the regret due to the estimation of the parameters similarly to what is done in Theorem~\ref{thm:constant}.
\begin{pf}\textit{(Theorem~\ref{thm:constant.ql})}
\noindent\textbf{Step 1: payments and the regret.}
Similar to the proof of Theorem~\ref{thm:constant.l.baba}, we use the form of the VCG payments as in (\ref{eq:pay.vcg.emp.tardos}):
\begin{align*}
p^*_i(\hat{\mathbf{v}}) = \Lambda_{\pi(i;f^*(\hat{\mathbf{v}}))} q_i \hat{v}_i - \int_{0}^{\hat{v}_i} \Lambda_{\pi(i;f^*(\hat{\mathbf{v}}_{-i},u))} q_i du,
\end{align*}
while A--VCG3 uses the contingent payments in (\ref{eq:pay.babaioff.ppc}), which in expectation become
\begin{align}\label{eq:pay.babaioff.exp.tilde}
\tilde p_i^{B}(\hat{\mathbf{v}}) &= \mathbb{E}_{\mathbf{x}}\big[\Lambda_{\pi(i;\tilde f(\mathbf{x}))}|\hat{\mathbf{v}}\big]q_i \hat{v}_i - \int_{0}^{\hat{v}_i} \mathbb{E}_{\mathbf{x}}\big[\Lambda_{\pi(i;\tilde f(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u\big]q_i du.
\end{align}
We also need to introduce the expected payments
\begin{align*}
\tilde p_i(\hat{\mathbf{v}}) = \Lambda_{\pi(i;\tilde f(\hat{\mathbf{v}}))} q_i \hat{v}_i - \int_{0}^{\hat{v}_i} \Lambda_{\pi(i;\tilde f(\hat{\mathbf{v}}_{-i},u))} q_i du,
\end{align*}
which correspond to the VCG payments except from the use of the estimated allocation function $\tilde f$ instead of $f^*$.
Initially, we compute an upper bound over the per--ad regret $r_i = p_i^* - p_i$ for each round of the exploitation phase and we later use this result to compute the upper bound for the regret over the whole time interval ($R_T$). We divide the per--ad regret in two different components:
\begin{align}
r_i(\hat{\mathbf{v}}) &= p_i^*(\hat{\mathbf{v}}) - \tilde{p}^B_i(\hat{\mathbf{v}}) \\
&= \underbrace{p_i^*(\hat{\mathbf{v}}) - p_i^{B,*}(\hat{\mathbf{v}})}_{\text{cSRP regret}} + \underbrace{p_i^{B,*}(\hat{\mathbf{v}}) - \tilde{p}^B_i(\hat{\mathbf{v}})}_{\text{learning regret}} = r_i^B(\hat{\mathbf{v}}) + r_i^L(\hat{\mathbf{v}}) \nonumber,
\end{align}
where
\begin{itemize}
\item $r_i^B(\hat{\mathbf{v}})$ is the regret due to the use of the approach proposed in~\cite{babaioff_impl_pay} instead of the VCG payments, when all the parameters are known;
\item $r_i^L(\hat{\mathbf{v}})$ is the regret due to the uncertainty on the parameters when the payments defined in~\cite{babaioff_impl_pay} are considered.
\end{itemize}
For the definitions of $\mathbf{s}$ and $\mathbb{E}_{\mathbf{x}|\mathbf{s}}[\Lambda_{\pi(i; f(\mathbf{x}))}|\hat{\mathbf{v}}]$ refer to the proof of Theorem~\ref{thm:constant.l.baba}.
\noindent\textbf{Step 2: the cSRP regret.}
We can reuse the result obtained in the proof of Theorem~\ref{thm:constant.l.baba}. In particular, we can use the bound in (\ref{bnd:rSRP}), i.e. $r_i^B(\hat{\mathbf{v}}) \leq \left(K+1\right) \mu v_{\max}$. Given that we have assumed $N > K$, in the remaining parts of this proof we will use the following upper bound: $r_i^B(\hat{\mathbf{v}}) \leq \left(K+1\right) \mu v_{\max} \leq N \mu v_{\max} $.
\noindent\textbf{Step 3: the learning regret.}
Similar to the previous step, we write the learning expected payments based on the cSRP in (\ref{eq:pay.babaioff.exp.tilde}) as
\begin{align*}
\tilde p^B_i(\hat{\mathbf{v}}) =\mathbb{P}[\mathbf{s}=\boldsymbol{1}] \tilde{p}_i(\hat{\mathbf{v}}) + \mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] \bigg(\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\pi(i; \tilde{f}(\mathbf{x}))}|\hat{\mathbf{v}}] q_i \hat{v}_i - \int_{0}^{\hat{v}_i}\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\pi(i; \tilde{f}(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u] q_i du\bigg).
\end{align*}
Then the per-ad regret is
\begin{align*}
r_i^L(\hat{\mathbf{v}}) &= p_i^{B,*}(\hat{\mathbf{v}}) - \tilde p_i^B(\hat{\mathbf{v}})\\
&= \mathbb{P}[\mathbf{s}=\boldsymbol{1}] (p_i^*(\hat{\mathbf{v}})- \tilde{p}_i(\hat{\mathbf{v}})) +\\
&\quad + \mathbb{P}[\mathbf{s}\neq\boldsymbol{1}] \bigg(\underbrace{\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}]q_i \hat{v}_i - \int_{0}^{\hat{v}_i}\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\pi(i;f^*(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u] q_i du}_{\leq v_{\max}} +\\
&\quad\quad\quad\quad\quad\quad\quad \underbrace{-\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\pi(i;\tilde{f}(\mathbf{x}))}|\hat{\mathbf{v}}]q_i \hat{v}_i + \int_{0}^{\hat{v}_i}\mathbb{E}_{\mathbf{x}|\mathbf{s}\neq\boldsymbol{1}}[\Lambda_{\pi(i;\tilde{f}(\mathbf{x}))}|\hat{\mathbf{v}}_{-i},u] q_i du}_{= - r_{i,1}^B \leq 0}\bigg)\\
&\leq p_i^*(\hat{\mathbf{v}})- \tilde{p}_i(\hat{\mathbf{v}}) + N\mu v_{\max}.
\end{align*}
We now simply notice that payments $\tilde{p}_i$ are WVCG payments corresponding to the estimated allocation function $\tilde{f}$ and can be written as
\begin{align*}
\tilde{p}_i(\hat{\mathbf{v}}) = \frac{q_i}{\tilde{q}^+_i} \Big[\widetilde{SW}\big(\tilde{f}_{-i}\left(\hat{\mathbf{v}}\right),\hat{\mathbf{v}}\big) - \widetilde{SW}_{-i}\big(\tilde{f}\left(\hat{\mathbf{v}}\right), \hat{\mathbf{v}}\big)\Big],
\end{align*}
which allows us to use the results stated in proof of Theorem~\ref{thm:constant} and from (\ref{eq:boundVCG.exactvsest}) we can conclude that
\begin{align*}
\sum_{i: \pi(i; f^*(\hat{\mathbf{v}})\leq K)} \left(p_i^*\left(\hat{\mathbf{v}}\right)- \tilde{p}_i\left(\hat{\mathbf{v}}\right)\right) \leq 2v_{\max}\eta \left( \sum_{m=1}^K \Lambda_m \right) \leq 2Kv_{\max}\eta.
\end{align*}
\noindent\textbf{Step 4: the global regret.} We now bring together the two instantaneous regrets and we have that at each round of the the exploitation phase we have the regret $r = \sum_{i=1}^N r_i$.
We first notice that the expected instantaneous regret $r_i$ for each ad $a_i$ is defined as the difference between the VCG payment $p_i^*(\hat{\mathbf{v}})$ and the (expected) payments computed by the estimated randomized mechanism $p_i(\hat{\mathbf{v}})$. We notice that $p_i^*(\hat{\mathbf{v}})$ can be strictly positive only for the $K$ displayed ads, while $p_i(\hat{\mathbf{v}}) \geq 0 \ \forall i \in \mathcal{N}$, due to the mechanism randomization. Thus, $p_i^*(\hat{\mathbf{v}}) - p_i(\hat{\mathbf{v}}) > 0$ only for at most $K$ ads. Thus we obtain the per--round regret
\begin{align*}
r &\leq \sum_{i: \pi(i;f^*(\hat{\mathbf{v}})) \leq K} r_i = \sum_{i: \pi(i;f^*(\hat{\mathbf{v}})) \leq K} \left(r_i^B + r_i^L\right)\\
& \leq K N \mu v_{\max} + \sum_{i: \pi(i;f^*(\hat{\mathbf{v}})) \leq K} \left(p_i^*\left(\hat{\mathbf{v}}\right)- \tilde{p}_i\left(\hat{\mathbf{v}}\right) + N\mu v_{\max} \right)\\
&\leq K N \mu v_{\max} + 2 K v_{\max} \eta + K N \mu v_{\max} = 2 K v_{\max} \eta + 2 K N \mu v_{\max}.
\end{align*}
Finally, the global regret becomes
\begin{align*}
R_T & \leq v_{\max} K \left[\left( T-\tau \right) \left(2 \sqrt{\frac{N}{\tau} \log \frac{2N}{\delta}} + 2 \mu N \right) + \tau + \delta T \right].
\end{align*}
\noindent\textbf{Step 5: parameters optimization.}
We first simplify further the previous bound as
\begin{align}\label{eq:global.regret.simplified}
R_T & \leq v_{\max} K \left[ T\left(2 \sqrt{\frac{N}{\tau} \log \frac{2N}{\delta}} + 2 \mu N \right) + \tau + \delta T \right].
\end{align}
We first optimize the value of $\tau$, take the derivative of the previous bound w.r.t. $\tau$ and set it to zero and obtain
\begin{align*}
v_{\max} K\Big(-\tau^{-\frac{3}{2}} T \sqrt{N \log \frac{2N}{\delta}} + 1\Big) = 0,
\end{align*}
which leads to
\begin{equation*}
\tau = T^{\frac{2}{3}} N^{\frac{1}{3}} \left( \log{\frac{2N}{\delta}}\right)^{\frac{1}{3}}.
\end{equation*}
Once replaced into (\ref{eq:global.regret.simplified}) we obtain
\begin{align*}
R_T & \leq v_{\max} K \left[ 3T^\frac{2}{3}N^\frac{1}{3}\Big( \log{\frac{2N}{\delta}}\Big)^{\frac{1}{3}} + 2 T\mu N + \delta T \right].
\end{align*}
The optimization of the asymptotic order of the bound can then be obtained by setting $\mu$ and $\delta$ so as to equalize the second and third term in the bound. In particular by setting
\begin{align*}
\mu = T^{-\frac{1}{3}} N^{-\frac{2}{3}} \quad \text{ and }\quad \delta=T^{-\frac{1}{3}}N^\frac{1}{3},
\end{align*}
we obtain the final bound
\begin{align*}
R_T & \leq 6v_{\max} K T^\frac{2}{3}N^\frac{1}{3}\Big( \log \big(2N^\frac{2}{3}T^\frac{1}{3}\big)\Big)^{\frac{1}{3}}.
\end{align*}
\end{pf}
\section{Proof of Revenue Regret in Theorem~\ref{thm:extern}}
Before deriving the proof of Theorem~\ref{thm:extern}, we prove two lemmas that we use in the following proofs.
\begin{lemma}\label{lem:welfare.mod}
Let $\mathcal{G}$ be an arbitrary space of allocation functions, then for any $g\in\mathcal{G}$, when $| q_i - \tilde{q}^+_i |\leq \eta$ with probability $1 - \delta$, we have
\begin{align*}
-2K v_{\max} \eta \leq \text{SW}(g(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \widetilde{\text{SW}}(g(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \frac{q_i}{\tilde{q}^+_{i}} \leq \frac{2K v_{\max}}{q_{\min}} \eta,
\end{align*}
with probability $1-\delta$.
\end{lemma}
\begin{pf}
By using the definition of $\text{SW}$ and $\widetilde{\text{SW}}$ we have the following sequence of inequalities
\begin{align*}
\text{SW}(g(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - &\widetilde{\text{SW}}(g(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \frac{q_i}{\tilde{q}^+_{i}} \\
&\leq \sum_{j: \pi(j; g(\hat{\mathbf{v}}))\leq K} \Gamma_{\pi(j; g(\hat{\mathbf{v}}))} \hat{v}_j \left( q_j - \tilde{q}^+_{j} \frac{q_i}{\tilde{q}^+_{i}}\right) \\
& \leq v_{\max} \sum_{j: \pi(j; g(\hat{\mathbf{v}}))\leq K} \left( q_j - q_j \frac{q_i}{\tilde{q}^+_{i}} + q_j \frac{q_i}{\tilde{q}^+_{i}} - \tilde{q}^+_{j} \frac{q_i}{\tilde{q}^+_{i}} \right) \\
&= v_{\max} \sum_{j: \pi(j; g(\hat{\mathbf{v}}))\leq K} \Bigg[ q_j \left( \frac{\tilde{q}^+_{i} - q_i}{\tilde{q}^+_{i}} \right) + \underbrace{(q_j - \tilde{q}^+_{j})}_{\leq 0} \frac{q_i}{\tilde{q}^+_{i}} \Bigg] \\
& \leq \frac{v_{\max}}{q_{\min}} \sum_{j: \pi(j; g(\hat{\mathbf{v}}))\leq K} \left( \tilde{q}_i - q_i + \eta \right) \leq \frac{2K v_{\max}}{q_{\min}} \eta.
\end{align*}
The second statement follows from
\begin{align*}
\widetilde{\text{SW}}(g(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \frac{q_i}{\tilde{q}^+_{i}} - &\text{SW}(g(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \\
&= \sum_{j: \pi(j; g(\hat{\mathbf{v}}))\leq K} \Gamma_{\pi(j; g(\hat{\mathbf{v}}))}(g(\hat{\mathbf{v}})) \hat{v}_j \left( \tilde{q}^+_{j} \frac{q_i}{\tilde{q}^+_{i}} - q_j \right) \\
& \leq v_{\max} \sum_{j: \pi(j; g(\hat{\mathbf{v}}))\leq K} \left( \tilde{q}^+_{j} \frac{q_i}{\tilde{q}^+_{i}} - q_j \right) \\
& \leq v_{\max} \sum_{j: \pi(j; g(\hat{\mathbf{v}}))\leq K} (\tilde{q}^+_j - q_j) \leq 2K v_{\max} \eta.
\end{align*}
\qed
\end{pf}
\begin{lemma}\label{lem:welfare}
Let $\mathcal{G}$ be an arbitrary space of allocation functions, then for any $g\in\mathcal{G}$, when $| q_i - \tilde{q}^+_i |\leq \eta$ with probability $1 - \delta$, we have
\begin{align*}
0 \leq \left( \widetilde{\text{SW}}(g(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \text{SW}(g(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \right) \leq 2K v_{\max} \eta,
\end{align*}
with probability $1-\delta$.
\end{lemma}
\begin{pf}
The first inequality follows from
\begin{align*}
\text{SW}(g(\hat{\mathbf{v}}), \hat{\mathbf{v}}) &- \widetilde{\text{SW}}(g(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \\
& = \sum_{j: \pi(j; g(\hat{\mathbf{v}}))\leq K} \Gamma_{\pi(j; g(\hat{\mathbf{v}}))}(g(\hat{\mathbf{v}})) \hat{v}_j \left( q_j - \tilde{q}^+_j \right) \\
&\leq v_{\max} \sum_{j: \pi(j; g(\hat{\mathbf{v}}))\leq K} ( q_j - \tilde{q}^+_j ) \leq 0,
\end{align*}
while the second inequality follows from
\begin{align*}
\widetilde{\text{SW}}(g(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - &\text{SW}(g(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \\
& = \sum_{j: \pi(j;g(\hat{\mathbf{v}}))\leq K} \Gamma_{\pi(j;g(\hat{\mathbf{v}}))}(g(\hat{\mathbf{v}})) \hat{v}_j \left( \tilde{q}^+_{j} - q_j \right) \\
&\leq v_{\max} \sum_{j: \pi(j; g(\hat{\mathbf{v}}))\leq K} \big( \tilde{q}^+_j - q_j \big) \\
&= v_{\max} \sum_{j: \pi(j; g(\hat{\mathbf{v}}))\leq K} \left(\tilde{q}_j + \eta - q_j \right) \leq 2K v_{\max} \eta.
\end{align*}
\qed
\end{pf}
We are now ready to proceed with the proof of Theorem~\ref{thm:extern}.
\begin{pf}\textit{(Theorem~\ref{thm:extern})}
\noindent \textbf{Step 1: per--ad regret.}
We first compute the instantaneous per--ad regret $r_i = p^*_i(\hat{\mathbf{v}}) - \tilde{p}_i(\hat{\mathbf{v}})$ at each round of the exploitation phase for each ad $a_i$. According to the definition of payments we have
\begin{align*}
r_i = \underbrace{\text{SW}(f^*_{-i}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \widetilde{\text{SW}}(\tilde{f}_{-i}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \frac{q_i}{\tilde{q}^+_{i}}}_{r^1_i} + \underbrace{\widetilde{\text{SW}}_{-i}(\tilde{f}(\hat{\mathbf{v}}),\hat{\mathbf{v}}) \frac{q_i}{\tilde{q}^+_{i}} - \text{SW}_{-i}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}})}_{r^2_i}.
\end{align*}
We bound the first term through Lemma~\ref{lem:welfare.mod} and the following inequalities
\begin{align*}
r^1_i &= \text{SW}(f^*_{-i}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \widetilde{\text{SW}}(f^*_{-i}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \frac{q_i}{\tilde{q}^+_{i}} + \widetilde{\text{SW}}(f^*_{-i}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \frac{q_i}{\tilde{q}^+_{i}} - \widetilde{\text{SW}}(\tilde{f}_{-i}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \frac{q_i}{\tilde{q}^+_{i}} \\
&\leq \max_{f \in \mathcal{F}_{-i}} \left( \text{SW}(f(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \widetilde{\text{SW}}(f(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \frac{q_i}{\tilde{q}^+_{i}} \right) + \underbrace{\left( \widetilde{\text{SW}}(f^*_{-i}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \max_{f \in \mathcal{F}_{-i}} \widetilde{\text{SW}}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \right)}_{\leq 0} \frac{q_i}{\tilde{q}^+_{i}} \\
& \leq \frac{2K v_{\max}}{q_{\min}} \eta,
\end{align*}
\noindent with probability $1-\delta$. We rewrite $r^2_i$ as
\begin{align*}
r^2_i &= \left( \widetilde{\text{SW}}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \Gamma_{\pi(i; \tilde{f}(\hat{\mathbf{v}}))}(\tilde{f}(\hat{\mathbf{v}})) \tilde{q}^+_{i} \hat{v}_i \right)\frac{q_i}{\tilde{q}^+_{i}} - \text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) + \Gamma_{\pi(i; f^*(\hat{\mathbf{v}}))}(f^*(\hat{\mathbf{v}})) q_i \hat{v}_i \\
&= \underbrace{\widetilde{\text{SW}}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \frac{q_i}{\tilde{q}^+_{i}} - \text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}})}_{r^{3}_i} + \left(\Gamma_{\pi(i; f^*(\hat{\mathbf{v}}))}\left(f^*(\hat{\mathbf{v}})\right) - \Gamma_{\pi(i; \tilde{f}(\hat{\mathbf{v}}))}(\hat{f}(\hat{\mathbf{v}}))\right) q_i \hat{v}_i.
\end{align*}
\noindent We now focus on the term $r^3_i$ and use Lemma~\ref{lem:welfare.mod} to bound it as
\begin{align*}
r^3_i &= \widetilde{\text{SW}}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \frac{q_i}{\tilde{q}^+_{i}} - \text{SW}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) + \underbrace{\text{SW}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \max_{f \in \mathcal{F}} \text{SW}(f(\hat{\mathbf{v}}), \hat{\mathbf{v}})}_{\leq 0} \\
&\leq \max_{f \in \mathcal{F}} \left(\widetilde{\text{SW}}(f(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \frac{q_i}{\tilde{q}^+_{i}} - \text{SW}(f(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \right) \\
&\leq 2K v_{\max} \eta.
\end{align*}
\noindent\textbf{Step 2: exploitation and global regret.}
We define $I = \{i |\pi(i; f^∗(\hat{\mathbf{v}})) \leq K \lor \pi(i; \tilde{f}(\hat{\mathbf{v}})) \leq K, i \in \mathcal{N}\}$, $|I| \leq 2K$. It is clear that only the ads $a_i$ s.t. $i \in I$ have a regret $r_i \not = 0$. The other ads, $i \not \in I$, have both $p^*_i(\hat{\mathbf{v}}) = 0$ and $\tilde{p}_i(\hat{\mathbf{v}})=0$. Thus, we can bound the regret $r$, at each exploitative round, in the following way
\begin{align*}
r &= \sum_{i \in I} (r^1_i + r^2_i) \\
& \leq \sum_{i \in I} \Big( \frac{2K v_{\max}}{q_{\min}} \eta + 2K v_{\max} \eta \Big) + \sum_{i \in I} \left(\Gamma_{\pi(i; f^*(\hat{\mathbf{v}}))}(f^*(\hat{\mathbf{v}})) - \Gamma_{\pi(i;\tilde{f}(\hat{\mathbf{v}}))}(\tilde{f}(\hat{\mathbf{v}})) \right)q_i \hat{v}_i \\
& = \sum_{i \in I} \Big( \frac{2K v_{\max}}{q_{\min}} \eta + 2K v_{\max} \eta \Big) + \sum_{i=1}^N \left(\Gamma_{\pi(i; f^*(\hat{\mathbf{v}}))}(f^*(\hat{\mathbf{v}})) - \Gamma_{\pi(i;\tilde{f}(\hat{\mathbf{v}}))}(\tilde{f}(\hat{\mathbf{v}})) \right)q_i \hat{v}_i \\
&\leq \frac{8K^2 v_{\max}}{q_{min}} \eta + \text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \text{SW}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \\
& = \frac{8K^2 v_{\max}}{q_{min}} \eta + \text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \widetilde{\text{SW}}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}})+\\
& + \underbrace{\widetilde{\text{SW}}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \max_{f \in \mathcal{F}} \widetilde{\text{SW}}(f)}_{\leq 0} + \widetilde{\text{SW}}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \text{SW}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) \\
& \leq \frac{8K^2 v_{\max}}{q_{min}} \eta + \underbrace{\text{SW}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \widetilde{\text{SW}}(f^*(\hat{\mathbf{v}}), \hat{\mathbf{v}})}_{r^1} + \underbrace{\widetilde{\text{SW}}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}}) - \text{SW}(\tilde{f}(\hat{\mathbf{v}}), \hat{\mathbf{v}})}_{r^2}
\end{align*}
The remaining terms $r^1$ and $r^2$ can be easily bounded using Lemma~\ref{lem:welfare} as
\begin{align*}
r^1 \leq 0 \quad \text{ and }\quad r^2 \leq 2K v_{\max} \eta.
\end{align*}
Summing up all the terms we finally obtain $$r \leq \frac{10K^2 v_{\max}}{q_{\min}} \eta$$ with probability $1-\delta$. Now, considering the instantaneous regret of the exploration and exploitation phases, we obtain the final bound on the cumulative regret $R_T$ as follows
\begin{align*}
R_T \leq v_{\max} K \left[ (T - \tau) \left( \frac{10 K}{\Gamma_{\min}q_{\min}} \sqrt{\frac{N}{2K\tau} \log \frac{N}{\delta}} \right) + \tau + \delta T \right].
\end{align*}
\noindent\textbf{Step 3: parameter optimization.} Let $c := \frac{5}{\sqrt{2} \Gamma_{\min} q_{\min}}$, then we first simplify the previous bound as
\begin{align*}
R_T \leq v_{\max} K \left[ 2c T\sqrt{\frac{NK}{\tau} \log \frac{N}{\delta}} + \tau + \delta T \right].
\end{align*}
Taking the derivative with respect to $\tau$ leads to
\begin{align*}
v_{\max} K\Big(-\tau^{-\frac{3}{2}} cT \sqrt{NK \log \frac{N}{\delta}} + 1\Big) = 0,
\end{align*}
which leads to
\begin{equation*}
\tau = c^\frac{2}{3}T^{\frac{2}{3}} K^{\frac{1}{3}} N^{\frac{1}{3}} \Big( \log{\frac{N}{\delta}}\Big)^{\frac{1}{3}}.
\end{equation*}
Once replaced in the bound, we obtain
\begin{align*}
R_T & \leq v_{\max} K \left[ 3T^\frac{2}{3}c^\frac{2}{3}N^\frac{1}{3}K^\frac{1}{3}\Big( \log{\frac{N}{\delta}}\Big)^{\frac{1}{3}} + \delta T \right].
\end{align*}
Finally, we choose $\delta$ to optimize the asymptotic order by setting
\begin{align*}
\delta = K^\frac{1}{3} N^\frac{1}{3} c^\frac{2}{3} T^{-\frac{1}{3}},
\end{align*}
which leads to the final bound
\[
R_T \leq 4v_{\max} K^\frac{4}{3} c^\frac{2}{3} T^\frac{2}{3} N^\frac{1}{3} \left(\log{\frac{N^\frac{2}{3} T^\frac{1}{3}}{K^\frac{1}{3} c^\frac{2}{3}}}\right)^\frac{1}{3}
\]
Notice that this bound imposes constraints on the value of $T$, indeed, $T>\tau$, thus $T > c^\frac{2}{3} K^\frac{1}{3} T^\frac{2}{3} N^\frac{1}{3} \left( \log{\frac{N}{\delta}} \right)^\frac{1}{3}$ and $\delta < 1$, thus $T > c^2 K N$, leading to:
$$
T > c^2 K N \max\left\lbrace\log{\frac{N}{\delta}}, 1\right\rbrace.
$$
The problem of the previous bound is that $\tau$ and $\delta$ depends on $q_{\min}$, which is an unknown quantity. Thus actually choosing this values to optimize the bound may be unfeasible. An alternative choice of $\tau$ and $\delta$ is obtained by optimizing the bound removing the dependency on $q_{\min}$. Let $d = \frac{5}{\sqrt{2} \Gamma_{\min}}$, then we choose
\[
\tau = d^{\frac{2}{3}} K^{\frac{1}{3}} T^{\frac{2}{3}} N^{\frac{1}{3}} \left( \log{\frac{N}{\delta}} \right)^{\frac{1}{3}},
\]
and
\[
\delta = K^\frac{1}{3} N^\frac{1}{3} d^\frac{2}{3} T^{-\frac{1}{3}},
\]
which leads to the final bound
\[
R_T \leq 4 v_{\max} K^\frac{4}{3} T^\frac{2}{3} N^\frac{1}{3} \frac{d^\frac{2}{3}}{q_{\min}} \left(\log{\frac{N^\frac{2}{3} T^\frac{1}{3}}{K^\frac{1}{3} d^\frac{2}{3}}}\right)^\frac{1}{3}
\]
under the constraint that $T \geq K N d^2$.
\qed
\end{pf}
\section{Deviation Regret}\label{app:deviation.regret}
The definition of regret in (\ref{eq:regret}) measures the cumulative difference between the revenue of a VCG compared to the one obtained by \avcg1\ over $T$ rounds. Upper--bounds over this quantity guarantees that the loss in terms of revenue does not linearly increase with $T$. As illustrated in the previous sections, the key passage in the proofs is the upper--bounding of the regret at each round of the exploitation phase (i.e., $r = \sum_{i=1}^N (p_i^* - \tilde{p}_i)$). Nonetheless, we notice that this quantity could be negative. In this section we introduce a different notion of regret ($\tilde{R}_T$) that we study only for \avcg1, leaving for the future a more detailed analysis. Let us consider the following simple example. Let $N=3$, $K=1$, $\hat{v}_i=1$ for all the ads, and $q_1=0.1$, $q_2=0.2$, and $q_3=0.3$. Let assume that after the exploration phase we have $\tilde{q}^+_1 = 0.1$, $\tilde{q}^+_2 = 0.29$, $\tilde{q}^+_3 = 0.3$. A standard VCG mechanism allocates ad $a_3$ and asks for a payment $p_3^*(\hat{\mathbf{v}})=0.2$. During the exploitation phase \avcg1\ also allocates $a_3$ but asks for an (expected) payment $\tilde{p}_3(\hat{\mathbf{v}}) = (\tilde{q}^+_2 / \tilde{q}^+_3) q_3 = 0.29$. Thus, the regret in each exploitation round is $r = p^*_3(\hat{\mathbf{v}}) - \tilde{p}_3(\hat{\mathbf{v}}) = -0.09$. Although this result might seem surprising, it is due to the fact that while both \avcg1\ and VCG are truthful, in general \avcg1\ is not efficient. We recall that a mechanism is efficient if for any set of advertisers it always maximizes the social welfare. In the example, if for instance the true quality of ad $a_3$ is $q_3 = 0.28$, then the allocation induced by $\tilde{q}^+$s is not efficient anymore. By dropping the efficiency constraint, it is possible to design mechanisms with larger revenues than the VCG. For this reason, we believe that a more complete characterization of the behavior of \avcg1\ compared to the VCG should consider the \textit{deviation} between their payments and not only the loss in the revenue.
In particular, let us define the regret as the deviation between the VCG and the approximated VCG:
\begin{align}\label{eq:aregret}
\tilde{R}_T(\mathfrak{A}) = \sum_{t=1}^T \Big| \sum_{i=1}^N (p^*_i - \tilde{p}_{it}) \Big|,
\end{align}
We prove an upper--bound for the single--slot case (the extension of the multi--slot results is straightforward).
\begin{theorem}\label{thm:a-constant}
Let us consider a sequential auction with $N$ advertisers, $K$ slots, and $T$ rounds with position--dependent cascade model with parameters $\{\Lambda_m\}_{m=1}^K$ and accuracy $\eta$ as defined in~(\ref{eq:eta}). For any parameter $\tau \in \{0, \ldots, T\}$ and $\delta \in [0,1]$, the A--VCG1 achieves a regret:
\begin{align}\label{eq:a-regret.const.exact}
\tilde{R}_T \leq K v_{\max} \left( \tau + \left( T-\tau \right) \frac{2\eta}{q_{\min}} + \delta T \right)
\end{align}
where $q_{\min} = \min_{i \in \mathcal{N}} q_i$. By setting the parameters to
\begin{align*}
\delta &= N^\frac{1}{3} K^{-\frac{1}{3}} T^{-\frac{1}{3}} \\
\tau &= 2^\frac{1}{3} \frac{ K^{-\frac{1}{3}} N^\frac{1}{3} T^\frac{2}{3}}{\Lambda_{\min}^\frac{2}{3}} \left(\log{\frac{N}{\delta}}\right)^\frac{1}{3},
\end{align*}
the regret is
\begin{align}\label{eq:a-regret.const}
\tilde{R}_T \leq 4 \cdot 2^\frac{1}{3} \frac{ K^{-\frac{1}{3}} N^\frac{1}{3} T^\frac{2}{3}}{q_{\min} \Lambda_{\min}^\frac{2}{3}} \left(\log{N^\frac{2}{3} K^\frac{1}{3} T^\frac{1}{3}}\right)^\frac{1}{3}.
\end{align}
\end{theorem}
\begin{pf}
We initially provide a bound over the instantaneous regret during the exploitation phase. We consider the two sides of the bound separately. We have that for the first side of the bound we can use the result provided in Step~3 in the proof of Theorem~\ref{thm:constant}, i.e.,
\begin{align*}
r_1 &= \sum_{m = 1}^K ( p^*_{\alpha(m;\theta^*)}(\hat{\mathbf{v}}) - \tilde{p}_{\alpha(m;\tilde{\theta})}(\hat{\mathbf{v}}) ) \\
& \leq 2 Kv_{\max} \eta,
\end{align*}
with probability $1-\delta$.
Now we bound the other side.
\begin{align*}
r_2 &= \sum_{m = 1}^K \left(\tilde{p}_{\alpha(m;\tilde{\theta})}(\hat{\mathbf{v}}) - p^*_{\alpha(m;\theta^*)}(\hat{\mathbf{v}}) \right) \\
&= \sum_{m = 1}^K \sum_{l = m}^{K} \Delta_l \left( \frac{ \max\limits_{i\in \mathcal{N}}(\tilde{q}^+_i \hat{v}_i; l+1) }{\tilde{q}^+_{\alpha(m;\tilde{\theta})}} q_{\alpha(m;\tilde{\theta})} - \max\limits_{i\in \mathcal{N}}(q_i \hat{v}_i; l+1) \right) \\
&\leq \max\limits_{i\in \mathcal{N}}(q_i \hat{v}_i; l+1) \sum_{m = 1}^K \sum_{l = m}^{K} \Delta_l \left( \frac{ \max\limits_{i\in \mathcal{N}}(\tilde{q}^+_i \hat{v}_i; l+1) }{\max\limits_{i\in \mathcal{N}}(q_i \hat{v}_i; l+1) } - 1 \right)
\end{align*}
In order to proceed with the bound, notice that, for a generic ad $a_i$ we have that $\tilde{q}^+_i \hat{v}_i = \left(\tilde{q}_i + \eta\right) \hat{v}_i \leq \left(q_i + 2\eta\right) \hat{v}_i \leq q_i \hat{v}_i + \frac{2\eta}{q_{\min}} q_i \hat{v}_i $.
Now, consider $i'=\arg\max\limits_{j\in \mathcal{N}}(q_j \hat{v}_j; l+1)$, the ad displayed in $s_{l+1}$ when the true qualities are known, we can face two different situation:
\begin{itemize}
\item $\pi\left(i'; \tilde{f}(\hat{\mathbf{v}})) \geq \pi(i'; f^*(\hat{\mathbf{v}})\right)$: in this case we can easily conclude that $\tilde{q}^+_{\alpha(l+1; \tilde{f}(\hat{\mathbf{v}}))} \hat{v}_{\alpha(l+1; \tilde{f}(\hat{\mathbf{v}}))} \leq \tilde{q}^+_{i'} \hat{v}_{i'} \leq q_{i'} \hat{v}_{i'} + \frac{2\eta}{q_{\min}} q_{i'} \hat{v}_{i'} $;
\item $\pi\left(i'; \tilde{f}(\hat{\mathbf{v}})\right) < \pi\left(i'; f^*(\hat{\mathbf{v}})\right)$: in this case we can observe that $ q_{i'} \hat{v}_{i'} + \frac{2\eta}{q_{\min}} q_{i'} \hat{v}_{i'} \geq q_j \hat{v}_j + \frac{2\eta}{q_{\min}} q_j \hat{v}_j$ $\forall j \in \mathcal{N}$ s.t. $\pi(j; f^*(\hat{\mathbf{v}})) < \pi(i'; f^*(\hat{\mathbf{v}}))$. Thus, considering that $\exists j \in \mathcal{N}$ s.t. $\pi(j; f^*(\hat{\mathbf{v}})) < \pi(i'; f^*(\hat{\mathbf{v}}))$ and $\pi(j; \tilde{f}(\hat{\mathbf{v}})) \geq l+1$, we can conclude $ \tilde{q}^+_{\alpha(l+1; \tilde{f}(\hat{\mathbf{v}}))} \hat{v}_{\alpha(l+1; \tilde{f}(\hat{\mathbf{v}}))} \leq \tilde{q}^+_j \hat{v}_j \leq q_j \hat{v}_j + \frac{2\eta}{q_{\min}} q_j \hat{v}_j \leq q_{i'} \hat{v}_{i'} + \frac{2\eta}{q_{\min}} q_{i'} \hat{v}_{i'} $.
\end{itemize}
Using these results we obtain
\begin{align*}
&\max\limits_{i\in \mathcal{N}}(\tilde{q}^+_i \hat{v}_i; l+1) = \tilde{q}^+_{\alpha(l+1; \tilde{f}(\hat{\mathbf{v}}))} \hat{v}_{\alpha(l+1; \tilde{f}(\hat{\mathbf{v}}))} \leq \\
&\leq q_{i'} \hat{v}_{i'} + \frac{2\eta}{q_{\min}} q_{i'} \hat{v}_{i'} = \max\limits_{i\in \mathcal{N}}(q_i \hat{v}_i; l+1) +\frac{1}{q_{\min}} 2\eta \max\limits_{i\in \mathcal{N}}(q_i \hat{v}_i; l+1)
\end{align*}
and thus
\begin{align*}
r_2&\leq v_{\max} \sum_{m = 1}^K \sum_{l = m}^{K} \Delta_l \left( \frac{ \max\limits_{i\in \mathcal{N}}(q_i \hat{v}_i; l+1) +\frac{1}{q_{\min}} 2\eta \max\limits_{i\in \mathcal{N}}(q_i \hat{v}_i; l+1) }{\max\limits_{i\in \mathcal{N}}(q_i \hat{v}_i; l+1)} -1 \right) \\
&\leq v_{\max} \sum_{m = 1}^K \sum_{l = m}^{K} \Delta_l \left( 1 +\frac{1}{q_{\min}} 2\eta -1 \right) \\
&\leq v_{\max} \frac{1}{q_{\min}} 2 \eta \sum_{m = 1}^K \underbrace{\sum_{l = m}^{K} \Delta_l}_{=\Lambda_m} \leq v_{\max} \frac{1}{q_{\min}} 2 \eta K.
\end{align*}
with probability $1-\delta$. As a result we have
\begin{align*}
\left|\sum_{m = 1}^K ( p^*_{\alpha(m;\theta^*)}(\hat{\mathbf{v}}) - \tilde{p}_{\alpha(m;\tilde{\theta})}(\hat{\mathbf{v}}) ) \right| \leq 2v_{\max} K \frac{\eta}{q_{\min}},
\end{align*}
with probability $1-\delta$. The final bound on the expected regret is thus
\begin{align}
\tilde{R}_T \leq K v_{\max} \left( \tau + \left( T-\tau \right) \frac{2\eta}{q_{\min}} + \delta T \right)
\end{align}
We first simplify the previous bound as
\begin{align*}
\tilde{R}_T & \leq K v_{\max} \left( \tau + \frac{2T}{q_{\min}} \sqrt{\left(\sum_{m=1}^{K} \frac{1}{\Lambda_{m}^2}\right) \frac{2N}{K^2 \tau} \log \frac{N}{\delta}} + \delta T \right)\\
& \leq K v_{\max} \left( \tau + \frac{2T}{q_{\min} \Lambda_{\min}} \sqrt{\frac{2N}{K \tau} \log \frac{N}{\delta}} + \delta T \right)
\end{align*}
and choosing the parameters
$$\tau = 2^\frac{1}{3} \frac{ K^{-\frac{1}{3}} N^\frac{1}{3} T^\frac{2}{3}}{\Lambda_{\min}^\frac{2}{3}} \left(\log{\frac{N}{\delta}}\right)^\frac{1}{3} $$
$$ \delta = N^\frac{1}{3} K^{-\frac{1}{3}} T^{-\frac{1}{3}}$$
the final bound is
$$ \tilde{R}_T \leq 4 \cdot 2^\frac{1}{3} \frac{ K^{-\frac{1}{3}} N^\frac{1}{3} T^\frac{2}{3}}{q_{\min} \Lambda_{\min}^\frac{2}{3}} \left(\log{N^\frac{2}{3} K^\frac{1}{3} T^\frac{1}{3}}\right)^\frac{1}{3}$$
\end{pf}
\myremark{(the bound).} We notice that the bound is very similar to the bound for the regret $R_T$ but now an inverse dependency on $q_{\min}$ appears. This suggests that bounding the deviation between the two mechanisms is more difficult than bounding the revenue loss and that as the qualities become smaller, the \avcg1\ could be less and less efficient and, thus, have a larger and larger revenue. This result has two important implications. \textit{(i)} If social welfare maximization is an important requirement in the design of the learning mechanism, we should analyze the loss of \avcg1\ in terms of social welfare and provide (probabilistic) guarantees about the number of rounds the learning mechanism need in order to be efficient (see~\cite{gonen2007incentive-compatible} for a similar analysis). \textit{(ii)} If social welfare is not a priority, this result implies that a learning mechanism could be preferable w.r.t. to a standard VCG mechanism. We believe that further theoretical analysis and experimental validation are needed to understand better both aspects.
\section{Learning with Position--Dependent Externalities}\label{s:constant}
In this section we study the multi--slot auctions with only position--dependent cascade model. The CTRs depend only on the quality of the ads and on the position of the slots in which the ads are allocated. Formally, parameters $\gamma_{m,i}$ are such that they coincide with the prominence parameter (i.e., $\gamma_{m,i}=\lambda_m$ for every $m$ and $i$). As a result, the cumulative probability of observation, defined in~(\ref{eq:coeff2}), reduces to
\begin{align} \label{eq:coeff}
\Lambda_m = \Gamma_m(\theta) = \left\{
\begin{array}{ll}
1 & \text{if } m=1 \\
\prod\limits_{l=1}^{m-1} \lambda_{l} & \text{if } 2 \leq m\leq K\\
0 & \text{otherwise}
\end{array} \right.,
\end{align}
where we use $\Lambda_m$ instead of $\Gamma_m(\theta)$ for consistency with most of the literature on position--dependent externalities and to stress the difference with respect to the general case.
When all the parameters are known by the auctioneer, the efficient allocation $\theta^*$ prescribes that the ads are allocated to the slots in decreasing order w.r.t. their expected reported value $q_i \hat{v}_i$. More precisely, for any $k\in \mathcal{K}'$, let $\max\limits_{i \in \mathcal{N}} (q_i \hat{v}_i; k)$ be the operator returning the $k$--th largest value in the set $\{q_1 \hat{v}_1, \ldots, q_N \hat{v}_N\}$, then $\theta^*$ is such that, for every $m\in \mathcal{K}'$, the ad displayed at slot $m$ is
\begin{align}\label{eq:pos.dep.efficient.alloc}
\alpha(m; \theta^*) \in \arg\max\limits_{i \in \mathcal{N}} (q_i \hat{v}_i; m).
\end{align}
This condition also simplifies the definition of the efficient allocation $\theta^*_{-i}$ when $a_i$ is removed from $\mathcal{N}$. In fact, for any $i,j\in \mathcal{N}$, if $\pi(j; \theta^*)<\pi(i; \theta^*)$ (i.e., ad $a_j$ is displayed before $a_i$) then $\pi(j; \theta^*_{-i}) = \pi(j;\theta^*)$, while if $\pi(j; \theta^*)>\pi(i; \theta^*)$ then $\pi(j; \theta^*_{-i}) = \pi(j; \theta^*)-1$ (i.e., ad $j$ is moved one slot upward), and $\pi(i; \theta^*_{-i}) = N$. By recalling the definition of VCG payments $p^*_i$ in (\ref{eq:pay.opt.vcg}), in case of position--dependent externalities we obtain the simplified formulation:
\begin{align}\label{eq:pay.opt.click.vcg.posdep.ad}
p^*_i(\hat{\mathbf{v}}) = \begin{cases}\sum\limits_{l=\pi(i; \theta^*)+1}^{K+1} \left[(\Lambda_{l-1} - \Lambda_l) \max\limits_{j\in \mathcal{N}}(q_j \hat{v}_j; l)\right] & \text{if } \pi(i; \theta^*)\leq K\\
0 & \text{otherwise}\end{cases},
\end{align}
which can be easily written as a per--slot payment as:
\begin{align}\label{eq:pay.slot.vgc.posdep.slot}
p^*_{\alpha(m; \theta^*)}(\hat v) = \begin{cases}\sum\limits_{l=m+1}^{K+1} \left[(\Lambda_{l-1} - \Lambda_l) \max\limits_{i\in \mathcal{N}}(q_i \hat{v}_i; l)\right] & \text{if } m\leq K\\
0 & \text{otherwise}\end{cases}.
\end{align}
In the following sections we study the problem of designing incentive compatible mechanisms under different conditions of lack of information over the parameters $\{q_i\}_{i \in \mathcal{N}}$ and $\{\Lambda_m\}_{m \in \mathcal{K}}$. In particular, in Section~\ref{ssec:uq}, we assume that the actual values of $\{q_i\}_{i \in \mathcal{N}}$ are unknown by the auctioneer, while those of $\{\Lambda_m\}_{m \in \mathcal{K}}$ are known. In Section~\ref{ssec:ul}, we assume that the actual values of $\{\Lambda_m\}_{m \in \mathcal{K}}$ are unknown by the auctioneer, while those of $q_i$s are known. Finally, in Section~\ref{ssec:uql}, we assume that the actual values of both $\{q_i\}_{i \in \mathcal{N}}$ and $\{\Lambda_m\}_{m \in \mathcal{K}}$ are unknown.
\subsection{Unknown qualities $\{q_i\}_{i \in \mathcal{N}}$} \label{ssec:uq}
In this section we assume that the qualities of the ads ($\{q_i\}_{i \in \mathcal{N}}$) are unknown, while $\{\Lambda_m\}_{m \in \mathcal{K}}$ are known. We initially focus on DSIC mechanisms and subsequently we discuss about mechanisms IC in expectation.
As in~\cite{devanur2009price,babaioff2008characterizing}, we formalize the problem as a multi--armed bandit problem and we study the properties of a learning mechanism where the exploration and exploitation phases are separated, such that during the exploration phase, we estimate the values of $\{q_i\}_{i \in \mathcal{N}}$ and during the exploitation phase we use the estimated qualities $\{\tilde{q}_i\}_{i \in \mathcal{N}}$ to implement an IC mechanism.
The pseudo code of the algorithm A--VCG1 (Adaptive VCG1) is given in Fig.~\ref{f:alg}. The details of the algorithm follow.
\begin{figure}[t]
\bookbox{
\begin{algorithmic}
\STATE \textbf{Input:} Length of exploration phase $\tau$, confidence $\delta$, position--dependent parameters $\{\Lambda_m\}_{m \in \mathcal{K}}$
\STATE
\STATE \textit{Exploration phase}
\FOR{$t = 1,\ldots,\tau$}
\STATE Allocate ads according to (\ref{eq:pos.dep.efficient.alloc})
\STATE Ask for no payment
\STATE Observe the clicks $\{click_{\pi(i;\theta_t)}^i(t)\}_{i=1}^{N}$
\ENDFOR
\STATE Compute the estimated quality $\tilde{q}_i = \frac{1}{|B_i|}\sum_{t \in B_i} \frac{click_{\pi(i; \theta_t)}^i(t)}{\Lambda_{\pi(i; \theta_t)}}$
\STATE Compute $\tilde{q}^+_i = \tilde{q}_i + \eta$ where $\eta$ is given by (\ref{eq:eta})
\STATE
\STATE \textit{Exploitation phase}
\FOR{$t = \tau+1,\ldots, T$}
\STATE Allocate ads according to $\tilde{f}$ defined in~(\ref{eq:optimalallocationestimatedq})
\IF{Ad $a_i$ is clicked}
\STATE Ask for payment $\tilde p^c_i$ defined in (\ref{eq:hpay.const.ppc})
\ENDIF
\ENDFOR
\end{algorithmic}}
\caption{Pseudo--code for the A--VCG1 mechanism.}\label{f:alg}
\end{figure}
\paragraph*{\indent Exploration phase} The exploration phase takes $\tau\geq N/K$ rounds.\footnote{Notice that we need $\tau > N/K$ in order to guarantee that all the ads have at least one sample to initialize the estimates $\tilde{q}_i$.} During this phase, the algorithm receives as input the parameters $\{\Lambda_m\}_{m \in \mathcal{K}}$ and collects data to estimate the quality of each ad. Unlike the single--slot case, where we collect only one sample of click or no--click events per round, here we can exploit the fact that each ad $a_i$ has a non--zero CTR whenever it is allocated to a slot $s_m$ with $m\leq K$. As a result, at each round of the exploration phase, we collect $K$ samples (click or no--click events), one from each slot. Let $\theta_t$ (for $t \leq \tau$) be a sequence of (potentially arbitrary) allocations independent from the advertisers' bids. The set $B_i = \{t: \pi(i; \theta_t) \leq K, t\leq\tau\}$ contains all the time instants when ad $a_i$ is allocated to a valid slot, so that $|B_i|$ corresponds to the total number of (click/no--click) samples available for ad $a_i$. We denote by $click_{\pi(i; \theta_t)}^i(t)\in\{0,1\}$ the click event at time $t$ for ad $a_i$ when displayed at slot $\pi(i; \theta_t)$. Depending on the slot in which the click event happens, the ad $a_i$ has different CTRs, thus we weigh each click sample by the probability of observation $\Lambda_m$ related to the slot in which the ad was allocated. The estimated quality $\tilde{q}_i$ is computed as
\begin{align}\label{eq:est.q}
\tilde{q}_i = \frac{1}{|B_i|}\sum_{t \in B_i} \frac{click_{\pi(i; \theta_t)}^i(t)}{\Lambda_{\pi(i; \theta_t)}},
\end{align}
which is an unbiased estimate of $q_i$ (i.e., $\mathbb{E}_{click} [\tilde{q}_i] = q_i$, where $\mathbb{E}_{click}$ is the expectation w.r.t. the realization of the clicks). By applying the Hoeffding's inequality~\cite{hoeffding1963probability}, we obtain a bound over the error of the estimated quality $\tilde{q}_i$ for each ad $i$.
\begin{proposition}\label{p:hoeffding}
For any ad $i \in \mathcal{N}$
\begin{align}\label{eq:hoeffproposition1}
| q_i - \tilde{q}_i | \leq \sqrt{\Bigg(\sum_{t \in B_i} \frac{1}{\Lambda_{\pi(i; \theta_t)}^2}\Bigg) \frac{1}{2 |B_i|^2} \log \frac{2N}{\delta}},
\end{align}
with probability $1-\delta$ (w.r.t. the click events).
\end{proposition}
During the exploration phase, at each round $t=1,\ldots,\tau$, we adopt the following sequence of allocations
\begin{equation}\label{eq:explorativeallocations}
\theta_t=\{\langle s_1, a_{(t \text{ mod } N) + 1} \rangle, \ldots, \langle s_{N}, a_{(t+N-1 \text{ mod } N) + 1} \rangle\},
\end{equation}
obtaining $|B_i| = \lfloor K \tau / N \rfloor$ for all the ads $a_i$. Thus, given that $\lfloor K \tau / N \rfloor \geq \frac{\tau K}{2N}$, Equation~(\ref{eq:hoeffproposition1}) becomes
\begin{align}\label{eq:eta}
| q_i - \tilde{q}_i | \leq \sqrt{\Bigg(\sum_{m=1}^{K} \frac{1}{\Lambda_{m}^2}\Bigg) \frac{2 N}{K^2 \tau} \log \frac{2 N}{\delta}} =: \eta.
\end{align}
During this phase, in order to guarantee DSIC, the advertisers cannot be charged with any payment, i.e. all the payments in rounds $t\leq \tau$ are set to 0. In fact, as shown in~\cite{babaioff2008characterizing}, any bid--dependent payment could be easily manipulated by bidders with better estimates of the CTRs, thus obtaining a non--truthful mechanism, whereas non--bid--dependent payments could make the mechanism not to be IR and thus bidders may prefer not to participate to the mechanism.
\paragraph*{\indent Exploitation phase} Once the exploration phase is concluded, an upper--confidence bound over each quality is computed as $\tilde{q}^+_i = \tilde{q}_i + \eta$ and the exploration phase is started and run for the remaining $T-\tau$ rounds. We define the \emph{estimated social welfare} as:
\begin{align*}
\widetilde{\text{SW}}(\theta,\hat{\mathbf{v}})= \sum_{i=1}^N \Lambda_{\pi(i;\theta)} \tilde{q}^+_i \hat{v}_i
\end{align*}
and we define $\tilde{f}$ as the allocation function that displays ads in decreasing order of $\tilde{q}^+_i \hat{v}_i$. $\tilde{f}$ returns the efficient allocation $\tilde{\theta}$ on the basis of the estimated qualities as
\begin{align}\label{eq:optimalallocationestimatedq}
\tilde{\theta}=\tilde{f}(\hat{\mathbf{v}}) \in \arg~\max\limits_{\theta \in \Theta}~ \{\widetilde{\text{SW}}(\theta,\hat{\mathbf{v}})\}
\end{align}
Our mechanism adopts $\tilde{f}$ during all the steps of the exploitation phase. Notice that $\tilde{f}$ is an affine maximizer, given that
\[
\tilde{f}(\hat{\mathbf{v}}) \in \arg\max\limits_{\theta \in \Theta} \sum_{i=1}^N \Lambda_{\pi(i; \theta)} \tilde{q}^+_i \hat{v}_i = \arg\max\limits_{\theta \in \Theta} \sum_{i=1}^N \frac{\tilde{q}^+_i }{q_i} \Lambda_{\pi(i; \theta)} q_i \hat{v}_i = \arg\max\limits_{\theta \in \Theta} \sum_{i=1}^N w_i \Lambda_{\pi(i; \theta)} q_i \hat{v}_i
\]
where each weight $w_i= \frac{\tilde{q}^+_i }{q_i} $ is independent of the advertisers' types $v_i$. Hence, we can apply the WVCG (weighted--VCG) payments (here denoted by $\tilde{p}$ because based on estimated parameters) satisfying the DSIC property. In particular, for any $i$, such that $\pi(i; \tilde\theta) \leq K$, we define the payment
\begin{align}\label{eq:hpay.const}
\tilde p_i (\hat{\mathbf{v}}) &= \frac{1}{w_i} \sum_{l=\pi(i; \tilde{\theta})+1}^{K+1} (\Lambda_{l-1} - \Lambda_l) \max\limits_{j \in \mathcal{N}}(\tilde{q}^+_j \hat{v}_j; l) \nonumber\\
&=\frac{q_i}{\tilde{q}^+_i} \sum_{l=\pi(i;\tilde{\theta})+1}^{K+1} (\Lambda_{l-1} - \Lambda_l) \max\limits_{j \in \mathcal{N}}(\tilde{q}^+_j \hat{v}_j; l).
\end{align}
These payments cannot be computed by the auctioneer, since the actual $\{q_i\}_{i \in \mathcal{N}}$ are unknown. However, we can resort to the \emph{pay--per--click} payments
\begin{align}\label{eq:hpay.const.ppc}
\tilde p^c_i (\hat{\mathbf{v}},click^i_{\pi(i; \tilde{\theta})}) = \frac{1}{\Lambda_{\pi(i; \tilde{\theta})}\tilde{q}^+_i}\bigg(\sum\limits_{l=\pi(i; \tilde{\theta})+1}^{K+1} (\Lambda_{l-1} - \Lambda_l) \max\limits_{j \in \mathcal{N}}(\tilde{q}^+_j \hat{v}_j; l)\bigg)\mathbb{I}\{click^i_{\pi(i; \tilde{\theta})}\}.
\end{align}
which in expectation coincide with the WVCG payments $\tilde{p}_i (\hat{\mathbf{v}}) = \mathbb{E}[\tilde p^c_i(\hat{\mathbf{v}},click^i_{\pi(i; \tilde{\theta})})]$. Unlike the payments $\tilde{p}_i (\hat{\mathbf{v}})$, these payments can be computed simply relying on the estimates $\tilde{q}^+_i$ and on the knowledge of the probabilities $\Lambda_m$.
We can state the following.
\begin{proposition}
The A--VCG1 is DSIC, IR \emph{a posteriori}, and WBB \emph{a posteriori}.
\end{proposition}
\begin{pf}
It trivially follows from the fact that the mechanism is a WVCG mechanism and that the payments are pay--per--click.\qed
\end{pf}
We now move to the analysis of the performance of A--VCG1 in terms of regret the mechanism cumulates through $T$ rounds.
\begin{theorem}\label{thm:constant}
Let us consider a sequential auction with $N$ advertisers, $K$ slots, and $T$ rounds with position--dependent cascade model with parameters $\{\Lambda_m\}_{m=1}^K$ and accuracy $\eta$ as defined in~(\ref{eq:eta}). For any parameter $\tau \in \{0, \ldots, T\}$ and $\delta \in [0,1]$, the A--VCG1 achieves a regret:
\begin{align}\label{eq:regret.const.exact}
R_T &\leq v_{\max} \left( \sum_{m = 1}^K \Lambda_m \right)\Big( 2(T - \tau) \eta + \tau + \delta T \Big).
\end{align}
\noindent By setting the parameters to
\begin{align*}
\delta &= K^{-\frac{1}{3}} T^{-\frac{1}{3}} N^{\frac{1}{3}}\\
\tau &= 2^{\frac{1}{3}} K^{-\frac{1}{3}} T^{\frac{2}{3}} N^{\frac{1}{3}} \Lambda_{\min}^{-\frac{2}{3}} \left[ \log \left( K^\frac{1}{3} T^\frac{1}{3} N^\frac{2}{3} \right) \right]^{\frac{1}{3}},
\end{align*}
where $\displaystyle \Lambda_{\min} = \min_{m \in \mathcal{K}} \Lambda_m, \ \Lambda_{\min} > 0$, then the regret is
\begin{align}\label{eq:regret.const}
R_T \leq 4 \cdot 2^\frac{1}{3} v_{\max} \Lambda_{\min}^{-\frac{2}{3}} K^\frac{2}{3} T^\frac{2}{3} N^\frac{1}{3} \left[ \log \big( K^\frac{1}{3} T^\frac{1}{3} N^\frac{2}{3} \big)\right]^{\frac{1}{3}}
\end{align}
\end{theorem}
We initially introduce some remarks about the above results, and subsequently discuss the sketch of the proof of the theorem.
\myremark{1 (The bound).} Up to numerical constants and logarithmic factors, the previous bound~(\ref{eq:regret.const}) is $R_T \leq \tilde O(T^\frac{2}{3} K^\frac{2}{3} N^\frac{1}{3})$.
We first notice that A--VCG1 is a no--regret algorithm since its per--round regret ($R_T/T$) decreases to 0 as $T^{-\frac{1}{3}}$, thus implying that it asymptotically achieves the same performance as the VCG. Furthermore, we notice that for $K=1$ the bound reduces (up to constants) to the single--slot case analyzed in~\cite{devanur2009price}. Unlike the standard bound for multi--armed bandit algorithms, the regret scales as $\tilde O(T^\frac{2}{3})$ instead of $\tilde O(T^\frac{1}{2})$. As pointed out in~\cite{devanur2009price} and \cite{babaioff2008characterizing} this is the unavoidable price the bandit algorithm has to pay to be DSIC.
Finally, the dependence of the regret on $N$ is sub--linear ($N^\frac{1}{3}$) and therefore an increase of the number of advertisers does not significantly worsen the regret. The dependency on the number of slots $K$ is similar: according to the bound (\ref{eq:regret.const}) the regret has a sublinear dependency $\tilde O(K^\frac{2}{3})$, meaning that whenever one slot is added to the auction, the performance of the algorithm does not significantly worsen. By analyzing the difference between the payments of the VCG and A--VCG1, we notice that during the exploration phase the regret is $O(\tau K)$ (e.g., if all the ads allocated into the $K$ slots are clicked at each explorative round), while during the exploitation phase the error in estimating the qualities sum over all the $K$ slots, thus suggesting a linear dependency on $K$ for this phase as well. Nonetheless, as $K$ increases, the number of samples available per ad increases as $\tau K/N$, thus improving the accuracy of the quality estimates by $\tilde O(K^{-\frac{1}{2}})$ (see Proposition~\ref{p:hoeffding}). As a result, as $K$ increases, the exploration phase can be shortened (the optimal $\tau$ actually decreases as $K^{-\frac{1}{3}}$), thus reducing the regret during the exploration, and still have accurate enough estimations to control the regret of the exploitation phase.
\myremark{2 (Distribution--free bound).} The bound derived in Theorem~\ref{thm:constant} is a \textit{distribution--free} (or worst--case) bound, since it holds for any set of advertisers (i.e., for any $\{q_i\}_{i\in\mathcal{N}}$ and $\{v_i\}_{i\in\mathcal{N}}$). This generality comes at the price that, as illustrated in other remarks and in the numerical simulations (see Section~\ref{s:experiments}), the bound could be inaccurate for some specific sets of advertisers. On the other hand, distribution--dependent bounds (see e.g., the bounds of UCB~\cite{auer2002finite-time}), where $q$ and $v$ appear explicitly, would be more accurate in predicting the behavior of the algorithm. Nonetheless, they could not be used to optimize the parameters $\delta$ and $\tau$, since they would then depend on unknown quantities.
\myremark{3 (Parameters).} The choice of parameters $\tau$ and $\delta$ reported in Theorem~\ref{thm:constant} is obtained by rough minimizing the upper--bound (\ref{eq:regret.const.exact}). Each parameter can be computed by knowing the characteristics of the auction (number of rounds $T$, number of slots $K$, number of ads $N$, and $\Lambda_m$). Moreover, since the values are obtained optimizing an upper--bound of the regret and not directly the true global regret, these values can provide a good guess for the parametrization, but there could be other values that better optimize the regret. Thus, in practice, the regret could be optimized by searching the space of the parameters around the values suggested in Theorem~\ref{thm:constant}.
\myremark{4 (IC in expectation).} Two interesting problems we do not solve in this paper once IC in expectation (over the click realizations and/or realizations of the random component of the mechanism) is adopted are whether or not it is possible to avoid the separation of the exploration and exploitation phases and whether it is possible to obtain a regret of $O(T^{1/2})$ as it is possible in the case of $K=1$~\cite{babaioff_impl_pay}. Any attempt we tried to extend the result presented in~\cite{babaioff_impl_pay} to the multi--slot case conducted us to a non--IC mechanism. We briefly provide some examples of adaptation to our framework of the two MAB presented~\cite{babaioff_impl_pay}. None of these attempts provided a monotone allocation function. We have tried to extend the UCB1 in different ways, e.g. introducing $N \cdot K$ estimators, one for each ad for each slot, or maintaining $N$ estimators weighting in different ways click obtained in different slots. The second MAB algorithm, called NewCB, is based on the definition of a set of active ads, the ones that can be displayed. We have considered extensions with a single set for all the slots and with multiple sets, one for each slot, without identifying monotone allocation algorithms.
\textit{(Comments to the proof).} The proof uses relatively standard arguments to bound the regret of the exploitation phase. As discussed in Remark 2, the bound is distribution--free and some steps in the proof are conservative upper--bounds on quantities that might be smaller for specific auctions. For instance, the inverse dependency on the smallest cumulative discount factor $\Lambda_{\min}$ in the final bound could be a quite inaccurate upper--bound on the quantity $\sum_{m=1}^{K} 1/ \Lambda_{m}^2$. In fact, the parameter $\tau$ itself could be optimized as a direct function of $\sum_{m=1}^{K} 1 /\Lambda_{m}^2$, thus obtaining a more accurate tuning of the length of the exploration phase and a slightly tighter bound (in terms of constant terms). Furthermore, we notice that the step $\max\limits_{i \in \mathcal{N}}(\tilde{q}^+_{i} v_i;h) / \max\limits_{i \in \mathcal{N}}(\tilde{q}^+_{i} v_i;m) \leq 1$ is likely to become less accurate as the difference between $h$ and $m$ increases (see Eq.~\ref{eq:step.loose} in the proof). For instance, if the qualities $q_i$ are drawn from a uniform distribution in $(0,1)$, as the number of slots increases this quantity reduces as well (on average) thus making the upper--bound by $1$ less and less accurate. The accuracy of the proof and the corresponding bound are further studied in the simulations in Section~\ref{s:experiments}.
In a similar way, adopting the same mechanism as before, it is also possible to derive an upper--bound over the global regret, when the regret, as in~\cite{babaioff_impl_pay} is computed over the social welfare of the allocation. In particular we obtain, that, even in this case, A--VCG1 is a no--regret algorithm and $R^{SW}_T\leq\tilde{O}(T^\frac{2}{3})$.
\begin{theorem} \label{th:pd_q_sw}
Let us consider a sequential auction with $N$ advertisers, $K$ slots, and $T$ rounds with position--dependent cascade model with parameters $\{\Lambda_m\}_{m=1}^K$ and $\eta$ as defined in~(\ref{eq:eta}). For any parameter $\tau \in \{0, \ldots, T\}$ and $\delta \in [0,1]$, the A--VCG1 achieves a regret:
\begin{align}
R^{SW}_T &\leq v_{\max} K \left( 2 \left(T - \tau \right) \eta + \tau + \delta T \right).
\end{align}
\noindent By setting the parameters to
\begin{align*}
\delta &= \left( \frac{\sqrt{2}}{\Lambda_{\min}} \right)^\frac{2}{3} K^{-\frac{1}{3}} N^\frac{1}{3} T^{-\frac{1}{3}} \\
\tau &= \left( \frac{\sqrt{2}}{\Lambda_{\min}} \right)^\frac{2}{3} T^\frac{2}{3} N^\frac{1}{3} K^{-\frac{1}{3}} \left( \log 2^\frac{2}{3} \Lambda_{\min}^\frac{2}{3} N^\frac{2}{3} K^\frac{1}{3} T^\frac{1}{3} \right)^\frac{1}{3},
\end{align*}
where $\displaystyle \Lambda_{\min} = \min_{m \in \mathcal{K}} \Lambda_m, \ \Lambda_{\min} > 0$, then the regret is
\begin{align}
R_T^{SW} \leq 4 v_{\max} \left( \frac{\sqrt{2}}{\Lambda_{\min}} \right)^\frac{2}{3} K^\frac{2}{3} N^\frac{1}{3} T^\frac{2}{3} \left( \log 2^\frac{2}{3} \Lambda_{\min}^\frac{2}{3} N^\frac{2}{3} K^\frac{1}{3} T^\frac{1}{3} \right)^\frac{1}{3}
\end{align}
\end{theorem}
Notice that using $\tau$ and $\delta$ defined in Theorem~\ref{thm:constant}, the bound for $R_T^{SW}$ is $\tilde{O}(T^\frac{2}{3})$, even if the parameters are not optimal for this second framework.
\subsection{Unknown $\{\Lambda_m\}_{m \in \mathcal{K}}$} \label{ssec:ul}
We now focus on the situation when the auctioneer knows $\{q_i\}_{i \in \mathcal{N}}$, while $\{\Lambda_m\}_{m \in \mathcal{K}}$ are unknown. By definition of cascade model, $\{\Lambda_m\}_{m \in \mathcal{K}}$ are strictly non--increasing in $m$. This dramatically simplifies the allocation problem since the optimal allocation can be found without knowing the actual values of $\{\Lambda_m\}_{m \in \mathcal{K}}$. Indeed, allocation $\theta^*$ such that $\alpha(m; \theta^*) \in \arg\max\limits_{i \in \mathcal{N}} (q_i \hat{v}_i; m)$ is optimal for all possible $\{\Lambda_m\}_{m \in \mathcal{K}}$. However, the lack of knowledge about $\{\Lambda_m\}_{m \in \mathcal{K}}$ makes the design of a truthful mechanism not straightforward because they appear in the calculation of the payments. Differently from what we presented in the previous section, here we initially focus on IC in expectation mechanisms, providing two mechanisms (the first is IC in expectation over the click realizations and the second is IC in expectation over the realizations of the random component of the mechanism), and subsequently we produce some considerations about DSIC mechanisms.
\subsubsection{IC in expectation over the click realizations mechanism}
\begin{figure}[t]
\bookbox{
\begin{algorithmic}
\STATE \textbf{Input:} Qualities parameters $\{q_i\}_{i \in \mathcal{N}}$
\STATE
\FOR{$t = 1,\ldots, T$}
\STATE Allocate ads according to $f^*$ as prescribed by~(\ref{eq:pos.dep.efficient.alloc})
\IF{Ad $a_i$ is clicked}
\STATE Ask for payment $p^c_i$ defined in (\ref{eq:olppc})
\ENDIF
\ENDFOR
\end{algorithmic}}
\caption{Pseudo--code for the A--VCG2 mechanism.}\label{f:alg2}
\end{figure}
In this case, we do not need any estimation of the parameters $\{\Lambda_m\}_{m \in \mathcal{K}}$ and therefore we do not resort to the multi--armed bandit framework and the mechanism does not present separate phases. The pseudo code of the algorithm A--VCG2 (Adaptive VCG2) is given in Fig.~\ref{f:alg2}. On the basis of the above considerations, we can adopt the allocatively efficient allocation function $f^*$ as prescribed by~(\ref{eq:pos.dep.efficient.alloc}) even if the mechanism does not know the actual values of the parameters $\{\Lambda_m\}_{m \in \mathcal{K}}$. Nonetheless, the VCG payments defined in (\ref{eq:pay.opt.click.vcg.posdep.ad}) cannot be computed, since $\{\Lambda_m\}_{m \in \mathcal{K}}$ not being known by the mechanism. However, by resorting to execution--contingent payments (generalizing the pay--per--click approach\footnote{In pay--per--click payments, an advertiser pays only once its ad is clicked; in our execution--contingent payments, an advertiser pays also once the ads of other advertisers are clicked.}), we can impose computable payments that, in expectation, are equal to (\ref{eq:pay.opt.click.vcg.posdep.ad}). More precisely, the contingent payments are computed given the bids $\hat{\mathbf{v}}$ and all click events over the slots and take the form:
\begin{align} \label{eq:olppc}
p_{i}^c&(\hat{\mathbf{v}},\{click_{\pi(j; \theta^*)}^j\}_{j=1}^K) \\
&=\sum\limits_{\pi(i;\theta^*) \leq m \leq K } click_m^{\alpha(m;\theta^*)} \cdot \frac{q_{\alpha(m;\theta^*_{-i})} \cdot \hat{v}_{\alpha(m;\theta^*_{-i})}}{q_{\alpha(m;\theta^*)}} \nonumber\\
&\quad\quad\quad- \sum\limits_{\pi(i;\theta^*) < m \leq K } click_m^{\alpha(m;\theta^*)} \cdot \hat{v}_{\alpha(m;\theta^*)}\nonumber
\end{align}
Notice that the payment $p_{i}^c$ depends not only on the click of ad $a_i$, but also on the clicks of all the ads displayed in the slots below. In expectation, the two terms of $p_i^c$ are:
\begin{align*}
\mathbb{E}_{click}\left[\sum\limits_{\pi(i;\theta^*) \leq m \leq K } click_m^{\alpha(m;\theta^*)} \cdot \frac{q_{\alpha(m;\theta^*_{-i})} \cdot \hat{v}_{\alpha(m;\theta^*_{-i})}}{q_{\alpha(m;\theta^*)}}\right] & = \sum_{\pi(j;\theta^*) \geq \pi(i;\theta^*)} \Lambda_{\pi(j;\theta^*_{-i})} q_j \hat{v}_j \\
\mathbb{E}_{click}\left[\sum\limits_{\pi(i;\theta^*) < m \leq K } click_m^{\alpha(m;\theta^*)} \cdot \hat{v}_{\alpha(m;\theta^*)}\right] & = \sum_{\pi(j;\theta^*) > \pi(i;\theta^*)} \Lambda_{\pi(j;\theta^*)} q_j \hat{v}_j
\end{align*}
and therefore, in expectation, the payment equals to (\ref{eq:pay.opt.click.vcg.posdep.ad}). Thus, we can state the following.
\begin{proposition}
The A--VCG2 is IC, IR, WBB in expectation (over click realizations) and AE.
\end{proposition}
\begin{pf}
It trivially follows from the fact that the allocation function is AE and the payments in expectation equal the VCG payments. \qed
\end{pf}
We discuss further properties of the mechanism in what follows.
\begin{proposition}
The A--VCG2 is not DSIC \emph{a posteriori} (w.r.t. click realizations).
\end{proposition}
\begin{pf}
The proof is by counterexample. Consider an environment with 3 ads $\mathcal{N}=\{a_1, a_2, a_3\}$ and 2 slots $S=\{s_1,s_2\}$ s.t. $q_1=0.5$, $v_1=4$, $q_2=1$, $v_2=1$, $q_3=1$, $v_3=0.5$, which correspond to expected values of $2$, $1$, and $0.5$.
The optimal allocation $\theta^*$ consists in allocating $a_1$ in $s_1$ and $a_2$ in $s_2$. Consider a time $t$ when both ad $a_1$ and $a_2$ are clicked, from Eq.~\ref{eq:olppc} we have that the payment of $a_2$ is:
\[
p_{2}^c = \frac{1}{q_2}q_3v_3 = 0.5
\]
If ad $a_2$ reports a value $\hat{v}_2=3$, the optimal allocation is now $a_2$ in $s_1$ e $a_1$ in $s_2$. In the case both $a_1$ and $a_2$ are clicked, the payment of $a_2$ is:
\[
p_{2}^c = \frac{1}{q_2} q_1 v_1 + \frac{1}{q_1} q_3 v_3 - v_1 = 2 + 1 - 4 = -1
\]
Given that, in both cases, the utility is $u_2 = v_2 - p_{2}^c$, reporting a non--truthful value is optimal. Thus, we can conclude that the mechanism is not DSIC.
\end{pf}
\begin{proposition}
The A--VCG2 is IR \emph{a posteriori} (w.r.t. click realizations). \label{prop:AVGC2IRaposteriori}
\end{proposition}
\begin{pf}
Rename the ads $\{a_1, \ldots, a_N\}$ such that $q_1 v_1 \geq q_2 v_2 \geq \ldots \geq q_N v_N$. We can write payments~(\ref{eq:olppc}) as:
\[
\tilde p_{i}^c = \sum_{j=i}^K \frac{click_{j}^j}{q_j} q_{j+1} v_{j+1} - \sum_{j=i+1}^K click_{j}^j v_j
\]
Thus, the utility for advertiser $a_i$ is:
\begin{align*}
u_i &= click_{j}^j v_i + \sum_{j=i+1}^K click_{j}^j v_j - \sum_{j=i}^K \frac{click_{j}^j}{q_j} q_{j+1} v_{j+1}\\
&= \sum_{j=i}^K click_{j}^j v_j - \sum_{j=i}^K \frac{click_{j}^j}{q_j} q_{j+1} v_{j+1}\\
&= \sum_{j=i}^K \left( click_{j}^j v_j - \frac{click_{j}^j}{q_j} q_{j+1} v_{j+1} \right) \\
&= \sum_{j=i}^K click_{j}^j v_j - \frac{click_{j}^j}{q_j} q_{j+1} v_{j+1} \\
&= \sum_{j=i}^K \frac{click_{j}^j}{q_j} ( q_j v_j - q_{j+1} v_{j+1}).
\end{align*}
Since $\frac{click_{j}^j}{q_j} \geq 0$ by definition and $q_j v_j - q_{j+1} v_{j+1} \geq 0$ because of the chosen ordering of the ads, then the utility is always positive and we can conclude the mechanism is IR \emph{a posteriori}. \qed
\end{pf}
\begin{proposition}
The A--VCG2 is not WBB \emph{a posteriori} (w.r.t. click realizations).
\end{proposition}
\begin{pf}
The proof is by counterexample. Consider an environment with 3 ads $\mathcal{N}=\{a_1, a_2, a_3\}$ and 2 slots $S=\{s_1,s_2\}$ s.t. $q_1=1$, $v_1=2$, $q_2=0.5$, $v_2=1$, $q_3=1$, $v_3=\epsilon$, where $\epsilon > 0$ is a small number.
The optimal allocation $\theta^*$ consists in allocating $a_1$ in $s_1$ e $a_2$ in $s_2$. Consider a time instant $t$ when both ad $a_1$ and $a_2$ are clicked, their payments are:
\[
p_{1}^c = \frac{1}{q_1}q_2v_2 + \frac{1}{q_2} q_3 v_3 - v_2 = 0.5 + 2 \epsilon - 1 = 2 \epsilon - 0.5 < 0
\]
\[
p_{2}^c = \frac{1}{q_2} q_3 v_3 = 2 \epsilon
\]
Thus, $\sum_{i=1}^3 p_{i}^c = 4 \epsilon - 0.5 < 0$, and we can conclude that the mechanism is not WBB \emph{a posteriori}.\qed
\end{pf}
Now we state the following theorem, whose proof is straightforward.
\begin{theorem}\label{thm:constant.l}
Let us consider an auction with $N$ advertisers, $K$ slots, and $T$ rounds, with position--dependent cascade model with parameters $\{\Lambda_m\}_{m=1}^K$. The A--VCG2 achieves an expected regret $R_T=0$.
\end{theorem}
An important property of this mechanism is that the expected payments are exactly the VCG payments for the optimal allocation when all the parameters are known. Moreover, the absence of an exploration phase allows us to obtain an instantaneous expected regret of zero and, thus, the cumulative regret over the $T$ rounds of auction $R_T=0$. Similar considerations can be applied to the study of the regret over the social welfare, obtaining the following.
\begin{corollary}
The A--VCG2 has an expected regret over the social welfare of zero.
\end{corollary}
\subsubsection{IC in expectation over random component realizations mechanism} \label{sssec:l.uc.m}
\begin{figure}[t]
\bookbox{
\begin{algorithmic}
\STATE \textbf{Input:} Length of exploration phase $\tau$, confidence $\delta$
\STATE
\STATE \textit{Exploitation phase}
\FOR{$t = \tau+1,\ldots, T$}
\STATE Allocate ads according to $f^{*'}$ as prescribed by Algorithm~1
\IF{Ad $a_i$ is clicked}
\STATE Ask for payment $p^{B,*,x}_i$ defined in (\ref{eq:pay.opt.babaioff.click})
\ENDIF
\ENDFOR
\end{algorithmic}}
\caption{Pseudo--code for the A--VCG2$^\prime$ mechanism.}\label{f:alg22}
\end{figure}
As for the previous mechanism, here we have only the exploitation phase. Differently from the previous mechanism, the mechanism has a random component as proposed in~\cite{babaioff_impl_pay}. The mechanism, called A--VCG2$^\prime$ is reported in Fig.~\ref{f:alg22}. It is obtained applying the approach described in~\cite{babaioff_impl_pay} to allocation function $f^*$.
Since $f^*$ is monotonic~(see~\ref{ap:monotonicity}) and the problem is with single parameter and linear utilities, payments assuring DSIC can be written as~\cite{tardos_sp}:
\begin{equation} \label{eq:pay.vcg.emp.tardos}
p_i^*(\hat{\mathbf{v}}) = \Lambda_{\pi(i;f^*(\hat{\mathbf{v}}))} q_i \hat{v}_i - \int_{0}^{\hat{v}_i} \Lambda_{\pi(i;f^*(\hat{\mathbf{v}}_{-i},u))} q_i du,
\end{equation}
which coincide with the VCG payments defined in~\ref{eq:pay.opt.vcg} (hence the use of the same notation $p_i^*$). This is justified by the fact that when a mechanism is AE, IR and WBB the only payments that lead to a DSIC mechanism are the VCG payments with Clacke's pivot~\cite{greenLaffont}, thus (\ref{eq:pay.vcg.emp.tardos}) must coincide
However, these payments are not directly computable, because parameters $\{\Lambda_m\}_{m \in \mathcal{K}}$ in the integral are unknown (and, as in the case discussed in Section~\ref{ssec:ul}.1, we cannot replace them by empirical estimates). We could obtain these payments in expectation by using execution--contingent payments associated with non--optimal allocations where the report $\hat{v}_i$ is modified between 0 and the actual value. This can be obtained by resorting to the approach proposed in~\cite{babaioff_impl_pay}. More precisely, the approach proposed in~\cite{babaioff_impl_pay} takes in input a generic allocation function $f$ and introduces a randomized component into it, producing a new allocation function that we denote by $f'$. This technique, at the cost of reducing the efficiency of $f$, allows the computation of the allocation and the payments at the same time even when payments described in~\cite{tardos_sp} cannot be computed directly.
We apply the approach proposed in~\cite{babaioff_impl_pay} to our $f^*$ obtaining a new allocation function $f^{*'}$. With $f^{*'}$, the advertisers' reported values $\{\hat{v}_i\}_{i \in \mathcal{N}}$ are modified, each with a (small) probability $\mu$. The (potentially) modified values are then used to compute the allocation (using $f^*$) and the payments. More precisely, with a probability of $(1-\mu)^N$, $f^{*'}$ returns the same allocation $f^*$ would return, while it does not with a probability of $1 - (1-\mu)^N$. The reported values $\{\hat{v}_i\}_{i \in \mathcal{N}}$ are modified through the \emph{canonical self--resampling procedure} (cSRP) described in~\cite{babaioff_impl_pay} that generates two samples: $x_i(\hat{v}_i,\omega_i)$ and $y_i(\hat{v}_i,\omega_i)$, where $\omega_i$ is the random seed. We sketch the result of cSRP where the function `rec' is defined in~\cite{babaioff_impl_pay}:
\begin{align*}
(x_i,y_i) = cSRP(\hat{v}_i)=\begin{cases} (\hat{v}_i,\hat{v}_i) & \mbox{w.p. } 1-\mu \\ (\hat{v}''_i,\hat{v}'_i) & \mbox{otherwise }\end{cases},
\end{align*}
where $\hat{v}_i'\sim\mathcal{U}([0,\hat{v}_i])$ and $\hat{v}_i''=\text{rec}(\hat{v}_i')$.
\begin{algorithm}
\begin{algorithmic}[1]
\begin{scriptsize}
\FORALL {$a_i \in N$}
\STATE $(x_i, y_i)=cSRP(\hat{v}_i)$ \label{s:csrp}
\STATE $\mathbf{x}=(x_1,\ldots,x_N)$
\ENDFOR
\STATE $\theta = f^*(\mathbf{x})$ \label{s:alloc}
\end{scriptsize}
\end{algorithmic}
\caption{$f^{*'}(\hat{\mathbf{v}})$}
\label{alg:babalg}
\end{algorithm}
Algorithm~\ref{alg:babalg} shows how $f^{*'}$ works when the original allocation function is $f^*$. The reported values $\{\hat{v}_i\}_{i \in \mathcal{N}}$ are perturbed through the canonical self--resampling procedure~(Step~\ref{s:csrp}) and then it returns the allocation found by applying the original allocation function $f^*$ to the new values $\mathbf{x}$ (Step~\ref{s:alloc}).
Finally, the payments are computed as
\begin{multline} \label{eq:pay.opt.babaioff.click}
p^{B,*,c}_i(\mathbf{x}, click_{\pi(i; f^*(\mathbf{x}))}^{i}) = \begin{cases} \frac{p_i^{B,*}(\mathbf{x},\mathbf{y};\hat{\mathbf{v}})}{\Lambda_{\pi(i; f^*(\mathbf{x}))} q_i} & \mbox{if } click_{\pi(i; f^*(\mathbf{x}))}^{i}=1 \\ 0 & \mbox{otherwise} \end{cases} \\
= \begin{cases} \hat{v}_i -
\begin{cases} \frac{1}{\mu} \hat{v}_i &\mbox{if $y_i<\hat{v}_i$} \\
0 & \mbox{otherwise}, \end{cases}
& \mbox{if } click_{\pi(i; f^*(\mathbf{x}))}^{i}=1 \\ 0 & \mbox{otherwise} \end{cases}
\end{multline}
where
\begin{align}
p_i^{B,*}(\mathbf{x},\mathbf{y};\hat{\mathbf{v}}) = \Lambda_{\pi(i; f^*(\mathbf{x}))}q_i \hat{v}_i -
\begin{cases} \frac{1}{\mu}\Lambda_{\pi(i; f^*(\mathbf{x}))}q_i \hat{v}_i &\mbox{if $y_i<\hat{v}_i$} \\
0 & \mbox{otherwise}, \end{cases}
\end{align}
$\mathbf{y}=(y_1,\ldots,y_N)$ and the expected value of payments~(\ref{eq:pay.opt.babaioff.click}) w.r.t. the randomization of the mechanism are the payments~\cite{tardos_sp} for the randomized allocation function $f^{*'}$. The result presented in~\cite{babaioff_impl_pay} assures that the resulting mechanism is IC in expectation over the realizations of the random component and \emph{a posteriori} w.r.t. the click realizations.
We state the following results on the properties of the above mechanism.
\begin{theorem}\label{thm:constant.l.baba}
Let us consider an auction with $N$ advertisers, $K$ slots, and $T$ rounds, with position--dependent cascade model with parameters $\{\Lambda_m\}_{m=1}^K$. The A--VCG2 $^\prime$ achieves an expected regret $R_T \leq 2 K^2 \mu v_{\max} T$.
\end{theorem}
Adopting $\mu = \frac{1}{T^\alpha}$ with $\alpha>1$ then $R_T \rightarrow 0$, but, as we will show in Section~\ref{s:experiments}, the smaller $\mu$ the larger the variance of the payments. We provide a similar result for the regret over the social welfare.
\begin{theorem}\label{thm:constant.l.sw.baba}
Let us consider an auction with $N$ advertisers, $K$ slots, and $T$ rounds, with position--dependent cascade model with parameters $\{\Lambda_m\}_{m=1}^K$. The A--VCG2 $^\prime$ achieves an expected regret $R^{SW}_T \leq K^2 \mu v_{\max} T$.
\end{theorem}
\subsubsection{Considerations about DSIC mechanisms}
At the cost of worsening the regret, one may wonder whether there exists some no--regret DSIC mechanism. In what follows, resorting to the same arguments used in~\cite{sarma2010multi-armed}, we show that the answer to such question is negative.
\begin{theorem}\label{thm:constant.l}
Let us consider an auction with $N$ advertisers, $K$ slots, and $T$ rounds, with position--dependent cascade model with parameters $\{\Lambda_m\}_{m=1}^K$ whose value are unknown. Any online learning DSIC \textit{a posteriori} (w.r.t. click realizations) mechanism achieves an expected regret $R_T=\Theta(T)$.
\end{theorem}
\begin{pf}\textbf{(sketch)} Basically, the A--VCG2 mechanism is only IC in expectation (and not DSIC) because it adopts execution--contingent payments in which the payment of advertiser $a_i$ depends also on the clicks over ads different from $a_i$. The above payment technique---i.e., payments reported in~(\ref{eq:olppc})---is necessary to obtain in expectation the values $\text{SW}(\theta^*_{-i},\hat{\mathbf{v}}_{-i})$ and $\text{SW}_{-i}(\theta^*,\hat{\mathbf{v}})$, since parameters $\{\Lambda_m\}_{m \in \mathcal{K}}$ are not known. In order to have DSIC \textit{a posteriori} (i.e., truthful for any realization of the clicks), we need payments $p_i$ that are deterministic w.r.t. the clicks over other ads different from $a_i$ (i.e., pay--per--click payments are needed).
We notice that even if $\Lambda_m$ have been estimated (e.g., in an exploitation phase), we cannot have payments leading to DSIC. Indeed, with estimates $\tilde{\Lambda}_m$, the allocation function maximizing $\widetilde{SW}$ (computed with $\tilde{\Lambda}_m$) is not an affine maximizer and therefore the adoption of WVCG mechanism would not guarantee DSIC. As a result, only mechanisms with payments defined as in~\cite{tardos_sp} can be used. However, these payments, if computed exactly (and not estimated in expectation), require the knowledge about the actual $\Lambda_m$ related to each slot $s_m$ in which an ad can be allocated for each report $\hat{v}\leq v$.
To prove the theorem, we provide a characterization of DSIC mechanisms. Exactly, we need a monotonic allocation function and the payments defined in~\cite{tardos_sp}. These payments, as said above, require the knowledge about the actual $\Lambda_m$ related to the slot $s_m$ in which an ad can be allocated for each report $\hat{v}\leq v$. Thus we have two possibilities:
\begin{itemize}
\item In the first case, an ad can be allocated only in one slot and its report determines only whether it is displayed or not. That is, the ads are partitioned and each partition is associated with a slot and the ad with the largest expected valuation is chosen at each slot independently. This case is equivalent to multiple separate--single slot auctions and therefore each auction is DSIC as shown in~\cite{devanur2009price}. However, as shown in~\cite{sarma2010multi-armed}, this mechanism would have a regret $\Theta(T)$.
\item In the second case, an ad can be allocated in more than one slot on the basis of its report. In this case, to compute the payments, it would be necessary to know the exact CTRs of the ad for each possible slot, but this is possible only in expectation either by using the above execution--contingent as we do in Section~4.2.1 or by generating non--optimal allocation as we do in Section~4.2.2.
\end{itemize}
Thus, in order to have DSIC, we need to adopt the class of mechanisms described in the first case, obtaining $R_T=\Theta(T)$.\qed
\end{pf}
\subsection{Unknown $\{\Lambda_{m}\}_{m \in \mathcal{K}}$ and $\{q_i\}_{i \in \mathcal{N}}$} \label{ssec:uql}
In this section we study the situation in which both $\{q_i\}_{i \in \mathcal{N}}$ and $\{\Lambda_{m}\}_{m \in \mathcal{K}}$ are unknown. From the results discussed in the previous section, we know that adopting DSIC as solution concept we would obtain $R_T=\Theta(T)$. Thus, we focus only on IC in expectation.
First of all, we remark that the mechanisms presented in Sections~\ref{ssec:uq} and~\ref{ssec:ul} cannot be adopted here, but the study of a new mechanism is required. The mechanism we design is given by the combination of A--VCG1 and A--VCG2$^\prime$. The pseudo code of the algorithm A--VCG3 (Adaptive VCG3) is given in Fig.~\ref{f:alg3}. As in the case in which only $\{q_i\}_{i \in \mathcal{N}}$ are unknown, we formalize the problem as a multi--armed bandit where the exploration and exploitation phases are separate and where, during the exploration phase, we estimate the values of $\{q_i\}_{i \in \mathcal{N}}$. Details of the algorithm follow.
\begin{figure}[t]
\bookbox{
\begin{algorithmic}
\STATE \textbf{Input:} Length of exploration phase $\tau$, confidence $\delta$
\STATE
\STATE \textit{Exploration phase}
\FOR{$t = 1,\ldots,\tau$}
\STATE Allocate ads according to (\ref{eq:explorativeallocations})
\STATE Ask for no payment
\STATE Observe the clicks $\{click_{1}^i(t)\}_{i=1}^{N}$
\ENDFOR
\STATE Compute the estimated quality $\tilde{q}_i = \frac{1}{|B_i|}\sum_{t \in B_i} click_{1}^i(t)$
\STATE Compute $\tilde{q}^+_i = \tilde{q}_i + \eta$ where $\eta$ is given by (\ref{eq:hoeff})
\STATE
\STATE \textit{Exploitation phase}
\FOR{$t = \tau+1,\ldots, T$}
\STATE Allocate ads according to $\tilde{f}'$ as prescribed by Algorithm~1
\IF{Ad $a_i$ is clicked}
\STATE Ask for payment $\tilde{p}^{B,c}_i$ defined in (\ref{eq:pay.babaioff.ppc})
\ENDIF
\ENDFOR
\end{algorithmic}}
\caption{Pseudo--code for the A--VCG3 mechanism.}\label{f:alg3}
\end{figure}
\paragraph*{\indent Exploration phase} During the first $\tau$ rounds of the auction, estimates of $\{q_i\}_{i \in \mathcal{N}}$ are computed. We use the same exploration policy of Section~\ref{ssec:uq}, but the estimations are computed just using samples from the first slot, since $\Lambda_m$ with $m>1$ are unknown.\footnote{In the following, we report some considerations about the case in which also the samples from the slots below the first are considered.} Define $B_i = \{t: \pi(i; \theta_t) = 1, t\leq\tau\}$ the set of rounds $t\leq \tau$ where $a_i$ is displayed in the first slot, the number of samples collected for $a_i$ is $|B_i| = \lfloor \frac{\tau}{N} \rfloor \geq \frac{\tau}{2N}$. The estimated value of $q_i$ is computed as:
\begin{align*}
\tilde{q}_i = \frac{1}{|B_i|}\sum_{t \in B_i} click_1^i(t).
\end{align*}
such that $\tilde{q}_i$ is an unbiased estimate of $q_i$ (i.e., $\mathbb{E}_{click} [\tilde{q}_i] = q_i$, where $\mathbb{E}_{click}$ is in expectation w.r.t. the realization of the clicks). By applying the Hoeffding's inequality we obtain an upper bound over the error of the estimated quality $\tilde{q}_i$ for each ad $a_i$.
\begin{proposition}\label{p:hoeffding.ql}
For any ad $\{a_i\}_{i \in \mathcal{N}}$
\begin{align}\label{eq:hoeff}
| q_i - \tilde{q}_i | \leq \sqrt{\frac{1}{2 |B_i|} \log \frac{2N}{\delta}} \leq \sqrt{\frac{N}{\tau} \log \frac{2N}{\delta}} =: \eta,
\end{align}
with probability $1-\delta$ (w.r.t. the click events).
\end{proposition}
After the exploration phase, an upper--confidence bound over each quality is computed as $\tilde{q}^+_i = \tilde{q}_i + \eta$.
\paragraph*{\indent Exploitation phase} We first focus on the allocation function. During the exploitation phase we want to use an allocation $\tilde{\theta}=\tilde{f}(\hat{\mathbf{v}})$ maximizing the estimated social welfare with estimated $\{\tilde{q}^+_i\}_{i \in \mathcal{N}}$ and the parameters $\{\Lambda_m\}_{m \in \mathcal{K}}$. Since the actual parameters $\{\Lambda_m\}_{m \in \mathcal{K}}$ are monotonically non--increasing we can use an allocation $\{\langle s_m, a_{\alpha(m; \tilde{\theta})} \rangle\}_{m \in \mathcal{K}'}$, where
\begin{align*}
\alpha(m; \tilde{\theta}) \in \arg\max_{i \in \mathcal{N}} (\tilde{q}^+_i \hat{v}_i; m) = \arg\max_{i \in \mathcal{N}} (\tilde{q}^+_i \Lambda_m \hat{v}_i; m).
\end{align*}
We now focus on payments. Allocation function $\tilde{f}$ is an affine maximizer (due to weights depending on $\tilde{q}_i$ as in Section~\ref{ssec:uq}), but WVCG payments cannot be computed given that parameters $\{\Lambda_m\}_{m \in \mathcal{K}}$ are unknown. Neither the adoption of execution--contingent payments, like in~(\ref{eq:olppc}), is allowed, given that $q_i$ is unknown and only estimates $\tilde{q}_i$ are available.
Thus, we resort to implicit payments as in Section~4.2.2. More precisely, we use the same exploitation phase we used in Section~4.2.2 except that we adopt $\tilde{f}$ in place of $f^*$. In this case, we have that the per--click payments are:
\begin{multline} \label{eq:pay.babaioff.ppc}
\tilde p^{B,c}_i(\mathbf{x}, click_{\pi(i; \tilde{f}(\mathbf{x}))}^{i}) = \begin{cases} \frac{\tilde p_i^B(\mathbf{x},\mathbf{y};\hat{\mathbf{v}})}{\Lambda_{\pi(i; \tilde{f}(\mathbf{x}))} q_i} & \mbox{if } click_{\pi(i; \tilde{f}(\mathbf{x}))}^{i}=1 \\ 0 & \mbox{otherwise} \end{cases}
= \\ \begin{cases} \hat{v}_i -
\begin{cases} \frac{1}{\mu} \hat{v}_i &\mbox{if $y_i<\hat{v}_i$} \\
0 & \mbox{otherwise}, \end{cases}
& \mbox{if } click_{\pi(i; \tilde{f}(\mathbf{x}))}^{i}=1 \\ 0 & \mbox{otherwise} \end{cases}
\end{multline}
where
\begin{align}
\tilde p_i^B(\mathbf{x},\mathbf{y};\hat{\mathbf{v}}) = \Lambda_{\pi(i; \tilde{f}(\mathbf{x}))}q_i \hat{v}_i -
\begin{cases} \frac{1}{\mu}\Lambda_{\pi(i; \tilde{f}(\mathbf{x}))}q_i \hat{v}_i &\mbox{if $y_i<\hat{v}_i$} \\
0 & \mbox{otherwise}, \end{cases}
\end{align}
We can state the following.
\begin{theorem}
The A--VCG3 is IC and WBB in expectation (over the realizations of the random component of the mechanism) and IR \emph{a posteriori} (w.r.t. the random component of the mechanism). These properties hold \emph{a posteriori} w.r.t. the click realizations.
\end{theorem}
\begin{pf}
The proof of IC in expectation and WBB in expectation easily follows from the definition of the adopted mechanism as discussed in~\cite{babaioff_impl_pay}. The proof of IR \emph{a posteriori} is similar to the proof of Proposition~\ref{prop:AVGC2IRaposteriori}. The fact that the properties hold \emph{a posteriori} w.r.t. the click realizations follows from~\cite{babaioff_impl_pay}.\qed
\end{pf}
Now we want to analyze the performance of the mechanism in terms of regret cumulated through $T$ rounds. Notice that in this case we have to focus on two different potential sources of regret: the adoption of a sub--optimal (randomized) allocation function and the estimation of the unknown parameters.
\begin{theorem}\label{thm:constant.ql}
Let us consider an auction with $N$ advertisers, $K$ slots, and $T$ rounds, with position--dependent cascade model with parameters $\{\Lambda_m\}_{m=1}^K$. For any parameter $\tau$ and $\delta$, the A--VCG3 achieves a regret
\begin{align*}
R_T & \leq v_{\max} K \left[\left( T-\tau \right) \left(2 \eta + 2 \mu N \right) + \tau + \delta T \right]
\end{align*}
\noindent By setting the parameters to
\begin{itemize}
\item $\mu = N^{-\frac{2}{3}} T^{-\frac{1}{3}}$. $\mu$ is always $\leq 1$
\item $\delta = N^\frac{1}{3} T^{-\frac{1}{3}}$. $\delta \leq 1$, thus $T \geq N$
\item $\tau = T^\frac{2}{3} N^\frac{1}{3} \left( \log{\frac{2N}{\delta}} \right)^\frac{1}{3}$
\end{itemize}
then the regret is
\begin{align}
R_T & \leq 6v_{\max} K T^\frac{2}{3} N^\frac{1}{3} \Big( \log \big(2 N^\frac{2}{3} T^\frac{1}{3} \big)\Big)^\frac{1}{3} \label{eq:regret.posdep.qlu}
\end{align}
\end{theorem}
\myremark{1 (The bound).} Up to numerical constants and logarithmic factors, the previous bound is $R_T \leq \tilde O(T^\frac{2}{3} K N^\frac{1}{3})$.
We first notice we match the lowest possible complexity for the parameter $T$ when exploration and exploitation phases are separate. Moreover observe that the proposed mechanism is a no--regret algorithm, thus asymptotically it achieves the same performances of VGC (when all the parameter are known), since its per--round regret ($R_T/T$) decreases to 0 as $T^{-\frac{1}{3}}$.
We can observe that, with respect to the case of Section~\ref{ssec:uq}, the dependence of the cumulative regret in the parameter $K$ is augmented by a factor $K^\frac{1}{3}$. The reason resides in the exploration phase, indeed, in this last case, we cannot take advantage of all data we can collect, given that we estimate the qualities only on the basis of their visualization in the first slot. Instead, the dependency on $N$ is the same of the one in the case studied in Section~\ref{ssec:uq}.
\myremark{2 (Non--separate phases and $O(T^{1/2})$).} The questions whether or not it is possible to avoid the separation of the exploration and exploitation phases preserving IC in expectation (in some form) and whether or not it is possible to obtain a regret of $O(T^{1/2})$ are open. We conjecture that, if it is possible to have $R_T=O(T^{1/2})$ when only $\{q_i\}_{i \in \mathcal{N}}$ are unknown, then it is possible to have $R_T=O(T^{1/2})$ also when $\{q_i\}_{i \in \mathcal{N}}$ and $\{\Lambda_m\}_{m \in \mathcal{K}}$ are unknown. However, such a problem is still open.
\myremark{3 (Using samples from multiple slots).} The question whether it is possible to exploit the samples from the slots below the first one to improve the accuracy of the estimates and to reduce the length of the exploration phase is open. The critical issue here is that the samples from those slots are about the product of two random variables, i.e., $\Lambda_s$ and $q_i$, and it is not trivial to find a method to use these samples to improve the esteems. However, in the case it is possible to exploit these samples, we would obtain a reduction of the regret bound of at most $K^{1/3}$, given that the dependency from $K$ cannot be better than in the case discussed in Section~\ref{ssec:uq} (i.e., $O(K^{\frac{2}{3}})$).
A--VCG3 allows also the identification of an upper--bound over the regret on the social welfare. The derivation is not straightforward with respect to the bound over the regret on the payments, but, using the value of the parameters identified in Theorem~\ref{thm:constant.ql}, the bound is $\tilde{O}(T^\frac{2}{3})$. Optimising the parameters w.r.t. to the regret over the social welfare, we obtain the following.
\begin{theorem} \label{th:pd_lq_sw}
Let us consider an auction with $N$ advertisers, $K$ slots, and $T$ rounds, with position--dependent cascade model with parameters $\{\Lambda_m\}_{m=1}^K$. For any parameter $\tau$ and $\delta$, the A--VCG3 achieves a regret
\begin{align*}
R_T^{SW} & \leq v_{\max} K \left[ (T - \tau) ( 2 \eta + N \mu) + \tau + \delta T \right]\\
& \leq v_{\max} K \left[ (T - \tau) \left( 2 \sqrt{\frac{N}{\tau} \log{\frac{2N}{\delta}}} + N \mu \right) + \tau + \delta T \right]
\end{align*}
\noindent By setting the parameters to
\begin{align*}
\mu &= K^{-1} N^\frac{1}{3} T^{-\frac{1}{3}}.\ \mu \leq 1 \textit{ when } T > \frac{N}{K^3}\\
\delta &= N^\frac{1}{3} T^{-\frac{1}{3}}\\
\tau &= T^\frac{2}{3} N^\frac{1}{3} \left( \log{\frac{2N}{\delta}} \right)^\frac{1}{3}
\end{align*}
then the regret is
\begin{align*}
R_T^{SW} &\leq 5 \cdot v_{\max} K N^\frac{1}{3} T^\frac{2}{3} \left( \log{N^\frac{2}{3} T^\frac{1}{3}} \right)^\frac{1}{3}.
\end{align*}%
\end{theorem}
\section{Problem statement}\label{s:statement}
In this section we introduce all the notation used throughout the rest of the paper. In particular, we formalize the sponsored search auction model, we define the mechanism design problem, and we introduce the learning process.
\subsection{Sponsored search auction model}
We resort to the standard model of sponsored search auctions~\cite{Narahari2009}. We denote by $\mathcal{N}=\{1,\ldots,N\}$ the set of ads indexes and by $a_i$ with $i \in \mathcal{N}$ the $i$--th ad (we assume w.l.o.g. each advertiser has only one ad and therefore we can identify by $a_i$ the $i$--th ad and the $i$--th advertiser indifferently). Each ad~$a_i$ is characterized by a \emph{quality} $q_i$ corresponding to the probability that $a_i$ is clicked once observed by the user, and by a \emph{value} $v_i\in\mathcal V$, with $\mathcal{V}=[0,V]$ and $V\in \mathbb{R}^+$, which $a_i$ receives when clicked ($a_i$ receives a value of zero if not clicked). We denote by $\mathbf{v}$ the profile $(v_1,\ldots,v_N)$ and, as customary in game theory, by $\mathbf{v}_{-i}$ the profile obtained by removing $v_i$ from $\mathbf{v}$. While qualities $\{q_i\}_{i \in \mathcal{N}}$ are commonly known by the auctioneer, values $\{v_i\}_{i \in \mathcal{N}}$ are private information of the advertisers. We denote by $\mathcal{K}=\{1,\ldots,K\}$ with $K < N$,\footnote{Although $K<N$ is the most common case, the results could be smoothly extended to $K>N$.} the set of slot indexes and by $s_m$ with $m\in \mathcal{K}$ the $m$--th slot from top to bottom. For notational convenience, we also define the extended set of slots indexes $\mathcal{K}'=\mathcal{K}\cup\{K+1,\ldots,N\}$.
We denote by the ordered pair $\langle s_m, a_i\rangle$ that ad $a_i$ is allocated into slot $s_m$, by $\theta$ a generic \emph{allocation} and by $\Theta$ the set of all the possible allocations. Although in an auction only $K$ ads can be actually displayed, we define an allocation as $\theta=\{\langle m,i\rangle: m\in \mathcal{K}',i \in \mathcal{N}\}$ where both $m$ and $i$ occur exactly once and any ad assigned to a slot $m>K$ is not displayed. We define two maps $\pi:\mathcal{N}\times\Theta \rightarrow \mathcal{K}'$ and $\alpha:\mathcal{K}'\times\Theta \rightarrow \mathcal{N}$ such that $\pi(i;\theta)$ returns the slot in which $a_i$ is displayed in allocation $\theta$ and $\alpha(m;\theta)$ returns the ad allocated in slot~$s_m$ in allocation $\theta$. Given $\theta \in \Theta$, we have that $\pi(i;\theta)=m$ if and only if $\alpha(m;\theta)=i$.
With more than one slot, it is necessary to adopt a model of the user describing how the expected value of an advertiser varies over the slots. We assume that the user behaves according to the popular \emph{cascade model} defined by~\cite{Kempe2008,Aggarwal2008}. In particular, the user's behavior can be modeled as a Markov chain whose states correspond to the slots, which are observed sequentially from the top to the bottom, and the transition probability corresponds to the probability of observing the ad $a_i$ displayed in the next slot; with the remaining probability the user stops observing the ads. This probability may depend on the index of the slot (i.e., $\pi(i;\theta)$), in this case the externalities are said \emph{position--dependent}, and/or on the ad that precedes $a_i$ in the current allocation $\theta$ (i.e., $\alpha(\pi(i;\theta)-1;\theta)$), in this case the externalities are said \emph{ad--dependent}.
In the general case, the cascade model can be described by introducing parameters $\gamma_{m,i}$ defined as the probability that a user observing ad~$a_i$ in slot~$s_{m}$ observes the ad in the next slot $s_{m+1}$. It can be easily seen that there are $KN$ different parameters $\gamma_{m,i}$. The (cumulative) probability that a user observes the ad displayed at slot $s_m$ in allocation $\theta$ is denoted by $\Gamma_m(\theta)$ and it is defined as:
\begin{align} \label{eq:coeff2}
\Gamma_m(\theta) = \left\{
\begin{array}{ll}
1 & \text{if } m=1 \\
\prod\limits_{l=1}^{m-1} \gamma_{l,\alpha(l;\theta)} & \text{if } 2 \leq m\leq K\\
0 & \text{otherwise}
\end{array} \right.
\end{align}
Given an allocation $\theta$, the \emph{click through rate} (CTR) of ad $a_i$ is the probability to be clicked once allocated according to $\theta$ and it is equal to $\Gamma_{\pi(i;\theta)}(\theta) q_{i}$. Similarly, the CTR of the ad displayed at slot $m$ can be computed as $\Gamma_m(\theta) q_{\alpha(m;\theta)}$. We notice that, according to this model, the user might click multiple ads at each impression. Given an allocation $\theta$, the \emph{expected value} (w.r.t. the user's clicks) of advertiser $a_i$ from $\theta$ is $\Gamma_{\pi(i;\theta)}(\theta) q_{i} v_i$, that is, the product of the CTR $\Gamma_{\pi(i;\theta)}(\theta) q_{i}$ by the value of the advertiser $v_i$. The advertisers' cumulative expected value from allocation $\theta$, commonly referred to as \emph{social welfare}, is:
\begin{align*}
\text{SW}(\theta,\mathbf{v})= \sum_{i=1}^N \Gamma_{\pi(i;\theta)}(\theta) q_{i} v_i
\end{align*}
In~\cite{Kempe2008,Aggarwal2008}, the authors factorize the probability $\gamma_{m,i}$ as the product of two independent terms: the \emph{prominence} $\lambda_m$, which only depends on the slot $s_m$, and the \emph{continuation probability} $c_i$, which only depends on the ad $a_i$. This leads to a reduction of the number of the parameters from $KN$ to $K+N$.\footnote{The allocation problem when either all the prominence probabilities $\lambda_m$s or all the continuation probabilities $c_i$s are equal to one can be solved in polynomial time, while, although no formal proof is known, the allocation problem with $\lambda_m$s and $c_i$s different from one is commonly believed to be $\mathcal{NP}$--hard~\cite{Kempe2008}. However, the allocation problem can be solved exactly for concrete settings and for very large settings approximation algorithms can be adopted as shown in~\cite{aamas2013}. In this paper, we just focus on optimal allocation functions.}
Finally, we denote by $click^i_{m}\in \{0,1\}$ the click/no--click event for ad $a_i$ allocated in slot $m$.
\subsection{Mechanism design problem} \label{ssec:md}
A direct--revelation economic mechanism for sponsored search auctions is formally defined as a tuple $(\mathcal{N},\mathcal V,\Theta,f,\{p_i\}_{i\in \mathcal{N}})$ where $\mathcal{N}$ is the set of agents (i.e., the advertisers), $\mathcal V$ is the set of possible actions available to the agents (i.e., the possible reported values), $\Theta$ is the set of the outcomes (i.e., the allocations), $f$ is the allocation function $f:\mathcal V^{N}\rightarrow \Theta$, and $p_i$ is the payment function of advertiser $a_i$ defied as $p_i:\mathcal V^{N}\rightarrow \mathbb R$. We denote by $\hat{v}_i$ the value reported by advertiser $a_i$ to the mechanism, by $\hat{\mathbf{v}}$ the profile of reported values and $\hat{\mathbf{v}}_{-i}$ the profile obtained by removing $\hat{v}_i$ from $\hat{\mathbf{v}}$.
At the beginning of an auction, each advertiser $a_i$ reports its value $\hat{v}_i$. The mechanism chooses the allocation on the basis of the advertisers' reports as $f(\hat{\mathbf{v}})$ and subsequently computes the payment of each advertiser $a_i$ as $p_i(\hat{\mathbf{v}})$. The expected utility of advertiser $a_i$ is defined as $\Gamma_{\pi(i;f(\hat{\mathbf{v}}))}f(\hat{\mathbf{v}}) q_{i} v_i - p_i(\hat{\mathbf{v}})$. Since each advertiser is an expected utility maximizer, it will misreport its value (i.e., $\hat v_i \neq v_i$) whenever this may lead its utility to increase. Mechanism design aims at finding an allocation function $f$ and a vector of payments $\{p_i\}_{i \in \mathcal{N}}$ such that some desirable properties---discussed in Section~2.1---are satisfied~\cite{mas-colell1995microeconomic}.
When all the parameters $q_i$ and $\gamma_{m,i}$ are known, the VCG mechanism satisfies IC in expectation (over click realizations), IR in expectation (over click realizations), WBB \emph{a posteriori} (w.r.t. click realizations), and AE. In the VCG mechanism, the allocation function, denoted by $f^*$, maximizes the social welfare given the reported types as:
\begin{align}
\label{eq:efficient-alloc}
\theta^*=f^*(\hat{\mathbf{v}}) \in \arg\max_{\theta \in \Theta}~ \{\text{SW}(\theta,\hat{\mathbf{v}})\}
\end{align}
\noindent and payments are defined as
\begin{align}\label{eq:pay.opt.vcg}
p^*_i(\hat{\mathbf{v}}) = \text{SW}(\theta^*_{-i},\hat{\mathbf{v}}_{-i}) - \text{SW}_{-i}(\theta^*,\hat{\mathbf{v}}),
\end{align}
where:
\begin{itemize}
\item $\theta^*_{-i}=f^*(\hat{\mathbf{v}}_{-i})$, i.e., the optimal allocation when advertiser $a_i$ is not present,
\item $\text{SW}_{-i}(\theta^*,\hat{\mathbf{v}})=\sum\limits_{j=1,j\neq i}^N \Gamma_{\pi(j;\theta^*)}(\theta^*) q_{j} \hat{v}_j$, i.e., the cumulative expected value of the optimal allocation $\theta^*$ minus the expected value of advertiser $a_i$.
\end{itemize}
In words, the payment of advertiser $a_i$ is the difference between the social welfare that could be obtained from allocation $\theta_{-i}^*$ computed removing ad $a_i$ from the auction and the social welfare of the efficient allocation $\theta^*$ without the contribution of advertiser $a_i$. The extension of the VCG mechanism do weighted ads (the WVCG mechanism) is straightforward. The weighted social welfare is $\text{SW}^w(\theta,\mathbf{v})= \sum_{i=1}^N \Gamma_{\pi(i;\theta)}(\theta) q_{i} v_i w_i$ where $w_i$ is the weight of advertiser $i$. In the WVCG, the allocation maximizing the weighted social welfare is chosen, while the payment is defined as $ p^w_i(\hat{\mathbf{v}}) = \frac{1}{w_i}(\text{SW}^w(\theta^*_{-i},\hat{\mathbf{v}}_{-i}) - \text{SW}^w_{-i}(\theta^*,\hat{\mathbf{v}}))$.
The previous mechanism is IC and IR in expectation, but it is not DSIC and IR \emph{a posteriori} w.r.t. the clicks (an advertiser may have a positive payment even when its ad has not been clicked). Nonetheless, the mechanism can be easily modified to satisfy DSIC and IR \emph{a posteriori} w.r.t. the clicks by using \emph{pay--per--click} payments $p^{*,c}_i$ as follows:
\begin{align}\label{eq:pay.opt.click.vcg}
p^{*,c}_i(\hat{\mathbf{v}},click^i_{\pi(i; \theta^*)}) = \frac{\text{SW}(\theta^*_{-i},\hat{\mathbf{v}}_{-i}) - \text{SW}_{-i}(\theta^*,\hat{\mathbf{v}})}{\Gamma_{\pi(i;\theta^*)}(\theta^*) q_i}\mathbb{I}\{click^i_{\pi(i; \theta^*)}\},\end{align}
where $\mathbb{I}\{\cdot\}$ denotes the indicator function. The contingent formulation of the payments is such that $\mathbb E[ p^c_i(\hat{\mathbf{v}},click^i_{\pi(i; \theta^*)})] = p^*_i(\hat{\mathbf{v}})$, where the expectation is w.r.t. the click event, which is distributed as a Bernoulli random variable with parameter coinciding with the CTR of ad $a_i$ in allocation $\theta^*$, i.e., $\Gamma_{\pi(i; \theta^*)}q_i$. Similar definitions hold for the WVCG.
\subsection{Online learning mechanism design problem}\label{ss:online.mechanism}
In many practical problems, the parameters (i.e., $q_i$ and $\gamma_{m,i}$) are not known in advance by the auctioneer and must be estimated at the same time as the auction is deployed. This introduces a tradeoff between \textit{exploring} different possible allocations so as to collect information about the parameters and \textit{exploiting} the estimated parameters so as to implement a truthful high--revenue auction (i.e., a VCG mechanism). This problem could be easily casted as a multi--arm bandit problem \cite{robbins1952some} and standard techniques could be used to solve it, e.g., \cite{auer2002finite-time}. Nonetheless, such an approach would completely overlook the strategic dimension of the problem: advertisers may choose their reported values at each round $t$ to influence the outcome of the auction at $t$ and/or in future rounds after $t$ in order to increase the cumulative utility over all the rounds of the horizon $T$. Thus, in this context, truthfulness requires that reporting the truthful valuation maximizes the cumulative utility over all the horizon $T$. The truthfulness can be: in dominant strategies if advertisers know everything (including, e.g., the ads that will be clicked at each round $t$ if displayed) or in expectation. As customary, we adopt three forms of truthfulness in expectation: IC in expectation over the click realizations and \emph{a posteriori} w.r.t. the realizations of the random component of the mechanism (if such a component is present), IC in expectation over the realizations of the random component of the mechanism and \emph{a posteriori} w.r.t. the click realizations, and, finally, IC in expectation over both randomizations. We consider IC in expectation over the click realizations weaker than IC in expectation over the realizations of the random mechanism since each advertiser could control the clicks by using software bots.
Thus, here we face the more challenging problem where the exploration--exploitation dilemma must be solved so as to maximize the revenue of the auction under the hard constraint of incentive compatibility. Let $\mathfrak{A}$ be an IC mechanism run over $T$ rounds. We assume, as it is common in practice, that the advertisers' reports can change during these $T$ rounds. At each round~$t$, $\mathfrak{A}$ defines an allocation $\theta_t$ and prescribes an expected payment $p_{i,t}(\hat{\mathbf{v}})$ for each ad $a_i$. The objective of $\mathfrak{A}$ is to obtain a revenue as close as possible to a VCG mechanism computed on the basis of the actual parameters.\footnote{We refer the reader to~\ref{app:deviation.regret} for a slightly different definition of regret measuring the deviation from the revenue of a VGC mechanism.} More precisely, we measure the performance of $\mathfrak{A}$ as its cumulative regret over $T$ rounds:
\begin{align*}
\mathcal R_T(\mathfrak{A}) = T \sum_{i=1}^n p_i^*(\hat{\mathbf{v}}) - \sum_{t=1}^T \sum_{i=1}^n p_{i,t}(\hat{\mathbf{v}}).
\end{align*}
We remark that the regret is not defined on the basis of the pay--per--click payments asked on a specific sequence of clicks but on the expected payments $p_{i,t}(\hat{\mathbf{v}})$. Furthermore, since the learning mechanism $\mathfrak{A}$ estimates the parameters from the observed (random) clicks, the expected payments $p_{i,t}(\hat{\mathbf{v}})$ are random as well. Thus, in the following we will study the expected regret:
\begin{align}\label{eq:regret}
R_T(\mathfrak{A}) = \mathbb E[\mathcal R_T(\mathfrak{A})],
\end{align}
where the expectation is taken w.r.t. random sequences of clicks and possibly the randomness of the mechanism.
The mechanism $\mathfrak{A}$ is a \textit{no--regret} mechanism if its per--round regret $R_T(\mathfrak{A})/T$ decreases to 0 as $T$ increases, i.e., $\lim\limits_{T\rightarrow \infty} R_T(\mathfrak{A}) / T = 0$. Another popular definition of performance \cite{gonen2007incentive-compatible,babaioff2008characterizing} is the social welfare regret, denoted by $R_T^{SW}$ and measured as the difference between the (expected) social welfare of the optimal allocation $\theta^*$ and the (expected) social welfare of the best allocation $\tilde{\theta}$ found with the estimated parameters (i.e., $\text{SW}(\theta^*,\hat{\mathbf{v}}) - \text{SW}(\tilde{\theta},\hat{\mathbf{v}})$). We notice that minimizing the social welfare regret does not coincide with minimizing $R_T$. In fact, once the quality estimates are accurate enough, such that $\theta_t$ is equal to $\theta^*$, the social welfare regret drops to zero. On the other hand, since $p_{i,t}(\hat{\mathbf{v}})$ is defined according to the estimated qualities, $R_T(\mathfrak{A})$ might still be positive even if $\theta_t = \theta^*$. In addition, we believe that in practical applications providing a theoretical bound over the regret of the auctioneer's revenue is more important rather than a bound on the regret of the social welfare.\footnote{However, we show that our bounds over the regret of auctioneer's revenue can be easily extended also to the regret of the social welfare.}
The study of the problem when $K=1$ is well established in the literature.
More precisely, the properties required to have a DSIC mechanism are studied in~\cite{devanur2009price} and it is shown that any learning algorithm must split the exploration and the exploitation in two separate phases in order to incentivize the advertisers to report their true values.
This condition has a strong impact on the regret $R_T(\mathfrak{A})$ of the mechanism. In fact, while in a standard bandit problem the distribution--free regret is of order $\Omega(T^{1/2})$, in single--slot auctions, DSIC mechanisms cannot achieve a regret smaller than $\Omega(T^{2/3})$. In~\cite{devanur2009price} a truthful learning mechanism is designed with a nearly optimal regret of order $\tilde O(T^{2/3})$.\footnote{The $\tilde O$ notation hides both constant and logarithmic factors, that is $R_T \leq \tilde O(T^{2/3})$ if there exist $a$ and $b$ such that $R_T \leq a T^{2/3} \log^b T$.} Similar structural properties for DSIC mechanisms are also studied in~\cite{babaioff2008characterizing} and similar lower--bounds are derived for the social welfare regret.
The authors show in~\cite{babaioff_impl_pay} that, by introducing a random component in the allocation function and resorting to truthfulness in expectation over the realizations of the random component of the mechanism, the separation of exploration and exploitation phases can be avoided. In this case, the upper bound over the regret over the social welfare is $O(T^{1/2})$ matching the best bound of standard distribution--free bandit problems. However, the payments of this mechanism suffer of potentially high variance. Although it is expected that with this mechanism also the regret over the auctioneer revenue is of the order of $O(T^{1/2})$, no formal proof is known.
On the other hand, the study of the problem when $K>1$ is still mostly open. In this case, a crucial role is played by the CTR model. While with only one slot, the advertisers' CTRs coincide to their qualities $q_i$, with multiple slots the CTRs may also depend on the slots and the allocation of the other ads. The only results on learning mechanisms for sponsored search auction with $K>1$ are described in~\cite{sarma2010multi-armed}, where the authors characterize DSIC mechanisms and provide theoretical bounds over the social welfare regret. More precisely, the authors assume a simple CTR model in which the CTR itself depends on the ad $i$ and the slot $m$. This model differs from the cascade model (see Section~2.1) where the CTR is a more complex function of the quality $q_i$ of an ad and the cumulative probability of observation $\Gamma_m(\theta)$ which, in general, depends on both the slot $m$ and the full allocation $\theta$ (i.e., the ads allocated before slot $s_m$). It can be easily shown that the model studied in~\cite{sarma2010multi-armed} does not include and, at the same time, is not included by the cascade model. However, the two models correspond when the CTRs are separable in two terms in which the first is the agents' quality and the second is a parameter in $[0,1]$ monotonically decreasing in the slots (i.e., only--position--dependent cascade model). Furthermore, while the cascade model is supported by an empirical activity which confirms its validity as a model of the user behavior~\cite{Craswell,Joachims}, the model considered in~\cite{sarma2010multi-armed} has not been empirically studied. In~\cite{sarma2010multi-armed}, the authors show that when the CTRs are unrestricted (e.g., they are not strictly monotonically decreasing in the slots), then the regret over the social welfare is $\Theta(T)$ and therefore at every round (of repetition of the auction) a non--zero regret is accumulated. In addition, the authors provide necessary and, in some situations, sufficient conditions to have DSIC in restricted environments (i.e., higher slot higher click probability, separable CTRs in which only ads qualities need to be estimated), without presenting any bound over the regret (except for reporting an experimental evidence that the regret is $\Omega(T^{2/3})$ when the CTRs are separable).
We summarize in Tab.~\ref{tab::results} the known results from the literature and, in bold font, the original results provided in this paper.
\begin{table}[h]
\begin{scriptsize}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
slots & CTR model & unknown & solution & regret over & regret over \\
& & parameters & concept & social welfare & auctioneer revenue \\ \hline \hline
1 & -- & $q_i$ & DSIC & $\Theta(T^{2/3})$ & $\Theta(T^{2/3})$ \\ \cline{4-6}
& & & IC in exp. & $O(T^{1/2})$ & $O(T^{2/3})$ \\ \hline
$>1$ & (unconstrained) $CTR_{i,m}$ & $CTR_{i,m}$ & DISC & $\Theta(T)$ & unknown \\ \cline{2-6}
& (unfactorized) cascade & $q_i$ & DISC & $\mathbf{O(T^{2/3})}$ & $\mathbf{\Theta(T^{2/3})}$ \\ \cline{3-6}
& & $\gamma_{i,s}$ & DISC & $\mathbf{\Theta(T)}$ & $\mathbf{\Theta(T)}$ \\ \cline{2-6}
& position--dep. cascade / & $\lambda_m$ & DSIC & $\mathbf{\Theta(T)}$ & $\mathbf{\Theta(T)}$ \\ \cline{4-6}
& separable $CTR_{i,m}$ & & IC in exp. & $\mathbf{0}$ & $\mathbf{0}$ \\
& & & (w.r.t. clicks) & & \\ \cline{4-6}
& & & IC in exp. & $O(1)$ & $O(1)$ \\
& & & (w.r.t. mechanism) & & \\ \cline{3-6}
& & $q_i$, $\lambda_m$ & DSIC & $\mathbf{\Theta(T)}$ & $\mathbf{\Theta(T)}$ \\ \cline{4-6}
& & & IC in exp. & $\mathbf{O(T^{2/3})}$ & $\mathbf{O(T^{2/3})}$ \\ \cline{2-6}
& ad--dependent cascade & $c_i$ & DSIC & $\mathbf{\Theta(T)}$ & $\mathbf{\Theta(T)}$ \\ \cline{3-6}
& & $q_i$, $c_i$ & DSIC & $\mathbf{\Theta(T)}$ & $\mathbf{\Theta(T)}$ \\ \hline
\end{tabular}
\end{center}
\end{scriptsize}
\caption{Known results on regret bounds for sponsored search auction. We remark with bold font the results provided in this paper.}
\label{tab::results}
\end{table}
\section{Learning with Position-- and Ad--Dependent Externalities}\label{s:externalities}
In this section we deal with the general model where both position-- and ad--dependent externalities are present, as formalized in~(\ref{eq:coeff2}), and we provide several partial results. In Section~\ref{sse:uqpad}, we analyze the problem of designing a DSIC mechanism when only the qualities of the ads are unknown. In Section~\ref{sse:pepad} we highlight some problems that rise when also other parameters are uncertain.
\subsection{Unknown quality} \label{sse:uqpad}
\begin{figure}[t]
\bookbox{
\begin{algorithmic}
\STATE \textbf{Input:} Length of exploration phase $\tau$, confidence $\delta$, position--dependent parameters $\{\Gamma_m\}_{m \in \mathcal{K}}$
\STATE
\STATE \textit{Exploration phase}
\FOR{$t = 1,\ldots,\tau$}
\STATE Allocate ads according to (\ref{eq:explorativeallocations})
\STATE Ask for no payment
\STATE Observe the clicks $\{click_{\pi(i;\theta_t)}^i(t)\}_{i=1}^{N}$
\ENDFOR
\STATE Compute the estimated quality $\tilde{q}_i = \frac{1}{|B_i|}\sum_{t \in B_i} \frac{click_{\pi(i; \theta_t)}^i(t)}{\Gamma_{\pi(i; \theta_t)(\theta_t)}}$
\STATE Compute $\tilde{q}^+_i = \tilde{q}_i + \eta$ where $\eta$ is given by (\ref{eq:eta2})
\STATE
\STATE \textit{Exploitation phase}
\FOR{$t = \tau+1,\ldots, T$}
\STATE Allocate ads according to $\tilde{f}$
\IF{Ad $a_i$ is clicked}
\STATE Ask for payment $\tilde p^c_i$ defined in (\ref{eq:hpay.extern})
\ENDIF
\ENDFOR
\end{algorithmic}}
\caption{Pseudo--code for the PAD--A--VCG\ mechanism.}\label{f:algpad}
\end{figure}
In this section we analyze the problem where the only unknown parameters are the qualities $\{q_i\}_{i \in \mathcal{N}}$ of the ads and the externality model includes position-- and ad--dependent externalities. As we do in Section~\ref{ssec:uq}, we focus on DSIC mechanisms and we leave open the question whether better bounds over the regret can be found by employing IC in expectation. Therefore we study MAB algorithms that separate the exploration and exploitation phases. The structure of the mechanism we propose, called PAD--A--VCG, is similar to the \avcg1 and is reported in Fig.~\ref{f:algpad}.
\paragraph{\indent Exploration phase.} During the exploration phase with length $\tau \leq T$ steps we collect $K$ samples of click or no--click events. Given a generic exploration policy $\{\theta_t\}_{0 \leq t \leq \tau}$, the estimate quality $\tilde{q}_i$ is computed as:
\begin{align*}
\tilde{q}_i = \frac{1}{|B_i|}\sum_{t \in B_i} \frac{click_{\pi(i; \theta_t)}^i(t)}{\Gamma_{\pi(i; \theta_t)}(\theta_t)},
\end{align*}
\noindent where we identify the set $B_i = \{t: \pi(i; \theta_t) \leq K, t \leq \tau\}$.
The explorative allocations $\theta_t$ have an impact on the discount $\Gamma_m(\theta_t)$ and thus a variation of Proposition~\ref{p:hoeffding} holds in which (\ref{eq:hoeffproposition1}) is substituted by:
\begin{align*
| q_i - \tilde{q}_i | \leq \sqrt{\Bigg(\sum_{t \in B_i} \frac{1}{\Gamma_{\pi(i; \theta_t)}(\theta_t)^2}\Bigg) \frac{1}{2 |B_i|^2} \log \frac{2N}{\delta}}.
\end{align*}
For each exploration policy such that $|B_i| = \lfloor K\tau / N \rfloor$ $\forall i \in \mathcal{N}$, e.g. policy (\ref{eq:explorativeallocations}), we redefine $\eta$ as
\begin{align}\label{eq:eta2}
| q_i - \tilde{q}_i | \leq \frac{1}{\Gamma_{\min}}\sqrt{\frac{N}{2 K \tau} \log \frac{N}{\delta}} := \eta,
\end{align}
where $\Gamma_{\min} = \min\limits_{\theta \in \Theta, m \in \mathcal{K}} \{\Gamma_m(\theta)\}$. We define the upper--confidence bound $\tilde{q}^+_i = \tilde{q}_i + \eta$.
During the exploration phase, in order to preserve the DSIC property, the allocations $\{\theta_t\}_{0 \leq t \leq \tau}$ do not depend on the reported values of the advertisers and no payments are imposed to the advertisers.
\paragraph{\indent Exploitation phase} We define the estimated social welfare as
\begin{align*
\widetilde{\text{SW}}(\theta, \hat{\mathbf{v}}) = \sum_{i=1}^N \Gamma_{\pi(i; \theta)}(\theta) \tilde{q}^+_i \hat{v}_i = \sum_{m=1}^K \Gamma_m(\theta) \tilde{q}^+_{\alpha(m;\theta)} \hat{v}_{\alpha(m;\theta)} .
\end{align*}
We denote by $\tilde{\theta}$ the allocation maximizing $\widetilde{\text{SW}}(f(\hat{\mathbf{v}}), \hat{\mathbf{v}})$ and by $\tilde{f}$ the allocation function returning $\tilde{\theta}$:
\[
\tilde{\theta} = \tilde{f}(\hat{\mathbf{v}}) \in \arg\max_{\theta\in\Theta}\widetilde{\text{SW}}(\theta, \hat{\mathbf{v}}).
\]
Once the exploration phase is over, the ads are allocated on the basis of $\tilde{f}$. Since $\tilde{f}$ is an affine maximizer, the mechanism can impose WVCG payments to the advertisers satisfying the DSIC property. In a \emph{pay--per--click} fashion, if ad $a_i$ is clicked, the advertiser is charged
\begin{align}\label{eq:hpay.extern}
\tilde p_i^c(\hat{\mathbf{v}}, click_{\pi(i;\tilde{\theta})}^i) = \frac{\widetilde{\text{SW}}(\tilde{\theta}_{-i}) - \widetilde{\text{SW}}_{-i}(\tilde{\theta})}{\Gamma_{\pi(i; \tilde{\theta})}(\tilde{\theta}) \tilde{q}^+_i}
\end{align}
which corresponds, in expectation, to the WVCG payment $\tilde p_i = \tilde p_i^c \Gamma_{\pi(i; \tilde{\theta})}(\tilde{\theta}) q_i$.
We are interested in bounding the regret of the auctioneer's revenue due to PAD--A--VCG\ compared to the auctioneer's revenue of the VCG mechanism when all the parameters are known.
\begin{theorem}\label{thm:extern}
Let us consider an auction with $N$ advs, $K$ slots, and $T$ rounds. The auction has position/ad--dependent externalities and cumulative discount factors $\{\Gamma_m(\theta)\}_{m=1}^K$ and $\eta$ defined as in (\ref{eq:eta2}). For any parameter $\tau \in \{0, \ldots, T\}$ and $\delta \in [0,1]$, the PAD--A--VCG\ achieves a regret:
\begin{align}\label{eq:regret.extern.exact}
R_T \leq v_{\max} K \left[ (T - \tau) \left( \frac{3\sqrt{2}n}{\Gamma_{\min}q_{\min}} \sqrt{\frac{N}{K\tau} \log \frac{N}{\delta}} \right) + \tau + \delta T \right],
\end{align}
\noindent where $q_{\min} = \min_{i \in \mathcal{N}} q_i$. By setting the parameters to
\begin{align*}
\delta &=K^\frac{1}{3} N^\frac{1}{3} \left( \frac{5}{\sqrt{2} \Gamma_{\min}} \right)^\frac{2}{3} T^{-\frac{1}{3}},\\
\tau &= \left( \frac{5}{\sqrt{2} \Gamma_{\min}} \right)^{\frac{2}{3}} K^{\frac{1}{3}} T^{\frac{2}{3}} N^{\frac{1}{3}} \left( \log{\frac{N}{\delta}} \right)^{\frac{1}{3}},
\end{align*}
the regret is
\begin{align}\label{eq:regret.extern}
R_T \leq 4 v_{\max} K^\frac{4}{3} T^\frac{2}{3} N^\frac{1}{3} \frac{5^\frac{2}{3}}{2^\frac{1}{3} \Gamma_{\min}^\frac{2}{3} q_{\min}} \left(\log{\frac{2^\frac{1}{3} \Gamma_{\min}^\frac{2}{3} N^\frac{2}{3} T^\frac{1}{3}}{K^\frac{1}{3} 5^\frac{2}{3}}}\right)^\frac{1}{3}.
\end{align}
\end{theorem}
\myremark{1 (Differences w.r.t. position--dependent externalities.)} Up to constants and logarithmic factors, the previous distribution--free bound is $R_T\leq \tilde O(T^\frac{2}{3} N^\frac{1}{3} K^\frac{4}{3})$.\footnote{We notice that in~\cite{glt} the authors provide a bound $O(T^\frac{2}{3} N K^\frac{2}{3})$ that does not match with their numerical simulations and thus they conjecture that the actual bound is $O(T^\frac{2}{3} N^\frac{1}{3} K^\frac{4}{3})$. Here we show that the conjecture is correct.} We first notice that moving from position-- to position/ad--dependent externalities does not change the dependency of the regret on both the number of rounds $T$ and the number of ads $N$. Moreover, the per--round regret still decreases to 0 as $T$ increases.
The main difference w.r.t. the bound in Theorem~\ref{thm:constant} is in the dependency on $K$ and on the smallest quality $q_{\min}$. We believe that the augmented dependence in $K$ is mostly due to an intrinsic difficulty of the position/ad--dependent externalities. The intuition is that now, in the computation of the payment for each ad $a_i$, the errors in the quality estimates cumulate through the slots (unlike the position--dependent case where they are scaled by $\Gamma_{k}-\Gamma_{k+1}$). This cumulated error should impact only on a portion of the ads (i.e., those which are actually impressed according to the optimal and the estimated optimal allocations) whose cardinality can be upper--bounded by $2K$. Thus we observe that the bound shows a super--linear dependency in the number of slots.
The other main difference is that now the regret has an inverse dependency on the smallest quality $q_{\min}$. Inspecting the proof, this dependency appears because the error of a quality estimation for an ad $a_i$ might be amplified by the inverse of the quality itself $\frac{1}{q_i}$. As discussed in Remark 2 of Theorem~\ref{thm:constant}, this dependency might follow from that fact the we have a distribution--free bound.
We investigate whether this dependency is an artifact of the proof or it is intrinsic in the algorithm in the numerical simulations reported in Section~\ref{s:experiments}.
\myremark{2 (Optimization of the parameter $\tau$).} We are considering an environment where $\{q_i\}_{i \in \mathcal{N}}$ are unknown, but if, at least, a guess about the value of $q_{\min}$ is available, it could be used to better tune $\tau$ by multiplying it by $(q_{\min})^{-\frac{2}{3}}$, thus reducing the regret from $\tilde O((q_{\min})^{-1})$ to $\tilde O((q_{\min})^{-\frac{2}{3}})$.
\myremark{3 (Externalities--dependent bound).} We notice that the above bound does not reduce to the bound (\ref{eq:regret.const}) in which only position--dependent externalities are present even disregarding the constant terms. Indeed, the dependency on $K$ is different in the two bounds: in (\ref{eq:regret.const}) we have $K^{\frac{2}{3}}$ while in (\ref{eq:regret.extern}) we have $K^{\frac{4}{3}}$. This means that bound (\ref{eq:regret.extern}) over--estimates the dependency on $K$ whenever the auction has position--dependent externalities. It is an interesting open question whether it is possible to derive an \textit{auction--dependent} bound where the specific values of the discount factors $\gamma_k(f)$ explicitly appear in the bound and that it reduces to (\ref{eq:regret.const}) for position--dependent externalities.
\textit{(Comment to the proof).}
While the proof of Thm.~\ref{thm:constant} could exploit the specific definition of the payments for position--dependent slots and it is a fairly standard extension of~\cite{devanur2009price}, in this case the proof is more complicated because of the dependency of the discount factors on the actual allocations and decomposes the regret of the exploitation phase in components due to the different allocations ($\tilde{f}$ instead of $f^*$) and the different qualities as well ($\tilde{q}^+$ instead of $q$).
Using the mechanism described before, it is possible to derive an upper--bound over the global regret, when the regret, as in~\cite{babaioff_impl_pay}, is computed over the social welfare of the allocation. We obtain the same dependence over $T$, as for the regret on the payment. Thus $R^{SW}_T\leq\tilde{O}(T^\frac{2}{3})$. In particular notice that PAD--A--VCG\ is a zero--regret algorithm.
\begin{theorem} \label{th:pad_q_sw}
Let us consider an auction with $N$ advs, $K$ slots, and $T$ rounds. The auction has position/ad--dependent externalities and cumulative discount factors $\{\Gamma_m(\theta)\}_{m=1}^K$ and $\eta$ defined as in (\ref{eq:eta2}). For any parameter $\tau \in \{0, \ldots, T\}$ and $\delta \in [0,1]$, the PAD--A--VCG\ achieves a regret:
\begin{align}
R^{SW}_T \leq v_{\max} K \left[ (T - \tau) \frac{2}{\Gamma_{\min}} \sqrt{\frac{N}{2K\tau} \log \frac{N}{\delta}} + \tau + \delta T \right],
\end{align}
By setting the parameters to
\begin{align*}
\delta & = \left( \frac{\sqrt{2}}{\Gamma_{\min}} \right)^\frac{2}{3} K^{-\frac{1}{3}} N^\frac{1}{3} T^{-\frac{1}{3}}\\
\tau & = \left( \frac{\sqrt{2}}{\Gamma_{\min}} \right)^\frac{2}{3} T^\frac{2}{3} N^\frac{1}{3} K^{-\frac{1}{3}} \left( \log \frac{2N}{\delta} \right)^\frac{1}{3},
\end{align*}
the regret is
\begin{align}
R^{SW}_T \leq 4 v_{\max} \left( \frac{\sqrt{2}}{\Gamma_{\min}} \right)^\frac{2}{3} K^\frac{2}{3} N^\frac{1}{3} T^\frac{2}{3} \left( \log 2^\frac{2}{3} \Gamma_{\min}^{-\frac{2}{3}} N^\frac{2}{3} K^\frac{1}{3} T^\frac{1}{3} \right)^\frac{1}{3}.
\end{align}
\end{theorem}
Notice that using $\tau$ and $\delta$ defined in Theorem~\ref{thm:extern}, the bound for $R_T^{SW}$ is $\tilde{O}(T^\frac{2}{3})$, even if the parameters are not optimal for this second framework.
\subsection{Further extensions} \label{sse:pepad}
In this section we provide a negative, in terms of regret, result under DSIC truthfulness when the parameter $\gamma_{i,m}$ depends only on the ad $i$ (as in~\cite{Kempe2008}, we denote it by $c_i$) and this parameter is the only uncertain parameter.
We focus on the exploitation phase, supposing the exploration phase has produced the estimates $\{\tilde{c}^+_i\}_{i \in \mathcal{N}}$ for the continuation probabilities $\{c_i\}_{i \in \mathcal{N}}$. The allocation function $f$ presented in~\cite{Kempe2008} is able to compute the optimal allocation when $\{c_i\}_{i \in \mathcal{N}}$ values are known, but it is not an affine maximizer when applied to the estimated values $\{\tilde{c}^+_i\}_{i \in \mathcal{N}}$. Indeed, we call this allocation function $\tilde{f}$:
\begin{equation} \label{eq:tf}
\tilde{f}(\hat{\mathbf{v}}) \in \arg\max_{\theta \in \Theta} \sum_{m=1}^K q_{\alpha(m;\theta)} \hat{v}_{\alpha(m;\theta)} \prod_{h=1}^{m-1} \tilde{c}^+_{\alpha(h; \theta)}.
\end{equation}
In this case, a weight depending only on a single ad cannot be isolated. Furthermore, we show also that this allocation function is not monotonic.
\begin{proposition}
The allocation function $\tilde{f}$ is not monotonic.
\end{proposition}
\begin{pf}
The proof is by counterexample
Consider an environment with 3 ads and 2 slots such that:
\begin{table}[!h]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
ad & $v_i$ & $\tilde{c}^+_i$ & $c_i$\\
\hline
$a_1$ & $0.85$ & $1$ & $0.89$\\
\hline
$a_2$ & $1$ & $0.9$ & $0.9$\\
\hline
$a_3$ & $1.4$ & $0$ & $0$\\
\hline
\end{tabular}
\end{center}
\end{table}
\noindent and $q_i = 1$ $\forall i \in \mathcal{N}$. The optimal allocation $\tilde{\theta}$ found by $\tilde{f}$ when agents declare their true values $\mathbf{v}$ is: ad $a_2$ is allocated in the first slot and $a_3$ in the second one. We have $CTR_{a_3}(\tilde{\theta}) = 0.9$.
If advertiser $a_3$ reports a larger value: $\hat{v}_3 = 1.6$, in the allocation $\hat{\theta}$ found by $\tilde{f}(\hat{v}_3, \mathbf{v}_{-3})$, ad $a_1$ is displayed into the first slot and $a_3$ into the second one. In this case $CTR_{a_3}(\hat{\theta}) = 0.89 < CTR_{a_3}(\tilde{\theta})$, thus the allocation function $\tilde{f}$ is not monotonic. \qed
\end{pf}
On the basis of the above result, we can state the following theorem.
\begin{theorem}\label{thm:regretsocialwelfarec}
Let us consider an auction with $N$ advertisers, $K$ slots, and $T$ rounds, with ad--dependent cascade model with parameters $\{c_i\}_{i=1}^N$ whose value are unknown. Any online learning DSIC mechanism achieves an expected regret $R^{SW}_T=\Theta(T)$ over the social welfare.
\end{theorem}
\begin{pf}
Call $f(\hat{\mathbf{v}}| \mathbf{c})$ the allocation function maximizing the social welfare given parameters $\mathbf{c}$. As shown above, $f(\hat{\mathbf{v}}| \tilde{\mathbf{c}})$ cannot be adopted in the exploitation phase, the mechanism would not be DSIC otherwise. However, it can be easily observed that a necessary condition to have a no--regret algorithm is that the allocation function used in the exploitation phase, say $g(\hat{\mathbf{v}}| \tilde{\mathbf{c}})$, is such that $g(\hat{\mathbf{v}}| \mathbf{c}) = f(\hat{\mathbf{v}}| \mathbf{c})$ for every $\hat{\mathbf{v}}$ and $\mathbf{c}$ (that is, they always return the same allocation) given that $\tilde{\mathbf{c}}$ are consistent estimates and $\tilde{\mathbf{c}}\rightarrow \mathbf{c}$ as $T\rightarrow +\infty$. Otherwise, since allocations are finite and the difference between the values of the allocations is generically strictly positive, the algorithm would suffer from a strictly positive regret when $T\rightarrow +\infty$ and therefore it would not be a no--regret mechanism. However, any such a $g$ would not be monotonic and therefore it cannot be adopted in a DSIC mechanism. As a result, any online learning DSIC mechanism is not a no--regret mechanism.
To complete the proof, we need to provide a mechanism with regret $\Theta(T)$. Such a mechanism can be easily obtained by partitioning ads in groups such that in each group the ads compete only for a single slot. Therefore, each ad can appear in only one slot. \qed
\end{pf}
The above result shows that no approach similar to the approach described in~\cite{babaioff_impl_pay} can be adopted even for IC in expectation. Indeed, the approach described in~\cite{babaioff_impl_pay} requires in input a monotonic allocation function. This would suggest a negative result in terms of regret also when IC in expectation. However, in this paper we leave the study of this case open.
Finally, we provide a result on the regret over the auctioneer's revenue, whose proof is straightforward given that the (W)VCG cannot be adopted due to the above result and therefore the regret over the payments cannot go to zero as $T$ goes to $\infty$.
\begin{theorem}\label{thm:constant.l}
Let us consider an auction with $N$ advertisers, $K$ slots, and $T$ rounds, with ad--dependent cascade model with parameters $\{c_i\}_{i=1}^N$ whose value are unknown. Any online learning DSIC mechanism achieves an expected regret over the auctioneer's revenue $R_T=\Theta(T)$.
\end{theorem}
\section{Monotonicity and Myerson's payments} \label{ap:monotonicity}
Consider a generic direct--revelation mechanism $M = (\mathcal{N}, \Theta, V, f, \{p_i\}_{i \in \mathcal{N}})$ as defined in Section~\ref{ssec:md}. A single--parameter linear environment is such that
\begin{itemize}
\item the type of each agent $i$ is a scalar $v_i$ (single--parameter assumption),
\item the utility function of agent~$i$ is $u_i(\hat{\mathbf{v}}) = z_i(f(\hat{\mathbf{v}})) v_i - p_i(\hat{\mathbf{v}})$ where $z_i: \Theta \rightarrow \Re$ is a function of the allocation (linear assumption).
\end{itemize}
An allocation function $f$ is \textit{monotone} in a single--parameter linear environment if
\begin{align*}
z_i(f(\hat{\mathbf{v}}_{-i}, v_i'')) \geq z_i(f(\hat{\mathbf{v}}_{-i}, v_i'))
\end{align*}
for any $v_i''\geq v_i'$. Essentially, $z_i$ is monotonically increasing in $v_i$ once $\hat{\mathbf{v}}_{-i}$ has been fixed. In such environments, it is always possible to design a DSIC mechanism imposing the following payments~\cite{tardos_sp}:
\begin{equation} \label{eq:p_tardos}
p_i(\hat{\mathbf{v}}) = h_i(\hat{\mathbf{v}}_{-i}) + z_i(f(\hat{\mathbf{v}})) \hat{v}_i - \int_{0}^{\hat{v}_i} z_i(f(\hat{\mathbf{v}}_{-i}, u)) du
\end{equation}
\noindent where $h_i(\hat{\mathbf{v}}_{-i})$ is a generic function not depending on the type of agent~$i$.
\section{Conclusions and Future Work}\label{s:conclusions}
In this paper, we studied the problem of learning the click through rates of ads in sponsored search auctions with truthful mechanisms. This problem is highly challenging, combining online learning tools (i.e., regret minimization algorithms) together with economic tools (i.e., truthful mechanisms). While almost all the literature focused on single--slot scenarios, here we focused on multi--slot scenarios. With multiple slots it is necessary to adopt a user model to characterize the valuations of the users over the different slots. Here, we adopted the cascade model, that is the most common model used in the literature. In the paper, we studied a number of scenarios, each with a specific information setting of unknown parameters. For each scenario, we designed a truthful learning mechanism, studied its economic properties, derived an upper bound over the regret, and, for some mechanisms, also a lower bound. We considered both the regret over the auctioneer's revenue and the social welfare.
We showed that for the cascade model with only position--dependent externalities it is possible to design a truthful no--regret learning mechanism for the general case in which all the parameters are unknown. Our mechanism presents a regret $O(T^{2/3})$ and it is incentive compatible in expectation over the random component of the mechanism. However, it remains open whether or not it is possible to obtain a regret $O(T^{1/2})$. For specific sub cases, in which some parameters are known to the auctioneer, we obtained better results in terms of either incentive compatibility, obtaining dominant strategy truthfulness, or regret, obtaining a regret of zero. We showed that for the cascade model with the position-- and ad--dependent externalities it is possible to design a dominant strategy truthful mechanism with a regret $O(T^{2/3})$ when only the quality is unknown. Instead, even when the cascade model is only with ad--dependent externalities and no parameter is known it is not possible to obtain a no--regret dominant strategy truthful mechanism. The proof of this result would seem to suggest that the same result holds also when truthfulness is in expectation. However, we did not produce any proof for that, leaving it for future works. Finally, we empirically evaluated the bounds we provided, showing that the dependency of each bound from the parameters is empirically confirmed.
Two main questions deserve future investigation. The first question concerns the study of a lower bound for the case in which there are only position--dependent externalities for different notions of truthfulness in expectation, e.g., both in expectation over the click realizations and in expectation over the random component of the mechanism. Furthermore, it is open whether the separation of exploration and exploitation phases is necessary and, in the negative case, whether it is possible to obtain a regret $O(T^{1/2})$. The second question concerns a similar study related to the case with only ad--dependent externalities.
\section{Introduction}\label{s:introduction}
Sponsored search auctions (SSAs) constitute one of the most successful applications of \emph{microeconomic mechanisms}, producing a revenue of about \$6 billion dollars in the US alone in the first half of 2010~\cite{IABreport2010}. In a SSA, a number of \emph{advertisers} bid to have their \emph{sponsored links} (from here on \textit{ads}) displayed in some slot alongside the search results of a keyword. Sponsored search auctions currently adopt a \emph{pay--per--click} scheme, requiring positive payments to an advertiser only if its ad has been clicked. Given an allocation of ads over the slots, each ad is associated with a \emph{click--through--rate} (CTR) defined as the probability that such ad will be clicked by the user. CTRs are estimated by the auctioneer and play a crucial role in the auction, since they are used by the auctioneer to find the optimal allocation (in expectation) and to compute the payments for each ad.
There is a large number of works formalizing SSAs as a \emph{mechanism design} problem~\cite{Narahari2009}, where the objective is to design an auction mechanism that incentivizes advertisers to bid their \emph{truthful} valuations (needed for \emph{economic stability}) and that assures both the advertisers and the auctioneer to have a non--negative utility. The most common SSA mechanism is the \emph{generalized second price} (GSP) auction~\cite{Edelman2007,Varian2007}. This mechanism is proved not to be truthful and advertisers may implement bidding strategies that gain more than bidding their truthful valuations as shown in~\cite{Edelman2007}. While in complete information settings the worst Nash equilibrium in the GSP gives a revenue to the auctioneer equal to the revenue given by the Vickrey--Clarke--Groves (VCG) equilibrium~\cite{Edelman2007}, in Bayesian settings the worst Bayes--Nash equilibrium in the GSP can provide a much smaller revenue than the VCG---a lower bound of $\frac{1}{8}$ is provided in~\cite{paes}. The implementation of the VCG mechanism (assuring truthfulness) for SSAs has been investigated in~\cite{Narahari2009}. Although the VCG mechanism is not currently adopted by the search engines (but it is, e.g., by Facebook), a number of scientific theoretical results builds upon it.
In this paper, we focus on the problem of designing truthful mechanisms when the CTRs are not known and need to be estimated in SSAs with multiple slots. This problem is particularly relevant in practice because the assumption that all the CTRs are known beforehand is rarely realistic. Furthermore, it also poses interesting scientific challenges since it represents one of the first examples where learning theory is paired with mechanism design techniques to obtain effective methods to learn under equilibrium constraints (notably the truthfulness property). Another field where these ideas have been used is crowdsourcing~\cite{crowdsourcing}.
The problem of estimating the CTRs and to identify the best allocation of ads is effectively formalized as a \textit{multi--arm bandit problem}~\cite{robbins1952some} where each ad is an arm and the objective is to minimize the cumulative regret (i.e., the revenue loss w.r.t. an optimal allocation defined according to the exact CTRs). The problem of budgeted advertisers (i.e., auctions where the total amount of money each advertiser is willing to pay is limited) with multiple queries is considered in~\cite{pandey2006handling}. This problem is formalized as a budgeted multi--bandit multi--arm problem, where each bandit corresponds to a query, and an algorithm is proposed with explicit bounds over the regret on the revenue. Nonetheless, the proposed method works in a non--strategic environment, where the advertisers do not try to influence the outcome of the auction and always bid their true values. The strategic dimension of SSAs is partially taken into consideration in~\cite{langford2010maintaining} where the advertisers are assumed to play a bidding strategy at the equilibrium w.r.t. a set of estimated CTRs which are available to both the auctioneer and the advertisers. The authors introduce a learning algorithm which explores different rankings on the ads so as to improve the CTR estimates and, at the same, not to introduce incentives for the advertisers to deviate from the previous equilibrium strategy. A more complete notion of truthfulness for bandit algorithms in multi--slot SSAs is studied in~\cite{gonen2007incentive-compatible}. In particular, they build on the action elimination algorithm in~\cite{even-dar2006action} and they report a probably approximately correct (PAC) analysis of its performance. Unfortunately, as pointed in~\cite{devanur2009price} and \cite{babaioff2008characterizing} the mechanism is not guaranteed to be truthful and thus it only works when the advertisers bid their true values. An extension to the action elimination algorithm is also proposed in~\cite{gonen2007an-adaptive} for the more general setting where budgeted advertisers are allowed to enter and exit the auction at different time instants that they declare along with their bid. The authors derive an algorithm that approximately achieves the best social welfare under the assumption that the gain of untruthful declarations is limited. Finally, single--slot online advertising is studied also in~\cite{nazerzadeh2008dynamic} where the notion of Bayesian incentive compatibility (BIC) is taken into consideration and an asymptotically BIC and \textit{ex ante} efficient mechanism is introduced. The most complete study of truthful bandit mechanisms so far is reported in~\cite{devanur2009price} and \cite{babaioff2008characterizing}. These works first provided a complete analysis on the constraints truthfulness forces on the multi--arm bandit algorithm with single--slot SSAs, showing that no \emph{dominant--strategy} truthful bandit mechanism can achieve a regret (over the social welfare and over the auctioneer's revenue) smaller than $\tilde\Omega(T^\frac{2}{3})$ and that the exploration and exploitation phases must be separate. Furthermore, they also suggest nearly--optimal algorithms. Instead, when the notion of truthfulness is relaxed, adopting truthfulness \emph{in expectation} w.r.t. click (and possibly mechanism) randomness, it is possible to obtain a regret $\tilde O(T^\frac{1}{2})$ (over the social welfare) without separating the exploration and exploitation phases in the case of single--slot SSAs~\cite{babaioff_impl_pay}.
When multiple slots are present, a user model is needed to describe how the valuations of the advertisers change over the slots. All the models available in the literature assume the separation of the CTR as the product of two terms, the first capturing the probability that an ad will be clicked once observed by the user, while the second capturing the probability that the user will observe such an ad given the displayed allocation. The basic model (commonly referred to as \emph{separability model}) prescribes that the probability of observing an ad depends only on its position~\cite{Narahari2009}. Recently, more accurate models have been proposed and the most famous model is the \emph{cascade model} according to which the user scans the slots from top to bottom and the probability with which the user moves from a slot to the next one depends on the ad and on the slot (this kind of user is commonly called \emph{Markovian user})~\cite{Kempe2008,Aggarwal2008}, while with the remaining probability the user stops to observe ads. As a result, the probability of observing an ad depends on position of the ad and on all the ads allocated above. The validity of the cascade model has been evaluated and supported by a wide range of experimental investigations~\cite{Craswell,Joachims}. The only results on learning mechanisms for SSAs with multiple slots are described in~\cite{sarma2010multi-armed}, where the authors characterize dominant--strategy truthful mechanisms and provide theoretical bounds over the social welfare regret for the separability model. However, these results are partial, e.g., they do not solve the common case in which the slot--dependent parameters are monotonically decreasing in the slots, and they cannot easily be extended to the more challenging case of the cascade model (see discussion in Section~\ref{ss:online.mechanism}).
In the present paper, we build on the results available in the literature and we provide a number of contributions when the separability model and the cascade model are adopted. More precisely, our results can be summarized as follow.
\begin{itemize}
\item \emph{Separability model with monotone parameters/only position--dependent cascade model}: in this case, there are two groups of parameters, one related to the ads (called \emph{quality}) and one to the slots (called \emph{prominence}). We studied all configurations of information incompleteness. When only qualities are unknown, we provide a regret analysis in dominant--strategy truthfulness obtaining a regret of $\tilde O(T^\frac{2}{3})$ (while it is open whether it is possible to obtain a better upper bound adopting truthfulness in expectation). When only prominences are unknown, we provide a regret analysis in truthfulness in expectation obtaining a regret of $0$, whereas we show that any dominant--strategy truthful learning mechanism would have a regret of $\tilde{\Theta}(T)$. When both groups of parameters are unknown, we provide a regret analysis in truthfulness in expectation obtaining a regret of $\tilde {O}(T^\frac{2}{3})$ (while it is open whether it is possible to obtain a better upper bound adopting truthfulness in expectation), whereas any dominant--strategy truthful learning mechanism would have a regret of $\tilde{\Theta}(T)$.
\item \emph{Cascade model}: in the non--factorized cascade model (i.e., when the observation probabilities can be any) we show that it is possible to obtain a regret of $\tilde {O}(T^\frac{2}{3})$ in dominant--strategy truthful learning mechanisms when only the qualities of the ads are unknown. We show also that in the factorized cascade model (i.e., when the observation probabilities are the products of terms depending only on the position or on the ads as used in~\cite{Kempe2008}), in the very special case in which the ad--dependent parameters are unknown we obtain a regret of $\tilde{\Theta}(T)$ in dominant--strategy truthful learning mechanisms (while it is open whether it is possible to obtain a better upper bound adopting truthfulness in expectation).
\item \emph{Learning parameters}: for each setting of uncertainty we study we provide functions, to be used in practice, to set the learning parameters in order to minimize the bound over the regret given the parameters in input.
\item \emph{Numerical simulations}: we investigate the accuracy of all the bounds we provide in the paper in predicting the dependency of the regret on the auction parameters by numerical simulations. We show that the theoretical dependency matches the actual dependency we observed by simulation.
\end{itemize}
The paper is organized as follows. In Section~\ref{s:notation} we briefly review the basics of mechanism design and multi--armed bandit learning. Section~\ref{s:statement} formalizes sponsored search auctions and introduces the corresponding online learning mechanism design problem. In Section~\ref{s:statement} we also provide a more formal overview of existing results in comparison with the findings of this paper. In Sections~\ref{s:constant} and~\ref{s:externalities} we report and discuss the main regret bounds in the case of position--dependent and position-- and ad--dependent externalities. In Section~\ref{s:experiments} we report numerical simulations aiming at testing the accuracy of the theoretical bounds. Section~\ref{s:conclusions} concludes the paper and proposes future directions of investigation. The detailed proofs of the theorems are reported in Appendix.
\section{Preliminaries}\label{s:notation}
\subsection{Economic Mechanisms}
\input{sec/021economicMechanisms}
\subsection{Multi--Armed Bandit}
\input{sec/022MultiArmedBandit}
|
3,212,635,537,939 | arxiv | \section{Introduction}
Let $G$ be a simple graph on the vertex set $V(G) = [n]$ with the edge set $E(G)$.
$S \subset V(G)$ is said to be {\em stable} if $\{i, j\} \not\in E(G)$ for all $i$, $j \in S$.
Note that $\emptyset$ is stable.
For each stable set $S$ of $G$, we define $\rho(S) = \sum_{i \in S} {\bf e}_{i} \in \bb{R}^{n}$, where ${\bf e}_{i}$ is the $i$-th unit coordinate vector
in $\bb{R}^{n}$.
The convex hull of $\{\rho(S) \mid S \text{\ is a stable set of\ } G \}$ is called the {\em stable set polytope} of $G$ (see \cite{C} ) , denoted by $\mathcal{Q}_{G}$.
$\mathcal{Q}_{G}$ is a kind of $(0, 1)$-polytope.
For this polytope, we define the subring of $k[T, X_{1}, \ldots, X_{n}]$ as follows:
\begin{center}
$k[\mathcal{Q}_{G}] := k[T\cdot X_{1}^{a_{1}} \cdots X_{n}^{a_{n}} \mid (a_{1}, \ldots, a_{n}) \text{\ is a vertex of } \mathcal{Q}_{G}]$,
\end{center}
\noindent where $k$ is a field.
$k[\mathcal{Q}_{G}]$ is called the {\em toric ring associated with the stable set polytope of } $G$.
We can regard $k[\mathcal{Q}_{G}]$ as a graded $k$-algebra by setting $\deg T\cdot X_{1}^{a_{1}} \cdots X_{n}^{a_{n}} = 1$.
In the theory of graded algebras, the notion of Koszulness (introduced by Priddy \cite{P} ) plays an important role and is closely related to the Gr\"{o}bner basis theory.
Let $\mathcal{P}$ be an integral convex polytope (i.e., a convex polytope each of whose vertices has integer coordinates)
and $k[\mathcal{P}] := k[T\cdot X_{1}^{a_{1}} \cdots X_{n}^{a_{n}} \mid (a_{1}, \ldots, a_{n}) \text{\ is a vertex of } \mathcal{P}]$ be the toric ring associated with $\mathcal{P}$.
In general, it is known that
\begin{center}
The defining ideal of $k[\mathcal{P}]$ possesses a quadratic Gr\"{o}bner basis
$\Downarrow$
$k[\mathcal{P}]$ is Koszul
$\Downarrow$
The defining ideal of $k[\mathcal{P}]$ is generated by quadratic binomials
\end{center}
\vspace{2mm}
\noindent follows from general theory (for example, see \cite{BHeV}).
In this note, we study the notion of a {\em strongly Koszul} algebra.
In \cite{HeHiR}, Herzog, Hibi, and Restuccia introduced this concept
and discussed the basic properties of strongly Koszul algebras.
Moreover, they proposed the conjecture that the strong Koszulness of $R$ is at the top of the above hierarchy, that is,
\begin{conj}[see \cite{HeHiR}]
The defining ideal of a strongly Koszul algebra $k[\mathcal{P}]$ possesses a quadratic Gr\"{o}bner basis.
\end{conj}
A ring $R$ is {\em trivial} if $R$ can be constructed by
starting from polynomial rings and repeatedly applying tensor and Segre products.
In this note, we propose the following conjecture.
\begin{conj}
Let $\mathcal{P}$ be a $(0, 1)$-polytope and $k[\mathcal{P}]$ be the toric ring generated by $\mathcal{P}$.
If $k[\mathcal{P}]$ is strongly Koszul, then $k[\mathcal{P}]$ is trivial.
\end{conj}
In the case of a $(0, 1)$-polytope, Conjecture 1.2 implies Conjecture 1.1.
If $\mathcal{P}$ is an order polytope or an edge polytope of bipartite graphs, then Conjecture 1.2 holds true \cite{HeHiR}.
In this note, we prove Conjecture 1.2 for stable set polytopes.
The main theorem of this note is the following:
\begin{thm}
Let $G$ be a graph.
Then the following assertions are equivalent:
\begin{enumerate}
\item $k[\mathcal{Q}_{G}]$ is strongly Koszul.
\item $G$ is a trivially perfect graph.
\end{enumerate}
In particular, if $k[\mathcal{Q}_{G}]$ is strongly Koszul, then $k[\mathcal{Q}_{G}]$ is trivial.
\end{thm}
Throughout this note, we will use the standard terminologies of graph theory in \cite{Diest}.
\section{Strongly Koszul algebra}
Let $k$ be a field, $R$ be a graded $k$-algebra,
and $\mf{m} = R_{+}$ be the homogeneous maximal ideal of $R$.
\begin{defn}[\cite{HeHiR}]
A graded $k$-algebra $R$ is said to be {\em strongly Koszul} if $\mf{m}$ admits a minimal system of generators $\{u_{1}, \ldots, u_{t}\}$
which satisfies the following condition:
\begin{quote}
For all subsequences $u_{i_{1}}, \ldots, u_{i_{r}}$ of $\{u_{1}, \ldots, u_{t}\}$ $(i_{1} \le \cdots \le i_{r})$ and for all $j = 1, \ldots, r - 1$,
$(u_{i_{1}}, \ldots, u_{i_{j - 1}}) : u_{i_{j}}$ is generated by a subset of elements of $\{u_{1}, \ldots, u_{t}\}$.
\end{quote}
\end{defn}
A graded $k$-algebra $R$ is called Koszul if $k = R/\mf{m}$ has a linear resolution.
By the following theorem, we can see that a strongly Koszul algebra is Koszul.
\begin{prop}[{[HeHiR, Theorem 1.2]}]
If $R$ is strongly Koszul with respect to the minimal homogeneous generators $\{u_{1}, \ldots, u_{t}\}$ of $\mf{m} = R_{+}$,
then for all subsequences $\{u_{i_{1}}, \ldots, u_{i_{r}}\}$ of $\{u_{1}, \ldots, u_{t}\}$, $R/(u_{i_{1}}, \ldots, u_{i_{r}})$ has a linear resolution.
\end{prop}
The following proposition plays an important role in the proof of the main theorem.
\begin{thm}[{[HeHiR, Proposition 2.1]}]
Let $S$ be a semigroup and $R = k[S]$ be the semigroup ring generated by $S$.
Let $\{u_{1}, \ldots, u_{t}\}$ be the generators of $\mf{m} = R_{+}$ which correspond to the generators of $S$.
Then, if $R$ is strongly Koszul, then for all subsequences $\{u_{i_{1}}, \ldots, u_{i_{r}}\}$ of $\{u_{1}, \ldots, u_{t}\}$,
$R/(u_{i_{1}}, \ldots, u_{i_{r}})$ is also strongly Koszul.
\end{thm}
By this theorem, we have
\begin{cor}
If $k[\mca{Q}_{G}]$ is strongly Koszul, then $k[\mca{Q}_{G_{W}}]$ is strongly Koszul for all induced subgraphs $G_{W}$ of $G$.
\end{cor}
\section{Hibi ring and comparability graph}
In this section, we introduce the concepts of a Hibi ring and a comparability graph.
Both are defined with respect to a partially ordered set.
Let $P = \{p_{1}, \ldots, p_{n}\}$ be a finite partially ordered set consisting of $n$ elements, which is referred to as a {\em poset}.
Let $J(P)$ be the set of all poset ideals of $P$, where a poset ideal of $P$ is a subset $I$ of $P$
such that if $x \in I$, $y \in P$, and $y \le x$, then $y \in I$.
Note that $\emptyset \in J(P)$.
First, we give the definition of the Hibi ring introduced by Hibi.
\begin{defn}[\cite{Hib}]
For a poset $P = \{p_{1}, \ldots, p_{n}\}$, the {\em Hibi ring} $\mca{R}_{k}[P]$ is defined as follows:
\vspace{3mm}
\[
\mca{R}_{k}[P] := k[T\cdot \prod_{i \in I}X_{i} \mid I \in J(P)] \subset k[T, X_{1}, \ldots, X_{n}]
\]
\end{defn}
\begin{ex}
Consider the following poset $P = (1 \le 3, 2 \le 3$ and $2 \le 4)$.
\vspace{2mm}
\begin{xy}
\ar@{} (0, 0); (10, -16) = "A",
\ar@{} "A" *{P = }; (24, -24) *++!R{1} *\dir<4pt>{*} = "B",
\ar@{-} "B"; (24, -8) *++!R{3} *\dir<4pt>{*} = "C",
\ar@{-} "C"; (48, -24) *++!L{2} *\dir<4pt>{*} = "D",
\ar@{-} "D"; (48, -8) *++!L{4} *\dir<4pt>{*} = "E",
\ar@{} "E"; (72, -16) *{J(P) = } = "F",
\ar@{} "F"; (108, -16) *++!R{\{1, 2\}\ } *\dir<4pt>{*} = "G",
\ar@{} "G"; (132, -16) *++!L{\{2, 4\}} *\dir<4pt>{*} = "H",
\ar@{-} "G"; (120, -8) *++!L{\ \{1, 2, 4\}} *\dir<4pt>{*} = "I",
\ar@{-} "H"; "I",
\ar@{} "I"; (120, -24) *++!L{\ \{2\}} *\dir<4pt>{*} = "J",
\ar@{-} "G"; "J",
\ar@{-} "H"; "J",
\ar@{} "I", (96, -8) *++!R{\{1, 2, 3\}} *\dir<4pt>{*} = "K",
\ar@{-} "G"; "K",
\ar@{} "K", (108, 0) *++!D{\{1, 2, 3, 4\}} *\dir<4pt>{*} = "L",
\ar@{-} "I"; "L",
\ar@{-} "K"; "L",
\ar@{-} "G"; (96, -24) *++!R{\{1\}} *\dir<4pt>{*} = "M",
\ar@{-} "M"; (108, -32) *++!U{\emptyset} *\dir<4pt>{*} = "N",
\ar@{-} "J"; "N",
\end{xy}
\noindent Then we have
\[
\mathcal{R}_{k}[P] = k[T, TX_{1}, TX_{2}, TX_{1}X_{2}, TX_{2}X_{4}, TX_{1}X_{2}X_{3}, TX_{1}X_{2}X_{4}, TX_{1}X_{2}X_{3}X_{4}].
\]
\end{ex}
Hibi showed that a Hibi ring is always normal.
Moreover, a Hibi ring can be represented as a factor ring of a polynomial ring: if we let
\[
I_{P} := (X_{I}X_{J} - X_{I \cap J}X_{I \cup J} \mid I, J \in J(P), I \not\subseteq J \ \text{and} \ J \not\subseteq I)
\]
be the binomial ideal in the polynomial ring $k[X_{I} \mid I \in J(P)]$ defined by a poset $P$, then $\mca{R}_{k}[P] \cong k[X_{I} \mid I \in J(P)] / I_{P}$.
Hibi also showed that $I_{P}$ has a quadratic Gr\"{o}bner basis for any term order which satisfies the following condition: the initial term of $X_{I}X_{J} - X_{I \cap J}X_{I \cup J}$ is
$X_{I}X_{J}$.
Hence a Hibi ring is always Koszul from general theory.
Next, we introduce the concept of a comparability graph.
\begin{defn}
A graph $G$ is called a {\em comparability graph} if there exists a poset $P$ which satisfies the following condition:
\[
\{i, j\} \in E(G) \iff i \ge j \hspace{3mm} \text{or} \hspace{3mm} i \le j \hspace{3mm} \text{in} \hspace{3mm} P.
\]
We denote the comparability graph of $P$ by $G(P)$.
\end{defn}
\begin{ex}
The lower-left poset $P$ defines the comparability graph $G(P)$.
\vspace{8mm}
\begin{xy}
\ar@{} (0, 0) ; (30, 10) *{P = } = "A",
\ar@{} (0, 0) ; (44, 0) *\dir<4pt>{*} = "B",
\ar@{-} "B"; (54, 10) *\dir<4pt>{*} = "C",
\ar@{-} "C"; (44, 20) *\dir<4pt>{*} = "D",
\ar@{-} "C"; (64, 0) *\dir<4pt>{*} = "E",
\ar@{-} "C"; (64, 20) *\dir<4pt>{*} = "F",
\ar@{} (0, 0) ; (80, 10) *{G(P) = } = "G",
\ar@{} (0, 0) ; (104, 0) *\dir<4pt>{*} = "H",
\ar@{-} "H" ; (114, 0) *\dir<4pt>{*} = "I",
\ar@{} "H" ; (98, 10) *\dir<4pt>{*} = "J",
\ar@{-} "H" ; (120, 10) *\dir<4pt>{*} = "K",
\ar@{-} "H" ; (114, 0) *\dir<4pt>{*} = "I",
\ar@{-} "H" ; (109, 20) *\dir<4pt>{*} = "L",
\ar@{-} "I" ; "J",
\ar@{-} "J" ; "K",
\ar@{-} "J" ; "L",
\ar@{-} "I" ; "L",
\ar@{-} "L" ; "K",
\end{xy}
\end{ex}
\vspace{3mm}
\begin{rem}
It is possible that $P \neq P^{'}$ but $G(P) = G(P^{'})$.
Indeed, for the following poset $P^{'}$, $G(P^{'})$ is identical to $G(P)$ in the above example.
\vspace{5mm}
\begin{xy}
\ar@{} (0, 0) ; (52, 10) *{P^{'} = } = "A",
\ar@{} (0, 0) ; (66, 0) *\dir<4pt>{*} = "B",
\ar@{} "B"; (76, 20) *\dir<4pt>{*} = "C",
\ar@{-} "C"; (66, 10) *\dir<4pt>{*} = "D",
\ar@{} "C"; (86, 0) *\dir<4pt>{*} = "E",
\ar@{-} "C"; (86, 10) *\dir<4pt>{*} = "F",
\ar@{-} "B" ; "D" ;
\ar@{-} "B" ; "F" ;
\ar@{-} "D" ; "E" ;
\ar@{-} "E" ; "F" ;
\end{xy}
\vspace{5mm}
\end{rem}
Complete graphs are comparability graphs of totally ordered sets.
Bipartite graphs and trivially perfect graphs (see the next section) are also comparability graphs.
Moreover, if $G$ is a comparability graph, then the suspension (e.g., see [HiNOS, p.4]) of $G$ is also a comparability graph.
Recall the following definitions of two types of polytope which are defined by a poset.
\begin{defn}[see {[St1]}]
Let $P = \{p_{1}, \ldots, p_{n}\}$ be a finite poset.
\begin{enumerate}
\item The {\em order polytope} $\mca{O} (P)$ of $P$ is the convex polytope which consists of
$(a_{1}, \ldots, a_{n}) \in \bb{R}^{n}$ such that $0 \le a_{i} \le 1$ with $a_{i} \ge a_{j}$ if $p_{i} \le p_{j}$ in $P$.
\item The {\em chain polytope} $\mca{C} (P)$ of $P$ is the convex polytope which consists of
$(a_{1}, \ldots, a_{n}) \in \bb{R}^{n}$ such that $0 \le a_{i} \le 1$ with $a_{i_{1}} + \cdots + a_{i_{k}} \le 1$
for all maximal chain $p_{i_{1}} < \cdots < p_{i_{k}}$ of $P$.
\end{enumerate}
\end{defn}
Let $\mca{C} (P)$ and $\mca{O} (P)$ be the chain polytope and order polytope of a finite poset $P$, respectively.
In \cite{St1}, Stanley proved that
\vspace{3mm}
\begin{center}
$\{\text{The vertices of\ } \mca{O} (P)\} = \{\rho(I) \mid I \text{\ is a poset ideal of\ } P\},$
\vspace{2mm}
\hspace{2mm} $\{\text{The vertices of\ } \mca{C} (P)\} = \{\rho(A) \mid A \text{\ is an anti-chain of\ } P\}$,
\end{center}
\vspace{3mm}
\noindent where $A = \{p_{i_{1}}, \ldots, p_{i_{k}}\}$ is an anti-chain of $P$ if $p_{i_{s}} \not\le p_{i_{t}}$ and $p_{i_{s}} \not\ge p_{i_{t}}$
for all $s \neq t$.
Hence we have $\mca{Q}_{G(P)} = \mca{C} (P)$.
In \cite{HiL}, Hibi and Li answered the question of when $\mca{C} (P)$ and $\mca{O} (P)$ are unimodularly equivalent.
From their study, we have the following theorem.
\begin{thm}[{[HiL, Theorem 2.1]}]
Let $P$ be a poset and $G(P)$ be the comparability graph of $P$.
Then the following are equivalent:
\begin{enumerate}
\item The X-poset in Example 3.4 does not appear as a subposet (refer to [St2, Chapter 3]) of $P$.
\item $\mca{R}_{k}[P] \cong k[\mca{Q}_{G(P)}]$.
\end{enumerate}
\end{thm}
\newpage
\begin{ex}
The cycle of length $4$ $C_{4}$ and the path of length $3$ $P_{4}$ are comparability graphs of $Q_{1}$ and $Q_{2}$, respectively.
\vspace{8mm}
\begin{xy}
\ar@{} (0, 0) ; (30, 0) *{Q_{1} = } ,
\ar@{} (0, 0) ; (44, -10) *\dir<4pt>{*} = "A",
\ar@{} "A"; (64, -10) *\dir<4pt>{*} = "B",
\ar@{-} "A"; (44, 10) *\dir<4pt>{*} = "C",
\ar@{-} "B"; (64, 10) *\dir<4pt>{*} = "D",
\ar@{-} "B" ; "C",
\ar@{-} "A" ; "D",
\ar@{} (0, 0) ; (86, 0) *{Q_{2} = } ,
\ar@{} (0, 0) ; (100, -10) *\dir<4pt>{*} = "E",
\ar@{} "E" ; (120, -10) *\dir<4pt>{*} = "F",
\ar@{-} "E" ; (100, 10) *\dir<4pt>{*} = "G",
\ar@{-} "F" ; (120, 10) *\dir<4pt>{*} = "H",
\ar@{-} "F" ; "G",
\end{xy}
\vspace{8mm}
\noindent Hence $k[\mca{Q}_{C_{4}}] \cong \mca{R}_{k}[Q_{1}]$ and $k[\mca{Q}_{P_{4}}] \cong \mca{R}_{k}[Q_{2}]$.
\end{ex}
A ring $R$ is {\em trivial} if $R$ can be constructed by
starting from polynomial rings and repeatedly applying tensor and Segre products.
Herzog, Hibi and Restuccia gave an answer for the question of when is a Hibi ring strongly Koszul.
\begin{thm}[see {[HeHiR, Theorem 3.2]}]
Let $P$ be a poset and $R = \mca{R}_{k}[P]$ be the Hibi ring constructed from $P$.
Then the following assertions are equivalent:
\begin{enumerate}
\item $R$ is strongly Koszul.
\item $R$ is trivial.
\item The N-poset as described below does not appear as a subposet of $P$.
\end{enumerate}
\vspace{5mm}
\begin{xy}
\ar@{} (0, 0) ; (54, -10) *\dir<4pt>{*} = "A",
\ar@{-} "A" ; (54, 10) *\dir<4pt>{*} = "B",
\ar@{-} "B" ; (62, 6) *\dir<4pt>{} = "C",
\ar@{.} "C" ; (86, -6) *\dir<4pt>{} = "D",
\ar@{-} "D" ; (94, -10) *\dir<4pt>{*} = "E",
\ar@{-} "E" ; (94, 10) *\dir<4pt>{*} = "F",
\end{xy}
\vspace{5mm}
\end{thm}
By this theorem, Corollary 2.4, and Example 3.8, we have
\begin{cor}
If $G$ contains $C_{4}$ or $P_{4}$ as an induced subgraph, then $k[\mca{Q}_{G}]$ is not strongly Koszul.
\end{cor}
\section{trivially perfect graph}
In this section, we introduce the concept of a trivially perfect graph.
As its name suggests, a trivially perfect graph is a kind of perfect graph; it is also a kind of comparability graph, as described below.
\begin{defn}
For a graph $G$, we set
\[
\alpha(G) := \max\{\#S \mid S \text{\ is a stable set of\ }G\},
\]
\[
m(G) := \#\{ \text{the set of maximal cliques of } G\}.
\]
\vspace{3mm}
We call $\alpha(G)$ the {\em stability number} (or {\em independence number}) of $G$.
\end{defn}
In general, $\alpha(G) \le m(G)$.
Moreover, if $G$ is chordal, then $m(G) \le n$ by Dirac's theorem \cite{Dir}.
In \cite{G}, Golumbic introduced the concept of a trivially perfect graph.
\begin{defn}[\cite{G}]
We say that a graph $G$ is {\em trivially perfect} if $\alpha(G_{W}) = m(G_{W})$ for any induced subgraph $G_{W}$ of $G$.
\end{defn}
For example, complete graphs and star graphs (i.e., the complete bipartite graph $K_{1, r}$) are trivially perfect.
We define some additional concepts related to perfect graphs.
Let $C_{G}$ be the set of all cliques of $G$.
Then we define
\[
\omega(G) := \max\{\#C \mid C \in C_{G}\},
\]
\[
\theta(G) := \min\{s \mid C_{1} \coprod \cdots \coprod C_{s} = V(G), C_{i} \in C_{G} \},
\]
\[
\chi(G) := \theta(\overline{G}),
\]
\vspace{3mm}
\noindent where $\overline{G}$ is the complement of $G$.
These invariants are called the {\em clique number}, {\em clique covering number}, and {\em chromatic number} of $G$, respectively.
In general, $\alpha(G) = \omega(\overline{G})$, $\theta(G) \le m(G)$ and $\omega(G) \le \chi(G)$.
The definition of a perfect graph is as follows.
\begin{defn}
We say that a graph $G$ is {\em perfect} if $\omega(G_{W}) = \chi(G_{W})$ for any induced subgraph $G_{W}$ of $G$.
\end{defn}
Lov\'{a}sz proved that $G$ is perfect if and only if $\overline{G}$ is perfect \cite{Lo}.
The theorem is now called the weak perfect graph theorem.
With it, it is easy to show that a trivially perfect graph is perfect.
\begin{prop}
Trivially perfect graphs are perfect.
\end{prop}
\begin{proof}
Assume that $G$ is trivially perfect.
By \cite{Lo}, it is enough to show that $\overline{G}$ is perfect.
For all induced subgraphs $\overline{G}_{W}$ of $\overline{G}$, we have
\[
m(G_{W}) = \alpha(G_{W}) = \omega(\overline{G_{W}}) \le \chi(\overline{G_{W}}) = \theta(G_{W}) \le m(G_{W})
\]
\noindent by general theory (note that $\overline{G}_{W} = \overline{G_{W}}$).
\end{proof}
Golumbic gave a characterization of trivially perfect graphs.
\begin{thm}[{[G, Theorem 2]}]
The following assertions are equivalent:
\begin{enumerate}
\item $G$ is trivially perfect.
\item $G$ is {\em $C_{4}, P_{4}$-free}, that is, $G$ contains neither $C_{4}$ nor $P_{4}$ as an induced subgraph.
\end{enumerate}
\end{thm}
\begin{proof}
$(1) \Rightarrow (2)$:
It is clear since $\alpha(C_{4}) = 2$, $m(C_{4}) = 4$, and $\alpha(P_{4}) = 2$, $m(P_{4}) = 3$.
$(2) \Rightarrow (1)$:
Assume that $G$ contains neither $C_{4}$ nor $P_{4}$ as an induced subgraph.
If $G$ is not trivially perfect, then there exists an induced subgraph $G_{W}$ of $G$
such that $\alpha(G_{W}) < m(G_{W})$.
For this $G_{W}$, there exists a maximal stable set $S_{W}$ of $G_{W}$ which satisfies the following:
\vspace{2mm}
There exists $s \in S_{W}$ such that $s \in C_{1} \cap C_{2}$ for some distinct pair of cliques $C_{1}$, $C_{2} \in C_{G_{W}}$.
\vspace{2mm}
\noindent Note that $\#S_{W} > 1$ since $G_{W}$ is not complete.
Then there exist $x \in C_{1}$ and $y \in C_{2}$ such that $\{x, s\}$, $\{y, s\} \in E(G_{W})$ and $\{x, y\} \not\in E(G_{W})$.
Let $u \in S_{W} \setminus \{s\}$.
If $\{x, u\} \in E(G_{W})$ or $\{y, u\} \in E(G_{W})$, then the induced graph $G_{\{x, y, s, u\}}$ is $C_{4}$ or $P_{4}$, a contradiction.
Hence $\{x, u\} \not\in E(G_{W})$ and $\{y, u\} \not\in E(G_{W})$.
Then $\{x, y\} \cup \{S \setminus \{s\}\}$ is a stable set of $G_{W}$, which contradicts that $S$ is maximal.
Therefore, $G$ is trivially perfect.
\end{proof}
Next, we show that a trivially perfect graph is a kind of comparability graph.
First, we define the notion of a tree poset.
\begin{defn}[see \cite{W}]
A poset $P$ is a {\em tree} if it satisfies the following conditions:
\begin{enumerate}
\item Each of the connected components of $P$ has a minimal element.
\item For all $p$, $p^{'} \in P$, the following assertion holds: if there exists $q \in P$ such that $p, p^{'} \le q$, then $p \le p^{'}$ or $p \ge p^{'}$.
\end{enumerate}
\end{defn}
\begin{ex}
The following poset is a tree:
\vspace{8mm}
\begin{xy}
\ar@{} (0, 0) ; (75, 0) *\dir<4pt>{*} = "A",
\ar@{-} "A" ; (60, 10) *\dir<4pt>{*} = "B",
\ar@{-} "A" ; (75, 10) *\dir<4pt>{*} = "C",
\ar@{-} "A" ; (90, 10) *\dir<4pt>{*} = "D",
\ar@{-} "C" ; (75, 20) *\dir<4pt>{*} = "E",
\ar@{-} "C" ; (90, 20) *\dir<4pt>{*} = "F",
\ar@{-} "D" ; (105, 20) *\dir<4pt>{*} = "G",
\end{xy}
\vspace{5mm}
\end{ex}
Tree posets can be characterized as follows.
\begin{prop}
Let $P$ be a poset.
Then the following assertions are equivalent:
\begin{enumerate}
\item $P$ is a tree.
\item Neither the X-poset in Example 3.4, the N-poset in Theorem 3.9, nor the diamond poset as described below
appears as a subposet of $P$.
\end{enumerate}
\vspace{5mm}
\begin{xy}
\ar@{} (0, 0) ; (75, 0) *\dir<4pt>{*} = "A",
\ar@{-} "A" ; (72, 2) *\dir<4pt>{} = "B",
\ar@{.} "B" ; (63, 8) *\dir<4pt>{} = "C",
\ar@{-} "C" ; (60, 10) *\dir<4pt>{*} = "D",
\ar@{-} "A" ; (78, 2) *\dir<4pt>{} = "E",
\ar@{.} "E" ; (87, 8) *\dir<4pt>{} = "F",
\ar@{-} "F" ; (90, 10) *\dir<4pt>{*} = "G",
\ar@{-} "D" ; (63, 12) *\dir<4pt>{} = "H",
\ar@{.} "H" ; (72, 18) *\dir<4pt>{} = "I",
\ar@{-} "I" ; (75, 20) *\dir<4pt>{*} = "J",
\ar@{-} "G" ; (87, 12) *\dir<4pt>{} = "K",
\ar@{.} "K" ; (78, 18) *\dir<4pt>{} = "L",
\ar@{-} "L" ; "J",
\end{xy}
\vspace{3mm}
\end{prop}
In \cite{W}, Wolk discussed the properties of the comparability graphs of a tree poset
and showed that such graphs are exactly the graphs that satisfy the ``diagonal condition".
This condition is equivalent to being $C_{4}$, $P_{4}$-free, and hence we have
\begin{cor}
Let $G$ be a graph.
Then the following assertions are equivalent:
\begin{enumerate}
\item $G$ is trivially perfect.
\item $G$ is a comparability graph of a tree poset.
\item $G$ is $C_{4}$, $P_{4}$-free.
\end{enumerate}
\end{cor}
\begin{rem}
A graph $G$ is a {\em threshold graph} if it can be constructed from a one-vertex graph by repeated applications of the following two operations:
\begin{enumerate}
\item Add a single isolated vertex to the graph.
\item Take a suspension of the graph.
\end{enumerate}
\end{rem}
The concept of a threshold graph was introduced by Chv\'{a}tal and Hammer \cite{CHam}.
They proved that $G$ is a threshold graph if and only if $G$ is $C_{4}$, $P_{4}$, $2K_{2}$-free.
Hence a trivially perfect graph is also called a {\em quasi-threshold graph}.
\newpage
\section{Proof of Main theorem}
In this section, we prove the main theorem.
\begin{thm}
Let $G$ be a graph.
Then the following assertions are equivalent:
\begin{enumerate}
\item $k[\mathcal{Q}_{G}]$ is strongly Koszul.
\item $G$ is trivially perfect.
\end{enumerate}
\end{thm}
\begin{proof}
We assume that $G$ is trivially perfect.
Then there exists a tree poset $P$ such that $G = G(P)$
from Corollary 4.9.
This implies that neither the X-poset in Example 3.4 nor the N-poset in Theorem 3.9 appears as a subposet of $P$ by Proposition 4.8 , and hence
$k[\mca{Q}_{G(P)}] \cong \mca{R}_{k}[P]$ is strongly Koszul by Theorems 3.7 and 3.9.
Conversely, if $G$ is not trivially perfect, $G$ contains $C_{4}$ or $P_{4}$ as an induced subgraph by Corollary 4.9.
Therefore, we have that $k[\mca{Q}_{G}]$ is not strongly Koszul by Corollary 3.10.
\end{proof}
\section{Remark on usual Koszulness of $k[\mca{Q}_{G}]$}
It seems to be difficult to give a complete characterization of when $k[\mca{Q}_{G}]$ is Koszul.
However, it is known that $k[\mca{Q}_{G}]$ is Koszul for many graphs $G$.
\begin{thm}[\cite{EN}]
If $G$ is an almost bipartite graph, then $k[\mca{Q}_{G}]$ is Koszul, where a graph $G$ is almost bipartite
if there exists a vertex $v \in [n]$ such that the induced subgraph $G_{[n] \setminus v}$ is bipartite, that is, does not contain
induced odd cycles.
\end{thm}
\begin{rem}
An almost bipartite graph is one such that all its odd cycles share a common vertex.
Hence if $G$ is almost bipartite, then $G$ is $K_{4}$-free, that is, $\omega(G) \le 3$.
In the case of $n \le 5$, $G$ is almost bipartite if and only if $G$ is $K_{4}$-free.
\end{rem}
Next, we recall the theorem of Hibi and Li (Theorem 3.7).
A graph $G$ is an {\em HL-comparability} graph if it is the comparability graph of a poset $P$ which
does not contain the X-poset in Example 3.4 as a subposet.
From their theorem, we have
\begin{thm}
If $G$ is an HL-comparability graph, then $k[\mca{Q}_{G}]$ is Koszul.
\end{thm}
\begin{rem}
\begin{enumerate}
\item If $n \le 5$, the notion of HL-comparability is equivalent to the usual comparability.
\item Bipartite graphs are comparability graphs defined by posets with $\rank P \le 1$.
Hence bipartite graphs are HL-comparability graphs.
\item Let $G$ be a complete $r$-partite graph with $V(G) = \prod_{i = 1}^{r} V_{i}$.
Then $G$ is an HL-comparability graph if and only if $\#\{V_{i} \mid \#V_{i} = 1\} \ge r - 2$.
\item Let $G$ be a closed graph (see \cite{HeHiHrKR}) which satisfies the following condition:
for all $C_{1}, C_{2} \in C_{G}$, $\#\{C_{1} \cap C_{2}\} \le 1$.
Then $G$ is an HL-comparability graph.
\end{enumerate}
\end{rem}
As the end of this note, we give a classification table of connected six-vertex graphs using Harary \cite{Har}.
\newpage
\begin{xy}
\ar@{} (0, 0);(100, 0) *\txt{Classification - six-vertex (112 items)};
\ar@{-} (0, 0);(65, 0);
\ar@{-} (134, 0);(150, 0) = "A";
\ar@{-} (0, 0);(0, -220) = "H";
\ar@{-} "A";(150, -220) = "I";
\ar@{-} "H";"I";
\textcolor{green}{\ar@{-} (8, -6);(20, -6);}
\textcolor{green}{\ar@{} (0, 0);(36, -6) *\txt{Almost Bipartite};}
\textcolor{green}{\ar@{-} (52, -6);(85, -6);}
\textcolor{green}{\ar@{-} (4, -6);(4, -182);}
\textcolor{green}{\ar@{-} (82, -6);(82, -182);}
\textcolor{green}{\ar@{-} (1, -182);(81, -182);}
\ar@{} (0, 0);(8, -12) *\dir<2pt>{*} = "B13";
\ar@{-} "B13";(6, -16) *\dir<2pt>{*} = "C13";
\ar@{} "C13";(8, -20) *\dir<2pt>{*} = "D13";
\ar@{-} "D13";(12, -20) *\dir<2pt>{*} = "E13";
\ar@{} "E13";(14, -16) *\dir<2pt>{*} = "F13";
\ar@{-} "F13";(12, -12) *\dir<2pt>{*} = "G13";
\ar@{-} "B13";"G13";
\ar@{-} "B13";"D13";
\ar@{} "B13";"E13";
\ar@{} "B13";"F13";
\ar@{} "C13";"E13";
\ar@{} "C13";"F13";
\ar@{} "C13";"G13";
\ar@{} "D13";"F13";
\ar@{-} "D13";"G13";
\ar@{} "E13";"G13";
\ar@{} (0, 0);(20, -12) *\dir<2pt>{*} = "B18";
\ar@{-} "B18";(18, -16) *\dir<2pt>{*} = "C18";
\ar@{-} "C18";(20, -20) *\dir<2pt>{*} = "D18";
\ar@{-} "D18";(24, -20) *\dir<2pt>{*} = "E18";
\ar@{-} "E18";(26, -16) *\dir<2pt>{*} = "F18";
\ar@{-} "F18";(24, -12) *\dir<2pt>{*} = "G18";
\ar@{} "B18";"G18";
\ar@{} "B18";"D18";
\ar@{} "B18";"E18";
\ar@{-} "B18";"F18";
\ar@{} "C18";"E18";
\ar@{} "C18";"F18";
\ar@{} "C18";"G18";
\ar@{} "D18";"F18";
\ar@{} "D18";"G18";
\ar@{} "E18";"G18";
\ar@{} (0, 0);(32, -12) *\dir<2pt>{*} = "B35";
\ar@{-} "B35";(30, -16) *\dir<2pt>{*} = "C35";
\ar@{-} "C35";(32, -20) *\dir<2pt>{*} = "D35";
\ar@{-} "D35";(36, -20) *\dir<2pt>{*} = "E35";
\ar@{} "E35";(38, -16) *\dir<2pt>{*} = "F35";
\ar@{-} "F35";(36, -12) *\dir<2pt>{*} = "G35";
\ar@{-} "B35";"G35";
\ar@{} "B35";"D35";
\ar@{-} "B35";"E35";
\ar@{} "B35";"F35";
\ar@{} "C35";"E35";
\ar@{} "C35";"F35";
\ar@{} "C35";"G35";
\ar@{} "D35";"F35";
\ar@{} "D35";"G35";
\ar@{-} "E35";"G35";
\ar@{} (0, 0);(44, -12) *\dir<2pt>{*} = "B38";
\ar@{-} "B38";(42, -16) *\dir<2pt>{*} = "C38";
\ar@{-} "C38";(44, -20) *\dir<2pt>{*} = "D38";
\ar@{-} "D38";(48, -20) *\dir<2pt>{*} = "E38";
\ar@{-} "E38";(50, -16) *\dir<2pt>{*} = "F38";
\ar@{-} "F38";(48, -12) *\dir<2pt>{*} = "G38";
\ar@{-} "B38";"G38";
\ar@{} "B38";"D38";
\ar@{} "B38";"E38";
\ar@{} "B38";"F38";
\ar@{} "C38";"E38";
\ar@{} "C38";"F38";
\ar@{} "C38";"G38";
\ar@{} "D38";"F38";
\ar@{} "D38";"G38";
\ar@{-} "E38";"G38";
\ar@{} (0, 0);(56, -12) *\dir<2pt>{*} = "B39";
\ar@{-} "B39";(54, -16) *\dir<2pt>{*} = "C39";
\ar@{-} "C39";(56, -20) *\dir<2pt>{*} = "D39";
\ar@{-} "D39";(60, -20) *\dir<2pt>{*} = "E39";
\ar@{-} "E39";(62, -16) *\dir<2pt>{*} = "F39";
\ar@{-} "F39";(60, -12) *\dir<2pt>{*} = "G39";
\ar@{} "B39";"G39";
\ar@{} "B39";"D39";
\ar@{} "B39";"E39";
\ar@{-} "B39";"F39";
\ar@{} "C39";"E39";
\ar@{} "C39";"F39";
\ar@{-} "C39";"G39";
\ar@{} "D39";"F39";
\ar@{} "D39";"G39";
\ar@{} "E39";"G39";
\ar@{} (0, 0);(68, -12) *\dir<2pt>{*} = "B58";
\ar@{-} "B58";(66, -16) *\dir<2pt>{*} = "C58";
\ar@{-} "C58";(68, -20) *\dir<2pt>{*} = "D58";
\ar@{-} "D58";(72, -20) *\dir<2pt>{*} = "E58";
\ar@{-} "E58";(74, -16) *\dir<2pt>{*} = "F58";
\ar@{-} "F58";(72, -12) *\dir<2pt>{*} = "G58";
\ar@{-} "B58";"G58";
\ar@{} "B58";"D58";
\ar@{} "B58";"E58";
\ar@{-} "B58";"F58";
\ar@{} "C58";"E58";
\ar@{} "C58";"F58";
\ar@{} "C58";"G58";
\ar@{} "D58";"F58";
\ar@{-} "D58";"G58";
\ar@{} "E58";"G58";
\textcolor{blue}{\ar@{-} (-6, -26);(86, -26);}
\textcolor{blue}{\ar@{} (0, 0);(100, -26) *\txt{Comparability};}
\textcolor{blue}{\ar@{-} (114, -26);(136, -26);}
\textcolor{blue}{\ar@{-} (-10, -26);(-10, -202);}
\textcolor{blue}{\ar@{-} (133, -26);(133, -202);}
\textcolor{blue}{\ar@{-} (-13, -202);(132, -202);}
\ar@{} (0, 0);(0, -46) *\dir<2pt>{*} = "B33";
\ar@{} "B33";(-2, -50) *\dir<2pt>{*} = "C33";
\ar@{-} "C33";(0, -54) *\dir<2pt>{*} = "D33";
\ar@{-} "D33";(4, -54) *\dir<2pt>{*} = "E33";
\ar@{-} "E33";(6, -50) *\dir<2pt>{*} = "F33";
\ar@{} "F33";(4, -46) *\dir<2pt>{*} = "G33";
\ar@{-} "B33";"G33";
\ar@{-} "B33";"D33";
\ar@{} "B33";"E33";
\ar@{-} "B33";"F33";
\ar@{} "C33";"E33";
\ar@{} "C33";"F33";
\ar@{} "C33";"G33";
\ar@{} "D33";"F33";
\ar@{} "D33";"G33";
\ar@{-} "E33";"G33";
\ar@{} (0, 0);(12, -46) *\dir<2pt>{*} = "B36";
\ar@{-} "B36";(10, -50) *\dir<2pt>{*} = "C36";
\ar@{} "C36";(12 , -54) *\dir<2pt>{*} = "D36";
\ar@{-} "D36";(16, -54) *\dir<2pt>{*} = "E36";
\ar@{-} "E36";(18, -50) *\dir<2pt>{*} = "F36";
\ar@{} "F36";(16, -46) *\dir<2pt>{*} = "G36";
\ar@{-} "B36";"G36";
\ar@{-} "B36";"D36";
\ar@{} "B36";"E36";
\ar@{-} "B36";"F36";
\ar@{} "C36";"E36";
\ar@{} "C36";"F36";
\ar@{} "C36";"G36";
\ar@{} "D36";"F36";
\ar@{} "D36";"G36";
\ar@{-} "E36";"G36";
\ar@{} (0, 0);(24, -46) *\dir<2pt>{*} = "B49";
\ar@{-} "B49";(22, -50) *\dir<2pt>{*} = "C49";
\ar@{-} "C49";(24, -54) *\dir<2pt>{*} = "D49";
\ar@{-} "D49";(28, -54) *\dir<2pt>{*} = "E49";
\ar@{-} "E49";(30, -50) *\dir<2pt>{*} = "F49";
\ar@{} "F49";(28, -46) *\dir<2pt>{*} = "G49";
\ar@{} "B49";"G49";
\ar@{} "B49";"D49";
\ar@{-} "B49";"E49";
\ar@{} "B49";"F49";
\ar@{} "C49";"E49";
\ar@{-} "C49";"F49";
\ar@{-} "C49";"G49";
\ar@{} "D49";"F49";
\ar@{} "D49";"G49";
\ar@{-} "E49";"G49";
\ar@{} (0, 0);(36, -46) *\dir<2pt>{*} = "B57";
\ar@{-} "B57";(34, -50) *\dir<2pt>{*} = "C57";
\ar@{-} "C57";(36, -54) *\dir<2pt>{*} = "D57";
\ar@{-} "D57";(40, -54) *\dir<2pt>{*} = "E57";
\ar@{-} "E57";(42, -50) *\dir<2pt>{*} = "F57";
\ar@{-} "F57";(40, -46) *\dir<2pt>{*} = "G57";
\ar@{-} "B57";"G57";
\ar@{} "B57";"D57";
\ar@{-} "B57";"E57";
\ar@{} "B57";"F57";
\ar@{} "C57";"E57";
\ar@{} "C57";"F57";
\ar@{} "C57";"G57";
\ar@{} "D57";"F57";
\ar@{-} "D57";"G57";
\ar@{} "E57";"G57";
\ar@{} (0, 0);(48, -46) *\dir<2pt>{*} = "B69";
\ar@{-} "B69";(46, -50) *\dir<2pt>{*} = "C69";
\ar@{-} "C69";(48, -54) *\dir<2pt>{*} = "D69";
\ar@{-} "D69";(52, -54) *\dir<2pt>{*} = "E69";
\ar@{-} "E69";(54, -50) *\dir<2pt>{*} = "F69";
\ar@{-} "F69";(52, -46) *\dir<2pt>{*} = "G69";
\ar@{-} "B69";"G69";
\ar@{} "B69";"D69";
\ar@{-} "B69";"E69";
\ar@{} "B69";"F69";
\ar@{} "C69";"E69";
\ar@{-} "C69";"F69";
\ar@{} "C69";"G69";
\ar@{} "D69";"F69";
\ar@{-} "D69";"G69";
\ar@{} "E69";"G69";
\ar@{} (0, 0);(0, -58) *\dir<2pt>{*} = "B14";
\ar@{-} "B14";(-2, -62) *\dir<2pt>{*} = "C14";
\ar@{-} "C14";(0, -66) *\dir<2pt>{*} = "D14";
\ar@{-} "D14";(4, -66) *\dir<2pt>{*} = "E14";
\ar@{-} "E14";(6, -62) *\dir<2pt>{*} = "F14";
\ar@{} "F14";(4, -58) *\dir<2pt>{*} = "G14";
\ar@{-} "B14";"G14";
\ar@{} "B14";"D14";
\ar@{-} "B14";"E14";
\ar@{} "B14";"F14";
\ar@{} "C14";"E14";
\ar@{} "C14";"F14";
\ar@{} "C14";"G14";
\ar@{} "D14";"F14";
\ar@{} "D14";"G14";
\ar@{} "E14";"G14";
\ar@{} (0, 0);(12, -58) *\dir<2pt>{*} = "B15";
\ar@{-} "B15";(10, -62) *\dir<2pt>{*} = "C15";
\ar@{-} "C15";(12, -66) *\dir<2pt>{*} = "D15";
\ar@{-} "D15";(16, -66) *\dir<2pt>{*} = "E15";
\ar@{-} "E15";(18, -62) *\dir<2pt>{*} = "F15";
\ar@{-} "F15";(16, -58) *\dir<2pt>{*} = "G15";
\ar@{} "B15";"G15";
\ar@{} "B15";"D15";
\ar@{-} "B15";"E15";
\ar@{} "B15";"F15";
\ar@{} "C15";"E15";
\ar@{} "C15";"F15";
\ar@{} "C15";"G15";
\ar@{} "D15";"F15";
\ar@{} "D15";"G15";
\ar@{} "E15";"G15";
\ar@{} (0, 0);(24, -58) *\dir<2pt>{*} = "B16";
\ar@{-} "B16";(22, -62) *\dir<2pt>{*} = "C16";
\ar@{-} "C16";(24, -66) *\dir<2pt>{*} = "D16";
\ar@{-} "D16";(28, -66) *\dir<2pt>{*} = "E16";
\ar@{} "E16";(30, -62) *\dir<2pt>{*} = "F16";
\ar@{} "F16";(28, -58) *\dir<2pt>{*} = "G16";
\ar@{-} "B16";"G16";
\ar@{} "B16";"D16";
\ar@{-} "B16";"E16";
\ar@{-} "B16";"F16";
\ar@{} "C16";"E16";
\ar@{} "C16";"F16";
\ar@{} "C16";"G16";
\ar@{} "D16";"F16";
\ar@{} "D16";"G16";
\ar@{} "E16";"G16";
\ar@{} (0, 0);(36, -58) *\dir<2pt>{*} = "B17";
\ar@{} "B17";(34, -62) *\dir<2pt>{*} = "C17";
\ar@{-} "C17";(36, -66) *\dir<2pt>{*} = "D17";
\ar@{-} "D17";(40, -66) *\dir<2pt>{*} = "E17";
\ar@{} "E17";(42, -62) *\dir<2pt>{*} = "F17";
\ar@{-} "F17";(40, -58) *\dir<2pt>{*} = "G17";
\ar@{-} "B17";"G17";
\ar@{-} "B17";"D17";
\ar@{} "B17";"E17";
\ar@{} "B17";"F17";
\ar@{} "C17";"E17";
\ar@{} "C17";"F17";
\ar@{} "C17";"G17";
\ar@{} "D17";"F17";
\ar@{} "D17";"G17";
\ar@{-} "E17";"G17";
\ar@{} (0, 0);(48, -58) *\dir<2pt>{*} = "B19";
\ar@{-} "B19";(46, -62) *\dir<2pt>{*} = "C19";
\ar@{-} "C19";(48, -66) *\dir<2pt>{*} = "D19";
\ar@{-} "D19";(52, -66) *\dir<2pt>{*} = "E19";
\ar@{-} "E19";(54, -62) *\dir<2pt>{*} = "F19";
\ar@{-} "F19";(52, -58) *\dir<2pt>{*} = "G19";
\ar@{-} "B19";"G19";
\ar@{} "B19";"D19";
\ar@{} "B19";"E19";
\ar@{} "B19";"F19";
\ar@{} "C19";"E19";
\ar@{} "C19";"F19";
\ar@{} "C19";"G19";
\ar@{} "D19";"F19";
\ar@{} "D19";"G19";
\ar@{} "E19";"G19";
\ar@{} (0, 0);(60, -58) *\dir<2pt>{*} = "B31";
\ar@{-} "B31";(58, -62) *\dir<2pt>{*} = "C31";
\ar@{-} "C31";(60, -66) *\dir<2pt>{*} = "D31";
\ar@{-} "D31";(64, -66) *\dir<2pt>{*} = "E31";
\ar@{-} "E31";(66, -62) *\dir<2pt>{*} = "F31";
\ar@{-} "F31";(64, -58) *\dir<2pt>{*} = "G31";
\ar@{-} "B31";"G31";
\ar@{} "B31";"D31";
\ar@{} "B31";"E31";
\ar@{} "B31";"F31";
\ar@{} "C31";"E31";
\ar@{} "C31";"F31";
\ar@{} "C31";"G31";
\ar@{} "D31";"F31";
\ar@{-} "D31";"G31";
\ar@{} "E31";"G31";
\ar@{} (0, 0);(0, -70) *\dir<2pt>{*} = "B1";
\ar@{-} "B1";(-2, -74) *\dir<2pt>{*} = "C1";
\ar@{-} "C1";(0, -78) *\dir<2pt>{*} = "D1";
\ar@{-} "D1";(4, -78) *\dir<2pt>{*} = "E1";
\ar@{-} "E1";(6, -74) *\dir<2pt>{*} = "F1";
\ar@{} "F1";(4, -70) *\dir<2pt>{*} = "G1";
\ar@{-} "B1";"G1";
\ar@{} "B1";"D1";
\ar@{} "B1";"E1";
\ar@{} "B1";"F1";
\ar@{} "C1";"E1";
\ar@{} "C1";"F1";
\ar@{} "C1";"G1";
\ar@{} "D1";"F1";
\ar@{} "D1";"G1";
\ar@{} "E1";"G1";
\ar@{} (0, 0);(12, -70) *\dir<2pt>{*} = "B2";
\ar@{-} "B2";(10, -74) *\dir<2pt>{*} = "C2";
\ar@{-} "C2";(12, -78) *\dir<2pt>{*} = "D2";
\ar@{-} "D2";(16, -78) *\dir<2pt>{*} = "E2";
\ar@{-} "E2";(18, -74) *\dir<2pt>{*} = "F2";
\ar@{} "F2";(16, -70) *\dir<2pt>{*} = "G2";
\ar@{} "B2";"G2";
\ar@{} "B2";"D2";
\ar@{} "B2";"E2";
\ar@{} "B2";"F2";
\ar@{} "C2";"E2";
\ar@{} "C2";"F2";
\ar@{} "C2";"G2";
\ar@{} "D2";"F2";
\ar@{} "D2";"G2";
\ar@{-} "E2";"G2";
\ar@{} (0, 0);(24, -70) *\dir<2pt>{*} = "B3";
\ar@{-} "B3";(22, -74) *\dir<2pt>{*} = "C3";
\ar@{} "C3";(24, -78) *\dir<2pt>{*} = "D3";
\ar@{-} "D3";(28, -78) *\dir<2pt>{*} = "E3";
\ar@{} "E3";(30, -74) *\dir<2pt>{*} = "F3";
\ar@{-} "F3";(28, -70) *\dir<2pt>{*} = "G3";
\ar@{-} "B3";"G3";
\ar@{-} "B3";"D3";
\ar@{} "B3";"E3";
\ar@{} "B3";"F3";
\ar@{} "C3";"E3";
\ar@{} "C3";"F3";
\ar@{} "C3";"G3";
\ar@{} "D3";"F3";
\ar@{} "D3";"G3";
\ar@{} "E3";"G3";
\ar@{} (0, 0);(36, -70) *\dir<2pt>{*} = "B4";
\ar@{-} "B4";(34, -74) *\dir<2pt>{*} = "C4";
\ar@{-} "C4";(36, -78) *\dir<2pt>{*} = "D4";
\ar@{-} "D4";(40, -78) *\dir<2pt>{*} = "E4";
\ar@{} "E4";(42, -74) *\dir<2pt>{*} = "F4";
\ar@{} "F4";(40, -70) *\dir<2pt>{*} = "G4";
\ar@{} "B4";"G4";
\ar@{} "B4";"D4";
\ar@{} "B4";"E4";
\ar@{} "B4";"F4";
\ar@{} "C4";"E4";
\ar@{} "C4";"F4";
\ar@{} "C4";"G4";
\ar@{-} "D4";"F4";
\ar@{-} "D4";"G4";
\ar@{} "E4";"G4";
\ar@{} (0, 0);(48, -70) *\dir<2pt>{*} = "B5";
\ar@{-} "B5";(46, -74) *\dir<2pt>{*} = "C5";
\ar@{} "C5";(48, -78) *\dir<2pt>{*} = "D5";
\ar@{} "D5";(52, -78) *\dir<2pt>{*} = "E5";
\ar@{} "E5";(54, -74) *\dir<2pt>{*} = "F5";
\ar@{-} "F5";(52, -70) *\dir<2pt>{*} = "G5";
\ar@{-} "B5";"G5";
\ar@{-} "B5";"D5";
\ar@{} "B5";"E5";
\ar@{} "B5";"F5";
\ar@{} "C5";"E5";
\ar@{} "C5";"F5";
\ar@{} "C5";"G5";
\ar@{} "D5";"F5";
\ar@{} "D5";"G5";
\ar@{-} "E5";"G5";
\ar@{} (0, 0);(60, -70) *\dir<2pt>{*} = "B6";
\ar@{} "B6";(58, -74) *\dir<2pt>{*} = "C6";
\ar@{} "C6";(60, -78) *\dir<2pt>{*} = "D6";
\ar@{} "D6";(64, -78) *\dir<2pt>{*} = "E6";
\ar@{} "E6";(66, -74) *\dir<2pt>{*} = "F6";
\ar@{-} "F6";(64, -70) *\dir<2pt>{*} = "G6";
\ar@{-} "B6";"G6";
\ar@{} "B6";"D6";
\ar@{} "B6";"E6";
\ar@{} "B6";"F6";
\ar@{} "C6";"E6";
\ar@{} "C6";"F6";
\ar@{-} "C6";"G6";
\ar@{} "D6";"F6";
\ar@{-} "D6";"G6";
\ar@{-} "E6";"G6";
\ar@{} (0, 0);(60, -86) *\dir<2pt>{*} = "B7";
\ar@{-} "B7";(58, -90) *\dir<2pt>{*} = "C7";
\ar@{-} "C7";(60, -94) *\dir<2pt>{*} = "D7";
\ar@{} "D7";(64, -94) *\dir<2pt>{*} = "E7";
\ar@{} "E7";(66, -90) *\dir<2pt>{*} = "F7";
\ar@{} "F7";(64, -86) *\dir<2pt>{*} = "G7";
\ar@{-} "B7";"G7";
\ar@{-} "B7";"D7";
\ar@{-} "B7";"E7";
\ar@{-} "B7";"F7";
\ar@{} "C7";"E7";
\ar@{} "C7";"F7";
\ar@{} "C7";"G7";
\ar@{} "D7";"F7";
\ar@{} "D7";"G7";
\ar@{} "E7";"G7";
\ar@{} (0, 0);(60, -98) *\dir<2pt>{*} = "B27";
\ar@{-} "B27";(58, -102) *\dir<2pt>{*} = "C27";
\ar@{-} "C27";(60, -106) *\dir<2pt>{*} = "D27";
\ar@{-} "D27";(64, -106) *\dir<2pt>{*} = "E27";
\ar@{-} "E27";(66, -102) *\dir<2pt>{*} = "F27";
\ar@{} "F27";(64, -98) *\dir<2pt>{*} = "G27";
\ar@{} "B27";"G27";
\ar@{-} "B27";"D27";
\ar@{} "B27";"E27";
\ar@{} "B27";"F27";
\ar@{} "C27";"E27";
\ar@{} "C27";"F27";
\ar@{} "C27";"G27";
\ar@{-} "D27";"F27";
\ar@{-} "D27";"G27";
\ar@{} "E27";"G27";
\ar@{} (0, 0);(60, -110) *\dir<2pt>{*} = "B28";
\ar@{-} "B28";(58, -114) *\dir<2pt>{*} = "C28";
\ar@{-} "C28";(60, -118) *\dir<2pt>{*} = "D28";
\ar@{-} "D28";(64, -118) *\dir<2pt>{*} = "E28";
\ar@{-} "E28";(66, -114) *\dir<2pt>{*} = "F28";
\ar@{} "F28";(64, -110) *\dir<2pt>{*} = "G28";
\ar@{} "B28";"G28";
\ar@{} "B28";"D28";
\ar@{-} "B28";"E28";
\ar@{} "B28";"F28";
\ar@{-} "C28";"E28";
\ar@{} "C28";"F28";
\ar@{} "C28";"G28";
\ar@{} "D28";"F28";
\ar@{} "D28";"G28";
\ar@{-} "E28";"G28";
\ar@{} (0, 0);(60, -122) *\dir<2pt>{*} = "B43";
\ar@{-} "B43";(58, -126) *\dir<2pt>{*} = "C43";
\ar@{-} "C43";(60, -130) *\dir<2pt>{*} = "D43";
\ar@{-} "D43";(64, -130) *\dir<2pt>{*} = "E43";
\ar@{-} "E43";(66, -126) *\dir<2pt>{*} = "F43";
\ar@{-} "F43";(64, -122) *\dir<2pt>{*} = "G43";
\ar@{} "B43";"G43";
\ar@{} "B43";"D43";
\ar@{-} "B43";"E43";
\ar@{} "B43";"F43";
\ar@{-} "C43";"E43";
\ar@{} "C43";"F43";
\ar@{} "C43";"G43";
\ar@{} "D43";"F43";
\ar@{} "D43";"G43";
\ar@{-} "E43";"G43";
\ar@{} (0, 0);(60, -134) *\dir<2pt>{*} = "B46";
\ar@{-} "B46";(58, -138) *\dir<2pt>{*} = "C46";
\ar@{-} "C46";(60, -142) *\dir<2pt>{*} = "D46";
\ar@{-} "D46";(64, -142) *\dir<2pt>{*} = "E46";
\ar@{-} "E46";(66, -138) *\dir<2pt>{*} = "F46";
\ar@{} "F46";(64, -134) *\dir<2pt>{*} = "G46";
\ar@{} "B46";"G46";
\ar@{} "B46";"D46";
\ar@{-} "B46";"E46";
\ar@{} "B46";"F46";
\ar@{-} "C46";"E46";
\ar@{} "C46";"F46";
\ar@{-} "C46";"G46";
\ar@{} "D46";"F46";
\ar@{} "D46";"G46";
\ar@{-} "E46";"G46";
\ar@{} (0, 0);(60, -146) *\dir<2pt>{*} = "B65";
\ar@{-} "B65";(58, -150) *\dir<2pt>{*} = "C65";
\ar@{-} "C65";(60, -154) *\dir<2pt>{*} = "D65";
\ar@{} "D65";(64, -154) *\dir<2pt>{*} = "E65";
\ar@{-} "E65";(66, -150) *\dir<2pt>{*} = "F65";
\ar@{-} "F65";(64, -146) *\dir<2pt>{*} = "G65";
\ar@{} "B65";"G65";
\ar@{} "B65";"D65";
\ar@{} "B65";"E65";
\ar@{-} "B65";"F65";
\ar@{-} "C65";"E65";
\ar@{-} "C65";"F65";
\ar@{-} "C65";"G65";
\ar@{-} "D65";"F65";
\ar@{} "D65";"G65";
\ar@{} "E65";"G65";
\ar@{} (0, 0);(76, -70) *\dir<2pt>{*} = "B98";
\ar@{-} "B98";(74, -74) *\dir<2pt>{*} = "C98";
\ar@{-} "C98";(76, -78) *\dir<2pt>{*} = "D98";
\ar@{-} "D98";(80, -78) *\dir<2pt>{*} = "E98";
\ar@{-} "E98";(82, -74) *\dir<2pt>{*} = "F98";
\ar@{} "F98";(80, -70) *\dir<2pt>{*} = "G98";
\ar@{} "B98";"G98";
\ar@{-} "B98";"D98";
\ar@{-} "B98";"E98";
\ar@{} "B98";"F98";
\ar@{-} "C98";"E98";
\ar@{} "C98";"F98";
\ar@{} "C98";"G98";
\ar@{} "D98";"F98";
\ar@{} "D98";"G98";
\ar@{-} "E98";"G98";
\ar@{} (0, 0);(76, -86) *\dir<2pt>{*} = "B63";
\ar@{-} "B63";(74, -90) *\dir<2pt>{*} = "C63";
\ar@{-} "C63";(76, -94) *\dir<2pt>{*} = "D63";
\ar@{-} "D63";(80, -94) *\dir<2pt>{*} = "E63";
\ar@{-} "E63";(82, -90) *\dir<2pt>{*} = "F63";
\ar@{-} "F63";(80, -86) *\dir<2pt>{*} = "G63";
\ar@{} "B63";"G63";
\ar@{-} "B63";"D63";
\ar@{-} "B63";"E63";
\ar@{} "B63";"F63";
\ar@{-} "C63";"E63";
\ar@{} "C63";"F63";
\ar@{} "C63";"G63";
\ar@{} "D63";"F63";
\ar@{} "D63";"G63";
\ar@{-} "E63";"G63";
\ar@{} (0, 0);(76, -98) *\dir<2pt>{*} = "B67";
\ar@{-} "B67";(74, -102) *\dir<2pt>{*} = "C67";
\ar@{-} "C67";(76, -106) *\dir<2pt>{*} = "D67";
\ar@{-} "D67";(80, -106) *\dir<2pt>{*} = "E67";
\ar@{-} "E67";(82, -102) *\dir<2pt>{*} = "F67";
\ar@{} "F67";(80, -98) *\dir<2pt>{*} = "G67";
\ar@{-} "B67";"G67";
\ar@{-} "B67";"D67";
\ar@{-} "B67";"E67";
\ar@{} "B67";"F67";
\ar@{-} "C67";"E67";
\ar@{} "C67";"F67";
\ar@{} "C67";"G67";
\ar@{} "D67";"F67";
\ar@{} "D67";"G67";
\ar@{-} "E67";"G67";
\ar@{} (0, 0);(76, -110) *\dir<2pt>{*} = "B85";
\ar@{-} "B85";(74, -114) *\dir<2pt>{*} = "C85";
\ar@{-} "C85";(76, -118) *\dir<2pt>{*} = "D85";
\ar@{-} "D85";(80, -118) *\dir<2pt>{*} = "E85";
\ar@{-} "E85";(82, -114) *\dir<2pt>{*} = "F85";
\ar@{-} "F85";(80, -110) *\dir<2pt>{*} = "G85";
\ar@{} "B85";"G85";
\ar@{} "B85";"D85";
\ar@{} "B85";"E85";
\ar@{-} "B85";"F85";
\ar@{-} "C85";"E85";
\ar@{-} "C85";"F85";
\ar@{-} "C85";"G85";
\ar@{-} "D85";"F85";
\ar@{} "D85";"G85";
\ar@{} "E85";"G85";
\ar@{} (0, 0);(76, -122) *\dir<2pt>{*} = "B86";
\ar@{-} "B86";(74, -126) *\dir<2pt>{*} = "C86";
\ar@{-} "C86";(76, -130) *\dir<2pt>{*} = "D86";
\ar@{-} "D86";(80, -130) *\dir<2pt>{*} = "E86";
\ar@{-} "E86";(82, -126) *\dir<2pt>{*} = "F86";
\ar@{} "F86";(80, -122) *\dir<2pt>{*} = "G86";
\ar@{-} "B86";"G86";
\ar@{-} "B86";"D86";
\ar@{-} "B86";"E86";
\ar@{} "B86";"F86";
\ar@{-} "C86";"E86";
\ar@{} "C86";"F86";
\ar@{} "C86";"G86";
\ar@{} "D86";"F86";
\ar@{-} "D86";"G86";
\ar@{-} "E86";"G86";
\ar@{} (0, 0);(76, -134) *\dir<2pt>{*} = "B95";
\ar@{-} "B95";(74, -138) *\dir<2pt>{*} = "C95";
\ar@{-} "C95";(76, -142) *\dir<2pt>{*} = "D95";
\ar@{-} "D95";(80, -142) *\dir<2pt>{*} = "E95";
\ar@{-} "E95";(82, -138) *\dir<2pt>{*} = "F95";
\ar@{} "F95";(80, -134) *\dir<2pt>{*} = "G95";
\ar@{-} "B95";"G95";
\ar@{-} "B95";"D95";
\ar@{-} "B95";"E95";
\ar@{} "B95";"F95";
\ar@{-} "C95";"E95";
\ar@{} "C95";"F95";
\ar@{-} "C95";"G95";
\ar@{} "D95";"F95";
\ar@{-} "D95";"G95";
\ar@{-} "E95";"G95";
\ar@{} (0, 0);(76, -146) *\dir<2pt>{*} = "B96";
\ar@{-} "B96";(74, -150) *\dir<2pt>{*} = "C96";
\ar@{-} "C96";(76, -154) *\dir<2pt>{*} = "D96";
\ar@{-} "D96";(80, -154) *\dir<2pt>{*} = "E96";
\ar@{-} "E96";(82, -150) *\dir<2pt>{*} = "F96";
\ar@{-} "F96";(80, -146) *\dir<2pt>{*} = "G96";
\ar@{-} "B96";"G96";
\ar@{} "B96";"D96";
\ar@{} "B96";"E96";
\ar@{-} "B96";"F96";
\ar@{-} "C96";"E96";
\ar@{-} "C96";"F96";
\ar@{-} "C96";"G96";
\ar@{-} "D96";"F96";
\ar@{} "D96";"G96";
\ar@{} "E96";"G96";
\ar@{} (0, 0);(88, -70) *\dir<2pt>{*} = "B98";
\ar@{-} "B98";(86, -74) *\dir<2pt>{*} = "C98";
\ar@{-} "C98";(88, -78) *\dir<2pt>{*} = "D98";
\ar@{-} "D98";(92, -78) *\dir<2pt>{*} = "E98";
\ar@{-} "E98";(94, -74) *\dir<2pt>{*} = "F98";
\ar@{-} "F98";(92, -70) *\dir<2pt>{*} = "G98";
\ar@{} "B98";"G98";
\ar@{} "B98";"D98";
\ar@{-} "B98";"E98";
\ar@{-} "B98";"F98";
\ar@{-} "C98";"E98";
\ar@{-} "C98";"F98";
\ar@{-} "C98";"G98";
\ar@{-} "D98";"F98";
\ar@{} "D98";"G98";
\ar@{} "E98";"G98";
\ar@{} (0, 0);(88, -86) *\dir<2pt>{*} = "B105";
\ar@{-} "B105";(86, -90) *\dir<2pt>{*} = "C105";
\ar@{-} "C105";(88, -94) *\dir<2pt>{*} = "D105";
\ar@{-} "D105";(92, -94) *\dir<2pt>{*} = "E105";
\ar@{-} "E105";(94, -90) *\dir<2pt>{*} = "F105";
\ar@{-} "F105";(92, -86) *\dir<2pt>{*} = "G105";
\ar@{-} "B105";"G105";
\ar@{-} "B105";"D105";
\ar@{-} "B105";"E105";
\ar@{-} "B105";"F105";
\ar@{-} "C105";"E105";
\ar@{-} "C105";"F105";
\ar@{} "C105";"G105";
\ar@{-} "D105";"F105";
\ar@{} "D105";"G105";
\ar@{} "E105";"G105";
\ar@{} (0, 0);(88, -98) *\dir<2pt>{*} = "B106";
\ar@{-} "B106";(86, -102) *\dir<2pt>{*} = "C106";
\ar@{-} "C106";(88, -106) *\dir<2pt>{*} = "D106";
\ar@{-} "D106";(92, -106) *\dir<2pt>{*} = "E106";
\ar@{-} "E106";(94, -102) *\dir<2pt>{*} = "F106";
\ar@{-} "F106";(92, -98) *\dir<2pt>{*} = "G106";
\ar@{-} "B106";"G106";
\ar@{-} "B106";"D106";
\ar@{-} "B106";"E106";
\ar@{-} "B106";"F106";
\ar@{} "C106";"E106";
\ar@{-} "C106";"F106";
\ar@{} "C106";"G106";
\ar@{-} "D106";"F106";
\ar@{-} "D106";"G106";
\ar@{} "E106";"G106";
\ar@{} (0, 0);(88, -110) *\dir<2pt>{*} = "B109";
\ar@{-} "B109";(86, -114) *\dir<2pt>{*} = "C109";
\ar@{-} "C109";(88, -118) *\dir<2pt>{*} = "D109";
\ar@{-} "D109";(92, -118) *\dir<2pt>{*} = "E109";
\ar@{-} "E109";(94, -114) *\dir<2pt>{*} = "F109";
\ar@{-} "F109";(92, -110) *\dir<2pt>{*} = "G109";
\ar@{-} "B109";"G109";
\ar@{} "B109";"D109";
\ar@{-} "B109";"E109";
\ar@{} "B109";"F109";
\ar@{-} "C109";"E109";
\ar@{-} "C109";"F109";
\ar@{-} "C109";"G109";
\ar@{-} "D109";"F109";
\ar@{-} "D109";"G109";
\ar@{-} "E109";"G109";
\ar@{} (0, 0);(88, -122) *\dir<2pt>{*} = "B111";
\ar@{-} "B111";(86, -126) *\dir<2pt>{*} = "C111";
\ar@{-} "C111";(88, -130) *\dir<2pt>{*} = "D111";
\ar@{-} "D111";(92, -130) *\dir<2pt>{*} = "E111";
\ar@{-} "E111";(94, -126) *\dir<2pt>{*} = "F111";
\ar@{-} "F111";(92, -122) *\dir<2pt>{*} = "G111";
\ar@{-} "B111";"G111";
\ar@{} "B111";"D111";
\ar@{-} "B111";"E111";
\ar@{-} "B111";"F111";
\ar@{-} "C111";"E111";
\ar@{-} "C111";"F111";
\ar@{-} "C111";"G111";
\ar@{-} "D111";"F111";
\ar@{-} "D111";"G111";
\ar@{-} "E111";"G111";
\ar@{} (0, 0);(88, -134) *\dir<2pt>{*} = "B112";
\ar@{-} "B112";(86, -138) *\dir<2pt>{*} = "C112";
\ar@{-} "C112";(88, -142) *\dir<2pt>{*} = "D112";
\ar@{-} "D112";(92, -142) *\dir<2pt>{*} = "E112";
\ar@{-} "E112";(94, -138) *\dir<2pt>{*} = "F112";
\ar@{-} "F112";(92, -134) *\dir<2pt>{*} = "G112";
\ar@{-} "B112";"G112";
\ar@{-} "B112";"D112";
\ar@{-} "B112";"E112";
\ar@{-} "B112";"F112";
\ar@{-} "C112";"E112";
\ar@{-} "C112";"F112";
\ar@{-} "C112";"G112";
\ar@{-} "D112";"F112";
\ar@{-} "D112";"G112";
\ar@{-} "E112";"G112";
\ar@{} (0, 0);(-2, -86) *\dir<2pt>{*} = "B8";
\ar@{-} "B8";(-4, -90) *\dir<2pt>{*} = "C8";
\ar@{-} "C8";(-2, -94) *\dir<2pt>{*} = "D8";
\ar@{} "D8";(2, -94) *\dir<2pt>{*} = "E8";
\ar@{} "E8";(4, -90) *\dir<2pt>{*} = "F8";
\ar@{-} "F8";(2, -86) *\dir<2pt>{*} = "G8";
\ar@{-} "B8";"G8";
\ar@{-} "B8";"D8";
\ar@{-} "B8";"E8";
\ar@{} "B8";"F8";
\ar@{} "C8";"E8";
\ar@{} "C8";"F8";
\ar@{} "C8";"G8";
\ar@{} "D8";"F8";
\ar@{} "D8";"G8";
\ar@{} "E8";"G8";
\ar@{} (0, 0);(10, -86) *\dir<2pt>{*} = "B9";
\ar@{-} "B9";(8, -90) *\dir<2pt>{*} = "C9";
\ar@{-} "C9";(10, -94) *\dir<2pt>{*} = "D9";
\ar@{} "D9";(14, -94) *\dir<2pt>{*} = "E9";
\ar@{} "E9";(16, -90) *\dir<2pt>{*} = "F9";
\ar@{-} "F9";(14, -86) *\dir<2pt>{*} = "G9";
\ar@{-} "B9";"G9";
\ar@{-} "B9";"D9";
\ar@{} "B9";"E9";
\ar@{} "B9";"F9";
\ar@{} "C9";"E9";
\ar@{} "C9";"F9";
\ar@{} "C9";"G9";
\ar@{} "D9";"F9";
\ar@{} "D9";"G9";
\ar@{-} "E9";"G9";
\ar@{} (0, 0);(22, -86) *\dir<2pt>{*} = "B10";
\ar@{-} "B10";(20, -90) *\dir<2pt>{*} = "C10";
\ar@{-} "C10";(22, -94) *\dir<2pt>{*} = "D10";
\ar@{-} "D10";(26, -94) *\dir<2pt>{*} = "E10";
\ar@{} "E10";(28, -90) *\dir<2pt>{*} = "F10";
\ar@{} "F10";(26, -86) *\dir<2pt>{*} = "G10";
\ar@{-} "B10";"G10";
\ar@{-} "B10";"D10";
\ar@{} "B10";"E10";
\ar@{-} "B10";"F10";
\ar@{} "C10";"E10";
\ar@{} "C10";"F10";
\ar@{} "C10";"G10";
\ar@{} "D10";"F10";
\ar@{} "D10";"G10";
\ar@{} "E10";"G10";
\ar@{} (0, 0);(34, -86) *\dir<2pt>{*} = "B11";
\ar@{-} "B11";(32, -90) *\dir<2pt>{*} = "C11";
\ar@{-} "C11";(34, -94) *\dir<2pt>{*} = "D11";
\ar@{-} "D11";(38, -94) *\dir<2pt>{*} = "E11";
\ar@{} "E11";(40, -90) *\dir<2pt>{*} = "F11";
\ar@{-} "F11";(38, -86) *\dir<2pt>{*} = "G11";
\ar@{-} "B11";"G11";
\ar@{-} "B11";"D11";
\ar@{} "B11";"E11";
\ar@{} "B11";"F11";
\ar@{} "C11";"E11";
\ar@{} "C11";"F11";
\ar@{} "C11";"G11";
\ar@{} "D11";"F11";
\ar@{} "D11";"G11";
\ar@{} "E11";"G11";
\ar@{} (0, 0);(46, -86) *\dir<2pt>{*} = "B12";
\ar@{-} "B12";(44, -90) *\dir<2pt>{*} = "C12";
\ar@{-} "C12";(46, -94) *\dir<2pt>{*} = "D12";
\ar@{-} "D12";(50, -94) *\dir<2pt>{*} = "E12";
\ar@{-} "E12";(52, -90) *\dir<2pt>{*} = "F12";
\ar@{-} "F12";(50, -86) *\dir<2pt>{*} = "G12";
\ar@{} "B12";"G12";
\ar@{-} "B12";"D12";
\ar@{} "B12";"E12";
\ar@{} "B12";"F12";
\ar@{} "C12";"E12";
\ar@{} "C12";"F12";
\ar@{} "C12";"G12";
\ar@{} "D12";"F12";
\ar@{} "D12";"G12";
\ar@{} "E12";"G12";
\ar@{} (0, 0);(-2, -98) *\dir<2pt>{*} = "B21";
\ar@{-} "B21";(-4, -102) *\dir<2pt>{*} = "C21";
\ar@{-} "C21";(-2, -106) *\dir<2pt>{*} = "D21";
\ar@{-} "D21";(2, -106) *\dir<2pt>{*} = "E21";
\ar@{-} "E21";(4, -102) *\dir<2pt>{*} = "F21";
\ar@{} "F21";(2, -98) *\dir<2pt>{*} = "G21";
\ar@{} "B21";"G21";
\ar@{-} "B21";"D21";
\ar@{} "B21";"E21";
\ar@{} "B21";"F21";
\ar@{} "C21";"E21";
\ar@{} "C21";"F21";
\ar@{} "C21";"G21";
\ar@{} "D21";"F21";
\ar@{-} "D21";"G21";
\ar@{-} "E21";"G21";
\ar@{} (0, 0);(10, -98) *\dir<2pt>{*} = "B22";
\ar@{-} "B22";(8, -102) *\dir<2pt>{*} = "C22";
\ar@{-} "C22";(10, -106) *\dir<2pt>{*} = "D22";
\ar@{-} "D22";(14, -106) *\dir<2pt>{*} = "E22";
\ar@{} "E22";(16, -102) *\dir<2pt>{*} = "F22";
\ar@{-} "F22";(14, -98) *\dir<2pt>{*} = "G22";
\ar@{} "B22";"G22";
\ar@{-} "B22";"D22";
\ar@{-} "B22";"E22";
\ar@{} "B22";"F22";
\ar@{} "C22";"E22";
\ar@{} "C22";"F22";
\ar@{} "C22";"G22";
\ar@{} "D22";"F22";
\ar@{} "D22";"G22";
\ar@{-} "E22";"G22";
\ar@{} (0, 0);(22, -98) *\dir<2pt>{*} = "B23";
\ar@{-} "B23";(20, -102) *\dir<2pt>{*} = "C23";
\ar@{} "C23";(22, -106) *\dir<2pt>{*} = "D23";
\ar@{-} "D23";(26, -106) *\dir<2pt>{*} = "E23";
\ar@{-} "E23";(28, -102) *\dir<2pt>{*} = "F23";
\ar@{} "F23";(26, -98) *\dir<2pt>{*} = "G23";
\ar@{-} "B23";"G23";
\ar@{-} "B23";"D23";
\ar@{} "B23";"E23";
\ar@{} "B23";"F23";
\ar@{} "C23";"E23";
\ar@{} "C23";"F23";
\ar@{} "C23";"G23";
\ar@{} "D23";"F23";
\ar@{-} "D23";"G23";
\ar@{-} "E23";"G23";
\ar@{} (0, 0);(34, -98) *\dir<2pt>{*} = "B25";
\ar@{-} "B25";(32, -102) *\dir<2pt>{*} = "C25";
\ar@{} "C25";(34, -106) *\dir<2pt>{*} = "D25";
\ar@{-} "D25";(38, -106) *\dir<2pt>{*} = "E25";
\ar@{} "E25";(40, -102) *\dir<2pt>{*} = "F25";
\ar@{-} "F25";(38, -98) *\dir<2pt>{*} = "G25";
\ar@{-} "B25";"G25";
\ar@{-} "B25";"D25";
\ar@{} "B25";"E25";
\ar@{} "B25";"F25";
\ar@{} "C25";"E25";
\ar@{} "C25";"F25";
\ar@{} "C25";"G25";
\ar@{} "D25";"F25";
\ar@{-} "D25";"G25";
\ar@{-} "E25";"G25";
\ar@{} (0, 0);(46, -98) *\dir<2pt>{*} = "B26";
\ar@{-} "B26";(44, -102) *\dir<2pt>{*} = "C26";
\ar@{-} "C26";(46, -106) *\dir<2pt>{*} = "D26";
\ar@{-} "D26";(50, -106) *\dir<2pt>{*} = "E26";
\ar@{} "E26";(52, -102) *\dir<2pt>{*} = "F26";
\ar@{-} "F26";(50, -98) *\dir<2pt>{*} = "G26";
\ar@{-} "B26";"G26";
\ar@{-} "B26";"D26";
\ar@{-} "B26";"E26";
\ar@{} "B26";"F26";
\ar@{} "C26";"E26";
\ar@{} "C26";"F26";
\ar@{} "C26";"G26";
\ar@{} "D26";"F26";
\ar@{} "D26";"G26";
\ar@{} "E26";"G26";
\ar@{} (0, 0);(-2, -110) *\dir<2pt>{*} = "B29";
\ar@{} "B29";(-4, -114) *\dir<2pt>{*} = "C29";
\ar@{-} "C29";(-2, -118) *\dir<2pt>{*} = "D29";
\ar@{-} "D29";(2, -118) *\dir<2pt>{*} = "E29";
\ar@{} "E29";(4, -114) *\dir<2pt>{*} = "F29";
\ar@{-} "F29";(2, -110) *\dir<2pt>{*} = "G29";
\ar@{-} "B29";"G29";
\ar@{-} "B29";"D29";
\ar@{} "B29";"E29";
\ar@{} "B29";"F29";
\ar@{} "C29";"E29";
\ar@{} "C29";"F29";
\ar@{} "C29";"G29";
\ar@{} "D29";"F29";
\ar@{-} "D29";"G29";
\ar@{-} "E29";"G29";
\ar@{} (0, 0);(10, -110) *\dir<2pt>{*} = "B30";
\ar@{-} "B30";(8, -114) *\dir<2pt>{*} = "C30";
\ar@{-} "C30";(10, -118) *\dir<2pt>{*} = "D30";
\ar@{} "D30";(14, -118) *\dir<2pt>{*} = "E30";
\ar@{} "E30";(16, -114) *\dir<2pt>{*} = "F30";
\ar@{-} "F30";(14, -110) *\dir<2pt>{*} = "G30";
\ar@{-} "B30";"G30";
\ar@{-} "B30";"D30";
\ar@{} "B30";"E30";
\ar@{} "B30";"F30";
\ar@{} "C30";"E30";
\ar@{} "C30";"F30";
\ar@{} "C30";"G30";
\ar@{} "D30";"F30";
\ar@{-} "D30";"G30";
\ar@{-} "E30";"G30";
\ar@{} (0, 0);(22, -110) *\dir<2pt>{*} = "B32";
\ar@{-} "B32";(20, -114) *\dir<2pt>{*} = "C32";
\ar@{-} "C32";(22, -118) *\dir<2pt>{*} = "D32";
\ar@{-} "D32";(26, -118) *\dir<2pt>{*} = "E32";
\ar@{-} "E32";(28, -114) *\dir<2pt>{*} = "F32";
\ar@{-} "F32";(26, -110) *\dir<2pt>{*} = "G32";
\ar@{} "B32";"G32";
\ar@{} "B32";"D32";
\ar@{-} "B32";"E32";
\ar@{} "B32";"F32";
\ar@{} "C32";"E32";
\ar@{} "C32";"F32";
\ar@{} "C32";"G32";
\ar@{} "D32";"F32";
\ar@{} "D32";"G32";
\ar@{-} "E32";"G32";
\ar@{} (0, 0);(34, -110) *\dir<2pt>{*} = "B34";
\ar@{-} "B34";(32, -114) *\dir<2pt>{*} = "C34";
\ar@{-} "C34";(34, -118) *\dir<2pt>{*} = "D34";
\ar@{-} "D34";(38, -118) *\dir<2pt>{*} = "E34";
\ar@{-} "E34";(40, -114) *\dir<2pt>{*} = "F34";
\ar@{} "F34";(38, -110) *\dir<2pt>{*} = "G34";
\ar@{-} "B34";"G34";
\ar@{-} "B34";"D34";
\ar@{} "B34";"E34";
\ar@{} "B34";"F34";
\ar@{} "C34";"E34";
\ar@{} "C34";"F34";
\ar@{} "C34";"G34";
\ar@{} "D34";"F34";
\ar@{} "D34";"G34";
\ar@{-} "E34";"G34";
\ar@{} (0, 0);(46, -110) *\dir<2pt>{*} = "B37";
\ar@{-} "B37";(44, -114) *\dir<2pt>{*} = "C37";
\ar@{-} "C37";(46, -118) *\dir<2pt>{*} = "D37";
\ar@{-} "D37";(50, -118) *\dir<2pt>{*} = "E37";
\ar@{-} "E37";(52, -114) *\dir<2pt>{*} = "F37";
\ar@{} "F37" ;(50, -110) *\dir<2pt>{*} = "G37";
\ar@{-} "B37";"G37";
\ar@{} "B37";"D37";
\ar@{-} "B37";"E37";
\ar@{} "B37";"F37";
\ar@{} "C37";"E37";
\ar@{} "C37";"F37";
\ar@{} "C37";"G37";
\ar@{} "D37";"F37";
\ar@{} "D37";"G37";
\ar@{-} "E37";"G37";
\ar@{} (0, 0);(-2, -122) *\dir<2pt>{*} = "B42";
\ar@{-} "B42";(-4, -126) *\dir<2pt>{*} = "C42";
\ar@{-} "C42";(-2, -130) *\dir<2pt>{*} = "D42";
\ar@{-} "D42";(2, -130) *\dir<2pt>{*} = "E42";
\ar@{} "E42";(4, -126) *\dir<2pt>{*} = "F42";
\ar@{-} "F42";(2, -122) *\dir<2pt>{*} = "G42";
\ar@{-} "B42";"G42";
\ar@{-} "B42";"D42";
\ar@{-} "B42";"E42";
\ar@{} "B42";"F42";
\ar@{} "C42";"E42";
\ar@{} "C42";"F42";
\ar@{} "C42";"G42";
\ar@{} "D42";"F42";
\ar@{} "D42";"G42";
\ar@{-} "E42";"G42";
\ar@{} (0, 0);(10, -122) *\dir<2pt>{*} = "B44";
\ar@{-} "B44";(8, -126) *\dir<2pt>{*} = "C44";
\ar@{-} "C44";(10, -130) *\dir<2pt>{*} = "D44";
\ar@{-} "D44";(14, -130) *\dir<2pt>{*} = "E44";
\ar@{-} "E44";(16, -126) *\dir<2pt>{*} = "F44";
\ar@{} "F44";(14, -122) *\dir<2pt>{*} = "G44";
\ar@{-} "B44";"G44";
\ar@{-} "B44";"D44";
\ar@{-} "B44";"E44";
\ar@{} "B44";"F44";
\ar@{} "C44";"E44";
\ar@{} "C44";"F44";
\ar@{} "C44";"G44";
\ar@{} "D44";"F44";
\ar@{} "D44";"G44";
\ar@{-} "E44";"G44";
\ar@{} (0, 0);(22, -122) *\dir<2pt>{*} = "B45";
\ar@{-} "B45";(20, -126) *\dir<2pt>{*} = "C45";
\ar@{-} "C45";(22, -130) *\dir<2pt>{*} = "D45";
\ar@{-} "D45";(26, -130) *\dir<2pt>{*} = "E45";
\ar@{-} "E45";(28, -126) *\dir<2pt>{*} = "F45";
\ar@{} "F45";(26, -122) *\dir<2pt>{*} = "G45";
\ar@{-} "B45";"G45";
\ar@{} "B45";"D45";
\ar@{-} "B45";"E45";
\ar@{} "B45";"F45";
\ar@{-} "C45";"E45";
\ar@{} "C45";"F45";
\ar@{} "C45";"G45";
\ar@{} "D45";"F45";
\ar@{} "D45";"G45";
\ar@{-} "E45";"G45";
\ar@{} (0, 0);(34, -122) *\dir<2pt>{*} = "B48";
\ar@{-} "B48";(32, -126) *\dir<2pt>{*} = "C48";
\ar@{-} "C48";(34, -130) *\dir<2pt>{*} = "D48";
\ar@{-} "D48";(38, -130) *\dir<2pt>{*} = "E48";
\ar@{-} "E48";(40, -126) *\dir<2pt>{*} = "F48";
\ar@{} "F48";(38, -122) *\dir<2pt>{*} = "G48";
\ar@{-} "B48";"G48";
\ar@{-} "B48";"D48";
\ar@{-} "B48";"E48";
\ar@{} "B48";"F48";
\ar@{} "C48";"E48";
\ar@{} "C48";"F48";
\ar@{} "C48";"G48";
\ar@{} "D48";"F48";
\ar@{-} "D48";"G48";
\ar@{} "E48";"G48";
\ar@{} (0, 0);(46, -122) *\dir<2pt>{*} = "B50";
\ar@{-} "B50";(44, -126) *\dir<2pt>{*} = "C50";
\ar@{-} "C50";(46, -130) *\dir<2pt>{*} = "D50";
\ar@{-} "D50";(50, -130) *\dir<2pt>{*} = "E50";
\ar@{-} "E50";(52, -126) *\dir<2pt>{*} = "F50";
\ar@{-} "F50";(50, -122) *\dir<2pt>{*} = "G50";
\ar@{-} "B50";"G50";
\ar@{-} "B50";"D50";
\ar@{-} "B50";"E50";
\ar@{} "B50";"F50";
\ar@{} "C50";"E50";
\ar@{} "C50";"F50";
\ar@{} "C50";"G50";
\ar@{} "D50";"F50";
\ar@{} "D50";"G50";
\ar@{} "E50";"G50";
\ar@{} (0, 0);(-2, -134) *\dir<2pt>{*} = "B51";
\ar@{-} "B51";(-4, -138) *\dir<2pt>{*} = "C51";
\ar@{-} "C51";(-2, -142) *\dir<2pt>{*} = "D51";
\ar@{-} "D51";(2, -142) *\dir<2pt>{*} = "E51";
\ar@{-} "E51";(4, -138) *\dir<2pt>{*} = "F51";
\ar@{} "F51";(2, -134) *\dir<2pt>{*} = "G51";
\ar@{-} "B51";"G51";
\ar@{-} "B51";"D51";
\ar@{} "B51";"E51";
\ar@{} "B51";"F51";
\ar@{-} "C51";"E51";
\ar@{} "C51";"F51";
\ar@{} "C51";"G51";
\ar@{} "D51";"F51";
\ar@{} "D51";"G51";
\ar@{-} "E51";"G51";
\ar@{} (0, 0);(10, -134) *\dir<2pt>{*} = "B52";
\ar@{-} "B52";(8, -138) *\dir<2pt>{*} = "C52";
\ar@{-} "C52";(10, -142) *\dir<2pt>{*} = "D52";
\ar@{-} "D52";(14, -142) *\dir<2pt>{*} = "E52";
\ar@{} "E52";(16, -138) *\dir<2pt>{*} = "F52";
\ar@{-} "F52";(14, -134) *\dir<2pt>{*} = "G52";
\ar@{-} "B52";"G52";
\ar@{} "B52";"D52";
\ar@{-} "B52";"E52";
\ar@{} "B52";"F52";
\ar@{-} "C52";"E52";
\ar@{} "C52";"F52";
\ar@{} "C52";"G52";
\ar@{} "D52";"F52";
\ar@{-} "D52";"G52";
\ar@{} "E52";"G52";
\ar@{} (0, 0);(22, -134) *\dir<2pt>{*} = "B53";
\ar@{-} "B53";(20, -138) *\dir<2pt>{*} = "C53";
\ar@{-} "C53";(22, -142) *\dir<2pt>{*} = "D53";
\ar@{-} "D53";(26, -142) *\dir<2pt>{*} = "E53";
\ar@{-} "E53";(28, -138) *\dir<2pt>{*} = "F53";
\ar@{-} "F53";(26, -134) *\dir<2pt>{*} = "G53";
\ar@{} "B53";"G53";
\ar@{} "B53";"D53";
\ar@{} "B53";"E53";
\ar@{-} "B53";"F53";
\ar@{} "C53";"E53";
\ar@{} "C53";"F53";
\ar@{-} "C53";"G53";
\ar@{-} "D53";"F53";
\ar@{} "D53";"G53";
\ar@{} "E53";"G53";
\ar@{} (0, 0);(34, -134) *\dir<2pt>{*} = "B55";
\ar@{-} "B55";(32, -138) *\dir<2pt>{*} = "C55";
\ar@{-} "C55";(34, -142) *\dir<2pt>{*} = "D55";
\ar@{-} "D55";(38, -142) *\dir<2pt>{*} = "E55";
\ar@{-} "E55";(40, -138) *\dir<2pt>{*} = "F55";
\ar@{} "F55";(38, -134) *\dir<2pt>{*} = "G55";
\ar@{-} "B55";"G55";
\ar@{} "B55";"D55";
\ar@{-} "B55";"E55";
\ar@{} "B55";"F55";
\ar@{} "C55";"E55";
\ar@{} "C55";"F55";
\ar@{} "C55";"G55";
\ar@{} "D55";"F55";
\ar@{-} "D55";"G55";
\ar@{-} "E55";"G55";
\ar@{} (0, 0);(46, -134) *\dir<2pt>{*} = "B56";
\ar@{-} "B56";(44, -138) *\dir<2pt>{*} = "C56";
\ar@{-} "C56";(46, -142) *\dir<2pt>{*} = "D56";
\ar@{-} "D56";(50, -142) *\dir<2pt>{*} = "E56";
\ar@{-} "E56";(52, -138) *\dir<2pt>{*} = "F56";
\ar@{-} "F56";(50, -134) *\dir<2pt>{*} = "G56";
\ar@{} "B56";"G56";
\ar@{} "B56";"D56";
\ar@{} "B56";"E56";
\ar@{-} "B56";"F56";
\ar@{} "C56";"E56";
\ar@{-} "C56";"F56";
\ar@{-} "C56";"G56";
\ar@{} "D56";"F56";
\ar@{} "D56";"G56";
\ar@{} "E56";"G56";
\ar@{} (0, 0);(-2, -146) *\dir<2pt>{*} = "B59";
\ar@{-} "B59";(-4, -150) *\dir<2pt>{*} = "C59";
\ar@{-} "C59";(-2, -154) *\dir<2pt>{*} = "D59";
\ar@{-} "D59";(2, -154) *\dir<2pt>{*} = "E59";
\ar@{-} "E59";(4, -150) *\dir<2pt>{*} = "F59";
\ar@{-} "F59";(2, -146) *\dir<2pt>{*} = "G59";
\ar@{-} "B59";"G59";
\ar@{-} "B59";"D59";
\ar@{} "B59";"E59";
\ar@{-} "B59";"F59";
\ar@{} "C59";"E59";
\ar@{} "C59";"F59";
\ar@{} "C59";"G59";
\ar@{} "D59";"F59";
\ar@{} "D59";"G59";
\ar@{} "E59";"G59";
\ar@{} (0, 0);(10, -146) *\dir<2pt>{*} = "B66";
\ar@{-} "B66";(8, -150) *\dir<2pt>{*} = "C66";
\ar@{-} "C66";(10, -154) *\dir<2pt>{*} = "D66";
\ar@{-} "D66";(14, -154) *\dir<2pt>{*} = "E66";
\ar@{-} "E66";(16, -150) *\dir<2pt>{*} = "F66";
\ar@{-} "F66";(14, -146) *\dir<2pt>{*} = "G66";
\ar@{-} "B66";"G66";
\ar@{-} "B66";"D66";
\ar@{-} "B66";"E66";
\ar@{-} "B66";"F66";
\ar@{} "C66";"E66";
\ar@{} "C66";"F66";
\ar@{} "C66";"G66";
\ar@{} "D66";"F66";
\ar@{} "D66";"G66";
\ar@{} "E66";"G66";
\ar@{} (0, 0);(22, -146) *\dir<2pt>{*} = "B68";
\ar@{-} "B68";(20, -150) *\dir<2pt>{*} = "C68";
\ar@{-} "C68";(22, -154) *\dir<2pt>{*} = "D68";
\ar@{-} "D68";(26, -154) *\dir<2pt>{*} = "E68";
\ar@{-} "E68";(28, -150) *\dir<2pt>{*} = "F68";
\ar@{-} "F68";(26, -146) *\dir<2pt>{*} = "G68";
\ar@{} "B68";"G68";
\ar@{} "B68";"D68";
\ar@{} "B68";"E68";
\ar@{-} "B68";"F68";
\ar@{} "C68";"E68";
\ar@{-} "C68";"F68";
\ar@{-} "C68";"G68";
\ar@{-} "D68";"F68";
\ar@{} "D68";"G68";
\ar@{} "E68";"G68";
\ar@{} (0, 0);(34, -146) *\dir<2pt>{*} = "B72";
\ar@{-} "B72";(32, -150) *\dir<2pt>{*} = "C72";
\ar@{-} "C72";(34, -154) *\dir<2pt>{*} = "D72";
\ar@{-} "D72";(38, -154) *\dir<2pt>{*} = "E72";
\ar@{-} "E72";(40, -150) *\dir<2pt>{*} = "F72";
\ar@{-} "F72";(38, -146) *\dir<2pt>{*} = "G72";
\ar@{-} "B72";"G72";
\ar@{-} "B72";"D72";
\ar@{} "B72";"E72";
\ar@{} "B72";"F72";
\ar@{} "C72";"E72";
\ar@{-} "C72";"F72";
\ar@{} "C72";"G72";
\ar@{-} "D72";"F72";
\ar@{} "D72";"G72";
\ar@{} "E72";"G72";
\ar@{} (0, 0);(46, -146) *\dir<2pt>{*} = "B73";
\ar@{-} "B73";(44, -150) *\dir<2pt>{*} = "C73";
\ar@{-} "C73";(46, -154) *\dir<2pt>{*} = "D73";
\ar@{-} "D73";(50, -154) *\dir<2pt>{*} = "E73";
\ar@{-} "E73";(52, -150) *\dir<2pt>{*} = "F73";
\ar@{-} "F73";(50, -146) *\dir<2pt>{*} = "G73";
\ar@{} "B73";"G73";
\ar@{} "B73";"D73";
\ar@{} "B73";"E73";
\ar@{-} "B73";"F73";
\ar@{-} "C73";"E73";
\ar@{} "C73";"F73";
\ar@{-} "C73";"G73";
\ar@{-} "D73";"F73";
\ar@{} "D73";"G73";
\ar@{} "E73";"G73";
\ar@{} (0, 0);(-2, -158) *\dir<2pt>{*} = "B74";
\ar@{-} "B74";(-4, -162) *\dir<2pt>{*} = "C74";
\ar@{-} "C74";(-2, -166) *\dir<2pt>{*} = "D74";
\ar@{-} "D74";(2, -166) *\dir<2pt>{*} = "E74";
\ar@{-} "E74";(4, -162) *\dir<2pt>{*} = "F74";
\ar@{} "F74";(2, -158) *\dir<2pt>{*} = "G74";
\ar@{-} "B74";"G74";
\ar@{-} "B74";"D74";
\ar@{} "B74";"E74";
\ar@{} "B74";"F74";
\ar@{-} "C74";"E74";
\ar@{} "C74";"F74";
\ar@{} "C74";"G74";
\ar@{} "D74";"F74";
\ar@{-} "D74";"G74";
\ar@{-} "E74";"G74";
\ar@{} (0, 0);(10, -158) *\dir<2pt>{*} = "B75";
\ar@{-} "B75";(8, -162) *\dir<2pt>{*} = "C75";
\ar@{-} "C75";(10, -166) *\dir<2pt>{*} = "D75";
\ar@{-} "D75";(14, -166) *\dir<2pt>{*} = "E75";
\ar@{-} "E75";(16, -162) *\dir<2pt>{*} = "F75";
\ar@{-} "F75";(14, -158) *\dir<2pt>{*} = "G75";
\ar@{-} "B75";"G75";
\ar@{} "B75";"D75";
\ar@{-} "B75";"E75";
\ar@{} "B75";"F75";
\ar@{} "C75";"E75";
\ar@{} "C75";"F75";
\ar@{} "C75";"G75";
\ar@{} "D75";"F75";
\ar@{-} "D75";"G75";
\ar@{-} "E75";"G75";
\ar@{} (0, 0);(22, -158) *\dir<2pt>{*} = "B76";
\ar@{-} "B76";(20, -162) *\dir<2pt>{*} = "C76";
\ar@{-} "C76";(22, -166) *\dir<2pt>{*} = "D76";
\ar@{-} "D76";(26, -166) *\dir<2pt>{*} = "E76";
\ar@{-} "E76";(28, -162) *\dir<2pt>{*} = "F76";
\ar@{} "F76";(26, -158) *\dir<2pt>{*} = "G76";
\ar@{-} "B76";"G76";
\ar@{} "B76";"D76";
\ar@{-} "B76";"E76";
\ar@{} "B76";"F76";
\ar@{-} "C76";"E76";
\ar@{} "C76";"F76";
\ar@{} "C76";"G76";
\ar@{} "D76";"F76";
\ar@{-} "D76";"G76";
\ar@{-} "E76";"G76";
\ar@{} (0, 0);(34, -158) *\dir<2pt>{*} = "B78";
\ar@{-} "B78";(32, -162) *\dir<2pt>{*} = "C78";
\ar@{-} "C78";(34, -166) *\dir<2pt>{*} = "D78";
\ar@{-} "D78";(38, -166) *\dir<2pt>{*} = "E78";
\ar@{-} "E78";(40, -162) *\dir<2pt>{*} = "F78";
\ar@{-} "F78";(38, -158) *\dir<2pt>{*} = "G78";
\ar@{-} "B78";"G78";
\ar@{-} "B78";"D78";
\ar@{-} "B78";"E78";
\ar@{} "B78";"F78";
\ar@{} "C78";"E78";
\ar@{-} "C78";"F78";
\ar@{} "C78";"G78";
\ar@{} "D78";"F78";
\ar@{} "D78";"G78";
\ar@{} "E78";"G78";
\ar@{} (0, 0);(46, -158) *\dir<2pt>{*} = "B89";
\ar@{-} "B89";(44, -162) *\dir<2pt>{*} = "C89";
\ar@{-} "C89";(46, -166) *\dir<2pt>{*} = "D89";
\ar@{-} "D89";(50, -166) *\dir<2pt>{*} = "E89";
\ar@{-} "E89";(52, -162) *\dir<2pt>{*} = "F89";
\ar@{-} "F89";(50, -158) *\dir<2pt>{*} = "G89";
\ar@{-} "B89";"G89";
\ar@{} "B89";"D89";
\ar@{-} "B89";"E89";
\ar@{} "B89";"F89";
\ar@{-} "C89";"E89";
\ar@{} "C89";"F89";
\ar@{} "C89";"G89";
\ar@{} "D89";"F89";
\ar@{-} "D89";"G89";
\ar@{-} "E89";"G89";
\ar@{} (0, 0);(-2, -170) *\dir<2pt>{*} = "B91";
\ar@{-} "B91";(-4, -174) *\dir<2pt>{*} = "C91";
\ar@{-} "C91";(-2, -178) *\dir<2pt>{*} = "D91";
\ar@{-} "D91";(2, -178) *\dir<2pt>{*} = "E91";
\ar@{-} "E91";(4, -174) *\dir<2pt>{*} = "F91";
\ar@{-} "F91";(2, -170) *\dir<2pt>{*} = "G91";
\ar@{-} "B91";"G91";
\ar@{} "B91";"D91";
\ar@{-} "B91";"E91";
\ar@{} "B91";"F91";
\ar@{} "C91";"E91";
\ar@{-} "C91";"F91";
\ar@{-} "C91";"G91";
\ar@{} "D91";"F91";
\ar@{} "D91";"G91";
\ar@{-} "E91";"G91";
\ar@{} (0, 0);(10, -170) *\dir<2pt>{*} = "B93";
\ar@{-} "B93";(8, -174) *\dir<2pt>{*} = "C93";
\ar@{-} "C93";(10, -178) *\dir<2pt>{*} = "D93";
\ar@{-} "D93";(14, -178) *\dir<2pt>{*} = "E93";
\ar@{-} "E93";(16, -174) *\dir<2pt>{*} = "F93";
\ar@{-} "F93";(14, -170) *\dir<2pt>{*} = "G93";
\ar@{-} "B93";"G93";
\ar@{} "B93";"D93";
\ar@{-} "B93";"E93";
\ar@{} "B93";"F93";
\ar@{} "C93";"E93";
\ar@{-} "C93";"F93";
\ar@{} "C93";"G93";
\ar@{} "D93";"F93";
\ar@{-} "D93";"G93";
\ar@{-} "E93";"G93";
\ar@{} (0, 0);(22, -170) *\dir<2pt>{*} = "B99";
\ar@{-} "B99";(20, -174) *\dir<2pt>{*} = "C99";
\ar@{-} "C99";(22, -178) *\dir<2pt>{*} = "D99";
\ar@{-} "D99";(26, -178) *\dir<2pt>{*} = "E99";
\ar@{-} "E99";(28, -174) *\dir<2pt>{*} = "F99";
\ar@{-} "F99";(26, -170) *\dir<2pt>{*} = "G99";
\ar@{-} "B99";"G99";
\ar@{} "B99";"D99";
\ar@{-} "B99";"E99";
\ar@{} "B99";"F99";
\ar@{-} "C99";"E99";
\ar@{-} "C99";"F99";
\ar@{-} "C99";"G99";
\ar@{} "D99";"F99";
\ar@{-} "D99";"G99";
\ar@{} "E99";"G99";
\ar@{} (0, 0);(-2, -190) *\dir<2pt>{*} = "B87";
\ar@{-} "B87";(-4, -194) *\dir<2pt>{*} = "C87";
\ar@{-} "C87";(-2, -198) *\dir<2pt>{*} = "D87";
\ar@{-} "D87";(2, -198) *\dir<2pt>{*} = "E87";
\ar@{-} "E87";(4, -194) *\dir<2pt>{*} = "F87";
\ar@{-} "F87";(2, -190) *\dir<2pt>{*} = "G87";
\ar@{-} "B87";"G87";
\ar@{} "B87";"D87";
\ar@{} "B87";"E87";
\ar@{-} "B87";"F87";
\ar@{-} "C87";"E87";
\ar@{} "C87";"F87";
\ar@{-} "C87";"G87";
\ar@{} "D87";"F87";
\ar@{} "D87";"G87";
\ar@{-} "E87";"G87";
\ar@{} (0, 0);(10, -190) *\dir<2pt>{*} = "B100";
\ar@{-} "B100";(8, -194) *\dir<2pt>{*} = "C100";
\ar@{-} "C100";(10, -198) *\dir<2pt>{*} = "D100";
\ar@{-} "D100";(14, -198) *\dir<2pt>{*} = "E100";
\ar@{-} "E100";(16, -194) *\dir<2pt>{*} = "F100";
\ar@{-} "F100";(14, -190) *\dir<2pt>{*} = "G100";
\ar@{-} "B100";"G100";
\ar@{} "B100";"D100";
\ar@{} "B100";"E100";
\ar@{-} "B100";"F100";
\ar@{-} "C100";"E100";
\ar@{} "C100";"F100";
\ar@{-} "C100";"G100";
\ar@{-} "D100";"F100";
\ar@{-} "D100";"G100";
\ar@{} "E100";"G100";
\ar@{} (0, 0);(22, -190) *\dir<2pt>{*} = "B108";
\ar@{-} "B108";(20, -194) *\dir<2pt>{*} = "C108";
\ar@{-} "C108";(22, -198) *\dir<2pt>{*} = "D108";
\ar@{-} "D108";(26, -198) *\dir<2pt>{*} = "E108";
\ar@{-} "E108";(28, -194) *\dir<2pt>{*} = "F108";
\ar@{-} "F108";(26, -190) *\dir<2pt>{*} = "G108";
\ar@{-} "B108";"G108";
\ar@{-} "B108";"D108";
\ar@{} "B108";"E108";
\ar@{-} "B108";"F108";
\ar@{-} "C108";"E108";
\ar@{} "C108";"F108";
\ar@{-} "C108";"G108";
\ar@{-} "D108";"F108";
\ar@{} "D108";"G108";
\ar@{-} "E108";"G108";
\ar@{} (0, 0);(-2, -206) *\dir<2pt>{*} = "B60";
\ar@{-} "B60";(-4, -210) *\dir<2pt>{*} = "C60";
\ar@{-} "C60";(-2, -214) *\dir<2pt>{*} = "D60";
\ar@{-} "D60";(2, -214) *\dir<2pt>{*} = "E60";
\ar@{-} "E60";(4, -210) *\dir<2pt>{*} = "F60";
\ar@{-} "F60";(2, -206) *\dir<2pt>{*} = "G60";
\ar@{-} "B60";"G60";
\ar@{} "B60";"D60";
\ar@{} "B60";"E60";
\ar@{-} "B60";"F60";
\ar@{} "C60";"E60";
\ar@{} "C60";"F60";
\ar@{-} "C60";"G60";
\ar@{} "D60";"F60";
\ar@{} "D60";"G60";
\ar@{} "E60";"G60";
\ar@{} (0, 0);(10, -206) *\dir<2pt>{*} = "B70";
\ar@{-} "B70";(8, -210) *\dir<2pt>{*} = "C70";
\ar@{-} "C70";(10, -214) *\dir<2pt>{*} = "D70";
\ar@{-} "D70";(14, -214) *\dir<2pt>{*} = "E70";
\ar@{-} "E70";(16, -210) *\dir<2pt>{*} = "F70";
\ar@{-} "F70";(14, -206) *\dir<2pt>{*} = "G70";
\ar@{-} "B70";"G70";
\ar@{} "B70";"D70";
\ar@{} "B70";"E70";
\ar@{} "B70";"F70";
\ar@{-} "C70";"E70";
\ar@{} "C70";"F70";
\ar@{-} "C70";"G70";
\ar@{} "D70";"F70";
\ar@{} "D70";"G70";
\ar@{-} "E70";"G70";
\ar@{} (0, 0);(22, -206) *\dir<2pt>{*} = "B71";
\ar@{-} "B71";(20, -210) *\dir<2pt>{*} = "C71";
\ar@{-} "C71";(22, -214) *\dir<2pt>{*} = "D71";
\ar@{-} "D71";(26, -214) *\dir<2pt>{*} = "E71";
\ar@{-} "E71";(28, -210) *\dir<2pt>{*} = "F71";
\ar@{-} "F71";(26, -206) *\dir<2pt>{*} = "G71";
\ar@{-} "B71";"G71";
\ar@{-} "B71";"D71";
\ar@{} "B71";"E71";
\ar@{} "B71";"F71";
\ar@{} "C71";"E71";
\ar@{-} "C71";"F71";
\ar@{} "C71";"G71";
\ar@{} "D71";"F71";
\ar@{} "D71";"G71";
\ar@{-} "E71";"G71";
\ar@{} (0, 0);(34, -206) *\dir<2pt>{*} = "B80";
\ar@{-} "B80";(32, -210) *\dir<2pt>{*} = "C80";
\ar@{-} "C80";(34, -214) *\dir<2pt>{*} = "D80";
\ar@{-} "D80";(38, -214) *\dir<2pt>{*} = "E80";
\ar@{-} "E80";(40, -210) *\dir<2pt>{*} = "F80";
\ar@{-} "F80";(38, -206) *\dir<2pt>{*} = "G80";
\ar@{-} "B80";"G80";
\ar@{-} "B80";"D80";
\ar@{-} "B80";"E80";
\ar@{} "B80";"F80";
\ar@{} "C80";"E80";
\ar@{} "C80";"F80";
\ar@{-} "C80";"G80";
\ar@{} "D80";"F80";
\ar@{} "D80";"G80";
\ar@{} "E80";"G80";
\ar@{} (0, 0);(46, -206) *\dir<2pt>{*} = "B94";
\ar@{-} "B94";(44, -210) *\dir<2pt>{*} = "C94";
\ar@{-} "C94";(46, -214) *\dir<2pt>{*} = "D94";
\ar@{-} "D94";(50, -214) *\dir<2pt>{*} = "E94";
\ar@{-} "E94";(52, -210) *\dir<2pt>{*} = "F94";
\ar@{-} "F94";(50, -206) *\dir<2pt>{*} = "G94";
\ar@{-} "B94";"G94";
\ar@{} "B94";"D94";
\ar@{-} "B94";"E94";
\ar@{} "B94";"F94";
\ar@{-} "C94";"E94";
\ar@{} "C94";"F94";
\ar@{} "C94";"G94";
\ar@{-} "D94";"F94";
\ar@{} "D94";"G94";
\ar@{-} "E94";"G94";
\ar@{} (0, 0);(104, -36) *\dir<2pt>{*} = "B20";
\ar@{-} "B20";(102, -40) *\dir<2pt>{*} = "C20";
\ar@{-} "C20";(104, -44) *\dir<2pt>{*} = "D20";
\ar@{-} "D20";(108, -44) *\dir<2pt>{*} = "E20";
\ar@{-} "E20";(110, -40) *\dir<2pt>{*} = "F20";
\ar@{-} "F20";(108, -36) *\dir<2pt>{*} = "G20";
\ar@{} "B20";"G20";
\ar@{-} "B20";"D20";
\ar@{-} "B20";"E20";
\ar@{} "B20";"F20";
\ar@{} "C20";"E20";
\ar@{} "C20";"F20";
\ar@{} "C20";"G20";
\ar@{} "D20";"F20";
\ar@{} "D20";"G20";
\ar@{-} "E20";"G20";
\ar@{} (0, 0);(104, -48) *\dir<2pt>{*} = "B24";
\ar@{-} "B24";(102, -52) *\dir<2pt>{*} = "C24";
\ar@{-} "C24";(104, -56) *\dir<2pt>{*} = "D24";
\ar@{-} "D24";(108, -56) *\dir<2pt>{*} = "E24";
\ar@{-} "E24";(110, -52) *\dir<2pt>{*} = "F24";
\ar@{-} "F24";(108, -48) *\dir<2pt>{*} = "G24";
\ar@{} "B24";"G24";
\ar@{-} "B24";"D24";
\ar@{} "B24";"E24";
\ar@{} "B24";"F24";
\ar@{} "C24";"E24";
\ar@{} "C24";"F24";
\ar@{} "C24";"G24";
\ar@{} "D24";"F24";
\ar@{} "D24";"G24";
\ar@{-} "E24";"G24";
\ar@{} (0, 0);(104, -60) *\dir<2pt>{*} = "B40";
\ar@{} "B40";(102, -64) *\dir<2pt>{*} = "C40";
\ar@{-} "C40";(104, -68) *\dir<2pt>{*} = "D40";
\ar@{-} "D40";(108, -68) *\dir<2pt>{*} = "E40";
\ar@{-} "E40";(110, -64) *\dir<2pt>{*} = "F40";
\ar@{} "F40";(108, -60) *\dir<2pt>{*} = "G40";
\ar@{-} "B40";"G40";
\ar@{-} "B40";"D40";
\ar@{-} "B40";"E40";
\ar@{} "B40";"F40";
\ar@{} "C40";"E40";
\ar@{} "C40";"F40";
\ar@{} "C40";"G40";
\ar@{} "D40";"F40";
\ar@{-} "D40";"G40";
\ar@{-} "E40";"G40";
\ar@{} (0, 0);(104, -72) *\dir<2pt>{*} = "B41";
\ar@{-} "B41";(102, -76) *\dir<2pt>{*} = "C41";
\ar@{-} "C41";(104, -80) *\dir<2pt>{*} = "D41";
\ar@{-} "D41";(108, -80) *\dir<2pt>{*} = "E41";
\ar@{} "E41";(110, -76) *\dir<2pt>{*} = "F41";
\ar@{-} "F41";(108, -72) *\dir<2pt>{*} = "G41";
\ar@{-} "B41";"G41";
\ar@{-} "B41";"D41";
\ar@{-} "B41";"E41";
\ar@{} "B41";"F41";
\ar@{-} "C41";"E41";
\ar@{} "C41";"F41";
\ar@{} "C41";"G41";
\ar@{} "D41";"F41";
\ar@{} "D41";"G41";
\ar@{} "E41";"G41";
\ar@{} (0, 0);(104, -84) *\dir<2pt>{*} = "B54";
\ar@{-} "B54";(102, -88) *\dir<2pt>{*} = "C54";
\ar@{-} "C54";(104, -92) *\dir<2pt>{*} = "D54";
\ar@{-} "D54";(108, -92) *\dir<2pt>{*} = "E54";
\ar@{-} "E54";(110, -88) *\dir<2pt>{*} = "F54";
\ar@{-} "F54";(108, -84) *\dir<2pt>{*} = "G54";
\ar@{-} "B54";"G54";
\ar@{-} "B54";"D54";
\ar@{} "B54";"E54";
\ar@{} "B54";"F54";
\ar@{} "C54";"E54";
\ar@{} "C54";"F54";
\ar@{} "C54";"G54";
\ar@{} "D54";"F54";
\ar@{} "D54";"G54";
\ar@{-} "E54";"G54";
\ar@{} (0, 0);(104, -96) *\dir<2pt>{*} = "B61";
\ar@{-} "B61";(102, -100) *\dir<2pt>{*} = "C61";
\ar@{-} "C61";(104, -104) *\dir<2pt>{*} = "D61";
\ar@{-} "D61";(108, -104) *\dir<2pt>{*} = "E61";
\ar@{-} "E61";(110, -100) *\dir<2pt>{*} = "F61";
\ar@{} "F61";(108, -96) *\dir<2pt>{*} = "G61";
\ar@{-} "B61";"G61";
\ar@{-} "B61";"D61";
\ar@{-} "B61";"E61";
\ar@{} "B61";"F61";
\ar@{} "C61";"E61";
\ar@{} "C61";"F61";
\ar@{} "C61";"G61";
\ar@{} "D61";"F61";
\ar@{-} "D61";"G61";
\ar@{-} "E61";"G61";
\ar@{} (0, 0);(104, -108) *\dir<2pt>{*} = "B62";
\ar@{-} "B62";(102, -112) *\dir<2pt>{*} = "C62";
\ar@{-} "C62";(104, -116) *\dir<2pt>{*} = "D62";
\ar@{-} "D62";(108, -116) *\dir<2pt>{*} = "E62";
\ar@{-} "E62";(110, -112) *\dir<2pt>{*} = "F62";
\ar@{-} "F62";(108, -108) *\dir<2pt>{*} = "G62";
\ar@{-} "B62";"G62";
\ar@{-} "B62";"D62";
\ar@{} "B62";"E62";
\ar@{} "B62";"F62";
\ar@{} "C62";"E62";
\ar@{} "C62";"F62";
\ar@{} "C62";"G62";
\ar@{} "D62";"F62";
\ar@{-} "D62";"G62";
\ar@{-} "E62";"G62";
\ar@{} (0, 0);(104, -120) *\dir<2pt>{*} = "B64";
\ar@{-} "B64";(102, -124) *\dir<2pt>{*} = "C64";
\ar@{-} "C64";(104, -128) *\dir<2pt>{*} = "D64";
\ar@{-} "D64";(108, -128) *\dir<2pt>{*} = "E64";
\ar@{-} "E64";(110, -124) *\dir<2pt>{*} = "F64";
\ar@{} "F64";(108, -120) *\dir<2pt>{*} = "G64";
\ar@{-} "B64";"G64";
\ar@{-} "B64";"D64";
\ar@{} "B64";"E64";
\ar@{} "B64";"F64";
\ar@{} "C64";"E64";
\ar@{} "C64";"F64";
\ar@{-} "C64";"G64";
\ar@{} "D64";"F64";
\ar@{-} "D64";"G64";
\ar@{-} "E64";"G64";
\ar@{} (0, 0);(104, -132) *\dir<2pt>{*} = "B77";
\ar@{-} "B77";(102, -136) *\dir<2pt>{*} = "C77";
\ar@{-} "C77";(104, -140) *\dir<2pt>{*} = "D77";
\ar@{-} "D77";(108, -140) *\dir<2pt>{*} = "E77";
\ar@{-} "E77";(110, -136) *\dir<2pt>{*} = "F77";
\ar@{-} "F77";(108, -132) *\dir<2pt>{*} = "G77";
\ar@{-} "B77";"G77";
\ar@{-} "B77";"D77";
\ar@{} "B77";"E77";
\ar@{} "B77";"F77";
\ar@{} "C77";"E77";
\ar@{} "C77";"F77";
\ar@{-} "C77";"G77";
\ar@{} "D77";"F77";
\ar@{} "D77";"G77";
\ar@{-} "E77";"G77";
\ar@{} (0, 0);(104, -144) *\dir<2pt>{*} = "B79";
\ar@{-} "B79";(102, -148) *\dir<2pt>{*} = "C79";
\ar@{-} "C79";(104, -152) *\dir<2pt>{*} = "D79";
\ar@{-} "D79";(108, -152) *\dir<2pt>{*} = "E79";
\ar@{-} "E79";(110, -148) *\dir<2pt>{*} = "F79";
\ar@{-} "F79";(108, -144) *\dir<2pt>{*} = "G79";
\ar@{-} "B79";"G79";
\ar@{-} "B79";"D79";
\ar@{} "B79";"E79";
\ar@{} "B79";"F79";
\ar@{} "C79";"E79";
\ar@{} "C79";"F79";
\ar@{-} "C79";"G79";
\ar@{} "D79";"F79";
\ar@{-} "D79";"G79";
\ar@{} "E79";"G79";
\ar@{} (0, 0);(104, -156) *\dir<2pt>{*} = "B81";
\ar@{-} "B81";(102, -160) *\dir<2pt>{*} = "C81";
\ar@{-} "C81";(104, -164) *\dir<2pt>{*} = "D81";
\ar@{-} "D81";(108, -164) *\dir<2pt>{*} = "E81";
\ar@{-} "E81";(110, -160) *\dir<2pt>{*} = "F81";
\ar@{-} "F81";(108, -156) *\dir<2pt>{*} = "G81";
\ar@{-} "B81";"G81";
\ar@{-} "B81";"D81";
\ar@{-} "B81";"E81";
\ar@{} "B81";"F81";
\ar@{} "C81";"E81";
\ar@{} "C81";"F81";
\ar@{} "C81";"G81";
\ar@{} "D81";"F81";
\ar@{-} "D81";"G81";
\ar@{-} "E81";"G81";
\ar@{} (0, 0);(104, -168) *\dir<2pt>{*} = "B82";
\ar@{-} "B82";(102, -172) *\dir<2pt>{*} = "C82";
\ar@{-} "C82";(104, -176) *\dir<2pt>{*} = "D82";
\ar@{-} "D82";(108, -176) *\dir<2pt>{*} = "E82";
\ar@{-} "E82";(110, -172) *\dir<2pt>{*} = "F82";
\ar@{-} "F82";(108, -168) *\dir<2pt>{*} = "G82";
\ar@{-} "B82";"G82";
\ar@{-} "B82";"D82";
\ar@{-} "B82";"E82";
\ar@{} "B82";"F82";
\ar@{-} "C82";"E82";
\ar@{} "C82";"F82";
\ar@{} "C82";"G82";
\ar@{} "D82";"F82";
\ar@{} "D82";"G82";
\ar@{-} "E82";"G82";
\ar@{} (0, 0);(116, -36) *\dir<2pt>{*} = "B83";
\ar@{-} "B83";(114, -40) *\dir<2pt>{*} = "C83";
\ar@{-} "C83";(116, -44) *\dir<2pt>{*} = "D83";
\ar@{-} "D83";(120, -44) *\dir<2pt>{*} = "E83";
\ar@{-} "E83";(122, -40) *\dir<2pt>{*} = "F83";
\ar@{} "F83";(120, -36) *\dir<2pt>{*} = "G83";
\ar@{-} "B83";"G83";
\ar@{-} "B83";"D83";
\ar@{} "B83";"E83";
\ar@{} "B83";"F83";
\ar@{-} "C83";"E83";
\ar@{} "C83";"F83";
\ar@{-} "C83";"G83";
\ar@{} "D83";"F83";
\ar@{-} "D83";"G83";
\ar@{-} "E83";"G83";
\ar@{} (0, 0);(116, -48) *\dir<2pt>{*} = "B84";
\ar@{-} "B84";(114, -52) *\dir<2pt>{*} = "C84";
\ar@{-} "C84";(116, -56) *\dir<2pt>{*} = "D84";
\ar@{-} "D84";(120, -56) *\dir<2pt>{*} = "E84";
\ar@{-} "E84";(122, -52) *\dir<2pt>{*} = "F84";
\ar@{-} "F84";(120, -48) *\dir<2pt>{*} = "G84";
\ar@{-} "B84";"G84";
\ar@{} "B84";"D84";
\ar@{-} "B84";"E84";
\ar@{} "B84";"F84";
\ar@{-} "C84";"E84";
\ar@{} "C84";"F84";
\ar@{-} "C84";"G84";
\ar@{} "D84";"F84";
\ar@{} "D84";"G84";
\ar@{-} "E84";"G84";
\ar@{} (0, 0);(116, -60) *\dir<2pt>{*} = "B88";
\ar@{-} "B88";(114, -64) *\dir<2pt>{*} = "C88";
\ar@{-} "C88";(116, -68) *\dir<2pt>{*} = "D88";
\ar@{-} "D88";(120, -68) *\dir<2pt>{*} = "E88";
\ar@{-} "E88";(122, -64) *\dir<2pt>{*} = "F88";
\ar@{-} "F88";(120, -60) *\dir<2pt>{*} = "G88";
\ar@{-} "B88";"G88";
\ar@{} "B88";"D88";
\ar@{} "B88";"E88";
\ar@{-} "B88";"F88";
\ar@{-} "C88";"E88";
\ar@{} "C88";"F88";
\ar@{-} "C88";"G88";
\ar@{-} "D88";"F88";
\ar@{} "D88";"G88";
\ar@{} "E88";"G88";
\ar@{} (0, 0);(116, -72) *\dir<2pt>{*} = "B90";
\ar@{-} "B90";(114, -76) *\dir<2pt>{*} = "C90";
\ar@{-} "C90";(116, -80) *\dir<2pt>{*} = "D90";
\ar@{-} "D90";(120, -80) *\dir<2pt>{*} = "E90";
\ar@{-} "E90";(122, -76) *\dir<2pt>{*} = "F90";
\ar@{-} "F90";(120, -72) *\dir<2pt>{*} = "G90";
\ar@{-} "B90";"G90";
\ar@{} "B90";"D90";
\ar@{} "B90";"E90";
\ar@{-} "B90";"F90";
\ar@{-} "C90";"E90";
\ar@{} "C90";"F90";
\ar@{} "C90";"G90";
\ar@{-} "D90";"F90";
\ar@{} "D90";"G90";
\ar@{-} "E90";"G90";
\ar@{} (0, 0);(116, -84) *\dir<2pt>{*} = "B92";
\ar@{-} "B92";(114, -88) *\dir<2pt>{*} = "C92";
\ar@{-} "C92";(116, -92) *\dir<2pt>{*} = "D92";
\ar@{-} "D92";(120, -92) *\dir<2pt>{*} = "E92";
\ar@{-} "E92";(122, -88) *\dir<2pt>{*} = "F92";
\ar@{-} "F92";(120, -84) *\dir<2pt>{*} = "G92";
\ar@{-} "B92";"G92";
\ar@{} "B92";"D92";
\ar@{-} "B92";"E92";
\ar@{} "B92";"F92";
\ar@{-} "C92";"E92";
\ar@{-} "C92";"F92";
\ar@{} "C92";"G92";
\ar@{-} "D92";"F92";
\ar@{} "D92";"G92";
\ar@{} "E92";"G92";
\ar@{} (0, 0);(116, -96) *\dir<2pt>{*} = "B97";
\ar@{-} "B97";(114, -100) *\dir<2pt>{*} = "C97";
\ar@{-} "C97";(116, -104) *\dir<2pt>{*} = "D97";
\ar@{-} "D97";(120, -104) *\dir<2pt>{*} = "E97";
\ar@{-} "E97";(122, -100) *\dir<2pt>{*} = "F97";
\ar@{-} "F97";(120, -96) *\dir<2pt>{*} = "G97";
\ar@{} "B97";"G97";
\ar@{-} "B97";"D97";
\ar@{-} "B97";"E97";
\ar@{-} "B97";"F97";
\ar@{} "C97";"E97";
\ar@{-} "C97";"F97";
\ar@{-} "C97";"G97";
\ar@{-} "D97";"F97";
\ar@{} "D97";"G97";
\ar@{} "E97";"G97";
\ar@{} (0, 0);(116, -108) *\dir<2pt>{*} = "B101";
\ar@{-} "B101";(114, -112) *\dir<2pt>{*} = "C101";
\ar@{-} "C101";(116, -116) *\dir<2pt>{*} = "D101";
\ar@{-} "D101";(120, -116) *\dir<2pt>{*} = "E101";
\ar@{-} "E101";(122, -112) *\dir<2pt>{*} = "F101";
\ar@{-} "F101";(120, -108) *\dir<2pt>{*} = "G101";
\ar@{-} "B101";"G101";
\ar@{-} "B101";"D101";
\ar@{-} "B101";"E101";
\ar@{} "B101";"F101";
\ar@{} "C101";"E101";
\ar@{-} "C101";"F101";
\ar@{} "C101";"G101";
\ar@{} "D101";"F101";
\ar@{-} "D101";"G101";
\ar@{-} "E101";"G101";
\ar@{} (0, 0);(116, -120) *\dir<2pt>{*} = "B102";
\ar@{-} "B102";(114, -124) *\dir<2pt>{*} = "C102";
\ar@{-} "C102";(116, -128) *\dir<2pt>{*} = "D102";
\ar@{-} "D102";(120, -128) *\dir<2pt>{*} = "E102";
\ar@{-} "E102";(122, -124) *\dir<2pt>{*} = "F102";
\ar@{-} "F102";(120, -120) *\dir<2pt>{*} = "G102";
\ar@{} "B102";"G102";
\ar@{-} "B102";"D102";
\ar@{-} "B102";"E102";
\ar@{-} "B102";"F102";
\ar@{-} "C102";"E102";
\ar@{} "C102";"F102";
\ar@{-} "C102";"G102";
\ar@{-} "D102";"F102";
\ar@{} "D102";"G102";
\ar@{} "E102";"G102";
\ar@{} (0, 0);(116, -132) *\dir<2pt>{*} = "B103";
\ar@{-} "B103";(114, -136) *\dir<2pt>{*} = "C103";
\ar@{-} "C103";(116, -140) *\dir<2pt>{*} = "D103";
\ar@{-} "D103";(120, -140) *\dir<2pt>{*} = "E103";
\ar@{-} "E103";(122, -136) *\dir<2pt>{*} = "F103";
\ar@{-} "F103";(120, -132) *\dir<2pt>{*} = "G103";
\ar@{-} "B103";"G103";
\ar@{} "B103";"D103";
\ar@{} "B103";"E103";
\ar@{-} "B103";"F103";
\ar@{-} "C103";"E103";
\ar@{-} "C103";"F103";
\ar@{} "C103";"G103";
\ar@{-} "D103";"F103";
\ar@{-} "D103";"G103";
\ar@{} "E103";"G103";
\ar@{} (0, 0);(116, -144) *\dir<2pt>{*} = "B104";
\ar@{-} "B104";(114, -148) *\dir<2pt>{*} = "C104";
\ar@{-} "C104";(116, -152) *\dir<2pt>{*} = "D104";
\ar@{-} "D104";(120, -152) *\dir<2pt>{*} = "E104";
\ar@{-} "E104";(122, -148) *\dir<2pt>{*} = "F104";
\ar@{-} "F104";(120, -144) *\dir<2pt>{*} = "G104";
\ar@{-} "B104";"G104";
\ar@{} "B104";"D104";
\ar@{-} "B104";"E104";
\ar@{-} "B104";"F104";
\ar@{-} "C104";"E104";
\ar@{-} "C104";"F104";
\ar@{-} "C104";"G104";
\ar@{-} "D104";"F104";
\ar@{} "D104";"G104";
\ar@{} "E104";"G104";
\ar@{} (0, 0);(116, -156) *\dir<2pt>{*} = "B107";
\ar@{-} "B107";(114, -160) *\dir<2pt>{*} = "C107";
\ar@{-} "C107";(116, -164) *\dir<2pt>{*} = "D107";
\ar@{-} "D107";(120, -164) *\dir<2pt>{*} = "E107";
\ar@{-} "E107";(122, -160) *\dir<2pt>{*} = "F107";
\ar@{-} "F107";(120, -156) *\dir<2pt>{*} = "G107";
\ar@{-} "B107";"G107";
\ar@{} "B107";"D107";
\ar@{-} "B107";"E107";
\ar@{} "B107";"F107";
\ar@{-} "C107";"E107";
\ar@{-} "C107";"F107";
\ar@{} "C107";"G107";
\ar@{-} "D107";"F107";
\ar@{-} "D107";"G107";
\ar@{-} "E107";"G107";
\ar@{} (0, 0);(116, -168) *\dir<2pt>{*} = "B110";
\ar@{-} "B110";(114, -172) *\dir<2pt>{*} = "C110";
\ar@{-} "C110";(116, -176) *\dir<2pt>{*} = "D110";
\ar@{-} "D110";(120, -176) *\dir<2pt>{*} = "E110";
\ar@{-} "E110";(122, -172) *\dir<2pt>{*} = "F110";
\ar@{-} "F110";(120, -168) *\dir<2pt>{*} = "G110";
\ar@{-} "B110";"G110";
\ar@{} "B110";"D110";
\ar@{-} "B110";"E110";
\ar@{-} "B110";"F110";
\ar@{-} "C110";"E110";
\ar@{-} "C110";"F110";
\ar@{-} "C110";"G110";
\ar@{-} "D110";"F110";
\ar@{-} "D110";"G110";
\ar@{} "E110";"G110";
\textcolor{red}{\ar@{} (0, 0);(-13, -32);}
\textcolor{red}{\ar@{-} (-13, -32);(2, -32);}
\textcolor{red}{\ar@{} (0, 0);(20, -32) *\txt{HL-Comparability};}
\textcolor{red}{\ar@{-} (37, -32);(124, -32);}
\textcolor{red}{\ar@{-} (-17, -32);(-17, -186);}
\textcolor{red}{\ar@{-} (121, -32);(121, -186);}
\textcolor{red}{\ar@{-} (-20, -186);(120, -186);}
\textcolor{black}{\ar@{} (0, 0);(-17, -40);}
\textcolor{black}{\ar@{-} (-17, -40);(12, -40);}
\textcolor{black}{\ar@{} (0, 0);(20, -40) *\txt{Bipartite};}
\textcolor{black}{\ar@{-} (29, -40);(55, -40);}
\textcolor{black}{\ar@{-} (-21, -40);(-21, -82);}
\textcolor{black}{\ar@{-} (52, -40);(52, -82);}
\textcolor{black}{\ar@{-} (-24, -82);(51, -82);}
\textcolor{cyan}{\ar@{} (0, 0);(35, -68);}
\textcolor{cyan}{\ar@{-} (35, -68);(78, -68);}
\textcolor{cyan}{\ar@{} (0, 0);(62, -60) *\txt{Trivially};}
\textcolor{cyan}{\ar@{} (0, 0);(60, -64) *\txt{Perfect};}
\textcolor{cyan}{\ar@{-} (31, -68);(31, -158);}
\textcolor{cyan}{\ar@{-} (72.5, -68);(72.5, -158);}
\textcolor{cyan}{\ar@{-} (28, -158);(71, -158);}
\end{xy}
\par \vspace{2mm}
${\mb{Acknowledgement.}}$ I wish to thank Professor Hidefumi Ohsugi
for many valuable comments.
This research was supported by the JST (Japan Science and Technology Agency)
CREST (Core Research for Evolutional Science and Technology) research project
Harmony of Gr\"{o}bner Bases and the Modern Industrial Society in the framework of the
JST Mathematics Program ``Alliance for Breakthrough between Mathematics and
Sciences."
|
3,212,635,537,940 | arxiv | \section{Introduction}
Reasoning is an important and challenging task in artificial intelligence and natural language processing, which is ``\textit{the process of drawing conclusions from the principles and evidence}'' \cite{wason1972psychology}.
The ``\textit{evidence}'' is the fuel and the ``\textit{principle}'' is the machine that operates on the fuel to make predictions.
The majority of studies typically only take the current datapoint as the input, in which case the important ``\textit{evidence}'' of the datapoint from background knowledge is ignored.
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{intro-crop.pdf}
\caption{An example from the CommonsenseQA dataset which requires multiple external knowledge to make the correct prediction. ConceptNet evidence helps pick up choices (A, C) and Wikipedia evidence helps pick up choices (C, E). Combining both evidence will derive the right answer C. Words in blue are the concepts in the question. Words in green are the relations from ConceptNet. Words in red are the choices picked up by evidence.}
\label{fig:intro_example}
\end{figure}
In this work, we study commonsense question answering, a challenging task which requires machines to collect background knowledge and reason over the knowledge to answer questions. For example, an influential dataset CommonsenseQA \cite{talmor2019commonsenseqa} is built in a way that the answer choices share the same relation with the concept in the question while annotators are asked to use their background knowledge to create questions so that only one choice is the correct answer. Figure~\ref{fig:intro_example} shows an example which requires multiple external knowledge sources to make the correct predictions.
The structured evidence from ConcepNet can help pick up the choices (A, C), while evidence from Wikipedia can help pick up the choices (C, E).
Combining both evidence will derive the correct answer (C).
Approaches have been proposed in recent years for extracting evidence and reasoning over evidence. Typically, they either generate evidence from human-annotated evidence~\cite{RajaniMXS19} or extract evidence from a homogeneous knowledge source like structured knowledge ConceptNet \cite{kag2019,bauer2018commonsense,mihaylov2018knowledgeable} or Wikipedia plain texts \cite{ryu2014open,yang2015wikiqa,chen2017reading}, but they fail to take advantages of both knowledge sources simultaneously. Structured knowledge sources contain valuable structural relations between concepts, which are beneficial for reasoning. However, they suffer from low coverage. Plain texts can provide abundant and high-coverage evidence, which is complementary to the structured knowledge.
In this work, we study commonsense question answering by
using automatically
collected evidence from heterogeneous external knowledge.
Our approach consists of two parts: knowledge extraction and graph-based reasoning. In the knowledge extraction part, we automatically extract graph paths from ConceptNet and sentences from Wikipedia.
To better use the relational structure of the evidence, we construct graphs for both sources, including extracted graph paths from ConceptNet and
triples derived from Wikipedia sentences by Semantic Role Labeling (SRL).
In the graph-based reasoning part, we propose a graph-based approach to make better use of the graph information.
We contribute by developing two graph-based modules, including (1) a graph-based contextual word representation learning module, which utilizes graph structural information to re-define the distance between words for learning better contextual word representations, and (2) a graph-based inference module, which first
adopts Graph Convolutional Network \cite{kipf2016semi} to encode neighbor information into the representations of nodes, followed by a graph attention mechanism for evidence aggregation.
We conduct experiments on the CommonsenseQA benchmark dataset.
Results show that both the graph-based contextual representation learning module and the graph-based inference module boost the performance.
We also demonstrate that incorporating both knowledge sources can bring further improvements.
Our approach
achieves the state-of-the-art accuracy (75.3\%) on the CommonsenseQA dataset.
Our contributions of this paper can be summarized as follows:
\begin{itemize}
\item We introduce a graph-based approach to leverage evidence from heterogeneous knowledge sources
for commonsense question answering.
\item We propose a graph-based contextual representation learning module and a graph-based inference module to make better use of the graph information for commonsense question answering.
\item Results show that our model achieves a new state-of-the-art performance on the CommonsenseQA dataset.
\end{itemize}
\section{Task Definition and Dataset}
This paper utilizes CommonsenseQA \cite{talmor2019commonsenseqa}, an influential dataset for commonsense question answering task for experiments. Formally, given a natural language question $Q$ containing $m$ tokens $\{ q_1, q_2, \cdots, q_m\}$, and $5$ choices $\{ a_1, a_2, \cdots, a_5\}$, the target is to distinguish the right answer from the wrong ones and accuracy is adopted as the metric.
Annotators are required to utilize their background knowledge to write questions in which only one of them is correct, thus making the task more challenging.
The lack of evidence requires the model to have strong commonsense knowledge extraction and reasoning ability to get the right results.
\section{Approach Overview}
In this section, we give an overview of our approach. As shown in Figure \ref{fig:overview}, our approach contains two parts: knowledge extraction and graph-based reasoning. In the knowledge extraction part, we extract knowledge from structured knowledge base ConcpetNet and Wikipedia plain texts according to the given question and choices. We construct graphs to utilize the relational structures of both sources.
In the graph-based reasoning part, we propose two graph-based modules which consists of a graph-based contextual word representation learning module and a graph-based inference module to infer final answers. We will describe each part in detail in the following sections.
\begin{figure}[H]
\centering
\includegraphics[width=0.45\textwidth]{overview-crop.pdf}
\caption{An overview of our approach.}
\label{fig:overview}
\end{figure}
\section{Knowledge Extraction}
In this section, we provide the methods to extract evidence from ConceptNet and Wikipedia given the question and choices. Furthermore, we describe the details of constructing graphs for both sources.
\subsection{Knowledge Extraction from ConceptNet}
ConceptNet is a large-scale commonsense knowledge base, containing millions of nodes and relations. The triple in ConceptNet contains four parts: two nodes, one relation, and a relation weight.
For each question and choice, we first identify their entities in the given ConceptNet graph. Then we search for the paths (less than 3 hops) from question entities to choice entities and merge the covered triples into a graph where nodes are triples and edges are the relation between triples. If two triples $s_i$, $s_j$ contain the same entity, we will add an edge from the previous triple $s_i$ to the next triple $s_j$. In order to obtain contextual word representations for ConceptNet nodes, we transfer the triple into a natural language sequence according to the relation template in ConceptNet.
An example is shown in Figure~\ref{fig:concept_graph}. We denote the graph as Concept-Graph.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.35\textwidth]{ConceptNet-crop.pdf}
\caption{An example of constructed Concept-Graph from the ConceptNet evidence. The underlined words are the concepts in ConceptNet.}
\label{fig:concept_graph}
\end{figure}
\subsection{Knowledge Extraction from Wikipedia}
We extract 107M sentences from Wikipedia\footnote{Wikipedia version enwiki-20190301} by Spacy\footnote{https://spacy.io/} and adopt Elastic Search tools\footnote{https://www.elastic.co/} to index the Wikipedia sentences. We first remove stopwords in the given question and choices then concatenate the words as queries to search from the Elastic Search engine. The engine ranks the matching scores between queries and all the Wikipedia sentences. We select top $K$ sentences as the Wikipedia evidence. Here we adopt $K$=10 in experiments.
To discover the structure information in Wikipedia evidence, we construct a graph for Wikipedia evidence. We utilize Semantic Role Labeling (SRL) to extract triples (subjective, predicate, objective) in one sentence. Both arguments and predicates are the nodes in the graph. We add two edges $<$subjective, predicate$>$ and $<$predicate, objective$>$ in the graph. In order to enhance the connectivity of the graph. We remove stopwords and add an edge from node $a$ to node $b$ according to the following enhanced rules: (1) Node $a$ is contained in node $b$ and the number of words in $a$ is more than 3; (2) Node $a$ and node $b$ only have one different word and the numbers of words in $a$ and $b$ are both more than 3. An example is shown in Figure~\ref{fig:wiki_graph}. We denote the graph as Wiki-Graph.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.4\textwidth]{Wikipedia-crop.pdf}
\caption{An example of constructed Wiki-Graph from the Wikipedia evidence ``He began making music when he started guitar lessons'' and ``Making music and playing guitar are his hobbies''. The dotted line denotes the added edge according to our enhanced rules (1).}
\label{fig:wiki_graph}
\end{figure}
\section{Graph-Based Reasoning}
In this section, we present the model architecture of graph-based reasoning over the extracted evidence, shown in Figure \ref{fig:model}. Our graph-based model consists of two modules: a graph-based contextual representation learning module and a graph-based inference module.
The first module learns better contextual word representations by using graph information to re-define the distance between words. The second module gets node representations via Graph Convolutional Network \cite{kipf2016semi} by using neighbor information and aggregates graph representations to make final predictions.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.45\textwidth]{model-crop.pdf}
\caption{An overview of our proposed graph-based reasoning model.}
\label{fig:model}
\end{figure}
\subsection{Graph-Based Contextual Representation Learning Module}
It is well accepted that pre-trained models have a strong text understanding ability and have achieved state-of-the-art results on a variety of natural language processing tasks.
We use XLNet \cite{xlnet2019yang} as the backbone here, which is a successful pre-trained model with the advantage of capturing long-distance dependency.
A simple way to get the representation of each word is to concatenate all the evidence as a single sequence and feed the raw input into XLNet.
However, this would assign a long distance for the words mentioned in different evidence sentences, even though they are semantically related.
Therefore, we use the graph structure to re-define the relative position between evidence words. In this way, semantically related words will have shorter relative position and the internal relational structures in evidence are used to obtain better contextual word representations.
Specifically, we develop an efficient way of utilizing topology sort algorithm\footnote{We also try to re-define the relative positions between two word tokens and get a position matrix according to the token distances in the graph. However, it consumes too much memory and cannot be executed efficiently.} to re-order the input evidence according to the constructed graphs.
For structured knowledge, ConceptNet triples are not represented as natural language. We use the relation template provided by ConceptNet to transfer a triple into a natural language text sentence.
For example, ``mammals HasA hair'' will be transferred to ``mammals has hair''. In this way, we can get a set of sentences $S_T$ based on the triples in the extracted graph. Then we can get the re-ordered evidence for ConceptNet $S_T'$ with the method shown in Algorithm~\ref{alg:training}. The output of Figure 3 is $<$``people has eyes'', ``eyes is related to cry'', ``people can do singing'', ``cry is a kind of sound'', ``singsing requires sound'', ``sound is related to playing guitar''$>$, which will shorten the distances between triples which are more similar to each other.
For Wikipedia sentences, we construct a sentence graph. The evidence sentences $S$ are nodes in the graph. For two sentences $s_i$ and $s_j$, if there is an edge $<$$p$, $q$$>$ in Wiki-Graph where $p$, $q$ are in $s_i$ and $s_j$ respectively, there will be an edge $<$$s_i$, $s_j$$>$ in the sentence graph. We can get a sorted evidence sequence $S'$ by the method in Algorithm~\ref{alg:training}. In Algorithm~\ref{alg:training}, the relations $R$ is a set of edges, and an edge $r$=$<$$x$,$y$$>$ means the edge from node $x$ to node $y$. The incident edges for $s_i$ represent edges from other nodes to the node $s_i$.
Formally, the input of XLNet is the concatenation of sorted ConceptNet evidence sentences $S_T'$, sorted Wikipedia evidence sentences $S'$, question $q$, and choice $c$. The output of XLNet is contextual word piece representations and the input representation $<$cls$>$. By transferring the extracted graph into natural language texts, we can fuse these two different heterogeneous knowledge sources into the same representation space.
\begin{algorithm}[t]
\centering
\footnotesize
\begin{algorithmic}[1]
\Require
A sequence of nodes $S = \{s_i, s_2, \cdots, s_n\}$; A set of relations $R = \{r_1, r_2, \cdots, r_m\}$.
\Function{dfs}{node, visited, sorted\_sequence}
\For{each child $s_c$ in node's children}
\If{$s_c$ has no incident edges and visited[$s_c$]==0}
\State
visited[$s_c$]=1
\State
sorted\_sequence.append(0, $s_c$)
\State Remove the incident edges of $s_c$
\State DFS($s_c$, visited, sorted\_sequence)
\EndIf
\EndFor
\EndFunction
\State
sorted\_sequence = []
\State
visited = [0 for i in range(n)]
\State
S,R = to\_acyclic\_graph(S,R)
\For{each node $s_i$ in $S$}
\If{$s_i$ has no incident edges and visited[i] == 0}
\State
visited[i] = 1
\State sorted\_sequence.append($s_i$)
\State
DFS($s_i$, visited, sorted\_sequence)
\EndIf
\EndFor
\State
\Return sorted\_sequence
\end{algorithmic}
\caption{Topology Sort Algorithm.}
\label{alg:training}
\end{algorithm}
\subsection{Graph-Based Inference Module}
The XLNet-based model mentioned in the previous subsection provides effective word-level clues for making predictions.
Beyond that, the graph provides more semantic-level information of evidence at a more abstract layer, such as the subject/object of a relation.
A more desirable way is to aggregate evidence at the
graph-level to make final predictions.
Specifically, we regard the two evidence graphs Concept-Graph and Wiki-Graph as one graph and adopt Graph Convolutional Networks (GCNs) \cite{kipf2016semi} to obtain node representations by encoding graph-structural information.
To propagate information among evidence and reason over the graph, GCNs update node representations by pooling features of their adjacent nodes. Because relational GCNs usually over-parameterize the model \cite{marcheggiani2017encoding,zhang2018graph}, we apply GCNs on the undirected graph.
The $i$-th node representation $h^{0}_i$ is obtained by averaging hidden states of the corresponding evidence in the output of XLNet and reducing dimension via a non-linear transformation:
\begin{equation}\label{hidden_state}
h^{0}_i=\sigma(W\sum_{w_j\in s_i}\frac{1}{|s_i|}h_{w_j}) \,.
\end{equation}
where $s_i=\{w_0,\cdots,w_t\}$ is the corresponding evidence to the $i$-th node, $h_{w_j}$ is the contextual token representation of XLNet for the token $w_j$, $W \in R^{d\times k}$ is to reduce high dimension $d$ into low dimension $k$, and $\sigma$ is an activation function.
In order to reason over the graph, we propagate information across evidence via two steps: aggregation and combination \cite{hamilton2017inductive}. The first step aggregates information from neighbors of each node. The aggregated information $z^{l}_i$ for $i$-th node can be formulated as Equation \ref{aggregation}, where $N_i$ is the neighbors of $i$-th node and $h^{l}_j$ is the $j$-th node representation at the layer $l$. The representation $z^{l}_i$ contains neighbors information for $i$-th node at the layer $l$, and we can combine it with the transformed $i$-th node representation to get the updated node representation $h^{l+1}_i$:
\begin{equation}\label{aggregation}
z^{l}_i=\sum_{j\in N_i}\frac{1}{|N_i|}V^{l}h^{l}_j \,,
\end{equation}
\begin{equation}\label{GCN}
h^{l+1}_i=\sigma(W^{l}h^{l}_i+z^{l}_i) \,.
\end{equation}
We utilize graph attention to aggregate graph-level representations to make the prediction. The graph representation is computed the same as the multiplicative attention \cite{luong2015effective}, where $h^{L}_i$ is the $i$-th node representation at the last layer, $h^c$ is the input representation $<$cls$>$, $\alpha_{i}$ is the importance of the $i$-th node, and $h^g$ is the graph representation:
\begin{gather}
\alpha_{i}=\frac{h^c\sigma(W_1h^{L}_i)}{\sum_{j\in N}h^c\sigma(W_1h^{L}_j)} \,, \\
h^g=\sum_{j\in N}\alpha_{j}^{L}h^{L}_j \,.
\end{gather}
We concatenate the input representation $h^c$ with the graph representation $h^g$ as the input of a Multi-Layer Perceptron (MLP) to compute the confidence score $score{(q,a)}$. The probability of the answer candidate $a$ to the question $a$ can be computed as follows, where $A$ is the set of candidate answers:
\begin{equation}
p(q,a)=\frac{e^{score{(q,a)}}}{\sum_{a^{'} \in A}e^{score{(q,a^{'})}}} \,.
\end{equation}
Finally, we select the answer with the highest confidence score as the predicted answer.
\section{Experiments}
In this section, we conduct experiments to prove the effectiveness of our proposed approach.
To dig into our approach, we perform ablation studies to explore the different effects of heterogeneous knowledge sources and graph-based reasoning models.
We study a case to show how our model can utilize the extracted evidence to get the right answer.
We also show some error cases to point directions to improve our model.
\subsection{Experiment Settings}
The CommonsenseQA \cite{talmor2019commonsenseqa} dataset contains 12,102 examples, include 9,741 for training, 1,221 for development and 1,140 for test.
We select XLNet large cased \cite{xlnet2019yang} as the pre-trained model. We concatenate ``The answer is'' before each choice to change each choice to a sentence. The input format for each choice is ``$<$evidence$>$ $<$sep$>$ question $<$sep$>$ The answer is $<$choice$>$ $<$cls$>$''. Totally, we get 5 confidences scores for all the choices then we adopt the softmax function to calculate the loss between the predictions and the ground truth. We adopt cross-entropy loss as our loss function. In our best model on the development dataset, we set the batch size to 4 and learning rate to 5e-6. We set max length of input to 256. We use Adam \cite{kingma2014adam} with $\beta_1$ = 0.9, $\beta_2$ = 0.999 for optimization. We set GCN layer to 1. We train our model for 2,800 steps (about one epoch) and get the results 79.3\% on development dataset and 75.3\% on blind test dataset.
\subsection{Baselines}
For the compared methods, we select models and classify them into 4 groups. \textbf{Group 1}: models without descriptions or papers, \textbf{Group 2:} models without extracted knowledge, \textbf{Group 3}: models with extracted structured knowledge and \textbf{Group 4}: models with extracted unstructured knowledge.
\begin{itemize}
\item \textbf{Group 1}: models without description or papers. These models include SGN-lite, BECON (single), BECON (ensemble), CSR-KG and CSR-KG (AI2 IR).
\item \textbf{Group 2}: models without extracted knowledge, including BERT-large \cite{DevlinCLT19}, XLNet-large \cite{xlnet2019yang} and RoBERTa \cite{roberta2019liu}. These models adopt pre-trained language models to finetune on the training data and make predictions directly on the test dataset without extracted knowledge.
\item \textbf{Group 3}: models with extracted structured knowledge, including KagNet~\cite{kag2019}, BERT + AMS \cite{align2019zhi} and BERT + CSPT. These models utilize structured knowledge ConceptNet to enhance the model to make predictions. KagNet extracts schema graphs from ConceptNet and utilize hierarchical path-based attention mechanism to infer answers. BERT + AMS constructs a commonsense-related multi-choice question answering dataset according to ConcepNet and pre-train on the generated dataset. BERT + CSPT first trains a generation model to generate synthetic data from ConceptNet, then finetunes RoBERTa on the synthetic data and Open Mind Common Sense (OMCS) corpus.
\item \textbf{Group 4}: models with extracted unstructured knowledge, including CoS-E \cite{RajaniMXS19}, HyKAS, BERT + OMCS, AristoBERTv7, DREAM, RoBERT + KE, RoBERTa + IR and RoBERTa + CSPT. Cos-E \cite{RajaniMXS19} constructs human-annotated evidence for each question and generates evidence for test data. HyKAS and BERT + OMCS models pre-train BERT whole word masking model on the OMCS corpus. AristoBERTv7 utilizes the information from machine reading comprehension data RACE \cite{LaiXLYH17} and extracts evidence from text sources such as Wikipedia, SimpleWikipedia, etc. DREAM adopts XLNet-large as the baseline and extracts evidence from Wikipedia. RoBERT + KE, RoBERTa + IR and RoBERTa + CSPT adopt RoBERTa as the baseline and utilize the evidence from Wikipedia, search engine and OMCS, respectively.
\end{itemize}
It should be noted that these methods either utilize evidence from structured or unstructured knowledge sources, failing to take advantages of both sources simultaneously. RoBERT + CSPT adopts knowledge from ConceptNet and OMCS, but the model pre-trains on the sources without explicit reasoning over the evidence, which is different from our approach.
\subsection{Experiment Results and Analysis}
\begin{table}[htbp]
\centering
\begin{tabular}{c|l|c|c}
\toprule
Group & Model & Dev Acc &Test Acc \\
\midrule
\multirowcell{4}{\textbf{Group 1}} & SGN-lite & - & 57.1 \\
& BECON (single) & - & 57.9 \\
& BECON (ensemble) & - & 59.6 \\
& CSR-KG & - & 61.8 \\
& CSR-KG (AI2 IR) & - & 65.3 \\
\hline
\hline
\multirowcell{3}{\textbf{Group 2}} & BERT-large & - & 56.7 \\
& XLNet-large & - & 62.9 \\
& RoBERTa(single) & 78.5 & 72.1 \\
& RoBERTa(ensemble) & - & 72.5 \\
\hline
\hline
\multirowcell{2}{\textbf{Group 3}} & KagNet & - & 58.9 \\
& BERT + AMS & - & 62.2 \\
& RoBERTa + CSPT & 76.2 & 69.6 \\
\hline
\hline
\multirowcell{9}{\textbf{Group 4}} & Cos-E & - & 58.2 \\
& BERT + OMCS & 68.8 & 62.5 \\
& HyKAS & - & 62.5 \\
& AristoBERTv7 & - & 64.6 \\
& DREAM & 73.0 & 66.9 \\
& RoBERT + KE & 77.5 & 68.4 \\
& RoBERTa + CSPT & 76.2 & 69.6 \\
& RoBERTa + IR & 78.9 & 72.1 \\
\hline
\hline
& Our Model & \textbf{79.3} & \textbf{75.3} \\
\bottomrule
\end{tabular}
\caption{Results on CommonsenseQA development and blind test dataset.}
\label{table:results}
\end{table}
The results on CommonsenseQA development dataset and blind test dataset are shown in Table \ref{table:results}. Our model achieves the best performance on both datasets. In the following comparisons we focus on the results on test dataset. Compared with the model in group 1, we can get more than 10\% absolute accuracy than these methods. Compared with models without extracted knowledge in group 2, our model also enjoys 2.8\% absolute gain over the strong baseline RoBERTa (ensemble). XLNet-large is our baseline model and our approach can get 12.4\% absolute improvement over the baseline and this approves the effectiveness of our approach. Compared to models with extracted structured knowledge in group 3, our model extracts graph paths from ConceptNet for graph-based reasoning rather than for pre-training, and we also extract evidence from Wikipedia plain texts, which brings 13.1\% and 5.7\% gains over BERT + AMS and ROBERTa + CSPT respectively. Group 4 contains model which utilizes unstructured knowledge such as Wikipedia or OMCS, etc. Compared with these methods, we not only utilize Wikipedia to provide unstructured evidences but also construct graphs to get the structural information. We also utilize the evidence from structure knowledge base ConceptNet. Our model achieves 3.2\% absolute improvement over the best model RoBERTa + IR in this part.
\subsection{Ablation Study}
In this section, we perform ablation studies on the development dataset\footnote{The dataset restricts to submit the results no more than every two weeks.} to dive into the effectiveness of different components in our model.
We first explore the effect of different components in graph-based reasoning. Then we dive into the heterogeneous knowledge sources and see their effects.
In the graph-based reasoning part, we dive into the effect of topology sort algorithm for learning contextual word representations and graph inferences with GCN and graph attention. We select XLNet + Evidence as the baseline. In the baseline, we simply concatenate all the evidence into XLNet and adopt the contextual representation for prediction. By adding topology sort, we can obtain a 1.9\% gain over the baseline. This proves that topology sort algorithm can fuse the graph structure information and change the relative position between words for better contextual word representation. The graph inference module brings 1.4\% benefit, showing that GCN can obtain proper node representations and graph attention can aggregate both word and node representations to infer answers. Finally, we add topology sort, graph inference module together to get a 3.5\% improvement, proving these models can be complementary and achieve better performance.
\begin{table}[h]
\centering
\resizebox{0.45 \textwidth}{1.2cm}{
\begin{tabular}{l|c}
\toprule
Model & Dev Acc \\
\midrule
XLNet + E & 75.8 \\
XLNet + E + Topology Sort & 77.7 \\
XLNet + E + Graph Inference& 77.2 \\
XLNet + E + Topology Sort + Graph Inference & \textbf{79.3} \\
\bottomrule
\end{tabular}}
\caption{Ablation studies on reasoning components in our model. E denotes evidence.}
\label{table:model_ablation}
\end{table}
Then we perform ablations studies on knowledge sources to see the effectiveness of ConceptNet and Wikipedia sources. The results are shown in Table \ref{table:source_ablation}, ``None'' represents we only adopts the XLNet \cite{xlnet2019yang} large model as the baseline. When we add one knowledge source, the corresponding graph-based reasoning models are also added. From the results, we see that the structured knowledge ConceptNet can bring 6.4\% absolute improvement and the Wikipedia source can bring 4.6\% absolute improvement. This proves the benefits of ConceptNet or Wikipedia source. When combining ConceptNet and Wikipedia, we can enjoy a 9.4\% absolute gain over the baseline. This proves that heterogeneous knowledge sources can achieve better performance than single one and different sources in our model and they are complementary to each other.
\begin{table}[h]
\centering
\begin{tabular}{l|c}
\toprule
Knowledge Sources & Dev Acc \\
\midrule
None & 68.9 \\
ConceptNet & 75.3 \\
Wikipedia & 73.5 \\
ConceptNet + Wikipedia & \textbf{79.3}\\
\bottomrule
\end{tabular}
\caption{Ablation studies on heterogeneous knowledge sources. ``None'' represents we only use XLNet baseline and do not utilize knowledge sources.}
\label{table:source_ablation}
\end{table}
\subsection{Case Study}
In this section, we select a case to show that our model can utilize the heterogeneous knowledge sources to answer questions. As shown in Figure \ref{fig:case_study}, the question is ``Animals who have hair and don't lay eggs are what?'' and the answer is ``mammals''. The first three nodes are from ConceptNet evidence graph. We can see that ``mammals is animals'' and ``mammals has hair'' can provide information about the relation between ``mammals'' and two concepts ``animals'' and ``hair''. More evidence is needed to show the relation between ``lay eggs'' and ``mammals''. The last three nodes are from Wikipedia evidence graph and they can provide the information that ``very few mammals lay eggs''. The examples also show that both sources are necessary to infer the right answer.
\begin{figure}[h]
\centering
\includegraphics[width=0.45\textwidth]{case_study-crop.pdf}
\caption{An attention heat-map for the question ``Animals who have hair and don't lay eggs are what?'' and the answer ``mammals''. The nodes in ConpcetNet are in natural language format and the template is: IsA (is a kind of), HasA (has).}
\label{fig:case_study}
\end{figure}
\subsection{Error Analysis}
We randomly select 50 error examples from the development dataset and the reasons are classified into three categories: the lack of evidence, similar evidence and dataset noise. There are 10 examples which are lack of evidence.
For example, the first example in Figure \ref{fig:error_case} extracts no triples from ConceptNet and the evidence from Wikipedia does not contain enough information to get the right answer.
This problem can be alleviated by utilizing more advanced extraction strategies and adding more knowledge sources. There are 38 examples which extract enough evidence but the evidence are too similar to distinguish between choices.
For example, the second example in Figure \ref{fig:error_case} has two choices ``injury'' and ``puncture wound'', the evidence from both sources provides similar information.
More evidence from other knowledge sources is needed to alleviate this problem. We also find there are 2 error examples which have 2 same choices\footnote{example id: e5ad2184e37ae88b2bf46bf6bc0ed2f4, fa1f17ca535c7e875f4f58510dc2f430}.
\begin{figure}[h]
\centering
\includegraphics[width=0.45\textwidth]{error_analysis-crop.pdf}
\caption{Error cases of our model on the development dataset.}
\label{fig:error_case}
\end{figure}
\section{Related Work}
\textbf{Commonsense Reasoning} Commonsense reasoning is a challenging direction since it requires reasoning over external knowledge beside the inputs to predict the right answer. Various downstream tasks have been released to address this problem like ATOMIC\cite{sap2019atomic}, Event2Mind\cite{rashkin2018event2mind}, MCScript 2.0\cite{ostermann2019mcscript2}, SWAG\cite{ZellersBSC18}, HellaSWAG\cite{ZellersHBFC19} and Story Cloze Test\cite{MostafazadehCHP16}.
Recently proposed CommonsenseQA\cite{talmor2019commonsenseqa} dataset derived from ConceptNet\cite{SpeerCH17} and the choices have the same relation with the concept in the question.
Recently, \citet{RajaniMXS19} explores adding human-written explanations to solve the problem.
\citet{kag2019} extracts evidence from ConceptNet to study this problem. This paper focuses on automatically extracting evidence from heterogeneous external knowledge and reasoning over the extracted evidence to study this problem.
\textbf{Knowledge Transfer in NLP} Transfer learning has played a vital role in the NLP community.
Pre-trained language models from large-scale unstructured data like ELMo \cite{PetersNIGCLZ18}, GPT \cite{radford2018improving}, BERT \cite{DevlinCLT19}, XLNet \cite{xlnet2019yang}, RoBERTa \cite{roberta2019liu} have achieved significant improvements on many tasks.
This paper utilizes XLNet \cite{xlnet2019yang} as the backend and propose our approach to study the commonsense question answering problem.
\textbf{Graph Neural Networks for NLP} Recently, Graph Neural Networks (GNN) has been utilized widely in NLP. For example, \citet{SunGWGJLSD19} utilizes Graph Convolutional Networks (GCN) to jointly extract entity and relation. \citet{zhang2018graph} applies GNN to relation extraction over pruned dependency trees and achieves remarkable improvements.
GNN has also been applied into muli-hop reading comprehension tasks \cite{TuWHTHZ19,KunduKSC19,JiangJCB19}. This paper utilizes GCN to represent graph nodes by utilizing the graph structure information, followed by graph attention which aggregates the graph representations to make the prediction.
\section{Conclusion}
In this work, we focus on commonsense question answering task and select CommonsenseQA \cite{talmor2019commonsenseqa} dataset as the testbed.
We propose an approach consisting of knowledge extraction and graph-based reasoning. In the knowledge extraction part, we extract evidence from heterogeneous external knowledge including structured knowledge source ConceptNet and Wikipedia plain texts.
In the graph-based reasoning part, we propose a graph-based approach consisting of graph-based contextual word representation learning module and graph-based inference module to select the right answer. Results show that our model achieves state-of-the-art on CommonsenseQA\cite{talmor2019commonsenseqa} dataset.
\section{Acknowledgement}
Songlin Hu is the corresponding author. We thank the anonymous reviewers for providing valuable suggestions.
|
3,212,635,537,941 | arxiv | \section{Introduction}
\subsection*{Implementation}
\label{ssec:parameters}
For T5 training, we add the prefix ``reasoning:'' in front of every concatenation of question and answer, then ask the model to predict ``1'' for true, and ``2'' for false.\\
Regarding libraries, we used python 3.7.10, pytorch 1.9.0 and transformers 4.11.3.\\
Among all the training sets, we are using learning rate of $1e^{-5}$ , batch size of 32, weight decay 0.01, training epochs of 5, adam-epsilon of $1e^{-6}$, $\beta1=0.9,\beta2=0.98$, warm-up proportion of 0.05, margin of 1.0.\\
For CPUs, we used Intel(R) Xeon(R) Gold 5217 CPU @ 3.00GHz (32 CPUs, 8 cores per sockets, 263GB ram).\\
For GPUs, we used Nvidia Quadro RTX 8000, and Nvidia Geforce 2080Ti.
\section{Conclusions}
\label{sec:conclusion}
This paper studied the impact of the knowledge, model, and task considerations on the self-supervision performance of LMs over KGs. We investigated different knowledge sizes and samples, as well as model architectures and sizes. We compared these against four task properties: domain overlap, answer similarity, answer length, and vocabulary overlap. We observed that the optimal knowledge size and sampling strategy is model-dependent, with encoder-only models learning quicker from less data than encoder-decoder models. Among the sampling strategies, random sampling performed best, closely followed with dimension-based sampling with temporal, quality, and desire/goal knowledge. Most of the improvement of the largest generative model comes from questions with short answers and dissimilar answer candidates, which is expected, given that the synthetic data generated from the KG has these properties.
These findings point to three key directions for future work that uses self-supervision with large KGs to create generalizable commonsense reasoning agents. The abilities of complementary sampling strategies can be exploited by combining them into a single model. To enable robustness and trust, next-generation zero-shot models should be able to explain their reasoning explicitly. Finally, subsequent work should generalize the self-supervision KG-based method to more advanced tasks, such as story completion~\cite{kalyanpur2020braid} and embodied QA~\cite{das2018embodied}. All our code and data is available at \url{https://drive.google.com/file/d/1FE1_moXTK2Zpe4HQat78L6cXyrwtK4lO/view?usp=sharing}.
\section{Discussion}
Our experiments show that the choice of LM size and architecture, as well as knowledge size and sampling strategy, affects the ability of models to answer commonsense questions across benchmarks. Encoder-decoder models benefit from more data to learn from, whereas sampling strategies that balance across different aspects yield best performance. The performance gain of knowledge-based pretraining is due to questions with short and dissimilar answers.
Next, we revisit three key assumptions of this study, and provide an alternative inspired by the results of our experiments.
\noindent \textbf{1. From a single model to mixture of models} Random sampling is the optimal single sampling strategy, as it balances well between the different properties of the data. More specialized strategies, e.g., focusing on knowledge dimensions, perform well on subsets of the task, but underperform on other subsets. This model specialization questions the assumption that a single zero-shot model is sufficient to perform optimally on different aspects of common sense, and suggests that model combinations, such as Mixture of models~\cite{gururangan2021demix}, might provide a more comprehensive and trustworthy commonsense model. This would entail, e.g., combining models from different dimensions, or models that capture complementary training dynamics.
\noindent \textbf{2. From implicit to explicit zero-shot commonsense reasoning} In our current framework, the rich and diverse commonsense knowledge is taught to an LM through a large set of QA pairs. Given the simplicity of these questions, an implicit assumption of this study is that LMs can reverse engineer these questions to learn commonsense knowledge implicitly, and apply this newly acquired knowledge on unforeseen benchmarks, whose surface properties may be different, but their underlying commonsense knowledge may be largely shared. While this is a reasonable assumption, our commonsense models are black boxes, and they do not provide an explicit justification for their decisions.
A natural extension of this work is to devise \textit{explainable models}, i.e., models whose output includes the explicit reasoning steps associated with a predicted answer.
\noindent \textbf{3. From question answering to more realistic tasks} A key aspect of our zero-shot framework is its generalization across QA tasks. The gap between zero-shot and fine-tuning performance is closing down, which brings a natural question: are zero-shot models, adapted through an adaptation of neural techniques with background knowledge, able to generalize across tasks and domains, or have they merely learned how to answer questions convincingly? To address this question, we propose a shift towards more realistic tasks that rely on common sense, such as
story understanding~\cite{kalyanpur2020braid}, dialogue modeling~\cite{ghosal2021cider}, and embodied QA~\cite{das2018embodied}. The rich prior work that focuses on these tasks has assumed the existence of benchmark-specific training data; zero-shot models have not been thoroughly explored.
\section{Introduction}
\label{sec:intro}
Common sense is the human knowledge about the world and the methods for making inferences from this knowledge~\cite{davis2014representations}. Commonsense knowledge includes the basic facts about events (including actions) and their effects, facts about knowledge and how it is obtained, facts about beliefs and desires, as well as the basic facts about material objects and their properties~\cite{mccarthy1989artificial}. Artificial Intelligence (AI) agents that are equipped with common sense are expected to possess a wide range of everyday knowledge about naive physics, folk psychology, and causality. Rich commonsense knowledge can be found in public knowledge graphs (KGs), like ConceptNet~\cite{10.5555/3298023.3298212}, ATOMIC~\cite{DBLP:conf/aaai/SapBABLRRSC19}, and Visual Genome~\cite{krishna2017visual}.
State-of-the-art commonsense reasoning systems are largely fueled by language models (LMs), as LMs are able to adapt (\textit{fine-tune}) to
benchmarks effectively, insofar as training data is available~\cite{devlin2018bert,liu2019roberta}.
Recent work has proposed lightweight alternatives to fine-tuning, such as Prefix-tuning~\cite{li2021prefixtuning} and AutoPrompt~\cite{autoprompt:emnlp20}, however, these methods still rely on training data being available and have limited generalizability to other tasks~\cite{ma2021exploring}, given that the data distribution varies greatly across tasks. Recognizing that the assumption of always having benchmark-specific training data is unrealistic for open-domain reasoning, recent work has increasingly focused on zero- and few-shot tasks and reasoning models. Common methods for zero-shot reasoning rely on careful pre-training of LMs with external resources: commonsense KGs~\cite{Banerjee2020SelfsupervisedKT, Ma2021}, elicitation of pre-existing knowledge in the LM~\cite{Shwartz2020UnsupervisedCQ,paranjape-etal-2021-prompting}, or instruction-prompted training with a diverse set of tasks \cite{sanh2021multitask}. While pre-training with commonsense knowledge has been shown to improve model performance~\cite{Banerjee2020SelfsupervisedKT,Ma2021}, prior work has not investigated how different architectural and data decisions affect model accuracy and generalization across tasks.
This paper conducts an empirical investigation of commonsense self-supervision of state-of-the-art language models with knowledge graphs. We study the interplay between methods for data generation from knowledge graphs, choices of language models, and task properties. Our contributions are as follows:
\begin{enumerate}
\item We formalize the self-supervision of language models with synthetic data extracted from knowledge graphs, as a joint framework. We consider the interplay between the selected knowledge as synthetic data, the language model, and the properties of the task.
\item We pose five research questions, which have not been answered so far for commonsense QA models adapted with KGs under zero-shot evaluation setting. We study the following aspects: (i) knowledge sampling size and strategy; (ii) language model architecture and size; and (iii) task properties: domain overlap, vocabulary overlap, answer similarity, and answer length.
\item We gain insight into these research questions, through rigorous experimentation: we evaluate seven sampling strategies with seven knowledge sizes, which are used to adapt five models belonging to two representative model architectural classes. We define fourteen different task partitions, for five different benchmarks. Our experiments show that: (i) the best sampling strategy balances across different properties of the data, (ii) the optimal amount of data is model-dependent, (iii) most of the improvement occurs on questions with short answers and dissimilar answer candidates.
\end{enumerate}
\section{Method}
\label{sec:method}
We follow the task formulation of \textit{generalizable commonsense reasoning} proposed by \citet{Ma2021}. The input consists of a natural language question $Q$ and $n$ candidate answers $A_i$, $|A_i|=n$. Exactly one of the candidate answers, marked with $A$, is correct. The remaining $(n-1)$ candidate answers serve as distractors. As we assume a zero-shot setup, the models have no access to the benchmark-specific training data.
Each model is adapted once, after which they are fixed, and directly applied on test partitions of various benchmarks.
To address the task of generalizable commonsense reasoning, we assume a knowledge-driven QA framework, where pre-trained LMs are adapted with artificial QA sets derived from KG data.
We create artificial QA sets with thousands of questions by sampling statements from the recently-introduced CommonSense Knowledge Graph (CSKG)~\cite{ilievski2021cskg}, and transforming them into multiple-choice questions.
Each question corresponds to a particular knowledge dimension (e.g., temporal or spatial knowledge)~\cite{ilievski2021dimensions}. We define \textit{domain} as the dimensions of common sense necessary for solving a particular set of tasks.
Given a natural language question $Q$, and $n$ candidate answers $\{A_{1}, ..., A_{n}\}$, the LM has to select the most probable answer $A$ during training. Once the LM adaptation is done, the updated LM is applied across QA tasks in a zero-shot manner.
The setup of this study is visualized in Figure~\ref{fig:overview}. We investigate the performance of self-supervision of language models with knowledge graphs, in relation to: 1) size and architecture of the language model; 2) size and sampling strategies of the knowledge used for model adaptation; and 3) properties of the task, such as overlap with knowledge and answer length. We describe the language models, knowledge sampling, and task properties in this section.
\subsection{Language Models}
\para{Model architectures}
We adopt two widely-used pre-trained models: RoBERTa~\cite{Liu2019RoBERTaAR} and T5~\cite{raffel2019exploring}. RoBERTa is an encoder-only masked language model (MLM), whereas T5 is an encoder-decoder model which converts tasks into text-to-text format.
Following~\citet{Ma2021}, for RoBERTa each input sequence is a concatenation of the question and one of its answer candidates. We mask one non-stop token in the sequence at a time, and compute the masked token's loss. We then take the averaged loss for the sequence and this is repeated for every answer candidates. We then train the model with the margin loss:
$$L=\frac{1}{n} \sum_{\mbox{\tiny$\begin{array}{c}
i=1\\
i\neq y\end{array}$}}^{n}max(0,\eta-S_y+S_i)$$
where $S_y$ and $S_i$ are the negative averaged loss for correct answer and distractor respectively. During inference, we take the candidate with highest score $S$ as the answer.
For T5, each input sequence is the same as RoBERTa except that we add a task-specific prefix, ''reasoning:'', to the front following the adaptation of T5 to downstream tasks in the original paper~\cite{raffel2019exploring}. The model is pre-trained to generate "true" or "false" token. $L_{true}$ represents the loss of the "true" token logits, while $L_{false}$ represents the "false" token logits. For each answer candidate, we compute the score $S=L_{true}-L_{false}$ and use the same margin loss function as in RoBERTa to jointly predict the optimal candidate.\footnote{We also tried to score the answers individually, or to concatenate the question with all answer candidates, and teach the model to predict the position or make a copy of the right candidate, following \cite{khashabi2020unifiedqa}. These loss strategies performed consistently worse, and we leave them out of the paper.}
\para{Model sizes} We use RoBERTa's base and large models, which have 125M and 355M parameters, respectively. We experiment with three T5 models of different sizes: small (60M parameters), large (740M), and 3b (2.85B). More details about the model training can be found in the Appendix of this paper.
\subsection{Knowledge sources and sampling}
We subselect knowledge from the CommonSense Knowledge Graph (CSKG)~\cite{ilievski2021cskg}, which combines seven commonsense knowledge resources under a shared representation, including ConceptNet~\cite{10.5555/3298023.3298212}, ATOMIC~\cite{DBLP:conf/aaai/SapBABLRRSC19}, and Visual Genome~\cite{krishna2017visual}. In total, CSKG contains over 7 million commonsense statements, which are used to describe over 2 million nodes with 58 properties. Each statement in CSKG consists of a head ($h$), relation ($r$), tail ($t$), and additional qualifiers ($q$).
Each knowledge statement in CSKG is categorized into one of the following 13 dimensions: lexical, similarity, distinctness, taxonomic, part-whole, creation, utility, comparative, quality, temporal, spatial, motivational, and relational-other~\cite{ilievski2021dimensions}.
\para{Sample sizes.} We use the subset of CSKG which combines ATOMIC, ConceptNet, WordNet~\cite{miller1995wordnet}, Wikidata~\cite{vrandevcic2014wikidata}, and Visual Genome. We use all relations in this subset. We sample QA sets that correspond to a percentage $K$ from this knowledge set. We experiment with $K \in \{1, 5, 10, 33, 50, 100\}$. In comparison, Ma et al. \cite{Ma2021} use 100\% of the data for fourteen manually-selected semantic relations.
\para{Sampling strategies.} The sampling of $K\%$ artificial examples brings up the question: how do we optimally select these questions? We experiment with seven selection strategies that are based on training indicators and the KG structure.
(i) \textit{Random} draws $K\%$ of the question pool by chance, without replacement.
(ii) \textit{Dimension} selects the questions that belong to a knowledge dimension. Out of the thirteen dimensions defined in~\cite{ilievski2021dimensions}, we evaluate the five best populated dimensions: temporal, desire/goal, taxonomic, quality, and relational-other. Here we are interested to see if there is a particular knowledge dimension that is more beneficial than others. For fair comparison between sampling strategies, we limit the questions selected for a dimensions to the equivalent of $K\%$ of the entire question set.
(iii) \textit{Uniform} selects an equal number of questions from each of the thirteen dimensions. We limit the total number of questions to the equivalent of $K\%$ of the entire question set.
(iv) \textit{Vanilla-confidence} selects questions based on the model confidence before any adaptation, i.e. we run the vanilla LM on the entire QA sets to get its confidence scores. We experiment with two variants, which select the questions with lowest and highest confidence, respectively. We would like to test if the model benefits more from questions it considers easy or hard.
(v) \textit{Confidence} selects questions based on the mean model confidence for the true label across the adaptation epochs. Specifically, we first train a model on entire QA sets and record each questions' training statistics as in ~\cite{swayamdipta2020dataset}. Then we select the subsets of the data to train our actual task model. This method applies to the last two strategies as well. Here we use two variants: confidence-low and confidence-high, analogous to the vanilla-confidence strategy.
(vi) \textit{Variability} detects the $K\%$ of the questions with extreme standard deviation for the true label across the adaptation epochs~\cite{swayamdipta2020dataset}. We experiment with variability-low and variability-high sampling.
(vii) \textit{Margin} selects the $K\%$ with the most extreme mean difference between the confidence of the correct answer and the incorrect ones~\cite{pleiss2020identifying}. We consider margin-low and margin-high sampling.
For every strategy, $K=100$ corresponds to the entire synthetic QA pool, while $K=0$ is the vanilla pre-trained LM without adaptation.
\subsection{Tasks}
We evaluate on five benchmarks for multiple-choice commonsense question answering. \textit{CommonsenseQA (CSQA)}~\cite{talmor-etal-2019-commonsenseqa} is a five-choice question answering benchmark which evaluates a broad range of common sense aspects. \textit{SocialIQA (SIQA)}~\cite{sap-etal-2019-social} is a three-choice QA benchmark that requires reasoning about social interactions.
\textit{Abductive NLI (aNLI)}~\cite{bhagavatula2019abductive} is formalized as natural language inference, where, given the beginning and the ending of a story, the task is to choose the more plausible hypothesis out of two options.
\textit{PhysicalIQA (PIQA)}~\cite{Bisk2020} is a binary choice task, which tests the ability of models to employ physical reasoning.
\textit{WinoGrande (WG)}~\cite{sakaguchi2019winogrande} is a binary choice anaphora resolution task.
We measure the \textit{accuracy} of a language model on a benchmark as the ratio between the correctly-answered questions and the total number of questions.
\para{Task properties} We measure granular model performance by computing its accuracy on task partitions. We partition commonsense tasks based on four properties.
First, we consider model accuracy in relation to the \textit{domain overlap (DO)} between the KG and the task. Two of the five benchmarks are known to have high domain overlap (HDO) with existing KGs~\cite{mitra2019exploring,Ma2021}: CSQA has been devised based on knowledge in ConceptNet, while SocialIQA has been created based on the ATOMIC KG~\cite{DBLP:conf/aaai/SapBABLRRSC19}. The remaining three benchmarks have been created independently of the KGs, therefore, we consider them to have low domain overlap (LDO) with our KGs. We compare the model performance on the benchmarks with HDO and LDO.
Second, we partition the task into quartiles based on the \textit{answer similarity (AS)} between the candidates. Here, we compute the answer similarity for a question $q$ through the Jaccard similarity between the tokens of the candidates $A_i$ and $A_j$:
$
AS (q)=\frac{|T_{A_i}\cap T_{A_j}|}{|T_{A_i}\cup T_{A_j}|}
$. Here,
$T_{A_i}$ and $T_{A_j}$ are the set of tokens of candidates $A_i$ and $A_j$, respectively. We limit the answer similarity analysis to the PIQA benchmark, which has two candidates for each question.
Third, we partition a task into quartiles based on the \textit{answer length (AL)} of its questions. We compute the answer length of a question $q$ by summing the tokens $T_{A_i}$ of the candidates $A_i$:
$
AL(q) = \sum_{i=1}^{n}{|T_{A_i}|}
$.
Fourth, we partition the task into quartiles based on the \textit{vocabulary overlap (VO)} between the task questions and the synthetic QA set. We first compute the frequency of every token that appears in the synthetic data. Given a task question, we compute the average frequency of the candidate tokens in the synthetic data.
To increase the effect of the tokens with low frequency, we use the reciprocal value of the token frequencies:
$VO(q)=\frac{1}{m} \sum_{
k=1}^{m}\frac{1}{f(t_k)}$.
Here, $m$ is the number of tokens in the combination of the candidates ($|\bigcup_i^n{T_{A_i}}|$ for each question, $t_k$ is the $k$-th token in the answer candidates, and $f(t_k)$ is its frequency in the synthetic data.
When splitting the task based on \textit{answer simillarity}, \textit{answer length}, and \textit{vocabulary overlap}, we use RoBERTa's tokenizer, and we focus on the PIQA benchmark which has a variety of properties among its questions.
\section{Related Work}
\para{Generalizable Commonsense Reasoning.}
UNICORN~\cite{lourie2021unicorn} investigates continual learning of commonsense knowledge from multiple benchmarks, ultimately aiming to perform well on all of the benchmarks. This work demonstrates that LMs can learn from commonsense benchmarks effectively and efficiently, reaching relatively high accuracy based on little training examples. Prefix-tuning~\cite{li2021prefixtuning} adapts language models, by keeping the model parameters intact, but extending them with a small set of additional parameters tuned separately for each benchmark.
Rather than updating model parameters, Autoprompt~\cite{autoprompt:emnlp20} extends the model input with trigger tokens, which are updated during training.
Such efforts share our vision to develop models that can generalize to multiple commonsense benchmarks simultaneously. However, these works assume availability of training data, while we focus on zero-shot commonsense QA models.
\para{Zero-shot Commonsense Reasoning.}
Zero-shot commonsense reasoning methods often elicit knowledge from pre-trained LMs, by using self-talk clarification prompts~\cite{Shwartz2020UnsupervisedCQ} or asking LMs to generate contrastive explanations~\cite{paranjape-etal-2021-prompting}. Models can be taught to answer questions by adapting to an external dataset~\cite{abdou-etal-2020-sensitivity}. We differ from these approaches because our models are based on knowledge available in large KGs, rather than mere distillation from language models. As shown in prior work~\cite{Ma2021}, KG-based approaches achieve superior performance compared to pure LM-based methods for zero-shot commonsense QA.
To use KGs for zero-shot pretraining and evaluation, \citet{Banerjee2020SelfsupervisedKT} pre-train an LM to perform knowledge completion, whereas \citet{bosselut2020dynamic} enhance the question based on knowledge completion models, and score an answer candidate in relation to the context, question, and generated knowledge.
Our work is based on the framework of~\citet{Ma2021}, which generates synthetic QA pairs from a consolidated KG to pre-train LMs.~\citet{Ma2021} investigate the impact of different loss functions and knowledge sources, showing that margin loss performs better than masked language modeling, and that more knowledge generally performs better, though this might change depending on the knowledge-task alignment. Follow-up work by~\citet{dou2022zero} extends the framework of~\citet{Ma2021}. with several data transformation methods, out of which measuring consistency between different prompt versions performs best. These data transformation efforts yielded notable performance gains, leading to new state-of-the-art results. Our work is orthogonal to these efforts, because we perform a systematic study of model size and architecture, knowledge sampling and size, and task properties, which have not been investigated in detail in prior work on zero-shot commonsense QA with knowledge graphs~\cite{Ma2021,dou2022zero}.
\para{Model Generalization and Data Selection.}
\citet{sen-saffari-2020-models} analyzed LM's ability to generalize across five different QA datasets.
\citet{ma2021exploring} showed that models can have drastically different performances by fine-tuning on different subset of the data.
\citet{swayamdipta2020dataset} proposed to select training instances based on models' confidence and variability, and they show that training on less-confident examples is more beneficial for generalization.
Follow-up work~\cite{ethayarajh2021information} proposed an information-theoretic metric for estimating the difficult of a training example, treating as difficult the examples for which information is missing.
\citet{pleiss2020identifying} propose to identify erroneous data points based on their rank in the area under the margin of a machine learning model.
While prior work analyses model robustness by sub-sampling instances from the task's training set, we investigate the impact of knowledge selection, model selection, and task properties when models are adapted on large KGs for commonsense QA. \citet{ilievski2021dimensions} split synthetic data from KGs into 12 commonsense dimensions, revealing that some kinds of knowledge are much more useful for pre-training compared to others. Our study provides a comprehensive investigation of these prior efforts on the task of zero-shot commonsense QA with KGs. As such, prior work on subsetting training data based on knowledge dimensions or training indicators is integrated into our study.
\section{Results}
\label{sec:results}
We provide results based on applying the methodology presented in Section~\ref{sec:method} to address the research questions in Section~\ref{sec:hypotheses}.
\noindent \textbf{What is the overall impact of model and knowledge choice on the generalizability of self-supervision of LMs with KGs (RQ1)?} Table~\ref{tab:Results} shows the results obtained with the best performing knowledge sampling strategy (random) and the best data size per model architecture (5\% for RoBERTa models, 33\% for T5 models). The results show that the best performance is clearly obtained with self-supervision of the model T5-3b with 33\% of the data. The zero-shot result of this model is 71.9 on average over the five benchmarks, which is 15.3 points higher than the vanilla RoBERTa-large model and 5.1 points higher than the previous state-of-the-art result of Ma et al. This result is especially encouraging in comparison with the supervised RoBERTa-large LM, which is now only 7.9 points higher than our result, despite relying on benchmark-specific training data. Our second best model is RoBERTa-large, which is able to outperform the RoBERTa-large model in Ma et al. despite relying on only 5\% of the training data. As expected, the accuracy of the Roberta and T5 models shows a clear positive impact of the model size, as T5-3b $>$ T5-large $>$ T5-small and RoBERTa-large $>$ RoBERTa-base.
\noindent \textbf{How much data is needed to adapt LMs to commonsense reasoning tasks (RQ2)?}
We investigate the impact of the synthetic data size on the five models (RoBERTa-base and large; T5-small, large, and 3b) in detail. Figure~\ref{fig:data_sizes} shows the average accuracy for the five models across the five benchmarks. We observe that models have different optima in terms of the data size that they are pre-trained with, which is largely determined by the model architecture. The encoder-only model, RoBERTa, performs best with about 5\% of the synthetic data. Specifically, RoBERTa-large achieves its optimum of 67.4\% accuracy with only 5\% of the data, whereas the performance of RoBERTa-base increases only marginally with more than 5\% of the data.
Meanwhile, the generative encoder-decoder model, T5, benefits from more commonsense data. This is especially the case for T5-3b, our largest model, whose accuracy grows around 7\% by switching from 5 to 33\% of the data. Yet, the performance of the models generally peaks around 33\%, and plateaus with the increase of the data size. In our experiments, we use 5\% of the data for RoBERTa and 33\% for T5.
The need for more data of T5 models, as well as their ability to benefit less from it given same-sized LMs, can be attributed to their architecture. The learning curves in Figure~\ref{fig:training_curves} show that RoBERTa can adapt quicker to the task, which is because the adaptation loss corresponds to its pre-training loss. T5 starts with a high loss value because it has to learn the new prefix introduced in our work. This difference in the LM architecture also explains why RoBERTa-large outperforms T5-large, despite it being a smaller LM.
\begin{figure}[!t]
\centering
\includegraphics[width=0.95\linewidth]{figures/curves.pdf}\\
\caption{Training curves of the models: RoBERTa-base, RoBERTa-large, T5-small, T5-large, and T5-3b. We use all of the models with 100\% of our training data}
\label{fig:training_curves}
\end{figure}
\begin{table*}[!t]
\centering
\small
\caption{Evaluation results on the five benchmarks of {\bf RoBERTa-large} with different sampling strategies. All samples have equivalent sizes, corresponding to 5\% of the training data. The best result per column is marked in bold.}
\label{tab:sampling_roberta}
\begin{tabular}{cc | rrr | rr | rrr}
\hline
\multirow{2}*{\bf Strategy}&&\multicolumn{3}{c|}{\bf LDO}& \multicolumn{2}{c|}{\bf HDO}&\multirow{2}*{\bf Avg(LDO)}&\multirow{2}*{\bf Avg(HDO)}&\multirow{2}*{\bf Avg}\\
&&{\bf aNLI}&{\bf WG}&{\bf PIQA}&{\bf SIQA}&{\bf CSQA} & \\\hline
{\bf Random}&5\%&72.0&60.2&{\bf 72.5}&{\bf 65.4}&{\bf 66.9}&68.2&{\bf 66.2}&{\bf 67.4}\\\hline
\multirow{5}*{\bf Dimension}&temporal&{\bf 72.7}&61.1&72.1&62.3&65.8&{\bf 68.6}& 64.1&66.8\\
&desire&70.2&59.5&72.4&60.9&64.3&67.4&62.6&65.5\\
&taxonomic&67.0&58.0&69.2&0.51&59.0&64.7&55.0&60.8\\
&quality&71.3&{\bf 61.8}&72.0&58.5&64.6&68.4&61.6&65.6\\
&rel-other&65.3&55.5&69.7&51.5&58.1&63.5&54.8&60.0\\\hline
{\bf Uniform}&&69.6&58.0&72.4&61.7&64.3&66.7&63.0&65.2\\\hline
\multirow{2}*{\bf Vanilla-conf}&high&63.3&59.1&67.6&49.4&47.2&63.3&48.3&57.3\\
&low&57.9&51.9&55.6&33.1&21.7&55.1&27.4&44.0\\\hline
\multirow{2}*{\bf Conf}&high&66.2&58.9&70.3&59.4&62.2&65.1&60.8&63.4\\
&low&71.4&59.2&72.1&62.6&65.7&67.6&64.2&66.2\\\hline
\multirow{2}*{\bf Varibility}&high&67.4&56.8&65.5&48.2&44.0&63.2&46.1&56.4\\
&low&65.4&56.0&68.6&54.4&61.0&63.3&57.7&61.1\\\hline
\multirow{2}*{\bf Margin}&high&67.1&58.2&70.7&60.1&62.3&65.3&61.2&63.7\\
&low&72.3&60.5&71.2&62.7&65.0&68.0&63.9&66.3\\\hline
\end{tabular}
\end{table*}
\begin{table*}[!ht]
\centering
\small
\caption{Evaluation results on the five benchmarks of {\bf T5-large} with different sampling strategies. All samples have equivalent sizes, corresponding to 5\% of the training data. The best result per column is marked in bold.}
\label{tab:sampling_t5}
\begin{tabular}{cc | rrr | rr | rrr}
\hline
\multirow{2}*{\bf Strategy}&&\multicolumn{3}{c|}{\bf LDO}& \multicolumn{2}{c|}{\bf HDO}&\multirow{2}*{\bf Avg(LDO)}&\multirow{2}*{\bf Avg(HDO)}&\multirow{2}*{\bf Avg}\\
&&{\bf aNLI}&{\bf WG}&{\bf PIQA}&{\bf SIQA}&{\bf CSQA} & \\\hline
{\bf Random}&5\%&65.9&56.5& 70.5&55.4&61.9& 64.3&58.7&62.0\\\hline
\multirow{5}*{\bf Dimension}&temporal& 66.6&56.4&{\bf 71.2}&54.9&{\bf 63.4}&{\bf 64.7}&{\bf 59.2}&{\bf 62.5}\\
&desire&64.4&57.9&69.6&55.9&62.2&64.0&59.1&62.0\\
&taxonomic&61.8&54.0&66.8&52.8&57.5&60.9&55.2&58.6\\
&quality&{\bf 66.8}&{\bf 58.4}&70.0&{\bf 56.4}&59.6&65.1&58.0&62.2\\
&rel-other&61.0&52.5&65.9&51.7&54.0&59.8&52.9&57.0\\\hline
{\bf Uniform}&&65.3&57.5&69.2&56.6&62.7&64.0&59.7&62.3\\\hline
\multirow{2}*{\bf Vanilla-conf}&high&65.3&56.8&69.0&55.5&57.5&63.7&56.5&60.8\\
&low&64.0&56.0&68.1&52.0&59.6&62.7&55.8&59.9\\\hline
\multirow{2}*{\bf Conf}&high&62.9&53.8&66.5&53.9&57.0&61.1&55.5&58.8\\
&low&41.8&48.5&42.0&24.7&07.7&44.1&16.2&32.9\\\hline
\multirow{2}*{\bf Varibility}&high&64&54.6&65.1&51.1&54.5&61.2&52.8&57.9\\
&low&61.7&54.9&66.8&52.7&55.9&61.1&54.3&58.4\\\hline
\multirow{2}*{\bf Margin}&high&63.8&54.5&67.2&52.8&56.9&61.8&54.9&59.0\\
&low&41.5&45.0&43.7&24.1&09.1&43.4&16.6&32.7\\\hline
\end{tabular}
\end{table*}
\begin{table}[!t]
\small
\caption{Examples of benchmark questions that are correctly answered with only one model, which is adapted with dimension-based knowledge. (*) denotes the correct answer.}
\begin{tabular}{l}
\tabucline[1.1pt]\\
dimension: temporal\\
Q:Jan went out with Quinn's friends and had a great time.\\What does Jan
need to do before this?\\
A1:get dressed(*); A2:cancel her plans; A3:see Quinn's Friends again\\\tabucline[1.1pt]\\
dimension: desire\\
Q:Robert has no regret for punching Justin in the nose\\ because \_ was the
victim of injustice.\\
A1:Robert(*); A2:Justin\\\tabucline[1.1pt]\\
dimension: quality\\
Q:What can machines do that humans cannot?\\
A1:fail to work; A2:perform work; \\A3:answering questions; A4:see work\\
A5:fly(*)\\\tabucline[1.2pt]\\
\end{tabular}
\label{tab:examples}
\end{table}
\noindent \textbf{How to best sample questions for model adaptation (RQ3)?} We study the impact of different sampling strategies on RoBERTa-large and T5-large, given that they are in the same order of magnitude (hundreds of millions of parameters). We focus on 5\% of the data for computational reasons. The results for RoBERTa-large (Table \ref{tab:sampling_roberta}) and T5-large (Table \ref{tab:sampling_t5}) reveal that the random sampling strategy is surprisingly robust. Random sampling performs best for RoBERTa and it comes close to the best performing strategy (temporal) for T5-large. Besides random sampling, we observe consistently strong performance when training with some dimensions of knowledge: temporal, desire, and quality knowledge. Adapting the RoBERTa-large model with almost any dimension, except low vanilla confidence and high variability, performs better than the vanilla RoBERTa-large baseline.
The finding that random sampling leads to a strong and balanced model is consistent with the finding that random sampling of distractors is better than heuristic- and embedding-based strategies~\cite{Ma2021}. Both of these findings show that the natural distribution of the data provides diversity which is difficult to match with more focused strategies. Yet, the strong performance of the dimension-based strategies, whose data samples are disjoint by design, indicates that models trained with these dimensions capture complementary knowledge. This is confirmed in Table~\ref{tab:examples}, which shows three benchmark questions which are only answered correctly with one of the dimension-based models. The temporal model correctly solves the example that requires temporal ordering, while the desire/goal and quality models uniquely solve the questions that require reasoning about human psychology. This result shows that, while random sampling might be the optimal strategy to create a single model, multiple complementary models could be combined to achieve better accuracy.
\noindent \textbf{Do models generalize well to tasks with low domain overlap (R4)?}
Table~\ref{tab:Results} shows that the average improvement of T5-3b is mostly due to its improved performance on LDO benchmarks. T5-3b's improvement over RoBERTa is on average 6.5\% on the LDO benchmarks, but only 1.4\% on the HDO benchmarks. This generalization ability of T5-3b can largely be attributed to the larger capacity of T5-3b, which allows it to represent additional knowledge and associations between terms. In addition, this Table shows that the HDO benchmarks have been much more popular in prior work, and much larger gains over the vanilla LM have been reported on them (up to 15.1 points on SIQA and 22.4 points on CSQA). Conversely, results on the LDO benchmark have rarely been reported in prior work on zero-shot commonsense reasoning, and the maximum improvement obtained in prior work is only 4.4 points on average across these benchmarks. Therefore, our accuracy improvement of 0.3 points for RoBERTa and 6.8 points for T5-3b is a notable leap towards robust performance on domains with low overlap.
\begin{figure*}[!t]
\centering
\includegraphics[width=\linewidth]{figures/main3.pdf}\\
\caption{Accuracy of the best performing RoBERTa-large and T5-3b models in relation to the answer similarity, answer length, and vocabulary overlap between the data used for pretraining and testing.}
\label{fig:analysis}
\end{figure*}
\begin{table*}[!t]
\centering
\small
\caption{Evaluation results on the similarity, length, and vocabulary overlap quartiles of PIQA data for the models RoBERTa and T5-3b with different data sizes. Best results per model and similarity quartile are marked in bold. }
\label{tab:Data_size_analysis}
\begin{tabular}{cc |rrrr|rrrr|rrrr}
\hline
\multirow{2}*{\bf Model}&\multirow{2}*{\bf Data Size}&\multicolumn{4}{c|}{\bf Similarity}& \multicolumn{4}{c|}{\bf Length}&\multicolumn{4}{c}{\bf Vocabulary overlap}\\
&&{\bf 25\%}&{\bf 50\%}&{\bf 75\%}&{\bf 100\%}&{\bf 25\%}&{\bf 50\%}&{\bf 75\%}&{\bf 100\%}&{\bf 25\%}&{\bf 50\%}&{\bf 75\%}&{\bf 100\%}\\\hline
\multirow{6}*{\bf Roberta}
&0\%&53.6&63.9&71.9&\bf 80.9&63.0&66.3&65.1&75.9&62.3&71.3&69.9&66.7\\
&1\%&56.6&\bf 73.5&75.6&78.7&\bf 68.8&\bf 69.6&68.0&78.0&\bf 68.6&71.3&74.3&70.2\\
&5\%&\bf 60.3&72.4&\bf 76.5&80.4&66.7&68.3&\bf 72.1&\bf 82.6&\bf 68.6&\bf 73.5&\bf 74.9&\bf 72.6\\
&10\%&58.2&71.1&73.2&79.8&67.1&65.9&70.2&79.1&68.0&72.4&70.6&71.3\\
&33\%&58.8&72.0&74.7&78.0&68.0&68.7&70.6&76.3&66.9&71.7&72.8&72.2\\
&50\%&57.1&70.4&73.9&80.2&66.4&66.1&71.2&77.8&65.8&70.0&\bf 74.9&70.9\\
&100\%&55.6&68.3&69.3&74.8&62.7&65.2&66.9&73.0&63.4&65.9&71.9&66.7\\\hline
\multirow{6}*{\bf T5-3b}
&0\%&48.8&48.3&51.9&51.5&50.1&50.2&50.1&50.0&47.1&52.6&51.2&49.6\\
&1\%&61.7&73.9&73.6&78.7&71.2&70.0&70.2&76.5&68.2&70.4&76.9&72.4\\
&5\%&60.3&72.4&71.9&79.3&68.4&68.7&67.8&79.1&64.1&70.7&77.6&71.7\\
&10\%&65.4&75.9&75.4&82.6&70.6&74.8&72.5&81.3&69.9&74.1&77.8&77.4\\
&33\%&67.3&\bf 77.2&80.0&82.4&\bf73.4&74.6&\bf 77.8&81.1&70.4&78.7& 79.7&78.0\\
&50\% &\bf68.8&\bf77.2&\bf 80.2&\bf 84.3&73.2&\bf76.3&77.3&\bf83.7&72.1&\bf79.1&\bf80.2&\bf79.1\\
&100\%&68.2&76.1&78.2&84.1&72.3&71.9&76.5&79.3&\bf78.9& 77.2&75.8&76.7\\
\hline
\end{tabular}
\end{table*}
\noindent \textbf{What is the connection between model’s accuracy and partitions of the task (R5)?}
Figure~\ref{fig:analysis} (left) shows that both RoBERTa-large and T5-3b perform better on questions with similar answers. Interestingly, vanilla RoBERTa already achieves high performance on this set, and pre-training only improves the performance on the questions with dissimilar answers. Given that the data used for pre-training is designed to only include questions with non-overlapping answers, this finding is intuitive, and explains the source of improvement of performance reported in prior work~\cite{Ma2021,dou2022zero}. T5-3b's accuracy gain over RoBERTa-large also owes to this property of the synthetic data, together with T5-3b's larger capacity to learn commonsense knowledge.
Moreover, Table~\ref{tab:Data_size_analysis} shows that both models perform better on the questions with dissimilar answers when they are trained with more data. The models perform optimal on the questions with similar answers with less data. This confirms our explanation that the knowledge used for pre-training directs the models towards better performance on the questions with dissimilar answers.
Both models perform best on questions with longer answers, while T5-3b is advantageous for short answers (Figure~\ref{fig:analysis}, middle). Notably, the synthetic QA data mostly consists of short answers, showing again that the performance gain of T5-3b owes to its capacity to extend original knowledge during the commonsense adaptation stage.
Furthermore, we see that T5 is able to exploit maximum amount of data for short answers (Table~\ref{tab:Data_size_analysis}), which is expected, given that most of the synthetic questions are relatively short. When it comes to longer answers, T5 performs best with less data, which indicates that the pre-training data has limited utility for this set of questions. Curiously, this pattern is not observed for RoBERTa - RoBERTa is unable to leverage more than 1\% of the data to improve its performance on the questions with short answers. We hypothesize that this is due to the limited model capacity of RoBERTa, causing limited ability to store additional knowledge from the synthetic data.
We do not see a clear correlation between vocabulary overlap and model accuracy, both overall (Figure \ref{fig:analysis} right), as well as across different data sizes (Table \ref{tab:Data_size_analysis}). Further experiments are needed to explain this finding.
\section{Research Questions}
\label{sec:hypotheses}
We study five questions which have not been fully answered by prior work on zero-shot question answering with knowledge graphs. We motivate each question and indicate the novelty introduced by studying the question in our setting.
\noindent \textit{RQ1: What is the overall impact of model and knowledge choices on the generalizability of self-supervision of LMs with KGs?} Prior work on zero-shot commonsense reasoning with KGs~\cite{Ma2021,dou2022zero} has reported large gains across benchmarks, over LM baselines. Yet, the gap between these results and the performance of supervised models remains large. It is unclear how much this gap can be bridged by tuning the knowledge and model selection in the self-supervision method.
\noindent \textit{RQ2: How much data is needed to adapt LMs to commonsense reasoning tasks?} Finding a right number of QA pairs to adapt a model with is crucial to reach optimal performance, prevent overfitting, and optimize efficiency. Ma et al.~\cite{Ma2021} report accuracy gains with a hand-selected subset of the CSKG graph, whereas Ilievski et al.~\cite{ilievski2021dimensions} show that adapting language models with questions from certain knowledge dimensions is much more beneficial than others, and sometimes, even better than using the entire set of questions. No prior work has performed systematic analysis of the relation between knowledge sample size and the model performance.
\noindent \textit{RQ3: How to best sample questions for model adaptation?} Even after detecting the optimal data size, it remains unclear how to best select the needed data points. For instance, sampling can focus on optimizing diversity across different knowledge types, or it can focus on questions where the model exhibits low confidence or high fluctuations during adaptation. Such sampling strategies based on training dynamics exist~\cite{swayamdipta2020dataset,pleiss2020identifying}, but they have not been applied to the task of zero-shot QA with KGs.
\noindent \textit{RQ4: Do models generalize well to tasks with low domain overlap?} Models generally perform better on tasks that require knowledge similar to that in the training data, i.e., tasks with high domain overlap (HDO). This is confirmed by the relatively larger gains obtained when using ConceptNet for CSQA and ATOMIC for SocialIQA, compared to using these sources on datasets like WinoGrande~\cite{mitra2019exploring,Ma2021}. Whether models can generalize well to questions with low domain overlap (LDO) is an open question.
\noindent \textit{RQ5: What is the connection between model’s accuracy and properties of the task?}
We analyze the impact of answer candidate similarity, answer candidate length, and vocabulary overlap on the model accuracy. We expect that the questions with different answer similarity and length would require different reasoning. For example, similar answer candidates would require models to focus on the relatively small differences between answers. Similarly, task partitions with higher vocabulary overlap to the synthetic data would intuitively be easier for the models. Prior work has reported that models rely on spurious correlations, such as lexical properties, to answer questions~\cite{gururangan2018annotation,mccoy2019right}.
Li et al.~\cite{li2021systematic} show that only considering the answer candidates may bring high performance on some tasks, indicating that the properties of the answer candidates have a large impact on the model. Similarly, fine-tuned models perform much better on questions that resemble the training data~\cite{ma2021exploring}. Yet, these investigations have not been conducted in a zero-shot setting.
|
3,212,635,537,942 | arxiv | |
3,212,635,537,943 | arxiv | \section{Introduction}
As scientific and technological interest focuses on increasingly smaller
length scales, tools for visualizing and characterizing nanoscale systems are
needed. An important technological step towards this goal is the development
of techniques that allow precise and quantitative analysis of materials on a
nanometer scale. In this context, Scanning Probe Microscopy, and in particular
Scanning Force Microscopy (SFM) \cite{SFMBinnig} has proved to be a very
powerful tool for Nanotechnology. SFM allows not only the visualization of
surfaces on a nanometer scale, but also its modification and the
characterization of material properties. SFM is based on the interaction of a
very sharp tip with the sample to be studied. Therefore, a deep understanding
of tip-sample interaction is fundamental in SFM. A variety of forces may act
between tip and sample: dispersion, electrostatic, chemical, elastic as well
as adhesion and friction forces. In addition, when tip and sample are composed
of magnetic materials, also magnetic forces act. Each of these forces in
principle opens the way for measuring the corresponding physical property of
the sample.
The variety of different forces - and thus different material properties -
that can be measured is one of the reasons for the great versatility of SFM.
Correspondingly, at present SFM is used very successfully and extensively in a
variety of scientific areas. However, we believe that it is still not the
precise and quantitative tool required by the Nanotechnology community. One of
the difficulties for quantitative measurements with SFM is -in our opinion-
precisely the wealth of interactions that may act in a SFM set up. Since, a
priory, only the total force is measured, it is difficult to discriminate
between the contributions of the different kind of forces. However, to obtain
quantitative measurements and for a complete characterization of material
properties on an nanometer scale, as well as in general for the correct
interpretation of SFM experiments, the determination of the origin and the
relative strength of the measured forces is fundamental.
Due to its importance, tip-sample interaction has been the topic of a variety
of studies. Some of these studies are focused on the modeling and analysis of
the dynamics of the oscillating tip within the (non-linear) surface
potential\cite{TappingAncowski,TappingOthmar,TappingGissibl,TappingAlvaro},
while others are devoted to the physics of the interaction
itself\cite{InteractionChen,Argento,InteractionStark}. In this context, the
modeling and measurement of electrostatic interaction within a SFM set-up has
received particular
attention\cite{ESFMTipConeMole,ESFMTipCone,moleAPLnew,ESFMReview}. A variety
of methods have been used to experimentally characterize the interaction
between tip and sample in a SFM-setup. The simplest is based on the
acquisition of force versus distance (\textit{FvsD})
curves\cite{FvsDWeissenhorn}. This mode measures the static deflection of the
cantilever as tip-sample distance is varied and is easy to implement but is
not sensitive enough to analyze weak forces. A more precise method measures
variations of the oscillation of the cantilever, that is, of its amplitude,
phase or resonance frequency (amplitude, phase or frequency vs. distance
curves). Dynamic methods are more sensitive due to signal enhancement induced
by the resonance of the cantilever\cite{ResonanceAlbrecht}. More sophisticated
methods are based on the multidimensional acquisition of data, where the
interaction is measured not only as a function of tip-sample distance, but as
a function of at least a second variable. In one of those modes the mechanical
spectrum of the cantilever is determined as a function of tip-sample distance,
in this case the second variable is the excitation frequency. The oscillation
of the tip is induced either by thermal fluctuation\cite{DuerigResNoise}, or
by external excitation\cite{InteractionDucker,InteractionRon,InteracionYo}.
SFM experiments can be performed in Ultra High Vacuum (UHV) conditions or in
liquids and air. While the first kind of experiments are more demanding, the
latter are more difficult to understand due to less controlled surface
conditions: in air adsorbed liquid films and different kind of contamination
may complicate data interpretation. Moreover, while in UHV conditions the tip
can approach the surface as near as atomic distances and "feel" chemical
interactions, in air the minimum tip-sample distance is about one magnitude
larger, since the tip snaps to the surface not due to the instability produced
by typical surface potentials, but due to the (spontaneous) formation of
liquid necks\cite{InteracionYo}, occurring at distances between 2 and 5 nm.
Correspondingly, quantitative determination of tip-sample interaction in air
is more challenging since signals are typically weaker and interaction data
has to be analyzed and interpreted with
care\cite{CuellosMonica,CuellosHerminghaus}. In many cases the correct
determination of tip-sample distance and of the "true" non-contact regime is
crucial\cite{CommentNonContact1}.
In the "true" non-contact regime dispersion forces, which are always present,
and electrostatic forces, which are the strongest forces and have long
interaction range, are the most relevant forces in a typical SFM set up.
Separation of these two kind of interactions is fundamental for a better
understanding of tip-sample interaction, for adjusting of optimum imaging
conditions as well as for the quantitative determination of material
properties on a nanometer scale. In principle, as recognized in previous
works, variation of tip-sample bias results in turning \textquotedblleft
on\textquotedblright\ and \textquotedblleft off\textquotedblright\ the
electrostatic interaction, allowing to separate the dispersion interaction
from the electrostatic interaction. In fact, to experimentally verify the
quadratic dependence of tip-sample interaction on on tip-sample voltage and to
characterize the dielectric properties of different samples Hu et
al.\cite{MiguelAPL} have acquired interaction versus voltage curves. In a
similar approach, Guggisberg et al.\cite{MeyerComparisonInteractions} have
measured tip-sample interaction as a function of tip-sample voltage and
tip-sample distance to determine and compensate the contact potential between
tip and sample. The goal of that work performed in ultra high vaccum was the
precise discrimination and control of dispersion and electrostatic
interactions in order to measure short ranged chemical forces. In the present
work we pretend to further develop these techniques for separation and
measurement of electrostatic and dispersion interaction. As in the works just
discussed data is acquired as a function of tip-sample distance and of
tip-sample voltage. The corresponding experimental data sets are stored and
visualized as "interaction images". The to obtain a complete characterization
of tip-sample interaction cantilever deflection (force), oscillation amplitude
as well as frequency shift are acquired simultaneously and processed using
appropriate data processing algorithms. The measurement of resonance frequency
shift yields very high signal to noise ratio and the absolute calibration of
the measured quantities, acquisition of oscillation amplitude allows to
recognize the "true" non-contact regime, and from the cantilever deflection
tip-sample distance is determined. Precise values for the dispersion
interaction, the contact potential and the tip-sample
capacity\cite{CommentCapacity} are obtained as a function of tip-sample
distance. Furthermore, this method allows to characterize parameters such as
the tip radius and the tip-sample distance. We note that this method requires
no previous information about the electric properties of the tip-sample system
-and in particular about the contact potential- since this information is
obtained \textquotedblleft self-consistently\textquotedblright\ by the
algorithm processing the interaction images. In particular for experiments
performed in air, but also for UHV applications we are convinced that the
method presented in this work will result in improved data acquisition and
data interpretation. Moreover, we believe that this method will contribute to
obtain the profound understanding of tip-sample interaction needed to
quantitatively determine electrostatic properties of samples on a true
nanometer scale.
\section{Theoretical Background}
In a typical SFM experiment tip-sample interaction $I(d)$ (that is, energy) is
not measured directly. Instead the force $F(d)=-I^{\prime}(d)$ or the
resonance frequency are determined from experiment. The resonant frequency of
the tip-sample system and the curvature $I^{\prime\prime}(d)$ of the
interaction potential are related by%
\begin{equation}
\upsilon_{0}(d)=\sqrt{\frac{c_{lev}+I^{\prime\prime}(d)}{m_{eff}}}%
/(2\pi)\simeq\upsilon_{00}(1+\frac{1}{2}\frac{I^{\prime\prime}(d)}{c_{lev}})
\label{ResFreq}%
\end{equation}
with $\upsilon_{00}$ free resonance frequency of the cantilever, $c_{lev}$ its
force constant and $m_{eff}$ the effective mass of the cantilever. We note
that this relation is only correct if the oscillation amplitude is
sufficiently small in order to avoid non-linearities of the
interaction\cite{FrequencyApprox}. The approximation is valid as long as
$I^{\prime\prime}(d)\ll c_{lev}$ or, equivalently, for small shifts
$\Delta\upsilon_{0}(d)=\upsilon_{00}-\upsilon_{0}(d)\ll\upsilon_{00}$ of the
resonance frequency. For a detailed discussion of the mechanical behavior of a
SFM setup see Duerig\cite{InteractionDuerig}.
To simplify the argumentation, in the context of the present work we will
assume a homogeneous sample. We note, however, that the results discussed here
are also relevant for heterogeneous samples as long as the sample is "locally"
homogeneous, which means that\ the material properties of the sample are
homogeneous on a length scale larger than the resolution of the SFM-system,
that is, its effective aperture function. We will also assume that in the true
non-contact regime\cite{CommentNonContact1} tip-sample interaction is governed
only by dispersion and electrostatic interaction:%
\begin{equation}
I(d)=I^{dis}(d)+I^{estat}(d) \label{eqInteraction}%
\end{equation}
with $I^{estat}(d)$ electrostatic interaction and $I^{dis}(d)$\ dispersion
interaction. From a fundamental point of view, both kind of interactions can
be considered ultimately of electronic origin. While the electrostatic
interaction arises "directly" from charges, dispersion forces arise
"indirectly" through the residual interaction of fluctuating dipoles within
matter (Van der Waals forces) or fluctuating electric fields in vacuum
(Casimir forces)\cite{DispersionGeneral}. Van der Waals and Casimir forces are
of quantum mechanical origin. In the context of\ the present work, we will
assume that the dispersion forces within a SFM-setup are well described by the
relation\cite{Argento,Israelachvili}%
\begin{equation}
F^{vdW}(d)=\frac{A_{tms}~R}{6~d^{2}} \label{VanDerWaals}%
\end{equation}
which describes the Van der Waals force between a tip of radius R and the
sample surface by means of the Hamaker constant $A_{tms}$\ ($t$ip interacting
with $s$ample through $m$edium, see Israelachvili\cite{Israelachvili} for a
detailed discussion).
If tip and sample are electrical conductors the electrostatic force between
tip and sample can be written as a surface integral over the electric field on
the sample surface\cite{Jackson}. In addition, the electric field lines can be
approximated by segments of circles connecting tip and sample. To a good
approximation, the electric potential decays linearly along these circular
segments. Within this approach, the electrostatic force between tip and sample
can be calculated as \cite{Guthman}:
\begin{equation}
F\left( d\right) =\int_{S}dS\,\frac{\varepsilon_{0}}{2}\;E\left(
x,y,d\right) ^{2}\simeq\frac{\varepsilon_{0}U_{0}^{2}}{2}\int_{S}dS\,\frac
{1}{a(x,y,d)^{2}} \label{eqForce}%
\end{equation}
where $E(x,y,d)$ is the electric field on the surface for a certain tip-sample
distance $d$, $U_{0}$ is the effective voltage between tip and sample and
$a(x,y,d)$ the arc length of the circular segment coming from the probe and
ending on a point $(x,y)$ of the surface. When nanoscale dielectric systems
are adsorbed on a conducting surface, it can be shown within a perturbative
approach\cite{moleperturbative} that tip-sample interaction depends also on
the electrical properties of these systems, which are the objects to be
characterized by SFM.
Many SFM experiments are performed with a conducting or semiconducting tip in
order to control its potential. It would seem reasonable to assume that
grounding the tip with respect to the sample surface would imply vanishing of
electrostatic forces. This is, however, only correct if the work function (or
contact potential) of tip and sample are equal. Otherwise, differences in work
function induce transfer of charges that result in electrostatic fields and
thus, according to equation \ref{eqForce}, in electrical forces even if no
external bias is applied. The electrostatic force between tip and sample is
therefore a quadratic function of tip voltage with its minimum shifted by an
amount $U_{cp}$ with respect to the origin due to contact potential
difference\cite{WickramasingeKPM,KPMAbraham}:%
\begin{equation}
F_{el}(d)=C^{\prime}(d)(U_{tip}-U_{cp})^{2}/2 \label{SFMforce}%
\end{equation}
where $C^{\prime}(d)$ is the derivative of the tip-sample capacity. For a
conductive tip and sample, this derivative of the\ capacitance can be
estimated by the surface integral of equation \ref{eqForce}%
\begin{equation}
C^{\prime}(d)=\varepsilon_{0}\int_{S}dS\,\frac{1}{a(x,y,d)^{2}}
\label{approxCapacitance}%
\end{equation}
For the case of Van der Waals interaction we assume a specific tip geometry
-namely a parabola described by a tip radius $R$ , see relation
\ref{VanDerWaals}- while for the case of the electrostatic interaction
equation \ref{eqForce} is generic, that is, in principle any tip-sample
geometry can be described using an appropriate tip-sample capacity $C(d)$.
This different treatment is justified because of the different distance
dependencies of Van der Waals and electrostatic interactions. Due to its
faster decay Van der Waals interaction is less sensitive to the large scale
geometry of the probe (see Argento et. al.\cite{Argento} and\ Colchero et.
al.\cite{modeloPRB} for a more detailed discussion). In fact, for the
experiments described in the present work we find that the commonly used
approximation - $C(d)=2\pi~\varepsilon_{0}~R\ln(d)$ - for the tip-sample
capacitance\cite{ESFMSpere} does not describe tip-sample interaction
satisfactory within our experimental error. Therefore, the relation%
\begin{equation}
C(d)=2\pi\varepsilon_{0}R\ln\left( \frac{d}{d+R(1-\sin\vartheta_{0})}\right)
\label{CapacityComplex}%
\end{equation}
for the tip-sample capacity will be used, which results from modeling the tip
as a truncated cone of opening angle $\vartheta_{0}$ ending smoothly in a
spherical tip apex of radius $R$ (see Hudlet et. al. \cite{Guthman} for a
detailed discussion of this model probe tip).
As a general result from this theoretical section we conclude that for a fixed
distance $d_{0}$ the total tip-sample interaction as a function of voltage and
tip-sample distance is of the form%
\begin{equation}
i(U,d_{0})=\alpha(d_{0})+\gamma(d_{0})(U-\kappa(d_{0}))^{2}/2
\label{eqParabola}%
\end{equation}
where the constant term $\alpha$ describes the Van der Waals interaction,
$\gamma$\ the curvature of the parabola induced by electrostatic interaction
and $\kappa$ the position of the minimum. Note that this general relation
applies to the force as well as for the force gradient, and thus also for the
resonance frequency. This quadratic dependence of tip-sample interaction on
voltage is essentially the basis for the separation of Van der Waals and
electrostatic interaction in this work (see also Hu et al. \cite{MiguelAPL} as
well as Guggisberg et al. \cite{MeyerComparisonInteractions}).
\section{Experimental Method}
Usually either the force, the resonance frequency or the oscillation amplitude
are measured to characterize the tip-sample interaction. In the present work,
all three quantities are measured simultaneously. By measuring the force the
first derivative of the interaction (equation \ref{eqInteraction}) is
determined experimentally while the resonance frequency is related to the
second derivative of the interaction. As will be discussed in more detail
below, the method presented here allows the precise determination of all
parameters relevant for tip-sample interaction. By measuring a family of
curves $i(U,d_{0})$ as a function not only of tip voltage $U$ but also of
tip-sample distance \textit{interaction images}\ $i(U,d)$ can be obtained.
Each horizontal line can then be adjusted to the parabola defined by equation
\ref{eqParabola} to obtain, for each tip-sample distance, the contribution of
the Van der Waals interaction, the contact potential and the tip-sample
capacitance. From a whole \textit{interaction image}\ $i(U,d)$\ three curves
are obtained, one, $\alpha(d)$, characterizing the Van der Waals interaction
as a function of distance, another one, $\gamma(d)$, describing the
capacitance of the tip-sample system as a function of distance and the third
one, $\kappa(d)$, corresponding to the contact potential as a function of
tip-sample distance.
\subsection{Data acquisition}
Figure 1 schematically shows the experimental implementation of the setup used
to acquire the \textit{interaction images}. The experiments were performed
with a NanoTec SFM system composed of SFM head, high voltage controller and
PLL/dynamic measurement board\cite{nanotec}. For our experiments, we find that
cantilevers with long tips, located at its very end are ideal, since these
kind of SFM\ probes minimize the effect of the interaction between the sample
and the (macroscopic) cantilever, which can result in uncontrolled
electrostatic interaction\cite{EstatNanotechnology} and (long-range) viscous
forces\cite{LangmuirViscoso} that decrease the quality factor of the
oscillation and thus the sensitivity in dynamic detection modes.\ Olympus
OMCL-AC-type cantilevers have been used in all experiments\cite{Cantilever}.
The \textit{interaction images}\ were acquired using the
"3D-Mode"\cite{3Dmodes}, which allows fast switching from normal imaging to a
series of extended acquisition modes. "3D-Mode" is a generalized acquisition
mode where data -that is, some input channel- is acquired as a function of two
output channels that are user selected: $datain=datain(output_{1},output_{2}%
)$. "3D-Mode" images are acquired in a raster scan mode where one channel is
varied "fast" at a fixed value of the "slow" output, then the "slow" output is
varied by one step and a new line of data is acquired by varying the "fast"
output. For the experimental data shown here, the \textquotedblleft%
3D-Mode\textquotedblright\ was set in order to have the tip voltage as fast
scan and the tip-sample distance as slow scan, that is, the tip-sample voltage
is ramped fast to acquire, for each distance, interaction data as a function
of tip-sample voltage while tip-sample distance is varied slowly to bring the
tip into and out of contact with the sample\cite{PNAS,Estat3D}. For each set
of experiments, the normal force, the frequency shift and the oscillation
amplitude are measured as a function of tip-sample voltage and tip-sample distance.
To measure the resonance frequency of the tip-sample system the cantilever is
excited near its resonance frequency by a small piezoelectric element and a
Phase Locked Loop (PLL)\cite{ResonanceAlbrecht} is used to track the resonance
frequency as bias voltage and tip-sample distance are varied. The use of a
PLL-circuit allows the direct measurement of the resonance frequency in Hertz,
and thus the determination of interaction data in physically meaningful units.
Calibration of force data was performed using the nominal force constant of
the cantilevers to convert deflection into force and by calibrating the
photodiode signal with a \textit{FvsD} curve\cite{CommentCali}. Typical
oscillation amplitudes were 0.1 to 2.5$%
\operatorname{nm}%
$ (0.2 to 5$%
\operatorname{nm}%
$ peak to peak). The lower threshold of the oscillation is set by thermal
noise of the cantilever, while the higher value was chosen in order to avoid
non-linearities of the interaction. As a reasonable criterion we have chosen
amplitudes that are smaller than the minimum tip-sample distance at the snap
to contact point.
\subsection{Data processing}
Data is processed as follows: in a first step, each force line is averaged to
obtain a \textit{FvsD} curve. This \textit{FvsD} curve is then analyzed by
means of an appropriate algorithm that determines the snap to contact point in
order to separate the contact and non-contact regimes of the curve. We note
that, since data is acquired in air, the snapping instability is induced by
the spontaneous condensation of a liquid neck between tip and sample at rather
large distances of 2-5$%
\operatorname{nm}%
$\cite{InteracionYo,CuellosMonica}. In order to obtain meaningful interaction
data the jump distance, which can only be measured if the cantilever
deflection (force) is acquired, has to be taken into account.\ From the
\textit{FvsD} curve the snapping distance is measured and the position of the
surface as well as the true tip-sample distance is determined. All curves
calculated from the same set of \textit{interaction images}\ are then shifted
horizontally by a constant amount corresponding to the position of the surface
as "seen" by the \textit{FvsD} curve. Then, the origin of the horizontal axis
(piezo displacement) corresponds to the displacement of the piezo where the
cantilever would touch the surface if no attractive forces where present. In a
second step, force and frequency data corresponding to the non-contact regime,
that is, before the snapping instability, are adjusted to a quadratic function
according to equation \ref{eqParabola}. From the force \textit{interaction
image} three curves $f^{vdW}(d)$, $C^{\prime}(d)$ and $U_{cp}^{force}(d)$ are
obtained describing the Van der Waals force, the first derivative of the
tip-sample capacitance, and the contact potential as "seen" by the force data.
In the same way the frequency \textit{interaction image} yields three
corresponding curves $\Delta\omega_{0}^{vdW}(d)$, $C^{\prime\prime}(d)$ and
$U_{cp}^{freq}(d)$; the frequency shift induced by the Van der Waals
interaction, the second derivative of the capacitance and the contact
potential as "seen" by the frequency data. Finally, the three most relevant
curves - $\Delta\omega_{0}^{vdW}(d)$, $C^{\prime}(d)$ and $C^{\prime\prime
}(d)$ - are compared with the relations%
\begin{equation}
\Delta\omega_{0}^{vdW}(d)=\frac{1}{2}\frac{\omega_{00}}{c_{lev}}%
V_{vdW}^{\prime\prime}(d)=\frac{1}{6}\frac{\omega_{00}}{c_{lev}}%
\frac{A~R_{vdW}}{(d-d_{0}^{vdW})^{3}} \label{freqVdW}%
\end{equation}
\begin{equation}
C^{\prime}(d)=2\pi\varepsilon_{0}R_{estat}\frac{R_{estat}(1-\sin\left(
\vartheta\right) )}{(d-d_{0}^{est1})\left( d-d_{0}^{est1}+R_{estat}%
(1-\sin\vartheta\right) }+b_{1} \label{CapacityForce}%
\end{equation}%
\begin{equation}
C^{\prime\prime}(d)=2\pi\varepsilon_{0}R_{estat}\left( \frac{1}%
{(d-d_{0}^{est2})^{2}}-\frac{1}{\left( d-d_{0}^{est2}+R_{estat}%
(1-\sin\vartheta)\right) ^{2}}\right) +b_{2} \label{CapacityFrequency}%
\end{equation}
which describe the frequency shift expected for a pure Van der Waals
interaction of a tip with radius $R_{vdW}$, and the first and second
derivatives of the capacitance predicted for a truncated cone ending in a
spherical tip apex of radius $R_{estat}$. The two constants $b_{1}$ and
$b_{2}$ are introduced to account for the possible effect of long range
contributions to the electrostatic interaction due to the tip cone or the
cantilever (see \cite{EstatNanotechnology,EstatSadewasser,MoleAPLTipConstant}%
). Each adjustment is allowed to find its own tip radius. In addition, each
adjustment is allowed to find the best value for $d_{0}$. Note that the
parameters $d_{0}$ represent mathematically the poles of the interaction, that
is, the position where the interaction diverges. From a physical point of
view, these poles correspond to the position of the surface. Complete
agreement with the surface position obtained from the \textit{FvsD} curve
would require $d_{0}=0$, since, as described previously, all curves have been
shifted to this position. From a physical point of view different tip radii
$R$ and different distances $d_{0}$ imply that the corresponding curves
\textquotedblleft see\textquotedblright\ different tips and different surface positions.
\section{Experimental Results}
Measurements have been performed on a variety of samples under different
experimental conditions. As representative cases two experiments performed on
a Au(111) surface are presented here, one taken with a highly doped silicon
tip at a relatively large oscillation amplitude of 2.5$%
\operatorname{nm}%
$ (5$%
\operatorname{nm}%
$ peak to peak), and another one taken with a Pt-coated tip at a significantly
lower amplitude of 1$%
\operatorname{nm}%
$. We consider the first case to be representative for typical experiments in
SFM and Electrostatic Force Microscopy, while the second case is closer to the
ideal situation where tip and sample are truly metal surfaces and the
oscillation amplitude is sufficiently small to neglect non-linear effects in
the tip-sample interaction.
Figure 2 shows a typical set of force and frequency \textit{interaction
images} as well as an oscillation amplitude image acquired on an Au(111)
surface with a Si tip\cite{Cantilever} in ambient air (temperature $\simeq$22$%
{{}^\circ}%
$C, relative humidity $\simeq$50\%). These images show pure raw data, no plane
or other form of data-processing has been applied. The lower part of the
images corresponds to small tip-sample distances (near the surface), the
lowest lines that appear flat correspond to data taken with tip and sample in
mechanical contact. In this regime, none of the \textit{interaction images}
show the quadratic dependence on the tip-sample voltage that is visible in the
non-contact regime. The frequency shift image is completely saturated because
the elastic interaction drives the resonance frequency of the tip-sample
system far away from the free resonance frequency\cite{CommentLockRange}. In
the normal force the\ quadratic dependence is observed for small and large
tip-sample distances, while in the\ amplitude and in the frequency
\textit{interaction image}\ the quadratic dependence is observed only very
near the surface ($\lesssim$10$%
\operatorname{nm}%
$). As will be discussed in more detail below, this indicates a very short
interaction range for the frequency and the amplitude signal. The parabolic
dependence of the amplitude is attributed to increased dissipation in the
tip-sample system, but at the moment the precise origin of this dissipation is
still unknown. In the normal force image the jump to contact is recognized as
a step towards lower force values followed by a continues increase of the
normal force.
Figure 3 shows the \textit{FvsD} curve obtained by averaging each line of the
force \textit{interaction image}. In addition, the amplitude image has been
processed to calculate the oscillation amplitude for each tip-sample distance,
which allows to determine the upper and lower turning point of the tip motion
during oscillation shown in figure 3. As discussed above, the oscillation
amplitude ($\simeq2.5%
\operatorname{nm}%
$) has been chosen to be smaller than the distance corresponding to the jump
to contact regime. The mean jump distance measured from the force curve is
about $\overline{d}_{jump}\simeq4%
\operatorname{nm}%
$, however, we believe that the effective jump distance is
\[
d_{eff}=\overline{d}_{jump}-a_{osci}%
\]
where $a_{osci}\simeq2%
\operatorname{nm}%
$ is the oscillation amplitude just before the jump to contact instability.
The effective jump distance is then about 2$%
\operatorname{nm}%
$. The reasoning behind this correction is that the jump to contact is
believed to be induced by condensation of liqid necks at the lowest turning
point of the oscillation, that is, at the smallest tip-sample
distance\cite{InteracionYo}. Jump to contact then occurs at the smallest
tip-sample distance rather than from the mean tip-sample distance. The force
curve shows very low adhesion and very little hysteresis: the area enclosed
during the whole forward and backward cycle is the energy dissipated during
the acquisition process and corresponds to about $10^{-17}$~$%
\operatorname{J}%
$ ($\simeq60%
\operatorname{eV}%
$). Acquisition of this curve is therefore quite gentle. The measured adhesion
force allows to estimate the product of tip radius and contact angle according
to\cite{Israelachvili,AdhesionOthmar}%
\[
F_{ad}=4\pi\gamma R\cos\left( \varphi\right)
\]
where $R$ is the tip radius, $\gamma$ the surface energy of water and
$\varphi$ the effective contact angle of water on the tip-sample system. With
a force constant $c=2%
\operatorname{N}%
$/$%
\operatorname{m}%
$ and assuming a wetting sample\cite{CommentWetting}, that is, a contact angle
$\varphi\simeq0-30%
{{}^\circ}%
$, we obtain an estimated tip radius of 10-15$%
\operatorname{nm}%
$, in good agreement with results obtained from the non-contact measurements
to be discussed below.
The force curve together with the frequency and amplitude distance behavior
allows to separate four regimes that can be present during tip-sample
approximation (see also Fig. 3): first, a true non-contact regime where the
cantilever essentially oscillates with its free resonance frequency, second, a
\textquotedblleft tapping non-contact\textquotedblright\ regime where the
oscillation amplitude decreases even though no mechanical contact between tip
and sample is formed, third, an intermittent contact regime where the tip
still oscillates and dynamically touches the surface, and forth a continuous
contact regime where the cantilever no longer oscillates. In our experiments
we use rather soft cantilevers at small oscillation amplitudes, therefore we
do not observe the intermittent contact regime (see figure 3), in this
case\ the oscillation stops as soon as the tip mechanically touches the
surface for the first time since the adhesion force is larger than the
restoring force of the cantilever\cite{LangmuirGotas,GiessiblParameters}. With
very small oscillation amplitudes also the second \textquotedblleft tapping
non-contact\textquotedblright\ regime may not be observed, then the tip
directly snaps on to the surface before any significant dissipation is detected.
In the non-contact regimes the \textit{interaction images} are processed as
discussed in the previous section to obtain, for each \textit{interaction
image}, three curves describing the Van der Waals interaction, the capacitance
and the surface potential. Figure 4 shows the result of this process for the
\textit{interaction images} shown in Fig. 2. The graphs in Fig. 4a, 4c and 4e
represent the curves obtained from the force \textit{interaction image} while
the graphs in Fig. 4b, 4d and 4f have been obtained from the frequency
\textit{interaction image}. The Van der Waals curve obtained from the force
\textit{interaction image} essentially reproduces the \textit{FvsD} curve
obtained by simple averaging, a small difference is due to the effect of the
electrostatic interaction: while the curve in Fig. 3 is a mixed
electrostatic/Van der Waals curve\cite{CommentMeanForce}, the curve in Fig. 4a
is a \textquotedblleft true\textquotedblright\ Van der Waals \textit{FvsD}
curve. The parameters obtained from the adjustment of the experimental data to
the relations \ref{freqVdW}-\ref{CapacityFrequency} are summarized in Table 1.
To investigate the precision of the method described in the present work two
test have been made. First, the data describing the interaction has been
adjusted to the relations \ref{freqVdW} and \ref{CapacityFrequency} allowing
the fit to find the exponents $n_{vdW}$ (which was $n_{vdW}=$3 in relation
\ref{freqVdW}) and $n_{el}$ (which was $n_{el}=$2 in relation
\ref{CapacityFrequency}, in the present context only the first term with the
pole $1/(d-d_{0}^{est2})^{2}$ is considered relevant). The results of the
corresponding fits are listed in table 2, confirming that the data indeed
reproduces correctly the exponent of Van der Waals as well as electrostatic interaction.
For the second test the data corresponding to tip-sample capacity (that is,
$C^{\prime\prime}(d)$) was adjusted using two different functions in order to
check whether the data is able to \textquotedblleft see\textquotedblright\ the
difference. One relation is obtained from the usual spherical approximation
which gives $C^{\prime\prime}\left( d\right) =2\pi~\epsilon_{0}~R/d^{2}$,
and the second relation is obtained using the more sophisticated model leading
to equation \ref{CapacityFrequency}. In this second model, the cone angle was
fixed to $\vartheta_{0}=20%
{{}^\circ}%
$, as specified by the manufacturer. Both models therefore have the tip-radius
as only free parameter. The corresponding fits together with their errors are
shown in Fig. 4d. Even though both fits yield parameters that are compatible
one with the other, the first model for the tip-sample capacitance shows a
clear tendency of the error not observed in the second one.
Figure 5 shows the result of the processing of force and frequency
\textit{interaction images} as well as an oscillation amplitude image acquired
on an Au(111) surface with a Pt-coated Silicon tip. As in the previous case,
data was acquired in ambient air. The graph in Fig. 5a, and 5c represent the
curves obtained from the force \textit{interaction image} while the graph in
Fig. 5b and 5d have been obtained from the frequency \textit{interaction
image}. Table 3 summarizes the values for tip radius and surface position
obtained from the adjustment of the curves 5b-5d to the relations
\ref{freqVdW}-\ref{CapacityFrequency}. Due to the lower oscillation amplitude
(1nm), no "tapping non-contact" range is observed. The mean jump distance is
again about 3.5 nm, and the effective jump distance is 2.5 nm, somewhat larger
than in the previous case. The results obtained from this second experiment
are similar to those obtained in the first one, as can be verified by
comparing the parameters for tip radius and surface position listed in Table 1
and 3. The slightly higher noise level in the second experiment as compared to
the first one is attributed to the lower oscillation amplitude. Nevertheless,
at present we believe that for low oscillation amplitudes our measurements are
limited by technical noise rather than fundamental limits such as shot noise
or thermal movement of the cantilever. In the case of the Si-tip, the Van der
Waals radius ($R^{vdW}$ =5.8$\pm$0.4$%
\operatorname{nm}%
$) is considerably smaller than the two electrostatic radii ($R^{estat1}%
$=12$\pm$1$%
\operatorname{nm}%
$, $R^{estat2}$=21$\pm$4$%
\operatorname{nm}%
$). Moreover, these two electrostatic radii do not agree within the measured
experimental error. One possible explanation for this may be that, at the
level of precision of the method, the Si-tip is not a sufficiently ideal tip
due to the presence of oxide layers on its surface and/or to band bending
effects. In the case of the Pt-tip, the Van der Waals radius ($R^{vdW}$
=14$\pm$5$%
\operatorname{nm}%
$) as well as both electrostatic radii ($R^{estat1}$=16$\pm$12$%
\operatorname{nm}%
$, $R^{estat2}$=21$\pm$6$%
\operatorname{nm}%
$) are compatible one with another, indicating that Van der Waals and
electrostatic interaction "see" the same geometry of the tip. Finally, we note
that the Van der Waals radii obtained for the Si-tip and for the Pt-tip are in
good agreement with the nominal values of the manufacturer: the Si-tip has a
nominal radius better than 10$%
\operatorname{nm}%
$, while the Pt-tip, consisting of a thin 10-15$%
\operatorname{nm}%
$ thin Pt-film on the same kind of Si-tip should have a tip radius of 15-25$%
\operatorname{nm}%
$\cite{Cantilever}.
As a general result from both experiments we find that the data obtained from
the frequency image has less dispersion and less error in the estimation of
the parameters describing the tip-sample interaction. The noise in the data
characterizing the surface potential decreases for small tip-sample distances
only for the data obtained from the frequency \textit{interaction imag}e
(compare figures 4f and 4e). The offset $b_{1}$ obtained from the force
interaction image is clearly different from zero, in fact, it accounts for
almost 50\% of the electrostatic interaction at the smallest tip-sample
distance, while the corresponding offset $b_{2}$ obtained from the frequency
\textit{interaction image} is essentially compatible with $b_{2}=0$. These
offsets reflect the amount of electrostatic interaction that does not change
on a length scale comparable to the distance sampled by an \textit{interaction
image} (100$%
\operatorname{nm}%
$). Therefore, as discussed previously, we conclude that the force signal with
the large value of the offset $b_{1}$ has a significant long range component
of electrostatic interaction while the frequency, with its vanishing offset
$b_{2}$, has a truly short range behavior. As described in detail
elsewhere\cite{modeloPRB}, the long range component of the electrostatic
interaction is due to the mesoscopic tip cone or the macroscopic cantilever,
while the short range interaction is induced by the nanoscopic tip apex,
usually described by means of an (effective) tip radius. In the frequency
(that is, force gradient) signal, the long range component of the interaction
is "derived away" and only the interaction with the nanoscopic tip apex is
relevant. In our opinion, this explains the large value of $b_{1}$ obtained
from the force \textit{interaction image} as compared to the vanishing of
$b_{2}$ obtained from the frequency \textit{interaction image}.
For small indentation forces, the experiments described here are quite
reproducible. This is seen in Fig. 6, which shows the result of processing
three different \textit{interaction images} taken consecutively at a same spot
of an Au(111) surface. The corresponding \textit{FvsD} curves obtained from
force \textit{interaction images} are shown with approach and retraction cycle
(Fig. 6a). The second derivative of the capacitance as well as the contact
potential were obtained from frequency \textit{interaction images} and only
approach curves are presented. Within the experimental error the three
indentations lead to essentially the same results. In particular, the
\textit{FvsD} curves are almost absolutely equivalent, the three curves show
not only the same jump to contact distance and the same adhesion force, also
the details of how the tip-sample contact breaks are very similar. Images
taken before and after the acquisition of these curves showed no variation of
the surface, confirming that the surface was not modified during the
indentation experiments. When thermal drift is sufficiently low, an even more
gentle measurement of tip-sample interaction is possible by acquiring
\textit{interaction images} without bringing the tip into mechanical contact
with the sample. Tip-sample distance can then in principle be estimated from
the values of the poles $d_{0}$ of tip-sample interaction (see relations
\ref{freqVdW}-\ref{CapacityFrequency}). In this case, the tip is maintained
pristine. In particular when very sharp tips are used, this is a significant
advantage over other techniques for measuring tip-sample interaction. In our
opinion, the results shown in Fig. 6 demonstrate the potential and precision
of the method. The method is sufficiently sensitive to detect possible
variations of the tip-sample system, which may be detected in any of the
measured parameters, that is, not only in the \textit{FvsD} curve (and in
particular in the adhesion force), but also in the surface potential or in the
capacitance of the tip-sample system.
\section{Conclusion}
In the present work, the Van der Waals and electrostatic contribution to the
interaction in a SFM set-up have been separated and analyzed quantitatively.
The method described is based on the acquisition of \textit{interaction
images}\ where the force, the oscillation amplitude and the resonance
frequency are acquired as a function of tip-sample distance and tip-sample
voltage. Using appropriate processing algorithms, the Van der Waals
interaction, the electrostatic interaction and the contact potential are
determined from the force as well as from the frequency data. We note in this
context that other methods for measuring tip-sample interaction do not
directly allow the separation of Van der Waals and electrostatic interaction.
In particular, when the materials of tip and sample are different -which is
usually the case- contact potentials may significantly contribute to the total
tip-sample interaction. If pure Van der Waals interaction is to be measured,
the contact potential has to be compensated and if electrostatic interaction
is measured, the contact potential has to be taken into account. With the
method described here, Van der Waals interaction, electrostatic interaction -
described by means of a tip-sample capacitance or some of its derivatives -
and contact potentials are determined \textquotedblleft self
consistently\textquotedblright, that is, the method automatically takes
account of the effect induced by each of these three phenomena. More
technically, our method assumes no prior information about the physical
properties of the tip-sample system, instead it "finds" the combination of
parameters corresponding to Van der Waals and electrostatic interaction that
best describe the measured data.
Our experiments performed with (highly doped) semi-conducting as well as with
metalized tips demonstrate the potential of the method and yield high
precision, quantitative and reproducible interaction data. The high precision
of the method is due to the fitting process which implies averaging of
experimental data. Typically 100 to 1000 data points are acquired (one line of
an \textit{interaction image}) to obtain three physically relevant parameters,
therefore the signal to noise ratio of these curves increases as compared to
the usual acquisition of interaction versus distance curves. Assuming that
data fitting increases the signal to noise ratio as $\sqrt{n}$, with $n$
number of data points, the improvement of signal to noise ratio is about one
order of magnitude. Another feature of the method presented is that the
processing algorithms can be developed further to test whether the
electrostatic interaction follows the quadratic dependence with bias voltage
described by relation \ref{SFMforce}. This is particularly relevant for
semiconducting tips or samples as well as for strong electric fields between
tip and sample where departures from the quadratic behavior could be expected.
In addition, also inhomogeneities of the tip surface potential could be
recognized due to non-ideal behavior of the interaction with tip-sample
voltage as well as due to a distance dependence of the measured surface
potential (see also\cite{modeloPRB}).
We have shown that the technique is able to experimentally determine the
exponent of the Van der Waals and the electrostatic interaction. In addition,
the technique is also able to differentiate between two models for
electrostatic tip-sample interaction, showing that an accurate description of
electrostatic interaction needs to go beyond the simple sphere-sample
approximation used generally.
We believe that the method presented in this work will significantly
contribute to improve the measurement of interaction in a SFM setup and aid
the correct interpretation of SFM experiments. In addition, an improved
characterization of interaction will allow quantitative determination of
material properties on a nanometer scale, an issue that recently is receiving
more and more attention but that, in our opinion, still needs some important
efforts from the SPM community.
\section{Acknowledgments}
The authors acknowledge stimulating discussions with J. Abellan, A. Urbina, C.
Munuera, J. G\'{o}mez, C. G\'{o}mez-Navarro, A.M. Bar\'{o} and A. Gil. The
authors also thank Atomic Force F\&E GmbH, and in particular Mr. Ludger
Weisser,\ for supplying the cantilevers\ used. M. Cuenca, I. Horcas, P.
Colilla and R. Fernandez from the Nanotec team contributed to this work by
adapting instrumental design and software routines. And finally, the work was
supported by the Spanish Ministry of Science and Technology as well as by the
Fundaci\'{o}n S\'{e}neca-CARM through the projects MAT2002-01084,
NAN-2004-09183C10-03 and "Crecimiento de Nanoestructuras....".
\section{Tables}
\begin{tabular}
[c]{||l|l|l|l||}\hline\hline
curve & tip radius $R$ & surface position $d_{0}$ & offset $b$\\\hline
force $F(d)$ & $10-15nm$ (from $F_{ad}$) & $0$ (by definition) & $-$\\\hline
frequency $\nu(d)$ & $R^{vdW}=5.8\pm0.4nm$ & $d_{0}^{VdW}=3.1\pm0.2nm$ &
$-$\\\hline
force-capacitance $C^{\prime}(d)$ & $R^{estat1}=12\pm4nm$ & $d_{0}%
^{estat1}=2\pm2nm$ & $b_{1}=-(7\pm4)\ast10^{-3}$\\\hline
frequency-capacitance $C^{\prime\prime}(d)$ & $R^{estat2}=21\pm1nm$ &
$d_{0}^{estat2}=1.2\pm0.9nm$ & $b_{2}=-(37\pm14)\ast10^{-6}$\\\hline\hline
\end{tabular}
Table 1
Parameters describing tip-sample interaction obtained after processing the
\textit{interaction images} shown in Fig. 2, and the curves shown in Figs.
4b), 4c) and 4d). Data acquired with a Si-tip on an Au (111) surface.
Resonance frequency: 73.67 kHz, force constant 2N/m, free oscillation
amplitude $2.5%
\operatorname{nm}%
$.\bigskip
\begin{tabular}
[c]{||l|l|l|l|l||}\hline\hline
Curve & tip radius $R$ & surface position $d_{0}$ & exponent experiment &
exponent theory\\\hline
$\nu(d)$ & $15\pm30nm$ & $3\pm2nm$ & $3.3\pm0.5$ & $3$\\\hline
$C^{\prime\prime}(d)$ & $17\pm1nm$ & $2.2\pm0.2nm$ & $1.9\pm0.5$ &
$2$\\\hline\hline
\end{tabular}
\bigskip
Table 2
Parameters describing tip-sample interaction obtained after processing the
\textit{interaction images} shown in Fig. 2, and the curves shown in Figs.
4b), 4c) and 4d). In addition to the parameters $R$ and $d_{0}$, also the
exponents describing the variation with tip-sample distance in relations
\ref{CapacityFrequency} and \ref{freqVdW} were allowed to vary.\bigskip
\begin{tabular}
[c]{||l|l|l|l||}\hline\hline
curve & tip radius $R$ & surface position $d_{0}$ & offset $b$\\\hline
force $F(d)$ & $10-15nm$ (from $F_{ad}$) & $0$ (by definition) & $-$\\\hline
frequency $\nu(d)$ & $R^{vdW}=14\pm5nm$ & $d_{0}^{VdW}=1.7\pm0.5nm$ &
$-$\\\hline
force-capacitance $C^{\prime}(d))$ & $R^{estat1}=16\pm12nm$ & $d_{0}%
^{estat1}=2.1\pm1.1nm$ & $b_{1}=-0.1\pm0.3$\\\hline
frequency-capacitance $C^{\prime\prime}(d)$ & $R^{estat2}=21\pm6nm$ &
$d_{0}^{estat2}=1.2\pm0.7nm$ & $b_{2}=-(1.7\pm1.3)\ast10^{-4}$\\\hline\hline
\end{tabular}
\bigskip
Table 3
Parameters describing tip-sample interaction obtained after processing
\textit{interaction images} acquired with a Pt-tip on an Au (111) surface.
Resonance frequency: 68.80 kHz, force constant 2N/m, oscillation amplitude $1%
\operatorname{nm}%
$.\bigskip
\bigskip
\section{Figure Captions}
\begin{center}
\bigskip%
\raisebox{-0cm}{\includegraphics[
trim=-1.061882in 0.000000in -1.061881in 0.000000in,
natheight=6.496500in,
natwidth=13.719400in,
height=6.6689cm,
width=16.1671cm
]%
{Figs/Fig1_New.jpg}%
}%
\end{center}
Figure 1
Schematic description of the experimental setup used to acquire
\textquotedblleft interaction images\textquotedblright. The lateral position
of the tip over the sample is fixed. The \textquotedblleft
fast\textquotedblright\ and \textquotedblleft slow\textquotedblright\ outputs
of the electronics are used to vary the tip-sample voltage and the tip-sample
distance. A Phase Locked Loop is used to mechanically oscillate the tip-sample
system always at resonance. Frequency shifts $\Delta v$ are measured with
respect to a reference frequency -usually the free resonance frequency- set by
the electronic system. In our experiments, the force, the frequency shifts and
the variation of oscillation amplitude induced by the tip-sample interaction
are measured simultaneously.
\begin{center}
\bigskip%
{\includegraphics[
natheight=14.513300in,
natwidth=4.506500in,
height=5.8323in,
width=1.8308in
]%
{Figs/Fig2_New.jpg}%
}%
\end{center}
Figure 2
Typical set of interaction images acquired by varying the bias voltage
(\textquotedblleft fast\textquotedblright\ scan, corresponding to a horizontal
scan line) and the tip-sample distance (\textquotedblleft
slow\textquotedblright\ scan, vertical direction). The upper part of the
images corresponds to large tip-sample distances, the lower part to small
distances. At the bottom of the images, tip and sample are in mechanical
contact. Resonance frequency: 73.67 kHz, force constant 2N/m, oscillation
amplitude $2.5%
\operatorname{nm}%
$.
a) Force \textquotedblleft interaction image\textquotedblright, total gray
scale corresponds to about 1 nN.
b) Frequency \textquotedblleft interaction image\textquotedblright, total gray
scale corresponds to about 300 Hz.
c) Oscillation amplitude \textquotedblleft interaction image\textquotedblright%
, total gray scale corresponds to a variation of 0.5~nm (that is about 20\% of
the total oscillation amplitude). All images have been taken simultaneously
with a voltage scan of $\pm$1.5V, and the total tip movement is 100nm.
Clearly, the parabolic dependence of interaction on the bias voltage is
recognized best in the frequency image for small tip-sample distances.
\begin{center}
\bigskip%
\raisebox{-0cm}{\includegraphics[
trim=-0.847359in 0.000000in -0.847359in 0.000000in,
natheight=9.052900in,
natwidth=13.386400in,
height=4.67cm,
width=7.7321cm
]%
{Figs/Fig3_New.jpg}%
}%
\end{center}
Figure 3
Force vs. distance curve calculated from the force and amplitude
\textit{interaction images}\ shown in Fig. 2. From the force
\textit{interaction image} the mean deflection measured by the detection
system during the corresponding line of the \textit{interaction image}\ is
shown as solid circles (approach) and triangles (retraction). From the
amplitude \textit{interaction image}\ the oscillation amplitude is calculated
and represented as the upper and lower lines in the graph showing the upper
and lower turning point of the cantilever during the oscillation. As discussed
in the main text, the piezo displacement has been adjusted in order to have
the position $\Delta=0$ at the point where the deflection of the cantilever is
zero when tip and sample are in contact. The (apparent) jump distance is about
$4%
\operatorname{nm}%
$ and the adhesion force is about 9 nN (=4.5nm x 2N/m). The different
interaction regimes described in the main text are separated with dashed
lines: (I) "true non-contact regime", (II) "tapping non-contact" and (IV)
"continuous contact regime". Due to the soft cantilevers and the low
oscillation amplitude used, the "intermittent contact regime" (referred to as
region III in the main text) does not appear.
\begin{center}
\bigskip%
\raisebox{-0cm}{\includegraphics[
natheight=13.594000in,
natwidth=15.046900in,
height=11.4664cm,
width=12.6855cm
]%
{Figs/Fig4_New.jpg}%
}%
\end{center}
Figure 4
Several interaction\ vs. distance curves calculated from the force and
frequency \textit{interaction images}\ as discussed in detail in the main
text. Graphs b), c) and d) show not only the data points, but also the best
fit to the data points using the relations \ref{freqVdW}%
-\ref{CapacityFrequency} as well as the error to the best fit (smaller points
around the dashed horizontal line). For the frequency data, this error is very
small, therefore an amplification factor of 10 has been applied as compared to
the data points.
Left graphs: Curves calculated from the force \textit{interaction image}: a)
Van der Waals force vs. distance curve, c) first derivative of the tip-sample
capacitance vs. distance together with the best fit to the data and the
corresponding error and e) surface potential vs. distance.
Right graphs: Curves calculated from the frequency \textit{interaction image}:
b) Van der Waals frequency vs. distance curve together with the best fit to
the data and the corresponding error (x10), d) second derivative of the
tip-sample capacitance vs. distance together with the best fit to the data and
the corresponding error (x10), and f) surface potential vs. distance. For the
capacity data calculated from the frequency \textit{interaction image} (graph
d)) the error is shown for two different relations, corresponding to two
different descriptions of the tip-sample system. One of the two models (the
empty circles) shows a small but clear tendency of the error.
In all graphs, the piezo displacement has been adjusted with the value used in
Fig. 3. Note that only data points corresponding to the non-contact regime are
shown, therefore no data is shown for tip-sample distances $\Delta<5$nm.
Resonance frequency: 73.67 kHz, force constant 2N/m, oscillation amplitude
$2.5%
\operatorname{nm}%
$.
\begin{center}
\bigskip%
\raisebox{-0cm}{\includegraphics[
trim=-2.212757in 0.000000in -2.212758in 0.000000in,
natheight=10.346600in,
natwidth=15.313200in,
height=8.7426cm,
width=16.6153cm
]%
{Figs/Fig5_New.jpg}%
}%
\end{center}
Figure 5
Interaction vs. distance curves calculated from force and frequency
\textit{interaction images}\ (not shown) acquired with a Pt-tip on an Au(111)
surface in air. As in Figure 4, graphs b), c) and d) show not only the data
points, but also the best fit to these data points using the relations
\ref{freqVdW}-\ref{CapacityFrequency}. The error to the best fit is
represented as smaller points around the dashed horizontal line. In this case,
no amplification factor has been applied to the errors.
Left graphs: Curves calculated from the force \textit{interaction image}: a)
(mean) \textit{FvsD} curve, as in Figure 3 forward and backward cycle are
shown. The thicker points correspond to the calculated mean force while the
two lines below and over these data points show the oscillation amplitude
calculated from the amplitude \textit{interaction image}. Note that, in
contrast to figure 3, only the "true non-contact regime" (region I in figure
3)\ and the continuous contact regime (region IV in figure 3) are measured due
to the small oscillation amplitude. c) first derivative of the tip-sample
capacitance vs. distance.
Right graphs: Curves calculated from the frequency \textit{interaction image}:
b) (Van der Waals) frequency vs. distance curve and d) second derivative of
the tip-sample capacitance vs. distance. Resonance frequency: 68.80 kHz, force
constant 2N/m, oscillation amplitude 1nm.
\begin{center}
\bigskip%
\raisebox{-0cm}{\includegraphics[
natheight=14.579900in,
natwidth=10.199600in,
height=11.1808cm,
width=7.8419cm
]%
{Figs/Fig6_New.jpg}%
}%
\end{center}
Figure 6
Interaction vs. distance data calculated from three consecutive
\textquotedblleft interaction images\textquotedblright\ acquired with a Pt-tip
on a Au(111) surface. The data obtained from the three interaction images;
the\ first indentation correspond to circles, the second to diamonds and the
third one to triangles. Top image (a): (mean) \textit{FvsD} curve: data
corresponding to tip-sample approach and tip-sample retraction is shown. Note
that the data corresponding to tip-sample retraction, that is, the part
showing adhesion, is represented without filling to aid the eye. Central image
(b): Second derivative of tip-sample capacitance obtained from the frequency
\textit{interaction images}. Only approach data is shown, the solid circles
correspond to the first indentation, the gray diamonds to the second and the
white triangles to the third one. Lower image (c): Surface potential
calculated from the frequency \textit{interaction images}. Again only approach
data is shown and the coding for the different indentations is the same as for
graph (b). Resonance frequency: 68.80 kHz, force constant 2N/m, oscillation
amplitude 1nm.
\section{References}
|
3,212,635,537,944 | arxiv | \section{Introduction}
\label{sec:intro}
BRST symmetry \cite{brst} is by now a familiar concept associated to
to the quantization of any gauge theory. It reflects in an deep
way the gauge symmetry at the quantum level. For QCD in a covariant
gauge there is a standard form for the BRST transformations
\begin{eqnarray}
\label{brst-standard}
\delta A^a_\mu &=& D_\mu c^a \nonumber \\
\delta c^a &=& - \frac{1}{2} g f^{abc} c^b c^c \nonumber \\
\delta \overline{c}^a &=& - \frac{i}{\xi} \partial_\mu A^{\mu a} \nonumber \\
\delta \psi &=& - g c^a \lambda^a \psi
\end{eqnarray}
This form however is not unique. In fact several new symmetries of the
quantum action of QED in Feynman gauge have been reported
\cite{lm,tang,yang}. It has
been pointed out however that they are just BRST transformations in
non standard form \cite{mine}. In this paper we will elaborate on this
point to extended it to the non-abelian case. We will show how the
search for non standard BRST transformations, and by this we mean non
local and/or not manifestly covariant BRST transformations, can be
performed in a systematic way.
A reason for the existence of different forms of the BRST
transformations is the fact that there is a set of changes of
variables, not
necessarily local, that leaves the ghost action
invariant. There is also some freedom in the canonical
formalism. For example, in the Batalin-Fradkin-Vilkovisky (BFV) formalism
\cite{bfv} we usually perform the path integration over the ghosts momenta
to arrive at the transformations Eqs.(\ref{brst-standard}). We could instead
perform the integration over the ghosts themselves and leave the ghost
momenta in the action. The resulting action is local after some
changes of variables but the BRST transformations are in general
non-local and not manifestly covariant.
In Section \ref{sec:1} we make a brief presentation of the BFV
formalism applied to QCD to set up our conventions and set the stage
for the next section. Then in Section \ref{sec:2} we derive the BRST
transformations when we perform the alternative ghost path integration
mentioned above. We show that there are two sets of BRST transformations,
the usual one which is covariant and local Eqs.(\ref{brst-standard})
and another one which is not manifestly covariant and non local.
In the next section we consider changes of ghost variables
which leave the action and the path integral measure invariant. We
particularize to the abelian case to avoid the unnecessary algebraic
complications of the non-abelian structure. We show in particular that
there exist changes of ghost variables which lead to BRST and
anti-BRST transformations which do not anticommute. In the BFV
formalism these changes of variables correspond to canonical
transformations in the ghost sector. At last in Section
\ref{sec:end} we make some final comments.
\section{Resum\'e of the BFV Formalism for QCD}
\label{sec:1}
We will present in a very summarized way how the usual BRST transformations
are obtained in BFV formalism \cite{bfv}.
The first step is to find out the constraint structure of QCD. After
that we introduce the ghosts and their momenta to build up the BRST
charge $Q$.
The quantum QCD action is then\cite{conv}
\begin{equation}
\label{action0}
S_q = \int d^4 x \,\,\, ( \Pi^{ia} \dot A^a_i + \Pi^{0a} \dot A^a_0 +
i \overline \psi \gamma^0 \dot \psi + \dot{\cal P}^a \overline{c}^a +
\dot{c}^a \overline{\cal P}^a - H_c - \{ \Psi, Q \} )
\end{equation}
where $\Pi^{\mu a}$ are the momenta conjugated to $A^a_\mu$; ${\cal
P}^a, \overline{\cal P}^a$ are the ghost momenta conjugated to the
ghosts $\overline{c}^a, c^a$; $H_c$ is the canonical QCD Hamiltonian and
$\Psi$ is the gauge fixing fermion. The BRST charge is given by
\begin{equation}
\label{Q}
Q = \int d^3 x \,\,\, [ ( D_i F^{i0 a} + i g J^a_0 ) c^a + \frac{1}{2}
f^{abc} \overline{\cal P}^a c^b c^c - i {\cal P}^a \Pi^{0 a} ]
\end{equation}
The main assertion of the BFV formalism is that the
path integral $Z = \int D[\phi] \,\,\, \exp i S_q $ is independent of the gauge
fixing fermion. Here $D[ \phi ]$ is the usual Liouville measure over all
fields and ghosts.
Proper choices of $\Psi$ allows to recover the
usual gauge conditions. Covariant gauges are implemented by the
choice
\begin{equation}
\label{psi}
\Psi = \int d^3 x \,\,\, [ i \overline{c}^a (
\frac{1}{2} \xi \Pi^{0a} + \partial_i A^{ia} ) + \overline{\cal P}^a
A^a_0 ]
\end{equation}
For $\xi = 1$ we get the
Feynman gauge, $\xi = 0$ the Landau gauge and $\xi \rightarrow
\infty$ the unitary gauge. By performing the
functional integration over the momenta $\Pi^{\mu a}$ we arrive at the
effective action $S$
\begin{eqnarray}
\label{action1}
S &=& S_{QCD} + S_{gf} + S_{gh} \nonumber \\
S_{QCD} &=& \int d^4x \,\,\, [ - \frac{1}{4} F^a_{\mu \nu} F^{\mu \nu a} +
\overline{\psi} ( i \gamma^\mu D_\mu - m ) \psi ] \nonumber \\
S_{gf} &=& - \int d^4x \,\,\, \frac{1}{2 \xi} ( \partial_\mu A^{\mu a})
( \partial_\nu A^{\nu a} ) \nonumber \\
S_{gh} &=& \int d^4x \,\,\, ( i \overline{c}^a \partial_i D^i c^a +
\dot{\cal P}^a \overline{c}^a - \overline{\cal P}^a D_0 c^a +
i \overline{\cal P}^a {\cal P}^a )
\end{eqnarray}
where $S_{QCD}$ is the classical QCD action, $S_{gf}$ is the gauge
fixing action and $S_{gh}$ is the ghost action. The effective action
$S$ is then
invariant under the BRST
transformations generated
by $Q$ which have the form
\begin{eqnarray}
\label{brst1}
&& \delta A^a_i = D_i c^a, \,\,\,\,\,\, \delta A^a_0 = i {\cal P}^a
\nonumber \\
&& \delta c^a = - \frac{1}{2} g f^{abc} c^b c^c, \,\,\,\,\,\, \delta
\overline{c}^a = - \frac{i}{\xi} \partial_\mu A^{\mu a} \nonumber \\
&& \delta {\cal P}^a = 0, \,\,\,\,\,\, \delta \overline{\cal P}^a =
D_i F^{0i a} -
i g J^a_0 - g f^{abc} \overline{\cal P}^b c^c \nonumber \\
&& \delta \psi = - g c^a \lambda^a \psi
\end{eqnarray}
where $J^a_0 = \overline{\psi} \lambda^a \gamma_0 \psi$ is the time
component of the current.
At this stage we usually perform the path integration over ${\cal P}^a$ and
$\overline{\cal P}^a$. Integrating over ${\cal P}^a$ gives a delta
functional $\delta( i \overline{\cal P}^a + \dot{\overline{c}}^a)$
and integrating over $\overline{\cal P}^a$ allow us to replace
$\overline{\cal P}^a$ by $i \dot{\overline{c}}^a$.
Then $S_{gh}$ in Eqs.(\ref{action1}) gets its usual Faddeev-Popov form
\begin{equation}
\label{action-gh1}
S_{gh} = \int d^4x \,\,\,i \overline{c}^a \partial^\mu D_\mu c^a
\end{equation}
and we get the usual BRST transformations
Eqs.(\ref{brst-standard}). Notice
the importance
of this last two integrals. We restore manifest covariance in
the BRST transformations Eqs.(\ref{brst1})
and we get a local and covariant ghost action
Eq.(\ref{action-gh1}). In the next section we
will describe the
results when we perform the integrations on other pair of ghost variables.
As usual the BRST transformations are nilpotent
on-shell. The nilpotency fails only on
$\overline{c}^a$ and is proportional to the $c^a$ field
equation. Usually this can be overcome by the introduction of an
auxiliary field. In the BFV formalism, however, it corresponds to the
situation where it is not performed the path integration over
$\Pi^a_0$. If we call $\lambda^a = \Pi^a_0 + \frac{1}{\xi}
\partial_\mu A^{\mu a}$ then the gauge fixed action has one more term
and becomes
\begin{equation}
\label{action-aux}
S_{gf} = \int d^4 x \,\,\, [ - \frac{1}{2 \xi} (\partial_\mu A^{\mu
a})^2 + \frac{1}{2} \xi \lambda^a \lambda^a ]
\end{equation}
and the BRST transformations are the same as Eqs.(\ref{brst-standard})
before except for
$\overline{c}^a$. The new BRST transformations are then
\begin{eqnarray}
\label{brst-aux}
&& \delta \overline{c}^a = - \frac{i}{\xi} \partial_\mu A^{\mu a} + i
\lambda^a \nonumber \\
&& \delta \lambda^a = \frac{1}{\xi} \partial^\mu D_\mu c^a
\end{eqnarray}
In Section \ref{sec:3} we will consider the case of QED with this
extra field.
Besides the BRST transformations the quantum action is also invariant under
anti-BRST transformations. In the BFV formalism the anti-BRST charge
can be obtained from the BRST charge by interchanging ghosts by
anti-ghosts in such a way that both charges anticommute. We then find
that the anti-BRST transformations in a covariant gauge are
\begin{eqnarray}
\label{antibrst1}
\overline{\delta} A^a_\mu &=& D_\mu \overline{c}^a \nonumber \\
\overline{\delta} c^a &=& - \frac{i}{\xi} \partial_\mu A^{\mu a} \nonumber \\
\overline{\delta} \overline{c}^a &=& \frac{1}{2} g f^{abc}
\overline{c}^b \overline{c}^c \nonumber \\
\overline{\delta} \psi &=& - g \overline{c}^a \lambda^a \psi
\end{eqnarray}
It should be noticed that the anti-BRST symmetry is not a new
symmetry. The anti-BRST charge has the same information content as the
BRST charge and we could use anyone to generate physical states or
to obtain Ward identities. Often both are used.
\section{Non-local BRST transformations}
\label{sec:2}
Consider the BFV formalism in the previous section up to the
point where the integration over $\Pi^{\mu a}$ was performed and
Eqs.(\ref{action1},\ref{brst1}) were obtained.
We will now perform the integration over the ghost fields instead of
their momenta. Integrating over $\overline{c}^a$ gives $\delta( i
\partial_i D^i c^a - \dot{\cal P}^a) = det( \partial_i D^i ) \,\,\,
\delta( i c^a - \frac{1}{\partial_i D^i} \dot{\cal P}^a)$. Integrating
now over $c^a$ we can replace $c^a = - i \frac{1}{\partial_i D^i}
\dot{\cal P}^a$. Then the ghost action in Eqs.(\ref{action1}) becomes
\begin{equation}
\label{action2}
S_{gh} = \int d^4x \,\,\, ( i \overline{\cal P}^a D_0
\frac{1}{\partial_i D^i} \dot{\cal P}^a + i \overline{\cal P}^a {\cal
P}^a )
\end{equation}
Notice the appearance of the non-local term in the ghost action as a result
of this unusual integration. Notice also that the path integral
measure has now an extra term $det( \partial_i D^i )$ which should be
taken into account. We can overcome these two troubles by making
judicious changes of variables. First perform the change of variables
$\dot{\cal P}^a \rightarrow i c^a, \overline{\cal P}^a \rightarrow
i \dot{\overline{c}}^a$ whose Jacobian is one. Then perform a second
change of variables $c^a \rightarrow - \partial_i D^i c^a$ whose
Jacobian is $det^{-1}(\partial_i D^i)$. As a result the contribution
from the Jacobian to the path integral measure
cancels out the contribution from the ghost integration. Also the
non-local ghost
action Eq.(\ref{action2}) becomes local and it takes the usual
Faddeev-Popov form
Eq.(\ref{action-gh1}). The BRST transformations can now be obtained from
Eqs.(\ref{brst1}). They are
\begin{eqnarray}
\label{brst3}
&& \delta A^a_i = D_i c^a, \,\,\,\,\,\, \delta A^a_0 = - \frac{1}{\partial_0}
\partial_j D^j c^a \nonumber \\
&& \delta c^a = - \frac{1}{2} g f^{abc} c^b c^c \nonumber \\
&& \delta \overline{c}^a = - i \frac{1}{\partial_0} D_i F^{0i a}
- g f^{abc} \frac{1}{\partial_0} ( \dot{\overline{c}^b} c^c ) - g
\frac{1}{\partial_0} J^a_0 \nonumber \\
&& \delta \psi = - g c^a \lambda^a \psi
\end{eqnarray}
We end up then with a local ghost action and a set of non-local and
not manifestly covariant BRST transformations by performing the integration
over
the ghost fields instead of their momenta. The transformations
Eqs.(\ref{brst3}) leave the effective action invariant and are nilpotent
as any good BRST transformation.
There are two other possibilities to perform the ghost integrations.
If we perform the path integration over $\overline{c}^a$ and
${\cal P}^a$ we obtain the same result as before after proper changes
of variables. If we integrate over $\overline{\cal P}^a$ and $ c^a$
instead
we get the usual BRST transformations Eqs.(\ref{brst-standard})
also after proper change of variables.
Then in the BFV formalism we can arrive at a local ghost action, the
Faddeev-Popov action, and
two standard sets of BRST
transformations which can be either covariant and local
Eqs.(\ref{brst-standard}) or not manifestly covariant and non-local
Eqs.(\ref{brst3}). It should also be noticed that both sets of BRST
transformations reduce to each other on-shell. If we use the $c^a$
and $A^a_0$ field equations we can turn the non-local BRST
transformations Eqs.(\ref{brst3}) into the local ones
Eqs.(\ref{brst-standard}).
\section{Changes of Variables in the Ghost Action}
\label{sec:3}
{}From now on let us consider just the abelian case for the sake of
simplicity. The ghost action is then $S_{gh} = \int d^4x \,\,\, i
\overline{c} \Box c$. This action allows a huge freedom to
perform changes of variables which leave the path integration measure
and the action itself
invariant.
Let us consider first some cases with the non-local form of the BRST
transformations.
If we perform the following change of variables $c \rightarrow
i \frac{\partial_0}{ \nabla^2 } c $ and $ \overline{c} \rightarrow i
\frac{\nabla^2 }{\partial_0} \overline{c}$ whose Jacobian is one,
then Eqs.(\ref{brst3}) in the abelian case reduce to
\begin{eqnarray}
\label{brst-lm}
&&\delta A_i = i \frac{\partial_i}{\nabla^2} \dot{ c}, \,\,\,\,\,\,
\delta A_0 = i c \nonumber \\
&&\delta c = 0, \,\,\,\,\,\, \delta \overline{c} = \frac{1}{\nabla^2}
\partial_i \dot{A}_i - A_0 + i g \frac{1}{\nabla^2} J_0 \nonumber \\
&&\delta \psi = - i g (\frac{1}{\nabla^2}\dot{ c}) \psi
\end{eqnarray}
These are precisely the transformations found in Ref.\cite{lm}. It
has also been pointed out that they can be found by a canonical
transformation in the ghost sector before any integration is performed
\cite{gaete}.
Another change of variables $c \rightarrow - \dot{c} $ and $
\dot{\overline{c}} \rightarrow - \overline{c}$ (which also has
Jacobian one) reduces the abelian form of Eqs.(\ref{brst3}) to
\begin{eqnarray}
\label{brst-yang}
&& \delta A_i = - \partial_i \dot{c}, \,\,\,\,\, \delta A_0 = -
\nabla^2 c \nonumber \\
&&\delta c = 0, \,\,\,\,\,\, \delta \overline{c} = i \partial_i F^{0i}
+ g J_0 \nonumber \\
&&\delta \psi = g \dot{c} \psi
\end{eqnarray}
We then get the local but not manifestly covariant transformations found in
Ref.\cite{yang}.
Still another change of variables $c \rightarrow
\frac{\partial_0}{\nabla^2} c $ and $ \overline{c} \rightarrow
\frac{\nabla^2}{\partial_0} \overline{c}$ this time in the abelian
form of the usual BRST
transformations Eqs.(\ref{brst-standard}) produces
\begin{eqnarray}
\label{brst-fink}
&& \delta A_\mu = D_\mu \frac{1}{\nabla^2} \dot{c} \nonumber \\
&& \delta c = 0, \,\,\, \delta \overline{c} = -\frac{i}{\xi}
\frac{1}{\nabla^2} \partial_\mu \dot{A}^\mu \nonumber \\
&& \delta \psi = - g ( \frac{1}{\nabla^2} \dot{c} ) \psi
\end{eqnarray}
These are the non-local transformations found in
Ref.\cite{tang}.
This procedure of performing changes of variables on the ghost fields
can be easily generalized. Assume that $F, G, \dots$ are operators
which possess an adjoint $F^t, G^t, \dots$ in the sense that
\begin{equation}
\label{adj}
\int dx \,\,\, \phi \,\, F[ \psi ] = \int dx \,\,\, F^t[ \phi ] \,\,
\psi
\end{equation}
Notice that the effective action in the abelian case defines a bilinear
metric $\int dx \,\, \phi \,\, \psi$. Assume also that these operators
are field independent so that they
commute with $\partial_\mu$.
Let us consider also the extra field which makes the BRST
transformations nilpotent off-shell as mentioned at the end of
Section \ref{sec:1}. Consider now the change of variables
\begin{eqnarray}
\label{F}
&& A_\mu \rightarrow A_\mu, \,\,\,\,\,\, \lambda \rightarrow \lambda,
\,\,\,\,\,\, \psi \rightarrow \psi \nonumber \\
&& c \rightarrow F[c],\,\,\, \overline{c} \rightarrow (F^{-1})^t [
\overline{c} ]
\end{eqnarray}
whose Jacobian
is one. Let us call this change of variables a $F$
transformation. Then the abelian ghost action remains
invariant under this $F$ transformation and the local abelian BRST
transformations Eqs.(\ref{brst-standard}, \ref{brst-aux})
become
\begin{eqnarray}
\label{brst-f}
&& \delta A_\mu = \partial_\mu F[c], \,\,\,\,\,\, \delta \lambda =
\frac{1}{\xi} \Box F[c] \nonumber \\
&& \delta c = 0, \,\,\,\,\,\, \delta \overline{c} = F^t[ - \frac{i}{\xi}
\partial^\mu A_\mu + i \lambda ] \nonumber \\
&& \delta \psi = g \psi F[c]
\end{eqnarray}
and clearly generalizes Eqs.(\ref{brst-fink}).
If instead we perform the change of variables in the
abelian non-local BRST transformations Eqs.(\ref{brst3}) we get the
generalization of the BRST transformations Eqs.(\ref{brst-lm},
\ref{brst-yang}).
We can now consider the following
situation. Consider a $F$ transformation and the resulting BRST
transformations Eqs.(\ref{brst-f}). Consider a $G$ transformation
defined as
\begin{eqnarray}
\label{G}
&& A_\mu \rightarrow A_\mu, \,\,\,\,\,\, \lambda \rightarrow
\lambda,
\,\,\,\,\,\, \psi \rightarrow \psi \nonumber \\
&& c \rightarrow ( G^{-1} )^t[ c ], \,\,\,\,\, \overline{c} \rightarrow
G [ \overline{c} ]
\end{eqnarray}
and the resulting {\bf anti}-BRST transformations, that is,
\begin{eqnarray}
\label{brst-g}
&& \overline{\delta} A_\mu = \partial_\mu G [ c ], \,\,\,\,\,\,
\overline{\delta} \lambda = \frac{1}{\xi} \Box G [ \overline{c} ]
\nonumber \\
&& \overline{\delta} c = G [ - \frac{i}{\xi}
\partial^\mu A_\mu + i \lambda ] \,\,\,\,\,\, \overline{\delta}
\overline{c} = 0 \nonumber \\
&& \overline{\delta} \psi = g \psi G [ \overline{c} ]
\end{eqnarray}
This $G$ transformation also leaves the effective action invariant.
We then end up with a set of $F$ transformed BRST transformations
Eqs.(\ref{brst-f})
and a set of $G$ transformed anti-BRST transformations
Eqs.(\ref{brst-g}) and the abelian effective action.
It is clear that the $F$ transformed BRST and the $G$ transformed
anti-BRST
transformations are nilpotent. However the anticommutator of a $F$
transformed BRST
transformation Eqs.(\ref{brst-f}) with the $G$ transformed {\bf
anti}-BRST transformations Eqs.(\ref{brst-g}) does not vanish. If
we denote this anticommutator by $\Delta$ we obtain
\begin{eqnarray}
\label{delta}
&& \Delta A_\mu = i \partial_\mu (G F^t - F G^t)[ \lambda - \frac{1}{\xi}
\partial^\nu A_\nu ] \nonumber \\
&& \Delta c = \Delta \overline{c} = 0 \nonumber \\
&& \Delta \lambda = i \frac{1}{\xi} \Box ( G F^t - F G^t )[ \lambda -
\frac{1}{\xi} \partial^\mu A_\mu ] \nonumber \\
&& \Delta \psi = - i g \psi ( G F^t - F G^t ) [ \lambda - \frac{1}{\xi}
\partial^\mu A_\mu ]
\end{eqnarray}
Notice that these transformations have ghost number zero. They do not
act on the ghosts and behave as gauge transformations on $A_\mu$ and
$\psi$.
We easily verify that Eqs.(\ref{delta}) are a symmetry of the
effective action as well. We also find that they correspond to a new
change of
variables defined by
\begin{eqnarray}
\label{h-transf}
&& A_\mu \rightarrow A_\mu + i \partial_\mu (H - H^t)[ \lambda -
\frac{1}{\xi} \partial^\nu A_\nu ] \nonumber \\
&& c \rightarrow c, \,\,\,\,\,\, \overline{c} \rightarrow
\overline{c} \nonumber \\
&& \lambda \rightarrow \lambda + \frac{i}{\xi} \Box ( H - H^t ) [
\lambda - \frac{1}{\xi} \partial^\nu A_\nu ] \nonumber \\
&& \psi \rightarrow \psi - i g \psi ( H - H^t ) [\lambda - \frac{1}{\xi}
\partial^\nu A_\nu ]
\end{eqnarray}
Let us call this change of variables an $H$ transformation. They
correspond to the transformations Eqs.(\ref{delta}) with $H = G F^t -
F G^t$. This change of variables has Jacobian equals to one and also
leave the effective action invariant.
Then the freedom to perform changes of variables which leave
the ghost action and the path integral measure invariant reflects itself in
the BRST symmetry by allowing another set of transformations of the type
Eqs.(\ref{h-transf}). As a consequence the
anticommutator of the BRST and anti-BRST transformations does not need
to vanish and is proportional to an $H$ transformation Eqs.(\ref{h-transf}).
This can also be seen when we build the BRST charge in the BFV
formalism. In the abelian
case the BRST charge and the anti-BRST charge are
\begin{eqnarray}
\label{q}
Q &=& \int d^3 x \,\,\, ( \partial^i A_i \,\, c - i \Pi_0 {\cal P} )
\nonumber \\
\overline{Q} &=& \int d^3x \,\, ( \partial^i A_i \,\,\,\,
\overline{c} + i \Pi_0
\overline{{\cal P}} )
\end{eqnarray}
respectively. It is easily verified that they anticommute.
Now we can build the $F$ transformed BRST charge where the $F$
transformation on the ghost momenta is
defined as ${\cal P} \rightarrow F[ {\cal P} ] $ and $ \overline{\cal P}
\rightarrow ( F^{-1} )^t [ \overline{\cal P} ]$. We easily verify that
the $F$ transformation is a canonical transformation.
Now because the $F$ and $G$ transformations leave the effective action
invariant we could build the BRST charge with the $F$ transformed
ghosts and the anti-BRST charge with the $G$ transformed ghosts (the
$G$ transformation on
$\cal{P}$ and $ \overline{\cal P}$ being defined in a similar way to
the $F$ transformation). Then
the anticommutator of the BRST and anti-BRST charges no longer
vanishes and is proportional to $F G^t - F^t G$. We then see that in
the BFV formalism the changes in the ghosts variables we have been working
with correspond to canonical transformations on the ghosts. The
non vanishing of the
anticommutator of the BRST and anti-BRST transformations is due to
the fact that the charges are build with ghosts which have been
subject to different canonical transformations.
Since the effective action is invariant under BRST and anti-BRST
transformations it is also invariant under any linear combination of
them. If we take the original transformations the combined
transformation is nilpotent since each
of the original transformations is by itself nilpotent as is their
anticommutator. We could however consider the sum of the $F$
transformed BRST with
the $G$ transformed anti-BRST
transformations. We then have a set of
transformations which do not have a
well defined ghost number, leave the
effective action invariant and are not nilpotent because of
Eqs.(\ref{delta}).
This is the origin of the non-nilpotent symmetry found in
Ref.\cite{tang}.
\section{Conclusions}
\label{sec:end}
We have shown that the effective action of QCD in a covariant gauge
is invariant under non-local and even
not manifestly covariant BRST transformations either as a result of
the BFV path
integral formulation or as a change of ghost variables which leave the
effective action and the path integral measure invariant. They just
reflect the
basic BRST symmetry in different forms. They are symmetries of
the full interacting quantum theory (in the absence of anomalies) but
do not entail any new
Ward identity besides those obtained from the usual BRST
transformations. It also shows the power of the Hamiltonian
formalism. Although the non standard form of the BRST transformations
can be found in the Lagrangian formalism its origin remains obscure
and its dependence on other known symmetries can not be traced. In the
BFV formalism all these issues can be clearly analysed.
It is worth remarking that it is possible for a gauge fixed action to
have further symmetries besides the BRST symmetry. One well known example
is the non-abelian Chern-Simons theory in Landau gauge in $2 + 1$
dimensions. It has a rigid vector supersymmetry \cite{birm} which is
independent of the BRST symmetry. In fact this vector
supersymmetry and the BRST symmetry
are part of a more general algebra which is a contraction of the
exceptional Lie superalgebra $D ( 2 | 1; \alpha )$ \cite{me}.
That this vector supersymmetry is a truly new symmetry manifests
itself in the existence of a new Ward identity which relates the
gauge and ghost inverse propagators \cite{birm}.
Our discussion on changes of ghost variables in Section \ref{sec:3}
were done only for the
abelian case and in the situation where the ghost transformations do
not
involve any field. In trying to consider a field dependent $F$
transformation we were led to very complicated expressions. Also the
non-abelian case became rather involved. We still miss a suitable
formalism to handle such situations.
\acknowledgments
The author wishes to thank P.van Nieuwenhuizen for conversations. This
work was partially supported by CNPq.
|
3,212,635,537,945 | arxiv | \section{Introduction}
Measurement-based quantum computation (MBQC) offers a new platform of quantum information (QI) processing. Quantum algorithms are performed by sequential single-qubit measurements in multipartite entangled states initially (e.g., cluster states \cite{Cluster}) instead of massive controls of individual qubits during the whole information processing \cite{MBQC1,MBQC2}. This advantage is, however, only beneficial for QI processing if the specific multipartite entangled state can be initially well-prepared and the capability of fast and precise single-qubit measurements are viable. For example, a two-qubit cluster state is the simplest resource state for MBQC given by $\ket{2CS}_{AB} = (\ket{0}_A \ket{+}_B + \ket{1}_A \ket{-}_B)/\sqrt{2}$ with $\ket{ \pm} = (\ket{0} \pm \ket{1})/\sqrt{2}$.
If the operation angle $\theta$ is chosen for the measurement basis vectors in qubit $A$, $\ket{ \pm \theta} = (\ket{0} \pm e^{-i \theta} \ket{1})/\sqrt{2}$, the resultant state in $B$ after the measurement $\ket{\pm \theta} \bra{\pm \theta}$ on $A$ becomes a single-qubit operated state such as ${}_{A} \bra{ \pm \theta} 2CS \rangle_{AB} \propto \hat{e}^{\pm} e^{\pm i {\theta \over 2}} H R^z (\theta) \ket{+}_{B}$, for Hadamard gate $H= ( X + Z )/\sqrt{2}$, $z$-axis rotation operator $R^z(\theta) = e^{-i{\theta \over 2} } \ket{0}\bra{0} + e^{i{\theta \over 2} } \ket{1}\bra{1} $ and $\hat{e}^{\pm}= \{ \leavevmode\hbox{\small1\kern-2.8pt\normalsize1}, X\}$ with Pauli operators $X, Z$. Thus, it is interpreted as the single-qubit gate $H R^z (\theta)$ is performed on $\ket{+}$ by the measurement of qubit $A$ with the chosen angle $\theta$ on $\ket{2CS}_{AB}$. It is therefore of essence to demonstrate efficiently building such a useful entangled resource state and performing single-qubit measurements on the resource state for practical MBQC.
The MBQC in continuous variables (CVs) has been firstly well developed in quantum optics since such CV cluster states are achievable using traveling squeezed states through optical parametric amplifiers \cite{CV-MBQC1, Sam_Braunstein1, Sam_Braunstein2, Sam_Braunstein3}. For example, the recent development of creating one-dimensional (1D) and 2D CV cluster states has been demonstrated in quantum optics using quantum memory and in time/frequency domain \cite{CV-MBQC_ex1,CV-MBQC_ex2}. In these methods, a phase-space translation operator is in general used for single-qubit gates while a two-qubit controlled-Z gate is implemented in a sequence of beam-splitters \cite{Furusawa2011}. Toward fault-tolerant CV MBQC using this approach, a scheme of high squeezing photons (20.5 dB) has been required to reach the error tolerance threshold with $10^{-6}$ through concatenated codes \cite{FT_CV_MBQC}, and is very challenging with the state-of-the-art experiments in quantum optics. Recently, an alternative method of creating four-qubit CV cluster states has been suggested in a circuit quantum electrodynamics (circuit-QED \cite{BlaisPRA}) system \cite{CV-cluster-CQED}.
One of the advantages of using CVs is that the optical cluster states are built in a deterministic manner and can in principle store information in infinite dimension \cite{GKP, Jeong02-1, CVentanglement, New_archive} while alternative optical methods of creating discrete-variable cluster states have been in general generated in polarization or spatial modes probabilistically by using parametric down conversion \cite{DVPhotonicMBQC}. We will in particular use a specific logical qubit encoded in generalised Schr\"odinger cat states, which are the superposition of phase-rotated coherent states \cite{Schro01}.
It is known that the specific CV-qudit states can be used for the error-correctable QI unit against particle-loss and have been successfully demonstrated in circuit-QED for practical quantum memory \cite{qcMAP13,NewYale16,NewYale17,Yale_FTcat}. This circuit-QED approach could thus be advantageous for error-correctable quantum computing equipped with photon-loss resilience in the CV-qudit code \cite{catcode, Yale-QECC}.
We here propose a novel circuit-QED scheme of performing logical qubit gates and the desired outcome is achieved by cavity measurements from a tripartite CV-qudit cluster state as a single-qubit operated state in the CV-qudit code. Because it might be concerned how to initially implement the complex multipartite cluster state by the manual controls of cavity states, we first suggest a circuit-QED architecture capable of building the target CV-qudit entangled state using an induced cross-Kerr interaction, which naturally provides an entangling gate between neighbouring cavity qudits. It is known that one can in principle engineer cross-Kerr interaction in the multiple-cavity architecture with tunable self-Kerr interaction \cite{Matt17}. Then, after we define the CV-qudit and its cluster states, we present a new protocol for a logical single-qubit gate in MBQC using three specific circuit-QED techniques such as a coherent-state measurement, parity measurement, and a selective number-dependent arbitrary phase (SNAP) gate. All these techniques have been well developed and demonstrated in theory and experiment \cite{1-photon_Kerr, EranPRB, SNAPgate}. We finally examine the cross-Kerr entangling scheme of builiding two CV-qudit cluster states with an intermediary superconducting qubit and this circuit-QED architecture would enable to investigate not only QI processing but also more broader sciences including many-body physics \cite{manybody} and quantum chemistry \cite{quan_chem} in the future.
\section{Results}
\begin{figure}[t]
\includegraphics[width=9cm,trim=3cm 5.5cm 3cm 0.5cm]{NewNewFig1a.pdf}
\includegraphics[width=7.5cm,trim=0.5cm 6cm 0cm 0.5cm]{NewFig1b.pdf}
\caption{Schematics of logical MBQC in a circuit-QED architecture.
(Left) Three cavities ($A, B, C$) have the intersected superconducting qubits $M_1$ and $M_2$ used for inducing the Kerr interactions between cavities. When a 3-qudit logical cluster state is built in the cavities by cross-Ker interaction ($K_{ij}$), logical MBQC is performed by a sequential measurement of each cavity.
The colours of transmon\rq{}s energy states represent the anharmonicity of the energy levels in a transmon. (Right) the tunability of Kerr effects between the neighbouring cavities provided with the help of tunable on-site superconducting qubits and an extra (tunable) intermediary qubit in the same architecture (the details are shown in \cite{Joo_SR,Matt17}). For example, the self-Kerr effects can be only reduced by shifting energy levels in on-site qubits at point (a) and the simultaneous entangling gates are performed by cross-Kerr $K_{ij}$ between (a) and (b). From (b) to (c), the cavities are uncoupled and the sequential measurements of each cavity are performed for MBQC.
\label{fig:02} }
\end{figure}
\subsection{Circuit-QED architecture for entangled cavity states}
The platform of superconducting circuits has been rapidly developed for QI processing over two decades \cite{Nakanmura}. The artificial qubits are intrinsically scalable and manufacturable in the forms of different qubit types with precise control of desired parameters \cite{transmon, flux}. In experiment, one utilises only superconducting qubits (mainly transmon qubits \cite{transmon}) for QI unit while it has also been successfully shown that a coupled system of superconducting qubits and 3D cavities offers excellent capability of creating quantum cavity states through the nonilnearity of an intermediary superconducting qubit, e.g., deterministic generation of Schr\"odinger cat states and entangling CV states inside the cavities \cite{Yale_teleport, Yale_big_cat}.
As shown in the left figure of Fig.~\ref{fig:02}, we consider a circuit-QED architecture for creating entangled microwave states and the neighboring cavities are connected with each other via a middle transmon qubit $M_i$ ($i=1,2$) enabling to entangle cavity states. This approach shows a unique advantage that a massive 1D CV-qudit cluster state can be built in one step as the key resource state for MBQC. Since two cavities are simple harmonic oscillators, a superconducting qubit inserted in between two cavities brings induced Kerr effects on the joint cavity modes. For an ideal case, it is assumed that two neighboring cavities are only coupled by a cross-Kerr interaction, which is induced by the intermediary superconducting qubit.
In a real circuit-QED setup, this architecture might cause unwanted nonlinear effects over the cavities (e.g., self-Kerr distortion effects and non-identical cross-Kerr effects). In general, the cavity self-Kerr effect makes the amount of distortion in the cavity state and could prevent building ideal CV-qudit entangled states and to measure the cavity qubit accurately at an appropriate time.
For example, let us consider the Jaynes-Cummings (JC) Hamiltonian for two cavities with an intermediary transmon is given by
\begin{eqnarray}
\label{total_Ham01}
\hspace{-1cm} \hat{H}^{JC}_{ABM} &&= \sum_{c=A,B,M} \omega_c \hat{a}^{\dag}_{c} \hat{a}_{c} + {K_M} \hat{a}^{\dag}_{M} \hat{a}_{M} \hat{a}^{\dag}_{M} \hat{a}_{M} + \sum_{c=A,B} \lambda^{M}_{c} ( \hat{a}^{\dag}_{M} \hat{a}_{c} + \hat{a}_{M} \hat{a}^{\dag}_{c}),
\end{eqnarray}
with creation operator $\hat{a}^{\dag}$ and $\hbar =1$.
It is experimentally confirmed that self- and cross-Kerr effects exist in the cavities coupled with a superconducting qubit \cite{1-photon_Kerr,SNAPgate} and theoretically the adiabatic elimination theory can show the existence of these effects (upto the fourth order in the JC Hamiltonian \cite{Matt17,AdiabaticRef}). We will examine the validity of the induced cross-Kerr interaction in this architecture to build a two CV-qudit cluster state in Section \ref{Implement_s02}.
Fortunately, a Kerr-engineering scheme has been recently proposed to amend self- and cross-Kerr effects in a qubit-cavity array and is applicable for creating a desired 1D CV-qudit entangled state with the help of extra tunable superconducting qubits in a similar architecture \cite{Matt17}. For example, suppose that a flux qubit is additionally attached on each cavity. In Ref.~\cite{Matt17}, it is shown that the controls of energy levels of the flux qubit diminish the amount of self-Kerr interaction $K_j$ in each cavity, but the cross-Kerr interaction still survives between neighbouring cavities. As shown in the right figure of Fig.~\ref{fig:02}, two cavity states starts to be entangled with $K_j \approx 0$ during the period between $(a)$ and $(b)$. After the entangling period, the cross-Kerr interaction can be also reduced in a similar technique and the cavity states can be effectively decoupled for better performance of individual cavity measurements between $(b)$ and $(c)$ (see Fig.~5 in \cite{Matt17}). For logical MBQC, we need to perform a type of quantum-non-demolition (QND) measurements on each cavity and their details are addressed in Section \ref{Sec:operation}.
\subsection{Cat qudits}
\label{2-1}
We first introduce the definition of CV qudits (with $d=4$) written in the superposition of phase-encoded coherent states. The CV qudits are defined by
\begin{eqnarray}
\label{Simple_03-1}
\ket{{0}_4} && = M^{0}_{\alpha} \left( \ket{\alpha} + \ket{i \alpha} + \ket{-\alpha} + \ket{-i \alpha} \right) = \sum_{m=0}^{\infty} c_{0m} \ket{4m}, \\
\label{Simple_03-3}
\ket{{1}_4} && = M^{1}_{\alpha} \left( \ket{\alpha} - i \ket{i \alpha} - \ket{-\alpha}+ i \ket{-i \alpha} \right) = \sum_{m=0}^{\infty} c_{1m} \ket{4m+1}, \\
\label{Simple_03-4}
\ket{{2}_4} && = M^{2}_{\alpha} \left( \ket{\alpha} - \ket{i \alpha} + \ket{-\alpha} - \ket{-i \alpha} \right) = \sum_{m=0}^{\infty} c_{2m} \ket{4m+2}, \\
\label{Simple_03-2}
\ket{{3}_4} && = M^{3}_{\alpha} \left( \ket{\alpha} +i \ket{i \alpha} - \ket{-\alpha} - i \ket{-i \alpha} \right) = \sum_{m=0}^{\infty} c_{3m} \ket{4m+3},
\end{eqnarray}
where a coherent state with real values $\alpha$ and $\phi$ is $\ket{\alpha e^{i\phi}} = e^{-|\alpha|^2/2} \sum_{n=0}^{\infty} {\alpha^n e^{i \phi n} \over \sqrt{n!}} \ket{n}$ and $\ket{4m+j}$ is a Fock state with $4m+j$ photons ($M_{\alpha}$ as a normalisation factor).
Note that their complementary qudits are defined as $\ket{\tilde{0}_4} = \ket{\alpha}$, $\ket{\tilde{1}_4} = \ket{i \alpha}$, $\ket{\tilde{2}_4} = \ket{-\alpha}$, and $\ket{\tilde{3}_4} = \ket{-i \alpha}$ \cite{Kim15}.
The generalised Pauli operators for the qudits are defined by $\hat{Z}_4\ket{\tilde{k}_4} = \ket{\widetilde{(k+1)}_4}$ and $\hat{X}_4 \ket{{(k+1)}_4} = \ket{{k}_4}$. The qudit Pauli operators can be physically implemented by phase rotation $\hat{Z}_4 = e^{ i {\pi \over 2} (\hat{a}^{\dag} \hat{a}) }$ and photon addition $\hat{X}_4 \approx \hat{a}^{\dag} / \sqrt{ \langle \hat{a}^{\dag}\hat{a} \rangle }$ (or photon subtraction $\hat{a} / \sqrt{ \langle \hat{a}^{\dag}\hat{a} \rangle }$). Note that the normalisation coefficients $M^{i}_{\alpha}$ are approximately equal to 1/2 for $\alpha \ge 2$, which implies the validity of orthogonality in qudit $\ket{k_4}$ for QI unit ($k=1,2,3,4$). In other words, if the average photon number should be large enough to distinguish between coherent states, the qudits can be used for logical qubits against photon-loss errors in Section \ref{sec:LogicalQubit} \cite{catcode}.
\subsection{How to create ideal three CV-qudit cluster states}
We here show an mathematical description of how to build 1D CV-qudit cluster states with an ideal cross-Kerr interaction \cite{Sanders12,Sanders92}. The cross-Kerr interaction shows a natural way to entangle two coherent states (see the details in Section \ref{Sec:Entangle-two-cat}). For a three-cavity case, an initial state $\ket{\psi^{int}}_{ABC}$ is prepared in three cavities and a time-evolved state at time $t$ is given by
\begin{eqnarray}
\label{total_Ham00}
\ket{\psi (t)}_{ABC} = \exp \left( i \hat{H}_{ABC} \, t \right) \ket{\psi^{int}}_{ABC}.
\end{eqnarray}
The cross-Kerr Hamiltonian is ideally given in
$\hat{H}^{tot}_{ABC} = K_{AB} (\hat{a}^{\dag}_A \hat{a}_A) (\hat{a}^{\dag}_B \hat{a}_B) + K_{BC} (\hat{a}^{\dag}_B \hat{a}_B) (\hat{a}^{\dag}_C \hat{a}_C)$. With the assumption $K_{AB} = K_{BC}$ for simplicity, the three CV-qudit state at a quarter of the revival time is written in
\begin{eqnarray}
\label{3QuditCS01}
\hspace{-2cm} \ket{\psi^{ideal} ( {\tau_r / 4} ) }_{ABC} = \exp \left( i {\tau_r \over 4} \hat{H}^{tot}_{ABC} \right) \ket{\alpha}_{A} \ket{\alpha}_{B} \ket{\alpha}_{C} = {1 \over 2} \sum_{k=0}^3 \ket{\tilde{k}_4}_A \ket{{k}_4}_B \ket{\tilde{k}_4}_C.
\end{eqnarray}
It could be crucial to match the strength values of two cross-Kerr interactions between neighbouring cavities ($K_{AB} = K_{BC}$) to create the target state in Eq.~(\ref{3QuditCS01}). Otherwise, the cavity state becomes maximally entangled in $A$ and $B$ at a certain time but it does not in $B$ and $C$. In Ref.~\cite{Matt17}, a slight modification of the circuit-QED architecture has been investigated with additional superconducting qubits to control self- and cross-Kerr interactions independently. This modified architecture might thus be beneficial for building a multi-partite entangled state in many cavities at once toward practical MBQC.
\subsection{Three single-qudit gates in cavity states}
\label{Sec:operation}
For logical MBQC, three specific single CV-qudit operations are required in each cavity such as (1) coherent-state projection $\hat{P}^{Coh}$, (2) parity measurement $\hat{P}^{Par}$, and (3) SNAP phase gates. Note that all the gates have already been demonstrated in a qubit-cavity architecture experimentally.
In a dispersive regime of the JC Hamiltonian, which is defined by much smaller coupling strength than the difference between cavity and qubit frequencies, it is feasible to perform the projection measurement on Fock states in a cavity-transmon coupled system (see the details in Section \ref{sec:coherent-measurement}).
To describe the operations, we define an arbitrary CV-qudit state $\ket{\Psi_4}_A$ given in cavity $A$ by
\begin{eqnarray}
\label{arbit_CVqudit01}
\ket{\Psi_4}_A && = a \ket{{0}_4}_A + b \ket{{1}_4}_A + c \ket{{2}_4}_A + d \ket{{3}_4}_A .
\end{eqnarray}
First, the projection set of a coherent-state is given by $\hat{P}^{Coh} (\alpha) = \{ \ket{\tilde{0}_4}\bra{\tilde{0}_4}, \, \leavevmode\hbox{\small1\kern-2.8pt\normalsize1} -\ket{\tilde{0}_4}\bra{\tilde{0}_4}\}$ and is viable in a microwave cavity coupled with a superconducting qubit and a readout resonator \cite{NewYale16}. For example, $\hat{P}^{Coh} (\alpha)\ket{\Psi_4}_A \approx \ket{\tilde{0}_4} = \ket{\alpha} $ for $\ket{\tilde{0}_4}\bra{\tilde{0}_4}$ and the definitions and details are presented in Section \ref{sec:coherent-measurement}.
Second, a QND parity measurement of cavity states has been successfully demonstrated with the assistance of an ancillary superconducting qubit in Ref.~\cite{Exp_parity}. The cavity state is projected on the even- or odd-photon subspace such as $\hat{P}^{Par} (even, odd) = \{ \ket{{0}_4} \bra{{0}_4} + \ket{{2}_4} \bra{{2}_4} , \ket{{1}_4} \bra{{1}_4}+ \ket{{3}_4} \bra{{3}_4} \}$ and its parity is imprinted in the state of an ancillary readout qubit. For example, the state $\ket{\Psi_4}_A$ is collapsed by the parity measurement into
$\hat{P}^{Par} (even) \Big( \ket{\Psi_4}_A\ket{g} \Big) \propto a \ket{{0}_4}_A + c \ket{{2}_4}_A $ with the outcome of the qubit state in $\ket{e}$ or
$\hat{P}^{Par} (odd) \Big( \ket{\Psi_4}_A\ket{g} \Big) \propto b \ket{{1}_4}_A + d \ket{{3}_4}_A$ with $\ket{g}$. Therefore, the cavity state is projected in either the even- or odd-photon subspace through the parity measurement performed by the readout qubit. (see details in Section \ref{sec:parity}).
Finally, the SNAP gate is essential for performing photon-phase operations for CV-qudits and originally designed for the correction of phase distortion induced by self-Kerr effects \cite{SNAPgate}. To inject a group of microwaves into a cavity induces a sum of the phase-rotation gates on each photon-Fock state $\ket{m}$ given by
\begin{eqnarray}
\label{SNAP01}
\hat{S} = \sum_{m} \exp(i \Phi_m) \ket{m}\bra{m}.
\end{eqnarray}
In our scheme, four groups of microwaves are applied due to $d=4$ to obtain the same phase rotations on each $\ket{k_4}$ ($k=0,1,2,3$) and the grouped phase gate is acheived on each $\ket{{k}_4}$ independently. For example, if we apply the SNAP operation with four-group phase gates, e.g., $\Phi_{4m} = \phi_0$, $\Phi_{4m+1} = \phi_1$, $\Phi_{4m+2} = \phi_2$, and $\Phi_{4m+3} = \phi_3$ on $\ket{\Psi_4}$, the phase-operated qudit is given in
\begin{eqnarray}
\label{SNAP02}
\hat{S} ( \phi_0, \phi_{1} , \phi_{2}, \phi_{3}) \ket{\Psi_4} && = a e^{i \phi_0} \ket{{0}_4} + b e^{i \phi_{1}} \ket{{1}_4} + c e^{i \phi_{2}} \ket{{2}_4} + d e^{i \phi_{3}} \ket{{3}_4}.
\end{eqnarray}
In particular, we utilise two specific SNAP gates for logical phase gates. The first is a parity-conditional phase gate $\hat{S}^{p1} (\phi) = S ( \phi , -\phi , \phi, -\phi)$ applied to only selected photon states with $ \phi_0 = \phi_2 =\phi$ and $\phi_1 = \phi_3 = -\phi$. For example, $\hat{S}^{p1} (\phi) \, (\ket{\alpha} \pm \ket{- \alpha} ) = e^{ \pm i \phi} (\ket{\alpha} \pm \ket{- \alpha} )$.
The other gate is given by $\hat{S}^{p2} (\phi) = \hat{S} ( 0 , 0 , \phi, \pi + \phi)$, which is applied to only selected Fock states with $ \phi_0 = \phi_1 =0$, $\phi_2 = \phi$, and $\phi_3 = \pi + \phi$. Simple examples are $\hat{S}^{p2} (\phi)\, (\ket{\alpha} + \ket{- \alpha} ) = (\ket{{0}_4} + e^{ i \phi} \ket{{2}_4} )$ and $\hat{S}^{p2} (\phi) \, (\ket{\alpha} - \ket{- \alpha} ) = (\ket{{1}_4} - e^{i \phi} \ket{{3}_4} )$.
The details of the operations are represented in Section \ref{useful_SNAP}.
\subsection{Logical CV qubit under the presence of photon-loss}
\label{sec:LogicalQubit}
The logical qubits for even photon states are defined in
\begin{eqnarray}
\label{Logical0}
\ket{{0}^L_e} &&= {1\over \sqrt{2}} (\ket{{0}_4} + \ket{{2}_4}) = N^{+}_{\alpha} (\ket{\alpha} + \ket{-\alpha}) = \ket{SCS^+_{\alpha}}, \\
\label{Logical1}
\ket{{1}^L_e} &&= {1\over \sqrt{2}} (\ket{{0}_4} - \ket{{2}_4}) = N^{+}_{\alpha} (\ket{i \alpha} + \ket{-i \alpha})= \ket{SCS^+_{i \alpha}},
\end{eqnarray}
where Schr\"odinger cat states are given with $N^{\pm}_{\alpha} = 1/\sqrt{2 (1+e^{-2|\alpha|^2} )}$ in
\begin{eqnarray}
\label{SCS01}
\ket{SCS^{\pm}_{\alpha}} = N^\pm_{\alpha} \left( \ket{\alpha} \pm \ket{-\alpha} \right).
\end{eqnarray}
Note that $\ket{{+}^L_e} \equiv \ket{0_4}$ and $\ket{{-}^L_e} \equiv \ket{2_4}$.
Similarly, for the odd-photon subspace, $\ket{{0}^L_o} = \ket{SCS^-_{\alpha}}$ and $\ket{{1}^L_o}= -i \ket{SCS^-_{i \alpha}}$.
The two types of logical qubits span only either even- or odd-photon states and a photon-loss error can be monitored and corrected by the real-time parity measurement on the final state \cite{Exp_parity}.
For example, let us assume that a logical qubit is encoded in $\ket{\Psi^L_e} = a_0 \ket{0^L_e} + a_1 \ket{1^L_e}$, which implies that the information of an arbitrary single qubit can be written in even photon subspace as a logical state. By real-time parity measurements, the cavity state is monitored through a superconducting qubit coupled with a readout resonator. Before cavity photon-loss, the parity measurement always results in the even state $\ket{\Psi^L_e}$. If the parity changes from even to odd, the updated logical state is equivalent to $\hat{a} \ket{\Psi^L_e} \propto \ket{\Psi^L_o} = a_0 \ket{0^L_o} - a_1 \ket{1^L_o}$. Thus, the parity change tells us that the quantum information is preserved against photon-loss but the relative phase is altered.
\subsection{Logical single-qubit gates in a three-qudit cluster state}
\label{Sec:mMBQC}
The essence of MBQC is to create a designed multipartite entangled state initially and to apply sequential measurements on individual qubits will operate one- and two-qubit gates for universal quantum computing \cite{MBQC1, MBQC2}. We now propose a specific protocol to perform a modified MBQC protocol from a three CV-qudit entangled state $\ket{3CS_4}_{ABC}$ given in Eq.~(\ref{3QuditCS01})
and its original MBQC from a three-qubit cluster state is described in Section \ref{sec:Original_MBQC}. The CV-qudit measurement schemes are all experimentally viable for logical MBQC using the photon-loss error-correcting code \cite{catcode, Yale-QECC}.
The first step is to determine the photon parity in the cavity state of the final outcome using the parity measurement on $B$ from Eq.~(\ref{3QuditCS01}).
Although any alternative implementation of building $\ket{\psi^{ideal} ( {\tau_r / 4} ) }_{ABC}$ is applicable for our initial CV-qudit states (e.g., a scheme in Ref.~\cite{Yale_teleport}), we simply assume that $\ket{\psi^{ideal} ( {\tau_r / 4} ) }_{ABC}$ is initially prepared by a cross-Kerr interaction among the cavities. Then, after the decoupling of all the Kerr-interactions (see Fig.~\ref{fig:02}), the middle cavity state is projected by the parity measurement such as $\hat{P}^{par}_{B} \ket{\psi^{ideal} ( {\tau_r / 4} ) }_{ABC}$ and is given in the even or odd parity state on $B$ such as
\begin{eqnarray}
\label{Special_even01}
\ket{3CS^e_4 }_{ABC} && = {1\over \sqrt{2}} \Big( \ket{\tilde{0}_4 }_A \ket{{0}_4}_B \ket{\tilde{0}_4 }_C + \ket{\tilde{2}_4 }_A \ket{{2}_4}_B \ket{\tilde{2}_4 }_C \Big), ~~~~{\rm (for~even)} \\
\label{Special_even02}
\ket{3CS^o_4 }_{ABC} && ={1\over \sqrt{2}} \Big( \ket{\tilde{1}_4}_A \ket{{1}_4}_B \ket{\tilde{1}_4}_C + \ket{\tilde{3}_4 }_A \ket{{3}_4}_B \ket{\tilde{3}_4 }_C \Big). ~~~~{\rm (for~odd)}
\end{eqnarray}
Note that this is the only intialisation operation on $B$ to choose the parity of the outcome state, and we do not touch the cavity state in $B$ afterwards.
Without loss of generality, we will assume that the state is subjected in $\ket{3CS^e_4 }_{ABC}$, however, the odd parity case is identical except the definition of logical qubits given in $\ket{{0}^L_o} = \ket{SCS^-_{\alpha}}$ and $\ket{{1}^L_o} = -i \ket{SCS^-_{i \alpha}}$.
We now perform the cavity operations in $A$ and $C$ with two parameters ($\theta_1$ and $\theta_2$) to obtain desired single-qubit gates on $B$. Because of the lack of cavity measurement capability, we cannot directly perform single-cavity measurement in $\ket{\pm \theta}\bra{\pm \theta}$, however, we alternatively suggest logical single-qubit phase operation first and cavity measurement along the logical $Z$-axis because $\ket{\pm \theta}\bra{\pm \theta} \propto {R}^Z (-\theta) \ket{\pm}\bra{\pm} ({R}^Z (-\theta))^\dag$. To implement a logical phase gate, SNAP gates are used for encoding the desired operations on logical qubits. More precisely, two SNAP gates, $\hat{S}^{p1} ({\theta_1 / 2})$ on qubit $A$ and $\hat{S}^{p1} (-\theta_1/ 2)$ on $C$, are applied for mimicking a single-qubit phase gate with $\theta_1$. Note that $\hat{S}^{p1}$ is a parity-conditional phase gate as shown in Section \ref{useful_SNAP} and the phase information is embeded in the three CV-qudit state
\begin{eqnarray}
\label{Special_even02-2}
\hspace{-2cm} && \ket{Out_2 (\theta_1)}_{ABC} = \left( \hat{S}^{p1}_A \left({\theta_1 \over 2} \right) \, \hat{S}^{p1}_C \left(-{\theta_1 \over 2} \right) \right) \ket{3CS^e_4 }_{ABC}, \\
\hspace{-2cm}&& ~~~~~~= {1\over 2} \Big( \left( \ket{0^L_e}_A \ket{0^L_e}_C + \ket{0^L_o}_A \ket{0^L_o}_C \right) \ket{{0}^L_e} _B + e^{i\theta_1} \left( \ket{0^L_e}_A \ket{0^L_o}_C + \ket{0^L_o}_A \ket{0^L_e}_C \right) \ket{{1}^L_e}_B \Big). \nonumber
\end{eqnarray}
Because the SNAP gates of $\theta_1$ are QND operations, the total cavity stat is not collapsed into a single cavity state yet.
In the next step, phase $\theta_2$ is imprinted by $\hat{S}^{p2} (\theta_2)$ on $C$ in $\ket{Out_2}_{ABC}$ such as
\begin{eqnarray}
\label{Special_even03}
\hspace{-2.5cm}&& \ket{Out_3 (\theta_1, \theta_2) }_{ABC} = \hat{S}^{p2}_C (\theta_2) \ket{Out_2 (\theta_1)}_{ABC}, \\
\hspace{-2.5cm} &&~~~~~= {1\over 2 \sqrt{2}} \Big[ \left[ \ket{0^L_e}_A ( \ket{{+}^L_e}_C + e^{i\theta_2} \ket{{-}^L_e}_C ) + \ket{0^L_o}_A ( \ket{{+}^L_o}_C - e^{i\theta_2} \ket{{-}^L_o}_C ) \right] \ket{{0}^L_e} _B , \nonumber \\
\hspace{-2.5cm} && ~~~~~~~~~ + e^{i\theta_1} \left[ \ket{0^L_e}_A ( \ket{{+}^L_o}_C - e^{i\theta_2} \ket{{-}^L_o}_C ) + \ket{0^L_o}_A ( \ket{{+}^L_e}_C + e^{i\theta_2} \ket{{-}^L_e}_C ) \right] \ket{{1}^L_e}_B \Big]. \nonumber
\end{eqnarray}
Although we showed a preferred sequence of SNAP gates performed by $\hat{S}^{p1}$ on $A$ and $C$ first and $\hat{S}^{p2}$ on $C$ second, one can choose an alternative sequence depending on each cavity (e.g., $\hat{S}^{p1}$ on $A$ first and $\hat{S}^{p2}\hat{S}^{p1}$ on $C$ second).
Finally, we are ready to perform the parity and coherent-state measurements on $\ket{Out_3 (\theta_1, \theta_2) }_{ABC}$ to gain the designed local qubit in $B$ given by cavity projections on $A$ and $C$, which is equivalent to the total operation of the original MBQC in Eq.~(\ref{final_qubit}). When we perform the parity measurement on $A$, the resultant state is equal to $\ket{Out_4 (\theta_1, \theta_2) }_{BC} \propto \hat{P}^{Par}_A \ket{Out_3}_{ABC}$.
The state for the even parity is given in
\begin{eqnarray}
\label{Special_even03}
\hspace{-2cm} \ket{Out_4^e}_{BC} && = {1\over 2 } \Big[ ( \ket{{+}^L_e}_C + e^{i\theta_2} \ket{{-}^L_e}_C ) \ket{{0}^L_e} _B + e^{i\theta_1}( \ket{{+}^L_o}_C - e^{i\theta_2} \ket{{-}^L_o}_C ) \ket{{1}^L_e}_B \Big],
\end{eqnarray}
while the odd one is
\begin{eqnarray}
\label{Special_odd03}
\hspace{-2cm} \ket{Out_4^o}_{BC} && = {1\over 2 } \Big[ ( \ket{{+}^L_o}_C - e^{i\theta_2} \ket{{-}^L_o}_C ) \ket{{0}^L_e} _B + e^{i\theta_1} ( \ket{{+}^L_e}_C + e^{i\theta_2} \ket{{-}^L_e}_C ) \ket{{1}^L_e}_B \Big].
\end{eqnarray}
\begin{table}[b]
\centering
\begin{tabular}{|l|l|l|l|}
\hline
Outcome & Logical gate & Outcome & Logical gate \\ \hline
$\ket{even}_A \ket{\alpha}_C$ & $f_{12} \, {\cal R}^{z} (\theta_1) {{ \cal H}}\, { \cal R}^{z} (\theta_2)$ &
$\ket{odd}_A \ket{\alpha}_C$& $f_{12} {\cal X} {\cal R}^{z} (-\theta_1) {{ \cal H}}\, { \cal R}^{z} (\theta_2)$ \\ \hline
$\ket{even}_A \ket{-\alpha}_C$ & $f_{12} \, {\cal Z} {\cal R}^{z} (\theta_1) {{ \cal H}}\, { \cal R}^{z} (\theta_2)$ &
$\ket{odd}_A \ket{-\alpha}_C$ & $f_{12} {\cal X Z} {\cal R}^{z} (-\theta_1) {{ \cal H}}\, { \cal R}^{z} (\theta_2)$ \\ \hline
$\ket{even}_A \ket{i\alpha}_C$ & $f\rq{}_{12} {\cal Z} {\cal X} {\cal R}^{z} (-\theta_1) {{ \cal H}}\, { \cal R}^{z} (\theta_2)$ &
$\ket{odd}_A \ket{i\alpha}_C$ & $f\rq{}\rq{}_{12} {\cal R}^{z} (\theta_1) {{ \cal H}}\, { \cal R}^{z} (\theta_2)$ \\ \hline
$\ket{even}_A \ket{-i\alpha}_C$ & $f\rq{}_{12} {\cal X} {\cal R}^{z} (-\theta_1) {{ \cal H}}\, { \cal R}^{z} (\theta_2)$ &
$\ket{odd}_A \ket{-i\alpha}_C$& $f\rq{}\rq{}_{12} e^{i \pi } {\cal Z} {\cal R}^{z} (\theta_1) {{ \cal H}}\, { \cal R}^{z} (\theta_2)$ \\ \hline
\end{tabular}
\caption{Table for measurement outcomes in $A$ and $C$ and the performed logical single-qubit gates ($f\rq{}_{12} = f_{12} \, e^{i \frac{ \pi }{4} }\, {\cal R}^{z} ({\pi \over 2})$ and $f\rq{}\rq{}_{12} =f_{12} \, e^{-i \frac{ \pi }{4} }\, {\cal R}^{z} ({\pi \over 2})$).}
\label{Table2}
\end{table}
Then, if we project the qubit $C$ by the coherent state-measurement $\{\ket{\alpha}, \ket{i\alpha}, \ket{-\alpha}, \ket{-i\alpha}\}$ as shown in Section \ref{sec:coherent-measurement}, the successful detection gives the logical qubit in
\begin{eqnarray}
\label{Special_even04}
&& \hspace{-1.5cm} \ket{Out_5^{\alpha,e}}_{B} = \sqrt{2} \, {}_{C} \langle \alpha \ket{Out_4^{e}}_{BC} = e^{\frac{i }{2} (\theta_1+\theta_2) } {\cal R}^{z} (\theta_1) {{ \cal H}}\, { \cal R}^{z} (\theta_2) \ket{{+}^L_e}_B, \\
&&\hspace{-1.5cm} \ket{Out_5^{-\alpha,e}}_{B} =e^{\frac{i }{2} (\theta_1+\theta_2) } {\cal Z} {\cal R}^{z} (\theta_1) {{ \cal H}}\, { \cal R}^{z} (\theta_2) \ket{{+}^L_e}_B, \\
&&\hspace{-1.5cm} \ket{Out_5^{i\alpha,e}}_{B} =e^{\frac{i }{2} (\theta_1+\theta_2+{\pi \over 2}) } {\cal R}^{z} ({\pi \over 2}) {\cal Z} {\cal X} {\cal R}^{z} (-\theta_1) {{ \cal H}}\, { \cal R}^{z} (\theta_2) \ket{{+}^L_e}_B,\\
&& \hspace{-1.5cm} \ket{Out_5^{-i\alpha,e}}_{B} =e^{\frac{i }{2} (\theta_1+\theta_2+{\pi \over 2}) }{\cal R}^{z} ({\pi \over 2}) {\cal X} {\cal R}^{z} (-\theta_1) {{ \cal H}}\, { \cal R}^{z} (\theta_2) \ket{{+}^L_e}_B,
\end{eqnarray}
and
\begin{eqnarray}
\label{Special_even05}
&& \hspace{-1.5cm} \ket{Out_5^{\alpha,o}}_{B} = e^{\frac{i }{2} (\theta_1+\theta_2) } {\cal X} {\cal R}^{z} (-\theta_1) {{ \cal H}}\, { \cal R}^{z} (\theta_2) \ket{{+}^L_e}_B, \\
&& \hspace{-1.5cm} \ket{Out_5^{-\alpha,o}}_{B} = e^{\frac{i }{2} (\theta_1+\theta_2) } {\cal X} {\cal Z} {\cal R}^{z} (-\theta_1) {{ \cal H}}\, { \cal R}^{z} (\theta_2) \ket{{+}^L_e}_B, \\
&& \hspace{-1.5cm} \ket{Out_5^{i\alpha,o}}_{B} =e^{\frac{i }{2} (\theta_1+\theta_2- {\pi \over 2}) } {\cal R}^{z} ({\pi\over 2}) {\cal R}^{z} (\theta_1) {{ \cal H}}\, { \cal R}^{z} (\theta_2) \ket{{+}^L_e}_B , \\
&& \hspace{-1.5cm} \ket{Out_5^{-i\alpha,o}}_{B} = e^{\frac{i }{2} (\theta_1+\theta_2+ 3{\pi \over 2}) } {\cal R}^{z} ({\pi \over 2}) {\cal Z} {\cal R}^{z} (\theta_1) {{ \cal H}}\, { \cal R}^{z} (\theta_2) \ket{{+}^L_e}_B,
\end{eqnarray}
where ${ \cal R}^{z}$ and ${{ \cal H}}$ are a logical rotation gate on z-axis and a logical Hadamard gate defined by logical qubits in $\ket{{0}^L}$ and $\ket{{1}^L}$. Note that a repeat-until-success method can be used for approximated orthogonal projection of the cavity states on the measurement set of $\{ \ket{\alpha}\bra{\alpha}, \ket{i\alpha}\bra{i\alpha}, \ket{-\alpha}\bra{-\alpha}, \ket{-i\alpha}\bra{-i\alpha} \}$ for large $\alpha \ge 2$.
The details of logical gates with respect to each outcome are presented in Table \ref{Table2}.
Note that logical Pauli operators ${\cal Z} \equiv \left( \hat{X}_4 \right)^2$ and ${\cal X} \equiv \hat{Z}_4$ can be defined by CV-qudit Pauli gates in Section \ref{2-1}.
Therefore, it is shown that the specific logical operation of mMBQC is performed by the sequential operations and measurements in the cavities of $A$ and $C$.
\subsection{Implementation of a two CV-qudit state in the JC Hamiltonian}
\label{Implement_s02}
We here mainly examine how to build two-qudit entangled states in the model of the JC generalised Hamiltonian ($\hat{H}^{JC}_{ABM} $), which describes the nonlinear effects given from the contribution of the intermediary transmon qubit (upto the third level). From the JC Hamiltonian in Eq.~(\ref{total_Ham01}) with two coherent states, the total state in the two cavities with the qubit evolves in time and the state of two cavities are given by
\begin{eqnarray}
\label{Time-evol-sim02}
\ket{\psi^{JC} (t)}_{ABM} = \exp \left( i \, \hat{H}^{JC}_{ABM} \, t \right) \ket{\alpha}_{A} \ket{\alpha}_{B} \ket{g}_{M} , \\
\label{Time-evol-sim03}
\rho^{JC}_{AB} (t) = Tr_{M} \Big( \ket{\psi^{JC} (t)}_{ABM} \bra{\psi^{JC} (t)} \Big).
\end{eqnarray}
\begin{figure}[t]
\includegraphics[width=14cm,trim= 0cm 0cm 2cm 0cm] {20180620_expect_g1215_Km06_3lv_0to250000.png}
\center
\includegraphics[width=12cm,trim= 3cm 0cm 2cm 0cm]{Fig2_low.pdf}
\caption{(Top) $|\langle a_A \rangle |$, $|\langle a_B \rangle |$ and $|\langle a^\dag_M a_M \rangle|$ are shown from the initial state $\ket{\alpha}_{A} \ket{\alpha}_{B}$ ($\alpha = 2.0$) under the generalised JC Hamiltonian in Eq.~(\ref{total_Ham01}) with the parameters of $\omega_A = 5.5$ GHz, $\omega_B = 8.5$ GHz, $\omega_{M} = 4.0$ GHz, $\lambda_{AM} = 0.12$ GHz, $\lambda_{BM} = 0.15$ GHz and $K_M = - 0.6$ GHz. While the orange line shows the revival of mode $B$ at $\tau_r \approx 160\,\mu$s, the green line indicates the expectation value $|\langle a^\dag_M a_M \rangle| \approx 0$, which shows the ground state $\ket{g}_M$ mostly as predicted in the adiabatic method.
(Bottom) In (a), a mixture of four coherent states is given by $tr_B \Big( \rho^{JC}_{AB} (t_0) \Big)$ at $t_0 =39.45\,\mu$s ($\approx \tau_r /4$) while the Wigner plot in (b) indicates that the evolved state $ \rho^{JC}_A = \left( {}_{B} \bra{\alpha} \rho^{JC}_{AB} (t_0)\ket{\alpha}_{B} \right)$ is also very close to the state $\ket{{0}_4}_A$ with $F\approx0.978$. From (c) to (f), we project the state on the Fock states from $\ket{0}_A\bra{0}$ to $\ket{3}_A\bra{3}$ and the Wigner plots of $ \rho^{JC, k}_B$ are shown as coherent states in $\ket{\tilde{k}_4}$ ($k=0,1,2,3$).
}
\label{fig:03}
\end{figure}
In Fig.~\ref{fig:03}, we numerically illustrate the dynamics of cavity states evolved by the JC Hamiltonian $\hat{H}^{JC}_{ABM}$ to create the two CV-qudit cluster state given in Eq.~(\ref{Simple_02}). The realistic parameters are chosen in $\omega_A = 5.5$ GHz, $\omega_B = 8.5$ GHz, $\omega_{M} = 4.0$ GHz, $\lambda_{AM} = 0.12$ GHz, $\lambda_{BM} = 0.15$ GHz and $K_M = - 0.6$ GHz. In the top of Fig.~\ref{fig:03}, the revival picks appear at around $t \approx 160 \mu$s with $\alpha = 2.0$ as given in the values of $|\langle a_A \rangle |$ (blue) and $|\langle a_B \rangle |$ (orange). Note that $|\langle a (t) \rangle | =0$ implies that the cavity states are the evenly distributed coherent states in phase space while $|\langle a^\dag_M a_M \rangle| \approx 0$ does that the transmon qubit is almost nearly in $\ket{g}_M$ such that $\ket{\psi^{JC} (t)}_{ABM} \bra{\psi^{JC} (t)}\approx \rho^{JC}_{AB} ({t} ) \otimes \ket{g}_M\bra{g}$.
What we would like to find is that the state $\rho^{JC}_{AB} (t_0) \approx \ket{\psi^{ideal} ( {\tau_r / 4} ) }_{AB} \bra{\psi^{ideal} ( {\tau_r / 4} ) }$ at certain time $t_0$ (see the details in Eq.~(\ref{Simple_02})). To compare $\rho^{JC}_{AB} (t_0)$ with the ideal two-qudit state (given in Eq.~(\ref{ideal2qudit}), one may obtain the fidelity between the two states, however, this value might not represent the characteristics of the time-evolved state $\rho^{JC}_{AB} (t_0)$ because the distortion of the cavity state from the self-Kerr effects suppress the fidelity very low.
In the spirit of MBQC, one of the simple verifications of the measured states is to compare between the projected cavity states of $\rho^{JC}_{AB} $ and of $\ket{\psi^{ieal} }_{AB}$. In Fig.~\ref{fig:03}(a), the state in $A$ is given by $tr_{B} \left( \rho^{JC}_{AB} (t_0) \right)$, in which we expect to obtain the mixture of four coherent states at $t_0=39.45\,\mu$s. From ($b$) to ($f$), we plot the Wigner functions of the cavity state in mode $A$ ($B$) at $t_0$ given by the projection of the certain states in mode $B$ ($A$) such as
\begin{eqnarray}
\label{Outcome-sim01}
\hspace{-1cm} \rho^{JC}_A \propto tr_B \Big( \left( \leavevmode\hbox{\small1\kern-2.8pt\normalsize1}_A \otimes \ket{\alpha}_{B} \bra{ \alpha}\right) \rho^{JC}_{AB} ( t_0 ) \Big) \approx \ket{{0}_4}_A \bra{{0}_4}, \\
\hspace{-1cm} \rho^{JC, k}_B \propto tr_A \Big(( \ket{k}_{A} \bra{ k} \otimes \leavevmode\hbox{\small1\kern-2.8pt\normalsize1}_B )\, \rho^{JC}_{AB} (t_0) \Big) \approx \ket{\tilde{k}_4}_B \bra{\tilde{k}_4} = \ket{\alpha e^{i k {\pi / 2}}}_B \bra{\alpha e^{i k {\pi / 2}}},
\end{eqnarray}
for $k=0,1,2,3$. In the bottom of Fig.~\ref{fig:03}, we show that the maximum fidelity $F= \left| {}_A \bra{{0}_4} \rho^{JC}_A \ket{{0}_4}_A \right|$ is approximately 0.978 at $t_0 \approx 40\,\mu$s in (b) and some levels of self-Kerr distortions occur during the time evolution from (c) to (f). We neglect decoherence processes in the cavities since the state-of-the-art lifetime of a 3D cavity is above 1.2 ms and the decoherence is expected to be not dominant until the period $t_0 \approx 40\,\mu$s. Apparently, this period of creating multi-partite microwave entangled state could not grow up much with increasing the number of cavities.
\section{Conclusion}
In summary, we introduce a new-type of CV logical MBQC in three microwave cavities coupled with superconducting qubits in a circuit-QED system. After the CV-qudits are defined, three specific circuit-QED gates are introduced to realise logical gate operations for the protocol of logical MBQC. We deliver the method of a logical single-qubit gate in photon-loss correcting codes from the three CV-qudit entangled state. Finally, the implementation of the two CV-qudit state and measured cavity states are numerically investigated under the JC Hamiltonian in a two-cavity system coupled with a superconducting qubit. The results show that the entangled CV-qudit states can be efficiently built with high fidelity (above 0.97) via the cross-Kerr effect induced by the intermediary superconducting qubit between cavities.
\section{Methods}
\subsection{How to build two CV-qudit states}
\label{Sec:Entangle-two-cat}
When an initial state $\ket{\psi^{int}}_{AB}$ is prepared in cavities $A$ and $B$, the time-evolved state at time $t$ is given by
\begin{eqnarray}
\label{total_Ham00}
\ket{\psi (t)}_{AB} = \exp \left( i \hat{H}_{AB} \, t \right) \ket{\psi^{int}}_{AB},
\end{eqnarray}
where the cross-Kerr Hamiltonian is $\hat{H}_{AB} = K_{AB} (\hat{a}^{\dag}_A \hat{a}_A) (\hat{a}^{\dag}_B \hat{a}_B)$ and $K_{AB}$ is the strength of cross-Kerr interaction.
The initial state is fully revived at $t = \tau_r = {2 \pi / \,K_{AB}}$, and the evolved state is in general written in an entangled (inseparable) state between two modes at $t \neq \tau_r $. For $t = \tau_r /d$, it is given by
\begin{eqnarray}
\label{Simple_00}
\ket{\psi^{ideal} ( {\tau_r / d} ) } = \exp \left( i {2 \pi \over d} (\hat{a}^{\dag}_A \hat{a}_A) (\hat{a}^{\dag}_B \hat{a}_B) \right) \ket{\psi^{int}}_{AB}.
\end{eqnarray}
For example, for $t = {\tau_r / 2} $ with $\ket{\psi^{int}}_{AB} = \ket{\alpha}_{A} \ket{\alpha}_{B}$, the state evolves such as
\begin{eqnarray}
\label{Simple_01}
\ket{\psi^{ideal} ( {\tau_r / 2} ) }_{AB} &&= {1 \over \sqrt{2}} \left( \ket{SCS^{+}_{\alpha}}_A \ket{\alpha }_B + \ket{SCS^{-}_{\alpha}}_A \ket{-\alpha }_B \right).
\end{eqnarray}
This state is known as an entangled coherent state \cite{Sanders12, Sanders92}, which is also of excellence for quantum metrology and other QI processing methods \cite{JooPRL11, Ralph03} and has been recently demonstrated in a deterministic method in circuit-QED \cite{Yale_ECS} and probabilistically in quantum optics \cite{Paris_ECS}.
In fact, the entangled coherent state can be used as a simplest resource state for MBQC with no error-correction because CV quantum teleportation, which is the building block for MBQC, has been demonstrated in quantum optics \cite{Vaidman, compare_CV_tele1, CV_RMP2} and investigated in circuit-QED \cite{Joo_SR}. The similar method of implementing the states has been suggested with the assumption of the cross-Kerr interaction in a circuit-QED system \cite{newarxiv_ECS}.
For $d=4$, the time evolution time is the half period of $\ket{\psi^{ideal} ( {\tau_r / 2} ) }_{AB} $. The evolved state at $t = {\tau_r / 4} $ is written by
\begin{eqnarray}
\label{Simple_02}
\hspace{-1cm} \ket{\psi^{ideal} ( {\tau_r / 4} ) }_{AB} && = {1\over 2} \Big( \ket{{0}_4}_A \ket{\alpha }_B +\ket{{1}_4}_A \ket{i \alpha }_B + \ket{{2}_4}_A \ket{-\alpha }_B +\ket{{3}_4}_A \ket{-i\alpha }_B \Big),
\nonumber \\
&&= {1 \over 2} \sum_{k=0}^3 \ket{{k}_4}_A \ket{\tilde{k}_4}_B. \label{ideal2qudit}
\end{eqnarray}
This state $\ket{\psi^{ideal} ( {\tau_r / 4} ) }_{AB}$ is a CV version of a two-qudit cluster state. Alternatively, the equivalent CV-qudit state has been very recently realised for qudit quantum teleportation \cite{Yale_teleport}.
\subsection{Fock- and coherent-state projections on a cavity state}
\label{sec:coherent-measurement}
One of the important techniques in circuit-QED is based on a conditional qubit-rotation depending on a chosen Fock state $\ket{m}_A$ and the projection measurement set is given by $\hat{P}^{Foc} (m) = \{ \ket{m}\bra{m}, \leavevmode\hbox{\small1\kern-2.8pt\normalsize1} - \ket{m}\bra{m} \}$ \cite{BlaisPRA}.
For example, let us assume that a coherent state $\ket{\alpha}$ is prepared in cavity $A$ with the ground state $\ket{g}$ such as $\ket{\alpha}_A \ket{g}_{J} = \sum_{m} c_m \ket{m}_{A} \ket{g}_{J}$ for $c_m = \bra{m} \alpha \rangle$.
A conditional qubit-rotation gate is effectively performed on photon state $\ket{m}$ represented by
\begin{eqnarray}
\label{total_StateProj01}
\hat{R}^{y}_{A\, J} (m, \phi) = \sum_{n \ne m} e^{i \eta_n} \ket{n}_{A}\bra{n} \otimes \leavevmode\hbox{\small1\kern-2.8pt\normalsize1}_{J} + \ket{m}_{A}\bra{m} \otimes \hat{R}^y_{J} (\phi),
\end{eqnarray}
where $\hat{R}^y (\phi) = \cos {\phi\over 2} \, \leavevmode\hbox{\small1\kern-2.8pt\normalsize1} - i \sin {\phi\over 2} \, Y = \left(
\begin{array}{cc}
\cos {\phi\over 2} & - \sin {\phi\over 2} \\
\sin {\phi\over 2} & \cos {\phi\over 2} \\
\end{array}
\right)$.
For $\phi = {\pi }$, the state becomes
$\hat{R}^{y}_{A\, J} (m, {\pi }) \, \ket{\alpha}_A \ket{g}_{J} = \sum_{n \neq m} c_n e^{i \eta_n} \ket{n}_{A} \ket{g}_{J} + c_m \ket{m}_{A} \ket{e}_{J}$ where $ e^{i \eta_n}$ is an undesired operation in $\hat{R}^{y}_{A\, J} (m, \pi)$ due to self-Kerr interaction but does not influence our result because we only use the outcome state $\ket{e}$ in a heralded way \cite{Yale_big_cat}.
Then, when the outcome is measured in $\ket{e}_{J}$, the cavity state is also projected in $\ket{m}_A$ and the operator of this Fock-state projection on the $m$-th photon is given by
\begin{eqnarray}
\label{total_StateProj02}
\hat{P}^{Foc}_{A} (m) = \left( \leavevmode\hbox{\small1\kern-2.8pt\normalsize1}_{A} \otimes \ket{e}_{J} \bra{e} \right) \, \hat{R}^{y}_{A\,J} (m, \pi).
\end{eqnarray}
In the unsuccessful case of measurement in $\ket{g}$, the cavity state is projected by the operator $\left( \leavevmode\hbox{\small1\kern-2.8pt\normalsize1}_A - \ket{m}_A \bra{m} \right) $ and we can perform the repeat-until-success protocol $\hat{P}^{Foc}_{A} (p)$ for $p \neq m$.
A coherent-state projection can be also performed by adding displacement operation $\hat{D}^{-\alpha} = e^{\alpha^* a - \alpha a^{\dag}}$ on cavity states \cite{Yale_big_cat}.
the coherent-state projection on $\ket{\alpha}$ is given in
\begin{eqnarray}
\label{total_StateProj03}
\hspace{-1.5cm} \hat{P}^{Coh}_{A} (\alpha) = \hat{P}^{Foc}_{A} (0) \left( \hat{D}^{-\alpha}_A \otimes \leavevmode\hbox{\small1\kern-2.8pt\normalsize1}_{J} \right)
= \left( \leavevmode\hbox{\small1\kern-2.8pt\normalsize1}_{A} \otimes \ket{e}_{J} \bra{e} \right) \, \hat{R}^{y}_{A\,J} (0, \pi) \, \left( \hat{D}^{-\alpha}_A \otimes \leavevmode\hbox{\small1\kern-2.8pt\normalsize1}_{J} \right).
\end{eqnarray}
\subsection{Parity measurement on a cavity state}
\label{sec:parity}
When we first perform the operation $\hat{R}^y_J (\pi/2) $ on the initial transmon state $\ket{g}_{J}$, a conditional cavity-rotation gate $\hat{C}^{p} (\varphi)$ is given by
\begin{eqnarray}
\label{arbit_CVqudit02}
\hat{C}^{p}_{A\,J} (\varphi) \left[ {1\over \sqrt{2} } \ket{\alpha}_A (\ket{g}_{J} + \ket{e}_{J}) \right] = {1\over \sqrt{2} } \left( \ket{\alpha}_A \ket{g}_{J} + \ket{\alpha\ e^{i \varphi} }_A \ket{e}_{J} \right) ,
\end{eqnarray}
and the operated state with $\varphi=\pi$ is represented by
\begin{eqnarray}
\label{arbit_CVqudit03}
\hspace{-1cm}
\hat{C}^{p}_{A\,J} (\pi) \, \hat{R}^y_{J} (\pi/2) \, \ket{\Psi_4}_A \ket{g}_{J} &=& \ket{\Psi_4}_A \ket{g}_{J} \nonumber \\
&& + \left( a \ket{{0}_4}_A - b \ket{{1}_4}_A + c \ket{{2}_4}_A - d \ket{{3}_4}_A\right) \ket{e}_{J}.
\end{eqnarray}
Finally, if we apply an additional $\hat{R}^y_{J} (\pi/2)$ and measure the superconducting qubit in $\{ \ket{g}_{J}\bra{g}, \ket{e}_{J}\bra{e} \}$, the cavity state is projected on the even- or odd-photon subspace such as parity measurement $\hat{P}^{Par} (even/odd) = \{ \ket{{0}_4} \bra{{0}_4} + \ket{{2}_4} \bra{{2}_4} , \ket{{1}_4} \bra{{1}_4}+ \ket{{3}_4} \bra{{3}_4} \}.$
For example, if the superconducting qubit is measured in $\ket{e}$ (or $\ket{g}$), the total state is projected in even (odd) photon numbers and the parity measurement is represented by
\begin{eqnarray}
\label{Parity02}
\hat{P}^{Par} (even/odd) = \ket{e/g}_{J} \bra{e/g}\, \hat{R}^y_{J} (\pi/2) \, \hat{C}^{p}_{A\,J} (\pi) \, \hat{R}^y_{J} (\pi/2).
\end{eqnarray}
\subsection{SNAP gate for a logical single-qudit phase gate}
\label{useful_SNAP}
The original motivation of SNAP gate was to cancel out the self-Kerr defect in each cavity independently because self-Kerr effects dominantly influence the shape of the cavity state in a physical setup if the evolution time is not short. This unique circuit-QED technique works in a dispersively coupled cavity-transmon system \cite{SNAPgate} and has been demonstrated to minimize phase distortions acquired during the self-Kerr interaction period. The dispersive energy shifts of the cavity system allow a phase gate in individual Fock states to be addressed by driven microwaves.
For $\hat{S}^{p1}$, the outcome state from $\ket{\Psi_4}$ is given by
a grouped phase gate dependent on photon parities such as
\begin{eqnarray}
\label{SNAP03}
\hat{S}^{p1} (\phi) \ket{\Psi_4} && = e^{i \phi} \left( a \ket{{0}_4} + c \ket{{2}_4} \right) + e^{-i \phi} \left( b \ket{{1}_4} + d \ket{{3}_4} \right),
\end{eqnarray}
while that for $\hat{S}^{p2}$
\begin{eqnarray}
\label{SNAP04}
\hat{S}^{p2} (\phi) \ket{\Psi_4} && = a \ket{{0}_4} + b \ket{{1}_4} + e^{ i \phi} c \ket{{2}_4} - d e^{ i \phi} \ket{{3}_4}.
\end{eqnarray}
\begin{table}[b]
\centering
\begin{tabular}{|c|c|}
\hline
Outcome state in $C$ & Single-qubit operations \\ \hline
$\ket{Out^{+_1+_2} (\theta_1, \theta_2) }_{C}$ & $ f_{12} R^z (\theta_1) H\, R^z (\theta_2) $ \\ \hline
$\ket{Out^{-_1+_2} (\theta_1, \theta_2) }_{C}$ & $f_{12} \,Z\, R^z (\theta_1) H\, R^z (\theta_2) $ \\ \hline
$\ket{Out^{+_1-_2} (\theta_1, \theta_2) }_{C}$ & $f_{12} \,X\, R^z (-\theta_1) H\, R^z (\theta_2)$ \\ \hline
$\ket{Out^{-_1 -_2} (\theta_1, \theta_2) }_{C}$ & $f_{12} \,ZX\, R^z (-\theta_1) H\, R^z (\theta_2) $ \\ \hline
\end{tabular}
\caption{Table for outcomes and performed gates in Section \ref{sec:Original_MBQC} ( $f_{12} =e^{\frac{i }{2} (\theta_1+\theta_2) }$).}
\label{Table1}
\end{table}
\subsection{MBQC in a three-qubit cluster state}
\label{sec:Original_MBQC}
We here describe the original MBQC protocol in a three-qubit cluster state.
If three qubits are initially prepared in $\ket{+}$ in $A$, $B$, and $C$, two CZ gates between $A$ and $B$ as well as $B$ and $C$, which construct a three-qubit cluster state given in
\begin{eqnarray}
\label{GHZ_01}
\ket{3CS}_{ABC} && = {1\over \sqrt{2}} ( \ket{0 }_A \ket{+}_B \ket{0 }_C + \ket{1 }_A \ket{-}_B \ket{1 }_C ).
\end{eqnarray}
In the frame of MBQC, qubits are sequentially measured in the basis vectors of $ \ket{\pm \theta} = (\ket{0} \pm e^{-i \theta} \ket{1})/\sqrt{2}$. For example, if $\ket{\pm \theta_1}$ is measured in qubit $A$ in Eq.~(\ref{GHZ_01}), the resultant state is given by
\begin{eqnarray}
\label{GHZ_02}
\hspace{-2.5cm}
\ket{Out^{\pm}}_{BC} &&= \sqrt{2}\, {}_{A} \langle \pm \theta_1 \ket{3CS}_{ABC}= {1\over 2} \left[ \ket{0}_B ( \ket{0 }_C \pm e^{i \theta_1} \ket{1 }_C) + \ket{1}_B ( \ket{0 }_C \mp e^{i \theta_1} \ket{1 }_C) \right].
\end{eqnarray}
In the case that the outcome is $\ket{+\theta_2}_{B}$, the final outcome state is equal to
\begin{eqnarray}
\hspace{-2cm}
\ket{Out^{+_1 +_2} (\theta_1, \theta_2) }_{C} && = \sqrt{2} \,{}_{B} \langle +\theta_2 \ket{Out^+ (\theta_1) }_{BC}= e^{\frac{i }{2} (\theta_1+\theta_2) } R^z (\theta_1) H\, R^z (\theta_2) \ket{+}_C. \label{final_qubit}
\end{eqnarray}
As shown in Table \ref{Table1}, this protocol is equivalent to two single-qubit rotations and two sequential projective measurements on $A$ and $B$. Thus, this procedure of MBQC is equivalent to the operation of two single-qubit gates with phases $\theta_1$ and $\theta_2$.
\section*{Reference}
|
3,212,635,537,946 | arxiv | \section{Introduction}
In this paper,
we consider the following problems:
$$
(-\Delta)^{s} u
=
\sum_{i=1}^{k}
\left(
\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{\alpha_{i}}}}{|x-y|^{\alpha_{i}}}
\mathrm{d}y
\right)
|u|^{2^{*}_{\alpha_{i}}-2}u
+
|u|^{2^{*}_{s}-2}u
,
\mathrm{~in~}
\mathbb{R}^{N},
\eqno(\mathcal{P}_{1})
$$
and
$$
(-\Delta)^{s} u
-\frac{\zeta u}{|x|^{2s}}
=
\left(
\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}y
\right)
|u|^{2^{*}_{\alpha}-2}u
+
\sum_{i=1}^{k}
\frac{|u|^{2^{*}_{s,\theta_{i}}-2}u}
{|x|^{\theta_{i}}}
,
\mathrm{~in~}
\mathbb{R}^{N},
\eqno(\mathcal{P}_{2})
$$
where
$N\geqslant3$,
$s\in(0,1)$,
$\zeta\in
\left[
0,4^{s}\frac{\Gamma(\frac{N+2s}{4})}{\Gamma(\frac{N-2s}{4})}
\right)$,
$\alpha\in(0,N)$,
$2^{*}_{s}=\frac{2N}{N-2s}$
is the critical Sobolev exponent,
$2^{*}_{s,\theta_{i}}=\frac{2(N-\theta_{i})}{N-2s}$
are the critical Hardy--Sobolev exponents,
$2^{*}_{\alpha_{i}}=
\frac{2N-\alpha_{i}}{N-2s}$
are the Hardy--Littlewood--Sobolev upper critical exponents,
the parameters
$\alpha_{i}$
and
$\theta_{i}$
satisfy the assumptions:
\noindent
($H_{1}$)
$0<\alpha_{1}<\alpha_{2}<\cdots<\alpha_{k}<N$
($k\in \mathbb{N}$,
$2\leqslant k<\infty$);
\noindent
($H_{2}$)
$0<\theta_{1}<\cdots<\theta_{k}<2s$
($k\in \mathbb{N},~2\leqslant k<\infty$),
and
$2\theta_{k}-\theta_{1}\in(0,2s)$.
The fractional Laplacian $(-\Delta)^{s}$
of a function
$u: \mathbb{R}^{N}\rightarrow\mathbb{R}$
can be defined as
$$(-\Delta)^{s}u=\mathcal{F}^{-1}(|\xi|^{2s}\mathcal{F}(u)(\xi)),~\mathrm{for~all}~\xi\in \mathbb{R}^{N},$$
and for $u\in C^{\infty}_{0}(\mathbb{R}^{N})$,
where $\mathcal{F}(u)$ denotes the Fourier transform of $u$.
The operator
$(-\Delta)^{s}$
in
$\mathbb{R}^{N}$
is a nonlocal pseudo--differential operator taking the form
\begin{equation*}
\begin{aligned}
(-\Delta)^{s}u(x)
=
C_{N,s}\mathrm{P. V.}
\int_{\mathbb{R}^{N}}
\frac{u(x)-u(y)}{|x-y|^{N+2s}}
\mathrm{d}y,
\end{aligned}
\end{equation*}
where
$\mathrm{P. V.}$
is the Cauchy principal value and
$C_{N,s}$
is a normalization constant.
The fractional
power of Laplacian is the infinitesimal generator of L\'{e}vy stable diffusion process and arise
in anomalous diffusion in plasma, population dynamics, geophysical fluid dynamics, flames
propagation, minimal surfaces and game theory
(see
\cite{Applebaum2004,Caffarelli2012,Garroni2005}).
Problem
$(\mathcal{P}_{1})$
and
$(\mathcal{P}_{2})$
are related to the nonlinear Choquard equation as follows:
\begin{equation}\label{1}
-\Delta u
+
V(x)u
=
\left(
|x|^{\alpha}*|u|^{q}
\right)
|u|^{q-2}u,
\mathrm{~in~}
\mathbb{R}^{N},
\end{equation}
where
$\frac{2N-\alpha}{N}\leqslant q\leqslant\frac{2N-\alpha}{N-2}$
and
$\alpha\in(0,N)$.
For
$q=2$
and
$\alpha=1$,
the problem
(\ref{1})
goes back to the description of the quantum theory of a polaron at rest by Pekar in 1954
\cite{Pekar1954}
and the modeling of an electron trapped in its own hole in
1976
in the work of Choquard,
as a certain approximation to Hartree--Fock theory of one--component plasma
\cite{Penrose1996}.
The existence and qualitative properties of solutions of Choquard type equations
(\ref{1})
have been widely studied in the last decades
(see \cite{Moroz2016}).
For Laplacian with nonlocal Hartree type nonlinearities,
Gao and Yang \cite{Gao2016} investigated the following critical Choquard equation:
\begin{equation}\label{2}
\begin{aligned}
-\Delta u
=
\left(
\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{h,\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}y
\right)
|u|^{2^{*}_{h,\alpha}-2}u
+\lambda u,
\mathrm{~in~}
\Omega,
\end{aligned}
\end{equation}
where
$\Omega$
is a bounded domain of
$\mathbb{R}^{N}$,
with lipschitz boundary,
$N\geqslant3$,
$\alpha\in(0,N)$
and
$\lambda>0$.
By using variational methods,
they established the existence, multiplicity and nonexistence of nontrivial solutions to equation (\ref{2}).
For details and recent works we refer to \cite{O.Alves2017,Gao2017JMAA,Mercuri2016} and the references therein.
For fractional Laplacian with nonlocal Hartree--type nonlinearities,
D'Avenia, Siciliano and Squassina
\cite{d'Avenia2015}
considered the following fractional Choquard equation:
\begin{equation}\label{3}
\begin{aligned}
(-\Delta)^{s} u
+
\omega u
=
\left(
\mathcal{K}_{\alpha}\ast|u|^{q}
\right)
|u|^{q-2}u,
\mathrm{~in~}
\mathbb{R}^{N},
\end{aligned}
\end{equation}
where
$N\geqslant3$,
$s\in(0,1)$,
$\omega\geqslant0$,
$\alpha\in(0,N)$
and
$q\in(\frac{2N-\alpha}{N},\frac{2N-\alpha}{N-2s})$.
In particularly,
when
$\omega=0$,
$\alpha=4s$
and
$q=2$,
then peoblem
(\ref{3})
become a fractional Choquard euqation with upper critical exponent in the sense of Hardy--Littlewood--Sobolev inequatlity as follows:
\begin{equation}\label{4}
\begin{aligned}
(-\Delta)^{s} u
=
\left(
\int_{\mathbb{R}^{N}}
\frac{|u|^{2}}{|x-y|^{4s}}
\mathrm{d}y
\right)
u,
\mathrm{~in~}
\mathbb{R}^{N}.
\end{aligned}
\end{equation}
D'Avenia, Siciliano and Squassina in
\cite{d'Avenia2015}
obtained regularity, existence, nonexistence of nontrivial solutions to problem (\ref{3}) and problem (\ref{4}).
Mukherjee and Sreenadh \cite{Mukherjee2017Fractional}
extended the study of problem \eqref{2} to fractional Laplacian equation.
Recently,
Yang and Wu \cite{Yang2017}
studied the following nonlocal elliptic problems:
\begin{equation}\label{7}
\begin{aligned}
(-\Delta)^{s} u
-\frac{\zeta u}{|x|^{2s}}
=
\left(
\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}y
\right)
|u|^{2^{*}_{\alpha}-2}u
+
\left(
\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{\beta}}}{|x-y|^{\beta}}
\mathrm{d}y
\right)
|u|^{2^{*}_{\beta}-2}u
,
\mathrm{~in~}
\mathbb{R}^{N},
\end{aligned}
\end{equation}
and
\begin{equation}\label{8}
\begin{aligned}
(-\Delta)^{s} u
-\frac{\zeta u}{|x|^{2s}}
=
\left(
\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}y
\right)
|u|^{2^{*}_{\alpha}-2}u
+
\frac{|u|^{2^{*}_{s,\theta}-2}u}{|x|^{\theta}},
\mathrm{~in~}
\mathbb{R}^{N},
\end{aligned}
\end{equation}
where
$N\geqslant3$,
$s\in(0,1)$,
$\zeta\in
\left[
0,4^{s}\frac{\Gamma(\frac{N+2s}{4})}{\Gamma(\frac{N-2s}{4})}
\right)$,
$\alpha,\beta\in(N-2s,N)$,
$\theta\in(0,2s)$,
$2^{*}_{\alpha}=\frac{2N-\alpha}{N-2s}$
and
$2^{*}_{s,\theta}=\frac{2(N-\theta)}{N-2s}$.
Using the refinement of the Sobolev inequality which is related to
the Morrey space,
they showed the existence of nontrivial solutions for problem
(\ref{7})
and
problem
(\ref{8}).
In \cite{ZhangBL2017},
Wang, Zhang and Zhang extended the study of problem (\ref{8}) to the fractional Laplacian system.
By using variational methods,
they investigated the extremals of the corresponding best fractional Hardy--Sobolev constant and established the existence of solutions to the fractional Laplacian system.
Moreover,
there are many other kinds of problem involving two critical nonlinearities,
such as the Laplacian
$-\Delta$
(see \cite{Li2012,Seok2018,Zhong2016}),
the p--Laplacian
$-\Delta_{p}$
(see \cite{Pucci2009}),
the biharmonic operator
$\Delta^{2}$ (see \cite{Bhakta2015}),
and the fractional operator
$(-\Delta)^{s}$
(see \cite{Ghoussoub2016,Chen2018}).
There are two questions arise:
{\bf Question 1:
For
$\zeta=0$,
can we extend the study of problem
(\ref{7})
in the finite many critical nonlinearities?}
{\bf Question 2:
Can we extend the studies of problem
(\ref{8})
and
problem
(\ref{3})
in the finite many critical nonlinearities?}
We answer above questions in this paper.
To our knowledge, there are no results in these senses.
The variational approach that we adopt here, relies on the following
inequalities:
\begin{lemma}\label{lemma1}
$\left.\right.$\cite[Hardy-Littlewood-Sobolev~inequality]{Lieb2001}
Let
$t,r>1$
and
$0<\alpha<N$
with
$\frac{1}{t}+\frac{1}{r}+\frac{\alpha}{N}=2$,
$f\in L^{t}(\mathbb{R}^{N})$
and
$h\in L^{r}(\mathbb{R}^{N})$.
There exists a sharp constant
$C(N,\alpha,t,r)>0$,
independent of
$f,g$
such that
$$\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}
\frac{|f(x)||h(y)|}
{|x-y|^{\alpha}}
\mathrm{d}x\mathrm{d}y
\leqslant
C(N,\alpha,t,r)
\|f\|_{t}
\|h\|_{r}.$$
If
$t=r=\frac{2N}{2N-\alpha}$,
then
$$C(N,\alpha,t,r)
=
C(N,\alpha)
=
\pi^{\frac{\alpha}{2}}
\frac{\Gamma(\frac{N}{2}-\frac{\alpha}{2})}{\Gamma(N-\frac{\alpha}{2})}
\left\{
\frac{\Gamma(\frac{N}{2})}{\Gamma(N)}
\right\}^{\frac{\alpha-N}{N}}.$$
\end{lemma}
\begin{lemma}\label{lemma2}
$\left.\right.$\cite[Endpoint refined Sobolev inequality]{Bellazzini2016}
Let
$s\in(0,\frac{N}{2})$
and
$\alpha\in(0,N)$.
Then there exists a constant
$C_{1}>0$
such that the inequality
$$
\|u\|_{L^{2^{*}_{s}}(\mathbb{R}^{N})}
\leqslant
C_{1}
\|u\|_{D}^{\frac{(N-\alpha)(N-2s)}{N(N+2s-\alpha)}}
\left(
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\alpha}}|u(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
\right)
^{\frac{s(N-2s)}{N(N+2s-\alpha)}},$$
holds for all
$u\in \mathcal{E}^{s,\alpha,2^{*}_{\alpha}}(\mathbb{R}^{N})$.
\end{lemma}
In particular,
the Coulomb--Sobolev space and endpoint refined Sobolev inequality play the key roles in this paper.
For
$s=1$,
Mercuri, Moroz and Schaftingen \cite{Mercuri2016} introduced the Coulomb--Sobolev space and a family of associated optimal interpolation inequalities (include endpoint refined Sobolev inequality).
They studied the existence of solutions of the nonlocal Schr\"{o}dinger--Poisson--Slater type equation by Coulomb--Sobolev space and endpoint refined Sobolev inequality.
For $s\not=1$,
Bellazzini, Ghimenti, Mercuri, Moroz and Schaftingen \cite{Bellazzini2016} studied the fractional Coulomb--Sobolev space and endpoint refined Sobolev inequality.
The first
result of this paper is as follows.
\begin{theorem}\label{theorem1}
Let
$N\geqslant3$,
$s\in(0,1)$
and
$( H_{1})$
hold.
Then
problem
$(\mathcal{P}_{1})$
has a nonnegative solution $\bar{v}(x)$.
Moreover,
set
\begin{equation*}
\begin{aligned}
\bar{\bar{v}}(x)
=
\frac{1}{|x|^{N-2s}}
\bar{v}
\left(
\frac{x}{|x|^{2}}
\right).
\end{aligned}
\end{equation*}
Then
$\bar{\bar{v}}(x)$
is a nonnegative solution of the problem
\begin{equation*}
\begin{aligned}
(-
\Delta)^{s}
\bar{\bar{v}}
=&
\sum\limits^{k}_{i=1}
\left(
\int_{\mathbb{R}^{N}}
\frac{|\bar{\bar{v}}|^{\frac{2N-\alpha_{i}}{N-2s}}}
{|x-y|^{\alpha_{i}}}
\mathrm{d}y
\right)
\left|
\bar{\bar{v}}
\right|^{\frac{4s-\alpha_{i}}{N-2s}}
\bar{\bar{v}}
+
\left|
\bar{\bar{v}}
\right|^{\frac{4s}{N-2s}}
\bar{\bar{v}}
,
~\mathrm{in}~\mathbb{R}^{N}\backslash\{0\}.
\end{aligned}
\end{equation*}
\end{theorem}
\begin{remark}
Problem
$(\mathcal{P}_{1})$ is invariant under the weighted dilation
$$u\mapsto \tau^{\frac{N-2s}{2}}u(\tau x).$$
Therefore, it is well known that the mountain pass theorem does not yield critical points,
but only the Palais--Smale sequences.
In this type of situation,
it is necessary to show the non--vanishing of Palais--Smale sequences.
There are finite many Hardy--Littlewood--Sobolev critical exponents in problem
$(\mathcal{P}_{1})$,
it is difficult to show the non--vanishing of Palais--Smale sequences.
By using fractional Coulomb--Sobolev space, endpoint refined Sobolev inequality and Lemma \ref{lemma6},
we overcome this difficult in Lemma \ref{lemma19}.
\end{remark}
The second
result of this paper is as follows.
\begin{theorem}\label{theorem2}
Let
$N\geqslant3$,
$\alpha\in(0, N)$,
$s\in(0,1)$
and
$( H_{2})$
hold.
Then
problem
$(\mathcal{P}_{2})$
has a nonnegative solution $\tilde{u}(x)$.
Moreover,
set
\begin{equation*}
\begin{aligned}
\tilde{\tilde{u}}(x)
=
\frac{1}{|x|^{N-2s}}
\tilde{u}
\left(
\frac{x}{|x|^{2}}
\right).
\end{aligned}
\end{equation*}
Then
$\tilde{\tilde{u}}(x)$
is a nonnegative solution of the problem
\begin{equation*}
\begin{aligned}
(-
\Delta)^{s}
\tilde{\tilde{u}}
-\zeta
\frac{\bar{\bar{v}}}{|x|^{2s}}
=&
\left(
\int_{\mathbb{R}^{N}}
\frac{|\tilde{\tilde{u}}|^{\frac{2N-\alpha}{N-2s}}}
{|x-y|^{\alpha}}
\mathrm{d}y
\right)
\left|
\tilde{\tilde{u}}
\right|^{\frac{4s-\alpha}{N-2s}}
\tilde{\tilde{u}}
+
\sum\limits^{k}_{i=1}
\frac{
\left|
\tilde{\tilde{u}}
\right|^{\frac{4s-2\theta_{i}}{N-2s}}
\tilde{\tilde{u}}}
{|x|^{\theta_{i}}},
~\mathrm{in}~\mathbb{R}^{N}\backslash\{0\}.
\end{aligned}
\end{equation*}
\end{theorem}
\begin{remark}
This paper not only extends the studies of problem (\ref{7}) and problem (\ref{8}) in the finite many critical nonlinearities,
but also extends $\alpha\in(N-2s,N)$ to $\alpha\in(0,N)$.
In \cite{ZhangBL2017} and \cite{Yang2017},
the authors just studied the case of $\alpha\in(N-2s,N)$.
It is nature to ask the case of $\alpha\in(0,N-2s)$.
In order to overcome this difficult,
we show the refinement of Hardy--Littlewood--Sobolev inequality for the case of $\alpha\in(0,N)$ (see Lemma \ref{lemma5}),
and
the endpoint refined Hardy--Sobolev inequality (see Lemma \ref{lemma7}).
\end{remark}
\section{Preliminaries}
The space
$H^{s}(\mathbb{R}^{N})$
is defined as
$$H^{s}(\mathbb{R}^{N})=\{u\in L^{2}(\mathbb{R}^{N})
|(-\Delta)^{\frac{s}{2}}u\in L^{2}(\mathbb{R}^{N})\}.$$
This space is endowed with the norm
$$\|u\|_{H}^{2}=\|(-\Delta)^{\frac{s}{2}}u\|_{2}^{2}+\|u\|_{2}^{2}.$$
The space
$D^{s,2}(\mathbb{R}^{N})$
is the completion of
$C^{\infty}_{0}(\mathbb{R}^{N})$
with respect to the norm
$$
\|u\|_{D}^{2}=\|(-\Delta)^{\frac{s}{2}}u\|_{2}^{2}.
$$
It is well known that
$\Lambda=4^{s}\frac{\Gamma^{2}(\frac{N+2s}{4})}{\Gamma^{2}(\frac{N-2s}{4})}$
is the best constant in the Hardy inequality
$$
\Lambda
\int_{\mathbb{R}^{N}}
\frac{ u^{2}}{|x|^{2s}}
\mathrm{d}x
\leqslant
\|u\|_{D}^{2}
,~~
\mathrm{for~any~}
u\in
D^{s,2}(\mathbb{R}^{N}).
$$
By Hardy inequality and $\zeta\in[0,\Lambda)$,
we derive that
$$
\|u\|_{\zeta}^{2}
=
\|u\|_{D}^{2}
-
\zeta
\int_{\mathbb{R}^{N}}
\frac{ |u|^{2}}{|x|^{2s}}
\mathrm{d}x,
$$
is an equivalent norm in
$D^{s,2}(\mathbb{R}^{N})$,
since the following inequalities hold:
$$\left(1-\frac{\zeta}{\Lambda}\right)\|u\|_{D}^{2}\leqslant\|u\|_{\zeta}^{2}\leqslant\|u\|_{D}^{2}.$$
For
$\alpha\in(0,N)$
and
$s\in(0,1)$,
the fractional Coulomb--Sobolev space \cite{Bellazzini2016} is defined by
$$
\mathcal{E}^{s,\alpha,2^{*}_{\alpha}}(\mathbb{R}^{N})=
\left\{
\|u\|_{D}<\infty
~\mathrm{and}~
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\alpha}}|u(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y<\infty
\right\}.
$$
We endow the space
$\mathcal{E}^{s,\alpha,2^{*}_{\alpha}}(\mathbb{R}^{N})$
with the norm
$$
\|u\|_{\mathcal{E},\alpha}^{2}
=
\|u\|_{D}^{2}
+
\left(
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\alpha}}|u(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
\right)^{\frac{1}{2^{*}_{\alpha}}}.
$$
For
$\alpha\in[0,N)$,
$\zeta\in[0,\Lambda)$
and
$s\in(0,1)$,
we define the best constant:
\begin{equation}\label{10}
\begin{aligned}
S_{\zeta,\alpha}:=
\inf_{u\in D^{s,2}(\mathbb{R}^{N})\setminus\{0\}}
\frac{\|u\|_{D}^{2}
-
\zeta
\int_{\mathbb{R}^{N}}
\frac{ |u|^{2}}{|x|^{2s}}
\mathrm{d}x}
{\left(
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\alpha}}|u(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
\right)^{\frac{1}{2^{*}_{\alpha}}}}.
\end{aligned}
\end{equation}
We know that
$S_{\zeta,\alpha}$
is attained in
$\mathbb{R}^{N}$
(see \cite{Yang2017}).
For
$s\in(0,1)$
and
$\theta\in(0,2s)$,
we define the best constant:
\begin{equation}\label{11}
\begin{aligned}
H_{\zeta,\theta}:=
\inf_{u\in D^{s,2}(\mathbb{R}^{N})\setminus\{0\}}
\frac{\|u\|_{D}^{2}
-
\zeta
\int_{\mathbb{R}^{N}}
\frac{ |u|^{2}}{|x|^{2s}}
\mathrm{d}x}
{\left(
\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{s,\theta}}}{|x|^{\theta}}
\mathrm{d}x
\right)^{\frac{2}{2^{*}_{s,\theta}}}}
\end{aligned}
\end{equation}
where
$H_{\theta}$
is attained in
$\mathbb{R}^{N}$
(see \cite{Yang2015}).
A measurable function
$u:\mathbb{R}^{N}\rightarrow \mathbb{R}$
belongs to the Morrey space
$\|u\|_{\mathcal{L}^{p,\varpi}}(\mathbb{R}^{N})$
with
$p\in[1,\infty)$
and
$\varpi\in(0,N]$
if and only if
$$
\|u\|^{p}_{\mathcal{L}^{p,\varpi}(\mathbb{R}^{N})}
=
\sup_{R>0,x\in\mathbb{R}^{N}}
R^{\varpi-N}
\int_{B(x,R)}
|u(y)|^{p}
\mathrm{d}y
<\infty.
$$
\begin{lemma}
\label{lemma4}
$\left.\right.$
\cite[Theorem 1]{Palatucci2014}
For
$s\in (0,\frac{N}{2})$,
there exists
$C_{2}>0$
such that
for
$\iota$
and
$\vartheta$
satisfying
$\frac{2}{2^{*}_{s}}\leqslant\iota<1$,
$1\leqslant \vartheta<2^{*}_{s}=\frac{2N}{N-2s}$,
we have
\begin{align*}
\left(
\int_{\mathbb{R}^{N}}
|u|^{2^{*}_{s}}
\mathrm{d}x
\right)^{\frac{1}{2^{*}_{s}}}
\leqslant
C_{2}
\|u\|_{D}^{\iota}
\|u\|_{\mathcal{L}^{\vartheta,\frac{\vartheta(N-2s)}{2}}(\mathbb{R}^{N})}^{1-\iota},
\end{align*}
for any
$u\in D^{s,2}(\mathbb{R}^{N})$.
\end{lemma}
We introduce the energy functionals associated to problems $(\mathcal{P}_{i})$ $(i=1,2,3)$ by
\begin{equation*}
\begin{aligned}
I_{1}(u)
=&
\frac{1}{2}
\|u\|_{D}^{2}
-
\sum_{i=1}^{k}
\frac{1}{2\cdot2^{*}_{\alpha_{i}}}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\alpha_{i}}}|u(y)|^{2^{*}_{\alpha_{i}}}}{|x-y|^{\alpha_{i}}}
\mathrm{d}x
\mathrm{d}y
-
\frac{1}{2^{*}_{s}}
\int_{\mathbb{R}^{N}}
|u|^{2^{*}_{s}}
\mathrm{d}x,\\
I_{2}(u)
=&
\frac{1}{2}
\|u\|_{\zeta}^{2}
-
\frac{1}{2\cdot2^{*}_{\alpha}}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\alpha}}|u(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
-
\sum_{i=1}^{k}
\frac{1}{2^{*}_{s,\theta_{i}}}
\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{s,\theta_{i}}}}
{|x|^{\theta_{i}}}
\mathrm{d}x,\\
I_{3}(u)
=&
\frac{1}{2}
\|u\|_{D}^{2}
-
\frac{1}{2\cdot2^{*}_{\alpha}}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\alpha}}|u(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y.
\end{aligned}
\end{equation*}
The Nehari manifolds associated with
problem $(\mathcal{P}_{i})$
$(i=1,2,3)$,
which are defined by
$$\mathcal{N}^{i}=\{u\in D^{s,2}(\mathbb{R}^{N})|\langle I^{'}_{i}(u),u\rangle=0,~u\not=0 \},$$
and
$$
c_{0}^{i}
=\inf_{u\in\mathcal{N}^{i}}
I_{i}(u),
~
c_{1}^{i}
=\inf_{u\in D^{s,2}(\mathbb{R}^{N})}\max_{t\geqslant0}
I_{i}(tu)
~\mathrm{and}~
c^{i}
=
\inf_{\Upsilon^{i}\in\Gamma^{i}}
\max_{t\in [0,1]}
I_{i}(\Upsilon^{i}(t)),$$
where
$
\Gamma^{i}=
\{
\Upsilon^{i}\in C([0,1],D^{s,2}(\mathbb{R}^{N}))
:
\Upsilon^{i}(0)=0,
I_{i}(\Upsilon^{i}(1))<0
\}
$.
\section{Some key Lemmas}
\noindent
We show the refinement of Hardy-Littlewood-Sobolev inequality.
\begin{lemma}\label{lemma5}
For any
$s\in(0,\frac{N}{2})$
and
$\alpha\in(0,N)$,
there exists
$C_{3}>0$
such that
for
$\iota$
and
$\vartheta$
satisfying
$\frac{2}{2^{*}_{s}}\leqslant\iota<1$,
$1\leqslant \vartheta<2^{*}_{s}=\frac{2N}{N-2s}$,
we have
\begin{align*}
\left(
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\alpha}}|u(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
\right)^{\frac{1}{2^{*}_{\alpha}}}
\leqslant
C_{3}
\|u\|_{D}^{2\iota}
\|u\|_{\mathcal{L}^{\vartheta,\frac{\vartheta(N-2s)}{2}}(\mathbb{R}^{N})}^{2(1-\iota)},
\end{align*}
for any
$u\in D^{s,2}(\mathbb{R}^{N})$.
\end{lemma}
\noindent
{\bf Proof.}
Let
$\frac{2}{2^{*}_{s}}\leqslant\iota<1$
and
$1\leqslant \vartheta<2^{*}_{s}=\frac{2N}{N-2s}$
.
By Hardy-Littlewood-Sobolev inequality
and
Lemma \ref{lemma4},
we obtain
\begin{equation*}
\begin{aligned}
\left(
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\alpha}}|u(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
\right)^{\frac{1}{2^{*}_{\alpha}}}
\leqslant&
C(N,\alpha)^{\frac{1}{2^{*}_{\alpha}}}
\|u\|_{L^{2^{*}_{s}}(\mathbb{R}^{N})}^{2}\\
\leqslant&
C(N,\alpha)^{\frac{1}{2^{*}_{\alpha}}}
C_{2}^{2}
\|u\|_{D}^{2\iota}
\|u\|_{\mathcal{L}^{\vartheta,\frac{\vartheta(N-2s)}{2}}(\mathbb{R}^{N})}^{2(1-\iota)}.
\end{aligned}
\end{equation*}
\qed
We show some properties of fractional Coulomb--Sobolev space
$\mathcal{E}^{s,\alpha,2^{*}_{\alpha}}(\mathbb{R}^{N})$.
\begin{lemma}\label{lemma6}
Let
$( H_{1})$
hold.
If
$u\in \mathcal{E}^{s,\alpha_{j},2^{*}_{\alpha_{j}}}(\mathbb{R}^{N})$
$(j=1,\ldots,k)$,
then
\noindent
(i) $\|\cdot\|_{D}$
is an equivalent norm in
$\mathcal{E}^{s,\alpha_{j},2^{*}_{\alpha_{j}}}(\mathbb{R}^{N})$;
\noindent
(ii)
$
u\in
\bigcap_{i=1,i\not=j}^{k}
\mathcal{E}^{s,\alpha_{i},2^{*}_{\alpha_{i}}}(\mathbb{R}^{N})$;
\noindent
(iii)
$\|\cdot\|_{\mathcal{E},\alpha_{i}}$
are equivalent norms in
$\mathcal{E}^{s,\alpha_{j},2^{*}_{\alpha_{j}}}(\mathbb{R}^{N})$,
where
$i\not=j$
and
$i=1,\ldots,k$.
\end{lemma}
\noindent
{\bf Proof.}
\noindent
{\bf (1).}
Set
$j=1,\ldots,k$.
For any $u\in\mathcal{E}^{s,\alpha_{j},2^{*}_{\alpha{j}}}(\mathbb{R}^{N})$,
applying the definition of fractional Coulomb--Sobolev space,
we know
\begin{align}\label{14a}
\|u\|_{D}^{2}
\leqslant
\|u\|_{\mathcal{E},\alpha_{j}}^{2}
<\infty.
\end{align}
This implies that
$\mathcal{E}^{s,\alpha_{j},2^{*}_{\alpha_{j}}}(\mathbb{R}^{N})\subset D^{s,2}(\mathbb{R}^{N})$.
\noindent
According to $\mathcal{E}^{s,\alpha_{j},2^{*}_{\alpha_{j}}}(\mathbb{R}^{N})\subset D^{s,2}(\mathbb{R}^{N})$
and \eqref{10},
we have
\begin{align}\label{14b}
\|u\|_{\mathcal{E},\alpha_{j}}^{2}
\leqslant
\left(
1
+
\frac{1}{S_{0,\alpha_{j}}}
\right)
\|u\|_{D}^{2}.
\end{align}
Combining
\eqref{14a}
and
\eqref{14b},
we obtain
\begin{align}\label{14c}
\|u\|_{D}^{2}
\leqslant
\|u\|_{\mathcal{E},\alpha_{j}}^{2}
\leqslant
\left(
1
+
\frac{1}{S_{0,\alpha_{j}}}
\right)
\|u\|_{D}^{2}.
\end{align}
These imply that
$\|\cdot\|_{D}$
is an equivalent norm in
$\mathcal{E}^{s,\alpha_{j},2^{*}_{\alpha_{j}}}(\mathbb{R}^{N})$.
\noindent
{\bf (2).}
For any $u\in\mathcal{E}^{s,\alpha_{j},2^{*}_{\alpha{j}}}(\mathbb{R}^{N})\subset D^{s,2}(\mathbb{R}^{N})$,
by using \eqref{14a}
and
\eqref{10},
we know
\begin{align}\label{14d}
S_{0,\alpha_{i}}
\left(
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\alpha_{i}}}|u(y)|^{2^{*}_{\alpha_{i}}}}{|x-y|^{\alpha_{i}}}
\mathrm{d}x
\mathrm{d}y
\right)^{\frac{1}{2^{*}_{\alpha_{i}}}}
\leqslant
\|u\|_{D}^{2}
\leqslant
\|u\|_{\mathcal{E},\alpha_{j}}^{2}
<\infty,
\end{align}
where
$i\not=j$
and
$i=1,\ldots,k$.
The inequality
\eqref{14d}
gives that
\begin{align*}
\|u\|_{\mathcal{E},\alpha_{i}}^{2}
=
\|u\|_{D}^{2}
+
\left(
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\alpha_{i}}}|u(y)|^{2^{*}_{\alpha_{i}}}}{|x-y|^{\alpha_{i}}}
\mathrm{d}x
\mathrm{d}y
\right)^{\frac{1}{2^{*}_{\alpha_{i}}}}
<\infty.
\end{align*}
This implies that $u\in\bigcap_{i=1,i\not=j}^{k}\mathcal{E}^{s,\alpha_{i},2^{*}_{\alpha{i}}}(\mathbb{R}^{N})$.
\noindent
{\bf (3).}
For any $u\in\mathcal{E}^{s,\alpha_{j},2^{*}_{\alpha{j}}}(\mathbb{R}^{N})$,
by using \eqref{14b},
we have
\begin{equation*}
\begin{aligned}
\|u\|_{\mathcal{E},\alpha_{j}}^{2}
\leqslant
\left(
\frac{S_{0,\alpha_{j}}+1}{S_{0,\alpha_{j}}}
\right)
\|u\|_{D}^{2}
\leqslant
\left(
\frac{S_{0,\alpha_{j}}+1}{S_{0,\alpha_{j}}}
\right)
\|u\|_{\mathcal{E},\alpha_{i}}^{2},
\end{aligned}
\end{equation*}
which imply that
\begin{equation}\label{15}
\begin{aligned}
\left(
\frac{S_{0,\alpha_{j}}}{S_{0,\alpha_{j}}+1}
\right)
\|u\|_{\mathcal{E},\alpha_{i}}^{2}
\leqslant
\|u\|_{\mathcal{E},\alpha_{j}}^{2}
\leqslant
\left(
\frac{S_{0,\alpha_{j}}+1}{S_{0,\alpha_{j}}}
\right)
\|u\|_{\mathcal{E},\alpha_{i}}^{2},
\end{aligned}
\end{equation}
where
$0<\frac{S_{0,\alpha_{j}}}{S_{0,\alpha_{j}}+1}
<1<
\frac{S_{0,\alpha_{j}}+1}{S_{0,\alpha_{j}}}<\infty
$.
\qed
\begin{lemma}\label{lemma7}
[Endpoint refined Hardy--Sobolev inequality]
Let
$s\in(0,\frac{N}{2})$,
$\theta\in(0,2s)$
and
$\alpha\in(0,N)$.
Then there exists a constant
$C_{4}>0$
such that the inequality
\begin{equation*}
\begin{aligned}
\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{s,\theta}}}
{|x|^{\theta}}
\mathrm{d}x
\leqslant
C_{4}
\|u\|_{D}^{\frac{2(N+\theta-\alpha)}{N+2s-\alpha}}
\left(
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\alpha}}|u(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
\right)
^{\frac{2s-\theta}{N+2s-\alpha}},
\end{aligned}
\end{equation*}
holds for all
$u\in \mathcal{E}^{s,\alpha,2^{*}_{\alpha}}(\mathbb{R}^{N})$.
\end{lemma}
\noindent
{\bf Proof.}
For any
$u\in \mathcal{E}^{s,\alpha,2^{*}_{\alpha}}(\mathbb{R}^{N})$.
By using H\"{o}lder inequality and fractional Hardy inequality,
we obtain
\begin{equation}\label{16}
\begin{aligned}
\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{s,\theta}}}
{|x|^{\theta}}
\mathrm{d}x
=&
\int_{\mathbb{R}^{N}}
\frac{|u|^{\frac{\theta}{s}}}
{|x|^{\theta}}
\cdot
|u|^{\frac{2(N-\theta)}{N-2s}-\frac{\theta}{s}}
\mathrm{d}x\\
\leqslant&
\left(
\int_{\mathbb{R}^{N}}
\frac{|u|^{\frac{\theta}{s}\cdot\frac{2s}{\theta}}}
{|x|^{\theta\cdot\frac{2s}{\theta}}}
\mathrm{d}x
\right)
^{\frac{\theta}{2s}}
\left(
\int_{\mathbb{R}^{N}}
|u|^{\frac{(2s-\theta)N}{(N-2s)s}\cdot\frac{2s}{2s-\theta}}
\mathrm{d}x
\right)
^{1-\frac{\theta}{2s}}
\\
=&
\left(
\int_{\mathbb{R}^{N}}
\frac{|u|^{2}}
{|x|^{2s}}
\mathrm{d}x
\right)
^{\frac{\theta}{2s}}
\left(
\int_{\mathbb{R}^{N}}
|u|^{\frac{2N}{N-2s}}
\mathrm{d}x
\right)
^{\frac{2s-\theta}{2s}}
\\
\leqslant&
\left(
\frac{1}{\Lambda}
\right)
^{\frac{\theta}{2s}}
\|u\|_{D}
^{\frac{\theta}{s}}
\|u\|_{L^{2^{*}_{s}}(\mathbb{R}^{N})}^{\frac{N(2s-\theta)}{s(N-2s)}}.
\end{aligned}
\end{equation}
According to
Lemma \ref{lemma2}
and
(\ref{16}),
we know
\begin{equation*}
\begin{aligned}
&\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{s,\theta}}}
{|x|^{\theta}}
\mathrm{d}x\\
\leqslant&
C_{4}
\|u\|_{D}
^{\frac{\theta}{s}}
\|u\|_{D}^{\frac{(N-\alpha)(2s-\theta)}{s(N+2s-\alpha)}}
\left(
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\alpha}}|u(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
\right)
^{\frac{2s-\theta}{N+2s-\alpha}}\\
=&
C_{4}
\|u\|_{D}^{\frac{2(N+\theta-\alpha)}{N+2s-\alpha}}
\left(
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\alpha}}|u(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
\right)
^{\frac{2s-\theta}{N+2s-\alpha}}.
\end{aligned}
\end{equation*}
\qed
We study the refinement of Hardy--Sobolev inequality.
In \cite{Yang2015,Yang2017},
the authors also obtained the Refinement of Hardy--Sobolev inequality.
However,
their parameter $\tilde{\vartheta}$
satisfying (see \cite[Theorem 1]{Yang2015})
$$1\leqslant \tilde{\vartheta}<2^{*}_{s,\theta}.$$
It is easy to see that
$$
2^{*}_{s,\theta}
=\frac{2(N-\theta)}{N-2s}
<
\frac{2N}{N-2s}
=
2^{*}_{s},$$
for
$s\in (0,\frac{N}{2})$
and
$\theta\in(0,2s)$.
{\bf It is natural to ask the case of
$\tilde{\vartheta}\in[2^{*}_{s,\theta},2^{*}_{s})$}.
Our method extends the parameter
$\tilde{\vartheta}$
from
$[1,2^{*}_{s,\theta})$
to
$[1,2^{*}_{s})$
.
\begin{lemma}\label{lemma8}
$\left.\right.$
[Refinement of Hardy--Sobolev inequality]
For
$s\in (0,\frac{N}{2})$
and
$\theta\in(0,2s)$,
there exists
$C_{5}>0$
such that
for
$\iota$
and
$\vartheta$
satisfying
$\frac{2}{2^{*}_{s}}\leqslant\iota<1$,
\textcolor{red}{$1\leqslant \vartheta<2^{*}_{s}$},
we have
\begin{align*}
\left(
\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{s,\theta}}}{|x|^{\theta}}
\mathrm{d}x
\right)^{\frac{1}{2^{*}_{s,\theta}}}
\leqslant
C_{5}
\|u\|_{D}
^{\frac{\theta(N-2s)+\iota N(2s-\theta)}{2s(N-\theta)}}
\|u\|_{\mathcal{L}^{\vartheta,\frac{\vartheta(N-2s)}{2}}(\mathbb{R}^{N})}^{\frac{N(1-\iota)(2s-\theta)}{2s(N-\theta)}},
\end{align*}
for any
$u\in D^{s,2}(\mathbb{R}^{N})$.
\end{lemma}
\begin{proof}
Combining (\ref{16}) and Lemma \ref{lemma4},
we have
\begin{equation*}
\begin{aligned}
\left(
\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{s,\theta}}}{|x|^{\theta}}
\mathrm{d}x
\right)^{\frac{1}{2^{*}_{s,\theta}}}
\leqslant&
\left(
\frac{1}{\Lambda}
\right)
^{\frac{\theta(N-2s)}{4s(N-\theta)}}
\|u\|_{D}
^{\frac{\theta(N-2s)}{2s(N-\theta)}}
\|u\|_{L^{2^{*}_{s}}(\mathbb{R}^{N})}^{\frac{N(2s-\theta)}{2s(N-\theta)}}\\
\leqslant&
\left(
\frac{1}{\Lambda}
\right)
^{\frac{\theta(N-2s)}{4s(N-\theta)}}
\|u\|_{D}
^{\frac{\theta(N-2s)}{2s(N-\theta)}}
\left(
C_{2}
\|u\|_{D}^{\iota}
\|u\|_{\mathcal{L}^{\vartheta,\frac{\vartheta(N-2s)}{2}}}^{1-\iota}
\right)^{\frac{N(2s-\theta)}{2s(N-\theta)}}\\
=&
C_{5}
\|u\|_{D}
^{\frac{\theta(N-2s)+\iota N(2s-\theta)}{2s(N-\theta)}}
\|u\|_{\mathcal{L}^{\vartheta,\frac{\vartheta(N-2s)}{2}}(\mathbb{R}^{N})}
^{\frac{N(1-\iota)(2s-\theta)}{2s(N-\theta)}}.
\end{aligned}
\end{equation*}
\end{proof}
\begin{lemma}\label{lemma9}
Let
$s\in(0,\frac{N}{2})$
and
$0<\theta<\tilde{\theta}<2s$.
Then the inequality
\begin{equation*}
\begin{aligned}
\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{s,\theta}}}
{|x|^{\theta}}
\mathrm{d}x
\leqslant
\left(
\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{s,\tilde{\theta}}}}
{|x|^{\tilde{\theta}}}
\mathrm{d}x
\right)
^{\frac{\theta}{\tilde{\theta}}}
\left(
\int_{\mathbb{R}^{N}}
|u|^{2^{*}_{s}}
\mathrm{d}x
\right)
^{\frac{\tilde{\theta}-\theta}{\tilde{\theta}}},
\end{aligned}
\end{equation*}
holds for all
$u\in D^{s,2}(\mathbb{R}^{N})$.
\end{lemma}
\noindent
{\bf Proof.}
For any
$u\in D^{s,2}(\mathbb{R}^{N})$.
By using H\"{o}lder inequality and $0<\theta<\tilde{\theta}<2s$,
we obtain
\begin{equation*}
\begin{aligned}
\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{s,\theta}}}
{|x|^{\theta}}
\mathrm{d}x
=&
\int_{\mathbb{R}^{N}}
\frac{|u|^{\frac{\theta}{\tilde{\theta}}\cdot\frac{2(N-\tilde{\theta})}{N-2s}}}
{|x|^{\theta}}
\cdot
|u|^{\frac{2N}{N-2s}\cdot\frac{\tilde{\theta}-\theta}{\tilde{\theta}}}
\mathrm{d}x\\
\leqslant&
\left(
\int_{\mathbb{R}^{N}}
\frac{|u|^{\frac{\theta}{\tilde{\theta}}\cdot\frac{2(N-\tilde{\theta})}{N-2s}\cdot\frac{\tilde{\theta}}{\theta}}}
{|x|^{\theta\cdot\frac{\tilde{\theta}}{\theta}}}
\mathrm{d}x
\right)
^{\frac{\theta}{\tilde{\theta}}}
\left(
\int_{\mathbb{R}^{N}}
|u|^{\frac{2N}{N-2s}\cdot\frac{\tilde{\theta}-\theta}{\tilde{\theta}}\cdot\frac{\tilde{\theta}}{\tilde{\theta}-\theta}}
\mathrm{d}x
\right)
^{1-\frac{\theta}{\tilde{\theta}}}
\\
=&
\left(
\int_{\mathbb{R}^{N}}
\frac{|u|^{\frac{2(N-\tilde{\theta})}{N-2s}}}
{|x|^{\tilde{\theta}}}
\mathrm{d}x
\right)
^{\frac{\theta}{\tilde{\theta}}}
\left(
\int_{\mathbb{R}^{N}}
|u|^{\frac{2N}{N-2s}}
\mathrm{d}x
\right)
^{\frac{\tilde{\theta}-\theta}{\tilde{\theta}}}.
\end{aligned}
\end{equation*}
\qed
\begin{lemma}\label{lemma10}
Let
$s\in(0,\frac{N}{2})$,
$0<\bar{\theta}<\theta<2s$
and
$2\theta-\bar{\theta}<2s$.
Then the inequality
\begin{equation*}
\begin{aligned}
\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{s,\theta}}}
{|x|^{\theta}}
\mathrm{d}x
\leqslant
\left(
\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{s,\bar{\theta}}}}
{|x|^{\bar{\theta}}}
\mathrm{d}x
\right)
^{\frac{1}{2}}
\left(
\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{s,2\theta-\bar{\theta}}}}
{|x|^{2\theta-\bar{\theta}}}
\mathrm{d}x
\right)
^{\frac{1}{2}},
\end{aligned}
\end{equation*}
holds for all
$u\in D^{s,2}(\mathbb{R}^{N})$.
\end{lemma}
\noindent
{\bf Proof.}
For any
$u\in D^{s,2}(\mathbb{R}^{N})$.
By using H\"{o}lder inequality and $0<\bar{\theta}<\theta<2s$,
we obtain
\begin{equation*}
\begin{aligned}
\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{s,\theta}}}
{|x|^{\theta}}
\mathrm{d}x
=&
\int_{\mathbb{R}^{N}}
\frac{|u|^{\frac{N-\bar{\theta}}{N-2s}}}
{|x|^{\frac{\bar{\theta}}{2}}}
\cdot
\frac{|u|^{\frac{N-(2\theta-\bar{\theta})}{N-2s}}}
{|x|^{\theta-\frac{\bar{\theta}}{2}}}
\mathrm{d}x\\
\leqslant&
\left(
\int_{\mathbb{R}^{N}}
\frac{|u|^{\frac{2(N-\bar{\theta})}{N-2s}}}
{|x|^{\bar{\theta}}}
\mathrm{d}x
\right)
^{\frac{1}{2}}
\left(
\int_{\mathbb{R}^{N}}
\frac{|u|^{\frac{2[N-(2\theta-\bar{\theta})]}{N-2s}}}
{|x|^{2\theta-\bar{\theta}}}
\mathrm{d}x
\right)
^{\frac{1}{2}}.
\end{aligned}
\end{equation*}
Since
$0<2\theta-\bar{\theta}<2s$,
we get
\begin{equation*}
\begin{aligned}
\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{s,\theta}}}
{|x|^{\theta}}
\mathrm{d}x
\leqslant
\left(
\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{s,\bar{\theta}}}}
{|x|^{\bar{\theta}}}
\mathrm{d}x
\right)
^{\frac{1}{2}}
\left(
\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{s,2\theta-\bar{\theta}}}}
{|x|^{2\theta-\bar{\theta}}}
\mathrm{d}x
\right)
^{\frac{1}{2}}.
\end{aligned}
\end{equation*}
\qed
\begin{lemma}\label{lemma22}
Let
$N\geqslant3$,
$\alpha\in(0, N)$,
$s\in(0,1)$
and
$\zeta\in[0,\Lambda)$
hold.
Then we have
\begin{equation*}
\begin{aligned}
S_{0,\alpha}\geqslant
S_{\zeta,\alpha}
\geqslant
\left(1-\frac{\zeta}{\Lambda}\right)
S_{0,\alpha},
\end{aligned}
\end{equation*}
and
\begin{equation*}
\begin{aligned}
H_{0,\theta}
\geqslant
H_{\zeta,\theta}
\geqslant
\left(1-\frac{\zeta}{\Lambda}\right)
H_{0,\theta}.
\end{aligned}
\end{equation*}
\end{lemma}
\noindent
{\bf Proof.}
For
$\zeta\in[0,\Lambda)$
and
$u\in D^{s,2}(\mathbb{R}^{N})$,
$u\not\equiv0$.
We set
\begin{equation*}
\begin{aligned}
F_{\zeta}(u):=
\frac{\|u\|_{\zeta}^{2}}
{\left(
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\alpha}}|u(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
\right)^{\frac{1}{2^{*}_{\alpha}}}},
\end{aligned}
\end{equation*}
clearly,
for a fixed $u$,
$F_{\zeta}(u)$
is decreasing with respect to
$\zeta$.
Moreover, for any fixed
$\zeta\in[0,\Lambda)$,
we denote by
$u_{\zeta}\in D^{s,2}(\mathbb{R}^{N})$
a function such that \eqref{10} is achieved,
that is
$S_{\zeta,\alpha}=F_{\zeta}(u_{\zeta})$.
Let
$0<\zeta_{1}<\zeta_{2}<\Lambda$.
Then
\begin{equation*}
\begin{aligned}
S_{\zeta_{1},\alpha}=
F_{\zeta_{1}}(u_{\zeta_{1}})
=&
\frac{\|u_{\zeta_{1}}\|_{D}^{2}
-
\zeta_{1}
\int_{\mathbb{R}^{N}}
\frac{ |u_{\zeta_{1}}|^{2}}{|x|^{2s}}
\mathrm{d}x}
{\left(
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u_{\zeta_{1}}(x)|^{2^{*}_{\alpha}}|u_{\zeta_{1}}(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
\right)^{\frac{1}{2^{*}_{\alpha}}}}\\
>&
\frac{\|u_{\zeta_{1}}\|_{D}^{2}
-
\zeta_{2}
\int_{\mathbb{R}^{N}}
\frac{ |u_{\zeta_{1}}|^{2}}{|x|^{2s}}
\mathrm{d}x}
{\left(
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u_{\zeta_{1}}(x)|^{2^{*}_{\alpha}}|u_{\zeta_{1}}(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
\right)^{\frac{1}{2^{*}_{\alpha}}}}
\geqslant
S_{\zeta_{2},\alpha}.
\end{aligned}
\end{equation*}
Let
$0<\zeta<\Lambda$.
Since the best constant in the Hardy inequality is not achieved,
we get
\begin{equation*}
\begin{aligned}
S_{\zeta,\alpha}=
F_{\zeta}(u_{\zeta})
>&
\left(1-\frac{\zeta}{\Lambda}\right)
\frac{\|u_{\zeta}\|_{D}^{2}}
{\left(
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u_{\zeta}(x)|^{2^{*}_{\alpha}}|u_{\zeta}(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
\right)^{\frac{1}{2^{*}_{\alpha}}}}\\
\geqslant&
\left(1-\frac{\zeta}{\Lambda}\right)
S_{0,\alpha}.
\end{aligned}
\end{equation*}
\qed
\section{The proof of theorem \ref{theorem2}}
In this section, we show the existence of nonnegative solution of problems $(\mathcal{P}_{2})$.
In Lemma \ref{lemma11}--Lemma \ref{lemma14},
we will prove some properties of the Nehari manifolds associated with
problems $(\mathcal{P}_{2})$ and $(\mathcal{P}_{4})$.
\begin{lemma}\label{lemma11}
Assume that the assumptions of Theorem
\ref{theorem2}
hold.
Then
$$c_{0}^{2}=\inf_{u\in\mathcal{N}^{2}}I_{2}(u)>0.$$
\end{lemma}
\noindent
{\bf Proof.}
\noindent
{\bf Step 1.}
We claim that
any limit point of a sequence in
$\mathcal{N}^{2}$
is different from zero.
According to
$\langle I^{'}_{2}(u),u\rangle=0$,
(\ref{10})
and
(\ref{11}),
for any $u\in\mathcal{N}^{2}$,
we obtain
\begin{equation*}
\begin{aligned}
0
=
\langle I^{'}_{2}(u),u\rangle
\geqslant&
\|u\|^{2}_{\zeta}
-
\frac{1}{S_{\zeta,\alpha} ^{2^{*}_{\alpha}}}
\|u\|_{\zeta}^{2\cdot 2^{*}_{\alpha} }
-
\sum_{i=1}^{k}
\frac{1}{H_{\zeta,\theta_{i}}^{\frac{2^{*}_{s,\theta_{i}} }{2}}}
\|u\|_{\zeta}^{2^{*}_{s,\theta_{i}} }.
\end{aligned}
\end{equation*}
From above expression,
we have
\begin{equation}\label{17}
\begin{aligned}
\|u\|^{2}_{\zeta}
\leqslant
\frac{1}{S_{\zeta,\alpha} ^{2^{*}_{\alpha}}}
\|u\|_{\zeta}^{2\cdot 2^{*}_{\alpha} }
+
\sum_{i=1}^{k}
\frac{1}{H_{\zeta,\theta_{i}}^{\frac{2^{*}_{s,\theta_{i}} }{2}}}
\|u\|_{\zeta}^{2^{*}_{s,\theta_{i}} }.
\end{aligned}
\end{equation}
Set
$$\kappa
:=
\frac{1}{S_{\zeta,\alpha} ^{2^{*}_{\alpha}}}
+
\sum_{i=1}^{k}
\frac{1}{H_{\zeta,\theta_{i}}^{\frac{2^{*}_{s,\theta_{i}} }{2}}}
.$$
Applying
(\ref{10})
and
(\ref{11}),
we get
$$
0<\kappa<\infty.$$
From
($H_{1}$),
we know
$$2^{*}_{s,\theta_{k}}<\cdots<2^{*}_{s,\theta_{1}}<2\cdot2^{*}_{\alpha}.$$
Now the proof of Step 1 is divided into two cases:
(i)
$\|u\|_{\zeta}\geqslant1$;
(ii)
$\|u\|_{\zeta}<1$.
\noindent
{\bf Case (i)}$\|u\|_{\zeta}\geqslant1$.
From
(\ref{15}),
we have
\begin{equation*}
\begin{aligned}
\|u\|_{\zeta}^{2}
\leqslant&
\kappa
\|u\|_{\zeta}^{2\cdot2^{*}_{\alpha}},
\end{aligned}
\end{equation*}
which implies that
\begin{equation}\label{18}
\begin{aligned}
\|u\|_{\zeta}
\geqslant
\kappa
^{\frac{1}{2-2\cdot2^{*}_{\alpha}}}.
\end{aligned}
\end{equation}
\noindent
{\bf Case (ii)}$\|u\|_{\zeta}<1$.
From
(\ref{17}),
we know
\begin{equation}\label{19}
\begin{aligned}
\|u\|_{\zeta}
\geqslant
\kappa
^{\frac{1}{2-2^{*}_{s,\theta_{k}} }}.
\end{aligned}
\end{equation}
Combining the Cases (i) and (ii),
according to
(\ref{18})
and
(\ref{19}),
we deduce that
\begin{equation}\label{20}
\begin{aligned}
\|u\|_{\zeta}
\geqslant
\begin{cases}
\kappa
^{\frac{1}{2-2\cdot2^{*}_{\alpha} }},
&\kappa<1,\\
\kappa
^{\frac{1}{2-2^{*}_{s,\theta_{k}} }},
&\kappa\geqslant1.
\end{cases}
\end{aligned}
\end{equation}
Hence,
we know that
any limit point of a sequence in $\mathcal{N}^{2}$ is different from zero.
\noindent
{\bf Step 2.}
Now,
we claim that
$I_{2}$
is bounded from below on
$\mathcal{N}^{2}$.
For any
$u\in \mathcal{N}^{2}$,
by using
(\ref{20}),
we get
\begin{equation*}
\begin{aligned}
I_{2}(u)
\geqslant&
\left(
\frac{1}{2}
-
\frac{1}{2^{*}_{s,\theta_{k}}}
\right)
\|u\|^{2}_{\zeta}
\geqslant
\begin{cases}
\left(
\frac{1}{2}
-
\frac{1}{2^{*}_{s,\theta_{k}}}
\right)
\kappa
^{\frac{2}{2-2\cdot2^{*}_{\alpha} }},
&\kappa\leqslant1,\\
\left(
\frac{1}{2}
-
\frac{1}{2^{*}_{s,\theta_{k}}}
\right)
\kappa
^{\frac{2}{2-2^{*}_{s,\theta_{k}}}},
&\kappa>1.
\end{cases}
\end{aligned}
\end{equation*}
Therefore,
$I_{2}$
is bounded from below on
$\mathcal{N}^{2}$,
and
$c_{0}^{2}>0$.
\qed
\begin{lemma}\label{lemma12}
Assume that the assumptions of Theorem
\ref{theorem2}
hold.
Then
\noindent
(i)
for each
$u\in D^{s,2}(\mathbb{R}^{N})\setminus\{0\}$,
there exists a unique
$t_{u}>0$
such that
$t_{u}u\in \mathcal{N}^{2}$;
\noindent
(ii)
$c_{0}^{2}=c_{1}^{2}=c^{2}>0$.
\end{lemma}
\noindent
{\bf Proof.}
The proof is standard, so we sketch it. Further details can be derived as in the proofs of Theorem 4.1 and 4.2 in
\cite{Willem1996},
we omit it.
\qed
Similar to Lemma \ref{lemma11} and Lemma \ref{lemma12},
we also have the following result.
\begin{lemma}\label{lemma14}
Assume that the assumptions of Theorem
\ref{theorem2}
hold.
For any
$u\in\mathcal{N}^{3}$,
we have
$$\|u\|_{D}^{2}\geqslant S_{0,\alpha}^{\frac{2N-\alpha}{N+2s-\alpha}},$$
and
$$c_{0}^{3}=\inf_{u\in\mathcal{N}^{3}}I_{3}(u)>0.$$
\end{lemma}
We show that the functional $I_{2}$ satisfies the Mountain--Pass geometry, and
estimate the Mountain--Pass levels.
\begin{lemma}\label{lemma15}
Assume that the assumptions of Theorem
\ref{theorem2}
hold.
Then
there exists a
$(PS)_{c^{2}}$
sequence of
$I_{2}$
at level
$c^{2}$,
where
$$0<c^{2}<c^{2,*}=
\min
\left\{
\frac{N+2s-\alpha}{2(2N-\alpha)}
S_{\zeta,\alpha}^{\frac{2N-\alpha}{N+2s-\alpha}}
,
\frac{2s-\theta_{1}}{2(N-\theta_{1})}
H_{\zeta,\theta_{1}} ^{\frac{N-\theta_{1}}{2s-\theta_{1}}}
,
\ldots
,
\frac{2s-\theta_{k}}{2(N-\theta_{k})}
H_{\zeta,\theta_{k}} ^{\frac{N-\theta_{k}}{2s-\theta_{k}}}
\right\}.$$
\end{lemma}
\noindent
{\bf Proof.}
The proof is standard, so we sketch it. Further details can be derived as in the proofs of Theorem 2 in \cite{Pucci2009}, we omit it.
\qed
The following result implies the non--vanishing of
$(PS)_{c^{2}}$
sequence.
\begin{lemma}\label{lemma16}
Assume that the assumptions of Theorem
\ref{theorem2}
hold.
Let
$\{u_{n}\}$
be a
$(PS)_{c^{2}}$
sequence of
$I_{2}$
with
$c^{2}\in(0,c^{2,*})$,
then
\begin{equation*}
\begin{aligned}
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}(x)|^{2^{*}_{\alpha}}|u_{n}(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y>0,
\end{aligned}
\end{equation*}
and
\begin{equation*}
\begin{aligned}
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}|^{2^{*}_{s,\theta_{i}}}}
{|x|^{\theta_{i}}}
\mathrm{d}x
>0,(i=1,\ldots,k).
\end{aligned}
\end{equation*}
\end{lemma}
\noindent
{\bf Proof.}
The proof of this Lemma is divided into four cases:
\noindent
(1)
$
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}|^{2^{*}_{s,\theta_{1}}}}
{|x|^{\theta_{1}}}
\mathrm{d}x
>0
$;
\noindent
(2)
$
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}|^{2^{*}_{s,\theta_{k}}}}
{|x|^{\theta_{k}}}
\mathrm{d}x
>0
$;
\noindent
(3)
$
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}|^{2^{*}_{s,\theta_{j}}}}
{|x|^{\theta_{j}}}
\mathrm{d}x
>0
$,
$(j=2,\ldots,k-1)$;
\noindent
(4)
$
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}(x)|^{2^{*}_{\alpha}}|u_{n}(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y>0
$.
\noindent
{\bf Case 1.}
It is easy to see that
$\{u_{n}\}$
is uniformly bounded in
$D^{s,2}(\mathbb{R}^{N})$.
Suppose on the contrary that
\begin{equation}\label{21}
\begin{aligned}
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}|^{2^{*}_{s,\theta_{1}}}}
{|x|^{\theta_{1}}}
\mathrm{d}x
=0.
\end{aligned}
\end{equation}
From
($H_{2}$),
we know
\begin{equation}\label{22}
\begin{aligned}
0<2\theta_{2}-\theta_{1}<\cdots<2\theta_{k}-\theta_{1}<2s.
\end{aligned}
\end{equation}
Since
$\{u_{n}\}$
is uniformly bounded in
$D^{s,2}(\mathbb{R}^{N})$,
there exists a constant
$0<C<\infty$
such that
$\|u_{n}\|_{D}\leqslant C$.
Applying
(\ref{22})
and
(\ref{11}),
we obtain
\begin{equation}\label{23}
\begin{aligned}
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}|^{2^{*}_{s,2\theta_{i}-\theta_{1}}}}
{|x|^{2\theta_{i}-\theta_{1}}}
\mathrm{d}x
\leqslant C,
(i=2,\ldots,k).
\end{aligned}
\end{equation}
According to
Lemma
\ref{lemma10},
(\ref{21})
and
(\ref{23}),
we obtain
\begin{equation}\label{24}
\begin{aligned}
&\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}|^{2^{*}_{s,\theta_{i}}}}
{|x|^{\theta_{i}}}
\mathrm{d}x\\
\leqslant&
\left(
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}|^{2^{*}_{s,\theta_{1}}}}
{|x|^{\theta_{1}}}
\mathrm{d}x
\right)
^{\frac{1}{2}}
\left(
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}|^{2^{*}_{s,2\theta_{i}-\theta_{1}}}}
{|x|^{2\theta_{i}-\theta_{1}}}
\mathrm{d}x
\right)
^{\frac{1}{2}}\\
=&0
~(i=2,\ldots,k).
\end{aligned}
\end{equation}
By using
(\ref{21}),
(\ref{24})
and the definition of
$(PS)_{c^{2}}$
sequence,
we obtain
$$
c^{2}
+
o(1)
=
\frac{1}{2}
\|u_{n}\|_{\zeta}^{2}
-
\frac{1}{2\cdot2^{*}_{\alpha}}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}(x)|^{2^{*}_{\alpha}}|u_{n}(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y,$$
and
$$o(1)
=
\|u_{n}\|_{\zeta}^{2}
-
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}(x)|^{2^{*}_{\alpha}}|u_{n}(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y.
$$
These yield
$$
c^{2}
+
o(1)
=
\frac{N+2s-\alpha}{2(2N-\alpha)}
\|u_{n}\|_{\zeta}^{2}.$$
Moreover,
\begin{equation*}
\begin{aligned}
S_{\zeta,\alpha}
\left(
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}(x)|^{2^{*}_{\alpha}}|u_{n}(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
\right)^{\frac{1}{2^{*}_{\alpha}}}
\leqslant
\|u_{n}\|^{2}_{\zeta},
\end{aligned}
\end{equation*}
which implies that
\begin{equation*}
\begin{aligned}
S_{\zeta,\alpha}^{\frac{2N-\alpha}{N+2s-\alpha}}
\leqslant
\|u_{n}\|^{2}_{\zeta}.
\end{aligned}
\end{equation*}
Therefore,
we obtain
\begin{equation*}
\begin{aligned}
\frac{N+2s-\alpha}{2(2N-\alpha)}
S_{\zeta,\alpha}^{\frac{2N-\alpha}{N+2s-\alpha}}
\leqslant
c^{2}.
\end{aligned}
\end{equation*}
This is a contradiction.
\noindent
{\bf Case 2.}
Suppose on the contrary that
\begin{equation}\label{25}
\begin{aligned}
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}|^{2^{*}_{s,\theta_{k}}}}
{|x|^{\theta_{k}}}
\mathrm{d}x
=0.
\end{aligned}
\end{equation}
By using
(\ref{10})
and
$\|u_{n}\|_{D}\leqslant C$,
we have
\begin{equation}\label{26}
\begin{aligned}
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
|u_{n}|^{2^{*}_{s}}
\mathrm{d}x
\leqslant C.
\end{aligned}
\end{equation}
Applying
($H_{2}$),
Lemma
\ref{lemma9},
(\ref{25})
and
(\ref{26}),
we obtain
\begin{equation}\label{27}
\begin{aligned}
&
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}|^{2^{*}_{s,\theta_{i}}}}
{|x|^{\theta_{i}}}
\mathrm{d}x\\
\leqslant&
\left(
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}|^{2^{*}_{s,\theta_{k}}}}
{|x|^{\theta_{k}}}
\mathrm{d}x
\right)
^{\frac{\theta_{i}}{\theta_{k}}}
\left(
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
|u_{n}|^{2^{*}_{s}}
\mathrm{d}x
\right)
^{\frac{\theta_{k}-\theta_{i}}{\theta_{k}}}\\
=&0
~(i=1,\ldots,k-1).
\end{aligned}
\end{equation}
By using
(\ref{25}),
(\ref{27})
and the definition of
$(PS)_{c^{2}}$
sequence,
similar to Case 1,
we get
$
\frac{N+2s-\alpha}{2(2N-\alpha)}
S_{\alpha}^{\frac{2N-\alpha}{N+2s-\alpha}}
\leqslant
c^{2}.
$
This is a contradiction.
\noindent
{\bf Case 3.}
Set
$j\in [2,k-1]$.
Suppose on the contrary that
\begin{equation}\label{}
\begin{aligned}
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}|^{2^{*}_{s,\theta_{j}}}}
{|x|^{\theta_{j}}}
\mathrm{d}x
=0.
\end{aligned}
\end{equation}
From
($H_{2}$),
we know
\begin{equation*}
\begin{aligned}
0<2\theta_{j+1}-\theta_{j}<\cdots<2\theta_{k}-\theta_{j}<2\theta_{k}-\theta_{1}<2s.
\end{aligned}
\end{equation*}
Similar to
\eqref{24},
we obtain
\begin{equation*}
\begin{aligned}
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}|^{2^{*}_{s,\theta_{i}}}}
{|x|^{\theta_{i}}}
\mathrm{d}x
=0
~(i=j+1,\ldots,k).
\end{aligned}
\end{equation*}
Similar to \eqref{27},
we have
\begin{equation*}
\begin{aligned}
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}|^{2^{*}_{s,\theta_{i}}}}
{|x|^{\theta_{i}}}
\mathrm{d}x
=&0
~(i=1,\ldots,j-1).
\end{aligned}
\end{equation*}
Similar to Case 1,
we get
$
\frac{N+2s-\alpha}{2(2N-\alpha)}
S_{\alpha}^{\frac{2N-\alpha}{N+2s-\alpha}}
\leqslant
c^{2}.
$
This is a contradiction.
\noindent
{\bf Case 4.}
Suppose on the contrary that
\begin{equation}\label{28}
\begin{aligned}
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}(x)|^{2^{*}_{\alpha}}|u_{n}(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
=0.
\end{aligned}
\end{equation}
By using
$\|u_{n}\|_{D}\leqslant C$,
(\ref{10})
and the definition of
fractional Coulomb--Sobolev space,
we obtain
$u_{n}\in \mathcal{E}^{s,\alpha,2^{*}_{\alpha}}(\mathbb{R}^{N})$.
Applying Lemma \ref{lemma7} and (\ref{28}),
we have
\begin{equation}\label{29}
\begin{aligned}
&
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}|^{2^{*}_{s,\theta_{i}}}}
{|x|^{\theta_{i}}}
\mathrm{d}x\\
\leqslant&
C
\left(
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}(x)|^{2^{*}_{\alpha}}|u_{n}(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
\right)
^{\frac{2s-\theta_{i}}{N+2s-\alpha}}
=0
~(i=1,\ldots,k).
\end{aligned}
\end{equation}
Applying
(\ref{28})
and
(\ref{29}),
we get
$$
c^{2}
+
o(1)
=
\frac{1}{2}
\|u_{n}\|_{\zeta}^{2},
$$
and
$$o(1)
=
\|u_{n}\|_{\zeta}^{2},
$$
which imply that
$c^{2}=0$.
This contradicts with $c^{2}>0$.
\qed
In next lemma,
we show that
$c^{3}_{0}>c_{0}^{2}$.
This result plays a key role in the proof of Theorem \ref{theorem2}.
\begin{lemma}\label{lemma17}
Assume that the assumptions of Theorem
\ref{theorem2}
hold.
Then $c^{3}_{0}>c_{0}^{2}$.
\end{lemma}
\begin{proof}
Consider the family of functions
$\{U_{\sigma}\}$
defined as
\begin{equation*}
\begin{aligned}
U_{\sigma}(x)
=
\sigma^{-\frac{N-2s}{2}}
\left(
\frac{u^{*}(\frac{x}{\sigma})}{\|u^{*}\|_{2^{*}_{s}}}
\right),
\end{aligned}
\end{equation*}
where
$$u^{*}(x)=\frac{\varpi}
{(1+|x|^{2})^{\frac{N -2s}{2}}},
~\varpi\in \mathbb{R}\setminus\{0\}.$$
Then,
for each
$\sigma>0$,
$U_{\sigma}$
(see \cite{Mukherjee2017Fractional})
satisfies
\begin{equation*}
\begin{aligned}
(-\Delta)^{s}
U_{\sigma}
=
|U_{\sigma}|^{2^{*}_{s}-2}U_{\sigma},
\mathrm{~in~}
\mathbb{R}^{N}.
\end{aligned}
\end{equation*}
Set
\begin{equation*}
\begin{aligned}
U_{\sigma,\alpha}
(x)
=
S_{0,0}
^{\frac{(N-\alpha)(2s-N)}{4(N-\alpha+2s)}}
C(N,\alpha)
^{\frac{2s-N}{2(N-\alpha+2s)}}
U_{\sigma}(x).
\end{aligned}
\end{equation*}
Then,
for each
$\sigma>0$,
$U_{\sigma,\alpha}$
satisfies
\begin{equation*}
\begin{aligned}
(-\Delta)^{s}
U_{\sigma,\alpha}
=
\left(
\int_{\mathbb{R}^{N}}
\frac{|U_{\sigma,\alpha}|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}y
\right)
|U_{\sigma,\alpha}|^{2^{*}_{\alpha}-2}U_{\sigma,\alpha},
\mathrm{~in~}
\mathbb{R}^{N}.
\end{aligned}
\end{equation*}
Hence,
we know that
$U_{\sigma,\alpha}\in \mathcal{N}^{3}$,
and
\begin{equation}\label{30}
\begin{aligned}
\|U_{\sigma,\alpha}\|_{D}^{2}
=
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{
|U_{\sigma,\alpha}(y)|^{2^{*}_{\alpha}}
|U_{\sigma,\alpha}(x)|^{2^{*}_{\alpha}}
}
{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
=
S_{0,\alpha}^{\frac{2N-\alpha}{N-\alpha+2s}}.
\end{aligned}
\end{equation}
Now,
we show that
$$c^{3}_{0}=\frac{N+2s-\alpha}{2(2N-\alpha)}S_{0,\alpha}^{\frac{2N-\alpha}{N+2s-\alpha}}.$$
Suppose on the contrary that
$c^{3}_{0}<\frac{N+2s-\alpha}{2(2N-\alpha)}S_{0,\alpha}^{\frac{2N-\alpha}{N+2s-\alpha}}$.
Then there exists
$\ddot{u}$
satisfies
$I_{3}(\ddot{u})=c_{0}^{3}$
and
$\ddot{u}\in \mathcal{N}^{3}$.
We get
\begin{equation}\label{31}
\begin{aligned}
c_{0}^{3}
=
I_{3}(\ddot{u})
-
\frac{1}{2\cdot 2^{*}_{\alpha}}
\langle I_{3}^{'}(\ddot{u}),\ddot{u}\rangle
=
\left(
\frac{1}{2^{*}}
-
\frac{1}{2\cdot 2^{*}_{\alpha}}
\right)
\|\ddot{u}\|^{2}_{D}.
\end{aligned}
\end{equation}
Combining
\eqref{30}
and
\eqref{31},
we know
\begin{equation*}
\begin{aligned}
\left(
\frac{1}{2^{*}}
-
\frac{1}{2\cdot 2^{*}_{\alpha}}
\right)
\|U_{\sigma,\alpha}\|^{2}_{D}
=
\frac{N+2s-\alpha}{2(2N-\alpha)}S_{0,\alpha}^{\frac{2N-\alpha}{N+2s-\alpha}}
>
c_{0}^{3}
=
\left(
\frac{1}{2^{*}}
-
\frac{1}{2\cdot 2^{*}_{\alpha}}
\right)
\|\ddot{u}\|^{2}_{D},
\end{aligned}
\end{equation*}
which implies that
\begin{equation*}
\begin{aligned}
S_{0,\alpha}^{\frac{2N-\alpha}{N-\alpha+2s}}
=
\|U_{\sigma,\alpha}\|^{2}_{D}
>
\|\ddot{u}\|^{2}_{D}.
\end{aligned}
\end{equation*}
This contradicts with
$
\|\ddot{u}\|^{2}_{D}
\geqslant
S_{0,\alpha}^{\frac{2N-\alpha}{N-\alpha+2s}}
$(see Lemma \ref{lemma14}).
Hence,
we know that
$$c^{3}_{0}=\frac{N+2s-\alpha}{2(2N-\alpha)}S_{0,\alpha}^{\frac{2N-\alpha}{N+2s-\alpha}}\geqslant\frac{N+2s-\alpha}{2(2N-\alpha)}S_{\zeta,\alpha}^{\frac{2N-\alpha}{N+2s-\alpha}}>c^{2}=c^{2}_{0}.$$
\end{proof}
\noindent
{\bf The proof of Theorem \ref{theorem2}:}
We divide our proof into five steps.
\noindent
{\bf Step 1.}
Since
$\{u_{n}\}$
is a bounded sequence in
$D^{s,2}(\mathbb{R}^{N})$,
up to a subsequence,
we can assume that
\begin{align*}
&
u_{n}\rightharpoonup u,
~
\mathrm{in}
~
D^{s,2}(\mathbb{R}^{N}),
~~
u_{n}\rightarrow u,
~
\mathrm{a.e. ~in}
~
\mathbb{R}^{N},\\
&u_{n}\rightarrow u,
~
\mathrm{in}
~
L^{r}_{loc}(\mathbb{R}^{N})
~
\mathrm{for~all}
~
r\in[1,2^{*}_{s}).
\end{align*}
According to
Lemma
\ref{lemma5},
Lemma
\ref{lemma8}
and
Lemma
\ref{lemma16},
there exists
$C>0$
such that
$$
\|u_{n}\|_{\mathcal{L}^{2,N-2s}(\mathbb{R}^{N})}\geqslant C>0.
$$
On the other hand,
since the sequence is bounded in
$D^{s,2}(\mathbb{R}^{N})$
and
$D^{s,2}(\mathbb{R}^{N})\hookrightarrow L^{2^{*}_{s}}(\mathbb{R}^{N})\hookrightarrow \mathcal{L}^{2,N-2s}(\mathbb{R}^{N})$,
we have
$$
\|u_{n}\|_{\mathcal{L}^{2,N-2s}(\mathbb{R}^{N})}\leqslant C,
$$
for some
$C>0$
independent of $n$.
Hence, there exists a positive constant which we denote again by $C$ such
that for any $n$ we obtain
$$
C
\leqslant
\|u_{n}\|_{\mathcal{L}^{2,N-2s}(\mathbb{R}^{N})}\leqslant C^{-1}.
$$
So we may find
$\sigma_{n} > 0$
and
$x_{n}\in \mathbb{R}^{N}$
such that
$$
\frac{1}{\sigma_{n}^{2s}}
\int_{B(x_{n},\sigma_{n})}
|u_{n}(y)|^{2}
\mathrm{d}y
\geqslant
\|u_{n}\|_{\mathcal{L}^{2,N-2s}(\mathbb{R}^{N})}^{2}
-
\frac{C}{2n}
\geqslant
C_{6}>0.
$$
Let
$\bar{u}_{n}(x)=\sigma_{n}^{\frac{N-2s}{2}}u_{n}(x_{n}+\sigma_{n}x)$.
We may readily verify that
$$\widetilde{I_{2}}(\bar{u}_{n})=I_{2}(u_{n})\rightarrow c^{2}
~\mathrm{and}~
\widetilde{I_{2}}^{'}(\bar{u}_{n})
\rightarrow0
~\mathrm{as}~n\rightarrow\infty,$$
where
\begin{equation*}
\begin{aligned}
\widetilde{I_{2}}(\bar{u}_{n})
=&
\frac{1}{2}
\|\bar{u}_{n}\|_{D}^{2}
-
\frac{1}{2}
\int_{\mathbb{R}^{N}}
\frac{|\bar{u}_{n}|^{2}}
{|x+\frac{x_{n}}{\sigma_{n}}|^{2s}}
\mathrm{d}x\\
&-
\frac{1}{2\cdot2^{*}_{\alpha}}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|\bar{u}_{n}(x)|^{2^{*}_{\alpha}}|\bar{u}_{n}(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
-
\sum_{i=1}^{k}
\frac{1}{2^{*}_{s,\theta_{i}}}
\int_{\mathbb{R}^{N}}
\frac{|\bar{u}_{n}|^{2^{*}_{s,\theta_{i}}}}
{|x+\frac{x_{n}}{\sigma_{n}}|^{\theta_{i}}}
\mathrm{d}x.
\end{aligned}
\end{equation*}
Now,
for all
$\varphi\in D^{s,2}(\mathbb{R}^{N})$,
we obtain
\begin{equation*}
\begin{aligned}
|\langle \widetilde{I_{2}}^{'}(\bar{u}_{n}),\varphi\rangle|
=&
|\langle I_{2}^{'}(u_{n}),\bar{\varphi}\rangle|\\
\leqslant&
\|I_{2}^{'}(u_{n})\|_{D^{-1}}
\|\bar{\varphi}\|_{D}\\
=&
o(1)
\|\bar{\varphi}\|_{D},
\end{aligned}
\end{equation*}
where
$\bar{\varphi}=\sigma_{n}^{-\frac{N-2s}{2}}\varphi(\frac{x-x_{n}}{\sigma_{n}})$.
Since
$\|\bar{\varphi}\|_{D}=\|\varphi\|_{D}$,
we get
$$\widetilde{I_{2}}^{'}(\bar{u}_{n})\rightarrow0~
\mathrm{as}~
n\rightarrow\infty.$$
Thus there exists
$\bar{u}$
such that
\begin{align*}
&
\bar{u}_{n}\rightharpoonup \bar{u},
~
\mathrm{in}
~
D^{s,2}(\mathbb{R}^{N}),
~~
\bar{u}_{n}\rightarrow \bar{u},
~
\mathrm{a.e. ~in}
~
\mathbb{R}^{N},\\
&\bar{u}_{n}\rightarrow \bar{u},
~
\mathrm{in}
~
L^{r}_{loc}(\mathbb{R}^{N})
~
\mathrm{for~all}
~
r\in[1,2^{*}_{s}).
\end{align*}
Then
$$
\int_{B(0,1)}
|\bar{u}_{n}(y)|^{2}
\mathrm{d}y
=
\frac{1}{\sigma_{n}^{2s}}
\int_{B(x_{n},\sigma_{n})}
|u_{n}(y)|^{2}
\mathrm{d}y
\geqslant
C_{6}>0.$$
As a result,
$\bar{u}\not\equiv0$.
\noindent
{\bf Step 2.}
Now,
we claim that
$\{\frac{x_{n}}{\sigma_{n}}\}$
is bounded.
If
$\frac{x_{n}}{\sigma_{n}}\rightarrow\infty$,
then for any
$\varphi\in D^{s,2}(\mathbb{R}^{N})$,
we get
\begin{equation}\label{32}
\begin{aligned}
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\frac{\bar{u}_{n}\varphi}
{|x+\frac{x_{n}}{\sigma_{n}}|^{2s}}
\mathrm{d}x
=
0
~\mathrm{and}~
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\frac{|\bar{u}_{n}|^{2^{*}_{s,\theta_{i}}-2}\bar{u}_{n}\varphi}
{|x+\frac{x_{n}}{\sigma_{n}}|^{\theta_{i}}}
\mathrm{d}x
=0.
\end{aligned}
\end{equation}
We will show that
\begin{equation*}
\begin{aligned}
\langle I^{'}_{3}(\bar{u}),\varphi\rangle=0.
\end{aligned}
\end{equation*}
Since
$\bar{u}_{n}\rightharpoonup \bar{u}$
weakly in
$D^{s,2}(\mathbb{R}^{N})$,
we know
\begin{equation}\label{33}
\begin{aligned}
&\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{(\bar{u}_{n}(x)-\bar{u}_{n}(y))(\varphi(x)-\varphi(y))}
{|x-y|^{N+2s}}
\mathrm{d}x
\mathrm{d}y\\
=&
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{(\bar{u}(x)-\bar{u}(y))(\varphi(x)-\varphi(y))}
{|x-y|^{N+2s}}
\mathrm{d}x
\mathrm{d}y.
\end{aligned}
\end{equation}
By the Hardy--Littlewood--Sobolev inequality,
the Riesz potential defines a linear continuous map from
$L^{\frac{2N}{2N-\alpha}}(\mathbb{R}^{N})$
to
$L^{\frac{2N}{\alpha}}(\mathbb{R}^{N})$.
Since
$|\bar{u}_{n}|^{2^{*}_{\alpha}}\rightharpoonup|\bar{u}|^{2^{*}_{\alpha}}$
weakly in
$L^{\frac{2^{*}}{2^{*}_{\alpha}}}(\mathbb{R}^{N})$,
it follows that
as
$n\rightarrow\infty$,
\begin{equation}\label{34}
\int_{\mathbb{R}^{N}}
\frac{|\bar{u}_{n}(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}y
\rightharpoonup
\int_{\mathbb{R}^{N}}
\frac{|\bar{u}(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}y
~\mathrm{weakly~in}~L^{\frac{2N}{\alpha}}(\mathbb{R}^{N}).
\end{equation}
Now,
we show that
$|\bar{u}_{n}|^{2^{*}_{\alpha}-2}
\bar{u}_{n}
\varphi
\rightarrow
|\bar{u}|^{2^{*}_{\alpha}-2}
\bar{u}\varphi$
in
$L^{\frac{2N}{2N-\alpha}}(\mathbb{R}^{N})$.
For any $\varepsilon>0$,
there exists $R>0$ large enough such that
\begin{equation}\label{35}
\begin{aligned}
&
\lim_{n\rightarrow\infty}
\int_{|x| >R}
\left|
|\bar{u}_{n}|^{2^{*}_{\alpha}-2}
\bar{u}_{n}
\varphi
\right|^{\frac{2N}{2N-\alpha}}
-
\left|
|\bar{u}|^{2^{*}_{\alpha}-2}
\bar{u}\varphi
\right|^{\frac{2N}{2N-\alpha}}
\mathrm{d}x\\
\leqslant&
\lim_{n\rightarrow\infty}
\int_{|x| >R}
\left|
\bar{u}_{n}
\right|^{(2^{*}_{\alpha}-1)\cdot\frac{2^{*}_{s}}{2^{*}_{\alpha}}}
\left|
\varphi
\right|^{\frac{2^{*}_{s}}{2^{*}_{\alpha}}}
\mathrm{d}x
+
\int_{|x| >R}
\left|
\bar{u}
\right|^{(2^{*}_{\alpha}-1)\cdot\frac{2^{*}_{s}}{2^{*}_{\alpha}}}
\left|
\varphi
\right|^{\frac{2^{*}_{s}}{2^{*}_{\alpha}}}
\mathrm{d}x\\
\leqslant&
\lim_{n\rightarrow\infty}
\left(
\int_{|x| >R}
\left|
\bar{u}_{n}
\right|^{2^{*}_{s}}
\mathrm{d}x
\right)^{1-\frac{1}{2^{*}_{\alpha}}}
\left(
\int_{|x| >R}
\left|
\varphi
\right|^{2^{*}_{s}}
\mathrm{d}x
\right)^{\frac{1}{2^{*}_{\alpha}}}\\
&+
\left(
\int_{|x| >R}
\left|
\bar{u}
\right|^{2^{*}_{s}}
\mathrm{d}x
\right)^{1-\frac{1}{2^{*}_{\alpha}}}
\left(
\int_{|x| >R}
\left|
\varphi
\right|^{2^{*}_{s}}
\mathrm{d}x
\right)^{\frac{1}{2^{*}_{\alpha}}}\\
\leqslant&
C
\left(
\int_{|x| >R}
\left|
\varphi
\right|^{2^{*}_{s}}
\mathrm{d}x
\right)^{\frac{1}{2^{*}_{\alpha}}}\\
<&
\frac{\varepsilon}{2}.
\end{aligned}
\end{equation}
On the other hand,
by the boundedness of
$\{\bar{u}_{n}\}$,
one has
\begin{equation*}
\begin{aligned}
\left(
\int_{|x|\leqslant R}
\left|
\bar{u}_{n}
\right|^{2^{*}_{s}}
\mathrm{d}x
\right)^{1-\frac{1}{2^{*}_{\alpha}}}
\leqslant&
M.
\end{aligned}
\end{equation*}
where $M>0$ is a constant.
Let
$\Omega=\{x\in\mathbb{R}^{N}||x|\leqslant R\}$.
For any $\tilde{\varepsilon}>0$,
there exists
$\delta>0$,
when
$E\subset\Omega$
with
$|E|<\delta$.
We obtain
\begin{align*}
\int_{E}
\left|
|\bar{u}_{n}|^{2^{*}_{\alpha}-2}
\bar{u}_{n}
\varphi
\right|^{\frac{2N}{2N-\alpha}}
\mathrm{d}x
=&
\int_{E}
\left|
\bar{u}_{n}
\right|^{(2^{*}_{\alpha}-1)\cdot\frac{2^{*}_{s}}{2^{*}_{\alpha}}}
\left|
\varphi
\right|^{\frac{2^{*}_{s}}{2^{*}_{\alpha}}}
\mathrm{d}x\\
\leqslant&
\left(
\int_{E}
\left|
\bar{u}_{n}
\right|^{2^{*}_{s}}
\mathrm{d}x
\right)^{1-\frac{1}{2^{*}_{\alpha}}}
\left(
\int_{E}
\left|
\varphi
\right|^{2^{*}_{s}}
\mathrm{d}x
\right)^{\frac{1}{2^{*}_{\alpha}}}\\
<&M\tilde{\varepsilon},
\end{align*}
where the last inequality is from the absolutely continuity of
$\int_{E}
\left|
\varphi
\right|^{2^{*}_{s}}
\mathrm{d}x$.
Moreover,
$|\bar{u}_{n}|^{2^{*}_{\alpha}-2}
\bar{u}_{n}
\varphi
\rightarrow
|\bar{u}|^{2^{*}_{\alpha}-2}
\bar{u}
\varphi$
a.e. in $\mathbb{R}^{N}$ as $n\rightarrow\infty$.
Thus,
by the Vitali convergence Theorem,
we get
\begin{equation}\label{36}
\begin{aligned}
\lim_{n\rightarrow\infty}
\int_{|x|\leqslant R}
\left|
|\bar{u}_{n}|^{2^{*}_{\alpha}-2}
\bar{u}_{n}
\varphi
\right|^{\frac{2N}{2N-\alpha}}
\mathrm{d}x
=
\int_{|x|\leqslant R}
\left|
|\bar{u}|^{2^{*}_{\alpha}-2}
\bar{u}
\varphi
\right|^{\frac{2N}{2N-\alpha}}
\mathrm{d}x.
\end{aligned}
\end{equation}
It follows from
\eqref{35}
and
\eqref{36}
that
\begin{equation*}
\begin{aligned}
&
\lim_{n\rightarrow\infty}
\left|
\int_{\mathbb{R}^{N}}
\left|
|\bar{u}_{n}|^{2^{*}_{\alpha}-2}
\bar{u}_{n}
\varphi
\right|^{\frac{2N}{2N-\alpha}}
-
\left|
|\bar{u}|^{2^{*}_{\alpha}-2}
\bar{u}
\varphi
\right|^{\frac{2N}{2N-\alpha}}
\mathrm{d}x
\right|\\
\leqslant&
\lim_{n\rightarrow\infty}
\left|
\int_{|x|\leqslant R}
\left|
|\bar{u}_{n}|^{2^{*}_{\alpha}-2}
\bar{u}_{n}
\varphi
\right|^{\frac{2N}{2N-\alpha}}
-
\left|
|\bar{u}|^{2^{*}_{\alpha}-2}
\bar{u}
\varphi
\right|^{\frac{2N}{2N-\alpha}}
\mathrm{d}x
\right|\\
&+
\lim_{n\rightarrow\infty}
\left|
\int_{|x|> R}
\left|
|\bar{u}_{n}|^{2^{*}_{\alpha}-2}
\bar{u}_{n}
\varphi
\right|^{\frac{2N}{2N-\alpha}}
-
\left|
|\bar{u}|^{2^{*}_{\alpha}-2}
\bar{u}
\varphi
\right|^{\frac{2N}{2N-\alpha}}
\mathrm{d}x
\right|\\
<&
\varepsilon.
\end{aligned}
\end{equation*}
This implies that
\begin{equation}\label{37}
\begin{aligned}
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\left|
|\bar{u}_{n}|^{2^{*}_{\alpha}-2}
\bar{u}_{n}
\varphi
\right|^{\frac{2N}{2N-\alpha}}
\mathrm{d}x
=
\int_{\mathbb{R}^{N}}
\left|
|\bar{u}|^{2^{*}_{\alpha}-2}
\bar{u}
\varphi
\right|^{\frac{2N}{2N-\alpha}}
\mathrm{d}x
\end{aligned}
\end{equation}
Combining
\eqref{34}
and
\eqref{37},
we have
\begin{equation}\label{38}
\begin{aligned}
&\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{
|\bar{u}_{n}(y)|^{2^{*}_{\alpha}}
|\bar{u}_{n}(x)|^{2^{*}_{\alpha}-2}
\bar{u}_{n}(x)
\varphi(x)
}{|x-y|^{\alpha}}
\mathrm{d}y
\mathrm{d}x\\
=&
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{
|\bar{u}(y)|^{2^{*}_{\alpha}}
|\bar{u}(x)|^{2^{*}_{\alpha}-2}
\bar{u}(x)
\varphi(x)
}{|x-y|^{\alpha}}
\mathrm{d}y
\mathrm{d}x.
\end{aligned}
\end{equation}
Applying
$\lim\limits_{n\rightarrow\infty}\langle \widetilde{I_{2}}^{'}(\bar{u}_{n}),\varphi\rangle\rightarrow0$,
\eqref{32},
\eqref{33}
and
\eqref{38}
we know
\begin{equation}\label{39}
\begin{aligned}
\langle I^{'}_{3}(\bar{u}),\varphi\rangle=0.
\end{aligned}
\end{equation}
Moreover,
according to \eqref{39}
and
$\bar{u}\not\equiv0$,
we get that
$$\bar{u}\in \mathcal{N}^{3}.$$
By Br\'{e}zis--Lieb lemma \cite[Lemma 2.2]{Gao2016},
we have
\begin{equation*}
\begin{aligned}
&\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|\bar{u}_{n}(x)|^{2^{*}_{\alpha}}|\bar{u}_{n}(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
-
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|\bar{u}_{n}(x)-\bar{u}(x)|^{2^{*}_{\alpha}}|\bar{u}_{n}(y)-\bar{u}(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y\\
=&
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|\bar{u}(x)|^{2^{*}_{\alpha}}|\bar{u}(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
+
o(1),
\end{aligned}
\end{equation*}
which implies that
\begin{equation}\label{40}
\begin{aligned}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|\bar{u}_{n}(x)|^{2^{*}_{\alpha}}|\bar{u}_{n}(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
\geqslant
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|\bar{u}(x)|^{2^{*}_{\alpha}}|\bar{u}(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
+
o(1).
\end{aligned}
\end{equation}
Similarly,
we get
\begin{equation}\label{41}
\begin{aligned}
\int_{\mathbb{R}^{N}}
\frac{|\bar{u}_{n}|^{2^{*}_{s,\theta_{i}}}}
{|x|^{\theta_{i}}}
\mathrm{d}x
\geqslant
\int_{\mathbb{R}^{N}}
\frac{|\bar{u}|^{2^{*}_{s,\theta_{i}}}}
{|x|^{\theta_{i}}}
\mathrm{d}x
+
o(1).
\end{aligned}
\end{equation}
Applying
Lemma \ref{lemma17},
Lemma \ref{lemma12},
(\ref{40}),
(\ref{41}),
$\bar{u}\in \mathcal{N}^{3}$
and
Lemma \ref{lemma14},
we obtain
\begin{align*}
c_{0}^{3}
>
c_{0}^{2}
=c^{2}
=&
I_{2}(\bar{u}_{n})
-
\frac{1}{2}
\langle I^{'}_{2}(\bar{u}_{n}),\bar{u}_{n}\rangle\\
\geqslant&
\left(
\frac{1}{2}
-
\frac{1}{2\cdot2^{*}_{\alpha}}
\right)
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|\bar{u}(x)|^{2^{*}_{\alpha}}|\bar{u}(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
+o(1)\\
=&
I_{3}(\bar{u})
-
\frac{1}{2}
\langle I_{3}^{'}(\bar{u}),\bar{u}\rangle
=
I_{3}(\bar{u})
\geqslant
c_{0}^{3},
\end{align*}
which yields a contradiction.
Hence, $\{\frac{x_{n}}{\sigma_{n}}\}$ is bounded.
\noindent
{\bf Step 3.}
In this step,
we study another
$(PS)_{c^{2}}$
sequence of
$I_{2}$.
Let
$\tilde{u}_{n}(x)=\sigma_{n}^{\frac{N-2s}{2}}u_{n}(\sigma_{n}x)$.
Then we can
verify that
$$I_{2}(\tilde{u}_{n})
=
I_{2}(u_{n})\rightarrow c^{2}
,~
I^{'}_{2}(\tilde{u}_{n})\rightarrow 0
~\mathrm{as}~n\rightarrow\infty.$$
Arguing as before, we have
\begin{align*}
&\tilde{u}_{n}\rightharpoonup \tilde{u}
~
\mathrm{in}
~
D^{1,2}(\mathbb{R}^{N}),~
\tilde{u}_{n}\rightarrow \tilde{u}
~
\mathrm{a.e. ~in}
~
\mathbb{R}^{N},\\
&\tilde{u}_{n}\rightarrow \tilde{u}
~
\mathrm{in}
~
L^{r}_{loc}(\mathbb{R}^{N})~~\mathrm{for~all~}r\in[1,2^{*}_{s}).
\end{align*}
By using $\{\frac{x_{n}}{\sigma_{n}}\}$ is bounded,
there exists
$\tilde{R}>0$ such that
$$
\int_{B(0,\tilde{R})}
|\tilde{u}_{n}(y)|^{2}
\mathrm{d}y
>
\int_{B(\frac{x_{n}}{\sigma_{n}},1)}
|\tilde{u}_{n}(y)|^{2}
\mathrm{d}y
=
\frac{1}{\sigma_{n}^{2s}}
\int_{B(x_{n},\sigma_{n})}
|u_{n}(y)|^{2}
\mathrm{d}y
\geqslant
C_{6}>0.$$
As a result,
$\tilde{u}\not\equiv0$.
\noindent
{\bf Step 4.}
In this step,
we show
$\tilde{u}_{n}\rightarrow \tilde{u}$
strongly in
$D^{s,2}(\mathbb{R}^{N})$.
Set
\begin{align*}
K(u)
=
\left(
\frac{1}{2}
-
\frac{1}{2\cdot2^{*}_{\alpha}}
\right)
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\alpha}}|u(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
+
\sum_{i=1}^{k}
\left(
\frac{1}{2}
-
\frac{1}{2^{*}_{s,\theta_{i}}}
\right)
\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{s,\theta_{i}}}}
{|x|^{\theta_{i}}}
\mathrm{d}x.
\end{align*}
Similar to Step 2,
we know that
\begin{equation}\label{42}
\begin{aligned}
\langle I^{'}_{2}(\tilde{u}),\varphi\rangle=0.
\end{aligned}
\end{equation}
Applying
Lemma \ref{lemma11},
Lemma \ref{lemma12},
$\tilde{u}\in \mathcal{N}^{2}$
and
(\ref{40})
--
(\ref{42}),
we obtain
\begin{equation}\label{43}
\begin{aligned}
c_{0}^{2}
=c^{2}
=&
I_{2}(\tilde{u}_{n})
-
\frac{1}{2}
\langle I^{'}_{2}(\tilde{u}_{n}),\tilde{u}_{n}\rangle\\
=&
\lim_{n\rightarrow\infty}K(\tilde{u}_{n})+o(1)\\
\geqslant&
K(\tilde{u})
+o(1)\\
=&
I_{2}(\tilde{u})
-
\frac{1}{2}
\langle I_{2}^{'}(\tilde{u}),\tilde{u}\rangle
=
I_{2}(\tilde{u})
\geqslant
c_{0}^{2}.
\end{aligned}
\end{equation}
\textcolor{red}{Therefore, the inequalities above have to be equalities}.
We know
\begin{align*}
\lim\limits_{n\rightarrow\infty}
K(\tilde{u}_{n})
=
K(\tilde{u}).
\end{align*}
By using Br\'{e}zis--Lieb lemma again,
we have
\begin{align*}
\lim\limits_{n\rightarrow\infty}
K(\tilde{u}_{n})
-
\lim\limits_{n\rightarrow\infty}
K(\tilde{u}_{n}-\tilde{u})
=
K(\tilde{u})+o(1).
\end{align*}
Hence,
we deduce that
\begin{align*}
\lim\limits_{n\rightarrow\infty}
K(\tilde{u}_{n}-\tilde{u})
=
0,
\end{align*}
which implies that
\begin{equation}\label{44}
\begin{aligned}
&
\lim\limits_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|\tilde{u}_{n}(x)-\tilde{u}(x)|^{2^{*}_{\alpha}}|\tilde{u}_{n}(y)-\tilde{u}(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
=0,
\\
&
\lim\limits_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\frac{
|\tilde{u}_{n}-\tilde{u}|^{2^{*}_{s,\theta_{i}}}
}
{|x|^{\theta_{i}}}
\mathrm{d}x
=0,~\mathrm{for~all}~
i=1,\ldots,k.
\end{aligned}
\end{equation}
According to
$\langle I^{'}_{2}(\tilde{u}_{n}),\tilde{u}_{n}\rangle=o(1)$,
$\langle I^{'}_{2}(\tilde{u}),\tilde{u}\rangle=0$
and Br\'{e}zis--Lieb lemma,
we obtain
\begin{equation*}
\begin{aligned}
o(1)=&
\langle I^{'}_{2}(\tilde{u}_{n}),\tilde{u}_{n}\rangle
-
\langle I^{'}_{2}(\tilde{u}),\tilde{u}\rangle\\
=&
\|\tilde{u}_{n}-\tilde{u}\|_{\zeta}^{2}
-
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|\tilde{u}_{n}(x)-\tilde{u}(x)|^{2^{*}_{\alpha_{i}}}|\tilde{u}_{n}(y)-\tilde{u}(y)|^{2^{*}_{\alpha_{i}}}}{|x-y|^{\alpha_{i}}}
\mathrm{d}x
\mathrm{d}y\\
&-\sum_{i=1}^{k}
\int_{\mathbb{R}^{N}}
\frac{
|\tilde{u}_{n}-\tilde{u}|^{2^{*}_{s,\theta_{i}}}
}
{|x|^{\theta_{i}}}
\mathrm{d}x
+o(1),
\end{aligned}
\end{equation*}
which implies that
\begin{equation}\label{45}
\begin{aligned}
&
\lim\limits_{n\rightarrow\infty}
\|\tilde{u}_{n}-\tilde{u}\|_{\zeta}^{2}\\
=&
\lim\limits_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|\tilde{u}_{n}(x)-\tilde{u}(x)|^{2^{*}_{\alpha}}|\tilde{u}_{n}(y)-\tilde{u}(y)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y\\
&+
\lim\limits_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\frac{
|\tilde{u}_{n}-\tilde{u}|^{2^{*}_{s,\theta_{i}}}
}
{|x|^{\theta_{i}}}
\mathrm{d}x+o(1).
\end{aligned}
\end{equation}
Combining
(\ref{44})
and
(\ref{45}),
we get
\begin{equation*}
\begin{aligned}
\lim\limits_{n\rightarrow\infty}
\|\tilde{u}_{n}-\tilde{u}\|_{\zeta}^{2}
=
0.
\end{aligned}
\end{equation*}
Since
$\tilde{u}\not\equiv0$,
we know that
$\tilde{u}_{n}\rightarrow \tilde{u}$
strongly in
$D^{s,2}(\mathbb{R}^{N})$.
\noindent
{\bf Step 5.}
By using \eqref{43} again,
we know that
$I_{2}(\tilde{u})=c^{2}$,
which means that $\tilde{u}$ is a nontrivial solution of problem $(\mathcal{P}_{2})$ at the
energy level $c^{2}$.
Then we have just to prove
that we can choose
$\tilde{u}\geqslant0$.
We know that
\begin{equation*}
\begin{aligned}
0=&
\langle I^{'}_{2}(\tilde{u}),\tilde{u}^{-}\rangle\\
=&
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{(\tilde{u}(x)-\tilde{u}(y))(\tilde{u}^{-}(x)-\tilde{u}^{-}(y))}{|x-y|^{N+2s}}
\mathrm{d}x
\mathrm{d}y
-
\zeta
\int_{\mathbb{R}^{N}}
\frac{\tilde{u}\tilde{u}^{-}}{|x|^{2s}}
\mathrm{d}x
\\
&-
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|\tilde{u}(y)|^{2^{*}_{\alpha}}|\tilde{u}(x)|^{2^{*}_{\alpha}-2}\tilde{u}(x)\tilde{u}^{-}(x)}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
-
\sum_{i=1}^{k}
\int_{\mathbb{R}^{N}}
\frac{|\tilde{u}|^{2^{*}_{s,\theta_{i}}-2}\tilde{u}\tilde{u}^{-}}
{|x|^{\theta_{i}}}
\mathrm{d}x,
\end{aligned}
\end{equation*}
where
$\tilde{u}^{-}=\max\{0,-\tilde{u}\}$.
For a.e.
$x,y\in \mathbb{R}^{N}$,
we have
$$(\tilde{u}(x)-\tilde{u}(y))(\tilde{u}^{-}(x)-\tilde{u}^{-}(y))\leqslant -|\tilde{u}^{-}(x)-\tilde{u}^{-}(y)|^{2}.$$
Then,
we get
\begin{equation*}
\begin{aligned}
0=&
-
\|\tilde{u}^{-}\|_{D}^{2}
-
\zeta
\int_{\mathbb{R}^{N}}
\frac{|\tilde{u}^{-}|^{2}}{|x|^{2s}}
\mathrm{d}x
-
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|\tilde{u}(y)|^{2^{*}_{\alpha}}|\tilde{u}^{-}(x)|^{2^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
-
\sum_{i=1}^{k}
\int_{\mathbb{R}^{N}}
\frac{|\tilde{u}^{-}|^{2^{*}_{s,\theta_{i}}}}
{|x|^{\theta_{i}}}
\mathrm{d}x\\
\leqslant&
-\|\tilde{u}^{-}\|_{D}^{2}.
\end{aligned}
\end{equation*}
Thus,
$\|\tilde{u}^{-}\|_{D}^{2}=0$.
Hence,
we can choose
$\tilde{u}\geqslant0$.
By using the fractional Kelvin transformation
\begin{equation}\label{46}
\begin{aligned}
\tilde{\tilde{u}}(x)
=
\frac{1}{|x|^{N-2s}}
\tilde{u}
\left(
\frac{x}{|x|^{2}}
\right).
\end{aligned}
\end{equation}
It is well known that
\begin{equation}\label{47}
\begin{aligned}
(-
\Delta)^{s}
\tilde{\tilde{u}}(x)
=
\frac{1}{|x|^{N+2s}}
(-
\Delta)^{s}
\tilde{u}
\left(
\frac{x}{|x|^{2}}
\right).
\end{aligned}
\end{equation}
The following identity is very useful.
For
$\forall x,y\in \mathbb{R}^{N}\backslash\{0\}$,
we get
\begin{equation}\label{48}
\begin{aligned}
\frac{1}
{
\left|
\frac{x}{|x|^{2}}-\frac{y}{|y|^{2}}
\right|^{\alpha}}
\cdot
\frac{1}{|xy|^{\alpha}}
=&
\frac{1}
{
\left|
\frac{x\cdot y^{2}-y\cdot x^{2}}{(xy)^{2}}
\right|^{\alpha}}
\cdot
\frac{1}{|xy|^{\alpha}}\\
=&
\frac{1}
{
\left|
\frac{x\cdot y^{2}-y\cdot x^{2}}{xy}
\right|^{\alpha}}\\
=&
\frac{1}
{
\left|
y-x
\right|^{\alpha}}.
\end{aligned}
\end{equation}
Set
$z=\frac{y}{|y|^{2}}$.
Applying
(\ref{46})
and
(\ref{48}),
we have
\begin{equation}\label{49}
\begin{aligned}
\int_{\mathbb{R}^{N}}
\frac{|\tilde{\tilde{u}}(y)|^{\frac{2N-\alpha}{N-2s}}}
{|x-y|^{\alpha}}
\mathrm{d}y
=&
\int_{\mathbb{R}^{N}}
\frac{|\tilde{u}
\left(
\frac{y}{|y|^{2}}
\right)|^{\frac{2N-\alpha}{N-2s}}}
{|x-y|^{\alpha}}
\cdot
\frac{1}{|y|^{2N-\alpha}}
\mathrm{d}y
~(\mathrm{by}~(\ref{46}))
\\
=&
\int_{\mathbb{R}^{N}}
\frac{|\tilde{u}
\left(
\frac{y}{|y|^{2}}
\right)|^{\frac{2N-\alpha}{N-2s}}}
{|\frac{x}{|x|^{2}}-\frac{y}{|y|^{2}}|^{\alpha}}
\cdot
\frac{1}{|xy|^{\alpha}\cdot|y|^{2N-\alpha}}
\mathrm{d}y
~(\mathrm{by}~(\ref{48}))\\
=&
\frac{1}{|x|^{\alpha}}
\int_{\mathbb{R}^{N}}
\frac{|\tilde{u}
\left(
\frac{y}{|y|^{2}}
\right)|^{\frac{2N-\alpha}{N-2s}}}
{|\frac{x}{|x|^{2}}-\frac{y}{|y|^{2}}|^{\alpha}}
\cdot
\frac{1}{|y|^{2N}}
\mathrm{d}y\\
=&
\frac{1}{|x|^{\alpha}}
\int_{\mathbb{R}^{N}}
\frac{|\tilde{u}
\left(
z
\right)|^{\frac{2N-\alpha}{N-2s}}}
{|\frac{x}{|x|^{2}}-z|^{\alpha}}
\mathrm{d}z
~(\mathrm{set}~z=\frac{y}{|y|^{2}}).
\end{aligned}
\end{equation}
By using
\eqref{46},
we get
\begin{equation}\label{50}
\begin{aligned}
\frac{
\left|
\tilde{\tilde{u}}
\right|^{\frac{4s-2\theta_{i}}{N-2s}}
\tilde{\tilde{u}}}
{|x|^{\theta_{i}}}
=
\frac{1}{|x|^{N+2s}}
\frac{
\tilde{u}
\left(
\frac{x}{|x|^{2}}
\right)
}
{\left|\frac{x}{|x|^{2}}\right|^{\theta_{i}}},
\end{aligned}
\end{equation}
and
\begin{equation}\label{59}
\begin{aligned}
\frac{\tilde{\tilde{u}}(x)}{|x|^{2s}}
=
\frac{1}{|x|^{N+2s}}
\frac{
\tilde{u}
\left(
\frac{x}{|x|^{2}}
\right)
}
{\left|\frac{x}{|x|^{2}}\right|^{2s}}.
\end{aligned}
\end{equation}
Therefore,
by using
(\ref{47})
and
(\ref{49})
--
(\ref{59}),
we get
\begin{equation*}
\begin{aligned}
(-
\Delta)^{s}
\tilde{\tilde{u}}
-\zeta
\frac{\tilde{\tilde{u}}}{|x|^{2s}}
=&
\left(
\int_{\mathbb{R}^{N}}
\frac{|\tilde{\tilde{u}}|^{\frac{2N-\alpha}{N-2s}}}
{|x-y|^{\alpha}}
\mathrm{d}y
\right)
\left|
\tilde{\tilde{u}}
\right|^{\frac{4s-\alpha}{N-2s}}
\tilde{\tilde{u}}
+
\sum\limits^{k}_{i=1}
\frac{
\left|
\tilde{\tilde{u}}
\right|^{\frac{4s-2\theta_{i}}{N-2s}}
\tilde{\tilde{u}}}
{|x|^{\theta_{i}}},
~\mathrm{in}~\mathbb{R}^{N}\backslash\{0\}.
\end{aligned}
\end{equation*}
\qed
\section{The proof of Theorem \ref{theorem1}}
In this section,
we study the existence of nonnegative solution of problem $(\mathcal{P}_{1})$.
\begin{lemma}\label{lemma18}
Assume that the assumptions of Theorem
\ref{theorem1}
hold.
Then
there exists a
$(PS)_{c^{1}}$
sequence of
$I_{1}$
at level
$c^{1}$,
where
$$0<c^{1}<c^{1,*}=
\min
\left\{
\frac{N+2s-\alpha_{1}}{2(2N-\alpha_{1})}S_{0,\alpha_{1}}^{\frac{2N-\alpha_{1}}{N+2s-\alpha_{1}}}
,
\ldots
,
\frac{N+2s-\alpha_{k}}{2(2N-\alpha_{k})}S_{0,\alpha_{k}}^{\frac{2N-\alpha_{k}}{N+2s-\alpha_{k}}}
,
\frac{s}{N}
S_{0,0} ^{\frac{N}{2s}}
\right\}.$$
\end{lemma}
\begin{lemma}\label{lemma13}
Assume that the assumptions of Theorem
\ref{theorem1}
hold.
Then
$$c_{1}^{1}=c^{1}=c_{0}^{1}=\inf_{u\in\mathcal{N}^{1}}I_{1}(u)>0.$$
\end{lemma}
The following result implies the non--vanishing of
$(PS)_{c^{1}}$
sequences.
\begin{lemma}\label{lemma19}
Assume that the assumptions of Theorem
\ref{theorem1}
hold.
Let
$\{u_{n}\}$
be a
$(PS)_{c^{1}}$
sequence of
$I_{1}$
with
$c^{1}\in(0,c^{1,*})$,
then
\begin{equation*}
\begin{aligned}
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
|u_{n}|^{2^{*}_{s}}
\mathrm{d}x>0,
\end{aligned}
\end{equation*}
and
\begin{equation*}
\begin{aligned}
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}(x)|^{2^{*}_{\alpha_{i}}}|u_{n}(y)|^{2^{*}_{\alpha_{i}}}}{|x-y|^{\alpha_{i}}}
\mathrm{d}x
\mathrm{d}y>0,(i=1,\ldots,k).
\end{aligned}
\end{equation*}
\end{lemma}
\noindent
{\bf Proof.}
Let
$\{u_{n}\}$
be a
$(PS)_{c^{1}}$
sequence of
$I_{1}$
with
$c^{1}\in(0,c^{1,*})$,
It's easy to see that
$\{u_{n}\}$
is uniformly bounded in
$D^{s,2}(\mathbb{R}^{N})$.
The proof of this Lemma is divided into three cases:
\noindent
(1)
$
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}(x)|^{2^{*}_{\alpha_{1}}}|u_{n}(y)|^{2^{*}_{\alpha_{1}}}}{|x-y|^{\alpha_{1}}}
\mathrm{d}x
\mathrm{d}y>0
$;
\noindent
(2)
$
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}(x)|^{2^{*}_{\alpha_{j}}}|u_{n}(y)|^{2^{*}_{\alpha_{j}}}}{|x-y|^{\alpha_{j}}}
\mathrm{d}x
\mathrm{d}y>0,
$
for
$j=2,\ldots,k$;
\noindent
(3)
$
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
|u_{n}|^{2^{*}_{s}}
\mathrm{d}x>0.
$
\noindent
{\bf Case 1.}
Suppose that
\begin{equation}\label{51}
\begin{aligned}
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}(x)|^{2^{*}_{\alpha_{1}}}|u_{n}(y)|^{2^{*}_{\alpha_{1}}}}
{|x-y|^{\alpha_{1}}}
\mathrm{d}x
\mathrm{d}y=0.
\end{aligned}
\end{equation}
Since
$\{u_{n}\}$
is uniformly bounded in
$D^{s,2}(\mathbb{R}^{N})$,
there exists a constant
$0<C<\infty$
such that
$\|u_{n}\|_{D}\leqslant C$.
By using (\ref{51})
and the definition of
fractional Coulomb--Sobolev space,
we obtain
$u_{n}\in \mathcal{E}^{s,\alpha_{1},2^{*}_{\alpha_{1}}}(\mathbb{R}^{N})$.
Applying Lemma \ref{lemma2} and (\ref{51}),
we have
\begin{equation}\label{52}
\begin{aligned}
&
\lim_{n\rightarrow\infty}
\|u_{n}\|_{L^{2^{*}_{s}}(\mathbb{R}^{N})}\\
\leqslant&
C
\left(
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}(x)|^{2^{*}_{\alpha_{1}}}|u_{n}(y)|^{2^{*}_{\alpha_{1}}}}{|x-y|^{\alpha_{1}}}
\mathrm{d}x
\mathrm{d}y
\right)
^{\frac{s(N-2s)}{N(N+2s-\alpha_{1})}}
=0.
\end{aligned}
\end{equation}
Combining Hardy--Littlewood--Sobolev inequality
and
(\ref{52}),
for all
$i=2,\ldots,k$,
we know
\begin{equation}\label{53}
\begin{aligned}
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}(x)|^{2^{*}_{\alpha_{i}}}|u_{n}(y)|^{2^{*}_{\alpha_{i}}}}{|x-y|^{\alpha_{i}}}
\mathrm{d}x
\mathrm{d}y
\leqslant&
C
\lim_{n\rightarrow\infty}
\|u_{n}\|_{L^{2^{*}_{s}}(\mathbb{R}^{N})}^{2\cdot2^{*}_{\alpha_{i}}}
=0.
\end{aligned}
\end{equation}
According to
(\ref{51})
--
(\ref{53})
and the definition of
$(PS)_{c^{1}}$
sequence
,
we obtain
$$
c^{1}
+
o(1)
=
\frac{1}{2}
\|u_{n}\|_{D}^{2},$$
and
$$o(1)
=
\|u_{n}\|_{D}^{2},
$$
these imply that
$c^{1}=0$,
which contradicts as $0<c^{1}$.
\noindent
{\bf Case 2.}
From Case 1,
we have
$u_{n}\in \mathcal{E}^{s,\alpha_{1},2^{*}_{\alpha_{1}}}(\mathbb{R}^{N})$.
Applying the result of (ii) in Lemma \ref{lemma6},
we know that
$u_{n}\in \bigcap_{i=2}^{k}
\mathcal{E}^{s,\alpha_{i},2^{*}_{\alpha_{i}}}(\mathbb{R}^{N})$.
Similar to Case 1,
for all
$i=2,\ldots,k$,
we prove that
$$\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}(x)|^{2^{*}_{\alpha_{i}}}|u_{n}(y)|^{2^{*}_{\alpha_{i}}}
}{|x-y|^{\alpha_{i}}}
\mathrm{d}x
\mathrm{d}y>0.$$
\noindent
{\bf Case 3.}
Suppose that
\begin{equation}\label{54}
\begin{aligned}
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
|u_{n}|^{2^{*}_{s}}
\mathrm{d}x
=0,
\end{aligned}
\end{equation}
By using Lemma \ref{lemma1}
and
(\ref{54}),
for all
$i=1,\ldots,k$,
we have
\begin{equation}\label{55}
\begin{aligned}
\lim_{n\rightarrow\infty}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u_{n}(x)|^{2^{*}_{\alpha_{i}}}|u_{n}(y)|^{2^{*}_{\alpha_{i}}}}{|x-y|^{\alpha_{i}}}
\mathrm{d}x
\mathrm{d}y
\leqslant&
C
\lim_{n\rightarrow\infty}
\|u_{n}\|_{L^{2^{*}_{s}}(\mathbb{R}^{N})}
^{2\cdot2^{*}_{\alpha_{i}}}=0.
\end{aligned}
\end{equation}
Applying
(\ref{54})
and
(\ref{55}),
we get
$$
c^{1}
+
o(1)
=
\frac{1}{2}
\|u_{n}\|_{D}^{2},$$
and
$$o(1)
=
\|u_{n}\|_{D}^{2},
$$
these imply that
$c^{1}=0$,
which contradicts as $0<c^{1}$.
\qed
We are now ready to prove the existence of nonnegative solution for problem
$(\mathcal{P}_{1})$.
\noindent
{\bf Proof of Theorem \ref{theorem1}:}
\noindent
{\bf Step 1.}
Since
$\{u_{n}\}$
is a bounded sequence in
$D^{s,2}(\mathbb{R}^{N})$,
up to a subsequence,
we can assume that
\begin{align*}
&u_{n}\rightharpoonup u
~
\mathrm{in}
~
D^{s,2}(\mathbb{R}^{N}),~
u_{n}\rightarrow u
~
\mathrm{a.e. ~in}
~
\mathbb{R}^{N},\\
&u_{n}\rightarrow u
~
\mathrm{in}
~
L^{r}_{loc}(\mathbb{R}^{N})
~
\mathrm{for~all}
~
r\in[1,2^{*}_{s}).
\end{align*}
According to
Lemma
\ref{lemma4},
Lemma
\ref{lemma5}
and
Lemma
\ref{lemma19},
there exists
$C>0$
such that
$$
\|u_{n}\|_{\mathcal{L}^{2,N-2s}(\mathbb{R}^{N})}\geqslant C>0.
$$
On the other hand,
since the sequence is bounded in
$D^{s,2}(\mathbb{R}^{N})$
and
$D^{s,2}(\mathbb{R}^{N})\hookrightarrow L^{2^{*}_{s}}(\mathbb{R}^{N})\hookrightarrow \mathcal{L}^{2,N-2s}(\mathbb{R}^{N})$,
we have
$$
\|u_{n}\|_{\mathcal{L}^{2,N-2s}(\mathbb{R}^{N})}\leqslant C,
$$
for some
$C>0$
independent of $n$.
Hence, there exists a positive constant which we denote again by $C$ such
that for any $n$ we obtain
$$
C
\leqslant
\|u_{n}\|_{\mathcal{L}^{2,N-2s}(\mathbb{R}^{N})}\leqslant C^{-1}.
$$
So we may find
$\sigma_{n} > 0$
and
$x_{n}\in \mathbb{R}^{N}$
such that
$$
\frac{1}{\sigma_{n}^{2s}}
\int_{B(x_{n},\sigma_{n})}
|u_{n}(y)|^{2}
\mathrm{d}y
\geqslant
\|u_{n}\|_{\mathcal{L}^{2,N-2s}(\mathbb{R}^{N})}^{2}
-
\frac{C}{2n}
\geqslant
C_{7}>0.
$$
Let
$\bar{v}_{n}(x)=\sigma_{n}^{\frac{N-2s}{2}}u_{n}(x_{n}+\sigma_{n}x)$.
We could verify that
$$I_{1}(\bar{v}_{n})=I(u_{n})\rightarrow c^{1},~
\langle I^{'}_{1}(\bar{v}_{n}),\varphi\rangle
~\mathrm{as}~n\rightarrow\infty.$$
It's obviously that
$\{\bar{v}_{n}\}$
is uniformly bounded in
$D^{s,2}(\mathbb{R}^{N})$.
Thus there exists
$v$
such that
\begin{align*}
&\bar{v}_{n}\rightharpoonup \bar{v}
~
\mathrm{in}
~
D^{s,2}(\mathbb{R}^{N}),~
\bar{v}_{n}\rightarrow \bar{v}
~
\mathrm{a.e. ~in}
~
\mathbb{R}^{N},\\
&\bar{v}_{n}\rightarrow \bar{v}
~
\mathrm{in}
~
L^{r}_{loc}(\mathbb{R}^{N})~~\mathrm{for~all~}r\in[1,2^{*}_{s} ).
\end{align*}
Then
$
\int_{B(0,1)}
|\bar{v}_{n}(y)|^{2}
\mathrm{d}y
\geqslant
C_{7}>0$.
As a result,
$\bar{v}\not\equiv0$.
\noindent
{\bf Step 2.}
Similar to \eqref{39},
we get
\begin{equation*}
\begin{aligned}
\langle I^{'}_{1}(\bar{v}),\varphi\rangle=0.
\end{aligned}
\end{equation*}
Similar to the proof of Theorem \ref{theorem2},
we know that
$\bar{v}_{n}\rightarrow \bar{v}$
strongly in
$D^{s,2}(\mathbb{R}^{N})$,
and
$I_{1}(\bar{v})=c^{1}$.
Moreover,
we can choose
$\bar{v}\geqslant0$.
By using the fractional Kelvin transformation
\begin{equation*}
\begin{aligned}
\bar{\bar{v}}(x)
=
\frac{1}{|x|^{N-2s}}
\bar{v}
\left(
\frac{x}{|x|^{2}}
\right).
\end{aligned}
\end{equation*}
We obtain
\begin{equation*}
\begin{aligned}
(-
\Delta)^{s}
\bar{\bar{v}}
=&
\sum\limits^{k}_{i=1}
\left(
\int_{\mathbb{R}^{N}}
\frac{|\bar{\bar{v}}|^{\frac{2N-\alpha_{i}}{N-2s}}}
{|x-y|^{\alpha_{i}}}
\mathrm{d}y
\right)
\left|
\bar{\bar{v}}
\right|^{\frac{4s-\alpha_{i}}{N-2s}}
\bar{\bar{v}}
+
\left|
\bar{\bar{v}}
\right|^{\frac{4s}{N-2s}}
\bar{\bar{v}}
,
~\mathrm{in}~\mathbb{R}^{N}\backslash\{0\}.
\end{aligned}
\end{equation*}
\qed
\section*{Open Problem}
During the preparation of the manuscript we faced one problem which is worth to be tackled in forthcoming investigation.
We want to generalize the study of problem $(\mathcal{P}_{1})$ to the following problem:
$$
(-\Delta)^{s} u
-\zeta\frac{ u}{|x|^{2s}}
=
\sum_{i=1}^{k}
\left(
\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{\alpha_{i}}}}{|x-y|^{\alpha_{i}}}
\mathrm{d}y
\right)
|u|^{2^{*}_{\alpha_{i}}-2}u
+
|u|^{2^{*}_{s}-2}u
,
\mathrm{~in~}
\mathbb{R}^{N},
\eqno(\mathcal{P}_{4})
$$
where
$N\geqslant3$,
$s\in(0,1)$,
$\zeta\in
\left(
0,4^{s}\frac{\Gamma(\frac{N+2s}{4})}{\Gamma(\frac{N-2s}{4})}
\right)$
and
$0<\alpha_{1}<\alpha_{2}<\cdots<\alpha_{k}<N$
($k\in \mathbb{N}$,
$2\leqslant k<\infty$).
If
$\zeta=0$,
then problem $(\mathcal{P}_{4})$
goes back to
problem $(\mathcal{P}_{1})$.
However,
they are very different from each other.
|
3,212,635,537,947 | arxiv | \section{Introduction}
Invariance under Lorentz transformations constitutes one of the fundamental
principles of modern physics. A more fundamental theory including a kind of
quantum theory for gravity, however, implies the possibility or even necessity
for Lorentz symmetry violation (LV) \cite{dm05,sl13,jdt14}. The study about
this violation has attracted extensive attention over the past twenty years,
and a nice theoretical framework called the Standard-Model Extension (SME)
\cite{cm97,cm98,kr11} was developed for the analysis of LV. This framework
formally contains all possible Lorentz-symmetry breaking terms which are
generated by the couplings between the standard fields and vacuum expectations
of the tensor fields which parameterize symmetry violations (more definitely,
it concerns only the violation of invariance under particle Lorentz
transformations, see the Ref. \cite{cm98}). The violation is always studied by
using certain terms in the presumed absence of all other symmetry breaking terms.
Although the Lorentz symmetry remains a property of the underlying fundamental
theory \cite{cm98}, it might be violated at experimentally accessible energy
scales due\ to spontaneous symmetry breaking, hence the development of
experimental tests for Lorentz invariance
\cite{kl99,kr99,mhs04,hsm09,hlb13,kv15,prp15,dfh16,vvf16}. Of particular
relevance to this aspect are several recent efforts using atoms and ions to
test LV through precision measurements. For the LV term that we are interested
in, it is usually tested by the isotropy of the speed of light (for recent
results, please refer to Ref. \cite{hsm09}). In this direction, the most
sensitive tests made use of neutral Dy atoms \cite{hlb13}, Ca$^{+}$ ions
\cite{prp15}, and Yb$^{+}$ ions \cite{dfh16}. A new method employing dynamic
decoupling was suggested to be implementable in current atomic clock
experiments \cite{sos18}, with which an especially high sensitivity to LV is
predicted. In this recent proposal \cite{sos18}, if the total angular momentum
of a physical system is fixed, the LV term can be regarded as an equivalent
term proportional to the square of the $z$-component of total angular moment
operator, a typical local interaction. In this paper, we will investigate the
related effect from such an term. Apart from revisiting this issue in the
current studies, we attempt to elucidate the meaning of locality clearly.
This interesting LV term concerns the dependence of physical results on the
moving direction and the momentum of a particle, resulting in the violation of
local Lorentz invariance (LLI) \cite{mtw70}, which also represents the
violation of Einstein equivalence principle (EEP) \cite{td12,cls15}. Since the
LV effect is always studied in those experiments made with the quantum systems
even using the obvious quantum property (i.e. entanglement in Ref.
\cite{prp15,dfh16}), we hope to investigate whether the above mentioned LV
effect is classical or quantum. This could be done by the recent suggestion
about a quantum formulation of EEP \cite{zb15,omm16}, in which the violations
of LLI can happen as either classical or quantum effects but a criteria was
given to distinguish them.
This paper is organized as follows. First, we revisit the LV term and reveal
its relation to spatial anisotropy in the second section. This is followed by
an analysis of it as a type of local interaction in the third section. In the
fourth section, we compared this LV term with the description about LLI in the
quantum formulation of EEP, and we also study the property of LV effect and
estimate the corresponding violation parameter of LLI according to the related
LV experiments. Finally, we end with a conclusion in the fifth section.
\section{Lorentz symmetry violation (LV)}
Start with the action of SME for QED \cite{vak04},
\begin{equation}
S_{SME}=\in
\mathcal{L
_{SME}d^{4}x,
\end{equation}
where
\mathcal{L
_{SME}
\mathcal{L
_{\psi}
\mathcal{L
_{A}
\mathcal{L
_{g}$.
\mathcal{L
_{\psi}$ is the Lagrangian density of the fermion part and contains terms
dominating at low energies that involve ordinary matter such as protons,
neutrons and electrons and their minimal coupling with gravity and gauge
particles,
\mathcal{L
_{A}$ contains gauge fields for the photon and the fields describing particles
that are not `ordinary', like muons, mesons, neutrinos and so on and the
minimal couplings to gravity, and
\mathcal{L
_{g}$ is the gravitational part that includes the terms consisting of the
vierbein and the spin connection. When LV is considered for the electron
sector within the framework of SME, the Lagrangian density we are concerned is
the fermion part (i.e. electron) up to the first-order expansion
\cite{cm97,cm98},
\begin{equation
\mathcal{L
_{\psi}=\frac{1}{2}i\,\overline{\psi}\left( \gamma_{\nu}+c_{\mu\nu
\gamma^{\mu}\right) \overleftrightarrow{D}^{\nu}\psi-\overline{\psi
\,m_{e}\psi, \label{lag
\end{equation}
where $m_{e}$ denotes the electron mass, $\psi$ is a Dirac spinor,
$\gamma^{\mu}$ are the usual Dirac matrices and $\overline{\psi
\overleftrightarrow{D}^{\nu}\psi\equiv\overline{\psi}D^{\nu}\psi-\psi
D^{\upsilon}\overline{\psi}$ with $D^{\upsilon}$ the shorthand for the
covariant derivative. When the other backgrounds of spacetime like
Riemann-Cartan spacetime is considered, the vierbein and spin connection would
enter into the Eq. (\ref{lag}), i.e. see Ref. \cite{vak04} for the complete
expression. The $c_{\mu\nu}$ tensor in the above Eq. (\ref{lag}) quantifies
the strength of LV for the electron sector by the frame dependent interaction
term, which gives an energy level shift \cite{kl99,kl992,kt11},
\begin{equation}
\delta H_{\mathrm{LSV}}=-[C_{0}^{(0)}-\frac{{2U}}{{3c^{2}}}c_{00
]\frac{{\mathbf{p}^{2}}}{2}-\frac{C_{0}^{(2)}T_{0}^{(2)}}{6}. \label{glv
\end{equation}
where $\mathbf{p}$ is the momentum of the (lone) bound (valance) electron, $U$
is the Newtonian gravitational potential and the specific parameters
$C_{0}^{(0)}$ and $c_{00}$ quantifying the strength of LV have been discussed
and tested before \cite{hlb13}. It represents the violation of Lorentz
symmetry and EEP in bound electronic states. The violation of EEP within SME
frame stemmed from the dependence of effective mass on the gravitational
potential and can be tested using the transition between atomic energy levels.
In particular, the kinetic energy $\frac{{\mathbf{p}^{2}}}{2}$ would also be
changed from one energy level to another one \cite{vvf16}. But in several
other experiments, the different spin states are considered for the test of LV
\cite{prp15,sos18} and the kinetic energy remains fixed when the spin states
are changed. Thus, the first terms in Eq. (\ref{glv}) would not be observed in
these experiments. From the perspective of EEP, the first term represents the
violation of local position invariance (LPI) which usually tested by the
experiments of gravitational redshift \cite{hcm11}. The second term represents
the violation of local Lorentz symmetry.
The relativistic form of rank 2 irreducible tensor operator $T_{0}^{(2)}$ is
$T_{0}^{(2)}=c\gamma_{0}(\mathbf{\gamma p}-3\gamma_{z}p_{z})$, with $p_{z}$
the momentum component along the quantization axis fixed in the laboratory
frame. Its non-relativistic form becomes $T_{0}^{(2)}=({\mathbf{p}^{2
-3p_{z}^{2}})/{m_{e}}$, where $p_{z}$ is the component of the electron's
momentum along the quantization axis which is fixed in the laboratory, $m_{e}$
is the mass of the electron. Thus the second LV term $\propto C_{0}^{(2)}$ of
$\delta H_{\mathrm{LSV}}$ reduces to,
\begin{equation}
\delta H=-C_{0}^{(2)}\frac{\left( \mathbf{p}^{2}-3p_{z}^{2}\right) }{6m_{e
}, \label{hlv
\end{equation}
where the parameter $C_{0}^{(2)}$ characterizes the violation of Lorentz
symmetry we focus on. In relativistic physics, Lorentz symmetry implies an
equivalence of observation or observational symmetry due to special
relativity, or stated more formally that the laws of physics stay the same for
all observers moving with constant velocities with respect to one another
within an inertial frame. It has also been described sometimes as the
independence of all experimental results on the orientation or the boost
velocity of the laboratory through space \cite{mtw70}, which is seen evidently
in Eq. (\ref{hlv})
The matrix element of the $T_{0}^{(2)}$ operator is calculated using the
Wigner-Eckart theorem as \cite{prp15}
\begin{equation}
\left\langle J,m\left\vert T_{0}^{(2)}\right\vert J,m\right\rangle
=\frac{-J(J+1)+3m^{2}}{\sqrt{(2J+3)(J+1)(2J+1)J(2J-1)}}\left\langle
J\left\vert T^{(2)}\right\vert J\right\rangle ,
\end{equation}
where $J$ and $m$ denote the quantum numbers of the total electronic angular
momentum and its projection along the quantization axis, respectively. The
term proportional to $m^{2}$ can be taken as a part of the signal for LV. If a
physical system with a fixed total angular momentum $J$ is considered, as for
the case first suggested in Ref. \cite{sos18}, the dynamics of LV term we
discuss is described equivalently by the Hamiltonian
\begin{equation}
H_{V}=\kappa J_{z}^{2},
\end{equation}
which is the very term we are concerned in this paper. In the following two
sections, we will show that this kind of interaction is local and represents
only a kind of classical violation based on the quantum formulation of EEP.
\section{Local interaction}
In earlier studies on the LV effect, different eigenstates with distinct
absolute values of the angular momentum $J_{z}$ are chosen in order to extract
the relative phase from a coherent superposition state of the eigenstates,
\textit{i.e.} in Ref. \cite{sos18}, the selected states are $\left\vert
\frac{7}{2},-\frac{7}{2}\right\rangle $ and $\left\vert \frac{7}{2},-\frac
{1}{2}\right\rangle $. The phase measurement is implemented through the
influence of the LV term on Ramsey interferometry augmented by dynamical
decoupling (DD).
The standard Ramsey interferometry consists of three parts: two $\frac{\pi
{2}$ pulses, and a free evolution in between. In terms of the pseudo-spin
angular momentum operators, its time evolution is described as
\begin{equation}
U_{\phi}=e^{-i\pi J_{x}/2}e^{i\phi J_{z}}e^{i\pi J_{x}/2}=e^{-i\phi J_{y}},
\end{equation}
where the two $\frac{\pi}{2}$ pulses act as 50:50 beamsplitters, and the
middle term $e^{i\phi J_{z}}$ denotes the free evolution or the free rotation
circling the z-axis (defined by a polarizing magnetic field) over an angle
$\phi$. When the LV term is present, it adds to the above free evolution and
give
\begin{equation}
U_{V\phi}=e^{-i\pi J_{x}/2}e^{i\phi J_{z}}e^{-i\kappa tJ_{z}^{2}}e^{i\pi
J_{x}/2}=e^{-i\phi J_{y}-i\kappa tJ_{y}^{2}},
\end{equation}
which is analogous to the nonlinear Ramsey interferometer
\cite{uf03,cb05,cs08} and provided a way for estimating the LV parameter
$\kappa$.
The effect discussed above is generally interpreted as a local (to the atom
considered) interaction. For an ensemble of $N$ atoms, the LV Hamiltonian can
be expressed as
\begin{equation}
H_{V}=\kappa J_{1z}^{2}+\kappa J_{2z}^{2}+\cdots+\kappa J_{Nz}^{2},
\end{equation}
which leads to
\begin{equation}
U_{V\phi}=\sum_{i=1}^{N}e^{-i\phi J_{yi}-i\kappa tJ_{yi}^{2}}.
\end{equation}
Evidently, this remains local, as it cannot generate any coherent results
among atoms. Consequently, such a form of local unitary interaction cannot
alter entanglement of the quantum state for the ensemble \cite{nc00}. This
also means that although the operation described by $H_{V}$ is for an ensemble
of $N$ atomic spins, the uncertainty in evaluating $\kappa$ scales as
$\propto1/\sqrt[.]{N}$, or follows the standard quantum limit (SQL). It would
be desirable to develop ideas based on entangled quantum states, such as spin
squeezed state \cite{ku93,wbi94}, the twin-fock state atomic Bose-Einstein
condensate (BEC) \cite{hb93,lpb98,lsk11,lzw17,zwl18}, the ground state of an
antiferromagnetic spin-1 atomic condensate \cite{hy00,wy16}, and so on for
improved scaling beyond the SQL.
Since the LV effect described by $H_{V}$ is local, it implies that this effect
might represent only a type of classical violation although the discussion
above is made under a quantum evolution. In the next section, we will
investigate it using a recent framework developed for testing the Einstein
equivalence principle (EEP) \cite{zb15}.
\section{Classical violation}
With the development of atomic precision measurements, some classical physical
effects that are minute and difficult to be observed, such as the violation of
weak equivalence principle (WEP), has been measured experimentally using
quantum systems. Naturally, one wants to understand whether the influence or
violation of these classical effects like WEP is actually caused by the
quantum property. To address this, Zych and Brukner \cite{zb15} constructed a
new framework to test the quantum aspects of the EEP. For a non-relativistic
quantum system with its Hamiltonian given by $H_{\mathrm{nr}}=mc^{2
+\frac{P^{2}}{2m}+mU\left( Q\right) $, where $m$ is the mass of the system,
$Q$, $P$ are position and momentum operators, respectively, for the center of
mass, and $U$ denotes the gravitational potential. In order to distinguish the
influences of different internal energies on the EEP, they suggested a quantum
formulation of the mass-energy equivalence principle by extending the mass
expression $m$ in $H_{\mathrm{nr}}$ int
\begin{equation}
M_{k}=m_{k}I^{\mathrm{int}}+\frac{H_{k}^{\mathrm{int}}}{c^{2}},
\end{equation}
where $k=\{r,i,g\}$ represents quantities related to rest, inertial, and
gravitational masses, respectively, $I^{\mathrm{int}}$ is the identity
operator in the space of internal degrees of freedom, $H^{\mathrm{int}}$ is
the internal energy operator that can contribute to the mass, and $m$ is the
mass of the corresponding ground state for $H^{\mathrm{int}}$. Thus, the total
Hamiltonian up to the lowest order in relativistic corrections is expressed
as
\begin{equation}
H_{\mathrm{test}}=m_{r}c^{2}+H_{r}^{\mathrm{int}}+\frac{P^{2}}{2m_{i}
+m_{g}U\left( Q\right) -H_{i}^{\mathrm{int}}\frac{P^{2}}{2m_{i}^{2}c^{2
}+H_{g}^{\mathrm{int}}\frac{U\left( Q\right) }{c^{2}},\label{the
\end{equation}
where the validity of the EEP is guaranteed by $H_{r}^{\mathrm{int}
=H_{i}^{\mathrm{int}}=H_{g}^{\mathrm{int}}$ or $M_{r}=M_{i}=M_{g}$. Actually,
according to the purpose of measurements, the EEP can be divided into three
classes. One is the validity of WEP which requires the condition $M_{i}=M_{g
$, or $H_{i}^{\mathrm{int}}=H_{g}^{\mathrm{int}}$ if the classical WEP holds
by $m_{i}=m_{g}$. $H_{r}^{\mathrm{int}}=$ $H_{i}^{\mathrm{int}}$ indicates the
validity of LLI and $H_{r}^{\mathrm{int}}=$ $H_{g}^{\mathrm{int}}$ indicates
the validity of LPI. Therefore, if the classical WEP is not violated, the
violation from any one condition does not imply the violation of any other
one, but once any two of the three conditions is broken, the third must be
broken. Comparing Eq. (\ref{glv}) and Eq. (\ref{the}), it is seen easily that
the reasons of the violation of LPI are different. The former violation in Eq.
(\ref{glv}) stemmed from the anomalous coupling between the gravitational
potential and the external kinetic energy, and the latter violation in Eq.
(\ref{the}) is due to the anomalous coupling between the gravitational
potential and the internal energy levels. So in the future, it deserves to
make a further investigation for the possible Lorentz-breaking modifications
in the presence of gravity from the perspective of experimental observation,
since EEP is the concept of general relativity in essence. However, the
related experiments \cite{sos18} involved in this paper cannot be discussed
for the violation of LPI under the background of quantum formulation of EEP
\cite{zb15}. Further, it is noted that the violation of LLI, irrespective of
that described in Eq. (\ref{hlv}) from SME or in Eq. (\ref{the}) from quantum
formulation of EEP, is both derived from the anomalous coupling between
internal energy levels and the external kinetic energy. Thus, in order to
study the behaviour of LV using the quantum formulation of EEP, it is
convenient to assume that LPI is valid since the violation of LLI does not
imply the violation of LPI.
LLI can be tested with such a Hamiltonian \cite{zb15}
\begin{equation}
H_{LLI}=m_{r}c^{2}+H_{r}^{\mathrm{int}}+\frac{P^{2}}{2m_{i}}-H_{i
^{\mathrm{int}}\frac{P^{2}}{2m_{i}^{2}c^{2}}, \label{lpv
\end{equation}
which constitutes an important part of the test framework for the EEP and the
violation is caused only by $H_{r}^{\mathrm{int}}\neq H_{i}^{\mathrm{int}}$ or
$M_{r}\neq M_{i}$. The terms related to the gravitational potential in Eq.
(\ref{the}) are ignored because they contribute equally to the evolution of
different energy levels (that means these terms would be subtracted when
calculating the difference of energy levels) if the LPI preserves.
After the test of the quantum aspects of EEP was suggested, some feasible
experiments were also analyzed \cite{omm16} or made \cite{rdt16}. In Ref.
\cite{omm16}, a harmonically trapped spin-$\frac{1}{2}$ atom is proposed to
test the Hamiltonian $H_{\mathrm{test}}$ and the crucial point lies in the
for
\begin{equation}
H_{k}^{\mathrm{int}}=H_{r}^{\mathrm{int}}+\mu B\xi_{k}\left( H_{r
^{\mathrm{int}}\right) ,\text{ \ }\xi_{k}\left( H_{r}^{\mathrm{int}}\right)
=\left(
\begin{array}
[c]{cc
a_{k} & b_{k}\\
b_{k}^{\ast} & c_{k
\end{array}
\right) ,\text{ }\label{eps
\end{equation}
where $k=i,g$ represents the violation of LLI, LPI, $B$ is the external
magnetic field, and $\mu$ is the atomic magnetic moment. $H_{r}^{\mathrm{int
}=\frac{1}{2}\mu B\left( \left\vert +\frac{1}{2}\right\rangle \left\langle
+\frac{1}{2}\right\vert -\left\vert -\frac{1}{2}\right\rangle \left\langle
-\frac{1}{2}\right\vert \right) =\mu B\left(
\begin{array}
[c]{cc
\frac{1}{2} & 0\\
0 & -\frac{1}{2
\end{array}
\right) $ and $\xi_{k}\left( H_{r}^{\mathrm{int}}\right) $ quantifies the
deviation from the EEP, and in particular, $\xi_{i}\left( H_{r
^{\mathrm{int}}\right) $ quantifies the deviation from the LLI, but the
required measurement was designed \cite{omm16} in the presence of gravitation
and external magnetic field. When the off-diagonal terms $b_{k}$ and its
complex conjugate are not assumed to exist, the violation of EEP parameterized
by $\xi_{k}$ is called the classical violation; otherwise, the violation is
regarded as quantum.
It is noted that $H_{k}^{\mathrm{int}}$ is actually not the test Hamiltonian,
and the complete Hamiltonian we need is obtained by inserting $H_{i
^{\mathrm{int}}$ into the Eq. (\ref{lpv})
\begin{equation}
H_{LLI}=\frac{P^{2}}{2m_{i}}+H_{r}^{\mathrm{int}}-H_{i}^{\mathrm{int}
\frac{P^{2}}{2m_{i}^{2}c^{2}}=\frac{P^{2}}{2m_{i}}+\mu B\left(
\begin{array}
[c]{cc
\frac{1}{2} & 0\\
0 & -\frac{1}{2
\end{array}
\right) +\frac{P^{2}}{2m_{i}^{2}c^{2}}\left(
\begin{array}
[c]{cc
a_{i}^{\prime} & b_{i}^{\prime}\\
b_{i}^{\prime\ast} & c_{i}^{\prime
\end{array}
\right) ,\label{lli
\end{equation}
where the static mass term has been ignored since it contributes only a
unrelated constant phase for the evolution of atoms. From the Eq. (\ref{lli}),
the first and second terms indicate that the atom moves in a magnetic field
freely and the $z$ direction of atomic spin is consistent with that of the
magnetic field. For example, the atom with spin-up (the eigenvalue of the
spin-$z$ component is $\frac{1}{2}$) state initially will keep in this state
during the later evolution if no extra interaction is added. The third term
represents the violation of LLI. Different from Ref. \cite{omm16}, we delete
the \textquotedblleft$\mu B$\textquotedblright\ in the third term since no
reason to imply that the assumption of LLI violation should be related to the
presence of the external magnetic field. In other words, the term
\textquotedblleft$\mu B$\textquotedblright\ is absorbed to the definition of
the violation parameters $a_{i}^{\prime}$ ($=\left( a_{i}+\frac{1}{2}\right)
\mu B$) and $c_{i}^{\prime}$ ($=\left( c_{i}-\frac{1}{2}\right) \mu B$). For
the off-diagonal terms in the third term of Eq. (\ref{lli}), it will lead to
the change of the initial spin state, \textit{i.e.} the atom with spin-up
state initially will change into spin-down state (the eigenvalue of the
spin-$z$ component is $-\frac{1}{2}$). For the diagonal term, it will lead to
the shift of energy level. The corresponding shift of energy level induced by
$\xi_{i}\left( H_{R}^{\mathrm{int}}\right) $ has been calculated and can be
approximated as \cite{omm16
\begin{equation}
\Delta E_{i}^{LLI}\sim\left\langle H_{i}^{\mathrm{int}}\frac{P^{2}}{2m_{i
^{2}c^{2}}\right\rangle \sim\frac{\hbar\omega_{0}\left( 2n+1\right)
{m_{i}c^{2}}a_{i}^{^{\prime}},\label{ess
\end{equation}
where the perturbation calculation gives the first order modification of the
energy level by $\left\langle n\right\vert P^{2}\left\vert n\right\rangle
=\frac{m_{i}\hbar\omega_{0}}{2}(2n+1)$, $n$ is an integer and represents the
energy level of the atom, and $\omega_{0}$ is the equivalent frequency of the
oscillator corresponding to the energy level used experimentally.
Then, we will investigate whether the LV model in the second and third section
can be put into the frame of the recent quantum formulation of EEP based on
the concrete form (\ref{lli}) constructed by the previous suggestion
\cite{omm16}. While considering the LV effect, the crucial part of the
Hamiltonian is expressed as \cite{sos18},
\begin{equation}
H_{LV}=\frac{P^{2}}{2m_{i}}+\mu BJ_{z}+\kappa J_{z}^{2}=\frac{P^{2}}{2m_{i
}+\mu B\left(
\begin{array}
[c]{cc
\frac{1}{2} & 0\\
0 & -\frac{1}{2
\end{array}
\right) +\kappa\left(
\begin{array}
[c]{cc
\frac{1}{4} & 0\\
0 & \frac{1}{4
\end{array}
\right) ,\label{lvc
\end{equation}
where the representation of spin-$\frac{1}{2}$ is taken to compare with the
Hamiltonian (\ref{lli}). As discussed in the last section, the LV effect was
designed to be measured using atomic Ramsey interferometry and the dynamical
decoupling method. It is easy to see from Eqs. (\ref{lli}) and (\ref{lvc})
that the above LV effect only causes the shift of energy levels but do not
cause transitions to different energy levels due to the absence of
off-diagonal terms. Hence, it only constitutes a classical violation of EEP.
Consider the example discussed before for Yb atom, and one can estimate the
parameter $a_{i}^{^{\prime}}$. As tabulated in Table I of the Ref.
\cite{sos18}, the shift of energy level caused by LV effect i
\begin{equation}
\Delta E_{i}^{LV}\sim hC_{0}^{(2)}\times3.9\times10^{16}J\sim10^{-40}J,
\end{equation}
where $h$ is the Planck's constant and $C_{0}^{(2)}\sim10^{-23}$ according to
Ref. \cite{dfh16}. By identifing $\Delta E_{i}^{LV}$ with $\Delta E_{i}^{LLI}$
in Eq. (\ref{ess}), we obtai
\begin{equation}
a_{i}^{^{\prime}}\sim10^{-19},\label{pv
\end{equation}
where the same parameters as in Ref. \cite{omm16} are adopted for the Eq.
(\ref{ess}), i.e. $n=2$, $\mu=0.429\mu_{N}$, $B=1T$, $\omega_{0}=10^{4}Hz$.
This result shows that the present experimental observation for the shift of
energy level is less sensitive than that expected in Ref. \cite{omm16} where
$a_{i}\sim1$ (the violation parameter was defined as ($a_{i}+\frac{1}{2}$)$\mu
B$ in Ref. \cite{omm16}) or $a_{i}^{\prime}\sim10^{-24}$ (according to the
definition here for the violation parameters in Eq. (\ref{lli})) is taken by
hand. A subtle point has to be clarified. For the definition of the previous
violation parameters, i.e. ($a_{i}+\frac{1}{2}$)$\mu B$, it seems that the
value of $a_{i}$ should be at the level of $1$ which is just the value taken
in Ref. \cite{omm16} to estimate the transition probability. In fact,
($a_{i}+\frac{1}{2}$) can take any value since the violation is added into the
Hamitonian by hand. So it is better to introduce a new parameter
$a_{i}^{\prime}$ to replace the previously whole expression ($a_{i}+\frac
{1}{2}$)$\mu B$ without influencing any discussion about the violation of LLI
in the quantum formulation of EEP. In this way, it is seen that the violation
parameter $a_{i}$ that is assumed to be $1$ in Ref. \cite{omm16} is equivalent
to $a_{i}^{\prime}\sim10^{-24}$. Although it is smaller than that given by Eq.
(\ref{pv}), it cannot be a constraint since the estimated transition
probability is too small to be observed at the present experimental presicion
to the best of our knowledge. Therefore, we can declare that the Eq.
(\ref{pv}) represents a constraint for the violation parameter from the
present experiments \cite{dfh16,sos18}, unless a better constraint is found
from any other experiments.
It is noted that when we constrain the violation parameter $a_{i}^{\prime}$
with the atomic experiments related to the LV test (it is easy to repeat the
above related discussion to give the constraint for the violation parameter
$c_{i}^{\prime}$ $\sim10^{-19}$), we have assumed that the discussion about
the LV is based on one specific model for LLI violation in the quantum
formulation of EEP, in which the specific model is given by the atom of
spin-$\frac{1}{2}$ state placing in an external magnetic field. Whether the
phenomenological description for the quantum formulation of EEP is consistent
with SME theory in essence required a further investigation.
\section{Conclusion}
In this paper, we revisit the LV effect and its influence on the Ramsey
interferometry. We provide a clear expression for the atomic Ramsey signal in
the absence or presence of the LV effect. It is emphasized that this type of
the LV effect discussed presents itself only as a local influence which limits
its detectability using the experiments of quantum entanglement beyond the
standard quantum limit unless a new understanding is found or suggested. The
LV effect can be regarded as equivalent to a Hamiltonian proportional to the
square of the angular momentum component along the z-axis, which can be
compared with the recently suggested quantum formulation of the EEP. We
compare the LV effect and the description for the LLI in the quantum
formulation of EEP, and find that the LV effect only amounts to a kind of
classical effect which cannot generate the non-local influence unitarily to
change entanglement among atoms. Based on the spin-$\frac{1}{2}$ system, we
build the relation to constrain the classical violation within the new EEP
test framework by the precision experiments performed to test the LV. Although
the built relation requires the further confirmation, in particular in the
presence of gravity, our work still provides an interesting and novel
exploration for the connection between LLI violation in the quantum
formulation of EEP and LV in SME.
\section{Acknowledge}
This work is supported by National Natural Science Foundation of China (NSFC)
with No. 91636213 and No. 11654001. We also want to thank Li You and Lingna Wu
for their suggestions and discussions.
|
3,212,635,537,948 | arxiv | \section{Introduction}
\subsection{General setting}
This article is devoted to a general analysis of free boundary and free transmission hyperbolic problems
in the one dimensional case.
It is mainly motivated by a new kind of free boundary problem arising in the study of wave-structure
interactions and for which the evolution of the free boundary is governed by a singular equation.
\medbreak
In order to explain the singular structure of this problem, let us recall some results on hyperbolic
initial boundary value problems (a good reference on this subject is the book \cite{benzoniserre2007}).
Let us for instance consider a general quasilinear equation of the form
\[
\partial_t U+ A(U)\partial_x U=0
\]
for $t>0$ and $x\in {\mathbb R}$. It is well known that if the system is Friedrichs symmetrizable, i.e.,
if there exists a positive definite matrix $S(u)$ such that $S(u)A(u)$ is symmetric,
then the associated initial value problem is well-posed in $C([0,T];H^s({\mathbb R}))$ if $s>d+1/2$
(with $d=1$ is the space dimension).
The proof is based on the study of the linearized system and an iterative scheme.
If we consider the same equation on ${\mathbb R}_+$, and impose a boundary condition on $U$ at $x=0$,
then the corresponding initial boundary value problem might not be well-posed,
even if the system is Friedrichs symmetrizable.
Well-posedness is however ensured if there exists a Kreiss symmetrizer which, as the Friedrichs symmetrizer,
transforms the system into a symmetric system, but with the additional property that the boundary condition
for this symmetric system is striclty dissipative (roughly speaking, this means that the trace of the solution
at the boundary is controled by the natural energy estimate).
The construction of such a Kreiss symmetrizer in extremely delicate and is usually done under the so-called
uniform Lopatinski\u{\i} condition which can formally be derived as a stability condition
for the normal mode solutions of the linearized equations with frozen coefficients.
Under such a condition (and additional compatibility conditions between the boundary and initial data),
a unique solution can again be constructed (though with many more technical issues) via estimates
on the linearized system and an iterative scheme.
The typical result for quasilinear initial boundary value problems satisfying the aforementioned condition,
as announced in \cite{RauchMassey} and proved in \cite{mokrane1987}, is that the equations are well-posed
but with higher regularity requirements, and more importantly, with a loss of half a derivative
with respect to the initial and boundary data.
\medbreak
In some situation, the boundary of the domain on which the equations are cast depends on time.
In dimension $d=1$ for instance, this means that instead of working on ${\mathbb R}_+$, one works on $(\underline{x}(t),+\infty)$,
where the function $\underline{x}$ is either a known function (boundary in forced motion) or an unknown function
determined by an equation involving the solution $U$ of the hyperbolic system, typically,
\[
\dot \underline{x}(t)=\chi(U_{\vert_{x=\underline{x}(t)}})
\]
for some smooth function $\chi$ (we shall say that this kind of boundary evolution of ``kinematic type''
because, as for kinematic boundary conditions, the regularity of $\dot\underline{x}$ is the same as the regularity
of the solution at the boundary).
Such problems are called free boundary hyperbolic problems.
\medbreak
It is noteworthy that, up to a doubling of the dimension of the system of equations under consideration,
the considerations above can be extended to transmission problems,
where two possibly different hyperbolic systems are considered on the two different sides of an interface,
and where the boundary condition is replaced by a condition involving the traces of the solution on both sides.
One of the most famous transmission problems with a free boundary is the stability of shocks.
The problem consists in finding solutions to a quasilinear hyperbolic system that are smooth on both sides
of a moving interface and whose traces on the interface satisfy the Rankine--Hugoniot condition.
In dimension $d=1$, this latter condition provides an evolution equation for the interface of the same form as above.
Showing the well-posedness of free boundary hyperbolic problems requires new ingredients and in particular,
\begin{itemize}
\item
A diffeomorphism must be used to transform the problem into a boundary value problem with a fixed boundary.
\item
A change of unknown must be introduced to study the linearized equation.
Indeed, with the standard linearization procedure, a derivative loss occurs due to the dependence
of the transformed problem on the diffeomorphism. This loss is removed by working with
so-called Alinhac's good unknown.
\end{itemize}
The proof of the stability of multidimensional shocks is a celebrated achievement of Majda
\cite{majda1,majda2,majda3}, with improvements in \cite{metivier2001}.
Since the proof relies on the theory of initial boundary value problems, the same loss of half a derivative
with respect to the initial and boundary data is observed.
\medbreak
The free boundary problem that motivates this work is the evolution of the contact line between
a floating object and the water, in the situation where the motion of the waves is assumed to be
governed by the (hyperbolic) nonlinear shallow water equations, and in horizontal dimension $d=1$.
In a simplified version, this problem can be reduced to a free boundary hyperbolic problem,
but with a more singular evolution equation for the free boundary, which is of the form
\[
U(t,\underline{x}(t))=U_{\rm i}(t,\underline{x}(t)),
\]
where $U_{\rm i}$ is a known function (for the contact line problem, this condition expresses the fact
that the surface elevation and the horizontal flux of the water are continuous across the contact point).
Time differentiating this condition yields an evolution equation for $\underline{x}$ of the form
\[
\dot \underline{x}(t) = \chi\bigl( (\partial_t U)_{\vert_{x=\underline{x}(t)}}, (\partial_x U)_{\vert_{x=\underline{x}(t)}},
(\partial_t U_{\rm i})_{\vert_{x=\underline{x}(t)}}, (\partial_x U_{\rm i})_{\vert_{x=\underline{x}(t)}} \bigr).
\]
The standard procedure for free boundary hyperbolic problems descrived above does not work with such a
boundary equation, because there is obviously a loss of one derivative in the estimates:
the boundary condition is fully nonlinear.
In order to handle this new difficulty without using a Nash--Moser type scheme,
we propose to work with a second order linearization and introduce a second order Alinhac's good unknown
in order to cancel out the terms responsible for the derivative losses.
\medbreak
Proving the well-posedness of this fully nonlinear free boundary hyperbolic problem also requires sharp and
new estimates for one-dimensional hyperbolic initial boundary values problems that are of independent interest.
One-dimensional hyperbolic boundary value problems are generally dealt with using the method of
characteristics \cite{li1985boundary}.
In the Sobolev setting, there is no specific work dealing with the one-dimensional setting, and the general
multidimensional results are used, with their drawbacks:
high regularity requirements and derivative loss with respect to the boundary and initial data.
These drawbacks however can easily be bypassed by taking advantage of the specificities of the one-dimensional case,
and in particular of the explicit construction of the Kreiss symmetrizers.
For this reason, we propose in this article a general study of initial boundary value problems
(as well as transmission problems) for fixed, moving, and free boundaries.
This study is based on the new sharp estimates developed to solve the fully nonlinear free boundary problem
mentioned above and fully exploits the specificities of the one-dimensional case.
In particular, the high regularity requirements and the derivative loss of the general theory are removed.
This is for instance of interest to solve the problem of transparent conditions for hyperbolic systems.
We use this general approach to solve several problems coming from wave-structure interactions,
as well as other problems such as conservation laws with a discontinuous flux and the stability
of one-dimensional standards and nonstandards shocks.
Another advantage of our approach is that it is much more elementary than the general results,
and does not require refined paradifferential calculus for instance.
\subsection{Organization of the paper}
Section \ref{sect2} is devoted to the study of several kinds of free boundary problems for
$2\times2$ quasilinear (strictly) hyperbolic systems.
The case of non homogeneous linear initial boundary value problems with variable coefficients and
a fix boundary is considered first in \S \ref{sect2VC}.
The main focus is the derivation of a sharp estimate, given in Theorem \ref{theoIBVP1}, which requires only
a weak control in time of the source term (weaker than $L^1(0,T)$, which is itself weaker than the standard
$L^2(0,T)$ that can be found in the literature \cite{benzoniserre2007}) and which provides a better control
of the trace of the solution at the boundary.
We first assume the existence of a Kreiss symmertrizer and derive a priori weighted $L^2$-estimates in
\S \ref{sectapL2}, and higher order estimates in \S \ref{sect2HO}.
In order to complete the proof of Theorem \ref{theoIBVP1}, the main step,
performed in \S \ref{secttheoIBVP1} is the explicit construction of a Kreiss symmetrizer under an explicit
Lopatinski\u{\i} condition.
In \S \ref{sectapplQL}, these linear estimates are used to prove the well-posedness of quasilinear systems;
Theorem \ref{theoIBVP2} provides a sharp result for such systems, which takes advantage of the specifities
of the one-dimensional case and improves the results provided by the general (multi-dimensional) theorems.
It can for instance be used to improve the existing results concerning transparent boundary conditions
for the nonlinear shallow water equations.
In \S \ref{sectVCm} we go back to the analysis of linear initial boundary value problems,
but this time on a moving domain, i.e., in the case where the domain on which the equations are cast is
$(\underline{x}(t),\infty)$, with $\underline{x}$ assumed here to be a known function. Using a diffeomorphism that maps
${\mathbb R}_+$ to $(\underline{x}(t),\infty)$ for all times, this problem is transformed into an initial boundary value problem
with fix boundary, but whose coefficients depend on the diffeomorphism.
One could apply Theorem \ref{theoIBVP1} to this problem, but would lose an unecessary derivative
in the dependence on the diffeomorphism.
This loss is avoided in Theorem \ref{theoIBVP3} by applying Theorem \ref{theoIBVP1} to the system satisfied
by Alinhac's good unknown;
in order to get a sharp result in terms of regularity requirements on the initial data, the sharp dependence
on the source terms proved in Theorem \ref{theoIBVP1} is necessary at this point.
These linear estimates are then used in \S \ref{sectFB1} to study quasilinear initial boundary value problems
with free boundary, i.e., where the function $\underline{x}(t)$ is no longer assumed to be known,
but satisfies an evolution equation.
The case of an evolution equation of ``kinematic'' type is considered first, so that a diffeomorphism of
``Lagrangian'' type can be used and a solution constructed by an iterative scheme based on the linear estimates
of Theorem \ref{theoIBVP3}.
The more complicated case of fully nonlinear boundary conditions of the type mentioned above is addressed
in \S \ref{sectVCm2}.
To handle this problem, another kind of diffeomorphism must be used and a generalization of Alinhac's good
unknown to the second order must be introduced to remove the loss of derivative induced by the fully nonlinear
boundary condition.
A more general type of fully nonlinear condition is also considered in \S \ref{sectext},
where a coupling with a system of ODEs is allowed.
\medbreak
As an illustration of the fact that the theory developed above for $2\times 2$ initial boundary value problems
can be generalized to systems involving a higher number of equations, we propose in Section \ref{secttransmission}
a rather detailed study of transmission problems.
More precisely, we consider two $2\times 2$ hyperbolic systems cast on both sides of an interface,
and coupled through transmission conditions at the interface.
Such transmission problems can be transformed into $4\times 4$ initial boundary value problems to which
the above theory can be adapted.
Linear transmission problems are first considered in \S \ref{sectVCtransm}, the main step being the construction
of a Kreiss symmetrizer whose nature depends on the number of characteristics pointing towards the interface;
the nonlinear case is then considered in \S \ref{sectappltransmQL}.
Moving interfaces are then treated in \S \ref{secttransmmov} for linear systems and an application to
free boundary transmission problems with ``kinematic'' boundary condition is given in \S \ref{secttransmkin}.
\medbreak
A first application of the general theory described above to wave-structure interactions is given in
Section \ref{sectlatpis}.
The problem consists in studying the interaction of waves in shallow water with a lateral piston.
The nonlinear shallow water equations are a quasilinear hyperbolic problem that falls into the class studied above.
The domain is a half-line delimited by a piston which can move under the pressure force exerted by the wave.
Its motion (and therefore the position of the boundary) is given by the resolution of a second order ODE in time
(Newton's equation) coupled with the nonlinear shallow water equations.
The key step is to show that this evolution equation is essentially of ``kinematic'' type so that
the results of \S \ref{sectFB1} can be applied.
\medbreak
In Section \ref{sectfloat} we present the problem that motivated this work, namely, the description of the
evolution of the contact line between a floating body and the surface of the water in the shallow water regime.
We recall in \S \ref{sectpresfloat} the derivation of the equations proposed in \cite{Lannes2017} to describe
this problem and investigate first, in \S \ref{sectfixfloat}, the case of a fixed floating body.
We show that the problem can be reduced to an initial boundary value problem with free boundary governed
by a fully nonlinear equation, which allows us to use the results of \S \ref{sectVCm2}.
The extension to the case of a floating object with a prescribed motion is then presented in
\S \ref{sectprescfloat} and the more complicated case of a freely floating object is studied in
\S \ref{sectfreefloat}.
For this latter case, the evolution of the contact point is more complicated because it is coupled with
the three dimensional Newton equation for the solid (on the vertical and horizontal coordinates of
the center of mass and on the rotation angle).
Technical computations are postponed to Appendix \ref{appreform}.
\medbreak
We finally present in Section \ref{sectsev} several applications of our results on transmission problems.
The first one, considered in \S \ref{sectdiscf} is a general $2\times 2$ system of conservation laws
with a discontinuous flux (a typical example is provided by the nonlinear shallow water equations
over a discontinuous topography).
We then investigate in \S \ref{sectshocks} the stability of one-dimensional shocks
(both classical and undercompressive); using our sharp one-dimensional results,
we are able to improve the results one would obtain by considering the one-dimensional case
in the general multi-dimensional theory of \cite{majda1,majda2,majda3,metivier2001} for classical shocks
and \cite{coulombel2003} for undercompressive shocks.
\subsection{General notations}\label{sectnot}
- We write $\Omega_T = (0,T)\times {\mathbb R}_+$.\\
- The notation $\partial$ stands for either $\partial_x$ or $\partial_t$, so that $\partial f\in L^\infty(\Omega_T)$ for instance,
means $\partial_x f\in L^\infty(\Omega_T)$ and $\partial_t f\in L^\infty(\Omega_T)$.\\
- We denote by $\cdot$ the ${\mathbb R}^2$ scalar product and by $(\cdot,\cdot)_{L^2}$ the $L^2({\mathbb R}_+)$ scalar product. \\
- If $A$ is a vector or matrix, and $X$ a functional space, we simply write $A\in X$ to express the fact that
all the elements of $A$ belong to $X$. \\
- In order to define smooth solutions of hyperbolic systems in $\Omega_T=(0,T)\times {\mathbb R}_+$,
it is convenient to introduce the space ${\mathbb W}^m(T)$ as
\[
{\mathbb W}^m(T)=\bigcap_{j=0}^l C^j([0,T];H^{m-j}({\mathbb R}_+)),
\]
with associated norm
\[
\| u \|_{{\mathbb W}^m(T)} = \sup_{t\in[0,T]} \@ifstar\@opnorms\@opnorm{ u(t) }_{m}
\quad \mbox{ with }\quad
\@ifstar\@opnorms\@opnorm{ u(t) }_m = \sum_{j=0}^m \| \partial_t^j u(t) \|_{H^m({\mathbb R}_+)}.
\]
We have in particular $H^{m+1}(\Omega_T)\subset {\mathbb W}^m(T) \subset H^m(\Omega_T)$. \\
- In order to control the boundary regularity of the solution, it is convenient to use the norm
\[
|u_{\vert_{x=0}}|_{m,t} = \biggl( \sum_{j=0}^m|(\partial_x^ju)_{\vert_{x=0}}|_{H^{m-j}(0,t)}^2 \biggr)^\frac12
= \biggl( \sum_{|\alpha| \leq m}|(\partial^\alpha u)_{\vert_{x=0}}|_{L^2(0,t)}^2 \biggr)^\frac12.
\]
- We also use weighted norms with an exponential function $e^{-\gamma t}$ for $\gamma>0$ defined by
\begin{align*}
& |g|_{L_\gamma^2(0,t)} = \biggl( \int_0^t e^{-2\gamma t'}|g(t')|^2{\rm d}t' \biggr)^\frac12, \qquad
|g|_{H_\gamma^m(0,t)} = \biggl( \sum_{j=0}^m |\partial_t^jg|_{L_\gamma^2(0,t)}^2 \biggr)^\frac12, \\
& \@ifstar\@opnorms\@opnorm{ u(t) }_{m,\gamma} = e^{-\gamma t}\@ifstar\@opnorms\@opnorm{ u(t) }_m, \qquad
\| u \|_{{\mathbb W}_\gamma^m(T)} = \sup_{t\in[0,T]} \@ifstar\@opnorms\@opnorm{ u(t) }_{m,\gamma}, \\
& |u_{\vert_{x=0}}|_{m,\gamma,t} = \biggl( \sum_{j=0}^m|(\partial_x^ju)_{\vert_{x=0}}|_{H_\gamma^{m-j}(0,t)}^2 \biggr)^\frac12. \\
\end{align*}
\medskip
\noindent
{\bf Acknowledgement} \
This work was carried out when T. I. was visiting Universit\'e de Bordeaux
on his sabbatical leave during the 2017 academic year.
He is very grateful to the members of Institut de Math\'ematiques de Bordeaux
for their kind hospitality and for fruitful discussions.
T. I. was partially supported by JSPS KAKENHI Grant Number JP17K18742 and JP17H02856.
D. L. is partially supported by the Del Duca Fondation, the Conseil R\'egional d'Aquitaine
and the ANR-17-CE40-0025 NABUCO.
\section{Hyperbolic initial boundary value problems with a free boundary}\label{sect2}
This section is devoted to the analysis of a general class of initial boundary value problems,
with a boundary that can be either fixed, in prescribed motion, or freely moving.
We refer to \S \ref{sectnot} for the notations used, and in particular for the definition of the functional spaces.
\subsection{Variable coefficients linear $2\times 2$ initial boundary value problems}\label{sect2VC}
The aim of this section is to provide an existence theorem with sharp estimates for
a general linear initial boundary value problem with variable coefficients of the following form,
\begin{equation}\label{systVC}
\begin{cases}
\partial_t u + A(t,x)\partial_x u + B(t,x) u = f(t,x) &\mbox{in}\quad \Omega_T, \\
u_{\vert_{t=0}} = u^{\rm in}(x) & \mbox{on}\quad {\mathbb R}_+, \\
\nu(t) \cdot u_{\vert_{x=0}} = g(t)& \mbox{on}\quad (0,T),
\end{cases}
\end{equation}
where $u$, $u^{\rm in}$, $f$, and $\nu$ are ${\mathbb R}^2$-valued functions and $g$ is real-valued function,
while $A$ and $B$ take their values in the space of $2\times2$ real-valued matrices.
We also make the following assumption on the hyperbolicity of the system and on the boundary condition.
\begin{assumption}\label{asshyp}
There exists $c_0>0$ such that the following assertions hold.
\begin{enumerate}
\item[{\bf i.}]
$A\in W^{1,\infty}(\Omega_T), \; B\in L^\infty(\Omega_T), \; \nu \in C([0,T])$.
\item[{\bf ii.}]
For any $(t,x) \in \Omega_T$, the matrix $A(t,x)$ has eigenvalues $\lambda_+(t,x)$ and $-\lambda_-(t,x)$ satisfying
\[
\lambda_\pm(t,x)\geq c_0.
\]
\item[{\bf iii.}]
(The uniform Kreiss--Lopatinski\u{\i} condition.)
Denoting by ${\bf e}_+(t,x)$ a unit eigenvector associated to the eigenvalue $\lambda_+(t,x)$ of $A(t,x)$,
for any $t\in[0,T]$ we have
\[
|\nu(t,0) \cdot \mathbf{e}_{+}(t,0)| \geq c_0.
\]
\end{enumerate}
\end{assumption}
\begin{example}\label{ex1}
A typical example of application is to consider the linearized shallow water equations
with a boundary condition on the horizontal water flux $q$.
This system has the form
\[
\begin{cases}
\partial_t \zeta + \partial_x q = 0, \\
\partial_t q +2 \frac{\underline{q}}{\underline{h}}\partial_x q
+ \Bigl( \mathtt{g}\underline{h}- \frac{\underline{q}^2}{\underline{h}^2} \Bigr)\partial_x \zeta = 0
\end{cases}
\]
with initial and boundary conditions
\[
(\zeta,q)_{\vert_{t=0}} = (\zeta^{\rm in},q^{\rm in})
\quad\mbox{ and }\quad q_{\vert_{x=0}} = g,
\]
where $\mathtt{g}$ is the gravitational constant.
This problem is of the form \eqref{systVC} with $u = (\zeta,q)^{\rm T}$, $B=0$, $f=0$, $\nu=(0,1)^{\rm T}$, and
\begin{equation}\label{eqASW}
A(t,x)=A(\underline{u})=
\begin{pmatrix}
0 & 1 \\
\mathtt{g}\underline{h}-\frac{\underline{q}^2}{\underline{h}^2} & 2\frac{\underline{q}}{\underline{h}}
\end{pmatrix}.
\end{equation}
The eigenvalues $\pm\lambda_{\pm}$ and the corresponding unit eigenvectors $\mathbf{e}_{\pm}$ of $A$
are given by $\lambda_{\pm} = \sqrt{\mathtt{g}\underline{h}}\pm \frac{\underline{q}}{\underline{h}}$
and ${\bf e}_{\pm} = \frac{1}{\sqrt{1+\lambda_{\pm}^2}}(1,\pm\lambda_{\pm})^{\rm T}$, so that
Assumption \ref{asshyp} is satisfied provided that $\underline{h},\underline{q} \in W^{1,\infty}(\Omega_T)$, and
\[
\underline{h}(t,x) \geq c_0, \qquad
\sqrt{\mathtt{g}\underline{h}(t,x)} \pm \frac{\underline{q}(t,x)}{\underline{h}(t,x)}\geq c_0
\]
with some positive constant $c_0$ independent of $(t,x)\in\Omega_T$.
\end{example}
\begin{notation}\label{dualnorm}
In order to define an appropriate norm to the source term $f(t,x)$ in \eqref{systVC},
it is convenient to use the following norm to functions of $t$
\[
S_{\gamma,T}^*(f) = \sup_{\varphi} \biggl\{ \biggl| \int_0^T e^{-2\gamma t}f(t)\varphi(t){\rm d}t \biggr|
\,;\, \sup_{t\in[0,T]}e^{-\gamma t}|\varphi(t)|
+ \biggl( \gamma \int_0^T e^{-2\gamma t}|\varphi(t)|^2{\rm d}t\biggr)^\frac12 \leq 1 \biggr\},
\]
which is the norm of the dual space to $L_\gamma^{\infty}(0,T) \cap L_\gamma^2(0,T)$ equipped with the norm
\[
\sup_{t\in[0,T]}e^{-\gamma t}|\varphi(t)|
+ \biggl( \gamma \int_0^T e^{-2\gamma t}|\varphi(t)|^2{\rm d}t\biggr)^\frac12
\]
associated to the inner product of $L_\gamma^2(0,T)$.
\end{notation}
It is easy to check that $S_{\gamma,t}^*(f)$ is a nondecreasing function of $t\geq0$ for each fixed $f$
and that $S_{\gamma,t}^*(f)$ is monotone with respect to $f$ in the sense that if $0\leq f_1(t) \leq f_2(t)$
for $t\in[0,T]$, then we have $S_{\gamma,t}^*(f_1) \leq S_{\gamma,t}^*(f_2)$ for $t\in[0,T]$.
Moreover, we have
\[
S_{\gamma,T}^*(f) \leq \int_0^T e^{-\gamma t}|f(t)| {\rm d}t \quad\mbox{and}\quad
S_{\gamma,T}^*(f) \leq \biggl( \frac{1}{\gamma} \int_0^T e^{-2\gamma t}|f(t)|^2{\rm d}t\biggr)^\frac12.
\]
\begin{remark}\label{rememb}
The first of these two inequalities implies an $L^2$-type control through the Cauchy--Schwarz inequality,
\[
\int_0^T e^{-\gamma t}|f(t)| {\rm d}t \leq \sqrt{T}\biggl( \int_0^T e^{-2\gamma t}|f(t)|^2{\rm d}t\biggr)^\frac12,
\]
but with a right-hand side involving a factor ${\sqrt{T}}$. This is not the case for the $L^2$-type control
(with respect to time) deduced from $S_{\gamma,T}^*(f) $ and this improvement allows to derive energy estimates
with an exponential growth in Theorems \ref{theoIBVP1}, \ref{theoIBVP3}, and \ref{theoIBVP1transm} for instance.
\end{remark}
The main result of this section is the following theorem (see \S \ref{sectnot} for the definition of .$ {\mathbb W}^{m-1}(T)$ and of the various weighted norms used in the statement).
\begin{theorem}\label{theoIBVP1}
Let $m\geq1$ be an integer, $T>0$, and assume that Assumption \ref{asshyp} is satisfied for some $c_0>0$.
Assume moreover that there are constants $0<K_0\leq K$ such that
\[
\begin{cases}
\frac{1}{c_0}, \Vert A \Vert_{L^\infty(\Omega_T)}, \abs{\nu}_{L^\infty(0,T)} \leq K_0, \\
\Vert A\Vert_{W^{1,\infty}(\Omega_T)}, \Vert B \Vert_{L^\infty(\Omega_T)},
\Vert(\partial A,\partial B)\Vert_{ {\mathbb W}^{m-1}(T)}, \abs{\nu}_{W^{m,\infty}(0,T)} \leq K.
\end{cases}
\]
Then, for any data $u^{\rm in} \in H^m({\mathbb R}_+)$, $g\in H^m(0,T)$, and $f\in H^m(\Omega_T)$ satisfying
the compatibility conditions up to order $m-1$ in the sense of Definition \ref{defcompVC} below,
there exists a unique solution $u \in {\mathbb W}^m(T)$ to the initial boundary value problem \eqref{systVC}.
Moreover, the following estimate holds for any $t\in[0,T]$ and any $\gamma \geq C(K)$:
\begin{align*}
& \@ifstar\@opnorms\@opnorm{ u(t) }_{m,\gamma} + \biggl( \gamma\int_0^t\@ifstar\@opnorms\@opnorm{ u(t') }_{m,\gamma}^2{\rm d}t' \biggr)^\frac12
+ \vert u_{\vert_{x=0}} \vert_{m,\gamma,t} \\
&\leq C(K_0)\bigl( \@ifstar\@opnorms\@opnorm{ u(0) }_{m} + \vert g \vert_{H_\gamma^m(0,t)}
+ |f_{\vert_{x=0}}|_{m-1,\gamma,t} + S_{\gamma,t}^*(\@ifstar\@opnorms\@opnorm{ \partial_t f(\cdot) }_{m-1}) \bigr).
\end{align*}
Particularly, we have
\begin{align*}
& \@ifstar\@opnorms\@opnorm{ u(t) }_{m} + \vert u_{\vert_{x=0}} \vert_{m,t} \\
&\leq C(K_0)e^{C(K)t} \biggl( \@ifstar\@opnorms\@opnorm{ u(0) }_{m} + \vert g \vert_{H^m(0,t)}
+ |f_{\vert_{x=0}}|_{m-1,t} + \int_0^t \@ifstar\@opnorms\@opnorm{ \partial_t f(t') }_{m-1}{\rm d}t' \biggr).
\end{align*}
\end{theorem}
\begin{remark}
The estimates provided by the theorem are a refinement of classical estimates that can be found
in the extensive literature on initial boundary value problems
(see for instance \cite{schochet1986, metivier2001, benzoniserre2007, metivier2012}).
{\bf i.} \
With the exception of \cite{metivier2001}, these references provide a control of the source term
in $L^2$-norm with respect to time;
it turns out that such a control is not enough to handle ``fully nonlinear'' boundary conditions
as in \S \ref{sectVCm2} below.
In \cite{metivier2001}, a more precise upper bound involving only the $L^1$-norm in time of $f$ is provided,
but only for constant coefficient symmetric systems.
The above theorem extends this result to variable coefficients systems and also refines it
since it provides a control in terms of $S^*_{\gamma,t}$ instead of $L^1$.
This latter refinement is important for instance to get low regularity results
-- ${\mathbb W}^2(T)$ instead of ${\mathbb W}^3(T)$ -- in Theorems \ref{theoIBVP2}, \ref{theoIBVP4}, \ref{theoIBVP5},
\ref{theoIBVP6}, and \ref{theoIBVP2transm}.
{\bf ii.} \
The estimates of the theorem provide a control of $\abs{u_{\vert_{x=0}}}_{m,t}$ and not only of $\abs{u_{\vert_{x=0}}}_{H^m(0,t)}$.
{\bf iii.} \
In addition to the classical $L^\infty(0,T)$ upper bound on $t\mapsto \@ifstar\@opnorms\@opnorm{ u(t) }_m$,
our estimates provide a control of its $L^1(0,T)$-norm which is uniform with respect to $t$
(see the comments in Remark \ref{rememb} above) which is typical of weghted estimates
\cite{metivier2012,benzoniserre2007}.
This term is essential in the derivation of the higher-order estimates
(see the proof of Proposition \ref{propVC2}).
\end{remark}
\begin{remark}
The assumption $\vert \nu \vert_{W^{m,\infty}(0,T)}\leq K$ can be weakened into
$\vert \nu \vert_{W^{1,\infty} \cap W^{m-1,\infty}(0,T)} \leq K$ and $\vert \partial_t^m \nu \vert_{L^2(0,T)}\leq K$
(this is a particular case of Theorem \ref{theoIBVP3} below with $\underline{x}\equiv 0$).
\end{remark}
\subsubsection{Compatibility conditions}
From the interior equations, denoting $u_k=\partial_t^k u$, we have
\[
u_1=-A\partial_x u -Bu +f.
\]
More generally, differentiating the equation $k$-times with respect to $t$, we have
a recursion relation
\[
u_{k+1} = -\sum_{j=0}^k \begin{pmatrix} k \\ j \end{pmatrix}\{
(\partial_t^{k-j}A)\partial_x u_j + (\partial_t^{k-j}B)u_j \} + \partial_t^kf.
\]
For a smooth solution $u$, $u_k^{\rm in} = {u_k}_{\vert_{t=0}}$ is therefore given inductively by
$u_0^{\rm in} = u^{\rm in}$ and
\begin{equation}\label{defu0k}
u_{k+1}^{\rm in} =-\sum_{j=0}^k \begin{pmatrix} k \\ j \end{pmatrix}\{
(\partial_t^{k-j}A)_{\vert_{t=0}}\partial_x u_j^{\rm in} + (\partial_t^{k-j}B)_{\vert_{t=0}}u_j^{\rm in} \} + (\partial_t^k f)_{\vert_{t=0}}.
\end{equation}
The boundary condition $\nu(t)\cdot u_{\vert_{x=0}}=g$ also implies that
\[
\partial_t^k \big(\nu(t)\cdot {u}_{\vert_{x=0}}\big)=\partial_t^k g.
\]
On the edge $\{t=0,x=0\}$, smooth enough solutions must therefore satisfy
\begin{equation}\label{compkVC}
\sum_{j=0}^k \binom{k}{j} (\partial_t^j \nu)_{\vert_{t=0}}\cdot {u^{\rm in}_{k-j}}_{\vert_{x=0}} = (\partial_t^k g)_{\vert_{t=0}}.
\end{equation}
\begin{definition}\label{defcompVC}
Let $m\geq1$ be an integer.
We say that the data $u^{\rm in}\in H^m({\mathbb R}_+)$, $f\in H^m(\Omega_T)$, and $g \in H^m(0,T)$
for the initial boundary value problem \eqref{systVC} satisfy the compatibility condition at order $k$
if the $\{u_j^{\rm in}\}_{j=0}^{m}$ defined in \eqref{defu0k} satisfy \eqref{compkVC}.
We also say that the data satisfy the compatibility conditions up to order $m-1$ if they satisfy the
compatibility conditions at order $k$ for $k=0,1,\ldots,m-1$.
\end{definition}
\subsubsection{A priori $L^2$-estimate}\label{sectapL2}
We prove here an $L^2$ a priori estimate using the following assumption,
which will be verified later as a consequence of Assumption \ref{asshyp}.
\begin{assumption}\label{assVC}
There exists a symmetric matrix $S(t,x) \in {\mathcal M}_2({\mathbb R})$ such that
for any $(t,x)\in\Omega_T$ $S(t,x)A(t,x)$ is symmetric and the following conditions hold.
\begin{enumerate}
\item[{\bf i.}]
There exist constants $\alpha_0,\beta_0>0$ such that for any
$(v,t,x)\in {\mathbb R}^2\times \Omega_T$ we have
\[
\alpha_0 |v|^2 \leq v^{\rm T} S(t,x) v \leq \beta_0 |v|^2.
\]
\item[{\bf ii.}]
There exist constants $\alpha_1,\beta_1>0$ such that for any
$(v,t)\in {\mathbb R}^2\times (0,T)$ we have
\[
v^{\rm T} S(t,0)A(t,0) v \leq -\alpha_1 |v|^2 + \beta_1 |\nu(t) \cdot v|^2.
\]
\item[{\bf iii.}]
There exists a constant $\beta_2$ such that
\[
\| \partial_t S + \partial_x (SA) - 2SB \|_{L^2(\Omega_T)\to L^2(\Omega_T)} \leq \beta_2.
\]
\end{enumerate}
\end{assumption}
\begin{notation}\label{notin}
We denote by $\beta_0^{\rm in}\leq \beta_0$ any constant such that
the inequality in {\bf i} of the assumption is satisfied at $t=0$.
\end{notation}
In the $L^2$ a priori estimate provided by the proposition, the control of the source term by
$S_{\gamma,t}^*( \|f(\cdot)\|_{L^2} )$ is crucial to get the refined higher order estimates of
Theorem \ref{theoIBVP1}.
\begin{proposition}\label{propNRJ1}
Under Assumption \ref{assVC}, there are constants
\[
\mathfrak{c}_0 = C\Bigl( \frac{\beta_0^{\rm in}}{\alpha_0},\frac{\beta_0^{\rm in}}{\alpha_1} \Bigr)
\quad\mbox{ and }\quad
\mathfrak{c}_1 = C\Big( \frac{\beta_0}{\alpha_0},\frac{\beta_1}{\alpha_0},\frac{\alpha_0}{\alpha_1} \Big)
\]
such that for any $u \in H^1(\Omega_T)$ solving \eqref{systVC}, any $t\in [0,T]$, and any
$\gamma\geq\frac{\beta_2}{\alpha_0}$,
the following inequality holds.
\begin{align*}
&\@ifstar\@opnorms\@opnorm{u(t)}_{0,\gamma} + \biggl(\gamma \int_0^t \@ifstar\@opnorms\@opnorm{u(t')}_{0,\gamma}^2 {\rm d}t' \biggr)^\frac12
+ |u_{\vert_{x=0}}|_{L_\gamma^2(0,t)} \\
&\leq \mathfrak{c}_0\|u^{\rm in}\|_{L^2}
+ \mathfrak{c}_1\bigl( |g|_{L_\gamma^2(0,t)} + S_{\gamma,t}^*( \|f(\cdot)\|_{L^2} ) \bigr),
\end{align*}
where we recall that $S_{\gamma,t}^*( \|f(\cdot)\|_{L^2} )$ is defined in Notation \ref{dualnorm}.
\end{proposition}
\begin{proof}
Multiplying the first equation of \eqref{systVC} by $S$ and taking the $L^2(\Omega_t)$ scalar product with
$e^{-2\gamma t}u$,
we get after integration by parts,
\begin{align*}
& e^{-2\gamma t}(Su(t),u(t))_{L^2} + 2\gamma\int_0^t e^{-2\gamma t'}(Su,u)_{L^2}{\rm d}t'
- \int_0^t e^{-2\gamma t'}(SA u\cdot u )_{\vert_{x=0}}{\rm d}t' \\
&= (S_{\vert_{t=0}}u^{\rm in},u^{\rm in})_{L^2}
+ \int_0^t e^{-2\gamma t'}((\partial_t S+\partial_x (SA)-2 SB)u + 2Sf,u)_{L^2}{\rm d}t'.
\end{align*}
Using Assumption \ref{assVC} with Notation \ref{notin}, this yields
\begin{align*}
&\alpha_0 \@ifstar\@opnorms\@opnorm{u(t)}_{0,\gamma}^2 + (2\alpha_0\gamma - \beta_2)\int_0^t \@ifstar\@opnorms\@opnorm{u(t')}_{0,\gamma}^2{\rm d}t'
+ \alpha_1 |u_{\vert_{x=0}}|^2_{L_\gamma^2(0,t)} \\
&\leq \beta_0^{\rm in} \|u^{\rm in}\|_{L^2}^2 + \beta_1 |g|^2_{L_\gamma^2(0,t)}
+ 2\beta_0 \int_0^t e^{-2\gamma t'}\|f(t')\|_{L^2} \|u(t')\|_{L^2} {\rm d}t'.
\end{align*}
We evaluate the last term as
\begin{align*}
& \int_0^t e^{-2\gamma t'}\|f(t')\|_{L^2} \|u(t')\|_{L^2} {\rm d}t' \\
&\leq S_{\gamma,t}^*(\|f(\cdot)\|_{L^2}) \biggl\{ \|u\|_{{\mathbb W}_\gamma^0(t)}
+ \biggl(\gamma\int_0^t \@ifstar\@opnorms\@opnorm{u(t')}_{0,\gamma}^2{\rm d}t' \biggr)^\frac12 \biggr\} \\
&\leq S_{\gamma,t}^*(\|f(\cdot)\|_{L^2}) \|u\|_{{\mathbb W}_\gamma^0(t)}
+ \frac{\beta_0}{\alpha_0}S_{\gamma,t}^*(\|f(\cdot)\|_{L^2})^2
+ \frac14\frac{\alpha_0}{\beta_0}\gamma \int_0^t\@ifstar\@opnorms\@opnorm{u(t')}_{0,\gamma}^2{\rm d}t'
\end{align*}
and we deduce that
\begin{align}\label{estuuu}
&\@ifstar\@opnorms\@opnorm{u(t)}_{0,\gamma}^2
+ \frac{\gamma}{2}\int_0^t \@ifstar\@opnorms\@opnorm{u(t')}_{0,\gamma}^2{\rm d}t'
+ \frac{\alpha_1}{\alpha_0} |u_{\vert_{x=0}}|^2_{L_\gamma^2(0,t)} \\
&\leq \frac{\beta_0^{\rm in}}{\alpha_0} \|u^{\rm in}\|_{L^2}^2 + \frac{\beta_1}{\alpha_0} |g|^2_{L_\gamma^2(0,t)}
+ 2\frac{\beta_0}{\alpha_0}S_{\gamma,t}^*(\|f(\cdot)\|_{L^2}) \|u\|_{{\mathbb W}_\gamma^0(t)}
+ 2\biggl( \frac{\beta_0}{\alpha_0}S_{\gamma,t}^*(\|f(\cdot)\|_{L^2}) \biggr)^2 \nonumber \\
&\leq \frac{\beta_0^{\rm in}}{\alpha_0} \|u^{\rm in}\|_{L^2}^2 + \frac{\beta_1}{\alpha_0} |g|^2_{L_\gamma^2(0,t)}
+ \frac12\|u\|_{{\mathbb W}_\gamma^0(t)}^2
+ 4\biggl( \frac{\beta_0}{\alpha_0}S_{\gamma,t}^*(\|f(\cdot)\|_{L^2}) \biggr)^2 \nonumber
\end{align}
for $\gamma\geq\frac{\beta_2}{\alpha_0}$.
Particularly, we have
\[
\frac12\|u\|_{{\mathbb W}_\gamma^0(t)}^2
\leq \frac{\beta_0^{\rm in}}{\alpha_0} \|u^{\rm in}\|_{L^2}^2 + \frac{\beta_1}{\alpha_0} |g|^2_{L_\gamma^2(0,t)}
+ 4\biggl( \frac{\beta_0}{\alpha_0}S_{\gamma,t}^*(\|f(\cdot)\|_{L^2}) \biggr)^2.
\]
Plugging this into \eqref{estuuu}, we obtain the desired estimate.
\end{proof}
\subsubsection{Product and commutator estimates}
To obtain higher order a priori estimates, we need to use calculus inequalities.
By the standard Sobolev imbedding theorem $H^1({\mathbb R}_+) \subseteq L^\infty({\mathbb R}_+)$,
we can easily obtain the following lemma.
\begin{lemma}\label{ineq1}
Let $m\geq1$ be an integer.
There exists a constant $C$ such that the following inequalities hold:
\begin{enumerate}
\setlength{\itemsep}{3pt}
\item[{\bf i.}]
$\@ifstar\@opnorms\@opnorm{ u(t)v(t) }_m \leq C(\|u(t)\|_{L^\infty({\mathbb R}_+)} + \@ifstar\@opnorms\@opnorm{ \partial u(t) }_{m-1}) \@ifstar\@opnorms\@opnorm{ v(t)}_m$,
\item[{\bf ii.}]
$\|[\partial^\alpha,u(t)]v(t)\|_{L^2({\mathbb R}_+)}
\leq C(\|\partial u(t)\|_{L^\infty({\mathbb R}_+)} + \@ifstar\@opnorms\@opnorm{ \partial u(t) }_{m-1}) \@ifstar\@opnorms\@opnorm{ v(t) }_{m-1}$
if \ $|\alpha| \leq m$,
\item[{\bf iii.}]
$\|\partial[\partial^\alpha,u(t)]v(t)\|_{L^2({\mathbb R}_+)}
\leq C(\|\partial u(t)\|_{L^\infty({\mathbb R}_+)} + \@ifstar\@opnorms\@opnorm{ \partial u(t) }_{m-1}) \@ifstar\@opnorms\@opnorm{ v(t) }_{m-1}$
if \ $|\alpha| \leq m-1$,
\item[{\bf iv.}]
$\|\partial[\partial^\alpha;u(t),v(t)]\|_{L^2({\mathbb R}_+)}
\leq C\@ifstar\@opnorms\@opnorm{ \partial u(t) }_{m-2} \@ifstar\@opnorms\@opnorm{ \partial v(t) }_{m-2}$ if \ $2\leq |\alpha| \leq m-1$,
\end{enumerate}
where $[\partial^\alpha;u,v] = \partial^\alpha(uv)-(\partial^\alpha u)v-u(\partial^\alpha v)$ is a
symmetric commutator.
\end{lemma}
The following Moser-type inequality is a direct consequence of the above lemma.
\begin{lemma}\label{ineq2}
Let $\mathcal{U}$ be an open set in ${\mathbb R}^N$, $F \in C^\infty(\mathcal{U})$, and $F(0)=0$.
If $m \in {\mathbb N}$ and $u \in {\mathbb W}^m(T)$ takes its value in a compact set
$\mathcal{K} \subset \mathcal{U}$, then for any $t\in[0,T]$ we have
\[
\@ifstar\@opnorms\@opnorm{ (F(u))(t) }_m \leq C(\|u\|_{W^{[m/2],\infty}(\Omega_t)}) \@ifstar\@opnorms\@opnorm{ u(t) }_m,
\]
where $[m/2]$ is the integer part of $m/2$.
\end{lemma}
We also need Moser-type inequalities for the trace at the boundary of the nonlinear terms,
as in the following lemma.
\begin{lemma}\label{ineq3}
Let $\mathcal{U}$ be an open set in ${\mathbb R}^N$, $F \in C^\infty(\mathcal{U})$, and $F(0)=0$.
If $m \in {\mathbb N}$ and $u=u(t,x)$ takes its value in a compact set $\mathcal{K} \subset \mathcal{U}$,
then we have
\begin{enumerate}
\setlength{\itemsep}{3pt}
\item[{\bf i.}]
$|F(u)_{\vert_{x=0}}|_{m,t}
\leq C( \sum_{|\alpha| \leq [m/2]}|(\partial^\alpha u)_{\vert_{x=0}}|_{L^\infty(0,t)} )|u_{\vert_{x=0}}|_{m,t}$,
\item[{\bf ii.}]
$|F(u)_{\vert_{x=0}}|_{m,t} \leq C( \|u\|_{{\mathbb W}^{[m/2]+1}(t)} )|u_{\vert_{x=0}}|_{m,t}$,
\item[{\bf iii.}]
$|\partial_t(F(u))_{\vert_{x=0}}|_{m,t}
\leq C( \|u\|_{{\mathbb W}^{m}(t)}, \|u\|_{L^\infty(\Omega_T)})
(|(\partial_t u)_{\vert_{x=0}}|_{m,t} + \|\partial_t u\|_{{\mathbb W}^{m}(t)}|u_{\vert_{x=0}}|_{m,t})$,
\end{enumerate}
where $[m/2]$ is the integer part of $m/2$.
\end{lemma}
\begin{proof}
The proof of {\bf i} is straightforward and {\bf i} together with the Sobolev imbedding theorem
$H^1({\mathbb R}_+) \subseteq L^\infty({\mathbb R}_+)$ yields {\bf ii}.
We will prove {\bf iii}.
The case $m=0$ is obvious so that we assume $m\geq1$.
In view of $\partial^\alpha\partial_t(F(u)) = F'(u)\partial^\alpha\partial_t u + [\partial^\alpha,F'(u)]\partial_t u$, we have
\begin{align*}
|\partial_t(F(u))_{\vert_{x=0}}|_{m,t}
&\leq C|(\partial_t u)_{\vert_{x=0}}|_{m,t}
+ C\|\partial_t u\|_{W^{m-1,\infty}(\Omega_t)}\sum_{1 \leq |\alpha| \leq m}|\partial^\alpha F'(u)|_{L^2(0,t)} \\
&\leq C|(\partial_t u)_{\vert_{x=0}}|_{m,t}
+ C(\|u\|_{{\mathbb W}^{[m/2]+1}(t)})\|\partial_t u\|_{{\mathbb W}^m(t)}|u_{\vert_{x=0}}|_{m,t}.
\end{align*}
Since $[m/2]+1 \leq m$, we obtain the desired inequality.
\end{proof}
\begin{lemma}\label{ineq4}
There exists an absolute constant $C$ such that for any $\gamma>0$ and any integer $m\geq1$ we have
\begin{align}
\label{ineq5}
& e^{-\gamma t}|u(t)| + \biggl( \gamma\int_0^t e^{-2\gamma t'}|u(t')|^2{\rm d}t' \biggr)^\frac12
\leq C\bigl( |u(0)| + S_{\gamma,t}^*(|\partial_t u|) \bigr), \\
\label{ineq6}
& |u_{\vert_{x=0}}|_{m-1,\gamma,t}
\leq C( \gamma^{-\frac12}\@ifstar\@opnorms\@opnorm{u(0)}_m + \gamma^{-1}|u_{\vert_{x=0}}|_{m,\gamma,t} ), \\
\label{ineq7}
& \@ifstar\@opnorms\@opnorm{u(t)}_{m-1,\gamma} + \biggl( \gamma\int_0^t \@ifstar\@opnorms\@opnorm{u(t')}_{m-1,\gamma}^2{\rm d}t' \biggr)^\frac12
\leq C\bigl( \@ifstar\@opnorms\@opnorm{u(0)}_{m-1} + S_{\gamma,t}^*(\@ifstar\@opnorms\@opnorm{\partial_t u(\cdot)}_{m-1}) \bigr).
\end{align}
\end{lemma}
\begin{proof}
Integrating the identity
\[
\frac{\rm d}{{\rm d}t}(e^{-2\gamma t}|u(t)|^2) + 2\gamma e^{-2\gamma t}|u(t)|^2
= 2e^{-2\gamma t}u(t)\cdot\partial_t u(t),
\]
we have
\[
e^{-2\gamma t}|u(t)|^2 + 2\gamma \int_0^t e^{-2\gamma t'}|u(t')|^2{\rm d}t'
= |u(0)|^2 + 2\int_0^t e^{-2\gamma t'}u(t')\cdot\partial_t u(t'){\rm d}t'.
\]
The last term is evaluated as
\begin{align*}
2\int_0^t e^{-2\gamma t'}u(t')\cdot\partial_t u(t'){\rm d}t'
&\leq 2\int_0^t e^{-2\gamma t'}|u(t')||\partial_t u(t')|{\rm d}t' \\
&\leq 2S_{\gamma,t}^*(|\partial_t u|)\biggl\{ \sup_{t'\in[0,t]}e^{-\gamma t'}|u(t')|
+ \biggl(\gamma \int_0^t e^{-2\gamma t'}|u(t')|^2{\rm d}t' \biggr)^\frac12 \biggr\} \\
&\leq \frac12\sup_{t'\in[0,t]}e^{-2\gamma t'}|u(t')|^2
+ \gamma \int_0^t e^{-2\gamma t'}|u(t')|^2{\rm d}t'
+3S_{\gamma,t}^*(|\partial_t u|)^2,
\end{align*}
so that we obtain \eqref{ineq5}.
Similarly, we can show \eqref{ineq7}.
As a corollary of \eqref{ineq5}, we have
\[
|u|_{L_\gamma^2(0,t)} \leq C( \gamma^{-\frac12}|u(0)| + \gamma^{-1}|\partial_t u|_{L_\gamma^2(0,t)} ).
\]
Applying this inequality to $(\partial^\alpha u)_{\vert_{x=0}}$, summing the resulting inequality over
$|\alpha| \leq m-1$, and using the Sobolev imbedding theorem $H^1({\mathbb R}_+) \subseteq L^\infty({\mathbb R}_+)$,
we obtain \eqref{ineq6}.
\end{proof}
\subsubsection{Higher order a priori estimate}\label{sect2HO}
We can now state the generalization of Proposition \ref{propNRJ1} to higher order Sobolev spaces.
\begin{proposition}\label{propVC2}
Let $m\geq1$ be an integer, $T>0$, and assume that Assumption \ref{assVC} is satisfied.
Assume moreover that there are two constants $0<K_0\leq K$ such that
\[
\begin{cases}
\mathfrak{c}_0, \mathfrak{c}_1, \|A\|_{L^\infty(\Omega_T)},
\|A^{-1}\|_{L^\infty(\Omega_T)}, |\nu|_{L^\infty(0,T)} \leq K_0, \\
\frac{\beta_2}{\alpha_0}, \|A\|_{W^{1,\infty}(\Omega_T)}, \|B\|_{L^\infty(\Omega_T)},
\|(\partial A,\partial B)\|_{{\mathbb W}^{m-1}(T)}, |\nu|_{W^{m,\infty}(0,T)} \leq K,
\end{cases}
\]
where $\mathfrak{c}_0$ and $\mathfrak{c}_1$ are as in Proposition \ref{propNRJ1}.
Then, every solution $u\in H^{m+1}(\Omega_{T})$ to the initial boundary value problem \eqref{systVC} satisfies,
for any $t\in[0,T]$ and any $\gamma \geq C(K)$,
\begin{align*}
&\@ifstar\@opnorms\@opnorm{ u(t) }_{m,\gamma} + \biggl( \gamma\int_0^t \@ifstar\@opnorms\@opnorm{ u(t') }_{m,\gamma}^2{\rm d}t' \biggr)^\frac12
+ |u_{\vert_{x=0}}|_{m,\gamma,t} \\
&\leq C(K_0)\bigl( \@ifstar\@opnorms\@opnorm{ u(0) }_{m} + \abs{g}_{H_\gamma^m(0,t)}
+ \abs{f_{\vert_{x=0}}}_{m-1,\gamma,t} + S_{\gamma,t}^*(\@ifstar\@opnorms\@opnorm{ \partial_t f(t') }_{m-1}) \bigr).
\end{align*}
\end{proposition}
\begin{proof}
Let $u_m = \partial_t^m u$.
Then, $u_m$ solves
\[
\begin{cases}
\partial_t u_m + A(t,x)\partial_x u_m + B(t,x) u_m = f_m & \mbox{in}\quad \Omega_T, \\
{u_m}_{\vert_{t=0}} = (\partial_t^m u)_{\vert_{t=0}} & \mbox{on}\quad {\mathbb R}_+, \\
\nu(t) \cdot {u_m}_{\vert_{x=0}}= g_m(t) & \mbox{on}\quad (0,T),
\end{cases}
\]
where
\[
\begin{cases}
f_m = \partial_t^m(f-Bu) - [\partial_t^m,A]\partial_x u, \\
g_m = \partial_t^m g- [\partial_t^m,\nu] \cdot u_{\vert_{x=0}}.
\end{cases}
\]
Applying Proposition \ref{propNRJ1} we obtain
\begin{align*}
&\@ifstar\@opnorms\@opnorm{u_m(t)}_{0,\gamma} + \biggl(\gamma \int_0^t \@ifstar\@opnorms\@opnorm{u_m(t')}_{0,\gamma}^2 {\rm d}t' \biggr)^\frac12
+ |{u_m}_{\vert_{x=0}}|_{L_\gamma^2(0,t)} \\
&\leq \mathfrak{c}_0\@ifstar\@opnorms\@opnorm{u(0)}_m
+ \mathfrak{c}_1\bigl( |g_m|_{L_\gamma^2(0,t)} + S_{\gamma,t}^*( \|f_m(\cdot)\|_{L^2} ) \bigr).
\end{align*}
On the other hand, it follows from Lemma \ref{ineq1} that
\[
\begin{cases}
\|f_m(t)\|_{L^2} \leq \@ifstar\@opnorms\@opnorm{ \partial_t f(t) }_{m-1} + C(K)\@ifstar\@opnorms\@opnorm{ u(t) }_m, \\
|g_m|_{L_\gamma^2(0,t)} \leq |g|_{H_\gamma^m(0,t)} + C(K)|u_{\vert_{x=0}}|_{m-1,\gamma,t}.
\end{cases}
\]
Therefore, we obtain
\begin{align}\label{hestpre1}
&\@ifstar\@opnorms\@opnorm{u_m(t)}_{0,\gamma} + \biggl(\gamma \int_0^t \@ifstar\@opnorms\@opnorm{u_m(t')}_{0,\gamma}^2 {\rm d}t' \biggr)^\frac12
+ |{u_m}_{\vert_{x=0}}|_{L_\gamma^2(0,t)} \\
&\leq C(K_0)\bigl( \@ifstar\@opnorms\@opnorm{ u(0) }_m + |g|_{H_\gamma^m(0,t)}
+ S_{\gamma,t}^*( \@ifstar\@opnorms\@opnorm{\partial_t f(\cdot)}_{m-1} ) \bigr) \nonumber \\
&\quad
+ C(K)\bigl( |u_{\vert_{x=0}}|_{m-1,\gamma,t} + S_{\gamma,t}^*(\@ifstar\@opnorms\@opnorm{ u(t') }_m) \bigr). \nonumber
\end{align}
We proceed to control the other derivatives.
Let $k$ and $l$ be nonnegative integers satisfying $k+l \leq m-1$.
Applying $\partial_t^k\partial_x^l$ to the equation, we get
\[
\partial_t^{k+1}\partial_x^l u + A \partial_t^k \partial_x^{l+1} u = \partial_t^k\partial_x^l(f-Bu)-[\partial_t^k\partial_x^l,A]\partial_x u=:f_{k,l}.
\]
By using these two expressions of $f_{k,l}$ together with Lemma \ref{ineq1} we see that
\[
\begin{cases}
\|f_{k,l}(0)\|_{L^2} \leq C(K_0)\@ifstar\@opnorms\@opnorm{u(0)}_m, \\
\|\partial_t f_{k,l}(t)\|_{L^2} \leq \@ifstar\@opnorms\@opnorm{\partial_t f(t)}_{m-1} + C(K)\@ifstar\@opnorms\@opnorm{u(t)}_m, \\
|f_{k,l \vert_{x=0}}|_{L_\gamma^2(0,t)} \leq |f_{\vert_{x=0}}|_{m-1,\gamma,t} + C(K)|u_{\vert_{x=0}}|_{m-1,\gamma,t}.
\end{cases}
\]
We have now the relation $\partial_t^k \partial_x^{l+1} u = A^{-1}(f_{k,l}-\partial_t^{k+1}\partial_x^l u)$ so that
\[
\begin{cases}
\|\partial_t^k \partial_x^{l+1} u(t)\|_{L^2} \leq C(K_0)( \|\partial_t^{k+1} \partial_x^l u(t)\|_{L^2} + \|f_{k,l}(t)\|_{L^2}), \\
|(\partial_t^k \partial_x^{l+1} u)_{\vert_{x=0}}|_{L_\gamma^2(0,t)} \leq C(K_0)(
|(\partial_t^{k+1} \partial_x^l u)_{\vert_{x=0}}|_{L_\gamma^2(0,t)} + |f_{k,l \vert_{x=0}}|_{L_\gamma^2(0,t)} ).
\end{cases}
\]
Therefore,
\begin{align*}
& \@ifstar\@opnorms\@opnorm{ \partial_t^k \partial_x^{l+1} u(t) }_{0,\gamma}
+ \biggl(\gamma\int_0^t\@ifstar\@opnorms\@opnorm{ \partial_t^k \partial_x^{l+1} u(t') }_{0,\gamma}^2{\rm d}t'\biggr)^\frac12
+ |(\partial_t^k \partial_x^{l+1} u)_{\vert_{x=0}}|_{L_\gamma^2(0,t)} \\
&\leq C(K_0)\biggl\{
\@ifstar\@opnorms\@opnorm{ \partial_t^{k+1} \partial_x^l u(t) }_{0,\gamma}
+ \biggl(\gamma\int_0^t\@ifstar\@opnorms\@opnorm{ \partial_t^{k+1} \partial_x^l u(t') }_{0,\gamma}^2{\rm d}t'\biggr)^\frac12
+ |(\partial_t^{k+1} \partial_x^l u)_{\vert_{x=0}}|_{L_\gamma^2(0,t)} \\
&\phantom{ \leq C(K_0)\biggl\{ }
+ \@ifstar\@opnorms\@opnorm{ f_{k,l}(t) }_{0,\gamma}
+ \biggl(\gamma\int_0^t\@ifstar\@opnorms\@opnorm{ f_{k,l}(t') }_{0,\gamma}^2{\rm d}t'\biggr)^\frac12
+ |f_{k,l \vert_{x=0}}|_{L_\gamma^2(0,t)} \biggr\}.
\end{align*}
Here, by Lemma \ref{ineq4} we have
\begin{align*}
& \@ifstar\@opnorms\@opnorm{ f_{k,l}(t) }_{0,\gamma}
+ \biggl(\gamma\int_0^t\@ifstar\@opnorms\@opnorm{ f_{k,l}(t') }_{0,\gamma}^2{\rm d}t'\biggr)^\frac12 \\
&\leq C\bigl( \|f_{k,l}(0)\|_{L^2} + S_{\gamma,t}^*(\|\partial_t f_{k,l}(\cdot)\|_{L^2}) \bigr) \\
&\leq C(K_0)\bigl( \@ifstar\@opnorms\@opnorm{u(0)}_m + S_{\gamma,t}^*(\@ifstar\@opnorms\@opnorm{\partial_t f(\cdot)}_{m-1}) \bigr)
+ C(K)S_{\gamma,t}^*(\@ifstar\@opnorms\@opnorm{u(\cdot)}_m).
\end{align*}
By using the above inequality inductively, we obtain
\begin{align*}
& \@ifstar\@opnorms\@opnorm{ u(t) }_{m,\gamma}
+ \biggl(\gamma\int_0^t\@ifstar\@opnorms\@opnorm{ u(t') }_{m,\gamma}^2{\rm d}t'\biggr)^\frac12
+ |u_{\vert_{x=0}}|_{m,\gamma,t} \\
&\leq C(K_0)\biggl\{
\@ifstar\@opnorms\@opnorm{u(0)}_m + S_{\gamma,t}^*(\@ifstar\@opnorms\@opnorm{\partial_t f(\cdot)}_{m-1}) + |f_{\vert_{x=0}}|_{m-1,\gamma,t} \\
&\phantom{ \leq C(K_0)\biggl\{ }
+ \@ifstar\@opnorms\@opnorm{ u_m(t) }_{0,\gamma}
+ \biggl(\gamma\int_0^t\@ifstar\@opnorms\@opnorm{ u_m(t') }_{0,\gamma}^2{\rm d}t'\biggr)^\frac12
+ |u_{m \vert_{x=0}}|_{L_\gamma^2(0,t)} \\
&\phantom{ \leq C(K_0)\biggl\{ }
+ \@ifstar\@opnorms\@opnorm{ u(t) }_{m-1,\gamma}
+ \biggl(\gamma\int_0^t\@ifstar\@opnorms\@opnorm{ u(t') }_{m-1,\gamma}^2{\rm d}t'\biggr)^\frac12 \biggr\} \\
&\quad
+ C(K)\bigl( |u_{\vert_{x=0}}|_{m-1,\gamma,t} + S_{\gamma,t}^*(\@ifstar\@opnorms\@opnorm{u(\cdot)}_m) \bigr).
\end{align*}
This together with \eqref{hestpre1} and Lemma \ref{ineq4} implies
\begin{align*}
& \@ifstar\@opnorms\@opnorm{ u(t) }_{m,\gamma}
+ \biggl(\gamma\int_0^t\@ifstar\@opnorms\@opnorm{ u(t') }_{m,\gamma}^2{\rm d}t'\biggr)^\frac12
+ |u_{\vert_{x=0}}|_{m,\gamma,t} \\
&\leq C(K_0)\bigl( \@ifstar\@opnorms\@opnorm{ u(0) }_m + |g|_{H_\gamma^m(0,t)} + |f_{\vert_{x=0}}|_{m-1,\gamma,t}
+ S_{\gamma,t}^*( \@ifstar\@opnorms\@opnorm{\partial_t f(\cdot)}_{m-1} ) \bigr) \nonumber \\
&\quad
+ C(K)\bigl( |u_{\vert_{x=0}}|_{m-1,\gamma,t} + S_{\gamma,t}^*(\@ifstar\@opnorms\@opnorm{ u(t') }_m) \bigr) \\
&\leq C(K_0)\bigl( \@ifstar\@opnorms\@opnorm{ u(0) }_m + |g|_{H_\gamma^m(0,t)} + |f_{\vert_{x=0}}|_{m-1,\gamma,t}
+ S_{\gamma,t}^*( \@ifstar\@opnorms\@opnorm{\partial_t f(\cdot)}_{m-1} ) \bigr) \nonumber \\
&\quad
+ C(K)\biggl\{ \gamma^{-\frac12}\@ifstar\@opnorms\@opnorm{ u(0) }_m
+ \gamma^{-1}\biggl( \gamma\int_0^t \@ifstar\@opnorms\@opnorm{u(t')}_{m,\gamma}^2{\rm d}t'\biggr)^\frac12
+ \gamma^{-1}|u_{\vert_{x=0}}|_{m,\gamma,t} \biggr\}.
\end{align*}
Therefore, by taking $\gamma$ sufficiently large compared to $C(K)$, we obtain the desired estimate
(note that this would not be possible without the second term of the left-hand side).
\end{proof}
\subsubsection{Proof of Theorem \ref{theoIBVP1}}\label{secttheoIBVP1}
Under Assumption \ref{assVC}, the existence and uniqueness of a solution $u\in {\mathbb W}^m(T)$ to \eqref{systVC}
can be deduced from Proposition \ref{propVC2} and the compatibility condition along classical lines
(see for instance \cite{metivier2001,metivier2012,benzoniserre2007}).
We still have to prove that the assumptions made in the statement of Theorem \ref{theoIBVP1} imply that
Assumption \ref{assVC} is satisfied.
This is given by the following lemma.
\begin{lemma}\label{lemsymmetrizer}
Let $c_0>0$ be such that Assumption \ref{asshyp} is satisfied.
There exist a symmetrizer $S\in W^{1,\infty}(\Omega_T)$ and constants
$\alpha_0,\alpha_1$ and $\beta_0,\beta_1,\beta_2$ such that Assumption \ref{assVC} is satisfied.
Moreover, we have
\[
\mathfrak{c}_0 \leq C\Bigl( \frac{1}{c_0}, \| A_{\vert_{t=0}} \|_{L^\infty({\mathbb R}_+)} \Bigr)
\quad\mbox{and}\quad
\mathfrak{c}_1 \leq C\Bigl( \frac{1}{c_0},\|A\|_{L^{\infty}(\Omega_T)} \Bigr),
\]
where $\mathfrak{c}_0$ and $\mathfrak{c}_1$ are as defined in Proposition \ref{propNRJ1}, and we also have
\[
\frac{\beta_2}{\beta_0} \leq C\Bigl( \frac{1}{c_0},\Vert A\Vert_{W^{1,\infty}(\Omega_T)},
\Vert B\Vert_{L^\infty(\Omega_T)}\Bigr).
\]
\end{lemma}
This lemma is a simple consequence of the following proposition and its proof,
which characterizes the uniform Kreiss--Lopatinski\u{\i} condition {\bf iii} in Assumption \ref{asshyp}.
\begin{proposition}\label{propBC}
Suppose that the condition {\bf ii} in Assumption \ref{asshyp}, $\vert \nu(t) \vert \geq c_0$,
and $\vert A(t,x) \vert \leq 1/c_0$ hold for some positive constant $c_0$.
Then, the following four statements are all equivalent.
\begin{enumerate}
\setlength{\itemsep}{3pt}
\item[{\bf i.}]
There exist a symmetrizer $S\in W^{1,\infty}(\Omega_T)$ and positive constants $\alpha_0$ and $\beta_0$
such that $\alpha_0{\rm Id} \leq S(t,x) \leq \beta_0{\rm Id}$ and that for any $v \in {\mathbb R}^2$ satisfying
$\nu(t)\cdot v = 0$ we have
\[
v^{\rm T}S(t,0)A(t,0)v \leq 0.
\]
\item[{\bf ii.}]
There exist a symmetrizer $S\in W^{1,\infty}(\Omega_T)$ and positive constants $\alpha_0$, $\beta_0$,
$\alpha_1$, and $\beta_1$ such that $\alpha_0{\rm Id} \leq S(t,x) \leq \beta_0{\rm Id}$ and that for any
$v \in {\mathbb R}^2$ we have
\[
v^{\rm T}S(t,0)A(t,0)v \leq -\alpha_1|v|^2 + \beta_1|\nu(t)\cdot v|^2.
\]
\item[{\bf iii.}]
There exists a positive constant $\alpha_0$ such that
\[
|\pi_{-}(t,0)\nu(t)^\perp| \geq \alpha_0,
\]
where $\pi_{\pm}(t,x)$ is the eigenprojector associated to the eigenvalue $\pm\lambda_{\pm}(t,x)$ of $A(t,x)$.
\item[{\bf iv.}]
There exists a positive constant $\alpha_0$ such that
\[
|\nu(t)\cdot\mathbf{e}_{+}(t,0)| \geq \alpha_0,
\]
where $\mathbf{e}_{\pm}(t,x)$ is the unit eigenvector associated to the eigenvalue $\pm\lambda_{\pm}(t,x)$ of $A(t,x)$.
\end{enumerate}
\end{proposition}
\begin{proof}
We note that the eigenprojector $\pi_{\pm}(t,x)$ is given explicitly by
\[
\pi_{+}(t,x) = \frac{A(t,x)+\lambda_{-}(t,x){\rm Id}}{\lambda_{+}(t,x)+\lambda_{-}(t,x)}, \qquad
\pi_{-}(t,x) = -\frac{A(t,x)-\lambda_{+}(t,x){\rm Id}}{\lambda_{+}(t,x)+\lambda_{-}(t,x)}
\]
and that under the assumption $\lambda_{\pm}(t,x)$ and $|\pi_{\pm}(t,x)|$ are bounded from above by a constant
depending on $c_0$.
We see that
\begin{align*}
|\nu(t) \cdot \mathbf{e}_{+}(t,0)|
&= |\nu(t)^\perp \cdot \mathbf{e}_{+}(t,0)^\perp|
= |(\pi_-(t,0)\nu(t)^\perp) \cdot \mathbf{e}_{+}(t,0)^\perp|
\leq |\pi_-(t,0)\nu(t)^\perp|
\end{align*}
and that
\begin{align*}
|\pi_-(t,0)\nu(t)^\perp|
&= |(\nu(t)^\perp \cdot \mathbf{e}_{+}(t,0)^\perp)\pi_-(t,0)\mathbf{e}_{+}(t,0)^\perp|
\leq |\pi_-(t,0)||\nu(t) \cdot \mathbf{e}_{+}(t,0)|.
\end{align*}
These imply the equivalence of {\bf iii} and {\bf iv}.
Obviously, {\bf ii} implies {\bf i}.
We proceed to show that {\bf i} implies {\bf iii}.
By the assumption we have
\[
(\nu(t)^\perp)^{\rm T} S(t,0)A(t,0)\nu(t)^\perp \leq 0,
\]
which together with the spectral decomposition
\[
A(t,x) = \lambda_{+}(t,x)\pi_{+}(t,x) - \lambda_{-}(t,x)\pi_{-}(t,x)
\]
implies
\begin{align*}
c_0\alpha_0 |\pi_{+}(t,0)\nu(t)^\perp|^2
\leq & \lambda_{+}(t,0)(\pi_{+}(t,0)\nu(t)^\perp)^{\rm T}S(t,0)\pi_{+}(t,0)\nu(t)^\perp \\
\leq & (\lambda_{-}(t,0)-\lambda_{+}(t,0))(\pi_{+}(t,0)\nu(t)^\perp)^{\rm T}S(t,0)\pi_{-}(t,0)\nu(t)^\perp \\
& +\lambda_{-}(t,0)(\pi_{-}(t,0)\nu(t)^\perp)^{\rm T}S(t,0)\pi_{-}(t,0)\nu(t)^\perp \\
\leq & \beta_0|\lambda_{-}(t,0)-\lambda_{+}(t,0)| |\pi_{+}(t,0)\nu(t)^\perp| |\pi_{-}(t,0)\nu(t)^\perp| \\
& + \beta_0\lambda_{-}(t,0)|\pi_{-}(t,0)\nu(t)^\perp|^2.
\end{align*}
Particularly, we have
\[
c_0\alpha_0 |\pi_{+}(t,0)\nu(t)^\perp|^2
\leq \biggl( \frac{\beta_0^2|\lambda_{-}(t,0)-\lambda_{+}(t,0)|^2}{c_0\alpha_0}
+ 2\beta_0\lambda_{-}(t,0) \biggr)|\pi_{-}(t,0)\nu(t)^\perp|^2.
\]
Therefore, in view of $c_0 \leq |\nu(t)| \leq |\pi_{-}(t,0)\nu(t)^\perp| + |\pi_{+}(t,0)\nu(t)^\perp|$
we obtain the desired inequality in the statement {\bf iii}.
Finally, we will show that {\bf iii} implies {\bf ii}.
This is the most important part of this proposition.
We want to show that for a suitably large $M>1$, a symmetrizer $S(t,x)$ satisfying the conditions
in the statement {\bf ii} is provided by the formula
\[
S(t,x) = \pi_+(t,x)^{\rm T}\pi_+(t,x) + M\pi_-(t,x)^{\rm T}\pi_-(t,x),
\]
so that the first point of {\bf ii} is satisfied with $\alpha_0=1$ and $\beta_0=M$.
By the definition of $\pi_\pm$, we compute indeed that
\[
SA = \lambda_+ \pi_+^{\rm T}\pi_+ - M \lambda_- \pi_-^{\rm T}\pi_-,
\]
which is obviously symmetric.
For the second point of {\bf ii}, just remark that
\[
v^{\rm T}SAv =\lambda_+ |\pi_+ v|^2 - M\lambda_- |\pi_- v|^2.
\]
We need to show that this quantity is negative on the kernel ${\mathbb R} \nu^\perp$ of the boundary condition.
Under the hypothesis we can assume that $|\nu(t)|=1$ without loss of generality.
Then, we see that
\begin{align*}
-|\pi_{-}v|^2
&= -|(\nu^\perp \cdot v)\pi_{-}\nu^\perp + (\nu \cdot v)\pi_{-}\nu|^2 \\
&\leq -\frac12|\nu^\perp \cdot v|^2|\pi_{-}\nu^\perp|^2 + |\nu \cdot v|^2|\pi_{-}\nu|^2 \\
&\leq -\frac12|\pi_{-}\nu^\perp|^2|v|^2 + (|\pi_{-}\nu|^2+|\pi_{-}\nu^\perp|^2)|\nu \cdot v|^2
\end{align*}
and that
\begin{align*}
|\pi_{+}v|^2
&= |(\nu^\perp \cdot v)\pi_{+}\nu^\perp + (\nu \cdot v)\pi_{+}\nu|^2 \\
&\leq 2|\pi_{+}\nu^\perp|^2|\nu^\perp \cdot v|^2 + 2|\pi_{+}\nu|^2|\nu \cdot v|^2 \\
&\leq 4|\pi_{+}\nu^\perp|^2|v|^2 + 4(|\pi_{+}\nu^\perp|^2+|\pi_{+}\nu|^2)|\nu \cdot v|^2.
\end{align*}
Therefore, we obtain
\begin{align*}
v^{\rm T}SAv
\leq & -\lambda_{-}|\pi_{-}\nu^\perp|^2\biggl(
\frac{M}{2} - 4\frac{\lambda_{+}}{\lambda_{-}}\frac{|\pi_{+}\nu^\perp|^2}{|\pi_{-}\nu^\perp|^2} \biggr)|v|^2 \\
& + \bigl\{ \lambda_{-}M(|\pi_{-}\nu|^2+|\pi_{-}\nu^\perp|^2) + 4\lambda_{+}(|\pi_{+}\nu^\perp|^2+|\pi_{+}\nu|^2)
\bigr\} |\nu \cdot v|^2
\end{align*}
Taking for instance
$M = 2+8\sup_{\Omega_T}\frac{\lambda_+}{\lambda_-}\frac{|\pi_+ \nu^\perp|^2}{|\pi_- \nu^\perp|^2}$,
we easily obtain the desired inequality in the statement {\bf ii}.
\end{proof}
\subsection{Application to quasilinear $2\times 2$ initial boundary value problems}\label{sectapplQL}
The aim of this section is to use the results of the previous section to handle general quasilinear
boundary value problems of the form
\begin{equation}\label{systQL}
\begin{cases}
\partial_t u + A(u)\partial_x u + B(t,x)u = f(t,x) & \mbox{in}\quad \Omega_T, \\
u_{\vert_{t=0}} = u^{\rm in}(x) & \mbox{on}\quad {\mathbb R}_+,\\
\Phi(t,u_{\vert_{x=0}})= g(t) & \mbox{on}\quad (0,T),
\end{cases}
\end{equation}
where $u$, $u^{\rm in}$, and $f$ are ${\mathbb R}^2$-valued functions, $g$ and $\Phi$ are real-valued functions,
while $A$ and $B$ take their values in the space of $2\times2$ real-valued matrices.
We also make the following assumption on the hyperbolicity of the system and on the boundary condition.
\begin{assumption}\label{asshypQL}
Let $\mathcal{U}$ be an open set in ${\mathbb R}^2$, which represents a phase space of $u$.
The following conditions hold.
\begin{enumerate}
\setlength{\itemsep}{3pt}
\item[{\bf i.}]
$A \in C^\infty({\mathcal U})$.
\item[{\bf ii.}]
For any $u \in {\mathcal U}$, the matrix $A(u)$ has eigenvalues $\lambda_+(u)$ and $-\lambda_-(u)$ satisfying
\[
\lambda_{\pm}(u) > 0.
\]
\item[{\bf iii.}]
There exist a diffeomorphism $\Theta: {\mathcal U}\to \Theta({\mathcal U})\subset {\mathbb R}^2$
and $\nu\in C([0,T])$ such that for any $t\in [0,T]$ and any $u\in {\mathcal U}$ we have
\[
\Phi(t,u) = \nu(t) \cdot \Theta(u)
\quad\mbox{and}\quad
\abs{ \nabla_u \Phi(t,u) \cdot {\bf e}_+(u) } > 0,
\]
where ${\bf e}_+ (u)$ is a unit eigenvector associated to the eigenvalue $\lambda_+(u)$ of $A(u)$.
\end{enumerate}
\end{assumption}
\begin{remark}\label{remphi}
In the case of a linear boundary condition as the we considered for Theorem \ref{theoIBVP1},
we have $\Phi(t,u) = \nu(t)\cdot u$ so that by taking $\Theta(u)=u$, the third point of the assumption reduces to
\[
|\nu(t) \cdot \mathbf{e}_{+}(u)| > 0.
\]
\end{remark}
\begin{remark}
If $\Phi(t,u)=\Phi(u)$ is independent of $t$ and if for some $u^0$ we have
$\abs{\nabla_{u} \Phi(t,u^0)\cdot {\bf e}_+(u^0)}>0$,
then by the inverse function theorem and up to shrinking ${\mathcal U}$ to a sufficiently small neighborhood of $u^0$,
the existence of a diffeomorphism $\Theta$ satisfying the properties of point {\bf iii} is automatic.
\end{remark}
\begin{example}\label{ex2}
For the nonlinear shallow water equations
\[
\partial_t u +A(u)\partial_x u=0
\]
with $u=(\zeta,q)^{\rm T}$ and $A(u)$ as given by \eqref{eqASW},
whose linear version has been considered in Example \ref{ex1},
the first two points of the assumption are equivalent to
\[
h>0, \qquad
\sqrt{\mathtt{g}{h}} \pm \frac{{q}}{{h}}>0 \qquad (\mbox{with } h=h_0+\zeta).
\]
The condition {\bf iii} of the assumption depends of course on the boundary condition under consideration.
Let us consider here two important examples:
\begin{itemize}
\item
Boundary condition on the horizontal water flux, that is, $q_{\vert_{x=0}}=g$.
As seen in Example \ref{ex1} and Remark \ref{remphi}, this corresponds to $\Phi(t,u) = \nu \cdot u$ with
$\nu=(0,1)^{\rm T}$, and the condition {\bf iii} of the assumption is satisfied.
\item
Boundary condition on the outgoing Riemann invariant, that is,
$2(\sqrt{{\mathtt g}h}-\sqrt{{\mathtt g}h_0})+q/h=g$.
We then have $\Phi(t,u) = \Phi(u) = 2(\sqrt{{\mathtt g}h}-\sqrt{{\mathtt g}h_0})+q/h$
and we can take the diffeomorphism defined on ${\mathcal U}=\{(h,q)\in {\mathbb R}^2\,;\, h>0\}$ by
\[
\Theta(h,q) = \big( 2(\sqrt{{\mathtt g}h}-\sqrt{{\mathtt g}h_0})+q/h,
2(\sqrt{{\mathtt g}h}-\sqrt{{\mathtt g}h_0})-q/h\big)^{\rm T},
\]
where $2(\sqrt{{\mathtt g}h}-\sqrt{{\mathtt g}h_0})-q/h$ is the incoming Riemann invariant.
Then, $\Phi(u) = \nu\cdot \Theta(u)$ with $\nu = (1,0)^{\rm T}$;
moreover, we compute $\nabla_u \Phi = (1/h)(\lambda^-, 1)^{\rm T}$ so that all the conditions of the third point
of the assumption are satisfied.
\end{itemize}
\end{example}
The main result is the following.
\begin{theorem}\label{theoIBVP2}
Let $m\geq 2$ be an integer, $B\in L^\infty(\Omega_T)$, $\partial B\in {\mathbb W}^{m-1}(T)$, and
assume that Assumption \ref{asshypQL} is satisfied with $\Theta \in C^\infty({\mathcal U})$ and
$\nu\in W^{m,\infty}(0,T)$.
If $u^{\rm in }\in H^m({\mathbb R}_+)$ takes its values in a compact and convex set $\mathcal{K}_0 \subset {\mathcal U}$
and if the data $u^{\rm in}$, $f\in H^m(\Omega_T)$, and $g\in H^m(0,T)$ satisfy the compatibility conditions
up to order $m-1$ in the sense of Definition \ref{defcompQL} below,
then there exist $T_1 \in (0,T]$ and a unique solution $u\in {\mathbb W}^m(T_1)$
to the initial boundary value problem \eqref{systQL}.
Moreover, the trace of $u$ at the boundary $x=0$ belongs to $H^m(0,T_1)$ and $\abs{u_{\vert_{x=0}}}_{m,T_1}$ is finite.
\end{theorem}
\begin{remark}
There is a wide literature devoted to the analysis of quasilinear hyperbolic initial boundary value problems.
For the general multi-dimensional case, assuming that the uniform Kreiss--Lopatinski\u{\i} condition holds,
the existence is obtained for $m>(d+1)/2+1$, with a loss of $1/2$ derivative with respect to the boundary
and initial data \cite{RauchMassey, mokrane1987} (see also \cite{benzoniserre2007}).
Existence for $m>d/2+1$ without loss of derivative is obtained under the additional assumption that
the system is Friedrichs symmetrizable \cite{schochet1986, metivier2012} but one cannot expect in general
an $H^m(0,T_1)$ estimate for the trace of the solution at the boundary.
In the particular one-dimensional case, a $C^1$ solution is constructed in \cite{li1985boundary}
using the method of characteristics;
more recently, in the Sobolev setting, it is shown in \cite{petcu2013one} that the general procedure of
\cite{RauchMassey, mokrane1987} can be implemented in the particular case of the shallow water equations
with transparent boundary conditions, that is, a boundary data on the outgoing Riemann invariant (see Example \ref{ex2} above):
for data in $H^{7/2}$, a solution is constructed in ${\mathbb W}^3(T)$.
As said in Example \ref{ex2}, our result covers this situation and, by taking advantage of the specificities
of the one-dimensional case proves existence in ${\mathbb W}^m(T)$, with $m\geq 2$ and without loss of derivative,
and provides an $H^m(0,T_1)$ trace estimate.
\end{remark}
\subsubsection{Compatibility conditions}
From the interior equations, denoting $u_k=\partial_t^k u$, we have
\[
u_1 = -A(u)\partial_x u - Bu + f.
\]
More generally, by induction, we have
\[
u_{k}=c_k(u,B,f),
\]
where $c_k(u,B,f)$ is a smooth function of $u$ and of its space derivatives of order at most $k$,
and of the time and space derivatives of order lower than $k-1$ of $B$ and $f$.
For a smooth solution $u$ to \eqref{systQL}, $u_k^{\rm in} = {u_k}_{\vert_{t=0}}$ is therefore given by
\begin{equation}\label{defu0kter}
u_{k}^{\rm in}=c^{\rm in}_k(u,B,f),
\end{equation}
where $c^{\rm in}_k(u,B,f)=c_k(u,B,f)_{\vert_{t=0}}$.
The boundary condition $\Phi( t,u_{\vert_{x=0}})=g$ also implies that
\[
\partial_t^k\Phi( t,u_{\vert_{x=0}}) = \partial_t^k g.
\]
On the edge $\{t=0,x=0\}$, smooth enough solutions must therefore satisfy
\[
\begin{cases}
\Phi(0,{u^{\rm in}}_{\vert_{x=0}})=g_{\vert_{t=0}} & k=0, \\
{u_1^{\rm in}}_{\vert_{x=0}}\cdot \nabla_u \Phi(0,{u^{\rm in}}_{\vert_{x=0}})+\partial_t \Phi (0,{u^{\rm in}}_{\vert_{x=0}})
= (\partial_t g)_{\vert_{t=0}} & k=1,
\end{cases}
\]
and more generally, for any $k\geq 1$,
\begin{equation}\label{compkQL}
{u_k^{\rm in}}_{\vert_{x=0}} \cdot \nabla_u \Phi(0,{u^{\rm in}}_{\vert_{x=0}})
+ F_k({u^{\rm in}_{0\leq j\leq k-1}}_{\vert_{x=0}})=(\partial_t^k g)_{\vert_{t=0}},
\end{equation}
where $F_k({u^{\rm in}_{1\leq j\leq k}}_{\vert_{x=0}})$ is a smooth function of its arguments
that can be computed explicitly by induction.
\begin{definition}\label{defcompQL}
Let $m\geq1$ be an integer.
We say that the data $u^{\rm in}\in H^m({\mathbb R}_+)$, $f\in H^m(\Omega_T)$, and $g \in H^m(0,T)$
for the initial boundary value problem \eqref{systQL} satisfy the compatibility condition at order $k$
if the $\{u_j^{\rm in}\}_{j=0}^m$ defined in \eqref{defu0kter} satisfy \eqref{compkQL}.
We also say that the data satisfy the compatibility conditions up to order $m-1$ if they satisfy
the compatibility conditions at order $k$ for $k=0,1,\ldots,m-1$.
\end{definition}
\subsubsection{Proof of Theorem \ref{theoIBVP2}}
Without loss of generality, we can assume that $\Theta(0)=0$.
The first step is to linearize the boundary condition.
Under Assumption \ref{asshypQL}, this is possible by introducing
\[
v=\Theta(u),\qquad J(v)=d_{v} (\Theta^{-1}(v)),
\quad\mbox{ and }\quad {A}^\sharp(v)=J(v)^{-1} A(\Theta^{-1}(v)) J(v).
\]
Then, $u$ is a classical solution to \eqref{systQL} if and only if $v$ is a classical solution of
\begin{equation}\label{systQLred}
\begin{cases}
\partial_t {v} + {A}^\sharp(v)\partial_x v + J(v)^{-1}B(t,x) \Theta^{-1}(v) = J(v)^{-1}f(t,x) & \mbox{in}\quad \Omega_T, \\
v_{\vert_{t=0}} = \Theta(u^{\rm in}(x)) & \mbox{on}\quad {\mathbb R}_+,\\
\nu(t)\cdot v_{\vert_{x=0}}= g(t) & \mbox{on}\quad (0,T)
\end{cases}
\end{equation}
with $\nu(t)$ as in Assumption \ref{asshypQL}.
Let $\mathcal{K}_1$ be a compact and convex set in ${\mathbb R}^2$ satisfying
$\mathcal{K}_0 \Subset \mathcal{K}_1 \Subset \mathcal{U}$.
Then, there exists a constant $c_0 > 0$ such that for any $u \in \mathcal{K}_1$ and any $t\in[0,T]$
we have
\begin{align*}
\lambda_{\pm}(u) \geq c_0 , \qquad |\nabla_u\Phi(t,u)\cdot \mathbf{e}_{+}(u)| \geq c_0.
\end{align*}
Note that there exists a constant $\delta_0>0$ such that
$\| v - \Theta(u^{\rm in})\|_{L^\infty} \leq \delta_0$ implies that
$u = \Theta^{-1}(v)$ takes its values in $ \mathcal{K}_1$.
We therefore construct a solution $v$ to \eqref{systQLred} satisfying
$\| v(t)-\Theta(u^{\rm in})\|_{L^\infty} \leq \delta_0$ for $0\leq t\leq T_1$.
The solution is classically constructed using the iterative scheme
\begin{equation}\label{systiter}
\begin{cases}
\partial_t v^{n+1} + {A}^\sharp(v^n)\partial_x v^{n+1} = f^n& \mbox{in}\quad \Omega_T, \\
{v^{n+1}}_{\vert_{t=0}} = \Theta(u^{\rm in}(x)) & \mbox{on}\quad {\mathbb R}_+, \\
\nu(t)\cdot {v^{n+1}}_{\vert_{x=0}} = g(t) & \mbox{on}\quad (0,T),
\end{cases}
\end{equation}
for all $n\in {\mathbb N}$ and with
$$
f^n(t,x)=J(v^n)^{-1}f(t,x) - J(v^n)^{-1}B(t,x)\Theta^{-1}(v^{n}).
$$
For the first iterate $u^0$, we choose a function $u^0\in H^{m+1/2}({\mathbb R}\times {\mathbb R}_+)$ such that
\[
(\partial_t^k u^0)_{\vert_{t=0}} = u_k^{\rm in} \quad\mbox{for}\quad k=0,1,\ldots,m
\]
with $u_k^{\rm in}$ as defined in \eqref{defu0kter}.
Such a choice ensures along a classical procedure \cite{metivier2001,metivier2012} that
the data $(\Theta(u^{\rm in}), f^n, g)$ are compatible for the linear initial boundary value problem
\eqref{systiter} in the sense of Definition \ref{defcompVC}.
Moreover, $\@ifstar\@opnorms\@opnorm{ v^n (0) }_m$ is independent of $n$, and there exists therefore $K_0$ such that
\[
\frac{1}{c_0}, \@ifstar\@opnorms\@opnorm{ v^n(0) }_m, \| {A}^\sharp(v^n)\|_{L^\infty(\Omega_{T_1})},
\| {A}^\sharp(v^n)^{-1}\|_{L^\infty(\Omega_{T_1})} \leq K_0,
\]
as long as $v^n$ satisfies $\|v^n(t) - \Theta(u^{\rm in})\|_{L^\infty} \leq \delta_0$ for $0\leq t\leq T_1$.
We prove now that for $M$ large enough and $T_1$ small enough, for any $n \in {\mathbb N}$ we have
\begin{equation}\label{assert}
\begin{cases}
\| v^n\|_{{\mathbb W}^m(T_1)} + |{ v^n}_{\vert_{x=0}}|_{m,T_1} \leq M, \\
\| v^n(t)-\Theta(u^{\rm in})\|_{L^\infty} \leq \delta_0 \quad\mbox{for}\quad 0 \leq t\leq T_1.
\end{cases}
\end{equation}
The main tool to prove this assertion is to apply Theorem \ref{theoIBVP1} to \eqref{systiter}.
In order to do so, we first need to check that Assumption \ref{asshyp} is satisfied.
The only non trivial point to check is the third condition of this assumption.
The fact that this is a consequence of Assumption \ref{asshypQL} for the original system \eqref{systQL}
is proved in the following lemma.
\begin{lemma}
For any $ v \in \Theta({\mathcal U})$, the matrix ${A}^\sharp( v)$ has two eigenvalues $\pm {\lambda}^\sharp_\pm(v)$
and associated eigenvectors ${{\bf e}}^\sharp_\pm(v)$ given by
\[
{\lambda}^\sharp_\pm(v) = \lambda_\pm(\Theta^{-1}(v))\quad \mbox{ and }\quad
{{\bf e}}^\sharp_\pm(v) = J(v)^{-1}{\bf e}_\pm(\Theta^{-1}(v)).
\]
Moreover, denoting $u=\Theta^{-1}(v)$ we have
\[
\nu(t)\cdot {\bf e}^\sharp_+(v) = \nabla_u\Phi(t,u)\cdot {\bf e}_+(u).
\]
\end{lemma}
\begin{proof}[Proof of the lemma]
The first part of the lemma is straightforward.
For the second point, just notice that by definition of $\Theta$, one has
$\nabla_u\Phi(t,u) = (\Theta'(u))^{\rm T}\nu(t)$.
Since moreover $\Theta'(u) = (d_v (\Theta^{-1}(v)) )^{-1} = J(v)^{-1}$, we have
\[
\nabla_u\Phi(t,u) \cdot {\bf e}_+(u) = \nu(t) \cdot J(v)^{-1} {\bf e}_+(\Theta^{-1}(v))
\]
and the result follows from the first point.
\end{proof}
We can therefore use Theorem \ref{theoIBVP1} to prove \eqref{assert} by induction.
Since it is satisfied for $n=0$ for a suitable $M$ and $T_1$,
we just need to prove that it holds at rank $n+1$ if it holds at rank $n$.
There is $K=K(M)$ such that
\[
\|{A}^\sharp(v^n)\|_{W^{1,\infty}(\Omega_{T_1})}, \|\partial ({A}^\sharp(v^n)) \|_{{\mathbb W}^{m-1}(T_1)} \leq K.
\]
Taking a greater $K$ if necessary, we can assume also that
$\|B\|_{L^\infty(\Omega_T)}$ and $\|\partial B\|_{{\mathbb W}^{m-1}(T)} \leq K$ and therefore that
\[
\@ifstar\@opnorms\@opnorm{f^n(t)}_m\leq C(K)(1+\@ifstar\@opnorms\@opnorm{f(t)}_m).
\]
It follows therefore from Theorem \ref{theoIBVP1} that
\begin{align*}
& \|v^{n+1} \|_{{\mathbb W}^m(T_1)} + |{v^{n+1}}_{\vert_{x=0}}|_{m,T_1} \\
&\leq C(K_0)e^{C(K)T_1} \Big( 1 + |g|_{H^m(0,T_1)}
+ |f_{\vert_{x=0}}|_{m-1,T_1} + C(K)\int_0^{T_1} (1+\@ifstar\@opnorms\@opnorm{ f(t) }_m) {\rm d}t \Big).
\end{align*}
We also have
\[
\|v^{n+1}(t)-\Theta(u^{\rm in})\|_{L^\infty} \leq \|\partial_t v^{n+1}\|_{L^\infty(\Omega_{T_1})}T_1
\leq C\|v^{n+1}\|_{{\mathbb W}^2(T_1)}T_1.
\]
Therefore, by choosing $M$ large enough and $T_1$ small enough the claim is proved.
The convergence is classically obtained by proving that $\{v^n\}_n$ is a Cauchy sequence
and, therefore, convergent in $L^2$, and that the limit is actually in ${\mathbb W}^m(T)$.
We omit the details.
\subsection{Variable coefficients $2\times 2$ boundary value problems on moving domains}\label{sectVCm}
We now turn to consider initial boundary value problems that are still cast on a half-line,
but instead of ${\mathbb R}_+$, we now consider $(\underline{x}(t),+\infty)$,
where the left boundary $\underline{x}(t)$ is a time dependent function.
We consider first linear problems with variable coefficients.
For the sake of simplicity and to prepare the ground for applications to quasilinear systems,
we consider a slightly less general system of equations than in \eqref{systVC}:
the variable coefficient matrix $A(t,x)$ is of the form $A(\underline{U}(t,x))$.
More precisely,
\begin{equation}\label{IBVPm}
\begin{cases}
\partial_t U + A(\underline{U})\partial_x U + {\mathtt B} U = F & \mbox{in}\quad (\underline{x}(t),\infty) \quad\mbox{for}\quad t\in(0,T), \\
U_{\vert_{t=0}} = u^{\rm in}(x) & \mbox{on}\quad (0,\infty), \\
\nu(t)\cdot U_{\vert_{x=\underline{x}(t)}} = g(t) & \mbox{on}\quad (0,T),
\end{cases}
\end{equation}
where without loss of generality we assumed $\underline{x}(0)=0$.
The first thing to do is of course to transform this initial boundary value problem on a moving domain
into another one cast on a fix domain, say, ${\mathbb R}_+$.
This is done through a diffeomorphism $\varphi(t,\cdot)$ that maps at all times ${\mathbb R}_+$ onto $(\underline{x}(t),\infty)$
and such that for any $t$, we have $\varphi(t,0)=\underline{x}(t)$.
Several choices are possible for $\varphi$ and shall be discussed later.
At this point, we just assume that $\varphi\in C^1(\Omega_T)$ and that $\varphi(0,x)=x$.
Composing the interior equation in \eqref{IBVPm} with the diffeomorphism $\varphi$
to work on the fix domain $(0,\infty)$, introducing the notations
\[
u = U\circ \varphi, \qquad \underline{u} = \underline{U}\circ \varphi, \qquad \partial_t^\varphi u = (\partial_t U)\circ\varphi,
\qquad \partial_x^\varphi u = (\partial_x U)\circ\varphi,
\]
so that, in particular,
\begin{equation}\label{dtphi}
\partial_x^\varphi=\frac{1}{\partial_x \varphi}\partial_x, \qquad \partial_t^\varphi= \partial_t - \frac{\partial_t \varphi}{\partial_x \varphi}\partial_x,
\end{equation}
and writing $B = {\mathtt B}\circ \varphi$ and $f=F\circ\varphi$, we obtain the following equation for $u$
\begin{equation}\label{equ}
\partial_t^\varphi u + A(\underline{u})\partial_x^\varphi u + B(t,x) u = f(t,x).
\end{equation}
The initial boundary value problem on a moving domain \eqref{IBVPm} can therefore be recast as
an initial boundary value problem on a fix domain
\begin{equation}\label{IBVPmT}
\begin{cases}
\partial_t u + \mathcal{A}(\underline{u},\partial\varphi)\partial_x u + B(t,x)u = f(t,x) & \mbox{in}\quad \Omega_T, \\
u_{\vert_{t=0}} = u^{\rm in}(x) & \mbox{on}\quad {\mathbb R}_+, \\
\nu(t)\cdot u_{\vert_{x=0}} = g(t) & \mbox{on}\quad (0,T),
\end{cases}
\end{equation}
with
$$
\mathcal{A}(\underline{u},\partial\varphi)=\frac{1}{\partial_x \varphi}\big( A(\underline{u}) - (\partial_t\varphi) \mbox{Id}\big).
$$
If we want to apply Theorem \ref{theoIBVP1} to construct solutions to \eqref{IBVPmT},
it is necessary to get some information on the regularity of $\varphi$,
which is of course related to the properties of the boundary coordinate $\underline{x}(t)$.
A direct application of Theorem \ref{theoIBVP1} requires that $\partial\varphi$ be in ${\mathbb W}^m(T)$
in order to get solutions $u$ in ${\mathbb W}^m(T)$.
Using Alinhac's good unknown \cite{alinhac1989}, it is however possible to obtain refined regularity estimates,
as shown in the following theorem which requires only the following assumption.
\begin{assumption}\label{asshypm}
We have $\underline{u} \in W^{1,\infty}(\Omega_T)$, $\underline{x}\in C^1([0,T])$, $\underline{x}(0)=0$,
and the diffeomorphism $\varphi$ is in $C^1(\Omega_T)$.
Moreover, there exists a constant $c_0>0$ such that the following three conditions hold.
\begin{enumerate}
\setlength{\itemsep}{3pt}
\item[{\bf i.}]
There exists an open set ${\mathcal U}\subset {\mathbb R}^2$ such that $A \in C^\infty({\mathcal U})$ and that for any $u\in{\mathcal U}$,
the matrix $A(u)$ has eigenvalues $\lambda_+(u)$ and $-\lambda_-(u)$.
Moreover, $\underline{u}$ takes its values in a compact set $\mathcal{K}_0 \subset {\mathcal U}$ and for any $(t,x)\in\Omega_T$ we have
\[
\lambda_\pm(\underline{u}(t,x))\mp \partial_t \varphi (t,x)\geq c_0
\quad\mbox{ and }\quad \lambda_\pm(\underline{u}(t,x))\geq c_0.
\]
\item[{\bf ii.}]
Denoting by ${\bf e}_+ (u)$ a unit eigenvector associated to the eigenvalue $\lambda_+(u)$ of $A(u)$,
for any $t \in [0,T]$ we have
\[
|\nu(t) \cdot {\bf e}_+(\underline{u}(t,0))| \geq c_0.
\]
\item[{\bf iii.}]
The Jacobian of the diffeomorphism is uniformly bounded from below and from above, that is,
for any $(t,x)\in\Omega_T$ we have
\[
c_0 \leq \partial_x \varphi (t,x) \leq \frac{1}{c_0}.
\]
\end{enumerate}
\end{assumption}
\begin{example}\label{ex3}
Considering as in Example \ref{ex1} the linearized shallow water equations,
but this time on a moving domain, Assumption \ref{asshypm} reduces to the conditions
$\underline{h},\underline{q} \in W^{1,\infty}(\Omega_T)$ and
\[
\underline{h}(t,x) \geq c_0, \quad
\sqrt{\mathtt{g}\underline{h}(t,x)} \pm \Bigl(
\frac{\underline{q}(t,x)}{\underline{h}(t,x)} - \partial_t \varphi(t,x) \Bigr) \geq c_0, \quad
\sqrt{\mathtt{g}\underline{h}(t,x)} \pm
\frac{\underline{q}(t,x)}{\underline{h}(t,x)} \geq c_0
\]
with some positive constant $c_0$ independent of $(t,x) \in \Omega_T$.
\end{example}
\begin{theorem}\label{theoIBVP3}
Let $m\geq1$ be an integer, $T>0$, and assume that Assumption \ref{asshypm} is satisfied for some $c_0>0$.
Assume moreover that there are two constants $0<K_0\leq K$ such that
\[
\begin{cases}
\frac{1}{c_0}, \@ifstar\@opnorms\@opnorm{ \partial\widetilde{\varphi}(0) }_{m-1}, |\nu|_{L^\infty(0,T)},
\|\partial\varphi\|_{L^\infty(\Omega_T)}, \|A\|_{L^\infty(\mathcal{K}_0)} \leq K_0, \\
\|\partial \widetilde\varphi \|_{{\mathbb W}^{m-1}(T)}, \|\partial_t \varphi \|_{H^m(\Omega_T)},
|(\partial^m\varphi)_{\vert_{x=0}}|_{L^\infty(0,T)} \leq K, \\
\|\underline{u} \|_{W^{1,\infty}(\Omega_T) \cap {\mathbb W}^{m}(T)}, \|B\|_{W^{1,\infty}(\Omega_T)},
\|\partial B\|_{{\mathbb W}^{m-1}(T)}, |\nu|_{W^{1,\infty} \cap W^{m-1,\infty}(0,T)},
|\partial_t^m\nu|_{L^2(0,T)} \leq K,
\end{cases}
\]
where $\widetilde \varphi(t,x) = \varphi(t,x)-x$.
Then, for any data $u^{\rm in }\in H^m({\mathbb R}_+)$, $f\in H^m(\Omega_T)$, and $g\in H^m(0,T)$ satisfying the
compatibility conditions up to order $m-1$ in the sense of Definition \ref{defcompVC},
there exists a unique solution $u\in {\mathbb W}^m(T)$ to \eqref{IBVPmT}.
Moreover, the following estimate holds for any $t\in[0,T]$ and any $\gamma \geq C(K)$:
\begin{align*}
& \@ifstar\@opnorms\@opnorm{ u(t) }_{m,\gamma} + \biggl(\gamma\int_0^t\@ifstar\@opnorms\@opnorm{ u(t') }_{m,\gamma}^2{\rm d}t'\biggr)^\frac12
+ |u_{\vert_{x=0}}|_{m,\gamma,t} \\
&\leq C(K_0)\bigl(
(1+|\partial_t^m\nu|_{L^2(0,t)})\@ifstar\@opnorms\@opnorm{ u(0) }_m + |g|_{H_\gamma^m(0,t)} + |f_{\vert_{x=0}}|_{m-1,\gamma,t}
+ S_{\gamma,t}^*(\@ifstar\@opnorms\@opnorm{ f(\cdot) }_m) \bigr).
\end{align*}
Particularly, we have
\begin{align*}
& \@ifstar\@opnorms\@opnorm{ u(t) }_m + |u_{\vert_{x=0}}|_{m,t} \\
&\leq C(K_0)e^{C(K)t}\biggl(
(1+|\partial_t^m\nu|_{L^2(0,t)})\@ifstar\@opnorms\@opnorm{ u(0) }_m + |g|_{H^m(0,t)} + |f_{\vert_{x=0}}|_{m-1,t}
+ \int_0^t \@ifstar\@opnorms\@opnorm{ f(t') }_m{\rm d}t' \biggr).
\end{align*}
\end{theorem}
\subsubsection{Proof of Theorem \ref{theoIBVP3}}
A direct estimate in ${\mathbb W}^m(T)$ for the solution of \eqref{IBVPmT} through Theorem \ref{theoIBVP1}
is not possible because it would require that $\partial^2 \varphi\in {\mathbb W}^{m-1}(T)$ while,
under the assumptions made in the statement of the theorem, we only have $\partial^2 \varphi\in {\mathbb W}^{m-2}(T)$.
The key step is to derive a ${\mathbb W}^{m-1}(T)$ estimate on $u$ as well as on
$\partial_t^\varphi u = \partial_t u - (\partial_t\varphi) \partial_x^\varphi u$.
\begin{proposition}\label{propAl}
Under the assumptions of Theorem \ref{theoIBVP3}, there is a unique solution $u\in {\mathbb W}^{m-1}(T)$ to
\eqref{IBVPmT} satisfying
\begin{align}\label{eqmm1}
\@ifstar\@opnorms\@opnorm{ u(t) }_{0} + |u_{\vert_{x=0}}|_{0,t}
&\leq C(K_0)e^{C(K)t}\biggl( \@ifstar\@opnorms\@opnorm{ u(0) }_{0} + |g|_{H^0(0,t)}
+ \int_0^t \@ifstar\@opnorms\@opnorm{ f(t') }_{0} {\rm d}t' \biggr)
\end{align}
in the case $m=1$ and
\begin{align}\label{eqmm1.5}
&\@ifstar\@opnorms\@opnorm{ u(t) }_{m-1} + |u_{\vert_{x=0}}|_{m-1,t} \\
&\leq C(K_0)e^{C(K)t}\biggl( \@ifstar\@opnorms\@opnorm{ u(0) }_{m-1} + |g|_{H^{m-1}(0,t)} + |f_{\vert_{x=0}}|_{m-2,t}
+ \int_0^t \@ifstar\@opnorms\@opnorm{ \partial_t f(t') }_{m-2} {\rm d}t' \biggr) \nonumber
\end{align}
in the case $m \geq 2$.
Moreover, $\partial_t^\varphi u\in {\mathbb W}^{m-1}(T)$ and we have
\begin{align}\label{eqmm2}
& \@ifstar\@opnorms\@opnorm{ \partial_t^\varphi u(t) }_{m-1,\gamma}
+ \biggl( \gamma\int_0^t\@ifstar\@opnorms\@opnorm{ \partial_t^\varphi u(t') }_{m-1,\gamma}^2{\rm d}t' \biggr)^\frac12
+ |(\partial_t^\varphi u)_{\vert_{x=0}}|_{m-1,\gamma,t} \\
&\leq C(K_0)\bigl(
(1+|\partial_t^m\nu|_{L^2(0,t)})\@ifstar\@opnorms\@opnorm{ u(0) }_m + |g|_{H_\gamma^m(0,t)} + |f_{\vert_{x=0}}|_{m-1,\gamma,t}
+ S_{\gamma,t}^*(\@ifstar\@opnorms\@opnorm{ f(\cdot) }_m) \bigr) \nonumber \\
&\quad + C(K)\bigl( S_{\gamma,t}^*(\@ifstar\@opnorms\@opnorm{ u(\cdot) }_m) + |u_{\vert_{x=0}}|_{m-1,\gamma,t} \bigr). \nonumber
\end{align}
\end{proposition}
\begin{proof}[Proof of the proposition]
{\bf Step 1.}
We first show that there exists a solution $u\in {\mathbb W}^{m-1}(T)$ to \eqref{IBVPmT} satisfying
\eqref{eqmm1}--\eqref{eqmm1.5}.
A direct application of Theorem \ref{theoIBVP1} almost yields the result, but with a constant $C(K')$
bigger than $C(K)$ in the sense that it depends on $\|\partial\varphi\|_{W^{1,\infty}(\Omega_T)}$
instead of $\|\partial\varphi\|_{L^{\infty}(\Omega_T)}$.
The improved estimate claimed in \eqref{eqmm1}--\eqref{eqmm1.5} is made possible by the particular structure of
the matrix $\mathcal{A}(\underline{u},\partial\varphi)$, as shown in the following lemma which improves Lemma \ref{lemsymmetrizer}.
\begin{lemma}\label{lemsymmetrizerbis}
Suppose that Assumption \ref{asshypm} is satisfied.
Then, there exist a symmetrizer ${\mathcal S}\in W^{1,\infty}(\Omega_T)$ and constants
$\alpha_0,\alpha_1$ and $\beta_0,\beta_1,\beta_2$ such that Assumption \ref{assVC}
is satisfied for the initial boundary value problem \eqref{IBVPmT}.
Moreover, we have
\begin{align*}
& \mathfrak{c}_0 \leq C\Bigl(\frac{1}{c_0}, \|A(\underline{u}^{\rm in})\|_{L^\infty({\mathbb R}_+)},
\|(\partial_t \varphi)_{\vert_{t=0}}\|_{L^\infty({\mathbb R}_+)} \Bigr), \\
& \mathfrak{c}_1 \leq C\Bigl(\frac{1}{c_0}, \|A(\underline{u})\|_{L^\infty(\Omega_T)},
\|\partial_t\varphi\|_{L^\infty(\Omega_T)}\Bigr),
\end{align*}
where $\underline{u}^{\rm in}=\underline{u}_{\vert_{t=0}}$ and ${\mathfrak c}_0$ and ${\mathfrak c}_1$ are as defined in
Proposition \ref{propNRJ1}, and
\[
\frac{\beta_2}{\beta_0} \leq C\Bigl( \frac{1}{c_0}, \|A(\underline{u})\|_{W^{1,\infty}(\Omega_T)},
\|\partial_t\varphi\|_{L^{\infty}(\Omega_T)}, \|B\|_{L^\infty(\Omega_T)} \Bigr).
\]
\end{lemma}
\begin{proof}[Proof of the lemma]
The proof is an adaptation of the proof of Lemma \ref{lemsymmetrizer}.
We still denote by $\pi_\pm$ the eigenprojector associated to the eigenvalues $\pm\lambda_{\pm}$ of $A(\underline{u})$.
As a symmetrizer for $\mathcal{A}(\underline{u},\varphi)$, we choose
\[
{\mathcal S} = (\partial_x \varphi) \bigl( \pi_+^{\rm T}\pi_+ + M \pi_-^{\rm T} \pi_- \bigr)
\]
with sufficiently large $M$.
Since we have
\begin{align*}
\beta_2
&= \|\partial_t {\mathcal S} + \partial_x ({\mathcal S}\mathcal{A}) - 2{\mathcal S} B\|_{L^\infty(\Omega_T)} \\
&= \|(\partial_x \varphi)\partial_t S + \partial_x (SA) - (\partial_t \varphi) \partial_x S - 2(\partial_x \varphi)SB\|_{L^\infty(\Omega_T)},
\end{align*}
where we denoted $S= \pi_+^{\rm T}\pi_+ + M \pi_-^{\rm T}\pi_-$, and since $\pi_\pm$ depends only on $A(\underline{u})$,
we deduce the desired results.
\end{proof}
Using Lemma \ref{lemsymmetrizerbis} instead of Lemma \ref{lemsymmetrizer} in the proof of
Theorem \ref{theoIBVP1} in the particular case of the initial boundary value problem \eqref{IBVPmT},
we get \eqref{eqmm1}--\eqref{eqmm1.5}.
\medskip
\noindent
{\bf Step 2.}
We prove here an extra regularity on $\partial_t^\varphi u $ that implies the inequality stated in the theorem.
The main tool to get this extra regularity is Alinhac's good unknown \cite{alinhac1989},
which removes the loss of derivative due to the dependence on $\varphi$ in the coefficients
of the initial boundary value problem \eqref{IBVPmT}.
Differentiating with respect to time the interior equation in \eqref{IBVPmT},
and writing $\dot u=\partial_t u$, $\dot f=\partial_t f$, etc., we get
\begin{equation}\label{eqbad}
\partial_t\dot{u} + \mathcal{A}(\underline{u},\partial \varphi)\partial_x\dot{u} + A'(\underline{u})[\dot{\underline{u}}]\partial_x^\varphi u
+ \mathcal{M}(\underline{u},\partial\varphi,\partial_x u)\partial\dot{\varphi} + B\dot{u} = \dot{f} - \dot{B}u
\end{equation}
with
\[
\mathcal{M}(u,\partial\varphi,\partial_x u)\partial \dot \varphi
= - \bigl( (\partial_x\dot{\varphi}) \mathcal{A}(\underline{u},\partial \varphi)
+ (\partial_t\dot{\varphi}) \mbox{Id} \bigr) \partial_x^\varphi u.
\]
Obviously, the term $\mathcal{M}(\underline{u},\partial\varphi,\partial_x u)\partial\dot{\varphi}$ is responsible
for the loss of one derivative, in the sense that a control of $\varphi $ in $ {\mathbb W}^{m+1}(T)$
is required to control the ${\mathbb W}^m(T)$ norm of $u$.
This singular dependence is removed by working with Alinhac's good unknown
$\dot{u}^\varphi = \dot{u} - \dot{\varphi} \partial_x^\varphi u$ instead of $\dot{u}$.
The notations $\dot{f}^\varphi$ and $\dot{B}^\varphi$ are defined similarly. The following lemma is due to Alinhac \cite{alinhac1989} and can be checked by simple computations.
\begin{lemma}\label{lemeq}
With $\dot{u}^\varphi = \dot{u} - \dot{\varphi}\partial_x^\varphi u$,
the equation \eqref{eqbad} can be rewritten under the form
\[
\partial_t\dot{u}^\varphi + \mathcal{A}(\underline{u},\partial \varphi)\partial_x\dot{u}^\varphi + A'(\underline{u})[\dot{\underline{u}}^\varphi] \partial_x^\varphi u
+ B\dot{u}^\varphi = \dot{f}^\varphi - \dot{B}^\varphi u.
\]
\end{lemma}
\begin{remark}
We use the notations $\dot{u} = \partial_t u$ and $\dot{u}^\varphi = \partial_t^\varphi u$ to underline the fact that
this is a general procedure that works for any linearization operator, not only time differentiation.
\end{remark}
We can use \eqref{equ} to write
\[
\partial_x^\varphi u = A(\underline{u})^{-1}(f-Bu-\dot u^\varphi),
\]
so that the lemma yields
\[
\partial_t\dot{u}^\varphi + \mathcal{A}(\underline{u},\partial \varphi)\partial_x\dot{u}^\varphi + B_{(1)}\dot{u}^\varphi = f_{(1)},
\]
where
\begin{equation}\label{equx}
\begin{cases}
B_{(1)} = B - A'(\underline{u})[\dot{\underline{u}}^\varphi]A(\underline{u})^{-1}, \\
f_{(1)} = \dot{f}^\varphi - A'(\underline{u})[\dot{\underline{u}}^\varphi] A(\underline{u})^{-1}f -
(\dot{B}^\varphi - A'(\underline{u})[\dot{\underline{u}}^\varphi] A(\underline{u})^{-1}B)u.
\end{cases}
\end{equation}
Therefore, $\dot{u}^\varphi = \partial_t^\varphi u$ solves an interior equation similar to those considered
in Theorem \ref{theoIBVP1}.
Let us now consider the initial and boundary conditions for $\dot{u}^\varphi$.
For the initial condition, we have
\[
(\dot u^\varphi)_{\vert_{t=0}} = u^{\rm in}_{(1)}
\quad\mbox{with}\quad
{u^{\rm in}_{(1)}} = (\partial_t u)_{\vert_{t=0}} - (\partial_t\varphi)_{\vert_{t=0}}\partial_x u^{\rm in}.
\]
For the boundary condition, let us differentiate with respect to time the boundary condition in \eqref{IBVPmT}
to obtain $\nu(t) \cdot \partial_t u_{\vert_{x=0}} = \partial_t g - \nu'(t)\cdot u_{\vert_{x=0}}$ or equivalently
\[
\nu(t) \cdot (\dot u^\varphi +\dot \underline{x} \partial_x^\varphi u)_{\vert_{x=0}} = \partial_t g - \nu'(t) \cdot u_{\vert_{x=0}}.
\]
Using \eqref{equ}, this yields
\[
\nu(t) \cdot \bigl((\mbox{Id}-\dot{x} A(\underline{u})^{-1})\dot u^\varphi\bigr)_{\vert_{x=0}}
= \partial_t g - \nu'(t)\cdot u_{\vert_{x=0}}-\dot{x} \nu(t) \cdot A(\underline{u})^{-1} (f-Bu)_{\vert_{x=0}}.
\]
It follows that $\dot u^\varphi$ satisfies an initial boundary value problem of the form \eqref{systVC}, namely,
\begin{equation}\label{systVCm}
\begin{cases}
\partial_t\dot{u}^\varphi + \mathcal{A}(\underline{u},\partial\varphi)\partial_x\dot{u}^\varphi + B_{(1)} \dot{u}^\varphi = f_{(1)}
& \mbox{in}\quad \Omega_T, \\
\dot{u}^\varphi_{\vert_{t=0}} = u^{\rm in}_{(1)} & \mbox{on}\quad {\mathbb R}_+, \\
\nu_{(1)}(t) \cdot \dot u^\varphi_{\vert_{x=0}} = g_{(1)} & \mbox{on}\quad (0,T),
\end{cases}
\end{equation}
where $f_{(1)}$ and $B_{(1)}$ are as in \eqref{equx} and
\begin{equation}\label{nu1}
\begin{cases}
g_{(1)} = \partial_t g - (\partial_t \nu) \cdot u_{\vert_{x=0}} - \dot \underline{x} \nu \cdot A(\underline{u})^{-1} (f-Bu)_{\vert_{x=0}}, \\
\nu_{(1)} = (\mbox{Id}-\dot \underline{x} A(\underline{u}_{\vert_{x=0}})^{-1})^{\rm T}\nu.
\end{cases}
\end{equation}
Concerning the boundary condition, we have the following lemma which shows that
the initial boundary value problem \eqref{systVCm} satisfies condition {\bf iii} in Assumption \ref{asshyp}.
\begin{lemma}\label{mdBC}
Under Assumption \ref{asshypm}, for any $t\in[0,T]$ we have
\[
|\nu_{(1)}(t) \cdot \mathbf{e}_+(\underline{u}(t,0))| \geq \frac{c_0^2}{\lambda_{+}(\underline{u}(t,0))}.
\]
\end{lemma}
\begin{proof}
We see that
\begin{align*}
\nu_{(1)}(t) \cdot \mathbf{e}_+(\underline{u}(t,0))
&= \nu(t) \cdot (\mbox{Id}-\dot{x}(t) A(\underline{u}(t,0))^{-1})\mathbf{e}_+(\underline{u}(t,0)) \\
&= \Bigl( 1-\frac{\dot{x}(t)}{\lambda_{+}(\underline{u}(t,0))} \Bigr) \nu(t) \cdot \mathbf{e}_+(\underline{u}(t,0)).
\end{align*}
Since $\dot{x}(t)=(\partial_t\varphi)(t,0)$,
this gives the desired inequality.
\end{proof}
Here, we see that
\[
|\nu_{(1)}|_{L^\infty(0,T)} \leq C(K_0), \qquad
\|B_{(1)}\|_{L^\infty(\Omega_T)} \leq C(K)
\]
and that in the case $m\geq2$
\[
\|\partial B_{(1)}\|_{{\mathbb W}^{m-2}(T)}, |\nu_{(1)}|_{W^{m-1,\infty}(0,T)} \leq C(K).
\]
Therefore, we can apply the result in Step 1 to obtain
\begin{align}\label{eqmm3}
& \@ifstar\@opnorms\@opnorm{ \dot{u}^\varphi(t) }_{m-1,\gamma}
+ \biggl( \gamma\int_0^t \@ifstar\@opnorms\@opnorm{ \dot{u}^\varphi(t') }_{m-1,\gamma}^2{\rm d}t' \biggr)^\frac12
+ |{\dot{u}^\varphi}_{\;\,\vert_{x=0}}|_{m-1,t} \\
&\leq C(K_0)\bigl( \@ifstar\@opnorms\@opnorm{ \dot{u}^\varphi(0) }_{m-1} + |g_{(1)}|_{H_\gamma^{m-1}(0,t)}
+ |f_{(1) \vert_{x=0}}|_{m-2,\gamma,t} + S_{\gamma,t}^*(\@ifstar\@opnorms\@opnorm{ f_{(1)}(\cdot) }_{m-1}) \bigr), \nonumber
\end{align}
where the term $|f_{(1) \vert_{x=0}}|_{m-2,\gamma,t}$ is dropped in the case $m=1$.
Here, we have
\[
\begin{cases}
\@ifstar\@opnorms\@opnorm{ \dot{u}^\varphi(0) }_{m-1} \leq C(K_0) \@ifstar\@opnorms\@opnorm{ u(0) }_m, \\
\@ifstar\@opnorms\@opnorm{ f_{(1)}(t) }_{m-1} \leq C(K)( \@ifstar\@opnorms\@opnorm{ f(t) }_m + \@ifstar\@opnorms\@opnorm{ u(t) }_{m-1} ), \\
|f_{(1) \vert_{x=0}}|_{m-2,\gamma,t} \leq C(K)( |f_{\vert_{x=0}}|_{m-1,\gamma,t} + |u_{\vert_{x=0}}|_{m-1,\gamma,t} ).
\end{cases}
\]
Concerning the term $|g_{(1)}|_{H^{m-1}(0,t)}$, especially, the term $(\partial_t\nu) \cdot u_{\vert_{x=0}}$
we need to estimate it carefully, because we do not assume $\nu \in W^{m,\infty}(0,T)$.
In the case $m=1$, we estimate it directly as
\[
|(\partial_t\nu) \cdot u_{\vert_{x=0}}|_{L_\gamma^2(0,t)} \leq C(K)|u_{\vert_{x=0}}|_{L_\gamma^2(0,t)}.
\]
In the case $m\geq2$, we see that
\begin{align*}
|(\partial_t\nu) \cdot u_{\vert_{x=0}}|_{H_\gamma^{m-1}(0,t)}
&\leq |\nu|_{W^{m-1,\infty}(0,t)}|u_{\vert_{x=0}}|_{m-1,\gamma,t}
+ |\partial_t^m\nu|_{L^2(0,t)} \sup_{t'\in[0,t]}e^{-\gamma t'}|u(t',0)| \\
&\leq C(K)|u_{\vert_{x=0}}|_{m-1,\gamma,t} + C|\partial_t^m\nu|_{L^2(0,t)} \@ifstar\@opnorms\@opnorm{ u(0) }_{m-1},
\end{align*}
where we used $\sup_{t'\in[0,t]}e^{-\gamma t'}|u(t',0)| \leq C( \|u(0)\|_{H^1} + \gamma^{-\frac12}|u_{\vert_{x=0}}|_{1,\gamma,t})$,
which is a simple consequence of \eqref{ineq5} in Lemma \ref{estuuu}.
In any case, we have
\begin{align*}
|g_{(1)}|_{H_\gamma^{m-1}(0,t)} \leq
& |g|_{H_\gamma^m(0,t)} + C|\partial_t^m\nu|_{L^2(0,t)} \@ifstar\@opnorms\@opnorm{ u(0) }_{m-1} + C(K)(|u_{\vert_{x=0}}|_{m-1,t}+|f_{\vert_{x=0}}|_{m-1,t}).
\end{align*}
Therefore, by \eqref{eqmm3} we obtain
\begin{align*}
& \@ifstar\@opnorms\@opnorm{ \dot{u}^\varphi(t) }_{m-1,\gamma}
+ \biggl( \gamma\int_0^t \@ifstar\@opnorms\@opnorm{ \dot{u}^\varphi(t') }_{m-1,\gamma}^2{\rm d}t' \biggr)^\frac12
+ |{\dot{u}^\varphi}_{\;\,\vert_{x=0}}|_{m-1,t} \\
& \leq C(K_0)\bigl( (1+|\partial_t^m\nu|_{L^2(0,t)})\@ifstar\@opnorms\@opnorm{ u(0) }_m + |g|_{H^m(0,t)} \bigr) \\
&\quad
+ C(K)\bigl( |f_{\vert_{x=0}}|_{m-1,t} + |u_{\vert_{x=0}}|_{m-1,t}
+ S_{\gamma,t}^*( \@ifstar\@opnorms\@opnorm{ f(\cdot) }_m) + S_{\gamma,t}^*(\@ifstar\@opnorms\@opnorm{ u(\cdot) }_{m-1}) \bigr),
\end{align*}
which shows $\partial_t^\varphi u \in {\mathbb W}^{m-1}(T)$.
\medskip
\noindent
{\bf Step 3.}
Finally, we improve the above inequality to show \eqref{eqmm2}.
It follows directly from Lemma \ref{lemeq} that we have also the equation for $\dot{u}^\varphi$ of the form
\[
\partial_t\dot{u}^\varphi + \mathcal{A}(\underline{u},\partial\varphi)\partial_x\dot{u}^\varphi = \widetilde{f}_{(1)}
\]
with
\[
\widetilde{f}_{(1)} = \partial_t^\varphi f - A'(\underline{u})[\partial_t^\varphi\underline{u}]\partial_x^\varphi u - \partial_t^\varphi(Bu).
\]
Moreover, we have \eqref{eqmm3} with $f_{(1)}$ replaced by $\widetilde{f}_{(1)}$.
In order to give modified estimates for $\widetilde{f}_{(1)}$ and $g_{(1)}$, in the case of $m\geq2$
we use the following expressions
\begin{align*}
\partial^\alpha \widetilde{f}_{(1)}
&= \partial_t^\varphi\partial^\alpha f
+ [\partial^\alpha,\partial_t^\varphi](\partial_t^\varphi u + A(\underline{u})\partial_x^\varphi u + Bu) \\
&\quad
- \partial^\alpha(A'(\underline{u})[\partial_t^\varphi\underline{u}]\partial_x^\varphi u + \partial_t^\varphi(Bu)), \\
\partial_t^k g_{(1)}
&= \partial_t^k(\partial_t g - (\partial_t \nu) \cdot u_{\vert_{x=0}})
- \dot \underline{x} \nu \cdot A(\underline{u})^{-1} \partial_t^k (f-Bu)_{\vert_{x=0}} \\
&\quad
- [\partial_t^k, \dot \underline{x} \nu \cdot A(\underline{u})^{-1}](\partial_t^\varphi u + A(\underline{u})\partial_x^\varphi u)_{\vert_{x=0}},
\end{align*}
where we used \eqref{equ}.
These expressions together with Lemma \ref{ineq1} give
\begin{align*}
& \@ifstar\@opnorms\@opnorm{ \widetilde{f}_{(1)}(t) }_{m-1} \leq C(K_0)\@ifstar\@opnorms\@opnorm{ f(t) }_m + C(K)\@ifstar\@opnorms\@opnorm{ u(t) }_m, \\
& |g_{(1)}|_{H_\gamma^{m-1}(0,t)} + |\widetilde{f}_{(1) \vert_{x=0}}|_{m-2,\gamma,t} \\
& \leq C(K_0)( |\partial_t^m\nu|_{L^2(0,t)}\@ifstar\@opnorms\@opnorm{ u(0) }_{m-1} + |g|_{H^m(0,t)} + |f_{\vert_{x=0}}|_{m-1,t})
+ C(K)|u_{\vert_{x=0}}|_{m-1,t},
\end{align*}
which yields \eqref{eqmm2}.
The proof of Proposition \ref{propAl} is complete.
\end{proof}
In order to conclude the proof of Theorem \ref{theoIBVP3}, we need to show that Proposition \ref{propAl}
provides a control of $u$ in ${\mathbb W}^{m}(T)$.
\begin{lemma}\label{lemequivn}
Under the assumptions of Theorem \ref{theoIBVP3}, if $u$ solves \eqref{IBVPmT}, then we have
\begin{align*}
&\@ifstar\@opnorms\@opnorm{ \partial u(t) }_{m-1,\gamma}
+ \biggl( \gamma\int_0^t \@ifstar\@opnorms\@opnorm{ \partial u(t') }_{m-1,\gamma}^2{\rm d}t' \biggr)^\frac12
+ |(\partial u)_{\vert_{x=0}}|_{m-1,t} \\
&\leq C(K_0)\biggl\{ \@ifstar\@opnorms\@opnorm{ u(0) }_m + |f_{\vert_{x=0}}|_{m-1,\gamma,t} + S_{\gamma,t}^*(\@ifstar\@opnorms\@opnorm{ \partial_t f(\cdot) }_{m-1}) \\
&\phantom{ \leq C(K_0)\biggl\{ }
+ \@ifstar\@opnorms\@opnorm{ \partial_t^\varphi u(t) }_{m-1,\gamma}
+ \biggl( \gamma\int_0^t \@ifstar\@opnorms\@opnorm{ \partial_t^\varphi u(t') }_{m-1,\gamma}^2{\rm d}t' \biggr)^\frac12
+ |(\partial_t^\varphi u)_{\vert_{x=0}}|_{m-1,t} \biggr\} \\
&\quad + C(K)\biggl\{ \biggl( \int_0^t \@ifstar\@opnorms\@opnorm{ u(t') }_{m,\gamma}^2{\rm d}t' \biggr)^\frac12
+ |u_{\vert_{x=0}}|_{m-1,t} \biggr\}.
\end{align*}
\end{lemma}
\begin{proof}
We will use the same notation $\dot{u}^\varphi=\partial_t^\varphi u$ in the proof of Proposition \ref{propAl}.
Then, \eqref{equ} can be written as
\begin{equation}\label{eqn2}
\dot{u}^\varphi + A(\underline{u})\partial_x^\varphi u = f-Bu =: f_0.
\end{equation}
We first consider the case $m=1$.
Here, it holds that
\[
\begin{cases}
\|f_0(0)\|_{L^2} \leq C(K_0)\@ifstar\@opnorms\@opnorm{ u(0) }_1, \\
\|\partial_t f_0(t)\|_{L^2} \leq \|\partial_t f(t)\|_{L^2} + C(K)\@ifstar\@opnorms\@opnorm{ u(t) }_1, \\
|{f_0}_{\vert_{x=0}}|_{L_\gamma^2(0,t)} \leq |f_{\vert_{x=0}}|_{L_\gamma^2(0,t)} + C(K)|u_{\vert_{x=0}}|_{L_\gamma^2(0,t)}.
\end{cases}
\]
It follows from \eqref{eqn2} that
\[
\partial_x u = (\partial_x \varphi)A(\underline{u})^{-1}(f_0 - \dot{u}^\varphi).
\]
We also have
\[
\partial_t u = \dot{u}^\varphi - \frac{\partial_t\varphi}{\partial_x\varphi}\partial_x u.
\]
Therefore, we obtain
\[
|\partial u(t,x)| \leq C(K_0)( |\dot{u}^\varphi(t,x)| + |f_0(t,x)|).
\]
By Lemma \ref{estuuu} we have
\begin{align*}
& \@ifstar\@opnorms\@opnorm{ f_0(t) }_{0,\gamma}
+ \biggl( \gamma\int_0^t \@ifstar\@opnorms\@opnorm{ f_0(t') }_{0,\gamma}^2{\rm d}t' \biggr)^\frac12 \\
&\leq C\bigl( \|f_0(0)\|_{L^2} + S_{\gamma,t}^*(\|\partial_t f_0(\cdot)\|_{L^2}) \bigr) \\
&\leq C(K_0)\bigl( \@ifstar\@opnorms\@opnorm{ u(0) }_1 + S_{\gamma,t}^*(\|\partial_t f(\cdot)\|_{L^2}) \bigr)
+ C(K)S_{\gamma,t}^*(\@ifstar\@opnorms\@opnorm{ u(\cdot) }_1).
\end{align*}
Using the above inequalities, we get the desired estimate in the case $m=1$.
We proceed to consider the case $m\geq2$.
Applying $\partial^\alpha$ with a multi-index $\alpha$ satisfying $|\alpha| \leq m-1$ to \eqref{eqn2}
and using the identity
\begin{equation}\label{id1}
\partial_x^\varphi\partial^\alpha u
= \partial^\alpha\partial_x^\varphi u + (\partial_x^\varphi\partial^\alpha\varphi)\partial_x^\varphi u
+ (\partial_x\varphi)^{-1}[\partial^\alpha; \partial_x\varphi, \partial_x^\varphi u]
\end{equation}
with a symmetric commutator
$[\partial^\alpha; v, w] = \partial^\alpha(vw)-(\partial^\alpha v)w-v(\partial^\alpha w)$,
we obtain
\begin{align*}
A(\underline{u})\partial_x^\varphi\partial^\alpha u + \partial^\alpha\dot{u}^\varphi
&= \partial^\alpha(f-Bu)-[\partial^\alpha,A(\underline{u})]\partial_x^\varphi u \\
&\quad
+A(\underline{u})( (\partial_x^\varphi\partial^\alpha\varphi)\partial_x^\varphi u
+ (\partial_x\varphi)^{-1}[\partial^\alpha; \partial_x\varphi, \partial_x^\varphi u] ) \\
&=: f_{1,\alpha}.
\end{align*}
Here, by Lemma \ref{ineq1} it holds that
\[
\begin{cases}
\|f_{1,\alpha}(0)\|_{L^2} \leq C(K_0)\@ifstar\@opnorms\@opnorm{ u(0) }_m, \\
\|\partial_t f_{1,\alpha}(t)\|_{L^2} \leq C(K_0)\@ifstar\@opnorms\@opnorm { \partial_t f(t) }_{m-1}
+ C(K)(1+\@ifstar\@opnorms\@opnorm{ \partial_t\varphi(t) }_m)\@ifstar\@opnorms\@opnorm{ u(t) }_m, \\
|{f_{1,\alpha}}_{\vert_{x=0}}|_{L_\gamma^2(0,t)}
\leq |f_{\vert_{x=0}}|_{m-1,\gamma,t} + C(K)|u_{\vert_{x=0}}|_{m-1,\gamma,t}.
\end{cases}
\]
We also have
\[
\partial^\alpha\partial_x u = (\partial_x\varphi)A(\underline{u})^{-1}(f_{1,\alpha}-\partial^\alpha\dot{u}^\varphi),
\]
which will be used to evaluate $\partial_x u$.
Applying $\partial^\alpha$ to the identity
$\partial_t u = \dot{u}^\varphi + (\partial_t\varphi)\partial_x^\varphi u$
and using \eqref{id1} we obtain
\begin{align*}
& \partial^\alpha\partial_t u - \partial^\alpha\dot{u}^\varphi - (\partial_t\varphi)(\partial_x\varphi)^{-1}\partial^\alpha\partial_x u \\
&= (\partial^\alpha\partial_t\varphi)\partial_x^\varphi u + [\partial^\alpha; \partial_t\varphi, \partial_x^\varphi u]
-(\partial_t\varphi)(\partial_x\varphi)^{-1}( (\partial^\alpha\partial_x\varphi)\partial_x^\varphi u
+ [\partial^\alpha; \partial_x\varphi, \partial_x^\varphi u] ) \\
&=: f_{2,\alpha}.
\end{align*}
Here, by Lemma \ref{ineq1} it holds that
\[
\begin{cases}
\|f_{2,\alpha}(0)\|_{L^2} \leq C(K_0)\@ifstar\@opnorms\@opnorm{ u(0) }_m, \\
\|\partial_t f_{2,\alpha}(t)\|_{L^2} \leq C(K)(1+\@ifstar\@opnorms\@opnorm{ \partial_t\varphi(t) }_m)\@ifstar\@opnorms\@opnorm{ u(t) }_m, \\
|{f_{2,\alpha}}_{\vert_{x=0}}|_{L_\gamma^2(0,t)} \leq C(K)|u_{\vert_{x=0}}|_{m-1,\gamma,t}.
\end{cases}
\]
We also have
\[
\partial^\alpha\partial_t u
= \partial^\alpha\dot{u}^\varphi + (\partial_t\varphi)(\partial_x\varphi)^{-1}\partial^\alpha\partial_x u + f_{2,\alpha},
\]
which will be used to evaluate $\partial_t u$.
Therefore, we obtain
\[
|\partial^\alpha\partial u(t,x)|
\leq C(K_0)( |\partial^\alpha\dot{u}^\varphi(t,x)| + |f_{1,\alpha}(t,x)| + |f_{2,\alpha}(t,x)| ),
\]
so that
\begin{align*}
&\@ifstar\@opnorms\@opnorm{ \partial u(t) }_{m-1,\gamma}
+ \biggl( \gamma\int_0^t \@ifstar\@opnorms\@opnorm{ \partial u(t') }_{m-1,\gamma}^2{\rm d}t' \biggr)^\frac12
+ |(\partial u)_{\vert_{x=0}}|_{m-1,t} \\
&\leq C(K_0)\biggl\{ \@ifstar\@opnorms\@opnorm{ \dot{u}^\varphi(t) }_{m-1,\gamma}
+ \biggl( \gamma\int_0^t \@ifstar\@opnorms\@opnorm{ \dot{u}^\varphi(t') }_{m-1,\gamma}^2{\rm d}t' \biggr)^\frac12
+ |{\dot{u}^\varphi}_{\;\,\vert_{x=0}}|_{m-1,t} \\
&\quad
+ \sum_{|\alpha| \leq m-1,j=1,2}\biggl( \@ifstar\@opnorms\@opnorm{ f_{j,\alpha}(t) }_{0,\gamma}
+ \biggl( \gamma\int_0^t \@ifstar\@opnorms\@opnorm{ f_{j,\alpha}(t') }_{0,\gamma}^2{\rm d}t' \biggr)^\frac12
+ |f_{j,\alpha \vert_{x=0}}|_{L_\gamma^2(0,t)} \biggr) \biggr\}.
\end{align*}
Here, by Lemma \ref{estuuu} we see that
\begin{align*}
& \@ifstar\@opnorms\@opnorm{ f_{j,\alpha}(t) }_{0,\gamma}
+ \biggl( \gamma\int_0^t \@ifstar\@opnorms\@opnorm{ f_{j,\alpha}(t') }_{0,\gamma}^2{\rm d}t' \biggr)^\frac12 \\
&\leq C\bigl( \|f_{j,\alpha}(0)\|_{L^2} + S_{\gamma,t}^*(\|\partial_t f_{j,\alpha}(\cdot)\|_{L^2}) \bigr) \\
&\leq C(K_0)\bigl( \@ifstar\@opnorms\@opnorm{ u(0) }_m + S_{\gamma,t}^*(\@ifstar\@opnorms\@opnorm{ \partial_t f(\cdot) }_{m-1}) \bigr)
+ C(K)S_{\gamma,t}^*( (1+\@ifstar\@opnorms\@opnorm{ \partial_t\varphi(\cdot) }_m)\@ifstar\@opnorms\@opnorm{ u(\cdot) }_m )
\end{align*}
and that
\begin{align*}
& S_{\gamma,t}^*( (1+\@ifstar\@opnorms\@opnorm{ \partial_t\varphi(\cdot) }_m)\@ifstar\@opnorms\@opnorm{ u(\cdot) }_m ) \\
&\leq \biggl(\frac{1}{\gamma}\int_0^t\@ifstar\@opnorms\@opnorm{ u(t') }_{m,\gamma}^2{\rm d}t'\biggr)^\frac12
+ \int_0^te^{-\gamma t'}\@ifstar\@opnorms\@opnorm{ \partial_t\varphi(t') }_m \@ifstar\@opnorms\@opnorm{ u(t') }_m{\rm d}t' \\
&\leq \biggl(\frac{1}{\gamma}\int_0^t\@ifstar\@opnorms\@opnorm{ u(t') }_{m,\gamma}^2{\rm d}t'\biggr)^\frac12
+ \|\partial_t\varphi\|_{H^m(\Omega_t)} \biggl(\int_0^t\@ifstar\@opnorms\@opnorm{ u(t') }_{m,\gamma}^2{\rm d}t'\biggr)^\frac12.
\end{align*}
Summarizing the above inequalities, we obtain the desired estimate.
\end{proof}
Now, it follows from the estimates in Proposition \ref{propAl} and Lemma \ref{lemequivn} together with
Lemma \ref{ineq4} that
\begin{align*}
&\@ifstar\@opnorms\@opnorm{ u(t) }_{m,\gamma}
+ \biggl( \gamma\int_0^t \@ifstar\@opnorms\@opnorm{ u(t') }_{m,\gamma}^2{\rm d}t' \biggr)^\frac12 + |u_{\vert_{x=0}}|_{m,t} \\
&\leq \@ifstar\@opnorms\@opnorm{ \partial u(t) }_{m-1,\gamma}
+ \biggl( \gamma\int_0^t \@ifstar\@opnorms\@opnorm{ \partial u(t') }_{m-1,\gamma}^2{\rm d}t' \biggr)^\frac12
+ |(\partial u)_{\vert_{x=0}}|_{m-1,t} \\
&\quad
+ \@ifstar\@opnorms\@opnorm{ u(t) }_{m-1,\gamma}
+ \biggl( \gamma\int_0^t \@ifstar\@opnorms\@opnorm{ u(t') }_{m-1,\gamma}^2{\rm d}t' \biggr)^\frac12 + |u_{\vert_{x=0}}|_{m-1,t} \\
&\leq C(K_0)\bigl( (1+|\partial_t^m\nu|_{L^2(0,t)})\@ifstar\@opnorms\@opnorm{ u(0) }_m + |g|_{H_\gamma^m(0,t)}
+ |f_{\vert_{x=0}}|_{m-1,\gamma,t} + S_{\gamma,t}^*(\@ifstar\@opnorms\@opnorm{ \partial_t f(\cdot) }_{m-1}) \bigr) \\
&\quad
+ C(K)\biggl\{ \gamma^{-\frac12}\biggl( \gamma\int_0^t \@ifstar\@opnorms\@opnorm{ u(t') }_{m,\gamma}^2{\rm d}t' \biggr)^\frac12
+ \gamma^{-\frac12}\@ifstar\@opnorms\@opnorm{ u(0) }_m + \gamma^{-1}|u_{\vert_{x=0}}|_{m,\gamma,t} \biggr\}.
\end{align*}
Therefore, by taking $\gamma$ sufficiently large compared to $C(K)$,
we obtain the desired estimate in Theorem \ref{theoIBVP3}.
The proof of Theorem \ref{theoIBVP3} is complete.
\subsection{Application to free boundary problems with a boundary equation of ``kinematic'' type}\label{sectFB1}
We investigate here a general class of free boundary problems.
We consider a quasilinear hyperbolic system cast on a moving domain $(\underline{x}(t),\infty)$,
\begin{equation}\label{IBVPfb}
\begin{cases}
\partial_t U + A(U)\partial_x U = 0 & \mbox{in}\quad (\underline{x}(t),\infty) \quad\mbox{for}\quad t\in(0,T), \\
U_{\vert_{t=0}} = u^{\rm in}(x) & \mbox{on}\quad (\underline{x}(0),\infty), \\
\underline{\nu}\cdot U_{\vert_{x=\underline{x}(t)}} = g(t) & \mbox{on}\quad (0,T),
\end{cases}
\end{equation}
and assume that the evolution of the boundary is governed by a nonlinear equation of the form
\begin{equation}\label{eqFB}
\dot{\underline{x}} = \mathcal{X}(U_{\vert_{x=\underline{x}(t)}})
\end{equation}
for some smooth function ${\mathcal X}$.
The set of equations \eqref{IBVPfb}--\eqref{eqFB} is a free boundary problem.
In the following, without loss of generality we assume $\underline{x}(0)=0$.
Using as in \S \ref{sectVCm} a diffeomorphism $\varphi(t,\cdot) : {\mathbb R}_+ \to (\underline{x}(t),\infty)$,
and recalling the notations
\[
u = U\circ \varphi, \qquad \partial_x^\varphi = \frac{1}{\partial_x \varphi}\partial_x, \qquad
\partial_t^\varphi = \partial_t - \frac{\partial_t \varphi}{\partial_x \varphi}\partial_x,
\]
the free boundary problem \eqref{IBVPfb}--\eqref{eqFB} can therefore be recast as
an initial boundary value problem on a fixed domain,
\begin{equation}\label{IBVPfbbis}
\begin{cases}
\partial_t u + \mathcal{A}(u,\partial\varphi)\partial_x u = 0 & \mbox{in}\quad \Omega_T, \\
u_{\vert_{t=0}} = u^{\rm in}(x) & \mbox{on}\quad {\mathbb R}_+, \\
\underline{\nu}\cdot u_{\vert_{x=0}} = g(t) & \mbox{on}\quad (0,T),
\end{cases}
\end{equation}
where $\underline{\nu}\in {\mathbb R}^2$ is a constant vector and
\[
\mathcal{A}(u,\partial\varphi) = \frac{1}{\partial_x \varphi}\bigl( A(u) - (\partial_t\varphi) \mbox{Id}\bigr),
\]
complemented by the evolution equation
\begin{equation}\label{eqFBbis}
\dot{\underline{x}} = {\mathcal X}(u_{\vert_{x=0}}),\qquad \underline{x}(0) = 0.
\end{equation}
As shown in \S \ref{sectVCm}, the regularity of $\varphi$ plays an important role in the analysis
of the initial boundary value problem \eqref{IBVPfbbis}.
It is therefore important to make an appropriate choice for the diffeomorphism.
For a boundary equation of the form \eqref{eqFBbis} which is of ``kinematic'' type,
a ``Lagrangian'' diffeomorphism is appropriate.
In particular, in the second point of the lemma, the structure of $\varphi$ allows the control of
$\partial_t \varphi$ in ${\mathbb W}^m(T)$ (which involves $m+1$ derivatives of $\varphi$) by $u$ in ${\mathbb W}^m(T)$
(which involves only $m$ derivative of $u$).
\begin{lemma}\label{lemdiffeo}
Let $\mathcal{U}$ be an open set in ${\mathbb R}^2$ and $\mathcal{X} \in C^\infty(\mathcal{U})$.
Suppose that $u\in W^{1,\infty}(\Omega_T)$ takes its values in a compact and convex set
$\mathcal{K}_1 \subset \mathcal{U}$ and that
\[
\|u\|_{W^{1,\infty}(\Omega_T)}, \|\mathcal{X}\|_{W^{1,\infty}(\mathcal{K}_1)} \leq K.
\]
Then, $\underline{x} \in C^1([0,T])$ can be defined by the ODE
\[
\begin{cases}
\dot{\underline{x}}(t) = {\mathcal X}(u_{\vert_{x=0}}(t)) \quad\mbox{for}\quad t\in(0,T), \\
\underline{x}(0) = 0.
\end{cases}
\]
Moreover, there exists $T_1 \in (0,T]$ depending on $K$ such that the mapping
$\varphi:\overline{\Omega_T}\to {\mathbb R}$ defined by
\begin{equation}\label{diffeo}
\varphi(t,x)=x+\int_0^t {\mathcal X}(u(t',x)){\rm d}t'
\end{equation}
satisfies the following properties:
\begin{enumerate}
\setlength{\itemsep}{3pt}
\item[{\bf i.}]
We have $\varphi(t,0) = \underline{x}(t)$ and that for any $t\in[0,T_1]$,
$\varphi(t,\cdot)$ is a diffeomorphism mapping ${\mathbb R}_+$ onto $(\underline{x}(t),\infty)$ and satisfying
$\frac12 \leq \partial_x\varphi(t,x) \leq 2$.
\item[{\bf ii.}]
If moreover $m\geq2$, $u\in {\mathbb W}^m(T_1)$, and $\mathcal{X}(0)=0$, then we have,
with $\widetilde{\varphi}(t,x)=\varphi(t,x)-x$,
\begin{align*}
& \@ifstar\@opnorms\@opnorm{ \partial\widetilde{\varphi}(0) }_{m-1}, \|\partial\varphi\|_{L^\infty(\Omega_{T_1})}
\leq C( \@ifstar\@opnorms\@opnorm{ u(0) }_m ), \\
& \|\widetilde{\varphi}\|_{{\mathbb W}^m(T_1)}, \|\partial_t\varphi\|_{{\mathbb W}^m(T_1)},
|(\partial^m\varphi)_{\vert_{x=0}}|_{L^\infty(0,T_1)}
\leq C\bigl( \|u\|_{{\mathbb W}^m(T_1)}, |u_{\vert_{x=0}}|_{m,T_1} \bigr).
\end{align*}
\end{enumerate}
\end{lemma}
We can now state the main result of this section, which holds under the following assumption.
\begin{assumption}\label{asshypQLFB}
Let $\mathcal{U}$ be an open set in ${\mathbb R}^2$, which represents a phase space of $u$.
The following conditions hold.
\begin{enumerate}
\setlength{\itemsep}{3pt}
\item[{\bf i.}]
$A,\mathcal{X} \in C^\infty(\mathcal{U})$, $\mathcal{X}(0)=0$.
\item[{\bf ii.}]
For any $u\in{\mathcal U}$, the matrix $A(u)$ has eigenvalues $\lambda_+(u)$ and $-\lambda_-(u)$ satisfying
\[
\lambda_\pm(u) > 0 \quad\mbox{and}\quad \lambda_\pm(u)\mp {\mathcal X}(u) > 0.
\]
\item[{\bf iii.}]
Denoting by ${\bf e}_+ (u)$ a unit eigenvector associated to the eigenvalue $\lambda_+(u)$ of $A(u)$,
for any $u\in\mathcal{U}$ we have
\[
|\underline{\nu}\cdot {\bf e}_+(u)| > 0.
\]
\end{enumerate}
\end{assumption}
\begin{theorem}\label{theoIBVP4}
Let $m\geq 2$ be an integer.
Suppose that Assumption \ref{asshypQLFB} is satisfied.
If $u^{\rm in}\in H^m({\mathbb R}_+)$ takes its values in a compact and convex set ${\mathcal K}_0\subset {\mathcal U}$ and
if the data $u^{\rm in}$ and $g \in H^m(0,T)$ satisfy the compatibility conditions up to order $m-1$
in the sense of Definition \ref{defcompfbp} below, then there exist $T_1 \in (0,T]$ and a unique solution
$(u,\underline{x})$ to \eqref{IBVPfbbis}--\eqref{eqFBbis} with $u\in {\mathbb W}^m(T_1)$, $\underline{x}\in H^{m+1}(0,T_1)$, and
$\varphi$ given by Lemma \ref{lemdiffeo}.
\end{theorem}
\subsubsection{Compatibility conditions}
For the free boundary problem, $\underline{x}(t)$ and $\varphi(t,x)$ are unknowns so that the interior equation
$\partial_t u + \mathcal{A}(u,\partial\varphi)\partial_x u = 0$ does not determine $(\partial_t^ku)_{\vert_{x=0}}$ directly in terms
of the initial data $u^{\rm in}$ and its derivatives.
In order to determine them, we need to use \eqref{diffeo}, or equivalently,
the evolution equation $\partial_t\varphi=\mathcal{X}(u)$ at the same time.
Suppose that $u$ is a smooth solution to \eqref{IBVPfbbis}--\eqref{eqFBbis}.
We note that the interior equation in \eqref{IBVPfbbis} can be written as
\[
\partial_t^\varphi u + A(u)\partial_x^\varphi u =0
\]
and that $\partial_t^\varphi$ and $\partial_x^\varphi$ commute.
Therefore, denoting $u_{(k)} = (\partial_t^\varphi)^k u$ and using the above equation inductively,
we have
\[
u_{(k)} = c_{1,k}(u,\partial_x^\varphi u,\ldots,(\partial_x^\varphi)^ku),
\]
where $c_{1,k}$ is a smooth function of its arguments.
In view of this, we define $u_{(k)}^{\rm in}$ by
\begin{equation}\label{ukin}
u_{(k)}^{\rm in} = c_{1,k}(u^{\rm in},\partial_x u^{\rm in},\ldots,\partial_x^k u^{\rm in})
\end{equation}
for $k=1,2,\ldots$.
Using the relation $\partial_t = \partial_t^\varphi+(\partial_t\varphi)\partial_x^\varphi$ inductively,
we see that
\[
\partial_t^k = (\partial_t^\varphi)^k + (\partial_t^k\varphi)\partial_x^\varphi
+ \sum_{l=2}^k\sum_{\substack{j_0+j_1+\cdots+j_l=k \\ 1\leq j_1,\ldots,j_l}}
c_{l,j_0,\ldots,j_l}(\partial_t^{j_1}\varphi)\cdots(\partial_t^{j_l}\varphi)(\partial_t^\varphi)^{j_0}(\partial_x^\varphi)^l,
\]
so that denoting $u_{k}=\partial_t^k u$ and $\varphi_k=\partial_t^k\varphi$ we have
\[
u_{k} = u_{(k)} + \varphi_k\partial_x^\varphi u
+ \sum_{l=2}^k\sum_{\substack{j_0+j_1+\cdots+j_l=k \\ 1\leq j_1,\ldots,j_l}}
c_{l,j_0,\ldots,j_l}\varphi_{j_1}\cdots\varphi_{j_l}(\partial_x^\varphi)^lu_{(j_0)}.
\]
Particularly, denoting $u_{k}^{\rm in}=(\partial_t^k u)_{\vert_{t=0}}$ and
$\varphi_k^{\rm in}=(\partial_t^k \varphi)_{\vert_{t=0}}$ we obtain
\begin{equation}\label{dtku}
u_{k}^{\rm in} = u_{(k)}^{\rm in} + \varphi_k^{\rm in}(\partial_x u^{\rm in})
+ \sum_{l=2}^k\sum_{\substack{j_0+j_1+\cdots+j_l=k \\ 1\leq j_1,\ldots,j_l}}
c_{l,j_0,\ldots,j_l}\varphi_{j_1}^{\rm in}\cdots\varphi_{j_l}^{\rm in}\partial_x^l u_{(j_0)}^{\rm in}.
\end{equation}
This implies that $u_{k}^{\rm in}$ is written in terms of $\varphi_j^{\rm in}$ and
$\partial_x^j u^{\rm in}$ for $0\leq j\leq k$.
On the other hand, differentiating the evolution equation $\partial_t\varphi = \mathcal{X}(u)$
$k$-times with respect to $t$, we have
\[
\varphi_{k+1} = c_{2,k}(u,\partial_t u,\ldots,\partial_t^k u),
\]
where $c_{2,k}$ is a smooth function of its arguments.
Therefore, we get
\begin{equation}\label{dtkx}
\varphi_{k+1}^{\rm in} = c_{2,k}({u^{\rm in}},u_{1}^{\rm in},\ldots,u_{k}^{\rm in}).
\end{equation}
Using \eqref{dtku} and \eqref{dtkx} alternatively we can determine $u_{k}^{\rm in}$ and $\varphi_k^{\rm in}$.
Now, the boundary condition $\underline{\nu}\cdot u_{\vert_{x=0}}=g$ implies that
\[
\underline{\nu} \cdot \partial_t^k u_{\vert_{x=0}} = \partial_t^k g.
\]
On the edge $\{t=0,x=0\}$, smooth enough solutions must therefore satisfy
\begin{equation}\label{compfbp}
\underline{\nu}\cdot {u_{k}^{\rm in}}_{\vert_{x=0}} = (\partial_t^k g)_{\vert_{t=0}}.
\end{equation}
\begin{definition}\label{defcompfbp}
Let $m\geq1$ be an integer. We say that the data $u^{\rm in}\in H^m({\mathbb R}_+)$ and $g \in H^m(0,T)$ for the
initial boundary value problem \eqref{IBVPfbbis}--\eqref{eqFBbis} satisfy the compatibility condition at order $k$
if the $\{u_{j}^{\rm in}\}_{j=0}^m$ defined by \eqref{ukin}--\eqref{dtkx} satisfy \eqref{compfbp}.
We also say that the data satisfy the compatibility conditions up to order $m-1$ if they satisfy the
compatibility conditions at order $k$ for $k=0,1,\ldots,m-1$.
\end{definition}
\begin{remark}
These compatibility conditions do not depend on the particular choice of the diffeomorphism $\varphi$
such as \eqref{diffeo}.
The other choice of the diffeomorphism $\varphi : {\mathbb R}_+ \to (\underline{x}(t),\infty)$ will give the same conditions.
\end{remark}
\subsubsection{Proof of Theorem \ref{theoIBVP4}}
Let $\mathcal{K}_1$ be a compact and convex set in ${\mathbb R}^2$ satisfying
$\mathcal{K}_0 \Subset \mathcal{K}_1 \Subset \mathcal{U}$.
Then, there exists a constant $c_0 > 0$ such that for any $u \in \mathcal{K}_1$ we have
\begin{align*}
\lambda_{\pm}(u) \geq c_0, \qquad \lambda_\pm(u)\mp {\mathcal X}(u) \geq c_0,
\qquad |\underline{\nu} \cdot \mathbf{e}_{+}(u)| \geq c_0.
\end{align*}
We will construct the solution $u$ with values in $\mathcal{K}_1$.
Note that there exists a constant $\delta_0>0$ such that
$\|u-u^{\rm in}\|_{L^\infty} \leq \delta_0$ implies $u(x) \in \mathcal{K}_1$ for all $x\in{\mathbb R}_+$.
Therefore, it is sufficient to construct the solution $u$ satisfying
$\|u(t) - u^{\rm in}\|_{L^\infty} \leq \delta_0$ for $0\leq t\leq T_1$.
The solution is classically constructed using the iterative scheme
\begin{equation}\label{eqFB_n}
\varphi^n(t,x) = x + \int_0^t {\mathcal X}(u^n(t',x)){\rm d}t'
\end{equation}
and
\begin{equation}\label{IBVPfbbis_n}
\begin{cases}
\partial_t u^{n+1} + \mathcal{A}(u^n,\partial\varphi^n)\partial_x u^{n+1} = 0 & \mbox{in}\quad \Omega_T, \\
{u^{n+1}}_{\vert_{t=0}} = u^{\rm in}(x) & \mbox{on}\quad {\mathbb R}_+, \\
\underline{\nu}\cdot {u^{n+1}}_{\vert_{x=0}} = g(t) & \mbox{on}\quad (0,T)
\end{cases}
\end{equation}
for all $n\in{\mathbb N}$.
For the first iterate $u^0$, we choose a function $u^0 \in H^{m+1/2}({\mathbb R}\times{\mathbb R}_+)$ such that
$(\partial_t^k u^0)_{\vert_{t=0}}=u_{k}^{\rm in}$ for $0\leq k\leq m$ with $u_{k}^{\rm in}$ defined by
\eqref{ukin}--\eqref{dtkx}.
Then, for the initial boundary value problem \eqref{IBVPfbbis_n} to the unknowns $u^{n+1}$
the data $(u^{\rm in},g)$ satisfy the compatibility conditions up to order $m-1$ in the sense of
Definition \ref{defcompVC}.
Moreover, $\@ifstar\@opnorms\@opnorm{ u^n(0) }_m$ is independent of $n$, and there exists therefore $K_0$ such that
\[
\frac{1}{c_0}, \@ifstar\@opnorms\@opnorm{ u^n(0) }_m, \@ifstar\@opnorms\@opnorm{ \partial\widetilde{\varphi}(0) }_{m-1},
\|\partial\varphi^n\|_{L^\infty(\Omega_{T_1})},|\underline{\nu}|,\|A\|_{L^\infty(\mathcal{K}_1)} \leq K_0,
\]
as long as $\|u^n\|_{W^{1,\infty}(\Omega_T)} \leq K$ and $T_1\in (0,T]$ sufficiently small depending on $K$.
We prove now that for $M$ large enough and $T_1$ small enough, for any $n \in {\mathbb N}$ we have
\[
\begin{cases}
\|u^n\|_{{\mathbb W}^m(T_1)} + |{u^n}_{\vert_{x=0}}|_{m,T_1} \leq M, \\
\|u^n(t)-u^{\rm in}\|_{L^\infty} \leq \delta_0 \quad\mbox{for}\quad 0 \leq t\leq T_1.
\end{cases}
\]
We prove this assertion by induction. Since it is satisfied for $n=0$ for a suitable $M$ and $T_1$,
we just need to prove that if holds at rank $n+1$ if it holds at rank $n$.
By the Sobolev imbedding theorem and Lemma \ref{lemdiffeo}, we have
\[
\|u^n\|_{W^{1,\infty}(\Omega_{T_1})}, \|\widetilde{\varphi}^n\|_{{\mathbb W}^m(T_1)}, \|\partial_t\varphi^n\|_{{\mathbb W}^m(T_1)},
|(\partial^m\varphi^n)_{\vert_{x=0}}|_{L^\infty(0,T_1)} \leq K(M).
\]
It follows therefore from Theorem \ref{theoIBVP3} that
\[
\| u^{n+1}(t) \|_{{\mathbb W}^m(T_1)} + |{u^{n+1}}_{\vert_{x=0}}|_{m,T_1} \leq
C(K_0)e^{C(M)t}( 1 + |g|_{H^m(0,T_1)} ).
\]
Choosing $M = 2C(K_0)( 1 + |g|_{H^m(0,T)} )$, it is possible to choose $T_1$ small enough to get that the
right-hand side is smaller than $M$.
We also have $\|u^{n+1}(t)-u^{\rm in}\|_{L^\infty} \leq C\|u^{n+1}\|_{{\mathbb W}^2(T_1)}T_1 \leq \delta_0$ for
$0\leq t\leq T_1$.
Therefore, the claim is proved.
We proceed to show that the sequence of approximate solutions $\{(u^n,\varphi^n)\}_n$ converges to the solution
$(u,\varphi)$ to \eqref{IBVPfbbis}--\eqref{eqFBbis} satisfying $u \in {\mathbb W}^m(T_1)$ and
$\underline{x}=\varphi_{\vert_{x=0}} \in H^{m+1}(0,T_1)$.
We have
\[
\begin{cases}
\partial_t (u^{n+2}-u^{n+1}) + \mathcal{A}(u^n,\partial\varphi^n) \partial_x(u^{n+2}-u^{n+1}) = f^n & \mbox{in}\quad \Omega_T, \\
(u^{n+2}-u^{n+1})_{\vert_{t=0}} = 0 & \mbox{on}\quad {\mathbb R}_+, \\
\underline{\nu} \cdot (u^{n+2}-u^{n+1})_{\vert_{x=0}} = 0 & \mbox{on}\quad (0,T)
\end{cases}
\]
with
\[
f^n = -(\mathcal{A}(u^{n+1},\partial\varphi^{n+1}) - \mathcal{A}(u^n,\partial\varphi^n)) \partial_x u^{n+1}.
\]
It follows therefore from \eqref{eqmm1.5} in Proposition \ref{propAl} that
\begin{align*}
& \@ifstar\@opnorms\@opnorm{ (u^{n+2}-u^{n+1})(t) }_{m-1} + |(u^{n+2}-u^{n+1})_{\vert_{x=0}}|_{m-1,t} \\
&\leq C(M)\Bigl( |{f^n}_{\vert_{x=0}}|_{m-2,t} + \int_0^t \@ifstar\@opnorms\@opnorm{ \partial_t f^n(t') }_{m-2}{\rm d}t' \Bigr) \\
&\leq C(M)\int_0^t ( \@ifstar\@opnorms\@opnorm{ \partial_t f^n(t') }_{m-2} + |(\partial_t f^n)_{\vert_{x=0}}|_{m-2,t'} ){\rm d}t'
\end{align*}
for $0\leq t\leq T_1$, where we used Lemma \ref{ineq4} and the fact that
$(\partial_t^k u^n)_{\vert_{t=0}} = u_{k}^{\rm in}$ does not depend on $n$.
Here, we see that
\begin{align*}
\|\partial_t f^n\|_{{\mathbb W}^{m-2}(T_1)}
&\leq C(M) \|(u^{n+1}-u^{n}, \varphi^{n+1}-\varphi^{n}, \partial_t(\varphi^{n+1}-\varphi^{n}))\|_{{\mathbb W}^{m-1}(T_1)} \\
&\leq C(M) \|u^{n+1}-u^{n}\|_{{\mathbb W}^{m-1}(T_1)}
\end{align*}
and that
\begin{align*}
|(\partial_t f^n)_{\vert_{x=0}}|_{m-2,T_1}
&\leq C(M)\bigl( \|(u^{n+1}-u^{n}, \varphi^{n+1}-\varphi^{n}, \partial_t(\varphi^{n+1}-\varphi^{n}))\|_{{\mathbb W}^{m-1}(T_1)} \\
&\quad
+ |(u^{n+1}-u^{n}, \varphi^{n+1}-\varphi^{n}, \partial_t(\varphi^{n+1}-\varphi^{n}))_{\vert_{x=0}}|_{m-1,T_1} \bigr) \\
&\leq C(M)\bigl( \|u^{n+1}-u^{n}\|_{{\mathbb W}^{m-1}(T_1)} + |(u^{n+1}-u^{n})_{\vert_{x=0}}|_{m-1,T_1} \bigr),
\end{align*}
where we used Lemma \ref{ineq3}.
Note that in the above inequalities, the quantity $ \partial_t(\varphi^{n+1}-\varphi^{n})$ has been controled in
${\mathbb W}^{m-1}(T_1)$; a similar control of $ \partial_x(\varphi^{n+1}-\varphi^{n})$ is not possible and
this is the reason why it is important to have $\@ifstar\@opnorms\@opnorm{\partial_t f(t)}_{m-2}$ rather than $\@ifstar\@opnorms\@opnorm{f(t)}_{m-1}$
in the right-hand side of \eqref{eqmm1.5} in Proposition \ref{propAl}.
Therefore, by taking $T_1$ sufficiently small if necessary, we obtain
\begin{align*}
& \|u^{n+2}-u^{n+1}\|_{{\mathbb W}^{m-1}(T_1)} + |(u^{n+2}-u^{n+1})_{\vert_{x=0}}|_{m-1,T_1} \\
&\leq \frac12 \bigl( \|u^{n+1}-u^{n}\|_{{\mathbb W}^{m-1}(T_1)} + |(u^{n+1}-u^{n})_{\vert_{x=0}}|_{m-1,T_1} \bigr).
\end{align*}
This together with an interpolation inequality
$\|u\|_{W^{1,\infty}(\Omega_{T_1})}^2 \leq C\|u\|_{{\mathbb W}^{m-1}(T_1)}\|u\|_{{\mathbb W}^{m}(T_1)}$
shows that $\{(u^n,\widetilde{\varphi}^n)\}_n$ converges to $(u,\widetilde{\varphi})$ in
${\mathbb W}^{m-1}(T_1) \cap W^{1,\infty}(\Omega_{T_1})$, so that $(u,\widetilde{\varphi})$ is a solution to
\eqref{IBVPfbbis}--\eqref{eqFBbis}.
Moreover, by standard compactness arguments we see that
\[
\|u\|_{{\mathbb W}^m(T_1)} + |u_{\vert_{x=0}}|_{m,T_1} \leq M.
\]
The regularity and the uniqueness of the solution stated in the theorem is obtained by standard
arguments so we omit them.
The proof of Theorem \ref{theoIBVP4} is complete.
\subsection{Application to free boundary problems with a fully nonlinear boundary equation}
\label{sectVCm2}
We now consider a $2\times2$ quasilinear hyperbolic system on a moving domain $(\underline{x}(t),\infty)$:
\begin{equation}\label{IBVPfb2}
\partial_t U+A(U)\partial_x U =0 \quad\mbox{in}\quad (\underline{x}(t),\infty)
\end{equation}
with a fully nonlinear boundary condition
\begin{equation}\label{fbBC}
U = U_{\rm i} \quad\mbox{on}\quad x=\underline{x}(t),
\end{equation}
where $U_{\rm i} = U_{\rm i}(t,x)$ is a given ${\mathbb R}^2$-valued function, whereas $\underline{x}(t)$ is unknown function.
Compared to the free boundary problem \eqref{IBVPfb}--\eqref{eqFB},
the evolution equation of the boundary is implicitly contained in the above boundary condition.
In fact, differentiating the boundary condition $U(t,\underline{x}(t)) = U_{\rm i}(t,\underline{x}(t))$ with respect to $t$
and taking the Euclidean inner product of the resulting equation with $\partial_x U - \partial_x U_{\rm i}$,
we obtain
\begin{equation}\label{eqFB2}
\dot{\underline{x}} = \chi( (\partial U)_{\vert_{x=\underline{x}}},(\partial U_{\rm i})_{\vert_{x=\underline{x}}} ),
\end{equation}
where
\[
\chi( \partial U, \partial U_{\rm i} )
= - \frac{( \partial_x U - \partial_x U_{\rm i} )\cdot( \partial_t U - \partial_t U_{\rm i} ) }{ |\partial_x U - \partial_x U_{\rm i}|^2 }.
\]
In view of this, a discontinuity of the spatial derivative $\partial_x U$ on the free boundary is crucial
to the free boundary problem \eqref{IBVPfb2}--\eqref{fbBC} whereas $U$ itself is continuous.
Compared to the boundary equation \eqref{eqFB} of kinematic type,
\eqref{eqFB2} does not depend on $U$ itself but on its derivative $\partial U$.
Therefore, \eqref{IBVPfb2}--\eqref{eqFB2} is more difficult than \eqref{IBVPfb}--\eqref{eqFB} in
the previous subsection.
We will use again a diffeomorphism $\varphi(t,\cdot): {\mathbb R}_+ \to (\underline{x}(t),\infty)$ and put
$u = U\circ\varphi$ and $u_{\rm i} = U_{\rm i}\circ\varphi$.
Then, the free boundary problem \eqref{IBVPfb2}--\eqref{fbBC} is recast as a problem on the
fixed domain:
\begin{equation}\label{IBVPfbbis2}
\begin{cases}
\partial_t^\varphi u + A(u)\partial_x^\varphi u =0 & \mbox{in}\quad \Omega_T, \\
u_{\vert_{x=0}} = {u_{\rm i}}_{\vert_{x=0}} & \mbox{on}\quad (0,T).
\end{cases}
\end{equation}
We impose the initial conditions of the form
\begin{equation}\label{IC}
u_{\vert_{t=0}} = u^{\rm in}(x) \quad\mbox{on}\quad {\mathbb R}_+, \qquad \underline{x}(0)=0.
\end{equation}
We also note that the equation \eqref{eqFB2} for the free boundary is then reduced to
\begin{equation}\label{eqFB3}
\dot{\underline{x}} = \chi( (\partial^\varphi u)_{\vert_{x=0}},(\partial^\varphi u_{\rm i})_{\vert_{x=0}} ).
\end{equation}
\begin{assumption}\label{asshypNLFB}
Let ${\mathcal U}$ be an open set in ${\mathbb R}^2$, which represents a phase space of $u$.
\begin{enumerate}
\item[{\bf i.}]
$A \in C^\infty({\mathcal U})$.
\item[{\bf ii.}]
There exists $c_0>0$ such that for any $u \in {\mathcal U}$, the matrix $A(u)$ has eigenvalues
$\lambda_+(u)$ and $-\lambda_-(u)$ satisfying $\lambda_{\pm}(u) \geq c_0$.
\end{enumerate}
\end{assumption}
As before, this condition ensures that the system is strictly hyperbolic.
We denote by ${\bf e}_{\pm}(u)$ normalized eigenvectors associated to the eigenvalues
$\pm\lambda_{\pm}(u)$ of $A(u)$.
They are uniquely determined up to a sign.
Since both eigenvalues are simple, we have $\lambda_{\pm}, {\bf e}_{\pm} \in C^\infty({\mathcal U})$
under an appropriate choice of the sign of ${\bf e}_{\pm}$.
As mentioned above, a discontinuity of $\partial_x U$ at the free boundary is crucial so that
we will work in a class of solutions satisfying
\begin{equation}\label{disconti}
|(\partial_x^\varphi u - \partial_x^\varphi u_i)_{\vert_{x=0}}| \geq c_0
\end{equation}
for some positive constant $c_0$.
The interior equation in \eqref{IBVPfbbis2} can be written as
\[
\partial_t u + \mathcal{A}(u,\partial\varphi) \partial_x u = 0,
\]
where $\mathcal{A}(u,\partial\varphi) = (\partial_x\varphi)^{-1}(A(u) - (\partial_t\varphi){\rm Id})$.
The eigenvalues of this matrix are $(\partial_x\varphi)^{-1}(\pm\lambda_{\pm}(u) - \partial_t\varphi)$, whereas the
corresponding eigenvectors are ${\bf e}_{\pm}(u)$ which does not depend on $\partial\varphi$.
In view of {\bf i} in Assumption 1, we also restrict a class of solution by
\begin{equation}\label{egenv}
\lambda_{\pm}(u) \mp \partial_t\varphi \geq c_0 \quad\mbox{in}\quad (0,T)\times{\mathbb R}_+.
\end{equation}
We note that the boundary equation \eqref{eqFB3} is not of the kinematic type considered in \S \ref{sectFB1}
so that we need to use a different diffeomorphism from the one given by Lemma \ref{lemdiffeo}.
Let $\psi \in C_0^\infty({\mathbb R})$ be a cut-off function such that
$\psi(x)=1$ for $|x| \leq 1$ and $=0$ for $|x| \geq 2$.
We define the diffeomorphism by
\begin{equation}\label{diffeo2}
\varphi(t,x) = x + \psi\Bigl(\frac{x}{\varepsilon}\Bigr)\underline{x}(t),
\end{equation}
where $\varepsilon>0$ is a small parameter which will be determined later.
As we will see below, under this choice of the diffeomorphism, \eqref{egenv} would be satisfied if
the solution satisfies
\begin{equation}\label{egenv2}
\lambda_{\pm}(u_{\vert_{x=0}}) \mp \dot{\underline{x}} \geq 2c_0 \quad\mbox{on}\quad (0,T).
\end{equation}
The following lemma shows that this choice of diffeomorphism behaves differently than the Lagrangian
diffeomorphism studied in Lemma \ref{lemdiffeo}; in particular, the latter has a better time regularity,
while the former has a better space regularity.
\begin{lemma}\label{lemdiffeo2}
Suppose that $\underline{x} \in C^1([0,T])$ satisfies $\underline{x}(0)=0$ and $|\dot{\underline{x}}|_{L^2(0,T)} \leq K$.
Then, there exists $T_1 \in (0,T]$ depending on $\varepsilon$ and $K$ such that the mapping
$\varphi:\overline{\Omega_T}\to {\mathbb R}$ defined by \eqref{diffeo2} satisfies the following properties:
\begin{enumerate}
\item[{\bf i.}]
We have $\varphi(t,0) = \underline{x}(t)$ and $\varphi(0,x)=x$ and for all $0\leq t \leq T_1$,
$\varphi(t,\cdot)$ is a diffeomorphism mapping ${\mathbb R}_+$ onto $(\underline{x}(t),\infty)$ and satisfying
$\frac12 \leq \partial_x\varphi(t,x) \leq 2$.
\item[{\bf ii.}]
For any nonnegative integers $k$ and $l$, we have
\[
\|\partial_t^l\partial_x^k\widetilde{\varphi}(t)\|_{L^1\cap L^\infty({\mathbb R}_+)}
\leq C(\varepsilon,k)|\partial_t^l\underline{x}(t)|,
\]
where $\widetilde{\varphi}(t,x)=\varphi(t,x)-x$.
Particularly, if moreover $m\geq2$ and $\underline{x} \in H^m(0,T_1)$, then we have
\begin{align*}
& \@ifstar\@opnorms\@opnorm{ \partial\widetilde{\varphi}(0) }_{m-2}, \|\partial\varphi\|_{L^\infty(\Omega_{T_1})}
\leq C(\varepsilon)\biggl( \sum_{j=0}^{m-1}|(\partial_t^j\underline{x})_{\vert_{t=0}}|
+ \sqrt{T_1}|\dot{\underline{x}}|_{H^2(0,T_1)} \biggr), \\
& \|\widetilde{\varphi}\|_{{\mathbb W}^{m-1}(T_1)}, \|\partial_t\varphi\|_{{\mathbb W}^{m-1}(T_1)},
|(\partial^{m-1}\varphi)_{\vert_{x=0}}|_{L^\infty(0,T_1)}
\leq C(\varepsilon)|\underline{x}|_{W^{m-1,\infty} \cap H^m(0,T_1)}.
\end{align*}
\end{enumerate}
\end{lemma}
\begin{theorem}\label{theoIBVP5}
Let $m\geq 2$ be an integer.
Suppose that Assumption \ref{asshypNLFB} is satisfied.
If $u^{\rm in}\in H^m({\mathbb R}_+)$ takes its values in a compact and convex set ${\mathcal K}_0\subset {\mathcal U}$ and
if the data $u^{\rm in}$ and $U_{\rm i} \in W^{m,\infty}((0,T)\times(-\delta,\delta))$ satisfy
\begin{enumerate}
\item[{\bf i.}]
$\lambda_{\pm}({u^{\rm in}}_{\vert_{x=0}}) \mp \underline{x}_1^{\rm in} >0$,
\item[{\bf ii.}]
$(\partial_x u^{\rm in})_{\vert_{x=0}} - (\partial_x U_{\rm i})_{\vert_{t=x=0}} \ne0$,
\item[{\bf iii.}]
$((\partial_x u^{\rm in})_{\vert_{x=0}} - (\partial_x U_{\rm i})_{\vert_{t=x=0}})^\perp \cdot
\mathbf{e}_{+}({u^{\rm in}}_{\vert_{x=0}}) \ne0$,
\end{enumerate}
where $\underline{x}_1^{\rm in}=(\partial_t\underline{x})_{\vert_{t=0}}$ will be determined by \eqref{xkin} below,
and the compatibility conditions up to order $m-1$ in the sense of Definition \ref{defCC} below,
then there exist $T_1 \in (0,T]$ and a unique solution
$(u,\underline{x})$ to \eqref{IBVPfbbis2}--\eqref{IC} with
$u, \partial_x u \in {\mathbb W}^{m-1}(T_1)$, $\underline{x}\in H^m(0,T_1)$, and $\varphi$ given by Lemma \ref{lemdiffeo2}.
\end{theorem}
\begin{remark}\label{remarkIC}
Thanks to Proposition \ref{conalg} below, the condition {\bf iii} in the theorem can be replaced by
\begin{enumerate}
\item[{\bf iii$'$.}]
$\mu_0 \cdot \mathbf{e}_{+}({u^{\rm in}}_{\vert_{x=0}}) \ne0$,
\end{enumerate}
where $\mu_0$ is the unit vector satisfying
$\mu_0 \cdot (\partial_t U_{\rm i} + A(U_{\rm i})\partial_x U_{\rm i})_{\vert_{t=x=0}} =0$.
This unit vector $\mu_0$ is uniquely determined up to the sign under the other assumptions
of the theorem.
\end{remark}
\subsubsection{Compatibility conditions}\label{sscomp}
Suppose that $u$ is a smooth solution to \eqref{IBVPfbbis2}--\eqref{IC}.
We note that $\partial_t^\varphi$ and $\partial_x^\varphi$ commute.
Denoting $u_{(k)} = (\partial_t^\varphi)^ku$ and using the interior equation in \eqref{IBVPfbbis2} inductively,
we have
\[
u_{(k)} = c_{1,k}(u,\partial_x^\varphi u,\ldots,(\partial_x^\varphi)^ku),
\]
where $c_{1,k}$ is a smooth function of its arguments.
In view of this, we define $u_{(k)}^{\rm in}$ by
\begin{equation}\label{ukin2}
u_{(k)}^{\rm in} = c_{1,k}(u^{\rm in},\partial_x u^{\rm in},\ldots,\partial_x^k u^{\rm in})
\end{equation}
for $k=1,2,\ldots$.
We proceed to express $(\partial_t^k\underline{x})_{\vert_{t=0}}$ in terms of the initial data.
Differentiating the boundary condition in \eqref{IBVPfbbis2} with respect to $t$, we have
$\partial_t^ku=\partial_t^ku_{\rm i}$ on $x=0$.
Using the relation $\partial_t = \partial_t^\varphi+(\partial_t\varphi)\partial_x^\varphi$ inductively,
we see that
\[
\partial_t^k = (\partial_t^\varphi)^k + (\partial_t^k\varphi)\partial_x^\varphi
+ \sum_{l=2}^k\sum_{\substack{j_0+j_1+\cdots+j_l=k \\ 1\leq j_1,\ldots,j_l}}
c_{l,j_0,\ldots,j_l}(\partial_t^{j_1}\varphi)\cdots(\partial_t^{j_l}\varphi)(\partial_t^\varphi)^{j_0}(\partial_x^\varphi)^l,
\]
so that denoting $\underline{x}_{k} = \partial_t^k\underline{x}$ we have
\begin{align*}
& u_{(k)} - (\partial_t^\varphi)^ku_{\rm i} + \underline{x}_{k}(\partial_x^\varphi u - \partial_x^\varphi u_{\rm i}) \\
& + \sum_{l=2}^k\sum_{\substack{j_0+j_1+\cdots+j_l=k \\ 1\leq j_1,\ldots,j_l}}
c_{l,j_0,\ldots,j_l}\underline{x}_{(j_1)}\cdots\underline{x}_{j_l} (\partial_x^\varphi)^l(u_{(j_0)} - (\partial_t^\varphi)^{j_0}u_{\rm i})
= 0 \quad\mbox{on}\quad x=0.
\end{align*}
Decomposing this relation into the direction $\partial_x^\varphi u - \partial_x^\varphi u_{\rm i}$ and
its perpendicular direction, we obtain
\begin{align*}
\underline{x}_{k}
&= -\frac{\partial_x^\varphi u - \partial_x^\varphi u_{\rm i}}{|\partial_x^\varphi u - \partial_x^\varphi u_{\rm i}|^2}
\cdot \biggl\{ u_{(k)} - (\partial_t^\varphi)^ku_{\rm i} \\
&\qquad
+ \sum_{l=2}^k\sum_{\substack{j_0+j_1+\cdots+j_l=k \\ 1\leq j_1,\ldots,j_l}}
c_{l,j_0,\ldots,j_l}\underline{x}_{j_1}\cdots\underline{x}_{j_l} (\partial_x^\varphi)^l(u_{(j_0)} - (\partial_t^\varphi)^{j_0}u_{\rm i})
\biggr\}_{\vert_{x=0}}
\end{align*}
and
\begin{align*}
& (\partial_x^\varphi u - \partial_x^\varphi u_{\rm i})^\perp \cdot \biggl\{
u_{(k)} - (\partial_t^\varphi)^ku_{\rm i} \\
& + \sum_{l=2}^k\sum_{\substack{j_0+j_1+\cdots+j_l=k \\ 1\leq j_1,\ldots,j_l}}
c_{l,j_0,\ldots,j_l}\underline{x}_{j_1}\cdots\underline{x}_{j_l} (\partial_x^\varphi)^l(u_{(j_0)} - (\partial_t^\varphi)^{j_0}u_{\rm i})
\biggr\}_{\vert_{x=0}}
= 0,
\end{align*}
respectively.
In view of this, we define $\underline{x}_{k}^{\rm in}$ inductively by $\underline{x}_{0}^{\rm in} = 0$ and
\begin{align}\label{xkin}
\underline{x}_{k}^{\rm in}
&= -\frac{\partial_x u^{\rm in} - (\partial_x U_{\rm i})_{\vert_{t=0}}}{|\partial_x u^{\rm in} - (\partial_x U_{\rm i})_{\vert_{t=0}}|^2}
\cdot \biggl\{ u_{(k)} ^{\rm in} - (\partial_t^k U_{\rm i})_{\vert_{t=0}} \\
&\qquad
+ \sum_{l=2}^k\sum_{\substack{j_0+j_1+\cdots+j_l=k \\ 1\leq j_1,\ldots,j_l}}
c_{l,j_0,\ldots,j_l}\underline{x}_{j_1}^{\rm in}\cdots\underline{x}_{j_l}^{\rm in}
\partial_x^l(u_{(j_0)}^{\rm in} - (\partial_t^{j_0}U_{\rm i})_{\vert_{t=0}}) \biggr\}_{\vert_{x=0}} \nonumber
\end{align}
for $k=1,2,\ldots$.
\begin{definition}\label{defCC}
Let $m\geq1$ be an integer.
We say that the data $u^{\rm in} \in H^m({\mathbb R}_+)$ and $U_{\rm i} \in W^{m,\infty}((0,T)\times(-\delta,\delta))$
for the initial boundary value problem
\eqref{IBVPfbbis2}--\eqref{IC} satisfy the compatibility condition at order $k$ if
$\{u_{(j)}^{\rm in}\}_{j=0}^m$ and $\{\underline{x}_{(j)}^{\rm in}\}_{j=0}^{m-1}$ defined by \eqref{ukin2}--\eqref{xkin}
satisfy ${u^{\rm in}}_{\vert_{x=0}} = U_{{\rm i} \, \vert_{t=x=0}}$ in the case $k=0$ and
\begin{align*}
& (\partial_x u^{\rm in} - (\partial_x U_{\rm i})_{\vert_{t=0}})^\perp \cdot \biggl\{
u_{(k)}^{\rm in} - (\partial_t^k U_{\rm i})_{\vert_{t=0}}\\
& + \sum_{l=2}^k\sum_{\substack{j_0+j_1+\cdots+j_l=k \\ 1\leq j_1,\ldots,j_l}}
c_{l,j_0,\ldots,j_l}\underline{x}_{(j_1)}^{\rm in}\cdots\underline{x}_{(j_l)}^{\rm in}
\partial_x^l(u_{(j_0)}^{\rm in} - (\partial_t^{j_0}U_{\rm i})_{\vert_{t=0}} )
\biggr\}_{\vert_{x=0}}
= 0
\end{align*}
in the case $k\geq 1$.
We say also that the data $u^{\rm in}$ and $U_{\rm i}$ for \eqref{IBVPfbbis2}--\eqref{IC}
satisfy the compatibility conditions up to order $m-1$ if they satisfy the compatibility
conditions at order $k$ for $k=0,1,\ldots,m-1$.
\end{definition}
Roughly speaking, the definition of $\underline{x}_{k}^{\rm in}$ ensures the equality
$\partial_t^k u =\partial_t^k u_{\rm i}$ at $x=t=0$ in the direction $\partial_x^\varphi u - \partial_x^\varphi u_i$,
whereas the compatibility conditions ensure it in the perpendicular direction
$(\partial_x^\varphi u - \partial_x^\varphi u_i)^\perp$.
We shall need to approximate $u^{\rm in}$ and $U_{\rm i}$ by more regular data which satisfy
higher order compatibility conditions.
Such an approximation is given by the following proposition.
\begin{proposition}\label{appdata}
Let $m$ and $s$ be integers satisfying $s>m\geq2$ and let $A \in C^\infty({\mathcal U})$.
If $u^{\rm in} \in H^m({\mathbb R}_{+})$ takes its value in ${\mathcal U}$ and if the data $u^{\rm in}$ and
$U_{\rm i} \in W^{m,\infty}((0,T)\times(-\delta,\delta))$ satisfy
\[
(\partial_x u^{\rm in})_{\vert_{x=0}} - (\partial_x U_{\rm i})_{\vert_{t=x=0}} \ne0
\]
and the compatibility conditions up to order $m-1$, then there exists
$\{(u^{{\rm in},(n)},U_{\rm i}^{(n)})\}_n$ a sequence of data such that
$(u^{{\rm in},(n)},U_{\rm i}^{(n)}) \in H^s({\mathbb R}_{+}) \times W^{s,\infty}((0,T)\times(-\delta,\delta))$
converges to $(u^{\rm in},U_{\rm i})$ in $H^m({\mathbb R}_{+})\times B^{m-1}([0,T]\times[-\delta,\delta])$ and
satisfies the compatibility conditions up to order $s-1$.
\end{proposition}
\begin{proof}
Once we fix $U_{\rm i}$, the compatibility condition at order $k$ is a nonlinear relation among
$(\partial_x^j u^{\rm in})_{\vert_{x=0}}$ for $j=0,1,\ldots,k$.
We need to know the explicit dependence of the highest order term $(\partial_x^k u^{\rm in})_{\vert_{x=0}}$
of the compatibility condition to show this proposition.
The compatibility conditions at order $0$ and $1$ are given by
$(u^{\rm in})_{\vert_{x=0}} = {U_{\rm i}}_{\vert_{t=x=0}}$ and
\[
((\partial_x u^{\rm in})_{\vert_{x=0}}-(\partial_x U_{\rm i})_{\vert_{t=x=0}})^\perp \cdot
(A({u^{\rm in}}_{\vert_{x=0}})(\partial_x u^{\rm in})_{\vert_{x=0}} + (\partial_t U_{\rm i})_{\vert_{t=x=0}}) = 0,
\]
respectively.
We proceed to consider the compatibility condition at order $k$ in the case $k\geq2$.
We will denote simply by LOT the terms containing $\partial_x^j u^{\rm in}$ for $j=0,1,\ldots,k-1$,
$U_{\rm i}$, and its derivatives only, and not containing $\partial_x^k u^{\rm in}$.
Then, we have
\[
u_{(k)}^{\rm in} = (-A(u^{\rm in}))^k\partial_x^k u^{\rm in} + \mbox{LOT}
\]
and $\underline{x}_j^{\rm in}=\mbox{LOT}$ for $0 \leq j\leq k-1$.
Denoting $u_k^{\rm in}=(\partial_t^k u)_{\vert_{t=0}}$ and using the relation
$\partial_t=\partial_t^\varphi+(\partial_t\varphi)\partial_x^\varphi$ inductively, we obtain
\begin{align*}
u_{k}^{\rm in}
&= \sum_{j=0}^k \binom{k}{j} ((\partial_t\varphi)_{t=0})^j\partial_x^ju_{(k-j)}^{\rm in}
+ (\partial_t^k\varphi)_{\vert_{t=0}}\partial_x u^{\rm in} + \mbox{LOT} \\
&= ((\partial_t\varphi)_{t=0}\mbox{Id}-A(u^{\rm in}))^k \partial_x^k u^{\rm in}
+ (\partial_t^k\varphi)_{\vert_{t=0}}\partial_x u^{\rm in} + \mbox{LOT},
\end{align*}
so that
\[
u_{k \vert_{x=0}}^{\rm in} = (\underline{x}_1^{\rm in}\mbox{Id}-A({u^{\rm in}}_{\vert_{x=0}}))^k (\partial_x^k u^{\rm in})_{\vert_{x=0}}
+ \underline{x}_k^{\rm in}(\partial_x u^{\rm in})_{\vert_{x=0}} + \mbox{LOT}.
\]
We also have
\[
(\partial_t^k u_{\rm i})_{\vert_{t=x=0}} = \underline{x}_k^{\rm in}(\partial_x U_{\rm i})_{\vert_{t=x=0}}+ \mbox{LOT}.
\]
Therefore, the compatibility condition at order $k$ is given by
\[
((\partial_x u^{\rm in})_{\vert_{x=0}}-(\partial_x U_{\rm i})_{\vert_{t=x=0}})^\perp \cdot
\{ (\underline{x}_1^{\rm in}\mbox{Id}-A({u^{\rm in}}_{\vert_{x=0}}))^k (\partial_x^k u^{\rm in})_{\vert_{x=0}}
+ \mbox{LOT} \} = 0.
\]
Once we obtain these expressions to the compatibility conditions,
the approximation stated in the proposition is obtained along classical lines.
See for instance \cite{RauchMassey}.
\end{proof}
\subsubsection{Reduction to a system with quasilinear boundary conditions}
At first glance the boundary condition in \eqref{IBVPfbbis2} is nothing but a nonhomogeneous
Dirichlet boundary condition.
However, $u_{\rm i}(t,0) = U_{\rm i}(t,\underline{x}(t))$ depends on the unknown free boundary $\underline{x}$, which would be determined
from the unknown $\partial^\varphi u$ through the evolution equation \eqref{eqFB3}.
Therefore, the boundary condition represents implicitly a nonlinear relation between $u$ and
its derivatives, so that we will reduce \eqref{IBVPfbbis2}
to a system with standard quasilinear boundary conditions to solve the initial value problem
\eqref{IBVPfbbis2}--\eqref{IC}.
Now, suppose that $u$ is a solution to \eqref{IBVPfbbis2}.
Putting
\begin{equation}\label{u2}
u_{(2)} = \partial_t^\varphi \partial_t^\varphi u,
\end{equation}
we will derive a system for $u$ and $u_{(2)}$ with quasilinear boundary conditions
together with a quasilinear evolution equation for $\underline{x}$.
We note that $\partial_t^\varphi$ and $\partial_x^\varphi$ commute.
Applying differential operators $\partial_t^\varphi$ and $\partial_x^\varphi$ to the first equation in \eqref{IBVPfbbis2},
we can express $\partial_t^\varphi \partial_x^\varphi u$ and $\partial_x^\varphi \partial_x^\varphi u$ in terms of $u_{(2)}$,
$u$, and $\partial^\varphi u$ as
\begin{equation}\label{d2u}
\begin{cases}
\partial_t^\varphi \partial_x^\varphi u
= (-A(u)^{-1}) ( u_{(2)} + A'(u)[\partial_t^\varphi u]\partial_x^\varphi u ), \\
\partial_x^\varphi \partial_x^\varphi u
= (-A(u)^{-1})^2 ( u_{(2)} + A'(u)[\partial_t^\varphi u]\partial_x^\varphi u )
+ (-A(u)^{-1})A'(u)[\partial_x^\varphi u]\partial_x^\varphi u.
\end{cases}
\end{equation}
Applying $\partial_t^\varphi \partial_t^\varphi$ to the first equation in \eqref{IBVPfbbis2} and using the above relations,
we obtain
\[
\partial_t^\varphi u_{(2)} + A(u)\partial_x^\varphi u_{(2)} + B(u,\partial^\varphi u) u_{(2)}
= f_{(2)}(u,\partial^\varphi u),
\]
where
\begin{align*}
& B(u,\partial^\varphi u) u_{(2)}
= A'(u)[u_{(2)}]\partial_x^\varphi u - 2A'(u)[\partial_t^\varphi u]A(u)^{-1}u_{(2)}, \\
& f_{(2)}(u,\partial^\varphi u)
= 2A'(u)[\partial_t^\varphi u]A(u)^{-1}A'(u)[\partial_t^\varphi u]\partial_x^\varphi u
- 2 A''(u)[\partial_t^\varphi u,\partial_t^\varphi u]\partial_x^\varphi u.
\end{align*}
This is an equation for $u_{(2)}$.
We proceed to derive a boundary condition for $u_{(2)}$ and an evolution equation for $\underline{x}$.
Differentiating the boundary condition $u=u_{\rm i}$ on $x=0$ with respect to $t$ twice and
using the relation $\partial_t = \partial_t^\varphi + (\partial_t\varphi)\partial_x^\varphi$, we have
\[
\partial_t^\varphi\partial_t^\varphi u + 2\dot{\underline{x}} \partial_t^\varphi\partial_x^\varphi u + \dot{\underline{x}}^2 \partial_x^\varphi\partial_x^\varphi u
+ \ddot{\underline{x}}\partial_x^\varphi u
= \partial_t^\varphi\partial_t^\varphi u_{\rm i} + 2\dot{\underline{x}} \partial_t^\varphi\partial_x^\varphi u_{\rm i}
+ \dot{\underline{x}}^2 \partial_x^\varphi\partial_x^\varphi u_{\rm i} + \ddot{\underline{x}}\partial_x^\varphi u_{\rm i}
\]
on $x=0$, where we used $\partial_t\varphi(t,0)=\dot{\underline{x}}(t)$.
This together with \eqref{d2u} implies
\[
({\rm Id} - \dot{\underline{x}}A(u)^{-1})^2 u_{(2)} + \ddot{\underline{x}}( \partial_x^\varphi u - \partial_x^\varphi u_{\rm i} )
= g_1(\dot{\underline{x}},u,\partial^\varphi u,\partial^\varphi\partial^\varphi u_{\rm i}),
\]
where
\begin{align*}
&g_1(\dot{\underline{x}},u,\partial^\varphi u,\partial^\varphi\partial^\varphi u_{\rm i}) \\
&= \bigl( 2\dot{\underline{x}}A(u)^{-1} - \dot{\underline{x}}^2(A(u)^{-1})^2 \bigr)A'(u)[\partial_t^\varphi u]\partial_x^\varphi u
+ \dot{\underline{x}}^2A(u)^{-1}A'(u)[\partial_x^\varphi u]\partial_x^\varphi u \\
&\quad\;
+ \partial_t^\varphi\partial_t^\varphi u_{\rm i} + 2\dot{\underline{x}}\partial_t^\varphi\partial_x^\varphi u_{\rm i}
+ \dot{\underline{x}}^2 \partial_x^\varphi\partial_x^\varphi u_{\rm i}.
\end{align*}
Decomposing this relation into the direction $\partial_x^\varphi u - \partial_x^\varphi u_{\rm i}$ and
its perpendicular direction, we obtain an evolution equation for $\underline{x}$ as
\[
\ddot{\underline{x}} = \chi(\dot{\underline{x}},u,u_{(2)},\partial^\varphi u,\partial^\varphi u_{\rm i},
\partial^\varphi\partial^\varphi u_{\rm i}),
\]
where
\begin{align*}
& \chi(\dot{\underline{x}},u,u_{(2)},\partial^\varphi u,\partial^\varphi u_{\rm i},\partial^\varphi\partial^\varphi u_{\rm i}) \\
&= \frac{(\partial_x^\varphi u - \partial_x^\varphi u_{\rm i}) \cdot
\bigl( g_1(\dot{\underline{x}},u,\partial^\varphi u,\partial^\varphi\partial^\varphi u_{\rm i})
- ({\rm Id} - \dot{\underline{x}}A(u)^{-1})^2 u_{(2)} \bigr)}{|\partial_x^\varphi u - \partial_x^\varphi u_{\rm i}|^2}
\end{align*}
and a boundary condition for $u_{(2)}$ as
\[
\nu_{(2)} \cdot u_{(2)}
= g_{(2)},
\]
where $\nu_{(2)}=\nu_{(2)}(\dot{\underline{x}},u,\partial_x^\varphi u,\partial_x^\varphi u_{\rm i})$ and
$g_{(2)} = g_{(2)}(\dot{\underline{x}},u,\partial^\varphi u,\partial^\varphi u_{\rm i},
\partial^\varphi\partial^\varphi u_{\rm i})$ are defined by
\begin{equation}\label{nu2g2}
\begin{cases}
\nu_{(2)} = (({\rm Id} - \dot{\underline{x}}A(u)^{-1})^2)^{\rm T}(
(\partial_x^\varphi u - \partial_x^\varphi u_{\rm i})^\perp), \\
g_{(2)} = (\partial_x^\varphi u - \partial_x^\varphi u_{\rm i})^\perp \cdot
g_1(\dot{\underline{x}},u,\partial^\varphi u,\partial^\varphi\partial^\varphi u_{\rm i}).
\end{cases}
\end{equation}
Concerning a boundary condition for $u$, we would like to write it in the form $\nu \cdot u = g$.
However, we have a high degree of freedom for choosing the vector $\nu$.
From the point of view of the maximal dissipativity in the sense of {\bf ii} in Assumption 1,
the most convenient choice is $\nu = \underline{\nu}$, where
\[
\underline{\nu} = {\bf e}_+(u^{\rm in}(0)).
\]
As before, we introduce the matrix $\mathcal{A}(u,\partial\varphi) = (\partial_x\varphi)^{-1}(A(u) - (\partial_t\varphi){\rm Id})$.
The eigenvalues of this matrix are $(\partial_x\varphi)^{-1}(\pm\lambda_{\pm}(u) - \partial_t\varphi)$, whereas the
corresponding eigenvectors are ${\bf e}_{\pm}(u)$ which does not depend on $\partial\varphi$.
Summarizing the above arguments, the initial value problem \eqref{IBVPfbbis2}--\eqref{IC} yields the following:
\begin{equation}\label{qleq1}
\begin{cases}
\partial_t u + \mathcal{A}(u,\partial\varphi) \partial_x u = 0 & \mbox{in}\quad \Omega_T, \\
u_{\vert_{t=0}} = u^{\rm in}(x) & \mbox{on}\quad {\mathbb R}_+, \\
\underline{\nu} \cdot u_{\vert_{x=0}} = \underline{\nu} \cdot {u_{\rm i}}_{\vert_{x=0}} & \mbox{on}\quad (0,T),
\end{cases}
\end{equation}
together with
\begin{equation}\label{qleq2}
\begin{cases}
\partial_t u_{(2)} + \mathcal{A}(u,\partial\varphi) \partial_x u_{(2)}
+ B(u,\partial^\varphi u)u_{(2)} = f_{(2)}(u,\partial^\varphi u) & \mbox{in}\quad \Omega_T, \\
u_{(2) \vert_{t=0}} = u_{(2)}^{\rm in}(x) & \mbox{on}\quad {\mathbb R}_+, \\
\nu_{(2)} \cdot u_{(2) \vert_{x=0}} = g_{(2) \vert_{x=0}} & \mbox{on}\quad (0,T),
\end{cases}
\end{equation}
and an equation for the evolution of the free boundary given by
\begin{equation}\label{qleq3}
\begin{cases}
\ddot{\underline{x}} = \chi(\dot{\underline{x}},u,u_{(2)},\partial^\varphi u,\partial^\varphi u_{\rm i},
\partial^\varphi\partial^\varphi u_{\rm i})_{\vert_{x=0}}
& \mbox{for}\quad t\in(0,T), \\
\underline{x}(0)=0, \quad \dot{\underline{x}}(0)=x_{(1)}^{\rm in},
\end{cases}
\end{equation}
where the initial data $u_{(2)}^{\rm in}$ and $x_{(1)}^{\rm in}$ should be chosen appropriately
for the equivalence of \eqref{qleq1}--\eqref{qleq3} with \eqref{IBVPfbbis2}--\eqref{IC}
and will be given in the next subsection.
\begin{remark} \
{\bf i.}
In place of $\partial_t^\varphi \partial_t^\varphi u$ we can also use $\partial_t^2 u - (\partial_t^2 \varphi)\partial_x^\varphi u$ as $u_{(2)}$.
An advantage of the choice \eqref{u2} is that the reduction and calculations become a little bit simpler.
{\bf ii.}
It is essential to differentiate \eqref{IBVPfbbis2} twice in time to derive a system
with quasilinear boundary conditions.
For example, the first derivative $u_{(1)}=\partial_t^\varphi u$ satisfies a boundary condition
\[
(A(u)^{-1}u_{(1)} + \partial_x^\varphi u_{\rm i})^\perp \cdot (u_{(1)} - \partial_t^\varphi u_{\rm i})_{\vert_{x=0}}=0
\quad\mbox{on}\quad (0,T),
\]
which is still nonlinear in $u_{(1)}$.
\end{remark}
Then, we will analyze maximal dissipativity for \eqref{qleq2} in the sense of {\bf ii} in Assumption 1,
that is, the positivity of $|\nu_{(2)} \cdot {\bf e}_+|$.
The following proposition characterizes this condition algebraically under the restrictions
\eqref{disconti} and \eqref{egenv}.
\begin{proposition}\label{conalg}
Suppose that $u$ together with $\underline{x}$ is a smooth solution to \eqref{IBVPfbbis2} satisfying
\eqref{disconti} and \eqref{egenv} and that $\nu_{(2)}$ is defined by \eqref{nu2g2}.
Then, there exists a unique unit vector $\mu=\mu(t)$ up to the sign such that
\[
\mu \cdot (\partial_t^\varphi u_{\rm i} + A(u_{\rm i})\partial_x^\varphi u_{\rm i})_{\vert_{x=0}} = 0.
\]
Moreover, we have the following identity on $x=0$:
\[
|\nu_{(2)} \cdot {\bf e}_+|
= \frac{(\lambda_{+} - \dot{\underline{x}})^3}{\lambda_+^2}
\frac{|\partial_x^\varphi u - \partial_x^\varphi u_{\rm i}|}{|(\dot{\underline{x}}{\rm Id}-A(u))^{\rm T}\mu|}
|\mu \cdot {\bf e}_+|.
\]
\end{proposition}
This proposition implies that the positivity of $|\nu_{(2)} \cdot {\bf e}_+|$ is essentially equivalent to
the positivity of $|\mu \cdot {\bf e}_+|$, where $\mu$ is a unique direction that the quantity
$\partial_t^\varphi u+A(u)\partial_x^\varphi u$ is continuous across the boundary.
\begin{proof}[Proof of the proposition]
Differentiating the boundary condition in \eqref{IBVPfbbis2} with respect to $t$ and using the
relation $\partial_t=\partial_t^\varphi+(\partial_t\varphi)\partial_x^\varphi$, we have
$\partial_t^\varphi u + \dot{\underline{x}}\partial_x^\varphi u = \partial_t^\varphi u_{\rm i} + \dot{\underline{x}}\partial_x^\varphi u_{\rm i}$ on $x=0$.
This and the interior equation in \eqref{IBVPfbbis2} imply
\begin{equation}\label{rel1}
(\dot{\underline{x}}{\rm Id} - A(u))(\partial_x^\varphi u - \partial_x^\varphi u_{\rm i})
= \partial_t^\varphi u_{\rm i} + A(u_{\rm i})\partial_x^\varphi u_{\rm i}
\quad\mbox{on}\quad x=0.
\end{equation}
Since the matrix $\dot{\underline{x}}{\rm Id} - A(u)$ is invertible, it should hold that
$(\partial_t^\varphi u_{\rm i} + A(u_{\rm i})\partial_x^\varphi u_{\rm i})_{\vert_{x=0}} \ne 0$.
Therefore, the direction $\mu$ is uniquely determined up to the sign as
\[
\mu = \frac{ ((\partial_t^\varphi u_{\rm i} + A(u_{\rm i})\partial_x^\varphi u_{\rm i})_{\vert_{x=0}})^\perp }{
|(\partial_t^\varphi u_{\rm i} + A(u_{\rm i})\partial_x^\varphi u_{\rm i})_{\vert_{x=0}}| }.
\]
By taking the Euclidean inner product of \eqref{rel1} with $\mu$, we have
\[
(\dot{\underline{x}}{\rm Id} - A(u_{\vert_{x=0}}))^{\rm T}\mu
\cdot (\partial_x^\varphi u - \partial_x^\varphi u_{\rm i})_{\vert_{x=0}} = 0.
\]
Since both vectors $(\dot{\underline{x}}{\rm Id} - A(u_{\vert_{x=0}}))^{\rm T}\mu$ and
$(\partial_x^\varphi u - \partial_x^\varphi u_{\rm i})_{\vert_{x=0}}$ are nonzero, so that
\[
(\partial_x^\varphi u - \partial_x^\varphi u_{\rm i})_{\vert_{x=0}}^\perp
= \pm\frac{|(\partial_x^\varphi u - \partial_x^\varphi u_{\rm i})_{\vert_{x=0}}|}{|(\dot{\underline{x}}{\rm Id}-A(u_{\vert_{x=0}}))^{\rm T}\mu|}
(\dot{\underline{x}}{\rm Id} - A(u_{\vert_{x=0}}))^{\rm T}\mu.
\]
Particularly, we see on $x=0$ that
\begin{align*}
\nu_{(2)} \cdot {\bf e}_+
&= (\partial_x^\varphi u - \partial_x^\varphi u_{\rm i})^\perp \cdot ({\rm Id} - \dot{\underline{x}}A(u)^{-1})^2 {\bf e}_+ \\
&= (1-\dot{\underline{x}}\lambda_+^{-1})^2 (\partial_x^\varphi u - \partial_x^\varphi u_{\rm i})^\perp \cdot {\bf e}_+ \\
&= \pm (1-\dot{\underline{x}}\lambda_+^{-1})^2
\frac{|\partial_x^\varphi u - \partial_x^\varphi u_{\rm i}|}{|(\dot{\underline{x}}{\rm Id}-A(u))^{\rm T}\mu|}
(\dot{\underline{x}}-\lambda_+) \mu \cdot {\bf e}_+,
\end{align*}
which gives the desired identity.
\end{proof}
Once the diffeomorphism $\varphi$ is given, we can regard the initial boundary value problems
\eqref{qleq1} and \eqref{qleq2} as the same type of problem considered in the previous sections.
Concerning the compatibility conditions for the problems,
it is straightforward to show the following lemma.
\begin{lemma}\label{comcon}
Suppose that the data $u^{\rm in} \in H^m({\mathbb R}_+)$ and $U_{\rm i} \in W^{m,\infty}((0,T)\times(-\delta,\delta))$
for the initial boundary value problem \eqref{IBVPfbbis2}--\eqref{IC} satisfy the compatibility conditions
up to order $m-1$ in the sense of Definition \ref{defCC} and
that the diffeomorphism $\varphi$ satisfies $\varphi(0,x)=x$ and
$(\partial_t^k\varphi)(0,0) = \underline{x}_{(k)}$ for $k=1,\ldots,m-1$.
\begin{enumerate}
\item[{\bf i.}]
The compatibility conditions for the initial boundary value problem \eqref{qleq1} are
satisfied up to order $m-1$ in the sense of Definitions \ref{defcompVC}--\ref{defcompQL}.
\item[{\bf ii.}]
Let $m\geq3$.
If the initial datum $u_{(2)}^{\rm in}$ is given by \eqref{ukin2} and $u$ satisfies
$((\partial_t^\varphi)^ku)_{\vert_{t=0}}=u_{(k)}^{\rm in}$ for $k=0,1,\ldots,m-1$,
then the compatibility conditions for the initial boundary value problem \eqref{qleq1} are
satisfied up to order $m-3$ in the sense of Definition \ref{defcompVC}.
\end{enumerate}
\end{lemma}
\subsubsection{Proof of Theorem \ref{theoIBVP5}}
We will first show the existence of the solution $(u,u_{(2)},\underline{x})$ to the reduced system
\eqref{qleq1}--\eqref{qleq3} with the diffeomorphism $\varphi$ given by \eqref{diffeo2}
under an additional assumption $m\geq4$.
Then, we will show that $(u,\underline{x})$ is in fact the solution to the original problem
\eqref{IBVPfbbis2}--\eqref{IC}.
In order to reduce the condition on $m$, we will derive an a priori estimate for the solution
$(u,\underline{x})$ under the weaker assumption $m\geq2$, which together with Proposition \ref{appdata}
and standard approximation technique gives the result stated in the theorem.
\medskip
\noindent
{\bf Step 1.}
Let $\mathcal{K}_1$ be a compact and convex set in ${\mathbb R}^2$ satisfying
$\mathcal{K}_0 \Subset \mathcal{K}_1 \Subset \mathcal{U}$.
We will construct the solution $(u,\underline{x})$ satisfying $u(t,x)\in\mathcal{K}_1$ and
\eqref{disconti}--\eqref{egenv}.
\begin{lemma}\label{prepa}
Under the assumptions of Theorem \ref{theoIBVP5}, there exist positive constants
$c_0,\varepsilon_0,\delta_0,C_0$, and $T_0 \in (0,T]$ such that if $u(t,x)$ and $\underline{x}(t)$ satisfy
\begin{equation}\label{precond}
\|u(t)-u^{\rm in}\|_{L^\infty}, |(\partial_x u (t,\cdot)- \partial_x u^{\rm in})_{\vert_{x=0}}|,
|\underline{x}(t)-\underline{x}_0^{\rm in}|, |\partial_t\underline{x}(t)-\underline{x}_1^{\rm in}| \leq \delta_0,
\end{equation}
and if $\varphi(t,x)$ is given by \eqref{diffeo2} with the choice $\varepsilon=\varepsilon_0$,
then for $0\leq t\leq T_0$ we have
\begin{enumerate}
\setlength{\itemsep}{2pt}
\item[{\bf i.}]
$u(t,x) \in \mathcal{K}_1$,
\item[{\bf ii.}]
$\lambda_{\pm}(u(t,x)) \geq c_0$, $\lambda_{\pm}(u(t,x)) \mp \partial_t\varphi(t,x) \geq c_0$,
\item[{\bf iii.}]
$c_0 \leq |(\partial_x^\varphi u(t,\cdot) - \partial_x^\varphi u_{\rm i}(t,\cdot))_{\vert_{x=0}}| \leq C_0$,
\item[{\bf iv.}]
$|\nu_{(2)}(t) \cdot \mathbf{e}_{+}(u(t,\cdot)_{\vert_{x=0}})| \geq c_0$,
\item[{\bf v.}]
$\frac12 \leq \partial_x\varphi(t,x) \leq 2$, $|\partial_t\varphi(t,x)| \leq C_0$,
\end{enumerate}
where $\nu_{(2)}$ is given by \eqref{nu2g2}.
\end{lemma}
\begin{proof}
It follows from the assumptions that there exists $c_0>0$ such that
\[
\begin{cases}
\lambda_{\pm}(u^{\rm in}(x)) \geq 2c_0, \quad
\lambda_{\pm}({u^{\rm in}}_{\vert_{x=0}}) \mp \underline{x}_1^{\rm in} \geq 4c_0, \\
|(\partial_x u^{\rm in})_{\vert_{x=0}} - (\partial_x U_{\rm i})_{\vert_{t=x=0}}| \geq 2c_0, \\
\bigl( 1 - \frac{\underline{x}_1^{\rm in}}{\lambda_{+}({u^{\rm in}}_{\vert_{x=0}})} \bigr)^2
| ((\partial_x u^{\rm in})_{\vert_{x=0}} - (\partial_x U_{\rm i})_{\vert_{t=x=0}})^\perp \cdot
\mathbf{e}_{+}({u^{\rm in}}_{\vert_{x=0}}) | \geq 2c_0.
\end{cases}
\]
In view of $\partial_t\varphi(t,x)=\psi(\frac{x}{\varepsilon})\partial_t\underline{x}(t)$,
we proceed to show that if we choose $\varepsilon_0$ sufficiently small, then we have
\[
\textstyle
\lambda_{\pm}(u^{\rm in}(x)) \mp \psi(\frac{x}{\varepsilon_0})\underline{x}_1^{\rm in} \geq 2c_0.
\]
Since $\psi(\frac{x}{\varepsilon_0})=0$ for $x\geq 2\varepsilon_0$,
it is sufficient to show this inequality for $0\leq x\leq 2\varepsilon_0$.
In the case $\underline{x}_1^{\rm in} \leq 0$ we easily get
\[
\textstyle
\lambda_{+}(u^{\rm in}(x))-\psi(\frac{x}{\varepsilon_0})\underline{x}_1^{\rm in} \geq \lambda_{+}(u^{\rm in}(x)) \geq 2c_0.
\]
In the case $\underline{x}_1^{\rm in} > 0$, for $0\leq x\leq 2\varepsilon_0$ we see that
\begin{align*}
\textstyle
\lambda_{+}(u^{\rm in}(x)) - \psi(\frac{x}{\varepsilon_0})\underline{x}_1^{\rm in}
&\geq \lambda_{+}(u^{\rm in}(x)) - \underline{x}_1^{\rm in} \\
&= \lambda_{+}({u^{\rm in}}_{\vert_{x=0}}) - \underline{x}_1^{\rm in}
+ ( \lambda_{+}(u^{\rm in}(x)) - \lambda_{+}({u^{\rm in}}_{\vert_{x=0}}) ) \\
&\geq 4c_0 - 2\varepsilon_0 \|\nabla u^{\rm in}\|_{L^\infty}\max_{u \in \mathcal{K}_0}|\nabla_u\lambda_+(u)|.
\end{align*}
Therefore, if we choose $\varepsilon_0>0$ so small that
$\varepsilon_0 \|\nabla u^{\rm in}\|_{L^\infty}\max_{u \in \mathcal{K}_0}|\nabla_u\lambda_+(u)| \leq c_0$,
then we obtain $\lambda_{+}(u^{\rm in}(x)) - \psi(\frac{x}{\varepsilon_0})\underline{x}_1^{\rm in} \geq 2c_0$.
Similarly, we can show $\lambda_{-}(u^{\rm in}(x)) + \psi(\frac{x}{\varepsilon_0})\underline{x}_1^{\rm in} \geq 2c_0$
so that the claim is proved.
Now, we note that
\[
\nu_{(2)}(0) \cdot \mathbf{e}_{+}(u_{\vert_{t=x=0}})
= \biggl( 1 - \frac{(\partial_t\underline{x})_{\vert_{t=0}}}{\lambda_{+}(u_{\vert_{t=x=0}})} \biggr)^2
( (\partial_x u)_{\vert_{t=x=0}} - (\partial_x U_{\rm i})_{\vert_{t=0,x=\underline{x}(0)}} )^\perp
\cdot \mathbf{e}_{+}(u_{\vert_{t=x=0}}),
\]
where we used $(\partial_x\varphi)_{\vert_{x=0}}=1$.
Therefore, by taking $\delta_0$ and $T_0$ sufficiently small, we obtain the desired results.
\end{proof}
We will construct the solution $(u,u_{(2)},\underline{x})$ as a limit of a sequence of approximate solutions
$\{(u^n,u_{(2)}^n,\underline{x}^n)\}_n$, which is defined as follows.
We start to construct $\underline{x}^1$ by
\[
\underline{x}^1(t) = \sum_{k=0}^{m-1}\frac{t^k}{k!} \underline{x}_k^{\rm in}.
\]
Suppose that $\underline{x}^n$ is given so that $(\partial_t^k\underline{x}^n)_{\vert_{t=0}}=\underline{x}_k^{\rm in}$ for $0\leq k\leq m-1$.
We define the diffeomorphism $\varphi^n$ by \eqref{diffeo2} with the choice $\varepsilon=\varepsilon_0$,
where $\varepsilon_0>0$ is the constant stated in Lemma \ref{prepa}.
Thanks to Theorem \ref{theoIBVP3} together with Lemma \ref{comcon}, using the standard arguments such as
those in the proof of Theorems \ref{theoIBVP2} and \ref{theoIBVP4}, we can define $u^n$ on a maximal time interval
$[0,T_*^n)$ as a unique solution to
\begin{equation}\label{qleq1n}
\begin{cases}
\partial_t u^n + \mathcal{A}(u^n,\partial\varphi^n) \partial_x u^n = 0 & \mbox{in}\quad (0,T_*^n)\times{\mathbb R}_+, \\
{u^n}_{\vert_{t=0}} = u^{\rm in}(x) & \mbox{on}\quad {\mathbb R}_+, \\
\underline{\nu} \cdot {u^n}_{\vert_{x=0}} = \underline{\nu} \cdot u_{\rm i}^n & \mbox{on}\quad (0,T_*^n),
\end{cases}
\end{equation}
where $u_{\rm i}^n=U_{\rm i}(t,\underline{x}^n(t))$.
Then, we see that $((\partial_t^{\varphi^n})^k u^n)_{\vert_{t=0}}=u_{(k)}^{\rm in}$ for $0\leq k\leq m-1$.
Therefore, by Theorem \ref{theoIBVP3} together with Lemma \ref{comcon} again,
we can define $u_{(2)}^n$ as a unique solution to
\begin{equation}\label{qleq2n}
\begin{cases}
\partial_t u_{(2)}^n + \mathcal{A}(u^n,\partial\varphi^n) \partial_x u_{(2)}^n
+ B(u^n,\partial^{\varphi^n} u^n)u_{(2)}^n = f_{(2)}^n
& \mbox{in}\quad (0,T_*^n)\times{\mathbb R}_+, \\
u_{(2) \vert_{t=0}}^n = u_{(2)}^{\rm in}(x) & \mbox{on}\quad {\mathbb R}_+, \\
\nu_{(2)}^n \cdot u_{(2) \vert_{x=0}}^n = g_{(2)}^n(t) & \mbox{on}\quad (0,T_*^n),
\end{cases}
\end{equation}
where $f_{(2)}^n=f_{(2)}^n(u^n,\partial^{\varphi^n} u^n)$ and
\[
\begin{cases}
\nu_{(2)}^n = \nu_{(2)}(\partial_t\underline{x}^n,u^n,\partial_x^{\varphi^n} u^n,\partial_x^{\varphi^n}u_{\rm i}^n)_{\vert_{x=0}}, \\
g_{(2)}^n = g_{(2)}(\partial_t\underline{x}^n,u^n,\partial^{\varphi^n} u^n,\partial^{\varphi^n}u_{\rm i}^n,
\partial^{\varphi^n}\partial^{\varphi^n}u_{\rm i}^n)_{\vert_{x=0}}.
\end{cases}
\]
Then, we define $\underline{x}^{n+1}$ as a unique solution to
\begin{equation}\label{qleq3n}
\begin{cases}
\partial_t^2\underline{x}^{n+1} = \chi^n \quad\mbox{for}\quad t\in(0,T_*^n), \\
\underline{x}^{n+1}(0) = 0, \quad (\partial_t\underline{x}^{n+1})(0) = x_1^{\rm in},
\end{cases}
\end{equation}
where
\[
\chi^n = \chi(\partial_t\underline{x}^n,u^n,u_{(2)}^n,\partial^{\varphi^n}u^n,\partial^{\varphi^n}u_{\rm i}^n,
\partial^{\varphi^n}\partial^{\varphi^n}u_{\rm i}^n)_{\vert_{x=0}}.
\]
We see that $(\partial_t^k\underline{x}^{n+1})_{\vert_{t=0}}=\underline{x}_k^{\rm in}$ for $0\leq k\leq m-1$, so that
we can define $(\underline{x}^n,u^n,u_{(2)}^n)$ on a time interval $[0,T_*^n)$ for all $n\geq1$.
We prove now that for $M_1,M_2,M_3$ large enough and $T_1$ small enough independent of $n$,
we have $T_1\leq T_*^n$ and
\begin{equation}\label{unifest}
\begin{cases}
\@ifstar\@opnorms\@opnorm{u^n}_{{\mathbb W}^{m-1}(T_1)} + |{u^n}_{\vert_{x=0}}|_{m-1,T_1} \leq M_1, \\
\@ifstar\@opnorms\@opnorm{u_{(2)}^n}_{{\mathbb W}^{m-2}(T_1)} + |u_{(2) \vert_{x=0}}|_{m-2,T_1} \leq M_2, \\
|\underline{x}^{n}|_{H^m(0,T_1)} \leq M_3.
\end{cases}
\end{equation}
Here, by taking $T_1=T_1(M_1,M_2,M_3)$ small enough again we see that $u^n(t,x)$ and $\underline{x}^n(t)$
satisfy \eqref{precond} so that we can apply Lemma \ref{prepa}.
In the following, we denote inessential constants independent of $M_1,M_2,M_3$, and $n$
by the same symbol $C$, which may change from line to line.
By \eqref{unifest}, without loss of generality we have also
\begin{equation}\label{unifest2}
\|u^n\|_{W^{m-2,\infty}(\Omega_{T_1})}, \|u_{(2)}^n\|_{W^{m-3,\infty}(\Omega_{T_1})},
\|\widetilde{\varphi}^n\|_{W^{m-1,\infty}(\Omega_{T_1})} \leq C,
\end{equation}
where $\widetilde{\varphi}^n(t,x)=\varphi^n(t,x)-x=\psi(\frac{x}{\varepsilon_0})\underline{x}^n(t)$,
so that
\[
\begin{cases}
\|B(u^n,\partial^{\varphi^n}u^n)\|_{{\mathbb W}^{m-2}(T_1)}, |\partial_t^{m-2}\nu_{(2)}^n|_{L^2(0,T_1)} \leq CM_1, \\
|\nu_{(2)}^n|_{W^{m-3,\infty}(0,T_1)} \leq C.
\end{cases}
\]
Therefore, it follows from Lemmas \ref{lemdiffeo2}, \ref{prepa}, and Theorem \ref{theoIBVP3} that
\begin{align*}
\@ifstar\@opnorms\@opnorm{u^n(t)}_{m-1} + |{u^n}_{\vert_{x=0}}|_{m-1,t}
&\leq Ce^{C(M_1,M_3)t}( 1 + |u_{\rm i}^n|_{H^{m-1}(0,t)} ), \\
\@ifstar\@opnorms\@opnorm{u_{(2)}^n(t)}_{m-2} + |u_{(2) \vert_{x=0}}^n|_{m-2,t}
&\leq Ce^{C(M_1,M_3)t}\biggl( 1 + |\partial_t^{m-2}\nu_{(2)}^n|_{L^2(0,t)} \\
&\qquad
+ |g_{(2)}^n|_{H^{m-2}(0,t)} + |f_{(2) \vert_{x=0}}^n|_{m-3,t}
+ \int_0^t \@ifstar\@opnorms\@opnorm{f_{(2)}^n(t')}_{m-2}{\rm d}t' \biggr).
\end{align*}
It is easy to see that
\[
|x^{n+1}|_{H^m(0,T_1)} \leq C(1+|\chi^n|_{H^{m-2}(0,T_1)}).
\]
Here, by \eqref{unifest}--\eqref{unifest2} we have
\[
\begin{cases}
|u_{\rm i}^n|_{H^{m-1}(0,T_1)}, |f_{(2) \vert_{x=0}}^n|_{m-3,T_1} \leq C, \\
|g_{(2)}^n|_{H^{m-2}(0,T_1)},\|f_{(2)}^n\|_{{\mathbb W}^{m-2}(T_1)} \leq C(1+M_1), \\
|\chi^n|_{H^{m-2}(0,T_1)} \leq C(1+M_1+M_2).
\end{cases}
\]
Therefore, we obtain
\[
\begin{cases}
\@ifstar\@opnorms\@opnorm{u^n}_{{\mathbb W}^{m-1}(T_1)} + |{u^n}_{\vert_{x=0}}|_{m-1,T_1} \leq Ce^{C(M_1,M_3)T_1}, \\
\@ifstar\@opnorms\@opnorm{u_{(2)}^n}_{{\mathbb W}^{m-2}(T_1)} + |u_{(2) \vert_{x=0}}^n|_{m-2,T_1} \leq Ce^{C(M_1,M_3)T_1}(1+M_1), \\
|\underline{x}^{n}|_{H^m(0,T_1)} \leq C(1+M_1+M_2).
\end{cases}
\]
Putting $M_1=2C$, $M_2=2C(1+M_1)$, and $M_3=C(1+M_1+M_2)$, and taking $T_1$ sufficiently small,
we see that \eqref{unifest} holds for all $n$.
Once we have such uniform bounds for the approximate solutions, by considering the equations for
$(u^{n+1}-u^n,u_{(2)}^{n+1}-u_{(2)}^n,\underline{x}^{n+1}-\underline{x}^n)$ as in the proof of Theorem \ref{theoIBVP4}
and by taking $T_1$ sufficiently small, we can show that $\{(u^n,u_{(2)}^n,\underline{x}^n)\}_n$ converges to
$(u,u_{(2)},\underline{x})$ in $({\mathbb W}^{m-2}(T_1)\cap W^{1,\infty}(\Omega_{T_1})) \times {\mathbb W}^{m-3}(T_1) \times H^m(0,T_1)$
and that the limit is a solution to \eqref{qleq1}--\eqref{qleq3}.
Moreover, by the standard compactness and regularity arguments we see that the solution satisfies
$(u,u_{(2)}) \in {\mathbb W}^{m-1}(T_1) \cap {\mathbb W}^{m-2}(T_1)$.
\medskip
\noindent
{\bf Step 2.}
We will show that the solution $(u,u_{(2)},\underline{x})$ to \eqref{qleq1}--\eqref{qleq3} constructed in Step 1
is in fact a solution to \eqref{IBVPfbbis2}--\eqref{IC} and satisfies $\partial_t^\varphi\partial_t^\varphi u=u_{(2)}$.
Putting $\widetilde{u}_{(2)}=\partial_t^\varphi\partial_t^\varphi u$, it is sufficient to show that
$\widetilde{u}_{(2)}=u_{(2)}$ and the boundary condition $u=u_{\rm i}$ on $x=0$.
Clearly, $u$ satisfies \eqref{d2u} with $u_{(2)}$ replaced by $\widetilde{u}_{(2)}$ so that
$\widetilde{u}_{(2)}$ satisfies the same interior equation in \eqref{qleq2} as $u_{(2)}$.
The boundary condition in \eqref{qleq2} for $u_{(2)}$ and the equation in \eqref{qleq3} for $\underline{x}$
are equivalent to
\begin{equation}\label{equiBC}
({\rm Id} - \dot{\underline{x}}A(u)^{-1})^2 u_{(2)} + \ddot{\underline{x}}( \partial_x^\varphi u - \partial_x^\varphi u_{\rm i} )
= g_1(\dot{\underline{x}},u,\partial^\varphi u,\partial^\varphi\partial^\varphi u_{\rm i})
\quad\mbox{on}\quad x=0.
\end{equation}
On the other hand, by differentiating the boundary condition in \eqref{qleq1} for $u$ twice
with respect to $t$ we see that
\begin{align*}
0 &= \underline{\nu} \cdot \partial_t^2(u-u_{\rm i})_{\vert_{x=0}} \\
&= \underline{\nu} \cdot \bigl( ({\rm Id} - \dot{\underline{x}}A(u)^{-1})^2 \widetilde{u}_{(2)}
+ \ddot{\underline{x}}( \partial_x^\varphi u - \partial_x^\varphi u_{\rm i} )
- g_1(\dot{\underline{x}},u,\partial^\varphi u,\partial^\varphi\partial^\varphi u_{\rm i}) \bigr)_{\vert_{x=0}}.
\end{align*}
Eliminating $\ddot{\underline{x}}$ from these two equations, we obtain
\[
\underline{\nu} \cdot ({\rm Id} - \dot{\underline{x}}A(u)^{-1})^2 ( \widetilde{u}_{(2)} - u_{(2)})_{\vert_{x=0}} = 0.
\]
Therefore, $v_{(2)} = \widetilde{u}_{(2)} - u_{(2)}$ is a solution to the initial boundary value problem
\[
\begin{cases}
\partial_t v_{(2)} + \mathcal{A}(u,\partial\varphi) \partial_x v_{(2)} + B(u,\partial^\varphi u)v_{(2)} = 0
& \mbox{in}\quad \Omega_{T_1}), \\
v_{(2) \vert_{t=0}} = 0 & \mbox{on}\quad {\mathbb R}_+, \\
\widetilde{\nu}_{(2)} \cdot v_{(2) \vert_{x=0}} = 0 & \mbox{on}\quad (0,T_1),
\end{cases}
\]
where $\widetilde{\nu}_{(2)}= (({\rm Id} - \dot{\underline{x}}A(u_{\vert_{x=0}})^{-1})^2)^{\rm T}\underline{\nu}$.
Here, we have
\[
\widetilde{\nu}_{(2)} \cdot \mathbf{e}_{+}(u_{\vert_{x=0}})
= \Bigl( 1 - \frac{\dot{\underline{x}}}{\lambda_{+}(u_{\vert_{x=0}})} \Bigr)
\mathbf{e}_{+}({u^{\rm in}}_{\vert_{x=0}}) \cdot \mathbf{e}_{+}(u_{\vert_{x=0}}),
\]
which is not zero.
Therefore, we can apply Theorem \ref{theoIBVP3} to the above problem and the uniqueness of the solution
gives $v_{(2)}=0$, that is, $\widetilde{u}_{(2)} = u_{(2)}$.
Particularly, \eqref{equiBC} holds with $u_{(2)}$ replaced by $\widetilde{u}_{(2)}$.
We proceed to show the boundary condition in \eqref{IBVPfbbis2}.
Putting $w(t)=(u-u_{\rm i})_{\vert_{x=0}}$ we have
\[
\ddot{w} = \bigl( ({\rm Id} - \dot{\underline{x}}A(u)^{-1})^2 \widetilde{u}_{(2)}
+ \ddot{\underline{x}}( \partial_x^\varphi u - \partial_x^\varphi u_{\rm i} )
- g_1(\dot{\underline{x}},u,\partial^\varphi u,\partial^\varphi\partial^\varphi u_{\rm i}) \bigr)_{\vert_{x=0}}
= 0.
\]
The compatibility conditions implies $w_{\vert_{t=0}}=\dot{w}_{\vert_{t=0}}=0$.
Therefore, we obtain $w=0$, that is, $u=u_{\rm i}$ on $x=0$, so that
$(u,\underline{x})$ is in fact the solution to \eqref{IBVPfbbis2}--\eqref{IC}.
Uniqueness of the solution follows from that of the reduced problem \eqref{qleq1}--\eqref{qleq3}.
\medskip
\noindent
{\bf Step 3.}
In order to reduce the condition $m\geq4$ to $m\geq2$, we will derive an a priori estimate for
the solution $(u,\underline{x})$ under this weaker assumption.
Although we will again use the reduced system \eqref{qleq1}--\eqref{qleq3},
we can now use the relation $\partial_t^\varphi\partial_t^\varphi u = u_{(2)}$ to obtain an additional regularity of $u$.
We will prove again that for $M_1,M_2,M_3$ large enough and $T_1$ small enough, we have
\begin{equation}\label{unifest3}
\begin{cases}
\@ifstar\@opnorms\@opnorm{u}_{{\mathbb W}^{m-1}(T_1)} + |u_{\vert_{x=0}}|_{m-1,T_1} \leq M_1, \\
\@ifstar\@opnorms\@opnorm{u_{(2)}}_{{\mathbb W}^{m-2}(T_1)} + |u_{(2) \vert_{x=0}}|_{m-2,T_1} \leq M_2, \\
|\underline{x}|_{H^m(0,T_1)} \leq M_3.
\end{cases}
\end{equation}
Let $c_0$ and $C_0$ be the constants in Lemma \ref{prepa}.
By Lemma \ref{lemdiffeo2}, there exists $K_0$ independent of $M_1,M_2,M_3$ such that
\[
\frac{1}{c_0}, C_0, \@ifstar\@opnorms\@opnorm{\partial\widetilde{\varphi}(0)}_{m-2}, |\underline{\nu}|,
\@ifstar\@opnorms\@opnorm{u(0)}_{m-1}, \@ifstar\@opnorms\@opnorm{u_{(2)}(0)}_{m-2}, \sum_{j=0}^{m-1}|\underline{x}_j^{\rm in}| \leq K_0.
\]
Moreover, by taking $T_1=T_1(M_1,M_2,M_3)$ sufficiently small if necessary, we have
\begin{equation}\label{estbyK0}
|\nu_{(2)}|_{L^\infty(0,T_1)}, |\underline{x}|_{W^{m-1,\infty}(0,T_1)},
\|\widetilde{\varphi}\|_{W^{m-1,\infty}(\Omega_{T_1})}, \|\partial_x\widetilde{\varphi}\|_{W^{m-1,\infty}(\Omega_{T_1})}
\leq C(K_0).
\end{equation}
Let $K$ be a constant such that $K_0,M_1,M_2,M_3 \leq K$.
\begin{lemma}\label{addreg}
For a smooth solution $(u,\underline{x})$ to \eqref{IBVPfbbis2} with $\varphi$ given by \eqref{diffeo2}
satisfying \eqref{unifest3} and \eqref{estbyK0}, we have
\[
\|\partial_x u\|_{{\mathbb W}^{m-1}(T_1)}, \|u\|_{W^{m-1,\infty}(\Omega_{T_1})}, |u_{\vert_{x=0}}|_{m,T_1} \leq C(K).
\]
\end{lemma}
\begin{proof}
We begin to evaluate $\@ifstar\@opnorms\@opnorm{\partial_x u(t)}_{m-1}$.
In view of the identities
\begin{equation}\label{iden}
\begin{cases}
\partial_x^2u = (\partial_x\varphi)^2\partial_x^\varphi\partial_x^\varphi u + (\partial_x^2\varphi)\partial_x^\varphi u, \\
\partial_t\partial_x u = (\partial_x\varphi)\{ \partial_t^\varphi\partial_x^\varphi u + (\partial_t\varphi)\partial_x^\varphi\partial_x^\varphi u
+ (\partial_x^\varphi\partial_t\varphi)\partial_x^\varphi u\},
\end{cases}
\end{equation}
we see that
\begin{align}\label{estdxu}
\@ifstar\@opnorms\@opnorm{\partial_x u(t)}_{m-1}
&\leq \@ifstar\@opnorms\@opnorm{\partial_x^2u(t)}_{m-2} + \@ifstar\@opnorms\@opnorm{\partial_t\partial_x u(t)}_{m-2} + \@ifstar\@opnorms\@opnorm{\partial_x u(t)}_{m-2} \\
&\leq C(K_0)( \@ifstar\@opnorms\@opnorm{\partial_x^\varphi\partial_x^\varphi u(t)}_{m-2} + \@ifstar\@opnorms\@opnorm{\partial_t^\varphi\partial_x^\varphi u(t)}_{m-2}
+ \@ifstar\@opnorms\@opnorm{u(t)}_{m-1}). \nonumber
\end{align}
We note that $u$ satisfies \eqref{d2u}.
In the case $m\geq3$, by Lemmas \ref{ineq1}--\ref{ineq2} we have
\[
\@ifstar\@opnorms\@opnorm{\partial_x^\varphi\partial_x^\varphi u(t)}_{m-2} + \@ifstar\@opnorms\@opnorm{\partial_t^\varphi\partial_x^\varphi u(t)}_{m-2}
\leq C(\@ifstar\@opnorms\@opnorm{u(t)}_{m-2})( \@ifstar\@opnorms\@opnorm{u_{(2)}(t)}_{m-2} + \@ifstar\@opnorms\@opnorm{\partial^\varphi u(t)}_{m-2}^2 ),
\]
which together with \eqref{estdxu} implies $\@ifstar\@opnorms\@opnorm{\partial_x u(t)}_{m-1} \leq C(K)$.
In the case $m=2$, by using the Sobolev imbedding theorem
$\|u\|_{L^\infty} \leq \sqrt{2}\|u\|_{L^2}^{1/2}\|\partial_x u\|_{L^2}^{1/2}$ we have
\begin{align*}
\|\partial_x^\varphi\partial_x^\varphi u(t)\|_{L^2} + \|\partial_t^\varphi\partial_x^\varphi u(t)\|_{L^2}
&\leq C(K_0)( \|u_{(2)}(t)\|_{L^2} + \|\partial u(t)\|_{L^2}\|\partial_x u(t)\|_{L^\infty} ) \\
&\leq C(K_0)( \|u_{(2)}(t)\|_{L^2} + \@ifstar\@opnorms\@opnorm{u(t)}_1^{3/2}\@ifstar\@opnorms\@opnorm{\partial_x u(t)}_1^{1/2} ),
\end{align*}
which together with \eqref{estdxu} implies
\[
\@ifstar\@opnorms\@opnorm{\partial_x u(t)}_1 \leq C(K_0)( \|u_{(2)}(t)\|_{L^2} + \@ifstar\@opnorms\@opnorm{u(t)}_1 + \@ifstar\@opnorms\@opnorm{u(t)}_1^3 ) \leq C(K).
\]
Therefore, in any case we have $\@ifstar\@opnorms\@opnorm{\partial_x u(t)}_{m-1} \leq C(K)$,
which together with the Sobolev imbedding theorem yields
\[
\|u\|_{W^{m-1,\infty}(\Omega_{T_1})}
\leq C \|u\|_{{\mathbb W}^{m-1}(T_1)}^{1/2}\|\partial_x u\|_{{\mathbb W}^{m-1}(T_1)}^{1/2}
\leq C(K).
\]
We proceed to evaluate $|u_{\vert_{x=0}}|_{m,t}$.
In view of \eqref{iden} and the identity
\[
\partial_t^2 u = u_{(2)} + (\partial_t^2\varphi)\partial_x^\varphi u + 2(\partial_t\varphi)\partial_t^\varphi\partial_x^\varphi u
+ (\partial_t\varphi)^2\partial_x^\varphi\partial_x^\varphi u,
\]
we see that
\begin{align*}
|u_{\vert_{x=0}}|_{m,t}
&\leq |(\partial_t^2 u)_{\vert_{x=0}}|_{m-2,t} + |(\partial_t\partial_x u)_{\vert_{x=0}}|_{m-2,t} + |(\partial_x^2 u)_{\vert_{x=0}}|_{m-2,t} + |u_{\vert_{x=0}}|_{m-1,t} \\
&\leq C(K_0)\bigl( |u_{(2) \vert_{x=0}}|_{m-2,t} + |u_{\vert_{x=0}}|_{m-1,t} \\
&\quad
+ |(\partial_t^2\varphi)_{\vert_{x=0}}|_{m-2,t}\|\partial_x u\|_{L^\infty(\Omega_t)}
+ |(\partial_x^\varphi\partial_x^\varphi u)_{\vert_{x=0}}|_{m-2,t} + |(\partial_t^\varphi\partial_x^\varphi u)_{\vert_{x=0}}|_{m-2,t} \bigr).
\end{align*}
Here, we have $|(\partial_t^2\varphi)_{\vert_{x=0}}|_{m-2,t} \leq C|\underline{x}|_{H^m(0,t)}$.
Noting again that $u$ satisfies \eqref{d2u} and using Lemma \ref{ineq2} we have
\[
|(\partial_x^\varphi\partial_x^\varphi u)_{\vert_{x=0}}|_{m-2,t} + |(\partial_t^\varphi\partial_x^\varphi u)_{\vert_{x=0}}|_{m-2,t}
\leq C(K)( |u_{(2) \vert_{x=0}}|_{m-2,t} + 1 ) \leq C(K).
\]
Therefore, we obtain $|u_{\vert_{x=0}}|_{m,T_1} \leq C(K)$.
\end{proof}
Thanks of this lemma, by taking $T_1$ sufficiently small we have \eqref{precond} and
\[
\|u\|_{W^{m-2,\infty}(\Omega_{T_1})} \leq C(K_0).
\]
Without loss of generality we can also assume $\|U_{\rm i}\|_{W^{m,\infty}((0,T)\times(-\delta,\delta))} \leq K_0$.
Since $u$ is a solution to \eqref{qleq1}, we can apply Theorem \ref{theoIBVP3} with $m$ replaced by
$m-1$ to $u$ and obtain
\begin{align*}
\@ifstar\@opnorms\@opnorm{u(t)}_{m-1} + |u_{\vert_{x=0}}|_{m-1,t}
&\leq C(K_0)e^{C(K)t}( \@ifstar\@opnorms\@opnorm{u(0)}_{m-1} + |u_{\rm i}|_{H^{m-1}(0,t)} ) \\
&\leq C(K_0)e^{C(K)t}( \@ifstar\@opnorms\@opnorm{u(0)}_{m-1} + 1 ).
\end{align*}
We note that $u_{(2)}$ is a solution to \eqref{qleq2} and that in the case of $m\geq3$ we have
\[
\|B(u,\partial^\varphi u)\|_{{\mathbb W}^{m-2}(T_1)}, |\nu_{(2)}|_{W^{1,\infty} \cap W^{m-3,\infty}(0,T_1)},
|\partial_t^{m-2}\nu_{(2)}|_{L^2(0,T_1)} \leq C(K).
\]
Therefore, thanks of Lemma \ref{prepa} we can apply Theorem \ref{theoIBVP3} with $m$ replaced
by $m-2$ in the case $m\geq3$ and Proposition \ref{propNRJ1} together with Lemma \ref{lemsymmetrizerbis}
in the case $m=2$ to $u_{(2)}$ and obtain
\begin{align*}
\@ifstar\@opnorms\@opnorm{u(t)}_{m-2} + |u_{\vert_{x=0}}|_{m-2,t}
&\leq C(K_0) e^{C(K)t} \biggl( (1+|\partial_t^{m-2}\nu_{(2)}|_{L^2(0,t)})\@ifstar\@opnorms\@opnorm{u_{(2)}(0)}_{m-2} \\
&\quad
+ |g_{(2)}|_{H^{m-2}(0,t)} + |f_{(2) \vert_{x=0}}|_{m-3,t} + \int_0^t \@ifstar\@opnorms\@opnorm{f_{(2)}(t')}_{m-2}{\rm d}t' \biggr),
\end{align*}
where the term $|f_{(2) \vert_{x=0}}|_{m-3,t}$ is dropped in the case $m=2$.
Here, we have
\[
|\nu_{(2)}|_{W^{m-2,\infty}(0,T_1)}, |g_{(2)}|_{W^{m-2,\infty}(0,T_1)},
\|f_{(2)}\|_{W^{m-2,\infty}(\Omega_{T_1}) \cap {\mathbb W}^{m-2}(T_1)} \leq C(K),
\]
so that
\[
\@ifstar\@opnorms\@opnorm{u(t)}_{m-2} + |u_{\vert_{x=0}}|_{m-2,t} \leq C(K_0)e^{C(K)t}( 1+C(K)\sqrt{t} ) ( \@ifstar\@opnorms\@opnorm{u_{(2)}(0)}_{m-2} + 1 ).
\]
Since $\underline{x}$ is a solution to \eqref{qleq3}, we see that
\[
|\underline{x}|_{H^m(0,T_1)} \leq C(K_0)( 1 + |u_{(2) \vert_{x=0}}|_{m-2,t} + |u_{\vert_{x=0}}|_{m-1,t} ).
\]
Therefore, if we define the constants $M_1,M_2,M_3$ by
\[
\begin{cases}
M_1 = 2C(K_0)( \@ifstar\@opnorms\@opnorm{u(0)}_{m-1} + 1 ), \\
M_2 = 2C(K_0)( \@ifstar\@opnorms\@opnorm{u_{(2)}(0)}_{m-2} + 1 ), \\
M_3 = C(K_0)( 1 + M_1 + M_2 ),
\end{cases}
\]
and if we take $T_1=T_1(K)$ sufficiently small, then \eqref{unifest3} holds.
The proof of Theorem \ref{theoIBVP5} is complete.
\subsubsection{An extension to a system coupled with ODEs}\label{sectext}
In application to physical and engineering problems, the free boundary problem
\eqref{IBVPfb2}--\eqref{fbBC} appears coupled with a system of ordinary differential equations for the
unknown $W=W(t)$, which takes its value in ${\mathbb R}^N$.
We will extend Theorem \ref{theoIBVP5} to such a problem.
More precisely, we consider \eqref{IBVPfb2}--\eqref{fbBC} with the boundary data $U_{\rm i}$ of the form
$U_{\rm i}(t,x)=G_{\rm i}(W(t),x)$, where $G_{\rm i}(W,x)$ is a given function whereas $W(t)$ satisfies
\begin{equation}\label{ODE}
\begin{cases}
\dot{W} = F(W,\underline{x}) & \mbox{in}\quad (0,T), \\
W = W^{\rm in} & \mbox{on}\quad \{t=0\}.
\end{cases}
\end{equation}
As before, we will use the diffeomorphism $\varphi(t,\cdot) : {\mathbb R}_{+} \to (\underline{x}(t),\infty)$ given by
Lemma \ref{lemdiffeo2} and put $u=U\circ\varphi$.
Then, the problem is recast as
\begin{equation}\label{nlfbp}
\begin{cases}
\partial_t^\varphi u + A(u)\partial_x^\varphi u = 0 & \mbox{in}\quad \Omega_T, \\
u_{\vert_{t=0}} = u^{\rm in}(x) & \mbox{on}\quad {\mathbb R}_{+}, \\
u_{\vert_{x=0}} = u_{\rm i}(t) & \mbox{on}\quad (0,T)
\end{cases}
\end{equation}
with $\underline{x}(0)=0$,
where $u_{\rm i}(t) = G_{\rm i}(W(t),\underline{x}(t))$.
\begin{assumption}\label{assFG}
Let $\mathcal{W}$ be an open set in ${\mathbb R}^N$, which represents a phase space of $W$.
We have $G_{\rm i}, F \in W^{m,\infty}(\mathcal{W}\times(-\delta,\delta))$.
\end{assumption}
\begin{theorem}\label{theoIBVP6}
Let $m\geq 2$ be an integer.
Suppose that Assumptions \ref{asshypNLFB}--\ref{assFG} are satisfied.
If $u^{\rm in}\in H^m({\mathbb R}_+)$ takes its values in a compact and convex set ${\mathcal K}_0\subset {\mathcal U}$ and
if the data $u^{\rm in}$ and $W^{\rm in} \in \mathcal{W}$ satisfy
\begin{enumerate}
\item[{\bf i.}]
$\lambda_{\pm}({u^{\rm in}}_{\vert_{x=0}}) \mp \underline{x}_1^{\rm in} >0$,
\item[{\bf ii.}]
$(\partial_x u^{\rm in})_{\vert_{x=0}} - (\partial_x G_{\rm i})_{\vert_{W=W^{\rm in},x=0}} \ne0$,
\item[{\bf iii.}]
$((\partial_x u^{\rm in})_{\vert_{x=0}} - (\partial_x G_{\rm i})_{\vert_{W=W^{\rm in},x=0}})^\perp \cdot
\mathbf{e}_{+}({u^{\rm in}}_{\vert_{x=0}}) \ne0$,
\end{enumerate}
where $\underline{x}_1^{\rm in}=(\partial_t\underline{x})_{\vert_{t=0}}$ will be determined by \eqref{xkin2} below,
and the compatibility conditions up to order $m-1$ in the sense of Definition \ref{defCC2} below,
then there exist $T_1 \in (0,T]$ and a unique solution
$(u,\underline{x})$ to \eqref{ODE}--\eqref{nlfbp} with
$u, \partial_x u \in {\mathbb W}^{m-1}(T_1)$, $\underline{x}\in H^m(0,T_1)$, $W\in H^{m+1}(0,T_1)$,
and $\varphi$ given by Lemma \ref{lemdiffeo2}.
\end{theorem}
\begin{remark}\label{remarkIC2}
As stated in Remark \ref{remarkIC}, the condition {\bf iii} in the theorem can be replaced by
\begin{enumerate}
\item[{\bf iii$'$.}]
$\mu_0 \cdot \mathbf{e}_{+}({u^{\rm in}}_{\vert_{x=0}}) \ne0$,
\end{enumerate}
where $\mu_0$ is the unit vector satisfying
$\mu_0 \cdot (\partial_t U_{\rm i} + A(U_{\rm i})\partial_x U_{\rm i})_{\vert_{t=x=0}} =0$ with
$U_{\rm i}(t,x)=G_{\rm i}(W(t),x)$.
This unit vector $\mu_0$ is uniquely determined up to the sign under the other assumptions
of the theorem.
\end{remark}
\begin{proof}[Outline of the proof of Theorem \ref{theoIBVP6}]
The solution $(u,\underline{x},W)$ can be constructed as a limit of a sequence of approximate solutions
$\{(u^n,\underline{x}^n,W^n)\}_n$, which are defined by
\[
\begin{cases}
\partial_t u^n + \mathcal{A}(u^n,\partial\varphi^n)\partial_x u^n = 0 & \mbox{in}\quad \Omega_T, \\
{u^n}_{\vert_{t=0}} = u^{\rm in}(x) & \mbox{on}\quad{\mathbb R}_{+}, \\
{u^n}_{\vert_{x=0}} = u_{\rm i}^n(t) & \mbox{on}\quad (0,T)
\end{cases}
\]
with $\underline{x}^n(0)=0$,
where $u_{\rm i}^n(t)=G_{\rm i}(W^n(t),\underline{x}^n(t))$ and $\varphi^n$ is given by \eqref{diffeo2}
with $\varepsilon=\varepsilon_0$ and $\underline{x}$ replaced by $\underline{x}^n$, and
\[
\begin{cases}
\dot{W}^{n+1} = F(W^n,\underline{x}^n) & \mbox{for}\quad t\in(0,T), \\
W^{n+1}(0) = W^{\rm in}.
\end{cases}
\]
Under the condition $|W^n|_{W^{m-1,\infty}(0,T)}, |\underline{x}^n|_{W^{m-1,\infty}(0,T)} \leq C(K_0)$ we have
\[
|W^{n+1}|_{H^{m+1}(0,T)} \leq C(K_0)(|W^n|_{H^m(0,T)} + |\underline{x}^n|_{H^m(0,T)} + 1).
\]
Therefore, we can apply Theorem \ref{theoIBVP5} for the existence of the solution $(u^n,\underline{x}^n)$ with
uniform bounds in appropriate function spaces, so that we can pass to the limit $n\to\infty$ to
obtain the desired solution.
\end{proof}
\subsubsection{Compatibility conditions}\label{sectCC}
Suppose that $(u,\underline{x},W)$ be a smooth solution to \eqref{ODE}--\eqref{nlfbp}.
As in \S \ref{sscomp}, we define $u_{(k)}^{\rm in}=((\partial_t^\varphi)^ku)_{\vert_{t=0}}$ by \eqref{ukin2}.
We denote $W_k^{\rm in}=(\partial_t^k W)_{\vert_{t=0}}$ and $\underline{x}_k^{\rm in}=(\partial_t^k\underline{x})_{\vert_{t=0}}$ as before.
It follows from $\dot{W}=F(W,\underline{x})$ that
\begin{equation}\label{Wkin}
W_{k+1}^{\rm in} = c_{3,k}(W_0^{\rm in},W_1^{\rm in},\ldots,W_k^{\rm in},
\underline{x}_0^{\rm in},\underline{x}_1^{\rm in},\ldots,\underline{x}_k^{\rm in})
\end{equation}
Using the relation $U_{\rm i}(t,x)=G_{\rm i}(W(t),x)$, we have
\[
(\partial_t^k\partial_x^l U_{\rm i})_{\vert_{t=x=0}} = c_{2,k,l}(W_0^{\rm in},W_1^{\rm in},\ldots,W_k^{\rm in}).
\]
This together with \eqref{xkin} yields
\begin{align}\label{xkin2}
\underline{x}_{k}^{\rm in}
&= -\frac{\partial_x u^{\rm in} - (\partial_x G_{\rm i})_{\vert_{W=W^{\rm in}}}}{
|\partial_x u^{\rm in} - (\partial_x U_{\rm i})_{\vert_{W=W^{\rm in}}}|^2}
\cdot \biggl\{ u_{(k)} ^{\rm in} - c_{2,k,0}(W_0^{\rm in},W_1^{\rm in},\ldots,W_k^{\rm in}) \\
&\qquad
+ \sum_{l=2}^k\sum_{\substack{j_0+j_1+\cdots+j_l=k \\ 1\leq j_1,\ldots,j_l}}
c_{l,j_0,\ldots,j_l}\underline{x}_{j_1}^{\rm in}\cdots\underline{x}_{j_l}^{\rm in}
\bigl( \partial_x^l u_{(j_0)}^{\rm in} - c_{2,j_0,l}(W_0^{\rm in},W_1^{\rm in},\ldots,W_{j_0}^{\rm in}) \bigr)
\biggr\}_{\vert_{x=0}}. \nonumber
\end{align}
Now, we can calculate $\underline{x}_k^{\rm in}$ and $W_k^{\rm in}$ inductively by
$\underline{x}_0^{\rm in}=0$, $W_0^{\rm in}=W^{\rm in}$, and \eqref{Wkin}--\eqref{xkin2}
in terms of the data $u^{\rm in}$ and $W^{\rm in}$.
\begin{definition}\label{defCC2}
Let $m\geq1$ be an integer.
We say that the data $u^{\rm in} \in H^m({\mathbb R}_+)$ and $W^{\rm in}$ for the problem
\eqref{ODE}--\eqref{nlfbp} satisfy the compatibility condition at order $k$ if
$\{u_{(j)}^{\rm in}\}_{j=0}^m$ and $\{\underline{x}_{(j)}^{\rm in}\}_{j=0}^{m-1}$ defined by \eqref{ukin2}
and \eqref{xkin2} satisfy $u^{\rm in}(0) = G_{\rm i}(W^{\rm in},0)$ in the case $k=0$ and
\begin{align*}
& (\partial_x u^{\rm in} - (\partial_x G_{\rm i})_{\vert_{W=W^{\rm in}}})^\perp \cdot \biggl\{
u_{(k)}^{\rm in} - c_{2,k,0}(W_0^{\rm in},W_1^{\rm in},\ldots,W_k^{\rm in}) \\
& + \sum_{l=2}^k\sum_{\substack{j_0+j_1+\cdots+j_l=k \\ 1\leq j_1,\ldots,j_l}}
c_{l,j_0,\ldots,j_l}\underline{x}_{(j_1)}^{\rm in}\cdots\underline{x}_{(j_l)}^{\rm in}
\bigl( \partial_x^l u_{(j_0)}^{\rm in} - c_{2,j_0,l}(W_0^{\rm in},W_1^{\rm in},\ldots,W_{j_0}^{\rm in}) \bigr)
\biggr\}_{\vert_{x=0}}
= 0
\end{align*}
in the case $k\geq 1$.
We say also that the data $u^{\rm in}$ and $W_k^{\rm in}$ for the problem \eqref{ODE}--\eqref{nlfbp}
satisfy the compatibility conditions up to order $m-1$ if they satisfy the compatibility
conditions at order $k$ for $k=0,1,\ldots,m-1$.
\end{definition}
Roughly speaking, the definition of $\underline{x}_{k}^{\rm in}$ ensures the equality
$\partial_t^k u =\partial_t^k u_{\rm i}$ at $x=t=0$ in the direction $\partial_x^\varphi u - \partial_x^\varphi u_i$,
whereas the compatibility conditions ensure it in the perpendicular direction
$(\partial_x^\varphi u - \partial_x^\varphi u_i)^\perp$.
\section{Transmission problems}\label{secttransmission}
We proposed in Section \ref{sect2} a general approach to study initial boundary value problems with a possibly
free boundary for $2\times 2$ hyperbolic systems.
Our results can easily be extended to systems involving more equations, provided that the diaganalizability
properties used in Proposition \ref{propBC} to construct the Kreiss symmetrizer are still valid.
This is for instance the case for transmission problems involving the coupling of two $2\times2$ hyperbolic
systems across an interface.
Such problems can be transformed into a $4\times 4$ initial boundary value problems that have the required
diagonalizability properties.
Transmission problems being relevant for many applications, we devote this section to their study.
\subsection{Variable coefficients linear $2\times 2$ transmission problems}\label{sectVCtransm}
We consider here a linear transmission problem, where we seek a solution $u$ solving a linear hyperbolic
system on $\Omega_T^- = (0,T)\times {\mathbb R}_-$, another one (possibly the same) for $\Omega_T^+ = (0,T)\times {\mathbb R}_+$,
assuming that a transmission condition is provided at the interface $\{x=0\}$
\begin{equation}\label{transmVC}
\begin{cases}
\partial_t u + \widetilde A(t,x)\partial_x u + \widetilde B(t,x) u = \widetilde f(t,x) &\mbox{in}\quad \Omega_T^-, \\
\partial_t u + A(t,x)\partial_x u + B(t,x) u = f(t,x) &\mbox{in}\quad \Omega_T^+, \\
u_{\vert_{t=0}} = u^{\rm in}(x) & \mbox{on}\quad {\mathbb R}_-\cup {\mathbb R}_+, \\
N_p^{\rm r}(t)u_{\vert_{x=+0}} - N_p^{\rm l}(t) u_{\vert_{x=-0}} = \mybf{g}(t)& \mbox{on}\quad (0,T),
\end{cases}
\end{equation}
where $u$, $u^{\rm in}$, $f$, and $\widetilde{f}$ are ${\mathbb R}^2$-valued functions,
$\mybf{g}$ is a ${\mathbb R}^p$-valued function, while $A$, $\widetilde A$, $B$, and $\widetilde B$ take their values
in the space of $2\times2$ real-valued matrices.
The matrices $N_p^{\rm l}$ and $N_p^{\rm l}$ that appear in the transmission condition are of size $p\times 2$,
where $p$ (the number of scalar transmission conditions) depends on the sign of the eigenvalues of
$\widetilde{A}$ and $A$.
\begin{notation}\label{notanumberp}
We shall consider three possibilities corresponding to the following cases,
where $\widetilde{\lambda}_{\pm,j}(t,-x)$ and ${\lambda}_{\pm,j}(t,x)$ ($j=1,2,\emptyset$)
are assumed to be strictly positive for all $(t,x)\in \Omega_T$:
\begin{itemize}
\item
{\bf Case $p=1$.} There is one outgoing characteristic, that is, one of the following two situations holds:
\begin{itemize}
\item
The matrices $\widetilde{A}(t,-x)$ and $A(t,x)$ have eigenvalues $\pm\widetilde{\lambda}_\pm(t,-x)$ and
$-\lambda_{-,j}(t,x)$ ($j=1,2$), respectively.
\item
The matrices $\widetilde{A}(t,-x)$ and $A(t,x)$ have eigenvalues $\widetilde{\lambda}_{+,j}(t,-x)$ ($j=1,2$)
and $\pm\lambda_{\pm}(t,x)$, respectively.
\end{itemize}
\item
{\bf Case $p=2$.} There are two outgoing characteristics, that is, the matrices $\widetilde{A}(t,-x)$ and
$A(t,x)$ have eigenvalues $\pm\widetilde{\lambda}_\pm(t,-x)$ and $\pm\lambda_{\pm}(t,x)$, respectively.
\item
{\bf Case $p=3$.} There are three outgoing characteristics, that is, one of the following two situations holds:
\begin{itemize}
\item
The matrices $\widetilde{A}(t,-x)$ and $A(t,x)$ have eigenvalues $\pm\widetilde{\lambda}_\pm(t,-x)$ and
$\lambda_{+,j}(t,x)$ ($j=1,2$), respectively.
\item
The matrices $\widetilde{A}(t,-x)$ and $A(t,x)$ have eigenvalues $-\widetilde{\lambda}_{-,j}(t,-x)$ ($j=1,2$)
and $\pm\lambda_{\pm}(t,x)$, respectively.
\end{itemize}
\end{itemize}
Denoting by $\widetilde{\bf e}_{\pm,j}(t,-x)$ and ${\bf e}_{\pm,j}(t,x)$ unit eigenvectors associated to
the eigenvalues $\widetilde{\lambda}_{\pm,j}(t,-x)$ and ${\lambda}_{\pm,j}(t,x)$ ($j=1,2,\emptyset$),
we define a $4\times p$ matrix $\mybf{E}_p(t)$ by
\[
\mybf{E}_p(t) = \left(\begin{array}{cc} \widetilde{\mybf{E}}_-(t) & 0_{2\times p^{\rm r}} \\
0_{2\times p^{\rm l}} & \mybf{E}_+(t)\end{array}\right),
\]
where $0\leq p^{\rm l}\leq 2$ (resp. $0\leq p^{\rm r}\leq 2$) denotes the number of negative eigenvalues of
$\widetilde{A}(t,0)$ (resp. positive eigenvalues of $A(t,0)$), and $\widetilde{\mybf{E}}_-(t)$ and
$\mybf{E}_+(t)$ the matrix formed by the corresponding eigenvectors.
\end{notation}
\begin{remark}
Here we did not list any possible cases, that is, the cases $p=0,4$ are omitted.
Moreover, even in the case $p=2$ there are two other posibilities.
Such cases can be treated in the same way so we omit them.
\end{remark}
It is convenient to recast \eqref{transmVC} as a $4\times 4$ initial boundary value problem by setting
\begin{equation}\label{leftright}
\begin{array}{llll}
A^{\rm r}(t,x)=A(t,x), & B^{\rm r}(t,x)=B(t,x), & f^{\rm r}(t,x)=f(t,x), & u^{\rm r}(t,x)=u(t,x), \\
A^{\rm l}(t,x)=\widetilde A(t,-x), & B^{\rm l}(t,x)=\widetilde B(t,-x), & f^{\rm l}(t,x)=\widetilde f(t,-x),
& u^{\rm l}(t,x)=u(t,-x),
\end{array}
\end{equation}
and
\begin{equation}\label{notaAB}
\mybf{A} = \left(\begin{array}{cc} -A^{\rm l} & 0_{2\times 2} \\ 0_{2\times 2} & A^{\rm r} \end{array}\right), \qquad
\mybf{B} = \left(\begin{array}{cc} B^{\rm l} & 0_{2\times 2} \\ 0_{2\times 2} & B^{\rm r} \end{array}\right), \qquad
\mybf{u} = \left( \begin{array}{c} u^{\rm l} \\ u^{\rm r} \end{array}\right), \qquad
\mybf{f} = \left( \begin{array}{c} f^{\rm l} \\ f^{\rm r} \end{array}\right).
\end{equation}
The transmission problem \eqref{transmVC} is equivalent to the following initial boundary value problem
\begin{equation}\label{transmref}
\begin{cases}
\partial_t \mybf{u} + \mybf{A}(t,x)\partial_x \mybf{u} + \mybf{B}(t,x)\mybf{u} = \mybf{f}(t,x) &\mbox{in}\quad \Omega_T, \\
\mybf{u}_{\vert_{t=0}} = \mybf{u}^{\rm in}(x) & \mbox{on}\quad {\mathbb R}_+, \\
\mybf{N}_p(t)\mybf{u}_{\vert_{x=0}} = \mybf{g}(t) & \mbox{on}\quad (0,T),
\end{cases}
\end{equation}
where $\mybf{u}^{\rm in}(x) = (u^{\rm in}(-x),u^{\rm in}(x))^{\rm T}$ and $\mybf{N}_p$ is the $p\times 4$ matrix
\begin{equation}\label{notaN}
\mybf{N}_p(t) = \left( \begin{array}{cc} -N_p^{\rm l}(t) & N_p^{\rm r}(t) \end{array} \right).
\end{equation}
This initial boundary value problem has a block structure.
In order to ensure its well-posedness, we shall make the following assumption,
which ensures that the sytem of equations is strictly hyperbolic.
Note that the condition on the invertibility of $\mybf{N}_p(t)\mybf{N}_p(t)^{\rm T}$ in the first point is here
to ensure that $\mybf{N}_p$ is uniformly of rank $p$.
\begin{assumption}\label{asshyptransm}
There exists $c_0>0$ such that the following assertions hold.
\begin{enumerate}
\item[{\bf i.}]
$A^{\rm l}, A^{\rm r}\in W^{1,\infty}(\Omega_T)$ and $ B^{\rm l}, B^{\rm r}\in L^\infty(\Omega_T)$.
Moreover, $\mybf{N}_p \in C([0,T])$ and for any $t\in[0,T]$ we have
\[
\det(\mybf{N}_p(t)\mybf{N}_p(t)^{\rm T}) \geq c_0.
\]
\item[{\bf ii.}]
One of the three cases stated in Notation \ref{notanumberp} holds. Moreover,
\begin{align*}
& \widetilde{\lambda}_{\pm,j}(t,-x),\lambda_{\pm,j}(t,x)\geq c_0 \quad(j=1,2,\emptyset), \\
& |\widetilde{\lambda}_{\pm,1}(t,-x) - \widetilde{\lambda}_{\pm,2}(t,-x)|,
|\lambda_{\pm,1}(t,x) - \lambda_{\pm,2}(t,x)| \geq c_0.
\end{align*}
\item[{\bf iii.}]
With $\mybf{E}_p(t)$ in Notation \ref{notanumberp}, the $p\times p$ Lopatinski\u{\i} matrix
$\mybf{L}_p(t) = \mybf{N}_p(t)\mybf{E}_p(t)$ is invertible and for any $t\in [0,T]$ we have
\[
\Vert \mybf{L}_p(t)^{-1}\Vert_{{\mathbb R}^p\to{\mathbb R}^p} \leq \frac{1}{c_0}.
\]
\end{enumerate}
\end{assumption}
We can then derive sharp estimates similar to those derived in Theorem \ref{theoIBVP1}
for initial boundary value problems.
The compatibility conditions are not made explicit because they can be obtained as for Definition \ref{defcompVC}.
\begin{theorem}\label{theoIBVP1transm}
Let $m\geq1$ be an integer, $T>0$, and assume that Assumption \ref{asshyptransm} is satisfied for some $c_0>0$.
Assume moreover that there are constants $0<K_0\leq K$ such that
\[
\begin{cases}
\frac{1}{c_0}, \| \mybf{A} \|_{L^\infty(\Omega_T)}, |\mybf{N}_p|_{L^\infty(0,T)} \leq K_0, \\
\| \mybf{A} \|_{W^{1,\infty}(\Omega_T)}, \| \mybf{B} \|_{L^\infty(\Omega_T)},
\|(\partial \mybf{A},\partial \mybf{B})\|_{ {\mathbb W}^{m-1}(T)}, |\mybf{N}_p|_{W^{m,\infty}(0,T)} \leq K.
\end{cases}
\]
Then, for any data $\mybf{u}^{\rm in} \in H^m({\mathbb R}_+)$, $\mybf{g}\in H^m(0,T)$, and $\mybf{f}\in H^m(\Omega_T)$
satisfying the compatibility conditions up to order $m-1$,
there exists a unique solution $\mybf{u} \in {\mathbb W}^m(T)$ to the transmission problem \eqref{transmref}.
Moreover, the following estimate holds for any $t \in[0,T]$ and any $\gamma \geq C(K)$:
\begin{align*}
& \@ifstar\@opnorms\@opnorm{ \mybf{u}(t) }_{m,\gamma}
+ \biggl( \gamma\int_0^t\@ifstar\@opnorms\@opnorm{ \mybf{u}(t') }_{m,\gamma}^2{\rm d}t' \biggr)^\frac12
+ | \mybf{u}_{\vert_{x=0}} |_{m,\gamma,t} \\
&\leq C(K_0)\bigl( \@ifstar\@opnorms\@opnorm{ \mybf{u}(0) }_{m} + | \mybf{g} |_{H_\gamma^m(0,t)}
+ |\mybf{f}_{\vert_{x=0}}|_{m-1,\gamma,t} + S_{\gamma,t}^*(\@ifstar\@opnorms\@opnorm{ \partial_t \mybf{f}(\cdot) }_{m-1}) \bigr).
\end{align*}
Particularly, we have
\begin{align*}
& \@ifstar\@opnorms\@opnorm{ \mybf{u}(t) }_{m} + | \mybf{u}_{\vert_{x=0}} |_{m,t} \\
&\leq C(K_0)e^{C(K)t} \biggl( \@ifstar\@opnorms\@opnorm{ \mybf{u}(0) }_{m} + | \mybf{g} |_{H^m(0,t)}
+ |\mybf{f}_{\vert_{x=0}}|_{m-1,t} + \int_0^t \@ifstar\@opnorms\@opnorm{ \partial_t \mybf{f}(t') }_{m-1}{\rm d}t' \biggr).
\end{align*}
\end{theorem}
\subsubsection{A priori estimates}
We prove here an $L^2$ a priori estimate using the following assumption,
which is the natural generalization of Assumption \ref{assVC} to $4\times 4$ systems.
\begin{assumption}\label{assVCtransm}
There exists a symmetric matrix $\mybf{S}(t,x) \in {\mathcal M}_4({\mathbb R})$ such that
for any $(t,x)\in\Omega_T$ $\mybf{S}(t,x)\mybf{A}(t,x)$ is symmetric and
the following conditions hold.
\begin{enumerate}
\item[{\bf i.}]
There exist constants $\alpha_0,\beta_0>0$ such that for any
$(\mybf{v},t,x)\in {\mathbb R}^4\times \Omega_T$ we have
\[
\alpha_0 |\mybf{v}|^2 \leq \mybf{v}^{\rm T} \mybf{S}(t,x) \mybf{v} \leq \beta_0 |\mybf{v}|^2.
\]
\item[{\bf ii.}]
There exist constants $\alpha_1,\beta_1>0$ such that for any
$(\mybf{v},t)\in {\mathbb R}^2\times (0,T)$ we have
\[
\mybf{v}^{\rm T} \mybf{S}(t,0)\mybf{A}(t,0) \mybf{v} \leq -\alpha_1 |\mybf{v}|^2 + \beta_1 |\mybf{N}_p(t) \mybf{v}|^2.
\]
\item[{\bf iii.}]
There exists a constant $\beta_2$ such that
\[
\| \partial_t \mybf{S} + \partial_x (\mybf{SA}) - 2\mybf{SB} \|_{L^2\to L^2} \leq \beta_2.
\]
\end{enumerate}
\end{assumption}
Under this assumption,
the $L^2$ a priori estimates of Proposition \ref{propNRJ1} can be straightforwardly generalized.
\begin{proposition}\label{propNRJ1transm}
Under Assumption \ref{assVCtransm}, there are constants
\[
\mathfrak{c}_0 = C\Bigl( \frac{\beta_0^{\rm in}}{\alpha_0},\frac{\beta_0^{\rm in}}{\alpha_1} \Bigr)
\quad\mbox{ and }\quad
\mathfrak{c}_1 = C\Big( \frac{\beta_0}{\alpha_0},\frac{\beta_1}{\alpha_0},\frac{\alpha_0}{\alpha_1} \Big)
\]
such that for any $\mybf{u} \in H^1(\Omega_T)$ solving \eqref{transmref}, any $t\in [0,T]$, and any
$\gamma\geq\frac{\beta_2}{\alpha_0}$, the following inequality holds.
\begin{align*}
&\@ifstar\@opnorms\@opnorm{\mybf{u}(t)}_{0,\gamma} + \biggl(\gamma \int_0^t \@ifstar\@opnorms\@opnorm{\mybf{u}(t')}_{0,\gamma}^2 {\rm d}t' \biggr)^\frac12
+ |\mybf{u}_{\vert_{x=0}}|_{L_\gamma^2(0,t)} \\
&\leq \mathfrak{c}_0\|\mybf{u}^{\rm in}\|_{L^2}
+ \mathfrak{c}_1\bigl( |\mybf{g}|_{L_\gamma^2(0,t)} + S_{\gamma,t}^*( \|\mybf{f}(\cdot)\|_{L^2} ) \bigr).
\end{align*}
\end{proposition}
Similarly, the following generalization of Proposition \ref{propVC2} does not raise any difficulty,
and we therefore omit the proof.
\begin{proposition}\label{propVC2transm}
Let $m\geq1$ be an integer, $T>0$, and assume that Assumption \ref{assVCtransm} is satisfied.
Assume moreover that there are two constants $0<K_0\leq K$ such that
\[
\begin{cases}
\mathfrak{c}_0, \mathfrak{c}_1, \|\mybf{A}\|_{L^\infty(\Omega_T)},
\|\mybf{A}^{-1}\|_{L^\infty(\Omega_T)}, |\mybf{N}_p|_{L^\infty(0,T)}\leq K_0, \\
\frac{\beta_2}{\alpha_0}, \|\mybf{A}\|_{W^{1,\infty}(\Omega_T)}, \|\mybf{B}\|_{L^\infty(\Omega_T)},
\|(\partial \mybf{A},\partial \mybf{B})\|_{{\mathbb W}^{m-1}(T)}, |\mybf{N}_p|_{W^{m,\infty}(0,T)}\leq K,
\end{cases}
\]
where $\mathfrak{c}_0$ and $\mathfrak{c}_1$ are as in Proposition \ref{propNRJ1transm}.
Then, every solution $\mybf{u}\in H^{m+1}(\Omega_{T})$ to the initial boundary value problem
\eqref{transmref} satisfies, for any $t \in [0,T]$ and any $\gamma \geq C(K)$,
\begin{align*}
&\@ifstar\@opnorms\@opnorm{ \mybf{u}(t) }_{m,\gamma}
+ \biggl( \gamma\int_0^t \@ifstar\@opnorms\@opnorm{ \mybf{u}(t') }_{m,\gamma}^2{\rm d}t' \biggr)^\frac12
+ |\mybf{u}_{\vert_{x=0}}|_{m,\gamma,t} \\
&\leq C(K_0)\bigl( \@ifstar\@opnorms\@opnorm{ \mybf{u}(0) }_{m} + |\mybf{g}|_{H_\gamma^m(0,t)}
+ |\mybf{f}_{\vert_{x=0}}|_{m-1,\gamma,t} + S_{\gamma,t}^*(\@ifstar\@opnorms\@opnorm{ \partial_t \mybf{f}(t') }_{m-1}) \bigr).
\end{align*}
\end{proposition}
\subsubsection{Proof of Theorem \ref{theoIBVP1transm}}
As for the proof of Theorem \ref{theoIBVP1transm}, we just have to prove that the assumptions made
in the statement of Theorem \ref{theoIBVP1transm} imply that Assumption \ref{assVCtransm} is satisfied.
This is what the following lemma claims;
its proof requires the construction of a Kreiss symmetrizer yielding maximal dissipativity on the boundary.
\begin{lemma}\label{lemsymmetrizertransm}
Let $c_0>0$ be such that Assumption \ref{asshyptransm} is satisfied.
There exist a symmetrizer $\mybf{S}\in W^{1,\infty}(\Omega_T)$ and constants
$\alpha_0,\alpha_1$ and $\beta_0,\beta_1,\beta_2$ such that Assumption \ref{assVCtransm} is satisfied.
Moreover, we have
\[
\mathfrak{c}_0 \leq C\Bigl( \frac{1}{c_0}, \| \mybf{A}_{\vert_{t=0}} \|_{L^\infty({\mathbb R}_+)} \Bigr)
\quad\mbox{and}\quad
\mathfrak{c}_1 \leq C\Bigl( \frac{1}{c_0},\|\mybf{A}\|_{L^{\infty}(\Omega_T)}, |\mybf{N}_p|_{L^\infty(0,T)}\Bigr),
\]
where $\mathfrak{c}_0$ and $\mathfrak{c}_1$ are as defined in Proposition \ref{propNRJ1transm}, and we also have
\[
\frac{\beta_2}{\beta_0} \leq C\Bigl( \frac{1}{c_0}, \|\mybf{A}\|_{W^{1,\infty}(\Omega_T)},
\|\mybf{B}\|_{L^\infty(\Omega_T)}\Bigr).
\]
\end{lemma}
\begin{proof}
Most of the proof is similar to the proof of Lemma \ref{lemsymmetrizer} and Proposition \ref{propBC}
and we therefore omit the details.
The only new point is to show that it is possible to construct a symmetrizer $\mybf{S}$ satisfying
${\bf ii}$ in Assumption \ref{asshyptransm}.
We show here how to prove this point, namely, that there exist constants $\alpha_1,\beta_1>0$ such that for any
$(\mybf{v},t)\in {\mathbb R}^4\times (0,T)$ we have
\[
\mybf{v}^{\rm T} \mybf{S}(t,0)\mybf{A}(t,0) \mybf{v}
\leq -\alpha_1 |\mybf{v}|^2 + \beta_1 |\mybf{N}_p(t) \mybf{v}|^2.
\]
Let us denote by $\widetilde{\boldsymbol{\pi}}_{\pm,j}(t,x)$ and $\boldsymbol{\pi}_{\pm,j}(t,x)$
the eigenprojectors associated to the eigenvalues $\widetilde{\lambda}_{\pm,j}$ and $\lambda_{\pm,j}$
(with $j=1,2,\emptyset$); they are of the form
\[
\widetilde{\boldsymbol{\pi}}_{\pm,j} = \left(
\begin{array}{cc}
\widetilde{\pi}_{\pm,j} & 0_{2\times 2} \\
0_{2\times 2} & {0}_{2\times 2}
\end{array}\right)
\quad\mbox{ and }\quad
\boldsymbol{\pi}_{\pm,j} = \left(
\begin{array}{cc}
0_{2\times 2} & 0_{2\times 2} \\
0_{2\times 2} & \pi_{\pm,j}
\end{array}\right),
\]
where $\widetilde{\pi}_{\pm,j}(t,x)$ and $\pi_{\pm,j}(t,x)$ are the corresponding eigenprojectors of
$\widetilde{A}(t,x)$ and $A(t,x)$.
Distinguishing the three cases stated in Notation \ref{notanumberp} and writing as in \eqref{leftright}
\begin{align*}
& \lambda_{\pm,j}^{\rm l}(t,x) = \widetilde{\lambda}_{\pm,j}(t,-x), \quad
\lambda_{\pm,j}^{\rm r}(t,x) = \lambda_{\pm,j}(t,x), \\
& \boldsymbol{\pi}_{\pm,j}^{\rm l}(t,x) = \widetilde{\boldsymbol{\pi}}_{\pm,j}(t,-x), \quad
\boldsymbol{\pi}_{\pm,j}^{\rm r}(t,x) = \boldsymbol{\pi}_{\pm,j}(t,x),
\end{align*}
the spectral decomposition of the matrix $\mybf{A}$ is given by
\[
\mybf{A} =
\begin{cases}
\lambda_-^{\rm l}{\boldsymbol \pi}_-^{\rm l} - \lambda_+^{\rm l}{\boldsymbol \pi}_+^{\rm l}
- \lambda_{-,1}^{\rm r}{\boldsymbol \pi}_{-,1}^{\rm r} - \lambda_{-,2}^{\rm r}{\boldsymbol \pi}_{-,2}^{\rm r}
& (\mbox{frist case of $p=1$}), \\
\lambda_+^{\rm r}{\boldsymbol \pi}_+^{\rm r} - \lambda_{+,1}^{\rm l}{\boldsymbol \pi}_{+,1}^{\rm l}
- \lambda_{+,2}^{\rm l}{\boldsymbol \pi}_{+,2}^{\rm l} - \lambda_{-}^{\rm r}{\boldsymbol \pi}_{-}^{\rm r}
& (\mbox{second case of $p=1$}), \\
\lambda_-^{\rm l}{\boldsymbol \pi}_-^{\rm l} + \lambda_{+}^{\rm r}{\boldsymbol \pi}_{+}^{\rm r}
- \lambda_+^{\rm l}{\boldsymbol \pi}_+^{\rm l} - \lambda_{-}^{\rm r}{\boldsymbol \pi}_{-}^{\rm r} & (p=2), \\
\lambda_{-}^{\rm l}{\boldsymbol \pi}_{-}^{\rm l} + \lambda_{+,1}^{\rm r}{\boldsymbol \pi}_{+,1}^{\rm r}
+ \lambda_{+,2}^{\rm r}{\boldsymbol \pi}_{+,2}^{\rm r} - \lambda_+^{\rm l}{\boldsymbol \pi}_+^{\rm l}
& (\mbox{first case of $p=3$}), \\
\lambda_{-,1}^{\rm l}{\boldsymbol \pi}_{-,1}^{\rm l} + \lambda_{-,2}^{\rm l}{\boldsymbol \pi}_{-,2}^{\rm l}
+ \lambda_{+}^{\rm r}{\boldsymbol \pi}_{+}^{\rm r} - \lambda_-^{\rm r}{\boldsymbol \pi}_-^{\rm r}
& (\mbox{second case of $p=3$}).
\end{cases}
\]
We construct the symmetrizer $\mybf{S}$ in the form
\[
\mybf{S} =
\begin{cases}
({\boldsymbol \pi}_-^{\rm l})^{\rm T}{\boldsymbol \pi}_-^{\rm l} + M\bigl\{ ({\boldsymbol \pi}_+^{\rm l})^{\rm T}{\boldsymbol \pi}_+^{\rm l}
+ ({\boldsymbol \pi}_{-,1}^{\rm r})^{\rm T}{\boldsymbol \pi}_{-,1}^{\rm r} + ({\boldsymbol \pi}_{-,2}^{\rm r})^{\rm T}{\boldsymbol \pi}_{-,2}^{\rm r} \bigr\}
& (\mbox{frist case of $p=1$}), \\
({\boldsymbol \pi}_+^{\rm r})^{\rm T}{\boldsymbol \pi}_+^{\rm r} + M\bigl\{ ({\boldsymbol \pi}_{+,1}^{\rm l})^{\rm T}{\boldsymbol \pi}_{+,1}^{\rm l}
+ ({\boldsymbol \pi}_{+,2}^{\rm l})^{\rm T}{\boldsymbol \pi}_{+,2}^{\rm l} + ({\boldsymbol \pi}_{-}^{\rm r})^{\rm T}{\boldsymbol \pi}_{-}^{\rm r} \bigr\}
& (\mbox{second case of $p=1$}), \\
({\boldsymbol \pi}_-^{\rm l})^{\rm T}{\boldsymbol \pi}_-^{\rm l} + ({\boldsymbol \pi}_+^{\rm r})^{\rm T}{\boldsymbol \pi}_+^{\rm r}
+ M\bigl\{ ({\boldsymbol \pi}_+^{\rm l})^{\rm T}{\boldsymbol \pi}_+^{\rm l} + ({\boldsymbol \pi}_-^{\rm r})^{\rm T}{\boldsymbol \pi}_-^{\rm r} \bigr\} & (p=2), \\
({\boldsymbol \pi}_{-}^{\rm l})^{\rm T}{\boldsymbol \pi}_{-}^{\rm l} + ({\boldsymbol \pi}_{+,1}^{\rm r})^{\rm T}{\boldsymbol \pi}_{+,1}^{\rm r}
+ ({\boldsymbol \pi}_{+,2}^{\rm r})^{\rm T}{\boldsymbol \pi}_{+,2}^{\rm r} + M ({\boldsymbol \pi}_+^{\rm l})^{\rm T}{\boldsymbol \pi}_+^{\rm l}
& (\mbox{first case of $p=3$}), \\
({\boldsymbol \pi}_{-,1}^{\rm l})^{\rm T}{\boldsymbol \pi}_{-,1}^{\rm l} + ({\boldsymbol \pi}_{-,2}^{\rm l})^{\rm T}{\boldsymbol \pi}_{-,2}^{\rm l}
+ ({\boldsymbol \pi}_{+}^{\rm r})^{\rm T}{\boldsymbol \pi}_{+}^{\rm r} + M ({\boldsymbol \pi}_-^{\rm r})^{\rm T}{\boldsymbol \pi}_-^{\rm r}
& (\mbox{second case of $p=3$}),
\end{cases}
\]
where $M>0$ will be determined later.
{\it From now on, we focus on the case $p=2$, the adaptations to the cases $p=1$ and $p=3$ being straightforward.}
Then, we have
\[
\mybf{SA} = \lambda_-^{\rm l}({\boldsymbol \pi}_-^{\rm l})^{\rm T}{\boldsymbol \pi}_-^{\rm l}
+ \lambda_+^{\rm r}({\boldsymbol \pi}_+^{\rm r})^{\rm T}({\boldsymbol \pi}_+^{\rm r})
- M\bigl\{ \lambda_+^{\rm l}({\boldsymbol \pi}_+^{\rm l})^{\rm T}{\boldsymbol \pi}_+^{\rm l}
+ \lambda_-^{\rm r}({\boldsymbol \pi}_-^{\rm r})^{\rm T}{\boldsymbol \pi}_-^{\rm r} \bigr\}.
\]
We begin to show that for $\mybf{v} \in \ker \mybf{N}_p(t)$ we have
\[
|\mybf{v}|^2 \leq -C\mybf{v}^{\rm T}(\mybf{SA})(t,0)\mybf{v}.
\]
For any $\mybf{v} = \begin{pmatrix} v^{\rm l} \\ v^{\rm r} \end{pmatrix} \in {\mathbb R}^4$, we have
\begin{align*}
-\mybf{v}^{\rm T} \mybf{SA} \mybf{v}
&= - \lambda_-^{\rm l}({\boldsymbol \pi}_-^{\rm l}\mybf{v})^{\rm T}{\boldsymbol \pi}_-^{\rm l}\mybf{v}
- \lambda_+^{\rm r}({\boldsymbol \pi}_+^{\rm r}\mybf{v})^{\rm T}{\boldsymbol \pi}_+^{\rm r}\mybf{v}
+ M\bigl\{ \lambda_+^{\rm l}({\boldsymbol \pi}_+^{\rm l}\mybf{v})^{\rm T}{\boldsymbol \pi}_+^{\rm l}\mybf{v}
+ \lambda_-^{\rm r}({\boldsymbol \pi}_-^{\rm r}\mybf{v})^{\rm T}{\boldsymbol \pi}_-^{\rm r}\mybf{v} \bigr\} \\
&= - \lambda_-^{\rm l}|\pi_-^{\rm l}v^{\rm l}|^2 - \lambda_+^{\rm r}|\pi_+^{\rm r}v^{\rm r}|^2
+ M\bigl\{ \lambda_+^{\rm l}|\pi_+^{\rm l}v^{\rm l}|^2
+ \lambda_-^{\rm r}|\pi_-v^{\rm r}|^2 \bigr\}.
\end{align*}
We decompose $v^{\rm l}$ and $v^{\rm r}$ as
\begin{equation}\label{decom}
\begin{cases}
v^{\rm l} = c_+^{\rm l}{\bf e}_+^{\rm l} + c_-^{\rm l}{\bf e}_-^{\rm l}, \\
v^{\rm r} = c_+^{\rm r}{\bf e}_+^{\rm r} + c_-^{\rm r}{\bf e}_-^{\rm r},
\end{cases}
\end{equation}
where $\pi_{\pm}^{\rm l}v^{\rm l} = c_{\pm}^{\rm l}{\bf e}_{\pm}^{\rm l}$ and
$\pi_{\pm}^{\rm r}v^{\rm r} = c_{\pm}^{\rm r}{\bf e}_{\pm}^{\rm r}$.
Particularly, we have $|\pi_{\pm}^{\rm l}v^{\rm l}| = |c_{\pm}^{\rm l}|$ and
$|\pi_{\pm}^{\rm r}v^{\rm r}| = |c_{\pm}^{\rm r}|$, so that
\[
-\mybf{v}^{\rm T} \mybf{SA}\mybf{v}
= - \lambda_-^{\rm l}|c_-^{\rm l}|^2 - \lambda_+^{\rm r}|c_+^{\rm r}|^2
+ M\bigl\{ \lambda_+^{\rm l}|c_+^{\rm l}|^2 + \lambda_-^{\rm r}|c_-^{\rm r}|^2 \bigr\}.
\]
Now, suppose that $\mybf{v} \in \ker \mybf{N}_p(t)$.
Then, we have
\[
\mybf{N}_p\mybf{v} = - N_p^{\rm l}v^{\rm l} + N_p^{\rm r}v^{\rm r} = 0.
\]
Plugging \eqref{decom} into the above relation, we have
\[
- c_+^{\rm l}N_p^{\rm l}{\bf e}_+^{\rm l} - c_-^{\rm l}N_p^{\rm l}{\bf e}_-^{\rm l}
+ c_+^{\rm r}N_p^{\rm r}{\bf e}_+^{\rm r} + c_-^{\rm r}N_p^{\rm r}{\bf e}_-^{\rm r} = 0,
\]
which we can rewrite, using the Lopatinski\u{\i} matrix,
\[
\mybf{L}_p(t)
\begin{pmatrix} c_-^{\rm l} \\ c_+^{\rm r} \end{pmatrix}
= \begin{pmatrix} N_p^{\rm l}{\bf e}_+^{\rm l} & -N_p^{\rm r}{\bf e}_-^{\rm r} \end{pmatrix}
\begin{pmatrix} c_+^{\rm l} \\ c_-^{\rm r} \end{pmatrix}.
\]
Under the uniform Kreiss--Lopatinski\u{\i} condition made in Assumption \ref{asshyptransm}, we deduce
\[
|c_-^{\rm l}|^2 + |c_+^{\rm r}|^2
\leq C( |c_+^{\rm l}|^2 + |c_-^{\rm r}|^2 ),
\]
where $C$ depends only on $| \mybf{N}_p |_{L^\infty(0,T)}$ and $1/c_0$, or equivalently,
\[
|\pi_-^{\rm l}v^{\rm l}|^2 + |\pi_+^{\rm r}v^{\rm r}|^2
\leq C( |\pi_+^{\rm l}v^{\rm l}|^2 + |\pi_-^{\rm r}v^{\rm r}|^2 ).
\]
Therefore, if we take $M$ sufficiently large, then for any $\mybf{v} \in \ker\mybf{N}_p(t)$ we have
\[
|\mybf{v}|^2 \leq -C\mybf{v}^{\rm T}(\mybf{SA})(t,0)\mybf{v}.
\]
Next, we will show that for any $\mybf{v} \in {\mathbb R}^4$ we have
\[
\mybf{v}^{\rm T}(\mybf{SA})(t,0)\mybf{v} \leq -\alpha_1|\mybf{v}|^2 + \beta_1|\mybf{N}_p(t)\mybf{v}|^2.
\]
To this end, we use the assumption that
\begin{equation}
| \det ( \mybf{N}_p(t) \mybf{N}_p(t)^{\rm T} ) | \geq c_0.
\end{equation}
This condition means that the $2\times4$ matrix $\mybf{N}_p(t)$ has rank 2 uniformly in time.
For any $\mybf{v} \in {\mathbb R}^4$, we decompose it as
\[
\mybf{v} = \mybf{v}_1 + \mybf{v}_2 \quad\mbox{with}\quad
\mybf{v}_2 = \mybf{N}_p^{\rm T}( \mybf{N}_p\mybf{N}_p^{\rm T} )^{-1} \mybf{N}_p\mybf{v}.
\]
Then, we have
\[
\mybf{v}_1 \in \ker \mybf{N}_p, \qquad \mybf{N}_p\mybf{v} = \mybf{N}_p\mybf{v}_2,
\]
so that
\begin{align*}
|\mybf{v}|^2 &\leq C(|\mybf{v}_1|^2 + |\mybf{v}_2|^2) \\
&\leq -C\mybf{v}_1^{\rm T}\mybf{SA}\mybf{v}_1 + C|\mybf{v}_2|^2 \\
&= -C(\mybf{v}-\mybf{v}_2)^{\rm T}\mybf{SA}(\mybf{v}-\mybf{v}_2) + C|\mybf{v}_2|^2 \\
&\leq -C\mybf{v}^{\rm T}\mybf{SA}\mybf{v} + \frac12|\mybf{v}|^2 + C|\mybf{v}_2|^2.
\end{align*}
Since $|\mybf{v}_2| \leq C|\mybf{N}_p\mybf{v}|$, we obtain the desired estimate.
\end{proof}
\subsection{Application to quasilinear $2\times 2$ transmission problems}\label{sectappltransmQL}
As done in \S \ref{sectapplQL} in the case of initial boundary value problems,
we can use the linear estimates of Theorem \ref{theoIBVP1transm} to solve quasilinear problems.
More precisely, after reduction to a $4\times 4$ initial boundary value problem as indicated
in \S \ref{sectVCtransm}, let us consider
\begin{equation}\label{systQLtransm}
\begin{cases}
\partial_t \mybf{u} + \mybf{A}(\mybf{u})\partial_x \mybf{u} + \mybf{B}(t,x)\mybf{u} = \mybf{f}(t,x) & \mbox{in}\quad \Omega_T, \\
\mybf{u}_{\vert_{t=0}} = \mybf{u}^{\rm in}(x) & \mbox{on}\quad {\mathbb R}_+,\\
\mybf{N}_{p}(t) \mybf{u}_{\vert_{x=0}} = \mybf{g}(t) & \mbox{on}\quad (0,T),
\end{cases}
\end{equation}
where $\mybf{u} = (u^{\rm l},u^{\rm r})^{\rm T}$, $\mybf{u}^{\rm in}$, and $\mybf{f}$ are ${\mathbb R}^4$-valued functions,
and $\mybf{g}$ is a ${\mathbb R}^p$-valued function,
while $\mybf{A}(\mybf{u}) = \mbox{diag}(-\widetilde{A}(\rm u^l), A(u^{\rm r}))$ and
$\mybf{B} = \mbox{diag}(B^{\rm l},B^{\rm r})$ take their values in the space of $4\times4$ real-valued matrices
and $\mybf{N}_p$ is a $p\times 4$ matrix, where $p$ is the number of outgoing characteristics
(i.e., the number of positive eigenvalues of $\mybf{A}(\mybf{u})$).
\begin{notation}\label{notanumberp2}
Adaptating Notation \ref{notanumberp} in a straightforward way,
we consider three different possibilities ($p=1,2,3$) depending on the sign of the eigenvalues of
$\widetilde{A}(u^{\rm l})$ and $A(u^{\rm r})$.
Correspondingly, a $4\times p$ matrix $\mybf{E}_p(\mybf{u}_{\vert_{x=0}})$ is formed as in
Notation \ref{notanumberp} with the eigenvectors associated to the eigenvalues defining outgoing characteristics,
and we define the Lopatinski\u{\i} matrix by
$\mybf{L}_p(t,\mybf{u}_{\vert_{x=0}}) = \mybf{N}_p(t)\mybf{E}_p(\mybf{u}_{\vert_{x=0}})$.
\end{notation}
We also make the following assumption on the hyperbolicity of the system and on the boundary condition.
\begin{assumption}\label{asshypQLtransm}
Let $\widetilde{{\mathcal U}}$ and $\mathcal{U}$ be open sets in ${\mathbb R}^2$ and $p\in \{1,2,3\}$ such that the following
conditions hold with $\boldsymbol{{\mathcal U}} = \widetilde{{\mathcal U}}\times {\mathcal U}$ representing a phase space of $\mybf{u}$.
\begin{enumerate}
\setlength{\itemsep}{3pt}
\item[{\bf i.}]
$\mybf{A} \in C^\infty(\boldsymbol{{\mathcal U}})$.
\item[{\bf ii.}]
The integer $p$ is such that for any $\mybf{u} = (u^{\rm l},u^{\rm r})^{\rm T} \in \boldsymbol{{\mathcal U}}$ the matrices
$\widetilde{A}(u^{\rm l})$ and $A(u^{\rm r})$ satisfy one of the three conditions of Notation \ref{notanumberp}.
\item[{\bf iii.}]
For any $t\in[0,T]$ and any $\mybf{u} \in \boldsymbol{{\mathcal U}}$,
the Lopatinski\u{\i} matrix $\mybf{L}_p(t,\mybf{u})$ is invertible.
\end{enumerate}
\end{assumption}
The main result is the following.
The compatibility conditions mentioned in the statement of the theorem can be obtained as for
Definition \ref{defcompQL}.
It can be deduced from Theorem \ref{theoIBVP1transm} in the same way that
Theorem \ref{theoIBVP2} was deduced from Thoerem \ref{theoIBVP1} and we therefore omit the proof.
\begin{theorem}\label{theoIBVP2transm}
Let $m\geq 2$ be an integer and assume that Assumption \ref{asshypQLtransm} is satisfied with some $p\in\{1,2,3\}$.
Assume moreover that $\mybf{B}\in L^\infty(\Omega_T)$, $\partial \mybf{B}\in {\mathbb W}^{m-1}(T)$,
and $\mybf{N}_p\in W^{m,\infty}(0,T)$.
If $\mybf{u}^{\rm in }\in H^m({\mathbb R}_+)$ takes its values in $\widetilde{\mathcal{K}}_0\times \mathcal{K}_0$
with $\widetilde{\mathcal{K}}_0 \subset \widetilde{{\mathcal U}}$ and ${\mathcal{K}}_0 \subset {\mathcal U}$ compact and convex sets,
and if the data $\mybf{u}^{\rm in}$, $\mybf{f}\in H^m(\Omega_T)$, and $\mybf{g}\in H^m(0,T)$
satisfy the compatibility conditions up to order $m-1$,
then there exist $T_1 \in (0,T]$ and a unique solution $\mybf{u}\in {\mathbb W}^m(T_1)$
to the transmission problem \eqref{systQLtransm}.
Moroever, the trace of $\mybf{u}$ at $x=0$ belongs to $H^m(0,T_1)$ and $|\mybf{u}_{\vert_{x=0}}|_{m,T_1}$ is finite.
\end{theorem}
\subsection{Variable coefficients $2\times2$ transmission problems on moving domains}\label{secttransmmov}
As for the initial boundary value problems considered previously, we consider here the case of variable coefficients
transmission problems on a moving domain as a preliminary step to treat free boundary transmission problems.
We consider therefore a transmission problem with transmission conditions given at a moving boundary located at
$x=\underline{x}(t)$ with $\underline{x}(\cdot)$ a given function.
As in \S \ref{sectVCm}, we consider variable coefficients matrices of the form $A(t,x)=A(\underline{U}(t,x))$, etc.
Let us consider therefore
\begin{equation}\label{transmmov}
\begin{cases}
\partial_t U + \widetilde{A}({\underline{U}})\partial_x U + \widetilde{{\mathtt B}}U = \widetilde{F}
& \mbox{in }\quad (-\infty,\underline{x}(t)) \quad \mbox{ for } \quad t\in(0,T), \\
\partial_t U + {A}(\underline{U})\partial_x U + {{\mathtt B}}U = {F}
& \mbox{in }\quad (\underline{x}(t),+\infty) \quad \mbox{ for } \quad t\in(0,T), \\
U_{\vert_{t=0}} = u^{\rm in}(x) & \mbox{on }\quad \mathbb{R}_-\cup\mathbb{R}_+, \\
N_p^ {\rm r}(t)U_{\vert_{x=\underline{x}(t)+0}}-N_p^{\rm l}(t)U_{\vert_{x=\underline{x}(t)-0}} = \mybf{g}(t)
& \mbox{on } \quad (0,T),
\end{cases}
\end{equation}
where, without loss of generality, we assumed that $\underline{x}(0)=0$,
and with notations inherited from the previous sections.
As in \S \ref{sectVCm}, we use a diffeomorphism $\varphi(t,\cdot): {\mathbb R}\to {\mathbb R}$ such that
$\varphi(0,\cdot) = \mbox{Id}$ and that for any $t\in[0,T]$ we have
\[
\varphi(t,0)=\underline{x}(t), \qquad \varphi(t,\cdot): {\mathbb R}_- \to (-\infty,\underline{x}(t)), \quad\mbox{ and }\quad
\varphi(t,\cdot): {\mathbb R}_+ \to (\underline{x}(t),+\infty).
\]
Writing as before $u=U\circ \varphi$, $\partial_t^\varphi u=(\partial_t U)\circ \varphi$, etc., and with $\partial_x^\varphi$ and
$\partial_t^\varphi$ as defined in \eqref{dtphi}, we transform \eqref{transmmov} into a transmission problem with
a fix interface located at $x=0$.
Using the same procedure as in \S \ref{sectVCtransm} and with the same notations as in \eqref{leftright}
(writing also $\varphi^{\rm l}(t,x) = \varphi(t,-x)$ and $\varphi^{\rm r}(t,x) = \varphi(t,x)$ for $x>0$),
this transmission problem can be recast as a $4\times 4$ initial boundary value problem on $(0,T)\times {\mathbb R}_+$,
namely
\begin{equation}\label{bigIBVP}
\begin{cases}
\partial_t \mybf{u} +{\bm{\mathcal{A}}}(\underline{{\boldsymbol{u}}},\partial {\boldsymbol \varphi})\partial_x \mybf{u} + \mybf{B}(t,x)\mybf{u} = \mybf{f}(t,x)
& \mbox{ in }\quad \Omega_T, \\
\mybf{u}_{\vert_{t=0}} = \mybf{u}^{\rm in}(x) & \mbox{ on }\quad {\mathbb R}_+, \\
\mybf{N}_p(t)\mybf{u}_{\vert_{x=0}} = \mybf{g}(t) & \mbox{ on }\quad (0,T),
\end{cases}
\end{equation}
with $\mybf{u} = (u^{\rm l},u^{\rm r})^{\rm T}$, ${\boldsymbol \varphi} = (\varphi^{\rm l},\varphi^{\rm r})^{\rm T}$, and
\[
{\bm{\mathcal{A}}}(\underline{{\boldsymbol{u}}},\partial {\boldsymbol \varphi}) = \left(
\begin{array}{cc}
-{\mathcal A}^{\rm l}(\underline{u}^{\rm l},\partial\varphi^{\rm l}) & 0_{2\times2} \\
0_{2\times2} &{\mathcal A}^{\rm r}(\underline{u}^{\rm r},\partial\varphi^{\rm r})
\end{array}\right)
\]
as well as
\[
{\mathcal A}^{\rm l}(\underline{u}^{\rm l},\partial\varphi^{\rm l})
= \frac{1}{\abs{\partial_x \varphi^{\rm l}}}\big( \widetilde{A}({\underline{u}}^{\rm l})-(\partial_t \varphi^{\rm l})\mbox{Id}\big), \qquad
{\mathcal A}^{\rm r}(\underline{u}^{\rm r},\partial\varphi^{\rm r})
= \frac{1}{\partial_x \varphi^{\rm r}}\big( A(\underline{u}^{\rm r})-(\partial_t \varphi^{\rm r})\mbox{Id}\big),
\]
while $\mybf{B}$ and $\mybf{f}$ as in \S \ref{sectVCtransm}.
The matrix $\mybf{N}_p$ is as in \eqref{notaN} and still denotes a $p\times 4$ matrix,
but the difference is that the value of $p$ depends not only on the eigenvalues of $\widetilde{A}(u)$ and $A(u)$,
but also on the speed $\dot \underline{x}$ of the interface.
For the sake of simplicity, we shall consider here the case where $\widetilde{A}(u)$ and $A(u)$ have both
a positive and a negative eigenvalue, and shall consider two cases depending on the speed of the interface.
\begin{definition}\label{defLax}
Denoting by $\pm\widetilde{\lambda}_\pm(\underline{u}^{\rm l})$ and $\pm \lambda_\pm(\underline{u}^{\rm r})$ the eigenvalues of
$\widetilde{A}(\underline{u}^{\rm l})$ and $A(\underline{u}^{\rm r})$, respectively
(with $\widetilde{\lambda}_\pm(\underline{u}^{\rm l}),\lambda_\pm(\underline{u}^{\rm r})>0)$, we define two regimes:
\begin{itemize}
\item
{\bf Subsonic regime.} We say that $\underline{{\boldsymbol{u}}} = (\underline{u}^{\rm l},\underline{u}^{\rm r})^{\rm T}$ and ${\chi}\in {\mathbb R}$ are in the
\emph{subsonic regime} if the following condition holds.
\[
\widetilde{\lambda}_\pm(\underline{u}^{\rm l})\mp\chi>0
\quad\mbox{ and }\quad
{\lambda}_\pm(\underline{u}^{\rm r})\mp\chi>0.
\]
\item
{\bf Lax regime.} We say that $\underline{{\boldsymbol{u}}} = (\underline{u}^{\rm l},\underline{u}^{\rm r})^{\rm T}$ and ${\chi}\in {\mathbb R}$ are in the
\emph{Lax regime} if the following condition holds.
\[
\widetilde{\lambda}_\pm(\underline{u}^{\rm l})\mp\chi>0
\quad\mbox{ and }\quad -\lambda_+(\underline{u}^{\rm r})+\chi>0,
\]
or
\[
-\widetilde{\lambda}_-(\underline{u}^{\rm l})-\chi>0
\quad\mbox{ and }\quad{\lambda}_\pm(\underline{u}^{\rm r})\mp\chi>0.
\]
\end{itemize}
\end{definition}
\begin{remark}
This terminology is of course inherited from the study of shocks \cite{Lax}.
The linearized equations around a shock can indeed be put under the form \eqref{transmmov}.
We refer to \S \ref{sectshocks} where we prove the stability of one-dimensional shocks
for nonlinear $2\times2$ hyperbolic systems.
\end{remark}
Since the eigenvalues of the matrix ${\bm{\mathcal{A}}}(\underline{{\boldsymbol{u}}},\partial {\boldsymbol \varphi})$ are given by
\[
\frac{1}{|\partial_x \varphi^{\rm l}|}\bigl( \pm\widetilde{\lambda}_\mp(\underline{u}^{\rm l}) + \partial_t \varphi^{\rm l} \bigr)
\quad\mbox{ and }\quad
\frac{1}{\partial_x \varphi^{\rm r}}\bigl( \pm{\lambda}_\pm(\underline{u}^{\rm r})-\partial_t \varphi^{\rm r} \bigr),
\]
the number $p$ of outgoing characteristics for \eqref{bigIBVP} is equal to $2$ in the subsonic regime,
and to $1$ in the Lax regime.
As in Notation \ref{notanumberp}, we form a $4\times p$ matrix $\mybf{E}_p (\underline{{\boldsymbol{u}}}_{\vert_{x=0}})$ given by
\begin{align*}
\mybf{E}_2 (\underline{{\boldsymbol{u}}}_{\vert_{x=0}}) = \left(
\begin{array}{cc}
\widetilde{\bf e}_-({\underline{u}^{\rm l}}_{\vert_{x=0}}) & 0_{2\times1} \\
0_{2\times1} & {\bf e}_+({\underline{u}^{\rm r}}_{\vert_{x=0}})
\end{array}\right)
\end{align*}
in the subsonic regime, and
\begin{align*}
\mybf{E}_1 (\underline{{\boldsymbol{u}}}_{\vert_{x=0}}) = \left(\begin{array}{c}
\widetilde{\bf e}_-({\underline{u}^{\rm l}}_{\vert_{x=0}}) \\ 0_{2\times1} \end{array}\right)
\quad\mbox{ or }\quad
\mybf{E}_1 (\underline{{\boldsymbol{u}}}_{\vert_{x=0}}) = \left(\begin{array}{c}
0_{2\times1} \\ {\bf e}_+({\underline{u}^{\rm r}}_{\vert_{x=0}}) \end{array}\right)
\end{align*}
(depending on which of the two conditions in Definition \ref{defLax} is satisfied) in the Lax regime.
As in Assumption \ref{asshyptransm}, we define a Lopatinski\u{\i} matrix
$\mybf{L}_p(t,\underline{{\boldsymbol{u}}}_{\vert_{x=0}})$ by
\begin{equation}\label{interfmatr}
\mybf{L}_p(t,\underline{{\boldsymbol{u}}}_{\vert_{x=0}}) = \mybf{N}_p(t)\mybf{E}_p (\underline{{\boldsymbol{u}}}_{\vert_{x=0}}).
\end{equation}
In order to be able to apply Theorem \ref{theoIBVP1transm} to this initial boundary value problem,
we make the following assumption.
It is the natural generalization of Assumption \ref{asshypm} to transmission problems.
\begin{assumption}\label{asstransmred}
We have $\underline{{\boldsymbol{u}}} = (\underline{u}^{\rm l},\underline{u}^{\rm r})^{\rm T} \in W^{1,\infty}(\Omega_T)$, $\underline{x}\in C^1([0,T])$, $\underline{x}(0)=0$,
and the diffeomorphisms $\varphi^{\rm l}$ and $\varphi^{\rm r}$ are in $C^1(\Omega_T)$.
Moreover, there exists $c_0>0$ such that the following three conditions hold.
\begin{enumerate}
\setlength{\itemsep}{3pt}
\item[{\bf i.}]
There exist open sets $\widetilde{{\mathcal U}},{\mathcal U} \subset {\mathbb R}^2$ such that, with
$\boldsymbol{{\mathcal U}} = \widetilde{{\mathcal U}}\times {\mathcal U}$, we have $\mybf{A} \in C^\infty(\boldsymbol{\mathcal U})$ and for any
$\mybf{u} = (u^{\rm l},u^{\rm r})^{\rm T} \in \boldsymbol{\mathcal U}$, the matrices $\widetilde{A}({u^{\rm l}})$ and $A(u^{\rm r})$
have eigenvalues $\widetilde{\lambda}_+(u^{\rm l}), -\widetilde{\lambda}_-(u^{\rm l})$ and
$\lambda_+(u^{\rm r}), -\lambda_-(u^{\rm r})$, respectively.
Moreover, $\underline{{\boldsymbol{u}}}$ takes its values in a compact set $\boldsymbol{\mathcal{K}}_0 \subset \boldsymbol{\mathcal U}$ and for any
$(t,x)\in\Omega_T$ we have
\[
\widetilde{\lambda}_\pm(\underline{u}^{\rm l}(t,x))\geq c_0\quad\mbox{ and }\quad \lambda_\pm(\underline{u}^{\rm r}(t,x))\geq c_0,
\]
and one of the following conditions holds
\begin{align*}
a) \qquad \widetilde{\lambda}_\pm(\underline{u}^{\rm l}(t,x))\mp \partial_t \varphi^{\rm l} (t,x)\geq c_0
& \quad\mbox{ and }\quad \lambda_\pm(\underline{u}^{\rm r}(t,x))\mp \partial_t \varphi^{\rm r} (t,x)\geq c_0, \\
b) \qquad\widetilde{\lambda}_\pm(\underline{u}^{\rm l}(t,x))\mp \partial_t \varphi^{\rm l} (t,x)\geq c_0
& \quad\mbox{ and }\quad -\lambda_+(\underline{u}^{\rm r}(t,x))+\partial_t \varphi^{\rm r} (t,x) \geq c_0, \\
c) \,\,\,\,\, -\widetilde{\lambda}_-(\underline{u}^{\rm l}(t,x))- \partial_t \varphi^{\rm l} (t,x)\geq c_0
& \quad\mbox{ and }\quad \lambda_\pm(\underline{u}^{\rm r}(t,x))\mp \partial_t \varphi^{\rm r} (t,x)\geq c_0.
\end{align*}
\item[{\bf ii.}]
The Lopatinski\u{\i} matrix $\mybf{L}_p(t,\underline{{\boldsymbol{u}}}_{\vert_{x=0}})$ associated to the condition $a)$, $b)$, or $c)$
constructed in \eqref{interfmatr} is invertible and for any $t \in [0,T]$ we have
\[
\| \mybf{L}_p(t,\underline{{\boldsymbol{u}}}_{\vert_{x=0}}(t))^{-1} \|_{{\mathbb R}^p \to {\mathbb R}^p} \leq \frac{1}{c_0}.
\]
\item[{\bf iii.}]
The Jacobian of the diffeomorphism is uniformly bounded from below and from above, that is,
for any $(t,x)\in\Omega_T$ we have
\[
c_0 \leq - \partial_x \varphi^{\rm l} (t,x) \leq \frac{1}{c_0} \quad \mbox{ and }\quad
c_0 \leq \partial_x \varphi^{\rm r} (t,x) \leq \frac{1}{c_0}.
\]
\end{enumerate}
\end{assumption}
The equivalent of Theorem \ref{theoIBVP3} for transmission problems is the following.
We do not make explicit the compatibility condition in the statement of the theorem
because they are obtained along a procedure similar to the one used for Definition \ref{defcompVC}.
\begin{theorem}\label{theoIBVP3transm}
Let $m\geq1$ be an integer, $T>0$, and assume that Assumption \ref{asstransmred} is satisfied for some $c_0>0$.
Assume moreover that there are constants $0<K_0\leq K$ such that
\[
\begin{cases}
\frac{1}{c_0}, \@ifstar\@opnorms\@opnorm{\partial \varphi^{\rm l,r}(0) }_{m-1}, \|\partial \varphi^{\rm l,r}\|_{L^\infty(\Omega_T)},
\|\mybf{A}\|_{L^\infty(\bm{{\mathcal K}}_0)}, |\mybf{N}_p|_{L^\infty(0,T)} \leq K_0, \\
\| \partial\widetilde{\varphi}^{\rm l,r} \|_{{\mathbb W}^{m-1}(T)}, \| \partial_t\varphi^{\rm l,r} \|_{H^m(\Omega_T)},
| (\partial^m \varphi^{\rm l,r})_{\vert_{x=0}} |_{L^\infty(0,T)}\leq K, \\
\| \underline{{\boldsymbol{u}}} \|_{W^{1,\infty}(\Omega_T)\cap {\mathbb W}^m(T)}, \| \mybf{B} \|_{W^{1,\infty}(\Omega_T)},
\| \partial \mybf{B} \|_{ {\mathbb W}^{m-1}(T)}, |\mybf{N}_p|_{W^{1,\infty}\cap W^{m-1,\infty}(0,T)},
|\partial_t^m\mybf{N}_p|_{L^2(0,T)} \leq K,
\end{cases}
\]
where $\widetilde{\varphi}^{\rm r}(t,x)=\varphi^{\rm r}(t,x)-x$ and
$\widetilde{\varphi}^{\rm l}(t,x)=\varphi^{\rm l}(t,x)+x$.
Then, for any data $\mybf{u}^{\rm in} \in H^m({\mathbb R}_+)$, $\mybf{g}\in H^m(0,T)$, and $\mybf{f}\in H^m(\Omega_T)$
satisfying the compatibility conditions up to order $m-1$,
there exists a unique solution $\mybf{u} \in {\mathbb W}^m(T)$ to the transmission problem \eqref{transmref}.
Moreover, the following estimate holds for any $t \in [0,T]$ and any $\gamma \geq C(K)$:
\begin{align*}
& \@ifstar\@opnorms\@opnorm{ \mybf{u}(t) }_{m,\gamma}
+ \biggl( \gamma\int_0^t\@ifstar\@opnorms\@opnorm{ \mybf{u}(t') }_{m,\gamma}^2{\rm d}t' \biggr)^\frac12
+ | \mybf{u}_{\vert_{x=0}} |_{m,\gamma,t} \\
&\leq C(K_0)\bigl( (1 + |\partial_t^m \mybf{N}_p|_{L^2(0,t)})\@ifstar\@opnorms\@opnorm{ \mybf{u}(0) }_{m}
+ | \mybf{g} |_{H_\gamma^m(0,t)}
+ | \mybf{f}_{\vert_{x=0}} |_{m-1,\gamma,t} + S_{\gamma,t}^*(\@ifstar\@opnorms\@opnorm{ \mybf{f}(\cdot) }_{m}) \bigr).
\end{align*}
Particularly, we also have
\begin{align*}
& \@ifstar\@opnorms\@opnorm{ \mybf{u}(t) }_{m} + | \mybf{u}_{\vert_{x=0}} |_{m,t} \\
&\leq C(K_0)e^{C(K)t} \biggl( (1 + |\partial_t^m \mybf{N}_p|_{L^2(0,t)})\@ifstar\@opnorms\@opnorm{ \mybf{u}(0) }_{m}
+ | \mybf{g} |_{H^m(0,t)}
+ | \mybf{f}_{\vert_{x=0}} |_{m-1,t} + \int_0^t \@ifstar\@opnorms\@opnorm{ \mybf{f}(t') }_{m}{\rm d}t' \biggr).
\end{align*}
\end{theorem}
\subsubsection{Proof of Theorem \ref{theoIBVP3transm}}
As for Theorem \ref{theoIBVP3transm}, we do not seek a direct estimate on $\mybf{u} = (u^{\rm l},u^{\rm r})$
in ${\mathbb W}^m(T)$, but ${\mathbb W}^{m-1}(T)$ estimates of $\mybf{u}$ and
$\dot{\mybf{u}}^{\boldsymbol \varphi} = (\partial_t^{\varphi^{\rm l}}u^{\rm l}, \partial_t^{\varphi^{\rm r}}u^{\rm r})$.
The ${\mathbb W}^{m-1}(T)$ estimate of $\mybf{u}$ is obtained exactly as in Step 1 of the proof of Proposition \ref{propAl}
and requires a variant of Lemma \ref{lemsymmetrizerbis} which is easily obtained by choosing a symmetrizer
$\boldsymbol{\mathcal S}$ given in the subsonic case $p=2$
(with straightforward adadptation in the Lax regime $p=1$) by
\begin{equation}\label{defbigsym}
\boldsymbol{\mathcal S}
= (-\partial_x\varphi^{\rm l})\bigl[ ({\boldsymbol \pi}_-^{\rm l})^{\rm T}{\boldsymbol \pi}_-^{\rm l} + M({\boldsymbol \pi}_+^{\rm l})^{\rm T}{\boldsymbol \pi}_+^{\rm l}\bigr]
+ (\partial_x\varphi^{\rm r})\bigl[({\boldsymbol \pi}_+^{\rm r})^{\rm T}{\boldsymbol \pi}_+^{\rm r}+M({\boldsymbol \pi}_-^{\rm r})^{\rm T}{\boldsymbol \pi}_-^{\rm r}\bigr]
\end{equation}
and by using Theorem \ref{theoIBVP1transm}.
In order to obtain the ${\mathbb W}^{m-1}(T)$ estimates of $\dot{\mybf{u}}^{\boldsymbol \varphi}$,
we first remark that $\dot{\mybf{u}}^{\boldsymbol \varphi}$ solves
\begin{equation}\label{bigIBVP1}
\begin{cases}
\partial_t \dot{\mybf{u}}^{\boldsymbol \varphi} + {\bm{\mathcal{A}}}(\underline{{\boldsymbol{u}}},\partial {\boldsymbol \varphi})\partial_x \dot{\mybf{u}}^{\boldsymbol \varphi}
+ \mybf{B}_{(1)}\dot{\mybf{u}}^{\boldsymbol \varphi} = \mybf{f}_{(1)} & \mbox{ in }\quad \Omega_T, \\
{ \dot{\mybf{u}}^{\boldsymbol \varphi} }_{\ \; \vert_{t=0}} = \mybf{u}^{\rm in}_{(1)} & \mbox{ on }\quad {\mathbb R}_+, \\
\mybf{N}_{(1)}(t){\dot{\mybf{u}}^{\boldsymbol \varphi}}_{\ \; \vert_{x=0}} = \mybf{g}_{(1)}(t) & \mbox{ on }\quad (0,T),
\end{cases}
\end{equation}
where $\mybf{B}_{(1)} = \mbox{diag}(B_{(1)}^{\rm l},B_{(1)}^{\rm r})$ and
$\mybf{f}_{(1)} = (f^{\rm l}_{(1)},f^{\rm r}_{(1)} )$ are straightforwardly deduced from \eqref{equx}
while $\mybf{g}_{(1)} = (g_{(1)}^{\rm l}, g_{(1)}^{\rm r})$ and
$\mybf{N}_{(1)} = \bigl(-N^{\rm l}_{(1)}(t) \quad N^{\rm r}_{(1)}(t)\bigr)$
are obtained using a procedure similar to the one used to derive \eqref{nu1}.
In particular
\[
N^{\rm l}_{(1)}(t) = N_p^{\rm l}\bigl(1-\dot\underline{x} \widetilde{A}({\underline{u}^{\rm l}}_{\vert_{x=0}})^{-1}\bigr), \qquad
N^{\rm r}_{(1)}(t) = N_p^{\rm r}\bigl(1-\dot\underline{x} {A}({\underline{u}^{\rm r}}_{\vert_{x=0}})^{-1}\bigr).
\]
In order to apply Theorem \ref{theoIBVP1transm} to \eqref{bigIBVP1}, it is necessary to show that
the third point in Assumption \ref{asshyptransm} is satisfied.
We therefore consider the Lopatinski\u{\i} matrix $\mybf{L}_{(1)}(t,\underline{{\boldsymbol{u}}}_{\vert_{x=0}})$
associated to \eqref{bigIBVP1}, namely,
\[
\mybf{L}_{(1)}(t,\underline{{\boldsymbol{u}}}_{\vert_{x=0}}) =
\begin{pmatrix} -N^{\rm l}_{(1)}(t) & N^{\rm r}_{(1)}(t)\end{pmatrix} \mybf{E}_p(\underline{{\boldsymbol{u}}}_{\vert_{x=0}}).
\]
When $p=2$ (the case $p=1$ is a straightforward adaptation), one has therefore
\[
\mybf{L}_{(1)}(t,\underline{{\boldsymbol{u}}}_{\vert_{x=0}}) = \mybf{L}_{p}(t,\underline{{\boldsymbol{u}}}_{\vert_{x=0}})
\begin{pmatrix}
1 - \frac{\dot{\underline{x}}}{\widetilde{\lambda}_-({\underline{u}^{\rm l}}_{\vert_{x=0}})} & 0 \\
0 & 1 - \frac{\dot{\underline{x}}}{{\lambda}_+({\underline{u}^{\rm r}}_{\vert_{x=0}})}
\end{pmatrix}
\]
and the required bound on $\mybf{L}_{(1)}(t,\underline{{\boldsymbol{u}}}_{\vert_{x=0}})^{-1}$ is therefore a direct consequence of
Assumption \ref{asstransmred}.
It is therefore possible to apply Theorem \ref{theoIBVP1transm} and to obtain an ${\mathbb W}^{m-1}(T)$ bound on
$\dot{\mybf{u}}^{\boldsymbol \varphi}$ by a close adaptation of the proof of Proposition \ref{propAl}.
Thanks to the block structure of the equations, the end of the proof follows the same lines as the proof of
Theorem \ref{theoIBVP3}, and we therefore omit the details.
\subsection{Application to free boundary transmission problems with a transmission condition of ``kinematic'' type}
\label{secttransmkin}
We consider here a general class of free boundary quasilinear transmission problem in which two quasilinear
hyperbolic systems at the left and at the right of a moving interface located at $x=\underline{x}(t)$ on which
transmission conditions are provided
\begin{equation}\label{transmmovQL}
\begin{cases}
\partial_t U + \widetilde{A}({U})\partial_x U = 0 & \mbox{in }\quad (-\infty,\underline{x}(t)) \quad \mbox{ for } \quad t\in(0,T), \\
\partial_t U + {A}(U)\partial_x U = 0 & \mbox{in }\quad (\underline{x}(t),+\infty) \quad \mbox{ for } \quad t\in(0,T), \\
U_{\vert_{t=0}} = u^{\rm in}(x) & \mbox{on }\quad \mathbb{R}_-\cup\mathbb{R}_+, \\
\underline{N}_p^{\rm r}U_{\vert_{x=\underline{x}(t)+0}} - \underline{N}_p^{\rm l}U_{\vert_{x=\underline{x}(t)-0}}
= \mybf{g}(t) & \mbox{on } \quad (0,T),
\end{cases}
\end{equation}
where we assumed that $\underline{x}(0)=0$ without loss of generality.
Moreover, we assume that the position of the interface is given through a nonlinear equation of the form
\begin{equation}\label{eqinterf}
\dot\underline{x}=\chi(U_{\vert_{x=\underline{x}(t)-0}},U_{\vert_{x=\underline{x}(t)+0}})
\end{equation}
for some smooth function $\chi$ defined on a domain of ${\mathbb R}^2\times {\mathbb R}^2$.
The same reduction as in \S \ref{secttransmmov}, and using the same notations, leads us to consider the
$4\times 4$ initial boundary value problem
\begin{equation}\label{bigIBVPQL}
\begin{cases}
\partial_t \mybf{u} + {\bm{\mathcal{A}}}(\mybf{u},\partial {\boldsymbol \varphi})\partial_x \mybf{u} = 0 & \mbox{ in }\quad \Omega_T, \\
\mybf{u}_{\vert_{t=0}} = \mybf{u}^{\rm in}(x) & \mbox{ on }\quad {\mathbb R}_+, \\
\underline{\mybf{N}}_p\mybf{u}_{\vert_{x=0}} = \mybf{g}(t) & \mbox{ on }\quad (0,T),
\end{cases}
\end{equation}
where $\underline{\mybf{N}}_p = \big(-\underline{N}_p^{\rm l} \ \ \underline{N}_p^{\rm r}\big)$ is here,
for the sake of simplicity, a {\it constant} $p\times 4$ matrix (the value of $p$ is discussed below).
These equations are complemented by the evolution equation
\begin{equation}\label{eqinterf2}
\dot\underline{x} = \chi(\mybf{u}_{\vert_{x=0}}).
\end{equation}
This boundary condition, of ``kinematic'' type, leads us to work with the following generalization of the
``Lagrangian'' diffeomorphism \eqref{diffeo},
\begin{equation}\label{choicediffeo}
\varphi(t,x) = x + \psi\Bigl(\frac{x}{\varepsilon}\Bigr)\int_0^t \chi(\mybf{u}(t',|x|)){\rm d}t',
\end{equation}
where $\psi\in C_0^\infty({\mathbb R})$ is an even cut-off function such that $\psi(x)=1$ for $\abs{x}\leq1$ and $=0$ for
$\abs{x}\geq 2$, while $\varepsilon$ is chosen small enough to have $\mybf{u}$ close enough to its initial boundary
value when $x$ is in the support of $\psi$ and $t$ small enough.
Contrary to \eqref{diffeo}, this cut-off is necessary here because $\chi$ might not be defined at the origin
(this is for instance the case in \S \ref{sectshocks} for the evolution of shocks).
In particular, we have
\[
\varphi^{\rm l}(t,x) = -x+\psi\Bigl(\frac{x}{\varepsilon}\Bigr)\int_0^t \chi(\mybf{u}(t',x)){\rm d}t' \quad\mbox{ and }\quad
\varphi^{\rm r}(t,x) = x+\psi\Bigl(\frac{x}{\varepsilon}\Bigr)\int_0^t \chi(\mybf{u}(t',x)){\rm d}t',
\]
and $\varphi^{\rm l,r}$ satisfy the same kind of bounds as those given in Lemma \ref{lemdiffeo}
(with $\widetilde{\varphi}^{\rm r}(t,x) = \varphi^{\rm r}(t,x)-x$ and
$\widetilde{\varphi}^{\rm l}(t,x) = \varphi^{\rm l}(t,x)+x$).
The well-posedness of \eqref{bigIBVPQL}--\eqref{choicediffeo} also requires the following assumption.
\begin{assumption}\label{asshypQLFBtransm}
Let $\widetilde{{\mathcal U}}$ and ${\mathcal U}$ be two open sets in ${\mathbb R}^2$ and let $\boldsymbol{{\mathcal U}}=\widetilde{{\mathcal U}}\times{\mathcal U}$
representing a phase space of $\mybf{u}$.
Let $\widetilde{{\mathcal U}}_I \subset \widetilde{{\mathcal U}}$ and ${\mathcal U}_I \subset {\mathcal U}$ be also open sets and let
$\boldsymbol{{\mathcal U}}_I = \widetilde{{\mathcal U}}_I\times{\mathcal U}_I$ representing a phase space of $\mybf{u}_{\vert_{x=0}}$.
The following conditions hold:
\begin{enumerate}
\setlength{\itemsep}{3pt}
\item[{\bf i.}]
$\mybf{A} \in C^\infty(\boldsymbol{{\mathcal U}})$ and $\chi\in C^\infty(\boldsymbol{{\mathcal U}}_I)$.
\item[{\bf ii.}]
For all $\mybf{u} = (u^{\rm l},u^{\rm r})^{\rm T}\in\boldsymbol{\mathcal U}$, the matrices $\widetilde{A}({u^{\rm l}})$
and $A(u^{\rm r})$ have eigenvalues $\widetilde{\lambda}_+(u^{\rm l}), -\widetilde{\lambda}_-(u^{\rm l})$
and $\lambda_+(u^{\rm r}), -\lambda_-(u^{\rm r})$, respectively, satisfying
\[
\widetilde{\lambda}_\pm(u^{\rm l})>0 \quad\mbox{ and }\quad {\lambda}_\pm(u^{\rm r})>0;
\]
moreover, one of the following situations
for any $\mybf{u} = (u^{\rm l},u^{\rm r})^{\rm T}\in\boldsymbol{\mathcal U}_I$ holds:
\begin{align*}
a) \qquad \widetilde{\lambda}_\pm(u^{\rm l})\mp \chi(\mybf{u}) > 0
& \quad\mbox{ and }\quad
\lambda_\pm(u^{\rm r})\mp \chi({\mathbf u}) > 0, \\
b) \qquad\widetilde{\lambda}_\pm(u^{\rm l}) \mp \chi(\mybf{u}) > 0
& \quad\mbox{ and }\quad \lambda_+(u^{\rm r}) - \chi(\mybf{u}) < 0, \\
c)\qquad \widetilde{\lambda}_-(u^{\rm l})+\chi(\mybf{u}) < 0
& \quad\mbox{ and }\quad \lambda_\pm(u^{\rm r}) \mp \chi(\mybf{u}) > 0.
\end{align*}
\item[{\bf iii.}]
For any $\mybf{u}\in \boldsymbol{\mathcal U}_I$, the Lopatinski\u{\i} matrix $\mybf{L}_p(\mybf{u})$ associated to
the condition $a)$, $b)$, or $c)$ constructed in \eqref{interfmatr} is invertible
(note that $p=2$ under condition $a)$ and $p=1$ under conditions $b)$ and $c)$).
\end{enumerate}
\end{assumption}
\begin{remark}
With the terminology introduced in the previous section, condition $a)$ corresponds to an interface moving
at subsonic speed, while conditions $b)$ and $c)$ correspond to interfaces moving at supersonic speed
(to the right for condition $a)$ and to the left for condition $b)$) and satisfying Lax's conditions.
\end{remark}
We can now state the following theorem, which can be deduced from Theorem \ref{theoIBVP3transm} in exactly
the same way as Theorem \ref{theoIBVP4} is deduced from Theorem \ref{theoIBVP3} for free boundary initial
value problem with an evolution equation of kinematic type for the location of the boundary.
\begin{theorem}\label{theoIBVP4transm}
Let $m\geq 2$ be an integer.
Suppose that Assumption \ref{asshypQLFBtransm} is satisfied.
If $\mybf{u}^{\rm in}\in H^m({\mathbb R}_+)$ takes its values in $\widetilde{\mathcal K}_0\times {\mathcal K}_0$ with
$\widetilde{\mathcal K}_0\subset\widetilde{{\mathcal U}}$ and $ {\mathcal K}_0\subset {\mathcal U}$ compact and convex sets,
if $\mybf{u}^{\rm in}(0) \in \boldsymbol{\mathcal U}_I$,
and if the data $\mybf{u}^{\rm in}$ and $\mybf{g} \in H^m(0,T)$ satisfy the compatibility conditions up to
order $m-1$, then there exist $T_1 \in (0,T]$ and a unique solution
$(\mybf{u},\underline{x})$ to \eqref{transmmovQL}--\eqref{eqinterf} with $\mybf{u}\in {\mathbb W}^m(T_1)$, $\underline{x}\in H^{m+1}(0,T_1)$,
and $\varphi$ given by \eqref{choicediffeo}.
\end{theorem}
\section{Waves interacting with a lateral piston}\label{sectlatpis}
We analyze here a particular example of wave-structure interaction in which the fluid occupies
a semi-infinite canal over a flat bottom which is delimited by a lateral wall that can move horizontally.
When the wall is in forced motion, this situation corresponds to a wave-maker device
often used to generate waves in wave-flumes \cite{katell2002accuracy,orszaghova2012paddle}.
We are more interested here in the case where the lateral wall moves under the action of the hydrodynamic force
created by the waves and of a spring force that tends to bring it back to its equilibrium position.
This configuration corresponds to a wave absorption mechanism and can also be seen as a simplified model
of wave energy convertor, such as the Oyster.
Such a configuration has been studied numerically in various references \cite{he2009nonlinear, korobkin2009motion,
khakimzyanov2017numerical}, but there is no mathematical result available yet.
Note also that this problem is related to the piston problem for isentropic gas dynamics whose linear analysis
can be found in \cite{gerlach1984two} and weak solutions constructed in \cite{takeno1995free}.
Our goal in this section is to provide a well-posedness result for this wave-structure interaction under
the shallow water approximation, i.e., assuming that the evolution of the free surface is governed by
the nonlinear shallow water equations.
The configuration under study here is described in Figure \ref{fpiston}.
\medskip
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.7\linewidth]{Piston.eps}
\end{center}
\setlength{\unitlength}{1pt}
\begin{picture}(0,0)
\put(-55,6){$\underline{x}(t)$}
\put(35,84){$z=Z(t,x)$}
\end{picture}
\caption{Waves interacting with a lateral piston}\label{fpiston}
\end{figure}
\subsection{Presentation of the problem}\label{sectprespb1}
In the canal, of mean depth $h_0$ and delimited on the left by the moving wall located at $x=\underline{x}(t)$,
the waves are described by the nonlinear shallow water equations.
It is convenient to write them in $(H,\overline{V})$ variables, where $H(t,x)=h_0+Z(t,x)$ is the water depth,
$Z(t,x)$ is the surface elevation of the water, and $\overline{V}(t,x)$ is the vertically averaged horizontal velocity
\begin{equation}\label{SW}
\begin{cases}
\partial_t H + \partial_x (H\overline{V}) = 0 & \mbox{in}\quad (\underline{x}(t),\infty), \\
\partial_t \overline{V} + \overline{V}\partial_x\overline{V} + \mathtt{g}\partial_x H = 0 & \mbox{in}\quad (\underline{x}(t),\infty),
\end{cases}
\end{equation}
where $\mathtt{g}$ is the gravitational constant;
with this formulation, the boundary condition at the left boundary at the canal will be imposed as the kinematic type:
the velocity $\overline{V}$ matches the velocity $\dot{\underline{x}}$, that is,
\begin{equation}\label{BCSW}
\overline{V}(t,\underline{x}(t)) = \dot{\underline{x}}(t).
\end{equation}
Since the wall moves under the action of the hydrodynamic force exerted by the fluid and of the spring force,
its position $\underline{x}(t)$ satisfies Newton's equation
$$
\mathtt{m}\ddot{\underline{x}} = -\mathtt{k}(\underline{x}-\underline{x}_0) + F_{\rm hyd},
$$
where $\mathtt{m}$ is the mass of the moving wall, $\mathtt{k}$ the stiffness of the spring force,
$\underline{x}_{0}$ its reference position, and $F_{\rm hyd}$ the hydrodynamic force.
This force corresponds to the horizontal pressure forces integrated on the vertical wall.
Assuming, in accordance with the modeling of the flow by the nonlinear shallow water equations,
that the pressure is hydrostatic, we get
\begin{align*}
F_{\rm hyd} &= \int_{-h_0}^{Z(t,\underline{x}(t))} \rho\mathtt{g} (Z(t,\underline{x}(t)) - z'){\rm d}z' \\
&= \frac{1}{2}\rho\mathtt{g} (h_0 + Z(t,\underline{x}(t)))^2.
\end{align*}
At rest, we have $H=h_0$ and the equilibrium position $\underline{x}_{\rm eq}$ is therefore given by
$$
\underline{x}_{\rm eq} - \underline{x}_0 = \frac{1}{2}\frac{\rho\mathtt{g} h_0^2}{\mathtt{k}}
$$
so that Newton's equation can be put under the form
\begin{equation}\label{NewtonSW}
\mathtt{m}\ddot{\underline{x}} = - \mathtt{k} (\underline{x}-\underline{x}_{\rm eq})
+ \frac{1}{2}\rho\mathtt{g} \bigl( (h_0+Z_{\vert_{x=\underline{x}}})^2 - h_0^2 \bigr).
\end{equation}
The free boundary problem we have to solve consists therefore in the equations \eqref{SW}--\eqref{NewtonSW}
complemented by the initial conditions
\begin{equation}\label{ICSW}
(Z,\overline{V})_{\vert_{t=0}} = (Z^{\rm in},\overline{V}^{\rm in}) \quad\mbox{on}\quad {\mathbb R}_+, \qquad
(\underline{x},\dot{\underline{x}})_{\vert_{t=0}} = (0,\underline{x}_1^{\rm in}),
\end{equation}
where we assumed without loss of generality that the wall is initially located at $x=0$.
\subsection{Reformulation of the equations}
As in \S \ref{sectVCm}, the first step is to use a diffeomorphism $\varphi(t,\cdot): {\mathbb R}_+\to (\underline{x}(t),\infty)$
and to work with the transform variables
$$
\zeta(t,x) = Z(t,\varphi(t,x)), \qquad \overline{v}(t,x) = \overline{V}(t,\varphi(t,x))
$$
with $h=h_0+\zeta$.
The boundary condition \eqref{BCSW} which can be rewritten as
$$
\dot{\underline{x}}(t) = \overline{v}(t,0)
$$
leads us to work with the Lagrangian diffeomorphism
\begin{equation}\label{diffL}
\varphi(t,x) = x + \int_0^t \overline{v}(t',x){\rm d}t',
\end{equation}
which satisfies the properties stated in Lemma \ref{lemdiffeo}.
After composition with $\varphi$, the problem under consideration is reduced to the initial boundary value problem
\begin{equation}\label{SW2}
\begin{cases}
\displaystyle \partial_t \zeta + h \partial_x^\varphi \overline{v} = 0 & \mbox{in}\quad \Omega_T, \\
\displaystyle \partial_t \overline{v} + \mathtt{g} \partial_x^\varphi \zeta = 0 & \mbox{in}\quad \Omega_T, \\
(\zeta,\overline{v})_{\vert_{t=0}} = (\zeta^{\rm in},\overline{v}^{\rm in}) & \mbox{on}\quad {\mathbb R}_{+}, \\
\overline{v}_{\vert_{x=0}} = \dot{x} & \mbox{on}\quad (0,T),
\end{cases}
\end{equation}
coupled to the ODE
\begin{equation}\label{NewtonSW2}
\begin{cases}
\mathtt{m}\ddot{\underline{x}} = -\mathtt{k}(\underline{x}-\underline{x}_{\rm eq})
+ \frac{1}{2}\rho\mathtt{g} \bigl( (h_0+\zeta_{\vert_{x=0}})^2 - h_0^2 \bigr)
\quad\mbox{for}\quad t\in(0,T), \\
(\underline{x},\dot{\underline{x}})_{\vert_{t=0}} = (0,\underline{x}_1^{\rm in}),
\end{cases}
\end{equation}
where we used the same notation as in \eqref{dtphi}, that is, $\partial_x^\varphi = \frac{1}{\partial_x\varphi}\partial_x$.
The initial boundary value problem \eqref{SW2} is of course of the form \eqref{IBVPmT} with $u=(\zeta,\overline{v})^{\rm T}$, $\nu=(0,1)^{\rm T}$, and
\begin{equation}\label{Au}
A(u) = \begin{pmatrix} \overline{v} & h \\ \mathtt{g} & \overline{v} \end{pmatrix},
\end{equation}
whose eigenvalues $\pm\lambda_{\pm}(u)$ and the corresponding unit eigen vectors $\mathbf{e}_{\pm}(u)$
are given by
\[
\lambda_{\pm}(u) = \sqrt{\mathtt{g}h} \pm \overline{v}, \qquad
\mathbf{e}_{\pm}(u) =\frac{1}{\sqrt{\mathtt{g}+h}
\binom{\sqrt{h}}{\pm\sqrt{\mathtt{g}}}. }
\]
Therefore, the positivity of $|\nu\cdot\mathbf{e}_{+}(u_{\vert_{x=0}})|$ stated in Assumption \ref{asshypm}
is automatically satisfied under the positivity of $h$.
Here, we will show another equivalent formulation to \eqref{SW2}--\eqref{NewtonSW2}.
The following lemma shows that \eqref{NewtonSW2} provides an expression for $\dot\underline{x}$
in terms of $\zeta_{\vert_{x=0}}$.
\begin{lemma}\label{lemmaG}
Let $m\geq1$ be an integer, $\underline{x}_1^{\rm in}\in {\mathbb R}$, and assume that $\zeta_{\rm b}\in H^m(0,T)$.
Then there exists a unique solution $\underline{x} \in H^{m+2}(0,T)$ to
\[
\begin{cases}
{\mathtt m}\ddot{\underline{x}} = -{\mathtt k} (\underline{x} - \underline{x}_{\rm eq})
+ \frac12 \rho {\mathtt g} \bigl( \zeta_{\rm b}^2 + 2h_0\zeta_{\rm b} \bigr), \\
(\underline{x},\dot \underline{x})_{\vert_{t=0}} = (0,\underline{x}_1^{\rm in}),
\end{cases}
\]
so that we can define a mapping $\mathcal{G} : H^m(0,T) \ni \zeta_{\rm b} \mapsto \dot{\underline{x}} \in H^{m+1}(0,T)$,
which satisfies
\[
|\mathcal{G}(\zeta_{\rm b})|_{H^{m+1}(0,t)}
\leq C\bigl( \sqrt{t}(|\underline{x}_{\rm eq}|+|\underline{x}_1^{\rm in}|)
+ (1+t)( 1 + |\zeta_{\rm b}|_{W^{[m/2],\infty}(0,t)} ) |\zeta_{\rm b}|_{H^m(0,t)} \bigr)
\]
for any $t\in[0,T]$, where $C>0$ is a constant depending only on $\mathtt{m},\mathtt{k},\rho\mathtt{g},h_0$,
and $m$.
\end{lemma}
\begin{proof}
The existence and uniqueness of the solution $\underline{x}$ is obvious, so that we focus on the derivation of the estimate.
Replacing $\underline{x}$ with $\underline{x}+\underline{x}_{\rm eq}$, it is sufficient to consider the problem
\[
\begin{cases}
{\mathtt m}\ddot{\underline{x}} = -\mathtt{k}\underline{x} + f, \\
(\underline{x},\dot \underline{x})_{\vert_{t=0}} = (\underline{x}_{\rm eq},\underline{x}_1^{\rm in}),
\end{cases}
\]
where $f=\frac12 \rho {\mathtt g} \bigl( \zeta_{\rm b}^2 + 2h_0\zeta_{\rm b} \bigr)$.
Then, we see that
\[
\frac12\frac{\rm d}{{\rm d}t}( \mathtt{m} \dot{\underline{x}}(t)^2 + \mathtt{k} \underline{x}(t)^2 ) = f(t)\dot{x}(t),
\]
from which we deduce that
\begin{align*}
|\dot{\underline{x}}(t)| + |\underline{x}(t)|
&\leq C\Bigl( |\underline{x}_1^{\rm in}| + |\underline{x}_{\rm eq}| + \int_0^t|f(t')|{\rm d}t' \Bigr) \\
&\leq C( |\underline{x}_1^{\rm in}| + |\underline{x}_{\rm eq}| + \sqrt{t}|f|_{L^2(0,t)} ),
\end{align*}
so that
\[
|\underline{x}|_{H^1(0,t)} \leq C\bigl( \sqrt{t}(|\underline{x}_1^{\rm in}| + |\underline{x}_{\rm eq}|) + t|f|_{L^2(0,t)} \bigr).
\]
On the other hand, it follows from the equation directly that
\[
|\partial_t^{k+2}\underline{x}|_{L^2(0,t)} \leq C(|\partial_t^k\underline{x}|_{L^2(0,t)}+|\partial_t^kf|_{L^2(0,t)})
\]
for $k=0,1,2,\ldots$.
Using these inductively, we obtain
\[
|\underline{x}|_{H^{m+2}(0,t)}
\leq C\bigl( \sqrt{t}(|\underline{x}_1^{\rm in}| + |\underline{x}_{\rm eq}|) + t|f|_{L^2(0,t)} + |f|_{H^m(0,t)} \bigr),
\]
which together with
$|f|_{H^m(0,t)} \leq C( 1 + |\zeta_{\rm b}|_{W^{[m/2],\infty}(0,t)} ) |\zeta_{\rm b}|_{H^m(0,t)}$
gives the desired estimate.
\end{proof}
It follows from the lines above that the problem presented in \S \ref{sectprespb1} can be recast
under the following form
\begin{equation}\label{pb1recast}
\begin{cases}
\partial_t u + \mathcal{A}(u,\partial\varphi)\partial_x u = 0 & \mbox{in}\quad \Omega_T, \\
u_{\vert_{t=0}} = u^{\rm in} & \mbox{on}\quad {\mathbb R}_{+}, \\
\underline{\nu}\cdot u_{\vert_{x=0}} = \mathcal{G}(\underline{\nu}^\perp\cdot u_{\vert_{x=0}}) & \mbox{on}\quad (0,T),
\end{cases}
\end{equation}
where $\underline{\nu}=(0,1)^{\rm T}$ and $\varphi$ is given by \eqref{diffL}, with a boundary equation given by
\begin{equation}\label{BCSWt}
\dot\underline{x}=\underline{\nu}\cdot u_{\vert_{x=0}}, \qquad \underline{x}_{\vert_{t=0}}=0.
\end{equation}
Here, we emphasize that the notation for the matrix $\mathcal{A}(u,\varphi)$ is the same as in \eqref{IBVPmT} with
the matrix $A(u)$ defined by \eqref{Au}.
However, thanks to our choice of the Lagrangian diffeomorphism $\varphi$,
the term $\partial_t\varphi$ is cancelled and does not appear in the equation.
The problem is therefore a small variant of the free boundary problem considered in \S \ref{sectFB1},
the difference being that the boundary condition $\underline{\nu}\cdot u_{\vert_{x=0}}=g(t)$ is replaced by a semi-linear
and nonlocal boundary condition $\underline{\nu}\cdot u_{\vert_{x=0}} = {\mathcal G}(\nu^\perp\cdot u_{\vert_{x=0}})$.
Of course, \eqref{pb1recast}--\eqref{BCSWt} is equivalent to \eqref{SW2}--\eqref{NewtonSW2}.
\subsection{Compatibility condition}
As usual, compatibility conditions are required to have regular solutions.
However, we can derive the conditions easier than the problem considered in \S \ref{sectFB1} because
the equation does not contain the term $\partial_t\varphi$.
Denoting $u_k=\partial_t^k u$, we get classically by induction that $u_k$ is a polynomial expression of space derivatives
of $u$ of order at most $k$, and of space and time derivatives of $(\partial_x\varphi)^{-1}$ of order at most $k-1$.
Remarking further that $\partial_x^{j}\partial_t^{l+1} \varphi= \partial_x^j\partial_t^l\overline{v}$ and $\partial_x^{j+1}\varphi_{\vert_{t=0}}=\delta_{j,0}$,
where $\delta_{j,0}$ is the Kronecker symbol, it follows that at $t=0$, we have an expression for
$u_k^{\rm in}={u_k}_{\vert_{t=0}}$ as
\begin{equation}\label{pb1comp}
u_k^{\rm in}=c_{1,k}(u^{\rm in},\partial_x u^{\rm in},\dots, \partial_x^k u^{\rm in})
\end{equation}
with $c_{1,k}$ a polynomial expression of its arguments such that the total number of derivatives of $u^{\rm in}$
involved in each monomial is at most $k$.
Using the equation in \eqref{NewtonSW2} we can express $\underline{x}_k^{\rm in}$ for $k\geq2$ in terms of the initial data as
\begin{equation}\label{xk+2in}
\underline{x}_{k+2}^{\rm in} = c_{2,k}(\underline{x}_1^{\rm in},\zeta^{\rm in},\zeta_1^{\rm in},\ldots,\zeta_k^{\rm in})_{\vert_{x=0}}
\end{equation}
with $c_{2,k}$ a polynomial expression of its arguments.
The compatibility condition is obtained by differentiating the boundary condition
$\overline{v}_{\vert_{x=0}} = \dot{\underline{x}}$ with respect to $t$ and taking its trace at $t=0$.
\begin{definition}\label{defcomppb1}
Let $m\geq1$ be an integer.
We say that the initial data $u^{\rm in} = (\zeta^{\rm in},\overline{v}^{\rm in})^{\rm T} \in H^m({\mathbb R}_+)$ and
$\underline{x}_1^{\rm in}\in {\mathbb R}$ for the initial boundary value problem \eqref{SW2}--\eqref{NewtonSW2} satisfy
the compatibility condition at order $k$ if $\{u_j^{\rm in}\}_{j=0}^m$ and $\{\underline{x}_j^{\rm in}\}_{j=1}^{m+1}$
defined by \eqref{pb1comp}--\eqref{xk+2in} satisfy
\[
{\overline{v}_k^{\rm in}}_{\vert_{x=0}} = \underline{x}_{k+1}^{\rm in}.
\]
We also say that the initial data $u^{\rm in}$ and $\underline{x}_1^{\rm in}$ satisfy the compatibility conditions
up to $m-1$ if they satisfy the compatibility conditions at order $k$ for $k=0,1,\ldots,m-1$.
\end{definition}
\begin{remark}
The local existence theorem given below requires that the compatibility conditions are satisfied at order $m-1$ with $m\geq 2$. In the case $m=2$, the compatibility conditions are
$$
{\overline{v}^{\rm in}}_{\vert_{x=0}} = \underline{x}_1^{\rm in} \quad\mbox{and}\quad
-\mathtt{g} (\partial_x \zeta^{\rm in})_{\vert_{x=0}}
= \mathtt{k}\underline{x}_{\rm eq} + \frac{\rho\mathtt{g}}{2\mathtt{m}}
\bigl( (\zeta^{\rm in})^2 + 2h_0\zeta^{\rm in} \bigr)_{\vert_{x=0}}.
$$
\end{remark}
\subsection{Local well-posedness}
We can now state the main result of this section, which shows the local well-posedness of the wave-structure
interaction problem presented in \S \ref{sectprespb1}.
\begin{theorem}
Let $m\geq 2$ be an integer.
If the initial data $(\zeta^{\rm in},\overline{v}^{\rm in})^{\rm T}\in H^m({\mathbb R}_+)$ and $\underline{x}_1^{\rm in} \in {\mathbb R}$ satisfy
\[
\inf_{x\in{\mathbb R}_{+}}\bigl( \sqrt{\mathtt{g}(h_0+\zeta^{\rm in}(x))} - |\overline{v}^{\rm in}(x)| \bigr) > 0
\]
and the compatibility conditions up to order $m-1$ in the sense of Definition \ref{defcomppb1},
then there exist $T>0$ and a unique solution $(\zeta,\overline{v},\underline{x})$ to \eqref{SW2}--\eqref{NewtonSW2} with
$(\zeta,\overline{v})\in {\mathbb W}^m(T)$ and $\underline{x}\in H^{m+2}(0,T)$, and $\varphi$ given by \eqref{diffL}.
\end{theorem}
\begin{proof}
The proof is a small variant of the proof of Theorem \ref{theoIBVP4}.
We define the phase space ${\mathcal U}$ of $u=(\zeta,\overline{v})^{\rm T}$ by
\[
{\mathcal U} = \{ u=(\zeta,\overline{v})^{\rm T} \in {\mathbb R}^2 \,|\, \sqrt{\mathtt{g}(h_0+\zeta)} - |\overline{v}|>0 \}.
\]
Then, we can readily check that all the conditions in Assumption \ref{asshypQLFB} are satisfied with
$\chi(u)=\overline{v}$ and $\underline{\nu}=(0,1)^{\rm T}$.
Moreover, once $u^n=(\zeta^n,\overline{v}^n)^{\rm T} \in {\mathbb W}^m(T)$ is given so that
\begin{equation}\label{unifest4}
\begin{cases}
(\partial_t^k u^n)_{\vert_{t=0}}=u_k^{\rm in} \quad\mbox{for}\quad k=0,1,\ldots,m-1, \\
\|u^n\|_{{\mathbb W}^m(T)} + |{u^n}_{\vert_{x=0}}|_{m,T} \leq M_1,
\end{cases}
\end{equation}
we can check that the data $u^{\rm in}$ and $g^n(t)=\mathcal{G}(\underline{\nu}^\perp\cdot {u^n}_{\vert_{x=0}})$ for the problem
\[
\begin{cases}
\partial_t u + \mathcal{A}(u,\partial\varphi)\partial_x u = 0 & \mbox{in}\quad \Omega_T, \\
u_{\vert_{t=0}} = u^{\rm in}(x) & \mbox{on}\quad {\mathbb R}_{+}, \\
\underline{\nu}\cdot u_{\vert_{x=0}} = g^n(t) & \mbox{on}\quad (0,T),
\end{cases}
\]
\[
\dot\underline{x} = \underline{\nu}\cdot u_{\vert_{x=0}}, \qquad \underline{x}_{\vert_{t=0}}=0,
\]
satisfy the compatibility conditions up to order $m-1$ in the sense of Definition \ref{defcompfbp},
and we can apply Theorem \ref{theoIBVP4} to show a unique existence of the solution
$u=(\zeta,\overline{v})^{\rm T} \in {\mathbb W}^m(T_1)$ and $\underline{x} \in H^{m+1}(0,T_1)$ to this problem for some $T_1 \in (0,T]$
depending on $M_1$.
We denote by $u^{n+1}$ this solution $u$.
Furthermore, we see that $u^{n+1}$ satisfies $(\partial_t^k u^{n+1})_{\vert_{t=0}}=u_k^{\rm in}$ for $k=0,1,\ldots,m-1$ and
\[
\|u^{n+1}\|_{{\mathbb W}^m(T_1)} + |{u^{n+1}}_{\vert_{x=0}}|_{m,T_1} \leq C_1(|\mathcal{G}(\underline{\nu}^\perp\cdot {u^n}_{\vert_{x=0}})|_{H^m(0,T_1)}).
\]
Here, by Lemma \ref{lemmaG} we have
\[
|\mathcal{G}(\underline{\nu}^\perp\cdot {u^n}_{\vert_{x=0}})|_{H^{m+1}(0,T_1)} \leq C(M_1,T_1).
\]
On the other hand, we have
\begin{align*}
|\mathcal{G}(\underline{\nu}^\perp\cdot {u^n}_{\vert_{x=0}})|_{H^m(0,T_1)}
&\leq \sqrt{T_1}\sum_{j=1}^{m+1}|\underline{x}_j^{\rm in}|
+ T_1|\mathcal{G}(\underline{\nu}^\perp\cdot {u^n}_{\vert_{x=0}})|_{H^{m+1}(0,T_1)},
\end{align*}
where we used $(\partial_t^k \mathcal{G}(\underline{\nu}^\perp\cdot {u^n}_{\vert_{x=0}}))_{\vert_{t=0}} = \underline{x}_{k+1}^{\rm in}$
for $k=0,1,\ldots,m$.
Therefore, for any fixed $M_0>0$ if we define $M_1>0$ by $M_1=C_1(M_0)$ and choose $T_1=T_1(M_0)$
sufficiently small, then we have
\[
|\mathcal{G}(\underline{\nu}^\perp\cdot {u^n}_{\vert_{x=0}})|_{H^m(0,T_1)} \leq M_0,
\]
so that $u^{n+1}$ satisfies \eqref{unifest4} with $T$ replaced by $T_1$.
Now, we can iterate the above procedure to construct a sequence of approximate solutions
$\{(\zeta^n,\overline{v}^n,\underline{x}^n)\}_n$, which satisfy the uniform bounds.
As in the proof of Theorem \ref{theoIBVP4}, we can prove the convergence of these approximate solutions
to the solution $(\zeta,\overline{v},\underline{x})$ to \eqref{pb1recast}--\eqref{BCSWt}.
This solution satisfies $\dot{\underline{x}} = \mathcal{G}(\underline{\nu}^\perp\cdot {u}_{\vert_{x=0}}) \in H^{m+1}(0,T_1)$,
so that we have the regularity $\underline{x} \in H^{m+2}(0,T_1)$.
\end{proof}
\section{Shallow water model with a floating body on the water surface} \label{sectfloat}
We turn to analyze other examples of wave-structure interaction in which the fluid occupies an infinite canal
and a floating rigid body is placed on the water surface.
We follow the approach proposed in \cite{Lannes2017} where the free surface Euler equations are reformulated
in terms of the free surface elevation and of the horizontal water flux.
Under this approach, the pressure exerted by the fluid on the floating body can be viewed as the Lagrange
multiplier associated to the constraint that, under the body, the surface of the fluid coincides
with the bottom of the body.
As shown in \cite{Lannes2017}, this approach can be used also in the shallow water approximation,
replacing the free surface Euler equations by the much simpler nonlinear shallow water equations.
This is the framework that we shall consider here, addressing three cases; the floating body is fixed,
the motion of the body is prescribed, and the body moves freely according to Newton's laws
under the action of the gravitational force and the pressure from the air and from the water.
The case of a floating body moving only vertically and with vertical lateral walls has been considered in
\cite{Lannes2017} in $1D$, in \cite{Bocchi} for a $2D$ configuration with radial symmetry, and numerical computations have been proposed in \cite{Bosi}.
For such configurations, the horizontal projection of the portion of the solid in contact with the water is
independent of time.
We consider here the more complex situation of nonvertical lateral walls:
even in the case of a fixed object, determining the portion of the solid in contact with the water
is then a free boundary problem that is difficult to handle; in the numerical study \cite{Godlewski} for instance, the authors use a compressible approximation of the equations in order to remove this issue.
The configuration under study here is described in Figure \ref{ffloating}.
\bigskip
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.7\linewidth]{Floating.eps}
\end{center}
\setlength{\unitlength}{1pt}
\begin{picture}(0,0)
\put(-75,6){$x_{-}(t)$}
\put(60,6){$x_{+}(t)$}
\put(-5,10){$\mathcal{I}(t)$}
\put(-130,10){$\mathcal{E}_{-}(t)$}
\put(110,10){$\mathcal{E}_{+}(t)$}
\put(-25,54){$z=Z_{\rm i}(t,x)$}
\put(110,85){$z=Z_{\rm e}(t,x)$}
\put(-160,85){$z=Z_{\rm e}(t,x)$}
\end{picture}
\caption{Waves interacting with a floating body}\label{ffloating}
\end{figure}
\subsection{Presentation of the equations for the water}\label{sectpresfloat}
We consider the two-dimensional water waves over a flat bottom with a floating body on the water surface
under the assumption that there are only two contact points where the water, the air, and the body meet.
These contact points at time $t$ are denoted by $x_{-}(t)$ and $x_{+}(t)$, which satisfy $x_{-}(t)<x_{+}(t)$.
Let $\mathcal{I}(t)$ and $\mathcal{E}(t)$ be the projections on the horizontal line of the parts where
the water surface contacts with the floating structure and the air, respectively, that is,
\[
\begin{cases}
\mathcal{I}(t) = (x_{-}(t),x_{+}(t)), \\
\mathcal{E}(t) = \mathcal{E}_{-}(t) \cup \mathcal{E}_{+}(t), \quad
\mathcal{E}_{-}(t)=(-\infty,x_{-}(t)), \quad \mathcal{E}_{-}(t)=(x_{+}(t),\infty).
\end{cases}
\]
The corresponding water regions to $\mathcal{I}(t)$ and $\mathcal{E}(t)$ will be called the interior and
the exterior regions, respectively.
We consider the case where overhanging waves do not occur and suppose that the surface elevation of the water
in the exterior region is denoted by $Z_{\rm e}(t,x)$ and that the underside of the floating body is
parameterized by $Z_{\rm i}(t,x)$, where $x$ is the horizontal coordinate.
Let $h_0$ be the mean depth of the water, so that the water depth in the interior and exterior regions are
given by $H_{\rm i}(t,x) = h_0 + Z_{\rm i}(t,x)$ and $H_{\rm e}(t,x) = h_0 + Z_{\rm e}(t,x)$, respectively.
We denote by $\overline{V}(t,x)$ the vertically averaged horizontal velocity and put $Q = H\overline{V}$, which is the
horizontal flux of the water.
The restrictions of $Q$ to the interior and the exterior regions will be denoted by
$Q_{\rm i}$ and $Q_{\rm e}$, respectively.
Let $\underline{P}_{\rm i}(t,x)$ be the pressure of the water at the underside of the floating body.
This pressure is an important unknown quantity and should be determined together with the motion of the water.
In the case where the floating body moves freely,
the body interacts with the water through the force exerted by this pressure.
The shallow water model was derived from the full water wave equations by using the assumption that
$\partial_x \big( \int_{-h_0}^\zeta V(t,x,z)^2 dz\big) \approx \partial_x \big( H \overline{V}^2\big)$,
where $V(t,x,z)$ denotes the horizontal component of the velocity field in the fluid,
and that the pressure $P(t,x,z)$ can be approximated by the hydrostatic pressure, that is,
\[
P(t,x,z) =
\begin{cases}
P_{\rm atm} - \rho\mathtt{g}(z-Z_{\rm e}(t,x)) & \mbox{in}\quad \mathcal{E}(t), \\
\underline{P}_{\rm i}(t,x) - \rho\mathtt{g}(z-Z_{\rm i}(t,x)) & \mbox{in}\quad \mathcal{I}(t),
\end{cases}
\]
where $\rho$ is the density of the water, $\mathtt{g}$ the gravitational constant, and
$P_{\rm atm}$ the atmospheric pressure (see \cite{Lannes2017}).
Then, the shallow water model for the water has the form
\begin{equation}\label{eqext}
\begin{cases}
\partial_t Z_{\rm e} + \partial_x Q_{\rm e} = 0 & \mbox{in}\quad \mathcal{E}(t), \\
\partial_t Q_{\rm e} + \partial_x \bigl( \frac{Q_{\rm e}^2}{H_{\rm e}} + \frac12\mathtt{g}H_{\rm e}^2 \bigr) = 0
&\mbox{in}\quad \mathcal{E}(t),
\end{cases}
\end{equation}
in the exterior region, while under the object we have
\begin{equation}\label{eqint}
\begin{cases}
\partial_t Z_{\rm i} + \partial_x Q_{\rm i} = 0 & \mbox{in}\quad \mathcal{I}(t), \\
\partial_t Q_{\rm i} + \partial_x \bigl( \frac{Q_{\rm i}^2}{H_{\rm i}} + \frac12\mathtt{g}H_{\rm i}^2 \bigr)
= -\frac{1}{\rho}H_{\rm i}\partial_x \underline{P}_{\rm i}
&\mbox{in}\quad \mathcal{I}(t),
\end{cases}
\end{equation}
with transmission conditions
\begin{equation}\label{BC1}
H_{\rm e} = H_{\rm i}, \quad Q_{\rm e} = Q_{\rm i}, \quad \underline{P}_{\rm i} = P_{\rm atm}
\quad \mbox{on} \quad \Gamma(t),
\end{equation}
where $\Gamma(t)=\partial\mathcal{I}(t)=\partial\mathcal{E}(t)$ denotes the contact points.
We also need to prescribe equations of the motion of the floating body.
Such equations will be given in the following sections according to the cases where the floating body
is fixed, the motion of the body is prescribed, or the body moves freely.
\subsubsection{Basic structure of the equations}\label{sectbseq}
Once the equations of the motion of the floating body are given, as we will see in the following sections,
we can solve the equations in the interior region \eqref{eqint} and the problem will be reduced to
the type considered in \S \ref{sectVCm2} with $U=(Z_{\rm e},Q_{\rm e})^{\rm T}$.
We note that \eqref{eqext} can be written in the matrix form
\[
\partial_t U + A(U) \partial_x U = 0.
\]
As was explained in Example \ref{ex1}, the eigenvalues $\lambda_{\pm}(U)$ of the coefficient matrix
$A(U)$ and the corresponding unit eigenvectors $\mathbf{e}_{\pm}(U)$ are given by
\[
\lambda_{\pm}(U) = \sqrt{\mathtt{g}H_{\rm e}} \pm \frac{Q_{\rm e}}{H_{\rm e}}, \qquad
\mathbf{e}_{\pm}(U) = \frac{1}{\sqrt{1+\lambda_{\pm}(U)^2}}
\binom{1}{\pm\lambda_{\pm}(U)}.
\]
Moreover, the unit vector $\mu_0$ defined in Remark \ref{remarkIC2} is in this case given by
$\mu_0=(1,0)^{\rm T}$, so that the condition $\mu_0 \cdot \mathbf{e}_{+}(U) \ne0$ is automatically satisfied.
As was explained in \S \ref{sectVCm2}, the discontinuity of $\partial_x U$ at the contact points plays an
important role to determine the contact points $x_{\pm}$.
Concerning this discontinuity condition, we have the following proposition.
\begin{proposition}\label{propdisconti}
Suppose that $U_{\rm e} = (Z_{\rm e},Q_{\rm e})^{\rm T}$, $U_{\rm i} = (Z_{\rm i},Q_{\rm i})^{\rm T}$,
$\underline{P}_{\rm i}$, and $x_{\pm}$ satisfy \eqref{eqext}--\eqref{BC1}.
Then, the condition $\partial_x U_{\rm e} - \partial_x U_{\rm i} \ne 0$ on $\Gamma(t)$
is equivalent to $\partial_x Z_{\rm e} - \partial_x Z_{\rm i} \ne 0$ on $\Gamma(t)$.
\end{proposition}
\begin{proof}
Differentiating the boundary condition $Z_{\rm e}(t,x_{\pm}(t)) = Z_{\rm i}(t,x_{\pm}(t))$ with respect to $t$,
we obtain
\[
\partial_t Z_{\rm e} + \dot{x}_{\pm}\partial_x Z_{\rm e}
= \partial_t Z_{\rm i} + \dot{x}_{\pm}\partial_x Z_{\rm i} \quad\mbox{on}\quad \Gamma(t).
\]
By the continuity equations in the interior and the exterior regions, we have
$\partial_t Z_{\rm e} = -\partial_x Q_{\rm e}$ and $\partial_t Z_{\rm i} = -\partial_x Q_{\rm i}$, so that
\[
\dot{x}_{\pm} ( \partial_x Z_{\rm e} - \partial_x Z_{\rm i} ) = \partial_x Q_{\rm e} - \partial_x Q_{\rm i} \quad\mbox{on}\quad \Gamma(t).
\]
This gives the desired result.
\end{proof}
\subsection{The case of a fixed floating body}\label{sectfixfloat}
In the case where the body is fixed, we impose the condition
\begin{equation}\label{body1}
Z_{\rm i}=Z_{\rm lid} \quad\mbox{on}\quad \mathcal{I}(t),
\end{equation}
where $Z_{\rm lid}=Z_{\rm lid}(x)$ is a given function defined on an open interval $I_{\rm f}$.
\subsubsection{Reformulation of the equations}\label{sectreform}
We begin to solve the equations in the interior region \eqref{eqint}.
It follows from \eqref{body1} that $H_{\rm i}(t,x)=h_0+Z_{\rm lid}(x)$ does not depend on $t$, so that
the continuity equation in \eqref{eqint} yields $\partial_x Q_{\rm i}=0$.
This means that $Q_{\rm i}$ does not depend on $x$, so that we can write $Q_{\rm i}(t,x)=q_{\rm i}(t)$.
Plugging this into the momentum equation in \eqref{eqint} we have
\[
\dot{q}_{\rm i} + \partial_x\Bigl( \frac{q_{\rm i}^2}{H_{\rm i}} + \frac12\mathtt{g}H_{\rm i}^2 \Bigr)
= -\frac{1}{\rho}H_{\rm i} \partial_x\underline{P}_{\rm i},
\]
which is equivalent to
\[
\frac{\dot{q}_{\rm i}}{H_{\rm i}}
+ \partial_x\Bigl( \frac12\frac{q_{\rm i}^2}{H_{\rm i}^2} + \mathtt{g}H_{\rm i} \Bigr)
= -\frac{1}{\rho} \partial_x\underline{P}_{\rm i}.
\]
Therefore, $\underline{P}_{\rm i}$ satisfies a simple boundary value problem
\begin{equation}\label{bvpp1}
\begin{cases}
\partial_x\underline{P}_{\rm i} = -\rho \bigl( \frac{\dot{q}_{\rm i}}{H_{\rm i}}
+ \partial_x\bigl( \frac12\frac{q_{\rm i}^2}{H_{\rm i}^2} + \mathtt{g}H_{\rm i} \bigr) \bigr)
& \mbox{in}\quad \mathcal{I}(t), \\
\underline{P}_{\rm i} = P_{\rm atm} & \mbox{on}\quad \Gamma(t).
\end{cases}
\end{equation}
\begin{notation}
For a function $F=F(t,x)$, we put $\jump{F}=F(t,x_{-}(t))-F(t,x_{+}(t))$.
\end{notation}
Integrating the first equation in \eqref{bvpp1} and using the boundary condition, we obtain
\begin{equation}\label{eqqi}
\dot{q}_{\rm i}\int_{\mathcal{I}(t)}\frac{1}{H_{\rm i}}
+ \jump{ \frac12\frac{q_{\rm i}^2}{H_{\rm i}^2} + \mathtt{g}H_{\rm i} } = 0,
\end{equation}
which is a solvability condition of the boundary value problem \eqref{bvpp1} for $\underline{P}_{\rm i}$.
Conversely, once $q_{\rm i}$ and $x_{\pm}$ are given so that \eqref{eqqi} holds,
we can resolve \eqref{bvpp1} for the pressure $\underline{P}_{\rm i}$ explicitly as
\begin{align*}
\underline{P}_{\rm i}(t,x)
&= P_{\rm atm} -\rho\biggl\{ \dot{q}_{\rm i}(t) \int_{x_{-}(t)}^x\frac{{\rm d}x'}{H_{\rm i}(x')} \\
&\quad
+ \frac12q_{\rm i}(t)^2\biggl( \frac{1}{H_{\rm i}(x)^2} - \frac{1}{H_{\rm i}(x_{-}(t))^2} \biggr)
+ \mathtt{g}( H_{\rm i}(x) - H_{\rm i}(x_{-}(t)) ) \biggr\}.
\end{align*}
Therefore, the equations in the interior region \eqref{eqint} are reduced to a scalar ordinary
differential equation \eqref{eqqi}.
We turn to reformulate the equations in the exterior region \eqref{eqext}.
As in \S \ref{sectVCm2}, we will use a coordinate transformation to reduce the equations on
the unknown region $\mathcal{E}(t)$ to those on a fixed region $\underline{\mathcal{E}}$.
Let $\underline{x}_{-}^{\rm in}$ and $\underline{x}_{+}^{\rm in}$ be the initial contact points at time $t=0$ such that
$\underline{x}_{-}^{\rm in} < \underline{x}_{+}^{\rm in}$ and put $\underline{\mathcal{E}}_{-} = (-\infty,\underline{x}_{-}^{\rm in})$,
$\underline{\mathcal{E}}_{+} = (\underline{x}_{+}^{\rm in},\infty)$,
and $\underline{\mathcal{E}} = \underline{\mathcal{E}}_{-} \cup \underline{\mathcal{E}}_{+}$.
We use a diffeomorphism $\varphi(t,\cdot) : \underline{\mathcal{E}} \to \mathcal{E}(t)$ and put
$\zeta_{\rm e} = Z_{\rm e}\circ\varphi$, $h_{\rm e} = H_{\rm e}\circ\varphi$,
$q_{\rm e} = Q_{\rm e}\circ\varphi$, and $\zeta_{\rm i} = Z_{\rm i}\circ\varphi$.
Such a diffeomorphism $\varphi$ can be constructed as in \eqref{diffeo2}, that is,
\begin{equation}\label{diffeo3}
\varphi(t,x)=
\begin{cases}
x + \psi(\frac{x-\underline{x}_{-}^{\rm in}}{\varepsilon})(x_{-}(t)-\underline{x}_{-}^{\rm in})
& \mbox{for}\quad x\in \underline{E}_{-}, \\
x + \psi(\frac{x-\underline{x}_{+}^{\rm in}}{\varepsilon})(x_{+}(t)-\underline{x}_{+}^{\rm in})
& \mbox{for}\quad x\in \underline{E}_{+},
\end{cases}
\end{equation}
with an appropriate choice of $\varepsilon=\varepsilon_0$ and a cut-off function $\psi \in C_0^\infty({\mathbb R})$
satisfying $\psi(x)=1$ for $|x| \leq 1$.
As before, we will use the notation $\partial_x^\varphi$ and $\partial_t^\varphi$ which were defined by \eqref{dtphi}.
Now, the problem under consideration is reduced to
\begin{equation}\label{teqext}
\begin{cases}
\partial_t^\varphi \zeta_{\rm e} + \partial_x^\varphi q_{\rm e} = 0 & \mbox{in}\quad \underline{\mathcal{E}}, \\
\partial_t^\varphi q_{\rm e} + 2 \frac{q_{\rm e}}{h_{\rm e}}\partial_x^\varphi q_{\rm e}
+ \Bigl( \mathtt{g}h_{\rm e} - \frac{q_{\rm e}^2}{h_{\rm e}^2} \Bigr)\partial_x^\varphi \zeta_{\rm e} = 0
& \mbox{in}\quad \underline{\mathcal{E}}, \\
\zeta_{\rm e} = \zeta_{\rm i}, \quad q_{\rm e} = q_{\rm i} & \mbox{on}\quad \partial\underline{\mathcal{E}},
\end{cases}
\end{equation}
with the interior value $q_i$ of the horizontal water flux given by
\begin{equation}\label{eqqi2}
\dot{q}_{\rm i} = -\frac{1}{\int_{\mathcal{I}(t)}\frac{1}{H_{\rm i}}}
\jump{ \frac12\frac{q_{\rm i}^2}{H_{\rm i}^2} + \mathtt{g}H_{\rm i} }.
\end{equation}
We impose the initial conditions of the form
\begin{equation}\label{ICs1}
(\zeta_{\rm e},q_{\rm e})_{\vert_{t=0}} = (\zeta_{\rm e}^{\rm in},q_{\rm e}^{\rm in})
\quad\mbox{in}\quad \underline{\mathcal{E}}, \qquad {x_{\pm}}_{\vert_{t=0}} = \underline{x}_{\pm}^{\rm in},
\qquad {q_{\rm i}}_{\vert_{t=0}} = q_{\rm i}^{\rm in}.
\end{equation}
\subsubsection{Local well-posedness}
The equations in \eqref{teqext} can be written in the matrix form
\[
\partial_t^\varphi u + A(u) \partial_x^\varphi u = 0,
\]
where $u=(\zeta_{\rm e},q_{\rm e})^{\rm T}$, so that \eqref{teqext}--\eqref{ICs1} is almost
the same type as the problem \eqref{ODE}--\eqref{nlfbp} considered in \S \ref{sectext}.
Therefore, the compatibility conditions for \eqref{teqext}--\eqref{ICs1} can be defined in the
same way as Definition \ref{defCC2} in \S \ref{sectCC}.
Here, we calculate $\underline{x}_{\pm,1}^{\rm in} = (\partial_t x_{\pm})_{\vert_{t=0}}$ in terms of the initial data.
Differentiating the boundary condition $\zeta_{\rm e}=\zeta_{\rm i}$ with respect to $t$, we have
$\partial_t \zeta_{\rm e} = \partial_t \zeta_{\rm i}$ on $\partial\underline{\mathcal{E}}$, which is equivalent to
$\partial_t^\varphi \zeta_{\rm e} + \dot{x}_{\pm}\partial_x^\varphi \zeta_{\rm e}
= \partial_t^\varphi \zeta_{\rm i} + \dot{x}_{\pm}\partial_x^\varphi \zeta_{\rm i}$ on $\partial\underline{\mathcal{E}}$.
By using $\partial_t^\varphi\zeta_{\rm e}=-\partial_x^\varphi q_{\rm e}$ and $\partial_t^\varphi\zeta_{\rm i}=0$, we see that
$(\partial_x^\varphi \zeta_{\rm e}-\partial_x^\varphi \zeta_{\rm i})\dot{x}_{\pm} = \partial_x^\varphi q_{\rm e}$ on
$\partial\underline{\mathcal{E}}$.
Therefore, we obtain
\begin{equation}\label{dtuxin}
\underline{x}_{\pm,1}^{\rm in} = \biggl( \frac{\partial_x q_{\rm e}^{\rm in}}{\partial_x \zeta_{\rm e}^{\rm in}
- \partial_x Z_{\rm lid}} \biggr)_{\vert_{\partial\underline{\mathcal{E}}_{\pm}}}.
\end{equation}
In view of this and the consideration in \S \ref{sectbseq}, we impose the following assumption on the data.
\begin{assumption}\label{assondata}
The data $(\zeta_{\rm e}^{\rm in},q_{\rm e}^{\rm in})$, $\underline{x}_{\pm}^{\rm in}$, and $Z_{\rm lid}$
satisfy the following conditions.
\begin{enumerate}
\item[{\bf i.}]
$\underline{x}_{-} < \underline{x}_{+}$,
\item[{\bf ii.}]
$\inf_{x \in I_{\rm f}}( h_0 + Z_{\rm lid}(x)) > 0$, \;
$\inf_{x \in \underline{\mathcal{E}}}( h_0 + \zeta_{\rm e}^{\rm in}(x)) > 0$,
\item[{\bf iii.}]
$\inf_{x \in \underline{\mathcal{E}}}\bigl( \sqrt{\mathtt{g}(h_0 + \zeta_{\rm e}^{\rm in}(x))}
- \frac{|q_{\rm e}^{\rm in}(x)|}{h_0 + \zeta_{\rm e}^{\rm in}(x)} \bigr) > 0$,
\item[{\bf iv.}]
$\bigl( \sqrt{ \mathtt{g}(h_0 + \zeta_{\rm e}^{\rm in}) }
- \bigl| \frac{q_{\rm e}^{\rm in}}{h_0 + \zeta_{\rm e}^{\rm in}} - \underline{x}_{\pm,1}^{\rm in} \bigr|
\bigr)_{\vert_{\partial\underline{\mathcal{E}}}} > 0$,
\item[{\bf v.}]
$(\partial_x Z_{\rm lid} - \partial_x \zeta_{\rm e}^{\rm in})_{\vert_{\partial\underline{\mathcal{E}}}} \ne 0$
\end{enumerate}
\end{assumption}
We can now state one of our main result in this section, which shows the well-posedness of the shallow water
model with a fixed floating structure on the water surface.
\begin{theorem}\label{theoIBVP8}
Let $m\geq2$ be an integer and $I_{\rm f}$ an open interval.
If the initial data $(\zeta_{\rm e}^{\rm in},q_{\rm e}^{\rm in}) \in H^m(\underline{\mathcal{E}})$,
$\underline{x}_{\pm}^{\rm in} \in I_{\rm f}$,
$q_{\rm i}^{\rm in} \in {\mathbb R}$, and $Z_{\rm lid} \in W^{m,\infty}(I_{\rm f})$ satisfy the conditions in
Assumption \ref{assondata}, where $\underline{x}_{\pm,1}^{\rm in}$ is defined by \eqref{dtuxin},
and the compatibility conditions up to order $m-1$,
then there exist $T>0$ and a unique solution $(\zeta_{\rm e},q_{\rm e},x_{\pm},q_{\rm i})$ to
\eqref{teqext}--\eqref{ICs1} with $\varphi$ given by \eqref{diffeo3} in the class
$\zeta_{\rm e},q_{\rm e} \in
\cap_{j=0}^{m-1}C^j([0,T];H^{m-j}(\underline{\mathcal{E}}))$, $x_{\pm} \in H^m(0,T)$, and
$q_{\rm i} \in H^{m+1}(0,T)$.
\end{theorem}
\begin{proof}
Given $q_{\rm i} \in W^{m,\infty}(0,T)$, \eqref{teqext} forms the same type problem in each exterior regions
$\underline{\mathcal{E}}_{-}$ and $\underline{\mathcal{E}}_{+}$ as the problem considered in \S \ref{sectVCm2},
so that we can apply Theorem \ref{theoIBVP5} to show the existence of the solution
$(\zeta_{\rm e},q_{\rm e},x_{\pm})$ to \eqref{teqext} under the initial conditions in \eqref{ICs1}
satisfying $x_{\pm} \in H^m(0,T_1)$ for some $T_1 \in (0,T]$.
Conversely, given $x_{\pm} \in H^m(0,T)$, we can easily show the existence of the solution
$q_{\rm i} \in H^{m+1}(0,T_1)$ to \eqref{eqqi2} under the initial condition in \eqref{ICs1}
for some $T_1 \in (0,T]$.
Iterating this procedure as in the proof of Theorem \ref{theoIBVP6} we can construct a sequence of approximate
solutions, which converges to the desired solution.
\end{proof}
\subsection{The case of a floating body with a prescribed motion}\label{sectprescfloat}
Since the floating body is allowed only to a solid motion, its motion is completely determined by
$(x_G(t),z_G(t))$ the coordinates of the center of mass and $\theta(t)$ the rotational angle of the body.
Without loss of generality, we have $\theta_{\vert_{t=0}}=0$.
Suppose that the underside of the floating body is initially parameterized by $Z_{\rm lid}(x)$ on an open
interval $I_{\rm f}$, that is, ${Z_{\rm i}}_{\vert_{t=0}}=Z_{\rm lid}$.
Consider a point of the underside of the body and denote the coordinates of the point at $t=0$ by $(X,Z)$.
Let the coordinates of the point at time $t$ be $(x,z)$.
Then, it holds that
\[
Z = Z_{\rm lid}(X), \qquad z = Z_{\rm i}(t,x),
\]
and that
\[
\begin{pmatrix} x - x_G(t) \\ z - z_G(t) \end{pmatrix}
=
\begin{pmatrix}
\cos\theta(t) & -\sin\theta(t) \\
\sin\theta(t) & \cos\theta(t)
\end{pmatrix}
\begin{pmatrix} X - x_G(0) \\ Z - z_G(0) \end{pmatrix}.
\]
Therefore, we obtain
\begin{align}\label{expression1}
& (Z_{\rm i}(t,x) - z_G(t))\cos\theta(t) - (x - x_G(t))\sin\theta(t) + z_G(0) \\
&= Z_{\rm lid}\bigl( (x - x_G(t))\cos\theta(t) + (Z_{\rm i}(t,x) - z_G(t))\sin\theta(t) + x_G(0) \bigr).
\nonumber
\end{align}
This is the equation for the motion of the body and gives an expression of $Z_{\rm i}$ implicitly
in terms of $x_G,z_G,\theta$, and $Z_{\rm lid}$.
\subsubsection{Reformulation of the equations}\label{sectreform2}
Proceeding as in \S \ref{sectreform}, it is possible to reformulate the equations in compact form.
Due to the various degrees of freedom of the solid,
the computations are a bit technical and are postponed to Appendix \ref{appreform}.
It is shown there that the surface elevation and the horizontal water flux
in the interior region are given by
$$
\begin{cases}
Z_{\rm i}(t,x) = \psi_{\rm lid}\bigl( x-x_G(t),\theta(t) \bigr) + z_G(t),\\
Q_{\rm i}(t,x) = \begin{pmatrix} \mathbf{U}_G(t) \\ \omega(t) \end{pmatrix} \cdot
\mathbf{T}( \mathbf{r}_G(t,x) ) + \overline{q}_{\rm i}(t),
\end{cases}
$$
for some smooth enough function $\psi_{\rm lid}$ and some function $\overline{q}_{\rm i}(t)$ of $t$
solving an ODE of the form
\[
\partial_t \overline{q}_{\rm i}
= F(\overline{q}_{\rm i},x_G,z_G,\theta,\mathbf{U}_G,\omega,\partial_t\mathbf{U}_G,\partial_t\omega,x_{-},x_{+})
\]
with $F$ in the class $W^{m,\infty}$ under the assumption $Z_{\rm lid} \in W^{m,\infty}(I_{\rm f})$.
As in the previous section, we use the same diffeomorphism
$\varphi(t,\cdot) : \underline{\mathcal{E}} \to \mathcal{E}(t)$ defined by \eqref{diffeo3} to transform
the equations in exterior region \eqref{eqext} and put
$\zeta_{\rm e} = Z_{\rm e}\circ\varphi$, $h_{\rm e} = H_{\rm e}\circ\varphi$,
$q_{\rm e} = Q_{\rm e}\circ\varphi$, $\zeta_{\rm i} = Z_{\rm i}\circ\varphi$, and
$q_{\rm i} = Q_{\rm i}\circ\varphi$.
Now, the problem under consideration is reduced to
\begin{equation}\label{teqext2}
\begin{cases}
\partial_t^\varphi \zeta_{\rm e} + \partial_x^\varphi q_{\rm e} = 0 & \mbox{in}\quad \underline{\mathcal{E}}, \\
\partial_t^\varphi q_{\rm e} + 2 \frac{q_{\rm e}}{h_{\rm e}}\partial_x^\varphi q_{\rm e}
+ \Bigl( \mathtt{g}h_{\rm e} - \frac{q_{\rm e}^2}{h_{\rm e}^2} \Bigr)\partial_x^\varphi \zeta_{\rm e} = 0
& \mbox{in}\quad \underline{\mathcal{E}}, \\
\zeta_{\rm e} = \zeta_{\rm i}, \quad q_{\rm e} = q_{\rm i} & \mbox{on}\quad \partial\underline{\mathcal{E}},
\end{cases}
\end{equation}
and
\begin{equation}\label{eqqi3}
\partial_t \overline{q}_{\rm i}
= F(\overline{q}_{\rm i},x_G,z_G,\theta,\mathbf{U}_G,\omega,\partial_t\mathbf{U}_G,\partial_t\omega,x_{-},x_{+}).
\end{equation}
We also impose the initial conditions of the form
\begin{equation}\label{ICs2}
(\zeta_{\rm e},q_{\rm e})_{\vert_{t=0}} = (\zeta_{\rm e}^{\rm in},q_{\rm e}^{\rm in})
\quad\mbox{in}\quad \underline{\mathcal{E}}, \qquad {x_{\pm}}_{\vert_{t=0}} = \underline{x}_{\pm}^{\rm in},
\qquad {\overline{q}_{\rm i}}_{\vert_{t=0}} = \overline{q}_{\rm i}^{\rm in}.
\end{equation}
\subsubsection{Local well-posedness}
\eqref{teqext2}--\eqref{ICs2} is again almost the same type as the problem \eqref{ODE}--\eqref{nlfbp}
considered in \S \ref{sectext}.
Therefore, the compatibility conditions for \eqref{teqext2}--\eqref{ICs2} can be defined in the
same way as Definition \ref{defCC2} in \S \ref{sectCC}.
Here, we calculate $\underline{x}_{\pm,1}^{\rm in} = (\partial_t x_{\pm})_{\vert_{t=0}}$ in terms of the initial data.
Differentiating the boundary condition $Z_{\rm e}(t,x_{\pm}(t))=Z_{\rm i}(t,x_{\pm}(t))$ with respect to
$t$ and using the equation $\partial_t Z_{\rm e} + \partial_x Q_{\rm e}=0$, we obtain
$(\partial_x Z_{\rm e} - \partial_x Z_{\rm i})_{\vert_{\partial\underline{\mathcal{E}}_{\pm}}} \partial_t x_{\pm}
= (\partial_x Q_{\rm e} + \partial_t Z_{\rm i})_{\vert_{\partial\underline{\mathcal{E}}_{\pm}}}$,
so that
\begin{equation}\label{dtuxin2}
\underline{x}_{\pm,1}^{\rm in} = \biggl(
\frac{Z_{\rm i,1}^{\rm in} + \partial_x q_{\rm e}^{\rm in}}{\partial_x \zeta_{\rm e}^{\rm in} - \partial_x Z_{\rm lid}}
\biggr)_{x=x_{\pm}},
\end{equation}
where $Z_{\rm i,1}^{\rm in}=(\partial_t Z_{\rm i})_{\vert_{t=0}}$ is given by
\[
Z_{\rm i,1}^{\rm in}(x) = \biggl( \mathbf{U}_{G}^{\rm in} + \omega^{\rm in}
\begin{pmatrix} Z_{\rm lid}(x) - z_G^{\rm in} \\ -(x-x_G^{\rm in}) \end{pmatrix} \biggr) \cdot
\begin{pmatrix} -\partial_x Z_{\rm lid}(x) \\ 1 \end{pmatrix}.
\]
with $(x_G^{\rm in},z_G^{\rm in},\mathbf{U}_{G}^{\rm in},\omega^{\rm in})
= (x_G,z_G,\mathbf{U}_{G},\omega)_{\vert_{t=0}}$.
Here, we used \eqref{zi2}.
We can now state one of our main result in this section, which shows the well-posedness of the shallow water
model with a floating body on the water surface whose motion is prescribed.
\begin{theorem}\label{theoIBVP9}
Let $m\geq2$ be an integer and $I_{\rm f}$ an open interval.
If the data $(\zeta_{\rm e}^{\rm in},q_{\rm e}^{\rm in}) \in H^m(\underline{\mathcal{E}})$,
$\underline{x}_{\pm}^{\rm in} \in I_{\rm f}$, $\overline{q}_{\rm i}^{\rm in} \in {\mathbb R}$, $Z_{\rm lid} \in W^{m,\infty}(I_{\rm f})$,
and $x_G,z_G,\theta \in H^{m+2}(0,T)$ satisfy the conditions in
Assumption \ref{assondata}, where $\underline{x}_{\pm,1}^{\rm in}$ is defined by \eqref{dtuxin2},
and the compatibility conditions up to order $m-1$,
then there exist $T_1 \in (0,T]$ and a unique solution $(\zeta_{\rm e},q_{\rm e},x_{\pm},\overline{q}_{\rm i})$ to
\eqref{teqext2}--\eqref{ICs2} with $\varphi$ given by \eqref{diffeo3} in the class
$\zeta_{\rm e},q_{\rm e} \in
\cap_{j=0}^{m-1}C^j([0,T_1];H^{m-j}(\underline{\mathcal{E}}))$, $x_{\pm} \in H^m(0,T_1)$, and
$\overline{q}_{\rm i} \in H^{m+1}(0,T_1)$.
\end{theorem}
\subsection{The case of a freely floating body}\label{sectfreefloat}
Finally, we consider the case where the floating body moves freely according to the Newton's laws
under the action of the gravitational force and the pressure from the air and from the water.
Let $\mathfrak{m}$ and $\mathfrak{i}_0$ be the mass and the inertia coefficient of the body.
Then, Newton's laws for the conservation of linear and angular momentum have the form
\begin{equation}\label{Newton's law}
\begin{cases}
\mathfrak{m} \partial_t \mathbf{U}_G
= -\mathfrak{mg}\mathbf{e}_z + \int_{\mathcal{I}(t)}( \underline{P}_{\rm i} - P_{\rm atm} )N_{\rm lid}, \\
\mathfrak{i}_0 \partial_t \omega = - \int_{\mathcal{I}(t)}( \underline{P}_{\rm i} - P_{\rm atm} )
\mathbf{r}_G^{\perp} \cdot N_{\rm lid},
\end{cases}
\end{equation}
which together with \eqref{expression1} constitute the equations of motion for the floating body.
\subsubsection{Reformulation of the equations}
Proceeding as in \S \ref{sectreform} and \S \ref{sectreform2}, and with the same notations,
the problem under consideration can be reduced to
\begin{equation}\label{teqext3}
\begin{cases}
\partial_t^\varphi \zeta_{\rm e} + \partial_x^\varphi q_{\rm e} = 0 & \mbox{in}\quad \underline{\mathcal{E}}, \\
\partial_t^\varphi q_{\rm e} + 2 \frac{q_{\rm e}}{h_{\rm e}}\partial_x^\varphi q_{\rm e}
+ \Bigl( \mathtt{g}h_{\rm e} - \frac{q_{\rm e}^2}{h_{\rm e}^2} \Bigr)\partial_x^\varphi \zeta_{\rm e} = 0
& \mbox{in}\quad \underline{\mathcal{E}}, \\
\zeta_{\rm e} = \zeta_{\rm i}, \quad q_{\rm e} = q_{\rm i} & \mbox{on}\quad \partial\underline{\mathcal{E}},
\end{cases}
\end{equation}
and with $W=(\overline{q}_{\rm i},x_G,z_G,\theta,\mathbf{U}_G,\omega)$ solving an ordinary differential equation
of the form
\begin{equation}\label{eqdtW}
\partial_t W = F(W,x_{-},x_{+})
\end{equation}
with $F$ in the class $W^{m,\infty}$ under the assumption $Z_{\rm lid} \in W^{m,\infty}(I_{\rm f})$
(see \eqref{eqqi4bis}--\eqref{eqbodybis} for more precisions).
The details of this technical reduction, which takes advantage of the so-called added mass effect,
are postponed to Appendix \ref{appreform2}.
We also impose the initial conditions of the form
\begin{equation}\label{ICs3}
\begin{cases}
(\zeta_{\rm e},q_{\rm e})_{\vert_{t=0}} = (\zeta_{\rm e}^{\rm in},q_{\rm e}^{\rm in})
\quad\mbox{in}\quad \underline{\mathcal{E}}, \qquad {x_{\pm}}_{\vert_{t=0}} = \underline{x}_{\pm}^{\rm in}, \\
{\overline{q}_{\rm i}}_{\vert_{t=0}} = \overline{q}_{\rm i}^{\rm in},
\qquad (x_G,z_G,\theta,\mathbf{U}_G,\omega)_{\vert_{t=0}}
= (x_G^{\rm in},z_G^{\rm in},0,\mathbf{U}_G^{\rm in},\omega^{\rm in}).
\end{cases}
\end{equation}
\subsubsection{Local well-posedness}
Therefore, \eqref{teqext3}--\eqref{ICs3} is again almost the same type as the problem
\eqref{ODE}--\eqref{nlfbp} considered in \S \ref{sectext}.
Therefore, the compatibility conditions for \eqref{teqext3}--\eqref{ICs3} can be defined in the
same way as Definition \ref{defCC2} in \S \ref{sectCC}.
Moreover, $\underline{x}_{\pm,1}^{\rm in} = (\partial_t x_{\pm})_{\vert_{t=0}}$ can be given by \eqref{dtuxin2}.
We can now state one of our main result in this section, which shows the well-posedness of the shallow water
model with a freely floating body on the water surface.
\begin{theorem}\label{theoIBVP10}
Let $m\geq2$ be an integer and $I_{\rm f}$ an open interval.
If the data $(\zeta_{\rm e}^{\rm in},q_{\rm e}^{\rm in}) \in H^m(\underline{\mathcal{E}})$,
$\underline{x}_{\pm}^{\rm in} \in I_{\rm f}$,
$(q_{\rm i}^{\rm in},x_G^{\rm in},z_G^{\rm in},\mathbf{U}_G^{\rm in},\omega^{\rm in}) \in {\mathbb R}^6$,
and $Z_{\rm lid} \in W^{m,\infty}(I_{\rm f})$ satisfy the conditions in
Assumption \ref{assondata}, where $\underline{x}_{\pm,1}^{\rm in}$ is defined by \eqref{dtuxin2},
and the compatibility conditions up to order $m-1$,
then there exist $T>0$ and a unique solution
$(\zeta_{\rm e},q_{\rm e},x_{\pm},\overline{q}_{\rm i},x_G,z_G,\theta)$ to
\eqref{teqext3}--\eqref{ICs3} with $\varphi$ given by \eqref{diffeo3} in the class
$\zeta_{\rm e},q_{\rm e} \in
\cap_{j=0}^{m-1}C^j([0,T];H^{m-j}(\underline{\mathcal{E}}))$, $x_{\pm} \in H^m(0,T)$,
$\overline{q}_{\rm i} \in H^{m+1}(0,T)$, and
$x_G,z_G,\theta \in H^{m+2}(0,T)$.
\end{theorem}
\section{Several examples of transmission problems}\label{sectsev}
We present here several applications of the results proved in Section \ref{secttransmission}
on transmission problems.
The first one, in \S \ref{sectdiscf}, is a transmission problem with a fixed interface:
it corresponds to a conservation law with a flux which is discontinuous across the interface.
A typical example of application is given by the propagation of shallow water waves over a step-like
discontinuous topography.
The second application, in \S \ref{sectshocks}, is a very classical free boundary transmission problem:
we show how the issue of the stability of one-dimensional shocks for $2\times2$ conservations laws
falls in the general framework of \S \ref{secttransmkin}.
This provides an elementary proof of these results, with an improved regularity threshold.
The case of classical (Lax) shock is considered in \S \ref{sectLax}, while nonclassical, undercompressive,
shocks are dealt with in \S \ref{sectunder}.
\subsection{Systems of conservation laws with discontinuous flux}\label{sectdiscf}
Let us consider here a system of two conservation laws, with a flux depending on the position.
For instance, let us consider a flux $\widetilde{f}$ on ${\mathbb R}^-$, and $f$ on ${\mathbb R}_+$, that is,
\begin{equation}\label{discconserv}
\begin{cases}
\partial_t u + \partial_x \widetilde{f}(u) = 0 & \mbox{ in }\quad (0,T)\times{\mathbb R}_-, \\
\partial_t u + \partial_x {f}(u) = 0 & \mbox{ in }\quad (0,T)\times{\mathbb R}_+,
\end{cases}
\end{equation}
where $\widetilde{f}: \widetilde{{\mathcal U}} \to {\mathbb R}^2$ and ${f}: {{\mathcal U}} \to {\mathbb R}^2$ are smooth mappings
defined on open subsets $\widetilde{{\mathcal U}}$ and ${\mathcal U}$ of ${\mathbb R}^2$.
In addition, $p$ transmission conditions are given at $x=0$ ($p=1,2,3$),
\begin{equation}\label{transmappl}
N_p^{\rm r}(t) u_{\vert_{x=+0}} - N_p^{\rm l}(t)u_{\vert_{x=-0}} = \mybf{g}(t),
\end{equation}
where $N_p^{\rm l}$ and $N_p^{\rm r}$ are $p\times 2$ matrices.
\begin{remark}
A natural condition is to impose the continuity of the fluxes at the interface,
$\widetilde{f}({u^{\rm l}}_{\vert_{x=0}}) = f({u^{\rm r}}_{\vert_{x=0}})$,
which is a nonlinear transmission condition.
One can in general use a nonlinear change of variables as in \S \ref{sectapplQL} or \S \ref{sectshocks}
to reduce to the case of a linear transmission condition.
\end{remark}
Denoting $\widetilde{A}(u) = \widetilde{f}'(u)$ and $A(u) = f'(u)$, and using the same notations
as in \S \ref{sectappltransmQL}, the system takes the form \eqref{systQLtransm}, namely,
\begin{equation}\label{systQLtransmappl}
\begin{cases}
\partial_t \mybf{u} + \mybf{A}(\mybf{u})\partial_x \mybf{u} = 0 & \mbox{in}\quad \Omega_T, \\
\mybf{u}_{\vert_{t=0}} = \mybf{u}^{\rm in}(x) & \mbox{on}\quad {\mathbb R}_+, \\
\mybf{N}_{p}(t) \mybf{u}_{\vert_{x=0}} = \mybf{g}(t) & \mbox{on}\quad (0,T),
\end{cases}
\end{equation}
and Theorem \ref{theoIBVP2transm} can therefore be applied.
\begin{example}[Shallow water equations with a discontinuous topography]
Let us consider the shallow water equations with a depth at rest $\widetilde{h}_0$ for $x<0$ and $h_0$ for $x>0$.
\medskip
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.7\linewidth]{SWDCBT.eps}
\end{center}
\setlength{\unitlength}{1pt}
\begin{picture}(0,0)
\put(-65,55){$\widetilde{h}_0$}
\put(63,63){$h_0$}
\end{picture}
\vspace{-5mm}
\caption{Shallow water with a discontinuous topography}\label{FigSW}
\end{figure}
The configuration under study here is described in Figure \ref{FigSW}.
This is a particular example of \eqref{discconserv} with
\[
\widetilde{f}(\zeta,q)
= (q,\frac{1}{\widetilde{h}_0+\zeta}q^2+\frac{1}{2}{\mathtt g}(\widetilde{h}_0+\zeta)^2)^{\rm T}
\quad\mbox{ and }\quad
f(\zeta,q) = (q,\frac{1}{{h}_0+\zeta}q^2+\frac{1}{2}{\mathtt g}({h}_0+\zeta)^2)^{\rm T}.
\]
If $\widetilde{\lambda}_\pm(u^{\rm l})
= \sqrt{\mathtt{g}(\widetilde{h}_0+\zeta^{\rm l})}\pm \frac{q^{\rm l}}{\widetilde{h}_0+\zeta^{\rm l}} >0$
and $\lambda_\pm(u^{\rm r}) = \sqrt{\mathtt{g}({h}_0+\zeta^{\rm r})}\pm \frac{q^{\rm r}}{h_0+\zeta^{\rm r}} >0$,
then one has $p=2$ in Assumption \ref{asshypQLtransm} and two transmission conditions are needed;
they are naturally given by the continuity of the surface elevation $\zeta$ and of
the horizontal water flux $q$, that is,
\[
{u^{\rm l}}_{\vert_{x=0}} = {u^{\rm r}}_{\vert_{x=0}}.
\]
In order to apply Theorem \ref{theoIBVP2transm}, we need to check the invertibility of the Lopatinski\u{\i} matrix
(third point in Assumption \ref{asshypQLtransm}), which is given here by
\[
\mybf{L}(\mybf{u}_{\vert_{x=0}}) = \begin{pmatrix} -\widetilde{{\bf e}}_-({u^{\rm l}}_{\vert_{x=0}})
& {\bf e}_+({u^{\rm r}}_{\vert_{x=0}}) \end{pmatrix},
\]
where $\widetilde{{\bf e}}_-(u)$ denotes a unit eigenvector associated to the eigenvalue $-\widetilde{\lambda}_-(u)$
of $\widetilde{A}(u)$ and ${\bf e}_+(u)$ a unit eigenvector associated to the eigenvalue ${\lambda}_+(u)$
of ${A}(u)$.
Using the expression for the eigenvectors provided in Example \ref{ex1}, the invertibility of the Lopatinski\u{\i}
matrix reduces to the condition
$| \widetilde{\lambda}_-({u^{\rm l}}_{\vert_{x=0}}) + {\lambda}_+({u^{\rm r}}_{\vert_{x=0}}) | > 0$,
which is always satisfied.
One can therefore apply Theorem \ref{theoIBVP2transm}.
\end{example}
\subsection{Stability of one-dimensional shocks}\label{sectshocks}
Let us consider again a system of two conservation laws
\begin{equation}\label{conserv}
\partial_t f_0 (U) + \partial_x f(U) = 0,
\end{equation}
where $f_0,f: {\mathcal U} \to {\mathbb R}^2$ are smooth mappings defined on an openset ${\mathcal U}$ in ${\mathbb R}^2$ and a $2\times2$ matrix
$f_0'(U)$ is assumed to be invertible.
The problem of showing the stability of shocks for \eqref{conserv} consists in finding a
curve $\underline{x}: [0,T]\to {\mathbb R}$ and $U$ such that $U$ is $C^1$ and solve \eqref{conserv} on
$\{(t,x)\in (0,T)\times {\mathbb R}\,;\, x< \underline{x}(t)\}$ and $\{(t,x)\in (0,T)\times {\mathbb R} \,;\, x> \underline{x}(t)\}$,
and satisfy the Rankine--Hugoniot condition
\begin{equation}\label{RH}
\dot\underline{x} \bigl( f_0(U_{\vert_{x=\underline{x}(t)+0}}) - f_0(U_{\vert_{x=\underline{x}(t)-0}}) \bigr)
= f(U_{\vert_{x=\underline{x}(t)+0}}) - f(U_{\vert_{x=\underline{x}(t)-0}}).
\end{equation}
This condition can be split into a nonlinear transmission condition
\[
\Phi( U_{\vert_{x=\underline{x}(t)-0}},U_{\vert_{x=\underline{x}(t)+0}} ) = 0 \quad \mbox{ with } \quad
\Phi(u^{\rm l},u^{\rm r}) = \bigl[ f(u^{\rm r}) - f(u^{\rm l}) \bigr] \cdot
\bigl[ f_0(u^{\rm r}) - f_0(u^{\rm l}) \bigr]^\perp
\]
and the evolution equation $\dot\underline{x} = \chi\bigl( U_{\vert_{x=\underline{x}(t)-0}},U_{\vert_{x=\underline{x}(t)+0}} \bigr)$ with
\begin{equation}\label{defchishock}
\chi(u^{\rm l},u^{\rm r}) = \bigl[ f(u^{\rm r}) - f(u^{\rm l}) \bigr] \cdot
\frac{f_0(u^{\rm r}) - f_0(u^{\rm l})}{ |f_0(u^{\rm r}) - f_0(u^{\rm l})|^2 }.
\end{equation}
Denoting $A(U) = \bigl( f_0'(U) \bigr)^{-1} f'(U)$, we are therefore led to consider the transmission problem
\[
\begin{cases}
\partial_t U + A(U)\partial_x U = 0 & \mbox{in }\quad (-\infty,\underline{x}(t)) \quad \mbox{ for } \quad t\in(0,T), \\
\partial_t U + A(U)\partial_x U = 0 & \mbox{in }\quad (\underline{x}(t),+\infty) \quad \mbox{ for } \quad t\in(0,T), \\
U_{\vert_{t=0}} = u^{\rm in}(x) & \mbox{on }\quad {\mathbb R}, \\
\Phi\bigl(U_{\vert_{x=\underline{x}(t)-0}},U_{\vert_{x=\underline{x}(t)+0}}\bigr) = 0 &\mbox{on } \quad (0,T).
\end{cases}
\]
As for \eqref{bigIBVPQL}, we use the diffeomorphism \eqref{choicediffeo} to recast this transmission problem
as an initial boundary value problem
\begin{equation}\label{bigIBVPshock}
\begin{cases}
\partial_t \mybf{u} + {\bm{\mathcal{A}}}(\mybf{u},\partial {\boldsymbol \varphi})\partial_x \mybf{u} = 0 & \mbox{ in }\quad \Omega_T, \\
\mybf{u}_{\vert_{t=0}} = \mybf{u}^{\rm in} & \mbox{ on }\quad {\mathbb R}_+, \\
\Phi(\mybf{u}_{\vert_{x=0}}) = 0 & \mbox{ on }\quad (0,T)
\end{cases}
\end{equation}
with $\underline{x}$ given by the resolution of
\begin{equation}\label{uxeq}
\dot\underline{x} = \chi( \mybf{u}_{\vert_{x=0}} ), \qquad \underline{x}(0) = 0,
\end{equation}
where $\chi$ given by \eqref{defchishock}.
There are several kinds of shock.
The most famous are the so-called Lax shocks which move at a supersonic speed;
more precisely, the number of positive eigenvalues for ${\bm{\mathcal{A}}}(\mybf{u},\partial {\boldsymbol \varphi})$
in \eqref{bigIBVPshock} is equal to one and one boundary condition is needed;
it is provided by the condition $\Phi(\mybf{u}_{\vert_{x=0}}) = 0$ in \eqref{bigIBVPshock}.
There are also undercompressive shocks that travel at a subsonic speed.
The number of positive eigenvalues for ${\bm{\mathcal{A}}}(\mybf{u},\partial {\boldsymbol \varphi})$
in \eqref{bigIBVPshock} is then equal to two and {\it two} boundary conditions are therefore necessary.
One needs therefore an additional boundary condition to the condition $\Phi(\mybf{u}_{\vert_{x=0}}) = 0$
that comes from the Rankine--Hugoniot condition.
\subsubsection{The stability of Lax shocks}\label{sectLax}
As said above, for Lax shocks, the number of positive eigenvalues for
${\bm{\mathcal{A}}}(\mybf{u},\partial {\boldsymbol \varphi})$ in \eqref{bigIBVPshock} is equal to one;
this correponds to $p=1$ and condition $b)$ or $c)$ in Assumption \ref{asshypQLFBtransm}.
The Kreiss--Lopatinski\u{\i} condition in the third point of Assumption \ref{asshypQLFBtransm} is therefore scalar.
It is explicited in the assumption below for right-going and left-going Lax shocks where for all function $g$
defined on $\boldsymbol{\mathcal U}$, we use the notation
\[
\jump{g}=g(u^{\rm r})-g(u^{\rm l}).
\]
\begin{assumption}\label{asshypQLFBshock}
Let $\widetilde{{\mathcal U}}$ and ${\mathcal U}$ be open sets in ${\mathbb R}^2$ and put $\boldsymbol{{\mathcal U}}=\widetilde{{\mathcal U}}\times{\mathcal U}$
representing a phase space of $\mybf{u}$.
Let $\widetilde{{\mathcal U}}_I \subset \widetilde{{\mathcal U}}$ and ${\mathcal U}_I \subset {\mathcal U}$ be also open sets and put
$\boldsymbol{{\mathcal U}}_I = \widetilde{{\mathcal U}}_I\times{\mathcal U}_I$ representing a phase space of $\mybf{u}_{\vert_{x=0}}$.
The following conditions hold:
\begin{enumerate}
\setlength{\itemsep}{3pt}
\item[{\bf i.}]
$\mybf{A}(\mybf{u}) = \mbox{\rm diag}(-A(u^{\rm l}),A(u^{\rm r})) \in C^\infty(\boldsymbol{{\mathcal U}})$ and
$\Phi,\chi\in C^\infty(\boldsymbol{{\mathcal U}}_I)$.
\item[{\bf ii.}]
For any $\mybf{u} = (u^{\rm l},u^{\rm r})^{\rm T}\in\boldsymbol{{\mathcal U}}$, the matrix $A(u^{\rm l,r})$
has eigenvalues $\lambda_+(u^{\rm l,r}) $ and $ -\lambda_-(u^{\rm l,r})$ with $\lambda_\pm(u^{\rm l,r})>0$.
Moreover, one of the following conditions
for all $\mybf{u}= (u^{\rm l},u^{\rm r})^{\rm T}\in\boldsymbol{{\mathcal U}}_I$ holds:
\begin{enumerate}
\item[-]
Right-going Lax shock
\[
\begin{cases}
\lambda_\pm(u^{\rm l}) \mp \chi(\mybf{u}) >0 \quad\mbox{ and }\quad
\lambda_+(u^{\rm r}) - \chi(\mybf{u}) < 0, \\
\bigl|\bigl(f_0'(u^{\rm l}){\bf e}_-(u^{\rm l})\bigr) \cdot \jump{f_0}^\perp\bigr| > 0.
\end{cases}
\]
\item[-] Left-going Lax shock
\[
\begin{cases}
\lambda_-(u^{\rm l}) + \chi(\mybf{u}) < 0 \quad\mbox{ and }\quad
\lambda_\pm(u^{\rm r}) \mp \chi(\mybf{u}) >0, \\
\bigl|\bigl(f_0'(u^{\rm r}){\bf e}_+(u^{\rm r})\bigr) \cdot \jump{f_0}^\perp\bigr| > 0.
\end{cases}
\]
\end{enumerate}
\item[{\bf iii.}]
There exists a $C^\infty$-mapping $\Theta : \boldsymbol{{\mathcal U}} \to {\mathbb R}^4$ such that it defines a diffeomorphism from
$\boldsymbol{{\mathcal U}}$ onto its image and for any $\mybf{u} = (u^{\rm l},u^{\rm r})^{\rm T}\in\boldsymbol{{\mathcal U}}_I$
we have
\[
\Theta(\mybf{u}) = \bigl( \Phi(\mybf{u}),\chi(\mybf{u}),u^{\rm r}\bigr)^{\rm T}.
\]
\end{enumerate}
\end{assumption}
\begin{remark}\label{remdiffeo}
Up to shrinking $\widetilde{{\mathcal U}}$ and ${\mathcal U}$, the third point is always satisfied.
Indeed, as remarked in \cite{metivier2001}, this follows from the local inversion theorem
since $\Theta'(\mybf{u})$ is invertible at any point $\mybf{u}$ satisfying $\Phi(\mybf{u})=0$.
In order to check this point, it is enough to prove that the partial derivative of the mapping
$\mybf{u} \mapsto (\Phi(\mybf{u}),\chi(\mybf{u}))$ with respect to $u^{\rm l}$ is invertible.
Denoting by $W(\mybf{u})$ a $2\times2$ matrix defined by
\[
W(\mybf{u})F = \Bigl(F \cdot \jump{f_0}^\perp, \frac{1}{|\jump{f_0}|^2} F \cdot \jump{f_0} \Bigr)^{\rm T},
\]
this partial derivative is given by the linear mapping
\begin{align*}
\dot u^{\rm l}\mapsto
& (d_{u^{\rm l}} W(\mybf{u})[\dot u^{\rm l}])\jump{f} - W(\mybf{u})f'(u^{\rm l})\dot u^{\rm l} \\
&= \chi({\mathbf u})(d_{u^{\rm l}} W(\mybf{u})[\dot u^{\rm l}])\jump{f_0}
- W(\mybf{u})f_0'(u^{\rm l})A(u^{\rm l})\dot u^{\rm l};
\end{align*}
observing by differentiating the identity $W(\mybf{u})\jump{f_0} = (0,1)^{\rm T}$ that
\[
d_{u^{\rm l}} W(\mybf{u})[\dot u^{\rm l}]\jump{f_0} = W(\mybf{u})f_0'(u^{\rm l})\dot u^{\rm l},
\]
the partial derivative can be written as
\begin{align*}
\dot u^{\rm l}\mapsto & W(\mybf{u})f_0'(u^{\rm l})
\bigl(\chi(\mybf{u})\mbox{\rm Id} - A(u^{\rm l}) \bigr)\dot u^{\rm l},
\end{align*}
which is invertible by the second point of Assumption \ref{asshypQLFBtransm}.
\end{remark}
We can now state the following stability result for Lax shocks.
\begin{theorem}\label{theoshock}
Let $m\geq 2$ be an integer.
Suppose that Assumption \ref{asshypQLFBshock} is satisfied.
If $\mybf{u}^{\rm in}\in H^m({\mathbb R}_+)$ takes its values in $\widetilde{{\mathcal K}}_0\times {\mathcal K}_0$ with
$\widetilde{{\mathcal K}}_0\subset\widetilde{{\mathcal U}}_0$ and ${\mathcal K}_0\subset{\mathcal U}_0$ compact and convex sets,
if $\mybf{u}^{\rm in}(0) \in \boldsymbol{{\mathcal U}}_I$,
and if it satisfies the compatibility conditions at order $m-1$,
then there exists $T>0$ and a unique solition $(\mybf{u},\underline{x})$ to \eqref{bigIBVPshock}--\eqref{uxeq} with
$\mybf{u} \in {\mathbb W}^m(T)$ and $\underline{x}\in H^{m+1}(0,T)$, and $\varphi$ given by \eqref{choicediffeo}.
Moreover, $\mybf{u}_{\vert_{x=0}}\in H^m(0,T)$.
\end{theorem}
\begin{remark}\label{remimp}
The stability of multidimensional shocks was proved in \cite{majda1,majda2,majda3},
with improvements in \cite{metivier2001}.
In space dimension one, this result shows the stability in ${\mathbb W}^m(T)$ for $m\geq 3$
provided that the data is in $H^{m+1/2}({\mathbb R}_+)$.
Our proof, which takes advantage of the specificities of the one-dimensional case, is much more elementary
and provides an improvement of these classical results since we only need $m\geq 2$
(and therefore one compatibility condition less) with data in $H^m({\mathbb R}_+)$ (and therefore no loss of regularity).
\end{remark}
\begin{proof}
There are two steps in the proof.
We first transform the problem \eqref{bigIBVPshock} into an initial boundary value problem with a
{\it linear} boundary condition, and we then prove that Assumption \ref{asshypQLFBtransm} is satisfied
so that we can conclude with Theorem \ref{theoIBVP4transm}.
Using the third point of Assumption \ref{asshypQLFBshock}, it is equivalent to solve the initial boundary
value problem satisfied by $\mybf{v} = \Theta(\mybf{u})$, namely,
\begin{equation}\label{bigIBVPshocklin}
\begin{cases}
\partial_t \mybf{v} + {\bm{\mathcal{A}}}^\sharp(\mybf{v},\partial {\boldsymbol \varphi})\partial_x \mybf{v} = 0
& \mbox{ in }\quad \Omega_T, \\
\mybf{v}_{\vert_{t=0}} = \mybf{v}^{\rm in} & \mbox{ on }\quad {\mathbb R}_+, \\
{\bf e}_1^\sharp \cdot \mybf{v}_{\vert_{x=0}} = 0 & \mbox{ on }\quad (0,T),
\end{cases}
\end{equation}
with $\underline{x}$ given by the resolution of
\begin{equation}\label{uxeqb}
\dot\underline{x} = {\bf e}_2^\sharp \cdot \mybf{v}_{\vert_{x=0}}, \qquad \underline{x}(0)=0,
\end{equation}
where $({\bf e}^\sharp_1,{\bf e}^\sharp_2,{\bf e}^\sharp_3,{\bf e}^\sharp_4)$
denotes the canonical basis of ${\mathbb R}^4$ and
\[
\boldsymbol{{\mathcal A}}^\sharp(\mybf{v}, \partial {\boldsymbol \varphi})
= \bigl( d_{\boldsymbol{v}}\Theta^{-1}(\mybf{v}) \bigr)^{-1}
{\bm{\mathcal{A}}}(\Theta^{-1}(\mybf{v}),\partial {\boldsymbol \varphi})\bigl( d_{\boldsymbol{v}}\Theta^{-1}(\mybf{v}) \bigr).
\]
In particular, the eigenvalues of $\boldsymbol{{\mathcal A}}^\sharp(\mybf{v}, \partial {\boldsymbol \varphi})$
are the same as those of $\boldsymbol{{\mathcal A}}(\mybf{u}, \partial {\boldsymbol \varphi})$ and
if $\mybf{E}$ is an eigenvector of $\boldsymbol{{\mathcal A}}(\mybf{u}, \partial {\boldsymbol \varphi})$,
the corresponding eigenvector of $\boldsymbol{{\mathcal A}}^\sharp(\mybf{v}, \partial {\boldsymbol \varphi})$
is $\mybf{E}^\sharp = \Theta'(\mybf{u})\mybf{E}$.
By the second point of Assumption \ref{asshypQLFBshock}, the system \eqref{bigIBVPshocklin} satisfies therefore
condition $b)$ or $c)$ in Assumption \ref{asshypQLFBtransm} and the Lopatinski\u{\i} matrix reduces to
a scalar denoted $L^\sharp(\mybf{v}_{\vert_{x=0}})$,
\[
L^\sharp(\mybf{v}_{\vert_{x=0}}) = {\bf e}^\sharp_1 \cdot \mybf{E}^\sharp_{\rm out}(\mybf{v}_{\vert_{x=0}}),
\]
where $\mybf{E}^\sharp_{\rm out}(\mybf{v})$ is the eigenvector of
$\boldsymbol{{\mathcal A}}^\sharp(\mybf{v}, \partial {\boldsymbol \varphi})$ associated to its unique positive eigenvalue.
From the discussion above, one has
$\mybf{E}^\sharp_{\rm out}(\mybf{v}) = \Theta'(\mybf{u})\mybf{E}_{\rm out}(\mybf{u})$,
where $\mybf{E}_{\rm out}(\mybf{u})$ is the eigenvector associated to the unique positive eigenvalue of
$\boldsymbol{{\mathcal A}}(\mybf{u}, \partial {\boldsymbol \varphi})$.
We have therefore
\begin{align*}
L^\sharp(\mybf{v})
&= \Theta'(\mybf{u})^{\rm T}{\bf e}^\sharp_1 \cdot \mybf{E}_{\rm out}(\mybf{u}), \\
&= \nabla_{\boldsymbol{u}}\Phi(\mybf{u}) \cdot \mybf{E}_{\rm out}(\mybf{u}).
\end{align*}
Let us assume for instance that the first condition holds in the second point of
Assumption \ref{asshypQLFBshock} (the adaptation if the second condition holds is straightforward).
One then has $\mybf{E}_{\rm out}(\mybf{u}) = \left(\begin{array}{c}{\bf e}_-(u^{\rm l}) \\ 0\end{array}\right)$
(where as usual ${\bf e}_-(u^{\rm l})$ is the eigenvector associate to the eigenvalue $-\lambda_-(u^{\rm l})$
of $A(u^{\rm l})$) and, with computations similar to those performed in Remark \ref{remdiffeo}, we obtain
\begin{align*}
L^\sharp(\mybf{v})
&= \jump{f_0}^\perp\cdot f_0'(u^{\rm l})(\chi(\mybf{u})\mbox{Id}-A(u^{\rm l})){\bf e}_-(u^{\rm l}) \\
&= (\chi(\mybf{u}) + \lambda_-(u^{\rm l})) \jump{f_0}^\perp \cdot f_0'(u^{\rm l}){\bf e}_-(u^{\rm l});
\end{align*}
the second point of the assumption implies that this quantity is nonzero, and we can therefore conclude with Theorem \ref{theoIBVP4transm}.
\end{proof}
\subsubsection{The stability of undercompressive shocks}\label{sectunder}
In some applications, one can encounter shock waves that violate Lax's conditions.
This is for instance the case for magnetohydrodynamics, or phase transitions in elastodynamics,
or van der Waals fluids.
In the particular case of {\it undercompressive shocks}, Lax's conditions are violated but
condition $a)$ is satisfied in Assumption \ref{asshypQLFBtransm}.
This means that $p=2$ (the number of positive eigenvalues for ${\bm{\mathcal{A}}}(\boldsymbol{u},\partial {\boldsymbol \varphi})$
in \eqref{bigIBVPshock} is equal to two)
and therefore that the system of equations \eqref{bigIBVPshock}--\eqref{uxeq} is now {\it underdeterminated}.
An additional boundary condition is therefore necessary.
This additional condition requires some additional modeling and depends on the context:
it often comes from considerations based on the theory of viscosity-capillarity,
see for instance \cite{slemrod1983,truskinovsky1994} for isothermal phase transitions
or \cite{abeyaratne1991} for elastic rods.
If such an additional boundary condition is provided and if it satisfies an appropriate stability condition as in
\S \ref{secttransmkin} then the undercompressive shocks are stable.
This extension of Majda's work on Lax's shock was proposed in \cite{freistuhler1998,freistuhler1998},
and studied in \cite{colombo1999} in the one-dimensional case.
The extension to several dimensions was performed in \cite{benzoni1998}
(derivation of the Kreiss--Lopatinski\u{\i} condition), \cite{benzoni1999} (linear estimates) and
\cite{coulombel2003} (nonlinear estimates).
We show here that the framework developed in \S \ref{secttransmkin} can be used to improve these results
for the stability of one-dimensional undercompressive shocks.
We shall consider here an general framework where the additional boundary conditions we use to complement
\eqref{bigIBVPshock}--\eqref{uxeq} is of the form
\begin{equation}\label{eqPsi}
\Psi(\mybf{u}_{\vert_{x=0}}) = 0,
\end{equation}
where $\Psi$ is a smooth function satisfiying the assumption below.
Note in particular that for undercompressive shocks, the Lopatinski\u{\i} matrix in the third point of
Assumption \ref{asshypQLFBtransm} is a $2\times2$ matrix;
its invertibility corresponds to the condition stated in the second point of the assumption below.
\begin{assumption}\label{asshypQLFBshockunder}
Let $\widetilde{{\mathcal U}}$ and ${\mathcal U}$ be open sets in ${\mathbb R}^2$ and put $\boldsymbol{{\mathcal U}}=\widetilde{{\mathcal U}}\times{\mathcal U}$
representing a phase space of $\mybf{u}$.
Let $\widetilde{{\mathcal U}}_I \subset \widetilde{{\mathcal U}}$ and ${\mathcal U}_I \subset {\mathcal U}$ be also open sets and put
$\boldsymbol{{\mathcal U}}_I = \widetilde{{\mathcal U}}_I\times{\mathcal U}_I$ representing a phase space of $\mybf{u}_{\vert_{x=0}}$.
The following conditions hold:
\begin{enumerate}
\setlength{\itemsep}{3pt}
\item[{\bf i.}]
$\mybf{A}(\mybf{u}) = \mbox{\rm diag}(-A(u^{\rm l}),A(u^{\rm r})) \in C^\infty(\boldsymbol{{\mathcal U}})$ and
$\Phi,\Psi,\chi\in C^\infty(\boldsymbol{{\mathcal U}}_I)$.
\item[{\bf ii.}]
For any $\mybf{u} = (u^{\rm l},u^{\rm r})^{\rm T} \in \boldsymbol{{\mathcal U}}$, the matrix $A(u^{\rm l,r})$
has eigenvalues $\lambda_+(u^{\rm l,r}) $ and $ -\lambda_-(u^{\rm l,r})$ with $\lambda_\pm(u^{\rm l,r})>0$.
Moreover, for any $\mybf{u} = (u^{\rm l},u^{\rm r})^{\rm T} \in \boldsymbol{{\mathcal U}}_I$
the following conditions hold:
\[
\lambda_\pm(u^{\rm l})\mp\chi(\boldsymbol{u}) > 0 \quad\mbox{ and }\quad
\lambda_\pm(u^{\rm r})\mp\chi(\boldsymbol{u}) > 0
\]
and the Lopatinski\u{\i} matrix
\[
\left(
\begin{array}{cc}
\bigl( \chi(\mybf{u})+\lambda_-(u^{\rm l}) \bigr)
\bigl( f_0'(u^{\rm l}){\bf e}_-(u^{\rm l}) \bigr) \cdot \jump{f_0}^\perp
& -\bigl( \chi(\mybf{u})-\lambda_+(u^{\rm r}) \bigr)
\bigl( f_0'(u^{\rm r}){\bf e}_+(u^{\rm r}) \bigr) \cdot \jump{f_0}^\perp \\
\nabla_{u^{\rm l}}\Psi \cdot {\bf e}_-(u^{\rm l}) & \nabla_{u^{\rm r}}\Psi \cdot {\bf e}_+(u^{\rm r})
\end{array}
\right)
\]
is invertible.
\item[{\bf iii.}]
There exists a $C^\infty$-mapping $\Theta: \boldsymbol{\mathcal U}\to {\mathbb R}^4$ such that it defines a diffeomorphism from $\boldsymbol{\mathcal U}$ onto
its image and for all $\mybf{u} = (u^{\rm l},u^{\rm r})^{\rm T} \in \boldsymbol{{\mathcal U}}_I$ we have
\[
\Theta(\mybf{u}) = \big(\Phi(\mybf{u}),\Psi(\mybf{u}),\theta(\mybf{u}) \big)^{\rm T}
\]
with a mapping $\theta: \boldsymbol{\mathcal U}\to {\mathbb R}^2$.
\end{enumerate}
\end{assumption}
\begin{remark}\label{remdiffeo2}
Up to shrinking $\widetilde{{\mathcal U}}$ and ${\mathcal U}$, the third point is always satisfied.
Indeed, the second point of the assumption shows that $d_{\boldsymbol{u}}(\Phi,\Psi)$ has rank $2$ so that
$\mybf{u}\mapsto (\Phi(\mybf{u}),\Psi(\mybf{u}))$ can be completed to form a local diffeomorphism.
\end{remark}
An easy adaptation of the proof of Theorem \ref{theoshock} yields the following stability result
for undercompressive shocks.
The same improvements as those described in Remark \ref{remimp} hold with respect the result obtained
by considering the one-dimensional case in \cite{coulombel2003}.
\begin{theorem}\label{theoshockunder}
Let $m\geq 2$ be an integer.
Suppose that Assumption \ref{asshypQLFBshockunder} is satisfied.
If $\mybf{u}^{\rm in} \in H^m({\mathbb R}_+)$ takes its values in $\widetilde{{\mathcal K}}_0\times {\mathcal K}_0$
with $\widetilde{{\mathcal K}}_0\subset\widetilde{{\mathcal U}}_0$ and ${\mathcal K}_0\subset{\mathcal U}_0$ compact and convex sets,
if $\mybf{u}^{\rm in}(0) \in \boldsymbol{\mathcal U}_I$,
and if it satisfies the compatibility conditions at order $m-1$,
then there exists $T>0$ and a unique solition $(\mybf{u},\underline{x})$ to \eqref{bigIBVPshock}--\eqref{uxeq}
complemented by \eqref{eqPsi}, with $\mybf{u} \in {\mathbb W}^m(T)$ and $\underline{x}\in H^{m+1}(0,T)$,
and $\varphi$ given by \eqref{choicediffeo}. Moreover, $\boldsymbol{u}_{\vert_{x=0}}\in H^m(0,T)$.
\end{theorem}
|
3,212,635,537,949 | arxiv | \section{Introduction}
As the application of speech signal processing becoming more and more popular, the technologies, such as Automatic Speech Recognition (ASR), speaker diarization are facing many challenges in real-world scenarios. In particular, meeting scenario is one of the most challenging and valuable because of its complexity and diversity, including overlapped speech, unknown number of speakers, reverberation, etc. Multi-channel Multi-party Meeting Transcription Challenge (M2MeT) \cite{yu2021m2met} focuses on addressing the “who speaks what at when” problem in real-world multi-speaker meetings. This challenge consists of two tracks, namely speaker diarization and multi-speaker ASR. The audios of M2MeT corpus named AliMeeting are collected by microphone array.
\begin{figure}[htb]
\centering
\centerline{\includegraphics[width=8.5cm]{f1_fusion_system.png}}
\caption{Fusion speaker diarization system overview}
\label{fig:fusion_system}
\end{figure}
\begin{figure*}[!htb]
\centering
\centerline{\includegraphics[width=19cm]{f3_DMSNET.png}}
\caption{DMSNet is a multi-channel sequence-to-sequence architecture used for sequence labeling.}
\label{fig:DMSNet}
\end{figure*}
One of the difficulties in M2Met is overlapped speech. There are a lot of overlapped regions in the meeting audios, which make this task more challenging. Overlapped speech not only cause wrong speakers number of clustering, but also increases the DER of system directly. Bredin H et al.in \cite{bredin2020pyannote} apply a Bi-LSTM based OSD named PyanNet to address single-channel overlapped speech. However, the audios of AliMeeting is 8-channel have rich spatial information, and the performance of Bi-LSTM based OSD still have room to improve. In this paper, inspired by filter-and-sum beamforming (FSB) algorithm\cite{frost1972algorithm}, we propose a novel architecture named Discriminative Multi-stream Neural Network (DMSNet) for overlapped speech detection. DMSNet is a novel multi-channel sequence-to-sequence architecture used for sequence labeling which consists of AFSB block and Conformer\cite{gulati2020conformer} encoder. Compared with Bi-LSTM based OSD model, DMSNet reduces Detection Error Rate (DetER) from 42.57\% to 32.47\%.What’s more, The audios of AliMeeting are collected from different rooms which is full of noise and reverberation. It is worth noting that the participants are required to remain in the same position during recording. In order to make full use of spatial information provided by the microphone array, we propose a multi-channel speaker diarization fusion system which combines spatial embedding and speaker embedding. In our systems, we perform Direction-of-Arrival (DOA) technology to extract spatial embedding, and combine x-vector to achieve better performance. The speaker diarization subsystem consists of speaker embedding extractor, spatial embedding extractor, clustering module, and OSD module. Compared with diarization system without OSD module, Applying DMSNet as OSD, the diarization error rate (DER) of speaker diarization system reduces from 13.44\% to 7.63\%. We fused different subsystems with different OSD modules and time scales to achieve better performance. Finally, we achieve 7.09\% and 9.80\% DER on AliMeeting evaluation set and test set.
\section{Data Preparation}
AliMeeting corpus supported by M2MeT contains 118.75 hours of speech data in total. The dataset includes 240 meeting audios collected by 8-channel microphone array, of which 212 audios are training set (Train), 8 audios are evaluation set (Eval) and the rest are test sets (Test). In particular, the average speech overlap ratio of training and evaluation set are 42.27\% and 34.76\%, respectively. We used the same systems which was trained with fixed training set in sub-track1 and sub-track2. AliMeeting, Aishell-4\cite{fu2021aishell} and CN-Celeb\cite{fan2020cn} as fixed training set can be used to train models. Our use of training set in this challenge is as follows:
\begin{itemize}
\item Speaker embedding extractor: We take CN-celeb1 (793 speakers) and CN-Celeb2 (2000 speaker) as the training set containing 2793 speakers in total (the CN-Celeb1-test part is excluded from training).
\end{itemize}
\begin{itemize}
\item Overlapped speech detection: We use AliMeeting training set to train OSD models and evaluation set for validation.
\end{itemize}
\begin{itemize}
\item Data augmentation: We use the MUSAN\cite{snyder2015musan} (including music and noise) and RIRs \cite{ko2017study} to perform data augmentation.
\end{itemize}
\begin{table*}[]
\centering
\caption{The performance comparisons of OSD modules with different blocks on AliMeeting evaluation set. Detection Error Rate (DetER) is the main index which define the "miss error" and "false alarm error" DER.}
\label{tab:osd}
\begin{tabular}{llllllll}
\hline
ID & Architecture & Extract block & Encoder & DetER(\%) & Accuracy(\%) & Precision(\%) & Recall(\%) \\ \hline
M1 & PyanNet{\cite{bredin2020pyannote}} & SinConv(FSB) & Bi-LSTM & 42.57 & 91.61 & 85.23 & 70.06 \\
M2 & - & SinConv(FSB) & Conformer & 35.48 & 92.97 & 87.22 & 75.58 \\
M3 & - & AFSB(Shared) & Bi-LSTM & 36.50 & 92.72 & 87.47 & 74.12 \\
M4 & - & AFSB(Shared) & Conformer & 36.73 & 92.68 & 86.41 & 75.09 \\
M5 & - & AFSB(Discriminative) & Bi-LSTM & 36.68 & 92.69 & 88.18 & 73.12 \\
M6 & DMSNet & AFSB(Discriminative) & Confromer & \textbf{32.47 }& \textbf{93.53} & \textbf{89.22} &\textbf{ 76.81} \\ \hline
\end{tabular}
\end{table*}
\section{System Description}
\subsection{Task1 sub-track1\&2}
Our system speaker diarization system consists of speaker embedding extractor, spatial embedding extractor, clustering modules and OSD. The fusion system is illustrated in Figure \ref{fig:fusion_system}. We will describe each module in detail as follows:
\subsubsection{Segmentation }
M2MeT provide segments file to participants to get oracle VAD labels. The performance of fusion system which composed of subsystems with different time scale (including window length and time shift) will be improved. According to segments file we split segments of audio into different time scale.
\subsubsection{Speaker embedding}
The ResNet34-SE \cite{zhou2019deep} is employed as the x-vectors extractor with additive margin Softmax loss \cite{wang2018additive} which learns a segment-level representation from the input acoustic feature. The dimension of x-vector is reduced from 256 to 128 by Linear Discriminant Analysis (LDA). In our system, the input is 81-dimensional filter-banks extracted from the original 16kHz audio with a window size of 25ms and a 10ms shift.
\begin{figure}[htb]
\centering
\centerline{\includegraphics[width=8.5cm]{f2_spatical_extractor.png}}
\caption{A diagram of spatial embedding extractor.}
\label{fig:spatial_vector}
\end{figure}
\subsubsection{Spatial embedding}
The multi-channel signal provides spatial information compared to the single-channel signal. As show in Figure \ref{fig:spatial_vector}, FSB algorithm is used to extract spatial embedding named s-vectors. The signal of the \textit{M}-channel microphone array is denoted as \textit{x}$_{m}$(\textit{t})(\textit{m}=1,2,...,\textit{M}). We input the received signal through the corresponding Finite Impulse Response filter(FIR) and convert to single channel:
\begin{center}
\begin{equation}
y(t)=\sum_{m=1}^{M}\sum_{k=-K}^{K}a_{m,k}x_{m}(t-k)
\end{equation}
\end{center}
where \textit{K} is the order of the filter, and \textit{a}$_{m,k}$ is the \textit{k}$_{th}$ order weight of the \textit{m}$_{th}$ channel FIR filter \cite{zitouni2013simulated}.Convert Eq.1 to the frequency domain:
\begin{center}
\begin{equation}
\mathit{\mathbf{Y}}(e^{j\omega })=\mathit{\mathbf{F}}(\theta ,\omega )\mathit{\mathbf{S}}(e^{j\omega })
\end{equation}
\end{center}
where $\theta$ is the incident angle, and \textbf{F}($\theta$,$\omega$) is the spatial transmission response of the FSB algorithm in the direction $\theta$. The minimum mean square error (MMSE) criterion is used as the cost function to optimize the filter coefficients. Construct the following optimization problem
\begin{center}
\begin{equation}
\underset{a}{min}\iint\left| \mathbf{\mathit{F}}_{\mathbf{\mathit{d}}}(\theta-\varphi)-\mathbf{\mathit{F}}(\theta )\right|^2d\omega d\theta
\end{equation}
\end{center}
where $\varphi$ represents the angle away from the desired direction. \textbf{F}($\theta$,$\omega$) is denoted as:
\begin{center}
\begin{equation}
\mathit{\mathbf{F}}(\theta ,\omega )=\sum_{m=1}^{M}a_{m}z_{m}^{H}(\theta ,\omega )=\mathbf{a}\mathbf{z}^{H}(\theta ,\omega )
\end{equation}
\end{center}
Then the coefficients of spatial filter can be obtained as:
\begin{center}
\begin{equation}
\mathbf{a}=\left (\sum_{l=1}^{L}\mathbf{z}_{l}\mathbf{z}_{l}^{H} \right )^{-1}\left ( \mathbf{b}_{\varphi }\sum_{l=1}^{L}\mathbf{z}_{l}^{H} \right )
\end{equation}
\end{center}
where $\mathbf{z}_{l}=\left\{ z^{H}(\theta _{1},\omega _{l}), z^{H}(\theta _{2},\omega _{l}),\cdots ,z^{H}(\theta _{D},\omega _{l})\right\}$, $\mathbf{b}_{\varphi }=\left\{ \mathbf{F}_{d}(\theta _{1}-\varphi ),\mathbf{F}_{d}(\theta _{2}-\varphi ),\cdots ,\mathbf{F}_{d}(\theta _{N}-\varphi )\right\}$, \textit{L} and \textit{N} are the dimensions for dividing the frequency domain and the spatial domain, respectively.
In this paper, considering the complexity of the algorithm, 128-order filter coefficients are used, and 24, 36, 72, 120, 240, etc. are used for spatial domain division, respectively. The corresponding resolutions are 15°, 10°, 5°, 3° and 1.5°, the dimension of the filter is (\textit{N}, 8, 128), which is (\textit{N}, 8, 128) in this paper. This filter coefficient is placed in the neural network as an artificially designed \textit{N} convolution kernels of dimension (8, 128). As shown in Figure \ref{fig:spatial_vector}, when the 8-channel raw audio is sent to the network, single-channel signals in \textit{N} directions can be obtained. By normalizing the energy of signals in different \textit{N }directions, we can obtain spatial embedding named s-vector. The physical meaning is the spatial distribution of energy, and its statistical meaning is the probability of the existence of signals in \textit{N} directions. By increasing \textit{N}, higher resolution can be obtained in the physical sense, and more features can be seen from the perspective of neural network. Finally, more dimensional vectors are used for clustering, which can improve the robustness of s-vector.
\begin{table*}[]
\centering
\caption{DER(\%) of speaker diarization systems on evaluation set. M1$\sim$M6 are different OSD modules mention in section \ref{subsubsection:osd}}.
\label{tab:fusion}
\begin{tabular}{ccccccccccc}
\hline
ID & Time scales/s & Embedding & NME-SC & \multicolumn{6}{c}{OSD} & Fusion M1$\sim$M6 \\ \cline{5-10}
& & & & M1 & M2 & M3 & M4 & M5 & M6 & \\ \hline
S1 & 1/0.5 & x-vector & 13.68 & 9.46 & 8.84 & 8.75 & 9.22 & 8.94 & \textbf{8.04} & 7.94 \\
S2 & 1.2/0.6 & x-vector & 13.53 & 9.16 & 8.41 & 8.25 & 8.69 & 8.40 & \textbf{7.84} & 7.73 \\
S3 & 1.5/0.75 & x-vector & 13.59 & 9.11 & 8.60 & 8.50 & 8.89 & 8.54 & \textbf{7.91} & 7.81 \\
S4 & 1/0.5 & sx-vector & 13.44 & 8.81 & 8.18 & 8.01 & 8.38 & 8.14 & \textbf{7.63} & 7.54 \\
S5 & 1.2/0.6 & sx-vector & 13.44 & 9.05 & 8.37 & 8.48 & 8.52 & 8.33 & \textbf{7.77 }& 7.63 \\
S6 & 1.5/0.75 & sx-vector & 13.57 & 9.09 & 8.50 & 8.45 & 8.92 & 8.51 & \textbf{7.81} & 7.64 \\ \hline
& \multicolumn{9}{c}{Fusion S3 S4 S5} &\textbf{ 7.09} \\ \hline
\end{tabular}
\end{table*}
\subsubsection{Later fusion}
In the embedding fusion module, we perform late fusion method \cite{kang2020multimodal} to construct a separate similarity matrix (denoted below by \textbf{\textit{A$_{x}$}} and \textbf{\textit{A}}$_{s}$ for the x-vector and s-vector, respectively). We score the cosine similarity of embeddings in pairs to get similarity matrix. We use the following formula to yield the fused similarity matrix which is the input of clustering module. After tuning on evaluation set, we set the \textit{a} to 0.95.
\begin{center}
\begin{equation}
{\mathbf{A}}=a {\mathbf{A_{d}}} +(1-a) {\mathbf{A_{s}}}
\end{equation}
\end{center}
\subsubsection{Clustering}
In this stage, we perform normalized maximum eigengap spectral clustering (NME-SC) \cite{park2019auto} to obtain speaker labels of segments. NME-SC is a kind of spectral clustering algorithm, which can automatically estimate the number of clusters. During initial clustering, we found that if we tune the time scale of segments, the speaker diarization system will estimate the number of speakers more accurately.
\subsubsection{Overlapped speech detection} \label{subsubsection:osd}
The duration of overlapped speech accounts for a large proportion in AliMeeting corpus. In our system, we take two-stage OSD method to solve overlapped speech problem. The first stage is to detects overlapped region of speech. Then, In the second stage, we use the output of heuristic algorithm \cite{otterson2007efficient} to obtain secondary speaker labels. The secondary speaker labels of non-overlapped region will be removed. In the first stage, we propose a novel neural network architecture named DMSNet to detect overlapped speech of multi-channel audio. DMSNet is a sequence-to-sequence architecture which can be addressed as a sequence labeling task which matches a feature sequence X to the corresponding label sequence y where \textit{\textbf{X}} = {x$_1$,x$_2$,...,x$_T$} and label y = {y$_1$, y$_2$,...,y$_T$}. We use pyannote.audio toolkit \cite{bredin2020pyannote} to built this module. As shown in figure \ref{fig:DMSNet}, DMSNet consists of Attention Filter-and-sum Beamforming (AFSB) block and Conformer encoder. Inspired by FSB algorithm which design different artificial filters for channels to enhance the signal, we propose a learnable AFSB block to overcomes many shortcomings of the FSB. In AFSB block, the 8-channels raw audio is split into segments by sliding window and the feature of segments is extracted by SincNet \cite{ravanelli2018speaker} and convolutional layers. The weights of each channels are different, which make the module learn the spatial information of speech better. Squeeze-and-excitation (SE) block \cite{hu2018squeeze} is applied to learn the weights of each channel. Then, we use 1D convolution whose kernel size is 1×1 to reduce the number of channel to 1. We also adopt 24 layers Conformer as encoder to extract the speaker and spatial information of signal. Finally the classification layer including linear and softmax layers output the label. We used AliMeeting to train the module with Binary Cross Entropy (BCE).
As reported in Table \ref{tab:osd}, DMSNet achieve the best performance among these architecture. AFSB block is a necessary components of DMSNet. We perform FSB to enhance the 8-channel audio on AliMeeting evaluation set in module 1 and module 2. The weights of convolution layer in AFSB (shared) block is shared by each channel. Considering that the filter and time delay of each channel is discriminative in FSB, we make the weight of each channel different in AFSB (Discriminative), which improve the performance of OSD module by comparison between module 4 and module 6. We take Conformer (24 layers) as encoder to replace the function of Bi-LSTM and reduce the DetER of OSD module from 42.57\% to 35.48\%. After replacing PyanNet with DMSNet, the DetER of the model is reduced from 42.57\% to 32.47\%.
\subsubsection{Fusion method}
DOVER-Lap \cite{raj2021dover} is a overlap-aware fusion method for speaker diarization systems. We take two tricks to improve the performance of speaker diarization system.
The first trick is called multi time scales subsystems fusion. If we fuse subsystems with different time scales, the DER of the final fusion system can be improved. The shorter the segments, the higher temporal resolution of diarization system. In contrast, if we increase the length of segments, the x-vector of segments will become more robust.The second trick is to fuse subsystems with different OSD module to improve the performance of fusion system. As show in Figure \ref{tab:fusion}, we performed those ticks in the final system, which make the final fusion diarization system achieve the lowest DER of all our systems.
\subsection{Task2 sub-track1\&2}
For the multi-speaker ASR track with restricted datasets, we use near-field and far-field data from the AliMeeting dataset, and AISHELL-4 as the training set with triple speed perturbation data augmentation. The multi-speaker speech recognition model uses the Serialized Output Training (SOT) algorithm with the same configuration as the baseline, and we achieve a CER of 29\% on the validation set.
\section{Results and Analysis}
The performance of our system on evaluation set is shown in Table \ref{tab:fusion}. A total of 6 different OSD modules were applied on speaker diarization systems. DMSNet based OSD module (M6) achieve the best performance of all OSD modules. By comparing S1 and S4, we found that the performance of sx-vector which combine spatial embedding and speaker embedding is better than x-vector on different time scales. We performance tow-step system fusion. In the first step, we fused sub-systems with different OSD modules and the results was improved. We further fused the S3, S4 and S5 whose time scales are different and the DER of system is reduced to 7.09\%. We performed the fused system on test set and achieved 9.80\% DER.
\section{Conclusions}
This paper described the XMUSPEECH system submitted to the M2MeT. We proposed a sequence-to-sequence architecture named DMSNet to detect overlapped speech of multi-channels audio. We designed a multi-channels speaker diarization system consists of ResNet34-SE based x-vector extractor, FSB based s-vector extractor, NME-SC module, and DMSNet based OSD. Finally, we fuse sub-systems with different OSD module and time scales to achieved 7.09\% and 9.80\% DER on AliMeeting evaluation set and test set.
\bibliographystyle{IEEEbib}
|
3,212,635,537,950 | arxiv | \section{Introduction}
In recent years much attention has been given to the study of gravitational
models appearing in superstring theory low energy limit
\cite {gsw}--\cite {mah}.
Einstein--Maxwell
theory with dilaton and axion fields (EMDA) is one of such models.
It appears in the
frames of heterotic string theory as a result of omission of
a part of vector and
scalar fields arising during extra dimensions compactification. As it has
been established earlier, the theory under consideration leads to
three--dimensional $\sigma$--model with symmetric target space which
possesses an isometry group locally isomorphic to $Sp(4,R)$
\cite {jmp}, \cite {pr1}, and the model
admits a null--curvature $Sp(4,R)/U(2)$ coset representation
\cite {prl}--\cite {pl}. The brief
description of matrix formalism is given in the following section.
Subsequently the class of exact solutions to the motion equations
written in matrix
form is constructed. Using the Kramer--Neugebauer approach \cite {kn},
we consider coset space matrix $M$
dependance on one space coordinate function $\lambda (x^i)$.
The found solutions are corresponding to
isotropic geodesic lines family in the target space and to the set of
point centers in the coordinate three--dimensional space. In case of
magnetic, axion and NUT charges absense the represented class transforms
into the earlier obtained by Gibbons \cite {gib}. The connection with other
known special solutions \cite {ren2}, \cite {jm}
is established during the study of the case of null--curvature matrix $M$ linear
dependance on the function $\lambda$. Then
Majumdar--Papapetrou--like black hole solutions family and massless naked
singularities can be obtained from the general one. A comlete list of these
particular solutions is given in the last section of the article.
\section{Matrix Representation of the Stationary String Gravity Equations}
Let us discuss low energy effective four--dimensional action, which
describes the bosonic sector of the heterotic string, taking into account the
contribution of gravitational, Abelian vector, dilaton and axion fields:
\begin{eqnarray}\label{e1}
S = \int d^4x {\mid g \mid}^{\frac {1}{2}} ( && -R+2{\partial \phi}^2+
\frac {1}{2}
e^{4\phi }{\partial \kappa}^2 \nonumber \\ && -e^{-2\phi}F^2 -
\kappa F\tilde {F}),
\end{eqnarray}
where $R=R^{\mu \nu}_{..\mu \nu}$ is the Ricci scalar
$(R^{\mu}_{.\nu \lambda \sigma} =
\partial _{\lambda}\Gamma ^{\mu}_{\nu \sigma}...)$
of the 4-metric $g_{\mu \nu}$, signature $+ - - -$, $\mu = 0,...,3$ and
\begin{eqnarray}\label{e2}
F_{\mu \nu}&=\partial _{\mu}A_{\nu}-\partial _{\nu}A_{\mu},
\nonumber \\
\tilde {F}^{\mu \nu}&=\frac {1}{2} E^{\mu \nu \lambda \sigma}F_{\lambda \sigma}.
\end{eqnarray}
In doing so we consider that the scalar field $\phi$ is the dilaton one, and
the axion is written in the form of pseudoscalar field $\kappa$.
As it has been done \cite{kn}, \cite{iw}, the four--dimensional line
element can be parametrized according to
\begin{equation}\label{e3}
ds^2=f(dt-\omega _idx^i)^2-f^{-1}h_{ij}dx^idx^j,
\end{equation}
where $i=1,2,3$. Below we will study the stationary case when both the metric
and the matter fields are time independent. It has been shown before
\cite {jmp}
that in this case part of the Euler-Lagrange equations can be used for the
transition from both spatial components of the vector potential $A_i$
and entered
in (\ref{e3}) functions $\omega _i$ to the magnetic $u$ and rotation $\chi$
potentials respectively. The new and old variables are connected by
differential relations:
\begin{equation}\label{e4}
\nabla u=fe^{-2\phi}(\sqrt{2}\nabla \times \vec A+\nabla v \times \vec \omega)
+\kappa \nabla v,
\end{equation}
\begin{equation}\label{e5}
\nabla \chi =u\nabla v-v\nabla u -f^2\nabla \times \vec \omega.
\end{equation}
The new notation $v=\sqrt{2}A_0$ is entered and the three--dimensional operator
$\nabla$ is
corresponded to the three--dimensional metric $h_{ij}$. Also it has been found
that expressed in terms of $f,\chi, u, v, \phi, \kappa$ variational equations
for the action (\ref{e1}) are at the same time Euler-Lagrange equations for the three
dimensional action
\begin{equation}\label{e6}
^3S=\int d^3x h^{\frac {1}{2}}(-^3R+^3L),
\end{equation}
where $^3R$ is the curvature scalar constructed according to 3--metric
$h_{ij}$ and
\begin{eqnarray}\label{e7}
^3L=&\frac {1}{2}f^{-2}[(\nabla f)^2+(\nabla \chi +v\nabla u-u\nabla v)^2]
\nonumber \\
&-f^{-1}[e^{2\phi}(\nabla u-\kappa \nabla v)^2+e^{-2\phi}(\nabla v)^2]
\nonumber \\
&+ 2(\nabla \phi)^2 + \frac {1}{2}e^{4\phi}(\nabla \kappa)^2
\end{eqnarray}
Thus in the stationary case the string gravity appears to be the nonlinear
$\sigma$-model. As it was shown \cite {jmp}--\cite {pr2},
the three--dimensional Lagrangian $^3L$
is invariant under the ten--parametric continuous transformation
group isomorphic to $Sp(4,R)$. Then it was established that $^3L$ can be
rewritten with the aid of the four--dimensional matrix $M$ in the form
\begin{equation}\label{e8}
^3L=\frac {1}{4}Trj^2,\qquad j=\nabla M M^{-1},
\end{equation}
and $M$, being the matrix of the coset $Sp(4,R)/U(2)$, satisfies the
symplectic and symmetric properties,
\begin{equation}\label{e9}
M^TJM=J,\qquad M^T=M,
\end{equation}
where
\begin{eqnarray}\label{e10}
J=\left (\begin{array}{crc}
0&-I\\
I&0\\
\end{array}\right ).
\end{eqnarray}
It's easy to see that any symplectic matrix $G$ defines automorphism
$M\rightarrow G^TMG$ for the coset under consideration.
The relations (\ref{e9}) allow to parametrize the matrix by six
independent functions
which can be chosen as potentials $f, \chi, u, v, \phi, \kappa$.
Here it is convenient to use the two--dimensional matrices
$P$ and $Q$ which define the Gauss decomposition
\begin{eqnarray}
M=\left (\begin{array}{crc}
P^{-1}&P^{-1}Q\\
QP^{-1}&P+QP^{-1}Q\\
\end{array}\right ).
\end{eqnarray}
Their evident form is \cite {pl}:
\begin{eqnarray}
P=\left (\begin{array}{crc}
f-v^2e^{-2\phi}&-ve^{-2\phi}\\
-ve^{-2\phi}&-e^{-2\phi}\\
\end{array}\right ),
\end{eqnarray}
\begin{eqnarray}
Q=\left (\begin{array}{crc}
-\chi +vw&w\\
w&-\kappa\\
\end{array}\right ),
\end{eqnarray}
where $w=u-\kappa v$.
\section{The General Geodesic Isotropic Solution}
The appropriate to the three--dimensional
action motion equations have the standart form
\begin{equation}
\nabla j=0,
\end{equation}
\begin{equation}
^3R_{ik}=\frac {1}{4}Tr(j_ij_k),
\end{equation}
and admit the procedure of exact solution construction stated before by
Kramer and Neugebauer for arbitrary $\sigma$-models \cite {kn}
and developed later by Clement for the case of $SL(3,R)/SO(2,1)$ matrix
representation of Kaluza--Klein five--dimensional theory
\cite {cl1}--\cite {cl3}.
We consider the ansatz for which the matrix
$M$ is determined by the aid of one space coordinate function $\lambda$
\begin{equation}
M=M(\lambda),\qquad \lambda =\lambda(x^i),
\end{equation}
when $\lambda(x^i)$ is supposed to satisfy the Laplace equation:
\begin{equation}
\nabla ^2\lambda =0.
\end{equation}
It is not difficult to prove that the `material' equation (14) turns
into a relation, determining the dependance of $M$ on $\lambda$:
\begin{equation}
\frac {d}{d\lambda}\left (\frac {dM}{d\lambda}M^{-1}\right )=0.
\end{equation}
The sense of (18) becomes clear after the introduction of the so called target
space, i.e., of the metric space with the coordinates
$f, \chi, u, v, \phi, \kappa$ and the linear element
uniquely connected with the Lagrangian $^3L$:
\begin{equation}
dS^2=\frac {1}{4}Tr(dM M^{-1} dM M^{-1}).
\end{equation}
Then the formula $M=M(\lambda)$ determines a line in the target space which
according to (18) is a geodesic one \cite {kn}.
The solution of the equation (18) is
\begin{equation}
M=SM_0,
\end{equation}
where
\begin{equation}
S=e^{\lambda A}=\sum _{0}^{\infty}\frac {(\lambda A)^n}{n!},
\end{equation}
$A=const$ and $M_0=M\mid _{\lambda = 0}$.
The matrix $S$, so far is only formally determined,
later on by natural causes it will be called the evolutionary operator.
The three--dimensional Einstein equations (15) in view of (20) and (21)
can be rewritten
\begin{equation}
^3R_{ik}=\frac {1}{4}\lambda _{,i}\lambda _{,k}TrA^2
\end{equation}
and form together with (17) the complete system of equations which determines
the three--dimensional metric $h_{ik}$ and the scalar function $\lambda$.
Let us now establish the conditions for the evolutionary operator $S$
and the matrix $A$ determined by it. Their fulfilment ensures that $M$ belongs
to the coset space $Sp(4,R)/U(2)$ along the whole geodesic line,
only if it is true for $\lambda =0$, i.e., for the matrix $M_0$.
It is evident that if the matrices $M_0$ and
$M$ are symplectic ones, the operator $S$ should possess the same feature,
and it is convenient to rewrite the first of the relations (\ref{e9}) for it
in the form of
\begin{equation}
S^T=-JS^{-1}J,
\end{equation}
hence $A$ can be immediately determined:
\begin{equation}
A^T=JAJ.
\end{equation}
Thus, $A$ is an element of $sp(4,R)$ algebra; and after the
solution of (24), it can be represented as
\begin{eqnarray}
A=\left (\begin{array}{crc}
-s^T&r\\
l&s\\
\end{array}\right ),
\end{eqnarray}
where $l^T=l,\quad r^T=r$ and $s$ are the two--dimensional matrices
which in sum define ten independent parameters.
Then, in order that $M^T=M$ followed from $M_0^T=M_0$, the evolutionary
operator should satisfy the (nongroup) condition
\begin{equation}
S^T=M_0^{-1}SM_0,
\end{equation}
which imposes on $A$ the following restriction:
\begin{equation}
A^T=M_0^{-1}AM_0.
\end{equation}
Here we can finish the general analysis and turn our attention to the
solutions determined by isotropic geodesic lines in the target space.
From (19)--(21) follows that condition $dS^2=0$ is equivalent to
\begin{equation}
TrA^2=0.
\end{equation}
Then from (17) and (22) immediately follows that $h_{ik}$ and $\lambda (x^i)$
can be taken in the form
\begin{equation}
h_{ik}=\delta _{ik},\qquad
\lambda =\sum \frac {\lambda _n}{\mid \vec r-\vec r_n\mid}
\end{equation}
where $\vec r$ is as usual connected to $x^i$ and $\vec r_n$ is considered
as the position of the n--center characterized by $\lambda _n$.
We will assume that $\sum \lambda _n \ne 0$ (the dropped
special case can be investigated in the same way), then in view of (21),
without loss of generality, it is possible to impose on
$\lambda _n$ the normalization condition
\begin{equation}
\sum _{n} \lambda _n=1.
\end{equation}
It is evident that $\lambda \rightarrow 0$ when $r\rightarrow \infty$,
thus $M_0=M_{\infty}$. Let us naturally determine
the asymptotic values of physical fields assuming that
\begin{equation}
f_{\infty}=1,\quad \chi _{\infty}=u_{\infty}=v_{\infty}=\phi _{\infty}=
\kappa _{\infty}=0;
\end{equation}
then from (11)--(13) we obtain
\begin{eqnarray}
M_0=\left (\begin{array}{crc}
\sigma _3&0\\
0&\sigma _3\\
\end{array}\right )\equiv \Sigma _3,
\end{eqnarray}
and $\sigma _3$ is one of the Pauli matrices. By substituting the found
value $M_0$ in (27), $A$ can be calculated as
\begin{eqnarray}
A=\left (\begin{array}{crc}
-\tilde s&r\\
\tilde r&s\\
\end{array}\right ),
\end{eqnarray}
where $s^T=\tilde s$ and for any two--dimensional matrix $m$ we define
$\tilde m=\sigma _3m\sigma _3$.
From (21), (29), (30) it follows that at $r\rightarrow \infty$
\begin{equation}
S\longrightarrow I+\frac {A}{r}
\end{equation}
and because of (20)
\begin{equation}
M\longrightarrow \Sigma _3+\frac {A\Sigma _3}{r}.
\end{equation}
After that, applying (11)--(13) and (25) it is easy to show that
the main parts of the asymptotic decomposition of the functions
$f-1, \chi, u, v, \phi, \kappa$ are proportional to $\frac {1}{r}$. In this
case, six components of matrices $s$ and $r$ act as coefficients which
in this way determine six physical charges of the system. By entering the mass
$M$, the parameter NUT $N$ and also the electric $Q_e$, magnetic $Q_m$,
dilaton $D$ and axion $A$ charges according to formulae
\begin{eqnarray}
f& \rightarrow 1-\frac{2M}{r},\qquad \chi \rightarrow \frac{2N}{r},
\nonumber \\
v& \rightarrow \frac{\sqrt 2 Q_e}{r},\qquad u \rightarrow
\frac{\sqrt 2 Q_m}{r}&,
\nonumber \\
\phi & \rightarrow \frac{D}{r},\qquad \kappa \rightarrow \frac{2A}{r},
\end{eqnarray}
$s$ and $r$ are found as follows:
\begin{eqnarray}
s=\left (\begin{array}{crc}
-2M&\sqrt 2Q_e\\
-\sqrt 2Q_e&-2D\\
\end{array}\right ),
\noindent \\
r=\left (\begin{array}{crc}
2N&-\sqrt 2Q_m\\
-\sqrt 2Q_m&-2A\\
\end{array}\right ).
\end{eqnarray}
Let us determine now the evident form of the evolutionary operator $S$, which
was written before with the aid of the formal exponential
series. The calculation
of $A^2$ in view of (33) gives:
\begin{equation}
A^2=\alpha ^{\mu}T_{\mu},
\end{equation}
where parameters $\alpha ^{\mu}$ are of the second order with respect
to charges, $T_0$ is
the unit matrix and three traceless matrices $T_i$ are
\begin{eqnarray}
T_1=\left (\begin{array}{crc}
\sigma _2&0\\
0&-\sigma _2\\
\end{array}\right ),
\qquad
T_2=\left (\begin{array}{crc}
0&\sigma _2\\
\sigma _2&0\\
\end{array}\right ),
\qquad
T_3=\Sigma _3.
\end{eqnarray}
It is convenient to unite six real charges into three complex ones:
\begin{eqnarray}
{\cal M}&=M+iN,
\nonumber \\
{\cal D}&=D+iA,
\nonumber \\
{\cal Q}&=Q_e+iQ_{m},
\end{eqnarray}
in terms of which
\begin{eqnarray}
\alpha ^0&=&2(\bar {\cal M}{\cal M}+\bar {\cal D}{\cal D}-
\bar {\cal Q}{\cal Q}),
\nonumber \\
\alpha ^1+i\alpha ^2&=&-2\sqrt 2({\cal M}\bar {\cal Q}+\bar {\cal D}{\cal Q}),
\nonumber \\
\alpha ^3&=&2(\bar {\cal M}{\cal M}-\bar {\cal D}{\cal D}).
\end{eqnarray}
In doing so the isotropic condition (28) can be
rewritten as
\begin{equation}
\bar {\cal M}{\cal M}+\bar {\cal D}{\cal D}=\bar {\cal Q}{\cal Q}
\end{equation}
and generalizes the known relations in the Einstein-Maxwell theory
\cite {per}--\cite {m}.
It is easy to verify that the commutators of the matrices $T_i$ are
not their linear
combinations, i.e., these matrices do not form the basis of a three--dimensional
Lie algebra. But the calculation of the corresponding anticommutators leads
to the following result:
\begin{equation}
\{ T_i,T_j\} =T_iT_j+T_jT_i=-\eta _{ij},
\end{equation}
where $\eta _{ij}=diag(1, 1, -1)$, thus in view of (39) and (40) it appears that
\begin{equation}
A^4=-\eta _{ij}\alpha ^i\alpha ^j.
\end{equation}
The application of the relation (43) also allows to determine the fact that the
quadratic form $\eta _{ij}\alpha ^i\alpha ^j$ is not negative and enter
a new parameter $\alpha$ according to the definition
\begin{equation}
\alpha ^4=\eta _{ij}\alpha ^i\alpha ^j
\end{equation}
The relation (46) allows to sum the series which correspond to the items with
$n=4k$ from the exponent decomposition (21). Now it is not difficult to find the
evident form for the remaining three series with
$n=4k+1$, $n=4k+2$ and $n=4k+3$. The mentioned four
series just compose the evolutionary operator and its expression in
terms of the charge matrix $A$, defined by
(33), (37), (38) and (43), and by the
function $\lambda$ (29) is
\begin{equation}
S=\sum _0^3S_{\mu}A^{\mu},
\end{equation}
where $A^{\mu}$ is the matrix $A$ to the $\mu$ power and
\begin{eqnarray}
S_0&=&\cosh (\alpha \lambda / \sqrt 2)\cos (\alpha \lambda / \sqrt 2),
\nonumber \\
S_1&=\frac {1}{\sqrt 2\alpha}
&(\sinh (\alpha \lambda / \sqrt 2)\cos (\alpha \lambda / \sqrt 2)
\nonumber \\
&+&\cosh (\alpha \lambda / \sqrt 2)\sin (\alpha \lambda / \sqrt 2)),
\nonumber \\
S_2&=\frac {1}{\alpha ^2}
&\sinh (\alpha \lambda / \sqrt 2)\sin (\alpha \lambda / \sqrt 2),
\nonumber \\
S_3&=\frac {1}{\sqrt 2\alpha ^3}
&(\cosh (\alpha \lambda / \sqrt 2)\sin (\alpha \lambda / \sqrt 2)
\nonumber \\
&-&\sinh (\alpha \lambda / \sqrt 2)\cos (\alpha \lambda / \sqrt 2)).
\end{eqnarray}
The constructed solution (29) and (47)--(48)
defines the system of interacting point centers which satisfies
the restriction (43).
Let us turn our attention to the group nature of the matrix $S$. From (33)
it is evident that the determining $S$ matrix $A$ differs from the belonging
to the $sp(4,R)$ algebra general matrix by
\begin{eqnarray}
\Gamma =\left (\begin{array}{crc}
\tilde \tau &\rho \\
-\tilde \rho &\tau \\
\end{array}\right ),
\end{eqnarray}
where $\tilde \tau =-\tau ^T$, and $\rho$ is symmetric.
Entered here $\Gamma$ has four independent parameters, the corresponding linear
independent matrices (generators) can be written as
\begin{eqnarray}
\Gamma _0=\left (\begin{array}{crc}
0&\sigma _3\\
-\sigma _3&0\\
\end{array}\right ),
\Gamma _1=\left (\begin{array}{crc}
0&\sigma _1\\
\sigma _1&0\\
\end{array}\right ),
\nonumber \\
\Gamma _2=\left (\begin{array}{crc}
-\sigma _1&0\\
0&\sigma _1\\
\end{array}\right ),
\Gamma _3=\left (\begin{array}{crc}
0&I\\
-I&0\\
\end{array}\right ).
\end{eqnarray}
It is easy to prove that
\begin{equation}
[\Gamma _0, \Gamma _i]=0
\end{equation}
and pair products for $\Gamma _i$ are
\begin{equation}
\Gamma _i\Gamma _j=I\eta _{ij}+\epsilon _{ijk}\eta ^{kl}\Gamma _l
\end{equation}
where $\eta ^{kl}=\eta _{kl}$.
Because of the resulting from (52) commutation relations,
the isomorphism between
algebra of $\Gamma _i$ and that of two--dimensional Pauli matrices $\sigma _i$
\begin{eqnarray}
\sigma _1=\left (\begin{array}{crc}
0&1\\
1&0\\
\end{array}\right ),
\sigma _2=\left (\begin{array}{crc}
0&-1\\
1&0\\
\end{array}\right ),
\sigma _3=\left (\begin{array}{crc}
1&0\\
0&-1\\
\end{array}\right ),
\end{eqnarray}
belonging to $sl(2,R)$, can be determined as
\begin{equation}
\Gamma _1\sim \sigma _1,\quad \Gamma _2\sim \sigma _3,\quad
\Gamma _3\sim \sigma _2.
\end{equation}
If we also identify $\Gamma _0\sim I$,
it is easy to notice that the algebra of matrices $T_{\mu}$
appears to be isomorphic to $sl(2,R)\bigoplus R \sim gl(2,R)$.
So the part of the group omitted while constructing the evolutionary operator
is locally isomorphic to $GL(2,R)$ and hence
$S\in Sp(4,R)/GL(2,R)$.
Later on it can be seen that the matrix
\begin{equation}
G=e^{\Gamma}=e^{\gamma ^{\mu}\Gamma _{\mu}}
\end{equation}
in view of (49) satisfies the relation
\begin{equation}
G^T\Sigma _3G=\Sigma _3.
\end{equation}
This allows to interpret $G$ as a general matrix of belonging to $Sp(4,R)$
transformation which preserves the asymptotical vacuum values for the
system of physical fields.
It is necessary to remark that the formal expression (55) as the corresponding
one for $S$ can be obtained in the evident form. Namely, let us denote the
matrix constructed on $\Gamma _0$ by $G_{(0)}$ and that of constructed on
$\Gamma _i$ by $G_{(3)}$.
Then in view of (51)
\begin{equation}
G=G_{(0)}G_{(3)}=G_{(3)}G_{(0)}.
\end{equation}
Employing the relation
\begin{equation}
\Gamma _0^2=-I
\end{equation}
we have that
\begin{equation}
G_{(0)}=I\cos \gamma ^0+\Gamma _0\sin \gamma ^0.
\end{equation}
Then, noticing that from (52) follows the anticommutation relations
$\{ \Gamma _i\Gamma _j\} =2I\eta _{ij}$
the expression for $G_{(3)}$ matrix can be found:
\begin{eqnarray}
2G_{(3)}=I&[(1+\sigma)\cosh \gamma +(1-\sigma)\cos \gamma]
\nonumber \\
+\frac {\gamma ^i\Gamma _i}{\gamma}
&[(1+\sigma)\sinh \gamma +(1-\sigma)\sin \gamma]
\end{eqnarray}
where the parameter $\gamma$ is determined as
\begin{equation}
(\gamma)^2=\sigma \eta _{ij}\gamma ^i \gamma ^j
\end{equation}
and $\sigma =sign(\eta _{ij}\gamma ^i \gamma ^j)$.
\section{Black Holes and Naked Singularities}
The general geodesic isotropic solution of the string gravity obtained
in the previous part has intricate dependence from the function $\lambda$,
which satisfies Laplace equation and hence, from the space coordinates.
It is easy to verify that in the case when $\alpha=0$, i.e., if
\begin{equation}
\alpha_i\alpha_j\eta^{ij}=0
\end{equation}
the evolutionary operator $S$ becomes the polynomial of third power on
$\lambda$ which considerably facilitates the solution analysis. The greatest
simplification is obtained at the simultaneous imposure of the
set of three additional relations $\alpha^i=0$
to the physical
charges, which according
to (42) are equivalent to
\begin{eqnarray}
{\cal M}\bar {\cal Q}+{\cal Q}\bar {\cal D}=0,
\nonumber \\
{\cal M}\bar {\cal M}-{\cal D}\bar {\cal D}=0.
\end{eqnarray}
In doing so, in view of (21) and (39) the evolutionary operator
and the null-curvature
matrix $M$ occure to be the linear functions of $\lambda$ and satisfy,
according to (17), the Laplace equation. The result can be investigated and,
as it is further demonstrated, it contains interesting physical solutions.
At first and foremost the independent `coordinates' can be entered in the
charge space. The number of such `coordinates' appears to be equal to three,
as it immediately follows from (43) and (63). By applying the complex form of
transcription (41) we have:
\begin{eqnarray}
{\cal M} &=& \rho e^{2i\delta_1},\qquad
{\cal D} = \rho e^{2i\delta_2},
\nonumber \\
{\cal Q} &=& -i\sqrt{2}\sigma\rho e^{i(\delta_1+\delta_2)},
\end{eqnarray}
or, going to the real charges
\begin{eqnarray}
M & = & \rho \cos{2\delta_1}, \nonumber \\
N & = & \rho \sin{2\delta_1}, \nonumber \\
D & = & \rho \cos{2\delta_2}, \nonumber \\
A & = & \rho \sin{2\delta_2}, \\
Q_e & = & \sqrt{2}\sigma \rho \sin{(\delta_1-\delta_2)}, \nonumber \\
Q_m & = & -\sqrt{2}\sigma \rho \cos{(\delta_1-\delta_2)}, \nonumber
\end{eqnarray}
and $\sigma =\pm 1$.
Turning back to relation (20) it is possible to calculate the matrix $M$.
After that, employing the Gauss decomposition formula (11) and formulae (12)
and (13),
which determine the explicit dependance of matrix elements from six
independent $\sigma$--model functions, the expressions can be found as follows:
\begin{eqnarray}
f & = & (1+2M\lambda)^{-1}, \nonumber \\
\chi & = & -2N\lambda (1+2M\lambda)^{-1}, \nonumber \\
v & = & \sqrt{2}Q_e\lambda (1+2M\lambda)^{-1}, \nonumber \\
u & = & \sqrt{2}Q_m\lambda (1+2M\lambda)^{-1}.
\end{eqnarray}
In this case the expressions for axion and dilaton appear to be rather
cumbersome, but by entering the complex variable
\begin{equation}
z=\kappa+ie^{-2\phi}
\end{equation}
which, according to \cite {pl}, is one of the Ernst potentials
for the stationary
system (\ref{e6})--(\ref{e7}) the following compact result, which generalizes expressions
found in \cite {ren1} and \cite {ren2}, can be obtained
\begin{equation}
z=i\frac{1+\lambda({\cal M}-{\cal D})}{1+\lambda({\cal M}+{\cal D})}.
\end{equation}
In this case, when the solution is determined only by one center, i.e., when
$\lambda=\frac{1}{r}$, it is convenient to turn to a new radial coordinate
$R=r+2M$. It is easy to show that the expressions for electric and magnetic
potentials transform to the most simple Coulomb form
\begin{eqnarray}
v &=& \frac{\sqrt{2}Q_e}{R},\qquad u=\frac{\sqrt{2}Q_m}{R},\nonumber \\
z &=& i\frac{R-\bar {\cal M}-{\cal D}}{R-\bar {\cal M}+{\cal D}}.
\end{eqnarray}
The employment of (\ref{e5}) and (66) allows to determine the obvious form of
four--dimensional space--time metric:
\begin{eqnarray}
ds^2 & = & (1-\frac{2M}{R})(dt-2Ncos{\theta } d\phi)^2
\nonumber \\
& - & (1-\frac{2M}{R})^{-1}dR^2 - R(R-2M)d\Omega ^2.
\end{eqnarray}
In the constructed solution the mass, which causes the horizon appearance at
$R=R_H=2M$, and the parameter NUT which makes the spatial interval
asymptotically
different from Minkowski metric, appear to be independent parameters.
This means that exist special solutions, which are asimptotically flat and
have the
horizon (black holes) and also solutions, possessing everywhere the regular
but asimptotically unflat four--dimensional metric with the Coulomb-like
expressions for the material fields (the naked singularities). It is important
to note that the presence of the naked singularities in the string gravity
appears to be possible due to the existence of the scalar sector in the
theory, i.e., dilaton and axion fields.
From here on while investigating the special cases, both the results for the
multicenter system and formulae (69) and (70) describing isolated
singular object,
will be taken into account.
So, let us discuss the case $N=0$, that according to (65) and condition $M>0$ is
equivalent to the relation $\delta_1=\pi n$. The corresponding formulae for
the charges lead to the following expressions:
\begin{equation}
D \sim Q_m^2-Q_e^2, \qquad A \sim Q_mQ_e
\end{equation}
It is seen that the dilaton black holes (with $A=0$) have one of the
electromagnetic charges equal to zero while the axion black holes
(for them $D=0$) have equal in absolute magnitude electric and magnetic
charges. Hence from the constructed before family of solutions (66), (68) and
(69)--(70)
naturally four black holes subfamilies stand out:
dilaton magnetic, for which $\delta_2=\pi k$,
\begin{equation}
D=M,\quad Q_m=\sqrt{2}\sigma M,
\end{equation}
and all the other charges are equal to zero;
dilaton electric, for which $\delta_2=\pi (k+1/2)$
\begin{equation}
D=-M,\quad Q_e=\sqrt{2}\sigma M;
\end{equation}
axion with $Q_e=-Q_m$ when $\delta_2 =\pi (k+1/4)$
\begin{equation}
A=M,\quad Q_e=-Q_m=\sigma M;
\end{equation}
and, at last, axion with $Q_e=Q_m$ appearing at
$\delta_2=\pi (k+3/4)$ and possessing
\begin{equation}
A=-M,\quad Q_e=Q_m=\sigma M.
\end{equation}
One can notice that the discret transformation $Q_m\rightarrow Q_e,\quad
Q_e\rightarrow -Q_m$ transforms magnetic dilaton solution (72) into
electric one (73) with simultaneous change $D\rightarrow -D$. In doing
so the appropriate axion solutions transform one into another, if
besides of the above mentioned electromagnetic transformation
$A\rightarrow -A$ taking place.
Now we can study the massless solution families having, in accordance with (65)
the NUT parameter value not equal to zero. From (70) it can be seen that
the space metric is regular everywhere and only matter
fields have physical peculiarities.
So let us examine the case $M=0$. As the parameter $N$ can be of different
sign, from (65) follows that $\delta_1 =\pi (2n+1)/4$. Omitting the
technical details we will turn our attention to the discussion of the main
results. It turns out that the family of naked singularities under
investigation, just as described above the black holes family, has the four
most simple solution classes. Namely, the case when $\delta_1 =\pi (l+1/4),
\delta_2=\pi (k+1/4)$ and $\delta_1 =\pi (l+3/4), \delta_2 =\pi (k+3/4)$
corresponds to the axion magnetic solution (constructed earlier in
\cite {pr2}):
\begin{equation}
A=N,\quad Q_m=\sqrt{2}\sigma N;
\end{equation}
when $\delta_1=\pi (l+1/4), \delta_2=\pi (k+3/4)$ and
$\delta_1=\pi (l+3/4), \delta_2=\pi (k+1/4)$ we get
dilaton electric singularity:
\begin{equation}
A=-N,\quad Q_e=\sqrt{2}\sigma N;
\end{equation}
when $\delta_1=\pi (l+3/4), \delta_2=\pi (k+1/2)$ and $\delta_1=\pi (l+1/4),
\delta_2= \pi k$ it appears that
\begin{equation}
D=N,\quad Q_e=-Q_m=\sigma N,
\end{equation}
and we get dilaton singularity with electromagnetic charges of different sign;
and, finally, when $\delta_1 = \pi (l+1/4), \delta_2 =\pi (k+1/2)$ and
$\delta_1=\pi (l+3/4), \delta_2 =\pi k$
the fields configuration of
the dilaton singularity is determined by equal values of electric and magnetic
charges, which are connected with NUT parameter as follows:
\begin{equation}
D=-N,\quad Q_e=Q_m=\sigma N.
\end{equation}
Similarly to the situation with black holes, the determined above discret
transformations acting in the charge space connect axion singularities (76)
and (77) and also (with the corresponding change of axion charge to dilaton one)
transfer dilaton solutions (78)--(79) one into another. It is important to
stress the resulting from formulae (72)--(75)
and (76)--(79) formal analogy between
dilaton black holes and naked axion singularities from one hand and axion
black holes and dilaton naked singularities from the other.
Turning back to the solution (65), (69), (70) describing singular object
with mass
and parameter NUT it can be pointed out that the solutions describing
asimptotic flat black holes can be transformed into solutions for the
horizonless asymptotic nonflat naked singularities with the aid of continuous
transformation of the parameter $\delta_1$.
\section{Conclusion}
Using the Kramer--Neugebauer method for the null--curvature matrix
Sp(4,R)/U(2) coset formulation of the stationary D=4 EMDA theory we
have constructed a new class of solutions which describe a system of interacting
point centers. These centers describe a set of Majumdar--Papapetrou--like
black holes in a special case and massless naked singularities in another one.
A general class is connected with a complete family of isotropic geodesic
lines which are crossing in a Minkowski vacuum point of the target space.
As it has
been shown, the evolutionary operator transforming vacuum solution to nontrivial
one belongs to Sp(4,R)/GL(2,R) coset. The ommited four generators of GL(2,R)
subgroup defines the general automorphism for the Sp(4,R)/U(2) target space
which preserves asymptotic flatness.
Used formalism admits the natural generalization when solutions are defined by
extremal area surfaces in the potential space. It gives the possibility to
construct Israel--Wilson--like sourses for the theory under consideration.
\acknowledgments
This work was supported in part by the ISF Grant No. M79000.
|
3,212,635,537,951 | arxiv | \section{Definitions and Preliminaries} \label{sec:pre}
\subsection{PAC learning}
Let $\mathcal{X}$ be an \textit{instance space}, $\mathcal{Y}=\{-1,1\}$ be a \textit{label set}, and let $\mathcal{D}$ be an (unknown) distribution
over $\mathcal{X}\times\mathcal{Y}$. An ``$\mathcal{X}\to\mathcal{Y}$'' function is called a concept/hypothesis. The goal here is to design
a learning algorithm, which given a large enough input sample $S=((x_{1},y_{1})),\ldots,(x_{m},y_{m}))$
drawn i.i.d.$\,$from $\mathcal{D}$, outputs an hypothesis $h:{\mathcal{X}}\to\mathcal{Y}$ whose \textit{expected risk} is small compared to the best hypothesis in a \emph{hypothesis class} $\mathcal{H}$, which is a fixed and
known to the algorithm. That is,
\[
L_{\mathcal{D}}(h)\lesssim\inf_{h'\in\mathcal{H}}L_{\mathcal{D}}(h')\quad\textrm{where}~~L_{\mathcal{D}}(h):=\mathbb{E}_{(x,y)\sim\mathcal{D}}[\ell(h(x),y)]~~\quad\ell(a,b)=\mathbf{1}_{a\neq b}\,.
\]
The distribution $\mathcal{D}$ is said to be realizable with respect to $\mathcal{H}$ if there exists $h^{\star}\in\mathcal{H}$ such that $L_{\mathcal{D}}(h)=0$. We also define the empirical risk of an hypothesis $h$ with respect to a sample $S=((x_{1},y_{1}),\ldots,(x_{m},y_{m}))$ as $L_{S}(h)=\frac{1}{m}\sum_{i=1}^{m}\ell(h(x_{i}),y_{i})$.
\begin{defn} \textbf{(PAC learning})
An hypothesis class $\mathcal{H}$ is PAC learnable with sample complexity $m(\alpha,\beta)$ if there exists
an algorithm $\mathcal{A}$ such that for any distribution $\mathcal{D}$ over $\mathcal{X}$, an accuracy and confidence parameters $\alpha,\beta\in(0,1)$,
if $\mathcal{A}$ is given an input sample $S=((x_{1},y_{m}),\ldots,(x_{m},y_{m}))\sim\mathcal{D}^{m}$ such that $m\ge m(\alpha,\beta)$, then it outputs an hypothesis $h:\mathcal{X}\rightarrow\mathcal{Y}$ satisfying $L_{\mathcal{D}}(h)\le\alpha$ with probability at least $1-\beta$. The class $\mathcal{H}$ is efficiently PAC learnable
if the runtime of $\mathcal{A}$ (and thus its sample complexity) are polynomial in $1/\alpha$ and $1/\beta$. If the above holds only
for realizable distributions then we say that $\mathcal{H}$ is PAC learnable in the realizable setting.
\end{defn}
\subsection{Differentially private PAC learning}
In some important learning tasks (e.g. medical analysis, social networks,
financial records, etc.) the input sample consists of sensitive data
that should be kept private. Differential privacy (\cite{Dinur2003,dwork2006calibrating})
is a by-now standard formalism that captures such requirements.
The definition of differentially private algorithms is as follows.
Two samples $S',S''\in(\mathcal{X}\times\mathcal{Y})^{m}$ are called \textit{neighbors} if there exists at most one $i\in[m]$
such that the $i$'th example in $S'$ differs from the $i$'th example
in $S''$.
\begin{defn} \textbf{(Differentially private learning)} A learning algorithm $\mathcal{A}$ is said to be $\epsilon$-differentially private\footnote{The algorithm is said to be $(\epsilon,\delta)$-approximate differentially
private if the above inequality holds up to an additive factor $\delta$.
In this work we focus on the so-called pure case where $\delta=0$. } (DP) if for any two neighboring
samples and for any measurable subset $\mathcal{F}\in\mathcal{Y}^{\mathcal{X}}$,
\begin{align*}
&Pr[\mathcal{A}(S)\in\mathcal{F}]\le\exp(\epsilon)Pr[\mathcal{A}(S')\in\mathcal{F}]~~\textrm{and}\\ & Pr[\mathcal{A}(S')\in\mathcal{F}]\le\exp(\epsilon)Pr[\mathcal{A}(S)\in\mathcal{F}]
\end{align*}
\end{defn}
\emph{Group privacy} is a simple extension of the above definition~\cite{dwork2014algorithmic}:
Two samples $S,S'$ are $q$-neighbors if they differ in at most $q$ of their pairs.
\begin{lem}
Let $\mathcal{A}$ be a DP learner. Then for any $q\in\mathbb{N}$
and any two $q$-neighboring samples $S,S'$ and any subset $\mathcal{F}\in\mathcal{Y}^{\mathcal{X}}\cap\mathrm{range}(\mathcal{A})$,
$Pr[\mathcal{A}(S)\in\mathcal{F}]\le\exp(\epsilon q)Pr[\mathcal{A}(S')\in\mathcal{F}]$
\end{lem}
Combining the requirements of PAC and DP learnability yields the definition
of private PAC (PPAC) learner.
\begin{defn}
\textbf{(PPAC Learning)} A concept class $\mathcal{H}$ is differentially
private PAC learnable with sample complexity $m(\alpha,\beta)$
if it is PAC learnable with sample complexity $m(\alpha,\beta)$
by an algorithm $\mathcal{A}$ which is an $\epsilon=0.1$-differentially private.
\end{defn}
\paragraph{Remark.}
Setting $\epsilon=0.1$ is without loss of generality;
the reason is that there are efficient methods to boost the value of $\epsilon$
to arbitrarily small constants, see~\cite{Vadhan2017} and references within.
\subsection{Online Learning}
The online model can be seen as a repeated game between a learner $\mathcal{A}$ and an environment (a.k.a. adversary) $\mathcal{E}$.
Let $T$ be a (known\footnote{Standard doubling techniques allow the learner to cope with scenarios
where $T$ is not known.}) horizon parameter.
On each round $t\in[T]$ the adversary decides on a pair $(x_{t},y_{t})\in\mathcal{X}\times\mathcal{Y}$, and the learner decides on a prediction rule $h_t:\mathcal{X}\to\{0,1\}$.
Then, the learner suffers the loss $|y_{t}-\hat{y}_{t}|$, where $\hat y_t = h(x_t)$.
Both players may base their decisions on the entire history and may use randomness.
Unlike in the statistical setting, the adversary $\mathcal{E}$ can generate the examples in an adaptive manner.
In this work we focus on the {\it realizable} setting where it is assumed that
the labels are realized by some target concept $c\in \mathcal{H}$, i.e., for
all $t\in[T]$, $y_{t}=c(x_{t})$.\footnote{However, the adversary does not need to decide on the identity of
$c$ in advance.} The measure of success is the expected number of mistakes done by
the learner:
\[
\mathbb{E}[M_{\mathcal{A}}]=\mathbb{E}\bigl[\sum_{t=1}^{T}\ell(\hat{y}_{t},y_{t})\bigr],
\]
where the expectation is taken over the randomness of the learner and the adversary.
An algorithm $\mathcal{A}$ is a (strong) online learner if for any horizon parameter $T$ and any realizable sequence
$((x_{1},y_{1}),\ldots,(x_{T},y_{T}))$, the expected number of mistakes made by $\mathcal{A}$ is sublinear in $T$.
\subsubsection{Weak Online Learning}
We describe an extension due to \cite{Beygelzimer} of the boosting framework
(\cite{Schapire2012}) (from the statistical setting) to the online.
\begin{defn}
\textbf{(Weak online learning) }An online learner $\mathcal{A}$ is
called a weak online learner for a class $\mathcal{H}$ with an \emph{edge} parameter $\gamma\in(0,1/2)$
and \textit{excess loss} parameter $T_{0}>0$ if for any horizon parameter
$T$ and every sequence $((x_{1},y_{1}),\ldots,(x_{T},y_{T}))$ realized
by some target concept $c\in\mathcal{H}$, the expected number of
mistakes done by $\mathcal{A}$ satisfies
\[
\mathbb{E}[M_{\mathcal{A}}]\le\left(\frac{1}{2}-\gamma\right)T+T_{0}~.
\]
\end{defn}
\subsubsection{Oblivious vs. Non-oblivious Adversaries}
The adversary described above is adaptive in the sense that it can
choose the pair $(x_{t},y_{t})$ based on the actual predictions $\hat{y}_{1},\ldots,\hat{y}_{t-1}$
made by the learner on rounds $1,\ldots,t-1$. An adversary is called
\textit{oblivious} if it chooses the entire sequence $((x_{1},y_{1}),\ldots,(x_{T},y_{T}))$
in advance.
\subsubsection{Regret bounds using Multiplicative Weights}
Although we focus our attention on the realizable setting, our development
also requires working in the so-called agnostic setting, where the
sequence $((x_{1},y_{1}),\ldots,(x_{T},y_{T}))$ is not assumed to
be realized by some $c\in\mathcal{H}$. The standard measure of success
in this setting is the expected \textit{regret} defined as
\[
\mathbb{E}[\mathrm{Regret}_{T}]=\mathbb{E}\,\sum_{t=1}^{T}\ell(\hat{y}_{t},y_{t})-\inf_{h\in\mathcal{H}}\sum_{t=1}^{T}\ell(h(x_{t}),y_{t}).
\]
Accordingly, an online learner in this context needs to achieve a sublinear regret in terms of
the horizon parameter $T$.
When the class $\mathcal{H}$ is finite, there is a well-known
algorithm named \textit{Multiplicative Weights} (MW) which maintains
a weight $w_{t,j}$ for each hypothesis (a.k.a. \textit{expert} in
the online model) $h_{j}$ according to
\[
w_{1,j}=1~,\quad w_{t+1,j}=w_{t,j}\exp(-\eta\ell(h_{j}(x_{t}),y_{t})))
\]
where $\eta>0$ is a step-size parameter. At each time $t$, MW predicts
with $\hat{y}_{t}=h_{j}(x_{t})$ with probability proportional to
$w_{t,j}$. We refer to \cite{Arora2012} for an extensive survey
on Multiplicative Weights and its many applications. The following
theorem establishes an upper bound on the regret of MW.
\begin{thm}\label{thm:mw}
\textbf{(Regret of MW) }If the class $\mathcal{H}$ is finite then
the expected regret of MW with step size parameter $\eta=\sqrt{\log(|H|)/T}$
is at most $\sqrt{2T\log|H|}$.
\end{thm}
\section{Discussion} \label{sec:discussion}
We have considered online learning in the presence of a private learning oracle, and gave an efficient reduction from online learning to private learning.
We conclude with two questions for future research.
\begin{itemize}
\item
Can our result can be extended to the approximate case?
That is, does an efficient approximately differentially private learner for a class $\mathcal{H}$,
implies an efficient online algorithm with sublinear regret? Can the online learner be derived
using only an oracle access to the private learner?
\item
Can our result be extended to the agnostic setting?
That is, does an efficient agnostic private learner for a class $\mathcal{H}$ implies an efficient agnostic online learner for it?
\end{itemize}
\section{Introduction}
\emph{Differential Private Learning} and \emph{Online Learning} are two well-studied areas in machine learning.
While at a first glance these two subjects may seem disparate, recent works gathered a growing amount of evidence which suggests otherwise.
For example, {\it Adaptive Data Analysis}~\cite{Dwork2015,Dwork2015a,Hardt2014,Feldman2017a,Bassily2015} shares strong similarities with adversarial frameworks studied in online learning, and on the other hand exploits ideas and tools from differential privacy.
A more formal relation between private and online learning is manifested by the following fact:
\begin{center}
{\it Every privately learnable class is online learnable}.
\end{center}
This implication and variants of it were derived by several recent works~\cite{Feldman2014,bun2015differentially,Alon2018} (see the related work section for more details).
One caveat of the latter results is that they are non-constructive: they show that every privately learnable class has a finite {\it Littlestone dimension}. Then, since the Littlestone dimension is known to capture online learnability~\cite{Littlestone1986,benagnostic}, it follows that privately learnable classes are indeed online learnable. Consequently, the implied online learner is not necessarily {\it efficient}, even if the assumed private learner is.
Thus, the following question emerges:
\begin{center}
Does efficient differentially private learning imply efficient online learning?
\end{center}
This question was explicitly raised by Neel, Roth and Wu \cite{NeelAaronRoth2018}.
In this work we resolve this question affirmatively under the assumption that the given private learner satisfies {\it \underline{Pure} Differential Privacy} (the case of {\it \underline{Approximate} Differential Privacy} remains open: see Section~\ref{sec:discussion} for a short discussion).
We give an efficient black-box reduction which transforms an efficient pure private learner to an efficient online learner. Our reduction exploits a characterization of private learning due to \cite{Beimel2014}, together with tools from online boosting \cite{Beygelzimer},
and a lemma which converts oblivious online learning to adaptive online learning. The latter lemma is novel and may be of independent interest.
\subsection{Main result}
\begin{thm}\label{thm:main}
Let $\mathcal{A}$ be a differentially private learning algorithm for an hypothesis class $\mathcal{H}$ in the realizable setting. Denote its sample complexity by $m(\cdot,\cdot)$ and denote by $m_0:=m(1/4,1/2)$. Then, Algorithm \ref{alg:privateToStrongOnline} is an efficient online learner for $\mathcal{H}$ in the realizable setting which attains an expected regret of at most $O(m_0\ln(T))$.
\end{thm}
The (standard) notation used in the theorem statment is detailed in Section \ref{sec:pre}.
\paragraph{Agnostic versus Realizable.}
It is natural to ask whether Theorem~\ref{thm:main} can be generalized to the agnostic setting, namely,
whether Algorithm~\ref{alg:privateToStrongOnline} can be extended to an (efficient) online learner which
achieves a sublinear regret against arbitrary adversaries. It turns out,
that the answer is no, at least if one is willing to assume certain customary complexity theoretical assumptions and consider a non-uniform\footnote{Complexity theory distinguishes between uniform and non-uniform models, such as Turing machines vs.\ arithmetic circuits. In this paper we consider the uniform model. However, the lower bound we sketch applies to non-uniform computation.} model of computation.
Specifically, consider the class of all halfspaces over the domain~$\{0,1\}^n\subseteq \mathbb{R}^n$ whose margin is at least $\mathsf{poly}(n)$.
This class satisfies: (i) it is efficiently learnable by a pure differentially private algorithm~\cite{Blum2005,Feldman2017SODA,Nguyen2019}.
(ii) Conditioned on certain average case hardness assumptions,
there is no efficient online learner\footnote{The result in~\cite{Daniely2016} is in fact stronger: it shows that there exists no efficient agnostic PAC learner for this class (see Theorem 1.4 in it). } for this class which achieves sublinear regret against arbitrary adversaries~\cite{Daniely2016}. We note that this argument only invalidates the possibility of reducing agnostic online learning to realizable private learning. The question of whether there exists an efficient reduction from agnostic online learning to \emph{agnostic} private learning remains open.
\paragraph{Proof overview.}
Here is a short outline of the proof.
A characterization of differentially private learning due to \cite{Beimel2014} implies that if $\mathcal{H}$ is privately learnable in the pure setting, then the \textit{representation dimension} of $\mathcal{H}$ is finite. Roughly, this means that for any fixed distribution $\mathcal{D}$ over labeled examples, by repeatedly sampling the (random) outputs of the algorithm $\mathcal{A}$ on a ``dummy" input sample, we eventually get an hypothesis that performs well with respect to $\mathcal{D}$.
In more detail, if one samples (roughly) $\exp(1/\alpha)$ random hypotheses, then with high probability
one of them will have excess population loss $\leq\alpha$ with respect to $\mathcal{D}$.
This suggests the following approach: sample $\exp(1/\alpha)$ random hypotheses ($\alpha$ will be specified later)
and treat them an a class of experts, denoted by $\mathcal{H}_{\alpha}$;
then, use {\it Multiplicative Weights} to online learn $\mathcal{H}_\alpha$ with regret (roughly) $\sqrt{T\log\lvert H_\alpha\rvert} \approx \sqrt{T/\alpha}$, and thus the total regret will be
\[\alpha\cdot T + \sqrt{T/\alpha},\]
which is at most $T^{2/3}$ if we set $\alpha=T^{-1/3}$.
There are two caveats with this approach: i) the number of experts in $H_\alpha$ is $\exp(T^{1/3})$, which is too large for applying Multiplicative Weights efficiently .
ii) A more subtle issue is that the above regret analysis only applies in the {\it oblivious} setting: an adaptive adversary may ``learn'' the random class $\mathcal{H}_\alpha$ from the responses of our online learner, and eventually produce a (non-typical) sequence of examples for which it is no longer the case that the best expert in $\mathcal{H}_\alpha$ has loss $\leq \alpha$.
To handle the first obstacle we only require a constant accuracy of $\alpha=1/4$, which we later reduce using online boosting from \cite{Beygelzimer}. As for the second obstacle, to cope with adaptive adversaries we propose a general reduction from the adaptive setting which might be of independent interest.
\subsection{Related work}
\paragraph{Online and private learning}
Feldman and Xiao \cite{Feldman2014} exploited techniques from communication complexity to show that every pure differentially private (DP) learnable class has a finite Littlestone dimension (and hence is online learnable). Their work actually proved that \emph{pure} private learning is strictly more difficult than online learning. That is, there exists classes with a finite Littlestone dimension which are not pure-DP learnable. More recently, Alon et al.~\cite{bun2015differentially,Alon2018} extended the former result to approximate differential privacy, showing that every approximate-DP learnable class has a finite Littlestone dimension.
It remains open whether the converse holds.
Another line of work by~\cite{NeelAaronRoth2018,BousquetLivniMoran19} exploit online learning techniques to derive results in differential privacy related to {\it sanitization} and {\it uniform convergence}.
\paragraph{Adaptive data analysis.} A growing area which intersects both fields of online learning and private learning is \textit{adaptive data analysis} (\cite{Dwork2015}, \cite{Dwork2015a},\cite{Hardt2014} \cite{Feldman2017a},\cite{Bassily2015}). This framework studies scenarios in which a data analyst wishes to test multiple hypotheses on a finite sample in an adaptive manner.
The adaptive nature of this setting resembles scenarios that are traditionally studied in online learning, and the connection with differential privacy is manifested in the technical tools used to study adaptive data analysis, many of which were developed in differential privacy (e.g.\ composition theorems).
\paragraph{Oracle complexity of online learning.}
One feature of our algorithm is that it uses an oracle access to a private learner.
Several works studied online learning in oracle model (\cite{Agarwal2019LearningOracle,Hazan2016a,Dudik2017}). This framework is natural in scenarios in which it is computationally hard to achieve sublinear regret in the worst case, but the online learner has access to an offline optimization and/or learning oracle. Our results fall into the same paradigm, where the oracle is a differentially private learner.
\section*{Acknowledgements}
We thank Amit Daniely for fruitful discussions on the aforementioned hardness results preventing an extension of our results to the agnostic setting.
\newpage
\bibliographystyle{plain}
\section{Proof of Lemma~\ref{lem:obliviousToAdaptive}}\label{sec:obliviousToAdaptiveProof}
The proof exploits Lemma 4.1 from~\cite{Cesa-Bianchi2006} which we explain next.
Let $\mathcal{A}$ be a (possibly randomized) online learner,
and let $u_t$ denote the response of $\mathcal{A}$ in time $t\leq T$.
Then, since $\mathcal{A}$ may be randomized,~$u_t$
is drawn from a random variable $U_t$ that may depend
on the entire history: namely, on \underline{both}
the responses of $\mathcal{A}$ as well as of the adversary up to time $t$.
So
\[
U_t= U_t( u_1\ldots u_{t-1}, v_1\ldots v_{t-1}),
\]
where $u_i\sim U_i$ denotes the response of $\mathcal{A}$
and $v_i\sim V_i$ denotes the response of the (possibly randomized) adversary on round $i < t$
(in the classifications setting, $v_i$ is the labelled example $(x_i,y_i)$,
and $ u_i$ is the prediction rule $h_i:\mathcal{X}\to\{0,1\}$ used by~$\mathcal{A}$).
Lemma~4.1 in~\cite{Cesa-Bianchi2006} asserts that if
$U_t$ is only a function of the $v_i$'s, namely
\begin{equation}\label{eq:cesa-bianci}
U_t= U_t(v_1\ldots v_{t-1}),
\end{equation}
then the expected regret of $\mathcal{A}$ in the adaptive setting is the same like
in the oblivious setting.
The proof now follows by noticing that Algorithm~\ref{alg:obliviousToAdaptive}
satisfies Equation~(\ref{eq:cesa-bianci}). To see this, note that at each round $t$,
Algorithm~\ref{alg:obliviousToAdaptive} uses the response of algorithm $A_o^{(t)}$ which only
{\it depends on the responses of the adversary and $A_o^{(t)}$ up to time $t$}.
In particular, it does not additionally depend the responses of Algorithm~\ref{alg:obliviousToAdaptive}
at times up to $t$.
Putting it differently, given the responses of the adversary $z_1\ldots z_{t-1}$,
one can produce the response of Algorithm~\ref{alg:obliviousToAdaptive}
at time $t$ by simulating $A_o^{(t)}$ on this sequence.
Thus, we may assume that the adversary is oblivious,
and therefore that the sequence of examples $(x_1,y_1)\ldots (x_t,y_t)$
is fixed in advance and independent from the algorithms $A_o^{j}$'s.
Now, since~$A_o^{(1)},\ldots,A_o^{(T)}$ are i.i.d.\ (i.e.\ have independent internal randomness),
the expected loss of Algorithm \ref{alg:obliviousToAdaptive} at time $t$ satisfies
$$
\mathbb{E} [\hat{\ell}_t] =\mathbb{E} [\hat{\ell}_t^{(t)}] =\mathbb{E} [\hat{\ell}_t^{(1)}] =\ldots = \mathbb{E} [\hat{\ell}_t^{(T)}] = \mathbb{E} \left [\frac{1}{T}\sum_{j=1}^T \hat{\ell}_t^{(j)} \right],
$$
where $\hat \ell_i^{(j)} = \ell(y_i, \hat y_i^{(j)})$. Thus, its expected number of mistakes is at most
$$
\mathbb{E} \left[\sum_{t=1}^T \hat{\ell}_t \right]= \mathbb{E} \left[ \sum_{t=1}^T \frac{1}{T}\sum_{j=1}^T\hat{\ell}_t^{(j)}\right]=\frac{1}{T}\sum_{j=1}^T \mathbb{E} \left[\sum_{t=1}^T\hat{\ell}_t^{(j)} \right].
$$
Therefore, the expected regret satisfies
\[
\mathbb{E}[\mathrm{Regret}_T] = \frac{1}{T} \sum_{j=1}^T\mathbb{E}[\mathrm{Regret}^{(j)}_T] \le R(T)~.
\]
\section{Online BBM} \label{sec:obbm}
\section{The Reduction and its Analysis}
In this section we formally present our efficient reduction from online learning to private PAC learning. Our reduction only requires a black-box oracle access to the the private learner.
The reduction can be roughly partitioned into 3 parts:
(i) We first use this oracle to construct an efficient weak online learner against oblivious adversaries. (ii) Then, we transform this learner so it also handles adaptive adversaries. This step is based on a general reduction which may be of independent interest. (iii) Finally, we boost the weak online learner to a strong one using online boosting.
\subsection{A Weak Online Learner in the Oblivious Setting}
Let $\mathcal{A}_p$ be a PPAC algorithm with sample complexity $m(\alpha,\beta)$ for $\mathcal{H}$ and denote by $m_0:=m(1/4,1/2)=\Theta(1)$. We only assume an oracle access to $\mathcal{A}_p$, and in the first part we use it to construct a distribution over hypotheses/experts.
Specifically, let $S_0$ be a dummy sample consisting of $m$ occurrences of the pair $(\bar{x},0)$ where $\bar{x}$ is an arbitrary instance from $\mathcal{X}$.
Note that the hypothesis/expert $\mathcal{A}_p(S_0)$ is random.\footnote{The definition of differential privacy implies that every private algorithm is randomized (ignoring trivialities).}
\begin{defn}
Let $P_0$ be the distribution over hypotheses/experts induced by applying $\mathcal{A}_p$ on the input sample $S_0$.
\end{defn}\label{def:prior}
\begin{lem}\label{lem:prior}
For any realizable distribution $\mathcal{D}$ over $\mathcal{X}\times\mathcal{Y}$, with probability at least~$15/16$ over the draw of $N=\Theta(\exp(m_0))=\Theta(1)$ i.i.d. hypothesis $h_1,\ldots, h_N\sim P_0$ , there exists $ i\in [N]$ such that $L_{\mathcal{D}}(h_i) \le 1/4$.
\end{lem}
\begin{proof}
Let $c \in \mathcal{H}$ be such that $L_\mathcal{D}(c)=0$, and denote by
$$
\mathcal{H}(\mathcal{D}) = \{h\in \mathrm{range}(\mathcal{A}):~L_{\mathcal{D}}(h)\le 1/4\}~.
$$
By assumption, if we feed the PPAC algorithm $\mathcal{A}$ with a sample $S$ drawn according to $\mathcal{D}^m$ and labeled by $c$, then with probability at least $1/4$ over both the internal randomness of $\mathcal{A}$ and the draw of $S$, the output of $\mathcal{A}$ belongs to $\mathcal{H}(\mathcal{D})$. It follows that there exists at least one sample, which we denote by $\bar{S}$, such that with probability at least $1/2$ over the randomness of $\mathcal{A}$, the output $h=\mathcal{A}(\bar{S})$ belongs to $\mathcal{H}(\mathcal{D})$. Since $\mathcal{A}$ is differentially private and $(\bar{S} ,S_0)$ are $m$-neighbors, we obtain that
$$
Pr[\mathcal{A}(S_0)\in \mathcal{H}(\mathcal{D},c)]\ge \frac{1}{2}\exp(-0.1m_0)~.
$$
Consequently if we draw $N=\Theta(\exp(m_0))$ hypotheses $h_j\sim P_0$ then with probability at least $15/16$, at least one of the $h_j$'s belongs to $\mathcal{H}(\mathcal{D})$. This completes the proof.
\end{proof}
Armed with this lemma, we proceed by applying the Multiplicative Weights method to the random class $H$ produced by the PPAC learner $\mathcal{A}_p$. The algorithm is detailed as Algorithm \ref{alg:weakOnlineOblivious}. The next lemma establishes its weak learnability in the oblivious setting.
\begin{algorithm}
\caption{Weak online learner for oblivious adversaries}
\label{alg:weakOnlineOblivious}
\begin{algorithmic}
\State \textbf{Oracle access:} Let $P_0$ denote the distribution from Definition \ref{def:prior}, and let $m_0=m(1/4,1/2)$, where $m(\alpha,\beta)$ is the sample complexity of the private learner $\mathcal{A}_p$.
\State {{\bf Set}:} $N = \Theta(\exp(m_0))$, $\eta=\sqrt{\frac{\log N}{T}}$.
\For {$j=1$ to $N$}
\State $h_j \sim P_0$
\State $w_{1,j} = 1 \qquad \forall j \in [N]$\Comment{Initializing MW w.r.t. $h_1,\ldots,h_N$}
\EndFor
\For {$t=1$ to $T$}
\State Receive an instance $x_t$
\State Predict $\hat{y}_t = h_j(x_t)$ with probability $w_{t,j}/\sum_{k=1}^N w_{t,k}$
\State Receive the true label $y_t$
\State $w_{t+1,j} = w_{t,j} \exp(-\eta|y_t - h_j(x_t)|)$
\EndFor
\end{algorithmic}
\end{algorithm}
\begin{lem} \label{lem:weakOblivious}
For any \underline{oblivious} adversary and horizon parameter $T$, the expected number of mistakes made by Algorithm \ref{alg:weakOnlineOblivious} is at most $O\left(\sqrt{T m_0 \log T} + \frac{T}{4} \right)$. In particular, the algorithm is a weak online learner with an edge parameter $1/8$ and excess loss $T_0=O(1)$.
\end{lem}
\begin{proof}
Since the adversary is oblivious, it chooses the (realizable) sequence $(x_1,y_1)\ldots,(x_T,y_T)$ in advance. In particular, these choices do not depend on the hypotheses $h_1,\ldots,h_N$ drawn from $P_0$. Define a distribution $\mathcal{D}$ over $\mathcal{X}\times\mathcal{Y}$ by
$$
\mathcal{D}[\{(x,y)\}] = \frac{|\{t \in [T]:~(x_t,y_t) = (x,y)\}|}{T}.
$$
By the previous lemma we have that with probability at least $15/16$, there exists $j \in [N]$ such that
$$
\frac{1}{T} \sum_{t=1}^T \ell(h_j(x_t), y_t ) =L_{\mathcal{D}}(h_i) \le 1/4.
$$
Using the standard regret bound of Multiplicative Weights (Lemma~\ref{thm:mw}), we obtain that the expected number of mistakes done by our algorithm is at most
$$
2\sqrt{T \log N} + \frac{T}{4}+T/16.
$$
(The $T/16$ factor is because the success probability of $A_{p}$ is $15/16$, see Lemma~\ref{lem:prior}).
In particular, set $T_0 = C\cdot\log N = O(m_0)$ for a sufficiently large constant $C$ such that,
\[
2\sqrt{T \log N} + \frac{T}{4}+\frac{T}{16} \leq \Bigl(\frac{1}{2} -\frac{1}{8} \Bigr)T + T_0.
\]
This concludes the proof.
\end{proof}
\subsection{General reduction from adaptive to oblivious environments}
In this part we describe a simple general-purpose extension from the oblivious setting to the adaptive setting. Let $\mathcal{A}_o$ be an online learner for $\mathcal{H}$ that handles oblivious adversaries. We may assume that $\mathcal{A}_o$ is random since otherwise any guarantee with respect to oblivious adversary holds also with respect to adaptive adversary. Given an horizon parameter $T$, we initialize~$T$ instances of this algorithm (each of with an independent random seed of its own). Finally, on round $t$ we follow the prediction of the $t$-th instance, $\mathcal{A}_o^{(t)}$.
\begin{algorithm}
\caption{Reduction from Oblivious to Adaptive Setting}
\label{alg:obliviousToAdaptive}
\begin{algorithmic}
\State \textbf{Oracle access:} Online algorithm $\mathcal{A}_o$ for the oblivious setting.
\State \textbf{Initialize} $T$ independent instances of $A_o$, denoted $A_o^{(1)},\ldots,A_o^{(T)}$.
\For{$t=1$ to $T$}
\State $\hat{y}_t^{(j)}:=$ prediction of $\mathcal{A}_o^{(j)}$, $j=1,\ldots,T$.
\State Predict $\hat{y}_t= \hat{y}_t^{(t)}$
\EndFor
\end{algorithmic}
\end{algorithm}
\begin{lem} \label{lem:obliviousToAdaptive}
Suppose that $\mathcal{A}_o$ is an online learner for a class $\mathcal{H}$ in the oblivious setting whose expected regret is upper bounded by $R(T)$. Then, the expected regret of Algorithm \ref{alg:obliviousToAdaptive} is also upper bounded by $R(T)$.
\end{lem}
\begin{proof}
The proof relies on a lemma by~\cite{Cesa-Bianchi2006}
which provides a reduction from the adaptive to the oblivious setting
given a certain condition on the responses of the online learner.
Since this lemma is somewhat technical, we defer the proof of the stated bound to the appendix
(Section \ref{sec:obliviousToAdaptiveProof}),
and prove here a slightly weaker bound, which is off by a factor of $\log T$.
This weaker bound however follows from elementary arguments in a self contained manner.
Note that the algorithms $A^{(j)}$'s for $j=1\ldots T$ are i.i.d.\ (i.e.\ have independent internal randomness).
Therefore, {\it the sequence of examples
chosen by the adversary up to time $t$ is independent
of the predictions of $A_o^{(j)}$ whenever~$j\geq t$},
and thus we can use the assumed guarantee for $A_o^{(j)}$ in the oblivious setting:
\begin{equation}\label{eq:1}
(\forall j \geq t):\mathbb{E}\Bigl[\sum_{i=1}^t \hat \ell_i^{(j)}\Bigr] \leq R(T),
\end{equation}
where $\hat \ell_i^{(j)} = \ell(y_i, \hat y_i^{(j)})$.
Similarly, it follows that
\begin{equation}\label{eq:2}
\mathbb{E} [\hat{\ell}_t] =\mathbb{E} [\hat{\ell}_t^{(t)}] =\mathbb{E} [\hat{\ell}_t^{(t+1)}] =\ldots = \mathbb{E} [\hat{\ell}_t^{(T)}] = \mathbb{E} \left [\frac{1}{T-t+1}\sum_{j=t}^T \hat{\ell}_t^{(j)} \right]~.
\end{equation}
Therefore,
\begin{align*}
\mathbb{E} \Bigl[\sum_{t=1}^T \hat{\ell}_t \Bigr] &= \mathbb{E} \Bigl[\sum_{t=1}^T \frac{1}{T-t+1}\sum_{j=t}^{T}\hat \ell_t^{(j)} \Bigr] \tag{by Equation~\ref{eq:2}}\\
&=\mathbb{E} \Bigl[\sum_{j=1}^T \sum_{t=1}^j \frac{\hat\ell_t^{(j)}}{T-t+1} \Bigr]\\
&\leq \mathbb{E} \Bigl[\sum_{j=1}^T \sum_{t=1}^j \frac{\hat\ell_t^{(j)}}{T-j+1} \Bigr]\\
&= \sum_{j=1}^T \frac{\mathbb{E}[\sum_{t=1}^j\hat\ell_t^{(j)}]}{T-j+1}\\
&\leq \sum_{j=1}^T \frac{R(T)}{T-j+1}\tag{by Equation~\ref{eq:1}}\\
&\leq R(T)\log T.\\
\end{align*}
\end{proof}
\subsection{Applying Online Boosting}
In this part we apply an online boosting algorithm due to \cite{Beygelzimer} to improve the accuracy of our weak learner. The algorithm is named Online Boosting-by-Majority (online BBM). We start by briefly describing online BBM and stating an upper bound on its expected regret.
The Online BBM can be seen as an extension
of Boosting-by-Majority algorithm due to \cite{Freund1995}.
Let $\texttt{WL}$ be a weak learner with an edge parameter $\gamma \in (0,1/2)$ and excessive loss $T_0$.
The online BBM algorithm maintains $N$ copies $\texttt{WL}$, denoted by
$\texttt{WL}^{(1)},\ldots,\texttt{WL}^{(N)}$. On each round $t$
it uses a simple (unweighted) majority vote over $\texttt{WL}^{(1)},\ldots,\texttt{WL}^{(N)}$
to perform a prediction $\hat{y}_{t}$.
The pair $(x_{t},y_{t})$ is passed to the weak learner $\texttt{WL}^{(j)}$
with probability that depends on the accuracy of the majority vote based on the weak
learners $\texttt{WL}^{(1)},\ldots,\texttt{WL}^{(j-1)}$ with respect to $(x_{t},y_{t})$.
Similarly to the well-known AdaBoost algorithm by~\cite{Adaboost},
the worse is the accuracy of the previous weak learners, the larger is
the probability that $(x_{t},y_{t})$ is passed to $\texttt{WL}^{(j)}$ (see Algorithm 1 in \cite{Beygelzimer}).
\begin{thm} \label{thm:beygelzimer}
(\cite{Beygelzimer}) For any $T$ and any $N$, the expected
number of mistakes made by the Online Boosting-by-Majority Algorithm is bounded by\footnote{The bound in \cite{Beygelzimer} is an high probability bound. It is easy to translate it to a bound in expectation.}
\[
\exp\left(-\frac{1}{2}N\gamma^{2}\right)T+\tilde{O}\left(\sqrt{N}(T_0+\frac{1}{\gamma})\right)
\]
In particular, if $\gamma$ and $T_0$ are constants then for any $\epsilon>0$, it suffices to pick $N=\Theta(\ln(1/\epsilon))$ weak learners to obtain an upper bound of
\begin{equation}\label{eq:boosting}
O(T\epsilon+\ln(1/\epsilon))
\end{equation}
on the expected number of mistakes.
\end{thm}
We have collected all the pieces of our algorithm.
\begin{algorithm}
\caption{Online Learning using a Private Oracle}
\label{alg:privateToStrongOnline}
\begin{algorithmic}
\State \textbf{Horizon parameter:} $T$
\State $\epsilon := 1/T$
\State Weak learner $\texttt{WL}$: Algorithm \ref{alg:obliviousToAdaptive} applied to Algorithm \ref{alg:weakOnlineOblivious}
\State Apply online BBM using $N=\Theta(\ln(1/\epsilon))=\Theta(\ln T)$ instances of $\texttt{WL}$
\end{algorithmic}
\end{algorithm}
\begin{proof} \textbf{(of Theorem \ref{thm:main})}
Combining Lemma \ref{lem:weakOblivious} and Lemma \ref{lem:obliviousToAdaptive}, we obtain that $\texttt{WL}$ is a weak online learner with an edge parameter $\gamma=1/8$ and constant excessive loss. Plugging $\epsilon=1/T$ in the accuracy parameter in Theorem \ref{thm:beygelzimer} (Equation~\ref{eq:boosting}) yields the stated bound.
\end{proof}
|
3,212,635,537,952 | arxiv | \section{Introduction}
The Coronavirus Disease 2019 SARS-CoV-2 (COVID-19) has become a global pandemic with an exponential growth and mortality rate.
The virus is harbored most commonly with little or no symptoms, but can also lead to a rapidly progressive and often fatal pneumonia \cite{sohrabi2020world,lai2020asymptomatic,hoehl2020evidence}.
It has become important to detect affected people as early as possible and isolate them to stop further spreading of the virus.
Various methods have been proposed to diagnose COVID-19, containing a variety of medical imaging techniques, blood tests and PCR.
COVID-19 pandemic has a very severe impact on the respiratory as well as other systems of the human body. Thus, medical imaging features of chest radiography is found to be useful for rapid COVID-19 detection. The imaging features of the chest can be obtained through medical imaging modalities like CT (Computed Tomography) scans. CT images can be used for precise COVID-19 detection \cite{alizadehsani2021risk}.
They provide: a) 3-D view formation of organs; CT scans provide a more detailed overview of the internal structure of lung parenchyma due to lack of overlapping tissues, b) convenient examination of disease and its location; CTs provide a window into pathophysiology that could shed light on several stages of disease detection and evolution. Radiologists report COVID-19 patterns of infection with typical features including ground glass opacities in the lung periphery, rounded opacities, enlarged intra-infiltrate vessels, and later more consolidations that are a sign of progressing critical illness.
At the time of CT scan recording, several slices are captured from each person suspected of COVID-19. The large volume of CT scan images calls for a high workload on physicians and radiologists to diagnose COVID-19. Taking this into account and also the rapid increase in number of new and suspected COVID-19 cases, it is evident that there is a need for using machine and deep learning for detecting COVID-19 in CT scans.
Such approaches require data to be trained on. Therefore, a few databases have been developed consisting of CT scans. However, new data sets with large numbers of 3-D CT scans are needed, so that researchers can train and develop COVID-19 diagnosis systems and trustfully evaluate their performance.
The current paper presents a baseline approach for the Competition part of the Workshop “AI-enabled Medical Image Analysis Workshop and Covid-19 Diagnosis Competition (MIA-COV19D)” which occurs in conjunction with the International Conference on Computer Vision (ICCV) 2021 in Montreal, Canada, October 11- 17, 2021.
The MIA-COV19D AI-enabled Medical Image Analysis (MIA) Workshop emphasizes on radiological quantitative image analysis for diagnosis of diseases. The focus is placed on Artificial Intelligence (AI), Machine and Deep Learning (ML, DL) approaches that target effective and adaptive diagnosis, as well as on approaches that enforce trustworthiness and create justifications of the decision making process.
The COV19D Competition is based on a new large database of chest CT scan series that is manually annotated for Covid-19/non-Covid-19 diagnosis. The training and validation partitions along with their annotations are provided to the participating teams to develop AI/ML/DL models for Covid-19/non-Covid-19 prediction. Performance of approaches will be evaluated on the test set.
The COV19-CT-DB is a new large database with about 5,000 3-D CT scans, annotated for COVID-19 infection.
The rest of the paper is as follows. Section 2 presents former work on which the presented baseline has been based. Section 3 presents the database created and used in the Competition. The ML approach and the pre-processing steps are described in Section 4. The obtained results, are presented in Section 5. Conclusions and future work are described in Section 6.
\section{Related Work}
In \cite{khadidos2020analysis} a CNN plus RNN network was used, taking as input CT scan images and discriminating
between COVID-19 and non-COVID-19 cases.
In \cite{li2020coronavirus}, the authors employed a variety of 3-D ResNet models for detecting COVID-19 and distinguishing it from other common pneumonia (CP) and normal cases, using volumetric 3-D CT scans.
In \cite{wang2020weakly}, a weakly supervised deep learning framework was suggested using 3-D CT volumes for COVID-19 classification and lesion localization. A pre-trained UNet was utilized for segmenting the lung region of each CT scan slice; the latter was fed into a 3-D DNN that provided the classification outputs.
The presented approach is based on a CNN-RNN architecture that performs 3-D CT scan analysis. The method follows our previous work \cite{Tailor, springer, cis, ijait} on developing deep neural architectures for predicting COVID-19, as well as neurodegenerative and other \cite{iet, cis, 48aposcholarmou, mdpi} diseases and medical situations.
These architectures have been applied for: a) prediction of Parkinson’s, based on datasets of MRI and DaTScans, either created in collaboration with the Georgios Gennimatas Hospital (GGH) in Athens \cite{cis}, or provided by the PPMI study sponsored by M. J. Fox for Parkinson’s Research \cite{iet}, b) prediction of COVID-19, based on CT chest scans, scan series, or x-rays, either collected from the public domain, or aggregated in collaboration with the Hellenic Ministry of Health and The Greek National Infrastructures for Research and Technology \cite{Tailor}.
\section{The COV19-CT-DB Database}
\begin{figure*}[h!]
\centering
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_13.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_14.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_15.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_16.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_16.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_17.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_18.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_18.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_19.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_20.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_20.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_21.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_22.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_23.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_24.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_25.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_25.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_26.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_27.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_28.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_29.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_30.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_31.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_31.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_32.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_32.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_33.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_34.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_34.jpg}
\adjincludegraphics[height=2.7cm]{non_covid_cluster_17/cam_series_1_slice_35.jpg}
\caption{Slices from a non COVID-19 CT scan.}
\label{full_non_covid_ct_scan}
\end{figure*}
\begin{figure*}[h!]
\centering
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_114.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_117.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_120.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_123.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_126.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_129.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_130.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_132.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_135.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_138.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_141.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_144.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_147.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_148.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_149.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_150.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_151.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_153.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_156.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_159.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_160.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_162.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_165.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_168.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_170.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_171.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_174.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_175.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_177.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_180.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_181.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_183.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_186.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_189.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_192.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_193.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_195.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_198.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_201.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_204.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_207.jpg}
\adjincludegraphics[height=2.7cm]{covid_cluster_6/cam_series_1_slice_210.jpg}
\caption{Slices from a COVID-19 CT scan.}
\label{full_covid_ct_scan}
\end{figure*}
The COVID19-CT-Database (COV19-CT-DB) consists of chest CT scans that are annotated for the existence of COVID-19. Data collection was conducted in the period from September 1, 2020 to March 31, 2021. Data were aggregated from many hospitals, containing anonymized human lung CT scans with signs of COVID-19 and without signs of COVID-19. Figure \ref{full_non_covid_ct_scan} shows some CT slices from a non-COVID-19 case and Figure \ref{full_covid_ct_scan} some CT slices from a COVID-19 case.
The COV19-CT-DB database consist of about 5000 chest CT scan series, which correspond to a high number of patients ($>$1000) and subjects ($>$2000). Annotation of each CT slice has been performed by 4 very experienced (each with over 20 years of experience) medical experts; two radiologists and two pulmonologists. Labels provided by the 4 experts showed a high degree of agreement (around 98\%).
One difference of COV19-CT-DB from other existing datasets is its annotation by medical experts (labels have not been created as a result of just positive RT-PCR testing).
Each of the 3-D scans includes different number of slices, ranging from 50 to 700.
The database has been split in training, validation and testing sets.
The training set contains, in total, 1560 3-D CT scans. These include 690 COVID-19 cases and 870 Non-COVID-19 cases. The validation set consists of 374 3-D CT scans. 165 of them represent COVID-19 cases and 209 of them represent Non-COVID-19 cases. Both include different numbers of CT slices per CT scan, ranging from 50 to 700.
\section{The Deep Learning Approach}
\subsection{3-D Analysis and COVID-19 Diagnosis}
The input sequence is a 3-D signal, consisting of a series of chest CT slices, i.e., 2-D images, the number of which is varying, depending on the context of CT scanning. The context is defined in terms of various requirements, such as the accuracy asked by the doctor who ordered the scan, the characteristics of the CT scanner that is used, or the specific subject’s features, e.g., weight and age.
The baseline approach is a CNN-RNN architecture, as shown in Figure \ref{cnn_rnn}. At first all input CT scans are padded to have length $t$ (i.e., consist of $t$ slices). The whole (unsegmented) sequence of 2-D slices of a CT-scan are fed as input to the CNN part. Thus the CNN part performs local, per 2-D slice, analysis, extracting features mainly from the lung regions. The target is to make diagnosis using the whole 3-D CT scan series, similarly to the annotations provided by the medical experts. The RNN part provides this decision, analyzing the CNN features of the whole 3-D CT scan, sequentially moving from slice $0$ to slice $t-1$. The outputs of the RNN part feed the output layer -with 2 units- that uses a softmax activation function and provides the final COVID-19 diagnosis.
In this way, the CNN-RNN network outputs a probability for each CT scan slice; the CNN-RNN is followed by a voting scheme that makes the final decision; the voting scheme can be either a majority voting or an at-least one voting (i.e., if at least one slice in the scan is predicted as COVID-19, then the whole CT scan is diagnosed as COVID-19, and if all slices in the scan are predicted as non-COVID-19, then the whole CT scan is diagnosed as non-COVID-19).
\begin{figure*}[h!]
\centering
\adjincludegraphics[height=11cm]{cnn-rnn.png}
\caption{The CNN-RNN model}
\label{cnn_rnn}
\end{figure*}
\subsection{Pre-Processing \& Implementation Details}
At first, CT images were extracted from DICOM files. Then, the voxel intensity values were clipped using a window/level of $350$ Hounsfield units (HU)/$-1150$ HU and normalized to the range of $[0, 1]$.
Regarding implementation of the proposed methodology: i) we utilized ResNet50 as CNN model, stacking on top of it a global average pooling layer, a batch normalization layer and dropout (with keep probability 0.8); ii) we used a single one-directional GRU layer consisting of 128 units as RNN model. The model was fed with 3-D CT scans composed of the CT slices; each slice was resized from its original size of $512 \times 512 \times 3$ to $224 \times 224 \times 3$. As a voting scheme, we used the at-least one.
Batch size was equal to 5 (i.e, at each iteration our model processed 5 CT scans) and the input length 't' was 700 (the maximum number of slices found across all CT scans). Softmax cross entropy was the utilized loss function for training the model. Adam optimizer was used with learning rate $10^{-4}$. Training was performed on a Tesla V100 32GB GPU.
\section{Experimental Results}
This section describes a set of experiments evaluating the performance of the baseline approach.
Table \ref{3dcnn_rnn} shows the performance of the network over the validation set, after training with the training dataset, in terms of macro F1 score. The macro F1 score is defined as the unweighted average of the class-wise/label-wise F1-scores, i.e., the unweighted average of the COVID-19 class F1 score and of the non-COVID-19 class F1 score.
The main downside of the model is that there exists only one label for the whole CT scan and there are no labels for each CT scan slice. Thus, the presented model analyzes the whole CT scan, based on information extracted from each slice.
\begin{table}[t]
\caption{Performance of the CNN-RNN network}
\label{3dcnn_rnn}
\centering
\scalebox{1.}{
\begin{tabular}{|c|c|}
\hline
Method & \multicolumn{1}{c|}{'macro' F1 Score}\\
\hline
\hline
ResNet50-GRU & 0.70 \\
\hline
\end{tabular}
}
\end{table}
\section{Conclusions and Future Work}
In this paper we have introduced a new large database of chest 3-D CT scans, obtained in various contexts and consisting of
different numbers of CT slices. We have also developed a deep neural network, based on a CNN-RNN architecture and used it for COVID-19 diagnosis on this database.
The scope of the paper is to present a baseline scheme regarding the performance that can be achieved based on analysis of the COV19-CT-DB database.
The model presented in the paper will be the basis for future expansion towards more transparent modelling of COVID-19 diagnosis.
{\small
\bibliographystyle{ieee_fullname}
|
3,212,635,537,953 | arxiv | \section{Introduction} \label{sec:intro}
Low-mass dwarfs are the most numerous stars in the Universe, and understanding them is thus clearly an important endeavour. Beyond their own interest, investigations by \citet{bonfils:2013:mdwarfsample}, \citet{dressing:2013:occurencerate} and \citet{kopparapu:2013:occurencerate} have shown that M-dwarfs\, may be the most abundant planet hosts in the Milky Way as well. The estimation of parameters and properties of an exoplanet are intimately
connected to the stellar host, e.g.\ the stellar mass determines the measured
semi-amplitude for radial velocity observations and hence influences the mass estimate of
the planet. In the case of transiting extrasolar planets (TEPs), their physical radii can be measured from the transit shape if the radii of the stellar hosts are known. In addition, the stellar radius and effective temperature are linked to the planet's surface temperature and the location of the habitable zone. All of these examples illustrate
how important stellar astrophysical properties are for the characterization of exoplanets in general. M-dwarfs are attractive targets to search for transiting exoplanets not only due to their numbers, but also due to the fact that for a given planetary size the transit depth is deeper around low-mass stars due to their smaller sizes. Also, the habitable zone around these stars is closer, resulting in shorter periods that make detection easier. Indeed, one of the main drivers for the upcoming TESS mission \citep{TESS} is to detect transiting exoplanets around low-mass stars.
A fundamental stellar property is the radius, and for low-mass stars its estimation has been done mostly through stellar models. Fortunately, considerable improvements in interferometric observation techniques allow us now to obtain stellar parameters such as the stellar radius directly. However, these measurements become more difficult as we go to cooler dwarf stars due to their inherently lower luminosity and smaller radii. Measured angular diameters of M-dwarfs\ are generally close to the current baseline limit of available interferometers. Up to now, extensive interferometric observations on M-dwarf\ stars have been done mainly from the Northern Hemisphere with the CHARA array \citep{ berger:2006, braun:2011:gj581, boyajian:2012:mdwarfs, braun:2014:diameters} and a few with the VLT-Interferometer (VLTI) from the South \citep{ segransan:2003, demory:2009}. These interferometric direct measurements showed a discrepancy with the parameters measured indirectly \citep{boyajian:2012}. The work of \citet{boyajian:2012} found in particular large disagreements for low-mass stars, where the radii measured by interferometers were more than 10\% larger than the ones based on models from \citet{chabrier:1997:stellarstructure}. Likewise, \citet{Kesseli:2018} found that this inflation of the M-dwarf radii extends down to the fully convective regime with a discrepancy of 13\% -- 18\%.
This discrepancy affects in turn other stellar parameters like surface temperature (\ensuremath{T_{\rm eff}}), gravities (\ensuremath{\log{g}}), masses, luminosities, and eventually also possible planetary parameters. Therefore, it is important to observe and re-evaluate the properties of more M-dwarf\, stars with interferometric observations, particularly towards the later spectral types which have not been extensively studied at all.
Theoretical stellar evolution models for low-mass stars predict a transition into the fully convective regime to occur somewhere between 0.2\,\ensuremath{M_{\sun}}{} \citep{dorman1989} and 0.35\,\ensuremath{M_{\sun}}{} \citep{chabrier:1997:stellarstructure}, depending on the underlying stellar model. For partially convective stars, the stellar structure is Sun-like, having a radiative zone and a convective envelope. The only previous observational indications for this transition in late-type stars are based on observations of magnetic fields and measurements of stellar rotational periods. \citet{browning:2008} showed that stars whose convective region extends to the core have strong large-scale magnetic fields and, in fact, we have observational evidence that the fraction of M-dwarfs{} with strong magnetic fields on a large scale is higher for mid- to late-type M-dwarfs{} than for early type ones \citep{donati:2008}. On the other hand, \citet{wright:2016} showed that rotation-dependent dynamos are very similar in both partially and fully convective stars. \citet{irwin:rotation} and \citet{newton:rotation} measured rotational spin-velocities of M-stars. The authors found two divergent populations of faster and slower rotators in the fully convective mass regime, which makes rotation measurements difficult to use in the determination of whether a late-type star is fully convective. Moreover, the rotation of fully convective stars depends on both age and mass. All former indications of fully convective stars have been done indirectly and are not unambiguous.
In this work we present directly measured physical parameters for a sample of 13 low-mass stars using observations with the VLT-Interferometer (VLTI). These observations are used to probe the transition between the partially and fully convective regimes and to identify the dependence of the stellar radii on other stellar properties. The paper is structured as follows. In \S\ref{sec:obs} we lay out the observational details. In \S\ref{sec:phys_params} we detail how we estimated the stellar physical parameters. Finally, we discuss the implication of the measured stellar parameters on stellar evolution and structure models in \S\ref{sec:discussion} and we conclude in \S\ref{sec:conclusion}.
\section{Observations and Data Reduction}
\label{sec:obs}
\subsection{PIONIER{} observations}
\label{subsec:pionierobs}
Our target sample is compiled from a list of M-dwarfs{} within $\sim 15$\,pc (so the stars are resolved within the given VLTI baseline) and with H-band magnitudes $<7$ (so that fringes will be easily visible and we can obtain a good signal-to-noise ratio).
In order to measure the angular diameter of our sample stars, we used the VLTI/PIONIER{} interferometer \citep{pionier}. PIONIER{} is an integrated optics four-beam combiner operating at the near-infrared{} wavelength range. We used the auxiliary telescopes (ATs) in a A1-G1-K0-J3 quadruplet configuration. This configuration gave us the longest VLTI baseline available (from 57 meters between the stations K0 and J3, up to 140 metres between A1 and J3) and we used the Earth's rotation to further fill the $(u, v)$ plane.
We observed our sample with a three-channel spectral dispersion (SMALL mode), whenever possible. In cases where this was not possible, due to low coherence time on a given night or the relative faintness of the target, we observed without spectral dispersion (FREE mode). Similarly, the number of scan steps were adjusted according to the objects' brightness and atmospheric conditions. As our sample stars were not too bright we were able to use the fast Fowler readout mode for all of our observations.
Our observing strategy was to bracket each science frame (SCI) with a calibrator star (CAL), observed with the same setup as the science object. The calibrators are chosen to be mostly point-like nearly unresolved stars \citep{vanbelle:calibrators}, so the uncertainties in their diameter will not influence our targets, but we also included calibrators with known diameter for verification proposes. We also made sure that the visibility precision of our calibrators was below 1\%. In order to search for suitable calibrators, we used the ASPRO2-tool and SearchCal\footnote{\url{http://www.jmmc.fr/aspro\_page.htm}}. For each science target we repeated around 11 times a CAL-SCI-CAL block, and in each block we used different calibrator stars. The same target was also observed on different nights. This strategy helped us to beat down the systematic noise from the instrument and atmosphere. We reduced our observed raw fringes to calibrated visibilities and closure phases with a modified version of the PIONIER{} data reduction software \citep[\emph{pndrs}, described in][]{Lachaume:2019}.
\subsection{Calibrated Visibilities and Angular diameters} \label{subsec:calvis}
Our modified data reduction with \emph{pndrs} is fully described in detail in a publication by \citet[]{Lachaume:2019}, where we also show a rigorous analysis of the interferometric measurement errors. Here we will give only a brief summary of the data reduction process and we refer interested readers to \citet[]{Lachaume:2019}, for more details on the data analysis. In the first step we calibrate the detector frames. This was done by dark correcting the detector data and from the kappa-matrix we calibrated the transmission of the respective baseline. Finally, we used frames illuminated by an internal light source to calibrate the wavelength. Basically, these calibrated frames will allow us to obtain the raw visibilities, which are in turn the product of true visibilities{} and the system transfer function. The system transfer function characterizes the response of the interferometer as a function of spatial frequency and in order to get the true visibilities{} it needs to be estimated by using calibrator stars. Assuming that all our calibrator stars have well known true visibilities, i.e. an unresolved calibrator has a known visibility{} of unity and a resolved star has a known diameter, either measured or from spectral typing. By further assuming a smooth transfer function, in theory this would allow us to calibrate our raw visibilities. Nevertheless, uncertainties in the assumed calibrators' diameters can impact all observations in a sequence due to systematic errors in the transfer function estimate \citep[][and references therein]{Lachaume:2019}. Further errors can be introduced through systematic uncertainties in the absolute wavelength calibration \citep{Gallenne:wavelength_cal:2018} and by several other random effects which will affect the different spectral channels in a similar or imbalanced manner, like e.g. atmospheric jitter or flux variations between the arms of the interferometer. In order to account for the correlation effects in our observations we apply a bootstrap method as described in \citet{Lachaume:2019,bootstrap}.
Generally, in a bootstrap one resamples several times new data sets from the empirical data itself by replacing parts of the original data. For each candidate, we started by picking randomly interferograms out of the parent population of $\sim 10^2$ interferograms. These interferograms are reduced and averaged to a single data set, which corresponds to the raw visibility. As mentioned before, uncertainties in the calibrators' diameter can cause correlated errors. Therefore, we choose arbitrarily a calibrator with a diameter, drawn randomly from a Gaussian distribution centered on the catalogued diameter and with a width corresponding to the error bars. We used 6 to 18 data sets and calibrators to replace the original data and to calculate the system transfer function and calibrated visibilities. We repeat this procedure to obtain 5,000 bootstrap realizations. These calibrated visibilities were fitted with a uniform disk (\ensuremath{\theta_{\rm UD}}) model to obtain a distribution of angular diameters for each star observed. In Fig.~\ref{fig:compareUD} we compare some of our measured \ensuremath{\theta_{\rm UD}}{} with the ones available in the literature. We find a good agreement between our measurements and the literature values.
\begin{figure}
\includegraphics[scale=0.27]{img/diam_lit_compareUD.pdf}
\caption{Our angular diameters \ensuremath{\theta_{\rm UD}}\, compared to literature values. We find good agreement between our measurements and the literature.
\label{fig:compareUD}}
\end{figure}
\section{Estimating the Physical Parameters from Interferometry} \label{sec:phys_params}
\subsection{Calculation of the stellar radius}
The limb darkened disk \ensuremath{\theta_{\rm LD}}{} is usually obtained by fitting directly a limb darkened disk model to the squared visibilities, assuming a certain limb darkening law and coefficient. Generally, a linear limb darkening law is assumed and tabulated values are used for the coefficients, see e.g.\ \citet{boyajian:2012}, \citet{braun:2014:diameters}, and \citet{braun:2011:gj581}. We note, that while \ensuremath{\theta_{\rm UD}} s are independent of stellar models, photospheric diameters, \ensuremath{\theta_{\rm LD}} s depend on stellar models as the limb-darkening coefficient are derived from them. However, the impact on the radius estimate by the limb-darkening in the near-infrared{} is small (2--4\%) and it is mostly dominated by the angular diameter measurement uncertainties and systematics.
In order to estimate the \ensuremath{\theta_{\rm LD}}, we used the \ensuremath{\theta_{\rm UD}}--\ensuremath{\theta_{\rm LD}}{} relation from \citet{hanbury_LD}:
\begin{equation}
\ensuremath{\theta_{\rm LD}}(\lambda)= \ensuremath{\theta_{\rm UD}}\, \sqrt[]{\frac{1-\frac 13\mu(\lambda,\ensuremath{T_{\rm eff}},\ensuremath{\log{g}})}{1-\frac7{15}\mu(\lambda,\ensuremath{T_{\rm eff}},\ensuremath{\log{g}})} },
\end{equation}
where \ensuremath{\theta_{\rm UD}}{} is the angular diameter we obtained from the calibrated visibilities{} and $\mu_\lambda$ is the linear limb darkening coefficient as function of wavelength, \ensuremath{T_{\rm eff}}{} and \ensuremath{\log{g}}. Rather than using tabulated coefficient, we calculated a grid of limb darkening coefficients following \citet{espinoza:2015} corresponding to the atmosphere grid with \ensuremath{T_{\rm eff}}{} in range 2300--4500\,K, \ensuremath{\log{g}}{} in range 4.0--6.0 and a fixed metallicity of 0.0. This allows us to have a conformity with the grid which will be used in Sect. \ref{fbol_estimate}. As filter transmission function of PIONIER, we used a top hat function between 1.5\ensuremath{\mu {\rm m}}{} and 1.8\ensuremath{\mu {\rm m}}.
\subsection{\ensuremath{T_{\rm eff}}{} estimate}
The measured diameters can be related to the effective temperature by
\begin{equation}
\ensuremath{T_{\rm eff}} = \sqrt[4]{\frac{4\ensuremath{F_{\rm bol}}}{\sigma\ensuremath{\theta_{\rm LD}}}},
\end{equation}
where \ensuremath{F_{\rm bol}}{} is the bolometric flux (obtained by e.g. fitting the spectral energy distribution with literature photometry to spectral templates), \ensuremath{\theta_{\rm LD}}{} is the limb darkened angular diameter and $\sigma$ is the Stefan-Boltzmann constant.
\subsection{Bolometric flux estimate}
\label{fbol_estimate}
In order to estimate the bolometric flux we started by using the PHOENIX atmosphere models from \citet{husser} to create a grid of synthetic photometric points for filters with available photometric observations of our sample stars. Their models are defined in the wavelength range from 0.05 to 5.5\ensuremath{\mu {\rm m}}. Our flux model grid runs \ensuremath{T_{\rm eff}}{} from 2300\,K to 4500\,K, \ensuremath{\log{g}}{} between 4.0 and 6.0 dex, and for a fixed metallicity of 0.0 dex. The flux was integrated over the respective band and convolved with the filter profiles from \citet{filtertransmission}. We linearly interpolated this grid of synthetic flux in-between.
The bolometric flux \ensuremath{F_{\rm bol}}\, of a given star is then defined as:
\begin{equation}
\ensuremath{F_{\rm bol}} = \int_0^{+\infty} F_\mathrm{model}(\lambda,\ensuremath{T_{\rm eff}},\ensuremath{\log{g}}) \frac{\ensuremath{R_\star}^2}{d^2} \mathrm{d}\lambda,
\end{equation}
where \ensuremath{R_\star}{} is the stellar radius and $d$ is the distance.
\subsection{Multinest fitting for \ensuremath{T_{\rm eff}}, \ensuremath{R_\star}, and \ensuremath{L_\star}}
\label{subsec:multinest}
We first collected observed fluxes for our stars using the VizieR photometric query. To these observed fluxes we fitted the model grid using the \emph{pymultinest} code \citep{pymultinest}. This program is a python code for multimodal nested sampling technique \citep{nestedsampling, multinest}. Our log-likelihood function is
\def\ensuremath{F_{i,\mathrm{obs}}}{\ensuremath{F_{i,\mathrm{obs}}}}
\def\ensuremath{F_{i,\mathrm{mod}}}{\ensuremath{F_{i,\mathrm{mod}}}}
\def\ensuremath{N_\mathrm{phot}}{\ensuremath{N_\mathrm{phot}}}
\begin{equation}
\mathcal{\log L}= -\sum_{i=1}^{\ensuremath{N_\mathrm{phot}}}
\left[
\frac{(\ensuremath{F_{i,\mathrm{obs}}} - \ensuremath{F_{i,\mathrm{mod}}})^2}{2\sigma_i^2}
- \log{ \frac 1 {\sigma_i\sqrt{2\pi}} }
\right],
\end{equation}
where $\ensuremath{F_{i,\mathrm{obs}}}$ is the observed flux in a given filter $i$, $\ensuremath{F_{i,\mathrm{mod}}}$ is the synthetic flux in that filter obtained from the atmosphere models, and $\sigma_i$ is the corresponding measurement error of the observed flux. The sum goes over the $\ensuremath{N_\mathrm{phot}}$ photometric measurements of a given star.
Our priors are \ensuremath{T_{\rm eff}}, \ensuremath{\log{g}}, distance, and angular diameter \ensuremath{\theta_{\rm UD}}. All our priors were drawn from a normal distribution centered at the literature value and with a dispersion corresponding to the respective error. We further repeated this process using M-dwarfs{} with measured diameters from \citet{braun:2012:gj436}, \citet{boyajian:2012:mdwarfs} and \citet{braun:2014:diameters}. Our final parameter estimates are shown in Table \ref{tab:multinest_results}. We compare our values with the ones from \citet{mann:2015} in Figure~\ref{fig:comparelit} and find good agreement with a mean difference of 3\% for all three parameters (from top to bottom: radius, \ensuremath{F_{\rm bol}}, \ensuremath{T_{\rm eff}}). In the same Figure~\ref{fig:comparelit} (bottom plot), we further compare our effective temperatures with the ones obtained by \citet{neves:2014:feh} through spectral type classification using optical spectroscopy and from Gaia DR2 using Apsis \citep{gaiadr2_teff:2018}. In the latter cases the relative difference for \ensuremath{T_{\rm eff}}{} is generally higher, with a mean difference of 5.4\% and $-8.2\%$ respectively. Therefore, spectral typing of M-dwarfs{} in the optical wavelength range generally overestimates \ensuremath{T_{\rm eff}} s, whereas Gaia DR2 \ensuremath{T_{\rm eff}} s are considerably underestimated.
\begin{figure}
\includegraphics[scale=0.67]{img/compare_mannetal.pdf}
\caption{We compare our calculated radius, \ensuremath{F_{\rm bol}}, and \ensuremath{T_{\rm eff}}{} with the ones from \citet{mann:2015}. Stars are ordered according to our calculated \ensuremath{T_{\rm eff}}{} from low (left) to high (right) temperature. The difference between the estimates is small, the mean difference for the radius is 2.9\%, \ensuremath{F_{\rm bol}}{} is 2.5\% and \ensuremath{T_{\rm eff}}{} is 1.4\%. However, by comparing our \ensuremath{T_{\rm eff}}{} with the ones from optical spectroscopy \citep{neves:2014:feh}, we find an higher mean difference of 5.4\% and -8.2\% for Gaia DR2 \ensuremath{T_{\rm eff}}. However, single \ensuremath{T_{\rm eff}} s from Gaia DR2 can have differences of up to $\sim 15 \%$. (See Sect. \ref{subsec:multinest} for details)
\label{fig:comparelit}}
\end{figure}
\begin{table*}
\caption{Final parameter estimates obtained through multi-modal nested sampling technique. (See Sect. \ref{subsec:multinest} for details)}
\label{tab:multinest_results}
\begin{tabular}{cccccccc}
\hline \hline
star & \ensuremath{\theta_{\rm UD}} & \ensuremath{\theta_{\rm LD}} & $\mu_{\lambda}$ & \ensuremath{F_{\rm bol}} & \ensuremath{R_\star} & parallax & calculated \\
name & [mas] & [mas] & & [$10^{-8}$ \ensuremath{\rm erg\,s^{-1}\,cm^{-2}}] & [\ensuremath{R_{\sun}}] & [mas] & \ensuremath{T_{\rm eff}} [K]\\
\hline
GJ~1 & $0.794\pm 0.005$ & $0.812\pm 0.005$ & 0.290 & $3.751\pm 0.072$ & $0.379\pm 0.002$ & $230.133\pm 0.059$ & $3616\pm 14$ \\
GJ~273 & $0.763\pm 0.010$ & $0.783\pm 0.010$ & 0.335 & $2.288\pm 0.118$ & $0.320\pm 0.005$ & $262.961\pm 1.387$ & $3253\pm 39$ \\
GJ~406 & $0.562\pm 0.020$ & $0.582\pm 0.020$ & 0.449 & $0.563\pm 0.044$ & $0.159\pm 0.006$ & $394.867\pm 7.893$ & $2657\pm 20$ \\
GJ~447 & $0.524\pm 0.029$ & $0.540\pm 0.029$ & 0.365 & $1.103\pm 0.091$ & $0.196\pm 0.010$ & $296.309\pm 0.069$ & $3264\pm 24$ \\
GJ~551 & $1.066\pm 0.007$ & $1.103\pm 0.007$ & 0.422 & $2.866\pm 0.210$ & $0.154\pm 0.001$ & $768.500\pm 0.203$ & $2901\pm 68$ \\
GJ~581 & $0.464\pm 0.007$ & $0.476\pm 0.007$ & 0.324 & $0.967\pm 0.039$ & $0.322\pm 0.005$ & $158.747\pm 0.051$ & $3366\pm 28$ \\
GJ~628 & $0.644\pm 0.014$ & $0.661\pm 0.014$ & 0.335 & $1.882\pm 0.068$ & $0.306\pm 0.007$ & $232.209\pm 0.063$ & $3372\pm 12$ \\
GJ~674 & $0.720\pm 0.037$ & $0.737\pm 0.037$ & 0.318 & $2.443\pm 0.232$ & $0.360\pm 0.018$ & $219.800\pm 0.047$ & $3409\pm 25$ \\
GJ~729 & $0.625\pm 0.020$ & $0.642\pm 0.020$ & 0.345 & $1.370\pm 0.096$ & $0.205\pm 0.006$ & $336.121\pm 0.064$ & $3162\pm 30$ \\
GJ~832 & $0.794\pm 0.010$ & $0.814\pm 0.010$ & 0.325 & $3.359\pm 0.113$ & $0.435\pm 0.005$ & $201.407\pm 0.043$ & $3512\pm 23$ \\
GJ~876 & $0.686\pm 0.009$ & $0.705\pm 0.009$ & 0.342 & $1.902\pm 0.058$ & $0.354\pm 0.005$ & $213.866\pm 0.078$ & $3275\pm 18$ \\
GJ~887 & $1.297\pm 0.005$ & $1.328\pm 0.004$ & 0.323 & $10.916\pm 0.657$ & $0.470\pm 0.001$ & $304.219\pm 0.044$ & $3692\pm 57$ \\
\hline
\multicolumn{2}{l}{Literature stars}\\
\hline
GJ~176 & $0.442\pm 0.020$ & $0.452\pm 0.020$ & 0.306 & $1.274\pm 0.099$ & $0.460\pm 0.020$ & $105.565\pm 0.069$ & $3700\pm 45$ \\
GJ~205 & $0.904\pm 0.003$ & $0.924\pm 0.003$ & 0.283 & $6.140\pm 0.400$ & $0.566\pm 0.002$ & $175.430\pm 0.069$ & $3835\pm 69$ \\
GJ~411 & $1.380\pm 0.013$ & $1.412\pm 0.013$ & 0.301 & $10.514\pm 0.515$ & $0.387\pm 0.004$ & $392.630\pm 0.675$ & $3547\pm 40$ \\
GJ~436 & $0.405\pm 0.013$ & $0.415\pm 0.013$ & 0.321 & $0.800\pm 0.053$ & $0.436\pm 0.013$ & $102.497\pm 0.093$ & $3436\pm 33$ \\
GJ~526 & $0.807\pm 0.013$ & $0.824\pm 0.013$ & 0.287 & $4.134\pm 0.183$ & $0.482\pm 0.008$ & $183.983\pm 0.051$ & $3677\pm 30$ \\
GJ~649 & $0.472\pm 0.012$ & $0.483\pm 0.012$ & 0.294 & $1.329\pm 0.072$ & $0.539\pm 0.013$ & $96.314\pm 0.031$ & $3619\pm 25$ \\
GJ~687 & $0.830\pm 0.013$ & $0.850\pm 0.013$ & 0.317 & $3.380\pm 0.145$ & $0.416\pm 0.007$ & $219.781\pm 0.033$ & $3443\pm 29$ \\
GJ~699 & $0.917\pm 0.005$ & $0.941\pm 0.005$ & 0.342 & $3.176\pm 0.120$ & $0.185\pm 0.001$ & $548.358\pm 1.513$ & $3221\pm 32$ \\
GJ~809 & $0.698\pm 0.008$ & $0.715\pm 0.008$ & 0.314 & $3.341\pm 0.148$ & $0.541\pm 0.006$ & $142.033\pm 0.030$ & $3743\pm 39$ \\
GJ~880 & $0.716\pm 0.004$ & $0.736\pm 0.004$ & 0.357 & $3.468\pm 0.084$ & $0.544\pm 0.003$ & $145.610\pm 0.038$ & $3724\pm 23$ \\
\hline
\end{tabular}
\end{table*}
\subsection{Mass estimates}
\label{subsec:mass_estimate}
The mass cannot be measured directly from interferometry. Therefore, we make use of a fully empirical model-independent mass-luminosity relation (MLR) from \citet{MLrelation:benedict} and \citet{Mann:MLR:2018}. In both cases we use their calibration relations in K-band, therefore
for all our stars we collected SAAO K-band magnitudes from \citet{koen:2010} and Ks-band magnitudes from \citet{mann:2015} and \citet{2003tmc..book.....C}. The corresponding magnitudes are given in Table \ref{tab:mass}. The SAAO K-band magnitudes were transformed to 2MASS Ks using the transformation\footnote{\url{http://http://www.astro.caltech.edu/~jmc/2mass/v3/transformations/}} $ {\rm Ks_{2MASS}} = {\rm K_{SAAO}} - (0.024 \pm 0.003) + (0.017 \pm 0.006) {\rm (J-K)}_{SAAO}$.
We converted the Ks-band magnitudes to absolute magnitudes using the respective parallax given in Table \ref{tab:multinest_results} and estimated the mass for a given star.
From the mass and radius, we were also able to calculate the surface gravity (\ensuremath{\log{g}}):
\begin{equation}
g_\star=\frac{G\ensuremath{M_\star}}{\ensuremath{R_\star}^2},
\end{equation}
where $G$ is the gravitational constant. Table \ref{tab:mass} shows a summary of the calculated mass, luminosity and \ensuremath{\log{g}}\, for our sample.
\begingroup
\renewcommand*{\thefootnote}{\alph{footnote}}
\begin{table*}
\caption{Calculated other distance-dependent stellar parameters. (See Sect. \ref{subsec:mass_estimate})}
\label{tab:mass}
\begin{tabular}{rccccc}
\hline \hline
\multicolumn{1}{c}{star} name & Ks [mag.] & M$_{\rm Ks}$ [mag.] & \ensuremath{M_\star} [\ensuremath{M_{\sun}}]\footnotemark[1] & \ensuremath{L_\star} [\ensuremath{L_{\sun}}] & \ensuremath{\log{g}} [dex] \\
\hline
GJ1 & 4.53$\pm $0.01\footnotemark[2] & 6.33$\pm $0.01 & 0.390$\pm $0.010 & 0.0220$\pm $0.0004 & 4.87 \\
GJ176 & 5.63$\pm $0.01\footnotemark[2] & 5.74$\pm $0.01 & 0.486$\pm $0.011 & 0.0356$\pm $0.0028 & 4.80 \\
GJ205 & 3.86$\pm $0.02\footnotemark[3] & 5.08$\pm $0.02 & 0.590$\pm $0.015 & 0.0621$\pm $0.0041 & 4.70 \\
GJ273 & 4.87$\pm $0.01\footnotemark[2] & 6.97$\pm $0.02 & 0.293$\pm $0.007 & 0.0103$\pm $0.0005 & 4.89 \\
GJ406 & 6.15$\pm $0.02\footnotemark[3] & 9.13$\pm $0.05 & 0.110$\pm $0.003 & 0.0011$\pm $0.0001 & 5.08 \\
GJ411 & 3.36$\pm $0.02\footnotemark[3] & 6.33$\pm $0.02 & 0.390$\pm $0.010 & 0.0212$\pm $0.0010 & 4.85 \\
GJ436 & 6.04$\pm $0.02\footnotemark[3] & 6.09$\pm $0.02 & 0.429$\pm $0.010 & 0.0237$\pm $0.0016 & 4.79 \\
GJ447 & 5.68$\pm $0.02\footnotemark[3] & 8.04$\pm $0.02 & 0.174$\pm $0.004 & 0.0039$\pm $0.0003 & 5.09 \\
GJ526 & 4.56$\pm $0.02\footnotemark[3] & 5.89$\pm $0.02 & 0.463$\pm $0.011 & 0.0380$\pm $0.0017 & 4.74 \\
GJ551 & 4.38$\pm $0.03\footnotemark[4] & 8.81$\pm $0.03 & 0.124$\pm $0.003 & 0.0015$\pm $0.0001 & 5.16 \\
GJ581 & 5.85$\pm $0.01\footnotemark[2] & 6.85$\pm $0.01 & 0.310$\pm $0.007 & 0.0119$\pm $0.0005 & 4.91 \\
GJ628 & 5.09$\pm $0.01\footnotemark[2] & 6.92$\pm $0.01 & 0.300$\pm $0.007 & 0.0109$\pm $0.0004 & 4.94 \\
GJ649 & 5.63$\pm $0.02\footnotemark[3] & 5.55$\pm $0.02 & 0.517$\pm $0.013 & 0.0446$\pm $0.0024 & 4.69 \\
GJ674 & 4.86$\pm $0.01\footnotemark[2] & 6.57$\pm $0.01 & 0.352$\pm $0.008 & 0.0157$\pm $0.0015 & 4.87 \\
GJ687 & 4.50$\pm $0.02\footnotemark[3] & 6.21$\pm $0.02 & 0.409$\pm $0.010 & 0.0218$\pm $0.0009 & 4.81 \\
GJ699 & 4.53$\pm $0.02\footnotemark[3] & 8.23$\pm $0.02 & 0.160$\pm $0.004 & 0.0033$\pm $0.0001 & 5.11 \\
GJ729 & 5.39$\pm $0.02\footnotemark[3] & 8.03$\pm $0.02 & 0.175$\pm $0.004 & 0.0038$\pm $0.0003 & 5.06 \\
GJ809 & 4.58$\pm $0.02\footnotemark[3] & 5.34$\pm $0.02 & 0.551$\pm $0.014 & 0.0515$\pm $0.0023 & 4.71 \\
GJ832 & 4.46$\pm $0.01\footnotemark[2] & 5.98$\pm $0.01 & 0.447$\pm $0.011 & 0.0258$\pm $0.0009 & 4.81 \\
GJ876 & 5.04$\pm $0.01\footnotemark[2] & 6.69$\pm $0.01 & 0.334$\pm $0.008 & 0.0129$\pm $0.0004 & 4.86 \\
GJ880 & 4.54$\pm $0.02\footnotemark[3] & 5.36$\pm $0.02 & 0.547$\pm $0.014 & 0.0509$\pm $0.0012 & 4.71 \\
GJ887 & 3.33$\pm $0.02\footnotemark[3] & 5.74$\pm $0.02 & 0.486$\pm $0.012 & 0.0367$\pm $0.0022 & 4.78 \\
\hline
\end{tabular}
\tablerefs{
\footnotemark[1] Estimated using MLR from \citet{Mann:MLR:2018}
\footnotemark[2] \citet{koen:2010};
\footnotemark[3] \citet{mann:2015};
\footnotemark[4] \citet{cutri:2003:2mass}}
\end{table*}
\endgroup
\section{Discussion} \label{sec:discussion}
In order to discuss the behaviour of our sample stars we investigate some relations between the available parameters. In the following analysis we also added stars from \citet{mann:2015} which have measured Gaia DR2 parallaxes \citep{gaiaDR2}. In order to avoid contamination, the population from \citet{mann:2015} has further been cleaned by removing double stars and variable stars (as, e.g., BY-Dra type). We start by constructing a relation between the stellar radius and stellar mass (MR-relation), shown in Figure~\ref{fig:MR-relation}. As pointed out by \citet{Mann:MLR:2018}, comparing their MLR to the one from \citet{MLrelation:benedict} resulted in a discrepancy of more than 10\% for stars with masses $>$0.3\ensuremath{M_{\sun}}. This discrepancy is also visible in Figure~\ref{fig:MR-relation}, where the black dots represents the masses calculated using the MLR relation from \citet{Mann:MLR:2018} and the grey dots using \citet{MLrelation:benedict}. Above 0.3\ensuremath{M_{\sun}}\ we get higher masses for the same star using \citet{MLrelation:benedict}, compared to \citet{Mann:MLR:2018}.
We also fitted polynomials of different degrees to each relation using the Levenberg-Marquardt algorithm. For each polynomial, we calculated the Akaike information criterion \citep[AIC]{akaike:1974} and the Bayesian Information criterion \citep[BIC]{schwarz197801}. We found, that by using the MLR relation from \citet{Mann:MLR:2018}, the best-fit polynomial for the MR relation is of 3rd order, whereas by using \citet{MLrelation:benedict}, it is a 5th order polynomial. The high order structure caused by the MLR from \citet{MLrelation:benedict} is also visible in Figure~\ref{fig:MR-relation}. Since \citet{Mann:MLR:2018} has been calibrated using accurate Gaia DR2 parallaxes, we continue to use their relation. We find that in this case the mass-radius relation is best characterized by a cubic order polynomial of the form:
\begin{equation}
\begin{split}
\frac\ensuremath{R_\star}\ensuremath{R_{\sun}} = & 0.013(\pm 0.010)
+1.238(\pm 0.117)\,\frac\ensuremath{M_\star}\ensuremath{M_{\sun}} \\
&-1.13(\pm 0.40)\,\left(\frac\ensuremath{M_\star}\ensuremath{M_{\sun}}\right)^2
+1.21(\pm 0.42)\,\left(\frac\ensuremath{M_\star}\ensuremath{M_{\sun}}\right)^3 \\
\end{split}
\end{equation}
The standard deviation of the residuals is $0.016\,\ensuremath{R_{\sun}}$ and the median absolute deviation (MAD) $0.008\,\ensuremath{R_{\sun}}$. The errors of the polynomial coefficients (closed brackets) are estimated from the covariance matrix.
\begin{figure}
\includegraphics[scale=0.23]{img/MR_plot.pdf}
\caption{Radius-mass relation for our sample (large dots) and a subset of data from \citet[][small dots]{mann:2015}, selected as described in the text. Black dots are based on mass estimates using the MLR from \citet{Mann:MLR:2018}, whereas grey dots are based on the one from \citet{MLrelation:benedict}. The lines show best-fit polynomial, as resulted from the respective MLR. (Details are discussed in Sect. \ref{sec:discussion})
\label{fig:MR-relation}}
\end{figure}
We also establish a relation between the stellar radius and its effective temperature, see Figure~\ref{fig:teffR}. Interestingly, in Figure~\ref{fig:teffR} we identified a discontinuous behaviour between 3200 and 3340 K (gray shaded area), where the radius spans a range from 0.18 to 0.42 \ensuremath{R_{\sun}}\, for similar effective temperatures. Considering that our mean measurement error for the radius is $\sim 0.006\,\ensuremath{R_{\sun}}$, this corresponds to a 40-$\sigma$ difference. We also note that we have done a detailed error analysis of our diameter measurements in \citet{Lachaume:2019}. We further find that this discontinuity corresponds to a mass of 0.23\,\ensuremath{M_{\sun}}, see filled and empty dots in Figure~\ref{fig:teffR}.
\begin{figure*}
\includegraphics[scale=0.47]{img/TeffR_M_plot.pdf}
\caption{ \ensuremath{T_{\rm eff}}{} versus radius plot with the stellar mass coded as filled and empty circles, respectively. The lines show fitted polynomials for the two different mass populations (red dashed line for stars with masses $< 0.23\,\ensuremath{M_{\sun}}$ and blue dotted line for stars with masses $\geq 0.23\,\ensuremath{M_{\sun}}$). Grey shaded area shows the region where we find a possible discontinuity and which we attribute to the transition between partially and fully convective stars. We also added M-dwarfs{} from \citet{mann:2015} (small dots), see text for details. The lower plot shows the residual after subtracting the polynomials in Eq. \ref{eq:RTeff} from the radius measurements. (Details are discussed in Sect. \ref{sec:discussion})
\label{fig:teffR}}
\end{figure*}
To the \ensuremath{T_{\rm eff}}-\ensuremath{R_\star}{} data we fitted two linear polynomials depending on the mass range, namely for stars with $\ensuremath{M_\star} \geq 0.23\,\ensuremath{M_{\sun}}$ and $\ensuremath{M_\star} < 0.23\,\ensuremath{M_{\sun}}$. We also tried higher order polynomials, but found in both cases that the higher order coefficients were consistent with zero. We conclude therefore, that for the two cases, the data are best described with two linear polynomials of the form
\begin{equation}
\frac\ensuremath{R_\star}\ensuremath{R_{\sun}} =
\begin{cases}
-1.223(\pm 0.085) + 2.700(\pm 0.138)\,\frac\ensuremath{T_{\rm eff}}\ensuremath{T_{\mathrm{eff},\sun}}\\
\qquad \text{for $\ensuremath{M_\star} \geq 0.23\,\ensuremath{M_{\sun}}$}, \\
-0.277(\pm 0.060) + 0.869(\pm 0.113)\,\frac\ensuremath{T_{\rm eff}}\ensuremath{T_{\mathrm{eff},\sun}}\\
\qquad \text{for $\ensuremath{M_\star} < 0.23\,\ensuremath{M_{\sun}}$}.
\end{cases}
\label{eq:RTeff}
\end{equation}
The standard deviation of the residuals are $0.051\,\ensuremath{R_{\sun}}$ for $\ensuremath{M_\star} \geq 0.23\,\ensuremath{M_{\sun}}$ and $0.016\,\ensuremath{R_{\sun}}$ for $\ensuremath{M_\star} < 0.23\,\ensuremath{M_{\sun}}$ and the respective MADs are $0.033\,\ensuremath{R_{\sun}}$ and $0.0107\,\ensuremath{R_{\sun}}$.
In Figure~\ref{fig:res_teffR} we show the residuals after subtracting Equation~\ref{eq:RTeff} as a function of metallicity. The slope in the data indicates a correlation between metallicity and radius, hence, we calculated the Pearson's correlation coefficient ($r$). For stars with $\ensuremath{M_{\sun}} \geq 0.23$ we get $r=0.69$ and for $\ensuremath{M_{\sun}} < 0.23$ $r=0.51$, respectively. We found that stars with higher metallicity have slightly lager radii, and sub-solar metallicity stars lower radii. This correlation is strong for partially convective stars and moderate for fully convective ones. \citet{burrows:2007} proposed that enhanced opacity in atmospheres due to enhanced metallicity could cause inflated radii in giant planets. Given that we find a correlation between metallicity and radius, it is possible to have a similar effect in M-dwarfs. The metallicity effect on the radius can be best described by two linear polynomials of the form:
\begin{equation}
\frac{\Delta\ensuremath{R_\star}}{\ensuremath{R_\star}} =
\begin{cases}
-0.0060(\pm 0.0093) + 0.4166(\pm 0.0462)\,\ensuremath{\rm [Fe/H]}\\
\qquad \text{ for $\ensuremath{M_\star} \geq 0.23\,\ensuremath{M_{\sun}}$}, \\
0.0187(\pm 0.0176) + 0.2504(\pm 0.0778)\,\ensuremath{\rm [Fe/H]}\\
\qquad \text{ for $\ensuremath{M_\star} < 0.23\,\ensuremath{M_{\sun}}$}.
\end{cases}
\label{eq:R_Feh}
\end{equation}
In Fig. \ref{fig:teffR_corrected} we show \ensuremath{T_{\rm eff}}-\ensuremath{R_\star}{}, where we corrected the stellar radius for possible metallicity effects using Eq. \ref{eq:R_Feh}. The best-fit polynomials in this case are:
\begin{equation}
\frac\ensuremath{R_\star}\ensuremath{R_{\sun}} =
\begin{cases}
-1.169(\pm 0.063) + 2.620(\pm 0.103)\,\frac\ensuremath{T_{\rm eff}}\ensuremath{T_{\mathrm{eff},\sun}}\\
\qquad \text{for $\ensuremath{M_\star} \geq 0.23\,\ensuremath{M_{\sun}}$}, \\
-0.367(\pm 0.050) + 1.041(\pm 0.094)\,\frac\ensuremath{T_{\rm eff}}\ensuremath{T_{\mathrm{eff},\sun}}\\
\qquad \text{for $\ensuremath{M_\star} < 0.23\,\ensuremath{M_{\sun}}$}.
\end{cases}
\label{eq:RTeff_corrected}
\end{equation}
The standard deviation of the residuals are $0.038\,\ensuremath{R_{\sun}}$ and $0.013\,\ensuremath{R_{\sun}}$ respectively, which is slightly lower compared to neglecting the influence of metallicity on the radius. The median absolute deviations of the residuals are $0.029\,\ensuremath{R_{\sun}}$ and $0.008\,\ensuremath{R_{\sun}}$.
\begin{figure*}
\includegraphics[scale=0.47]{img/res_TeffR_FeH_plot.pdf}
\caption{ Residuals after subtracting Eq. \ref{eq:RTeff} from the radius measurement as function of stellar metallicity. The Pearson correlation coefficient is $r=0.69$ for $\ensuremath{M_\star} \geq 0.23\,\ensuremath{M_{\sun}}$ and $r=0.51$ for $\ensuremath{M_\star} < 0.23\,\ensuremath{M_{\sun}}$. Horizontal line represents the case of zero residuals. (Details are discussed in Sect. \ref{sec:discussion})
\label{fig:res_teffR}}
\end{figure*}
\begin{figure*}
\includegraphics[scale=0.47]{img/TeffR_M_plot_corrected.pdf}
\caption{ Same as Fig. \ref{fig:teffR}, but correcting the radius for possible metallicity effects. (Details are discussed in Sect. \ref{sec:discussion})
\label{fig:teffR_corrected}}
\end{figure*}
Based on our observations and our inferred physical parameters, we further show in Figure~\ref{fig:lumteff} the empirical HR-diagram for the two different mass populations. We also can identify a transition region in the HR-diagram. We establish the following linear ($\log$ \ensuremath{L_\star})-\ensuremath{T_{\rm eff}}-relation for the two different populations
\begin{equation}
\log \ensuremath{L_\star} =
\begin{cases}
-6.710(\pm 0.179) + 8.318(\pm 0.290)\,\frac\ensuremath{T_{\rm eff}}\ensuremath{T_{\mathrm{eff},\sun}}\\
\qquad \text{for $\ensuremath{M_\star} \geq 0.23\,\ensuremath{M_{\sun}}$}, \\
-6.856(\pm 0.345) + 8.099(\pm 0.653)\,\frac\ensuremath{T_{\rm eff}}\ensuremath{T_{\mathrm{eff},\sun}}\\
\qquad \text{for $\ensuremath{M_\star} < 0.23\,\ensuremath{M_{\sun}}$}.\\
\end{cases}
\label{eq:RLum}
\end{equation}
\begin{figure*}
\includegraphics[scale=0.47]{img/LumR_M_plot.pdf}
\caption{Empirical HR diagram for two different stellar mass populations (filled and empty circles respectively). The red dashed line shows our fitted polynomials for stars with masses $< 0.23\,\ensuremath{M_{\sun}}$ and blue dotted one for masses $\geq 0.23\,\ensuremath{M_{\sun}}$). We identify a discontinuity (grey shaded area) reflecting the transition region between partially and fully convective stars. Our sample is depicted by the large dots. The sample from \citet{mann:2015} is represented by the smaller dots, which also have their error bars suppressed. Lower plot shows the relative residuals after subtracting Eq. \ref{eq:RLum} from the luminosity value (\ensuremath{ \frac{\rm calc.~luminosity - polynomial}{\rm calc.~luminosity} }). (Details are discussed in Sect. \ref{sec:discussion})
\label{fig:lumteff}}
\end{figure*}
\subsection{Transition into the fully convective regime\label{subsec:transition}}
Theoretical stellar evolution predict a transition from partially convective stars into the fully convective stellar regime to occur at stellar mass somewhere between 0.2\,\ensuremath{M_{\sun}}{} \citep{dorman1989} and 0.35\,\ensuremath{M_{\sun}}{} \citep{chabrier:1997:stellarstructure}, depending on the underlying stellar model. While a partially convective star still resembles a sun-like structure, having a radiative zone and a convective envelope, fully convective stars have no such zone. Our observations indicate that the limit between partially and fully convective regime is around $\approx 0.23\,\ensuremath{M_{\sun}}$ and between 3200 and 3340\,K. The lack of a detection of this transition in previous works can be explained mainly by the fact that very few single M-dwarfs{} with temperatures below 3270\,K have interferometrically measured radii. In fact, \citet{boyajian:2012:mdwarfs} shows only two M-dwarfs{} with temperatures below this value. Moreover, they include in their work mostly one of the stars (GJ~699) which is in the fully convective regime, as GJ~551 was excluded from most of their analyses. Another reason is that previous radius measurements of fully convective stars relied on eclipsing M-dwarf{} binaries, where the disentanglement of the respective components is not straightforward. Finally, most radius estimates rely on stellar evolution models rather than direct measurements, i.e.\ in many cases the radius has not been measured directly.
Furthermore, we find that the linear term of the polynomial in Equation~\ref{eq:RTeff} shows a steeper slope for stars with $\ensuremath{M_\star} \geq 0.23\,\ensuremath{M_{\sun}}$, than for $\ensuremath{M_\star} < 0.23\,\ensuremath{M_{\sun}}$. This is possibly due the fact that stars with $\ensuremath{M_\star} \geq 0.23\,\ensuremath{M_{\sun}}$ still have a radiative zone which decreases with shrinking \ensuremath{T_{\rm eff}}. For M-dwarfs{} with $\ensuremath{M_\star} < 0.23\,\ensuremath{M_{\sun}}$ the stars are fully convective, i.e.\ the convective zone extents towards the core. Therefore, the linear term for M-dwarfs{} with masses below 0.23\,\ensuremath{M_{\sun}}{} indicates a more flattened slope. The gentle slope for masses below 0.23\,\ensuremath{M_{\sun}}{} is consistent with the fact that fully convective stars have similar spectral types due to $\mathrm{H}_2$ formation, which also flattens the radius-temperature relation \citep{ChabrierBaraffe:2000}.
\subsection{M-dwarfs in the context of Gaia}
\label{sec:Gaia}
In Sect. \ref{subsec:multinest} we noticed a considerable difference between \ensuremath{T_{\rm eff}}\ for M-dwarfs inferred from Gaia three band photometry \citep{gaiadr2_teff:2018} and estimates found here and in the literature \citep{neves:2014:feh,mann:2015} . Therefore, we establish an empirical calibration relation for stars with very well measured G magnitudes and parallaxes from Gaia. We use these two measurements to calculate the absolute G magnitude $M_G$, which we relate to the \ensuremath{T_{\rm eff}}. In Fig.~\ref{fig:Gmag_Teff} we show \ensuremath{T_{\rm eff}}\ as a function of $M_G$. The empty circles show the stellar \ensuremath{T_{\rm eff}}\ as estimated by Gaia Apsis, whereas the filled circles show our \ensuremath{T_{\rm eff}}\ measurements and the ones from \citet{mann:2015}. The previously shown discrepancy is also visible in Fig.~\ref{fig:Gmag_Teff}. We determine an empirical relation to obtain \ensuremath{T_{\rm eff}}\ from $M_G$. In our attempt to find the best relation, we fitted polynomials of different degrees and we calculated their respective AIC and BIC, see Fig. \ref{fig:Gmag_Teff}. The best relation is described by a cubic polynomial of the form:
\begin{multline}
\ensuremath{T_{\rm eff}} = 10171.7(\pm 1449.6) -1493.4(\pm 410.8) M_G \\
+ 114.1(\pm 38.3) M_G^2 -3.2(\pm 1.2) M_G^3 \\
\end{multline}
The standard deviation of the residuals is $53\,\ensuremath{T_{\rm eff}}$ and the median absolute deviation (MAD) $36\,\ensuremath{T_{\rm eff}}$. The errors of the polynomial coefficients (closed brackets) are estimated from the covariance matrix.
\begin{figure}
\includegraphics[scale=0.23]{img/Gmag_Teff.pdf}
\caption{Effective temperature as function of absolute Gaia G magnitude. Empty circles show \ensuremath{T_{\rm eff}}\ estimated by Gaia DR2 Apsis. Filled dots show our measurements and the ones from \citet[]{mann:2015}. Lines show polynomial fits of different orders. Their respective AIC and BIC values are given in the legend. (Details are discussed in Sect. \ref{sec:Gaia})
\label{fig:Gmag_Teff}}
\end{figure}
Recently, \citet{jao:2018} presented an investigation showing a $\sim$0.05 mag gap in the HR diagram constructed from M-dwarfs using the Gaia DR2. The authors attributed this gap to a possible transition from partially to fully convective low-mass stars. However, recent simulations by \citet{MacDonald:2018} argued that this gap can be explained by $^{3}$He instabilities of low-mass stars rather than the before mentioned transition region. This $^{3}$He instabilities are caused by stars with a thin radiative zone, slightly above the transition to fully convective stars. These instabilities can produce energy fluctuations and a dip in the luminosity function \citep{vansaders:He:2012,MacDonald:2018}. In Fig. \ref{fig:G_BP_RP} we show $M_G$ over $G_{BP}-G_{RP}$ and mark the region where \citet{jao:2018} found their discontinuity (grey shaded area). The locus of our discovered discontinuity is slightly below the one from \citet{jao:2018}. This increases the likelihood of the finding from \citet{MacDonald:2018} and our claim having observed the transition region between fully and partially convective stars, which should occur slightly below the $^{3}$He instability region.
\begin{figure}
\includegraphics[scale=0.23]{img/Gmag_BPmRP.pdf}
\caption{Absolute Gaia G magnitude versus $G_{\rm BP} - G_{\rm RP}$ for M-dwarfs with different masses. Gray shaded area shows the region, where \citet{jao:2018} found a gap in their HR-diagram. (Details are discussed in Sect. \ref{sec:Gaia})
\label{fig:G_BP_RP}}
\end{figure}
\section{Conclusion} \label{sec:conclusion}
We have measured physical parameters of 13 M-stars covering the partially and fully convective regime using interferometric measurements from the VLTI and parallaxes from Gaia DR2. Our measurements extend to lower \ensuremath{T_{\rm eff}}{} than previous interferometric studies, and we use them augmented with literature data to present improved empirical relations between stellar radius and mass, and between stellar radius and luminosity as a function of \ensuremath{T_{\rm eff}}{}.
Analysing residuals to our relations, we identified a general trend that late-type stars with higher metallicity are slightly inflated, whereas for stars with lower metallicity we measure predominantly smaller radii. We find this correlation to be strong for stars with $\ensuremath{M_\star} \ge 0.23\,\ensuremath{M_{\sun}}$ and moderate for $\ensuremath{M_\star} < 0.23\,\ensuremath{M_{\sun}}$, respectively. We also found that Gaia \ensuremath{T_{\rm eff}}{} values are significantly underestimated ($\approx 8\%$) for M-dwarfs.
The most striking feature we identified in our data is a sharp transition in the relation between \ensuremath{R_\star}{} and \ensuremath{T_{\rm eff}}{}, as well as in the empirical HR diagram, which we identify as reflecting the transition between partially and fully convective stars. While previously only a hint for this change had been inferred indirectly, we now have a possible direct observation. We showed that this change happens at $\sim 0.23\,\ensuremath{M_{\sun}}$ and between 3200 and 3340 K. In this region we measure radii in the range from 0.18 to 0.42\,\ensuremath{R_{\sun}}. Thus, our findings put strong constraints on the stellar evolution and interior structure models.
\section*{Acknowledgements}
We thank the reviewer for their helpful comments
on the manuscript.
M.R. acknowledges support from CONICYT project Basal AFB-170002.
Partially based on observations obtained via ESO under program IDs 090.D-0917, 091.D-0584, 092.D-0647, 093.D-0471.
A.J.\ acknowledges support from FONDECYT project 1171208, CONICYT project BASAL AFB-170002, and by the Ministry for the Economy, Development, and Tourism's Programa Iniciativa Cient\'{i}fica Milenio through grant IC\,120009, awarded to the Millennium Institute of Astrophysics (MAS). R.B.\ acknowledges additional support from project IC120009 ``Millenium Institute of Astrophysics (MAS)'' of the Millennium Science Initiative, Chilean Ministry of Economy.
This work made use of the Smithsonian/NASA Astrophysics Data System (ADS) and of the Centre de Donn\'ees astronomiques de Strasbourg (CDS). This research made use of Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration, 2013).
This work has made use of data from the European Space Agency (ESA) mission
{\it Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the {\it Gaia}
Data Processing and Analysis Consortium (DPAC,
\url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for the DPAC
has been provided by national institutions, in particular the institutions
participating in the {\it Gaia} Multilateral Agreement.
\bibliographystyle{mnras}
|
3,212,635,537,954 | arxiv | \section*{Introduction}
For any integer $\mu \geqslant 1$ let $F_\mu$ be the free group on
$\mu$ generators $z_1,z_2,\ldots,z_\mu$. The classification theory of
high-dimensional $\mu$--component boundary links involves `Seifert
$\Z$--modules' and `Blanchfield $\Z[F_\mu]$--modules', corresponding to
the algebraic invariants obtained from $\mu$--component Seifert surfaces
and the boundary link exterior. This paper concerns the algebraic
relationship between f.g.~projective Seifert $A$--modules and h.d.~1
Blanchfield $A[F_\mu]$--modules for any ring $A$, extending the work of
Sheiham \cite{Sh2}. Part I deals with the algebraic $K$--theory of the
Seifert and Blanchfield modules. Part II will deal with the algebraic
$L$--theory of the Seifert and Blanchfield forms, such as arises in the
computation of the cobordism groups of boundary links.
The algebraic $K$-- and $L$--theory in the knot case $\mu = 1$
have already been done by Ranicki \cite{RBS}.
\subsection*{Combinatorial transversality}
\fullref{combinatorial transversality} develops a combinatorial
construction of fundamental domains for $F_\mu$--covers of $CW$ complexes
which will serve as a role model for the algebraic transversality
of $A[F_\mu]$--module chain complexes in the subsequent sections.
The $F_\mu$--covers $p\co \wwtilde{W}{\to}W$ of a space $W$ are
classified by the homotopy classes of maps
$$c\co W \longrightarrow BF_\mu = {\bigvee_\mu} S^1$$
with $\wwtilde{W} = c^*EF_\mu$ the pullback to $W$ of the universal cover
$EF_\mu$ of $BF_\mu$.
Let $0 \in BF_\mu$ be the point at which the circles $S^1$ are joined,
and choose points $1,2,\ldots,\mu \in BF_\mu\backslash \{0\}$, one in
each circle. If $W$ is a compact manifold then $c$ is homotopic to a map which
is transverse regular at $\{1,2,\ldots,\mu\} \subset BF_\mu$, so that
$$V = c^{-1}(\{1,2,\ldots,\mu\}) = V_1 \sqcup V_2 \sqcup \ldots \sqcup V_\mu
\subset W$$
is a disjoint union of $\mu$ codimension--1 submanifolds
$V_i = c^{-1}(\{i\})\subset W$ (which may be empty) and cutting $W$ at
$V$ there is obtained a fundamental domain $U \subset
\wwtilde{W}$, a compact manifold with boundary
$$\partial U = \bigsqcup\limits_{i = 1}^{\mu}(V_i \sqcup z_iV_i).$$
If $W$ is connected and $c_*\co \pi_1(W) \to F_\mu$ is surjective then $U$
is connected and $V_1,V_2,\ldots,V_\mu$ are non-empty, and may be chosen
to be connected. In the combinatorial version of transversality it is
only required that $W$ be a finite $CW$ complex, and $W$ may be
replaced by a simple homotopy equivalent finite $CW$ complex also
denoted by $W$, with disjoint subcomplexes $V_1,V_2,\ldots,V_\mu \subset
W$ and a fundamental domain $U \subset \wwtilde{W}$ which is a finite
subcomplex with a subcomplex
$$\partial U = \bigsqcup\limits_{i = 1}^{\mu}(V_i \sqcup z_iV_i) \subset U,
\quad V_i = U \cap z^{-1}_iU$$
such that
$$\bigcup\limits_{g \in F_\mu}gU = \wwtilde{W},~
gU \cap hU = \emptyset \text{ unless}~g^{-1}h \in \{1,z_1,z_1^{-1},
\ldots,z_{\mu},z_{\mu}^{-1}\}.$$
Ranicki \cite{RAC} developed combinatorial transversality at $Y \subset X$
for maps of finite $CW$ complexes
$$W\to X = X_1\cup_YX_2$$
with $X,X_1,X_2,Y$ connected and $\pi_1(Y) \to \pi_1(X_1)$, $\pi_1(Y) \to
\pi_1(X_2)$ injective. The essential difference from \cite{RAC} is that we
are here using the Cayley tree $EF_\mu = G_\mu$ of $F_{\mu}$ rather than
the Bass--Serre tree of the amalgamated free product given by the
Seifert--van Kampen Theorem
$$\pi_1(X) = \pi_1(X_1)*_{\pi_1(Y)}\pi_1(X_2)$$
for bookkeeping purposes. We show that $W$ can be replaced by a simple
homotopy equivalent finite $CW$ complex $W$ with disjoint subcomplexes
$V_1,V_2,\ldots,V_\mu \subset W$, such that the $F_\mu$--cover
$\wwtilde{W}$ can be constructed from a fundamental domain finite
subcomplex $U \subset \wwtilde{W}$ obtained by cutting $W$ at
$V = V_1\sqcup V_2 \sqcup \ldots \sqcup V_\mu \subset W$.
\subsection*{Algebraic transversality}
Let $A$ be an associative ring with 1. All $A$--modules will be understood
to be left $A$--modules, unless a right $A$--module structure is specified.
\fullref{algebraic transversality} develops an `algebraic
transversality' technique for cutting $A[F_\mu]$--modules along
$A$--modules, which mimics the geometric transversality method of
\fullref{combinatorial transversality}. In \fullref{algebraic
transversality} we shall prove:
\begin{athm}
\label{thm1}
Every $A[F_\mu]$--module
chain complex $E$ admits a `Mayer--Vietoris presentation'
$$\disablesubscriptcorrection\xysavmatrix{0 \ar[r] & \bigoplus\limits^{\mu}_{i = 1} C^{(i)}[F_\mu]
\ar[r]^-{\displaystyle{f}} & D[F_\mu] \ar[r]& E \ar[r] &0}$$
with $C^{(i)},D$ $A$--module chain complexes, and
$f = (f^+_1z_1-f_1^- \ldots f^+_\mu z_\mu -f_\mu^-)$
defined using $A$--module chain maps $f^+_i,f^-_i\co C^{(i)} \to D$.
If $E$ is a f.g.~free $A[F_\mu]$--module chain complex
then $C^{(i)}$, $D$ can be chosen to be f.g.~free $A$--module chain complexes,
with $D \subset E$ and $f^+_i,f^-_i\co C^{(i)} = D \cap z^{-1}_iD \to D$
given by $f^+_i(x) = x$, $f^-_i(x) = z_ix$.
\end{athm}
\begin{rem}
For $\mu = 1$ \fullref{thm1} was first proved by
Waldhausen \cite{Wald1}, being the chain complex version of the
Higman linearization trick for matrices with entries in the
Laurent polynomial extension $A[F_1] = A[z,z^{-1}]$. The algebraic
transversality theory of \cite{Wald1} applies to chain complexes
over the group rings $A[G_1*_HG_2]$ of injective amalgamated free
products $G_1*_HG_2$, using the Bass--Serre theory of groups acting
on trees. In principle, \fullref{thm1} for $\mu \geqslant 2$ could be proved by
applying \cite{Wald1} to the successive free products in
$$F_\mu = F_1*F_{\mu-1} = F_1*(F_1*F_{\mu-2}) = \cdots =
F_1*(F_1*(F_1*\cdots*(F_1)))$$
but this would be quite awkward in practice. In view of both the
geometric motivation and the algebraic applications it is better
to prove \fullref{thm1} (as will be done in \fullref{algebraic transversality}) using the Cayley tree of
$F_\mu$ with respect to the generator set $\{z_1,z_2,\ldots,z_{\mu}\}$.
\end{rem}
\subsection*{Boundary links}
A $\mu$--component link is a (locally flat, oriented) embedding
$$\ell\co \bigsqcup\limits_\mu S^n \subset S^{n+2}.$$
Every link admits a Seifert surface $V^{n+1} \subset S^{n+2}$, a
codimension--1 submanifold with boundary
$$\partial V = \ell\Bigl(\bigsqcup\limits_\mu S^n\Bigr) \subset
S^{n+2}.$$
By definition, $\ell$ is a \emph{$\mu$--component boundary link}
if there exists a $\mu$--component Seifert surface
$$V^{n+1} = V_1 \sqcup V_2 \sqcup \ldots \sqcup V_\mu \subset S^{n+2}.$$
The exterior of a link $\ell$ is the $(n{+}2)$--dimensional manifold
with boundary
$$(W^{n+2},\partial W) =
\Bigl({\rm cl}\Bigl(S^{n+2}-\Bigl(\ell\Bigl(\bigsqcup\limits_\mu
S^n\Bigr)\times D^2\Bigr)\Bigr), \ell\Bigl(\bigsqcup \limits_\mu S^n\Bigr)
\times S^1\Bigr).$$
In particular, a knot $S^n \subset S^{n+2}$ is a 1--component boundary link.
The trivial $\mu$--component boundary link
$$\ell_0\co \bigsqcup\limits_\mu S^n \subset S^{n+2}$$
is defined by the connected sum of $\mu$ copies of the trivial knot
$$S^n \subset (S^n \times D^2) \cup (D^{n+1} \times S^1) = S^{n+2},$$
so that
$$\ell_0\co \bigsqcup\limits_\mu S^n \subset \mathop{\#}\limits_\mu S^{n+2} =
S^{n+2} = \Bigl(\bigsqcup\limits_\mu S^n \times D^2\Bigr)\cup W_0$$
has Seifert surface and exterior
$$V_0 = \bigsqcup\limits_\mu D^{n+1} ,~
W_0 = \mathop{\#}\limits_\mu (D^{n+1} \times S^1)\subset
S^{n+2}.$$
The exterior $W_0$ has the homotopy type of
$\bigvee_\mu S^1\vee \bigvee_{\mu-1}S^{n+1}$,
with $\pi_1(W_0) = F_\mu$.
We shall make much use of the fact that the universal cover of
$BF_\mu = \bigvee_\mu S^1$ is the contractible
space with free $F_\mu$--action defined by the Cayley tree
$EF_\mu = G_\mu$ of $F_\mu$, with vertices $g \in F_\mu$ and edges
$(g,gz_i)$ $(g \in F_\mu,~1 \leqslant i \leqslant \mu)$. The
cellular chain complex $C(EF_\mu) = C(G_\mu)$ is the
standard 1--dimensional f.g.~free $\Z[F_\mu]$--module resolution of $\Z$
$$\disablesubscriptcorrection\xysavmatrix{
0 \ar[r] &} C_1(G_\mu) = \bigoplus\limits_{i = 1}^{\mu} \Z[F_\mu]
\disablesubscriptcorrection\xysavmatrix{\ar[r]^-{\displaystyle{d}}& C_0(G_\mu) = \Z[F_\mu]
\ar[r] & \Z \ar[r] &0},$$
the Mayer--Vietoris presentation with
$d = (z_1-1~z_2-1~\ldots~z_\mu-1)$.
The exterior $W$ of an $n$--dimensional link
$\ell\co \bigsqcup_\mu S^n \subset S^{n+2}$ is homotopy
equivalent to the complement $S^{n+2}\backslash
\ell\bigl(\bigsqcup_\mu S^n\bigr)$, so that
$$\begin{array}{ll}
H_*(W)& = H_*\Bigl(S^{n+2}\backslash \ell\Bigl(\bigsqcup \limits_\mu
S^n\Bigr)\Bigr)\\[2ex]
& = H^{n+2-*}\Bigl(S^{n+2},\ell\Bigl(\bigsqcup\limits_\mu S^n\Bigr)\Bigr)
= H^{n+1-*}\Bigl(\bigsqcup\limits_\mu S^n\Bigr)\quad(* \neq 0,n+2)
\end{array}$$
by Alexander duality. The homology groups $H_*(W),H_*(W_0)$ are thus
the same:
$$H_r(W) = H_r(W_0) = \begin{cases} \Z&\hbox{if $r = 0$} \\
\bigoplus\limits_\mu \Z&\hbox{if $r = 1$}\\
\bigoplus\limits_{\mu-1} \Z&\hbox{if $r = n+1$}\\
0&\hbox{otherwise}.
\end{cases}$$
The homotopy groups $\pi_*(W),\pi_*(W_0)$ are in general not the
same, on account of linking. By Smythe \cite{Sm} and Gutierrez
\cite{Gu} $\ell$ is a boundary link if and only if there exists a
surjection $\pi_1(W) \to \pi_1(W_0) = F_\mu$ sending the meridians
$m_1,m_2,\ldots,m_{\mu}\co S^1 \subset W$ around the $\mu$ components
$\ell_1,\ell_2,\ldots,\ell_\mu\co S^n \subset S^{n+2}$ of $\ell$ to
the generators $z_1,z_2,\ldots,z_\mu \in F_\mu$. We shall only be
considering boundary links $\ell$ with a particular choice of such
a surjection $\pi_1(W)\to F_\mu$, the \emph{$F_\mu$--links} of
Cappell and Shaneson \cite{CS2}. For any such $\ell$ there exists
a map $c\co W \to W_0$ which induces a surjection $c_*\co \pi_1(W) \to
\pi_1(W_0)$ and isomorphisms $c_*\co H_*(W)\cong H_*(W_0)$. Let
$\wwtilde{W} = c^*\wwtilde{W}_0$ be the pullback $F_\mu$--cover
of $W$, with a f.g.~free $\Z[F_\mu]$--module cellular chain
complex $C(\wwtilde{W})$. An $F_\mu$--equivariant lift
$\wtilde{c}\co \wwtilde{W} \to \wwtilde{W}_0$ of $c$ induces a
$\Z[F_\mu]$--module chain map $\wtilde{c}\co C(\wwtilde{W}) \to
C(\wwtilde{W}_0)$ and a $\Z$--module chain equivalence $c\co C(W)
\to C(W_0)$. A $\mu$--component Seifert surface $V = V_1\sqcup V_2
\sqcup \ldots \sqcup V_\mu \subset S^{n+2}$ for $\ell$ has a
neighbourhood $V \times [-1,1] \subset S^{n+2}$, with $V = V \times
\{0\}$. The $F_\mu$--cover $\wwtilde{W}$ can be constructed from
$F_\mu$ copies of $S^{n+2}\backslash V$, glued together using the
inclusions $f^+_i,f^-_i\co V_i \to S^{n+2}\backslash V$ defined by
$$f^{\pm}_i(v_i) = (v_i,\pm 1) \in V \times [-1,1] \subset S^{n+2}.$$
It follows that $C(\wwtilde{W})$ has a f.g.~free $\Z[F_\mu]$--module
Mayer--Vietoris presentation
$$\disablesubscriptcorrection\xysavmatrix@C-5pt{0 \ar[r] &}
\bigoplus\limits^\mu_{i = 1}C(V_i)[F_\mu]
\disablesubscriptcorrection\xysavmatrix@C-5pt{\ar[r]^-{\displaystyle{f}}&}
C(S^{n+2}\backslash V)[F_\mu] \disablesubscriptcorrection\xysavmatrix@C-5pt{\ar[r] &} C(\wwtilde{W})
\disablesubscriptcorrection\xysavmatrix@C-5pt{\ar[r]&0}$$
with $f = f^+z-f^- = (f^+_1z_1-f_1^-~\ldots~f^+_\mu z_\mu -f_\mu^-)$.
\subsection*{Seifert and Blanchfield modules}
There are four fundamental notions in our abstract version for any
ring $A$ of the Seifert and Blanchfield modules of $\mu$--component
boundary links:
\begin{itemize}
\item[(i)] A \emph{Seifert $A$--module} is a triple
$$(P,e,\{\pi_i\}) = (~\hbox{$A$--module},~\hbox{endomorphism},~
\{\pi_i\})$$
where $\{\pi_i\co P \to P\}$ is a system of idempotents expressing $P$ as
a $\mu$--fold direct sum, with
\begin{align*}
\pi_i\co P = P_1 \oplus P_2 \oplus \cdots \oplus P_{\mu} &\to P;\\[1ex]
(x_1,x_2,\ldots,x_\mu) &\mapsto (0,\ldots,0,x_i,0,\ldots,0).
\end{align*}
Let ${\mathcal{S}\mathit{ei}}_{\infty}(A)$ be the category of Seifert $A$--modules. A
Seifert $A$--module $(P,e,\{\pi_i\})$ is \emph{f.g.~projective} if
$P$ is a f.g.~projective $A$--module. Let ${\mathcal{S}\mathit{ei}}(A) \subset
{\mathcal{S}\mathit{ei}}_{\infty}(A)$ be the full subcategory of the f.g.~projective
Seifert $A$--modules.
\item[(ii)] A \emph{Blanchfield $A[F_\mu]$--module} $M$ is an
$A[F_\mu]$--module such that
$$\Tor^{A[F_\mu]}_*(A,M) = 0,$$
regarding $A$ as a right $A[F_\mu]$--module via the augmentation map
$$\epsilon\co A[F_\mu] \to A;~z_i \mapsto 1.$$
Let ${\mathcal{B}\mathit{la}}_{\infty}(A)$ be the category of Blanchfield $A[F_\mu]$--modules.
In \fullref{bla} Blanchfield $A[F_\mu]$--modules will be identified
with the \emph{$F_\mu$--link modules} in the sense of Sheiham \cite{Sh2},
that is $A[F_\mu]$--modules $M$ which admit an $A[F_\mu]$--module presentation
$$\disablesubscriptcorrection\xysavmatrix{0 \ar[r]& P[F_\mu] \ar[r]^-{\displaystyle{d}} & Q[F_\mu] \ar[r]& M \ar[r] & 0}$$
for $A$--modules $P,Q$ with the augmentation $\epsilon(d)\co P \to Q$ an
$A$--module isomorphism. Thus ${\mathcal{B}\mathit{la}}_{\infty}(A)$ is just the
$F_\mu$--link module category ${\mathcal{F}\mathit{lk}}_{\infty}(A)$ of \cite{Sh2}. A
Blanchfield $A[F_\mu]$--module $M$ has \emph{homological dimension $1$}
(or \emph{h.d.~1} for short) if it has a 1--dimensional f.g.~projective
$A[F_\mu]$--module resolution
$$\disablesubscriptcorrection\xysavmatrix{0 \ar[r]& K \ar[r]^-{\displaystyle{d}} & L \ar[r]& M \ar[r] & 0}$$
with (necessarily)
$\epsilon(d) = 1\otimes d\co A\otimes_{A[F_{\mu}]}K \to A\otimes_{A[F_{\mu}]}L$
an $A$--module isomorphism. Let ${\mathcal{B}\mathit{la}}(A)
\subset {\mathcal{B}\mathit{la}}_{\infty}(A)$ be the full subcategory of the h.d.~1
Blanchfield $A[F_\mu]$--modules. Let ${\mathcal{F}\mathit{lk}}(A) \subset {\mathcal{B}\mathit{la}}(A)$ be the full
subcategory of the h.d.~1 Blanchfield modules $M$ which admit a 1--dimensional
induced f.g.~projective $A[F_\mu]$--module resolution
$$\disablesubscriptcorrection\xysavmatrix{0 \ar[r]& P[F_\mu] \ar[r]^-{\displaystyle{d}} & Q[F_\mu] \ar[r]& M \ar[r] & 0}$$
with $P,Q$ f.g.~projective $A$--modules. As in \cite{Sh2}
the objects of ${\mathcal{F}\mathit{lk}}(A)$ will be called \emph{h.d.~1 $F_\mu$--link modules}.
\item[(iii)] The \emph{covering} of a Seifert $A$--module $(P,e,\{\pi_i\})$
is the Blanchfield $A[F_\mu]$--module
$$B(P,e,\{\pi_i\}) = \coker(1-e+ez\co P[F_\mu] \to
P[F_\mu])$$
with $z = \sum\limits^{\mu}_{i = 1}\pi_iz_i\co P[F_{\mu}] \to
P[F_{\mu}]$, defining functors
$$B_{\infty}\co {\mathcal{S}\mathit{ei}}_{\infty}(A) \to
{\mathcal{B}\mathit{la}}_{\infty}(A),~ B\co {\mathcal{S}\mathit{ei}}(A) \to {\mathcal{F}\mathit{lk}}(A).$$
\item[(iv)] A Seifert
$A$--module $(P,e,\{\pi_i\})$ is \emph{primitive} if
$B(P,e,\{\pi_i\}) = 0$. Let
$${\mathcal{P}\mathit{rim}}_{\infty}(A) = {\rm
ker}(B_{\infty}\co {\mathcal{S}\mathit{ei}}_{\infty}(A)\to {\mathcal{B}\mathit{la}}_{\infty}(A))$$
be the full subcategory of ${\mathcal{S}\mathit{ei}}_{\infty}(A)$ with objects the primitive Seifert
$A$--modules, and let
$${\mathcal{P}\mathit{rim}}(A) = \ker(B\co {\mathcal{S}\mathit{ei}}(A)\to {\mathcal{F}\mathit{lk}}(A)) \subset {\mathcal{S}\mathit{ei}}(A)$$
be the full subcategory of ${\mathcal{S}\mathit{ei}}(A)$ with objects the
primitive f.g.~projective Seifert $A$--modules.
\end{itemize}
\subsection*{Simple boundary links}
The motivational examples of f.g.~projective Seifert $\Z$--modules
and h.d.~1 $F_\mu$--link $\Z[F_\mu]$--modules come from the
$(2q{-}1)$--dimensional $\mu$--component boundary links
$\ell\co \bigsqcup_{\mu} S^{2q-1} \subset S^{2q+1}$ which are
\emph{simple}, meaning that the exterior $W$ has homotopy groups
$$\pi_r(W) = \begin{cases} F_\mu&{\rm if}~r = 1\\
0&{\rm if}~2 \leqslant r \leqslant q-1,
\end{cases}$$
so that the universal cover $\wwtilde{W}$ is $(q{-}1)$--connected.
These conditions are equivalent to the existence of a $\mu$--component Seifert
surface $V = V_1 \sqcup V_2 \sqcup \cdots \sqcup V_{\mu}$
with each component $V_i$ $(q{-}1)$--connected:
$$\pi_r(V_i) = 0\quad
(1 \leqslant i \leqslant \mu,~1 \leqslant r \leqslant q-1).$$
The homology of the Seifert surface defines a f.g.~projective
(actually f.g.~free) Seifert $\Z$--module $(P,e,\{\pi_i\})$, with
$$\pi_i = 0 \oplus \cdots \oplus 0 \oplus 1 \oplus 0 \oplus \cdots \oplus 0\co
P = \smash{\bigoplus\limits^\mu_{i = 1}}H_q(V_i)\to
P = \smash{\bigoplus\limits^\mu_{i = 1}}H_q(V_i)$$
and
\begin{multline*}
e = (f^+_1~f^+_2\ldots f^+_\mu)\co
P = H_q(V) = \bigoplus\limits^\mu_{i = 1}H_q(V_i)\longrightarrow \\
H_q(S^{2q+1}\backslash V) = H^q(V) = H_q(V) = P
\end{multline*}
the endomorphism induced by the inclusions $f_i^+\co V_i \to
S^{2q+1}\backslash V$, identifying
$$H_q(S^{2q+1}\backslash V) = H^q(V)$$
by Alexander duality and $H^q(V) = H_q(V)$ by Poincar\'e
duality. The covering of $(P,e,\{\pi_i\})$ is the h.d.~1 $F_\mu$--link
$\Z[F_\mu]$--module
$$B(P,e,\{\pi_i\}) = H_q(\wwtilde{W})$$
defined by the homology of the $F_\mu$--cover $\wwtilde{W}$ of the
exterior $W$. The f.g.~projective Seifert $\Z$--module
$(P,e,\{\pi_i\})$ is primitive if and only if
$H_q(\wwtilde{W}) = 0$; for $q \geqslant 2$ this is the case if
and only if $\ell$ is unlinked (Gutierrez \cite{Gu}).
\subsection*{Blanchfield = Seifert/primitive}
\fullref{modules} uses algebraic transversality to prove that
every h.d.~1 $F_\mu$--link module $M$ is isomorphic to the covering
$B(P,e,\{\pi_i\})$ of a f.g.~projective Seifert $A$--module
$(P,e,\{\pi_i\})$, and that morphisms of h.d.~1 $F_\mu$--link
modules can be expressed as fractions of morphisms of f.g.~projective
Seifert $A$--modules.
The algebraic relation between Seifert $A$--modules and Blanchfield
$A[F_\mu]$--modules for $A = \Z$ was first investigated
systematically in the knot case $\mu = 1$, by Levine \cite{L1,L2} and
Trotter \cite{T}, and for the link case $\mu \geqslant 1$ by
Farber \cite{Fa1,Fa3} and Sheiham \cite{Sh2}. In particular,
\cite{Sh2} expressed the Blanchfield module
category ${\mathcal{B}\mathit{la}}_\infty(A) = {\mathcal{F}\mathit{lk}}_\infty(A)$ as the quotient of the Seifert
$A$--module category ${\mathcal{S}\mathit{ei}}_{\infty}(A)$ by the primitive Seifert
$A$--module subcategory ${\mathcal{P}\mathit{rim}}_{\infty}(A)$, as we now recall.
Let $\mathcal{A}$ be an abelian category. By definition, a \emph{Serre
subcategory} $\C \subset \mathcal{A}$ is a non-empty full
subcategory such that for every exact sequence in $\mathcal{A}$
$$0 \to M' \to M \to M'' \to 0$$
$M$ is an object in $\C$ if and
only if $M',M''$ are objects in $\C$. Gabriel \cite{Ga}
defined the quotient abelian category $\mathcal{A}/\C$ with
the same objects as $\mathcal{A}$ but different morphisms: if
$M,N$ are objects in $\mathcal{A}$ then
$$\Hom_{\mathcal{A}/\C}(M,N) =
\varinjlim \Hom_\mathcal{A}(M',N'')$$
with the direct limit taken over all the exact sequences in $\mathcal{A}$
$$0 \to M' \to M \to M'' \to 0,~0 \to N' \to N \to N'' \to 0$$
with $M'',N'$ objects in $\C$. The canonical functor
$F\co \mathcal{A} \to \mathcal{A}/\C;A \mapsto A$ sends each object $C$ in $\C$ to $F(C) = 0$,
and has the universal property that for any exact functor
$G\co \mathcal{A} \to \mathcal{B}$ such that $G(C) = 0$ for all objects in $\C$
there exists a unique functor $\wwbar{G}\co \mathcal{A}/\C \to \mathcal{B}$ such that
$\wwbar{G}F = G$. In particular, if $\mathcal{B}$ is an exact category and
$G\co \mathcal{A} \to \mathcal{B}$ is an exact functor then the full subcategory $\C \subset \mathcal{A}$
with objects $C$ such that $G(C) = 0$ is a Serre subcategory, and there
is induced a functor $\wwbar{G}\co \mathcal{A}/\C \to \mathcal{B};A \mapsto G(A)$
such that $G = \wwbar{G}F$.
By definition, a category is \emph{small} if the class of morphisms is a set.
In order to avoid set-theoretic difficulties we shall only be
dealing with categories which are \emph{essentially small}, ie equivalent
to a small category.
Let $\mathcal{A}$ be an essentially small category, and let $\Sigma$ be a set
of morphisms in $\mathcal A$, e.g. the morphisms of a subcategory.
A \emph{category of fractions} $\Sigma^{-1}\mathcal{A}$
is a category with a universally $\Sigma$--inverting functor
$F\co \mathcal{A} \to \Sigma^{-1}\mathcal{A}$, meaning that:
\begin{itemize}
\item[(i)] $F$ sends each $f \in \Sigma$ to an isomorphism $F(f)$
in $\Sigma^{-1}\mathcal{A}$,
\item[(ii)] for any functor $G\co \mathcal{A} \to \mathcal{B}$
which sends each $f \in \Sigma$ to an isomorphism $G(f)$
there exists a unique functor $\wwbar{G}\co \Sigma^{-1}\mathcal{A} \to
\mathcal{B}$ such that $\wwbar{G}F = G$.
\end{itemize}
An essentially small category of fractions $\Sigma^{-1}\mathcal{A}$ exists,
with the same objects as $\mathcal{A}$, and such a category
is unique up to isomorphism (Gabriel and Zisman \cite{GZ},
Borceux \cite[5.2.2]{Bo}).
For example, if $\mathcal A$ is an abelian category and $\C
\subset \mathcal{A}$ is a Serre subcategory, then
$$\mathcal{A}/\C = \Sigma^{-1}\mathcal{A}$$
is a category of fractions inverting the set $\Sigma$ of morphisms
$f$ in $\mathcal{A}$ with $\ker(f)$ and $\coker(f)$ in $\C$.
An $A[F_\mu]$--module $M$ is Blanchfield if and only if the
$A$--module morphism
$$\gamma\co \bigoplus\limits_\mu M \to M;~
(m_1,m_2,\ldots,m_\mu) \mapsto \sum\limits^\mu_{i = 1}(z_i-1)m_i$$
is an isomorphism, called the \emph{Sato isomorphism} (after \cite{Sa},
the case $A = \Z$).
As in Sheiham \cite{Sh2}, for any Blanchfield
$A[F_\mu]$--module $M$ use the $A$--module morphisms
\begin{align*}
p_i\co \bigoplus\limits_\mu M &\to M;
\quad(m_1,m_2,\ldots,m_\mu) \mapsto m_i,\\
\omega\co \bigoplus\limits_\mu M &\to M;
\quad(m_1,m_2,\ldots,m_\mu) \mapsto \smash{\sum\limits^\mu_{i = 1}}m_i,\\
\pi_i = \gamma p_i \gamma^{-1}\co M &\to M,\\
e = \omega \gamma^{-1}\co M &\to M
\end{align*}
to define a Seifert $A$--module $U(M) = (M,e,\{\pi_i\})$.
The categories ${\mathcal{P}\mathit{rim}}_{\infty}(A)$, ${\mathcal{S}\mathit{ei}}_{\infty}(A)$ are abelian,
while ${\mathcal{B}\mathit{la}}_{\infty}(A)$ is in general only exact. The covering
functor
$B_{\infty}\co {\mathcal{S}\mathit{ei}}_{\infty}(A) \to {\mathcal{B}\mathit{la}}_{\infty}(A)$ was shown in
\cite[5.2]{Sh2} to be exact, so that ${\mathcal{P}\mathit{rim}}_{\infty}(A) \subset
{\mathcal{S}\mathit{ei}}_{\infty}(A)$ is a Serre subcategory. Thus if $\Xi_{\infty}$ is
the set of morphisms $f$ in ${\mathcal{S}\mathit{ei}}_{\infty}(A)$ such that $B(f)$ is an
isomorphism in ${\mathcal{B}\mathit{la}}_{\infty}(A)$, or equivalently $\ker(f)$ and
$\coker(f)$ are in ${\mathcal{P}\mathit{rim}}_{\infty}(A)$, then
$${\mathcal{S}\mathit{ei}}_{\infty}(A)/{\mathcal{P}\mathit{rim}}_{\infty}(A) = \Xi_{\infty}^{-1}{\mathcal{S}\mathit{ei}}_{\infty}(A).$$
The induced exact functor
$\wbar{B}_{\infty}\co {\mathcal{S}\mathit{ei}}_{\infty}(A)/{\mathcal{P}\mathit{rim}}_{\infty}(A) \to {\mathcal{B}\mathit{la}}_{\infty}(A)$
is such that
$$B_{\infty}\co {\mathcal{S}\mathit{ei}}_{\infty}(A) \to {\mathcal{S}\mathit{ei}}_{\infty}(A)/{\mathcal{P}\mathit{rim}}_{\infty}(A)
\disablesubscriptcorrection\xysavmatrix{\ar[r]^-{\displaystyle{\wbar{B}_\infty}}&} {\mathcal{B}\mathit{la}}_{\infty}(A)$$
and has the universal property of inverting $\Xi_{\infty}$. The
functor $\wbar{B}_{\infty}$ was shown to be an equivalence in
\cite[5.15]{Sh2} using the fact that the functor
$$U_{\infty}\co {\mathcal{B}\mathit{la}}_{\infty}(A) \to {\mathcal{S}\mathit{ei}}_{\infty}(A);~M \mapsto U(M)$$
is right adjoint to $B$:
for any Seifert $A$--module $V$ there is a natural isomorphism
$$\Hom_{{\mathcal{B}\mathit{la}}_{\infty}(A)}(B(V),M)\cong~\Hom_{{\mathcal{S}\mathit{ei}}_{\infty}(A)}(V,U(M)).$$
The functor $U_{\infty}$ is fully faithful, allowing ${\mathcal{B}\mathit{la}}_{\infty}(A)$
to be regarded as a full subcategory of ${\mathcal{S}\mathit{ei}}_{\infty}(A)$. By
\cite[5.15]{Sh2} $U_{\infty}$ induces a functor
$$\wwbar{U}_{\infty}\co {\mathcal{B}\mathit{la}}_{\infty}(A)\to {\mathcal{S}\mathit{ei}}_{\infty}(A)/{\mathcal{P}\mathit{rim}}_{\infty}(A)$$
which is an equivalence inverse to $\wbar{B}_{\infty}$. Thus up
to equivalence
$${\mathcal{S}\mathit{ei}}_{\infty}(A)/{\mathcal{P}\mathit{rim}}_{\infty}(A) = \Xi_{\infty}^{-1}{\mathcal{S}\mathit{ei}}_{\infty}(A) =
{\mathcal{B}\mathit{la}}_{\infty}(A).$$
\indent The categories ${\mathcal{P}\mathit{rim}}(A),{\mathcal{S}\mathit{ei}}(A),{\mathcal{F}\mathit{lk}}(A),{\mathcal{B}\mathit{la}}(A)$ are exact but
not in general abelian. As in \cite{Sh2} let
${\mathcal{S}\mathit{ei}}(A)/{\mathcal{P}\mathit{rim}}_{\infty}(A) \subset {\mathcal{S}\mathit{ei}}_{\infty}(A)/{\mathcal{P}\mathit{rim}}_{\infty}(A)$
be the full subcategory with objects in ${\mathcal{S}\mathit{ei}}(A)$. The equivalence
$$\wbar{B}_{\infty}\co {\mathcal{S}\mathit{ei}}_{\infty}(A)/{\mathcal{P}\mathit{rim}}_{\infty}(A)
\disablesubscriptcorrection\xysavmatrix{\ar[r]^-{\approx}&}{\mathcal{B}\mathit{la}}_{\infty}(A)$$
was shown in \cite[5.17]{Sh2} to restrict to an
equivalence of exact sequences
$$\wbar{B}\co {\mathcal{S}\mathit{ei}}(A)/{\mathcal{P}\mathit{rim}}_{\infty}(A) \disablesubscriptcorrection\xysavmatrix{\ar[r]^-{\approx}&}
{\mathcal{F}\mathit{lk}}(A)$$
with
$$B\co {\mathcal{S}\mathit{ei}}(A) \to {\mathcal{S}\mathit{ei}}(A)/{\mathcal{P}\mathit{rim}}_{\infty}(A)
\disablesubscriptcorrection\xysavmatrix{\ar[r]^{\displaystyle{\wbar{B}}}_{\approx}&} {\mathcal{F}\mathit{lk}}(A).$$
From the construction of ${\mathcal{S}\mathit{ei}}_{\infty}(A)/{\mathcal{P}\mathit{rim}}_{\infty}(A)$
a morphism in ${\mathcal{S}\mathit{ei}}(A)/{\mathcal{P}\mathit{rim}}_{\infty}(A)$ may involve objects
in ${\mathcal{S}\mathit{ei}}_{\infty}(A)$ which are not in ${\mathcal{S}\mathit{ei}}(A)$, so that the
equivalence $\wbar{B}$
cannot be used to relate the algebraic $K$--theories of
${\mathcal{S}\mathit{ei}}(A)$ and ${\mathcal{F}\mathit{lk}}(A)$.
A category of fractions $\Sigma^{-1}\mathcal{A}$ has a
\emph{left calculus of fractions} if:
\begin{itemize}
\item[(i)] $(1\co A \to A) \in \Sigma$ for every object $A$ in $\mathcal A$,
\item[(ii)] if $(s\co A \to B),(t\co B \to C) \in \Sigma$ then $(ts\co A \to C)
\in \Sigma$,
\item[(iii)] for any
$f\co A \to B$ in $\mathcal{A}$ and $s\co A \to D$ in $\Sigma$
there exist
$g\co D \to C$ in $\mathcal{A}$ and $t\co B \to C$ in $\Sigma$
such that $tf = gs\co A \to C$,
\item[(iv)] for any $f,g\co A \to B$ in $\mathcal{A}$ and
$s\co D \to A$ in $\Sigma$ with $fs = gs\co D \to B$ there
exists $(t\co B \to C) \in \Sigma$ with $tf = tg\co A \to C$.
\end{itemize}
It then follows that a morphism $A \to B$ in $\Sigma^{-1}\mathcal{A}$
can be regarded as an equivalence class $s^{-1}f$ of pairs
$(f\co A \to C,s\co B \to C)$ of morphisms in $\mathcal A$
with $s \in \Sigma$, where
$$\begin{array}{ll}
(f,s) \sim (f',s')&\hbox{\rm if there exist morphisms}~
g\co C \to D,~g'\co C' \to D~{\rm in}~\mathcal{A}\\[1ex]
&{\rm with}~(gs = g's'\co B \to D) \in \Sigma~{\rm and}~gf = g'f'\co A\to D
\end{array}$$
so that
$$s^{-1}f = (gs)^{-1}(gf) = (g's')^{-1}(g'f') = {s'}^{-1}f'\co
A \to B~{\rm in}~\Sigma^{-1}\mathcal{A}.$$
\indent
Let $\Xi$ be the set of morphisms $f$ in ${\mathcal{S}\mathit{ei}}(A)$ such that $B(f)$ is
an isomorphism in ${\mathcal{F}\mathit{lk}}(A)$, or equivalently such that $\ker(f)$
and $\coker(f)$ are in ${\mathcal{P}\mathit{rim}}_{\infty}(A)$.
In \fullref{modules} we shall prove:
\begin{athm}\label{thm2}
{\rm (i)}\qua The category of fractions
$\Xi^{-1}{\mathcal{S}\mathit{ei}}(A)$ has a left calculus of fractions, and the
covering functor $B\co {\mathcal{S}\mathit{ei}}(A) \to {\mathcal{F}\mathit{lk}}(A)$ induces an equivalence of exact
categories
$$\wbar{B}\co \Xi^{-1}{\mathcal{S}\mathit{ei}}(A) \disablesubscriptcorrection\xysavmatrix{\ar[r]^-{\approx}&} {\mathcal{F}\mathit{lk}}(A).$$
{\rm (ii)}\qua The h.d.~1 Blanchfield $A[F_\mu]$--module category
${\mathcal{B}\mathit{la}}(A)$ is the idempotent completion of the h.d.~1 $F_\mu$--link
module category ${\mathcal{F}\mathit{lk}}(A)$.
\end{athm}
The key step in the proof of \fullref{thm2} (i) is the use of the algebraic
transversality \fullref{thm1} to verify that for any h.d.~1 $F_\mu$--link
module $M$ the Seifert $A$--module $U(M)$ is a direct limit of morphisms
in $\Xi$.
\subsection*{Primitive = near-projection}
\fullref{kernel} gives an intrinsic characterization of the
primitive f.g.~projective Seifert $A$--modules $(P,e,\{\pi_i\})$ as
generalized near-projections.
An endomorphism $e\co P \to P$ of an $A$--module $P$ is {\it
nilpotent} if $e^N = 0$ for some $N \geqslant 0$.
An endomorphism $e\co P \to P$ is a \emph{near-projection}
if $e(1-e)\co P \to P$ is nilpotent (L\"uck and Ranicki \cite{LR}).
In \fullref{kernel} we shall prove:
\begin{athm}\label{thm3}
A f.g.~projective Seifert $A$--module
$(P,e,\{\pi_i\})$ is primitive if and only if it can be expressed
as
$$(P,e,\{\pi_i\}) = \biggl(P^+\oplus P^-,\begin{pmatrix}
e^{++} & e^{+-} \\
e^{-+} & e^{--}
\end{pmatrix},\{\pi_i^+\}\oplus \{\pi_i^-\}\biggr)$$
and the $2\mu$--component Seifert $A$--module
$$(P',e',\pi') = \left( P^+\oplus P^-\ ,\
\begin{pmatrix}
e^{++} & -e^{+-} \\
e^{-+} & 1-e^{--}
\end{pmatrix}\ ,\ \{\pi_i^+\}\oplus\{\pi_i^-\}\right)$$
is such that $e'z'\co P'[F_{2\mu}] \to P'[F_{2\mu}]$ is
nilpotent, with $F_{2\mu}$ the free group on $2\mu$ generators
$z'_1,\ldots,z'_{2\mu}$.
\end{athm}
For $\mu = 1$ the condition for a f.g.~projective Seifert $A$--module
$(P,e,\{\pi_i\})$ to be primitive is just that $e$ be a near-projection.
For $\mu = 1$ \fullref{thm3} is just the result of Bass, Heller and Swan \cite{BHS}
that $1-e+ez\co P[z,z^{-1}] \to P[z,z^{-1}]$ is an $A[z,z^{-1}]$--module
isomorphism if and only if $e$ is a near-projection, if and only if
$(P,e) = (P^+,e^{++})\oplus (P^-,e^{--})$ with
$e^{++}\co P^+ \to P^+$ and $1-e^{--}\co P^- \to P^-$ nilpotent.
\subsection*{Algebraic $K$--theory}
\fullref{ktheory} obtains results on the algebraic $K$--theory of
$A[F_\mu]$, ${\mathcal{P}\mathit{rim}}(A)$, ${\mathcal{S}\mathit{ei}}(A)$, ${\mathcal{F}\mathit{lk}}(A)$ and ${\mathcal{B}\mathit{la}}(A)$, using
the algebraic $K$--theory noncommutative localization exact sequences
of Schofield \cite{Sc} and Neeman--Ranicki \cite{NR1,NR2}.
The class group $K_0(\mathcal{E})$ of an exact category $\mathcal{E}$ is the Grothendieck
group with one generator $[M]$ for each object $M$ in $\mathcal{E}$, and one relation
$[K]-[L]+[M] = 0$ for each exact sequence in $\mathcal{E}$
$$0 \to K \to L \to M \to 0.$$
The algebraic $K$--groups $K_n(\mathcal{E})$ are defined by Quillen \cite{Q} for
$n \geqslant 1$ and by Schlichting \cite{Schl} for $n \leqslant -1$.
Write
\begin{align*}
{\rm Prim}_*(A) & = K_*({\mathcal{P}\mathit{rim}}(A)), &{\rm Sei}_*(A) & = K_*({\mathcal{S}\mathit{ei}}(A)),\\
{\rm Bla}_*(A) & = K_*({\mathcal{B}\mathit{la}}(A)), &{\rm Flk}_*(A) & = K_*({\mathcal{F}\mathit{lk}}(A)),
\end{align*}
noting that ${\rm Bla}_n(A) = {\rm Flk}_n(A)$ for $n \neq 0$.
\begin{athm}\label{thm4}
{\rm (i)}\qua The algebraic $K$--groups of $A[F_\mu]$ split as
$$K_*(A[F_\mu]) = K_*(A) \oplus \bigoplus\limits_{\mu} K_{*-1}(A)
\oplus \widetilde{{\rm Prim}}_{*-1}(A).$$
{\rm (ii)}\qua The sequence of functors
$$\disablesubscriptcorrection\xysavmatrix{{\mathcal{P}\mathit{rim}}(A) \ar[r]& {\mathcal{S}\mathit{ei}}(A) \ar[r]^-{\displaystyle{B}} & {\mathcal{B}\mathit{la}}(A)}$$
induces a long exact sequence of algebraic $K$--groups
$$\cdots \to {\rm Prim}_n(A) \to {\rm Sei}_n(A) \disablesubscriptcorrection\xysavmatrix{\ar[r]^-{\displaystyle{B}}&}
{\rm Bla}_n(A) \to {\rm Prim}_{n-1}(A) \to \cdots$$
with
$${\rm im}(B\co {\rm Sei}_0(A) \to {\rm Bla}_0(A)) = {\rm Flk}_0(A) \subseteq {\rm Bla}_0(A).$$
{\rm (iii)}\qua The exact sequence in {\rm (ii)} splits as a direct sum of
exact sequences
$$\begin{array}{l}
\cdots \to \bigoplus \limits_{2\mu} K_n(A) \to
\bigoplus \limits_\mu K_n(A) \disablesubscriptcorrection\xysavmatrix@C-10pt{\ar[r]^-{\displaystyle{0}}&}
\bigoplus \limits_\mu K_{n-1}(A) \to
\bigoplus \limits_{2\mu} K_{n-1}(A) \to \cdots,\\[2ex]
\cdots \to \widetilde{{\rm Prim}}_n(A) \to \widetilde{{\rm Sei}}_n(A) \to
\widetilde{{\rm Bla}}_n(A) \to \widetilde{{\rm Prim}}_{n-1}(A) \to \cdots.
\end{array}$$
\end{athm}
For $\mu = 1$ ${\mathcal{P}\mathit{rim}}(A)$ is the exact category of f.g.~projective
$A$--modules $P$ with a near-projection $e\co P \to P$, which is equivalent
to the product ${\mathcal{N}\mathit{il}}(A) \times {\mathcal{N}\mathit{il}}(A)$ of two copies of
the exact category ${\mathcal{N}\mathit{il}}(A)$ of f.g.~projective $A$--modules $P$
with a nilpotent endomorphism $e\co P \to P$, and
\begin{eqnarray*}
{\rm Prim}_*(A) & = & K_*({\mathcal{P}\mathit{rim}}(A)) = {\rm Nil}_*(A) \oplus {\rm Nil}_*(A),\\[1ex]
{\rm Nil}_*(A) & = & K_*({\mathcal{N}\mathit{il}}(A)) = K_*(A) \oplus \widetilde{{\rm Nil}}_*(A),\\[1ex]
\widetilde{{\rm Prim}}_*(A) & = & \widetilde{{\rm Nil}}_*(A) \oplus
\widetilde{{\rm Nil}}_*(A).
\end{eqnarray*}
Thus for $\mu = 1$ \fullref{thm4} (i) is just the splitting theorem
of Bass, Heller and Swan \cite{BHS}, \cite{B2} for $K_1(A[z,z^{-1}])$
and its generalization to the higher $K$--groups
$$K_*(A[z,z^{-1}]) = K_*(A) \oplus K_{*-1}(A) \oplus
\widetilde{{\rm Nil}}_{*-1}(A) \oplus \widetilde{{\rm Nil}}_{*-1}(A).$$
\fullref{thm4} (ii)--(iii) is new even in the case $\mu = 1$.
Let $\Sigma^{-1}A[F_\mu]$ be the noncommutative Cohn (ie universal)
localization of $A[F_\mu]$ inverting the set $\Sigma$ of the morphisms of
f.g.~projective $A[F_\mu]$--modules which induce isomorphisms of
f.g.~projective $A$--modules under the augmentation $\epsilon:
A[F_\mu] \to A$. The exact category $H(A[F_\mu],\Sigma)$
of h.d.~1 $\Sigma$--torsion $A[F_\mu]$--modules is such that
$$H(A[F_\mu],\Sigma) = {\mathcal{B}\mathit{la}}(A),~K_*(H(A[F_\mu],\Sigma)) = {\rm Bla}_*(A).$$
\begin{athm}\label{thm5}
{\rm (i)}\qua The localization exact sequence
$$K_1(A[F_\mu]) \to K_1(\Sigma^{-1}A[F_\mu])
\to K_0(H(A[F_\mu],\Sigma)) \to K_0(A[F_\mu]) \to \cdots $$
splits as a direct sum of the exact sequences
$$\begin{array}{l}
K_1(A)\oplus\bigoplus \limits_\mu K_0(A) \to K_1(A)
\disablesubscriptcorrection\xysavmatrix@C-10pt{\ar[r]^-{\displaystyle{0}}&} \bigoplus \limits_\mu K_{-1}(A) \to
K_0(A) \oplus\bigoplus \limits_\mu K_{-1}(A) \to \cdots,\\[1ex]
\widetilde{{\rm Prim}}_0(A) \to \widetilde{{\rm Sei}}_0(A)
\to \widetilde{{\rm Bla}}_0(A) \to \widetilde{{\rm Prim}}_{-1}(A) \to \cdots.
\end{array}$$
{\rm (ii)}\qua If $\Sigma^{-1}A[F_\mu]$ is stably flat (ie if
$\Tor^{A[F_\mu]}_*(\Sigma^{-1}A[F_\mu],\Sigma^{-1}A[F_\mu]) = 0$
for $* \geqslant 1$)
the exact sequences and the splitting in {\rm (i)} extend to the left,
involving the algebraic $K$--groups $K_n$ for $n \geqslant 2$, with
$$K_*(\Sigma^{-1}A[F_\mu]) = K_*(A) \oplus \widetilde{{\rm Sei}}_{*-1}(A).$$
\end{athm}
For $\mu = 1$ ${\mathcal{S}\mathit{ei}}(A)$ is the exact category ${\mathcal{E}\mathit{nd}}(A)$ of f.g.~projective
$A$--modules $P$ with an endomorphism $e\co P \to P$, and
$${\rm Sei}_*(A) = K_*({\mathcal{E}\mathit{nd}}(A)) = {\rm End}_*(A) = K_*(A) \oplus
\widetilde{{\rm End}}_*(A).$$
The special case of \fullref{thm5} (i)
$$K_1(\Sigma^{-1}A[z,z^{-1}]) = K_1(A) \oplus \widetilde{{\rm End}}_0(A)$$
is the splitting theorem of Ranicki \cite[10.21]{RHK}.
We are grateful to Pere Ara, Warren Dicks, Marco Schlichting and the
referee for helpful comments on the preprint version of the paper,
which have led to various improvements. In particular, it was Pere Ara
who pointed out that the Blanchfield $A[F_\mu]$--module category
${\mathcal{B}\mathit{la}}_{\infty}(A)$ is the same as the $F_\mu$--link module category
${\mathcal{F}\mathit{lk}}_{\infty}(A)$ of \cite{Sh2}.
\section{Combinatorial transversality for $F_\mu$--covers}
\label{combinatorial transversality}
For $\mu \geqslant 1$ let
$F_{\mu} = \langle z_1,z_2,\ldots,z_{\mu} \rangle$
be the free group with generators $z_1,z_2,\ldots,z_{\mu}$.
\subsection{$F_\mu$--covers}
\begin{definition} {\rm An \emph{$F_{\mu}$--cover} of a space $W$
is a regular covering $p\co \wwtilde{W}\to W$ with group of
covering translations $F_{\mu}$.}
\end{definition}
\indent A classifying space $BF_\mu$ for $F_\mu$--covers is a
connected space such that
$$\pi_j(BF_{\mu}) = \begin{cases}
F_{\mu}&{\rm if}~j = 1\\
0&{\rm if}~j \geqslant 2.
\end{cases}$$
The universal cover of $BF_\mu$ is an $F_\mu$--cover
$$p_{\mu}\co EF_{\mu} = \widetilde{BF}_{\mu}\to BF_{\mu}$$
with $EF_{\mu}$ a contractible space with a free $F_{\mu}$--action.
\begin{proposition}
{\rm (i)}\qua Given an $F_{\mu}$--cover $p\co \wwtilde{W} \to W$ and a
map $f\co V \to W$ there is defined a pullback square
$$\disablesubscriptcorrection\xysavmatrix@C+10pt@R+10pt{\wtilde{V} \ar[r]^-{\displaystyle\wtilde{f}}
\ar[d]_-{\displaystyle{f^*p}} &
\wwtilde{W} \ar[d]^-{\displaystyle{p}} \\
V \ar[r]^-{\displaystyle{f}} & W}$$ with
$$\begin{array}{l}
\wtilde{V} = f^*\wwtilde{W} = \bigl\{(x,y) \in V\times
\wwtilde{W}\,\vert\,
f(x) = p(y) \in W\bigr\},\\[1ex]
f^*p\co \wtilde{V} \to V;~(x,y) \mapsto x,\quad
\wtilde{f}\co \wtilde{V} \to \wwtilde{W};~(x,y) \mapsto y
\end{array}$$
such that $f^*p\co \wtilde{V} \to V$ is the pullback $F_{\mu}$--cover.
{\rm (ii)}\qua The $F_{\mu}$--covers $p\co \wwtilde{W} \to W$ of a space
$W$ are classified by the homotopy classes of maps $c\co W \to BF_{\mu}$
with
$$\begin{array}{l}
\wwtilde{W} = c^*EF_{\mu} = \bigl\{(x,y) \in W \times EF_{\mu}\,\vert\,
c(x) = [y] \in BF_{\mu}\bigr\},\\[1ex]
p(x,y) = c^*p_{\mu}(x,y) = x.
\end{array}$$
For a connected space $W$ the homotopy classes of maps $c\co W \to
BF_{\mu}$ are in one-one correspondence with the morphisms
$c_*\co \pi_1(W) \to F_{\mu}$; the connected $F_{\mu}$--covers
$\wwtilde{W}$ correspond to surjections $c_*\co \pi_1(W) \to F_{\mu}$.
\end{proposition}
\begin{proof} Standard.
\end{proof}
\subsection{The Cayley tree $G_\mu$}
We shall be working with the following explicit constructions of
$BF_{\mu}$ and $EF_{\mu}$, as well as the Cayley tree of $F_\mu$:
\begin{definition} \label{cayley}
{\rm The \emph{Cayley tree} $G_{\mu}$ is the tree with vertex set
$$G_{\mu}^{(0)} = F_{\mu}$$
and edge set
$$G_{\mu}^{(1)} = \{(g,gz_i)\,\vert\,g \in
F_{\mu},\,1 \leqslant i \leqslant \mu\} \subset G_{\mu}^{(0)} \times
G_{\mu}^{(0)}.$$
$$\Draw
\MoveTo(0,0) \Text(--$\bullet$--) \MoveTo(0,60)
\Text(--$\bullet$--) \MoveTo(0,-60) \Text(--$\bullet$--)
\MoveTo(60,0) \Text(--$\bullet$--) \MoveTo(-60,0)
\Text(--$\bullet$--) \LineAt(-90,0,90,0) \LineAt(0,-90,0,90)
\LineAt(-60,40,-60,-40) \LineAt(-40,-60,40,-60)
\LineAt(60,-40,60,40) \LineAt(-40,60,40,60) \MoveTo(-70,10)
\Text(--$z_i^{-1}$--) \MoveTo(-25,10) \Text(--$(z_i^{-1},1)$--)
\MoveTo(7,10) \Text(--$1$--) \MoveTo(35,10) \Text(--$(1,z_i)$--)
\MoveTo(70,7) \Text(--$z_i$--) \MoveTo(10,-70) \Text(--$z_j$--)
\MoveTo(16,-30) \Text(--$(1,z_j)$--) \MoveTo(20,30)
\Text(--$(z_j^{-1},1)$--) \MoveTo(12,70) \Text(--$z_j^{-1}$--)
\EndDraw$$
Define a transitive $F_{\mu}$--action on $G_\mu$
$$F_\mu \times G_\mu \to G_\mu ;~(g,x) \mapsto gx$$
with quotient the one-point union of $\mu$ circles
$$G_\mu/F_\mu = BF_\mu = \smash{\bigvee_{\mu}} S^1.$$}
\end{definition}
Let
$$I_{\mu} =
\bigcup^{\mu}_{i = 1} [e^-_i,e^+_i] \subset \mathbb{R}^{\mu}$$
with
\begin{align*}
e^+_i & = (0,\ldots,0,1,0,\ldots,0),~e^-_i = (0,\ldots,0,-1,0,\ldots,0)
\in \mathbb{R}^{\mu},\\
[e^-_i,e^+_i] & = \{(0,\ldots,0,t,0,\ldots,0)\,\vert\, -1 \leqslant t \leqslant 1\}
\subset \mathbb{R}^{\mu}.
\end{align*}
Thus $I_\mu$ is the one-point union of $\mu$ copies of the interval
$[-1,1] \subset \mathbb{R}$, identifying the $\mu$ copies of $0 \in [-1,1]$.
We regard $BF_{\mu}$ as the quotient space of $I_{\mu}$
$$BF_{\mu} = I_{\mu}/\{e^+_i \sim e^-_i\,\vert\,1 \leqslant i \leqslant \mu\} =
\bigvee_{\mu}S^1,$$
the one-point union of $\mu$ copies of the circle $S^1 = [-1,1]/(-1
\sim 1)$ in which the $\mu$ copies of $[0] \in S^1$ are
identified, with
$$e_i = [e^+_i] = [e^-_i]\neq [0] \in BF_{\mu}$$
a point in the $i^{th}$ circle. The universal cover $EF_{\mu}$ of
$BF_{\mu}$ is
$$EF_{\mu} = (F_{\mu} \times I_{\mu})/\{(g,e^+_i) \sim (gz_i,e^-_i)\,\vert\,g
\in F_{\mu},1 \leqslant i \leqslant \mu\},$$
a contractible space with a free $F_{\mu}$--action
$$F_{\mu} \times EF_{\mu} \to EF_{\mu};~(g,(h,x)) \mapsto (gh,x)$$
and covering projection
$$p_{\mu}\co EF_{\mu} \to BF_{\mu};~[g,x] \mapsto [x].$$
Define an $F_{\mu}$--equivariant homeomorphism
$\disablesubscriptcorrection\xysavmatrix{G_{\mu} \ar[r]^-{\cong} & EF_{\mu}}$
by sending the vertex $g \in \smash{G^{(0)}_{\mu}} = F_{\mu}$ to
the point $(g,0) \in EF_{\mu}$, and the edge $(g,gz_i) \in
\smash{G^{(1)}_{\mu}}$ to the line segment
$$\{(g,te^+_i)\,\vert\, 0 \leqslant t \leqslant 1\} \cup
\{(gz_i,te^-_i)\,\vert\, 0 \leqslant t \leqslant 1\} \subset
EF_{\mu}$$
with endpoints $(g,0)$, $(gz_i,0) \in EF_{\mu}$. The projection
$G_\mu \to G_\mu/F_\mu$ can thus be identified with the universal
cover $p_\mu\co EF_\mu \to BF_\mu$.
\subsection{Fundamental domains}
\begin{definition}{\rm
A \emph{fundamental domain} of an $F_{\mu}$--cover
$p\co \wwtilde{W}\to W$ is a closed subspace $U \subset
\wwtilde{W}$ such that
\begin{itemize}
\item[(a)] $F_{\mu}U = \wwtilde{W}$, or equivalently $p(U) = W$,
\item[(b)] for any $g,h \in F_{\mu}$
$$gU \cap hU = \begin{cases}
gV_i&{\rm if}~g = hz_i\\
hV_i&{\rm if}~g = hz^{-1}_i\\
gU&{\rm if}~g = h\\
\emptyset&{\rm otherwise}
\end{cases}$$
with $V_i = U \cap z_i^{-1}U$.
\end{itemize}}
\end{definition}
{\tiny
\tracingstats = 2
\newcounter{rot}
\newcounter{rot2}
\newcounter{posx}
\newcounter{posy}
\newcounter{psz}
\newcounter{pcentx}
\newcounter{pcenty}
\newcounter{subpcentx}
\newcounter{subpcenty}
\newcounter{subsubpcentx}
\newcounter{subsubpcenty}
\newcounter{ovalthk}
\newcounter{ovalthn}
\newcounter{subovalthk}
\newcounter{subovalthn}
\newcounter{subsubovalthk}
\newcounter{subsubovalthn}
\setcounter{psz}{50}
\setcounter{subsubpcentx}{\value{psz}*19/6}
\setcounter{subsubpcenty}{0}
\setcounter{subpcentx}{\value{psz}*21/10}
\setcounter{subpcenty}{0}
\setcounter{pcentx}{0}
\setcounter{pcenty}{0}
\setcounter{subsubovalthk}{\value{psz}/8}
\setcounter{subsubovalthn}{\value{psz}/10}
\setcounter{subovalthk}{\value{psz}/6}
\setcounter{subovalthn}{\value{psz}/8}
\setcounter{ovalthk}{\value{psz}/4}
\setcounter{ovalthn}{\value{psz}/5}
\begin{equation*}\label{picture}
\Draw(0.8pt,0.8pt)
\def\smallestpic{%
\setcounter{posx}{\value{subsubpcentx}-\value{subsubovalthk}}
\setcounter{posy}{\value{subsubpcenty}+\value{psz}*9/20}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(c+2-)
\setcounter{posx}{\value{subsubpcentx}-\value{subsubovalthk}}
\setcounter{posy}{\value{subsubpcenty}+\value{subsubovalthk}*5/4}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(C-+)
\setcounter{posx}{\value{subpcentx}+\value{psz}*2/3}
\setcounter{posy}{\value{subpcenty}+\value{subovalthk}}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(b+1+)
\Curve(b+1+,C-+,C-+,c+2-)
\setcounter{posx}{\value{subsubpcentx}-\value{subsubovalthk}}
\setcounter{posy}{\value{subsubpcenty}-\value{psz}*9/20}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(c-2-)
\setcounter{posx}{\value{subsubpcentx}-\value{subsubovalthk}}
\setcounter{posy}{\value{subsubpcenty}-\value{subsubovalthk}*5/4}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(C--)
\setcounter{posx}{\value{subpcentx}+\value{psz}*2/3}
\setcounter{posy}{\value{subpcenty}-\value{subovalthk}}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(b+1-)
\Curve(b+1-,C--,C--,c-2-)
\setcounter{posx}{\value{subsubpcentx}+\value{subsubovalthk}}
\setcounter{posy}{\value{subsubpcenty}+\value{psz}*9/20}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(c+2+)
\setcounter{posx}{\value{subsubpcentx}+\value{psz}*9/20}
\setcounter{posy}{\value{subsubpcenty}+\value{subsubovalthk}}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(c+1+)
\setcounter{posx}{\value{subsubpcentx}+\value{subsubovalthk}}
\setcounter{posy}{\value{subsubpcenty}+\value{subsubovalthk}*5/4}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(C++)
\Curve(c+2+,C++,C++,c+1+)
\setcounter{posx}{\value{subsubpcentx}+\value{subsubovalthk}}
\setcounter{posy}{\value{subsubpcenty}-\value{psz}*9/20}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(c-2+)
\setcounter{posx}{\value{subsubpcentx}+\value{psz}*9/20}
\setcounter{posy}{\value{subsubpcenty}-\value{subsubovalthk}}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(c+1-)
\setcounter{posx}{\value{subsubpcentx}+\value{subsubovalthk}}
\setcounter{posy}{\value{subsubpcenty}-\value{subsubovalthk}*5/4}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(C+-)
\Curve(c-2+,C+-,C+-,c+1-)
\setcounter{posx}{\value{subsubpcentx}+\value{psz}*9/20}
\setcounter{posy}{\value{subsubpcenty}}
\MoveTo(\value{posx},\value{posy})
\DrawOval(\value{subsubovalthn},\value{subsubovalthk})
\setcounter{posx}{\value{subsubpcentx}}
\setcounter{posy}{\value{subsubpcenty}-\value{psz}*9/20}
\MoveTo(\value{posx},\value{posy})
\DrawOval(\value{subsubovalthk},\value{subsubovalthn})
\setcounter{posx}{\value{subsubpcentx}}
\setcounter{posy}{\value{subsubpcenty}+\value{psz}*9/20}
\MoveTo(\value{posx},\value{posy})
\DrawOval(\value{subsubovalthk},\value{subsubovalthn})
}
\def\quartpic{
\setcounter{posx}{\value{pcentx}-\value{psz}}
\setcounter{posy}{\value{pcenty}}
\MoveTo(\value{posx},\value{posy})
\DrawOval(\value{ovalthn},\value{ovalthk})
\setcounter{posx}{\value{pcentx}+\value{psz}}
\setcounter{posy}{\value{pcenty}+\value{ovalthk}}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(a+1+)
\setcounter{posx}{\value{pcentx}+\value{ovalthk}}
\setcounter{posy}{\value{pcenty}+\value{psz}}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(a+2+)
\setcounter{posx}{\value{pcentx}+\value{ovalthk}}
\setcounter{posy}{\value{pcenty}+\value{ovalthk}}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(A++)
\Curve(a+1+,A++,A++,a+2+)
\setcounter{posx}{\value{pcentx}+\value{psz}}
\setcounter{posy}{\value{pcenty}-\value{ovalthk}}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(a+1-)
\setcounter{posx}{\value{pcentx}+\value{ovalthk}}
\setcounter{posy}{\value{pcenty}-\value{psz}}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(a-2+)
\setcounter{posx}{\value{pcentx}+\value{ovalthk}}
\setcounter{posy}{\value{pcenty}-\value{ovalthk}}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(A+-)
\Curve(a+1-,A+-,A+-,a-2+)
\setcounter{posx}{\value{subpcentx}-\value{subovalthk}}
\setcounter{posy}{\value{subpcenty}+\value{psz}*2/3}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(b+2-)
\setcounter{posx}{\value{subpcentx}-\value{subovalthk}}
\setcounter{posy}{\value{subpcenty}+\value{subovalthk}*5/4}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(B-+)
\Curve(a+1+,B-+,B-+,b+2-)
\setcounter{posx}{\value{subpcentx}-\value{subovalthk}}
\setcounter{posy}{\value{subpcenty}-\value{psz}*2/3}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(b-2-)
\setcounter{posx}{\value{subpcentx}-\value{subovalthk}}
\setcounter{posy}{\value{subpcenty}-\value{subovalthk}*5/4}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(B--)
\Curve(a+1-,B--,B--,b-2-)
\setcounter{posx}{\value{subpcentx}+\value{subovalthk}}
\setcounter{posy}{\value{subpcenty}+\value{psz}*2/3}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(b+2+)
\setcounter{posx}{\value{subpcentx}+\value{psz}*2/3}
\setcounter{posy}{\value{subpcenty}+\value{subovalthk}}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(b+1+)
\setcounter{posx}{\value{subpcentx}+\value{subovalthk}}
\setcounter{posy}{\value{subpcenty}+\value{subovalthk}*5/4}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(B++)
\Curve(b+2+,B++,B++,b+1+)
\setcounter{posx}{\value{subpcentx}+\value{subovalthk}}
\setcounter{posy}{\value{subpcenty}-\value{psz}*2/3}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(b-2+)
\setcounter{posx}{\value{subpcentx}+\value{psz}*2/3}
\setcounter{posy}{\value{subpcenty}-\value{subovalthk}}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(b+1-)
\setcounter{posx}{\value{subpcentx}+\value{subovalthk}}
\setcounter{posy}{\value{subpcenty}-\value{subovalthk}*5/4}
\MoveTo(\value{posx},\value{posy})
\MarkLoc(B+-)
\Curve(b-2+,B+-,B+-,b+1-)
\setcounter{posx}{\value{subpcentx}+\value{psz}*2/3}
\setcounter{posy}{\value{subpcenty}}
\MoveTo(\value{posx},\value{posy})
\DrawOval(\value{subovalthn},\value{subovalthk})
\setcounter{posx}{\value{subpcentx}}
\setcounter{posy}{\value{subpcenty}-\value{psz}*2/3}
\MoveTo(\value{posx},\value{posy})
\DrawOval(\value{subovalthk},\value{subovalthn})
\setcounter{posx}{\value{subpcentx}}
\setcounter{posy}{\value{subpcenty}+\value{psz}*2/3}
\MoveTo(\value{posx},\value{posy})
\DrawOval(\value{subovalthk},\value{subovalthn})
}
\setcounter{rot}{0} \Do(1,4){
\MoveTo(\value{pcentx},\value{pcenty})
\setcounter{rot2}{\value{rot}+90}
\RotatedAxes(\value{rot},\value{rot2}) \quartpic \smallestpic
\EndRotatedAxes \addtocounter{rot}{90} }
\MoveTo(\value{subpcentx},-\value{subpcentx}) \RotatedAxes(90,180)
\smallestpic \EndRotatedAxes
\MoveTo(\value{subpcentx},\value{subpcentx}) \RotatedAxes(180,270)
\smallestpic \EndRotatedAxes
\MoveTo(-\value{subpcentx},\value{subpcentx}) \RotatedAxes(270,0)
\smallestpic \EndRotatedAxes
\MoveTo(-\value{subpcentx},-\value{subpcentx}) \RotatedAxes(0,90)
\smallestpic \EndRotatedAxes
\MoveTo(-\value{subpcentx},\value{subpcentx}) \RotatedAxes(0,90)
\smallestpic \EndRotatedAxes
\MoveTo(\value{subpcentx},\value{subpcentx}) \RotatedAxes(270,0)
\smallestpic \EndRotatedAxes
\MoveTo(\value{subpcentx},-\value{subpcentx})
\RotatedAxes(180,270) \smallestpic \EndRotatedAxes
\MoveTo(-\value{subpcentx},-\value{subpcentx})
\RotatedAxes(90,180) \smallestpic \EndRotatedAxes
\MoveTo(0,0) \Text(--\mbox{$U$}--)
\MoveTo(-50,0) \Text(--\mbox{$V_i$}--)
\MoveTo(50,0) \Text(--\mbox{$z_iV_i$}--)
\MoveTo(0,-50) \Text(--\mbox{$V_j$}--)
\MoveTo(0,50) \Text(--\mbox{$z_jV_j$}--)
\MoveTo(105,0) \Text(--\mbox{$z_iU$}--)
\MoveTo(-105,0) \Text(--\mbox{$z^{-1}_iU$}--)
\MoveTo(0,105) \Text(--\mbox{$z_jU$}--)
\MoveTo(0,-105) \Text(--\mbox{$z^{-1}_jU$}--)
\EndDraw
\end{equation*}
}
Thus $U \subset \wwtilde{W}$ is sufficiently large for the translates
$gU \subset \wwtilde{W}$ ($g \in F_\mu$) to cover $\wwtilde{W}$, but
sufficiently small for the overlaps $gU \cap hU$ to be non-empty
only if $g^{-1}h = 1$ or $z_i$ or $z^{-1}_i$.
\begin{example} \label{funex}
(i)\qua The subspace $(1,I_\mu) \subset EF_\mu$ is a fundamental
domain of the universal cover $p_\mu\co EF_\mu \to BF_\mu$.
(ii)\qua Let $G'_\mu$ be the barycentric subdivision of the Cayley
tree $G_\mu$, the tree with
$$\begin{array}{l}
(G'_{\mu})^{(0)} = G_\mu^{(0)} \cup G_\mu^{(1)},\\[1ex]
(G'_{\mu})^{(1)} = \{(h,(g,gz_i))\,\vert\, h = g~{\rm or}~gz_i\}
\subset (G'_\mu)^{(0)} \times (G'_{\mu})^{(0)}.
\end{array}$$
The $F_\mu$--equivariant homeomorphism $G_\mu = G'_\mu\cong EF_\mu$
sends the vertex $(g,gz_i) \in (G'_\mu)^{(0)}$ to $(g,e^+_i) \in
EF_{\mu}$. The subgraph $U_{\mu} \subset G'_{\mu}$ defined by
$$\begin{array}{l}
U_{\mu}^{(0)} = \{1\} \cup \{(1,z_i)\} \cup
\{(z_i^{-1},1)\}\\[1ex]
U_\mu^{(1)} = \{(1,(1,z_i))\} \cup \{(z_i^{-1},(z_i^{-1},1))\}
\end{array}$$
is the fundamental domain of the cover $G_\mu \to G_\mu/F_\mu$
corresponding to $(1,I_\mu) \subset EF_\mu$ under the
$G_\mu$--equivariant homeomorphism $G_\mu\cong EF_\mu$.
{\small
$$\Draw
\MoveTo(0,0) \Text(--$\bullet$--)
\MoveTo(0,60) \Text(--$\bullet$--)
\MoveTo(0,-60) \Text(--$\bullet$--)
\MoveTo(60,0) \Text(--$\bullet$--)
\MoveTo(-60,0)\Text(--$\bullet$--)
\MoveTo(0,30) \Text(--$\bullet$--)
\MoveTo(0,-30) \Text(--$\bullet$--)
\MoveTo(30,0) \Text(--$\bullet$--)
\MoveTo(-30,0)\Text(--$\bullet$--)
\LineAt(0,-90,0,-30)
\LineAt(0,30,0,90)
\LineAt(-90,0,-30,0)
\LineAt(30,0,90,0)
{\PenSize(1.5pt)
\LineAt(-30,0,30,0)
\LineAt(0,-30,0,30)}
\LineAt(-60,40,-60,-40) \LineAt(-40,-60,40,-60)
\LineAt(60,-40,60,40) \LineAt(-40,60,40,60) \MoveTo(-70,10)
\Text(--$z_i^{-1}$--) \MoveTo(-25,10) \Text(--$(z_i^{-1},1)$--)
\MoveTo(7,10) \Text(--$1$--) \MoveTo(35,10) \Text(--$(1,z_i)$--)
\MoveTo(70,7) \Text(--$z_i$--) \MoveTo(10,-70) \Text(--$z_j$--)
\MoveTo(16,-30) \Text(--$(1,z_j)$--) \MoveTo(20,30)
\Text(--$(z_j^{-1},1)$--) \MoveTo(10,70) \Text(--$z_j^{-1}$--)
\EndDraw$$
}
\end{example}
\begin{proposition} \label{fund}
{\rm (i)}\qua Given an $F_\mu$--cover $p\co \wwtilde{W} \to W$ and a map
$f\co V \to W$ let $f^*p\co \wtilde{V} = f^*\wwtilde{W} \to V$ be the
pullback $F_\mu$--cover. If $U \subset \wwtilde{W}$ is a
fundamental domain of $p$ then
$$\wtilde{f}^{-1}(U) = \{(x,y)\,\vert\, x \in V,y \in U,f(x) = p(y) \in W\}
\subset \wtilde{V}$$
is a fundamental domain of $f^*p$.
{\rm (ii)}\qua Every $F_{\mu}$--cover $p\co \wwtilde{W} \to W$ has
fundamental domains.
\end{proposition}
\begin{proof} (i)\qua By construction.
(ii)\qua Apply (i), using the fundamental domain $U_\mu \subset
G'_\mu = EF_\mu$ for the cover
$$p_\mu\co EF_{\mu} \to EF_\mu/F_\mu = BF_\mu$$
given by \fullref{funex}, noting that
$$p = c^*p_{\mu}\co \wwtilde{W} = c^*EF_{\mu} \to W$$
is the pullback of the universal $F_{\mu}$--cover
$p_{\mu}\co EF_{\mu} \to BF_{\mu}$ along a classifying map $c\co W \to BF_{\mu}$
$$\disablesubscriptcorrection\xysavmatrix@C+10pt@R+10pt{\wwtilde{W} \ar[r]^-{\displaystyle\wtilde{c}}
\ar[d]_-{\displaystyle{p}} &
EF_{\mu} \ar[d]^-{\displaystyle{p_{\mu}}} \\
W \ar[r]^-{\displaystyle{c}} & BF_{\mu}}$$
The inverse image of $U_{\mu} \subset EF_{\mu}$
$$U = \wtilde{c}^{-1}(U_{\mu}) \subset \wwtilde{W}$$
is a fundamental domain of $c\co \wwtilde{W} \to W$.
\end{proof}
\subsection{Combinatorial transversality}
If $p\co \wwtilde{W} \to W$ is an $F_{\mu}$--cover of a space $W$
with an additional structure such as a manifold or finite $CW$
complex, we should like to have fundamental domains $U \subset
\wwtilde{W}$ with the additional structure. For manifolds this
is achieved by choosing a classifying map $c\co W \to BF_{\mu}$
transverse at $\{e_1,e_2,\ldots,e_{\mu}\} \subset BF_{\mu}$ -- see
\fullref{manifold transversality} below for a more detailed
discussion. For a finite $CW$ complex $W$ we shall develop a
combinatorial version of transversality,
constructing finite subcomplexes $X \subset X(\infty)$
of the Borel construction
$X(\infty) = \wwtilde{W}\times_{F_{\mu}}G_{\mu}$, such that the
projection $f(\infty)\co W(\infty) \to W$ restricts to
a simple homotopy equivalence $f\co X \to W$ such that the
pullback $F_{\mu}$--cover $\widetilde{X} = f^*\wwtilde{W} \to X$
has a fundamental domain $U \subset \widetilde{X}$ which is a
finite subcomplex.
\begin{proposition} \label{Borel} For any $F_{\mu}$--cover
$p\co \wwtilde{W} \to W$ let $F_{\mu}$ act diagonally on
$\wwtilde{W} \times G_{\mu}$
$$F_{\mu} \times (\wwtilde{W} \times G_{\mu}) \to (\wwtilde{W} \times G_{\mu});~
(g,(x,y)) \mapsto (gx,gy).$$
{\rm (i)}\qua The map
$$\pi\co X = \wwtilde{W}\times_{F_{\mu}}G_{\mu} \to W;~
[x,g] \mapsto p(x)$$
is the projection of a fibration
$$\disablesubscriptcorrection\xysavmatrix{G_{\mu} \ar[r] & X \ar[r]^-{\displaystyle{\pi}} & W}$$
with contractible point inverses;
for each $x \in \wwtilde{W}$ there is defined a homeomorphism
$$G_{\mu} \to \pi^{-1}p(x);~g \mapsto [x,g].$$
In particular, $\pi$ is a homotopy equivalence.
{\rm (ii)}\qua The pullback $F_{\mu}$--cover of $X$
$$\pi^*p\co \widetilde{X} = p^*\wwtilde{W} = \wwtilde{W}
\times G_{\mu} \to X = \wwtilde{W} \times_{F_{\mu}} G_{\mu}$$
has fundamental domain $\wwtilde{W} \times U \subset
\widetilde{X} = \wwtilde{W} \times G_{\mu}$, with $U \subset
G_\mu$ any fundamental domain.
\end{proposition}
\begin{proof} Standard.
\end{proof}
\begin{definition} \label{fsplit} {\rm
(i)\qua An \emph{$F_{\mu}$--splitting} $(X,Y,Z,h)$ of a space $W$ is
a homeomorphism $h\co X \to W$ from a space with a decomposition
$$X = Y \times [-1,1] \cup_{Y \times \{-1,1\}}Z$$
with $Y = Y_1\sqcup Y_2 \sqcup \ldots \sqcup Y_{\mu}$ the disjoint union
of spaces $Y_1,Y_2,\ldots,Y_{\mu}$ and
$Y \times [-1,1]$ attached to $Z$ along maps
$$\alpha^-_i\co Y_i \times \{-1\} \to Z,~
\alpha^+_i\co Y_i \times \{1\} \to Z.$$
(ii)\qua An $F_{\mu}$--splitting $(X,Y,Z,h)$ of a connected space $W$
is \emph{connected} if each of $Y_1,Y_2,\ldots,Y_{\mu},Z$ is
non-empty and connected.}
\end{definition}
\begin{proposition} \label{fcover}
Let $W$ be a space with an $F_{\mu}$--splitting $(X,Y,Z,h)$.
{\rm (i)}\qua The $F_\mu$--splitting determines an $F_{\mu}$--cover
$p\co \wwtilde{W} \to W$ with
$$\begin{array}{l}
\wwtilde{W} = \big(F_{\mu} \times (Y \times [-1,1] \sqcup
Z)\big)/\sim,\\[1ex]
\hspace*{50pt}(g,y_i,1) \sim (z_ig,\alpha^+_i(y_i,1)),\\[1ex]
\hspace*{50pt}(g,y_i,-1) \sim (g,\alpha^-_i(y_i,-1))\quad
(g \in F_{\mu}, y_i \in Y_i,1 \leqslant i \leqslant \mu),\\[1ex]
p\co \wwtilde{W} \to W;~(g,x) \mapsto [h(x)].
\end{array}$$
The subspace
$$Z' = (1,Y\times [0,1]) \cup (1,Z) \cup \bigcup\limits^{\mu}_{i = 1}
(z_i,Y_i \times [-1,0]) \subset \wwtilde{W}$$
is a fundamental domain of $p\co \wwtilde{W} \to W$.
{\rm (ii)}\qua If there exists a homeomorphism $\phi\co Z' \to Z$ such that
$$\phi(1,y_i,0) = \alpha^-_i(y_i,-1),~\phi(z_i,y_i,0) =
\alpha^+_i(y_i,1)\quad
(y_i \in Y_i,1 \leqslant i \leqslant \mu)$$
the identification space
$$\wwtilde{W}' = \big(F_{\mu} \times Z\big)/(g,\alpha^-_i(y_i))
\sim (z_ig,\alpha^+_i(y_i))$$
is such that there is defined a homeomorphism
$$(1,\phi)\co \wwtilde{W} \to \wwtilde{W}';~(g,x) \mapsto (g,\phi(x))$$
so that
$$p' = p(1,\phi)^{-1}\co \wwtilde{W}' \to W;~(g,x) \mapsto p\phi^{-1}(x)$$
is an $F_{\mu}$--cover of $W$ which is isomorphic to $p\co \wwtilde{W} \to W$,
with fundamental domain
$$(1,\phi)(Z') = (1,Z) \subset \wwtilde{W}'.$$
{\rm (iii)}\qua The fundamental group of a connected space $W$ with a
connected $F_{\mu}$--splitting $(X,Y,Z,h)$ is an amalgamated free
product
$$\pi_1(W) = \pi_1(Z)*F_{\mu}/\{\alpha^+_i(g_i)z_i = z_i\alpha^-_i(g_i)\,\vert\,
g_i \in \pi_1(Y_i),1 \leqslant i \leqslant \mu\}.$$
The surjection $\pi_1(W) \to F_{\mu}$ is induced by a map $c\co W \to
BF_{\mu}$ sending $h(Y_i \times \{0\})\subset W$ to $\{e_i\} \subset
BF_{\mu}$. The surjection $\pi_1(W) \to F_{\mu}$ classifies the
connected $F_{\mu}$--cover $p\co \wwtilde{W} \to W$ in {\rm (i)}.
\end{proposition}
\begin{proof} (i) and (ii) follow by construction.
(iii) follows from the Seifert--van Kampen theorem and obstruction theory.
\end{proof}
\begin{example} \label{J}
Define an $F_{\mu}$--splitting $(H_\mu,\{1,2,\ldots,\mu\},I_{\mu},f)$ of
$BF_{\mu}$ by
$$\begin{array}{l}
H_{\mu} = \{1,2,\ldots,\mu\} \times [-1,1] \cup_{(i,1)\sim
e^+_i,(i,-1)\sim e^-_i}I_{\mu},\\[1ex]
f\co H_{\mu} \to BF_{\mu};~
\begin{cases}
(i,t) \mapsto [(1-t/2)e^+_i]&{\rm for}~0 \leqslant t \leqslant 1\\[1ex]
(i,t) \mapsto [(1+t/2)e^-_i]&{\rm for}~-1 \leqslant t \leqslant 0\\[1ex]
u \mapsto u/2&{\rm for}~u \in I_{\mu}
\end{cases}
\end{array}$$
with
$$f(i,0) = e_i,~f(i,1) = e^+_i/2,~f(i,-1) = e^-_i/2.$$
The corresponding $F_{\mu}$--cover of $BF_{\mu}$ is the universal
$F_\mu$--cover $\widetilde{BF}_{\mu} = G_{\mu} \to BF_{\mu}$, with
fundamental domain $I_{\mu} = (1,I_{\mu}) \subset G_{\mu}$.
Note that $f(I_{\mu}) = J_{\mu}$, with $J_{\mu} \subset I_{\mu}$ the
homeomorphic copy of $I_{\mu}$ defined by
$$J_{\mu} = \{(0,\ldots,0,t,0,\ldots,0) \in I_{\mu}\,\vert\,
-1/2 \leqslant t \leqslant 1/2\}.$$
\end{example}
A subspace $Y \subset X$ is \emph{collared} if the inclusion $i\co Y \to X$
extends to an embedding $j\co Y\times [0,1] \to X$, with $i(y) = j(y,0)\in X$
for $y \in Y$. In particular, $\partial Z \subset Z$ is collared,
for any manifold with boundary $(Z,\partial Z)$.
\begin{example} \label{manifold transversality}
Use the $F_{\mu}$--splitting $(H_{\mu},\{1,2,\ldots,\mu\},I_{\mu},f)$
of $BF_{\mu}$ given by \fullref{J} to identify
$$BF_{\mu} = H_{\mu} = \{1,2,\ldots,\mu\} \times
[-1,1]\cup_{\{1,2,\ldots,\mu\}\times \{-1,1\}}I_{\mu}.$$
If $p\co \widetilde{X} \to X$ is an $F_{\mu}$--cover of a manifold
$X$ it is possible to choose a classifying map
$$c\co X \to BF_{\mu} = \{1,2,\ldots,\mu\} \times
[-1,1]\cup_{\{1,2,\ldots,\mu\}\times \{-1,1\}}I_{\mu}$$
which is transverse regular at $\{e_1,e_2,\ldots,e_{\mu}\} \subset
BF_{\mu}$, with the inverse images of $e_i = (i,0) \in BF_{\mu}$
disjoint framed codimension--1 submanifolds
$$Y_i = c^{-1}(e_i) \subset X\quad(1 \leqslant i \leqslant \mu).$$
Cutting $X$ along
$$Y = c^{-1}\{e_1,e_2,\ldots,e_\mu\} =
Y_1 \sqcup Y_2 \sqcup \ldots \sqcup Y_{\mu} \subset X$$
there is obtained an $F_{\mu}$--splitting $(X,Y,Z,{\rm id.})$ of $X$, so that
$$X = Y \times [-1,1] \cup_{Y \times \{-1,1\}}Z$$
with $Y = Y\times \{0\} \subset X$ a framed codimension--1
submanifold, and $Z = c^{-1}(I_{\mu}) \subset X$ a codimension--0
submanifold with
$$\alpha^+_i\co Y_i \times \{1\} \to Z,\quad \alpha^-_i\co Y_i \times \{-1\} \to Z$$
components of the inclusion of the boundary $\partial Z = Y \times \{-1,1\} \subset Z$.
Since $\partial Z \subset Z$ is collared
the fundamental domain of the $F_{\mu}$--cover $\widetilde{X} = c^*G_{\mu}$
$$Z' = (1,Y\times [0,1]) \cup (1,Z) \cup \bigcup\limits^{\mu}_{i = 1}
(z_i,Y_i \times [-1,0]) \subset \widetilde{X}$$
is such that there exists a homeomorphism $\phi\co Z' \to Z$ with
$$\phi(1,y_i,0) = \alpha^-_i(y_i,-1),~\phi(z_i,y_i,0) =
\alpha^+_i(y_i,1)\quad
(y_i \in Y_i,1 \leqslant i \leqslant \mu).$$
Thus by \fullref{fcover} (ii) $p\co \widetilde{X} \to X$ is isomorphic
to the $F_{\mu}$--cover $p'\co \widetilde{X}' \to X$ with
$$\begin{array}{l}
\widetilde{X}' = \big(F_{\mu} \times Z\big)/(g,\alpha^-_i(y_i))
\sim (z_ig,\alpha^+_i(y_i)),\\[1ex]
p' = p(1,\phi)^{-1}\co \widetilde{X}' \to X;~(g,x) \mapsto p\phi^{-1}(x).
\end{array}$$
If $X$ and $\widetilde{X}$ are connected it is possible to choose $c$ such
that each $Y_i = p^{-1}(e_i)$ is connected, with
$$p_* = p(Y,Z)_*\co \pi_1(X) \to F_{\mu}.$$
\end{example}
\begin{definition}
(i)\qua A \emph{homotopy $F_{\mu}$--splitting} $(X,Y,Z,h)$ of a space $W$ is
a homotopy equivalence $h\co X \to W$ from a space with an
$F_\mu$--splitting $(X,Y,Z,1)$, so that
$$X = Y \times [-1,1] \cup_{Y \times \{-1,1\}}Z,~
Y = Y_1 \sqcup Y_2 \sqcup \ldots \sqcup Y_{\mu}.$$
(ii)\qua A homotopy $F_{\mu}$--splitting $(X,Y,Z,h)$ of a finite $CW$
complex $W$ is \emph{simple} if $X$ is a finite $CW$ complex,
$Y_1,Y_2,\ldots,Y_{\mu},Z \subset X$ are subcomplexes and $h\co W \to X$ is a
simple homotopy equivalence.
\end{definition}
\begin{example}
Any finite $CW$ complex $W$ with an $F_{\mu}$--cover
$\wwtilde{W}\to W$ admits simple homotopy $F_{\mu}$--splittings $(X,Y,Z,h)$\,:
embed $W \subset S^N$ ($N$ large) with closed regular
neighbourhood $(X,\partial X)$ and apply the manifold
transversality of \fullref{manifold transversality} to
the $F_\mu$--cover $\widetilde{X} \simeq \wwtilde{W} \to W \simeq X$.
\end{example}
Working as in Ranicki \cite{RAC} we shall now develop a
combinatorial transversality construction of simple homotopy
$F_{\mu}$--splittings of $W$ using finite subcomplexes of
the Borel construction (\fullref{Borel})
$\wwtilde{W}\times_{F_{\mu}}G_{\mu}$, as follows.
\begin{definition} \label{canonical}
The \emph{canonical homotopy $F_{\mu}$--splitting}
$(X(\infty),Y(\infty),Z(\infty),h(\infty))$ of a space $W$
with an $F_{\mu}$--cover $p\co \wwtilde{W} \to W$ is given by
$$X(\infty) = Y(\infty) \times [-1,1] \cup_{Y(\infty)\times
\{-1,1\}}Z(\infty)$$
with
$$\begin{array}{l}
\alpha(\infty)^+_i\co Y(\infty)_i = \wwtilde{W} \to
Z(\infty) = \wwtilde{W}\times I_{\mu};~
x \mapsto (z_ix,e^+_i),\\[1ex]
\alpha(\infty)^-_i\co Y(\infty)_i = \wwtilde{W} \to
Z(\infty) = \wwtilde{W}\times I_{\mu};~ x \mapsto
(x,e^-_i),\\[1ex]
h(\infty)\co X(\infty) \to W;~(x,y) \mapsto p(x).
\end{array}$$
The map $h(\infty)$ is a homotopy equivalence since it is the
composite
$$h(\infty) = \pi\circ f\co X(\infty) \disablesubscriptcorrection\xysavmatrix{\ar[r]^-{\displaystyle{f}}&}
\wwtilde{W} \times_{F_{\mu}}G_{\mu}
\disablesubscriptcorrection\xysavmatrix{\ar[r]^-{\displaystyle{\pi}}&} W$$
of the homeomorphism
$$f\co X(\infty) \to \wwtilde{W} \times_{F_{\mu}}G_{\mu};~
\begin{cases}
(x,i,t) \mapsto (x,(1-t/2)e^+_i)&{\rm for}~0 \leqslant t \leqslant 1\\[1ex]
(x,i,t) \mapsto (z_ix,(1+t/2)e^-_i)&{\rm for}~-1 \leqslant t \leqslant 0\\[1ex]
(x,u) \mapsto (x,u/2)&{\rm for}~u \in I_{\mu}
\end{cases}$$
and the homotopy equivalence
$$\pi\co X = \wwtilde{W}\times_{F_{\mu}}G_{\mu} \to W$$
given by \fullref{Borel}. For every $y \in G_{\mu}$ there
is a unique $g \in F_{\mu}$ such that $gy \in I_{\mu} \backslash
\{e^+_1,e^+_2,\ldots,e^+_{\mu}\}$, so that either $gy = te^+_i$ with
$0 \leqslant t <1$, or $gy = te^-_i$ with $0 \leqslant t \leqslant
1$, and
$$\begin{array}{l}
f^{-1}\co \wwtilde{W} \times_{F_{\mu}}G_{\mu} \to X(\infty)~:\\[1ex]
[x,y] \mapsto
\begin{cases}
(gx,i,2(1-t))&{\rm if}~gy = te^+_i~{\rm with}~1/2 \leqslant t < 1\\[1ex]
(z^{-1}_igx,i,2(t-1))&{\rm if}~gy = te^-_i~{\rm with}~1/2 \leqslant t \leqslant 1\\[1ex]
(gx,gy)&{\rm if}~2gy \in I_{\mu}~(\hbox{\rm ie if $-1/2
\leqslant t \leqslant 1/2$}).
\end{cases}
\end{array}$$
\end{definition}
\begin{proposition}\label{xv}
Given a space $W$ with $F_{\mu}$--cover $p\co \wwtilde{W} \to W$ and a
subspace $V \subseteq \wwtilde{W}$ let
$$X(V) = Y(V) \times [-1,1] \cup_{Y(V) \times \{-1,1\}}Z(V)
\subseteq X(\infty)$$
with
$$\begin{array}{l}
\alpha(V)^+_i\co
Y(V)_i = V \cap z_i^{-1}V \to Z(V) = V \times I_{\mu};~
x \mapsto (z_ix,e^+_i),\\[1ex]
\alpha(V)^-_i\co Y(V)_i = V \cap z_i^{-1}V \to Z(V) = V \times
I_{\mu};~ x \mapsto (x,e^-_i),
\end{array}$$
and set
$$h(V) = h(\infty)\vert\co X(V) \to W;~(x,t) \mapsto p(x).$$
{\rm (i)}\qua For any $x \in V$
$$\begin{array}{l}
h(V)^{-1}(p(x)) = \{(x,y) \in \wwtilde{W}\times_{F_{\mu}}G_\mu\,\vert\,
y \in G_{\mu}(V,x)\}\\[1ex]
\hphantom{h(V)^{-1}(p(x))~} = \{x\} \times G_{\mu}(V,x)
\subseteq X(V) \subseteq X(\infty) = \wwtilde{W}\times_{F_{\mu}}G_\mu
\end{array}$$
with $G_{\mu}(V,x) \subseteq G_{\mu}$ the subgraph defined by
$$\begin{array}{l}
G_{\mu}(V,x)^{(0)} = \{g \in F_{\mu}\,\vert\,gx \in V\}
\subseteq G_{\mu}^{(0)} = F_{\mu},\\[1ex]
G_{\mu}(V,x)^{(1)} = \{(i,g) \,\vert\,gx,gz_ix \in V\}
\subseteq G_{\mu}^{(1)} = \{1,2,\ldots,\mu\} \times F_{\mu}.
\end{array}$$
{\rm (ii)}\qua The image of $h(V)$ is
$$h(V)(X(V)) = p(V) \subseteq W,$$
so that $h(V)$ is surjective if and only if $p(V) = W$, if and only if
$\bigcup_{g \in F_{\mu}}gV = \wwtilde{W}$.
\end{proposition}
\begin{proof} By construction.
\end{proof}
In particular, if $V = \wwtilde{W}$ then
$$(X(V),Y(V),Z(V),h(V)) = (X(\infty),Y(\infty),Z(\infty),h(\infty))$$
and $h(V)\co X(V) = X(\infty) \to W$ is a homotopy equivalence (since
it has contractible point inverses).
\begin{thm}[Combinatorial transversality]
\label{combtrans}
Let $W$ be a connected finite $CW$
complex with a connected $F_{\mu}$--cover $p\co \wwtilde{W}\to W$.
The canonical homotopy $F_{\mu}$--splitting
$(X(\infty),Y(\infty),Z(\infty),h(\infty))$ of $W$ is a union
$$(X(\infty),Y(\infty),Z(\infty),h(\infty)) = \bigcup\limits_{\{V\}}
(X(V),Y(V),Z(V),h(V))$$
of simple homotopy $F_{\mu}$--splittings $(X(V),Y(V),Z(V),h(V))$ of $W$,
with $\{V\}$ a collection of finite subcomplexes $V \subset \wwtilde{W}$
such that
$$p(V) = W,\qquad \bigcup\limits_{\{V\}} V = \wwtilde{W}.$$
In particular, there exist simple homotopy $F_{\mu}$--splittings of $W$.
\end{thm}
\begin{proof} Let
$$W = \bigcup D^0 \cup \bigcup D^1 \cup \ldots \cup \bigcup D^n$$
be the given cell structure of $W$, with skeleta
$$W^{(r)} = \bigcup D^0 \cup \bigcup D^1 \cup \ldots \cup \bigcup D^r.$$
The characteristic maps $D^r \to W$ of the $r$--cells restrict to embeddings
$D^r\backslash S^{r-1}\subset W$ on the interiors, and as a set
$W$ is the disjoint union of the interiors
$$W = \bigsqcup D^0 \sqcup \bigsqcup (D^1\backslash S^0) \sqcup
\ldots \sqcup \bigsqcup (D^n\backslash S^{n-1}).$$
Choose a lift of each $r$--cell $D^r$ in $W$ to an $r$--cell
$\widetilde{D}^r$ in $\wwtilde{W}$, so that
$$\wwtilde{W} = \bigcup\limits_{g \in F_\mu}\bigcup g\widetilde{D}^0 \cup
\bigcup\limits_{g \in F_\mu}\bigcup g\widetilde{D}^1 \cup \ldots \cup
\bigcup\limits_{g \in F_\mu}\bigcup g\widetilde{D}^n.$$
Write $\phi\co S^r \to W^{(r)}$ for the attaching maps of the
$(r{+}1)$--cells in $W$, and let $\widetilde{\phi}\co S^r \to \wwtilde{W}^{(r)}$
be the attaching maps of the chosen lifted $(r{+}1)$--cells in $\wwtilde{W}$.
For any subtree $T_n \subseteq G_{\mu}$ there exists a sequence
of subtrees $T_r \subseteq G_{\mu}$ for $r = n-1,n-2,\ldots,0$
such that
$$\widetilde{\phi}(S^r) \subseteq \wwtilde{W}^{(r-1)} \cup
\bigcup\limits_{g_r \in T^{(0)}_r}g_r\widetilde{D}^r.\eqno{(*)}$$
The sequence $T = (T_n,T_{n-1},\ldots,T_0)$ determines a subcomplex
$$V\langle T \rangle = \bigcup\limits_{g_0 \in T^{(0)}_0}\bigcup g_0\widetilde{D}^0 \cup
\bigcup\limits_{g_1 \in T^{(0)}_1}\bigcup g_1\widetilde{D}^1 \cup \ldots \cup
\bigcup\limits_{g_n \in T^{(0)}_n}\bigcup g_n\widetilde{D}^n \subseteq
\wwtilde{W}$$
such that $p(V\langle T \rangle) = W$.
The map $h(V\langle T \rangle)\co X(V\langle T \rangle) \to W$ constructed in
\fullref{xv} is surjective, with contractible point inverses
$$h(V\langle T \rangle)^{-1}(p(x)) = G_{\mu}(V,x) = T_r\hskip10pt
(p(x) \in D^r\backslash S^{r-1} \subset W),$$
so that it is a homotopy equivalence and
$(X(V\langle T \rangle),Y(V\langle T \rangle),Z(V\langle T \rangle),
h(V\langle T \rangle))$ is a homotopy $F_\mu$--splitting of $W$.
For the maximal sequence $T = (G_\mu,G_\mu,\ldots,G_\mu)$
$V\langle T\rangle = \wwtilde{W}$ and we have the canonical homotopy
$F_\mu$--splitting $(X(\infty),Y(\infty),
Z(\infty),\allowbreak h(\infty))$ of $W$. Any
finite subtree $T_n \subset G_\mu$ can be used to start a sequence
$T = (T_n,T_{n-1},\ldots,T_0)$ of finite subtrees $T_r \subset G_\mu$
satisfying $(*)$, since for each $r = n,n-1,\ldots,1$ the $r$--cells
$\widetilde{D}^r \to \wwtilde{W}$ are attached to a finite subcomplex
of the $(r{-}1)$--skeleton $\wwtilde{W}^{(r-1)}$. For a sequence $T$ of
finite subtrees $(X(V\langle T \rangle),Y(V\langle T
\rangle),\allowbreak Z(V\langle T \rangle),h(V\langle T \rangle))$ is a
simple homotopy $F_{\mu}$--splitting of $W$. Finally, note that $G_\mu$
is a union of finite subtrees $T_n \subset G_\mu$, so that
$(F_{\mu},F_{\mu},\ldots,F_{\mu})$ is a union of sequences
$T = (T_n,T_{n-1},\ldots,T_0)$ of finite subtrees $T_r \subset G_\mu$
satisfying $(*)$, with corresponding expressions
\begin{align*}
\wwtilde{W} & = \bigcup\limits_TV\langle T \rangle,\\
(X(\infty),Y(\infty),Z(\infty),h(\infty)) & =
\bigcup\limits_T (X(V\langle T \rangle),Y(V\langle T \rangle),
Z(V\langle T \rangle),h(V\langle T \rangle)).
\end{align*}
This completes the proof.
\end{proof}
\section{Algebraic transversality for $A[F_{\mu}]$--module complexes}
\label{algebraic transversality}
Algebraic transversality for $A[F_\mu]$--module chain complexes is
modelled on the combinatorial transversality for $F_\mu$--covers of
\fullref{combinatorial transversality}. The procedure replaces
matrices with entries in $A[F_\mu]$ by (in general larger) matrices
with entries of the linear type
$$a_1+\sum\limits_{i = 1}^\mu a_{z_i}z_i \in
A[F_\mu]\quad(a_1,a_{z_1},\ldots,a_{z_\mu} \in A).$$
Algebraic transversality can be traced back to the work of Higman,
Bass--Heller--Swan, Stallings, Casson and Waldhausen on the algebraic
$K$--theory of polynomial extensions and more general amalgamated
free products. See of Ranicki \cite[Chapter~7]{RHK} for a treatment
of algebraic transversality in the case $\mu = 1$ when
$A[F_\mu] = A[z,z^{-1}]$ is the Laurent polynomial extension of $A$.
\begin{definition}
Given an $A$--module $P$ and a set $F$ let
$$P[F] = \bigoplus\limits_{x \in F}xP$$
be the direct sum of copies $xP$ of $P$, consisting of the formal $A$--linear
combinations $\sum\limits_{x \in F}xa_x$ $(a_x \in P)$
with $\{x \in F\,\vert\, a_x \neq 0\}$ finite.
\end{definition}
In particular, if $F$ is a semigroup with 1 then $A[F]$ is a ring.
We shall be particularly concerned with the case of a free group
$F = F_\mu$ or the free semigroup $F^+_{\mu}$ on $\mu$ generators
$z_1,z_2,\ldots,z_{\mu}$. Thus $F^+_{\mu} \subset F_{\mu}$ consists
of all the products $z_{i_1}^{n_1}z_{i_2}^{n_2}\ldots
z_{i_k}^{n_k}$ with $n_1,n_2,\ldots,n_k \geqslant 0$. The rings
$A[F_\mu]$, $A[F^+_\mu]$ are free products
\begin{align*}
A[F_\mu] & = A[z_1,z_1^{-1}]*_AA[z_2,z_2^{-1}]*_A\ldots *_AA[z_{\mu},z_{\mu}^{-1}],\\
A[F^+_\mu] & = A[z_1]*_AA[z_2]*_A\cdots *_AA[z_{\mu}].
\end{align*}
For any ring morphism $k\co A \to B$ induction and restriction
define functors
$$\begin{array}{l}
k_!\co \{\hbox{\rm $A$--modules}\} \to \{\hbox{\rm
$B$--modules}\};~L \mapsto k_!L = B\otimes_AL,\\[1ex]
k^!\co \{\hbox{\rm $B$--modules}\} \to \{\hbox{\rm $A$--modules}\};~M
\mapsto k^!M = M
\end{array}$$
such that $k_!$ is left adjoint to $k^!$, with a natural isomorphism
$$\Hom_A(L,k^!M) \to \Hom_B(k_!L,M);~
f \mapsto (b \otimes x \mapsto bf(x)).$$
\begin{definition}
An $A[F]$--module is \emph{induced} if it is of the form
$$P[F] = k_!P = A[F]\otimes_AP$$
for an $A$--module $P$, with $k\co A \to A[F]$ the inclusion.
\end{definition}
\begin{proposition}\label{induced}
Let $P,Q$ be $A$--modules.
{\rm (i)}\qua There is defined a natural isomorphism of additive
groups
$$\Hom_A(P,Q[F]) \to \Hom_{A[F]}(P[F],Q[F]);~
f \mapsto \big(\sum\limits_{y \in F}yg_y \mapsto \sum\limits_{y \in F}yf(g_y)\big).$$
{\rm (ii)}\qua There is defined a natural injection of additive groups
$$\Hom_A(P,Q)[F] \to \Hom_A(P,Q[F]);~
\sum\limits_{x \in F} xf_x \mapsto \big(y \mapsto
\sum\limits_{x \in F}xf_x(y)\big).$$
{\rm (iii)}\qua If $P$ is a f.g.~projective $A$--module the injection
in {\rm (ii)} is also a surjection, so that the composite with
the isomorphism in {\rm (i)} is a natural isomorphism allowing the
identification
$$\Hom_A(P,Q)[F] = \Hom_{A[F]}(P[F],Q[F]).$$
\end{proposition}
\begin{proof} (i)\qua This is just the adjointness of $k_!$ and $k^!$,
with $k\co A \to A[F]$ the inclusion.
(ii)\qua Obvious.
(iii)\qua It is sufficient to consider the case $P = A$.
\end{proof}
\begin{definition} Let $P$ be an $A$--module which is given as a
$\mu$--fold direct sum
$$P = P_1 \oplus P_2 \oplus \cdots \oplus P_{\mu}$$
with idempotents $\pi_i\co P \to P_i \to P$.
(i)\qua Define the $A[F]$--module endomorphism
\begin{multline*}
z = \sum\limits^{\mu}_{i = 1}\pi_iz_i =
\begin{pmatrix} z_1 & 0 & \cdots & 0 \\
0 & z_2 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & z_{\mu} \end{pmatrix} \co
P[F] = P_1[F] \oplus P_2[F] \oplus \cdots \oplus P_{\mu}[F]\\
\longrightarrow P[F] = P_1[F] \oplus P_2[F]\oplus \cdots \oplus P_{\mu}[F].
\end{multline*}
For $F = F_{\mu}$ this is an automorphism, with inverse
{\small\begin{multline*}
z^{-1} = \sum\limits^{\mu}_{i = 1}\pi_iz_i^{-1} =
\begin{pmatrix} z_1^{-1} & 0 & \cdots & 0 \\
0 & z_2^{-1} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & z_{\mu}^{-1} \end{pmatrix}\co
P[F_{\mu}] = P_1[F_{\mu}] \oplus P_2[F_{\mu}] \oplus \cdots \oplus P_{\mu}[F_{\mu}]\\
\longrightarrow P[F_{\mu}] = P_1[F_{\mu}] \oplus P_2[F_{\mu}] \oplus
\cdots \oplus P_{\mu}[F_{\mu}].
\end{multline*}}%
(ii)\qua Given a collection of $A$--module morphisms
$$e = \{e_i \in \Hom_A(P_i,Q)\,\vert\, 1 \leqslant i \leqslant \mu\}$$
define the $A[F]$--module morphism
{\small$$
ez = \!\sum^{\mu}_{i = 1}\!e\pi_iz_i =
\begin{pmatrix} e_1 z_1 & e_2 z_2 & \cdots & e_{\mu}z_{\mu}
\end{pmatrix}\co
P[F] = P_1[F] \oplus P_2[F] \oplus \cdots \oplus P_{\mu}[F] \to Q[F].$$}%
(iii)\qua An $A[F]$--module morphism $f\co P[F] \to Q[F]$ is \emph{linear} if
$$\begin{array}{l}
f = f^+z - f^- = \begin{pmatrix} f^{+,1}z_1-f^{-,1} & \ldots &
f^{+,\mu}z_\mu-f^{-,\mu} \end{pmatrix}~:\\[1ex]
\hskip100pt
P[F] = P_1[F]\oplus P_2[F] \oplus \cdots \oplus P_{\mu}[F] \to Q[F]
\end{array}$$
for some $A$--module morphisms $f^{+,i},f^{-,i}\co P_i \to Q$.
\end{definition}
\begin{definition}
(i)\qua A \emph{Mayer--Vietoris presentation} of an
$A[F]$--module $E$ is an exact sequence of the type
$$\disablesubscriptcorrection\xysavmatrix{
0 \ar[r] & {{\bigoplus\limits_{i = 1}^\mu C^{(i)}[F]}}
\ar[r]^-{\displaystyle{f}} & D[F] \ar[r] & E \ar[r] & 0 }$$
with $C^{(i)},D$ $A$--modules and $f = f^+z-f^-$ a linear
$A[F]$--module morphism.
(ii)\qua A \emph{Mayer--Vietoris presentation} of an $A[F]$--module morphism
$\phi\co E \to E'$ is a morphism of Mayer--Vietoris presentations
$$\disablesubscriptcorrection\xysavmatrix{
0 \ar[r] & {{\bigoplus\limits_{i = 1}^\mu C^{(i)}[F]}}\ar[r]^-{\displaystyle{f}}
\ar[d]^{\displaystyle{\oplus g^{(i)}}} & D[F] \ar[r]\ar[d]^{\displaystyle{h}} &
E \ar[r]\ar[d]^{\displaystyle{\phi}} & 0 \\
0 \ar[r] & {\bigoplus\limits_{i = 1}^\mu
C^{\prime(i)} [F]}\ar[r]^-{\displaystyle{f'}} & D'[F] \ar[r] & E' \ar[r] & 0
}$$
where $g^{(i)}\co C^{(i)} \to {C'}^{(i)}$ and $h\co D \to D'$ $A$--module morphisms.
(iii)\qua A \emph{Mayer--Vietoris presentation} of an $A[F]$--module
chain complex $E$ is an exact sequence as in (i),
with $C^{(i)},D$ $A$--module chain complexes and
$f^{\pm,i}\co C^{(i)} \to D$ $A$--module chain maps.
Similarly for an $A[F]$--module chain map $\phi\co E \to E'$, with a
morphism of exact sequences as in (ii)
in which $g^{(i)}$, $h$ are $A$--module chain maps.
(iv)\qua A Mayer--Vietoris presentation of a finite induced
f.g.~projective $A[F_\mu]$--module
chain complex $E$ is \emph{finite} if $C^{(i)},D$ are finite f.g.~projective
$A$--module chain complexes.
\end{definition}
\begin{example} Let $X$ be the $CW$ complex
$$X = Z/\bigl\{x \sim \beta_i(x)\,\vert\,x \in Y^+_i,1 \leqslant i
\leqslant \mu\bigr\}$$
which is obtained from a $CW$ complex $Z$ and disjoint collared
subcomplexes
$$Y^+_1,~Y^+_2,\ldots,~Y^+_{\mu},~Y^-_1,~Y^-_2,\ldots,~Y^-_{\mu}
\subset Z$$
using cellular homeomorphisms $\beta_i\co Y^+_i \to Y^-_i$ as
identifications. As in \fullref{fsplit} there is an $F_\mu$--splitting
$(X,Y,Z,h)$, where $Y = Y^+_1\sqcup Y^+_2 \sqcup \ldots \sqcup
Y^+_\mu$ and
$$\begin{array}{l}
\alpha^+_i = {\rm inclusion}_{Y^+_i \subset Z}\co Y_i = Y^+_i \to Z,\\[1ex]
\alpha^-_i = ({\rm inclusion}_{Y^-_i \subset
Z})\beta_i\co Y_i = Y^+_i \to Z.
\end{array}$$
The cellular free $\Z[F_\mu]$--module chain complex $C(\widetilde{X})$
of the $F_\mu$--cover $\widetilde{X}$ of $X$ given by \fullref{fcover} (i)
has a Mayer--Vietoris presentation
$$\disablesubscriptcorrection\xysavmatrix{0 \ar[r]& C(Y)[F_\mu] \ar[r]^-{\displaystyle{\alpha}}&
C(Z)[F_\mu] \ar[r]& C(\widetilde{X}) \ar[r]& 0}$$
with $C(Y)^{(i)} = C(Y_i)$, $C(Z)$ free $\Z$--module chain complexes,
and $\alpha = \alpha^+z-\alpha^-$ a linear $\Z[F_\mu]$--module chain map.
If $Z$ is a finite $CW$ complex the
Mayer--Vietoris presentation is finite.
\end{example}
We shall construct Mayer--Vietoris presentations of free
$A[F_\mu]$--module chain complexes using the Cayley tree $G_{\mu}$
(\fullref{cayley}) and the subtree $G^+_{\mu} \subset G_{\mu}$
corresponding to $F^+_{\mu} \subset F_{\mu}$.
\begin{definition}
(i)\qua Let $G_{\mu}^+ \subset G_{\mu}$ be the subtree with
$$(G_{\mu}^+)^{(0)} = F^+_{\mu},\quad(G_{\mu}^+)^{(1)} = \{(g,gz_i)\,\vert\,g \in F^+_{\mu},
1 \leqslant i \leqslant \mu\}.$$
(ii)\qua For any subtree $T \subseteq G_{\mu}$ and
$i = 1,2,\ldots,\mu$ let $T^{(i,1)} \subseteq T^{(1)}$ be the set of
edges of type $(g,gz_i)$ with $g \in F_{\mu}$, such that
$$T^{(1)} = \smash{\coprod^{\mu}_{i = 1}}T^{(i,1)},$$
and let
$$T^+ = T \cap G_{\mu}^+ \subseteq T.$$
(iii)\qua For $F = F_\mu$ (resp. $F^+_\mu$) let $G = G_{\mu}$ (resp.
$G^+_{\mu}$).
\end{definition}
We shall only be considering subtrees $T \subseteq G$ containing
the vertex $1 \in G^{(0)}$.
\begin{proposition}\label{MVprop}
Given an $A$--module $P$ let $E = P[F]$ be the induced
$A[F]$--module, regarded as a 0--dimensional $A[F]$--module chain complex.
{\rm (i)}\qua For any subtree $T \subseteq G$ there is defined a
Mayer--Vietoris presentation of $E$
$$\disablesubscriptcorrection\xysavmatrix{E\langle T \rangle\co
0 \ar[r] & {\smash{\bigoplus\limits_{i = 1}^\mu C^{(i)}[F]}} \ar[r]^-{\displaystyle{f}} &
D[F] \ar[r] & E \ar[r] & 0 }$$ with
$$\begin{array}{l}
D = P[T^{(0)}],\quad C^{(i)} = D \cap
z^{-1}_iD = P[T^{(i,1)}] \subseteq E,\\[1ex]
f^{+,i}\co C^{(i)} \to D;~ xp \mapsto xp,\quad f^{-,i}\co C^{(i)}
\to D;~ xp \mapsto z_ixp.\end{array}$$
{\rm (ii)}\qua The Mayer--Vietoris presentations $E\langle T \rangle$
are such that
$$E\langle T \cap T' \rangle = E\langle T \rangle \cap E\langle T' \rangle,
\quad E\langle T \cup T' \rangle = E\langle T \rangle + E\langle T'
\rangle \quad(T,T' \subseteq G).$$
If $P$ is f.g.~projective and $T$ is finite then $C^{(i)}$, $D$
are f.g.~projective $A$--modules.
{\rm (iii)}\qua Given a morphism of induced $A[F]$--modules
$$\phi\co E = P[F]\to E' = P'[F]$$
and a subtree $T \subseteq G$ let $\phi_*T\subseteq G$ be the
smallest subtree such that
$$\phi(P) \subseteq P'[\phi_*T^{(0)}] \subseteq E'.$$
For any subtree $T' \subseteq G$ such that $\phi_*T \subseteq T'$
there is defined a morphism of Mayer--Vietoris presentations
$$\disablesubscriptcorrection\xysavmatrix{ E\langle T \rangle~:~ 0 \ar[r] &
{{\bigoplus\limits_{i = 1}^\mu C^{(i)}[F]}}\ar[r]^-{\displaystyle{f}}
\ar[d]^{\bigoplus \displaystyle{g^{(i)}}} & D[F] \ar[r]\ar[d]^{\displaystyle{h}} & E
\ar[r]\ar[d]^-{\displaystyle{\phi}} & 0 \\
E'\langle T' \rangle~:~ 0 \ar[r] & {\bigoplus\limits_{i = 1}^\mu
C^{\prime(i)} [F]}\ar[r]^-{\displaystyle{f'}} & D'[F] \ar[r] & E' \ar[r] &0}$$
with
$$g^{(i)} = \phi\vert\co C^{(i)} \to {C'}^{(i)},\quad h = \phi\vert\co D \to D'.$$
If $P$ is a f.g.~$A$--module and $T \subset G$ is finite, then so
is $\phi_*T \subset G$.
\end{proposition}
\begin{proof} By construction.
\end{proof}
\begin{example}
The Mayer--Vietoris presentation of $E$ associated to the
minimal subtree $T = \{1\}\subset G$ is
$$\disablesubscriptcorrection\xysavmatrix{0 \ar[r] & 0 \ar[r] & P[F] \ar[r]^-{\rm id.}& E \ar[r]
&0.}$$
\end{example}
\begin{definition} \label{algcan}
The \emph{canonical Mayer--Vietoris presentation} of an
$A[F]$--module chain complex $E$ with each $E_r = P_r[F]$ an
induced $A[F]$--module
$$\disablesubscriptcorrection\xysavmatrix{E\langle \infty \rangle~:~
0 \ar[r] & \bigoplus\limits^{\mu}_{i = 1} C^{(i)}[F] \ar[r]^-{\displaystyle{f}} &
D[F] \ar[r]& E \ar[r] &0}$$
is the Mayer--Vietoris presentation with $E_r\langle \infty \rangle
= E_r\langle T \rangle$ the Mayer--Vietoris presentation of $E_r$
associated to the maximal subtree $T = G \subseteq G$, where
$$f^{+,i} = {\rm id.},~f^- = z_i~:~C^{(i)} = k^!E \to D = k^!E$$
with $k\co A \to A[F]$ the inclusion.
\end{definition}
\begin{remark} (i)\qua The canonical Mayer--Vietoris presentation
can be written in terms of induction and restriction
$$\disablesubscriptcorrection\xysavmatrix{E\langle \infty \rangle~:~
0 \ar[r] & \lower16pt\hbox{$\bigoplus\limits_{\mu}k_!k^!E$}
\ar[r]^{\displaystyle{f}} & k_!k^!E \ar[r]& E \ar[r] &0}$$
with
$$\begin{array}{l}
f\co \bigoplus\limits_{\mu}k_!k^!E \to k_!k^!E;~
x_i \otimes y \mapsto x_iz_i\otimes y-x_i\otimes z_iy\quad (x_i \in A[F],y \in E),\\[1ex]
k_!k^!E \to E;~ x \otimes y \mapsto xy\quad (x \in A[F],y \in E).
\end{array}$$
(ii)\qua The canonical Mayer--Vietoris presentation for $F = F_\mu$ is the
algebraic analogue of the canonical homotopy $F_\mu$--splitting of a
space $W$ with an $F_\mu$--cover $\wwtilde{W}$ in \fullref{canonical}.
\end{remark}
\begin{thm}[Algebraic transversality for chain complexes]\label{MV}
Let $E$ be an $n$--dimensional $A[F]$--module chain complex
$$\disablesubscriptcorrection\xysavmatrix{E\co E_n \ar[r]^-{d_n} & E_{n-1} \ar[r] &\cdots \ar[r]&
E_1 \ar[r]^-{d_1} & E_0}$$
with each $E_r = P_r[F]$ induced from an $A$--module $P_r$.
{\rm (i)}\qua For any sequence $T = (T_n,T_{n-1},\ldots,T_0)$ of subtrees
$T_r \subseteq G$ such that
$$(d_r)_*(T_r) \subseteq T_{r-1}\quad(r = n,n-1,\ldots,1)\eqno{(*)}$$
there is defined a Mayer--Vietoris presentation
$$\disablesubscriptcorrection\xysavmatrix@C+15pt{E\langle T \rangle\co
0 \ar[r] & {{\bigoplus\limits_{i = 1}^\mu C^{(i)}[F]}}
\ar[r]^-{{f^+z{-}f^-}} & D[F] \ar[r] & E \ar[r] & 0
}$$
with
$$E\langle T \rangle_r = E_r \langle T_r \rangle
\quad(0 \leqslant r \leqslant n),\quad
E\langle T \rangle \subseteq E\langle \infty \rangle.$$
{\rm (ii)}\qua If the $A$--modules $P_r$ are f.g.~projective then for
any finite subtree $T_n \subseteq G$ there exists a sequence
$T = (T_n,T_{n-1},\ldots,T_0)$ of finite subtrees $T_r \subseteq G$
satisfying $(*)$, so that $E\langle T \rangle$ is a finite
Mayer--Vietoris presentation of $E$. Thus
$$E\langle \infty \rangle = \bigcup\limits_T E \langle T \rangle $$
with the union taken over all such sequences $T$. In particular,
$E$ admits a finite Mayer--Vietoris presentation.
\end{thm}
\begin{proof} By repeated applications of \fullref{MVprop},
with the sequences $T = (T_n,T_{n-1},\allowbreak\ldots,T_0)$ the
chain complex analogues of the sequences used to construct the homotopy
$F_\mu$--splittings of $CW$ complexes in the proof of \fullref{combtrans}.
\end{proof}
This completes the proof of \fullref{thm1} of the Introduction.
\section{Blanchfield and Seifert modules}\label{modules}
\subsection{The Magnus--Fox embedding}
This section obtains some technical results on the Magnus--Fox embedding
which we shall need to characterize Blanchfield $A[F_\mu]$--modules, and
to approximate h.d.~1 $F_\mu$--link modules by f.g.~projective Seifert
$A$--modules.
Let $A\langle\!\langle x_1,x_2,\ldots,x_{\mu}\rangle\!\rangle$ be the ring of
$A$--coefficient formal power series in non-commuting indeterminates
$x_1,x_2,\ldots,x_\mu$. The \emph{Magnus--Fox embedding} is defined by
$$i\co A[F_{\mu}] \to \widehat{A[F_\mu]} =
A\langle\!\langle x_1,x_2,\ldots,x_{\mu}\rangle\!\rangle ;~z_j \mapsto 1+x_j.$$
See the paper of Ara and Dicks \cite{AD} for a recent
account of the Magnus--Fox embedding, including the relationship with
noncommutative Cohn localization.
The augmentations $\epsilon(z_j) = 1$, $\widehat\epsilon(x_j) = 0$ give rise to a
commutative triangle of rings
$$\disablesubscriptcorrection\xysavmatrix{ A[F_{\mu}] \ar[rr]^-{\displaystyle{i}} \ar[dr]^-{\displaystyle{\epsilon}}
&& \widehat{A[F_\mu]} \ar[dl]_-{\displaystyle{\widehat{\epsilon}}}\\
&A &}$$
\begin{proposition}\label{Magnus-Foxon}
{\rm (i)}\qua For projective $\widehat{A[F_\mu]}$--modules
$\what{K},\what{L}$ the augmentation map
$$\what{\epsilon}\co
\Hom_{\scriptsize\widehat{A[F_\mu]}}\bigl(\what{K},\what{L}\bigr)
\to\Hom_A\bigl(A\otimes_{\scriptsize\widehat{A[F_\mu]}}\what{K},
A\otimes_{\scriptsize\widehat{A[F_\mu]}}\what{L}\bigr);~
\what{f} \mapsto 1 \otimes \what{f}$$
is surjective.
{\rm (ii)}\qua A morphism $\what{f}\co \what{K} \to \what{L}$ of
projective $\widehat{A[F_\mu]}$--modules is an isomorphism if and only
if the $A$--module morphism
$$1\otimes\what{f}\co A\otimes_{\scriptsize\widehat{A[F_\mu]}}\what{K}
\to A\otimes_{\scriptsize\widehat{A[F_\mu]}}\what{L}$$
is an isomorphism.
{\rm (iii)}\qua A morphism $f\co K \to L$ of projective $A[F_\mu]$--modules
induces an $\widehat{A[F_\mu]}$--module isomorphism
$$1\otimes f\co \widehat{A[F_\mu]}\otimes_{A[F_\mu]}K \to
\widehat{A[F_\mu]}\otimes_{A[F_\mu]}L$$
if and only if the $A$--module morphism
$$1\otimes f\co A\otimes_{A[F_\mu]}K \to A\otimes_{A[F_\mu]}L$$
is an isomorphism.
\end{proposition}
\begin{proof}
(i)\qua By additivity this reduces to the special case
$\what{K} = \what{L} = \widehat{A[F_\mu]}$, which is just the
fact that $\widehat{\epsilon}\co \widehat{A[F_\mu]} \to A$ is surjective.
(ii)\qua It suffices to prove that if $1\otimes\what{f}$ is an $A$--module isomorphism
then $\what{f}$ is an $\widehat{A[F_\mu]}$--module isomorphism.
Consider first the special case when $\what{K},\what{L}$
are free $\widehat{A[F_\mu]}$--modules, say $\smash{\widehat{A[F_\mu]}}^k$,
$\smash{\widehat{A[F_\mu]}}^\ell$ for some sets $k,\ell$. The augmentation map
$$\widehat{\epsilon}\co
\Hom_{\widehat{A[F_\mu]}}\bigl(\smash{\widehat{A[F_\mu]}}^k,
\smash{\widehat{A[F_\mu]}}^{\ell}\bigr)
\to\Hom_A\bigl(A^k,A^{\ell}\bigr);~
\what{f} \mapsto 1 \otimes \what{f}$$
has a canonical splitting. If $1\otimes \what{f}$ is an isomorphism then
all the entries in the matrix of the $\widehat{A[F_\mu]}$--module morphism
$$g = 1-(1\otimes \what{f})^{-1}\what{f}\co \smash{\widehat{A[F_\mu]}}^k
\to \smash{\widehat{A[F_\mu]}}^k$$
have constant term 0, so that $1-g = (1\otimes \what{f})^{-1}\what{f}$
is an $\widehat{A[F_\mu]}$--module isomorphism with inverse
$$(1-g)^{-1} = 1+g+g^2+g^3+g^4+\cdots\co \smash{\widehat{A[F_\mu]}}^k
\to \widehat{A[F_\mu]}^k,$$
and $\what{f} = (1\otimes \what{f})(1-g)$ is an isomorphism.
For the general projective case apply (i) to lift $(1\otimes\what{f})^{-1}$
to an $\widehat{A[F_\mu]}$--module morphism $\what{e}\co \what{L}
\to \what{K}$. Choose a projective $\widehat{A[F_\mu]}$--module
$\what{J}$ such that $\what{J} \oplus \what{K} \oplus
\what{L}$ is a free $\widehat{A[F_\mu]}$--module, and apply the
special case to the $\widehat{A[F_\mu]}$--module morphism
$$1 \oplus \begin{pmatrix} 0 & \smash{\what{e}} \\ \smash{\what{f}} & 0 \end{pmatrix}\co
\what{J} \oplus \what{K} \oplus
\what{L} \to \what{J} \oplus \what{K} \oplus
\what{L}.$$
(iii)\qua This is a special case of (ii).
\end{proof}
For $j = 1,2,\ldots,\mu$ let $y_j$ be a formal square root of $z_j$, so
that $(y_j)^2 = z_j$. Let $F_\mu(y)$ be the free group generated by
$y_1,y_2,\ldots,y_\mu$, so that $F_\mu \subset F_\mu(y)$ is the free
subgroup generated by $z_1,z_2,\ldots,z_\mu$. We can identify
$G_\mu^{(1,j)}$ with the subset $F_\mu y_j \subset F_\mu(y)$: the edge
$(g,gz_j) \in G_\mu^{(1,j)}$ ($g \in F_\mu$) is identified with the
element $gy_j^{-1} \in F_\mu(y)$.
\begin{lemma}\label{decompose_tree}
If $T\subset G_\mu$ is a finite subtree then
$$A[T^{(0)}] = A[\{1\}] \oplus \Bigl(\bigoplus\limits_{j = 1}^\mu
A[T^{(1,j)}](y_j^{-1}-y_j)\Bigr) \subset A[F_\mu].\eqno{(*)}$$
\end{lemma}
\begin{proof} If $w\in T^{(1,j)}$ then certainly
$w(y_j^{-1}-y_j)\in A[T^{(0)}]$.
Let us check linear independence of the generators on the right hand
side of~$(*)$. Assuming the contrary, let
$$a_1+
\sum\limits_{gy_j^{-1}\in U} a_g g(y_j^{-1}-y_j) = 0 \in A[F_\mu]
$$
be a non-trivial relation with $U\subset T^{(1)}$ non-empty and
minimal. We reach a contradiction by
observing that if $g(y_j)^{-1}\in U$ is a word of maximal length (in reduced
form) then $a_g = 0$.
We must also show that every $v\in T^{(0)}$ is an element of the
right-hand side of~$(*)$. Indeed there is a (unique)
path in the tree from $1$ to $v$ defined by a sequence of edges
$w_1,w_2,\ldots,w_n \in T^{(1)}$ and we have
$$v = 1+\sum_{i = 1}^nw_i(y_{j(i)}^{-1}-y_{j(i)})\eta_i \in A[F_\mu]$$
if the signs $\eta_i\in\{\pm1\}$ are chosen appropriately and $j(i)$
is such that $w_i\in T^{(1,j(i))}$.
\end{proof}
\begin{proposition} \label{Magnus-Foxtw}
For any finite subset $S \subset F_\mu$ the inclusion
$i\vert\co A[S] \to \widehat{A[F_\mu]}$ is a split $A$--module injection.
\end{proposition}
\begin{proof}
Since every finite $S$ is contained in the vertex set of
some finite tree we may assume that $S = T^{(0)}$ for some finite subtree
$T\subset G_\mu$. We proceed by induction on $|T^{(0)}|$.
If the tree $T$ has only one vertex then $T^{(0)} = \{1\}$
with $i(1) = 1 \in \widehat{A[F_\mu]}$ and
$$\widehat{A[F_\mu]} = A[\{1\}]\oplus \bigoplus\limits^{\mu}_{i = 1}
\widehat{A[F_{\mu}]}x_i~
= A\oplus \bigoplus\limits^{\mu}_{i = 1}\widehat{A[F_\mu]}(1-z_i^{\eta})
\eqno{(**)}$$
for any $\eta \in \{\pm 1\}$,
and $i|\co A[\{1\}] \to \widehat{A[F_\mu]}$ is a split injection.
Suppose now that $|T^{(0)}|\geqslant 2$. Let $v_0\in T^{(0)}$ be a leaf,
ie a vertex to which only one edge is incident. Let $T\backslash \{v_0\}$
denote the tree obtained by removing the vertex $v_0$ and the incident
edge. By the inductive hypothesis, $i|\co A[T^{(0)}\backslash \{v_0\}] \to
\widehat{A[F_\mu]}$ is a split injection; we denote the image by $P$.
Since $v_0$ is incident to precisely one edge then
$v_0 = w_0y_k^\eta$ for unique $\eta\in\{\pm1\}$,
$k\in\{1,\ldots,\mu\}$ and $w_0\in T^{(1,k)}$.
Now for every $j$ we have
$T^{(1,j)}y_j^{-\eta} \subset T^{(0)}\backslash \{v_0\}$. Thus
$$\begin{array}{ll}
T^{(1,j)}(y_j^{-1}-y_j)& = T^{(1,j)}y_j^{-\eta}(1-y_j^{2\eta})\eta\\[1ex]
& = T^{(1,j)}y_j^{-\eta}(1-z_j^\eta)\eta \subset
(T^{(0)}\backslash \{v_0\})(1-z_j^\eta)\eta.
\end{array}$$
It follows from~$(*)$ that $i(A[T^{(0)}])$ is a direct summand of
$$Ai(v_0)\oplus \bigoplus\limits_{j = 1}^\mu P(1-z_j^\eta)$$
and hence, by the following \fullref{decompose_widehatA[F]}, a direct
summand of $\widehat{A[F_\mu]}$.
\end{proof}
\begin{lemma}\label{decompose_widehatA[F]}
Suppose $P$ is an $A$--module which is a direct summand of $\widehat{A[F_\mu]}$.
If $\theta\in \widehat{A[F_\mu]}$ is an element such that
$\widehat{\epsilon}(\theta) = 1\in A$ and $\eta = 1$ or $-1$ then
$$A\theta\oplus\left(\bigoplus\limits_{j = 1}^\mu P(1-z_j^\eta) \right) \subset
\widehat{A[F_\mu]}$$
is again a direct summand.
\end{lemma}
\begin{proof}
We may write $\widehat{A[F_\mu]} = P\oplus Q$ for some $A$--module $Q$. Let
$\eta = 1$ or $-1$. Now it follows easily from $(**)$ that
$$\begin{array}{ll}
\widehat{A[F_\mu]}& = A\theta\oplus
\left(\bigoplus\limits_{j = 1}^\mu\widehat{A[F_\mu]}(1-z_j^\eta)\right)\\[3ex]
& = A\theta \oplus \left(\bigoplus\limits_{j = 1}^\mu
P(1-z_j^\eta)\right) \oplus
\left(\bigoplus\limits_{j = 1}^\mu Q(1-z_j^\eta)\right)
\end{array}$$
which completes the proof.\end{proof}
\subsection{Blanchfield modules}\label{bla}
\begin{definition} \label{Fdefinition}
(i)\qua A \emph{Blanchfield $A[F_\mu]$--module} $M$ is an $A[F_\mu]$--module
such that
$$\Tor_*^{A[F_\mu]}(A,M) = 0.$$
(ii)\qua (Sheiham \cite{Sh2}) An \emph{$F_\mu$--link module} $M$ is an $A[F_\mu]$--module
which has a 1--dimensional induced $A[F_\mu]$--module resolution
$$\disablesubscriptcorrection\xysavmatrix{
0 \ar[r]&P[F_\mu] \ar[r]^-{\displaystyle{d}}& P[F_\mu] \ar[r] & M \ar[r] & 0}$$
with $P$ an $A$--module and $d$ an $A[F_\mu]$--module morphism such that
the augmentation $A$--module morphism $\epsilon(d)\co P \to P$ is an isomorphism.
\end{definition}
As before, let $k\co A \to A[F_\mu]$ be the inclusion.
\begin{proposition} \label{B = F} The following conditions on
an $A[F_\mu]$--module $M$ are equivalent:
\begin{itemize}
\item[{\rm (i)}] $M$ is a Blanchfield module,
\item[{\rm (ii)}] $M$ is an $F_\mu$--link module,
\item[{\rm (iii)}] the $A$--module morphism
$$\gamma_M\co \bigoplus \limits_\mu k^!M \to k^!M;~
(m_1,m_2,\ldots,m_\mu)\mapsto \sum\limits^{\mu}_{i = 1}(z_i-1)m_i$$
is an isomorphism.
\end{itemize}
\end{proposition}
\begin{proof}
The canonical Mayer--Vietoris presentation (\fullref{algcan}) of any
$A[F_\mu]$--module $M$ is defined by
$$\disablesubscriptcorrection\xysavmatrix{0 \ar[r] & \lower16pt\hbox{$\bigoplus\limits_{\mu}k_!k^!M$}
\ar[r]^{\displaystyle{d}} & k_!k^!M \ar[r]& M \ar[r] &0}$$
with
$$\begin{array}{l}
d\co \bigoplus_{\mu}k_!k^!M \to k_!k^!M;~
x_i \otimes y \mapsto x_iz_i\otimes y-x_i\otimes z_iy
\quad (x_i \in A[F_\mu],y \in M),\\[1ex]
k_!k^!M = k^!M[F_\mu] \to M;~ x \otimes y \mapsto xy
\quad (x \in A[F_\mu],y \in M),
\end{array}$$
such that $d$ has augmentation $A$--module morphism
$$\epsilon(d) = -\gamma_M\co \bigoplus\limits_{\mu}k^!M \to k^!M.$$
Regarded as a right $A[F_\mu]$--module $A$ has a 1--dimensional f.g.~free resolution
$$\disablesubscriptcorrection\xysavmatrix@C+25pt{
0 \ar[r]& \bigoplus\limits_{i = 1}^{\mu} A[F_\mu] \ar[r]^-{\displaystyle{\oplus
(z_i-1)}}& A[F_\mu] \ar[r]^-{\displaystyle{\epsilon}} & A \ar[r] &0,}$$
so that for any $A[F_\mu]$--module $M$
$$\Tor^{A[F_\mu]}_n(A,M) = \begin{cases}
A\otimes_{A[F_{\mu}]}M = \coker(\gamma_M)&{\rm if}~n = 0,\\
\ker(\gamma_M)&{\rm if}~n = 1,\\
0&{\rm if}~n \geqslant 2.
\end{cases}$$
The equivalences (i) $\Longleftrightarrow$ (ii)
$\Longleftrightarrow$ (iii) are now clear.
\end{proof}
\begin{definition} \label{Bdefinition}
(i)\qua Let ${\mathcal{B}\mathit{la}}_\infty(A)$ be the category of Blanchfield
$A[F_\mu]$--modules, and let ${\mathcal{B}\mathit{la}}(A) \subset {\mathcal{B}\mathit{la}}_\infty(A)$ be the
full subcategory of the h.d.~1 Blanchfield $A[F_\mu]$--modules.
(In view of \fullref{B = F} ${\mathcal{B}\mathit{la}}_\infty(A)$ is the same as the $F_\mu$--link
module category ${\mathcal{F}\mathit{lk}}_{\infty}(A)$ of Sheiham \cite{Sh2}).
(ii)\qua Let ${\mathcal{F}\mathit{lk}}(A) \subset {\mathcal{B}\mathit{la}}(A)$ be the full
subcategory of the h.d.~1 Blanchfield $A[F_\mu]$--modules $M$
such that there exists a 1--dimensional induced $A[F_\mu]$--module resolution
$$\disablesubscriptcorrection\xysavmatrix{
0 \ar[r]&P[F_\mu] \ar[r]^-{\displaystyle{d}}& P[F_\mu] \ar[r] & M \ar[r] & 0}$$
with $P$ a f.g.~projective $A$--module.
\end{definition}
\begin{example}
(i)\qua For a principal ideal domain $A$
$$K_0(A[F_\mu]) = K_0(A) = \Z$$
(see Bass \cite{B1}) so that
$${\mathcal{B}\mathit{la}}(A) = {\mathcal{F}\mathit{lk}}(A).$$
(ii)\qua A finitely presented Blanchfield $\Z[F_\mu]$--module is a
`type $L$' $\Z[F_\mu]$--module in the sense of Sato \cite{Sa}.
(iii)\qua Given a $\mu$--component boundary link $\ell\co \bigsqcup_\mu
S^n \subset S^{n+2}$ let $c\co W \to W_0$ be a $\Z$--homology equivalence
from the exterior $W$ to the exterior $W_0$ of the trivial
$\mu$--component boundary link $\ell_0\co \bigsqcup_\mu S^n \subset
S^{n+2}$, with $F_\mu$--equivariant lift $\smash{\wtilde{c}\co \wwtilde{W}
\to \wwtilde{W}_0}$ to the $F_\mu$--covers.
The homology groups $\smash{\dot H_*\bigl(\wwtilde{W}\bigr) = H_{*+1}\bigl(
\wtilde{c}\co \wwtilde{W} \to \wwtilde{W}_0\bigr)}$
are Blanchfield $\Z[F_\mu]$--modules
of homological dimension $\leqslant 2$.
Each $\dot H_r\bigl(\wwtilde{W}\bigr)$ has a $\Z$--con\-trac\-ti\-ble
f.g.~free $\Z[F_\mu]$--module resolution of the type
$$0 \to \Z[F_\mu]^{a_r} \to \Z[F_\mu]^{b_r} \to \Z[F_\mu]^{c_r}
\to \dot H_r\bigl(\wwtilde{W}\bigr) \to 0\quad(0 \leqslant r \leqslant n+1)$$
with $a_r-b_r+c_r = 0$, and
$\dot H_r(\wwtilde{W})/\Z\hbox{\rm -torsion}$ is an h.d.~1 $F_\mu$--link
module (Levine \cite[3.5]{L2} for $\mu = 1$, Sato \cite[3.1]{Sa} and Duval
\cite[4.1]{Du} for $\mu \geqslant 2$).
See \fullref{chain} below for the construction of an
$(n{+}1)$--dimensional chain complex $C$ in ${\mathcal{S}\mathit{ei}}(\Z)$ such that
the covering $B(C)$ is an $(n{+}1)$--dimensional chain complex
in ${\mathcal{F}\mathit{lk}}(\Z)$ with $H_*(B(C)) = \dot H_*(\wwtilde{W})$.
\end{example}
The following \fullref{char} characterizes Blanchfield $A[F_\mu]$--modules
in terms of $A[F_\mu]$--modules $K$ such that
$$\Tor_1^{A[F_\mu]}(A,K) = 0.$$
If $K$ is a flat $A[F_\mu]$--module then ${\rm
Tor}_1^{A[F_\mu]}(B,K) = 0$ for any right $A[F_\mu]$--module $B$, and in
particular $B = A$.
If $K = P[F_\mu]$ is induced from an $A$--module $P$ then
$$\Tor_1^{A[F_\mu]}(A,P[F_\mu]) = \Tor_1^A(A,P) = 0.$$
\begin{proposition} \label{char}
{\rm (i)}\qua If $M$ is an $A[F_\mu]$--module with a resolution
$$\disablesubscriptcorrection\xysavmatrix{0 \ar[r] & K \ar[r]^-{\displaystyle{d}} &L\ar[r] & M\ar[r] & 0}$$
such that
$$\Tor^{A[F_{\mu}]}_1(A,K) = \Tor^{A[F_{\mu}]}_1(A,L) = 0$$
(e.g. the canonical Mayer--Vietoris presentation of \fullref{algcan}) then $M$
is Blanchfield if and only if the $A$--module morphism
$1\otimes d\co A\otimes_{A[F_\mu]}K \to A\otimes_{A[F_\mu]}L$
is an isomorphism.
{\rm (ii)}\qua A morphism $d\co K \to L$ of projective $A[F_\mu]$--modules is
injective and $M = \coker(d)$ is a Blanchfield $A[F_\mu]$--module
if and only if the
$A$--module morphism $1\otimes d\co A\otimes_{A[F_\mu]}K \to
A\otimes_{A[F_\mu]}L$ is an isomorphism.
\end{proposition}
\begin{proof}
(i)\qua It follows from \fullref{B = F} and
the commutative diagram with exact rows and columns
$$\disablesubscriptcorrection\xysavmatrix@C+10pt{ & 0 \ar[d] & 0 \ar[d] & & \\
\Tor_1^{A[F_\mu]}(A,K) = 0 \ar[r] & \bigoplus\limits_{i = 1}^{\mu}
K \ar[r]^-{\displaystyle{\gamma_K}} \ar[d]^-{\displaystyle{\oplus d}}& K \ar[r]\ar[d]^-{\displaystyle{d}} &
A\otimes_{A[F_\mu]}K \ar[r]\ar[d]^-{\displaystyle{1\otimes d}} & 0 \\
\Tor_1^{A[F_\mu]}(A,L) = 0 \ar[r] & \bigoplus\limits_{i = 1}^{\mu} L \ar[r]^-{\displaystyle{\gamma_L}} \ar[d]
& L \ar[r]\ar[d] & A\otimes_{A[F_\mu]}L \ar[r] & 0 \\
& \bigoplus\limits_{i = 1}^{\mu} M \ar[r]^-{\displaystyle{\gamma_M}} \ar[d]& M \ar[d]& &\\
& 0 & 0 & & }$$
that $M$ is Blanchfield if and only if $1\otimes d$ is an isomorphism.
(ii)\qua If $d$ is injective and $M$ is Blanchfield
then $1\otimes d$ is an isomorphism by (ii), since projective $A[F_\mu]$--modules
are flat. Conversely, if $1\otimes d\co A\otimes_{A[F_\mu]}K \to
A \otimes_{A[F_\mu]}L$ is an isomorphism then
$1\otimes d\co \smash{\widehat{A[F_\mu]}}\otimes_{A[F_\mu]}K \to
\widehat{A[F_\mu]} \otimes_{A[F_\mu]}L$ is an isomorphism
by \fullref{Magnus-Foxon} (iii), and it follows from the injectivity of
$K \to \smash{\widehat{A[F_\mu]}}\otimes_{A[F_\mu]}K$,
$L \to \smash{\widehat{A[F_\mu]}}\otimes_{A[F_\mu]}L$
and the commutative diagram
$$\disablesubscriptcorrection\xysavmatrix{K \ar[r]^-{\displaystyle{d}} \ar[d] & L \ar[d]\\
\widehat{A[F_\mu]}\otimes_{A[F_\mu]}K \ar[r]^-{\displaystyle{1\otimes d}}&
\widehat{A[F_\mu]}\otimes_{A[F_\mu]}L}$$
that $d\co K \to L$ is injective.
\end{proof}
The \emph{idempotent completion} ${\mathcal{P}}(\mathcal{E})$ of an additive category
$\mathcal{E}$ is the additive category with objects pairs $(M,p = p^2\co M \to M)$
defined by projections $p$ of objects $M$ in $\mathcal{E}$, and morphisms
$f\co (M,p) \to (N,q)$ defined by morphisms $f\co M \to N$ in $\mathcal{E}$ such that
$qfp = f\co M \to N$. As usual, $\mathcal{E}$ is \emph{idempotent complete} if
the functor $\mathcal{E} \to {\mathcal{P}}(\mathcal{E});M \mapsto (M,1)$ is an equivalence,
or equivalently if for every idempotent $p = p^2\co M \to M$ in $\mathcal{E}$
there exists a direct sum decomposition $M = P\oplus Q$ with
$$p = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}~:~
M = P\oplus Q \to M = P\oplus Q.$$
For any exact category $\mathcal{E}$ there exists a full embedding
$\mathcal{E} \subset \mathcal{A}$ in an abelian category $\mathcal{A}$ (Gabriel--Quillen),
and the idempotent completion
${\mathcal{P}}(\mathcal{E})$ is equivalent to the full exact subcategory of $\mathcal{A}$
with objects ${\rm im}(p)$ for objects $(M,p)$ in ${\mathcal{P}}(\mathcal{E})$.
For $\mathcal{E} = {\mathcal{F}\mathit{lk}}(A) \subset \mathcal{A} = {\mathcal{B}\mathit{la}}_{\infty}(A)$ we have that
${\mathcal{P}}({\mathcal{F}\mathit{lk}}(A)) \subset {\mathcal{B}\mathit{la}}_{\infty}(A)$. In fact, we have:
\begin{proposition} \label{idem}
{\rm (i)}\qua The exact categories ${\mathcal{P}\mathit{rim}}(A)$, ${\mathcal{S}\mathit{ei}}(A)$, ${\mathcal{B}\mathit{la}}(A)$ are
idempotent complete.
{\rm (ii)}\qua The idempotent completion of ${\mathcal{F}\mathit{lk}}(A)$ is equivalent to ${\mathcal{B}\mathit{la}}(A)$
$${\mathcal{P}}({\mathcal{F}\mathit{lk}}(A))~\approx~{\mathcal{B}\mathit{la}}(A).$$
\end{proposition}
\begin{proof} (i)\qua The exact categories ${\mathcal{P}\mathit{rim}}(A)$, ${\mathcal{S}\mathit{ei}}(A)$, ${\mathcal{B}\mathit{la}}(A)$ are
closed under direct summands.
(ii)\qua For any f.g.~projective
$A[F_\mu]$--modules $K,L$ the augmentation map
$$\epsilon\co \Hom_{A[F_\mu]}(K,L) \to \Hom_A(A\otimes_{A[F_\mu]}K,
A\otimes_{A[F_\mu]}L);~d \mapsto 1\otimes d$$
is surjective, by the following argument:
choose f.g.~projective $A[F_\mu]$--modules $K',L'$ such that
$$K\oplus K' = A[F_\mu]^k,~L\oplus L' = A[F_\mu]^{\ell}$$
for some $k,\ell \geqslant 0$, and note that the augmentation map
\begin{multline*}
\epsilon\co \Hom_{A[F_\mu]}\bigl(K\oplus K',L\oplus L'\bigr) =
\Hom_{A[F_\mu]}\bigl(A[F_\mu]^k,A[F_\mu]^{\ell}\bigr)\\
\longrightarrow \Hom_A\bigl(A\otimes_{A[F_\mu]}\bigl(K\oplus K'\bigr),
A\otimes_{A[F_\mu]}\bigl(L\oplus L'\bigr)\bigr) = \Hom_A\bigl(A^k,A^{\ell}\bigr)
\end{multline*}
is surjective. Given an h.d.~1 Blanchfield $A[F_\mu]$--module $M$ with
a f.g.~projective $A[F_\mu]$--module resolution
$$\disablesubscriptcorrection\xysavmatrix{0 \ar[r] & K \ar[r]^-{\displaystyle{d}} & L \ar[r]& M \ar[r]&0}$$
we know from \fullref{char} (i) that
$1\otimes d\co A\otimes_{A[F_\mu]}K \to A\otimes_{A[F_\mu]}L$ is an
$A$--module isomorphism. By \fullref{Magnus-Foxon} (i)
it is possible to lift $(1\otimes d)^{-1}$ to an $A[F_\mu]$--module
morphism $e\co L \to K$, so that by \fullref{char} (i) $e$ is an injection with
$$N = \coker(e)$$
an h.d.~1 Blanchfield $A[F_\mu]$--module. Let $J$ be a f.g.~projective
$A[F_\mu]$--module such that $J\oplus K \oplus L$ is f.g.~free,
say $A[F_\mu]^m$. The $A[F_\mu]$--module morphism
$$f = 1\oplus \begin{pmatrix}
0 & e \\ d & 0 \end{pmatrix}~:~
J \oplus K \oplus L = A[F_\mu]^m \to J \oplus K \oplus L = A[F_\mu]^m$$
is such that $1\otimes f\co A^m \to A^m$ is an isomorphism, so that
$\coker(f) = M\oplus N$ is an h.d.~1 $F_\mu$--link module.
The functor
$${\mathcal{F}\mathit{lk}}(A) \to {\mathcal{B}\mathit{la}}(A);~M \mapsto M$$
is a full embedding such that every object in ${\mathcal{B}\mathit{la}}(A)$ is a direct summand
of an object in ${\mathcal{F}\mathit{lk}}(A)$, so that ${\mathcal{B}\mathit{la}}(A)$ is (equivalent to) the
idempotent completion ${\mathcal{P}}({\mathcal{F}\mathit{lk}}(A))$.
\end{proof}
\subsection{Seifert modules}
Let $Q_\mu$ be the complete quiver which has $\mu$ vertices and
$\mu^2$ arrows, one arrow between each ordered pair of vertices.
The path ring is given by
$$Q_{\mu} =
\Z[e]*\Z\bigl[\pi_1,\pi_2,\ldots,\pi_{\mu}\,\vert\,\pi_i\pi_j = \delta_{ij}\pi_i,
{\textstyle\sum^{\mu}_{i = 1}}\pi_i = 1\bigr]$$
where $\pi_i e \pi_j$ corresponds to the unique path of length 1
from the $i$th vertex to the $j$th vertex. An $A$--module $P$
together with a ring morphism $\rho\co Q_\mu \to \End_A(P)$ is
essentially the same as a triple $(P,e,\{\pi_i\})$ with $e\co P \to
P$ an endomorphism, and $\{\pi_i\co P \to P\}$ a complete system of
$\mu$ idempotents. (Such representations of $Q_\mu$ were first
considered by Farber \cite{Fa3} for particular $A$.)
\begin{definition} \label{Sdefinition}
(i)\qua A \emph{Seifert $A$--module} $(P,e,\{\pi_i\})$ is an
$A$--module $P$ together with an endomorphism $e\co P \to P$,
and a system $\{\pi_i\co P \to P\}$ of idempotents expressing $P$
as a $\mu$--fold direct sum, with
$$\pi_i\co P = P_1 \oplus P_2 \oplus \cdots \oplus P_{\mu} \to P;~
(x_1,x_2,\ldots,x_\mu) \mapsto (0,\ldots,0,x_i,0,\ldots,0).$$
(ii)\qua A \emph{morphism} of Seifert $A$--modules
$$g\co (P,e,\{\pi_i\}) \to (P',e',\{\pi'_i\})$$
is an $A$--module morphism such that
$$ge = e'g,~g\pi_i = \pi'_ig\co P \to P'.$$
The conditions $g\pi_i = \pi'_ig$ are equivalent to $g$ preserving
the direct sum decompositions, so that
$$g = \begin{pmatrix} g_1 & 0 & \ldots & 0 \\
0 & g_2 & \ldots & 0 \\
0 & 0 & \ldots & 0 \\
\vdots & \vdots & \ddots & 0 \\
0 & 0 & \ldots & g_{\mu} \end{pmatrix}\co P = P_1 \oplus P_2 \oplus
\cdots \oplus P_{\mu} \to P' = P'_1 \oplus P'_2 \oplus \cdots \oplus
P'_{\mu}$$
with $g_i\co P_i \to P'_i$.
(iii)\qua The \emph{Seifert $A$--module category} ${\mathcal{S}\mathit{ei}}_{\infty}(A)$
has objects Seifert $A$--modules and morphisms as in (ii). Let ${\mathcal{S}\mathit{ei}}(A)
\subseteq {\mathcal{S}\mathit{ei}}_{\infty}(A)$ be the full subcategory of the Seifert
$A$--modules $(P,e,\{\pi_i\})$ with $P$ f.g.~projective.
\end{definition}
\subsection{The covering functor $B$}
Seifert modules determine $F_\mu$--link modules by:
\begin{definition} (i)\qua The \emph{covering} of a Seifert $A$--module
$(P,e,\{\pi_i\})$ is the $F_\mu$--link module
$$B(P,e,\{\pi_i\}) = \coker(
1-e+ez\co P[F_\mu] \to P[F_\mu])$$
with Mayer--Vietoris presentation
$$\disablesubscriptcorrection\xysavmatrix{0 \ar[r] &\bigoplus\limits_{i = 1}^\mu P_i[F_{\mu}]
\ar[r]^-{\displaystyle{d}} & P[F_{\mu}] \ar[r] & B(P,e,\{\pi_i\})\ar[r] & 0 ,}$$
where $d = 1-e+ez$.\\
(ii)\qua The \emph{covering} of a Seifert $A$--module
morphism $g\co (P,e,\{\pi_i\}) \to (P',e',\{\pi'_i\})$ is the
$F_\mu$--link module morphism
$$B(g)\co B(P,e,\{\pi_i\}) \to B(P',e',\{\pi'_i\});~x \mapsto g(x)$$
resolved by
$$\disablesubscriptcorrection\xysavmatrix{0 \ar[r] &P[F_\mu] \ar[r]^-{\displaystyle{d}}
\ar[d]^-{\displaystyle{g}} & P[F_\mu] \ar[d]^-{\displaystyle{g}} \ar[r] &
B(P,e,\{\pi_i\}) \ar[d]^-{\displaystyle{B(g)}} \ar[r] &0\\
0 \ar[r] & P'[F_\mu] \ar[r]^-{\displaystyle{d'}} & P'[F_\mu] \ar[r] &
B(P',e',\{\pi'_i\})\ar[r] & 0}$$
with $d = 1-e+ez$, $d' = 1-e'+e'z$.
\end{definition}
\begin{example} \label{chain}
Let $\ell\co \bigsqcup_\mu S^n \subset S^{n+2}$ be a
$\mu$--component boundary link with exterior $W$, so that there exists a
$\Z$--homology equivalence $c\co W \to W_0$ to the exterior $W_0$ of the trivial
$\mu$--component boundary link $\ell_0\co \bigsqcup_\mu S^n \subset
S^{n+2}$. The $(n{+}2)$--dimensional f.g.~free $\Z[F_\mu]$--module
chain complex
$$\dot C(\wwtilde{W}) = {\cal C}(\wtilde{c}\co C(\wwtilde{W}) \to
C(\wwtilde{W}_0))_{*+1}$$
is $\Z$--contractible.
For any $\mu$--component Seifert surface $V = V_1\sqcup V_2 \sqcup \ldots
\sqcup V_\mu \subset S^{n+2}$ for $\ell$ there exists a degree 1 map $V
\to V_0$ to the $\mu$--component Seifert surface
$V_0 = \bigsqcup_{\mu}D^{n+1} \subset S^{n+2}$ for $\ell_0$.
Let
$$\dot C(V_i) = {\mathcal C}(C(V_i) \to C(D^{n+1}))_{*+1},\quad
\dot C(V) = {\sum\limits^\mu_{i = 1}}\dot C(V_i).$$
The map $V \to S^{n+2}\backslash V$ pushing $V$ off itself in
the positive normal direction combines with chain level
Alexander duality to induce a $\Z$--module chain map
$$e\co \dot C(V) \to C\bigl(S^{n+2}\backslash V,
{\textstyle\bigsqcup_\mu} \{{\rm pt.}\}\bigr)~\simeq~ \dot C(V)^{n+1-*},$$
so that there is defined an $(n{+}1)$--dimensional chain complex
$(\dot C(V),e,\{\pi_i\})$ in ${\mathcal{S}\mathit{ei}}(\Z)$.
The covering $B(\dot C(V),e,\{\pi_i\})$ is an
$(n{+}1)$--dimensional chain complex in ${\mathcal{F}\mathit{lk}}(\Z)$,
with the projection
\begin{multline*}
{\mathcal C}(1-e+ez\co \dot C(V)[F_\mu] \to \dot C(V)[F_\mu]) =
\dot C(\wwtilde{W})\\
\longrightarrow B(\dot C(V),e,\{\pi_i\}) =
\coker(1-e+ez\co \dot C(V)[F_\mu] \to \dot C(V)[F_\mu])
\end{multline*}
a homology equivalence.
\end{example}
The covering construction defines a functor of exact categories
$$B_{\infty}\co {\mathcal{S}\mathit{ei}}_{\infty}(A) \to {\mathcal{B}\mathit{la}}_{\infty}(A);~
(P,e,\{\pi_i\}) \mapsto B(P,e,\{\pi_i\})$$
which restricts to a functor $B\co {\mathcal{S}\mathit{ei}}(A) \to {\mathcal{F}\mathit{lk}}(A)$.
\begin{definition} A morphism $f$ in ${\mathcal{S}\mathit{ei}}_{\infty}(A)$ is a
\emph{$B$--isomorphism} if $B(f)$ is an isomorphism in
${\mathcal{B}\mathit{la}}_{\infty}(A)$. Let $\Xi_{\infty}$ denote the set of
$B$--isomorphisms in ${\mathcal{S}\mathit{ei}}_{\infty}(A)$, and let $\Xi$ denote the
set of $B$--isomorphisms in ${\mathcal{S}\mathit{ei}}(A)$.
\end{definition}
\subsection{Blanchfield/Seifert algebraic transversality}
We shall now use the algebraic transversality of \fullref{algebraic
transversality} to establish that every h.d.~1 $F_\mu$--link module $M$ is
isomorphic to the covering $B(P,e,\{\pi_i\})$ of a f.g.~projective
Seifert $A$--module $(P,e,\{\pi_i\})$, uniquely up to morphisms in
$\Xi$.
We refer to Sheiham \cite{Sh2} for the proof that
$B_\infty\co {\mathcal{S}\mathit{ei}}_\infty(A) \to {\mathcal{F}\mathit{lk}}_\infty(A)$ induces an
equivalence of exact categories
$\wbar{B}_\infty\co \Xi_\infty^{-1}{\mathcal{S}\mathit{ei}}_\infty(A)
\approx {\mathcal{F}\mathit{lk}}_\infty(A)$.
Algebraic transversality will be used to prove
that the universal localization
${\mathcal{S}\mathit{ei}}(A) \to \Xi^{-1}{\mathcal{S}\mathit{ei}}(A)$ has a calculus of
fractions, and that the covering functor $B\co {\mathcal{S}\mathit{ei}}(A) \to
{\mathcal{F}\mathit{lk}}(A)$ induces an equivalence of exact categories
$\wbar{B}\co \Xi^{-1}{\mathcal{S}\mathit{ei}}(A) \approx {\mathcal{F}\mathit{lk}}(A)$.
Given an $F_\mu$--link module $M$ let $U(M) = (M,e_M,\{\pi_i\})$ be
the Seifert $A$--module defined in \cite{Sh2} -- the definition is
recalled in the Introduction of this paper, along with the fact
proved in \cite{Sh2} that $B_{\infty}$ is a left adjoint of
$$U_{\infty}\co {\mathcal{B}\mathit{la}}_{\infty}(A) \to {\mathcal{S}\mathit{ei}}_{\infty}(A);~
M \mapsto U(M).$$
The natural isomorphism of the adjointness
\begin{eqnarray*}
\Hom_{{\mathcal{B}\mathit{la}}_{\infty}(A)}(B(Q,f,\{\rho_i\}),M)
&\disablesubscriptcorrection\xysavmatrix{\ar[r]^-{\displaystyle{\cong}}&} &
\Hom_{{\mathcal{S}\mathit{ei}}_{\infty}(A)}((Q,f,\{\rho_i\}),U(M));\\
g &\longmapsto& \adj(g) = U(g) h
\end{eqnarray*}
is defined for any Seifert $A$--module $(Q,f,\{\rho_i\})$,
with
$$h\co Q \subset Q[F_\mu] \to UB(Q,f,\{\rho_i\})$$
the restriction of the canonical surjection $Q[F_\mu] \to B(Q,f,\{\rho_i\})$.
If $M$ is h.d.~1 and $(Q,f,\{\rho_i\})$ is f.g.~projective
the natural isomorphism can be written as
$$\Hom_{{\mathcal{F}\mathit{lk}}(A)}(B(Q,f,\{\rho_i\}),M)\cong~
\Hom_{{\mathcal{S}\mathit{ei}}_{\infty}(A)}((Q,f,\{\rho_i\}),U(M))$$
but note that in general $U(M)$ is not a f.g.~projective Seifert $A$--module.
The following result establishes that for an h.d.~1 $F_\mu$--link
module $M$ the Seifert $A$--module $U(M)$
is the direct limit of a directed system of f.g.~projective Seifert
$A$--modules $(P,e,\{\pi_i\})$ and morphisms in $\Xi$,
with isomorphisms $B(P,e,\{\pi_i\})\cong M$.
\begin{thm}[Blanchfield/Seifert algebraic transversality]
\label{Btrans}
Let $M$ be an h.d.~1 $F_\mu$--link module,
with a 1--dimensional induced f.g.~projective $A[F_\mu]$--module resolution
$$\disablesubscriptcorrection\xysavmatrix{0 \ar[r] & P[F_\mu] \ar[r]^-{\displaystyle{d}} &
P[F_\mu]\ar[r] & M \ar[r] &0}$$
such that $\epsilon(d)\co P \to P$ is an $A$--module isomorphism.
{\rm(i)}\qua
Let $I_\infty$ be the set of ordered pairs
$T = (T_0,T_1)$ of subtrees $T_0,T_1 \subseteq G_\mu$ such that
$d_*(T_1) \subseteq T_0$.
The set $I_\infty$ is partially ordered by inclusion, with maximal element
$$T_{\max} = \bigcup\limits_{T \in I_\infty}T = (G_\mu,G_\mu) \in I_{\infty}.$$
There is defined a directed system of Seifert $A$--modules
$(P\langle T \rangle,e\langle T\rangle,
\{\pi_i\langle T\rangle\})$ and morphisms in $\Xi_{\infty}$
$$\phi\langle T,T' \rangle\co
(P\langle T\rangle,e\langle T\rangle,\{\pi_i\langle T\rangle\})
\longrightarrow
(P\langle T'\rangle,e\langle T'\rangle,\{\pi_i\langle T'\rangle\})
\quad(T \subseteq T' \in I_\infty)$$
with direct limit
$$\varinjlim\limits_{T \in I_\infty}
(P\langle T\rangle,e\langle T\rangle,\{\pi_i\langle T\rangle\}) =
(P\langle T_{\max}\rangle,e\langle T_{\max}\rangle,\{\pi_i\langle T_{\max}\rangle\}) =
U(M).$$
For any $T = (T_0,T_1)\in I_\infty$ the morphism
$\phi\langle T,T_{\max} \rangle\co
(P\langle T\rangle,e\langle T\rangle,\{\pi_i\langle T\rangle\})\to U(M)$
is the adjoint $\phi\langle T,T_{\max} \rangle = \adj(\phi\langle T \rangle)$
of an isomorphism in ${\mathcal{F}\mathit{lk}}(A)$
$$\phi\langle T \rangle\co
B(P\langle T\rangle,e\langle T\rangle,\{\pi_i\langle T\rangle\})
\disablesubscriptcorrection\xysavmatrix{\ar[r]^-{\displaystyle{\cong}}&} M$$
such that for any $T \subseteq T' \in I_\infty$ there is defined a commutative
triangle of isomorphisms in ${\mathcal{F}\mathit{lk}}_\infty(A)$
$$\disablesubscriptcorrection\xysavmatrix@R+20pt{
B(P\langle T\rangle,e\langle T\rangle,\{\pi_i\langle T\rangle\})
\ar[dr]^-{\displaystyle{\phi\langle T \rangle}}_{\displaystyle{\cong}}
\ar[rr]^-{\displaystyle{B(\phi\langle T,T' \rangle)}}_{\displaystyle{\cong}}
& &B(P\langle T'\rangle,e\langle T'\rangle,\{\pi_i\langle T'\rangle\})
\ar[dl]_-{\displaystyle{\phi\langle T' \rangle}}^-{\displaystyle{\cong}}\\
&M&}$$
In particular, $\phi\langle T,T_{\max} \rangle \in \Xi_{\infty}$.
{\rm(ii)}\qua
Let $I\subset I_\infty$ be the subset of the ordered pairs
$T = (T_0,T_1)$ of finite subtrees $T_0,T_1 \subset G_\mu$ such that
$d_*T_1 \subseteq T_0$. For $T \in I$
$(P\langle T\rangle,e\langle T\rangle,\{\pi_i\langle T\rangle\})$ is
a f.g.~projective Seifert $A$--module, and
$$\varinjlim\limits_{T \in I}
(P\langle T\rangle,e\langle T\rangle,\{\pi_i\langle T\rangle\}) =
U(M)$$
with $\phi\langle T,T' \rangle \in \Xi$ $(T \subseteq T' \in I)$.
{\rm(iii)}\qua
For any f.g.~projective Seifert $A$--module $(Q,f,\{\rho_i\})$ every
morphism
$$g\co B(Q,f,\{\rho_i\}) \to M$$
in ${\mathcal{F}\mathit{lk}}(A)$ factors as
$$g\co B(Q,f,\{\rho_i\}) \disablesubscriptcorrection\xysavmatrix@C+20pt{\ar[r]^-{\displaystyle{B(g\langle T \rangle)}}&}
B(P\langle T \rangle,e\langle T \rangle,\{\pi_i\langle T \rangle\})
\disablesubscriptcorrection\xysavmatrix@C+10pt{\ar[r]_-{\displaystyle{\cong}}^-{\displaystyle{\phi\langle T \rangle}}&} M$$
for some $T \in I$, with
$g\langle T \rangle\co (Q,f,\{\rho_i\}) \to
(P\langle T\rangle,e\langle T\rangle,\{\pi_i\langle T\rangle\})$
a morphism in ${\mathcal{S}\mathit{ei}}(A)$.
\end{thm}
\begin{proof} (i)\qua The induced f.g.~projective
$A[F_\mu]$--module chain complex
$$\disablesubscriptcorrection\xysavmatrix{E\co E_1 = P[F_\mu] \ar[r]^-{{d}} & E_0 = P[F_\mu]}$$
is such that $H_0(E) = M$, $H_1(E) = 0$. By \fullref{MV} for any subtree
$T_1 \subseteq G_\mu$ there exists a subtree $d_*(T_1) \subseteq G_\mu$
such that for any subtree $T_0 \subseteq G_\mu$ with $d_*(T_1) \subseteq T_0$
$E$ admits a Mayer--Vietoris presentation
$$\disablesubscriptcorrection\xysavmatrix@C+20pt{
E_1\langle T_1 \rangle\co 0 \ar[r] & \bigoplus\limits_{i = 1}^\mu
C_1^{(i)} [F_{\mu}] \ar[r]^-{{f_1^+z-f_1^-}} \ar[d]^-{{d_C}}
& D_1[F_{\mu}] \ar[r]\ar[d]^-{{d_D}} & E_1 \ar[r]\ar[d]^-{{d}} & 0 \\
E_0\langle T_0 \rangle\co 0 \ar[r] & \bigoplus\limits_{i = 1}^\mu
C_0^{(i)} [F_{\mu}] \ar[r]^-{{f_0^+z-f_0^-}} & D_0[F_{\mu}] \ar[r] & E_0
\ar[r] & 0 }$$
\begin{align*}
&\text{with}\quad C^{(i)}_j = P\bigl[T_j^{(i,1)}\bigr],\quad
D_j = P\bigl[T_j^{(0)}\bigr]~\subseteq~ E_j = P[F_\mu]\quad(j = 0,1),\\
&\text{and}\quad d_C = {\textstyle\bigoplus_{i = 1}^\mu} d\vert\co
{\textstyle\bigoplus_{i = 1}^\mu} C^{(i)}_1 \to
{\textstyle\bigoplus_{i = 1}^\mu} C^{(i)}_0,\quad d_D = d\vert\co D_1 \to D_0.
\end{align*}
The $A$--modules defined by
\begin{align*}
P_i\langle T \rangle& =
\coker(d\vert\co C^{(i)}_1 \to C^{(i)}_0),\\
P\langle T \rangle& = \coker(d_C) =
{\textstyle\bigoplus_{i = 1}^\mu} P_i\langle T \rangle,\\
Q\langle T \rangle& = \coker(d_D)
\end{align*}
fit into a commutative diagram of $A[F_\mu]$--modules with exact rows
and columns
$$\disablesubscriptcorrection\xysavmatrix@C+10pt@R-5pt{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] & \\
0 \ar[r] & \bigoplus\limits_{i = 1}^\mu
C_1^{(i)} [F_{\mu}] \ar[r]^-{{f_1^+z-f_1^-}} \ar[d]^-{{d_C}}
& D_1[F_{\mu}] \ar[r]\ar[d]^-{{d_D}} & P[F_\mu] \ar[r]\ar[d]^-{{d}} & 0 \\
0 \ar[r] & \bigoplus\limits_{i = 1}^\mu
C_0^{(i)} [F_{\mu}] \ar[r]^-{{f_0^+z-f_0^-}} \ar[d]
& D_0[F_{\mu}] \ar[r]\ar[d] & P[F_\mu] \ar[r] \ar[d]& 0 \\
0 \ar[r] & P\langle T \rangle [F_\mu] \ar[r]^-{{f^+z-f^-}} \ar[d]& Q\langle T \rangle
[F_\mu] \ar[r] \ar[d]& M \ar[r] \ar[d]& 0\\
& 0 & 0 & 0 & }$$
with
$f^+,f^-\co P\langle T \rangle \to Q\langle T \rangle$
the $A$--module morphisms induced by
$$f^+_0,f^-_0\co {\textstyle\bigoplus^\mu_{i = 1}}
C^{(i)}_0 \to D_0.$$
It follows from $\Tor^{A[F_\mu]}_1(A,M) = 0$ that
$f^+-f^-\co P\langle T \rangle \to Q\langle T \rangle$
is an $A$--module isomorphism. The Seifert $A$--module
$(P\langle T \rangle,e\langle T \rangle,\{\pi_i\langle T \rangle\})$
defined by
$$
e\langle T \rangle = (f^+-f^-)^{-1}f^+\co P\langle T \rangle \to
P\langle T \rangle,\quad
\pi_i\langle T \rangle\co P \langle T \rangle\to
P_i\langle T \rangle \to P\langle T \rangle
$$
is such that $P\langle T \rangle[F_\mu]\cong Q\langle T \rangle[F_\mu] \to M$
induces the isomorphism of Blanchfield $A[F_\mu]$--modules
$\phi\langle T \rangle\co
B(P\langle T \rangle,e\langle T \rangle,\{\pi_i\langle T \rangle\})
\cong M$ adjoint to the natural map
$(P\langle T \rangle,e\langle T \rangle,\{\pi_i\langle T \rangle\})
\to U(M)$. (In particular,
$(P\langle T_{\max} \rangle,e\langle T_{\max} \rangle,\{\pi_i\langle T_{\max} \rangle\})
\allowbreak = U(M)$
and $\phi\langle T_{\max} \rangle\co BU(M)\cong M$ is the natural
isomorphism $\psi_M$ defined in \cite[5.10]{Sh2}.)
For $T \subseteq T' \in I$ the $B$--isomorphism $\phi \langle T,T' \rangle$
is induced by the inclusion $T \subseteq T'$.
(ii)\qua The augmentation of the $A[F_\mu]$--module morphism $d\co P[F_\mu] \to P[F_\mu]$
is an $A$--module isomorphism $\epsilon(d)\co P \to P$, so that the
induced $\smash{\widehat{A[F_\mu]}}$--module morphism $\widehat{d}\co
\smash{\widehat{P[F_\mu]}} \to \smash{\widehat{P[F_\mu]}}$ is an isomorphism, by
\fullref{Magnus-Foxon}. For any $T = (T_0,T_1) \in I$ the inclusion
$P\bigl[\smash{T_1^{(0)}}\bigr] \to \smash{\widehat{P[F_\mu]}}$ is a split $A$--module injection
by \fullref{Magnus-Foxtw}. Let $s\co \smash{\widehat{P[F_\mu]}} \to
P\bigl[\smash{T_1^{(0)}}\bigr]$
be a splitting $A$--module surjection. The anticlockwise composition
of the morphisms (inverting $\widehat{d}$) in the diagram
$$\disablesubscriptcorrection\xysavmatrix{
D_1 = P\bigl[T_1^{(0)}\bigr] \ar[d]_-{\displaystyle{d_D = d|}}\ar[r] & P[F_\mu]
\ar[r]\ar[d]^-{\displaystyle{d}} &
\widehat{P[F_\mu]}\ar@/_2pc/@{>}[ll]_-{\displaystyle{s}}
\ar[d]^-{\displaystyle{\widehat{d}}}_-{\displaystyle{\cong}} \\
D_0 = P\bigl[T_0^{(0)}\bigr] \ar[r] & P[F_\mu] \ar[r] & \widehat{P[F_\mu]}
}$$
defines an $A$--module surjection $P\bigl[\smash{T_0^{(0)}}\bigr] \to
P\bigl[\smash{T_1^{(0)}}\bigr]$
splitting $d\vert\co P\bigl[\smash{T_1^{(0)}}\bigr] \to
P\bigl[\smash{T_0^{(0)}}\bigr]$. Thus
$d\vert$ is a split injection of f.g.~projective $A$--modules and
$P\langle T \rangle = \coker(d\vert)$ is a f.g.~projective
$A$--module.
(iii)\qua The morphism $g\co B(Q,f,\{\rho_i\}) \to M$ in ${\mathcal{F}\mathit{lk}}(A)$
has a canonical resolution
$$\disablesubscriptcorrection\xysavmatrix@C+10pt@R+10pt{
0 \ar[r] & Q[F_{\mu}] \ar[d]^-{\displaystyle{\adj(g)}}\ar[r]^-{\displaystyle{1-f+fz}}
& Q[F_{\mu}] \ar[d]^-{\displaystyle{\adj(g)}}\ar[r] &
B(Q,f,\{\rho_i\}) \ar[d]^-{\displaystyle{g}}\ar[r] \ar[d]& 0 \\
0 \ar[r] & P\langle T_{\max} \rangle [F_\mu]
\ar[r]& P\langle T_{\max} \rangle [F_\mu]\ar[r] & M \ar[r]& 0}$$
with
$$\adj(g)\co (Q,f,\{\rho_i\}) \to
(P\langle T_{\max} \rangle,e\langle T_{\max} \rangle,
\{\pi_i\langle T_{\max} \rangle\}) = U(M)$$
the adjoint morphism in ${\mathcal{S}\mathit{ei}}_{\infty}(A)$.
Since $Q$ is f.g.~projective there exists $T \in I$ such that
$${\rm im}(g\co B(Q,f,\{\rho_i\}) \to M) \subseteq {\rm im}(
B(P\langle T \rangle,e\langle T \rangle,\{\pi_i\langle T \rangle\})
\to M)$$
with a lift of $g$ to an $A$--module morphism
$g\langle T \rangle\co Q \to P\langle T \rangle$
which preserves the direct sum structures.
The diagram of $A$--modules and morphisms
$$\disablesubscriptcorrection\xysavmatrix@C+10pt{
Q \ar[rr]^-{\displaystyle{f}} \ar[dd]^-{\displaystyle{g\langle T \rangle}}
\ar@/_4pc/[dddd]^{\displaystyle{g}}
&&
Q \ar[dd]^-{\displaystyle{g\langle T \rangle}}\ar@/^4pc/[dddd]^{\displaystyle{g}}
\\
&(*)& \\
P\langle T \rangle
\ar[rr]^-{\displaystyle{e\langle T \rangle}}
\ar[dd]^-{\displaystyle{\phi\langle T \rangle}} &&
P\langle T \rangle
\ar[dd]^-{\displaystyle{\phi\langle T \rangle}} \\
&&\\
U(M)\ar[rr]^-{\displaystyle{e}}&&U(M)}$$
commutes except possibly in $(*)$, and $(*)$ commutes if and only if
$$g\langle T \rangle\co (Q,f,\{\rho_i\}) \to (P\langle T \rangle,
e\langle T \rangle,\{\pi_i\langle T \rangle\})$$
is a morphism of Seifert $A$--modules. Since $Q$ is f.g.~projective and
the composite
$$\disablesubscriptcorrection\xysavmatrix@C+60pt{
Q \ar[r]^-{{g\langle T \rangle f-e\langle T \rangle g\langle T \rangle}} &
} P\langle T \rangle \disablesubscriptcorrection\xysavmatrix@C+15pt{\ar[r]^-{{\phi\langle T \rangle}} &}
U(M) = \varinjlim\limits_{T'\in I} P\langle T' \rangle$$
is 0 there exists $T' \in I$ such that $ T \subseteq T'$ and the composite
$$\disablesubscriptcorrection\xysavmatrix@C+60pt{g\langle T' \rangle f-e\langle T' \rangle g\langle T' \rangle\co
Q \ar[r]^-{{g\langle T \rangle f-e\langle T \rangle g\langle T \rangle}} &
} P\langle T \rangle \disablesubscriptcorrection\xysavmatrix{\ar[r]&} P\langle T' \rangle$$
is 0, so that
$$g\langle T' \rangle\co (Q,f,\{\rho_i\}) \to (P\langle T' \rangle,
e\langle T' \rangle,\{\pi_i\langle T' \rangle\})$$
is a morphism of Seifert $A$--modules as required (except that $T'$ has
to be called $T$).
\end{proof}
\begin{definition}
Let $M = B(P,e,\{\pi_i\})$ for a f.g.~projective Seifert $A$--module
$(P,e,\{\pi_i\})$.
(i)\qua For any $T \in I_{\infty}$ let
$$s\langle T \rangle\co (P,e,\{\pi_i\}) \to
(P\langle T \rangle ,e\langle T \rangle,\{\pi_i\langle T \rangle\})$$
be the $B$--isomorphism determined by the inclusion
$P = P[\{1\}] \subseteq P\bigl[\smash{T^{(0)}_0}\bigr]$.
(ii)\qua For $T = T_{\max}\in I_{\infty}$ write
$$s_M = s\langle T_{\max} \rangle\co
(P,e,\{\pi_i\}) \to
(P\langle T_{\max} \rangle ,e\langle T_{\max} \rangle,\{\pi_i\langle T_{\max} \rangle\}) =
U(M).$$
This is the $B$--isomorphism adjoint of $1\co M \to M$, such that
$$\disablesubscriptcorrection\xysavmatrix@C+10pt{s_M\co (P,e,\{\pi_i\}) \ar[r]^-{{s\langle T \rangle}}&
(P\langle T \rangle ,e,\{\pi_i\})
\ar[r]^-{{\phi \langle T \rangle}} & U(M)}$$
for any $T \in I_{\infty}$.
\end{definition}
Putting everything together:
\begin{thm} \label{calculus}
{\rm (i)}\qua Every h.d.~1 $F_\mu$--link module $M$ is isomorphic
to the covering $B(P,e,\{\pi_i\})$ of a f.g.~projective Seifert $A$--module
$(P,e,\{\pi_i\})$.
{\rm (ii)}\qua For any f.g.~projective Seifert $A$--modules $(P,e,\{\pi_i\})$,
$(Q,f,\{\rho_i\})$ every morphism
$g\co B(Q,f,\{\rho_i\})\to B(P,e,\{\pi_i\})$
in ${\mathcal{F}\mathit{lk}}(A)$ is of the form $g = B(s)^{-1}B(t)$
for some morphisms
$$s\co (P,e,\{\pi_i\}) \to (P',e',\{\pi'_i\}),~
t\co (Q,f,\{\rho_i\}) \to (P',e',\{\pi'_i\})$$
in ${\mathcal{S}\mathit{ei}}(A)$ with $s\in \Xi$.
{\rm (iii)}\qua If $u\co (Q,f,\{\rho_i\})\to (P,e,\{\pi_i\})$
is a morphism of f.g.~projective Seifert $A$--modules such that
$B(u) = 0$ there exists an element
$v\co (P,e,\{\pi_i\})\to (P',e',\{\pi'_i\})$ in $\Xi$
such that $vu = 0$.
{\rm (iv)}\qua The localization $\Xi^{-1}{\mathcal{S}\mathit{ei}}(A)$ has a left calculus of fractions, and the
covering construction defines an equivalence of exact categories
$$\wbar{B}\co
\Xi^{-1}{\mathcal{S}\mathit{ei}}(A)
\disablesubscriptcorrection\xysavmatrix{\ar[r]^-{\displaystyle{\approx}}&} {\mathcal{F}\mathit{lk}}(A);~(P,e,\{\pi_i\}) \mapsto B(P,e,\{\pi_i\}).$$
\end{thm}
\begin{proof} (i)\qua By \fullref{Btrans} (i)--(ii) $M$ is
isomorphic to $B(P\langle T \rangle,e\langle T \rangle,\{\pi_i\langle T \rangle\})$
for any $T \in I$, e.g. for the minimal element
$T_{\min} = (d_*\{1\},\{1\}) \in I$.
(ii)\qua By \fullref{Btrans} (iii) the adjoint of $g$ factors in
${\mathcal{S}\mathit{ei}}_{\infty}(A)$ as
$$\disablesubscriptcorrection\xysavmatrix@R+10pt@C-5pt{
(Q,f,\{\rho_i\}) \ar[rr]^-{\displaystyle{\adj(g)}}
\ar[dr]_-{\displaystyle{g\langle T \rangle}}
& &UB(P,e,\{\pi_i\})\\
&(P\langle T \rangle,e\langle T \rangle,\{\pi_i\langle T \rangle\})
\ar[ur]_-{\displaystyle{\adj(\phi\langle T \rangle)}}&}$$
for some $T\in I$. The morphisms in ${\mathcal{S}\mathit{ei}}(A)$ defined by
$$\begin{array}{l}
s = s\langle T \rangle\co (P,e,\{\pi_i\}) \to
(P',e',\{\pi'_i\}) =
(P\langle T \rangle,e\langle T \rangle,\{\pi_i\langle T \rangle\})\\[1ex]
t = g\langle T \rangle\co (Q,f,\{\rho_i\}) \to
(P',e',\{\pi'_i\}) =
(P\langle T \rangle,e\langle T \rangle,\{\pi_i\langle T \rangle\})
\end{array}$$
are such that $s$ is a $B$--isomorphism (ie $s \in \Xi$)
and $g = B(s)^{-1}B(t)$.
(iii)\qua Let $M = B(P,e,\{\pi_i\})$. We have a commutative diagram in ${\mathcal{S}\mathit{ei}}_{\infty}(A)$
$$\disablesubscriptcorrection\xysavmatrix@R+10pt@C-5pt{
(Q,f,\{\rho_i\}) \ar[rr]^-{\displaystyle{\adj(B(u)) = 0}}
\ar[dr]_-{\displaystyle{u}} & &U(M)\\
&(P,e,\{\pi_i\})
\ar[ur]_-{\displaystyle{\theta_M}}&}$$
Since $Q$ is f.g.~projective there exists $T \in I$ such that
$$v = s\langle T \rangle\co (P,e,\{\pi_i\}) \to (P\langle T \rangle,e\langle T \rangle,\{\pi_i\langle T \rangle\})$$
is a $B$--isomorphism in ${\mathcal{S}\mathit{ei}}(A)$ (ie $v \in \Xi$) with $vu = 0$.
(iv)\qua Immediate from (i)--(iii).
\end{proof}
This completes the proof of \fullref{thm2} of the Introduction.
\section{Primitive Seifert modules} \label{kernel}
This section is devoted to the kernel of the covering functor
$B\co {\mathcal{S}\mathit{ei}}(A) \to {\mathcal{F}\mathit{lk}}(A)$.
Following the terminology of Sheiham \cite{Sh2}:
\begin{definition}
(i)\qua A Seifert $A$--module $(P,e,\{\pi_i\})$ is {\it
primitive} if
$$B(P,e,\{\pi_i\}) = 0$$
or equivalently $1-e+ez\co P[F_\mu] \to P[F_\mu]$ is an $A[F_\mu]$--module
isomorphism.
(ii)\qua Let ${\mathcal{P}\mathit{rim}}(A) \subset {\mathcal{S}\mathit{ei}}(A)$ be the full subcategory with
objects the primitive f.g.~projective Seifert $A$--modules.
\end{definition}
We shall now obtain an intrinsic characterization of
the objects in ${\mathcal{P}\mathit{rim}}(A)$, generalizing the results for $\mu = 1$
recalled below.
\begin{definition}[{{L\"uck and Ranicki \cite[Section~5]{LR}}}]
A \emph{near-projection} $(P,e)$ is an $A$--module $P$
together with an endomorphism $e \in \End_A(P)$ such that
$e(1-e) \in \End_A(P)$ is nilpotent.
\end{definition}
\begin{proposition}
[Bass, Heller and Swan \cite{BHS}, L\"uck and Ranicki \cite{LR}]\quad
\label{split}
{\rm (i)}\qua A linear morphism of induced f.g.~projective $A[z]$--modules
$$f_0+f_1z\co P[z] \to Q[z]$$
is an isomorphism if and only if $f_0+f_1\co P \to Q$ is an isomorphism and
$$e = (f_0+f_1)^{-1}f_1\co P \to P$$
is nilpotent.
{\rm (ii)}\qua A linear morphism of induced f.g.~projective
$A[z,z^{-1}]$--modules
$$f_0+f_1z\co P[z,z^{-1}] \to Q[z,z^{-1}]$$
is an isomorphism if and only if $f_0+f_1\co P \to Q$ is an isomorphism and
$$e = (f_0+f_1)^{-1}f_1\co P \to P$$
is a near-projection.
{\rm (iii)}\qua Suppose that $(P,e)$ is a near-projection, or equivalently
that
$$1-e+ze\co P[z,z^{-1}] \to P[z,z^{-1}]$$
is an $A[z,z^{-1}]$--module automorphism. If $N \geqslant 0$ is so large
that $(e(1-e))^N = 0$ then
$$e^N+(1-e)^N\co P \to P$$
is an $A$--module automorphism, and the endomorphism
$$e_{\omega} = (e^N+(1-e)^N)^{-1}e^N\co P \to P$$
is a projection, with $e_{\omega}(1-e_{\omega}) = 0$.
The submodules of $P$
\begin{align*}
P^+& = (1-e_{\omega})(P) = (1-e)^N(P) = \{x \in P\,\vert\, (1-e+ez)^{-1}e(x)
\in P[z]\},\\[1ex]
P^-& = e_{\omega}(P) = e^N(P) = \{x \in P\,\vert\,
(1-e+ez)^{-1}(1-e)(x)\in z^{-1}P[z^{-1}]\}
\end{align*}
are such that
$$(P,e) = (P^+,e^+) \oplus (P^-,e^-)$$
with $e^+\co P^+ \to P^+$ and
$1-e^-\co P^- \to P^-$ nilpotent.
\end{proposition}
\begin{definition} {\rm
A f.g.~projective
Seifert $A$--module $(P,e,\{\pi_i\})$ is \emph{strongly nilpotent} if the
$A[F^+_{\mu}]$--module endomorphism
$$ez = \sum_{i = 1}^\mu e\pi_iz_i\co P[F_\mu^+]\to P[F_\mu^+]$$
is nilpotent, ie $(ez)^N = 0$ for some $N \geqslant 1$.}
\end{definition}
The condition for strong nilpotence is equivalent to the
$A[F_{\mu}]$--module endomorphism
$$ez = \sum_{i = 1}^\mu e\pi_iz_i\co P[F_\mu]\to P[F_\mu]$$
being nilpotent.
Expressed as a representation of the complete quiver $Q_{\mu}$, a
Seifert module $(P, \rho\co Q_{\mu}\to\End_AP)$ is strongly nilpotent
if and only if there exists $N \geqslant 1$ such that $\rho(p) = 0$
for every path $p\in Q_{\mu}$ of length $\geqslant N$.
\begin{proposition} The following conditions on a f.g.~projective
Seifert $A$--module $(P,e,\{\pi_i\})$
are equivalent:
\begin{itemize}
\item[\rm (i)] $(P,e,\{\pi_i\})$ is strongly nilpotent,
\item[\rm (ii)] the $A[F_{\mu}^+]$--module endomorphism
$$1-ez\co P[F^+_{\mu}] \to P[F^+_{\mu}]$$
is an automorphism,
\item[\rm (iii)] the $A[F_{\mu}^+]$--module endomorphism
$$1-e+ez\co P[F^+_{\mu}] \to P[F^+_{\mu}]$$
is an automorphism.
\end{itemize}
\end{proposition}
\begin{proof}
(i) $\Longrightarrow$ (ii)\qua If $(ez)^N = 0$ then $1-ez$ has inverse
$$\begin{array}{l}
(1-ez)^{-1} = 1+ez + (ez)^2 +\cdots + (ez)^{N-1}\\[1ex]
\hskip100pt
\in \Hom_{A[F^+_{\mu}]}(P[F^+_{\mu}],P[F^+_{\mu}]) =
\Hom_A(P,P)[F^+_{\mu}].
\end{array}$$
(ii) $\Longrightarrow$ (i)\qua The inverse of $1-ez$ is of the form
$$(1-ez)^{-1} = \sum_{\scriptsize
\begin{array}{c}
1 \leqslant i_1,i_2,\ldots,i_k \leqslant \mu\\
n_1,n_2,\ldots,n_k \geqslant 0\\
n_1+n_2+\cdots+n_k < N
\end{array}} f_{i_1 i_2 \ldots i_k} z_{i_1}^{n_1}z_{i_2}^{n_2}
\ldots z_{i_k}^{n_k}\co P[F^+_{\mu}] \to P[F^+_{\mu}]$$
for some $N \geqslant 1$. We have the identity
$$\begin{array}{l}
(1-ez)^{-1}-(1+ez+(ez)^2+\cdots+(ez)^{N-1}) = (1-ez)^{-1}(ez)^N\\[2ex]
\hskip100pt
\in \Hom_{A[F^+_{\mu}]}(P[F^+_{\mu}],P[F^+_{\mu}]) = \Hom_A(P,P)[F^+_{\mu}]
\end{array}$$
in which the left hand side is a sum of monomials in
$z_{i_1}z_{i_2}^{n_2}\ldots z_{i_k}^{n_k}$ of degree
$n_1+n_2+\cdots+n_k < N$ and the right hand side is a sum of monomials
of degree $\geqslant N$. Both sides of the identity are thus 0,
$$(ez)^N = 0\co P[F^+_{\mu}] \to P[F^+_{\mu}]$$
and $(P,e,\{\pi_i\})$ is strongly nilpotent.
(ii) $\Longleftrightarrow$ (iii)\qua Immediate from the identity
$$1-e+ez = 1-e(1-z)\co P[F^+_{\mu}] \to P[F^+_{\mu}]$$
and the change of variables $z_i \mapsto 1-z_i$.
\end{proof}
\begin{definition}
A $\mu$--component Seifert $A$--module $(P,e,\{\pi_i\})$ is a
\emph{near-projection} if it can be expressed as
$$(P,e,\{\pi_i\}) = \biggl(P^+\oplus P^-,\begin{pmatrix}
e^{++} & e^{+-} \\
e^{-+} & e^{--}
\end{pmatrix},\{\pi_i^+\}\oplus \{\pi_i^-\}\biggr)$$
and the $2\mu$--component Seifert $A$--module
$$(P',e',\pi') = \biggl( P^+\oplus P^-\ ,\
\begin{pmatrix}
e^{++} & -e^{+-} \\
e^{-+} & 1-e^{--}
\end{pmatrix}\ ,\ \{\pi_i^+\}\oplus\{\pi_i^-\}\biggr)$$
is strongly nilpotent.
\end{definition}
\begin{lemma} \label{near}
For a near-projection $(P,e,\{\pi_i\})$
the pairs $(P,e)$, $(P,e')$ are near-projections.
\end{lemma}
\begin{proof} We have a decomposition $P = P^+\oplus P^-$ with respect to
which $e'$ is strongly nilpotent. Now
\begin{align*}
e(1-e) & = \begin{pmatrix}
e^{++} & e^{+-} \\
e^{-+} & e^{--}
\end{pmatrix}\begin{pmatrix}
1-e^{++} & -e^{+-} \\
-e^{-+} & 1-e^{--}
\end{pmatrix} \\[1ex]
& = \begin{pmatrix}
e^{++}-(e^{++})^2-e^{+-}e^{-+} & -e^{++}e^{+-}+e^{+-}(1-e^{--}) \\
e^{-+}-e^{-+}e^{++}-e^{--}e^{-+} & -e^{-+}e^{+-}+e^{--}(1-e^{--})
\end{pmatrix} \\[1ex]
& = \begin{pmatrix}
e^{++}-(e^{++})^2-e^{+-}e^{-+} & -e^{++}e^{+-}+e^{+-}(1-e^{--}) \\
e^{-+} -e ^{-+}e^{++}-e^{--}e^{-+} &
-e^{-+}e^{+-}+(1-e^{--}) - (1-e^{--})^2
\end{pmatrix}.
\end{align*}
The matrix
$$e' = \left(\begin{array}{c|c}
e^{++} & -e^{+-} \\\hline
e^{-+} & 1-e^{--}
\end{array}\right)$$
denotes a strongly nilpotent representation of the complete quiver $Q_{2\mu}$
on $2 \mu$ vertices. In the following illustration $\mu = 1$:
$$\disablesubscriptcorrection\xysavmatrix{
\bullet \ar@/^/[r] \ar@(ul,dl)[] & \bullet \ar@/^/[l] \ar@(ur,dr)[]
}$$
Now each entry in the $2\mu\times 2\mu$ matrix $e(1-e)$ above
is (the image of) a linear combination of paths of length at least one in the
quiver. Hence each entry of $(e(1-e))^N$ is the image of a sum of
paths of length at least $N$. It follows that $(e(1-e))^N = 0$ for some
$N\geqslant 1$.
The pair $(P,e')$ is a near-projection since $e'\co P \to P$
is nilpotent.
\end{proof}
For $\mu = 1$ there is no difference between a near-projection
$(P,e,\{\pi_i\})$ and a near-projection $(P,e)$. For $\mu
\geqslant 2$ a near-projection $(P,e,\{\pi_i\})$ has $(P,e)$ a
near-projection (\fullref{near}) but the splitting
$(P,e) = (P^+,e^+) \oplus (P^-,e^-)$ given by \fullref{split} does not in general extend to a direct sum
decomposition of Seifert $A$--modules
$$(P,e,\{\pi_i\}) = (P^+,e^+,\{\pi^+_i\}) \oplus (P^-,e^-,\{\pi_i^-\}).$$
This is illustrated by the following example.
\begin{example}
Let $A$ be a field, and consider the $2$--component Seifert $A$--module
$(P,e,\{\pi_1,\pi_2\})$ given by
$$P = A^4,\quad e = \left(\begin{array}{cc|cc}
0 & 0 & 0 & 0 \\
0 & 1 & 1 & 0 \\ \hline
0 & 0 & 0 & 0 \\
1 & 0 & 0 & 1
\end{array}\right),~
\pi_1 = \begin{pmatrix} 1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \end{pmatrix},~
\pi_2 = \begin{pmatrix} 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \end{pmatrix}.$$
In this case $e\co P \to P$ is a projection, with $e(1-e) = 0$.
This f.g.~projective Seifert $A$--module has just one submodule
$$(\wwbar{P},\bar{e},\{\bar{\pi}_1,\bar{\pi}_2\})
\subseteq (P,e,\{\pi_1,\pi_2\})$$
namely
$$\wwbar{P} = e(P) = \{(0,x,0,y)\in P\,\vert\,(x,y) \in A^2\}.$$
It is not possible to decompose $(P,e,\{\pi_1,\pi_2\})$ as a direct
sum, since $(\wwbar{P},\bar{e},\{\bar{\pi}_1,\bar{\pi}_2\})$ is not
a summand. Neither $e$ nor $1-e$ is nilpotent but
{\small$$\begin{array}{ll}
1-e+ez& = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & z_1 & z_2-1 & 0 \\
0 & 0 & 1 & 0 \\
z_1-1 & 0 & 0 & z_2
\end{pmatrix}
= \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & z_2-1 & 0 \\
0 & 0 & 1 & 0 \\
z_1-1 & 0 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & z_1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & z_2
\end{pmatrix}
\end{array}$$}%
and
$$\begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & z_2-1 & 0 \\
0 & 0 & 0 & 0 \\
z_1-1 & 0 & 0 & 0
\end{pmatrix}^2 = 0$$
so $1-e+ez$ is invertible. Moreover, $(P,e,\{\pi_1,\pi_2\})$ is a near-projection, with
$$P_1^+ = A \oplus 0 \oplus 0 \oplus 0,~
P_1^- = 0 \oplus A \oplus 0 \oplus 0,~
P_2^+ = 0 \oplus 0 \oplus A \oplus 0,~
P_2^- = 0 \oplus 0 \oplus 0 \oplus A$$
such that
$$e' = \begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0\end{pmatrix}\co
P = P_1^+ \oplus P_1^- \oplus P_2^+ \oplus P_2^- \to
P = P_1^+ \oplus P_1^- \oplus P_2^+ \oplus P_2^-$$
is strongly nilpotent.
\end{example}
The main result of this section is:
\begin{thm}\label{characterize_primitives}
A f.g.~projective
Seifert $A$--module $(P,e,\{\pi_i\})$ is primitive if and only if it
is a near-projection.
\end{thm}
\begin{proof}
Suppose that $(P,e,\{\pi_i\})$ is a near-projection, with
$e' = \begin{pmatrix}
e^{++} & -e^{+-} \\
e^{-+} & 1-e^{--}
\end{pmatrix}$ strongly nilpotent. We have
{\small\begin{align*}
1{-}e{+}ez & = 1{-}e(1{-}z)\\[1ex]
& = \begin{pmatrix}
1{-}e^{++}(1{-}z) & -e^{+-}(1{-}z) \\
-e^{-+}(1{-}z) & 1{-}e^{--}(1{-}z)
\end{pmatrix} \\[1ex]
& = \begin{pmatrix}
1{-}e^{++}(1{-}z) & e^{+-}(1{-}z^{-1}) \\
-e^{-+}(1{-}z) & 1{-}(1{-}e^{--})(1{-}z^{-1})
\end{pmatrix}
\begin{pmatrix}
1 & 0 \\
0 & z
\end{pmatrix} \\[1ex]
& = \left(\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}{-}\begin{pmatrix}
e^{++} & -e^{+-} \\
e^{-+}& 1-e^{--}
\end{pmatrix}
\begin{pmatrix}
1{-}z & 0 \\
0 & 1{-}z^{-1}
\end{pmatrix}\right)
\begin{pmatrix}
1 & 0 \\
0 & z
\end{pmatrix}
\co(P^+ {\oplus} P^-)[F_{\mu}] \\
&\hskip242pt\longrightarrow (P^+ {\oplus} P^-)[F_{\mu}].
\end{align*}}
It follows from the strong nilpotence of $e'$ that
$e'((1-z)\oplus (1-z^{-1}))$ is nilpotent, and hence that
$$1-e(1-z) = (1-e'((1-z)\oplus (1-z^{-1})))(1\oplus z)\co
(P^+ \oplus P^-)[F_{\mu}] \to (P^+ \oplus P^-)[F_{\mu}]$$
is an isomorphism, so that $B(P,e,\{\pi_i\}) = 0$ and $(P,e,\{\pi_i\})$ is
primitive.
Conversely, suppose that $(P,e,\{\pi_i\})$ is a primitive
f.g.~projective Seifert $A$--module,
ie such that the $A[F_{\mu}]$--module morphism
$$1-e+ez\co P[F_{\mu}] \to P[F_{\mu}]$$
is an isomorphism. We shall use a variant $\wwbar{G}_\mu$ of
the Cayley tree $G_{\mu}$ (\fullref{cayley}) to prove that
$1-e+ez\co P[F_{\mu}] \to P[F_\mu]$ is a near-projection. Define
$$\wwbar{G}_{\mu}^{(0)} = F_\mu,
\quad\wwbar{G}_{\mu}^{(1)} = \bigl\{(w,z_iw)\,\vert\,
w \in F_{\mu},i \in \{1,2,\ldots,\mu\}\bigr\}$$
so that there is defined a right $F_\mu$--action
$$\wwbar{G}_{\mu} \times F_\mu \to \wwbar{G}_{\mu};~(w,g)
\mapsto wg.$$
For each $i = 1,2,\ldots,\mu$ partition $F_{\mu}$ as
$$F_{\mu} = F^{+,i}_{\mu} \sqcup F^{-,i}_{\mu} \sqcup \{1\}$$
with $F_{\mu}^{+,i}$ (resp. $F_{\mu}^{-,i}$) consisting of the
reduced words in $z_1,z_2,\ldots,z_{\mu}$ which start (resp. do
not start) with $z_i$. Removing the edge $(w,z_iw)$ disconnects
$\wwbar{G}_{\mu}$, and the complement is a disjoint union of
trees
$$\wwbar{G}_{\mu}-\{(w,z_iw)\} = \wwbar{G}_\mu^+(w,z_iw) \sqcup \wwbar{G}_\mu^-(w,z_iw)$$
with
$$\wwbar{G}_\mu^+(w,z_iw)^{(0)} = F^{+,i}w,~\wwbar{G}_\mu^-(w,z_iw)^{(0)} =
(F^{-,i}\cup \{1\})w.$$
In the diagram
\medskip
{\footnotesize
\Draw
\LineAt(-40,60,-40,20)
\LineAt(-40,20,40,20)
\LineAt(40,20,40,60)
\LineAt(80,60,80,20)
\LineAt(80,20,120,20)
\LineAt(120,-20,80,-20)
\LineAt(80,-20,80,-60)
\LineAt(80,-20,80,-60)
\LineAt(40,-60,40,-20)
\LineAt(40,-20,-40,-20)
\LineAt(-40,-20,-40,-60)
\LineAt(-80,-60,-80,-20)
\LineAt(-80,-20,-120,-20)
\LineAt(-120,20,-80,20)
\LineAt(-80,20,-80,60)
\MoveTo(33,3)
\Text(--$\disablesubscriptcorrection\xysavmatrix{\ar[r]^-{\displaystyle{ez}}&}$--)
\MoveTo(-87,3)
\Text(--$\disablesubscriptcorrection\xysavmatrix{\ar[r]^-{\displaystyle{ez}}&}$--)
\MoveTo(-33,-3)
\Text(--$\disablesubscriptcorrection\xysavmatrix{&\ar[l]^-{\displaystyle{1{-}e}}}$--)
\MoveTo(87,-3)
\Text(--$\disablesubscriptcorrection\xysavmatrix{&\ar[l]^-{\displaystyle{1{-}e}}}$--)
\MoveTo(72,30)
\Text(--$\disablesubscriptcorrection\xysavmatrix{\ar[d]^-{\displaystyle{ez}}&\\&}$--)
\MoveTo(67,-30)
\Text(--$\disablesubscriptcorrection\xysavmatrix{&\\ \ar[u]^-{\displaystyle{1{-}e}}&}$--)
\MoveTo(-48,30)
\Text(--$\disablesubscriptcorrection\xysavmatrix{\ar[d]^-{\displaystyle{ez}}&\\&}$--)
\MoveTo(-52,-30)
\Text(--$\disablesubscriptcorrection\xysavmatrix{&\\ \ar[u]^-{\displaystyle{1{-}e}}&}$--)
\MoveTo(0,0)
\Text(--$P_i$--)
\DrawOval(15,20)
\MoveTo(120,0)
\Text(--$z_iP_i$--)
\DrawOval(15,20)
\MoveTo(-120,0)
\Text(--$z^{-1}_iP_i$--)
\DrawOval(15,20)
\MoveTo(-60,60)
\Text(--$z^{-1}_jP_j$--)
\DrawOval(20,15)
\MoveTo(60,60)
\Text(--$z_iz^{-1}_jP_j$--)
\DrawOval(20,15)
\MoveTo(-60,-60)
\Text(--$P_j$--)
\DrawOval(20,15)
\MoveTo(60,-60)
\Text(--$z_iP_j$--)
\DrawOval(20,15)
\MoveTo(60,0)
\Text(--$z_iP$--)
\MoveTo(-60,0)
\Text(--$P$--)
\EndDraw
}
\medskip
we are placing the components of the range (resp.
domain) $P[F_{\mu}]$ at the vertices (resp. edges) of
$\wwbar{G}_{\mu}$, with the $A$--module $wP$ at $w \in
\wwbar{G}^{(0)}_{\mu}$, and the $A$--module $wP_i$ at
$(w,z_iw)\in \wwbar{G}^{(1)}_\mu$. An element
$$x \in P[F_{\mu}] = \!\!\!\!\sum\limits_{(w,z_iw)\in\wwbar{G}^{(1)}_{\mu}}
\!\!wP_i$$
is sent to
$$(1-e)(x) + ez(x) \in P[F_{\mu}] = \!\!\sum\limits_{w \in
\wwbar{G}^{(0)}_{\mu}}\!\!wP,$$
as indicated by the arrows in the diagram. For $i = 1,2,\ldots,\mu$
define the $A$--modules
\begin{align*}
P_i^+ & = \biggl\{x \in P_i\,\vert\,
(1-e+ez)^{-1}ez(x) \in \!\!\sum\limits_{w \in F^{+,i}_{\mu}}\!\!wP \biggr\},\\
P_i^- & = \biggl\{x \in P_i\,\vert\, (1-e+ez)^{-1}(1-e)(x)
\in\sum\limits_{j \neq i}P_j \oplus
\!\!\sum\limits_{w \in F^{-,i}_{\mu}}\!\!wP \biggr\}.
\end{align*}
An element $x^+ \in P_i$ belongs to $P_i^+$ if and only
if there exist elements $y^+(w) \in P$ ($w \in F^{+,i}_{\mu}$)
such that
$$
ez(x^+) = (1-e+ez)\biggl(\sum\limits_{w \in F^{+,i}_{\mu}}wy^+(w)\biggr) \in
\sum\limits_{w \in F_{\mu}^{+,i}}wP.\eqno{(*)}$$
There is one component $y^+(w)$ for each edge in
$\wwbar{G}^+_{\mu}(1,z_i)^{(1)}$, and one equation for each
vertex in $\wwbar{G}^+_{\mu}(1,z_i)^{(0)}$. Similarly, an
element $x^- \in P_i$ belongs to $P_i^-$ if and only if there
exist elements $y_j \in P_j$ ($j \neq i$) and $y^-(w) \in P$ ($w
\in F^{-,i}_{\mu}$) such that
$$((1-e)(x^-),0) =
(1-e+ez)\biggl(\sum\limits_{j\neq i}y_j+\sum\limits_{w \in
F^{-,i}_{\mu}}wy^-(w)\biggr) \in P\oplus\sum\limits_{w\in
F^{-,i}_{\mu}}wP.\eqno{(**)}$$ There is one component $y_j$ ($j
\neq i$) or $y^-(w)$ for each edge
$\wwbar{G}^-_{\mu}(1,z_i)^{(1)}$, and one equation for each
vertex in $\wwbar{G}^-_{\mu}(1,z_i)^{(0)}$. For
$i = 1,2,\ldots,\mu$ partition
$$F^{+,i}_{\mu} = F^{++,i}_{\mu} \sqcup F^{-+,i}_{\mu},\quad
F^{-,i}_{\mu} = F^{+-,i}_{\mu} \sqcup F^{--,i}_{\mu}$$ with
$F^{\alpha +,i}_{\mu}$ consisting of the words
$w = \smash{z_{i_0}^{\epsilon_0}}\ldots\smash{z_{i_k}^{\epsilon_k}} \in F_{\mu}$
with $(i_0,\epsilon_0) = (i,+)$, $\epsilon_k = \alpha$, and $F^{\alpha
-,i}_{\mu}$ consisting of the words $w = \smash{z_{i_0}^{\epsilon_0}}\ldots
\smash{z_{i_k}^{\epsilon_k}} \in F_{\mu}$ with $(i_0,\epsilon_0)\neq
(i,+)$, $\epsilon_k = \alpha$. For any $x^+ \in P^+_i$ and $w \in
F^{\alpha+,i}_{\mu}$ we have that $y^+(w) \in \smash{P^\alpha_j}$, as
given by all the terms in $(*)$ involving
$\wwbar{G}^+(w,z_iw)$. Similarly, for any $x^- \in P^-_i$ and
$w \in F^{\alpha-,i}_{\mu}$ we have that $y^-(w) \in P^\alpha_j$,
as given by all the terms in $(**)$ involving
$\wwbar{G}^-(w,z_iw)$.
Regarded as an $A$--module isomorphism
$1{-}e{+}ez\co P[F_{\mu}] \to P[F_{\mu}]$ can be expressed as
\begin{multline*}
1{-}e{+}ez = \\
\begin{pmatrix} ez\vert & (1{-}e{+}ez)\vert & 0 \\
(1{-}e)\vert & 0 & (1{-}e{+}ez)\vert \end{pmatrix}\co
P_i \oplus \biggl( \sum\limits_{w \in F^{+,i}_{\mu}}\!\!wP \biggr)\oplus
\biggl( \sum\limits_{j \neq i} P_j \oplus\!\! \sum\limits_{w \in
F^{-,i}_{\mu}}\!\!wP \biggr) \\
\longrightarrow \biggl(\sum\limits_{w \in F^{+,i}_{\mu}}\!\!wP \biggr)\oplus
\biggl(P\oplus\!\! \sum\limits_{w \in F^{-,i}_{\mu}}\!\!wP \biggr)
\end{multline*}
so that there is induced an $A$--module isomorphism
\begin{multline*}
\left[\begin{matrix} ez\vert \\ (1-e)\vert \end{matrix} \right]\co
P_i \to \biggl(\coker\biggl((1-e+ez)\vert\co \!\!\sum\limits_{w \in
F^{+,i}_{\mu}}\!\!wP
\to \!\!\sum\limits_{w \in F^{+,i}_{\mu}}\!\!wP\biggr)\biggr)\\
\oplus \biggl(\coker\biggl((1-e+ez)\vert\co
\sum\limits_{j \neq i} P_j \oplus \!\!\sum\limits_{w \in
F^{-,i}_{\mu}}\!\!wP
\to P\oplus \!\!\sum\limits_{w \in F^{-,i}_{\mu}}\!\!wP\biggr)\biggr)
\end{multline*}
and
$$P_i = P_i^+ \oplus P_i^-,$$
with
\begin{align*}
(1-e+ez)^{-1}ez(P^+_i) &\subseteq \sum\limits^{\mu}_{j = 1}
\sum\limits_{w \in F^{++,i}_{\mu}}\!\!wP^+_j \oplus
\sum\limits^{\mu}_{j = 1}
\sum\limits_{w \in F^{-+,i}_{\mu}}\!\!wP^-_j ,\\
(1-e+ez)^{-1}(1-e)(P^-_i) &\subseteq \sum\limits^{\mu}_{j = 1}
\sum\limits_{w \in F^{+-,i}_{\mu}}\!\!wP^+_j \oplus
\sum\limits^{\mu}_{j = 1} \sum\limits_{w \in F^{--,i}_{\mu}}\!\!wP^-_j.
\end{align*}
For $\alpha,\beta \in \{\pm\}$ let
$$e_{ji}^{\beta \alpha}\co P^{\alpha}_i \to P^{\beta}_j$$
be the $A$--module morphisms such that
$$e = \begin{pmatrix} e_{ji}^{++} & e_{ji}^{+-} \\ e_{ji}^{-+} &
e_{ji}^{--} \end{pmatrix}\co P = \sum\limits^{\mu}_{i = 1}(P_i^+ \oplus
P_i^-) \to P = \sum\limits^{\mu}_{j = 1}(P_j^+ \oplus P_j^-).$$
Let
$$\nu_{ji}^{\beta \alpha}(w)\co P^{\alpha}_i \to P^{\beta}_j$$
be the $A$--module morphisms such that
\begin{align*}
-(1{-}e{+}ez)^{-1}ez\vert & =
\begin{pmatrix} \sum\limits^{\mu}_{j = 1}
\sum\limits_{w \in F^{++,i}_{\mu}}w\nu^{++}_{ji}(w)\\
\sum\limits^{\mu}_{j = 1}
\sum\limits_{w \in F^{-+,i}_{\mu}}w\nu^{-+}_{ji}(w)\end{pmatrix}
\co P^+_i\\[-2ex]
&\hskip70pt\longrightarrow \sum\limits^{\mu}_{j = 1}
\sum\limits_{w \in F^{++,i}_{\mu}}wP^+_j \oplus
\sum\limits^{\mu}_{j = 1}
\sum\limits_{w \in F^{-+,i}_{\mu}}wP^-_j ,\\
-(1{-}e{+}ez)^{-1}(1-e)\vert & =
\begin{pmatrix} \sum\limits^{\mu}_{j = 1}
\sum\limits_{w \in F^{+-,i}_{\mu}}w\nu^{+-}_{ji}(w)\\
\sum\limits^{\mu}_{j = 1}
\sum\limits_{w \in F^{--,i}_{\mu}}w\nu^{--}_{ji}(w)\end{pmatrix}\co
P^-_i\\[-2ex]
&\hskip70pt\longrightarrow \sum\limits^{\mu}_{j = 1}
\sum\limits_{w \in F^{+-,i}_{\mu}}wP^+_j \oplus
\sum\limits^{\mu}_{j = 1}
\sum\limits_{w \in F^{--,i}_{\mu}}wP^-_j.
\end{align*}
Composing with $1-e+ez$ gives
\begin{equation}
\begin{aligned}
-e_{ji}^{++}z_i & = \sum\limits^{\mu}_{k = 1} \biggl(\sum\limits_{w \in
F^{++,i}_{\mu}}\!\!\!\!w(\delta_{jk}-e_{jk}^{++})\nu^{++}_{ki}(w)
+\!\!\!\! \sum\limits_{w \in
F^{-+,i}_{\mu}}\!\!\!\!wz_k(e_{jk}^{+-})\nu^{-+}_{ki}(w)\biggr)\co\\[-2ex]
&\hskip220pt P_i^+ \longrightarrow \!\!\!\!\sum\limits_{w \in
F^{++,i}_{\mu}}\!\!\!\!P_j^+,\\
-e_{ji}^{-+}z_i & = \sum\limits^{\mu}_{k = 1}
\biggl(
\sum\limits_{w \in F^{++,i}_{\mu}}\!\!\!\!we_{jk}^{-+}\nu^{++}_{ki}(w)
+\!\!\!\!\sum\limits_{w \in
F^{-+,i}_{\mu}}\!\!\!\!wz_ke_{jk}^{--}\nu^{-+}_{ki}(w)\biggr)\co\\[-2ex]
&\hskip220pt P_i^+ \longrightarrow \!\!\!\!\sum\limits_{w \in
F^{-+,i}_{\mu}}\!\!\!\!wP_j^-,\\
-(-e_{ji}^{+-}) & = \sum\limits^{\mu}_{k = 1} \biggl(\sum\limits_{w \in
F^{+-,i}_{\mu}}\!\!\!\!w(\delta_{jk}{-}e_{jk}^{++})\nu^{+-}_{ki}(w)
+ \!\!\!\!\sum\limits_{w \in
F^{-+,i}_{\mu}}\!\!\!\!wz_ke_{jk}^{+-}\nu^{--}_{ki}(w)\biggr)\co\\[-2ex]
&\hskip220pt P_i^- \longrightarrow
\!\!\!\!\sum\limits_{w \in F^{+-,i}_{\mu}}\!\!\!\!wP_j^+,\\
-(\delta_{ji}{-}e_{ji}^{--})& = \sum\limits^{\mu}_{k = 1}
\biggl(\sum\limits_{w \in F^{+-,i}_{\mu}}\!\!\!\!we_{jk}^{-+}\nu^{+-}_{ki}(w)
+\!\!\!\!\sum\limits_{w \in F^{--,i}_{\mu}}\!\!\!\!wz_ke_{jk}^{--}\nu^{--}_{ki}(w)
\biggr)\co\\[-2ex]
&\hskip220pt P_i^- \longrightarrow \!\!\!\!\sum\limits_{w \in
F^{--,i}_{\mu}}\!\!\!\!wP_j^-.
\end{aligned}
\tag{$*$}
\label{eqstar}
\end{equation}
Comparing the coefficients of $z_i$ and 1 gives
\begin{align*}
-\begin{pmatrix} e_{ji}^{++} \\ e_{ji}^{-+} \end{pmatrix} & =
\sum\limits^{\mu}_{k = 1}
\left(\begin{pmatrix} \delta_{jk}-e_{jk}^{++} \\ -e_{jk}^{-+} \end{pmatrix}
\nu^{++}_{ki}(z_i)+\begin{pmatrix} e_{jk}^{+-} \\ e_{jk}^{--} \end{pmatrix}
\nu^{-+}_{ki}(z_iz_k^{-1})\right)\co \\
&\hspace*{225pt}
P^+_i \to P^+_j \oplus P^-_j,\\
-\begin{pmatrix} -e_{ji}^{+-} \\ \delta_{ji}-e_{ji}^{--} \end{pmatrix} & =
\sum\limits^{\mu}_{k = 1}\left(
\begin{pmatrix} \delta_{jk}-e_{jk}^{++}\\ -e_{jk}^{-+} \end{pmatrix}
\nu^{+-}_{ki}(1)+
\begin{pmatrix} e_{jk}^{+-}\\ e_{jk}^{--} \end{pmatrix}
\nu^{--}_{ki}(z_k^{-1})\right)\co \\
&\hspace*{225pt}
P^-_i \to P^+_j \oplus P^-_j.
\end{align*}
Writing
$$\begin{pmatrix} \nu^{++} & \nu^{+-} \\ \nu^{-+} & \nu^{--} \end{pmatrix} =
\begin{pmatrix}
\nu_{ki}^{++}(z_i) & \nu_{ki}^{+-}(1) \\
\nu_{ki}^{-+}(z_iz_k^{-1}) & \nu_{ki}^{--}(z_k^{-1}) \end{pmatrix}\co
P^+ \oplus P^- \to P^+ \oplus P^-,$$
we thus have
$$-\begin{pmatrix} e^{++} & -e^{+-} \\ e^{-+} & 1-e^{--} \end{pmatrix} =
\begin{pmatrix} 1-e^{++} & e^{+-} \\ -e^{-+} & e^{--} \end{pmatrix}
\begin{pmatrix} \nu^{++} & \nu^{+-} \\ \nu^{-+} & \nu^{--} \end{pmatrix}.$$
Let $Q^{+,-}_{\mu}$ be the quiver with $2\mu$ vertices
$(i,\pm)_{1 \leqslant i \leqslant \mu}$ and one edge
$(i_0,\epsilon_0) \to (i_1,\epsilon_1)$ for each pair of vertices
with $(i_0,\epsilon_0) \neq (i_1,-\epsilon_1)$. (The path ring is given by
\begin{multline*}
Q^{+,-}_{\mu} = \\
\Z[s]*\Z\Bigl[\pi^+_1,,\ldots,\pi^+_{\mu},
\pi^-_1,\ldots,\pi^-_{\mu}\,\vert\,\pi^{\alpha}_i\pi^{\beta}_j
= \delta_{\alpha\beta}\delta_{ij}\pi^{\alpha}_i,
\sum^{\mu}_{i = 1}(\pi^+_i+\pi^-_i) = 1\Bigr] \\
/\bigl\{\pi^{\alpha}_is\pi^{-\alpha}_i\bigr\}
\end{multline*}
where $\pi^{\alpha}_i s \pi^{\beta}_j$ ($(i,\alpha) \neq (j,-\beta)$)
corresponds to the unique path of length 1 from $(i,\alpha)$ to
$(j,\beta)$.) In the illustration $\mu = 2$:
$$\disablesubscriptcorrection\xysavmatrix{
\bullet \ar@/^/[r] \ar@(ul,dl)[] \ar@/^/[d]
& \bullet \ar@/^/[l] \ar@(ur,dr)[] \ar@/^/[d] \\
\bullet \ar@/^/[r] \ar@(ul,dl)[] \ar@/^/[u]
& \bullet \ar@/^/[l] \ar@(ur,dr)[] \ar@/^/[u]}$$
Regard a word $w = z_{i_0}^{\epsilon_0}z_{i_1}^{\epsilon_1} \ldots
z_{i_k}^{\epsilon_k} \in F_{\mu}$ as a path of length $\vert w \vert = k$ in
$Q^{+,-}_{\mu}$
$$(i_0,\epsilon_0) \to (i_1,\epsilon_1) \to\cdots \to (i_k,\epsilon_k)$$
and for $k \geqslant 1$ define an $A$--module morphism
$\nu(w)\co P^{\epsilon_0}_{i_0} \to P^{\epsilon_1}_{i_1}$ as follows.
Define
$$[w] = \bigl[z_{i_0}^{\epsilon_0}z_{i_1}^{\epsilon_1}\bigr]
\bigl[z_{i_1}^{\epsilon_0}z_{i_2}^{\epsilon_1}\bigr] \ldots
\bigl[z_{i_{k-1}}^{\epsilon_{k-1}}z_{i_k}^{\epsilon_k}\bigr]\in F_{\mu}$$
with
$$\bigl[z_{i_0}^{\epsilon_0}z_{i_1}^{\epsilon_1}\bigr] =
\begin{cases}
z_{i_0}^{\epsilon_0}&\hbox{\rm if $(\epsilon_0,\epsilon_1) = (+,+)$}\\[1ex]
z_{i_0}^{\epsilon_0}z_{i_1}^{\epsilon_1}&
\hbox{\rm if $(\epsilon_0,\epsilon_1) = (+,-)$}\\[1ex]
z_{i_1}^{\epsilon_1}&\hbox{\rm if $(\epsilon_0,\epsilon_1) = (-,-)$}\\[1ex]
1&\hbox{\rm if $(\epsilon_0,\epsilon_1) = (-,+)$}.
\end{cases}$$
For $k = 1$ set
$$\nu\bigl(z_{i_0}^{\epsilon_0}z_{i_1}^{\epsilon_1}\bigr) =
\nu^{\epsilon_1\epsilon_0}_{i_1i_0}\bigl(\bigl[z_{i_1}^{\epsilon_1}z_{i_0}^{\epsilon_0}\bigr]\bigr)$$
and for $k \geqslant 2$ set
$$\nu\bigl(z_{i_0}^{\epsilon_0}z_{i_1}^{\epsilon_1}\ldots
z_{i_k}^{\epsilon_k}\bigr) =
\nu\bigl(z_{i_{k-1}}^{\epsilon_{k-1}}z_{i_k}^{\epsilon_k}\bigr)\ldots
\nu\bigl(z_{i_1}^{\epsilon_1}z_{i_2}^{\epsilon_2}\bigr)
\nu\bigl(z_{i_0}^{\epsilon_0}z_{i_1}^{\epsilon_1}\bigr).$$
The identities
$$\nu(w) = \nu^{\epsilon_k\epsilon_0}_{i_ki_0}([w])\co
P^{\epsilon_0}_{i_0} \to P^{\epsilon_k}_{i_k}$$
may be verified by induction on $k$,
since both sides satisfy the equations \eqref{eqstar} and so
\begin{align*}
-(1-e+ez)^{-1}ez\vert & = \sum\limits_{w\in F^{+,i}_{\mu}}\!\!w\nu(w)\co
P^+_i \to \!\!\sum\limits_{w \in F^{+,i}_{\mu}}wP,\\
-(1-e+ez)^{-1}(1-e)\vert & = \sum\limits_{j \neq i}\nu^{+-}_{ji} +
\!\!\sum\limits_{w \in F^{-,i}_{\mu}}\!\! w\nu(w)\co
P^-_i \to \sum\limits_{j \neq i}P_j \oplus \!\!\sum\limits_{w \in F^{-,i}_{\mu}}wP.
\end{align*}
For $\alpha,\beta \in \{\pm\}$ let
$F^{\beta\alpha}_{\mu}$ be the set of paths
$$(i_0,\epsilon_0) \to (i_1,\epsilon_1) \to\cdots \to (i_k,\epsilon_k)$$
in $Q^{+,-}_{\mu}$ with $\epsilon_0 = \alpha$, $\epsilon_k = \beta$.
The $A[F_{\mu}]$--module endomorphism
$$\nu' = \begin{pmatrix} z & 0 \\ 0 & 1 \end{pmatrix}
\begin{pmatrix} \nu^{++} & \nu^{+-} \\ \nu^{-+} & \nu^{--} \end{pmatrix}
\begin{pmatrix} 1 & 0 \\ 0 & z^{-1} \end{pmatrix}\co
(P^+ \oplus P^-)[F_{\mu}] \to (P^+ \oplus P^-)[F_{\mu}] $$
is such that for any $N \geqslant 1$
\begin{multline*}
(\nu')^N =
\begin{pmatrix} \sum\limits_{w\in F^{++}_{\mu},\vert w \vert = N}w\nu^{++}(w) &
\sum\limits_{w\in F^{+-}_{\mu},\vert w \vert = N}w\nu^{+-}(w)\\
\sum\limits_{w\in F^{-+}_{\mu},\vert w \vert = N}w\nu^{-+}(w) &
\sum\limits_{w\in F^{--}_{\mu},\vert w \vert = N}w\nu^{--}(w)
\end{pmatrix}\co\\
(P^+ \oplus P^-)[F_{\mu}] \to (P^+ \oplus P^-)[F_{\mu}].
\end{multline*}
If $N \geqslant 1$ is so large that
$$(1-e+ez)^{-1} = \sum\limits_{w \in F_{\mu},\vert w \vert<N}a_ww\co
P[F_{\mu}] \to P[F_{\mu}]\quad (a_w \in \Hom_A(P_{i_0},P_{i_k})),$$
then for any word $w \in F_{\mu}$ of length $\vert w \vert = k > N$
$$\nu(w) = 0\co P^{\epsilon_0}_{i_0} \to P^{\epsilon_k}_{i_k}.$$
The $2\mu$--component Seifert module
$$(P',\nu',\pi') = \biggl(P^+ \oplus P^-,
\begin{pmatrix} \nu^{++} & \nu^{+-} \\ \nu^{-+} & \nu^{--} \end{pmatrix},
\{\pi_i^+ \oplus \pi_i^-\}\biggr)$$
is strongly nilpotent, with $(\nu' z')^N = 0$, regarding
$F_{2\mu}$ as free group on $2\mu$ generators
$z'_1,z'_2,\ldots,z'_{2\mu}$ and letting
$$z' = \begin{pmatrix}
z'_1 & 0 & \ldots & 0 \\
0 & z'_2 & \ldots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \ldots & z'_{2\mu}
\end{pmatrix}\co P'[F_{2\mu}] \to P'[F_{2\mu}].$$
Define the $2\mu$--component Seifert module
$$(P',e',\pi') = \biggl(P^+\oplus P^-,\begin{pmatrix}
e^{++} & -e^{+-} \\
e^{-+} & 1-e^{--}
\end{pmatrix},\{\pi^+_i\oplus \pi^-_i\}\biggr).$$
Applying the augmentation $\epsilon\co z_i \mapsto 1$ to the
$A[F_{\mu}]$--module morphisms
\begin{align*}
-\sum\limits_{w \in F^{+,i}_{\mu}}w\nu(w)&\co
P^+_i[F_\mu] \disablesubscriptcorrection\xysavmatrix{\ar[r]^{ez}&} P[F_{\mu}]
\disablesubscriptcorrection\xysavmatrix@C+20pt{\ar[r]^{(1-e+ez)^{-1}}&} P[F_{\mu}],\\[1ex]
-\bigg(\sum\limits_{j \neq i}\nu^{+-}_{ji}+\sum\limits_{w \in
F^{-,i}_{\mu}}w\nu(w)\bigg)&\co
P^-_i[F_\mu] \disablesubscriptcorrection\xysavmatrix{\ar[r]^{1-e}&} P[F_{\mu}]
\disablesubscriptcorrection\xysavmatrix@C+20pt{\ar[r]^{(1-e+ez)^{-1}}&} P[F_{\mu}]
\end{align*}
shows that the components of $e'$ are given by linear combinations
of paths of length $\geqslant 1$
\begin{align*}
e^{++} & = -\sum\limits_{w \in F^{++}_{\mu}}\nu(w)\co P^+ \to P^+,\\
e^{-+} & = -\sum\limits_{w \in F^{+-}_{\mu}}\nu(w)\co P^+ \to P^-,\\
-e^{+-} & = -\sum\limits_{w \in F^{-+}_{\mu}}\nu(w)\co P^- \to P^+,\\
1-e^{--} & = -\sum\limits_{w \in F^{--}_{\mu}}\nu(w)\co P^- \to P^-.
\end{align*}
The $A[F_{2\mu}]$--module endomorphism $e'z'\co P'[F_{2\mu}] \to P'[F_{2\mu}]$
is nilpotent, with
$$(e'z')^N = 0,$$
so that $(P,e,\pi)$ is strongly nilpotent.
\end{proof}
This completes the proof of \fullref{thm3} of the Introduction.
\section{Algebraic $K$--theory} \label{ktheory}
We shall obtain our results on the algebraic $K$--theory of $A[F_\mu]$
and Blanchfield and Seifert modules using the Waldhausen \cite{Wald3}
algebraic $K$--theory of categories with cofibrations and weak
equivalences, and the noncommutative localization algebraic $K$--theory
exact sequence of Neeman and Ranicki \cite{NR1,NR2}.
\subsection{The algebraic $K$--theory of exact categories}
The higher algebraic $K$--groups $K_n(\mathcal{E})$ of an exact category $\mathcal{E}$ are
defined by Quillen \cite{Q} to be the homotopy groups of
a connective spectrum $K(\mathcal{E})$
$$\pi_n(K(\mathcal{E})) = K_n(\mathcal{E})\quad (n \geqslant 0)$$
with $K_0(\mathcal{E})$ the Grothendieck class group. The idempotent completion
$\mathcal{E} \to {\mathcal{P}}(\mathcal{E})$ induces an injection $K_0(\mathcal{E}) \to K_0({\mathcal{P}}(\mathcal{E}))$ and
isomorphisms $K_n(\mathcal{E}) \to K_n({\mathcal{P}}(\mathcal{E}))$ for $n \geqslant 1$, by the
cofinality theorem of Grayson \cite{Gr2}. The lower $K$--groups
$K_n(\mathcal{E})$ ($n \leqslant -1$) are defined by Schlichting \cite{Schl}
(following on from the definitions of Karoubi and Pedersen--Weibel for
the lower $K$--groups of filtered additive categories) to be the lower
homotopy groups of a nonconnective spectrum $K{\mathcal{P}}(\mathcal{E})$ such that
$$\pi_n(K{\mathcal{P}}(\mathcal{E})) = K_n({\mathcal{P}}(\mathcal{E}))\quad (n \in \Z),$$
with $K_n(\mathcal{E}) = K_n({\mathcal{P}}(\mathcal{E}))$ for $n \neq 0$.
The algebraic $K$--groups of a ring $R$ are the algebraic
$K$--groups of the idempotent complete exact category $\mathcal{E} = {\mathcal{P}\mathit{roj}}(R)$
of f.g.~projective $R$--modules
$$K_n(R) = K_n({\mathcal{P}\mathit{roj}}(R))\quad (n \in \Z),$$
as defined for $-\infty < n \leqslant 1$ in Bass \cite{B2}, and for $2
\leqslant n < \infty$ in Quillen \cite{Q}. The nonconnective spectrum
defined by $K(R) = K{\mathcal{P}}({\mathcal{P}\mathit{roj}}(R))$ has homotopy groups $\pi_*(K(R)) = K_*(R)$.
A \emph{Waldhausen category} $(\C,w)$ is a small category $\C$ with cofibrations
together with a subcategory $w \subset \C$ of weak equivalences satisfying
the axioms of \cite{Wald3}. As usual, there is defined a connective
algebraic $K$--theory spectrum
$$K(\C,w) = \Omega \vert w S_{\bullet} \C\vert $$
with homotopy groups the algebraic $K$--theory groups
$$K_n(\C,w) = \pi_n(K(\C,w))\quad (n \geqslant 0).$$
A functor $F\co (\C,w) \to (\C',w')$ of Waldhausen categories
induces a long exact sequence of algebraic $K$--groups
$$\disablesubscriptcorrection\xysavmatrix@C-10pt
{\cdots \ar[r]&K_{n+1}(F) \ar[r] & K_n(\C,w)
\ar[r]^-{\displaystyle{F}} & K_n(\C',w') \ar[r] & \cdots \ar[r]&K_0(F) \ar[r] &0}$$
with $K_n(F) = \pi_n(F\co K(\C,w) \to K(\C',w'))$ ($n \geqslant 0$).
As in Thomason and Trobaugh \cite[1.9]{TT} we shall only be considering
Waldhausen categories $(\C,w)$ which are `complicial biWaldhausen', so
that in particular $\C$ is a full subcategory of the category of chain
complexes in an abelian category ${\mathcal A}$, the cofibrations are
chain maps which are split injections in each degree, $w$ contains the
quasi-isomorphisms ( = the chain maps inducing isomorphisms in homology),
and which in addition are closed under the formation
of canonical homotopy pushouts and pullbacks.
The \emph{homotopy} (or \emph{derived}) \emph{category} \cite[page~269]{TT}
of a Waldhausen category $(\C,w)$ is the category of fractions
$$D(\C,w) = w^{-1}\C,$$
which is a triangulated category under the above hypotheses.
The idempotent completion ${\mathcal{P}} D(\C,w)$ is then
also triangulated (Balmer and Schlichting \cite{BS}),
and the class groups $K_0(D(\C,w))$, $K_0({\mathcal{P}} D(\C,w))$ are defined,
with $K_0(D(\C,w)) = K_0(\C,w)$.
Schlichting \cite{Schl} defined the lower $K$--groups $K_n({\mathcal{P}} D(\C,w))$
for $n \leqslant -1$ for Waldhausen categories as above,
and constructed a nonconnective spectrum $K{\mathcal{P}}(\C,w)$
with homotopy groups
$$\pi_n(K{\mathcal{P}}(\C,w)) = K{\mathcal{P}}_n(\C,w) = \begin{cases}
K_n(\C,w)&\hbox{\rm for $n \geqslant 1$}\\
K_0({\mathcal{P}} D(\C,w))&\hbox{\rm for $n = 0$}\\
K{\mathcal{P}}_n(\C,w)&\hbox{\rm for $n\leqslant -1$.}
\end{cases}$$
A functor $F\co (\C,w) \to (\C',w')$ of Waldhausen categories
induces a long exact sequence of algebraic $K$--groups
$$\disablesubscriptcorrection\xysavmatrix@C-10pt{\cdots \ar[r]&K{\mathcal{P}}_{n+1}(F) \ar[r] & K{\mathcal{P}}_n(\C,w)
\ar[r]^-{\displaystyle{F}} & K{\mathcal{P}}_n(\C',w') \ar[r] & K{\mathcal{P}}_n(F)\ar[r] &\cdots,}$$
with $K{\mathcal{P}}_n(F) = \pi_n(F\co K{\mathcal{P}}(\C,w) \to K{\mathcal{P}}(\C',w'))$ $(n \in \Z)$.
Given an exact category $\mathcal{E}$ let $C^b(\mathcal{E})$ be the
category of bounded chain complexes in $\mathcal{E}$ and chain maps.
An object $C$ in $C^b(\mathcal{E})$ is \emph{acyclic} (in the sense of Keller \cite[Chapter 11]{K})
if each differential $d\co C_r \to C_{r-1}$ factors as
$C_r \to Z_r \to C_{r-1}$ with
$$0 \to Z_{r+1} \to C_r \to Z_r \to 0$$
exact. A morphism $f\co C \to D$ in $C^b(\mathcal{E})$ is a \emph{quasi-isomorphism}
if the mapping cone $\C(f)$ is chain equivalent to an acyclic complex.
If $\mathcal{E}$ is fully embedded in an abelian category $\mathcal{A}$ with the embedding
closed under extensions and the idempotent completion ${\mathcal{P}}(\mathcal{E})$ is closed
under taking kernels of surjections then a quasi-isomorphism is the same
as a chain map inducing isomorphisms in homology in the ambient abelian
category $\mathcal{A}$ \cite[Appendix A]{TT}.
Let $(\C^b(\mathcal{E}),w_{\mathcal{E}})$ be the Waldhausen category with cofibrations
the chain maps which are degreewise split injections, and
$w_\mathcal{E}\subset\C^b(\mathcal{E})$ the subcategory of quasi-isomorphisms. The
derived category
$$D^b(\mathcal{E}) = D(\C^b(\mathcal{E}),w_{\mathcal{E}})$$
is the category of bounded chain complexes
in $\mathcal{E}$ and fractions of chain homotopy classes of chain maps,
with denominators quasi-isomorphisms. As usual, let $K^b(\mathcal{E})$ be the
category of bounded chain complexes in $\mathcal{E}$ and chain homotopy classes
of chain maps, and let $wK_{\mathcal{E}} \subset K^b(\mathcal{E})$ be the subcategory
of quasi-isomorphisms: the localization
$$D^b(\mathcal{E}) = (wK_{\mathcal{E}})^{-1}K^b(\mathcal{E})$$
has both a left and a right calculus of fractions. The derived
category $D^b(\mathcal{E})$ is a triangulated category \cite[1.9.6]{TT}.
Balmer and Schlichting \cite[2.12]{BS} prove that the idempotent
completion of the derived category is the derived category of the
idempotent completion
$${\mathcal{P}} D^b(\mathcal{E}) = D^b({\mathcal{P}}(\mathcal{E}))$$
and the algebraic $K$--groups are such that
$$\begin{cases}
K_n(\C^b(\mathcal{E}),w_\mathcal{E}) = K_n(\mathcal{E})&
\hbox{\rm for $n \geqslant 0$ (Gillet \cite{Gi})}\\[1ex]
K{\mathcal{P}}_n(\C^b(\mathcal{E}),w_\mathcal{E}) = K_n({\mathcal{P}}(\mathcal{E}))
&\hbox{\rm for $n \in \Z$ (Schlichting \cite{Schl})}.
\end{cases}$$
By \cite[1.9.2]{TT} the Waldhausen category defined in the same way but
with cofibrations the chain maps which are degreewise admissible
monomorphisms has the same algebraic $K$--theory.
\begin{definition} \label{w}
Let $F\co \mathcal{E} \to \D$ be a functor of exact categories.
{\rm (i)}\qua The \emph{algebraic $K$--groups} $K{\mathcal{P}}_*(\mathcal{E},\D)$ are
$$K{\mathcal{P}}_n(\mathcal{E},\D) = K{\mathcal{P}}_n(C^b(\mathcal{E},\D),w_{(\mathcal{E},\D)})\quad (n \in \Z)$$
with $(\C^b(\mathcal{E},\D),w_{(\mathcal{E},\D)}) \subset (\C^b(\mathcal{E}),w_{\mathcal{E}})$ the
Waldhausen subcategory with $\C^b(\mathcal{E},\D)\subset \C^b(\mathcal{E})$ the full
subcategory with objects the bounded chain complexes $C$ in $\mathcal{E}$
which are chain equivalent in $\D$ to acyclic complexes, and
$$w_{(\mathcal{E},\D)} = w_{\mathcal{E}}\cap \C^b(\mathcal{E},\D) \subset \C^b(\mathcal{E},\D)$$
the subcategory of the quasi-isomorphisms.
{\rm (ii)}\qua The \emph{algebraic $\Gamma K$--groups} of $F$ are
$$\Gamma K_n(F) = K{\mathcal{P}}_n(C^b(\mathcal{E}),w_{\D})\quad (n \in \Z)$$
with $w_{\D} \subset \C^b(\mathcal{E})$ the
subcategory with morphisms the chain maps in $\mathcal{E}$ which become
quasi-isomorphisms in $\D$, or equivalently such that
the mapping cones are in $\C^b(\mathcal{E},\D)$.
\end{definition}
The groups $\Gamma K_*(F)$ are the algebraic
$K$--theory analogues of the algebraic $L$--theory groups
$\Gamma_*(F)$ of Cappell and Shaneson \cite{CS1}.
\begin{thm} \label{long}
Let $F\co \mathcal{E} \to \D$ be a functor of exact categories.
{\rm (i)}\qua The algebraic $K$--groups fit into a commutative braid
of exact sequences
$$\disablesubscriptcorrection\xysavmatrix@C-15pt{
K{\mathcal{P}}_n(\mathcal{E},\D)\ar[dr] \ar@/^2pc/[rr]^{} && K{\mathcal{P}}_n(\mathcal{E}) \ar[dr]
\ar@/^2pc/[rr]^-{\displaystyle{F}} &&K{\mathcal{P}}_n(\D)\\&
K{\mathcal{P}}_{n+1}(F)\ar[ur] \ar[dr] &&
\Gamma K_n(F) \ar[ur]^-{\displaystyle{\Gamma F}}\ar[dr]&&\\
K{\mathcal{P}}_{n+1}(\D) \ar[ur] \ar@/_2pc/[rr]_-{}&&K{\mathcal{P}}_{n+1}(\Gamma F)
\ar[ur]\ar@/_2pc/[rr]_{}&&K{\mathcal{P}}_{n-1}(\mathcal{E},\D)}$$
with $\Gamma F\co (C^b(\mathcal{E}),w_{\D}) \to (C^b(\D),w_{\D})$ induced by $F$.
{\rm (ii)}\qua If $\Gamma F\co {\mathcal{P}} D(\C^b(\mathcal{E}),w_{\D}) \to {\mathcal{P}}
D(\C^b(\D),w_{\D})$ is an equivalence of categories then
$$K{\mathcal{P}}_*(\Gamma F) = 0,~K{\mathcal{P}}_{*+1}(F)\cong~K{\mathcal{P}}_*(\mathcal{E},\D),~
\Gamma K_*(F)\cong~K{\mathcal{P}}_*(\D)$$
and the braid of {\rm (i)} collapses to the exact sequence
$$\disablesubscriptcorrection\xysavmatrix{ \cdots \ar[r]& K{\mathcal{P}}_{n+1}(\D) \ar[r]&
K{\mathcal{P}}_n(\mathcal{E},\D) \ar[r] & K{\mathcal{P}}_n(\mathcal{E})
\ar[r]^-{\displaystyle{F}} & K{\mathcal{P}}_n(\D) \ar[r] &\cdots}$$
{\rm (iii)}\qua The hypothesis of {\rm (ii)} is satisfied if
$F\co \mathcal{E}\to\D = \Sigma^{-1}\mathcal{E}$ is the canonical functor to a
category of fractions and $\D$ has a calculus of left fractions.
\end{thm}
\begin{proof} (i)\qua The cases $n \geqslant 0$ are a direct application of
the version of the localization theorem of \cite[1.6.4]{Wald3} stated
in Theorem 2.3 and Lemma 2.5 of Neeman and Ranicki \cite{NR2}, with
$$\begin{array}{l}
{\mathcal R}^c = D(\C^b(\mathcal{E},\D),w_{(\mathcal{E},\D)})
\subset {\mathcal S}^c = D(\C^b(\mathcal{E}),w_{\mathcal{E}}),\quad
{\mathcal S}^c/{\mathcal R}^c \approx D(\C^b(\mathcal{E}),w_{\D}),\\
{\bf R} = (\C^b(\mathcal{E},\D),w_{(\mathcal{E},\D)}),\quad
{\bf S} = (\C^b(\mathcal{E}),w_\mathcal{E}),\quad {\bf T} = {\bf S}_{\bf R} = (\C^b(\mathcal{E}),w_\D)
\end{array}$$
giving a fibration sequence of connective spectra
$$K(\C^b(\mathcal{E},\D),w_{(\mathcal{E},\D)}) \to K(\C^b(\mathcal{E}),w_\mathcal{E}) \to K(\C^b(\mathcal{E}),w_{\D}).$$
The cases $n <0$ follow from Theorems 2.4, 3.7 of \cite{NR2} and
Schlichting \cite[Theorems 1,9]{Schl}, which give a fibration sequence of
nonconnective spectra
$$K{\mathcal{P}}(\C^b(\mathcal{E},\D),w_{(\mathcal{E},\D)}) \to K{\mathcal{P}}(\C^b(\mathcal{E}),w_\mathcal{E}) \to
K{\mathcal{P}}(\C^b(\mathcal{E}),w_{\D}).$$
(ii)\qua This is a direct application of the Approximation Theorem
of Waldhausen \cite[Theorem~1.6.7]{Wald3}: if $F\co (\C,w) \to (\C',w')$ is a functor which induces an equivalence
of the homotopy categories $F\co D(\C,w) \to D(\C',w')$ then
$F\co K(\C,w) \to K(\C',w')$ is a homotopy equivalence inducing isomorphisms
$F\co K_*(\C,w)\cong K_*(\C',w')$. Similarly, if $F\co {\mathcal{P}} D(\C,w) \to {\mathcal{P}}
D(\C',w')$ is an equivalence there are induced isomorphisms
$F\co K{\mathcal{P}}_*(\C,w)\cong K{\mathcal{P}}_*(\C',w')$ (\cite{Schl}).
(iii)\qua Every object $D$ in $\C^b(\D)$ is chain equivalent to $F(E)$ for an
object $E$ in $\C^b(\mathcal{E})$, and the functors $F\co C^b(\mathcal{E}) \to C^b(\D)$,
$F\co D(\C^b(\mathcal{E}),w_{\D}) \to D(\C^b(\D),w_{\D})$ are localizations.
\end{proof}
\begin{definition}
(i)\qua Write the algebraic $K$--groups of the exact categories
${\mathcal{P}\mathit{rim}}(A)$, ${\mathcal{S}\mathit{ei}}(A)$, ${\mathcal{B}\mathit{la}}(A)$, ${\mathcal{F}\mathit{lk}}(A)$ as
$$\begin{array}{l}
{\rm Prim}_*(A) = K_*({\mathcal{P}\mathit{rim}}(A)),~{\rm Sei}_*(A) = K_*({\mathcal{S}\mathit{ei}}(A)),\\[1ex]
{\rm Bla}_*(A) = K_*({\mathcal{B}\mathit{la}}(A)),~{\rm Flk}_*(A) = K_*({\mathcal{F}\mathit{lk}}(A)).
\end{array}$$
(ii)\qua Write the algebraic $K$--groups of the idempotent completion
of the homotopy category of
$(\C^b({\mathcal{S}\mathit{ei}}(A),{\mathcal{B}\mathit{la}}(A)),w_{({\mathcal{S}\mathit{ei}}(A),{\mathcal{B}\mathit{la}}(A))})$ as
$$({\rm Sei},{\rm Bla})_*(A) = K{\mathcal{P}}_*(\C^b({\mathcal{S}\mathit{ei}}(A),{\mathcal{B}\mathit{la}}(A)),w_{({\mathcal{S}\mathit{ei}}(A),{\mathcal{B}\mathit{la}}(A))}).$$
\end{definition}
\begin{proposition} \label{blink}
The covering functor $B\co {\mathcal{S}\mathit{ei}}(A) \to {\mathcal{B}\mathit{la}}(A)$
induces morphisms $B\co {\rm Sei}_*(A) \to {\rm Bla}_*(A)$ which fit into
a long exact sequence
$$\disablesubscriptcorrection\xysavmatrix@C-5pt{ \cdots \ar[r]& ({\rm Sei},{\rm Bla})_n(A) \ar[r]&
{\rm Sei}_n(A) \ar[r]^-{\displaystyle{B}} & {\rm Bla}_n(A)
\ar[r] & ({\rm Sei},{\rm Bla})_{n-1}(A) \ar[r] &\cdots}$$
with
$${\rm im}(B\co {\rm Sei}_0(A) \to {\rm Bla}_0(A)) = {\rm Flk}_0(A) \subseteq {\rm Bla}_0(A).$$
\end{proposition}
\begin{proof} Apply \fullref{long} (iii) with
$$F\co \mathcal{E} = {\mathcal{S}\mathit{ei}}(A) \to \D = \Xi^{-1}{\mathcal{S}\mathit{ei}}(A)~\approx~{\mathcal{F}\mathit{lk}}(A),$$
noting that ${\mathcal{S}\mathit{ei}}(A)$ is idempotent complete (\fullref{idem} (i)), that
$\Xi^{-1}{\mathcal{S}\mathit{ei}}(A)\approx {\mathcal{F}\mathit{lk}}(A)$ has a
left calculus of fractions by \fullref{calculus}, and
that ${\mathcal{B}\mathit{la}}(A)\approx {\mathcal{P}}({\mathcal{F}\mathit{lk}}(A))$ (\fullref{idem}(ii)).
\end{proof}
In the next section it will be shown that the functor
$${\mathcal{P}\mathit{rim}}(A) \to \C^b({\mathcal{S}\mathit{ei}}(A),{\mathcal{B}\mathit{la}}(A));~(P,e,\{\pi_i\}) \mapsto
(\cdots \to 0 \to (P,e,\{\pi_i\}))$$
induces isomorphisms of algebraic $K$--groups ${\rm Prim}_*(A)\cong
({\rm Sei},{\rm Bla})_*(A)$.
\subsection{The algebraic $K$--theory of noncommutative localizations}
Given a ring $R$ let ${\mathcal{M}\mathit{od}}(R)$ be the abelian category of $R$--modules,
so that ${\mathcal{P}\mathit{roj}}(R) \subset {\mathcal{M}\mathit{od}}(R)$ is an exact subcategory.
Write the Waldhausen category of ${\mathcal{P}\mathit{roj}}(R)$ as
$$(\C^b(R),w_R) = (\C^b({\mathcal{P}\mathit{roj}}(R)),w_{{\mathcal{P}\mathit{roj}}(R)}).$$
An object in $\C^b(R)$ is a bounded chain complex $C$ of f.g.~projective
$R$--modules; $C$ is acyclic if and only if $H_*(C) = 0$.
A morphism $f\co C \to D$ in $\C^b(R)$ is a chain map;
$f$ is in $w_R$ if and only if $f_*\co H_*(C) \to H_*(D)$ is an isomorphism.
The algebraic $K$--groups of $R$ are given by
$$K_*(R) = K_*({\mathcal{P}\mathit{roj}}(R)) = K{\mathcal{P}}_*(\C^b(R),w_R).$$
\indent
A ring morphism $\mathcal{F}\co R \to S$ induces a functor of abelian categories
$$\mathcal{F} = S\otimes_R-\co {\mathcal{M}\mathit{od}}(R) \to {\mathcal{M}\mathit{od}}(S);~P \mapsto S\otimes_RP$$
which restricts to an exact functor
$F\co {\mathcal{P}\mathit{roj}}(R) \to {\mathcal{P}\mathit{roj}}(S)$. There is also induced
a functor of Waldhausen categories
$$\mathcal{F}\co (\C^b(R),w_R) \to (\C^b(S),w_S);~C\mapsto S\otimes_RC.$$
The relative homotopy groups of $\mathcal{F}\co K(R) \to K(S)$
are the relative $K$--groups $K_*(\mathcal{F})$ in the long exact sequence
$$\disablesubscriptcorrection\xysavmatrix{\cdots \ar[r]& K_n(R) \ar[r]^{\mathcal{F} }& K_n(S) \ar[r] &
K_n(\mathcal{F}) \ar[r] & K_{n-1}(R) \ar[r] & \cdots.}$$
Let $R$ be a ring, and let $\Sigma$ be a set of morphisms
of f.g.~projective $R$--modules. A ring morphism $R\to T$ is
\emph{$\Sigma$--inverting}
if each $(s\co P \to Q) \in \Sigma$ induces a
$T$--module isomorphism $1\otimes s\co T\otimes_RP \to T\otimes_RQ$.
By Cohn \cite{Co} there exists a
\emph{universal $\Sigma$--inverting localization} ring morphism
$$\mathcal{F}\co R \to S = \Sigma^{-1}R$$
such that any $\Sigma$--inverting ring morphism $R \to T$ has a unique
factorization
$$\disablesubscriptcorrection\xysavmatrix{R\ar[r]^-{\mathcal{F} }& S \ar[r] &T.}$$
The category of fractions $\Sigma^{-1}{\mathcal{P}\mathit{roj}}(R)$ is equivalent to the
full subcategory
$${\mathcal{P}\mathit{roj}}_R(S) \subseteq {\mathcal{P}\mathit{roj}}(S)$$
with objects isomorphic to the f.g.~projective $S$--modules
$\Sigma^{-1}P = S\otimes_RP$ induced from f.g.~projective $R$--modules
$P$, and ${\mathcal{P}\mathit{roj}}(S) = {\mathcal{P}}({\mathcal{P}\mathit{roj}}_R(S))$ is the idempotent completion.
\begin{definition}
(i)\qua For any ring morphism $\mathcal{F}\co R \to S$
write the Waldhausen categories defined in \fullref{w} as
\begin{align*}
(\C^b({\mathcal{P}\mathit{roj}}(R),{\mathcal{P}\mathit{roj}}(S)),w_{({\mathcal{P}\mathit{roj}}(R),{\mathcal{P}\mathit{roj}}(S))}) & = (\C^b(R,S),w_{(R,S)}),\\
(\C^b({\mathcal{P}\mathit{roj}}(R)),w_S) & = (\C^b(R),w_S)
\end{align*}
with corresponding nonconnective algebraic $K$--theory spectra
$$K{\mathcal{P}}(\C^b(R,S),w_{(R,S)}) = K(R,S),~K{\mathcal{P}}(\C^b(R),w_S) = \Gamma K(\mathcal{F})$$
and algebraic $K$--groups $K_*(R,S)$, $\Gamma K_*(\mathcal{F})$.
An object in $\C^b(R,S)$ is a bounded chain complex $C$ of f.g.~projective
$R$--modules such that $H_*(S\otimes_RC) = 0$.
A morphism $f\co C \to D$ in $\C^b(R,S)$ is a chain map;
$f$ is in $w_{(R,S)}$ if and only if $f_*\co H_*(C) \to H_*(D)$ is an
isomorphism. A morphism $f\co C \to D$ in $\C^b(R)$ is
in $w_S$ if and only if $1\otimes f\co H_*(S\otimes_RC) \to H_*(S\otimes_RD)$ is
an isomorphism.
(ii)\qua For an injective universal localization $\mathcal{F}\co R \to S = \Sigma^{-1}R$
let $H(R,\Sigma)$ be the exact category of \emph{h.d.~1
$\Sigma$--torsion $R$--modules}, ie the cokernels of injective
morphisms $s\co P \to Q$ of f.g.~projective $R$--modules which induce
an $S$--module isomorphism $1\otimes s\co S\otimes_RP \to S\otimes_RQ$
(eg if $s \in \Sigma$).
\rm (iii)\qua (Neeman and Ranicki \cite{NR1,NR2})
A universal localization $\mathcal{F}\co R \to S = \Sigma^{-1}R$ is \emph{stably flat} if
$$\Tor^R_i(S,S) = 0 \quad(i \geqslant 1).$$
\end{definition}
In particular, a universal localization $\mathcal{F}\co R \to S$ is stably flat if
$S$ has flat dimension $\leqslant 1$ as an $R$--module, ie if there
exists a 1--dimensional
flat $R$--module resolution
$$0 \to F_1 \to F_0 \to S \to 0.$$
\begin{proposition} \label{Gamma}
{\rm (i)}\qua For any ring morphism $\mathcal{F}\co R \to S$ the functor
$${\Gamma \mathcal F}\co (\C^b(R),w_S) \to (\C^b(S),w_S);~C \mapsto S\otimes_RC$$
induces morphisms of algebraic $K$--groups
$\Gamma \mathcal{F}\co \Gamma K_*(\mathcal{F}) \to K_*(S)$ which fit into a
commutative braid of exact sequences
$$\disablesubscriptcorrection\xysavmatrix@C-5pt@R-20pt{
K_n(R,S)\ar[dr] \ar@/^2pc/[rr]^{} && K_n(R) \ar[dr]
\ar@/^2pc/[rr]^-{\displaystyle{\mathcal{F}}} &&K_n(S)\\&
K_{n+1}(\mathcal{F})\ar[ur] \ar[dr] &&
\Gamma K_n(\mathcal{F}) \ar[ur]^-{\displaystyle{\Gamma\mathcal F}} \ar[dr]&&\\
K_{n+1}(S) \ar[ur] \ar@/_2pc/[rr]_-{}&&K_{n+1}(\Gamma\mathcal{F})
\ar[ur]\ar@/_2pc/[rr]_{}&&K_{n-1}(R,S)}$$
{\rm (ii)}\qua For any universal localization $\mathcal{F}\co R \to S = \Sigma^{-1}R$
$$\begin{array}{l}
\Gamma K_n(\mathcal{F}) = K_n(S),~K_n(\mathcal{F}) = K_{n-1}(R,S),~
K_n(\Gamma\mathcal{F}) = 0\quad (n \leqslant 1).
\end{array}$$
{\rm (iii)}\qua For a stably flat universal localization
$\mathcal{F}\co R \to S = \Sigma^{-1}R$
$$\Gamma K_*(\mathcal{F}) = K_*(S),~K_{*+1}(\mathcal{F}) = K_*(R,S),~K_*(\Gamma\mathcal{F}) = 0,$$
and there is induced a localization exact sequence in the algebraic $K$--groups
$$\disablesubscriptcorrection\xysavmatrix@C-7pt{\cdots \ar[r]& K_n(R,S) \ar[r]&
K_n(R) \ar[r]^-{\displaystyle{\mathcal{F}}}& K_n(S) \ar[r]& K_{n-1}(R,S) \ar[r]& \cdots.}$$
{\rm (iv)}\qua For an injective universal localization
$\mathcal{F}\co R \to S = \Sigma^{-1}R$ there is defined an equivalence
of homotopy categories
$$D(\C^b(R,S),w_{(R,S)})~\approx~D(C^b(H(R,\Sigma)),w_{H(R,\Sigma)})$$
inducing isomorphisms
$$K_*(R,S)\cong~ K_*(H(R,\Sigma)).$$
{\rm (v)}\qua For an injective stably flat universal localization
$\mathcal{F}\co R \to S = \Sigma^{-1}R$ there is defined
a localization exact sequence in the algebraic $K$--groups
$$\disablesubscriptcorrection\xysavmatrix@C-7pt{\cdots \ar[r]& K_n(H(R,\Sigma)) \ar[r]&
K_n(R) \ar[r]^-{\displaystyle{\mathcal{F}}}& K_n(\Sigma^{-1}R) \ar[r]& K_{n-1}(H(R,\Sigma))
\ar[r]& \cdots}$$
\end{proposition}
\begin{proof} (i)\qua Immediate from \fullref{long} (i) and (ii) applied to
$\mathcal{F}\co \C^b(R) \to \C^b(S)$.
(ii)--(v)\qua See Neeman and Ranicki \cite{NR1,NR2}.
\end{proof}
\subsection{Triangular matrix rings}
We refer to Haghany and Varadarajan \cite{HV} for the
general theory of modules over triangular matrix rings,
and to Schofield \cite{Sc}, Ranicki \cite{RNLAT} and
Sheiham \cite{Sh4} for previous accounts of the universal
localization of triangular matrix rings.
\begin{proposition} \label{triangular}
Let
$$A = \begin{pmatrix} A_1 & B \\ 0 & A_2
\end{pmatrix}$$
be the triangular $2\times 2$ matrix ring
defined by rings $A_1,A_2$ and an $(A_1,A_2)$--bimodule $B$.
{\rm (i)}\qua An $A$--module $L = (L_1,L_2,\lambda)$ is defined by an $A_1$--module
$L_1$, an $A_2$--module $L_2$ and an $A_1$--module morphism
$\lambda\co B\otimes_{A_2}L_2 \to L_1$. As an additive group $L = L_1 \oplus L_2$,
written
$\bigl(\begin{smallmatrix} L_1 \\ L_2 \end{smallmatrix}\bigr)$,
with
$$\begin{pmatrix} A_1 & B \\ 0 & A_2 \end{pmatrix} \times
\begin{pmatrix} L_1 \\ L_2 \end{pmatrix} \to
\begin{pmatrix} L_1 \\ L_2 \end{pmatrix}\co
\biggl(\begin{pmatrix} a_1 & b \\ 0 & a_2 \end{pmatrix},
\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}\biggr) \to
\begin{pmatrix} a_1x_1+\lambda(b\otimes x_2) \\ a_2x_2 \end{pmatrix} .$$
{\rm (ii)}\qua An $A$--module morphism $(f_1,f_2)\co (L_1,L_2,\lambda) \to
(L'_1,L'_2,\lambda')$ is defined by an $A_1$--module morphism $f_1\co L_1 \to L'_1$,
and an $A_2$--module morphism $f_2\co L_2 \to L'_2$ such that the diagram
$$\disablesubscriptcorrection\xysavmatrix{
B\otimes_{A_2}L_2 \ar[r]^-{\displaystyle{\lambda}}\ar[d]_-{\displaystyle{1\otimes f_2}}
& L_1 \ar[d]^-{\displaystyle{f_1}}\\
B\otimes_{A_2}L'_2 \ar[r]^-{\displaystyle{\lambda'}}
& L'_1}$$
commutes.
{\rm (iii)}\qua An $A$--module $L = (L_1,L_2,\lambda)$ is
f.g.~projective if and only if $\lambda$ is injective,
$\coker(\lambda)$ is a f.g.~projective $A_1$--module, and $L_2$ is
a f.g.~projective $A_2$--module.
{\rm (iv)}\qua The projection
$${\mathcal{P}\mathit{roj}}(A) \to {\mathcal{P}\mathit{roj}}(A_1) \times {\mathcal{P}\mathit{roj}}(A_2);~
L = (L_1,L_2,\lambda) \mapsto (\coker(\lambda),L_2)$$
induces isomorphisms
$$K_*(A)\cong~ K_*(A_1) \oplus K_*(A_2).$$
{\rm (v)}\qua If an $A$--module $L = (L_1,L_2,\lambda)$ is h.d.~1 then
\begin{enumerate}
\item the 1--dimensional $A_1$--module chain complex
$$\disablesubscriptcorrection\xysavmatrix{K\co \cdots \ar[r] & 0 \ar[r] &
B\otimes_{A_2}L_2 \ar[r]^-{\displaystyle{\lambda}}& L_1}$$
is such that there exists a quasi-isomorphism ( = homology equivalence)
$J \to K$ for a 1--dimensional f.g.~projective $A_1$--module chain complex $J$,
and
\item $L_2$ is an h.d.~1 $A_2$--module.
\end{enumerate}
If $B$ is a flat right $A_2$--module the converse also holds:
an $A$--module $L$ is h.d.~1 if and only if conditions 1. and 2.
are satisfied.
{\rm (vi)}\qua The columns of $A$ are f.g.~projective $A$--modules
$$S_1 = (A_1,0,0),~S_2 = (B,A_2,1)$$
with
$$S_1\oplus S_2 = A,~{\rm End}(S_1) = A_1,~{\rm End}(S_2) = A_2.$$
The universal localization of $A$ inverting a non-empty subset
$\Sigma \subseteq \Hom_A(S_1,S_2)$ is a morphism of $2 \times 2$ matrix rings
$$A = \begin{pmatrix} A_1 & B \\ 0 & A_2 \end{pmatrix} \to
\Sigma^{-1}A = M_2(C) = \begin{pmatrix} C & C \\ C & C \end{pmatrix}$$
with $C$ the endomorphism ring of the induced f.g.~projective
$\Sigma^{-1}A$--module
$$\Sigma^{-1}S_1\cong \Sigma^{-1}S_2.$$
The composite of the functor
$$\Sigma^{-1}\co {\mathcal{M}\mathit{od}}(A) \to {\mathcal{M}\mathit{od}}(\Sigma^{-1}A);~P \mapsto
\Sigma^{-1}P = \Sigma^{-1}A\otimes_AP$$
and the Morita equivalence of categories
\begin{eqnarray*}
(C~C)\otimes_{\Sigma^{-1}A}-~:~
{\mathcal{M}\mathit{od}}(\Sigma^{-1}A) &\disablesubscriptcorrection\xysavmatrix{\ar[r]^-{\displaystyle{\simeq}}&}& {\mathcal{M}\mathit{od}}(C); \\
L = (L_1,L_2,\lambda) &\longmapsto& (C~C)\otimes_{\Sigma^{-1}A}L
\end{eqnarray*}
is the {\rm assembly} functor
\begin{align*}
{\mathcal{M}\mathit{od}}(A) &\longrightarrow {\mathcal{M}\mathit{od}}(C);\\
L = (L_1,L_2,\lambda) &\longmapsto
(C~C)\otimes_{\Sigma^{-1}A}\Sigma^{-1}L = (C~C)\otimes_AL\\
& = \coker\biggl(\begin{pmatrix} 1 \otimes \lambda \\ \kappa \otimes 1
\end{pmatrix}\co C \otimes_{A_1}B\otimes_{A_2}L_2 \to C \otimes_{A_1}L_1 \oplus
C\otimes_{A_2}L_2 \biggr)
\end{align*}
with
$$\kappa\co C \otimes_{A_1}B \to C;~x\otimes y \mapsto xy$$
the $(C,A_2)$--bimodule morphism defined by multiplication in $C$,
using the $A_1$--module morphism $B \to C$.
The assembly functor ${\mathcal{P}\mathit{roj}}(A) \to {\mathcal{P}\mathit{roj}}(C)$ induces the morphisms
$$\Sigma^{-1}\co K_*(A) = K_*(A_1)\oplus K_*(A_2) \to
K_*(\Sigma^{-1}A) = K_*(C).$$
{\rm (vii)}\qua If $B$ and $C$ are flat $A_1$--modules and $C$ is a flat
$A_2$--module then the $A$--module
$\bigl(\begin{smallmatrix}C \\ C \end{smallmatrix}\bigr)$
has a 1--dimensional flat $A$--module resolution
$$0 \to \begin{pmatrix} B \\ 0 \end{pmatrix}\otimes_{A_2}C
\to \begin{pmatrix} A_1 \\ 0 \end{pmatrix}\otimes_{A_1}C \oplus
\begin{pmatrix} B \\ A_2 \end{pmatrix}\otimes_{A_2}C
\to \begin{pmatrix} C \\ C \end{pmatrix} \to 0$$
so that $\Sigma^{-1}A = \bigl(\begin{smallmatrix} C \\ C
\end{smallmatrix}\bigr) \oplus \bigl(\begin{smallmatrix} C \\ C
\end{smallmatrix}\bigr)$ is stably flat.
An h.d.~1 $A$--module $L = (L_1,L_2,\lambda)$ is $\Sigma$--torsion if and only if
the $C$--module morphism
$$\begin{pmatrix} 1 \otimes \lambda \\ \kappa \otimes 1\end{pmatrix}\co
C \otimes_{A_1}B\otimes_{A_2}L_2 \to C \otimes_{A_1}L_1 \oplus
C\otimes_{A_2}L_2$$
is an isomorphism.
\end{proposition}
\begin{proof}
(i) and (ii)\qua Standard.
(iii)\qua For any $A$--module $L = (L_1,L_2,\lambda)$ there is defined an
exact sequence
$$0 \to (\ker(\lambda),0,0)\to
(B\otimes_{A_2}L_2,L_2,1) \disablesubscriptcorrection\xysavmatrix{\ar[r]^-{\displaystyle{(\lambda,1)}} &}
(L_1,L_2,\lambda) \to (\coker(\lambda),0,0) \to 0.$$
Now $(A_1,0,0) = \bigl(\begin{smallmatrix} A_1 \\ 0 \end{smallmatrix}\bigr)$ and
$(B,A_2,1) = \bigl(\begin{smallmatrix} B \\ A_2 \end{smallmatrix}\bigr)$ are
f.g.~projective $A$--modules, since
$$(A_1,0,0) \oplus (B,A_2,1) =
\biggl(A_1\oplus B,A_2,\begin{pmatrix} 0 \\ 1\end{pmatrix}\biggr) = A.$$
If $\ker(\lambda) = 0$ and $\coker(\lambda)$ is a f.g.~projective
$A_1$--module then
$(\coker(\lambda),0,0) = (A_1,0,0)\otimes_{A_1} \coker(\lambda)$
is a f.g.~projective $A$--module. If $L_2$ is a f.g.~projective $A_2$--module
then
$$(B\otimes_{A_2}L_2,L_2,1) = \begin{pmatrix} B \\ A_2 \end{pmatrix}\otimes_{A_2}L_2$$
is a f.g.~projective $A$--module. Thus if these two conditions are satisfied
then the exact sequence splits and $L$ is a f.g.~projective $A$--module.
Conversely, suppose that $(L_1,L_2,\lambda)$ is a f.g.~projective
$A$--module, so that there exists an $A$--module $(L'_1,L'_2,\lambda')$
with an $A$--module isomorphism
$$(L_1,L_2,\lambda) \oplus (L'_1,L'_2,\lambda')\cong~
\biggl((A_1)^k\oplus B^k,(A_2)^k,\begin{pmatrix} 0 \\ 1\end{pmatrix}\biggr) = A^k$$
for some $k \geqslant 0$. It follows from $\ker(\lambda \oplus
\lambda') = 0$ that $\ker(\lambda) = 0$, and from
$\coker(\lambda \oplus \lambda')\cong (A_1)^k$ that
$\coker(\lambda)$ is a f.g.~projective $A_1$--module.
Also, $L_2 \oplus L'_2\cong (A_2)^k$, so that $L_2$ is a f.g.~projective
$A_2$--module.
(iv)\qua The result that the inclusion and projection
$$i = A_1 \times A_2 \to A,\quad j\co A \to A_1 \times A_2$$
induce inverse isomorphisms
$$K_*(A_1 \times A_2) = K_*(A_1) \oplus K_*(A_2) \disablesubscriptcorrection\xysavmatrix{\ar@<0.5ex>[r]^{{i_*}} &
\ar@<0.5ex>[l]^{{j_*}}} K_*(A)$$
was first obtained by Berrick and Keating \cite{BK}.
Here is a proof using Waldhausen $K$--theory. It is immediate from $ji = 1$ that
$$j_*i_* = 1\co K_*(A_1 \times A_2) \to K_*(A) \to K_*(A_1 \times A_2).$$
Every f.g.~projective $A$--module $L = (L_1,L_2,\lambda\co B\otimes_{A_2}L_2 \to L_1)$
fits into a natural short exact sequence of f.g.~projective $A$--modules
$$0 \to (B\otimes_{A_2}L_2,L_2,1) \disablesubscriptcorrection\xysavmatrix{\ar[r]^-{{(\lambda,1)}}&}
(L_1,L_2,\lambda) \to (\coker(\lambda),0)\to 0.$$
The functors
$$\begin{array}{l}
F_1\co {\mathcal{P}\mathit{roj}}(A) \to {\mathcal{P}\mathit{roj}}(A);~ L \mapsto
A\otimes_{A_1}A_1\otimes_AL = (\coker(\lambda),0),\\[2ex]
F_2\co {\mathcal{P}\mathit{roj}}(A) \to {\mathcal{P}\mathit{roj}}(A);~ L \mapsto
A\otimes_{A_2}A_2\otimes_AL = (B\otimes_{A_2}L_2,L_2,1)
\end{array}$$
fit into a cofibration sequence
$$F_2 \to 1_{{\mathcal{P}\mathit{roj}}(A)} \to F_1,$$
and are such that
$$F_k\co K_*(A) \to K_*(A_k) \to K_*(A)\quad (k = 1,2).$$
Now apply the additivity theorem for Quillen $K$--theory
\cite[Proposition 1.3.2 (4)]{Wald3} to identify
$$i_*j_* = F_1 + F_2 = 1\co K_*(A) \to K_*(A),$$
so that $i_*$, $j_*$ are inverse isomorphisms.
(v)\qua If $L = (L_1,L_2,\lambda)$ is an h.d.~1 $A$--module there
exists a 1--dimensional f.g.~projective $A$--module resolution
$$\disablesubscriptcorrection\xysavmatrix@C+10pt{
0 \ar[r] & (P_1,P_2,f) \ar[r]^-{{(h_1,h_2)}}&
(Q_1,Q_2,g) \ar[r] & (L_1,L_2,\lambda) \ar[r] & 0,}$$
so that $\coker(f),\coker(g)$ are f.g.~projective $A_1$--modules
and $P_2,Q_2$ are f.g.~projective $A_2$--modules.
The 1--dimensional $A_1$--module chain complex
$$\disablesubscriptcorrection\xysavmatrix{K\co \cdots \ar[r] & 0 \ar[r] &
B\otimes_{A_2}L_2 \ar[r]^-{{\lambda}}& L_1}$$
and the 1--dimensional f.g.~projective $A_1$--module chain complex
$$J\co J_1 = \coker(f) \disablesubscriptcorrection\xysavmatrix{\ar[r]^-{{h_1}}&}
J_0 = \coker(g)$$
are related by a homology equivalence $J \to K$. Furthermore,
$L_2 = \coker(h_2)$ is an h.d.~1 $A_2$--module. Thus both conditions
1. and 2. are satisfied.
Conversely, suppose that $B$ is a flat right $A_2$--module and
that $L = (L_1,L_2,\lambda)$ is an $A$--module
such that there exists a homology equivalence $J \to K$ with $J$
a 1--dimensional f.g.~projective $A_1$--module chain complex and that
$L_2$ is an h.d.~1 $A_2$--module with a 1--dimensional
f.g.~projective $A_2$--module resolution
$$0 \to P_2 \to Q_2 \to L_2 \to 0.$$
There is induced a short exact sequence of $A_1$--modules
$$0 \to B\otimes_{A_2}P_2 \to B\otimes_{A_2}Q_2 \to B\otimes_{A_2}L_2 \to 0$$
and it follows from the 1--dimensional f.g.~projective $A$--module resolution of $L$
$$0\to (B\otimes_{A_2}P_2,P_2,1) \oplus (J_1,0,0) \to
(B\otimes_{A_2}Q_2,Q_2,1) \oplus (J_0,0,0) \to L \to 0$$
that $L$ is an h.d.~1 $A$--module.
(vi) and (vii)\qua See \cite[2.2]{RNLAT}.
\end{proof}
We shall actually be working with $(\mu+1) \times (\mu{+}1)$--matrix rings:
\begin{definition}
For any ring $A$ and $\mu \geqslant 1$ define
the triangular $(\mu+1)\times (\mu{+}1)$--matrix ring
$$T_\mu(A) = \begin{pmatrix} A & A \oplus A & A \oplus A & \ldots &
A\oplus A \\
0 & A & 0 & \ldots & 0 \\
0 & 0 & A & \ldots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 &\ldots & A
\end{pmatrix}.$$
\end{definition}
The ring $T_\mu(A)$ is the $A$--coefficient path algebra of the quiver
with vertices $0,1,\ldots,\mu$ and two arrows $s^+_i,s^-_i\co i \to 0$
for $i = 1,2,\ldots,\mu$. A $T_\mu(A)$--module $L = (L_i,f^+_i,f^-_i)$
consists of $A$--modules $L_0,L_1,\ldots,L_\mu$ and $A$--module morphisms
$f^+_i,f^-_i\co L_i \to L_0$ ($1 \leqslant i \leqslant \mu$).
Let $S_0,S_1,\ldots,S_\mu$ be the $T_\mu(A)$--modules defined
by the columns of $T_\mu(A)$, so that
\begin{align*}
S_0 & = (A,0,\ldots,0;0,\ldots,0),\\
S_i & = (A\oplus A,0,\ldots,0,A,0,\ldots,0;0,\ldots,0,{\rm id.},0,\ldots,0)
\quad (1 \leqslant i \leqslant \mu).
\end{align*}
It follows from
$$S_0 \oplus S_1 \oplus \cdots \oplus S_\mu = T_\mu(A)$$
that each $S_i$ is a f.g.~projective $T_\mu(A)$--module.
Let $\sigma = \{s^+_i,s^-_i\}$ be the set of
f.g.~projective $T_\mu(A)$--module morphisms
$$s^+_i = ((1~0),0,\ldots,0),~s^-_i = ((0~1),0,\ldots,0)\co
S_i \to S_0 \quad (1 \leqslant i \leqslant \mu).$$
\begin{proposition}\label{Floc}
{\rm (i)}\qua The universal $\sigma$--inverting localization of $T_\mu(A)$
is given by the inclusion
$$\mathcal{F}\co T_\mu(A) \to \sigma^{-1}T_\mu(A) = M_{\mu+1}(A[F_\mu])$$
with $M_{\mu+1}(A[F_\mu])$ the ring of all $(\mu+1)\times (\mu{+}1)$--matrices
with entries in $A[F_\mu]$.
The universal localization $\mathcal{F}$ is both injective and stably flat.
{\rm (ii)}\qua The composite
$$\disablesubscriptcorrection\xysavmatrix@C+10pt{
{\mathcal{M}\mathit{od}}(T_\mu(A)) \ar[r]^-{\displaystyle{\mathcal{F}}} &
{\mathcal{M}\mathit{od}}(M_{\mu+1}(A[F_\mu])) \ar[r]^-{\rm Morita}_-{\approx}& {\mathcal{M}\mathit{od}}(A[F_\mu])}$$
sends a $T_\mu(A)$--module $L = (L_i,f^+_i,f^-_i)$
to the {\rm assembly} $A[F_\mu]$--module
$$M = \coker\Bigl((f_1^+z_1{-}f_1^-~\ldots~ f_\mu^+z_\mu{-}f_\mu^-)\co
\bigoplus^\mu_{i = 1}L_i[F_\mu] \to L_0[F_\mu]\Bigr).$$
{\rm (iii)}\qua A $T_\mu(A)$--module $L = (L_i,f^+_i,f^-_i)$ is
f.g.~projective if and only if $L_0,\ldots,L_\mu$ are f.g.~projective
$A$--modules and the $A$--module morphism
$$\begin{pmatrix}
f^+_1 & f^-_1 & f^+_2 & f^-_2 & \ldots & f_{\mu}^+ & f_{\mu}^-
\end{pmatrix}\co \bigoplus^\mu_{i = 1}L_i \oplus L_i \to L_0$$
is a split injection. The projection
$${\mathcal{P}\mathit{roj}}(T_\mu(A)) \to \prod\limits_{\mu+1}{\mathcal{P}\mathit{roj}}(A);~
(L_i,f^+_i,f^-_i) \mapsto (L_0,L_1,L_2,\ldots,L_\mu)$$
induces isomorphisms in algebraic $K$--theory
$$K_*(T_\mu(A))\cong~\bigoplus\limits_{\mu+1}K_*(A).$$
{\rm (iv)}\qua A $T_\mu(A)$--module $L = (L_i,f^+_i,f^-_i)$ is h.d.~1
$\sigma$--torsion if and only if $L_0,\ldots,L_\mu$ are f.g.~projective
$A$--modules and the $A[F_\mu]$--module morphism
$$\begin{pmatrix} f^+_1z_1-f^-_1 & f^+_2z_2-f^-_2 & \ldots &
f^+_\mu z_\mu -f^-_\mu
\end{pmatrix}\co \bigoplus^\mu_{i = 1}L_i[F_\mu]\to L_0[F_\mu]$$
is an isomorphism. A f.g.~projective Seifert $A$--module $(P,e,\{\pi_i\})$ is
primitive if and only if $(P,P_i,f^+_i,f^-_i)$ is an h.d.~1
$\sigma$--torsion $T_\mu(A)$--module. The functor
$${\mathcal{P}\mathit{rim}}(A) \to H(T_\mu(A),\sigma);~(P,e,\{\pi_i\}) \mapsto
(P,P_i,e\pi_i,(e-1)\pi_i)$$
is an equivalence of exact categories, so that
$${\rm Prim}_*(A) = K_*(H(T_\mu(A),\sigma)).$$
The forgetful functor
\begin{multline*}
{\mathcal{P}\mathit{rim}}(A) \to\prod\limits_{2\mu}{\mathcal{P}\mathit{roj}}(A);\\
\biggl(P^+ \oplus P^-,\begin{pmatrix} e^{++} & e^{+-} \\ e^{-+} & e^{--}
\end{pmatrix},\{\pi^+_i\}\oplus \{\pi^-_i\}\biggr)
\mapsto (P^+_1,P^-_1,\ldots,P^+_\mu,P^-_\mu)
\end{multline*}
(defined using \fullref{characterize_primitives}) is split by
$$\prod\limits_{2\mu}{\mathcal{P}\mathit{roj}}(A) \to {\mathcal{P}\mathit{rim}}(A);\quad
(P^+_1,P^-_1,\ldots,P^+_\mu,P^-_\mu) \mapsto (P^+ \oplus P^-,0,\{\pi^+_i\}
\oplus \{\pi^-_i\}).$$
The reduced $K$--groups defined by
$$\widetilde{{\rm Prim}}_*(A) =
\ker({\rm Prim}_*(A) \to \bigoplus\limits_{2\mu} K_*(A))$$
are such that
$$K_*(H(T_\mu(A),\sigma)) =
{\rm Prim}_*(A) = \bigoplus\limits_{2\mu}K_*(A)
\oplus\widetilde{{\rm Prim}}_*(A).$$
\end{proposition}
\begin{proof} The universal localization $\sigma^{-1}T_{\mu}(A)$
is the $(\mu+1)\times (\mu{+}1)$--matrix
ring $M_{\mu+1}(R)$ with $R$ the endomorphism ring
of the induced f.g.~projective $\sigma^{-1}T_\mu(A)$--module $\sigma^{-1}S_0$,
and there is defined an isomorphism
$$A[F_\mu] \to R;~z_i \mapsto s^+_i(s^-_i)^{-1}.$$
The remaining parts are given by \fullref{triangular},
viewing the $(\mu+1)\times (\mu+1)$ matrix ring
$T_\mu(A)$ as a triangular $2 \times 2$ matrix ring
$$T_\mu(A) = \begin{pmatrix} A_1 & B \\ 0 & A_2 \end{pmatrix}$$
with
$$A_1 = A,\quad A_2 = \begin{pmatrix}
A & 0 & \ldots & 0 \\
0 & A & \ldots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 &\ldots & A
\end{pmatrix},\quad B = \begin{pmatrix} A\oplus A & \ldots & A\oplus A\end{pmatrix}$$
such that
$${\mathcal{M}\mathit{od}}(A_2) = \prod\limits_{\mu}{\mathcal{M}\mathit{od}}(A).$$
An $A_2$--module is just a $\mu$--tuple $(L_1,L_2,\ldots,L_{\mu})$
of $A$--modules. By the $2 \times 2$ theory a $T_\mu(A)$--module
$L$ just a $(\mu{+}1)$--tuple $(L_0,L_1,\ldots,L_{\mu})$ of $A$--modules,
together with $A$--module morphisms $f^+_i,f^-_i\co L_i \to L_0$
($1 \leqslant i \leqslant \mu$).
Note that $B$ is a flat right $A_2$--module, and that for an h.d.~1
$\sigma$--torsion $T_\mu(A)$--module $L = (L_i,f^+_i,f^-_i)$
each $L_i$ ($0 \leqslant i \leqslant \mu$) is a f.g.~projective
$A$--module, by the following argument.
The necessary and sufficient conditions of \fullref{triangular} (v) and (vii)
for a $T_\mu(A)$--module $L$ to be h.d.~1 $\sigma$--torsion
are:
\begin{itemize}
\item[(i)] there exists a 1--dimensional f.g.~projective $A$--module chain
complex $J\co J_1 \to J_0$ with a homology equivalence
$$\disablesubscriptcorrection\xysavmatrix@C+20pt{J_1 \ar[r] \ar[d] & J_0 \ar[d] \\
\bigoplus\limits^\mu_{i = 1}L_i \oplus L_i \ar[r]^-{\displaystyle{(f^+_i~f^-_i)}}&
L_0,}$$
\item[(ii)] each $L_i$ $(1 \leqslant i \leqslant \mu)$
is an h.d.~1 $A$--module,
\item[(iii)] the $A[F_\mu]$--module morphism
$$(f^+_iz_i-f^-_i)\co \bigoplus\limits^\mu_{i = 1}L_i[F_\mu]\to L_0[F_\mu]$$
is an isomorphism.
\end{itemize}
If $L$ satisfies these conditions there is defined a
commutative diagram of $A$--modules
$$\disablesubscriptcorrection\xysavmatrix@C+10pt{0 \ar[r] & \bigoplus\limits^\mu_{i = 1}L_i
\ar[r]^-{\bigl(\begin{smallmatrix} 1 \\ -1 \end{smallmatrix}\bigr)}
\ar[d]_{\displaystyle{\cong}}^-{{(f^+_i-f^-_i)}} &
\bigoplus\limits^\mu_{i = 1}L_i \oplus L_i \ar[d]^-{{(f^+_i~f^-_i) }}
\ar[r]^-{\displaystyle{(1~1)}} & \bigoplus\limits^{\mu}_{i = 1}L_i\ar[d] \ar[r]& 0
\\
0 \ar[r] & L_0\ar[r]^-{\displaystyle{1}} & L_0 \ar[r]& 0 \ar[r] & 0
}$$
with exact rows and with $(f^+_i-f^-_i)$ an isomorphism.
There are defined $A$--module isomorphisms
$$J_0 \oplus L_0\cong~ J_0 \oplus \bigoplus\limits^{\mu}_{i = 1}L_i\cong~J_1,$$
so that each $L_i$ ($0 \leqslant i \leqslant \mu$) is a f.g.~projective
$A$--module.
\end{proof}
\begin{example} The assembly of
$A[F_\mu]$--modules is an algebraic analogue of the geometric
construction of an $F_\mu$--cover $\wwtilde{W}$ of a space $W$
from a fundamental domain $U \subset \wwtilde{W}$. The subspaces
$$V_i = U \cap z^{-1}_iU,\quad z_iV_i = z_iU \cap U \subset
U \quad (1 \leqslant i \leqslant \mu)$$
are disjoint, with embeddings
$$f^+_i\co V_i \to U;~x \mapsto x,\quad f^-_i\co V_i \to U;~x \mapsto
z_ix \quad (1 \leqslant i \leqslant \mu),$$
and $\wwtilde{W}$ can be
constructed by glueing together $F_\mu$ copies of $U$
\begin{align*}
\wwtilde{W}& = (F_\mu \times U)/\{(g,f^+_i(x)) \sim (gz_i,f^-_i(x))\,\vert\,
g \in F_\mu,x \in V_i,1 \leqslant i \leqslant \mu\}\\[1ex]
& = \bigcup\limits_{g \in F_\mu} gU~{\rm with}~U \cap z_i^{-1}U = V_i.
\end{align*}
Such a situation arises if $W$ is a manifold with
a surjection $\pi_1(W) \to F_\mu$, eg a boundary link exterior.
The surjection is induced by a map
$$c\co W \to BF_\mu = \bigvee\limits_\mu S^1$$
which is transverse regular at $\{1,2,\ldots,\mu\} \subset BF_\mu$.
Cutting $W$ open at the inverse image codimension--1 submanifolds
$V_i = c^{-1}(\{i\}) \subset W$ there is obtained a fundamental
domain $U \subset \wwtilde{W}$ for the pullback
$\wwtilde{W} = c^*EF_\mu$ to $W$ of the universal cover $EF_{\mu}$ of $BF_\mu$.
More generally, suppose that $W$ is a finite $CW$ complex with
an $F_\mu$--cover $\wwtilde{W}$, and that $U \subset \wwtilde{W}$
is a fundamental domain which is a subcomplex. The embeddings
$f^+_i,f^-_i\co V_i\to U$ induce inclusions of the
cellular f.g.~free $\Z$--module chain complexes $f^+_i,f^-_i\co C(V_i)\to
C(U)$. The f.g.~projective $T_\mu(\Z)$--module chain complex
$C = (C(U),C(V_i),f^+_i,f^-_i)$ has assembly the cellular f.g.~free
$\Z[F_\mu]$--module chain complex of $\wwtilde{W}$
$$\coker\bigg((f_1^+z_1-f_1^- ~\ldots~f_\mu^+z_\mu-f_\mu^-)\co
\bigoplus^\mu_{i = 1}C(V_i)[F_\mu] \to C(U)[F_\mu]\bigg) = C(\wwtilde{W}),$$
such that
$$C(\wwtilde{W})_r = \coker\bigg((f_1^+ ~\ldots~
f_\mu^+)\co \bigoplus^\mu_{i = 1}C(V_i)_r \to C(U)_r\bigg)[F_\mu].$$
\end{example}
\begin{thm} The algebraic $K$--groups of $A[F_\mu]$ split as
$$K_*(A[F_\mu]) = K_*(A) \oplus \bigoplus\limits_{\mu} K_{*-1}(A) \oplus
\widetilde{{\rm Prim}}_{*-1}(A).$$
\end{thm}
\begin{proof} By \fullref{Floc} the universal localization
$$\mathcal{F}\co A[F_\mu]\to\sigma^{-1}T_\mu(A) = M_{\mu+1}(A)$$
is injective and stably flat.
The noncommutative localization exact sequence of Neeman and Ranicki
\cite{NR1,NR2}
\begin{multline*}
\cdots \longrightarrow K_{n+1}(\sigma^{-1}T_\mu(A)) \longrightarrow
K_n(H(T_\mu(A),\sigma)) \\
\longrightarrow K_n(T_\mu(A)) \longrightarrow
K_n(\sigma^{-1}T_\mu(A)) \longrightarrow \cdots
\end{multline*}
is given by
$$\cdots \to K_{n+1}(A[F_\mu]) \to {\rm Prim}_n(A) \to
\bigoplus\limits_{\mu+1} K_n(A) \to K_n(A[F_\mu]) \to \cdots$$
with ${\rm Prim}_n(A) \to K_n(T_\mu(A)) = \bigoplus\limits_{\mu+1}K_n(A)$
induced by
$${\mathcal{P}\mathit{rim}}(A) \to \prod\limits_{\mu+1} {\mathcal{P}\mathit{roj}}(A);~
(P,e,\{\pi_i\}) \mapsto (P,P_1,P_2,\ldots,P_\mu),$$
so that
\begin{align*}
&{\rm Prim}_n(A) = \bigoplus\limits_{2\mu} K_n(A) \oplus
\widetilde{{\rm Prim}}_n(A) \to \bigoplus\limits_{\mu+1}K_n(A);\\[1ex]
&(x^+_1,x^-_1,x^+_2,x^-_2,\ldots,x^+_\mu,x^-_\mu,\wtilde{x})\\
&\hspace{100pt}\longmapsto \Bigl(\sum\limits^\mu_{i = 1}(x^+_i+x^-_i),x^+_1+x^-_1,
x^+_2+x^-_2,\ldots,x^+_\mu+x^-_\mu\Bigr).
\end{align*}
This completes the proof.
\end{proof}
\begin{definition}{\rm
Let $\mathcal{G}\co A[F_\mu]\to\Sigma^{-1}A[F_\mu]$ be the universal
localization inverting the set $\Sigma$ of morphisms of f.g.~projective
$A[F_\mu]$--modules which induce an isomorphism of f.g.~projective
$A$--modules under the augmentation
$\epsilon\co A[F_\mu] \to A;z_i \mapsto 1$.}
\end{definition}
\begin{proposition} \label{Delta}
{\rm (i)}\qua The universal localization
$\mathcal{G}\co A[F_\mu]\to\Sigma^{-1}A[F_\mu]$ is injective.
The h.d.~1 $\Sigma$--torsion $A[F_\mu]$--module category is
$$H(A[F_\mu],\Sigma) = {\mathcal{B}\mathit{la}}(A).$$
{\rm (ii)}\qua The composite
$$\mathcal{G}\mathcal{F}\co T_\mu(A) \stackrel{\mathcal{F}}{\longrightarrow}
\sigma^{-1}T_\mu(A) = M_{\mu+1}(A[F_\mu])
\stackrel{\mathcal{G}}{\longrightarrow}
\tau^{-1}T_\mu(A) = M_{\mu+1}(\Sigma^{-1}A[F_\mu])$$
is the universal localization inverting the set $\tau$ of morphisms of
f.g.~projective $T_\mu(A)$--modules which become isomorphisms under the
composite
$$\disablesubscriptcorrection\xysavmatrix{\epsilon \mathcal{F}\co
T_\mu(A) \ar[r]^-{\displaystyle{\mathcal{F}}} & \sigma^{-1}T_\mu(A) =
M_{\mu+1}(A[F_\mu]) \ar[r]^-{\displaystyle{\epsilon}} & M_{\mu+1}(A).}$$
{\rm (iii)}\qua A $T_\mu(A)$--module $L = (L_i,f^+_i,f^-_i)$ is h.d.~1
$\tau$--torsion if and only if $L_0,\ldots,L_\mu$ are f.g.~projective
$A$--modules and the $A$--module morphism
$$f = \begin{pmatrix} f^+_1-f^-_1 & f^+_2-f^-_2 & \ldots & f^+_\mu-f^-_\mu
\end{pmatrix}\co L_1 \oplus L_2 \oplus \cdots \oplus L_\mu \to L_0$$
is an isomorphism, if and only if
$$(P,e,\{\pi_i\}) = \Bigl(\bigoplus\limits_{i = 1}^\mu L_i,
f^{-1}(f^+_1~f^+_2~\ldots~f^+_\mu),\{\pi_i\}\Bigr)$$
is a f.g.~projective Seifert $A$--module. The functor
$${\mathcal{S}\mathit{ei}}(A) \to H(T_\mu(A),\tau);~(P,e,\{\pi_i\}) \mapsto
(P,P_i,e\pi_i,(e-1)\pi_i)$$
is an equivalence of exact categories. The assembly of
$(L_i,f^+_i,f^-_i)$ is the covering Blanchfield $A[F_\mu]$--module of
$(P,e,\{\pi_i\})$
\begin{multline*}
\coker\Bigl((f_1^+z_1-f_1^-~\ldots~
f_\mu^+z_\mu-f_\mu^-)\co \bigoplus\limits^\mu_{i = 1}L_i[F_\mu] \to
L_0[F_\mu]\Bigr)\\
= \coker\big(1-e+ze\co P[F_\mu] \to P[F_\mu]\big)
= B(P,e,\{\pi_i\}),
\end{multline*}
so that up to equivalence
$$\mathcal{F} = B\co H(T_\mu(A),\tau) = {\mathcal{S}\mathit{ei}}(A) \to
H(M_{\mu+1}(A[F_\mu]),\tau) = {\mathcal{B}\mathit{la}}(A).$$
{\rm (iv)}\qua The forgetful functor
$${\mathcal{S}\mathit{ei}}(A) \to\prod\limits_{\mu}{\mathcal{P}\mathit{roj}}(A) ;~
(P,e,\{\pi_i\}) \mapsto (P_1,P_2,\ldots,P_\mu)$$
is split by
$$\prod\limits_{\mu}{\mathcal{P}\mathit{roj}}(A) \to {\mathcal{P}\mathit{rim}}(A);~
(P_1,P_2,\ldots,P_\mu) \mapsto
\Bigl(\bigoplus\limits_{i = 1}^{\mu}P_i,0,\{\pi_i\}\Bigr)$$
The reduced $K$--groups defined by
$$\widetilde{{\rm Sei}}_*(A) =
\ker\Bigl({\rm Sei}_*(A) \to \bigoplus\limits_{\mu} K_*(A)\Bigr)$$
are such that
$$K_*(H(T_\mu(A),\tau)) = {\rm Sei}_*(A) = \bigoplus\limits_{\mu}K_*(A)
\oplus\widetilde{{\rm Sei}}_*(A).$$
\end{proposition}
\begin{proof} (i)\qua The Magnus--Fox embedding
$A[F_\mu] \to A\langle\!\langle x_1,
\ldots,x_\mu \rangle\!\rangle$ is $\Sigma$--inverting, so that there is
a unique factorization
$$A[F_{\mu}] \to \Sigma^{-1}A[F_\mu] \to A\langle\!\langle x_1,x_2,
\ldots,x_\mu \rangle\!\rangle.$$
The identification $H(A[F_\mu],\Sigma) = {\mathcal{B}\mathit{la}}(A)$ is a formality,
as is the identification ${\mathcal{P}\mathit{roj}}(A[F_\mu]) = {\mathcal{P}}({\mathcal{P}\mathit{roj}}_A(A[F_\mu]))$
with ${\mathcal{P}\mathit{roj}}_A(A[F_\mu]) \subseteq {\mathcal{P}\mathit{roj}}(A[F_\mu])$ the full subcategory
with objects isomorphic to the f.g.~projective $A[F_\mu]$--modules $P[F_\mu]$ induced
from f.g.~projective $A$--modules $P$.
(ii)--(iv)\qua By construction, working as in the proof of
\fullref{Floc} (iv) to show that if $L = (L_i,f^+_i,f^-_i)$ is an h.d.~1
$\tau$--torsion $T_\mu(A)$--module then $L_0,L_1,\ldots,L_\mu$ are
f.g.~projective $A$--modules.
\end{proof}
\begin{thm} \label{final}
{\rm (i)}\qua The algebraic $K$--groups of ${\mathcal{P}\mathit{rim}}(A)$, ${\mathcal{S}\mathit{ei}}(A)$ and
${\mathcal{B}\mathit{la}}(A)$ fit into a commutative braid of exact sequences
$$\disablesubscriptcorrection\xysavmatrix@C-25pt@R-10pt{
{\rm Prim}_n(A)\ar[dr] \ar@/^2pc/[rr]^{} &&
\bigoplus\limits_{\mu+1}K_n(A) \ar[dr] \ar@/^2pc/[rr]
&&\Gamma K_n(\mathcal{G}) \\&
{\rm Sei}_n(A)\ar[ur] \ar[dr]^-{\displaystyle{B}} && K_n(A[F_\mu]) \ar[ur] \ar[dr]&&\\
\Gamma K_{n+1}(\mathcal{G}) \ar[ur]
\ar@/_2pc/[rr]_-{}&&{\rm Bla}_n(A)
\ar[ur]\ar@/_2pc/[rr]_{}&&{\rm Prim}_{n-1}(A)}$$
for $n \in \Z$, with $\mathcal{G}\co A[F_\mu] \to \Sigma^{-1}A[F_\mu]$ the
universal localization and
\begin{align*}
K_*(T_\mu(A))& = \bigoplus\limits_{\mu+1}K_*(A),\\[-1ex]
K_*(H(T_\mu(A),\sigma))& = ({\rm Sei},{\rm Bla})_*(A) = {\rm Prim}_*(A) =
\bigoplus\limits_{2\mu}K_*(A)\oplus \widetilde{{\rm Prim}}_*(A),\\[-1ex]
K_*(H(T_\mu(A),\tau))& = {\rm Sei}_*(A) = \bigoplus\limits_{\mu}K_*(A)
\oplus\widetilde{{\rm Sei}}_*(A),\\[-1ex]
K_*(H(A[F_\mu],\Sigma))& = {\rm Bla}_*(A) = \bigoplus\limits_{\mu}K_{*-1}(A)
\oplus \widetilde{{\rm Bla}}_*(A),\\[-1ex]
\Gamma K_*(\mathcal{G})& = K_*(A) \oplus \widetilde{{\rm Sei}}_{*-1}(A)~( = K_*(\Sigma^{-1}A[F_\mu])~
\emph{for}~* \leqslant 1)
\end{align*}
The reduced $K$--groups fit into a long exact sequence
$$\cdots \to \widetilde{{\rm Prim}}_n(A) \to
\widetilde{{\rm Sei}}_n(A) \to \widetilde{{\rm Bla}}_n(A)
\to \widetilde{{\rm Prim}}_{n-1}(A) \to \cdots.$$
{\rm (ii)}\qua If $\mathcal{G}\co A[F_\mu]\to\Sigma^{-1}A[F_\mu]$ is stably flat then
$$\Gamma K_n(\mathcal{G}) = K_n(\Sigma^{-1}A[F_\mu]) =
K_n(A) \oplus \widetilde{{\rm Sei}}_{n-1}(A)$$
for all $n \in \Z$.
\end{thm}
\begin{proof}
(i)\qua Consider the commutative square of Waldhausen categories
$$\disablesubscriptcorrection\xysavmatrix{
(\C^b(T_\mu(A)),w_{T_{\mu}(A)}) \ar[r] \ar[d]_-{\displaystyle{\mathcal{F}}} &
(\C^b(T_\mu(A)),w_{\tau^{-1}T_{\mu}(A)}) \ar[d] \\
(\C^b(T_\mu(A)),w_{\sigma^{-1}T_{\mu}(A)})\ar[r]^-{\displaystyle{\mathcal{G}}} &
(\C^b(\sigma^{-1}T_\mu(A)),w_{\tau^{-1}T_{\mu}(A)}).}$$
Since $\mathcal{F}\co T_\mu(A)\to\sigma^{-1}T_\mu(A) = M_{\mu+1}(A[F_\mu])$ is stably flat
there are defined equivalences
$$(\C^b(T_\mu(A)),w_{\sigma^{-1}T_{\mu}(A)})~\approx~
(\C^b(\sigma^{-1}T_\mu(A)),w_{\sigma^{-1}T_{\mu}(A)})~\approx~
(\C^b(A[F_\mu]),w_{A[F_\mu]})$$
which induce homotopy equivalences
$$K{\mathcal{P}}(\C^b(T_\mu(A)),w_{\sigma^{-1}T_{\mu}(A)})~\simeq~K(\sigma^{-1}T_\mu(A))~
\simeq~K(A[F_\mu]).$$
Also, since $\tau^{-1}T_\mu(A) = M_{\mu+1}(\Sigma^{-1}A[F_\mu])$
the functor
$$(\C^b(T_\mu(A)),w_{\tau^{-1}T_{\mu}(A)}) \to
(\C^b(\sigma^{-1}T_\mu(A)),w_{\tau^{-1}T_{\mu}(A)})$$
induces an equivalence of the homotopy categories
$$D(\C^b(T_\mu(A)),w_{\tau^{-1}T_{\mu}(A)})~\approx~
(\C^b(\sigma^{-1}T_\mu(A),w_{\tau^{-1}T_{\mu}(A)}).$$
The composite of this equivalence and the Morita equivalence
$$D(\C^b(\sigma^{-1}T_\mu(A)),w_{\tau^{-1}T_{\mu}(A)})~\approx~
D(\C^b(A[F_\mu]),w_{\Sigma^{-1}A[F_\mu]})$$
induces a homotopy equivalence
$$\begin{array}{ll}
K{\mathcal{P}}(\C^b(\sigma^{-1}T_\mu(A)),w_{\tau^{-1}T_{\mu}(A)})~
\simeq&K{\mathcal{P}}(\C^b(A[F_\mu]),w_{\Sigma^{-1}A[F_\mu]})\\[1ex]
& = \Gamma K(\mathcal{G}\co A[F_\mu] \to\Sigma^{-1}A[F_\mu]).
\end{array}$$
Thus Propositions \ref{Gamma}, \ref{Floc} and \ref{Delta}
give a braid of Waldhausen categories
$$\disablesubscriptcorrection\xysavmatrix@C-85pt{
(\C^b(T_\mu(A),\sigma),w_{(T_{\mu}(A),\sigma)})\ar[dr] \ar@/^2pc/[rr]^{} &&
(\C^b(T_\mu(A)),w_{T_{\mu}(A)})\quad \ar[dr] \ar@/^2pc/[rr]
&&(\C^b(A[F_\mu]),w_{\Sigma^{-1}A[F_\mu]})\\
&(\C^b(T_\mu(A),\tau),w_{(T_{\mu}(A),\tau)})\ar[ur] \ar[dr]&&
(\C^b(A[F_\mu]),w_{A[F_\mu]}) \ar[ur] &&\\
&&(\C^b(A[F_\mu],\Sigma),w_{\C^b(A[F_\mu],\Sigma)})\ar[ur]&&}$$
inducing a commutative braid of exact sequences
$$\disablesubscriptcorrection\xysavmatrix@C-40pt@R-10pt{
K_n(H(T_\mu(A),\sigma))\ar[dr] \ar@/^2pc/[rr]^{} &&
K_n(T_\mu(A)) \ar[dr] \ar@/^2pc/[rr]&&\Gamma K_n(\mathcal{G}) \\&
K_n(H(T_\mu(A),\tau))\ar[ur] \ar[dr] && K_n(A[F_\mu]) \ar[ur] \ar[dr]&&\\
\Gamma K_{n+1}(\mathcal{G}) \ar[ur] \ar@/_2pc/[rr]_-{}&&
K_n(H(A[F_\mu],\Sigma))
\ar[ur]\ar@/_2pc/[rr]_{}&&K_{n-1}(H(T_\mu(A),\sigma))}$$
Split off the reduced $K$--groups in
\begin{align*}
{\rm Prim}_*(A)& = \bigoplus\limits_{2\mu}K_*(A) \oplus \widetilde{{\rm Prim}}_*(A),\\
{\rm Sei}_*(A)& = \bigoplus\limits_{\mu}K_*(A) \oplus \widetilde{{\rm Sei}}_*(A)
\end{align*}
from the long exact sequence
$$\cdots \to {\rm Prim}_n(A) \to{\rm Sei}_n(A) \to {\rm Bla}_n(A)\to {\rm Prim}_{n-1}(A) \to
\cdots$$
to define the reduced $K$--groups in
$${\rm Bla}_*(A) = \bigoplus\limits_{\mu}K_{*-1}(A) \oplus \widetilde{{\rm Bla}}_*(A)$$
and to obtain the long exact sequence
$$\cdots \to \widetilde{{\rm Prim}}_n(A) \to
\widetilde{{\rm Sei}}_n(A) \to \widetilde{{\rm Bla}}_n(A)
\to \widetilde{{\rm Prim}}_{n-1}(A) \to \cdots.$$
(ii)\qua This is a special case of \fullref{Gamma} (ii).
\end{proof}
This completes the proofs of Theorems \ref{thm4} and \ref{thm5} of the
Introduction.
\begin{remark}
Unfortunately, we do not know if the universal localization
$\Sigma^{-1}A[F_\mu]$ is stably flat in general.
See Dicks and Sontag \cite{DS}, Farber and Vogel \cite{FV}
for proofs that $\Sigma^{-1}A[F_\mu]$ is stably flat
when $A$ is a principal ideal domain, and
Ara and Dicks \cite[Theorem 4.4]{AD} when $A$ is a von Neumann regular
ring or a commutative Bezout domain.
\end{remark}
\begin{remark}
\label{det}
Sheiham \cite{Sh3} computed
$$K_1(\Sigma^{-1}A[F_\mu]) = K_1(A) \oplus \epsilon_\Sigma^{-1}(1)/C$$
with $\epsilon_\Sigma\co \Sigma^{-1}A[F_\mu] \to A$ the
factorization of the augmentation map $\epsilon\co A[F_\mu] \to A$ and
$C \subseteq \epsilon_{\Sigma}^{-1}(1)$ the subgroup generated by the
commutators
$$(1+ab)(1+ba)^{-1}\qquad(a,b \in \Sigma^{-1}A[F_\mu],~
\epsilon(ab) = \epsilon(ba) = 0).$$
It follows from the splitting given by \fullref{final} (i)
$$K_1(\Sigma^{-1}A[F_\mu]) = K_1(A) \oplus \widetilde{{\rm Sei}}_0(A)$$
that there is defined an isomorphism
$$\widetilde{{\rm Sei}}_0(A) \disablesubscriptcorrection\xysavmatrix{\ar[r]^-{{\cong}}&}
\epsilon_{\Sigma}^{-1}(1)/C;~
(P,e,\{\pi_i\}) \mapsto D(1-e+ez\co P[F_\mu] \to P[F_\mu])$$
with $D$ the generalized Dieudonn\'e noncommutative determinant of
\cite[4.3]{Sh3}.
\end{remark}
\begin{example}
{\rm (i)}\qua The algebraic $K$--groups of $\Z[F_\mu]$ are such that
$$\begin{array}{l}
K_*(\Z[F_\mu]) = K_*(\Z) \oplus \bigoplus\limits_\mu K_{*-1}(\Z),\\[1ex]
K_n(\Z[F_\mu]) = K_n(\Z) = \begin{cases}
\Z&{\rm if}~n = 0\\
0&{\rm if}~n \leqslant -1
\end{cases}
\end{array}$$
by Stallings \cite{St}, Gersten \cite{Ge}, Bass \cite[XII]{B2} and
Waldhausen \cite{Wald2,Wald2a}, so that
\begin{align*}
&{\mathcal{F}\mathit{lk}}(\Z) = {\mathcal{B}\mathit{la}}(\Z), \qquad \widetilde{{\rm Prim}}_*(\Z) = 0,\\
&K_{*+1}(\Sigma^{-1}\Z[F_\mu])/K_{*+1}(\Z) =
\widetilde{{\rm Sei}}_*(\Z) = {\rm Flk}_*(\Z) = {\rm Bla}_*(\Z),\\
&K_*(H(\Z[F_\mu],\Sigma)) = \bigoplus\limits_{\mu}K_{*-1}(\Z)\oplus
\widetilde{{\rm Sei}}_*(\Z),\\
&K_n(H(\Z[F_\mu],\Sigma)) = \widetilde{{\rm Sei}}_n(\Z)\quad(n \leqslant 0).
\end{align*}
{\rm (ii)}\qua Given a $\mu$--component boundary link
$\ell\co \bigsqcup_\mu S^n \subset S^{n+2}$ with exterior $W$
and given a $\mu$--component Seifert surface $V = V_1 \sqcup V_2
\sqcup \ldots \sqcup V_\mu \subset S^{n+2}$ for $\ell$
let $\dot C(W)$, $(\dot
C(V),e,\{\pi_i\})$ be the chain complexes defined in \fullref{chain}. Thus $\dot C(\wwtilde{W})$ is a
$\Sigma^{-1}\Z[F_\mu]$--acyclic $(n{+}2)$--dimensional f.g.~free
$\Z[F_\mu]$--module chain complex, $(\dot C(V),e,\{\pi_i\})$ is an
$(n{+}1)$--dimensional chain complex in ${\mathcal{S}\mathit{ei}}(\Z)$, and $B(\dot
C(V),e,\{\pi_i\})$ is an $(n{+}1)$--dimensional chain complex in
${\mathcal{F}\mathit{lk}}(\Z)$ with a homology equivalence $\dot C(\wwtilde{W}) \to
B(\dot C(V),e,\{\pi_i\})$. The torsion
\begin{align*}
\tau(\ell)& = \tau(\Sigma^{-1}\dot C(\wwtilde{W}))\\
& = (\dot C(V),e,\{\pi_i\})
= \sum\limits^{n+1}_{r = 0}(-)^r(\dot C_r(V),e,\{\pi_i\}) = [\dot C(\wwtilde{W})]\\
&\hspace*{40pt}
\in K_1(\Sigma^{-1}\Z[F_\mu])/K_1(\Z) = K_0(H(\Z[F_\mu],\Sigma)) =
\widetilde{\rm {\rm Sei}}_0(\Z) = {\rm Bla}_0(\Z)
\end{align*}
is an isotopy invariant of $\ell$, given by
Sheiham \cite{Sh3} to be the generalized Dieudonn\'e determinant
$$\tau(\ell) =
\sum\limits^{n+1}_{r = 0}(-)^r
D(1-e+ez\co \dot C_r(V)[F_\mu] \to \dot C_r(V)[F_\mu])
\in \widetilde{{\rm Sei}}_0(\Z) = \epsilon_{\Sigma}^{-1}(1)/C$$
with $\epsilon_{\Sigma}\co \Sigma^{-1}\Z[F_\mu] \to \Z$ and
$C \subseteq \epsilon_{\Sigma}^{-1}(1)$ as recalled in \fullref{det}.
The $\Z[F_\mu]$--modules
$\dot H_r(\wwtilde{W})/\Z\hbox{\rm -torsion}$
($0 \leqslant r \leqslant n+1$) are h.d.~1 $F_\mu$--link modules, and
\begin{align*}
\tau(\ell)& = \sum\limits^{n+1}_{r = 0}(-)^r
D(1-e+ez\co \dot H_r(V)[F_\mu] \to \dot H_r(V)[F_\mu])\\
& = \sum\limits^{n+1}_{r = 0}(-)^r[\dot H_r(\wwtilde{W})/\Z\hbox{\rm--torsion}] \\
&\in
K_1(\Sigma^{-1}\Z[F_\mu])/K_1(\Z) = K_0(H(\Z[F_\mu],\Sigma))\\
&\hskip75pt
= \widetilde{\rm {\rm Sei}}_0(\Z) = {\rm Bla}_0(\Z) =
(\Sigma^{-1}\Z[F_\mu])^{\bullet}/\{\pm 1\}.
\end{align*}
For $\mu = 1$ this is just the Reidemeister torsion of a knot $\ell\co S^n
\subset S^{n+2}$, which is the alternating product of the Alexander
polynomials
$$\begin{array}{ll}
\tau(\ell)& = \sum\limits^{n+1}_{r = 0}(-)^r
{\rm det}(1-e+ez\co \dot H_r(V)[z,z^{-1}] \to \dot H_r(V)[z,z^{-1}])\\
& = \sum\limits^{n+1}_{r = 0}(-)^r[\dot H_r(\wwtilde{W})/\Z\hbox{\rm -torsion}]
\\
&\hskip20pt\in
K_1(\Sigma^{-1}\Z[z,z^{-1}])/K_1(\Z) = K_0(H(\Z[z,z^{-1}],\Sigma))\\
&\hskip40pt
= \widetilde{\rm {\rm Sei}}_0(\Z) = {\rm Bla}_0(\Z) = \widetilde{{\rm End}}_0(\Z)
= (\Sigma^{-1}\Z[z,z^{-1}])^{\bullet}/\{\pm 1\}
\end{array}$$
(Milnor \cite{M}, cf \cite[Example 17.11]{RHK}).
(iii)\qua The isotopy classes of simple $\mu$--component boundary links
$\ell\co \bigsqcup_{\mu}S^{2q-1} \subset S^{2q+1}$ for $q
\geqslant 3$ are in one-one correspondence with the
`$l$--equivalence classes of Seifert matrices' (Liang \cite{Li},
generalizing the case $\mu = 1$ due to Levine \cite{L1}), and also
with the `$R$--equivalence classes of $(-)^q$--symmetric isometry
structures of multiplicity $\mu$' (Farber \cite[4.7]{Fa2}). For
simple $\ell$ $H_q(\wwtilde{W})$ is an h.d.~1 $F_\mu$--link
module, and the torsion
$$\tau(\ell) = (-)^q[H_q(\wwtilde{W})] \in \widetilde{{\rm Sei}}_0(\Z) =
{\rm Flk}_0(\Z) = {\rm Bla}_0(\Z)$$
is just the $K$--theory part of these complete isotopy invariants for
$q \geqslant 3$.\end{example}
\bibliographystyle{gtart}
|
3,212,635,537,955 | arxiv |
\section{Introduction and Motivation}
The use of microarray technologies have become popular to monitor genome-wide expression changes in health and disease. Typically, a microarray data set is high dimensional in the sense, it usually has tens of thousands of gene expression profile(variables) but only tens or hundreds of subjects(observations). In microarray analysis, a group of genes sharing the same biological pathway tend to have highly correlated expression levels \cite{Segal} and the goal is to identify all(rather than a few) of them if they are related to the underlying biological process. This is one example where the need to select groups of correlated variables arises. In many applications it is required to identify all relevant correlated variables.
In this paper, we consider the problem of model selection and estimation in sparse high dimensional linear regression models with strongly correlated variables.
We start with the standard linear regression model as
\begin{align} \label{eq:lr}
\textbf{Y} &= \textbf{X} \beta+\epsilon,
\end{align}
with response vector $\textbf{Y}_{n \times 1}$, design matrix $\textbf{X}_{ n \times p}$, true underlying coefficient vector $\beta_{p \times 1}$ and error vector $\epsilon_{n\times 1} \sim N_n(0,I_n)$. In particular, we consider the case of sparse high dimensional linear model $(p \gg n)$ with strong empirical correlation among few variables. The Lasso is a widely used regularized regression method to find sparse solutions, the lasso estimator is defined as
\begin{align}
\hat{\beta}_{Lasso} = \arg\min_{\beta \in \mathbb{R}^p} \left\lbrace \frac{1}{2}\|{ \textbf{Y} -\textbf{X} \beta}\|_2^2 + \lambda \|\beta\|_1 \right\rbrace, \label{eq:lasso}
\end{align}
where $\lambda \geq 0$ is the regularization parameter that controls the amount of regularization. It is known that the Lasso tends to select a single variable from a group of strongly correlated variables even if many or all of these variables are important.
In presence of correlated predictors, the concept of clustering or grouping correlated predictors and then pursuing group-wise model fitting was proposed, see \cite{Buhlmann2} and \cite{Niharika}. When the dimension is very high or in case of overlapping clusters, finding an appropriate group structure remains as difficult as the original problem. We note that clustering followed by model fitting is computationally expensive, not reliable and do not scale for large, high-dimensional data sets, so we do not consider it further in this paper. An alternatively approach is simultaneous clustering and model fitting that involves combination of two different penalties. For example, Elastic Net (\cite{Hui}) is a combination of two regularization techniques, the $\ell_2$ regularization provides grouping effects and $\ell_1$ regularization produces sparse models. Therefore, the eNet selects or drops highly correlated variables together that depends on the amount of $\ell_1$ and $\ell_2$ regularization.
The influence of correlations on Lasso prediction has been studied in \cite{Hebiri} and \cite{vandegeer}, and it is shown that Lasso prediction works well in presence of any degree of correlations with an appropriate amount of regularization. However, studies show that correlations are problematic for parameter estimation and variable selection. It has been proven that the design matrix must satisfy the following two conditions for the Lasso to perform exact variable selection: irrepresentability(IC) condition(\cite{Zhao}) and beta-min condition(\cite{Buhlmann1}). Having highly correlated variables implies that the design matrix violates the IC, and the Lasso solution is not stable. When active covariates are highly correlated the Lasso solution is not unique and Lasso randomly selects one variable from correlated group. However, even in case of highly correlated variables the corresponding dual Lasso solution is always unique. The dual of the Lasso problem (\ref{eq:lasso}), as shown in \cite{Jie} is given by
\begin{align} \label{eq:dualLasso}
\sup_{\theta} \; & g(\theta) = \frac{1}{2} \|\textbf{Y} \|_2^2 - \| \theta - \textbf{Y} \|_2^2 \nonumber \\
\text{subject to } & |X_j^T \theta| \leq \lambda \; \text{ for all } j \in \{ 1,...,p \}
\end{align}
The intuition drawn from the articles \cite{Osborne} and \cite{Jie} further motivates us to consider the Lasso optimal and its dual optimal solution together, that yields in selecting correlated active predictors.
Exploiting the fact about uniqueness of the dual Lasso solution, we propose a new variable selection procedure, the Dual Lasso Selector (DLS). For a given $\lambda$ and a Lasso estimator $\hat{\beta}_{Lasso}$, we can compute the corresponding Dual Lasso solution using KKT conditions. Basically, the DLS active set corresponds to the predictors that satisfies dual Lasso feasible boundary conditions (we discuss it in details in later section). We argue that correlations among active predictors are not problematic, and we define a new weaker condition on the design matrix that allows for correlation among active predictors, called Pseudo Irrepresentable Condition (PIC). We prove that the Pseudo Irrepresentable Condition is a necessary and sufficient condition for the proposed dual Lasso selector to select the true active set (under assumption of beta-min condition) with high probability. Moreover, we use the $\ell_2$ penalty (the Ridge regression, \cite{Ridge}) which is known to perform best in case of correlated variables, to estimate the coefficients of the predictors selected by the dual Lasso selector. We call the combination of the two, the DLSelect+Ridge. Though, DLSelect+Ridge resembles the "Ridge post Lasso" but it is conceptually different and behaves differently than the Lasso followed by the Ridge, especially in the presence of highly correlated variables. For example DLSelect+Ridge looks like Elastic-net, since both are combination of $\ell_1$ and $\ell_2$ penalties but Elastic-net is a combination of the Ridge Regression followed by the Lasso. In addition, Enet needs to cross-validate on a two-dimensional surface $O(k^2)$ to select its the optimal regularization parameters, whereas DLSelect+Ridge needs to cross validated twice on one-dimensional surface $O(k)$, where k is the length of the search space for a regularization parameter.
Our contribution is summarized as follows:
\begin{enumerate}
\item We briefly review the state-of-the-art methods of simultaneous clustering and model fitting using combination of penalties such as Elastic-net, OSCAR and Fused Lasso etc.
\item We study the theoretical properties of the Lasso and its dual optimal solution together and we show that selection of active correlated variables is related to the dual feasible boundary conditions.
\item By further exploiting the uniqueness property of the dual Lasso solution, we develop a variable selection algorithm to efficiently select the true active predictors (including correlated active predictors). we call this selection technique as the \textit{Dual Lasso Selector}.
\item We derive the Pseudo Irrepresentable Conditions (PIC) for the design matrix that allow for the correlation between active covariates, and we show that under assumption of PIC the dual Lasso selector is variable selection consistent.
\item We propose a new combined approach, the DLSelect+Ridge: Dual Lasso selecting predictors and the Ridge estimating their coefficients.
\item We study the theoretical properties of the combination DLSelect+Ridge.
\item We implement the DLSelect+Ridge method and empirically compare it with existing methods like Lasso and Enet etc. in terms of variable selection consistency, prediction accuracy, estimation accuracy and time complexity (using various simulations and real data examples).
\end{enumerate}
We have organized the rest of the article in the following manner. We start with background in section 2. In section 3, we present Dual Lasso Selector. We define PIC and discuss variable selection consistency under this assumption on the design matrix, in section 4. Section 5 is concerned with illustration of the proposed method on real and simulated data sets. Section 6 gives computational details. We shall provide some concluding remarks in section 7.
\section{Notations and Background}
In this section, we state notations and assumptions, used throughout the paper.
We consider usual sparse high-dim linear regression model as given in \ref{eq:lr} with $p \gg n$. For the design matrix $\textbf{X} \in \mathcal{R}^{n \times p}$, we represent rows by $x_i^T \in \mathbb{R}^p, \ i= 1,...,n$, and columns by $X_j^T \in \mathbb{R}^n, \ i= 1,...,p$. We assume that the design matrix $\textbf{X}_{n\times p}$ is fixed, the data is centred and the predictors are standardized, so that $\sum_{i=1}^{n} \textbf{Y}_i = 0$, $\sum_{i=1}^{n} ({X}_{j}){i} = 0$ and $\frac{1}{n} \textbf{X}^T_{j}\textbf{X}_{j} = 1$ for all $j=1,...,p$. We denote by
\begin{align}
S = \{j \in \{ 1,...,p \} : \beta_{j} \neq 0\},
\end{align}
the true active set and cardinality of the set $s = |S|$, is called sparsity index. We assume that the true coefficient vector $\beta$ is sparse, that is $s \ll p$. We denote $\textbf{X}_S$ as the restriction of $\textbf{X}$ to columns in $S$, and $\beta_S$ is the vector $\beta$ restricted to the support $S$, with 0 outside the support $S$. Without loss of generality we can assume that the first $s$ variables are the active variables, and we partition the covariance matrix, $C = \frac{1}{n} \textbf{X}^{T}\textbf{X}$, for the active and the redundant variables as follows.
\begin{align} \label{eq:sigma_par}
C = \left[ \begin{array}{cc}
C_{11} & C_{12}\\
C_{21} & C_{22}
\end{array} \right]
\end{align}
Similarly the coefficient vector $\beta$ can be partitioned as $ \left[ \begin{array}{c}
\beta_1 \\
\beta_2
\end{array} \right]. $\\
The $\ell_1$-norm and $\ell_2$-norm (square) are defined as
\begin{align}
\|\beta\|_1 &= \textstyle \sum_{j=1}^p |\beta_j| \label{eq:l1} \\
\|\beta\|_2^2 &= \textstyle \sum_{j=1}^p \beta_j^2.
\end{align}
Throughout the paper, we use the notation $\lambda_1 > 0$ for $\ell_1$ penalty and $\lambda_2 > 0$ for other penalty functions.
For a vector $a \in \mathbb{R}^p$, we denote its sign vector as
\begin{equation}
\begin{aligned}
\mathbb{S}(a) = \left\lbrace \begin{array}{ll}
1 & \text{ if } a > 0 \\
-1 & \text{ if } a < 0 \\
0 & \text{ if } a = 0 \\
\end{array} \right.
\end{aligned}
\end{equation}
We denote sub-gradient of $\ell_1$-norm evaluated at $\beta \in \mathbb{R}^p$, as $ \tau \in \partial \|\beta\|_1$, where $\tau$ satisfies the following.
\begin{align}
\tau_i = \left\lbrace \begin{array}{ll}
1 & \text{ if } \beta_i > 0 \\
\left[-1,1 \right] & \text{ if } \beta_i = 0\\
-1 & \text{ if } \beta_i < 0 \\
\end{array} \right.
\end{align}
\section{Review of Relevant Work}
Given the huge literature on the use of Lasso-type penalties for variable selection, we provide only a brief overview here, with focus on previous approaches which are closely related to our work. In particular, we briefly review the Lasso, the Ridge and the state-of-the-art in simultaneous clustering and model fitting using combination of penalties for high-dimensional sparse linear models. In general, we define a penalized least squares method as follows.
\begin{align}
\min_{\beta \in \mathbb{R}^p} \left\lbrace \frac{1}{2}\|{ \textbf{Y} -\textbf{X} \beta}\|_2^2 + \mathcal{P}_{method}(\beta, .) \right\rbrace \label{eq:pen_ls}
\end{align}
where the penalty terms $ \mathcal{P}_{method}(\beta, .)$ can be different for different methods depending on the type and number of penalties used. In the following we define various penalized least squares estimators in terms of penalties used by them, and we also mention their computational complexity, variable selection consistency and grouping effects of selecting and dropping highly correlated predictors together.
\begin{enumerate}
\item Lasso: The Lasso method was proposed by \cite{Tibshirani} and the lasso penalty is defined as
\begin{align}
\mathcal{P}_{Lasso}(\beta, \lambda_1) = \lambda_1 \|\beta\|_1 \label{eq:lasso_pen}
\end{align}
It uses the single $\ell_1$ penalty, and due to nature of the $\ell_1$ penalty it simultaneously performs variable selection and estimation. The whole regularization path can be computed efficiently with the computational effort of a single OLS fit, by some modification of the LARS algorithm, see \cite{efron2004}. It does not provide grouping effect, in fact the Lasso tends to select a single predictor from a group of highly correlated predictors.
\item Ridge Regression (RR): The ridge method was proposed by \cite{Ridge} and the ridge penalty is defined as follows.
\begin{align}
\mathcal{P}_{Ridge}(\beta, \lambda_2) = \lambda_2 \|\beta\|_2 \label{eq:ridge}
\end{align}
It uses the single $L_2$ penalty, and it always has a unique solution for a fixed regularization parameter $\lambda_2$. Though it is known to correctly detect the variable signs with reduced mean square error with correlated variables, but does not provide variable selection. It provides grouping effect with the highly correlated variables, and the computational complexity of the ridge is same as the computational effort of a single OLS fit.
\item Elastic-net (Enet): The Enet method was introduced by \cite{Hui}. The Enet penalty is a combination of $\ell_1$ and $L_2$ penalties and it is defined as follows.
\begin{align}
\mathcal{P}_{Enet}(\beta, \lambda_1, \lambda_2) = \lambda_1 \|\beta\|_1 + \lambda_2 \|\beta\|_2 \label{eq:enet}
\end{align}
Enet addresses both the limitations of the Lasso, that is it can select correlated predictors as well as it can handle the $s>n$ case. It provides grouping effect, but requires to search in two-dimensional space for choosing optimal values of its regularization parameters. Hence its effective time complexity depends on the length of the search space for the regularization parameters $\lambda_1$ and $\lambda_2$.
\item Correlation Based Penalty (CP): The correlation based penalized least squares method was proposed by \cite{Tutz2009},
which uses the following correlation-based penalty term
\begin{align*}
\mathcal{P}_{CP}(\beta, \lambda_2) = \lambda_2 \sum_{i=1}^{p-1} \sum_{j > i} \left\lbrace \frac{(\beta_i - \beta_j)^2 }{1 - \rho_{ij}} +\frac{(\beta_i + \beta_j)^2 }{1 + \rho_{ij}} \right\rbrace
\end{align*}
It uses the single $CP$ penalty norm, and it always has a unique solution for a fixed regularization parameter and the grouping effect strongly depends on the convexity of the penalty term. It does not provide variable selection. However, a boosted version of the penalized estimator allows to select variables. But the major drawback is that it is not scalable for large high dimensional problems.
\item Fused Lasso: The Fused Lasso method was given by \cite{Fused}, and the Fused Lasso penalty is defined as
\begin{align}
\mathcal{P}_{Fused}(\beta, \lambda_1, \lambda_2) = \lambda_1 \|\beta\|_1 + \lambda_2 \sum_{j=2}^{p} | \beta_j - \beta_{j-1} | \label{eq:fused}
\end{align}
The first constraint encourages sparsity in the coefficients and the second constraint encourages sparsity in their differences.
The major drawback of this method is that it requires the covariates to be in some order.
It does not perform automated variable clustering to unordered features.
\item OSCAR: The OSCAR was invented by \cite{Oscar}, and the OSCAR penalty is given as follows.
\begin{align*}
\mathcal{P}_{Oscar}(\beta, \lambda_1, \lambda_2)= \lambda_1 \|\beta\|_1 + \lambda_2 \sum_{i<j} max\{| \beta_i| , | \beta_{j} | \}
\end{align*}
with $|\beta_1| \leq ... \leq |\beta_p|$.
The first constraint is to encourage sparsity in the coefficients and the second constraint encourages equi-sparsity in $|\beta|$.
The time complexity limits its scalability on ultra high-dimensional problems, moreover it requires two-dimensional grid search over the two parameters $(\lambda_1 , \lambda_2)$
\item L1CP : The L1CP penalty term is given by as follows, see \cite{L1CP}.
\begin{align}
\mathcal{P}_{L1CP}(\beta, \lambda_1, \lambda_2) = \lambda_1 \| \beta \|_1 + \lambda_2 \sum_{i=1}^{p-1} \sum_{j > i} \left\lbrace \frac{(\beta_i - \beta_j)^2 }{1 - \rho_{ij}} + \frac{(\beta_i + \beta_j)^2 }{1 + \rho_{ij}} \right\rbrace . \label{eq:l1cp}
\end{align}
It performs variable selection with grouping effect and estimation together but is not scalable to the large scale problems due to expensive computation time, and it also requires two-dimensional grid search over the two parameters $(\lambda_1 , \lambda_2)$.
\item Clustered Lasso:
The Clustered Lasso penalty is defined as
\begin{align*}
\mathcal{P}_{CL}(\beta, \lambda_1, \lambda_2) = \lambda_1 \|\beta\|_1 + \lambda_2 \sum_{i<j} | \beta_i - \beta_{j} |
\end{align*}
The first constraint encourages sparsity in the coefficients and the second constraint encourages equi-sparsity in $|\beta|$, It is similar as the Fused lasso but does not require ordering of variables, see \cite{she2010}. It provides grouping effect, but it requires two-dimensional grid search over the two parameters $(\lambda_1 , \lambda_2)$. It is computationally expensive since it has to check equi-sparsity pattern for each pair of variables.
\end{enumerate}
In the following table we summarize the properties discussed above for various regularization methods.
\begin{small}
\begin{table}[h!]
\begin{tabular}{ p{35mm} p{20mm} p{20mm} p{20mm} p{20mm} p{20mm} }
\toprule
Properties\;\;\;/\ \;\;Methods & Clustering/ Ordering Required & Variable Selection & Grouping Effect & Scalability & Grid Search\\
\toprule
Lasso & No & Yes & No & Yes & 1D \\
Ridge & No & No & Yes & Yes & 1D \\
PC & No & No & Yes & No & 1D \\
Elastic-Net & No & Yes & Yes & Yes & 2D \\
Fused-lasso & Yes & Yes & Yes & Yes & 2D \\
OSCAR & No & Yes & Yes & Yes & 2D \\
L1CP & No & Yes & Yes & Yes & 2D \\
Clustered-Lasso & No & Yes & Yes & Yes & 2D \\
DLSelect+Ridge & No & Yes & Yes & Yes & 1D \\
\bottomrule
\end{tabular}
\caption{Comparision Table}\label{table:block}
\end{table}
\end{small}
\end{document}
\section{Dual Lasso Selector}
In this section, we present the dual Lasso selector, a new variable selection method for sparse high-dim regression models with correlated variables. First, we study the theoretical properties of the Lasso and dual Lasso solutions. Then, we show that the magnitude of correlations between the predictors and the dual vector determines the set of active predictors. This is the basis for our correlated variable selection.
The dual problem of the Lasso problem (\ref{eq:lasso}) can be given as follows (we provide the detailed derivation of the Lasso's dual in the appendix A.1):
\begin{align}
\sup_{\theta} \; & \frac{1}{2} \|\textbf{Y} \|_2^2 - \| \theta - \textbf{Y} \|_2^2 \\
\text{subject to } & | X_j^T \theta|\leq \lambda \text{ for } j = 1,...,p ,
\end{align}
where $\theta$ is the dual vector, as defined in equation (\ref{eq:lang}). For a fixed $\lambda \geq 0$, let $\hat{\beta}_{lasso}(\lambda)$ and $\hat{\theta}(\lambda)$ denote the optimal solutions of the Lasso and its dual problem respectively. Since it is implicit that the Lasso and its dual optimal depends on the $\lambda$, we drop the term $\lambda$ from the expression for notational simplicity.
From KKT conditions (derivation of the KKT conditions is given in the appendix A.2) we get the following primal dual relationship:
\begin{align}
\hat{\theta} = \textbf{Y} - \textbf{X} \hat{\beta}_{lasso}. \label{eq:primal_dual}
\end{align}
It is worth mentioning the basic properties of the Lasso and its dual, which has already been derived and studied by various authors (see \cite{tibshirani2011} and \cite{Jie} for more insights).
\begin{enumerate}
\item \textbf{Uniqueness of the Lasso-fit}: There may not be a unique solution for the Lasso problem because for the criterion (\ref{eq:lasso_recall}) is not strictly convex in $\beta$. But the least square loss is strictly convex in $\textbf{X} \beta$, hence there is always a unique fitted value $\textbf{X} \hat{\beta}$.
\item \textbf{Uniqueness of the dual vector}: The dual problem is strictly convex in $\theta$, therefore the dual optimal $\hat{\theta}$ is unique. Another argument for the uniqueness of $\hat{\theta}$ is that it is a function of $\textbf{X} \hat{\beta}$ (\ref{eq:primal_dual}) which itself is unique.
The fact that the DLS can achieve consistent variable selection for situations (with correlated active predictors) when the Lasso is unstable for estimation of the true active set is related to the uniqueness of the dual Lasso solution.
\item \textbf{Uniqueness of the Sub-gradient}: Sub-gradient of $\ell_1$ norm of any Lasso solution $\hat{\beta}$ is unique because it is a function of $\textbf{X} \hat{\beta}$ (see Appendix A.2). More specifically, suppose $\hat{\beta}$ and $\tilde{\beta}$ are two lasso solutions for a fixed $\lambda$ value, then they must have the same signs $sign(\hat{\beta}) = sign(\tilde{\beta})$, it is not possible that $\hat{\beta}_j > 0 $ and $\hat{\beta}_j < 0 $ for some $j$.
\end{enumerate}
Let $\hat{S}_{lasso}$ denote the support set or active set of the Lasso estimator $\hat{\beta}$ which is given as
\begin{align}
\hat{S}_{lasso}(\lambda) = \{j \in \{ 1,...,p \} : (\hat{\beta}_{lasso})_{j} \neq 0\}
\end{align}
Similarly, we define the active set of the dual Lasso vector that corresponds to the active constraints of the dual optimization problem. We note that constraints are said to be active at a feasible point if that point lies on a boundary formed by the constraint.
\begin{align}
\hat{S}_{dual}(\lambda) = \{j \in \{ 1,...,p \} : \; | X_j^T \theta| = \lambda \}
\end{align}
Now, we define the following lemmas that will be used later for our mathematical derivations.
\begin{lemma} \label{lemma:lasso_in_dual}
The active set selected by the Lasso $\hat{S}_{lasso}(\lambda)$ is always contained in the active set selected by the dual Lasso $\hat{S}_{dual}(\lambda)$, that is
\[ \hat{S}_{lasso}(\lambda) \subseteq \hat{S}_{dual}(\lambda).
\]
\end{lemma}
\begin{proof}
The proof is rather easy. From KKT condition (dual feasibility condition, see Appendix A.2), we have
\begin{align}
| X_j^T \theta| < \lambda \implies \hat{\beta}_j = 0 \label{eq:inactive_feature}
\end{align}
The proof lies in the ``implication" in the above equation (\ref{eq:inactive_feature})(but not in equivalence).
\end{proof}
It is known that Irrepresentable condition (assuming beta-min
conditions holds) is necessary and sufficient condition for the Lasso to select true model, see \cite{Zhao} (for completeness we have proved it in appendix A.4).
\begin{lemma} \label{lemma:lasso_dual}
Under assumption of the Irrepresentable Condition (IC) on the design matrix, the active set selected by the Lasso $\hat{S}_{lasso}(\lambda)$ is equal to the active set selected by the dual Lasso $\hat{S}_{dual}(\lambda)$, that is
\[ \hat{S}_{lasso}(\lambda) = \hat{S}_{dual}(\lambda).
\]
\end{lemma}
The proof is worked out in Appendix A.3.
The IC may fail to hold due to violation of any one of (or both) the following two conditions:
\begin{enumerate}
\item When $C_{11}$ is not invertible, that implies there is strong correlation among variables of the true active set.
\item The active predictors are correlated with the noise features (this situation is better explained in terms of irrepresentable condition).
\end{enumerate}
When there is strong correlation among variables of the active set, then $C_{11}$ is not invertible and the IC does not hold, and the Lasso fails to do variable selection. But we argue that the dual Lasso can still perform variable selection consistently even when $C_{11}$ not invertible, when we impose some milder condition on the design matrix, we call it Pseudo Irrepresentable Condition (PIC). The Pseudo Irrepresentable Condition is defined as follows.
\begin{definition} [Pseudo Irrepresentable Condition(PIC)]
We partition the covariance matrix as in (\ref{eq:sigma_par}). Then the Pseudo Irrepresentable condition is said to be met for the set S with a constant $\eta >0 $, if the following holds:
\begin{align} \label{eq:PIC}
|X^T_j G \;sign(\beta_1) | \leq 1 - \eta, \text{ for all } j \in S^c,
\end{align}
where G is a generalized inverse of the form $\left[ \begin{array}{cc}
C_{A}^{-1} & 0 \\
0 & 0 \end{array} \right] $, and (\ref{eq:PIC}) holds for all $C_A \in C_R$, where $C_R$ is defined as $C_R := \{C_{rr}: rank(C_{rr}) = rank(C_{11})= r, C_{rr} \subset C_{11} \}$.
\end{definition}
The following lemma gives the sufficient condition for the dual Lasso for support recovery. This lemma is similar in spirit of the Lemma 2 define in \cite{Omidiran}. Here, we do not assume that $\Sigma_{11}$ is invertible.
\begin{lemma}[Primal-dual Condition for Variable Selection]
Suppose that we can find a primal-dual pair $(\hat{\beta}, \hat{\theta})$ that satisfy the KKT conditions
\begin{align}
\textbf{X}^T (\textbf{Y} - \textbf{X} \hat{\beta} )+ \lambda \hat{v} &= 0, \; \text{ where } \hat{v} = sign(\hat{\beta})\\
\hat{\theta} &= \textbf{Y} - \textbf{X} \hat{\beta},
\end{align}
and the signed support recovery conditions
\begin{align}
\hat{v}_j &= sign(\beta_j) \text{ for all } j \in S, \\
\hat{\beta}_j & = 0 \text{ for all } j \in S^c ,\\
|\hat{v}_j| & < 1 \text{ for all } j \in S^c \label{eq:dualSign}
\end{align}
Then $\hat{\theta}$ is the unique optimal solution to the dual Lasso and $\hat{S}_{dual}$ recovers the true active set.
\end{lemma}
We have shown that the dual Lasso optimal $\hat{\theta}$ is always unique, and it remains to show that the $\hat{S}_{dual}$ recovers the true active set. Under the assumption (\ref{eq:dualSign}), we can derive that $|X_j^T \hat{\theta}| < \lambda $ for all $j \in S^c $. Therefore $\hat{S}_{dual} = S$.
\begin{theorem}
Under assumption of the PIC on the design matrix $\textbf{X}$, the active set selected by the dual Lasso $\hat{S}_{dual}$, is the same as the true active set $S$ with high probability. that is
\[\hat{S}_{dual} = S.
\]
\end{theorem}
When $C_{11}$ is invertible the PIC coincides with the IC, and under assumption of the IC we have already shown that $\hat{S}_{dual} = \hat{S}_{lasso}$. In Appendix A.4, we prove that the PIC is necessary and sufficient condition (beta-min condition is implicit) for the dual Lasso to consistently select the true active set. The PIC may hold even when $C_{11}$ is not invertible, which implies that the PIC is weaker than the IC. We illustrate it with the following examples:
Let $S = \{1,2,3,4 \}$ be the active set, the covariance matrix $C = \frac{\textbf{X}^T \textbf{X}}{n} $ and is given as
\begin{align*}
C = \left[ \begin{array}{ccccc}
1 & 0 & 0 & 0 & \rho \\
0 & 1 & 0 & 0 & \rho \\
0 & 0 & 1 & 0 & \rho \\
0 & 0 & 0 & 1 & \rho \\
\rho & \rho & \rho& \rho & 1
\end{array} \right],
\end{align*}
where the active variables are uncorrelated and the noise variable is equally correlated with all active covariates.
First of all, it is easy to check that only for $ | \rho | \leq \frac{1}{2} $, $C$ is positive semi definite, and for $ | \rho | < \frac{1}{4}$, $C$ satisfies the IC.
Now, we augment this matrix with two additional columns, one copy of the first and second active variables, and we rearrange the columns such that we get the following covariance matrix, and we redefine the set of active variables as $S = \{1,2,3,4,5,6 \}$ and we assume that $ | \rho | < \frac{1}{4}$.
\begin{align*}
C = \left[ \begin{array}{ccccccccc}
1 & 1 & 0 & 0 & 0 & 0 & \rho \\
1 & 1 & 0 & 0 & 0 & 0 & \rho \\
0 & 0 & 1 & 1 & 0 & 0 & \rho \\
0 & 0 & 1 & 1 & 0 & 0 & \rho \\
0 & 0 & 0 & 0 & 1 & 0 & \rho \\
0 & 0 & 0 & 0 & 0 & 1 & \rho \\
\rho & \rho & \rho & \rho & \rho& \rho & 1
\end{array} \right].
\end{align*}
We partition $\Sigma$ as (\ref{eq:sigma_par}), and it is clear that $C_{11}$ is not invertible and IC does not hold, hence the Lasso does not perform variable selection. The rank of the $C_{11}$ is 4. Let us consider any $(4 \times 4)$ sub matrix of $C_{11}$ such that its rank is four ($ S_1 \subset S, and rank(S_1)= 4, S_1 = \{ \{1,3,5,6 \}, \{1,4,5,6 \}, \{2,3,5,6 \}, \{2,4,5,6 \}$). Further, we partition $C_{11}$ as
\begin{align} \label{eq:sigma11_par}
C_{11} = \left[ \begin{array}{cc}
C_{rr} & C_{rr'}\\
C_{r'r} & C_{r'r'}
\end{array} \right],
\end{align}
where $rank(C_{rr})$ has full column rank and $rank(C_{rr}) = rank(C_{11})$, let $C_R$ be set of are four such possible invertible sub matrices of $C_{11}$. Then considering the generalized inverses corresponding to them as
\begin{align} \label{eq:sigma11_inv}
C_{11}^{+} = \left[ \begin{array}{cc}
C_{A}^{-1} & 0 \\
0 & 0
\end{array} \right],
\end{align}
where $C_A \in C_R$ is invertible. With the above inverse $C_{11}^{+}$ the PIC holds for the design matrix $\textbf{X}$. It can be also viewed as the IC is satisfied for each reduced active set $S' \in S_1$ and the corresponding reduced design matrix $\textbf{X}_{S'}$, and hence the Lasso picks randomly one element from the set $S_1$ and sets the coefficient of the noise variable to zero (with high probability). Also, since PIC holds, the dual Lasso will select the true active set $S$ with high probability and will set zero for the coefficient of noise feature.
\subsection{Dual Lasso Selection and Ridge Estimation}
After proving that the joint consideration of the Lasso primal and its dual leads to correlated variable selection (under certain regularity condition), we now combine the dual Lasso selection with the Ridge estimation. Mainly, we consider the $\ell_2$ penalty (Ridge penalty) which is known to perform best in case of correlated variables, to estimate the coefficients of the predictors selected by the dual Lasso. We develop an algorithm called DLSelect+RR, which is a two stage procedure, the dual selection followed by the Ridge Regression.
\begin{algorithm}[h!]
\SetAlgoLined
\textbf{Input:} dataset $(\textbf{Y},\textbf{X})$\\
\textbf{Output:} $\hat{S}$:= the set of selected variables\\
$\hat{\beta}$ := the estimated coefficient vector\\
\caption{DLSelect+RR Algorithm}\label{algo:DLSelect}
\textbf{Steps:}\\
1. Perform Lasso on the data $(\textbf{Y}$, $\textbf{X})$.
Denote the Lasso estimator as $\hat{\beta}_{lasso}$.\\
2. Compute the dual optimal as
\[\hat{\theta} = \textbf{Y} - \textbf{X}\hat{\beta}_{lasso}. \]
Denote the dual Lasso active set as $\hat{S}_{dual}$\\
3. Compute the reduced design matrix as \[\textbf{X}_{red} = \{ {X}_j : j \in \hat{S}_{dual} \}. \]
4. Perform Ridge regression based on the data $(\textbf{Y}$, $\textbf{X}_{red})$ and obtain the ridge estimator $\beta_j$ for $j \in S_{dual} $. Set the remaining coefficients to zero.
\[ \hat{\beta}_j = 0 \text{ if } j \not\in S_{dual} \]
\textbf{return} $(\hat{S}, \hat{\beta})$
\end{algorithm}
If model selection works perfectly (under strong assumptions, i.e. IC), then the post-model selection estimators are the oracle estimators with well behaved properties (see \cite{Belloni}). In the following we argue that for the combination, dual selection followed by $\ell_2$ estimation, the prediction accuracy is at least as good as the Lasso
It has been already proven that the Lasso+OLS (\cite{Belloni}) estimator performs at least as good as Lasso in terms of the rate of convergence, and it has a smaller bias than the Lasso. Further Lasso+mLS (Lasso+ modified OLS) or Lasso+Ridge estimator have been also proven to be asymptotically unbiased under the Irrepresentable condition and other regularity conditions, see \cite{liu2013}. Under the Irrepresentable condition the Lasso solution is unique and the DLSelect+RR is the same as the Lasso+Ridge and the same argument holds for the DLSelect+RR. Also, In the following section we prove empirically that the prediction performance of the DLSelect+RR is at least as good as the Lasso.
\end{document}
\section{Numerical Studies}
In this section, we apply the DLSelect+Ridge for variable selection and estimation on simulations and real data and compare the results with that of the Lasso, Ridge and Elastic-net. We consider the True Positive Rate (power) and False Discovery Rate (FDR) as the measure of performances for variable selection, which are defined as follows.
\begin{equation}\label{eq:tpr}
\begin{aligned}
TPR &= \frac{ | \hat{S} \bigcap S |} {|S|}\\
FDR & = \frac{ | \hat{S} \bigcap S^c |} {|\hat{S}|}
\end{aligned}
\end{equation}
For prediction performance we consider the Mean Squared Prediction Error, which is defined as
\begin{align}
MSE &= \frac{1}{n}\|\textbf{Y} - \hat{\textbf{Y}}\|^2_2,
\end{align}
where $\hat{\textbf{Y}}$ is the predicted response vector or an estimate $\textbf{X} \hat{\beta}$ based on an estimator $\hat{\beta}$.
Since our aim is to avoid false negatives, we do not report false positives, and ridge does not perform variable selection therefore TPR is not reported for the Ridge. The Ridge is considered as a competitor because its prediction performance is better than the Lasso for correlated designs.
\subsection{Simulation Examples}
We consider five different simulation settings,
where simulate data from the linear model as in (\ref{eq:lr}) with fixed design matrix $\textbf{X}$, and $\sigma = 1$. We generate the design matrix $\textbf{X}$ once from a multivariate normal distribution $N_p(0, \Sigma)$ with different structures for $\Sigma$, and keep it fixed for all replications.
For each simulation example, $100$ data sets were generated, where each dataset consists of a training set used to fit the model, an independent validation set used for tuning the regularization parameter and an independent test set used for evaluation of the performance. We denote by $ \#/\#/\#$, the number of observation in training, validation and test set respectively.
For most of the simulation examples we fix the size of the active set to $s=20$ and the true coefficient vector as
\begin{align} \label{eq:true_beta}
\beta = \{ \underbrace{1,...,1}_{20}, \underbrace{0,...,0}_{480} \}.
\end{align}
We generate $100$ data sets with sample sizes $n/n/1000$ with $n = 100, 200, 400 , 600$. For each simulation example and each method the MSE and TPR are computed over $100$ data sets. A suitable grid of values for the tuning parameters is considered, and all reported results are based on the median of $100$ simulation runs.
\subsubsection{Block Diagonal Model}
Here we generate the fixed design matrix $\textbf{X} \sim N_p(0, \Sigma_1)$ with $p=500$, where $\Sigma_1$ is a block diagonal matrix. The matrix $\Sigma_1$ consists of $50$ independent blocks $B$ of size $10 \times 10$, defined as
\[
B_{j,k} = \left\lbrace \begin{array}{cc}
1, & j=k\\
.9, & otherwise
\end{array} \right. \]
This simulation example is considered to show that when the Lasso (due to collinearity) and Ridge (due to noise) do not perform well, the Enet and DLSelect+Lasso perform quite well.
From the table (\ref{table:BD}) it is easy to figure out that the Ridge performs poorly in terms of prediction performance for all simulation setting and the Lasso is not stable for variable selection. The Enet consistently selects the true active set, and DLSelect+Ridge completes with Enet in all settings.
\begin{table}[h!] \centering
\caption{Performance measures for block diagonal case}\label{table:BD}
\begin{tabular}{ p{10mm}p{30mm}p{25mm}p{25mm}}
\toprule
n & Method& MSE(SE) & TPR \\
\toprule
100 & Lasso & 22.37(1.31) & 0.45 \\
& Ridge & 565.58(3.31) & NA \\
& Enet& 22.17(1.2) & 1 \\
& DLSelect+Ridge & 18.92(1.2) & 0.6 \\
200 & Lasso & 11.52(0.67) & 0.6 \\
& Ridge & 466.35(2.26) & NA \\
& Enet& 11.37(0.63) & 1 \\
& DLSelect+Ridge & 8.45(0.55) & 1 \\
400 & Lasso & 6.88(0.31)& 0.55 \\
& Ridge & 417.59(2.07) & NA \\
& Enet& 6.85(0.32)& 1 \\
& DLSelect+Ridge & 5.42(0.37) & 1 \\
600 & Lasso & 5.54(0.28) & 0.65 \\
& Ridge & 5.87(0.31) & NA \\
& Enet& 5.34(0.25)& 1 \\
& DLSelect+Ridge & 3.53(0.22) & 1 \\
\bottomrule
\end{tabular}
\end{table}
\subsubsection{Single Block Model with Noise Features}
Here we generate the fixed design matrix $\textbf{X} \sim N_p(0, \Sigma_2)$ with $p=500$, where $\Sigma_2$ is almost an identity matrix except for the first $20 \times 20$ is a single highly correlated block. The matrix $\Sigma_2$ is defined as
\[
\Sigma_{j,k} = \left\lbrace \begin{array}{cc}
1, & j=k\\
.9, & j \neq k \textbf{ and } i,j \leq 20 \\
0, & otherwise
\end{array} \right. \]
In this setting, the first twenty variables are active predictors and they are highly correlated, and the remaining $480$ are independent noise variables. We generate $100$ data sets with sample sizes $n/n/1000$, where $n = 100, 200, 400 , 600$. The simulation results are reported in Table (\ref{table:SBN}).
\begin{table}[h!] \centering
\caption{Performance measures for single block with noise }\label{table:SBN}
\begin{tabular}{ p{10mm}p{30mm}p{25mm}p{25mm}}
\toprule
n & Method& MSE(SE) & TPR \\
\toprule
100 & Lasso & 101.37(2.26) & 0.4 \\
& Ridge & 922.41(4.36) & NA \\
& Enet& 102.91(2.37) & 1 \\
& DLSelect+Ridge & 88.00(3.2) & 1 \\
200 & Lasso & 15.55(0.59) & 0.25 \\
& Ridge & 627.66(2.67) & NA \\
& Enet& 15.92(0.57) & 1 \\
& DLSelect+Ridge & 8.26(0.56) & 1 \\
400 & Lasso & 2.67(0.17) & 0.2 \\
& Ridge & 456.17(1.95) & NA \\
& Enet& 2.50(0.15) & 1 \\
& DLSelect+Ridge & 1.16(0.081) & 1 \\
600 & Lasso & 1.46(0.06) & 0.15 \\
& Ridge & 5.87(0.26) & NA \\
& Enet& 1.12(0.06) & 1 \\
& DLSelect+Ridge & 2.29(0.13) & 1 \\
\bottomrule
\end{tabular}
\end{table}
From Table (\ref{table:SBN}), it is clear that the Lasso and Ridge performs poorly (one can give similar argument as Block diagonal model). The Enet and DLSelect+Ridge consistently selects true active set with reduced prediction error.
\subsubsection{Single Block Model without Noise Features}
Here we generate the fixed design matrix $\textbf{X} \sim N_p(0, \Sigma_3)$ with $p=20$, where $\Sigma_3$ is a single block of highly correlated variables. The matrix $\Sigma_3$ is defined as
\[
\Sigma_{j,k} = \left\lbrace \begin{array}{cc}
1, & j=k\\
.99, & otherwise
\end{array} \right. \]
The true coefficient vector is \[\beta = \{ \underbrace{1,...,1}_{20} \}. \]
We generate $100$ data sets with sample sizes $n/n/200$ with $n = 20, 200 $. The simulation results are reported in Table (\ref{table:SBWN}).
\begin{table}[h!] \centering
\caption{Performance measures for single block with noise }\label{table:SBWN}
\begin{tabular}{ p{10mm}p{30mm}p{25mm}p{25mm}}
\toprule
n & Method& MSE(SE) & TPR \\
\toprule
20 & Lasso & 245.95(8.91) & 0.25 \\
& Ridge & 246.23(8.97) & NA \\
& Enet& 244.95(8.85) & 1 \\
& DLSelect+Ridge & 249.99(7.15) & 1 \\
200 & Lasso & 4.75(0.42) & 0.1 \\
& Ridge & 4.64(0.43) & NA \\
& Enet& 4.75(0.40) & 1 \\
& DLSelect+Ridge & 1.00(0.10) & 1 \\
\bottomrule
\end{tabular}
\end{table}
From Table (\ref{table:SBWN}), it is apparent that the Lasso performs poorly in terms of variable selection as well as prediction accuracy. The Ridge gives the best predictive performs for the number of sample size increases.
The Enet and DLSelect+Ridge consistently selects true active set with, however the DLSelect+Ridge has better prediction accuracy for moderate sample size.
\subsubsection{Toeplitz Model}
Here we consider special case of a Toeplitz matrix $\Sigma_4$ to generate the fixed design matrix $\textbf{X} \sim N_p(0, \Sigma_4)$ with $p=500$. The matrix $\Sigma_4$ is defined as
\[
\Sigma_{j,k} = \left\lbrace \begin{array}{cc}
1, & j=k\\
\rho^{|i-j |} & otherwise
\end{array} \right. \], where $\rho = 0.9$.
The true coefficient vector is as defined in (\ref{eq:true_beta}), and we generate $100$ data sets with sample sizes $n/n/1000$ where $n = 100, 200, 400 , 600$. The Table \ref{table:TP} shows the simulation results.
\begin{table}[h!] \centering
\caption{Performance measures for Toeplitz settings}\label{table:TP}
\begin{tabular}{ p{10mm}p{30mm}p{25mm}p{25mm}}
\toprule
n & Method& MSE(SE) & TPR \\
\toprule
100 & Lasso & 10440.42(53.28) & 0.36 \\
& Ridge & 12478.77(23.98) & NA \\
& Enet& 10352.06(23.21) & 0.89 \\
& DLSelect+Ridge & 7789.103(22.61) & 1\\
200 & Lasso & 713.34(4.13) & 0.51 \\
& Ridge & 654.85(3.8) & NA \\
& Enet& 651.08(3.86) & 0.99 \\
& DLSelect+Ridge & 97.17(1.70) & 0.77 \\
400 & Lasso & 145.15(2.12)& 0.54 \\
& Ridge & 98.63(1.13) & NA \\
& Enet& 103.57(1.19) & 1 \\
& DLSelect+Ridge & 52.19(0.85) & 1 \\
600 & Lasso & 200.15(1.68) & 0.65 \\
& Ridge & 169.01(1.27) & NA \\
& Enet& 169.66(1.31)& 0.99 \\
& DLSelect+Ridge & 22.35(0.47)) & 1 \\
\bottomrule
\end{tabular}
\end{table}
The Table (\ref{table:SBWN}) shows that the Lasso and the Ridge performs poorly for all settings. The DLSelect+Ridge consistently selects the true active set, however the DLSelect+Ridge has better prediction accuracy for moderate sample size.
\subsubsection{Independent Predictor Model}
Finally we consider an identity matrix to generate the fixed design matrix $\textbf{X} \sim N_p(0, I)$ with $p=500$. In this setting all predictors and uncorrelated. The true coefficient vector is as defined in (\ref{eq:true_beta}), and we generate $100$ data sets with sample sizes $n/n/1000$, where $n = 100, 200, 400 , 600$.
The Table \ref{table:Indep} shows the simulation results.
\begin{table}[h!] \centering
\caption{Performance measures for independent settings}\label{table:Indep}
\begin{tabular}{ p{10mm}p{30mm}p{25mm}p{25mm}}
\toprule
n & Method& MSE(SE) & TPR \\
\toprule
100 & Lasso & 14.50(3.74) & 1 \\
& Ridge & 153.47(0.91) & NA \\
& Enet& 28.85(4.59) & 1 \\
& DLSelect+Ridge & 39.16(29.73) & 1 \\
200 & Lasso & 2.32(0.23) & 1 \\
& Ridge & 139.34(0.76) & NA \\
& Enet& 2.45(0.24) & 1 \\
& DLSelect+Ridge & 3.48(0.51) & 1 \\
400 & Lasso & 1.56(0.10) & 1 \\
& Ridge & 118.21(0.73) & NA \\
& Enet& 1.59(0.10) & 1 \\
& DLSelect+Ridge & 3.70(0.46) & 1 \\
600 & Lasso & 1.35(0.06) & 1 \\
& Ridge & 8.60(0.53) & NA \\
& Enet& 1.37(0.06)& 1 \\
& DLSelect+Ridge & 3.37(0.35) & 1 \\
\bottomrule
\end{tabular}
\end{table}
The Table (\ref{table:SBWN}) shows that the Lasso gives the best prediction accuracy and the Ridge performs poorly for all the settings. The Enet and DLSelect+Ridge competes each other.
\subsection{Real Data Example}
In this section, we consider five real world data to evaluate the prediction and variable selection performance of the proposed method $DLSelect+Ridge$. We randomly split the data sets into two halves for $100$ times, we use first half for training (using cross validation) and second half is used as a test set. For testing variable selection, For first two datasets (UScrime and Prostate) we consider all the variables as relevant variables and for the remaining datasets we select ten most variable which are highly correlated with the response and another ten variables which are correlated with the selected variables. Median MSE, standard error and median TPR are reported over $100$ splits for each example.
\subsubsection{USCrime Data}
This is a classical dataset collected in 1960 where criminologists are mainly interested in the effect of punishment on crime rates.
There are Independent $15$ independent variables and the response is rate of crimes in a particular category per head of population. For more details on this dataset we refer to \cite{UScrime}. The performance measures are reported in Table \ref{table:USCrime}.
\begin{table}[h!] \centering
\caption{Performance measures for UScrime data}\label{table:USCrime}
\begin{tabular}{ p{10mm}p{30mm}p{25mm}p{25mm}}
\toprule
& Method& MSE(SE) & TPR \\
\toprule
& Lasso & 87725(36371) & 0.54 \\
& Ridge & 77153(26118) & NA \\
& Enet& 83403(34342) & 0.45 \\
& DLSelect+Ridge & 78275(24625) & 0.54 \\
\bottomrule
\end{tabular}
\end{table}
Here, we have considered all covariates as important variables. The Ridge regression outperforms the other methods, and DLSelect+Ridge performs better than Lasso and the Enet in terms of prediction perform as well as variable selection.
\subsubsection{Prostate Data}
The Prostate dataset has $97$ observations and $9$ covariates.
This dataset is an outcome of a study that examined the correlation between the level of prostate specific antigen and a number of clinical measures in men who were about to receive a radical prostatectomy. For further details on the dataset we refer to \cite{Prostate}.
\begin{table}[h!] \centering
\caption{Performance measures for Prostate data}\label{table:Indep}
\begin{tabular}{ p{10mm}p{30mm}p{25mm}p{25mm}}
\toprule
& Method& MSE(SE) & TPR \\
\toprule
& Lasso & 0.56(0.09) & 0.63 \\
& Ridge & 0.56(0,08) & NA \\
& Enet& 0.55(0.09) & 0.63 \\
& DLSelect+Ridge & 0.56(0.07) & 1\\
\bottomrule
\end{tabular}
\end{table}
The performance measures are reported in Table \ref{table:USCrime}. Here, we have considered all covariates as important variables, From the table, it is clear that all method seems to report almost the same prediction error, and DLSelect+Ridge performs better than Lasso and the Enet in terms of variable selection.
\subsubsection{Riboflavin Data}
The dataset of riboflavin consists of, $n=71$ observations of $p=4088$ predictors (gene expressions) and univariate response, riboflavin production rate(log-transformed), see \cite{HDview} for details on riboflavin dataset. Since the ground truth is not available, we consider Riboflavin data for the design matrix $\textbf{X}$ with synthetic parameters $\beta$ and simulated Gaussian errors $\epsilon \sim \mathbb{N}_n(0, \sigma^2 I)$. We fix the size of the active set to $s = 20$ and $\sigma = 1$ and for the true active set, select ten predictors which are highly correlated with the response and another ten variables which are most correlated with those selected variables. The true coefficient vector is
\begin{align*}
\beta_j = \left\lbrace \begin{array}{ll}
1 & \text{ if } j \in S \\
0 & \text{ if } j \not\in S\\
\end{array} \right. .
\end{align*}
Then we compute the response using the Equation (\ref{eq:lr}).
The performance measures are reported in Table \ref{table:Ribo}.
\begin{table}[h!] \centering
\caption{Performance measures for Leukaemia data}\label{table:Ribo}
\begin{tabular}{ p{10mm}p{30mm}p{25mm}p{25mm}}
\toprule
& Method& MSE(SE) & TPR \\
\toprule
& Lasso & 96.69(63) & 0.27 \\
& Ridge & 290.98(138) & NA \\
& Enet& 92.44(65) & 0.44 \\
& DLSelect+Ridge & 88.31(54) & 0.38\\
\bottomrule
\end{tabular}
\end{table}
From the table (\ref{table:Ribo}), we conclude that Enet outperforms in terms of variable selection, whereas, DLSelect+Ridge performs better than others in terms of prediction performance.
\subsubsection{Myeloma Data}
We consider another real dataset, Myeloma $(n = 173, p = 12625)$ data for the design matrix $\textbf{X}$ with synthetic parameters $\beta$ and simulated Gaussian error
We refer to \cite{myeloma} for details on Myeloma dataset. In this example also, we set active set and generated response same as previous example (Riboflavin). The performance measures are reported in Table \ref{table:Myeloma}.
\begin{table}[h!] \centering
\caption{Performance measures for Leukaemia data}\label{table:Indep}
\begin{tabular}{ p{10mm}p{30mm}p{25mm}p{25mm}}
\toprule
& Method& MSE(SE) & TPR \\
\toprule
& Lasso & 68.29(27.73) & 0.35 \\
& Ridge & 239.58(58.25) & NA \\
& Enet& 70.37(29.16) & 0.58 \\
& DLSelect+Ridge & 75.25(28.70) & 0.52\\
\bottomrule
\end{tabular}
\end{table}
From the table (\ref{table:Ribo}), the Enet outperforms in terms of variable selection as well as prediction performance.
\subsubsection{Leukaemia Data}
We consider the famous dataset of Leukaemia Data \cite{leuk}. In this example also, we set active set and generated response same as previous examples. The performance measures are reported in Table \ref{table:leuk}.
\begin{table}[h!] \centering
\caption{Performance measures for Leukaemia data}\label{table:leuk}
\begin{tabular}{ p{10mm}p{30mm}p{25mm}p{25mm}}
\toprule
& Method& MSE(SE) & TPR \\
\toprule
& Lasso & 111.53(97.3) & 0.46 \\
& Ridge & 182.11(97.2) & NA \\
& Enet& 90.47(82.64) & 0.6 \\
& DLSelect+Ridge & 72.43(67.3) & 0.5\\
\bottomrule
\end{tabular}
\end{table}
From the table (\ref{table:leuk}), it is clear that the Enet gives the better prediction performance, and DLSelect+Ridge performs better than Lasso and the Enet in terms of variable selection.
\end{document}
\begin{table}[h!] \centering
\caption{Performance measures for block diagonal case}\label{table:BD}
\begin{tabular}{ p{10mm} p{15mm} p{20mm}p{20mm}p{20mm}p{25mm}}
\toprule
n& &Lasso & Ridge & Enet & DLSelect+Ridge \\
\toprule
$100$ & MSE(SE) & 22.37(1.31) & 565.58(3.31) & 22.17(1.2) & 18.92(1.2) \\
& TPR& 0.45 & 1& 1& 0.6\\
$200$ & MSE(SE) & 11.52(0.67) & 466.35(2.26)& 11.37(0.63) &8.45(0.55) \\
& TPR& 0.6 & 1& 1& 1\\
$400$ & MSE(SE) & 6.88(0.31)& 417.59(2.07) &6.85(0.32)& 5.42(0.37) \\
& TPR& 0.55 & 1& 1& 1\\
$600$ & MSE(SE) & 5.54(0.28) & 5.87(0.31)& 5.34(0.25) &3.53(0.22)\\
& TPR& 0.65 & 1& 1& 1\\
\bottomrule
\end{tabular}
\end{table}
We estimate the following for each method.\\
MSE (SD) : mean squared prediction error as define in \ref{eg:TBD}.\\
TPR(True Positive Rate) : The ratio of the total number of truly identified non-zero components of $\beta$ and sparsity index (s).
\section{Computational Details}
Statistical analysis was performed in R 3.2.2. We used the package ``glmnet" for penalized regression method(the Lasso).
\section{Concluding Remarks}
The main achievements of this work are summarized as follows: We argued that the correlations among active predictors is not problematic, as long as the PIC is satisfied by the design matrix. In particular, we proved that the dual Lasso performs consistent variable selection under assumption of PIC. Exploiting this result we proposed the dual Lasso+Ridge method. We illustrated DLSelection+Ridge method on simulated and real high dimensional data sets. The numerical studies based on the simulations and real examples show clearly that the proposed method is very competitive in terms of variable selection, prediction accuracy, estimation accuracy and computation speed.
\section*{Appendix A}
\subsection*{A.1 Derivation of the Dual Form of the Lasso}
In this section, we derive the Lagrange dual of the Lasso problem (\ref{eq:lasso}), which serves as the selection operator for our approach. That is, by considering the lasso and its dual simultaneously it is possible to identify the non-zero entries in the estimator. For more details on dual derivation and projection on polytope formed by the dual constraints, we refer to \cite{Jie}.
We recall that the Lasso problem is defined as the following convex optimization problem.
\begin{align}
\min_{\beta \in \mathbb{R}^p} \left\lbrace \frac{1}{2}\|{ \textbf{Y} -\textbf{X} \beta}\|_2^2 + \lambda \|\beta\|_1 \right\rbrace \label{eq:lasso_recall}
\end{align}
Since the above problem has no constraints, its dual problem is trivial. So we introduce a new vector $\textbf{r} = \textbf{Y} -\textbf{X} \beta$, then the Lasso problem can be written as:
\begin{align}
\min_{\beta \in \mathbb{R}^p} & \left\lbrace \frac{1}{2}\| \textbf{r} \|_2^2 + \lambda \|\beta\|_1 \right\rbrace \\
\text{subject to } & \textbf{r} = \textbf{Y} -\textbf{X} \beta \nonumber
\end{align}
Now, to account for the constraints we introduce the dual vector $\theta in \mathbb{R}^n$, then we get the following Lagrangian equation with $\beta$ and $r$ as primal variables.
\begin{align}
L(\beta, \textbf{r}, \theta) = \frac{1}{2}\| \textbf{r} \|_2^2 + \lambda \|\beta\|_1 + \theta^T(\textbf{Y} - \textbf{X}\beta - \textbf{r}) \label{eq:lang}
\end{align}
Then the dual function can be written as:
\begin{align*}
g(\theta) & = \inf_{\beta, \textbf{r}} L(\beta, \textbf{r}, \theta) \\
&= \frac{1}{2}\|\textbf{r}\|_2^2 + \lambda \|\beta\|_1 + \theta^T(\textbf{Y} - \textbf{X}\beta - z)\\
& = \theta^T y + \inf_{r} \left\lbrace \frac{1}{2}\| \textbf{r} \|_2^2 - \theta^T \textbf{r} \right\rbrace +\inf_{\beta} \left\lbrace \lambda \|\beta\|_1 - \theta^T \textbf{X}\beta \right\rbrace \\
& = \theta^T y + \inf_{r} L_1(r)+\inf_{\beta} L_2(\beta)
\end{align*}
After solving the first optimization problem, we get
\begin{align}
\inf_{r} L_1(r) = - \frac{1}{2} \| \textbf{r} \|_2^2 \label{eq:residual}
\end{align}
Since $L_1(r)$ is non-differentiable, we consider its subgradient
\begin{align*}
\partial L_1(\beta) = \lambda v -\textbf{X}^T \theta ,
\end{align*}
where v is the subgradient of $\|\beta\|_1$, and it satisfies $\|v \|_{\infty} \leq 1$ and $v^{T}\beta = \|\beta\|_1$. For $L_1$ to attain an optimum, the following must hold.
\begin{align*}
\lambda v -\textbf{X}^T \theta & = 0\\
\implies \textbf{X}^T \theta & = \lambda v
\end{align*}
\begin{align}
\therefore |X_j^T \theta| \leq \lambda \; \text{ for all } j \in \{ 1,...,p \} \label{eq:dual_con}
\end{align}
From (\ref{eq:residual}) and (\ref{eq:dual_con}), we get the dual objective function as:
\begin{align*}
g(\theta) &= \theta^T \textbf{Y} - \frac{1}{2} \theta^T\theta \\
g(\theta) & = \frac{1}{2} \|\textbf{Y} \|_2^2 - \| \theta - \textbf{Y} \|_2^2
\end{align*}
Then the dual problem is given as:
\begin{align} \label{eq:dual_lasso}
\sup_{\theta} \; & g(\theta) = \frac{1}{2} \|\textbf{Y} \|_2^2 - \| \theta - \textbf{Y} \|_2^2 \nonumber \\
\text{subject to } & \therefore |X_j^T \theta| \leq \lambda \; \text{ for all } j \in \{ 1,...,p \}
\end{align}
\subsection*{A.2 Relationship Between the Lasso and its Dual Optimal}
In this section, we derive the relationship between the Lasso optimal and its dual optimal.
For a fixed $\lambda$, the Lasso problem (\ref{eq:lasso}) is convex in $\beta$ and it is strictly feasible since it has no constraints, therefore by Slater’s condition, strong duality holds. Let us suppose that $\hat{\beta}, \hat{r}$ and $\hat{\theta} $ are optimal primal and dual variables, then by the KKT conditions the following must hold.
\begin{align}
0 & \in \partial_\beta L(\hat{\beta}, \hat{r}, \hat{\theta}) \label{eq:par_beta} \\
\Delta_z L(\hat{\beta}, \hat{r}, \hat{\theta}) & = \hat{r} - \hat{\theta }= 0 \label{eq:par_z}\\
\Delta_\theta L(\hat{\beta}, \hat{r}, \hat{\theta}) & = \textbf{Y} - \textbf{X}\hat{\beta }- \hat{r} = 0 \label{eq:par_theta}
\end{align}
From (\ref{eq:par_beta}) we get
\begin{align*}
x^T \hat{\theta} & = \lambda \hat{v} \\
|X_j^T \theta| & \leq \lambda \; \text{ for all } j \in \{ 1,...,p \}
\end{align*}
Or equivalently for all $j \in \{ 1,...,p \}$ the following must hold.
\begin{align}
X_j^T \hat{\theta} = \left\lbrace \begin{array}{ll}
\lambda & \text{ if } \hat{\beta} > 0 \\
\in \left[ -\lambda, \lambda \right] & \text{ if } \hat{\beta}=0 \\
-\lambda & \text{ if } \hat{\beta} < 0 \\
\end{array} \right. \label{eq:dual_feasibility}
\end{align}
From the above equation (\ref{eq:dual_feasibility}), we get the following important result.
\begin{align}
| X_j^T \hat{\theta} | < \lambda \implies \hat{\beta}=0 \label{eq:lasso_in_dual}
\end{align}
Finally, from (\ref{eq:par_z}) and (\ref{eq:par_theta}) we get the following equality.
\begin{align} \label{dual_lasso_fit}
\hat{\theta} = \textbf{Y} - \textbf{X} \hat{\beta}
\end{align}
and substituting value of $\hat{\theta}$ in (\ref{eq:dual_feasibility}) we get the following expression.
\begin{align} \label{eq:subgrad_lasso_fit}
\textbf{X}^T(\textbf{Y} - \textbf{X} \hat{\beta}) = \lambda v.
\end{align}
\section*{A.3 Proof of Lemma \ref{lemma:lasso_dual}}
\begin{proof}
Without loss of generality we can assume that the first $s = |S|$ variables are the active variables, and we partition the empirical covariance matrix as in Equation (\ref{eq:sigma_par}), $\hat{\beta }= (\beta_1 \; \beta_2)^T $ and $\hat{ v }= (v_1 \; v_2)^T $ accordingly. Let us recall the IC (for the noiseless case for simplicity), it is defined as follows.
\begin{definition} [Irrepresentable Condition(IC)]
The irrepresentable condition is said to be met for the set S with a constant $\eta >0 $, if the following holds:
\begin{align} \label{eq:IC}
\|C_{12} C_{11}^{-1}sign(\beta_1) \|_{\infty} \leq 1 - \eta.
\end{align}
\end{definition}
Under IC, the lasso solution is unique. If we further assume the beta-min condition then the following holds, see (\cite{Buhlmann1}) for the detailed proof.
\[ S = \hat{S}_{lasso}.
\]
The proof of the proposition (\ref{lemma:lasso_dual}) is fairly simple, we prove it by contradiction. Let us assume that $\hat{S}_{lasso} ! = \hat{S}_{dual}$, then from Proposition (\ref{pro:lasso_in_dual}) the Lasso active set $\hat{S}_{lasso}$ is a proper subset of the dual active set $\hat{S}_{dual}$ , and it follows that there exists some $j \in \hat{S}^c$ for which the following condition is satisfied.
\begin{align*}
\beta_j = 0 \text{ and } |X_j^{T} \hat{\theta}| = \lambda
\end{align*}
Substituting value of $\hat{\theta} = \textbf{Y} - \textbf{X} \hat{\beta}$ (see Appendix A.2) and $\textbf{Y} = \textbf{X} \beta$, we get the following.
\begin{align*}
|X_j^{T} (\textbf{Y} - \textbf{X}\hat{\beta})| &= \lambda \\
\implies |X_j^{T} \textbf{X} (\beta - \hat{\beta})| &= \lambda
\end{align*}
Under IC the Lasso selects the active sets, so we have $\beta_2 = \hat{\beta}_2 = 0$, some algebraic simplification gives the following equality.
\begin{align}
\implies |(C_{21})_j (\beta_1 - \hat{\beta}_1)| = \lambda \label{eq:c_21}
\end{align}
From the KKT condition (see AppendixA.2) we have:
\begin{align*}
X^T (\textbf{Y} - \textbf{X} \hat{\beta} )+ \lambda \hat{v} = 0
\end{align*}
where $\| v \|_{\infty} \leq 1$ and $\beta v = \| \beta \|_1 $.
, by substituting $\textbf{Y} = \textbf{X}\beta$, we get
\begin{align*}
\textbf{X}^T \textbf{X} (\hat{\beta }- \beta) &= - \lambda \hat{v }
\end{align*}
We can write the above equation in terms of partitions of $C = \hat{\Sigma}$ as follows.
\begin{align*}
\left[ \begin{array}{cc}
C_{11} & C_{12}\\
C_{21} & C_{22})
\end{array} \right] ( \begin{array}{c} \beta_1- \hat{\beta_{1}} \\ \beta_2- \hat{\beta_{2}} \end{array}) & = \lambda ( \begin{array}{c} v_1 \\ v_2 \end{array})
\end{align*}
Since $\beta_2 = \hat{\beta_{2}}=0$, therefore, we get the following equality.
\begin{align*}
\beta_1- \hat{\beta_{1}} = \lambda C_{11}^{-1} sign(\beta_1)
\end{align*}
Substituting value of $\beta_1- \hat{\beta_{1}}$ into the equation
(\ref{eq:c_21}) we get
\begin{align}
|(C_{21})_j \lambda C_{11}^{-1} sign(\beta_1)| = \lambda \\
|(C_{21})_j C_{11}^{-1} sign(\beta_1)| = 1
\end{align}
It violet the IC, hence under assumption of IC,
\[ \beta_j = 0 \implies |X_j^{T} \hat{\theta}| < \lambda .\] Therefore the following equality must hold, that completes the proof.
\[ \hat{S}_{lasso}(\lambda) = \hat{S}_{dual}(\lambda).
\]
\end{proof}
\section*{Appendix A.4 IC implies Lasso Variable Selection}
\begin{proof}
This result and proof are from \cite{Buhlmann1}. The IC depends on the covariance of the predictors $C = \hat{\Sigma}$ and the signs of the unknown true parameter $\beta$ (beta-min condition is implicit). For simplicity, we prove it for the noiseless case, where $\textbf{Y} = \textbf{X}\beta$. We first assume that the IC holds and we will show that Lasso correctly identifies the active set $S$. From KKT condition as in (\ref{eq:subgrad_lasso_fit}), and substituting $\textbf{Y} = \textbf{X}\beta$, we get
\begin{align*}
\textbf{X}^T \textbf{X} (\hat{\beta }- \beta) &= - \lambda v \\
\left[ \begin{array}{cc}
C_{11} & C_{12}\\
C_{21} & C_{22})
\end{array} \right] ( \begin{array}{c} \beta_1- \hat{\beta_{1}} \\ \beta_2- \hat{\beta_{2}} \end{array}) & = \lambda ( \begin{array}{c} v_1 \\ v_2 \end{array})
\end{align*}
We note that, for the true parameter vector, $\beta_{2} $ is a null vector by definition. We get the following two equations after some simplification:
\begin{align} \label{eq:kkt_1}
C_{11}(\beta_{1} - \hat{\beta_1}) - C_{12} \hat{\beta_2} &= \lambda v_1 \\
C_{21}(\beta_{1} - \hat{\beta_1}) - C_{22} \hat{\beta_2} &= \lambda v_2
\end{align}
After some algebraic simplification of the first equation we get
\begin{align*}
\hat{\beta_1}-\beta_{1} = C^{-1}_{11} ( C_{12} \hat{\beta_2} +\lambda v_1)\\
\end{align*}
Substituting value of $\hat{\beta_1}-\beta_{1}$, in the second equation
\begin{align*}
C_{21} C^{-1}_{11} ( C_{12} \hat{ \beta_2 } +\lambda v_1) - \Sigma_{22} \beta_2 &= \lambda v_2
\end{align*}
by multiplying both the sides with $\hat{\beta}_2^T$
\begin{align*}
\hat{\beta}_2^T (C_{22} - C_{21} C^{-1}_{11} C_{12} ) \hat{ \beta_2 } &= -\lambda \| \beta_2 \|_1 - \hat{ \beta_2 }^T C_{21} C^{-1}_{11} \lambda v_1
\end{align*}
Applying holders inequality for the term $\hat{ \beta_2 }^T C_{21} C^{-1}_{11}\lambda v_1$ on RHS we get
\begin{align*}
\hat{ \beta_2 }^T C_{21} C^{-1}_{11} \lambda v_1 & \leq \lambda \| \hat{ \beta_2 } \|_1 C_{21} C^{-1}_{11} v_1 \|_{\infty} \\
\implies & \leq \lambda \| \hat{ \beta_2 } \|_1.
\end{align*}
We get the following expression after substitution,
\begin{align*}
\hat{\beta}_2^T (C_{22} - C_{21} C^{-1}_{11} C_{12} ) \hat{ \beta_2 } \leq - \lambda \| \hat{ \beta_2 } \|_1.
\end{align*}
Since $\lambda \hat{ \beta_2 } \|_1 > 0 $ we get the following inequality,
\begin{align*}
\hat{\beta}_2^T (C_{22} - C_{21} C^{-1}_{11} C_{12} ) \hat{ \beta_2 } \leq 0
\end{align*}
The matrix $(C_{22} - C_{21} C^{-1}_{11} C_{12} )$ is a positive semi-definite, we have arrived at a contradiction. Therefore $\hat{\beta}_{S^c} = 0$, for any Lasso solution it is true. Hence the Lasso correctly identifies all the zero components, and $\hat{S}_{lasso} \subset S$.
Now, we assume that lasso selects the true active set, and we will show that the IC holds. Basically, It is given that $\hat{\beta}_2 = \hat{\beta}_{S^c} = 0$. Using the KKT condition again, and substituting $\hat{\beta}_2 = 0$ in (\ref{eq:kkt_1}), we get the following expression.
\begin{align*}
C_{11}(\beta_{1} - \hat{\beta_1}) &= \lambda v_1 \\
C_{21}(\beta_{1} - \hat{\beta_1}) &= \lambda v_2
\end{align*}
After solving the above we have
\begin{align*}
C_{21} C_{11}^{-1} \lambda v_1 &= \lambda v_2
\end{align*}
Since $\| v2 \|_{\infty} < 1$ and we have the following inequality.
\begin{align*}
\| C_{21} C_{11}^{-1} \lambda sign(\beta_1) \|_{\infty} & < \lambda \\
\implies \| C_{21} C_{11}^{-1} sign(\beta_1) \|_{\infty} < 1
\end{align*}
\end{proof}
\section*{Appendix A.5 PIC implies dual Lasso Variable selection}
\begin{proof}
The proof is similar to the proof given for IC (see Appendix A.4) except we replace $G_{11}^{-1}$ with the one of the generalized inverse . The PIC, like IC depends on the covariance of the predictors $C = \hat{\Sigma}$ and the signs of the unknown true parameter $\beta$. For simplicity, we prove it for the noiseless case, where $\textbf{Y} = \textbf{X}\beta$. We first assume that the PIC holds and we will show that dual Lasso correctly identifies the active set $S$. From KKT condition as in (\ref{eq:subgrad_lasso_fit}), and substituting $\textbf{Y} = \textbf{X}\beta$, we get
\begin{align*}
\textbf{X}^T \textbf{X} (\hat{\beta }- \beta) &= - \lambda v \\
\left[ \begin{array}{cc}
C_{11} & C_{12}\\
C_{21} & C_{22})
\end{array} \right] ( \begin{array}{c} \beta_1- \hat{\beta_{1}} \\ \beta_2- \hat{\beta_{2}} \end{array}) & = \lambda ( \begin{array}{c} v_1 \\ v_2 \end{array})
\end{align*}
We note that, for the true parameter vector, $\beta_{2} $ is a null vector, by definition. We get the following two equations after some simplification:
\begin{align} \label{eq:kkt_1}
C_{11}(\beta_{1} - \hat{\beta_1}) - C_{12} \hat{\beta_2} &= \lambda v_1 \\
C_{21}(\beta_{1} - \hat{\beta_1}) - C_{22} \hat{\beta_2} &= \lambda v_2
\end{align}
After simplification of the first equation we get
\begin{align*}
\hat{\beta_1}-\beta_{1} = C^{+}_{11} ( C_{12} \hat{\beta_2} +\lambda v_1)\\
\end{align*}
Substituting value of $\hat{\beta_1}-\beta_{1}$, in the second equation
\begin{align*}
C_{21} C^{+}_{11} ( C_{12} \hat{ \beta_2 } +\lambda v_1) - \Sigma_{22} \beta_2 &= \lambda v_2
\end{align*}
by multiplying both the sides with $\hat{\beta}_2^T$
\begin{align*}
\hat{\beta}_2^T (C_{22} - C_{21} C^{+}_{11} C_{12} ) \hat{ \beta_2 } &= -\lambda \| \beta_2 \|_1 - \hat{ \beta_2 }^T C_{21} C^{+}_{11} \lambda v_1
\end{align*}
Applying holders inequality for the term $\hat{ \beta_2 }^T C_{21} C^{+}_{11}\lambda v_1$ on RHS we get
\begin{align*}
\hat{ \beta_2 }^T C_{21} C^{+}_{11} \lambda v_1 & \leq \lambda \| \hat{ \beta_2 } \|_1 C_{21} C^{+}_{11} v_1 \|_{\infty} \\
\implies & \leq \lambda \| \hat{ \beta_2 } \|_1.
\end{align*}
We get the following expression after substitution,
\begin{align*}
\hat{\beta}_2^T (C_{22} - C_{21} C^{+}_{11} C_{12} ) \hat{ \beta_2 } \leq - \lambda \| \hat{ \beta_2 } \|_1.
\end{align*}
Since $\lambda \hat{ \beta_2 } \|_1 > 0 $ we get the following inequality,
\begin{align*}
\hat{\beta}_2^T (C_{22} - C_{21} C^{+}_{11} C_{12} ) \hat{ \beta_2 } \leq 0
\end{align*}
The matrix $(C_{22} - C_{21} C^{+}_{11} C_{12} )$ is a positive semi-definite, we have arrived at a contradiction. Therefore $\hat{\beta}_{S^c} = 0$, for any Lasso solution it is true. Hence the Lasso correctly identifies all the zero components.
Hence giving the similar argument as lemma (\ref{lemma:lasso_dual}), it can be shown that $|X_j^T \hat{\theta}| < \lambda$ for all $j \in S^c$. Therefore PIC implies dual lasso selects the true active set.
Now, we assume that dual lasso selects the true active set, and we will show that the PIC holds.
It is given that $|X_j^T \hat{\theta}| < \lambda$ for all $j \in S^c$. Therefore for any beta solution $\hat{\beta}_{S^c} = 0$. Using the KKT condition again, and substituting $\hat{\beta}_2 = 0$ in (\ref{eq:kkt_1}), we get the following expression.
\begin{align*}
C_{11}(\beta_{1} - \hat{\beta_1}) &= \lambda v_1 \\
C_{21}(\beta_{1} - \hat{\beta_1}) &= \lambda v_2
\end{align*}
After solving the above we have
\begin{align*}
C_{21} C_{11}^+ \lambda v_1 &= \lambda v_2
\end{align*}
Since $|X_j^T \hat{\theta}| < \lambda \; for j \in S^c$ , therefore $\| v2 \|_{\infty} < 1$ and we have the following PIC.
\begin{align*}
\| C_{21} C_{11}^+ \lambda sign(\beta_1) \|_{\infty} & < \lambda \\
\implies \| C_{21} C_{11}^+ sign(\beta_1) \|_{\infty} < 1
\end{align*}
\end{proof}
\end{document}
|
3,212,635,537,956 | arxiv | \section{Introduction}
Based on the large and still growing number of astronomical observations, one can agree that there is an exotic component
called dark energy which represents more than the $70\%$ of the total energy of the Universe. Its existence has radically changed our standard paradigm of cosmology mostly because of its visible effects on the current state of the Universe \cite{Book}. It turns out that dark energy is a repulsive fuel characterized by a strong negative pressure to overcome the slowing down effect of gravity, making the Universe exhibit an accelerated expansion state at the present time \cite{Book}. This tremendous fact has been confirmed by a plethora of observational tests such as the high redshift Hubble diagram of type Ia supernovae as standard candles \cite{obse1} and accurate measurements of cosmic microwave background anisotropies \cite{obse2}. According to the current observations, the present-day value of the dark energy density is about 120 orders of magnitude smaller than the energy scales at the end of inflation, so one of the main challenges in the modern cosmology is to understand such deep mismatch. One way to alleviate the aforesaid problem is working within the context of dynamical dark energy models, leaving aside the standard $\Lambda$CDM model. Besides, the necessity of a dark matter component comes from astrophysical evidences of colliding galaxies, gravitational lensing of mass distribution or a power spectrum of clustered matter \cite{Book}, \cite{DMobserva}, \cite{dme}. The first evidence of dark matter's existence stemmed from the studies performed by Zwicky in 1934 to the Coma cluster
of galaxies \cite{dmo} and since its discovery, dark matter has played an essential role for resolving the riddle of the missing mass in the Universe.
At the present moment, the astrophysical observations from the galactic to the cosmological scales suggest that dark matter is a substantial
component to the Universe's total matter density \cite{dme} and sustain that dark matter represents nearly $25\%$ of the total energy matter of the Universe;
this invisible and nonbaryonic component is the major agent responsible for the large-structure formation in the Universe \cite{Book}, \cite{DMobserva}. Motivated to understand more about the nature of both dark components, one could consider an exchange of energy between themselves, i.e., the dark matter not only can feel the presence of the dark energy through a gravitational expansion of the Universe but also can interact between them \cite{jefe1}. A coupling between dark energy and dark matter changes the background evolution of the dark sector, allowing us to constrain a particular type of interaction and giving rise to a richer cosmological dynamics compared with noninteracting models \cite{jefe1}. One way to extend the insight about the dark matter-dark energy interacting mechanism is to explore a bigger picture in which a third component is added, perhaps a weakly interacting radiation term as it occurs within a warm inflation paradigm \cite{WI}. A scenario in which dark energy interacts with both dark matter and radiation was explored in \cite{T1} whereas the validity of the generalized second law of thermodynamics was studied in \cite{T2} without using a particular kind of interaction. Other cases correspond to take the third component as an unparticle fluid \cite{T3a}, unparticle fluid in loop quantum cosmology \cite{T3b}, or a general unspecified fluid \cite{T4}. As a step forward to constraining dark matter and dark energy with the physic behind recombination or big-bang nucleosynthesis epochs, a decoupled radiation term was added to the interacting dark sector for taking into account the stringent bounds related to the behavior of dark energy at early times \cite{hmi1}, \cite{hmi2}.
Below we develop a model composed of three interacting fluids and introduce a 3-dimensional internal space where the three interaction terms and barotropic indexes are viewed as vectors in a vector space. We discuss
the existence of a transversal interaction and center our finding in a model with energy exchange proportional to a linear combination of the total energy density and its derivative up to third order.
We also study the stability of a power law solution (scaling solution) with the help of Lyapunov's theorem. Finally, we perform a cosmological constraint using the Hubble data and the severe bounds for dark energy at early times. We will use the units $8\pi G=1$ and signature $(-,+,+,+)$ for the metric of the spacetime.
\section{The model }
We consider a spatially flat homogeneous and isotropic Universe described by Friedmann-Robertson-Walker (FRW) spacetime with a line element given by $ds^{2}=-dt^{2}+a^{2}(t) (dx^{2}+dy^{2}+dz^{2})$ being $a(t)$ the scale factor. The Universe is filled with three interacting fluids, namely, dark energy, dark matter, and radiation so that the evolution of the FRW Universe is governed by the Friedmann and conservation equations, respectively,
\be
\n{01}
3H^{2}=\ro= \ro_{x}+\ro_{m}+\ro_{r},
\ee
\be
\n{02}
\dot{\ro}+3H(\ro_{x}+p_{x}+\ro_{m}+p_{m}+\ro_{r}+p_{r})=0,
\ee
where $H = \dot a/a$ is the Hubble expansion rate. Introducing the variable $\eta = \ln(a/a_0)^{3}$, with $a_0$ the present value of the scale factor and $' \equiv d/d\eta$, Eq. (\ref{02}) can be recast
\be
\n{03}
\ro'=-\gamma_{x}\ro_{x}-\gamma_{m}\ro_{m}- \gamma_{r}\ro_{r},
\ee
where $\gamma_{i}=1+p_{i}/\ro_{i}$ is the barotropic index of each component with $i=\{x,m,r\}$ and $0<\ga_x<\ga_m<\ga_r$. The interaction terms $3H Q_{i}$ between the components are introduced by splitting (\ref{03}) into three equations:
\be
\n{04}
\ro_x' + \ga_{x} \ro_x = Q_{x}.
\ee
\be
\n{05}
\ro_m' + \ga_{m} \ro_m = Q_{m},
\ee
\be
\n{06}
\ro_r' + \ga_{r} \ro_r = Q_{r},
\ee
where the $Q_{i}$ describe the energy transfer and satisfy the condition
\be
\n{06b}
Q_{x}+ Q_{m}+ Q_{r}=0,
\ee
to recover the whole conservation equation (\ref{03}) after having summed Eqs. (\ref{04})-(\ref{06}).
Also, we assume that all component energy densities are definite positive.
To investigate the proposed model, we introduce a 3-dimensional internal space with an orthonormal vector basis $\{\mathbf{e_t},\mathbf{e_o},\mathbf{n}\}$ defined in the following way: we set the coordinate origin at the intersection of the interaction plane (\ref{03}) with the vector formed with the barotropic indexes $\mbox{\boldmath ${\gamma}$}=(\ga_x,\ga_m,\ga_r)$, the pair of vectors $\{\mathbf{e_t},\mathbf{e_o}\}$ is contained in the interaction plane, $\mathbf{e_t}$ is orthogonal to the vector $\mbox{\boldmath ${\gamma}$}$, so $\mbox{\boldmath ${\gamma}$}\cdot\mathbf{e_t}=0$, the orthogonal projection of the vector $\mbox{\boldmath ${\gamma}$}$ on the interaction plane defines the direction of the vector $\mathbf{e_o}$ and the normal vector $\mathbf{n}=(1,1,1)/\sqrt{3}$ is orthogonal to the interaction plane. The interaction terms $Q_{i}$ are also viewed as the components of a vector $\mathbf{Q}=(Q_{x}, Q_{m}, Q_{r})$ that lives on the plane $ \Pi:Q_{x}+Q_{m}+Q_{r}=0$, meaning that
\bn{vQ}
\mathbf{Q}=q_t\,\mathbf{e_t}+q_o\,\mathbf{e_o},
\ee
where $q_t$ and $q_o$ are the components of the interaction vector $\mathbf{Q}$ on the plane $\Pi$ and $\mathbf{n}\cdot \mathbf{Q}=0$.
Taking into account that $\mathbf{e_t}$ is the unique vector of the basis with the property of being orthogonal to $\mbox{\boldmath ${\gamma}$}$, we adopt this property as a simple criteria for selecting only those interactions which are collinear with the aforesaid preferred direction in the plane $\Pi$ that we call ``transversal interaction'', so
\bn{qt}
\mathbf{Q_t}=q_t\,\mathbf{e_t}
\ee
with $Q_t=q_t$ ensuring that we can always take $\mbox{\boldmath ${\gamma}$}\cdot \mathbf{Q}=0$ uniquely. The transversal character of the interaction vector (\ref{qt}) will simplify enough the equations which determine the component energy densities as we will see below.
\begin{figure}[h!]
\begin{center}
\includegraphics[height=6.9cm,width=7.5cm]{1.eps}
\caption{The plot shows the orthonormal vector basis formed by $\{\mathbf{e_t},\mathbf{e_o},\mathbf{n}\}$, the interaction plane $\Pi: Q_{x}+Q_{m}+Q_{r}=0$, the interaction vector $\mathbf{Q}=q_t\,\mathbf{e_t}+q_o\,\mathbf{e_o}$ and the vector $\mbox{\boldmath ${\gamma}$}=\ga_n\mathbf{n}+\ga_o\mathbf{e_o}$.}
\label{F1}
\end{center}
\end{figure}
The two basis vectors that span the interaction plane $\Pi$ are given by
\be
\n{v1}
\mathbf{e_t}=\frac{(\ga_{m}-\ga_{r}, \ga_{r}-\ga_{x}, \ga_{x}-\ga_{m})}{e},
\ee
\be
\n{v2}
\mathbf{e_o}=\frac{(\ga_{m}+\ga_{r}-2\ga_{x}, \ga_{x}+\ga_{r}-2\ga_{m}, \ga_{x}+\ga_{m}-2\ga_{3})}{\sqrt{3}\,e},
\ee
where $e^2= 3\sum_{i}\ga^{2}_{i}-[\sum_{i}\ga_{i}]^{2}$ while the vector built with the barotropic indexes $\mbox{\boldmath ${\gamma}$}=\ga_n\mathbf{n}+\ga_o\mathbf{e_o}$ lives in the plane spanned by the normal $\mathbf{n}$ and $\mathbf{e_o}$ [see Fig.(\ref{F1})].
Now, we will construct an interacting three fluid model with the transversal interaction (\ref{qt}). After differentiating Eq. (\ref{03}) and using Eqs. (\ref{04})-(\ref{06}) we have
\be
\n{07}
\ro''=\ga^{2}_{x}\ro_{x}+\ga^{2}_{m}\ro_{m}+ \ga^{2}_{r}\ro_{r}.
\ee
Solving the algebraic system of equations in the $(\ro_{x}, \ro_{m}, \ro_{r})$ variables (\ref{01}), (\ref{03}), and (\ref{07}), we obtain $\ro_{x}$, $\ro_{m}$, and $\ro_{r}$ as a function of $\ro$, $\ro'$, and $\ro''$ only;
\be
\n{08}
\ro_x=\frac{\ga_{m}-\ga_{r}}{\Delta}\left[ \ga_{m}\ga_{r}\ro+(\ga_{m}+\ga_{r})\ro'+ \ro''\right],
\ee
\be
\n{09}
\ro_m=-\frac{\ga_{x}-\ga_{r}}{\Delta}\left[ \ga_{x}\ga_{r}\ro+ (\ga_{x}+\ga_{r})\ro'+ \ro''\right],
\ee
\be
\n{10}
\ro_r=\frac{\ga_{x}-\ga_{m}}{\Delta}\left[ \ga_{x}\ga_{m}\ro+ (\ga_{x}+\ga_{m})\ro'+\ro''\right],
\ee
where $\Delta=(\ga_{x}-\ga_{m}).(\ga_{x}-\ga_{r}).(\ga_{m}-\ga_{r})$ is the determinant of that algebraic system of equations. Equations (\ref{08}), (\ref{09}), and (\ref{10}) clearly represent the straightforward extension of the case studied in \cite{jefe1} where an interacting two-fluid scenario for the dark sector in the FRW Universe was investigated. Following Ref. \cite{jefe1}, we replace (\ref{08}) into (\ref{04}), (\ref{09}) into (\ref{05}), or (\ref{10}) into (\ref{06}) and obtain the same third order differential equation, that we call ``source equation'', for the total energy density;
$$
\ro'''+(\ga_{x}+\ga_{m}+\ga_{r})\ro''+
$$
\be
\n{11}
(\ga_{x}\ga_{r}+\ga_{x}\ga_{m}+\ga_{m}\ga_{r})\ro'+\ga_{x}\ga_{m}\ga_{r}\ro={\cal Q},
\ee
where its source term ${\cal Q}$ involves a linear combination of the interaction vector components $Q_{i}$
\be
\n{12}
{\cal Q}=\ga_{x}Q_{m}\ga_{r}+ \ga_{r}Q_{x}\ga_{m}+\ga_{m}Q_{r}\ga_{x}.
\ee
By combining the transversal interaction (\ref{qt}), the basis vector (\ref{v1}) and Eq. (\ref{12}) we find
\bn{qs}
{\cal Q}=-\Delta \frac{q_t}{e}
\ee
Finally, once the transversal interaction $\mathbf{Q_t}$ is specified we obtain the energy density $\ro$ by solving the source equation Eq. (\ref{11}) and the component energy densities $\ro_x$, $\ro_m$, and $\ro_r$ after inserting $\ro$ into Eqs. (\ref{08}), (\ref{09}), and (\ref{10}).
\subsection{Stability analysis}
Now, we will investigate the stability of power law solutions. To this end, we will assume that the source equation (\ref{11}) admits scaling solutions and subsequently we will look for the set of interaction terms that give rises to them. The knowledge of power law solutions is very useful because it determines the asymptotic behavior of the effective barotropic index $\ga=(\ga_x\ro_x+\ga_m\ro_m+\ga_r\ro_r)/(\ro_x+\ro_m+\ro_r)=-2\dot{H}/3H^{2}$, which ranges between $\ga_x<\ga<\ga_r$. These solutions represent a Universe approaching to a stationary stage characterized by $\ga=\ga_s$ and $a=t^{2/3\ga_{s}}$; then the existence of the attractor solution $\ga_{s}$ will imply that $\ga$ goes to the asymptotic constant value $\ga\to\ga_s$. So on the attractor
\be
\n{13}
\ga_{s}=\frac{\ga_x\ro_{xs}+\ga_m\ro_{ms}+\ga_r\ro_{rs}}{\ro_{xs}+\ro_{ms}+\ro_{rs}}=\frac{\ga_{x}+\ga_{m}r_{mx}+\ga_{r}r_{rx}}{1+r_{mx}+r_{rx}},
\ee
or
\be
\n{13b}
(\ga_{x}-\ga_{s})+ (\ga_{m}-\ga_{s})r_{mx}+(\ga_{r}-\ga_{s})r_{rx}=0,
\ee
where we have defined the ratios $r_{mx}=\ro_{ms}/\ro_{xs}$ and $r_{rx}=\ro_{rs}/\ro_{xs}$. From the positivity of the component energy densities, the small value of the ratio $r_{rs}=\Om_{rs}/\Om_{xs}$, the range of the effective barotropic index $\ga_x<\ga<\ga_r$, and tEq. (\ref{13b}), we determine that $\ga_s$ ranges between $\ga_{x}<\ga_{s}<\ga_{m}<\ga_{r}$ and the ratios $r_{mx}$ and $r_{rx}$ become asymptotically constant on the attractor stage, alleviating the cosmic coincidence problem. In the case of $\ga_{s}=0$, we have a final de Sitter regime, $H=cte$ with $r_{mx}=-(\ga_{x}/\ga_{m})-r_{rx}(\ga_{r}/\ga_{m})$.
To investigate the existence of a power law attractor solution $a=t^{2/3\ga_s}$, we use that $\ro'=-\ga\ro$, $\ro''=(\ga^{2}-\ga')\ro$, $\ro'''=-(\ga^{3}-3\ga\ga'+ \ga'')\ro$, change the source equation (\ref{11}) into
\be
\n{14a}
\ga''+(\ga_{x}+\ga_{m}+\ga_{r}-3\ga)\ga'- {\cal P}(\ga)=-\frac{{\cal Q}}{\ro},
\ee
\be
\n{14b}
{\cal P}(\ga)=-(\ga-\ga_{x})(\ga-\ga_{m})(\ga-\ga_{r}),
\ee
and impose both (i) that $\ga_{s}$ be a constant stationary solution of Eq. (\ref{14a}) and (ii) the stability condition so that $\ga_{s}$ is stable. The existence of $\ga_{s}$ implies $\ga_{s}'=0$, $\ga_{s}''=0$ and
\be
\n{15}
{\cal Q}(\ga_{s})={\cal P}(\ga_{s})\,\ro, \qquad {\cal Q}(\ga_{s})<0,
\ee
due to Eq. (\ref{13b}).
Taking into account the specific form of the interaction (\ref{15}), we will assume separability and the stability analysis will be performed for interactions of the form
\be
\n{16}
{\cal Q}(\ro, \ga, \ga', \ga'')={\cal P}(\ga){\cal K}(\ga, \ga', \ga'')\,\ro
\ee
Combining Eqs. (\ref{14a}) and (\ref{16}), we write the equation governing the dynamical evolution of the barotropic index in a simpler form
\be
\n{17}
\ga''+(\ga_{x}+\ga_{m}+\ga_{r}-3\ga)\ga'= -{\cal P}(\ga)[{\cal K}-1],
\ee
where the function ${\cal K}$ fulfills the condition,
\be
\n{18a}
{\cal K}(\ga=\ga_{s}, \ga'=0, \ga''=0)=1,
\ee
for assuring the existence of the constant stationary solutions $\ga_{s}$. Perturbing around the solution $\ga_{s}$ by taking $\ga=\ga_{s}+\epsilon$ with $|\epsilon/\ga_s|\ll 1$, Eq. (\ref{17}) can be recast as
\be
\n{18b}
\epsilon''+ (\ga_{x}+\ga_{m}+\ga_{r}-3\ga_{s})\epsilon'=-{\cal P}(\ga_{s})[{\cal K}_{\ga} \epsilon+{\cal K}_{\ga'} \epsilon'+ {\cal K}_{\ga''} \epsilon''],
\ee
where ${\cal K}_{\ga}$, ${\cal K}_{\ga'}$, and ${\cal K}_{\ga''}$ stand for the partial derivatives with respect to $\ga$, $\ga'$, and $\ga''$, respectively. These derivatives are evaluated at the point $(\ga=\ga_{s},\ga_{s}'=0, \ga_{s}''=0)$ and we have used that ${\cal K}(\ga=\ga_{s}+\epsilon, \epsilon')=1+{\cal K}_{\ga}\epsilon+{\cal K}_{\ga'}\epsilon'+{\cal K}_{\ga''}\epsilon''+{\cal O}(\epsilon^2, \epsilon'^2, \epsilon''^2)$. It turns out that (\ref{18b}) can be written as the equation of motion for a dissipative or antidissipative mechanical system. This resemblance emerges from the analogy with the classical potential problem
\be
\n{19}
\frac{d}{d \eta}\left[\frac{\epsilon'^{2}}{2} +{\cal V}(\epsilon)\right]=-\alpha\epsilon'^{2},
\ee
where $\alpha=[\ga_{x}+\ga_{m}+\ga_{r}-3\ga_{s}+{\cal P}(\ga_{s}) {\cal K}_{\ga'}]/[1+{\cal P}(\ga_{s}) {\cal K}_{\ga''}]$ and the potential
\be
\n{19b}
{\cal V}(\epsilon)=\beta\frac{\epsilon^{2}}{2},
\ee
with $\beta={\cal P}(\ga_{s}) {\cal K}_{\ga}/[1+{\cal P}(\ga_{s}) {\cal K}_{\ga''}]$. In order to assure the stability of the scaling solution $\ga=\ga_{s}$, we demand that both coefficients $\al$ and $\beta$ are positives. So, for any transversal interaction $\mathbf{Q_t}$ leading to $\al>0$, the potential ${\cal V}$ has a minimum at $\epsilon=0$ when $\beta>0$, the function inside the square bracket in Eq. (\ref{19}) is a Lyapunov function, and the perturbation decreases asymptotically reaching $\epsilon=0$ (attractor) in the limit $\eta \rightarrow \infty$; then the system is asymptotically stable in the sense of Lyapunov. In other words, when the condition (\ref{18a}) is fulfilled $\ga_{s}$ becomes a constant stationary solution of Eq. (\ref{17}) and it is stable whenever the stability conditions $\al>0$ and $\beta>0$ are satisfied. Note that the interaction (\ref{16}) depends complicatedly on $\ro$, $\ro'$, $\ro''$ and $\ro'''$ because the term ${\cal K}(\ga, \ga', \ga'')$ depends on $\ga=-\ro'/\ro$, $\ga'=(\ro'/\ro)^{2}-\ro''/\ro$, and $\ga''=-(\ro'''/\ro)-(\ro'/\ro)^{3}+3(\ro'/\ro)[\ro''/\ro-(\ro'/\ro)^{2}]$; then ${\cal K}$ contains some nontrivial terms implying that our findings are indeed valid even for a broad set of nonlinear interactions.
\subsection{Linear transversal interaction $\mathbf{Q_t}$ }
Following \cite{jefe1}, we assume a transversal interaction $\mathbf{Q_t}$, with $\mbox{\boldmath ${\gamma}$}\cdot \mathbf{Q_t}=0$, which is a linear combination of $\ro_{x}$, $\ro_{c}$, $\ro_{r}$, their derivatives up to first order, $\ro$, $\ro'$, $\ro''$, and $\ro'''$, so from Eq. (\ref{qs}) we have
\[{\cal Q}=\al_{1}\ro_{x}+\al_{2}\ro_{m}+\al_{3}\ro_{r}+\al_{4}\ro' \]
\be
\n{GG}
+\al_{5}\ro''+\al_{7}\ro'''+\al_{8}\ro'_{x}+\al_{9}\ro'_{m}+\al_{10}\ro'_{r}.
\ee
Using Eqs. (\ref{08}), (\ref{09}), and (\ref{10}), we can recast (\ref{GG}) as
\be
\n{GG2}
{\cal Q}=\beta_{1}\ro+\beta_{2}\ro'+\beta_{3}\ro''+\beta_{4}\ro'''.
\ee
where the new $\beta_i$ coefficients are written in terms of the old $\alpha_i$ ones. Eq. (\ref{GG2}) clearly shows that the most general linear interaction only requires a linear combination of $\ro$ and its derivatives up to third order. Combining Eqs. (\ref{16}) and (\ref{GG2}) along with $\ro'=-\ga\ro$, $\ro''=(\ga^{2}-\ga')\ro$, and $\ro'''=-(\ga^{3}-3\ga\ga'+ \ga'')\ro$, we obtain that
\be
\n{K}
{\cal K}=\frac{\beta_{1}-\beta_{4}(\ga^{3}+\ga''+3\ga\ga')+\beta_{3}\ga^{2}-\beta_{3}\ga'-\beta_{2}\ga}{{\cal P}(\ga)}.
\ee
Applying the condition (\ref{18a}) referring to the existence of the constant solution $\ga_{s}$ to (\ref{K}), we get a constraint
\be
\n{K2}
\beta_{1}-\beta_{4}\ga^{3}_{s}+\beta_{3}\ga^{2}_{s}-\beta_{2}\ga_{s}={\cal P}(\ga_{s}),
\ee
for the coefficients $\beta_{1}$, $\beta_{2}$, $\beta_{3}$, and $\beta_{4}$. By solving Eq.(\ref{K2}) for $\beta_{2}$ and inserting into Eq. (\ref{GG2}), we obtain the final form of the effective interaction term
\be
\n{GLI}
{\cal Q}=\beta_{1}\ro+\ga^{-1}_{s}[\beta_{1}-{\cal P}(\ga_{s}) -\beta_{4}\ga^{3}_{s}+\beta_{3}\ga^{2}_{s}]\ro'+\beta_{3}\ro''+\beta_{4}\ro''',
\ee
so in the end, the most general linear interaction (\ref{GG}) or (\ref{GLI}) only involves four parameters, namely, $\beta_{1}$, $\beta_{3}$, $\beta_{4}$, and $\ga_{s}$. Replacing (\ref{GLI}) into Eq. (\ref{14a}), we find the polynomial $\ga^{3}+A\ga^{2}+B\ga+C=0$ for constant $\ga$ solutions; one of its roots is $\ga_{s}$ , whereas the other two are given by
\be
\n{Ra1}
\ga_{-}=\left(\frac{A+\ga_{s}}{2}\right)\left[-1 \pm \sqrt{1+\frac{4C}{\ga_{s}(A+\ga_{s})^{2}}}~\right],
\ee
\be
\n{Ra2}
\ga_{+}=-\frac{C}{\ga_{-}\ga_{s}}
\ee
where the coefficients $A$, $B$, and $C$ turn out to be
\be
\n{R1}
A=\frac{\sum_{i}\ga_{i}-\beta_{3}}{\beta_{4}-1}, \qquad C=\frac{\ga_{x}\ga_{m}\ga_{r}-\beta_{1}}{\beta_{4}-1},
\ee
\be
\n{R2}
B=\frac{\sum_{i\neq j}{}\ga_{i}\ga_{j}-\ga^{-1}_{s}[\beta_{1}-{\cal P}(\ga_{s}) -\beta_{4}\ga^{3}_{s}+\beta_{3}\ga^{2}_{s}]}{\beta_{4}-1}.
\ee
The exact solution of the source equation (\ref{11}) for the general linear transversal interaction (\ref{GLI}) is given by
\be
\n{den1}
\ro=b_{1}a^{-3\ga_{s}}+b_{2}a^{-3\ga_{+}}+b_{3}a^{-3\ga_{-}}
\ee
whereas the component energy densities are obtained from Eqs. (\ref{08}), (\ref{09}), and (\ref{10}):
\[\ro_x=\frac{\ga_{m}-\ga_{r}}{\Delta}[ \ga_{m}\ga_{r}(b_{1}a^{-3\ga_{s}}+ b_{2}a^{-3\ga_{+}}+ b_{3} a^{-3\ga_{-}})\]
\[-(\ga_{m}+\ga_{r})(b_{1}\ga_{s}a^{-3\la_{s}}+ b_{2}\ga_{+}a^{-3\ga_{+}}+ b_{3}\ga_{-} a^{-3\ga_{-}})\]
\be
\n{08b}
+ (\ga^{2}_{s}b_{1} a^{-3\ga_{s}}+ \ga^{2}_{+}b_{2} a^{-3\ga_{+}}+\ga^{2}_{-}b_{3} a^{-3\ga_{-}})],
\ee
\[\ro_m=-\frac{\ga_{x}-\ga_{r}}{\Delta}[ \ga_{x}\ga_{r}(b_{1}a^{-3\ga_{s}}+ b_{2}a^{-3\ga_{+}}+ b_{3} a^{-3\ga_{-}})\]
\[-(\ga_{x}+\ga_{r})(b_{1}\ga_{s}a^{-3\ga_{s}}+ b_{2}\ga_{+}a^{-3\ga_{+}}+ b_{3}\ga_{-} a^{-3\ga_{-}})\]
\be
\n{09b}
+ (\ga^{2}_{s}b_{1} a^{-3\ga_{s}}+ \ga^{2}_{+}b_{2} a^{-3\ga_{+}}+\ga^{2}_{-}b_{3} a^{-3\ga_{-}})],
\ee
\[\ro_r=\frac{\ga_{x}-\ga_{m}}{\Delta}[ \ga_{m}\ga_{x}(b_{1}a^{-3\ga_{s}}+ b_{2}a^{-3\ga_{+}}+ b_{3} a^{-3\ga_{-}})\]
\[-(\ga_{x}+\ga_{m})(b_{1}\ga_{s}a^{-3\ga_{s}}+ b_{2}\ga_{+}a^{-3\ga_{+}}+ b_{3}\ga_{-} a^{-3\ga_{-}})\]
\be
\n{10b}
+ (\ga^{2}_{s}b_{1} a^{-3\ga_{s}}+ \ga^{2}_{+}b_{2} a^{-3\ga_{+}}+\ga^{2}_{-}b_{3} a^{-3\ga_{-}})],
\ee
In general, the total energy density in terms of the physical quantities such as density parameters is given by
\be
\n{DenZ}
\ro=3H^{2}_{0}\Big({\cal A}x^{3\ga_{s}}+{\cal B}x^{3\ga_{+}}+{\cal C}x^{3\ga_{-}}\Big)
\ee
with $x=z+1$ and the $z$ cosmological redshift while the integration constants are given by
\[{\cal A}=\frac{\Omega_{m0}(\ga_{+}-\ga_{m})(\ga_{-}-\ga_{m})+\Omega_{r0}(\ga_{+}-\ga_{r})(\ga_{-}-\ga_{r})}{(\ga_{s}-\ga_{+})(\ga_{s}-\ga_{-})}\]
\be
\n{A}
+\frac{\Omega_{x0}(\ga_{+}-\ga_{x})(\ga_{-}-\ga_{x})}{(\ga_{s}-\ga_{+})(\ga_{s}-\ga_{-})}
\ee
\[{\cal B}=\frac{\Omega_{m0}(\ga_{s}-\ga_{m})(\ga_{m}-\ga_{-})+\Omega_{0r}(\ga_{s}-\ga_{r})(\ga_{r}-\ga_{-})}{(\ga_{s}-\ga_{+})(\ga_{+}-\ga_{-})}\]
\be
\n{B}
+\frac{\Omega_{x0}(\ga_{s}-\ga_{x})(\ga_{x}-\ga_{-})}{(\ga_{s}-\ga_{+})(\ga_{+}-\ga_{-})}
\ee
\[{\cal C}=\frac{\Omega_{m0}(\ga_{s}-\ga_{m})(\ga_{m}-\ga_{+})+\Omega_{r0}(\ga_{s}-\ga_{r})(\ga_{r}-\ga_{+})}{(\ga_{s}-\ga_{-})(\ga_{-}-\ga_{+})}\]
\be
\n{C}
+\frac{\Omega_{x0}(\ga_{s}-\ga_{x})(\ga_{x}-\ga_{+})}{(\ga_{s}-\ga_{-})(\ga_{-}-\ga_{+})}
\ee
where $\Omega_{0i}=\ro_{0i}/3H^{2}_{0}$ are density parameters fulfilling the condition $\Omega_{x0}+\Omega_{r0}+\Omega_{m0}=1$ for a spatially flat FRW Universe. Additionally, we will choose $(\ga_{x}, \ga_{m}, \ga_{r})=(0,1,4/3)$ to recover the three self-conserved cosmic components in the limit of vanishing interaction.
\subsection{Observational constraints on a transversal interacting model}
We will provide a qualitative estimation of the cosmological paramaters by constraining them with the Hubble data \cite{obs3}- \cite{obs4} and the strict bounds for the behavior of dark energy at early times \cite{EDE1}-\cite{EDE2}. In the former case, the statistical analysis is based on the $\chi^{2}$ function of the Hubble data which is constructed as (e.g., \cite{Press})
\be
\n{c1}
\chi^2(\theta) =\sum_{k=1}^{12}\frac{[H(\theta,z_k) - H_{obs}(z_k)]^2}{\sigma(z_k)^2},
\ee
where $\theta$ stands for cosmological parameters, $H_{obs}(z_k)$ is the observational $H(z)$ data at the redshift $z_k$, $\sigma(z_k)$ is the corresponding $1\sigma$ uncertainty, and the summation is over the $12~$ observational $H(z)$ data. The Hubble function is not integrated over and it is directly related with the properties of the dark energy, since its value comes from the cosmological observations. Using the absolute ages of passively evolving galaxies observed at different redshifts, one obtains the differential ages $dz/dt$ and the function $H(z)$ can be measured through the relation $H(z)=-(1+z)^{-1}dz/dt$ \cite{obs3}, \cite{obs4}. The data $H_{obs}(z_i)$ and $H_{obs}(z_k)$ are uncorrelated because they were obtained from the observations of galaxies at different redshifts.
From Eq. (\ref{DenZ}), one finds that the Hubble expansion of the model becomes
\be
\n{Ht}
H(\theta| z)=H_{0} \Big( {\cal A}x^{3\ga_{s}}+{\cal B}x^{4}+{\cal C}x^{3}\Big)^{\frac{1}{2}}
\ee
where ${\cal A}$, ${\cal B}$, and ${\cal C}$ are obtained form (\ref{A}), (\ref{B}), and (\ref{C}), respectively. For practical reasons mentioned in the last section we have simply selected $\ga_{+}=4/3$, $\ga_{-}=1$.
Here, we consider $\theta=\{H_{0},\ga_{s}, \Omega_{x0},\Omega_{m0}\}$ plus the constraint on the density parameters to assure the flatness condition $(\Omega_{r0}=1-\Omega_{x0}-\Omega_{m0})$; then we have four independent parameters only. We will take two independent parameters and will fix the other ones along the statistic analysis until all parameters have been varied and estimated with the $\chi^{2}$ function. Then, for a given pair of $(\theta_{1}, \theta_{2})$, we are going to perform the statistical analysis by minimizing the $\chi^2$ function to obtain the best-fit values of the random variables $\theta_{c}=\{\theta_{1c}, \theta_{2c} \}$ that correspond to a maximum of Eq.(\ref{c1}). More precisely, the best-fit parameters $\theta_{c}$ are those values where $\chi^2_{min}(\theta_{c})$ leads to the local minimum of the $\chi^2(\theta)$ distribution. If $\chi^2_{d.o.f.}=\chi^2_{min}(\theta_{c})/(N -n) \leq 1$ the fit is good and the data are consistent with the considered model $H(z|\theta)$. Here, $N$ is the number of data and $n$ is the number of parameters \cite{Press}. The variable $\chi^2$ is a random variable that depends on $N$ and its probability distribution is a $\chi^2$ distribution for $N-n$ degrees of freedom. Besides, $68.3\%$ confidence contours in the 2D plane are made of the random data sets that satisfy the inequality $\Delta\chi^{2}=\chi^2(\theta)-\chi^{2}_{min}(\theta_{c})\leq 2.30$. The latter equation defines a bounded region by a closed area around $\theta_{c}$ in the two-dimensional parameter plane; thus, the $1\sigma$ error bar can be identified with the distance from the $\theta_{c}$ point to the boundary of the two-dimensional parameter plane. It can be shown that $95.4\%$ confidence contours with a $2\sigma$ error bar in the samples satisfy $\Delta\chi^{2}\leq 6.17$.
\begin{figure}[hbt!]
\begin{minipage}{1\linewidth}
\resizebox{1.6in}{!}{\includegraphics{2.eps}}
\resizebox{1.6in}{!}{\includegraphics{3.eps}}\hskip0.05cm
\resizebox{1.6in}{!}{\includegraphics{4.eps}}\hskip0.05cm
\resizebox{1.6in}{!}{\includegraphics{5.eps}}\hskip0.05cm
\resizebox{1.6in}{!}{\includegraphics{6.eps}}
\resizebox{1.6in}{!}{\includegraphics{7.eps}}
\caption{\scriptsize{Two-dimensional C.L. associated with $1\sigma$,$2\sigma$ for different $\theta$ planes.}}
\label{Fig1}
\end{minipage}
\end{figure}
\begin{figure}[hbt!]
\begin{minipage}{1\linewidth}
\resizebox{1.6in}{!}{\includegraphics{8.eps}}
\resizebox{1.6in}{!}{\includegraphics{9.eps}}\hskip0.05cm
\resizebox{1.6in}{!}{\includegraphics{10.eps}}\hskip0.05cm
\resizebox{1.6in}{!}{\includegraphics{11.eps}}
\resizebox{1.6in}{!}{\includegraphics{12.eps}}
\resizebox{1.6in}{!}{\includegraphics{13.eps}}
\caption{\scriptsize{Plot of $\Omega_{x}(z)$ $\Omega_{m}(z)$, $\Omega_{r}(z)$, $\omega_{eff}(z)$, and $q(z)$, using the best-fit values obtained with the Hubble data for different $\theta$ planes. These plots follows the same same order used of in two-dimensional C.L. of Fig.1}}
\label{Fig2}
\end{minipage}
\end{figure}
\begin{figure}[hbt!]
\begin{minipage}{1\linewidth}
\resizebox{1.6in}{!}{\includegraphics{14.eps}}
\resizebox{1.6in}{!}{\includegraphics{15.eps}}\hskip0.05cm
\resizebox{1.6in}{!}{\includegraphics{16.eps}}\hskip0.05cm
\resizebox{1.6in}{!}{\includegraphics{17.eps}}
\resizebox{1.6in}{!}{\includegraphics{18.eps}}
\resizebox{1.6in}{!}{\includegraphics{19.eps}}
\caption{\scriptsize{Plot of $\log \Omega_{x} (z)$ for $z \in [10^{2}, 10^{5}]$ using the best-fit values obtained with the Hubble data for different $\theta$ planes. These plots follows the same same order used in the two-dimensional C.L. of Fig.1}}
\label{Fig3}
\end{minipage}
\end{figure}
The two-dimensional C.L. obtained with the standard $\chi^{2}$ function
for two independent parameters is shown in Fig. (\ref{Fig1}), whereas the estimation
of these cosmic parameters is briefly summarized in Table (\ref{I}). We obtain that
$\ga_{s}$ varies from $10^{-4}$ to $10^{-3}$, so these values clearly fulfill
the constraint $\ga_{s}<2/3$ that assures the existence of the accelerated phase of
the Universe at late times. We find the best fit at
$(H_{0}, \Omega_{x0})=(74.32 {\rm km~s^{-1}\,Mpc^{-1}},0.77)$ with $\chi^{2}_{d.o.f}=0.783$
by using the priors $\Omega_{m0}=0.2$ and $\ga_{s}=10^{-3}$. These findings show, in broad terms,
that the estimated values of $H_{0}$ and $\Omega_{x0}$ are in agreement with
the standard ones reported by the WMAP-7 project \cite{WMAP7}. The value of $\Omega_{x0}$
is slightly greater than the standard one of $0.7$ with a discrepancy only of $0.1\%$.
Moreover, we find that using the priors $(H_{0}, \ga_{s})=(74.20 {\rm km~s^{-1}\,Mpc^{-1}}, 10^{-3})$
the best-fit values for the present-day density parameters are considerably improved, namely, these in turn give
$(\Omega_{x0},\Omega_{m0})=(0.74, 0.23)$ along with a lower goodness condition ($\chi^{2}_{d.o.f}=0.779$).
Regarding the estimated values of $\Omega_{m0}$, we find that it varies from
$0.24$ to $0.29$, without showing a significant difference with the standard ones \cite{WMAP7}. In
performing the statistical analysis, we find that $H_{0} \in [71.28, 74.32]{\rm km~s^{-1}\,Mpc^{-1}}$
so the estimated values are met within $1 \sigma$ C.L. reported
by Riess \emph{et al} \cite{H0}, to wit, $H_{0}=(72.2 \pm 3.6){\rm km~s^{-1}\,Mpc^{-1}}$.
\begin{center}
\begin{table}
\begin{minipage}{1\linewidth}
\begin{tabular}{|l|l|l|}
\hline
\multicolumn{3}{|c|}{2D Confidence level} \\
\hline
Priors & Best fits & $\chi^{2}_{d.o.f}$\\
\hline
{$(\Omega_{m0},\ga_{s})=(0.2, 10^{-3})$} & $( H_{0},\Omega_{x0} )=( 74.32, 0.77)$& $0.783$\\
\hline
{$(\Omega_{x0}, \ga_{s} )=(0.70, 10^{-3})$} & $( H_{0},\Omega_{m0} )=( 72.28, 0.29)$& $0.834$\\
\hline
{$(\Omega_{x0}, \Omega_{m0})=(0.7, 0.29)$} & $( H_{0},\ga_{s} )=( 71.74, 10^{-4})$& $0.890$\\
\hline
{$( H_{0},\ga_{s} )=( 74.20, 10^{-3})$} & $(\Omega_{x0}, \Omega_{m0})=(0.74, 0.23)$& $0.779$\\
\hline
{$( H_{0},\Omega_{m0} )=( 74.20, 0.24)$} & $(\Omega_{x0},\ga_{s})=(0.74, 10^{-4})$& $0.855$\\
\hline
{$( H_{0},\Omega_{x0} )=( 74.20, 0.75)$} & $(\ga_{s},\Omega_{m0})=(10^{-4}, 0.24)$& $0.776$\\
\hline
\end{tabular}
\caption{\scriptsize{ Observational bounds for the 2D C.L. obtained in Fig. (\ref{Fig1}) by varying two cosmological parameters.}}
\label{I}
\end{minipage}
\end{table}
\end{center}
\begin{center}
\begin{table}
\begin{minipage}{1\linewidth}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
\multicolumn{6}{|c|}{Bounds for cosmic parameters} \\
\hline
$\theta_{c}$&$z_{t}$ & $q(z=0)$ & $\omega_{eff}(z=0)$ & $\Omega_{r0}$ & $\Omega_{z}(z \simeq 1100)$\\
\hline
{$I$}& $0.75$& $-0.65$& $-0.77$ & $0.02$ & $1.6 \times 10^{-11}$\\
\hline
{$II$}& $0.66$& $-0.55$& $-0.70$& $ 10^{-9}$ & $2.6 \times 10^{-10}$\\
\hline
{$III$}&$0.63$& $-0.54$& $-0.69$ & $0.01$ & $4.5 \times 10^{-11}$\\
\hline
{$IV$}&$0.75$& $-0.62$& $-0.74$ & $0.01$& $4.3 \times 10^{-11}$\\
\hline
{$V$}&$0.68$& $-0.60$ & $-0.73$ & $0.02$ & $2.4 \times 10^{-11}$\\
\hline
{$VI$}&$0.75$& $-0.62$ & $-0.74$ & $0.01$ & $4.9 \times 10^{-11}$ \\
\hline
\end{tabular}
\caption{\scriptsize{ Derived bounds for cosmic parameters using the best fits value of 2D C.L. obtained in Table. (\ref{I}) by varying two cosmological parameters in six different cases: $I-$ $( H_{0},\Omega_{x0} )=( 74.32, 0.77)$, $II-$ $( H_{0},\Omega_{m0} )=( 72.28, 0.29)$, $III-$ $( H_{0},\ga_{s} )=( 71.74, 10^{-4})$, $IV-$ $(\Omega_{x0}, \Omega_{m0})=(0.74, 0.23)$, $V-$ $(\Omega_{x0},\ga_{s})=(0.74, 10^{-4})$, and $VI-$ $(\ga_{s},\Omega_{m0})=(10^{-4}, 0.24)$.}}
\label{II}
\end{minipage}
\end{table}
\end{center}
For the sake of completeness,
we also report bounds for other cosmological relevant parameters [see Table (\ref{II})], such as the fraction of radiation $\Omega_{r}(z=0)$, the effective equation
of state at $z=0$ ($\omega_{eff0}=\ga_{eff0}-1$), the decelerating parameter at the present time $q_{0}$, and the transition redshift ($z_t$); all these quantities are derived
using the six best-fit values reported in Table (\ref{I}). We find that the $z_{t}$ is of the order unity varying over the interval $[0.63, 0.75]$;such values are close
to $z_{t}=0.69^{+0.20}_{-0.13}$ reported in \cite{Zt1}, \cite{Ztn} quite recently. Moreover, taking into account
a $\chi^{2}$ statistical analysis made in the $(\omega_{0m}, z_{t})$ plane based on the supernova sample
(Union2), it has been shown that at $2 \sigma$ C.L.
the transition redshift varies from $0.60$ to $1.18$ \cite{Zt2}.
The behavior of a decelerating parameter with redshift is shown in Fig.3, in particular, its present-day
value varies as $-0.62<q_{0}< -0.54$ for the six cases mentioned in Table II; all these values are in perfectly
agreement with the one reported by the WMAP-7 project \cite{WMAP7}. In Fig. (\ref{Fig2}) we plot the effective equation of state as a function of
redshit for the best-fit value shown in Table (\ref{II}). In general, we find that $-1\leq \omega_{eff}\leq 0$ whereas their
present-day values cover the range $[-0.74, -0.69]$, so this does not exhibit a quintom phase \cite{NQ}. Regarding the behavior of density parameters $\Omega_x$, $\Omega_m$,
and $\Omega_r$, we find that nearly close to $z=0$, the dark energy is the main agent that speeds up the Universe, far away from $z=1$
the Universe is dominated by the dark matter and at very early times the radiation component enter in the action, controlling the entire dynamic
of the Universe around $z \simeq 10^3$[cf. Fig. (\ref{Fig3}) ]. As it was expected the fraction of radiation at the present moment is negligible; thus, its value varies over the range $ 10^{-8}\leq \Omega_{r0} \leq 10^{-3}$ [see Table (\ref{II})].
Now, we seek for another kind of constraint that comes form the physics at early times because this can be considered as a complementary tool for testing
our model. As is well known, the fraction of dark
energy at the recombination epoch should fulfill the bound $\Omega_{x}(z\simeq 1100)<0.1$ in order for the dark energy model be consistent
with the big bang nucleosynthesis (BBN) data. Some signals could arise from the early dark energy (EDE) models, uncovering the nature of DE as well as
their properties to high redshift, giving an invaluable guide to the physics behind the recent speed up of the Universe \cite{EDE1}. Then, the current and future data for constraining the amount of EDE was examined, and the cosmological data analyzed has led to an upper bound of $\Omega_{x}(z\simeq 1100)<0.043$ with $95\%$ confidence level in the case of relativistic EDE, while a quintessence type of EDE has given $\Omega_{x}(z\simeq 1100)<0.024$ although the EDE component is not preferred, it is also not
excluded from the current data \cite{EDE1}. Another forecast for the bounds of the EDE are obtained with the Planck and CMBPol experiments\cite{EDE2}; thus, assuming a $\Omega_{x}(a \simeq 10^{-3}) \simeq 0.03$ for studying
the stability of this value, it found that $1\sigma$ error coming from the Planck experiment
is $\sigma^{Planck}_{e} \simeq 0.004$ whereas the CMBPol improved this bound by a factor of 4 \cite{EDE2}.
Taking into account the best-fit values reported in Table (\ref{I}), we find that at early times the dark energy changes rapidly with the redshift $z$ over the interval
$[10^{3}, 10^{4}]$ [see Fig. (\ref{Fig3}) for more detail]. Indeed, Table (\ref{II}) shows that it varies as
$2.4 \times 10^{-11} \leq \Omega_{x} \leq 2.6 \times 10^{-10}$ around $z \simeq 1100$. Such findings point out that the model constructed here not only fulfills the severe bound
of $\Omega_{x}(z\simeq 1100)<0.1$ but it is also consistent with the future constraints
achievable by Planck and CMBPol experiments \cite{EDE2} as well, corroborating that the value of the cosmological parameters selected before,
through the statistical analysis made with Hubble data, are consistent with BBN constraints.
It is interesting to compare the former analysis,
where radiation was considered as a free evolving component which is decoupled from the dark sector in relation with
other cases, where a radiation component interacts with
the dark sector in order to provide a quantitative analysis of
the role played by the interaction. At present, dark energy dominates the whole dynamics of the Universe and
there is practically an obvious decoupling with radiation.
However, from a theoretical point of view, it is reasonable to expect that dark components can interact with other fluids of the Universe substantially in the very beginning of its evolution due to a process occurred in the early Universe. For instance, dark energy interacting with neutrinos was investigated in \cite{GK}.
The framework of many interacting components could provide a more natural arena for studying the stringent bounds of dark energy at the recombination epoch. There could be a signal in favor of having dark matter exchanging energy with dark energy while radiation is treated as a decoupled component \cite{hmi1}, \cite{hmi2} or the case where dark matter, dark energy, and radiation exchange energy. More precisely, when the Universe is filled with an interacting dark sector plus a decoupled radiation term, it was found that $\Omega_{x}(z\simeq 1100)=0.01$ \cite{hmi1} or $\Omega_{x}(z\simeq 1100)=10^{-8}$ \cite{hmi2} but if radiation is coupled to the dark sector, the amount of dark energy is drastically reduced, giving $\Omega_{x}(z\simeq 1100)\simeq {\cal{O}}(10^{-10})$ so the addition of an interacting radiation term is important for reducing the amount of early dark energy in 2 or 8 orders of magnitude.
\section{conclusion}
We have taken under study the dynamical behavior of a cosmological scenario in which the Universe contains three interacting components, namely, dark energy, dark matter, and radiation within the framework of the usual spatially flat FRW spacetime. We have worked within the case of a transversal linear interaction because it gives a unique preferred direction in the plane constraint $\sum_{i}{Q_{i}}=0$ ; then, for such a case we have obtained the partial energy densities $(\ro_{x}, \ro_{m}, \ro_{r})$ in terms of the total energy density, and its derivatives up to second order. Additionally, we have imposed the existence of a power law solution and investigated its stability finding that the system is asymptotically stable in the sense of Lyapunov for $\al>0$ and $\beta>0$ [see Eqs. (\ref{19})-(\ref{19b})]. Using a linear transversal interaction that depends on the total density and its derivatives up to third order only, we have found that the interaction between the three components makes the total energy density exhibit three different behaviors with the scale factors $a^{-3\ga_{s}}$, $a^{-3\ga_{+}}$, and $a^{-3\ga_{-}}$, altering the usual corresponding traits to have no interaction.
On the observational side, we have examined the previous model by constraining the cosmological parameters with the Hubble data and the well-known bounds for dark energy at the recombination era. In the case of 2D C.L., we have made six statistical constraints with the Hubble function [see Fig. (\ref{Fig2}) and Table (\ref{I})]. We have found that $\ga_{s}$ varies from $10^{-4}$ to $10^{-3}$, so these values clearly fulfill
the constraint $\ga_{s}<2/3$ for getting an accelerated phase of the Universe at late times. Using the priors $(H_{0}, \ga_{s})=(74.20 {\rm km~s^{-1}\,Mpc^{-1}}, 10^{-3})$, the best-fit values for the present-day density parameters are given by $(\Omega_{x0},\Omega_{m0})=(0.74, 0.23)$ along with a $\chi^{2}_{d.o.f.}=0.779<1$. We have obtained that the estimated values of $\Omega_{m0}$ vary from
$0.24$ to $0.29$, without showing a significant difference with the standard ones \cite{WMAP7}. Besides, it turned out that $H_{0} \in [71.28, 74.32]{\rm km~s^{-1}\,Mpc^{-1}}$ so the latter values are met within $1 \sigma$ C.L. reported
by Riess \emph{et al} \cite{H0}. Regarding the derived cosmological parameters, for instance, the transition $z_{t}$ turned to be of the order unity, varying over the interval $[0.63, 0.75]$; such values are in agreement with $z_{t}=0.69^{+0.20}_{-0.13}$ reported in \cite{Zt1}-\cite{Ztn} , and meet within the $2 \sigma$ C.L obtained with the supernovae (Union 2) data in \cite{Zt2}. Besides, the decelerating parameter $q(z=0) \in [-0.62, 0.54]$ and the equation of state $\omega_{eff}(z=0) \in [-0.74, -0.69]$; indeed, $-1\leq \omega_{eff}\leq 0$ [see Fig. (\ref{Fig3})], while the fraction of radiation at the present momment $\Omega_{r0}$ varies in the interval $ [10^{-3}, 10^{-9}]$ for the six cases mentioned in Table (\ref{II}). Further, the dark energy amount $\Omega_x(z)$ governs the dynamic of the Universe near $z=0$, whereas far away from $z=1$ the Universe is dominated by the fraction of dark matter $\Omega_m(z)$ and at very early times the fraction of radiation $\Omega_r(z)$ controls the entire dynamic of the Universe around $z \simeq 10^3$[cf. Fig. (\ref{Fig3})].
Finally, taking into account the best-fit values reported in Table (\ref{I}), we find that at early times the dark energy changes rapidly with the redshift $z$ over the interval $[10^{3}, 10^{4}]$ [cf. Fig. (\ref{Fig3})]; indeed, we have obtained a $2.4 \times 10^{-11} \leq \Omega_{x} \leq 2.6 \times 10^{-10}$ around $z \simeq 1100$ [ see Table (\ref{II})]. The latter results indicate that the model constructed fulfills the severe bound of $\Omega_{x}(z\simeq 1100)<0.1$ and is consistent with the future constraints
achievable by Planck and CMBPol experiments \cite{EDE2} as well, pointing that the cosmological parameters selected before are coherent with BBN constraints also. In a future investigation, we will perform a full study of the interaction vector proportional to orthonormal projection $\mathbf{e_o}$ that is related to the other direction defined in the constraint plane $\sum_{i}{Q_{i}}=0$.
\acknowledgments
We are grateful to the referee for his careful reading of the manuscript.
L.P.C thanks the University of Buenos Aires under Project No. 20020100100147 and the Consejo Nacional de Investigaciones Cient\'{\i}ficas y T\' ecnicas (CONICET) under Project PIP 114-200801-00328 for the partial support of this work during its different stages. M.G.R is partially supported by CONICET.
|
3,212,635,537,957 | arxiv | \section*{References}\list
{[\arabic{enumi}]}{\settowidth\labelwidth{#1}\leftmargin\labelwidth
\advance\leftmargin\labelsep
\usecounter{enumi}}
\def\hskip .11em plus .33em minus .07em{\hskip .11em plus .33em minus .07em}
\sloppy\clubpenalty4000\widowpenalty4000
\sfcode`\.=1000\relax}
\let\Large=\large
\def\op#1{\mathop{\fam0 #1}\limits}
\newcommand{{\rm Id\,}}{{\rm Id\,}}
\newcommand{{\rm Ker\,}}{{\rm Ker\,}}
\newcommand{\nm}[1]{\mid {#1}\mid}
\newcommand{\begin{equation}}{\begin{equation}}
\newcommand{\end{equation}}{\end{equation}}
\newcommand{\begin{eqnarray}}{\begin{eqnarray}}
\newcommand{\end{eqnarray}}{\end{eqnarray}}
\newcommand{\begin{eqnarray*}}{\begin{eqnarray*}}
\newcommand{\end{eqnarray*}}{\end{eqnarray*}}
\newcommand{\begin{eqalph}}{\begin{eqalph}}
\newcommand{\end{eqalph}}{\end{eqalph}}
\newcommand{{\cal A}}{{\cal A}}
\newcommand{{\cal T}}{{\cal T}}
\newcommand{{\cal P}}{{\cal P}}
\newcommand{{\cal D}}{{\cal D}}
\newcommand{{\cal R}}{{\cal R}}
\newcommand{{\cal V}}{{\cal V}}
\newcommand{{\cal C}}{{\cal C}}
\newcommand{{\cal L}}{{\cal L}}
\newcommand{{\cal E}}{{\cal E}}
\newcommand{{\cal H}}{{\cal H}}
\newcommand{{\cal F}}{{\cal F}}
\newcommand{{\cal S}}{{\cal S}}
\newcommand{{\cal O}}{{\cal O}}
\newcommand{{\bf L}}{{\bf L}}
\newcommand{{\bf R}}{{\bf R}}
\newcommand{{\bf C}}{{\bf C}}
\newcommand{{\bf Z}}{{\bf Z}}
\newcommand{{\bf H}}{{\bf H}}
\newcommand{{\bf T}}{{\bf T}}
\newcommand{{\bf E}}{{\bf E}}
\newcommand{\alpha}{\alpha}
\newcommand{\varrho}{\varrho}
\newcommand{\beta}{\beta}
\newcommand{\delta}{\delta}
\newcommand{\lambda}{\lambda}
\newcommand{\Lambda}{\Lambda}
\newcommand{\phi}{\phi}
\newcommand{\omega}{\omega}
\newcommand{\Omega}{\Omega}
\newcommand{\mu}{\mu}
\newcommand{\nu}{\nu}
\newcommand{\epsilon}{\epsilon}
\newcommand{\gamma}{\gamma}
\newcommand{\Gamma}{\Gamma}
\newcommand{\theta}{\theta}
\newcommand{\Theta}{\Theta}
\newcommand{\vartheta}{\vartheta}
\newcommand{\varphi}{\varphi}
\newcommand{\upsilon}{\upsilon}
\newcommand{\langle}{\langle}
\newcommand{\rangle}{\rangle}
\newcommand{{\rm dim\,}}{{\rm dim\,}}
\newcommand{\sigma}{\sigma}
\newcommand{\Sigma}{\Sigma}
\newcommand{\wedge}{\wedge}
\newcommand{\widetilde}{\widetilde}
\newcommand{\widehat}{\widehat}
\newcommand{\overline}{\overline}
\newcommand{\partial}{\partial}
\newcommand{\op\longrightarrow}{\op\longrightarrow}
\newcommand{\otimes}{\otimes}
\newcommand{\approx}{\approx}
\newcommand{\varepsilon}{\varepsilon}
\newcommand{\flat}{\flat}
\newcommand{\sharp}{\sharp}
\renewcommand{\theequation}{\arabic{equation}}
\newcounter{eqalph}
\newcounter{equationa}
\newcounter{theorem}
\newcounter{remark}
\newcounter{proposition}
\newcounter{lemma}
\newcounter{corollary}
\newcounter{definition}
\setcounter{remark}{0} \setcounter{theorem}{0}
\setcounter{proposition}{0} \setcounter{lemma}{0}
\setcounter{corollary}{0} \setcounter{definition}{0}
\def\arabic{remark}{\arabic{remark}}
\def\arabic{definition}{\arabic{definition}}
\def\arabic{definition}{\arabic{definition}}
\def\arabic{definition}{\arabic{definition}}
\def\arabic{definition}{\arabic{definition}}
\def\arabic{definition}{\arabic{definition}}
\newenvironment{rem}{\refstepcounter{remark}\medskip\noindent{\it Remark
\arabic{remark}.}}{\medskip}
\newenvironment{ex}{\refstepcounter{remark}\medskip\noindent{\it Example
\therexample.}}{\medskip}
\newenvironment{theo}{\refstepcounter{definition} \medskip
\noindent{\it Theorem \arabic{definition}.}}{\medskip}
\newenvironment{prop}{\refstepcounter{definition} \medskip
\noindent{\it Proposition \arabic{definition}.}}{\medskip}
\newenvironment{lem}{\refstepcounter{definition} \medskip
\noindent{\it Lemma \arabic{definition}.}}{\medskip}
\newenvironment{cor}{\refstepcounter{definition} \medskip
\noindent{\it Corollary \arabic{definition}.}}{\medskip}
\newenvironment{defi}{\refstepcounter{definition} \medskip
\noindent{\it Definition \arabic{definition}:}}{\medskip}
\newcommand{\mar}[1]{}
\hyphenation{ma-ni-fold La-gran-gi-ans di-men-si-o-nal
-di-men-si-o-nal La-gran-gi-an Ha-mil-to-ni-an multi-symplec-tic}
\begin{document}
\hbox{}
{\parindent=0pt
{\large\bf Space-time BRST symmetries}
\bigskip
{\sc D.Bashkirov and G.Sardanashvily}
{\sl Department of Theoretical Physics, Moscow State University,
Russia}
\bigskip
{\small {\bf Abstract} Bearing in mind BV quantization of gauge
gravitation theory, we extend general covariant transformations to
the BRST ones. }
}
\section{Introduction}
Let us consider a Lagrangian system on a smooth fiber bundle $Y\to
X$. It is called a gauge system if its Lagrangian $L$ admits a set
of symmetries depending on parameter functions $\xi^r$ and their
derivatives. BRST transformations come from gauge transformations
by replacement of gauge parameters with odd ghosts $c^r$.
Moreover, one completes these transformations with the terms
acting on ghosts such that the total ones become nilpotent.
In the case of general covariant transformations, parameter
functions are vector fields on $X$. We introduce the corresponding
ghosts and construct BRST transformations in the case of metric
and metric-affine gravitation theories.
Let $J^rY$, $r=1,\ldots$, be finite order jet manifolds of
sections of $Y\to X$. In the sequel, the index $r=0$ stands for
$Y$. Given bundle coordinates $(x^\lambda,y^i)$ on $Y$, jet manifolds
$J^rY$ are endowed with the adapted coordinates
$(x^\lambda,y^i,y^i_\Lambda)$, where $\Lambda=(\lambda_k...\lambda_1)$,
$k=1,\ldots,r$, is a symmetric multi-index. We use the notation
$\lambda+\Lambda=(\lambda\la_k...\lambda_1)$ and
\mar{5.177}\begin{equation}
d_\lambda = \partial_\lambda + \op\sum_{0\leq|\Lambda|} y^i_{\lambda+\Lambda}\partial_i^\Lambda,
\qquad d_\Lambda=d_{\lambda_r}\circ\cdots\circ d_{\lambda_1}. \label{5.177}
\end{equation}
In order to describe gauge transformations depending on
parameters, let us consider Lagrangian formalism on the bundle
product
\mar{0681}\begin{equation}
E=Y\op \times_X V, \label{0681}
\end{equation}
where $V\to X$ is a vector bundle whose sections are gauge
parameter functions \cite{noether}. Let $V\to X$ be coordinated by
$(x^\lambda,\xi^r)$. Then gauge transformations are represented by a
differential operator
\mar{0509}\begin{equation}
\upsilon= \op\sum_{0\leq|\Lambda|\leq
m}\upsilon^{i,\Lambda}_r(x^\lambda,y^i_\Sigma)\xi^r_\Lambda \partial_i \label{0509}
\end{equation}
on $E$ (\ref{0681}) which is linear on $V$ and takes its values
into the vertical tangent bundle $VY$ of $Y\to X$. By means of a
replacement of even gauge parameters $\xi^r$ and their jets
$\xi^r_\Lambda$ with the odd ghosts $c^r$ and their jets $c^r_\Lambda$,
the operator (\ref{0509}) defines a graded derivation
\mar{0680}\begin{equation}
\upsilon= \op\sum_{0\leq|\Lambda|\leq m}\upsilon^{i,\Lambda}_r(x^\lambda,y^i_\Sigma)c^r_\Lambda
\partial_i \label{0680}
\end{equation}
of the algebra of the original even fields and odd ghosts. Its
extension
\mar{0684}\begin{equation}
\upsilon= \op\sum_{0\leq|\Lambda|\leq m}\upsilon^{i,\Lambda}_r c^r_\Lambda \partial_i +
u^r\partial_r \label{0684}
\end{equation}
to ghosts is called the BRST transformation if it is nilpotent.
Such an extension exists if the original gauge transformations
form an algebra \cite{algebr}.
Note that, if gauge transformations are reducible, $(0\leq
k)$-stage ghosts are introduced \cite{brst}. Gauge theories in
question here are irreducible.
In the case of gravitation theory, a fiber bundle $Y\to X$ belongs
to the category of natural bundles, and it admits the canonical
lift of any vector field on $X$. One thinks of such a lift as
being an infinitesimal generator of general covariant
transformations of $Y$. In this case, the vector bundle $V\to X$
possesses the composite fibration $V\to TX\to X$, where $TX$ is
the tangent bundle of $X$.
Here, we consider the following three gauge theories: (i) the
gauge model of principal connections on a principal bundle (the
gauge symmetry (\ref{0660}), the BRST symmetry (\ref{gr20})), (ii)
this gauge model in the presence of a metric gravity (the gauge
symmetry (\ref{gr11}), the BRST symmetry (\ref{gr21})) and (iii)
in the presence of a metric-affine gravity (the gauge symmetry
(\ref{gr15}), the BRST symmetry (\ref{gr22})).
\section{Gauge systems on fiber bundles}
Lagrangian formalism on a fiber bundle $Y\to X$ is phrased in
terms of the following graded differential algebra (henceforth
GDA).
With the inverse system of jet manifolds
\mar{5.10}\begin{equation}
X\op\longleftarrow^\pi Y\op\longleftarrow^{\pi^1_0} J^1Y
\longleftarrow \cdots J^{r-1}Y \op\longleftarrow^{\pi^r_{r-1}}
J^rY\longleftarrow\cdots, \label{5.10}
\end{equation}
one has the direct system
\mar{5.7}\begin{equation}
{\cal O}^*X\op\longrightarrow^{\pi^*} {\cal O}^*Y
\op\longrightarrow^{\pi^1_0{}^*} {\cal O}_1^*Y \op\longrightarrow\cdots {\cal O}^*_{r-1}Y
\op\longrightarrow^{\pi^r_{r-1}{}^*}
{\cal O}_r^*Y \longrightarrow\cdots \label{5.7}
\end{equation}
of GDAs ${\cal O}_r^*Y$ of exterior forms on jet manifolds $J^rY$ with
respect to the pull-back monomorphisms $\pi^r_{r-1}{}^*$. Its
direct limit
${\cal O}_\infty^*Y$ is a GDA
consisting of all exterior forms on finite order jet manifolds
modulo the pull-back identification.
The projective limit $(J^\infty Y, \pi^\infty_r:J^\infty Y\to
J^rY)$ of the inverse system (\ref{5.10}) is a Fr\'echet manifold.
A bundle atlas $\{(U_Y;x^\lambda,y^i)\}$ of $Y\to X$ yields the
coordinate atlas
\mar{jet1}\begin{equation}
\{((\pi^\infty_0)^{-1}(U_Y); x^\lambda, y^i_\Lambda)\}, \qquad
{y'}^i_{\lambda+\Lambda}=\frac{\partial x^\mu}{\partial x'^\lambda}d_\mu y'^i_\Lambda, \qquad
0\leq|\Lambda|, \label{jet1}
\end{equation}
of $J^\infty Y$, where $d_\mu$ are the total derivatives
(\ref{5.177}). Then ${\cal O}^*_\infty Y$ can be written in a
coordinate form where the horizontal one-forms $\{dx^\lambda\}$ and
the contact one-forms $\{\theta^i_\Lambda=dy^i_\Lambda
-y^i_{\lambda+\Lambda}dx^\lambda\}$ are local generating elements of the
${\cal O}^0_\infty Y$-algebra ${\cal O}^*_\infty Y$. There is the canonical
decomposition ${\cal O}^*_\infty Y=\oplus{\cal O}^{k,m}_\infty Y$ of
${\cal O}^*_\infty Y$ into ${\cal O}^0_\infty Y$-modules ${\cal O}^{k,m}_\infty
Y$ of $k$-contact and $m$-horizontal forms together with the
corresponding projectors $h_k:{\cal O}^*_\infty Y\to {\cal O}^{k,*}_\infty
Y$ and $h^m:{\cal O}^*_\infty Y\to {\cal O}^{*,m}_\infty Y$. Accordingly,
the exterior differential on ${\cal O}_\infty^* Y$ is split into the
sum $d=d_H+d_V$ of the nilpotent total and vertical differentials
\begin{eqnarray*}
d_H(\phi)= dx^\lambda\wedge d_\lambda(\phi), \qquad d_V(\phi)=\theta^i_\Lambda \wedge
\partial^\Lambda_i\phi, \qquad \phi\in{\cal O}^*_\infty Y.
\end{eqnarray*}
Any finite order Lagrangian
\mar{0512}\begin{equation}
L={\cal L}\omega:J^rY\to \op\wedge^nT^*X, \qquad \omega=dx^1\wedge\cdots\wedge dx^n,
\qquad n={\rm dim\,} X, \label{0512}
\end{equation}
is an element of ${\cal O}^{0,n}_\infty Y$, while
\mar{0513}\begin{equation}
\delta L={\cal E}_i\theta^i\wedge\omega=\op\sum_{0\leq|\Lambda|}
(-1)^{|\Lambda|}d_\Lambda(\partial^\Lambda_i {\cal L})\theta^i\wedge\omega\in {\cal O}^{1,n}_\infty Y
\label{0513}
\end{equation}
is its Euler--Lagrange operator taking values into the vector
bundle
\mar{0548}\begin{equation}
T^*Y\op\wedge_Y(\op\wedge^n T^*X)= V^*Y\op\otimes_Y\op\wedge^n T^*X. \label{0548}
\end{equation}
A Lagrangian system on a fiber bundle $Y\to X$ is said to be a
gauge theory if its Lagrangian $L$ admits a family of variational
symmetries parameterized by elements of a vector bundle $V\to X$
and its jet manifolds as follows.
Let ${\protect\pgot d}{\cal O}^0_\infty Y$ be the ${\cal O}^0_\infty Y$-module of
derivations of the $\protect\pBbb R$-ring ${\cal O}^0_\infty Y$. Any $\vartheta\in
{\protect\pgot d}{\cal O}^0_\infty Y$ yields the graded derivation (the interior
product) $\vartheta\rfloor\phi$ of the GDA ${\cal O}^*_\infty Y$ given by the
relations
\begin{eqnarray*}
\vartheta\rfloor df=\vartheta(f), \qquad \vartheta\rfloor(\phi\wedge\sigma)=(\vartheta\rfloor
\phi)\wedge\sigma +(-1)^{|\phi|}\phi\wedge(\vartheta\rfloor\sigma), \quad f\in {\cal O}^0_\infty
Y, \quad \phi,\sigma\in {\cal O}^*_\infty Y,
\end{eqnarray*}
and its derivation (the Lie derivative)
\mar{0515}\begin{equation}
{\bf L}_\vartheta\phi=\vartheta\rfloor d\phi+ d(\vartheta\rfloor\phi), \qquad
{\bf L}_\vartheta(\phi\wedge\phi')={\bf L}_\vartheta(\phi)\wedge\phi' +\phi\wedge{\bf L}_\vartheta(\phi'), \qquad
\phi,\phi'\in {\cal O}^*_\infty Y. \label{0515}
\end{equation}
Relative to an atlas (\ref{jet1}), a derivation
$\vartheta\in{\protect\pgot d}{\cal O}^0_\infty$ reads
\mar{g3}\begin{equation}
\vartheta=\vartheta^\lambda \partial_\lambda + \vartheta^i\partial_i + \op\sum_{|\Lambda|>0}\vartheta^i_\Lambda
\partial^\Lambda_i, \label{g3}
\end{equation}
where the tuple of derivations $\{\partial_\lambda,\partial^\Lambda_i\}$ is defined
as the dual of that of the exterior forms $\{dx^\lambda, dy^i_\Lambda\}$
with respect to the interior product $\rfloor$ \cite{cmp}. Note
that the tuple of derivations $\{\partial^\Lambda_i\}$ is the dual of the
basis $\{\theta^i_\Lambda\}$ of contact forms.
A derivation $\vartheta$ is called contact if the Lie derivative
${\bf L}_\vartheta$ (\ref{0515}) preserves the contact ideal of the GDA
${\cal O}^*_\infty Y$ generated by contact forms. A derivation $\upsilon$
(\ref{g3}) is contact iff
\mar{g4}\begin{equation}
\vartheta^i_\Lambda=d_\Lambda(\vartheta^i-y^i_\mu\vartheta^\mu)+y^i_{\mu+\Lambda}\vartheta^\mu, \qquad
0<|\Lambda|. \label{g4}
\end{equation}
Any contact derivation admits the horizontal splitting
\mar{g5}\begin{equation}
\vartheta=\vartheta_H +\vartheta_V=\vartheta^\lambda d_\lambda + (\upsilon^i\partial_i + \op\sum_{0<|\Lambda|}
d_\Lambda \upsilon^i\partial_i^\Lambda), \qquad \upsilon^i= \vartheta^i-y^i_\mu\vartheta^\mu.
\label{g5}
\end{equation}
Its vertical part $\vartheta_V$ is completely determined by the first
summand
\mar{0641}\begin{equation}
\upsilon=\upsilon^i(x^\lambda,y^i_\Lambda)\partial_i, \qquad 0\leq |\Lambda|\leq k.
\label{0641}
\end{equation}
This is a section of the pull-back $VY\op\times_Y J^kY\to J^kY$,
i.e., a $k$-order $VY$-valued differential operator on $Y$. One
calls $\upsilon$ (\ref{0641}) a generalized vector field on $Y$. Any
vertical contact derivation $\vartheta$ satisfies the relations
\mar{g232}\begin{equation}
\vartheta\rfloor d_H\phi=-d_H(\vartheta\rfloor\phi), \qquad
{\bf L}_\vartheta(d_H\phi)=d_H({\bf L}_\vartheta\phi), \qquad \phi\in{\cal O}^*_\infty Y.
\label{g232}
\end{equation}
One can show that the Lie derivative of a Lagrangian $L$
(\ref{0512}) along a contact derivation $\vartheta$ (\ref{g5}) fulfills
the first variational formula
\mar{g8'}\begin{equation}
{\bf L}_\vartheta L= \upsilon\rfloor\delta L +d_H(h_0(\vartheta\rfloor\Xi_L)) +{\cal L} d_V
(\vartheta_H\rfloor\omega), \label{g8'}
\end{equation}
where $\Xi_L$ is a Lepagean equivalent of $L$ \cite{cmp}. A
contact derivation $\vartheta$ (\ref{g5}) is called variational if the
Lie derivative (\ref{g8'}) is $d_H$-exact, i.e., ${\bf L}_\vartheta
L=d_H\sigma$, $\sigma\in {\cal O}^{0,n-1}_\infty$. A glance at the expression
(\ref{g8'}) shows that: (i) $\vartheta$ (\ref{g5}) is variational only
if it is projected onto $X$; (ii) $\vartheta$ is variational iff its
vertical part $\vartheta_V$ is well; (iii) it is variational iff
$\upsilon\rfloor\delta L$ is $d_H$-exact. Therefore, we can restrict our
consideration to vertical contact derivations $\vartheta=\vartheta_V$. A
generalized vector field $\upsilon$ (\ref{0641}) is called a
variational symmetry of a Lagrangian $L$ if it generates a
variational contact derivation.
Turn now to the notion of a gauge symmetry \cite{noether}. Let us
consider the bundle product $E$ (\ref{0681}) coordinated by
$(x^\lambda,y^i,\xi^r)$. Given a Lagrangian $L$ on $Y$, let us
consider its pull-back, say again $L$, onto $E$. Let $\vartheta_E$ be a
contact derivation of the $\protect\pBbb R$-ring ${\cal O}^0_\infty E$, whose
restriction
\mar{0508}\begin{equation}
\vartheta=\vartheta_E|_{{\cal O}^0_\infty Y}=
\op\sum_{0\leq|\Lambda|}d_\Lambda\upsilon^i\partial_i^\Lambda \label{0508}
\end{equation}
to ${\cal O}^0_\infty Y\subset {\cal O}^0_\infty E$ is linear in coordinates
$\xi^r_\Xi$. It is determined by a generalized vector field
$\upsilon_E$ on $E$ whose projection
\begin{eqnarray*}
\upsilon:J^kE\op\longrightarrow^{\upsilon_E} VE\to E\op\times_Y VY
\end{eqnarray*}
is a linear $VY$-valued differential operator $\upsilon$ (\ref{0509})
on $E$. Let $\vartheta_E$ be a variational symmetry of a Lagrangian $L$
on $E$, i.e.,
\mar{0552}\begin{equation}
\upsilon_E\rfloor \delta L=\upsilon\rfloor \delta L=d_H\sigma. \label{0552}
\end{equation}
Then one says that $\upsilon$ (\ref{0509}) is a gauge symmetry of a
Lagrangian $L$.
In accordance with Noether's second theorem \cite{noether}, if a
Lagrangian $L$ (\ref{0512}) admits a gauge symmetry $\upsilon$
(\ref{0509}), its Euler--Lagrange operator (\ref{0513}) obeys the
Noether identity
\mar{0550}\begin{equation}
[\op\sum_{0\leq|\Lambda|\leq m} \Delta^{i,\Lambda}_r d_\Lambda {\cal E}_i]
\xi^r\omega=0, \label{0550}
\end{equation}
where
\mar{0511}\begin{equation}
\Delta^{i,\Lambda}_r =\op\sum_{0\leq|\Sigma|\leq
m-|\Lambda|}(-1)^{|\Sigma+\Lambda|}C^{|\Sigma|}_{|\Sigma+\Lambda|} d_\Sigma
\upsilon^{i,\Sigma+\Lambda}_r. \label{0511}
\end{equation}
For instance, if a gauge symmetry
\mar{0656}\begin{equation}
\upsilon=(\upsilon_r^i\xi^r +\upsilon^{i,\mu}_r\xi^r_\mu)\partial_i \label{0656}
\end{equation}
is of first jet order in parameters, the corresponding Noether
identity (\ref{0550}) reads
\mar{0657,8}\begin{eqnarray}
&& \Delta^i_r=\upsilon^i_r -d_\mu \upsilon^{i,\mu}_r,\qquad \Delta^{i,\mu}_r=-
\upsilon^{i,\mu}_r,\label{0657}\\
&& [\upsilon^i_r{\cal E}_i - d_\mu(\upsilon^{i,\mu}_r{\cal E}_i)]\xi^r\omega=0.
\label{0658}
\end{eqnarray}
Gauge models considered below are of this type.
\section{Space-time gauge symmetries}
We consider gravitation theory in the absence of spinor fields.
Its gauge symmetries are general covariant transformations (e.g.,
\cite{jack,fat94,book}). As was mentioned above, gravitation
theory is formulated on natural bundles which admit the canonical
lift of any vector field on $X$. One thinks of such a lift as
being an infinitesimal generator of general covariant
transformations. Natural bundles are exemplified by tensor
bundles, the bundles of world metrics and world connections. One
also considers non-vertical automorphisms of a principal bundle
$P\to X$ and their extensions to the gauge-natural prolongations
of $P$ and the associated natural-gauge bundles
\cite{fat,fat04,pal}.
We here address the gauge theory of principal connections on a
principal bundle $P\to X$ with a structure Lie group $G$. These
connections are represented by sections of the quotient
\mar{0654}\begin{equation}
C=J^1P/G\to X,\label{0654}
\end{equation}
called the bundle of principal connections \cite{book}. This is an
affine bundle coordinated by $(x^\lambda, a^r_\lambda)$ such that, given a
section $A$ of $C\to X$, its components $A^r_\lambda=a^r_\lambda\circ A$
are coefficients of the familiar local connection form (i.e.,
gauge potentials). We consider the GDA ${\cal O}^*_\infty C$.
Infinitesimal generators of one-parameter groups of automorphisms
of a principal bundle $P$ are $G$-invariant projectable vector
fields on $P\to X$. They are associated to sections of the vector
bundle $T_GP=TP/G\to X$. This bundle is endowed with the
coordinates $(x^\lambda,\tau^\lambda=\dot x^\lambda,\xi^r)$ with respect to
the fiber bases $\{\partial_\lambda, e_r\}$ for $T_GP$, where $\{e_r\}$ is
the basis for the right Lie algebra ${\protect\pgot g}$ of $G$ such that
$[e_p,e_q]=c^r_{pq}e_r.$ If
\mar{0652}\begin{equation}
u=u^\lambda\partial_\lambda
+u^r e_r, \qquad v=v^\lambda\partial_\lambda +v^r e_r, \label{0652}
\end{equation}
are sections of $T_GP\to X$, their bracket reads
\mar{0651}\begin{equation}
[u,v]=(u^\mu\partial_\mu v^\lambda -v^\mu\partial_\mu u^\lambda)\partial_\lambda +(u^\lambda\partial_\lambda
v^r - v^\lambda\partial_\lambda u^r +c^r_{pq}u^pv^q)e_r. \label{0651}
\end{equation}
Any section $u$ of the vector bundle $T_GP\to X$ yields the vector
field
\mar{0653}\begin{equation}
u_C=u^\lambda\partial_\lambda +(c^r_{pq}a^p_\lambda u^q +\partial_\lambda u^r -a^r_\mu\partial_\lambda
u^\mu)\partial^\lambda_r \label{0653}
\end{equation}
on the bundle of principal connections $C$ (\ref{0654})
\cite{book}.
Let us consider a subbundle $V_GP=VP/G\to X$ of the vector bundle
$T_GX$ coordinated by $(x^\lambda, \xi^r)$. Its sections $u=u^re_r$
are infinitesimal generators of vertical automorphisms of $P$.
There is the exact sequence of vector bundles
\begin{eqnarray*}
0\to V_GP \op\longrightarrow T_GP\to TX \to 0.
\end{eqnarray*}
Its pull-back onto $C$ admits the canonical splitting which takes
the coordinate form
\mar{gr9}\begin{equation}
\tau^\lambda\partial_\lambda +\xi^r e_r = \tau^\lambda(\partial_\lambda +a^r_\lambda e_r) +
(\xi^r - \tau^\lambda a^r_\lambda)e_r. \label{gr9}
\end{equation}
Let us consider the bundle product
\mar{0659}\begin{equation}
E=C\op\times_X T_GP, \label{0659}
\end{equation}
coordinated by $(x^\lambda, a^r_\lambda, \tau^\lambda, \xi^r)$. It can be
provided with the generalized vector field
\mar{0660}\begin{equation}
\upsilon_E= \upsilon=(c^r_{pq}a^p_\lambda \xi^q + \xi^r_\lambda
-a^r_\mu\tau^\mu_\lambda-\tau^\mu a_{\mu\lambda}^r)\partial^\lambda_r, \label{0660}
\end{equation}
taking the form
\mar{gr8}\begin{equation}
\upsilon=(c^r_{pq}a^p_\lambda \xi'^q + \xi'^r_\lambda +
\tau^\mu{\cal F}_{\lambda\mu}^r)\partial^\lambda_r, \qquad \xi'^r=\xi^r -\tau^\lambda
a^r_\lambda, \label{gr8}
\end{equation}
due to the splitting (\ref{gr9}). For instance, this is a gauge
symmetry of the global Chern--Simons Lagrangian in gauge theory on
a principal bundle with a structure semi-simple Lie group $G$ over
a three-dimensional base $X$ \cite{chern}. Given a section $B$ of
$C\to X$ (i.e., a background gauge potential), this Lagrangian
reads
\mar{r50}\begin{eqnarray}
&& L= [\frac12a^G_{mn} \varepsilon^{\alpha\beta\gamma}a^m_\alpha({\cal F}^n_{\beta\gamma}
-\frac13 c^n_{pq}a^p_\beta a^q_\gamma) -\frac12a^G_{mn}
\varepsilon^{\alpha\beta\gamma}B^m_\alpha(F(B)^n_{\beta\gamma} \label{r50}\\
&& \qquad -\frac13 c^n_{pq}B^p_\beta B^q_\gamma) -d_\alpha(a^G_{mn}
\varepsilon^{\alpha\beta\gamma}a^m_\beta B^n_\gamma)]d^3x, \nonumber\\
&& F(B)^r_{\lambda\mu}=\partial_\lambda B^r_\mu-\partial_\mu B^r_\lambda +c^r_{pq}B^p_\lambda
B^q_\mu, \qquad {\cal F}^r_{\lambda\mu}=a^r_{\lambda\mu}-a^r_{\mu\lambda}
+c^r_{pq}a^p_\lambda a^q_\mu, \nonumber
\end{eqnarray}
where $a^G$ is the Killing form. Its first term is the well-known
local Chern--Simons Lagrangian, the second one is a density on
$X$, and the Lie derivative of the third term is $d_H$-exact
because of the relations (\ref{g232}). The corresponding Noether
identities (\ref{0658}) read
\mar{gr1,2}\begin{eqnarray}
&& c^r_{pq}a^p_\lambda{\cal E}_r^\lambda - d_\lambda({\cal E}_q^\lambda)=0,
\label{gr1}\\
&& -a^r_{\mu\lambda}{\cal E}^\lambda_r +d_\lambda(a^r_\mu{\cal E}^\lambda_r)=0. \label{gr2}
\end{eqnarray}
The first one is the well-known Noether identity corresponding to
the vertical gauge symmetry
\mar{0662}\begin{equation}
\upsilon=(c^r_{pq}a^p_\lambda \xi^q + \xi^r_\lambda)\partial^\lambda_r. \label{0662}
\end{equation}
The second Noether identity (\ref{gr2}) is brought into the form
\begin{eqnarray*}
-a^r_\mu[c^r_{pq}a^p_\lambda{\cal E}_r^\lambda - d_\lambda({\cal E}_q^\lambda)]
+{\cal F}^r_{\lambda\mu}{\cal E}^\lambda_r=0,
\end{eqnarray*}
i.e., it is equivalent to the Noether identity
\mar{gr6}\begin{equation}
{\cal F}^r_{\lambda\mu}{\cal E}^\lambda_r=0, \label{gr6}
\end{equation}
which also comes from the splitting (\ref{gr9}) of the generalized
vector field $\upsilon$.
In the case of Chern--Simons Lagrangian, the Noether identity
(\ref{gr6}) is trivial since its coefficients ${\cal F}^r_{\lambda\mu}$
vanish on the kernel of the Euler--Lagrange operator
\begin{eqnarray*}
\delta L=a^G_{rn}\varepsilon^{\lambda\mu\nu}{\cal F}^n_{\mu\nu}\theta_\lambda^r\wedge\omega
\end{eqnarray*}
of the Chern--Simons Lagrangian (\ref{r50}).
In order to obtain a gauge symmetry of the Yang--Mills Lagrangian,
one should complete the generalized vector field (\ref{0660}) with
the term acting on a world metric.
Let $LX$ the fiber bundle of linear frames in the tangent bundle
$TX$ of $X$. It is a principal bundle with the structure group
$GL(n,\protect\pBbb R)$, $n={\rm dim\,} X$, which is reduced to its maximal compact
subgroup $O(n)$. Global sections of the quotient bundle
$\Sigma=LX/O(n)$ are Riemannian metrics on $X$. If $X$ obeys the
well-known topological conditions, pseudo-Riemannian metrics on
$X$ are similarly described. Being an open subbundle of the tensor
bundle $\op\vee^2 TX$, the bundle $\Sigma$ is provided with bundle
coordinates $\sigma^{\mu\nu}$. It admits the canonical lift
\mar{gr3}\begin{equation}
u_\Sigma=u^\lambda\partial_\lambda +(\sigma^{\nu\beta}\partial_\nu u^\alpha
+\sigma^{\alpha\nu}\partial_\nu u^\beta)\frac{\partial}{\partial \sigma^{\alpha\beta}}
\label{gr3}
\end{equation}
of any vector field $u=u^\lambda\partial_\lambda$ on $X$.
In order to describe the gauge theory of principal connections in
the presence of a dynamic metric field, let us consider the bundle
product
\mar{gr10}\begin{equation}
E=C\op\times_X \Sigma\op\times_X T_GP, \label{gr10}
\end{equation}
coordinated by $(x^\lambda,a^r_\lambda,\sigma^{\alpha\beta}, \tau^\lambda, \xi^r)$. It
can be provided with the generalized vector field
\mar{gr11}\begin{equation}
\upsilon=(c^r_{pq}a^p_\lambda \xi^q + \xi^r_\lambda -a^r_\mu\tau^\mu_\lambda-\tau^\mu
a_{\mu\lambda}^r)\partial^\lambda_r + (\sigma^{\nu\beta}\tau_\nu^\alpha +\sigma^{\alpha\nu}
\tau_\nu^\beta-\tau^\lambda\sigma_\lambda^{\alpha\beta})\frac{\partial}{\partial
\sigma^{\alpha\beta}}. \label{gr11}
\end{equation}
This is a gauge symmetry of the sum $L=L_{\rm YM} + L_g$ of the
Yang--Mills Lagrangian $L_{\rm YM}({\cal F}^r_{\alpha\beta}, \sigma^{\mu\nu})$
and a Lagrangian $L_g$ of a metric field. The corresponding
Noether identities read
\mar{gr12,3}\begin{eqnarray}
&& c^r_{pq}a^p_\lambda{\cal E}_r^\lambda - d_\lambda({\cal E}_q^\lambda)=0,
\label{gr12}\\
&& -a^r_{\mu\lambda}{\cal E}^\lambda_r +d_\lambda(a^r_\mu{\cal E}^\lambda_r)
-\sigma_\mu^{\alpha\beta}{\cal E}_{\alpha\beta} - 2d_\nu (\sigma^{\nu\beta}
{\cal E}_{\mu\beta})=0. \label{gr13}
\end{eqnarray}
The first one is the Noether identity (\ref{gr1}), while the
second identity is brought into the form
\begin{eqnarray*}
-a^r_{\mu\lambda}{\cal E}^\lambda_r +d_\lambda(a^r_\mu{\cal E}^\lambda_r) - 2\nabla_\nu
(\sigma^{\nu\beta} {\cal E}_{\mu\beta})=0,
\end{eqnarray*}
where $\nabla_\nu$ are covariant derivatives with respect to the
Levi--Civita connection
\begin{eqnarray*}
K=dx^\lambda\otimes (\partial_\lambda + K_\lambda{}^\mu{}_\nu \dot x^\nu\dot\partial_\mu),
\qquad K_\lambda{}^\mu{}_\nu =-\frac12\sigma^{\nu\beta}(\sigma_{\lambda\beta\mu} +
\sigma_{\mu\beta\lambda} -\sigma_{\beta\lambda\mu}).
\end{eqnarray*}
In metric-affine gravitation theory, dynamic variables are a world
metric and a world connection. World connections are principal
connections on the frame bundle $LX$, and they are represented by
sections of the quotient fiber bundle
\mar{gr14}\begin{equation}
C_K=J^1LX/GL(n,\protect\pBbb R). \label{gr14}
\end{equation}
This fiber bundle is provided with bundle coordinates
$(x^\lambda,k_\lambda{}^\nu{}_\alpha)$ such that, for any section $K$ of
$C_K\to X$, its coordinates $k_\lambda{}^\nu{}_\alpha\circ
K=K_\lambda{}^\nu{}_\alpha$ are coefficient of the linear connection
\begin{eqnarray*}
K=dx^\lambda\otimes (\partial_\lambda + K_\lambda{}^\mu{}_\nu \dot x^\nu\dot\partial_\mu)
\end{eqnarray*}
on $TX$. The bundle of world connections (\ref{gr14}) admits the
canonical lift
\begin{eqnarray*}
u_K=u^\mu\partial_\mu +[\partial_\nu u^\alpha k_\mu{}^\nu{}_\beta -\partial_\beta u^\nu
k_\mu{}^\alpha{}_\nu -\partial_\mu u^\nu k_\nu{}^\alpha{}_\beta
+\partial_{\mu\beta}u^\alpha]\frac{\partial}{\partial k_\mu{}^\alpha{}_\beta}.
\end{eqnarray*}
In order to describe the gauge theory of principal connections in
the presence of metric-affine gravity, let us consider the bundle
product
\mar{gr16}\begin{equation}
E=C\op\times_X \Sigma\op\times_X C_K\op\times_X T_GP, \label{gr16}
\end{equation}
coordinated by $(x^\lambda,a^r_\lambda,\sigma^{\alpha\beta}, k_\lambda{}^\nu{}_\alpha,
\tau^\lambda, \xi^r, )$. It can be provided with the generalized
vector field
\mar{gr15}\begin{eqnarray}
&& \upsilon=(c^r_{pq}a^p_\lambda \xi^q + \xi^r_\lambda
-a^r_\mu\tau^\mu_\lambda-\tau^\mu a_{\mu\lambda}^r)\partial^\lambda_r +
(\sigma^{\nu\beta}\partial_\nu \tau^\alpha +\sigma^{\alpha\nu}\partial_\nu
\tau^\beta-\tau^\lambda\sigma_\lambda^{\alpha\beta})\frac{\partial}{\partial \sigma^{\alpha\beta}}
\label{gr15}\\
&& \qquad (\tau_\nu^\alpha k_\mu{}^\nu{}_\beta -\tau_\beta^\nu
k_\mu{}^\alpha{}_\nu -\tau_\mu^\nu k_\nu{}^\alpha{}_\beta
+\tau_{\mu\beta}^\alpha-\tau^\lambda k_{\lambda\mu}{}^\alpha{}_\beta)\frac{\partial}{\partial
k_\mu{}^\alpha{}_\beta}. \nonumber
\end{eqnarray}
This is a gauge symmetry of the sum $L=L_{\rm YM} + L_{\rm MA}$ of
the Yang--Mills Lagrangian $L_{\rm YM}({\cal F}^r_{\alpha\beta},
\sigma^{\mu\nu})$ and a metric-affine Lagrangian $L_{\rm MA}$.
\section{BRST symmetries}
In order to introduce BRST symmetries, let us consider Lagrangian
systems of even and odd variables. We describe odd variables and
their jets on a smooth manifold $X$ as generating elements of the
structure ring of a graded manifold whose body is $X$
\cite{cmp,ijmp}. This definition reproduces the heuristic notion
of jets of ghosts in the field-antifield BRST theory
\cite{barn,bran01}.
Recall that any graded manifold $({\protect\pgot A},X)$ with a body $X$ is
isomorphic to the one whose structure sheaf ${\protect\pgot A}_Q$ is formed by
germs of sections of the exterior product
\mar{g80}\begin{equation}
\wedge Q^*=\protect\pBbb R\op\oplus_X Q^*\op\oplus_X\op\wedge^2
Q^*\op\oplus_X\cdots, \label{g80}
\end{equation}
where $Q^*$ is the dual of some real vector bundle $Q\to X$ of
fiber dimension $m$. In field models, a vector bundle $Q$ is
usually given from the beginning. Therefore, we consider graded
manifolds $(X,{\protect\pgot A}_Q)$ where the above mentioned isomorphism holds,
and call $(X,{\protect\pgot A}_Q)$ the simple graded manifold constructed from
$Q$. The structure ring ${\cal A}_Q$ of sections of ${\protect\pgot A}_Q$ consists of
sections of the exterior bundle (\ref{g80}) called graded
functions. Let $\{c^a\}$ be the fiber basis for $Q^*\to X$,
together with transition functions $c'^a=\rho^a_bc^b$. It is
called the local basis for the graded manifold $(X,{\protect\pgot A}_Q)$. With
respect to this basis, graded functions read
\begin{eqnarray*}
f=\op\sum_{k=0}^m \frac1{k!}f_{a_1\ldots a_k}c^{a_1}\cdots
c^{a_k},
\end{eqnarray*}
where $f_{a_1\cdots a_k}$ are local smooth real functions on $X$.
Given a graded manifold $(X,{\protect\pgot A}_Q)$, let ${\protect\pgot d}{\cal A}_Q$ be the
${\cal A}_Q$-module of $\protect\pBbb Z_2$-graded derivations of the $\protect\pBbb
Z_2$-graded ring of ${\cal A}_Q$, i.e.,
\begin{eqnarray*}
u(ff')=u(f)f'+(-1)^{[u][f]}fu (f'), u\in{\protect\pgot d}{\cal A}_Q, \qquad f,f'\in
{\cal A}_Q,
\end{eqnarray*}
where $[.]$ denotes the Grassmann parity. Its elements are called
$\protect\pBbb Z_2$-graded (or, simply, graded) vector fields on
$(X,{\protect\pgot A}_Q)$. Due to the canonical splitting $VQ= Q\times Q$, the
vertical tangent bundle $VQ\to Q$ of $Q\to X$ can be provided with
the fiber bases $\{\partial_a\}$ which is the dual of $\{c^a\}$. Then a
graded vector field takes the local form $u= u^\lambda\partial_\lambda +
u^a\partial_a$, where $u^\lambda, u^a$ are local graded functions. It acts
on ${\cal A}_Q$ by the rule
\mar{cmp50'}\begin{equation}
u(f_{a\ldots b}c^a\cdots c^b)=u^\lambda\partial_\lambda(f_{a\ldots b})c^a\cdots
c^b +u^d f_{a\ldots b}\partial_d\rfloor (c^a\cdots c^b). \label{cmp50'}
\end{equation}
This rule implies the corresponding transformation law
\begin{eqnarray*}
u'^\lambda =u^\lambda, \qquad u'^a=\rho^a_ju^j +
u^\lambda\partial_\lambda(\rho^a_j)c^j.
\end{eqnarray*}
Then one can show that graded vector fields on a simple graded
manifold can be represented by sections of the vector bundle
${\cal V}_Q\to X$, locally isomorphic to $\wedge Q^*\otimes_X(Q\oplus_X TX)$.
Accordingly, graded exterior forms on the graded manifold
$(X,{\protect\pgot A}_Q)$ are introduced as sections of the exterior bundle
$\op\wedge{\cal V}^*_Q$, where ${\cal V}^*_Q\to X$ is the $\wedge Q^*$-dual of
${\cal V}_Q$. Relative to the dual local bases $\{dx^\lambda\}$ for $T^*X$
and $\{dc^b\}$ for $Q^*$, graded one-forms read
\begin{eqnarray*}
\phi=\phi_\lambda dx^\lambda + \phi_adc^a,\qquad \phi'_a=\rho^{-1}{}_a^b\phi_b,
\qquad \phi'_\lambda=\phi_\lambda +\rho^{-1}{}_a^b\partial_\lambda(\rho^a_j)\phi_bc^j.
\end{eqnarray*}
The duality morphism is given by the interior product
\begin{eqnarray*}
u\rfloor \phi=u^\lambda\phi_\lambda + (-1)^{[\phi_a]}u^a\phi_a.
\end{eqnarray*}
Graded exterior forms constitute the bigraded differential algebra
(henceforth BGDA) ${\cal C}^*_Q$ with respect to the bigraded exterior
product $\wedge$ and the exterior differential $d$.
Since the jet bundle $J^rQ\to X$ of a vector bundle $Q\to X$ is a
vector bundle, let us consider the simple graded manifold
$(X,{\protect\pgot A}_{J^rQ})$ constructed from $J^rQ\to X$. Its local basis is
$\{x^\lambda,c^a_\Lambda\}$, $0\leq |\Lambda|\leq r$, together with the
transition functions
\mar{+471}\begin{equation}
c'^a_{\lambda +\Lambda}=d_\lambda(\rho^a_j c^j_\Lambda), \qquad d_\lambda=\partial_\lambda +
\op\sum_{|\Lambda|<r}c^a_{\lambda+\Lambda} \partial_a^\Lambda, \label{+471}
\end{equation}
where $\partial_a^\Lambda$ are the duals of $c^a_\Lambda$. Let ${\cal C}^*_{J^rQ}$
be the BGDA of graded exterior forms on the graded manifold
$(X,{\protect\pgot A}_{J^rQ})$. A linear bundle morphism $\pi^r_{r-1}:J^rQ \to
J^{r-1}Q$ yields the corresponding monomorphism of BGDAs
${\cal C}^*_{J^{r-1}Q}\to {\cal C}^*_{J^rQ}$. Hence, there is the direct
system of BGDAs
\mar{g205}\begin{equation}
{\cal C}^*_Q\op\longrightarrow^{\pi^{1*}_0} {\cal C}^*_{J^1Q}\cdots
\op\longrightarrow^{\pi^r_{r-1}{}^*}{\cal C}^*_{J^rQ}\op\longrightarrow\cdots. \label{g205}
\end{equation}
Its direct limit ${\cal C}^*_\infty Q$ consists of graded exterior
forms on graded manifolds $(X,{\protect\pgot A}_{J^rQ})$, $r\in\protect\pBbb N$, modulo
the pull-back identification, and it inherits the BGDA operations
intertwined by the monomorphisms $\pi^r_{r-1}{}^*$. It is a
$C^\infty(X)$-algebra locally generated by the elements $(1,
c^a_\Lambda, dx^\lambda,\theta^a_\Lambda=dc^a_\Lambda -c^a_{\lambda +\Lambda}dx^\lambda)$,
$0\leq|\Lambda|$.
In order to regard even and odd dynamic variables on the
same footing, let $Y\to X$ be hereafter an affine bundle, and let
${\cal P}^*_\infty Y\subset {\cal O}^*_\infty Y$ be the
$C^\infty(X)$-subalgebra of exterior forms whose coefficients are
polynomial in the fiber coordinates $y^i_\Lambda$ on jet bundles $J^r
Y\to X$. Let us consider the product
\mar{0670}\begin{equation}
{\cal S}^*_\infty={\cal C}_\infty^*Q\wedge{\cal P}^*_\infty Y \label{0670}
\end{equation}
of graded algebras ${\cal C}_\infty^*Q$ and ${\cal P}^*_\infty Y$ over their
common graded subalgebra ${\cal O}^*X$ of exterior forms on $X$
\cite{cmp}. It consists of the elements
\begin{eqnarray*}
\op\sum_i \psi_i\otimes\phi_i, \qquad \op\sum_i \phi_i\otimes\psi_i, \qquad
\psi\in {\cal C}^*_\infty Q, \qquad \phi\in {\cal P}^*_\infty Y,
\end{eqnarray*}
modulo the commutation relations
\mar{0442}\begin{eqnarray}
&&\psi\otimes\phi=(-1)^{|\psi||\phi|}\phi\otimes\psi, \qquad
\psi\in {\cal C}^*_\infty Q, \qquad \phi\in {\cal P}^*_\infty Y, \label{0442}\\
&& (\psi\wedge\sigma)\otimes\phi=\psi\otimes(\sigma\wedge\phi), \qquad \sigma\in {\cal O}^*X.
\nonumber
\end{eqnarray}
They are endowed with the total form degree $|\psi|+|\phi|$ and the
total Grassmann parity $[\psi]$. Their multiplication
\mar{0440}\begin{equation}
(\psi\otimes\phi)\wedge(\psi'\otimes\phi'):=(-1)^{|\psi'||\phi|}(\psi\wedge\psi')\otimes
(\phi\wedge\phi'). \label{0440}
\end{equation}
obeys the relation
\begin{eqnarray*}
\varphi\wedge\varphi' =(-1)^{|\varphi||\varphi'| +[\varphi][\varphi']}\varphi'\wedge \varphi, \qquad
\varphi,\varphi'\in {\cal S}^*_\infty,
\end{eqnarray*}
and makes ${\cal S}^*_\infty$ (\ref{0670}) into a bigraded $C^\infty
(X)$-algebra. For instance, elements of the ring $S^0_\infty$ are
polynomials of $c^a_\Lambda$ and $y^i_\Lambda$ with coefficients in
$C^\infty(X)$.
The algebra ${\cal S}^*_\infty$ is provided with the exterior
differential
\mar{0441}\begin{equation}
d(\psi\otimes\phi):=(d_{\cal C}\psi)\otimes\phi +(-1)^{|\psi|}\psi\otimes(d_{\cal P}\phi),
\qquad \psi\in {\cal C}^*_\infty, \qquad \phi\in {\cal P}^*_\infty,
\label{0441}
\end{equation}
where $d_{\cal C}$ and $d_{\cal P}$ are exterior differentials on the
differential algebras ${\cal C}^*_\infty Q$ and ${\cal P}^*_\infty Y$,
respectively. It obeys the relations
\begin{eqnarray*}
d(\varphi\wedge\varphi')= d\varphi\wedge\varphi' +(-1)^{|\varphi|}\varphi\wedge d\varphi', \qquad
\varphi,\varphi'\in {\cal S}^*_\infty,
\end{eqnarray*}
and makes ${\cal S}^*_\infty$ into a BGDA. Hereafter, let the
collective symbols $s^A_\Lambda$ stand both for even and odd
generating elements $c^a_\Lambda$, $y^i_\Lambda$ of the $C^\infty(X)$-ring
${\cal S}^0_\infty$. Then the BGDA ${\cal S}^*_\infty$ is locally generated
by $(1,s^A_\Lambda, dx^\lambda, \theta^A_\Lambda=ds^A_\Lambda -s^A_{\lambda+\Lambda}dx^\lambda)$,
$|\Lambda|\geq 0$. We agree to call elements of ${\cal S}^*_\infty$ the
graded exterior forms on $X$.
Similarly to ${\cal O}^*_\infty Y$, the BGDA ${\cal S}^*_\infty$ is
decomposed into ${\cal S}^0_\infty$-modules ${\cal S}^{k,r}_\infty$ of
$k$-contact and $r$-horizontal graded forms together with the
corresponding projections $h_k$ and $h^r$. Accordingly, the
exterior differential $d$ (\ref{0441}) on ${\cal S}^*_\infty$ is split
into the sum $d=d_H+d_V$ of the total and vertical differentials
\begin{eqnarray*}
d_H(\phi)=dx^\lambda\wedge d_\lambda(\phi), \qquad d_V(\phi)=\theta^A_\Lambda\wedge\partial^\Lambda_A
\phi, \qquad \phi\in {\cal S}^*_\infty.
\end{eqnarray*}
One can think of the elements
\begin{eqnarray*}
L={\cal L}\omega\in {\cal S}^{0,n}_\infty, \qquad \delta (L)= \op\sum_{|\Lambda|\geq
0}
(-1)^{|\Lambda|}\theta^A\wedge d_\Lambda (\partial^\Lambda_A L)\in {\cal S}^{0,n}_\infty
\end{eqnarray*}
as being a graded Lagrangian and its Euler--Lagrange operator,
respectively.
A graded derivation $\vartheta\in{\protect\pgot d} {\cal S}^0_\infty$ of the $\protect\pBbb R$-ring
${\cal S}^0_\infty$ is said to be contact if the Lie derivative
${\bf L}_\vartheta$ preserves the ideal of contact graded forms of the BGDA
${\cal S}^*_\infty$. With respect to the local basis $(x^\lambda,s^A_\Lambda,
dx^\lambda,\theta^A_\Lambda)$ for the BGDA ${\cal S}^*_\infty$, any contact graded
derivation takes the form
\mar{g105}\begin{equation}
\vartheta=\vartheta_H+\vartheta_V=\vartheta^\lambda d_\lambda + (\vartheta^A\partial_A +\op\sum_{|\Lambda|>0}
d_\Lambda\vartheta^A\partial_A^\Lambda), \label{g105}
\end{equation}
where $\vartheta^\lambda$, $\vartheta^A$ are local graded functions \cite{cmp}.
The interior product $\vartheta\rfloor\phi$ and the Lie derivative
${\bf L}_\vartheta\phi$, $\phi\in{\cal S}^*_\infty$, are defined by the formulae
\begin{eqnarray*}
&& \vartheta\rfloor \phi=\vartheta^\lambda\phi_\lambda + (-1)^{[\phi_A]}\vartheta^A\phi_A, \qquad
\phi\in {\cal S}^1_\infty,\\
&& \vartheta\rfloor(\phi\wedge\sigma)=(\vartheta\rfloor \phi)\wedge\sigma
+(-1)^{|\phi|+[\phi][\vartheta]}\phi\wedge(\vartheta\rfloor\sigma), \qquad \phi,\sigma\in
{\cal S}^*_\infty, \\
&& {\bf L}_\vartheta\phi=\vartheta\rfloor d\phi+ d(\vartheta\rfloor\phi), \qquad
{\bf L}_\vartheta(\phi\wedge\sigma)={\bf L}_\vartheta(\phi)\wedge\sigma
+(-1)^{[\vartheta][\phi]}\phi\wedge{\bf L}_\vartheta(\sigma).
\end{eqnarray*}
The Lie derivative ${\bf L}_\vartheta L$ of a Lagrangian $L$ along a contact
graded derivation $\vartheta$ (\ref{g105}) fulfills the first
variational formula
\mar{g107}\begin{equation}
{\bf L}_\vartheta L= \vartheta_V\rfloor\delta L +d_H(h_0(\vartheta\rfloor \Xi_L)) + d_V
(\vartheta_H\rfloor\omega){\cal L}, \label{g107}
\end{equation}
where $\Xi_L=\Xi+L$ is a Lepagean equivalent of a graded
Lagrangian $L$ \cite{cmp}.
A contact graded derivation $\vartheta$ is said to be variational if the
Lie derivative (\ref{g107}) is $d_H$-exact. A glance at the
expression (\ref{g107}) shows that: (i) a contact graded
derivation $\vartheta$ is variational only if it is projected onto $X$,
and (ii) $\vartheta$ is variational iff its vertical part $\vartheta_V$ is
well. Therefore, we restrict our consideration to vertical contact
graded derivations
\mar{0672}\begin{equation}
\vartheta=\op\sum_{0\leq|\Lambda|} d_\Lambda\upsilon^A\partial_A^\Lambda. \label{0672}
\end{equation}
Such a derivation is completely defined by its first summand
\mar{0673}\begin{equation}
\upsilon=\upsilon^A(x^\lambda,s^A_\Lambda)\partial_A, \qquad 0\leq|\Lambda|\leq k,
\label{0673}
\end{equation}
which is also a graded derivation of ${\cal S}^0_\infty$. It is called
the generalized graded vector field. A glance at the first
variational formula (\ref{g107}) shows that $\vartheta$ (\ref{0672}) is
variational iff $\upsilon\rfloor \delta L$ is $d_H$-exact.
A vertical contact graded derivation $\vartheta$ (\ref{0672}) is said to
be nilpotent if
\mar{g133}\begin{equation}
{\bf L}_\upsilon({\bf L}_\upsilon\phi)= \op\sum_{|\Sigma|\geq 0,|\Lambda|\geq 0 }
(\upsilon^B_\Sigma\partial^\Sigma_B(\upsilon^A_\Lambda)\partial^\Lambda_A +
(-1)^{[s^B][\upsilon^A]}\upsilon^B_\Sigma\upsilon^A_\Lambda\partial^\Sigma_B \partial^\Lambda_A)\phi=0
\label{g133}
\end{equation}
for any horizontal graded form $\phi\in S^{0,*}_\infty$ or,
equivalently, $(\vartheta\circ\vartheta)(f)=0$ for any graded function $f\in
{\cal S}^0_\infty$. One can show that $\vartheta$ is nilpotent only if it is
odd and iff the equality
\mar{0688}\begin{equation}
\vartheta(\upsilon^A)=\op\sum_{|\Sigma|\geq 0} \upsilon^B_\Sigma\partial^\Sigma_B(\upsilon^A)=0
\label{0688}
\end{equation}
holds for all $\upsilon^A$ \cite{cmp}.
Return now to the original gauge system on a fiber bundle $Y$ with
a Lagrangian $L$ (\ref{0512}) and a gauge symmetry $\upsilon$
(\ref{0509}). For the sake of simplicity, $Y\to X$ is assumed to
be affine. Let us consider the BGDA
${\cal S}^*_\infty[V;Y]={\cal C}^*_\infty V\wedge{\cal P}^*_\infty Y$ locally
generated by $(1,c^r_\Lambda, dx^\lambda, y^i_\Lambda, \theta^r_\Lambda, \theta^i_\Lambda)$.
Let $L\in {\cal O}^{0,n}_\infty Y$ be a polynomial in $y^i_\Lambda$, $0\leq
|L|$. Then it is a graded Lagrangian $L\in {\cal P}^{0,n}_\infty
Y\subset {\cal S}^{0,n}_\infty[V;Y]$ in ${\cal S}^*_\infty[V;Y]$. Its gauge
symmetry $\upsilon$ (\ref{0509}) gives rise to the generalized vector
field $\upsilon_E=\upsilon$ on $E$, and the latter defines the generalized
graded vector field $\upsilon$ (\ref{0673}) by the formula
(\ref{0680}). It is easily justified that the contact graded
derivation $\vartheta$ (\ref{0672}) generated by $\upsilon$ (\ref{0680}) is
variational for $L$. It is odd, but need not be nilpotent.
However, one can try to find a nilpotent contact graded derivation
(\ref{0672}) generated by some generalized graded vector field
(\ref{0684}) which coincides with $\vartheta$ on ${\cal P}^*_\infty Y$. Then
$\upsilon$ (\ref{0684}) is called a BRST symmetry.
In this case, the nilpotency conditions (\ref{0688}) read
\mar{0690,1}\begin{eqnarray}
&& \op\sum_\Sigma d_\Sigma(\op\sum_\Xi\upsilon^{i,\Xi}_rc^r_\Xi)
\op\sum_\Lambda\partial^\Sigma_i (\upsilon^{j,\Lambda}_s)c^s_\Lambda +\op\sum_\Lambda d_\Lambda
(u^r)\upsilon^{j,\Lambda}_r
=0, \label{0690}\\
&& \op\sum_\Lambda(\op\sum_\Xi d_\Lambda(\upsilon^{i,\Xi}_r c^r_\Xi)\partial^\Lambda_i
+d_\Lambda (u^r)\partial_r^\Lambda)u^q=0\label{0691}
\end{eqnarray}
for all indices $j$ and $q$. They are equations for graded
functions $u^r\in{\cal S}^0_\infty[V;Y]$. Since these functions are
polynomials
\mar{0693}\begin{equation}
u^r=u_{(0)}^r + \op\sum_\Gamma u_{(1)p}^{r,\Gamma} c^p_\Gamma +
\op\sum_{\Gamma_1,\Gamma_2} u_{(2)p_1p_2}^{r,\Gamma_1\Gamma_2}
c^{p_1}_{\Gamma_1}c^{p_2}_{\Gamma_2} +\cdots \label{0693}
\end{equation}
in $c^s_\Lambda$, the equations (\ref{0690}) -- (\ref{0691}) take the
form
\mar{0694}\begin{eqnarray}
&& \op\sum_\Sigma d_\Sigma(\op\sum_\Xi\upsilon^{i,\Xi}_rc^r_\Xi)
\op\sum_\Lambda\partial^\Sigma_i (\upsilon^{j,\Lambda}_s)c^s_\Lambda +\op\sum_\Lambda d_\Lambda
(u_{(2)}^r)\upsilon^{j,\Lambda}_r
=0, \label{0694a}\\
&& \op\sum_\Lambda d_\Lambda (u_{(k\neq 2)}^r)\upsilon^{j,\Lambda}_r =0,
\label{0694b}\\
&& \op\sum_\Lambda\op\sum_\Xi d_\Lambda(\upsilon^{i,\Xi}_r c^r_\Xi)\partial^\Lambda_i
u_{(k-1)}^q +\op\sum_{m+n-1=k}d_\Lambda (u_{(m)}^r)\partial_r^\Lambda u_{(n)}^q
=0. \label{0694c}
\end{eqnarray}
One can think of the equalities (\ref{0694a}) and (\ref{0694c})
as being the generalized commutation relations and
generalized Jacobi identities of original gauge transformations,
respectively \cite{algebr}.
\section{Space-time BRST symmetries}
Let us consider gauge symmetries (\ref{0660}), (\ref{gr11}) and
(\ref{gr15}). Following the procedure in Section 4, we replace
parameters $\xi^r$ and $\tau^\lambda$ with the odd ghosts $c^r$ and
$c^\lambda$, respectively, and obtain the generalized graded vector
fields
\mar{gr20-22}\begin{eqnarray}
&& \upsilon= (c^r_{pq}a^p_\lambda c^q + c^r_\lambda -a^r_\mu c^\mu_\lambda-c^\mu
a_{\mu\lambda}^r)\partial^\lambda_r +(-\frac12c^r_{pq}c^pc^q -c^\mu c^r_\mu)\partial_r
+ c^\lambda_\mu c^\mu\partial_\lambda, \label{gr20}\\
&& \upsilon= (c^r_{pq}a^p_\lambda c^q + c^r_\lambda -a^r_\mu c^\mu_\lambda-c^\mu
a_{\mu\lambda}^r)\partial^\lambda_r + (\sigma^{\nu\beta} c_\nu^\alpha +\sigma^{\alpha\nu}
c_\nu^\beta-c^\lambda\sigma_\lambda^{\alpha\beta})\frac{\partial}{\partial \sigma^{\alpha\beta}}
+ \label{gr21}\\
&& \qquad (-\frac12c^r_{pq}c^pc^q -c^\mu c^r_\mu)\partial_r + c^\lambda_\mu
c^\mu\partial_\lambda, \nonumber\\
&& \upsilon=(c^r_{pq}a^p_\lambda c^q + c^r_\lambda -a^r_\mu c^\mu_\lambda-c^\mu
a_{\mu\lambda}^r)\partial^\lambda_r + (\sigma^{\nu\beta} c_\nu^\alpha +\sigma^{\alpha\nu}
c_\nu^\beta-c^\lambda\sigma_\lambda^{\alpha\beta})\frac{\partial}{\partial \sigma^{\alpha\beta}}+
\label{gr22}\\
&& \qquad (c_\nu^\alpha k_\mu{}^\nu{}_\beta -c_\beta^\nu k_\mu{}^\alpha{}_\nu
-c_\mu^\nu k_\nu{}^\alpha{}_\beta +c_{\mu\beta}^\alpha-c^\lambda
k_{\lambda\mu}{}^\alpha{}_\beta)\frac{\partial}{\partial k_\mu{}^\alpha{}_\beta}
+(-\frac12c^r_{pq}c^pc^q -c^\mu c^r_\mu)\partial_r + c^\lambda_\mu
c^\mu\partial_\lambda. \nonumber
\end{eqnarray}
The vertical contact graded derivations (\ref{0672}) generated by
these generalized graded vector fields are nilpotent, i.e., these
generalized graded vector fields are BRST symmetries.
It should be noted that all the BRST symmetries (\ref{gr20}) --
(\ref{gr22}) possesses the same ghost term
\begin{eqnarray*}
(-\frac12c^r_{pq}c^pc^q -c^\mu c^r_\mu)\partial_r + c^\lambda_\mu c^\mu\partial_\lambda.
\end{eqnarray*}
It is not surprised because this term corresponds to the bracket
of the vector fields (\ref{0652}).
|
3,212,635,537,958 | arxiv | \section*{Biographies}
\noindent \textbf{Sriramya Bhamidipati} is a Ph.D. student in the Department of Aerospace Engineering at the University of Illinois at Urbana-Champaign, where she also received her master’s degree in 2017. She obtained her B.Tech. in Aerospace from the Indian Institute of Technology Bombay in 2015. Her research interests include GPS, power and control systems, artificial intelligence, computer vision and unmanned aerial vehicles.
~\\
\noindent \textbf{Grace Xingxin Gao} is an assistant professor in the Department of Aeronautics and Astronautics at Stanford University. Before joining Stanford University, she was an assistant professor at University of Illinois at Urbana-Champaign. She obtained her Ph.D. degree at Stanford University. Her research is on robust and secure positioning, navigation and timing with applications to manned and unmanned aerial vehicles, robotics and power systems.
\section*{Abstract}
Urban navigation using GPS and fish-eye camera suffers from multipath effects in GPS measurements and data association errors in pixel intensities across image frames.
We propose a Simultaneous Localization and Mapping (SLAM)-based Integrity Monitoring (IM) algorithm to compute the position protection levels while accounting for multiple faults in both GPS and vision.
We perform graph optimization using the sequential data of GPS pseudoranges, pixel intensities, vehicle dynamics and satellite ephemeris to simultaneously localize the vehicle as well as the landmarks, namely GPS satellites and key image pixels in the world frame. We estimate the fault mode vector by analyzing the temporal correlation across the GPS measurement residuals and spatial correlation across the vision intensity residuals. In particular, to detect and isolate the vision faults, we developed a superpixel-based piecewise Random Sample Consensus~(RANSAC) technique to perform spatial voting across image pixels. For an estimated fault mode, we compute the protection levels by applying worst-case failure slope analysis to the linearized Graph-SLAM framework.
We perform ground vehicle experiments in the semi-urban area of Champaign, IL and have demonstrated the successful detection and isolation of multiple faults. We also validate tighter protection levels and lower localization errors achieved via the proposed algorithm as compared to SLAM-based IM that utilizes only GPS measurements.
\section{Introduction} \label{sec:intro}
Integrity Monitoring~(IM) serves as a important performance metric to assess the navigation solution estimation~\cite{ochieng2003gps}. Vehicles operating in urban areas face challenges~\cite{joerger2017towards} due to static infrastructure, such as buildings and thick foliage, dynamic obstacles, such as traffic and pedestrians, and environmental conditions, such as shadows, sunlight and weather. GPS systems receive fewer measurements in urban environments due to degraded satellite visibility. They also suffer from received signal faults caused by multipath and satellite faults caused by anomalies in the broadcast navigation message.
To address the above-mentioned challenges, one possible solution is to incorporate additional redundancy through the sensor fusion of GPS and vision. Vision sensor performs well in urban areas due to the feature-rich surroundings~\cite{hol2011sensor}. Sensor fusion~\cite{krishnaswamy2008sensor} integrates measurements from multiple sensors to improve the accuracy of the vehicle and provide robust performance. Individual sensors, such as GPS and camera, have inherent limitations in operability that are reliably corrected by combining these complementary sensors in a sensor fusion framework. In particular, occlusion and illumination variations in multiple pixel intensities induce data association errors across images, thereby termed as vision faults~\cite{miro2006towards}. Therefore, there is a need for the development of sensor-fusion-based IM techniques that account for multiple faults in both GPS and vision.
Rich literature exists on urban IM approaches for GPS-based navigation systems that utilize external information sources. In~\cite{velaga2012map}, the authors developed a sequential map-aided IM technique that checks for outliers in position and Geographic Information System~(GIS) using traditional RAIM~\cite{walter2008shaping} and weight-based topological map-matching process, respectively. Another paper~\cite{binjammaz2013gps} developed three phases of integrity checks that include assessing the position quality via traditional Receiver Autonomous Integrity Monitoring~(RAIM), speed integrity via GPS Doppler and map matching accuracy via fuzzy inference system. However, these approaches have practical limitations because the offline map database is not always available and its accuracy cannot be guaranteed due to the dynamic changes in the urban surroundings. Another line of prior work~\cite{li2017lane,toledo2009lane} utilizes the odometry information obtained from Dead-Reckoning~(DR) sensors, such as inertial measurement units, wheel speed encoder and camera, to perform GPS-based IM. But the drawbacks of these approaches are that they do not address the faults associated with DR sensors, and also, do not account for the simultaneous occurrence of faults across multiple sensor sources.
In this paper, we leverage the generalized and flexible platform developed in our prior work~\cite{bhamidipati2018multiple}, which is Simultaneous Localization and Mapping~(SLAM)-based Fault Detection and Isolation~(FDI), as the basis for assessing the sensor fusion integrity. Another extension of the SLAM-based FDI platform, described in our prior work~\cite{bhamidipati2018receivers}, assesses the integrity of cooperative localization using a network of receivers. SLAM~\cite{cadena2016past}, a well-known technique in robotics, utilizes sensor measurements to estimate the landmarks in a three-dimensional~(3D) map while simultaneously localizing the robot within it. Analogous to this, our prior work~\cite{bhamidipati2018multiple} on SLAM-based FDI combines the sequential data of GPS measurements, receiver motion model and satellite orbital model in a graph framework to simultaneously localize the \textit{robot}, which is the GPS receiver, \textit{landmarks} in the map, which are the GPS satellites. A key feature of this platform is that it utilizes graph optimization techniques~\cite{latif2014robust} and therefore, does not require any prior assumption regarding the distribution of states. Given that we localize the landmarks as well, our SLAM-based FDI does not require any prior information regarding the surrounding 3D maps.
We propose SLAM-based IM algorithm using GPS and fish-eye camera to compute the error bounds, termed as \textit{protection levels}, of the estimated navigation solution by applying worst-case failure slope analysis~\cite{salos2013receiver,joerger2014solution} to the Graph-SLAM framework. In this work, we consider \textit{global landmarks} as the GPS satellites and additional \textit{local landmarks} as the key image pixels in the world frame. Here, world frame represents the Earth-Centered Earth-Fixed~(ECEF) frame. We simultaneously update the state vectors of the vehicle, GPS satellites and key image pixels and thereafter, perform multiple FDI. We constrain the graph via GPS pseudoranges, raw fish-eye images, vehicle dynamics and satellite ephemeris.
For vision measurements, we opt for a fish-eye camera mounted on an vehicle and point it upwards for the following reasons; Firstly, given its wide~$(\geq 180^{\circ})$ Field-Of-View~(FOV) the image pixels are spatially spread-out in different directions with respect to the vehicle, thereby, compensating for the restricted spatial geometry of the limited global landmarks, i.e., GPS satellites. Secondly, given that the camera is pointing upwards, the unstructured skyline of the buildings aids in resolving the attitude of the vehicle. Thirdly, the fish-eye image captures the open sky section with respect to the vehicle that is used to distinguish the Line-Of-Sight~(LOS) GPS satellites from that of the Non-Line-Of-Sight~(NLOS) ones~\cite{shytermeja2014proposed}.
\begin{figure}[H]
\setlength{\belowcaptionskip}{-4pt}
\centering \includegraphics[width=0.6\columnwidth]{Images/Architecture.jpg}
\caption{Architecture of our SLAM-based IM algorithm using GPS and fish-eye camera.}
\label{fig_algo:architecture}
\end{figure}
The rest of the paper is organized as follows:~Section~II describes our SLAM-based IM algorithm that utilizes GPS and fish-eye camera; Section~III experimentally validates the proposed algorithm in performing multiple FDI of GPS and vision faults and assessing the corresponding localization accuracy and size of protection levels;~Section~IV concludes the paper.
\section{SLAM-based IM using GPS and Fish-eye Camera}
We outline the high-level architecture and later, explain the details of the proposed algorithm.
In this work, we focus on the measurement faults that are more frequently observed in urban areas, namely GPS and vision faults. Even though the formulation of the proposed algorithm is capable of addressing other faults, for simplicity, we consider no measurement faults associated with the receiver motion model and satellite orbital model. For reference, details regarding addressing the satellite faults using SLAM-based FDI are described in our prior work~\cite{bhamidipati2018multiple}.
In Fig.~\ref{fig_algo:architecture}, we show the architecture of our SLAM-based IM algorithm using GPS and fish-eye camera that is summarized as follows:
\begin{enumerate}
\item During initialization, we initialize a 3D map using the PVT of the receiver and satellites computed via an established GPS receiver algorithm~\cite{lashley2010valid}. We set the initial value of all GPS measurement fault status to $0.5$ indicating neutrality. For the vision module, we perform initial calibration to estimate the scaling from image to global frame.
\item Firstly, we pre-process the raw image obtained from the fish-eye camera using our hybrid sky detection algorithm to distinguish the sky pixels from the non-sky pixels. The detected sky pixels are used to distinguish the LOS and NLOS satellites and thereafter, formulate the corresponding GPS measurement covariance.
\item We consider the non-sky pixels along with GPS pseudoranges and carrier-to-noise density~$(C/N_{0})$ values, receiver motion model and satellite orbital model as input measurements to our algorithm. We combine the measurements in an extended graph optimization module to estimate the overall state vector, which consists of the state vector of the vehicle, GPS satellites and key image pixels using M-estimator~\cite{shevlyakov2008redescending}-based Levenberg Marquardt algorithm~\cite{lourakis2005brief}.
\item We independently analyze the measurement residuals against an empirical distribution to detect and isolate GPS faults. We develop a superpixel~\cite{li2015superpixel}-based piecewise Random Sample Consensus~(RANSAC)~\cite{conte2009vision} to perform spatial voting for the detection and isolation of vision faults. Based on the estimated fault status of the measurements, we estimate the measurement fault mode, which has binary entries, such that $0$ indicates non-faulty and $1$ represents faulty.
\item Finally, utilizing the estimated fault mode and overall state vector, we formulate the failure slope for the Graph-SLAM framework and subsequently, compute the protection levels using worst-case failure mode slope analysis~\cite{salos2013receiver,joerger2014solution}.
\end{enumerate}
\begin{figure}[H]
\centering
\begin{subfigure}[b]{0.75\textwidth}
\includegraphics[width=\textwidth]{Images/HybridSky.png}
\caption{Our hybrid sky detection algorithm}
\label{fig_exp:hybrid_sky}
\end{subfigure}
\begin{subfigure}[b]{0.255\textwidth}
\includegraphics[width=\textwidth]{Images/RawImage.jpg}
\caption{Raw fish-eye image}
\label{fig_exp:raw}
\end{subfigure}
\begin{subfigure}[b]{0.255\textwidth}
\includegraphics[width=\textwidth]{Images/SkyBorder.jpg}
\caption{Hybrid optimal border}
\label{fig_exp:border}
\end{subfigure}
\begin{subfigure}[b]{0.255\textwidth}
\includegraphics[width=\textwidth]{Images/SkyArea.jpg}
\caption{Sky area}
\label{fig_exp:area}
\end{subfigure}
\caption{An example showing the pipeline of pre-processing the fish-eye image in the vision module}
\label{fig_algo:vision}
\end{figure}
The proposed SLAM-based IM algorithm using GPS and fish-eye camera consists of three main modules, namely measurement pre-processing, extended graph optimization and IM for Graph-SLAM. We describe the details as follows:
\subsection{Pre-processing the measurements}
We consider the following measurements as inputs to our SLAM-based IM algorithm: GPS pseudoranges and $C/N_{0}$ values from the GPS receiver, pixel intensities from the fish-eye image, control input obtained from the vehicle motion model and satellite ephemeris decoded from the navigation message.
~\\
\noindent \textit{1. Vision module: }~\\
\indent We pre-process the raw image obtained from fish-eye camera using hybrid sky detection algorithm, to distinguish the sky-pixels from the non-sky pixels. The pipeline of our hybrid sky detection is seen in Fig.~\ref{fig_exp:hybrid_sky}. Our hybrid sky detection takes into account not only the pixel intensities but also prior knowledge regarding the spatial location of the sky pixels.
We convert the raw image to gray scale and then perform median blur operation. The median blur~\cite{wang2010new} is a non-linear filter that reduces the noise in the image while keeping the edges relatively sharp. Next, we compute the gradient by combining the magnitude obtained via two Sobel operations~\cite{gao2010improved}, one executed in horizontal and the other in vertical direction. An example of the image obtained from fish-eye camera operating in urban areas and pointing upwards is seen in Fig.~\ref{fig_exp:raw}.
We observe that the probability of sky is highest close to the center and exponentially decreases outwards~\cite{haque2008hybrid}. Therefore, the corresponding location-based sky probability, denoted by~$p_{loc}$ is given by
\begin{align} \label{eq_algo:loc}
p_{loc}({\bm u}) = \textrm{exp}\Big(-\dfrac{2\big|\big| {\bm u}-{\bm c}_{loc}\big|\big|}{|\Pi|}\Big),
\end{align}
\noindent where ${\bm u}$ is the 2D image coordinates, such that ${\bm u}=[u,v]^{T}\in \Pi$, $\Pi$~represents the pre-defined domain of the image coordinates. $|\cdot|$~denotes the cardinality of the image domain and $||\cdot||$~denotes the 2-norm residual. ${\bm c}_{loc}\subset \Pi$ denotes the pre-defined center coordinates in the 2D image frame.
Combining the location probability with Otsu's method of intensity thresholding~\cite{moghaddam2012adotsu}, we compute the hybrid optimal border, seen in Fig.~\ref{fig_exp:border}, that separates the sky region, represented by subscript~$sky$, from that of the infrastructure, denoted by subscript~$inf$. We minimize the variance of sky and infrastructure to estimate the Otsu's intensity threshold~$I_{otsu}$ as
\begin{align} \label{eq_algo:otsu}
I_{otsu}= \argmin_{k\in\,\bm{I}}\Big(\omega_{sky}(k)\sigma _{sky}^{2}(k)+\omega_{inf}(k)\sigma_{inf}^{2}(k) \Big)~\textrm{from~\cite{moghaddam2012adotsu}},
\end{align}
\noindent where
\begin{itemize}
\item [--] ${\bm I}$~denotes the intensity vector that stacks all the pixel intensities in the fish-eye image, such that ${\bm I}=\{I({\bm u})\,\big|\, {\bm u}\in \Pi \}$, where $I({\bm u}): \Pi \rightarrow \mathbb{R}$~denotes the intensity of any 2D pixel coordinates~${\bm u}$;
\item [--] $\omega _{sky}(k)$ and $\omega _{inf}(k)$ denotes the weights associated with the sky and building infrastructure, respectively, such that $\omega _{sky}(k)=\dfrac{\sum_{{\bm u}\in |\Pi|}\mathds{1}\big\{I({\bm u})<k\big\}}{|\Pi|}$ and $\omega _{inf}(k)=\dfrac{\sum_{{\bm u}\in |\Pi|}\mathds{1}\big\{I({\bm u})>k\big\}}{|\Pi|}$;
\item[--] $\sigma_{sky}^{2}(k)$ and $\sigma_{inf}^{2}(k)$ denotes the variance of the pixel intensities associated with the sky and building infrastructure.
\end{itemize}
Utilizing Eqs.~\eqref{eq_algo:loc} and~\eqref{eq_algo:otsu}, we compute the hybrid sky probability, denoted by $p_{sky}$ at any 2D image coordinate~${\bm u},\,{\bm u}\in\Pi$ as
\begin{align} \label{eq_algo:hybrid}
p_{sky}({\bm u}) &= \textrm{exp}\bigg(\dfrac{-\big| I({\bm u})-I_{otsu}\big|}{\big|I_{max}-I_{min}\big|}\bigg)~p_{loc}({\bm u}),
\end{align}
\noindent where $I_{max}$ and $I_{min}$ are the maximum and minimum intensity values in the fish-eye image, such that $I_{max}=\max_{{\bm u}\in \Pi}I({\bm u})$ and $I_{min}=\min_{{\bm u}\in \Pi}I({\bm u})$, respectively. Considering $\eta$ as the pre-defined sky threshold, if $p_{sky}({\bm u})>\eta$, then it is categorized as sky pixel and non-sky pixel otherwise. The sky-enclosed area in the fish-eye image is seen in Fig.~\ref{fig_exp:area}.
Next, using the non-sky detected pixels, we describe the vision measurement model in Eq.~\eqref{eq_ion:cam_model} that is formulated via omni-directional camera model~\cite{caruso2015large} and direct image alignment~\cite{engel2014lsd}. Direct image alignment computes the depth maps in an incremental fashion and compares the pixel intensities across the image frames directly, such that the spatial context of the image is preserved. This vision measurement model is utilized later in our extended graph optimization module to formulate the corresponding vision odometry-based component of the cost function.
\begin{align}
\begin{split} \label{eq_ion:cam_model}
I_{kf}({\bm u}) &= I_{t}\Big(\pi\big(w(\Delta{\bm \mu}_{t},{\bm u})\big)\Big) + \eta_{vis}({\bm u})~\textrm{from~\cite{engel2014lsd}},
\end{split}
\end{align}
\noindent such that $\eta_{vis}({\bm u})$ is pixel noise and from~\cite{caruso2015large},
\begin{align*}
\begin{split}
w(\Delta{\bm \mu},{\bm u}) &= {\textbf R}(\Delta{\bm \mu})\,\pi^{-1}\Big({\bm u},d_{kf}({\bm u})\Big) + {\textbf t}(\Delta{\bm \mu}), \\
\pi(\textbf{p}) &=
\begin{bmatrix}
\vspace{0.5pc}
f_{x}\dfrac{p_{x}}{p_{z}+\big|\big|\textbf{p}\big|\big|\xi} \\
f_{y}\dfrac{p_{y}}{p_{z}+\big|\big|\textbf{p}\big|\big|\xi}
\end{bmatrix} +
\begin{bmatrix}
c_{x}\\
c_{y}
\end{bmatrix}, \\
\pi^{-1}\big({\bm u}, d\big) &= \dfrac{1}{d}\Bigg(\dfrac{\xi+\sqrt{1+(1-\xi)^2(\hat{u}^2+\hat{v}^2)}}{\hat{u}^2+\hat{v}^2+1}
\begin{bmatrix}
\hat{u}^2 \\
\hat{v}^2 \\
1
\end{bmatrix}-
\begin{bmatrix}
0 \\
0 \\
\xi
\end{bmatrix} \Bigg),
\end{split}
\end{align*}
\noindent where the subscript~$kf$ refers to keyframe,
\begin{enumerate}
\item [--] $I_{kf}({\bm u}): \Pi_{kf} \rightarrow \mathbb{R}$~denotes the intensity of any 2D pixel coordinates~${\bm u}$ in the keyframe and $\Pi_{kf} \subset \mathbb{R}^2$ denotes the image domain of keyframe; Detailed explanation regarding keyframe selection and estimation of semi-dense depth maps is given in prior literature~\cite{caruso2015large};
\item [--] $I_{t}({\bm u}): \Pi_{ns} \rightarrow \mathbb{R}$~denotes the intensity of any 2D pixel coordinates~${\bm u}$ in the current frame and $\Pi_{ns} \subseteq \Pi$ denotes the image domain consisting of non-sky pixels;
\item [--] $\pi: \mathbb{R}^3 \rightarrow \Pi_{ns}$ denotes the map from 3D world coordinates, denoted by ${\bm p}=[p_{x},p_{y}, p_{z}]$ to 2D pixel in image frame;
\item [--] $w(\Delta{\bm \mu},{\bm u})$ denotes the 3D warp function that unprojects the pixel coordinates~${\bm u}$ and transforms it by a relative state vector $\Delta{\bm \mu}$. The relative state vector~$\Delta{\bm \mu}$ indicates the difference between the current vehicle pose, denoted by ${\bm \mu}_{t}=[\textbf{x},{\bm \psi}]_{t}$ with respect to that of the keyframe, denoted by~${\bm \mu}_{kf}$; Here, $\textbf{x}$~denotes the 3D vehicle position and ${\bm \psi}$~denotes the 3D orientation; ${\textbf R}\in \textrm{SO(3)}$ and ${\textbf t}\in \mathbb{R}^{3}$ denotes the rotation matrix and translation vector of~${\bm \mu}$, respectively;
\item [--] $\pi^{-1}:\Pi_{ns} \times \mathbb{R}^{+}\rightarrow \mathbb{R}^{3}$ denotes the inverse mapping of 2D pixel coordinates to 3D world coordinates via an inverse distance represented by $d$. Here, $\hat{u} = (u-c_x)/f_x$ and $\hat{v} = (v-c_y)/f_y$ denotes the transformed 2D pixel coordinates. We calibrate the camera parameters, namely $f_{x}$, $f_{y}$, $c_{x}$, $c_{y}$ and $\xi$ during initialization;
\item [--] $d_{kf}({\bm u})$ denotes the inverse distance of the pixel coordinates in the keyframe.
\end{enumerate}
~\\
\noindent \textit{2. GPS module: }~\\
\indent In the GPS module, considering $N$~visible satellites, we describe the GPS measurement model as
\begin{align}
\begin{split}
\rho^{k} &=
\|\textbf{y}^{k}-\textbf{x}\|+\big(c\delta t-c\delta t^{k}\big)+\eta^{k}, \\
\end{split}
\label{eq:pseudo}
\end{align}
\noindent where $\textbf{x}$ and $\textbf{y}^{k}$~denotes the 3D position of the vehicle and $k^{th}$~satellite, respectively;~$c\delta t$ and $c\delta t^{k}$ represents the receiver clock bias and $k^{th}$~satellite clock bias corrections, respectively; $\eta^{k}$ represents the measurement noise related to $k^{th}$~satellite.
We also formulate the measurement covariance of $k^{th}$ satellite via the measured~$C/N_0$ values and the sky area detected via Eq.~\eqref{eq_algo:hybrid} in the vision pre-processing module. Note that the classification of the satellite as either LOS or NLOS depends on the unknown state vector of the vehicle and $k^{th}$ satellite. Therefore, the measurement covariance of $k^{th}$ satellite is given by
\begin{align} \label{eq:cov}
\big(\sigma^{k}({\bm x}_{t}, {\bm y}^{k}_{t})\big)^2= \sqrt{b^{k}+a^{k}\dfrac{1}{(C/N_{0})^{k}}}~\textrm{from~\cite{shytermeja2014proposed}},
\end{align}
\noindent where
\begin{itemize}
\item[--] ${\bm x}_{t}$ denotes the vehicle state vector at $t^{th}$~time instant comprising of 3D position, 3D velocity, clock bias, clock drift and 3D attitude, respectively, such that ${\bm x}_{t}=[\textbf{x},~c\delta t,~\dot{\textbf{x}},~c\dot{\delta t},~\bm{\psi}]_{t}$;
\item[--] ${\bm y}_{t}^{i}$ denotes the state vector of~$i^{th}$~satellite comprising of its 3D position, 3D velocity, clock bias and clock drift corrections, such that ${\bm y}_{t}^{k}=[\textbf{y}^{k},~c\delta t^{i},~\dot{\textbf{y}}^{k},~c\delta\dot{t}^{k}]_{t},~i\in\{1,\cdots,N\}$;
\item [--] $b^{k}$ and $a^{k}$ are the vision coefficients, such that $b^{k}=\dfrac{b_{LOS}}{p_{sky}\big(\pi(\textbf{y}^{k})\big)}$ and $a^{k}=\dfrac{a_{LOS}}{p_{sky}\big(\pi(\textbf{y}^{k})\big)}$ when $p_{sky}\big(\pi(\textbf{y}^{k})\big)>\eta$ and $b^{k}=\dfrac{b_{NLOS}}{p_{sky}\big(\pi(\textbf{y}^{k})\big)}$ and $a^{k}=\dfrac{a_{NLOS}}{p_{sky}\big(\pi(\textbf{y}^{k})\big)}$ otherwise; $\eta$ is the pre-defined threshold explained in Eq.~\eqref{eq_algo:hybrid}; $b_{LOS}$, $b_{NLOS}$, $a_{LOS}$ and $a_{NLOS}$ are constant pre-determined coefficients and $\pi(\textbf{y}^{k})$ denotes the projection of the state vector of $k^{th}$ satellite in the image frame.
\end{itemize}
\subsection{Extended graph optimization} \label{sec_algo:graph_opt}
In our extended graph optimization module, our cost function consists of four error terms, namely GPS pseudoranges, non-sky pixel intensities, receiver motion model and satellite orbital model, as follows:
\begin{align} \label{eq:track_err}
e_{t}(\bm{\theta}_{t}) &=\sum_{k=1}^{N}\Lambda\Big(\big((\bar{r}^{k}_{t}+1)\sigma_{t}^{k}\big)^{-1}\Big|\bm{\rho}_{t}^{k}-h({\bm x}_{t},{\bm y}^{k}_{t})\Big|\Big) + \sum_{k=1}^{N}\Lambda\Big(\big(\hat{\Omega}^{k}_{t}\big)^{-1}\big|\big|{\bm y}_{t}^{i}-f(u^{k}_{t},\bar{\bm y}^{k}_{t-1})\big|\big|\Big) \\
&~~~+\Lambda\Big(\big(\bar{\chi}_{t}{\bm I}+\hat{\Sigma}_{t}\big)^{-1}\big|\big|{\bm x}_{t}-g(u_{R,t},\bar{\bm x}_{t-1})\big|\big| \Big) + \sum_{{\bm u}\in \Pi_{ns}}\Lambda\Big(\big((\bar{s}_{t-1}({\bm u})+1)\omega_{t}({\bm u})\big)^{-1}\big|\bm{I}_{kf}({\bm u})-\bm{I}_{t}\Big(\pi\big(w(\Delta{\bm \mu}_{t},{\bm u})\big)\Big)\big|\Big), \nonumber
\end{align}
\noindent where
\begin{itemize}
\item[--] $\bm{\theta}_{t}$~denotes the overall state vector comprising of the state vector of the vehicle, GPS satellites and key image pixels in the world frame, given by~$\bm{\theta}_{t}=\{{\bm x}_{t},{\bm y}_{t}^{1},\cdots,{\bm y}_{t}^{N},{\bm p}_{t}^{j}~\forall j\in|\Pi|_{ns}\}$ and is estimated during the graph optimization;
\item[--] $\Lambda$~denotes the M-estimator used to transform the corresponding weighted residuals; Details regarding the choice of M-estimator used are explained in our prior work~\cite{bhamidipati2018multiple};
\item[--] $\bar{r}^{k}_{t-1}$~denotes the fault status associated with the GPS pseudorange of $k^{th}$ satellite and estimated at the past time instant; Similarly, $s_{t}({\bm u})$~denotes the estimated vision fault status of any 2D pixel~${\bm u}\in \Pi_{kf}$ at the previous time instant;
\item [--] $h$~denotes the GPS measurement model; $g$ denotes the motion model of the receiver and $f$ denotes the satellite orbital model;~$\bar{\bm x}_{t-1}$ and $\bar{\bm y}^{k}_{t-1}$ denotes the estimated state vector of the vehicle and $k^{th}$ satellite, respectively, at the previous time instant;~$u_{R,t}$ and $u^{k}_{t}$~denote the motion control inputs of the vehicle and $k^{th}$ satellite, respectively;
\item[--] $\hat{\Sigma}_{t}$ and $\hat{\Omega}^{k}_{t}$~denotes the predicted covariance matrix of the vehicle state vector and $k^{th}$ satellite state vector at the $t^{th}$ time instant; Explanation regarding estimating these covariances is given in our prior work~\cite{bhamidipati2018multiple};
\item[--] $\sigma_{t}^{k}$ denote the measurement covariance of the $k^{th}$ satellite and is estimated from Eq.~\eqref{eq:cov}; Similarly, $\omega_{t}({\bm u})$~denotes the covariance associated with the intensity of the non-sky pixel~${\bm u}$ and is estimated based on Section~$2.3$ of~\cite{engel2014lsd}.
\end{itemize}
The first three terms in the cost function~$\mathbf{e}_{t}$, given in Eq.~\eqref{eq:track_err}, correspond to the residuals associated with the GPS pseudoranges, satellite ephemeris and vehicle state vector, whose details are provided in our prior work~\cite{bhamidipati2018multiple}. The last term represents the summation of intensity residuals across non-sky pixels based on the vision measurement model explained in Eq.~\eqref{eq_ion:cam_model}. In particular, we perform sub-graph optimization at each instant, as seen in Eq.~\eqref{eq:track_error}, where the cost function is formulated using the past history of measurements.
\begin{align} \label{eq:track_error}
\bm{\bar{\theta}}_{t-T:t} &=\argmin_{\theta_{t-T:t}}\Bigg(\sum_{s=t-T}^{t} e_{s}(\bm{\theta}_{s})\Bigg),
\end{align}
\noindent where $T$~denotes the number of time instants utilized in the sub-graph optimization thread and $\bm{\bar{\theta}}_{t-T:t}$~denotes our SLAM-based IM estimate of the overall state vector computed during the sub-graph optimization. We estimate the key image pixels in the world frame, represented by ${\bm p}_{t}^{j}$, via inverse-mapping defined in Eq.~\eqref{eq_ion:cam_model}. Details regarding mapping that involves periodically executing full-graph optimization is given in our prior work~\cite{bhamidipati2018multiple}.
\subsection{IM for Graph-SLAM framework} \label{sec_ion:fdi_im}
We compute the protection levels associated with the estimated vehicle position using worst-case failure mode slope analysis~\cite{salos2013receiver,joerger2014solution}. This is justified because worst-case failure mode slope is derived for weighted least squares estimator and graph optimization via M-estimator-based Levenberg Marquardt algorithm is also a non-linear weighted least squares problem. However, there are certain design challenges involved in applying worst-case failure slope analysis for the protection level computation of Graph-SLAM framework. Firstly, given that the worst-case failure slope is derived for linear measurement model but the cost function associated with Graph-SLAM is non-linear, we linearize the formulation of graph optimization at the estimated overall state vector. Secondly, Graph-SLAM is a sequential methodology, whereas the worst-case failure slope falls under snapshot technique for integrity monitoring. Therefore, we linearize our graph formulation over not only the current time instant, but over the past time history of measurements so as to incorporate the temporal aspect in protection level computation. Thirdly, the graph optimization for SLAM framework consists of a large number of states and measurements. However, evaluating all possible fault modes associated with the measurements is computationally cumbersome. Therefore, we directly compute a single fault mode based on the measurement fault status estimated via multiple FDI module.
~\\
\textit{1) Multiple FDI module: } ~\\
\indent Based on the estimated overall state vector from the extended graph optimization explained in Section~\ref{sec_algo:graph_opt}, we independently compute the measurement residuals associated with GPS pseudoranges and non-sky pixel intensities. In our multiple FDI module, we evaluate the GPS residuals by analyzing the temporal correlation of their non-faulty error distribution and vision residuals using spatial correlation across image pixels.
~\\
\noindent \textbf{GPS faults: }To detect and isolate GPS faults in pseudoranges, we evaluate each residual against an empirical Gaussian distribution, which represents the measurement error distribution during non-faulty conditions. This is justified because we observe that the GPS measurements follow a Gaussian distribution during non-faulty conditions, as explained in our prior work~\cite{bhamidipati2018multiple}. We replicate the non-faulty conditions of GPS measurements by executing the initialization procedure in open-sky conditions. Thereafter, deviation of the measurement residual, denoted by $\Delta\rho^{k}$, from the Cumulative Distribution Function~(CDF) of its empirical Gaussian distribution, denoted by $\Phi_{\Delta\rho}^{k}$, is categorized as a fault and the corresponding fault status~$\bar{r}^{k}_{t}$ is computed in Eq.~\eqref{eq_algo:gps_faults}. The justification regarding the formulation of fault status is explained in our prior work~\cite{bhamidipati2018receivers}.
\begin{align} \label{eq_algo:gps_faults}
\bar{r}^{k}_{t} = 4\Big(\Phi_{\Delta\rho}^{k}(\Delta \rho^{k})-0.5\Big)^2~~~\forall~ k\in\{1,\cdots,N\}.
\end{align}
\noindent \textbf{Vision faults: }Unlike GPS faults, vision faults caused by data association errors exhibit high spatial correlation across image pixels and low temporal correlation. This is justified because the vision faults are localized to a group of neighboring pixels and are not isolated to a standalone pixel. We developed a superpixel-based piecewise RANSAC technique that performs spatial voting across the image pixels to detect and isolate vision faults. RANSAC~\cite{conte2009vision}, a popular outlier detection method in image processing, estimates the optimal fitting parameters of a model via random sampling of data containing both inliers and outliers.
\begin{figure}[H]
\setlength{\belowcaptionskip}{-4pt}
\centering \includegraphics[width=0.68\columnwidth]{Images/VisionFaults.png}
\caption{Pipeline for superpixel-based piecewise RANSAC technique used for estimating the vision fault status.}
\label{fig_algo:vision_faults}
\end{figure}
The steps involved in the superpixel-based piecewise RANSAC technique are described as follows: first, we segment the image into clusters, known as superpixels, based on the color similarity and space proximity between image pixels using superpixel segmentation~\cite{li2015superpixel}. We denote the number of superpixels depicting non-sky pixels to be $\Gamma$, where the total number of superpixels into which the image is segmented is pre-defined during initialization. For each non-sky superpixel, we denote the pixel intensity vector as ${\bm I}^{j}\,\forall j\in\{1,\cdots,\Gamma\}$, which stacks the intensities of pixels within the superpixel. We represent the received intensity, i.e., keyframe pixel intensities Vs expected intensity, i.e., transformed current pixel intensities as a two-Dimensional~(2D) plot. Next, we estimate the fitted line using RANSAC that passes through the optimal set of inliers and thereafter, compute the fraction of outliers in the superpixel, which is represented by $\nu^{j}_{j}$. Then, utilizing the estimated model parameters of the fitted line, we evaluate the corresponding fraction of outliers at all the other non-sky superpixels, denoted by $\nu^{j}_{k}\, \forall k\in\{1,\cdots,\Gamma\}-j$. Finally, the fault status at each superpixel is computed as the product of all the estimated outlier fractions, as seen in Eq.~\eqref{eq_algo:vision_flt}, and the same fault status is assigned to all the pixels within that superpixel. This procedure is repeated for all the non-sky superpixels to compute the fault status of all the non-sky pixels in the keyframe. Our algorithm considers an underlying assumption that there are sufficient number of superpixels to reach a common consensus. If the number of superpixels associated with non-sky pixels is less, such as in open-sky setting, a pre-defined penalty is assigned to the vision fault status.
\begin{align} \label{eq_algo:vision_flt}
\bar{s}_{t}({\bm u})= \nu^{j}_{1}\cdots \nu^{j}_{\Gamma}~~~\forall {\bm u}\in {\bm I}^{j}
\end{align}
~\\
\textit{2) Protection level computation}~\\
\indent In Eq.~\eqref{eq_ion:overall_meas}, based on the design solutions explained in Section~\ref{sec_ion:fdi_im}.1, we linearize the overall measurement model of the graph optimization framework using first-order approximation. For simplicity, we derive the protection levels using measurements of the current time instant, but the same formulation is applicable for extension to the past history of measurements.
\begin{align} \label{eq_ion:overall_meas}
\Delta{\bm z} = C\Delta {\bm \theta} + {\bm \eta} + {\bm f},
\end{align}
\noindent where
\begin{enumerate}
\item [--] $\Delta {\bm z}$~denotes the overall measurement vector that concatenates GPS pseudoranges, control input of vehicle, satellite ephemeris and keyframe pixel intensities against an estimated overall state vector~$\bar{\bm \theta}_{t}$; ${\bm \eta}$ denotes the overall measurement noise;
\item [--] $C$~denotes the linearized overall measurement model that vertically stacks the Jacobian associated with GPS pseudoranges, denoted by $H$, vehicle motion model and satellite orbital model, denoted by $A$ and non-sky pixel intensities, denoted by $J$, such that $C = [H,A,J]^{T}$;
\item[--] ${\bm f}$~denotes the overall fault vector associated with the overall measurement vector and thereby, stacks measurement faults obtained from individual sensor sources.
\end{enumerate}
As described in Eq.~\eqref{eq:track_err} of the graph optimization module, we express the M-estimator-based Levenberg Marquardt formulation, which is a weighted non-linear least squares problem, as
\begin{align} \label{eq_ion:overall_state}
\Delta \bar{\bm \theta}_{t} &= K_{t} \Delta{\bm z}_{t}, \nonumber \\
K_{t}&=[V_{t}H_{t}^{T}S_{t}^{-1}, V_{t}A_{t}^{T}R_{t}^{-1}, V_{t}J_{t}^{T}P_{t}^{-1}],
\end{align}
\noindent where
\begin{enumerate}
\item [--] $K_{t}$~denotes the estimation matrix of the graph-optimization framework and $V$~denotes the pseudo-inverse matrix, such that $V_{t}=\Big(H_{t}^TS_{t}^{-1}H_{t}+ A_{t}^TR_{t}^{-1}A_{t}+J_{t}^TP_{t}^{-1}J_{t}+\beta\,diag(H_{t}^{T}H_{t}+A_{t}^{T}A_{t}+J_{t}^{T}J_{t})\Big)^{-1}$;
\item [--] $S_{t},\,R_{t},\,P_{t}$~denotes the M-estimator-based weight functions for the GPS pseudoranges, vehicle motion model and satellite orbital model, non-sky image pixel intensities, respectively, and evaluated at ${\bm \theta}=\bar{\bm \theta}_{t}$; Details regarding the choice of M-estimator and the corresponding weight functions are explained in our prior work~\cite{bhamidipati2018multiple};
\item [--] $\beta_{n,t}$~denotes the iterative damping factor associated with the Levenberg Marquardt algorithm.
\end{enumerate}
Next, we define the overall test statistic, denoted by $\zeta$, as the summation of the weighted squared residuals across all the measurements. We consider an assumption that the overall test statistic is chi-square distributed, denoted by $\chi^{2}_{k}$ under non-faulty conditions and non-central chi-squared, denoted by $\chi^{2}_{k,\lambda}$, under the presence of GPS faults or vision faults or both.
\begin{align} \label{eq_algo:test_stat}
\zeta &= \big(\Delta {\bm z}-C\Delta \bar{\bm \theta}\big)^{T} \big(\Delta {\bm z}-C\Delta \bar{\bm \theta}\big),
\end{align}
\noindent such that
\begin{align}
\zeta = \begin{cases}
\chi^{2}_{k} & {\bm f}={\bm 0}~\textrm{or non-faulty},\\
\chi^{2}_{k,\lambda} & \textrm{otherwise}.
\end{cases}
\end{align}
\noindent where $k$~denotes the number of redundant measurements, i.e., difference between the number of overall measurements, denoted by $n$ and overall states, denoted by $l$, such that $k=n-l$. $\lambda$~indicates the non-centrality parameter associated with the overall test statistic during faulty conditions.
According to the worst-case failure mode slope analysis~\cite{salos2013receiver}, as seen in Fig.~\ref{fig_algo:prot_levels}, the protection level is calculated as the projection in the position domain of the measurement faults that would generate a non-centrality parameter~$\lambda=\lambda_{th}$ in the overall test statistic~$\zeta$ with the maximum slope. In particular, the non-centrality parameter~$\lambda_{th}$ is estimated from the false-alarm, denoted by~$p_{FA}$ and mis-detection rates, denoted by~$p_{MD}$, which are set according to the pre-defined integrity requirements.
\begin{figure}[H]
\setlength{\belowcaptionskip}{-4pt}
\centering \includegraphics[width=0.4\columnwidth]{Images/ProtectionLevels.png}
\caption{Protection levels computed as the intersection of worst-case failure mode slope and non-centrality parameter~\cite{salos2013receiver}.}
\label{fig_algo:prot_levels}
\end{figure}
In Eq.~\eqref{eq_algo:fault_mode}, we formulate the measurement fault mode, denoted by ${\bm b}_{t}$, using GPS and vision fault status estimated in Eqs.~\eqref{eq_algo:gps_faults} and~\eqref{eq_algo:vision_flt}. For this, we consider a pre-defined fault threshold, denoted by $\kappa$, such that if the fault status is above $\kappa$, the measurement is flagged as faulty in the computation of protection levels. Given that we consider measurement faults in only GPS and vision, the fault entries of receiver and satellite motion models are set to zero for this work. However, the corresponding fault vector, which comprises of the exact measurement fault magnitudes, is still unknown. According to~\cite{salos2013receiver}, for a given fault mode, the worst case fault direction that maximizes the integrity risk, is the one that maximizes the failure mode slope, which is seen in Fig.~\ref{fig_algo:prot_levels} and denoted by~$g_{\bm b}$. In this context, we define the square of failure mode slope, denoted by~$g^{2}_{\bm b}$, as the ratio of squared state estimation error in position of the vehicle over the overall test statistic. Using the linearized equations seen in Eqs.~\eqref{eq_ion:overall_meas},~\eqref{eq_ion:overall_state} and~\eqref{eq_algo:test_stat}, we derive the failure slope for the graph optimization framework in terms of unknown fault vector. For this, we consider $CKC\approx \textbf{I}$, which is valid approximation after the iterative convergence of the graph optimization at any time instant since $\beta<<0$.
\begin{align} \label{eq_algo:fault_mode}
{\bm b}_{t} = \bigg[\mathds{1}_{\big\{\bar{r}^{1}_{t}>\kappa\big\}},\cdots,\mathds{1}_{\big\{\bar{r}^{1}_{t}>\kappa\big\}},{\bm 0},\mathds{1}_{\big\{\bar{s}({\bm u})_{t}>\kappa\big\}}~\forall {\bm u}\in \Pi_{ns} \bigg]_{t}
\end{align}
Considering $n_{\bm b}$ to be the number of non-zero entries in the fault mode~${\bm b}$ estimated via multiple FDI module, we define fault matrix, denoted by $B_{\bm b}$, as $B_{\bm b}=[{\bm I}_{n_{\bm b}},{\bm 0}_{n-n_{\bm b}}]^{T}$ and next, re-arrange the rows of the $m_{\epsilon}$ and $M_{\zeta}$ matrices to match the rows of the fault matrix. Thereafter, we define a transformed fault vector, denoted by~${\bm f}_{\zeta}$, such that ${\bm f}=B_{\bm b}M_{\zeta}{\bm f}_{\zeta}$. Based on the above-mentioned steps, we describe the failure slope formulation of Graph-SLAM framework in Eq.~\eqref{eq_algo:fail_slp}.
\begin{align} \label{eq_algo:fail_slp}
g^{2}_{\bm b} &= \dfrac{{\bm \epsilon}^{T}{\bm \epsilon}}{\zeta} = \dfrac{\big(\Delta {\bm \theta}-\Delta \bar{\bm \theta}\big)^{T}\big(\Delta {\bm \theta}-\Delta \bar{\bm \theta}\big)}{\big(\Delta {\bm z}-C\Delta \bar{\bm \theta}\big)^{T} \big(\Delta {\bm z}-C\Delta \bar{\bm \theta}\big)},\nonumber \\
&= \dfrac{{\bm f}^{T}\Big[\big(\alpha^TK\big)^{T}\big(\alpha^TK\big)\Big]{\bm f}}{{\bm f}^{T}\Big[\big(\textbf{I}-CK\big)^{T}\big(\textbf{I}-CK\big)\Big] {\bm f}}, \\
&= \dfrac{{\bm f}_{\zeta}^{T}M^{T}_{\zeta}m_{\bm \epsilon}m_{\bm \epsilon}^{T}M_{\zeta}{\bm f}_{\zeta}}{{\bm f}_{\zeta}^{T}{\bm f}_{\zeta}}.\nonumber
\end{align}
\noindent where $\alpha$ extracts the vehicle 3D position from the overall state vector~${\bm \theta}$, such that $\alpha^{T}=[\bm{1}_{3\times 1},\bm{0}_{(l-3)\times 1}]$, $M_{\zeta}$~denotes the residual matrix, such that $M_{\zeta}=\Big(B_{\bm b}^{T}\Big[\big(\textbf{I}-CK\big)^{T}\big(\textbf{I}-CK\big)\Big]B_{\bm b}\Big)^{-1/2}$ and $m_{\bm \epsilon}$~represents the state gain matrix, such that $m_{\bm \epsilon}=B^{T}_{\bm b}\alpha^TK$.
Referring to~\cite{joerger2014solution}, for a given fault mode but unknown fault vector, the worst-case failure slope equals the maximum eigenvalue of the corresponding failure slope formulation. Therefore, we express the worst-case failure slope of the Graph-SLAM framework as
\begin{align}
\bar{g}^2_{\bm b} = m_{\bm \epsilon}^{T}M_{\zeta}M_{\zeta}^{T}m_{\bm \epsilon}.
\end{align}
Next, we compute protection level~$\bar{\chi}_{t}$, seen in Eq.~\eqref{eq_algo:pl} as the y-coordinate that corresponds to the integrity metric~$\lambda_{th}$ along the line passing through the origin and with slope given by~$\bar{g}_{\bm b}^{2}$.
\begin{align} \label{eq_algo:pl}
\bar{\chi}_{t} = \sqrt{\lambda_{th}\bar{g}_{\bm b}^{2}}
\end{align}
\begin{figure}[H]
\setlength{\belowcaptionskip}{-4pt}
\centering \includegraphics[width=0.6\columnwidth]{Images/RouteNewNew.png}
\caption{Route taken by a ground vehicle during the experiment conducted for $100~$s. Between $t=70-100~$s, the vehicle experiences GPS faults due to multipath and vision faults due to illumination variations. At $t=78~$s, the overlap of the skyplot of GPS satellites with the fish-eye image shows the multipath affected GPS measurements. }
\label{fig_exp:route}
\end{figure}
\section{Experiment Results}
We validate the performance of the proposed SLAM-based IM algorithm that utilizes both GPS and fish-eye camera. We conduct real-world experiments on a moving ground vehicle in the semi-urban area of Champaign, IL along the route shown in Fig.~\ref{fig_exp:route}. Our experimental setup comprises of a commercial off-the-shelf GPS receiver and a fish-eye camera fitted with $180^{\circ}$ FOV lens. During $t=0-70~$s, the ground vehicle operates in open-sky conditions, thereby experiencing no GPS faults but less visual features. In Fig.~\ref{fig_exp:route}, the blue highlighted region suffers from vision challenges, namely illumination variations due to sunlight and shadows, that causes data association errors across images. Similarly, the red highlighted region is enclosed with tall buildings that leads to multipath effects in the GPS measurements. For instance, at $t=78~$s we showcase the true overlap of the GPS satellite positions over the fish-eye image, where $3$ out of the $7$ visible satellites are affected by multipath.
Fig.~\ref{fig_exp:avg_fault_prob} shows the average fault status of GPS pseudoranges and vision superpixels, as indicated in red and blue, respectively. Given that the ground vehicle navigates in open-sky conditions for $t<70~$s, the average GPS fault status estimated via our multiple FDI module is low, whereas the average vision fault status is high due to the feature-less surroundings. As the vehicle passes through the red highlighted region shown in Fig.~\ref{fig_exp:route} that represents the semi-urban area, the average fault status of vision is low but that of GPS increases due to multipath.
We further analyze the performance of our multiple FDI module in the challenging semi-urban area, i.e., for $t>70~$s during which the ground vehicle experiences GPS faults due to multipath and vision faults due to illumination variations. Fig.~\ref{fig_exp:gps_faults} plots that the individual GPS fault status of $3$ out of the $7$ visible satellites with PRNs $6,\,12$, and $2$. In accordance with the skyplot shown in Fig.~\ref{fig_exp:route}, our proposed SLAM-based IM algorithm successfully flags the satellites with PRN $6$ and $12$ as faulty while accurately estimating the high-elevation satellite with PRN~$2$ as non-faulty. During the same duration, we also analyze the vision fault status associated with the superpixels. In Fig.~\ref{fig_exp:vision_faults}, at each time instant, we plot the top four fault status of the superpixels, such that each marker represents a superpixel. We observe that in urban region, the value of the associated vision fault status decreases due to feature-rich tall buildings in urban areas. However, when the vehicle enters the blue highlighted region seen in Fig.~\ref{fig_exp:vision_faults}, the illumination variations induced by the bright sunlight causes the fault status associated with certain superpixels to shown an increasing trend.
\begin{figure}[H]
\setlength{\belowcaptionskip}{-4pt}
\centering \includegraphics[width=0.6\columnwidth]{Images/my_avg_fault.jpg}
\caption{Performance of our multiple FDI module via average fault status of GPS pseudoranges, indicated in red and vision superpixels, indicated in blue. When the ground vehicle navigates through the semi-urban region, i.e., for $t>70~$s, the average fault status associated with GPS is high due to multipath, whereas vision is low due to rich features.}
\label{fig_exp:avg_fault_prob}
\end{figure}
\begin{figure}[h]
\centering
\begin{subfigure}[b]{0.355\textwidth}
\includegraphics[width=\textwidth]{Images/Newgps_fault.jpg}
\caption{GPS fault status of PRN $6,\,12,\,2$}
\label{fig_exp:gps_faults}
\end{subfigure}
\begin{subfigure}[b]{0.355\textwidth}
\includegraphics[width=\textwidth]{Images/Newvision_fault.jpg}
\caption{Vision fault status of superpixels}
\label{fig_exp:vision_faults}
\end{subfigure}
\caption{Estimated fault status of (a) GPS measurements and (b) vision superpixels during $t=70-100~$s, i.e., when the ground vehicle navigates through the semi-urban area. In (a), our multiple FDI module successfully detects satellites with PRN $6,\,12$ as faulty while accurately estimating the PRN~$2$ as non-faulty. In (b), where each marker indicates a superpixel, the trend of fault status associated with superpixels is low given the rich features but later, increases due to illumination changes. }
\end{figure}
We demonstrate the improved performance of the SLAM-based IM algorithm that utilizes GPS and fish-eye camera seen in Fig.~\ref{fig_exp:PL_multisensor}, as compared to the SLAM-based IM algorithm that utilizes GPS-only seen in Fig.~\ref{fig_exp:PL_GPSonly}. By utilizing GPS and fish-eye camera, we demonstrate higher localization accuracy, with an Root Mean Squared Error~(RMSE) of $8.8~$m and standard deviation of $1.73~$m, as compared to employing GPS-only that shows an RMSE of $16.2~$m and standard deviation of $2.86~$m. We also validate that the lower mean size of protection levels are estimated using GPS and fish-eye camera, i.e. $6.5~$m than using GPS-only, i.e., $10.5~$m thereby, achieving tighter protection levels.
\begin{figure}[h]
\centering
\begin{subfigure}[b]{0.425\textwidth}
\includegraphics[width=\textwidth]{Images/NewNewMultiSensor_PL.jpg}
\caption{SLAM-based IM via GPS and fish-eye camera}
\label{fig_exp:PL_multisensor}
\end{subfigure}
\begin{subfigure}[b]{0.425\textwidth}
\includegraphics[width=\textwidth]{Images/NewNewOnlyGPS_PL.jpg}
\caption{SLAM-based IM via GPS-only}
\label{fig_exp:PL_GPSonly}
\end{subfigure}
\caption{Comparison of SLAM-based IM: (a)~using GPS and fish-eye camera; (b)~using GPS-only. Lower localization errors and tighter protection levels are achieved via GPS and fish-eye camera as compared to GPS-only.}
\end{figure}
\section{Conclusions}
We proposed a Simultaneous Localization and Mapping~(SLAM)-based Integrity Monitoring~(IM) algorithm using GPS and fish-eye camera that estimates the protection levels of the Graph-SLAM framework while accounting for multiple faults in GPS and vision. We developed hybrid sky detection algorithm to distinguish the non-sky and sky pixels, which are later used in graph optimization and GPS measurement covariance, respectively. By utilizing the GPS pseudoranges, non-sky pixel intensities, receiver and satellite motion model, we performed graph optimization via M-estimator-based Levenberg Marquardt algorithm. We simultaneously estimated the state vector of the vehicle, GPS satellites and key image pixels in the world frame. We estimated the fault mode vector by independently evaluating the measurement residuals against an empirical Gaussian distribution for GPS faults and using our developed superpixel-based piecewise RANSAC for vision faults. We computed the protection levels via the worst-case failure slope analysis that estimates the maximum eigenvalue associated with the failure slope formulation of the linearized Graph-SLAM framework.
We conducted real-world experiments using a ground vehicle in a semi-urban region to analyze the performance our proposed SLAM-based IM algorithm that utilizes GPS and fish-eye camera. We successfully detected and isolated multiple measurement faults in GPS and vision. We demonstrated higher localization accuracy using our proposed algorithm with an RMSE of $8.8~$m and standard deviation of $1.73~$m, as compared to GPS-only that shows an RMSE of $16.2~$m and standard deviation of $2.86~$m. We also validated that the mean size of protection levels estimated using GPS and fish-eye camera, i.e. $6.5~$m is lower than using GPS-only, i.e., $10.5~$m.
\bibliographystyle{ieeetran}
|
3,212,635,537,959 | arxiv | \section{Introduction}
The extragalactic background light (EBL) at any epoch is a diffuse isotropic background
radiation, defined here over the wavelength range 0.1 to 1000$\mu$m excluding the cosmic
microwave background radiation (CMBR), believed to be contributed mainly by the sources
such as galaxies and QSOs. The knowledge of how the intensity and shape of the EBL evolves
is very important for understanding the galaxy evolution in the universe.
Direct measurements of the EBL are possible only in the local universe
\citep{Dwek98, Dole06,Matsuoka11}. However, there are large uncertainties associated with
the removal of foreground contributions from the unresolved point sources and zodiacal light
\citep[see][]{Hauser01}. The local EBL can be inferred by adding the light from resolved
sources \citep{Madau00, Xu05, Hopwood10} but the convergence of the number of sources is
in dispute \citep{Bernstein02,Levenson08}. However, with the aid of rapidly developing
$\gamma$-ray astronomy, in principle, it is possible to place strong constraints on the
intermediate redshift (z$<$2) EBL.
The high energy $\gamma$-rays by interacting with the EBL photons can annihilate themselves
and produce electron positron pairs. The byproduct of this interaction, ultra-relativistic
electron positron pairs, are expected to inverse Compton scatter the CMBR and produce
secondary $\gamma$-rays. These secondary $\gamma$-rays are not yet detected by the Fermi
satellite implying either the presence of a small intergalactic magnetic field that scatters
the produced pairs \citep[][]{Neronov10, Tavecchio11, Takahashi12, Arlen12} or the
electromagnetic pair cascade dissipating pair beam energy in to the intergalactic medium
\citep[IGM; see for e.g.,][]{Schlickeiser13, Miniati13}. Nevertheless, this process of
pair production attenuates the $\gamma$-rays originating from distant sources while
traveling through the IGM \citep{Gould66, Jelley66}. The amount of attenuation suffered by
the $\gamma$-rays emitted by sources at different emission redshifts depends on the number
density of the EBL photons encountered while traveling from the source to the earth.
Thus a well measured $\gamma$-ray attenuation ($\tau_{\gamma}$) can be used to put
constraints on the evolution of the shape and amplitude of the EBL.
This was initially suggested by \citet{Stecker92} and the first few limits on the IR part of
the EBL were placed by \citet{Dwek94} and \citet{deJager94} using TeV $\gamma$-ray
observations of the blazar Mrk 421.
With the aid of new generation ground based $\gamma$-ray Imagining Atmospheric Cherenkov
telescopes (IACT) and the Fermi satellite, many high energy $\gamma$-ray sources have been
detected. The observed spectrum of distant $\gamma$-ray sources are used to determine the
$\tau_{\gamma}$. However, the difficulty in doing so arises from the fact that the
intrinsic spectral energy distribution (SED) of each source is unknown.
Recently, \citet{GammaSci} have circumvent this difficulty and reported the measurements of
$\tau_{\gamma}$ up to $z\sim 1.5$ with the observed $\gamma$-ray energies from 10 to 500
GeV using the stacked spectra of $\gamma$-ray blazars selected from the sample of objects
observed with Large Area Telescope (LAT) on board the Fermi satellite. It is expected to
find the $\tau_{\gamma}$ increasing with increasing redshift since these $\gamma$-rays
travel longer distances through EBL photons to reach earth. The observation of this
cosmological evolution in $\tau_{\gamma}$ has been reported recently by \citet{Sanchez13}.
The $\gamma$-ray horizon for $\gamma$-ray photons with the observed energy $E_{\gamma}$
is defined as the emission redshift of $\gamma$-rays beyond which they encounter
$\tau_{\gamma}>1$. Recently, by using a physically motivated modeling of intrinsic SEDs
of 15 blazars, \citet{Dominguez13} reported the $\gamma$-ray horizon measurements.
\citet{Dominguez13a} using such $\gamma$-ray horizon measurements have demonstrated
the capability of $\gamma$-ray astronomy to measure the Hubble constant. Recently,
\citet{Scully14} and \citet{Stecker12} used their EBL model to constrain the redshift of
$\gamma$-ray blazars. In some sense, the backbone of this rapidly developing $\gamma$-ray
astronomy is the EBL. Therefore, it is very important to have an EBL estimates consistent
with different observations over a large redshift range.
To estimate the EBL one needs the specific emissivity (or some times referred as luminosity
density) at each frequency and redshift. The EBL in optical wavelengths is
predominantly contributed by the stellar emission and in the infra-red (IR)
wavelengths by dust emission from galaxies. Therefore,
for the correct estimation of the EBL one has to determine the comoving specific galaxy
emissivity at different frequencies and redshifts, $\rho_{\nu}(z)$, as accurately as
possible. The EBL models are generally classified in to different categories depending on
the method adopted to get the $\rho_{\nu}(z)$. For example, some of the models, say the
first kind of models, start with simulating the galaxy evolution in the framework of
standard cosmological model taking into account the dark matter halo formation and some
prescription to relate baryons to star formation in them. These models then predict
the $\rho_{\nu}(z)$ forward in time \citep{Primack05, Gilmore09, Gilmore12, Inoue13}. There
are second type of models which construct the grid of $\rho_{\nu}(z)$ measurements in
different wavebands and redshift and then apply the interpolation and the extrapolation to
get the $\rho_{\nu}(z)$ at each $\nu$ and $z$
\citep{Stecker06, Franceschini08, Dominguez11, Stecker12, Helgason12}. There are third kind
of models where the cosmic star formation history and the SED of stellar population of
galaxies are convolved to get the $\rho_{\nu}(z)$ \citep{Kneiske04, Finke10, HM2012}.
The $\rho_{\nu}(z)$ obtained in this way depends on star formation history of galaxies over
the cosmic time and absorption and scattering by the dust present in them. Main uncertainties
in the first and third approach are related to the the amount of dust corrections which is
usually quantified by the dust attenuation magnitude $A_{\nu_0}$ at frequency $\nu_0$ and a
wavelength dependent dust extinction curve. It is a general practice to assume a form of
$A_{\nu_0}(z)$ and an extinction curve to get the $\rho_{\nu}(z)$ from the star formation
history. Irrespective of the approach one adopts, all the methods are expected to reproduce
the measured $\rho_{\nu}(z)$ using observed luminosity functions.
Here, in this paper, we address the issue of self consistently determining the dust
correction and the star formation history which will reproduce the observed emissivity.
We present a novel `progressive fitting method' which by using the $\rho_{\nu}(z)$,
for a given extinction curve, determines a unique combination of cosmic star formation
rate density (SFRD) and $A_{\nu_0}(z)$. We apply this method on observationally determined
$\rho_{\nu}(z)$ using the available multi-wavelength multi-epoch galaxy data from the
literature. We determine the combinations of SFRD($z$) and $A_{\nu_0}(z)$ for a set of
five well known extinction curves and compare the results with different independent
measurements of SFRD($z$) and $A_{\nu_0}(z)$ available in the literature. This allows us to
determine the average extinction curve that can be used to convert the emissivity into
the SFRD($z$). We provide the simple fitting forms of these combinations of SFRD($z$)
and $A_{\nu_0}(z)$ for each extinction curve with their $1\sigma$ upper and lower limits.
We self-consistently determine the amount of stellar light absorbed by dust with the help
of these combinations of $A_{\nu_0}(z)$ and SFRD($z$) obtained for different extinction
curves and then estimate the far infra-red (FIR) emission from galaxies using the local
galaxy FIR templates and the energy conservation arguments. In this way we obtain the
specific emissivity from UV to FIR and then use standard prescription to calculate the EBL,
the $\tau_{\gamma}$ and the $\gamma$-ray horizon and compare these results with
different available measurements. We conclude that the combination of $A_{\nu_0}(z)$ and
SFRD($z$) obtained using the extinction curve of Large Magellanic cloud Supershell (LMC2)
and the inferred local FIR emissivity are consistent with the different measurements and
we call the EBL obtained using it as our fiducial model for the EBL. The EBL obtained in
this way, by exploring different well known extinction curves and corresponding combinations
of self-consistent $A_{\nu_0}(z)$ and SFRD($z$), includes better treatment of dust correction
and gives a general picture of how the FIR part of the EBL depends on it.
The outline of this paper is as follows. In section~\ref{sec.ebl} we present the standard
radiative transfer equation used for calculating the EBL from the inferred emissivities.
In section~\ref{sec.qso}, we describe the QSO contribution to the total emissivity used by us.
In section~\ref{sec.egal}, we describe the standard procedure to get the galaxy emissivity
using the SFRD($z$) and $A_{\nu_0}(z)$. In section~\ref{sec.method}, we summarize the galaxy
emissivity measurements from the literature that we use in our study and describe our
`progressive fitting' technique which determines a unique combination of $A_{\nu_0}(z)$
and SFRD($z$) for an assumed extinction curve. In section~\ref{sec.sfr}, we make a detailed
comparison of SFRD($z$) and $A_{\nu_0}(z)$ obtained using our technique for five different
extinction curves with those determined from the independent observations. We explain in
detail the method used by us to calculate the FIR emissivity from galaxies in
section~\ref{sec.fir}. Then we use these inferred galaxy emissivities to calculate the EBL
at different $z$. We present our EBL predictions and compare them with the other EBL
estimates from the literature in section~\ref{sec.eblcal}. In section~\ref{sec.gamma-tau},
we describe the basics of the pair production mechanism used for calculating the
$\tau_{\gamma}$ for our EBL models and compare our results with the other independent
measurements. We conclude with the discussion related to the uncertainties in estimating
the star formation history, $A_{\nu_0}(z)$ and the EBL in section~\ref{sec.res} and summarize
the results in section~\ref{sec.sum}. Throughout the paper we use cosmology with
$\Omega_{\lambda}$=0.7, $\Omega_{m}$=0.3 and $H_{0}$=70 km s$^{-1}$ Mpc$^{-1}$.
\section{Cosmological radiative transfer}\label{sec.ebl}
In this section, we provide a general outline for the basic EBL calculations.
The number density of background photons at a frequency $ \nu_{0}$ and redshift $z_{0}$
is given by,
\begin{equation}\label{Eq.num}
n(\nu_{0},z_{0})= \frac{4\pi J_{\nu_{0}}(z_{0})}{hc} \,\, ,
\end{equation}
where, $h$ is the Planck's constant and $J_{\nu_{0}}$ is the specific intensity of the EBL
(in units of erg cm$^{\text{-2}}$ s$^{\text{-1}}$ Hz$^{\text{-1}}$ sr$^{\text{-1}}$) at a
frequency $\nu_{0}$. Following the standard procedure \citep[see for example,][referred as
HM12 from now onwards]{HM2012}, we assume that the QSOs and galaxies are the sole
contributers to the EBL at all wavelengths. We do not consider contributions to the EBL
from the non-standard sources like decaying dark matter or dark energy. From the observed
luminosity functions of QSOs and galaxies at a redshift $z$ and a frequency $\nu$
one can calculate the proper space averaged specific volume emissivity $\epsilon_{\nu}(z)$
(in units of erg s$^{\text{-1}}$ Hz$^{\text{-1}}$ Mpc$^{\text{-3}}$).
Then, the radiative transfer equation, which gives the specific intensity
$J_{\nu_{0}}(z_0)$ of the EBL as seen by an observer at a redshift $z_{0}$ and a frequency
$\nu_{0}$, can be written as \citep{Peebles, HM96},
\begin{equation}\label{Eq.rad_t}
J_{\nu_{0}}(z_{0})=\frac{1}{4\pi}\int_{z_{0}}^{\infty}dz\,\frac{dl}{dz}\,\frac{(1+z_{0})^{3}}{(1+z)^{3}}\,\epsilon_{\nu}(z)\,e^{-\tau_{eff}(\nu_{0},z_{0},z)}.
\end{equation}
Here, $\frac{dl}{dz}$ is the cosmological Freidmann-Robertson-Walker (FRW) line element,
the $\nu=\nu_{0}(1+z)/(1+z_{0})$ is a frequency of the radiation originated from a
redshift $z$ and $\tau_{eff}(\nu_{0},z_{0},z)$ is the effective IGM optical depth
encountered by the radiation emitted at a frequency $\nu$ while traveling through the IGM
from an emission redshift $z$ to a redshift $z_{0}$ where it has been observed at a
frequency $\nu_{0}$. The hydrogen and helium gas present in the IGM and in galaxies
dominate $\tau_{eff}$ at $\lambda \leq 0.091\mu$m through the photo-absorption. However,
in the optical wavelengths it was believed that the main contribution to the opacity comes
from the attenuation by the dust associated with high H~{\sc i} column density intervening
systems. Based on the available QSO spectroscopic observations one can conclude that this
effect is indeed negligible \citep[see,][]{Srianand97, York06, Frank10, Khare12, Menard12}.
Here, as we are interested in calculating the EBL at $\lambda > $ 0.1$\mu m$,
we will consider $\tau_{eff}$=0 in Eq.~\ref{Eq.rad_t}. This assumption has negligible effect
on the computed EBL and it does not affect the $\tau_\gamma$ significantly over the
$\gamma$-ray energy range of our interest.
\subsection{QSO contribution to emissivity}\label{sec.qso}
The proper specific volume emissivity of the radiating sources can be written as,
\begin{equation}
\epsilon_{\nu}(z)=\epsilon_{\nu , \rm Q}(z)+ \epsilon_{\nu , \rm G}(z) \,\,,
\end{equation}
where, $\epsilon_{\nu , \rm Q}(z)$ and $\epsilon_{\nu , \rm G}(z)$ are the proper specific
volume emissivity of QSOs and galaxies, respectively.
For QSOs, we use the parametric form for $\epsilon_{\nu , \rm Q}(z)$ as given in HM12 at
$912\text{\AA}$ which is consistent with the QSO luminosity function of \citet{Hopkins},
\begin{equation}\label{Eqso}
\frac{\epsilon_{912, \rm Q}(z)}{(1+z)^{3}}=10^{24.6}\,(1+z)^{4.68}\,\frac{exp(-0.28z)}{exp(1.77z) + 26.3}\,\,\,,
\end{equation}
in units of ergs s$^{\text{-1}}$ Mpc$^{\text{-3}}$ Hz$^{\text{-1}}$. To get
$\epsilon_{\nu , \rm Q}$ at different wavelengths, we use a SED given by the broken power law,
$L_\nu\propto \nu^{-0.44}$ for $\lambda>$1300\AA~and $L_\nu\propto \nu^{-1.57}$ for
$\lambda<$1300\AA~\citep{vanden,tefler}. It is well known that the stellar emission
from galaxies dominate the EBL in the optical regime in all redshifts. Therefore, we place
more emphasis on the estimating $\epsilon_{\nu , \rm G}(z)$ accurately. We discuss this in
detail in the following section.
\subsection{Galaxy contribution to emissivity}\label{sec.egal}
\begin{table*}
\caption{Details of the observed galaxy luminosity functions used to get the $\rho_{\nu}$ in our study.}
\begin{center}
\begin{tabular}{l c c c c c c }
\hline
\hline
Reference & Waveband$^*$ & Redshift range & Plotting Symbol$^\dagger$ \\
\hline
\citet{Schiminovich05} & FUV & 0.2-2.95 & magenta triangle \\
\citet{Reddy09} & FUV & 1.9-3.4 & orange triangle \\
\citet{Bouwens07} & FUV & 3.8-5.9 & Red diamond \\
\citet{Bouwens11} & FUV & 6.8-8 & Red diamond \\
\citet{Dahlen07} & FUV & 0.92-2.37 & blue square \\
& NUV & 0.29-2.37 & blue square \\
\citet{Cucciati12} & FUV & 0.05-4.5 & green circle \\
& NUV & 0.05-3.5 & green circle \\
\citet{Tresse07} & FUV, NUV, U, V, B, R, I & 0.05-2 & red circle \\
\citet{Wyder05} & NUV & 0.055 & black triangle \\
\citet{Faber07} & B & 0.2-1.2 & orange star \\
\citet{Dahlen05} & U, B, R & 0.1-2 & blue square \\
& J & 0.1-1 & blue square \\
\citet{Stefanon13} & J, H & 1.5-3.5 & green square \\
\citet{Pozzetti03} & J, K & 0.2-1.3 & orange triangle \\
\citet{Arnouts07} & K & 0.2-2 & green diamond \\
\citet{Cirasuolo07} & K & 0.25-2.25 & blue triangle \\
\hline
\hline
\end{tabular}
\end{center}
\label{lf_data}
\begin{flushleft}
\footnotesize {$^*$Central wavelengths corresponding to different wavebands
are as follows: FUV=0.15$\mu$m, NUV=0.28$\mu$m, U=0.365$\mu$m, B=0.445$\mu$m, V=0.551$\mu$m,
R=0.658$\mu$m, I=0.806$\mu$m, J=1.27$\mu$m, H=1.63$\mu$m and K=2.2$\mu$m.\\
$^\dagger$ These plotting symbols are used in Fig.~\ref{fig.fuv} and Fig.~\ref{fig.A1}
for the $\rho_{\nu}$ obtained using different luminosity functions.} \\
\end{flushleft}
\label{table.main}
\end{table*}
We need to compute the galaxy emissivity, $\epsilon_{\nu,\rm G}(z)$, which is consistent with
the observed luminosity functions of galaxies at different wavelengths and redshifts. The
luminosity function, $\phi_{\nu}(L,z)$, observed at different $z$ and frequency $\nu$ is
usually specified in the form of Schechter function. The comoving luminosity density,
$\rho_{\nu}(z)$, for galaxies which is nothing but the space averaged comoving specific
emissivity,
\begin{equation*}
\rho_{\nu}(z)=\frac{\epsilon_{\nu,\rm G}(z)}{(1+z)^3} \,\, ,
\end{equation*}
is given by an integral,
\begin{equation}\label{rho}
\rho_{\nu}=\int_{L_{min}}^{\infty} {L \phi_{\nu}(L) dL}=\phi_{\nu}^{*} \,\, L^{*} \,\, \Gamma(\alpha+2, \,\,L_{min}/L^{*}).
\end{equation}
Here, $\phi_{\nu}^{*}$, $L^{*}$ and $\alpha$ are the Schechter parameters, $L_{min}$ is the
luminosity corresponding to faintest galaxy at a redshift $z$ and $\Gamma$ is the incomplete
gamma function. We dropped the subscript $z$ in above equation for clarity. The $\rho_{\nu}$
depends on the choice of $L_{min}$. In principle, one can always take $L_{min}=0$ for
$\alpha >-2.0$ where the integral in Eq.~\ref{rho} converges. Generally, for galaxies at
$z<2.5$, one finds $\alpha > -1.3$ \citep{Cucciati12}. In this case, the change in
$\rho_{\nu}$, when one changes the $L_{min}$ from 0 to 0.01$L^{*}$, is less than $10\%$. We
discuss the effect of adopting different the $L_{min}$ values in
section~\ref{sec.res}.
The $\rho_{\nu_{0}}(z)$ measurements are used to determine the global star formation
history \citep[see][]{Madau96, Lilly96} of the universe provided that the magnitude of the
dust attenuation, $A_{\nu_{0}}(z)$, at any frequency $\nu_{0}$ and redshift $z$ is known. The
average star formation rate density (SFRD), in units of M$_{\odot}$ yr$^{-1}$ Mpc$^{-3}$, is
connected to $\rho_{\nu_{0}}$ through the relationship \citep{Kennicutt98},
\begin{equation}\label{Eq.conversion}
\text{SFRD}(z)=\zeta_{\nu_{0}} \times\rho_{\nu_{0}}(z)10^{0.4A_{\nu_{0}}(z)} .
\end{equation}
Here, $\zeta_{\nu_{0}} $ is a constant conversion factor which depends on $\nu_{0}$ and the
initial mass function (IMF) assumed for galaxies. However, note that the relation given in
Eq.~\ref{Eq.conversion} is an approximation as $\rho_{\nu_0}(z)$ can also have contributions
from the old stellar population where the stars that are formed earlier are still shining at
$z$.
The derived SFRD($z$) and the SED produced from the instantaneous burst of star
formation can be used to get the luminosity
density at different $\nu$ and $z$. For an assumed initial mass function (IMF) and
metallicity $Z$, the population synthesis models provide a SED in terms of specific
luminosity, $l_{\nu}(\tau, Z)$, (in units of ergs s$^{-1}$ Hz$^{-1}$ per unit mass of stars
formed) at different age $\tau$ of the stellar population. Since the timescales involved in
the process of star formation (10$^5$ to 10$^7$ years) are relatively small, the SED from the
instantaneous star burst can be directly convolved with the global SFRD($z$) to get the
$\rho_{\nu}(z_0)$ by solving the following convolution integral
\citep[see for eg,][HM12]{Kneiske02},
\begin{equation}\label{Eq.convolution}
\rho_{\nu} (z_0)=C_{\nu}(z_0)\int_{z_0}^{z_{max}}{\rm SFRD}(z)\,\,l_{\nu}[t(z_0)-t(z), Z]\,\,\frac{dt}{dz}\,\,dz\, ,
\end{equation}
where, the $\tau=t(z_0)-t(z)$ is an age of the stellar population at
the redshift $z_{0}$ which went through an instantaneous burst of star formation at a
redshift $z$ , $C_{\nu}(z_0)$ is the dust correction factor at $z_{0}$ and
$\frac{dt}{dz}=[(1+z)H(z)]^{-1}$.
The fact that the burst of star formations occurs at all epochs, $t$, but with the
average star formation rates equals to the global SFRD$(t)$ is captured by the product
of ${\rm SFRD}(z)$ and $l_{\nu}[t(z_0)-t(z), Z]$ in the convolution integral. We use
$z_{max}=\infty$ as often used in the literature \citep[e.g.,][HM12]{Gilmore09, Inoue13}.
Later in section~\ref{sec.res}, we discuss the validity of the $z_{max}=\infty$
assumption and the effect of using different $z_{max}$. The dust correction factor,
$C_{\nu}(z_0)$, for $\lambda <912$\AA~ is assumed to be equal to the escape fraction of
hydrogen ionizing photons from galaxies as given in HM12. For $\lambda >912$\AA, we
use $C_{\nu}(z_0)=10^{-0.4\,A_{\nu}(z_{0})}$, where, $A_{\nu}(z_{0})$ which is normalized at
$\nu_{0}$ as given by,
\begin{equation}\label{Eq.k}
A_{\nu}(z_{0})=A_{\nu_{0}}(z_{0})\frac{k_{\nu}}{k_{\nu_0}}.
\end{equation}
Here, $k_{\nu}$ is a frequency dependent dust extinction curve.
\begin{figure*}
\centering
\includegraphics[bb=100 365 520 710,width=12cm,keepaspectratio,clip=true]{fig1.eps}
\caption{ The $\rho_{_{\rm FUV}}$ as a function of $z$ fitted with a functional form given in
Eq.~\ref{Eq.fit_form}. \emph{Solid, dashed} and \emph{dot-dash}
curves are the median high and low fits, respectively. Data is taken from the references listed in Table~\ref{lf_data}
for the FUV band.}
\label{fig.fit}
\end{figure*}
\section{Method to determine SFRD($z$) and A$_{\nu_0}(z)$}\label{sec.method}
In this sections, we summarize $\rho_{\nu}(z)$ measurements from the literature and the
progressive fitting method which determines a unique combination of SFRD($z$) and
A$_{\nu_0}(z)$ for an assumed extinction curve using $\rho_{\nu} (z)$. By construct,
this combination of SFRD($z$) and A$_{\nu_0}(z)$ reproduces the emissivity measurements.
\subsection{Compiled luminosity density measurements}
Motivated by the previous works of \citet{Stecker12} and \citet{Helgason12a}, we have
compiled available observations of the galaxy luminosity functions and the corresponding
$\rho_{\nu}$ at different rest wavelengths and $z$. In Table~\ref{lf_data}, we have
given references along with the rest waveband and a redshift range over which the
luminosity functions have been determined. In Table~\ref{lmin_table} in the Appendix,
we list the faint end slopes of luminosity functions with the rest waveband and redshift
along with the $L_{min}$ values we used to determine $\rho_{\nu}$. In general, we preferred
the references where luminosity functions are determined in different wavebands the
from the FUV (centered at $\lambda=0.15\mu$m) to K
(2.2$\mu$m) band and with the largest possible coverage
in redshift. This compilation has luminosity functions determined in the FUV band up to $z=8$,
in the NUV and H band up to $z=3.5$ and for all other bands the measurements are available
up to $z\sim2.5$. We take the $\rho_{\nu}(z)$ with the errors from the references where it is
explicitly calculated. We use luminosity functions given in other references and compute
$\rho_{\nu}$($z$) (using Eq.~\ref{rho}) with $L_{min}=0.01L^*$. Since there are more
measurements of the $\rho_{\nu}$ in the FUV band and covering a large $z$ range, we choose
$\nu_{0}=\nu_{\rm FUV}$, the frequency we use to determine SFRD($z$) as a frequency
corresponding to the FUV band.
\subsection{Progressive fitting method}\label{sec.pfmethod}
We use a population synthesis model `{\sc Starburst99}'
\footnote{http://www.stsci.edu/science/starburst99/docs/default.html} \citep{Leitherer99},
to get the specific luminosity from stellar population of a typical galaxy, $l_{\nu}(t, Z)$,
at an age $t$ and a metallicity $Z$ with an instantaneous burst of star formation.
In these simulations, we consider a constant metallicity of $Z=0.008$ over all $z$.
Later in section~\ref{sec.res}, we also discuss the effect of using different values of
metallicity. We use the Salpeter IMF with the exponent of 2.35 and the stellar mass range
from 0.1 to 100 M$_{\odot}$. For this particular galaxy model, we find the conversion factor
for connecting $\rho_{FUV}(z)$ and SFRD($z$) (see Eq.~\ref{Eq.conversion}) to be
$\zeta_{\nu_0}=1.25\times10^{-28}$. As described before, the reference frequency,
$\nu_0$ which we use corresponds to the frequency of the FUV band. Note that, this conversion
factor 1.25$\times10^{-28}$ is 11\% smaller than widely used, 1.4$\times10^{-28}$, quoted
by \citet{Kennicutt98}. This difference is mainly because of the updated population synthesis
model and the assumed metallicity.
We fit a functional form to the compiled $\rho_{\rm FUV}$ data and obtained its parameter
using the {\sc mpfit IDL} routine \footnote{{\sc mpfit} is a robust non-linear least square
fitting IDL program used to fit model parameters for a given data \citep{mpfit}.}
that uses $\chi^2$ minimization. At high redshifts, we take 20\% errors on the
$\rho_{\rm FUV}$ calculated from the luminosity function given by \citet{Bouwens11}.
We convert the asymmetric errors into symmetric errors by taking the average of them.
To fit $\rho_{\rm FUV}(z)$, we use a following functional form that was originally used by
\citet{Cole01} to fit the SFRD($z$),
\begin{equation}\label{Eq.fit_form}
\rho_{\rm FUV}(z)=\frac{a+bz}{1+(z/c)^{d}}.
\end{equation}
There is a large scatter in the $\rho_{\rm FUV}(z)$ data. Therefore, along with this fit
(hereafter, median fit) we construct 1-$\sigma$ upper and lower limit fits (hereafter we
refer to them as the high and low $\rho_{\rm FUV}$ fits, respectively). These $\rho_{\rm FUV}$
fits multiplied by $1.25\times10^{-28}$ are nothing but the different SFRD($z$) with
the $A_{\text{FUV}}(z)=0$ (see Eq.~\ref{Eq.conversion}) which are plotted in
Fig.~\ref{fig.fit}. The values of fitting parameters for the median
$\rho_{\rm FUV}\times1.25\times10^{-28}$ fit are $a=(6\pm1)\times 10^{-2}$,
$b=(11\pm2)\times 10^{-2}$, $c=4.41\pm0.58$ and $d=3.15\pm0.62$.
We construct 1-$\sigma$ high and low $\rho_{\rm FUV}$ fits by adding and subtracting
the error in each parameter from its best fit values, respectively (see, Fig.~\ref{fig.fit}).
We determine the combinations of SFRD($z$) and $A_{\rm FUV}(z)$ for all three (low, median
and high) $\rho_{\rm FUV}$ fits with different extinction curves as discussed below.
\begin{figure*}
\centering
\includegraphics[bb=100 365 520 710,width=12cm,keepaspectratio,clip=true]{fig2.eps}
\caption{ Extinction curves normalized at the FUV band for SMC, LMC, LMC Supershell (LMC2),
Milky-Way (MW) and nearby starburst galaxies by \citet{Calzetti}. Here, \emph{triangles,
squares} and \emph{diamonds} represent the mean extinction curve measurements from
\citet{Gordon03} normalized at the FUV band for the SMC, LMC2 and LMC, respectively.
Different wavebands are marked with the \emph{vertical dashed} lines to show the difference
in different extinction curves at those wavelengths. }
\label{fig.k}
\end{figure*}
The average extinction curve for high redshift galaxies is one of the key unknowns in the
astronomy. However, the mean extinction curves for our galaxy, Small and Large Megallenic
Clouds (SMC and LMC) and some low redshift starburst galaxies are well known
\citep{Lequeux82,Clayton85,Calzetti94}. It is a general practice to use the average
extinction curve determined for the nearby starburst galaxies by \citet{Calzetti}
\footnote{Note that, sometimes it is also called as an attenuation curve or an
obscuration curve \citep{Calzetti01}. However, in this paper, along with other four
extinction curves we call it as an extinction curve for the uniformity.}
for the high redshift galaxies. Here, along with
\citet{Calzetti} extinction curve, we use extinction curves determined for SMC, LMC and
LMC supershell (LMC2) from \citet{Gordon03} and Milky-Way (MW) from \citet{Misselt99}.
In particular, this set of extinction curves encompasses a wide range of dust properties
typically present in the astronomical domain. Since, we are using $\rho_{\rm FUV}$
measurements for determining the SFRD($z$), we normalize all the extinction curves $k_{\nu}$
at $\nu$ corresponding to the FUV band ($0.15\mu$m). In Fig.~\ref{fig.k}, we have plotted
the $k_{\nu}/k_{\text{FUV}}$ for different extinction curves as a function of $\lambda^{-1}$
along with the respective measured data points from \citet{Gordon03} for the SMC, the LMC and
the LMC2. In Fig.~\ref{fig.k}, we also mark the different $\lambda^{-1}$ for the wavebands at
which we have compiled the $\rho_{\nu}$ measurements to determine the $A_{\rm FUV}$ and
the SFRD.
From Eq.~\ref{Eq.conversion} it is clear that the SFRD($z$) and $A_{\text{FUV}}(z)$ are
degenerate quantities and different combinations of them can give the same
$\rho_{\text {FUV}}$. However, the measured $\rho_{\nu}$ values at different frequencies other
than the FUV band along with the assumed extinction curve break this degeneracy. Here we
introduce a novel method that, by using the multi-wavelength and multi-epoch luminosity
functions, determines the $A_{\rm FUV}(z)$ and SFRD($z$) uniquely for an assumed extinction
curve. In this method we initially fix the $A_{\rm FUV}$ and SFRD at some higher redshifts and
then using this we progressively determine $A_{\rm FUV}$ and SFRD at lower redshifts. This
`progressive fitting method' is described below in details.
Combining Eq.~\ref{Eq.conversion}, \ref{Eq.convolution} and \ref{Eq.k}, the
$\rho_{\nu}$($z_0$) can be written as,
\begin{eqnarray}
\lefteqn{\rho_{\nu} (z_0)= 1.25\times10^{-28}\times 10^{\big[-A_{\rm FUV}(z_{0})\frac{k_{\nu}}{k_{\rm FUV}}\big]}}\nonumber \\
& & \times \int_{z_0}^{\infty}\rho_{\rm FUV}(z)\; 10^{0.4A_{\rm FUV}(z)} \, l_{\nu}[t(z_0)-t(z), Z]\,\,\frac{dt}{dz}\,\,dz\,.
\label{Eq.tau}
\end{eqnarray}
For a given extinction curve, $k_{\nu}$, and our $\rho_{\rm FUV}(z)$ fits, the only unknown
in the above equation is $A_{\rm FUV}(z)$ for $z\ge z_0$. Therefore, to get the
$\rho_{\nu}$($z_0$) one needs to know the $A_{\rm FUV}(z)$ for all $z\ge z_0$. The procedure
we followed to get the $A_{\rm FUV}(z)$ for each extinction curve $k_{\nu}$ using the
$\rho_{\nu}(z)$ measurements is given below:
\begin{enumerate}
\item We choose the highest possible redshift $z_{th}$ where we have
$\rho_{\nu} (z_{th})$ measurements in most of the wavebands.
\item For all $z\ge z_{th}$, we assume a functional form for $A_{\rm FUV}(z)$.
\item We fix the normalization of this function and hence the value of
$A_{\rm FUV}(z_{th})$ by matching the predicted $\rho_{\nu}(z_{th})$ with the measured ones
at different wavebands (other than the FUV band) using the least square minimization. This
fixes the $A_{\rm FUV}(z)$ for $z\ge z_{th}$. Then we call $z_{th}$ as $z_1$.
\item We choose the next redshift $z_0<z_1$ which is the next nearby lower redshift
where we have multi-wavelength $\rho_{\nu}(z_0)$ measurements.
\item We assume $A_{\rm FUV}(z)$ is constant and equal to $A_{\rm FUV}(z_0)$ in
between the redshifts $z_0$ and $z_1$. For $z\ge z_1$ we use the $A_{\rm FUV}(z)$ as
determined earlier. Then we calculate the $\rho_{\nu}(z_0)$ for different values of
$A_{\rm FUV}(z_0)$.
\item We compare the resultant $\rho_{\nu}(z_0)$ with the measured one at different
wavebands and determine the best fit $A_{\rm FUV}(z_0)$ by the least square minimization.
This fixes the $A_{\rm FUV}(z)$ for $z\ge z_0$. Then we call this $z_0$ as $z_1$.
\item We repeat the steps 4 to 6 until we reach the lowest $z$ where we have
multi-wavelength $\rho_{\nu}(z)$ measurements. This provides us the best fit values of
$A_{\rm FUV}(z)$ and SFRD($z$) over the whole redshift range for a given extinction
curve.
\end{enumerate}
\begin{figure*}
\centering
\includegraphics[bb=68 360 540 690,width=12.6cm,keepaspectratio,clip=true]{fig3.eps}
\caption{ The $A_{\rm FUV}$ as a function of $z$. \emph{Solid black histogram} is a best
fit $A_{\rm FUV}$ determined by the method described in Section 5.1 for different extinction
curves appropriately labeled in each panel. \emph{Solid blue and green histograms} are the
best fit $A_{\rm FUV}$ determined using low and high $\rho_{\rm FUV}$ fits. The
$A_{\rm FUV}(z)$ is fitted with a functional form given in Eq.~\ref{Eq.fit_form}.
\emph{Solid red, dashed green} and \emph{dot-dash blue} curves are fit to the $A_{\rm FUV}$
obtained using median, high and low $\rho_{\rm FUV}$ fits, respectively.
The fitting parameters are given in Table~\ref{dust_parm}.}
\label{fig.dust_fit}
\end{figure*}
In Fig.~\ref{fig.dust_fit} we show the $A_{\rm FUV}(z)$ obtained (\emph {histograms})
using the progressive fitting method described above for different extinction curves.
We fit a continuous function through the resultant $A_{\rm FUV}(z)$ using a functional form
same as the one we used to fit $\rho_{\rm FUV}(z)$ measurements (given in
Eq.~\ref{Eq.fit_form}). For fitting this functional form we use {\sc mpfit idl} routine
by taking 10\% errors for all. To demonstrate the procedure described here, in
Fig.~\ref{fig.dust_fit}, we also show the resultant $A_{\rm FUV}$($z$) obtained using the
high, low and median $\rho_{\rm FUV}$ fits (\emph{histograms}) along with its fitted
functional form for different extinction curves. Since, we show Fig.~\ref{fig.dust_fit} for
the purpose of demonstrating our `progressive fitting method', for clarity, we do not show
$A_{\rm FUV}(z)$ obtained for Milky-Way extinction curve. Note that this resultant
$A_{\rm FUV}(z)$ will directly give the corresponding SFRD$(z)$ (see Eq.~\ref{Eq.conversion}).
We also fit SFRD$(z)$ using the same functional form (see Eq.~\ref{Eq.fit_form}).
Our aim is to get the combinations of $A_{\rm FUV}$($z$) and SFRD($z$) which will reproduce
the measured $\rho_{\nu}(z)$ obtained using the observed luminosity functions at different
wavebands and different $z$. The $\rho_{\nu}(z)$ measurements are taken from different
references and they have different biases and error estimates. Therefore, to minimize the
uncertainty and determine the $A_{\rm FUV}$ over large $z$ range uniquely, we have to choose
$\rho_{\nu}(z)$ measurements which span many wavebands and large $z$ range and possibly
reported by the same group so that the effect of various biases will be minimum. Fortunately
this requirement is satisfied by the $\rho_{\nu}$ measurements reported in \citet{Tresse07}
where the $\rho_{\nu}$ is measured over seven different wavebands (from FUV to I band)
and at the same redshift bins spanning up to $z=2$. Therefore, to get a robust
$A_{\rm FUV}(z)$ and SFRD($z$) combination we choose the observed $\rho_{\nu}(z)$ given by
\citet{Tresse07} and take $z_{th}=2$. We assume that the form of the $A_{\rm FUV}$($z$) for
$z \ge 2$ goes as $1/(1+z)$ and independent of the extinction curve used. We show later that
this assumed form gives the $A_{\rm FUV}$($z$) consistent with other independent measurements.
This trend of decreasing $A_{\text{FUV}}$ at higher $z$ has been previously observed
\citep[see for e.g][]{Burgarella13, Cucciati12, Bouwens09, Takeuchi05}.
This is consistent with the picture of gradual build up of dust in galaxies with cosmic time
as evident from the fact that galaxies at very high redshifts ($z>5$) are bluer than
the $z\sim2$ to $4$ galaxies \citep{Bouwens09}.
We calculate the $A_{\rm FUV}(z)$ and corresponding SFRD($z$) for all the five extinction
curves used in this paper using the low, high and median $\rho_{\rm FUV}(z)$ fits. As we show
later, we use the $\rho_{\nu}(z)$ obtained using the combinations of $A_{\rm FUV}(z)$ and
SFRD($z$) to estimate the $\rho_{\nu}(z)$ at FIR wavelengths and the EBL at different
redshifts. We denote the obtained combinations of $A_{\rm FUV}(z)$ and SFRD($z$), the
$\rho_{\nu}(z)$ and the EBL using different extinction curves as the `smc', `lmc', `lmc2',
`mw' and `cal' models based on the SMC, LMC, LMC2, Milky-Way and \citet{Calzetti} extinction
curves used, respectively. For most comparisons we use our default models which are obtained
using median fits through $\rho_{\rm FUV}$ points. We use the predictions of the high and the
low fits only when we discuss the spread. For clarity in the subsequent discussions, whenever
we use the `high (low) model' we mean the relevant quantity (like $\rho_{\nu}$, $A_{\rm FUV}$,
SFRD and EBL) obtained with the high (low) $\rho_{\rm FUV}$ fit and the `model' extinction
curve. When we denote only `model' we mean the relevant quantity obtained using the median
$\rho_{\rm FUV}$ fit and that `model' extinction curve.
In the following section we discuss the resultant $\rho_{\nu}(z)$, $A_{\text{FUV}}(z)$ and
SFRD(z) determined using the method described in this section.
\section{Dust attenuation and star formation history}
\label{sec.sfr}
\subsection{Reproducing $\rho_{\nu}(z)$ measurements}
\begin{figure*}
\centering
\includegraphics[bb=10 110 575 780, width=16cm, keepaspectratio,clip=true]{fig4.eps}
\caption{ \small{The average comoving galaxy emissivity, $\rho_{\nu}$, at different $z$
calculated using our best fit combination of the SFRD($z$) and $A_{\rm FUV}(z)$ obtained
for different models (with median $\rho_{\rm FUV}$ fits).The red diamonds are the $\rho_{\nu}$
measurements from \citet{Tresse07}. (See Fig.~\ref{fig.A1} in Appendix for $\rho_{\nu}$($z$)
obtained for the low and high models at different wavebands along with the compiled
luminosity density measurements.)}}
\label{fig.zband}
\end{figure*}
\begin{figure}
\centering
\includegraphics[bb=100 365 520 715,width=9cm,keepaspectratio,clip=true]{fig5.eps}
\caption{ The FUV band comoving luminosity density with $z$. Solid, dashed and dotted lines
represent $\rho_{\nu}$ calculated at the FUV band using the median, high and low `lmc2'
model, respectively. The plotting symbols and corresponding references are mentioned in
Table~\ref{lf_data} for the FUV band. The $\rho_{\rm FUV}$ calculated for different models
have negligible difference with respect to each other. Therefore, as a representative for all
other models, we show $\rho_{\rm FUV}$ only for the `lmc2' model.}
\label{fig.fuv}
\end{figure}
In Fig.~\ref{fig.zband}, we plot the $\rho_{\nu}$ obtained using convolution integral
(Eq.\ref{Eq.convolution}) for the best fit combinations of SFRD and $A_{\rm FUV}$ at different
$z$ along with the measurements of \citet{Tresse07}. Note that, to get the $A_{\rm FUV}$ by
least square minimization we use the $\rho_{\nu}$ measurements of \citet{Tresse07} in all
wavebands except at the FUV band. However, these $\rho_{\rm FUV}$ measurements along with
many other reported in the literature up to $z=8$ (see Table~\ref{lf_data} and
Table~\ref{lmin_table}) goes into fitting the $\rho_{\rm FUV}$ (as shown in
Fig.~\ref{fig.fit}). All our five models show very a good agreement with the measurements
of \citet{Tresse07} in all wavebands (including the FUV band). The difference in the
strength of 2175\AA~absorption feature arises because of using different extinction curves.
In Fig.~\ref{fig.fuv}, along with the compiled measurements, we plot the $\rho_{\nu}$ at the
FUV band obtained by using the combination of the SFRD$(z)$ and the $A_{\text{FUV}}(z)$ for
the low, median and high `lmc2' model. There are negligible differences in the
$\rho_{\rm FUV}$ obtained for different models. Therefore, for clarity, we do not show the
similar $\rho_{\rm FUV}$ plots for other models.
For the high $z$ and all other wavebands, we show our estimated $\rho_{\nu}$ along with the
compiled measurements in Fig.~\ref{fig.A1} in the appendix (see Table~\ref{lf_data} for
the references and plotting symbols). Even though we use measurements of $\rho_{\nu}$ up to
$z\sim2$ and up to wavelength corresponding to the I band to get the $A_{\rm FUV}(z)$, our
estimated $\rho_{\nu}$ matches well with various measurements up to $z\sim 4$ from the NUV
to K band. This implies that our determined combinations of the $A_{\rm FUV}(z)$ and the
SFRD($z$) are valid over a large $z$ range and suggests that our assumption of decreasing
dust attenuation at high $z$ is also valid. However, note that, at the high redshifts (i.e,
$z>4$) there are no measurements of $\rho_{\nu}$ except at the FUV band. In the H band, our
calculated $\rho_{\nu}$($z$) is slightly over-estimated than the measured ones. However, as
there are very few measurements we do not attempt to address this disagreement.
The good matching between the observations and the model predictions suggests that we have a
consistent combination of the $A_{\rm FUV}(z)$ and SFRD($z$) for each extinction curve under
consideration. The evolution of our best fit $A_{\rm FUV}$ and the corresponding SFRD with
$z$ is discussed in the next section.
\subsection{Redshift evolution of $A_{\text{FUV}}$}\label{sec.dust}
Understanding the dust attenuation and its wavelength and redshift dependences are very
important to derive the intrinsic SFRD($z$) accurately. Dust attenuation is measured by using
either of the SED fitting techniques, the Balmer decrement method or by comparing the FUV and
the IR luminosity function measurements. It has also been noticed that at any given
$z$, the derived $A_{\text{FUV}}$ may also depend on the galaxy luminosity and the stellar
mass of the galaxy \citep[see for e.g.,][]{Bouwens12}. Recently it has been shown
that the shape of the extinction curve strongly depends on the distribution of the dust in
the galaxies and the viewing geometries where scattering plays an important role
\citep{Chevallard13}. As our main purpose is to calculate the EBL, we are mainly
interested in the volume averaged star formation rates and emissivity. Therefore, to
calculate the average dust correction as a function of $z$, for simplicity we do
not consider the dependence of $A_{\text{FUV}}$ on galaxy luminosity or stellar mass
and the dependence of $k_{\nu}$ on scattering and viewing geometries. In this section,
we compare the $A_{\text{FUV}}(z)$ obtained for different extinction curves with the
$A_{\text{FUV}}$ measurements in the literature based on other independent approaches.
\begin{figure}
\centering
\includegraphics[bb=140 20 520 770,width=9.5cm, keepaspectratio,clip=true]{fig6.eps}
\caption{ \small{ Our best fit dust attenuation, $A_{\rm FUV}$, in magnitude as a
function of redshift calculated by using
different extinction curves. \emph{Dotted, solid} and
\emph{dashed} lines represent values of the best fit $A_{\rm FUV}$
obtained using the low, median and high models, respectively.
\emph{Green circles} represent the A$_{\rm FUV}$ determined
through the SED fitting by \citet{Cucciati12} using Calzetti extinction
curves. \emph{Red diamonds} and \emph{Blue squares} represents
the A$_{FUV}$ measured through $\rho_{\text{FIR}}$
to $\rho_{\text{FUV}}$ ratio by \citet{Burgarella13} and
\citet{Takeuchi05}, respectively.
\emph{Cyan triangles} are from \citet{Bouwens12}.}
}
\label{fig.dust_shade_all}
\end{figure}
\begin{figure}
\centering
\includegraphics[bb=140 20 520 770,width=9.5cm, keepaspectratio,clip=true]{fig7.eps}
\caption{ \small{ Our best fit SFRD($z$) in units
M$_{\odot}$ yr$^{-1}$ Mpc$^{-3}$ obtained using different extinction curves.
\emph{Dotted}, \emph{solid} and \emph{dashed} lines represent
values of best fit SFRD($z$) obtained using the low, median and
high models, respectively. Here, \emph{squares},
\emph{diamonds} and \emph{circles} represent the SFRD($z$) determined from
the radio, the H-$\alpha$ and the FIR observations, respectively.
References and plotting symbols used here are provided in Table~\ref{sfr_data}.
The SFRD($z$) obtained using the LMC2 extinction curve shows good agreement with the
different dust independent measurements.}}
\label{fig.sfrd_shade_all}
\end{figure}
\begin{table}
\caption{Fitting parameters for the $A_{\rm FUV}$$^*$}
\begin{tabular}{ l l c c c c c c}
\hline
\hline
Extinction curve & $\rho_{FUV}$ fit$^\dagger$ & a & b & c & d & \\
\hline
\hline
SMC & Low & 1.41 & 0.79 & 2.51 & 2.32 \\
& Median & 1.03 & 1.01 & 1.87 & 2.16 \\
& High & 0.85 & 0.86 & 1.77 & 2.16 \\
\hline
LMC2 & Low & 1.91 & 0.85 & 2.40 & 2.23 \\
& Median & 1.42 & 0.93 & 2.08 & 2.20 \\
& High & 1.00 & 1.16 & 1.66 & 2.14 \\
\hline
LMC & Low & 2.24 & 0.79 & 2.51 & 2.21 \\
& Median & 1.81 & 0.82 & 2.12 & 2.06 \\
& High & 1.53 & 0.73 & 1.93 & 2.05 \\
\hline
Milky-Way & Low & 2.76 & 0.44 & 2.98 & 2.14 \\
& Median & 2.39 & 0.48 & 2.44 & 1.97 \\
& High & 2.02 & 0.49 & 2.15 & 1.97 \\
\hline
Calzetti & Low & 2.45 & 0.79 & 2.48 & 2.15 \\
& Median & 1.96 & 1.00 & 1.94 & 2.02 \\
& High & 1.61 & 0.91 & 1.81 & 2.03 \\
\hline
\hline
\end{tabular}
\hfill
\label{dust_parm}
\begin{flushleft}
\footnotesize {*The fitting form is $A_{\rm FUV}(z)=\frac{a+bz}{1+(z/c)^{d}}$.}\\
\footnotesize {$^\dagger$Note that, the models with low (high) $\rho_{\rm FUV}$ fit give higher (lower)
A$_{\rm FUV}$ than the median model as explained in the text.}\\
\end{flushleft}
\end{table}
The fitting parameters for $A_{\text{FUV}}(z)$ for different extinction curves are given in
Table~\ref{dust_parm}. In Fig.~\ref{fig.dust_shade_all}, we plot the range of
$A_{\text{FUV}}(z)$ for different extinction curves along with the measurements of
\citet{Takeuchi05}, \citet{Cucciati12}, \citet{Burgarella13} and \citet{Bouwens12}.
\citet{Takeuchi05} and \citet{Burgarella13} determined the $A_{\text{FUV}}$ using the ratio of
the FUV to FIR band luminosity density. \citet{Cucciati12} have calculated the $A_{\rm FUV}$
using the \citet{Calzetti} extinction curve and used the SED fitting technique. At very high
redshifts \citet{Bouwens12} determined the effective dust extinction using the UV-continuum
slope $\beta$ distribution and the IRX-$\beta$ relationship \citep[see,][]{Meurer99}.
We take the effective extinction calculated for the luminosity function integrated up to -17.7
magnitude from \citet{Bouwens12} (from table 6 of their paper).
The shaded region in Fig.~\ref{fig.dust_shade_all} is obtained by using the low, high and
median $\rho_{\rm FUV}$ fits to determine the $A_{\rm FUV}$. Since the SFRD is directly
related to the $\rho_{\rm FUV}$ and $A_{\rm FUV}$, when we use the low (high) $\rho_{\rm FUV}$
fits, to get the same $\rho_{\nu}$ at different wavebands we need higher (lower) SFRD and
hence higher (lower) $A_{\rm FUV}$. In other words, the low (high) $\rho_{\rm FUV}$ implies
that the galaxies are more red (blue) which suggest that these galaxies should have more
(less) dust extinction. This trend is evident from Fig.~\ref{fig.dust_shade_all}, where the
dotted and dashed curves show the $A_{\rm FUV}$ obtained using the low and the high
$\rho_{\rm FUV}$ fits, respectively.
The shaded region in Fig.~\ref{fig.dust_shade_all} represents the allowed range of
$A_{\rm FUV}$. For each assumed extinction curve, we get a different allowed range
for the $A_{\rm FUV}$ and the difference is prominent at redshifts $z<1$. For redshifts
$1<z<2$, we find that the $A_{\rm FUV}$ values remain constant or show a mild decrease with
increase in $z$. At high redshifts, i.e $z>2$, our assumption of decreasing dust attenuation
plays a role in getting similar allowed range of the $A_{\rm FUV}$ for all assumed extinction
curves. As can be seen from Fig.~\ref{fig.dust_shade_all}, the allowed $A_{\rm FUV}$ range for
$z>2$ nicely follows that of other independent measurements rendering support to our
assumption. Apart from the $A_{\rm FUV}$ determined for the `mw' model, for all the other
models we find a moderate increase in the $A_{\rm FUV}$ with redshift up to $z=1$ from $z=0$.
This trend of increasing FUV band dust attenuation magnitude has been detected
previously \citep[see,][]{Takeuchi05, Cucciati12, Burgarella13} as shown in
Fig.~\ref{fig.dust_shade_all}. However, at $z\le0.8$, our estimated $A_{\text{FUV}}(z)$
for `cal', `mw', and `lmc' models are higher than these measurements. The $A_{\rm FUV}$
determined for `smc' model matches well in all redshifts expect that it under-predicts
$A_{\rm FUV}$ at $1<z<2$. Overall, a good match with these measurements of the $A_{\rm FUV}$
is obtained over the large $z$ range for the `lmc2' model.
From the very good agreement between the $A_{\rm FUV}(z)$ determined for the `lmc2' model and
the measurements of \citet{Burgarella13} and \citet{Takeuchi05}, we conclude that the
average extinction curve which is applicable for galaxies over wide range of redshifts is
most likely to be similar to LMC2 extinction curve.
Recently, \citet{Kriek13} using SED of the galaxies investigated the dust extinction curves
for 32 different spectral classes of galaxies over $0.5 \le z \le2$. They found that the
Milky-Way and Calzetti extinction curves provide poor fits to the UV wavelengths for all SEDs.
They concluded that the SED with the $2175\AA$ UV bump albeit weaker in strength compared to
the Milky-Way is preferred. \citet{Buat12} studied a sample of 751 galaxies with redshift
$0.95 < z < 2.2$ and found that the mean parameters describing the dust attenuation curves
are similar to those found for the LMC2 extinction curve. This is consistent with what we find
here. It is interesting to note that in the case of intervening absorption systems seen in the
QSO spectra, the high percentages of systems with the $2175\AA$ absorption feature detection
favors the LMC2 extinction curve \citep[see for e.g.][]{Srianand08, Noterdaeme09, Jiang13}.
It has also been observed that the extinction curves for individual galaxies depend on
the type and other galaxy properties \citep{Wild11, Chevallard13, Kriek13}. Therefore, single
universal extinction curve for all galaxies may not be realistic. However, our study suggests
that for estimating the volume average properties like the SFRD, $A_{\rm FUV}$ and EBL the
LMC2 extinction curve should be preferred.
\subsection{Redshift evolution of SFRD}\label{sec.sfrd}
We assume that the SFRD is a smooth and continuous function of $z$ and fit it with the
same functional form (using Eq.~\ref{Eq.fit_form}) we are using to fit the $\rho_{\rm FUV}$
and $A_{\rm FUV}$. The fitting parameters for the SFRD$(z)$ are given in Table~\ref{sfrd_parm}.
In Fig.~\ref{fig.sfrd_shade_all}, we plot the SFRD($z$) for all the five extinction curves
with their high and low models. As explained earlier we get the high (low) SFRD for low
(high) $\rho_{\rm FUV}$ fits. Since the SFRD is directly proportional to the $\rho_{\rm FUV}$
and $A_{\rm FUV}$, at $z>3$, where differences in the $A_{\rm FUV}$ for all high and low
models are small, the term $\rho_{\rm FUV}$ dominates and the SFRD($z$) curves cross each
other as demonstrated in Fig.~\ref{fig.sfrd_shade_all}.
\begin{table*}
\caption{SFRD from different observations}
\begin{center}
\begin{tabular}{l c c c c c c }
\hline
\hline
Reference & Technique & Redshift range & Plotting Symbols \\
\hline
\cite{Shim09} & H-$\alpha$ & 0.7-1.9 & cyan diamond \\
\cite{Tadaki11} & H-$\alpha$ & 2.2 & red diamond \\
\cite{Sobral13} & H-$\alpha$ & 0.4-2.3 & green diamond \\
\cite{Ly11} & H-$\alpha$ & 0.8 & orange diamond \\
\cite{Condon02} & 1.4 GHz & 0.02 & cyan squares \\
\citet{Smolcic09} & 1.4 GHz & 0.1-1.3 & red \& orange squares \\
\cite{Rujopokarn10} & FIR & 0-1.3 & blue circles \\
\cite{Burgarella13} & FIR & 0-4 & red circles \\
\hline
\hline
\end{tabular}
\end{center}
\hfill
\label{sfr_data}
\end{table*}
In Fig.~\ref{fig.sfrd_shade_all}, we also plot the SFRD determined through different
observations.There are different indicators of star formation but not all are independent of
the assumed dust correction. Therefore for comparison we use the SFRDs determined through the
radio, H-$\alpha$ and FIR emission from galaxies. We do not consider the SFRD determined with
observations like the UV luminosity where it mainly depends on the assumed values of
$A_{\rm FUV}$ and the extinction curve. We select the measurements where luminosity densities
are converted to the SFRD using conversion laws (see Eq.~\ref{Eq.conversion}) with assumed
Salpeter IMF with stellar mass with range 0.1 to 100 M$_\odot$, similar to the one we use.
Table~\ref{sfr_data} summarizes the references to such a data and indicates the plotting
symbols used for it in Fig.~\ref{fig.sfrd_shade_all}. Our inferred SFRD($z$) using the `cal',
`lmc' and `mw' models are higher than these SFRD measurements with different techniques at
$z<1$ (see, Fig.~\ref{fig.sfrd_shade_all}). The SFRD($z$) calculated for the `smc' model
is consistent with SFRD measurements using radio observations at low $z$ but under-predict
the SFRD at $2>z>1$. We find that the SFRD($z$) determined using the `lmc2' model fits th
e SFRD data well at all $z$. Like in the case of $A_{\rm FUV}$, determination of SFRD$(z)$
based on the LMC2 extinction curve provides the best fit to the different independent
measurements compared to those of other models.
From Fig.~\ref{fig.dust_shade_all} and Fig.~\ref{fig.sfrd_shade_all}, it is clear that,
irrespective of the extinction law used, we find the $z$ at which the SFRD($z$) peaks is
higher than the $z$ beyond which $A_{\rm FUV}$($z$) begins to decline. The similar trend in
peaks of SFRD and $A_{\rm FUV}$ has reported in \citet{Cucciati12} and \citet{Burgarella13}.
\begin{figure*}
\centering
\includegraphics[bb=80 370 545 710,width=12.0cm,keepaspectratio,clip=true]{fig8.eps}
\caption{ Our best fit SFRD($z$) in units M$_{\odot}$
yr$^{-1}$ Mpc$^{-3}$ obtained using the LMC2 extinction curve.
The \emph{gray shaded} region gives the range covered by
the low and high models. We also plot the SFRD($z$) determined
by \citet{Madau14}
using measurements of $\rho_{\rm FUV}$ for the same IMF
we use but with different metallicity.
To compare the general trend, we also plot the SFRD($z$) fit given
by \citet{Behroozi13} for the different observational determination
of the SFRD given in literature for the recent data and the old
data compiled by \citet{Hopkins06}. Both these fits are for
\citet{Chabrier03} IMF.
We scaled the \citet{Hopkins06} fit by 2.5 for clarity. }
\label{fig.sfrd_comp}
\end{figure*}
In Fig.~\ref{fig.sfrd_comp} we plot our SFRD obtained using the LMC2 extinction curve along
with the SFRD determined by \citet{Madau14}. Shaded region in Fig.~\ref{fig.sfrd_comp}
represents SFRD range covered when we use low and high `lmc2' models to determine the SFRD and
$A_{\rm FUV}$. \citet{Madau14} used the $\rho_{\rm FUV}$ measurements from the literature and
converted them in to the SFRD using the conversion constant $\zeta=1.15\times10^{-28}$ which
is 10\% smaller than what we use. For the dust correction they use the $A_{\nu}$ provided by
the different surveys from where the luminosity functions are used to get the
$\rho_{\rm FUV}$. They calculate the $\rho_{\rm FUV}$ by integrating the luminosity function
from $L_{min}=0.03L^*$, while in our case we directly take $\rho_{\rm FUV}$ given in different
references where it is often calculated with $L_{min}=0$ (for the $L_{min}$ values used here,
see Table~\ref{lmin_table} in appendix). \citet{Madau14} use the same IMF used by us but take
different metallicities and consider metallicity evolution with $z$. As compared to our
preferred SFRD($z$) for the LMC2 model the SFRD of \citet{Madau14} shows rapid increase and
decrease in low and high $z$, respectively. However, the difference between both is within
0.1 to 0.2 dex for $z<5$. The peak of SFRD($z$) of our preferred `lmc2' model matches exactly
with that of \citet{Madau14}. The peak of our SFRD($z$) is at $z=1.9^{+0.2}_{-0.3}$ which is
also consistent with the peak of SFRD reported by \citet{Cucciati12}.
\begin{table}
\caption{Fitting parameters for the SFRD($z$)$^*$}
\begin{tabular}{ l l c c c c c c}
\hline
\hline
Extinction & $\rho_{FUV}$ & a & b & c & d & \\
curve & fit$^\dagger$ & (10$^{-2}$) & (10$^{-2}$) & & & \\
\hline
\hline
SMC & Low & 1.55 & 7.14 & 2.53 & 3.10 \\
& Median & 1.38 & 6.24 & 2.65 & 3.01 \\
& High & 1.50 & 5.12 & 3.08 & 3.09 \\
\hline
LMC2 & Low & 2.54 & 10.9 & 2.22 & 3.07 \\
& Median & 2.01 & 8.48 & 2.50 & 3.09 \\
& High & 1.67 & 7.09 & 2.74 & 3.02 \\
\hline
LMC & Low & 3.57 & 13.6 & 2.15 & 3.13 \\
& Median & 3.13 & 9.88 & 2.37 & 3.03 \\
& High & 3.03 & 7.37 & 2.70 & 3.01 \\
\hline
Milky-Way & Low & 6.27 & 15.2 & 2.14 & 3.16 \\
& Median & 5.78 & 11.2 & 2.28 & 3.02 \\
& High & 5.03 & 8.33 & 2.59 & 2.99 \\
\hline
Calzetti & Low & 4.44 & 15.8 & 2.06 & 3.11 \\
& Median & 3.62 & 12.0 & 2.20 & 2.97 \\
& High & 3.23 & 8.78 & 2.54 & 2.97 \\
\hline
\hline
\end{tabular}
\hfill
\label{sfrd_parm}
\begin{flushleft}
\footnotesize {*The fitting form is ${\rm SFRD}(z)=\frac{a+bz}{1+(z/c)^{d}}$}
M$_{\odot}$ yr$^{-1}$ Mpc$^{-3}$ .\\
\footnotesize {$^\dagger$Note that, as explained in the text, the models
with low and high $\rho_{\rm FUV}$ fit
need not to give higher and lower SFRD($z$) than median model for all $z$, respectively.}\\
\end{flushleft}
\end{table}
For comparing the SFRD($z$) shapes, we also plot the fit to the SFRD measurements
compiled from the different observational data reported in the literature given by
\citet{Behroozi13} (see figure 2 and table 4 of their paper). \citet{Behroozi13} provides
the fit for the recent data and the old data used by \citet{Hopkins06}. Both of these fits
are obtained for \citet{Chabrier03} IMF and show rapid increase at low $z$ as compared to
our SFRD($z$). At high $z$, the fit through compiled SFRD data used by \citet{Hopkins06}
shows the slow decrease while \citet{Behroozi13} fit shows rapid decrease as compared to our
SFRD. The peak of the compiled SFRD measurements of \citet{Behroozi13} for the old and new
measurements is at $z=1.7$ which is consistent with our SFRD$(z)$ peak within the allowed
uncertainties. The main differences between our SFRD$(z)$ estimated here and the compiled
SFRD measurements of \citet{Behroozi13} and SFRD($z$) estimated by \citet{Madau14} is that
we self-consistently calculate the $A_{\rm FUV}(z)$ which gives SFRD$(z)$ consistent with
$\rho_{\nu}$ measurements at different wavebands and redshifts.
Having obtained the best fit $A_{\rm FUV}$($z$) and SFRD($z$), we use the stellar population
synthesis models to calculate the emissivity from UV to NIR regime. However, to generate the
complete EBL, in addition to this we need the IR emissivity. We predict the IR emissivity
using our best fit $A_{\rm FUV}$($z$) and SFRD($z$) which is explained in the following
section.
\section{Galaxy emissivity in infrared}\label{sec.fir}
The old stellar population, the interstellar gas and the dust are main sources which
contribute to the IR emission form galaxies. The IR emission from old stars peaks around 1 to
3$\mu$m and a very few per cent of the total IR output of a galaxy is emitted by atoms and
molecules that constitute the interstellar gas. The main source of the IR emission at
$\lambda > 3\mu$m is a thermal emission from the dust grains heated by the local stellar
light in the UV and optical wavelength range. The amplitude and shape of this emission from
the NIR to FIR wavelengths depend on the temperature, size distribution and the composition
of the dust grains \citep[e.g.][]{Dale12, Magdis12, Magdis13}. However, like the extinction
curves, these quantities are unknown for distant galaxies. Therefore, instead of assuming the
dust properties to model the NIR to FIR emission of a typical galaxy, we use the observed
IR templates and make use of the $A_{\rm FUV}$ for different models determined here.
We use the average IR templates of \citet{Rieke09} from 5$\mu$m to $30$cm obtained for
infrared galaxies having different total infrared luminosity, $L_{\rm TIR}$. They have
assembled the SEDs of 11 local luminous and ultra-luminous infrared galaxies and for
generating the templates at lower luminosity they have combined the templates of
\citet{Dale07} and \citet{Smith07}. The shape of each template is moderately different for
different $L_{\rm TIR}$. Therefore, we have to choose an appropriate template for calculating
the IR emission. Since at any $z$, most of the total luminosity is contributed by the galaxies
with luminosity $L^*(z)$, we choose templates of \citet{Rieke09} obtained for
$L_{\rm TIR}=L_{\rm TIR}^*(z)$. We use $L_{\rm TIR}^*(z)$ values for different redshift using
the total IR luminosity function given in \citet{Gruppioni13} up to $z=4$. To get the
$L_{\rm TIR}^*(z)$ values for the high redshifts (in units $L_{\odot}$) we fit a second
degree polynomial through $log(L_{\rm TIR}^*)$ using {\sc mpfit IDL} routine. The best fit
second degree polynomial is $log[L_{\rm TIR}^*(z)]=10.0+1.18z-0.18z^2$. To compute the FIR
spectrum we interpolate the templates of \citet{Rieke09} for corresponding values of
$L_{\rm TIR}^*(z)$ given by the above polynomial fit. For all redshifts $z>7$ we use
the interpolated template of \citet{Rieke09} with $L_{\rm TIR}^*=10^9 L_{\odot}$. However
this lower limit has no effect on the shape of IR template since it is just a template for
lowest luminosity ($10^{9.75} L_{\odot}$) given by \citet{Rieke09} scaled to give
the $L_{\rm TIR}=10^9 L_{\odot}$. \footnote{Note that the range of
wavelengths used to define $L_{\rm TIR}$ is different in the case of \citet{Rieke09} (5 to
1000 $\mu$m) and \citet{Gruppioni13} (8 to 1000$\mu$m). We take this into account and scale
the \citet{Rieke09} templates with the definition \citet{Gruppioni13} which we use for IR
emission from galaxies.}
We use the energy conservation to calculate the IR emission. We assume that the average
energy absorbed by the dust from FUV to NIR regime per Mpc$^{3}$ per $s$ at any redshift
$z_0$, $E_{abs}(z_0)$, is emitted in the NIR to FIR regime as a thermal emission with the
assumed SED taken from the appropriate IR template at $z_0$ as explained above. The
$E_{abs}(z_0)$ is given by,
\begin{equation}\label{Eq.Eabs}
E_{abs}(z_0)=\int_{\nu_i}^{\nu_f}d\nu\Big[1-C_{\nu}(z_0)\int_{z_0}^{\infty}{\rm SFRD}(z)\,\,l_{\nu}(t_{0}(z), Z)\,\,dz\,\Big],
\end{equation}
where, $C_{\nu}(z_0)=10^{-0.4\,A_{\nu}(z_{0})}$ and $\nu_i$ and $\nu_f$ are the frequencies
corresponding to 0.092$\mu$m and 10$\mu$m. Photons in this wavelength range heat the
interstellar dust effectively. Hard photons at $\lambda < 0.092\mu$m are mainly photo-absorbed
by the interstellar hydrogen and helium. We assume that $E_{abs}(z_0)$ is emitted at the same
time in IR from 5 to 1000$\mu$m. We scale the IR template with total IR luminosity
$L_{\rm TIR}^{*}(z_0)$ to match the value of $E_{abs}(z_0)$ in between wavelength from
5 to 1000$\mu$m. Then we do a power law extrapolation to this scaled IR template at
$\lambda < 5\mu$m. We use a second degree polynomial to smoothly connect the NIR part
of the SED with the IR part of the extrapolated template at the connecting points.
\footnote{ The connecting points are usually in between 2 to 4$\mu$m.} Note that, here we
assume the efficiency of dust to re-emit in the NIR to FIR wavelengths is 100\%.
\begin{figure*}
\centering
\includegraphics[bb=105 360 490 710,width=12 cm,keepaspectratio,clip=true]{fig9.eps}
\caption{ The galaxy emissivity at $z=0$ for different models. Apart from the
$2175$\AA~absorption feature, in the IR ($\lambda>1\mu$m) the difference between
different models can be easily seen. The data in the optical and NIR
($0.35 \le \lambda < 3 \mu$m) is taken from \citet{Montero09} and \citet{Jones06}.
Emissivity measurements in the NIR to FIR wavelengths ($8 \le \lambda \le 850 \mu$m)
are taken from \citet{Huang07}, \citet{Soifer91}, \citet{Takeuchi06} and \citet{Serjeant05} }
\label{fig.emis}
\end{figure*}
In Fig.~\ref{fig.emis}, we show the emissivity of galaxies from the UV to FIR
at $z=0$ for the different extinction curves assumed. As expected, apart from the
$2175 \rm \AA$ absorption feature the emissivity up to $\lambda=0.8\mu$m is quite same for all
models. However, from the NIR to FIR wavelengths the emissivity is different for different
models. This is because the absorbed energy by the interstellar dust depends on the extinction
curve and the $A_{\rm FUV}$. In Fig.~\ref{fig.emis}, we also show the local emissivity from
different surveys and at different wavelengths like in the optical from SDSS by
\citet{Montero09}, in the NIR wavelength from the 2MASS and 6dF by \citet{Jones06}, in the IR
to FIR from surveys like ISO FIRBACK, IRAS and SCUBA by \citet{Soifer91}, \citet{Takeuchi06}
and \citet{Serjeant05} and at 8$\mu$m from Spitzer space telescope survey by \citet{Huang07}.
Our predicted local emissivity from the NIR to FIR ($\lambda > 2\mu$m) range in case of the
`mw' and `cal' models give higher intensity by factor $\sim 2$ than the local emissivity
measurements from these different surveys. Our local emissivity for the `lmc' and `smc' models
are marginally higher and lower from these measurements, respectively. Our `lmc2' model gives
a local emissivity which is in very good agreement with these measurements.
In principle, the dust can be assumed to have lower efficiency to re-emit. In that case, the
models which give higher IR emissivity can be scaled down to reproduce the measurements. But
there is no room for scaling up the IR emission form the models which give lower IR
emissivity. Therefore, the `smc' model can not be scaled up to match the local emissivity
measurements.
We use the emissivity of galaxies from UV to FIR range obtained here to calculate the EBL
which is discussed in detail in the following section.
\section{EBL Calculation}\label{sec.eblcal}
We solve the cosmological radiative transfer equation (Eq.~\ref{Eq.rad_t}) numerically to
compute the EBL. In this equation the source term, $\epsilon_{\nu}$, is the sum of the QSO
and galaxy emissivities. We take the QSO emissivity, $\epsilon_{\nu , \rm Q}(z)$, and the
SED as given in Section~\ref{sec.ebl}. We use the galaxy emissivity from the combinations of
the SFRD($z$) and A$_{\rm FUV}(z)$ for different dust extinction curves and corresponding
dust emission as described in the previous sections. In the following sections, we compare our
estimated EBL from UV to FIR regime with the direct measurements of the local EBL and other
estimates of the EBL reported in the literature.
\subsection{The local EBL}\label{sec.firebl}
First, we compute the EBL at $z=0$ for our five different models which are plotted in
Fig.~\ref{fig.ebl}. Apart from the wavelengths near the 2175\AA~absorption, different EBLs
are quite indistinguishable from each other for $\lambda < 0.8\mu$m. As expected, we see a
clear differences appearing at higher wavelengths between different models. Apart from the
extinction curves, the difference is also because of the differences in $A_{\rm FUV}$ values
for the different models (see Fig.~\ref{fig.dust_shade_all}). Therefore, even though the
estimated EBL in the UV and optical parts are similar (because of the way we determine SFRD
and $A_{\rm FUV}$), there are clear differences in the IR wavelengths where the dust
re-emission is important.
\begin{figure*}
\centering
\includegraphics[bb=95 360 550 711,width=12 cm,keepaspectratio,clip=true]{fig10.eps}
\caption{ The EBL at $z=0$ for different models.
The shaded region represents the range of the allowed EBL intensity from observations.
The lower limits are determined by the intensity of the IGL (\emph{squares}).
While the upper limits are determined by the direct measurements of the EBL (\emph {circles}).
The data used here is taken from \citet{Dwek13} and references their in. The IR
predictions of the `smc'
model are inconsistent with the observations. The \emph{stars}
are the sum of the IGL background
compiled by \citet{Franceschini08} and the estimated intra-halo light
from the measured fluctuations of EBL by
\citet{Zemcov14}.}
\label{fig.ebl}
\end{figure*}
The shaded area in Fig.~\ref{fig.ebl} represents the range of the allowed EBL intensity
determined by the local EBL observations. The data used in Fig.~\ref{fig.ebl} is taken from
compilation of \citet{Dwek13} (see their Fig. 7). The lower limits are determined by the
intensity of integrated galaxy light (IGL). The IGL is obtained by adding the light emitted
by resolved galaxies in deep surveys. In principle the IGL should converge to the total EBL
at $z=0$. However, because of the problem in the convergence of number counts and sensitivity
of the surveys to resolve fainter galaxies, the IGL gives a lower limit to the EBL. Here, we
use the IGL measurements in the UV from Galex (Galaxy Evolution Explorer), in the optical
from HST and in the IR from Spitzer, ISO and Herschel \citep{Totani01, Xu05, Levenson07,
Fazio04, Bethermin10, Berta10}. The upper limits on the local EBL come from the direct
measurements of the EBL. Important uncertainty in the direct measurements is the removal of
strong foreground zodiacal light caused by the interplanetary dust and the stellar emission
from the Milky-way. Therefore, these measurements provide a strict upper limits on the local
EBL. In Fig.~\ref{fig.ebl}, we take the absolute measurements of EBL in the optical from
Pioneer 10/11 and in the IR from COBE
\citep{Fixsen98, Dwek98, Finkbeiner00, Dole06, Levenson07, Matsuoka11, Matsuura11}.
Our EBL estimate for the `smc' model goes below the lower limits at $\lambda>1\mu$m. While
for all other models, the estimated EBL is within the allowed range. If we strictly follow
the shaded region, we can rule out the `smc' model and conclude that the average extinction
curve for galaxies is inconsistent with the SMC type of dust extinction. However, the allowed
range is too large to distinguish between other models. All our EBL models in the UV, optical
and NIR ($\lambda <2 \mu$m) follow the lower limits of the EBL where we use the emissivity
consistent with the observed multi-wavelength galaxy luminosity functions. The `lmc2' model,
which also provides the $A_{\rm FUV}(z)$ and SFRD$(z)$ consistent with different independent
measurements, produces the IR background ($1<\lambda<100 \mu$m) consistent with the lower
limits while other models produce slightly higher background intensity but well within the
allowed range and closer to the lower limits. In the FIR regime ($\lambda >100 \mu$m), the
estimated background for the `lmc2' and `lmc' model goes through the observed points. The
EBL for the `mw' and `cal' models are just consistent or slightly higher than the observed
upper limits at $\lambda >100 \mu$m. In summary, the available local EBL measurements in
the NIR to FIR regime doest not support the average dust extinction similar to the one
observed in case of SMC. However, these measurements can not discriminate between the EBL
obtained with other extinction curves.
Recently, using the rocket-borne instrument Cosmic Infrared Background Experiment
(CIBER), \citet{Zemcov14} have measured the fluctuation amplitude of IR background at
$1.1\mu$m and $3.6\mu$m. One of the plausible explanations for the large fluctuation found
at these wavelengths is that it arises from the intra-halo light (IHL) produced by the
tidally striped old stars in the halo of the galaxy \citep{Cooray04, Thacker14}. Using these
fluctuations measured over the large scales \citet{Zemcov14} gives the model dependent
total EBL contributed by IHL. In Fig.~\ref{fig.ebl}, we show their computed values of the
total EBL which is sum of their estimated IHL and the compiled measurements of IGL by
\citet{Franceschini08}. As expected, since we do not include the additional contributions
like the IHL in our emissivities, we find that our estimated EBL for all models is lower
than the predicted by \citet{Zemcov14}. Except for the `smc' model, the EBL obtained for all
other models at $\lambda \le 2.4\mu$m are within 2-$\sigma$ lower than the total EBL
predicted by \citet{Zemcov14}. We find that only for the `smc' model it is more than
2.5-$\sigma$ lower at all wavelengths. At $\lambda \ge 3.6\mu$m our `lmc', `mw' and
`cal' models match with the EBL predicted by \citet{Zemcov14} within 1-$\sigma$.
In the light of these recent developments, it will be interesting to consider the IHL
contribution to IR, the signal arising from the epoch of re-ionization
\citep{Cooray04, Kashlinsky04} and the FIR light from dusty galaxies
\citep{Amblard10, Thacker13, Viero13} which we will attempt in the near future.
From the very good agreement with the local emissivity (see Fig.~\ref{fig.emis}) and with
different independent measurements of the $A_{\rm FUV}(z)$ and SFRD$(z)$ (see
Fig.~\ref{fig.dust_shade_all} and \ref{fig.sfrd_shade_all}), for the EBL calculations we
prefer our `lmc2' model over other models.
\subsection{High $z$ EBL}\label{sec.firebl}
In Fig.~\ref{fig.eblall}, we plot the EBL at redshifts 0, 0.5, 1 and 1.5 for our preferred
`lmc2' model. We also show the range covered by the high and low `lmc2' model by a gray
shaded region and the range covered by all five median models by a vertical striped region.
The fact that we made sure the SFRD($z$) and A$_{\rm FUV}(z)$ determined for different
extinction curves should give the same observed emissivity has resulted in the narrow spread
in the striped region for $\lambda <3\mu$m, especially at high redshifts. For comparison, in
Fig.~\ref{fig.eblall}, we also show the previous estimates of the EBL reported in the
literature \citep[][HM12]{Inoue13, Gilmore12, Finke10, Kneiske10, Franceschini08,
Dominguez11, Helgason12, Scully14}. Below we compare our EBL predictions with the previous
EBL estimates which use observational data like galaxy number counts and galaxy luminosity
functions to get the EBL directly.
\begin{figure*}
\centering
\includegraphics[bb=90 390 560 710, width=17.5cm, keepaspectratio, clip=true]{fig11.eps}
\caption{ The predicted EBL at different redshifts for our preferred
`lmc2' model. The \emph{gray shaded} region shows
the EBL covered by the high and low `lmc2' model. The \emph{vertical striped}
region gives the range covered
by all other median models with different extinction curves.
For comparison we plot other estimates of the EBL from literature.
In the top left panel the \emph{yellow diamonds} are the lower
limit data compiled by \citet{Kneiske10}.
For clarity the legends are distributed over the entire plot.
Note that the EBLs presented here are in proper units. }
\label{fig.eblall}
\end{figure*}
\citet{Helgason12} and \citep{Stecker12} reconstructed the EBL using the multi-wavelength and
multi-epoch luminosity functions. We also use a similar compilation of luminosity functions
but up to the $K$ band. Therefore, our EBL matches very well with the EBL predictions of
\citet{Helgason12} up to $\lambda \sim 3 \mu$m, as shown in Fig.~\ref{fig.eblall}. For
$\lambda > 3 \mu$m, \citet{Helgason12} predicts the higher EBL intensity than us.
The EBL model of \citep{Scully14} extend the model of \citep{Stecker12} upto 5$\mu$m and
provide 1-$\sigma$ upper and lower limits on the EBL as well as $\tau_{\gamma}$. Our
EBL predictions are consistent with their lower and upper limit EBL for $\lambda >1\mu$m.
The EBL estimated by \citet{Dominguez11} is based on the observed K-band luminosity function
with the galaxy SEDs based on the multi-wavelength observations from the SWIRE library.
For $z<1$ and $\lambda \sim 3 \mu$m, our EBL is consistent with the predictions of
\citet{Dominguez11} which shows slightly lower EBL intensities in the UV and slightly higher
EBL intensities in the NIR. At high $z$ this difference is more prominent.
\citet{Franceschini08} used different multi-wavelength survey data which includes
luminosity functions, number counts and the redshift distribution of different galaxy types
and the relevant data is fitted and interpolated to get the EBL. Our estimated EBL matches
well with the EBL of \citet{Franceschini08} up to NIR wavelengths. At $z>1$, in the UV
regime their EBL gives factor $\sim 1.5$ lower intensity than our EBL. However in the FIR
wavelengths, our EBL gives around factor $\sim ~2$ smaller EBL intensity as compared to the
EBL of \citet{Franceschini08} and \citet{Dominguez11}.
In summary, as expected our EBL at $\lambda <3 \mu$m is consistent with the models which use
direct observations to get the EBL but gives a lower EBL in the NIR to FIR regime.
Below we mention some of the general trends in different EBL estimates that can be seen
from Fig.~\ref{fig.eblall}.
Given that there are more observations of the EBL as well as the luminosity functions of
galaxies in the local Universe, almost all the different independent models including our
`lmc2' model converge very well, spans narrower range at $z=0$ in the UV to NIR regime and
pass through observed lower limits of EBL. However, at the NIR and FIR wavelengths the spread
between different estimates is relatively higher. All the EBL estimates differ from each
other at high $z$ and the difference is as high as factor $\sim$4. Our local EBL passes very well
through the lower limit EBL data compiled by \citet{Kneiske10}. Most of the EBL models for
$0.4 <\lambda < 2 \mu$m give similar intensity upto $z<1.5$.
Our local EBL is very much similar to the estimates of HM12. However, at higher $z$, the UV
background ($\lambda <0.4 \mu$m) intensity of HM12 is higher than our EBL. This will have
implications on the values of escape fraction for H~{\sc i} ionizing photons which indirectly
plays an important role in interpreting He~{\sc ii} Lyman-$\alpha$ effective optical depth
measurements near the epoch of Helium reionization \citep[see,][]{KS13}.
Having obtained the EBL at different $z$ from UV to FIR, we calculate its effect on the
transmission of high energy $\gamma$-rays through the IGM and compare it with the different
observations in the following section.
\section{Gamma ray attenuation}\label{sec.gamma-tau}
Two photons with sufficient energy upon collision can annihilate into an electron positron
pair. The condition on energies of photons ($E_1$ and $E_2$) for this process of pair
production is given by
\begin{equation}
\sqrt{2E_1 E_2 (1-\cos\theta)} \geq 2 m_e c^2,
\end{equation}
where, $\theta$ is the collision angle, $m_e$ is the mass of the electron and $c$ is the
speed of light. Thus $\gamma$-rays with an energy $E_\gamma$ can annihilate themselves with
the background extra-galactic photons having energy greater than a threshold energy $E_{th}$,
\begin{equation}
E_{th}=\frac{2m_e^2c^4}{E_\gamma (1-\cos\theta)}.
\end{equation}
The cross-section for this process is,
\begin{eqnarray}
\lefteqn{\sigma(E_1,E_2,\theta) = \frac{3\sigma_T}{16}(1-\beta^2)} \nonumber \\
& & \times \left[ 2\beta(\beta^2-2)+ (3-\beta^4)\ln \left( \frac{1+\beta}{1-\beta}\right)\right],
\label{Eq.cross}
\end{eqnarray}
where,
\[\beta = \sqrt{1-\frac{2m_e^2c^4}{E_1 E_2 (1-\cos\theta)}}\,\,,\]
and $\sigma_T$ is the Thompson scattering cross-section. The pair production cross-section
given in Eq.~\ref{Eq.cross} has a maximum value $\sigma(E_1,E_2,\theta)_{max}=0.25\,\sigma_T$
and the corresponding value of $\beta=0.7$.
If the number density of background photons at redshift $z$ and energy
$E_{bg}$ is $n(E_{bg},z)$ (from Eq.~\ref{Eq.num}), then as a result of pair production the
optical depth encountered by the $\gamma$-ray photons emitted at redshift $z_0$ and observed
at energy $E_\gamma$ on the Earth (i.e. at $z=0$) is given by
\begin{eqnarray}
\lefteqn{\tau_{\gamma}(E_\gamma,z_0) = \frac{1}{2}\int^{z_0}_0 dz\;\frac{dl}{dz}\int^1_{-1}d(\cos\theta) \; (1-\cos\theta)} \nonumber \\
& & \times \int^{\infty}_{E_{min}} dE_{bg}\; n(E_{bg},z)\;\sigma(E_\gamma (1+z),E_{bg},\theta).
\label{Eq.tau}
\end{eqnarray}
Here,
\begin{equation}
E_{min}=E_{th}\:(1+z)^{-1}=\frac{2m_e^2c^4}{E_\gamma (1+z)(1-\cos\theta)}\,\, .
\end{equation}
Above equation in terms of the maximum wavelength of the EBL, which is going to attenuate
observed $\gamma$-rays of energy $E_{\gamma}$, can be simplified as
$\lambda_{max}(z)=23.74 \text{\AA} E_{\gamma} (1+z)(1-\cos\theta)$ where
$E_{\gamma}$ is in GeV. The cross-section for the pair production will be maximum at
$\lambda(z)=12.06\text{\AA} E_{\gamma} (1+z) (1-\cos\theta)$.
The specific number density of the EBL photons is directly related to the optical depth
$\tau_{\gamma}$ encountered by $\gamma$-rays while traveling through the IGM as explained
above. For the EBL estimated here, we calculate $\tau_{\gamma}$ using Eq.~\ref{Eq.tau}
over a energy range from GeV to TeV. In the following sub-section, we compare our calculated
of $\tau_{\gamma}$ with those obtained using different EBL estimates reported in the
literature.
\begin{figure*}
\centering
\includegraphics[bb=52 375 650 710 width=15cm, clip=true]{fig12.eps}
\caption{ \emph{Top panel}: The $\gamma$-ray optical depth, $\tau_{\gamma}$, for different
emission redshifts for observed $\gamma$-ray energy using our preferred `lmc2' EBL
(\emph{red curve}) along with the $\tau_{\gamma}$ from other EBL estimates from the
literature. For clarity the legends are distributed over the entire plot. Dark gray shaded
region gives the range covered in $\tau_{\gamma}$ by the high and low `lmc2' EBL models. The
light gray shaded region gives the range covered by all five of our EBL models.
\emph{Bottom panel}: The ratio of $\tau_{\gamma}$ predictions of different models to the
$\tau_{\gamma}$ obtained for our `lmc2' model.}
\label{fig.tau_z}
\end{figure*}
\subsection{The $\gamma$-ray opacity at different $z$ }\label{sec.taucomp}
The optical depth encountered by $\gamma$-rays while traveling from the emission redshifts
$z_0$ to earth (i.e, $z=0$) and observed at $\gamma$-ray energies from GeV to TeV range is
plotted in Fig.~\ref{fig.tau_z} for different emission redshifts. We plot $\tau_{\gamma}$
calculated for our median `lmc2' model along with the $\tau_{\gamma}$ for its low and high
counterparts in a \emph{dark gray} shaded region. We also show the range of $\tau_{\gamma}$
covered by all five median models by a \emph{light gray} shaded region. The extent of the
\emph{light gray} shaded region for different $z$ up to the $\gamma$-ray energy of 0.6 TeV
shows that the $\tau_{\gamma}$ is indistinguishable for all five EBL models. This is because
the maximum effective wavelength of EBL photons which interact with $\gamma$-rays of energy
less than 0.6 TeV is less than $3\,\,\mu$m where, by construct, our different EBL models
predict similar intensities (see Fig.~\ref{fig.eblall}). The spread of shaded region for
$\gamma$-ray energy $>$ 0.6 TeV points to the fact that our EBL intensity at FIR wavelengths
is different for different models.
For comparison, in Fig.~\ref{fig.tau_z}, we also plot the $\tau_{\gamma}$ obtained by other
estimates of EBL given in the literature. To clearly show the differences between
$\tau_{\gamma}$ calculated for other EBL estimates and for our `lmc2' EBL model, we plot the
ratio of the former to the later in the bottom panel of Fig.~\ref{fig.tau_z}. Differences in
the $\tau_{\gamma}$ obtained for various EBL estimates can be directly understood from the
differences in the EBLs as shown in Fig.~\ref{fig.eblall}. Here, we mention some of the
general trends seen in the various $\tau_{\gamma}$ estimates and then compare our
$\tau_{\gamma}$ with the models which used direct observations of galaxy properties to
construct the EBL.
The difference between the $\tau_{\gamma}$ at energies from GeV to TeV for various EBL
estimates increases with $z$. The difference is more in the TeV energy range where the EBL
photons which effectively attenuate the $\gamma$-rays are from the FIR part of the EBL.
The large scatter in different EBL estimates in the FIR wavelengths (see,
Fig.~\ref{fig.eblall}) is responsible for that. However, for the observationally relevant
$\gamma$-rays which are the ones with $0.1<\tau_{\gamma}<2$ where in the corresponding energy
range the differences between $\tau_{\gamma}$ estimates are quite small. The difference in
$\tau_{\gamma}$ for $\gamma$-ray energies from 0.1 to 3 TeV is small and within 30\%
for various estimates for emission redshifts $z<1.5$. In general, our EBL gives lower
$\tau_{\gamma}$ compared to most of the other estimates. The $\tau_{\gamma}$ obtained for
lower limit EBL of \citet{Scully14} at $z=0.01$ is factor $\sim 2$ higher at
$E_{\gamma}<0.2$TeV. The $\tau_{\gamma}$ obtained for the EBL by \citet{Inoue13} at $z\ge 1$
is within the 10\% of our estimate at $0.2<E_{\gamma}<40$TeV. However, note that the
$\tau_{\gamma}$ at $E_{\gamma}>10$ TeV in our model is quite small because we do not consider
the CMBR in our EBL model \citep[see][]{Stecker06}. The $\tau_{\gamma}$ in the energy range
$0.05<E_{\gamma}<2$~TeV estimated by \citet{Helgason12} upto $25\mu$m is within 15\% of our
$\tau_{\gamma}$. The $\tau_{\gamma}$ calculated using the EBL generated by
\citet{Franceschini08} and \citet{Dominguez11} for $0.1<E_{\gamma}<4$~TeV is within the
30\% of our $\tau_{\gamma}$. However, at high energies, $\tau_{\gamma}$ differs more than
this which is evident from the differences in the FIR part of the EBL in between their
models and our `lmc2' model (see, Fig.~\ref{fig.eblall}).
\begin{figure*}
\centering
\includegraphics[bb=93 360 500 711,width=11 cm,keepaspectratio,clip=true]{fig13.eps}
\caption{ The $\gamma$-ray horizon plotted for different EBL models. Data points
are from \citet{Dominguez13}.
\emph{Gray area} shows range covered by EBL in the high and low `lmc2' models.}
\label{fig.horizon}
\end{figure*}
\subsection{ $\gamma$-ray horizon }\label{sec.horizon}
The $\gamma$-ray horizon is defined as the maximum redshift of the $\gamma$-ray source that
can be detected through $\gamma$-rays of observed energy $E_{\gamma}$ with
$\tau_{\gamma} \le 1$. This is nothing but the $\gamma$-ray source redshift $z_0$ for
$\gamma$-rays of observed energy $E_{\gamma}$ on earth corresponding to
$\tau_{\gamma}(E_{\gamma}, z_0)=1$. In Fig.~\ref{fig.horizon}, we plot the $\gamma$-ray
horizon for different source redshifts for our different EBL models where
the gray shaded area shows the range spanned by the high and low `lmc2' model.
It can be directly seen from the Fig.~\ref{fig.horizon} that our universe is transparent
for the $\gamma$-rays with energies less than 10 GeV. The $\gamma$-rays with energy
$E_{\gamma}<$ 30 GeV can be observed from sources at $z\sim 3$ without significant
attenuation. Due to the differences in the IR part of the EBL, the $\gamma$-ray horizon
redshift for our models differ in TeV energies. The well measured $\tau_{\gamma}$ at TeV
energies at low $z$ can, in principle, distinguish between different EBL models
presented here.
In Fig.~\ref{fig.horizon}, we also plot the measurements of $\gamma$-ray horizon by
\citet{Dominguez13} where they model the multi-wavelength observations of blazars to determine
$\gamma$-ray horizon which is EBL independent. Apart from the `smc' model, all our other
models show good agreement with these $\gamma$-ray horizon measurements. This is evident from
the $\chi^2$ statistics. The reduced $\chi^2$ values are 13.1, 1.7, 0.5, 0.4 and 0.4 for our
`smc', `lmc2', `lmc', `mw' and `cal' models, respectively. This indicates that for the `smc'
model, the predicted NIR to FIR part of the EBL intensity is less and inconsistent with the
available $\gamma$-ray horizon measurements.
Note that the reduced $\chi^2$ quoted above includes only observational and systematic errors.
We notice that even when we allow for the EBL obtained using high and low models for `smc', it
does not give $\gamma$-ray horizon consistent with the measurements. If we include the average
deviation in the $\gamma$-ray horizon in case of `lmc2' model due to its higher and lower
limits of the EBL obtained by using high and low models as the errors in the prediction of
$\gamma$-ray horizon, the reduced $\chi^2$ for `lmc2' becomes 0.92. It is evident
from Fig~\ref{fig.horizon} that, even though we could rule out the `smc' model,
purely based on the available $\gamma$-ray horizon measurements alone, we can not
distinguish between other four models. Given the uncertainties involved in the modeling
intrinsic SEDs of $\gamma$-ray sources, the contribution of IHL to the galaxy emissivity
in the NIR and the small spread of $\gamma$-ray horizon predicted in our remaining four
models, it may be challenging to distinguish them based on more of such measurements.
\subsection{ Fermi measurements of $\tau_{\gamma}$ }\label{sec.fermi}
\begin{figure}
\centering
\includegraphics[bb=160 360 500 730, width=11cm, clip=true]{fig14.eps}
\caption{The $\gamma$-ray transmission ($e^{-\tau_{\gamma}}$) for our
`lmc2' EBL model (\emph{solid blue curve})
along with the measurements of \citet{GammaSci} in the redshift
bins defined by them. We calculate $e^{-\tau_{\gamma}}$ for our EBL models
which mimic the method of calculating it by stacking the individual
blazar spectrum in a redshift bin as done by
\citet{GammaSci} (see text in section~\ref{sec.fermi}). The
range covered by all five EBL models along with their high and
low counterparts is shown by \emph{gray shade}.}
\label{fig.sci}
\end{figure}
Recently, \citet{GammaSci} reported the average measurements of $\tau_{\gamma}$ over a
large redshift range using a sample of 150 blazars observed with the Fermi-LAT. Since, it is
difficult to determine $\tau_{\gamma}$ in individual blazar spectrum, they divided their
blazar sample into three redshift bins $z<0.2$, $0.2<z<0.5$ and $0.5<z<1.6$. Then they stacked
spectra in each redshift bin and determined the intrinsic SED of blazars by extrapolating the
stacked spectrum from the lower energies where various EBL estimates suggests that
$\tau_{\gamma}$ is negligible. Using these stacked spectra, \citet{GammaSci} reported the
measurements of $\tau_{\gamma}$ for the observed $\gamma$-ray energies from 10 to 500 GeV in
the redshift range of $0 < z \le 1.6$. Based on the good agreement between the EBL measured
by Fermi and that expected from the lower limits determined from the IGL measurements, they
argued that there is a negligible room for residual emission from other sources. The
$\gamma$-rays in this observed energy range from 10 to 500 GeV will be attenuated effectively
by the EBL photons of rest wavelength $\lambda< 3.1\mu$m for $z<1.6$,
$\lambda< 1.8\mu$m for $z<0.5$ and $\lambda< 1.4\mu$m for $z<0.2$ where the cross-section of
pair-production is maximum (at $\theta=\pi$). This is the wavelength range where, by
construct, our all five models give similar EBL.
For comparison with the measurements of \citet{GammaSci}, we calculate the $\tau_{\gamma}$
in a way that mimics stacking of individual blazar spectra as done by them. We take the
number distribution of blazars as a function of redshift for 150 blazars used by
\citet{GammaSci} and calculate $e^{-\tau_{\gamma}}$ for each blazar at a corresponding
$z$ and at different energies $E_{\gamma}$. Then we take the average of
$e^{-\tau_{\gamma}}$ over the same number of blazars in the redshift bins. This average is
equivalent of getting $e^{-\tau_{\gamma}}$ by stacking the individual blazar spectrum in a
redshift bin. In Fig.~\ref{fig.sci}, we plot our estimates of $\gamma$-ray optical depth
along with the measurements of \citet{GammaSci}, in terms of the transmission
$e^{-\tau_{\gamma}}$ for our `lmc2' EBL model. The range in $e^{-\tau_{\gamma}}$ covered by
all five EBL models with their high and low counterparts is shown by gray shaded region
in Fig.~\ref{fig.sci}. For our all EBL models, the $e^{-\tau_{\gamma}}$ fits well in low
($z \le $0.2) and high (0.5$< z \le$1.6) redshift bins. However, in the intermediate
redshifts, our estimated $\tau_{\gamma}$ is slightly higher than that reported in
\citet{GammaSci}. This excess optical depth is not statistically significant given the large
uncertainties in the measurements of $\tau_{\gamma}$. However if these measurements are indeed
very accurate then to account for this we need the EBL intensity to be factor 2 lower than the
predicted by our models at $z<0.5$ in optical to NIR regime. This requires a factor 2
reduction in $\rho_{\nu}$ at $\lambda< 1.8\mu$m for $z<0.5$. One can, in principle, reduce the
$\rho_{\nu}$ by increasing $L_{min}$ (in Eq.~\ref{rho}). However, how much $\rho_{\nu}$ can
be reduced depends upon the $\alpha$ and the luminosity of the faintest galaxy detected to
determine the luminosity functions. It can be seen from Table~\ref{lmin_table} that the
values of $\alpha$ at $z<0.5$ are high and therefore to reduce $\rho_{\nu}$ by factor 2 one
needs to take $L_{min} >0.2L^*$. However, in this wavelength range (UV to NIR), given the fact
that our EBL models are consistent with the observational lower limits on local EBL, there is
not much room available to reduce the EBL intensity. This re-iterates the findings of
\citet{GammaSci} that the observed luminosity densities of galaxies are just sufficient to
reproduce the $\tau_{\gamma}$ measurements.
Note that to obtain intrinsic blazar spectra, the continuum extrapolation from the lower
energy to higher energy is a practical solution but may not be the valid one. Therefore, the
minor discrepancy mentioned above does not conclude anything significantly. However, it will
be more interesting for constraining various EBL models if the errors on $\tau_{\gamma}$ are
reduced significantly.
\section{Effect of uncertainties on model parameters}\label{sec.res}
In this paper, we use a progressive fitting method to determine the combinations of
$A_{\rm FUV}$($z$) and SFRD($z$) for five different extinction curves using multi-wavelength
multi-epoch luminosity functions. We use these $A_{\rm FUV}$($z$) and SFRD($z$) to get the IR
emissivity and generate the EBL. The progressive fitting method uses the convolution integral
(see Eq.~\ref{Eq.convolution}) to get the $\rho_{\nu}$. The convolution integral involves the
stellar emission from the population synthesis model which depends on the assumed input
parameters like metallicity, IMF and age of the galaxy. In this section, we investigate the
possible uncertainties arising from the scatter in these input parameters involved in the
modeling as compared to the that arising purely from uncertainties in the $\rho_{\nu}$
measurements. In particular, we concentrate on the assumed $z_{max}$ which corresponds to
the age of galaxies in convolution integral and the metallicity.
\begin{figure*}
\centering
\includegraphics[bb=65 370 545 710,width=12.6cm,keepaspectratio,clip=true]{fig15.eps}
\caption{ The $A_{\rm FUV}$($z$) (\emph{top panel}) and SFRD($z$) ( in units of M$_{\odot}$ yr$^{-1}$ Mpc$^{-3}$; \emph{bottom panel})
for different ages of the stellar populations contributing in the convolution integral (Eq.\ref{Eq.convolution}) for our `cal' and `lmc2' models.
The time $t_0<13.6$Gyr is the age with our fiducial $z_{max}=\infty$ limit. Different data points plotted in the \emph{top} and
\emph{bottom panel} are same as in the Fig.~\ref{fig.dust_shade_all} and Fig.~\ref{fig.sfrd_shade_all}, respectively.
Legends are scattered over entire plot.}
\label{fig.tmax}
\end{figure*}
\subsection{Maximum redshift $z_{max}$ in the convolution}
The convolution integral (Eq.~\ref{Eq.convolution}) gives the $\rho_{\nu}(z_0)$ by
convolving the SFRD($z$) with the intensity output of instantaneous burst of star formation
which has occurred at epoch $t$ corresponding to redshift $z$. The population synthesis
models give the specific intensity $l_{\nu}(\tau)$ where $\tau$ is an age of the stellar
population which goes through the starburst at $z$ and shining at $z_0$.
In the calculations discussed above we have taken the maximum redshift $z_{max}=\infty$,
which means, in principle, the $\rho_{\nu}(z_0)$ will have the contribution from very old
stars up to the ages of, $t_{max}$, equal to the light travel time from the Big-Bang to
redshift $z=z_0$ which is less than 13.6 Gyr depending on $z_0$. Note that the actual
contribution from very old stars (age $>$10 Gyr) which went through the starburst at
time $t$ is negligible because the SFRD($z$) at $z$ corresponding to these earlier
epochs $t$ is negligible.
Here, we are investigating the validity of $z_{max}=\infty$ assumption and the effect
of taking different values of $z_{max}$ (or corresponding maximum age of galaxy $t_{max}$)
on our derived quantities like $A_{\rm FUV}$ and SFRD$(z)$ mainly at low redshifts
where the age of the universe is large.
If we take sufficiently small values of $t_{max}$, a contribution from the old stellar
population, which shines at the optical and NIR wavelengths, will be less. Therefore,
galaxies will be bluer than one expects. It requires a large dust attenuation to make
them red and match the $\rho_{\nu}$ measurements at the NIR wavelengths. With this small
$t_{max}$, if we follow the progressive fitting method described in
section~\ref{sec.pfmethod}, we get large $A_{\rm FUV}$($z$) and hence large SFRD($z$) at low
redshifts. This is demonstrated in the case of `cal' and `lmc2' model in
Fig.~\ref{fig.tmax} where we show the SFRD($z$) and $A_{\rm FUV}(z)$ determined by
stopping the convolution integral after a time $t_{max}=t_0$ for different values of
$t_0$ ranging from 1 Gyr to 10 Gyr. We take $t_{max}$ equal to the age of the universe
when the age is smaller than $t_0$. We see similar trends for all five models but for the
sake of presentation we show results in Fig.~\ref{fig.tmax} only for `cal' and `lmc2' model.
It is clear from the Fig.~\ref{fig.tmax} that lower the value of $t_{max}$, higher will
be the values of inferred SFRD($z$) and $A_{\rm FUV}$($z$).
In Fig.~\ref{fig.tmax}, we also plot the independent measurements shown in
Fig.~\ref{fig.dust_shade_all} and Fig.~\ref{fig.sfrd_shade_all}. These measurements of the
$A_{\rm FUV}$ shows that it increases from $z=0$ to a peak at $z\sim 1$ and then decreases at
higher $z$ \citep{Burgarella13, Cucciati12, Takeuchi05}. To get such a shape of $A_{\rm FUV}$,
as shown in Fig.~\ref{fig.tmax},one needs $t_{max}>5$ Gyr. We find that the
SFRD($z$) and $A_{\rm FUV}$($z$) are insensitive to $t_{max}$ when it is $\ge 10$ Gyr.
Therefore, to get the SFRD($z$) and $A_{\rm FUV}$($z$) consistent with the trend seen in
different independent observations one needs to consider the stellar population ages
$\ge$ 10 Gyr which is consistent with $z_{max}=\infty$ and the estimated age 11 Gyr
of Milky-way \citep{Krauss03}.
\subsection{Metallicity}
Another source of possible uncertainty in determining the SFRD($z$) and $A_{\rm FUV}$($z$)
can be related to the redshift evolution of metallicity, which we do not consider. We use
constant $Z=0.008$ metallicity for all redshifts. HM12 and \citet{Madau14} use the
metallicity evolution as $Z(z)=Z_{\odot}10^{-0.15z}$ suggested by \citet{Kewley07} where
$Z_{\odot}=0.020$. This evolution gives $Z=0.008$ at $z=2.6$. To see the effect of using
different metallicity, we determine the SFRD($z$) and $A_{\rm FUV}$($z$) for metallicity
$Z=0.020$ and $Z=0.004$. Our results are plotted in Fig.~\ref{fig.metal} for the `cal' and
`lmc2' models. We also show the range covered by the $A_{\rm FUV}$ and SFRD when obtained for
respective high and low models for our fiducial metallicity $Z=0.008$
(same as in the Fig.~\ref{fig.dust_shade_all} and Fig.~\ref{fig.sfrd_shade_all}). Note that
the low and high models use 1-$\sigma$ low and high fit through $\rho_{\rm FUV}$
measurements used to determine the $A_{\rm FUV}$ and SFRD. It is clear from
Fig.~\ref{fig.metal} that the higher (lower) metallicity gives higher (lower)
$A_{\rm FUV}$($z$) and SFRD($z$). We see similar trends for all five models but for the sake
of presentation we show only for `cal' and `lmc2' model. The difference between the
$A_{\rm FUV}$($z$) obtained for these metallicities (as high as factor 5 in $Z$), are well
within the allowed range of $A_{\rm FUV}$ and SFRD obtained using our fiducial $Z=0.008$.
This suggests that the scatter in the $A_{\rm FUV}$($z$) and SFRD($z$) arising due to change
in metallicity (as high as factor 5 ) is smaller than the scatter one gets in the
$A_{\rm FUV}$($z$) and SFRD($z$) due to scatter in $\rho_{\rm FUV}$ measurements at a
constant metallicity.
The analysis presented here shows that the effect of the metallicity evolution is sub-dominant
as compared to those arising from uncertainties in $\rho_{\rm FUV}$ measurements. Therefore,
our assumption of constant metallicity is valid and compatible with the current $\rho_{\nu}$
measurements.
\begin{figure*}
\centering
\includegraphics[bb=65 370 545 710,width=12.6cm,keepaspectratio,clip=true]{fig16.eps}
\caption{The $A_{\rm FUV}$($z$) (\emph{top panel}) and SFRD($z$)
(in units of M$_{\odot}$ yr$^{-1}$ Mpc$^{-3}$; \emph{bottom panel})
for different metallicity $Z=0.004$, $Z=0.008$
(fiducial) and $Z=0.020$ of the stellar populations for our `cal' and `lmc2' models.
Gray shaded region represent the range covered in the $A_{\rm FUV}$
and SFRD determined using the high and low models for our fiducial $Z=0.008$
(same as shown in Fig.~\ref{fig.dust_shade_all} and Fig.~\ref{fig.sfrd_shade_all}). }
\label{fig.metal}
\end{figure*}
\subsection{IMF and $L_{min}$}
In this paper, we consider the \citet{Salpeter55} IMF in the population synthesis model with
stellar mass range from 0.1 to 100$M_{\odot}$. Even though there are other IMFs like
\citet{Kroupa03}, \citet{Chabrier03} and \citet{Baldry03}, since most of the star formation
rates reported in the literature use Salpeter IMF, we also prefer it for our work. Note that,
the different IMF can change the combination of $A_{\rm FUV}$($z$) and SFRD($z$) but the
fact that the progressive fitting method we use to get these combinations will ensure that
the emissivities and EBL will remain the same in UV to NIR wavelengths (the wavelengths where
we have multi-wavelength multi-epoch luminosity functions). However, the different IMF and
hence different $A_{\rm FUV}$($z$) and SFRD($z$) will give different FIR emissivity and
FIR part of the EBL \citep[see for e.g][]{Primack01}.
The luminosity densities calculated from the observed luminosity function depend on the
values of $L_{min}$ and the faint end slope $\alpha$. For most of the $\rho_{\nu}$ used here,
we use $L_{min}=0$. In Table~\ref{lmin_table} of Appendix, we list $\alpha$ and $L_{min}$
values used to get $\rho_{\nu}$. We also list the percentage decrease in the $\rho_{\nu}$ if
we use $L_{min}=0.01L^*$ and $L_{min}=0.03L^*$ instead of the fiducial $L_{min}$ (see the
column labeled as $\Delta_1$ and $\Delta_2$ in Table~\ref{lmin_table}). The difference is
large for the small $\alpha$ (i.e, $\alpha<-1.3$). The value $L_{min}=0.01L^*$ is used by
HM12 in their UV background calculations. The value of $L_{min}=0.03L^*$ is used by
\citet{Madau14} to get the SFRD. It can be directly seen from the Table~\ref{lmin_table} that
the choice of $L_{min}$ between 0 to 0.03$L^*$ can change the $\rho_{\nu}$ upto 30\%. This
difference is large for the $\rho_{\nu}$ measurements of \citet{Tresse07} at high $z$ and
higher wavebands where $\alpha$ is small. Because of the sensitivity limit of
survey, \citet{Tresse07} could not determine $\alpha$ for $z>1.2$. Therefore, at high $z$,
the $\alpha$ is extrapolated from the low redshift measurements. However, note that the
errors estimated on the $\rho_{\nu}$ values by \citet{Tresse07} include uncertainties
arising from the differer values of $\alpha$ which is larger than the difference due to
$L_{min}$ values mentioned here.
By increasing the values of $L_{min}$, we can get the lower $\rho_{\nu}$ which will give rise
to the EBL with lower intensity. Since our EBL generated using the $\rho_{\nu}$ with
$L_{min}=0$ passes through the lower limits on the local EBL obtained from the IGL
measurements in the FUV to NIR bands (see Fig.~\ref{fig.ebl}), we do not consider the higher
values of $L_{min}$.
\section{Summary}\label{sec.sum}
In this paper, we estimate the extragalactic background light (EBL) which is consistent with
the observed multi-wavelength and multi-epoch luminosity functions upto $z\sim 8$.
To achieve that we introduce a novel method which determines the unique combination of the
dust attenuation magnitude at FUV band, $A_{\rm FUV}(z)$, and the star formation rate density,
SFRD$(z)$, for a given extinction curve. It allows us to investigate the mean extinction curve
which can be used to determine the global average quantities like EBL, SFRD$(z)$ and
$A_{\rm FUV}(z)$. The main results of our work are summarized below.
\begin{enumerate}
\item We introduce a `progressive fitting method' which uses multi-wavelength and multi-epoch
luminosity functions to determine a unique combination $A_{\rm FUV}(z)$ and SFRD$(z)$ for a
given extinction curve. The combination of $A_{\rm FUV}(z)$ and SFRD$(z)$, by construct,
reproduces the emissivity consistent with the observed luminosity functions.
\item We compiled the observed luminosity functions from the FUV to K band and upto
$z \sim 8$. Using this we determine the combinations of $A_{\rm FUV}(z)$ and SFRD$(z)$ from
the `progressive fitting method' for a set of well known extinction curves observed for
Milky-Way, Small Megallenic Clouds (SMC), Large Megallenic Clouds (LMC), LMC supershell
(LMC2) and for the nearby starburst galaxies given by \citet{Calzetti}.
\item With the help of these combinations of $A_{\rm FUV}(z)$ and SFRD$(z)$, for each
extinction curve, we calculate the average energy absorbed by the inter-stellar dust in the
UV to NIR wavelengths. This allowed us to estimate the NIR to FIR emissivity using the
principle of energy conservation and the IR emission templates of local galaxies.
\item Out of all five extinction curves, we find that the $A_{\rm FUV}(z)$, SFRD$(z)$ and
local emissivity obtained using LMC2 extinction curve reproduces different independent
measurements. This enables us to conclude that, out of five well measured extinction curves
for nearby galaxies, the average extinction curve which is applicable for galaxies over wide
range of redshifts is most likely to be similar to LMC2 extinction curve.
\item We use the emissivity obtained for each extinction curve from UV to IR wavelengths and
calculate the EBL for each. We compare these with the different EBL estimates reported in
the literature and with the lower and upper limits placed on the local EBL from various
observations.
\item For different EBLs estimated here, we calculate the optical depths, $\tau_{\gamma}$,
encountered by the high energy $\gamma$-rays due to electron positron pair production upon
collision with the EBL photons. We compare the $\tau_{\gamma}$ computed for our EBL with
those from different EBL estimates reported in the literature and with the measurements of
\citet{GammaSci}. We also calculate the $\gamma$-ray horizon and compare with recently
reported measurements of \citet{Dominguez13}.
\item We find that the IR part of the local EBL and the corresponding $\gamma$-ray horizon in
TeV energies estimated using the SMC extinction curve are inconsistent with various
measurements. However, these measurements are consistent with results obtained from all other
extinction curves.
\item We discuss the uncertainties in $A_{\rm FUV}(z)$, SFRD$(z)$ and the EBL estimates
related to the standard assumptions like metallicity, faint end slope of the luminosity
function and age of the stellar population contributing to the emissivity.
\end{enumerate}
We fit the $A_{\rm FUV}(z)$ and SFRD($z$) using a functional form given in
Eq.~\ref{Eq.fit_form} and the fitting parameters obtained for each extinction curve are
provided in Table~\ref{dust_parm} \& Table~\ref{sfrd_parm}. From the very good agreement with
various measurements we conclude that the LMC2 extinction curve should be used to
calculate the global averaged quantities like the EBL, SFRD and $A_{\rm FUV}$.
The `progressive fitting method' used here to get the $A_{\rm FUV}(z)$ and SFRD$(z)$ requires
luminosity functions observed over different wavebands and redshifts. Therefore, the surveys
like \citet{Tresse07} and \citet{Ilbert05} are very important which determine the luminosity
functions uniformly over large redshifts and different wavebands. Currently our method is
limited by the lack of good observations in different wavebands and at high redshifts.
\section*{Acknowledgments}
We wish to thank Lawrence Tresse, Kari Helgason, Peter Behroozi, Marco Ajello, Hamsa Padmanabhan,
Tirthankar Roy Choudhury and Sunder Sahayanathan for
providing relevant data and useful discussions. We thank anonymous referee for useful comments
which helped us to improve the paper.
VK thanks CSIR for providing support for this work.
\def\apj{ApJ
\def\mnras{MNRAS
\def\aap{A\&A
\def\apjl{ApJ}
\def\aj{AJ}
\def\physrep{PhR}
\def\apjs{ApJS}
\def\pasa{PASA}
\def\pasj{PASJ}
\def\pasp{PASP}
\def\nat{Natur}
\def\araa{AR\&A}
\bibliographystyle{apj}
|
3,212,635,537,960 | arxiv |
\section{Introduction}
$\gamma$-ray astronomy has made major advances in the last decade.
The construction and operation of the current generation of very high
energy (VHE; E $>$ 100 GeV) instruments like VERITAS, HESS and MAGIC
have resulted in the detection of more than 120 VHE sources (see
Figure \ref{fig:skymap}). The launch of the {\it Fermi} satellite
\citep{Atwood2009} has opened new windows on the high energy (HE; 100
MeV $<$ E $<$ 100 GeV) sky. Individually, these instruments have
produced many good results, but the fact that they can perform
multiwavelength studies increases their scientific output
dramatically. It is important to remember that all of the work
presented here is the result of collaborations between many excellent
people, not only in VERITAS and {\em Fermi} but in MAGIC and HESS and
many, many radio scientists.
\begin{figure*}[t]
\centering
\includegraphics[width=110mm]{SkyMap}
\caption{As of this meeting (November 2011) There are currently over
120 VHE sources, about 40 of which are blazars (the red dots shown
in this figure). This contribution focuses on the three misaligned
AGN highlighted in the above figure. Figure from {\em
http://tevcat.uchicago.edu}.}
\label{fig:skymap}
\end{figure*}
The field of VHE astronomy is necessarily ground based (see
\citet{Hinton2009a} for a review). You cannot get enough effective
area at these energies with a space based instrument, while the ground
based techniques result in an effective area the size of a football
field for the current generation of telescopes. These large effective
areas are possible due to the Cherenkov technique employed by the VHE
community. There are three major arrays in use right now: VERITAS,
HESS and MAGIC and the field continues to move forward, there are
upgrades in progress (HESS2 \citep{Moudden2011}, VERITAS upgrade
\citep{KiedaD.B.2011}) and completed (MAGIC2 \citep{LombardiS.2011}).
Also, a new generation of instruments is coming online in the next
years (HAWC \citep{GonzalezM.M.2011}, CTA \citep{CTAConsortium2010}).
There are over 120 VHE sources in the catalog
(http://tevcat.uchicago.edu). The most populous class of objects
(about 40) are blazars and a handful of these are misaligned blazars
(see Figure \ref{fig:doppler}). It is important to note that all of
the misaligned blazars are also {\it Fermi} LAT sources, allowing a
simultaneous measurement of the $\gamma$-ray SED. This is not the
place for a review of AGN physics but the basic understanding is that
these objects contain a central supermassive black hole surrounded by
an accretion disk that powers a relativistic jet of photons and
particles. The orientation of this jet towards the observer
determines the source type: a jet pointed directly at the earth is
seen as a BL Lac object or FSRQ while one that is misaligned is seen
as some type of radio galaxy. There are many open questions that we
are trying to answer by studying AGN at $\gamma$-ray wavelengths
including: determining the emission mechanisms, understanding the
accretion physics, and finding the emission location.
\section{Misaligned Blazars}
Only a handful of the detected VHE AGN are misaligned. The reason for
this can be readily understood by looking at Figure \ref{fig:doppler}.
Blazar emission benefits from high Doppler and Lorentz factors which
boost the flux and energy of the detected $\gamma$-rays and increases
the detection probability. Jet emission is still possible at large
viewing angles but the Doppler factor will be small and thus these
types of objects are harder to detect at VHE. It is thought that the
same emission mechanism that occurs in blazars is what is seen in
these misaligned AGN and so the same types of models could be used to
understand the observed SED \citep{Urry1995}. Additionally, there is
also the possibility to see lobe emission from regions outside of the
core (as is seen in Centaurus A from the LAT \citep{Abdo2010h}).
\begin{figure}
\includegraphics[width=65mm]{DopplerVsAngle}
\caption{Shows the Doppler factor versus the viewing angle for AGN for
different Lorentz factors. Radio galaxies are usually thought to
occur in the green region while blazars are covered by the blue
region (Figure from \citet{Urry1995}).}
\label{fig:doppler}
\end{figure}
The complete study of misaligned AGN is only possible through
multiwavelength studies. The main reason for this is that these
sources emit photons throughout the electromagnetic spectrum and to
accurately understand the emission mechanisms, you need measurements
all the way from the radio to the $\gamma$-ray. A complication is
that these sources are known to be highly variable and thus, you not
only need multiwavelength studies, but simultaneous multiwavelength
studies. These are difficult, but the payoff is great since we can
learn about the AGN population as a whole by fitting these types of
objects into a general AGN emission model. We learn new things about
the diverse AGN class by observing AGN of different types.
\subsection{NGC1275}
\begin{figure}
\includegraphics[width=65mm]{NGC1275Spectrum}
\caption{VERITAS upper limit (99\% confidence) and LAT spectrum (solid
points are for the time period of the VERITAS observations and open
points are for the first year of LAT operations) of NGC 1275 with
models. The upper limit at VHE indicated that the spectrum is not
well fit by a simple power law (Figure from \citet{Acciari2009a}).}
\label{fig:ngc1275spectrum}
\end{figure}
\begin{figure}
\includegraphics[width=65mm]{NGC1275LightCurve}
\caption{LAT lightcurve of NGC1275. The VHE detection by MAGIC
occurred during the flare in 2010 (Figure from \citet{Brown2011}).}
\label{fig:ngc1275lightcurve}
\end{figure}
NGC 1275 is a radio galaxy at the core of the Perseus cluster. It was
initially detected at GeV energies in three months of {\it Fermi}
observations \citep{Abdo2009a}. This indicated strong evidence for
variability since it was detected by COS B \citep{StrongA.W.1982} but
not by EGRET. Observations with VERITAS did not yield a detection at
VHE energies but the upper limit combined with the LAT measurements
indicated that the full SED is not compatible with a single power law
(\citet{Acciari2009a}, Figure \ref{fig:ngc1275spectrum}).
\begin{figure}
\includegraphics[width=65mm]{CenAVHE}
\caption{The upper plot shows the spectra seen by the LAT (in black)
and HESS (in blue and red). The butterfly shows the overall fit of
the LAT data extended to the HESS energy range. In the lower plot,
are model fits to the nuclear region of Cen A. The green curve is a
synchrotron plus SSC fit to the entire data set. The dashed green
curve shows this model without $\gamma$-$\gamma$ attenuation. The
violet curve is a similar fit but is designed to under fit the X-ray
data, and the brown curve is designed to fit the HESS data while not
over-producing the other data in the SED. The blue curve is a
decelerating jet model fit (Figures from \citet{Abdo2010c}).}
\label{fig:cenavhe}
\end{figure}
\begin{figure*}[t]
\centering
\includegraphics[width=135mm]{CenASpectrum}
\caption{The spectrum of the North and South lobes of Centaurus A
based on the first 10 months of LAT observations (Figure from
\citet{Abdo2010h}).}
\label{fig:cenaspectrum}
\end{figure*}
The VERITAS observations were prompted by flaring seen in 2009 (Figure
\ref{fig:ngc1275lightcurve}) along with hardening in GeV
\citep{Acciari2009a}. There was another large flare seen in 2010 along
with a VHE detection ($\sim$60 - 400 GeV) by MAGIC also prompted by
the detection of GeV flaring by the LAT \citep{DonatoD.2010}. It is
thought that these flares might correlate with flares in the radio and
it will be important in the future to trigger TeV observations off of
radio/GeV flares since no VHE emission has been seen during low
states.
\subsection{Centaurus A}
Centaurus A at a distance of 3.7 Mpc is the nearest radio galaxy and
contains some impressive (10 degrees in extent) radio lobes. The LAT
has detected emission from both the lobes and the core
\citep{Abdo2010c,Abdo2010h} which indicated that inverse Compton
emission was the source of the MeV/GeV emission in the lobes (Figure
\ref{fig:cenaspectrum}). The basic understanding is that the
$\gamma$-ray emission is scattered CMB and EBL photons and this
determination allows the measurement of the lobe magnetic field
($\sim$ 1 $\mu$G, near equipartition). The dominant inverse Compton
(determined from the ratio of the energy desity of the CMB and the
lobe magnetic field: U$_{\rm cmb}$/U$_{\rm b}$ $\sim$ 10) component
indicates that the magnetic field is lower than that seen in other
radio sources. Looking at the model in Figure \ref{fig:cenaspectrum}
it is apparent that there should be detectable hard X-ray emission but
no evidence of such has been found \citep{Beckmann2011}. Archival hard
X-ray observations from SAS-3 \citep{Marshall1980} are shown as upper
limits in Figure \ref{fig:cenaspectrum}.
In addition to modelling the emission mechanism of the $\gamma$-ray
photons from the lobes of Cen A, the initial observations allowed the
LAT team to probe the EBL since the inverse Compton/EBL interaction
dominates above 1 GeV. Unfortunately, the statistics required to
differentiate models is greater than that allowed by the inital 10
month data set \citep{Abdo2010h}. Deeper analysis using the full 3
year data set is underway.
One of the most exciting developments in VHE astrophysics over the
last several years is the opening up of new populations with long-term
observations and novel analysis techniques. Centaurus A is one of
those sources. It was detected in a long (120 hour) observation by
HESS \citep{Aharonian2009}. The interesting thing is that the VHE and
HE spectra are barely consistent with each other (see Figure
\ref{fig:cenavhe}). The understanding of this issue is unknown at the
moment but one thing to note is that if FR I's are the parent
population of blazars, than an SSC model should fit the SED but a
simple SSC cannot explain the VHE emission (see Figure
\ref{fig:cenavhe}). If one allows for the optical and radio emission
to come from a different component than the high energy emission, an
SSC scenario can explain the X-ray to VHE emission \citep{Lenain2008}.
\subsection{M87}
\begin{figure}
\includegraphics[width=65mm]{M87SkyMap}
\caption{VLA image of M87 showing the core and the large radio halo.
The overlaid circles indicate the LAT 68\% and 95\% containment.
Many structures are resolved in the radio that cannot be
disentangled in $\gamma$-rays (Figure from \citet{Abdo2009b}).}
\label{fig:m87skymap}
\end{figure}
The view of M87 in the radio is impressive. At 90 cm, the jet
outflows terminate in a halo roughly 80 kpc from the core. At the
scale of a few kpc you can see several knots in X-rays, optical and
radio wavelengths. M87 is the only non-blazar AGN detected by the
previous generation of VHE instruments (a strong hint of emission at
the 4 sigma level was seen by HEGRA \citep{Aharonian2003b}). Since it
is close (16 Mpc ) you do not have to worry about EBL attenuation and
you can resolve the impressive jet structure in the radio. The mass
of the central black hole is assumed to be $(3 - 6) \times 10^9 {\rm
M}_\odot$. Due to the PSF of VHE and HE instruments, the jet
structure of M87 cannot be resolved at the highest energies (see
Figure \ref{fig:m87skymap}) but by leveraging multiwavelength
observations, flares can be associated with specific knots and other
emission regions. The LAT emission can be adequately modeled using a
1 zone SSC model assuming a moderate jet beaming of 2 - 4
\citep{Abdo2009b}.
\begin{figure}
\includegraphics[width=65mm]{M87LightCurveLong}
\caption{Full 10 year multiwavelength lightcurve of M87 (Figure from
\citet{Abramowski2011}).}
\label{fig:m87lightcurvelong}
\end{figure}
The history of determining the source of the VHE emission is muddled.
Figure \ref{fig:m87lightcurvelong} shows the full 10 year lightcurve
of M87 plotting joint observation campaigns (synchronized and ToO) of
VERITAS, MAGIC, HESS, Fermi, Chandra, HST, VLA and VLBA
\citep{Abramowski2011}. The coordination and effort needed to produce
such a figure is impressive. The first flare at VHE (indicated by the
first vertical grey line) was seen in 2005 and it coincided with
flaring at other wavelengths in the knot HST-1, located more than 120
pc from the core. The conclusion of these observations was the VHE
emission was most likely originating in that knot
\citep{Aharonian2006c}. It is important to note that when either the
HST-1 knot or the core is X-ray bright, it contaminates the flux of
the other feature.
In 2008 a large flare was seen by all three major VHE instruments
(VERITAS, MAGIC and HESS) and a dedicated multiwavelength campaign was
initiated (Figure \ref{fig:m87lightcurve}). During this flare, the
VHE emission rose to 10\% of the Crab Nebula's flux, the largest flux
from this galaxy to date and emission was seen to rise in the X-ray
and radio for the core. The conclusion of these observations was that
the emission originated from the core region of M87 and not from any
knots further down the jet \citep{Acciari2009b}.
\begin{figure}
\includegraphics[width=65mm]{M87LightCurve}
\caption{Multiwavelength lightcurve of M87 during the 2008 flaring
period (Figure from \citet{Acciari2009b}.).}
\label{fig:m87lightcurve}
\end{figure}
The story is further complicated by a large (20\% Crab) flare seen by
VERITAS in 2010 \citep{Ong2010}. This incident was also observed with the
LAT, VLBA and Chandra. Even though the core is seen to be brightening
in the X-ray, no increase in flux is seen in the GeV range and the
radio does not exhibit the same behavior as in previous flares.
However, the brightness of the flare in the VHE allows for detailed
studies of the spectral state of M87 during the flaring period.
\section{Conclusions}
It should be apparent that there is still a lot to learn before we can
understand the VHE and HE emission from misaligned blazars (and by
extension AGN in general). It should also be clear that the only
reason we know as much as we do at this point is due to the amazing
amount of effort put into multiwavelength studies. Without a full
view of the SED, we could not adequately model these systems. Without
a full view of the lightcurve we could not try and understand the
location of the emission. It is vitally important to continue
multiwavelength studies from the optical all the way to VHE.
\bigskip
\begin{acknowledgments}
VERITAS is supported by grants from the U.S. Department of Energy
Office of Science, the U.S. National Science Foundation and the
Smithsonian Institution, by NSERC in Canada, by Science Foundation
Ireland (SFI 10/RFP/AST2748) and by STFC in the U.K. We acknowledge
the excellent work of the technical support staff at the Fred Lawrence
Whipple Observatory and at the collaborating institutions in the
construction and operation of the instrument.
The \textit{Fermi} LAT Collaboration acknowledges support from a
number of agencies and institutes for both development and the
operation of the LAT as well as scientific data analysis. These
include NASA and DOE in the United States, CEA/Irfu and IN2P3/CNRS in
France, ASI and INFN in Italy, MEXT, KEK, and JAXA in Japan, and the
K.~A.~Wallenberg Foundation, the Swedish Research Council and the
National Space Board in Sweden. Additional support from INAF in Italy
and CNES in France for science analysis during the operations phase is
also gratefully acknowledged.
\end{acknowledgments}
\bigskip
|
3,212,635,537,961 | arxiv | \section{Introduction}
In the Randall-Sundrum (RS) scenario \cite{RS1,RS2}
the observable four-dimensional (4D) Universe is a 3-brane world embedded
in a $Z_2$ symmetric 5D anti-de Sitter (AdS) space. In the RS1 model
\cite{RS1} the fifth dimension is compact and there are two 3-brane
boundaries. The gravitational field is bound to the hidden positive
tension brane and decays towards the observable negative tension brane.
In this setting, the hierarchy problem is reformulated as an
exponential hierarchy between the weak and Planck scales \cite{RS1}.
In the RS2 model \cite{RS2}, the orbifold has an infinite fifth
dimension and just one observable positive tension brane near which
gravity is exponentially localized.
In the RS models, the classical field dynamics is defined by 5D
Einstein equations with a negative bulk cosmological constant
$\Lambda_{\rm B}$, Dirac delta sources standing for the branes and a
stress-energy tensor describing other fields propagating in the bulk
\cite{RS1}-\cite{GW}. A set of vaccum solutions is given by ${\rm d}{\tilde{s}_5^2}=
{\rm d}{y^2}+{{\rm e}^{-2|y|/l}}{\rm d}{s_4^2}$, where $y$ is the
cartesian coordinate representing the fifth dimension, the 4D line
element ${\rm d}{s_4^2}$ is Ricci flat, $l$ is
the AdS radius given by $l=1/\sqrt{-{\Lambda_{\rm
B}}{\kappa_5^2}/6}$ with
${\kappa_5^2}=8\pi/{{\rm M}_5^3}$ and ${\rm M}_5$ the fundamental 5D Planck
mass. In the RS1 model, the hidden Planck brane is located at $y=0$ and the
visible brane at $y'=\pi{r_{\rm c}}$, where
$r_{\rm c}$ is the RS compactification scale \cite{RS1}.
The brane tensions $\lambda>0$ and $\lambda'<0$ have the same absolute
value $|\lambda'|=\lambda$. In the vaccum $\lambda$ is given in terms of
$\Lambda_{\rm B}$ and
$l$ by $\lambda=-{\Lambda_{\rm B}}l$. In the RS2 model
\cite{RS2}, the visible brane is the one with positive tension
$\lambda$ located at $y=0$. The hidden brane is sent to infinity and
is physically decoupled.
The low energy theory of gravity on the observable brane is
4D general relativity and the cosmology may be
Friedmann-Robertson-Walker \cite{RS1}-\cite{TM}.
In the RS1 model, this requires the stabilization of
the radion mode using, for example, a 5D scalar
field \cite{GW,WFGK,CGRT,TM}. The gravitational
collapse of matter has also been analyzed in the RS
scenario \cite{CHR}-\cite{RC2}. However, an exact 5D geometry
describing a stable black hole localized on a 3-brane
has not yet been discovered. Indeed, non-singular localized black holes have
only been found in an $\rm{AdS}_4$ braneworld \cite{EHM}. A solution to this
problem requires a simultaneous localization of gravity and matter which avoids
unphysical divergences \cite{CHR,KOP}-\cite{RC2} and
could be related to quantum black holes on the brane \cite{EFK}. In
addition, the covariant Gauss-Codazzi approach \cite{BC,SMS}
has uncovered a rich set of braneworld solutions, many of which have
not yet been associated with exact 5D spacetimes \cite{COSp}-\cite{RC1}.
In this paper we report on research about the dynamics of a
spherically symmetric 3-brane in the presence of 5D
conformal matter fields \cite{RC2,RC3} (see also \cite{EONO}). In the previous work
\cite{RC2,RC3} we have discovered a new class of exact
5D dynamical solutions for which gravity is bound to the brane by the
exponential RS warp. These solutions were shown to be
associated with conformal bulk fields characterized by a stress-energy
tensor $\tilde{T}_\mu^\nu$ of weight -4 and by the equation of
state ${\tilde{T}_a^a}=2{\tilde{T}_5^5}$ (see also
\cite{KKOP} and \cite{IR}). They were also shown
to describe on the brane the dynamics of
inhomogeneous dust, generalized dark radiation and homogeneous
polytropic matter. However, the density and pressures
of the conformal bulk fluid increase with the coordinate of the fifth
dimension. Consequently and just like in the Schwarzschild black
string solution \cite{CHR}, the RS2 scenario is plagued with an
unphysical singularity at the AdS horizon. Such divergence does not
occur in the RS1 model because the compactified space ends before the
AdS horizon is reached. However, the radion
mode turns out to be unstable \cite{RN}. In this
work we discuss new exact 5D braneworld solutions which are stable under
radion field perturbations and still describe on the visible brane the
dynamics of inhomogeneous dust, generalized dark radiation and homogeneous
polytropic dark energy. We also consider the point of view of an effective
Gauss-Codazzi observer and show that the gravitational field is bound
to the vicinity of the brane.
\section{5D Einstein equations and conformal fields}
To map the ${\rm AdS}_5$ orbifold, consider the coordinates
$(t,r,\theta,\phi,z)$ where $z$ is related to the cartesian coordinate
$y$ by $z=l{{\rm e}^{y/l}}$, $y>0$. The most general non-factorizable
dynamical metric
consistent with the $Z_2$ symmetry in $z$ and
with 4D spherical symmetry on the brane is given by
\begin{equation}
{\rm d}{\tilde{s}_5^2}={\Omega^2}\left({\rm d}{z^2}-{{\rm e}^{2A}}{\rm
d}{t^2}+{{\rm e}^{2B}}{\rm d}{r^2}+{R^2}{\rm d}{\Omega_2^2}\right),\label{gm1}
\end{equation}
where $\Omega=\Omega(t,r,z)$, $A=A(t,r,z)$, $B=B(t,r,z)$ and
$R=R(t,r,z)$ are $Z_2$ symmetric functions. $R(t,r,z)$ represents the
physical radius of the 2-spheres and $\Omega$ is the warp factor
characterizing a global conformal transformation on the metric.
In the RS1 model, the classical dynamics is defined by the 5D
Einstein equations,
\begin{equation}
{\tilde{G}_\mu^\nu}=-{\kappa_5^2}\left\{{\Lambda_{\rm B}}{\delta_\mu^\nu}+
{1\over{\sqrt{\tilde{g}_{55}}}}\left[\lambda\delta
\left(z-{z_0}\right)+\lambda'\delta\left(z-{{z'}_0}\right)\right]\left({\delta_\mu^\nu}-{\delta_5^\nu}
{\delta_\mu^5}\right)-{\tilde{\mathcal{T}}_\mu^\nu}\right\},
\label{5DEeq}
\end{equation}
where ${\tilde{\mathcal{T}}_\mu^\nu}$ is
the stress-energy tensor of the matter fields which is
conserved in 5D,
\begin{equation}
{\tilde{\nabla}_\mu}{\tilde{\mathcal{T}}_\nu^\mu}=0\label{5Dceq}.
\end{equation}
For a general 5D metric $\tilde{g}_{\mu\nu}$, (\ref{5DEeq}) and
(\ref{5Dceq}) form an extremely complex system of differential equations.
To solve it we need simplifying assumptions about
the field varia\-bles involved in the problem. Let us first consider that the bulk matter is
described by conformal fields with weight
$s$. Under the conformal transformation
${\tilde{g}_{\mu\nu}}={\Omega^2}{g_{\mu\nu}}$, the stress-energy tensor
satisfies
${\tilde{\mathcal{T}}_\mu^\nu}={\Omega^{s+2}}{\mathcal{T}_\mu^\nu}$.
Consequently, (\ref{5DEeq}) and (\ref{5Dceq}) may be rewritten as \cite{RC2}
\begin{eqnarray}
&&{G_\mu^\nu}=-6{\Omega^{-2}}\left({\nabla_\mu}\Omega\right){g^{\nu\rho}}
{\nabla_\rho}\Omega+
3{\Omega^{-1}}{g^{\nu\rho}}{\nabla_\rho}{\nabla_\mu}\Omega
-3{\Omega^{-1}}{\delta_\mu^\nu}{g^{\rho\sigma}}{\nabla_\rho}{\nabla_\sigma}
\Omega\nonumber\\
&&-{\kappa_5^2}
{\Omega^2}\left\{{\Lambda_{\rm B}}{\delta_\mu^\nu}+{\Omega^{-1}}\left[\lambda
\delta(z-{z_0})+\lambda'\delta\left(z-{{z'}_0}\right)\right]\left(
{\delta_\mu^\nu}-{\delta_5^\nu}{\delta_\mu^5}\right)
-{\Omega^{s+2}}{\mathcal{T}_\mu^\nu}\right\},
\label{t5DEeq}
\end{eqnarray}
\begin{equation}
{\nabla_\mu}{\mathcal{T}_\nu^\mu}+{\Omega^{-1}}\left[(s+7){\mathcal{T}_\nu^\mu}{\partial_\mu}
\Omega-{\mathcal{T}_\mu^\mu}{\partial_\nu}\Omega\right]=0\label{t5Dceq}.
\end{equation}
If we separate the conformal tensor $\tilde{\mathcal{T}}_\mu^\nu$ in two sectors
$\tilde{T}_\mu^\nu$ and $\tilde{U}_\mu^\nu$ with the same weight $s$,
${\tilde{\mathcal{T}}_\mu^\nu}={\tilde{T}_\mu^\nu}+{\tilde{U}_\mu^\nu}$
where ${\tilde{T}_\mu^\nu}={\Omega^{s+2}}{T_\mu^\nu}$ and
${\tilde{U}_\mu^\nu}={\Omega^{s+2}}{U_\mu^\nu}$, and take $s=-4$ then it is possible to split (\ref{t5DEeq}) as follows
\begin{equation}
{G_\mu^\nu}={\kappa_5^2}{T_\mu^\nu},\label{r5DEeq}
\end{equation}
\begin{eqnarray}
&6{\Omega^{-2}}{\nabla_\mu}\Omega{\nabla_\rho}
\Omega{g^{\rho\nu}}-
3{\Omega^{-1}}{\nabla_\mu}{\nabla_\rho}\Omega{g^{\rho\nu}}+3{\Omega^{-1}}
{\nabla_\rho}{\nabla_\sigma}\Omega{g^{\rho\sigma}}{\delta_\mu^\nu}=\nonumber\\
&-{\kappa_5^2}
{\Omega^2}\left\{{\Lambda_{\rm B}}{\delta_\mu^\nu}+{\Omega^{-1}}\left[\lambda
\delta(z-{z_0})+\lambda'\delta\left(z-{{z'}_0}\right)\right]\left(
{\delta_\mu^\nu}-{\delta_5^\nu}{\delta_\mu^5}\right)\right\}+{\kappa_5^2}{U_\mu^\nu}.\label{5DEeqwf}
\end{eqnarray}
On the other hand, the Bianchi identity implies
\begin{equation}
{\nabla_\mu}{T_\nu^\mu}=0.\label{5DceqT}
\end{equation}
Then (\ref{t5Dceq}) is in fact
\begin{equation}
{\nabla_\mu}{U_\nu^\mu}+{\Omega^{-1}}\left(3{\mathcal{T}_\nu^\mu}{\partial_\mu}
\Omega-{\mathcal{T}_\mu^\mu}{\partial_\nu}\Omega\right)=0.\label{u5Dceq}
\end{equation}
Note that (\ref{r5DEeq}) and (\ref{5DceqT}) are 5D
Einstein equations with conformal bulk fields, but without
a brane or bulk cosmological constant. They do not depend on the
warp factor which is dynamically defined by (\ref{5DEeqwf}) and
(\ref{u5Dceq}). The warp is then the only effect reflecting the
existence of the brane or of the bulk cosmological constant. We
emphasize that this is only possible for the special set of bulk fields
which have a stress-energy tensor with conformal weight $s=-4$.
Although the system of dynamical equations is now partially decoupled,
it remains difficult to solve. Note for instance that $\Omega$ depends
non-linearly on $A$, $B$ and $R$. In addition, it is affected by
$T_\mu^\nu$ and $U_\mu^\nu$. So consider the special setting
$A=A(t,r)$, $B=B(t,r)$, $R=R(t,r)$ and $\Omega=\Omega(z)$. Then (\ref{r5DEeq}) and (\ref{5DEeqwf}) lead to
\begin{equation}
{G_a^b}={\kappa_5^2}{T_a^b},\label{4DECeq}
\end{equation}
\begin{equation}
{G_5^5}={\kappa_5^2}{T_5^5},\label{5DEeqz}
\end{equation}
\begin{equation}
6{\Omega^{-2}}{{({\partial_z}\Omega)}^2}+{\kappa_5^2}{\Omega^2}{\Lambda_{\rm
B}}={\kappa_5^2}{U_5^5},\label{rswf1}
\end{equation}
\begin{equation}
\left\{3{\Omega^{-1}}{\partial_z^2}\Omega+{\kappa_5^2}{\Omega^2}
\left\{{\Lambda_{\rm B}}+{\Omega^{-1}}\left[\lambda\delta(z-{z_0})+\lambda'\delta(z-{{z'}_0})\right]\right\}\right\}{\delta_a^b}={\kappa_5^2}{U_a^b},\label{rswf2}
\end{equation}
where the latin indices represent the 4D coordinates
$t$, $r$, $\theta$ and $\phi$. Since according to (\ref{4DECeq}) and
(\ref{5DEeqz}) $T_\mu^\nu$ depends only on $t$ and $r$, (\ref{5DceqT}) becomes
\begin{equation}
{\nabla_a}{T_b^a}=0.\label{5DceqT1}
\end{equation}
On the other hand, (\ref{rswf1}) and (\ref{rswf2}) imply that
$U_\mu^\nu$ must be diagonal,
${U_\mu^\nu}=diag(-\bar{\rho},{\bar{p}_{\rm r}}$,\\${\bar{p}_{\rm
T}},{\bar{p}_{\rm T}},{\bar{p}_5})$,
with the density $\bar{\rho}$ and pressures $\bar{p}_{\rm r}$,
$\bar{p}_{\rm T}$ satisfying
$\bar{\rho}=-{\bar{p}_{\rm r}}=-{\bar{p}_{\rm T}}$. In addition,
$U_\mu^\nu$ must only depend on $z$. Consequently,
${\nabla_a}{U_b^a}=0$ is an identity. Then using (\ref{u5Dceq}) and
noting that ${T_\mu^\nu}={T_\mu^\nu}(t,r)$, we find
\begin{equation}
{\partial_z}{\bar{p}_5}+{\Omega^{-1}}{\partial_z}\Omega\left(2{U_5^5}-{U_a^a}\right)=0,\quad
2{T_5^5}={T_a^a}.\label{u5Dceq1}
\end{equation}
If ${U_\mu^\nu}(z)$ is a conserved tensor field like $T_\mu^\nu$, then
$\bar{p}_5$ must be constant. So (\ref{u5Dceq1}) leads to the
following equations of state:
\begin{equation}
2{T_5^5}={T_a^a},\quad 2{U_5^5}={U_a^a}.\label{eqst2}
\end{equation}
Then we obtain ${\bar{p}_5}=-2\bar{\rho}$. $U_\mu^\nu$ is thus constant. On the other hand if ${T_\mu^\nu}=diag\left(-\rho,{p_{\rm r}},{p_{\rm T}},{p_{\rm T}},{p_5}\right)$ where
$\rho$, $p_{\rm r}$, $p_{\rm T}$ and $p_5$ are, respectively, the density and
pressures then its equation of state is rewritten as
\begin{equation}
\rho-{p_{\rm r}}-2{p_{\rm T}}+2{p_5}=0.\label{eqst3}
\end{equation}
Note that $\rho$, $p_{\rm r}$, $p_{\rm T}$ and $p_5$ must be
independent of $z$, but may be functions of $t$ and $r$. The bulk matter is, however,
inhomogeneously distributed along the fifth dimension because
the physical energy density, $\tilde{\rho}(t,r,z)$, and pressures,
$\tilde{p}(t,r,z)$, are related
to $\rho(t,r)$ and $p(t,r)$ by the scale factor $\Omega^{-2}(z)$. Also
note that $T_\mu^\nu$ determines the dynamics on the
branes and that in the RS1 model, the two branes have identical cosmological
evolutions. On the other hand, it is also important to note that the
warp factor depends on the
conformal bulk fields only through $U_\mu^\nu$. Consequently, the role of
$U_\mu^\nu$ is to influence how
the gravitational field is warped around the
branes. In our previous work $U_\mu^\nu$
was set to zero \cite{RC2,RC3,RN}. The corresponding
braneworld solutions were warped by the exponential RS scale factor
and turned out to be
unstable under radion field perturbations \cite{RN}. So we also introduce $U_\mu^\nu$ as a stabilizing sector.
\section{Exact 5D warped solutions}
The $\rm{AdS}_5$ braneworld dynamics is defined by the solutions of
(\ref{4DECeq}) to (\ref{5DceqT1}) and (\ref{eqst3}). Let us first solve (\ref{rswf1}) and
(\ref{rswf2}). As we have seen, $U_\mu^\nu$ is constant with
$\bar{\rho}=-{\bar{p}_{\rm r}}=-{\bar{p}_{\rm T}}=-{\bar{p}_5}/2$. If
${\bar{p}_5}=0$ then ${U_\mu^\nu}=0$, and we end up with the
usual RS warp equations. As is well known, a solution is the
exponential RS warp $\Omega(y)={\Omega_{\rm
RS}}(y)={{\rm e}^{-|y|/l}}$ \cite{RS1,RS2}. If $\bar{p}_5$ is non-zero then we find a new set of warp solutions.
Integrating (\ref{rswf1}) and taking into
account the $Z_2$ symmetry, we obtain (see figure~\ref{fig1:wfs})
\begin{equation}
\Omega(y)={{\rm e}^{-|y|/l}}\left(1+{p_{\rm B}^5}{{\rm e}^{2|y|/l}}\right),\label{wfp5y}
\end{equation}
where ${p_{\rm B}^5}={\bar{p}_5}/(4{\Lambda_{\rm B}})$. This set of
solutions must also satisfy (\ref{rswf2}) which contains the
Israel jump conditions. As a consequence, the brane tensions
$\lambda$ and $\lambda'$ are given by
\begin{equation}
\lambda={\lambda_{\rm RS}}{{1-{p_{\rm B}^5}}\over{1+{p_{\rm
B}^5}}},\quad \lambda'=-{\lambda_{\rm RS}}{{1-{p_{\rm
B}^5}\exp(2\pi{r_{\rm c}}/l)}\over{1+{p_{\rm
B}^5}\exp(2\pi{r_{\rm c}}/l)}},\label{wft4}
\end{equation}
where ${\lambda_{\rm RS}}=6/(l{\kappa_5^2})$.
Note that in the limit ${p_{\rm B}^5}\to 0$, we obtain the RS warp and
also the corresponding brane tensions.
To determine the dynamics on the brane we need to solve (\ref{4DECeq}) and
(\ref{5DEeqz}) when $T_\mu^\nu$
satisfies (\ref{5DceqT1}) and (\ref{eqst3}). Note that as long as
$p_5$ balances $\rho, {p_{\rm
r}}$ and $p_{\rm T}$ according to (\ref{5DEeqz}) and (\ref{eqst3}), the 4D equation of state is not constrained. Three
examples corresponding to inhomogeneous dust, generalized dark
radiation and homogeneous polytropic matter were analised in
\cite{RC2} and \cite{RC3}.
\begin{figure}[H]
\center{\psfig{file=rcqfext05fig1.eps,width=0.4\hsize}}
\vspace{-0.2cm}
\caption{Plots of $W=\ln\Omega(x)$, $x=y/{r_{\rm c}}$ for
$l/{r_{\rm c}}=5$. The dashed, thin and thick
lines correspond, respectively, to
$p_{\rm B}^5$ equal to $1.5,0.5$ and $0.15$.}
\label{fig1:wfs}
\end{figure}
\noindent The latter describes the
dynamics on the brane of dark energy in the form of a polytropic
fluid. The diagonal conformal matter may be defined by
\begin{equation}
\rho={\rho_{\rm P}},\;
{p_{\rm r}}+\eta{{\rho_{\rm P}}^\alpha}=0,\;{p_{\rm T}}={p_{\rm r}},\;{p_5}=-{1\over{2}}
\left({\rho_{\rm P}}
+3\eta{{\rho_{\rm P}}^\alpha}\right),
\label{dmeqst}
\end{equation}
where $\rho_{\rm P}$ is the polytropic energy
density and the parameters ($\alpha$, $\eta$) characterize
different polytropic phases.
Solving the conservation equations, we find \cite{RC3,BBS}
\begin{equation}
{\rho_{\rm P}}={{\left(\eta+{a\over{S^{3-3\alpha}}}\right)}^
{1\over{1-\alpha}}},\label{dmdena}
\end{equation}
where $\alpha\not=1$, $a$ is an integration constant and $S=S(t)$ is the
Robertson-Walker scale factor of the brane world which is related to
the physical radius by $R=rS$. For $-1\leq\alpha<0$, the fluid is in
its generalized Chaplygin phase (see also \cite{BBS}). With this
density, the Einstein equations lead to the following 5D dark energy polytropic solutions \cite{RC3}:
\begin{equation}
{\rm d}{\tilde{s}_5^2}={\Omega^2}\left[-{\rm d}{t^2}+{S^2}
\left({{{\rm d}{r^2}}\over{1-k{r^2}}}+{r^2}{\rm
d}{\Omega_2^2}\right)\right]+{\rm d}{y^2},
\label{dmsol1}
\end{equation}
where the brane scale factor $S$ satisfies ${\dot{S}^2}={\kappa_5^2}
{\rho_{\rm P}}{S^2}/3-k$. The global evolution of the observable
universe is then given by \cite{RC1,RC3}
\begin{equation}
S{\dot{S}^2}=V(S)={{\kappa_5^2}\over{3}}{{\left(\eta{S^{3-3\alpha}}+
a\right)}^{1\over{1-\alpha}}}-k S.\label{dmdineq}
\end{equation}
In figure~\ref{fig2:l0ppp}, we present some ilustrative examples.
\begin{figure}[H]
\center{\psfig{file=rcqfext05fig2.eps,width=0.4\hsize}}
\vspace{-0.2cm}
\caption{Plots of $V=S{\dot{S}^2},\;Z={S^{1-\alpha}}$ for
$k>0$, $\eta>0$ and $a>0$.
The dashed, thin and thick lines correspond, respectively, to
$\alpha$ equal to $-1/4,-1/2$ and $-1$.}
\label{fig2:l0ppp}
\end{figure}
\section{Radion stability}
To analyse how these solutions behave under radion field
perturbations, we consider the saddle point expansion of the RS action
\cite{RN,HKP,CGHW}
\begin{equation}
\tilde{S}=\int{{\rm d}^4}x{\rm d}y\sqrt{-\tilde{g}}
\left\{{\tilde{R}\over{2{\kappa_5^2}}}-
{\Lambda_{\rm B}}-{1\over{\sqrt{\tilde{g}_{55}}}}\left[\lambda\delta\left(y\right)+\lambda'\delta\left(y-\pi
{r_{\rm c}}\right)\right]+
{\tilde{L}_{\rm B}}\right\},\label{5Dact1}
\end{equation}
where $\tilde{L}_{\rm B}$ is the lagrangian characterizing the 5D
matter fields. The most general metric
consistent with the
$Z_2$ symmetry in $y$ and with 4D spherical symmetry on the
brane may be written in the form
\begin{equation}
{\rm d}{\tilde{s}^2}={a^2}{\rm d}{s_4^2}+{b^2}{\rm d}{y^2},\quad {\rm
d}{s_4^2}=-{\rm d}{t^2}
+{{\rm e}^{2B}}{\rm d}{r^2}+{R^2}{\rm d}{\Omega_2^2},\label{5Dmt}
\end{equation}
where the metric functions $a=a(t,r,y)$, $B=B(t,r,y)$,
$R=R(t,r,y)$ and $b=b(t,r,y)$ are $Z_2$ symmetric. Now $a$ is the warp
factor and $b$ is related to the radion field. Our braneworld
backgrounds correspond to $b=1$, $B=B(t,r)$, $R=R(t,r)$ and $a=\Omega(y)$.
Consider (\ref{5Dmt}) with $a(t,r,y)=\Omega(y){{\rm e}^{-\beta(t,r)}}$ and
$b(t,r)={{\rm e}^{\beta(t,r)}}$. Then the dimensional reduction of
(\ref{5Dact1}) in the Einstein frame leads to \cite{RN}
\begin{equation}
\tilde{S}=\int{{\rm d}^4}x\sqrt{-{g_4}}\left({{R_4}\over{2{\kappa_4^2}}}-{1\over{2}}{\nabla_c}\gamma{\nabla_d}\gamma{g_4^{cd}}-\tilde{V}\right),\label{DR5Dact}
\end{equation}
where $\gamma=\beta/({\kappa_4}\sqrt{2/3}\,)$ is the canonically
normalized radion field. The function $\tilde{V}=\tilde{V}(\gamma)$ is
the radion potential, and it is given by
\begin{eqnarray}
&\tilde{V}={2\over{\kappa_5^2}}{\chi^3}\int{\rm d}y{\Omega^2}\left[3
{{({\partial_y}\Omega)}^2}+2\Omega
{\partial_y^2}\Omega\right]+\chi\int{\rm d}
y{\Omega^4}\left({\Lambda_{\rm B}}-{\tilde{L}_{\rm B}}\right)\nonumber\\
&+{\chi^2}\int{\rm d}y{\Omega^4}\left[\lambda\delta\left(y\right)+\lambda'\delta\left(y-\pi
{r_{\rm c}}\right)\right],\label{rp}
\end{eqnarray}
where $\chi=\exp(-{\kappa_4}\gamma\sqrt{2/3}\,)$ and
we have chosen ${\int_{-\pi{r_{\rm c}}}^{\pi{r_{\rm c}}}}{\rm d} y{\Omega^2}={\kappa_5^2}/{\kappa_4^2}$.
To analyse the stability of the $\rm{AdS}_5$ braneworld solutions, we
consider the saddle point expansion of the radion field potential
$\tilde{V}$. If ${p_{\rm B}^5}=0$, then $\Omega={\Omega_{\rm{RS}}}$. The radion
potential has two critical extrema, ${\chi_1}=1$ and ${\chi_2}=1/3$ \cite{RN}. Our solutions correspond
to the first root ${\chi_1}=1$. The same happens if the bulk matter
is absent as in the RS vacuum solutions. Stable background solutions must be associated with a positive second variation of the radion potential.
If the equation of state of the conformal bulk fields is independent
of the radion perturbation, then for $\chi={\chi_1}=1$
the second variation is negative, and so the braneworld solutions
are unstable \cite{RN}.
If the equation of state is kept invariant
under the radion perturbations, it is possible to find stable solutions
at $\chi=1$ if the warp is changed. Indeed, the new
relevant warp functions are given in (\ref{wfp5y}). Consider
$\tilde{\mathcal{V}}=\int{{\rm d}^4}x\sqrt{-{g_4}}\tilde{V}$. With
$x=y/{r_{\rm c}}$ and ${r_{\rm c}}{\int_{-\pi}^\pi}
{\rm d}x{\Omega^2}={\kappa_5^2}/{\kappa_4^2}$, we find
\begin{equation}
{{{\delta^2}\tilde{\mathcal{V}}}\over{\delta{\gamma^2}}}
{{\Big |}_{\gamma=0}}=-{4\over{3}}{\kappa_4^2}{{\left({r_{\rm c}^2}\int
{\rm d}x{\Omega^2}\right)}^{-1}}\int{{\rm d}^4}x\sqrt{-{\tilde{g}_4}}M,
\end{equation}
where the dimensionless radion mass parameter $M$ is
\begin{equation}
M=\lambda{r_{\rm c}}{\kappa_5^2}{\Omega^4}(0)+\lambda'{r_{\rm c}}{\kappa_5^2}
{\Omega^4}(\pi)-{{6{r_{\rm c}^2}}\over{l^2}}\int{\rm d}x{\Omega^4}.
\end{equation}
Stable solutions correspond to $M<0$. Consequently, stability exists
for a range of the model parameters if ${p_{\rm B}^5}>0$ (see
figure~\ref{fig3:sta123}). For ${p_{\rm B}^5}\leq 0$, all solutions are unstable.
\begin{figure}[H]
\center{\psfig{file=rcqfext05fig3.eps,width=0.42\hsize}}
\vspace{-0.2cm}
\caption{Plot of radion mass parameter $M$ for
$l/{r_{\rm c}}=5$. Thick line, $0<{p_{\rm B}^5}
\leq{{\rm e}^{-2\pi/5}}: \lambda>0, \lambda'\leq 0$. Thin line,
${{\rm e}^{-2\pi/5}}
<{p_{\rm B}^5}\leq 1: \lambda\geq 0,\lambda'>0$. Dashed line,
${p_{\rm B}^5}>1:
\lambda<0, \lambda'>0$.}
\label{fig3:sta123}
\end{figure}
\noindent For ${p_{\rm B}^5}>0$, the stability of the $\rm{AdS}_5$ braneworlds
also depends on the dimensionless ratio $l/{r_{\rm c}}$. For $l/{r_{\rm
c}}<1.589\cdots$, all solutions turn out to be
unstable. Stable universes begin to appear at $l/{r_{\rm
c}}=1.589\cdots, {p_{\rm B}^5}=0.138\cdots$. For $l/{r_{\rm
c}}>1.589\cdots$, we find stable solutions for an interval of
$p_{\rm B}^5$ (see in figure \ref{fig3:sta123} the example of $l/{r_{\rm
c}}=5$) which increases with $l/{r_{\rm c}}$. For large
enough but finite $l/{r_{\rm c}}$, the stability interval
approaches the limit $\left]0.267\cdots,3.731\cdots\right[$. Naturally, $M\to 0$ if $l/{r_{\rm c}}\to\infty$.
\section{Gauss-Codazzi equations and localization of\\ gravity}
For an observer confined to the brane,
the effective 4D Einstein equations are given by \cite{RC2,BC,SMS,RM}
\begin{eqnarray}
&{\mathcal{G}_\mu^\nu}={{2{\kappa_5^2}}\over{3}}
\left[{\mathcal{U}_\alpha^\beta}
{q_\mu^\alpha}{q_\beta^\nu}+\left({\mathcal{U}_\alpha^\beta}{n^\alpha}
{n_\beta}-{1\over{4}}{\mathcal{U}_\alpha^\alpha}\right){q_\mu^\nu}
\right]+{\mathcal{K}_\alpha^\alpha}{\mathcal{K}_\mu^\nu}\nonumber\\
&-{\mathcal{K}_\mu^\alpha}{\mathcal{K}_\alpha^\nu}-{1\over{2}}{q_\mu^\nu}
\left({\mathcal{K}^2}-{\mathcal{K}_\alpha^\beta}{\mathcal{K}_\beta^\alpha}
\right)-{\mathcal{E}_\mu^\nu},
\end{eqnarray}
where
${\mathcal{G}_\mu^\nu}={G_\alpha^\beta}{q^\alpha_\mu}{q_\beta^\nu}$, ${n^\mu}={\delta_5^\mu}$ is the unit
normal to the brane and
${q_\mu^\nu}={\delta_\mu^\nu}-{n_\mu}{n^\nu}$.
The stress-energy tensor is
${\mathcal{U}_\mu^\nu}=-{\Lambda_{\rm B}}{\Omega^2}(0){\delta_\mu^\nu}+
{\mathcal{T}_\mu^\nu}$, ${\mathcal{K}_\mu^\nu}$ is the extrinsic
curvature and ${\mathcal{E}_\mu^\nu}$ the traceless
projection of the 5D Weyl tensor. The 4D observer finds the same
dynamics on the brane because \cite{RC2}
\begin{equation}
{\mathcal{E}_a^b}={{\kappa_5^2}\over{3}}
\left(-{T_a^b}+{1\over{2}}{T_5^5}{\delta_a^b}
\right)
\end{equation}
and
\begin{equation}
{4\over{3}}
\left({U_a^b}+{1\over{4}}{U_5^5}{\delta_a^b}\right)-\left({\Lambda_{\rm B}}+
{{{\kappa_5^2}{\lambda^2}}\over{6}}\right){\delta_a^b}\Omega^2(0)=0.
\end{equation}
Since the tidal acceleration \cite{RC2,RM} is
${a_{\rm T}}={\kappa_5^2}{\Lambda_{\rm B}}{{(1+{p_{\rm B}^5})}^2}/6<0$,
the gravitational field is bound to the vicinity of the brane.
\section{Conclusions}
In this paper we have analised exact
5D solutions describing the dynamics of $\mbox{AdS}_5$
braneworlds when conformal fields of weight -4 are present in the bulk.
We have discussed their behaviour under
radion field perturbations and shown
that if the equation of state characterizing the conformal fluid is
independent of the perturbation, then the radion may be stabilized by a sector of the conformal fields while
another sector of the same class of fields generates the dynamics
on the brane. Stabilization requires a bulk fluid sector with a constant
negative 5D pressure and involves new warp functions. On the brane
these solutions are able to describe, for example, the dynamics of
inhomogeneous dust, generalized dark radiation and homogeneous
polytropic dark energy. More general 4D equations of state may also be
considered. This analysis is left for future work. We have also shown
that an effective Gauss-Codazzi
observer sees gravity localized near the brane and deduces
the same dynamics on the brane if she makes the same hyphotesis about
the 5D fields. Whether gravity is suficiently bound to the brane and the
hierarchy strong enough are open problems for future research.
\vspace{1cm}
\leftline{\large \bf Acknowledgements}
\vspace{0.25cm}
We would like to thank the financial support of Funda\c {c}\~ao
para a Ci\^encia e a Tecnologia (FCT) and Fundo Social
Europeu (FSE) under the contract
SFRH/BPD/7182\\/2001 (III Quadro Comunit\'ario de Apoio), of Centro
Multidisciplinar de Astrof\'{\i}sica (CENTRA) with project
FJ01-CENTRA and Conselho de Reitores das Universidades
Portuguesas (CRUP) with project Ac\c {c}\~ao Integrada
Luso-Espanhola E-126/04.
|
3,212,635,537,962 | arxiv | \section{Introduction}
Let $(X,d)$ be a complete metric space and let $T$ be a mapping from $X$ to $X$. Then $T$ is called \emph{nonexpansive} if for all $x,y \in X$,
$$d(T(x), T(y) ) \leq d(x,y).$$ A point $x \in X$ is called a \emph{fixed point} of $T$ if $T(x) = x$. $Fix(T)$ denotes the set of fixed points of $T$. Kirk proved the existence of fixed points for nonexpansive mappings on \catzero\ spaces in \cite{k} and \catk\ spaces in \cite{k2}.
A \catk\ space is a metric space in which no triangle is fatter than the triangle with the same edge lengths in a model space, which is the 2-dimensional, complete, simply-connected space of constant curvature $K$ (see Definition \ref{de3}). A \catk\ space is a generalization of a simply-connected Riemannian manifold with sectional curvature $\leq K$; we will introduce generalized definitions of convergence and notations including the sum $\oplus$, which interpolates between a pair of points along a geodesic, in Section 2.
In a \catk\ space $X$, for $t_n, s_n \in [0,1]$ and $x_0 \in X$, the Ishikawa iteration $\{ x_n \}$ is defined by
\begin{equation}\label{e3}
x_{n+1} = t_n T (y_n) \oplus (1 - t_n) x_n , \;\; n \geq 0,
\end{equation}
where
$y_n = s_n T(x_n) \oplus (1-s_n) x_n$.
For a nonexpansive mapping $T$,
Dhompongsa and Panyanak in \cite{dp} obtained a $\Delta$-convergence result for Ishikawa iterations in complete \catzero\ spaces under the conditions $$\sum^{\infty}_{n=0} t_n(1-t_n) = \infty, \sum^{\infty}_{n=0} (1-t_n) s_n < \infty \;\;\textit{and}\;\; \limsup_{n} s_n < 1.$$
In \cite{pl}, a similar result was proved by Panyanak and Laokul under the other conditions $$\sum^{\infty}_{n=0} t_n(1-t_n) = \infty \;\; \textit{and} \;\; \sum^{\infty}_{n=0} t_n(1-t_n) s_n < \infty.$$
In this paper, we will obtain $\Delta$-convergence results for Ishikawa iterations in complete \catk\ spaces (see Theorem \ref{t1} and Theorem \ref{t2}).
In \cite{h}, He, Fang, L$\acute{\mathrm{o}}$pez and Li studied the $\Delta$-convergence of Mann iterations in complete \catk\ spaces with the condition $$\sum^{\infty}_{n=0} t_n(1-t_n) = \infty.$$ Since the Mann iteration is given by \eqref{e3} when $s_n =0$ for all $n$, we provide an alternative proof of the $\Delta$-convergence theorem for Mann iterations in complete \catk\ spaces.
\section{Preliminaries}
Let $(X,d)$ be a metric space. The open ball centered at $p$ with radius $r$ is denoted by $B_r(p)$. The closed ball centered at $p$ with radius $r$ is denoted by $B_r[p]$.
A curve $\gamma: I \to X$ is called a \emph{geodesic} if for any two $t, t^\prime \in I$, $d(\gamma(t),\gamma(t^\prime))=|t-t^\prime|$.
We denote by $[xy]$, a unit-speed geodesic $\gamma: I \to X$ from $x$ to $y$ defined on $I=[0,t]$, where $\gamma(0)=x$, $\gamma(t)=y$ and $t=d(x,y)$.
By $\triangle xyz$, we denote the geodesic triangle of geodesics $[x y]$,$[x z]$ and $[y z]$.
Let $C$ be a positive constant. A metric space $X$ is a \emph{geodesic space} if any two points are joined by a geodesic; and a \emph{C-geodesic space} if any two points with distance $< C$ are joined by a geodesic.
A set $Y \subset X$ is \emph{convex} if any two points $x,y \in Y$ can be joined by a geodesic and all geodesics joining them are contained in $Y$. If this condition holds for any two points in $Y$ with distance $<C$, $Y$ is said to be \emph{C-convex}.
For a constant $K$, we use $M_K$ to denote the 2-dimensional, complete, simply-connected space of constant curvature $K$.
Then $M_0 = \mathbb{E}^2$, $M_1 = \mathbb{S}^2$ and $M_{-1} = \mathbb{H}^2$.
Let $d_K$ be the metric of $M_K$.
$D_K$ denotes the diameter of $M_K$. Thus, $D_K = {\pi \over \sqrt{K} }$ if $K > 0$ and $D_K = \infty$ if $K \leq 0$.
A triangle $\triangle \widetilde{x}_1\widetilde{x}_2\widetilde{x}_3$ in $M_K$ is called a \emph{comparison triangle} for $\triangle x_1x_2x_3$ in $X$ if $d_K(\widetilde{x}_i,\widetilde{x}_j)=d(x_i,x_j)$ for $i,j \in \{1,2,3\}$.
\begin{definition}\label{de3}
Let $(X,d)$ be a metric space and let $K$ be a real constant. A $D_K$-geodesic space $X$ is a \emph{\catk\ space} if for any geodesic triangle $\triangle xy_1y_2$ of perimeter $< 2 D_K$, and its comparison triangle $\triangle \widetilde{x}\widetilde{y}_1\widetilde{y}_2$ in $M_K$, we have $$d(z_1,z_2) \leq d_K(\widetilde{z}_1,\widetilde{z}_2),$$ where $z_i$ is any point on $[x y_i]$ and $\widetilde{z}_i$ is the point on $[\widetilde{x} \widetilde{y}_i]$ such that $d_K(\widetilde{x},\widetilde{z}_i)=d(x,z_i)$ for $i \in \{1,2\}$.
\end{definition}
We now record a few lemmas about \catk\ spaces that we will need in the sequel.
\begin{lemma}\cite[Page 160]{bh}\label{l456}
Let $X$ be a \catk\ space. \\
$(1)$ For any two points $x,y$ in $X$ with distance less than $D_K$, there is a unique geodesic $[x y]$ connecting them. \\
$(2)$ Any ball in $X$ with radius less than $D_K/2$ is convex.
\end{lemma}
\begin{lemma}\cite[Page 178]{bh}\label{l123}
Let $(X,d)$ be a \catk\ space and let $F$ be a closed and $D_K$-convex subset of $X$. Then for each point $x \in X$ such that $d(x,F) < D_K/2$, there is a unique point $y \in F$ such that $d(x,y) = d(x,F)$.
\end{lemma}
Let $(X,d)$ be a \catk\ space and let $x, y \in X$ such that $d(x,y) < D_K$. Then
$ t x \oplus (1-t) y $ denotes the unique point on $[x y]$ for $t \in [0,1]$ such that $d( x , tx \oplus (1-t) y ) = (1-t) d(x,y) $ and $ d( y , tx \oplus (1-t) y ) = t d(x,y) $.
\begin{lemma}\cite[Lemma 3.3]{p}\label{l}
For a positive number $C \leq \pi/2$, let $(X,d)$ be a \catone\ space and let $p, x, y \in X$ such that $d(p,x) \leq C$, $d(p,y) \leq C$ and $d(x,y) \leq C$.
Then for any $t \in [0,1]$, $$d( (1-t) p \oplus t x , (1-t) p \oplus t y ) \leq {\sin t C \over \sin C } d(x,y).$$
\end{lemma}
We can get the next lemma by following the proof of Prop. 3.1 in \cite{o} with $\varepsilon=\pi/4$.
\begin{lemma}\label{l0}
Let $(X,d)$ be a \catone\ space. Then there is a constant $k>0$ such that
$$d^2(x, ty \oplus (1-t)z ) \leq t d^2 ( x,y) + (1-t) d^2 ( x,z) - { k \over 2} t(1-t) d^2 (y,z)$$ for any $t \in [0,1]$
and any points $x, y, z \in X$ such that $d(x,y) \leq \pi/4$, $d(x,z) \leq \pi/4$ and $d(y,z) \leq \pi/2$.
\end{lemma}
For a bounded sequence $\{x_n\}$ in $X$, define $$r(x,\{x_n\}) = \limsup_{ n \to \infty} d(x,x_n).$$ The \textit{asymptotic radius} of $\{x_n\}$ is defined by $$r(\{x_n\}) = \inf \{ r(x,\{x_n\}) : x \in X \}.$$
The \textit{asymptotic center} of $\{x_n\}$ is defined by $$A( \{x_n\} ) = \{ x \in X : r(x, \{x_n\} ) = r( \{x_n\} ) \}.$$
Now we can give a definition of $\Delta$-convergence, and list a few properties.
\begin{definition}
For a bounded sequence $\{x_n\}$ in $X$, the sequence $\{x_n\}$ is said to \emph{$\Delta$-converge to} $x \in X$ if $x$ is the unique asymptotic center of $\{u_n\}$ for every subsequence $\{u_n\}$ of $\{x_n\}$. In this case, we will write $\Delta-\lim_{n} x_n = x$ and call $x$ the \emph{$\Delta$-limit of} $\{x_n\}$.
\end{definition}
Then we have a lemma to show a property of a sequence which $\Delta$-converges.
\begin{lemma}\cite[Prop. 2.3]{h}\label{l000}
Let $(X,d)$ be a complete \catk\ space and let $p \in X$. Suppose that a sequence $\{ x_n \}$ $\Delta$-converges to $x$ such that $r(p,\{x_n\}) < D_K/2$. Then $$d(x,p) \leq \liminf_{ n \to \infty} d(x_n,p).$$
\end{lemma}
\begin{definition}
Let $(X,d)$ be a complete metric space and let $F$ be a nonempty subset of $X$. Then a sequence $\{x_n \}$ in $X$ is \emph{Fej$\acute{\textit{e}}$r monotone} with respect to $F$ if $$d( x_{n+1} , q) \leq d(x_n , q)$$ for all $n \geq 0$ and all $q \in F$.
\end{definition}
\begin{definition}\cite{h}\label{de5}
For a sequence $\{x_n\}$ in $X$, a point $x \in X$ is a \emph{$\Delta$-cluster point} of $\{x_n\}$ if there exists a subsequence of $\{ x_n \}$ that $\Delta$-converges to $x$.
\end{definition}
With Definition \ref{de5}, we will know when a sequence $\{ x_n \}$ in $X$ $\Delta$-converges to a point of $F$ if $\{ x_n \}$ is Fej$\acute{\textit{e}}$r monotone with respect to $F$.
\begin{lemma}\cite[Lemma 3.2]{h}\label{l00}
Let $X$ be a complete \catk\ space and let $F$ be a nonempty subset of $X$. Suppose that the sequence $\{ x_n \}$ of $X$ is Fej$\acute{\textit{e}}$r monotone with respect to $F$ and the asymptotic radius $r(\{x_n\})$ of $\{x_n\}$ is less than $D_K/2$. If any $\Delta$-cluster point $x$ of $\{ x_n \}$ belongs to $F$, then the sequence $\{x_n\}$ $\Delta$-converges to a point of $F$.
\end{lemma}
\begin{lemma}\label{l1}\cite{z1}\cite{z2}
Suppose that $\{ a_n \}$ and $\{ b_n \}$ satisfy that
$$ a_n \geq 0,\; b_n \geq 0 \;\; \textit{and} \;\; a_{n+1} \leq (1 + b_n)a_n$$
for all $n \geq 0$.
If $\sum^{\infty}_{n=0} b_n$ converges, then $\lim_{ n \to \infty} a_n$ exists. Additionally, if there is a subsequence of $\{ a_n \}$ which converges to $0$, then $\lim_{ n \to \infty} a_n=0$.
\end{lemma}
\section{Ishikawa iteration process on \catk\ spaces}
\begin{lemma}\label{l3}\cite[Theorem 3.4]{p}
Let $X$ be a complete \catone\ space and let $T: X \to X$ be a nonexpansive mapping such that $F:=\mathrm{Fix}(T) \neq \emptyset$. Then $F$ is closed and $\pi$-convex.
\end{lemma}
\begin{lemma}\label{l5}
Let $X$ be a complete \catone\ space and let $T: X \to X$ be a nonexpansive mapping such that $F:=\mathrm{Fix}(T) \neq \emptyset$. If $\{x_n \}$ is defined by \eqref{e3} for $x_0 \in X$ such that $d(x_0,F) \leq \pi/4$, then there is a unique point $p$ in $F$ such that $x_n$, $y_n$, $T(x_n)$ and $T(y_n)$ are at distance $\leq d(x_0,p)$ from $p$.
\end{lemma}
\begin{proof}
Since $d(x_0,F) \leq \pi/4$, by Lemma \ref{l123} and \ref{l3}, there is a unique point $p$ in $F$ such that $d(x_0,p) = d(x_0,F)$.
By induction, we want to show that $$d(p, y_n ) \leq d(p, x_n) \leq d(p, x_0).$$
For $n=0$, since $T$ is nonexpansive, we have $d(p, T(x_0) ) \leq d( p, x_0 ) \leq \pi/4$. Since $B_{\pi/4}[p]$ is convex, we get $$d(p,y_0) = d(p, s_0 T(x_0) \oplus (1-s_0) x_0) \leq d(p,x_0).$$
Suppose that $d(p, y_n) \leq d(p, x_n) \leq d(p, x_0)$.
Since $d(p, T(y_n) ) \leq d( p, y_n ) \leq \pi/4$ and $B_{\pi/4}[p]$ is convex, $$d(p,x_{n+1}) = d(p, t_n T(y_n) \oplus (1- t_n ) x_n) \leq d(p,x_n).$$
Since $d(p, T( x_{ n+1} ) ) \leq d( p, x_{n+1} ) \leq \pi/4$, $$d(p, y_{n+1}) = d(p, s_{n+1} T(x_{n+1}) \oplus (1-s_{n+1}) x_{n+1}) \leq d(p,x_{n+1}).$$
Therefore $$d(p, y_{n+1}) \leq d(p, x_{n+1}) \leq d(p, x_{n}).$$
\end{proof}
We will prove Lemma \ref{l2} and Lemma \ref{l4}, which are obtained by following the proofs in \cite{pl}.
If $d(x_0,F)=0$, then $x_0 \in F$ and hence by definition \eqref{e3}, $x_n=x_0$ for all $n$. So we just consider the case $d(x_0,F)>0$.
\begin{lemma}\label{l2}
Let $X$ be a complete \catone\ space and let $T: X \to X$ be a nonexpansive mapping such that $F:=\mathrm{Fix}(T) \neq \emptyset$. If $\{x_n \}$ is defined by \eqref{e3} for $x_0 \in X$ such that $d(x_0,F) \leq \pi/4$, then
$$d( T (x_{n+1}), x_{n+1} ) \leq [ 1 + 4 { C \over \sin C } t_n (1-t_n) s_n ] d( T (x_n) , x_n )$$ for all $n \geq 0$ where $C:= 2 d(x_0,F)$.
\end{lemma}
\begin{proof}
Since $d (T (x_n) , t_n T(x_n) \oplus (1-t_n)x_n ) = ( 1 - t_n )d(T(x_n), x_n)$ and $T$ is nonexpansive, we get
\begin{equation*}
\begin{split}
d( T (x_{n+1}), x_{n+1} ) &\leq d(T (x_{n+1}) , T( t_n T(x_n) \oplus (1-t_n) x_n) ) \\ &+ d( T( t_n T(x_n) \oplus (1-t_n) x_n) , T (x_n) ) \\
&+ d (T (x_n) , t_n T(x_n) \oplus (1-t_n)x_n ) \\ &+ d( t_n T(x_n) \oplus (1-t_n) x_n , x_{n+1} ) \\
&\leq 2 d( t_n T(x_n) \oplus (1-t_n) x_n , x_{n+1} ) \\ &+ d( t_n T(x_n) \oplus (1-t_n) x_n , x_n ) \\ &+ ( 1 - t_n )d(T(x_n), x_n). \\
\end{split}
\end{equation*}
Since $ d( t_n T(x_n) \oplus (1-t_n) x_n , x_n ) = t_n d(T(x_n), x_n)$,
it becomes
\begin{equation*}
d( T (x_{n+1}), x_{n+1} ) \leq 2 d( t_n T(x_n) \oplus (1-t_n) x_n , x_{n+1} ) + d(T(x_n), x_n).
\end{equation*}
Note that $d(x_n,T(x_n)) \leq C$, $d(x_n,T(y_n)) \leq C$ and $d(T(x_n),T(y_n)) \leq C$ from Lemma \ref{l5}. Since $d( x_n ,y_n) = s_n d( T (x_n), x_n)$ and $C \leq \pi /2$, by Lemma \ref{l},
\begin{equation*}
\begin{split}
d( T (x_{n+1}), x_{n+1} )
&\leq 2 {\sin t_n C \over \sin C } d( T(x_n) , T(y_n) ) + d(T(x_n), x_n) \\
&\leq 2 { t_n C \over \sin C } d( x_n ,y_n) + d(T(x_n), x_n) \\
&= (1 + 2 { t_n C \over \sin C } s_n) d( T (x_n), x_n). \\
\end{split}
\end{equation*}
Then, multiplying by $(1 - t_n)$, we have
\begin{equation}\label{e1}
(1 - t_n) d(T(x_{n+1}), x_{n+1} ) \leq [ 1 - t_n + 2 { C \over \sin C } t_n(1-t_n)s_n ] d( T (x_n), x_n).
\end{equation}
Also, since $d (T (y_n) , t_n T(y_n) \oplus (1-t_n)y_n ) = ( 1 - t_n )d(T(y_n), y_n)$ and $T$ is nonexpansive, we get
\begin{equation*}
\begin{split}
d( T (x_{n+1}), x_{n+1} ) &\leq d(T (x_{n+1}) , T( t_n T(y_n) \oplus (1-t_n) y_n) ) \\ &+ d( T( t_n T(y_n) \oplus (1-t_n) y_n) , T (y_n) ) \\
&+ d (T (y_n) , t_n T(y_n) \oplus (1-t_n)y_n ) \\ &+ d( t_n T(y_n) \oplus (1-t_n) y_n , x_{n+1} ) \\
&\leq 2 d( t_n T (y_n) \oplus (1-t_n) y_n , x_{n+1} ) \\ &+ d( t_n T(y_n) \oplus (1-t_n) y_n , y_n ) \\ &+ ( 1 - t_n )d(T(y_n), y_n). \\
\end{split}
\end{equation*}
Since $d( t_n T(y_n) \oplus (1-t_n) y_n , y_n ) = t_n d(T(y_n), y_n)$,
it becomes
\begin{equation*}
d( T (x_{n+1}), x_{n+1} ) \leq 2 d( t_n T (y_n) \oplus (1-t_n) y_n , x_{n+1} ) + d(T(y_n), y_n).
\end{equation*}
Since $d(x_n,T(y_n)) \leq C$, $d(y_n,T(y_n)) \leq C$ and $d(x_n,y_n) \leq C$, by Lemma \ref{l},
\begin{equation*}
\begin{split}
d( T (x_{n+1}), x_{n+1} )
&\leq 2 {\sin (1-t_n) C \over \sin C } d( x_n , y_n ) + d(T(y_n), y_n) \\
&\leq 2 { (1-t_n) C \over \sin C } d( x_n ,y_n) + d(T(y_n), y_n) \\
&\leq 2 { (1-t_n) C \over \sin C } d( x_n ,y_n) + d(T(y_n), T(x_n)) + d(T (x_n) , y_n) \\
&\leq 2 { (1-t_n) C \over \sin C } d( x_n ,y_n) + d( y_n, x_n) + d(T (x_n) , y_n) \\
&\leq 2 { (1-t_n) C \over \sin C } s_n d( x_n ,T (x_n) ) + s_n d( T (x_n), x_n) \\ &+ (1-s_n) d(T (x_n) , x_n) \\
&= [1 + 2 { (1-t_n) C \over \sin C } s_n ]d( T (x_n) , x_n ). \\
\end{split}
\end{equation*}
Then we have
\begin{equation}\label{e2}
t_n d( T (x_{n+1}), x_{n+1} ) \leq [t_n + 2 { C \over \sin C } t_n (1-t_n) s_n ]d( T (x_n) , x_n ).
\end{equation}
From \eqref{e1} and \eqref{e2}, we get
$$d( T (x_{n+1}), x_{n+1} ) \leq [ 1 + 4 { C \over \sin C } t_n (1-t_n) s_n ] d( T (x_n) , x_n ).$$
\end{proof}
\begin{lemma}\label{l4}
Let $X$ be a complete \catone\ space and let $T: X \to X$ be a nonexpansive mapping such that $F:=\mathrm{Fix}(T) \neq \emptyset$. Suppose that $\{t_n \}$ and $\{s_n \}$ satisfy that
$$\sum^{\infty}_{n=0} t_n(1-t_n) = \infty \;\; \textit{and} \;\;
\sum^{\infty}_{n=0} t_n(1-t_n) s_n < \infty.$$
If $\{x_n \}$ is defined by \eqref{e3} for $x_0 \in X$ such that $d(x_0,F) \leq \pi/4$, then $$\lim_{ n \to \infty} d(T (x_n) , x_n ) = 0.$$
\end{lemma}
\begin{proof}
By Lemma \ref{l1} and \ref{l2}, $\lim_{ n \to \infty} d(T (x_n) , x_n )$ exists.
Let $p$ be the unique point in $F$ such that $d(x_0,p) = d(x_0,F)$.
By Lemma \ref{l0}, we get
\begin{equation*}
\begin{split}
d^2(p,x_{n+1}) &= d^2(p, t_n T (y_n) \oplus (1 - t_n) x_n ) \\
&\leq t_n d^2(p, T (y_n) ) + (1-t_n) d^2 (p, x_n) - { k \over 2} t_n(1-t_n) d^2 (T (y_n) , x_n). \\
\end{split}
\end{equation*}
Then since $T$ is nonexpansive, we have
\begin{equation}\label{e4}
d^2(p,x_{n+1}) \leq t_n d^2(p, y_n ) + (1-t_n) d^2 (p, x_n) - { k \over 2} t_n(1-t_n) d^2 (T (y_n) , x_n).
\end{equation}
By Lemma \ref{l0}, also we get
\begin{equation*}
\begin{split}
d^2(p,y_{n}) &= d^2(p, s_n T (x_n) \oplus (1 - s_n) x_n ) \\
&\leq s_n d^2(p, T (x_n) ) + (1-s_n) d^2 (p, x_n) - { k \over 2} s_n(1-s_n) d^2 (T (x_n) , x_n). \\
\end{split}
\end{equation*}
Since $T$ is nonexpansive,
\begin{equation}\label{e5}
\begin{split}
d^2(p,y_{n})
&\leq s_n d^2(p, x_n ) + (1-s_n) d^2 (p, x_n) - { k \over 2} s_n(1-s_n) d^2 (T (x_n) , x_n) \\
&\leq d^2(p, x_n ). \\
\end{split}
\end{equation}
By \eqref{e4} and \eqref{e5}, we obtain $$ d^2(p,x_{n+1}) \leq d^2(p,x_{n}) - { k \over 2} t_n(1-t_n) d^2 (T (y_n) , x_n).$$
This implies
\begin{equation}\label{e6}
{ k \over 2} \sum^{\infty}_{n = 0} t_n(1-t_n) d^2 (T (y_n) , x_n) \leq d^2(p,x_0) < \infty.
\end{equation}
Since $\sum^{\infty}_{n=0} t_n(1-t_n) s_n < \infty$, from \eqref{e6}, we get
$$\sum^{\infty}_{n = 0} t_n(1-t_n) [ d^2 (T (y_n) , x_n) + s_n] < \infty.$$
Since $\sum^{\infty}_{n=0} t_n(1-t_n) = \infty$, it implies
$$ \liminf_{ n \to \infty} [ d^2 (T (y_n) , x_n) + s_n] =0.$$
Then there exists a subsequence $\{n_k \}$ of $\{n \}$ such that
\begin{equation}\label{e7}
\lim_{k \to \infty} d (T (y_{n_k}) , x_{n_k}) = 0 \;\;\;\;\mathrm{and} \;\; \lim_{k \to \infty} s_{n_k} = 0.
\end{equation}
Also,
\begin{equation*}
\begin{split}
d (T (x_{n_k}) , x_{n_k}) &\leq d (T (x_{n_k}) , T (y_{n_k})) + d (T (y_{n_k}) , x_{n_k}) \\
& \leq d ( x_{n_k} , y_{n_k}) + d (T (y_{n_k}) , x_{n_k}) \\
& = s_{n_k} d (T (x_{n_k}) , x_{n_k}) + d (T (y_{n_k}) , x_{n_k}).
\end{split}
\end{equation*}
Then it becomes
\begin{equation}\label{e8}
(1- s_{n_k}) d (T (x_{n_k}) , x_{n_k}) \leq d (T (y_{n_k}) , x_{n_k}).
\end{equation}
From \eqref{e7} and \eqref{e8}, we get
\begin{equation}\label{e14}
\lim_{k \to \infty} d (T (x_{n_k}) , x_{n_k}) = 0.
\end{equation}
Since $\lim_{ n \to \infty} d(T (x_n) , x_n )$ exists, \eqref{e14} implies that $$\lim_{ n \to \infty} d(T (x_n) , x_n )=0.$$
\end{proof}
We will prove Theorem \ref{t1}, which is obtained by following a first part of the proof in \cite[Theorem 3.1]{h}.
\begin{theorem}\label{t1}
Let $X$ be a complete \catk\ space and let $T: X \to X$ be a nonexpansive mapping such that $F:=\mathrm{Fix}(T) \neq \emptyset$.
Suppose that $\{x_n\}$ is defined by \eqref{e3} under the conditions $$\sum^{\infty}_{n=0} t_n(1-t_n) = \infty
\;\; \textit{and} \;\;
\sum^{\infty}_{n=0} t_n(1-t_n) s_n < \infty.$$ Then, for each $x_0 \in X$ with $d(x_0, F ) < D_K /4 $, the sequence $\{x_n\}$ $\Delta$-converges to a point of $F$.
\end{theorem}
\begin{proof}
Rescaling the metric by $1/\sqrt{K}$, we may assume that $K=1$.
Set $F_0:= F \cap B_{\pi/2}(x_0)$.
For any $q \in F_0$, since $d(T(x_0) ,q ) \leq d(x_0,q ) $ and since the open ball $B_r(q)$ in $X$ with radius $r < \pi/2$ is convex, we have $$d( y_0 , q) = d( s_0 T(x_0) \oplus (1-s_0) x_0 , q ) \leq d(x_0,q ).$$ Similarly, since $d(T(y_0) ,q ) \leq d(y_0,q )$ and since the open ball $B_r(q)$ is convex,
we have $$d(x_1 , q) = d( t_0 T (y_0) \oplus (1 - t_0) x_0 ,q) \leq d(x_0 ,q).$$
Using mathematical induction, we can easily get that $$d( x_{n+1} , q ) \leq d( x_{n} , q ) \leq d( x_0 , q )$$ for all $n \geq 0$.
Therefore the sequence $\{x_n\}$ is Fej$\acute{\mathrm{e}}$r monotone with respect to $F_0$.
Let $p$ be the unique point in $F$ such that $d(x_0,p) = d(x_0,F)$. Then $p \in F_0$. Also
we get
\begin{equation}\label{e11}
d( x_{n+1} , p ) \leq d( x_{n} , p ) \leq d( x_0 , p ) < \pi/4
\end{equation}
for all $n \geq 0$. This means that the asymptotic radius $r(\{x_n \})$ of $\{x_n\}$ is less than $\pi/4$.
By Lemma \ref{l00},
we only need to show that for each point $x$ such that there exists a subsequence $\{ x_{n_k} \}$ of $\{ x_n \}$ which $\Delta$-converges to $x$, it belongs to $F_0$. From \eqref{e11}, note that $r(p, \{x_n\}) \leq d(x_0,p) < \pi/4$.
By Lemma \ref{l000}, we obtain
$$d(x,x_0) \leq d(x,p) + d(x_0,p) \leq \liminf_{ k } d(x_{n_k},p) + d(x_0,p) < \pi/2,$$
that is,
\begin{equation}\label{e12}
x \in B_{\pi/2}(x_0).
\end{equation}
By Lemma \ref{l4}, we have
\begin{equation*}
\begin{split}
\limsup_{k} d( T(x) , x_{n_k}) &\leq \limsup_{k} d( T(x) , T (x_{n_k})) + \limsup_{k} d( T (x_{n_k}) , x_{n_k}) \\
&\leq \limsup_{k} d( x , x_{n_k}).
\end{split}
\end{equation*}
This implies that $T(x) \in A(\{x_{n_k}\})$ and $T(x)=x$. Therefore $x \in F$. With \eqref{e12}, $x$ belongs to $F_0$. By Lemma \ref{l00}, it is proved.
\end{proof}
Since the following corollary is a direct result by letting $K=0$ in Theorem \ref{t1}, Theorem \ref{t1} is the extended result of the corresponding one in \cite{pl}.
\begin{cor}
Let $X$ be a complete \catzero\ space and let $T: X \to X$ be a nonexpansive mapping such that $F:=\mathrm{Fix}(T) \neq \emptyset$.
Suppose that $\{x_n\}$ is defined by \eqref{e3} under the conditions $$\sum^{\infty}_{n=0} t_n(1-t_n) = \infty
\;\; \textit{and} \;\;
\sum^{\infty}_{n=0} t_n(1-t_n) s_n < \infty.$$ Then, for each $x_0 \in X$, the sequence $\{x_n\}$ $\Delta$-converges to a point of $F$.
\end{cor}
For the conditions $\sum^{\infty}_{n=0} t_n(1-t_n) = \infty$, $\sum^{\infty}_{n=0} (1-t_n) s_n < \infty$ and $\limsup_{n} s_n < 1$, we will get the following lemma, which is an analog of Lemma 2.12 in \cite{dp}.
\begin{lemma}\label{l7}
Let $X$ be a complete \catone\ space and let $T: X \to X$ be a nonexpansive mapping such that $F:=\mathrm{Fix}(T) \neq \emptyset$. Suppose that $\{t_n \}$ and $\{s_n \}$ satisfy that
$$\sum^{\infty}_{n=0} t_n(1-t_n) = \infty, \sum^{\infty}_{n=0} (1-t_n) s_n < \infty \;\; \textit{and} \;\; \limsup_{n} s_n < 1.$$
If $\{x_n \}$ is defined by \eqref{e3} for $x_0 \in X$ such that $d(x_0,F) < \pi/4$, then $$\lim_{ n \to \infty} d(T (x_n) , x_n ) = 0.$$
\end{lemma}
\begin{proof}
From Equation \eqref{e6} in Lemma \ref{l4}, $${ k \over 2} \sum^{\infty}_{n = 0} t_n(1-t_n) d^2 (T (y_n) , x_n) \leq d^2(p,x_0) < \infty$$ for the unique point $p$ in $F$ such that $d(x_0,p) = d(x_0,F)$. Since $\sum^{\infty}_{n=0} t_n(1-t_n)$ diverges, it means that $$\liminf_n d^2 (T (y_n) , x_n) = 0$$ and then
\begin{equation}\label{e9}
\liminf_n d (T (y_n) , x_n) = 0.
\end{equation}
Since $T$ is nonexpansive and $d( x_n, y_n )=s_n d( T(x_n) , x_n )$,
\begin{equation*}
\begin{split}
d( T(x_n) ,x_n) &\leq d( T(x_n),T(y_n) ) + d( T(y_n) , x_n) \\
& \leq d( x_n, y_n ) + d( T(y_n) , x_n) \\
& = s_n d( T(x_n) , x_n ) + d( T(y_n) , x_n).
\end{split}
\end{equation*}
Then we get $$d( T(x_n) ,x_n ) \leq { 1 \over 1 - s_n} d ( T(y_n) , x_n).$$
By \eqref{e9},
\begin{equation}\label{e10}
\liminf_n d (T (x_n) , x_n) = 0.
\end{equation}
Since $d(x_0,F) < \pi/4$, by Lemma \ref{l5}, $x_n$ and $T(y_n)$ are in the open ball centered at $T(x_{n+1})$ with radius $< \pi/2$.
Since this open ball is convex, we have
\begin{equation*}
\begin{split}
d( T(x_{n+1}) , x_{n+1} ) &\leq t_n d( T(x_{n+1}) , T(y_n) ) + (1-t_n) d( T(x_{n+1}) , x_n ) \\
&\leq t_n d( x_{n+1} , y_n ) + (1-t_n) [d( T(x_{n+1}) , x_{n+1} ) + d( x_{n+1} , x_n ) ] \\
&\leq t_n d( x_{n+1} , y_n ) + (1-t_n) [d( T(x_{n+1}) , x_{n+1} ) + t_n d( T(y_n) , x_n ) ]. \\
\end{split}
\end{equation*}
Dividing by $t_n$, this becomes
$$ d( T(x_{n+1}) , x_{n+1} ) \leq d( x_{n+1} , y_n ) + (1-t_n) d( T(y_n) , x_n ).$$
By Lemma \ref{l5}, $T(y_n)$ and $x_n$ are in the open ball centered at $y_n$ with radius $< \pi/2$, which is convex.
This yields
\begin{equation*}
\begin{split}
d( T(x_{n+1}) , x_{n+1} ) &\leq t_n d( T(y_n) , y_n ) + (1-t_n) d(x_n,y_n) + (1-t_n) d( T(y_n) , x_n ). \\
\end{split}
\end{equation*}
Since $T(x_n)$ and $x_n$ are in the open ball centered at $T(y_n)$ with radius $< \pi/2$, we have
\begin{equation*}
\begin{split}
d( T(x_{n+1}) , x_{n+1} ) &\leq t_n [ s_n d( T(y_n) , T(x_n) ) + ( 1- s_n) d( T(y_n) , x_n) ] \\ &+ (1-t_n) d(x_n,y_n) + (1-t_n) d( T(y_n) , x_n ). \\
\end{split}
\end{equation*}
Since $T$ is nonexpansive and $d( x_n, y_n )=s_n d( T(x_n) , x_n )$, we get
\begin{equation*}
\begin{split}
d( T(x_{n+1}) , x_{n+1} ) &\leq (1-t_n + t_n s_n ) d(x_n,y_n) + (1 - t_n s_n) d( T(y_n) , x_n ) \\
&\leq s_n (1-t_n + t_n s_n ) d(x_n, T(x_n) ) \\
&+ (1 - t_n s_n) [ d( T(y_n) , T(x_n) ) + d( T(x_n) , x_n ) ]\\
&\leq [s_n (1-t_n + t_n s_n ) + (1 - t_n s_n)(1 + s_n) ] d(x_n, T(x_n) ) \\
&= [ 1 + 2 s_n ( 1 - t_n) ] d(x_n, T(x_n) ).
\end{split}
\end{equation*}
Then we get the following inequality
\begin{equation}\label{e13}
d( T(x_{n+1}) , x_{n+1} ) \leq [ 1 + 2 s_n ( 1 - t_n) ] d(x_n, T(x_n) ).
\end{equation}
Since $\sum s_n ( 1 - t_n)$ converges, applying Lemma \ref{l1} to \eqref{e13}, $\lim_{ n \to \infty} d(T (x_n) , x_n ) $ exists. By \eqref{e10}, it is equal to zero.
\end{proof}
By following the same proof of Theorem \ref{t1} and using Lemma \ref{l7}, we obtain
\begin{theorem}\label{t2}
Let $X$ be a complete \catk\ space and let $T: X \to X$ be a nonexpansive mapping such that $F:=\mathrm{Fix}(T) \neq \emptyset$. Suppose that $\{x_n\}$ is defined by \eqref{e3} under the conditions
$$\sum^{\infty}_{n=0} t_n(1-t_n) = \infty, \sum^{\infty}_{n=0} (1-t_n) s_n < \infty \;\; \textit{and} \;\; \limsup_{n} s_n < 1.$$
Then, for each $x_0 \in X$ with $d(x_0, F ) < D_K /4 $, the sequence $\{x_n\}$ $\Delta$-converges to a point of $F$.
\end{theorem}
\section*{Acknowledgements}
\small The author would like to express his gratitude to Prof. R. Ghrist for his support.
He gratefully acknowledge support from the ONR Antidote MURI project, grant no. N00014-09-1-1031.
|
3,212,635,537,963 | arxiv |
\section{The Set Cover Problem}\label{sec:scp}
The set cover problem is one of the open problems in computer science and operations research.
It has many real-life applications, such as crew-scheduling for trains and airlines, nurse scheduling,
and location selection of facilities (e.g., fire stations and schools).
Given a set of $m$ instances $\mathbb{M}=\left\{e_1,e_2,\ldots,e_i,\ldots,e_m\right\}$ (the universe) and a family
of $n$ subsets $\mathbb{N}=\left\{S_1,S_2,\ldots,S_j,\ldots,S_n\right\}$, whose union is the universe,
i.e., $\cup_{S_j \in \mathbb{N}} S_j = \mathbb{M}$,
the set cover problem is to find the subfamily (or subfamilies)
from the $n$ subsets, $\mathbb{S} \subseteq \mathbb{N}$, with the {\it minimum} cost that covers the entire universe.
In other words, the problem is, among all the combinations of subsets from $\mathbb{N}$ whose union is the universe,
identify the one that has the minimum cost.
If the cost for each subset is the same, then SCP is equivalent to finding the smallest number of subsets
to include all the instances in $\mathbb{M}$. In the following formulation,
we treat weighted set cover problem, i.e., with non-uniform costs, as the default.
\subsection{The integer linear programming formulation}\label{sec:formalation}
Mathematically, SCP can be formulated as an integer linear programming problem as follows.
If there are $m$ rows (instances in the universe) to cover and there are $n$ columns (subsets in the family)
to select from, we define a $m \times n$ binary matrix $\bs{A}$ to represent their relationships.
If column $j$ covers row $i$, then the element $a_{ij} = 1$, while $a_{ij} = 0$ if otherwise.
If the cost of column $j$ is $c_j$, SCP is to find a set of columns $\mathbb{S}$ with the minimum cost $v({\rm SCP})$,
subject to that all rows must be covered. Formally, SCP is to
\begin{eqnarray}
\text{Minimize}\ & \bs{c}^T \bs{x} \equiv \sum\limits_{j \in N} c_j x_j \label{eq:SCP}\\
\text{subject to}\ & \nonumber \\
& \bs{A}_i \bs{x} \equiv \sum\limits_{j \in N} a_{ij} x_j \geqslant 1\mathrm{,} \ i \in M\mathrm{,} \label{eq:allrows} \\
& x_j \in \{0,1\}\mathrm{,}\ j \in N\mathrm{,} \label{eq:integer}
\end{eqnarray}
\noindent where we have used $M$ and $N$ to represent the set of indices of the instances in $\mathbb{M}$
and that of subsets in $\mathbb{N}$ for brevity. We express the solution to SCP (i.e., the minimum set) as
\begin{equation}
S=\left\{j: x_j=1\right\}\mathrm{,}
\end{equation}
\noindent and the total cost of this set as
\begin{equation}
v({\rm SCP}) = \sum\limits_{j \in S} c_j\mathrm{.}
\end{equation}
\noindent If the costs of columns are the same ($c$), then the minimum total cost is simply $v({\rm SCP})=c\,\lvert S \rvert$,
where $\lvert S \rvert$ is the cardinality (the number of elements) of $S$.
\begin{figure*}
\epsscale{0.95}
\plotone{SCP_example.pdf}
\caption{An example of the set cover problem: build schools in a county with the minimum cost, subject to that
no child is left behind, assuming each school costs the same and students can go to the school in their home precinct
or one in a directly neighboring precinct unless there is a natural barrier in between.
The left figure presents the geographical layout and the right figure shows the matrix form of
the relationships between the schools and (the students in) the precincts,
with an element value $a_{ij}=1$ (\texttt{True}) indicating that a given column (the school) $j$
can cover the row (precinct) $i$. There are multiple solutions to this problem, e.g., \{2,4,10\},
\{3,7,10\}.\footnote{Are these solutions optimal? Are there other optimal solutions? We leave these interesting questions to the reader.}
}
\vspace{0.2cm}
\label{fig:schools}
\end{figure*}
While analyzing the problem, we need to consider the reciprocal relationship between rows and columns.
We use $I_j$ to denote the set of rows covered by column $j$,
\begin{equation}
I_j = \left\{ i \in M: a_{ij} =1 \right\}\mathrm{,}\ j \in N\mathrm{,}
\end{equation}
and $J_i$ to represent the set of columns that cover row $i$,
\begin{equation}
J_i = \left\{ j \in N: a_{ij} =1 \right\}\mathrm{,}\ i \in M\mathrm{.}
\end{equation}
\vspace{0.1in}
\subsection{A simple example}\label{sec:buildschool}
To understand the simplicity of the set cover problem and the complexity of its solution,
it is instructive to consider a simple yet concrete case.
We consider an example in which we seek to build a minimum number of schools in a county.\footnote{This example is
a modified version of a problem from the lecture notes by Michael A. Trick, CMU (1997):
\texttt{http://mat.gsia.cmu.edu/orclass/integer/node8.html}.}
We illustrate the problem in Figure~\ref{fig:schools}.
In this example, there are $11$ precincts in the county. If we build a school in a precinct,
students in this precinct and its bordered precincts can attend this school,
unless there is a natural barrier in between, such as a dangerous river.
Under these restrictions, a school in precinct $10$ covers precincts $8$, $9$, $10$ and $11$,
and a school in precinct $11$ covers precincts $10$ and $11$, or
$I_{10}=\left\{8,9,10,11\right\}$ and $I_{11} = \left\{10,11\right\}$ using the notations above.
We can then define a binary matrix $\bs{A}$ to describe the relationship between schools (the columns)
and precincts (rows), which we show in the right panel of Figure~\ref{fig:schools}.
If we further assume each school costs the same,
then the problem becomes how to select a minimum number of precincts to build a school,
subject to the condition that no child is left behind.
We will refer back to this school-location example while discussing some key aspects of SCP below.
We invite the reader to think how they would solve the problem before move on to the rest of the paper.
The caption includes more information regarding the solution.
We would like to note that the examples we describe in this paper all have a symmetric, square
binary relationship matrix and uniform cost for all the subsets (columns) for simplicity.
However, the set cover problem and the algorithms we describe below do not have such restrictions.
In the school-location example, if we assume one of the precincts (say $9$) is not eligible and
needs to be excluded and then $\bs{A}$ would become an $11\times10$ matrix, and if the costs
of schools in different precincts are different, all the discussions and methods still apply.
\subsection{The NP completeness}\label{sec:NPcomplete}
In principle, SCP can be solved by an exhaustive search, i.e., a search over all the combinations of
the subsets. In the school-location example, one can select one precinct, or a combination of two, or three,
and see if any of the combinations can cover all the $m$ (11) precincts in the whole county.
The worst-case scenario is that we need to go through all the combinations,
in which case the total number of combinations is given by the sum of the binomial coefficients,
\begin{equation}
\sum\limits_{k=1}^{k=n} \binom{n}{k} = 2^{n}-1\,\mathrm{.}
\end{equation}
Such a brute-force approach therefore has an exponential time complexity $\mathcal{O}(m\,2^n)$.
If we have $10^3$ instances in the dataset, the number of required operations is
of the order of $2^{1000} \sim 10^{300}$, which is more than the total number of atoms in the whole observable Universe.
Although some techniques of data pre-processing, such as removing obvious redundant columns, can
reduce the complexity by a small factor, there is no existing exact algorithm with polynomial time complexity.
The problem was proved to be NP-complete by \citet{karp1972a},
where NP refers to {\it non-deterministic polynomial time},
and is directly related to one of the millennium prize problems, P$\stackrel{?}{=}$NP.
A detailed discussion of the P-versus-NP problem is beyond the scope of this paper.
However, we would like to point out an important theorem, the Cook-Levin theorem \citep[][]{cook1971a, levin1973a},
which states that any problem in the NP class can be reduced in polynomial time to an NP-complete problem,
the {\it Boolean satisfiability problem}. Based on this theorem, if one finds an efficient algorithm
with polynomial time complexity for any of the proven NP-complete problems, such as the set cover problem,
then \textit{there exists an efficient algorithm for all the NP-complete problems, and therefore P$=$NP},
and vise versa, if P$=$NP, then all the NP-complete problems can be solved efficiently.
For SCP specifically, since \citet{karp1972a}, we did not find a reference that studies
the best time complexity of an exact algorithm.\footnote{
We refer the reader to \citet[][]{woeginger2003a},
who conducted a survey of exact algorithms for some other NP-complete problems, such as the traveling salesman problem.}
Instead, theoretical investigations have focused on approximation algorithms \citep[e.g., ][]{lund1994a, feige1998a}
and tried to address what the best solution an approximation algorithm (with polynomial time complexity) can achieve.
For instance, the greedy algorithm we discuss below has time complexity $\mathcal{O}(\log n)$ and gives
an {\it approximation ratio}, the total cost of its solution divided by that of the optimal solution,
of about $\log(n)/2$ \citep[e.g., ][]{johnson1974a, chvatal1979a}.
\citet{alon2006a} showed that the best approximation ratio any approximation algorithm can achieve within
polynomial time is $a\,\log(n)$, where the constant $a$ is $\lesssim0.25$.
\section{A heuristic SCP solver}\label{sec:scpsolver}
As there is no known exact algorithm to solve the set cover problem efficiently,
we resort to heuristic approximation methods, aiming at finding a (near-)optimal
solution in a short amount of time.
The simplest heuristic approximation algorithm is the greedy algorithm, which we describe in more detail below.
Most recent effective approximation algorithms are based on {\it Linear Programming} (LP)
relaxation \citep[][]{balinski1964a, hochbaum1982a} or {\it Lagrangian Relaxation}
\citep[LR, e.g., ][]{held1970a, held1971a, geoffrion1974a}.
The main idea of LP relaxation is to relax the integer constraint on $x$ (Equation~\ref{eq:integer}),
allowing it to be any number between $0$ and $1$: $x_j \in \left[0,1\right]$.
The relaxed LP problem can then be solved efficiently using well-known
methods \citep[e.g., ][]{khachiyan1980a, karmarkar1984a}.
Starting with the optimal solution to the LP problem, one then uses heuristic methods
such as branch-and-bound and cutting-planes to find the integer version of
the solution \citep[e.g., ][]{little1963a}.\footnote{Many linear programming commercial software packages, e.g., CPLEX,
include these heuristic methods for integer linear programming problems.}
In our work, we choose to adopt the LR approach, which has provided the best existing solution
to the standard test problems \citep[e.g., ][]{caprara2000a}. It is also easy to combine LR with
the greedy algorithm and other techniques for significant improvement of the (near-)optimal solution.
We describe the essential ingredients of our algorithms in more detail below.
\subsection{The greedy algorithm}\label{sec:greedy}
The basic idea of the greedy algorithm is as follows. We start with an empty solution set and
select the subset (column) that covers the largest number of rows and costs the least to add to the solution.
Then at each stage, we select the column that covers the largest number of \textit{uncovered} rows and costs the least.
We repeat the operation until the union of the solution set covers all the rows.
Let $S^*$ be the current set of columns already included in the solution set and $M^*$ be the set of uncovered rows,
We start with $S^* = \O$ and $M^* = M$. At every stage,
we define a score for every remaining column $j \in N\setminus S^*$ and select the one with the \textit{minimum} score
to the solution set $S^*$. A natural choice for the score is the ratio between the cost of the column and
the number of remaining rows covered by the column,
\begin{equation}
\sigma_j = c_j/\mu_j\mathrm{,}
\label{eq:score}
\end{equation}
\noindent where $\mu_j$ is the number of remaining rows in set $M^*$ covered by column $j$,
\begin{eqnarray}
\mu_j & = & \lvert I_j^* \rvert\mathrm{.} \nonumber \\
& = & \lvert I_j \cap M^* \rvert\mathrm{.}
\end{eqnarray}
\noindent When combined with LR, we will modify this score definition to include the Lagrangian multiplier,
which we describe below.
\subsection{The Lagrangian relaxation method}\label{sec:lagrangian}
The main difficulty of SCP arises from the condition that all the rows need to covered, i.e.,
the inequality constraint in the integer linear programming formulation (Equation~\ref{eq:allrows}).
The goal of the Lagrangian relaxation method is to first relax this constraint by adding it to the
cost function with a Lagrangian multiplier vector $\bs{u}$,
which penalizes the violations of the constraint, and turn the original problem into an easier one.
We refer the reader to \citet{fisher2004a} for a recent review on the LR method.
Below we describe the main ideas.
\subsubsection{The relaxed Lagrangian subproblem}\label{sec:lagrangiansubprob}
The relaxed Lagrangian subproblem reads
\begin{eqnarray}
\text{Minimize}\ & ~~\bs{c}^T \bs{x} + \bs{u}^T\,(\bs{\mathrm{I}} - \bs{A}\bs{x}) \label{eq:LagrangianSubproblem} \\
\text{subject to}\ & \nonumber \\
& u_i \geqslant 0\mathrm{,}\ i \in M\mathrm{,}\ \\
& x_j \in \{0,1\}\mathrm{,}\ j \in N\mathrm{,}
\end{eqnarray}
\noindent where $\bs{\mathrm{I}}$ is an identity vector with $m$ values all equal to one.
We require the Lagrangian multiplier vector $\bs{u}$, an $m$-element vector, to be composed
of \textit{nonnegative} values, and when the original constraint (Equation~\ref{eq:allrows})
is violated for a given row $i$, i.e., when $1-\bs{A}_i\bs{x} = 1$,
the cost function we want to minimize increases,
which in turn penalizes the current solution to the Lagrangian subproblem.
The first insight why LR is an effective approximation algorithm is that
the solution to the Lagrangian subproblem is a lower bound to the original SCP.
To see this, assume the solution to the original SCP is $\bs{\hat{x}}$ and the solution to
the Lagrangian subproblem is $\bs{\bar{x}}$, then
\begin{equation}
\bs{c}^T \bs{\bar{x}} + \bs{u}^T\,(\bs{\mathrm{I}} - \bs{A} \bs{\bar{x}}) \leqslant
\bs{c}^T \bs{\hat{x}} + \bs{u}^T\,(\bs{\mathrm{I}} - \bs{A} \bs{\hat{x}}) \leqslant
\bs{c}^T \bs{\hat{x}}\,\mathrm{.}
\label{eq:LR_SCP}
\end{equation}
\noindent The first relation is true because $\bs{\bar{x}}$ is the solution to the Lagrangian subproblem,
and the second inequality is true because $\bs{\hat{x}}$ is the solution to the original SCP, which
requires $(\bs{\mathrm{I}} - \bs{A} \bs{\hat{x}}) \leqslant 0$. In reality, it is very rare that
the last two terms are equal as it requires every row is covered by exactly one column.
The second insight is that given a Lagrangian multiplier vector $\bs{u}$,
the Lagrangian subproblem has a simple solution.
We can simply reorganize the terms and rewrite the subproblem as
\begin{eqnarray}
\text{Minimize}\ & ~~\bs{c^T(\bs{u})}\,\bs{x} + \bs{u}^T\,\bs{\mathrm{I}}\,\mathrm{,}
\label{eq:LR_SCP_costform}
\end{eqnarray}
\noindent where the new cost vector $\bs{c(\bs{u})}$, termed the {\it Lagrangian cost} vector, is given by
\begin{equation}
\bs{c(\bs{u})} = \bs{c}-\bs{A}^T\bs{u}\,\mathrm{.}
\end{equation}
\noindent For a given Lagrangian multiplier vector, the solution to the Lagrangian subproblem is
\begin{eqnarray}
x_j = 0\ & {\rm if}\ c_j - (\bs{A^T}\bs{u})_j \geqslant 0 \mathrm{,} \nonumber \\
x_j = 1\ & {\rm if}\ c_j - (\bs{A^T}\bs{u})_j < 0 \mathrm{,}
\label{eq:Lsub_sol}
\end{eqnarray}
\noindent since the second term $\bs{u}^T\,\bs{\mathrm{I}}$ is a constant and $x_j$ can only be $0$ or $1$.
The minimum objective function given by this solution, which we label as $L(\bs{u})$, is then a lower bound to
the original SCP.
\subsubsection{The Lagrangian Dual}\label{sec:lagrangiandual}
Since the solution to the Lagrangian subproblem, $L(\bs{u})$, is a lower bound to the original SCP
for a given multiplier, the goal of LR is now to find the Lagrangian multiplier $\bs{u}$ that
\textit{maximizes} $L(\bs{u})$, so that the three quantities in Equation~\ref{eq:LR_SCP}
are (nearly) equal to each other. And this defines the Lagrangian Dual problem to the original SCP,
\begin{eqnarray}
\text{Maximize}\ & L(\bs{u}) \\
\text{subject to}\ & \nonumber \\
& u_i \geqslant 0\mathrm{,}\ i \in M\mathrm{.}\
\end{eqnarray}
\subsubsection{The subgradient method}\label{sec:subgradient}
One of the popular approaches to solving the Lagrangian Dual optimization problem is the iterative method using the subgradient vector,
\begin{equation}
\bs{s} = \bs{\mathrm{I}} - \bs{A} \bs{x}\,\mathrm{,}
\label{eq:subgradient}
\end{equation}
\noindent which is a generalization of the well-known gradient descent method for differentiable cost functions.
In practice, we adopt the update rule first proposed by \citet{held1971a},
\begin{equation}
u_i^{k+1} = {\rm max}\biggl(u_i^k + \lambda \frac{{\rm UB} - L(\bs{u}^k)}{\norm{\bs{s}(\bs{u}^k)}^2} s_i(\bs{u}^k), 0\biggr),\,i\in M\mathrm{.}
\label{eq:subgupdate}
\end{equation}
\noindent where ${\rm UB}$ is the current known upper bound, i.e., the best known solution, to the original SCP,
$\norm{\bs{s}(\bs{u})}$ is the Euclidean ($L^2$) norm of the subgradient, and $\lambda$ is the \textit{adaptive} step size
parameter, which can be increased or decreased depending on the rate of change in the last few iterations.
Starting with an initial guess of the solution multiplier $\bs{u}^0$, we repeat the update rule until it converges or
a maximum number of iterations has been reached.
We discuss how to choose the initial guess $\bs{u}^0$ in Section~\ref{sec:initialization}.
\subsubsection{New scores for the greedy algorithm}\label{sec:newscore}
Once we find the (near-)optimal solution to the Lagrangian Dual problem with the subgradient method,
there are two ways to find the solution to the original SCP.
One is to start with the solution $\bs{x}$ to the Lagrangian subproblem defined by the solution
$\bs{u}$ to the Lagrangian Dual problem, and apply the greedy algorithm to the
uncovered rows, if there is any. The other is to replace the cost in the
original SCP in the score definition (Equation~\ref{eq:score}) with the following
Lagrangian cost \citep[e.g., ][]{fisher1990a} at each stage,
\begin{equation}
\gamma_j = c_j - \sum\limits_{i \in I_j^*} u_i^k\ \mathrm{,}
\end{equation}
\noindent where $I_j^*$ is the remaining (uncovered) rows covered by column $j$.
We then apply the greedy algorithm with the following new score definition,
\begin{eqnarray}
\sigma_j = \gamma_j/\mu_j,\ & \text{if}\ \gamma_j>0\ \mathrm{,} \\
\sigma_j = \gamma_j\,\mu_j,\ & \text{if}\ \gamma_j<0\ \mathrm{.}
\end{eqnarray}
\noindent The new greedy algorithm is equivalent to solving the relaxed Lagrangian subproblem
without the constant term ($\bs{u}^T\bs{\mathrm{I}}$ in Equation~\ref{eq:LR_SCP_costform}) for the given (near-)optimal Lagrangian multiplier $\bs{u}$,
but in addition subject to the constraint that all the rows must be covered (Equation~\ref{eq:allrows}).
We find this approach is particularly effective and adopt it in our solver.
\subsubsection{Iterations with new initial Lagrangian multipliers}\label{sec:initialization}
The iterative subgradient approach to the Lagrangian Dual problem may find a local instead of a global maximum.
To circumvent this issue and find a solution that is as good as possible, we can iterate all the steps with different
initial Lagrangian multipliers $\bs{u}^0$.
We alternate two methods to select the initial $\bs{u}^0$. In the first one,
we generate a vector with values randomly distributed between 0 and 1.
In the second approach, we define $\bs{u}^0$ in a greedy way, following \citet{caprara1999a},
\begin{equation}
u^0_i = {\rm min}\,\frac{c_j}{\lvert I_j\rvert},\ j \in J_i\ \mathrm{.}
\end{equation}
\noindent This choice is motivated by that columns with minimum score (low cost and many covered rows) are more likely to be in the solution,
and the rows they cover tend to have smaller multipliers to \textit{maximize} the solution to the Lagrangian Dual
(see Equation~\ref{eq:Lsub_sol}).
In each iteration, we also add a perturbation vector with small random values and generate a new $\bs{u}^0$ vector,
with $u^0_i \rightarrow (1+\delta)\,u^0_i$, where the random value $\delta \in [-0.1, 0.1]$.
This again is to maximize the chance for the subgradient optimization iterations to escape from a local maximum.
We iterate the entire procedure for a maximum 20 times or when a convergence criterion is reached.
Our experience with the cases in the test bed shows that only in a few cases could we find a marginally better solution
with more iterations.
\begin{figure*}
\vspace{0.2cm}
\epsscale{0.85}
\plotone{Archetype_Flowchart3.pdf}
\caption{The flow chart of the Archetype technique. Once one defines a distance metric, the minimum distance is the only free parameter.
The final criterion, the set size, is an example for how to investigate the final basis set.
}
\vspace{0.3cm}
\label{fig:flowchart}
\end{figure*}
\subsection{The code}\label{sec:code}
We have implemented the algorithms described above in Python.
We test our code, named SetCoverPy, on the standard test problems from Beasley's Operations Research Library \citep[][]{beasley1990a}.
We test the code on a Macbook Pro laptop with a moderate configuration of $16\,$GB RAM and $2.8\,$GHz Intel Core i7 (Quad Core).
For all the cases (4,5,6, and A-H categories), our code yields a solution that is on average $99\%$ optimal in terms of the final cost.
We provide all the test cases in convenient data format for interested readers and publish our code on the PyPI package management system.
We briefly discuss the code and the test, and demonstrate how to install and use the code in Appendix~\ref{app:code}.
\vspace{0.1in}
\section{The Archetype technique for classification}\label{sec:archetype}
How can we adopt the set cover problem for classification purposes?
Back to the school-location example, we can think about it in a different way.
Instead of selecting a minimum number of precincts to build schools so that
every student in the whole county has a school to attend, we select a minimum number
of precincts to represent all the precincts in the whole county,
assuming that neighboring precincts are similar to each other, either in geographic
distance or by some other criteria, and they can represent each other.
Generalizing this methodology to any sample of any objects, such as animals, plants, galaxies, stars or
planets, if we can define a distance between any pair of instances in the sample,
we can apply the same SCP solver to the data and select a minimum subset of instances,
which we call \textit{archetypes}, that represent the whole sample.
We introduce this generic Archetype technique for classification and describe
the key steps below.
\subsection{The technique}\label{sec:archetypemethod}
We present the flowchart for the steps of the Archetype technique in Figure~\ref{fig:flowchart},
which we describe in detail below.
As we will demonstrate how to use the technique with a spectroscopic dataset of extragalactic sources later,
when necessary, we will assume the properties of a given instance are measured by the spectrum $f(\lambda)$,
the flux vector as a function of wavelength. However, we would like to stress that the discussions below are
generic and can be applied to any dataset. In other words, we can treat $\lambda$ as dimension instead of wavelength,
and $f(\lambda)$ as the location in the given dimension rather than the flux value at the wavelength.
\vspace{0.1in}
\noindent [1]. Define and apply the distance metric. The first immediate question in the technique asks
how to measure the distance, or similarity, between a pair of instances in the dataset.
The best distance metric depends on the application, the dimensions interested, and the purpose of the distance.
For spectral analysis, a choice often used in astrophysics is the chi-squared $\chi^2$,
which can be considered as \textit{weighted squared Euclidean distance}.
If we choose to scale two spectra to the same normalization with a scaling factor $a$, then the $\chi^2$ is given by
\begin{equation}
\chi^2_{ij} = \sum\limits_{l=1}^{l=d} \frac{(f_i(\lambda_l) - af_j(\lambda_l))^2}{\sigma_{i}^2(\lambda_l) + a^2\sigma_{j}^2(\lambda_l)}\mathrm{,}
\end{equation}
\noindent and the reduced $\chi_{\rm red} ^2$ is given by $\chi^2/(d-1)$, where $d$ is the number of dimensions.
In practice, we can obtain $\chi^2$ and $a$ simultaneously by fitting the two spectra with an iterative maximum likelihood method
or a Bayesian estimator to take into account the uncertainties in both vectors \citep[e.g., ][]{hogg2010a, ivezic2014a}.
If we want to include the normalization (the flux level) in the metric, we can also choose to fix the scaling factor $a=1$.
A thorough discussion of distance metric is beyond the scope of this paper as
distance metric learning itself is an active field in machine learning \citep[e.g., ][]{xing2003a, weinberger2009a, kulis2012a}.
We here comment on the specific usage of the (squared) Euclidean distance.
First, it is worth pointing out that using \textit{weighted} $\chi^2$ as the distance metric works the best
on a dataset with a narrow distribution of relative errors (at all dimensions).
This is because a vector with very small errors compared to the rest of the dataset will yield very large distances to
all other vectors by definition, while one with very large errors will yield very small distances to all other vectors,
both of which are more likely to be selected as archetypes for the opposite
reasons.\footnote{It therefore depends on specific applications whether to use weighted or unweighted $\chi^2$.}
Second, the definition above uses all the $d$ dimensions in the input data, in our example, all the wavelengths in the spectrum.
In the case of spectral analysis (of extragalactic sources), it is known that some wavelength regions are more informative
about some intrinsic physical properties than the others. For example, the regions where the strong stellar absorption lines
are particularly revealing about the stellar age and heavy element abundances in stars in the galaxy \citep[e.g., ][]{worthey1994a},
while those where the recombination and nebular lines are located informs mostly on the instantaneous star formation rate
and heavy element abundances in the interstellar medium \citep[e.g., ][]{kennicutt1998a, kewley2008a}.
In principle, we can use a distance metric defined in any combination of the dimensions,
or on any hyperplane (i.e., any projection),
or in some reduced-dimension subspace (e.g., in the first few PCA component space).
It is therefore often desirable to pre-process the data and reduce the dimensionality first.
We can achieve this by upweighting or selecting the most informative dimensions \citep[e.g., ][]{yip2014a},
or projecting many correlated dimensions onto a few new dimensions defined by the most important basis components
determined from PCA \citep[e.g., ][]{budavari2000a, wild2006a} or matrix factorization \citep[][]{blanton2007a, zhu2013a},
and defining a new distance metric with the reduced dimensionality.
In our example, however, we will consider all the dimensions (wavelengths) provided by the observation
for simplicity.
After defining a distance metric, we compute the distance between every pair of instances in the dataset
and obtain a (symmetric, square) distance matrix $\bs{D}$ ($\bs{\chi}^2$ in our example).
\vspace{0.1in}
\noindent [2]. Define and apply the minimum distance. To apply the SCP solver to the dataset,
we need to select a minimum distance, within which we consider two instances are similar
and thus can represent each other. With a chosen distance metric, this minimum distance parameter is
the \textit{only} free parameter in the Archetype technique.
In our example, we select a minimum chi-squared $\chi_{\rm min}^2$
and turn the $\bs{\chi}^2$ distance matrix into the binary relationship matrix $\bs{A}$:
\begin{eqnarray}
a_{ij} = 1\ ({\rm T})\ & {\rm if}\ \chi_{ij}^2 \leqslant \chi_{\rm min}^2\mathrm{,} \nonumber\\
a_{ij} = 0\ ({\rm F})\ & {\rm if}\ \chi_{ij}^2 > \chi_{\rm min}^2\mathrm{.}
\end{eqnarray}
The freedom of choosing the minimum distance offers a degree of flexibility in the Archetype technique.
If the minimum distance is large, then the number of archetypes will be small.
Although in this paper, for simplicity and demonstration purposes, we focus only on a given minimum distance,
we note that varying the minimum distance can reveal, level by level, the hierarchy of the dataset.
In astrophysics in particular, it can reveal the physical mechanisms that are responsible to different degrees
for the cosmic evolution of the astronomical systems (such as galaxies, stars and planets).
We will discuss this further in Step 4.
A question related to specifics of the distance metric above is that whether there can be a
forbidden connection between two instances, such as due to a natural barrier as in the school-location
example.\footnote{I thank Adrian Liu for this interesting question.}
In the context of the distance metric (Step 1), it is not straightforward to select or add a
dimension in which the projected distance of any pair of instances is either zero or infinity,
because a barrier is restricted to certain pairs of instances.
In practice, we can instead form a binary barrier matrix $\bs{B}$, in which an element is 0 (False)
if we do not consider the corresponding pair can represent each other,
and perform a logical AND operation between the two binary matrices.
\vspace{0.1in}
\noindent [3]. Apply the SCP solver. Once we have chosen a minimum distance and turn the distance matrix
$\bs{D}$ into the binary relationship matrix $\bs{A}$, it is now straightforward
to apply the SCP solver (to $\bs{A}$).
One variable while applying the SCP solver is the cost to each instance.
We here assume every instance in the dataset is equally valuable and their costs in the SCP context
are the same. In practice, it is conceivable that some instances are more valuable than the others.
However, the assignment of cost would often be \textit{ad hoc} and subjective.
For example, considering astronomical observations, we may want to select sources with lots of
auxiliary data for in-depth investigations and therefore we may assign a smaller cost to them.
We consider how to assign cost an open question in the Archetype technique.
\vspace{0.1in}
\noindent [4]. Investigate the basis set of archetypes and iterate the procedure.
The flowchart shown in Figure~\ref{fig:flowchart} presents only a simple way of
investigating the basis set by using the number of archetypes.
In real applications, it is desirable to investigate the results more
carefully, e.g., using pairs whose relationship is well understood
(\textit{labeled}, as in the context of supervised machine learning).
As mentioned in Step 2,
once we have chosen a distance metric, the only free parameter of the Archetype technique is the minimum distance
and consequently the final basis set of archetypes strongly depends on the choice.
We can view this freedom in two complementary ways.
First, ideally we would like to select the minimum distance in an \textit{ab initio} way,
according to some (known) strict criteria or based on our understanding of the underlying physics.
However, in most cases, we do not know if there is such an ideal minimum distance and it is often one of the goals
to find out if such a golden separation exists.
We therefore can consider in an alternative way that the freedom of choosing the minimum distance offers a degree of flexibility
and can be used to learn the different degrees of similarity among the instances and
what different physical mechanisms are responsible at different levels.
As common in nature, any group of objects can often be classified into a hierarchical structure.
For example, we separate plants into kingdoms, phya, classes, series, families, genera and species.
We can iterate the procedure with different minimum distance choices and build a hierarchical classification system.
Starting with a large minimum distance, there are two options to achieve this.
One is simply to decrease the minimum distance and apply the SCP solver to the whole dataset in each iteration,
and the other is to decrease the minimum distance but apply the SCP solver to subsamples represented
by each archetype in the previous iteration.
A subtle aspect of the Archetype technique is that an instance can often be represented by more than
one archetype, which is by design (see Figure~\ref{fig:schools}).
When it is desirable to select \textit{the} archetype for a given instance,
one can simply choose the one with the smallest distance.
\vspace{0.1in}
\noindent [Note]. Finally, we would like to note that,
if computing resources are limited, we recommend to pre-process
the initial dataset and select a subsample to choose the archetypes from.
How to select the subsample depends on the specific application, but
ideally the subsample should still span the whole space.
In practice, however, our code can work on a subsample of several thousands of instances on
a typical personal computer with an average configuration (as of 2016) and yield a (near-)optimal solution
within an hour or so (for a sample of about $3000$ instances).
\subsection{Relationships with other machine learning techniques}\label{sec:othertechnique}
The archetype technique we developed has close relationships to some of the well-known machine learning techniques,
especially in clustering analysis, such as $k$-means clustering, $k$-nearest neighbors ($k$NN) and
friends-of-friends, and in dimensionality reduction.
We first discuss a comparison of the Archetype technique with the $k$-means clustering problem.
The $k$-means clustering problem aims at partitioning all the instances into $k$ clusters
by minimizing the within-cluster sum of squares, i.e., sum of squared distances of each point
in a cluster to the cluster center. The means (centers) of the clusters can also
be interpreted as archetypes, which are usually called prototypes in the context of $k$-means clustering.
It should not be surprising that the $k$-means clustering problem is also NP-hard
and no optimal solution can be found within polynomial time \citep[e.g., ][]{aloise2009a}.
The main difference between the archetype technique and the $k$-means clustering problem is
in the free parameters, the minimum distance as opposed to the number of clusters ($k$).
As a consequence, the clusters in $k$-means can have a wide range of scopes,
depending on the exact way how instances are connected to each other.
In the Archetype technique, the maximum distance within a group represented by an archetype
cannot be larger than the minimum distance,
while the number of archetypes depends on the overall scale of the parent sample.
Another difference is that in the Archetype technique, an instance can be represented (covered) by
more than one archetype, while in $k$-means, as well as many other clustering/classification schemes,
one instance only belongs to one group.
The friends-of-friends method is another popular clustering technique used in astronomy,
especially in dark matter cosmological simulations
\citep[][]{davis1985a}.\footnote{I thank Peter Behroozi for very useful discussions on this comparison.}
It shares the same free parameter with the Archetype technique, a minimum distance within which
two instances (e.g., dark matter particles) are considered connected to each other and belong
in the same group (e.g., dark matter halos). However, it uses a chain connection to form
the groups: if one instance is connected to another (a friend), then it is connected to all the other
instances connected to that friend, and all the friends connected belong to the same group (thus the name of the method).
In this regard, the friends-of-friends method, and many other distribution/connection-based clustering methods
(such as Gaussian mixture models and $k$-means clustering),
can form groups with a wide range of scopes, as opposed to the uniform scope of groups in the Archetype technique.
As mentioned earlier, when the dimensionality is high,
we can first pre-process the data with dimensionality-reduction
techniques, such as PCA, and define the distance metric in the reduced-dimension space
and apply the Archetype technique.
We refer the reader to Section~\ref{sec:archetypemethod} for a brief discussion.
\begin{figure*}
\vspace{-0.4cm}
\epsscale{1.15}
\plotone{GalaxyArchetypes_edited.jpg}
\vspace{0.0cm}
\caption{The optical spectra of the common extragalactic source archetypes that can represent more sources than themselves,
ordered by the continuum slope (as indicated by the color).
The number $N$ shows how many sources in the parent dataset the archetype can represent, i.e., with distances shorter than the minimum distance.}
\label{fig:spec1}
\end{figure*}
\begin{figure*}
\vspace{-0.4cm}
\epsscale{1.20}
\plotone{Archetypes.jpg}
\vspace{0.5cm}
\caption{The composite pseudo-color images of the extragalactic source archetypes that can represent more sources than themselves, ordered by the continuum slope as in Figure~\ref{fig:spec1}.
For display purposes, we have scaled down images in the last two rows to accommodate 12 archetypes.}
\label{fig:image1}
\end{figure*}
\begin{figure*}
\vspace{0.1cm}
\epsscale{1.12}
\plotone{GalaxyArchetypes_pec_edited.jpg}
\vspace{0.1cm}
\caption{The optical spectra of the peculiar extragalactic source archetypes that can only represent themselves,
ordered by the continuum slope (as indicated by the color).}
\label{fig:spec2}
\end{figure*}
\begin{figure*}
\vspace{0.3cm}
\epsscale{1.20}
\plotone{Archetypes_pec.jpg}
\vspace{0.3cm}
\caption{The composite pseudo-color images of the peculiar extragalactic source archetypes that can only represent themselves, ordered by the continuum slope as in Figure~\ref{fig:spec2}.}
\vspace{0.2cm}
\label{fig:image2}
\end{figure*}
\subsection{A test case with extragalactic sources}\label{sec:archetypeexample}
\subsubsection{The parent dataset}
To further discuss the Archetype technique and illustrate how to use the method,
we use an optical spectroscopic dataset of extragalactic sources as an example.
We select the sources and their spectra from the seventh
data release \citep[DR7,][]{abazajian2009a} of the SDSS legacy survey \citep{york2000a}.
In addition, we use the measurements of emission line strength and estimates of
intrinsic properties such as stellar mass ($M^*$) and star formation rate (SFR) from the MPA-JHU
value-added catalog \citep[e.g., ][]{kauffmann2003a, brinchmann2004a}.\footnote{\texttt{http://wwwmpa.mpa-garching.mpg.de/SDSS/DR7/}}
\vspace{0.1in}
We first select sources that meet the following criteria:
\begin{itemize}
\item[1.] $0.050 < z\, ({\rm redshift}) < 0.052$,
\item[2.] $15 < {\rm S/N} < 30$ and with no significant missing data.
\end{itemize}
\noindent The first selection is a compromise between the following two requirements.
We would like to select sources at sufficiently high redshift so that the SDSS $3\arcsec$ fiber
encompass a reasonably large area. At $z\sim0.05$, the fiber covers about $3\,{\rm\,kpc}$.
On the other hand, we also want to select closer systems in order to investigate
the morphology confidently from the shallow imaging data.
We impose the second criterion because we use the weighted $\chi^2$ as the distance metric,
and for the reasons mentioned in Step $1$ in the previous section, we try to avoid
selecting instances with very small or large measurement uncertainties.
Our parent test dataset includes $2820$ extragalactic sources.
We calculate the distance matrix with the weighted $\chi^2$ by performing a least squares fitting
to every pair of spectra between $3700\,$\AA\ and $7000\,$\AA\ for the scaling factor $a$.
Note that including the scaling factor $a$ in the $\chi^2$ calculation means we \textit{exclude}
the normalization in the comparison, so mass or luminosity of the sources will not be
an important dimension in our analysis.
We then choose a minimum (reduced) $\chi^2_{\rm red, min}=15$
to convert the distance matrix into a binary relationship matrix.
We choose this minimum distance to be concordant with the S/N selection criterion.
However, we would like to remind the reader that the minimum distance is a free parameter in the technique
and varying it can help construct a hierarchical classification system and
reveal different degrees of similarity among the instances and
physical processes that are responsible at different levels.
As our goal is to describe the key steps involved in the technique,
we have selected this particular minimum distance for the convenience of presentation.
Finally, we treat all the sources equally and assign equal cost.
Applying the SCP solver to the dataset, we establish a basis set of $42$ extragalactic source archetypes.
We investigate these archetypes and their relationships with the instances in the parent sample below.
\begin{figure*}
\vspace{0.1cm}
\epsscale{0.36}
\plotone{GalaxyArchetypes_colorha.pdf}
\epsscale{0.36}
\plotone{GalaxyArchetypes_BPT.pdf}
\epsscale{0.36}
\plotone{GalaxyArchetypes_msfr.pdf}
\vspace{0.1cm}
\caption{The basis set of archetypes for extragalactic sources.
The color indicates the continuum slope and the symbol size shows how many other sources the archetype can represent,
with the open diamonds showing separately peculiar archetypes that can only represent themselves.
The gray scales show the density distribution of the parent dataset.
\textit{Left}: the distribution of the continuum slope and the [\ion{O}{3}]$\,\lambda5008$ line flux.
\textit{Middle}: the BPT classification diagram for AGN and star-forming galaxies.
\textit{Right}: the distribution of two derived parameters, stellar mass ($M^*$) and star formation rate (SFR).
Note we included a scale (normalization) factor in the $\chi^2$ distance metric,
so mass is not an important dimension in this particular classification scheme.
Most of the quiescent galaxies are represented by the three archetypes (23, 25, 28) with slope $\sim1.2$ for the particular $\chi^2_{\rm min}$ choice
in this example.
}
\vspace{0.6cm}
\label{fig:GalaxyArchetypeProp}
\end{figure*}
\subsubsection{The basis set: SED and morphology}
To investigate the basis set of the archetypes, we divide the archetypes into two subsets.
One subset includes common archetypes that can represent more than themselves in the parent sample,
and the other includes peculiar archetypes that can only represent themselves.
Figure~\ref{fig:spec1} shows the spectra of the subset of common archetypes,
in the order of the slope of the underlying continuum, the ratio between the fluxes
at $6100\,$\AA\ and $4600\,$\AA\ ($f_{\lambda6100}/f_{\lambda4600}$).
We choose the two wavelengths to be where the continuum is relatively smooth.
For normal galaxies, a bluer continuum (more flux at shorter wavelength) roughly means the
there are more younger stars in the galaxy.
Note we have normalized the spectra since we do not consider the overall brightness.
These archetypes span a variety of spectral types, as indicated by the diverse array of continuum shapes and
the emission line strengths and ratios. An interesting observation is that the three archetypes that cover most
instances, with ID $23$, $25$, and $28$ as shown in the figure, are moderately red spectra that show little line emission.
This means the emission lines are a major source of the spectral variety and must be responsible for
a large fraction of the (high) dimensionality in the spectral space.
Figure~\ref{fig:image1} presents the pseudo-color composite images of these archetypes from $g$-, $r$-, $i$-band
imaging data.\footnote{Retrieved from \texttt{http://skyserver.sdss.org/}.}
Unsurprisingly, they display different morphologies, including irregular, spiral, lenticular and elliptical shapes.
We now take a look at the subset of peculiar archetypes that can only represent themselves
in Figure~\ref{fig:spec2} and \ref{fig:image2}, again ordered by the continuum slope.
The first one turns out to be an error of the observation.
It is actually a high-redshift quasar with an strong absorption system induced by
a foreground gaseous cloud.\footnote{For interested readers, the SDSS Plate-MJD-Fiber of this object
is 352-51694-380 and its equatorial coordinates are (${\rm RA}=258.20837\,{\rm deg}$, ${\rm DEC}=64.05295\,{\rm deg}$).
To learn more about intervening absorption-line systems, see \citet{zhu2013a} and references therein.}
The reason it was identified as a $z\sim0.05$ galaxy is because one of the absorption lines, the \ion{Mg}{2}\,$\lambda\lambda2796,2804$\ doublet,
is treated as (negative) H$\alpha$\ emission by the SDSS reduction pipeline. The pipeline fit the spectra with
linear combinations of PCA components, without nonnegativity constraint. It is interesting to see that the Archetype technique
can automatically pick out such misidentifications. We exclude this source in the further discussion below.
The second and third peculiar archetypes (with ID $34$ and $35$) exhibit extremely strong emission lines,
indicating extreme star-bursting behaviors.
Other peculiar archetypes share a common trait that they tend to have a red continuum but also strong and often broad emission lines,
which indicates the presence of an AGN in an otherwise quiescent galaxy.
Their colors and morphologies shown in Figure~\ref{fig:image2} also support this proposition.
The main differences among them are the different line strengths and ratios.
\subsubsection{The basis set: the distribution in the reduced-dimension space}
To further investigate the basis set of archetypes, we compare them with the parent sample
in the reduced-dimension space. We do not perform complex dimensionality reduction,
but rather define/pick the dimensions manually according to our understanding of the spectral energy distribution (SED) and the underlying physics.
Figure~\ref{fig:GalaxyArchetypeProp} shows the comparison, where the color indicates the continuum slope
and the symbol size represents the number of instances the archetype can represent, with the open diamonds separately
showing the peculiar archetypes that can only represent themselves.
The gray scale shows the distribution of the parent sample.
In the left panel, we plot the distribution of the continuum slope and the [\ion{O}{3}]$\,\lambda5008$ emission line strength.
These are likely the two of the most informative dimensions in our analysis, i.e., they affect the $\chi^2$ value the most.
The archetypes that can represent the largest number of sources are located at the densest regions of this diagram,
while peculiar archetypes and those that can only represent a small number of sources have strongest emission lines,
suggesting a wide range of emission line strengths/ratios are responsible for driving the dissimilarity
(larger $\chi^2$ distance) between them and the others.
What are the blue archetypes with strong emission lines that can only represent a small sample of similar sources?
We can find the answer in the middle panel, where we present the AGN-star-forming galaxy diagnostic
diagram, the so-called BPT diagram \citep{baldwin1981a}.
The BPT diagram compares the ratio of [\ion{O}{3}]$\,\lambda5008$ to H$\beta$\ and
that of [\ion{N}{2}]$\,\lambda6584$ to H$\alpha$\ and can efficiently distinguish different types of
extragalactic sources with different physical properties.
We have overplotted the demarcation lines for star-forming galaxies empirically-defined
by \citet[][dotted line]{kauffmann2003a} and theoretically-defined by \citet[][dashed line]{kewley2001a},
and a horizontal line at [\ion{O}{3}]$\,\lambda5008$/H$\beta$$=0.3$, the conventional criterion to separate
Seyfert 2 galaxies \citep[][]{seyfert1943a} and low-ionization nuclear emission
regions \citep[LINERs,][]{heckman1980a}.\footnote{The nature of LINERs is still under debate.}
Looking at the blue archetypes first, we see they basically follow the left star-forming sequence.
The blue archetypes that can only represent a few sources are mostly metal-poor star-forming galaxies
with large [\ion{O}{3}]$\,\lambda5008$-to-H$\beta$\ line ratios. For the peculiar archetypes with red continuum slope but strong
and often broad emission lines, they are located in the upper right corner,
identified as Seyfert 2 galaxies, galaxies that host Type 2 AGN at the center.
The common archetypes with weak emission that can represent many sources are mostly distributed
in between, representing metal-rich star-forming galaxies, composite (with both moderate star formation and AGN activities),
and LINERs.
The right panel shows two \textit{derived} properties of the extragalactic sources,
the stellar mass ($M^*$) and instantaneous star formation rate (SFR).
We remind the reader that we fit for the scaling factor when comparing
every pair of sources, thus the stellar mass, roughly proportional to the luminosity,
is not an important dimension.
The metal-poor star-forming galaxies tend to be low-mass systems that are undergoing star-bursts (strong star formation events).
The peculiar archetypes are mostly massive galaxies.
However, their SFRs are likely severely over-estimated due to AGN contribution to the emission lines.
Their correct positions in this diagram are likely an order-of-magnitude down.
With our minimum distance choice, the quiescent galaxies can be represented by the few archetypes with
little line emission (${\rm ID} = 23$, $25$ and $28$).
We note that SFRs (for relatively quiescent galaxies) derived from weak line emission are uncertain
and should not be taken at the face value.
In order to select a basis set of archetypes that can properly represent quiescent galaxies in this space,
one also needs to include the luminosity as a factor in the distance metric, i.e., use the absolute flux
without the scaling factor $a$, and also use more robust estimates of the intrinsic properties.
To summarize, we have demonstrated that the Archetype technique we developed can naturally select a basis set
of physically-motivated archetypes to represent the whole universe of extragalactic sources.
In the next section, we discuss further the potential applications of this new technique.
\section{Further Discussions}\label{sec:discussion}
We have introduced a new generic classification technique, the Archetype technique, by adopting the set cover problem.
As we discussed earlier in Section~\ref{sec:othertechnique}, there are many different ways of categorizing a sample of objects.
Nature is often more complex than what Occam's razor suggests, and there is not a golden method that works for everything.
For different purposes, it is therefore often necessary to use different approaches or certain combinations of them.
We briefly discuss some of the astrophysical applications in which the Archetype technique can be particularly useful.
[1]. Classification and identification.
The obvious application is classification and identification of any astronomical sources: galaxies, stars or planets.
As we discussed earlier, we can also vary the minimum distance and/or combine with other techniques,
such as $k$-means or PCA, to study different aspects of the data or build a hierarchical system.
For example, in the test case with the extragalactic sources, we can separate the sources into
star-forming galaxies and non-star-forming systems, and then divide star-forming galaxies into
metal-poor and metal-rich sub-populations. We can then study the intrinsic properties and different physical mechanisms
responsible for the different types of systems and their cosmic formation history.
[2]. Redshift determination in future dark-energy surveys.
Future dark-energy surveys, including SDSS-IV/eBOSS \citep{dawson2016a},
DESI \citep{schlegel2011a, levi2013a}, and PFS \citep{takada2014a},
aim to obtain optical spectra in the observer frame for tens of millions of galaxies
at redshift $0.6\lesssim z \lesssim 2.0$ and measure the scale of the baryon acoustic oscillation \citep[BAO, e.g., ][]{cole2005a, eisenstein2005a}
as a function of cosmic time and chronicle the expansion history of the Universe.
A primary large-scale structure tracer these surveys will target is
the emission-line galaxies \citep[e.g., ][]{zhu2009a, comparat2016a},
for which the redshift can in principle be well-determined by their strong emission lines.
For DESI and PFS, the line of interest is [\ion{O}{2}]\,$\lambda\lambda3727,3730$\ and spectroscopy with
resolution $\mathcal{R}\gtrsim4000$ can tell apart the two components and identify the doublet.
For eBOSS, the resolution is lower, but because the targets are at lower redshift,
other lines at longer wavelength can be used to help measure the redshift.
All these surveys are only interested in the redshift and pushing the exposure time
to the shortest limit in order to maximize the survey efficiency and minimize the cost.
The low S/N of the spectroscopy makes even the determination of the redshifts
difficult, especially due to the strong telluric emission lines (e.g., hydroxyl lines)
in the background that can mimic the emission lines.
We have learned that the emission lines strengths and ratios span a wide range and are
some of the most important dimensions in the spectral space (see the above section).
Redshift surveys usually fit the observed spectra with linear combinations of some
basis templates, such as components from PCA or matrix factorization analysis,
that are shifted to different redshifts,
and find the redshift and the combination that give the least $\chi^2$.
The low S/N of the spectroscopy and the contamination from telluric lines,
however, means it is very easy to find a good fit to the data with a wrong combination of the templates.
In other words, the method could find a wrong answer by overfitting the low-S/N data
with a completely unrealistic model that does not even exist in the real Universe.
One of the advantages of the Archetype technique is that the archetypes
would be all real systems that actually exist in the Universe.
If we fit the observations with realistic archetypal spectra, then we do not need to
worry that we would be overfitting the data with unrealistic instances in the unoccupied space.
An earlier example of using archetypes for redshift determination is the deep low-resolution prism
spectroscopic PRIMUS survey \citep{coil2011a}.
To overcome the challenge caused by the low resolution and low S/N of their prism spectra,
they selected archetypes at low redshift from the AGES survey \citep[e.g., ][]{moustakas2011a} with the CPLEX package,
manually modified the basis set based on the understanding of the evolution of the emission lines and shapes
of the galaxy SEDs, and used the modified set of spectra to fit the observed spectra.
Compared to using linear combinations of templates from PCA or matrix factorization analysis,
this approach has helped improve the success rate of redshift determination \citep{zhu2011thesis, cool2013a}.
For dark-energy surveys, we expect to apply the Archetype technique iteratively to the redshift determination.
We start from the sample of objects with robust redshift measurements and
select a first basis set of archetypes. We then iterate the archetype set construction and redshift determination,
each time with a better and more complete set of archetypes, until we reach the maximum success rate.
At each stage, instead of using the low S/N spectrum of a given archetype,
we can construct a high S/N composite spectrum of the sources that can be represented by this archetype
and use this composite as the new archetype.
We are currently investigating this new method in the eBOSS survey,
and we expect that it will also help maximize the efficiency of future dark-energy surveys.
\vspace{0.1in}
[3]. Astrophysical sciences with low-S/N data.
The large amount of data provided by the ongoing and future surveys offers a good
opportunity for making unexpected discoveries. One of the major challenges to extract
the scientific information is the low S/N of the observation of any individual source.
We will have to resort to composite analysis in order to enhance the S/N by orders-of-magnitude for any robust measurement.
For example, \citet{zhu2015a} stacked the \textit{entire} dataset of about $9000$ spectra
from the pilot observation of emission-line galaxies in eBOSS in order to
robustly measure the resonant absorption and nonresonant fluorescent emission in the near-ultraviolet,
the signatures of galactic-scale outflows associated with star formation.
While statistical stacking has its merits, it also has some major caveats
as one may be averaging over instances that are distinctly different systems.\footnote{As the old joke says:
how many legs do you get if you average over humans and dogs?}
We can use the Archetype technique to mitigate the selection bias
in composite analysis and construct more sensible subsets of same type of sources for further investigations.
More specifically, we can select the most informative dimensions for the physics we are
interested in (with the previous knowledge as a prior),
define a distance metric in the reduced-dimension space, build a basis set of archetypes, and then perform
composite analysis for each group of sources that can be represented by a given archetype.
Again, take the high-redshift emission-line galaxies from the eBOSS survey as an example,
since we have learned that the emission lines are some of the most important dimensions (e.g., see the above section),
we can select the wavelength regions where the lines are located and then apply the Archetype technique.
In a sense, stacking the observations of sources represented by a given archetype
is equivalent to taking many exposures (or a very long exposure) of a single source.
We can then use the composite observations to study the underlying physical mechanisms
and the formation histories of different types of sources (Zhu et al. in prep).
We expect that this new way of analyzing the big data will be particularly useful for
the low S/N data from future dark-energy surveys.
\vspace{0.1in}
\section{Summary}\label{sec:summary}
Astronomy is a data science. As in many other fields, the amount of data has grown drastically in astronomy
and will continue to increase exponentially thanks to all the ongoing and upcoming large programs.
How to efficiently extract scientific information from the unstructured data
has now become one of the major challenges in the field.
One of the important and interesting problems is to classify and identify sources
into families or types, ideally based on intrinsic properties.
A proper classification scheme can further the understanding of the roles of different physical processes
that govern the formation and evolution of different types of astronomical sources.
We have introduced a novel classification method, the Archetype technique,
based on the NP-complete set cover problem (SCP) in computer science and operations research.
We first introduced SCP and in particular discussed the simplicity of
the problem and the complexity of its solution, the NP-completeness.
We have developed a heuristic solver in Python,
by combining the greedy algorithm and the Lagrangian Relaxation approximation approach.
We have tested the performance of our code on the standard test cases from Beasley's OR Library and
shown that our code can efficiently produce solutions that are on average $99\%$ optimal.
Adopting SCP for classification purposes, we introduced the Archetype technique.
Based on how similar the sources are to each other, the Archetype technique finds a basis set of
archetypes to represent the whole universe of the sources.
We described the steps of the technique, paying special attention to the distance metric and
the only free parameter, the minimum distance within which two sources can represent each other.
We used a spectroscopic sample of extragalactic sources from the SDSS survey as
an example to illustrate how to apply the technique and how to interpret the results.
We showed that the technique naturally selected a basis set of
physically-motivated archetypes for the extragalactic sources.
The archetypes include different types of sources, such as metal-poor/rich star-forming galaxies,
AGN and composite systems, and span a wide range in the spectral energy distribution and morphology.
We show that the line emission strengths and ratios are important dimensions in the spectral energy distribution
and suggest that dark-energy surveys targeting emission-line galaxies can use the Archetype
technique to improve the survey efficiency.
We further discuss the potential future applications of our technique.
We discuss how to apply it to the low-S/N spectroscopic data and maximize the potential
for astrophysical sciences of future dark-energy surveys,
Our technique is generic and is easy to use and expand, and we expect that it can find applications in many fields of astronomy,
including the formation and evolution of a variety of astrophysical systems,
such as galaxies, stars and planets.
\acknowledgments
G.B.Z. wishes to thank Sam Roweis and David Hogg for introducing the set cover problem to him and
for very illuminating discussions.
He also thanks John Moustakas for very useful discussions and comments that helped improve the clarity of the paper.
He would also like to thank Mike Blanton for very useful discussions.
G.B.Z. acknowledges support provided by NASA through Hubble Fellowship grant \#HST-HF2-51351 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under contract NAS 5-26555.
This paper uses the public data from the SDSS survey. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.
|
3,212,635,537,964 | arxiv |
\section{Introduction}
SELEX (Fermilab Experiment 781)~\cite{inst}
employs beams of $\Sigma^-$, $\pi^-$, and
protons at around $600\,\mbox{GeV}/c$ to study production and decay properties
of charmed baryons. It took data in the 1996/7 fixed target run and is
currently analyzing those data.
Here we will focus on recently obtained results concerning
the $\Omega_c^0$ lifetime and the doubly-charmed baryons $\Xi_{cc}^+$
and $\Xi_{cc}^{++}$.
\section{New Results on the {\mbox{\boldmath$\Omega_c^0$}}}
SELEX observes the $\Omega_c^0$ in three decay modes, namely
$\Omega_c^0\to\Omega^-\pi^+$,
$\Omega_c^0\to\Omega^-\pi^+\pi^+\pi^-$, and
$\Omega_c^0\to\Xi^-K^-\pi^+\pi^+$. The invariant mass distributions
for these modes are shown in fig.~\ref{omcmass}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.32\textwidth]{Imagen1.eps}
\includegraphics[width=0.32\textwidth]{Imagen2.eps}
\includegraphics[width=0.32\textwidth]{Imagen3.eps}
\end{center}
\caption{
Invariant mass distributions for different decay modes of
the $\Omega_c^0$.
Left: $\Omega^-\pi^+$, Signal: $35\pm12$ events;
center: $\Omega^-\pi^+\pi^+\pi^-$, $44\pm14$ events;
right: $\Xi^-K^-\pi^+\pi^+$, $28\pm12$ events
}
\label{omcmass}
\end{figure}
The total sample contains $107\pm22$ events,
nearly half of them in $\Omega3\pi$. At this moment we are working on the
systematics of the mass and branching ratio measurements of these
modes~\cite{sedat}.
We use the $\Omega\pi$ and $\Omega3\pi$ channels to determine the
lifetime of the $\Omega_c^0$.
We calculate the reduced proper time $ct$, given by
$ct=(L-N\sigma)/\gamma$, requiring $L/\sigma>N$ with $N=6$, for each
event within the mass region of the $\Omega_c^0$. The proper lifetime
resolution is $\sim 20\,\mbox{fs}$.
We make a maximum likelihood fit to
a probability distribution having an exponential decay
for the signal and two exponentials for the fast and slow
components of the background:
\begin{displaymath}
N_s(1-\alpha)f(t)\tau^{-1} e^{-t/\tau} +
\alpha N_B(\beta\tau_1^{-1} e^{-t/\tau_1} +
(1-\beta)\tau_2^{-1} e^{-t/\tau_2})
\end{displaymath}
where $\tau$, $\alpha$, $\beta$, $\tau_1$, $\tau_2$
are the fit parameters describing the lifetime and the relative
contributions of the background to the $ct$ distribution, and
$f(t)$ is the acceptance function.
We do this for each mode separately, and obtain
for the $\Omega\pi$ mode $\tau=62.6 \pm 22.0\,\mbox{fs}$ and for the
$\Omega3\pi$ mode $\tau=65.8 \pm 16.0\,\mbox{fs}$. Combining the two
results yields
$\tau_{\Omega_c}=65 \pm 13\,(stat) \pm 9\,(sys)\,\mbox{fs}$.
More details can be found in~\cite{omclife}.
This result should be compared to the current
PDG average~\cite{pdg}
of $69\pm12\,\mbox{fs}$, using a total of 175~events
from three different experiments.
\section{Doubly Charmed Baryons}
\subsection{The Discovery of Double Charm Baryons}
In 2002 the SELEX collaboration reported the first observation of a
candidate for a double charm baryon,
decaying as $\Lambda_{c}^+ K^-\pi^+$~\cite{prl,Thesis}. The
state had a mass of $3519\pm2\,\mbox{MeV}/c^2$,
and its observed width was consistent
with experimental resolution, less than $5\,\mbox{MeV}/c^2$. The final state
contained a charmed baryon and negative strangeness ($\Lambda_c^+$ and $K^-$),
consistent with the Cabibbo-allowed decay of a $\Xi_{cc}^+$ configuration.
In order to confirm the interpretation of this state as a double charm
baryon, it is essential to observe the same state in some other way.
Other experiments with large charm baryon samples, e.g., the FOCUS~\cite{FOCUS}
and E791 fixed target charm experiments at Fermilab or the
B-factories,
have not confirmed the double charm signal. This is not inconsistent
with the SELEX results. The report in Ref.~\cite{prl} emphasized that
this new state was produced by the baryon beams ($\Sigma^-$, proton) in
SELEX, but not by the $\pi^-$ beam. It also noted that the apparent lifetime
of the state was significantly shorter than that of the $\Lambda_c^+$, which
was not expected in model calculations~\cite{guberina}.
A more detailed discussion can be found in~\cite{goteborg}.
\subsection{Features and Problems in the Original Analysis,
and Possible Solutions}
All the signals observed so far are statistically significant,
but have only a few signal events.
The signals are clean, e.g.\ there is very little
background, but the background itself is also difficult to estimate.
SELEX only observes events from the baryon ($\Sigma^-$, proton) beams,
and the number of observed events is larger than some
production models (see for
example~\cite{murray,kiselev}) predict. As mentioned before, the lifetime seems
to be very short, and no other experiment has confirmed our observations.
Another way to confirm the $\Xi_{cc}^+$ is to observe
it in a different decay mode that also involves a final state with baryon
number and charm (not anti-charm). One such mode involving only stable charged
particles is the channel $pD^+K^-$,
another one $\Xi_c^+\pi^+\pi^-$. SELEX developed a new method for
a more reliable background determination. We also improved the resolution
on the secondary vertex position by including the single-charm track into
the vertex fit, and we redid our full analysis chain to increase our
statistics. In the following we will describe these step in details.
\subsection{New Analysis Features within SELEX}
The Cabibbo-allowed decay of the $\Xi_{cc}^+$ is shown in the following figure.
\begin{center}
\begin{picture}(20000,15000)
\THICKLINES
\drawline\fermion[\E\REG](1000,3000)[15000]
\drawline\photon[\NE\REG](\particlemidx,\particlemidy)[6]
\drawline\fermion[\N\REG](\photonbackx,\photonbacky)[3000]
\put(\particlebackx,\particlebacky){$u$}
\global\Xthree=\particlebackx
\global\advance\Xthree by 500
\global\Ythree=\particlebacky
\global\advance\Ythree by 400
\drawline\fermion[\E\REG](\photonbackx,\photonbacky)[3000]
\put(\particlebackx,\particlebacky){$\overline{d}$}
\put(0,2600){$c$}
\put(16500,2600){$s$}
\put(8200,5000){$W^+$}
\drawline\fermion[\E\REG](1000,0)[15000]
\put(0,-400){$d$}
\put(16500,-400){$d$}
\drawline\fermion[\E\REG](1000,1500)[15000]
\put(0,1100){$c$}
\put(16500,1100){$c$}
\end{picture}
\end{center}
In the final state we expect a baryon,
and the quarks $csdu\bar{d}$ plus eventually some pairs from the sea.
We also expect a cascaded decay chain, with the first, and later the
second charm quark undergoing a weak decay.
For SELEX, the easily accessible decay modes for the different
doubly charmed baryons are:
$\Xi_{cc}^+\to \Lambda_c^+ K^- \pi^+$,
$\Xi_{cc}^+\to p D^+ K^-$,
$\Xi_{cc}^+\to \Xi_c^+ \pi^- \pi^+$,
$\Xi_{cc}^{++}\to \Lambda_c^+ K^- \pi^+\pi^+$,
$\Xi_{cc}^{++}\to p D^+ K^-\pi^+$
(depending on the mass of the $\Xi_{cc}^{++}$),
$\Xi_{cc}^{++}\to \Xi_c^+ \pi^+$,
$\Xi_{cc}^{++}\to \Xi_c^+ \pi^+ \pi^+\pi^-$,
$\Omega_{cc}^+\to \Xi_c^+ K^- \pi^+$, and
$\Omega_{cc}^+\to \Xi_c^+ K^- \pi^+\pi^+\pi^-$.
The first two modes are already published~\cite{prl,SELEX2} by SELEX, and
work on the other modes is in progress; here we will report on a first
observation of the third decay mode listed.
For the background determination, we developed an event mixing method.
The first decay vertex is close to the primary vertex, and we assume
that all the background is purely combinatoric. We make combinatoric
backgrounds by taking the first decay vertex from one event, and the second
vertex from another event; to increase statistics, we use the single-charm
vertex 25~times. The resulting combinatoric background is absolutely
normalized. We employed this method already in~\cite{SELEX2}.
\subsection{{\mbox{\boldmath$\Xi_{cc}^+\to \Lambda_c^+ K^- \pi^+$}} --
New Analysis}
\begin{figure}[ht]
\hfill
\includegraphics[clip,width=6cm]{lc-old.eps.fixed}
\hfill
\includegraphics[width=8.0cm]{lcp2.eps}
\hfill
\caption{$\Lambda_c^+\to pK^-\pi^+$ data sets of original (left) and new
(right) analysis.}
\label{lamc}
\end{figure}
To increase our statistics, we re-analyzed our full data set with some
softer cuts and with improved tracking software. In fig.~\ref{lamc}
we show a comparison of the $\Lambda_c^+$ data set used for the
analysis.
The number of $\Lambda_c\to pK^-\pi^+$ candidates increased from 1630 to 2140.
We also improved the resolution of the decay vertex position of
the $\Xi_{cc}^+$ candidate
by including the vector of the $\Lambda_c^+$ into the
vertex fit. This improved resolution reduces the background when applying
a cut in $L/\sigma$, while keeping more signal events. It also
increases the possibility of measuring the lifetime of double charm baryons.
\begin{figure}[ht]
\centerline{\includegraphics[width=0.8\textwidth]{ccd_lc_ls.eps}}
\caption{$\Lambda_c^+K^-\pi^+$ ($\Lambda_c^+\to pK^-\pi^+$)
invariant mass distributions (blue) for
various cuts in $L/\sigma$ on the first decay vertex. In green we show
the estimated combinatoric background from the event mixing procedure
described in the text.}
\label{lcls}
\end{figure}
Figure~\ref{lcls} shows the results of our new analysis, for various
cuts in $L/\sigma$ of the first decay vertex.
Re-analyzing and relaxing some cuts in the single charm sample
increased the number of signal events, but also
resulted in a somewhat higher background level;
but the background is nicely reproduced and well understood from the
combinatoric analysis. The improved secondary vertex resolution yields in
cleaner signals and allows access to other decay modes, which we will
pursue in the future. Measuring the lifetime now seems possible, but is
still challenging. As seen from the yields for different cuts in $L/\sigma$,
the lifetime seems to be around $1\,\sigma$.
\subsection{{\mbox{\boldmath$\Xi_{cc}(3780)^{++}\to\Lambda_c^+K^-\pi^+\pi^+$}}}
\begin{figure}[ht]
\centerline{\includegraphics[width=0.6\textwidth]{ccu_slp.eps}}
\caption{The $\Lambda_c^+K^-\pi^+\pi^+$ invariant mass distribution,
for $\Sigma^-$ beam only.}
\label{xicc3780}
\end{figure}
We also revisited with our re-analyzed data set the first double-charm
baryon state we found in SELEX~\cite{Thesis}, the $\Xi_{cc}(3780)^{++}$.
In fig.~\ref{xicc3780} is shown the $\Lambda_c^+K^-\pi^+\pi^+$
invariant mass distribution, restricting ourselves to $\Sigma^-$ induced
events. The peak at $3780\,\mbox{MeV}/c^2$ is statistically significant,
and is wider than our experimental resolution, as shown by Monte Carlo.
The background is well described by our mixed event procedure. By removing
the slower of the $\pi^+$'s, we observe that about half of the
$\Xi_{cc}(3780)^{++}$ decay to $\Xi_{cc}^+(3520)$. At this moment we
are finishing up the analysis for this state.
\clearpage
\section{First Observation of
{\mbox{\boldmath$\Xi_{cc}^+\to \Xi_c^+ \pi^+ \pi^-$}}}
SELEX published~\cite{sun} the first observation of the
Cabibbo-suppressed decay
of $\Xi_c^+\to pK^-\pi^+$; this is the same final state as we used
before for the reconstruction of the $\Lambda_c^+$. Our sample of
$\Xi_c^+$ in the mode is much smaller than our $\Lambda_c^+$ sample,
but the branching fraction of $\Xi_{cc}^+\to\Xi_c^+\pi^+\pi^-$ should
be larger than to $\Lambda_c^+K^-\pi^+$. We applied the same cuts
and procedure as to the previously described analyzes, and
obtained~\cite{ibrahim}
the $\Xi_c^+\pi^+\pi^-$ invariant mass distribution shown in
fig.~\ref{xiccc}.
\begin{figure}[ht]
\centerline{\includegraphics[width=0.8\textwidth]{ccd_xic400_ls.eps}}
\caption{Left: $pK^-\pi^+$ invariant distribution and $\Xi_c^+$ sample
(yellow) used. Right: $\Xi_c^+ \pi^+ \pi^-$ invariant mass distribution.
The green histogram is our estimate of the combinatoric background.}
\label{xiccc}
\end{figure}
A clear peak at about $3520\,\mbox{MeV}/c^2$ is seen in the figure. This
constitutes the first observation of this decay mode of the $\Xi_{cc}^+(3520)$.
\section{Summary}
SELEX is still the only experiment observing double charm baryons. We
published observations on two different decays modes,
$\Xi_{cc}^+\to \Lambda_c^+ K^- \pi^+$~\cite{prl} and
$\Xi_{cc}^+\to pD^+K^-$~\cite{SELEX2}.
After a re-analysis of our full data set, with improved efficiency
and resolution, we presented here a higher-statistics observation of
$\Xi_{cc}^+\to \Lambda_c^+ K^- \pi^+$, and a re-analysis of the
$\Xi_{cc}(3780)^{++}$. The new analysis also allows access to additional
decay modes, and we presented here the
first observation of $\Xi_{cc}^+\to \Xi_c^+ \pi^- \pi^+$.
SELEX will continue the line of analysis, by first publishing these
preliminary results. We will try to measure the lifetime of the $\Xi_{cc}^+$.
We will also seek the isospin-partner of the $\Xi_{cc}^+$, the $\Xi_{cc}^{++}$
in all corresponding decay modes around $3500\,\mbox{MeV}/c$.
\section{Acknowledgment}
The author thanks the organizers for the invitation to present these
results at the conference. This work was supported in part by
the Consejo Nacional de Ciencia y Tecnolog\'{\i}a (CONACyT), Mexico,
and by a special research grant of UASLP.
|
3,212,635,537,965 | arxiv | \section{Introduction}
Sequence alignment is one of the most commonly used computational
tools of molecular biology. Its applications range from the
identification of the function of newly sequenced genes to the
construction of phylogenic trees~\cite{wate94,dool96}. Beyond its
practical importance, it is one of the simplest model systems for
pattern matching. In computational biology, sequences are routinely
compared via a transfer matrix algorithm to find the ``optimal''
alignment. Recently, it has been noted that this transfer matrix
algorithm is the same as the one used to calculate the partition
function or optimal energy of a directed polymer in a random
medium~\cite{hwa96}. This problem is known to belong to the
universality class of surface growth as described by the
Kardar-Parisi-Zhang (KPZ) equation~\cite{kard86}. From the assignment
of sequence alignment to the KPZ universality class various {\em
scaling laws} characterizing sequence alignment have been deduced.
They have been used in order to answer questions of practical
importance to sequence alignment, e.g., the optimal choice of
alignment parameters~\cite{dras98a,hwa98,dras98b}. But there are also
{\em non universal} features which are of great importance for
practical applications. They cannot be extracted from the knowledge of
the universality class alone, but have to be evaluated by a
microscopic study taking into account all the details of the given
sequence alignment algorithm. In this paper, we will perform such a
study for a certain choice of parameters for which sequence alignment
maps {\em exactly} onto the asymmetric exclusion
process~\cite{krug91a,derr98a}, which is the best studied
nonequilibrium system of the KPZ universality class, equivalent also
to the six vertex model~\cite{kand90,gwa92}.
We will apply this mapping to address the central question in the
biological application of sequence alignment, namely the assessment of
alignment significance: The problem is that an ``optimal'' alignment,
i.e., the best possible alignment of two given sequences according to
some scoring function, does not necessarily reflect sequence homology.
A sequence alignment algorithm will produce an ``optimal'' alignment
for any pair of sequences, including randomly chosen ones. The
important question is whether the alignment produced reflects an
underlying similarity of the two sequences compared. A common way to
address this question is to evaluate the probability of getting a
certain alignment score by chance. This requires the knowledge of the
distribution of alignment scores for random sequences. This
distribution turns out to obey a universal (Gumbel) form with two
non-universal parameters. In this paper, we will derive the Gumbel
distribution and characterize some of its properties by relating them
to the corresponding asymmetric exclusion process. In particular, we
show how the tail of this distribution can be obtained from the
generating function for the total number of hopped particles. The
latter is also the generating function of the average surface height
in the equivalent surface growth formulation of the asymmetric
exclusion process. This important quantity has been calculated for the
case of continuous time and continuous space using the replica
trick~\cite{kard85} a long time ago. More recently, it has been
obtained for the case of continuous time and discrete space in the
scaling regime~\cite{derr98b}. Here, we will calculate this quantity
in discrete time and discrete space as necessary for the mapping to
sequence alignment, in the asymptotic large size limit which is {\em
beyond} the scaling limit. Our calculation does not make use of the
replica trick and leads to a very simple closed form expression. It
explicitly contains the anomalous $t^{1/3}$ scaling of the surface
height fluctuations of KPZ surface growth in one dimension. We use
this generating function to give an explicit expression for the
significance of sequence alignments.
The paper is organized as follows: First, we will give a
self-contained introduction to sequence alignment in
Sec.~\ref{sec_seqalign}. This familiarizes the reader with the
sequence alignment algorithm and gives us a chance to develop the
notations to be used later. In Sec.~\ref{sec_localtoglobal}, we will
reduce the problem of assessing the statistical significance of the
widely used {\em local} alignment to a quantity defined in terms of
the simpler {\em global} alignment. Readers more interested in the
properties of the discrete asymmetric exclusion process can skip these
two sections and go directly to Sec.~\ref{sec_maptoasep}, which
describes the simplest version of the global alignment problem. Here,
the exact mapping to the asymmetric exclusion process in discrete time
and space with sublattice-parallel updating is described.
Sec.~\ref{sec_evalgenfunc} is devoted to the calculation of the
generating function of interest for the asymmetric exclusion process.
In Sec.~\ref{sec_implications}, we discuss the result obtained, apply
it to the assessment of alignment significance, and verify the
analytical predictions numerically. In Sec.~\ref{sec_generalize}, we
consider more general scoring systems and map them onto generalized
asymmetric exclusion processes. The final section gives a short
summary of the paper and points towards several future directions. A
number of technical details are given in the appendices.
\section{Review of Sequence Alignment}\label{sec_seqalign}
\subsection{Gapless Alignment}
Sequence alignment algorithms come in different levels of
sophistication. The simplest alignment algorithm is {\em gapless}
alignment. It is not only extremely fast but also very well understood
theoretically. Thus, it has been very widely used, e.g., in its
implementation of the program BLAST~\cite{alts90}.
Gapless alignment looks for similarities between two sequences
${\vec a} = \{a_1 a_2 \ldots a_M\}$,
and ${\vec b}=\{b_1 b_2 \ldots b_{N}\}$
of length $M$ and $N\sim M$ respectively. The letters $a_i$
and $b_j$ are taken from an alphabet of size $c$. This may be
the four letter alphabet $\{A,C,G,T\}$ of DNA sequences or the twenty
letter alphabet of protein sequences with the letters distributed
according to the natural frequencies of the twenty amino acids. A
local gapless alignment ${\cal A}$ of these two sequences consists of
a substring $a_{i-\ell+1}\dots a_{i-1}a_i$
of length $\ell$ of sequence ${\vec a}$ and a substring
$b_{j-\ell+1}\dots b_{j-1}b_j$ of sequence
${\vec b}$ of the same length. Each such alignment is assigned a
score
\begin{equation}
S[{\cal A}]=S(i,j,\ell)=\sum_{k=0}^{\ell-1}
s_{a_{i-k},
b_{j-k}},
\end{equation}
where $s_{a,b}$ is some given ``scoring
matrix'' measuring the mutual degree of similarity of the different
letters of the alphabet. A simple example of such a scoring matrix is
the match--mismatch matrix
\begin{equation}\label{eq_idmatrix}
s_{a,b}=
\left\{\begin{array}{ll}1&a=b\\
-\mu&a\not=b\end{array}\right.
\end{equation}
which is used for DNA sequence comparisons~\cite{need70}. For protein
sequences, the more complicated $20\times20$ PAM~\cite{dayh78} or
BLOSUM matrices~\cite{heni92} are used to account for the variable
degrees of similarity (e.g., hydrophobicity, size) among the $20$
amino acids. The computational task is to find the $i$, $j$, and
$\ell$ which give the {\em highest} total score
\begin{equation}\label{eq_maxscore}
\Sigma\equiv\max_{\cal A} S[{\cal A}]
\end{equation}
for a given scoring matrix $s_{a,b}$.
The optimization task called for in gapless alignment can be easily
accomplished by introducing an auxiliary quantity,
$S_{i,j}$, which is the optimal score of the above
consecutive subsequences ending at $(i,j)$ (optimized over $\ell$.)
It can be conveniently calculated in $O(N^2)$ instead of the expected
$O(N^3)$ steps using the transfer matrix algorithm
\begin{equation}\label{eq_gaplessevolv0}
S_{i,j}=
\max\{S_{i-1,j-1}+s_{a_i,b_j},0\},
\end{equation}
with the initial condition $S_{0,k}=0=S_{k,0}$.
This recursion equation reflects that for a given $(i,j)$ the optimal
$\ell$ is either zero or larger than zero. If the optimal $\ell$ is
zero the corresponding score is zero as well. If the optimal $\ell$ is
at least one, the pair $(a_i,b_j)$ certainly belongs
to the optimal alignment together with whatever has been chosen to be
optimal up to the point $(i-1,j-1)$. Eq.~(\ref{eq_gaplessevolv0}) is
basically a random walk with increments
$s_{a,b}$ which is cut off if it falls below
zero. The global optimal score is obtained as
\begin{equation}\label{eq_Sigma}
\Sigma = \max_{1\le i \le M, 1 \le j \le N} S_{i,j}.
\end{equation}
In order to characterize the statistical significance of the
alignment, it is necessary to know the distribution of $\Sigma$ for
gapless alignments of two {\em random} sequences, whose elements
$a_k$'s are generated independently from the same frequencies
$p_a$ as the query sequences, and scored with the same matrix
$s_{a,b}$. This distribution of $\Sigma$
has been worked out rigorously~\cite{karl92,karl93}. For suitable
scoring parameters, it is a Gumbel or extreme value distribution given
by
\begin{equation}\label{eq_gumbeldist}
\Pr\{\Sigma<S\}=\exp(-\kappa e^{-\lambdaS}).
\end{equation}
This distribution is characterized by the two parameters $\lambda$ and
$\kappa$ with $\lambda$ giving the tail of the distribution and
$\lambda^{-1}\log\kappa$ describing the mode. For gapless alignment,
these non universal parameters can be explicitly
calculated~\cite{karl92,karl93} from the scoring matrix
$s_{a,b}$ and the letter frequencies
$p_a$. For example, $\lambda$ is the unique positive solution
of the equation
\begin{equation}\label{eq_lambdacond}
\langle \exp(\lambda s)\rangle\equiv
\sum_{a,b}p_a p_b
\exp(\lambda s_{a,b})=1.
\end{equation}
The other parameter $\kappa$ is given by $\kappa=KMN$, where $K$ is a
more complicated function of the scoring matrix and the letter
frequencies. Instead of reviewing the full derivation of the
distribution Eq.~(\ref{eq_gumbeldist}) and its parameters, we will
below give some heuristic arguments which yield the known
result. These can later be generalized to the more relevant case of
alignment with gaps.
For random sequences, one can take $j=i$ in (\ref{eq_gaplessevolv0})
without loss of generality. Eq.~(\ref{eq_gaplessevolv0}) then becomes
a discrete Langevin equation, with
\begin{equation}
S_{i,i} \equivS(i)=\max\{S(i-1)+s(i),0\},
\label{eq_gaplessevolv}
\end{equation}
where the ``noise'' $s(i)\equiv
s_{a,b}$ is uncorrelated and given by the
distribution
\begin{equation}\label{eq_scoredist}
\Pr\{s_i>s\}=
\sum_{\{a,b|s_{a,b}>s\}}
p_a p_b.
\end{equation}
The dynamics of the evolution equation (\ref{eq_gaplessevolv}) can be
in two distinct phases. The quantity which distinguishes these two
phases is the expected local similarity score
\begin{equation}\label{eq_averscoreneg}
\langle s\rangle\equiv\sum_{a,b}
p_a p_b s_{a,b}.
\end{equation}
If it is positive, the score $S(i)$ will increase on
average. After a while, it becomes positive enough that the maximum in
Eq.~(\ref{eq_gaplessevolv}) will never be given by the zero
option. This option could thus be omitted which corresponds to {\em
global} gapless alignment. The dynamics is then a random walk
$S(i)=S(i-1)+s(i)$ with an average upward
drift $\langle s\rangle$. The maximal score will be close to
the end of the sequences and will be given by $\Sigma\approx
N\cdot\langle s\rangle$. Since it is linear in the length of
the sequences, this is called the {\em linear phase} of local
alignment. It is obviously not suited to identify matches of {\em
subsequences}, and the distribution of the maximal score $\Sigma$
is not an extreme value distribution. (It is just a sum of many
independent local scores $s(i)$ and therefore obeys a Gaussian
distribution according to the central limit theorem.)
The situation is dramatically different if $\langles\rangle$ is
negative. In this case the dynamics is qualitatively as follows: The
score $S(i)$ starts at zero. If the next local score
$s(i+1)$ is negative --- which is the more typical case in this
regime --- then $S$ remains zero. But if the next local
score is positive, then $S$ will increase by that
amount. Once it is positive, $S(i)$ performs a random walk
with independent increments $s(i)$. Since
$\langles\rangle$ is negative, there is a {\em negative
drift\/} which forces $S(i)$ to eventually return to
zero. After it is reset to zero, the whole process starts over again.
The qualitative ``temporal'' behavior of the score $S(i)$ is
depicted in Fig.~\ref{fig_gaplessscore}.
\begin{figure}[ht]
\begin{center}
\epsfig{figure=figure1.eps,width=0.8\columnwidth}
\caption{Sketch of the total score as a function of sequence
position in gapless local alignment.}
\label{fig_gaplessscore}
\end{center}
\end{figure}
{From} the figure, it is clear that the score landscape can be divided
into a series of ``{\em islands\/}'' of positive scores, separated by
``oceans'' where $S=0$. Each such island originates from a
single jump out of the zero-score state and terminates when the
zero-score state is reached again. Since each of these islands
depends on a different subset of independent random numbers
$s(i)$, the islands are {\em statistically independent} of each
other. If we let the maximal score of the $k^{\rm th}$ island be
$\sigma_k$, then these $\sigma_k$ are independent random
variables. Calculating the probability for the maximum score
$\sigma_k$ of an island of length $L$ in a saddle point
approximation and optimizing over the length $L$ of the islands, we
asymptotically obtain a Poisson distribution
\begin{equation}\label{eq_pislandpeak}
\Pr(\sigma_k>\sigma)\approx C_*e^{-\lambda\sigma}
\end{equation}
for the maximal island scores $\sigma_k$ (see
App.~\ref{app_gllambda}.) The parameter $\lambda$ which gives the
typical scale of the maximal island score is given by the
drift-diffusion balance of the underlying Brownian process. If the
local scores $s(i)$ were Gaussian variables with average $v<0$
and variance $D$, this drift-diffusion balance would yield
\begin{equation}\label{eq_diffusivelambda}
\lambda=2\frac{|v|}{D}.
\end{equation}
For an arbitrary discrete or continuous distribution of the local
scores $s(i)$, it turns out to be given by the more general
condition~(\ref{eq_lambdacond}), which reduces to
Eq.~(\ref{eq_diffusivelambda}) in the limit
$\langles\rangle\to0^-$ where the central limit theorem takes
hold.
Since the global optimal score $\Sigma$ can be expressed by the maximal
island scores as
\begin{equation}\label{eq_sigmamax}
\Sigma = \max_k \{\sigma_k\},
\end{equation}
the distribution of $\Sigma$ can be calculated from the
distribution of the $\sigma_k$. The connection is covered by the
theory of extremal statistics as developed by
Gumbel~\cite{gumb58,gala78}. In the case of a large number $K_*\sim N$
of independent island peak scores each of which asymptotically obeys
the Poisson distribution Eq.~(\ref{eq_pislandpeak}), the connection is
especially simple and we get
\begin{eqnarray}\label{eq_derivegumbel}
\Pr\{\Sigma<S\}&=&
\Pr\{\max\{\sigma_1,\ldots,\sigma_{K_*}\}<S\}
=\Pr\{\sigma_1<S\}^{K_*}\\\nonumber
&=&(1-C_*e^{-\lambdaS})^{K_*}
\approx[\exp(-C_*e^{-\lambdaS})]^{K_*}
=\exp(-\kappa e^{-\lambdaS})
\end{eqnarray}
with $\kappa\equiv C_*K_*$, i.e., the parameter $\lambda$ of the
island peak score distribution Eq.~(\ref{eq_pislandpeak}) is the same
as the parameter $\lambda$ in the Gumbel distribution
Eq.~(\ref{eq_gumbeldist}) of the maximal alignment scores.
\subsection{Alignment with Gaps}
In order to detect weak similarities between sequences separated by a
large evolutionary distance, ``gaps'' have to be allowed within an
alignment to compensate for insertions or deletions occurred during
the course of evolution~\cite{pear91}. Here, we will specifically
consider Smith-Waterman local alignment~\cite{smit81}. In this case,
a possible alignment ${\cal A}$ still consists of two substrings of
the two original sequences ${\vec a}$ and ${\vec b}$. But
now, these subsequences may have different lengths, since gaps may be
inserted in the alignment. For example the two subsequences {\tt
GATGC} and {\tt GCTC} may be aligned as {\tt GATGC} and {\tt GCT-C}
using one gap. Each such alignment ${\cal A}$ is assigned a score
according to
\begin{equation}
S[{\cal A}]=
\sum_{(a,b)\in{\cal A}}
s_{a,b}-\delta N_{\mathrm g}
\end{equation}
where the sum is taken over all pairs of aligned letters, $N_{\mathrm
g}$ is the total number of gaps in the alignment, and $\delta$ is an
additional scoring parameter, the ``gap cost''. In practice more
complicated gap scores may be used, but we will concentrate on this
version.
The task of local alignment is again to find the alignment ${\cal A}$
with the highest score as in Eq.~(\ref{eq_maxscore}), in this enlarged
class of possible alignments. This can be very efficiently done by a
transfer matrix method which becomes obvious in the alignment path
representation~\cite{need70}. In this representation, the two sequences
to be compared are written on the edges of a square lattice as the one
shown in Fig.~\ref{fig_alpath} where we chose for simplicity
$N=M$. Each directed path on this lattice represents one
possible alignment. The score of this alignment is the sum over the
local scores of the traversed bonds. Diagonal bonds correspond
to gaps and carry the score $-\delta$. Horizontal bonds are assigned
the similarity scores
\begin{equation}
s(r,t)\equiv s_{a_i,b_j}
\end{equation}
where $a_i$ and $b_j$ are the letters of the two
sequences belonging to the position $(r,t)=(i-j,i+j-1)$ as
shown in Fig.~\ref{fig_alpath}.
\begin{figure}[htbp]
\begin{center}
\epsfig{figure=figure2.eps,width=0.7\columnwidth}
\caption{Local alignment of two sequences \protect$CGATGCT$ and
\protect$TGCTCGA$ represented as a directed path on the alignment
lattice: the diagonal bonds correspond to gaps in the alignment. The
horizontal bonds represent aligned pairs. Alignments of identical
letters (matches) are shown as solid lines; alignments of different
letters (mismatches) are shown dashed. The highlighted alignment path
\protect$r(t)$ corresponds to one possible alignment of two
subsequences, \protect$GATGC$ to \protect$GCT\!-\!C$. This path
contains one gap. It is also shown how the coordinates $r$ and $t$
are used to identify the nodes of the lattice.}
\label{fig_alpath}
\end{center}
\end{figure}
If we were interested in finding the highest scoring {\em global}
alignment of the two sequences ${\vec a}$ and ${\vec b}$, this
corresponds to finding the best scoring path connecting the beginning
$(0,0)$ with the end $(0,2N)$ of the lattice. To find this path
effectively, we define the auxiliary quantity $h(r,t)$ to be
the score of the best path ending in the lattice point
$(r,t)$. This quantity can be calculated by the
Needleman-Wunsch transfer matrix algorithm~\cite{need70}
\begin{equation}\label{eq_globrecurs}
h(r,t+1)=
\max\{h(r,t-1)+s(r,t),\
h(r+1,t)-\delta,
h(r-1,t)-\delta\}.
\end{equation}
This is easily recognized~\cite{hwa96} as the algorithm used to
calculate the zero temperature configuration and energy of a directed
polymer in a random potential given by the local scores
$s(r,t)$. The scores $h(r,t)$ represent the
(negative) energy of the optimally chosen polymer configuration ending
in the point $(r,t)$. Alternatively, the $h(r,t)$
can also be interpreted as the spatial height profile of a growing
surface through the well known relation between the directed polymer
and the KPZ equation.
If we are interested in {\em local} alignments, we can use the same
trick as in the gapless case~(\ref{eq_gaplessevolv0}). Cutting off
unfavorable scores by adding the choice of zero in the maximum of
Eq.~(\ref{eq_globrecurs}) leads to the Smith-Waterman
algorithm~\cite{smit81}
\begin{equation}\label{eq_swrecursion}
S(r,t+1)=\max\left\{\begin{array}{l}
S(r,t-1)+s(r,t)\\
S(r+1,t)-\delta\\S(r-1,t)-\delta\\0
\end{array}\right\}.
\end{equation}
The score of the best local alignment is then given by
\begin{equation}
\Sigma=\max_{r,t}S(r,t).
\end{equation}
In the presence of gaps, we can still distinguish a linear and a
logarithmic phase. If the global alignment score tends to grow, the
zero option of the local alignment algorithm does not play any role.
We effectively revert to global alignment and get a maximum score
which is linear in the length of the sequences. Contrary to gapless
alignment, it is not enough to have a negative expectation value of
the local scores $\langles\rangle$ in order to prevent
this. This is due to the fact that the alignment algorithm uses gaps
to connect random stretches of good matches to optimize the score. The
average score grows by a gap dependent amount
$u(\{s_{a,b}\},\delta)$ faster compared to
the expectation value $\langles\rangle$. The log-linear
transition occurs now at
$u(\{s_{a,b}\},\delta_{\mathrm
c})+\langles\rangle=0$. For the simple scoring system
Eq.~(\ref{eq_idmatrix}) this corresponds to a line $\delta_{\mathrm
c}(\mu)$ in the two dimensional space of the parameters $\mu$ and
$\delta$ shown in Fig.~\ref{fig_ptline}. Even for this simple scoring
system, the loci of the phase transition are only known
approximately~\cite{bund99}; for more complicated scoring systems,
only numerical results are available.
\begin{figure}[ht]
\begin{center}
\epsfig{figure=figure3.eps,angle=270,width=0.5\columnwidth}
\caption{Loci of the log-linear phase transition for alignment with
the scoring system Eq.~(\protect\ref{eq_idmatrix}) for an alphabet of
\protect$c=4$ letters in terms of the mismatch cost \protect$\mu$ and
the gap cost \protect$\delta$. Useful alignments can only be
obtained in the logarithmic phase above the phase transition line. The
diamonds are numerically estimated points on the phase transition
line; the solid line is the approximate locus calculated
in~\protect\cite{bund99}. Below the dashed line the alignments do not
depend on the mismatch cost \protect$\mu$ any more and the phase
transition line is known to be strictly horizontal.}
\label{fig_ptline}
\end{center}
\end{figure}
If the parameters are chosen such that $u+\langles\rangle<0$,
i.e., such that the expected global alignment score drifts downwards
on average, then the average maximum score $\langle\Sigma\rangle$
is proportional to the logarithm of the sequence length as in the
logarithmic phase of gapless alignment. The reduced value of
$\langle\Sigma\rangle$ in the logarithmic phase makes it the regime of
choice for the purpose of homology detection. Again, the distribution
of $\Sigma$ must be known for local alignments of random sequences
in order to characterize the statistical significance of local
alignment. There is no rigorous theory of this distribution in the
presence of gaps. However, there is a lot of empirical evidence that
the distribution is again of the Gumbel
form~\cite{smit85,coll88,mott92,wate94a,wate94b,alts96}. The values
of the parameters $\kappa$ and $\lambda$ are only known approximately
for a few cases close to the gapless limit~\cite{mott99,sieg99}. In
practice, they are determined empirically by time consuming
simulations. Below we will present an explicit calculation of the
parameter $\lambda$ for a simple scoring system.
\section{Significance Estimation using Global
Alignment}\label{sec_localtoglobal}
As a first step, we want to show that the parameter $\lambda$, which
describes the tail of the Gumbel distribution, can be derived solely
from studying the much simpler {\em global}
alignment~(\ref{eq_globrecurs}). Later, we will see that global
alignment is in certain cases equivalent to the asymmetric exclusion
process. We will derive an explicit formula for $\lambda$ by studying
the corresponding asymmetric exclusion process.
Let us define the generating function
\begin{equation}\label{eq_defgf}
Z(\lambda;L)\equiv\langle\exp[\lambda h(r=0,L)]\rangle
\end{equation}
where the brackets $\langle\cdot\rangle$ denote the ensemble average
over all possible realizations of the disorder, i.e., over all choices
of random sequences ${\vec a}$ and ${\vec b}$ and
$h(0,L)$ is the {\em global} alignment score at the end of a
lattice of length $L$ as shown in Fig.~\ref{fig_triangle}(a). It can
be obtained from the recursion relation~(\ref{eq_globrecurs}) with the
initial condition
$h(2k,t=0)=h(2k\!+\!1,t\!=\!1)=0$. It will turn out
that the parameter $\lambda$ of the Gumbel distribution is obtained
from
\begin{equation}\label{eq_glambda}
\lim_{L\to\infty}Z(\lambda;L)=1.
\end{equation}
Note, that this condition reduces simply to Eq.~(\ref{eq_lambdacond})
in the case of gapless alignment, since for infinite gap cost $\delta$,
we have
\begin{equation}
\langle\exp[\lambda h(0,L)]\rangle=
\langle\exp[\lambda\sum_{k=1}^{L/2}s(0,2k-1)]\rangle=
\langle\exp[\lambda s]\rangle^{L/2}.
\end{equation}
\begin{figure}[ht]
\begin{center}
\epsfig{figure=figure4.eps,width=0.6\columnwidth}
\caption{Global alignment lattice used for significance estimation.
(a) shows the right half of the lattice from
Fig.~\protect\ref{fig_alpath}. It can represent all possible paths of
length \protect$L$ which end at the point
\protect$(r,t)=(0,L)$ and start at \protect$(r,0)$ for an
arbitrary \protect$r$. (b) shows such a path schematically. It
represents the ``rim'' of an island with its high score denoted by the
filled dot at the tip of the triangle. The open dot at
$(r_0,t_0)$ represents the corresponding island initiation
event.}
\label{fig_triangle}
\end{center}
\end{figure}
The key observation which leads to the result~(\ref{eq_defgf})
and~(\ref{eq_glambda}) is the fact that similar to the case of gapless
alignment discussed in the last section, the points on the alignment
lattice can be grouped together as {\em islands}~\cite{olse99}. By the
construction of the local alignment algorithm~(\ref{eq_swrecursion}),
many points on the alignment lattice have a score of zero in the
logarithmic alignment regime. As for gapless alignment, a positive
score will be generated out of this ``sea'' of zeroes, if a good match
occurs by chance. This positive score can then imply further positive
scores via the recursion relation~(\ref{eq_swrecursion}). For every
point $(r,t)$ on the lattice which has a positive score, we can
define a restricted optimal path $\widehat{r}^*_{r,t}(\tau)$,
which is the highest scoring path out of all paths $\widehat{r}(\tau)$
with an end fixed at $\widehat{r}(t)=r$; see the example in
Fig.~\ref{fig_alpath}. The path must start at some point
$(r_0,t_0)$ where a positive score is created out of the zero
sea by a good match. An island is then defined to be the collection of
points $(r,t)$ with positive score, i.e.,
$S(r,t)>0$, and whose restricted optimal path
$\widehat{r}^*_{r,t}(\tau)$ originates at the same point
$(r_0,t_0)$. A sketch of these islands is shown in
Fig.~\ref{fig_islands}. Each of these islands has a maximum score
which we denote by $\sigma_k$ as we did in the gapless case. By
this definition, every lattice point with a positive score belongs to
exactly one island. Thus, the maximal score $\Sigma$ on the total
lattice is given by Eq.~(\ref{eq_sigmamax}). Since large islands are
well separated by a sea of points with score zero, they are
statistically independent clusters on the alignment lattice. Thus,
their maximal scores $\sigma_k$ are again independent
identically distributed random variables which yield a Gumbel
distribution of $\Sigma$ via Eq.~(\ref{eq_derivegumbel}). Our task
is thus to calculate the distribution of the island peak scores
$\sigma_k$ in the presence of gaps.
\begin{figure}[ht]
\begin{center}
\epsfig{figure=figure5.eps,width=0.5\columnwidth}
\caption{Sketch of some islands on the local alignment lattice. The
lattice sites with a positive score are marked with dots. The bonds
which have been chosen in the maximization
process~(\protect\ref{eq_swrecursion}) are highlighted. Together they
are the restricted optimal path associated with each point with a
positive score. Each of these paths goes back to an island initiation
event which is marked by an open dot. The large filled dots mark the
positions of the highest scoring point on each island.}
\label{fig_islands}
\end{center}
\end{figure}
This distribution of maximal island scores can be derived analogously
to the gapless case (App.~\ref{app_gllambda}.) While a single gapless
island is described by a random walk of some optimized length $L$, an
island with gaps corresponds to a {\em global gapped alignment} of some
optimized length $L$ as the one shown schematically in
Fig.~\ref{fig_triangle}(b). Using this replacement, the maximal island
distribution again has an asymptotically Poissonian
form~(\ref{eq_pislandpeak}) with the decay constant $\lambda$ given by
Eq.~(\ref{eq_glambda}). An approximate interpretation for the
result~(\ref{eq_glambda}) is the following: Due to the choice of
scoring parameters in the logarithmic phase of local alignment, the
average score $\langleh(0,L)\rangle$ of global alignment with
the same choice of parameters decreases linearly with the length $L$
of the alignment. Thus, typical configurations of the disorder have a
strongly negative score $h(0,L)$ and hardly contribute to
$Z(\lambda;L)=\langle\exp[\lambdah(0,L)]\rangle$. Only on
very rare occasions, $h(0,L)$ is positive for large $L$ and
contributes significantly to $Z(\lambda;L)$. The fact that
there is a choice of $\lambda$ with $Z(\lambda;L)=1$ for large
$L$ implies that these configurations with positive $h(0,L)$
are {\em exponentially rare}. It is thus necessary to weight these
configurations with the exponential factor
$\exp[\lambdah(0,L)]$, and choose $\lambda$ to match the decay
constant of the probability of finding such rare events.
As already noted in the analogy between the directed polymer and
sequence alignment, the score corresponds to the (negative of the)
free energy. Thus the quantity
$Z(\lambda;L)=\langle\exp[\lambda h(0,L)]\rangle$ can be
interpreted as the disorder-averaged (zero temperature) partition
function\footnote{However, $Z(\lambda;L)$ should {\em not} be
interpreted as the partition function at temperature $\lambda^{-1}$.}
of $\lambda$ ``replicas'' of a directed polymer of length $L$. Note
that the replica number given by $\lambda$ need not be integer. In the
surface growth interpretation, $Z(\lambda;L)$ is the
generating function for the space averaged surface height. While many
of the universal features of global and local sequence alignment
(e.g., its scaling behavior in the logarithmic phase and upon
approaching the phase transition line) can be understood merely from
the knowledge that sequence alignment belongs to the KPZ universality
class~\cite{hwa96,dras98a,hwa98,dras98b} or from the limit
$Z(\lambda\to0;L)$, a solution of Eq.~(\ref{eq_glambda}) for
the non universal quantity $\lambda$ requires the knowledge of the
large $L$ behavior of the entire function $Z(\lambda;L)$ and
hence a more detailed microscopic calculation for the given model.
\section{Global Alignment as an Asymmetric Exclusion
Process}\label{sec_maptoasep}
\subsection{A Simple Model of Sequence Alignment}
{From} now on we will focus on {\em global} alignment as described by
Eq.~(\ref{eq_globrecurs}), and use Eq.~(\ref{eq_glambda}) to infer the
value of the parameter $\lambda$ characterizing local alignment. In
order to simplify the presentation, we restrict ourselves here to a
very simple scoring system. As will be discussed in latter sections,
our formulation as well as some of the results can be generalized to
the more complicated scoring systems.
We will study the scoring system in which the local similarity scores
$s_{a,b}$ can take on only two possible
values,
\begin{equation}
s_{a,b}=
\left\{\begin{array}{ll}1&a=b\\0&a\not=b
\end{array}\right..
\end{equation}
Moreover we will choose the gap cost to be $\delta=0$. With this
choice of the scoring parameters, the score $h$ has the additional
interpretation of being the length of the {\em longest common
subsequence} of the two sequences ${\vec a}$ and
${\vec b}$. This longest common subsequence problem has a long
history\footnote{The longest common subsequence model is in the limit
of an alphabet size equal to the sequence length also related to the
{\em longest increasing subsequence} model which has been of recent
interest in connection with surface growth processes~\cite{prae99}.}
as a toy model for sequence comparisons~\cite{chva75,danc94,monv99}.
Additionally, we will neglect correlations between the local scores
$s(r,t)$, which arise from the fact that all $M\times
N$ local scores are generated by the $M+N$ randomly
drawn letters. Instead of taking these correlations into account, we
will assume that $s(r,t)=\eta(r,t)$ with
independent random variables $\eta(r,t)$ given by
\begin{equation}\label{eq_etadist}
\eta(r,t)=\left\{\begin{array}{ll}1&\mbox{with probab. $p$}\\
0&\mbox{with probab. $1-p$}\end{array}\right.
\end{equation}
with
\begin{equation}\label{eq_etaindep}
\Pr\{\forall_{r,t}\,\eta(r,t)=\eta_{r,t}\}=
\prod_{r,t}\Pr\{\eta(r,t)=\eta_{r,t}\}.
\end{equation}
To model sequences randomly drawn with equal probability from an
alphabet of size $c$, we take $p=1/c$. The
approximation~(\ref{eq_etaindep}) is known to change characteristic
quantities of sequence alignment only slightly~\cite{dras98a}. We
will confirm numerically at the end of this paper, that this also
holds for the values of $\lambda$ which we are mainly interested in
here. For our choices of parameters, the global alignment
algorithm~(\ref{eq_globrecurs}) reads
\begin{equation}\label{eq_lcsrecurs}
h(r,t+1)=
\max\{h(r,t-1)+\eta(r,t),
h(r+1,t),h(r-1,t)\}.
\end{equation}
\subsection{Choice of the alignment lattice geometry}
In order to handle finite size effects better, we will use a
rectangular geometry (Fig.~\ref{fig_rect}) for the alignment lattice,
instead of the triangular geometry shown in
Fig.~\ref{fig_triangle}(a). We will further apply periodic boundary
condition to the top and bottom edges of the lattice, i.e.,
$h(0,t)=h(2W,t)$ for a rectangular
lattice of width $2W$, and will start on the left edge with the
initial conditions $h(2k+1,t=0)=h(2k,t=1)=0$.
Note that despite the different lattice geometries, the score
$h(r,t)$ for all points with $t\leW$ on the
rectangular lattice will be {\em identical} to the score at the same
$(r,t)$ coordinate on the triangular lattice\footnote{Since
directed polymers in a random medium are known to have a wandering
exponent $\zeta=2/3$ this actually still holds for
$t<W^{3/2}$.}; see Fig.~\ref{fig_rect}.
\begin{figure}[ht]
\begin{center}
\epsfig{figure=figure6.eps,width=0.5\columnwidth}
\caption{Rectangular alignment lattice of width \protect$2W$ with
periodic boundary conditions in the spatial (vertical) direction. We
use this lattice instead of the triangular lattice shown in
Fig.~\protect\ref{fig_triangle}(a) in order to simplify the handling of
finite-size effects. As indicated by the thick gray lines, the score at a
point with \protect$t\leW$ as the one at the tip of the
triangle is identical with the corresponding score calculated on a
triangular lattice as the one shown in
Fig.~\protect\ref{fig_triangle}(a).}
\label{fig_rect}
\end{center}
\end{figure}
\subsection{The dynamics of sequence alignment as an asymmetric exclusion
process}
In this section we will perform a change of variables on the sequence
alignment algorithm~(\ref{eq_lcsrecurs}) for the rectangular lattice
shown in Fig.~\ref{fig_rect}. We will find that the resulting problem
is equivalent to an asymmetric exclusion process on a one-dimensional
lattice of width $2W$. As a guidance towards the choice of
suitable variables, we take the knowledge from the (continuous) KPZ
equation that the gradient of the surface height is an especially
simple quantity. At a fixed time, the gradients at different positions
become uncorrelated and Gaussian
distributed~\cite{kard86,krug91b}. Thus, we will look at their
discrete analogs in the alignment problem. They are the score
differences between neighboring lattice points and thus located on the
diagonal bonds of the lattice. We will parameterize these score
differences by the bond variables $n(r,t)$. They
will later turn out to be the occupation numbers of the sites of an
asymmetric exclusion process. With the choice of coordinates as
illustrated in Fig.~\ref{fig_element}(a), we define them to
be\footnote{Note, that the $n(r,t)$ are not literally
score differences but suitably chosen parameterizations of these score
differences. This complication is necessary in order to enable the
interpretation as the particle occupation numbers in the asymmetric
exclusion process.}
\begin{equation}\label{eq_occupdiff}
n(r,t)\equiv\left\{
\begin{array}{ll}h(r+1,t)-h(r,t+1)+1&
\mbox{for $r+t$ even}\\[5pt]
h(r+1,t+1)-h(r,t)&
\mbox{for $r+t$ odd}\end{array}\right.
\end{equation}
As explained in detail in App.~\ref{app_vtransform}, rewriting the time
evolution equation~(\ref{eq_lcsrecurs}) in terms of the variables
$n(r,t)$ leads to a time evolution equation of
$n(r,t)$ alone, without any reference to the absolute
scores $h(r,t)$. Moreover, this time evolution equation
implies that the score differences take only the values
$n(r,t)\in\{0,1\}$. By the structure of the alignment
lattice as a composition of elements as the one shown in
Fig.~\ref{fig_element}(a), the resulting time evolution for the
$n(r,t)$ transforms a pair
$(n(r-1,t-1),n(r,t-1))\in\{|00\rangle,
|01\rangle,|10\rangle,|11\rangle\}$ into the new pair
$(n(r-1,t),n(r,t))\in\{|00\rangle,
|01\rangle,|10\rangle,|11\rangle\}$ independently from all the other
$n(r^\prime,t-1)$. This transformation only depends
on the single random variable $\eta(r,t)$ and can be
expressed by the transfer matrix
\begin{equation}\label{eq_transfermatrix}
{\sf T}_1(0)\equiv\left(\begin{array}{cccc}1&0&0&0\\0&1&1-p&0\\0&0&p&0\\0&0&0&1
\end{array}\right)
\end{equation}
in the basis $|00\rangle$, $|01\rangle$, $|10\rangle$, $|11\rangle$.
We can thus interpret the action of the lattice element shown in
Fig.~\ref{fig_element}(a) as a ``device'' like the one shown in
Fig.~\ref{fig_element}(b) which takes a pair of variables
$(n_1^\prime,n_2^\prime)$ as its inputs, applies
the transfer matrix ${\sf T}_1(0)$, and generates a new pair of
variables $(n_1,n_2)$ as its outputs.
\begin{figure}[htbp]
\begin{center}
\epsfig{figure=figure7.eps,width=0.7\columnwidth}
\caption{One building block of the alignment lattice. By our numbering
scheme of the lattice $r$ and $t$ are either both even or both
odd. (a) shows the scores at the lattice points and the bond variables
\protect$n(r,t)$. (b) shows this building block as a
``device'', which takes two incoming bond variables
\protect$n_1^\prime$ and \protect$n_2^\prime$ and transforms them with
the help of the transfer matrix \protect${\sl T}_0$ into the new bond
variables \protect$n_1$ and \protect$n_2$.}
\label{fig_element}
\end{center}
\end{figure}
We recognize the action of the transfer matrix ${\sf T}_1(0)$ as the
elementary time step of an asymmetric exclusion process, if we
interpret the $n(r,t)$ as particle occupation numbers
on a one-dimensional lattice of $2W$ sites with periodic boundary
conditions as the one shown in Fig.~\ref{fig_singleparticles}. Each
of these sites can either be empty or occupied by a single particle.
In each time step for each pair of neighboring sites, a particle hops
to the right with some probability $1-p$, if the site to its right is
empty according to the non vanishing entry $|10\rangle\to|01\rangle$
of the transfer matrix ${\sf T}_1(0)$. If there is no particle or if
the site on the right is already occupied the configuration remains
unchanged.
\begin{figure}[htbp]
\begin{center}
\epsfig{figure=figure8.eps,width=0.5\columnwidth}
\caption{Interpretation of the transfer matrix \protect${\sf T}_1(0)$ as given
in Eq.~(\protect\ref{eq_transfermatrix}) as
an asymmetric exclusion process. A configuration of the
local score differences is represented by particles on a
one-dimensional lattice of width \protect$2W$.
At an odd time step for each even site \protect$r-1$
a particle hop is attempted with probability \protect$1-p$. In our
example, the particle at site \protect$0$ cannot hop, since site \protect$1$
is already occupied. The particle on site \protect$2$ can hop to site
\protect$3$ as indicated by the dashed square.}
\label{fig_singleparticles}
\end{center}
\end{figure}
In terms of the elementary devices shown in Fig.~\ref{fig_element}(b)
the lattice structure of Fig.~\ref{fig_rect} can be depicted
schematically as shown in Fig.~\ref{fig_network}. Thus, the process of
hopping a particle to the right is attempted for each even numbered
site at odd time steps and for each odd numbered site at even time
steps. This hopping dynamics is exactly the asymmetric exclusion
process with sublattice-parallel updating with periodic boundary
conditions\footnote{If we had chosen the ``hard wall'' boundary
conditions $h(-1,t)=h(2W,t)=\infty$ instead of the
periodic boundary conditions $h(2W,t)=h(0,t)$ for the
score, we would have arrived at the asymmetric exclusion process with
sublattice-parallel updating and {\em open} boundary conditions at a
feeding and extinction rate of $\alpha=\beta=1-p$ at the two ends of
the lattice with $2W-1$ sites
respectively.}~\cite{kand90,raje98}.
\begin{figure}[htbp]
\begin{center}
\epsfig{figure=figure9.eps,width=0.5\columnwidth}
\caption{Schematic representation of the alignment lattice of
Fig.~\protect\ref{fig_rect} as an ``electric circuit''. The boxes
represent elements of the type shown in
Fig.~\protect\ref{fig_element}(b). They take two particle occupation
numbers as their ``inputs'' and generate two new particle occupation
numbers as their ``outputs''. Their interconnection into a layered
structure as shown here with a shifted pairing scheme in every time
step leads to the non-trivial behavior of sequence alignment.}
\label{fig_network}
\end{center}
\end{figure}
In reducing the dynamics from a dynamics of scores into a dynamics of
the occupation numbers $n(r,t)$, one has to pay
attention to the boundary conditions. Periodic boundary conditions for
the $n(r,t)$ do not automatically lead to meaningful
periodic boundary conditions for the scores $h(r,t)$. We
thus have to impose the additional constraint that the total sum of
the local score differences across the whole lattice vanishes. In
terms of our bond variables $n(r,t)$ this translates
into the condition
\begin{equation}\label{eq_initialconstraint}
\frac{1}{2W}\sum_{r=0}^{2W-1}n(r,t)=\frac{1}{2},
\end{equation}
i.e., the system of hopping particles is at half filling. Since the
number of particles is conserved under the dynamics described by the
transfer matrix ${\sf T}_1(0)$, the condition~(\ref{eq_initialconstraint})
is guaranteed to hold if we choose the initial conditions
$\sum_{r=0}^{2W-1}n(r,t=0)/2W=1/2$.
Particle densities different from one half would correspond to a
tilted ``score profile'' $h(r,t)$ at each fixed time
$t$.
\section{The Generating Function}\label{sec_evalgenfunc}
\subsection{Expressing the generating function in terms of the hopping
process}
We now want to apply the mapping between sequence alignment and the
asymmetric exclusion process to the practical problem of assessing
alignment significance. As noted in Sec.~\ref{sec_localtoglobal}, this
amounts to calculating the generating function
\begin{equation}
Z_0(\lambda;N)\equiv\langle\exp[\lambdah(0,N)]\rangle_0,
\end{equation}
where $\langle\ldots\rangle_0$ denoted the average over the ensemble
of uncorrelated disorder defined by Eqs.~(\ref{eq_etadist})
and~(\ref{eq_etaindep}). Thus, we first need to express the total
score $h(0,N)$ in terms of the occupation numbers
$n(r,t)$. As explained in more detail in
App.~\ref{app_vtransform}, $h(0,t)$ is on average
incremented by $1/2W$ every time the transfer matrix ${\sf
T}_1(0)$ is applied except for the transition
$|01\rangle\to|10\rangle$. Thus, $Z_0(\lambda;N)$ can be
expressed as
\begin{equation}\label{eq_explambdas}
Z_0(\lambda;N)=
\exp[\lambda N/2]\langle\exp[-\lambdaJ]\rangle_0
\end{equation}
in terms of the total number of particle hops per lattice site
\begin{equation}
J\equiv\frac{1}{2W}\sum_{l=1}^{N/2}\sum_{k=0}^{W-1}
(j(2k+1,2l-1)+j(2k,2l)),
\end{equation}
where $j(r,t)\in\{0,1\}$ is the number of particle
hops at lattice site $(r,t)$. We thus need to
determine the generating function
\begin{equation}\label{eq_genfuncsbyeachother}
Q(\lambda;W,N)\equiv\langle\exp[-\lambdaJ]\rangle_0
\end{equation}
for the asymmetric exclusion process. Note, that this is different
from the generating function of the local current
$j(r,t)$: since $J/N$ is the {\em time and
space averaged} current, $Q$ contains information on spatial
and temporal {\em correlations} in the number of hopping particles
which the generating function for the local current does not.
\subsection{The Generating function as an eigenvalue
problem}\label{sec_eigenvalue}
Now we will reformulate the calculation of the generating function
$Q(\lambda;W,N)$ for the asymmetric exclusion process as
an eigenvalue problem. As already mentioned,
$\exp[-\lambdaJ]$ is a product of factors
$\exp[-\lambda/2W]$ for every particle that hops. Since the
dynamics of the hopping process is described by the transfer matrix
${\sf T}_1(0)$ defined in Eq.~(\ref{eq_transfermatrix}), we can
calculate $Q(\lambda;W,N)$ by associating a weight
$\exp[-\lambda/2W]$ to the element of the transfer matrix ${\sf
T}_1(0)$ which corresponds to a hop. This can be derived more formally
from a dynamics path integral representation of
$Z_0(\lambda;N)$ as detailed in App.~\ref{app_pathintegral}. We
get the modified local transfer matrix
\begin{equation}\label{eq_deftoflambda}
{\sf T}_1\left(\frac{\lambda}{W}\right)\equiv\left(\begin{array}{cccc}
1&0&0&0\\0&1&(1-p)e^{-\frac{\lambda}{2W}}&0\\0&0&p&0\\0&0&0&1
\end{array}\right)
\end{equation}
in the basis $|00\rangle,|01\rangle,|10\rangle,|11\rangle$ of a pair
of neighboring lattice sites.
Next, we need to take into account the special lattice structure of
Fig.~\ref{fig_network}. We note that at every even time step the
lattice is decomposed into $W$ of the building blocks described
by ${\sf T}_1$. Thus, a single time step of the total system
at even time is described by the matrix
\begin{equation}
{\sf T}_{\mathrm even}\equiv
{\sf T}_W(\lambda)\equiv
\bigotimes_{k=1}^W {\sf T}_1\left(\frac{\lambda}{W}\right).
\end{equation}
At odd times the dynamics is the same, but the pairing of neighboring
sites is shifted. To generate the time evolution at odd time steps, we
can thus shift all particles to the right, apply the dynamics of even time
steps and then shift all particles back to the left. Let
${\sf C}$ be the translation operator such that
\begin{equation}
{\sf C}|n_0n_1\ldotsn_{2W-1}\rangle
\equiv|n_1\ldotsn_{2W-1}n_0\rangle
\end{equation}
which shifts all particles by one site to the left taking into account
the periodic boundary conditions. With this definition we can write
${\sf T}_{\mathrm odd}={\sf C}{\sf T}_W(\lambda){\sf C}^{-1}$.
The sublattice-parallel updating procedure (i.e., the structure of the
lattice as depicted by Fig.~\ref{fig_network}) finally leads to
\begin{equation}\label{eq_genasmatrixproduct}
Q(\lambda;W,N)=
\langle\psi_1|({\sf T}_{\mathrm even}
{\sf T}_{\mathrm odd})^{N/2}|\psi_0\rangle=
\langle\psi_1|({\sf T}_W(\lambda){\sf C}{\sf T}_W(\lambda)
{\sf C}^{-1})^{N/2}|\psi_0\rangle
\end{equation}
where $|\psi_0\rangle$ is a $4^{W}$ dimensional state vector
representing the initial conditions, and $\langle\psi_1|$ is the
$4^{W}$ dimensional vector whose entries are all $1$, used here
to denote a summation over all possible final configurations. In the
limit of large $N\ggW$, this obviously becomes
\begin{equation}\label{eq_rhodef}
Q(\lambda;W,N)=\rho_W^{N}(\lambda)
\end{equation}
where $\rho_W^2(\lambda)$ is the eigenvalue of ${\sf
T}_W(\lambda){\sf C}{\sf T}_W(\lambda){\sf C}^{-1}$ with the
largest real part. Since this matrix has no negative entries and is
irreducible for non-pathological choices of the scoring matrix (while
restricted to the physical sector of half filling), the largest
eigenvalue of this matrix is guaranteed by the Perron Frobenius
theorem to be non degenerate and real, and its eigenvector can be
chosen without negative entries. When $\lambda=0$, we have
$\rho(0)=1$ and its eigenvector is the stationary distribution of
the asymmetric exclusion process, which is a simple tensor product of
independent occupation numbers. This is no longer the case for
$\lambda\not=0$.
\subsection{Calculating the largest eigenvalue}
For a finite $W$, it is in principle possible to solve for the
largest eigenvalue of the $4^W$ dimensional matrix ${\sf
T}_W(\lambda){\sf C}{\sf T}_W(\lambda){\sf C}^{-1}$ by
directly diagonalizing the matrix. It is convenient to reduce the
size of this matrix by exploiting some symmetries. Since the lattice
is translationally invariant with respect to shifts in $r$ by $2$, we
expect the same symmetry of the largest eigenvalue of ${\sf
T}_W(\lambda){\sf C}{\sf T}_W(\lambda){\sf C}^{-1}$. Thus,
for the purpose of computing the largest eigenvalue we can restrict
ourselves to the subspace ${\cal C}$ of translationally invariant
vectors
\begin{equation}
{\cal C}\equiv\Big\{|\psi\rangle\Big|C^2|\psi\rangle=|\psi\rangle\Big\}.
\end{equation}
This corresponds to a discrete Fourier transform of the matrix ${\sf
T}_W(\lambda){\sf C}{\sf T}_W(\lambda){\sf C}^{-1}$ and
choosing the $k=0$ component. On ${\cal C}$, we have ${\sf
C}^{-1}={\sf C}$ by definition. Thus, it is enough to look for the
largest eigenvalue $\rho_W(\lambda)$ of the matrix ${\sf
T}_W(\lambda){\sf C}$ restricted to ${\cal C}$. A further
restriction which helps reducing the size of the matrix is the mirror
symmetry of the lattice which has to be respected by the eigenvector
as well. Additionally, ${\sf T}_W(\lambda){\sf C}$ has to be
restricted onto the physical subspace of half filling.
After applying these simplifications, the largest eigenvalue can be
calculated for small widths $W$ using computer algebra. Although
the matrix ${\sf T}_W(\lambda){\sf C}$ explicitly contains the
quantity $\exp[-\lambda/2W]$, it turns out that the
characteristic polynomial depends only on $\exp[-\lambda/2]$. This is
a consequence of the translational invariance of the
lattice\footnote{Instead of looking at the average score
$\overline{h}(N)=\frac{1}{2W}\sum_rh(r,N)$ as we do in
the derivation of Eq.~(\ref{eq_genasmatrixproduct}) in
App.~\ref{app_pathintegral}, we could also have chosen a specific
position, say $r=0$ and $r=1$, and monitored the behavior of the score
$\widetilde{h}(N)\equiv\frac{1}{2}[h(1,N)+h(0,N-1)]$.
Since the differences between scores at the same time are bounded,
these two quantities must have the same generating function for large
$N$. The transfer matrix which calculates the generating function for
$\widetilde{h}(N)$ is $\widetilde{{\sf T}}(\lambda)\equiv {\sf
T}_1(\lambda)\otimes \bigotimes_{k=2}^{W}{\sf T}_1(0)$ instead of
${\sf T}_W(\lambda)$. It has the technical disadvantage that it
breaks the translational invariance, but it explicitly depends only on
$\exp[-\lambda/2]$ instead of $\exp[-\lambda/2W]$.}. In order to
reveal the underlying structure of the largest eigenvalues for
different $W$, it is very useful to {\em expand} the resulting
largest eigenvalues $\rho_W(\lambda)$ in powers of this
quantity $e^{-\lambda/2}$. We get
\begin{eqnarray*}
W=1:\quad \rho_1(\lambda)&=&\sqrt{p}+O(e^{-\frac{\lambda}{2}})\\
W=2:\quad \rho_2(\lambda)&=&\sqrt{p}-(p-1)e^{-\frac{\lambda}{2}}
+O((e^{-\frac{\lambda}{2}})^2)\\
W=3:\quad \rho_3(\lambda)&=&\sqrt{p}-(p-1)e^{-\frac{\lambda}{2}}
+(p-1)\sqrt{p}(e^{-\frac{\lambda}{2}})^2+O((e^{-\frac{\lambda}{2}})^3)\\
W=4:\quad \rho_4(\lambda)&=&\sqrt{p}-(p-1)e^{-\frac{\lambda}{2}}
+(p-1)\sqrt{p}(e^{-\frac{\lambda}{2}})^2
-(p-1)\sqrt{p}^2(e^{-\frac{\lambda}{2}})^3+
O((e^{-\frac{\lambda}{2}})^4),
\end{eqnarray*}
where the $O((e^{-\lambda/2})^k)$ terms denote terms of the given
order with prefactors which are different for different $W$. We
can see that the coefficients up to order
$(e^{-\lambda/2})^{W-1}$ remain unchanged upon increasing
$W$ and they constitute the beginning of a simple geometric
series. Assuming that this pattern holds for arbitrary orders, we can
resum the series for any {\em fixed} $\lambda>0$ and get
\begin{equation}
\rho(\lambda)\equiv\lim_{W\to\infty}\rho_W(\lambda)=
\frac{\sqrt{p}+e^{-\frac{\lambda}{2}}}{1+\sqrt{p}e^{-\frac{\lambda}{2}}}.
\end{equation}
Combined with Eqs.~(\ref{eq_explambdas}),
(\ref{eq_genfuncsbyeachother}), and~(\ref{eq_rhodef}) this yields the
generating function
\begin{equation}\label{eq_sfunclcs}
Z_0(\lambda;N)=\exp[\lambda N/2]\rho^{N}(\lambda)=
\left(\exp[\lambda/2]\rho(\lambda)\right)^{N}=
\left(\frac{1+\sqrt{p}\exp[\frac{\lambda}{2}]}%
{1+\sqrt{p}\exp[-\frac{\lambda}{2}]}\exp[-\frac{\lambda}{2}]\right)^{N}
\end{equation}
in the limit of large $N$.
A related generating function has also recently~\cite{derr98b} been
calculated in the simpler case of a discrete space and continuous time
asymmetric exclusion process. In this case, the full dependence on the
finite width $W$ can be computed. While Derrida {\it et
al.}~\cite{derr98b} find that the generating function takes a {\em
universal} form in the limit $W\to\infty$ with
$\lambdaW^{1/2}$ kept constant, the problem of assessing
statistical significance of sequence alignments calls for the limit
$W\to\infty$ at a fixed $\lambda>0$. This limit goes {\em beyond}
the universal regime. For the asymmetric exclusion process with
sublattice-parallel updating it is given by our
Eq.~(\ref{eq_sfunclcs})
Eq.~(\ref{eq_sfunclcs}) can be generalized~\cite{bund99b} to the
match-mismatch scoring system given in Eq.~(\ref{eq_idmatrix}) with a
gap cost $\delta=\mu/2$ for an arbitrary value of $\mu$. If we denote
the score in this scoring system by $h^\prime(r,t)$ it is
connected to the score $h(r,t)$ of the scoring system with
$\mu=\delta=0$ by the simple global rescaling and shifting
\begin{equation}
h^\prime(r,t)=(1+\mu)h(r,t)-\frac{\mu}{2}t.
\end{equation}
Thus the corresponding generating function is given by
\begin{equation}
Z(\lambda,\mu;N)\equiv\langle e^{\lambdah^\prime(0,N)}\rangle
=e^{-\mu N}\langle e^{\lambda(1+\mu)h(0,N)}\rangle.
\end{equation}
If we again neglect correlations and use uncorrelated random variables
\begin{equation}
\eta(r,t)=\left\{\begin{array}{ll}1&\mbox{with probab. $p$}\\
-\mu&\mbox{with probab. $1-p$}\end{array}\right.
\end{equation}
the same rescaling and shifting leads to
\begin{equation}\label{eq_gfresult}
Z_0(\lambda,\mu;N)\equiv
\langle e^{\lambdah^\prime(0,N)}\rangle_0=
\left(\frac{1+\sqrt{p}\exp[\frac{\lambda}{2}(1+\mu)]}%
{1+\sqrt{p}\exp[-\frac{\lambda}{2}(1+\mu)]}\exp[-\frac{\lambda}{2}\mu]
\right)^{N}.
\end{equation}
\section{Implications on directed polymers and sequence
alignment}\label{sec_implications}
Now, we will study the consequences of our main result
Eq.~(\ref{eq_gfresult}). First, we will discuss the general properties
of the generating function and its implications on the physics of
directed polymers in a random medium. Then, we will come back to our
original question of the assessment of sequence alignment
significance. We find, that Eq.~(\ref{eq_gfresult}) is an explicit
expression for the significance assessment parameter $\lambda$. It
reproduces known limiting cases and we will demonstrate that our
result agrees well with numerical simulations.
\subsection{Properties of the generating function}
The most notable property of the generating function of the connected
moments of the average score (or average height)
\begin{equation}
\log\langle\exp[\lambdah^\prime(0,N)]\rangle_0=
\logZ_0(\lambda,\mu;N)
\end{equation}
is that it is an {\em odd} function of $\lambda$. The first two terms
of its expansion are
\begin{equation}\label{eq_freeenergy}
\frac{\logZ_0(\lambda,\mu;N)}{N}=
v(\mu)\lambda+\frac{1}{6}b(\mu)\lambda^3+O(\lambda^5).
\end{equation}
where
\begin{equation}\label{eq_velocity}
v(\mu)=\left.\frac{\mathrm d}{{\mathrm d}\lambda}\right|_{\lambda=0}
[Z_0(\lambda,\mu;N)]^{\frac{1}{N}}=
-\frac{\mu}{2}+(1+\mu)\frac{\sqrt{p}}{1+\sqrt{p}}
\end{equation}
and $b(\mu)=\left(\frac{1+\mu}{1+\sqrt{p}}\right)^3
\frac{(1-\sqrt{p})\sqrt{p}}{4}>0$. As already mentioned, we can regard
the generating function $Z_0(\lambda,\mu;N)$ as the ensemble
averaged partition function of $\lambda$ replicas of a directed
polymer in a random medium. In this sense, Eq.~(\ref{eq_freeenergy})
is the free energy per length of this $\lambda$ replica system. It has
the same form (with a vanishing quadratic term) as the result of an
earlier explicit replica calculation in continuous time and continuous
space~\cite{kard85}. However, our analysis is directly on the
discrete model and is not plagued by the difficulty of taking the
continuum limit in~\cite{kard85}. Moreover, we do not have to rely on
a questionable analytic continuation in the replica number since our
calculation is valid for an arbitrary $\lambda>0$ from the
very beginning.
The vanishing of the second order term in $\lambda$ will not even be
affected by the universal contributions to our result for small
$\lambda$ which have been found in~\cite{derr98b} using the explicit
dependence on the width $W$, since its second order coefficient
vanishes as $W^{-1/2}$ in the limit of large width. The
consequence of this vanishing second order term in $\lambda$ is that
the second connected moment of the average height, i.e., the height
fluctuations, scales sublinear in $N$. Instead the {\em third} moment
of the height fluctuations scales linearly with $N$. This is a
signature of the presence of the anomalous $N^{1/3}$ fluctuations of
the average surface height characteristic for the KPZ universality
class.
\subsection{Statistical significance and the log-linear transition}
According to Eqs.~(\ref{eq_glambda}) and~(\ref{eq_gfresult}) the
parameter $\lambda$ which characterizes the statistical significance
of local alignments with the match-mismatch scoring scheme
Eq.~(\ref{eq_idmatrix}) and gap cost $\delta=\mu/2$ is given by the
unique positive solution of the equation
\begin{equation}\label{eq_lambdaresult}
\frac{1+\sqrt{p}\exp[\frac{\lambda}{2}(1+\mu)]}%
{1+\sqrt{p}\exp[-\frac{\lambda}{2}(1+\mu)]}\exp[-\frac{\lambda}{2}\mu]=1.
\end{equation}
In the limit of large $\mu$, the solution of
Eq.~(\ref{eq_lambdaresult}) converges to $\lambda=-\log{p}$. This is
the value which we expect since this limit corresponds to the case of
gapless alignment (recall that $\delta=\mu/2$ here),
and $\lambda=-\log{p}$ is the solution of the large
$\mu$ limit of Eq.~(\ref{eq_lambdacond}). If the gap cost is
decreased, $\lambda$ is reduced, too. At some critical value of $\mu$
there will not be any positive solution of Eq.~(\ref{eq_lambdaresult})
any more, i.e., islands of all sizes are equally probable. This indicates
a phase transition between the logarithmic and the linear alignment
phase. The approach of this phase transition is especially interesting.
Close to the phase transition, we can use the expansion~(\ref{eq_freeenergy})
and rewrite Eq.~(\ref{eq_lambdaresult}) as
\begin{equation}\label{eq_logcondition}
v(\mu)\lambda+\frac{1}{6}b(\mu)\lambda^3+O(\lambda^5)=0.
\end{equation}
{From} this expansion the origin of the phase transition is very
clear: If $v(\mu)>0$, the right hand side of
Eq.~(\ref{eq_logcondition}) is a monotonously increasing function of
$\lambda$. Thus, $\lambda=0$ is the only solution of
Eq.~(\ref{eq_logcondition}). This corresponds to a flat distribution
of island sizes, i.e., the linear alignment phase. If $v(\mu<0)$, the
shape of the right hand side of Eq.~(\ref{eq_logcondition}) changes
and there are three roots, one of which is the positive solution
\begin{equation}\label{eq_lambdaapprox}
\lambda\approx\left(-6\frac{v(\mu)}{b(\mu)}\right)^{1/2}.
\end{equation}
This indicates that we are in the logarithmic alignment phase. Thus,
the phase transition occurs at the critical mismatch cost
$\mu_{\mathrm c}$ which is defined by the condition
\begin{equation}
v(\mu_{\mathrm c})=0.
\end{equation}
Using the explicit form~(\ref{eq_velocity}) of $v(\mu)$, we get the
critical mismatch cost
\begin{equation}
\mu_{\mathrm c}=\frac{2\sqrt{p}}{1-\sqrt{p}}.
\end{equation}
This reproduces the already known result~\cite{bund99} for the phase
transition point of this model. As the mismatch cost $\mu$ approaches
this critical value from above, $\lambda$ vanishes as
\begin{equation}
\lambda\approx\left(\frac{6(1-\sqrt{p})^3}{\sqrt{p}(1+\sqrt{p})}\right)^{1/2}
(\mu-\mu_{\mathrm c})^{1/2}.
\end{equation}
In the case of finite width $W$, the above expression is valid down to
$\lambda\simW^{-1}$. This confirms the characteristic universal
power law $|\mu-\mu_{\mathrm c}|^{1/2}$
proposed previously~\cite{dras98b} by scaling arguments.
\subsection{Numerical Verification}
In order to test the approximation of uncorrelated local
disorder~(\ref{eq_etaindep}) and the heuristic elements of the
derivation of Eq.~(\ref{eq_lambdaresult}), we performed extensive
numerical simulations to corrobate our result. We used the DNA
alphabet of size $c=4$ with identical frequencies for all four
letters, i.e., $p=1/4$. For different choices of the mismatch cost $\mu$
with corresponding gap cost $\delta=\mu/2$, we used the island
method~\cite{olse99} to find the values of $\lambda$ as a function of
$\mu$ numerically. For each value of $\delta$ several billion
islands have been generated using sequences of $N=25,000$ in order to
achieve relative errors of approximately $1\%$. We used completely
uncorrelated local scores chosen as
\begin{equation}
s(r,t)=\left\{\begin{array}{ll}1&\mbox{with probab. $p$}\\
-\mu&\mbox{with probab. $1-p$}\end{array}\right.
\end{equation}
with $p=1/4$. The resulting values of $\lambda$ are shown in
Fig.~\ref{fig_lambdanumerics}. The solid line is the solution of
Eq.~(\ref{eq_lambdaresult}) and the circles represent the values of
$\lambda$ for uncorrelated local scores~(\ref{eq_etaindep}). As shown
in Fig.~\ref{fig_lambdanumerics} the observed $\lambda$'s follow the
analytic solution very closely, thereby confirming
Eq.~(\ref{eq_lambdaresult}). We also included the values of $\lambda$
which result from correlated local scores generated from aligning
randomly chosen sequences according to Eq.~(\ref{eq_idmatrix}). As
one can see, they deviate only slightly from the analytical result for
uncorrelated disorder. This deviation is strongest close to the
log-linear phase transition, which for uncorrelated disorder happens
at $\mu_{\mathrm c}=2$. The difference of $\sim2\%$ in $\mu_{\mathrm
c}$ between the correlated and the uncorrelated case rapidly becomes
much smaller for larger alphabet sizes $c$~\cite{monv99}.
\begin{figure}[htbp]
\begin{center}
\epsfig{figure=figure10.eps,angle=270,width=0.6\columnwidth}
\caption{Dependence of the significance parameter \protect$\lambda$ on
the scoring parameter \protect$\mu$. The circles represent the
numerically obtained values of \protect$\lambda$ for uncorrelated
local disorder~(\ref{eq_etaindep}) with match probability
\protect$p=1/4$ for which Eq.~(\protect\ref{eq_lambdaresult}) (the
solid line) has been derived. They agree well with the analytical
result. The diamonds correspond to local disorder generated by
comparing two randomly chosen sequences over an alphabet of size
\protect$c=4$. The values of \protect$\lambda$ obtained from the two
ensembles differ from each other only very close to the phase
transition point \protect$\mu_{\mathrm c}=2$.}
\label{fig_lambdanumerics}
\end{center}
\end{figure}
\section{More general scoring systems}\label{sec_generalize}
While the approximation of the ensemble of random sequences by the ensemble
of independent local scores appears to have negligible effects, our
treatment is so far limited to the special scoring system
Eq.~(\ref{eq_etadist}). While the computation of the generating
function $\langle\exp[-\lambdaJ]\rangle_0$ seems feasible
only for this special scoring system, the mapping to an asymmetric
exclusion process and the reformulation as an eigenvalue problem is
still possible for more general scoring systems~\cite{bund99b}.
We consider here scoring systems satisfying the following two
conditions: First, the differences between the possible values
$s_{a,b}$ of the scoring matrix are multiples
of some score unit $\Delta$. Second, the gap costs $\delta$ is such
that $2\delta+s_0$ is also an integer multiple of $\Delta$, with
\begin{equation}
s_0\equiv
\max_{a,b}\{s_{a,b}\}
\end{equation}
being the maximal entry of the scoring matrix
$s_{a,b}$. These two conditions are easily
satisfied (with $\Delta=1$) by the most frequently used protein
scoring systems~\cite{dayh78,heni92} which use integer scores and gap
costs for performance reasons. For the match--mismatch scoring
system~(\ref{eq_idmatrix}), the first condition is satisfied with
$\Delta=1+\mu$, while the second condition applies only to a discrete
set of $\delta$'s. However, it is possible in principle to interpolate
to arbitrary gap costs~\cite{bund99}.
Mapping to an asymmetric exclusion process is possible for scoring systems
satisfying the above two conditions. It will be convenient to
express the gap cost $\delta$ in the following way,
\begin{equation}\label{eq_ndef}
2\delta={n_{\mathrm max}}\Delta-s_0\quad\mbox{with ${n_{\mathrm max}}\in{\bf N}$}.
\end{equation}
As before, we shall ignore correlations between the local scores
$s(r,t)$ and introduce uncorrelated random variables
$\eta(r,t)\in\{0,1,\ldots\}$ such that
\begin{equation}
s(r,t)\equiv s_0-\eta(r,t)\Delta,
\end{equation}
i.e.,
\begin{equation}
\Pr\{\forall_{r,t}\,\eta(r,t)=\eta_{r,t}\}=
\prod_{r,t}\Pr\{\eta(r,t)=\eta_{r,t}\}
\end{equation}
with
\begin{equation}\label{eq_fulletadist}
\Pr\{\eta(r,t)=\eta\}=\sum_{a,b}
p_a p_b
\delta_{s_{a,b},s_0-\eta\Delta}.
\end{equation}
Note, that these random variables $\eta(r,t)$ only take on
a finite number of different positive integer values, since the
scoring matrix $s_{a,b}$ itself has only a
finite number of entries.
A derivation analogous to the one given above for the longest common
subsequence problem again maps the dynamics of the alignment algorithm
onto the dynamics of particles on a one-dimensional lattice. The state
of the system is still given by the number of particles
$n(r,t)$ at each lattice site, but now these
occupation numbers are defined as
\begin{equation}
n(r,t)\equiv\left\{
\begin{array}{ll}\frac{1}{\Delta}[h(r+1,t)-h(r,t+1)+
\delta+s_0]&
r+t{\mathrm\ even}\\[5pt]
\frac{1}{\Delta}[h(r+1,t+1)-h(r,t)+\delta]&
r+t{\mathrm\ odd}\end{array}\right.
\end{equation}
and can take any integer value between $0$ and ${n_{\mathrm max}}$. The
dynamics is given by the relations
\begin{eqnarray}\label{eq_generalasepfirst}
n(r-1,t)&=&
n(r-1,t-1)-j(r,t)\mbox{\ and}\\
n(r,t)&=&n(r,t-1)+j(r,t)
\end{eqnarray}
for even $r+t$, where
\begin{equation}\label{eq_generalaseplast}
j(r,t)\equiv
\min\{\eta(r,t),{n_{\mathrm max}}-n(r,t-1),
n(r-1,t-1)\}
\end{equation}
and the total number of particles is fixed to be
\begin{equation}
\frac{1}{2W}\sum_{r=0}^{2W-1}n(r,t)
=\frac{{n_{\mathrm max}}}{2}.
\end{equation}
Eqs.~(\ref{eq_generalasepfirst})-(\ref{eq_generalaseplast}) can be
equally expressed as the following cellular automata: For each time
step and for each pair of neighboring sites of the one-dimensional
lattice the particles live on,
\begin{enumerate}
\item choose an integer number $\eta\ge0$ of particles to hop
from site $r-1$ to site $r$ according to the
distribution~(\ref{eq_fulletadist})
\item if there are fewer particles than $\eta$ on site $r-1$,
then reduce $\eta$ to the number of particles on site $r-1$
\item if there are fewer free spaces than $\eta$ on site $r$,
then reduce $\eta$ to the number of free spaces on site $r$
\item move $\eta$ particles from site $r-1$ to site $r$
\end{enumerate}
This updating rule is to be applied sublattice-parallel as for the
simpler scoring system. The process is illustrated in
Fig.~\ref{fig_particles}.
\begin{figure}[htbp]
\begin{center}
\epsfig{figure=figure11.eps,width=0.5\columnwidth}
\caption{Interpretation of
Eqs.~(\protect\ref{eq_generalasepfirst})-(\protect\ref{eq_generalaseplast}) as
a generalized asymmetric exclusion process. A configuration of the
local score differences is represented by particles on a
one-dimensional lattice of width \protect$2W$. Each lattice site
can accommodate up to \protect${n_{\mathrm max}}$ particles (here
\protect${n_{\mathrm max}}=4$.) At an odd time step for each even site
\protect$r-1$, a number of particles is chosen to attempt hopping to
the right. If there are enough particles at site \protect$r-1$ and
enough space on site \protect$r$, the chosen number hops. In the
example shown, the filled particles are the ones to hop and the dashed
boxes show their positions after the time step. No particle which
could hop is on site \protect$6$. The particle on site \protect$0$
cannot hop since its destination site is already fully occupied. For
site \protect$2$, one particle has been chosen. On site \protect$4$, at
least two particles tried to hop. If the number chosen was larger, it
would have been cut down to two since there are only two particles on
site \protect$4$ and since there are only two spaces left at site
\protect$5$.}
\label{fig_particles}
\end{center}
\end{figure}
The more complicated hopping process is reflected in a different
matrix ${\sf T}_1(\lambda/W)$ without changing anything else in
the calculations. Thus, the significance assessment constant $\lambda$
is still given by the generating function of the space and time
averaged current as
\begin{equation}\label{eq_lambdaimplicit}
\exp[\lambda s_0/2]
\langle\exp[-\lambda\DeltaJ]\rangle_0^{\frac{1}{N}}=1
\end{equation}
but the calculation of this generating function for an arbitrary
distribution~(\ref{eq_fulletadist}) becomes much more difficult for
the generalized asymmetric exclusion process than for the case
${n_{\mathrm max}}=1$ of the original asymmetric exclusion process.
Already, the knowledge of the dependence of the average current on the
scoring parameters would be very helpful to biologists, since this
determines the position of the log-linear phase transition. As
discussed in the case of the simpler scoring system, the phase
transition occurs, if the first moment of the score distribution
vanishes, i.e., for
\begin{equation}
0=\left.\frac{\mathrm d}{{\mathrm d}\lambda}\right|_{\lambda=0}
\exp[\lambda s_0/2]\langle\exp[-\lambda\DeltaJ]\rangle_0^{\frac{1}{N}}
=s_0/2-\frac{\langleJ\rangle_0}{N}\Delta=
s_0/2-\langlej\rangle_0\Delta
\end{equation}
The average current is much easier to calculate, since in contrast to
the generating function, it is independent of temporal
correlations. Thus, it can be calculated from the knowledge of the
stationary state alone. For the original asymmetric exclusion process,
the occupation numbers of the stationary state become independent
random variables. For the generalized asymmetric exclusion process
presented here, this is not the case any more. If the number of
particles which hop in one move is at most one (as for the scoring
system~(\ref{eq_idmatrix}) with arbitrary gap costs) approximating the
stationary state as a product state still yields reasonable values of
$\langlej\rangle_0$ and hence the phase transition point
$(\delta_{\mathrm c},\mu_{\mathrm c})$~\cite{bund99}. Nevertheless,
exact results or at least systematic improvements taking into account
the spatial correlations of the occupation numbers would be desirable.
For the more general case allowing for an arbitrary number of
particles to hop at a given time, no analytical result is known.
\section{Concluding Remarks}
In this paper, we have shown how a question of great practical
importance to molecular biologists, like the significance assessment
of local sequence alignment results, can be answered by studying the
asymmetric exclusion process, an exactly solvable model of the KPZ
universality class. Conversely, in trying to answer this question for
biologists, we derived an important physical quantity like the
generating function $Z$ for the corresponding physical system
in discrete time and discrete space. This complements the existing
solutions in continuous time and space~\cite{kard85} and in continuous
time and discrete space~\cite{derr98b}. Our result is the first
successful analytical approach to assessing the statistical
significance of sequence alignment with gaps.
Future work of practical importance includes solving the
generalizations of the asymmetric exclusion process described in
Sec.~\ref{sec_generalize} and studying the effect of the widely used
``affine gap cost'', where a contiguous gap of length $\ell$ is
assigned some gap cost $\delta+(\ell-1)\epsilon$ instead of simply
$\delta\ell$. A general expression which gives $\lambda$ as a function
of an arbitrary scoring system should finally give rise to a deeper
understanding of the role of the gap cost and lead to better choices
of scoring systems for alignments of biological sequences.
\section*{Acknowledgments}
The author gratefully acknowledges discussions with S. Altschul,
T. Hwa, R. Olsen, and N. Rajewsky, and the hospitality of the Center
for Studies in Physics and Biology at Rockefeller University where
this work was completed. This work is supported in part by a
Hoch\-schul\-son\-der\-pro\-gramm~III fellowship of the DAAD and by
the NSF through Grant No. DMR-9971456.
|
3,212,635,537,966 | arxiv |
\section{Motivation}
\section{Introduction}
Covariance estimation is a fundamental problem in the field of statistical signal
processing.
Many algorithms for detection and estimation rely on accurate covariance matrix estimation
\cite{covest_imp1,covest_imp2}.
Roughly speaking, the problem is tractable as long as the global maximum likelihood
solution can be efficiently found (or approximated). Thus, it is important to understand
whether the associated negative-log-likelihood minimization problem is convex. Following this line of thought, we
combine two ideas.
First, there is an increasing interest in covariance estimation in non-Gaussian
distributions which are typically non-convex
but have been shown to be geodesically convex \cite{ami1,ami3}. Second, many problems
adhere to known symmetry constraints which can be exploited in the estimation.
Recently, \cite{venkat} addressed such structures in the Gaussian setting. In this paper,
we will consider them in non-Gaussian covariance estimation using the theory of geodesic
convexity.
In many applications, the assumption of normal data is not realistic \cite{abr,pascal1}.
In such scenarios, improved performance may be obtained by resorting to more general
distributions, such as Generalized Gaussian and Elliptical distributions
\cite{ellipt,ami4}. The associated Maximum Likelihood optimization usually do not lead to closed form
solutions and iterative algorithms are required \cite{ami1,pascal1}.
One of the most prominent robust methods is the Tyler's method for covariance
matrix estimation in scaled Gaussian models, which has been successfully applied to
different practical applications ranging from array processing to sensor networks \cite{tyler}.
It has been extended to other settings involving
regularization and incomplete data \cite{abr} - \cite{ami2}.
Recently, it was shown that the underlying principle behind these successful non-convex
optimizations is the geodesic convexity \cite{rapcsak, ami}. This principle provides more insight on the analysis and design of robust covariance estimation
methods, and paves the road to numerous extensions based on $g$-convexity, e.g.,
regularization \cite{ami1} and their combination with Kronecker structures \cite{ami}.
Over the last years, many works have been developed in the area of estimating covariance matrices
possessing some additional knowledge such as sparsity or structure \cite{levina}. Our work is motivated by \cite{venkat} which considered group symmetry structures. In particular, \cite{venkat} addressed symmetry constraints in random fields
of physical phenomena, Bayesian models and cyclostationary processes. In addition, it is well known that
circulant matrices are invariant to shifts \cite{dembo,circ}. Symmetric persymmetric (bisymmetric) matrices are
invariant under the exchange-operator \cite{pascal_per,dimaio}. Proper complex normal
distributions are defined via their invariance to rotations with respect to the real and
imaginary axis \cite{kay}. Proper quaternion distributions follow invariances with respect
to isoclinic rotations \cite{quat_properness,alba}. All of these properties have been
successfully exploited in covariance estimation in the multivariate Gaussian distribution.
Many of them have also been considered in non-Gaussian distributions via problem-specific
fixed point iterations and algorithm-dependent existence, uniqueness and convergence
proofs.
The main result in this paper is that the set of positive definite matrices which are
invariant under a conjugation action of a subgroup of orthogonal transformations is
$g$-convex on their respective manifold. Together with the $g$-convexity of various
non-Gaussian negative-log-likelihoods,
this implies that the global constrained maximum likelihood solution can be efficiently
found using standard descent algorithms. This provides a unified framework for robust
covariance estimation with group symmetry constraints.
Unlike previous approaches, our results are not specific to any distribution, symmetry set
or even numerical algorithm. As a byproduct, we provide a few results on
specific symmetry groups and reformulate proper complex and quaternion structures using
a finite number of rotation-invariant constraints. For completeness, we also propose a
simple numerical method for solving these problems, although we emphasize that other
descent algorithms can be used instead. Finally, we demonstrate the performance advantage
of our framework via synthetic simulations in a non-Gaussian proper quaternion
environment.
The paper is organized in the following form. First, we give an outline of $g$-convexity and matrix group symmetry. Then the main result is formulated and examples of symmetry matrix classes are given. Finally, we provide a computational algorithm and numerical results.
\section{Geodesic Convexity}
Geodesic convexity is a generalization of the notion of convexity in linear spaces.
We therefore begin with a brief review on $g$-convexity on the manifold $\mathcal{P}(p)$
of positive definite matrices $p \times p$.
More details are available in \cite{rapcsak}, \cite{ami2}. With each $\mathbf{Q}_0,\mathbf{Q}_1\in
\mathcal{P}(p)$ we associate the following geodesic
\begin{eqnarray}\label{geodesics}
\mathbf{Q}_t=\mathbf{Q}_0^{\frac{1}{2}}\(\mathbf{Q}_0^{-\frac{1}{2}}\mathbf{Q}_1\mathbf{Q}_0^{-\frac{1}{2}}\)^t\mathbf{Q}_0^{\frac{1}{2}},\quad
t\in[0,1].
\end{eqnarray}
\begin{definition}
A set ${\mathcal{N}}\in \mathcal{P}(p)$ is $g$-convex if for any $\mathbf{Q}_0,\mathbf{Q}_1\in
\mathcal{N}$ the geodesic $\mathbf{Q}_t$ lies in $\mathcal{N}$.
\end{definition}
\begin{definition}
Given a $g$-convex subset $\mathcal{N} \subset \mathcal{P}(p)$,
we say that a function $f$ is $g$-convex on $\mathcal{N}$ if for any two points
$\mathbf{Q}_0,\mathbf{Q}_1 \in \mathcal{N}, f(\mathbf{Q}_t)\leq tf(\mathbf{Q}_0)+(1-t)f(\mathbf{Q}_1), \forall t \in [0,1]$.
\end{definition}
The advantage of $g$-convexity stems from the following result \cite{ami2}
\begin{prop}
Any local minimum of a $g$-convex function over a $g$-convex set is a global minimum.
\label{general_convexity}
\end{prop}
Finding local minimum is usually easy and hence $g$-convexity guarantees that a global
solution can also be efficiently found.
Recently, it was shown that the negative-log-likelihoods of many popular non-Gaussian
distributions are $g$-convex. Two examples are:
\begin{itemize}
\item Tyler's \cite{ami1}
\begin{eqnarray}\label{ml1}
L\(\{\mathbf{x}_i\}_{i=1}^n;\mathbf{Q}\)=\frac{p}{n}\sum_{i=1}^n \log(\mathbf{x}_i^T
\mathbf{Q}^{-1}\mathbf{x}_i) + \log|\mathbf{Q}|,
\end{eqnarray}
\item Mutlivariate Generalized Gaussian Distribution \cite{ami3}
\begin{eqnarray}\label{ml2}
L\(\{\mathbf{x}_i\}_{i=1}^n;\mathbf{Q}\)= \frac{1}{n} \sum_{i=1}^n (\mathbf{x}_i^T
\mathbf{Q}^{-1}\mathbf{x}_i)^\beta + \log|\mathbf{Q}|,
\end{eqnarray}
\end{itemize}
where $\beta$ is the shape parameter.
Together with Proposition \ref{general_convexity} above, \cite{ami1,ami3} proved that
simple descent algorithm converge to the
global estimate in these distributions. In the next section, we will show that this is
also true when using symmetry invariance constraints which are also $g$-convex.
\section{Matrix group symmetry}
In order to improve the accuracy of covariance estimators it is common to add constraints
based on prior knowledge. Of course, this priors can only be exploited if the constraints
are convex
and the associated optimization can be efficiently solved.
Recently, \cite{venkat} proposed the use of group symmetry constraints which are indeed
convex (actually linear)
and can be incorporated into a Gaussian setting. The main result in this paper is that
such sets are
also $g$-convex and can also be utilized in non-Gaussian settings.
Let $\mathcal{K}$ be a set\footnote{Here we treat the case of finite $\mathcal{K}$, but the result can be easily generalized to the infinite case.} of orthogonal matrices. Following \cite{venkat}, we formally assume that this set is actually a multiplicative group. Associated with $\mathcal{K}$, we define the fixed-point subset $\mathcal{F} \subset
\mathcal{P}(p)$ of
matrices that are invariant with respect to the conjugation by each element of
$\mathcal{K}$:
\begin{eqnarray}\label{FK}
\mathcal{F}(\mathcal{K}) = \{ \mathbf{Q} \in \mathcal{P}(p) | \mathbf{Q} = \L \mathbf{Q} \L^T, \forall \L \in
\mathcal{K} \}.
\end{eqnarray}
\begin{theorem} \label{main_theorem}
The set $\mathcal{F}(\mathcal{K})$ in (\ref{FK}) is $g$-convex.
\begin{proof}
First note that $\mathbf{Q} = \L \mathbf{Q} \L^T$ is equivalent to $\mathbf{Q}\L = \L\mathbf{Q}$. Now, assume $\mathbf{Q}_0, \mathbf{Q}_1 \in \mathcal{F}(\mathcal{K})$.
Let us show that the geogesic (\ref{geodesics}) lies in $\mathcal{F}(\mathcal{K})$.
Choose $\L \in \mathcal{K}$, $\L \mathbf{Q}_0 = \mathbf{Q}_0 \L, \L \mathbf{Q}_1 = \mathbf{Q}_1 \L$. Let $\mathbf{M}$ be a
diagonalizable matrix and $f$ a smooth function, then we can think of $f(\mathbf{M})$ as of $f$ acting
on the eigenvalues of $\mathbf{M}$ in the orthonormal eigenbasis of $\mathbf{M}$. For any diagonalizable
matrix $\mathbf{M}$ it commutes with $\P$ iff $f(\mathbf{M})$ commutes with $\P$ for any smooth function $f$,
also if two matrices $\mathbf{M}_1$ and $\mathbf{M}_2$ commute with $\P$, then their product $\mathbf{M}_1 \mathbf{M}_2$
commutes with $\P$.
This implies that $\mathbf{Q}_0^{-\frac{1}{2}}\mathbf{Q}_1\mathbf{Q}_0^{-\frac{1}{2}}$ commutes with $\L$,
thus $\(\mathbf{Q}_0^{-\frac{1}{2}}\mathbf{Q}_1\mathbf{Q}_0^{-\frac{1}{2}}\)^t$ also commutes with $\L$ and the
whole $\mathbf{Q}_t$ commutes with $\L$. Thus the geodesic (\ref{geodesics}) lies in
$\mathcal{F}(\mathcal{K})$ and the set
$\mathcal{F}(\mathcal{K})$ is $g$-convex.
\end{proof}
\end{theorem}
\section{Examples and Applications}
In this section we provide examples of group symmetry constraints which appear in real
world covariance estimation problems.
\subsection{Circulant}
A common class of symmetry constrained covariances is the set of positive definite
circulant matrices:
$$
\mathbf{C} =
\begin{pmatrix}
c_0 & c_1 & c_2 & \dots & c_{n-1} \\
c_{n-1} & c_0 & c_1 & \dots & c_{n-2}\\
\vdots & \vdots & \vdots & \ddots & \vdots \\
c_1 & c_2 & c_3 & \dots & c_0
\end{pmatrix}.
$$
Such matrices are typically used as an approximation to Toeplitz structured matrices which
are associated with signal processing in stationary environments \cite{dembo},
\cite{circ}. It is easy to see that the set of circulant matrices can be expressed as
$\mathcal{F}(\mathcal{K})$ with $\mathcal{K}$ being the cyclic group of order $n$ which
acts on the rows of the
matrix by shifts. Thus, an immediate corollary of Theorem \ref{main_theorem} it that the
set of circulant matrices is $g$-convex.
\subsection{Persymmetric}
Another class of symmetry constrained covariances is the set of positive definite
persymmetric matrices, i.e., matrices which are symmetric in the northeast-to-southwest diagonal $\P\mathbf{J}_n=\mathbf{J}_n\P^T$, where $\mathbf{J}$ is the exchange $n \times n$ matrix containing ones only on the northeast-to-southwest diagonal. Since we deal with symmetric matrices the constraint becomes $\P\mathbf{J}_n=\mathbf{J}_n\P$ and the matrix form is:
$$
\P =
\begin{pmatrix}
p_{11} & p_{12} & \dots & p_{1n} \\
p_{12} & p_{22} & \dots & p_{1n-1}\\
\vdots & \vdots & \ddots & \vdots \\
p_{1n-1} & p_{2n-1} & \dots & p_{12}\\
p_{1n} & p_{1n-1} & \dots & p_{11}
\end{pmatrix}.
$$
Such matrices are commonly encountered in radar systems using a symmetrically spaced
linear array with constant pulse repetition interval \cite{pascal_per}. This structure
information could be exploited to improve detection performance \cite{dimaio},
\cite{pascal_per}. This set can be expressed as $\mathcal{F}(\mathcal{K})$ with $\mathcal{K}$ consisting of $\mathbf{I}_n$ and $\mathbf{J}_n$. Thus, an immediate corollary of Theorem \ref{main_theorem} it that the set of persymmetric matrices is also $g$-convex.
Recently, \cite{pascal_per} extended the Tyler's covariance estimator to the case of
persymmetric matrices, proposed and analyzed the asymptotic behaviour of the fixed point
estimator. Theorem \ref{main_theorem} generalizes this result to other $g$-convex
optimizations, independent of the algorithm that finds the local minimum.
\subsection{Proper Complex}
An important class of matrices is known as proper complex, or circularly symmetric covariance
matrices. In most radar and communication problems it is typical to work with complex
valued random variables which are invariant to rotations. A $p$-dimensional complex vector can be expressed as a $2p$-dimensional
real valued vector. Due to the symmetries, the associated
$2p\times 2p$ covariances belong to $\mathcal{F}(\mathcal{K})$ with $\mathcal{K}$ being an infinite set of rotations of the form \cite{kay}
\begin{eqnarray}
\L_\theta=\begin{pmatrix}
\cos\theta & \sin\theta\\
-\sin\theta & \cos\theta \end{pmatrix}\otimes \mathbf{I}_p,
\end{eqnarray}
which must hold for any $\theta$. This result already shows that the set is $g$-convex. However, in order to efficiently exploit it, we also need a finite characterization.
\begin{prop}
The set of proper complex $2p\times 2p$ covariance matrices is equivalent
to $\mathcal{F}(\mathcal{K})$ with $\mathcal{K}$ consisting of $\L_0=\mathbf{I}_{2p}$ and
$\L_1=\left(\begin{smallmatrix} 0&1\\ -1&0 \end{smallmatrix}\right)\otimes \mathbf{I}_p$.
\end{prop}
\begin{proof}
This is a particular case of the Proposition \ref{eq_quat} below.
\end{proof}
Thus, $g$-convex maximum likelihood problems with proper complex constraints can be globally and efficiently solved. As special cases this includes proper complex versions of Tyler's estimator and MGGD solutions. We note that this result is not surprising. Recently, most of these complex multivariate settings have been analyzed \cite{poor,ollila}. However, previous approaches were highly specific, and relied on defining new complex distributions. Our framework allows a unified treatment based on the real valued distributions with a single additional $g$-convex constraint.
\subsection{Proper quaternion}\label{section_0}
Another modern class of covariance matrices is known as proper quaternion \cite{quat_properness}. Quaternions are a generalization of complex numbers and is a $4$-dimensional vector space over reals, so that a length $p$ quaternion vector can be dealt with as a length $4p$ real vector. Typical applications are complex electromagnetic signals with two polarizations \cite{quaternion_polar,alba}. Similarly to the complex case, here too it is common to consider proper distributions, which are invariant to specific quaternion rotations. A $4p\times 4p$ proper quaternion covariance belongs to $\mathcal{F}(\mathcal{K})$ with $\mathcal{K}$ being an infinite set of rotations of the form
\begin{eqnarray}\label{real_matrix}
\L_{\theta \alpha \beta \gamma}={\scriptsize{\[
\begin{array}
{cccc} \cos({\theta}) & \alpha \sin({\theta}) & \beta \sin({\theta}) & \gamma
\sin({\theta}) \\
-\alpha \sin({\theta}) & \cos({\theta}) & - \gamma \sin({\theta}) & \beta \sin({\theta})
\\
-\beta \sin({\theta}) & \gamma \sin({\theta}) & \cos({\theta}) & -\alpha \sin({\theta})
\\
-\gamma \sin({\theta}) & -\beta \sin({\theta}) & \alpha \sin({\theta}) & \cos({\theta})
\\
\end{array}
\]}}\otimes \mathbf{I}_p,
\end{eqnarray}
which must hold for $\theta, \alpha, \beta, \gamma$ satisfying $\alpha^2+\beta^2+\gamma^2 = 1$.
The next result characterizes this set using a finite number of constraints.
\begin{prop}\label{eq_quat}
The set of proper quaternion $4p\times 4p$ covariance matrices is equivalent
to $\mathcal{F}(\mathcal{K})$ with $\mathcal{K}$ consisting of $\L_0=\mathbf{I}_{4p}$, $\L_1=\mathbf{R}_1 \otimes \mathbf{I}_p$, $\L_2=\mathbf{R}_2 \otimes \mathbf{I}_p$, $\L_3=\mathbf{R}_3 \otimes \mathbf{I}_p$, $\L_4=-\L_0$, $\L_5=-\L_1$, $\L_6=-\L_2$ and $\L_7=-\L_3$, where
\begin{equation}
\mathbf{R}_1=\left(\begin{smallmatrix}
0&1&0&0\\-1&0&0&0\\mathbf{0}&0&0&-1\\mathbf{0}&0&1&0\end{smallmatrix}\right),
\mathbf{R}_2=\left(\begin{smallmatrix}
0&0&1&0\\mathbf{0}&0&0&1\\-1&0&0&0\\mathbf{0}&-1&0&0\end{smallmatrix}\right),
\mathbf{R}_3=\left(\begin{smallmatrix}
0&0&0&1\\mathbf{0}&0&-1&0\\mathbf{0}&1&0&0\\-1&0&0&0\end{smallmatrix}\right).
\end{equation}
\end{prop}
\begin{proof} The matrices $\L_i$ for $i=0,\dots,7$ are particular cases of (\ref{real_matrix}),
so the necessity is obvious. Assume now that $\mathbf{Q}$ is invariant under $\L_i$
conjugation, meaning that $\mathbf{Q}$ commutes with them: $\mathbf{Q} \L_i = \L_i \mathbf{Q}$ and we are given some matrix $\mathbf{R}$ of the form (\ref{real_matrix}). Take the equalities $\mathbf{Q} \L_i = \L_i \mathbf{Q}, i=0,1,2,3$, multiply them by $\cos(\theta), \alpha \sin(\theta), \beta \sin(\theta), \gamma \sin(\theta)$
correspondingly and add them up to get: $\mathbf{Q} \mathbf{R} = \mathbf{R} \mathbf{Q}$.
\end{proof}
In other words, the set of proper quaternion covariance matrices is $g$-convex. Thus, we can easily extend the $g$-convex estimates of Tyler and MGGD to the quaternion case, and guarantee that any descent algorithm will converge to the global solution.
\section{Minimization algorithm}
In this section, we address the numerical optimization of the above minimizations. Various numerical techniques can be used to find local minimas. Since the problems are $g$-convex these local minimas will also be the global solution. The negative-log-likelihoods in (\ref{ml1})-(\ref{ml2}) have the form \cite{ami3}:
\begin{equation}
\L(\mathbf{Q})=\frac{1}{n}\sum_{i=1}^n\rho(\mathbf{s}_i^T \mathbf{Q}^{-1} \mathbf{s}) + \log |\mathbf{Q}|.
\end{equation}
For simplicity, we consider the classical iterative reweighed scheme:
\begin{equation} \label{iter_scheme}
\mathbf{Q}_{k+1}=\frac{1}{n}\sum_{i=1}^nu(\mathbf{s}_i^T \mathbf{Q}^{-1} \mathbf{s}_i)\mathbf{s}_i\mathbf{s}_i^T,
\end{equation}
where $u(x)=\rho'(x)$.
Following \cite{venkat}, we note that adding the $g$-convex constraints in the form of symmetry is equivalent to replicating the sample measurements. Given $n$
$p$-dimensional measurements $\{s_i\}_{i=1}^n$ the symmetrization is equivalent to
generating synthetically $|\mathcal{K}|$ new measurements from each one, thus getting
$|\mathcal{K}|n$ samples $\{\L s_i\}_{i=1, \L\in\mathcal{K}}^n$ instead of $n$. This generalizes the iterative scheme \label{iter_scheme} as follows:
\begin{equation} \label{iter_scheme}
\mathbf{Q}_{k+1}=\frac{1}{|\mathcal{K}|n}\sum_{\L \in \mathcal{K}} \sum_{i=1}^nu((\L \mathbf{s}_i)^T \mathbf{Q}^{-1} (\L \mathbf{s}_i))(\L \mathbf{s}_i)(\L \mathbf{s}_i)^T.
\end{equation}
A simple minimization majorization argument can be used to show that this iteration leads to a descent method, see for example \cite{ami1}.
\section{Numerical Results}
For numerical simulations, we chose Tyler's scatter estimate in proper quaternion distributions. We have generated a proper real covariance
matrix $\mathbf{Q}_0$ and generated elliptically distributed $10$-dimensional quaternion random vectors as $s_i = \sqrt{\tau} \mathbf{v}$, where $\tau \sim \chi^2$ and $\mathbf{v}$ is zero-mean normally distributed with covariance matrix $\mathbf{Q}_0$. We choose $\rho(x)=p \log(x)$ to get the Tyler's covariance estimator \cite{tyler}.
\begin{center}
\includegraphics[scale=0.4]{convergence_nonlog.jpg}
\end{center}
We compare four different covariance estimators:
\begin{itemize}
\item Sample Covariance
\begin{equation}
\mathbf{Q}_{SC}=\frac{1}{n}\sum_{i=1}^n \mathbf{s}_i \mathbf{s}_i^T,
\end{equation}
\item Proper Sample Covariance
\begin{equation}
\mathbf{Q}_{PSC} = \frac{1}{|\mathcal{K}|n}\sum_{\L \in \mathcal{K}} \sum_{i=1}^n \L \mathbf{s}_i \mathbf{s}_i^T \L^T,
\end{equation}
\item Tyler Covariance Estimator Iteration
\begin{equation}\label{tyler_iter}
\mathbf{Q}_{k+1} = \frac{p}{n} \sum_{i=1}^n \frac{\mathbf{s}_i \mathbf{s}_i^T}{\mathbf{s}_i^T
\mathbf{Q}_{k}^{-1} \mathbf{s}_i}.
\end{equation}
\item Tyler Proper Covariance Estimator Iteration
\begin{eqnarray}\label{tyler_proper_iter}
\begin{split}
& \mathbf{Q}_{k+1} = \frac{p}{|\mathcal{K}|n} \sum_{\L \in \mathcal{K}} \sum_{i=1}^n \frac{\L
\mathbf{s}_i \mathbf{s}_i^T \L^T}{\mathbf{s}_i^T \L^T \mathbf{Q}_{k}^{-1} L \mathbf{s}_i} \\
& = \frac{p}{|\mathcal{K}|n} \sum_{\L \in \mathcal{K}} \sum_{i=1}^n \frac{(\L \mathbf{s}_i) (\L \mathbf{s}_i)^T}{(\L
\mathbf{s}_i)^T \mathbf{Q}_{k}^{-1} (\L \mathbf{s}_i)}.
\end{split}
\end{eqnarray}
\end{itemize}
We repeat the computations for $100$ times for the four estimators with $150-600$ samples. In order to make the results consistent we divide all the matrices by their traces.
\section{Acknowledgement}
This work was partially supported by Israel Science Foundation Grant No. 786/11 and Kaete Klausner Scholarship. The authors would like to thank Alba Sloin for numerous and helpful discussions.
|
3,212,635,537,967 | arxiv | \section{Introduction}
{\it Quantum coherent electronics} \cite{Splettstoesser:2009im,Haack:2013ch}, also known as electron quantum optics \cite{Bocquillon:2013fp}, and single electron electronics \cite{Bauerle:2018ct}, is an actively developing platform for quantum information processing \cite{Bennett:2000kl}, which is aimed at creating, manipulating, and detecting individual electrons as carriers of information.
Recently, quite a lot of single-electron sources on-demand have been experimentally realized.\cite{Blumenthal:2007ho,Feve:2007jx,Kaestner:2008gv,{Fujiwara:2008gt},{Roche:2013jw},Dubois:2013ul,{Rossi:2014kp},{Tettamanzi:2014gx},{dHollosy:2015ez},{vanZanten:2016fl},{Gabelli:2016uz},{Rossi:2018wj}}
One of the crucial tests, this source has to pass through, is the verification of a single-particle emission regime.
In quantum optics, the single-photon emission regime is verifies via the measurement of the second-order correlation function, $g ^{(2)}$, which characterizes the probability of joint detection of two photons.\cite{Walls2008}
Such a verification is universal and it does not rely on any specific properties of the source.
If the stream generated by a periodically working source consists of non overlapping single photons, then the function $g ^{(2)}\left( \tau \right)$ vanishes at zero time delay between the two detections, $ \tau=0$.
In contrast, if there are multi-photon wave packets in the stream, the two photons can be detected simultaneously and the function $g ^{(2)}_{}$ is finite at $ \tau=0$.
The measurement of the joint detection probability in the optical frequencies range is possible due to availability of efficient single photon detectors.
In the microwave frequencies range, no efficient detectors are available.
Nevertheless, the single-particle emission regime for a source of microwave photons \cite{Houck:2007dy} can be demonstrated via the linear amplification of the magnitude of an electromagnetic field \cite{Bozyigit:2010bw}.
There are no efficient on-fly detectors available for single electrons so far, and there is no way to measure the magnitude of a fermionic field.
This is why for the verification of a single-electron emission regime the various nonuniversal methods were used.
The nonuniversality in this context means that the given method can be good for one system, but not for another.
In particular, a strong decrease in an electrical noise was used as an indicator of the single-particle emission regime for a dynamical quantum dot \cite{{Maire:2008hx}} and for a quantum capacitor \cite{{Mahe:2010cp},{Albert:2010co},{Parmentier:2012ed}}, while this method does not work in the case of the source of levitons\cite{Dubois:2013ul}.
Another method for validation of the single-electron injection regime, which is relied on the partition noise\cite{{Reznikov:1998kn},Blanter:2000wi} of an electron beam splitter, was demonstrated in Refs.~\onlinecite{Bocquillon:2012if} and \onlinecite{Dubois:2013ul}.
Nevertheless, in some systems it is possible to measure directly the second-order correlation function for injected electrons, $G ^{(2)}_{}$, which vanishes identically in the case of a single-particle injection.
Generally, in the case of electrons injected into an electron waveguide, the second order correlation function contains several contributions: (i) One is due to electrons belonging to the Fermi sea of the waveguide, (ii) one more is due to the injected electrons, that is $G ^{(2)}_{}$, (iii) and, finally, the last contribution is due to the joint contribution of the injected electrons and Fermi-sea electrons. \cite{Moskalets:2014ea,Thibierge:2015up}
As it is pointed out in Ref.~\onlinecite{Thibierge:2015up}, when electrons are injected into one of the two incoming channels of an electron beam splitter, the cross-correlation noise of currents after the beam splitter is directly related to the function $G ^{(2)}_{}$.
The Fermi sea electrons do not contribute to the cross-correlation noise either directly or in conjunction with injected electrons.
For this to be true, the two conditions must be met.
First, the Fermi seas in both incoming channels have the same temperature and the same chemical potential.
Second, the incoming and outgoing channels are spatially separated, which can be achieved using chiral or helical edge states \cite{Buttiker:2009bg} as electron waveguides.
Here I focus on the effect of temperature on the second-order correlation function, $G ^{(2)}_{}$, of electrons injected on top of the Fermi sea in conductors.
The fact, that at nonzero temperatures the quantum state of injected electrons is a mixed state,\cite{Moskalets:2015ub,Moskalets:2017vk} leads to existence of a purely single-particle contribution to the correlation function $G ^{(2)}_{}$.
This contribution puts the lower limit to the second-order correlation function, and it must be taken into account when the function $G ^{(2)}_{}$ is used to distinguish single-electron and multi-electron quantum states.
The paper is organized as follows:
In Sec.~\ref{sec2} I discuss in detail how the correlation function $G ^{(2)}_{}$ changes when a pure quantum state becomes a mixed quantum state.
In Sec.~\ref{sec3} the relation between the function $G ^{(2)}_{}$ and the current correlation function is given in frequency domain.
The temperature dependence of the functions $G ^{(2)}$ for single- and two-electron excitations are contrasted in Sec.~\ref{sec4}.
The conclusion is given Sec.~\ref{sec5}.
Some details of calculations are preseted in the Appendixes \ref{ap01} and \ref{ap02}.
\section{Correlation function of electrons injected at nonzero temperatures}
\label{sec2}
A convenient quantity for characterizing the excitations injected by the electron source into the electron waveguide is the excess first-order correlation function, $G ^{(1)}_{}$.\cite{Grenier:2011js,Grenier:2011dv,Moskalets:2013dl}
To get rid of the contribution of the underlying Fermi sea and keep track of the contribution of injected electrons only, this function is defined as the difference of the two terms, evaluated with the source being switched on and off, respectively, $G ^{(1)}\left( 1;2 \right) = \langle \hat\Psi^{\dag}_{ }(1) \hat\Psi_{ }(2) \rangle_{on} - \langle \hat\Psi^{\dag}_{ }(1) \hat\Psi_{ }(2) \rangle_{off}$.
Here $\hat\Psi^{}_{ }(j)$ is an electron field operator in second quantization evaluated at time $t_{j}$ and point $x_{j}$, $j = 1, 2$, after the source.
The quantum statistical average, $\langle \dots \rangle$, is performed over the state of electrons in the waveguide before the source.
In this work I suppose that the waveguide is one-dimensional, and before the source an electron system is in equilibrium state, which is characterized by the Fermi distribution function with a temperature $ \theta$ and a chemical potential $ \mu$.
Since I am interested in time dependence, rather than spatial dependence, below I keep argument $t_{j}$ only.
Note, the function $G ^{(1)}_{}$ for a stream of identically prepared separated electrons in a ballistic conductor was measured in Ref.~\onlinecite{Jullien:2014ii} using the tomography protocol suggested in Ref.~\onlinecite{Grenier:2011dv}.
For noninteracting electrons, the first-order correlation function determines the higher-order correlation functions through the corresponding Slater determinants.
For example, the second-order correlation function, $G ^{(2)}\left( t_{1}, t_{2}; t_{3}, t_{4} \right)$, is determined as follows,
\begin{eqnarray}
G ^{(2)}\left( t_{1},t_{2};t_{3},t_{4} \right) =
\det
\begin{pmatrix}
G^{(1)}\left( t_{1};t_{4} \right) &
G^{(1)}\left( t_{1};t_{3} \right) \\
G^{(1)}\left( t_{2};t_{4} \right) &
G^{(1)}\left( t_{2};t_{3} \right) \end{pmatrix} .
\label{gn01}
\end{eqnarray}
\ \\ \indent
First let us consider a pure quantum state.
For a single-particle state ($N=1$) with wave function $ \Psi_{1}\left( t \right)$, the first-order correlation function is factorized into the product of two terms that depend on one time each, $G_{N=1} ^{(1)}\left( t_{1}; t_{2}\right) = \Psi_{1}^{*}\left( t_{1} \right) \Psi_{1}\left( t_{2} \right)$.
Apparently, that in this case the second order-correlation function vanishes identically, $G_{N=1} ^{(2)}=0$.
However, already for a two-particle state ($N=2$), when $G_{N=2} ^{(1)}\left( t_{1}; t_{2} \right) = \sum_{j=1}^{2} \Psi_{j}^{*}\left( t_{1} \right) \Psi_{j}\left( t_{2} \right)$, the second-order correlation function is not zero.
It is represented as follows, $G_{N=2} ^{(2)}\left( t_{1},t_{2};t_{3},t_{4} \right) = \Psi_{1,2}^{(2) *}\left( t_{1},t_{2} \right) \Psi_{1,2}^{(2) }\left( t_{4},t_{3} \right)$, where the two-particle wave function,
\begin{eqnarray}
\Psi_{1,2} ^{(2)}\left( t_{1},t_{2} \right) =
\det
\begin{pmatrix} \Psi^{}_{1}\left( t_{1}\right) & \Psi^{}_{2}\left( t_{1} \right) \\ \Psi^{}_{1 }\left( t_{2} \right) & \Psi^{}_{2}\left( t_{2} \right) \end{pmatrix} ,
\label{gn02}
\end{eqnarray}
\ \\ \noindent
is the Slater determinant composed of wave functions of both particles, $ \Psi_{1}$ and $ \Psi_{2}$.
In contrast, in the case of a mixed state, the above relations become essentially modified.
The quantum state is now characterize by the density matrix rather than the wave function.
As an example, I consider a mixed state characteristic for electrons injected on top of the Fermi sea at finite temperatures.\cite{Moskalets:2017vk}
The components of such a mixed state are parametrized by a continuous variable, the energy $ \epsilon$, with the probability density $p _{ \epsilon} = - \partial f\left( \epsilon \right)/ \partial \epsilon$, where $f\left( \epsilon \right) = \left( 1 + e^{\frac{ \mu + \epsilon }{ k_{B} \theta }} \right)$ is the Fermi distribution function, $k_{B}$ is the Boltzmann constant.
So, for the mixed two-particle state, the first-order correlation function reads,\cite{Moskalets:2017vk}
\begin{eqnarray}
G_{N=2} ^{(1)}\left( t_{1}; t_{2} \right) =
\int _{}^{ } d \epsilon p_{ \epsilon}
\sum_{j=1}^{2} \Psi_{j \epsilon}^{*}\left( t_{1} \right) \Psi_{j \epsilon}\left( t_{2} \right) .
\label{gn03}
\end{eqnarray}
\ \\ \noindent
Accordingly to Eq.~(\ref{gn01}), the second-order correlation function becomes,
\begin{eqnarray}
G_{N=2} ^{(2)}\left( t_{1},t_{2};t_{3},t_{4} \right) =
\int _{}^{ } d \epsilon p_{ \epsilon}
\int _{}^{ } d \epsilon ^{\prime} p_{ \epsilon ^{\prime}}
\Bigg\{
\nonumber \\
\Psi_{1 \epsilon, 2 \epsilon ^{\prime}}^{(2) *}\left( t_{1},t_{2} \right)
\Psi_{1 \epsilon, 2 \epsilon ^{\prime}}^{(2) }\left( t_{4},t_{3} \right)
\label{gn04} \\
+ \sum_{j=1}^{2}
\Psi_{ j\epsilon, j\epsilon ^{\prime}}^{*}\left( t_{1},t_{2} \right)
\Psi_{ j\epsilon, j\epsilon ^{\prime}}^{ }\left( t_{4},t_{3} \right)
\Bigg\} .
\nonumber
\end{eqnarray}
\ \\ \noindent
Here the two-particle wave function $\Psi_{1 \epsilon, 2 \epsilon ^{\prime}}^{(2) }$ is determined by Eq.~(\ref{gn02}) with $ \Psi_{1}$ being replaced by $ \Psi_{1 \epsilon}$ and $ \Psi_{2}$ being replaced by $ \Psi_{2 \epsilon ^{\prime}}$.
In addition, we have a new function $\Psi_{ j\epsilon, j\epsilon ^{\prime}}$ dependent of two times, which is determined by the Slater determinant composed of different components of the same single-particle mixed state,
\begin{eqnarray}
\Psi ^{}_{ j\epsilon, j\epsilon ^{\prime}}\left( t_{1},t_{2} \right) =
\frac{ 1 }{ \sqrt{2} }
\det
\begin{pmatrix} \Psi^{}_{j\epsilon}\left( t_{1}\right) & \Psi^{}_{j \epsilon ^{\prime}}\left( t_{1} \right) \\ \Psi^{}_{j \epsilon }\left( t_{2} \right) & \Psi^{}_{j \epsilon ^{\prime}}\left( t_{2} \right) \end{pmatrix} .
\label{gn05}
\end{eqnarray}
\ \\ \noindent
I name it {\it the two-time wave function}.
Note, that at coincident times, $t_{1}=t_{2}$, this function is zero, $\Psi ^{}_{ j\epsilon, j\epsilon ^{\prime}}\left( t_{},t_{} \right)=0$, which is a manifestation of the fermionic nature of an electron.
The contribution to $G ^{(2)}$ due to the two-time wave function is present even in the case of a single-particle, but {\it mixed} state $\left(N=1 \right)$,
\begin{eqnarray}
G_{N=1} ^{(2)}\left( t_{1},t_{2};t_{3},t_{4} \right) &=&
\int _{}^{ } d \epsilon p_{ \epsilon}
\int _{}^{ } d \epsilon ^{\prime} p_{ \epsilon ^{\prime}}
\label{gn06} \\
&&
\Psi_{ 1\epsilon, 1\epsilon ^{\prime}}^{*}\left( t_{1},t_{2} \right)
\Psi_{ 1\epsilon, 1\epsilon ^{\prime}}^{ }\left( t_{4},t_{3} \right) .
\nonumber
\end{eqnarray}
\ \\ \noindent
Note, at zero temperature the probability density becomes the delta function of energy, $p_{ \epsilon} = \delta\left( \epsilon - \mu \right)$, and the only component with Fermi energy, $ \epsilon = \mu$, survives.
Since $\Psi ^{}_{ j \mu, j \mu}\left( t_{1},t_{2} \right) =0$, the second-order correlation function vanishes, $G_{N=1} ^{(2)}=0$, at zero temperature, as expected for a (pure) single-particle state.
In contrast, at nonzero temperatures, when $p_{ \epsilon} \ne \delta\left( \epsilon - \mu \right)$, a single-particle state demonstrates some degree of second-order coherence, which is quantified by $G ^{(2)}_{N=1} \ne 0$.
This is somewhat counter-intuitive, since the quantities like $G ^{(2)}$ are considered essentially multi-particle in nature.
To resolve this seeming paradox, let us recall the physical meaning of $G ^{(2)}$, specifically with pairwise equal arguments, $t_{1}=t_{4}$ and $t_{2}=t_{3}$.
For a pure state, it is represented by the square of a two-particle wave function, $G ^{(2)}\left( t_{1}, t_{2}; t_{2}, t_{1} \right) = \left | \Psi^{(2)}\left( t_{1}, t_{2} \right) \right |^{2}$.
The conventional meaning of the wave function square is the detection probability, the probability of a strong, projective measurement.
In our case, it is the joint probability of two detections, at time $t_{1}$ and at time $t_{2}$.
In the case of a state with two particles, say, with wave functions $ \Psi_{1}(t)$ and $ \Psi_{2}(t)$, both detections are possible.
Let us suppose that in the first measurement we detect a particle with wave function $ \Psi_{1}$ at time $t = t_{1}$.
The projective measurement means, that after the detection the wave function is changed, it is reduced to the delta function, $ \Psi_{1}(t) \sim \delta\left( t - t_{1} \right)$.
Therefore, the original wave function cannot be measured at any other times.
However, there is a second particle with wave function $ \Psi_{2}(t)$, which can be detected in the second measurement, say, at time $t= t_{2} \ne t_{1}$.
Hence, we are able to perform two measurements, and the probability for such a joint measurement is given by $G_{N=2} ^{(2)}\left( t_{1}, t_{2}; t_{2}, t_{1} \right) \ne 0$.
The case with a single-particle state is different.
As I mentioned above, we can perform a projective measurement on a single particle state only once, say, at $t= t_{1}$, and cannot measure it again $t= t_{2} \ne t_{1}$.
This fact is manifested as $G_{N=1} ^{(2)}=0$.
However, this logic fails in the case of a mixed state.
The reason for this is that a particle in a mixed state can be in several quantum states, components of a mixed state, appearing with some probabilities.
In Eqs.~(\ref{gn06}) and (\ref{gn05}) these states are $ \Psi_{1 \epsilon}$ for various $ \epsilon$.
When we detect a particle at time $t = t_{1}$, we detect it in some particular component state, say, in the state with $ \epsilon = \epsilon_{0}$.
As a result this component is reduced to the delta function, $ \Psi_{ 1\epsilon_{0}}(t) \sim \delta\left( t - t_{1} \right)$, and the original wavefunction cannot be measured at any other times.
But there are many other components of the mixed state with $ \epsilon \ne \epsilon_{0}$.
Any of them is available for the next detection, say, at time $t= t_{2} \ne t_{1}$.
This is why for a mixed single-particle state the second-order correlation function is not vanishing, $G_{N=1} ^{(2)} \ne 0$.
Obviously, all higher-order correlation functions are also not vanishing.
The ability of a (mixed) single-particle state to demonstrate the second-order coherence can be verified (or refuted) experimentally.
In particular, the function $G ^{(2)}$ with pairwise equal arguments is directly accessible through the cross-correlation noise measurement.
\section{ $G ^{(2)}$ and the cross-correlation noise}
\label{sec3}
As it was pointed-out in Ref.~\onlinecite{Thibierge:2015up}, the second-order correlation function with pairwise equal arguments, $G ^{(2)}\left( t_{1}, t_{2}; t_{2}, t_{1} \right)$, is directly related to the cross-correlation symmetrized noise\cite{Buttiker:1990tn,Samuelsson:2004uv,Neder:2007jl,McClure:2007gl}.
More precisely, it is related to the currents, $I_{3}\left( t_{1} \right)$ and $I_{4}\left( t_{2} \right)$, and their correlation function, ${ P}_{34}\left( t_{1}, t_{2} \right)$, which are measured at the outputs of an electronic interferometer \cite{Henny:1999tb,Oliver:1999ws,Oberholzer:2000wx}, analogous to the Hanbury Brown and Twiss (HBT) interferometer \cite{HanburyBrown:1956bi} known in optics, see Fig.~\ref{fig1}.
The source of electrons is placed in one of the inputs.
The temperature of both input channels $1$ and $2$, with and without an electron source, should be the same, $ \theta_{1} = \theta_{2} \equiv \theta$.
Since the source injects particles periodically with period $ {\cal T} _{0}$, the resulting currents are periodic functions of time, $ I_{ \alpha}\left( t \right) = I_{ \alpha}\left(t + {\cal T} _{0} \right)$, $ \alpha = 3, 4$.
\begin{figure}[b]
\includegraphics[width=70mm, angle=0]{fig1.pdf}
\caption{
Scheme of an electron HBT interferometer, where the quantum state injected into one of input channels (marked by $G ^{(2)}_{}$) is transmitted through and reflected at the wave splitter (shown as a shaded thin rectangle).
As a result the outgoing current $I_{3}$ and $I_{4}$ are generated.
These currents, together with their cross-correlation function ${ P}_{34}$, define the second-order correlation function of injected state, $G ^{(2)}_{}$, according to Eq.~(\ref{n03}).
The arrows show the direction of propagation of electrons.
}
\label{fig1}
\end{figure}
Since the measurement of a time-resolved noise is challenging, below I focus on a frequency-resolved noise\cite{Buttiker:1992vn,Liu:1994fy,Lesovik:1994tq,Pedersen:1998uc,Salo:2006ie,Braggio:2016ki,Dittmann:2018ve}, which was measured more than once, see, e.g., Refs.~\onlinecite{{Schoelkopf:1997cc},DiCarlo:2006ju,ZakkaBajjani:2007gg} and also Ref.~\onlinecite{Mahe:2010cp}, where, as I already mentioned, a frequency-resolved noise was used for validation of the single-electron injection regime.
Let us introduce the following Fourier transform,
\begin{eqnarray}
G ^{(2)}_{ \ell}\left( \omega \right) =
\int\limits _{0}^{ {\cal T} _{0} } dt e^{i \Omega \ell t}
\int\limits _{- \infty}^{ \infty } d \tau e^{i \omega \tau}
G ^{(2)}\left( t + \tau, t; t, t + \tau \right) ,
\label{n01}
\end{eqnarray}
\ \\ \noindent
where $ \Omega = 2 \pi/ {\cal T} _{0}$ and $ \ell$ is an integer.
Then, $ G ^{(2)}_{ \ell}\left( \omega \right)$ is expressed in terms of the finite-frequency cross-correlation noise power, $ {\cal P}_{ 34,\ell}\left( \omega \right)$ and outgoing currents $I_{3}$ and $I_{4}$, as follows, (see the Appendix~\ref{ap01} for the precise definition of $ {\cal P}_{ 34,\ell}\left( \omega \right)$, Eqs.~(\ref{a01}) and (\ref{a02}), and for the corresponding derivation within the Floquet scattering matrix approach)
\begin{eqnarray}
v_{ \mu}^{2} G ^{(2)}_{ \ell}\left( \omega \right) =
\frac{{\cal P}_{34, \ell}(\omega) }{ e^{2} RT/ {\cal T} _{0} }
\label{n02} \\
+
\frac{1 }{ e^{2} RT }
\int\limits _{0}^{ {\cal T} _{0} } e^{i \Omega \ell t} dt
\int\limits _{- \infty}^{ \infty } d \tau e^{i \omega \tau}
I_{3}\left( t + \tau \right) I_{4}\left( t \right) .
\nonumber
\end{eqnarray}
\ \\ \noindent
Here $T$ and $R = 1-T$ are the transmission and reflection probabilities of a wave splitter of an electron HBT interferometer, $v_{ \mu}$ is the Fermi velocity of electrons in a waveguide, $e$ is an electron charge.
In the special case, when the excitations produced during different periods do not overlap, the equation (\ref{n02}) can be simplified.
We take into account explicitly the fact that the current is periodic, set $ {\cal T} _{0} \to \infty$, and introduce a continuous frequency $ \omega_{ \ell} = \ell \Omega$ instead of the series of discrete frequencies $ \ell \Omega$, see Appendix~\ref{ftg1} for details.
Then the equation (\ref{n02}) becomes,
\begin{eqnarray}
v_{ \mu}^{2} G ^{(2)}_{ \ell}\left( \omega \right) &=&
\frac{{\cal P}_{34, \ell}(\omega) }{ e^{2} RT/ {\cal T} _{0} } + \frac{ I_{3}\left( \omega \right) I_{4}\left( \omega_{ \ell} - \omega \right) }{ e^{2} RT/ {\cal T} _{0}^{2}} .
\label{n03}
\end{eqnarray}
\ \\ \noindent
This equation
resembles Eq.~(22) of Ref.~\onlinecite{Moskalets:2014ea}, where the two-energy distribution function was related to the zero-frequency noise power and DC currents in the circuit with two energy filters, quantum dots each with one working resonant quantum level.
Below, I will use Eq.~(\ref{n03}) and address the temperature dependence of $G ^{(2)}_{ \ell}\left( \omega \right)$ for electrons injected by some particular source, namely the source of levitons \cite{Dubois:2013ul}, that is capable of generating single- as well as multi-particle excitations \cite{Dubois:2013fs,Gaury:2014jz,Hofer:2014jb,Belzig:2016jza,Suzuki:2017er,Glattli:2017vp}.
\section{Example: The source of levitons}
\label{sec4}
The sequence of the Lorentzian voltage pulses,
\begin{eqnarray}
eV(t) = N\sum\limits_{m=-\infty}^{\infty} \frac{2 \hbar \Gamma _{\tau} }{\left( t - m {\cal T} _{0} \right)^{2} + \Gamma _{\tau}^{2} }.
\label{L01}
\end{eqnarray}
\ \\ \noindent
applied to a metallic contact, generates the stream of excitations with charge $eN$ each in a ballistic channel attached to the contact. \cite{Levitov:1996,Ivanov:1997,Keeling:2006}
Here $ \Gamma _{\tau}$ is the half-width of a voltage pulse.
These excitations are named $N$-electron levitons or $N$-levitons \cite{Ronetti:2017vd}.
\subsection{Correlation functions}
In the regime, when the period is much larger then the width of a voltage pulse, $ {\cal T} _{0} \gg \Gamma _{\tau}$, the excitations created at different periods do not overlap.
Then, we can restrict ourselves to a single period only, say, $m=0$, and send $ {\cal T} _{0} \to \infty$ in the integrals we need to evaluate.
In this case the first-order correlation function of excitations injected by the source of levitons is represented as follows,
\begin{eqnarray}
v_{ \mu} G^{(1)}_{ }( t_{1};t_{2} ^{}) =
\int\limits_{ }^{ } d \epsilon^{}
p_{ \theta} \left( \epsilon \right)
\sum\limits_{j=1}^{N}
\Psi^{*}_{j, \epsilon}(t_{1}) \Psi_{j, \epsilon}(t_{2}^{}) .
\label{L02}
\end{eqnarray}
\ \\ \noindent
Here $\Psi_{j, \epsilon}(t) = e^{-i t \frac{ \mu + \epsilon }{ \hbar} } \psi_{j}\left( t \right)$ is the wave function of the $j$th particles comprising an $N$-electron leviton ($j = 1, \dots, N$).
The corresponding envelope function is the following, \cite{Grenier:2013,Moskalets:2015vr,glattli2016method}
\begin{eqnarray}
\psi_{j}\left( t \right) &=&
\sqrt{\frac{\Gamma _{\tau}}{ \pi } }
\frac{ 1 }{ t - i \Gamma _{\tau} }
\left( \frac{ t + i \Gamma _{\tau} }{ t - i \Gamma _{\tau} } \right) ^{j-1} .
\label{L03}
\end{eqnarray}
\ \\ \noindent
Using the fact that the envelope wave functions, $ \psi_{j}$, are independent of energy, we can integrate $ \epsilon$ out in Eq.~(\ref{L02}) and get,
\begin{eqnarray}
v_{ \mu} G^{(1)}_{ }( t_{1};t_{2} ^{}) =
\eta\left( \frac{ t_{1} - t_{2} }{ \tau_{ \theta} } \right)
\sum\limits_{j=1}^{N}
\psi^{*}_{j}(t_{1}) \psi_{j}(t_{2}^{}) ,
\label{L04}
\end{eqnarray}
\ \\ \noindent
where $ \eta(x) = x/\sinh(x)$ and the thermal coherence time is $ \tau_{ \theta} = \hbar/( \pi k_{B} \theta)$.
Substituting the above equation into Eq.~(\ref{gn01}), one can calculate the second-order correlations function.
For $t_{1} = t_{4}$ and $t_{2} = t_{3}$ we have,
\begin{eqnarray}
v_{ \mu}^{2}G ^{(2)}\left( t_{1},t_{2};t_{2},t_{1} \right) =
\sum\limits_{j=1}^{N}
\sum\limits_{k=1}^{N}
\Bigg\{
\left | \psi^{}_{j}(t_{1}) \right |^{2}
\left | \psi^{}_{k}(t_{2}) \right |^{2}
\nonumber \\
\label{L05} \\
-
\eta^{2}\left( \frac{ t_{1} - t_{2} }{ \tau_{ \theta} } \right)
\psi^{*}_{j}(t_{1})
\psi^{}_{j}(t_{2})
\psi^{*}_{k}(t_{2})
\psi^{}_{k}(t_{1})
\Bigg\}.
\nonumber
\end{eqnarray}
Now I will analyze the above equation in two cases, $N=1$ and $N=2$.
\subsection{A single-electron leviton, $N=1$}
For a single-particle leviton, $N=1$, the function $G ^{(2)}$ becomes,
\begin{eqnarray}
v_{ \mu}^{2} G_{N=1} ^{(2)}\left( t_{1},t_{2};t_{2},t_{1} \right) =
\frac{ \Gamma _{\tau}^{2} }{ \pi^{2} }
\frac{ 1 - \eta^{2}\left( \frac{ t_{1} - t_{2} }{ \tau_{ \theta} } \right) }{ \left( t_{1}^{2} + \Gamma _{\tau}^{2} \right) \left( t_{2}^{2} + \Gamma _{\tau}^{2} \right) } .
\label{L06}
\end{eqnarray}
\ \\ \noindent
From this equation, we can conclude the following.
First, when the time difference is smaller then the thermal coherence time, $\left | t_{1} - t_{2} \right | \ll \tau_{ \theta}$, the function $ \eta = 1$, and the second order correlation function vanishes, $G_{N=1} ^{(2)} =0$.
This fact is a manifestation of a single-particle nature of a quantum state in question.
Second, at larger time difference, $\left | t_{1} - t_{2} \right | \gg \tau_{ \theta}$, the function $ \eta = 0$, and the second order correlation function is factorized into the product of two terms, each of which depends only on one time, $G_{N=1} ^{(2)}\left( t_{1},t_{2};t_{2},t_{1} \right) = \left | \psi_{1}\left( t_{1} \right) \right |^{2} \left | \psi_{1}\left( t_{2} \right) \right |^{2}$.
Namely, the two-particle detection probability becomes the product of two statistically independent single-particle detection probabilities.
Such a property is expected for a classical rather than a quantum state.
Nevertheless, the state of a leviton remains quantum and respects the Pauli exclusion principle, which requires that the function $G ^{(2)}$ strictly vanishes at equal times (at any temperature), $G_{N=1} ^{(2)}\left( t,t;t,t \right) =0$.
\subsubsection{The frequency representation}
Let us perform the Fourier transformation defined in Eq.~(\ref{n01}) on the function $G ^{(2)}$ of a single leviton, Eq.~(\ref{L06}).
Using the fact that $ {\cal T} _{0} \gg \Gamma _{\tau}$, we get,
\begin{eqnarray}
v_{ \mu}^{2} G_{N=1, \ell} ^{(2)}\left( \omega \right) =
e^{- \left | \omega \right | \Gamma _{\tau}}
e^{- \left | \omega - \omega_{\ell} \right | \Gamma _{\tau}}
-
e^{- \left | \omega_{ \ell} \right | \Gamma _{\tau}}
\nonumber \\
\times
\frac{ \Gamma _{\tau} }{ \pi }
\int\limits _{- \infty}^{ \infty } d \tau
\frac{ \eta^{2}\left( \frac{ \tau }{ \tau_{ \theta} } \right) }{ \tau^{2} + 4 \Gamma _{\tau}^{2} }
\bigg\{
\cos\left( \omega \tau \right)
+
\cos\left( \left [ \omega - \omega_{ \ell} \right] \tau \right)
\nonumber \\
+ \frac{ 2 \Gamma _{\tau} }{ \tau } {\rm sgn}( \omega_{ \ell})
\left[
\sin\left( \omega \tau \right)
-
\sin\left( \left [ \omega - \omega_{ \ell} \right] \tau \right)
\right]
\bigg\} .
\nonumber \\
\label{Lom01}
\end{eqnarray}
As I already mentioned, this function is experimentally accessible through the finite-frequency noise measurement, see Eq.~(\ref{n03}).
In Fig.~\ref{fig2} I show $G_{N=1, \ell} ^{(2)}\left( \omega \right)$, Eq.~(\ref{Lom01}), as a function of temperature for several fixed frequencies.
My aim is to show that the function $G ^{(2)}_{N=1}$ is capable of demonstrating a crossover from a single-particle behaviour at zero temperature to a multi-particle-like behaviour at nonzero temperatures.
Indeed, at zero temperature $G_{N=1, \ell} ^{(2)}\left( \omega \right)=0$ for any frequencies, demonstrating that the state in question is a true single-particle state.
At nonzero temperatures, the function $G ^{(2)}$ becomes different from zero, indicating that the state of a single leviton demonstrates rather multi-particle behaviour.
At high temperatures, when the thermal coherence time becomes smaller than the width of a voltage pulse, $\tau_{ \theta} \ll \Gamma _{\tau}$, the second-order correlation function achieves its high-temperature asymptotic behaviour, $v_{ \mu}^{2}\lim\limits_{ \theta\to \infty} G_{N=1, \ell} ^{(2)}\left( \omega \right)=e^{- \left | \omega \right | \Gamma _{\tau}} e^{- \left | \omega - \omega_{\ell} \right | \Gamma _{\tau}}$.
This product form is characteristic of a completely classical state.
Note, that the growth of $G ^{(2)}_{N=1}$ with temperature is manifested in the reduction of single-particle shot noise\cite{Dubois:2013fs,Moskalets:2015ub,Moskalets:2017dy}.
The shot noise decrease with increasing temperature was reported in Refs.~\onlinecite{Bocquillon:2012if,{Bocquillon:2013fp},Glattli:2016wl,Glattli:2016tr}.
\begin{figure}[t]
\includegraphics[width=85mm, angle=0]{fig2.pdf}
\caption{
The second-order correlation function of a single leviton, Eq.~(\ref{Lom01}), normalized to its high-temperature asymptotics, $g_{N=1} ^{(2)}\left( \omega_{ \ell}, \omega \right) = v_{ \mu}^{2} G_{N=1, \ell} ^{(2)}\left( \omega \right)/\left( e^{- \left | \omega \right | \Gamma _{\tau}} e^{- \left | \omega - \omega_{\ell} \right | \Gamma _{\tau}} \right) $, is given as a function of temperature.
The temperature, $k_{B} \theta$, and frequencies, $ \hbar \omega$ and $ \hbar \omega_{ \ell}$, are given in units of the energy of a leviton, $ {\cal E}_{L} = \hbar / \left( 2 \Gamma _{\tau} \right)$.
}
\label{fig2}
\end{figure}
The temperature dependence of the function $G ^{(2)}$ for a multi-electron state is remarkably different.
Namely, its zero temperature limit is not universal.
On contrary, such a limit depends strongly on frequency.
To illustrate this statement, let us consider the case of a $2$-electron leviton.
\subsection{A two-electron leviton, $N=2$}
The correlation function of a $2$-electron leviton, $G_{N=2} ^{(2)}$, is given in Eq.~(\ref{L02}) with $N=2$.
The corresponding wave functions are presented in Eq.~(\ref{L03}).
After performing the Fourier transformation according to Eq.~(\ref{n01}), we obtain $G_{N=2, \ell} ^{(2)}\left( \omega \right)$, see Eq.~(\ref{b04}), which is shown in Fig.~\ref{fig3} as a function of temperature for several fixed frequencies.
\begin{figure}[t]
\includegraphics[width=85mm, angle=0]{fig3.pdf}
\caption{
The second-order correlation function of a $2$-electron leviton, Eq.~(\ref{b04}), normalized to its high-temperature asymptotics, $g_{N=2} ^{(2)}\left( \omega_{ \ell}, \omega \right) = v_{ \mu}^{2} G_{N=2, \ell} ^{(2)}\left( \omega \right)/\left(4 e^{- \left | \omega \right | \Gamma _{\tau}} e^{- \left | \omega - \omega_{\ell} \right | \Gamma _{\tau}} \right) $, is given as a function of temperature.
The temperature, $k_{B} \theta$, and frequencies, $ \hbar \omega$ and $ \hbar \omega_{ \ell}$, are given in units of the energy of a leviton, $ {\cal E}_{L} = \hbar / \left( 2 \Gamma _{\tau} \right)$.
}
\label{fig3}
\end{figure}
We can see, that at zero temperature, the magnitude of $G_{N=2, \ell} ^{(2)}\left( \omega \right)$ is not zero, unlike the case of a plain leviton, $N=1$.
Indeed, it strongly depends on the frequencies $ \omega_{ \ell}$ and $ \omega$.
Such a nonuniversal frequency-dependent behaviour is characteristic of a multi-particle state, the two-particle state in the present case.
Interestingly, the high-temperature asymptotics of the second-order correlation function of a multi-electron leviton is universal: It is determined by the corresponding asymptotics of the function $G ^{(2)}$ of a single-electron leviton,
\begin{eqnarray}
\lim\limits_{ \theta\to \infty} G_{N=N_{0}} ^{(2)} = N_{0}^{2} \lim\limits_{ \theta\to \infty} G_{N=1} ^{(2)} .
\label{L07}
\end{eqnarray}
\ \\ \noindent
This is a manifestation of the high-temperature fusion effect (when the multi-electron system behaves like one particle of the total charge) discussed in Ref.~\onlinecite{Moskalets:2018ch}.
Note, that the low-temperature regime, for which Figs.~\ref{fig2} and \ref{fig3} show different behaviour, is achievable in present day experiment.
So, in Ref.~\onlinecite{Glattli:2016tr} the voltage pulses with width $2 \Gamma _{\tau} = 75$~ps were user to generate levitons with energy $ {\cal E}_{L} \approx 320$~mK.
The experimental data on shot noise were reported for the temperature range from $ \theta_{1} = 40$~mK to $ \theta_{2} = 138$~mK.
Correspondingly, the ratio $ \theta / {\cal E}_{L}$ is changed from $ \theta_{1} / {\cal E}_{L} \approx 0.125$ to $ \theta_{2} / {\cal E}_{L} \approx 0.43$.
From Figs.~\ref{fig2} and \ref{fig3} we see that for these parameters the function $G ^{(2)}_{}$ allows one to uniquely distinguish single-particle and multi-particle states: The second-order correlation function, $G ^{(2)}_{}$, is almost constant in the case of a $2$-electron leviton, while it decreases rapidly to zero with decreasing temperature in the case of a single-electron leviton.
\section{Conclusion}
\label{sec5}
I have discussed the effect of temperature on the second-order correlation function of electrons, $G ^{(2)}$, which are injected by an on-demand source on top of the Fermi sea in conductors.
The second-order correlation function is a universal tool that is able to distinguish between single- and multi-particle injection regime of an electron source.
The function $G ^{(2)}$ is accessible via the cross-correlation electrical noise measurement at the exit of an electron Hanbury Brown and Twiss interferometer.
At zero temperature, the function $G ^{(2)}$ is vanishing in the case of a single-electron injection and does not vanish in the case of multi-particle injection.
In contrast, at nonzero temperatures, the function $G ^{(2)}$ does not vanish even in the case of a single-electron injection.
The reason is that at nonzero temperatures, the single-particle quantum state is a mixed state that demonstrates some degree of second-order coherence, which is quantified by $G ^{(2)} \ne 0$.
Therefore, the existence of this single-particle contribution has to be taken into account, when the second-order correlation function is used for the verification of a single-particle injection into conductors at nonzero temperatures.
\acknowledgments
I appreciate the warm hospitality of the Department of Applied Physics, Aalto University, Finland, where this project was started.
I am grateful to Christian Flindt and Pablo Burset for numerous discussions.
|
3,212,635,537,968 | arxiv | \section{Introduction}
\IEEEPARstart{P}{ET} is a functional imaging technique with a variety of applications in both preclinical as well as clinical research and practice \cite{Phelps2004, Myers2001}. Two \SI{511}{\kilo\eV} gamma particles originating from a positron-electron annihilation are registered by radiation detectors arranged in a ring geometry. State-of-the-art radiation detectors consist of scintillation crystals (e.g., BGO, LSO, LYSO) converting the gamma particles to optical photons and photosensor arrays with multiple channels detecting the optical photons.
The key challenge in PET detector instrumentation is to detect the gamma particles with high sensitivity and with good spatial, energy and time resolution. The knowledge of \acused{DOI}\ac{DOI} reduces the parallax error (radial astigmatism) at off-center positions within a PET ring \cite{Saha2016}. \ac{DOI} information is especially important for PET systems with a small ring diameter such as preclinical devices or organ-specific applications imaging the brain or female breast \cite{Gonzalez2016a, Lee2017, Krishnamoorthy2018}. Recently, the advantages of \ac{DOI} information have also been experimentally validated for a clinical case with a \SI{70}{\centi\metre} diameter tomographic setup \cite{Borghi2018}. Furthermore, \ac{DOI} is expected to also improve the performance of long-axial field-of-view PET systems (Total Body PET) \cite{Zhang2018}.
Several detector concepts have been demonstrated in literature which can be mainly divided up into two groups: pixelated and monolithic detectors. Most state-of-the-art clinical PET scanners are operated with pixelated detector designs. These detectors consist of arrays of scintillator needles and are read out employing light-sharing techniques or one-to-one coupling. None of the currently used clinical detectors for whole-body PET provide \ac{DOI} information \cite{Vandenberghe2016}. Several adaptations of pixelated detectors have been presented to obtain \ac{DOI} information. Among others, multiple-layer designs with small shifts between each layer \cite{Ito2010}, light-sharing techniques encoding the \ac{DOI} information into the light distribution \cite{Lee2015}, combination of scintillation layers with different properties (phoswich detectors) \cite{Seidel1999}, sub-surface laser engraving \cite{Uchida2016} and dual-sided readout \cite{Kang2015} were proposed. However, often these concepts come at the cost of other key performance parameters or introduce additional complexity and cost to the detector design.
As an alternative, monolithic detectors consist of scintillators without any segmentation coupled directly to a photosensor array. Monoliths are able to provide good spatial, timing and energy resolutions as widely shown in literature \cite{Borghi2016,VanDam2013,Bruyndonckx2008,marcinkowski2016sub,Gonzalez2016}. Furthermore, monolithic detectors enable intrinsic DOI encoding. For example, the three-dimensional position can be estimation based on a fit of a model of the light distribution \cite{Li2008}. Instead of fitting to a model, maximum likelihood searches based on three-dimensional training data have also been successfully demonstrated \cite{Krishnamoorthy2018, Marcinkowski2016}. A different approach employed an analytic expression based on the attenuation of the scintillator and the expected light-transport \cite{Gonzalez-Montoro2018, Gonzalez-Montoro2017}. After fitting two free parameters, the ratio of event energy to the maximum local intensity was utilized as DOI observable. A method based on a single \ac{DOI} observable only including the photon counts - the sum of the squared pixel intensities - was applied to both single and dual-sided read-out \cite{Borghi2016, Borghi2016a}. The expected distribution of events according to the attenuation was simulated and then matched to the distribution of the \ac{DOI} observable. The main principle of this method testing several \ac{DOI} observables was first published in \cite{vanDam.2011_practicalMethod}. Besides the presented methods, also neural network estimators based on training data acquired by a side irradiation of the scintillator were shown \cite{Wang2013}. To widely translate monolithic scintillators into applications with a large number of detectors, time-efficient and easy calibration methods are required. Furthermore, all employed algorithms such as position estimation need to be scalable for a large number of detectors.
Recently, we presented a novel planar positioning algorithm based on the supervised machine learning technique \ac{GTB} \cite{Muller2018a}. Here, we present the adaptation and application of this algorithm to \ac{DOI} calibration by a side irradiation of the scintillator. \ac{GTB} builds a set of sequential binary decisions (decision trees) which are evaluated as simple comparisons with two possible outcomes. The algorithm handles different sets of input features and their combinations as well as partially missing data. Trained \ac{GTB} models are shown to be implementable in FPGA if the memory requirement does not exceed the available resources \cite{Kuaga, VanEssen2012}. Besides a general description of the hyperparameter tuning of \ac{GTB} models, we present two optimization scenarios to find the best possible positioning performance: one restricting the available memory to enable a future FPGA implementation and one without any restrictions. The positioning performance is compared to a DOI-approach using a single \ac{DOI} observable similar to \cite{Borghi2016} as well as results presented in literature.
\section{Materials}
As the same materials and detector were used as presented in \cite{Muller2018a}, only a brief description of the single components is given in the corresponding sections. We utilized the \ac{PDPC} technology evaluation kit (TEK) with two sensor tiles of DPC 3200-22 photon counter as a coincidence setup. For calibration, a fan beam collimator providing adjustable beam widths for both coincidence and detector under study was employed. The fan beam collimator consists of a fixed bottom shielding, a tool housing up to two sources, and two adjustable top shielding units defining the beam (see Fig. \ref{fig:sketch_setup}). The whole setup was operated in a light-tight temperature chamber at approx. \SI{5}{\celsius} sensor temperature.
\subsection{Photodetector}
We used an array, referred to as tile, of \num{4x4} digital Silicon Photomultiplier (dSiPM) DPC 3200-22 of \ac{PDPC} \cite{Degenhardt2009, Frach2009}. Each DPC is an independent trigger region consisting of \num{4} pixels with a pitch of \SI{4}{\milli\metre} resulting in a total of \num{64} pixels per tile. Every DPC provides a customizable two-level trigger scheme: After the first trigger signal is generated, the second, higher threshold needs the be reached within a given time interval as well. For our settings, on average \num{2.33} and \num{17} photons need to be detected to reach the first and second trigger threshold. In case the second trigger condition is met, the integration phase starts and all \num{4} pixels of the DPC are read out afterward. The collected information of a DPC is referred to as a hit. The entity of all hits corresponding to one gamma particle interaction is called a cluster as described in more detail in Sec. \ref{subsec:data_acqusition}. For a single gamma particle interaction, not all \num{16} DPCs of the tile will certainly output hit data. This leads to clusters with missing hits as reported in Sec. \ref{sec:results_data_acquisition}, especially if the photon densities are low. Further information regarding the photosensor can be found in \cite{Schug2012, Tabacchini2014, Marcinkowski2013, Schaart2016}.
\subsection{Scintillator Crystal and Wrapping}
We studied a monolithic LYSO scintillator of dimensions \SI{32x32x12}{\milli\metre} matching the active sensor area of the tile. The monolith was wrapped in highly reflective Teflon\texttrademark{} tape (Klinger, Idstein, Germany) and coupled to the photosensor using a two-component dielectric silicon gel (Sylgard 527, Dow Corning, Midland, Michigan, USA). We chose the reflective wrapping to achieve a high light-output of the scintillator. An optical simulation studying the influence of several scintillator wrappings (e.g., black tape) on the position performance of \ac{GTB}-models can be found in \cite{Grahe2017a}. To register coincidences, we employed a \SI{12}{\milli\metre}-high pixelated array with \SI{1}{\milli\metre} pitch also utilized in \cite{Schug2015, Gross-Weege2016}.
\subsection{Collimator Setup}
\begin{figure}[t!]
\center
\includegraphics[width=2.7in]{sketch_setup}
\caption{Sketch of the setup containing the fan beam collimator, coincidence detector and detector under test. For \ac{DOI} calibration, a side irradiation of the detector under test is performed. The irradiation position is determined by the feed-back loop of the stepper motor.}
\label{fig:sketch_setup}
\end{figure}
The setup contained a fan beam collimator equipped with two $^{22}\text{Na}$ sources of approximately \SI{10}{\mega\becquerel} each and an electrically driven translation stage (LIMES 90, Owis, Staufen im Breisgau, Germany) (see Fig. \ref{fig:sketch_setup}). All measurements were conducted at a sensor temperature of approx. \SI{5}{\celsius} achievable on system level as demonstrated in \cite{weissler2015hyperionIID}. The fan beam collimator reached a coincidence rate of \SI{199}{\hertz} for slit widths of \SI{5}{\milli\metre} and \SI{0.25}{\milli\metre} for the coincidence detector and the detector under study, respectively. We chose a much larger slit width for the coincidence detector to avoid losing coincidence events due to geometrical effects and uncertainties of the collimator setup. The slit width of \SI{0.25}{\milli\metre} for the detector under study translates to a beam width of \SI{0.42}{\milli\metre} FWHM at the crystal surface determined by stepping a crystal edge through the beam \cite{Muller2018a}. The translation stage was connected to a control PC and the current position of the detector under test was recorded. This irradiation position was employed as reference in later analysis.
\section{Methods}
\subsection{Data Acquisition and Preprocessing}
\label{subsec:data_acqusition}
The data acquisition aimed to create datasets for establishing and testing \ac{DOI} positioning models based on a side irradiation of the detector. However, a side irradiation leads to an exponential distribution of the gamma particle interactions along the propagation direction according to Lambert-Beer's law. For the employed setup (see Fig. \ref{fig:sketch_setup}), we expected an exponential distribution along the x-direction and a uniform distribution along the y-direction. To achieve reliable \ac{DOI} positioning models and estimates of the positioning performance, datasets with uniformly distributed events in both x- and y-direction are strongly beneficial. Otherwise, the exponential distribution may mislead machine learning-based algorithms during training of DOI models and weights the positioning performance achieved over the whole crystal volume nonproportional. Therefore, we need a planar positioning algorithm to estimate the gamma interaction position of the events obtained with the side irradiation. Subsequently, the full calibration process included irradiations of the crystal's top surface for planar positioning calibration and a side irradiation for \ac{DOI} positioning calibration. Both x- and y-direction of the crystal were irradiated at parallel lines with a pitch of \SI{0.75}{\milli\metre} for the planar positioning calibration. The side irradiation was performed with a pitch of \SI{0.25}{\milli\metre}.
All conducted measurements shared the initial preprocessing based on a tool developed by Schug et al. \cite{Schug2015}: The collected optical photons of one gamma particle interaction create up to \num{16} hits in each of the detectors. All hits associated to one gamma particle interaction were identified and merged to clusters. We applied a cluster window of \SI{40}{\nano\second}. The timestamp of the earliest hit determined the timestamp of the cluster. Afterward, coincident clusters were searched using a sliding coincidence window of \SI{20}{\nano\second}. Pixels missing in a cluster were marked with a negative value to distinguish them from zero photon counts.
Clusters with a total photon count fewer than \num{700} photons were rejected to remove noisy events from the data. Based on the total photon distribution, this photon cut translates to an energy cut of around \SI{300}{\kilo\eV}. No further quality cuts were applied to the data. Accepted clusters are called events.
In the next step, \ac{GTB}-based planar positioning models were established employing the irradiations of the crystal's top surface as described in \cite{Muller2018a}. All events acquired by the side irradiation were positioned along the planar directions. Then, a subset uniformly distributed along x- and y-direction was chosen for every irradiation position of the side irradiation as motivated above. The resulting data were split up into three data sets: 1) training data for building the \ac{DOI} positioning models; 2) validation data for optimization of the hyperparameter of the \ac{DOI} positioning models; 3) test data for the final evaluation of the positioning performance. Each data set contained \num{10000} events per irradiation position of the side irradiation. For the training data, both the binning of the training data as well as the number of events per irradiation position were varied later on aiming for short calibration times. We selected a minimum number of \num{100} training events per irradiation position due to practical reasons: For \num{100} training events, the required time for moving the translation stage and operations of the control PC are already the dominating factor compared to an irradiation time of around \SI{0.5}{\second}.
\subsection{Performance Parameters}
All following positioning performance parameters are based on the positioning error distribution (irradiation position - estimated position) calculated for every single irradiation position. If a single value is given, the performance parameter is averaged for all irradiation positions.
\begin{enumerate}
\item Bias Vector: The bias vector is defined as the mean positioning error for a given position. Due to the edge effects found in monolithic crystals, the bias vector magnitude distribution is non-Gaussian. 50th and 90th percentile of the bias vector magnitude distribution ($\text{Bias}_{50}$ and $\text{Bias}_{90}$) are given to account for both the central part and tails of the distribution.
\item \ac{SR}: The \ac{SR} is defined as the FWHM of the positioning error distribution. We calculate the \ac{SR} in accordance to the NEMA NU 4-2008 procedure \cite{Of2010}. The \ac{SR} is not corrected for the finite beam width of the collimator.
\item Root Mean Squared Error (RMSE): The RMSE is the root of the mean squared positioning error.
\item Mean Absolute Error \acused{MAE}(\ac{MAE}): The \ac{MAE} is the mean of the absolute positioning error.
\item Percentile Distance $d_x$: The percentile distance is defined as the distance enclosing the given percentile $x$ of all events around an irradiation position. We report the 50th and 90th percentile distance.
\item Score of distance \SI{1.5}{\milli\metre}: The score of distance \SI{1.5}{\milli\metre} is the fraction of events which are assigned a position within \SI{1.5}{\milli\metre} around the irradiation position.
\end{enumerate}
While \ac{MAE}, RMSE, percentile distances, and the score value are sensitive to bias effects, SR is not prone to a global shift of the positioning error distribution.
\subsection{Single Observable DOI Estimation}
This \ac{DOI} estimation method utilizes a single observable \acused{SO}(\ac{SO}) calculated based on the measured light distribution as demonstrated in \cite{vanDam.2011_practicalMethod, Borghi2016a}. In contrast to methods presented in literature, the observable must not include more than the four pixels of one DPC to avoid jitter caused by missing hits in an event. We defined a set of possible observables and examined the correlation with the \ac{DOI} position. The set included \mbox{1) the} fraction of the highest photon count to the total photon count of the hottest DPC, and 2) the sum of the squared pixel intensities of the hottest DPC. The crystal was divided into equally sized segments with their own calibration to account for differences in the detector response. The number of segments ranged from \numrange{1}{32} along both planar directions.
To match the observable to the \ac{DOI} position, multiple methods such as lookup tables and fits to polynomials of higher order are possible. For this work, we employed \ac{IR}. \ac{IR} minimizes the mean squared error for monotone data without assuming any form of target function \cite{Kakade, Pedregosa2011}. Thus, this method is preferable if no physical model is present.
As this method is a benchmark to compare the \ac{GTB}-based \ac{DOI} estimation models, only results of the best performing observable and segmentation are shown. The averaged performance parameters as well as the spatial distributions of bias vector, MAE and SR are reported. Tests reducing the number of training events and irradiation positions aiming for short calibration times are out of the scope of this paper and are not presented.
\subsection{Gradient Tree Boosting DOI Estimation}
A detailed description of the \ac{GTB} algorithm for planar positioning is given in \cite{Muller2018a}. Thus, only the main characteristics of the algorithm and hyperparameters used later on are described.
As part of supervised machine learning techniques, \ac{GTB} utilizes training data with known irradiation positions to establish predictive regression models. The algorithm handles missing data for training and evaluation and can be used with arbitrary input features \cite{Chen}. \ac{GTB} builds a set of sequential binary decisions (decision trees) which are evaluated as simple comparisons with two possible outcomes \cite{Kotsiantis2013, Natekin2013}. The ensemble is trained in an additive manner: The first decision tree is based on the given irradiation position. Every following decision tree is trained on the positioning error (irradiation position – estimated position) of the previous ensemble. We employed RMSE as training loss of the objective function.
Four hyperparameters of \ac{GTB} models are of particular importance for the following optimizations: 1) Number of decision trees of the ensemble. 2) Maximum depth: The maximum number of comparisons in a single decision tree. 3) Learning rate: The learning rate multiplicates the positioning error of the already established ensemble with a constant factor less or equal \num{1} for training of the next decision tree as described in \cite{Muller2018a}. This allows to reduce the number of decision trees while reducing the highest achievable positioning performance \cite{Friedman2001}. 4) Features of the input set: In addition to the \num{64} raw photon counts, further features motivated by the physical properties of the problem can be added to improve the positioning performance. Used features and input sets are defined in the following section.
Evaluation of trained \ac{GTB} models is possible in CPU-, GPU- and FPGA-based architectures \cite{VanEssen2012}. All listed architectures should allow for real-time event processing. This work focusses on the possibility of an FPGA implementation because FPGAs are widely employed at several points in the current and future architecture developed in this group \cite{weissler2015hyperionIID, GebhardtdigitalFPGAPipeline2012}. FPGA-based processing combined with the data acquisition allows to significantly decrease the amount of data which needs to be sent to the connected server reducing the bandwidth requirements. Trained \ac{GTB} models are implementable in FPGA if the memory requirement does not exceed the available resources \cite{Kuaga, VanEssen2012}. The memory requirement (MR) of a single decision tree of a \ac{GTB} model can be estimated by
\begin{equation}
\text{MR(\textit{d})} = \left(2^d - 1\right)\cdot \SI{11}{\byte} + 2^d \cdot\SI{6}{\byte}
\end{equation}
with $d$ the maximum depth \cite{Muller2018a}.
We present a general optimization protocol examining the influence of the single hyperparameters. Furthermore, two optimization scenarios of GTB models for the best possible positioning performance are demonstrated: One restricting the available memory to enable a future FPGA implementation and one without any restrictions.
\subsubsection{General Optimization Process}
We adapted the developed optimization protocol presented in \cite{Muller2018a} to the \ac{DOI} problem. First, a suitable start point was searched testing similar settings found for the planar positioning as in \cite{Muller2018a}. Aiming for short calibration times, we studied the influence of the binning of the irradiation positions as well as of the number of training events per irradiation position. The binning of the irradiation positions ranged from \SIrange{0.25}{3}{\milli\metre} and the number of events per irradiation position from \numrange{100}{10000}. Then, always one of the hyperparameters introduced in the previous section was varied keeping all other parameters constant. We tested maximum depths from \numrange{4}{12}, learning rates from \numrange{0.05}{0.7} and three different input sets defined below. The parameter ranges were chosen according to the results of the planar positioning optimization and suggestions found in literature \cite{Muller2018a, Natekin2013}. In general, \ac{GTB} models were trained for \num{1000} decision trees and tested with the validation data set.
To study the influence of the input sets, we validated three possible sets of input features: 1) Raw data: the 64 photon counts; 2) Raw data and calculated features (CF): The calculated features included the index numbers of the hottest pixel and DPC, first and second moment of the light distribution, both defined \ac{DOI} observables, the total photon sum and projections of the photon counts along both planar directions; 3) Raw data, CF and estimated planar interactions positions. In contrast to input set 3), input sets 1) and 2) do not require the estimated planar interaction position for evaluating events. Thus, \ac{GTB} models trained with these input sets can be fully parallelized together with the planar positioning models which could be beneficial for processing data of a full PET system. We present results for the different input sets for all tested maximum depths ranging from \numrange{4}{12}. To represent models with low and high memory requirements and study the influence of the input sets during the training process, ensembles of \num{50} and \num{1000} decision trees were validated.
For all optimization steps, we chose to present the averaged \ac{MAE} as validation metric to account for bias effects and allow a direct comparison with other publications \cite{Borghi2016, Borghi2016a}. Additionally, an overlay and a difference plot of the spatial distribution of the \ac{MAE} for the varied binning of irradiation positions are shown. The spatial distribution indicates if \ac{GTB} models have working regression capabilities: For a working regression model, no bias towards the irradiation positions used for training should be observed.
\subsubsection{High-Performance Optimization}
No memory restrictions were applied to select the best-performing \ac{DOI} positioning models. We elected to pick those models with the best averaged RMSE value for all three tested input sets as the RMSE is used as loss function during model training as well. First, the minimum averaged RMSE was searched for every \ac{GTB} model trained during the general optimization process. In case the minimum RMSE was found for the maximum trained number of decision trees (1000 trees), we continued training until the respective \ac{GTB} model tend to overfit. The \ac{GTB} model was assumed to be overfitting, if the RMSE started to worsen. Second, the best averaged RMSE of all models was searched. The averaged performance parameters of these \ac{GTB} models are presented. Furthermore, the spatial distributions of bias vector, MAE, and SR are plotted for the best-performing model with raw data and calculated features as input.
\subsubsection{Memory-Requirement Optimization}
To enable an FPGA implementation, the \ac{GTB} model with the best possible positioning performance for a given memory restriction is searched. We trained the \ac{GTB} models for combinations of maximum depth and learning rate while an empirically chosen convergence criterion determined the number of decision trees. No further decision trees were added if the averaged MAE did not improve more than \SI{0.0001}{\milli\metre/decision\,tree}. For high learning rates, the chosen convergence criterion may be too conservative and takes action in the overfitting regime. Thus, the found positioning performance is compared to those found for the high-performance optimization. In case the found number of decision trees is larger than those of the high-performance model, the cut gets discarded and the high-performance model of same maximum depth and learning rate is selected.
\section{Results}
\subsection{Data Acquisition}
\label{sec:results_data_acquisition}
The number of read out DPCs per gamma interaction followed a Gaussian distribution with a mean of \num{9.8} DPCs and a standard deviation of \num{1.8} DPCs. An irradiation time per irradiation position of around \SI{5}{\min} was required to measure \num{10000} events uniformly distributed along the planar directions.
\subsection{Single Observable DOI Estimation}
\begin{figure}[!t]
\center
\includegraphics[width=2.5in]{cmp_IR_lin_fit}
\caption{Course of the DOI observable squared pixel intensities (SPI) and \ac{IR}. The \ac{DOI} observable monotonely increases with the \ac{DOI} position. \ac{DOI} position \SI{0}{\milli\metre} represents the top surface of the crystal and \SI{12}{\milli\metre} the photosensor.}
\label{fig:comparison_IR_lin_fit}
\end{figure}
\begin{table}[t!]
\center
\caption{Overview of averaged performance parameters for the best found single observable (SO) model and high performance \ac{GTB} models for all three input sets (raw data (r); raw data and calculated features (r+CF); raw data, calculated features and estimated planar interaction positions (r+CF+pos).}
\label{tab:overview_results}
\begin{tabular}{@{}lS[round-mode=places, round-precision=2]S[round-mode=places, round-precision=2]S[round-mode=places, round-precision=2]S[round-mode=places, round-precision=2]@{}}
\toprule
& {SO} & \multicolumn{3}{c}{GTB} \\ \midrule
& & {r} & {r+CF} & {r+CF+pos} \\\midrule
RMSE / \si{\milli\metre} & 2,2207 & 1,8635 & 1,814625 & 1,809 \\
MAE / \si{\milli\metre} & 1,7256 & 1,3194 & 1,283739 & 1,277968 \\
SR / \si{\milli\metre} & 2,1487 & 2,2584 & 2,120161 & 2,09335 \\
$d_{50}$/ \si{\milli\metre} & 1,3457 & 0,922421 & 0,844323 & 0,877167 \\
$d_{90}$ / \si{\milli\metre} & 3,595 & 3,0074 & 2,911626 & 2,90467 \\
$\text{Bias}_{50}$/ \si{\milli\metre} & 1,0559 & 0,540936 & 0,549904 & 0,5474 \\
$\text{Bias}_{90}$ / \si{\milli\metre} & 2,6198 & 1,737724 & 1,672499 & 1,6639 \\
Score of \SI{1.5}{\milli\metre} & 0,5421 & 0,702886 & 0,714351 & 0,7164 \\ \bottomrule
\end{tabular}
\end{table}
\begin{figure}[t!]
\center
\subfloat[]{\includegraphics[width=2.4in]{result_IR_GTB_bias}%
\label{fig:result_IR_GTB_bias}}\hfill
\subfloat[]{\includegraphics[width=2.4in]{result_IR_GTB_MAE}%
\label{fig:result_IR_GTB_MAE}}\hfill
\subfloat[]{\includegraphics[width=2.4in]{result_IR_GTB_SR}%
\label{fig:result_IR_GTB_SR}}
\caption{Spatial distribution of three performance parameters for the best performing single observable models (denoted as SO) and \ac{GTB} model with raw data and calculated features as input (denoted as GTB r+CF). \ac{DOI} position \SI{0}{\milli\metre} represents the top surface of the crystal and \SI{12}{\milli\metre} the photosensor. (a) Distribution of the bias vector. (b) Distribution of the \ac{MAE}. (c) Distribution of the \ac{SR}.}
\label{fig:result_IR_GTB}
\end{figure}
Both tested \ac{DOI} observables led to similar results. In general, the sum of squared pixel intensities performed a bit better (approx. \SI{3}{\%}). Also, the segmentation along the planar directions improved the positioning performance with an optimum at \num{8} segments along both directions. The difference of positioning performance between the best and worst \ac{DOI} model was less than \SI{2.5}{\%}. Fig. \ref{fig:comparison_IR_lin_fit} shows the course of the chosen \ac{DOI} observable. Assuming a monotone behavior of the \ac{DOI} observable is justified. An averaged \ac{MAE} and \ac{SR} of \SI{1.73}{\milli\metre} and \SI{2.15}{\milli\metre} FWHM were achieved (see Tab. \ref{tab:overview_results}). Fig. \ref{fig:result_IR_GTB} displays the spatial distribution of bias vector, \ac{MAE} and \ac{SR}. Over the whole crystal depth, a bias vector ranging from approx. \SI{\pm 3}{\milli\metre} close to the edges to approx. \SI{1}{\milli\metre} in the central area is observed. The \ac{MAE} deteriorates towards the edges and shows an additional decrease of the positioning performance between \SIrange{4}{10}{\milli\metre} \ac{DOI} position. The \ac{SO} method achieves a \ac{SR} better than \SI{2}{\milli\metre} FWHM for \SIrange{0}{4.5}{\milli\metre} and \SIrange{8.9}{12}{\milli\metre} \ac{DOI} position. However, the positioning performance significantly worsens for the other \ac{DOI} positions to a maximum of \SI{5.4}{\milli\metre} FWHM.
\subsection{Gradient Tree Boosting DOI Estimation}
\subsubsection{General Optimization Process}
\newcommand{2.4in}{2.4in}
\begin{figure*}[!t]
\center
\subfloat[]{\includegraphics[width=2.4in]{general_reference_binning}%
\label{fig:genereal_reference_binning}}
\subfloat[]{\includegraphics[width=2.4in]{general_reference_binning_overlay}%
\label{fig:general_reference_binning_overlay}}
\subfloat[]{\includegraphics[width=2.4in]{general_reference_binning_overlay_diff}%
\label{fig:general_reference_binning_overlay_diff}}
\hfill
\subfloat[]{\includegraphics[width=2.4in]{general_training_events}%
\label{fig:general_training_events}}
\subfloat[]{\includegraphics[width=2.4in]{general_max_depth}%
\label{fig:general_max_depth}}
\subfloat[]{\includegraphics[width=2.4in]{general_learning_rate}%
\label{fig:general_learning_rate}}
\hfill
\subfloat[]{\includegraphics[width=2.4in]{general_input_sets_small}%
\label{fig:general_input_sets_small}}
\subfloat[]{\includegraphics[width=2.4in]{general_input_sets_large}%
\label{fig:general_input_sets_large}}
\caption{General optimization process for \ac{GTB} models. We employed a maximum depth of \num{10}, learning rate \num{0.1}, raw data of \num{5000} training events per irradiation position as input set and pitch of \SI{1}{\milli\metre} of the calibration grid unless stated otherwise. The \ac{MAE} is calculated employing the validation data set with \num{10000} test events and \SI{0.25}{\milli\metre} pitch of the irradiation grid. For the figures which display the number of decision trees, the abscissa is linearly scaled up to \num{100} decision trees and logarithmically afterwards. In case a spatial distribution is shown, \ac{DOI} position \SI{0}{\milli\metre} represents the top surface of the crystal and \SI{12}{\milli\metre} the photosensor. Plots regarding the binning of the calibration grid show only selected pitches of the calibration grid due to reasons of clarity. (a) Averaged \ac{MAE} against the number of decision trees and binning of the calibration grid. (b) Spatial distribution of the \ac{MAE} for different pitches of the calibration grid. (c) Difference of the spatial distribution of the \ac{MAE} for the tested calibration grids. (d) Averaged \ac{MAE} against number of decision trees and number of training events per irradiation position. (e) Averaged \ac{MAE} against number of decision trees and maximum depth. (f) Averaged \ac{MAE} against number of decision trees and learning rate. (g) and (h) Averaged \ac{MAE} against maximum depth and different input sets [raw data (r); raw data and calculated features (r+CF); raw data, calculated features and estimated planar interaction positions (r+CF+pos)]. Ensembles of \num{50} decision trees and \num{1000} decision trees are displayed, respectively.}
\label{fig:general_optimization}
\end{figure*}
We found a maximum depth of \num{10} and a learning rate of \num{0.1} with raw data of \num{5000} training events at \SI{1}{\milli\metre} binning of the calibration grid as a suitable start point. The full optimization process is displayed in Fig. \ref{fig:general_optimization}.
Fig. \subref*{fig:genereal_reference_binning} shows the averaged \ac{MAE} against the number of decision trees and different pitches of the calibration grid. In general, the \ac{MAE} improves with an increasing number of decision trees. Only the model based on \SI{3}{\milli\metre} binning shows slight overfitting effects starting at around \num{400} decision trees. The positioning performance difference is less than \SI{13}{\%} between the \SI{0.25}{\milli\metre} and \SI{3}{\milli\metre} calibration grid evaluated at their respective optimum. For the calibration grids of \SI{0.25}{\milli\metre} and \SI{1}{\milli\metre}, a deviation of the positioning performance of \SI{6}{\%} is found. The course of the spatial distribution of the \ac{MAE} is generally the same for all tested calibration grids [see Fig. \subref*{fig:general_reference_binning_overlay}]: the \ac{MAE} is nearly constant in the central region of the crystal and deteriorates towards the crystal edges. Significant higher order effects such as additional periodicity in the spatial distribution of the \ac{MAE} start to get visible for the \SI{3}{\milli\metre} calibration grid. These effects are not visible for the \SI{1}{\milli\metre} calibration grid; the \ac{MAE} worsens globally. However, the positioning performance difference close to the crystal edges is less dominant [see Fig. \subref*{fig:general_reference_binning_overlay_diff}]. Due to reasons of clarity not shown in Fig. \subref*{fig:genereal_reference_binning} to Fig. \subref*{fig:general_reference_binning_overlay_diff}, further pitches of the calibration grid beside the shown ones were tested - namely \SI{0.5}{\milli\metre}, \SI{0.75}{\milli\metre}, \SI{1.5}{\milli\metre} and \SI{2}{\milli\metre}. All obtained results are in congruence with the plotted pitches of the calibration grid and follow the behavior of the shown curves of averaged and spatial MAE.
Fig. \subref*{fig:general_training_events} displays the averaged \ac{MAE} against the number of decision trees for different numbers of training events per irradiation position. The positioning performance increases if more events are employed for training the \ac{GTB} models. However, only small improvements can be achieved after a sufficient amount of training events is present. Increasing the number of training events per irradiation position from \num{2500} to \num{5000} or \num{10000} improves the positioning performance less than \SI{3.4}{\%} or \SI{6}{\%}, respectively.
The maximum depth strongly influences the positioning performance for ensembles with a low number of decision trees [see Fig. \subref*{fig:general_max_depth}]; increasing the maximum depth improves the averaged \ac{MAE}. Using higher maximum depths than \num{10} show no significant enhancement. The positioning performance difference between the \ac{GTB} models with different maximum depths vanishes for ensembles with a high number of decision trees and converges to an optimum.
Increasing the learning rate, the optimum of the \ac{GTB} models is found for ensembles with less decision trees before overfitting occurs [see Fig. \subref*{fig:general_learning_rate}]. For example, overfitting starts after less than \num{10} decision trees for learning rate \num{0.7} while no overfitting is observed up to \num{1000} decision trees for learning rate \num{0.05}. At the same time, the best possible positioning performance decreases for higher learning rates.
The influence of different input sets at several maximum depths is shown for ensembles of \num{50} and \num{1000} decision trees in Fig. \subref*{fig:general_input_sets_small} and Fig. \subref*{fig:general_input_sets_large}, respectively. For the \ac{GTB} model with \num{50} decision trees, the input sets including the calculated features and planar position improve the averaged \ac{MAE} for all tested maximum depths. The difference between the \ac{MAE} gets smaller increasing the maximum depth. The maximum positioning performance gain by additionally adding the estimated planar interaction positions is smaller than \SI{0.7}{\%} for every maximum depth. The behavior regarding the benefit of adding calculated features and estimated planar interaction positions is similar for \ac{GTB} models with \num{1000} decision trees. In contrast to the \ac{GTB} models with fewer decision trees, the averaged \ac{MAE} slightly deteriorates for higher maximum depths as the models are already in the overfitting regime.
\subsubsection{High-Performance Optimization}
The averaged performance parameters of the best found \ac{GTB} models for all three input sets are displayed in Tab. \ref{tab:overview_results}. A \ac{GTB} model of maximum depth \num{8}, learning rate \num{0.05} and around \num{1550} decision trees performed best for raw data as input. The best models with inputs containing the calculated features for both excluding and including the planar interaction position were found at maximum depth \num{6}, learning rate \num{0.05} and around \num{1000} decision trees. These correspond to memory requirements of \SI{6.7}{\mega\byte} and \SI{1.1}{\mega\byte} according to equation (1), respectively. Furthermore, adding calculated features improves the positioning performance by \SIrange{1.6}{8}{\%} compared to raw data as input while $r_{50}$ and \ac{SR} are affected most. No significant deviation of the positioning performance is induced by adding the planar interaction position.
Fig. \ref{fig:result_IR_GTB} shows the spatial distribution for bias vector, \ac{MAE} and \ac{SR} for raw data and calculated features as input set. The bias vector points towards the crystal center close to both edges. In the region ranging from \SIrange{2}{10}{\milli\metre}, a bias vector magnitude below \SI{1}{\milli\metre} is observed. The \ac{MAE} deteriorates towards the crystal edges and is nearly constant at around \SI{1.2}{\milli\metre} for \ac{DOI} positions between \SIrange{2}{10}{\milli\metre}. The \ac{SR} is nearly constant over the whole crystal and shows a maximum drop of approx. \SI{24.7}{\%} close to \SIrange{6}{8}{\milli\metre}.
\subsubsection{Memory-Requirement Optimization}
\begin{figure}[!t]
\center
\includegraphics[width=3.2in]{memory_optimization}
\caption{Memory-requirement optimization: \ac{MAE} against memory requirement for all three input sets [raw data (r); raw data and calculated features (r+CF); raw data, calculated features and estimated planar interaction positions (r+CF+pos)]. For reasons of clarity, only selected maximum depths are shown. Adding the CF to the input of the \ac{GTB} models improves the positioning performance and allows GTB models with lower memory requirements.}
\label{fig:memory-optimization}
\end{figure}
Fig. \ref{fig:memory-optimization} shows the averaged \ac{MAE} against the memory requirement of \ac{GTB} models. The number of decision trees was determined by the convergence criterion for all three input sets. The lines represent models with the maximum depth denoted next to them while every marker indicates a tested learning rate. The plot allows choosing the best performing \ac{GTB} models for given restrictions on the memory. High learning rates generally led to models with low memory requirements and vice versa. Adding calculated features to the input is beneficial for the positioning performance for models of both low and high memory requirement. The positioning performance improves from \SIrange{0.1}{0.9}{\%} comparing the corresponding \ac{GTB} models with and without estimated planar interaction position of same maximum depth and learning rate.
\section{Discussion}
The photon densities in the employed monolithic crystal are too low to generate a significant amount of events with all \num{16} hits present. This emphasizes the need for calibration and positioning algorithms able to deal with missing hit information. Although the presented results are obtained with the DPC photosensor, both \ac{DOI} estimation methods can be applied to all kinds of photosensors.
The \ac{SO} model can be successfully employed as a \ac{DOI} estimator. In contrast to lookup table-approaches as employed in \cite{vanDam.2011_practicalMethod, Borghi2016, Borghi2016a}, the \ac{IR} model allows a continuous \ac{DOI} estimation. In general, no huge influence on the positioning performance was observed for varying the segmentation of the scintillator because the main differences of the photon detection efficiency between single DPCs are canceled out by the normalization of the \ac{DOI} observable. Here, \ac{SO} models with a segmentation matching the pixel pitch performed best. For more segments, the number of training events per segment decreases below \num{150} per irradiation position which increases the uncertainty of the \ac{IR}. Thus, a finer segmentation may lead to an improved performance in case more training events are available. The positioning performance deteriorates for all performance parameters in the central DOI region (see Fig. \ref{fig:result_IR_GTB}). We assume that the variation of the light distribution is lower in this area reducing the specificity of the used \ac{DOI} observable. A similar course of the positioning performance employing the same \ac{DOI} observable is reported for a \SI{22}{\milli\metre} thick monolithic scintillator \cite{Borghi2016}. The deterioration in the central \ac{DOI} region is significantly reduced if a dual-sided read-out is utilized \cite{Borghi2016a}. In contrast to this work, the calculation of the \ac{DOI} observable in \cite{Borghi2016} is based on fully read out events. The authors presented an adpatation and performance of their method for up to \num{4} missing DPCs. Thus, the observed deterioration could be intensified as the \ac{DOI} observable needs to be limited to one DPC to handle events with missing hits. Van Dam et al. studied a monolithic scintillator of identical height as used in this study testing multiple \ac{DOI} observables \cite{vanDam.2011_practicalMethod}. For all \ac{DOI} observables, a \ac{SR} ranging from \SIrange{1}{5}{\milli\metre} FWHM is reported while a \ac{SR} better than \SI{2}{\milli\metre} FWHM is only shown within \SI{2}{\milli\metre} distance to the photosensor. As far as the \ac{SR} alone allows a comparison, the presented \ac{SO} approach performs better than the presented estimation by van Dam et al. and will be used as one benchmark for the \ac{GTB} models.
\ac{GTB} models were successfully adapted to the \ac{DOI} estimation problem. The general optimization process shows the influence of the hyperparameters and represents a protocol applicable to further scintillator geometries as well. \ac{GTB} models show well-working regression capabilities allowing to significantly reduce the number of irradiation positions required for training [see Fig. \subref*{fig:genereal_reference_binning} to Fig. \subref*{fig:general_reference_binning_overlay_diff}]. Thus, we were able to reduce the binning of the calibration grid to \SI{1}{\milli\metre} corresponding to \num{12} irradiation positions for a full calibration. Furthermore, \ac{GTB} creates \ac{DOI} positioning models with a comparably small amount of training data in the order of a few thousand events per irradiation position [see \subref*{fig:general_training_events}]. Based on these results, we chose \num{5000} events per irradiation position for all trained models including the high-performance optimization. The selected calibration grid and number of training events per irradiation position lead to a calibration time of less than \SI{30}{\min} for the side irradiation a single detector block. This calibration time seems practical for calibrating a large number of detectors as well. If calibration time is a critical issue, the \ac{GTB} models allow a further reduction of irradiation positions and training events without compromising much on the positioning performance.
After a sufficient maximum depth is reached, a further increase of the maximum depth did not yield to a significantly better positioning performance [see Fig. \subref*{fig:general_max_depth}]. In case a single decision tree corrected the given input during the training process with a smaller depth than the allowed maximum depth, all further added nodes are based on statistical fluctuations found in the training data. As these additional nodes do not contribute to the general predictivity of the \ac{GTB} model, the positioning performance is not improved.
Maximum depth and learning rate are the most important hyperparameters to tune both the positioning performance and memory requirement of the \ac{GTB} models. As the memory requirement is $\mathcal{O}(2^d)$ with $d$ the maximum depth, this is the most important hyperparameter for an FPGA implementation. High learning rates allow a significant reduction of the number of decision trees at the cost of a decreased positioning performance [see Fig. \subref*{fig:general_learning_rate}].
Adding calculated features to the input features improves the positioning performance for \ac{GTB} models with both small and high number of decision trees [see Fig. \subref*{fig:general_input_sets_small} and Fig. \subref*{fig:general_input_sets_large}]. The physically motivated features such as the defined \ac{DOI} observable are easily interpretable and allow faster learning. Also, geometrical information as encoded in the projections has a high information content which is not directly accessible by the raw data. This also leads to less complex \ac{GTB} models with smaller memory requirement as shown for the high-performance optimization. Due to differences in the light response of the detector, the \ac{GTB} models need to distinguish between different planar interaction positions. Thus, it is of interest if the \ac{GTB} models benefit from the planar interaction position concerning a better positioning performance or reduction of required memory. As a consequence, adding the planar interaction position would require a sequential positioning process in a system architecture. As shown in Fig. \subref*{fig:general_input_sets_small} and Fig. \subref*{fig:general_input_sets_large}, only a small positioning performance difference of less than \SI{0.7}{\%} can be observed when adding the planar interaction position to the calculated features. The same observation holds true for the high-performance and memory-requirement optimization (see Tab. \ref{tab:overview_results} and Fig. \ref{fig:memory-optimization}). The estimated planar interaction position does not provide a significant higher information content than the already added calculated features such as center of gravity.
Both presented optimization scenarios focus on different use-cases. In case the calculated features are added to the input set, the high-performance optimization leads to \ac{GTB} models of around \SI{1}{\mega\byte} memory requirement which are easy to handle for modern computers. However, the memory requirement is the limiting factor for currently available FPGA in case no external memory is added. The memory-requirement optimization allows selecting the best-performing model for given resources. For example, the memory requirement can be reduced down to \SI{91}{\kilo\byte} with a reduction of the \ac{MAE} of less than \SI{3}{\%} compared to the high-performance model requiring \SI{1}{\mega\byte}. This memory requirement can be easily handled by currently available FPGA families.
Comparing the \ac{SO} models and high-performance \ac{GTB} models, the \ac{GTB} models have a better positioning performance of up to \SI{50}{\%} for the averaged performance parameters (see Tab. \ref{tab:overview_results}). Particularly, the bias vector and bias-sensitive performance parameters are improved; the averaged \ac{SR} shows no significant difference. However, the differences get clearly visible comparing the spatial distribution of the performance parameters (see Fig. \ref{fig:result_IR_GTB}): In contrast to the \ac{SO} model, the \ac{GTB} models provide a nearly uniform positioning performance over the whole crystal depth. For example, the \ac{SR} ranges between \SIrange{1.9}{2.5}{\milli\metre} FWHM for \ac{GTB} models and between \SIrange{1.0}{5.3}{\milli\metre} for \ac{SO} models. The bias-sensitive performance parameters such as \ac{MAE} [see Fig. \subref*{fig:result_IR_GTB_MAE}] stays constant over the whole crystal except close to the edges for \ac{GTB} models. As a main difference between \ac{SO} and \ac{GTB}, the \ac{GTB} algorithm has access and utilizes all available information of the raw data and caluclated features. To our best knowledge, the \ac{GTB} models achieve the best reported positioning performance for \ac{DOI} estimation with respect to averaged performance parameters and uniformity for the tested scintillator geometry. For example, Li et al. achieve an averaged \ac{SR} of \SI{2.6}{\milli\metre} FWHM (corrected for finite beam width) using a parametric fitting model for a \SI{10}{\milli\metre} thick scintillator \cite{Li2008}. As already discussed, van Dam et al. presented a \ac{SR} ranging from \SIrange{1}{5}{\milli\metre} not providing an uniform positioning performance \cite{vanDam.2011_practicalMethod}. Comparable to our results, Wang et al. demonstrated an averaged \ac{SR} of around \SI{2}{\milli\metre} employing a neural network \cite{Wang2013}. The reported \ac{SR} deteriorated after applying a correction for bias effects. In contrast to the \ac{GTB} method, the monolithic crystal was segmented into \num{35} cuboids with their own neural network \ac{DOI} estimation models. Thus, the planar interaction position needs to be estimated before the \ac{DOI} position.
\section{Conclusion}
We presented two \ac{DOI} positioning methods based on a side irradiation conducted with a fan beam collimator. Both \ac{SO} and \ac{GTB} models handle missing hit information. Due to the regression capabilities of the \ac{GTB} models, the required time for the side irradiation can be reduced to less than \SI{30}{\min} without compromising too much on the positioning performance. The \ac{GTB} model is able to employ the raw photon counts as input set. Adding physically motivated features to the raw data improves the positioning performance and allows models with smaller memory requirements. For \ac{DOI} estimation, the \ac{GTB} models do not require information of the planar interaction position. Thus, a full parallelization of both planar and \ac{DOI} positioning is feasible on system level. The developed memory optimization process allows training \ac{GTB} models suitable for future FPGA implementation of the algorithm. In contrast to the \ac{SO} models and other methods presented in literature, \ac{GTB} models provide a nearly uniform positioning performance over the whole crystal depth. We achieved an averaged MAE of \SI{1.28}{\milli\metre} and SR of \SI{2.12}{\milli\metre} FWHM for the \SI{12}{\milli\metre}-high crystal, respectively. Future research will translate and evaluate the \ac{GTB} algorithm to monolithic scintillators of higher depth. Furthermore, alternatives replacing the side irradiation by means of optical simulation to create the training data are under investigation.
\section*{Acknowledgment}
The authors want to acknowledge the open-source software packages NumPy \cite{VanderWalt2011}, Matplotlib \cite{Hunter2007} and ROOT \cite{Brun1997} heavily used in the analysis of the presented data.
\renewcommand*{\bibfont}{\footnotesize}
\printbibliography[heading=bibnumbered]
\begin{acronym}
\acro{CF}{calculated features}
\acro{COG}{center of gravity}
\acro{DOI}{depth of interaction}
\acro{GTB}{gradient tree boosting}
\acro{IR}{Isotonic Regression}
\acro{kNN}{$k$ nearest neighbors}
\acro{LSF}{line spread function}
\acro{MAE}{mean absolute error}
\acro{ML}{maximum likelihood}
\acro{SO}{single observable}
\acro{SR}{Spatial Resolution}
\acro{SPAD}{single photon avalanche diode}
\acro{PCA}{principal component analysis}
\acro{PDPC}{Philips Digital Photon Counting}
\acro{PSF}{point spread function}
\end{acronym}
\end{document}
|
3,212,635,537,969 | arxiv | \section{Introduction}
In many applications, capacitances have a remarkable influence on the speed and power dissipation of the devices, and also set an upper frequency limit (cutoff frequency) for their correct operations.~\cite{Neamen2012,Theis2010} The effects of the capacitances on spintronic devices have also been observed and analyzed recently. The magnetocapacitance or magnetoimpedance has been studied, for example, in magnetic tunnel junctions (MTJs)~\cite{Xiao01,Chui02,Kaiju2002,Chien2006,Gupta07,Chang2010,Kaiju2015,Parui2016} and in a single electron transistor.~\cite{Lee15}
Using a time-dependent approach, Rashba has studied the frequency-dependent impedance of a junction composed of a ferromagnetic (FM) conductor, a spin-selective tunnel or Schottky contact, and a nonmagnetic (NM) conductor.~\cite{Rashba02a} The imaginary part of the impedance was attributed to a diffusion capacitance $C_\mathrm{diff}$, which can be compared to (but is certainly different from) that of a p-n junction.~\cite{Neamen2012} However, Rashba's derivation involved the quasi-neutrality condition, which assumes that the charge accumulation is negligible everywhere. Thus it seems difficult to reconcile this condition with the charge storage that is usually associated with the diffusion capacitance.~\cite{Neamen2012} In the present paper, we try to resolve this problem by introducing a \emph{spin-accumulation} (SA) capacitance for each layer. The two plates of the SA capacitor model the two spin channels, since spin accumulation manifests itself as an excess of electrons in one spin channel and an equal deficiency in the other under the quasi-neutrality condition. Thus the charge storage of the SA capacitor happens in the spin degree of freedom rather than the coordinate space, and a spatial charge storage is not essential to it. We also prove that the SA capacitances lead to the same low-frequency reactance as $C_\mathrm{diff}$. Then the conflict mentioned above is resolved.
The form of energy storage associated with the diffusion capacitance is another basic question, because it indicates the type and origin of the capacitance. However, this issue has not been addressed to our knowledge.~\cite{Neamen2012,Rashba02a} On the other hand, most studies on the magnetoimpedance of the MTJs assumed that the energy is stored in an extra electrostatic field like usual geometric capacitances.~\cite{Xiao01,Chien2006} Here, we also show that the energy storage associated with the magnetoimpedance of the FM/NM junction is equal to that stored in the SA capacitors. It is the splitting energy of the spin-dependent chemical potentials instead of the electrostatic energy. This characteristic is similar to that of a quantum capacitance, which stores energy in forms other than the electrostatic field.~\cite{Luryi88,Datta2005,Kopp2009}
We study the SA capacitance using a method similar to that for dealing with quantum capacitance.~\cite{Luryi88} This method is quite different from Rashba's time-dependent approach.~\cite{Rashba02a} It enables us to find the connections between the SA capacitance, the diffusion capacitance, and the quantum capacitance. Moreover, this method has the potential to be used in more complicated situations.
This paper is organized as follows. In section~\ref{theory}, we introduce the SA capacitance for the FM/NM junction. Then the model is used to reinterpret the magnetoimpedance in section~\ref{discussion}. The main conclusions are given in section~\ref{conclusion}.
\section{Spin-accumulation capacitance\label{theory}}
To illustrate the concept of the SA capacitance, we consider a magnetic multilayer with current perpendicular to the plane, which is the well-known configuration giving rise to spin accumulation.~\cite{vf93} To be specific, we consider the same FM/NM junction studied by Rashba.~\cite{Rashba02a} The junction is composed of a ferromagnetic layer occupying $z<0$, a spin-selective contact at $z=0$, and a nonmagnetic layer occupying $z>0$. The $z$ axis is set to be perpendicular to the layer plane as shown in Fig.~\ref{fig_fin}. Without loss of generality, the magnetization of the FM is set to be ``up'' and the current is flowing in the positive direction of the $z$ axis.
Although the importance of capacitance usually shows up in time-dependent transport, it is also present in the constant DC situation, which is much easier to deal with. The spin accumulation is usually described by the splitting of the spin-dependent chemical potentials
\begin{equation}\label{spinaccu}
\mu_\mathrm{m}(z)=2\Delta\mu(z)=\mu_{+}(z)-\mu_{-}(z),
\end{equation}
where the subscripts ``$+$'' and ``$-$'' stand for the absolute spin directions ``up'' and ``down'', respectively.~\cite{vf93} In each layer, $\Delta\mu(z)$ satisfies the well-established spin diffusion equation
\begin{equation}
\frac{\partial^2\Delta\mu(z)}{\partial{z}^2}=
\frac{\Delta\mu(z)}{l_\mathrm{sf}^2},\label{new6a}
\end{equation}
where $l_\mathrm{sf}$ is the spin-diffusion length (see Appendix~\ref{appfn}). The solution to Eq.~(\ref{new6a}) in each layer is of exponential form as shown by Eqs.~(\ref{deltamuf}) and (\ref{deltamun}). Introducing the average chemical potential $\mu(z)=[\mu_{+}(z)+\mu_{-}(z)]/2$ and using Eq.~(\ref{spinaccu}), we can write the spin-dependent chemical potentials as
\begin{equation}
\mu_\pm(z)=\mu(z)\pm\Delta\mu(z).\label{averagemu}
\end{equation}
Since we are concerned with capacitance, it is more convenient to use the charge density in the two spin channels
\begin{equation}
\rho_\pm(z)=-eN_s\left[\mu_\pm(z)-\mu^0\right],\label{deltans}
\end{equation}
where $-e$ is the charge of an electron and $N_s$ the density of states at the Fermi level $\mu^0$.~\cite{zhu08,zhu14} According to the Valet-Fert theory,~\cite{vf93} $N_s$ is assumed to be the same for the two spin directions and also for both layers. We have $\mu(z)=\mu^0$ in the NM layer, whereas $\mu(z)$ is unequal to $\mu^0$ in general for the FM layer.~\cite{zhu14} However, it can be justified for our problem to assume
\begin{equation}
\mu(z)=\mu^0\label{qnd1}
\end{equation}
even in the FM layer using the quasi-neutrality approximation (see Appendix~\ref{qnd}). Then substituting Eqs.~(\ref{averagemu}) and (\ref{qnd1}) into Eq.~(\ref{deltans}), we have
\begin{equation}\label{rhomu}
\rho_\pm(z)=\mp{e}N_s\Delta\mu(z),
\end{equation}
which is valid in both FM and NM layers. Therefore, the electron excess in one spin channel cancels the deficiency in the other, $\rho_{+}(z)+\rho_{-}(z)=0$, in both layers.
\begin{figure}
\includegraphics[width=0.48\textwidth]{figure1}
\caption{\label{fig_fin} Sketch of the spin-dependent charge density $\rho_\pm(z)$ in the FM/NM junction. We plot the curves using Eq.~(\ref{rhomu}) but without realistic parameters, because a demonstration of the basic profile is enough for our current purpose. The units for $\rho_\pm(z)$ and $z$ are arbitrary. The constant DC current has a density of $J_0$. The solid curves on the left and right sides of the interface at $z=0$ stand for $\rho_+(z)$ in the FM and NM layers, respectively. Similarly, the dashed curves are for $\rho_-(z)$. The vertical dotted lines indicate the scale of the spin-diffusion lengths, $l_\mathrm{sf}^\mathrm{F}$ and $l_\mathrm{sf}^\mathrm{N}$, in the FM and NM layers, respectively. The capacitances $C_\mathrm{sa}^\mathrm{F}$ and $C_\mathrm{sa}^\mathrm{N}$ are defined by Eq.~(\ref{sc}), and will be connected in the circuit shown by Fig.~\ref{fig_circuit_fin}(b).}
\end{figure}
The basic idea of the SA capacitance arises initially from the resemblance of charge storage between the two spin channels and the two plates of a normal capacitor, as shown in Fig.~\ref{fig_fin}. Further analysis shows that the spin accumulation also bears similarity to a capacitor in other aspects, including energy storage, leakage current, and heat generation.
\subsection{Charge storage}
When the two spin channels are modeled as the two plates of the SA capacitor, the charge storage of the capacitor can be defined as the absolute value of the charge accumulation in either spin channel. Treating the FM and NM layers separately, we can write the charge storage as
\begin{align}\label{chargefn}
Q_\mathrm{sa}^\mathrm{F}&=\int_{-\infty}^0\left|\rho_\pm(z)\right|\mathrm{d}z
=\int_{-\infty}^0{e}N_s\left|\Delta\mu(z)\right|\mathrm{d}z,\\
Q_\mathrm{sa}^\mathrm{N}&=\int_0^\infty\left|\rho_\pm(z)\right|\mathrm{d}z
=\int_0^\infty{e}N_s\left|\Delta\mu(z)\right|\mathrm{d}z,
\end{align}
where Eq.~(\ref{rhomu}) has been used. Following the procedure detailed in Appendix~\ref{appfn}, we can write $Q_\mathrm{sa}^\mathrm{F}$ and $Q_\mathrm{sa}^\mathrm{N}$ in a form resembling a normal capacitor
\begin{equation}\label{qs}
Q_\mathrm{sa}^\mathrm{F}=C_\mathrm{sa}^\mathrm{F}V_\mathrm{c}^\mathrm{F},\qquad
Q_\mathrm{sa}^\mathrm{N}=C_\mathrm{sa}^\mathrm{N}V_\mathrm{c}^\mathrm{N},
\end{equation}
where we have introduced the SA capacitances
\begin{equation}\label{sc}
C_\mathrm{sa}^\mathrm{F}=\frac{T_1^\mathrm{F}}{2r_\mathrm{F}},\qquad
C_\mathrm{sa}^\mathrm{N}=\frac{T_1^\mathrm{N}}{2r_\mathrm{N}},
\end{equation}
and the corresponding effective voltages
\begin{equation}\label{ep}
V_\mathrm{c}^\mathrm{F}=\frac{\left|\Delta\mu(0^-)\right|}{e},\qquad
V_\mathrm{c}^\mathrm{N}=\frac{\left|\Delta\mu(0^+)\right|}{e},
\end{equation}
for the \emph{semi-infinite} FM and NM layers, respectively. In the expression of $C_\mathrm{sa}^\mathrm{F(N)}$ given in Eq.~(\ref{sc}), $T_1^\mathrm{F}$ ($T_1^\mathrm{N}$) is the spin-relaxation time in the FM (NM) layer.~\cite{zhu08} The resistances $r_\mathrm{F}$ and $r_\mathrm{N}$ are defined as~\cite{fert01}
\begin{equation}\label{rfn}
r_\mathrm{F}=\rho^\ast_\mathrm{F}l_{\mathrm{sf}}^\mathrm{F},\qquad
{r}_\mathrm{N}=\rho^\ast_\mathrm{N}l_{\mathrm{sf}}^\mathrm{N},
\end{equation}
where $\rho^\ast_\mathrm{F}$ ($\rho^\ast_\mathrm{N}$) is the FM (NM) resistivity. Using the expressions of $\Delta\mu(0^-)$ and $\Delta\mu(0^+)$ in Eq.~(\ref{mu0pm}), we can rewrite Eq.~(\ref{ep}) in a form similar to Ohm's law
\begin{equation}\label{ep2}
V_\mathrm{c}^\mathrm{F}=r^\mathrm{sf}_\mathrm{F}\left|\alpha_\mathrm{F}\right|\frac{J_0}{2},\qquad
V_\mathrm{c}^\mathrm{N}=r^\mathrm{sf}_\mathrm{N}\left|\alpha_\mathrm{N}\right|\frac{J_0}{2},
\end{equation}
where we have defined the dimensionless parameters
\begin{equation}\label{alphafn}
\alpha_\mathrm{F}=\frac{\beta(r_\mathrm{N}+r_\mathrm{b}^\ast)-\gamma{r}_\mathrm{b}^\ast}
{r_\mathrm{F}+r_\mathrm{b}^\ast+r_\mathrm{N}},\qquad
\alpha_\mathrm{N}=\frac{\beta{r}_\mathrm{F}+\gamma{r}_\mathrm{b}^\ast}
{r_\mathrm{F}+r_\mathrm{b}^\ast+r_\mathrm{N}},
\end{equation}
and the characteristic (spin-flip) resistances
\begin{equation}\label{rsr}
r^\mathrm{sf}_\mathrm{F}=2r_\mathrm{F},\qquad{r}^\mathrm{sf}_\mathrm{N}=2r_\mathrm{N},
\end{equation}
for the FM and NM layers, respectively. In Eq.~(\ref{alphafn}), the bulk spin asymmetry coefficient $\beta$ in the FM layer is defined by the relation
\begin{equation}
1/\sigma_{\uparrow(\downarrow)}=2\rho_\mathrm{F}^\ast[1-(+)\beta],
\end{equation}
where $\sigma_{\uparrow(\downarrow)}$ is the conductivity for the majority (minority) spin direction. Similarly, we have $1/\sigma_{\pm}=2\rho_\mathrm{N}^\ast$ in the NM layer. The interfacial resistance $r_\mathrm{b}^\ast$ and its spin asymmetry coefficient $\gamma$ are defined by
\begin{equation}\label{bcr}
r_{\uparrow\left(\downarrow\right)}=2r_{b}^{*}\left[1-\left(+\right)\gamma\right],
\end{equation}
where $r_{\uparrow\left(\downarrow\right)}$ is the resistance of the majority (minority) spin channel. In general, the effective voltages $V_\mathrm{c}^\mathrm{F}$ and $V_\mathrm{c}^\mathrm{N}$ are not equal to each other due to the interface resistance. In the configuration specified above (``up'' magnetization and positive DC current), $\alpha_\mathrm{N}$ is always positive, whereas $\alpha_\mathrm{F}$ may be negative. Their interpretation will be given by Eq.~(\ref{jleak}) in combination with the effective circuit shown in Fig.~\ref{fig_circuit_fin}(b).
The expression of $C_\mathrm{sa}^\mathrm{F(N)}$ in Eq.~(\ref{sc}) is similar to that of the diffusion capacitance in a p-n junction.~\cite{Neamen2012} However, $C_\mathrm{sa}^\mathrm{F(N)}$ does not require a spatial charge storage, which is essential for the diffusion capacitance in a p-n junction. Therefore, $C_\mathrm{sa}^\mathrm{F(N)}$ measures the ability of a material to store spins instead of charge. Spin accumulating and dissipating correspond to charging and discharging of the SA capacitor, respectively. Note that $C_\mathrm{sa}^\mathrm{F(N)}$ cannot be compared directly with the diffusion capacitance $C_\mathrm{diff}$ in Eq.~(\ref{cdiff}), which was introduced by Rashba.~\cite{Rashba02a} One distinction between them is the independence of $C_\mathrm{sa}^\mathrm{F(N)}$ on the spin asymmetry coefficient $\beta$ or $\gamma$.
\subsection{Energy storage}
Since $\mu_\mathrm{m}(z)$ is the splitting between the chemical potentials of the two spin channels, spin accumulation also accompanies an increase in energy in comparison to the equilibrium state.~\cite{Tulapurkar11,Juarez16} The energy stored in the differential $\mathrm{d}z$ is equivalent to the energy required to shift a number of electrons, $N_s|\Delta\mu(z)|\mathrm{d}z$, from the spin channel with lower chemical potential to the other. Because the average energy increase per electron is just $|\Delta\mu(z)|$, the energy storage in $\mathrm{d}z$ is $N_s[\Delta\mu(z)]^2\mathrm{d}z$. The total energy storage is the sum of the following two terms
\begin{align}
W_\mathrm{sa}^\mathrm{F}=\int_{-\infty}^0{N}_s\left[\Delta\mu(z)\right]^2\mathrm{d}z,\\
W_\mathrm{sa}^\mathrm{N}=\int_0^\infty{N}_s\left[\Delta\mu(z)\right]^2\mathrm{d}z,
\end{align}
where $W_\mathrm{sa}^\mathrm{F}$ and $W_\mathrm{sa}^\mathrm{F}$ are the contributions from the FM and NM layers, respectively. Using Eqs.~(\ref{deltamuf}) and (\ref{deltamun}), we can write them in a form resembling the energy stored in a capacitor
\begin{equation}\label{esc}
W_\mathrm{sa}^\mathrm{F}=\frac{1}{2}C_\mathrm{sa}^\mathrm{F}\left(V_\mathrm{c}^\mathrm{F}\right)^2,\qquad
W_\mathrm{sa}^\mathrm{N}=\frac{1}{2}C_\mathrm{sa}^\mathrm{N}\left(V_\mathrm{c}^\mathrm{N}\right)^2,
\end{equation}
where Eqs.~(\ref{sc}) and (\ref{ep}) have also been used.
The energy stored in an SA capacitor is essentially the splitting energy of the spin-dependent chemical potentials instead of the electrostatic energy associated with a geometric capacitance. This character is similar to the quantum capacitance, which stores energy in the form of the Fermi degeneracy energy.~\cite{Luryi88} In general, quantum capacitance appears when the spatial charge accumulation on at least one plate of the capacitor induces a change in chemical potential. This is also true for the SA capacitance according to the discussion above. Moreover, substituting Eq.~(\ref{csa}) into Eq.~(\ref{sc}), we can also rewrite the SA capacitance as
\begin{equation}\label{qc}
C_\mathrm{sa}^\mathrm{F(N)}=C_\mathrm{Q}l_\mathrm{sf}^\mathrm{F(N)},
\end{equation}
where we define $C_\mathrm{Q}$ as the quantum capacitance per unit volume and spin channel
\begin{equation}
C_\mathrm{Q}=e^2N_s,
\end{equation}
following Ref.~\onlinecite{Luryi88}. Thus we can also interpret the SA capacitance as the quantum capacitance due to spin accumulation on the scale of the spin-diffusion length. It has the characteristics of the diffusion capacitance as well as the quantum capacitance as shown by Eqs.~(\ref{sc}) and (\ref{qc}). However, the SA capacitance is different from the usual quantum capacitance: its change in chemical potential happens in the spin degree of freedom instead of the coordinate space owing to the relation $\mu(z)=\mu^0$.
\subsection{Leakage current}
Spin accumulation coexists with spin relaxation, in which electrons undergo transitions from the spin channel with higher chemical potential to the other via spin-flip scattering. This process can be modeled as the leakage of the SA capacitor (spin flux in Ref.~\onlinecite{Wegrowe2011}). If we set the positive direction of leakage current to be from the spin-down channel to the spin-up channel, the leakage current can be written as
\begin{align}
J_\mathrm{sf}^\mathrm{F}=\int_{-\infty}^0\frac{eN_{s}\mu_\mathrm{m}(z)}
{\tau_\mathrm{sf}^\mathrm{F}}\mathrm{d}z,\label{jsff}\\
J_\mathrm{sf}^\mathrm{N}=\int_0^\infty\frac{eN_{s}\mu_\mathrm{m}(z)}
{\tau_\mathrm{sf}^\mathrm{N}}\mathrm{d}z,\label{jsfn}
\end{align}
for the FM and NM layers, respectively. Here $\tau_{\rm{sf}}=2T_{1}$ is the spin-flip scattering time.~\cite{fl96} Using Eqs.~(\ref{deltamuf}) and (\ref{deltamun}), we can write $J_\mathrm{sf}^\mathrm{F}$ and $J_\mathrm{sf}^\mathrm{N}$ in a form resembling Ohm's law
\begin{equation}\label{jleak}
J_\mathrm{sf}^\mathrm{F}=\frac{V_\mathrm{c}^\mathrm{F}}{r^\mathrm{sf}_\mathrm{F}}
=\alpha_\mathrm{F}\frac{J_0}{2},\qquad
J_\mathrm{sf}^\mathrm{N}=\frac{V_\mathrm{c}^\mathrm{N}}{r^\mathrm{sf}_\mathrm{N}}
=\alpha_\mathrm{N}\frac{J_0}{2}
\end{equation}
where Eq.~(\ref{ep2}) has been used. Comparing Eqs.~(\ref{qs}) and (\ref{jleak}), one can identify $r^\mathrm{sf}_\mathrm{F}$ ($r^\mathrm{sf}_\mathrm{N}$) as the resistor in parallel with the SA capacitor $C_\mathrm{sa}^\mathrm{F}$ ($C_\mathrm{sa}^\mathrm{N}$) as shown in Fig.~\ref{fig_circuit_fin}(b). Using $\alpha_\mathrm{F}+\alpha_\mathrm{N}=\beta$, we have
\begin{equation}\label{leakcurrent}
\beta\frac{J_0}{2}=J_\mathrm{sf}^\mathrm{F}+J_\mathrm{sf}^\mathrm{N},
\end{equation}
which can also be derived directly by integrating Eq.~(\ref{jspinderivative2}) from $-\infty$ to $\infty$. In fact, $\beta{J}_0/2$ is half of $J_\mathrm{m}(-\infty)$, which is the spin current density produced by the bulk FM layer. It decreases to zero exponentially at $z=\infty$ after the spin relaxation around the interface. During this process, half of $J_\mathrm{m}(-\infty)$, that is $\beta{J}_0/2$, leaks from spin-down channel to spin-up channel in the FM and NM layers.
\begin{figure}
\includegraphics[width=0.48\textwidth]{figure2}
\caption{\label{fig_circuit_fin}(a) Sketch of the FM/NM junction with the spin-selective contact, which is signified by the thick vertical line at $z=0$. The vertical dotted lines indicate the scales of spin-diffusion length in the two layers. (b) An effective circuit of the spin transport (ECST) in the junction shown by (a). The SA capacitors in Fig.~\ref{fig_fin} are connected in parallel with two characteristic resistors to model the leakage due to spin relaxation. The resistances labeled by $r^\mathrm{sf}_\mathrm{F}$ ($r^\mathrm{sf}_\mathrm{N}$) and $r^\mathrm{sd}_\mathrm{F}$ ($r^\mathrm{sd}_\mathrm{N}$) have the same value $2r_\mathrm{F}$ ($2r_\mathrm{N}$). The superscripts ``sf'' and ``sd'' stand for spin flip and spin diffusion [see the text right before Eq.~(\ref{rsd})], respectively. The current densities $J_\mathrm{sf}^\mathrm{F}$, $J_\mathrm{sf}^\mathrm{N}$, and $J_\mathrm{sd}^\mathrm{C}$ are given by Eqs.~(\ref{jleak}) and (\ref{jc}), respectively. The current densities $\beta{J}_0/2$ and $\gamma{J}_0/2$ are generated by two current sources, the FM layer [see Eq.~(\ref{leakcurrent})] and the spin-selective contact [see Eq.~(\ref{efc})], respectively.}
\end{figure}
\subsection{Heat generation}
The spin relaxation also generates heat as the spin-flip scattering causes dissipation of the energy stored in the chemical-potential splitting. The heat generation rate due to spin-flip scattering can be written as
\begin{align}
\Sigma_\mathrm{heat}^\mathrm{F,sf}&=\int_{-\infty}^0\frac{N_{s}\left[\mu_\mathrm{m}(z)\right]^2}
{\tau_\mathrm{sf}^\mathrm{F}}\mathrm{d}z,\\
\Sigma_\mathrm{heat}^\mathrm{N,sf}&=\int_0^\infty\frac{N_{s}\left[\mu_\mathrm{m}(z)\right]^2}
{\tau_\mathrm{sf}^\mathrm{N}}\mathrm{d}z,
\end{align}
for the FM and NM layers, respectively. Using Eqs.~(\ref{deltamuf}) and (\ref{deltamun}), we can write $\Sigma_\mathrm{heat}^\mathrm{F,sf}$ and $\Sigma_\mathrm{heat}^\mathrm{N,sf}$ formally as
\begin{align}
\Sigma_\mathrm{heat}^\mathrm{F,sf}&=r^\mathrm{sf}_\mathrm{F}\left(J_\mathrm{sf}^\mathrm{F}\right)^2
=\frac{1}{2}r_\mathrm{F}\alpha_\mathrm{F}^2J_0^2,\label{leakpf}\\
\Sigma_\mathrm{heat}^\mathrm{N,sf}&=r^\mathrm{sf}_\mathrm{N}\left(J_\mathrm{sf}^\mathrm{N}\right)^2
=\frac{1}{2}r_\mathrm{N}\alpha_\mathrm{N}^2J_0^2,\label{leakpn}
\end{align}
where Eq.~(\ref{jleak}) has been used. Thus the heat generation due to spin relaxation can be modeled as the Joule heat of the resistors in parallel with the SA capacitors.
\subsection{An effective circuit for spin transport}
We have constructed an effective circuit, shown in Fig.~\ref{fig_circuit_fin}(b), to model the spin transport, which takes place on the scales of spin-diffusion length around the interface. We will explain briefly how we build the circuit on the basis of the physical analysis above. Meanwhile, we will also point out the differences between this circuit and those in the Valet-Fert theory.~\cite{vf93}
In Fig.~\ref{fig_circuit_fin}(b), the SA capacitors ($C_\mathrm{sa}^\mathrm{F}$ and $C_\mathrm{sa}^\mathrm{N}$) are connected in parallel with the resistors ($r_\mathrm{F}^\mathrm{sf}$ and $r_\mathrm{N}^\mathrm{sf}$), respectively. The two capacitors model the charge storage in the two spin channels due to spin accumulation as shown by Eq.~(\ref{qs}). The leakage current of the SA capacitors models the electron flow from one spin channel to the other due to spin relaxation as shown by Eqs.~(\ref{jsff}) and (\ref{jsfn}). Comparing Eqs.~(\ref{qs}) and (\ref{jleak}), one can see that the two capacitors have the same voltages as the two resistors, respectively. Therefore, the leakage can be represented equivalently by connecting the two finite resistors in parallel with the SA capacitors, respectively. The voltages here are defined by Eq.~(\ref{ep}) and stand for the effective voltages of the chemical-potential splitting, which may be measured by some spin-resolved optical method. This circuit has one obvious difference from those in the Valet-Fert theory: the capacitors and resistors here are connected between the effective electrodes for the two spin channels instead of the real electrodes.
The current sources of this circuit supply only the spin-polarized part of the current in each spin channel. This feature also makes the circuit different from those in the Valet-Fert theory.~\cite{vf93} There are two sources of the spin-polarized current in the junction: the FM layer with bulk spin asymmetry coefficient $\beta\neq{0}$ and the contact with interfacial spin asymmetry coefficient $\gamma\neq{0}$. They are represented by two independent current sources in the circuit: $\beta{J}_0/2$ and $\gamma{J}_0/2$. The current source $\beta{J}_0/2$ is based on Eq.~(\ref{leakcurrent}) and the explanation following it. The introduction of the current source $\gamma{J}_0/2$ can be understood as follows. If we consider the current driven by the electric field (ef) alone (without spin accumulation) at the spin-selective contact, we have
\begin{equation}
J_\pm^\mathrm{ef}(z_\mathrm{C})=\left(1\mp\gamma\right){J_0}/2,\label{efc}
\end{equation}
for the two spin channels, where $J_0$ flows through $r_+=2r_\mathrm{b}^\ast(1+\gamma)$ and $r_-=2r_\mathrm{b}^\ast(1-\gamma)$ in parallel. Here the spin-up electrons are assumed to be in the minority channel without loss of generality. The spin-selective contact makes $J_\pm^\mathrm{ef}(z_\mathrm{C})$ deviate from its unpolarized value $J_0/2$ by an amount of $\gamma{J}_0/2$, and thus it can be modeled as a spin-current source.
The capacitors are charged by the two current sources. This models the formation of spin accumulation under the drive of the spin current. If the interface resistance is negligible, the two capacitors will be charged simultaneously by the current source $\beta{J}_0/2$ and thus they should be connected in parallel as shown in Fig.~\ref{fig_circuit_fin}(b). When the current source due to the interface is taken into account, we usually hope that it can enhance the voltage (or equivalently spin accumulation) of $C_\mathrm{sa}^\mathrm{N}$. Thus it is reasonable to connect it in the way shown in Fig.~\ref{fig_circuit_fin}(b).
Two additional resistances, $r_\mathrm{F}^\mathrm{sd}$ and $r_\mathrm{N}^\mathrm{sd}$, need to be introduced in the effective circuit to model the heat generation due to the spin diffusion in the FM and NM layers, denoted by $\Sigma_\mathrm{heat}^\mathrm{F,sd}$ and $\Sigma_\mathrm{heat}^\mathrm{N,sd}$, respectively. Using a macroscopic approach based on the Boltzmann equation, we can prove in general that the spin diffusion leads to the same heat generation as the spin relaxation (or the spin-flip scattering) in semi-infinite layers.~\cite{Zhang17cpl} To meet this requirement, we connect $r_\mathrm{F}^\mathrm{sd}$ and $r_\mathrm{N}^\mathrm{sd}$ in series with $r_\mathrm{F}^\mathrm{sf}$ and $r_\mathrm{N}^\mathrm{sf}$, respectively, as shown in Fig.~\ref{fig_circuit_fin}(b). Thus $r_\mathrm{F}^\mathrm{sd}$ and $r_\mathrm{N}^\mathrm{sd}$ should be defined as
\begin{equation}\label{rsd}
r_\mathrm{F}^\mathrm{sd}=r_\mathrm{F}^\mathrm{sf},\qquad r_\mathrm{N}^\mathrm{sd}=r_\mathrm{N}^\mathrm{sf},
\end{equation}
to satisfy $\Sigma_\mathrm{heat}^\mathrm{F,sd}=\Sigma_\mathrm{heat}^\mathrm{F,sf}$ and $\Sigma_\mathrm{heat}^\mathrm{N,sd}=\Sigma_\mathrm{heat}^\mathrm{N,sf}$. Then using Eqs.~(\ref{leakpf}) and (\ref{leakpn}), we can write the spin-dependent heat generation as
\begin{align}
\Sigma_\mathrm{heat}^\mathrm{F}&=\Sigma_\mathrm{heat}^\mathrm{F,sd}+\Sigma_\mathrm{heat}^\mathrm{F,sf}
=r_\mathrm{F}\alpha_\mathrm{F}^2J_0^2,\label{hgf}\\
\Sigma_\mathrm{heat}^\mathrm{N}&=\Sigma_\mathrm{heat}^\mathrm{N,sd}+\Sigma_\mathrm{heat}^\mathrm{N,sf}
=r_\mathrm{N}\alpha_\mathrm{N}^2J_0^2,\label{hgn}
\end{align}
for the FM and NM layers, respectively.
The currents $J_\mathrm{sf}^\mathrm{F}$ and $J_\mathrm{sf}^\mathrm{N}$ given by Eq.~(\ref{jleak}) can also be derived by applying the superposition theorem of electrical circuits to the circuit in Fig.~\ref{fig_circuit_fin}(b). Both current sources drive current to flow from the spin-down to the spin-up channel in the NM layer. However, they drive opposite currents in the FM layer and the net current $J_\mathrm{sf}^\mathrm{F}$ can be negative in some cases. The effective voltage on the resistance $r_\mathrm{F}^\mathrm{sd}$ ($r_\mathrm{N}^\mathrm{sd}$) is also equal to that on $r_\mathrm{F}^\mathrm{sf}$ ($r_\mathrm{N}^\mathrm{sf}$), which is consistent with Eq.~(\ref{ep}). Moreover, using Kirchhoff's current law, one can calculate the current density flowing through the resistance $4r_\mathrm{b}^\ast$
\begin{equation}
J_\mathrm{sd}^\mathrm{C}=\gamma\frac{J_0}{2}-J_\mathrm{sf}^\mathrm{N}
=\alpha_\mathrm{C}\frac{J_0}{2},\label{jc}
\end{equation}
where we have introduced the dimensionless parameter
\begin{equation}\label{alphac}
\alpha_\mathrm{C}=\gamma-\alpha_\mathrm{N}
=\frac{\gamma(r_\mathrm{F}+r_\mathrm{N})-\beta{r}_\mathrm{F}}
{r_\mathrm{F}+r_\mathrm{b}^\ast+r_\mathrm{N}}.
\end{equation}
Then the effective voltage on the resistance $4r_\mathrm{b}^\ast$ can be written as $2(V_\mathrm{c}^\mathrm{N}-V_\mathrm{c}^\mathrm{F})$ or $[\mu_\mathrm{m}(0^+)-\mu_\mathrm{m}(0^-)]/e$. Using Joule's law, we can write the heat generation of the resistance $4r_\mathrm{b}^\ast$ as
\begin{equation}\label{leakpb}
\Sigma_\mathrm{heat}^\mathrm{C}=4r_\mathrm{b}^\ast\left(J_\mathrm{sd}^\mathrm{C}\right)^2
=\frac{\left(2V_\mathrm{c}^\mathrm{N}-2V_\mathrm{c}^\mathrm{F}\right)^2}{4r_\mathrm{b}^\ast}
=r_\mathrm{b}^\ast\alpha_\mathrm{C}^2J_0^2,
\end{equation}
which is equal to the result derived by using a more microscopic approach.~\cite{Zhang17pb} It has been shown, in Ref.~\onlinecite{Zhang17pb}, that $\Sigma_\mathrm{heat}^\mathrm{C}$ results only from the spin diffusion across the interface because the spin relaxation at the interface is negligible.~\cite{vf93} Combining Eqs.~(\ref{hgf}), (\ref{hgn}), and (\ref{leakpb}), we can write the total heat generation due to the spin relaxation and the spin diffusion as
\begin{equation}\label{hgfn}
\Sigma_\mathrm{heat}^\mathrm{FN}=\Sigma_\mathrm{heat}^\mathrm{F}
+\Sigma_\mathrm{heat}^\mathrm{N}+\Sigma_\mathrm{heat}^\mathrm{C}
=r_\mathrm{FN}^\ast{J}_0^2,
\end{equation}
where $r_\mathrm{FN}^\ast$ is defined as $r_\mathrm{FN}^\ast=r_\mathrm{F}\alpha_\mathrm{F}^2+r_\mathrm{N}\alpha_\mathrm{N}^2 +r_\mathrm{b}^\ast\alpha_\mathrm{C}^2$. One can verify that $r_\mathrm{FN}^\ast$ is equal to the spin-coupled interface resistance of the FM/NM junction.~\cite{vf93,Zhang17pb}
\begin{figure}
\includegraphics[width=0.48\textwidth]{figure3}
\caption{\label{fig_circuit_fin2} (a) An equivalent circuit unit (ECU) for the ECST shown in Fig.~\ref{fig_circuit_fin}(b). In the ECU, the current flowing through the various resistances is the total charge current $J_0$, instead of the spin currents in ECST. The equivalent resistances $r_\mathrm{F}^\ast$, $r_\mathrm{N}^\ast$, and $r_\mathrm{C}^\ast$ are given in Eqs.~(\ref{req}) and (\ref{reqc}), respectively. The ESA capacitances $C_\mathrm{sa}^{\mathrm{F},\ast}$ and $C_\mathrm{sa}^{\mathrm{N},\ast}$ are defined in Eq.~(\ref{ceq}). (b) An overall equivalent circuit of the FM/NM junction shown in Fig.~\ref{fig_circuit_fin}(a). The block labeled by ``ECU'' stands for the circuit unit in Fig.~\ref{fig_circuit_fin2}(a). The FM (NM) layer thickness $t_\mathrm{F}$ ($t_\mathrm{N}$) is much larger than the spin diffusion length $l_\mathrm{sf}^\mathrm{F}$ ($l_\mathrm{sf}^\mathrm{N}$).}
\end{figure}
\subsection{An overall equivalent circuit}
To make our results comparable with Rashba's theory and experimental results, we have also constructed an overall equivalent circuit, which includes the ECST shown in Fig.~\ref{fig_circuit_fin}(b). In the equivalent circuit shown in Fig.~\ref{fig_circuit_fin2}, the current source is the total charge current $J_0$ instead of the spin current sources $\beta{J}_0/2$ and $\gamma{J}_0/2$. This makes the circuit similar to those in the Valet-Fert theory. However, we have combine the two spin channels to make the circuit more comparable to the practical situation. The two terminals in Fig.~\ref{fig_circuit_fin2}(b) are supposed to be connected to the real electrodes, which are not shown for simplicity. Thus its voltage can be measured directly in experiments.
We require that the equivalent circuit has the same energy storage $W_\mathrm{sa}^\mathrm{F(N)}$ and heat generation $\Sigma_\mathrm{sa}^\mathrm{F(N)}$ as the ECST, which in turn has the same $W_\mathrm{sa}^\mathrm{F(N)}$ and $\Sigma_\mathrm{sa}^\mathrm{F(N)}$ as the real physical system. We first determine the equivalent resistances by requiring the same heat generation. Using Eqs.~(\ref{hgf}) and (\ref{hgn}), we can set the equivalent resistances to be
\begin{equation}\label{req}
r_\mathrm{F}^\ast=r_\mathrm{F}\alpha_\mathrm{F}^2,\qquad
r_\mathrm{N}^\ast=r_\mathrm{N}\alpha_\mathrm{N}^2,
\end{equation}
for the FM and NM layers, respectively. Moreover, using Eq.~(\ref{leakpb}), we can set the equivalent resistance for $4r_\mathrm{b}^\ast$ in Fig.~\ref{fig_circuit_fin}(b) to be
\begin{equation}\label{reqc}
r_\mathrm{C}^\ast=r_\mathrm{b}^\ast\alpha_\mathrm{C}^2,
\end{equation}
which satisfies $r_\mathrm{FN}^\ast=r_\mathrm{F}^\ast+r_\mathrm{N}^\ast+r_\mathrm{C}^\ast$. Then the equivalent voltages on them can be written as
\begin{equation}\label{veq}
V_\mathrm{F}^\ast=r_\mathrm{F}^\ast{J}_0,\qquad
V_\mathrm{N}^\ast=r_\mathrm{N}^\ast{J}_0.
\end{equation}
For simplicity, we assume that the equivalent spin-accumulation (ESA) capacitance $C_\mathrm{sa}^{\mathrm{F},\ast}$ ($C_\mathrm{sa}^{\mathrm{N},\ast}$) is connected in parallel with the equivalent resistance $r_\mathrm{F}^\ast$ ($r_\mathrm{N}^\ast$). Then, $V_\mathrm{F}^\ast$ ($V_\mathrm{N}^\ast$) is also the voltage on $C_\mathrm{sa}^{\mathrm{F},\ast}$ ($C_\mathrm{sa}^{\mathrm{N},\ast}$). We require that $C_\mathrm{sa}^{\mathrm{F},\ast}$ and $C_\mathrm{sa}^{\mathrm{N},\ast}$ have the same energy storage as $C_\mathrm{sa}^\mathrm{F}$ and $C_\mathrm{sa}^\mathrm{N}$
\begin{equation}\label{esceq}
W_\mathrm{sa}^\mathrm{F}=\frac{1}{2}C_\mathrm{sa}^{\mathrm{F},\ast}
\left(V_\mathrm{F}^\ast\right)^2,\qquad
W_\mathrm{sa}^\mathrm{N}=\frac{1}{2}C_\mathrm{sa}^{\mathrm{N},\ast}
\left(V_\mathrm{N}^\ast\right)^2.
\end{equation}
Substituting Eqs.~(\ref{esc}) and (\ref{veq}) into Eq.~(\ref{esceq}), we can find
\begin{equation}\label{ceq}
C_\mathrm{sa}^{\mathrm{F},\ast}=\frac{T_1^\mathrm{F}}{2r_\mathrm{F}^\ast}
=\frac{C_\mathrm{sa}^\mathrm{F}}{\alpha_\mathrm{F}^2},\qquad
C_\mathrm{sa}^{\mathrm{N},\ast}=\frac{T_1^\mathrm{N}}{2r_\mathrm{N}^\ast}
=\frac{C_\mathrm{sa}^\mathrm{N}}{\alpha_\mathrm{N}^2},
\end{equation}
for the FM and NM layers, respectively. Finally, we can construct an overall equivalent circuit shown in Fig.~\ref{fig_circuit_fin2}(b), which also include the normal resistances $(1-\beta^2)\rho_\mathrm{F}^\ast{t}_\mathrm{F}$, $(1-\gamma^2)r_\mathrm{b}^\ast$, and $\rho_\mathrm{N}^\ast{t}_\mathrm{N}$.
\section{Reinterpretation of the magnetoimpedance\label{discussion}}
As an application of the SA or ESA capacitances, we revisit the magnetoimpedance of the FM/NM junction by using the equivalent circuit shown in Fig.~\ref{fig_circuit_fin2}. The reactance in this configuration has been interpreted by Rashba in terms of the diffusion capacitance $C_\mathrm{diff}$.~\cite{Rashba02a}
The SA capacitance is approximately a constant for time-varying DC and (sinusoidal) AC signals in the regime for the quasi-static approximation to be valid. When the junction is driven by a constant DC current as discussed in the previous section, the spin accumulation is an exponential function in each layer and the SA capacitance is a constant independent on the current density. In general, the SA capacitance will change if the driving current becomes time-dependent, for example, a time-varying DC or AC signal. However, if the charge current $J(t)$ changes with time slowly enough that the solutions to the time-dependent equations are still exponential functions approximately at each moment, they can be replaced by the static solutions for the DC current of the same density $J(t)$. Then the SA capacitance can be assumed to be a constant for simplicity. This is the quasi-static approximation, which has been widely used in circuit analysis and also applied to the diffusion capacitance of a p-n junction.~\cite{Neamen2012}
The valid range for the quasi-static approximation can be determined as follows. Since we are considering the impedance, it is enough to focus on the sinusoidal AC signal. The characteristic time scale of the spin dynamics is the spin relaxation time $T_1^\mathrm{N}$, if we have the relation $T_1^\mathrm{N}>T_1^\mathrm{F}$ as usual.~\cite{zhu08,zhang02} Therefore, the quasi-static approximation is valid if the frequency of the driving current satisfies $\omega\ll{1/T_1^\mathrm{N}}$ or equivalently $\omega{C}_\mathrm{sa}^{\mathrm{N},\ast}\ll{1/\left(2r_\mathrm{N}^\ast\right)}$. In practice, most experiments on time-dependent transport use slowly-varying or low-frequency AC signals.~\cite{Xiao01,Chien2006} In the high-frequency regime, the quasi-static approximation breaks down because the spin dynamics cannot follow the variation of the driving signal and the spin accumulation deviates from the exponential form severely.~\cite{zhu08} In this case, the spin wave-diffusion theory should be used instead. Consequently, the inductive terms becomes remarkable and may be dominant over the capacitive terms if the frequency is high enough.~\cite{zhu09}
If the AC current density is set to be $J(t)=J_0\exp\left(-\mathrm{i}\omega{t}\right)$, the impedances of $C_\mathrm{sa}^{\mathrm{F},\ast}$ and $C_\mathrm{sa}^{\mathrm{N},\ast}$ in Fig.~\ref{fig_circuit_fin2}(a) can be written, respectively, as $\mathrm{i}/\left(\omega{C}_\mathrm{sa}^{\mathrm{F},\ast}\right)$ and $\mathrm{i}/\left(\omega{C}_\mathrm{sa}^{\mathrm{N},\ast}\right)$ using the quasi-static approximation. Using usual circuit theorems, we can calculate the impedance, $Z_\mathrm{ECU}$, of the ECU in Fig.~\ref{fig_circuit_fin2}(a). Then expanding $Z_\mathrm{ECU}$ in terms of $\omega$ and keeping up to the first-order terms, we have
\begin{equation}\label{zecu}
Z_\mathrm{ECU}=r_\mathrm{FN}^\ast+\mathrm{i}X_\mathrm{sa}^\mathrm{FN},
\end{equation}
where $r_\mathrm{FN}^\ast$ is defined in Eq.~(\ref{hgfn}) and the reactance $X_\mathrm{sa}^\mathrm{FN}$ can be written as
\begin{equation}\label{xfn}
X_\mathrm{sa}^\mathrm{FN}=\omega{C}_\mathrm{sa}^{\mathrm{F},\ast}\left(r_\mathrm{F}^\ast\right)^2
+\omega{C}_\mathrm{sa}^{\mathrm{N},\ast}\left(r_\mathrm{N}^\ast\right)^2.
\end{equation}
We will prove that the impedance given in Eq.~(\ref{zecu}) is equal to that derived by Rashba ($\mathcal{Z}_\mathrm{n-eq}$) in the low-frequency regime.~\cite{Rashba02a} Rashba's basic results are outlined with our notations in Appendix~\ref{fnmi}. Using Eqs.~(\ref{alphafn}) and (\ref{alphac}) in the present paper, we can rewrite the real part of the impedance, $R_\mathrm{n-eq}$ given in Eq.~(\ref{rneq}), in a form with more transparent physical interpretation
\begin{equation}
R_\mathrm{n-eq}=r_\mathrm{F}^\ast+r_\mathrm{N}^\ast+r_\mathrm{C}^\ast=r_\mathrm{FN}^\ast,
\end{equation}
which is equal to the real part of $Z_\mathrm{ECU}$ in Eq.~(\ref{zecu}). The imaginary part of the impedance, $\mathrm{Im}(\mathcal{Z}_\mathrm{n-eq})$ in Eq.~(\ref{xfnrashba}), can be rewritten as
\begin{equation}\label{xfn2} \mathrm{Im}(\mathcal{Z}_\mathrm{n-eq})=\omega\frac{T_1^\mathrm{F}}{2r_\mathrm{F}}
(r_\mathrm{F}\alpha_\mathrm{F})^2+
\omega\frac{T_1^\mathrm{N}}{2r_\mathrm{N}}(r_\mathrm{N}\alpha_\mathrm{N})^2,
\end{equation}
where we have used Eqs.~(\ref{alphafn}) and (\ref{alphac}). Then by using the SA capacitances defined in Eq.~(\ref{sc}), we can rewrite Eq.~(\ref{xfn2}) as
\begin{equation}\label{xfns}
\mathrm{Im}(\mathcal{Z}_\mathrm{n-eq})=\omega{C}_\mathrm{sa}^\mathrm{F}
\left(r_\mathrm{F}\alpha_\mathrm{F}\right)^2
+\omega{C}_\mathrm{sa}^\mathrm{N}\left(r_\mathrm{N}\alpha_\mathrm{N}\right)^2.
\end{equation}
Finally, by using Eqs.~(\ref{req}) and (\ref{ceq}), we can prove that $\mathrm{Im}(\mathcal{Z}_\mathrm{n-eq})$ is equal to $X_\mathrm{sa}^\mathrm{FN}$ in Eq.~(\ref{xfn}). Therefore, the impedance $Z_\mathrm{ECU}$ is the same as $\mathcal{Z}_\mathrm{n-eq}$ derived by Rashba and the ECU shown in Fig.~\ref{fig_circuit_fin2}(a) can also be used to interpret $\mathcal{Z}_\mathrm{n-eq}$.
In comparison to Rashba's theory, we wrote the reactance of the FM/NM junction in terms of the SA or ESA capacitances instead of the diffusion capacitance $C_\mathrm{diff}$ defined in Eq.~(\ref{cdiff}). One benefit is that the form of the SA or ESA capacitances resemble obviously that of the diffusion capacitance in a p-n junction. More importantly, our results revealed the form of the energy storage associated with the reactance. In the low-frequency limit, the energy storage associated with $\mathcal{Z}_\mathrm{n-eq}$ is given by Eq.~(\ref{esc}) or (\ref{esceq}), which is the splitting energy of the spin-dependent chemical potentials instead of the electrostatic energy. Our results also suggest a way to calculate the reactance without using the time-dependent equations.
One characteristic of the reactance, $X_\mathrm{sa}^\mathrm{FN}$, is that it is zero if $\beta=0$ and $\gamma=0$. This means that a spin-current source is indispensable to the reactance we consider. Multilayers with $\beta=0$ and $\gamma=0$ may still have parasitic capacitance in parallel with resistance as long as the material resistivity varies from layer to layer. It results from the induced charge accumulation at the interfaces, which adjusts the electric field in different layers to maintain a homogeneous current across the entire multilayer. The energy stored in this kind of capacitance is essentially electrostatic energy and it will not be discussed in this paper.
It is worthwhile to compare the magnetoimpedance in the present paper with that reviewed in Ref.~\onlinecite{Phan2008}. Although the same term ``magnetoimpedance'' is used in the two papers, their configurations are different in the following three ways. First, Ref.~\onlinecite{Phan2008} discussed the change of impedance with an applied magnetic field, whereas such an external field is not essential in our problem and the change of impedance here depends on the spin polarization as shown by Eq.~(\ref{xfns}). Second, the impedance in Ref.~\onlinecite{Phan2008} is mostly inductive. On the contrary, our impedance is capacitive. Finally, Ref.~\onlinecite{Phan2008} was mainly concerned with the impedance of a single FM conductor. However, we are interested in magnetic multilayers, such as an FM/NM junction. Therefore, the imaginary part of the magnetoimpedance reviewed in Ref.~\onlinecite{Phan2008} arises from quite different origin and is beyond the scope of the current paper.
\section{Conclusions\label{conclusion}}
In summary, we model the two spin channels of a metal or degenerate semiconductor as the two plates of a capacitor. By using a method similar to that for studying quantum capacitance, we show that the SA capacitance can be introduced for each layer to measure its ability to store spins rather than charge and a spatial charge storage is not essential to it. The energy stored in the SA capacitors is the splitting energy of the spin-dependent chemical potentials instead of the electrostatic energy. The leakage of the SA capacitor models the spin relaxation. The heat generation due to the spin relaxation can be written formally as the Joule heat of the resistance in parallel with the SA capacitor. Equivalent circuits can be constructed to give a transparent interpretation of the model for a typical FM/NM junction.
As an application of the model, we derive the magnetoimpedance of the FM/NM junction directly from the equivalent circuit using the quasi-static approximation. We rewrite the reactance in terms of the SA capacitances, whose formula is obviously similar to that of the diffusion capacitance in a p-n junction. Our results also reveal the form of the energy storage associated with the reactance. We expect that the SA capacitance can be generalized to other structures, such as MTJs, and used to interpret the ongoing experiments on the magnetoimpedance.
\begin{acknowledgments}
We thank Prof.~H. C. Schneider and Y. Suzuki for fruitful discussions. This work was supported by the National Natural Science Foundation of China under Grant Nos~11404013, 11605003, 61405003, and 11474012.
\end{acknowledgments}
|
3,212,635,537,970 | arxiv | \section{Introduction}\label{intro}
\noindent
Let~$M$ be an even-dimensional manifold endowed with a complex structure $J$ and a symplectic
form~$F$. When~$J$ and~$F$ are compatible, i.e. the symplectic form $F$ is $J$-invariant, and the associated metric~$g$ is Riemannian, then
the manifold $(M,J,F)$ is K\"ahler. In the compact K\"ahler case, the positive-definiteness of~$g$
imposes strong topological conditions
on the manifold~$M$; for instance, its Betti numbers
$b_{2k+1}(M)$ are even and the manifold is formal \cite{DGMS}.
Since
there are compact complex manifolds
with no K\"ahler metrics, many efforts have been done in understanding the properties of manifolds endowed with a pair $(J,F)$ satisfying weaker conditions than those of a K\"ahler structure.
On the one hand, if we drop the closedness condition for the 2-form $F$, other special Hermitian structures arise, such as the
strong K\"ahler with torsion (SKT) or balanced Hermitian structures. On the other hand, if we no longer require the positive definiteness of the metric $g$ but preserve the compatibility of the symplectic form $F$ with $J$,
then a structure called \emph{pseudo-K\"ahler} is obtained.
In this paper we focus on compact manifolds~$M$ endowed with a
pseudo-K\"ahler structure, i.e. a pair $(J,F)$ where~$J$ is a complex structure
and~$F$ is a non-degenerate closed $2$-form such that $F(JU,JV)=F(U,V)$,
for any smooth vector fields~$U,V$ on~$M$. This is equivalent to $J$ being parallel, i.e. $\nabla J=0$, where~$\nabla$ is the Levi-Civita connection of the pseudo-Riemmanian metric $g(U,V)=F(JU,V)$ (see \cite{AFGM}).
If the real dimension of~$M$ is~$2n$, then the pseudo-K\"ahler metric has signature $(2k,2n-2k)$, where~$k=n$ corresponds to the K\"ahler case.
In other words, we have a compact complex manifold $X=(M,J)$ with a symplectic form~$F$ of bidegree $(1,1)$ with respect to the complex structure~$J$.
Notice that there are many compact pseudo-K\"ahler manifolds not admitting any K\"ahler metric, the simplest example being the compact complex surface known as the Kodaira-Thurston manifold~\cite{Thu}.
Pseudo-K\"ahler structures appear in relation to other interesting structures on manifolds.
For instance, compact complex homogeneous manifolds endowed with a pseudo-K\"ahler structure are classified in \cite{DGuan, Guan}.
Pseudo-K\"ahler Einstein metrics on compact complex surfaces
are studied by Petean \cite{Petean}.
More recently, it is proved in~\cite{KL} that there is a natural pseudo-K\"ahler structure on the
universal intermediate $G_2$-Jacobian $\mathcal{J}$
of the moduli space of torsion-free $G_2$-structures
on a fixed compact $7$-manifold.
Furthermore, any $4n$-dimensional hypersymplectic manifold
in particular has
a neutral Calabi-Yau structure
\cite{Hitchin} (see also \cite{DS}), which is a special type of pseudo-K\"ahler structure
whose underlying metric is Ricci-flat and has signature $(2n,2n)$.
Although several aspects in pseudo-K\"ahler geometry have been investigated, the stability of these structures under small holomorphic deformations of the complex manifold is, to our knowlegde, only known in the positive-definite case. Indeed, if $X$ is a compact K\"ahler manifold, then any sufficiently small deformation of $X$ admits a K\"ahler metric due to a well-known result by Kodaira and Spencer~\cite{KS60}. In this paper we focus on the stability properties of compact
pseudo-K\"ahler manifolds, as well as manifolds with related neutral (K\"ahler and Calabi-Yau) metrics.
We next explain in more detail the contents of the paper.
In Section~\ref{general-results} we firstly construct compact pseudo-K\"ahler manifolds of complex dimension $n\geq 3$ that are not stable under small holomorphic deformations of the
complex structure. This result motivates the study of sufficient conditions
for the stability of the pseudo-K\"ahler property.
Our stability result involves the Bott-Chern cohomology of complex manifolds. We prove that if~$X_0$ is pseudo-K\"ahler and the upper-semi-continuous function $t\mapsto h^{1,1}_{\rm BC}(X_t)$ is constant, then the compact complex manifold $X_t$ admits a pseudo-K\"ahler metric for any small enough $t$ (see Proposition~\ref{BC-constant}). Combining this with a result of Teleman~\cite{Tel06} on the complex invariant
$\Delta^2$
introduced by Angella and Tomassini in~\cite{AT4},
we prove in Theorem~\ref{stability-p-K-surfaces} that
compact pseudo-K\"ahler surfaces are stable.
The results of Section~\ref{general-results} are illustrated with explicit constructions of pseudo-K\"ahler nilmanifolds and solvmanifolds, together with their small holomorphic deformations. In particular, we prove in Proposition~\ref{iwasawa} that the Iwasawa manifold and its small deformations do not admit any pseudo-K\"ahler metric. In contrast,
the holomorphically parallelizable Nakamura manifold $X$ is pseudo-K\"ahler, as proved by Yamada in~\cite{Yamada-1}, and there exists a small deformation $X_{\mathbf{t}}$ of $X$ admitting pseudo-K\"ahler metrics for every $\mathbf{t}$
(see Proposition~\ref{nakamura}).
Section~\ref{neutralCY} is devoted to a special class
of pseudo-K\"ahler manifolds, namely, neutral Calabi-Yau manifolds.
They are neutral K\"ahler manifolds, i.e. manifolds with even complex dimension $2m$ and a metric $g$
of signature $(2m,2m)$, that additionally have
a nowhere vanishing form $\Phi$ of bidegree $(2m,0)$
satisfying $\nabla \Phi=0$, where $\nabla$ is the Levi-Civita connection of $g$. Observe that neutral Calabi-Yau manifolds are Ricci-flat.
In Section~\ref{neutralCY-surfaces} we prove the stability of neutral K\"ahler and neutral Calabi-Yau structures on compact complex surfaces.
However,
our Theorem~\ref{deform-pK} shows
that such structures are not stable on compact complex manifolds of
any complex dimension $n\geq 4$.
This result is in deep contrast to the case of K\"ahler Calabi-Yau manifolds, whose deformation space is unobstructed by the well-known Bogomolov-Tian-Todorov theorem~\cite{Bogomolov,Tian,Todorov}.
For the proof of Theorem~\ref{deform-pK} we first construct a complex nilmanifold $X$ of complex dimension~$4$ with (non-flat) neutral Calabi-Yau structures (see Proposition~\ref{neutralCY-nil}). An interesting class of neutral Calabi-Yau nilmanifolds, called Kodaira manifolds, was constructed by Fino, Pedersen, Poon and S{\o}rensen in \cite{FPPS}. It is worth to note that the complex structure of the nilmanifold $X$ is of a very different kind. Indeed, $X$ has the special feature that it is 4-step and the center of its underlying Lie algebra has minimal dimension, which implies that $X$ is far from being the total space of a torus bundle over a torus. Moreover, in Proposition~\ref{pK-en-SnN} we show that
$X$ provides counterexamples to a conjecture in~\cite{CFU} about the type
of invariant complex structures on nilmanifolds that admit compatible pseudo-K\"ahler metrics.
By appropriately deforming the complex nilmanifold $X$, we construct
a holomorphic family of compact complex manifolds
$\{X_\mathbf{t}\}_{\mathbf{t}\in\Delta}$, with $X_0=X$,
showing that the neutral Calabi-Yau and
neutral K\"ahler properties
are unstable.
In Section~\ref{pseudo-HS} we consider pseudo-Hermitian-symplectic structures, which are an indefinite version of Hermitian-symplectic structures. This geometry naturally arises from small deformations of pseudo-K\"ahler manifolds (see Proposition~\ref{openness-p-HS}). Motivated by a question of Streets and Tian in the positive-definite case~\cite[Question 1.7]{ST}, we prove that there are
compact complex manifolds admitting pseudo-Hermitian-symplectic structures but
no pseudo-K\"ahler metrics.
\section{Pseudo-K\"ahler manifolds}\label{general-results}
\noindent
This section starts showing that compact pseudo-K\"ahler manifolds of complex dimension $n\geq 3$ are not stable under small holomorphic deformations of the
complex structure. This motivates the study of conditions under which a sufficiently small deformation $X_{\mathbf t}$ of a pseudo-K\"ahler manifold $X$ is again pseudo-K\"ahler. Here,
we present a stability result related to the
Bott-Chern cohomology of complex manifolds.
Several explicit constructions of pseudo-K\"ahler nilmanifolds and solvmanifolds
are provided along the section,
illustrating the behaviour of the pseudo-K\"ahler property under their small holomorphic deformations.
In the final part of the section,
we focus on the behaviour of
small deformations of
pseudo-K\"ahler compact complex surfaces.
\smallskip
Let us recall that a \emph{complex analytic family}, or \emph{holomorphic family}, of compact complex manifolds is a proper holomorphic submersion $\pi\colon {\mathcal X} \longrightarrow \Delta$ between two complex manifolds~${\mathcal X}$ and~$\Delta$~\cite{Kod86}. This implies that the fibres $X_{\mathbf{t}} = \pi^{-1}(\mathbf{t})$ are compact complex manifolds of the same dimension. By a classical result of Ehresmann \cite{Ehr47}, any such family is locally $\mathcal{C}^\infty$ trivial (globally, if $\Delta$ is contractible), so all the fibres $X_{\mathbf{t}}$ have the same underlying $\mathcal{C}^\infty$ manifold $M$.
Consequently, the holomorphic family can be viewed as a collection $\{X_{\mathbf{t}}\}_{\mathbf{t}\in \Delta}$ of complex manifolds $X_{\mathbf{t}}=(M, J_{\mathbf{t}})$, where $J_{\mathbf{t}}$ is the complex structure of $X_{\mathbf{t}}$ for $\mathbf{t}\in \Delta$.
A classical result of Kodaira and Spencer~\cite{KS60} asserts that
the property of being a compact K\"ahler manifold is stable
under holomorphic deformations. In the following section we prove that such a stability result cannot be extended to compact pseudo-K\"ahler manifolds.
\subsection{Instability of the pseudo-K\"ahler property}\label{ii-1}
Here we prove that compact pseudo-K\"ahler manifolds of complex dimension $n\geq 3$ are
in general not stable under small deformations of the
complex structure.
In the proof we will consider
an appropriate holomorphic family consisting of complex nilmanifolds.
We recall that a nilmanifold is a compact quotient $N=\nilm$ of a
connected and
simply connected nilpotent Lie group~$G$
by a lattice $\Gamma$ of maximal rank in~$G$.
A \emph{complex nilmanifold} $X=(N,J)$ is a nilmanifold $N=\nilm$ endowed with an invariant complex structure~$J$, i.e.~$J$ comes from a left-invariant complex structure on~$G$ by passing to the quotient~$\nilm$.
(For results on complex nilmanifolds see for instance \cite{Ang-libro,COUV,Salamon} and the references therein.)
\begin{proposition}\label{deform-pK-dim6}
There is a holomorphic family of compact complex manifolds
$\{X_\mathbf{t}\}_{\mathbf{t}\in\Delta}$ of complex dimension $3$,
where $\Delta=\{ \mathbf{t}\in \mathbb{C}\mid |\mathbf{t}|< 1 \}$, such that:
\begin{enumerate}
\item[\rm (i)] $X_0$ is a pseudo-K\"ahler manifold;
\item[\rm (ii)] $X_\mathbf{t}$ does not admit pseudo-K\"ahler metrics for any
$\mathbf{t}\not= 0$.
\end{enumerate}
Hence, the pseudo-K\"ahler property is not stable
under deformations of the complex structure.
\end{proposition}
\begin{proof}
Let $X$ be the complex nilmanifold defined by the following complex structure equations
\begin{equation}\label{ecccus}
d\omega^1 = d\omega^2=0, \quad \ d\omega^3= \omega^{1\bar{2}}.
\end{equation}
Here $\omega^k$ has bidegree (1,0) and $\omega^{j\bar{k}}$ denotes the (1,1)-form $\omega^{j}\wedge\overline{\omega^{k}}$.
Observe that the compact complex manifold $X$ has pseudo-K\"ahler metrics. For instance,
the 2-form
$$
F= i\,\omega^{1\bar{1}} + \omega^{2\bar{3}} - \omega^{3\bar{2}}
$$
satisfies that $F=\bar F$, i.e. the 2-form $F$ is real, and $F^3 \not= 0$, i.e. it is non-degenerate. The form $F$ is compatible with the complex structure $J$ of $X$ because it is of pure type (1,1) on $X$.
Moreover, from \eqref{ecccus} we get that $dF=0$, so the form $F$ is symplectic.
Thus, $F$ defines a pseudo-K\"ahler metric on $X$.
Notice that the (0,1)-form $\omega^{\bar{3}}$ is $\overline{\del}$-closed on $X$.
We will use the Dolbeault cohomology class
$[\omega^{\bar{3}}] \in H^{0,1}_{\db}(X)$
to perform an appropriate holomorphic deformation of $X$.
For each $\mathbf{t}\in \mathbb{C}$ such that $|\mathbf{t}|<1$, let us consider the complex nilmanifold $X_{\mathbf{t}}$
defined by the following complex basis of (1,0)-forms:
\begin{equation}\label{rel-tt}
\omega_{\mathbf{t}}^1:=\omega^1,\quad \omega_{\mathbf{t}}^2:=\omega^2,\quad \omega_{\mathbf{t}}^3:=\omega^3 + \mathbf{t}\,\omega^{\bar{3}}.
\end{equation}
It follows from \eqref{ecccus} and \eqref{rel-tt} that the complex structure equations for $X_{\mathbf{t}}$ are
\begin{equation}\label{ecccus-t}
d\omega_{\mathbf{t}}^1=d\omega_{\mathbf{t}}^2=0,\quad \
d\omega_{\mathbf{t}}^3= \omega_{\mathbf{t}}^{1\bar2} - \mathbf{t}\,\omega_{\mathbf{t}}^{2\bar1}.
\end{equation}
In order to prove that $X_{\mathbf{t}}$ has no pseudo-K\"ahler metrics for any $\mathbf{t}\not= 0$, we use
Nomizu's theorem \cite{Nomizu} for the de Rham cohomology of nilmanifolds together with the symmetrization
process introduced in \cite{Bel}. More concretely, we take into account that, by \cite[Remark 5]{U}, any closed $k$-form
$\alpha$ on a nilmanifold is cohomologous to the invariant $k$-form $\widetilde\alpha$
obtained by the symmetrization process.
Fix any ${\mathbf{t}}\in\Delta$.
Since the complex structure on $X_{\mathbf{t}}$ is invariant,
if there exists a pseudo-K\"ahler structure~$\Theta$ on $X_{\mathbf{t}}$, then the
form~$\widetilde\Theta$,
obtained by symmetrization of $\Theta$, would be an invariant closed real 2-form of bidegree (1,1)
such that $[\widetilde\Theta]=[\Theta]$ in the second de Rham cohomology group $H_{\rm dR}^2(X_{\mathbf{t}};\mathbb{R})$.
In particular, this would imply $[{\widetilde\Theta}^3]=[\Theta]^3 \not=0$ in the de Rham cohomology group $H_{\rm dR}^6(X_{\mathbf{t}};\mathbb{R})$, due to the non-degeneracy of $\Theta$ on
the compact complex manifold
$X_{\mathbf{t}}$.
However, for $\mathbf{t}\not= 0$, it follows from \eqref{ecccus-t} that any invariant closed (1,1)-form $\zeta$ on $X_{\mathbf{t}}$
satisfies
$$
\zeta\ \in\ \C\langle \omega_{\mathbf{t}}^{1\bar1},\ \omega_{\mathbf{t}}^{1\bar2},\ \omega_{\mathbf{t}}^{2\bar1},\ \omega_{\mathbf{t}}^{2\bar2} \rangle,
$$
which implies that $\zeta^3=\zeta\wedge\zeta\wedge\zeta=0$, i.e. $\zeta$ is degenerate.
In conclusion, there are no (invariant or not) pseudo-K\"ahler metrics on $X_{\mathbf{t}}$ for $\mathbf{t}\not= 0$.
\end{proof}
\begin{remark}\label{remark-ii-1}
{\rm
Let us consider $Y_\mathbf{t}=X_\mathbf{t}\times {\mathbb T}^m$, where
$\{X_\mathbf{t}\}_{\mathbf{t}\in\Delta}$
is the holomorphic family given in Proposition~\ref{deform-pK-dim6}
and ${\mathbb T}^m$ denotes the $m$-dimensional complex torus endowed with its standard K\"ahler metric.
A similar argument as the one in the proof of the previous proposition shows that $Y_\mathbf{t}$ does not admit any pseudo-K\"ahler metric for
$\mathbf{t}\not= 0$. Hence, the pseudo-K\"ahler property is not stable in any complex dimension $n\geq 3$
(see Theorem~\ref{deform-pK} for an irreducible example in complex dimension 4).
In contrast, we will prove in Section~\ref{surfaces} that compact pseudo-K\"ahler surfaces are stable.
}
\end{remark}
The following example illustrates that, although the pseudo-K\"ahler property is in general not stable, one can find certain deformations
where the existence of pseudo-K\"ahler metrics is preserved.
\begin{example}\label{ejemplo2-full}
{\rm
Let us consider the differentiable family $\{X_t\}_{t\in (-1,1)}$ of compact complex nilmanifolds
defined by the complex structure equations
$$
d\omega_t^1=0,\quad d\omega_t^2=\omega_t^{1\bar{1}},\quad d\omega_t^3= \omega_t^{12} + t\,\omega_t^{1\bar2}.
$$
We note that this is a differentiable family of deformations of the compact complex nilmanifold $X=X_0$ determined by the equations
\begin{equation}\label{X-gen-ecus}
d\omega^1=0,\quad d\omega^2=\omega^{1\bar{1}},\quad d\omega^3= \omega^{12}.
\end{equation}
The manifolds $X_t$ in this family are all pseudo-K\"ahler since, for instance,
$F_t=i\,(\omega_t^{1\bar{3}}+\omega_t^{3\bar{1}}) +i\,(1+t)\,\omega_t^{2\bar{2}}$
defines
a pseudo-K\"ahler metric on $X_t$ for every $t\in (-1,1)$.
}
\end{example}
Hence, one would like to study additional conditions under which the pseudo-K\"ahler property becomes stable. We next establish a condition in terms of the Bott-Chern cohomology.
\subsection{Bott-Chern cohomology and stability of pseudo-K\"ahler manifolds}\label{ii-5}
Here, we show that the stability of the pseudo-K\"ahler property
is closely related to the variation of the Bott-Chern cohomology.
We recall that the Bott-Chern and the Aeppli cohomologies \cite{Aeppli, Bott-Chern}
(see also~\cite{Ang-libro}) of a compact complex manifold $X$
are defined, respectively, by
$$
H^{\bullet,\bullet}_{\rm BC}(X):=\frac{\ker\partial\cap\ker\overline{\del}}{\imm\partial\overline{\del}}
\qquad
\text{ and }
\qquad
H^{\bullet,\bullet}_{\rm A}(X):=\frac{\ker\partial\overline{\del}}{\imm\partial+\imm\overline{\del}} .
$$
The dimensions of these cohomology groups will be denoted by $h^{p,q}_{\rm BC}(X) = \dim_{\C} H^{p,q}_{\rm BC}(X)$ and $h^{p,q}_{\rm A}(X) = \dim_{\C} H^{p,q}_{\rm A}(X)$.
Suppose that $X$ admits a pseudo-K\"ahler metric
defined by $F$.
Since the real form $F^k$ is closed and has bidegree $(k,k)$, it defines a Bott-Chern class $[F^k] \in H^{k,k}_{\rm BC}(X)$ for any $1 \leq k \leq n$.
Moreover:
\begin{lemma}\label{necessary-BC-pK-cond}
Let $X$ be a compact complex manifold with $\dim_{\mathbb C} X=n$.
If $X$ admits a pseudo-K\"ahler metric, then
$h^{k,k}_{\rm BC}(X)\geq 1$
for any $1 \leq k \leq n$.
\end{lemma}
\begin{proof}
Let us consider a pseudo-K\"ahler metric on $X$ defined by $F$
and suppose that the class $[F^k] \in H^{k,k}_{\rm BC}(X)$ is zero for some $1 \leq k \leq n$.
This fact implies that $F^k=\partial\overline{\del} \beta$ for some form $\beta \in \Omega^{k-1,k-1} (X)$, so
$$
F^n=F^{n-k} \wedge F^k = F^{n-k} \wedge \partial\overline{\del} \beta = \partial (F^{n-k} \wedge \overline{\del} \beta)
= d (F^{n-k} \wedge \overline{\del} \beta),
$$
which contradits the non-degeneracy of the form $F$.
\end{proof}
The following proposition is an extension
of the Kodaira-Spencer stability result \cite[Theorem~15]{KS60} for K\"ahler metrics.
We recall that, for every $(p,q)$, the function $t\mapsto h^{p,q}_{\rm BC}(X_t)$ is upper-semi-continuous~\cite{schweitzer}.
\begin{proposition}\label{BC-constant}
Let $X$ be a compact pseudo-K\"ahler manifold, and let $\{X_t\}_{t\in(-\varepsilon,\varepsilon)}$ be a differentiable family of deformations of $X=X_0$,
where $\varepsilon>0$. Suppose that the
upper-semi-continuous
function $t\mapsto h^{1,1}_{\rm BC}(X_t)$ is constant.
Then, the compact complex manifold $X_t$ admits a pseudo-K\"ahler metric for any $t$ close enough to~$0$.
\end{proposition}
\begin{proof}
Let $\{\omega_t\}_{t}$ be a family of Hermitian metrics on $X_t$ for $t\in(-\varepsilon,\varepsilon)$.
For each $t$, we consider the Bott-Chern Laplacian $\Delta^{\rm BC}_t$ associated to the Hermitian metric $\omega_t$ on $X_t$ and the corresponding Green operator~$G_t$~\cite{schweitzer}. Denote by $H_t\colon \Omega^{*}_{\C}(X_t) \to \ker\Delta^{\rm BC}_t$ the projection onto the space of harmonic forms with respect to $\Delta^{\rm BC}_t$ (and with respect to the
$L^2_{\omega_t}$-orthogonal decomposition induced by $\omega_t$~\cite{schweitzer}),
and by $\pi^{1,1}_t \colon \Omega^{*}_{\C}(X) \to \Omega^{1,1}(X_t)$
the projection of the space of complex forms on $X$ onto the space of $(1,1)$-forms on $X_t$.
Now, for any $t\in(-\varepsilon,\varepsilon)$, the operator $\Pi_t$ defined by
$$
\Pi_t \;:=\; \left( H_t + \partial_t\overline{\del}_t\left(\partial_t\overline{\del}_t\right)^{*_t}G_t \right) \circ \pi^{1,1}_t \colon \Omega^{*}_{\C}(X) \longrightarrow \ker\partial_t\cap\ker\overline{\del}_t \;,
$$
gives the projection of the space of complex forms on $X$ onto the space of $\partial_t$-closed and $\overline{\del}_t$-closed $(1,1)$-forms on the compact complex manifold $X_t$. Here, $*_t$ is the Hodge-operator with respect to the Hermitian metric $\omega_t$ on $X_t$.
Since the function $t\mapsto h^{1,1}_{\rm BC}(X_t)$ is constant, by elliptic theory \cite[Theorem 7]{KS60} one has that the family $\{\Pi_t\}_{t}$ is smooth in $t$. Let $F_0$ be a pseudo-K\"ahler metric on $X_0=X$.
For $t\in(-\varepsilon,\varepsilon)$, we set
$$
F_t \;:= \frac{\Pi_t F_0 + \overline{\Pi_t F_0}}{2}.
$$
Then, the family $\{F_t\}_t$ is smooth in $t$,
and each $F_t$ is a real (1,1)-form on $X_t$ which is $d$-closed, because it is closed by $\partial_t$ and $\overline{\del}_t$.
Since $F_0^n$ is a nowhere vanishing $(n,n)$-form, we have that $F_t$ is non-degenerate for $t$ close enough to $0$.
Therefore, the form $F_t$ defines a pseudo-K\"ahler metric on the compact complex manifold $X_t$ for any $t$ close enough to $0$.
\end{proof}
\begin{example}\label{ejemplo1-BC}
{\rm
We here show that for the holomorphic family
$\{X_{\mathbf{t}}\}_{\mathbf{t}\in\Delta}$
constructed in the proof of Proposition~\ref{deform-pK-dim6},
the upper-semi-continuous function $\mathbf{t}\mapsto h^{1,1}_{\rm BC}(X_{\mathbf{t}})$ is not constant.
According to Proposition~\ref{BC-constant}, this fact gives a reason for the instability
of the pseudo-K\"ahler property along the deformation
of $X_0$.
Indeed,
for the compact complex manifold $X_0$ we have
$$
H^{1,1}_{\rm BC}(X_0)= \langle \,[\,i\omega^{1\bar{1}}\,],\,[\,i\omega^{2\bar{2}}\,],\,
[\,\omega^{1\bar{2}}-\omega^{2\bar{1}}\,],\,[\,i\,(\omega^{1\bar{2}}+\omega^{2\bar{1}})\,],\,
[\,\omega^{2\bar{3}}-\omega^{3\bar{2}}\,],\,
[\,i\,(\omega^{2\bar{3}}+\omega^{3\bar{2}})\,] \, \rangle,
$$
and,
for any ${\mathbf{t}} \in \Delta-\{0\}$, the Bott-Chern cohomology group of bidegree (1,1) of $X_{\mathbf{t}}$ is
$$
H^{1,1}_{\rm BC}(X_{\mathbf{t}}) = \langle \,[\,i\omega_{\mathbf{t}}^{1\bar{1}}\,],\,[\,i\omega_{\mathbf{t}}^{2\bar{2}}\,],\,
[\,\omega_{\mathbf{t}}^{1\bar{2}}-\omega_{\mathbf{t}}^{2\bar{1}}\,],\,[\,i\,(\omega_{\mathbf{t}}^{1\bar{2}}+\omega_{\mathbf{t}}^{2\bar{1}})\,] \, \rangle.
$$
Therefore, $h^{1,1}_{\rm BC}(X_0)=6$, and $h^{1,1}_{\rm BC}(X_{\mathbf{t}})=4$, for any ${\mathbf{t}} \not=0$.
}
\end{example}
\begin{example}\label{ejemplo2-BC}
{\rm
Let us consider the differentiable family $\{X_t\}_{t\in (-1,1)}$ of compact pseudo-K\"ahler manifolds given in Example~\ref{ejemplo2-full}.
The Bott-Chern cohomology group of bidegree (1,1) of~$X_t$ is
$$
H^{1,1}_{\rm BC}(X_{t}) = \langle \,[\,i\,\omega_{t}^{1\bar{1}}\,],\, [\,\omega_{t}^{1\bar{2}}-\omega_{t}^{2\bar{1}}\,],\,[\,i\,(\omega_{t}^{1\bar{2}}+\omega_{t}^{2\bar{1}})\,],\,
[\,i\,(\omega_t^{1\bar{3}}+\omega_t^{3\bar{1}}) +i\,(1+t)\,\omega_t^{2\bar{2}}\,]\,\rangle.
$$
Therefore, $h^{1,1}_{\rm BC}(X_{t})=4$ for every $t$, i.e. the function $t\mapsto h^{1,1}_{\rm BC}(X_t)$ is constant.
}
\end{example}
We next show that the result in Example~\ref{ejemplo2-BC} can indeed be extended to any family
of deformations of~$X=X_0$.
\begin{proposition}\label{example-general}
Let $X$ be the
compact pseudo-K\"ahler manifold in Example~$\ref{ejemplo2-full}$ defined by \eqref{X-gen-ecus},
and
let $\{X_t\}_{t\in(-\varepsilon,\varepsilon)}$ be any differentiable family of deformations of $X=X_0$, where $\epsilon>0$.
Then, $X_t$ admits a pseudo-K\"ahler metric for every $t$ close enough to $0$.
\end{proposition}
\begin{proof}
Since $X$ is a nilmanifold endowed with an invariant complex structure,
one has by~\cite[Theorem 2.6]{rollenske} that the complex structure $J_t$
of any sufficiently small deformation $X_t$ of $X$ is also invariant.
The dimension of the Bott-Chern cohomology groups of any invariant complex structure $J_t$
is given in \cite[Table 2]{AFR} and \cite[Appendix 6]{LUV}. Note that the Lie algebra underlying the nilmanifold~$X$ is precisely $\frh_{15}$, and $h^{1,1}_{\rm BC}(\frh_{15},J)\geq 4$ for any complex structure $J$ on $\frh_{15}$.
Since $h^{1,1}_{\rm BC}(X_{t})=h^{1,1}_{\rm BC}(\frh_{15},J_t)\geq 4$ varies upper-semi-continuously along differentiable families~\cite{schweitzer}, for any $t$ close enough to $0$ we have
$$
4=h^{1,1}_{\rm BC}(X_{0}) \geq h^{1,1}_{\rm BC}(X_{t})\geq 4,
$$
and thus $h^{1,1}_{\rm BC}(X_{t})= 4$.
Hence, the function $t\mapsto h^{1,1}_{\rm BC}(X_t)$ is constant and, by Proposition~\ref{BC-constant},
$X_t$ admits a pseudo-K\"ahler metric for any $t$ close enough to $0$.
\end{proof}
We observe that the condition on $h_{\text{BC}}^{1,1}$ given in Proposition~\ref{BC-constant}
is sufficient but not necessary. To
illustrate this fact, we next study the existence of pseudo-K\"ahler metrics on the small
deformations of the well-known Nakamura manifold. This manifold is a
holomorphically parallelizable solvmanifold, i.e. a compact quotient $X=\Gamma\backslash G$
where~$G$ is a simply connected \emph{complex}
solvable Lie group and~$\Gamma$ a lattice in~$G$.
Let $G$ be the simply-connected complex solvable Lie group given by
$$
G =\left\{
\begin{pmatrix}e^{z_1}&0&0 &z_2 \\
0&e^{-z_1}&0& z_3 \\
0&0& 1 & z_1 \\
0& 0& 0& 1 \end{pmatrix} \ \mid\ z_1,z_2,z_3 \in \mathbb{C}\right\},
$$
i.e. $G$ is the semi-direct product $G = \mathbb{C} \ltimes_{\varphi} \mathbb{C}^2$, where
$(z_2,z_3)$ are the coordinates on $\mathbb{C}^2$ and
$$
\varphi(z_1) =
\begin{pmatrix}e^{z_1}&0\\0&e^{-z_1}\end{pmatrix}, \quad
z_1 \in \mathbb{C}.
$$
One can consider the symplectic form on the Lie group $G$ defined by
\begin{equation}\label{yamada-F-1}
F= i\,dz_1 \wedge d{\bar z}_1 + dz_2 \wedge d{\bar z}_3 + d{\bar z}_2 \wedge dz_3,
\end{equation}
which clearly has bidegree (1,1) with respect to the complex structure of $G$.
That is to say, $F$ is a pseudo-K\"ahler metric on $G$. Notice that $F$ is not left-invariant on $G$, indeed the forms
$$
\omega^1=dz_1,\quad\ \omega^2=e^{-z_1}\,dz_2,\quad\ \omega^3=e^{z_1}\,dz_3,
$$
constitute a basis of left-invariant
forms of bidegree (1,0) on the complex Lie group $G$.
Yamada proved in~\cite{Yamada-1} that there is a lattice $\Gamma$ of maximal rank in $G$ such that $F$ descends to a
pseudo-K\"ahler metric on $X=\nilm$.
Let $A \in {\rm SL}(2,\mathbb{Z})$ be a unimodular matrix with distinct real eigenvalues $\lambda$ and $\lambda^{-1}$, and take $a=\log\lambda$.
One can consider a lattice $\Gamma$ in $G$ of the form $\Gamma = \Gamma_1 \ltimes_{\varphi} \Gamma_{\mathbb{C}^2}$,
with $\Gamma_1=a\,\mathbb{Z}+ 2\,\pi i\,\mathbb{Z}$ and $\Gamma_{\mathbb{C}^2}$ a lattice in $\mathbb{C}^2$.
The compact complex (solv)manifold $X=\nilm$ is known as the (holomorphically parallelizable) Nakamura manifold~\cite{Nakamura}.
Yamada proved in \cite[Theorem 2.1]{Yamada-1} that the symplectic form $F$ given in \eqref{yamada-F-1} descends to $X$,
providing in this way the first example of a non-toral compact holomorphically parallelizable pseudo-K\"ahler solvmanifold.
In fact, in terms of the (1,0)-basis $\{\omega^k\}_{k=1}^3$, the form $F$ expresses as
$$
F = i\,\omega^{1}\wedge\omega^{\bar{1}} + e^{2i\,\Imag z_1}\,\omega^{2}\wedge\omega^{\bar{3}}
+ e^{-2i\,\Imag z_1}\,\omega^{\bar{2}}\wedge\omega^{3},
$$
where the functions $e^{2i\,\Imag z_1}$ and $e^{-2i\,\Imag z_1}$
are $\Gamma$-invariant. Therefore, $F$ induces a pseudo-K\"ahler metric on
the Nakamura manifold $X$.
In \cite{Hasegawa-DGA} Hasegawa extended Yamada's result to any compact holomorphically parallelizable solvmanifold of complex dimension 3,
determining all the
lattices of simply-connected unimodular complex solvable Lie groups.
Hasegawa proves that such a complex solvmanifold admits a pseudo-K\"ahler metric if and only if the Hodge number $h_{\db}^{0,1}=3$ \cite[Theorem 2]{Hasegawa-DGA}.
Notice that for the Nakamura manifold $X$ one has
$H_{\db}^{0,1}(X)=\langle [\omega^{\bar{1}}], [e^{-2i\,\Imag z_1}\,\omega^{\bar{2}}], [e^{2i\,\Imag z_1}\,\omega^{\bar{3}}] \rangle$.
In the following proposition
we show that there is a small deformation $X_{\mathbf{t}}$ of $X$
admitting pseudo-K\"ahler metrics. Note that by~\cite{AK} the Nakamura manifold
has $h_{\text{BC}}^{1,1}(X)=7$, whereas our small deformation satisfies
$h_{\text{BC}}^{1,1}(X_{\mathbf{t}})=3$ for every $\mathbf{t}\neq 0$.
\begin{proposition}\label{nakamura}
There exists a small deformation $X_{\mathbf{t}}$ of the holomorphically parallelizable Nakamura manifold $X$ admitting pseudo-K\"ahler metrics for any ${\mathbf{t}}$.
\end{proposition}
\begin{proof}
In \cite[Section 4]{AK} Angella and Kasuya studied
some
deformations of the Nakamura manifold $X=\nilm$. We
here
consider the deformation given by ${\mathbf{t}}\, \frac{\partial}{\partial z_1}\otimes d\bar{z}_1 \in H^{0,1}(X; T^{1,0}X)$, which corresponds
to the \emph{case (1)} in their paper.
Note that this deformation defines a holomorphic family $\{X_{\mathbf{t}}\}_{\mathbf{t} \in \Delta}$,
where $\Delta=\{\mathbf{t}\in \mathbb{C} \mid \, |\mathbf{t}|< 1 \}$, such that $X_0=X$.
We have the (1,0)-forms on $X_{\mathbf{t}}$ given by
$$
\omega_{\mathbf{t}}^1=
dz_1-\mathbf{t}\,d\bar{z}_1, \quad
\omega_{\mathbf{t}}^2=
e^{-z_1}dz_2, \quad
\omega_{\mathbf{t}}^3=
e^{z_1}dz_3,
$$
whose differentials satisfy
$$
\left\{
\begin{array}{llll}
d\omega_{\mathbf{t}}^1 \!\!&\!\!=\!\!&\!\! 0,\\[4pt]
d\omega_{\mathbf{t}}^2 \!\!&\!\!=\!\!&\!\! -\frac{1}{1-|\mathbf{t}|^2}\,\omega_{\mathbf{t}}^{12}
+ \frac{\mathbf{t}}{1-|\mathbf{t}|^2}\,\omega_{\mathbf{t}}^{2\bar{1}},\\[6pt]
d\omega_{\mathbf{t}}^3 \!\!&\!\!=\!\!&\!\! \frac{1}{1-|\mathbf{t}|^2}\,\omega_{\mathbf{t}}^{13}
- \frac{\mathbf{t}}{1-|\mathbf{t}|^2}\,\omega_{\mathbf{t}}^{3\bar{1}}.
\end{array}
\right.
$$
A direct calculation shows that the real 2-form $F_{\mathbf{t}}$ defined on $X_{\mathbf{t}}$ by
$$
F_{\mathbf{t}} = i\,\omega_{\mathbf{t}}^{1\bar{1}} + e^{2i\,\Imag z_1}\,\omega_{\mathbf{t}}^{2\bar{3}}
+ e^{-2i\,\Imag z_1}\,\omega_{\mathbf{t}}^{\bar{2}3}
$$
is closed and non-degenerate. Since $F_{\mathbf{t}}$ has bidegree (1,1) with respect to the complex structure on $X_{\mathbf{t}}$, we get a pseudo-K\"ahler metric on the compact complex manifold $X_{\mathbf{t}}$, for any $\mathbf{t} \in \Delta$.
\end{proof}
In complex dimension $3$, there is
another well-known complex holomorphically parallelizable solvmanifold,
namely,
the Iwasawa (nil)manifold.
Although this manifold is symplectic,
it is proved in~\cite[Theorem 3.2]{CFU-Lefschetz}
(see also \cite{Yamada-1})
that a holomorphically
parallelizable nilmanifold is pseudo-K\"ahler if and only if it is a complex torus. Hence, the
Iwasawa manifold does not admit pseudo-K\"ahler metrics. Moreover:
\begin{proposition}\label{iwasawa}
The Iwasawa manifold and its small deformations do not admit any pseudo-K\"ahler metric.
\end{proposition}
\begin{proof}
Recall that the Iwasawa manifold $X$ is the compact complex manifold obtained as a quotient of the $3$-dimensional complex Heisenberg group.
As we have previously stated, by~\cite[Theorem 3.2]{CFU-Lefschetz}
the Iwasawa manifold is not pseudo-K\"ahler.
Nakamura studied in \cite{Nakamura} the small deformations of the Iwasawa manifold (see also \cite{Ang-libro} for more details).
It turns out that the complex structure equations of any sufficiently small deformation $X_\mathbf{t}$ of the Iwasawa manifold $X=X_0$
can be written as
$$
\left\{
\begin{array}{llll}
d\omega_{\mathbf{t}}^1 \!\!&\!\!=\!\!&\!\! d\omega_{\mathbf{t}}^2 = 0,\\[4pt]
d\omega_{\mathbf{t}}^3 \!\!&\!\!=\!\!&\!\! \sigma_{12}\,\omega_{\mathbf{t}}^{12}
+ \sigma_{1\bar{1}}\,\omega_{\mathbf{t}}^{1\bar{1}} + \sigma_{1\bar{2}}\,\omega_{\mathbf{t}}^{1\bar{2}}
+ \sigma_{2\bar{1}}\,\omega_{\mathbf{t}}^{2\bar{1}} + \sigma_{2\bar{2}}\,\omega_{\mathbf{t}}^{2\bar{2}},
\end{array}
\right.
$$
where the coefficients
$\sigma_{12},\sigma_{1\bar{1}},\sigma_{1\bar{2}},\sigma_{2\bar{1}},\sigma_{2\bar{2}} \in \mathbb{C}$ only depend on
the parameter $\mathbf{t}$ in the deformation space
$\Delta=\{\mathbf{t}=(t_{11},t_{12},t_{21},t_{22},t_{31},t_{32})\in \mathbb{C}^6 \mid \ |\mathbf{t}|< \varepsilon \}$
for a sufficiently small $\varepsilon>0$.
Let $H^+(X_{\mathbf{t}})
\subset H_{\rm dR}^2(X_{\mathbf{t}};\mathbb{R})$ be the subspace
determined by the second de Rham cohomology classes that can be represented by closed real forms
of bidegree $(1,1)$ on the compact complex manifold $X_{\mathbf{t}}$.
As proved in
\cite[Proposition 3.4]{LU-ProcAMS}, this subspace is given by
$$
H^+(X_{\mathbf{t}})=\langle\,[i\,\omega_{\mathbf{t}}^{1\bar{1}}],\,[i\,\omega_{\mathbf{t}}^{2\bar{2}}],
\,[\omega_{\mathbf{t}}^{1\bar{2}}-\omega_{\mathbf{t}}^{2\bar{1}}],\,
[i\,(\omega_{\mathbf{t}}^{1\bar{2}}+\omega_{\mathbf{t}}^{2\bar{1}})]\,\rangle.
$$
It is then clear that any de Rham cohomology class in $H^+(X_{\mathbf{t}})$ is degenerate, so $X_{\mathbf{t}}$
does not admit any pseudo-K\"ahler metric.
\end{proof}
\subsection{Compact pseudo-K\"ahler surfaces}\label{surfaces}
We start this section applying our Proposition~\ref{BC-constant} to provide
a stability result for the pseudo-K\"ahler property in terms of
the complex invariant~$\Delta^2(X)$ introduced in~\cite{AT4}.
Let us recall that, for each $k\in\mathbb N$, Angella and Tomassini introduced the complex invariant
\begin{equation}\label{def-Deltas}
\Delta^k(X) \;:=\; \sum_{p+q=k} \left( h^{p,q}_{\rm BC}(X) + h^{p,q}_{A}(X) \right) - 2\, b_k,
\end{equation}
which is a non-negative integer~\cite[Theorem A]{AT4}.
Furthermore, by \cite[Theorem B]{AT4}, a compact complex manifold $X$ satisfies the $\partial\overline{\del}$-Lemma
if and only if $\Delta^k(X)=0$ for any $k$. We are interested in the term $\Delta^2(X)$.
\begin{corollary}\label{cor1}
Let $X$ be a compact pseudo-K\"ahler manifold, and let $\{X_t\}_{t\in(-\varepsilon,\varepsilon)}$ be a differentiable family
of deformations of $X=X_0$, where $\varepsilon>0$. If the upper-semi-continuous
function $t\mapsto \Delta^2(X_t)$ is constant,
then $X_t$ admits a pseudo-K\"ahler metric for any $t$ close enough to~$0$.
\end{corollary}
\begin{proof}
Suppose $\Delta^2(X_t)=c$ for any $t\in(-\varepsilon,\varepsilon)$, where $c$ is a non-negative integer.
Then, expanding~\eqref{def-Deltas} for $k=2$, we have
\begin{eqnarray*}
&& h^{2,0}_{\rm BC}(X_0) + h^{1,1}_{\rm BC}(X_0) + h^{0,2}_{\rm BC}(X_0)
+ h^{2,0}_{A}(X_0) + h^{1,1}_{A}(X_0) + h^{0,2}_{A}(X_0) \\[5pt]
&& = \Delta^2(X_0) + 2\, b_{2} = c + 2\, b_{2} = \Delta^2(X_t) +2\, b_{2} \\[5pt]
&& = h^{2,0}_{\rm BC}(X_t) + h^{1,1}_{\rm BC}(X_t) + h^{0,2}_{\rm BC}(X_t)
+ h^{2,0}_{A}(X_t) + h^{1,1}_{A}(X_t) + h^{0,2}_{A}(X_t)\;.
\end{eqnarray*}
Since the functions $t\mapsto h^{p,q}_{\rm BC}(X_t)$ and $t\mapsto h^{p,q}_{A}(X_t)$ are upper-semi-continuous for any $(p,q)$,
they all must be constant.
In particular, the function $t\mapsto \dim_\C H^{1,1}_{\rm BC}(X_t)$ is constant, so
Proposition~\ref{BC-constant} implies that
the compact complex manifold $X_t$ admits a pseudo-K\"ahler metric for any $t$ close enough to~$0$.
\end{proof}
The following result comes as a direct consequence.
\begin{corollary}\label{cor2}
Any sufficiently small deformation of a compact pseudo-K\"ahler manifold $X$ satisfying
$\Delta^2(X)=0$ admits a pseudo-K\"ahler metric. In particular, any sufficiently small deformation
of a compact pseudo-K\"ahler $\partial\overline{\del}$-manifold
is pseudo-K\"ahler.
\end{corollary}
Let us recall that a compact complex surface is K\"ahler if and only if its first Betti number $b_1$ is even (see Kodaira's classification of surfaces, \cite{Miy74} and \cite{Siu83}, or \cite{Buc99,Lam99} for a direct proof).
To our knowledge, there is no classification of compact complex surfaces admitting
a pseudo-K\"ahler metric.
Petean found in \cite{Petean} some
obstructions to the existence of \emph{indefinite K\"ahler} metrics on
compact complex surfaces in terms of Seiberg-Witten invariants.
Here ``indefinite K\"ahler'' means that the signature of the metric is (2,2), i.e. the metric is pseudo-K\"ahler but non-K\"ahler
(see \cite[Theorem 4]{Petean} for a list of the possible compact complex surfaces
that might admit an indefinite K\"ahler metric).
\vskip.1cm
Next, we make use of our previous results and a result by Teleman~\cite{Tel06} to prove the stability of the pseudo-K\"ahler property on compact complex surfaces.
\begin{theorem}\label{stability-p-K-surfaces}
Any sufficiently small deformation of a compact pseudo-K\"ahler surface
admits a pseudo-K\"ahler metric.
\end{theorem}
\begin{proof}
Let $X$ be a compact pseudo-K\"ahler surface. By Teleman's result \cite[Lemma 2.3]{Tel06},
the invariant $\Delta^2(X)$
is either $0$ or $2$.
Moreover, Teleman also proves that $\Delta^2(X)=0$ if and only if the first Betti number $b_1$ is even, which is equivalent to the existence of a K\"ahler metric on $X$.
Hence, in the case $\Delta^2(X)=0$, any sufficiently small deformation of $X$ is again K\"ahler by \cite{KS60}, so in particular pseudo-K\"ahler.
Let us now focus on the case $\Delta^2(X)=2$. By \cite{Tel06} this is equivalent to $X$ having odd first Betti number, so
the condition $\Delta^2(X)=2$ is a topological property. Thus,
$\Delta^2(X_t)=2$ for any differentiable family $\{X_t\}_{t\in(-\varepsilon,\varepsilon)}$
of deformations of the compact pseudo-K\"ahler surface $X=X_0$.
Since the upper-semi-continuous
function $t\mapsto \Delta^2(X_t)$ is constant, by Corollary~\ref{cor1}
the compact complex surface
$X_t$ admits a pseudo-K\"ahler metric for any $t$ close enough to~$0$.
\end{proof}
\vskip.1cm
To finish this section we show that there is only one
compact complex non-K\"ahler surface diffeomorphic to a solvmanifold that admits pseudo-K\"ahler metrics, namely the Kodaira-Thurston manifold.
Hasegawa classified in~\cite{Hasegawa} the compact complex surfaces $X$ that are diffeomorphic to a
4-dimensional solvmanifold. Moreover, he proved that the complex structures on such surfaces are invariant
(see \cite{Ovando} for a study of 4-dimensional Lie algebras admitting pseudo-K\"ahler metrics).
In fact, in~\cite[Theorem 1]{Hasegawa} it is shown
that $X$ must be one of the following surfaces:
complex torus, hyperelliptic surface,
Inoue surface of type $\mathcal{S}_M$, primary
Kodaira surface, secondary Kodaira surface, or Inoue surface of type $\mathcal{S}^{\pm}$.
Only the first two are K\"ahler, whereas the other ones have vanishing second Betti number,
with the only exception of a primary Kodaira surface.
It is well-known that the latter admits symplectic forms \cite{Thu}.
Consequently, a compact complex non-K\"ahler surface diffeomorphic to a solvmanifold
admits a symplectic form if and only if it is a primary Kodaira surface.
We recall that a primary Kodaira surface, which we will denote by $KT$, admits pseudo-K\"ahler metrics. By \cite{Hasegawa}, for any complex structure on $KT$
there is a global basis $\{\omega^1, \omega^2\}$ of $(1,0)$-forms satisfying
\begin{equation}\label{KT-ecus}
d \omega^1 = 0, \quad d \omega^2 = \omega^{1\bar{1}}.
\end{equation}
A real $(1,1)$-form $F$ on $KT$ is closed if and only if
\begin{equation}\label{KT-pseudo-K}
F= i r\, \omega^{1\bar{1}} + u\, \omega^{1\bar{2}} - \bar{u}\, \omega^{2\bar{1}},
\end{equation}
for some $r\in {\mathbb R}$ and $u\in {\mathbb C}$.
Since $F^2=-2|u|^2\omega^{12\bar{1}\bar{2}}$, we have that $F$ is non-degenerate if and only if $u\not=0$.
Thus, there are pseudo-K\"ahler metrics on $KT$.
As a consequence of our previous discussion, we have:
\begin{proposition}\label{cor3-deform-c-pK-stable}
Let $X$ be a compact complex non-K\"ahler surface diffeomorphic to a solvmanifold.
If $X$ admits a pseudo-K\"ahler metric, then $X$ is a primary Kodaira surface $KT$.
\end{proposition}
\section{Neutral Calabi-Yau manifolds}\label{neutralCY}
\noindent
In this section we focus our attention on a special kind of pseudo-K\"ahler manifolds, namely, neutral Calabi-Yau manifolds. Moreover, the intermediate
class constituted by
neutral K\"ahler manifolds
is also studied.
We first prove that compact neutral, K\"ahler or Calabi-Yau, surfaces are stable by small deformations of the complex structure.
In higher dimensions, we construct
an 8-dimensional nilmanifold endowed with a neutral Calabi-Yau metric that
allows
us to prove the instability of the neutral K\"ahler and neutral Calabi-Yau properties in any even complex dimension $n\geq 4$.
It is worth to note that such nilmanifold also provides a counterexample to a conjecture in~\cite{CFU}.
\vskip.1cm
We first recall some definitions. Let $X=(M,J)$ be a complex manifold of complex dimension $n=2m$.
Following~\cite{FPPS}, a \emph{neutral K\"ahler} structure on $X$
is a neutral metric $g$, i.e. of signature $(2m,2m)$, such that
\medskip
$\bullet$ $g$ is compatible with $J$, i.e. $g(JU,JV)=g(U,V)$ for any vector fields $U,V$ on $M$; and
\medskip
$\bullet$ $J$ is parallel with respect to the Levi-Civita connection $\nabla$ of $g$, i.e. $\nabla J=0$.
\medskip
These conditions imply that the 2-form $F(U,V)=g(U,JV)$ is closed, i.e. a neutral K\"ahler structure is in particular pseudo-K\"ahler.
A neutral K\"ahler structure is said to be \emph{neutral Calabi-Yau} if
there exists a nowhere vanishing form $\Phi$ of bidegree $(2m,0)$ with respect to $J$
satisfying $\nabla \Phi=0$. Neutral Calabi-Yau manifolds are Ricci-flat.
\medskip
\subsection{Stability on compact complex surfaces}\label{neutralCY-surfaces}
In this section we study the stability of the neutral K\"ahler and neutral Calabi-Yau properties in complex dimension 2.
\begin{proposition}\label{stability-neutral-Kahler-surfaces}
Let $X$ be a compact complex non-K\"ahler surface.
Suppose that $X$ admits a neutral K\"ahler metric. Then,
any sufficiently small deformation of $X$ also
admits neutral K\"ahler metrics.
\end{proposition}
\begin{proof}
By Theorem~\ref{stability-p-K-surfaces}, any sufficiently small deformation $X_\mathbf{t}$ of $X$ admits pseudo-K\"ahler metrics. Since $X$ is non-K\"ahler by hypothesis, the first Betti number of $X_\mathbf{t}$ is odd, so a pseudo-K\"ahler metric on $X_\mathbf{t}$ cannot have signature (4,0) or (0,4). Hence, the pseudo-K\"ahler metrics are necessarily neutral.
\end{proof}
Let us now observe the following. On the one hand, Petean proves in \cite[Proposition 5]{Petean} that if a compact complex surface $X$ admits a Ricci-flat neutral K\"ahler metric, then its Kodaira dimension
and its first Chern class are both zero. Moreover,
$X$ must be a complex torus, a hyperelliptic surface,
or a primary Kodaira surface \cite[Corollary 2]{Petean}.
On the other hand, any compact complex surface with holomorphically trivial canonical bundle is isomorphic to a
K3 surface, a torus, or a primary Kodaira surface. Hence, as a consequence of these results,
one can ensure that
the only compact complex surfaces that can admit
neutral Calabi-Yau structures are a complex torus and
a primary Kodaira surface.
We know by \cite{Hasegawa} that
these two
are diffeomorphic to a 4-dimensional nilmanifold
with an invariant complex structure. It is easy to check that they are both neutral Calabi-Yau.
In fact, the result is clear for the torus, whereas for a primary Kodaira surface it suffices
to remark that, according to the complex equations \eqref{KT-ecus}, the $(2,0)$-form
$\Phi=\omega^{12}$ is nowhere vanishing and parallel with respect to the Levi-Civita connection of any neutral K\"ahler metric \eqref{KT-pseudo-K}.
Concerning deformations, it is well-known that the
small deformations of the invariant complex structures on the complex torus or on the primary Kodaira surface are again invariant,
so they admit neutral Calabi-Yau structures.
We sum up our previous discussion in the following result.
\begin{proposition}\label{stability-neutral-CY-surfaces}
Let $X$ be a compact neutral Calabi-Yau surface. Then, $X$ is a complex torus or a primary Kodaira surface.
Moreover, any sufficiently small deformation of $X$
admits neutral Calabi-Yau structures.
\end{proposition}
Our next goal is to prove the instability of the neutral Calabi-Yau and neutral K\"ahler properties
in complex dimension $n\geq 4$. We begin constructing an 8-dimensional nilmanifold endowed with neutral Calabi-Yau metrics.
To our knowledge, this neutral Calabi-Yau nilmanifold is new and, in addition, it provides counterexamples to a conjecture on pseudo-K\"ahler nilpotent Lie algebras, as we will shortly see.
\subsection{A neutral Calabi-Yau nilmanifold in eight dimensions}\label{neutralCY-8dim-nil}
Let $M$ be a nilmanifold endowed with an invariant complex structure $J$. Let $n$ be the complex dimension of $X=(M,J)$, and suppose that $F$ is an invariant pseudo-K\"ahler metric on $X$. By~\cite{Salamon}, there always exists a closed (non-zero) invariant form $\Phi$ of bidegree $(n,0)$ with respect to $J$, so $\nabla \Phi=0$ for the Levi-Civita connection~$\nabla$ of the invariant metric.
Therefore, any invariant pseudo-K\"ahler metric~$F$ on a complex nilmanifold~$X$ is Ricci-flat (see \cite{FPS}).
\begin{proposition}\label{ejemplo-dim8}
Let $X=(M,J)$ be the $8$-dimensional nilmanifold $M$ endowed with the invariant complex structure $J$ defined by a basis
of $(1,0)$-forms $\{\omega^k\}_{k=1}^4$ satisfying the structure equations
\begin{equation}\label{eleccion}
\left\{\begin{array}{rcl}
d\omega^1 &\!\!\!=\!\!\!& 0,\\[3pt]
d\omega^2 &\!\!\!=\!\!\!& -i\,\omega^{14} + i\,\omega^{1\bar 4},\\[4pt]
d\omega^3 &\!\!\!=\!\!\!& \omega^{12} + \omega^{1\bar 2} - \omega^{2\bar1},\\[4pt]
d\omega^4 &\!\!\!=\!\!\!& - \omega^{1\bar 3} + \omega^{3\bar 1}.
\end{array}\right.
\end{equation}
Then, $X$ admits pseudo-K\"ahler metrics. Moreover, any invariant pseudo-K\"ahler metric $F$ on $X$ is given by
\begin{equation}\label{pK-concreta}
F=i\,(r\,\omega^{1\bar 1} + s\,\omega^{4\bar 4}) + u\,(\omega^{1\bar 2} - \omega^{2\bar 1})
+ v\,(\omega^{1\bar 3} - \omega^{3\bar 1}) - s\,(\omega^{2\bar 3} - \omega^{3\bar 2}),
\end{equation}
where $r,s,u,v\in\mathbb R$ and $rs\neq 0$.
\end{proposition}
\begin{proof}
It is easy to check that the equations~\eqref{eleccion} satisfy the Jacobi identity, i.e. $d^2 \omega^k=0$ for $1\leq k\leq 4$. Thus, they define a simply-connected, connected, nilpotent Lie group $G$ of real dimension $8$. Moreover, since the structure constants belong to ${\mathbb Q}[i]$, the well-known Malcev's theorem \cite{Malcev} ensures the existence of a lattice $\Gamma$ of maximal rank in $G$.
We consider the nilmanifold $M=\nilm$, which is endowed by construction with the complex structure $J$ defined by the $(1,0)$-forms $\{\omega^k\}_{k=1}^4$.
Any invariant real $2$-form $F$ of bidegree $(1,1)$ on $X=(M,J)$ can be written as
\begin{equation}\label{pK-dim8}
F \ = \ \sum_{k=1}^4 i\,x_{k\bar k}\,\omega^{k\bar k}
\ +\sum_{1\leq k<l\leq 4}\big( x_{k\bar l}\,\omega^{k\bar l}-\bar x_{k\bar l}\,\omega^{l\bar k} \big),
\end{equation}
where $x_{k\bar k}\in\mathbb R$ and $x_{k\bar l}\in\mathbb C$, for $1\leq k, l\leq 4$. Since we are looking for pseudo-K\"ahler metrics, we will study the condition
$dF=0$ and the non-degeneracy condition $F^4\neq 0$.
By a direct computation using the complex structure equations~\eqref{eleccion} we get
$$
dF=\Theta+\overline\Theta,
$$
where $\Theta=\partial F$ is the complex $3$-form of bidegree $(2,1)$
given by
\begin{equation*}
\begin{split}
\Theta =& \ 2\,i\,\Imag x_{1\bar3}\,\omega^{12\bar1} + 2\,i\,\Imag x_{2\bar3}\,\omega^{12\bar2}
- (x_{2\bar4}-i\,x_{3\bar3})\,\omega^{12\bar3} + x_{3\bar4}\,\omega^{12\bar4}
- x_{1\bar4}\,\omega^{13\bar1} + i\,x_{3\bar3}\,\omega^{13\bar2} \\[3pt]
& - x_{3\bar4}\,\omega^{13\bar3} + 2\,\Imag x_{1\bar2}\,\omega^{14\bar1}
+ (x_{2\bar2} - \bar x_{3\bar4})\,\omega^{14\bar2} - i\,(x_{2\bar3} + x_{4\bar4})\,\omega^{14\bar3}
- i\,x_{2\bar4}\,\omega^{14\bar4} \\[3pt]
& - (x_{2\bar4} + i\,x_{3\bar3})\,\omega^{23\bar1} + (x_{2\bar2} + \bar x_{3\bar4})\,\omega^{24\bar1} + i\,(x_{4\bar4} + \bar x_{2\bar3})\,\omega^{34\bar1}.
\end{split}
\end{equation*}
Now, the closedness condition $dF=0$ is equivalent to $\Theta =0$.
It is straightforward to check that the latter is satisfied if and only if
$$
x_{2\bar2}=x_{3\bar3}=x_{1\bar4}=x_{2\bar4}=x_{3\bar4}=\Imag x_{1\bar2}=\Imag x_{1\bar3} = \Imag x_{2\bar3}=0,
\quad \mbox{ and }
\quad x_{2\bar3}=-x_{4\bar4}.
$$
Replacing these values in~\eqref{pK-dim8} and denoting $x_{1\bar1}=r$, $x_{4\bar4}=s$, $x_{1\bar2}=u$, $x_{1\bar3}=v$,
with $r,s,u,v\in\mathbb R$, one directly gets the expression~\eqref{pK-concreta}.
We finally need to ensure the non-degeneracy condition for $F$. Using~\eqref{pK-concreta}, it is easy to see that
$$
F^4 = - 24\,r s^3\omega^{1234\bar1\bar2\bar3\bar4}.
$$
Therefore, $F^4\neq 0$ if and only if $rs\neq 0$, as stated in the proposition.
\end{proof}
In the following result we prove that the family of pseudo-K\"ahler metrics \eqref{pK-concreta}
provides neutral Calabi-Yau metrics in eight dimensions that are not flat (although they all are Ricci flat).
\begin{proposition}\label{neutralCY-nil}
The complex nilmanifold $X=(M,J)$ constructed in Proposition~$\ref{ejemplo-dim8}$ has non-flat neutral Calabi-Yau structures.
\end{proposition}
\begin{proof}
Let us take the basis of real $1$-forms
$\{e^1,\ldots,e^8\}$ on $X$ defined by $e^{2k-1}+i\, e^{2k}=\omega^k$, for $1 \leq k \leq 4$, where $\{\omega^1,\ldots,\omega^4\}$ is the basis of $(1,0)$-forms in Proposition~\ref{ejemplo-dim8}
satisfying the complex structure equations~\eqref{eleccion}.
In terms of this real basis, the complex structure $J$ and the pseudo-K\"ahler metrics $F$ given in \eqref{pK-concreta}
express as
$$
\begin{array}{rl}
& J e^1 = -e^2, \quad J e^3 = -e^4, \quad J e^5 = -e^6, \quad J e^7 = -e^8, \\[8pt]
& F = \, 2r\, e^{12} + 2u\, e^{13} + 2v\, e^{15} + 2u\, e^{24} + 2v\, e^{26} - 2s\, e^{35} - 2s\, e^{46} + 2s\, e^{78},
\end{array}
$$
where $r,s,u,v\in\mathbb R$ with $rs\neq 0$. The pseudo-Riemannian metric $g(x,y)=F(Jx,y)$ is
then given in terms of this real basis by the matrix
\begin{equation}\label{neutral-g}
(g_{ij})_{i,j}= \left(
\begin{array}{cccccccc}
2r & 0 & 0 & -2u & 0 & -2v & 0 & 0 \\[2pt]
0 & 2r & 2u & 0 & 2v & 0 & 0 & 0 \\[2pt]
0 & 2u & 0 & 0 & 0 & 2s & 0 & 0 \\[2pt]
-2u & 0 & 0 & 0 & -2s & 0 & 0 & 0 \\[2pt]
0 & 2v & 0 & -2s & 0 & 0 & 0 & 0 \\[2pt]
-2v & 0 & 2s & 0 & 0 & 0 & 0 & 0 \\[2pt]
0 & 0 & 0 & 0 & 0 & 0 & 2s & 0 \\[2pt]
0 & 0 & 0 & 0 & 0 & 0 & 0 & 2s
\end{array}
\right).
\end{equation}
It is easy to see that there are metrics in the family~\eqref{neutral-g} with neutral signature.
In fact, taking for instance $u=v=0$, one has that $rs<0$ is equivalent to the signature being $(4,4)$.
Since the $(4,0)$-form $\Phi=\omega^{1234}$ is parallel,
we conclude that $X$ has neutral Calabi-Yau structures.
Let us now prove that any pseudo-K\"ahler metric~\eqref{pK-concreta} on $X$ is non-flat.
Since the pseudo-K\"ahler structures are invariant, the (complexified) Koszul formula for the Levi-Civita connection $\nabla$
of the metric $g$ reduces to
$$
2 g(\nabla_UV,W)=g([U,V],W)-g([V,W],U)+g([W,U],V),
$$
for (invariant) complex vector fields $U,V,W$ on the complex nilmanifold $X$.
Let $\{Z_j\}_{j=1}^4$ denote the basis of complex vector fields of bidegree (1,0) dual to the
basis $\{\omega^j\}_{j=1}^4$.
Notice that, by complex conjugation, it suffices to compute
$\nabla_{Z_k}Z_j$ and $\nabla_{\overline{Z}_k}Z_j$ for
$1 \leq j,k \leq 4$.
Furthermore, since $\nabla J=0$,
one has that $\nabla_UV$ is of bidegree (1,0) whenever $V$ is.
In particular, $\nabla_U Z_j$ has type $(1,0)$ for every $1 \leq j \leq 4$.
Let $R$ be the curvature tensor of the pseudo-K\"ahler metric, i.e. $R$ is
given by
$$
R(U,V,W,T)=g\big(\nabla_U\nabla_VW-\nabla_V\nabla_UW-\nabla_{[U,V]}W, \, T\big),
$$
for $U,V,W,T$ complex vector fields on $X$.
Taking into account the observation in the previous paragraph, complex conjugation and the symmetries of the curvature tensor, one concludes that
the metric $g$ is non-flat if and only if
$R(Z_i,\bar{Z}_j,Z_k,\bar{Z}_l)\not=0$ for some $i,j,k,l$.
In our case, we will prove that $R(Z_2,\bar{Z}_2,Z_2,\bar{Z}_2)\not=0$.
A direct calculation shows:
$$
\nabla_{\bar{Z}_2}Z_1 = Z_3, \quad\quad \nabla_{\bar{Z}_2}Z_2 = 0, \quad\quad \nabla_{\bar{Z}_2}Z_3 = 0,
\quad
\mbox{ and }
\quad
\nabla_{Z_2}Z_2 = -\frac{is}{r}\, Z_1 -\frac{iv}{r}\, Z_2 +\frac{iu}{r}\, Z_3.
$$
From the complex equations~\eqref{eleccion} we have $[Z_2,\bar{Z}_2]=0$.
Hence,
$$
\begin{array}{rl}
R(Z_2,\bar{Z}_2,Z_2,\bar{Z}_2)
\!\!&\!\!= \, g(\nabla_{Z_2}\nabla_{\bar{Z}_2}Z_2-\nabla_{\bar{Z}_2}\nabla_{Z_2}Z_2-\nabla_{[Z_2,\bar{Z}_2]}Z_2, \, \bar{Z}_2) \\[5pt]
&= -g(\nabla_{\bar{Z}_2}\nabla_{Z_2}Z_2, \, \bar{Z}_2) \\[5pt]
&=\, g\left(\frac{is}{r} Z_3, \, \bar{Z}_2\right) \\[5pt]
&= - \frac{s^2}{r} \not= 0.
\end{array}
$$
\vskip-.4cm
\end{proof}
Note that in~\cite{FPPS} neutral Calabi-Yau structures on a specific
class of nilmanifolds
are constructed. This class is given by the so-called
\emph{Kodaira manifolds}, which are $4m$-dimensional $2$-step nilmanifolds
whose underlying Lie algebras have center of dimension $2m$.
Moreover, their complex structure
is invariant and preserves the center. Kodaira manifolds are a generalization of the
Kodaira-Thurston manifold $KT$, and they have the structure of a
principal torus bundle over a torus, with fiber the central torus.
It is worth to remark that, for the neutral Calabi-Yau nilmanifold $X=(M,J)$ constructed in Proposition~\ref{neutralCY-nil},
the center of the Lie algebra $\frg$ underlying $M$ is 1-dimensional. Hence, it is not invariant under the complex structure $J$ (see
Section~\ref{counterexample} for more details).
Furthermore, $\frg$ has nilpotency step equal to~$4$. By \cite{PS}, this implies that
the neutral Calabi-Yau nilmanifold $X=(M,J)$ is far from being the total space of a principal torus bundle over a torus.
\subsubsection{Counterexamples to a conjecture on pseudo-K\"ahler nilmanifolds}\label{counterexample}
\noindent Here we show that the new pseudo-K\"ahler nilmanifold constructed in Proposition~\ref{ejemplo-dim8} provides counterexamples to a conjecture in~\cite{CFU}.
The conjecture states that an invariant complex structure~$J$ on a nilmanifold $M=\nilm$
must satisfy a certain property so that $(M,J)$ admits pseudo-K\"ahler metrics.
Let us first recall some results on complex structures on nilpotent Lie algebras (which can be found
in~\cite{LUV-SnN} and the references therein) and then formulate the conjecture in precise terms.
Let $\frg$ be a nilpotent Lie algebra.
Complex structures
on $\frg$
can be classified into different types attending to
the behaviour of the \emph{ascending $J$-compatible series} of~$\frg$,
which is defined inductively as
$$
\fra_0(J)=\{0\}, \quad
\quad
\fra_k(J)=\big\{X\in\frg \mid [X,\frg]\subseteq \fra_{k-1}(J)\ {\rm and\ } [JX,\frg]\subseteq \fra_{k-1}(J)\big\}, \ \text{ for } k\geq 1.
$$
Note that $\fra_k(J)$ is an even-dimensional $J$-invariant ideal of $\frg$.
In particular,
$\fra_1(J)$ is the largest subspace
of the center of $\frg$ which is $J$-invariant.
Unlike the usual ascending central series $\{\frg_k\}_k$ of $\frg$, the series $\{\fra_k(J)\}_k$ is adapted to the complex
structure $J$, and it allows to introduce the following partition of the space of complex structures:
\begin{definition}\label{tipos_J}\cite{LUV-SnN}
A complex structure $J$ on a nilpotent Lie algebra $\frg$ is said to be
\begin{itemize}
\item[(i)] \emph{nilpotent}, if there exists an integer~$t>0$ such that~$\fra_t(J)=\frg$;
\item[(ii)] \emph{non-nilpotent}, if $\fra_t(J)\neq \frg$ for every $t\geq 0$; moreover, $J$ is called
\begin{itemize}
\item[(ii.1)] \emph{strongly non-nilpotent},
if $\fra_1(J)=\{0\}$ $($which implies $\fra_t(J)=\{0\}$ for every~$t)$;
\item[(ii.2)] \emph{weakly non-nilpotent}, if there is an integer~$t>0$ satisfying~$\{0\}\neq \fra_t(J)=\fra_l(J) \neq \frg$, for every~$l\geq t$.
\end{itemize}
\end{itemize}
\end{definition}
Notice that $\fra_1(J)\neq \{0\}$ for any nilpotent or weakly non-nilpotent
complex structure $J$. This allows to construct such structures from other complex
structures defined on lower dimensional nilpotent Lie algebras
(see \cite{LUV-SnN} for details).
This fact leaves strongly non-nilpotent complex structures as
the essentially new complex structures that arise in each even real dimension.
In \cite[Section 3.1]{LUV-SnN} it is proved that if $\frg$ admits a
strongly non-nilpotent complex structure $J$, then the nilpotency step of $\frg$ is at least $3$
(see \cite{LUV-SnN} for other general properties on Lie algebras with strongly non-nilpotent complex structures and structure results up to real dimension 8).
\vskip.15cm
We can now formulate the following conjecture proposed in \cite{CFU}:
\vskip.15cm
\noindent\textbf{Conjecture \cite[page 123]{CFU}}: \emph{a complex
structure on a $2n$-dimensional nilpotent Lie algebra must be of nilpotent type in the presence of a
compatible symplectic form}.
\vskip.15cm
It is proved in \cite{CFU} that the conjecture holds for $n\leq 3$.
However, the complex structure $J$ given in Proposition~\ref{ejemplo-dim8}
provides a \emph{counterexample} for $n=4$.
In fact, from the equations \eqref{eleccion} it is straightforward to
prove that $\fra_1(J)=\{0\}$, that is to say, $J$ is strongly non-nilpotent
according to Definition~\ref{tipos_J}, and
it admits the compatible symplectic forms given in \eqref{pK-concreta}.
Furthermore,
in every complex dimension $n \geq 4$ we have the following result.
\begin{proposition}\label{pK-en-SnN}
For each $n \geq 4$, there exists a $2n$-dimensional nilmanifold endowed with a non-nilpotent complex structure
that admits pseudo-K\"ahler metrics.
\end{proposition}
\begin{proof}
Let us consider $Y=X\times {\mathbb T}^k$, where $X=(M,J)$ is the $8$-dimensional pseudo-K\"ahler nilmanifold given in Proposition~\ref{ejemplo-dim8} and ${\mathbb T}^k$ denotes the $k$-dimensional complex torus endowed with any invariant pseudo-K\"ahler metric. Then, $Y$ is a pseudo-K\"ahler nilmanifold of real dimension $2n=8+2k$ with invariant complex structure
$J_{Y}=J\times J_{{\mathbb T}^k}$
satisfying
$
\{0\}\neq \frb= \fra_t(J_{Y}) \neq \frg\times \frb$,
for every~$t>0$, where $\frb$ denotes the abelian Lie algebra underlying the torus ${\mathbb T}^k$, and $\frg$
the Lie algebra of~$M$.
In particular, $J_{Y}$ is (weakly) non-nilpotent.
\end{proof}
\subsection{Instability in complex dimension $n\geq 4$}\label{sect4}
\noindent In contrast to the stability results for compact complex surfaces proved in
Section~\ref{neutralCY-surfaces}, we will next
show that the neutral K\"ahler and neutral Calabi-Yau properties are both unstable in every complex dimension $n\geq 4$.
This constitutes a deep difference with the K\"ahler Calabi-Yau case, for which the deformation space is unobstructed by the well-known Bogomolov-Tian-Todorov theorem \cite{Bogomolov,Tian,Todorov}.
We begin with the following result in complex dimension 4.
\begin{theorem}\label{deform-pK}
There exists a holomorphic family of compact complex manifolds
$\{X_\mathbf{t}\}_{\mathbf{t}\in\Delta}$
of complex dimension $4$,
where $\Delta=\{ \mathbf{t}\in \mathbb{C}\mid |\mathbf{t}|< 1 \}$, satisfying the following properties:
\begin{enumerate}
\item[\rm (i)] $X_0$ is a neutral Calabi-Yau manifold;
\item[\rm (ii)] $X_\mathbf{t}$ does not admit any pseudo-K\"ahler structure for
$\mathbf{t}\in\Delta\setminus \mathcal{C}$, where $\mathcal{C}$ is the real curve through $0$ given by
$\mathcal{C}=\{ \mathbf{t}\in\Delta \mid \Real\mathbf{t}=0 \}$.
\end{enumerate}
Therefore, neither the neutral Calabi-Yau property nor the neutral K\"ahler property are stable
under deformations of the complex structure.
\end{theorem}
\begin{proof}
The proof is based on an appropriate deformation of the neutral Calabi-Yau
nilmanifold $X=(M,J)$ found in Proposition~\ref{neutralCY-nil}.
Let $\{\omega^k\}_{k=1}^4$
be the basis of $(1,0)$-forms on $X$ satisfying \eqref{eleccion}.
Observe that the (0,1)-form $\omega^{\bar{1}}$ defines a Dolbeault cohomology class on $X$. We consider the class
$[\omega^{\bar{1}}] \in H^{0,1}_{\db}(X)$
to perform an appropriate holomorphic deformation of $X$.
For each $\mathbf{t}\in \mathbb{C}$ such that $|\mathbf{t}|<1$, we define the complex structure
$J_\mathbf{t}$ on~$M$ given by the following basis $\{\eta_\mathbf{t}^k\}_{k=1}^4$ of $(1,0)$-forms:
\begin{equation}\label{ec-h-epsilon-deform}
\eta_\mathbf{t}^1:=\omega^1,\quad \eta_\mathbf{t}^2:=\omega^2-\mathbf{t}\,\omega^{\bar1},\quad
\eta_\mathbf{t}^3:=\omega^3,\quad \eta_\mathbf{t}^4:=\omega^4.
\end{equation}
The complex structure equations for $X_\mathbf{t}=(M,J_\mathbf{t})$ are
\begin{equation}\label{ec-h-epsilon-llegada}
\left\{\begin{array}{rcl}
d\eta_\mathbf{t}^1 &\!\!\!=\!\!\!& 0,\\[3pt]
d\eta_\mathbf{t}^2 &\!\!\!=\!\!\!& -i\,\eta_\mathbf{t}^{14}+i\,\eta_\mathbf{t}^{1\bar{4}},\\[4pt]
d\eta_\mathbf{t}^3 &\!\!\!=\!\!\!& \eta_\mathbf{t}^{12}+\mathbf{t}\,\eta_\mathbf{t}^{1\bar1}+\eta_\mathbf{t}^{1\bar2} - \eta_\mathbf{t}^{2\bar1},\\[4pt]
d\eta_\mathbf{t}^4 &\!\!\!=\!\!\!& -\eta_\mathbf{t}^{1\bar3}+\eta_\mathbf{t}^{3\bar1}.
\end{array}\right.
\end{equation}
Observe that the initial structure $J$ is recovered for $\mathbf{t}=0$. Therefore, $X_0=X$ and one immediately
gets part~\textrm{(i)} of the statement.
To prove~\textrm{(ii)}, since $X_\mathbf{t}$ is a complex nilmanifold, a similar argument as in the proof of Proposition~\ref{deform-pK-dim6} allows us to reduce the study of existence of pseudo-K\"ahler metrics on $X_\mathbf{t}$ to the study of invariant ones.
We first analyze the existence of invariant
closed $2$-forms $\Omega$ on~$M$ compatible with~$J_\mathbf{t}$, not necessarily real.
Any such $\Omega$ belongs to $\bigwedge^{1,1}_{J_\mathbf{t}}(\frg^*)$, where $\frg$ denotes the Lie algebra underlying the nilmanifold $M$, and it is given by
$$
\Omega=\sum_{k=1}^4\big(a_k\,\eta_\mathbf{t}^1+b_k\,\eta_\mathbf{t}^2+c_k\,\eta_\mathbf{t}^3+f_k\,\eta_\mathbf{t}^4\big)\wedge\eta_\mathbf{t}^{\bar k},
$$
where $a_k$, $b_k$, $c_k$, $f_k\in\mathbb C$, for $1\leq k\leq 4$.
Making use of the complex structure equations \eqref{ec-h-epsilon-llegada}, one has
$$
d\Omega=\partial_{\mathbf{t}}\Omega + \db_{\mathbf{t}}\Omega,
$$
where
\begin{equation*}
\begin{split}
\partial_{\mathbf{t}}\Omega &= (a_3-\bar{\mathbf{t}}\,b_3+c_1)\,\eta_\mathbf{t}^{12\bar1} + (b_3+c_2)\,\eta_\mathbf{t}^{12\bar2}
+ (c_3-b_4)\,\eta_\mathbf{t}^{12\bar3} + c_4\,\eta_\mathbf{t}^{12\bar4}
- (a_4+\bar{\mathbf{t}}\,c_3)\,\eta_\mathbf{t}^{13\bar1} \\
&\ \ + c_3\,\eta_\mathbf{t}^{13\bar2} - c_4\,\eta_\mathbf{t}^{13\bar3} - i\,(a_2+b_1-i\,\bar{\mathbf{t}}\,f_3)\,\eta_\mathbf{t}^{14\bar1}
+ (f_3-i\,b_2)\,\eta_\mathbf{t}^{14\bar2} - (f_4 + i\,b_3)\,\eta_\mathbf{t}^{14\bar3} \\
&
\ \ - i\,b_4\,\eta_\mathbf{t}^{14\bar4} - (b_4+c_3)\,\eta_\mathbf{t}^{23\bar1} - (f_3+i\,b_2)\,\eta_\mathbf{t}^{24\bar1} + (f_4-i\,c_2)\,\eta_\mathbf{t}^{34\bar1}.
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
\db_{\mathbf{t}}\Omega & = - (a_3+c_1-\mathbf{t}\,c_2)\,\eta_\mathbf{t}^{1\bar1\bar2}
+ (f_1+\mathbf{t}\,c_3)\,\eta_\mathbf{t}^{1\bar1\bar3} -i\,(a_2+b_1+i\,\mathbf{t}\,c_4)\,\eta_\mathbf{t}^{1\bar1\bar4}
+ (c_3+f_2)\,\eta_\mathbf{t}^{1\bar2\bar3} \\
&
\ \ + (c_4-i\,b_2)\,\eta_\mathbf{t}^{1\bar2\bar4}- (f_4+i\,b_3)\,\eta_\mathbf{t}^{1\bar3\bar4}
- (b_3+c_2)\,\eta_\mathbf{t}^{2\bar1\bar2} - c_3\,\eta_\mathbf{t}^{2\bar1\bar3}
- (c_4+i\,b_2)\,\eta_\mathbf{t}^{2\bar1\bar4} \\
& \ \ + (f_2-c_3)\,\eta_\mathbf{t}^{3\bar1\bar2}
+f_3\,\eta_\mathbf{t}^{3\bar1\bar3} + (f_4-i\,c_2)\,\eta_\mathbf{t}^{3\bar1\bar4}
- f_3\,\eta_\mathbf{t}^{4\bar1\bar2} - i\,f_2\,\eta_\mathbf{t}^{4\bar1\bar4}.
\end{split}
\end{equation*}
From these expressions we see that $\Omega$ is closed, i.e. $\partial_{\mathbf{t}}\Omega=0=\db_{\mathbf{t}}\Omega$, if and only if
$$
a_4=b_2=b_4=c_3=c_4=f_1=f_2=f_3=0,
$$
and
$$
b_1=-a_2, \quad c_2=-b_3, \quad f_4=-i\,b_3,
\quad -a_3+\bar{\mathbf{t}}\,b_3 =c_1=-a_3-\mathbf{t}\,b_3.
$$
Now, the latter two equalities imply the equation
$$
b_3\,\Real \mathbf{t}=0,
$$
which gives rise to the following two cases.
If $\Real \mathbf{t}=0$, then pseudo-K\"ahler metrics exist on $X_\mathbf{t}$; for instance
$$
F=i\,\eta_\mathbf{t}^{1\bar 1} - i\,\eta_\mathbf{t}^{4\bar 4}
-\frac{\mathbf{t}}{2}\,\eta_\mathbf{t}^{1\bar 3}
+ \eta_\mathbf{t}^{2\bar3}
+ \frac{\bar{\mathbf{t}}}{2}\,\eta_\mathbf{t}^{3\bar 1}
-\eta_\mathbf{t}^{3\bar 2}
$$
is a real closed $(1,1)$-form which is non-degenerate.
However, if we assume $\Real \mathbf{t}\neq 0$, then one has $b_3=0$ and
consequently $c_2=f_4=0$. Thus, every closed $2$-form $\Omega$ of bidegree (1,1) with respect to the complex structure $J_\mathbf{t}$ is given by
$$
\Omega=a_1\,\eta_\mathbf{t}^{1\bar1} + a_2\,(\eta_\mathbf{t}^{1\bar2}-\eta_\mathbf{t}^{2\bar1}) + a_3\,(\eta_\mathbf{t}^{1\bar3}-\eta_\mathbf{t}^{3\bar1}),
$$
with $a_1, a_2, a_3\in\mathbb C$.
Hence, when $\Real \mathbf{t}\neq 0$, the space of real closed $2$-forms on $\frg$ compatible with $J_\mathbf{t}$, namely
$\mathcal Z^{+}_{J_\mathbf{t}}(\frg)=\{ \alpha\in\wedge^2 \frg^* \mid d\alpha=0 \text{ and }
J_\mathbf{t}\alpha=\alpha\} = \{ \alpha\in\wedge^{1,1}_{J_\mathbf{t}}\frg_{\mathbb C}^* \mid d\alpha=0 \text{ and }
\bar{\alpha}=\alpha\}$,
is generated by
\begin{equation}\label{Re-t-no-nula}
\mathcal Z^{+}_{J_\mathbf{t}}(\frg)=
\langle i\,\eta_\mathbf{t}^{1\bar1},\, \eta_\mathbf{t}^{1\bar2}-\eta_\mathbf{t}^{2\bar1},\, \eta_\mathbf{t}^{1\bar3}-\eta_\mathbf{t}^{3\bar1}\rangle.
\end{equation}
Since
every element in $\mathcal Z^{+}_{J_\mathbf{t}}(\frg)$ is degenerate, no pseudo-K\"ahler metrics
exist on $X_\mathbf{t}=(M, J_\mathbf{t})$ when $\Real \mathbf{t}\neq 0$.
This clearly implies that $X_\mathbf{t}$ cannot admit any neutral K\"ahler or neutral Calabi-Yau structure for $\mathbf{t}\in\Delta\setminus \mathcal{C}$, where
$\mathcal{C}=\{ \mathbf{t}\in\Delta \mid \Real\mathbf{t}=0 \}$. Hence, both properties are unstable by small deformations.
\end{proof}
In the following theorem we sum up the main results about instability found along the preceding sections.
\begin{theorem}\label{NCY-no-estable}
On compact complex manifolds of complex dimension $\geq 3$, the properties of \emph{``being
pseudo-K\"ahler''}, \emph{``being neutral K\"ahler''} and \emph{``being neutral Calabi-Yau''}
are not stable under small deformations of the complex structure.
\end{theorem}
\begin{proof}
For the pseudo-K\"ahler property, the result follows from
Proposition~\ref{deform-pK-dim6} and Remark~\ref{remark-ii-1}.
For the neutral K\"ahler and neutral Calabi-Yau properties in complex dimension 4, the result is given in Theorem~\ref{deform-pK}. In higher dimensions, it suffices to consider the product $Y_\mathbf{t}=X_\mathbf{t}\times {\mathbb T}^{2m}$, where $X_\mathbf{t}=(M, J_\mathbf{t})$
is the holomorphic family given in Theorem~\ref{deform-pK}
and ${\mathbb T}^{2m}$ the $2m$-dimensional complex torus endowed with an invariant neutral Calabi-Yau metric. If $\frg$ and $\frb$ denote the Lie algebras underlying $M$ and ${\mathbb T}^{2m}$, respectively, then the space of
invariant closed real $2$-forms on $Y_\mathbf{t}$ compatible with the product complex structure $J_{Y_\mathbf{t}}=J_{\mathbf{t}}\times J_{{\mathbb T}^{2m}}$ is given by
$$
\mathcal Z^{+}_{J_{Y_\mathbf{t}}}(\frg\times\frb) =
\mathcal Z^{+}_{J_\mathbf{t}}(\frg) \oplus
\left\{
\eta_\mathbf{t}^{1} \wedge \bar{\alpha} - \alpha\wedge\eta_\mathbf{t}^{\bar1} \mid \alpha\in\Lambda^{1,0}_{J_{{\mathbb T}^{2m}}}\, \frb^* \right\}
\oplus \mathcal Z^{+}_{J_{{\mathbb T}^{2m}}}(\frb),
$$
where the space $\mathcal Z^{+}_{J_\mathbf{t}}(\frg)$ is described in \eqref{Re-t-no-nula} for any $\mathbf{t} \in \Delta$ such that $\Real \mathbf{t}\neq 0$.
It is easy to see that any element in $\mathcal Z^{+}_{J_{Y_\mathbf{t}}}(\frg\times\frb)$ is degenerate, so
$Y_\mathbf{t}$ does not admit pseudo-K\"ahler metrics for any
$\mathbf{t}\in\Delta\setminus \mathcal{C}$.
Since $Y_0=X_0\times {\mathbb T}^{2m}$ is neutral Calabi-Yau, the result follows immediately.
\end{proof}
\section{Pseudo-Hermitian-symplectic structures}\label{pseudo-HS}
\noindent
In this section we consider an indefinite version of the Hermitian-symplectic geometry. The motivation comes from the fact that the small deformations of any pseudo-K\"ahler manifold always posses
what we will call a pseudo-Hermitian-symplectic structure.
We will show that there are
compact complex manifolds with pseudo-Hermitian-symplectic structure but
not admitting any pseudo-K\"ahler metric.
Recall that a complex structure~$J$ on a symplectic manifold $(M,\Omega)$ is said
to be \emph{tamed} by the symplectic form $\Omega$ if the condition
$\Omega(x,Jx)>0$ is satisfied for all non-zero tangent vectors~$x$.
Following the terminology of~\cite{ST}, we
will refer to the pair $(\Omega, J)$ as a \emph{Hermitian-symplectic structure}.
Note that the tamed condition is equivalent to require that the $(1,1)$-component $\Omega^{1,1}$ of the symplectic form $\Omega$ is
positive, i.e. $\Omega^{1,1}$ is a Hermitian metric on the complex manifold $X=(M,J)$.
No example of a non-K\"ahler compact complex manifold admitting
a
Hermitian-symplectic structure is known
(see \cite[page 678]{LZ} and \cite[Question~1.7]{ST}).
By analogy, in the pseudo-Hermitian setting, we introduce the following notion:
\begin{definition}\label{def-p-H-S}
A complex manifold $X=(M,J)$ is called \emph{pseudo-Hermitian-symplectic} if there exists a symplectic form
$\Omega$ on $M$ such that its component of bidegree (1,1) with respect to $J$ is non-degenerate. In such case we will say that the pair $(\Omega, J)$ is a \emph{pseudo-Hermitian-symplectic structure}.
\end{definition}
From the definition, it is clear that any pseudo-K\"ahler manifold is pseudo-Hermitian-symplectic.
\begin{example}\label{ejemplos-KT}
{\rm
For a primary Kodaira surface, with equations given by \eqref{KT-ecus},
the pseudo-Hermitian-symplectic (indeed pseudo-K\"ahler) structures are defined in \eqref{KT-pseudo-K}. Observe that the form $\Omega= \omega^{12} + \omega^{\bar{1}\bar{2}}$ is symplectic, but the pair $(\Omega,J)$ is not pseudo-Hermitian-symplectic
because the $(1,1)$-component of $\Omega$ is identically zero.
}
\end{example}
\begin{remark}\label{impli}
{\rm In the positive-definite case, \cite[Proposition 2.1]{EFV} provides an important
characterization of the Hermitian-symplectic condition. More precisely,
the existence of such a structure
on a complex manifold $X$ is shown to be equivalent to the existence of
a Hermitian metric~$F$ satisfying
$\partial F =\overline \partial\alpha$, for some $\partial$-closed $(2,0)$-form $\alpha$ on $X$. In the following example
we illustrate that a similar result does not hold in the pseudo-Hermitian-symplectic setting.
}\end{remark}
\begin{example}
{\rm
Let us consider a compact complex nilmanifold defined by the complex equations
$$
d\omega^1=d\omega^2=0, \quad d\omega^3= \rho\,\omega^{12}+\omega^{1\bar{1}}+\rho\,\omega^{1\bar{2}}+D\,\omega^{2\bar{2}},
$$
for some $D \in \mathbb{C}$ and $\rho \in \{0,1\}$.
Let $F=\frac{i}{2} r\,\omega^{1\bar{1}}-\omega^{2\bar{3}}+\omega^{3\bar{2}}$, where $r \in \mathbb{R}^*$.
Then, $F$ is a real form of bidegree $(1,1)$ and
$F^3=3i\, r\, \omega^{1\bar{1}2\bar{2}3\bar{3}} \not=0$.
Moreover, the $(2,0)$-form $\alpha=-\omega^{23}$ is $\partial$-closed
and satisfies
$$
\partial F= \omega^{12\bar{1}} + \rho\,\omega^{12\bar{2}}=-\db \alpha.
$$
Take now the real 2-form $\Omega=\alpha+F+\bar{\alpha}$. By the previous condition we have $\partial\alpha=\db \alpha+\partial F=0$, which implies $d\Omega =0$.
However, the closed 2-form $\Omega$ is not symplectic. In fact,
$$
\Omega^3=(\alpha+F+\bar{\alpha})^3= F^3 + 6 \,\alpha\wedge\bar{\alpha}\wedge F =0.
$$
Even more, if $\rho=0$ and $D\in \mathbb{R}^*$, then the corresponding nilmanifold does not admit any symplectic structure
(indeed, the underlying Lie algebra is $\mathfrak{h}_3$ in the notation of \cite{U}).
For the other values of
$\rho$ and $D$
we have by \cite[Proposition 2.4]{COUV} that the nilmanifold has underlying Lie algebra
$\mathfrak{h}_2$, $\mathfrak{h}_4$, $\mathfrak{h}_6$ or $\mathfrak{h}_8$. These nilmanifolds are symplectic.
}
\end{example}
We next prove that, similarly to the Hermitian case~\cite[Proposition~2.4]{Yang}, the pseudo-Hermitian-symplectic property is open under holomorphic deformations.
\begin{proposition}\label{openness-p-HS}
For compact complex manifolds, the pseudo-Hermitian-symplectic property is stable.
Therefore, any sufficiently small deformation of a pseudo-K\"ahler manifold is pseudo-Hermitian-symplectic.
\end{proposition}
\begin{proof}
Let us denote by $M$ the real manifold underlying a complex manifold $X$ and by $J$ the complex structure on $M$ such that $X=(M,J)$.
Let $\Omega$ be a symplectic form on $M$ whose (1,1)-component with respect to $J$ is non-degenerate, i.e. $(\Omega,J)$ is a pseudo-Hermitian-symplectic structure. We consider a holomorphic family of compact complex manifolds $\{X_{\mathbf{t}}=(M, J_{\mathbf{t}})\}_{\mathbf{t}\in \Delta}$, with $\Delta$ containing $0$, such that $X_0=X$.
The symplectic form $\Omega$ decomposes on the compact complex manifold $X_{\mathbf{t}}$ as
$$
\Omega = \alpha_{\mathbf{t}} + F_{\mathbf{t}} + \beta_{\mathbf{t}},
$$
where $\alpha_{\mathbf{t}}$ has bidegree (2,0), $F_{\mathbf{t}}$ is the (1,1) component of $\Omega$, and $\beta_{\mathbf{t}}=\bar\alpha_{\mathbf{t}}$.
By hypothesis, for~$\mathbf{t}=0$ the form~$F_0$ is non-degenerate.
Hence, one concludes that $F_{\mathbf{t}}$ is also non-degenerate
for any $\mathbf{t} \in \Delta$ sufficiently close to $0 \in \Delta$, so
$X_{\mathbf{t}}$ satisfies the pseudo-Hermitian-symplectic property.
The second assertion in the proposition is clear since any pseudo-K\"ahler manifold is pseudo-Hermitian-symplectic.
\end{proof}
As we recalled above,
Streets and Tian pose in \cite[Question 1.7]{ST} the following question, which is still an open problem:
\emph{Does there exist a compact complex manifold, of complex dimension $\geq 3$, admitting a Hermitian-symplectic structure but no K\"ahler metrics?}
In the following result we prove that the indefinite counterpart of this problem has an affirmative answer.
\begin{proposition}\label{pseudo-ST}
There exist compact complex manifolds with pseudo-Hermitian-symplectic structure but not admitting any pseudo-K\"ahler metric.
\end{proposition}
\begin{proof}
Let us consider the family of compact complex manifolds
$\{X_\mathbf{t}\}_{\mathbf{t}\in\Delta}$, of complex dimension~$3$, constructed in Proposition~\ref{deform-pK-dim6}.
We have:
\begin{enumerate}
\item[\rm (i)] $X_0$ is a pseudo-K\"ahler manifold,
\item[\rm (ii)] $X_\mathbf{t}$ does not admit any pseudo-K\"ahler metric for $\mathbf{t}\not= 0$.
\end{enumerate}
Since $X_0$ is a pseudo-K\"ahler manifold, by Proposition~\ref{openness-p-HS} and (ii) we conclude that, for sufficiently small values of
$\mathbf{t}\not= 0$, the compact complex manifold $X_\mathbf{t}$ is a pseudo-Hermitian-symplectic manifold with no pseudo-K\"ahler metrics.
\end{proof}
In the next examples we consider pseudo-K\"ahler structures
on a compact complex manifold $X_0$ and illustrate their behaviour along a small holomorphic deformation $X_{\mathbf{t}}$ of $X_0$.
\begin{example}\label{contraej-p-HS-dim3}
{\rm
Let us consider the family of compact complex manifolds
$\{X_\mathbf{t}\}_{\mathbf{t}\in\Delta=\{ \mathbf{t}\in \mathbb{C} \mid |\mathbf{t}|< 1 \}}$ of complex dimension~$3$ constructed in Proposition~\ref{deform-pK-dim6}.
Using the complex equations \eqref{ecccus} of the complex nilmanifold $X_0$, we have that any invariant pseudo-K\"ahler metric $F$ on $X_0$ is given by
$$
F=i\,(r\,\omega^{1\bar 1} + s\,\omega^{2\bar 2}) + u\,\omega^{1\bar 2} - \bar{u}\,\omega^{2\bar 1}
+ v\,\omega^{2\bar 3} - \bar{v}\,\omega^{3\bar 2},
$$
for some $r,s\in\mathbb R$ and $u,v\in\mathbb C$ satisfying $rv\neq 0$.
A direct calculation using \eqref{rel-tt} shows
that the 2-form $F$ decomposes along the deformation~$X_\mathbf{t}$~as
$$
F= \alpha_{\mathbf{t}} + F_{\mathbf{t}} + \beta_{\mathbf{t}}
= - \frac{v\,\bar{\mathbf{t}}}{1\!-\!|\mathbf{t}|^2}\,\omega_\mathbf{t}^{23}
+i\,(r\,\omega_\mathbf{t}^{1\bar 1} + s\,\omega_\mathbf{t}^{2\bar 2}) + u\,\omega_\mathbf{t}^{1\bar 2} - \bar{u}\,\omega_\mathbf{t}^{2\bar 1} +
\frac{1}{1\!-\!|\mathbf{t}|^2} (v\,\omega_\mathbf{t}^{2\bar 3} - \bar{v}\,\omega_\mathbf{t}^{3\bar 2})
- \frac{\bar{v}\,\mathbf{t}}{1\!-\!|\mathbf{t}|^2}\,\omega_\mathbf{t}^{\bar 2\bar 3},
$$
where $\alpha_{\mathbf{t}}=- \frac{v\,\bar{\mathbf{t}}}{1-|\mathbf{t}|^2}\,\omega_\mathbf{t}^{23}$ is the (2,0)-component of $F$, and $\beta_{\mathbf{t}}=\bar\alpha_{\mathbf{t}}$.
The real (1,1)-form $F_{\mathbf{t}}$ is non-degenerate on the manifold $X_\mathbf{t}$,
i.e. $F$ defines a pseudo-Hermitian-symplectic structure on $X_\mathbf{t}$, which is in accord to Proposition~\ref{openness-p-HS}.
Notice that $d F_{\mathbf{t}}=\frac{1}{1-|\mathbf{t}|^2} (v\,\bar{\mathbf{t}}\,\omega_\mathbf{t}^{12\bar 2}
+\bar{v}\,\mathbf{t}\,\omega_\mathbf{t}^{2\bar 1\bar 2}) \not=0$ for any $\mathbf{t}\in\Delta\setminus \{0\}$, i.e. the (1,1)-form $F_{\mathbf{t}}$ is not closed.
Furthermore, by Proposition~\ref{deform-pK-dim6}~(ii), $X_\mathbf{t}$ does not admit any pseudo-K\"ahler structure for
$\mathbf{t}\in\Delta\setminus \{0\}$.
}
\end{example}
\begin{example}\label{contraej-p-HS-dim4}
{\rm
Let $\{X_\mathbf{t}\}_{\mathbf{t}\in\Delta=\{ \mathbf{t}\in \mathbb{C}\mid |\mathbf{t}|< 1 \}}$ be the family of compact complex manifolds of complex dimension~$4$ constructed in Theorem~\ref{deform-pK}.
The manifold $X_0$ is neutral Calabi-Yau, hence pseudo-K\"ahler, and by \eqref{pK-concreta} any invariant pseudo-K\"ahler metric $F$ on $X_0$ is given by
$$
F=i\,(r\,\omega^{1\bar 1} + s\,\omega^{4\bar 4}) + u\,(\omega^{1\bar 2} - \omega^{2\bar 1})
+ v\,(\omega^{1\bar 3} - \omega^{3\bar 1}) - s\,(\omega^{2\bar 3} - \omega^{3\bar 2}),
$$
for some $r,s,u,v\in\mathbb R$ with $rs\neq 0$. A direct calculation using \eqref{ec-h-epsilon-deform} shows that the 2-form $F$ decomposes as follows along the deformation $X_\mathbf{t}$:
$$
F= \alpha_{\mathbf{t}} + F_{\mathbf{t}} + \beta_{\mathbf{t}}
= - s\,\bar{\mathbf{t}}\,\eta_\mathbf{t}^{13}
+i\,(r\,\eta_\mathbf{t}^{1\bar 1} + s\,\eta_\mathbf{t}^{4\bar 4}) + u\,(\eta_\mathbf{t}^{1\bar 2} - \eta_\mathbf{t}^{2\bar 1})
+ v\,(\eta_\mathbf{t}^{1\bar 3} - \eta_\mathbf{t}^{3\bar 1}) - s\,(\eta_\mathbf{t}^{2\bar 3} - \eta_\mathbf{t}^{3\bar 2})
- s\,\mathbf{t}\,\eta_\mathbf{t}^{\bar 1\bar 3},
$$
where $\alpha_{\mathbf{t}}=- s\,\bar{\mathbf{t}}\,\eta_\mathbf{t}^{13}$ is the (2,0)-component of the form $F$, and $\beta_{\mathbf{t}}=\bar\alpha_{\mathbf{t}}$.
The real (1,1)-form $F_{\mathbf{t}}$ defines a pseudo-Hermitian-symplectic structure on $X_\mathbf{t}$ as it is non-degenerate, accordingly to Proposition~\ref{openness-p-HS}.
Note that $F_{\mathbf{t}}$ is not closed for any $\mathbf{t}\in\Delta\setminus \{0\}$, since $d F_{\mathbf{t}}= s (\bar{\mathbf{t}}\,\eta_\mathbf{t}^{12\bar 1}
+\mathbf{t}\,\eta_\mathbf{t}^{1\bar 1\bar 2})$.
Moreover, by Theorem~\ref{deform-pK}~(ii), the compact complex manifold $X_\mathbf{t}$ does not admit any pseudo-K\"ahler structure for every
$\mathbf{t}\in\Delta\setminus \mathcal{C}$, where
$\mathcal{C}=\{ \mathbf{t}\in\Delta \mid \Real\mathbf{t}=0 \}$.
}
\end{example}
\section*{Acknowledgments}
\noindent
This work has been partially supported by the projects MTM2017-85649-P (AEI/FEDER, UE),
and E22-17R ``Algebra y Geometr\'{\i}a'' (Gobierno de Arag\'on/FEDER).
\smallskip
|
3,212,635,537,971 | arxiv | \section{Introduction}
\label{intro}
Laser spectroscopy provides a detailed insight into atomic structure including all subtle effects that contribute to the exact energy and the splittings of individual energy levels. Many of these effects are of great relevance in fundamental physics problems, as for example quantum electrodynamics, nuclear structure and weak interaction. Nowadays, laser spectroscopy combined with theoretical calculations is an indispensable tool to explore many-body QED in weak and strong fields and the search for a time or spatial dependence of fundamental constants like the fine structure constant. It provides important information for the analysis of spectra from stars and quasars, for studies of the nuclear structure and for determining the weak charge of a nucleus.
The technique we present here, has provided new data in two of the mentioned fields, namely the determination of nuclear charge radii and moments of beryllium isotopes \cite{Noe09,Kri12} and the test of many-body bound-state QED calculations in three-electron systems \cite{Noe15}. It is based on collinear laser spectroscopy, a technique that has been contributing to these fields considerably and is one of the workhorses for investigations of nuclear spins and moments, which is witnessed by a long series of review papers \cite{Neu85,Ott89,Bil95,Neu02,Neu06,Che10,Bla13,Cam15} over the last decades. In parallel, it has also been used to investigate the fine structure splittings in helium-like ions as a test of bound-state QED. Such tests were carried out using boron B$^{3+}$ \cite{Din91}, nitrogen N$^{5+}$ \cite{Tho98} and fluorine F$^{7+}$ \cite{Mye99}. In these experiments counter- and copropagating beams have been used to determine absolute frequencies, while for the spectroscopy of short-lived neon isotopes a similar approach was used to calibrate the acceleration voltage of the ions \cite{Gei99,Mar11}.
For the measurements on beryllium isotopes we have further developed this technique and combined it with a frequency comb to provide high-precision measurements of the transition frequencies. A photon-ion coincidence detection provided the sensitivity required for the detection of the 20-ms isotope $^{12}$Be. These investigations were motivated twofold, by the nuclear structure aspect and the possibility to provide an important benchmark for bound-state QED calculations in three-electron systems.
For nuclear structure physics the nuclear charge radius is an important observable. Its change along a chain of isotopes is extracted with high precision from optical isotope shifts. This provides insight into differences of the radial distribution of protons and the underlying collective effects of soft or rigid deformation or cluster structures, which are often observed for the few-nucleon systems of light nuclei. Only during the last decade new experimental techniques and precise atomic structure calculations for few-electron systems gave access to the determination of charge radii of low-$Z$ nuclei ($Z < 10$) with unprecedented precision. In 2000, first calculations of the mass shift in three-electron systems \cite{Yan00} provided sufficient accuracy to extract the very small finite nuclear size effect from high-precision isotope shift measurements. Since then, calculational precision for three-electron systems has been improved by two orders of magnitude \cite{Yan03,Puc06,Yan08,Yan08b,Noe11}. Pachucki {\it{et al.}} published first results for four-electron systems \cite{Pac04,Puc13} and recently even showed results that pave the way towards boron-like five-electron systems \cite{Puc15b}.
Laser spectroscopy experiments on helium and lithium isotopes were strongly motivated by the existence of so-called halo nuclei. These are nuclear systems with the last neutron(s) being bound by only a few 100\,keV, compared to typical nuclear binding energies of the order of 5--7\,MeV/nucleon. Due to this weak binding, the neutrons are allowed to tunnel far away from the central core, having a large part of their wavefunctions beyond the classical interaction length of the strong force. These nuclei have been a hot topic in nuclear structure research since their discovery in 1983 \cite{Tan85}. Isotope shifts for such systems were measured previously in helium and lithium isotopes. Single atoms of the short-lived two-neutron and four-neutron halo nuclei $^{6,8}$He were confined in a magneto-optical trap and probed by laser light \cite{Wan04,Mue07}. The lithium isotopes including the two-neutron halo nucleus $^{11}$Li were investigated by applying two-photon resonance ionization spectroscopy \cite{Noe11,Ewa04,San06}. The beryllium isotope chain contains the one-neutron halo nucleus $^{11}$Be and the isotope $^{12}$Be which in the traditional shell model should have a closed neutron shell.
With regard to atomic structure, the vast progress in nonrelativistic few-electron bound-state QED of has opened the possibility of additional tests of many-body QED of in these light systems. The helium fine structure was recently calculated up to the order $m\alpha^7$ and is now one of the most precise QED tests in two-electron systems \cite{Pac10}. The extension of such calculations to three-electron systems proved to be much harder since the extension of the respective computational methods with explicitly correlated functions turned out to be considerably more difficult. Only recently it became possible to perform a complete calculation of $m \alpha^6$ and $m \alpha^7 \ln \alpha$ contributions to the fine structure \cite{Puc14} of a three-electron atom.
On the experimental side, measurements of the $2p$ fine structure splitting in light three-electron systems are limited in accuracy for isotopes with non-zero nuclear spin due to the unresolved hyperfine structure (hfs) in the $2p_{\nicefrac{3}{2}}$ level. This has been the reason for the fluctuating results on the fine structure splittings in lithium \cite{Noe11,Nob06} being reported for a long time. These turned out to be caused by quantum interference effects in the observation of the unresolved resonance lines \cite{San11}. Once this issue had been resolved experimentally, good agreement with ab initio calculations was obtained \cite{Bro13}. Since relativistic and QED contributions grow in size with increasing $Z$, it became important to study the fine structure splitting also in Be$^+$ to further test bound-state QED.
Both aspects have been addressed with the technique described in this paper. Besides giving a detailed description of the experiment implemented at ISOLDE (CERN), we will present an overview of the spectroscopic results obtained in two beamtimes (called Run I and Run II). Compared with the techniques used to study helium and lithium isotopes, the collinear approach has the advantage of being more generally applicable and providing high-precision isotope shift data for short-lived isotopes of elements in the so-far inaccessible region $4<Z<10$.
\section{Theory}
\label{sec:Theory}
\begin{table*}
\begin{center}
\caption{Theoretical mass shifts $\delta\nu_{\rm{MS}}^{9,A}$ and field shift factors $F^{9,A}$ for the D1 and D2 transitions \ensuremath{2s\;^2\mathrm{S}_{\nicefrac{1}{2}} \rightarrow 2p\;^2\mathrm{P}_{\nicefrac{1}{2},\, \nicefrac{3}{2}}}\ in Be$^+$ with respect to $^9$Be$^+$ obtained in two independent calculations \cite{Yan08,Yan08b,Puc08,Puc10} with updated values presented in \cite{Kri12} based on \cite{Dra10,Pac11}. The listed uncertainties are an estimation of unknown higher order terms. The calculations from \cite{Puc10} include another uncertainty that originates from the atomic mass. All values are given in MHz. The deciphered contributions to the mass shift can be found for example in \cite{Puc08}.}
\label{tab:propIS}
\vspace{2mm}
\begin{footnotesize}
\begin{tabular}{r r @{} l r @{} l r @{} l r}
\hline\hline
Isotope & $\delta\nu_{\rm{MS}}^{9,A}$ D1 & & $\delta\nu_{\rm{MS}}^{9,A}$ D2 & & $F^{9,A}$ & & Reference\\
\hline
$^7$Be$^+$ & -49~225.744(35)&(9) & -49~231.779(35)&(9) &-17.&021(31) & \cite{Puc08,Puc10,Pac11}\\
& -49~225.779(38)& & -49~231.828(38)& & -16.&912& \cite{Yan08,Yan08b,Dra10} \\
$^{10}$Be$^+$ & 17~310.459(13)&(11) & 17~312.553(13)&(11) &-17.&027(31)&\cite{Puc08,Puc10,Pac11}\\
& 17~310.442(12)& & 17~312.569(12)& & -16.&912& \cite{Yan08,Yan08b,Dra10}\\
$^{11}$Be$^+$ & 31~560.245(31)&(12) & 31~564.207(31)&(12) &-17.&020(31)&\cite{Puc08,Puc10,Pac11}\\
& 31~559.990(24)& & 31~563.868(24)& &-16.&912 & \cite{Yan08,Yan08b,Dra10}\\
$^{12}$Be$^+$ & 43~390.180(30)&(180) & 43~395.480(30)&(180) &-17.&022(31)&\cite{Puc08,Puc10,Pac11}\\
& 43~390.168(39)& & 43~395.499(39)& & -16.&912 & \cite{Yan08,Yan08b,Dra10}\\
\hline\hline\\
\end{tabular}\\
\end{footnotesize}
\end{center}
\end{table*}
It is well known that the isotope shift $\delta\nu^{A,A'}$ between two isotopes $A$ and $A'$ can be separated into the mass shift $\delta\nu_{\rm{MS}}^{A,A'}$ and the field shift $\delta\nu_{\rm{FS}}^{A,A'}$ according to
\begin{eqnarray}
\label{eq:IS}
\delta \nu_{\mathrm{IS}}^{A,A'}&=&\nu^{A'}-\nu^{A} \\
&=& \underbrace{K_{\mathrm{MS}}\frac{M_{A'}-M_{A}}{M_AM_{A'}}}_{\delta\nu_{\mathrm{MS}}^{A,A'}} + \underbrace{F^{A,A'} \delta\left\langle r_{\mathrm{c}}^2\right\rangle^{A,A'}}_{\delta\nu_{\mathrm{FS}}^{A,A'}}.
\end{eqnarray}
The mass shift contribution (MS) is related to the center-of-mass motion of the atomic nucleus. For light elements this is the major part of the isotope shift, while the small nuclear volume shift $\delta\nu_{\mathrm{FS}}^{A,A'}$, being typically at the $10^{-5}$ level of the mass shift, contains the information about the change $\delta\left\langle r_{\mathrm{c}}^2\right\rangle$ in the mean square nuclear charge radius. Extraction of nuclear charge radii from experimental isotope shifts in the lightest elements requires accurate mass shift calculations. Semi-empirical techniques that have often been applied for heavier elements to evaluate the atomic parameters $K_{\mathrm{MS}}$ and $F$ are not sufficiently accurate. Only state-of-the-art ab-initio calculations can provide the accurate mass shift and field shift coefficients. Detailed descriptions of these calculations can be found, e.g., in \cite{Yan00,Puc06,Noe11,Lu13}. Briefly, the starting point is the non-relativistic Schr\"odinger equation which is solved with high numerical accuracy in the basis of Hylleraas coordinates that explicitly take electron-electron correlations into account. The wavefunctions obtained are then used to calculate relativistic and QED corrections perturbatively as a power series in terms of the fine structure constant $\alpha$. The results for the Be$^+$ isotopes as taken from \cite{Yan08,Yan08b,Dra10,Puc10} are listed in Table~\ref{tab:propIS}. It is worthwhile to note that the calculations performed by two independent groups agree within uncertainties for all isotopes. The only significant difference concerns the case of $^{11}$Be, where the nuclear polarizability correction of 211\,kHz has been calculated and included in \cite{Puc10} but not in \cite{Dra10}. The field shift factor $F^{9,A}$ has been calculated for each isotope individually and is almost constant along the isotopic chain, besides a small difference in the relativistic correction. Using the mass shift values from the table and the measured isotope shifts, the change in the mean square nuclear charge radius can be determined using
\begin{equation}
\delta \langle r_{\mathrm{c}}^2 \rangle^{9,A} = \frac{\delta \nu_{\mathrm{IS}}^{9,A}-\delta \nu_{\mathrm{MS}}^{9,A}}{F^{9,A}}.
\label{eq:changer}
\end{equation}
The absolute charge radius $R_{\mathrm{c}}(A) = \sqrt{\langle r_{\mathrm{c}}^2 \rangle^A}$ of at least one (or more) stable isotope(s) determined by other methods is required to obtain absolute charge radii of the radioactive isotopes. In the case of beryllium the nuclear charge radius of the stable $^9$Be nucleus was determined from elastic electron scattering \cite{Jan72} and thus
\begin{equation}
R_{\mathrm{c}}(^{A}\mathrm{Be}) = \sqrt{R_{\mathrm{c}}^2(^9\mathrm{Be}) + \delta \langle r_{\mathrm{c}}^2 \rangle^{9,A}}.
\label{eq:absoluter}
\end{equation}
\begin{figure*}[bth]
\includegraphics[width=\linewidth]{BetinaExpSetup_Overview.pdf}
\caption{Experimental setup for the beryllium measurements at ISOLDE. Two dye laser systems were used to excite the \ensuremath{2s\;^2\mathrm{S}_{\nicefrac{1}{2}} \rightarrow 2p\;^2\mathrm{P}_{\nicefrac{1}{2},\, \nicefrac{3}{2}}}\ transitions in Be$^+$. The dye laser for collinear excitation (left) was operated at a fundamental wavelength of 624\,nm and stabilized to a hyperfine transition of molecular iodine. The output beam is frequency-doubled to 312\,nm and guided into the beam line. The other laser (right) is locked to a frequency comb. After frequency-doubling to 314\,nm the UV laser beam is anticollinearly superposed with the ion beam. The resonance fluorescence is detected by a pair of photomultipliers. A photon-ion coincidence detection unit increases the detection efficiency if the ion beam rate is low (not shown, see Fig.\,\ref{fig:COLLAPSel}).}
\label{fig:COLLAPSlasersetup}
\end{figure*}
The many-electron Dirac equation poses some difficulties for the inclusion of relativistic effects and correlations between electrons in atomic systems. According to QED the equation has to include multiple electron-positron pairs, which leads to numerical instabilities. This problem limited the relativistic calculation of the lithium
$2p\,^2\mathrm{P}_{\nicefrac{1}{2}}$ -- $2p\,^2\mathrm{P}_{\nicefrac{3}{2}}$ splitting to one significant digit \cite{Der08}. Forty years after first numerical calculations using explicitly correlated basis sets with Hylleraas and
Gaussian functions for two electrons \cite{Dou74}, Puchalski and Pachucki extended such calculations to three-electron systems \cite{Puc14}. Nonrelativistic QED can perturbatively account for relativistic, retardation, electron self-inter\-action, and vacuum polarization contributions by an expansion of the level energy in
powers of the fine structure constant $\alpha$
\begin{equation}
E = m\alpha^2 E^{(2)} + m\alpha^4 E^{(4)} + m\alpha^5 E^{(5)} + m\alpha^6 E^{(6)} + ... \, ,
\end{equation}
where the expansion coefficients $E^{(i)}$ may include powers of $\ln \alpha$. In this expansion, the fine structure arises at the order of $m\alpha^4$, together with the nuclear recoil term, which in this order is comparable in size to $m\alpha^6$ contributions, but of opposite sign. For all details of the calculations and the individual contributions we refer to \cite{Puc15}.
In the splitting isotope shift (SIS), i.e. the difference in fine structure splitting between isotopes, all mass-independent terms cancel and only the mass-dependent terms remain, which can be calculated with very high accuracy.
The SIS therefore provides a valuable consistency check of the experimental results \cite{Yan02}. For isotopes with nuclear spin, hyperfine-induced fine-structure mixing can lead to an additional level shift that also contributes to the SIS. This in combination with the unresolved hyperfine splittings in the $2p\,^2\mathrm{P}_{\nicefrac{3}{2}}$ level in light three-electron systems makes even-even isotopes with nuclear spin $I=0$ an exceptionally suitable case to perform tests of the calculations. While there is no such isotope for lithium, the beryllium chain with $^{10}$Be and $^{12}$Be includes two spinless isotopes that are accessible to the measurement.
\section{Experimental Setup}
A schematic overview of the experimental setup applied for collinear laser spectroscopy on Be$^+$ ions in the \ensuremath{2s\,^2{\rm{S}}_{\nicefrac{1}{2}} \rightarrow 2p\,^2{\rm{P}}_{\nicefrac{1}{2}}}\ (D1) and \ensuremath{2s\,^2{\rm{S}}_{\nicefrac{1}{2}} \rightarrow 2p\,^2{\rm{P}}_{\nicefrac{3}{2}}}\ (D2) transitions is shown in Fig.~\ref{fig:COLLAPSlasersetup}. A mass-separated ion beam of a stable or a radioactive beryllium isotope at an energy up to 60~keV was transported to the laser beam line. Two frequency-stabilized dye laser systems delivered UV beams that superposed the beryllium ion beam in opposite directions. The resonance fluorescence photons were detected via photomultipliers. The resonance condition was established by tuning the Doppler-shifted frequency with an electrical potential applied to the fluorescence detection chamber.
The individual parts of the experimental setup as well as the scanning procedure are described in detail in the following subsections.
\subsection{Production of Radioactive Beryllium Isotopes}
The stable and radioactive beryllium isotopes were produced at the on-line isotope separator facility ISOLDE at CERN. High-energy (1.4~GeV) protons from the PS-Booster synchrotron impinge on a uranium carbide target. The atoms are photo-ionized using the resonance ionization laser ion source RILIS \cite{Fed08}. Resonant excitation at 234.9~nm from the atomic ground state in the $2s^2$ $^{1}{\rm{S}}_0 \rightarrow 2s 2p$ $^{1}{\rm{P}}_1$ transition, followed by excitation at 297.3~nm to the auto-ionizing $2p^2$ $^{1}{\rm{S}}_0$ level was employed to ionize the Be atoms which have the rather large ionization potential of 9.4~eV.
Table~\ref{tab:isoldeyield} lists the ion beam intensities decreasing from $^{7}$Be to $^{12}$Be by seven orders of magnitude \cite{Koe98}.
In the final stage of our experiment an upgraded solid-state pump laser system \cite{Fed08} was used. This gave a $^{11}$Be yield of up to $2.7 \cdot 10^7$ ions/s, about 4 times larger than reported previously.
The yields are sufficient to perform collinear laser spectroscopy on $^{7-11}$Be solely based on a standard fluorescence detection system. However, for a beam intensity of less than 10$^4$~ions/s, i.e. for measurements on $^{12}$Be, the sensitivity had to be enhanced. This was achieved by detecting ion-photon coincidences and thus rejecting the stray light background which usually determines the sensitivity limit. Coincidence detection requires an isobarically clean ion beam. For that reason the pulse structure and possible contamination of the beam was investigated and optimized for $^{12}$Be.
\begin{table}
\begin{center}
\caption{Nuclear properties and production rates of the beryllium isotopes at the ISOLDE facility at CERN. The table includes the half-life ($T_{1/2}$), nuclear spin $I$, magnetic dipole moment ($\mu_I$) in nuclear magnetons ($\mu_{\mathrm{N}}$) and the yields using a 1.4-GeV proton beam from the PS booster and RILIS for ionization \cite{Iso13}.}
\label{tab:isoldeyield}\vspace{2mm}
\begin{tabular}{l l c r l c}
\hline\hline
& T$_{1/2}$ & I & \multicolumn{1}{c}{$\mu_{I}/\mu_{\mathrm{N}}$}& $({\rm{ions}}/\mu {\rm{C}})$ \\
\hline
~$^{7}$Be & 53 d & 3/2 & --1.39928(2)\,\cite{Oka08} & $1.4 \cdot 10^{10}$ \\
~$^{9}$Be & stable & 3/2 & --1.177432(3)\,\cite{Win83} & \\
$^{10}$Be & $1.6\cdot10^6$\,a&0& ~~~--~~~ & $6.0 \cdot 10^{9}$\\
$^{11}$Be & 13.8 s & 1/2 & -1.6813(5)~~\cite{Noe09} & $7.0 \cdot 10^{6}$\\
$^{12}$Be & 23.8 ms & 0 & ~~~--~~~ & $1.5 \cdot 10^{3}$\\
$^{14}$Be & 4.35 ms & 0 & ~~~--~~~ & $4.0 \cdot 10^{0}$\\
\hline\hline \\
\end{tabular}
\end{center}
\end{table}
\subsection{Beryllium Ion Beam Structure} \label{sec:ionprop}
During our experiment at ISOLDE, pulses of $3 \cdot 10^{13}$ protons impinged on a UC$_x$ target typically every 4\,s. The release of resonantly ionized $^{12}$Be was tracked using a secondary electron multiplier installed at the end of the laser spectroscopy beam line.
The proton pulses triggered a multichannel analyzer that recorded the ion events as a function of time. Figure\,\ref{fig:release} shows such a release curve summed over 100 proton pulses with a resolution of 0.2\,ms/channel. The integral corresponds to a release of 12\,000\,ions per proton pulse. This is almost a factor of 10 more than listed in the yield table (Tab.\,\ref{tab:isoldeyield}). During the measurements on $^{12}$Be the typical ion yield was about 8\,000\,ions/pulse.
The release curve of Fig.\,\ref{fig:release} demonstrates a characteristic feature of the ISOLDE HV supply: First ions are detected about 2-3\,ms after the proton pulse hit the target. This delay is determined by the recovery time of the high voltage, which is pulsed down right before a proton pulse hits the target, in order to reduce the current load from ionized air \cite{Fia92}. After an initial steep rise the release curve follows essentially the exponential decay of $^{12}$Be. The extracted half-life of $T_{1/2} \approx 21.9(8)$\,ms agrees well with the literature value of 21.50(4)\,ms \cite{Aud97}. The single exponential does not exhibit any significant offset. This demonstrates that practically no beam contamination from the isobar $^{12}$C$^+$ is present and $^{12}$B$^+$ (having a similar lifetime as $^{12}$Be) is also not expected due to the relatively high ionization potential. This situation is prerequisite for the application of a photon-ion coincidence technique, which otherwise would suffer from random coincidences between scattered laser light and isobaric ions. With the rapid decay of $^{12}$Be the fluorescence detection can be limited to about 100~ms after the proton pulses.
\begin{figure}
\begin{center}
\includegraphics[width=0.9\linewidth]{releasecurve.pdf}
\end{center}
\caption{Release of beryllium ions (solid blue line) from ISOLDE as a function of time after the proton pulse hit the target container, measured with a secondary electron multiplier at the end of the COLLAPS beam line. The release curve, integrated over 100 proton pulses with a resolution
of 0.2\,ms/channel, is modelled with an exponential decay curve (dash-dotted red line).(Color online)}
\label{fig:release}
\end{figure}
\subsection{Experimental Beam Line} \label{sec:COLLAPSsetup}
The COLLAPS collinear spectroscopy beam line at the ISOLDE facility was commissioned in the early eighties \cite{Neu81,Buc82,Mue83} and has been improved continuously \cite{Neu06,Gei99,Neu86,Gei05,Ney05,Kow05} with the objective of widening the range of accessible elements and isotopes. An important aspect was the development of highly sensitive alternatives to the traditional flourescence photon detection technique.
\begin{figure*}
\begin{center}
\includegraphics[width=0.85\linewidth,clip=true]{COLLAPS_APB.pdf}
\end{center}
\caption{Schematic view of the COLLAPS beam line at ISOLDE/CERN and corresponding high-voltage circuits: A mass-separated ion beam is directed along the axis of the vacuum beam line with the help of an electrostatic deflector. An electric dipole and quadrupole collimates and steers the beam through the apparatus. The fluorescence detection region is a cage floated on a variable potential against ground to enable Doppler-tuning. At the end of the beam line a photon-ion coincidence detection chamber is installed, whereby a secondary electron multiplier is used to count the ions. The generation and measurement of the high-voltage potential is explained in the text.}
\label{fig:COLLAPSel}
\end{figure*}
For conventional collinear spectroscopy the ions are accelerated to a beam energy of typically 50\,keV, with the corresponding positive potential applied to the ion source, while the mass separator and the experimental beam line are on ground potential. The ion beam is merged with a laser beam by a pair of deflector plates as shown in Figs.\,\ref{fig:COLLAPSlasersetup} and \ref{fig:COLLAPSel}.
A quadrupole triplet collimates the ion beam, matching it to the laser beam profile, and a second set of deflector plates aligns it with the laser beam axis which is defined by two apertures at a distance of about 2\,m. Two UV-sensitive photomultiplier tubes with 45\,mm active aperture and 15\,\% quantum efficiency at 313\,nm are used for fluorescence detection. The light collection system consists of two fused-silica lenses of 75\,mm diameter and a cylindrical mirror opposite to them.
Stray laser light is suppressed by sets of apertures with diameters decreasing with distance from the optical detection region and Brewster-angle quartz windows at both ends.
Collinear laser spectroscopy is usually performed with the laser running at a fixed frequency, while the absorption frequency of the ions is tuned by changing their velocity (Doppler-tuning). This means that a variable electrical potential has to be applied to the interaction region. For applying post-acceleration/deceleration voltages up to 10\,kV a set of four electrodes provides a smoothly variable potential along the beam axis. In order to avoid optical pumping into dark states, the final ion velocity is reached just in front of the detection region by applying a small fixed offset voltage between the last electrode and the detection chamber.
The lower part of Fig.~\ref{fig:COLLAPSel} illustrates the generation of the voltage between the detection region and ground as a combination of a static high voltage in the range of $\pm 10$~kV and a scanning voltage of $\pm 500$\,V. The latter is created by amplification ($\times 50$) of the $\pm$ 10\,V dc output of an 18-bit DAC controlled by the measuring computer.
This voltage defines the floating offset potential of a stabilized $\pm 10$\,kV power supply. The combination of power supplies makes it possible to perform measurements on a series of isotopes with different Doppler shifts and for each of them scan small frequency ranges covering the hyperfine structure with high resolution.
Both the static high voltage and the scanning voltage are measured with a high-precision 1:1000 voltage divider and a digital voltmeter. A comparison of the measured voltages with those obtained using a
precision voltage divider calibrated at PTB (Braunschweig, Germany)
\cite{Thu09} has demonstrated an uncertainty of $\nicefrac{\Delta U}{U} < 3 \cdot 10^{-5}$ which corresponds to about 0.3\,V at a maximum
voltage of 10\,kV applied to the excitation region. Still, the knowledge
of the ion beam velocity is limited by the uncertainty of the ion source
potential which is determined by the main acceleration voltage power
supply. Also the specified accuracy $\nicefrac{\Delta U}{U} < 1 \cdot
10^{-4}$ of the voltage measurement on the operational voltage of
60\,kV was verified by calibration with the precision voltage divider \cite{Kri11}.
It translates to less than 6\,eV uncertainty in the ion beam energy.
As in the laboratory frame the transition frequency of the ions in collinear geometry scales as
\begin{equation}
\nu_{c} = \nu_0 \, \gamma \, \left( 1 + \beta \right)
\label{eq:vnu}
\end{equation}
where the dimensionless ion velocity is $\beta = v/c$, the relativistic factor $\gamma = \sqrt{1/(1-\beta^2)}$ and the transition frequency $\nu_0$, any uncertainty in $\beta$ arising from the ion source potential results in an uncertainty of measured absolute transition frequencies or isotope shifts, especially for light ions. In the particular case of beryllium a deviation of 6\,V from the measured voltage result in an artificial isotope shift of $\delta \nu^{9,11} \left(^{9} \textrm{Be},^{11}\textrm{Be} \right) = 18$\,MHz.
To overcome these limitations, we have introduced the (quasi-)simultaneous excitation by a collinear and an anticollinear laser beam. The method is based on the fact that in this geometry the measured resonance frequencies, $\nu_{c} = \nu_0 \gamma (1+\beta)$ for collinear and $\nu_a = \nu_0 \gamma (1-\beta)$ for anticollinear excitation are simply related to the rest frame frequency $\nu_0$ by
\begin{equation}
\nu_{c} \cdot \nu_{a} = \nu^2_0 \gamma^2 \cdot \left( 1 + \beta \right)\left( 1 - \beta \right) = \nu^2_0.
\label{eq:vnull}
\end{equation}
This provides a method to determine the transition frequency independently of the knowledge of the ion beam energy which depends on assumptions about the ion source potential and on measured voltages. However, in contrast to conventional collinear laser spectroscopy, this approach requires two laser systems instead of one and, additionally, the capability to determine the absolute laser frequencies with an accuracy better than $10^{-9}$. Similar approaches were proposed and demonstrated for the measurement of absolute transition frequencies \cite{Pou88} and used for, e.g., precision spectroscopy in the fine structure of helium-like Li$^+$, yielding an accurate value of the Lamb shift \cite{Rii94}. Here we have developed a procedure which is widely applicable in cases where high precision is required for the spectroscopy of unstable isotopes.
\subsection{Setup and Specification of the Frequency-Comb-Based Laser System} \label{sec:COLLAPSlaser}
The transition wavelength of the \ensuremath{2s\;^2\mathrm{S}_{\nicefrac{1}{2}} \rightarrow 2p\;^2\mathrm{P}_{\nicefrac{1}{2},\, \nicefrac{3}{2}}}\ transitions in Be$^+$ is about 313\,nm corresponding to an energy splitting of $\approx 4$\,eV. The laser system installed at COLLAPS is schematically shown in Fig.\,\ref{fig:COLLAPSlasersetup}. For anticollinear excitation a frequency doubled Nd$:$YVO$_4$ laser (Verdi V18) was operated at 9\,W to pump a Coherent 699-21 dye laser. Using a dye solution of Sulforhodamine B in ethylene glycol, a typical output power of 700\,mW was achieved at the wavelength of 628\,nm. Another dye laser, Sirah MATISSE DS, was installed for collinear excitation and operated with a dye solution of DCM in 2-phenoxy-ethanol. With the 8-W pump beam from a Verdi V8, about 1.2\,W were achieved at the fundamental wavelength of 624\,nm. Each laser beam was then coupled into a 25-m long photonic crystal fiber (LMA-20) to transport the laser light to one of the second harmonic generators installed in the ISOLDE hall. A two-mirror delta cavity (Spectra Physics Wavetrain) and a four-mirror bow-tie cavity (Tekhnoscan FD-SF-07) were located nearby the COLLAPS beam line. A Brewster-cut and an anti-reflection coated BBO crystal, respectively, converted the laser beams of 628\,nm and 624\,nm into their second harmonics at 314\,nm and 312\,nm, in both cases with an output of more than 10\,mW. The elliptical UV beams were reshaped to circular beams with diameters of 3-4\,mm to match the transversal profile of the ion beam and finally attenuated to powers below 5\,mW. Two remote-controlled fast beam shutters blocked alternatively the collinear or the anticollinear laser beam. This enabled us to perform scans of 3-30\,s duration in collinear or anti-collinear configuration in a fast sequence.
\begin{figure}
\includegraphics[width=\linewidth,clip=true]{stability_frequencycomb_196HZ_V2.pdf}
\caption{Beat frequency histograms of the frequency-stabilized dye lasers. The beat was averaged for 1\,s and the distribution of the beat frequencies over a period of 2 hours is shown in the histogram. Graph (A) shows the histogram of the MATISSE DS laser stabilized to a hyperfine transition of molecular iodine. The corresponding uncertainty was estimated as the FWHM of a Gaussian fit (red) of about 75\,kHz. The results of the frequency-comb stabilized Coherent 699-21 dye laser is depicted in Graph (B). In this case the FWHM is approximately 400\,Hz.}
\label{fig:coherentdw}
\end{figure}
The backbone of the laser system was the precise frequency stabilization and frequency measurement required for the application of Eq.\,(\ref{eq:vnull}). In practice, the transition rest-frame frequency $\nu_0$ depends on the absolute output frequencies of both dye lasers which have to be known with a relative accuracy better than $\Delta\nu/\nu \leq 10^{-9}$ to yield the isotope shifts with an accuracy better than $10^{-5}$.
Therefore, a Menlo Systems frequency comb (FC 1500) with a repetition frequency of 100\,MHz was employed. A Stanford Research rubidium clock (PRS10) provided the 10-MHz reference for the stabilization of the carrier-envelope-offset (CEO) and the repetition frequency. This clock was long-term stabilized using a GPS receiver tracking the 1-pps signal.
The MATISSE dye laser for collinear excitation was stabilized to its internal reference cavity for short-term stability. In this case frequency drifts were further reduced by locking the laser to a hyperfine transition in molecular iodine using frequency-modulated saturation spectroscopy. In total 12 hyperfine transitions of $^{127}$I$_2$ match the desired Doppler-shifted frequencies for a wide range of acceleration voltages between 30--60\,kV. The demodulated dispersion signal from the phase-sensitive detection was fed into a 16-bit National Instruments DAQ card (NI-DAQ 6221) and further processed with the MATISSE control software to provide a counter-drift for the MATISSE reference cavity. In regular time intervals the laser frequency was measured with the frequency comb and recorded for a few 100\,s to ensure the stability of the locking point and to provide the absolute frequency for the application of Eq.\,(\ref{eq:vnull}). A histogram of 1-s averaged beat signals measured over 2 hours is depicted in Fig.~\ref{fig:coherentdw}(A). It exhibits a Gaussian distribution with standard deviation of about 75~kHz.
The frequency of the Matisse laser stabilized to the various iodine lines was repeatedly measured during the beamtimes. The averaged results are listed in Table~\ref{tab:AF1} and compared with the calculated frequencies from \cite{Kno04}. Reasonable agreement is obtained in all cases.
\begin{table*}
\begin{center}
\caption{Frequencies of the $a_1$ hyperfine component in various transitions of iodine $^{127}$I$_2$ utilized and determined during the experiment. The total uncertainty of the experimental values is about 190\,kHz. The calculated frequencies (theory) are afflicted with an uncertainty of approximately 3\,MHz \cite{Kno04}.}
\label{tab:AF1}
\begin{tabular}{llll}
\hline\hline
HFS a$_1$ & Frequency & Frequency & Deviation\\
transition &(theory) (MHz) & (experiment) (MHz) & (MHz)\\
\hline
R(62)(8-3) & 479~804~354.67 & 479~804~355.09 &--0.42\\
R(70)(10-4) & 479~823~072.75 & 479~823~072.58 &~0.17\\
P(64)(10-4) & 479~835~709.4 & 479~835~708.96 &~0.44\\
R(60)(8-3) & 479~870~011.92 & 479~870~012.20 &--0.28\\
R(58)(8-3) & 479~933~416.07 & 479~933~416.36 &--0.29\\
R(56)(8-3) & 479~994~568.08 & 479~994~568.11 &--0.03\\
R(54)(8-3) & 480~053~468.95 & 480~053~469.06 &--0.11\\
R(52)(8-3) & 480~110~119.57 & 480~110~119.59 &--0.02\\
R(50)(8-3) & 480~266~578.9 & 480~266~578.59 &~0.31\\
R(48)(8-3) & 480~314~237.19 & 480~314~236.83 &~0.36\\
R(42)(8-3) & 480~359~649.42 & 480~359~649.13 &~0.29\\
R(40)(8-3) & 480~402~816.3 & 480~402~815.78 &~0.52\\
\hline\hline\\
\end{tabular}\\
\end{center}
\end{table*}
The Coherent 699 dye laser for anticollinear excitation was internally stabilized to its own reference cavity of Fabry-Perot type, long-term frequency drifts were corrected by an additional stabilization to the frequency comb. Therefore the beat signal between the dye laser and the nearest frequency comb mode was detected on a fast photo diode and fed into the Menlo Systems phase comparator DXD 100. A low-noise PI regulator (PIC 210) processed the signal from the phase comparator and provided a servo-voltage to counteract all frequency excursions of the dye laser by correcting the length of the reference cavity.
As a measure of the long-term stability a beat signal with the frequency comb was detected. The result is shown in Fig.\,\ref{fig:coherentdw}(B). The standard deviation over 2 hours measuring time and 1-s averaging time is about 400~Hz.
\section{Measurement Procedure} \label{sec:procedureberyllium}
The ion beam acceleration voltage at the ISOLDE front end was fixed to
40\,kV.
A suitable iodine line is chosen such that the isotope under investigation
can be recorded by applying an offset voltage in the available range of
$U_{{\rm{Offset}}}=\pm 10$\,kV at the fluorescence detection region. For
example, when choosing the $a_1$ hyperfine component in the transition
R(56)(8-3) as a reference, the isotopes $^{9-12}$Be can be addressed.
The scan voltage range $U_{\rm{scan}}$ of up to $\pm 500$\,V is then
adjusted to cover the full hyperfine structure in the collinear
direction and the expected position of the center of gravity is
calculated. The required offset voltage as well as the scan voltage
range was estimated based on previous measurements of the beryllium
absolute transition frequency \cite{Bol85} and nuclear moments
\cite{Gei99} in combination with the precisely calculated mass shift
\cite{Dra10,Puc10}. Once the resonance position of $^9$Be was found, the
laser frequencies $\nu_c$ and $\nu_a$ could be predicted for all
radioactive beryllium isotopes with an accuracy of a few MHz,
which is the size of the expected field shift contribution.
This knowledge allowed us to calculate the required frequency of the
second dye laser to simultaneously cover the full hyperfine structure in
anticollinear geometry within the same Doppler tuning voltage range and
even to ensure that the centers of gravity of both hyperfine spectra
practically coincide within a few 100\,mV, corresponding to the size
of the field shift contribution. This is only possible because this dye
laser is locked to the frequency comb and thus can be stabilized at any
arbitrarily chosen frequency.
Fast laser beam shutters placed in front of the Brewster windows of the
apparatus were controlled by the data acquisition software in order to
allow only one of the two laser beams to enter.
For the isotopes with half-lives longer than the typical 4-s repetition time of proton pulses, fast scans of the Doppler-tuning voltage $U_{{\rm{scan}}}$ were performed with alternating laser beams. The scanning range was chosen depending on the hyperfine splitting of the respective isotope and spectra were taken in 200 channels for $^{10}$Be and up to 800 channels for the odd-$A$ isotopes $^{7,9,11}$Be. The common dwell time was 22\,ms per voltage step.
Depending on the ion beam intensity, a single spectrum is the
sum of 50--800 individual scans for each direction.
This procedure was applied using about 3--4 different iodine lines for
each isotope.
Because of the short 21.5-ms half-life of $^{12}$Be, photon counts had to be accumulated for typically 60\,ms after each proton pulse. The laser shutters for collinear and anticollinear beams were switched between consecutive pulses and the voltage steps were triggered by every second pulse. Given the extremely low ion beam intensity, the single-line spectrum of $^{12}$Be was taken in only 20 channels with a total measuring time of about 8 hours, corresponding to 200 scans.
\begin{figure}[bth]
\begin{center}
{\includegraphics[width=\linewidth]{Coincidencespec.pdf}}
\caption{Comparison between conventional optical fluorescence
detection (upper trace, right $y$-axis) and photon ion coincidence
detection (lower trace, left $y$-axis) at an ion beam rate of
30\,000\,$^{10}$Be$^+$ ions/s. The optical spectrum (black circles) in the
conventional detection is covered by stray light of the laser beam. In
the photon-ion coincidence spectrum a clear resonance (blue circles) is
observed, fitted with a Voigt profile.}
\label{Fig:bepicoincidence}
\end{center}
\end{figure}
Detection of the weak $^{12}$Be signals required the additional rejection of background from scattered laser light reaching the photomultiplier tubes. This was achieved by implementing
a photon-ion coincidence:
Photomultiplier signals were accepted only if an ion was simultaneously
traversing the detection unit. Downstream of the photon detection chamber the ions were deflected onto the cathode of a secondary
electron multiplier (SEM) installed off-axis. Discriminated pulses from the photomultipliers were delayed by the appropriate time of flight (TOF) (3--4\,$\mu$s) of the ion to the SEM.
To avoid electronic dead times, the delay was realized logically in a
first-in first-out (FIFO) queue structure on a field-programmable gate
array (FPGA) with a resolution of 10\,ns, based on the FPGA's internal
clock. Signals leaving the queue were transformed back into a TTL pulse
and fed together with the SEM pulses into a standard coincidence unit.
The photon-ion coincidence detection was optimized using a $^{10}$Be$^+$
ion beam, attenuated to about 30.000\,ions/s by detuning the RILIS
laser. The time of flight for $^9$Be was determined with a multi-channel
analyzer and the respective TOF for $^{10}$Be was calculated.
Figure~\ref{Fig:bepicoincidence} shows a comparison between the
conventional ungated spectrum (grey circles) and the optical spectrum
detected in delayed coincidence (blue circles). The resonance peak is
only visible in the gated spectrum. The background induced by laser
stray-light was reduced by a factor of 35. However, it must be noted
that this reduction factor strongly increases with a reduction of the
ion beam rate.
\section{Analysis and Results}
Two beam times were performed to investigate first the isotopes
$^{7-11}$Be (Run I) \cite{Noe09} and then concentrate on
$^{12}$Be (Run II) \cite{Kri12} after installing the
ion-photon coincidence setup. The stable isotope $^9$Be and the
even-even isotope $^{10}$Be were used as reference isotopes,
respectively. We concentrate here on results and procedures from Run II
and provide differences to Run I only when it is of importance.
\subsection{Line Shape Studies on $^9$Be and $^{10}$Be} \label{sec:refscans}
Resonance spectra of $^9$Be in the \ensuremath{2s\,^2{\rm{S}}_{\nicefrac{1}{2}} \rightarrow 2p\,^2{\rm{P}}_{\nicefrac{1}{2}}} tran\-sition are shown in the
upper trace of Fig.~\ref{Fig:9benew} taken in collinear (left) and
anticollinear geometry (right) as a function of the Doppler-tuned laser
frequency. Each spectrum is the sum of 20 individual scans. To avoid
saturation broadening, the laser beam was attenuated to 3\,mW and
collimated to a beam diameter of about 3--4\,mm to match approximately
the size of the ion beam. Similar spectra for $^{10}$Be are shown
in the lower traces. These are the integral of 50 single scans
at an ion beam current of 10\,pA. A best fit of the resonance was obtained for a
double Voigt profile with a full
width half maximum (FWHM) of 40~MHz. It
becomes apparent that each peak in the hyperfine structure is actually
a composition of two components: A satellite peak, with a
typical intensity of less than 5\% of the corresponding main peak,
appears on the low-energy tail in each spectrum independent of the
direction of excitation. It is induced by a class of ions which have
lost some of their kinetic energy. The loss is
almost exactly 4\,eV and can be explained by inelastic collisions with residual gas atoms that lead to excitations into the $2p$ states. The energy required for this excitation is taken from the kinetic energy of the ion
and is lost when the excited ion decays to the ground state by
emitting a photon. The overall line shape is
reasonably well fitted using a Voigt doublet and only small structures remain in the residua,
depicted below each spectrum. The remaining small asymmetry seen in this structure is similar for the different peaks. It is an asset of the technique that asymmetries in the collinear and
the anticollinear spectra shift the peak center to slightly lower and
slightly larger frequencies, respectively. Hence, these shifts
largely cancel when calculating the rest frame frequency.
\subsection{Hyperfine Fitting Procedure}
\label{sec:lineshapefitting}
Fitting was performed as follows: Each voltage information was converted
into the corresponding Doppler-shifted laser frequency to account for
the small nonlinearities in the voltage-frequency relation. Hyperfine
peak positions relative to the center of gravity $\nu_{\mathrm{cg}}$
were calculated based on the Casimir formula:
\begin{figure}
\begin{center}
\includegraphics[width=\linewidth]{Be9_Be10_2010_Spectra.pdf}
\caption{Top: Optical hyperfine spectrum of the \ensuremath{2s\,^2{\rm{S}}_{\nicefrac{1}{2}} \rightarrow 2p\,^2{\rm{P}}_{\nicefrac{1}{2}}}\ transition in
$^{9}$Be$^+$ for collinear (left) and anticollinear excitation (right).
Spectra were taken at 35-kV ISOLDE voltage and are the sum of 20
individual scans with a resolution of 400 channels. Hyperfine spectra
are fitted using a multiple Voigt profile for each component (red line,
for further details see text). Striking is the appearance of a small
satellite peak on the left of each component which is ascribed to energy
loss of the ions in inelastic collisions in flight. The differential
Doppler-tuning parameter is about 39\,MHz/V. Bottom: Resonance spectra of the \ensuremath{2s\,^2{\rm{S}}_{\nicefrac{1}{2}} \rightarrow 2p\,^2{\rm{P}}_{\nicefrac{1}{2}}}\ transition in $^{10}$Be$^+$ again
in collinear and anticollinear geometry also fitted with a Voigt-doublet. A small structure in the residua (shown on the bottom of each graph) remains in all cases, which is discussed in the text.}
\label{Fig:9benew}
\end{center}
\end{figure}
The position of each hfs sublevel with total angular momentum
$\vec{F}=\vec{I}+\vec{J}$, composed of electronic angular momentum $J$
and nuclear spin $I$, and $C=F(F+1)-I(I+1)-J(J+1)$
is given to first order by the hyperfine energy
\begin{equation}
\Delta
E_{\rm{hfs}}=\frac{A}{2}C+B\cdot\frac{\frac{3}{4}C(C+1)-I(I+1)J(J+1)}{2I(2I-1)J(2J-1)}
\ .
\label{eq:hfsForm}
\end{equation}
In the fit function, these shifts determine the spectral line positions
relative to the center of gravity. The factors $A$ and $B$ (only for
$2p\,^2\mathrm{P}_{\nicefrac{3}{2}}$) of the upper and the lower fine
structure state of the transition and the center of gravity
$\nu_\mathrm{cg}$ are the free fitting parameters for the peak
positions. The line shape of each component was modelled by two Voigt
resonance terms representing the main peak and the satellite peak
as discussed above. The
distance between the two peaks was fixed to $4$\,V on the voltage axis.
The Gaussian (Doppler) line width parameter and the intensity ratio
between the main peak and the satellite were free parameters but
constrained to be identical for all hyperfine components, while the
total intensity of each component was also a free parameter. The Lorentzian
line width was kept fixed at the natural line width of 19.64\,MHz since
significant saturation broadening was not observed.
Non\-linear least-square minimization of $\chi^2$ was performed using a
Levenberg-Marquardt algorithm.
Fitting the collinear and the anticollinear spectra independently, we
obtain in both cases the centroid frequency $\nu_{\mathrm{cg}}$ of the hyperfine structure.
However, for calculating the absolute transition rest-frame frequency $\nu_0$ we
must take into account that Eq.\,(\ref{eq:vnull})
requires $\nu_c$ and $\nu_a$ to be measured at the same ion velocity.
This is only the case if the center of gravity appears in both spectra
at the same voltage. This was accomplished approximately by changing
the frequency of the laser used for anticollinear excitation until the
deviation between the corresponding centers of gravity was typically smaller than
3\,V, facilitated by the fact that the comb-stabilized laser
can be locked at any arbitrary frequency. The remaining small shift
$\delta U$ was considered in the analysis by correcting the collinear frequency
using the linear approximation $\delta \nu = \nicefrac{\partial
\nu}{\partial U} \cdot \delta U$, where $U$ is the total acceleration
voltage of the ions which have entered the optical detection
region. Hence, the transition rest-frame frequency was calculated
according to
\begin{equation}
\label{eq:diffdopplertransition}
\nu_0 = \sqrt{\left(\nu_c - \frac{\partial \nu_{\rm{D}}}{\partial U}
\cdot \delta U \right) \cdot \nu_{a}} - \delta \nu_{\rm rec}.
\end{equation}
with the differential Doppler shift
\begin{equation}
\frac{\partial \nu_{\rm{D}}}{\partial U} = \frac{\nu_0}{mc^2}\left(
e+\frac{e(mc^2+eU)}{\sqrt{eU(2mc^2+eU)}} \right)
\end{equation}
and the recoil correction term
\begin{equation}
\delta \nu_{\rm{rec}} = \frac{h \nu^2_{\rm{photon}}}{m_0c^2}.
\end{equation}
The latter takes energy and momentum conservation during the
absorption/emission process into account. It contributes with about
200\,kHz to the absolute transition frequency and is slightly isotope-dependent.
Each measurement of $\nu_0$ was repeated at least five times for each isotope. Statistical fitting uncertainty of the center of gravity was usually less than 100\,kHz. For each pair of collinear/anti\-collinear spectra the absolute transition frequency was calculated and the final statistical uncertainty was then derived as the standard error of the mean of all measurements being usually of the order of 100--500\,kHz.
\subsection{Investigations of Systematic Uncertainties}
\label{sec:systematicuncertainty}
Sources of systematic errors were investigated on-line in Run I and Run
II as well as in an additional test run, when only a previously
irradiated target was used to extract the long-lived isotope $^{10}$Be.
For each isotope about 3--4 different iodine hyperfine transitions were
used as reference points for the collinear laser frequency, which
implies different locking frequencies of the comb-locked anticollinear
laser as well as different offset voltages at the fluorescence detection
region. It should be noted that the actual locking frequency of the
iodine-locked laser was regularly checked with the frequency comb during
each block of measurements.
In Run I, the Rb reference clock for the frequency comb was not long-term stabilized on the 1-pps signal and contributed with about 350\,kHz to the systematic uncertainty of the absolute transition frequency \cite{Noe09}.
Additional uncertainties related to
the applied acceleration voltages could only arise from the center-of-gravity
correction according to Eq.\,(\ref{eq:diffdopplertransition}), which was
typically less than 3\,V. The HV-amplification
factor, calibrated regularly to better than
$3 \cdot 10^{-4}$, leads to uncertainties clearly below the 3-mV level corresponding to approximately 100\,kHz in transition frequency. This contribution can be safely neglected compared to other systematic
uncertainties discussed below.
Additionally, ion and laser beam properties were modified on purpose for
investigating a possible influence on the measured transition frequencies
and isotope shifts. In deviation from a parallel collimation the ion beam
was focused close to
the fluorescence detection region with the available electrostatic
quadrupole lenses. Similarly, additional convex lenses were added into
the light path to focus the laser beams inside the beam line. It was
found that these modifications merely changed the signal-to-noise ratio
but had no significant influence on the determined resonance frequencies.
\subsubsection{Laser-Ion-Beam Alignment} \label{sec:alignment}
The parallel and antiparallel alignment of the respective laser beams
with the ion beam was ensured using two apertures inside the beam line.
Hence, the range of a possible angle misalignment between laser and ion
beam was estimated taking the full aperture of 5\,mm and their distance
to each other of 2\,m into account. A conservative estimate with beam
diameters of about 4\,mm results in an angle of
$\alpha = \arctan\left(\Delta z / \Delta x\right) \approx
1$~mrad. However, during the preparation of the experiment, both laser
beams were superimposed 2~m after the collinear and anticollinear exit
windows, respectively. Two extreme cases are to be discussed: The
Doppler-shifted frequencies $\nu_{c,a}$ get angle-dependent if both
laser beams are well superposed, but are misaligned relative to the
ion beam
\begin{equation}
\label{eq:dopplerangle}
\nu_{c,a} = \nu_0 \gamma ( 1 \pm \beta \cdot \cos\alpha).
\end{equation}
Then the transition rest-frame frequency $\nu_{0}$
becomes angle-dependent as well
\begin{eqnarray}
\label{eq:dopplerangledoble}
\nu_{0} & = & \frac{1}{\gamma} \sqrt{\frac{\nu_c \cdot \nu_{ac}}{1
- \beta^2 \cdot \cos^2\alpha}} \\
&=& \sqrt{\frac{1 - \beta^2}{1 - \beta^2 \cos^2 \alpha}}
\sqrt{\nu_c \cdot \nu_{ac}} \\
&\approx& \left( 1 - \beta^2 \alpha^2 \right)\sqrt{\nu_c \cdot
\nu_{ac}}.
\end{eqnarray}
Even though the Doppler shift is reduced for both beams, the
collinear-anticollinear geometry almost leads to a cancellation of the
effect, the anticollinear resonance is less blue shifted while the
collinear resonance is less red shifted. For an angle misalignment of 1 mrad, $\nu_a$ and $\nu_c$ will each be shifted by as much as 1.4\,MHz, whereas the effect on $\nu_0$ is only of the order of about 1\,kHz.
The other extreme is a misalignment of one laser relative to a well
superposed laser-ion-beam pair. Then the transition frequency $\nu_0$ becomes
\begin{eqnarray}
\label{eq:dopplerangledoble2}
\nu_{0} &=& \frac{1}{\gamma} \sqrt{\frac{\nu_c \cdot \nu_{ac}}{(1 -
\beta \cdot \cos\alpha) ( 1+\beta)}} \vspace{3mm} \\
&=& \sqrt{\frac{ 1 - \beta}{(1 - \beta
\cos\alpha})}\sqrt{\nu_c \cdot \nu_{ac}} \vspace{3mm}\\
&\approx& \left( 1 - \beta \alpha^2 \right)\sqrt{\nu_c \cdot
\nu_{ac}}.
\end{eqnarray}
In contrast to Eq.\,(\ref{eq:dopplerangledoble}) this angle-dependence
can lead under unfavourable conditions to an appreciable shift.
The influence of the laser-ion-beam alignment was extensively
studied using a stable $^9$Be ion beam by misaligning one of the
laser beams so that a deviation was clearly visible in the horizontal or
vertical direction. With the typical beam diameter a deviation of about 2\,mm across a distance of $\approx 8$\,m ($\alpha=0.25$\,mrad) was detectable, corresponding to a total effect of about 600\,kHz. Misalignment and realignments where repeated
several times but the results of the measurements with misalignment
scattered similarly as the measurements with optimized alignment and in both
cases the scatter was in accordance with the standard deviation of all
regular $^9$Be measurements. During the experiment, the counterpropagating alignment of
the laser beams was inspected visually several times per day. A systematic uncertainty of 300\,kHz, corresponding to half the full scattering amplitude was conservatively estimated.
\subsubsection{Photon Recoil Shift}
\label{sec:prs}
\begin{figure*}
\begin{center}
{\includegraphics[width=0.98\linewidth,clip=true, trim=5mm 90mm 0mm 0mm]
{BePowerdependenceD1D2.pdf}}
\caption{
Power dependence of the extracted frequency of
$^{10}$Be$^+$ in the D1 (left) and the D2 transition (right). Plotted
are the differences to the total mean frequency as a function of
laser power. Copropagating and counterpropagating laser beams were
adjusted to approximately equal power. Measurements were performed in
three series varying the power from the highest to the lowest values.
Uncertainties of the individual data points were estimated as the
standard deviation for each individual set of three measurements at
approximately the same power from this series.
}
\label{Fig:bepowerdependence}
\end{center}
\end{figure*}
Repeated interaction with a laser beam can influence the external degrees of motion of an ion or atom as it is well known from laser cooling and
laser deceleration in a Zeeman slower. In collinear laser spectroscopy
the repeated directed absorption and isotropic re-emission of photons
will have the consequence that the ions are either accelerated
(collinear excitation) or decelerated (anticollinear excitation). With
every absorbed photon, the Doppler-shifted resonance frequency is
shifted towards higher frequencies ($\nu_{c}$ and $\nu_{a}$) for both directions and
this systematic shift results in a transition frequency
$\nu_0$ that is too large. The combination of light ions and
ultraviolet photons leads to an exceptionally large photon recoil and
the possible influence of this effect must be studied.
Due to the absence of hyperfine splitting the $2s \rightarrow 2p$
transitions in the even isotopes $^{10,12}$Be$^+$ are closed two-level
systems. Hence, the possibility of repeated photon scattering is enhanced
compared to the odd-mass isotopes which are pumped into a dark hyperfine state after a few
absorption-emission cycles. To investigate whether the photon recoil has
a measurable effect, the power dependence of the transition frequency of
$^{10}$Be was determined as a function of the laser power.
The laser power in both beams was increased stepwise and simultaneously
from below 1\,mW up to 6\,mW.
The deviation of the extracted transition frequencies from the mean
frequency determined for the D1- and D2-transition are plotted in
Fig.~\ref{Fig:bepowerdependence} as a function of laser power. Each data
point is associated with an uncertainty estimated as the standard
deviation of a block of three measurements at similar power. In both
transitions the peak positions scatter but do not show a common trend
upwards or downwards. It appears, however, that we observe at
photomultipier tube 2 (PMT2), located about 15\,cm downstream from PMT1,
resonances that are systematically higher in frequency than at PMT1.
As a consequence of this observation we have included only data from
PMT1 in the analysis and have
estimated an additional uncertainty for the remaining effect.
The average difference between PMT1 and PMT2 is about 300\,kHz and 450\,kHz in the D1 and D2 transition, respectively. Since the distance between the two PMTs is slightly larger than the path of the ions before reaching PMT1, we estimate conservatively a maximal shift of about 400\,kHz for the systematic uncertainty $\Delta \nu_{{\rm{Ph}}}$ caused by photon recoil.
\subsection{Spectra of the Short-Lived Isotopes $^{7,11,12}$Be}
\label{sec:radioactiveberyllium}
A typical spectrum of $^{7}$Be is depicted in Fig.\,\ref{Fig:be7}. It is
the sum of 50 individual scans, taken in the first beam time in 2008.
Here, line shapes are slightly broader than observed in the second
beam time \cite{Zak10}. Since the measurements on $^{7}$Be were not repeated in 2010,
the uncertainties of the fitted line positions are larger
than for the other isotopes.
\begin{figure*}
\begin{center}
{\includegraphics[width=\linewidth]{Be7.pdf}}
\caption{Spectra obtained in the \ensuremath{2s\,^2{\rm{S}}_{\nicefrac{1}{2}} \rightarrow 2p\,^2{\rm{P}}_{\nicefrac{1}{2}}} transition in $^{7}$Be$^+$ for
copropagating (left) and counterpropagating laser excitation (right) as
a sum of 50 individual scans. The data points are fitted with a multiple
Voigt profile as discussed in the text. In the lower trace the residua
of the fit are displayed. For more details see \cite{Zak10}.}
\label{Fig:be7}
\end{center}
\end{figure*}
The reduced line width in the second beam time is clearly visible in the
spectrum of $^{11}$Be depicted in Fig.\,\ref{Fig:be11}. This is the sum of
400 individual scans.
\begin{figure*}
\begin{center}
{\includegraphics[width=\linewidth]{be11.pdf}}
\caption{Resonance spectra of the \ensuremath{2s\,^2{\rm{S}}_{\nicefrac{1}{2}} \rightarrow 2p\,^2{\rm{P}}_{\nicefrac{1}{2}}} transition in $^{11}$Be$^+$
in collinear (left) and anticollinear direction (right). The production
rate was about $10^6$\,ions$/$pulse and thus 400 individual scans were
accumulated. The data points are fitted with a multiple Voigt profile as
discussed in the text. Fitting residua are displayed in the lower trace.}
\label{Fig:be11}
\end{center}
\end{figure*}
Spectra of the unresolved hyperfine structure in the \ensuremath{2s\,^2{\rm{S}}_{\nicefrac{1}{2}} \rightarrow 2p\,^2{\rm{P}}_{\nicefrac{3}{2}}} transitions
of both these isotopes can be found in \cite{Zak10}.
For the investigation of $^{12}$Be$^+$, proton pulses impinged on the
target every 3--5 seconds. In this case, spectra in co- and
counterpropagating geometry were taken by switching the laser beams
after each proton pulse and photon detection was limited to 60\,ms
($\approx 3\,T_{\nicefrac{1}{2}}(^{12}\mathrm{Be})$) after the
pulse to reduce the number of random coincidence events.
\begin{figure*}
\begin{center}
{\includegraphics[width=0.95\textwidth]{be12.pdf}}
\caption{Resonance spectra of the \ensuremath{2s\,^2{\rm{S}}_{\nicefrac{1}{2}} \rightarrow 2p\,^2{\rm{P}}_{\nicefrac{1}{2}}} transition in $^{12}$Be$^+$
for collinear (left) and anticollinear excitation (right) plotted as a
function of the Doppler-tuning frequency. In total 180 single scans were
summed up for 8 hours. The symmetric spectra were fitted with a single
Voigt profile (red line). Residua are displayed in the lower trace.}
\label{Fig:be12coincidence}
\end{center}
\end{figure*}
Figure~\ref{Fig:be12coincidence} shows a typical spectrum which is
accumulated over 180 individual scans. Here a detection efficiency of about
1\,photon per 800\,ions was obtained. The Doppler-tuning
voltage range was restricted to 6\,V corresponding to about 200\,MHz
frequency span around the main peak, to limit the time required to record
a single resonance to a few hours. Reference measurements of
$^{10}$Be$^+$ were interspersed after every 40 single scans to ensure
stability of all conditions.
Due to the limited statistics not allowing the observation of the small
satellite peak, only a single Voigt profile was used for fitting the
resonances. Statistical uncertainty obtained from the fit was usually less than 100\,kHz.
\subsection{Absolute Transition Frequencies}
\label{sec:absolutefrequencies}
\begin{table*}
\begin{center}
\caption{Absolute transition frequencies $\nu_0$ for the \ensuremath{2s\,^2{\rm{S}}_{\nicefrac{1}{2}} \rightarrow 2p\,^2{\rm{P}}_{\nicefrac{1}{2}}} (D1) and
the \ensuremath{2s\,^2{\rm{S}}_{\nicefrac{1}{2}} \rightarrow 2p\,^2{\rm{P}}_{\nicefrac{3}{2}}} (D2) transition in beryllium isotopes obtained in Run I
(Beamtime 2008) and Run II (Beamtime 2010). The first uncertainty
represents the statistical, the second one the total uncertainty
including systematic uncertainties as discussed in
Sec.\,\ref{sec:systematicuncertainty}. Systematic uncertainties of the
results of Run I may be reduced from those published in
\cite{Noe09,Zak10} due to the information gained in Run II. Final
results for all isotopes are printed bold. All values are in MHz.}
\label{tab:expfrequency}
\vspace{2mm}
\begin{tabular}{c c r@{.}l r@{.}l}
\hline\hline
Isotope & Run & \multicolumn{2}{c}{$\nu_0$} &\multicolumn{2}{c}{$\nu_0$} \\
& & \multicolumn{2}{c}{D1} &\multicolumn{2}{c}{D2} \\
\hline
~$^{7}$Be & I & \bf 957\,150\,316&\bf2 (0.8) (0.9) & \bf
957\,347\,374&\bf5 (0.9) (1.1) \\
\hline
~$^{9}$Be & I & 957\,199\,552&9 (0.8) (1.0) & 957\,396\,616&6
(1.4)(1.5) \\
& II & 957\,199\,553&40 (0.12)(0.52) & \multicolumn{2}{c}{--} \\
& \bf comb. & \bf 957\,199\,553&\bf28 (0.12)(0.52) &\bf
957\,396\,616&\bf6 (1.4)(1.5) \\
& \cite{Bol85} & \multicolumn{2}{l}{957\,199\,652 (120)} &
\multicolumn{2}{l}{957\,396\,802 (135)} \\
\hline
$^{10}$Be & I & 957\,216\,876&9 (1.4)(1.5) & 957\,413\,943&9 (0.8)
(1.0) \\
$^{10}$Be & II & 957\,216\,876&84 (0.42)(0.66) & 957\,413\,942&17
(0.44)(0.70) \\
& \bf comb. & \bf 957\,216\,876&\bf85 (0.42)(0.66) &\bf
957\,413\,942&\bf74 (0.44)(0.67) \\
\hline
$^{11}$Be & I & 957\,231\,118&1 (1.1)(1.2) & 957\,428\,185&2
(1.5)(1.6) \\
$^{11}$Be & II & 957\,231\,118&11 (0.10)(0.52) &
\multicolumn{2}{c}{--} \\
& \bf comb. & \bf 957\,231\,118&\bf11 (0.10)(0.52) &\bf
957\,428\,185&\bf2 (1.5)(1.6) \\
\hline
$^{12}$Be & II & \bf 957\,242\,944&\bf86 (0.33)(0.61) &\bf
957\,440\,013&\bf60 (0.28)(0.58) \\
\hline\hline\\
\end{tabular}
\end{center}
\end{table*}
Each pair of spectra was fitted as discussed in
Sec.~\ref{sec:lineshapefitting} to determine the centers of gravity and
to extract the respective rest frame transition frequency $\nu_0$
according to Eq.\,(\ref{eq:diffdopplertransition}). For each isotope and beamtime the weighted mean of all measurements was calculated and results are
listed in Tab.\,\ref{tab:expfrequency}. All absolute frequencies from
both beam times agree within their 1-$\sigma$ error bars confirming the
reproducibility of the measurement. There is a small change compared to
Table\,I of \cite{Noe15}, namely a slightly larger uncertainty for
$^{10}$Be from Run II in the D2 line due to a transfer error in the
statistical uncertainty. Statistical uncertainties in Run II are based
on the standard error of the mean for typically 4--5 measurements for $^{11,12}$Be and 20--30 measurements for the less exotic isotopes $^{9,10}$Be.
Where available, values from both runs are combined weighted with the
respective uncertainty. Systematic uncertainties from the second run cannot be further reduced.
These absolute transition frequencies can now be used to evaluate
differential observables, like the isotope shift, the fine structure
splitting and the splitting isotope shift.
\subsection{Isotope Shifts and Nuclear Charge Radii} \label{sec:chargeradii}
\begin{table*}
\begin{center}
\caption{Compilation of experimental isotope shifts $\delta
\nu_{IS}^{9,A}$, obtained from the absolute transition frequencies
$\nu_0$ in Tab.\,\ref{tab:expfrequency}, field shift
$\delta\nu_{\rm{FS}}$ extracted as the difference to the theoretical
mass shifts listed in Tab.\,\ref{tab:propIS} \cite{Puc10} and the
corresponding change in the mean square nuclear charge radius $\delta
\left\langle r^2 \right\rangle$ according to Eq.\,(\ref{eq:changer}).
Results from the \ensuremath{2s\,^2{\rm{S}}_{\nicefrac{1}{2}} \rightarrow 2p\,^2{\rm{P}}_{\nicefrac{1}{2}}} (D1) and the \ensuremath{2s\,^2{\rm{S}}_{\nicefrac{1}{2}} \rightarrow 2p\,^2{\rm{P}}_{\nicefrac{3}{2}}} (D2) transitions are
compatible and were combined before the absolute charge radius $R_c$ is
calculated according to Eq.\,(\ref{eq:absoluter}).}
\label{tab:IsotopeShift}
\vspace{2mm}
\begin{tabular}{l c c c c}
\hline\hline
Isotope & $\delta \nu_{\rm{IS}}^{9,A}$ & $\delta \nu_{\rm{FS}}$ &
$\delta \left\langle r^2 \right\rangle $ & $R_c$ \\
and transition & /MHz & /MHz & /fm$^2$ & /fm \\
\hline
~$^{7}$Be$^+$ D1 & -49~237.1~(1.1) & -11.4(1.1) & 0.67~(6) & \\
~$^{7}$Be$^+$ D2 & -49~242.1~(1.8) & -10.3(1.8) & 0.61(11) & \\
~$^{7}$Be$^+$ Mean & & & 0.65~(5) &
2.646\,(15)\\
~$^{9}$Be$^+$ D1/D2 & 0 & 0 & 0 & 2.519\,(12)\\
$^{10}$Be$^+$ D1 & 17~323.57(84) & 13.11(84) & -0.77~(5) & \\
$^{10}$Be$^+$ D2 & 17~326.1~(1.6) & 13.6~(1.6) & -0.80~(10)& \\
$^{10}$Be$^+$ Mean & & & -0.78~(4) &
2.360\,(14)\\
$^{11}$Be$^+$ D1 & 31~564.82\,(74) & ~4.58(74) & -0.27~(4) & \\
$^{11}$Be$^+$ D2 & 31~568.6~\,(2.2) & ~4.4~(2.2) & -0.26(13) & \\
$^{11}$Be$^+$ Mean & & & -0.27~(4) &
2.465\,(15)\\
$^{12}$Be$^+$ D1 & 43~391.58\,(80) & ~1.40(82) & -0.08~(5) & \\
$^{12}$Be$^+$ D2 & 43~397.0~\,(1.6) & ~1.5~(1.6) & -0.09~(10)& \\
$^{12}$Be$^+$ Mean & & & -0.08~(4) &
2.502\,(15)\\
\hline\hline\\
\end{tabular}\\
\end{center}
\end{table*}
Isotope shifts are easily obtained as difference of the absolute
transition frequency of the isotope of interest and the reference
isotope, which in our case is the stable isotope $^9$Be. The field shift
$\delta \nu_{\rm{FS}}$, also known as the finite nuclear size or nuclear
volume effect, can then be extracted according to Eq.\,(\ref{eq:IS}). The
corresponding mass shifts $\delta \nu_{\rm{MS}}^{9,A}$ as well as the
field shift constant $F^{9,A}$ were theoretically evaluated to an
accuracy that exceeds the experimental uncertainty by about an order of
magnitude and are compiled in Tab.~\ref{tab:propIS}. We have been using
the results from \cite{Puc10,Puc09} to calculate $\delta
\nu_{\rm{FS}}$, which is then combined with the reference radius
$R_{\mathrm{c}} = 2.519(12)$~fm \cite{Jan72} of
$^9$Be to provide absolute charge radii along the chain using
Eq.\,(\ref{eq:absoluter}). It should be noted that the uncertainty of $R_{\mathrm{c}}(^9\mathrm{Be})$ is probably underestimated since C2 scattering from the quadrupole distribution has been omitted, which might change the radius by about 3\% \cite{Sic13}.
The results are listed in Tab.~\ref{tab:IsotopeShift}. Changes in the
mean-square charge radii deduced from the isotope shifts in the D1-
and D2-transition are of similar accuracy in case of even isotopes,
while for odd-mass isotopes the unresolved hyperfine
splitting in the D2 lines leads to larger uncertainties.
Figure~\ref{Fig:be12radii} depicts the development of the rms charge radius along
the isotopic chain as extracted from the experiment
($\bullet$) by combining all available data. We have also included
results from Fermionic Molecular Dynamics (FMD) calculations \cite{Kri12} that follow
the observed trend quite closely, but the charge radii are generally somewhat
too small. The two triangles shown for $^{12}$Be are results of two additional
calculations performed under the assumption that the two outermost
neutrons occupy either a pure
$p^2$ state ($\bigtriangledown$), as expected in the traditional shell
model or a pure $(sd)^2$ state ($\bigtriangleup$). The latter is
expected to contribute to the ground state only if the $N=8$ shell gap
between the $p$-shell and the $sd$-shell is significantly reduced. This
prediction shows that the charge radius of $^{12}$Be is extremely
sensitive to the admixture of $sd$-shell states to the ground state,
which was a strong motivation for the measurement of the $^{12}$Be
isotope shift.
\begin{figure}
\begin{center}
{\includegraphics[width=\linewidth,clip=true]{ChargeRadii_7-12_v2.pdf}}
\caption{Nuclear charge radii along the beryllium isotopic chain. The
reference radius of $^{9}$Be was determined from electron scattering
experiments \cite{Jan72}. The red bullets ($\bullet$)
represent the experimental results with uncertainties dominated by
the uncertainty of the reference charge radius of $^{9}$Be. Therefore
all uncertainties are similar in size. Additionally shown are results of
Fermionic Molecular Dynamic (FMD) calculations \cite{Kri12}. For
$^{10}$Be, calculations were also performed forcing the neutrons into a
$p^2$ and an $(sd)^2$ orbit. The bottom row shows the structure of the
isotopes interpreted in a cluster picture. For details see text. }
\label{Fig:be12radii}
\end{center}
\end{figure}
The trend of the charge radii along the isotopic chain can be understood
in a simplified picture based on the cluster structure of light
nuclei \cite{Ash04}. This is visualized in the small panels below the
graph in Fig.\,\ref{Fig:be12radii}. $^7$Be can be thought of as a
two-body cluster consisting of an $\alpha$ particle and a helion
(pnp = $^3$He) nucleus that are bound together and exhibit a considerable
center-of-mass motion. This motion blurs the proton
distribution and leads to an increased charge radius.
$^8$Be is missing since the two $\alpha$ particles
constituting this nucleus are not bound and the nucleus only exists
as a resonance. The stable isotope $^9$Be, which has a
$\alpha+\alpha+n$ structure, is more compact than $^7$Be because the
$\alpha$ particles themselves are very compact and well bound by the
additional neutron. This effect is even enhanced with the second neutron
added in $^{10}$Be. The sudden upward trend to $^{11}$Be is attributed
to the one-neutron halo character of $^{11}$Be which can be disentangled
into a $^{10}$Be core and a loosely bound neutron. This halo character
not only increases the matter radius, but also affects the charge
radius due to the center-of-mass motion of the core caused by
the halo neutron. The fact that the charge radius of $^{12}$Be is even
larger has been related to the fact that the two outermost
neutrons exhibit a strongly mixed $sd$ character rather than belonging
to the $p$ shell as expected in the simplified shell-model picture. This
mixture leads to an increased probability density outside the
$^{10}$Be core, pulling the $\alpha$ particle apart due to the
attractive n--$\alpha$ interaction. Theory predicts an $(sd)^2$ admixture of about
70\% for this nucleus, being a clear
indication for the disappearance of the classical $N=8$ shell closure.
For a more detailed discussion of the nuclear charge radii, the
comparison with ab-initio microscopic nuclear structure calculations
and the conclusions about the shell closure see
Refs.\,\cite{Noe09,Kri12,Zak10}.
\subsection{Fine Structure Splitting and Splitting Isotope Shifts}
\label{sec:FineStructure}
\begin{table*}
\begin{center}
\caption{Fine structure splittings $\Delta \nu_{\mathrm{fs}}$, the
experimental and theoretical \cite{Puc09} splitting isotope shifts
$\delta \nu_{\mathrm{sis}}$, and the transferred fine structure
splittings $\Delta \nu_{\mathrm{fs,^ABe\rightarrow\,^9Be}}$ for $^9$Be
based on the measured splittings in the radioactive isotopes according
to Eq.\,(\ref{eq:fs_projection}) are listed. The bottom row shows the
splitting isotope shift between the two even isotopes $^{10}$Be and
$^{12}$Be. For $\delta \nu^{A,10}_{\mathrm{sis}}$ and $\Delta
\nu_{\mathrm{fs,^ABe\rightarrow\,^9Be}}$ for $^{7,9,11}$Be after Run II
required information from Run I since D2 lines of these isotopes were
not measured in Run II. All values are in MHz.}
\label{tab:fs_splitting}
\vspace{2mm}
\begin{tabular}{c r@{.}l r@{.}l r@{.}l r@{.}l}
\hline\hline
Isotope & \multicolumn{2}{c}{$\Delta \nu_\mathrm{fs}$} &
\multicolumn{4}{c}{$\delta \nu^{A,9}_{\mathrm{sis}}$} &
\multicolumn{2}{c}{$\Delta \nu_{\mathrm{fs,^ABe\rightarrow\,^9Be}}$} \\
& \multicolumn{2}{c}{D2--D1} & \multicolumn{2}{c}{Exp} &
\multicolumn{2}{c}{Theory}\\
\hline
~$^{7}$Be & 197\,058&4 (1.4) & 5&0 (2.1) & 6&036(1) & 197\,064&4
(1.4) \\
~$^{9}$Be & 197\,063&2 (1.6) & 0&0 & 0&0 &
197\,063&3 (1.6) \\
~$^{9}$Be\,$^a$ &\multicolumn{2}{l}{197\,150 (64)} \\
$^{10}$Be & 197\,065&3 (0.9) & --2&0 (18) & --2&096(1) & 197\,063&2
(0.9) \\
$^{11}$Be & 197\,067&1 (1.7) & --3&8 (23) & --3&965(1) & 197\,063&1
(1.7) \\
$^{12}$Be & 197\,068&7 (0.9) & --5&4 (18) & --5&300(1) & 197\,063&4
(0.8) \\
$^{12-10}$Be & \multicolumn{2}{r}{$\delta
\nu^{12,10}_{\mathrm{sis}}=$} & --3&4 (6) & --3&203 \\
\hline\hline\\
\multicolumn{4}{l}{$^a$ Bollinger \textit{et al.} \cite{Bol85}}\\
\end{tabular}
\end{center}
\end{table*}
From the information provided in Tab.\,\ref{tab:expfrequency} we can
furthermore extract the fine structure splitting as a function of the
atomic number. The splitting has previously been measured for $^9$Be with a
relative uncertainty of about $3 \cdot 10^{-4}$ \cite{Bol85}. The present
value obtained simply from the difference in $\nu_\mathrm{D1}$ and
$\nu_\mathrm{D2}$ of this isotope is already 40 times more accurate although its
accuracy is limited by the unresolved hyperfine structure in the D2
transition. Only recently the fine structure splitting in
three-electron atoms became calculable with high precision as
demonstrated for the case of lithium \cite{Puc14}. Our result confirmed first
calculations for the $Z=4$ three-electron system of Be$^+$ \cite{Noe15}.
Moreover, the change in the fine structure splitting along the chain of
isotopes provides a useful check of the so-called splitting isotope
shift which can be calculated theoretically to very high accuracy, because the value is
nearly independent of both QED and nuclear volume effects.
This mass dependence is shown in Fig.\,\ref{fig:fs_sis}: the blue data
points represent the experimental fine structure splittings and the red
line with red crosses the theoretically expected mass dependence based
on the calculated splitting isotope shift and the measured splitting
of $^9$Be. The excellent agreement between the
theoretical curve and the experimental data can be interpreted as a
reliable check of the consistency of the mass shift calculations. On the
other hand, it proves the consistency of the experimental data, because
the fine structure splittings are based on the combination of absolute
transition frequencies obtained independently in two beam times, even with
independent optimization of the experimental conditions. The observation
that the data points scatter much less than expected from their error
bars can be ascribed to the fact that our systematic uncertainties largely
cancel out in considering differential effects.
\begin{figure}
\begin{center}
{\includegraphics[width=0.9\linewidth,clip=true]{be_fs_splitting.pdf}}
\caption{Left: Mass dependence of the fine structure splitting along
the berylium isotopic chain. The red curve with crosses shows the
theoretically expected mass dependence -- the splitting isotope shift
(sis) -- with respect to $^9$Be. The experimental values (blue circles)
and the sis-corrected values according to Eq.\,(\ref{eq:fs_projection})
(magenta dots) are included. The solid and dotted lines represent the
mean and standard deviation of the corrected values, respectively.
Right: Fine structure splitting in $^9$Be from experiment (line) and theory (data point).}
\label{fig:fs_sis}
\end{center}
\end{figure}
The fine structure splitting can most reliably be determined for the
even-even isotopes $^{10}$Be and $^{12}$Be since there is no hyperfine
structure which in the D2 transition obscures the
determination of the center of gravity. This is clearly visible
in Fig.\,\ref{fig:fs_sis} where the error bars are smallest for these
two cases. For these isotopes, the fine structure splitting was
determined sequentially in a short time interval and therefore the best
cancellation of all systematic uncertainties should occur. Indeed, the
splitting isotope shift between the two isotopes
$\delta\nu_\mathrm{sis}^{10,12}=3.43(78)$\,MHz agrees very well with the
theoretical value of 3.203\,MHz. Here the
uncertainty of the experimental value is purely statistical assuming
that all dominant systematic contributions are cancelled out.
With the confidence gained about the reliability of the theoretical
estimates, we can combine the calculated splitting isotope shifts of
all isotopes to extract an improved value for the fine structure
splitting of $^9$Be. To this aim, we correct all measured fine structure
splittings by subtracting the theoretical splitting isotope shift
\begin{equation}
\Delta \nu_{\mathrm{fs,^ABe\rightarrow\,^9Be}} = \Delta \nu_{\rm
fs}(^A{\rm Be}) - \delta \nu^{A,9}_{\mathrm{sis, Theory}}.
\label{eq:fs_projection}
\end{equation}
The results are included in Tab.\,\ref{tab:fs_splitting} and plotted in
Fig.\,\ref{fig:fs_sis} as magenta bullets. The weighted mean for the
"isotope-projected" fine structure splitting of $^9$Be represented by the horizontal line is 197\,063.47\,(53)\,MHz. The uncertainty is
shown by the dashed lines. We believe that a small remaining
systematic uncertainty of the fine structure splitting is still covered
by the size of this uncertainty.
In the right hand part of the figure, the theoretical prediction represented
by the filled red triangle is
compared with experiment. The calculated splitting in $^9$Be amounts to
197\,068.0\,(25)\,MHz, which is about 4.5\,MHz larger than the
experimental value. This difference corresponds to about $1.5\,\sigma$
of the combined uncertainties. The theoretical uncertainty is based on
an estimation of the size of the uncalculated nonlogarithmic terms in
$m\alpha^7$. These terms are expected to be less than 50\% of the
calculated leading logarithmic terms.
To further test the QED calculations in lithium-like light systems, similar measurements on ions with higher $Z$, e.g., in boron B$^{2+}$ or carbon C$^{3+}$ are of great interest. At least for B$^{2+}$ the wavelength of about 206\,nm is still achievable and measurements with the collinear laser spectroscopy technique presented here are planned.
\section{Summary}
We have demonstrated that quasi-simultaneous collinear-anticollinear laser
spectroscopy on stable and radioactive isotopes can be used to perform
precision measurements from which nuclear charge radii even of light and
very short-lived species can be extracted. At the same time the data can
be used to perform high-precision tests of fundamental atomic structure
calculations and bound-state QED. The experimental technique relies on
the accurate determination of the absolute laser frequency, which has
become possible with the invention and availability of
frequency combs. We will apply the technique for further studies: an
important case is the determination of the charge radius of the
proton-halo candidate $^8$B for which an experiment is currently under
preparation at the ATLAS facility at the Argonne National Laboratory.
\begin{acknowledgements}
This work was supported by the Helm\-holtz Association (VH-NG148), the German Ministry for Science and Education (BMBF) under contracts 05P12RDCIC and 05P15RDCIA, the Helmholtz International Center for FAIR (HIC for FAIR) within the LOEWE program by the State of Hesse, the Max-Planck Society, the European Union 7$^\mathrm{th}$ Framework through ENSAR, and the BriX IAP Research Program No. P6/23 (Belgium). A.~Krieger acknowledges support from the Carl-Zeiss-Stiftung (AZ:21-0563-2.8/197/1).
\end{acknowledgements}
|
3,212,635,537,972 | arxiv | \section{Introduction}
\indent
The gravitational waves (\textbf{GW}) are spacetime curvature disturbances which propagate at speed of light. One of the familiar sources is rapidly accelerated massive objects which make ripples in their surrounding spacetime. These waves were first predicated by Einstein as approximative solutions of the field equations of General
Relativity in 1916~\cite{GWs 2016}. In recent decays, study of GW becomes an interesting topic in theoretical and experimental physics such as GW produced during inflationary period or the compact binary system~\cite{Luc Blanche, mirza04}. The first direct detections of GW was from the merging of black holes and neutron stars reported by the LIGO/VIRGO~\cite{Abbott 2016, Abbott 20171, Abbott 2017}.\\
GW can give information from regions of spacetime of universe that are not accessible by other source of information as electromagnetic radiation.
From the cosmological standpoint, the GW play a potentially role to extract or infer the information about theories as Cosmic Inflation. This is due to the tensorial nature of GW, because some information cannot be extracted from the scalar perturbations.\\
In this work, we are interested in studying and analysing the GW originating in the early universe. To understand and describe of its physical conditions of the early universe, the inflationary paradigm is a very successful theory. This theory was welcomed due to solving the major cosmological problems, because it is well--supported by observational data such as Large Scale Structure of the present universe (this theory is well--described by the scalar perturbations). On the other hand, the tensorial perturbations predict the existence of a uniform background of gravitational radiation in the universe. Despite the very low intensity of GW, observational evidences and technological tools have succeeded in detecting them, the hopes for future are that, by improved technologies one should be able to test the prediction from inflation which this would be a huge success in confirming the theory.\\
During the inflationary epoch, the other possible physical processes and matter--field fluctuations are also involved in the generation of GW and can be considered as sources of the wave production. These processes can cause the noise in the wave propagation which makes a random character for the GW (what is known as stochastic background GW). The study and investigation of such a background can help us to understand and explore the early universe events and high--energy physics. In fact, the energy scale for which the inflation can occur is one of the main challenge for theoretical and experimental physics and in this way the GW generated at that time can play an important role to probe the such energy scale. In fact, it is believed that since there are no gravitational waves found at the 5 percent level, thus the energy scale of an inflationary epoch was below the Planck scale.\\
In general case when one deals with a stochastic (radiation) signal, the signal hasn't a meaningful phase data information (due to the random noise effects) and it is common to use the spectral methods. The spectral energy density is an important quantity in analysis of the (periodic) signals to extract information. It identifies at what frequency (or frequencies) is the signal power or, what form does the energy density function over frequency have?. Therefore, to study and analysis of the primordial GW, the power spectrum can be used as an useful tool for extracting the information which can not be read from the time domain of the wave. Indeed, these information (as magnitudes of the frequencies components) can be directly related to the energy scale of inflation.\\
In the present article, we deal with the tensorial perturbations amplified by inflation leading to primordial GW, by special focus on their power spectrum. We calculate the spectral energy in a different way from the one used in the previous works, e. g. ~\cite{mirza04, Latham 2005, Yuki 2006}. The spectral energy density (SED) is obtained directly in terms of scale factor, from the beginning of the universe until the present age. In the mentioned works, the spectrum is obtained
for each dynamical regime of the universe (that is radiation, matter--radiation and matter regimes) individually and then patch them together. The corresponding spectrum diagrams are also plotted which through them some physics are deduced.\\
The work is organized as follows:
In section 2, we obtain the wave equation governing the perturbations in terms of the scale factor using the Lagrangian formalism. In section 3, the power spectrum of gravitational waves and corresponding diagrams are presented. Conclusions are given in section 4.
\section{Perturbations in \textbf{FRW} Background}
In the beginning of this section, let's give a brief overview of the time evolution of gravitational perturbations, needed for the work.
Consider a perturbed FRW spacetime whose line element can be written in the form~\cite{ mirza04, Latham 2005}
\begin{equation}\label{N7}
ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}=\left(\bar{g}_{\mu\nu}+h_{\mu\nu}\right)dx^{\mu}dx^{\nu} = a^{2}\left\{-d\tau^2+ \left[\delta_{ij}+h_{ij}(t,x)\right]dx^i dx^j\right\}
\end{equation}
where $\tau$ is the conformal time, $\bar{g}_{\mu\nu}= diag\{-a^{2}, a^{2},a^{2},a^{2}\}$ is the unperturbed FRW background metric and $h_{\mu\nu}=h_{\mu\nu}(x)$ are the perturbations satisfying $|h_{\mu\nu}|<<1$,\hspace{2mm}$h_{00}=h_{0i}=0$ and traceless ($h^i_{i} = 0)$ and transverse $(h^j_{i,j} = 0)$ conditions.
The linearization of Einstein equations (in presence of an isotropic and perfect fluid) for the spectrum of perturbations lead to the following equation~\cite{mirza04, Yuki 2006}:
\begin{equation}\label{C00}
h^{''}_{k}+2\left(\frac{a^{'}}{a}\right)h^{'}_{k}+k^{2}h_{k}=0,
\end{equation}
where $h_k=h(\tau,k)=\int d^{3}\textbf{x} e^{-i\textbf{k}\cdot\textbf{x}}h(\tau, \textbf{x})$ is the Fourier transform of perturbations and prime denotes derivative with respect to the conformal time $\tau$. For later purposes, we obtain equation (2) by the Lagrangian formalism in the following discussion.
The gravitational action describing the tensor perturbations is given by\cite{Latham 2005}:
\begin{equation}\label{C00}
S =\int d\tau d\textbf{x}\sqrt{-\bar{g}}\left(R+\frac{1}{2}\Pi_{ij}h_{ij}\right)=\int d\tau d\textbf{x}\sqrt{-\bar{g}}\left(\frac{-\bar{g}^{\mu\nu}}{64\pi G } \partial_{\mu}h_{ij}\partial_{\nu}h_{ij} +\frac{1}{2}\Pi_{ij}h_{ij}\right),
\end{equation}
where $R$ is the Ricci scalar, $\Pi_{ij}$ is the anisotropic stress tensor and $\bar{g}$ is the determinant of $\bar{g}_{\mu\nu}$\rlap.\footnote{Since, the first order of $h_{ij}$ are considered and also $h_{ii} = 0$, then $ \sqrt{-g} = \sqrt{-\bar{g}}$ .} For an isotropic and perfect fluid ($h_{ij}(x)=h(x)$ and $\Pi_{ij}= 0$), the action (3) reduces to
\begin{equation}\label{C00}
S=\int d\tau d\textbf{x}\sqrt{-\bar{g}}\left(\frac{\bar{-g^{\mu\nu}}}{64\pi G } \partial_{\mu}h\partial_{\nu}h\right)=\int d\tau d\textbf{x}\mathcal{L},
\end{equation}
which gives the Lagrangian as
\begin{equation}\label{C00}
\mathcal{L} =\frac{\bar{-g^{\mu\nu}}}{64\pi G } \partial_{\mu}h\partial_{\nu}h \sqrt{-\bar{g}}.
\end{equation}
This Lagrangian is independent of the perturbation $h(x)$, then the Lagrange equations
\begin{equation}\label{C00}
\frac{\partial \mathcal{L}}{\partial h}-\partial_{\mu}\left(\frac{\partial \mathcal{L}}{\partial(\partial_{\mu}h)}\right) =0,
\end{equation}
reduce to
\begin{equation}\label{C00}
\partial_{\mu}\left(\sqrt{-\bar{g}}\frac{\partial \mathcal{L}}{\partial(\partial_{\mu}h)}\right)=0,
\end{equation}
which by substituting (5), we get
\begin{equation}\label{C00}
\partial_{\mu}\left(a^{4}{\bar g^{\mu\nu}}\partial_{\nu} h(\tau, \textbf{x})\right)=0.
\end{equation}
By inserting the (inverse) background metric in the last equation and taking the Fourier transform on both sides (8), one gets
\begin{equation}\label{C00}
h^{''}(\tau, k) + 2\left(\frac{a^{'}(\tau)}{a(\tau)}\right)h^{'}(\tau, k)+ k^{2}h(\tau, k)=0,
\end{equation}
where this is the same equation (2). Equation (9) has two independent variables (that is $a(\tau), h(\tau, k)$) and therefore to solve it, one first have to specify the scale factor (as known time function) for each evolutionary stage of the universe and matching the solutions at the epoch of the
transition between the different stages. Instead of this method, we replace the derivative with respect to conformal time by derivative with respect to scale factor, we come in to an independent equation. In other words, if the time dependence is replaced by the scale factor dependence through the familiar replacement $d\tau=\frac{dt}{a}$, then equation (9) takes the following form
\begin{equation}\label{C00}
a^{4}H^{2}h_{aa}(a, k) + \left(4 a^{3}H^{2}+\frac{a^{4}}{2}\frac{dH^{2}}{da}\right)h_{a}(a, k)+ k^{2}h(a, k)=0,
\end{equation}
where $h_{a}(a, k)=\frac{dh(a, k)}{da}$, $h_{aa}(a, k)=\frac{d^{2}h(a, k)}{da^{2}}$ and $H=\frac{\dot{a}}{a}$ is the Hubble parameter. Since the Hubble parameter can be usually described as function of scale factor, that is $H=H(a)$, then equation (10) describes the evolution of perturbations in terms of the scale factor. It is true that the above equation has a relatively more complex appearance than the equation (9), but as mentioned above, it is an autonomous equation which by specifying the Hubble rate function $H(a)$, one can proceed to solve it exactly or numerically.\\
\section{The Power Spectrum}
In this section, we are going to compute SED corresponding to the perturbations satisfying the equation (10). This is done by numerical instructions without restoring to the direct solution of (10). In the first step, we should specify the Hubble parameter as function of scale factor. This parameter as expansion rate in terms of present--day measurable quantities is given by the Friedmann--Lemaître equation:
\begin{equation}\label{C00}
H^{2}(a)=H_{0}^{2}\left(\Omega_{r}a^{-4}+\Omega_{m}a^{-3}+\Omega_{k}a^{-2}+\Omega_{\Lambda}\right),
\end{equation}
where $\Omega_{r}=9.4\times10^{-5}\simeq 10^{-4}$, $\Omega_{m}=0.3$, $|\Omega_{k}|\leq 0.01$ and $\Omega_{\Lambda}=0.7$ are radiation, matter, curvature and dark energy density parameters, respectively. For a flat background $\Omega_{k}=0$, thus (11) reads
\begin{equation}\label{C00}
H^{2}(a)=H_{0}^{2}\left(\Omega_{r}a^{-4}+\Omega_{m}a^{-3}+\Omega_{\Lambda}\right).
\end{equation}
In the second step, we introduce quantities necessary to describe the relationships between perturbations and their corresponding spectrum. The first quantity is the spectral amplitude $\Delta_{h}^{2}(\tau,k)$ defined by
\begin{equation}\label{N7}
<h_{ij}(\tau,\textbf{x})h^{ij}(\tau,\textbf{x})>=\int\frac{dk}{k}\Delta_{h}^{2}(\tau,k),
\end{equation}
which can be written in the reverse form as
\begin{equation}\label{N7}
\Delta_{h}^{2}(\tau,k)=\frac{2k^{3}}{2\pi^{2}} <|h(\tau, k)|^{2}>,
\end{equation}
or, in terms of scale factor
\begin{equation}\label{N7}
\Delta_{h}^{2}(a,k)=\frac{2k^{3}}{2\pi^{2}} <|h(a, k)|^{2}>.
\end{equation}
The spectral amplitude relates the spectral distribution of the amplified fluctuations and
the cosmological kinematic parameters. Indeed, in the two end of inflation time interval, one deals with the amplified modes in outside and inside horizon and the spectral amplitude is useful to describe the distribution of the modes in outside the horizon.\\
The second quantity is SED that introduced by
\begin{equation}\label{N7}
\Omega_{h}(a,k)=\frac{1}{\rho_c}\frac{d\rho}{d\ln k },
\end{equation}
where $\rho$ and $\rho_c$ are energy density and critical energy density respectively. Since, the relic GW (amplified by inflation) with mode inside horizon should be still present today, they must be accessible to direct observations (by the high improved gravitational detectors). SED characterises the spectrum of the relic waves and thus it is useful to discuss a possible their direct detection. For the mode inside horizon, SED is related to the spectral amplitude through the relatively simple relation, that is
\begin{equation}\label{N7}
\Omega_{h}(a,k)=k^2\frac{\Delta_{h}^{2}}{12\hspace{0.5mm}a^2H^{2}(a)}=\frac{k^5}{12\pi^{2}}\frac{<|h(a, k)|^{2}>}{\hspace{0.5mm}a^2H^{2}(a)},
\end{equation}
finally, to obtain the spectral energy for the present time, by substituting the (conventional) present value of scale factor ($a(\tau_0)=a_0=1$), one gets
\begin{equation}\label{N7}
\Omega(k)=\Omega_{h}(1, k)=\frac{k^5}{12\pi^{2}}\frac{<|h(1, k)|^{2}>}{\hspace{0.5mm}H_0^{2}}=\frac{k^5}{3\pi^{2}}\frac{|h(k)|^{2}}{\hspace{0.5mm}H_0^{2}},
\end{equation}
where $|h(k)|^{2}=\frac{1}{2}\left(\langle |h_{+}|^{2}\rangle+\langle |h_{\times}|^{2}\rangle\right)=\frac{1}{4}\langle h^{ij} h_{ij}\rangle=\frac{1}{4}<|h(1, k)|^{2}>$. The latter relations mean that the contributions of the two polarization states ($+,\times$) of GW is taken to be equal\cite{Spergel 2003}.\\
Let's remember that discussion and investigation of the equation (18) is conditional on obtaining the perturbation $h(k)$ satisfying (10) which is generally not exactly solvable. Thus we have to use and implement the numerical recipes which requires the appropriate initial conditions. Knowing that PGW are originated from inflationary period, the appropriate initial conditions can be considered as
\begin{eqnarray}
h\left(a_{e}, k\right)=1 \\ \nonumber
h_{a}\left(a_{e}, k\right)= 0,\\ \nonumber
\end{eqnarray}
where $a_{e}\simeq 10^{-27}$ is the scale factor at the end of inflation \cite{Scott 2003}.\\
In order to have an idea of the behavior of the spectral density (18), the result of numerical calculations is illustrated in Fig.1. The three (color) graphs shown in the figure are the spectrum diagrams corresponding to three cases of the Hubble rate function (12). The three cases of the Hubble function and corresponding spectrum graphs (in Fig.1) are stated below:\vspace{2mm}\\
1) General case: $H^{2}(a)=H_{0}^{2}\left(\Omega_{r}a^{-4}+\Omega_{m}a^{-3}+\Omega_{\Lambda}\right)$ corresponds to the blue graph, \vspace{3mm}\\
2) Matter--Dark Energy dominate case: $H^{2}(a)\approx H_{0}^{2}\left(\Omega_{m}a^{-3}+\Omega_{\Lambda}\right)$ corresponds to the yellow graph, \vspace{3mm}\\
3) Matter dominate case: $H^{2}(a)\approx H_{0}^{2}\Omega_{m}a^{-3}$ corresponds to the red graph.\vspace{3mm}\\
Based on the graphs illustrated in Fig.1, we can deduce the followings:\vspace{2mm}\\
1) All cases are compatible with the Big Bang Nucleosynthesis (BBN) bounds.\footnote{In the frequency band around 100 Hz or $k = \frac{2\pi f}{c} \simeq 628 \hspace{0.5mm} m^{-1}$ (in $c = 1$ unity), $\Omega_{gw}(f)\leq 6.9\times 10^{-6}$ \cite{Abbott 2009}.}\\
2) The graphs corresponding to the cases (1), (2) (yellow and blue graphs) are not much different and this is to be expected due to the very small contribution of radiation term ($\Omega_{r}\simeq 10^{-4}$) in comparison with matter and dark energy contributions.\vspace{1mm}\\
3) The wave power with the dark energy contributions is significantly enhanced.\vspace{1mm}\\
4) The phenomenological and astrophysical allowed constraints on $\Omega(k)$ (as BBN or observations of millisecond pulsars) can restrict the inflationary parameters as expansion rate in the inflationary epoch. On the other hand, dark energy has an increasing effect on the spectrum, thus the experimental and observational constraints on the spectrum can impose limits on the values of the Cosmological Constant. In other words, SED can say not only important facts about the primordial universe, but also it can carry information about the values of the Cosmological Constant.\vspace{1mm}\\
5) The numerical values of SED at k=628 are written in Table 1, these results show that the values obtained for SED are compatible with BBN bounds.\\
\begin{figure}[!htb]
\center{\includegraphics[width=12cm]
{Fig-1.eps}}
\caption{\label{Fig.1}Three spectrum diagrams of $\Omega(k)$ corresponding to three cases of the Hubble parameter. $H^{2}(a)\approx H_{0}^{2}\Omega_{m}a^{-3}$ (red graph), $H^{2}(a)\approx H_{0}^{2}\left(\Omega_{m}a^{-3}+\Omega_{\Lambda}\right)$ (yellow graph) and $H^{2}(a)=H_{0}^{2}\left(\Omega_{r}a^{-4}+\Omega_{m}a^{-3}+\Omega_{\Lambda}\right)$ (blue graph). The difference between the figures in the top and bottom rows is only in the presented frequency ranges.}
\end{figure}
\begin{table}[h]
\caption{Numerical values of SED at $k\simeq 628$}
\centering
\begin{tabular}{c c }
\hline\hline
$\frac{H^{2}(a)}{H_{0}^{2}}$ & $\Omega(k\simeq 628)$ \\ [0.5ex]
\hline
$ \Omega_{m}a^{-3}$ & \hspace{2mm}$ 5.83\times10^{-10}$ \\
$\Omega_{m}a^{-3}+\Omega_{\Lambda}$ &\hspace{0.1mm} $2.29\times10^{-9} $ \\
$\Omega_{r}a^{-4}+\Omega_{m}a^{-3}+\Omega_{\Lambda}$ & $2.28\times 10^{-9}$ \\
\hline
\end{tabular}
\label{table:nonlin}
\end{table}
\section{Conclusions}
In the very early universe, tensorial perturbations was amplified during the inflationary epoch leading to primordial radiation. The remnants of this radiation
must exist in the present age as well and thus its detection can tell us the facts about the past history of the universe. An efficient quantity associated with the gravity waves detection is the radiation spectrum (or what is called in breif SED). In this work, SED of primordial gravity waves is calculated and presented by their corresponding diagrams in three cases of the Hubble expansion rate. The observations (data coming from the phenomenological and
experimental bounds on the spectral energy) predict mostly growing character for the spectrum and this character is confirmed by the illustrated diagrams here.
The graphs of spectrum, which are also consistent with the observational data (as BBN bounds), indicate the increasing effect of a constant cosmological presence on growing spectrum.
\vspace{1 cm}\\
\textbf{Acknowledgments}: We thank K. Rezazadeh for useful discussions.
|
3,212,635,537,973 | arxiv | \section{Introduction}
Facial affective behavior recognition plays an important role in human-computer interaction \cite{kollias2021analysing}. It allows computer systems to understand human feelings and behaviors, which makes human computer interaction more applicable. Existing research used different approaches to represent human emotions, such as valence-arousal estimation (VA), facial action unit (AU) detection, and facial expression (Expr) classification.
In the challenges of the 2nd Affective Behavior Analysis in-the-wild (ABAW2) Competition \cite{kollias2021analysing, kollias2020analysing,kollias2021distribution,kollias2021affect,kollias2019expression,kollias2019face,kollias2019deep,zafeiriou2017aff}, the organizers collect a large scale in-the-wild database Aff-Wild2 to provide a benchmark for the three emotion representation tasks. There are strong correlations between the three different tasks. Multi-task learning can extract joint features from the correlated tasks and provide better performance than training on a single task. In the last year’s competition, some teams proposed multi-task learning model to explore the learning of multiple correlated tasks simultaneously. For example, Two-Stream Aural-Visual model (TSAV) \cite{Kuhnke_2020} achieved superior performance in a multi-task manner.
However, the labels of ABAW2 Competition database are incomplete. Even through the three tasks share the same video database, most videos in the dataset are only labeled for one or two tasks. During the multi-task training process, only tasks with labels can be trained while other tasks without labels are ignored, which is quite inefficient. Previous studies also faced this challenge and had to treat different tasks independently. Those methods can only make use of limited labeled data while ignoring abundant incomplete labeled data. Hence, it is highly desirable to leverage additional unlabeled data to improve the performance.
To tackle this problem, we develop a multi-task mean teacher \cite{tarvainen2017mean} framework to boost affective behavior recognition performance in a semi-supervised manner. We firstly proposed an audio-video model to learn the three tasks mutually. Our model shares the same backbone with TSAV while differing in a preprocessing step and output layers. Second, we take this multi-task model as both student network and teacher network. For labeled tasks, the supervised losses on all labeled tasks are integrated as multi-task supervised loss. For unlabeled tasks, we enforce the outputs of the student network and the teacher network to be consistent with each other using consistency loss. By adding the supervised loss and the consistency loss together, our network can be trained with both labeled and unlabeled data. Howerver, prediction results of the teacher network could be inconsistent or incorrect, which is harmful to model training. To address this problem, we employ self-attention importance weighting, ranking regularization modules described in self-cure network \cite{wang2020suppressing}, to suppress the impact of uncertainties and to prevent deep networks from over-fitting uncertain facial expression.
With these improvements, our model achieves a competitive result in the competition. At the second ABAW competition, Our proposed method ranked fourth for both valence-arousal estimation and expression classification tasks. It is worth mentioning that our model was trained on the competition database only but achieved comparable performance with large scale pretrained models with extra datasets proposed by other teams.
Our major contributions are summarized as:
• First, we developed a multi-task multi-modal model for simultaneously analyzing valence-arousal estimation, facial action unit detection, and expression classification.
• Second, we designed a mean teacher framework to fuse consistency loss of incomplete labeled data with suoervised loss from labeled data. We also made a step forward to adopt self-attention importance weighting, ranking regularization to solve the uncertainty problem of pseudo labels. In this way, our proposed model can effectively leverage both labeled and unlabeled data.
\section{Related works}
Previous studies on the Aff-Wild2 have proposed some effective facial affective behavior analysis models, especially to explore inter-task correlations using multi-task learning. In \cite{kollias2021distribution}, Kollias \etal proposed FaceBehaviorNet for large-scale face analysis, by jointly learning multiple facial affective behavior tasks and a distribution matching approach. Kuhnke \etal~\cite{Kuhnke_2020} proposed a two-stream aural-visual network to combine vision and audio information for multi-task emotion recognition.
Even though the mentioned multi-task methods obtained promising results, they did not solve incomplete label problem of Aff-Wild2 dataset. Deng \etal~\cite{deng2020multitask} proposed a data-driven teacher model to fill in the missing labels. They trained a teacher model firstly. After that, the distillation knowledge technique is applied to train a student model. Inspired by their work, in this paper we propose a mean teacher model to leverage incomplete labeled data. Different from Deng’s work, the teacher model and student model in our framework are trained at the same time. Such an end-to-end training process is more flexible than Deng’s method.
\section{Methodology}
\subsection{Multi-task Affective Behavior Recognition Model}
Figure~\ref{fig:model} shows the framework of our multi-modal affective behavior analysis model. The audio-video dual branch architecture is inspired by TSAV. The multi-modal model fuse features of two branches to give prediction on three different emotion representation tasks.
For the Visual stream, the input clips are composed of cropped aligned images and corresponding face masks. The usage of the face mask in TSAV is believed to be the most helpful for its performance. We use HRNet \cite{cheng2020higherhrnet} to detect 106 facial landmarks and render a face segmentation mask for every face image. As shown in Figure~\ref{fig:mask}, comparing to the mask rendering method of TSAV, which can only render contours image using 68 landmarks, our method can provide more semantic information.
\begin{figure}
\begin{center}
\includegraphics[width=1.0\linewidth]{model.png}
\end{center}
\caption{Framework of multi-task affective behavior analysis model.}
\label{fig:model}
\end{figure}
After preprocessing, each frame image has 4 channels (RGB + mask). These frames are sampled from video with dilation of 6, and constitute input clips with 8 frames. We employ pre-trained (R2+1)D \cite{tran2018closer} as a visual model to extract spatio-temporal information from the visual stream.
As for audio stream, we followed the setting of TSAV. We compute a mel spectrogram for all audio stream extracted from the video using TorchAudio \cite{paszke2019pytorch} package. For each clip, a spectrogram is cut into a smaller sub-spectrogram with the center of sub-spectrogram aligning with the current frame at time $t$. A ResNet-18 \cite{he2016deep} is used for mel spectrogram analysis.
Finally, the output features extracted by the video branch and audio branch are merged. Prediction heads share the same fused features and give final predictions on the three expression representation tasks. AU and VA heads are fully connected layers that map features to AU classification and valence-arousal regression respectively. Expr head contains fully connected layers with self-attention module for expression classification.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.7\linewidth]{mask.png}
\end{center}
\caption{Left: Mask of TSAV; Right: Proposed mask.}
\label{fig:mask}
\end{figure}
\subsection{Mean Teacher}
In the Aff-Wild2 dataset, only 144 videos contain labels for all the three different tasks. The rest 297 videos only have labels for one or two tasks, which brought a challenge for multi-task model training. Specifically, different tasks cannot be supervised at the same time during the multi-task training process. There are different numbers of labeled data for the three tasks, which could lead to imbalanced performance among different tasks. In order to train three tasks at the same time, we introduced the mean teacher \cite{tarvainen2017mean} to take advantage of semi-supervised learning, as shown in Figure~\ref{fig:mean_teacher}.
\begin{figure*}
\begin{center}
\includegraphics[width=0.8\linewidth]{mean_teacher.png}
\end{center}
\caption{Framework for Mean Teacher.}
\label{fig:mean_teacher}
\end{figure*}
The mean teacher framework is extended from a supervised architecture by making a copy of the original model. The original model is called a student and the new one is called the teacher. The parameters of a teacher network are updated by computing the exponential moving average (EMA) of student model’s parameters. Updating parameters of the teacher model and student model by interleaving can reduce overfitting due to the additional unlabeled data.
At each mini-batch in training process, the same batch data is the input to both the student and the teacher. Random noise is added to input data of the teacher to enforce the model to keep consistency under random disturbances. Here we apply random brightness augmentation for each input clip of the teacher model.
For tasks with labels, we can calculate the supervision loss with ground truth. For the unlabeled tasks, we take the predictions of the teacher network as a hard label and then enforce the predictions of student network to be consistent with the hard label. In this way, the optimizer can update the weights of the student network normally for both labeled and unlabeled tasks.
After each training step, the weights of the teacher network are updated by calculating the exponential moving average (EMA) of the student weights, which can be understood as ensemble of student models. At the $t$ training iteration, the parameters of the teacher network are
\begin{equation}
\theta_{t}^{\prime}=\eta \theta_{t-1}^{\prime}+(1-\eta) \theta_{t}
\end{equation}
where $\theta_{t}$ represents the parameters of model, $\eta$ is a hyper parameter of moving average. Here we choose $\eta=0.99$, as suggested in \cite{tarvainen2017mean}.
\subsection{Self-Cure module for uncertainty suppression}
Comparing to ground truth labels, pseudo labels predicted by the teacher model could be inconsistent or incorrect. Training with these uncertain labels may cause model overfitting to incorrect samples, especially in the expression recognition task. We introduce self-cure module described in self-cure network \cite{wang2020suppressing} to solve this problem in expression classification head. The key idea of the self-cure module is to allow network to learn to decide which labels are correct. It consists of two parts: a self-attention importance weighting module and a rank regularization module.
The self-attention importance weighting module can learn the importance of each sample by predicting an attention weight.
Attention weights are predicted by a fully connected layer with sigmoid functions. Samples with higher weights are more reliable and more important for training. For each input sample, its expression classification output is multiplied to the importance weight predicted by self-attention weighting module and give final prediction.
With predicted weights, the ranking regularization module ranks these weights and split them into two groups, which are a high-importance group and a low-importance group. The two groups are regularized by forcing a margin between the two groups with a rank regularization loss (RR-Loss):
\begin{equation}
\mathcal{L}_{R R}=\max \left\{0, \delta-\left(\alpha_{H}-\alpha_{L}\right)\right\}
\end{equation}
where $\alpha_{H}$ is the mean weight of high-importance group, $\alpha_{L}$ is the mean weight of low-importance group. $\delta$ represents a margin, which is set to be $0.15$.
Ranking regularization module and self-attention weighting module together ensures model learns meaningful training data by highlighting certain samples and suppressing uncertain samples in our semi-supervised learning framework.
\subsection{Loss Function}
Loss function for our semi-supervised model consists of two parts: multi-task supervised loss and multi-task consistency loss. Supervised loss is computed with ground truth if label is available. When a label is missing, we take the prediction results of the teacher model as a hard label. Then we calculate consistency losses between the hard label and the prediction of student model in the same way as supervised.
\begin{equation}
L_{t}=\left\{\begin{array}{c}
L_{t}^{s}=L_{t}(S, G), \text {\emph{label available}} \\
L_{t}^{c}=L_{t}(S, T), \text {\emph{label missing}}
\end{array}\right.
\end{equation}
where $L_{t}$ denotes a loss for task $t$, $L^{s}$ is a supervised loss, $L^{c}$ is a consistency loss; $S$ and $T$ represent the predictions of the student network and teacher network respectively, and $G$ is ground truth.
For expression classification, we used the sum of the cross entropy and rank regularization loss as objective function:
\begin{equation}
L_{E x p r}=L_{C E}+L_{R R}
\end{equation}
For AU task, we define the total binary cross entropy loss by
\begin{equation}
L_{A U}=-\sum_{i}^{12}\left\{y_{i} \cdot \log \left(o_{i}\right)+\left(1-y_{i}\right) \cdot \log \left(1-o_{i}\right)\right\}
\end{equation}
where $y$ is the 12 dimensional label vector, $o$ is the corresponding prediction vector.
The concordance correlation coefficient (CCC) loss \cite{kollias2019expression} is used for valence and arousal estimation:
\begin{equation}
L_{V A}=\frac{1}{2} \times(C C C_{V}+C C C_{A})
\end{equation}
The final total loss for current batch is the sum of losses for expression, action unit, and valence and arousal estimation tasks, defined as follows:
\begin{equation}
L_{total}=w_{1} L_{Expr}+w_{2} L_{AU}+w_{3} L_{VA}
\end{equation}
In this paper, we set $w{1} = 1.0$, $w{2} = w{3} = 0.3$.
\section{Experiments}
\subsection{Dataset}
The proposed model was trained on the large-scale in-the-wild Aff-Wild2 dataset only. This dataset contains 564 videos with frame-level annotations for valence-arousal estimation, facial action unit detection, and expression classification tasks. We use the official provided cropped and aligned images in the Aff-wild2 dataset. Additionally, we rendered corresponding facial masks for all the cropped images as described in Section 3.1.
We split the training and validation set by ourselves instead of using official validation set. The official database do not have the same training and validation split among the three tasks. For example, some videos belonging to validation set of AU task also appear in training set of expression classification and valence-arousal estimation tasks. When validating AU task using the inconsistent split data, prior knowledge from training other task will affect the evaluation. Thus, we create a custom training and validation split to ensure that three tasks share consistent split. Moreover, we keep the samples in each task to be split into training and validation set at a ratio of 8:2.
\subsection{Training Setup}
A model was trained with our training split dataset only. We used the pretrained weight from TSAV to initialize the backbone for audio and video branch. The model was optimized using Adam optimizer and a learning rate of 0.0005. Random brightness augmentation was applied for each input clip. The mini-batch size was set to 32. The training and validating processes were performed using two GPU to allocate each of the teacher and student networks to one GPU.
\subsection{Results}
We used the same metrics as suggested in \cite{kollias2021analysing} to evaluate performance.
The metrics of the three tasks are defined as follows:
\begin{equation}
M_{VA}=\frac{1}{2} \times(C C C_{V} + C C C_{A})
\end{equation}
\begin{equation}
M_{Expr}=0.67 \times F_{1}+0.33 \times A c c
\end{equation}
\begin{equation}
M_{AU}=\frac{1}{2} \times (F_{1} + A c c)
\end{equation}
\begin{table}
\begin{center}
\begin{tabular}{|c|ccc|}
\hline
Method & $M_{Expr}$ & $M_{VA}$ & $M_{AU}$ \\
\hline\hline
Basic model & 0.475 & 0.513 & 0.623 \\
Basic model+MT & 0.489 & 0.566 & 0.674\\
Basic model+SC+MT & \textbf{0.501} & \textbf{0.568} & \textbf{0.675}\\
\hline
\end{tabular}
\end{center}
\caption{Performance of our models on validation set. Expr, VA, and AU mean the score for each task. MT denotes the mean teacher, SC denotes the self-cure module.}
\label{table:evaluation}
\end{table}
\begin{table*}
\begin{center}
\resizebox{\textwidth}{!}{
\begin{tabular}{|c|ccc|ccc|ccc|}
\hline
Method & $F_{1}$(Expr) & Acc(Expr) & $M_{Expr}$ & $CCC_{V}$ &$CCC_{A}$ & $M_{VA}$&$F_{1}$(AU)&Acc(AU)&$M_{AU}$ \\
\hline\hline
Baseline \cite{kollias2021analysing} & 0.260 & 0.460 & 0.326 & 0.200 & 0.190 & 0.195 & 0.367 & 0.193 & 0.280\\
NISL-2021 & 0.431 & 0.654 & 0.505& \textbf{0.533} & 0.454 & \textbf{0.494} & 0.451 & 0.847 & 0.653 \\
Netease Fuxi Virtual Human\cite{zhang2021prior} & \textbf{0.763} & \textbf{0.807} & \textbf{0.778} & 0.486 & 0.495 & 0.491 & \textbf{0.506} & \textbf{0.888} & \textbf{0.697}\\
Our & 0.476 & 0.732 & 0.560& 0.478 & \textbf{0.498} & 0.488 & 0.394 & 0.875 & 0.634\\
\hline
\end{tabular}}
\end{center}
\caption{Results on the test set of the Aff-Wild2 dataset. The best result is indicated in bold}
\label{table:test}
\end{table*}
In order to analyze the effects of our proposed framework design, we conducted ablation studies to compare performance with or without proposed components. The results on the validation set can be seen in Table \ref{table:evaluation}. Note that the basic multi-task model is trained with complete labeled data only, while the model with a mean teacher (MT) framework is trained with incomplete labeled data. The usage of the mean teacher allows the model to learn from the unlabeled data which incurs a superior affective behavior recognition performance on each task. However, performance improvement in expression task is not as significant as the other two tasks. We made further investigation by using self-cure mechanism to suppress the impact of uncertainties and further improve expression recognition performance. Experiment indicates that the self-cure module resolves the problem and achieves best performance on three benchmarks.
We also evaluated our model on the official test set. The results on the test set can be seen in Table \ref{table:test}. Our model outperforms the baseline model of \cite{kollias2021analysing} a lot. As for VA track, our model achieve comparable performance comparing to model of team NISL-2021, especially in CCC Arousal metric.As for Expr and AU tracks, our accuracy is comparable in contrast to model of \cite{zhang2021prior}, which verifies that the proposed network can effectively learn from incomplete datasets. But our $F_{1}$ score is much lower. The first reason is that they use large scale pre-trained model on additional dataset, whereas we only use the Aff-Wild2 database for training. Another reason is that we did not use data balancing strategy, which lead to poor $F_{1}$ score. We will investigate data balancing strategies in future research.
\section{Conclusions}
This paper presents a semi-supervised facial affective behavior recognition model by developing a multi-task mean teacher framework. Our key idea is to firstly develop a multi-modal model to recognize the three emotion representation tasks. Then we employ the mean teacher with semi-supervised learning to learn from additional unlabeled data for further improving the recognition performance. Experimental results on validation datasets show that our semi-supervised model outperforms the original supervised model in all tasks, which verifies the effectiveness of the proposed method. For future work, we plan to resolve the problem of data imbalance to further boost the performance.
{\small
\bibliographystyle{unsrt}
|
3,212,635,537,974 | arxiv | \section{Introduction}
\label{Introduction}
Over the past decade, synthesis and exploration of atomically thin two-dimensional (2D) materials have almost revolutionised our common understandings
of condensed matter systems and opened a new era in nanosciences \cite{novoselov-PNAS2005,xu_ms-ChemRev2013,choi-mattoday2017}. In particular, 2D
materials usually show drastically different electronic properties compared to their corresponding bulk structures composed of van der Waals (vdW)
coupled atomic layers \cite{butler-ACSNano2013,geim-Nature2013,novoselov-Science2016}. Transition metal dichalcogenides (TMDCs) are among vdW layered
materials which show a wide range of interesting phenomena and applications due to the tunability of their electronic structures
\cite{splendiani-NanoL2010.4,mak-PRL.105.136805,kuc-PRB.83.245213,wang_qh-nnano2012}. Moreover, they can host exotic phases, such as superconductivity
(SC), charge density waves (CDW), and even topologically non-trivial states
\cite{chhowalla-NatChem2013,frindt-PRL.28.299,wilson-AdvPhys2001,gong-SciRep2017,qi-ncomm2016}. Strongly correlated phases in bulk TMDCs have been well
studied for decades, and recent observations of CDWs and Ising SC \cite{lu_jm-Science2015} in few-layer films have revived interest in these materials
\cite{xi-nnano2015,xi-nphys2016,ugeda-nphys2016,xi-PRL.117.106801,zhu-ncomm2016}.
\par
Among TMDCs, NbSe$_2$ has been considered as a prototype material for investigation of CDW orders and SC \cite{bevolo-JAP1974,corcoran-JPCM1994}.
Bulk 2H-NbSe$_2$ hosts a CDW phase with a $3\times3$ periodicity below 33K which can coexist with an s-wave superconducting phase below
7K \cite{wilson-PRL.32.882,moncton-PRL.34.734,revolinsky-JPCS1965,rahn-PRB.85.224532,castroneto-PRL.86.4382}, and it is known for its high
magnetic anisotropy \cite{galvis-ncommPhys2018}. By reducing its thickness down to single layers, the SC weakens but still survives, whereas
the CDW transition temperature increases \cite{frindt-PRL.28.299,xi-nnano2015,ugeda-nphys2016}. The SC-CDW coexistence in NbSe$_2$ is due to
a momentum dependent gap opening \cite{rahn-PRB.85.224532,zheng-PRB.97.081101} accompanied by an electronic reconstruction over a wide energy
range \cite{arguello-PRB.89.235115}, leaving enough electronic states available for the superconducting transition. Such SC-CDW coexistence
is maintained with charge doping, and their order parameters vary in the same fashion \cite{xi-PRL.117.106801}. The origin of the CDWs has
been intensively debated for some TMDCs, including NbSe$_2$, mostly because of controversies over the role of Fermi-surface nesting
\cite{rice-PRL.35.120,liu_r-PRL.80.5762,straub-PRL.82.4504,inosov-NJP2008,rossnagel-PRB.64.235119,rossnagel-PRB.76.073102,johannes-PRB.73.205102,johannes-PRB.77.165135}.
Nevertheless, in recent years theoretical and experimental evidences has been accumulated in support of momentum-dependent electron-phonon
coupling as a key mechanism in the formation of CDWs \cite{calandra-PRB.80.241108,valla-PRL.92.086401,zhu-PNAS2015,arguello-PRL.114.037001}.
Turning to 2D single-layers, the lack of inversion symmetry, the disappearance of the coterminous of vdW interactions and the interplay with
many-body strong-correlation effects may lead to the aforementioned drastic changes \cite{kotov-RMP.84.1067,guinea-ANDP2014,mak-nmat2012}.
On the other hand, the structure of CDWs in bulk and single-layer NbSe$_2$ is controversial \cite{skripov-SSC1985.53,malliakas-JACS2013,silvaguillen-2DMat2016}. While
a recent detailed experimental work reported evidence of a $3\times3$ commensurate modulation of the crystal structure, the case of the single-layer still needs to be clarified
\cite{calandra-PRB.80.241108,malliakas-JACS2013}. In addition, impurities or gate doping can play a major role on the CDW behaviour. For example, long-range CDW phase
coherence can be suppressed in NbSe$_2$ by a moderate percentage of Co or Mn intercalated at the surface \cite{chatterjee-ncomm2015}, Na intercalated in a bi-layer
\cite{lian-PRB.96.235426} and electron doping \cite{xi-PRL.117.106801}, whereas it can be increased by hole doping \cite{xi-PRL.117.106801}. Besides, Bi adsorption can
also lead to a transition to a stripe phase \cite{fang-ScienceAdv2018}, previously observed in accidentally doped samples \cite{soumyanarayanan-PNAS2013}.
In this work, performing an exhaustive first principle calculations, we reveal possible structures of CDWs, particularly at the presence of certain types of
impurities. Among three different modulated structures, those two with triangular Nb-Nb clusters are found to be energetically favoured in clean NbSe$_2$ 2D
sheet. These results are consistent with experimental evidence \cite{skripov-SSC1985.53,malliakas-JACS2013}. As a key finding we demonstrate that CDWs with
hexagonal modulation can be established by adsorption of certain atoms such as Co and Mn. This type of CDW phase is in fact hidden in pristine NbSe$_2$ because
it has a higher energy compared to triangularly-modulated CDWs. In addition, it is uncovered that the presence of the transition metal impurities induces
magnetism and promotes modulated phases with reduced symmetry of the charge density distribution compared to pure CDW structures. Other types of metallic
ad-atoms, namely K and Ga, allow the same ground state as pristine single-layer NbSe$_2$, but Ga also supports the hexagonally modulated structure. The
current manuscript aims at steering future research towards a new interpretation of the experimental evidence on the effect of impurities
\cite{fang-ScienceAdv2018,chatterjee-ncomm2015,soumyanarayanan-PNAS2013}, and represents an important contribution in the field of the interplay between CDWs
and SC, as a recent work strongly points out \cite{cho-ncomm2018}.
This paper is organised as follows. In Sec.\ \ref{comp}, we will briefly introduce the computational method based on density functional theory
which is used for investigation of CDW phases. Thereafter, we go through the results of \emph{ab-initio} calculations for CDWs in pristine
NbSe$_2$, Sec.\ \ref{pristine}, where the relaxed CDW structures and the profiles of charge densities are presented both in real space and in
Fourier transformed form. The core of our work is found in Sec.\ \ref{impure}, which shows the CDWs in the presence of various impurities. In
particular, we show how the energy hierarchy of the CDWs can be different from the pristine system and a hidden order arise as a new ground
state by adding Co or Mn ad-atoms. Finally, after a discussion over the results, the conclusions are presented in Sec.\ \ref{conc}.
\section{Computational details}
\label{comp}
Our results are obtained by means of density-functional theory (DFT). We employ the projected augmented wave (PAW) method with Perdew-Burke-Ernzerhof (PBE) pseudopotentials,
as implemented in the Vienna {\itshape{ab initio}} Simulation Package (VASP) \cite{blochl-PRB.50.17953,kresse-PRB.59.1758}. Accordingly, the exchange-correlation functional
is treated in the generalised gradient approximation in the PBE parametrisation \cite{perdew-PRL.77.3865,perdew-PRLerratum.78.1396}.
The basis set consists of plane waves, with the explicit treatment of 13, 6, 9, 15, 9 and 13 valence electrons for Nb, Se, Co, Mn, K and Ga states, respectively.
As previously suggested \cite{wehling-PRB.84.235110}, standard local and semi-local exchange-correlation functionals may not offer a proper description of the partially
filled $3d$ and $4f$ shells of TM adatoms on 2D materials. Therefore, calculations involving Co and Mn are performed in the DFT+U approach,
using the rotationally invariant formulation of Lichtenstein {\itshape{et al.}} \cite{liechtenstein-PRB.52.R5467}.
As in ref.\ \onlinecite{wehling-PRB.84.235110}, we use a generalised value of $U = 4.00$ eV and
$J = 0.90$ eV for the $3d$ orbitals of both Co and Mn. Small variations of these parameters (within a reasonable
range) are unlikely to change the physical picture outlined in the present paper. For sake of completeness we
analyse this issue for Co adatoms, in the Appendix.
For all calculations, the cutoff energy of the plane waves is 400 eV, while the energy tolerance on the electronic loops for the
relaxation and for the electronic properties are set to 10$^{-6}$ eV and 10$^{-7}$ eV, respectively; a conjugate gradient algorithm
is employed for structural relaxation. Structures are considered relaxed when the forces on each atom are smaller than 2 meV/\AA.
The simulations are run in $3\times3\times1$, $6\times6\times1$ and $9\times9\times1$ replicas of the NbSe$_2$ single-layer unit cell. After convergence tests
on the {\bf k}-meshes for the $3\times3\times1$ and $6\times6\times1$ replicas were performed, $15\times15\times1$, $7\times7\times1$ and $5\times5\times1$
grids of {\bf k}-points were used to sample their Brillouin Zones for the total energy calculations; $11\times11\times1$ and $5\times5\times1$ {\bf k}-meshes
were used for the partial charge density calculations in the $6\times6\times1$ and $9\times9\times1$ replicas, respectively; a $45\times45\times1$
{\bf k}-mesh was used for the DOS calculations in the $3\times3\times1$ replica.
In modelling metal adsorption on NbSe$_2$, the concentration of one ad-atom in a $6\times6\times1$ replica of the unit cell was adopted, corresponding to 0.0278
impurities/f.u., which allows for a description of a $3\times3\times1$ CDW while minimising the interaction between impurities and their images. The adsorbates
taken into account in the present study are Co, Mn, K and Ga.
\begin{figure*}[t]
\centering
{\includegraphics[trim = 0 0 0 0,width=\textwidth,clip]{Fig1.jpg}}
\caption{(Colour online) First row: relaxed CDW structures for pristine NbSe$_2$ (top view):
T-U (a), T-C (b) and HX (c) CDWs (see text for details); Second row: charge densities of
the respective CDW structures. Volumetric data are represented as blue surfaces enclosing
points whose electronic density is greater than or equal to 0.0075 electrons/Bohr$^3$.
Atoms are represented by spheres, as illustrated in the legends (magnified); Nb-Nb bonds
shorter than the equilibrium distance (3.45 \AA) are represented by solid lines, in order
to help visualising the CDW structure pattern. Dashed lines mark the supercells borders.}
\label{prstStrcts-fig}
\end{figure*}
\begin{figure*}[t]
\centering
{\includegraphics[trim = 0 0 0 0,width=\textwidth,clip]{Fig2.jpg}}
\caption{(Colour online) Differences between the Fourier Transform (FT) of the charge density
distributions for the pristine CDW structures and the non-modulated structure. The plots of
T-U (a), T-C (b) and HX (c) CDWs charge density correspond to those in Fig.\ \ref{prstStrcts-fig}.
The large dashed hexagon with vertices at ${\left|\mathbf q\right|} = \frac{2\pi}{a}$
(with $a = 3.45$ \AA) marks the characteristic Bragg peaks; the medium size dashed hexagon with
vertices at ${\left|\mathbf q\right|} = \frac{2\pi}{3a}$ marks the (shorter) CDW peaks; the small
green shaded hexagon (${\left|\mathbf q\right|} = \frac{2\pi}{6a}$) maps points beyond the supercell
borders.}
\label{FFTprst-fig}
\end{figure*}
\section{CDW phases in pristine single layers}\label{pristine}
We begin with the study of the single-layer NbSe$_2$ in a $3\times3\times1$ supercell, which is the minimal size cell for a CDW in NbSe$_2$. Pristine single-layer NbSe$_2$
is known to be metallic, non magnetic and hosts CDWs below 145 K \cite{wilson-AdvPhys1969,moncton-PRL.34.734,wilson-AdvPhys2001}. Our models include three structures
obtained according to existing work \cite{skripov-SSC1985.53,malliakas-JACS2013,xi-nnano2015}, which are shown in Fig.\ \ref{prstStrcts-fig}; the Nb atoms cluster
in triangular patterns - see Figs.\ \ref{prstStrcts-fig}(a~and~b) - and hexagonal ones - see Fig.\ \ref{prstStrcts-fig} (c); the two triangular patterns differ by the
position of the Se atoms with respect to the triangle composed by the Nb-Nb bonds; accordingly, the CDW structures are named T-U, T-C and HX, respectively (T-U and T-C
stand for triangle-uncentred and triangle-centred, whereas HX stands for hexagonal). The two structures T-U and T-C are related to each other by a mirror reflection of
the Nb sublattice. It is also known that the Se-Se bond patterns (analogous to Nb-Nb bond patterns) accompany those of Nb-Nb \cite{silvaguillen-2DMat2016}. The T-U and
T-C CDWs differ for the Se-Se pattern, see the Appendix for details. Their total energies were compared in $3\times3\times1$ and $6\times6\times1$ replicas of the NbSe$_2$
unit cell, and suggest that the formation of all of them is favoured. The T-U is the lowest energy CDW structure; calculations in the $3\times3\times1$ supercell yield
differences of 3.9 meV, 0.5 meV and 1.3 meV/f.u. with respect to the undistorted structure, T-C and HX, respectively. In the $6\times6\times1$ supercell, slight changes
are observed in the energy differences between the CDW structures: the T-U is favoured by 0.4 meV and 1.1 meV over the T-C and HX, respectively; these differences are
maintained for a $9\times9\times1$ supercell. The three CDW structures were recently investigated elsewhere on $3\times3\times1$ supercells \cite{lian-NanoL2018.5}.
Here, we demonstrate our agreement with ref.\ \onlinecite{lian-NanoL2018.5}, and we make use of the results for the pristine to compare the metal-adsorbed NbSe$_2$.
The effect of the CDW distortions on the density of states (DOS) is analysed in the Appendix. For what concerns the electronic reconstruction following the
CDW formation, our calculations are in agreement with the literature \cite{shen-PRL.99.216404,calandra-PRB.80.241108}. The charge distributions are computed
integrating the charge density over the occupied Nb band (the band crossing Fermi level, see the Appendix) and they are shown in Fig.\ \ref{prstStrcts-fig},
second row. The integration over the whole occupied band simulates a topography retaining the symmetry of the Nb band only. The charge density clusters in
patches, with different shape and patterns for each CDW; the lowest energy CDWs, T-U and T-C, have three-fold symmetric patterns and the HX CDW has six-fold
symmetric ones. The patches of T-U are on the vertices of a hexagon (with no element on the centre), whereas those of T-C and HX are placed on the vertices
and centre of a larger hexagon.
The Fourier Transform (FT) of the charge density distribution allows to recognise more clearly the symmetry of the modulation patterns and trace their length
scale. Due to the three-dimensional periodic boundary conditions, the three-dimensional data is originally computed as a function of the $(h, k, l)$ Miller
indices; the subset with $l = 0$ is analysed to track modulations of the CDW charge distributions only along the plane. Thus, the FT is mapped as a function
of $h$ and $k$ ($k_x$ and $k_y$ in the relative plots). The computed FT plots of the charge density distributions of the three CDW structures in pristine
NbSe$_2$ are reported in Fig.\ \ref{prstFFT-fig}, in Appendix A.
More significant information can be obtained by considering the difference between those plots and the FT plot of the charge
density distribution of the non-modulated structure, shown in Fig.\ \ref{FFTprst-fig}. In these (and following) plots, the
vertices of the large hexagon, at ${\left|\mathbf q\right|} = \frac{2\pi}{a}$, mark the position of the characteristic Bragg
peaks, which are evident in Fig.\ \ref{prstFFT-fig}. The vertices of the medium size hexagon,
at ${\left|\mathbf q\right|} = \frac{2\pi}{3a}$, mark instead the position of the shorter peaks associated to the CDW. Both
hexagons are emphasised with a dashed line, as a guide for the eye. The small green shaded hexagon, whose vertices are at
${\left|\mathbf q\right|} = \frac{2\pi}{6a}$, map points beyond the supercell borders, and thus are not meaningful. Due
to the size of the supercell, the width of the spots denoting Bragg peaks or CDW peaks is large because different but close
modulation frequencies cannot be resolved, even with a relatively dense k-mesh. Fig.\ \ref{FFTprst-fig} show the relative
differences between the three CDWs and the non-modulated structure. The CDW main peaks are not sensibly different from plot
to plot, neither are satellite peaks at $(2/3) \Gamma K$ and at $(1/3) K K'$. However, a sizeable satellite peak appears at
$2/3 \Gamma M$ in the T-U CDW, but not (as large) in the T-C nor in the HX CDWs. Such configuration of peaks mirrors the
configuration of the electronic patches in Fig.\ \ref{prstStrcts-fig}, which in the T-U (T-C and HX) are placed at the vertices
of a small (large) hexagon, without (with) a central element. In fact, two different vectors map equivalent patterns in T-U.
Naming the lattice vectors of the unit cell $\mathbf a$ and $\mathbf b$, the mapping vectors are $3\mathbf{a}$ (and equivalently
$3\mathbf{b}$) and $\mathbf a + \mathbf b$; the latter does not map equivalent patches in
T-C nor in HX.
\begin{table}[t]
\centering
\caption{Energetics and magnetism for M$\vert$NbSe$_2$ (M = Co, K, Ga, Mn).
Energy differences are computed with respect to the lowest configuration
and are expressed in meV. Magnetic moments are expressed in $\mu_B$.}
\label{tab-ads}
\begin{tabular}{r|r||r|r|r}
& & $\Delta$E & $\mu^{TM}$ & $\mu_{tot}$ \\
\hline\hline
& hollow & 16 & 1.9 & 2.2 \\
Co & top Nb & 0 & 1.9 & 2.0 \\
& top Se & 3551 & 2.0 & 3.4 \\
\hline
& hollow & 91 & 4.4 & 5.8 \\
Mn & top Nb & 0 & 4.4 & 4.3 \\
& top Se & 1291 & 4.9 & 4.1 \\
\hline
& hollow & 4 & 0.0 & 5.9 \\
K & top Nb & 0 & 0.0 & 0.0 \\
& top Se & 241 & 0.0 & 1.2 \\
\hline
& hollow & 183 & 0.0 & 5.9 \\
Ga & top Nb & 0 & 0.0 & 0.0 \\
& top Se & 605 & 0.0 & 0.2
\end{tabular}
\end{table}
\begin{table}[t]
\centering
\caption{Energetics for the CDWs in M$\vert$NbSe$_2$ (M = Co, Mn, K, Ga); the energy for each column
(adsorbate) is given as differences with respect to the ground state. Different position and resulting
structure are possible for each metal M adsorbed on a CDW; we report those which evolve to structures
relatively close in energy to the ground state, mentioning the type of structure, when it is similar to
one of the three pristine CDWs, as well as the energy difference.
The difference in energy in the 9x9 supercell (1 Co) is 0.3 meV/f.u. in favour of the HX CDW.}
\label{tab-cdwia}
\begin{tabular}{lr||lr|lr|lr|lr}
\multicolumn{2}{c}{pristine} & \multicolumn{2}{c|}{Co} & \multicolumn{2}{c|}{Mn} & \multicolumn{2}{c}{K} & \multicolumn{2}{|c}{Ga} \\
\hline\hline
T-U & (0.0) & T-U & (0.7) & T-U & (1.5) & T-Uh & (0.0) & T-U & (0.0) \\
& & T-U & (2.6) & & & T-UN & (0.2) & & \\
\hline
T-C & (0.4) & HX-A & (0.0) & HX-A & (0.0) & T-CN & (1.5) & HX-A & (0.1) \\
& & & & & & T-Ch & (1.3) & & \\
\hline
HX & (1.1) & HX-S & (0.0) & HX-S & (0.0) & HX-S & (0.7) & HX-S & (0.0) \\
& & HX-A & (0.0) & HX-A & (0.0) & & & &
\end{tabular}
\end{table}
\begin{table}[t]
\centering
\caption{Energetics and magnetism for M$\vert$NbSe$_2$ (M = Co, K, Ga, Mn).
Energies are given as differences with respect to the lowest energy solution
and are expressed in meV; magnetic moments are expressed in $\mu_B$.}
\label{tab-adscdw}
%
\begin{tabular}{r|r||r|r|r}
& & $\mu^{TM}$ & $\mu_{tot}$ & $E^{ad}$ \\
\hline\hline
& T-U & 1.9 & 2.0 & 3.440 \\
Co & HX-A & 1.9 & 2.0 & 3.504 \\
& HX-S & 1.9 & 1.9 & 3.504 \\
\hline
& T-U & 4.4 & 4.4 & 2.472 \\
Mn & HX-A & 4.4 & 4.4 & 2.559 \\
& HX-S & 4.4 & 4.3 & 2.561 \\
\hline
& T-Uh & 0.0 & 0.0 & 3.045 \\
K & T-UN & 0.0 & 0.0 & 3.038 \\
& HX-S & 0.0 & 0.1 & 3.060 \\
\hline
& T-U & 0.0 & 0.0 & 2.907 \\
Ga & HX-A & 0.0 & 0.0 & 2.946 \\
& HX-S & 0.0 & 0.0 & 2.944 \\
\end{tabular}
\end{table}
\section{CDW phases in the presence of impurities}\label{impure}
\begin{figure*}[t]
\centering
{\includegraphics[trim = 0 0 0 0,width=\textwidth,clip]{Fig3.jpg}}
\caption{(Colour online) First row: ground state Co$\vert$NbSe$_2$ HX-S (a), HX-A (b) and T-U (c)
CDW structure; Second row: charge densities of the respective CDW structures. The isosurface
value for the volumetric data is set in agreement with Fig.\ \ref{prstStrcts-fig}. Atoms are
represented by spheres, as illustrated in the legends; Nb-Nb bonds shorter than the equilibrium
distance (3.45 \AA) are represented by solid lines, to help visualising the CDW structure pattern.
Dashed lines mark the supercells borders.}
\label{CoStrcts-fig}
\end{figure*}
\begin{figure*}[t]
\centering
{\includegraphics[trim = 0 0 0 0,width=\textwidth,clip]{Fig4.jpg}}
\caption{(Colour online) Magnetisation densities of Co$\vert$NbSe$_2$ HX-S (a), Co$\vert$NbSe$_2$
HX-A (b), Co$\vert$NbSe$_2$ T-U at 1 (c), Co$\vert$NbSe$_2$ T-U at 3 (d), Mn$\vert$NbSe$_2$ HX-A (e)
and Mn$\vert$NbSe$_2$ T-U at 1 (f). The magnetisation densities in (c) and (d) are different although
they have virtually the same structure and total energy; the magnetisation in (c) is juxtaposed to (f)
to compare structures with the same adsorption site. Volumetric data are represented as blue (red)
surfaces enclosing points whose magnetisation density is greater than or equal to 0.0075 (smaller
than or equal to -0.0075) $\mu_B$/Bohr$^3$. Atoms are represented by spheres, as illustrated in
the legends; Nb-Nb bonds shorter than the equilibrium distance (3.45 \AA) are represented by solid
lines to help visualising the CDW structure pattern. Dashed lines mark the supercells borders.}
\label{CoMnMgn-fig}
\end{figure*}
\begin{figure*}[t]
\centering
{\includegraphics[trim = 0 0 0 0,width=\textwidth,clip]{Fig5.jpg}}
\caption{(Colour online) Differences between charge density distributions. Co$\vert$NbSe$_2$
T-U minus NbSe$_2$ T-U (a), Co$\vert$NbSe$_2$ HX-A minus NbSe$_2$ T-U (b) and Mn$\vert$NbSe$_2$ HX-A
minus NbSe$_2$ HX (c). Volumetric data are represented as blue (red) surfaces enclosing points whose
electronic density is greater than or equal to 0.0025 (smaller than or equal to -0.0025)
electrons/Bohr$^3$. Atoms are represented by spheres, as illustrated in the legends; Nb-Nb bonds
shorter than the equilibrium distance (3.45 \AA) are represented by solid lines to help visualising
the CDW structure pattern. Dashed lines mark the supercells borders.}
\label{CoMnChgDiffs-fig}
\end{figure*}
\begin{figure*}[t]
\centering
{\includegraphics[trim = 0 0 0 0,width=\textwidth,clip]{Fig6.jpg}}
\caption{(Colour online) Differences between FT of the charge density
distribution of Co-adsorbed and Mn-adsorbed NbSe$_2$ CDWs and the T-U CDW in
pristine NbSe$_2$. Ground state Co$\vert$NbSe$_2$ HX-A (a), Co$\vert$NbSe$_2$
HX-S (b) and Mn$\vert$NbSe$_2$ HX-A (c). The dashed hexagons mark the same
regions as in Fig.\ \ref{FFTprst-fig} (see caption).}
\label{FFTCoMn-fig}
\end{figure*}
Adsorption of the adsorbates Co, Mn, K and Ga on a fully symmetric structure induces structural distortions breaking the symmetry of the NbSe$_2$ layer
according to the site: the adsorption on the Nb site, hollow site and Se site induces CDWs of HX type (i.e. having similar Nb-Nb distance patterns to the
pristine HX), T-U type and T-C type, respectively. By total energy calculations, the likelihood of the adsorption sites is analysed. The energies
for different adsorption sites of the investigated ad-atoms are reported in Table \ref{tab-ads}. These results are consistent with
those reported in a recent work on MoS$_2$ \cite{wei-PRB.95.075419}.
The preferred adsorption site is on top of a Nb atom for Co, K, Ga and Mn. The competition between hollow site
and Nb site is strong in the case of K (4 meV difference). The Se site remains the most unfavoured for single atoms. However, it competes with Nb for larger
molecules. In order to highlight this trend, we computed the energy difference for Co-(OH)$_2$, which can be considered as a prototype of a small molecule
and a possible impurity. In the case of a single Co, the Nb site is preferred to the hollow site and Se site by 16 meV and 3551 meV, respectively; in the case
of Co-(OH)$_2$, the corresponding energy difference are $\sim$~670 meV and 166 meV, suggesting a hindrance to the Co-Se bonds and Co-Nb bonds or a decrease
of the Co charge state and in turn of its coordination. In summary, single Co is preferably adsorbed on the Nb site, with the hollow site relatively close
in energy; large molecules favour adsorption on the Se site and unfavour adsorption at the hollow site, a trend in line with combined theoretical and
experimental results \cite{kezilebieke-NanoL2018.4}. After establishing the preferred adsorption sites on the NbSe$_2$ non modulated structure for each
adsorbate by total energy calculations, the adsorption on the different CDW structures is modelled (it involves different inequivalent adsorption sites
due to the lower symmetry).
\begin{table}[th]
\centering
\caption{Occupancies on Co and Mn $3d$ orbitals, which split according to a trigonal prismatic
environment, namely $\mathit{e''}$, $\mathit{e'}$ and $\mathit{a'_{1}}$, where $\mathit{a'_{1}}
= d_{z^2 - 3r^2}$. The coefficients $u_{\uparrow}$, $v_{\uparrow}$, $u_{\downarrow}$ and
$v_{\downarrow}$ are .90, .43, .62 and .78 for Co and .99, .16, .78 and .62 for Mn, respectively.}
\label{tab-occTM}
\begin{tabular}{r||c|c|c|c}
& \multicolumn{2}{c|}{Co} & \multicolumn{2}{c}{Mn} \\
\hline
& $\sigma = \uparrow$ & $\sigma = \downarrow$ & $\sigma = \uparrow$ & $\sigma = \downarrow$ \\
\hline\hline
$\mathit{e'_{1,\sigma}}$ & 0.93 & 0.07 & 0.93 & 0.03 \\
$\mathit{a'_{1}}$ & 0.90 & 0.79 & 0.92 & 0.08 \\
$\mathit{e''_{1,\sigma}}$ & 0.95 & 0.92 & 0.93 & 0.09
\end{tabular}
\end{table}
\subsection{Adsorption of Co}
Upon adsorption of Co, the energy and state of CDWs are modified. In general, the solutions are a combination of the states found in the pristine system.
Pure solutions of the T-U CDW are found and are referred to as such in the remainder. The HX CDW solutions are found in mixed states or pure states; those
with lowest energy are grouped according to the symmetry of their structures and charge densities, and named HX-S and HX-A, for symmetric and asymmetric
ones, respectively. The T-C CDW solutions are found only mixed (with the HX-A CDW). The HX-S, HX-A and T-U CDWs structures are shown in Figs.\
\ref{CoStrcts-fig} (a), (b) and (c), respectively. The HX-S features the characteristic hexagonal patches of the pristine HX CDW together with a tri-fold
symmetric star of Nb-Nb bonds, compare Figs.\ \ref{CoStrcts-fig} (a) and \ref{prstStrcts-fig} (c). Their charge density distributions are not remarkably
different from their pristine counterparts, compare Figs.\ \ref{CoStrcts-fig} (d) and \ref{prstStrcts-fig} (f). The flatness of the (multi-dimensional)
potential energy surface allows adsorption on different sites of the underlying CDW structure to give different solutions. As Table \ref{tab-cdwia} reports,
the HX-A CDW results from the relaxation of the adsorption of Co on the vertex of the large triangle of a T-C CDW structure, compare Fig.\ \ref{prstStrcts-fig} (b),
and a mixing between the T-C and the HX CDWs occurs. The patches in the charge density distribution, Fig.\ \ref{CoStrcts-fig} (e), recall those in both the
pristine T-U and the pristine T-C, Figs.\ \ref{prstStrcts-fig} (e) and (f), supporting the previous observation. Finally, the structure and charge density
distribution of the T-U solution are virtually identical to those in the pristine T-U CDW, with minor differences around the adsorption site, compare Figs.\
\ref{CoStrcts-fig} (f) and \ref{prstStrcts-fig} (a).
The magnetic density distributions exhibit antiferromagnetic coupling between Co and Nb in the occupied Nb band; moreover the magnetisation around Co in that
energy range is opposite to the total magnetisation on Co, compare Fig.\ \ref{CoMnMgn-fig}, first row, with Table \ref{tab-adscdw}. The magnetic moment on Co
(value integrated over all the occupied states) is around 2.0 $\mu_B$ for every CDW, and the total magnetisation over the whole NbSe$_2$ layer vanishes. The
same observation is valid for Mn, see Table \ref{tab-adscdw}, and the discussion in the Appendix. The modulation of the magnetisation density
in the HX-A CDW is larger than that in the HX-S (and that in T-U) CDW, compare Figs.\ \ref{CoMnMgn-fig} (a) and (b), first row, suggesting that larger mixing
of different CDWs supports the formation of a spin density wave (SDW).
Moreover, a different modulation of the magnetisation densities of Co$\vert$NbSe$_2$ T-U with Co at different adsorption CDW site is observed, compare
Figs.\ \ref{CoMnMgn-fig} (c) and (d). These two structures are at the same energy, suggesting that a magnetic order transition is still incipient. The
magnetisation data obtained by a site-by-site analysis point to a ferromagnetic coupling, in agreement with ref.\ \onlinecite{zhou-ACSNano2012}, where an
incipient magnetic transition is achieved by tensile strain.
The variation of the charge modulation can be analysed looking at the difference in the charge density distribution between
Co$\vert$NbSe$_2$ CDWs and the pristine CDWs in the direct space, see Fig.\ \ref{CoStrcts-fig}. In these plots, blue (red)
lobes denote injection (depletion) of charge with respect to the pristine CDW charge distribution. The adsorption of Co modifies
the T-U CDW only in the neighbourhood of the adsorption site, see Fig.\ \ref{CoMnChgDiffs-fig} (a), as expected from Fig.\
\ref{CoStrcts-fig} (f); charge depletion (injection) occurs out-of-plane (in-plane) in correspondence to the Nb at the adsorption
site and in a more complex pattern on the surrounding Nb atoms; in general, red lobes point towards the adsorbate. The charge
difference between the HX-A and the T-U CDWs features a constant charge difference - due to the misalignment between the two
charge densities (in NbSe$_2$ and Co$\vert$NbSe$_2$) - and an enhancement of in-plane modulations identified by the isolated
blue lobes in Fig.\ \ref{CoMnChgDiffs-fig} (b); these may be more relevant in comparison to the out-of-plane modulations
represented by the out-of-plane red lobes on isolated Nb atoms in Fig.\ \ref{CoMnChgDiffs-fig} (c), which refer to Mn-adsorbed
NbSe$_2$.
Eliminating charge displacements (due to the different adsorption sites) helps analysing the symmetry of the charge distribution. In analogy to
what done for the pristine case (Fig.\ \ref{FFTprst-fig}), we do not show the FT plots of the charge density distributions directly. Instead, we
focus on the differences between these plots and the FT plot obtained for the T-U CDW in the pristine case. In order to identify variations in
the CDW signal, the marks of Bragg peaks, CDW peaks and the region corresponding to points in the direct space beyond the supercell size are used
in agreement with Figs.\ \ref{FFTprst-fig}.
The FT plots relative to Co$\vert$NbSe$_2$ HX-A, Co$\vert$NbSe$_2$ HX-S and Mn$\vert$NbSe$_2$ HX-A are shown in Figs.\ \ref{FFTCoMn-fig} (a)-(c); the FTs shown
are computed difference with respect to the pristine T-U CDW. In these cases with adsorbates, a peak in the neighbourhood of ${\left|\mathbf q\right|} = 0$ appears,
because of the background charge (uniformly) injected into the system. All plots in Fig.\ \ref{FFTCoMn-fig} point to a considerable suppression (enhancement) of the
CDW intensity along $\Gamma K$ at $\left|\mathbf q \right| = 2\pi/3a$ (at $\left|\mathbf q \right| = \pi/a$); such suppression/enhancement is anisotropic for Co HX-A,
being large along the line $\mathbf k_y = 0$ and small along the other two $\Gamma K$ lines; the other cases are isotropic (Co HX-S) or virtually isotropic (Mn HX-A).
Furthermore, large depletion of intensity at $\left|\mathbf q \right| = 4\pi/3a$ along $\Gamma M$ as well as at $\left|\mathbf q \right| = 2\pi/3a, 4\pi/3a$ along
$K K'$, illustrates again the difference between the pristine HX CDW (T-C as well) and the pristine T-U CDW observed above. Finally, intensity enhancement occurs
also within the small hexagon marking the $3\times3$ CDW peaks (but with no leading ${\mathbf q}$-vector), suggesting a competition between modulations with different
wavelengths, as previously discussed \cite{calandra-PRB.80.241108,lian-NanoL2018.5}. Indeed, strain-induced modifications of the $\mathbf q$ vector were recently
found \cite{gao-PNAS2018}. In general, the asymmetric form of the CDW intensity (with respect to pristine T-U) suggests a connection with recently observed stripe
phases \cite{soumyanarayanan-PNAS2013,fang-ScienceAdv2018,gao-PNAS2018} of which it could be a precursor. The character of the asymmetry in the adsorbates systems is
further treated in the Appendix, with reference to Fig.\ \ref{otherFFT-fig}.
\subsection{Adsorption of Mn}
In the case of Mn adsorption, magnetism plays a major role. A high magnetic moment, which correctly describes Mn, favours the Nb site more than in
the case of Co adsorption (the energy difference between the Nb site and the hollow site is 5 times larger), compare their energy differences in Table
\ref{tab-ads}. The ordering of the CDW structures, assessed by total energy calculations, follows the same pattern as in the case of Co adsorption,
having a family of ground state HX CDWs, including a symmetric one and an asymmetric one, a solution of T-U at a higher energy and the absence of a
T-C solution. The energy difference between the ground state HX CDW and the T-U solution is 1.5 meV/f.u., see Table \ref{tab-cdwia}, which compared
to the case of Co, it suggests that Mn drives a slightly stronger transition to a HX CDW. The charge density distributions of the HX-S and HX-A show
no essential difference, being also very similar to that of Co$\vert$NbSe$_2$ HX-S. Therefore, the relative figures are omitted, and the remaining
discussion is limited to the magnetisation density distributions and the FT of the charge density distributions.
The magnetisation densities of the HX-S, HX-A the T-U are shown in Fig.\ \ref{CoMnMgn-fig}, second row. The two HX CDWs are similar also in their
magnetisation density distribution - compare with the case of Co adsorption, Fig.\ \ref{CoMnMgn-fig}, second row. The structure of HX-A has a reflection
symmetry through the top-left to bottom-right diagonal as represented in Fig.\ \ref{CoMnMgn-fig}. The T-U CDW structure is modified slightly in the
vicinity of the adsorption site. The magnetisation density in all of the CDWs is negative around Mn, unlike the case of Co, where a negative cloud
surrounding Co is neighboured by a positive cloud around the closest Nb atoms, compare also Figs.\ \ref{CovsMnMgn-fig} (a) and (b) in the Appendix,
showing a detailed view on the vicinity of the adsorption sites.
Charge density differences in direct space are analysed in comparison with Co adsorption. The main difference between the two adsorbates is that with Mn adsorption
the out-of-plane charge modulations are not suppressed. In fact, note that the in-plane red lobes in Fig.\ \ref{CoMnChgDiffs-fig} (e) replace the out-of-plane red
lobes in Fig.\ \ref{CoMnChgDiffs-fig} (c), point in three directions, symmetrically. In fact, the similarity between Figs.\ \ref{CoMnChgDiffs-fig} (d) and (f)
confirms that the HX-S has the same features in the case of Co and Mn, with blue lobes pointing out-of-plane.
The analysis of the FT plots was mentioned previously with reference to Fig.\ \ref{FFTCoMn-fig}. The main difference with the case of Co adsorption is that the HX-A and HX-S
do not differ much, i.e. the intensity variation shows little contrast between the $C_6$ and the $C_2$ symmetries, compare the HX-A and HX-S CDWs in Figs.\ \ref{FFTCoMn-fig}
(c) and \ref{otherFFT-fig} (c). The preference of the TM for the HX CDWs seems to be at variance with their tendency of reducing the symmetry from $C_6$ to $C_2$. The lowest
energy solutions for Co$\vert$NbSe$_2$ (and Mn$\vert$NbSe$_2$) are mixed state of HX and T-C CDWs, compare Fig.\ \ref{CoStrcts-fig} (b) with Fig.\ \ref{prstStrcts-fig} (b).
However the Co$\vert$NbSe$_2$ HX-A CDW solution has a consistent T-C component, whereas the Mn$\vert$NbSe$_2$ HX-A CDW solution has a small T-C component, see the underlying
structure in the magnetisation density plot, Fig.\ \ref{CoMnMgn-fig} (e).
The CDW solutions for TM adsorbates on NbSe$_2$ are in fact mixed states; where the mixing between HX and T-C is high, the symmetry of the charge density
distribution is reduced, whereas predominant HX CDW solutions keep a $C_6$ symmetry. In the case of Mn, the preference for HX solutions is higher than in
the case of Co, and as a result the T-C component in the mixed solutions is smaller and in turn the $C_6$ symmetry is virtually maintained. Finally, the
observation that the T-U / T-C have a reduced symmetry, raises the question on pristine T-U/C and the symmetry of its charge density distribution, which
has a $C_6$ symmetry; perhaps the adsorbates induces some symmetry breaking which allows the T-U to undergo a transition.
Table \ref{tab-occTM} shows the occupancies of Co and Mn $3d$ orbitals. The environment around the TM in the NbSe$_2$ layer is trigonal
prismatic, and therefore orbitals split into three groups due to the crystal field; these are classified according to their symmetry into
$\mathit{e''}$, $\mathit{e'}$ and $\mathit{a'_{1}}$. The orbital splitting is valid also for Co and Mn $3d$ orbitals, but the intensity
of the splitting is reduced because the environment is incomplete. Therefore, the spin splitting, which depends on the $l$-character of
the orbitals is larger than or comparable to the crystal field splitting, and orbitals are ordered by increasing energy as follows: in the
majority spin channel, $\mathit{e''}$, $\mathit{e'}$ and $\mathit{a'_{1}}$; in the minority spin channel the order of $\mathit{e'}$ and
$\mathit{a'_{1}}$ is inverted. Furthermore, the orbital projections onto cubic (real spherical) harmonics ($d_{xy}$, $d_{yz}$, $d_{z^2 - 3r^2}$,
$d_{zx}$ and $d_{x^2 - y^2}$, ordered by increasing $m$ value) are spin dependent. The orbitals of the eigenbasis can be written as
$\mathit{e''}^1_{\sigma} = v_{\sigma} d_{xy} + u_{\sigma} d_{yz}$, $\mathit{e''}^2_{\sigma} = u_{\sigma} d_{zx} + v_{\sigma} d_{x^2 - y^2}$,
$\mathit{e'}^1_{\sigma} = u_{\sigma} d_{xy} + v_{\sigma} d_{yz}$ and $\mathit{e'}^2_{\sigma} = v_{\sigma} d_{zx} + u_{\sigma} d_{x^2 - y^2}$.
(Freedom in the choice of the basis set allows one to set $\mathit{a'_{1}} = d_{z^2 - 3r^2}$.) The $l$-character of the $\mathit{e''}^{\uparrow}$
orbitals is close to 1 (i.e. the $u$ values are larger than the $v$ values); conversely, the $\mathit{e'}^{\uparrow}$ orbitals have a prevalent
$l = 2$ character ($v$ values are larger); overall, this fact holds for the $\mathit{e''}^{\downarrow}$ and $\mathit{e'}^{\downarrow}$ orbitals.
\begin{figure*}[t]
\centering
{\includegraphics[trim = 0 0 0 0,width=\textwidth,clip]{Fig7.jpg}}
\caption{(Colour online) First row: ground state CDW structure, with K adsorbed on the hollow
position (a), its charge density distribution (b), and the charge density for K adsorbed on
the Nb site (c); Second row: Difference in the charge densities between the K$\vert$NbSe$_2$
ground state and NbSe$_2$ T-U (d), ground state CDW structures for two Ga$\vert$NbSe$_2$,
namely T-U (e) and HX-A (f). Atoms are represented by spheres, as illustrated in the legends;
Nb-Nb bonds shorter than the equilibrium distance (3.45 \AA) are represented by solid lines to
help visualising the CDW structure pattern. Dashed lines mark the supercells borders. The
isosurface value for the volumetric data is set in agreement with Fig.\ \ref{prstStrcts-fig}.}
\label{KStrctsChg-fig}
\end{figure*}
\begin{figure*}[t]
\centering
{\includegraphics[trim = 0 0 0 0,width=\textwidth,clip]{Fig8.jpg}}
\caption{(Colour online) Difference in the FT of the charge density
distributions between the K$\vert$NbSe$_2$ on the hollow site (a),
and Nb site (b) and the T-U pristine CDW. The charge distribution has
a $C_2$ symmetry for the adsorption on the hollow site and a $C_6$
symmetry for the adsorption on the Nb site. In (c), the FT difference
between K$\vert$NbSe$_2$ on the hollow site and the pristine T-C CDW
is shown.}
\label{FFTK-fig}
\end{figure*}
In summary, TM adsorption favours HX CDW, weakening the CDW signals for the T-U and T-C CDWs, and the symmetry of the charge distribution is reduced from
$C_6$ to $C_2$, especially in the case of Co, where mixing between HX and T-C occur. The modulation of the magnetic density is dependent on the symmetry of
the charge distribution: the higher the symmetry, the weaker the modulation. Also, due to a weaker crystal field splitting, the orbital order of the $3d$
of Co and Mn in the two spin channels is different. Finally, a probe for Mn and Co is given in terms of $l$-character of their electrons.
\subsection{Adsorption of K and Ga}
The adsorption of K is different from the other cases, since the energy difference between having an adatom
at the hollow site or in top of a Nb site (see Table \ref{tab-ads}) is small, and therefore can be reversed
by the presence of a CDW. This is, in fact, what happens. While on a non modulated NbSe$_2$ structure, the
hollow site stands 4 meV above the Nb adsorption site, on the CDW modulated structures the hollow site becomes
more favourable, of about 0.2 meV/f.u. (their respective ground state are compared). The solutions obtained
starting from the T-C converge to mixed states between T-C itself and HX and they are found at a high energy,
while the T-U CDW is still favoured by 0.7 meV/f.u. over the HX CDW (the energy difference slightly changes
with respect to the pristine case). The CDW structure and charge density distribution (in the T-U) on the
hollow site (the ground state) and on the Nb site look very similar, see Fig.\ \ref{KStrctsChg-fig}, and trace
back to the pristine CDW ground state, compare Fig.\ \ref{prstStrcts-fig}. The CDW is slightly enhanced inside
the hexagon delimiting the CDW peaks, but without a leading $\mathbf q$. The symmetry of the FT plot for K on
the Nb (hollow) site is reduced from C$_6$ to C$_2$ at the points
$\mathbf q = (\pm \pi/3a,\pi/\sqrt{3}a)$ ($\mathbf q = (\pm 2 \pi/3a,0)$), see Fig.\ \ref{FFTK-fig}; however,
the intensity is one order of magnitude smaller than that of the Nb and Mn cases. Note also the difference with
the T-C CDW, showing that the signal of K-adsorbed T-U on the hollow site is significantly enhanced with respect
to pristine T-C at distinct $\mathbf q$ vectors, see Fig.\ \ref{FFTK-fig} (c).
The case of Ga is interesting in comparison with K because Ga has a fully occupied $s$ shell and a single electron in the $p$ shell. In this case, the adsorption on the
Nb site is favoured by 183 meV over the adsorption at the hollow site. The T-U CDW is in strong competition or coexists with the HX CDW, and the T-C converges to a mixed
solution between T-C and HX, analogously to the case of K. The ground state structures are shown in Fig.\ \ref{KStrctsChg-fig}. The charge density distributions and their
FTs are not particularly different from the case with K adsorption. Both K and Ga cases are related to gate doping, because they consist of an electron per 36 f.u. injected
in the system. However, structural reconstructions due to the chemical adsorption must be considered for the enhancement or suppression of the CDW order. Compared to
experimental observations with gate doping \cite{xi-PRL.117.106801}, where the case of K and Ga adsorption suppress the CDW order, the FT plots suggest that a considerable
order remains (no depletion is seen at any $\mathbf q_{CDW}$). Therefore, a doping with a hole carrying atom could enhance the CDW signal.
Overall, adsorption of atoms on single-layers NbSe$_2$ suppresses the T-C CDW and promotes the HX CDWs in all cases; in particular, with Co and Mn the HX CDWs become
the ground state for all the coverage considered in this study, whereas in the case of Ga the HX CDWs are at the same energy of the T-U CDW. Adsorption of K does not
change the ground state (T-U) but does suppress the T-C CDW; the HX CDW solutions are the closest ones to the ground state (0.7 meV/f.u. above it), within thermal
fluctuations, and therefore they are very likely to be seen by Scanning Tunnelling Microscopy in real samples. In fact, as several STM data are becoming available, a
guide on the CDW hierarchy may be very useful to correctly identify and locate metallic impurities in TMDCs.
\begin{figure*}[t]
\centering
{\includegraphics[trim = 0 0 0 0,width=\textwidth,clip]{Fig9.jpg}}
\caption{(Colour online) Relaxed structures of the pristine CDWs as they appear in Fig.\
\ref{prstStrcts-fig}:
1\ in the main text: T-U (a), T-C (b) and HX (c). Atoms are represented by
spheres, as illustrated in the legends; Se-Se bonds shorter than the equilibrium distance
(3.45 \AA) are represented by solid lines, in order to help visualising the CDW structure
pattern. Dashed lines mark the supercells borders.}
\label{prstStrctsipSe-fig}
\end{figure*}
\begin{figure}[t]
\centering
{\includegraphics[trim = 0 0 0 0,width=\linewidth,clip]{Fig10.jpg}}
\caption{(Colour online) Total DOS of NbSe$_2$ with the fully symmetric
structure and with the HX, the T-C and the T-U CDW structures, obtained
for a $3\times3\times1$ supercell.}
\label{CDWDOS-fig}
\end{figure}
\begin{figure*}[t]
\centering
{\includegraphics[trim = 0 0 0 0,width=\textwidth,clip]{Fig11.jpg}}
\caption{(Colour online) Fourier Transform (FT) plots of the charge density distributions for the
pristine T-U (a) T-C (b) and HX (c) CDW structures, as given in Fig.\ \ref{prstStrcts-fig}. The
dashed hexagons mark the same regions as in Fig.\ \ref{FFTprst-fig} (see caption).}
\label{prstFFT-fig}
\end{figure*}
\begin{figure*}[t]
\centering
{\includegraphics[trim = 0 0 0 0,width=\textwidth,clip]{Fig12.jpg}}
\caption{(Colour online) Relaxed structures of a variant of the HX CDW in a $9\times9\times1$
supercell, highlighting its Nb-Nb (a) and Se-Se (b) distance pattern. Atoms are represented
by spheres, as illustrated in the legends; in (a) and (b) respectively, Nb-Nb or Se-Se bonds
shorter than the equilibrium distance (3.45 \AA) are represented by solid lines, in order to
help visualising the CDW structure pattern. Dashed lines mark the supercells borders.}
\label{HXoreStrcts9x9-fig}
\end{figure*}
\begin{figure*}[t]
\centering
{\includegraphics[trim = 0 0 0 0,width=\textwidth,clip]{Fig13.jpg}}
\caption{(Colour online) Total DOS of Co$\vert$NbSe$_2$ and K$\vert$NbSe$_2$
with symmetric structures and with their CDW structures.}
\label{TMCDWDOS-fig}
\end{figure*}
\begin{figure*}[t]
\centering
{\includegraphics[trim = 0 0 0 0,width=\textwidth,clip]{Fig14.jpg}}
\caption{(Colour online) Differences of the FT of the charge density
distributions, done with respect to pristine T-U, of Co-adsorbed (a),
Mn-adsorbed (b) T-U, and Mn-adsorbed (c) HX-S.}
\label{otherFFT-fig}
\end{figure*}
\begin{figure*}[t]
\centering
{\includegraphics[trim = 0 0 0 0,width=\textwidth,clip]{Fig15.jpg}}
\caption{(Colour online) Relaxed structures of Co$\vert$NbSe$_2$ in a $9\times9\times1$
supercell. Atoms are represented by spheres, as illustrated in the legends; Nb-Nb bonds
shorter than the equilibrium distance (3.45 \AA) are represented by solid lines, in
order to help visualising the CDW structure pattern. Dashed lines mark the supercells
borders.}
\label{CoStrcts9x9-fig}
\end{figure*}
\begin{figure*}[t]
\centering
{\includegraphics[trim = 0 0 0 0,width=\textwidth,clip]{Fig16.jpg}}
\caption{(Colour online) Direct space charge density distribution of Co$\vert$NbSe$_2$
in a $9\times9\times1$ supercell; (a) and (b) correspond to the respective structures
in Fig.\ \ref{CoStrcts9x9-fig}. The isosurface value for the volumetric data is set in
agreement with the relative figures in the main text. Atoms are represented by spheres,
as illustrated in the legends; Nb-Nb bonds shorter than the equilibrium distance (3.45
\AA) are represented by solid lines, in order to help visualising the CDW structure
pattern. Dashed lines mark the supercells borders.}
\label{CoChg9x9-fig}
\end{figure*}
\begin{figure*
\centering
{\includegraphics[trim = 0 0 0 0,width=\textwidth,clip]{Fig17.jpg}}
\caption{(Colour online) Magnetisation densities of Co$\vert$NbSe$_2$ T-U (a) and
Mn$\vert$NbSe$_2$ T-U (b) with details around Co and Mn, respectively. The sets of
axes represent the unitary lattice vectors in red, green and blue colour, respectively.
Volumetric data representation are set in agreement to the relative figures in the main
text.}
\label{CovsMnMgn-fig}
\end{figure*}
A detailed comparison between theory and experiment requires also an analysis of the role of the substrate,
which may induce a variety of effects, as e.g. in-plane strain and charge transfer. Recently, single layers
were grown on bi-layer graphene, and a CDW order slightly weaker compared to the bulk was reported
\cite{ugeda-nphys2016}; however, the phase competition between different modulations was not investigated.
Our study suggests, based on the few examples analysed, that one CDW phase (T-U) is suppressed with respect
to other hidden ones (HX). As a concluding remark, we observe that the symmetry of the charge density
distribution is reduced from $C_6$ to $C_2$, hinting to a weakening of a $\mathbf q$-vector, which may be the
precursor of a stripe phase recently observed by STM \cite{fang-ScienceAdv2018,soumyanarayanan-PNAS2013}.
\section{Conclusions}
\label{conc}
By means of {\it ab-initio} calculations based on total energy and direct space charge computation,
we have investigated the existence and competition of CDWs in single-layer NbSe$_2$ without and with
impurities. The T-C CDW is suppressed in all cases, suggesting that its observation in STM images is
unlikely in non-passivated samples, due to the high reactivity of NbSe$_2$ and, in general, metallic
TMDCs. Transition metal adsorbates invert the energy hierarchy between CDWs, favouring the HX CDWs
over the T-U and the T-C. Adsorption of K keep the T-U CDW as the ground state, although the HX CDW
is preferred to the T-C CDW; adsorption of Ga equally favours the T-U and the HX CDWs pointing to a
coexistence. In general, adsorption of atoms, changing the local symmetry mixes the `pristine' CDWs,
in particular the HX and the T-C. The symmetry of the charge density distribution is reduced from
$C_6$ to $C_2$ upon Co or Mn adsorption. Future research will be focused on understanding the role
of the substrate in the stabilisation of the CDWs, in order to have a better correspondence
between theory and experiment.
\section*{Acknowledgements}
We are grateful to B.\ I.\ Min, V.\ Fiorentini, E.\ Tosatti and D.\ Payne for fruitful
discussions.
This research work was supported by the Ministry of Education, Gyeongsangbuk-do and Pohang City, through the National Research Foundation of Korea (Grant Nos.:
2015R1C1A1A01052411 and 2017R1D1A1B03033465).
K.\ K. and A.\ A. acknowledge the Max Planck POSTECH / KOREA Research Initiative
programs through the National Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (Grant No. 2016K1A4A4A01922028).
%
The computational work was performed on resources provided
by the Swedish National Infrastructure for Computing (SNIC) at the High Performance Computing Center
North (HPC2N), at the PDC center for High Perfomance Computing and on the National Computational
Infrastructure (NCI) of Australia. Further computational resources were provided by the IBS centre at
POSTECH and the Korean Institute of Science and Technology Information (KISTI).
|
3,212,635,537,975 | arxiv |
\subsection{Curve Discrepancy}
Note from Fig.~\ref{fig:smoothed} that although channel reciprocity is clearly apparent for the naked eye, the frequency response curves are more or less shifted or zoomed versions at corresponding frequencies. Moreover, distinct local fluctuations exist. These discrepancies are unavoidable because they spontaneously result from the hardware imperfections and environment interferences. This shows that direct quantization and mapping of the frequency response can lead to high mismatch rates. We, therefore, develop a shape-based approach to solve the encoding problem.
\begin{algorithm}[h]
\caption{CurveCoding}
\KwIn{\\complex samples $a[0,\cdots,n]$\;number of segments $m$;}
\KwOut{\\code $[C_{1},C_{2},\cdots,C_{m}]$}
\SetKwFunction{Max}{Max1}
\SetKwFunction{Min}{Min1}
\SetKwFunction{PatternGeneration}{PatternGeneration}
\SetKw{KwInit}{Initialization}
\KwInit{
\\divide $a[0,\cdots,n]$ into $m$ segments $b_{1}$,$b_{2}$,$\cdots$,$b_{m}$\; $peak =\ \{\ $\Max{$a[0, \cdots, n]$} - \Min{$a[0, \cdots, n]$}$\}$
\PatternGeneration{$\lfloor n/m\rfloor$,$m$, $peak$}:\\
\quad generate 3 patterns of size $\lfloor n/m\rfloor$: $p_{1}$,$p_{2}$,$p_{3}$\;
}
\For{$i\leftarrow 1$ \KwTo $m$}
{$temp$ = $\infty$\;
\For{$j = 1 \to 3$}
{$dis$ = Fr\'echet($b_{i}$,$p_{j}$)\;
\If{$temp$ $>$ $dis$}
{$temp$ = $dis$\;
$C_{i}$ = $j$\;}
}
}
\label{alg:coding}
\end{algorithm}
\begin{algorithm}[h]
\caption{PatternGeneration}
\KwIn{\\$k$, $m$, $peak$;}
\KwOut{\\ 3 patterns $p_{1}[1,\cdots,k]$, $p_{2}[1,\cdots,k]$, $p_{3}[1,\cdots,k]$}
\For{$i\leftarrow 1$ \KwTo $k$}
{$p_{1}[i]\ =\ \frac{peak\ \times\ i}{k}$\;
$p_{2}[i]\ =\ -\frac{peak\ \times\ i}{k}$\;
$p_{3}[i]\ =\ \frac{peak}{m/2}$\;}
\label{alg:pattern}
\end{algorithm}
\subsection{Curve Smoothing}\label{sec:curve_smoothing}
As mentioned above, even though local details of a power spectral density pair are significantly different, channel reciprocity manifests itself by the similarity of the overall shapes between the pair. By plotting smoothed points, conformal information about the overall shape is extracted despite the local variations. In our algorithm, we adopt Locally Weighted Scatter Plot (Lowess) smoothing \cite{Clev:1979}, a curve fitting method that calculates the smoothed value by applying locally weighted regression over a span. Fig.~\ref{fig:smoothed} depicts two PSD curves obtained by two communicating wireless nodes and their corresponding curves after applying Lowess smoothing with a span of 0.4. From Fig.~\ref{fig:smoothed}, we can see that the Lowess curves coincide with each other almost exactly and the overall shapes are preserved, even though the original ones differ from each other in most of the locations.
\subsection{Curve Encoding}\label{sec:curve_encoding}
By using curve smoothing, we obtain two highly similar curves.
\iffalse
Feedback to correct the errors can be introduced in the later stage of information reconciliation, but is not permitted in this stage of advantage distillation.
\fi
To solve the encoding problem, let us first briefly consider several alternative methods: 1)~encode in accordance with an approximation function that describes the curve; 2)~encode in accordance with the statistical properties of the curve; 3)~encode by describing the shape of the response. We adopt the third one for the following reason. As mentioned in Section~\ref{sec:curve_smoothing}, channel reciprocity is readily seen by the similarity of the overall shapes between curves. Hence, encoding by describing the shape should preserve most of the information shared by the two ends. By way of contrast, extracting secrets from the statistical properties definitely suffers from losing much of the mutual information. And the approximation function does not tolerate even small deviations, but measurement error and interference make such deviations quite common.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\columnwidth]{coding2.eps}
\caption{An example of curve encoding.}
\label{fig:coding}
\end{figure}
Fig.~\ref{fig:coding} gives an example of curve coding. The curve obtained in a certain band is treated as a block, which can be divided into varying number of segments of equal length, and then the segments are mapped to one of three curve patterns which are of the same length, as shown in Fig.~\ref{fig:coding}. These three patterns are indexed as 0, 1, and 2. The three ``predetermined'' patterns describe the ascending, descending and steady trend of the curves respectively. By ``predetermined'', we mean that the indices and the shapes of the patterns are well known to all wireless nodes. The gradient of the ascending and descending lines, however, is decided by each node according to the maximum and minimum values of the smoothed curve, and the length of the segment. We have designed that pattern generation thus to tolerate measurement errors and different device settings. For example, two communicating nodes may wish to use different tx/rx gains that would amplify the signals differently. Since each pattern is related to the locally received signals, it describes the shape correctly without the need to negotiate with the other node. We set the gradient of the ascending pattern to be relative to $\frac{max-min}{\text{\# of samples in each segment}}$, and likewise for the descending pattern is relative to $-\frac{max-min}{\text{\# of samples in each segment}}$. The segment is then mapped to the most similar of the three patterns by measuring the discrete Fr\'echet distance \cite{Wien94computingdiscrete} $\delta_{dF}$ between the segment and the patterns, which measures the similarity of two polygonal curves while taking the location and ordering of the points along the curves into consideration. The smaller the distance, the more is the similarity the two curves share.
\iffalse
If the smallest $\delta _{dF}$ is the one of the segment and Pattern 1, the segment must be more similar to Pattern 1 than other two patterns. Thus each segment is encoded as the index of the pattern that has the smallest distance from it.
\fi
The complete algorithm is presented in Algorithm~\ref{alg:coding} and Algorithm~\ref{alg:pattern}.
\subsection{Physical Layer Model}
\subsubsection{Channel model}
\setlength{\belowdisplayskip}{2pt} \setlength{\belowdisplayshortskip}{2pt}
\setlength{\abovedisplayskip}{2pt} \setlength{\abovedisplayshortskip}{2pt}
Assume Alice and Bob operate in a Time-Division Duplexing (TDD) system. If they talk to each other in coherence time, the observed signals of Alice and Bob are represented by
\begin{equation} \label{eq:ob_A}
y_{A}(t)\ =\ (h*x_{A})(t)+n_{A}(t)
\end{equation}
\begin{equation} \label{eq:ob_B}
y_{B}(t)\ =\ (h*x_{B})(t)+n_{B}(t)
\end{equation}
where $h(t)$ is the channel impulse response, which is identical in both directions by virtue of channel reciprocity, $x_{A}$ and $x_{B}$ are the signals transmitted by Alice and Bob respectively, $n_{A}(t)$ and $n_{B}$ are additive white Gaussian noise with the same variance $N$, and ``$*$" indicates convolution. In the frequency domain, the equations above are rewritten as
\begin{equation}\label{eq:Y_A'}
Y_{A}(f)\ =\ H(f)\cdot X_{A}(f)+N_{A}(f),\ \frac{-W}{2}+f_{c}<f<\frac{W}{2}+f_{c}
\end{equation}
\begin{equation}\label{eq:Y_B'}
Y_{B}(f)\ =\ H(f)\cdot X_{B}(f)+N_{B}(f),\ \frac{-W}{2}+f_{c}<f<\frac{W}{2}+f_{c}
\end{equation}
where $W$ is the transmission bandwidth, $f_{c}$ is the center frequency, and $H(f)$ is the channel frequency response.
\subsubsection{Channel Frequency Response}\label{sec:reciprocity}
In this section, we propose two ways to extract the channel frequency response $H(f)$.
\begin{itemize}
\item \textbf{Direct calculation}:
By using pre-defined training signals or decoding the received signals, Alice and Bob know the frequency components $X_{A}(f)$ and $X_{B}(f)$ of the transmitted signals. Therefore, they can calculate $H(f)$ easily, assuming that noise can be ignored.
\item \textbf{PSD based method}:
Let $\{\ x_{0},x_{1},...,x_{N-1}\ \}$ be a complex sample sequence. Since the sequence is stationary and random, the auto-correlation of the sequence is
\begin{equation}\label{eq:covariance}
R(t_{1},t_{2})\ =\ \frac{P}{N}\times\delta(t_{2}-t_{1})
\end{equation}
where $P$ is the power contained by the signal sequence.
Then, the PSD of the sequence is
\begin{equation}\label{eq:fft}
F[R(\tau)]\ =\ \int_{-\infty}^{+\infty}\frac{P}{N}\times\delta(\tau)e^{-j\omega\tau}d\tau = \frac{P}{N}
\end{equation}
From Equation~\ref{eq:fft}, we know that
\begin{equation}\label{eq:X_A}
X_{A}(f)\ =\ \frac{P_{A}}{W},\quad
X_{B}(f)\ =\ \frac{P_{B}}{W}
\end{equation}
Combining Equations \ref{eq:Y_A'} through \ref{eq:X_A} we get
\begin{equation}\label{eq:Y_A}
Y_{A}(f)\ \approx\ \frac{H(f)\cdot P_{A}}{W}+N,\
Y_{B}(f)\ \approx\ \frac{H(f)\cdot P_{B}}{W}+N
\end{equation}
According to the above equations, we conclude that the PSD of $y_{A}(t)$ is the same as that of $y_{B}(t)$ as long as $P_{A}\ =\ P_{B}$.
It is worth noting that even if $P_{A} \ \neq \ P_{B}$, the shape of Alice's and of Bob's PSD are still similar. This property is remarkable because it can be extended to the case in which Alice and Bob experience different levels of transmission power, noise or cross-band interference. Even in such cases, the shapes still don't change significantly.
\end{itemize}
\subsection{Threat Model}
Eve is motivated to derive the shared secret generated by Alice and Bob. There are two main ways of achieving this.
\subsubsection{Eavesdropping}
Eve can attempt to derive $Ch_{AB}$ from $Ch_{AE}$ or $Ch_{BE}$, where $Ch_{AB}$, $Ch_{AE}$, and $Ch_{BE}$ denote the channel from Alice to Bob, Alice to Eve, and Bob to Eve, respectively. This may be possible if Eve has full knowledge of the environment. In general, however, full knowledge of the environment is a rather unrealistic assumption, so we do not regard it as the main threat to our system. Instead, we focus on the threat of spatial correlation of the secrets produced by our algorithm. We assume that Eve cannot stalk Alice or Bob to being within half of a wave length of either of them. This assumption is reasonable since close eavesdroppers suffer from a high exposure risk. Recall that theory \cite{Rappaport:2001:WCP:559977} supports that channels decorrelate beyond half a wavelength.
\subsubsection{Planned movement}
Eve can move in between Alice and Bob to block and unblock their transmissions. Planned movements can thus introduce predictable increase or decrease of RSS at Alice and Bob. Note that while this attack is harmful to RSS-based methods, without the full knowledge about the environment, Eve cannot, however, predict the impact of the planned move on the frequency response of the channel.
\subsection*{ABSTRACT}
\input{abstract}
\section{Introduction}\label{introduction}
\input{introduction}
\section{System Model}\label{model}
\input{model}
\section{Secret Generation}\label{generation}
\input{generation}
\section{Experimental Validation}\label{validation}
\input{validation}
\setlength{\bibsep}{0pt plus 0.2ex}
{\small \bibliographystyle{acm}
\subsection{Environment and System}
\label{sub:environment}
The measurement environment is a lab where there are 6 cubicles. Data were collected during daytime (from 7:00 am to 6:00 pm).
Human activities introduced a certain level of interference in the channel, but generally speaking,
the environment is quite stable.
We conducted the experiment in such a stable environment because we wanted to see clearly the performance comparisons without risking mismatches caused by the changes of the channel itself. In theory, further implementation in mobile environment would give both higher mismatch rate and higher secret bit extraction rate.
The communication system consists of three software-defined transceivers. Each of their RF chains contains an XCVR2450 (RF front end), a NI-5781 (data converter module) and an NI PXIe-7965R (a Xilinx Virtex-5 FPGA). Two of the three transceivers transmit at 2.45 GHz with 20MHz bandwidth. We call these two transceivers Alice and Bob. The third transceiver, Eve, overhears the communication. During reception, each transceiver records the I and Q samples at a sampling rate of 100 MHz and down converts to the baseband. The received samples are then sent to the NI PXIe-8133, an RTOS-based controller, through two direct-memory-access (DMA) channels, which have a data streaming rate that is as high as 800 MB/s. Except for the experiment done in Section~\ref{exp:CSI}, all the results of Puzzle are obtained based on the PSD of 10240 received samples with QPSK modulation.
\subsection{Performance Evaluation}
\subsubsection{Entropy and mismatch rate}\label{exp:CSI}
We first compare Puzzle with the frequency domain secret key generation method with 2-bit quantization \cite{freq}, which in the rest of this paper we refer to as the CSI-2bit. We choose CSI-2bit as the basis for bit mismatch rate and entropy comparison because, to the best of our knowledge, it achieves the highest bit generation rate along with a low mismatch rate. Coarse-grained method like RSS-based ones achieve only 1$\sim$3 bits per packet. We conducted an experiment where packets were transmitted over coherence time using OFDM in a 20MHz band, with each OFDM symbol consisting of of 72 subcarriers. A channel frequency response is extracted from each OFDM subcarrier. The same channel frequency response was used in both Puzzle (to construct curves) and CSI-2bit (to quantize the response). By dividing the curve composed of the 72 channel frequency responses into a certain number of segments of even length for Puzzle, and by selecting a certain number of frequency responses evenly from all the 72 subcarriers for CSI-2bit, we extracted the respective secrets from each packet for the two methods, thus obtaining secrets of different lengths. No device calibration is done as it is orthogonal to the secrecy metrics we wish to compare.
Fig.~\ref{fig:MisMatch} shows that Puzzle outperforms CSI-2bit in bit mismatch rate for bit generate rates from 8bit/pkt to 56bit/pkt. On average, Puzzle has a 63\% lower bit mismatch rate than CSI-2bit. It is worth noting CSI-2bit has an option of online device calibration but that procedure requires the two communicating nodes collect CSI over hundreds of coherence intervals, therefore it has high overhead and is not practical for fast secret sharing.
From Fig.~\ref{fig:entropy}, we first see that Puzzle produces a comparable amount of entropy as CSI-2bit does. It implies that shape-based method does not harm the entropy compared to quantization-based method. Another thing we notice is that, the entropy of the generated bits does not increase linearly with the number of bits used to encode them. This is caused by the fact that neighboring subcarriers are correlated. Therefore, the claim made by Liu et al \cite{freq} that, CSI-2bit can generate 60-bit secret per packet by using 30 subcarriers in a 20MHz band, is not accurate. Fig.~\ref{fig:entropy} shows that for a 14-bit code generated by Puzzle or CSI-2bit, the real secret contained in it, is not longer than 5-bit. And the entropy is saturating as the bit generation rate increases.
\begin{figure}[!htbp]
\centering
\begin{minipage}[t]{\columnwidth}
\begin{subfigure}[t]{.49\columnwidth}
\centering
\includegraphics[width = \columnwidth]{MisMatch.eps}
\caption{ Mismatch rate of different bit generation rates.}
\label{fig:MisMatch}
\end{subfigure}
\begin{subfigure}[t]{.49\columnwidth}
\centering
\includegraphics[width = \columnwidth]{entropyOfChannel.eps}
\caption{
Puzzle produces a comparable amount of entropy as CSI-2bit.}
\label{fig:entropy}
\end{subfigure}
\vspace{0\baselineskip}
\caption{Bit Mismatch Rate and Entropy
\end{minipage}
\end{figure}
\subsubsection{Correlation of codes relative to distance}
To evaluate the resistance to an eavesdropping attacker, we establish the correlation of bits generated by two receivers at different distances. We performed an experiment where we fixed the distance between one transmitter and one receiver, and then placed another receiver at a certain distance away from the first receiver along 6 orientations as shown in Fig.~\ref{fig:deployment}. Each frequency response curve is segmented into 4 pieces. We measured the correlation between the codes produced by the two receivers at distances ranging from 5cm away to 45cm away. To be more specific, we measured the correlation of 6 pairs of locations by fixing the first receiver and moving the second one $60^{\circ}$ apart at each distance. Figure~\ref{fig:correlation} shows the result. We see that the correlation decreases rapidly as the distance between two receivers increases. In practice, it is reasonable to assume that eavesdroppers are beyond one meter away, otherwise they suffer from high risk of exposures. Therefore Puzzle is robust against eavesdropping.
\begin{figure}[!t]
\centering
\begin{minipage}[t]{\columnwidth}
\begin{subfigure}[t]{.49\columnwidth}
\centering
\includegraphics[width = .95\columnwidth]{deployment_new.eps}
\caption{ Deployment of correlation experiment.}
\label{fig:deployment}
\end{subfigure}
\begin{subfigure}[t]{.49\columnwidth}
\centering
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip, width = \columnwidth]{correlation2.eps}
\caption{ The correlation of two codes generated by Puzzle relative to distance.}
\label{fig:correlation}
\end{subfigure}
\caption{Deployment and result of correlation experiment}
\end{minipage}
\end{figure}\label{fig:correlationexp}
\begin{figure}[h]
\centering
\begin{subfigure}[t]{\columnwidth}
\centering
\includegraphics[trim=0cm 1.5cm 0cm 0cm, clip=true, width=.85\columnwidth]{movingObject_new.eps}
\caption{An object moves between Alice and Bob with a certain temporal pattern and Eve
overhears the transmission from Alice to Bob.}
\label{fig:MovingObj}
\end{subfigure}
\begin{minipage}{\columnwidth}
\begin{subfigure}[t]{.49\columnwidth}
\centering
\includegraphics [trim = 0.5mm 1mm 0.5mm 0.5mm, clip, width=\columnwidth]{leakage_new.eps}
\caption{Leakage relative to distance between Bob and Eve. Puzzle has a stable low leakage rate irrespective of the distance.}
\label{fig:subfigLeakage}
\end{subfigure}
\begin{subfigure}[t]{.49\columnwidth}
\centering
\includegraphics [ trim = 0.5mm 1mm 0.5mm 0.5mm, clip, width=\columnwidth]{secrecy_new.eps}
\caption{Non-leaked secrete bits produced by Puzzle and ASBG per packet, relative to the distance between Bob and Eve.}
\label{fig:subfigLeakEntropy}
\end{subfigure}
\caption{Performance: Leakage}\label{fig:leakperformance}
\end{minipage}
\end{figure}
\subsubsection{Leakage}\label{sub:comparison}
Towards validating the resistance to the planned movement attacker (cf.~Section~\ref{model}), we
\iffalse
In this section we show that Puzzle is robust against to such attackers.
\fi
compared the leakage performance of a the state-of-the-art RSS-based method ABSG and Puzzle by moving an object across the transmission path between Alice and Bob, while placing an eavesdropper near Bob, as shown in Figure~\ref{fig:MovingObj}.
Since ABSG like many other RSS-based methods asks the two communicating ends to drop some RSS values based on certain thresholds and to exchange the indices of those values, Eve knows exactly which RSS probe is used by Bob but dropped by herself. In this case, we assume that Eve makes a random guess as to the quantization result with a success rate of 50\%. We calculate the mismatch rate of Eve's and Bob's bits to be the combination of the actual mismatch rate between them and the failure rate of the random guess. And again, we segment the frequency response curves into four pieces.
Fig~\ref{fig:subfigLeakage} shows the leakage of our algorithm against that of ABSG over a distance from $10$ cm to $50$ cm. It is clear that Puzzle is much more insensitive to the threat of planned movement. Furthermore, due to the fact that Puzzle has a much higher secret generation rate ($4*log(3) \approx 6.3$ bits/pkt) than ASBG (1 bits/pkt), the non-leaked secret produced by Puzzle is much larger. Fig~\ref{fig:subfigLeakEntropy} shows the result. It is worth noting that although 4 wavelength might not sound like a large distance in practice, our blocking objects are not large either. The variations induced by larger obstacles, like a train passing by or the example mentioned above, might impact a much larger distance in practice.
|
3,212,635,537,976 | arxiv | \section{Introduction}
Multiparticle correlation analyses have proven to be a powerful tool in exploring the
underlying mechanism of particle productions in high energy hadronic collisions. Both short-
and long-range correlations have been discovered in the past decades \cite{UA5_3energy,ISR_twolowenergy}
which have been given a simple interpretation via the concept of clustering \cite{cluster_model,cluster_fit}.
In a cluster model, clusters of hadrons are formed first and emitted independently according
to some basic scheme. They then decay isotropically in their center of mass frame into the
observed hadrons. Two-particle angular correlations provide detailed information about the
cluster properties, e.g. their multiplicity (``size'') and extent in phase space (``width'').
In heavy ion collisions at RHIC, the expected formation of a Quark Gluon Plasma (QGP) could
lead to a modification of the clusters relative to p+p collisions \cite{AAcluster_prediction}.
This study should provide a useful baseline measurement for understanding hadronization stage
in A+A collisions.
\section{Analysis method}
Covering pseudorapidity range ($\eta=-\ln(\tan(\theta/2))$) $-3<\eta<3$ over almost full
azimuthal angle, the PHOBOS Octagon detector \cite{phobos_detector} is ideally suited for
direct study of the angular correlations of the particles emitted from clusters.
Following a similar approach as in Ref. \cite{ISR_twolowenergy}, the inclusive two-particle
correlation function in ($\Delta \eta,\Delta \phi$) space is defined as follows:
\begin{equation}
\label{2pcorr_incl}
R(\Delta \eta,\Delta \phi)=\left<(n-1)\left(\frac{F_{n}
(\Delta \eta,\Delta \phi)}{B_{n}(\Delta \eta,\Delta \phi)}-1\right)\right>
\end{equation}
\noindent where $F_{n}(\Delta \eta,\Delta \phi)$ is the foreground distribution determined
by taking two-particle pairs from the same event and $B_{n}(\Delta \eta,\Delta \phi)$ is the
background distribution constructed by randomly selecting single particles from two different
events with similar vertex position and centrality. The track multiplicity, $n$, is introduced
to compensate for the trivial dilution effects from uncorrelated particles. $R(\Delta \eta,\Delta \phi)$
is defined such that if a heavy ion collision is simply a superposition of individual p+p
collisions, and thus has the same local correlations, the same correlation function should be observed.
\section{Two-particle correlations in p+p collisions}
\vspace{-0.5cm}
\begin{figure}[t]
\captionsetup[figure]{margin=0.1cm,font=small}
\begin{minipage}[t]{0.45\textwidth}
\centerline{
\includegraphics[width=\textwidth]{pp200_2D_corrected.eps}
}
\vspace{-0.3cm}
\caption{Final fully corrected two-particle correlation
function in ($\Delta \eta$, $\Delta \phi$) for
p+p collisions at $\sqrt{s}$ = 200~GeV. A small
region near ($\Delta \eta$, $\Delta \phi$)
of (0,0) has been removed due to its large experimental uncertainty
arising from the secondary effects.}
\label{pp200_2D_corrected}
\end{minipage}
\hspace{\fill}
\begin{minipage}[t]{0.45\textwidth}
\centerline{
\includegraphics[width=\textwidth]{pp200_eta_corrected_fit.eps}
}
\vspace{-0.3cm}
\caption{1D two-particle rapidity correlation function in
($\Delta \eta$) for p+p collisions at $\sqrt{s}$ = 200~GeV
together with a fit from a cluster model. The fit
parameters are: $K_{\rm eff}$=2.44$\pm$0.08 and
$\delta$=0.66$\pm$0.03 with 90\% C.L. errors}
\label{pp200_eta_corrected_fit}
\end{minipage}
\end{figure}
Fig.~\ref{pp200_2D_corrected} shows the two-particle inclusive correlation function in p+p
collisions at $\sqrt{s}$ = 200~GeV as a function of $\Delta \eta$ and $\Delta \phi$.
To achieve the correlations between primary particles, a set of correction procedures has
been applied, based on MC simulations. The complex correlation structure suggests that the
short range correlation is approximately Gaussian in $\Delta \eta$ and persists over the full
$\Delta \phi$ range, becoming broader toward higher $\Delta \phi$. If clusters are the precursors
to the final measured hadrons, a high $p_{T}$ cluster would contribute to a narrow peak at
the near-side ($\Delta \phi$ near $0^{o}$) region of the correlation function in Fig.~\ref{pp200_2D_corrected},
while a lower $p_{T}$ cluster will contribute to the broader away-side.
To study these features quantitatively, the two-dimensional (2D) correlation function is
projected into a one-dimensional (1D) correlation function $R(\Delta \eta)$ shown in
Fig.~\ref{2pcorr_clusterfitting_incl}. It takes a form derived in Ref. \cite{cluster_fit}
in an independent cluster emission model:
\begin{equation}
\label{2pcorr_clusterfitting_incl}
R(\Delta \eta)=\alpha\left[\frac{\Gamma(\Delta \eta)}{B(\Delta \eta)}-1\right]
\end{equation}
\noindent where $B(\Delta \eta)$ is the background distribution obtained by event-mixing.
Here the multiplicity dependence of $B(\Delta \eta)$ is not considered since the multiplicity
distribution in p+p collisions is relatively narrow with $\sigma(n)/<n>$ of around 0.25.
The parameter $\alpha=\frac{<K(K-1)>}{<K>}$ contains the information about the cluster
size $K$ and $\Gamma(\Delta \eta)$ is a Gaussian function $\propto exp{(- (\Delta \eta)^{2}/(4\delta^{2}))}$
characterizing the correlation of particles produced by a single cluster, where $\sqrt{2}\delta$
corresponds to the decay width of the clusters. The effective cluster multiplicity,
or ``size'' is defined to be $K_{\rm eff}=\frac{<K(K-1)>}{<K>}+1=<K>+\frac{\sigma_{K}^{2}}{<K>}$.
Of course, without any knowledge of the distribution of $K$, it is impossible to directly
measure the average cluster size $<K>$. However, by a $\chi^{2}$ fit of Eq.~\ref{2pcorr_clusterfitting_incl}
to the measured two-particle correlation function, an example of which shown in Fig.~\ref{pp200_eta_corrected_fit},
the effective cluster size $K_{\rm eff}$ can be estimated, as well as the cluster decay width $\sqrt{2}\delta$.
A $K_{\rm eff}$ of about 2.5 indicates that on average every charged particle is strongly
correlated with another 1.5 particles, if it is assumed that $\sigma_{K}^{2}$=0.
Fig.~\ref{cluster_sqrts} shows the collision energy dependence of $K_{\rm eff}$ and $\delta$.
PHOBOS data at $\sqrt{s}$ = 200~GeV and 410~GeV are consistent with the previous
UA5 measurements \cite{UA5_3energy} and show a significant increase of the cluster size
with energy. The cluster decay width is essentially constant with collision energy.
The observed cluster size exceeds the expectation from resonance decay (about 1.7) \cite{UA5_3energy}
even at very low collision energy \cite{ISR_twolowenergy, ISR_63GeV}, suggesting additional
sources of short-range correlations are required to account for the experimental results.
\begin{figure}[t]
\captionsetup[figure]{margin=0.1cm,font=small}
\vspace{-1.0cm}
\begin{minipage}[t]{0.40\textwidth}
\centerline{
\includegraphics[width=\textwidth]{cluster_sqrt_paper.eps}
}
\vspace{-0.5cm}
\caption{$K_{\rm eff}$ (top) and $\delta$ (bottom) as a function of $\sqrt{s}$
measured by PHOBOS (solid circle), UA5 \cite{UA5_3energy}
(solid square) and ISR \cite{ISR_twolowenergy, ISR_63GeV}
(solid star) experiments in p+p and p+\={p} collisions.}
\label{cluster_sqrts}
\end{minipage}
\hspace{\fill}
\begin{minipage}[t]{0.48\textwidth}
\centerline{
\includegraphics[width=\textwidth]{k_npart_CuCu_ppline_phobos.eps}
}
\vspace{-0.5cm}
\caption{ Effective cluster size $K_{\rm eff}$ as a function
of $N_{part}$ in Cu+Cu collisions at $\sqrt{s_{_{NN}}}$ =
200~GeV (solid circle).The dashed line shows the corresponding
value in p+p collisions measured by PHOBOS.}
\label{k_npart_CuCu_ppline}
\end{minipage}
\end{figure}
\vspace{-0.5cm}
\section{Two-particle correlations in Cu+Cu collisions}
In Cu+Cu collisions, a $\cos(2\Delta \phi)$ dependence is observed, and attributed to
collective flow. However, in this work, the $\Delta \phi$ dependence is integrated out
in order to just study the cluster properties in pseudorapidity. As was explained above,
the two-particle pseudorapidity correlation function in Cu+Cu is fit by
Eq.~\ref{2pcorr_clusterfitting_incl}. The resulting effective cluster size as a function
of participating nucleons ($N_{part}$) is shown in Fig.~\ref{k_npart_CuCu_ppline}
for Cu+Cu collisions at $\sqrt{s_{_{NN}}}$ = 200~GeV. The dashed line indicates the
value found in $\sqrt{s}$ = 200~GeV p+p collisions, and suggests that the cluster properties
are similar in the two systems. This seems reasonable if clusters are produced in the
late stages of the collision evolution and mainly reflect the hadronization process.
However, it is observed that the cluster size decreases with increasing collision centrality.
In comparing the data with dynamical models, it is found that AMPT gives the same
qualitative trend as the data, but is systematically lower in magnitude. By contrast,
HIJING remains constant with increasing centrality. Further comparison of Cu+Cu and
Au+Au data with p+p should provide more information on the dynamical origins of the centrality
dependence seen in the Cu+Cu data.
\section{Conclusion}
In conclusion, the two-particle correlation function for inclusive charged particles
has been studied over broad range in $\eta$ and $\phi$. In particular, it is shown that
the observed short-range correlations in pseudorapidity have a natural interpretation
in terms of clusters. In this approach, multiple particles are understood to be emitted
close together in phase space, with a typical cluster size of 2-3 in p+p collisions.
In Cu+Cu, clusters have a similar size but show a non-trivial decrease in size with
increasing centrality. Future work will extend this study by providing a comprehensive
study of two-particle correlations in p+p, d+Au, Cu+Cu and Au+Au reactions.
\section*{Acknowledgments}
This work was partially supported by U.S. DOE grants
DE-AC02-98CH10886,
DE-FG02-93ER40802,
DE-FG02-94ER40818,
DE-FG02-94ER40865,
DE-FG02-99ER41099, and
DE-AC02-06CH11357, by U.S.
NSF grants 9603486,
0072204,
and 0245011,
by Polish KBN grant 1-P03B-062-27(2004-2007),
by NSC of Taiwan Contract NSC 89-2112-M-008-024, and
by Hungarian OTKA grant (F 049823).
\section*{References}
|
3,212,635,537,977 | arxiv | \section{Introduction}
\label{intro}
A quantum mechanical system is completely described by the Hilbert space and by the Hamiltonian $H$.
However, in practical calculations, the infinite dimensional Hilbert space is often truncated
and the calculations are performed on a finite subset of the basis. This truncation is variational and
the exact results are reached as the truncated basis approaches the complete basis set.
Green's operators are defined by $G(z)=(z-H)^{-1}$, and they can also be used to describe the
quantum system. The poles of the Green's operator are the eigenvalues of the Hamiltonian and the
residue are the projection onto the subspace spanned by the eigenfunctions. However, if at some
energy, the Green's operator is singular, then it is
singular in any representation and thus also singular in a finite subspace representation. The poles of the
Green's operator are insensitive to the truncation of the Hilbert space.
In general, it is much harder to work with the Green's operator than with the Hamiltonian. The Hamiltonian can be represented by differential operators, while the Green's operator, the inverse of the Hamiltonian, is an integral operator.
If the Hamiltonian is represented in a countable infinite basis by an $\infty \times \infty$ matrix, the Green's operator is the inverse of an $\infty \times \infty$ matrix.
Working with $\infty \times \infty$ matrices is not very encouraging, but they can be inverted in special cases. If the matrix $J=z-H$ is of Jacobi type,
i.e. it is an infinite symmetric tridiagonal matrix, then an $N\times N$ representation of the Green's operator, $G^{N}$, can be constructed from the
$N\times N$ representation of $J$ plus a continued fraction.
In fact $(G_{N})^{-1}$ is
almost identical to $J_{N}$,
the only difference being a continued fraction is added to the $N\times N$ term. This is the way
the Coulomb Green's operator has been determined in the Coulomb-Sturmian basis representation.
\cite{Konya:1997JMP,Konya:1999by,PRADemir2006}.
The approach has been extended to infinite symmetric band matrices. An infinite symmetric band matrix can be considered as a Jacobi matrix of block matrices.
Therefore
\begin{equation}\label{gm1}
(G^{N})^{-1}_{i,j} = J_{i,j}^{N} - \delta_{i,N}\delta_{j,N} J_{N,N+1} C_{N+1} J_{N+1,N},
\end{equation}
where $J_{i,j}$ are block matrices and $C_{N+1}$ is a matrix continued fraction.
Using this method, the Green's operators for Hamiltonians
containing kinetic energy, Coulomb and polynomial potentials have been evaluated in Ref.\ \cite{kelbert2007green}.
At eigenvalue energies the determinant of the $G_{N}(E)$ is singular and the
determinant of $(G_{N}(E))^{-1}$ vanishes. So, we can consider $ (G_{N}(E))^{-1}$ as an improved Hamiltonian, which irrespective of $N$, provides the correct eigenvalues. This way we accomplished a kind
of ``packing'' of the $\infty \times \infty$ matrix into an $N \times N$ matrix, where $N$ is not necessarily big.
A general Hamiltonian, besides the Coulomb and and some polynomial potential, may contain a short-range potential as well.
A general short-range potential in a discrete basis representation is certainly not tridiagonal, not even
block-tridiagonal. But for any reasonable potential, the matrix representation looks like a ridge: the
matrix elements are much bigger if $n$ and $n'$ are close and become negligible otherwise, just
like a band matrix. Therefore, the technique developed before may be applicable for finding a faithful matrix representation of a general short-range potential.
The aim of this paper is find a low-$N$ representation of the short-range potential such
that it carries the information of the whole Hilbert-space representation.
We accomplish our goal through the inverse of the potential operator.
We present our results in the Coulomb-Sturmian basis representation, but we believe that the results are valid for
any discrete basis provided the matrix exhibits a ridge-like structure.
The method of approximating the potential on Coulomb-Sturmian basis has quite a long history. It has successfully been applied to various problems, including the solution of the Faddeev equations with a Coulomb \cite{PhysRevC.54.50,PhysRevC.55.1080} and confining \cite{PhysRevC.62.044004} potential. In Sec.\ \ref{sec2}, we outline the technique for solving the Lippmann-Schwinger equation. Then in Sec.\ \ref{sec3} we introduce the physical system and outline previous schemes for approximating the potential. Then in Sec.\ \ref{sec4} we apply the
inverse matrix idea to matrices with ridge-like structure. Finally we summarize our findings in Sec.\ \ref{summary}.
\section{Solution of the Lippmann-Schwinger equations}
\label{sec2}
We consider a Hamiltonian with a Coulomb $v^{C}$ plus short-range $v^{(s)}$ potential.
The bound states are the solutions of the homogeneous
\begin{equation}\label{lsb}
|\psi \rangle = g_{l}^{C}(E) v_{l}^{(s)}|\psi \rangle,
\end{equation}
while the scattering states are the solutions of the inhomogeneous
\begin{equation}\label{lssc}
|\psi^{(\pm)} \rangle = |\phi_{l}^{C(\pm)}\rangle + g_{l}^{C}(E\pm i\epsilon) v_{l}^{(s)}|\psi^{(\pm)} \rangle
\end{equation}
Lippmann-Schwinger equations. Here $E$ is the energy, $l$ is the angular momentum,
$g_{l}^{C}(E)=(E-h_{l}^{0}-v^{C})^{-1}$ is the
Coulomb Green's operator, $h_{l}^{0}$ is the kinetic energy and $\phi_{l}^{C}$
is the Coulomb scattering state.
The scattering state $\psi^{(\pm)}$ is related to the scattering amplitude by
\begin{equation}\label{al}
a_{l}= \langle \phi_{l}^{C(-)} | v_{l}^{(s)} |\psi^{(+)} \rangle = \frac{\exp(i(2\eta_{l}+\delta_{l}))}{k} \sin \delta_{l}~,
\end{equation}
where $k$ is the wave number, $\eta_{l}$ is the Coulomb phase shift and $\delta_{l}$ is the Coulomb-modified nuclear phase shift.
The Coulomb-Sturmian basis, in angular momentum $l$, is defined by
\begin{equation}
\langle r | n l \rangle = \frac{\sqrt{n!}}{\sqrt{(n+2l+1)!}} \exp(-br) (2br)^{l+1} L_{n}^{2l+1}(2br)
\end{equation}
and
\begin{equation}
\langle p | n l \rangle = \frac{\sqrt{2} \sqrt{n!}(n+l+1) l! (4bp)^{l+1}}{\sqrt{\pi} \sqrt{(n+2l+1)! } (p^{2} +b^{2})^{l+2}} G_{n}^{l+1}\left (\frac{p^{2}-b^{2}}{p^{2}+b^{2}}\right),
\end{equation}
in configuration and momentum space, respectively. Here $L$ and $G$ are the Laguerre and Gegenbauer polynomials, respectively, and $b$ is a parameter. Together with
\begin{equation}
\langle r | \widetilde{nl} \rangle = \langle r | {nl} \rangle /r
\end{equation}
and
\begin{equation}
\langle p | \widetilde{nl} \rangle = \langle p | {nl} \rangle \frac{p^{2}+b^{2}}{2 b (n+l+1)}
\end{equation}
these functions are orthonormal
\begin{equation}
\langle nl | \widetilde{n' l} \rangle = \delta_{n n'}
\end{equation}
and form a complete set
\begin{equation}
\lim_{N\to\infty}\sum_{n=0}^{N} | nl \rangle \langle \widetilde{n' l} | = 1~.
\end{equation}
The finite dimensional representation of the short-range potential is given by
\begin{equation}\label{vapprox}
v_{l}^{(s)} \approx v^{N,N} = \sum_{n n'}^{N} | \widetilde{nl}\rangle \underline{v}^{(s) N,N}_{l,n n'} \langle \widetilde{n' l} |~,
\end{equation}
where $ \underline{v}^{(s)N,N}_{l,n n'} = \langle{nl} | v_{l}^{(s)} | n' l \rangle$.
Now the Lippmann-Schwinger equations (\ref{lsb}) and (\ref{lssc}) become matrix
equations
\begin{equation}\label{lsbm}
\underline{\psi} = \underline{g}_{l}^{C} \underline{v}_{l}^{(s)} \underline{\psi}
\end{equation}
and
\begin{equation}\label{lsscm}
\underline{\psi} = \underline{\phi}_{l}^{C} + \underline{g}_{l}^{C} \underline{v}_{l}^{(s)} \underline{\psi},
\end{equation}
respectively, where the matrices and vectors are underlined.
Some rearrangement gives
\begin{equation} \label{homae}
( ( \underline{g}_{l}^{C})^{-1} - \underline{v}_{l}^{(s)} ) \underline{\psi} = 0
\end{equation}
and
\begin{equation}\label{inhome}
( ( \underline{g}_{l}^{C})^{-1} - \underline{v}_{l}^{(s)} ) \underline{\psi} = (\underline{g}_{l}^{C})^{-1} \underline{\phi}_{l}^{C}~,
\end{equation}
i.e.\ the homogeneous Lippmann-Schwinger equation becomes a homogeneous algebraic equation and the
inhomogeneous Lippmann-Schwinger equation becomes an inhomogeneous algebraic equation. The homogeneous algebraic equation is solvable if the determinant is zero
\begin{equation}
|( \underline{g}_{l}^{C})^{-1}(E) - \underline{v}_{l}^{(s)} | =0.
\end{equation}
This condition provides the eigenvalues and the solution of (\ref{homae}) provides the eigenvectors. The solution of the inhomogeneous algebraic equation gives the scattering state $\underline{\psi}$, which, with the help of Eq.\ (\ref{al}), can provide us with the phase shift.
The matrix $(\underline{g}_{l}^{C})^{-1}$ can be calculated by using (\ref{gm1})
\begin{equation}
(\underline{g}_{l}^{C})^{-1} = \underline{J}^{C} - \delta_{i,N}\delta_{j,N} J^{2}_{N,N+1} C_{N+1}~.
\end{equation}
Here $J(E)=(E-h_{l}^{0}-v^{C})$ and $v^{C}=Z/r$. The matrix $\underline{J}^{C}$ is symmetric tridiagonal, and the nonzero elements are given by
\begin{equation}
\underline{J}_{i,i}^{C} = 2(i+l+1) \frac{\hbar^{2}(k^{2}-b^{2})}{4\mu b}-Z
\end{equation}
and
\begin{equation}
\underline{J}_{i,i+1}^{C} = -\sqrt{(i+1)(i+2l+2)} \frac{\hbar^{2}(k^{2}+b^{2})}{4\mu b}~,
\end{equation}
where $\mu$ is the reduced mass and $k=\sqrt{2\mu/\hbar^{2} \:E}$.
In this particular case the continued fraction can be summed up to a ratio of hypergeometric functions
\begin{eqnarray}
C_{N+1} &&= -\frac{4m/\hbar^{2}\:b }{(b-ik)^{2}(N+l+2+i\gamma) } \\
&& \times \frac{_{2}F_{1}\left(-l+i\gamma,N+2;N+l+3+i\gamma;\left( \frac{b+ik}{b-ik} \right)^{2} \right) }
{ _{2}F_{1}\left(-l+i\gamma,N+1;N+l+2+i\gamma;\left( \frac{b+ik}{b-ik} \right)^{2} \right) }~, \nonumber
\end{eqnarray}
where $\gamma=Z\mu/(\hbar^{2}k)$ \cite{PRADemir2006}.
The analytic evaluation of $\underline{\phi}_{l}^{C}$ has been presented
before in Ref.\ \cite{PhysRevC.38.2457}.
This representation of $(\underline{g}_{l}^{C})^{-1}$ is exact and analytic. Even a very low-rank matrix gives an account
for the complete spectrum of the Coulomb Hamiltonian. Fig.\ \ref{fighyd} shows the determinant of the $3 \times 3$
$(\underline{g}_{l}^{C})^{-1}$ matrix for Coulomb Hamiltonian with $l=0$, $Z=-1$, $\mu=1$ and $\hbar=1$. The exact eigenvalues
are $E_{n}=-1/(2 n^{2})$. The figure shows the energy range corresponding to the $E_{90}-E_{100}$ interval. We can
see that the numerical zeros are at the exact locations even in this extreme case.
\begin{figure
\centering
\includegraphics[width=8.5cm]{hyd-90-100.png}
\caption{The zeros of $\underline{g}_{0}^{C}(E)$ for a hydrogen system in atomic units in the energy
range $E_{90}-E_{100}$. The large dots represent the exact eigenvalues.}
\label{fighyd}
\end{figure}
In this approach the only approximation is the finite-basis representation of the potential, since the evaluation
of $\underline{g}_{l}^{C}$ and $\underline{\phi}_{l}^{C}$ is exact and analytic. Therefore both the bound and scattering state
wave function $\psi$ and $\psi^{\pm}$
possess the exact Coulomb-like asymptotic behavior \cite{PhysRevA.46.4437}.
Finite-rank potentials have a long history in physics (see eg.\ Ref.\ \cite{adhikari1991dynamical}). Various schemes have been proposed.
Most of them use some form factors which allow for an easy and exact
evaluation of the matrix elements of the Green's operator. The use of Coulomb-Sturmian functions offers several advantages.
Since they form a basis, the convergence of the approximation is guaranteed. More importantly,
it works with Coulomb-like potentials, unlike the majority of approaches.
\section{The example problem}
\label{sec3}
To illustrate the method we consider a typical nucleon-nucleon
potential, the Malflet-Tjon potential.
This potential has a strong repulsive core and an attractive tail, like most of the potentials in physics.
The Malflet-Tjon potential is given by
\begin{equation}
v^{s}= v_{1} \exp( - \beta_{1} r )/r + v_{2} \exp( - \beta_{2} r )/r
\end{equation}
with $v_{1}=1438.720\: \mbox{MeV}$, $\beta_{1}= 3.11\: \mbox{fm}^{-1}$,
$v_{2}= -626.885 \:\mbox{MeV}$, $\beta_{2}= 1.55\: \mbox{fm}^{-1}$.
The other parameters in the model are charge parameter $Z=e^{2}=1.44 \:\mbox{MeV fm}$,
$\hbar^{2}/m = 41.47\: \mbox{MeV / amu}$ and nucleon reduced mass $\mu=1/2\: \mbox{amu}$.
We used $b=3\: \mbox{fm}^{-1}$, which is around the optimum. We note that the rate of convergence is rather insensitive
to the choice of $b$ within a rather broad interval.
Figure \ref{step0} shows the $\underline{v}^{20,20}$ matrix. We can see that the matrix representation
exhibits a ridge-like structure and the dominant matrix elements decrease only very slowly. So, if we
truncate the basis to this size, we chop down the tail of the matrix and we neglect terms which are not
small at all. Consequently, this representation results in a slow convergence.
\begin{figure
\centering
\includegraphics[width=8.5cm]{stepout_zero_nobox.png}
\caption{$\underline{v}^{20,20}$ Coulomb-Sturmian matrix elements of the Malflet-Tjon potential.}
\label{step0}
\end{figure}
We have to note here that there had been approaches before to improve the situation. Inspired by
Lanczos filtering, it has been proposed to multiply the potential matrix by some function which suppress the higher elements \cite{revai,gyarmati1979rigorous}
\begin{equation}
\underline{\tilde{v}}^{N}_{i,j} = \sigma_{i}^{N} \underline{v}^{N,N} \sigma_{j}^{N},
\end{equation}
where
\begin{equation}\label{sig}
\sigma_{i}^{N} = \frac{ 1-\exp(-[\alpha(i-N-1)/(N+1)]^{2}) }{1-\exp(-\alpha^{2})}
\end{equation}
with $\alpha\sim 6$. This approach results in a transformed matrix shown in Fig.\ \ref{sigma}.
\begin{figure
\centering
\includegraphics[width=8.5cm]{sigma_nobox.png}
\caption{Potential matrix $\underline{\tilde{v}}^{20}$. The matrix of Fig.\ \ref{step0} has been modified by the $\sigma$ factors of Eq.\ (\ref{sig}).}
\label{sigma}
\end{figure}
In the other approach two Hilbert-space bases has been adopted
\cite{PhysRevC.36.1275,PhysRevC.63.057001}
\begin{equation}\label{2b}
\underline{\hat{v}}^{N} = \underline{O} \: \underline{v}^{N,N} \: \underline{O}',
\end{equation}
where $\underline{v}^{N,N}$ has been calculated with basis parameter $b_{1}$ and
$\underline{O}= (\langle \widetilde{ nl;b_{1}}| n'l;b_{2} \rangle)^{-1}$.
The
a potential matrix $\underline{\hat{v}}$ with $b_{1}=2.5\:\mbox{fm}^{-1}$ and $b_{2}=3.5\:\mbox{fm}^{-1}$ is shown in Fig.\ \ref{2bases}. It is interesting to note that this approach also utilizes the inverse of the potential operator \cite{PhysRevC.36.1275}. Our approach is however different, as we are going to see below.
\begin{figure
\centering
\includegraphics[width=8.5cm]{original_nobox.png}
\caption{Potential matrix $\underline{\hat{v}}^{20}$ from double-basis representation of Eq.\ (\ref{2b}).}
\label{2bases}
\end{figure}
We can see that both methods basically suppress the higher index elements of the matrix.
Now the transition to the neglected terms is smooth. We found that among these two methods, the one with
two bases gives a faster convergence \cite{PhysRevC.63.057001}.
\section{ Approximation through the inverse}
\label{sec4}
We saw before that we could achieve a good approximation of the Hamiltonian by considering
it on an infinite
basis representation and then by rolling up the tail of the band matrix into a matrix continued fraction.
Here we try a similar
procedure with the potential operator. First we calculate the Coulomb-Sturmian matrix elements of ${v}^{s}$ on a basis of $N'$ size, invert the matrix, then
truncate it to $N\le N'$, and finally invert the matrix again. We denote the resulting matrix by
$\underline{v}^{N,N'}$.
Figures \ref{step1}, \ref{step2} and \ref{step7} display $\underline{v}^{20,21}$, $\underline{v}^{20,22}$ and
$\underline{v}^{20,27}$, respectively. We can see that as $N'$ increases, the matrix elements around the corner
become more and more suppressed. Thus if we truncate the matrix to $N\times N$ size, we neglect
terms which are small. We can also see from these pictures that this procedure
in accordance with
Eq.\ (\ref{gm1}) modifies mostly the lower right corner of the potential matrix. It is also interesting to see
that in Fig.\ \ref{step1}, even stepping out just by one basis state, and truncating back, results in a dramatic reduction
of the matrix elements around the lower right corner.
\begin{figure
\centering
\includegraphics[width=8.5cm]{stepout_one_nobox.png}
\caption{$\underline{v}^{20,21}$ Coulomb-Sturmian matrix elements of the Malflet-Tjon potential.}
\label{step1}
\end{figure}
\begin{figure
\centering
\includegraphics[width=8.5cm]{stepout_two_nobox.png}
\caption{$\underline{v}^{20,22}$ Coulomb-Sturmian matrix elements of the Malflet-Tjon potential.}
\label{step2}
\end{figure}
\begin{figure
\centering
\includegraphics[width=8.5cm]{stepout_seven_nobox.png}
\caption{$\underline{v}^{20,27}$ Coulomb-Sturmian matrix elements of the Malflet-Tjon potential.}
\label{step7}
\end{figure}
Figure \ref{deut} shows the convergence of the deuteron binding energy with
$\underline{v}^{20,N'}$ and $\underline{v}^{N',N'}$
as a function of $N'$. We can see that our approach of inverting and cutting back the potential matrix
is more advantageous than keeping the original bigger matrix.
We can also see that beyond $N'=N+4 \to N'=N+7$ there is no further improvement.
We found the same effect with other $N$ values and for scattering states as well.
So, we fix $N'=N+7$.
\begin{figure}[!ht]
\label{convfig}
\centering
\includegraphics[width=8.5cm]{convegence-20-30.png}
\caption{Convergence of the deuteron bound state energy with potential matrices
$\underline{v}^{20,N'}$ and $\underline{v}^{N',N'}$.}
\label{deut}
\end{figure}
Table \ref{table2} shows the convergence of the deuteron states and $p-p$ scattering phase shifts at
low, intermediate and high energies with increasing $N$.
We can observe excellent results even with very low $N$. We observe four digit accuracy with basis representation as low as $N=5$ and eight or nine digit accuracy with $N=20$. The rate of convergence is better than with the double basis method and higher accuracy can be achieved.
\begin{table
\caption{The convergence of the deuteron bound state energy and $p-p$ scattering phase shifts at low, intermediate and high energies. The $N\times N$ representation of the potential was calculated from an $N+7 \times N+7$ representation. }
\label{table2}
\begin{tabular}{| l | | c | c | c | c | }
\hline
N & E$_{\mbox{d}}$ & E = 0.1 MeV & E = 1.0 MeV & E = 100 MeV \\
\hline
\hline
3& -2.14459561 & -0.121277396 & -0.708187417 & 0.378017184 \\
4& -2.23413092 & -0.119071149 & -0.701325274 & 0.400336999 \\
5& -2.22996195 & -0.119221696 & -0.701844279 & 0.406882711 \\
6& -2.22826603 & -0.119221074 & -0.701711392 & 0.406989848 \\
7& -2.22954511 & -0.119209853 & -0.701708254 & 0.406858120 \\
8& -2.23027115 & -0.119172562 & -0.701542559 & 0.407291677 \\
9& -2.23060304 & -0.119165639 & -0.701519683 & 0.407488351 \\
10& -2.23068178 & -0.119161793 & -0.701503458 & 0.407494569 \\
11& -2.23069092 & -0.119161850 & -0.701503994 & 0.407494183 \\
12& -2.23068711 & -0.119161857 & -0.701503930 & 0.407498147 \\
13& -2.23068566 & -0.119161942 & -0.701504323 & 0.407498408 \\
14& -2.23068594 & -0.119161930 & -0.701504270 & 0.407499317 \\
15& -2.23068671 & -0.119161903 & -0.701504180 & 0.407499210 \\
16& -2.23068728 & -0.119161894 & -0.701504142 & 0.407499294 \\
17& -2.23068757 & -0.119161880 & -0.701504093 & 0.407499379 \\
18& -2.23068769 & -0.119161881 & -0.701504095 & 0.407499377 \\
19& -2.23068773 & -0.119161878 & -0.701504085 & 0.407499387 \\
20& -2.23068774 & -0.119161880 & -0.701504090 & 0.407499381 \\
21& -2.23068774 & -0.119161879 & -0.701504089 & 0.407499381 \\
22& -2.23068774 & -0.119161880 & -0.701504090 & 0.407499382 \\
23& -2.23068774 & -0.119161880 & -0.701504090 & 0.407499380 \\
24& -2.23068774 & -0.119161880 & -0.701504091& 0.407499380 \\
25& -2.23068774 & -0.119161880 & -0.701504091& 0.407499380 \\
\hline
\end{tabular}\label{table:accuracy_check}
\end{table}
\section{Summary and conclusions}
\label{summary}
In this work we propose a new finite-basis representation for the potential operator.
The approach is inspired by our recent finding concerning Green's operators. If the asymptotic
Hamiltonian is represented in a discrete basis, then for the resolvent the $\infty \times \infty$
symmetric band matrix is inverted by a matrix continued fraction.
A general potential operator is not exactly an infinite band matrix, but it is similar.
The potential matrix exhibits a ridge-like structure which
looks like a band matrix.
We propose a numerical procedure for a finite-basis representation of the potential such that it retains
some information about the whole Hilbert space.
We need to calculate the matrix elements of the potential in a
slightly larger basis, about $5-7$ terms larger, invert the matrix numerically, then truncate the matrix to
the desired size, and finally invert again. This procedure is very straightforward, automatic and
results in a fast convergence in $N$.
|
3,212,635,537,978 | arxiv | \section*{Abstract}
Behavior is characterized by sequences of goal oriented
conducts, such as food uptake, socializing and resting.
Classically, one would define for each task a corresponding
satisfaction level, with the agent engaging, at a given time,
in the activity having the lowest satisfaction level.
Alternatively, one may consider that the agent follows the
overarching objective to generate sequences of distinct
activities. To achieve a balanced distribution of activities
would then be the primary goal, and not to master a specific
task. In this setting the agent would show two types of
behaviors, task-oriented and task-searching phases, with
the latter interseeding the former.
We study the emergence of autonomous task switching
for the case of a simulated robot arm. Grasping one
of several moving objects corresponds in this
setting to a specific activity. Overall, the arm should
follow a given object temporarily and then move away,
in order to search for a new target and reengage. We show that
this behavior can be generated robustly when modeling
the arm as an adaptive dynamical system. The dissipation
function is in this approach time dependent. The arm
is in a dissipative state when searching for a nearby
object, dissipating energy on approach. Once close,
the dissipation function starts to increase, with the
eventual sign change implying that the arm will take
up energy and wander off. The resulting explorative
state ends when the dissipation function becomes again
negative and the arm selects a new target. We believe
that our approach may be generalized to generate
self-organized sequences of activities in general.
\section*{Introduction}
Besides their industrial and practical applications,
real and simulated robots are used increasingly to study
the principles underlying embodied cognition
\cite{martius2013information} and locomotion \cite{ijspeert2014biorobotics},
together with the self organization of critical sensorimotor states
\cite{aguilera2015self} and motor primitives \cite{tani2003self}.
Simulated robots may be considered in addition as proxies for cognitive
and information processing agents \cite{beer2015information}.
It is well known that gaits and other regular muscle contractions,
like breathing \cite{arshavsky2016central}, are induced in many
cases by central pattern generators \cite{marder2001central,ijspeert2008central},
even though it is currently controversial whether this is the
case for biped locomotion \cite{minassian2017human}, viz for
human walking. Abstracting from animal models, one may ask conversely to which
extent compliant locomotion may be generated via self-organizing
principles \cite{sandor2015sensorimotor}, that is in the absence
of top-down control in the form of a central pattern generator.
One talks in this context of `embodiment' \cite{pfeifer2007self},
when part of the computation generating locomotion is carried
out by the elasto-mechanical properties of the constituting body
\cite{aguilar2016review}. For quadruped robots with legs that
are independently controlled by single non-linear phase
oscillators \cite{owaki2013simple}, it has been shown that
the limb-specific sensorimotor feedback derived form pressure
sensors leads to self-organized interlimb communications, with
emerging gaits that correspond to walking, trotting and
galloping \cite{owaki2017quadruped}.
Self-organizing principles may be implemented within the
sensorimotor loop \cite{sandor2015sensorimotor}, which
is comprised of environment, body, actuator and sensory
readings, with the latter being restricted in the pure case
to propiosensation, viz to the internal state of the robot.
The attractors self-stabilizing in the sensorimotor loop
may then give rise to complex patterns of regular and of
chaotic motion primitives \cite{martin2016closed}, which
can be selected in a second step using `kick control'
\cite{sandor2018kick}. From a general perspective, kick
control is an instance of a higher-level control mechanism
exploiting the reduction in control complexity provided by
morphologically computing robots
\cite{muller2017morphological,ghazi2017morphological}.
These approaches are hence different from other works
where closed-loop policies are applied on the top of
open-loop gait cycles
\cite{sprowitz2013towards,travers2016dynamical}.
Alternatively, sequential switching between self-organizing behaviors
in the combined phase space of the controller, body and
environment can also be generated via
self-exploration of the attractor landscape using an
adaptive repelling potential \cite{pinneri2018systematic}.
Motor primitives and their generating guidelines are part
of the basic constituents of a cognitive system
\cite{gros2010cognition}. Here we investigate
whether self-organizing principles may be used also
on a higher level. As a background we consider a
setting where an agent has to follow a certain number
of goals successively, with a typical example being that
of an animal needing to forage, to watch out for predators,
to rest and to socialize \cite{sibly1976fitness}.
The agent is hence confronted with tasks that can be
tackled only sequentially, a problem that may be cast
into the framework of multi objective optimization
\cite{deb2014multi}, an approach which is however not
taken in the present study. We examine instead to
which extend a self-organized dynamical system may
solve the time allocation problem implicitly.
\begin{figure}[!t]
\centering
\includegraphics[width=0.75\textwidth]{twoArmRobot.pdf}
\caption{{\bf The simulated robot arm.}
The two angles $\alpha$ and $\beta$ are actuated,
with (\ref{dot_alpha_v}) governing the evolution
of $\alpha$. An equivalent dynamical system is
in place for $\beta$. The arm has the task to catch one
of the slowly moving objects $\mathbf{m}_i$, to
follow it for a while, with $\mathbf{r}\approx\mathbf{m}_i$,
and to switch autonomously to a distinct object.
}
\label{fig_arm_sketch}
\end{figure}
As a basic protocol we consider an agent having to
solve a series of indistinguishable tasks, with the
agent being given by a simulated two-dimensional robot
arm, as depicted in Fig.~\ref{fig_arm_sketch}. Within
the reach of the arm there are a number of slowly
moving objects the end actuator needs to reach and follow.
Upon success, the self-organized dynamics of the arm should
become 'bored' of the object, move away and search for a
new one. We consider this protocol as a proxy for an
agent showing a non-trivial sequence of behaviors
generated not by top-down commands, but that emerges from
underlying self-organizing principles.
\section*{Materials and methods}
The simulated robot arm sketched in Fig.~\ref{fig_arm_sketch}
has two degrees of freedom, the angles $\alpha$ and $\beta$,
with the position $\mathbf{r}=(r_1,r_2)$ of the end effector,
the hand, being given by
\begin{align}
r_1 &= l_1 \cos (\alpha) - l_2 \cos (\beta - \alpha) \\
r_2 &= l_1 \sin (\alpha) + l_2 \sin (\beta - \alpha)\,,
\label{r_1_2}
\end{align}
where $l_1$ and $l_2$ are the respective arm lengths.
We define a generalized potential $U$ as
\begin{equation}
U = U_{m}\prod_i T^2\big(R_i\big),\qquad\quad
R_i = \sqrt{\left( \mathbf{r} -\mathbf{m}_{i} \right)^2},
\label{U_R_i}
\end{equation}
where $R_i$ is the Euclidean distance between the position
$\mathbf{m}_{i}$ of the $i$th target object and
$\mathbf{r}=\mathbf{r}(\alpha,\beta)$.
In (\ref{U_R_i}) we used a squashing function $T$,
\begin{equation}
T \left( z \right) = \kappa_z \tanh \left( z / s_z \right),
\qquad\quad
\frac{\partial T}{\partial \theta} =
\frac{\kappa_z}{s_z}
\left( 1 - \dfrac{T^2}{\kappa^2_z} \right) \frac{\partial z}{\partial \theta}~,
\label{T_squashing}
\end{equation}
which is characterized by a maximal value $\kappa_z$ and a scale
$s_z$. We use $T(z)$ throughout this study for the renormalization
of several dynamical quantities, with the purpose to avoid exceedingly
large forces or velocities. For the case of the distance we select
a maximum value $\kappa_R\to1$, such that we have
$T(R_i)=\tanh(R_i/s_R)$, as entering (\ref{U_R_i}). $U_m$
is then the maximal value for the potential $U=U(\alpha,\beta)$.
\subsection*{Robot arm dynamics}
The dynamics of the angle $\alpha$ is controlled by
\begin{align}
\dot{\alpha} &= T \left( v_\alpha \right),
\qquad\quad
\dot{v}_\alpha = f(U) \, T \left( v_\alpha \right) -
\nabla_{\alpha} \, U (\alpha, \beta)\,,
\label{dot_alpha_v}
\end{align}
where the objective function $U(\alpha, \beta)$ has the form
of a mechanical potential, with $\nabla_{\alpha}$ denoting the
gradient with respect to $\alpha$. Equivalent equations govern
the time evolution of $\beta$. Eq.~(\ref{dot_alpha_v}) corresponds
to a mechanical system with a potential $U$ and a dissipation
function $f(U)$, for which the velocity $v_\alpha$ has been
renormalized by $T(z)$.
Mechanical systems with dissipation functions $f(U)$ depending
exclusively on the potential $U$, as in (\ref{dot_alpha_v}),
can be considered on a general level as versatile prototype dynamical
systems which exhibit, beside other, complex bifurcation
cascades \cite{sandor2015versatile}.
Several forms may be selected for the dissipation function $f(U)$, as
proposed further below. The system is adaptive \cite{gros2015complex},
dispersing and taking up energy respectively for $f<0$ and $f>0$.
\begin{itemize}
\item In the dissipative stage, when $f(U)<0$, the arm will follow a
damped trajectory towards the next minimum of the potential $U=U(R)$,
that is towards the next object $\mathbf{m}_i$.
\item For a dynamical dissipation function $f(U)$, that is for a $f=f(U)$
which depends functionally but not necessarily explicitly on the potential
$U$, one can achieve that the state $\mathbf{r}\approx\mathbf{m}_i$
becomes progressively unstable, such that the arm eventually moves
away from the object upon taking up energy after $f(U)$ becomes positive.
\end{itemize}
The mechanical potential in (\ref{dot_alpha_v}) treats all targets
$\mathbf{m}_i$ on an equal footing, the setup studied here.
\subsection*{Dissipation function dynamics}
The generic principle for selecting the dissipation function
$f(U)$ is that the system needs to be dissipative when
far away from all objects $\mathbf{m}_i$, with the
configuration $\mathbf{r}\approx\mathbf{m}_i$ becoming
unstable once a specific target has been reached and
followed for a certain time. Distinct ways to implement
this principle are conceivable, here we study
three possibilities.
\begin{itemize}
\item {\bf Exponentially damped (ED).}
One may presume that the dissipation should become
small far away from the objects, viz for large
potentials $U$, as expressed by the ansatz
\begin{equation}
f(U) = f_0 \exp (- \mu \, U),\qquad\quad
\tau_f \, \dot{f}_0 = E_t - U\,.
\label{f_U_ED}
\end{equation}
The prefactor $f_0$ changes sign when the potential
$U$ stays below the reference energy $E_t$ for a period
comparable to $\tau_f$, viz when the end effector remains
close to an object. Once $f_0$ turns positive, the arm will
start to move away from the current object $\mathbf{m}_i$.
\item {\bf Trailing potential (TP).}
In this setup the dissipation function is explicitly
time dependent, with the evolution equation being
determined by the trailing potential $U_T=U_T(t)$,
\begin{equation}
\tau_f \, \dot{f} = E_t - U_T,\qquad\quad
\tau_{T} \, \dot{U}_T = U - U_T\,,
\label{f_U_TP}
\end{equation}
where the integration time scales are regulated
by $\tau_f$ and $\tau_T$. The system is dissipative
when $U_T$ is large, taking up energy once it
falls below the reference energy $E_t$.
\item {\bf Adapting threshold (AT).}
One postulates that $f(U)$ becomes positive when the
potential $U$ falls below a time dependent threshold
$U_\theta=U_\theta(t)$:
\begin{equation}
f(U) = f_0 \, \left( U_\theta - U \right)\exp(-\mu U),
\qquad\quad
\tau_\theta \, \dot{U}_\theta = E_t - U\,,
\label{f_U_AT}
\end{equation}
where $E_t$ is a reference energy. The overall
scale for $f(U)$ is regulated by $f_0$, with
$\tau_\theta$ determining the time needed for
starting to take up energy, after the target
has been reached dissipatively.
\end{itemize}
Further below we will present comparative
results for the above three types of dissipation
function dynamics, with in-detail investigations
of robustness and other dynamical properties
concentrating on ED.
\begin{figure}[!t]
\centering
\includegraphics[width=.8\textwidth]{R_statistics_ED_TP_AT.pdf}
\caption{{\bf Distance statistics.}
The probability distribution $\rho(R_i)$ for he distance
$R_i$ between the end effector and a selected object $i$,
as averaged over time. The targets are indistinguishable,
which implies that $\rho(R_i)=\rho(R_j)$ for all $i,j\in[1,n]$,
where $n=3$ is the number of moving objects. Shown are the results
for three different dissipation functions dynamics, ED
(top,
\href{https://doi.org/10.6084/m9.figshare.7706630.v2}
{click for animation} to see \nameref{S1_Video}),
TP (middle,
\href{https://doi.org/10.6084/m9.figshare.7706624.v1}
{click for animation} to see \nameref{S2_Video}),
and AT (bottom,
\href{https://doi.org/10.6084/m9.figshare.7706627.v1}
{click for animation} to see \nameref{S3_Video}),
as defined respectively by (\ref{f_U_ED}),
(\ref{f_U_TP}) and (\ref{f_U_AT}).
The parameters are listed in Table.~\ref{tableParameters}.
}
\label{fig_distance_probabilities}
\end{figure}
\subsection*{Moving objects}
For the dynamics of the moving objects, the robot arm
has to grab, we used two closely related algorithms.
\begin{itemize}
\item {\bf Polar representation of the velocity (M-PV).}
In the first case the absolute velocity $|v_i|$ of
an object $\mathbf{m}_i$ is drawn from an uniform
distribution in $[0,a]$, with the angle $\varphi_i$
being drawn from $[0,2\pi]$.
\item {\bf Cartesian representation of the velocity (M-CV).}
In the second approach the Cartesian xy-components of
$\mathbf{v}_i$ are drawn independently from an uniform
distribution in $[-b, b]$.
\end{itemize}
The resulting velocity $\mathbf{v}_i$ is applied in both cases
for a time span $t_i$ which is drawn uniformly from $[0, t_{max}]$.
The diffusion of the object is restricted in addition to a circular
area of radius $r_{area}$, reflecting at the boundary. We generally
selected $r_{area}$ to coincide with the reach of the robot arm.
For the other parameters we took $a=b=0.001$ and $t_{max}=10$.
As the simulation results for M-PV and M-CV are very similar,
we show in the following the ones for M-PV.
\begin{table}[!b]
\caption{{\bf Simulation parameters.} The parameters
$\kappa_v$ and $s_v$ entering the renormalization
of the velocity of the mechanical system (\ref{dot_alpha_v})
have been adapted slightly for the three different dissipation
function dynamics, ED, TP and AT. Listed are furthermore
all parameters entering the respective defining equations
(\ref{f_U_ED}), (\ref{f_U_TP}) and (\ref{f_U_AT}). Note
that $\mu$ is given in units of $1/U_m$.}
\smallskip
\centering
\begin{tabular}{l|lc|ccccc}
& $\kappa_v$ & $s_v$ & $\mu U_m$ & $\tau_f$ &
$\tau_T$ & $\tau_\theta$ & $f_0$ \\
\hline
{\bf ED} & 2.8 & 1 & 25 & 1.2 & $\bullet$ & $\bullet$ & $\bullet$\\
{\bf TP} & 4.3 & 3 & $\bullet$ & 6.0 & 4.0 & $\bullet$ & $\bullet$\\
{\bf AT} & 4.0 & 2 & 34 & $\bullet$ & $\bullet$ & 1 & 0.5\\
\end{tabular}
\label{tableParameters}
\end{table}
\begin{figure}[!t]
\centering
\includegraphics[width=.8\textwidth]{timeLine_ED_normal.pdf}
\caption{{\bf Time series for three moving objects.}
As a function of simulation time~$t$, the evolution of
key variables for the ED dissipation-function dynamics,
compare (\ref{f_U_ED}).
(top) The angular velocities $v_\alpha$ and $v_\beta$.
(second from top) The modulus $|v_{arm}|$ of velocity
$v_{arm}$ of the end effector.
(second from bottom) The dissipation function $f$ and
the potential $U$, see (\ref{U_R_i}), with the shading
indicating that the criterion (\ref{following_criterion}) is fulfilled.
The separation of time scales characterizing the dynamics
of $f$, for which a fast drop to negative values is
followed by a slow recovery, drives the distinction
between irregular searching phases and the laminar
flow observed when the end-effector is close to a
specific target.
(bottom) The distances $R_i$ to the $n=3$ moving objects.
}
\label{fig_timeline_ED_normal}
\end{figure}
\subsection*{Parameters}
The overall length $L=l_1+l_2$ of the arm is set to $L=2$,
with the lengths of the two segments being identical,
$l_1 = l_2 = 1$. The parameters for the squashing
function (\ref{T_squashing}) for the distance are
$\kappa_R = 1$ and $s_R = \sqrt{3/n}\,L/2$. For
$n=3$ moving objects we have hence $s_R=L/2=1$.
For the maximum of the potential $U_m$ and for the reference
energy $E_t$ we used $U_{m}=17$ and $E_t=0.05U_m$, respectively,
with all other parameters being listed in Table \ref{tableParameters}.
For the simulation a time step of $dt=0.01$ has been used.
\section*{Results}
For the parameters given in Table~\ref{tableParameters}
we find transients in which the arm tends to stay close
to a target it has approached. The flow in phase space is
laminar when the arm is close to a target, accelerating
however considerably once the dissipation function
$f(U)$ turns positive, compare (\ref{dot_alpha_v})
together with (\ref{f_U_ED}), (\ref{f_U_TP}) and (\ref{f_U_AT}).
For a first understanding we present in Fig.~\ref{fig_distance_probabilities}
the probability $\rho(R_i)$ to observe the distance $R_i$
between the end effector and a given target $i$, see
(\ref{U_R_i}). With all $n=3$ targets being equivalent,
one has $\rho(R_i)=\rho(R_j)$, for all $i,j\in[1,n]$.
\subsection*{Following vs.\ explorative phase}
The distribution of the distance $R_i$ presented in
Fig.~\ref{fig_distance_probabilities} shows that the motion
of the arm can be subdivided into a phase of small $R_i$
and a phase of medium to large distances of all sizes,
modulo fine details. That this is the case for three
different types of dissipation function dynamics proves
that the underlying generating principles is both robust
and versatile. For the three variants considered here,
(\ref{f_U_ED}), (\ref{f_U_TP}) and (\ref{f_U_AT}),
the arm will start to take up energy whenever it did hover
for a certain time close to a target, dissipating on the other
side energy when far away.
The evolution of key variables as a function of simulation
time is presented in Fig.~\ref{fig_timeline_ED_normal}. Shown are,
for the ED dissipation function dynamics, the velocities
$v_\alpha$, $v_\beta$ and $v_{arm}$, of the actuators
and respectively of the arm, together with the evolution
of the dissipation function $f$, of the potential $U$,
and of the distances $R_i$ between the hand of the arm
and the individual objects.
One can distinguish in Fig.~\ref{fig_timeline_ED_normal}
laminar `following phases' and highly irregular
`explorative phases'. Particularly
evident is the driving role of the dissipation
function, which remains negative for most of the smooth
following phase. Visible is also a certain time lag
between the crossing of $f$ from negative to
positive values, which results from the time the system
needs to take up enough energy for the angular velocities
$v_\alpha$ and $v_\beta$, and the potential $U$ to become
visible.
\begin{figure}[!t]
\centering
\includegraphics[width=1.0\textwidth]{parameterSweep_Um_sR_kv_sv.pdf}
\caption{{\bf Parameter sweep.}
For the ED dissipation function dynamics, the probability
$P_{close}$ for the arm to be close to one of the $n=3$ targets
(red circles), as defined by (\ref{following_criterion}), and
$P_{new}$, which measures the chance that two targets approached
one after the another are different (green triangles). With respect
to the reference values $U_m=17$, $s_R=L/2=1$, $\kappa_v=2.8$
and $s_v=1$, the values of the parameters have been changed
individually.
}
\label{fig_parSweep_Um_sR_kv_sv}
\end{figure}
\subsection*{Robustness with respect to parameter changes}
For a criterion that determines whether the end
effector follows a given target we use
\begin{equation}
U<E_t,\qquad\quad
f(U)<0, \qquad\quad
|v_{arm}|<v_{tar}^{max}~,
\label{following_criterion}
\end{equation}
which demands that the potential $U$ is small with respect
to the threshold energy $E_t$ and that the system is
momentarily dissipative, viz that the dissipation function $f(U)$
is negative. The last term in (\ref{following_criterion})
rules out coincidental crossings at high velocities,
which occur when magnitude of the velocity $v_{arm}$ of
the end effector is larger than the maximal velocity
$v_{tar}^{max}$ of the targets. With the dynamics of the
targets being generated, as described, $v_{tar}^{max}$
is known. For practical applications it would be in any
case sufficient to use an empirical estimate for $v_{tar}^{max}$.
Using the criterion (\ref{following_criterion}), one can
define a probability $P_{close}$ that measures the relative
fraction of time the arm follows a target, with following
and the exploration being the two dominant states of the
system, as evident from Fig.~\ref{fig_timeline_ED_normal}.
In Fig.~\ref{fig_parSweep_Um_sR_kv_sv} we present for
the ED dissipation function dynamics the numerical
result for $P_{close}$. Starting from the reference
set of parameters $U_m=17$, $s_R=L/2=1$, $\kappa_v=2.8$
and $s_v=1$, compare also Table~\ref{tableParameters},
the parameters have been modified one by one and
the probability for the arm to follow a target evaluated.
Also included in Fig.~\ref{fig_parSweep_Um_sR_kv_sv} is
the probability $P_{new}$, namely that two targets approached
successively differ.
\begin{figure}[!t]
\centering
\includegraphics[width=1.0\textwidth]{parameterSweep_Et_tf_mu_n.pdf}
\caption{{\bf Robustness of the dissipation function dynamics.}
For the ED dissipation function dynamics, the probability
$P_{close}$ for the arm to be close to a target (red circles),
as defined by (\ref{following_criterion}), and $P_{new}$, which
measures the chance that two targets approached one after
the another are different (green triangles). With respect to
the reference values $E_t=0.05U_m=0.85$, $\mu=25/U_m=1.47$
$\tau_f=1.2$, the values of the parameters have been changed
individually for $n=3$. Also included are the values
of $P_{close}$ and $P_{new}$ upon changing the number $n$
of targets. Here $s_R = \sqrt{3/n}L/2$.
}
\label{fig_parSweep_Et_tf_mu_n}
\end{figure}
\begin{itemize}
\item The probability $P_{close}$ for the arm to be
in the following phase increases monotonically with
the strength $U_m$ of the potential, an intuitive
result. $P_{new}$ decreases conversely, with the reason
being that a larger $U_m$ makes it more difficult to
escape the local potential well.
\item Increasing the characteristic length $s_R$ for
the distance between the arm and a target,
which enters the squashing function (\ref{T_squashing}),
decreases $P_{close}$ dramatically. This is because
the local potential wells attracting the end actuator
to a target in first place tend to disappear for large
$s_R$. $P_{new}$ increases on the other side.
\item The squashing parameters $\kappa_v$ and $s_v$ for
the velocity of the actuators can be changed considerable
without affecting either $P_{close}$ or $P_{new}$,
implying that the system is robust with respect to
both $\kappa_v$ and $s_v$.
\end{itemize}
The data shown in Fig.~\ref{fig_parSweep_Um_sR_kv_sv}
describes the influence of global parameters. In
Fig.~\ref{fig_parSweep_Et_tf_mu_n} we present for
completeness the effect of changing the parameters
$E_t$, $\mu$ and $\tau_f$ of the ED dissipation
function dynamics, see (\ref{f_U_ED}). We find
the generating principle to be robust, viz that
the dependency of $P_{close}$ and $P_{new}$ on
$E_t$, $\mu$ and $\tau_f$ is moderate.
Also included in Fig.~\ref{fig_parSweep_Et_tf_mu_n} are
the values of $P_{close}$ and $P_{new}$ obtained upon
changing the number $n$ of targets. One observes that the
relative fraction of time $P_{close}$ the arm spends close
to a target remains flat. For $n=1$ the probability to
change targets vanishes, as it must, becoming on the
other side substantial for large numbers of targets
$n$.
The here presented sequential task-switching behavior,
generated by the prototype dynamical system (\ref{dot_alpha_v})
does not rely on the particular choice of the generalized
dissipation function dynamics. As demonstrated by
Fig.~\ref{fig_distance_probabilities}, similar distance
distributions $\rho(R_i)$ may result from very
different dissipation function implementations.
This is also reflected by the fraction of time spent
with following and the probability of switching targets,
$P_{close} = 0.44/0.69/0.44$ and $P_{new} = 0.17/0.07/0.14$,
when comparing the dissipation functions
ED/TP/AT see Eqs.~(\ref{f_U_ED}), (\ref{f_U_TP}) and
(\ref{f_U_AT}) respectively, for the parameters given in
Table~\ref{tableParameters}.
\subsection*{Robustness with respect to target properties}
It is clear that the arm would not be able to follow
a target if the maximal velocity $v_{tar}^{max}$ is
too large. We find, however, that the here proposed
generating principle works for a substantial range of
$v_{tar}^{max}$. For the ED dissipation function
dynamics we present in Fig.~\ref{fig_timeline_ED_special}
the time series of the dissipation function and of the
potential both for the case of $v_{tar}^{max}=0.1$,
as used hitherto, and for $v_{tar}^{max}=0.5$.
We find that only details of the overall dynamics
change. This holds also when increasing
the number of moving objects from $n=3$ to $n=8$.
\subsection*{A single non-moving target}
From the dynamical system perspective it is of
interest to investigate the case of a single
stationary target. With noise being absent,
the system is deterministic.
\begin{itemize}
\item \textbf{Fixpoints.} In case of a purely
dissipative dynamics, with $f(U)=f_0<0$, the system disposes
of two stable fixpoints, defined by vanshing angular
velocities $v_{\alpha},v_\beta\to0$, that correspond
to a right- and respectively to a left bend.
\item \textbf{Limit cycle attractors.} With the dynamical
dissipation function ED, it is evident that the
robot arm settles into a limit cycle in which the
destabilized fixpoints are revisited, see
Fig.~\ref{fig_timeline_ED_fixed}. There exist,
hence, multiple symmetry related limit cycles even
for a single resting target (only one of them is shown).
\end{itemize}
Therefore, in the presence of multiple fixed targets, several
different activity sequences may be generated, even for
the same starting position~$\mathbf{r}(0)$ of the arm,
viz for different initial conditions of the internal variables.
\begin{figure}[!t]
\centering
\includegraphics[width=.8\textwidth]{timeLine_ED_special.pdf}
\caption{{\bf Variable object characteristics.}
As a function of simulation time~$t$, the evolution of
the dissipation function $f$ (red) and of the potential
$U$ (blue) for the ED dissipation-function dynamics.
The shaded regions indicate that the criterion
(\ref{following_criterion}) for the arm to be in
the following phase is fulfilled.
(top) For $n=3$ objects for which the maximal velocity is $0.5$, viz
five times larger than in Fig.~\ref{fig_timeline_ED_normal}
(\href{https://doi.org/10.6084/m9.figshare.7706618.v2}
{click for animation} to see \nameref{S4_Video}).
(middle) For $n=8$ objects with a maximal velocity $0.1$
(\href{https://doi.org/10.6084/m9.figshare.7706621.v2}
{click for animation} to see \nameref{S5_Video}).
(bottom) For $n=8$ objects with a maximal velocity $0.5$
(\href{https://doi.org/10.6084/m9.figshare.7706633.v2}
{click for animation} to see \nameref{S6_Video}).
}
\label{fig_timeline_ED_special}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=.8\textwidth]{timeLine_ED_fixed.pdf}
\caption{{\bf A single non-moving object.}
As a function of simulation time~$t$, the evolution of
key variables for the ED dissipation-function dynamics,
as for Fig.~\ref{fig_timeline_ED_normal}, but here for
a single non-moving object located at $(3/8,3/8)\,L$.
The system is fully deterministic, with the robot arm
settling into a limit cycle. The criterion
(\ref{following_criterion}) for the arm to
be close to the object is not applicable, as
$v_{tar}^{max}=0$.
}
\label{fig_timeline_ED_fixed}
\end{figure}
\section*{Discussion}
Action switching in embodied agents may be guided
by fitness considerations, f.i.\ when the task is
to collect a series of different food sources
\cite{agmon2014evolution}. Typically, the action
selected at a given time will be then the one
with the most pressing need. We have followed here
a different approach, examining an overarching generation
principle and not the generation of action sequences
driven by an utility optimization that is local in time.
\subsection*{The stationarity principle}
The question how to decide in which action to
engage has been termed the motivational problem
\cite{gros2012emotional}. The utility of many
activities, like foraging, socializing and resting,
that are regularly repeated, address distinct
needs, which implies that they cannot be lumped
together into an overarching utility function.
In terms of multi-objective optimization \cite{deb2014multi}
the agent must dedicate time to a range of
activities, with the constraint that the resulting
distribution of utilities remains within a given
range. This constraint may be expressed as a
stationarity principle, namely that the statistical
properties of the time series of activities should
become stationary for extended time spans.
The result presented here for the self-organized
robot arm can be viewed as an implementation of
the stationarity principle. With the dynamics being
irregular, viz chaotic, in the explorative phase,
the exact sequence of objects followed is not
pre-determined. The long term statistics, such
as the distance distribution presented in
Fig.~\ref{fig_distance_probabilities}, is
however stationary.
The stationarity principle is a guiding principle
that can be used in various settings. Statistical
learning, e.g.\ of receptive fields \cite{brito2016nonlinear},
is characterized by statistically stationary sensory inputs,
with learning continuing until the statistics of the
output activity becomes also stationary \cite{echeveste2015fisher}.
It has been shown, that one can use the Fisher information
of the neural firing rate to encode the stationarity
principle \cite{echeveste2014generating} and that one
obtains Hebbian learning when minimizing the Fisher information,
viz when the stationarity condition is enforced.
\subsection*{Transient-state dynamics}
With the agent being formulated in term of a mechanical
system, see Eq.~(\ref{dot_alpha_v}), one can abstract from
the behavioral level and describe the robot arm within
dynamical system theory \cite{gros2015complex}. The
striking alternation of dynamical states, as visible in
Fig.~\ref{fig_timeline_ED_normal}, can be interpreted in this
context as an example of transient-state dynamics \cite{gros2007neural}.
The following phase corresponds on a dynamical level
to a transient attractor that becomes unstable on an
extended time scale, namely when the dissipation function
turns positive.
The here discussed mechanism, the coupling of an attracting
state to a slow variable, is the core route for generating
transient-state dynamics in general \cite{gros2009cognitive},
with the flow being laminar during the transient dynamics,
and irregular during the transition periods.
We note that transient-state dynamics may be viewed as
a form of metastability, which may arise either from the
brain dynamics as such \cite{kelso2012multistability},
or from sensorimotor couplings in response to tasks demanding
behavioral flexibility \cite{aguilera2016extended}.
\subsection*{Distinguishable vs.\ non-distinguishable targets}
It would be possible to introduce a bias $b_i=b_i(t)$ that
allows to differentiate between distinct objects. In this
case one would work with the generalized Euclidean distance
\begin{equation}
R_i \ \to\ \sqrt{R_i^2+b_i^2}\,.
\label{RR_cc}
\end{equation}
instead of (\ref{U_R_i}),
for which the bias $b_i$ encodes the depth of the potential, and with
this indirectly also the relative importance of the respective
object. For an appropriate evolution equation for $b_i(t)$, the
respective target would become repelling once the end effector
of the robot has reached it. Two routes on how the dynamical
system (\ref{dot_alpha_v}) induces an autonomously
generated sequence of behaviors are hence possible.
\begin{itemize}
\item {\bf Distinguishable targets.} One works with a constant
dissipation function, $f(U)\to f_0$, with every object
being characterized by a time-dependent attribute, namely
$b_i=b_i(t)$.
\item {\bf Indistinguishable targets.} When all $b_i\equiv0$
there is no variable distinguishing the individual objects.
The sequence of behaviors is then a consequence of
dynamical instabilities resulting from the dynamics of
the dissipation function.
\end{itemize}
In this study we concentrate on the second case as the basic generative
mechanism, noting that the resulting residence times, viz when
$\mathbf{r}\approx\mathbf{m}_i$, could be fine-tuned in a
second step by allowing the $b_i$ to be weakly time dependent.
This protocol is left for future studies.
\section*{Conclusion}
One of the biggest challenges in the design of controllers
for autonomous agents is the combination of different
goal oriented behaviors into a series of self-organized
activities \cite{bekey2005autonomous}.
Here, we investigated how such a higher order controller
may be constructed within a dynamical systems framework,
by adapting a recently introduced versatile prototype
system \cite{sandor2015versatile} to the problem of
an object-following arm. By introducing a model with
a dynamically changing generalized dissipation function
we provide a proof of concept demonstration
of how target following can be turned into a sequential
task switching behavior in terms of transient-state
dynamics \cite{gros2007neural}.
Within this framework the goal oriented activities
are represented by a target-following behavior of
a simulated arm, while the switching dynamics between targets
corresponds to an explorative phase upon getting bored
of the respective task.
Such a self-organized behavior can be generated
both at the level of motion primitives, in case of robotic
locomotion \cite{sandor2015sensorimotor}, and on the
level of action selection \cite{agmon2014evolution},
as demonstrated here. The resulting behavior is robust
within a wide range of parameters, as it does not require
precise fine tuning, which simplifies the selection of
an adequate parameter set with, e.g., machine learning
techniques. Being based on self-organized attractors in
the overarching phase space of agent and environment,
the sensorimotor loop, our approach is resistant to
external noise, retaining at the same time the
flexibility to adapt to the environment or to interact
with other agents \cite{martin2016closed}.
The proposed framework can be generalized
to produce series of activities with a well-defined
order or a given multi-modal probability distribution
by modulating the Euclidean distance as a function
of the actual importance of the respective task --
a research direction left for future studies.
\section*{Acknowledgments}
None.
\nolinenumbers
|
3,212,635,537,979 | arxiv | \section{INTRODUCTION}
Interesting, unconventional phenomena
such as the stochastic resonance (SR) and the noise-induced
phase transition are created by noise. Theoretical studies on noise
in nonlinear dynamical systems
have usually adopted Gaussian white (or colored) noise.
In recent years, there is a growing interest in studying dynamical
systems driven by non-Gaussian noise.
This is motivated by the fact that non-Gaussian noise
with random amplitudes following the power-law distribution
is quite ubiquitous in natural phenomena.
For example, experimental results for crayfish and rat skin
offer strong indication that there could be
non-Gaussian noise in these sensory systems
\cite{Wiesen94}\cite{Nozaki99}.
A simple mechanism has been proposed
to generate the non-Gaussian noise \cite{Borland98}.
With the use of such a theoretical model,
the SR induced by non-Gaussian colored noise
has been investigated \cite{Fuentes01}.
It has been shown that the peak in the signal-to-noise ratio (SNR)
for non-Gaussian noise becomes broader than that for Gaussian noise.
This result has been confirmed by an analog experiment \cite{Castro01}.
Stochastic systems with non-Gaussian colored noise
are originally expressed by the non-Markovian process.
This problem is transformed into a Markovian one
by extending the number of variables and equations.
The relevant Fokker-Planck equation (FPE) includes
the probability distribution expressed in terms of multi-variables.
We may transform this FPE for multivariate probability to
the effective single-variable FPE, or obtain
one-variable differential equation (DE)
with the use of some approximation methods like
the universal colored noise approximation (UCNA)
\cite{Jung87,Hanggi95}
and the functional-integral
methods \cite{Fuentes02}\cite{Wu07}.
The obtained results, however, do not agree each other,
depending on the adopted approximations,
as will be explained in Sec. 2.2 (Table 1).
It is not easy to
trace the origin of this discrepancy because of the
complexity in adopted procedures.
The purpose of
the present paper is to discuss the non-Gaussian noise
and to make a comparison among various methods,
by employing the second-order moment method
which is simple and transparent, and which is exact
in the weak-noise limit.
The paper is organized as follows.
We have applied the second-moment method to the Langevin model
subjected to non-Gaussian noise which is generated
by two kinds of models.
In Sec. 2, non-Gaussian noise is generated
by the specific function which was proposed by Borland \cite{Borland98}
and which has been adopted in several studies
\cite{Fuentes01}\cite{Fuentes02}\cite{Wu07}.
In contrast, in Sec. 3,
non-Gaussian noise is generated by multiplicative noise
\cite{Sakaguchi01}-\cite{Hasegawa07}.
We derive the effective one-variable DE, from which
the stationary distribution is calculated with the use of
the FPE.
A comparison among various methods generating the
non-Gaussian noise is made is Sec. 4, where contributions
from higher moments than the second moment
are also discussed.
The final Sec. 5 is devoted to our conclusion.
\section{Models A$_0$ and A}
\subsection{Moment method}
We have adopted the Langevin model subjected to
non-Gaussian colored noise ($\epsilon$)
and Gaussian white noise ($\psi \xi$),
as given by \cite{Borland98}
\begin{eqnarray}
\dot{x}&=& F(x) + \epsilon(t) + \psi \xi(t)+I(t), \\
\tau \dot{\epsilon} & =& K(\epsilon)
+ \phi \eta(t),
\hspace{2cm}\mbox{(model A$_0$)}
\end{eqnarray}
with
\begin{equation}
K(\epsilon)=- \frac{\epsilon}{[1+(q-1)(\tau/\phi^2) \epsilon^2]},
\end{equation}
which is referred to as the model A$_0$.
In Eqs. (1)-(3), $F(x)$ is an arbitrary function of $x$,
$I(t)$ stands for an external input,
$q$ is a parameter expressing a departure from
the Gaussian distribution which is realized for $q=1$,
$\tau$ denotes the characteristic time of colored noise,
and $\eta$ and $\xi$ the zero-mean white noises
with correlations:
$\langle \eta(t) \eta(t') \rangle=\delta(t-t')$,
$\langle \xi(t) \xi(t') \rangle=\delta(t-t')$ and
$\langle \eta(t) \xi(t') \rangle=0$.
First, we briefly discuss the non-Gaussian colored noise
generated by Eqs. (2) and (3),
which yield
the stationary distribution given by \cite{Borland98}
\cite{Tsallis88,Tsallis98}
\begin{equation}
p_q(\epsilon) \propto \left[1+(q-1)
\left(\frac{\tau}{\phi^2} \right)\epsilon^2\right]_{+}^{-\frac{1}{q-1}},
\end{equation}
with $[x]_{+}=x$ for $x \geq 0$ and zero otherwise.
For $q=1$, Eq. (3) reduces to
\begin{equation}
K(\epsilon)=- \epsilon,
\end{equation}
which leads to the Gaussian distribution given by
\begin{equation}
p_1(\epsilon) \propto e^{-(\tau/\phi^2)\epsilon^2}.
\end{equation}
For $q > 1$ and $q < 1$, Eq. (4) yields long-tail and
cut-off distributions, respectively.
Thus Eqs. (2) and (3) generate the Gaussian and non-Gaussian noises,
depending on the value of parameter $q$.
Expectation values of $\epsilon$ and $\epsilon^2$ are given by
\begin{eqnarray}
\langle \epsilon \rangle &=& 0,\\
\langle \epsilon^2 \rangle
&=& \frac{\phi^2}{\tau (5-3 q)},
\end{eqnarray}
which shows that $\langle \epsilon^2 \rangle$
diverges at $q=5/3$.
In order to make our calculation tractable, we
replace the $\epsilon^2$ term in the denominator of Eq. (3)
by its expectation value:
$\epsilon^2 \simeq \langle \epsilon^2 \rangle$,
to get \cite{Wu07}
\begin{eqnarray}
K(\epsilon) &\simeq& - \frac{\epsilon}{r_q},
\end{eqnarray}
by which Eq. (2) becomes
\begin{eqnarray}
\tau \dot{\epsilon} & =& - \left( \frac{1}{r_q} \right)\:\epsilon
+ \phi \:\:\eta(t),
\hspace{2cm}\mbox{(model A)}
\end{eqnarray}
with
\begin{eqnarray}
r_q &=& \frac{2(2-q)}{(5-3q)}.
\end{eqnarray}
A model given by Eq. (1) with Eq. (10) is hereafter
referred to as the model A, which is discriminated
from the model A$_0$ given by Eqs. (1)-(3).
The solid curve in Fig. 1 expresses $r_q$.
We note that we get $r_q=1$
for $q=1$, and
$r_q < 1$ ($r_q > 1$) for $q < 1$ ($1 < q < 5/3$).
The dashed curve will be discussed in Sec. 3.1.
Now we discuss
the FPE of the distribution $p(x, \epsilon, t)$ for
Eqs. (1) and (10), which are regarded as
the coupled Langevin model.
We get
\begin{eqnarray}
\frac{\partial}{\partial t} p(x,\epsilon,t) &=&
-\frac{\partial }{\partial x}
\{[F(x)+ \epsilon + I] p(x,\epsilon,t)\}
+ \frac{\psi^2}{2} \frac{\partial^2}{\partial x^2}
p(x,\epsilon,t) \nonumber \\
&+&\frac{1}{r_q \tau}\frac{\partial}{\partial \epsilon}
[\epsilon p(x,\epsilon,t)]
+ \frac{1}{2}\left(\frac{\phi}{\tau} \right)^2
\frac{\partial}{\partial \epsilon} \left[\epsilon
\frac{\partial}{\partial \epsilon} \epsilon p(x,\epsilon,t) \right].
\end{eqnarray}
We define means, variances and covariances by
\begin{eqnarray}
\langle x^m \epsilon^n \rangle
&=& \int dx \int d \epsilon \; x^m \epsilon^n p(x,\epsilon,t).
\hspace{1cm}\mbox{($m, n$: integer)}
\end{eqnarray}
By using the moment method for the coupled Langevin model
\cite{Hasegawa06,Hasegawa07},
we get their equations of motion given by
\begin{eqnarray}
\frac{d \langle x \rangle}{d t}
&=& \langle F(x) + \epsilon + I \rangle, \\
\frac{d \langle \epsilon \rangle}{d t}
&=& - \frac{1}{r_q \tau}\langle \epsilon \rangle, \\
\frac{d \langle x^2 \rangle}{d t}
&=& 2 \langle x[F(x) + \epsilon + I] \rangle + \psi^2, \\
\frac{d \langle \epsilon^2 \rangle}{d t}
&=& - \frac{2}{r_q \tau}\langle \epsilon^2 \rangle
+\left( \frac{\phi}{\tau} \right)^2, \\
\frac{d \langle x \epsilon \rangle}{dt}
&=& \langle \epsilon [F(x) + \epsilon + I] \rangle
-\frac{1}{r_q \tau}\langle x \epsilon \rangle.
\end{eqnarray}
We consider means, variances and covariance defined by
\begin{eqnarray}
\mu &=& \langle x \rangle, \\
\nu &=& \langle \epsilon \rangle, \\
\gamma &=& \langle x^2 \rangle - \langle x^2 \rangle^2, \\
\zeta &=& \langle \epsilon^2 \rangle - \langle \epsilon \rangle^2, \\
\chi &=& \langle x \epsilon \rangle
- \langle x \rangle \langle \epsilon \rangle.
\end{eqnarray}
When we expand Eqs. (14)-(18)
as $x=\mu+\delta x$ and $\epsilon=\nu+\delta \epsilon$
around the mean values of $\mu$ and $\nu$,
and retaining up to their second order
contributions such as $ \langle (\delta x)^2 \rangle $,
equations of motion become \cite{Hasegawa06,Hasegawa07}
\begin{eqnarray}
\frac{d \mu}{dt} &=& f_0 + f_2 \gamma
+ \nu +I(t), \\
\frac{d \nu}{dt} &=& - \frac{\nu}{r_q \tau}, \\
\frac{d \gamma}{dt} &=& 2(f_1 \gamma + \phi)
+ \psi^2, \\
\frac{d \zeta}{dt} &=& - \left( \frac{2}{r_q \tau} \right) \zeta
+ \left( \frac{\phi}{\tau} \right)^2, \\
\frac{d \chi}{dt} &=& \left(f_1 -\frac{1}{r_q \tau} \right) \chi+ \zeta,
\end{eqnarray}
with
\begin{eqnarray}
f_{\ell} &=& \frac{1}{\ell !}
\frac{\partial^{\ell} F(\mu)}{\partial x^{\ell}}.
\end{eqnarray}
When we adopt the stationary values
for $\nu$, $\zeta$ and $\phi$:
\begin{eqnarray}
\nu& \simeq &\nu_s =0, \\
\zeta & \simeq&\zeta_s =\frac{r_q \phi^2}{2 \tau}, \\
\chi &\ \simeq&\chi_s = \frac{r_q^2 \phi^2}{2(1-r_q\tau f_1)},
\end{eqnarray}
equations of motion for
$\mu$ and $\gamma$ become
\begin{eqnarray}
\frac{d \mu}{dt} &=& f_0 + f_2 \gamma + I(t), \\
\frac{d \gamma}{dt} &=& 2 f_1 \gamma
+ \frac{r_q^2 \phi^2}{(1-r_q \tau f_1)}+\psi^2,
\end{eqnarray}
where $r_q$ is given by Eq. (11).
It is noted that the stationary value of
$\zeta_s=(2-q)\phi^2/\tau(5-3q)$ in Eq. (31) is a little different
from $\langle \epsilon^2 \rangle=\phi^2/\tau(5-3q)$ in Eq. (8),
which is due to an introduced approximation.
We may express the effective DE for $x$ as
\begin{eqnarray}
\dot{x}&=& F_{eff}(x) + I_{eff}(t) + \alpha_{eff} \:\eta(t)
+ \psi \xi(t),
\end{eqnarray}
with
\begin{eqnarray}
F_{eff}(x) &=& F(x), \\
I_{eff}(t) &=& I(t), \\
\alpha_{eff} &=& \frac{\phi_q}{\sqrt{1- \tau_q f_1}}, \\
\phi_q &=& r_q \phi, \\
\tau_q &=& r_q \tau,
\end{eqnarray}
from which Eqs. (33) and (34) are derived \cite{Hasegawa06,Hasegawa07}.
Equations (35) and (38) clearly express the effect of non-Gaussian
colored noise. The effective magnitude of noise $\alpha_{eff}$
is increased with increasing $q$ (Fig. 1). In contrast,
with increasing $\tau$, the effective $\alpha_{eff}$ value is
decreased for $f_1 < 0$ which is usually realized.
The FPE of $P(x,t)$ for Eq. (35) is expressed by
\begin{eqnarray}
\frac{\partial }{\partial t}P(x,t) &=&
- \frac{\partial}{\partial \epsilon}
\{ (F_{eff}+I) P(x,t) \}
+ \frac{1}{2}
\frac{\partial}{\partial x} \left[ \alpha_{eff} \frac{\partial}
{\partial x} \alpha_{eff} P(x,t) \right] \nonumber \\
&+& \frac{\psi^2}{2}
\frac{\partial^2}{\partial x^2}P(x,t),
\end{eqnarray}
which may be applicable to $\alpha_{eff}$ depending on $x$
[{\it i.e.} Eqs. (50) and (59)].
The stationary distribution is given by
\begin{eqnarray}
\ln P(x) &=& 2 \int \:dx
\left( \frac{F_{eff}+I}{\alpha_{eff}^2+\psi^2} \right)
-\frac{1}{2} \ln \left(\frac{\alpha_{eff}^2+\psi^2}{2} \right).
\end{eqnarray}
For $F(x)=-\lambda x$, we get
\begin{eqnarray}
P(x) &\propto & \exp\left[-\left(\frac{\lambda}
{\left[\phi_q^2/(1+\lambda \tau_q)+\psi^2 \right]}\right)
\left( x-\frac{I}{\lambda} \right)^2\right].
\end{eqnarray}
For $F(x)=a x - b x^3$, we get
\begin{eqnarray}
P(x) &\propto & \exp\left[\frac{1}
{[\phi_q^2/(1-\tau_q(a - 3 b\mu^2))+\psi^2]}\right]
\left(ax^2-\frac{bx^4}{2}+2 Ix \right).
\end{eqnarray}
\subsection{Comparison with other methods}
We will compare the result of the moment method with those of several
analytical methods: the universal colored noise approximation
(UCNA) and functional-integral methods (FI-1 and FI-2).
\noindent
{\bf (a) UCNA}
The universal colored noise approximation (UCNA)
was proposed by
Jung and H\"{a}nggi \cite{Jung87,Hanggi95}
by interpolating between
the two limits of $\tau=0$ and $\tau=\infty$ of colored noise,
and it has been widely adopted for a study
of effects of Gaussian and non-Gaussian colored noises.
By employing the UCNA,
we may derive the effective DE for the variable $x$.
Taking the time derivative of Eq. (1) with $\psi=0$, and
using Eq. (10) for $\dot{\epsilon}$, we get
\begin{eqnarray}
\ddot{x} &=& F' \dot{x}+ \dot{\epsilon} + \dot{I}, \\
&=& \left(F'-\frac{1}{\tau_q}\right) \dot{x}
+ \left(\frac{F+I}{\tau_q} \right)
+\dot{I}+\left( \frac{r_q \phi}{\tau_q}\right) \eta.
\end{eqnarray}
When we neglect the $\ddot{x}$ term after the UCNA,
we get the effective DE for $x$ given by
\begin{eqnarray}
\dot{x}&=& F_{eff}(x)+ I_{eff}(t) + \alpha_{eff} \:\eta(t),
\end{eqnarray}
with
\begin{eqnarray}
F^U_{eff}(x) &=& \frac{F(x)}{(1 - \tau_q F')}, \\
I^U_{eff}(t) &=& \frac{(I+\tau_q \dot{I})}{(1 - \tau_q F')}, \\
\alpha^U_{eff} &=& \frac{r_q \phi}{(1 - \tau_qF')},
\end{eqnarray}
where $F=F(x)$, $F'=F'(x)$, and
$\tau_q$ and $r_q$ are given by Eqs. (40) and (11), respectively.
It is noted that $\alpha^U_{eff}$ given by Eq. (50)
generally depends on $x$, yielding the multiplicative
noise in Eq. (47).
For $F(x)=-\lambda x$, the stationary distribution is given by
\begin{eqnarray}
P^U(x) &\propto & \exp\left[-\frac{\lambda (1+\lambda \tau_q)}{\phi_q^2}
\left(x-\frac{I_c}{\lambda} \right)^2 \right],
\end{eqnarray}
which agrees with the result of Eq. (43) with $\psi=0$.
For $F(x)=ax-b x^3$, we get
\begin{eqnarray}
P^U(x) &\propto & [1-\tau_q(a-3bx^2)]
\exp \left( \left[\frac{a(1-a\tau_q)x^2}{\phi_q^2}
-\frac{b(1-4a\tau_q)x^4}{2\phi_q^2}
-\frac{b^2\tau_q x^6}{\phi_q^2} \right] \right)
\nonumber \\
&&\times \exp\left( \left[\frac{2I_c(1-a\tau_q)x}{\phi_q^2}
+\frac{2I_cb\tau_q x^2}{\phi_q^2} \right] \right),
\end{eqnarray}
whose functional form is rather different from that given by Eq. (44).
\noindent
{\bf (b) Functional-integral method (FI-1)}
Wu, Luo and Zhu \cite{Wu07} started from the formally
exact expression for $P(x,t)$
of Eqs. (1) and (10) with $I(t)=0$ given by
\begin{eqnarray}
\frac{\partial}{\partial t} P(x,t)
&=& -\frac{\partial}{\partial x} [F(x) P(x,t)]
-\frac{\partial}{\partial x}
\langle \epsilon(t) \delta(x(t)-x) \rangle
-\psi \frac{\partial}{\partial x}
\langle \xi(t) \delta(x(t)-x) \rangle,
\end{eqnarray}
where $\langle \cdot \rangle $ denotes the average
over the probability $P(x,t)$ to be determined.
Employing the Novikov theorem \cite{Novikov65} and
the functional-integral method, they obtained
the effective FPE for $P(x,t)$ which yields Eq. (47) but with
\begin{eqnarray}
F^W_{eff}(x) &=& F(x), \\
\alpha^W_{eff} &=& \frac{r_q \phi}
{\sqrt{1- \tau_q F_s'} },
\end{eqnarray}
where $r_q$ and $\tau_q$ are given by Eqs. (40) and (11),
respectively,
$F'=dF/dx$ and
$F_s$ {\it et al.} denote the steady-state values at $x=x_s$.
For $F(x)=-\lambda x$, we get
\begin{eqnarray}
P^W(x) &\propto & \exp\left[-\frac{\lambda (1+\lambda \tau_q)}{\phi_q^2}
x^2 \right].
\end{eqnarray}
For $F(x)=ax-b x^3$, we get
\begin{eqnarray}
P^W(x) &\propto&
\exp\left[\frac{[1-\tau_q(a-3b\mu^2)]}{\phi_q^2}
\left(ax^2-\frac{bx^4}{2} \right) \right].
\end{eqnarray}
\noindent
{\bf (c) Functional-integral method (FI-2)}
By applying the alternative functional-integral method
to the FPE for $p(x,\epsilon,t)$ given by Eqs. (1) and (10)
with $\psi=I(t)=0$,
Fuentes, Toral and Wio \cite{Fuentes01}
derived the FPE of $P(x,t)$, which leads to
the effective DE given by Eq. (47), but with
\begin{eqnarray}
F^F_{eff}(x) &=& \frac{F}{(1-s_q \tau F') }, \\
\alpha^F_{eff} &=& \frac{ s_q \phi}
{(1- s_q \tau F')},
\end{eqnarray}
with
\begin{equation}
s_q= \left[1+(q-1)\left(\frac{\tau}{2\phi^2}\right)F^2\right].
\end{equation}
We note that $\alpha^F_{eff}$
generally depends on $x$, yielding the multiplicative noise in Eq. (47).
For $F(x)=-\lambda x$, we get
\begin{eqnarray}
P^F(x) &\propto &
(1+\lambda \tau s_q ) s_q^{-[2/(q-1)+1]}
\exp\left[ \frac{2}{\lambda \tau (q-1) s_q} \right],
\end{eqnarray}
with
\begin{equation}
s_q= \left[1+(q-1)\left(\frac{\tau \lambda^2}{2\phi^2}\right)x^2 \right].
\end{equation}
For $F(x)=a x - b x^3$, it is necessary to numerically
evaluate the distribution $P(x)$
with the use of Eqs. (42) and (58)-(60).
A comparison among various methods is summarized in the Table 1.
We note that the result of our moment method agrees with that of
FI-1, but disagrees with those of UCNA and FI-2.
The result of UCNA is not identical with that
of FI-2, although they are consistent each other
if the identity of $s_q=r_q$ holds,
which is realized for $q=1$ with $s_1=r_1=1$.
\subsection{Numerical calculations}
We present some numerical examples to make a comparison
with direct simulation (DS), which has been performed
for Eqs. (1)-(3) by the fourth-order Runge-Kutta method
with a time step of 0.01 for 1000 trials.
Figures 2(a)-2(f) show the stationary probability
calculated by various methods for $F(x)=-\lambda x$
with changing
parameters of $q$ and $\tau$ for fixed $\phi=0.5$
and $\psi=0$.
A comparison between Fig. 2(c) and 2(d) shows that
the width of the distribution for $\tau=1.0$ is narrower
than that for $\tau=0.5$.
This is explained by the reduced effective strength of
$\alpha_{eff}=\phi/(1+\lambda \tau)$ by an increased $\tau$.
We note that for $q=1.0$,
results of all methods are in good agreement each other.
Comparing Fig. 2(a) to Fig. 2(c) [and Fig. 2(b) to Fig. 2(d)],
we note that the width of the distribution for $q=0.8$ is
a little narrower than that for $q=1.0$.
This is due to the fact that the $r_q$ value is reduced
to 0.82 from unity.
An agreement among various methods is good for $q=0.8$.
In contrast, Figs. 2(e) and 2(f) show that for $q=1.5$,
the width of $P(x)$ becomes wider because of the increased
$r_{1.5}=2.0$.
The results of the moment method, UCNA and FI-1
are in fairly good agreement.
On the contrary, the distribution calculated by FI-2 is
sharper than that of DS.
Figures 3(a)-3(f) show the stationary probability
calculated by various methods for $F(x)=x-x^3$
with changing
parameters of $q$ and $\tau$ for fixed $\phi=0.5$
and $\psi=0.0$.
The general trend realized in Figs. 3(a)-3(f) is the same
as in Figs.2(a)-2(f).
The result of FI-2 for $q=1.5$ is not so bad compared to
those of other approximation methods. However, the result
of FI-2 for $q=0.8$ and $\tau=1.0$ is worse than
other methods.
\section{Model B}
\subsection{Moment method}
In order to generate non-Gaussian noise,
we may employ an alternative model
(referred to as the model B) given by
\begin{eqnarray}
\dot{x}&=& F(x) + \epsilon(t) +I(t), \\
\tau \dot{\epsilon} & =& -\epsilon
+ \epsilon \:\alpha \eta(t) + \beta \xi(t),
\hspace{2cm}\mbox{(model B)}
\end{eqnarray}
where $F(x)$ expresses an arbitrary function of $x$,
$I(t)$ an external input,
$\tau$ the characteristic time of colored noise, and
$\alpha$ and $\beta$ denote magnitudes of
additive and multiplicative noises, respectively, given by
zero-mean white noises, $\eta$ and $\xi$, with
correlations:
$\langle \eta(t) \eta(t') \rangle
= \langle \xi(t) \xi(t') \rangle= \delta(t-t')$ and
$\langle \eta(t) \xi(t') \rangle=0$.
The FPE for the distribution $p(\epsilon, t)$ for Eq. (64)
in the Stratonovich representation is given by
\begin{eqnarray}
\frac{\partial}{\partial t} p(\epsilon,t) &=&
\frac{1}{\tau}\frac{\partial}{\partial \epsilon}
[\epsilon p(\epsilon,t)]
+ \frac{1}{2}\left(\frac{\alpha}{\tau}\right)^2
\frac{\partial}{\partial \epsilon}\left( \epsilon
\frac{\partial}{\partial \epsilon} [\epsilon p(\epsilon,t)] \right)
+ \frac{1}{2}\left(\frac{\beta}{\tau}\right)^2
\frac{\partial^2}{\partial \epsilon^2}
p(\epsilon,t).
\end{eqnarray}
The stationary distribution of $\epsilon$
has been extensively discussed
\cite{Sakaguchi01}-\cite{Hasegawa07}
in the context of the
nonextensive statistics \cite{Tsallis88,Tsallis98}.
It is given by
\cite{Sakaguchi01}-\cite{Hasegawa07}
\begin{eqnarray}
p_q(\epsilon)
&\propto& \left[1 + \left(\frac{\alpha^2}{\beta^2}\right)
\epsilon^2 \right]_{+}^{-(\tau/\alpha^2+1/2)}, \\
&\propto&
\left[1 + (q-1)
\left(\frac{\tau}{\kappa \beta^2} \right)
\epsilon^2\right]_{+}^{-\frac{1}{q-1}},
\end{eqnarray}
with
\begin{eqnarray}
q&=& 1+ \left( \frac{2 \alpha^2}{2 \tau+\alpha^2} \right), \\
\kappa &=& \left( \frac{3-q}{2} \right)
= \left( \frac{2 \tau}{2 \tau+\alpha^2} \right),
\end{eqnarray}
where $[x]_{+}=x$ for $x \geq 0$ and zero otherwise.
In the limit of $\alpha=0.0$ ($q=1$), the distribution given by Eq. (67)
reduces to the Gaussian distribution given by
\begin{eqnarray}
p(\epsilon) &\propto& \exp\left(-\frac{\tau}{\beta^2} \epsilon^2\right).
\end{eqnarray}
In the opposite limit of $\beta=0.0$, Eq. (67) leads to the power-law
distribution given by
\begin{eqnarray}
p(\epsilon) &\propto& \epsilon^{- \delta},
\end{eqnarray}
with
\begin{eqnarray}
\delta &=&1+ \frac{2 \tau }{\alpha^2} = \frac{2}{q-1}.
\end{eqnarray}
The expectation values of $\epsilon$ and $\epsilon^2$ are given by
\begin{eqnarray}
\langle \epsilon \rangle &=& 0, \\
\langle \epsilon^2 \rangle
&=& \frac{\kappa \beta^2}{\tau (5-3q)}
= \frac{\beta^2}{2(\tau-\alpha^2)}.
\end{eqnarray}
The second moment is finite for $\alpha^2 < \lambda$ ($q < 5/3$).
It is expected that
Eq. (64) leads to the non-Gaussian colored noise
with the correlation given by
\begin{equation}
\langle \epsilon(t) \epsilon(t') \rangle
= \frac{\beta^2}{2 (\tau-\alpha^2)}
\exp\left[-\frac{\mid t-t' \mid}{\tau} \right].
\end{equation}
By applying the moment method to Eqs. (63) and (64),
we may obtain the effective one-variable DE for $x$
given by
\begin{eqnarray}
\dot{x} &=& F_{eff}+ I(t)+ \beta_{eff}\; \xi(t),
\end{eqnarray}
with
\begin{eqnarray}
F_{eff} &=& F(x), \\
\beta_{eff} &=& \frac{\beta_q}{\sqrt{1-\tau f_1}}, \\
\beta_q &=& \beta u_q, \\
u_q &=& \sqrt{\frac{1}{(1-\alpha^2/\tau)}}
= \sqrt{ \frac{3-q}{5-3q}},
\end{eqnarray}
details of calculations being explained in the Appendix.
The $q$ dependence of $u_q$ is plotted by
the dashed curve in Fig. 1, where
$u_q < 1$, $u_q=1$ and $u_q > 1$
for $q <1$, $q=1$ and $1 < q < 5/3$, respectively.
We note that $u_q$ has a similar $q$ dependence
as $r_q$ shown by the solid curve.
The FPE of $P(x,t)$ for Eq. (76) is given by
\begin{eqnarray}
\frac{\partial }{\partial t}P(x,t) &=&
- \frac{\partial}{\partial \epsilon}
\left[ (F_{eff}+I) P(x,t) \right]
+ \left(\frac{1}{2} \right)
\frac{\partial}{\partial x}\left[\beta_{eff} \frac{\partial}
{\partial x}\beta_{eff} P(x,t)\right].
\end{eqnarray}
The stationary distribution is given by
\begin{eqnarray}
\ln P(x) &=& 2 \int \:dx
\left( \frac{F_{eff}+I}{\beta_{eff}^2} \right)
-\frac{1}{2} \ln \left(\frac{\beta_{eff}^2}{2} \right).
\end{eqnarray}
For $F(x)=-\lambda x$, we get
\begin{eqnarray}
P(x) &\propto & \exp\left[-\left(\frac{\lambda \left(1+\lambda \tau \right)}
{\beta_q^2}\right)
\left( x-\frac{I}{\lambda} \right)^2\right].
\end{eqnarray}
For $F(x)=a x - b x^3$, we get
\begin{eqnarray}
P(x) &\propto & \exp\left[\frac{[1-\tau (a - 3 b\mu^2)]}
{\beta_q^2}\right]
\left(a x^2-\frac{bx^4}{2}+2 Ix \right).
\end{eqnarray}
Equations (83) and (84) are similar to Eqs. (43) and (44)
(with $\psi=0.0$), respectively, for the model A,
although $u_q$ and $\tau$ in the former are
different from $r_q$ and $\tau_q$ in the latter.
It would be interesting to compare the result of the moment
method for the model B with those of the UCNA and FI method,
as we have made for the model A in Sec. 2.2.
Unfortunately the UCNA method cannot be applied to
the model B because Eq. (64)
includes the multiplicative noise \cite{Note1}.
It is very difficult to apply the FI method to the model B
including both additive and multiplicative noises:
such calculations have not been reported as far as
we are aware of.
Then we will make a comparison of the result
of the moment method
only with that of DS in the next subsection 3.2.
\subsection{Numerical calculations}
We present some numerical examples to make a comparison
with DS, which has been performed
for Eqs. (63) and (64) by the Heun method
with a time step of 0.001 for 1000 trials.
Figures 4(a)-4(d) show the stationary probability $P(x)$
calculated for $F(x)=-\lambda x$ with changing
parameters of $\alpha$ and $\tau$ for a fixed $\beta=0.5$.
In Figs. 4(a) and 4(b) for $\alpha=0.0$ ($q=1.0$),
we observe that the width of $P(x)$ is decreased with increasing $\tau$,
as shown in Figs. 2(c) and 2(d). Figures 4(c) and 4(d) show that
with increasing $\alpha$ to 0.5, we get wider width in $P(x)$ because
we get $q=1.40$ and 1.22 for $\tau=0.5$ and 1.0, respectively [Eq. (68)].
Similarly, Figs. 5(a)-5(d) show $P(x)$ for $F(x)=x-x^3$.
Results of the moment method are in good agreement with those
of DS for $\alpha=0.0$ ($q=1.0$) as shown in Figs. 5(a) and 5(b).
The width of $P(x)$ in Fig. 5(c)
for $\alpha=0.5$ and $\tau=0.5$ ($q=1.40$)
is wider than that in Fig. 5(a)
for $\alpha=0.0$ and $\tau=0.5$ ($q=1.0$),
but it is narrower than that in Fig. 5(d)
for $\alpha=0.5$ and $\tau=1.0$ ($q=1.22$).
\section{Discussion}
We will make a comparison among the various methods
for generating non-Gaussian noise given by
\begin{eqnarray}
\dot{x}&=& F(x) + \epsilon(t) +I(t),
\end{eqnarray}
with
\begin{eqnarray}
\tau \dot{\epsilon} & =& K(\epsilon)
+ \phi \eta(t),
\hspace{2cm}\mbox{(model A$_0$)} \\
\tau \dot{\epsilon} &=& -\left( \frac{\epsilon}{r_q} \right)
+\phi \eta(t),
\hspace{2cm}\mbox{(model A)} \\
\tau \dot{\epsilon} & =& -\epsilon
+ \epsilon \:\alpha \eta(t) + \beta \xi(t),
\hspace{1cm}\mbox{(model B)}
\end{eqnarray}
where $\eta$ and $\xi$ are white noises, $r_q$ is given by Eq. (11),
and $K(\epsilon)$ is given by Eq. (3) or
\begin{eqnarray}
K(\epsilon) &=& - \frac{d U(\epsilon)}{d \epsilon}, \\
U(\epsilon) &=& \frac{\phi^2}{2 \tau (q-1) }
\ln \left[1 + (q-1) \left( \frac{\tau}{\phi^2} \right)
\epsilon^2 \right].
\end{eqnarray}
Note that the model A is derived from the model A$_0$
with the approximation:
$K(\epsilon) \simeq -\epsilon/r_q$
and $U(\epsilon) \simeq \epsilon^2/2 r_q$ [Eq. (9)].
Noises in the models A$_0$ and A are generated by a motion
under the potentials given by Eq. (90)
and $U(\epsilon) = \epsilon^2/2 r_q$,
respectively, subjected to additive noise.
In contrast, noise in the model B is generated
by a motion under the potential of $U(\epsilon)=\epsilon^2/2$
subjected to additive and multiplicative noises.
We note from Eqs. (4) and (67) that
the stationary distributions of $\epsilon$
in the models A$_0$ and B become the equivalent non-Gaussian
distribution if the
parameters in the two models satisfy the relation:
\begin{eqnarray}
\phi^2 &=& \kappa \beta^2
= \frac{\beta^2}{(1+ \alpha^2/2 \tau)}.
\hspace{1cm}\mbox{for $q \geq 1$}
\end{eqnarray}
This equivalence, however, does no hold
between the models A and B, because
the stationary distribution of the model A
is not the non-Gaussian but the Gaussian given by
\begin{eqnarray}
p_q(\epsilon) \propto
\exp \left[-\left(\frac{\tau}{r_q \phi^2} \right) \epsilon^2 \right].
\hspace{1cm}\mbox{(model A)}
\end{eqnarray}
As for the dynamical properties,
equations of motion for $\langle \epsilon^2 \rangle$
in the moment method are given by
Eqs. (17) and (A6):
\begin{eqnarray}
\frac{d \langle \epsilon^2 \rangle}{dt}
&=& - \left( \frac{2}{r_q \tau} \right) \langle \epsilon^2 \rangle
+ \left( \frac{\phi}{\tau} \right)^2,
\hspace{1cm}\mbox{(model A)} \\
\frac{d \langle \epsilon^2 \rangle}{dt}
&=& - \frac{2}{\tau}
\left(1- \frac{\alpha^2}{\tau} \right) \langle \epsilon^2 \rangle
+ \left( \frac{\beta}{\tau} \right)^2,
\hspace{1cm}\mbox{(model B)}
\end{eqnarray}
Equations of motion for $\mu$ and $\gamma$
are given by Eqs. (33), (34), (A16) and (A17):
\begin{eqnarray}
\frac{d \mu}{dt} &=& f_0 + f_2 \gamma + I(t),
\hspace{1cm}\mbox{(models A and B)} \\
\frac{d \gamma}{dt} &=& 2 f_1 \gamma
+ \frac{r_q^2 \phi^2}{(1-r_q \tau f_1)},
\hspace{1cm}\mbox{(model A)} \\
\frac{d \gamma}{dt} &=& 2 f_1 \gamma
+ \frac{u^2_q \:\beta^2}{(1-\tau f_1)},
\hspace{1cm}\mbox{(model B)}
\end{eqnarray}
where $u_q=\sqrt{(3-q)/(5-3 q)}$ [Eq. (A18)].
In the model A, we have adopted the approximation:
$K(\epsilon) \simeq -\epsilon/r_q$ [Eq. (9)],
without which reasonable results are not obtainable
in the moment approach (see the discussion below).
Equations (93)-(97) show that equations of motion
for the models A and B have the same structure.
In the case of weak noise and small $\tau$,
for which the second-moment approach is expected to be valid,
the dynamical properties of the models A and B
(as well as the model A$_0$) are qualitatively the same,
although there are some quantitative
difference among them:
{\it i.e.} the stationary value of $\gamma$ of the model A
is different from that of the model B.
Our discussion presented in this paper
is based on the second-order moment method.
Effects of higher-order moment neglected in our method
are examined in the following.
The equation of motion
for the $k$th moment with even $k$
of the model A$_0$ is formally given by
\begin{eqnarray}
\frac{d \langle \epsilon^k \rangle }{d t}
&=& \left( \frac{k}{\tau} \right)
\langle \epsilon^{(k-1)} K(\epsilon) \rangle
+ \frac{k(k-1)}{2}\left( \frac{\phi}{\tau} \right)^2
\langle \epsilon^{(k-2)} \rangle,
\hspace{1cm}\mbox{(model A$_0$)}
\end{eqnarray}
though an evaluation of the first term of Eq. (98)
is very difficult.
In order to get a meaningful result within the moment method,
we have assumed
$K(\epsilon) \simeq -\epsilon/r_q$ (the model A) to get
\begin{eqnarray}
\frac{d \langle \epsilon^k \rangle }{d t}
&=& - \left( \frac{k}{r_q \tau} \right)
\langle \epsilon^{k} \rangle
+ \frac{k(k-1)}{2}\left( \frac{\phi}{\tau} \right)^2
\langle \epsilon^{(k-2)} \rangle,
\hspace{1cm}\mbox{(model A)}
\end{eqnarray}
from which we may recurrently calculate
the stationary $k$th moment as
\begin{eqnarray}
\langle \epsilon^{k} \rangle
&=& \frac{(k-1) r_q}{2} \left( \frac{\phi^2}{\tau} \right)
\langle \epsilon^{(k-2)} \rangle, \\
&=& \frac{(k-1)!! \: r_q^{k/2}}{2^{k/2}}
\left(\frac{\phi^2}{\tau} \right)^{k/2}.
\hspace{1cm}\mbox{(model A)}
\end{eqnarray}
The stationary distribution in the model A$_0$
given by Eq. (4)
leads to the second- and fourth-order moments:
\begin{eqnarray}
\langle \epsilon^2 \rangle
&=& \frac{\phi^2}{(5-3 q)\tau}, \\
\langle \epsilon^4 \rangle
&=& \frac{3 \phi^4}{ (5-3q)(7-5q)\tau^2}.
\hspace{1cm}\mbox{(model A$_0$)}
\end{eqnarray}
In contrast, the stationary distribution in the model A
given by Eq. (92) yields
\begin{eqnarray}
\langle \epsilon^2 \rangle
&=& \frac{(2-q)\phi^2}{(5-3 q)\tau}, \\
\langle \epsilon^4 \rangle
&=& 3 \langle \epsilon^2 \rangle^2
= \frac{3 (2-q)^2 \phi^4}{ (5-3q)^2 \tau^2}.
\hspace{1cm}\mbox{(model A)}
\end{eqnarray}
The expression of Eq. (104) is different
from that of Eq. (102) by a factor of $(2-q)$.
The ratio of Eq. (105) to Eq. (103) becomes
$(2-q)^2(7-5q)$, which is less than unity for $1 < q < 5/3$.
These show that the distribution
given by Eq. (92) in the model A
underestimates the effective
width of the distribution of $\epsilon$ compared to that
in the model A$_0$.
In order to include the higher-order moment in an appropriate way,
we have to go beyond the approximation
with $K(\epsilon) \simeq - \epsilon/r_q $ [Eq. (9)].
With the model B, we may obtain the equation of motion
for the $k$th moment with even $k$, as given by
\begin{eqnarray}
\frac{d \langle \epsilon^k \rangle }{d t}
&=& -\left[\frac{k}{\tau}
-\frac{k^2}{2} \left( \frac{\alpha}{\tau} \right)^2 \right]
\langle \epsilon^k \rangle
+ \frac{k(k-1)}{2}\left( \frac{\beta}{\tau} \right)^2
\langle \epsilon^{(k-2)} \rangle.
\hspace{1cm}\mbox{(model B)}
\end{eqnarray}
The stationary value of the $k$th moment is given by
\begin{eqnarray}
\langle \epsilon^k \rangle
&=& \frac{(k-1) \beta^2}{2(\tau - k \alpha^2/2)}
\langle \epsilon^{(k-2)} \rangle, \\
&=& \frac{(k-1)!!\:\beta^k}
{2^{k/2} \: \Pi_{\ell=1}^{k/2}(\tau - \ell \alpha^2)}.
\end{eqnarray}
For example, second- and fourth-moments are given by
\begin{eqnarray}
\langle \epsilon^2 \rangle
&=& \frac{\beta^2}{2(\tau-\alpha^2)}, \\
\langle \epsilon^4 \rangle
&=& \frac{3 \beta^4}{4(\tau-2\alpha^2)(\tau-\alpha^2)},
\hspace{1cm}\mbox{(model B)}
\end{eqnarray}
which agree with the result obtained from
the stationary distribution given by Eq. (66) or (67).
We get the positive definite
$\langle \epsilon^{k} \rangle$ for $\alpha^2 < 2 \tau/k$.
This suggests that for $2 \tau/k < \alpha^2 < \tau $
with $k \geq 4$, the $k$th moment diverges even if
the second moment remains finite.
This might throw some doubt on the validity of the
second-moment approach.
Equation (106) expresses
that the motion of $\langle \epsilon^{k} \rangle$
depends on those of its lower moments ($\leq k-2$),
but it is independent of its higher moments ($\geq k+2$).
For example, even if $\langle \epsilon^{4} \rangle$
diverges, it has no effects on the motion of
$\langle \epsilon^{2} \rangle$ for $\tau/2 < \alpha^2 < \tau$.
It is promising to take into account
contributions from higher-order moments in the model B,
although its validity range becomes narrower
because $\alpha$ has to satisfy the condition:
$\alpha^2 < 2 \tau/k $ for the $k$th moment
to remain finite.
Our discussions presented in the preceding sections
are confined to the stationary properties of the Langevin model
subjected to non-Gaussian noise. It is possible
to discuss its dynamics,
by solving equations of motion for $\mu$ and $\gamma$.
Numerical calculations for the model B are plotted in
Figure 6(a) and 6(b), which show the time dependences of
$\mu$ and $\gamma$, respectively.
We apply an external pulse
input given by $I(t)=A \:\Theta(t-100)\Theta(200-t)$
with $A=1.0$, which is plotted at the bottom
of Fig. 6(a), $\Theta(x)$ denoting the Heaviside function.
Figure 6(a) shows that $\mu(t)$ of the moment method
is in good agreement with the result of DS.
Figure 6(b) shows that $\gamma(t)$ is independent of
an input pulse [Eq. (A17)].
With increasing $\alpha$, a steady value of $\gamma$
is increased. The result of the moment method for $\alpha=0.0$
is in good agreement with that of DS although
for $\alpha=0.5$, the former is underestimated
compared to the latter.
The overall behavior of the stationary distribution
is fairly well reproduced by all the approximations mentioned in Sec. 2.2.
Tails of $P(x)$ are, however, not satisfactorily described,
in particular, in calculations of the model A.
This is partly due to the fact that the approximate
Eq. (10) yields the Gaussian stationary distribution
given by Eq. (92),
though Eqs. (2) and (3) are originally introduced to
generate non-Gaussian noise.
This point is improved in the model B, in which
the stationary distribution given by Eq. (64) is non-Gaussian
as expressed by Eqs. (66) and (67).
Indeed, tails of $P(x)$ of Fig. 5 for the model B
are slightly well reproduced than those of Fig. 3 for the model A.
\section{Conclusion}
To summarize,
we have studied effects of non-Gaussian noise on the
Langevin model, by using the second-order moment approach.
The obtained result is summarized as follows.
\noindent
(1) With increasing $\tau$, the width of the
stationary distribution $P(x)$ is decreased.
\noindent
(2) For $q>1$ ($q < 1$), the width of $P(x)$ is
increased (decreased) compared to that for $q=1$.
\noindent
(3) The prefactor of $F_{eff}$ for the model A
in the moment method agrees with that in FI-1,
but disagrees with that in the UCNA and FI-2 (Table 1).
\noindent
The items (1) and (2) are realized in both the
models A and B. This may be explained by the $q$-
and $\tau$-dependent $\alpha_{eff}$ [Eq. (38)]
or $\beta_{eff}$ [Eq. (78)].
As for the item (3), it is necessary to point out that
although the UCNA \cite{Jung87,Hanggi95} exactly interpolates
between the two limits of $\tau=0$ and $\tau=\infty$,
it is not exact for $O(\tau)$ \cite{Mallick06}.
The functional integral method is a formally exact transformation
if the functional integral is correctly performed.
In the actual applications, however, it is inevitable to adopt
some kinds of approximation, with which the final result
depends on the adopted approximation.
The difference between the results of
FI-1 \cite{Wu07} and FI-2 \cite{Fuentes01} arise
from the difference between the adopted approximations
in performing the functional integral.
These yield the difference in the results
listed in the Table 1.
As for the models A and B, we get
\noindent
(i) although the stationary distribution of $p(\epsilon)$
in the model A is the Gaussian,
the effect of the non-Gaussian distribution
of the original model A$_0$ is fairly well taken into
account by a factor of $r_q$, and
\noindent
(ii) the newly introduced model B, which yields
the stationary non-Gaussian
$p(\epsilon)$ equivalent to that of
the model A$_0$, is expected to be a promising model
generating non-Gaussian noise.
It is possible to apply the moment approach to
a wide class of stochastic systems subjected to non-Gaussian noise,
because its calculation is simple and transparent.
It would be interesting to investigate effects of non-Gaussian noise
on the synchronization in coupled nonlinear
systems with the use of the model B,
which is left as our future study.
\section*{Acknowledgements}
This work is partly supported by
a Grant-in-Aid for Scientific Research from the Japanese
Ministry of Education, Culture, Sports, Science and Technology.
\vspace{1cm}
\newpage
|
3,212,635,537,980 | arxiv | \section{Introduction}
In recent years, there has been a renewed interest in theories of
gravity in higher dimensions. The motivation arises from string
theory. As a possibility, the Einstein-Gauss-Bonnet (EGB) gravity
theory is selected by the low energy limit of the string
theory~\cite{cuerdas1,cuerdas2,cuerdas3}. In this theory appears
corrective terms to Einstein gravity which are quadratic in the
curvature of the space-time. For 4-dimensional gravity, these
terms result in a topological invariant so they will have no
consequences in the field equations of the theory unless a surface
term is involved. However, it was shown in ~\cite{Olea1,Olea2},
that such terms modify the conserved current of the theory in four
dimensions. Moreover, the effect of those Gauss Bonnet terms is
nontrivial for higher dimensions, so the theory of gravity, which
includes Gauss-Bonnet terms, is called Einstein-Gauss-Bonnet (EGB)
gravity. Studies in this area were made in ~\cite{ZW}.
Perhaps black holes are the most striking prediction of any theory
of gravity. This issue was also discussed in EGB gravity. Efforts
were addressed to the understanding of properties of isolated
black holes in equilibrium. In particular, for spherically
symmetric space-time, solutions describing static charged black
holes for both Maxwell and Born-Infeld electrodynamics and other
fields were found in the
literature~\cite{BD,WH1,WH2,WSH1,Giribet}. Thermodynamic
properties of these solutions were also studied
in~\cite{WSH2,Ca1,Ca2}. On physical grounds, one would expect that
an interesting solution should be stable under non-spherically
symmetric perturbations of the state of the black hole represented
by this solution. This subject was investigated by Dotti and
Gleiser in~\cite{DG1,DG2,GD}. Furthermore, the causal structure of
these solutions has been studied in detail by Torii and Maeda in
~\cite{MaedaN,MaedaC}.
However, nature behaves in more complex way: black holes are dynamic
systems, which are seldom in equilibrium. Then, from a physical
point of view, the above cited static solutions should represent the
eventually steady state of dynamic evolution of black holes. This
kind of solutions -- dynamical black holes-- has a twofold value.
First, they allow us to model more realistic physical situations
associated with the black hole dynamic, such as processes of
formation and evaporation of black holes. Second, these solutions
can be used to evaluate the cosmic censorship
hypothesis~\cite{Penrose} which states that naked singularity are
forbidden in physical gravitational collapses.
In the framework of GR, the Vaidya metric~\cite{Vaidya} represents
a radial null fluid which can be used to model dynamic processes
associated with black holes. There exists many possible
generalizations of the Vaidya metric -- see~\cite{Ghosh, Mark} and
references therein--. At this point, it is important to note that
some dynamic solutions describing the collapse of certain kinds of
matter seems to evolve to the formation of naked singularities.
Unfortunately, no much information is available about the
stability of these solutions.
Recently, Dadwood and Ghosh~\cite{Ghosh} have proved a theorem which
is readily seen to generate dynamic solutions of black holes in GR,
by imposing some conditions on energy-momentum tensor. This theorem
is a generalization of a previous one obtained by Salgado in
~\cite{Salgado} and that was generalized for $n$-dimension by Gallo
~\cite{Gallo}.
Unfortunately, in EGB gravity a few solutions representing dynamic
black holes are known. Effectively, a recent Vaidya-type solutions
in the context of EGB gravity were independently obtained by
Kobayashi and Maeda in~\cite{Kobayashi,MaedaN2}. The possibility
that black hole space-times evolve to the formation of naked
singularity were mentioned in ~\cite{MaedaC,MaedaN2}. On the other
hand, Grain, Barrau and Kanti have recently calculated the greybody
factors associated to Einstein-Gauss-Bonnet black holes, which are
needed to describe the evaporation spectrum by Hawking
radiation~\cite{Grain}. Konoplya studied in ~\cite{Kona1} the
quasinormal modes of charged EGB black holes, and in ~\cite{Kona2},
he analyzed the evolution of a scalar field in EGB.
The aim of this work is to prove two similar theorems that were
presented in~\cite{Ghosh} but in the framework of EGB gravity. The
first one is an extension of that theorem for EGB gravity; the
second one is deduced from the first one but relaxing some
conditions on energy-momentum tensor, in order to include more
realistic physical situations. As a consequence of these theorems,
we find the analogous solutions, in EGB gravity, of many well
known solutions for GR (Vaidya, Bonnor-Vaidya, Husain, Global
Monopole). Also, we obtain the Vaidya-type generalization for
static black holes in EGB gravity, such as Born-Infeld solutions.
In section II, we review the EGB gravity and recalled some of the
well known static spherically symmetric black hole solutions. In
section III, we prove the former theorem and we show some
solutions resulting from this theorem. In section IV, we analyzed
the imposition of the energy conditions on these metrics. Finally,
in section V we present a more general version of the latter
theorem and we apply it to obtain the metric of Born-Infeld
dynamic black hole. In the conclusions we discuss possible future
works in this area.
\section{The Einstein-Gauss-Bonnet gravity}
The action which describe Einstein-Gauss-Bonnet gravity coupled
with matter fields reads,
\begin{equation*}
S = \frac {1}{16\pi } \int d^nx \sqrt {-g} \left[ R - 2\Lambda +
\alpha (R_{a b c d } R^{a b c d } + \right.
\end{equation*}
\begin{equation*}
\left. + R^2 - 4R_{a b } R^{a b} )\right]+S_{\text{matt}},
\end{equation*}
where $S_{\text{matt}}$ is the action associated to the matter
fields, and $\alpha $ is the Gauss-Bonnet coupling constant
associated in the string models, with the tension of these
strings. This constant introduce a length scale. In fact, the
correction that these theory produce to GR, are noted in short
distance, given by the scale $l=\sqrt{4 \alpha}$.
The equations of motion resulting from $\delta S=0$ are
\begin{eqnarray*}
8\pi T_{a b } &=& \mathcal{G}_{a b}=G^{(0)}_{a b}+G^{(1)}_{a
b}+G^{(2)}_{a b},
\end{eqnarray*}
where $T_{a b}$ is the energy-momentum tensor, representing the
matter-field distribution resulting from the variation $\delta
S_{\text{matt}}/\delta g^{a b},$ and
\begin{eqnarray*}
G^{(0)}_{a
b}&=&\Lambda g_{a b} \\
G^{(1)}_{a
b}&=& R_{a b }-\frac{1}{2}Rg_{a b}\\
G^{(2)}_{a b}&=&-\alpha \left[\frac {1}{2} g_{a b} (R_{ c j e
k}R^{c
j e k }-4R_{c j }R^{c j }+R^2) \right. - \\
&-&\left. 2RR_{a b}+4R_{a c}R^{c}_{b}+4R_{c j} R^{c j}_{ \ \ a
b}-2R_{a c j e}R_{b}^{ \ c j e} \right ].
\end{eqnarray*}
The static, spherically symmetric metric which satisfies these
vacuum equations of fields for n-dimensional was obtained by
Boulware and Deser~\cite{BD} and in standard spherical coordinates
$(t,r,\theta_1 ,\cdots,\theta_{n-2})$ reads
\begin{equation}
ds^2=-f(r,t)dt^2+f^{-1}(r,t)dr^2
+r^2d\Omega^2_{n-2} \; ,
\end{equation}
where $f(r,t)$ is given by
\begin{equation}
f_{\pm}(r,v)= 1+\frac{r^2}{2\widehat{\alpha} }\left \{1\pm
\sqrt{1+\frac{8\widehat{\alpha}}{n-2}\left
[\frac{\Lambda}{n-1}+\frac{2M}{r^{n-1}}\right ]}\right \},\\
\end{equation}
with $\widehat{\alpha}=\alpha (n-3)(n-4)$ and $d\Omega^2_{n-2}$
being the metric of the $(n-2)$-sphere
\begin{equation}
d\Omega^2_{n-2} = d\theta^2_1 + \sum^{n-2}_{i=2}\prod^{i-1}_{j=1}
\sin^{2}\theta_j\;d\theta^2_i \; ,
\end{equation}
It is straightforward to prove that only the minus-branch
solution, $f_{-}(r,v)$, has limit for $\alpha\rightarrow 0$,
namely n-dimensional GR.
In the next section we will find that this metric is a particular
case of a large family of metrics. Although it is not crucial for
the proof of the theorem, we adopt $\alpha$ positive since this
condition arises from the strings theory. Hereafter we will assume
that both $\Lambda$ and $\alpha$ are fixed quantities.
\section{Radiating black holes in Einstein-Gauss-Bonnet gravity}
\textbf{Theorem 1:} \textit{Let $(\mathcal{M},g_{ab})$ a
n-dimensional space-time such that: i) it satisfies the EGB
equations, ii) it is spherically symmetric, iii) In the
Eddington-Bondi coordinates, where the metric reads
$\,ds^2=-h^2(r,v)f(r,v)dv^2+2\epsilon h(r,v)dvdr+r^2d\Omega^2_{n-2}$,
the energy-momentum tensor $T^a_b$ satisfies the conditions
$T^v_r=0$, and $T^{\theta_1}_{\theta_1}=kT^r_r$,
$(k=\text{const}\in \mathbb{R})$ iv) If $\alpha\rightarrow 0$, the
solution converges to the General Relativity limit. Then the
metric of the space-time is given by
\begin{eqnarray} ds^2=-f(r,v)dv^2+2\epsilon
dvdr+r^2d\Omega^2_{n-2},\,~~~~~~(\epsilon=\pm1),\label{metrica}\end{eqnarray}
where}
\begin{widetext}
\begin{equation}
f(r,v)=\left\{
\begin{array}{rll}
1+\frac{r^2}{2\widehat{\alpha}}\left \{1-
\sqrt{1+\frac{8\widehat{\alpha}}{n-2}\left
[\frac{\Lambda}{n-1}+\frac{2M(v)}{r^{n-1}}-\frac{8\pi
C(v)}{[k(n-2)+1]r^{(1-k)(n-2)}}\right ]}\right \}\;&\text{if}&\;\; k\neq-\frac{1}{n-2},\\
1+\frac{r^2}{2\widehat{\alpha}}\left \{1-
\sqrt{1+\frac{8\widehat{\alpha}}{n-2}\left
[\frac{\Lambda}{n-1}+\frac{2M(v)}{r^{n-1}}-\frac{8\pi
C(v)\ln(r)}{r^{n-1}}\right ]}\right \} \;\;\;\;~\;~~~\;\;\;&\text{if}&\;\; k=-\frac{1}{n-2}.\\
\end{array}\\\right.\label{lametrica}
\end{equation}
\end{widetext}
with the diagonal components of $T^a_b$ given by \begin{equation}
T^a_{b\text{(Diag)}}=\frac{C(v)}{r^{(n-2)(1-k)}}\text{diag}[1,1,k,\cdots,k],\label{Trr}
\end{equation}
and an unique non-vanishing off-diagonal element
\begin{equation} T^r_v=\left\{
\begin{array}{lll}
\frac{1}{4\pi
r^{n-2}}\frac{dM}{dv}-\frac{r^{(k-1)(n-2)+1}}{k(n-2)+1}\frac{dC}{dv}\;\;
&\text{if}&\;\; k\neq -\frac{1}{n-2}\\
\frac{1}{4\pi
r^{n-2}}\frac{dM}{dv}-\frac{\ln(r)}{r^{n-2}}\frac{dC}{dv} \;\;
&\text{if}&\;\; k= -\frac{1}{n-2}\\
\end{array}\\\right.
\end{equation}
with $M(v)\;,\;C(v)$ two arbitraries functions depending of the
distribution of the energy-matter.$\blacksquare$\\
\textbf{Proof:} By the hypothesis iii) of the theorem $1$, the
metric under discussion reads
\begin{equation}
ds^2=-h^2(r,v)f(r,v)dv^2+2\epsilon h(r,v)dvdr+r^2d\Omega^2_n,
\end{equation}
which due to hypothesis ii) must satisfy the EGB equations. The
only nontrivial components of the EGB tensor are
\begin{widetext}
\begin{eqnarray}
\mathcal{G}^v_r&=& (n-2) \left [r^2+2( 1-f)\widehat{\alpha}
\right ]\frac{h_r}{{r}^{3}
{\epsilon} h^{2}},\label{Gvr}\\
\nonumber\\
\mathcal{G}^r_v&=&-(n-2)\left [ r^2+2(1-f)
\widehat{\alpha} \right ]\frac {f_v}{2{ r}^{3}},\label{Grv}\\
\nonumber\\
\mathcal{G}^r_r&=&\Lambda+(n-2)\left [ r^2+2( 1-f)
\widehat{\alpha}
\right ]\frac{f_r}{2r^3}
-(n-2)\left [ (n-3)r^2+( n-5)(1-f)\widehat{\alpha}
\right ]\frac{1-f}{2r^4}\nonumber\\
&&+(n-2)\left [ r^2+4(1-f) \widehat{\alpha}
\right ]f\frac{h_r}{r^3h},\label{Grr}\\\nonumber\\
\mathcal{G}^v_v&=& \Lambda+(n-2)\left [ r^2+2( 1-f)
\widehat{\alpha}
\right ]\frac{f_r}{2r^3}-(n-2)\left [(n-3) r^2+( n-5)(1-f)\widehat{\alpha}
\right ]\frac{1-f}{2r^4},\label{Gvv}\\
\nonumber\\
\mathcal{G}^{\theta_i}_{\theta_i}&=&
\Lambda-\frac{\widehat{\alpha}}{r^4}\left
[r^2f^2_r-2(n-6)(n-5)(1-f^2)\right ]
-(n-3)(n-4)\frac{1-f}{2r^2}\nonumber\\
&&+\left [r^2+2\widehat{\alpha}(1-f)r^2\right ]\left [(fh^2h_{rr}+
hh_{rv}-h^3 f_{rr})-\epsilon h_rh_v\right ]\frac{1}{r^2h^3}\nonumber\\
&&+\left [(n-3)r^2+2(n-5)\widehat{\alpha}(1-f)\right
]\frac{f_r}{r^3}\nonumber\\
&&+\left \{6r^3h+4\widehat{\alpha}\left [
(3-5f)rh+4(n-5)(1-f)hf-4\epsilon rf_v\right
]+(n-3)\frac{f}{r}\right
\}\frac{h_r}{4r^3h^2}.\label{Gtt}\end{eqnarray}
\end{widetext}
By using hypothesis $iii)$, $T^v_r=0$, in Eq.(\ref{Gvr}), we
deduce that: either $h(r,v)$ is independent of $r$, i.e.,
$$h(v,r)=h(v),$$ or $f(v,r)$ is independent of $v$, and
satisfies \begin{equation} r^2+2(
1-f)\widehat{\alpha}=0.\label{fotrocaso}\end{equation}
However, this last equation leads to non-radiating metrics (
because Eq.(\ref{Grv}) implies $T^r_v=0$), which are isometric to
AdS/dS Gauss-Bonnet solutions, and naturally contained in the
metrics Eq.(\ref{lametrica}) as particular cases with
$M(v)=C(v)=0$. We show this in the Appendix A. Then, we conclude
that $h(r,v)=h(v)$, and redefining to the null coordinate
$\widehat{v}=\int h(v)dv,$ we can take $h(v)=1$, without loss of
generality.
Now, from Eq.(\ref{Grr}) and Eq.(\ref{Gvv}), we obtain that
$G^v_v=G^r_r$, and then $$T^v_v=T^r_r.$$
If we impose the conservation laws, $\nabla_a T^a_b=0,$ and using
again the hypothesis $iii)$, $T^{\theta_i}_{\theta_i}=kT^r_r$, we
have that
\begin{eqnarray}
0&=&\frac{\partial }{\partial v}{ T^v_v} +\frac {\partial
}{\partial r} T^r_v +\frac{1}{2\epsilon}\left (T^r_r- T^v_v\right
){\frac
{\partial }{\partial r}f}+ \frac {( n-2)}{r}T^r_v,\nonumber\\
&&\label{Bt11}\\
0&=&\frac {\partial }{\partial r}T^r_r +\frac {( n-2)(1-k)}
{r}T^r_r. \label{Bt22}
\end{eqnarray}
Solving Eq.(\ref{Bt22}) for $T^r_r$, we obtain:
\begin{equation}
T^r_r=\frac{C(v)}{r^{(n-2)(1-k)}},\label{Trr}
\end{equation}
where $C(v)$ is an arbitrary function.
By joining these results, we can write the diagonal elements of
$T^a_b$ as
$$ T^a_{b\text{(Diag)}}=\frac{C(v)}{r^{(n-2)(1-k)}}\text{diag}[1,1,k,\cdots,k].$$
Now, we can find $f(r,v)$ by replacing Eq.(\ref{Grr}) and
Eq.(\ref{Trr}) in the suitable component of the EGB equations
$$\mathcal{G}^r_r=8\pi T^r_r,$$ resulting after solving this differential equation and making some
algebraic simplifications
\begin{widetext}
\begin{equation}
f_{\pm}(r,v)=\left\{
\begin{array}{rll}
1+\frac{r^2}{2\widehat{\alpha}}\left \{1\pm
\sqrt{1+\frac{8\widehat{\alpha}}{n-2}\left
[\frac{\Lambda}{n-1}+\frac{2M(v)}{r^{n-1}}-\frac{8\pi
C(v)}{[k(n-2)+1]r^{(1-k)(n-2)}}\right ]}\right \}\;&\text{if}&\;\; k\neq-\frac{1}{n-2},\\
1+\frac{r^2}{2\widehat{\alpha}}\left \{1\pm
\sqrt{1+\frac{8\widehat{\alpha}}{n-2}\left
[\frac{\Lambda}{n-1}+\frac{2M(v)}{r^{n-1}}-\frac{8\pi
C(v)\ln(r)}{r^{n-1}}\right ]}\right \} \;\;\;\;~\;~~~\;\;\;&\text{if}&\;\; k=-\frac{1}{n-2}.\\
\end{array}\\\right.
\end{equation}
\end{widetext}
where $M(v)$ is another arbitrary function (which can be shown to
be proportional to the mass of the underlying matter ).
At this point, it is important to note that we have obtained two
branches of solutions, namely, $f_+$ and $f_-$, which correspond to $\pm$ signs
in front of the square root term. However, the positive branch, $f_+$,
does not converge to GR.
Then, if we impose the hypothesis $iv)$, then it can be shown that
the only possible solutions are the following:
\begin{widetext}
\begin{equation}
f(r,v)=\left\{
\begin{array}{lll}
1+\frac{r^2}{2\widehat{\alpha}}\left \{1-
\sqrt{1+\frac{8\widehat{\alpha}}{n-2}\left
[\frac{\Lambda}{n-1}+\frac{2M(v)}{r^{n-1}}-\frac{8\pi
C(v)}{[k(n-2)+1]r^{(1-k)(n-2)}}\right ]}\right \}\;&\text{if}&\;\; k\neq-\frac{1}{n-2},\\
1+\frac{r^2}{2\widehat{\alpha}}\left \{1-
\sqrt{1+\frac{8\widehat{\alpha}}{n-2}\left
[\frac{\Lambda}{n-1}+\frac{2M(v)}{r^{n-1}}-\frac{8\pi
C(v)\ln(r)}{r^{n-1}}\right ]}\right \} \;\;\;\;~\;~~~\;\;\;&\text{if}&\;\; k=-\frac{1}{n-2}.\\
\end{array}\\\right.
\end{equation}
\end{widetext}
In this case, the limit $\alpha\rightarrow 0$ reduces $f(r,v)$ to
\begin{widetext}
\begin{equation}
f(r,v)=\left\{
\begin{array}{lll}
1-\frac{4M(v)}{(n-2)r^{n-3}}-\frac{2\Lambda
r^2}{(n-1)(n-2)}+\frac{16\pi
C(v)}{(n-2)[k(n-2)+1]r^{(1-k)(n-2)-2}}\;&\text{if}&\;\; k\neq-\frac{1}{n-2},\\\\
1-\frac{4M(v)}{(n-2)r^{n-3}}-\frac{2\Lambda
r^2}{(n-1)(n-2)}+\frac{16\pi C(v) ln(r)}{(n-2)r^{n-3}}\;\;\;\;~\;~~~\;\;\;&\text{if}&\;\; k=-\frac{1}{n-2}.\\\\
\end{array}\\\right.
\end{equation}
\end{widetext}
These are the expressions that we would have found if we had
started with the $n$-dimensional Einstein version of the theorem
(General Relativity with a cosmological constant) instead of the
Einstein-Gauss-Bonnet gravity. In fact, if $n=4$, these
expressions are the same that those found in~\cite{Ghosh}.
Finally, from Eq.(\ref{Grv}) and the appropriated EGB equation,
$$\mathcal{G}^r_v=8\pi T^r_v,$$
we obtain that the only non-vanishing off-diagonal el\-ement of
$T^a_b,$ reads
\begin{equation}
T^r_v=\left\{
\begin{array}{lll}
\frac{1}{4\pi
r^{n-2}}\frac{dM}{dv}-\frac{r^{(k-1)(n-2)+1}}{k(n-2)+1}\frac{dC}{dv}\;\;
&\text{if}&\;\; k\neq -\frac{1}{n-2}\\
\frac{1}{4\pi
r^{n-2}}\frac{dM}{dv}-\frac{\ln(r)}{r^{n-2}}\frac{dC}{dv} \;\;
&\text{if}&\;\; k= -\frac{1}{n-2}\\
\end{array}\\\right.
\end{equation}
Q.E.D.\\\\
This family of metrics contains many potentially interesting
solutions. Some of these space-times are shown in the Table 1.
These solutions could be very useful to study the collapse of
different matter fields, or the formation of naked singularities.
On the other hand, we also think that they can be used to analyzed
semiclassical approaches to the evaporation of black holes.
However, the imposed conditions on the energy-momentum tensor are
rather restrictive so other important solutions are not allowed.
In the section V we generalize the theorem $1$ in order to get new
exact solutions.
\begin{table*}
\caption{\label{tab:table3}Some space-times satisfying the
conditions of the Theorem $1$ and $2$, obtained with particular
values of $k$, $C(v)$ and $M(v).$ }
\begin{ruledtabular}
\begin{tabular}{llll}
$T^a_b$&Space-Time&M(v) and C(v) &k-index\\ \hline\\
$T^a_{b\text{(Diag)}}=0\;,\;T^r_v=\frac{1}{4\pi r^{n-2}}\frac{dM}{dv}$&Kobayashi-Maeda&$C(v)=0$
&\\\\\hline\\
$T^a_{b\text{(Diag)}}=-\frac{Q^2(v)}{4\pi r^{n-2}}\text{diag}[1,1,-1,\cdots,-1]$&Bonnor-Vaidya-EGB
&$C(v)=-\frac{Q^2(v)}{8\pi}$&$k=-1$\\\\ $T^r_v=\frac{1}{4\pi
r^{2n-5}}\left[r^{n-3}\frac{dM}{dv}-\frac{Q}{(n-3)}\frac{dQ}{dv}\right]$&&&
\\\\\hline\\
$T^a_{b\text{(Diag)}}=-\frac{g^2(v)}{4\pi r^{(n-2)(m+1)}}\text{diag}[1,1,-m,\cdots,-m]$&Husain-EGB&$C(v)=
-\frac{g^2(v)}{4\pi}$&$k=-m$\\\\ $ T^r_v=\frac{1}{4\pi}
\left[r^{2-n}\frac{dM}{dv}-r^{(m+1)(2-n)+1}\frac{2g}{1-m(n-2)}\frac{dg}{dv}\right]$&&&
\\\\\hline\\
$ T^v_v=T^r_r=-\frac{a}{8\pi r^{n-2}}$& Global monopole-EGB
&$M(v)=0, \;\;C(v)=-\frac{a}{4\pi}$&$k=0$\\\\\hline\\
$T^a_b=0$&Boulware-Deser-Wheeler
&$M(v)=M_0$, $C(v)=0$&
\\\\\hline\\
$T^r_r=T^v_v=\frac{Q^2}{4\pi
L^2r^{n-2}}\left(r^{n-2}-\sqrt{r^{2n-4}+L^2}\right)$&BI-Vaidya-EGB&$C(v)=-\frac{Q^2(v)}{4\pi
|L|}$& $k(r,v)=-\frac{r^{n-2}}{\sqrt{r^{2(n-2)}+L^2}}$\\\\
$T^{\theta_i}_{\theta_i}=k(r,v)T^r_r$&&&
\\\\
$T^r_v=\frac{1}{4\pi r^{n-2}}\left[\frac{dM}{dv}-br\left(\frac{dQ}{dv}F+Q\frac{\partial F}{\partial v}\right)\right]$&&&\\
\end{tabular}
\end{ruledtabular}
\end{table*}
\section{Energy conditions}
In this section we discuss the restrictions on the energy-momentum
tensor based in the dominant energy conditions. These will impose
restrictions on $k$, $M(v)$ and $C(v)$, defined in the last
section. The case of weak energy conditions is totally analogous
to that discussed in~\cite{Ghosh} for General Relativity theory.
The covariant components of the energy-momentum tensor can be
written with the help of two future null vectors, $l_a$, $n_a$
(the vector $l^a$ is tangent to the null surface generated by $v$,
and $n^a$ is an independent null vector such as $l_an^a=-1$; in
Eddington-Bondi coordinates $\{v,r,\theta_i\}$, the components of
these vectors are $l_a=(1,0,\cdots,0)$, $n_a=(f/2,-1,0,\cdots,0)$)
as follows,
$$T_{ab}=\epsilon \mu
l_al_b-P_r(l_an_b+l_bn_a)+P_{\theta}(g_{ab}+l_an_b+l_bn_a),$$
where
\begin{eqnarray}
\mu&=&T^r_v\\
P_r&=&-T^r_r=-C(v)r^{(2-n)(1-k)}\\
P_{\theta}&=&kP_r.
\end{eqnarray}
Physically $\mu$ represents the radiating energy along the null
direction given by $l^a$; $P_r$ denotes the radial pressure
generated for the charges of the fluids and $P_{\theta}$ are the
transversal pressures. All these quantities are measured by an
observer moving along a time-like direction $u^a$ given by
$$u^a=\frac{1}{\sqrt{2}}(l^a+n^a).$$ This observer will measure an
energy density given by
$$\rho=-P_r=-T_{ab}u^au^b.$$ Note that the energy-momentum tensor
corresponds to a null fluid Type II.
As it is well known, the dominant energy conditions implies that
for all timelike vector $t^a$, $T_{ab}t^at^b\geq 0$ and also
$T_{ab}t^b$ is non-spacelike vector. Then, we must require:
$$\mu\geq 0\;~~~~~~\; \text{and} \;~~~~~~~\rho\geq P_{\theta}\geq 0.$$
The first condition, in the case of a radiating fluid $\mu>0$, is
equivalent to
\begin{equation}
\frac{1}{4\pi
r^{n-2}}\frac{dM}{dv}-\frac{r^{(k-1)(n-2)+1}}{k(n-2)+1}\frac{dC}{dv}>
0\;\; \text{if}\;\; k\neq
-\frac{1}{n-2},\label{conkdis}\end{equation} and
\begin{equation}
\frac{1}{4\pi
r^{n-2}}\frac{dM}{dv}-\frac{\ln(r)}{r^{n-2}}\frac{dC}{dv}> 0
\;\;\text{if}\;\; k= -\frac{1}{n-2}.\label{conkigual}
\end{equation}
The Eq.(\ref{conkdis}) is satisfied if $\frac{dM}{dv}>0$, and
either $\frac{dC}{dv}>0$ with $k< -\frac{1}{n-2},$ or
$\frac{dC}{dv}<0$ with $k> -\frac{1}{n-2}.$
On the other hand, the Eq.(\ref{conkigual}) is satisfied if
$\frac{dM}{dv}>4\pi \ln(r)\frac{dC}{dv}.$
Finally, the conditions $\rho\geq P_{\theta}\geq 0$ are satisfied
only if $C(v)\leq 0$ and $-1\leq k \leq 0$.
\section{A more general version of the Theorem}
In this section, we present a slightly different version of the
theorem, which allows more general distributions of matter fields.
\\\\
\textbf{Theorem 2:}\textit{ Let $(\mathcal{M},g_{ab})$ a
n-dimensional space-time such that: i) it satisfies the EGB
equations, ii) it is spherically symmetric, iii) In the
Eddington-Bondi coordinates, where the metric reads
$\,ds^2=-h^2(r,v)f(r,v)dv^2+2\epsilon h(r,v)dvdr+r^2d\Omega^2_n$,
the energy-momentum tensor $T^a_b$ satisfies the conditions
$T^v_r=0$, and $T^{\theta_1}_{\theta_1}=k(r,v)T^r_r$, with
$k(r,v)$ a real function, iv) If $\alpha\rightarrow 0$, then the
solution converges to the General Relativity limit. Then the
metric of the space-time is given by
$$ds^2=-f(r,v)dv^2+2\epsilon
dvdr+r^2d\Omega^2_n,\,~~~~~~(\epsilon=\pm1),$$ where
\begin{widetext}
\begin{equation}
f(r,v)=
1+\frac{r^2}{2\widehat{\alpha}}\left \{1-
\sqrt{1+\frac{8\widehat{\alpha}}{n-2}\left
[\frac{\Lambda}{n-1}+\frac{2M(v)}{r^{n-1}}-\frac{8\pi
C(v)}{r^{n-1}}I(r,v)\right ]}\right \}
\end{equation}
\end{widetext}
with
\begin{equation}
I(r,v)=\int^r_0{e^{(n-2)\int^s{k(t,v)t^{-1}}dt}ds},
\end{equation}
and the diagonal components of $T^a_b$ given by
\begin{equation}
T^a_{b\text{(Diag)}}=\frac{C(v)e^{(n-2)\int{k(r,v)r^{-1}}dr}}{r^{(n-2)}}\text{\textrm{diag}}[1,1,k,\cdots,k],
\end{equation}
with an unique non-vanishing off-diagonal element
\begin{equation}
T^r_v=\frac{1}{
r^{n-2}}\left[\frac{1}{4\pi}\frac{dM}{dv}-\frac{dC}{dv}I(r,v)-C\frac{\partial
I}{\partial v}\right ]
\end{equation}
where $M(v)\;,\;C(v)$ are two arbitraries functions depending of
the distribution of the energy-matter.}$\blacksquare$\\
The proof is completely analogous to that of the previous theorem.
Note that if $k=\text{const},$ we recover the results of that
Theorem. In the next example, we apply this theorem in order to
obtain the metric of a null charged fluid in Born-Infeld nonlinear
electrodynamics.
\section{An example: The radiating Born-Infeld black
hole} Since Thompson discovered the electron, the physicists were
very worried about the divergence of the self-energy of point-like
charges in the Maxwell electrodynamics. There were a lot of
proposals to solve this problem, and one of the most prominent
theories compatible with the spirit of Lorentz invariance was
created by Born and Infeld between 1932 and 1934~\cite{BI1}. They
proposed a nonlinear and Lorentz invariant Lagrangian, which depends
of one parameter $b$, such as, if $b\rightarrow\infty$, then its
Lagrangian tends to the Maxwell Lagrangian. The electromagnetic
field $F_{ab}$ can be obtained from a 1-form $A_a$ in a similar way
to Maxwell theory, $F=dA$. In the last years, the interest in this
theory was renewed due to the fact that it appears as a low energy
limit of string theory~\cite{cuerdas1,cuerdas2}.
The free Lorentz invariant Lagrangian suggested by Born-Infeld
reads
$$\frak{L}=\frac{b^2\sqrt{-g}}{4\pi}\left (1-\sqrt{1+\frac{F^{ab}F_{ab}}{2b^2}}\right
),$$ where $g$ is the determinant of the metric $g_{ab}$. If we
add an interaction term in the form of $J^aA_a$ to the Born-Infeld
Lagrangian, then the equations of motions that follows from
$\frak{L}$ are
\begin{equation}
\frac{1}{\sqrt{-g}}\frac{\partial}{\partial{x^a}}\left
(\frac{\sqrt{-g}F^{ab}}{\sqrt{1+\frac{F^{ab}F_{ab}}{2b^2}}}\right
)=4\pi J^b.
\end{equation}
In addition to the \textit{Bianchi identities}
$$F_{ab,c}+F_{ca,b}+F_{bc,a}=0.$$
Now, we are interested in finding a spherically symmetric
Vaidya-like solution with a charged null fluid in Born-Infeld
electrodynamics.
Let us consider a spherically symmetric and non-stationary
electromagnetic field of the form
$$F=E(r,v)dv\wedge dr.$$ In this case, the contravariant
components of $F_{ab}$ in the metric $g_{ab}$ (
Eq.(\ref{metrica})) are
\begin{eqnarray}
F^{rv}&=&-\epsilon F_{rv}=\epsilon E(r,v)\\
F^{vr}&=&\epsilon F_{rv}=-\epsilon E(r,v)
\end{eqnarray}
and the determinant of the metric reads
\begin{equation}
g=-r^{2(n-2)}\prod^{n-2}_{i=3}\left[\sin(\theta_{i})\right]^{n-i}.
\end{equation}
Note, because we are considering a null fluid, there will be a
null current given by
\begin{equation}
J^a=J(r,v)\delta^a_v.
\end{equation}
Having all these expressions into account, the equations of
motions for $F_{ab}$ are,
\begin{eqnarray}
\frac{1}{\sqrt{-g}}\frac{\partial}{\partial{r}}\left
(\frac{\sqrt{-g}E(r,v)}{\sqrt{1-\frac{E^2(r,v)}{b^2}}}\right
)&=&0,\\
\frac{1}{\sqrt{-g}}\frac{\partial}{\partial{v}}\left
(\frac{\sqrt{-g}E(r,v)}{\sqrt{1-\frac{E^2(r,v)}{b^2}}}\right
)&=&4\pi J(r,v).
\end{eqnarray}
Solving this equations we find
\begin{equation}
E(r,v)=\frac{Q(v)}{\sqrt{r^{2(n-2)}+L^2(v)}}
\end{equation}
where
\begin{equation}
L(v)=\frac{Q(v)}{b},
\end{equation}
and $Q(v)$ an arbitrary function of $v$, representing the charge
of the fluid. The null current results, $$J^a=\frac{1}{4\pi
r^{n-2}}\frac{dQ}{dv}\delta^a_v.$$ Note that if
$b\rightarrow\infty$ then $E(v,r)$ tends to its Maxwell
expression. Note also that the 1-form $A_a$ from this problem has
the form:
$$A=\left[\int_0^r{\frac{dr}{\sqrt{{r^{2(n-2)}}+L^2}}}\right]Q(v)dv.$$
The energy-momentum tensor associated to the field $F_{ab}$ is,
$$T_{ab}=\frac{1}{4\pi}\left(\frac{b^2g_{ab}-F_{ad}F_b\,^d-\frac{1}{2}g_{ab}F_{cd}F^{cd}}{\sqrt{1+\frac{1}{2b^2}F_{cd}F^{cd}}}-b^2g_{ab}\right).$$
From this and from the form of the current, the non-radiative part
of the energy-momentum tensor of the charged null fluid results
\begin{eqnarray}
T^r_r&=&\frac{Q^2}{4\pi
L^2r^{n-2}}\left(r^{n-2}-\sqrt{r^{2(n-2)}+L^2}\right),\\\nonumber\\
T^v_v&=&T^r_r,\\
T^{\theta_i}_{\theta_i}&=&-\frac{r^{n-2}}{\sqrt{r^{2(n-2)}+L^2}}T^r_r,
\end{eqnarray}
and the other off-diagonal components vanishes (with the exception
to $T^r_v$).
Then, we can see that this null charged fluid satisfies the
conditions of the Theorem 2, with
$$k(r,v)=-\frac{r^{n-2}}{\sqrt{r^{2(n-2)}+L^2}},$$ and
$$C(v)=-\frac{Q^2}{4\pi |L|}=-\frac{|bQ(v)|}{4\pi},$$ because it can be easily computed that
$$e^{(n-2)\int{k(r,v)r^{-1}dr}}=-\frac{1}{|L|}\left(r^{n-2}-\sqrt{r^{2(n-2)}+L^2}\right).$$
If we replace these expressions of $k(r,v)$ and $C(v)$ into
$f(r,v)$, we obtain the metric for a Born-Infeld dynamic black
hole:
\begin{widetext}
\begin{equation}
f(r,v)=
1+\frac{r^2}{2\widehat{\alpha}}\left \{1-
\sqrt{1+\frac{8\widehat{\alpha}}{n-2}\left
\{\frac{\Lambda}{n-1}+\frac{2M(v)}{r^{n-1}}-\frac{2
b^2}{(n-1)}+\frac{2
b^2}{r^{n-1}}\int^r_0{\sqrt{r^{2(n-2)}+L^2}dr}\right ]}\right \}
\end{equation}
\end{widetext}
In the case of $Q(v)=Q_0$, $M(v)=M_0$, we recover the static
Born-Infeld solution, first obtained by Wiltshire~\cite{WSH1} and
reexamined in~\cite{Giribet}.
It can be shown that the integral factor
$$\mathcal{I}=\int^r_0{\sqrt{r^{2(n-2)}+L^2}dr},$$ can be written
in terms of a generalized hypergeometric function
$F([a_i],[b_j],z)$ (dependent of $p$ parameters $a_i$ and $q$
parameters $b_j$) defined by
$$F([a_i],[b_j],z)=\sum _{k=0}^{\infty}\prod _{i=1}^{p}\prod _{j=1
}^{q}{\frac {\Gamma \left( a_{{i}}+k
\right)\Gamma(b_j) }{\Gamma \left( a_{{i}} \right){\Gamma \left( b_{{j}}+k
\right) }}}\frac{{z}^{k}}{k!}.$$
The result is:
\begin{equation}\ \mathcal{I}= |L|\,r F\left( \left[-\frac{1}{2}, \frac{1}{2n-4}\right],\left[
\frac{2n-3}{2n-4}\right],-{\frac {r^{2(n-2)}}{L^2}}
\right).\label{hyp}\end{equation}
Finally, if we compute the $T^r_v$ component of the
energy-momentum tensor using the theorem $2$, we get
$$T^r_v=\frac{1}{4\pi r^{n-2}}\left[\frac{dM}{dv}-br\left(\frac{dQ}{dv}F+Q\frac{\partial F}{\partial v}\right)\right],$$
where $F$ denotes the hypergeometric function defined in
Eq.(\ref{hyp}).
\section{Conclusions}
In this work, we have proved two theorems, which allow us to
obtain exact solutions to the Einstein-Gauss-Bonnet equations.
This solutions represent dynamic black holes, and we thought that
these solutions are of interest, because they generalize some
known solutions of GR to the context to Einstein-Gauss-Bonnet
gravity which has been proved, is the first nontrivial term of low
energy limit of string theory.
It is important to note that with plus sign in front of the square
root, there are black hole solution even when the
$(n-2)$-dimensional submanifold is plane or hyperbolically
symmetric. The analogous theorem seems to be hold in these cases.
In a future work, we will make a detailed study of these metrics,
analyzing the causal structure and its thermodynamics.
Also, it should be very interesting to apply these metrics to
study the matter collapse, the naked singularities formations, and
semiclassical analysis of evaporation of black holes. These works,
are now in progress.
\begin{acknowledgments}
E.G. would like to thank CONICET for support. A. E. D. would like
to thank Instituto Universitario Aeron\'autico for support. We
also thank an anonymous referee for their interesting comments and
suggestions which helped us to improve this work.
\end{acknowledgments}
|
3,212,635,537,981 | arxiv | \section{The Basic Quantum Mechanics of Neutrino Oscillations}
\subsection {Coherence and the momentum uncertainty}
Coherence and interference in neutrino oscillations have been extensively
discussed and clarified
\cite{Dost,QM,NeutHJL,Kayser,GoldS,Pnonexp,MMNIETO,GrossLip,Leo,okun1,pnow98}
but there is still considerable confusion. The standard textbook neutrino wave
function, a coherent linear combination of states with different energies,
never exists in the real world. Elementary quantum mechanics and quantum
statistical mechanics tell us that components with different energies in an
initial state are never coherent\cite{Leo}while components with different momenta must
be coherent. The probability must vanish for finding a neutrino source
outside the tiny region of space where the source is known to exist.
Any wave packet or density matrix describing the source as a linear
combination of plane wave momentum eigenstates which exist over all space with
constant amplitudes must somehow conspire to produce this cancellation outside
the source. This coherence between states having the same energy and different
momenta produces coherence between neutrino states with the same energy and
different masses.
\subsection{Simple Quantum Mechanics and Super-Kamiokande}
Simple quantum mechanics alone, without the full apparatus of the standad
model, shows that the Super-Kamiokande results\cite{SuperK} require the
existence of two different mass eigenstates for neutrinos. The energy spectrum
of atmospheric neutrinos cannot change between their source at the top of the
atmosphere and their detection in a detector on earth if neutrinos are not
absorbed and do not decay en route and any interactions en route conserve
energy. The momentum spectrum for neutrinos of a given energy is a set of delta
functions, one for each neutrino mass value. If there is only one mass value,
the energy and momentum spectra will be identical for the upward and downward
going neutrinos incident on the detector and no difference between them can be
observed. The observation of such a difference\cite{SuperK} therefore
indicates that there are at least two different mass eigenstates, and that the
difference can arise from interference between the waves of states having
different masses and therefore different momenta if they have the same
energy.
\subsection {The Static Point Source Approximation}
If a neutrino is emitted from a point source which
is at a definite position in the laboratory for all time, the neutrino energy
can be determined precisely by measuring the energy of the source before and
after the neutrino emission. But the point source localization
introduces an infinite momentum uncertainty. In a realistic case, the source
still remains undisturbed
for a sufficiently long time on the relevant time scale, and its finite size
is still very much smaller than the wave lengths in space of any neutrino
oscillation and the distance between the source and the detector.
Thus the static point source provides a very good approximation for determining
which amplitudes are
coherent and which are incoherent.
Amplitudes describing neutrino states with the same energy and different
momenta are coherent and must be summed before squaring, while amplitudes
having different energy are incoherent and are squared before summation.
This is discussed quantitatively below.
\section{The Analog with Two-Slit and Bragg Scattering Experiments}
The wave-particle duality and quantum mechanics inherent in a neutrino-oscillation
experiment can be clarified by considering it as a typical ``which-path"
experiment\cite{ADY}. Just as in the two-slit electron diffraction experiment and in
coherent Bragg scattering of photons by a crystal, the neutrino oscillation
experiment describes the emission of a particle from a source and its detection by a
detector separated from the source by a macroscopic distance. There there is no
measurement of the precise path taken by the particle from the source to the
detector. The amplitude at the detector is the coherent sum of the amplitudes from
all allowed paths.
In the Bragg scattering experiment, the photon may be scattered by any one of
the atoms in the crystal, transfering momentum and energy, but which atom
scattered the photon is not known. In a neutrino oscillation experiment, the
neutrino carrying momentum and energy from the source to detector may be any
one of the allowed neutrino mass eigenstates, but which mass eigenstate carries
this momentum and energy is not known. Here the relevant paths are in
energy-momentum space, rather than configuration space. It is not simple
ignorance which conceals the information on the neutrino mass. Simple ignorance
of which path is taken by a particle does not introduce coherence between
amplitudes.
Coherence results only from an uncertainty required by quantum mechanics.
Both in Bragg scattering and neutrino oscillations there
would be no coherence if the energy and momenta of all relevant particles were
measured precisely. The positions both of the atoms in the crystal and of the neutrino
source in the laboratory are known to a precision which produces a sufficiently
large momentum uncertainty to prevent the identification of the scattering atom
or of the neutrino mass. These uncertainties prevent the use of momentum conservation
to distinguish between different possible amplitudes leading to the same final
state at the detector. Because the experimental setup is crucial to the
determination of which amplitudes are coherent, the relevant conditions
determined by the experimental setup must be introduced into any calculation
from the beginning. It is thus desirable to work at all times in the laboratory
system, where the source, detector and scattering apparatus are not moving and
the constraints from the uncertainty principle are most simply described.
\section {Which Path or Witch Craft?}
Further insight into the physics of Which-Path experiments is given by noting
the existence of quantum detectors and including the quantum mechanics of the
detector in the analysis of the experiemnt.
\subsection {Classical and Quantum Detectors}
A classical detector in one path of a two-path experiment determines which
path was taken and destroys all coherence and interference.
A quantum detector is a quantum system in one path of a which-path experiment.
If a particle passes through its path, it undergoes a transition denoted by
$\ket{D_i}\rightarrow \ket{D_f}$, where $\ket{D_i}$ and $\ket{D_f}$ denote
the initial and final states of the detector.
Consider a simple ``two-slit" which-path experiment in which a quantum
detector is introduced\cite{ADY,LipAB} into one path. A particle beam is split
into two paths and the two amplitudes, denoted by $\ket{L(x)}$ and
$\ket{R(x)}$ are then recombined at a point $x$ on a screen.
If no path detector is present
the wave function and the intensity at the point $x$ are
\beq{WP11}
\Psi(x) = \ket{L(x)} + \ket{R(x)}
\end{equation}
\beq{WP12}
I(x) = |\Psi(x)|^2 = |\ket{L(x)}|^2 + |\ket{R(x)}|^2 +
2 Re [\langle {L(x)} \ket{R(x)}]
\end{equation}
This can be rewritten
\beq{WP17}
I(x) = |\ket{L(x)}|^2 + |\ket{R(x)}|^2 +
2 Re [|\langle {L(x)} \ket{R(x)}|\cdot e^{i\theta (x)}]
\end{equation}
\beq{WP18}
I(x) = |\ket{L(x)}|^2 + |\ket{R(x)}|^2 +
2 |\langle {L(x)} \ket{R(x)}| \cos \theta (x)
\end{equation}
where $\theta (x)$ is relative phase of $\ket{L(x)}$ and $\ket{R(x)}$
If there is a quantum detector in the ``R" path,
the wave function for the combined system of the particle and the detector and
the intensity observed at $x$ are
\beq{WP13}
\Psi(x,D) = \ket{L(x),D_i} + \ket{R(x),D_f}
\end{equation}
\beq{WP14}
I(x) = |\ket{L(x)}|^2 + |\ket{R(x)}|^2 +
2 Re [\langle {L(x)} \ket{R(x)} \cdot \langle D_i \ket {D_f}]
\end{equation}
The quantum detector introduces an additional factor in the interference term,
the detector overlap $\langle D_i \ket {D_f} $ . It can also have an additional
phase introduced by the phase of $\langle D_i \ket {D_f} $
The Bragg scattering process is an example of a which-path experiment with
many paths, one for each scattering atom, and a quantum detector in each path.
The detector is the full lattice and each interference term between two paths
contains two coherence factors $\langle D_i \ket {D_f}, $ one for each path,
that depend on the lattice dynamics. The probability $P_{DW}$ that the
scattering is coherent is called the ``Debye-Waller" factor\cite{Mossb,LipAB} and is
just given by
\beq{WP94}
P_{DW} = |\langle D_i \ket {D_f} |^2
\end{equation}
\subsection {A Simple Toy Model for a Quantum Detector}
Consider the following modification of the toy model of Stern et al\cite{ADY}
in which the particle moving in the ``R" path interacts with an
external spin- one-half object and produces a rotation of this external spin by exactly
$180^o$ about the $z$-axis, while if the particle passes through the ``L" path
there is no effect. Then
\beq{WP15}
\ket {D_f} = e^{i\pi s_z} \ket {D_i} =
e^{i\pi \sigma_z/2} \ket {D_i}
\end{equation}
\beq{WP16}
\langle D_f \ket {D_i} =
\bra { D_i} e^{i\pi \sigma_z/2} \ket {D_i}
=\bra { D_i} i\sigma_z \ket {D_i} = i \langle \sigma_z \rangle_i
\end{equation}
The wave function and intensity at the point $x$ on the screen are now
\beq{WP19}
\Psi(x,D) = [\ket{L(x),D_i} + i\langle \sigma_z \rangle_i \ket{R(x),D_i}]
\end{equation}
\beq{WP20}
I
(x) = |\ket{L(x)}|^2 + |\ket{R(x)}|^2 -
2 |\langle {L(x)} \ket{R(x)}| \sin \theta (x)]\cdot
\langle \sigma_z \rangle _i
\end{equation}
The interference term with quantum detector contains an
additional factor $ \bra { D_i} \sigma_z \ket {D_i} =
\langle \sigma_z \rangle _i $
with an extra $90^o$ phase.
If the spin is initially polarized in the any direction normal to the $z$
axis, then $ \langle \sigma_z \rangle _i =0$ and there is no interference
between the two paths. One path flips the external spin; the other does not,
and the detector determines the path.
If the spin is initially polarized in the $z$-direction, then
$\langle \sigma_z \rangle _i =\pm 1$ and the rotation
does not change the spin state; it only introduces a phase. There is no
dephasing, just the addition of a constant relative phase.
The interesting case which illustrates the difference between classical and
quantum detectors is when the spin is initially polarized in another
direction; e.g. at $45^o$ relative to the $z$-axis in the $x-z$ plane, with
the $z$ and $x$ components both positive. Here $\langle \sigma_z \rangle _i =
1/2$.
Classically it is always possible to know the path taken by the particle.
If the spin is rotated the ``R" path has been taken; if the spin is not rotated,
the ``L" path has been taken. The rotation brings the spin into a direction in the
$x-z$ plane which is still at $45^o$ relative to the $z$ axis and normal to the
original direction. The $z$-component is still positive but the $x$ component is
negative.
This rotation is easily detected classically but not
quantum-mechanically. The initial spin state which is 100\% polarized positive
relative to an axis at $45^o$ with respect to the $z$ axis with both $x$ and
$z$ components positive is a 50-50 mixture of both positive and negative
polarizations relative to an axis normal to the initial polarization direction
with the $x$ component negative.
Thus if we know that the initial spin is polarized as above, and we now
measure the polarization in the direction of the classically expected final
polarization, we will indeed find that the final spin is 100\% polarized as
expected from the classical analysis if the particle went through the path that
interacts with the spin. But if the particle went through the other path and
did not affect the spin at all, the spin is completely unpolarized with respect
to this new axis, 50\% positive and 50\% negative.
This is thus only a ``partial which path" experiment with partial
dephasing.. The initial and final states of the spin before and after the
rotation are very different and distinguishable classically. But
quantum-mechanically they are not orthogonal. The overlap defines a domain
where it is impossible to determine ``which path" and interference will still
be observed.
\section {Detailed Quantum Mechanics of Neutrino Detector}
We now apply the general which-path formalism develped above to a neutrino -
detector system. The wave function for the initial state of neutrino and detector
can be written
\beq{WP21}
\Psi_i(\nu,D) =
\sum_{k=1}^{N_\nu} \ket{\nu(E_\nu,m_k,\vec P_k),D_i(E_i)}
\end{equation}
where $N_\nu$ is the number of neutrino mass states,
$E_\nu$, $m_k$ and $\vec P_k$ denote the neutrino energy, mass and momentum and
$D_i(E_i)$ is
the initial state of the detector with energy
$E_i$.
If the detector is a muon detector the final detector state after neutrino
absorption is
\beq{WP23}
\Psi_f(\mu^\pm,D) =
\sum_{k=1}^{N_\nu} \ket{\mu^\pm(E_\mu,\vec P_\mu),D^\mp_{kf}(E - E_\mu)}
\end{equation}
where
$E_\mu$ and $\vec P_\mu$ denote the muon energy and momentum, $D^\mp_{kf}$ is
the final detector state produced in the ``path $k$"; i.e. after the absorption
of a neutrino with mass $m_k$ and emission of a $\mu^\pm$, and
$E=E_\nu + E_i$ is the total energy which is conserved in the transition.
The transition in the detector occurs on a nucleon, whose co-ordinate is
denoted by by $\vec X$, and involves a charge exchange denoted by the isospin
operator $I_{\mp}$ and a mementum transfer $\vec P_k -\vec P_\mu$.
The detector transition matrix element is therefore given by
\beq{WP24}
\bra {D^\mp_{kf}} T^{\mp}\ket {D_i} =
\bra {D^\mp_{kf}}I_{\mp}e^{i(\vec P_k -\vec P_\mu) \cdot \vec X} \ket {D_i}
\end{equation}
The overlap between the final detector wave functions after the transitions
absorbing neutrinos with masses $m_k$ and $m_j$ is then
\beq{WP25}
\langle {D^\mp_{kf}}\ket {D^\mp_{jf}} =
\bra {D_i} e^{i(\vec P_j-\vec P_k) \cdot \vec X} \ket{D_i} \end{equation}
If the quantum fluctuations in the position of the active nucleon in the
initial state of the detector are small in comparison with the oscillation wave
length, $\hbar /(\vec P_j-\vec P_k)$,
\beq{WP26}
|\vec P_j -\vec P_k|^2 \cdot \bra {D_i} |\vec X^2|\ket {D_i} \ll 1
\end{equation}
\beq{WP27}
\langle {D^\mp_{kf}} \ket {D^\mp_{jf}} \approx
1 - (1/2)\cdot|\vec P_j -\vec P_k|^2 \cdot \bra {D_i} |\vec X^2|\ket {D_i}
\approx 1
\end{equation}
There is thus effectively a full overlap between the final detector states after
absorption of different mass neutrinos, and a full coherence between the neutrino
states with the same energy and different momenta.
The total energies of the final muon and detector produced after absorption of
neutrinos with different energies are different. These muon-detector states are thus
orthogonal to one another and there is no coherence between detector states produced
by the absorption of neutrins with different energies.
\section{Conclusions}
\subsection {What we know from simple quantum mechanics}
Neutrinos propagate from the source to the detector as ordinary Dirac particles
moving freely in space if they are not interacting with matter. They do not get lost
in transit and the relative number of the different mass eigenstates is the same at
the detector as at the source. Only the relative phase between the different mass
eigenstates can change in the propagation from the source to the detector .
The observation of a difference between upward and downward going atmospheric
neutrinos measured in the same detector can have only two possible explanations.
1. At least two different neutrinos with different masses are emitted
from the source and observed in the detector, and the detector is sensitive to
the relative phases of the waves arriving from neutrinos with different masses.
These relative phases increase monotonically with distance as a well-known
function of the unknown neutrino mass differences, thereby producing
oscillations in the signal observed at the detector as a function of distance.
The experimental results place constraints on the values of the neutrino
mass differences and the couplings of the different neutrino mass states to the
source and the detector (mixing angles in the language of the standard model).
2. The neutrinos traveling through the earth do not propagate freely, but
interact with matter. This is generally known as the MSW effect. These
interactions can change the relative magnitudes as well as the relative phases
of the neutrino mass eigenstates reaching the detector. They can transfer
momentum, but they conserve energy like a ball bouncing elastically against
the earth.
All these conclusions depend only upon quantum mechanics.
\subsection {What we think we know from the standard model}
In the standard model all the neutrinos observed so far in experiments
originate in a source from weak decays or $W$ and $Z$ exchanges and are
detected via $W$ or $Z$ exchange in a detector. The couplings of the three
neutrino mass eigenstates to the three charged leptons and the $W$ is described
by a $3 \times 3$ unitary matrix analogous to the CKM matrix in the quark
sector. These are usually described in terms of mixing angles.
\subsection {What we don't know and need to determine from experiment}
The masses of the three types of neutrinos and the mixing angles describing
their couplings to the $W$ are completely unknown and are free parameters in
the standard model. We really do not know if the standard model relations
between couplings are really valid and whether new physics beyond the standard
model might influence these relations. However, we emphasize that there is no
justificaton for believing that new physics beyond the standard model can
violate quantum mechanics. Thus the conclusions from quantum mechanics
described above hold even if the standard model is not valid.
\subsection{Energy-Momentum Kinematics}
We now use the above considerations to specify the relevant scales in neutrino
oscillation experiments. Consider a neutrino emitted from a macroscopic source
whose size is described by a linear dimension $S$, and detected by a
macroscopic detector at a distance $D \gg S$ from the source. The knowledge of
the source position leads to uncertainties in the initial source momentum, the
momentum transfer and the neutrino momentum.
\beq{WP2}
\delta p_\nu \approx {{\hbar}\over{S}}
\end{equation}
The energy of the source before the emission of the neutrino can be measured in
principle with arbitrary precision. The energy after the emission can be
measured during the time of flight of the neutrino from source to detector.
This leads to an uncertainty in energy transfer and the neutrino energy
\beq{WP3} \delta E_\nu \approx {{\hbar c}\over{D}} \ll c \delta p_\nu \end{equation}
The uncertainty in the square of the neutrino mass is then given by
\beq{WP4} \delta (m_\nu)^2 \cdot c^4= 2 p_\nu \cdot \delta p_\nu \cdot c^2+ 2
E_\nu \cdot \delta E_\nu \approx 2 p_\nu \cdot c^2 \cdot
\left({{\hbar}\over{S}} + {{\hbar}\over{D}}\right) \approx 2 p_\nu \cdot c^2
\cdot {{\hbar}\over{S}} \end{equation} Interference effects can be observed at the
detector between the contributions from neutrino states with different masses
if the squared mass difference is less than this value (\ref{WP4}). The
uncertainty in neutrino mass arises from the uncertainty in the neutrino
momentum. Eq. (\ref{WP3}) shows that the uncertainty in the neutrino energy is
negligible. Thus any coherence observed at the detector between amplitudes
from neutrinos with different masses must come from states with the same energy
and different momenta.
The relative phase between two neutrino waves with the same energy, masses
$m_1$ and $m_2$ and momenta $p_1$ and $p_2$ changes in traversing a distance
D by the amount
\beq{WP5}
\delta \phi(D)= (p_1 - p_2)\cdot{{D}\over{\hbar}} =
{{(p_1^2 - p_2^2)}\over{(p_1 + p_2)}} \cdot {{D}\over{\hbar}} \approx
{{(m_2^2 - m_1^2)\cdot c^4}\over{2p_\nu}}\cdot {{D}\over{\hbar}}
\end{equation}
For this phase to be of order unity and give rise to observed neutrino
oscillations,
\beq{WP6}
m_2^2 - m_1^2 \approx {{2 \hbar p_\nu}\over{D c^4 }} \ll
{{2 \hbar p_\nu}\over{S c^4 }} \approx
\delta (m_\nu)^2
\end{equation}
This squared-mass difference (\ref{WP6}) is much less than
the lower limit on detectable mass difference imposed by the uncertainty
condition (\ref{WP4}). The momentum difference between the different mass
eigenstates having the same energy is much smaller than the momentum
uncertainty produced by the localization of the source.
Thus the neutrino mass difference needed to produce oscillations with wave
lengths of the order of the source-detector distance $D$ cannot be detected in
any experiment in which the distance from the source to the detector is much
larger than the size of the source.
The wave length of the neutrino oscillations is given directly by eq.
(\ref{WP5}).
For an example showing characteristic numbers, the neutrino momentum from a
pion decay at rest is $\approx 30$ MeV/c or $3 \times 10^7$ ev/c. If there are
two neutrino masses of 1 and 2 ev. their momentum difference if they have the
same energy is
\beq{WP7}
p_1 - p_2 \approx {{(m_2^2 - m_1^2)c^2}\over p} = 10^{-7} \rm{ev/c}
\end{equation}
Since $\hbar c \approx 2 \times 10^{-7} $ ev $\times$ meters, the oscillation
wave length will be of order 2 meters and knowing the source position with a
precision of more than two meters will prevent the measurement of this
momentum difference. If the two neutrino masses are 0.1 and 0.2 ev. these
numbers scale by a factor of 100 and $p_1 - p_2 \approx 10^{-9} $ ev/c and the
oscillation wave length is 200 meters.
This effectively says it all for neutrino propagation in free space between a
source and detector whose size and distance satisfy the condition
that the distance between source and detector is much greater than the size of
the source. The point source approximation is good. Except for the case of
matter-induced oscillations and the MSW effect or for the case of propagation
through external fields there is no need to engage in more complicated
descriptions to obtain the desired results.
\section{acknowledgments}
It is a pleasure to thank Maury Goodman, Yuval Grossman,
Boris Kayser, Lev Okun and Leo Stodolsky for helpful discussions and
comments.
|
3,212,635,537,982 | arxiv | \section{Introduction}
Extended lobes of radio galaxies are interesting as a probe of the energetics of active galaxies.
In the standard model of radio galaxies of Fanaroff-Riley type II \citep[FR-II;][]{fanaroff1974}, radio
lobes are formed by shock interactions of the jets with the surrounding intergalactic medium at the jet extremities \citep{scheuer1974,blandford1974}.
In this scenario, radio lobes consist mainly of the jet-supplied matter that passed
the termination shock.
The gas density in the radio lobes is sufficiently low that radiative cooling is ineffective,
resulting in the energy supplied to the radio lobes to be well conserved for the lifetime of the
radio galaxy.
The lobe has a higher pressure than the ambient intergalactic medium (IGM)
due to shock-compression, and it can expand supersonically into the ambient IGM
(``overpressured cocoon": \citealp{blandford1974,begelman1989, nath1995, heinz1998, yamada1999}).
It is expected to form a shell of IGM matter compressed by the external shock
exterior to the radio lobes.
Detections of inverse Compton scattered photons in the X-ray energy bands of radio galaxies
can drastically improve our understanding of the properties of hotspots and radio lobes.
The distribution of energy among the magnetic field component and in the non-thermal
electrons has been investigated by examining the spectral energy distribution from the radio to
X-ray bands (e.g., for Cygnus A, \citealt{harris1994,wilson2000}; for 3C123, \citealt{hardcastle2001};
for 3C295, \citealt{harris2000}; for Pictor A, \citealt{wilson2001}; for 3C120, \citealt{harris1999};
for 3C390.3, \citealt{harris1998}). The results of these studies revealed that high energy electrons
of $\gamma\approx 10^3 - 10^5$ have an energy comparable to or greater than that in the magnetic field
\citep[e.g.,][]{tashiro1998, grandi2003, croston2005, isobe2006,isobe2009}.
If the energy spectrum of non-thermal electrons is given by a power-law ($dN(E)/dE \propto E^{-\eta}$),
most of the non-thermal electron energy is carried by those electrons characterized by energies
near the lower cut-off energy of $N(E)$ for $\eta>1$.
However, direct measurements of the thermal and/or low energy ($\gamma\approx 10^3$ or less)
electrons are quite difficult, since the plasma density in a radio galaxy is too low to emit
detectable radiation.
The standard overpressured cocoon models of FR-II radio galaxies are characterized by two types
of shocks, i.e., the external and the internal (jet-terminal) shock, where the latter is believed
to be observed as hotspots.
In this model, radio galaxies emit synchrotron radiation by shock accelerated electrons with
an acceleration efficiency $\xi_e\equiv U_\mathrm{syn}/U_\mathrm{tot}$, which is
inferred to be low ($U_\mathrm{syn}$ and $U_\mathrm{tot}$ are
energy of synchrotron-emitting electrons and total internal energy, respectively).
If we adopt the standard model and assume the low acceleration
efficiency in both shocks (low $\xi_e$), we can expect that a large amount of thermal electron
energy, as well as non-thermal, synchrotron-emitting electron energy.
In order to investigate the energetics of active galaxies, it is quite important
to measure all of the energy in the electron component, including those that emitting strongly.
In this paper we employ the Sunyaev-Zel'dovich effect (SZE) as a tool to measure the energy of the
electrons in a radio galaxy. The SZE represents the spectral deformation of the Cosmic Microwave
Background (CMB) radiation due to the inverse Compton scattering of these photons by the energetic
electrons \citep{zeldovich1969} in the galaxy. The intensity change of the thermal SZE is classically
described as follows \citep{zeldovich1969};
\begin{eqnarray}
\frac{\Delta I_{x}}{I_{x}} &= &\frac{xe^x}{(e^x-1)}
\left[ x\left(\frac{e^x+1}{e^x-1}\right)-4\right] y, \label{eq:SZ} \\
y &=& \int \frac{k_BT_e}{m_ec^2}\sigma_Tn_e dl \propto\int p_edl, \label{eq:press}
\end{eqnarray}
where $x\equiv h\nu/k_BT_r$ is the non-dimensional frequency, $T_r$ is the temperature of the CMB,
$k_B$ is the Boltzmann constant, $T_e$ is the electron kinetic temperature,
$\sigma_T$ is the cross section of Thomson scattering, $n_e$ is the electron number density,
$m_e$ is the electron mass, and $p_e$ is the electron thermal pressure, respectively,
integrated along the line of sight.
Equation (\ref{eq:press}) shows that the Compton parameter $y$ is proportional to the
sum of the thermal pressure of electrons along the line of sight.
Therefore we can estimate the thermal energy of the electrons contained in a radio galaxy
with the thermal SZE.
Similarly, the decrease in CMB intensity by SZE in the Rayleigh-Jeans regime reflects all of the electron
energy, not only the high energy electrons which generate X-ray photons,
but also the lower energy electrons and the thermal electrons.
Although the energy distribution of the electrons is lost in the SZE (Eq.[\ref{eq:press}]), it
measures the total energy deposited in the radio galaxy.
In this paper, we ignore the non-thermal and kinetic SZE as well as relativistic corrections and focus
mainly on thermal SZE as the first trial observation \citep{birkinshaw1999,ensslin2000,yamada2001}.
The study of the SZE has been directed toward understanding the thermal properties of the intra
cluster medium (ICM), and it has been detected in dozens of clusters of galaxies (see for recent reviews, \citealt{birkinshaw1999}; \citealt{carlstrom2001}; \citealt{rephaeli2002}).
Among the numerous SZE detections in clusters of galaxies, \citet{mckinnon91} first tried to detect the
SZE in the radio lobes with the NRAO 12-m telescope at 90 GHz using the double-subtraction method.
They observed 4 FR-II radio galaxies whose lobe sizes were smaller than the beam size, and obtained upper limits on the antenna temperature fluctuation ($\approx 0.1$ mK, or $y\approx 10^{-4}$).
In this paper, we report on the results of the imaging observation of a giant radio galaxy B1358+305,
extending much further in angular size than the beam size ($\sim 80^{\prime\prime}$ in our observation) using the Nobeyama 45-m telescope at 21 GHz.
A two dimensional imaging study of a region in a radio galaxy is expected to provide
the most reliable limit
on the SZE in a radio galaxy by resolving the substructure within and the emission sources in the field of view.
The organization of the paper is as follows.
In Section 2 we briefly review the overpressured cocoon model \citep[see for detail,][]{yamada1999}.
In Section 3 we describe the features of our target B1358+305, and the observation and analysis procedures.
In Section 4 the results are presented. We discuss the possible uncertainties in the estimation of the
SZE amplitude in B1358+305 and their implications for radio galaxy models in section 5. Finally, we
summarize our conclusions and future prospects in the last section.
\section{Overview of Overpressured Cocoon Model} \label{sect:model}
To provide an interpretative framework,
we review our model of the overpressured cocoon \citep[see for details][]{yamada1999},
which is an extension of the classical overpressured-cocoon model of \citet{begelman1989}.
In this model, the jets from the AGNs form external shocks at the interaction with the surrounding IGM.
Due to the action of the reverse shock (or internal shock), the jets are compressed. In the FR-II radio
galaxies, the compressed jet matter at the locations of the internal shock is believed to appear
as hotspots. The IGM compressed by the external shock is in pressure balance with the internal
shock compressed jet matter at a contact discontinuity. A high pressure gas clump forms around the
AGN, which expands supersonically into the IGM \citep{begelman1989,nath1995,heinz1998,yamada1999}.
Hereafter we use the term ``cocoon" as the region consisting of both the jet-supplied matter and the
shock-compressed IGM throughout this paper.\footnote{Note that the definition of the term ``cocoon"
can change with different authors, and should not be confused with that defined herein.}
The cocoon expands as its internal energy, supplied by the jet, increases with time.
In this paper, for simplicity, we first treat $\xi_e$ is a single free parameter that characterizes the cocoon,
and assume that the cocoon has an approximately spatially uniform thermal electron distribution.
Thus the parameter $\xi_e$ is to be taken as an appropriately intensity-weighted mean value over the cocoon, and
hereafter we denote the mean value as $\langle \xi_e\rangle$.
This assumption may not be fully consistent with current observations, but
its simplicity can provide a basis for feasibility evaluation and interpretation as a first
step (see Section \ref{sect:alt} for the discussion of this assumption).
In these models, most of the matter that has passed through the shock is assumed to be
almost fully thermalized except for a small fraction, which is assumed to be non-thermal.
The density in the cocoon is insufficient for
effective radiative cooling and, hence, the cocoon is expected to conserve nearly all of the
kinetic energy of the jets. Furthermore, the cocoon is expected to remain hot for a relatively
long time due to the low radiative cooling efficiency (an effective life time could be $t_\mathrm{life}
\sim 10^8$ yrs or more: see \citealp{yamada1999}). Even taking into account the $PdV$ work
against the surrounding IGM, the cocoon would remain hot for $\approx 10^8$ yrs \citep{yamada1999}.
Therefore the energy contained in a cocoon can be written as
\begin{equation}
U_\mathrm{tot} \approx L_\mathrm{jet}t_\mathrm{life}, \label{eq:etot}
\end{equation}
%
where $L_\mathrm{jet}$ is the kinetic luminosity of the jets, and $t_\mathrm{life}$ is
life time scale of the AGN (typically $\sim 10^7 - 10^8$ yrs).
For the thermal SZE as written (\ref{eq:press}) with the thermal pressure in the cocoon and the
low value of $\langle\xi_e\rangle$, the total energy of the cocoon becomes $U_\mathrm{tot}=
U_\mathrm{th}/(1-\langle\xi_e\rangle)=\int p_edV/(1-\langle\xi_e\rangle)$
which is proportional to $y$ (Eq.[\ref{eq:press}]).
This estimate indicates that SZE, which is proportional to $\int p_edl$, provides a
good measure of thermalized electrons and the jet-supplied energy contained in the cocoon.
We employ this interpretative framework and attempt to measure the energy contained in electrons
in a radio galaxy.
Since direct measurement of $L_\mathrm{jet}$ from observation is difficult, equation (\ref{eq:etot})
is not used to estimate the expected value for the Compton $y$ parameter. Instead, we use the
synchrotron emission argument and make the assumption that the synchrotron-emitting electrons are
generated by diffuse shock acceleration with a small acceleration efficiency $\xi_e$ (again, we
assume $\xi_e$ is the same for the internal and external shocks). The minimum energy of the
synchrotron-emitting electrons $U_\mathrm{syn}$ can be estimated from the synchrotron luminosity
$L_{\rm syn}$ using the minimum energy condition, which in the simplest form \citep{moffet1975} is
given by
\begin{equation}
U_\mathrm{syn} = \frac{1}{2}
\left\{ \frac{9a}{2} \left( \frac{\pi c}{3e^7} \right)^{1/2} m_ec^2
\frac{\alpha +1}{2\alpha +1}
\left( \frac{\nu_u^{\alpha+1/2}-\nu_l^{\alpha+1/2}}{\nu_u^{\alpha+1}-\nu_l^{\alpha+1}} \right)
\right\}^{4/7} V^{3/7} L_\mathrm{syn}^{4/7},
\end{equation}
where $a$ is the ratio of energies of non-thermal protons and non-thermal electrons,
$\alpha$ is the photon index of synchrotron emission, $\nu_u$ and $\nu_l$ are
the upper and the lower limits of observation frequencies, and $V$ is the volume of the
synchrotron-emitting region, respectively.
Once we obtain the minimum energy $U_\mathrm{syn}$, we can estimate
$U_\mathrm{tot}\gtrsim U_\mathrm{syn}/\langle\xi_e\rangle$, where the inequality takes into account the larger
internal energy compared with the minimum energy.
If we adopt the small acceleration efficiency of the shock-accelerated electrons $\langle\xi_e\rangle$,
the thermal electron energy contained in a cocoon is evaluated by Compton $y$ parameter as,
\begin{eqnarray}
y &=& \frac{\sigma_T}{m_ec^2}\int p_e dl, \nonumber \\
&\simeq& \frac{\sigma_T}{m_ec^2} \frac{U_{\rm th}}{V} l
= \frac{\sigma_T}{m_ec^2}(1-\langle\xi_e\rangle)\frac{U_{\rm tot}}{V} l
\gtrsim \frac{\sigma_T}{m_ec^2} \frac{U_{\rm syn}}{V}\frac{1-\langle\xi_e\rangle}{\langle\xi_e\rangle} l, \\
&\sim& 9.2\times 10^{-5} \left( \frac{l}{{\rm 300kpc}}\right)
\left( \frac{\langle\xi_e\rangle}{0.05}\right)^{-1} \left(\frac{U_\mathrm{syn}/V}{9.7\times 10^{-14}\, \mathrm{ergs ~ cm}^{-3}}\right),
\label{eq:expect}
\end{eqnarray}
%
along the line of the sight that passes through the center of the cocoon.
Numerical values are based on the assumption that the axial ratio of the cocoon is 1:3
(Figure \ref{fig:all}) and the adopted cosmological parameters ($H_0=72$ km s$^{-1}$ Mpc$^{-1}$,
$\Omega_M = 0.3$, $\Omega_\Lambda = 0.7$, \citealt{spergel2003}).
For the mean acceleration efficiency $\langle\xi_e\rangle$, we take the small value that is observationally suggested
($\langle\xi_e\rangle \sim 0.05 - 0.1$: \citealp[e.g.,][]{sturner1997,baring1999,loeb2000}).
The synchrotron-emitting electron energy in B1358+305 was estimated to be $U_\mathrm{syn}/V\sim 5.7-9.7
\times 10^{-14}$ ergs cm$^{-3}$ in the 10 MHz $\le \nu\le$ 10 GHz band by \citet{parma1996}
(Yamada \& Parma, private communication),
who assumed that the energy ratio of non-thermal protons and electrons, and the filling factor
of the synchrotron electrons are equal to be unity.
Note that equation (\ref{eq:expect}) indicates the Comptonization parameter $y$, or
the amplitude of the intensity decrement of SZE is proportional to the length within the
radio galaxy along the line of sight, so that a giant source can yield a large value of $y$.
\section{Observations and Data Reductions}
\subsection{Strategy of Search for SZE in a Radio Galaxy}
Our basic plan of observation in search for the SZE signal in a radio galaxy and the feasibility
evaluation are provided below. Compared to the case of galaxy clusters, the spatial extent of the
cocoon relative to the radio lobes is less understood both theoretically and observationally
\citep[see][and references therein]{mcnamara2007}. We, therefore, develop the basic observational
plan paying a special attention to the extent of the cocoon as follows.
\noindent
1) The amplitude of the intensity decrement of SZE is proportional to the length along the line of
sight (Eq.[\ref{eq:expect}]). Hence a large radio galaxy should be selected as a target.
\noindent
2) In order to reduce the cancellation of the intensity decrement of SZE by the emission from the radio
lobes, the observation frequency should be higher than that used in the observations of the synchrotron
emission of the radio lobes. Synchrotron emission has a power-law spectral energy distribution $\propto
\nu^{-\alpha}$ with a typical $\alpha$ about 0.7, and its intensity decreases with $\nu$ in the GHz regime.
\noindent
3) Taking into account the potentially wider spatial extent of the cocoon in comparison to the radio lobes
that is assumed in the classical hydrodynamic models \citep[e.g.,][]{begelman1989}, a wider region than the
radio lobes should be mapped.
\noindent
4) Since the expected signal of the SZE is weak and the cocoon structure is not well understood,
substructures and emission sources within the mapped region should be carefully resolved and removed.
To resolve the substructures and to take account of the proportionality of the $y$ parameter to the
length scale along the line of sight (Eq.[\ref{eq:expect}]), a giant radio galaxy whose angular size
is also large should be selected as a target.
In addition to the point 2, the amplitude of the thermal SZE is known to be maximum at $\nu\simeq 90$ GHz
\citep{birkinshaw1999}, hence the first choice for the observation frequency is 90 GHz. However, the beam
size decreases with the observation wavelength, and thus the observation frequency cannot be too high in
order to perform the imaging observation of a wide region (the point 4) in a reasonable timescale.
The beam size at 90 GHz of the NRO 45-m telescope is about 20$^{\prime\prime}$, which is too small to
image the large central region of B1358+305 ($\gtrsim 5^{\prime}$) to achieve a reasonable signal-to-noise
ratio within a realistic timescale (see discussion on the feasibility in Section \ref{sect:feasibility} below).
Additionally the atmospheric stability becomes worse for higher frequencies in
this waveband, or the typical system temperature $T_\mathrm{sys}\approx$ 250 - 900 K
at 80 - 90 GHz is much higher than that at 21GHz ($T_\mathrm{sys}\approx$ 100 - 150 K).
Hence, the required observational time becomes unrealistically long.
We chose the single-dish observation as the first trial rather than interferometric observation.
This follows from the fact that the hydrodynamic models predict a smoother distribution of SZE
compared to galaxy clusters \citep{begelman1989,yamada1999}. In other words, the smooth SZE signal
would be resolved out if an interferometric observation were carried out. However, interferometric
observations would be useful for detecting the SZE for more compact objects with known structures.
In addition, high angular resolution observation with interferometers can also be used to remove
emission source contaminations within a field of view. Both single-dish and interferometric observations
have their advantages and disadvantages. In order to avoid the risk of resolving out of the possibly
extended SZE signal, we use the single-dish observation strategy to measure the total flux.
Given these considerations and the low luminosity at 10 GHz, we decided to observe the central region
of a giant radio galaxy B1358+305 with the NRO 45-m telescope at 21GHz. The typical beam size of the 45-m
telescope is $\sim 80^{\prime\prime}$ at this frequency. The low system temperature (typically $\sim$
100 - 150 K) and a wide bandwidth (2 GHz) of the HEMT 22 receiver installed on the 45-m telescope led
to a decrease in the necessary observation time for an SZE search in a faint giant radio galaxy.
The detailed set-ups and observation method are described in Section \ref{sect:obs} below.
\subsubsection{Feasibility Estimation} \label{sect:feasibility}
The imaging sensitivity for the observation with the 45-m telescope at 21 GHz is about
$2T_\mathrm{sys}/\eta/\sqrt{B} =$ 7.55 mK $\mathrm{s}^{1/2}$, where the typical system
temperature $T_\mathrm{sys}$ is taken to be 135 K, the
aperture efficiency $\eta$ is estimated to be 0.8, and the bandwidth $B$
is taken to be 2 GHz for one of the dual channels, respectively.
In order to obtain a 3$\sigma$ detection of the expected intensity fluctuation from equation
(\ref{eq:expect}) or 0.30 mK with the beam width $\Delta\theta_\mathrm{HPBW}\simeq
80^{\prime\prime}$ in the central $6.7^{\prime}\times 6.7^{\prime}$ region, the
imaging sensitivity estimated above requires an observation time of about 67.3 hrs.
The estimated observation time implies a deep imaging.
As the expected SZE decrement in the B1358+305 is small, we examine the cumulative
contributions from background (or foreground) discrete sources to the intensity fluctuation.
\citet{franceschini89} calculated the expected fluctuations of the antenna temperature
($\Delta T_A/T_A$) as a function of beamwidth at 6\,cm, produced by randomly distributed
sources within a flux range $10^{-8}<S({\rm Jy})<10$ (see Figure\,\,3 in \citealt{franceschini89}).
We assume that contributing sources have a spectral index $\alpha=0.7$ on average ($I_{\nu}\propto
\nu^{-\alpha}$), and estimate the fluctuation of the brightness temperature $\Delta T_B/T_B (\sim \Delta T_A/T_A)$
at 21 GHz to be $1.1\times 10^{-4}$.
In terms of Compton $y$ parameter, this temperature fluctuation corresponds to $y=4.7\times 10^{-5}$ from equation (\ref{eq:SZ}).
The evaluated confusion limit is only marginally smaller than the expected
SZE (Eq.[\ref{eq:expect}]).
Hence in the data analysis procedure, emission sources in the field of view should be
carefully removed from the integration.
\subsection{The Giant Radio Galaxy B1358+305}
B1358+305 is one of the largest radio galaxies of FR-II type at a
moderate redshift ($z=0.206$, \citealt{parma1996}; \citealt{saripalli1996}).
Employing cosmological parameters for the $\Lambda$-dominated flat universe \citep{spergel2003},
its projected size amounts to $\sim$ 926 kpc, extending over $\sim 10^{\prime}$ in the plane
of the sky \citep{parma1996}.\footnote{\citet{parma1996} estimated the size of B1358+305 to be 1.34 Mpc making use of cosmological parameters $H_0=100$ km s$^{-1}$ Mpc$^{-1}$ and $q_0=1.0$.
We update cosmological parameters in this paper.}
It has a relatively low radio luminosity ($P_{\rm 1.4GHz}=1.9\times 10^{25}$W Hz$^{-1}$),
and the spectral age was estimated to about $\sim$2.5 -- $7.5\times10^7$ yrs from the multi-band imaging
observations \citep{parma1996}.
\citet{parma1996} further calculated the advance speed of the shock head of
B1358+305 as $R/\tau_\mathrm{spec}\simeq 0.02c$ -- $0.03c$ ($R$ is the lobe length, $\tau_\mathrm{spec}$
is the spectral age, $c$ is the speed of light, respectively).
From the balance of the external ram pressure and the jet ram pressure, they concluded that
the environment density of B1358+305 is quite low ($n_e\lesssim 10^{-6}$ cm$^{-3}$), and
B1358+305 is overpressured even for transverse direction, for an ambient temperature as
high as that of a typical poor cluster environment ($\sim 10^7$ K).
This makes B1358+305 a good target in the sense that it resembles the classical model.
B1358+305 has an FR-II morphology with two aligned lobes with a radio core,
which resides closer to the northern lobe.
As SZE appears as the decrement of the CMB in the frequency range $\nu\lesssim$ 200 GHz,
any compensating radio emission should be avoided.
Since the lobe intensity rapidly decreases towards the region slightly south of the core,
we focused on that region in order to avoid the lobe emission contamination as much as possible
(Figure\,\ref{fig:all}).
\subsection{Observations} \label{sect:obs}
Observations of B1358+305 were performed with the HEMT 22 receiver on the Nobeyama 45-m telescope from 2001
February 28 to March 18. The receiver has 2 GHz bandwidth, and is equipped with dual channels which can
simultaneously receive two circular polarization components, which enables the reduction of the required
observation time. The near-central region in B1358+305 of size $6.7^{\prime}\times 6.7^{\prime}$ around
($\alpha_{50}, \delta_{50})=(13^{\rm h}58^{\rm m}25.0^{\rm s}, +30\degr 32^{\prime}0.0^{\prime\prime}$) was
raster-scanned with $40^{\prime\prime}$ spacing, which created a map composed of $11\times 11$ pixels.
Since the jet axis is nearly along the declination, raster-scans were performed along the RA-DEC coordinates in turn.
The scan speed is about $25^{\prime\prime}$ s$^{-1}$, resulting in the acquisition of one map in $\sim 180$ s.
The exposure time was about 74.8 ksec in total, corresponding to 618 s for each pixel. The system noise
temperature was about $T_{\rm sys}=140$\,K on average, ranging from about just below 130 K to nearly 150 K.
The noise produced by the fluctuations in the atmosphere were reduced by the simultaneous observation of an
off-source point
via the beam switching technique. The off-source point
is about 400$^{\prime\prime}$ away in azimuth direction to the east and was simultaneously observed at 15 Hz.
The antenna temperature was calibrated using the chopper-wheel method, while the optical depths of the
atmosphere were measured by elevation scans. Since the atmospheric optical depths were about 0.04 throughout
the observing period, the flux was not corrected for atmospheric absorption.
The main beam size $\theta_{\rm HPBW}$ was estimated to be $81.2^{\prime\prime}$ by observing 3C273.
With this beam size we can image a large ($6.7^{\prime}\times 6.7^{\prime}$) area in a
reasonable timescale, simultaneously resolving the substructures in the mapped region.
For the pointing and flux calibrator, we used 3C286 about $7.5\degr$ away from the center of the image.
The pointing offset was monitored about every hour, and was typically smaller than $6^{\prime\prime}$
throughout the observing period. The flux of 3C286 was assumed to be $2.56\pm 0.2$Jy \citep{ott1994},
and it was used to evaluate the aperture efficiency of the telescope.
Due to the good weather conditions, the aperture efficiency was stable during the latter half
of the observation period (rms of the flux variation is about 5\%).
In the former period, the stability was slightly worse (the flux variation reached about 10\%), and we
carefully corrected each observation using the efficiency measured just before the observation.
\subsection{Data Reduction and Analysis} \label{sect:analys}
Raster-scans were performed in two orthogonal directions (along the right ascension and the declination coordinates)
in turn in order to obtain pairs of orthogonally scanned maps taken as close together in time as possible.
Within a raster-scanned map, antenna temperature fluctuations induced by the time variation of the atmospheric
conditions, instability of the detectors or other instruments and so on, appeared along the scanning direction
(scanning noise). The scanning noise in the data was reduced by combining the two orthogonally scanned maps
in the Fourier space via the basket-weaving method (\citealt{sieber1979}; \citealt{kuno1993}).
Although we tried to avoid contamination from lobe emissions into the mapped region, some of the scans suffered
from lobe emissions in the operation of beam switching. Due to the limited throw amplitude ($\sim 400^{\prime\prime}$)
and to the rotation of the position angle of the beam switch, emissions from one of the lobes entered into the
reference field, resulting in over-subtraction in a part of the mapped region (less than 1/4 of the mapped area at
most). This over-subtraction generated artificial intensity decrements.
We identified the affected pixels in each scan and removed them in the integration.
In the identification process, we made use of the 10.6 GHz map of \citet{saripalli1996}
(see Figure\,\ref{fig:all}), and set the position of the reference beam $400^{\prime\prime}$ away from the main beam.
Since at 21 GHz both lobes are expected to be dimmer than displayed in the map in
\citet{saripalli1996} (the spectrum is a decreasing function of frequency),
the procedure above does not overlook the affected pixels.
Due to this partial masking in the integration, the noise level is not uniform in the
obtained integrated map, ranging from 0.72 mJy beam$^{-1}$ to 1.2 mJy beam$^{-1}$
(1$\sigma$ level): in spite of the loss of some pixels, however, the root mean square of the
intensity in the integrated map reached 1.22 mJy beam$^{-1}$ with emission sources.
In order to evaluate the noise level without emission sources,
we constructed two independent maps by dividing the maps taken in the former
and the latter halves of the observing period, and subtracted the latter from the former.
In this process, emission from true sources should cancel out and
the resultant map should contain only random noise.
The rms fluctuation of the differential map constructed in this way is 1.01 mJy beam$^{-1}$.
\section{Results}
Figure \ref{fig:full_image} shows the integrated map of the central region of B1358+305.
One can easily identify the AGN component and the innermost edges of the two radio lobes
in this figure, which appear also in Figure \ref{fig:all}.
On the right edge of the map exists a bright region, with a peak intensity of
$\sim 6$ mJy beam$^{-1}$ (source ``D").
We failed to find any identified source corresponding to this bright spot in any
catalogue from the radio to X-ray energy band, though, it indeed appears on the VLA NVSS 1.4 GHz
map and the Effelsberg 10.6 GHz map obtained by \citet{saripalli1996}.
Taking into account a relatively large beam size ($\theta_{\rm HPBW}=81.2^{\prime\prime}$),
we recognize it as an unknown point source.
Since emissions from the sources smear out the SZ flux decrement, it is necessary to subtract
them from the map.
However, it is quite difficult to obtain the correct zero flux level for mainly two reasons.
One is related to the larger uncertainties in models of cocoons in comparison with clusters of galaxies.
In the case of a galaxy cluster, a well-established model obtained by X-ray observation (so-called $\beta$ model)
is available. Thus, one can set the baseline (zero flux level) at a sufficient distance ($\theta\gg \theta_{\rm core}$;
see e.g., \citealt{komatsu2001}) from the center with the help of the X-ray observation.
On the other hand, in the case of the theoretical model for the cocoon, there are many uncertainties in its
morphology. For example, the density distribution of the surrounding
medium \citep[e.g.,][]{kaiser1999} or the magnetic fields alter the shape of the cocoon
(see \citealt{burns1991} for a review of numerical simulations, and for examples of MHD simulations, see e.g., \citealt{clarke1986}; \citealt{lind1989}).
The other reason is that our deep imaging of this region approaches the confusion limit, so the
background fluctuation impedes the establishment of a definitive zero flux level (see Section \ref{sect:feasibility}).
Therefore, we could not ascertain the absolute intensity of the sources included in the map at 21 GHz, and
it is difficult to correctly subtract the emission from the true emission sources.
Instead we evaluated the intensity fluctuation in the mapped region, and re-defined
the baseline so that the average fluctuation became zero.
In order to calculate the intensity fluctuation, we masked the emission source regions.
These regions around the emission sources are indicated by white lines in Figure \ref{fig:mask}.
The baseline of the map is re-defined after the source removal so as to set the average
intensity of the remaining pixels to be zero.
\subsection{Structure of the Cocoon} \label{sect:structure}
In order to improve the effective signal-to-noise ratio, we constructed projections of the
two-dimensional map along two orthogonal directions.
We can also expect that these projections reflect the structure of the cocoon more clearly
than the original two-dimensional map if the data contains the SZE signal from the cocoon of B1358+305.
As can be seen in Figure \ref{fig:all}, the jet axis is almost perpendicular to the right ascension.
Thus, we can examine the structure perpendicular (parallel) to the jet axis by projecting the map along
the declination (right ascension) coordinates. In the process of projection, we average the pixels along
each column or row without using the masked pixels corresponding to the emission sources that are indicated
with the white lines in Figure \ref{fig:mask}.
Figure \ref{fig:proj} (a) shows the projection along the jet axis (the declination coordinate).
A relatively dark region appears at about $+2^{\prime}$ west of the central column.
Comparing Figure \ref{fig:proj} (a) with the original two-dimensional map (Figure\,\ref{fig:full_image}),
one can see the decrement appearing in Figure \ref{fig:proj} (a) mainly picks up
the dark region between the unidentified source labeled ``D" in the map and the AGN (labeled ``A").
However, as the statistical errors are large (error bars in Figure \ref{fig:proj} correspond to $1\sigma$)
even though they are reduced by $\sim 1/\sqrt{N_p}$ in averaging over each column ($N_p$ is the number of
pixels included), we cannot rule out the possibility that the projection obtained here is consistent
with a flat distribution (no SZE signal accompanied B1358+305).
As shown in Figure \ref{fig:all}, B1358+305 has a relatively large axis ratio, and it is reasonable to
approximate it as a cylinder with an axis along the jet axis.
Hydrodynamic simulations \citep[e.g.,][]{loken1992,kaiser1999,scheck2002} show that
the pressure $p_e$ is nearly uniform inside a cocoon
when the density distribution of the surrounding IGM is nearly flat
(there will be a difference in $\xi_e$ at internal and external shocks, and accordingly $p_e$
would not be spatially uniform as is suggested by observation, but this assumption is discussed
separately in Section \ref{sect:alt} below).
Since $y\propto p_el$ (Eq.[\ref{eq:press}], where $l$ is the length cut through the cocoon along the line
of sight), the projection along the jet axis is expected to present $l(\theta)$ in the isobaric cocoon,
where $\theta$ is the angular distance projected onto the sky from the map center.
For a cylindrical cocoon, therefore, the intensity decrement $\Delta I_{\nu}$ is
proportional to $-\sqrt{1-\theta^2}$.
We overplot the expected profile ($\Delta I_{\nu}\propto -\sqrt{1-\theta^2}$) based on equation (\ref{eq:expect})
with fiducial values on Figure \ref{fig:proj} (a) as a dashed line, offset so that the average intensity is
equal to zero.
The obtained distribution is not inconsistent with the expected one or with the flat distribution.
This implies the dominance of the statistical errors over the signal reflecting the possible cocoon
structure in the observed region.
We estimate the standard deviation from the flat distribution,
\begin{equation}
|\Delta I_{\nu}| = 0.28 ~ {\rm mJy ~ beam}^{-1}, \label{eq:xsigma}
\end{equation}
which sets the upper limit $y_{\rm upp}\lesssim 1.04\times 10^{-4}$ at the 95\% confidence level
(Eq.[\ref{eq:SZ}]).
We further examine the allowed upper limit for a Compton $y$ parameter by a chi-square test
of the data for the projection along the jet axis and the expected profile from a cylindrical cocoon
model $\Delta I_{\nu}\propto -\sqrt{1-\theta^2}$.
We find that as long as $\Delta I_\nu \le$ 0.32 mJy beam$^{-1}$ (or $y\le 1.19\times 10^{-4}$),
these two distributions can agree at the 95\% confidence level.
Herewith we denote these as $|\Delta I_\nu|^{\chi}$ and $y_\mathrm{upp}^{\chi}$.
We have also constructed the projection onto the jet axis.
If the cocoon has a prolate figure and lies in the plane of the sky,
the pressure scale length along the jet axis is larger than the declination range of
the observed field (see Figure\,\ref{fig:all}). Hence, the projection onto the jet axis would display
a more moderate decrement in contrast with the projection along
the jet axis.\footnote{If the jet axis is not straight or the inclination angle is not small,
the projected profile will be different from the one described here.
However, \citet{parma1996} concluded that a bend of the jet or the inclination are not so significant, from the contrast of the intensity of northern and southern lobes.}
Figure \ref{fig:proj} (b) shows the projection onto the jet axis.
There clearly exist a peak about $0.7^{\prime}$ north of the central row and two decrements toward both edges,
which are not consistent with the expectation from the isobaric cocoon model filled with thermal electrons (Section 2).
These decrements are due to the two dark regions north and south of the unidentified source ``D" (at
around $\alpha_{50}\sim 13^{\rm h} 58^{\rm m} 20^{\rm s}$: see Figure\,\ref{fig:full_image}).
The standard deviation from a flat distribution derived from the projection in this direction (onto the jet axis) is $|\Delta I_{\nu}| =$0.44 mJy beam$^{-1}$, which is larger than the statistical error estimated from the noise level of the original two-dimensional map in the process of the projection ($1.01/\sqrt{N_p}$, $\sim 0.27$\,mJy beam$^{-1}$ on average).
Even though the dark regions appearing in Figure \ref{fig:proj} may indeed suggest the existence of SZE
signals, we cannot rule out the possibility that these are not due to B1358+305.
In order to examine the dominant source of the fluctuations in Figures\,\ref{fig:proj} (a) and
\ref{fig:proj} (b), we analyzed the differential map in Section \ref{sect:analys} in the same
manner as the total map. Specifically, the differential map was constructed by dividing the observation period
into the former half and the latter half.
Figure\,\ref{fig:proj_diff} shows the fluctuations in the projections of the differential map
overplotted with dashed lines on the total map (solid lines, the same as Figure\,\ref{fig:proj}).
The projections on the both axes have similar amplitudes (the rms of the
projection along the jet axis is 0.27 mJy beam$^{-1}$ and the projection
onto the jet axis is 0.43 mJy beam$^{-1}$) as those of the total map, but
their distributions do not resemble the total map.
Taking into account the fact that real signals, including foreground or background
emission sources in the field of view, do not appear in the differential map,
we conclude that the fluctuations in Figure\,\ref{fig:proj_diff} are largely caused by the
excess atmospheric noises at 21 GHz.
\section{Discussion}\label{sect:discussion}
Due to the lack of definite information about the position of the edge of the cocoon, it is
difficult to define a concrete baseline. For this reason we
evaluated the relative variation of the intensity in the map. In order to improve the
signal-to-noise ratio, we made two projections on orthogonal directions. A differential map
analysis in Section \ref{sect:analys} shows that the main source of the relative fluctuations in
these projections is the excess atmospheric noise rather than the true SZE signal induced by the
hot electrons in B1358+305.
We discuss possible factors that may reduce the true amplitude of SZE in B1358+305 in comparison
to the estimate with equation (\ref{eq:expect}), including a non-uniform distribution of
thermal electrons.
In addition, the degree to which our results can affect the
previous studies on pressure balance of B1358+305 (Section \ref{sect:pressure}) is examined.
Finally, we direct attention on radio galaxies in clusters of galaxies to demonstrate the usefulness
of the SZE as a probe of the energetics of radio galaxy in a more general sense to guide
future observations. It is to be emphasized that the intensity fluctuations discussed in this
section should not be confused with a true SZE signal in B1358+305, but only as a rough guide
for a quantitative discussion.
\subsection{Thermal Evolution of the Cocoon} \label{sect:cooling}
The expected amplitude of SZE is estimated using a steady mean acceleration efficiency $\langle\xi_e\rangle$
(Eq.[\ref{eq:expect}]).
If $\langle\xi_e\rangle$ is significantly different from assumed, the expected value of $y$ is also affected.
In particular, different cooling timescales for the thermal and non-thermal electrons may alter
$\langle\xi_e\rangle$ from the fiducial value (=0.05 in Eq.[\ref{eq:expect}]) in time, especially in the central
region where the oldest population is contained. Since an underestimate of $\langle\xi_e\rangle$ leads to the
overestimate of the expected $y$ amplitude (Eq.\,[\ref{eq:expect}]), we examine whether the
different cooling timescales for the thermal and non-thermal electrons can significantly alter
$\langle\xi_e\rangle$ using a simple argument about exhaustion of the internal energy. Since the cooling
timescale is strongly dependent on the electron energy $\gamma$, we discuss synchrotron electrons
of high $\gamma$ ($\gamma\gtrsim 10^3$ or more) and thermal electrons ($\gamma\lesssim 10$ at
$T_e\lesssim 100$ keV) separately.
\citet{parma1996} derived the spectral ages of the non-thermal electrons for both the northern and
southern lobes using the compiled multi-frequency observation of B1358+305 from 325 MHz to 10550 MHz,
finding the break frequency to be 2.4 GHz from the integrated spectral energy distribution. Further
fitting of the multi-frequency images with the energy loss formula including synchrotron and
CMB inverse Compton losses leads to spectral ages of $\tau_\mathrm{spec}$ for B1358+305 in the range
$2.5\times 10^7$ yrs $\lesssim \tau_\mathrm{spec} \lesssim 7.5\times 10^7$ yrs.
Since the synchrotron cooling timescale and inverse Compton cooling timescales are proportional to
$\gamma^{-1}$, lower energy electrons have a longer cooling timescale.
The simple scaling of the above estimation by \citet{parma1996} indicates that an electron that
emits 325 MHz synchrotron light has a spectral age of about a few times of $\sim 10^8$ yrs, or even in the
extreme case $\sim 10^9$ yrs at best. We adopt $10^8$ yrs as the typical cooling time scale for
the non-thermal electrons.
On the other hand, a relativistic Maxwellian distribution of high temperature thermal electrons,
\begin{equation}
P_e(\gamma)d\gamma
= \frac{\gamma^5\beta^2\exp(-\gamma/\Theta)}{\Theta K_2(\Theta^{-1})}
d\gamma, \label{eq:thermal}
\end{equation}
spans over $\gamma\lesssim 10$ for $k_BT_e\lesssim $100 keV (where $K_2(x)$ is the second order modified
Bessel function, $\beta=v/c$, and $\Theta\equiv k_BT_e/m_ec^2$ is the non-dimensional electron temperature).
We estimate the cooling timescales following \citet{sarazin1999}.
Inverse Compton scattering of CMB photons, synchrotron self-Compton scattering, Coulomb cooling, and bremsstrahlung are taken into account as cooling processes. The density of the IGM and in B1358+305 is quite low
($n_e\lesssim 10^{-6}$ cm$^{-3}$; \citealt{parma1996}), and thus bremsstrahlung and
line cooling are negligibly small compared with the other processes.
In this low energy regime ($\gamma\lesssim 10$) and in the very poor environment (the density of the
surrounding IGM was estimated to be $1.4\times 10^{-7}$ cm$^{-3} \lesssim n_\mathrm{IGM} \lesssim
8.4\times 10^{-7}$ cm$^{-3}$; see Table 3 of \citealt{parma1996}), the major cooling mechanisms are
inverse Compton cooling and the Coulomb cooling, but the cooling time is longer than the Hubble time
($\tau_\mathrm{cool}\gtrsim 10^{10}$ yrs).
Therefore, the thermal energy of electrons is conserved for a typical cooling time of the
synchrotron electron ($\sim 10^8$ yrs).
Even if we adopt a cooling timescale for electrons that radiate low frequency 325 MHz photons
($\tau_\mathrm{cool}\lesssim 10^9$ yrs), it is unlikely that the cooling time of thermal electrons
is lower than that of non-thermal electrons.
In other words, as long as the radio lobes of B1358+305 are bright in synchrotron emission, the
thermal energy of electrons in B1358+305 does not change in time.
Therefore, it is unlikely that the lack of the detection of an intensity decrement in B1358+305 is
attributable to the loss of thermal electrons by cooling.
In addition to the cooling processes described above, some models emphasize the importance of adiabatic
expansion \citep{ito2008} or backflows of jet matter that escapes thermalization at the shock interaction
\citep{scheck2002}. These models predict a lower thermal pressure of electrons than the model of
\citet{begelman1989}. Quantitative evaluation of these effects has yet to be determined, but it is to be noted that our model
tends to overestimate the thermal pressure of the electrons in a cocoon.
If these effects that reduce the thermal pressure of electrons are significant
in the evolution of a radio galaxy, much more sensitive observations will be necessary.
\subsection{Non-Thermal Electrons in Radio Lobes} \label{sect:alt}
In this paper, we adopted a model based on the classical overpressured cocoon model with
a constant and uniform mean shock acceleration efficiency $\langle\xi_e\rangle$, and
a uniform distribution of thermal electrons within a cocoon as the simplest and most
optimistic interpretation of the observation (Section \ref{sect:model}).
However, in spite of the search for X-rays in radio galaxies in galaxy clusters,
the clear evidence for non-thermal electron acceleration at the external shock has not been
found except for objects such as Cen A \citep{croston2009}.
Although this lack of clear evidence may reflect the limited observational sensitivity or angular resolution,
the uncertainty in determining $\langle\xi_e\rangle$ is also likely to contribute.
Especially, at the internal shock or termination shock of the relativistic jets, $\xi_e$ can be
as large as close to unity, while $\xi_e$ at the non-relativistic external shock is much smaller.
In this case, the radio lobes within the shocked shell would have only a small population of
thermal electrons, whereas the shocked shell has a large fraction of thermal electrons as
in the first adopted model (Section \ref{sect:model}).
The spatial SZE profile of this model would differ significantly from that adopted (see Section
\ref{sect:structure}).
The amplitude of SZE caused by non-thermal power law electrons depends on the parameters
describing the electron energy distributions (e.g., the power law index, the lower and upper
cut-off energies, and the break energy if any).
The ratio of thermal and non-thermal SZE amplitudes is consequently dependent on these parameters.
If the non-thermal SZE amplitude is much smaller than thermal SZE in the radio lobes, the SZE spatial profile for this case
would not reflect the length passing through the cocoon, but the length passing through the shocked IGM shell.
Therefore, it may have a flatter SZE distribution as compared to that in Section \ref{sect:structure}
($\propto -\sqrt{1-\theta^2}$). This possibility makes the detection of SZE much more difficult
\citep[see e.g.,][]{pfrommer2005,colafrancesco2008,hardcastle2008}.
On the contrary, it may also be possible that the non-thermal SZE amplitude is similar to the thermal SZE.
In this case, the spatial profile does not differ from that in Section \ref{sect:structure}, and detection of SZE is
unlikely to be significantly more difficult compared with the cocoon model filled with thermal electrons (Section \ref{sect:model}).
Multi-frequency observation will be necessary to discriminate these distributions (see Section \ref{sect:xray} below).
Additionally, it is possible that a mature giant radio galaxy does not expand supersonically
in the lateral direction \citep[e.g.,][]{konar2009}.
In this case, the SZE signal peaks at the head of the external shock and decreases toward the central
region where we observe (see below).
Since the diffuse shock acceleration efficiency problem remains to be solved, we cannot
exclude either of these scenarios with different acceleration efficiencies.
In order to solve this problem and to obtain a deeper understanding of the subsequent dynamical
and thermal evolution of a radio galaxy, progress in both theoretical studies of shock acceleration
efficiency and higher sensitivity observations will be required.
Radio images and some X-ray images of radio galaxies suggest that the central
region close to the nucleus lack the energetic electrons that many hydrodynamic
models suggest.
If this applies to B1358+305, our results would correspond to genuine atmospheric noise.
However, the X-ray images of a FR-II radio galaxy MS0735+74221 and Cygnus A
show extended diffuse X-ray emissions that fill the central regions
\citep[see for a review,][and references therein]{mcnamara2007}. In addition,
\citet{hardcastle2008} observed bright FR-II galaxies at 90 GHz with the BIMA, and found
that some of their sample (3C 20 and 3C 388) have extended emissions that cover the
central regions.
Although there are ambiguities in the distributions of the thermal and synchrotron electrons,
these results imply that some FR-II radio galaxies seem to have electron populations
at the central regions away from the jets and hotspots.
We cannot tell if B1358+305 is deficient of electrons because of a small number of past observations
or limited sensitivity of our results.
Hence we conclude that our results are largely due to atmospheric noise, but we
cannot completely rule out the small SZE signal in B1358+305, and much higher sensitivity
is required in future observations.
\subsection{Expansion of B1358+305} \label{sect:pressure}
Since only upper limits on the $y$ parameter were obtained. we compare our results with past studies
of B1358+305 in the case where (1) the true pressure is close to that derived from the upper limit,
and (2) the true pressure is significantly lower than the obtained upper limit.
Consider the first case. Since the observed region does not extend over the entire cocoon, the
perpendicular projection onto the jet axis will show a flatter intensity
distribution than the parallel projection (i.e., the pressure scale length along the jet axis is
longer than the declination range of the observed field: see Figure\,\ref{fig:all}). Hence, it
is more easily subject to the contamination from the background emission sources and the excess
atmospheric noise. On the other hand, the right ascension range of the observation is expected to be
comparable to the cocoon width and the parallel projection along the jet axis is expected to represent
the structure of the cocoon (Section 4.2). Therefore, we employ the upper limit of the $y$ parameter
calculated with the projection along the jet axis ($|\Delta I_\nu|_{2\sigma}=0.56$ mJy beam$^{-1}$,
or $y_{\rm upp}\lesssim 1.04\times 10^{-4}$) in the discussion below.
If we assume that the obtained intensity fluctuation accompanies B1358+305, then
the electron pressure derives $p_e = 2.20\times 10^{-12}$ dyne cm$^{-2}$ (Eq.[\ref{eq:expect}]) for
$l$=300\,kpc.
If we take values obtained by the chi-square test $y_\mathrm{upp}^{\chi}$, the equivalent electron
pressure becomes $p_e = 2.51 \times 10^{-12}$ dyne cm$^{-2}$. On the other hand, \citet{parma1996}
deduced the jet ram-pressure to be $\ge 1.2\times 10^{-12}$ dyne cm$^{-2}$, which is comparable to the
equipartition value under the assumption that B1358+305 expands supersonically.
This result is quantitatively consistent with \citet{parma1996} in the sense that B1358+305
expands supersonically against the surrounding IGM.
However, the results obtained using the upper limit in our observation ($2.20-2.51\times 10^{-12}$ dyne cm$^{-2}$)
are almost double the result of \citet{parma1996} ($\ge 1.2\times 10^{-12}$ dyne cm$^{-2}$).
This difference would come from the assumed parameter value $\langle\xi_e\rangle=0.05$ (Eq.[\ref{eq:expect}]),
and may imply a larger mean acceleration efficiency ($\langle\xi_e\rangle \gtrsim 0.1$) in B1358+305
and significant contribution from non-thermal SZE of smaller amplitude than thermal SZE.
Since the intensity fluctuations obtained turned out to be dominated by the excess atmospheric
fluctuations, however, it is also possible that the true thermal electron pressure in B1358+305
is much lower than that calculated with $y_\mathrm{upp}$ or $y_\mathrm{upp}^{\chi}$ (the case (2)).
Because B1358+305 resides in a poor environment, it is difficult to directly measure the IGM pressure
around it. Instead we estimate the external pressure under the assumption that the environment resembles
an extremely poor galaxy cluster or a group of galaxies, and assume the IGM temperature to be
$\lesssim 10^7$\,K. For the density, we can take the cosmological baryon density at the redshift
$z=0.206$ as the reference value, and describe the IGM density with a density excess parameter
$\epsilon (\ge 1)$, $n_\mathrm{IGM}=\epsilon 4.3\times 10^{-7}$ cm$^{-3}$ \citep{spergel2003},
which sets the lower limit to the IGM pressure to be $\gtrsim \epsilon 5.9\times 10^{-16}$ dyne cm$^{-2}$.
In this case, if the true thermal electron pressure in B1358+305 is less than $10^{-4}\epsilon$ of
that inferred from our upper limit, it indicates that thermal electrons do not
play a major role in supporting the radio lobe against the external pressure. We note that this
conclusion strongly depends on the estimate of the IGM pressure and the SZE measurement.
Deeper X-ray observations for the pressure determination of the IGM and much sensitive SZE measurement
are required to be more conclusive.
\subsection{Comparison with X-ray Observations of Radio Galaxies in Clusters of Galaxies} \label{sect:xray}
Finally, we discuss the X-ray observations in conjunction with the SZE in radio galaxies as
effective means of probing the pressure of a cocoon \citep[also see][]{pfrommer2005}.
The observations by {\sl ROSAT} and {\sl Chandra} satellites have revealed that radio lobes of radio
galaxies centered at the cluster of galaxies generate cavities in the surface brightness distribution
of the X-ray emitting intracluster medium
\citep[e.g.,][]{bohringer1993,carilli1994,mcnamara2000,fabian2000,mcnamara2007}.
These are considered to be due to the pressure in the lobe overwhelming that of the surrounding hot ICM.
\citet{hardcastle2000} and \citeauthor{leahy2002} (2002), however, discovered that the ICM pressure
inferred by the X-ray emission obtained by {\sl ROSAT} observation of bright FR-II radio galaxies
is much higher than the minimum pressure estimated by the synchrotron luminosity in the radio band for
large ($\gtrsim 100$\,kpc) radio galaxies. A detailed study by \citet{leahy2001a} confirmed this
pressure discrepancy for 3C388. These results suggest the dominance of the other forms of pressure above
that due to the synchrotron electrons (and magnetic field) in the radio lobes.
\citet{hardcastle2000} argued that plausible candidates for carrying the ``invisible" pressure are
non-thermal protons, magnetic field with strength differing from that derived from the minimum
energy condition ($\simeq B_{\rm eq}$), or low energy electrons which do not emit strong radiation
in the observed energy bands.
If we assume that $\xi_e$ in the internal shock is as small as that in the external shock,
the radio lobes could have a significant amount of thermal electrons (Section\,\ref{sect:model}).
In this case, the pressure of the thermal electrons could be one of the candidates for
the origin of the ``invisible" pressure.
Since SZE is sensitive only to electron energy, it is a good probe of either the thermal electrons
or non-thermal electrons of low energy which do not emit high frequency emission in the observation bands.
The observational different signature between thermal and non-thermal SZE is the
frequency of zero amplitude \citep[e.g.,][]{birkinshaw1999}. Specifically, the thermal SZE has the null
frequency ($=218$\,GHz) which is almost independent of the gas temperature,
whereas the null frequency of non-thermal SZE changes with the electron energy distribution parameters.
Hence, future multi-frequency observations that covers the null frequency of thermal SZE will be a tool to
probe the electron contribution to the ``invisible" pressure components in the cocoon.
On the other hand, if $\xi_e$ of the internal shock is larger than that of the external shock and close to unity,
resulting in a significant difference in the fractions of thermal and non-thermal
electrons in the shocked shell and the radio lobes inside, SZE studies become much more difficult.
In this case, the multi-frequency observation should be carried out over more than one region.
For example, multi-frequency observations of the head-top region of the bow shock will be able to
determination the contribution of mainly thermal electrons formed at the external shock. This can be
used to correct the observation of the central region apart from the shock-head, where
the thermal and non-thermal SZE coexist.\footnote{For the radio galaxy
in a galaxy cluster, another difficult problem is to distinguish the contribution from the unshocked ICM.
In order to examine this contribution, multi-dimensional modeling of a radio galaxy embedded in the ICM
is necessary.}
In summary, the SZE can be a probe of electron contribution to the ``invisible" pressure in a radio
galaxy, but discrimination between thermal and non-thermal contributions requires sensitive
multi-frequency and multi-region observations in the future.
For the other sources of pressure, it is unlikely that the magnetic field is significantly different
from $B_\mathrm{eq}$. For example, X-ray observations have shown that majority of the radio galaxies
tend to have larger particle energy than that of the magnetic field \citep[for a recent results, see][and
references therein]{isobe2009}. Protons, as another possible candidate for the ``invisible" pressure, on
the other hand, would lead to an SZE much smaller than that given by equation (\ref{eq:expect}). The
existence of high-energy protons should be probed in the hard X-ray and/or $\gamma$-ray regime
\citep{scheck2002}.
\section{Conclusion}
We have reported the results of a trial observation of SZE in a giant radio galaxy B1358+305 with
the Nobeyama 45-m telescope at 21 GHz. By performing the imaging observation, we have obtained the most
stringent upper limit achieved for the Compton $y$ parameter in a radio galaxy. The obtained upper
limit is close to the expected value derived for a low acceleration efficiency of synchrotron electrons
at the shock (the low value $\langle \xi_e\rangle$, see Eq.\,[\ref{eq:expect}]), but detailed analysis shows that the
obtained intensity fluctuation is likely to be caused by the excess atmospheric noise.
If we assume that the obtained intensity fluctuations in the observed region were due to the thermal
electrons in the cocoon of B1358+305, our results are qualitatively consistent with the supersonic expansion of
B1358+305 but for the original assumption $\langle\xi_e\rangle$ being too small.
Alternatively, if the true pressure of thermal electrons is much lower than that derived from
the obtained upper limit, our observation is
not sensitive enough to derive any definitive conclusions on the
pressure balance of B1358+305.
Since the SZE is sensitive only to electron energy, it would serve as a probe of the pressure components
in radio galaxies as well as galaxy clusters. Future high sensitive multi-frequency SZE observations of
multi-regions will provide an important information about the contributions of thermal or low-energy
non-thermal electrons to the total pressure of a radio galaxy. This may provide a clue for disentangling
the pressure discrepancy between the surrounding ICM derived by X-ray observations and the minimum energy
of radio galaxies at the centers of clusters of galaxies \citep{hardcastle2000,leahy2002}.
Though we have failed to obtain a definitive SZE signal with the Nobeyama 45-m telescope, higher
frequency observations using either a large field-of-view (e.g., $\gtrsim 5^{\prime}$), multi-beam
receivers on large (e.g., $\gtrsim$ 30 m) single dish telescopes, or interferometers with a large number
of small dishes (e.g., $\lesssim 50$ cm) similar to the AMiBA
project\footnote{See {\tt http://amiba.asiaa.sinica.edu.tw/} for the AMiBA project.} might lead to
the detection of SZE of a cocoon. The SZE would provide a new observational tool to probe the energetics of
radio galaxies, along with the projects to detect low frequency radio emission from non-thermal low
energy electrons like the Long Wavelength Array (LWA: \citealp{harris2005}).
\acknowledgements
We greatly appreciate N. Kuno and H. Ezawa for their help in observations with the 45-m telescope at Nobeyama Radio Observatory, and the other members of NRO that supported us during the observation.
MY especially gives her thanks to E. Komatsu for providing his programs for data analysis and
to R. Taam for his critical reading of the manuscript.
N.S. is supported by the Sumitomo Foundation and Grant-in-Aid for scientific Research Fund (No.11640235).
{\it Facilities:} \facility{NRO(NAOJ)}.
|
3,212,635,537,983 | arxiv | \section{Introduction}
Magnetic switching is the single most important operation for any
modern magnetic storage device, where a magnetic field is employed to
switch microscopic spins from one direction to another. However, as
the areal density increases, the switching speed becomes a major
bottleneck for future technological advancement. A possible solution
emerged when Beaurepaire {\it et al. } \cite{eric} reported that a 60-fs laser
pulse reduced the spin moment of ferromagnetic nickel films within 1
ps. Their finding heralded the arrival of
femtomagnetism {\cite{ourreview,prl00,rasingreview}}, and
research efforts intensified immediately\cite{kimel,np09}. However,
for over a decade, the focus has been on demagnetization, not magnetic
switching. A major breakthrough came when Stanciu and coworkers
\cite{stanciu} demonstrated that a single laser pulse could
permanently switch the magnetic spin orientation in amorphous GdFeCo
samples. This all-optical helicity-dependent spin switching (AOS)
ignited the research community since it may be an alternative to the
current magnetic storage technology\cite{rasingreview}. However, most
AOS samples are amorphous \cite{hass,mangin} and are hard to simulate
without significant approximations. To this end, a unified
understanding is still missing, but several promising mechanisms have
been proposed, which include the inverse Faraday effect
\cite{stanciu,vahaplarprb}, spin-flip stimulated Raman scattering
\cite{gr,pop}, magnetic circular dichroism\cite{khorsand}, magnetic
sublattice competition\cite{mentink}, pure thermal effect
\cite{ostler,atxitia} and ultrafast exchange scattering\cite{bar}.
Recently, Lambert {\it et al. } \cite{lambert} reported AOS in an ultrathin
ferromagnetic [Co(0.4 nm)/Pt(0.7 nm)]$_3$ multilayer.
{Medapalli {\it et al. } \cite{med} demonstrated that the
helicity-dependent switching in Co/Pt proceeds in two steps
\cite{hadri}.} Such a system is much more amenable to the
simulation without any major approximation, and its magnetic
properties have been well known for some time\cite{sod}. It is likely
that a detailed study of such a system may shed new light on AOS.
\section{Spin reversal theory}
We employ a thin film of $101\times 101\times 4$ or 40,804 lattice
sites in a simple cubic structure (see the top half of
Fig. \ref{fig1}) {with an open boundary condition}. Each
site has a spin ${\bf S}_i$ which is exchange-coupled to the nearest
neighboring spins through the exchange interaction $J_{ex}$. {Our
Hamiltonian \cite{jpcm11,jpcm13,epl15,mplb16}}, which is
often used in magnetic multilayers\cite{hs}, is \begin{eqnarray} H&=&\sum_i \left
[\frac{{\bf p}_i^2}{2m}+V({\bf r}_i) +\lambda {\bf L}_i\cdot {\bf S}_i
-e {\bf E}({\bf r}, t) \cdot {\bf r}_i\right
]\nonumber\\ &-&\sum_{ij}J_{ex}{\bf S}_i\cdot {\bf S}_{j}, \label{ham}
\end{eqnarray} where the first term is the kinetic energy operator of the
electron, the second term is the potential energy operator, $\lambda$
is the spin-orbit coupling in units of eV/$\hbar^2$, $ {\bf L}_i$ and
$ {\bf S}_i $ are the orbital and spin angular momenta at site $i$ in
the unit of $\hbar$, respectively, and {\bf p} and {\bf r} are the
momentum and position operators of the electron, respectively. To
minimize the number of parameters, we choose a spherical harmonic
potential $V({\bf r}_i)=\frac{1}{2}m\Omega^2 {\bf r}_i^2$ with system
frequency $\Omega$, but this approximation can be lifted when accurate
potentials are known. {Our model represents a small step
towards a complete model}.
We assume that the electron moves along the $z$
axis with an initial velocity of 1 nm/fs in the harmonic potential, so
the initial orbital angular momentum is zero. The last term is the
exchange interaction, and $J_{ex}$ is the exchange integral in units
of eV/$\hbar^2$. Although our main interest is in ferromagnets, the
same Hamiltonian can describe both antiferromagnets and ferrimagnets.
Such a Hamiltonian contains the necessary ingredients for AOS.
Figure \ref{fig1} shows that a laser pulse propagates along the $+z$
axis; its amplitude is attenuated according to Beer's law ${e}^{-z/d}$
(along $+z$), where $d$ is the penetration depth. The bottom half of
Fig. \ref{fig1} illustrates our idea of spin torque to switch spins.
For convenience, the spatial dimension is measured in the unit of the
lattice site number along each direction, so that all the spatial
variables are dimensionless or in the unit of the site number. The
laser spot is centered at $x_c=51$ and $y_c=51$ with radius $r$ and
lateral spatial profile\cite{vahaplarprb}
$e^{-[(x-x_c)^2+(y-y_c)]^2/r^2}$ (in the $xy$ plane). The laser
electric field is described by \begin{equation} {\bf E}({\bf r},t)={\bf
A}(t)\exp[-\frac{(x-x_c)^2+(y-y_c)^2}{r^2}-\frac{z}{d}], \end{equation} where
$x$ and $y$ are the coordinates in the unit of the site number. Since
in the following our spins are all initialized along the $-z$ axis, we
choose a left-circularly polarized field ${\bf A}(t)$ which has a
Gaussian shape $ {\bf A}(t)=A_0{\rm e}^{-t^2/T^2}[- \sin(\omega t)
\hat{x}+\cos(\omega t) \hat{y} ], $ where $\omega$ is the laser
carrier frequency, $T$ is the laser pulse duration, $A_0$ is the laser
field amplitude, $t$ is time, $\hat{x}$ and $\hat{y}$ are unit
vectors, respectively. We choose $T=100$ fs. {We} only consider a
resonant excitation where the laser photon energy $\hbar\omega=1.6$ eV
matches the system energy $\hbar\Omega$; for an off-resonant
excitation, we refer the reader to a prior study\cite{epl15}. In
transition metals, the penetration depth is about 14 nm, which
corresponds to 30 layers, so we choose $d=30$. To compute the spin
evolution, we employ the Heisenberg's equation of motion,
$i\hbar\dot{A}=[A,H]$, {where we make the time-dependent Hartree-Fock
approximation, so that all the operators are replaced by their
respective expectation values}, and then we solve the equation
numerically. {Our calculation of the spin change is similar
to that of Wienholdt {\it et al. } \cite{wienholdt} though they used a thermal
field. }
\section{Dependence of spin switching on spin angular momentum}
We choose eight initial spin momenta $S_z(0)$ from $0.2\hbar$ to
1.6$\hbar$ in steps of 0.2$\hbar$, which covers most magnetic
materials. For each $S_z(0)$, we vary the laser field
amplitude {\cite{prl00,med}} $A_0$ from 0.01 to 0.08 $\rm V/\AA$ in steps
of 0.002 $\rm V/\AA$. This step is tedious but necessary, since
different $S_z(0)$ have different optimal field amplitudes for spin
reversal. We fix the spin-orbit coupling at $\lambda=0.06 {\rm
eV}/\hbar^2$, the exchange interaction $J_{ex}$ at 1 ${\rm
eV}/\hbar^2$, and the spot radius of $r=100$. { The spins are
initialized along the $-z$ axis, equivalent to applying a magnetic
uniaxial anisotropy.} A spin reversal is considered achieved if the
$z$ component spin angular momentum $S_z$ changes from a negative
value to a large and positive value at the end of the dynamics. Figure
\ref{fig20}(a) shows the normalized and system-averaged spin as a
function of time for each $S_z(0)$ at its respective optimal laser
field amplitude. All the curves, except $S_z(0)=0.2\hbar$, are
vertically shifted for clarity. The dotted horizontal lines denote
$0\hbar$. We start with $S_z(0)=0.2\hbar$, and we see that the spin
does not switch and only oscillates around $0\hbar$ with a period
determined by the product of $\lambda$ and
$S_z(0)$\cite{epl15,jap15}. When we increase $S_z(0)$ to 0.4$\hbar$,
the oscillation is attenuated and the final spin is barely above
$0\hbar$. And the situation does not change much for
$S_z(0)=0.6\hbar$. However, when we continue to increase $S_z(0)$
above $0.8\hbar$, the spin ringing is strongly reduced, and the final
spin settles down at a large positive value, an indication of spin
reversal. Above $0.8\hbar$, the situation gets better. For this
reason, we define a critical spin angular momentum $S_z^c=0.8\pm 0.2
\hbar$ for AOS.
{ To quantify AOS, we define the spin switchability as $
\eta=\frac{S_z^f}{S_z(0)}\times 100\%,$ where $S_z^f$ is the final
spin angular momentum. This definition is different from that of
Vahaplar {\it et al.}\cite{vahaplarprb}}. We fix $S_z(0)=1.2\hbar$, but
change the spin-orbit coupling $\lambda$. Note that our conclusions
are the same for different $S_z(0)$ as far as it is above $S_z^c$.
Figure \ref{fig20}(b) shows that a minimum $\lambda$ of 0.04
eV$/\hbar^2$ is required to reverse spins. Too small a $\lambda$ only
leads to a strong spin oscillation, regardless of the laser field
amplitude. This indicates a unique role of spin-orbit coupling (SOC)
in AOS. The roles of the exchange interaction and laser field
amplitude are shown in Fig. \ref{fig20}(c), where we fix
$S_z(0)=1.2\hbar$, $r=100$ and $\lambda=0.06 \rm~eV/\AA^2$. We notice
that as $A_0$ increases, $S_z$ sharply increases and reaches its
maximum. If we increase it further, $S_z$ is reduced since the spin
overshoots, and an asymmetric peak is formed. This constitutes our
first criterion that the laser amplitude must fall into a narrow
region for AOS to occur. {This is consistent with Medapalli
{\it et al.}'s finding (see Fig. 1(c) of their paper \cite{med}); such a
helicity-dependent switching also agrees with another study by El
Hadri {\it et al. } \cite{hadri}}. {These agreements do not
necessarily validate all the aspects of our model but instead they
simply suggest that our model may offer an alternative to the
existing models.}
If we increase $A_0$ further, a second
peak appears since the spin re-switching starts. These double peaks
do not appear for a smaller $S_z(0)$. We find that the exchange does
not change this dependence a lot.
\section{Phase diagram of spin reversal}
We construct a phase diagram of spin reversal ($\eta-S_z(0)$) in
Fig. \ref{fig3}(a) for thirteen $S_z(0)$'s and two radii of the laser
spot, $r=100$ and 50. For $\eta$ to exceed 50-60\%, $S_z(0)$ must be
higher than the critical value of $S_z^c=0.8\pm 0.2\hbar$. The
long-dashed line denotes $S_z^c$. We see that the nickel's spin
momentum is well below $S_z^c$, which explains why nickel has never
been used for AOS. Co is on the threshold. In Co-Pt granular
samples\cite{figueroa}, the effective spin magnetic moment per $3d$
hole is 0.77 $\mu_{\rm B}$; since there are 2.49-2.62 holes, the spin
angular momentum is 0.96$\hbar$, satisfying this criterion. In the
ultrathin ferromagnetic [Co(0.4 nm)/Pt(0.7 nm)]$_3$
films\cite{lambert}, due to the reduced dimensionality, the enhanced
spin moment greatly increases the chance for AOS. The empty boxes in
Fig. \ref{fig3}(a) represent the case with $r=50$ (which is close to
the switch limit), where only a small portion of the sample is exposed
to the laser light. We see that the switchability reduces sharply
since the laser fluence on lattice sites away from the center of the
laser beam becomes very weak and is not strong enough to reverse spins
on those sites. {Since the essence of AOS is rooted in
spin-orbit coupling and all the switchabilities are obtained at the
optimal field amplitude, we do not expect that a more accurate
potential would change the phase diagram strongly.} Our criterion
not only applies to ferromagnets, but also to ferrimagnets. Figure
\ref{fig3}(b) illustrates that each of the major elements in all the
11 GdFeCo and TbFe alloys\cite{hassdenteufel2015} has the effective
spin above $S_z^c$. This constitutes strong evidence that our finding
has a broader impact on the ongoing research in all-optical spin
switching.
\section{Emergence of optical spin-orbit torque}
While the effect of the laser field amplitude on AOS is obvious
\cite{vahaplarprl}, how the initial spin $S_z(0)$ affects the spin
switching is not obvious. We examine how the spin evolves with time.
For a spin at site $i$, the spin angular momentum ${\bf S}$ precedes
according to \begin{equation} \frac{d{\bf S}_i}{dt}= \sum_{j(i)} J_{ex} {\bf
S}_i\times {\bf S}_j +\lambda ({\bf L}_i \times {\bf S}_i),
\label{s} \end{equation} where the two driving terms on the right-hand
side represent two torques. The first is the Heisenberg exchange
torque $\tau_{ex}=\sum_{j(i)} J_{ex} {\bf S}_i\times {\bf S}_j$. Since
all the spins are ferromagnetically ordered, this torque is very
small. The second one is the spin-orbit torque (SOT),
$\tau_{soc}=\lambda ({\bf L}_i \times {\bf S}_i)$, {which may
serve as a source term for the inverse Faraday effect
\cite{john,berritta}.} Before the laser excitation, $\tau_{soc}$ is
small, since in solids the orbital angular momentum ${\bf L}$ is
largely quenched. With the arrival of the laser pulse, ${\bf L}$ is
boosted sharply {\cite{john}} (see Fig. \ref{orbital}) and
helicity-dependent, where $J_{ex}=1{\rm eV}/\hbar^2$, and
$S_z(0)=1.2\hbar$, but three components of the orbital angular
momentum behave differently. $L_x$ and $L_y$ are mostly negative, but
$L_z$ is positive. Around 50 fs, $L_x$ reaches $-0.24\hbar$, while
$L_y$ swings to $-0.16\hbar$ and the change in $L_z$ is smaller,
around $0.04\hbar$. All three components settle down to zero around
200 fs. This is very important, since if the orbital momentum were big
after the laser field is gone, the spin would oscillate very strongly
and could not be reversed faithfully. Thus, through the spin-orbit
coupling, the laser field increases the orbital angular momentum, and
subsequently $\tau_{soc}$ is boosted. For this reason, Tesavova
{\it et al. }\cite{tesarova} called $\tau_{soc}$ the optical spin-orbit torque,
or femtosecond spin-orbit torque by Lingos {\it et al. } \cite{lingos}.
We choose two initial spin momenta, $S_z(0)=0.3\hbar$ and 1.2$\hbar$,
with all the spins initialized along the $-z$ axis (see the light blue
arrows in Figs. \ref{fig4}(a) and (b)). Figure \ref{fig4}(a) shows
that at $0.3\hbar$ the spin undergoes strong oscillations and shows
many spirals, but does not settle down to the $+z$ axis after the
laser pulse is gone (see the red arrow). By contrast, at $1.2\hbar$
the spin flips over from the $-z$ to $+z$ axis within 110 fs, without
strong oscillation (see the solid red arrow). To understand why the
initial spin angular momentum has such a strong effect on AOS, Figure
\ref{fig4}(c) shows that $\tau_{soc}$ at $0.3\hbar$ is very weak,
around 0.01 $\hbar$/fs, and more importantly, it rapidly swings
between positive and negative values, both of which are detrimental to
the spin reversal. At $S_z(0)=1.2\hbar$, $\tau_{soc}$ is positive and
large, which allows the spin to switch over successfully. This
suggests that SOT offers an alternative path to AOS (see the bottom
figure of Fig. \ref{fig1}), and it acts like an effective magnetic
field, which has been sought after in the literature
\cite{vahaplarprb,ostler} for nearly a decade. At 1.2$\hbar$, we
time-integrate the torque from -200 to +200 fs and find that the
time-averaged torque corresponds to 65 T of a magnetic field. In
spintronics, the spin transfer torque heavily relies on the high
electric current\cite{hs,stiles2}. Such a large SOT, if implemented in
real experiments, should significantly reduce the requirement of huge
electric current for spintronics\cite{bokor}, and thus opens a door
for rapid applications in storage technology\cite{wolf}.
\section{Conclusion}
We have investigated all-optical spin switching in
40,000 ferromagnetic spins. We identify that it is the laser-induced
optical spin-orbit torque that determines the fate of spin switching.
The spin-orbit torque sensitively depends on the value of the initial
spin momentum of each active element in a sample, regardless of the
types of magnets. To switch, each active element must have its
effective spin angular momentum larger than $(0.8\pm
0.2)\hbar$. { This means that the switchability in Fe, Gd
and Tb is likely to be higher than Co and Ni. PMA observed in
various AOS materials \cite{mangin} seems to be an indication of
enhanced spin moment, which is in line with our theory. } The ps
all-optical spin switching observed in ferrimagnets is associated with
the weak exchange coupling; in ferromagnets, with a stronger coupling,
the switching is much faster. SOT is so large that it will
significantly reduce the electric current used in spintronics. After
our present study was finished, we noticed a recent publication by
Bokor's group \cite{bokor} to use a laser to assist magnetization
reversal. A combination of photonics and spintronics represents the
arrival of photospintronics\cite{mondal}.
\acknowledgments We would like to thank Dr. Hassdenteufel for sending
us the experimental results\cite{hassdenteufel2015}. This work was
solely supported by the U.S. Department of Energy under Contract
No. DE-FG02-06ER46304. Part of the work was done on Indiana State
University's quantum cluster and high-performance computers. The
research used resources of the National Energy Research Scientific
Computing Center, which is supported by the Office of Science of the
U.S. Department of Energy under Contract No. DE-AC02-05CH11231. This
work was performed, in part, at the Center for Integrated
Nanotechnologies, an Office of Science User Facility operated for the
U.S. Department of Energy (DOE) Office of Science by Los Alamos
National Laboratory (Contract DE-AC52-06NA25396) and Sandia National
Laboratories (Contract DE-AC04-94AL85000).
|
3,212,635,537,984 | arxiv | \section{Introduction}
Quantum tori (also known as noncommutative tori and irrational rotation algebras) are landmark examples in noncommutative geometry. These algebras have featured in many directions in physics, such as the study of the quantum Hall effect \cite{Bellissard-original, Bellissard-van-Elst-Schulz-Baldes, Xia-qhe}, Matrix theory \cite{Connes-Douglas-Schwarz-matrix-theory-1998}, string theory \cite{Seiberg-Witten} and deformation quantisation \cite{Rieffel-deformation-quantization}.
Quantum tori have been heavily studied from the perspective of operator algebras \cite{Effros-Hahn-memoirs-1967,Pimsner-Voiculescu-crossed-products-1980, Rieffel-1981} and were later taken as a fundamental example in noncommutative
geometry (see \cite{Connes1980}, \cite[Chapter 12]{green-book} and \cite{CM2014}). In the context of foliation theory, quantum tori are studied as the $C^*$-algebra associated to a Kronecker foliation \cite[Chapter 2, Section 9.$\beta$]{Connes1994}.
A. Connes introduced the quantised calculus in \cite{Connes-ncdg-1985} as an analogue of the algebra of differential forms in a noncommutative setting, and later explored the link with the action functional of Yang-Mills theory \cite{Connes1988}.
Connes successfully applied quantised calculus in computing the Hausdorff measure of Julia sets and limit sets of Quasi-Fuchsian groups in the plane \cite[Chapter 4, Section 3.$\gamma$]{Connes1994} (for a more recent exposition see \cite{CSZ,CMSZ}).
The core ingredients of the quantised calculus, as outlined in \cite{Connes-ncdg-1985}, are a separable Hilbert space $H$, a unitary self-adjoint operator $F$ on $H$ and a $C^*$-algebra $\mathcal{A}$ represented on $H$ such that
for all $a \in \mathcal{A}$ the commutator $[F,a]$ is a compact operator on $H$. Then the quantised differential of $a \in \mathcal{A}$ is defined to be the operator $\,{\mathchar'26\mkern-12mu d} a = {\rm{i}}[F,a]$. The compact operators
on $H$ are described by Connes as being analogous to infinitesimals, and the rate of decay of the sequence of singular values:
\begin{equation*}
\mu(n,T) := \inf\{\|T-R\|\;:\;\mathrm{rank}(R)\leq n\}
\end{equation*}
corresponds in some way to the ``size" of the infinitesimal $T$ (see \cite{Connes1995}). In this setting one can quantify the smoothness of an element $a \in \mathcal{A}$
in terms of the rate of decay of $\{\mu(n,\,{\mathchar'26\mkern-12mu d} a)\}_{n=0}^\infty$. Of particular interest are those elements $a \in \mathcal{A}$ which satisfy:
\begin{align*}
\mu(n,\,{\mathchar'26\mkern-12mu d} a) &= O((n+1)^{-1/p}),\quad n\to \infty,\text{ or,}\\
\sum_{n=0}^\infty \mu(n,\,{\mathchar'26\mkern-12mu d} a)^p &< \infty,\text{ or,}\\
\sup_{n \geq 1} \frac{1}{\log(n+2)} \sum_{k=0}^n \mu(k,\,{\mathchar'26\mkern-12mu d} a)^p &< \infty\,,
\end{align*}
for some $p \in (0,\infty)$. The first condition stated above is that $\,{\mathchar'26\mkern-12mu d} a$ is in the weak-Schatten ideal ${\mathcal L}_{p,\infty}$, the second condition
is for $\,{\mathchar'26\mkern-12mu d} a$ to be in the Schatten ideal ${\mathcal L}_p$, and the final condition is that $|\,{\mathchar'26\mkern-12mu d} a|^p$ is in the Macaev-Dixmier ideal $\mathcal{M}_{1,\infty}$ \cite[Chapter 4, Section 2.$\beta$]{Connes1994} (see also \cite[Example 2.6.10]{LSZ2012}).
The link between quantised calculus and geometry is discussed by Connes in \cite{Connes1988}. A model example for quantised calculus is to take a compact Riemannian spin manifold $M$ with Dirac operator $D$,
and define $H$ to be the Hilbert space of square integrable sections of the spinor bundle. The algebra $\mathcal{A} = C(M)$ of continuous functions on $M$ acts by pointwise multiplication on $H$, and one defines
\begin{equation*}
F := \chi_{[0,\infty)}(D)-\chi_{(-\infty,0)}(D).
\end{equation*}
One then has $\,{\mathchar'26\mkern-12mu d} f = {\rm{i}}[F,M_f]$, where $M_f$ is the operator on $H$ of pointwise multiplication by $f$.
In quantised calculus the immediate question is to determine the relationship between the degree of differentiability of $f \in C(M)$ and
the rate of decay of the singular values of $\,{\mathchar'26\mkern-12mu d} f$. In general, we have the following:
\begin{equation*}
f \in C^\infty(M) \Rightarrow |\,{\mathchar'26\mkern-12mu d} f|^d \in \mathcal{M}_{1,\infty},
\end{equation*}
where $d$ is the dimension of the manifold $M$ \cite[Theorem 3.1]{Connes1988}.
For certain special cases it is possible to obtain a far more precise understanding of the relationship between the smoothness of $f$ and the singular values of $\,{\mathchar'26\mkern-12mu d} f$.
The simplest example is to take the unit circle ${\mathbb T} = \{z \in {\mathbb C}\;:\; |z| = 1\}$,
with $\mathcal{A} = C({\mathbb T})$, $H = L_2({\mathbb T})$ and the standard choice of $F$ in this setting is the Hilbert transform. Then by a result of V. Peller \cite[Theorem 7.3]{Peller2003}, we have that
for any $p \in (0,\infty)$: $\,{\mathchar'26\mkern-12mu d} f \in {\mathcal L}_p$ if and only if $f$ is in the Besov space $B^{1/p}_{p,p}({\mathbb T})$. Peller's work has been extended to obtain even more precise relationships between $f$ and the singular values of $\,{\mathchar'26\mkern-12mu d} f$, for example
L. Gheorghe \cite{Gheorghe2001} found necessary and sufficient conditions on $f$ to ensure that $\,{\mathchar'26\mkern-12mu d} f$ is in an arbitrary Riesz-Fisher space. For more details from a quantised calculus perspective, see \cite[Chapter 4, Section 3.$\alpha$]{Connes1994}.
In higher dimensions, the relationship between $f$ and $\,{\mathchar'26\mkern-12mu d} f$ has also been studied \cite{JW1982,RS1989,CST1994}. To illustrate the situation, consider the $d$-dimensional torus ${\mathbb T}^d${, $d \geq 2$}. The appropriate Dirac operator in this setting
is:
\begin{equation*}
D = \sum_{j=1}^d -{\rm{i}}\gamma_j\otimes \partial_j,
\end{equation*}
where $\partial_j$ denotes differentiation with respect to the $j$th coordinate on ${\mathbb T}^d$, and $\{\gamma_1,\ldots,\gamma_d\}$ denotes the $d$-dimensional Euclidean gamma matrices,
which are self-adjoint $2^{\lfloor \frac{d}{2}\rfloor}\times 2^{\lfloor \frac{d}{2}\rfloor}$ complex matrices satisfying $\gamma_j\gamma_k+\gamma_k\gamma_j = 2\delta_{j,k}1$. The operator $D$ may be considered as an unbounded self-adjoint operator
on the Hilbert space $L_2({\mathbb T}^d,{\mathbb C}^{2^{\lfloor \frac{d}{2}\rfloor}})$.
The corresponding operator $F$ is a linear combination of Riesz transforms. The commutators of Riesz transforms and multiplication operators are studied in classical harmonic analysis: S. Janson
and T. Wolff \cite{JW1982} proved that for $\,{\mathchar'26\mkern-12mu d} f$ to be in ${\mathcal L}_p$ when $p > d$ it is necessary and sufficient that $f$ is in the Besov space $B^{\frac{d}{p}}_{p,p}({\mathbb T}^d)$. On the other hand, Janson and Wolff also proved that if $p \leq d$ then
$\,{\mathchar'26\mkern-12mu d} f \in {\mathcal L}_{p}$ if and only if $f$ is a constant.
A far more general characterisation of the spectral properties of commutators of Riesz transforms and multiplication operators was obtained by R. Rochberg and S. Semmes \cite{RS1989}.
To date, investigations on the relationship between $f$ and $\,{\mathchar'26\mkern-12mu d} f$ have been limited to the commutative case. To the best of our knowledge, the results treated in this paper are the
first concerning quantum differentiability in the strictly noncommutative setting.
A related direction of research concerning quantised differentials is trace formulae. As early as \cite{Connes1988} it was known that for functions on compact manifolds, it is possible
to express the Dixmier trace $\mathrm{tr}_\omega(|\,{\mathchar'26\mkern-12mu d} f|^p)$ as an integral of a derivative of $f$ (See Subsection \ref{operator notation subsection} for the relevant definitions, and \cite[Chapter 6]{LSZ2012} for details
on Dixmier traces).
If $f \in C^\infty({\mathbb T}^d)$, let $\nabla f = (\partial_1f,\partial_2f,\ldots,\partial_df)$ be the gradient vector of $f$, and let $\|\nabla f\|_2 = \left(\sum_{j=1}^d |\partial_j f|^2\right)^{\frac{1}{2}}$. Then as a special case of \cite[Theorem 3.3]{Connes1988} we have:
\begin{equation}\label{torus trace formula}
\mathrm{tr}_\omega(|\,{\mathchar'26\mkern-12mu d} f|^d) = k_d\int_{{\mathbb T}^d} \|\nabla f(t)\|_{2}^ddm(t),
\end{equation}
where $k_d$ is a constant, and $m$ denotes the flat measure on ${\mathbb T}^d$ (i.e., the Haar measure). From the perspective of noncommutative geometry this formula ``shows how to pass from quantized $1$-forms to ordinary forms, not by a classical limit, but by a direct application
of the Dixmier trace" \cite[Page 676]{Connes1988}. It is also possible to prove a similar formula for functions on the non-compact manifold ${\mathbb R}^d$, and indeed
to extend the class of traces on the left hand side of \eqref{torus trace formula} to the much larger class of all continuous normalised traces on ${\mathcal L}_{1,\infty}$ \cite{LMSZ2017}.
Recently there has been work on generalising the methods of harmonic analysis on tori to quantum tori.
On a noncommutative torus ${\mathbb{T}_\theta^d}$ (defined in terms of an arbitrary antisymmetric real $d\times d$ matrix $\theta$), it is possible to define analogues of many of the tools
of harmonic analysis, such as differential operators and function spaces \cite{XXY2018} (see Section \ref{nc tori subsection}). In this setting, there are analogues of all of the components of \eqref{torus trace formula}, although the integral on the right
must be replaced with the canonical trace associated to ${\mathbb{T}_\theta^d}$. However the most straightforward generalisation of \eqref{torus trace formula}
to ${\mathbb{T}_\theta^d}$ is actually false. In this paper we state and prove a correct version of \eqref{torus trace formula} for noncommutative tori (Theorem \ref{trace formula}). The formula is stated for an appropriate class of elements
$x \in { L_2({\mathbb{T}_\theta^d})}$ as:
\begin{equation}\label{nc torus trace formula}
\varphi(|\,{\mathchar'26\mkern-12mu d} x|^d) = c_d\,\int_{\mathbb{S}^{d-1}} \tau\Bigg(\Big(\sum_{j=1}^d |{\partial}_j x-s_j\sum_{k=1}^d s_k{\partial}_k x|^2\Big)^{\frac{d}{2}}\Bigg)\,ds.
\end{equation}
Here, $\tau$ is the canonical trace associated to the noncommutative torus, and $c_d$ is a certain constant depending on $d$ (different to the constant $k_d$ in \eqref{torus trace formula}). The integral
is over $s = (s_1,\ldots,s_d)$ in the $(d-1)$-dimensional sphere $\mathbb{S}^{d-1}$, with respect to its rotation-invariant measure $ds$.
The partial derivatives $\{\partial_1x,\ldots,\partial_dx\}$ are defined in Subsection \ref{calculus definition subsubsection}.
In the commutative case, the above formula reduces to \eqref{torus trace formula} (for a full comparison, see the discussion in Subsection \ref{commutative_discussion_subsection}). There are a number of nontrivial
corollaries to \eqref{nc torus trace formula}, which we describe in the section below.
\subsection{Main results}
We have three main results. We take $\theta$ to be an arbitrary $d\times d$ antisymmetric real matrix { where $d \geq 2$}, in particular $\theta=0$ is not excluded. For further explanation of the notation, see Section \ref{notation section} below.
Our first main result provides sufficient conditions for $\,{\mathchar'26\mkern-12mu d} x \in {\mathcal L}_{d,\infty}$:
\begin{thm}\label{sufficiency}
If { $x \in \dot{H}^1_d({\mathbb{T}_\theta^d})$}, then { $\,{\mathchar'26\mkern-12mu d} x$ has bounded extension, and the extension is in ${\mathcal L}_{d,\infty}$}.
\end{thm}
{ The space $\dot{H}^{1}_d({\mathbb{T}_\theta^d})$ is a noncommutative homogeneous Sobolev space }defined with respect to the partial derivatives $\partial_j$, $j=1,\ldots,d$ (these notions will be defined and discussed in Subsection \ref{calculus definition subsubsection}). We note that the above condition is similar to that in \cite[Theorem 11]{LMSZ2017}.
With Theorem \ref{sufficiency}, we can prove our second main result, the following trace formula:
\begin{thm}\label{trace formula}
Let $x\in \dot H^1_d({\mathbb{T}_\theta^d})$ be self-adjoint. Then there is a constant $c_d$ depending only on the dimension $d$ such that for any continuous normalised trace $\varphi$ on ${\mathcal L}_{1,\infty}$ we have:
\begin{equation*}
\varphi(|\,{\mathchar'26\mkern-12mu d} x|^d) = c_d\,\int_{\mathbb{S}^{d-1}} \tau\Bigg(\Big(\sum_{j=1}^d |{\partial}_j x-s_j\sum_{k=1}^d s_k{\partial}_k x|^2\Big)^{\frac{d}{2}}\Bigg)\,ds.
\end{equation*}
Here, the integral over $\mathbb{S}^{d-1}$ is taken with respect to the rotation-invariant measure $ds$ on $\mathbb{S}^{d-1}$, and $s = (s_1,\ldots,s_d)$.
\end{thm}
As an aside we note that it is possible to give a short argument that the integrand above is continuous as a function of $s \in \mathbb{S}^{d-1}$.
Theorem \ref{trace formula}, in addition to being of interest in its own right, has a couple of corollaries, which to the best of our knowledge are novel.
\begin{cor}\label{trace formula-bound}
Let $x\in \dot H^1_d({\mathbb{T}_\theta^d})$ be self-adjoint. Then there are constants $c_d$ and $C_d$ depending only on $d$ such that for any continuous normalised trace $\varphi$ on ${\mathcal L}_{1,\infty}$ we have
$$c_d \| x\|_{\dot{H}_d^1}^d \leq \varphi(|\,{\mathchar'26\mkern-12mu d} x|^d) \leq C_d \| x\|_{\dot{H}_d^1}^d .$$
\end{cor}
As a converse to Theorem \ref{sufficiency}, we prove our third main result: the necessity of the condition {$x \in \dot H^1_d({\mathbb{T}_\theta^d})$} for $\,{\mathchar'26\mkern-12mu d} x \in {\mathcal L}_{d,\infty}$.
\begin{thm}\label{necessity}
Let {$x \in L_2({\mathbb{T}_\theta^d})$}. If $\,{\mathchar'26\mkern-12mu d} x$ { has bounded extension in ${\mathcal L}_{d,\infty}$ then $x \in \dot H^1_d({\mathbb{T}_\theta^d})$.}
\end{thm}
{ The {\it a priori} assumption that $x \in L_2({\mathbb{T}_\theta^d})$ can be justified as follows: $L_2({\mathbb{T}_\theta^d})$ is the smallest class of $x$ where we can define $\,{\mathchar'26\mkern-12mu d} x$ in a natural way. Furthermore,
one can motivate this assumption by noting that an $L_2$-condition is necessary and sufficient for Connes' trace theorem to hold in the commutative setting, see \cite[Theorem 2.5]{LPS2010} for details.}
Since $\varphi$ vanishes on the trace class ${\mathcal L}_1$, Corollary \ref{trace formula-bound} immediately yields the following noncommutative version of the $p\leq d$ component of \cite[Theorem 1]{JW1982}:
\begin{cor}\label{triviality}
If $x \in { L_2({\mathbb{T}_\theta^d})}$ and $\,{\mathchar'26\mkern-12mu d} x \in {\mathcal L}_{p}$, for $p \leq d$, then $x$ is a constant.
\end{cor}
Indeed, the $p \leq d$ component of \cite[Theorem 1]{JW1982} is an immediate and simple consequence of Corollary \ref{triviality} when $\theta = 0$.
A further corollary of Theorem \ref{trace formula} is that $\varphi(|\,{\mathchar'26\mkern-12mu d} x|^d)$ does not depend on the choice of continuous normalised trace $\varphi$. This
implies certain asymptotic properties of the singular numbers of $\,{\mathchar'26\mkern-12mu d} x$, beyond being merely in ${\mathcal L}_{d,\infty}$ \cite{KLPS,SSUZ2015}.
\subsection{Comparison to the commutative case}\label{commutative_discussion_subsection}
Take $x \in \dot{H}^1_d({\mathbb{T}_\theta^d})$. Consider the right hand side of the trace formula in Theorem \ref{trace formula},
\begin{equation*}
c_d\int_{\mathbb{S}^{d-1}} \tau\Bigg(\Big(\sum_{j=1}^d |{\partial}_j x-s_j\sum_{k=1}^d s_k{\partial}_k x|^2\Big)^{\frac{d}{2}}\Bigg)\,ds\,.
\end{equation*}
Define $\nabla x = (\partial_1 x,\partial_2x,\ldots,\partial_d x)$, and
\begin{equation*}
\|\nabla x\|_{2} := \Big(\sum_{j=1}^d |{\partial}_j x|^2\Big)^{\frac{1}{2}}.
\end{equation*}
In the commutative case (when $\theta = 0$), $x$ is a scalar valued function and $\|\nabla x\|_2^d$ coincides with the integrand in \eqref{torus trace formula}. Assuming commutativity,
we can define the unit vector $u = \frac{\nabla x}{\|\nabla x\|_2}$ and take out a factor of $\|\nabla x\|_{2}^{\frac{d}{2}}$ to get:
\begin{equation*}
\tau\Bigg(\int_{\mathbb{S}^{d-1}} \Big(\sum_{j=1}^d |{\partial}_j x-s_j\sum_{k=1}^d s_k{\partial}_k x|^2\Big)^{\frac{d}{2}}\,ds\Bigg) = \tau\Bigg(\|\nabla x\|_2^{\frac{d}{2}}\int_{\mathbb{S}^{d-1}} \Big(\sum_{j=1}^d|u_j-s_j\sum_{k=1}^d s_k u_k |^2\Big)^{\frac{d}{2}}\,ds \Bigg)
\end{equation*}
(where $u = (u_1,u_2,\ldots,u_d)$, and the interchange of $\tau$ and the integral is easily justified by Fubini's theorem in the commutative case).
However, since the measure $ds$ on $\mathbb{S}^{d-1}$ is invariant under rotations, we can choose coordinates $\{e_1,\ldots,e_d\}$ for ${\mathbb R}^d$ so that $u = e_1$, and then:
\begin{equation*}
b_d := \int_{\mathbb{S}^{d-1}} \Big(\sum_{j=1}^d|u_j-s_j\sum_{k=1}^d s_k u_k |^2\Big)^{\frac{d}{2}}\,ds
\end{equation*}
is independent of $u$, and is a constant scalar. Thus in the commutative case we have:
\begin{equation*}
\varphi(|\,{\mathchar'26\mkern-12mu d} x|^d) = c_db_d\tau(\|\nabla x\|_2^d).
\end{equation*}
This recovers \eqref{torus trace formula} upon taking $k_d = c_db_d$.
In the noncommutative case, we cannot take out a factor of $\|\nabla x\|_2^2$, and this explains why the form of the right hand side
of Theorem \ref{trace formula} is more complicated than $k_d\tau(\|\nabla x\|_2^d)$.
\section{Notation}\label{notation section}
\subsection{Operators, Ideals and traces}\label{operator notation subsection}
The following material concerning operator ideals and traces is standard. For more details we refer the reader to \cite{LSZ2012, Simon1979}.
Let $H$ be a complex separable Hilbert space, and let $\mathcal{B}(H)$ denote the set of bounded operators on $H$, and let ${\mathcal K}(H)$ denote the ideal of compact operators on $H$. Given $T\in {\mathcal K}(H)$, the sequence of singular values $\mu(T) = \{\mu(k,T)\}_{k=0}^\infty$ is defined as:
\begin{equation*}
\mu(k,T) = \inf\{\|T-R\|\;:\;\mathrm{rank}(R) \leq k\}.
\end{equation*}
Equivalently, $\mu(T)$ is the sequence of eigenvalues of $|T|$ arranged in non-increasing order with multiplicities.
Let $p \in (0,\infty).$ The Schatten class ${\mathcal L}_p$ is the set of operators $T$ in ${\mathcal K}(H)$ such that $\mu(T)$ is $p$-summable, i.e. in the sequence space $\ell_p$. If $p \geq 1$ then the ${\mathcal L}_p$
norm is defined as:
\begin{equation*}
\|T\|_p := \|\mu(T)\|_{\ell_p} = \left(\sum_{k=0}^\infty \mu(k,T)^p\right)^{1/p}.
\end{equation*}
With this norm ${\mathcal L}_p$ is a Banach space, and an ideal of $\mathcal{B}(H)$.
Analogously, the weak Schatten class ${\mathcal L}_{p,\infty}$ is the set of operators $T$ such that $\mu(T)$ is in the weak $L_p$-space $\ell_{p,\infty}$, with quasi-norm:
\begin{equation*}
\|T\|_{p,\infty} = \sup_{k\geq 0} (k+1)^{1/p}\mu(k,T) < \infty.
\end{equation*}
As with the ${\mathcal L}_p$ spaces, ${\mathcal L}_{p,\infty}$ is an ideal of $\mathcal{B}(H)$. We also have the following form
of H\"older's inequality,
\begin{equation}\label{weak-type-holder}
\|TS\|_{r,\infty} \leq c_{p,q}\|T\|_{p,\infty}\|S\|_{q,\infty}
\end{equation}
where $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$, for some constant $c_{p,q}$.
Of particular interest is ${\mathcal L}_{1,\infty}$, and we are concerned with traces on this ideal. For more details, see \cite[Section 5.7]{LSZ2012} and \cite{SSUZ2015}. A functional $\varphi:{\mathcal L}_{1,\infty}\to \mathbb{C}$ is called a trace if it is unitarily invariant. That is, for all unitary operators
$U$ and $T\in {\mathcal L}_{1,\infty}$ we have that $\varphi(U^*TU) = \varphi(T)$. It can then be shown that for all bounded operators $B$ we have $\varphi(BT)=\varphi(TB).$
An important fact about traces is that any trace $\varphi$ on ${\mathcal L}_{1,\infty}$ vanishes on ${\mathcal L}_1$ \cite[Theorem~5.7.8]{LSZ2012}. A trace $\varphi$ is called continuous if it is continuous with respect to the ${\mathcal L}_{1,\infty}$ quasi-norm. It is known that not all traces on ${\mathcal L}_{1,\infty}$ are continuous \cite[Remark~3.1(3)]{LSZ2018}.
Within the class of continuous traces on ${\mathcal L}_{1,\infty}$ there are the well-known Dixmier traces \cite[Chapter 6]{LSZ2012}.
Finally, we say that a trace $\varphi$ on ${\mathcal L}_{1,\infty}$ is normalised if $\varphi$ takes the value $1$ on any compact positive operator with eigenvalue sequence $\{\frac{1}{n+1}\}_{n=0}^\infty$ (any two such
operators are unitarily equivalent, and so the particular choice of operator is inessential).
\subsection{Noncommutative Tori}\label{nc tori subsection}
Harmonic analysis on noncommutative tori is an established subject. The exposition here closely follows \cite{XXY2018}, and for sake of brevity we refer the reader to \cite{XXY2018}
for a detailed exposition of the topic and provide here only the definitions relevant to this text.
\subsubsection{Basic definitions}
We fix an integer $d > 1$ and $\theta = \{\theta_{j,k}\}_{j,k = 1}^d$, a $d\times d$ antisymmetric real matrix. The $C^*$-algebra of continuous functions on the noncommutative torus, denoted $C({\mathbb{T}_\theta^d})$, is the universal $C^*$-algebra on $d$ unitary generators $U_1,\ldots,U_d$
which satisfy:
\begin{equation*}
U_jU_k = e^{2\pi {\rm{i}} \theta_{j,k}}U_kU_j,\quad 1\leq j,k\leq d.
\end{equation*}
Given $n = (n_1,\ldots,n_d) \in {\mathbb Z}^d$, we adopt the shorthand notation:
\begin{equation*}
U^n := U_1^{n_1}U_2^{n_2}\cdots U_d^{n_d}.
\end{equation*}
There exists an action $\alpha$ of the torus group ${\mathbb T}^d$ on $C({\mathbb{T}_\theta^d})$, given on a generator $U_j$ by:
\begin{equation}\label{action-T-qt}
\alpha_z(U_j) = z_jU_j, \quad z = (z_1,z_2,\ldots, z_d)\in {\mathbb T}^d.
\end{equation}
The action $\alpha$ can be extended to a norm-continuous group of automorphisms of $C({\mathbb{T}_\theta^d})$. There is a distinguished trace state $\tau$
on $C({\mathbb{T}_\theta^d})$, which may be constructed in several ways, one of which is by averaging over $\alpha$ as follows:
It can be shown that the fixed point subalgebra of $C({\mathbb{T}_\theta^d})$ under the action of $\alpha$ is exactly the trivial subalgebra $\mathbb{C} 1$. Hence if $x \in C({\mathbb{T}_\theta^d})$ then
averaging over ${\mathbb T}^d$ with respect to the Haar measure $m$ on ${\mathbb T}^d$:
\begin{equation*}
\int_{{\mathbb T}^d} \alpha_z(x)\,dm(z)
\end{equation*}
yields a multiple of the identity element. Defining
\begin{equation*}
\tau(x)1 = \int_{{\mathbb T}^d} \alpha_z(x)\,dm(z)
\end{equation*}
yields the canonical trace state $\tau$ on $C({\mathbb{T}_\theta^d})$. Given $\tau$ we can now define the GNS Hilbert space $L_2(C({\mathbb{T}_\theta^d}),\tau)$, which we denote $L_2({\mathbb{T}_\theta^d})$, and we identify $C({\mathbb{T}_\theta^d})$ as an algebra of bounded
operators on $L_2({\mathbb{T}_\theta^d})$, where $x \in C({\mathbb{T}_\theta^d})$ acts on $\xi \in L_2({\mathbb{T}_\theta^d})$ by left multiplication. Taking the weak operator topology closure $C({\mathbb{T}_\theta^d})''$ in $\mathcal{B}(L_2({\mathbb{T}_\theta^d}))$ yields a von Neumann algebra, which we denote $L_\infty({\mathbb{T}_\theta^d})$.
The $L_p$-spaces for $p \in [1,\infty)$ on ${\mathbb{T}_\theta^d}$ are then defined as the operator $L_p$-spaces \cite{PX2003,LSZ2012} on $(L_\infty({\mathbb{T}_\theta^d}),\tau)$,
\begin{equation*}
L_p({\mathbb{T}_\theta^d}) := {\mathcal L}_p(L_\infty({\mathbb{T}_\theta^d}),\tau).
\end{equation*}
For $x \in L_1({\mathbb{T}_\theta^d})$ and $n\in {\mathbb Z}^d$, we define:
\begin{equation*}
\widehat{x}(n) = \tau(x(U^{n})^{*}).
\end{equation*}
By the definition of $\tau$, we see that $\tau(U_n) = \delta_{n,0}$, and then standard Hilbert space arguments show that any $x \in L_2({\mathbb{T}_\theta^d})$ can be written as an $L_2$-convergent
series:
\begin{equation*}
x = \sum_{n \in {\mathbb Z}^d} \widehat{x}(n)U^n,
\end{equation*}
with
\begin{equation}\label{Plancherel-qt}
\|x\|_2^2 = \sum_{n\in {\mathbb Z}^d} | \widehat x(n) |^2 .
\end{equation}
The space ${C^\infty}({\mathbb{T}_\theta^d})$ is defined to be the subset of $x\in C({\mathbb{T}_\theta^d})$ such that the sequence of Fourier coefficients $\{\widehat{x}(n)\}_{n\in {\mathbb Z}^d}$
has rapid decay (i.e., the sequence $\{|\widehat{x}(n)|\}_{n\in {\mathbb Z}^d}$ is eventually dominated by the reciprocal of any polynomial). We may consider ${C^\infty}({\mathbb{T}_\theta^d})$
as the space of smooth functions on ${\mathbb{T}_\theta^d}$, since in the commutative setting this space corresponds with the space of $C^\infty$ functions. There is also a canonical Fr\'echet
topology on ${C^\infty}({\mathbb{T}_\theta^d})$, and the space { ${\mathcal{D}}'({\mathbb{T}_\theta^d})$}, called the space of distributions on ${\mathbb{T}_\theta^d}$, is defined to be the topological dual of ${C^\infty}({\mathbb{T}_\theta^d})$.
\subsubsection{Calculus for quantum tori}\label{calculus definition subsubsection}
Many aspects of harmonic analysis on ${\mathbb T}^d$ carry over to ${\mathbb{T}_\theta^d}$. For example we may define the partial differentiation operators ${\partial}_j$, $j = 1,\cdots,d$ by:
\begin{equation*}
{\partial}_j (U^n) = 2\pi {\rm{i}} n_jU^n,\quad n = (n_1,\ldots,n_d) \in {\mathbb Z}^d.
\end{equation*}
Every partial derivation ${\partial}_j$ can be viewed a densely defined closed (unbounded)
operator on $L_2({\mathbb{T}_\theta^d})$, whose adjoint is equal to $-{\partial}_j$. Let $\Delta=\partial_1^2+\cdots+\partial_d^2$ be the Laplacian. Then $\Delta = - ({\partial}_1^* {\partial}_1 + \cdots +{\partial}_d^* {\partial}_d)$, so $-\Delta$ is a positive operator on $L_2({\mathbb{T}_\theta^d})$ with spectrum equal to $\{4\pi^2 |n|^2 : n \in {\mathbb Z}^d\}$. As in the Euclidean case, we let $D_j = -{\rm{i}} {\partial}_j$, which is then self-adjoint.
Given $n =(n_1,\cdots,n_d)\in \mathbb{N}_0^d$ ($\mathbb{N}_0$ denoting the set of nonnegative integers), the associated partial derivation $D^n$ is defined to be
$D_1^{n_1}\cdots D_d^{n_d}$. The order of $D^n$ is $|n|_1=n_1+\cdots+ n_d$.
By duality, the derivations transfer to ${\mathcal{D}}'({\mathbb{T}_\theta^d})$ as well.
For ${\alpha} \in {\mathbb R}$, denote by $J^{\alpha}$ the ${\alpha}$-order Bessel potential $(1 - \Delta )^{\frac{{\alpha}}{2}}$.
The potential (or fractional) Sobolev space of order ${\alpha} \in {\mathbb R}$ is defined to be
\begin{equation}\label{bessel-sobolev-space-definition}
H_p^\alpha(\mathbb{T}_{\theta}^d)=\big\{ x\in{\mathcal{D}}'({\mathbb{T}_\theta^d}) : J^{\alpha} x\in L_p(\mathbb{T}_{\theta}^d) \big\},
\end{equation}
equipped with the norm
$$\|x\|_{H_p^\alpha}=\|J^{\alpha} x\|_p\,.$$ Since $J^0$ is the identity, $H_p^0({\mathbb{T}_\theta^d}) = L_p({\mathbb{T}_\theta^d})$.
{
As in the classical case, if $\alpha$ is a non-negative integer then $H^{\alpha}_p({\mathbb{T}_\theta^d})$ admits an equivalent norm in terms of the sum of the $p$-norms of the partial derivatives of order up to $\alpha$.
To be explicit, the Sobolev space of order $k\in \mathbb{N}$ on $\mathbb{T}_{\theta}^d$ may be described as:
$$H_p^k(\mathbb{T}_{\theta}^d)= \big\{ x\in{\mathcal{D}}'({\mathbb{T}_\theta^d}) : D^n x \in L_p(\mathbb{T}_{\theta}^d) \textrm{ for each }n\in \mathbb {N}_0^d \textrm{ with } |n |_1\leq k \big\},$$
equipped with the norm
$$\|x\|_{H_p^k}=\Big(\sum_{0\leq |n |_1\leq k}\|D^n x\|_{p}^p\Big)^{\frac{1}{p}}.$$
The equivalence of the above norm and the Bessel potential norm $\|J^{k} x\|_p$ is a well-established fact in the theory of harmonic analysis on ${\mathbb{T}_\theta^d}$, being proved in the $p=2$ case by \cite[Theorem 2.1]{Spera1992}
and a later proof for general $p$ can be found as \cite[Theorem 2.9]{XXY2018}.
}
In this paper, we will mainly use the ``homogeneous" Sobolev space $\dot{H}^1_p ({\mathbb{T}_\theta^d})$ and the potential Sobolev spaces $H_2^\alpha(\mathbb{T}_{\theta}^d)$. The norm of $\dot{H}^1_p ({\mathbb{T}_\theta^d})$ with $ p \geq 2$, may be described in the following equivalent forms:
\begin{equation}\label{Sob-equi-norm}
\|x\|_{\dot{H}^1_p}
= \Big(\sum_{j=1}^d \|{\partial}_j x\|_p^p \Big)^{\frac{1}{p}}\approx \sum_{j=1}^d \|{\partial}_j x\|_p \approx \| (\sum_{j=1}^d |{\partial}_j x |^2 ) ^{\frac 1 2 } \|_p ,
\end{equation}
where the relevant constants depend only on $d$ and $p$. Then $\dot{H}^{1}_p({\mathbb{T}_\theta^d})$ may be defined as the subspace of ${\mathcal{D}}'({\mathbb{T}_\theta^d})$ for which the above norm is finite. Note that the difference between $\dot{H}^{1}_p({\mathbb{T}_\theta^d})$ and $H^1_p({\mathbb{T}_\theta^d})$
is that for $\dot{H}^{1}_p({\mathbb{T}_\theta^d})$ we do not assume that the $L_p$-norm is finite.
For any $x\in H^1_p ({\mathbb{T}_\theta^d})$, we have the following Poincar\'e type inequality
\begin{equation}\label{Poincare}
\|x- \widehat x (0) \|_p \leq C_{p,d} \|x\|_{\dot{H}^1_p} .
\end{equation}
See \cite[Theorem~2.12]{XXY2018}. For every ${\alpha} \in {\mathbb R}$, the space $H_2^{\alpha}({\mathbb T}_\theta^d)$ is a Hilbert space with the inner product
$${\langle} x , y {\rangle} = \tau ( J^{\alpha} y^* J^{\alpha} x ) .$$
{
It is proved in \cite[Theorem 3.3]{Spera1992} and \cite[Proposition 9.2]{HLP2018a} for arbitrary real ${\alpha}, \beta \in \mathbb{R}$ with ${\alpha} > \beta$, the embedding
\begin{equation}\label{Sob-cpt-emb}
H_2^{\alpha}({\mathbb T}_\theta^d) \hookrightarrow H_2^\beta ({\mathbb T}_\theta^d) \quad \mbox{is compact}.
\end{equation}
}
The Dirac operator ${D}$ { (more precisely, the spin-Dirac operator)} is defined in terms of $\gamma$ matrices in direct analogy to commutative tori. Define $N = 2^{\lfloor \frac{d}{2}\rfloor}$ and select $N\times N$ complex self-adjoint matrices
$\{\gamma_1,\ldots,\gamma_d\}$ satisfying $\gamma_j\gamma_k +\gamma_k\gamma_j = 2\delta_{j,k}1$, and define:
\begin{equation*}
{D} = \sum_{j=1}^d \gamma_j\otimes D_j
\end{equation*}
as an unbounded, densely defined linear operator on the Hilbert space $\mathbb{C}^N\otimes L_2({\mathbb{T}_\theta^d})$. { This definition coincides with \cite[Definition 12.14]{green-book}. }
Since all $D_j$'s
are self-adjoint, ${D}$ is also self-adjoint. We then define the sign of ${D}$ via the Borel functional calculus, which can be expressed as
$${{\rm{sgn}}} ({D}) = \sum_{j=1}^d {\gamma}_j {\otimes} \frac{D_j }{\sqrt{ D_1^2 + D_2^2 +\cdots + D_d^2}} \,.$$
Given $x\in L_{\infty}({\mathbb{T}_\theta^d})$, denote by $M_x: y \mapsto xy$ the operator of left multiplication on $L_2({\mathbb{T}_\theta^d})$. The operator $1{\otimes} M_x$ is a bounded linear operator on ${\mathbb C}^N {\otimes} L_2({\mathbb{T}_\theta^d}) $, where $1$ denotes the identity operator on ${\mathbb C}^N$. The commutator
$$\,{\mathchar'26\mkern-12mu d} x : = {\rm{i}} [{{\rm{sgn}}}({D}) , 1{\otimes} M_x],\quad x \in L_\infty({\mathbb{T}_\theta^d})$$
denotes the quantised differential on quantum tori.
{
On the other hand, if $x$ is not necessarily bounded we may still define $\,{\mathchar'26\mkern-12mu d} x$ on the dense subspace $C^\infty({\mathbb{T}_\theta^d})\otimes {\mathbb C}^N$ as follows. Suppose that $x \in L_2({\mathbb{T}_\theta^d})$. Then if $\eta \in C^\infty({\mathbb{T}_\theta^d})\otimes {\mathbb C}^N$, we will
have $(1\otimes M_x)\eta \in L_2({\mathbb{T}_\theta^d})\otimes {\mathbb C}^N$. Moreover, ${{\rm{sgn}}}(D)\eta$ is still in $C^{\infty}({\mathbb{T}_\theta^d})\otimes {\mathbb C}^N$ since by definition an element of $C^{\infty}({\mathbb{T}_\theta^d})$ has Fourier coefficients of rapid decay, and ${{\rm{sgn}}}(D)$ is represented as a
Fourier multiplier with bounded symbol.
Thus the expression:
\begin{equation*}
(\,{\mathchar'26\mkern-12mu d} x)\eta := {\rm{i}}{{\rm{sgn}}}(D)(1\otimes M_x)\eta - {\rm{i}}(1\otimes M_x){{\rm{sgn}}}(D)\eta
\end{equation*}
is a well-defined element of $L_2({\mathbb{T}_\theta^d})\otimes {\mathbb C}^N$ for all $\eta \in C^{\infty}({\mathbb{T}_\theta^d})\otimes {\mathbb C}^N$.
}
\subsubsection{Fourier multipliers for quantum tori}
Let $g$ be a bounded scalar function on ${\mathbb Z}^d$. For $x \in L_2({\mathbb{T}_\theta^d})$, the Fourier multiplier $T_g$ with symbol $g$ is defined
on $x$ by:
\begin{equation}\label{def-Fourier-Z}
T_gx = \sum_{n \in {\mathbb Z}^d} g(n)\widehat{x}(n)U^n.
\end{equation}
By virtue of the Plancherel identity \eqref{Plancherel-qt}, $T_g$ indeed defines a bounded linear operator on $L_2({\mathbb{T}_\theta^d})$ and the above series converges in the $L_2$-sense.
If $g$ is unbounded, we may define $T_g$ on the dense subspace of $L_2({\mathbb{T}_\theta^d})$ of those $x$ with finitely many non-zero Fourier coefficients.
An equivalent perspective on Fourier series is to consider a function $\phi \in L_1({\mathbb T}^d)$ on the commutative torus. We may then define
the convolution of $\phi$ with $x \in L_2({\mathbb{T}_\theta^d})$ by:
\begin{equation*}
\phi\ast x =\int_{{\mathbb T}^d}{\alpha}_w(x)\phi(w) dw.
\end{equation*}
In terms of Fourier coefficients, we have:
\begin{equation*}
\phi\ast x = T_{\widehat{\phi}}x.
\end{equation*}
Fourier multipliers for quantum tori were studied in detail in \cite[Chapter 7]{XXY2018} (there, $T_{g}$ was denoted by $M_g$).
From the perspective of functional calculus, we may also write:
\begin{equation*}
T_g = g(\frac{1}{2\pi {\rm{i}}}\partial_1, \frac{1}{2\pi {\rm{i}}}\partial_2,\ldots,\frac{1}{2\pi {\rm{i}}}\partial_d).
\end{equation*}
The above defined derivatives $D^{\alpha}$, Laplacian $\Delta$, and Bessel potential $J^{\alpha}$ may all be viewed as Fourier multipliers: the symbol of $D_j$ is $2\pi \xi_j$; the symbol of $\Delta$ is $-|2\pi \xi |^2$; and the symbol of $J^{\alpha}$ is $(1+ | 2\pi \xi |^2) ^{\frac{{\alpha}}{2}}$. We will denote by ${\langle \xi \rangle}$ the function $(1+ | \xi |^2) ^{\frac{1}{2}}$ in the sequel.
A far reaching extension of the notion of a Fourier multiplier is a pseudodifferential operator. We outline the pseudodifferential operator theory for the noncommutative torus in Section \ref{sec-pdo}.
\section{Cwikel-type estimates for quantum tori}
In the classical, commutative setting, Cwikel estimates are bounds on the singular values of operators of the form:
\begin{equation*}
M_fg(-{\rm{i}}\nabla)
\end{equation*}
where $f$ and $g$ are essentially bounded functions on ${\mathbb R}^d$, and $M_f$ and $g(-{\rm{i}}\nabla)$ denote pointwise multiplication and Fourier multiplication
on $L_2({\mathbb R}^d)$ respectively (see e.g. \cite[Chapter 4]{Simon1979} and \cite{Cwikel1977}).
In the setting of noncommutative tori, we instead consider operators of the form $M_xT_g$, where $x\in L_{\infty}({\mathbb{T}_\theta^d})$ and $g \in \ell_{\infty}({\mathbb Z}^d)$. We can obtain
the following as a special case of \cite{LeSZ2017}:
\begin{thm}\label{Cwikel-type}
\begin{enumerate}[\rm (i)]
\item\label{L_p cwikel} If $x\in L_p({\mathbb{T}_\theta^d})$ and $g\in \ell_p({\mathbb Z}^d)$ with $2\leq p <{\infty}$, then $M_x \,T_g$ is in ${\mathcal L}_p$ and
$$\|M_x\, T_g\|_{{\mathcal L}_p} \leq C_p \|x\|_p \|g\|_p.$$
\item\label{weak L_p cwikel} If $x\in L_p({\mathbb{T}_\theta^d})$ and $g\in \ell_{p,{\infty}}({\mathbb Z}^d)$ with $2< p <{\infty}$, then $M_x \,T_g$ is in ${\mathcal L}_{p,{\infty}} $ and
$$\|M_x\, T_g\|_{{\mathcal L}_{p,{\infty}}} \leq C_p \|x\|_p \|g\|_{p,{\infty}}.$$
\end{enumerate}
\end{thm}
\begin{proof}
We in fact prove the following far stronger estimate, stated in the language of symmetric function spaces \cite[Chapter 2]{LSZ2012}: For any symmetric function space $E$ whose norm satisfies the Fatou property\footnote{meaning that if $A_n$ is a sequence of positive operators with $A_n\uparrow A$ in the weak operator topology, then $\|A\|_{E} \leq \sup_{n} \|A_n\|_{E}$} and is an interpolation space of $L_2$ and $L_\infty$, if $x\otimes g \in E(L_{\infty}({\mathbb{T}_\theta^d})\otimes \ell_{\infty}({\mathbb Z}^d))$ then $M_xT_g$ is in $E(\mathcal{B}(L_2({\mathbb{T}_\theta^d})))$, with norm bound,
\begin{equation}\label{interpolated cwikel estimate}
\|M_xT_g\|_{E(\mathcal{B}(L_2({\mathbb{T}_\theta^d})))} \leq C_E \|x\otimes g\|_{E(L_\infty({\mathbb{T}_\theta^d})\otimes \ell_\infty({\mathbb Z}^d))}.
\end{equation}
After proving \eqref{interpolated cwikel estimate}, we explain how it entails the results in the statement of the theorem.
In fact \eqref{interpolated cwikel estimate} can be obtained by a direct application of \cite[Corollary 3.5]{LeSZ2017}. Here we have two von Neumann algebras $L_{\infty}({\mathbb{T}_\theta^d})$
and $\ell_{\infty}({\mathbb Z}^d)$ represented on the same Hilbert space $L_2({\mathbb{T}_\theta^d})$ by left multiplication and Fourier multiplication respectively. In this setting, we can use \cite[Corollary 3.5]{LeSZ2017} which states that if we have an estimate of the form:
\begin{equation}\label{L_2 cwikel}
\|M_xT_g\|_{{\mathcal L}_{2}(\mathcal{B}(L_{2}({\mathbb{T}_\theta^d})))} \leq \|x\|_{L_2({\mathbb{T}_\theta^d})}\|g\|_{\ell_2({\mathbb Z}^d)}
\end{equation}
then \eqref{interpolated cwikel estimate} follows.
To prove \eqref{L_2 cwikel}, we can express the Hilbert-Schmidt norm in terms of an expansion with respect to the basis $\{U^m\}_{ m \in {\mathbb Z}^d}$ of $L_2({\mathbb{T}_\theta^d})$,
\begin{eqnarray*}\begin{split}
\| M_x T_g \|_{{\mathcal L}_2}^2 & = \sum_{m, n \in {\mathbb Z}^d} \Big| \tau \Big(x \big(T_gU^m\big) (U^n)^* \Big) \Big|^2\\
& = \sum_{m, n \in {\mathbb Z}^d} \Big| \tau \Big(x g(m )U^m (U^n)^* \Big) \Big|^2 = \sum_{m, n \in {\mathbb Z}^d} |g(m)|^2 \Big| \tau \Big(x U^m (U^n)^* \Big) \Big|^2 \\
&= \sum_{m \in {\mathbb Z}^d} |g(m)|^2 \sum_{n \in {\mathbb Z}^d} \Big| \tau \Big(x U^m (U^n)^* \Big) \Big|^2 .
\end{split}\end{eqnarray*}
By the Plancherel formula \eqref{Plancherel-qt}, we have
$$\sum_{n \in {\mathbb Z}^d} | \tau (x U^m (U^n)^* ) |^2 = \|x U^m \|_2^2= \|x\|_2^2.$$
Thus,
$$
\| M_x T_g \|_{{\mathcal L}_2}^2 = \|x\|_2^2 \|g\|_2^2.$$
Hence, \eqref{L_2 cwikel} holds and thus by \cite[Corollory 3.5]{LeSZ2017} it follows that \eqref{interpolated cwikel estimate} holds.
Now, we take $E = L_p$ in \eqref{interpolated cwikel estimate} for $p \in (2,\infty)$. This is indeed an interpolation space between $L_2$ and $L_\infty$ whose norm satisfies the Fatou property.
Then combining \eqref{interpolated cwikel estimate} with the identity
\begin{equation*}
\|x\otimes g\|_{L_p(L_\infty({\mathbb{T}_\theta^d})\otimes \ell_\infty({\mathbb Z}^d))} = \|x\|_p \|g\|_p
\end{equation*}
yields \eqref{L_p cwikel}.
Finally, to obtain \eqref{weak L_p cwikel}, we take $E=L_{p,\infty}$ in \eqref{interpolated cwikel estimate} and use the estimate:
\begin{equation*}
\|x\otimes g\|_{L_{p,\infty}(L_\infty({\mathbb{T}_\theta^d})\otimes \ell_\infty({\mathbb Z}^d))}\leq \|x\|_p \|g\|_{p,\infty}.
\end{equation*}
This completes the proof of \eqref{weak L_p cwikel}.
\end{proof}
Consider the function on ${\mathbb Z}^d$, $n\mapsto (1+|n|^2)^{-\frac{d}{2}}$. When $|n| > 1 $, we have $(1+|n| ^2)^{-\frac{d}{2}} \leq |n|^{-d}$. For $|n| \leq 1$, $(1+|n| ^2)^{-\frac{d}{2}}$ is bounded from above by $1$. Hence $n \mapsto (1+|n| ^2)^{-\frac{d}{2}} \in \ell_{1,\infty} ({\mathbb Z}^d)$, and so $n\mapsto (1+|n| ^2)^{-\frac{{\beta}}{2}}\in \ell_{\frac{d}{{\beta}} , {\infty}} ({\mathbb Z}^d)$. Then it follows immediately from the above theorem that
\begin{cor}\label{Cwikel-type-cor}
Consider the linear operator $(1{\otimes} x) (1+{D}^2)^{-\frac \beta 2} $ on ${\mathbb C}^N {\otimes} L_2({\mathbb{T}_\theta^d})$. If $x\in L_{\frac d \beta}({\mathbb{T}_\theta^d})$ with $\frac d \beta >2$, then $(1{\otimes} M_x) (1+{D}^2)^{-\frac \beta 2} \in {\mathcal L}_{\frac d \beta, {\infty}} ,$ and
$$\|(1{\otimes} M_x) (1+{D}^2)^{-\frac \beta 2}\|_{{\mathcal L}_{\frac d \beta, {\infty}}} \leq C\|x\|_{\frac d \beta},$$
where the constant $C>0$ depends only on $d$ and $\beta$.
\end{cor}
At this point it is worth noting that since the function $n\mapsto (1+|n|^2)^{-\frac{{\alpha}}{2}}$ is in $\ell_{\frac{d}{{\alpha}},{\infty}}({\mathbb Z})$, for all ${\alpha} > 0$ we have:
\begin{equation}\label{bessel potential ideal}
{ J^{-\alpha} \in {\mathcal L}_{\frac{d}{{\alpha}},\infty}.}
\end{equation}
\section{Proof of Theorem \ref{sufficiency}}\label{sec-suff}
This section is devoted to the proof of Theorem \ref{sufficiency}, that is, that the condition $x \in { \dot{H}^{1}_d({\mathbb{T}_\theta^d})}$ is sufficient
for $\,{\mathchar'26\mkern-12mu d} x \in {\mathcal L}_{d,\infty}$, and with an explicit norm bound: { $$\|\,{\mathchar'26\mkern-12mu d} x\|_{d,\infty} \leq C_d\|x\|_{\dot{H}^{1}_d({\mathbb{T}_\theta^d})}.$$
Note that due to the Poincar\'e inequality \eqref{Poincare}, $\dot{H}^1_d({\mathbb{T}_\theta^d})$ is a subset of $L_d({\mathbb{T}_\theta^d})$, and thus in particular the operator $\,{\mathchar'26\mkern-12mu d} x$ is well-defined.}
The following lemma is a corollary of Theorem \ref{Cwikel-type}\eqref{L_p cwikel}.
\begin{lem}\label{Cwikel-xp}
Suppose that $p>\frac d 2$ and $x\in L_p({\mathbb{T}_\theta^d})$. If $p\geq 2$, then there exists a constant $C_{p,d}>0$ such that
$$ \big\|\big[{{\rm{sgn}}}({D}) - \frac{{D}}{\sqrt{1+{D}^2}}, 1{\otimes} M_x \big]\big\|_{{\mathcal L}_p} \leq C_{p,d} \|x\|_p , $$
{ meaning that, if $x \in L_p({\mathbb{T}_\theta^d})$ then the above commutator (initially defined on $C^\infty({\mathbb{T}_\theta^d})\otimes {\mathbb C}^N$) admits an extension to a bounded operator which is in the ideal ${\mathcal L}_p$ with the above norm bound.}
\end{lem}
\begin{proof}
Let $1\leq j \leq d$, and for $n \in {\mathbb Z}^d$ define
$$h_j(n) := \frac{n_j}{|n|} -\frac{n_j}{((2\pi)^{-2}+|n|^2)^{\frac{1}{2}}}.$$
Thus,
\begin{equation*}
T_{h_j} = h_j(-\frac{{\rm{i}}}{2\pi}\nabla) = \frac{-{\rm{i}}\partial_j}{\sqrt{-\Delta}}-\frac{-{\rm{i}}\partial_j}{(1-\Delta)^{\frac{1}{2}}}
\end{equation*}
and so,
\begin{align*}
{{\rm{sgn}}}({D}) - \frac{{D}}{\sqrt{1+{D}^2}} &= \sum_{j=1}^d \gamma_j\otimes \left( \frac{-{\rm{i}}\partial_j}{\sqrt{-\Delta}}-\frac{-{\rm{i}}\partial_j}{(1-\Delta)^{\frac{1}{2}}}\right)\\
&= \sum_{j=1}^d \gamma_j\otimes h_j(-\frac{{\rm{i}}}{2\pi}\nabla)\\
&= \sum_{j=1}^d \gamma_j\otimes T_{h_j}.
\end{align*}
One can easily check that $h_j \in \ell_p({\mathbb Z}^d)$ as $p> \frac d 2$. Expanding out the commutator,
\begin{eqnarray*}
\begin{split}
\big[{{\rm{sgn}}}({D}) - \frac{{D}}{\sqrt{1+{D}^2}}, 1{\otimes} M_x \big] &= \big[\sum_{j=1}^d {\gamma} _j {\otimes} T_{h_j}, 1{\otimes} M_x \big]\\
&=\sum_{j=1}^d {\gamma}_j {\otimes} [T_{h_j}, M_x].
\end{split}
\end{eqnarray*}
Hence,
\begin{eqnarray*}
\begin{split}
\|\big[{{\rm{sgn}}}({D}) - \frac{{D}}{\sqrt{1+{D}^2}}, 1{\otimes} M_x \big]\|_{{\mathcal L}_p} &\leq d \max_{1\leq j \leq d} \|\big[ T_{h_j}, M_x \big]\|_{{\mathcal L}_p}\\
&\leq d \max_{1\leq j \leq d} \big(\| T_{h_j} M_x \|_{{\mathcal L}_p}+\| M_x T_{h_j}\|_{{\mathcal L}_p}\big)\\
&= d \max_{1\leq j \leq d} \big(\| M_{x ^*} T_{h_j}\|_{{\mathcal L}_p}+ \|M_x T_{h_j}\|_{{\mathcal L}_p}\big).
\end{split}
\end{eqnarray*}
The desired conclusion follows then from Theorem \ref{Cwikel-type}.(i).
\end{proof}
The proof of the next lemma relies on the technique of double operator integrals (see \cite{PSW2002} and \cite{PS2009} and references therein).
Let $H$ be a (complex) separable Hilbert space. Let $D_0$ and $D_1$ be self-adjoint (potentially unbounded) operators on $H$, and $E^0$ and $E^1$ be the associated spectral measures. For all $x, y \in {\mathcal L}_2(H)$, the measure $(\lambda, \mu) \mapsto {\mathrm{Tr} }(x\, dE^0(\lambda) \, y \, dE^1(\mu) )$ is a countably additive complex valued measure on ${\mathbb R}^2$. We say that $\phi \in L_{\infty}({\mathbb R}^2)$ is $E^0 \otimes E^1$ integrable if there exists an operator $T_\phi ^{D_0, D_1} \in \mathcal{B} ({\mathcal L}_2(H))$ such that for all $x, y \in {\mathcal L}_2(H)$,
$${\mathrm{Tr} } (x\,T_\phi ^{D_0, D_1} y ) =\int _{{\mathbb R}^2} \phi(\lambda, \mu ) {\mathrm{Tr} }(x\, dE^0(\lambda) \, y \, dE^1(\mu) ). $$
The operator $T_\phi ^{D_0, D_1} $ is called the transformer. For $A\in {\mathcal L}_2(H) $, we define
\begin{equation}\label{doi-def}
T_\phi ^{D_0, D_1}(A)=\int _{{\mathbb R}^2} \phi(\lambda, \mu ) dE^0(\lambda) \, A \, dE^1(\mu) . \end{equation}
This is called a double operator integral.
\begin{lem}\label{commutator-Sob}
Let $x\in \dot H_d^1({\mathbb{T}_\theta^d})$. Then
$$ \big\|\big[\frac{{D}}{\sqrt{1+{D}^2}}, 1{\otimes} M_x \big]\big\|_{{\mathcal L}_{d,{\infty}} } \leq B_{d} \|x\|_{\dot H_d^1}$$
where the constant $B_d>0$ depends only on $d$.
{ As with Lemma \ref{Cwikel-xp}, the above commutator is interpreted as being initially defined on $C^\infty({\mathbb{T}_\theta^d})\otimes {\mathbb C}^N$.}
\end{lem}
\begin{proof}
Set $g(t)= t (1+t^2)^{-\frac{1}{2}}$ for $t\in {\mathbb R}$. { Suppose initially that $x \in C^{\infty}({\mathbb{T}_\theta^d})$. Under this assumption, $[D,1\otimes M_x]$ extends to a bounded operator, and thus we can apply \cite[Theorem~4.1]{BS1989} (see also Proposition 2.6 and Theorem 3.1 in \cite{PS2008}) to get}
\begin{equation}\label{repr-commutator}
[g({D}), 1{\otimes} M_x] = T_{g^{[1]}}^{{D},{D}} ([{D}, 1{\otimes} M_x]),
\end{equation}
where $g^{[1]}(\lambda,\mu ) := \frac{g(\lambda)-g(\mu )}{\lambda-\mu }$ for different $\lambda, \mu \in {\mathbb R}$. By \cite[Lemma 9]{LMSZ2017}, we have $g^{[1]} ={\psi}_1{\psi}_2 {\psi}_3$, with
$$\psi_1 = 1+ \frac{1-\lambda \mu }{(1+ \lambda^2)^{\frac 1 2 } (1+\mu ^2) ^{\frac 1 2 } },\;\; \psi_2 = \frac{(1+\lambda^2)^{\frac 1 4} (1+\mu ^2)^{\frac 1 4} }{(1+ \lambda^2)^{\frac 1 2 } + (1+\mu ^2) ^{\frac 1 2 } },\;\;\psi_3 = \frac{1 }{(1+ \lambda^2)^{\frac 1 4 } (1+\mu ^2) ^{\frac 1 4 } }.$$
It follows that
\begin{equation}\label{TDDg}
T_{g^{[1]}}^{{D},{D}} = T_{{\psi}_1}^{{D},{D}} T_{{\psi}_2}^{{D},{D}} T_{{\psi}_3}^{{D},{D}} .
\end{equation}
By \cite[Lemma 8]{LMSZ2017}, we see that the transformer $T_{{\psi}_2}^{{D},{D}} $ is bounded on both ${\mathcal L}_1$ and ${\mathcal L}_{\infty}$.
{ For $k=1,3$ the function $\psi_k$ can be written as a linear combination of products of bounded functions of $\lambda$ and of $\mu$, and from this it follows that $T_{{\psi}_k}^{{D},{D}}$ is a bounded linear map on ${\mathcal L}_{1}$
and ${\mathcal L}_{\infty}$. For further details, see e.g. \cite[Corollary 2]{PS2009} and \cite[Corollary 2.4]{RX2011}.}
Then by real interpolation of $({\mathcal L}_1, {\mathcal L}_{\infty})$ (see \cite{DDP1992} or \cite{Xu2007}), the transformers $T_{{\psi}_k}^{{D},{D}} $ with $k=1,2,3$ are bounded linear transformations from ${\mathcal L}_{d,{\infty}}$ to ${\mathcal L}_{d,{\infty}}$.
We now exploit the identity in \eqref{repr-commutator} and the product of terms in \eqref{TDDg}, noticing that
\begin{eqnarray*}\begin{split}
\|[g({D}), 1{\otimes} M_x]\|_{{\mathcal L}_{d,{\infty}}}&\leq \| T_{{\psi}_1}^{{D},{D}} \|_{{\mathcal L}_{d,{\infty}}{\rightarrow} {\mathcal L}_{d,{\infty}}} \| T_{{\psi}_2}^{{D},{D}} \|_{{\mathcal L}_{d,{\infty}}{\rightarrow} {\mathcal L}_{d,{\infty}}}\\
&\;\;\;\;\;\;\;\; \times \| T_{{\psi}_3}^{{D},{D}} ([{D}, 1{\otimes} M_x])\|_{ {\mathcal L}_{d,{\infty}}} \\
&\leq C_d \| T_{{\psi}_3}^{{D},{D}} ([{D}, 1{\otimes} M_x])\|_{ {\mathcal L}_{d,{\infty}}},
\end{split}\end{eqnarray*}
where the constant $C_d>0$ does not depend on $x$. Since ${\psi}_3(\lambda,\mu)=(1+\lambda^2)^{-1/4} (1+\mu^2)^{-1/4}$ is a product a function of $\lambda$ and a function of $\mu$, by \eqref{doi-def}, we have
$$T_{{\psi}_3}^{{D},{D}} ([{D}, 1{\otimes} M_x]) = (1+{D}^2)^{-1/4} [{D}, 1{\otimes} M_x](1+{D}^2)^{-1/4}. $$
Hence
$$\|[g({D}), 1{\otimes} M_x]\|_{{\mathcal L}_{d,{\infty}}} \leq C_d \| (1+{D}^2)^{-1/4} [{D}, 1{\otimes} M_x](1+{D}^2)^{-1/4}\|_{{\mathcal L}_{d,{\infty}}}.$$
Expanding out ${D}$ and using the quasi-triangle inequality for ${\mathcal L}_{d,{\infty}}$, we have
\begin{eqnarray*}\begin{split}
&\| (1+{D}^2)^{-1/4} [{D}, 1{\otimes} M_x](1+{D}^2)^{-1/4}\|_{{\mathcal L}_{d,{\infty}}} \\
&\leq K_d \sum_{j=1}^d \| (1+{D}^2)^{-1/4} [{\gamma}_j{\otimes} D_j, 1{\otimes} M_x](1+{D}^2)^{-1/4}\|_{{\mathcal L}_{d,{\infty}}},
\end{split}\end{eqnarray*}
where $K_d>0$ depends only on $d$. But $[{\gamma}_j{\otimes} D_j, 1{\otimes} M_x] =-{\rm{i}} {\gamma}_j {\otimes} M_{{\partial}_j x} $, thus we obtain
$$\| (1+{D}^2)^{-1/4} [{\gamma}_j{\otimes} D_j, 1{\otimes} M_x](1+{D}^2)^{-1/4}\|_{{\mathcal L}_{d,{\infty}}} = \| (1-\Delta)^{-1/4} M_{{\partial}_j x}(1- \Delta)^{-1/4}\|_{{\mathcal L}_{d,{\infty}}}.$$
Note that the first norm $\|\cdot\|_{{\mathcal L}_{d,{\infty}}}$ is the norm of ${\mathcal L}_{d,{\infty}}({\mathbb C}^N\otimes L_2({\mathbb{T}_\theta^d}))$, and the second one is the norm of ${\mathcal L}_{d,{\infty}}(L_2({\mathbb{T}_\theta^d}))$.
We are reduced to estimating the quantity $\| (1-\Delta)^{-1/4} M_{{\partial}_j x}(1- \Delta)^{-1/4}\|_{{\mathcal L}_{d,{\infty}}}$. By polar decomposition, for every $j$, there is a partial isometry $U_j$ such that
$${\partial}_j x = U_j |{\partial}_j x | = U_j |{\partial}_j x |^{\frac 1 2 } |{\partial}_j x |^{\frac 1 2 }.$$
Taking $\beta=\frac 1 2 $, and recalling that $x$ is such that $\|U_j |{\partial}_j x|^{\frac 1 2 } \|_{2d}\leq \|\, |{\partial}_j x|^{\frac 1 2 } \|_{2d} = \|{\partial}_j x \|_d^{\frac 1 2 }<{\infty}$, we apply Corollary \ref{Cwikel-type-cor} to get
(for some constant $Q_d$)
$$\| M_{|{\partial}_j x|^{\frac 1 2}}(1- \Delta)^{-1/4}\|_{{\mathcal L}_{2d,{\infty}}}= \| (1- \Delta)^{-1/4} M_{|{\partial}_j x|^{\frac 1 2}}\|_{{\mathcal L}_{2d,{\infty}}}\leq Q_d \|\, |{\partial}_j x|^{\frac 1 2 } \|_{2d}$$
and
$$\| (1- \Delta)^{-1/4} M_{U_j |{\partial}_j x|^{\frac 1 2}}\|_{{\mathcal L}_{2d,{\infty}}}\leq Q_d \|U_j |{\partial}_j x|^{\frac 1 2 } \|_{2d}\leq Q_d \|\, |{\partial}_j x|^{\frac 1 2 } \|_{2d}.$$
Thus, by the H\"{o}lder inequality \eqref{weak-type-holder},
$$\| (1-\Delta)^{-1/4} M_{{\partial}_j x}(1- \Delta)^{-1/4}\|_{{\mathcal L}_{d,{\infty}}}\leq c_{\frac{d}{2},\frac{d}{2}}Q_d^2\|\, |{\partial}_j x|^{\frac 1 2 } \|_{2d}^2=c_{\frac{d}{2},\frac{d}{2}}Q_d^2\|{\partial}_j x \|_d.$$
Taking $B_d = c_{\frac{d}{2},\frac{d}{2}}d Q_d^2 C_d K_d$, we conclude that
\begin{equation}\label{g(D)_sobolev_bound}
\|[g({D}), 1{\otimes} M_x]\|_{{\mathcal L}_{d,{\infty}}} \leq B_d \sum_{j=1}^d\|{\partial}_j x \|_d\leq B_d \|x\|_{\dot H_d^1}.
\end{equation}
{
We now remove the initial assumption that $x \in C^\infty({\mathbb{T}_\theta^d})$. Suppose that $x \in \dot H_d^1({\mathbb{T}_\theta^d})$. As $C^\infty({\mathbb{T}_\theta^d})$ is dense in $H_d^1({\mathbb{T}_\theta^d})$ \cite[Proposition 2.7]{XXY2018}, we may select a sequence $\{x_n\}_{n=0}^\infty \subset C^\infty({\mathbb{T}_\theta^d})$
such that $\lim_{n\to\infty} \|x_n-x\|_{H_d^1({\mathbb{T}_\theta^d})} = 0$. From \eqref{g(D)_sobolev_bound}, we have that the sequence $\{[g(D),1\otimes M_{x_n}]\}_{n=0}^\infty$ is Cauchy in the ${\mathcal L}_{d,\infty}$ topology. Hence there is a limit $T \in {\mathcal L}_{d,\infty}$.
On the other hand, if $\eta \in C^\infty({\mathbb{T}_\theta^d})\otimes {\mathbb C}^N$ from the H\"older inequality we have:
\begin{equation*}
\|(1\otimes M_{x_n})\eta-(1\otimes M_x)\eta\|_{L_2({\mathbb{T}_\theta^d})\otimes {\mathbb C}^N} \leq \|x_n-x\|_{L_d({\mathbb{T}_\theta^d})}\|\eta\|_{L_{2d/(d-2)}({\mathbb{T}_\theta^d})\otimes {\mathbb C}^N}
\end{equation*}
and similarly,
\begin{equation*}
\|(1\otimes M_{x_n})g(D)\eta-(1\otimes M_x)g(D)\eta\|_{L_2({\mathbb{T}_\theta^d})\otimes {\mathbb C}^N} \leq \|x_n-x\|_{L_d({\mathbb{T}_\theta^d})}\|g(D)\eta\|_{L_{2d/(d-2)}({\mathbb{T}_\theta^d})\otimes {\mathbb C}^N}.
\end{equation*}
Therefore for each fixed $\eta \in C^{\infty}({\mathbb{T}_\theta^d})\otimes {\mathbb C}^N$ we have:
\begin{equation*}
[g(D),1\otimes M_{x_n}]\eta \rightarrow [g(D),1\otimes M_x]\eta
\end{equation*}
in the $L_2({\mathbb{T}_\theta^d})\otimes {\mathbb C}^N$ sense. Thus,
\begin{equation*}
[g(D),1\otimes M_{x}]\eta = T\eta
\end{equation*}
for all $\eta \in C^{\infty}({\mathbb{T}_\theta^d})\otimes {\mathbb C}^N$. Therefore $T$ and $[g(D),1\otimes M_x]$ are equal, and we have:
\begin{equation*}
[g(D),1\otimes M_{x_n}]\to [g(D),1\otimes M_x]
\end{equation*}
in the ${\mathcal L}_{d,\infty}$ topology. Thus \eqref{g(D)_sobolev_bound} holds for all $x \in \dot H^1_d({\mathbb{T}_\theta^d})$.
}
\end{proof}
Now we are able to complete the proof of Theorem \ref{sufficiency}.
\begin{proof}[Proof of Theorem \ref{sufficiency}]
Let $x\in \dot H_d^1({\mathbb{T}_\theta^d})$. Combining Lemmas \ref{Cwikel-xp} and \ref{commutator-Sob}, we find that
$$ \big\|\big[{{\rm{sgn}}}({D}) , 1{\otimes} M_x \big]\big\|_{{\mathcal L}_{d,{\infty}}} \leq C_{d,d} \|x\|_d + B_d \|x\|_{\dot H_d^1} . $$
We can remove the dependence on $\|x\|_d$ on the right hand side by the aid of the Poincar\'e inequality \eqref{Poincare}. Since for constant operator $\widehat{x}(0)\in L_{\infty}({\mathbb{T}_\theta^d})$, it is obvious that $\big[{{\rm{sgn}}}({D}) , 1{\otimes} M_{\widehat{x}(0)} \big]=0$, we have
\begin{eqnarray*}\begin{split}
\big\|\big[{{\rm{sgn}}}({D}) , 1{\otimes} M_x \big]\big\|_{{\mathcal L}_{d,{\infty}}} & = \big\|\big[{{\rm{sgn}}}({D}) , 1{\otimes} M_{x-\widehat{x}(0)} \big]\big\|_{{\mathcal L}_{d,{\infty}}} \\
&\leq C_{d,d} \|x-\widehat{x}(0) \|_d + B_d \|x-\widehat{x}(0) \|_{\dot H_d^1}\\
&\leq C_d \|x\|_{ \dot H_d^1}.
\end{split}\end{eqnarray*}
The theorem is therefore proved.
\end{proof}
\section{Pseudodifferential operators on quantum tori}\label{sec-pdo}
In this section we give an introduction to some recent developments in pseudodifferential operators on quantum tori. The most important result stated in this section for us is Theorem \ref{Connes-qt}, which
is a form of Connes' trace formula obtained in \cite{MSZ2018}.
The theory of pseudodifferential operators goes back to Kohn-Nirenberg \cite{KN1965} and H\"{o}rmander \cite{Hor1965}. It has been extended to the noncommutative setting, especially the quantum torus case, by many authors; see for instance \cite{MPR2005,LMR2010,LJP2016,GJP2017,
Tao2018,XX2018}. Our main references of this part are \cite{Connes1980,Baaj1988} and \cite{CT2011}, while the details can be found in \cite{HLP2018,HLP2018a}. In the following, let us collect some definitions and well known properties of symbol classes and pseudodifferential operators on quantum tori.
Denote by ${\langle \xi \rangle}$ the function $(1+|\xi|^2) ^{\frac 1 2 }$ on ${\mathbb R}^d$.
For every $m\in {\mathbb R}$, the class $S^m({\mathbb R}^d; {\Sc}({\mathbb{T}_\theta^d}))$ consists of all maps $\rho \in C^{{\infty}}({\mathbb R}^d ; {\Sc}({\mathbb{T}_\theta^d}))$ such that, for all multi-indices $\alpha,\beta \in {\mathbb N}_0^d$, there exists $C_{\alpha, \beta} >0$ such that
$$\|D^\alpha D_\xi^\beta \rho(\xi) \| \leq C_{{\alpha},{\beta}} {\langle \xi \rangle} ^{m-|{\beta}|_1} ,\quad \forall \xi \in {\mathbb R}^d . $$
Endowed with the locally convex topology generated by the semi-norms
$$p_N^{(m)} (\rho) : = \sup_{|{\alpha}|_1 +|{\beta}|_1 \leq N } \sup _{\xi \in {\mathbb R} ^d} {\langle \xi \rangle} ^{-m+|{\beta}|_1} \|D^\alpha D_\xi^\beta \rho(\xi) \| ,\quad N\in {\mathbb N} _0, $$
$S^m({\mathbb R}^d; {\Sc}({\mathbb{T}_\theta^d}))$ is then a Fr\'echet space.
Let $\rho \in S^m({\mathbb R}^d; {\Sc}({\mathbb{T}_\theta^d}))$, $m\in {\mathbb R}$, and $\rho_j(\xi) \in S^{m-j}({\mathbb R}^d; {\Sc}({\mathbb{T}_\theta^d}))$ for each $j\in {\mathbb N}$. If for every $N\geq 1$,
$$\rho(\xi) - \sum_{j<N}\rho_j(\xi) \in S^{m-N}({\mathbb R}^d; {\Sc}({\mathbb{T}_\theta^d})),$$
we shall write $\rho(\xi) \sim \sum_{j\geq 0} \rho_j(\xi)$. This is referred to as an asymptotic expansion of the symbol $\rho$.
The homogeneous class of symbols $\dot S^m({\mathbb R}^d; {\Sc}({\mathbb{T}_\theta^d}))$ consists of maps $\rho \in C^{{\infty}}({\mathbb R}^d \setminus \{0\} ; {\Sc}({\mathbb{T}_\theta^d}))$ satisfying
$$\rho (\lambda \xi ) = \lambda^m \rho(\xi),\quad \forall \xi \in {\mathbb R}^d \setminus \{0\},\;\forall \lambda >0.$$
In this case, $\rho$ on ${\mathbb R}^d \setminus \{0\}$ is determined by its restriction to $\mathbb S^{d-1}$, the $d$-dimensional unit sphere.
If { a (not necessarily homogeneous) symbol} $\rho$ admits an asymptotic expansion $\rho \sim \sum_{j\geq 0} \rho_{m-j}$ with $ \rho_{m-j}\in \dot S^{m-j}({\mathbb R}^d; {\Sc}({\mathbb{T}_\theta^d}))$ for each $j\geq 0$, then $\rho$ is called a {\it classical }symbol, and the leading term $\rho_m$ is called the principal symbol of $\rho$.
\medskip
Let us turn to the definition of pseudodifferential operators with the above symbols on quantum torus. Let ${\alpha} _s$ be a $d$-parameter group of automorphisms given by
\begin{equation}\label{action-R-qt}
{\alpha}_s (U^n)= e^{2\pi {\rm{i}} s\cdot n} U^n,\end{equation}
which is a periodic version of the action in \eqref{action-T-qt} if we identify $[0,1]^d $ with ${\mathbb T}^d$ by the correspondence $(s_1,s_2,\cdots, s_d) \leftrightarrow (e^{2\pi {\rm{i}} s_1}, e^{2\pi {\rm{i}} s_2}, \cdots, e^{2\pi {\rm{i}} s_d})$. For $\rho \in C^{{\infty}}({\mathbb R}^d ; {\Sc}({\mathbb{T}_\theta^d}))$, let $P_\rho$ be the pseudodifferential operator sending arbitrary $a \in {\Sc}({\mathbb{T}_\theta^d})$ to
\begin{equation}\label{pdo-def}
P_\rho(a) := \int_{{\mathbb R}^d} \int _{{\mathbb R}^d} e ^{-2\pi {\rm{i}} s \cdot \xi } \rho (\xi) {\alpha}_s(a) ds \, d\xi .\end{equation}
Note that this integral does not converge absolutely; it is defined as an oscillatory integral. See \cite{CT2011,HLP2018,HLP2018a,Tao2018} for more information. By \cite[Proposition~5.9]{HLP2018}, if $a = \sum_{n\in {\mathbb Z}^d} a_n U^n\in {\Sc}({\mathbb{T}_\theta^d})$ and $\rho \in S^m({\mathbb R}^d ; {\Sc}({\mathbb{T}_\theta^d}))$ with { $m\in {\mathbb R}$}, then
\begin{equation}\label{pdo-def-ent}
P_\rho (a) = \sum_{n\in {\mathbb Z}^d} \rho(n) a_n U^n ,\end{equation}
where the sum converges in the operator norm to an element in ${\Sc}({\mathbb{T}_\theta^d})$.
In other words, the pseudodifferential operator on ${\mathbb{T}_\theta^d}$ with symbol $\rho$ is determined by the value of $\rho$ on ${\mathbb Z}^d$, which coincides with the definition given in \cite{LJP2016}.
If $\rho \in S^m({\mathbb R}^d;{\Sc}({\mathbb{T}_\theta^d}))$ with { $m \in {\mathbb R}$}, then $P_\rho$ is said to be a pseudodifferential operator
of order $m$.
Also note that, by the noncommutativity, if we change the order of $\rho(\xi)$ and ${\alpha}_s(a)$ in \eqref{pdo-def} (or $\rho(n) $ and $U^n$ in \eqref{pdo-def-ent}), we get another pseudodifferential operator with the same symbol. In \cite{XX2018}, these two operators are distinguished as column and row operators. But in this paper, we will not need to consider both kinds of operators, and so we focus only on those with the form \eqref{pdo-def} or \eqref{pdo-def-ent}.
\begin{ex}\label{ex-symbol}
Let us formulate some first examples of symbols defined above.
\begin{enumerate}[{\rm i)}]
\item Let $x\in {\Sc}( {\mathbb T}_\theta^d)$ and consider the constant function $\psi (\xi) \equiv x$, $\xi \in {\mathbb R}^d$. Obviously, $\psi \in S^0({\mathbb R}^d; {\Sc}({\mathbb{T}_\theta^d}))$. So by the above definition, the multiplier $M_x (y) =xy$ on $ {\mathbb T}_\theta^d $ is an order $0$ pseudodifferential operator. The principal symbol of this operator is $x$ itself.
\item Let $k\in {\mathbb N}_0^d$. The symbol of the $|k|_1$-order differential operator $D^k= D_1^{k_1}\cdots D_d^{k_d}$ is $\psi (\xi) = (2\pi \xi_1)^{k_1}(2\pi \xi_2)^{k_2}\cdots (2\pi \xi_d)^{k_d}$. It is easily checked that $\psi \in S^{|k|_1}({\mathbb R}^d; {\Sc}({\mathbb{T}_\theta^d}))$. Thus $D^k$ is a pseudodifferential operator of order $|k|_1$, and its principal symbol is $ (2\pi \xi)^k$.
\item Let $\alpha \in {\mathbb R}$, and consider the $\alpha$-order Bessel potential $J^\alpha= (1- \Delta )^{\frac{{\alpha}}{2}}$ on the quantum torus, which is a Fourier multiplier with symbol $\psi (\xi) = {\langle} 2\pi \xi {\rangle} ^\alpha = (1+|2\pi \xi|^2) ^{\frac {\alpha} 2 } \in S^\alpha({\mathbb R}^d; {\Sc}({\mathbb{T}_\theta^d}))$. Thus, $J^\alpha$ is an $\alpha$-order pseudodifferential operator. Moreover, as a scaler-valued function, $\psi (\xi) $ has the asymptotic expansion
$${\langle} 2\pi \xi {\rangle} ^\alpha \sim \sum_{j=0}^{{\infty}} \begin{pmatrix}
j \\
\frac{{\alpha} }{2}
\end{pmatrix} |2\pi \xi|^{\alpha-2j} .$$
Hence, $J^\alpha$ is classical with principal symbol $|2\pi \xi |^\alpha$. See \cite[Proposition~5.14]{HLP2018}.
\end{enumerate}
\end{ex}
The above examples illustrate that both pointwise multipliers $M_x$ and Fourier multipliers $T_g$ from \eqref{def-Fourier-Z} are considered as the special cases of pseudodifferential operators. For general symbol $\rho$, $P_\rho$ may be thought as a limit of linear combinations of operators composed by pointwise multipliers and Fourier multipliers.
\medskip
The composition of two pseudodifferential operators is again a pseudodifferential operator, and there is a method for computing an asymptotic expansion of its symbol.
The following proposition, which is the quantum analogue of the classical result in \cite[p. 237]{Stein1993}, first appears in \cite{CT2011}; a complete proof is given in \cite[Proposition~7.5]{HLP2018a}.
\begin{prop}\label{symbol-composition}
Let $\rho_1$, $\rho_2$ be two symbols in $S ^{n_1}({\mathbb R}^d; {\Sc}({\mathbb{T}_\theta^d}))$ and $S ^{n_2}({\mathbb R}^d; {\Sc}({\mathbb{T}_\theta^d}))$ respectively. Then there exists a symbol $\rho_3$ in $S ^{n_1+n_2}({\mathbb R}^d; {\Sc}({\mathbb{T}_\theta^d}))$ such that
$$
P_{\rho_3} =P_{\rho_1} P_{\rho_2} .
$$
Moreover,
\begin{equation}\label{composition-asym}
\rho_3-\sum_{|{\alpha} |_1<N_0} \frac{(2\pi {\rm i})^{-|{\alpha} |_1}}{{\alpha} !}D_\xi^{\alpha} \rho_1 D^{\alpha} \rho_2 \in S^{n_1+n_2- N_0}({\mathbb R}^d; {\Sc}({\mathbb{T}_\theta^d})), \quad \forall\, N_0\geq 0,
\end{equation}
where the first derivative $D_\xi^{\alpha} $ is the derivative of $\rho_1$ with respect to the variable $\xi \in {\mathbb R}^d$, and the second derivative $D^{\alpha} $ is the derivation on ${\Sc}({\mathbb{T}_\theta^d})$ described in Section \ref{calculus definition subsubsection}.
\end{prop}
{ Many authors have considered the question of the mapping properties of pseudodifferential operators on functions spaces on quantum tori \cite{XX2018,GJP2017,HLP2018a}. In this paper we are concerned solely with the boundedness
of a pseudodifferential operator on $L_2({\mathbb{T}_\theta^d})$.
The following proposition can be found in \cite[Proposition 10.1]{HLP2018a}, \cite[Corollary 6.6]{Tao2018}.
\begin{prop}\label{bdd-2-Sobolev}
Let $\rho\in S^0({\mathbb R}^d; {\Sc}({\mathbb{T}_\theta^d}))$. Then the pseudodifferential operator $P_\rho$ extends to a bounded operator from $L_2({\mathbb T}_\theta^d)$ to $L_2({\mathbb T}_\theta^d)$.
\end{prop}
Proposition \ref{bdd-2-Sobolev} is simply a special case of the general Sobolev space mapping property of pseudodifferential operators \cite[Proposition 6.6]{HLP2018a}.
Even greater generalisations to mapping properties of pseudodifferential operators on Sobolev spaces and Besov and Triebel-Lizorkiin spaces \cite[Section 6.2]{XX2018} are also known.
Symbols of negative order are in particular of order zero, and thus if $m > 0$ and $\rho \in S^{-m}({\mathbb R}^d,{\Sc}({\mathbb{T}_\theta^d}))$ then $P_{\rho}$ has bounded extension on $L_2({\mathbb{T}_\theta^d})$. However in the case of strictly negative order we can provide
more detailed information on $P_{\rho}$. The following is proved in \cite[Lemma~13.6]{HLP2018a}:
}
\begin{prop}\label{pdo-weak-L}
If $\rho\in S^{-m}({\mathbb R}^d; {\Sc}({\mathbb{T}_\theta^d}))$ with $m>0$, then $P_\rho$ is a compact operator on $L_2({\mathbb T}_\theta^d)$. Furthermore, $P_\rho \in \mathcal{L}_{\frac d m,{\infty}}$.
\end{prop}
{
The proof of Proposition \ref{pdo-weak-L} is a simple combination of the fact that since $P_{\rho}$ has order $-m$, and $J^m$ has order $m$, the product formula in Proposition \ref{symbol-composition} implies
that the composition $P_{\rho}J^m$ is of order zero. Hence by Proposition \ref{bdd-2-Sobolev}, $P_{\rho}J^{m}$ has bounded extension, and since $J^{-m} \in \mathcal{L}_{\frac{d}{m},\infty}$ \eqref{bessel potential ideal}, it follows immediately that $P_{\rho} \in \mathcal{L}_{\frac{d}{m},\infty}$.
Thanks to Proposition \ref{pdo-weak-L}, we can easily obtain from the symbol calculus the following:
\begin{cor}\label{commutator-MxJ}
Let $x\in {\Sc}({\mathbb{T}_\theta^d})$, and ${\alpha}>0$. Then
$$[M_x , (1-\Delta )^{-\frac{{\alpha}}{2}}] \in \mathcal{L}_{\frac{d}{{\alpha}+1} ,{\infty}}.$$
\end{cor}
Indeed, $[M_x , (1-\Delta )^{-\frac{{\alpha}}{2}}] $ is a pseudodifferential operator of order at most $-{\alpha}-1$, as can be seen by a short computation using Proposition \ref{symbol-composition}.
}
\medskip
Next, we are going to present Connes' trace formula on quantum torus in the specific form obtained in \cite[Theorem 6.5]{MSZ2018}. This trace formula will play a crucial role in the proof of the trace formula for a quantised differential. Recall that if $\rho$ is a homogeneous symbol of order $0$, then $\rho(\xi) =\rho(\frac{\xi}{|\xi|}) $ for every $\xi \neq 0$. So this $\rho$ could be viewed as a function on the $(d-1)$-dimensional sphere $\mathbb{S}^{d-1}$.
\begin{thm}\label{Connes-qt}
Let $A$ be a classical pseudodifferential operator on ${\mathbb{T}_\theta^d}$ of order $0$ with self-adjoint extension, and denote by $\rho_A$ its principal symbol. Then for any normalised trace ${\varphi}$ on $\mathcal{L}_{1,{\infty}}$, we have
$${\varphi} \big(|A|^d(1-\Delta )^{-\frac d 2}\big) =\frac 1 d \int_{\mathbb{S}^{d-1}} \tau(|\rho_A(s)|^d) ds .$$
\end{thm}
The reason to refer specifically to \cite{MSZ2018} is that if $d$ is odd then $|A|^d$ is not a pseudodifferential operator in the usual sense, and so it needs to be understood as an element
of the $C^*$-closure of the algebra of order $0$ pseudodifferential operators on $L_2({\mathbb{T}_\theta^d})$. It is proved in \cite{MSZ2018} that on the $C^*$-closure the principal symbol mapping extends
to a $C^*$-algebra homomorphism, and hence $\rho_{|A|^d} = |\rho_A|^d$.
\section{The trace formula}\label{sec-Ctf}
This section is devoted to the proofs of Theorem \ref{trace formula} and Corollary \ref{trace formula-bound}. That is, we show that for all $x \in \dot{H}^1_d({\mathbb{T}_\theta^d})$ and all
continuous normalised traces $\varphi$ on the ideal ${\mathcal L}_{1,{\infty}}$ that:
\begin{equation}\label{trace formula in section}
\varphi(|\,{\mathchar'26\mkern-12mu d} x|^d) = c_d\,\int_{\mathbb{S}^{d-1}} \tau\Bigg(\Big(\sum_{j=1}^d |{\partial}_j x-s_j\sum_{k=1}^d s_k{\partial}_k x|^2\Big)^{\frac{d}{2}}\Bigg)\,ds
\end{equation}
for a positive constant $c_d$. Moreover, there are positive constants $0 < c_d < C_d < \infty$ such that:
\begin{equation}\label{trace formula-bound in section}
c_d \| x\|_{\dot{H}_d^1}^d \leq \varphi(|\,{\mathchar'26\mkern-12mu d} x|^d) \leq C_d \| x\|_{\dot{H}_d^1}^d .
\end{equation}
Our strategy of proof is as follows: first, \eqref{trace formula in section} is proved for $x \in {\Sc}({\mathbb{T}_\theta^d})$ by aid of the theory of pseudodifferential operators developed in the preceding section.
Then by an approximation argument based on the density of ${\Sc}({\mathbb{T}_\theta^d})$ in $\dot{H}^{1}_d({\mathbb{T}_\theta^d})$, we complete the proof of \eqref{trace formula in section} in full generality.
Finally \eqref{trace formula-bound in section} is achieved by bounding the right hand side of \eqref{trace formula in section} from above and below by a constant multiple of $\|x\|_{\dot{H}^1_d}^d$.
To begin with, we explain how the operator $|\,{\mathchar'26\mkern-12mu d} x|^d$ can, up to trace class perturbations, be written in the form $|A|^d(1+{D}^2)^{-\frac{d}{2}}$ for a certain order zero pseudodifferential operator $A$.
Let $x\in {\Sc}({\mathbb{T}_\theta^d})$. For $j=1,\cdots, d$, we define the operators $\{A_j\}_{j=1}^d$ on $L_2({\mathbb{T}_\theta^d})$ by
$$A_j\eta := \left(M_{{\partial}_j x } -\frac 1 2 \sum_{k=1}^d \Big( \frac{D_j D_k}{1-\Delta} M_{{\partial}_k x } + M_{{\partial}_k x } \frac{D_j D_k}{1-\Delta} \Big)\right)\eta,\quad \eta \in L_2({\mathbb{T}_\theta^d}).$$
For each $j$, $A_j$ is defined initially on ${\Sc}({\mathbb{T}_\theta^d})$, but by functional calculus $A_j$ extends uniquely to a bounded operator on $L_2({\mathbb{T}_\theta^d})$ which we denote with the same symbol. We then define the operator $A$ on ${\mathbb C}^N \otimes L_2({\mathbb{T}_\theta^d})$ as
\begin{equation}\label{def-A}A:= \sum_{j=1}^d \gamma_j \otimes A_j . \end{equation}
If $x= x ^*$, since ${\partial}_j$ commutes with the adjoint operation $*$, we have for every $y_1, y_2 \in L_2({\mathbb{T}_\theta^d})$,
\begin{eqnarray*}
{\langle} ({\partial}_j x) y_1 , y_2 {\rangle} = \tau \big(({\partial}_j x ) y_1 y_2^*\big) = \tau \big(y_1({\partial}_j x^* y_2) ^*\big) = {\langle} y_1 , ({\partial}_j x^*)y_2 {\rangle},
\end{eqnarray*}
which yields $(M_{{\partial}_j x })^* = M_{{\partial}_j x^*} = M_{{\partial}_j x}$. Furthermore, since each $D_j$ is a self-adjoint operator on $ L_2({\mathbb{T}_\theta^d})$, we know that
$$\Big(\frac{D_j D_k}{1-\Delta} M_{{\partial}_k x } \Big)^* = M_{{\partial}_k x } \frac{D_j D_k}{1-\Delta} .$$
Therefore, each $A_j$ is a self-adjoint operator, and so is $A$.
We will now show that $|\,{\mathchar'26\mkern-12mu d} x|^d-|A|^d(1+{D}^2)^{-\frac{d}{2}} \in {\mathcal L}_1$. The following lemma is an important first step:
\begin{lem}\label{commutator-AJ}
Let ${\beta} \geq 0$, and ${\alpha} >0$ be such that ${\alpha}+1 <d$. Then for $A$ defined in \eqref{def-A}, we have
$$[|A|, (1+{D}^2 ) ^{-\frac {\alpha} 2} ] \, ( 1+{D}^2)^{-\frac{{\beta}}{2}}\in \mathcal{L}_{\frac{d}{{\alpha}+{\beta} +1} , {\infty} }. $$
\end{lem}
\begin{proof}
As { $(1+{D}^2)^{-\frac{1}{2}} \in {\mathcal L}_{d,\infty}$}, the operator $ ( 1+{D}^2)^{-\frac{{\beta}}{2}}$ is in $\mathcal{L}_{\frac{d}{{\beta}} , {\infty} }$. Thus by H\"older's inequality, it suffices to consider ${\beta} =0$. First, we shall prove that $[A_j , (1-\Delta)^{-\frac{{\alpha}}{2} } ] \in \mathcal{L}_{\frac{d}{{\alpha} +1} , {\infty} }$. From Corollary \ref{commutator-MxJ}, we have that
$$[M_{{\partial}_j x } , (1-\Delta)^{-\frac{{\alpha}}{2} } ] \in \mathcal{L}_{\frac{d}{{\alpha} +1} , {\infty} } .$$
Hence, by linearity, it suffices to prove that
\begin{equation}\label{commutator-DMJ}
\Big[ \frac{D_j D_k}{1-\Delta}M_{{\partial}_k x } , (1-\Delta)^{-\frac{{\alpha}}{2} } \Big] \in \mathcal{L}_{\frac{d}{{\alpha} +1} , {\infty} } \end{equation}
and
\begin{equation}\label{commutator-MDJ}
\Big[ M_{{\partial}_k x } \frac{D_j D_k}{1-\Delta} , (1-\Delta)^{-\frac{{\alpha}}{2} } \Big] \in \mathcal{L}_{\frac{d}{{\alpha} +1} , {\infty} } .\end{equation}
{ Note that \eqref{commutator-MDJ} follows from \eqref{commutator-DMJ} by taking the adjoint. So we prove only \eqref{commutator-DMJ}.}
However, since $\frac{D_j D_k}{1-\Delta}$ commutes with $ (1-\Delta)^{-\frac{{\alpha}}{2} } $, we have
$$\Big[ \frac{D_j D_k}{1-\Delta}M_{{\partial}_k x } , (1-\Delta)^{-\frac{{\alpha}}{2} } \Big] = \frac{D_j D_k}{1-\Delta}\Big[ M_{{\partial}_k x } , (1-\Delta)^{-\frac{{\alpha}}{2} } \Big]. $$
By functional calculus, $\frac{D_j D_k}{1-\Delta}$ is bounded on $L_2({\mathbb{T}_\theta^d})$. Then \eqref{commutator-DMJ} and \eqref{commutator-MDJ} follow from Corollary \ref{commutator-MxJ} and the boundedness of $\frac{D_j D_k}{1-\Delta}$ on $L_2({\mathbb{T}_\theta^d})$.
Thus, we have proved that
$$[A, (1+{D}^2)^{-\frac{{\alpha}}{2} } ] =\sum_{j=1}^d \gamma_j \otimes [A_j , (1-\Delta)^{-\frac{{\alpha}}{2} } ] \in \mathcal{L}_{\frac{d}{{\alpha} +1} , {\infty} }.$$
To complete the proof, we need to replace $A$ with $|A|$.
To this end, we use the result of \cite{PS2011}, which implies that if $1<p<{\infty}$ and $A$ and $B$ are self-adjoint operators such that $[A, B] \in \mathcal{L}_p$, then $[|A|, B] \in\mathcal{L}_p$; see also \cite[Corollary~3.5]{DDPS1997} for more general results. Since $\frac{d}{{\alpha} +1} >1$, the result follows from interpolation.
\end{proof}
\begin{lem}\label{T-AJ}
Let $T$ be a bounded operator on ${\mathbb C}^N \otimes L_2({\mathbb{T}_\theta^d})$, and suppose that
$$T\in A(1+{D}^2)^{-\frac 1 2 } +\mathcal{L}_{\frac{2d}{3}, {\infty}}, $$
where $A$ is given in \eqref{def-A}.
Then $|T|^d \in \mathcal{L}_{1,{\infty}}$ and for any continuous normalised trace ${\varphi}$ on $\mathcal{L}_{1,{\infty}}$, we have
$${\varphi} (|T|^d) ={\varphi} (|A|^d (1+{D}^2)^{-\frac d 2} ).$$
\end{lem}
\begin{proof}
By the aid of Lemma \ref{commutator-AJ}, the proof proceeds as in \cite [Lemma~14]{LMSZ2017}, and is therefore omitted.
\end{proof}
\begin{lem}\label{commutator-Dx}
For $x\in {\Sc}({\mathbb{T}_\theta^d})$ and $A$ defined in \eqref{def-A}, we have
$$\,{\mathchar'26\mkern-12mu d} x - A(1+{D}^2)^{-\frac 1 2} = [{{\rm{sgn}}}({D}), 1\otimes M_x] - A(1+{D}^2)^{-\frac 1 2} \in \mathcal{L}_{\frac{d}{2},{\infty}}.$$
\end{lem}
\begin{proof}
Let $g({D}) = {D} (1+{D}^2) ^{-\frac 1 2 }$. Then
\begin{eqnarray*}\begin{split}
{{\rm{sgn}}}({D}) - g({D}) &= {{\rm{sgn}}} ({D}) \Big(1 -\frac{|{D}|}{(1+{D}^2)^{\frac 1 2 }}\Big)\\
&= {{\rm{sgn}}} ({D}) \Big(\frac{1}{(1+{D}^2)^{\frac 1 2 } \big( (1+{D}^2)^{\frac 1 2 } +|{D}| \big) }\Big).
\end{split}\end{eqnarray*}
Since $(1+{D}^2)^{-\frac 1 2 } \in \mathcal{L}_{d,{\infty}}$, it follows that ${{\rm{sgn}}}({D}) - g({D}) \in \mathcal{L}_{\frac d 2 , {\infty}}$. Therefore,
$$[{{\rm{sgn}}}({D}), 1\otimes M_x] - [g({D}), 1\otimes M_x] \in \mathcal{L}_{\frac d 2 ,{\infty}} .$$
Thus, it suffices to prove
\begin{equation}\label{commutator-gDx}
[g({D}), 1\otimes M_x] - A(1+{D}^2)^{-\frac 1 2} \in \mathcal{L}_{\frac{d}{2},{\infty}}.\end{equation}
Now let us prove \eqref{commutator-gDx}. { By a short computation using Proposition \ref{symbol-composition}}, we see that the principal symbol of $[\frac{D_j}{(1-\Delta)^{\frac{1}{2}}}, M_x] $ is
\begin{equation}\label{ps-DM}
\frac{1}{|2\pi \xi| } {\partial}_j x - \sum_{k=1}^d \frac{2\pi \xi_k 2\pi \xi_j}{|2\pi\xi|^3} {\partial}_k x\,.\end{equation}
We also need to determine the principal symbol of $A_j (1-\Delta)^{-\frac 1 2 } $, and to this end we compute the principal symbol of $A_j$.
Recall that
$$A_j = M_{{\partial}_j x } -\frac 1 2 \sum_{k=1}^d \Big( \frac{D_j D_k}{1-\Delta} M_{{\partial}_k x } + M_{{\partial}_k x } \frac{D_j D_k}{1-\Delta} \Big).$$
It is evident that the symbol of $M_{{\partial}_k x } \frac{D_j D_k}{1-\Delta} $ is $\frac{2\pi\xi_j 2\pi \xi_k }{1+|2\pi\xi|^2 } {\partial}_k x$, so the principal symbol is $\frac{\xi_j \xi_k }{|\xi| ^2 } {\partial}_k x$.
By Proposition \ref{symbol-composition}, we know that
the symbol of $\frac{D_j D_k}{1-\Delta} M_{{\partial}_k x } $ has the asymptotic expansion
$$\sum _{\alpha \in \mathbb{N}_0^d} \frac{(2\pi {\rm i})^{-|{\alpha} |_1}}{{\alpha} !}D_\xi^{\alpha} \Big(\frac{2\pi\xi_j 2\pi \xi_k }{ (1+|2\pi\xi|^2)^{\frac 1 2 } }\Big) D^{\alpha} ({\partial}_k x) . $$
Thus, the principal symbol of $\frac 1 2 \sum_{k=1}^d \Big( \frac{D_j D_k}{1-\Delta} M_{{\partial}_k x } + M_{{\partial}_k x } \frac{D_j D_k}{1-\Delta} \Big)$ is $ \sum_{k=1}^d \frac{\xi_k \xi_j}{|\xi|^2} {\partial}_k x$, which ensures that the principal symbol of $A_j(1-\Delta) ^{-\frac 1 2 }$ is of order $-1$, given by
$$\frac{1}{2\pi |\xi| } {\partial}_j x - \sum_{k=1}^d \frac{\xi_k \xi_j}{2\pi |\xi|^3} {\partial}_k x\,,$$
the same as that of $[\frac{D_j}{(1-\Delta)^{\frac{1}{2}}}, M_x] $ given in \eqref{ps-DM}. Hence, the order of $[\frac{D_j}{(1-\Delta)^{\frac{1}{2}}}, M_x] - A_j (1-\Delta)^{-\frac 1 2} $ is $-2$. By Theorem \ref{pdo-weak-L}, we have
\begin{eqnarray*}
[\frac{D_j}{(1-\Delta)^{\frac{1}{2}}}, M_x] - A_j (1-\Delta)^{-\frac 1 2} \in \mathcal{L}_{\frac{d}{2},{\infty}}.\end{eqnarray*}
Since $[g({D}), 1\otimes M_x] = \sum_j \gamma_j \otimes [\frac{D_j}{(1-\Delta)^{\frac{1}{2}}}, M_x] $ and $A(1+{D}^2)^{-\frac 1 2} = \sum_j \gamma_j \otimes A_j (1-\Delta)^{-\frac 1 2} $, we obtain \eqref{commutator-gDx}. The lemma is thus proved.
\end{proof}
Based on the above lemmas, we are able to complete the proof of Theorem \ref{trace formula}.
\begin{proof}[Proof of Theorem \ref{trace formula}]
Assume initially that $x\in {\Sc}({\mathbb{T}_\theta^d})$. By Lemma \ref{commutator-Dx}, we have that:
\begin{equation*}
\,{\mathchar'26\mkern-12mu d} x\in A(1+{D}^2)^{-\frac{1}{2}} + {\mathcal L}_{\frac{d}{2},{\infty}}.
\end{equation*}
For any continuous normalised trace ${\varphi}$ on ${\mathcal L}_{1,{\infty}}$, we invoke Lemma \ref{T-AJ} to obtain that:\
\begin{equation*}
\varphi(|\,{\mathchar'26\mkern-12mu d} x|^d) = \varphi(|A|^d(1+{D}^2)^{-\frac{d}{2}}).
\end{equation*}
In the proof of Lemma \ref{commutator-Dx}, we have that the principal symbol of $A_j$ is $ {\partial}_j x - \sum_{k=1}^d \frac{\xi_k \xi_j}{|\xi|^2} {\partial}_k x$, which restricted to the unit sphere $\mathbb{S}^{d-1}$ is $ {\partial}_j x - \sum_{k=1}^d \xi_k \xi_j {\partial}_k x$. Now we appeal to Theorem \ref{Connes-qt} to conclude
\begin{equation*}
\varphi(|\,{\mathchar'26\mkern-12mu d} x|^d) = c_d\int_{\mathbb{S}^{d-1}} \tau\Bigg(\Big(\sum_{j=1}^d |{\partial}_j x-s_j\sum_{k=1}^d s_k{\partial}_k x|^2\Big)^{\frac{d}{2}}\Bigg)\,ds.
\end{equation*}
However the appeal to Theorem \ref{Connes-qt} relies on the assumption that $x \in {\Sc}({\mathbb{T}_\theta^d})$, so we remove this assumption by an approximation argument. Indeed, let $x\in \dot{H}_d^1 ({\mathbb{T}_\theta^d}) $. By Theorem \ref{sufficiency}, we have $\,{\mathchar'26\mkern-12mu d} x \in {\mathcal L}_{d,{\infty}}$. By the density of ${\Sc}({\mathbb{T}_\theta^d}) $ in $\dot H_d^1 ({\mathbb{T}_\theta^d})$ (see \cite[Proposition~2.7]{XXY2018}), we can choose a sequence $\{x_n\}_{n=1}^{\infty} \subset {\Sc}({\mathbb{T}_\theta^d})$ such that $x_n {\rightarrow} x$ in $\dot{H}_d^1({\mathbb{T}_\theta^d})$. We shall show that $\varphi(|\,{\mathchar'26\mkern-12mu d} x_n|^d) {\rightarrow} \varphi(|\,{\mathchar'26\mkern-12mu d} x|^d) $ and
\begin{equation}\label{convergence-Tx}
\int_{\mathbb{S}^{d-1}} \tau\Bigg(\Big(\sum_{j=1}^d |{\partial}_j x_n-s_j\sum_{k=1}^d s_k{\partial}_k x_n|^2\Big)^{\frac d 2}\Bigg)\,ds {\rightarrow} \int_{\mathbb{S}^{d-1}} \tau\Bigg(\Big(\sum_{j=1}^d |{\partial}_j x-s_j\sum_{k=1}^d s_k{\partial}_k x|^2\Big)^{\frac d 2}\Bigg)\,ds.
\end{equation}
{
Note that we have a bound:
\begin{equation*}
\int_{\mathbb{S}^{d-1}} \tau\Bigg(\Big(\sum_{j=1}^d |{\partial}_j x_n-s_j\sum_{k=1}^d s_k{\partial}_k x_n|^2\Big)^{\frac d 2}\Bigg)\,ds \leq C_d\|x_n\|_{\dot{H}^1_d}
\end{equation*}
for a certain constant $C_d$,
and hence \eqref{convergence-Tx} is immediate. On the other hand, using Theorem \ref{sufficiency}, we have:
\begin{equation*}
\|\,{\mathchar'26\mkern-12mu d} x-\,{\mathchar'26\mkern-12mu d} x_n\|_{{\mathcal L}_{d,{\infty}}} \leq C_d\|x-x_n\|_{\dot{H}^{1}_d({\mathbb{T}_\theta^d})} {\rightarrow} 0.
\end{equation*}
By a verbatim repetition of the argument in the proof of \cite[Theorem 17]{LMSZ2017}, we get
$$\||\,{\mathchar'26\mkern-12mu d} x|^d-|\,{\mathchar'26\mkern-12mu d} x_n|^d\|_{{\mathcal L}_{1,{\infty}}} {\rightarrow} 0.$$
Since the trace $\varphi$ is assumed to be continuous in the ${\mathcal L}_{1,\infty}$ quasi-norm, it follows that $\varphi(|\,{\mathchar'26\mkern-12mu d} x_n|^d){\rightarrow} \varphi(|\,{\mathchar'26\mkern-12mu d} x|^d)$.
}
\end{proof}
We are now concerned with relating the right hand side of the trace formula in Theorem \ref{trace formula} with the $\dot H^1_d$-norm of $x$.
\begin{proof}[Proof of Corollary \ref{trace formula-bound}]
We prove the upper bound first. { Denote
\begin{equation*}
T(x) := \Big(\sum_{j=1}^d |\partial_j x-s_j\sum_{k=1}^d s_k\partial_kx|^2\Big)^{\frac{1}{2}},\quad s \in \mathbb{S}^{d-1}.
\end{equation*}
Then
\begin{align*}
| T(x)|^2 &= \sum_{j=1}^d \big|\partial_j x-s_j\sum_{k=1}^d s_k\partial_k x \big|^2\\
&= \sum_{j=1}^d\Big( |\partial_j x|^2 -\sum_{k=1}^d (s_j\partial_j x^* \cdot s_k\partial_k x+s_k\partial_k x^*\cdot s_j\partial_j x)+s_j^2|\sum_{k=1}^d s_k\partial_k x|^2\Big)\\
&= \sum_{j=1}^d |\partial_j x|^2 +|\sum_{k=1}^d s_k\partial_kx|^2-\sum_{j,k=1}^d \big(s_j\partial_j x^*\cdot s_k\partial_k x+s_k\partial_k x^*\cdot s_j\partial_j x\big).
\end{align*}
However, observing that
\begin{equation*}
|\sum_{j=1}^d s_j\partial_j x|^2 = \sum_{j,k=1}^d s_j\partial_j x^*\cdot s_k\partial_k x\,,
\end{equation*}
we get
\begin{equation}\label{Tx-expand}
|T(x)|^2 = \sum_{j=1}^d |\partial_j x|^2-|\sum_{j=1}^d s_j\partial_j x|^2.
\end{equation}}
We then have have easily:
\begin{equation*}
|T(x)|^2 \leq \sum_{j=1}^d |\partial_j x|^2.
\end{equation*}
Therefore,
\begin{equation*}
\| |T(x)|^2 \|_{\frac d 2} \leq \|\sum_{j=1}^d |\partial_j x|^2 \|_{\frac d 2}.
\end{equation*}
Hence, by \eqref{Sob-equi-norm}, for every $s\in \mathbb{S}^{d-1}$, we have
\begin{equation*}
\tau\big( |T(x)|^d \big) = \| |T(x)|^2 \|_{\frac d 2}^{\frac d 2} \leq \|\sum_{j=1}^d |\partial_j x|^2 \|_{\frac d 2} ^{\frac d 2 }\leq C_d\|x\|_{\dot{H}^1_d}^d .
\end{equation*}
Thus the upper bound is proved.
Now we prove the lower bound. Since $|T(x)|^2 = \sum_{j=1}^d |\partial_j x-s_j\sum_{k=1}^d s_k\partial_kx|^2$, for each $j$ we have
\begin{equation*}
|\partial_j x-s_j\sum_{k=1}^d s_k\partial_k x|^2 \leq |T(x)|^2,
\end{equation*}
and therefore,
\begin{equation}\label{Xj-leq-T}
\|\partial_j x-s_j\sum_{k=1}^d s_k\partial_k x\|_d \leq \|T(x)\|_d.
\end{equation}
For brevity, define
\begin{equation*}
X_j = \|\partial_jx-s_j\sum_{k=1}^d s_k\partial_k x\|_d.
\end{equation*}
Then \eqref{Xj-leq-T} implies that
\begin{equation}\label{Xj-leq-T}
\big(\sum_{j=1}^d X_j\big)^d \leq d^d\|T(x)\|_d^d.
\end{equation}
By the triangle inequality,
\begin{align*}
X_j &= \|(1-s_j^2)\partial_j x-\sum_{k\neq j} s_js_k\partial_kx\|_d\\
&\geq (1-s_j^2)\|\partial_j x\|_d-\sum_{k\neq j} |s_js_k|\|\partial_kx\|_d.
\end{align*}
and therefore,
\begin{equation*}
\sum_{j=1}^d X_j \geq \sum_{j=1}^d\Big( (1-s_j^2)-\sum_{k\neq j} |s_js_k|\Big)\|\partial_j x\|_d.
\end{equation*}
Now, select $1\leq l\leq d$ such that $\|\partial_l x\|_{d}$ is the minimum of $\{\|\partial_1 x\|_d,\|\partial_2 x\|_d,\ldots,\|\partial_d x\|_d\}$. Denote by $e_l$ the $l$-th canonical basic vector of ${\mathbb R}^d$,
and assume that $s \in B(e_l,\varepsilon)\cap \mathbb{S}^{d-1}$.
We have:
\begin{equation*}
\big|(1-s_l^2)-\sum_{k\neq l}|s_ls_k|\,\big| \leq \max\big(1-s_l^2,\sum_{k\neq l}|s_ks_l |\big)
\end{equation*}
Hence,
$(1-s_l) \leq |s-e_l | \leq \varepsilon$, so $(1-s_l^2) = (1-s_l)(1+s_l) \leq 2\varepsilon$, and by the Cauchy-Schwarz inequality
\begin{align*}
\sum_{k\neq l} |s_k s_l| &\leq (\sum_{k\neq l} |s_k|^2)^{\frac 1 2 }d^{\frac 1 2 }|s_l|\\
&\leq |s-e_l | d^{\frac 1 2 } |s_l| \\
&\leq \sqrt{d}\varepsilon.
\end{align*}
So,
\begin{equation}\label{epsilon upper bound}
\big|(1-s_l^2)-\sum_{k\neq l}|s_ls_k|\,\big| \leq \max\{2,\sqrt{d}\}\varepsilon.
\end{equation}
On the other hand, if $j\neq l$, then $|s_j| \leq \varepsilon$ and so:
\begin{align}
(1-s_j^2)-\sum_{k\neq j} |s_ks_j| &= 1-|s_j|\sum_{k=1}^d |s_k|\nonumber\\
&\geq 1-\sqrt{d}\varepsilon \label{epsilon lower bound}.
\end{align}
If we select $\varepsilon$ sufficiently small, we have $1-\sqrt{d}\varepsilon\geq 3\max\{2,\sqrt{d}\}\varepsilon$.
Then combining \eqref{epsilon upper bound} and \eqref{epsilon lower bound}, we have that for all $j\neq l$:
\begin{equation*}
3\big|(1-s_l^2)-\sum_{k\neq l}|s_ls_k|\,\big| \leq (1-s_j^2)-\sum_{k\neq j} |s_ks_j|\,,
\end{equation*}
and thus,
\begin{equation*}
\big|(1-s_l^2)-\sum_{k\neq l} |s_ls_k|\,\big| \, \|\partial_l x\|_{d}\leq \frac{1}{3}\big((1-s_j^2)-\sum_{k\neq j}|s_ks_j|\big)\|\partial_j x\|_d.
\end{equation*}
Therefore, using the numerical inequality that if $|z| \leq \frac{1}{3}|w|$ then $|z-w| \geq \frac{2}{3}|w|$, we have
\begin{align*}
& \big((1-s_j^2)-\sum_{k\neq j}|s_ks_j|\big)\|\partial_j x\|_{d}+\big((1-s_l^2)-\sum_{k\neq l} |s_ls_k|\big)\|\partial_l x\|_{d}\\ &\geq \frac{2}{3} \big((1-s_j^2)-\sum_{k\neq j}|s_ks_j|\big)\|\partial_j x\|_{d}\\
&\geq \frac{1}{3}(1-\sqrt{d}\varepsilon)(\|\partial_j x\|_{d}+\|\partial_l x\|_{d}).
\end{align*}
Consequently, for $s \in B(e_l,\varepsilon)\cap \mathbb{S}^{d-1}$, we have,
\begin{align*}
\sum_{j=1}^d X_j \geq (1-\sqrt{d}\varepsilon)\sum_{j=1}^d \|\partial_j x\|_d.
\end{align*}
Now,
\begin{align*}
\int_{\mathbb{S}^{d-1}} \Big(\sum_{j=1}^d X_j\Big)^d\,ds &\geq \int_{B(e_l,\varepsilon)\cap \mathbb{S}^{d-1}} \Big(\sum_{j=1}^d X_j\Big)^d\,ds\\
&\geq c_{d,\varepsilon}\|x\|_{\dot{H}^1_d}.
\end{align*}
By virtue of \eqref{Xj-leq-T}, the desired conclusion is proved.
\end{proof}
\section{Proof of Theorem \ref{necessity}}\label{sec-nece}
In this section, we are going to give the proof of Theorem \ref{necessity}. We require a lemma on the quantised derivative of $x$ acting by a Fourier multiplier.
Recall that for a function $\psi \in L_1({\mathbb T}^d)$, the convolution with $x \in { L_2({\mathbb{T}_\theta^d})}$ is defined as:
\begin{equation*}
\psi \ast x = \int_{{\mathbb T}^d} \psi(w){\alpha}_{w^{-1}}(x)\,dm(w).
\end{equation*}
\begin{lem}\label{commutator-convo}
Let $\psi \in L_1({\mathbb T}^d)$. If $x\in { L_2({\mathbb{T}_\theta^d})}$ is such that $\,{\mathchar'26\mkern-12mu d} x$ { extends to a bounded operator in ${\mathcal L}_{d,{\infty}}$, then $\,{\mathchar'26\mkern-12mu d} (\psi \ast x)$ also extends to a bounded operator in ${\mathcal L}_{d,{\infty}}$ and we have:}
$$\|\,{\mathchar'26\mkern-12mu d}(\psi\ast x)\|_{{\mathcal L}_{d,{\infty}}} \leq { C_d}\|\,{\mathchar'26\mkern-12mu d} x \|_{{\mathcal L}_{d,{\infty}}} \|\psi\|_1 $$
{ for a certain constant $C_d$.}
\end{lem}
\begin{proof}
Let ${\alpha}$ be the $d$-parameter group of automorphisms given in \eqref{action-T-qt}, i.e. if $u \in {\mathbb T}^d$ then ${\alpha} _u( U^n) = u^nU^n$. Then for each $u\in {\mathbb T}^d$, ${\alpha}_u$ commutes with Fourier multipliers on ${\mathbb{T}_\theta^d}$, and $u\mapsto {\alpha}_u$ is a strongly continuous family of unitary operators on $L_2({\mathbb{T}_\theta^d})$. By the definition of convolution, we have
\begin{equation}\label{convo-translation}
\psi\ast x =\int_{{\mathbb T}^d } \psi(u)\, {\alpha}_u^{-1}( x) \, dm(u). \end{equation}
Since ${\alpha}_u$ and $\frac{D_j}{\sqrt{D_1^2 +D_2^2 +\cdots+ D_d^2}}$ commute, we see that $1\otimes {\alpha}_u$ commutes with ${{\rm{sgn}}}({D})$. Therefore, by the fact that $\big({\alpha}_u^{-1}(x)\big) y = {\alpha}_u^{-1} \Big( x \big({\alpha}_u( y)\big) \Big)$, we obtain
$$[{{\rm{sgn}}}({D}), 1\otimes M_{\psi\ast x} ] =\int _{{\mathbb T}^d}\psi (u)\, (1\otimes {\alpha}_{u^{-1}}) [{{\rm{sgn}}}({D}), 1\otimes M_{x} ] (1\otimes {\alpha}_u) \, dm(u) .$$
Applying \cite[Lemma~18]{LMSZ2017} to the finite Borel measure $\psi(u)\, dm(u)$ on ${\mathbb T}^d$, we get
$$\|[{{\rm{sgn}}}({D}), 1\otimes M_{\psi\ast x} ]\|_{{\mathcal L}_{d,{\infty}}} \leq { C_d}\|[{{\rm{sgn}}}({D}), 1\otimes M_{x} ]\|_{{\mathcal L}_{d,{\infty}}} \|\psi\|_1 $$
{ where the constant comes from the use of the quasi-triangle inequality in the ${\mathcal L}_{d,{\infty}}$ quasi-norm.}
This now completes the proof.
\end{proof}
\begin{proof}[Proof of Theorem \ref{necessity}]
Firstly, we prove the theorem for self-adjoint $x\in { L_2 ({\mathbb{T}_\theta^d})}$. If we show that $x\in \dot{H}_d^1({\mathbb{T}_\theta^d})$, then Corollary \ref{trace formula-bound} will ensure that there exists a constant $c_d >0 $ such that for all continuous normalised traces ${\varphi}$ on ${\mathcal L}_{1,{\infty}}$,
$$c_d \|x\|_{\dot{H}_d^1} \leq {\varphi}(|\,{\mathchar'26\mkern-12mu d} x|^d ) ^{\frac 1 d }\leq \|\,{\mathchar'26\mkern-12mu d} x\|_{d,{\infty}}.$$
Thus, we are reduced to proving $x\in \dot{H}_d^1({\mathbb{T}_\theta^d})$.
Consider the square Fej\'er mean
$$F_N(x) = \sum_{m\in {\mathbb Z}^d, \max_j |m_j| \leq N} \Big(1-\frac{ |m_1|}{N+1}\Big) \cdots \Big(1-\frac{ |m_d|}{N+1}\Big)\widehat{x}(m) U^m .$$
For every $N\in {\mathbb N}$, it is the convolution of $x$ with the periodic function
$$F_N(u) = \frac{1}{( N+1 )^d } \Big( \frac{\sin \big(\pi (N+1) u_1\big)}{\sin \big(\pi u_1\big)} \Big)^2 \cdots \Big( \frac{\sin \big(\pi (N+1) u_d\big)}{\sin \big(\pi u_d\big)} \Big)^2.$$
The family $\{F_N\}_{N\in {\mathbb N}}$ is an approximate identity of $L_1({\mathbb T}^d)$ (see \cite{Graf2008}), so we have uniform bound of $\| F_N\|_1$ in $N\in {\mathbb N}$. Thus we can apply Lemma \ref{commutator-convo} to $F_N$. The result is
$$\|\,{\mathchar'26\mkern-12mu d}\big( F_N(x)\big)\|_{{\mathcal L}_{d,{\infty}}} \leq \|\,{\mathchar'26\mkern-12mu d} x\|_{{\mathcal L}_{d,{\infty}}} \| F_N\|_1 \leq C \|\,{\mathchar'26\mkern-12mu d} x \|_{{\mathcal L}_{d,{\infty}}} .$$
Since each $F_N(x)$ is a polynomial in ${\mathbb{T}_\theta^d}$, Corollary \ref{trace formula-bound} yields
\begin{eqnarray*}
c_d \|F_N(x)\|_{\dot{H}_d^1} \leq {\varphi}(|\,{\mathchar'26\mkern-12mu d}\big(F_N(x)\big)|^d ) ^{\frac 1 d } \leq C\|\,{\mathchar'26\mkern-12mu d} x\|_{{\mathcal L}_{d,{\infty}}} . \end{eqnarray*}
Hence, for each $j $, we obtain a bounded sequence $\{ {\partial}_j F_N(x) \}_{N\in {\mathbb N}}$ in $L_d({\mathbb{T}_\theta^d})$. Moreover, since $L_d({\mathbb{T}_\theta^d})$ is reflexive, we may assume that ${\partial}_j F_N(x) $ converges to some $y_j \in L_d({\mathbb{T}_\theta^d})$. On the other hand, by \cite[Proposition~3.1]{CXY2013}, we have $\lim_N F_N(x) = x$ in {$L_2({\mathbb{T}_\theta^d})$}. Hence, ${\partial}_j F_N(x) $ converges to ${\partial}_j x$ in ${\mathcal{D}}'({\mathbb{T}_\theta^d})$. Therefore, we have $y_j = {\partial}_j x \in L_d({\mathbb{T}_\theta^d})$. Consequently, we conclude that $x\in \dot{H}_d^1({\mathbb{T}_\theta^d})$.
It remains to consider $x\in { L_2 ({\mathbb{T}_\theta^d})}$ which are not self-adjoint. Write $x= x_1 + {\rm{i}} x _2$ with
$$x_1 = \frac{1}{2} (x +x^*),\quad x_2 = \frac{1}{2{\rm{i}} } (x-x^*).$$
If $[{{\rm{sgn}}}({D}), 1\otimes M_{x} ] \in {\mathcal L}_{d, {\infty}}$, then $[{{\rm{sgn}}}({D}), 1\otimes M_{x^*} ]=- [{{\rm{sgn}}}({D}), 1\otimes M_{x} ]^* \in {\mathcal L}_{d,{\infty}}$. Then we have $[{{\rm{sgn}}}({D}), 1\otimes M_{x_1} ]\in {\mathcal L}_{d,{\infty}}$ and $[{{\rm{sgn}}}({D}), 1\otimes M_{x_2} ]\in {\mathcal L}_{d,{\infty}}$. By the above conclusion for self-adjoint elements, we know that $x_1 , x_2\in \dot{H}^1_d({\mathbb{T}_\theta^d})$, which implies $x= x_1+ {\rm{i}} x_2 \in\dot{H}^1_d({\mathbb{T}_\theta^d})$. More precisely,
\begin{eqnarray*}\begin{split}
\|x\|_{\dot{H}^1_d} &\leq \|x_1\|_{\dot{H}^1_d} +\|x_2\|_{\dot{H}^1_d} \\
&\leq C_1(\|[{{\rm{sgn}}}({D}), 1\otimes M_{x_1} ]\|_{ {\mathcal L}_{d,{\infty}}} +\|[{{\rm{sgn}}}({D}), 1\otimes M_{x_2} ]\|_{ {\mathcal L}_{d,{\infty}}} ) \\
& \leq C_2(\|[{{\rm{sgn}}}({D}), 1\otimes M_{x} ]\|_{ {\mathcal L}_{d,{\infty}}} +\|[{{\rm{sgn}}}({D}), 1\otimes M_{x} ]^*\|_{ {\mathcal L}_{d,{\infty}}} ) \\
& = 2C_2 \|[{{\rm{sgn}}}({D}), 1\otimes M_{x} ]\|_{ {\mathcal L}_{d,{\infty}}} .
\end{split}\end{eqnarray*}
The theorem is proved.
\end{proof}
{
\noindent{\bf Acknowledgements.} The authors wish to thank the anonymous referees for careful reading and useful suggestions; in particular one referee pointed out how our main results could be proved without an $L_\infty$ condition. We are also greatly indebted to Professor Rapha\"el Ponge for many helpful comments on the section of Pseudodifferential Operators. The authors are supported by Australian Research Council (grant no. FL170100052); X. Xiong is also partially supported by the National Natural Science Foundation of China (grant no. 11301401).
}
|
3,212,635,537,985 | arxiv |
\section{Conclusion}
\label{sec:conslusion}
In this paper we evaluated an alternative pipeline for decreasing the runtimes of object recognition when $k$NN queries are used for the generation of
tentative correspondences instead of Bags of Visual Words. While the reduction of query features can have negative effects on query performance,
especially if the unmodified standard recognition pipeline is used, some simple modifications in the pipeline aiming at feature ranking and match
expansion can already produce good results at only a fraction of $k$NN queries. Some challenges however, remain. First, more techniques for
feature ranking will have to be investigated that provide good results for any type of keypoint descriptor and extractor.
Additionally, improvements in the match expansion stage should aim at increasing efficiency and effectiveness. Due to the simple structure of the
pipeline used in this paper, these improvements can be easily integrated.
\section{Experiments}
\label{sec:experiments}
\subsection{Experimental Setup}
\textbf{Datasets. }We evaluated the modified recognition pipeline on four
datasets. The \textit{Oxford5k} (O5k) building dataset \cite{PhiChuIsaSivZis07} consists of 5063 images of
common tourist landmarks in Oxford. The authors of the benchmark also provide a
set of 55 queries including rectangular query regions and ground truth files
listing, for each query, the images that contain at least parts of the query.
Ground truth files are split into three categories: good, ok and junk. Good and
ok files are considered for computing the Mean Average Precision (MAP) of the
query. Junk images are neither scored as true hit nor as false hit and simply
discarded for computing the MAP. We also included \textit{Oxford105k} (O105k) in
our evaluation which consists of the Oxford5k dataset in combination with about
100k distractor images \cite{PhiChuIsaSivZis07} that do not contain images
related to the query.
The \textit{Paris6k} (P6k) dataset \cite{PhiChuIsaSivZis08}, conceptually
similar to the Oxford dataset, consists of 6412 images of common landmarks in
Paris, and has the same structure as the Oxford dataset. As a third dataset we
used the \textit{INRIA Holidays} (Hol) dataset \cite{JegDouSch08} which consists
of 1491 images including 500 queries and their corresponding ground truth. In
contrast to the Oxford and Paris dataset, Holidays contains more natural scenes
and a lower number of result images for each query. Images of the Holidays
dataset were scaled down to a maximum side length of 1024 before feature
extraction.
\textbf{Feature Extraction and Indexing. }We used two different feature extraction techniques: a rotation-variant version of SIFT using affine
invariant keypoints\footnote{https://github.com/perdoch/hesaff/} made available
by the authors of \cite{PerChuMat09} and, as an instance of state-of-the-art
binary descriptors, the BinBoost descriptor which is also publicly available \cite{TrzChrFuaLep13}. We decided to include binary features in our
evaluation as we see them as another mean of decreasing query complexity, however we will concentrate on SIFT features in our evaluation.
\begin{table}
\center
\caption{Database Statistics}
\label{tab:data_statistics}
\begin{tabular}[h!]{|c||c|c|c|c|}
\hline
Dataset & Extractor & Features & $\diameter$ \\
\hline
\hline
O5k & BinBoost & 10,640,081 & 2101.5\\
O105k & BinBoost & 195,068,373 & 1855.4\\
O5k & SIFT (Hess.-Aff.) & 13,516,675 & 2669.7\\
O105k & SIFT (Hess.-Aff.) & 253,761,866 & 2413.7 \\
P6k & SIFT (Hess.-Aff.) & 16,073,531 & 2506.8\\
Hol & SIFT (Hess.-Aff.) & 4,596,567 & 3082.9 \\
\hline
\end{tabular}
\vspace{-1em}
\end{table}
Concerning \textit{Hessian-affine SIFT}, scale was separated from the affinity
matrices according to \cite{PerChuMat09}, however for expanding matches we used
the square root of this scale which roughly corresponds to the radius of the
image patch used for SIFT extraction.
The parameters of the feature extraction stage have been left at the default
parameters. SIFT features are 128-dimensional real-valued vectors.
These vectors were square-root weighted similar to RootSift\cite{AraZis12},
however without $l_1$ normalization. The weighted features were then indexed using LOPQ in combination
with a multi-index \cite{KalAvr14}. We use a vocabulary of size $V=2*1024$ for
the inverted lists, and 8 subquantizers for vector quantization, each
subquantizer with a vocabulary of size of 256 clusters.
The corresponding source code has been kindly provided by the
authors\footnote{http://image.ntua.gr/iva/research/lopq/}. To compress the
feature vectors for the expansion phase, we again used 8 subquantizers
consisting of 256 clusters, reducing storage overhead of feature vectors to
6.25\% of their uncompressed memory footprint. Codebooks for the Oxford and
Holidays datasets were trained on Paris6k, and for Paris6k code books where
trained on Oxford5k. During query processing, we applied a simple means of
burst removal \cite{JegDouSch09}, scoring each query feature once even if it had more
than one match.
\begin{table*}
\center
\caption{Parameters for Match Expansion}
\begin{tabular}[h!]{|c||c|c||c|c|c|c|c|}
\hline
Extractor & Train & $\delta_{xy}$ & $\delta_s$ & $\delta_{d_v}$ & $\delta_\alpha$ & $\delta_r$ & $\delta_{d_{xy}}$\\
\hline
\hline
SIFT & P6k & 6 & 0.8 & 26.2 & 24.3 & -- & 0.49\\
\hline
SIFT & O5k & 6 & 0.8 & 26.9 & 18.9 & -- & 0.56\\
\hline
BinBoost & P6k & 4 & 0.8 & 73 & 21.1 & 26.0 & 0.46 \\
\hline
\end{tabular}
\label{tab:parameters}
\end{table*}
\textit{BinBoost} descriptors, i.e. 256-dimensional binary vectors, can be queried rather efficiently using exact indexing techniques optimized for
binary feature vectors, e.g. \cite{NorPunFle12}. We have used a publicly available implementation of their index during our experimental evaluation.
We applied burst removal when querying these features as well.
An overview of the extracted features can be found in Table~\ref{tab:data_statistics}. Note that the number of query features was different to the
number of database features on Oxford5k, Oxford105k and Paris6k due to the bounding boxes provided by the dataset authors, and for several queries the
number of query features was less than 1000. The average number of features available over all queries was 1371.4 (BinBoost,
$\sigma=612.3$) and 1452.8 (SIFT Hessian-Affine, $\sigma=950.2$) for queries on Oxford.
The code was written in C++ using OpenCV. Runtime experiments were conducted on
an off-the-shelf Linux Machine with i7-3770@3.40GHz CPU and 32GB of main memory without parallelization. During our experimental evaluation we
concentrate on analyzing the effectiveness of the approaches in terms of Mean Average Precision (MAP); we also provide numbers on the performance of the evaluated approaches concerning the runtime of the scoring, querying
and ranking stages.
\textbf{Parameters.} The parameters for query processing were set as follows. First, range multiplier $\delta_{xy}$, maximum
scale change $\delta_s$, $k$, and $n$ were set by hand with computational efficiency in mind, as a lower number of features considered during
expansion reduces the cost of this step. Given these manually set parameters, the remaining parameters of the expansion phase, i.e.
feature distance threshold $\delta_{d_v}$, angular threshold $\delta_\alpha$, gradient angle threshold $\delta_r$ and spatial distance ratio $\delta_{d_{xy}}$
were set to the outcome of a Nelder-Mead Downhill-Simplex optimization maximizing MAP; initialization was performed with reasonable seed values.
Minimization was done on the Paris6k dataset (with LOPQ and quantization code books trained on Paris6k as well) for the Oxford5k, Oxford105k and Holidays datasets. For the Paris6k dataset, we optimized these parameters on the Oxford5k dataset. The parameters were selected for each of the
descriptor types (SIFT and BinBoost) using ANMS ranking at $k=100$, number of keypoints $n=10$, recursively descending into every expanded match. The resulting parameters
were reused for the remaining ranking approaches, different $k$, $n$ and the non-recursive approach. An overview over the selected parameters
is shown in Table~\ref{tab:parameters}.
We varied each of the optimized parameters by $\pm 10\%$ separately on Oxford 5k (ANMS ranking with match expansion) to get insights into their
effect on MAP. The maximum deviation resulted from decreasing the feature distance threshold, which lead to a decrease in MAP of $-0.012$, indicating
that while there is an impact of the optimized parameters on the performance of match expansion, there is still a range of relatively ``good''
parameters.
\subsection{Experiments}
We evaluated the algorithm's performance by varying $k$ and $n$ as these parameters affect the number of initial seed points that are expanded later.
As a baseline for our experiments we implemented a scoring scheme based on \cite{JegDouSch11b} that considers the distances between features and the
number of features in the image for score computation.
\begin{table}
\center
\caption{SIFT, Oxford5k, k=100}
\label{tab:k100SIFT_HESAFF_OXFORD5K_SC6_LOPQ_RSNL1_SQ8}
\begin{tabular}[h!]{|c||c|c||c|c|}
\hline
$\downarrow$ Appr. $\rightarrow n$& 50 & 100 & 500 & 1000 \\
\hline
\hline
RND & .616 & .698 & .810 & .827 \\
\hline
RESP & .557 & .640 & .787 & .822 \\
\hline
ANMS & .676 & .727 & .825 & .836 \\
\hline
\hline
RND+ME & .679 & .749 & .829 & .838 \\
\hline
ANMS+ME & .741 & .780 & .843 & .844 \\
\hline
\hline
RND+MER & .686 & .752 & .826 & .832 \\
\hline
ANMS+MER & .752 & .786 & .837 & .838 \\
\hline
\end{tabular}
\end{table}
\begin{table}
\center
\caption{SIFT, Paris6k, k=100}
\label{tab:k100SIFT_HESAFF_PARIS6K_SC6_LOPQ_RSNL1_SQ8}
\begin{tabular}[h!]{|c||c|c||c|c|}
\hline
$\downarrow$ Appr. $\rightarrow n$& 50 & 100 & 500 & 1000 \\
\hline
\hline
RND & .566 & .652 & .770 & .786 \\
\hline
RESP & .519 & .594 & .743 & .775 \\
\hline
ANMS & .578 & .668 & .783 & .794 \\
\hline
\hline
RND+ME & .629 & .699 & .781 & .789 \\
\hline
ANMS+ME & .648 & .723 & .793 & .796 \\
\hline
\end{tabular}
\end{table}
\textbf{Keypoint Ranking.} In our first experiment (see
Table~\ref{tab:k100SIFT_HESAFF_OXFORD5K_SC6_LOPQ_RSNL1_SQ8} and
Table~\ref{tab:k100BIN_BOOST_OXFORD5K_SC4}) we wanted to evaluate the
performance difference in MAP when querying a low number of features (i.e. 50,
100, 500 and 1000 keypoints) with different keypoint ranking techniques,
providing a baseline for further experiments.
The simplest ranking (RND) takes random features from the extracted keypoints;
we averaged this approach over 5 runs to get accurate results. Furthermore we
evaluated a ranking based on keypoint responses (RESP), and a more sophisticated
approach called Adaptive Non-Maximal Suppression \cite{BroSzeWin05} (ANMS) that
aims at distributing keypoints relatively uniformly over the image. As expected,
considering only few keypoints significantly reduces the MAP of all approaches.
The MAP of the response-based ranking is worse or similar to the random
baseline: for SIFT, the response decreases performance compared to the random
approach, while for BinBoost (that is based on SURF Keypoints) results are
sometimes slightly better than the random baseline. The ANMS ranking increases
the MAP for all approaches.
Note that the gain resulting from using ANMS is rather astonishing for the
Oxford5k dataset; we can easily gain 0.03 ($n$=100) to 0.06 ($n$=50) points in
MAP without significant computational overhead if the number of features queried is
relatively low. Similar observations hold for Holidays
(Table~\ref{tab:k10SIFT_HESAFF_HOLIDAYS_SC6_LOPQ_RSNL1_SQ8}) but considering
Paris6k (Table~\ref{tab:k100SIFT_HESAFF_PARIS6K_SC6_LOPQ_RSNL1_SQ8}), the gain
resulting from using ANMS instead of a random ranking is lower. Our results with
BinBoost (Table~\ref{tab:k100BIN_BOOST_OXFORD5K_SC4}) on Oxford5k indicate that
ANMS without match expansion can increase performance by over 0.07
points in MAP ($n$=50), however its performance is generally lower than SIFT,
even if SIFT vectors are quantized as in our case; the memory overhead (8 bytes) for
quantized SIFT vectors is actually lower than for BinBoost (32 bytes) features.
\begin{table}
\center
\caption{SIFT, Holidays, k=10}
\label{tab:k10SIFT_HESAFF_HOLIDAYS_SC6_LOPQ_RSNL1_SQ8}
\begin{tabular}[h!]{|c||c|c||c|c|}
\hline
$\downarrow$ Appr. $\rightarrow n$& 50 & 100 & 500 & 1000 \\
\hline
\hline
RND & .600 & .662 & .765 & .792 \\
\hline
RESP & .571 & .630 & .735 & .770 \\
\hline
ANMS & .642 & .696 & .779 & .803 \\
\hline
\hline
RND+ME & .646 & .702 & .764 & .770 \\
\hline
ANMS+ME & .699 & .734 & .780 & .781 \\
\hline
\end{tabular}
\end{table}
\begin{table}
\center
\caption{BinBoost, Oxford5k, k=100}
\label{tab:k100BIN_BOOST_OXFORD5K_SC4}
\begin{tabular}[h!]{|c||c|c||c|c|}
\hline
$\downarrow$ Appr. $\rightarrow n$& 50 & 100 & 500 & 1000 \\
\hline
\hline
RND & .390 & .462 & .586 & .616 \\
\hline
RESP & .389 & .461 & .600 & .625 \\
\hline
ANMS & .461 & .508 & .614 & .620 \\
\hline
\hline
RND+ME & .469 & .529 & .625 & .638 \\
\hline
ANMS+ME & .542 & .588 & .648 & .644 \\
\hline
\hline
RND+MER & .481 & .539 & .626 & .634 \\
\hline
ANMS+MER & .551 & .591 & .648 & .640 \\
\hline
\end{tabular}
\end{table}
\textbf{Match expansion.} Our second experiment aims at evaluating the gain in MAP that can be achieved for a low number of $k$NN queries when
additional hypotheses are generated by match expansion (ME) and the same approach in its recursive version (MER). Affine-invariant SIFT
(ANMS+ME, $n$=50) achieves about 90\% of the random baseline (RND, $n$=1000) at
50 keypoints on Oxford5k, where the baseline only achieves 75\%. At the same time the results at 1000 keypoints are similar for all approaches, showing that match expansion does not considerably affect MAP if a high number of keypoints is
queried. This substantiates our statement made in the introduction: if a small number of features is queried, techniques that do
not achieve significant performance gain for a high number of features can achieve considerable gain in performance. Results are similar for
Holidays (88\% for ANMS+ME@$n=50$ vs. 76\% for the random Baseline) while for Paris6k the gain of match expansion is lower (82\% vs 72\% for the
random baseline). Further note that MAP for ANMS+ME decreases slower with
decreasing $n$ than without expansion (-0.003 (ANMS+ME) vs. -0.011 (ANMS) for
$n:1000\rightarrow500$ on Paris6k). Results for Oxford105k are shown in Table~\ref{tab:k100SIFT_HESAFF_OXFORD105K_SC6_LOPQ_RSNL1_SQ8}.
Using BinBoost (Table~\ref{tab:k100BIN_BOOST_OXFORD5K_SC4} and Table~\ref{tab:k100BIN_BOOST_OXFORD105K_SC4}) the results are similar. The combination
of ANMS ranking and match expansion at 100 keypoints performs similar to the
random baseline at 500 keypoints, which is especially interesting as the exact indexing techniques used for BinBoost already lead to a relatively high runtime.
While there is some gain for recursively descending (MER) into matches,
this additional step does not significantly improve the performance with both SIFT and BinBoost, while being computationally much more expensive.
Therefore we will concentrate on ME in the following.
ANMS+ME on Oxford using SIFT accepted about 16500 matches per query image ($n=100$, $k=100$), in
contrast to the approximately 8800 tentative correspondences (less than $n*k$ due to burst removal) that have been generated using $k$NN matching
alone (ANMS).
Note that we also evaluated the effect of a re-ranking stage, weak geometric
consistency (WGC) \cite{JegDouSch08} with 32 angular and 16 scale bins, on match
expansion using BinBoost. Both pipelines, with and without match expansion were
positively affected by WGC, gaining about 0.04 (ANMS and ANMS+ME) points in MAP
at $n=1000$, indicating that WGC is complementary to match expansion. We further
observed that WGC did not have a large effect with a low number of
features (0.013 with ANMS and 0.006 with ANMS+ME at $n=100$)
involved both with and without match expansion.
\begin{table}
\center
\vspace{-1em}
\caption{SIFT, Oxford 105k, k=100}
\label{tab:k100SIFT_HESAFF_OXFORD105K_SC6_LOPQ_RSNL1_SQ8}
\begin{tabular}[h!]{|c||c|c||c|c|}
\hline
$\downarrow$ Appr. $\rightarrow n$& 50 & 100 & 500 & 1000 \\
\hline
\hline
ANMS & .489 & .554 & .710 & .748 \\
\hline
ANMS+ME & .584 & .630 & .753 & .775 \\
\hline
\end{tabular}
\vspace{-1em}
\end{table}
\begin{table}
\center
\caption{BinBoost, Oxford 105k, k=100}
\label{tab:k100BIN_BOOST_OXFORD105K_SC4}
\begin{tabular}[h!]{|c||c|c||c|c|}
\hline
$\downarrow$ Appr. $\rightarrow n$& 50 & 100 & 500 & 1000 \\
\hline
\hline
ANMS & .369 & .412 & .527 & .558 \\
\hline
ANMS+ME & .445 & .477 & .576 & .590 \\
\hline
\end{tabular}
\end{table}
\textbf{Value of $k$.} Not only the number of queried keypoints can be used to increase the number of seed hypotheses and therefore the matching
quality, but also $k$. While increasing $k$ comes at a lower query cost, it also produces hypotheses of lower quality. However, as it is well known
for example in the context of $k$NN classification, reasonable values for $k$ can increase the query performance. In this experiment we evaluate the effect of $k$ if the
number of features to be queried is fixed to a given number. For a large number of keypoints, a very high value of $k$ adds a lot of false positives
such that the MAP decreases \cite{JegDouSch11b}. On the other hand, if $k$ is too small, only a small amount of correct hypotheses is found
\cite{JegDouSch11b}. We reproduced this result with the ANMS ranker at 1000
keypoints (see Figure~\ref{fig:hesaffSiftK}), as here the MAP at $k=100$ is
highest compared to $k=10$ or $k=1000$.
What happens when we decrease the number of keypoints? As shown in
Figure~\ref{fig:hesaffSiftK}, if a large number of keypoints is queried
($n=1000$), then for all of the evaluated approaches a value of $k=100$
performed better than $k=1000$. So match expansion does not greatly affect the
optimal value of $k$ in this case. However, if only very few keypoints are used
for query processing (e.g. $n=10$), a large $k$ performed better with match
expansion. Without this additional step, performance decreased for large $k$
(however at a larger $k$ than at a higher number of keypoints queried), most
likely because the additional noise introduced could not be out-weighted by
the higher number of correct matches. This leads us to the following results:
The best way to increase query performance, which is well known, is to increase
the number of keypoints queried. In order to increase query performance however,
it is possible to decrease the number of keypoints queried. In this case, some
of the performance loss resulting from a lower number of keypoints can be
compensated by a large $k$ in combination with match expansion (and, at a lower
degree, even without expanding matches).
\begin{figure}[t] \centering
\includegraphics[width=\columnwidth]{HesaffSiftK.pdf}
\vspace{-2em}
\caption{Performance for varying $k$ (Hessian-affine SIFT). Straight lines show the performance for 10 keypoints, dashed lines for 1000 keypoints.
Equivalent approaches have equivalent colors.}
\label{fig:hesaffSiftK}
\vspace{-1em}
\end{figure}
\textbf{Runtime.} The cost of the evaluated \textit{keypoint ranking} approaches is negligible for the random and response based ones, as these just
have to sort the query features, and about 7ms (SIFT) and 5ms (BinBoost) for the
ANMS ranker. For Hessian-affine SIFT on Oxford5k, scoring times (including ME)
were about 6ms for processing all $k$ results of a single $k$NN query ($k=100$, $n=100$), and therefore slightly lower than the runtimes of running a single $k$NN query which took about 7ms, at the possible gain of adding additional matches and a rough geometric check. The feature quantization needed for
match expansion took about 45ms for all features in a query image. For BinBoost features (including ME), the match expansion and scoring took less
than 4ms for processing a single $k$NN result. Runtimes increase with $k$, as more correspondences have to be expanded. The overall runtime for Hessian-Affine SIFT at
100 keypoints (ANMS+ME) was about 1.34s ($k=100$, $n=100$), while for binary
features it was higher (15s), as for this we used an exact,
though state-of-the-art, indexing technique.
Setting runtimes in relation to MAP, it is possible to beat an RND ranker
considering 100 keypoints with ANMS+ME considering 50 keypoints at a slightly
lower runtime 0.69s vs 0.75s and a higher MAP (see
Table~\ref{tab:k100SIFT_HESAFF_OXFORD5K_SC6_LOPQ_RSNL1_SQ8}, 0.741
vs. 0.698). For the holidays dataset runtimes of the random approach (RND) were
about 0.9s ($k=10$, $n=100$) and for ANMS+ME it was only approximately 0.6s
($k=10$, $n=50$) at a higher MAP.
The most time-consuming operation during match expansion is the search of
spatially close features. Therefore we think that the runtimes of match
expansion can be reduced significantly by optimizing this matching step, e.g. by
ordering features in a $k$d-tree which can be realized without additional space
overhead. This would also help on the Paris6k dataset where runtimes of the
random approach (RND) were about 0.8s ($k=100$, $n=100$) and for ANMS+ME
approximately 0.7s ($k=100$, $n=50$) at similar MAP.
Runtimes have been measured using only a single core. As each keypoint is
queried separately and match expansion is also achieved on a per-keypoint basis,
query processing can be easily extended to a multi-core setting.
For Oxford105k the runtime for match expansion was similar to Oxford5k: A single $k$NN query took slightly less than 8ms and expansion took about 7ms.
\textbf{Comparison to the State of the Art.} While the primary goal of this
research is not to increase the effectiveness of object recognition but rather
to reduce the number of features queried, let us still compare the results from
this paper to the state of the art in order to get insights into its
performance. We will compare to \cite{QinWenGoo13}, as the authors were using
the same Hessian-Affine SIFT features as we do and a similar recognition
pipeline involving Product Quantization. On the Oxford5k dataset the authors of
\cite{QinWenGoo13} achieved 0.78 points in MAP using 8 subquantizers for
indexing features using Product Quantization, i.e. a setting close to our
scenario. This corresponds to the performance we could achieve when querying 100
keypoints. However, using the pipeline from this chapter requires a larger
memory footprint; if we consider techniques with a memory footprint closer to ours,
\cite{QinWenGoo13} was able to achieve 0.83 points in MAP by approximating
features more accurately using 32 bytes per feature. On Paris6k, using 8
subquantizers, \cite{QinWenGoo13} achieved a MAP of 0.74, which is slightly
better than the performance of ANMS+ME at 100 keypoints (at a higher memory
footprint, performance of \cite{QinWenGoo13} was 0.76).
Concerning Oxford105k, a larger number of features (about 300) is needed to
achieve performance comparable to the state of the art (0.728
\cite{QinWenGoo13}). Finally note that the performance of our baseline is lower
for Holidays than the state of the art performance of $0.80$ ($0.84$
respectively) from \cite{QinWenGoo13}; this might be due to a different
quantization training set or related to their similarity measure, which is
however complementary to our approach and can be easily integrated into our
pipeline.
\section{Introduction}
While the development of the SIFT-Descriptor \cite{Low04} made effective object retrieval on a large scale feasible, its initial use of nearest
neighbor queries lead to slow runtimes even on relatively small data sets. In 2003, the invention of the Bag of Visual Words (BoVW) technique
\cite{SivZis03} aimed at solving this issue by roughly approximating the matching step using quantization, initiating a whole new area of research.
However soon the limitations of this rough approximation became obvious, enforcing the development of more accurate techniques for assigning query
vectors to database features. Whilst initial approaches aiming at increasing the accuracy of the matching step such as soft assignment
\cite{PhiChuIsaSivZis08} were relatively close to the BoVW approach, the focus in recent years turned back more and more to approximate $k$NN queries
\cite{KalAvr14,GeHeKeSun13,BabLem12,NorPunFle12} due to their possible gain in matching accuracy \cite{JegDouSch11b}: $k$NN queries provide
an accurate ranking of the matching candidates and a measure of proximity between feature vectors and query vectors. This additional information can
be exploited for weighting the scores of image matches, increasing retrieval accuracy considerably \cite{JegDouSch11b}.
Current research on $k$NN processing in the image retrieval community focuses on maximizing accuracy, on minimizing the memory footprint of index
structure and feature vectors, and on minimizing processing time. In recent years such techniques have received a vast amount of interest even in the
most prestigious conferences addressing image retrieval\cite{KalAvr14,GeHeKeSun13,BabLem12,JegDouSch11a,NorPunFle12}.
As a result, a remarkable leap in performance has been achieved concerning efficient and effective $k$NN query processing. However, with the vast
amount of features that have to be matched during recognition (up to a few thousand), even very fast $k$NN indexing techniques that can
provide approximate query results in under ten milliseconds (e.g. \cite{KalAvr14}), would yield recognition runtimes of many seconds.
We argue that the use of $k$NN queries for object recognition in large-scale systems cannot be achieved by developing efficient indexing techniques
alone. The problem of efficiency has to be approached from different research directions as well, such as the \textit{number} of $k$NN queries posed
on the system, as reducing the number of $k$NN queries linearly decreases the runtime of the matching step.
In this paper we aim at addressing this problem. We evaluate an alternative recognition pipeline that ranks features extracted from the query image by
assessing their matchability. Then, the most promising features in this ranking are matched against the database using traditional $k$NN queries.
However, despite gaining efficiency, the enforced reduction of $k$NN queries causes a reduction of feature matches, decreasing the quality of the
query result. While recall can be increased by increasing $k$, to increase Mean Average Precision (MAP) we propose to expand matches on the image
level:
Given a single seed feature match in a candidate image, this match is expanded by comparing its spatially neighboring keypoints. The idea of this
additional step is to push load from the matching step (with complexity mostly determined by the underlying index structure) to an additional step
that only has to consider the features stored in an image pair. The resulting enriched set of matches can then be processed equivalently to
techniques based on BoVW, e.g. by using query expansion \cite{ChuPhiSivIsaZis07} or geometric verification \cite{PhiChuIsaSivZis07}.
This work stands in contrast to research in the area of BoVW-based retrieval: Research involving the BoVW pipeline often assumes that the
matching step is relatively cheap, especially if approximate cluster assignment techniques such as hierarchical $k$-means \cite{NisSte06} or
approximate $k$-means \cite{PhiChuIsaSivZis07} are used. Therefore such research focused on increasing Mean Average Precision (MAP) at a large
number of query features.
In contrast, this paper aims at maximizing MAP for a small number of processed features. This different optimization criterion is
especially of interest as techniques that do not lead to significant gains in performance at a high number of features (where convergence to the
maximum possible MAP has already been achieved by other techniques) can lead to a remarkably higher MAP when only a low number of features is
queried.
The contribution of this paper is to provide a \textit{simple} and \textit{extensible} pipeline for \textit{large-scale} object retrieval based on
\textit{$k$NN queries} with \textit{all} of the following properties:
\begin{compactitem}
\item \textit{Reduction of the number of keypoints queried} by a general keypoint ranking scheme in order to reduce matching times. The pipeline is
not bound to a specific keypoint selection technique as long as keypoints can be ranked by their estimated quality.
\item Acceleration of the pipeline by state-of-the art index structures such as (Locally Optimized) Product quantization \cite{JegDouSch11a} or
Multi-Index-Hashing\cite{NorPunFle12}.
\item \textit{Geometric Match Expansion} to relieve the index structure and to increase query \textit{MAP}.
\item The use of many nearest neighbors ($k > 2$) to increase the number of seed hypotheses and therefore query \textit{recall}.
\item Consideration of \textit{distances} between features during score generation to allow accurate scoring of image features by their similarity.
\end{compactitem}
We further provide a thorough evaluation of this pipeline on a variety of well-known datasets, including Oxford5k, Oxford105k, Paris 6k, and
INRIA Holidays, provide insights into advantages and disadvantages of the approach, and show that such match expansion techniques can lead to
performance improvements. We also evaluate the effect of $k$ in relation to the number of keypoints queried on the systems performance, and the
pipeline's behaviour on different feature descriptors including real-valued (SIFT) and binary (BinBoost) features.
This paper is organized as follows. Section \ref{sec:problemdef} formally defines the problem addressed in this paper. We then review related work in
Section~\ref{sec:related}. In Section~\ref{sec:pipeline-all} we describe our solution to reducing the number of $k$NN queries during retrieval.
Section~\ref{sec:experiments} evaluates our solution on different feature types and datasets. Section~\ref{sec:conslusion}
concludes this work.
\endinput
\section{Pipeline}
\label{sec:pipeline-all}
The general retrieval pipeline from this paper follows the one used in the past for BoVW-based image retrieval, but in order to
incorporate $k$NN queries and reduce the number of query features we applied some changes. In this section, we first provide a theoretic overview over the pipeline. Then, as implementing the pipeline in such a naive way would lead to unacceptable overhead in terms of memory and computational
resources, we provide practical considerations about its implementation in a real-world setup.
\subsection{Theory}
\label{sec:pipeline}
We split our pipeline into the stages of feature detection and extraction, feature ranking, feature matching, match expansion, scoring, and
re-ranking. The pipeline was designed with extensibility in mind such that each stage, e.g. keypoint reduction and match expansion, can be easily
exchanged by different techniques.
\paragraph{1) Feature Extraction.}During feature extraction, given the query
image, we extract the set $I_q$ of keypoints and descriptors. Possible features include floating point features such as SIFT \cite{Low04} or binary features such as BinBoost and ORB \cite{TrzChrFuaLep13,RubRabKonBra11}.
The cardinality of $I_q$ depends on the used feature extractors and can range up to several thousand features.
\paragraph{2) Feature Ranking.}The next stage, feature ranking, is based on the
idea that some features in an image contain more information than others. For example, vegetation usually provides less information about a specific object contained in the image than the features of the object
itself. We aim at ordering the extracted features by a given quality measure, as we would like to query the most promising features first, i.e. the features
with the highest chance of providing good match hypotheses. There exist several techniques for feature ranking, and we will fall back to these instead
of developing a new approach. The only criterion such a technique needs to fulfill in order to be integrated into the recognition pipeline is that it
returns a quality score for each query feature. A simple baseline is a random ranking. Features can also be ranked by their response or size.
More sophisticated techniques include Adaptive Non-Maximal Suppression \cite{BroSzeWin05} and the use of decision trees involving additional training
\cite{HarHavSch14}, which has however neither been adapted to binary features nor to $k$NN-based matching, yet. The result of this feature ranking
step is a feature list, ordered such that the most promising features appear first.
\paragraph{3) Feature Matching.}The next step, feature matching, aims at finding
match hypotheses for the highest ranked features found during the last step. For each of the first $n$ features in the ranking, a $k$NN query is posed on the database.
The selection of the parameter $k$ of the $k$NN query is important for maximizing the quality of the query result \cite{JegDouSch11b}. On the one hand
side, a large $k$ decreases the quality of the query result, as this introduces a high number of erroneous correspondences which have to be filtered
out during a verification step later in the pipeline. On the other hand, a small $k$ also reduces the retrieval quality as many high-quality
hypotheses are left unconsidered. Basically, $k$ can be seen as a way to tweak \textit{recall} at a given number of query features, as the number of
images returned by the query is at most $n*k$. As a result, especially if a
very small number of $k$NN queries is used for correspondence generation, it is possible that an even larger $k$ increases effectiveness, as it allows for finding more initial correspondences
(however of lower quality). We refer to Section~\ref{sec:experiments} for an experimental analysis of this problem. The feature matching stage
provides a list of tentative matches (tuples) $(p_q^i,p_{x}^j)$.
\paragraph{4) Match expansion.}The match expansion phase is tightly interleaved
with the match generation phase. In our scenario where we want to pose a very small number of $k$NN queries on the system, we face the problem that even if we find some correspondences between the query and a database
image, their number will be relatively low, increasing the probability that a good match is outranked by an image containing common random matches
only. To resolve this problem, we shift the load of correspondence generation from the matching stage --that employs $k$NN queries-- to an
intermediate stage that avoids such queries.
Match expansion aims at reducing the runtimes of generating additional matches, which usually depend on the underlying index structure, to runtimes
depending on the features stored in a single image pair. When employing exhaustive search with product quantization for indexing, match expansion
therefore avoids additional linear scans over the feature database; as non-exhaustive variants of product quantization only consider a fraction of
features in the database, the gain of match expansion in this case depends on the desired recall of the index structure.
It is however important to realize that, while such a match expansion can
find additional hypotheses for candidate images, i.e. increase \textit{MAP}, it cannot retrieve any new candidates, i.e. increase recall. This
stage therefore aims at compensating for the loss in MAP due to querying less features.
\begin{figure*}[t] \centering
\includegraphics[width=\textwidth]{hypothesis_generation.pdf}
\caption{Generation of additional match hypotheses.}
\label{fig:hypothesis_generation}
\end{figure*}
Match expansion exploits the keypoint information of the seed matches that
provide scale, rotation, and possibly affine information.
These properties can be used to identify spatially close keypoints, adapting the
ideas of \cite{SchZis02,FerTuyGoo04,ChuPerMat09,WuKeIsaSun09}; we will use a
modified version of \cite{SchMoh96} for expanding matches.
Given that a match hypothesis is correct, not only the corresponding feature
pair should match, but also its spatial neighborhood, as an object is usually
not only described by a single but rather by multiple keypoints. The similarity
of a match's neighborhood is evaluated using the procedure visualized in
Figure~\ref{fig:hypothesis_generation}. The figure shows an initial seed match,
i.e. a $k$NN of a query feature, and keypoints surrounding the seed match. The
scale of each keypoint is represented by the keypoint's size, and the gradient
direction is represented by a line anchored in the keypoint's center. The top
row of this figure visualizes the features of the query image, while the bottom
row visualizes the image features of a tentative match.
Starting point is an initial correspondence pair $(p_q^i, p_{\ensuremath{D}}^j)$ established by $k$NN-search in feature space, see
Figure~\ref{fig:hypothesis_generation}~a). In a first step, features in a given spatial range are retrieved in the image $I_q$ for $p_q^i$ and in
Image $I_{\ensuremath{D}}$ for $p_{\ensuremath{D}}^j$, see Figure~\ref{fig:hypothesis_generation}~b); the spatial range is visualized by a dotted circle. Given the constant $\delta_{xy}$, the spatial range is given by $s_q^i\delta_{xy}$ for the query feature
and $s_{\ensuremath{D}}^j\delta_{xy}$ for the matching database feature, achieving scale invariance. Spatially close keypoints with a significantly different
scale (determined by the scale ratio threshold $\delta_{s}$) than their reference feature are discarded (see the small features in the figure) similar to
\cite{ChuPerMat09}, resulting in two sets of features $P_q$ and $P_{\ensuremath{D}}$. These remaining features are rotation-normalized using the reference
keypoint's gradient orientation information $r_q^i$ and $r_{\ensuremath{D}}^i$, rotating the set of keypoints and their corresponding gradient orientations, see
Figure~\ref{fig:hypothesis_generation}~c).
Then the two lists of keypoints are traversed in parallel. If the rotation-normalized angle $\alpha$ to the reference feature, the rotation-normalized
gradient angle $r$, and the feature-space distance of two features $d_v$ are within a predefined threshold ($\delta_\alpha$, $\delta_r$, and
$\delta_{d_v}$ respectively) and the ratio of their scale-normalized spatial distance is within given bounds $\delta_{d_{xy}}$, the corresponding
features are accepted as a matching pair (see Figure~\ref{fig:hypothesis_generation}~d)). The remaining features are discarded. Note
that, while the complexity of this step is $|P_q|*|P_{\ensuremath{D}}|$ in the worst case, it can be reduced by an efficient sweep-line implementation that sorts features by
their angle $\alpha$ and traverses both lists in parallel.
This technique of finding neighboring keypoints assumes that two images are only distorted by similarity transforms. To mitigate the effects of
non-similarity or even (small) non-rigid distortions, a recursive procedure (in our case with a maximum recursion depth of~2) can be chosen that
performs the same procedure on each of the resulting pairs. Moreover, by choosing the Mahalanobis distance using the affinity matrices of the seed
pair ($A_q^i$ and $A_{\ensuremath{D}}^i$ respectively) instead of Euclidean distances for finding spatially neighboring keypoints, the process can be extended to
affine-invariant features. This technique returns features within an elliptical region around the seed points, reducing performance loss from affine
distortions.
Result of the expansion phase is an extended list of match hypotheses.
\paragraph{5) Scoring.}Scoring is again tightly interleaved with match
generation.
In this phase, based on the expanded list of matches, a score is computed for every database image. In the simplest case, each hypothesis pair votes with a score of one for a given database image. This however, has
shown to have a relatively low performance \cite{JegDouSch11b}, as for example images containing many features would have higher scores than images
containing only a few features. For this purpose, more sophisticated scoring techniques have been developed. We will adapt some of the techniques from
\cite{JegDouSch11b}, weighting scores based on the distance of the candidate
feature to the query feature and the number of features in the image. For each
matched feature from image $I_x$ its score is increased by $\frac{\sqrt{d_{kNN}-d_{ref}}}{\sqrt{|I_q|}\sqrt{|I_x|}}$ with $d_{kNN}$ the $k$NN distance of the seed feature, and $d_{ref}$ the distance between the seed feature and its tentative match in the candidate image, i.e. features generated during match
expansion are assigned the same score as their seed match. This score is similar to the scores from \cite{JegDouSch11b}, however we have added
additional square root weighting which further increased effectiveness of these scores. For scoring we implemented a simple burst removal
\cite{JegDouSch09} scheme after match expansion that allows only for one correspondence per query feature.
\paragraph{6) Re-Ranking.}After building match hypotheses and scoring, the
ranked list can be processed equivalently to BoVW-based approaches. Further
steps can include geometric verification or query expansion techniques
\cite{ChuPhiSivIsaZis07}. As these techniques are complementary to the remaining
pipeline we will not further consider them in this chapter.
\subsection{Practical Considerations}
To enable efficient query processing using the pipeline summarized previously, three conditions must be fulfilled. First, it must be possible to
efficiently retrieve the $k$NN features of a query feature and their corresponding keypoints from the database. Second, to enable match expansion, it
must be possible to compute, given two keypoints, the distance of their corresponding feature vectors. Third, also concerning match expansion, it must
be possible to pose a range query on all keypoints from a given image, retrieving spatially close keypoints. In the most basic case, the image
database used for query processing can be seen of a list of tuples $(p_0^0, \ldots, p_0^{|I_0|}, \ldots, p_i^0, \ldots, p_i^{|I_i|}, \ldots)$
containing feature and keypoint information. The features in the list are ordered by their corresponding image to allow efficient match expansion.
However, in order to enable usability of this approach in a practical setup, special care has to be taken concerning computational and memory
efficiency and the thorough selection of parameters; we will address solutions for these challenges in the following section. Computational efficiency
can be achieved using indexing techniques such as Product Quantization or Multi-Index Hashing, while the memory footprint of the image database can be
reduced by compressing the feature vectors used during match expansion. Finally, the selection of parameters can be achieved using appropriate
optimization techniques.
\subsubsection{Indexing}
In order to improve the performance of of the pipeline in real-world
applications, fast (approximate) indexing techniques optimized for
high-dimensional data
\cite{KalAvr14,GeHeKeSun13,BabLem12,NorPunFle12,JegDouSch11a} can be employed.
In this research we focused on (Locally Optimized) Product Quantization for
real-valued features and Multi-Index Hashing for binary features; we will
summarize these techniques in the following paragraphs for the sake of
completeness.
\paragraph{Product Quantization.} Approximate nearest neighbor search based on
Product Quantization, initially proposed by J\'{e}gou et al.
\cite{JegDouSch11a} and further optimized e.g.
in \cite{KalAvr14,GeHeKeSun13,BabLem12}, is an elegant solution for indexing
high-dimensional real-valued features. During a training phase, features in the
database are clustered using $k$-means and the database features are assigned to
their closest cluster mean, partitioning the set of vectors into distinct cells,
similar to Locality-Sensitive Hashing \cite{IndMot98}. Then, for each feature
vector, the residual to its corresponding cluster mean is computed and the
resulting residuals are product quantized. Product quantization is achieved by
splitting a vector into a small number of subvectors (e.g. 8) and quantizing
each of these subvectors separately using a relatively small codebook of e.g.
256 centroids. Instead of storing the residuals themselves, only the cluster id
of the closest residual is stored in the index for each subvector, resulting in
a reduction in memory complexity. With product quantization using 8 subvectors
of 256 cluster centers, a SIFT vector could be compressed from 128 bytes to 8
bytes, resulting in a compression of nearly 95\%.
The index itself consists mostly of a list of outer clusters and for each of
these clusters an inverted list storing, for each feature assigned to this
cluster mean, its list of quantized subvectors.
During query evaluation, the query is first assigned to the closest outer
cluster mean (or possibly the closest $c$ means in the case of
multi-assignment). Then the inverted lists of these means are scanned, and a distance approximation is computed for each of the database vectors stored in this list: As vectors are represented as a list of their closest subvector-centroids, a distance approximation can be generated by summing the squared distances
of the corresponding centroids which can be sped up with the use of look-up
tables. The resulting distance approximations are then used to rank the feature vectors.
In the past, a variety of improvements of this approach have been proposed, for example the Inverted Multi-Index \cite{BabLem12}, Optimized Product
Quantization \cite{GeHeKeSun13}, and Locally Optimized Product Quantization (LOPQ) \cite{KalAvr14}. For our experiments we will use the most recent of
these approaches, namely LOPQ.
\paragraph{Multi-Index-Hashing.} While Product Quantization has been developed to support efficient query processing on real-valued and high-dimensional
feature vectors such as SIFT, Multi-Index Hashing (MIH) \cite{NorPunFle12} has been specifically designed for binary features, such as ORB or
BinBoost\cite{TrzChrFuaLep13,RubRabKonBra11}. It is based on the idea of Locality-Sensitive Hashing \cite{IndMot98}, however in contrast to this
approach it aims at exact query processing. The idea behind MIH is, similar to Product Quantization, to split a binary vector into a set of
subvectors. Each of these subvectors is indexed in a dedicated hash table with the subvectors' binary value directly representing the id of its hash
cell: A single cell of the index contains all database vectors that contain a given subvector.
During query processing, the query is split into subvectors as well. These
subvectors provide the hash cells that have to be looked up in order to find
vectors with similar values. Bit-flipping the query subvectors and retrieving
the corresponding hash cells allows retrieving features with similar, but not
equivalent subvectors. To allow exact $k$NN processing, Norouzi et al. developed
a retrieval strategy that enumerates all relevant bit-flip operations to
retrieve an exact query result. In our experimental evaluation, we will use this
index structure in combination with BinBoost\cite{TrzChrFuaLep13} features to
evaluate the pipeline from Section~\ref{sec:pipeline} on binary
features.
\subsubsection{Match Expansion}
Concerning the expansion of initial matches we face two challenges. First, we have to find the best parameters for the expansion step. Second, memory
consumption has to be minimized in order to store features in main memory and hence speed up query processing.
\textbf{Parameter Selection.}
Unfortunately it is a tedious task to determine the thresholds of the flood-filling procedure for match expansion, namely $\delta_{d_v}$,
$\delta_\alpha$, $\delta_r$, and $\delta_{d_{xy}}$, by hand. This problem can be solved by utilizing Nelder-Mead Simplex-Downhill optimization:
After selecting the distance multiplier $\delta_{xy}$ and the maximum scale change ratio $\delta_{s}$ by considering runtime constraints, the
remaining thresholds are automatically determined by the Simplex-Downhill approach. Optimization of these parameters should be conducted on a training
dataset different from the test set in order to avoid overfitting.
\textbf{Vector Compression.}
For compressing real-valued feature vectors, we consider Product Quantization as
well. In contrast to Product Quantization based indexing based on LOPQ, however,
we do not product quantize residual vectors, but rather the vectors themselves, as otherwise vectors belonging to different cells in the outer quantizer could not be compared efficiently. For compression, we split each feature vector in a set of $m=8$ subquantizers and for each of these subquantizers
build a codebook of $s=256$ centroids. The distance between feature vectors can then easily be approximated as the sum of squared distances between
the closest subquantizer centroids followed by a square root operation. As distances between cluster centroids can be stored in a lookup table
of size $m*s*s$, distance computations reduce to $m$ table look-ups and a single
square root operation.
\section{Problem Definition}
\label{sec:problemdef}
Let $DB = \{I_0, ..., I_{|DB|}\}$ denote a database of images $I_j$. Images are represented by a list of interest points and their corresponding
feature vectors, i.e. $I_j = \{p_j^0, ..., p_j^{|I_j|}\}$ with $p_j^i = (v_j^i,x_j^i,y_j^i,s_j^i,r_j^i,\sigma_j^i)$ for affine-variant interest point
descriptors and $p_j^i = (v_j^i,x_j^i,y_j^i,s_j^i,r_j^i,\sigma_j^i,A_j^i)$ for affine-invariant descriptors, with $v_j^i$ a (real-valued
or binary) feature vector, $(x_j^i,y_j^i)$ the coordinate of the interest point in the image, $s_j^i$ its scale, $r_j^i$ its rotation, $\sigma_j^i$
its response, and for affine-invariant descriptors $A_j^i$ the parameters of the ellipse describing its affine shape, see
\cite{PerChuMat09}.
Given a query image $I_q$ containing an object $o$, we would like to retrieve all images $I_n \in DB$ containing object $o$. This is usually achieved
by a combination of \textit{feature matching} and \textit{scoring}. During feature matching, we retrieve tuples of similar feature vectors $m(p_q^i) =
\{(p_q^i,p_{x_0}^{j_0}), \ldots, (p_q^i,p_{x_r}^{j_r})\}$, $\{I_{x_0}, ..., I_{x_r}\} \subseteq DB $ denoting that feature $i$ of the query is
visually similar to the features $\{p_{x_0}^{j_0}, ..., p_{x_r}^{j_r}\}$. This matching problem can for example be solved using the BoVW approach.
In recent years however, as mentioned in the introduction, there has been a shift away from BoVW towards more accurate, however less efficient $k$NN
queries, leading to $m(p_q^i) = \{(p_q^i,p_x^i) | p_x^i \in kNN(p_q^i, DB)\}$ where $kNN(p_q^i, DB)$ retrieves the $k$ tuples from the database whose
feature vectors are closest to the query feature vector given a pre-defined distance function, e.g Euclidean distance. Now let $M = \cup_{i =
0}^{|\Theta|}m(p_q^i)$, with $\Theta = \{p_q^0, \ldots, p_q^{|\Theta|}\} \subseteq I_q$ a subset of the query features. The score of database image
$I_x \in DB$ is computed as $\sum\limits_{\{(p_q^i,p_y^j) \in M | x=y \}}score(p_q^i,p_y^j)$. The most trivial solution would be to increase the score
of image $I_x$ by one for each tentative match, resulting in $\sum\limits_{\{(p_q^i,p_y^j) \in M | x=y \}}1$. More sophisticated scoring approaches
for $k$NN-based image retrieval can be found e.g. in \cite{JegDouSch11b}.
Accurate $k$NN queries are, even after astonishing research efforts in the last years, still relatively expensive. For example, running a 100NN-query
on 100 million binary features using Multi-Index Hashing \cite{NorPunFle14} would take about 100ms, summing up to 100 seconds in a scenario where 1000
features are queried to retrieve a single image\footnote{Note that the number of features extracted from an image is often even larger, see the
dataset statistics in our experimental evaluation}. SIFT features are generally queried approximately, runtimes vary significantly with recall and are
often between 8ms/query and 53 ms/query for a billion SIFT features at a recall below 0.5 \cite{KalAvr14}. Generally, achieving good recall over 0.5
for 1NN queries with such techniques is very expensive. We are not aware of recall evaluations of these techniques for $k\not = 1$ although it was
shown in \cite{JegDouSch11b} that a larger $k$ can notably boost recognition performance. Based on these observations we argue that in addition to
indexing efficiency, other possibilities must be considered to reduce the complexity of the feature matching phase. Generally, to achieve this
complexity reduction, different approaches are reasonable:
\begin{compactitem}
\item Reduce the \textit{dimensionality} of feature vectors. One well-known approach would be to apply PCA to SIFT features and drop the dimensions
with least variance. Another more desirable option would be to directly extract lower-dimensional features.
\item Reduce the \textit{cost} of distance functions, for example by binarization
\cite{TorFerWei08,JolBui11,ZhoLuLiTia12,HeoLeeHeChaYoo12,HeWenSun13} or by extracting binary features \cite{TrzChrFuaLep13,RubRabKonBra11} and using
the Hamming distance.
\item Reduce the \textit{accuracy} of a matching query. This has been widely used in the past, e.g BoVW \cite{SivZis03} can be seen as an extreme
case.
\item Reduce the \textit{cost for querying}. A variety of (exact) indexing techniques have been proposed, e.g. Multi-index-hashing \cite{NorPunFle12} for binary features.
\item Reduce the \textit{number} of $k$NN queries, e.g. \cite{HarHavSch14,HajZha13,LeeKimKimKimYoo10,BroSzeWin05}.
\end{compactitem}
In this paper we focus on the last approach: Let a database of images, represented by sets of features describing the neighborhood around interest points, be given.
Let $n$ denote the upper bound on the number of matching queries, constraining the number of $k$NN queries. The goal of this research is to
develop a retrieval algorithm that returns a list of images ranked by their visual similarity to the query. We aim at modifying the image recognition
pipeline such that a given performance measure (in our case MAP) is maximized for a given $n$.
The problem setting is similar to BoVW-based approaches, however in such a context it is usually assumed that $n=|I_q|$. In this paper we address the
opposite case where $n<<|I_q|$.
\section{Related Work}
\label{sec:related}
This section, addressing related research, follows the organization of the image processing pipeline used in
Section~\ref{sec:pipeline-all}.
\textbf{Keypoint reduction.} In order to reduce the number of extracted features that have to be matched, \cite{HarHavSch14} aimed at predicting the
matchability of features by interpreting the problem as a classification task. Keypoint reduction can also be achieved by employing the Adaptive
Non-Maxima suppression (ANMS) from Brown et al. \cite{BroSzeWin05}.
Their approach aims at finding interest points that are sufficiently distributed across the whole image and is computationally relatively inexpensive.
Hajebi and Zhang \cite{HajZha13} propose to keep track of the distribution of scores during query processing and stop the investigation of further
features as soon as the score difference between the best-scored image and the average score becomes large enough. Other approaches to rank features
are based on visual attention \cite{LeeKimKimKimYoo10}. In contrast to us, the authors query all features of higher scale levels to build a
coarse-grained (32x32) top-down attention map and combine it with a bottom-up saliency map. Then, in an iterative fashion, the features in the most
promising cells of these attention maps are queried. The authors perform some kind of geometric verification, but no match expansion.
\textbf{$k$NN indexing.}
As exact $k$NN query processing on high-dimensional features often cannot significantly decrease runtimes compared to a linear scan due to the curse
of dimensionality, indexing research in the image community concentrates on \textit{approximate} nearest neighbor search. Some well-known approximate
indexing techniques used in image retrieval are forests of randomized $k$D-trees\cite{SilHar08,MujLow09} and the $k$Means-tree
\cite{NisSte06,MujLow09}. These techniques however suffer either from high storage complexity if the database descriptors are needed for refinement,
or low-quality distance approximations. Recent research in $k$NN indexing aims
at providing low runtime and storage complexity while providing accurate distance approximations at the same time. One group of these techniques is
based on the Product Quantization approach from Jegou et al. \cite{JegDouSch11a}, a quantization-based approximate indexing technique distantly
related to the BoVW paradigm. Recent extensions of this approach include \cite{BabLem12,GeHeKeSun13,KalAvr14}. Another group of techniques aiming at
efficient indexing is built on the idea of generating distance-preserving binary codes from real-valued features, sometimes referred to as
\textit{binarization}. Recently developed binarization techniques include the approach
from \cite{TorFerWei08}, Random Maximum Margin Hashing \cite{JolBui11}, Scalar Quantization \cite{ZhoLuLiTia12}, Spherical Hashing
\cite{HeoLeeHeChaYoo12} and k-means hashing \cite{HeWenSun13}. In contrast to binarization techniques, binary keypoint descriptors such as BinBoost
and ORB \cite{TrzChrFuaLep13,RubRabKonBra11} can avoid the indirection of extracting real-valued (e.g.
SIFT) features first and then binarizing them.
Nearest Neighbor queries on databases of binary features can be speeded up by employing (approximate) LSH-based techniques \cite{IndMot98} or exact
indexing \cite{NorPunFle12} and are relatively fast due to them employing the Hamming
distance instead of the Euclidean distance.
\textbf{$k$NN-based Matching.} $k$NN-based matching techniques have a long history in the context of Image retrieval. One of the most famous
techniques using such approaches is Lowe's SIFT recognition pipeline \cite{Low04}. Lowe retrieved, for each query feature, the two nearest neighbors
from the database and accepted a feature as match if its distance ratio between 1NN and 2NN was above a given threshold.
J\'{e}gou et al. \cite{JegDouSch11b} evaluated $k$NN-based matching based on local features, especially SIFT. They proposed a voting scheme optimized
for $k$NN-based retrieval. This adaptive criterion basically scores matches relative to the distance of the $k$-th match. Furthermore, the authors
analyzed normalization methods for the resulting votes in order to reduce the negative effect of favouring images with many features over those with
only a few. They did however not consider reducing the number of query features. Qin et al. \cite{QinWenGoo13} proposed a normalization scheme for
SIFT-features that locally reweights their Euclidean distance, optimizing the separability of matching and non-matching features. Based on this
normalization, the authors developed a new similarity function and scoring scheme based on thresholding rather than $k$NN query processing.
\textbf{Match Expansion.}
As our technique aims at reducing the number of $k$NN queries during the matching step, the generation of a sufficient number of match hypotheses has
to be achieved in a different fashion. We do so by applying a flood-filling approach using $k$NN matches as seed points.
Match expansion has received quite some attention in the computer vision community \cite{SchMoh96,SchZis02,SivZis03,FerTuyGoo04,GuoCao12,CuiNga13},
and will most likely become more relevant again with the use of $k$NN-based matching techniques. One of the first technique in this area of research
has been proposed by Schmid and Mohr \cite{SchMoh96}.
They used the spatial neighbors of match candidates to increase the distinctiveness of features. They also considered the consistency of gradient angles between these
features to reject false-positive matches, however they did not consider the combination of their approach with feature reduction. Sivic and Zisserman
adapted the technique for Video Google \cite{SivZis03}. We however do not reject matches based on this technique but rather increase the score of a
given image by considering neighboring features.
Our work is also inspired by \cite{SchZis02}, where the authors used a region-growing approach for establishing correspondences in the context of
multi-view matching. After establishing a set of initial matches in a traditional index-supported manner, an affine transformation is estimated that
guides search of additional matches in a local neighborhood of the seed match. The authors, however, did not use this technique for reducing the
number of queries in the matching step, but rather to increase the result quality.
Ferrari et al. \cite{FerTuyGoo04} developed another related technique in order to achieve high invariance to perspective distortion and non-rigid
transformation; it further allowed to perform an accurate segmentation of objects during recognition. Their approach builds a dense grid of features
over the image; in contrast we use the initially provided keypoints and descriptors that are stored in the database nonetheless, reducing
computational overhead. A recent work related to this approach includes \cite{CuiNga13}. Guo and Cao \cite{GuoCao12} proposed to use Delaunay
triangulation to improve geometric verification.
Wu et al. \cite{WuKeIsaSun09} proposed to enrich visual words by their surrounding visual words, generating scores not only by the weight of a visual
word, but also the neighboring features; the authors however did not consider keypoint reduction.
Geometric min-Hashing \cite{ChuPerMat09}, based on the BoVW-paradigm, considers neighboring features as well, however in the context of
hashing. The approach aims at increasing precision at the cost
of recall, by dropping features that do not share a similar neighborhood. However, if we reduce the number of matching queries, one of the main
concerns is recall, such that our approach aims at increasing MAP without negatively affecting recall.
\endinput
|
3,212,635,537,986 | arxiv | \section{Introduction}
The material point method (MPM)~\cite{sulsky1994particle} is a hybrid Lagrangian--Eulerian numerical technique for continuum mechanics simulation, whereby physical quantities are traced via material points, or ``particles,'' and the governing equation (in its weak form) is solved in a background grid.
The particles and the grid exchange their information through projection operations which use basis (interpolation) functions associated with the grid.
Remarkably, MPM shares many features with the finite element method (FEM)---the most popular numerical method in solid mechanics---while it is rooted in the fluid implicit particle (FLIP) method~\cite{brackbill1986flip}---a particle-in-cell (PIC) method for fluid dynamics.
For this reason, MPM has been commonly used for simulating large deformation in a wide variety of solids and fluids alike ({\it e.g.}~\cite{gaume2018dynamic,fern2019material,zhao2020stabilized,li2021three}).
When modeling nearly incompressible materials ({\it e.g.}~water, rubber, and undrained porous media), MPM solutions are susceptible to volumetric locking, that is, overly stiff behavior with erroneous strain and stress fields.
Volumetric locking commonly arises in FEM and related methods when integration points pose excessive incompressibility constraints on the calculation of element stiffness.
Unfortunately, MPM is inherently vulnerable to volumetric locking, because it typically uses a combination of low-order basis functions and a large number of integration (material) points per element.
In the FEM literature, a number of approaches have been proposed for mitigating locking, relaxing the incompressibility constraints on element kinematics in different ways.
Among them, only those compatible with finite deformation kinematics as well as movable integration points may be adapted to MPM.
Locking-mitigation approaches that have been adapted to MPM can be categorized into the following four types.
The first type is mixed multi-field formulations.
For example, Love and Sulsky~\cite{love2006energy} and Mast {\it et al.}~\cite{mast2012mitigating} have used a three-field MPM formulation based on the Hu--Washizu variational principle, and Iaconeta {\it et al.}~\cite{iaconeta2019stabilized} have presented a two-field formulation with stabilization.
The second type is operator splitting algorithms explored by Zhang {\it et al.}~\cite{zhang2017incompressible} and Kularathna and Soga~\cite{kularathna2017implicit}, which are built on Chorin's projection method for incompressible fluid dynamics~\cite{chorin1968numerical}.
The third type is the assumed deformation gradient ($\bar{\bm{F}}$) method~\cite{de1996design}.
Coombs and coworkers~\cite{coombs2018overcoming,wang2021efficient} have developed implicit $\bar{\bm{F}}$ MPM formulations for quasi-static solid mechanics,
and Moutsanidis {\it et al.}~\cite{moutsanidis2020treatment} have proposed a different way to calculate $\bar{\bm{F}}$ in explicit particle methods.
The fourth one is the nonlinear $\bar{\bm{B}}$ method~\cite{simo1985variational}, which is a large-strain generalization of the $\bar{\bm{B}}$ method originally proposed by Hughes~\cite{hughes1980generalization} for small-strain FEM, extended to MPM recently by Bisht {\it et al.}~\cite{bisht2021simulating,bisht2021material}.
It is noted that the $\bar{\bm{F}}$ and nonlinear $\bar{\bm{B}}$ methods are highly related in that both methods rely on reduced integration of the volumetric part of a deformation measure (but their specific procedures are not exactly the same).
Very recently, Telikicherla and Moutsanidis~\cite{telikicherla2022treatment} have proposed a projection technique for reduced integration in MPM with high-order basis functions, whereby an additional background grid with low-order basis functions is introduced.
Nevertheless, none of the existing locking-mitigation approaches lends itself to straightforward application to standard explicit MPM formulations.
The mixed formulations and operator splitting algorithms require one to change the standard governing equations and time-stepping scheme, respectively, demanding significant cost for implementation and utilization.
Extension of these methods to multiphysical problems ({\it e.g.}~coupled deformation and flow) is also a challenging endeavor.
Regarding the $\bar{\bm{F}}$ and $\bar{\bm{B}}$ methods, their current MPM versions are either restricted to basis functions that are not associated with adjacent elements (undesirable due to cell-crossing errors), or require a non-trivial modification of basis functions if they are associated with adjacent elements.
As such, it is not straightforward to apply the existing $\bar{\bm{F}}$ or $\bar{\bm{B}}$ method to MPM formulations with different MPM basis functions such as B-splines~\cite{steffen2008analysis,gan2018enhancement}.
In this work, we present a new approach that is unprecedentedly simple, efficient, and general for circumventing volumetric locking in a family of standard explicit MPM formulations.
The key idea is to calculate the assumed deformation gradient, $\bar{\bm{F}}$, using standard particle--grid transfer schemes in MPM, instead of the element-wise averaging operation or multiple background grids used in the existing $\bar{\bm{F}}$ method for MPM.
Combining this idea with the standard explicit time discretization in MPM, we arrive at a new and simple algorithm for updating the deformation gradient.
The new algorithm neither changes any other parts of the existing MPM machinery nor introduces any additional parameter.
Therefore, this approach can be utilized in a straightforward manner, regardless of the MPM basis functions and material types.
We implement the proposed approach with two types of MPM basis functions, namely, GIMP's basis functions and B-splines, and verify it with various types of nearly incompressible problems arising in solid and fluid mechanics.
\section{Material point method formulation}
\label{section:mpm-formulation}
This section recapitulates the standard MPM formulation for a continuum body undergoing large deformation.
For more details of the formulation, the reader is referred to~\cite{jiang2016material,zhang2016material,de2020material}.
\subsection{Problem statement}
Consider a continuum body whose current configuration is denoted by $\Omega \in \mathbb{R}^{\dim}$, where ``$\dim$'' refers to the spatial dimension.
The boundary of $\Omega$ is denoted by $\partial \Omega$, and it is decomposed into the displacement (Dirichlet) boundary $\partial_{u}\Omega$ and the traction (Neumann) boundary $\partial_{t}\Omega$ such that $\partial_{u}\Omega \cap \partial_{t}\Omega = \emptyset$ and $\overline{\partial_{u}\Omega \cup \partial_{t}\Omega} = \partial\Omega$.
The time domain is denoted by $\mathcal{T}:=(0,T]$ with $T > 0$.
Finite deformation theory should be used to accurately describe nonlinear kinematics in large deformation.
Let us denote by $\tensor{X}$ and $\tensor{x}$ the position vectors of a material point in the reference and current configurations, respectively.
The displacement vector of the material point is then defined as $\tensor{u} := \tensor{x} - \tensor{X}$.
The velocity and acceleration vectors are given by $\tensor{v} := \dot{\bm{u}}$ and $\tensor{a} := \dot{\bm{v}} = \ddot{\bm{u}}$, where the dot denotes the material time derivative.
The deformation gradient is defined as
\begin{equation}
\tensor{F} := \dfrac{\partial \tensor{x}}{\partial \tensor{X}} = \tensor{1} + \dfrac{\partial \tensor{u}}{\partial \tensor{X}}\,,
\end{equation}
where $\tensor{1}$ is the second-order identity tensor.
The Jacobian is defined as
\begin{equation}
J := \det{(\tensor{F})} = \dfrac{\mathrm{d} v}{\mathrm{d} V},
\end{equation}
where $\mathrm{d} V$ and $\mathrm{d} v$ are the differential volumes in the reference and current configurations, respectively.
Balance of linear momentum provides the governing equation.
Since the standard MPM is built on the updated Lagrangian approach, we write the momentum balance equation in the current configuration as
\begin{align}
\diver\tensor{\sigma}(\bm{F}) + \rho\tensor{g} = \rho\dot{\tensor{v}} \quad
&\text{in} \:\: \Omega \times \mathcal{T},
\label{eq:strong-form}
\end{align}
where $\diver\,(\circ)$ denotes the divergence operator defined in the current configuration,
$\tensor{\sigma}$ is the Cauchy stress tensor,
$\rho$ is the mass density,
and $\tensor{g}$ is the gravitational acceleration vector.
To close the equation, a constitutive relation between $\bm{\sigma}$ and $\bm{F}$ should be introduced.
In this work, we will consider a range of commonly used constitutive relations to demonstrate the generality of the proposed method.
For brevity, we omit the details of these constitutive relations, referring to textbooks on this subject ({\it e.g.}~\cite{holzapfel2002nonlinear,de2011computational,borja2013plasticity}).
The initial--boundary-value problem of interest can be stated as follows.
Find $\bm{u}$ that satisfies Eq.~\eqref{eq:strong-form}, subject to the initial condition of $\tensor{u} = \tensor{u}_{0}$ and boundary conditions
\begin{align}
\tensor{u} = \hat{\tensor{u}} \quad
&\text{on}\:\: \partial_{u}\Omega \times \mathcal{T}, \\
\tensor{n}\cdot\tensor{\sigma} = \hat{\tensor{t}} \quad
&\text{on}\:\: \partial_{t}\Omega\times \mathcal{T},
\end{align}
where $\hat{\tensor{u}}$ and $\hat{\tensor{t}}$ are the boundary displacement and traction, respectively, and $\tensor{n}$ is the unit outward normal vector in the current configuration.
Through the standard weighted residual procedure, the variational form of the governing equation can be formulated as
\begin{equation}
\int_{\Omega} \tensor{\eta} \cdot \rho \dot{\tensor{v}} \, \mathrm{d} V
=
-\int_{\Omega} \grad^{\mathrm{s}} \tensor{\eta} :
\tensor{\sigma}(\bm{F}) \, \mathrm{d} V
+
\int_{\Omega} \tensor{\eta} \cdot \tensor{g} \, \mathrm{d} V
+
\int_{\partial_{t} \Omega} \tensor{\eta} \cdot \hat{\tensor{t}} \, \mathrm{d} A,
\label{eq:variational-equation}
\end{equation}
where $\tensor{\eta}$ denotes the variation of the displacement field, and $\grad^{\mathrm{s}}$ is the symmetric gradient operator defined in the current configuration.
\subsection{Material point method discretization}
For MPM discretization of the problem, we introduce a set of particles (material points) filling the domain and a background grid that accommodates the particles.
We then update the solution from the previous time step ($t^{n}$) to the next time step ($t^{n+1}$) through the procedure illustrated in Fig.~\ref{fig:mpm-procedure}.
The MPM procedure is described in the following.
Hereafter, we shall use subscript $(\circ)_{p}$ to denote quantities related to particles and use subscript $(\circ)_{i}$ to denote quantities related to nodes.
We shall also use superscripts $(\circ)^{n}$ and $(\circ)^{n+1}$ to denote quantities at $t^{n}$ and $t^{n+1}$, respectively.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.9\textwidth]{figures/mpm_calculation_procedure.pdf}
\caption{MPM update procedure.}
\label{fig:mpm-procedure}
\end{figure}
\paragraph{Particle to grid transfer (P2G)}
First, we map the mass and momentum of each particle to nodes in the background grid.
This process is called the particle-to-grid (P2G) transfer.
The particle mass and momentum are transferred to the nodes as
\begin{align}
m_i &= \sum_{p} w_{ip} m_p, \label{eq:P2G-mass}\\
m_i\bm{v}_i^{n} &= \sum_{p} w_{ip} m_p \bm{v}_p^{n}, \label{eq:P2G-momentum}
\end{align}
where $m_{i}$ and $m_p$ are the nodal and particle masses, respectively, and $\bm{v}_{i}$ and $\bm{v}_{p}$ are the particle and nodal velocity vectors, respectively.
Also, $w_{ip} := w_i (\tensor{x}_p)$ is the basis function for interpolating values at node $i$ to the position of particle $p$, and $\sum_{p}$ is the summation over particles supported by the basis function associated with node $i$.
In the MPM literature, a few types of basis functions have been employed.
Among them, here we consider two popular choices, namely, (i) generalized interpolation material point (GIMP)~\cite{bardenhagen2004generalized} and (ii) B-splines~\cite{steffen2008analysis}.
Note that these basis functions are free of the cell-crossing error problem which may arise when the original MPM basis functions (linear shape functions) are used.
\paragraph{Grid update}
Following the P2G transfer, we update the velocity vector at each node through the governing equation~\eqref{eq:variational-equation}.
As standard, we use the explicit Euler method to integrate the governing equation in time, and update the nodal velocity as
\begin{equation}
m_{i}\bm{v}_{i}^{n+1} = m_{i}\bm{v}_{i}^{n} + \Delta{t}(\bm{f}^{\mathrm{int}}_i + \bm{f}^{\mathrm{ext}}_i).
\label{eq:nodal-update}
\end{equation}
Here, $\Delta{t} := t^{n+1} - t^{n}$ is the time increment, and $\bm{f}^{\mathrm{int}}_i$ and $\bm{f}^{\mathrm{ext}}_i$ are the internal and external force vectors, respectively.
The internal force vector is calculated as
\begin{align}
\bm{f}^{\mathrm{int}}_i &= -\sum_{p} \grad^{\mathrm{s}} w_{ip} : \tensor{\sigma} (\bm{F}_p)\, V_p^{n},
\label{eq:internal-force-vector}
\end{align}
where $V_p^{n}$ is the current particle volume.
The external force vector is calculated from the body force and boundary traction vectors.
\paragraph{Grid to particle transfer (G2P)}
After updating the nodal velocity, we map it back to the particles and update the particle velocity.
This procedure is called the grid-to-particle (G2P) transfer.
Two schemes exist for updating the particle velocity:
(i) the fluid-implicit-particle (FLIP) method~\cite{brackbill1986flip} which transfers the velocity increment,
and (ii) the particle-in-cell (PIC) method~\cite{harlow1964particle} which transfers the updated velocity itself.
These two schemes can be written as
\begin{align}
(\tensor{v}_p^{n+1})_{\text{FLIP}} &= \tensor{v}_p^n + \sum_{i} w_{ip} (\tensor{v}^{n+1}_i - \tensor{v}_i^n), \label{eq:FLIP} \\
(\tensor{v}_p^{n+1})_{\text{PIC}} &= \sum_{i} w_{ip}\tensor{v}^{n+1}_i, \label{eq:PIC}
\end{align}
where $\sum_{i}$ is the summation over nodes supporting particle $p$.
While FLIP is less stable than PIC, it has significantly less numerical damping than PIC.
In general, the FLIP and PIC schemes can be blended as
\begin{equation}
\tensor{v}_p^{n+1} = \eta (\tensor{v}_p^{n+1})_{\text{FLIP}} + (1 - \eta)(\tensor{v}_p^{n+1})_{\text{PIC}}.
\end{equation}
where $\eta\in[0,1]$ is the blending coefficient. In this work, we shall use $\eta=1$ by default.
Subsequently, we update the deformation gradient, volume, stress, and position of each particle.
The deformation gradient is updated as
\begin{align}
\bm{F}_{p}^{n+1} = \left(\tensor{1} + \Delta t \sum_{i} \tensor{v}_i^{n+1}\dyadic\grad w_{ip}\right)\cdot\bm{F}_{p}^{n}.
\label{eq:deformation-gradient-update}
\end{align}
The particle volume is updated as
\begin{equation}
V_{p}^{n+1} = J_{p}^{n+1}V_{p}^{0},
\label{eq:particle-volume-update}
\end{equation}
where $J_{p}^{n+1} := \det{\tensor{F}}_{p}^{n+1}$, and $V_{p}^{0}$ is the initial (reference) particle volume.
For updating the stress tensor, we use the \emph{relative} deformation gradient---analogous to the incremental strain tensor in small-strain problems---to accommodate history-dependent material behavior.
The relative deformation gradient is defined and calculated as
\begin{equation}
\Delta \tensor{F}_p
:= {\tensor{F}}_{p}^{n+1} \cdot ({\tensor{F}}_{p}^{n})^{-1}
= \tensor{1} + \Delta t \grad \tensor{v}_p^{n+1}.
\label{eq:relative-deformation-gradient}
\end{equation}
By assigning the relative deformation gradient to the specific constitutive relation, the stress tensor is updated.
Lastly, the particle position is updated as
\begin{equation}
\tensor{x}_p^{n+1} = \tensor{x}_p^n + \Delta t \sum_{i} w_{ip}\tensor{v}^{n+1}_i.
\end{equation}
\paragraph{Grid reset and repeat}
After updating the state variables and positions of the particles, we reset the background grid, move to the next time step, and repeat the aforementioned procedure.
It would be worthwhile to note that the grid is not reset in total Lagrangian MPM formulations ({\it e.g.}~\cite{deVaucorbeil2020total,deVaucorbeil2021modelling}).
Here, we focus on standard MPM formulations built on an updated Lagrangian approach.
\section{Circumventing volumetric locking}
\label{section:locking-free-mpm}
In this section, we formulate a new approach to mitigating volumetric locking in MPM.
We first adopt the assumed deformation gradient ($\bar{\bm{F}}$) method---originally proposed for overcoming volumetric locking in FEM~\cite{de1996design}---in the context of MPM.
We then present a new way to calculate the assumed deformation gradient in MPM, which averages the volumetric part of the deformation gradient (the Jacobian) through particle--grid transfer operations.
Subsequently, we develop a detailed procedure to apply the proposed method in MPM.
\subsection{Assumed deformation gradient method}
The key idea of the assumed deformation gradient method is to replace the deformation gradient ($\bm{F}$) in the constitutive relation by an assumed deformation gradient ($\bar{\bm{F}}$), where the volumetric part of $\bm{F}$ ({\it i.e.}~the Jacobian, $J$) is volume-averaged in some manner.
When applied to the current formulation, it replaces $\bm{F}_{p}$ in the constitutive relation by
\begin{equation}
\bar{\tensor{F}}_p = \left( \dfrac{\bar{J}_p}{J_p} \right)^{1/\dim} \tensor{F}_p,
\label{eq:F-bar}
\end{equation}
where $\bar{J}_{p}$ denotes the averaged Jacobian, which is subject to fewer volumetric constraints than $J_p$.
It can be seen that $(\bar{J}_p/J_p)^{1/\dim}$ acts as a scaling term for the deformation gradient used for constitutive update.
In FEM, for which the $\bar{\bm{F}}$ method was originally proposed, $\bar{J}_p$ can be calculated straightforwardly as the average of $J_{p}$ in each element.
In MPM, however, such element/cell-wise averaging is not ideal, because the basis functions are often related to adjacent elements to avoid cell-crossing errors ({\it e.g.}~GIMP and B-splines).
As such, when Coombs~{\it et al.}~\cite{coombs2018overcoming} applied the $\bar{\bm{F}}$ method to GIMP, they had to introduce an additional basis function specialized to element-wise averaging.
Unfortunately, introducing such a special basis function is not only cumbersome but also restricted to a specific MPM scheme.
For example, the formulation in Coombs~{\it et al.}~\cite{coombs2018overcoming} is not compatible with other MPM basis functions.
\subsection{Volume averaging}
In this work, we present a new approach to volume-averaging $J_p$ in MPM, which builds on the existing particle--grid transfer schemes and hence preserves the existing basis functions.
Concretely, it first projects the Jacobians at the particles to the background grid in a volume-averaging manner, in a way similar to the P2G transfer.
Then, through the G2P transfer, the volume-averaged Jacobians at the nodes are mapped back to the particles, so that they can be used for constitutive updates at the individual particles.
To express the approach mathematically, let us define the volume-averaged projection of $J_p$ to the background grid, as
\begin{equation}
\bar{J}_{i} = \sum_{p} w_{ip} J_p V_p / V_i, \quad V_i := \sum_{p} w_{ip} V_p.
\label{eq:J_bar_node}
\end{equation}
It can be seen that this projection is more or less the same as the standard P2G transfer, except that the particle volume is considered for volume averaging.
Next, we project back $\bar{J}_{i}$---the volume-averaged Jacobian defined at the grid nodes---to the particles where $\bar{\bm{F}}$ is used for updating the stress tensor.
For this purpose, we use the standard G2P transfer as
\begin{equation}
\bar{J}_p = \sum_{i} w_{ip} \bar{J}_{i}.
\end{equation}
For notational simplicity in the succeeding formulations, we shall express the foregoing operation as the operator $\Pi(\circ)$, say,
\begin{equation}
\Pi(J_p) := \bar{J}_{p} = \sum_{i} w_{ip} \bar{J}_{i}.
\label{eq:volume-averaging-operator}
\end{equation}
\smallskip
\begin{remark}
While the projection-based averaging described above appears similar to that in Telikicherla and Moutsanidis~\cite{telikicherla2022treatment}, there are a couple of critical differences.
First, the projection operation in Telikicherla and Moutsanidis~\cite{telikicherla2022treatment} utilizes lower-order basis functions than their basis functions (B-splines) for other parts of MPM.
For this reason, their projection entails an additional background mesh dedicated to the projection.
Second, their methods applies the projection to the divergences of the velocity and stress fields in the variational equation, while we apply the projection to the Jacobian before stress update.
\end{remark}
\smallskip
\begin{remark}
The foregoing volume-averaging operation is analogous to the projection scheme utilized for mitigating locking in a different meshfree method in Ortiz-Bernardin {\it et al.}~\cite{ortiz2015improved}, where the authors adapted the assumed deformation gradient operation in Broccardo {\it et al.}~\cite{broccardo2009assumed} to the meshfree context.
It is noted, however, that the formulation of Ortiz-Bernardin {\it et al.}~\cite{ortiz2015improved} involves a modification of the strain--displacement matrix in addition to the volume-averaging projection.
This is different from the approach proposed herein, whereby the volume-averaging operation is used to evaluate $\bar{\bm{F}}$ only and the existing discretization is retained.
\end{remark}
\smallskip
\begin{remark}
Unlike the existing $\bar{\bm{F}}$ MPM formulations where the volume averaging of $J$ is performed inside individual cells~\cite{coombs2018overcoming,moutsanidis2020treatment,wang2021efficient}, here the volume averaging is done inside the support of the basis functions associated with individual particles.
This way allows us to accommodate particles that influence multiple cells without any change in the existing basis functions.
When an implicit integration is used, however, it may be less desirable than introducing a new basis function, because it would be more onerous to calculate the derivative of $\bar{\bm{F}}$.
For an explicit integration---dominant in the MPM community---the projection operation must be far simpler than modifying the basis functions.
\end{remark}
\subsection{Stress update procedure}
We now discuss how to update the stress tensor of a material point with the proposed approach.
Consider the stress update stage during an MPM update between $t^{n}$ (previous time step) and $t^{n+1}$ (next time step).
All the quantities of the particle at $t^{n}$, including the assumed deformation gradient, $\bar{\bm{F}}^{n}_{p}$, are known.
However, the quantities at $t^{n+1}$ are unknown, except the deformation gradient, $\bm{F}^{n+1}_{p}$, calculated from Eq.~\eqref{eq:deformation-gradient-update}.
Let us recall that the stress update is based on the relative deformation gradient, Eq.~\eqref{eq:relative-deformation-gradient}.
Since we use the $\bar{\bm{F}}$ method, we define the \emph{relative} $\bar{\bm{F}}$ as
\begin{equation}
\Delta \bar{\tensor{F}}_p := \bar{\tensor{F}}_{p}^{n+1} \cdot (\bar{\tensor{F}}_{p}^{n})^{-1}.
\label{eq:relative-assumed-deformation-gradient-v1}
\end{equation}
It is noted that while the (original) deformation gradient at $t^{n+1}$, ${\bm{F}}^{n+1}_{p}$, is given from the updated velocity, the assumed deformation gradient at $t^{n+1}$, $\bar{\tensor{F}}_{p}^{n+1}$, is not given directly.
So we first derive the following expression for $\bar{\tensor{F}}_{p}^{n+1}$:
\begin{align}
\bar{\tensor{F}}_{p}^{n+1}
&= \left( \dfrac{\bar{J}_{p}^{n+1}}{J_{p}^{n+1}} \right)^{1/\dim} \tensor{F}_{p}^{n+1} \nonumber \\
&= \left( \dfrac{\bar{J}_{p}^{n+1}}{J_{p}^{n} \Delta J_p} \right)^{1/\dim} \tensor{F}_{p}^{n+1} \nonumber \\
&= \left( \dfrac{\bar{J}_{p}^{n+1} \bar{J}_{p}^{n}}{J_{p}^{n} \bar{J}_{p}^{n} \Delta J_p } \right)^{1/\dim} \tensor{F}_{p}^{n+1} \nonumber \\
&= \left( \dfrac{\bar{J}_{p}^{n+1}}{\bar{J}_{p}^{n} \Delta J_p} \right)^{1/\dim} \Delta \tensor{F}_p \cdot \tensor{F}_{p}^{n} \left( \dfrac{\bar{J}_{p}^{n}}{J_{p}^{n}} \right)^{1/\dim} \nonumber \\
&= \left( \dfrac{\bar{J}_{p}^{n+1}}{\bar{J}_{p}^{n} \Delta J_p} \right)^{1/\dim} \Delta \tensor{F}_p \cdot\bar{\tensor{F}}_{p}^{n}
\label{eq:assumed-deformation-gradient-next-step}
\end{align}
where
\begin{equation}
\Delta J_p := \det{(\Delta \tensor{F}_p)}, \quad
\bar{J}_{p}^{n} := \det{(\bar{\tensor{F}}_{p}^{n})}, \quad
J_p^{n} := \det{(\tensor{F}_{p}^{n})}.
\label{eq:J-definitions}
\end{equation}
Equation~\eqref{eq:assumed-deformation-gradient-next-step} has replaced $\tensor{F}_{p}^{n+1}$ by the product of $\Delta \tensor{F}_{p}$, which is given in the G2P stage as in Eq.~\eqref{eq:relative-deformation-gradient}, and $\bar{\tensor{F}}_{p}^{n}$, which was used for evaluating the stress tensor at $t^{n}$.
In doing so, it has also replaced $J_{p}^{n+1}$ in the denominator of the scaling term by $\bar{J}_{p}^{n} \Delta J_p$.
Inserting Eq.~\eqref{eq:assumed-deformation-gradient-next-step} into Eq.~\eqref{eq:relative-assumed-deformation-gradient-v1} gives
\begin{align}
\Delta \bar{\tensor{F}}_p
&= \left( \dfrac{\bar{J}_{p}^{n+1}}{\bar{J}_{p}^{n} \Delta J_p} \right)^{1/\dim} \Delta \tensor{F}_p \cdot \bar{\tensor{F}}_{p}^{n} \cdot (\bar{\tensor{F}}_{p}^{n})^{-1} \nonumber \\
&= \left( \dfrac{\bar{J}_{p}^{n+1}}{\bar{J}_{p}^{n} \Delta J_p}\right)^{1/\dim} \Delta \tensor{F}_p.
\label{eq:relative-assumed-deformation-gradient-v2}
\end{align}
Comparing the above equation with Eq.~\eqref{eq:F-bar}, one can see that the above equation is an incremental version of the $\bar{\tensor{F}}$ method.
The final task is to evaluate $\bar{J}_{p}^{n+1}$ as a volume-averaged version of the term in the denominator of the scaling term.
To this end, we apply the volume-averaging operator proposed earlier, $\Pi(\circ)$, as
\begin{equation}
\bar{J}_{p}^{n+1} = \Pi (\bar{J}_{p}^{n}\Delta J_p).
\label{eq:J-approximation-explicit}
\end{equation}
Substituting Eq.~\eqref{eq:J-approximation-explicit} into Eq.~\eqref{eq:relative-assumed-deformation-gradient-v2} gives
\begin{equation}
\Delta \bar{\tensor{F}}_p
= \left( \dfrac{\Pi(\bar{J}_{p}^{n} \Delta J_p)}{\bar{J}_{p}^{n} \Delta J_p} \right)^{1/\dim}
\Delta \tensor{F}_p.
\label{eq:relative_bar_deformation_gradient_explicit}
\end{equation}
Algorithm~\ref{algo:explicit_Fbar} presents a detailed stress-update procedure in which the proposed locking-mitigation approach is applied.
We emphasize that the proposed approach only modifies the stress update during the G2P stage, preserving all the other aspects described in Section~\ref{section:mpm-formulation}.
For example, the particle volume update, Eq.~\eqref{eq:particle-volume-update}, remains unchanged.
\begin{algorithm}[h!]
\caption{Stress update procedure with the proposed locking-mitigation approach}
\begin{enumerate}
\item Calculate the (original) relative deformation gradient:
\begin{equation*}
\Delta \tensor{F}_p = \tensor{1} + \Delta t \sum_{i} \tensor{v}_i^{n+1}\dyadic\grad w_{ip}.
\end{equation*}
\item Compute $\Delta J_p = \det{(\Delta \tensor{F}_p)}$ and $\bar{J}_p^{n} = \det{(\bar{\boldsymbol{F}}_p^{n})}$.
\item Calculate volume-averaged Jacobians at nodes:
\begin{equation*}
\bar{J}_{i}^{n+1} = \sum_{p} w_{ip} V_p^{n} (\bar{J}_p^{n}\Delta{J}_{p}) / V_i^{n}, \quad V_i^{n} = \sum_{p} w_{ip} V_p^{n}.
\end{equation*}
\item Compute the relative assumed deformation gradient:
\begin{align*}
\Delta \bar{\tensor{F}}_p
&= \left( \dfrac{\Pi(\bar{J}_p^{n}\Delta J_p)}{\bar{J}_p^{n}\Delta J_p} \right)^{1/\dim} \Delta \tensor{F}_p \\
&= \left( \dfrac{\sum_{i} w_{ip} \bar{J}_{i}^{n+1}}{\bar{J}_p^{n}\Delta J_p} \right)^{1/\dim} \Delta \tensor{F}_p.
\end{align*}
\item Update the particle stress with the relative assumed deformation gradient, $\Delta \bar{\tensor{F}}_p$.
\end{enumerate}
\label{algo:explicit_Fbar}
\end{algorithm}
\section{Numerical examples}
\label{section:examples}
In this section, we verify and demonstrate the performance of the proposed locking-mitigation approach through four numerical examples involving various types of nearly incompressible materials.
The first example is Cook's membrane~\cite{cook1974improved}, which is a popular benchmark problem for incompressible elasticity.
The second example is the problem of a strip footing on an incompressible elastoplastic solid, for which an analytical solution (the Prandtl solution) is available for the bearing capacity.
The third example is the dam break problem in Mast {\it et al.}~\cite{mast2012mitigating}, where a nearly incompressible fluid (water) is allowed to flow freely.
The fourth and last example is a 3D landslide problem in which undrained clay---an elastoplastic and incompressible solid---collapses.
Except for the last 3D example, plane-strain conditions are considered.
Gravity is neglected in the first and second examples.
To confirm that the proposed approach works well regardless of the MPM basis functions, we simulate each example with two different basis functions, namely, GIMP and B-splines.
The specific GIMP scheme used herein is uGIMP~\cite{wallstedt2008evaluation} in which the influence domains of individual particles are fixed.
The particular B-splines used in the following results are quadratic B-splines.
While not presented for brevity, we have also found that the proposed method manifests similar performance when cubic B-splines are used.
The MPM results in this section are produced using the \texttt{Taichi} library~\cite{hu2019taichi}.
\subsection{Cook's membrane}
To verify our formulation in a simple setting, we first simulate Cook's membrane problem~\cite{cook1974improved}, which has widely been used as a benchmark problem for incompressible elasticity.
As shown in Fig.~\ref{fig:cooks_membrane_setup}, this problem considers a trapezoidal membrane subjected to a distributed shear traction on its right side and clamped on its left side.
To compare our $\bar{\tensor{F}}$-MPM results with those of $\bar{\tensor{B}}$-MPM results in Bisht {\it et al.}~\cite{bisht2021simulating}, we set the shear load and material parameters identical to those in the reference paper.
The load is set as 1 N.
The membrane is a Neo-Hookean solid with a Young's modulus of $E = 70$ Pa and a Poisson's ratio of $\nu = 0.499$ (nearly incompressible).
\begin{figure}[htbp]
\centering
\includegraphics[width=0.4\textwidth]{figures/cooks_membrane_setup.pdf}
\caption{Cook's membrane: problem geometry and boundary conditions.}
\label{fig:cooks_membrane_setup}
\end{figure}
To examine the convergence of the numerical solutions, we use two levels of spatial discretization: (i) a coarse discretization that uses 5,776 material points with a background grid comprised of 1-m-long square elements, and (ii) a fine discretization that uses 23,072 material points with a background grid comprised of 0.5-m-long square elements.
To simulate this quasi-static problem with the current explicit dynamic formulation, we adopt the local damping method in Al-Kafaji~\cite{alkafaji2013formulation}, with the same local damping factor used in Bisht {\it et al.}~\cite{bisht2021simulating}.
We calculate the time increment as $\Delta t = 0.3 (h/c)$, where $h$ is the element size and $c$ is the P-wave velocity.
This calculation gives $\Delta t = 2.773\times 10^{-3}$ s for the coarse grid and $\Delta t = 1.386\times 10^{-3}$ s for the fine grid.
Figure~\ref{fig:cooks_membrane_GIMP} presents the mean normal stress fields obtained by the standard and $\bar{\tensor{F}}$ MPM formulations with GIMP.
As can be seen, the standard MPM solutions are plagued by severe non-physical stress oscillations, which are not remedied by spatial refinement.
Such oscillations have been commonly observed in numerical solutions affected by volumetric locking.
Meanwhile, the $\bar{\tensor{F}}$ MPM solutions are free of non-physical oscillations in the stress fields.
This difference indicates that the proposed formulation does not suffer from volumetric locking.
\begin{figure}[htbp]
\centering
\subfloat[Coarse discretization (5,776 material points)]{\includegraphics[width=1.0\textwidth]{figures/cooks_membrane_GIMP_coarser.pdf}} \newline
\subfloat[Fine discretization (23,072 material points)]{\includegraphics[width=1.0\textwidth]{figures/cooks_membrane_GIMP_finer.pdf}}
\caption{Cook's membrane: mean normal stress fields in the standard and $\bar{\tensor{F}}$ MPM solutions, obtained with GIMP basis functions.}
\label{fig:cooks_membrane_GIMP}
\end{figure}
Figure~\ref{fig:cooks_membrane_Bspline} shows how the mean normal stress fields become different when the basis functions are changed to B-splines.
One can see that the B-spline MPM provides less oscillatory stress fields than GIMP, especially when the discretization is fine.
Still, however, the standard MPM solutions show undesirable oscillations.
Meanwhile, the same $\bar{\tensor{F}}$ MPM formulation continues to work well notwithstanding the change in the basis functions.
\begin{figure}[htbp]
\centering
\subfloat[Coarse discretization (5,776 material points)]{\includegraphics[width=1.0\textwidth]{figures/cooks_membrane_Bspline_coarser.pdf}} \newline
\subfloat[Fine discretization (23,072 material points)]{\includegraphics[width=1.0\textwidth]{figures/cooks_membrane_Bspline_finer.pdf}}
\caption{Cook's membrane: mean normal stress fields in the standard and $\bar{\tensor{F}}$ MPM solutions, obtained with B-splines basis functions.}
\label{fig:cooks_membrane_Bspline}
\end{figure}
For further verification, we compare our GIMP solution with the reference solution from Bisht {\it et al.}~\cite{bisht2021simulating}---obtained with a nonlinear $\bar{\tensor{B}}$ method specialized to GIMP---in terms of the mean normal stress field and the vertical displacement at the tip (upper right corner), respectively, in Figs.~\ref{fig:cooks_membrane_verification} and~\ref{fig:cooks_membrane_tip_displacement}.
From Fig.~\ref{fig:cooks_membrane_verification}, we can see that the mean normal stress fields in the two solutions are very similar.
Figure~\ref{fig:cooks_membrane_tip_displacement} also shows that the tip displacements in the $\bar{\tensor{F}}$ and $\bar{\tensor{B}}$ solutions are very close, while those in the standard MPM solutions are noticeably lower than the locking-free solutions.
It would be worthwhile to note that the nonlinear $\bar{\tensor{B}}$ MPM solution itself has also been verified with other results in the literature, see Bisht {\it et al.}~\cite{bisht2021simulating} for more details.
Taken together, it has been confirmed that the proposed method performs similarly to the nonlinear $\bar{\tensor{B}}$ method.
\begin{figure}[htbp]
\centering
\includegraphics[width=1.0\textwidth]{figures/cooks_membrane_verification.pdf}
\caption{Cook's membrane: comparison of our $\bar{\tensor{F}}$ GIMP solution with the $\bar{\tensor{B}}$ GIMP solution in Bisht {\it et al.}~\cite{bisht2021simulating}. Both solutions are produced with 5,776 material points.}
\label{fig:cooks_membrane_verification}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\textwidth]{figures/cooks_membrane_tip_displacement.pdf}
\caption{Cook's membrane: tip vertical displacements in our $\bar{\bm{F}}$ GIMP solution and the $\bar{\bm{B}}$ GIMP solution in Bisht {\it et al.}~\cite{bisht2021simulating}.}
\label{fig:cooks_membrane_tip_displacement}
\end{figure}
\subsection{Strip footing}
In our second example, we investigate the performance of the proposed approach when a nearly incompressible material is in contact with a rigid body---a common scenario in many engineering applications.
To this end, we simulate the problem of a strip footing on an incompressible elastoplastic solid, for which an analytical solution for the bearing capacity---the Prandtl solution---is available.
Figure~\ref{fig:strip_footing_setup} depicts the specific geometry and boundary conditions simulated herein.
As shown in the figure, only the right half of the problem is modeled taking advantage of symmetry.
Note that the footing is treated explicitly as a rigid body.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.45\textwidth]{figures/strip_footing_setup.pdf}
\caption{Strip footing: problem geometry and boundary conditions. $B$ refers to the width of the footing.}
\label{fig:strip_footing_setup}
\end{figure}
We treat the contact between the footing and the ground with a barrier method~\cite{li2020incremental,zhao2022barrier,li2022bfemp,jiang2022hybrid}, which guarantees non-interpenetration between the two objects.
The particular barrier method implemented in this example is based on the formulation specializes to MPM~\cite{jiang2022hybrid}.
The friction coefficient in the contact model is set to be sufficiently large to prevent slip between the footing and the ground.
The elastoplastic behavior of the ground is described by a combination of Hencky elasticity and J2 plasticity.
The elasticity parameters assigned are a Young's modulus of $E = 1000$ kPa and a Poisson's ratio $\nu = 0.49$.
The yield strength of J2 plasticity is set such that the (undrained) shear strength of the ground is $0.1$ kPa under plane strain.
These parameters are adopted from Bisht {\it et al.}~\cite{bisht2021simulating}.
Note that the high ratio between the Young's modulus and the shear strength allows the ground to be in the small deformation range, such that the bearing capacity can be estimated by Prandtl's analytical solution.
Similar to the previous example, we investigate the performance of the proposed approach under three different levels of discretization.
They are:
(i) $h=B/20$ (57,600 material points),
(ii) $h=B/40$ (230,400 material points),
and (iii) $h=B/80$ (921,600 material points).
The time increment is calculated as $\Delta t = 0.4 (h/c)$, which gives $\Delta t = 1.529 \times 10^{-4}$ s, $\Delta t = 7.644 \times 10^{-5}$ s, and $\Delta t = 3.822 \times 10^{-5}$ s, respectively, for the three levels of grid sizes.
To emulate a quasi-static condition, we apply the damping method used in the previous example (with a damping coefficient of 0.002), as well as pushing the footing slowly with the same penetration rate used in Bisht {\it et al.}~\cite{bisht2021simulating}.
Figure~\ref{fig:strip_footing_load_displacement} presents the normalized load--displacement curves produced from the standard and $\bar{\bm{F}}$ MPM, along with the analytical solution, 5.14. (The load is normalized by the shear strength, and the displacement is normalized by the footing width.)
As is well known, the standard MPM significantly overestimates the bearing capacity due to volumetric locking.
One can see that the proposed method well remedies the problem regardless of the basis functions used, giving numerical solutions converging to the analytical solution.
\begin{figure}[htbp]
\centering
\subfloat[GIMP]{\includegraphics[width=0.45\textwidth]{figures/strip_footing_load_displacement_GIMP.pdf}}\hspace{1em}
\subfloat[B-splines]{\includegraphics[width=0.45\textwidth]{figures/strip_footing_load_displacement_Bsplines.pdf}}
\caption{Strip footing: normalized load--displacement curves from the standard and $\bar{\bm{F}}$ MPM solutions, obtained with GIMP and B-splines basis functions.}
\label{fig:strip_footing_load_displacement}
\end{figure}
Figure~\ref{fig:strip_footing_mean_normal_stress} shows the mean normal stress fields in the standard and $\bar{\bm{F}}$ MPM solutions, when $h=B/40$.
As in Cook's membrane example, the standard MPM solutions show non-physical oscillations in the stress fields, which are particularly severe when GIMP is used.
The solutions obtained by the proposed $\bar{\bm{F}}$ MPM, however, are free of such oscillations.
Taking this result together with the bearing capacity results above, it can be concluded that the proposed method successfully alleviates volumetric locking in MPM where contact is involved.
\begin{figure}[htbp]
\centering
\subfloat[GIMP]{\includegraphics[width=1.0\textwidth]{figures/strip_footing_GIMP.pdf}}
\\
\subfloat[B-splines]{\includegraphics[width=1.0\textwidth]{figures/strip_footing_Bspline.pdf}}
\caption{Strip footing: mean normal stress fields in the standard and $\bar{\bm{F}}$ MPM solutions, obtained with GIMP and B-splines basis functions.}
\label{fig:strip_footing_mean_normal_stress}
\end{figure}
\subsection{Dam break}
Our third example is the dam break problem in Mast {\it et al.}~\cite{mast2012mitigating}, where a three-field mixed formulation is used for mitigating locking in MPM.
Figure~\ref{fig:dam_break_setup} depicts the geometry and boundary conditions of the problem.
As shown, it considers a 4-m-long and 2-m-high water reservoir that is initially constrained by a gate and starts to flow after the gate is removed.
It is noted that there is another gate on the right boundary, from which the water will bounce back.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.65\textwidth]{figures/dam_break_setup.pdf}
\caption{Dam break: problem geometry and boundary conditions.}
\label{fig:dam_break_setup}
\end{figure}
Following Mast {\it et al.}~\cite{mast2012mitigating}, the water is modeled as a nearly incompressible Newtonian fluid, with a bulk modulus of $K = 2.0$ GPa, a dynamic viscosity of $\mu = 0.001$ Pa$\cdot$s, and a density of $\rho$ = 0.9975 t/m$^3$.
For MPM discretization, we introduce a background grid comprised of 0.25-m long square elements and initialize each element with 25 material points.
This results in 3,200 material points in total.
We then apply gravity loading to the water reservoir until it reaches the hydrostatic state.
Subsequently, we remove the gate so that the water can flow freely.
We simulate the problem until $t=2.0$ s, setting the time increment as $\Delta t = 10^{-5}$ s.
Figure~\ref{fig:dam_break_verification} shows the flow snapshots simulated by the standard and $\bar{\bm{F}}$ MPM formulations (with both GIMP and B-splines), in comparison with the reference solutions from Mast {\it et al.}~\cite{mast2012mitigating}.
One can easily see that the standard MPM is subjected to severe volumetric locking.
Not only does the pressure field show non-physical oscillations, but the water also is unrealistically stiff.
The locking is less severe in B-splines MPM than GIMP, but it is still unacceptable.
When the $\bar{\bm{F}}$ MPM formulation is employed, however, the numerical solutions are free of the locking problem, irrespective of the basis functions.
It can also be seen that the numerical solutions produced by the $\bar{\bm{F}}$ MPM formulation are very similar to those by the three-field mixed MPM formulation~\cite{mast2012mitigating}.
It is reminded that the proposed method involves significantly less implementation effort and computational cost than the three-field mixed formulation.
\begin{figure}
\centering
\subfloat[GIMP]{\includegraphics[width=1.0\textwidth]{figures/dam_break_verification_GIMP.pdf}}\\
\subfloat[B-splines]{\includegraphics[width=1.0\textwidth]{figures/dam_break_verification_Bspline.pdf}}
\caption{Dam break: flow snapshots from the standard and $\bar{\bm{F}}$ MPM simulations, along with those from Mast {\it et al.}~\cite{mast2012mitigating} produced from a three-field mixed MPM formulation. Material points are colored by the water pressure.}
\label{fig:dam_break_verification}
\end{figure}
\subsection{3D landslide}
As our last example, we investigate the performance of the proposed approach for 3D large deformation in an elastoplastic material.
To this end, we simulate a 3D landslide process where a brittle clay slope fails in an undrained manner.
The slope geometry is depicted in Fig.~\ref{fig:slope_setup}.
The bottom boundary of the slope is fully fixed, while the three lateral boundaries are supported by rollers.
The elastoplastic behavior of the undrained clay is modeled through a combination of Hencky elasticity and J2 plasticity with a softening law.
The specific softening law adopted is $\kappa = \kappa_r + (\kappa_p - \kappa_r) e^{-\eta \varepsilon_q^{\rm p}}$, where $\kappa$ is the yield strength, $\kappa_r$ and $\kappa_p$ are the residual and peak strengths, respectively, $\eta$ is the softening parameter, and $\varepsilon_q^{\rm p}$ is the cumulative equivalent plastic strain.
The material parameters are assigned to be similar to the undrained sensitive clay modeled in Bui and Nguyen~\cite{bui2021smoothed}.
They are: Young's modulus $E = 25$ MPa, Poisson's ratio $\nu = 0.499$, peak strength $\kappa_p = 40.82$ kPa, residual strength $\kappa_r = 2.45$ kPa, softening coefficient $\eta = 5$, and the density $\rho = 2.15$ t/m$^3$.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{figures/slope_setup.pdf}
\caption{3D landslide: problem geometry.}
\label{fig:slope_setup}
\end{figure}
For MPM discretization, we introduce a 3D background grid comprised of mono-sized cubic elements whose length is 0.2 m.
For each element in the slope domain, we assign 8 material points, which results in a total of 311,250 material points.
We initialize the stress field in the material points through a gravity loading stage.
Then, to trigger the slope failure, we decrease the peak strength with a reduction factor of 1.65, as done in Bui and Nguyen~\cite{bui2021smoothed}.
We simulate the problem until $5.5$ s with a time increment of $\Delta t = 5 \times 10^{-5}$ s.
Figures~\ref{fig:slope_strain} and~\ref{fig:slope_stress} present snapshots of the landslide simulated by the standard and $\bar{\bm{F}}$ MPM, showing the equivalent plastic strain and mean normal stress fields, respectively.
As in the previous example, the standard MPM is subjected to severe volumetric locking, manifesting non-physical stress oscillations as well as overly stiff behavior.
The slope does not even fail when GIMP is used, while it shows diffusive plastic strains when B-splines are used.
However, the $\bar{\bm{F}}$ MPM results show retrogressive failure---a signature failure pattern in sensitive clay slopes---with both GIMP and B-splines.
Although there is no reference solution for this problem, it can be seen that the simulation results produced with the two types of basis functions are quite similar.
Thus the $\bar{\bm{F}}$ MPM solutions are believed to be reasonable.
\begin{figure}[htbp]
\centering
\subfloat[$t = 1.5$ s]{\includegraphics[width=0.85\textwidth]{figures/slope_combined_t=1.5s_strain.pdf}}\\
\subfloat[$t = 2.5$ s]{\includegraphics[width=0.85\textwidth]{figures/slope_combined_t=2.5s_strain.pdf}}\\
\subfloat[$t = 3.5$ s]{\includegraphics[width=0.85\textwidth]{figures/slope_combined_t=3.5s_strain.pdf}}\\
\subfloat[$t = 5.5$ s]{\includegraphics[width=0.85\textwidth]{figures/slope_combined_t=5.5s_strain.pdf}}
\caption{3D landslides: snapshots from the standard and $\bar{\bm{F}}$ MPM simulations. Material points are colored by the equivalent plastic strain.}
\label{fig:slope_strain}
\end{figure}
\begin{figure}[htbp]
\centering
\subfloat[$t = 1.5$ s]{\includegraphics[width=0.85\textwidth]{figures/slope_combined_t=1.5s_stress.pdf}}\\
\subfloat[$t = 2.5$ s]{\includegraphics[width=0.85\textwidth]{figures/slope_combined_t=2.5s_stress.pdf}}\\
\subfloat[$t = 3.5$ s]{\includegraphics[width=0.85\textwidth]{figures/slope_combined_t=3.5s_stress.pdf}}\\
\subfloat[$t = 5.5$ s]{\includegraphics[width=0.85\textwidth]{figures/slope_combined_t=5.5s_stress.pdf}}
\caption{3D landslides: snapshots from the standard and $\bar{\bm{F}}$ MPM simulations. Material points are colored by the mean normal stress.}
\label{fig:slope_stress}
\end{figure}
Figure~\ref{fig:slope_runout} demonstrates how the run-out distance, which is of primary interest in slope analysis, is different in the standard and $\bar{\bm{F}}$ MPM solutions.
Without a proper locking-mitigation approach, the standard MPM significantly underestimates the run-out distance, if not being unable to simulate the failure process at all.
This difference highlights why it is of critical importance to mitigate volumetric locking from the practical viewpoint.
\begin{figure}[htbp]
\centering
\subfloat[GIMP]{\includegraphics[width=1.0\textwidth]{figures/slope_GIMP_t=5.5s_runout.pdf}}\\
\subfloat[B-splines]{\includegraphics[width=1.0\textwidth]{figures/slope_Bspline_t=5.5s_runout.pdf}}
\caption{3D landslides: comparison of run-out distances in the standard and $\bar{\bm{F}}$ MPM simulations.}
\label{fig:slope_runout}
\end{figure}
\section{Closure}
\label{section:closure}
In this paper, we have proposed a simple and efficient approach for circumventing volumetric locking in MPM, which can be generally applied to a family of standard explicit MPM formulations regardless of basis functions and material types.
The key idea of the proposed approach is to evaluate the assumed deformation gradient ($\bar{\bm{F}}$) with a volume-averaging operation that resembles the standard particle--grid transfer schemes in MPM.
The approach can be implemented in a far simpler way than the existing approaches for mitigating locking in MPM, and it is independent of other parts such as the basis function and the constitutive behavior.
The results of the numerical examples have verified and demonstrated that the proposed approach performs well for various types of nearly incompressible problems arising in solid and fluid mechanics.
Taken together, it is believed that the proposed approach features unparalleled attractiveness for mitigating volumetric locking in explicit MPM.
\section*{Acknowledgments}
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. 2022R1F1A1065418).
The authors also wish to thank Dr. Vibhav Bisht for sharing his numerical solution to Cook's membrane problem obtained with the nonlinear $\bar{\bm{B}}$ MPM.
\section*{Data Availability Statement}
\label{sec:data-availability}
The data that support the findings of this study are available from the corresponding author upon reasonable request.
|
3,212,635,537,987 | arxiv |
\subsection{Network Architecture for Bidirectional Flow Estimation}
\begin{figure}[t!]
\centering
\includegraphics[width=0.9\linewidth, trim={5mm 35mm 0mm 10mm}, clip]{figures/BiSED2.pdf}
\caption{Symmetric epipolar distance.
}
\label{Fig:SED}
\vspace{-3mm}
\end{figure}
\subsection{Training Light-invariant Dense Flows from Camera Poses}
As camera poses between images with large appearance and geometric variations are easier to acquire than dense flows, our proposed LIFE learns from such camera poses as weak supervisions to predict lighting invariant dense flows.
\vspace{3pt}
\noindent \textbf{Symmetric Epipolar Distance (SED) loss.} Based on the epipolar geometry, we propose a symmetric epipolar distance (SED) loss to achieve this goal.
We compute the fundamental matrix $\mathbf{F}$ from $I_A$ to $I_B$ according to their camera intrinsic parameters and relative camera pose.
As shown in Fig.~\ref{Fig:SED},
$\mathbf{F}$ restricts that a pixel location $\mathbf{x}\in \mathbb{R}^2$ in image $I_A$ can only be mapped to one of the points on a line $l'=\mathbf{Fx}$, referred to as {\it epipolar line} in image $I_B$.
For each point $\mathbf{x}$ in image $I_A$, our network estimates its corresponding point in image $I_B$ as $\mathbf{x}'=f_{B\leftarrow A}(\mathbf{x})$.
If the $A$-to-$B$ flows are ideal, the distance of the corresponding pixel location $x'$ to the epipolar line $l'$, named epipolar distance~(ED), shall be zero.
Reversely, $x$ is also supposed to lie on the epipolar line $l$ derived from $x'$ in image $I_B$ if the reverse flows are ideal, and the distance from $x$ to the epipolar line $l$ should also be zero.
The sum of the two epipolar distances is defined as the symmetric epipolar distance~(SED).
According to the epipolar geometry, the inverted fundamental matrix equals the transpose of the fundamental matrix, we can therefore compute the SED as
\begin{equation}
SED(\mathbf{x}, \mathbf{x}', \mathbf{F})
= ED(\mathbf{x}, \mathbf{x}', \mathbf{F}) + ED(\mathbf{x}', \mathbf{x}, \mathbf{F}^T).
\end{equation}
Given the $A$-to-$B$ dense flows $f_{B\leftarrow A}$ predicted by our flow network, we can define the following SED loss to evaluate their accuracies by computing SED for all flows in $f_{B\leftarrow A}$.
\begin{equation}
L_{SED}
= \sum_{\mathbf{x}_i \in S}SED(\mathbf{x}_i, f_{B\leftarrow A}(\mathbf{x}_i),\mathbf{F}),
\end{equation}
where $S$ is the set containing all pixel locations in $I_A$.
Compared with photometric consistency loss used in existing unsupervised flow learning frameworks, which assumes constant lighting conditions, the proposed epipolar distance loss is only determined by the fundamental matrix (or relative camera pose). Therefore, the SED loss works even when there exist significant lighting variations between the two images.
\noindent \textbf{Cycle consistency regularization.} The cycle loss measuring flows' cycle consistency $d(\mathbf{x}) = ||f_{A\leftarrow B}(f_{B\leftarrow A}(\mathbf{x})) - \mathbf{x}||_2$ is a common regularization term in correspondence learning~\cite{rocco2018neighbourhood}.
However, pixels in occluded regions do not satisfy the cycle consistency assumption and might infer large cycle distance errors to overwhelm the cycle loss. Therefore, making all flows to be cycle consistent would actually hinder the training and generate over-smooth flow fields with degraded performances.
Inspired by unsupervised optical flow learning~\cite{meister2018unflow, yin2018geonet}, we filter out pixels with too large cycle distance errors and use the cycle loss from the kept pixels.
$$
L_{cyc} = \sum_{\mathbf{x}_i \in S}\mathbf{1}(d(\mathbf{x}_i) \leq \max\{\alpha, \beta ||f_{B\leftarrow A}(\mathbf{x}_i)||_2\})d(\mathbf{x}_i),
$$
where $\mathbf{1}$ denotes the indicator function. A pixel $\mathbf{x}$ whose cycle distance is larger than $\alpha$ and $\beta ||f_{B\leftarrow A}(\mathbf{x})||_2$ will be filtered in the cycle loss.
\vspace{3pt}
\noindent {\bf Synthetic dense-flow regularization.} Although the SED loss can work on image pairs of actual scenes with significant lighting and pose variations, it can only provide weak supervision on minimizing the distances from points to their epipolar lines. In other words, as long as the predicted flow aligns points to their epipolar lines in the other image, their SED loss is minimal. However, points' ground-truth correspondences should also be single points.
In order to improve the accuracy of the flow prediction, we propose to regularize the estimated flows with synthetic pixel-to-pixel supervisions.
Inspired by Rocco~\etal~\cite{rocco2017convolutional},
for each image pair, we randomly generate an affine or thin-plate spline transformation $\mathbf{T}$ to transform image $I_B$ of the image pair.
In this way, we can create a synthesized image $I_{B'}$ and a synthesized image pair $<I_B, I_{B}'>$~(Fig.~\ref{Fig:Triplet}) with accurate dense pixel-to-pixel correspondence ground truth.
Given the location of a pixel in $I_{B}$, we can compute its accurate corresponding location in $I_{B'}$ according to the synthesized geometric transformation $\mathbf{T}$, and vice versa via the inverse of the geometric transformation $\mathbf{T}^{-1}$.
We therefore additionally regularize our flow network with a bidirectional geometric transformation~(BiT) loss.
The BiT loss supervises the bidirectional flow with accurate pixel-to-pixel dense correspondences, but the image pair has a synthetic image transformation and zero lighting variation,
\begin{equation}
\begin{split}
L_{BiT} = & \sum_{\mathbf{x}_i\in S_{B}}||f_{B'\leftarrow B}(\mathbf{x}_i) - \mathbf{T}(\mathbf{x}_i)||_1 + \\
&\sum_{\mathbf{x}_i\in S_{B'}}||f_{B\leftarrow B'}(\mathbf{x}_i) - \mathbf{T}^{-1}(\mathbf{x}_i)||_1,
\end{split}
\end{equation}
where $S_{B}$ and $S_{B'}$ contains locations of valid pixels in $I_{B}$ and $I_{B'}$ respectively. For points that might be out of the visible region in the target images, we regard these flow as invalid and filter out them for loss computation.
\noindent \textbf{The overall loss.} For image $I_B$ and its synthetically transformed version $I_{B'}$,
$I_B$ is related to $I_A$ according to their fundamental matrix $\mathbf{F}$, while the dense correspondences between $I_B$ and $I_{B'}$ are precisely determined by the synthesized geometric transformation $\mathbf{T}$.
We apply the SED loss and the cycle loss to the bidirectional flow deduced from $I_A$ and $I_B$, and the BiT loss to the bidirectional flow deduced from $I_B$ and $I_{B'}$~(Fig.~\ref{Fig:Triplet}).
The SED loss supervises flows between actual image pairs with natural lighting and viewpoint variations but only provides weak supervision signals.
On the contrary, the BiT loss regularizes flow with strict dense correspondences with constant lighting conditions and synthesized image transformation.
Unifying the losses of the images can simultaneously mitigate both their drawbacks and contribute to training a robust lighting-invariant flow estimation network.
\begin{figure}[t!]
\centering
\includegraphics[width=0.9\linewidth, trim={10mm 10mm 50mm 0mm}, clip]{figures/Triplet.pdf}
\caption{The overall training loss. The top pair with large lighting and viewpoint variation is supervised by the proposed Symmetric Epipolar Distance (SED) loss and the cycle consistency loss. The bottom pair with a synthetic transformation is supervised by the bidirectional geometric transformation (BiT) loss.
}
\label{Fig:Triplet}
\end{figure}
\subsection{Finding Sparse Correspondences with LIFE}
Dense correspondences can be used in many applications but are not mandatory in whole-image geometric transformation estimation tasks, such as relative pose estimation and homography estimation.
In such tasks, we only need a small number of sparse but accurate correspondences to estimate whole-image transformation.
Sparse correspondences can be generally obtained by detecting locally salient feature points in the image pair with feature descriptors and establishing correspondences between feature points with similar descriptors.
However, descriptors encode contents of local image patches, which are inevitably ambiguous in terms of the global image context.
Erroneous matches caused by ambiguous descriptors is a long-standing problem even with learning-based descriptors, which both reduce inlier ratio and the number of effective correspondences for the follow-up transformation estimation step.
In contrast, our dense flows are less possible to be trapped by local ambiguous patterns thanks to the context information.
We design a simple but effective two-stage algorithm to identify sparse correspondences.
In stage 1, we identify correspondences with the assist of LIFE.
We believe the flows predicted by LIFE are confident, so we only need to find the corresponding features in local regions guided by the predicted flows.
The correspondences identified in stage 1 are of high quality but the correspondence number may be unsatisfactorily low because of the strict flow constraints.
Therefore, in stage 2, we try to identify more correspondences via the remaining feature points of stage 1.
\begin{figure}[t!]
\centering
\includegraphics[width=0.9\linewidth, trim={0mm 80mm 60mm 5mm}, clip]{figures/GuidedMatching.pdf}
\caption{With the flow predicted by LIFE, we find the feature point $\mathbf{b}_j$ whose descriptor is the closest to that of $\mathbf{a}_i$ in the circle as the corresponding point.
}
\label{Fig: GM}
\vspace{-3mm}
\end{figure}
\noindent \textbf{Stage 1.}
We denote the sets of sparse feature points detected in $I_A$ and $I_B$ by $\mathbf{a}_i \in A$ and $\mathbf{b}_j \in B$, and denote their descriptors by $\mathbf{q}^a_i$ and $\mathbf{q}^b_j$.
For a given query feature point $\mathbf{a}_i$, we first calculate its warped point $\mathbf{b}'_i = f_{B\leftarrow A}(\mathbf{a}_i)$ in $I_B$ according to the predicted flows $f_{B\leftarrow A}$.
Inside the circle centered at $\mathbf{b}_i'$,
we regard the feature point $\mathbf{b}_j$ whose descriptor is the closest to that of $\mathbf{a}_i$ as the corresponding point~(Fig.~\ref{Fig: GM}), \ie,
\begin{equation}
\centering
\begin{aligned}
&j = \underset{j}{\arg \max} \,\, \mathbf{q}^{a}{}_{i}^T\mathbf{q}^{b}_{j}, \\
&\mathrm{s.t.}~ ||\mathbf{b}_j - \mathbf{b}'_i||_2 \leq r.
\end{aligned}
\end{equation}
$r$ is the radius of the circle, which is set as 5 pixel.
In this way, we can try to identify a corresponding feature point $\mathbf{b}_j$ in $I_B$ for each $\mathbf{a}_i$ in $I_A$, and create a matching feature point set $M_{B\leftarrow A}$ that satisfy the above formula. Reversely, we can use the same strategy to establish a matching point set $M_{A\leftarrow B}$ in the reverse direction from $I_B$ to $I_A$.
Given the two matching point sets of the two matching directions, we only keep the final matching point pairs as those survive the cycle consistency check, \ie, $\mathbf{x}_i = M_{A\leftarrow B}(M_{B\leftarrow A}(\mathbf{x}_i))$
\noindent \textbf{Stage 2.}
We collect the features that are weeded out in stage 1 and denote them by $A_w$ and $B_w$ according to which image they belong to. More correspondences are tried to be established from them as supplements.
Given a feature point $\mathbf{a}_i \in A_w$ as a query feature, we directly find the feature point $\mathbf{b}_j \in B$ who has the closest descriptor in $I_B$ as its corresponding point.
Note that this not equivalent to establishing correspondences between all points $A$ and $B$ with feature descriptors, as $A_w$ and $B_w$ are smaller feature point sets after the stage-1 matching.
Similar to stage 1,
only the matched point pairs that satisfy the cycle consistency check would eventually be kept.
The survived matches are the outcome correspondences in stage 2.
Directly matching local features with descriptors would cause erroneous matches due to the ambiguity of descriptors, which can be corrected by the local match in stage 1 with the guidance of the flows, so the quality of the correspondences highly relies on the flows.
Unreliable flows can produce misleading guidance, which may even impact the matching.
As LIFE is able to predict accurate flows in challenging scenarios, we can identify more inlier correspondences through this algorithm.
\subsection{Flow Evaluation}
We evaluate our flows on the KITTI 2012 flow~(training), KITTI 2015 flow~(training)~\cite{geiger2013vision}, Hpatches~\cite{balntas2017hpatches}, RobotCar~\cite{maddern20171, larsson2019cross} and MegaDepth datasets~\cite{li2018megadepth}.
KITTI datasets annotate dense flows between consecutive images, which only have small motion and constant illumination.
The KITTI 2015 flow contains more dynamic objects than the KITTI 2012 flow.
HPatches contains image pairs with viewpoint and illumination variations of different levels.
The dense correspondences used in evaluation is computed from the provided homography.
However, these two datasets have certain limitations.
To compare the flow prediction performance in real scenes, we further evaluate LIFE on RobotCar~\cite{maddern20171, larsson2019cross} and MegaDepth datasets~\cite{li2018megadepth}.
Dense correspondence annotations are not available in such challenging scenes with large lighting and viewpoint variations.
We therefore evaluate the flow performance on provided sparse correspondences.
\noindent \textbf{Dense Flow Estimation on KITTI.}
We first test flow estimation on the lighting-constant KITTI.
We compare LIFE with state-of-the-art unsupervised flow learning methods that are not fine-tuned on KITTI.
Following the standard optical flow evaluation protocol, we use the Average End-Point Error~(AEPE) and F1 scores in Tab.~\ref{tab: kitti flow}.
LIFE achieves the best performance on the KITTI 2012 flow.
Compared with the KITTI 2012 flow, the KITTI 2015 flow contains more dynamic objects.
Truong \etal~\cite{truong2020gocor} introduced a \textit{dynamic} training strategy by augmenting images with random independently moving objects from the COCO~\cite{lin2014microsoft} so they achieve lower AEPE than our LIFE on the KITTI 2015 flow.
Nonetheless, LIFE still has smaller F1 error even without the {dynamic} training strategy.
Moreover, the experiments on KITTI can only demonstrate the flow estimation performance on cases of simple lighting-constant consecutive images, while LIFE focuses on addressing flow estimation in challenging lighting- and viewpoint-varying scenarios, which will be demonstrated in the following experiments.
\begin{figure*}[t]
\hsize=\textwidth
\centering
\begin{subfigure}[c]{0.7\linewidth}
\includegraphics[width=\linewidth, trim={20mm 0mm 20mm 0mm}, clip]{figures/mma.png}
\end{subfigure}
\begin{subfigure}[c]{0.25\linewidth}
\centering
\resizebox{1.0\linewidth}{!}{
\begin{tabular}{ccc}
\hline
& \#Feat & \#Match \\
\hline
D2-Net MS~\cite{Dusmanu2019CVPR} & 8.3K & 2.8K \\
LF-Net~\cite{ono2018lf} & 0.5K & 0.2K \\
DELF~\cite{noh2017large} & 4.6K & 1.9K\\
HesAff~\cite{mikolajczyk2004scale, arandjelovic2012three} & 6.7K & 2.8K \\
CAPS+SP kpt.~\cite{wang2020learning} & 4.4K & 1.5K \\
SuperPoint \cite{detone2018superpoint} & 1.7K & 0.9K\\
R2D2~\cite{NEURIPS2019_3198dfd0} & 5.0K & 1.8K \\
LIFE+SP & 1.7K & 1.0K \\
LIFE+R2D2 & 5.0K & 2.1K \\
\hline
\end{tabular}
}
\end{subfigure}
\caption{Sparse correspondence identification on HPatches.
}
\label{Fig: mma}
\vspace{-3mm}
\end{figure*}
\noindent \textbf{Dense Flow Estimation on Hpatches.}
We compute the AEPE and accuracy of compared methods in Tab.~\ref{tab: hpatches pck} with increasing levels of difficulty~(from I to V).
As the image pairs in the \textit{Illumination} subset share the same viewpoint, the ground truth flow is $\mathbf{0}$ for all pixels, which is too simple.
We augment the target images in the \textit{Illumination} subset with generated homographies, so the ground truth flows are no longer all-zero flows.
Here, the accuracy is calculated as the percentage of pixels whose endpoint errors are smaller than 5.
CAPS~\cite{wang2020learning} learns dense descriptors for feature matching, which establishes dense correspondences by finding the nearest descriptors.
It has low accuracy because it does not utilize context information.
RAFT~\cite{teed2020raft} is the state-of-the-art supervisory flow estimation method, which fails as well when encountering difficult cases~(II-V).
DGC-Net~\cite{melekhov2019dgc} and GLU-Net~\cite{truong2020glu} are trained by synthesized images.
LIFE consistently outperforms all previous methods and presents increasing superiority from I to V.
In Viewpoint~(V) and Illumination~(V), which include the most challenging cases, LIFE reduces 63.5\% and 61.4\% AEPE, and raises 20.4\% and 17.1\% accuracy.
These experiments show the remarkable performance of LIFE in the difficult cases with illumination and viewpoint variations.
\noindent \textbf{Sparse Flow Estimation on RobotCar and MegaDepth.}
These two datasets contain image pairs that correspond to various conditions such as dawn and night.
The accuracy of estimating sparse flows are reported in Tab.~\ref{tab: sparse flow}.
Following RANSAC-Flow~\cite{shen2020ransac}, a correspondence is deemed correct if its endpoint error is less than $\epsilon=1, 3, 5$ pixels.
RANSAC-Flow~(R-Flow) regularizes flows by RANSAC-based multiple homographies.
The images in the MegaDepth are captured at a long distance, such as bird's view images, which can fit the homography model well, so it presents remarkable performance on MegaDepth.
However, this model does not work in general scenes such as the RobotCar, which takes images with car-mounted cameras.
Moreover, we can also use the RANSAC-based multiple homographies as a post-processing strategy to refine flows predicted by other methods if the scenes actually conform to the model.
Compared with other methods that directly predict flows without the homography-based refinement, LIFE outperforms all of them on both RobotCar and MegaDepth.
\begin{table}[t]
\centering
\resizebox{1.0\linewidth}{!}{
\begin{tabular}{cccc|ccc}
\hline
& \multicolumn{3}{c|}{RobotCar} & \multicolumn{3}{c}{MegaDepth}\\
$\epsilon$ & 1 & 3 & 5 & 1 & 3 & 5\\
\hline
S-Flow~\cite{liu2010sift} & 1.12 & 8.13 & 16.45 & 8.70 & 12.19 & 13.30\\
DGC~\cite{melekhov2019dgc} & 1.19 & 9.35 & 20.17 & 3.55 & 20.33 & 34.28\\
GLU~\cite{truong2020glu} & 2.16 & 16.77 & 33.38 & 25.2 & 51.0 & 56.8\\
GLU-GOC$^+$~\cite{truong2020gocor} & - & - & - & 37.3 & 61.2 & 68.1\\
R-Flow$^*$~\cite{shen2020ransac} & 2.10 & 16.07 & 31.66 & $\mathbf{53.47}$ & $\mathbf{83.45}$ & $\mathbf{86.81}$ \\
\hline
LIFE & $\mathbf{2.30}$ & $\mathbf{17.40}$ & $\mathbf{34.30}$ & 39.98 & 76.14 & 83.14 \\
\hline
\end{tabular}
}
\caption{Sparse flow evaluation on RobotCar and MegaDepth. $^*$ denotes that RANSAC-Flow explicitly regularizes flows with RANSAC-based multiple homographies.
}
\label{tab: sparse flow}
\vspace{-3mm}
\end{table}
\begin{table}[t]
\centering
\resizebox{1.0\linewidth}{!}{
\begin{tabular}{cccc|c}
\hline
& \multicolumn{3}{c|}{MegaDepth} & HP\\
& easy & moderate & hard & $\epsilon=5$\\
\hline
SIFT~\cite{lowe2004distinctive} & 63.9/25.6 & 36.5/17.0 & 20.8/13.2 & 79.0\\
SuperPoint~\cite{detone2018superpoint} & 67.2/27.1 & 38.7/18.8 & 24.5/14.1 & 90.5\\
HardNet~\cite{mishchuk2017working} & 66.3/26.7 & 39.3/18.8 & 22.5/12.3 & - \\
D2-Net~\cite{Dusmanu2019CVPR} & 61.8/23.6 & 35.2/19.2 & 19.1/12.2 & - \\
CAPS+SP kpt.~\cite{wang2020learning} & 72.9/30.5 & 53.5/27.9 & 38.1/19.2 & 90.7 \\
R2D2~\cite{NEURIPS2019_3198dfd0} & 69.4/30.3 & 48.3/23.9 & 32.6/17.4 & 75.7 \\
\hline
LIFE+R2D2 & $\mathbf{78.7}$/$\mathbf{33.1}$ & $\mathbf{62.7}$/$\mathbf{28.7}$ & $\mathbf{45.8}$/$\mathbf{22.4}$ & $\mathbf{91.2}$ \\
\hline
\end{tabular}
}
\caption{Relative pose estimation on MegaDepth and homography estimation on HPatches.
}
\label{tab: relative pose}
\vspace{-1mm}
\end{table}
\subsection{Sparse Correspondence Identification.}
We evaluate LIFE's effectiveness on sparse correspondence identification based on the HPatches dataset.
Following the protocol in D2-Net~\cite{Dusmanu2019CVPR}, we use the mean matching accuracy~(MMA) and the number of matches as evaluation metrics.
As shown in Fig.~\ref{Fig: mma}, LIFE can significantly increase both of the MMA and match number of local features~(LIFE+R2D2 v.s. R2D2 and LIFE+SP v.s. SuperPoint).
LIFE+R2D2 is slightly inferior at smaller thresholds compared to other methods because the detector of R2D2 is not accurate enough.
In contrast, LIFE+SP shows remarkable performance at all thresholds.
\subsection{Whole-image Transformation Estimation}
In all downstream whole-image transformation estimation tasks,
we can see the significant improvement of LIFE+R2D2 based on R2D2, which demonstrates both of the remarkable flow prediction performance of LIFE and the practicability of LIFE-based sparse correspondence identification.
We evaluate the LIFE+R2D2 on homography estimation, relative pose estimation, and visual localization.
\noindent \textbf{Homography estimation on HPatches.}
We use the corner correctness metric introduced by SuperPoint \cite{detone2018superpoint}, which transforms the four corners of an image respectively using the estimated homography and the ground truth homography, and compute the average pixel error of the transformed four corners.
An estimated homography is deemed correct if the average pixel error is less than $\epsilon=5$ pixels.
LIFE increases the accuracy of R2D2 by 15.5\% and achieves state-of-the-art performance~(Tab.~\ref{tab: relative pose}).
\noindent \textbf{Relative pose estimation on MegaDepth.}
We divide the MegaDepth test set into three difficulty levels according to the ground truth relative rotation angle: easy([0$^\circ$, 15$^\circ$]), moderate~([15$^\circ$, 30$^\circ$]) and hard ([30$^\circ$, 60$^\circ$]).
We report the rotation/translation accuracy in Tab.~\ref{tab: relative pose}.
The relative pose is deemed correct if the angle deviation of its rotation or translation is less than $10^\circ$.
LIFE+R2D2 significantly improves R2D2 and outperforms other methods by large margins.
\noindent \textbf{Visual localization in Aachen.}
We evaluate the visual localization on the challenging Aachen DayNight benchmark~\cite{sattler2018benchmarking}.
The reference images used to build the SfM map are all taken during daytime while the query images are capture at nighttime.
We report the percentage of query images localized within three given translation and rotation thresholds at nighttime~(Tab.~\ref{tab: visual localization}).
LIFE+R2D2 improves the localization accuracy at all thresholds and ranks 1st on the benchmark at the threshold of (0.5m, $5^\circ$) and (5m, $10^\circ$).
\begin{table}[t]
\centering
\begin{tabular}{cccc}
\hline
& 0.25m,$2^\circ$ & 0.5m,$5^\circ$ & 5m,$10^\circ$\\
\hline
SuperPoint~\cite{sarlin2019coarse} & 75.5 & 86.7 & 92.9 \\
D2-Net~\cite{Dusmanu2019CVPR} & 84.7 & 90.8 & 96.9 \\
SuperGlue~\cite{sarlin2020superglue} & $\mathbf{86.7}$ & 93.9 & $\mathbf{100.0}$ \\
R2D2~\cite{NEURIPS2019_3198dfd0} & 80.6 & 90.8 & 96.9 \\
\hline
LIFE+R2D2 & 81.6 & $\mathbf{94.9}$ & $\mathbf{100.0}$\\
\hline
\end{tabular}
\caption{Visual localization on Aachen (night).
}
\label{tab: visual localization}
\vspace{-4mm}
\end{table}
\subsection{Ablation study}
\noindent \textbf{Training loss ablation.}
We investigate individual components of our LIFE on KITTI and Hpatches with the AEPE metric.
K-12 and K-15 denotes KITTI flow 2012 and KITTI flow 2015. HP-V and HP-I (T) denotes the \textit{Viewpoint} subset and the \textit{Illumination~(trans)} subset in HPatches.
We assess the effectiveness of our methods by sequentially adding the proposed training losses.
T(B'B) denotes that only supervising the network with the flows in one direction of $B'\leftarrow B$ from the synthetic geometric transformation $\mathbf{T}$.
We can see a significant error reduction from T(B'B) to the BiT loss, which indicates the significance of supervising flow prediction in both directions.
The KITTI contains consecutive images that share similar lighting conditions.
Compared with the SED loss, training with the BiT loss produces less errors on the KITTI because it can provide pixel-to-pixel ground truth.
However, it presents worse performance on the HPatches because the synthesized image does not change the illumination and is not in line with the actual situations.
In contrast, the SED loss works with image pairs captured in real scenes so it achieves less error on the HPatches.
We also test imposing cycle consistency on all flows (denoted by ``FC'') and adaptive cycle consistency only on small-error pairs (denoted by ``AC'').
FC impacts the flow estimation because occluded regions in the image do not satisfy cycle consistency, and AC can improve SED on viewpoint change cases.
The final model~(SED+BiT+AC) that unifies BiT, SED, and AC in the created image triplets achieves the best performance, which demonstrates the SED loss and the BiT loss mitigates their drawbacks well.
\noindent \textbf{LIFE with different local features.}
We test LIFE with different local features~(denoted by ``LIFE+*''), including SIFT, SuperPoint, and R2D2 on HPaches.
As LIFE-based sparse correspondences are computed in two stages, we also report the performance of outcome matches in stage 1~(denoted by ``LIFE+* local''), which are calculated with the guidance of flows predicted by LIFE.
We report the MMA, feature number, and match number of corresponding methods in Fig.~\ref{Fig: ablation}.
LIFE consistently improves the MMA and increases the match number.
\textit{LIFE+SIFT local}, \textit{LIFE+SP local}, and \textit{LIFE+R2D2 local} all achieves remarkable MMA scores and remains similar match number, which demonstrates the superior performance of LIFE.
After introducing the matches established in stage 2~(``LIFE+* local'' to ``LIFE+*''),
the match number increases while the MMA scores decreases because the matches calculated in stage 1 are better than
these supplemented matches.
\begin{figure}[t]
\centering
\begin{subfigure}[c]{0.7\linewidth}
\includegraphics[width=\linewidth, trim={0mm 0mm 0mm 0mm}, clip]{figures/mma-ablation.png}
\end{subfigure}
\begin{subfigure}[c]{0.9\linewidth}
\centering
\resizebox{1.0\linewidth}{!}{
\begin{tabular}{ccccc}
\hline
& \#Feat & \multicolumn{3}{c}{\#Matches} \\
& & raw & LIFE(local) & LIFE(all) \\
\hline
SIFT~\cite{lowe2004distinctive} & 4.4K & 1.8K & 1.8K & 2.2K\\
SuperPoint~\cite{detone2018superpoint} & 1.7K & 0.9K & 0.9K & 1.0K \\
R2D2~\cite{NEURIPS2019_3198dfd0} & 5K & 1.8K & 1.9K & 2.1K \\
\hline
\end{tabular}
}
\end{subfigure}
\caption{LIFE with different local features.
}
\label{Fig: ablation}
\end{figure}
\begin{table}[t]
\centering
\resizebox{0.9\linewidth}{!}{
\begin{tabular}{c|c|c|c|c}
Training loss & K-12 & K-15 & HP-V & HP-I(T)\\
\hline
T(B'B) & 9.03 & 27.94 & 61.03 & 48.15 \\
BiT & 3.71 & 11.83 & 16.03 & 37.6\\
SED & 5.62 & 16.26 & 11.69 & 15.17\\
SED+FC & 6.48 & 16.83 & 17.79 & 23.64\\
SED+AC & 4.89 & 14.99 & 12.57 & 15.47 \\
SED+BiT+AC~(Tri) & 2.59 & 8.30 & 5.90 & 9.19 \\
\hline
\end{tabular}
}
\caption{Ablation study of training losses.
}
\label{tab: flow ablation}
\vspace{-3mm}
\end{table}
\section{Introduction}
\input{01-introduction}
\section{Related Works}
\input{02-relatedworks}
\section{Method}
\input{03-method}
\section{Experiments}
\input{04-experiments}
\section{Conclusion}
\input{05-conclusion}
{\small
\bibliographystyle{ieee_fullname}
|
3,212,635,537,988 | arxiv | \section{Introduction}
In \cite{cagutt}, a moment map $\mu$ on the space of symplectic connections is introduced. The study of zeroes of $\mu$ and of the so-called critical symplectic connections was first proposed by D.J. Fox \cite{Fox} in analogy with the moment map picture for the Hermitian scalar curvature on almost-K\"ahler manifolds. Recently \cite{LLF}, we give additional motivations for the study of $\mu$, and its zeroes on K\"ahler manifolds, coming from the formal deformation quantization of symplectic manifolds.
Our goal is to exhibit an obstruction to the existence of zeroes of $\mu$ on closed K\"ahler manifolds in the spirit of Futaki invariants. We will consider closed K\"ahler manifolds $(M,\omega,J)$ with K\"ahler class $\Theta$ and denote by $\mathfrak{h}$ the Lie algebra of the reduced group of complex automorphisms of the K\"ahler manifold that is the Lie algebra of holomorphic vector fields of the form $Z=X_F+JX_H$, where we denoted by $X_K$ the Hamiltonian vector field defined by $i(X_K)\omega=dK$ for $K\in C^{\infty}(M)$ normalised by $\int_M K\frac{\omega^n}{n!}=0$. Our first result is:
\begin{theoremintro} \label{theorprinc:Futinvdef}
Let $(M,\omega,J)$ be a closed K\"ahler manifold with K\"ahler class $\Theta$ and Levi-Civita connection $\nabla$. Let $\mathfrak{h}$ be the Lie algebra of the reduced group of complex automorphisms of the K\"ahler manifold. Then, the map
\begin{equation*}
\mathcal{F}^{\omega}:\mathfrak{h} \rightarrow \mathbb{R} : Z \mapsto \int_M H\mu(\nabla)\frac{\omega^n}{n!},
\end{equation*}
for $Z=X_F+JX_H$ and $\mu$ is the Cahen-Gutt moment map on $\mathcal{E}(M,\omega)$, is a character that does not depend on the choice of a K\"ahler form in the K\"ahler class $\Theta$.
\end{theoremintro}
Deformation quantization as defined in \cite{BFFLS} is a formal associative deformation of the Poisson algebra $(C^{\infty}(M),.,\{\cdot,\cdot\})$ of a Poisson manifold $(M,\pi)$ in the direction of the Poisson bracket. The deformed algebra is the space $C^{\infty}(M)[[\nu]]$ of formal power series of smooth functions with composition law $*$ called star product.
On a symplectic manifold $(M,\omega)$ endowed with a symplectic connection $\nabla$ (i.e. torsion-free connection leaving $\omega$ parallel), one can associate the Fedosov's star product $*_{\nabla}$, \cite{fed2}. The moment map $\mu$ evaluated at $\nabla$ is the first non-trivial term in the expression of a trace density for the star product $*_{\nabla}$, see \cite{LLF}. So that, if the star product $*_{\nabla}$ is closed (in the sense of Connes-Flato-Sternheimer \cite{CFS}), then $\mu(\nabla)$ is the zero function which implies the following result.
\begin{cor} \label{cor:corprinc}
Let $(M,\omega,J)$ be a closed K\"ahler manifold with K\"ahler class $\Theta$, such that $\mathcal{F}^{\omega}$ is not identically zero, then, given any K\"ahler form $\widetilde{\omega} \in \mathcal{M}_{\Theta}$ with Levi-Civita connection $\widetilde{\nabla}$, the Fedosov's star product $*_{\widetilde{\nabla}}$ is not closed.
\end{cor}
Finally, we identify the character $\mathcal{F}^{\omega}$ with one of the so-called higher Futaki invariants. It enables us to exhibit an example of K\"ahler manifolds \cite{NP,Ono} admitting non-zero values of $\mathcal{F}$ and hence no closed Fedosov's star products as considered in the above Corollary \ref{cor:corprinc}.
\section{The moment map and Fedosov's star products} \label{sect:symplconn}
Consider a closed symplectic manifold $(M,\omega)$ of dimension $2n$. A symplectic connection $\nabla$ on $(M,\omega)$ is a torsion-free connection such that $\nabla \omega=0$. There always exists a symplectic connection on a symplectic manifold and the space $\mathcal{E}(M,\omega)$ of symplectic connections is the affine space
$$\mathcal{E}(M,\omega)=\nabla+ \Gamma(S^3T^*M) \textrm{ for some } \nabla \in \mathcal{E}(M,\omega),$$
where $S^3T^*M:=\{A\in \Lambda^1(M)\otimes End(TM) \ | \ \omega(A(\cdot)\cdot,\cdot) \textrm{ is completely symmetric}\}$. For $A\in S^3T^*M$, we set $\underline{A}(\cdot,\cdot,\cdot)$ for the symmetric $3$-tensor $\omega(A(\cdot)\cdot,\cdot)$.
There is a natural symplectic form on $\mathcal{E}(M,\omega)$. For $A, B \in T_{\nabla} \mathcal{E}(M,\omega)$, seen as element of $\Lambda^1 (M)\otimes End(TM,\omega)$, one defines
$$\Omega^{\mathcal{E}}_{\nabla}(A,B):= \int_M \mathrm{tr}(A\stackrel{\circ}{\wedge} B)\wedge \frac{\omega^{n-1}}{(n-1)!}= -\int_M \Lambda^{kl}\mathrm{tr}(A(e_k)B(e_l))\frac{\omega^n}{n!},$$
where $\stackrel{\circ}{\wedge}$ is the product on $\Lambda^1 (M)\otimes End(TM,\omega)$ induced by the usual $\wedge$-product on forms and the composition on the endomorphism part, $\Lambda^{kl}$ is defined by $\Lambda^{kl}\omega_{lt}=\delta^k_t$ for $\omega_{lt}:=\omega(e_l,e_t)$ for a frame $\{e_k\}$ of $T_xM$.
The $2$-form $\Omega^{\mathcal{E}}$ is a symplectic form on $\mathcal{E}(M,\omega)$.
\begin{rem}
The symplectic form $\Omega^{\mathcal{E}}$ can be written in coordinate as :
$$\Omega^{\mathcal{E}}_{\nabla}(A,B):=\int_M \Lambda^{i_1j_1}\Lambda^{i_2j_2}\Lambda^{i_3j_3}\underline{A}_{i_1i_2i_3}\underline{B}_{j_1j_2j_3}\frac{\omega^n}{n!},$$
for $A,B\in T_{\nabla}\mathcal{E}(M,\omega)$.
\end{rem}
There is a natural symplectic action of the group of symplectomorphisms on $\mathcal{E}(M,\omega)$. For $\varphi$, a symplectic
diffeomorphism, we define an action
\begin{equation} \label{eq:action}
(\varphi.\nabla)_X Y := \varphi_*(\nabla_{\varphi^{-1}_* X}\varphi^{-1}_* Y),
\end{equation}
for all $X,Y \in TM$ and $\nabla \in \mathcal{E}(M,\omega)$.
Recall that a Hamiltonian vector field is a vector field $X_F$ for $F\in C^{\infty}(M)$ such that
$i(X_F)\omega=dF.$
We denote by $\mathop{\mathrm{Ham}}\nolimits(M,\omega)$ the group of Hamiltonian diffeomorphisms of the symplectic manifold $(M,\omega)$ with Lie algebra the space $C^{\infty}_0(M)$ of smooth functions $F$ such that $\int_M F \frac{\omega^n}{n!}=0$.
The action defined in Equation (\ref{eq:action}) restricts to an action of the group $\mathop{\mathrm{Ham}}\nolimits(M,\omega)$. Let $X_F$ be a Hamiltonian vector field with $F\in C^{\infty}_0(M)$, the fundamental vector field on $\mathcal{E}(M,\omega)$ associated to this action is
\begin{equation*}
(X_F)^{*\mathcal{E}}(Y)Z:=\frac{d}{dt}|_0\varphi_{-t}^F.\nabla=(\mathscr{L}_{X_F}\nabla)(Y)Z=\nabla^2_{(Y,Z)}X_F + R^{\nabla}(X_F,Y)Z,
\end{equation*}
where $R^{\nabla}(U,V)W:=[\nabla_U,\nabla_V]W-\nabla_{[U,V]}W$ is the curvature tensor of $\nabla$.
Denote by $Ric^{\nabla}$ the Ricci tensor of $\nabla$ defined by $Ric^{\nabla}(X,Y):=\mathrm{tr}[V\mapsto R^{\nabla}(V,X)Y]$
for all $X,Y \in TM$.
\begin{theorem} [Cahen-Gutt \cite{cagutt}] \label{theor:momentE}
The map $\mu:\mathcal{E}(M,\omega) \rightarrow C^{\infty}_0(M)$ defined by
$$\mu(\nabla):=(\nabla^2_{(e_p,e_q)} Ric^{\nabla})(e^{p},e^q) + P(\nabla)-\mu_0$$
where $\{e_k\}$ is a frame of $T_{x}M$ and $\{e^l\}$ is the symplectic dual frame of $\{e_k\}$ (that is $\omega(e_k,e^l)=\delta_k^l$) and $P(\nabla)$ is the function defined by $P(\nabla)\frac{\omega^n}{n!}:=\frac{1}{2}\mathrm{tr}(R^{\nabla}(.,.)\stackrel{\circ}{\wedge} R^{\nabla}(.,.))\wedge \frac{\omega^{n-2}}{(n-2)!}$ with $\int_M P(\nabla)\frac{\omega^n}{n!}=:\mu_0$, is a moment map for the action of $\mathop{\mathrm{Ham}}\nolimits(M,\omega)$ on $\mathcal{E}(M,\omega)$, i.e.
\begin{equation*} \label{eq:momentmu}
\frac{d}{dt}|_{0} \int_M \mu(\nabla+tA)F\frac{\omega^n}{n!}=\Omega^{\mathcal{E}}_{\nabla}((X_F)^{*\mathcal{E}},A).
\end{equation*}
\end{theorem}
\noindent In \cite{LLF}, the moment map $\mu$ is related to the notion of trace density for Fedosov's star products. Also, the closedness (closedness in the sense of Connes-Flato-Sternheimer \cite{CFS}) of Fedosov's star product implies $\mu=0$. Let us recall briefly all those notions and results.
A {\bf star product}, as defined in \cite{BFFLS}, on $(M,\omega)$ is a $\mathbb{R}[[\nu]]$-bilinear associative law on the space $C^{\infty}(M)[[\nu]]$ of formal power series
of smooth functions :
\begin{equation} \nonumber
*: (C^{\infty}(M)[[\nu]] )^2 \rightarrow C^{\infty}(M)[[\nu]]: (H,K)\mapsto H*K:=\sum_{r=0}^{\infty} \nu^r C_r(H,K)
\end{equation}
where the $C_r$'s are bidifferential operators null on constants such that for all $H,\, K \in C^{\infty}(M)[[\nu]]$ :
$C_0(H,K)=HK$ and
$C_1(H,K)-C_1(K,H)=\{H,K\}$.
In \cite{fed2}, Fedosov gave a geometric construction of star products on symplectic manifolds using a symplectic connection $\nabla$ and a formal series of closed $2$-forms $\Omega\in \nu \Omega^2(M)[[\nu]]$. We will only consider Fedosov's star products build with $\Omega=0$ and denote them by $*_{\nabla}$.
Let $*$ be a star product on a symplectic manifold.
A {\bf trace} for $*$ is a $\mathbb{R}[[\nu]]$-linear map
$$\mathrm{tr} : C^{\infty}(M)[[\nu]] \rightarrow \mathbb{R}[[\nu]],$$
satisfying $\mathrm{tr}(F*H)=\mathrm{tr}(H*F)$ for all $F,H \in C^{\infty}(M)[[\nu]]$.
Any star product $*$ on a symplectic manifold $(M,\omega)$ admit a trace \cite{fed3,NT,gr}. More precisely, there exists $\kappa \in C^{\infty}(M)[[\nu]]$ such that
\begin{equation*}
\mathrm{tr}(F):= \int_M F\kappa \frac{\omega^n}{n!}
\end{equation*}
for all $F \in C^{\infty}(M)[[\nu]]$. The function $\kappa$ is called a {\bf trace density}. Moreover, any two traces for $*$ differ from each other by multiplication with a formal constant $C \in \mathbb{R}[\nu^{-1},\nu]]$.
A star product is called {\bf closed} \cite{CFS} if the map $F\mapsto \int_M F \frac{\omega^n}{n!}$ satisfies the trace property:
\begin{equation*}
\int_M F*H \frac{\omega^n}{n!} = \int_M H*F \frac{\omega^n}{n!}, \ \textrm{ for all } F,H \in C^{\infty}(M)[[\nu]].
\end{equation*}
In \cite{LLF}, we linked the moment map with the trace density $\kappa^{\nabla}$ of the Fedosov's star product $*_{\nabla}$ by the formula :
$$\kappa^{\nabla}:= 1+\frac{\nu^2}{24}\mu(\nabla)+O(\nu^3).$$
So that, if $*_{\nabla}$ is closed, then $\mu(\nabla)=0$.
\section{Futaki invariant for $\mu$}
\subsection{Definition and main Theorem}
We consider a closed K\"ahler manifold $(M,\omega,J)$. Let $\Theta$ be the K\"ahler class of $\omega$ and denote
by $\mathcal{M}_{\Theta}$ the set of K\"ahler forms in the class $\Theta$. By the classical $dd^c$-lemma,
$$\mathcal{M}_{\Theta}:= \{\omega_{\phi}=\omega+dd^c\phi \textrm{ s.t. } \phi\in C^{\infty}_0(M),\ \omega_{\phi}(\cdot,J\cdot) \textrm{ is positive definite }\},$$
where $d^cF:=-dF\circ J$ for $F\in C^{\infty}(M)$.
Consider the functional
\begin{equation*}
\omega_{\phi} \in \mathcal{M}_{\Theta} \mapsto \mu^{\phi}(\nabla^{\phi})\in C^{\infty}(M),
\end{equation*}
where $\mu^{\phi}$ is the moment map on $\mathcal{E}(M,\omega_{\phi})$ and $\nabla^{\phi}$ is the Levi-Civita connection of $g_\phi(\cdot,\cdot):= \omega_{\phi}(\cdot,J\cdot)$. Using the second Bianchi identity, one can write :
$$\mu^{\phi}(\nabla^{\phi})=-\frac{1}{2}\Delta^{\phi} Scal^{\nabla^{\phi}} + P(\nabla^{\phi})-\mu_0.$$
Note that $\mu_0$ does not depend on $\phi$ and that $\mu^{\phi}(\nabla^{\phi})$ is normalised with respect to the integral using the K\"ahler form $\omega_{\phi}$.
Let $\mathfrak{h}$ be the Lie algebra of all holomorphic vector fields having at least one zero on $M$. For any $Z\in \mathfrak{h}$ and $\omega_{\phi}\in \mathcal{M}_{\Theta}$, there are unique $F^{\phi},H^{\phi} \in C^{\infty}(M)$ (depending on $\omega_{\phi}$) whose integral with respect to $\frac{\omega_{\phi}^n}{n!}$ is zero such that $Z=X^{\omega_{\phi}}_{F^{\phi}}+JX^{\omega_{\phi}}_{H^{\phi}}$ where $X^{\omega_{\phi}}_K$ denotes the Hamiltonian vector field of $K\in C^{\infty}(M)$ with respect to the symplectic form $\omega_{\phi}$.
\begin{defi}
For $\omega_{\phi} \in \mathcal{M}_{\Theta}$, we define the map
\begin{equation} \label{eq:Futakiinv}
\mathcal{F}^{\omega^{\phi}}:\mathfrak{h}(M) \mapsto \mathbb{R}: Z \mapsto \int_M H^{\phi}\mu^{\phi}(\nabla^{\phi}) \frac{\omega_{\phi}^n}{n!},
\end{equation}
for $Z=X^{\omega_{\phi}}_{F^{\phi}}+JX^{\omega_{\phi}}_{H^{\phi}}$ as above.
\end{defi}
Though, the definition of $\mathcal{F}^{\omega_{\phi}}$ seems a priori to depend on the choice of a point in $\mathcal{M}_{\Theta}$, we will prove it is not the case.
\begin{theoremprinc}
Let $(M,\omega,J)$ be a closed K\"ahler manifold with K\"ahler class $\Theta$ and Levi-Civita connection $\nabla$. Let $\mathfrak{h}$ be the Lie algebra of the reduced group of complex automorphisms of the K\"ahler manifold. Then, the map
\begin{equation*}
\mathcal{F}^{\omega}:\mathfrak{h}(M) \rightarrow \mathbb{R} : Z \mapsto \int_M H\mu(\nabla)\frac{\omega^n}{n!},
\end{equation*}
for $Z=X_F+JX_H$ and $\mu$ is the Cahen-Gutt moment map on $\mathcal{E}(M,\omega)$, is a character that does not depend on the choice of a K\"ahler form in the K\"ahler class $\Theta$.
\end{theoremprinc}
The Theorem \ref{theorprinc:Futinvdef} implies that the non-vanishing of $\mathcal{F}^{\omega}$ is an obstruction to the existence of $\omega_{\phi}\in \mathcal{M}_{\Theta}$ such that $\mu^{\phi}(\nabla^{\phi})=0$.
\begin{proof}[Proof of Corollary \ref{cor:corprinc}]
For $\widetilde{\omega}\in \mathcal{M}_{\Theta}$ with Levi-Civita connection $\widetilde{\nabla}$, assume the Fedosov's star product $*_{\widetilde{\nabla}}$ is closed. Then $\mu^{\widetilde{\omega}}(\widetilde{\nabla})=0$ and hence $\mathcal{F}^{\omega}=0$. It concludes the proof.
\end{proof}
\subsection{The space $\mathcal{J}_{int}(M,\omega)$}
The goal of this subsection is to state the necessary formulas coming from \cite{LLF} in order to prove Theorem \ref{theorprinc:Futinvdef}.
\begin{defi}
We denote by $\mathcal{J}_{int}(M,\omega)$ the space of integrable complex structures on $M$ compatible with $\omega$, that is $J\in \mathcal{J}_{int}(M,\omega)$ is a complex structure such that $\omega(J.,J.)=\omega(.,.)$ and $\omega(.,J.)$ is a Riemannian metric.
\end{defi}
For $J_t\in \mathcal{J}_{int}(M,\omega)$ a smooth path and $A:=\frac{d}{dt}|_0J_t\in T_J \mathcal{J}_{int}(M,\omega)$. Then, $A$ is a section of the bundle $End(TM)$ satisfying $ AJ_0+J_0A=0$ and the $2$-tensor
$$J(\nabla A (X)Y)-(\nabla A)(JX)Y$$
is symmetric in $X, Y$.
Consider the map
$$\mathrm{lc}:\mathcal{J}_{int}(M,\omega) \rightarrow \mathcal{E}(M,\omega): J \mapsto \nabla^{J}$$
which associates to an integrable complex structure $J$ compatible with $\omega$, the Levi-Civita
connection $\nabla^{J}$ of the K\"ahler metric $g_J(.,.):= \omega(.,J.)$.
The map $\mathrm{lc}$ is equivariant with respect to the group of symplectic diffeomorphisms of $(M,\omega)$.
That is : for all $\varphi \in \mathop{\mathrm{Symp}}\nolimits(M,\omega)$ and $J\in \mathcal{J}_{int}(M,\omega)$ with $\varphi.J:=\varphi_*J\varphi^{-1}_*$:
$$\mathrm{lc}(\varphi.J)=\varphi.\mathrm{lc}(J) .$$
\begin{prop} \label{prop:LC*}
Let $A\in T_J \mathcal{J}_{int}(M,\omega)$ and write $B\in T_{\nabla}\mathcal{E}(M,\omega)$ such that $B=\mathrm{lc}_{*J}(A)$.
Then $B$ is the unique solution to the equation
\begin{equation*} \label{eq:BJlin}
B(X)Y+JB(X)JY=- \nabla J A(X)Y.
\end{equation*}
and if $JA \in T_J \mathcal{J}_{int}(M,\omega)$, then :
$$\mathrm{lc}_{*J}(JA)(X)Y=JB(JX)JY +\frac{1}{2}\left(J(\nabla A) (JX)Y)+(\nabla A)(X)Y\right).$$
\end{prop}
\noindent From those equations we obtain \cite{LLF}:
\begin{lemme} \label{lemme:Iinv}
If $A, A'$ and $JA, JA'\in T_J\mathcal{J}_{int}(M,\omega)$ then
$$(lc^*\Omega^{\mathcal{E}})_J(JA,JA')=(lc^*\Omega^{\mathcal{E}})_J(A,A').$$
\end{lemme}
\subsection{Proof of Theorem \ref{theorprinc:Futinvdef}} \label{sect:cal}
We will prove Theorem \ref{theorprinc:Futinvdef} in this section. For this, consider a smooth one-parameter family $\phi:\ ]-\epsilon,\epsilon[\rightarrow C^{\infty}_0(M):t\mapsto \phi(t)$ for some $\epsilon\in \mathbb{R}^+_0$ such that the $2$-form $\omega_{\phi(t)}:= \omega+ dd^c\phi(t)$ is a smooth path in $\mathcal{M}_{\Theta}$ passing through $\omega$. To prove the independence of $\mathcal{F}^{\omega^{\phi}}$, we will show that for all $Z\in \mathfrak{h}(M)$ :
\begin{equation*}
\frac{d}{dt}|_{0}\mathcal{F}^{\omega^{\phi(t)}}(Z)=0.
\end{equation*}
All the forms $\omega_{\phi(t)}$ are symplectomorphic to each other. Indeed, set $X_t:=-\textrm{grad}^{\phi(t)}(\dot{\phi})$ the gradient vector field of $\dot{\phi}(t)$ with respect to $g_{\phi(t)}$ (that is $g_{\phi(t)}(\textrm{grad}^{\phi(t)}(\dot{\phi}),\cdot)=d\dot{\phi}$). Then the one parameter family of diffeomorphisms $f_t$ integrating the time-dependent vector field $X_t$ satisfies
\begin{equation} \label{eq:Moser}
f_t^*\omega_{\phi(t)}=\omega.
\end{equation}
Consider $f_t$ as in the above equation (\ref{eq:Moser}). Then, the natural action of $f_t^{-1}$ on $J$ produces a path
$$J_t:=f_t^{-1}.J:= f_{t*}^{-1} J f_{t*} \in \mathcal{J}_{int}(M,\omega).$$
Define the associated K\"ahler metric $g_{J_t}(\cdot,\cdot):=\omega(\cdot,J_t\cdot)$ and denote by $\nabla^{J_t}$ its
Levi-Civita connection. Then, $\nabla^{J_t}$ and $\nabla^{\phi(t)}$ are related by the following formula :
$$\nabla^{J_t}=f_t^{-1}.\nabla^{\phi(t)},$$
where $(f_t^{-1}.\nabla^{\phi(t)})_YZ = f_{t*}^{-1}\nabla^{\phi(t)}_{f_{t*}Y}f_{t*}Z$. Then, their image by the moment map is related by :
\begin{equation} \label{eq:muJmuphi}
\mu(\nabla^{J_t})=f_t^*\mu^{\phi(t)}(\nabla^{\phi(t)}).
\end{equation}
Note that on the LHS the moment map is taken with respect to a fixed sympelctic form while on the RHS $\mu^{\phi(t)}$ is a function on $\mathcal{E}(M,\omega_{\phi(t)})$.
\begin{proof}[Proof of Theorem \ref{theorprinc:Futinvdef}]
We will use the notations introduced above. First, using Equations (\ref{eq:Futakiinv}), (\ref{eq:Moser}) and (\ref{eq:muJmuphi}), we have :
$$\mathcal{F}^{\omega^{\phi(t)}}(Z)=\int_M H^{\phi(t)}\mu^{\phi(t)}(\nabla^{\phi(t)}) \frac{\omega_{\phi(t)}^n}{n!}=\int_M f_t^*(H^{\phi(t)})\mu(\nabla^{J_t})\frac{\omega^n}{n!}.$$
We will differentiate at $t=0$. We will write $H$ for $H^{\phi(0)}$ :
$$\frac{d}{dt}|_0 f_t^*(H^{\phi(t)}) = X_0(H)+\frac{d}{dt}|_0 H^{\phi(t)}=X_0(H)+Z(\dot{\phi}(0))=X_F(\dot{\phi}(0)).$$
Using the fact that $\frac{d}{dt}|_0J_t=\mathscr{L}_{X_0} J=-\mathscr{L}_{JX_{\dot{\phi}(0)}}J=-J\mathscr{L}_{X_{\dot{\phi}(0)}}J$, we compute
\begin{eqnarray*}
\frac{d}{dt}|_0\mathcal{F}^{\omega^{\phi(t)}}(Z) & = & \int_M X_F(\dot{\phi}(0))\mu(\nabla) \frac{\omega^n}{n!} + \frac{d}{dt}|_0 \int_M H \mu(\nabla^{J_t})\frac{\omega^n}{n!}, \\
& = & \int_M X_F(\dot{\phi}(0))\mu(\nabla) \frac{\omega^n}{n!} + \Omega(\mathscr{L}_{X_H}\nabla, \mathrm{lc}_{*J}(-J\mathscr{L}_{X_{\dot{\phi}(0)}}J)).
\end{eqnarray*}
Using the equivariance of the map $\mathrm{lc}$ and Lemma \ref{lemme:Iinv}, we have
\begin{eqnarray*}
\Omega(\mathscr{L}_{X_H}\nabla, \mathrm{lc}_{*J}(-J\mathscr{L}_{X_{\dot{\phi}(0)}}J)) & = & \Omega(\mathrm{lc}_{*J}(\mathscr{L}_{X_H}J),\mathrm{lc}_{*J}(-J\mathscr{L}_{X_{\dot{\phi}(0)}}J)),\\
& = & \Omega(\mathrm{lc}_{*J}(J\mathscr{L}_{X_H}J),\mathrm{lc}_{*J}(\mathscr{L}_{X_{\dot{\phi}(0)}}J)).
\end{eqnarray*}
Finally, as $Z$ is holomorphic, $J\mathscr{L}_{X_H}J=-\mathscr{L}_{X_F}J$, so that :
\begin{eqnarray*}
\frac{d}{dt}|_0\mathcal{F}^{\omega^{\phi(t)}}(Z) & = & \int_M X_F(\dot{\phi}(0))\mu(\nabla) \frac{\omega^n}{n!} - \Omega(\mathrm{lc}_{*J}(\mathscr{L}_{X_F}J),\mathrm{lc}_{*J}(\mathscr{L}_{X_{\dot{\phi}(0)}}J)),\\
& = & \int_M X_F(\dot{\phi}(0))\mu(\nabla) \frac{\omega^n}{n!}+ \int_M \mu(\nabla) X_{\dot{\phi}(0)}(F)\frac{\omega^n}{n!} =0.
\end{eqnarray*}
The fact that $\mathcal{F}$ is a character, is a consequence of the above computations. Indeed, for $Y,Z \in \mathfrak{h}(M)$, one has $[Y,Z]=\frac{d}{dt}|_0\varphi^Y_{-t*}Z$, for $\varphi^Y_{t*}$ the flow of $Y$. Then, when $Z=X^{\omega}_F+JX^{\omega}_H$, one computes $\varphi^Y_{-t*}Z=X^{\varphi^{Y*}_{t}\omega}_{\varphi^{Y*}_{t}F}+JX^{\varphi^{Y*}_{t}\omega}_{\varphi^{Y*}_{t}H}$. Finally, one has
$$\mathcal{F}^{\omega}([Y,Z])=\frac{d}{dt}|_0\mathcal{F}^{\omega}(\varphi^Y_{-t*}Z) = \frac{d}{dt}|_0\mathcal{F}^{\varphi_t^{Y*}\omega}(\varphi^Y_{-t*}Z)= \frac{d}{dt}|_0\mathcal{F}^{\omega}(Z)=0$$
\end{proof}
\section{Generalised Futaki invariants}
\subsection{$\mathcal{F}^{\omega}$ is a generalised Futaki invariant}
In \cite{Fut}, Futaki generalised the Futaki invariant obstructing the existence of K\"ahler-Einstein metrics. One of these so-called generalised Futaki invariants is the invariant we define using the moment map.
Futaki's construction goes as follows. On a K\"ahler manifold $(M,\omega,J)$, consider the holomorphic bundle $T^{(1,0)}M$ of tangent vectors of type $(1,0)$. Choose any $(1,0)$-connection $\overline{\nabla}$ on $T^{(1,0)}M$ with curvature $R^{\overline{\nabla}}$. For $Z\in \mathfrak{h}(M)$, define $L(Z^{(1,0)}):=\overline{\nabla}_{Z^{(1,0)}}-\mathscr{L}_{Z^{(1,0)}}$, it is a section of the bundle $End(T^{(1,0)}M)$. Let $q$ be a $Gl(n,\mathbb{C})$-invariant polynomials on $\mathfrak{gl}(n,\mathbb{C})$ of degree $p$, Futaki defined in \cite{Fut}, the map $F_q:\mathfrak{h}\rightarrow \mathbb{C}$ by
\begin{equation*} \label{eq:Futakigendef}
F_q(Z):= (n-p+1) \int_M u_Z q(R^{\overline{\nabla}})\wedge \omega^{(n-p)}+\int_M q(L(Z^{(1,0)})+R^{\overline{\nabla}})\wedge \omega^{(n-p+1)},
\end{equation*}
where $u_Z$ is the complex valued function defined by $i(Z^{(1,0)})\omega=\overline{\partial} u_Z$.
Futaki shows that $F_q$ depends neither on the choice of the ${(1,0)}$-connection nor on the choice of the K\"ahler form in $\mathcal{M}_{\Theta}$, see \cite{Fut}. Moreover, if you take $q=c_k$ the polynomials defining the $k$-th Chern form, it is proved in \cite{Fut} that one recovers Bando's obstruction to the harmonicity of the k$^{th}$ Chern form:
\begin{equation} \label{eq:Fc2}
F_{c_k}(Z)=(n-k+1) \int_M u_Z c_k(R^{\nabla})\wedge \omega^{(n-k)}.
\end{equation}
\begin{prop} \label{prop:FequalF}
We have that $\mathcal{F}^{\omega}$ is the imaginary part of $F_{\frac{8\pi^2}{(n-1)!}(c_2-\frac{1}{2}c_1. c_1)}$
\end{prop}
\begin{proof}
The key of the computation is that the Pontryagin $4$-form defining $P(\nabla)$ satisfies:
\begin{equation*}
\mathrm{tr}(R^{\nabla}\stackrel{\circ}{\wedge} R^{\nabla})=16\pi^2(c_2-\frac{1}{2}c_1. c_1)(R^{\nabla}).
\end{equation*}
Then,
$$\mathcal{F}^{\omega}(Z)=-\frac{1}{2} \int_M H\Delta Scal \frac{\omega^n}{n!} +8\pi^2\int_M Hc_2(R^{\nabla})\wedge \frac{\omega^{n-2}}{(n-2)!} - 4\pi^2\int_M Hc_1.c_1(R^{\nabla}) \wedge \frac{\omega^{n-2}}{(n-2)!}.$$
As $u_z=F+iH$, Equation (\ref{eq:Fc2}) tells us that the imaginary part of $F_{\frac{8\pi^2}{(n-1)!}c_2}(Z)$ is:
$$8\pi^2\int_M Hc_2(R^{\nabla})\wedge \frac{\omega^{n-2}}{(n-2)!}.$$ It remains to compute de second term of $F_{\frac{4\pi^2}{(n-1)!}c_1c_1}$ :
\begin{eqnarray*}
F_{\frac{4\pi^2}{(n-1)!}c_1c_1}(Z)& = &4\pi^2\left(\int_M u_Zc_1.c_1(R^{\nabla})\wedge \frac{\omega^{n-2}}{(n-2)!}+\int_M c_1.c_1(L(Z^{(1,0)})+R^{\nabla})\wedge \frac{\omega^{n-1}}{(n-1)!}\right)\\
& = & 4\pi^2\int_M u_Zc_1.c_1(R^{\nabla})\wedge \frac{\omega^{n-2}}{(n-2)!}+ 2i\int_M \mathrm{tr}^{\mathbb{C}}(L(Z^{(1,0)}))\rho^{\nabla}\wedge \frac{\omega^{n-1}}{(n-1)!}.
\end{eqnarray*}
Since $\mathrm{tr}^{\mathbb{C}}(L(Z^{(1,0)}))= \frac{-i}{2}\left(\Delta F+i\Delta H\right)$, we have:
\begin{eqnarray*}
2i\int_M \mathrm{tr}^{\mathbb{C}}(L(Z^{(1,0)}))\rho^{\nabla}\wedge \frac{\omega^{n-1}}{(n-1)!} & = & \frac{1}{2}\int_M \left(\Delta F+i\Delta H\right)Scal^{\nabla} \frac{\omega^n}{n!},\\
& = & \frac{1}{2}\int_M \left( F+iH\right)\Delta Scal^{\nabla} \frac{\omega^n}{n!}.
\end{eqnarray*}
So, $\mathcal{F}^{\omega}$ is the imaginary part of $F_{\frac{8\pi^2}{(n-1)!}(c_2-\frac{1}{2}c_1. c_1)}$.
\end{proof}
\subsection{Example}
The computations of generalized Futaki invariants $F_q$ defined by (\ref{eq:Futakigendef}) is not an easy task. For $q=\mathrm{Td}_p$, the invariant polynomials defining the $p^{\mathrm{th}}$ Todd class, methods coming from algebraic geometry are developped to compute $F_{\mathrm{Td}_p}$, see \cite{VedZud,Ono}, in order to study the asymptotic semi-stability \cite{Fut} of the manifold. Those methods and this notion of asymptotic semi-stability are beyond the scope of this paper. However, when the manifold is K\"ahler-Einstein, as it is the case in \cite{Ono}, $F_{\mathrm{Td}_2}$ determines completely $\mathcal{F}^{\omega}$.
\begin{obs} \label{obs}
When $(M,\omega,J)$ is K\"ahler-Einstein, $\mathcal{F}^{\omega}$ is the imaginary part of $\frac{8\pi^2}{(n-1)!}F_{\mathrm{Td}_2}$.
\end{obs}
\begin{proof}
Recall that $\mathrm{Td}_2=c_2+c_1.c_1$. Now, because the Ricci form $\rho=\lambda \omega$, from the computations in the proof of Proposition \ref{prop:FequalF}, one has $F_{c_1.c_1}=0$. So that, $\frac{8\pi^2}{(n-1)!}F_{\mathrm{Td}_2}=F_{\frac{8\pi^2}{(n-1)!}(c_2-\frac{1}{2}c_1. c_1)}$ and its imaginary part is $\mathcal{F}^{\omega}$ by Proposition \ref{prop:FequalF}.
\end{proof}
In \cite{NP}, a 7-dimensional (complex dimension) smooth K\"ahler manifold $(V,\omega,J)$ is constructed, the so-called Nill-Paffenholz example. $V$ is a toric Fano manifold that is K\"ahler-Einstein, \cite{NP}. Moreover, Ono, Sano and Yotsutani \cite{Ono} showed that, on $V$, $F_{\mathrm{Td}_p} \ne 0$ for $2\leq p \leq 7$. Combined with the above Observation \ref{obs}, it means $\mathcal{F}^{\omega}\ne 0$. Consequently, Corollary \ref{cor:corprinc} implies:
\begin{theorem}
Let $(V,\omega,J)$ be the Nill-Paffenholz example \cite{NP} and $\Theta=[\omega]$, then there is no closed Fedosov's star products of the form $*_{\widetilde{\nabla}}$ for $\widetilde{\nabla}$ the Levi-Civita connection of some $\widetilde{\omega}\in \mathcal{M}_{\Theta}$.
\end{theorem}
|
3,212,635,537,989 | arxiv | \section{Introduction}
Heavy ion experiments at the Large Hadron Collider (LHC) and the Relativistic
Heavy Ion Collider (RHIC) have set out the goal to produce and study new forms
of strongly interacting matter, such as the quark gluon plasma.
Besides direct emissions, we can observe this matter at the point of break-up
through the hadrons leaving the system. Prominent approaches include
the hydrodynamical modelling of the angular distribution and the study
of the event-by-event distribution of conserved charges \cite{Hippolyte:2012yu}.
The chemical freeze-out, defined as the last inelastic scattering of
hadrons before detection, has already been studied in terms of the
statistical hadronization model by fitting a chemical potential and
a temperature parameter to the pion, kaon, proton and other accessible yields
from experiment \cite{Andronic:2005yp,Cleymans:2005xv}. For higher collision
energies smaller chemical potential are realized at freeze-out.
Repeating the analysis for a series of beam energies provide a manifold
of $(T-\mu)$ pairs on the phase diagram, the freeze-out curve
in Fig.~\ref{fig:phasediag}.
While we know from lattice simulations that the QCD transition is a crossover
at zero chemical potential \cite{Aoki:2006we}, a critical end point and a
first order transition line may exist in the ($T$-$\mu$) plane. Its
experimental search is based on the analysis of event-by-event fluctuations
\cite{Stephanov:1999zu}.
Parallel to the experimental effort lattice field theory has been able to
describe the QCD transition in an increasing detail. The transition temperature
has been determined \cite{Tctrilogy,Bazavov:2011nk}, and the curvature of the
transition line was also given \cite{Endrodi:2011gv}. The equation of state has
been calculated at zero \cite{Borsanyi:2010cj,Borsanyi:2013bia} and small
chemical potentials \cite{Borsanyi:2012cr}. Quark number susceptibilities have
also been determined both for strange as well as light flavors
\cite{Borsanyi:2011sw,Bazavov:2012jq}. All these results have been subject to a
continuum extrapolation.
\begin{figure}
\begin{center}
\includegraphics[width=3.5in]{phase_diag_cont}
\end{center}
\caption{\label{fig:phasediag}
The QCD phase diagram for small chemical potentials \cite{Endrodi:2011gv}.
A temperature and a chemical potential has been fitted in terms of the
statistical hadronization model for every collision energy
\cite{Andronic:2005yp,Cleymans:2005xv}. For comparison we show the crossover
lines based on two observables from lattice simulations \cite{Endrodi:2011gv}.
}
\end{figure}
The ever-increasing accuracy of fluctuation measurements at RHIC and LHC
allows us today to make direct comparisons of lattice results with data.
The STAR experiment has recently published the beam-energy and centrality
dependence of the net-proton distribution \cite{Adamczyk:2013dal}. For
the net electric charge distribution there are preliminary results available
both from the STAR \cite{McDonald:2012ts,Sahoo:2012bs} and from the
PHENIX collaboration \cite{Mitchell:2012mx}.
The strategy for a successful comparison between theory and experiment has been
long worked on \cite{Jeon:2000wg,Asakawa:2000wh,Karsch:2012wm}. Here we use
the observables suggested in Ref.~\cite{Bazavov:2012vg}.
The fluctuations for a conserved quantum number, such as electric charge,
are measured in a sub-system, small enough to behave like a grand canonical
ensemble, yet large enough to behave like an ensemble. The selection of a
subsystem is accomplished through cuts in rapidity and transverse momentum.
Still, the fluctuations or even the mean value of net charge depends on
the unknown subvolume. To cancel this factor ratios are considered, such
as mean/variance, which was described as a baryometer in
Refs.~\cite{Karsch:2012wm,Bazavov:2012vg}. Other relevant combinations
are listed in Eq.~(\ref{eq:observables}).
At zero chemical potential the mean and skewness vanish, leaving
us only with the kurtosis and variance to work with at the energies of LHC.
RHIC, however, works at non-zero chemical potentials. There we expand
the lattice results around zero chemical potential and extrapolate to small but
finite values and use then the mean and the skewness, which are now non-zero.
In Ref.~\cite{Bazavov:2012vg} these observables were used as
baryometer and thermometer, respectively.
The rules for such an extrapolation are given by the experimental setting:
there is no strangeness input in the colliding nuclei, and the ratio
of protons and neutrons in the gold or lead atoms predeterimne the
charge-to-baryon ratio in the outcoming hadrons as well. Thus:
\begin{eqnarray}
\left\langle S\right\rangle=0,&&
\left\langle Q\right\rangle=0.4\left\langle B\right\rangle.
\label{eq:constraint}
\end{eqnarray}
These conditions can be respected if we introduce a strange and electric charge
chemical potential in addition to the baryochemical potential, as it has already
been a method in the statistical hadronization model.
\section{Fluctuations from the lattice}
We generated finite temperature ensembles using the three-level Symanzik
improved gauge action with dynamical stout-improved staggered fermions (see
Ref.~\cite{Aoki:2005vt}. The temporal extent of the lattices determine
the lattice spacing at a given temperature, we use
$N_t=6,~8,~10,~12,~16$ (around $T_c$ these translate to the lattice spacings of
$a=0.22, 0.16, 0.13, 0.11$ and $0.08$~fm, respectively).
At every lattice spacing and temperature we stored and analyzed every 10th
configuration in the rational hybrid Monte Carlo streams.
\begin{figure}
\includegraphics[width=\textwidth]{kurtstatbins_16}
\caption{Statistics behind the fluctuation calculations.
The stored configurations have been separated by 10 HMC trajectories, each.
Each configuration was analyzed by $(128 \dots 256)\times 4$ random sources. \label{fig1}}
\end{figure}
In a grand canonical ensemble we obtain the fluctuations as derivatives of the partition function with respect to the chemical potentials:
\begin{eqnarray}
\frac{\chi_{lmn}^{BSQ}}{T^{l+m+n}}=\frac{\partial^{\,l+m+n}(p/T^4)}{\partial(\mu_{B}/T)^{l}\partial(\mu_{S}/T)^{m}\partial(\mu_{Q}/T)^{n}}.
\end{eqnarray}
and they are related to the moments of the distributions of the corresponding conserved charges
by
\begin{eqnarray}
\mathrm{ mean:}~~M=\chi_1~~&&~~\mathrm{ variance:}~~\sigma^2=\chi_2
\nonumber
\\
\mathrm{ skewness:}~~S=\chi_3/\chi_{2}^{3/2}
~~&&~~
\mathrm{kurtosis:}~~\kappa=\chi_4/\chi_{2}^{2}\,.
\label{eq:observables}
\end{eqnarray}
With these moments we can express the volume independent ratios
\begin{eqnarray}
~S\sigma=\chi_3/\chi_{2}
\quad&;&\quad
\kappa\sigma^2=\chi_4/\chi_{2}\nonumber\\
M/\sigma^2=\chi_1/\chi_2
\quad&;&\quad
S\sigma^3/M=\chi_3/\chi_1\,.
\label{moments}
\end{eqnarray}
The chemical potential dependence enters through the fermion determinant ($\det
M_i$), allowing for one $\mu_i$ parameter for each of the three dynamical
flavor $i=u,d,s$. The actual observables are based on the derivatives of the logarithm of these determinants:
\begin{eqnarray}
A_j&=\frac{d}{d\mu_j} \log(\det M_j)^{1/4} = &\trt M_j^{-1} M_j'\,,\\
B_j&=\frac{d^2}{(d\mu_j)^2} \log(\det M_j)^{1/4} =&\trt\left(
M_j'' M_j^{-1}
-M_j' M_j^{-1} M_j' M_j^{-1}
\right)\,,\\
C_j&=\frac{d^3}{(d\mu_j)^3} \log(\det M_j)^{1/4} =&\trt\left(
M_j' M_j^{-1}
-3 M_j'' M_j^{-1} M_j' M_j^{-1}\right.\nonumber\\
&&
\left.
+2 M_j' M_j^{-1} M_j' M_j^{-1} M_j' M_j^{-1}
\right)\,,\\
D_j&=\frac{d^4}{(d\mu_j)^4} \log(\det M_j)^{1/4} = &\trt\left(
M_j'' M_j^{-1}
-4 M_j' M_j^{-1} M_j' M_j^{-1}
+12 M_j'' M_j^{-1} M_j' M_j^{-1} M_j' M_j^{-1}\right.\nonumber\\
&&
\left.
-3 M_j'' M_j^{-1} M_j'' M_j^{-1}
-6 M_j' M_j^{-1} M_j' M_j^{-1} M_j' M_j^{-1}M_j' M_j^{-1}
\right)\,.
\end{eqnarray}
We calculate these traces for every configuration using
$(128\dots256)\times 4$ random sources. The final derivatives emerge as
connected and disconnected contributions, e.g. to second order we have
\begin{eqnarray}
\partial_i\partial_j \log Z&=&
\avr{A_iA_j}+\delta_{ij}\avr{B_i}\,.
\end{eqnarray}
Where products of diagrams appear, a disjoint set of random sources are
used, like here in $A_i$ and $A_j$, even when $i=j$. The first (disconnected)
term is responsible for most of the noise, lattice artefacts, on the other
hand, come mainly from the connected contributions.
\section{Results}
The quantities that we look at, in order to extract the freeze-out temperature
and baryon chemical potential, are the ratios
$R_{31}^Q(T,\mu_B)=\chi_3^Q/\chi_{1}^Q$ and
$R_{12}^Q(T,\mu_B)=\chi_1^Q/\chi_{2}^Q$ for small chemical potentials,
where $\mu_Q(\mu_B)$ and $\mu_S(\mu_B)$ are chosen to satisfy
Eqs.~(\ref{eq:constraint}). We also calculated the analogous baryon
fluctuations. For details, see the journal version of this work
\cite{Borsanyi:2013hza}.
\begin{figure}
\begin{center}
\includegraphics[width=2.9in]{Q3Q1_v2}\hfil
\includegraphics[width=2.9in]{B3B1}
\end{center}
\caption{
Lattice results on the skewness ratio for the charge (left) and the baryon
number (right). The colored symbols correspond to lattice QCD simulations at
finite-$N_t$. Black points correspond to the continuum extrapolation
\cite{Borsanyi:2013hza}; blue
pentagons are the $N_t=8$ results from the BNL-Bielefeld collaboration
\cite{Bazavov:2012vg}
\label{fig:RQ31}}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=2.9in]{Q1Q2mu_v2}\hfil
\includegraphics[width=2.9in]{B1B2mu}
\end{center}
\caption{$R_{12}^Q$ as a function of $\mu_B$: the different colors correspond
to the continuum extrapolated lattice QCD results, calculated in a range of
temperatures around the QCD crossover \cite{Borsanyi:2013hza}. \label{fig:R12Q}}
\end{figure}
In Fig. \ref{fig:RQ31} we show the ratios $R_{31}^Q$ (left) and $R_{31}^B$
(right) as a function of the temperature. The continuum extrapolations are
shown as black dots. For the charge fluctuations we used five lattice spacings.
Baryon fluctuations are plagued by greater noise, but are less sensitive to
cut-off effects, here we used four spacings. Charge fluctuation results from
the BNL-Bielefeld collaboration corresponding to $N_t=8$ (from
Ref.~\cite{Bazavov:2012vg}) are also shown for comparison.
In Fig. \ref{fig:R12Q} we show our results for $R_{12}^Q$ as a function of the
baryon chemical potential: the different curves correspond to different
temperatures, in the range where freeze-out is expected. Such expectations
may come from the arguments in Ref.~\cite{BraunMunzinger:2003zz} supporting
a freeze-out just below the transition. Alternative hints come from the
existing estimates from the statistical hadronization model
\cite{Andronic:2005yp,Cleymans:2005xv}. Similarly to the electric charge
fluctuations, $R_{31}^B$ will allow us to constrain the temperature and using
$R_{12}^B$ we can then obtain $\mu_B$.
Notice that the ordering of the temperatures in Fig.~\ref{fig:R12Q} (left) and
(right) is opposite. Thus, whether the chemical potentials from the charge
and the baryon (proton) fluctuations deliver consistent results will very
much depend on the associated temperature, which we can extract from the
skewness analysis. A possible source for inconsistencies might be
the comparison of proton fluctuation data with baryon fluctuations from the
lattice, and also the remnant effects of baryon number conservation
\cite{Bzdak:2012an}. A cross-check between the freeze-out parameters from
proton and electric charge data also test the basic assumption of equilibrium
at the time of freeze-out.
\begin{figure}
\begin{center}
\includegraphics[width=2.9in]{B4B2}\hfil
\includegraphics[width=2.9in]{kurt_LS_cont}
\end{center}
\caption{\label{fig:kurtosis}
The baryon number (left) and flavor specific (right) kurtosis
($\kappa\times \sigma^2$) prediction from lattice QCD. These parameters are
in principle accessible to LHC experiments, and may be used to define
the freeze-out temperature for specific flavors, or to the system as a whole.
}
\end{figure}
Finally we show the kurtosis data in the continuum limit in
Fig.~\ref{fig:kurtosis}. The kurtosis of baryon number and light vs. strange
quark numbers show different sensitivity to temperature, so are the maxima
and the deviation point from the hadron resonance gas prediction flavor
dependent. The great question that the experiment will have to decide is
whether the freeze-out temperatures themselves are flavor dependent
\cite{Bellwied:2013cta}.
\textbf{Acknowledgments:}
This project was funded by the DFG grant SFB/TR55.
The work of C. Ratti is supported by funds provided by the Italian Ministry of
Education, Universities and Research under the
Firb Research Grant RBFR0814TT.
S. D. Katz is funded by the ERC grant ((FP7/2007-2013)/ERC No 208740)
as well as the "Lend\"ulet" program of the Hungarian Academy of Sciences
((LP2012-44/2012).
The numerical simulations were in part performed the GPU cluster at the
Wuppertal University as well as on QPACE, funded by the DFG. We acknowledge
PRACE for awarding us access to the Blue Gene/Q system (JUQUEEN) at
Forschungszentrum J\"ulich, Germany.
|
3,212,635,537,990 | arxiv | \section{Introduction}
Network embedding (a.k.a. network representation learning) has gained a tremendous amount of interest among the researchers in the last few years \cite{perozzi2014deepwalk,grover2016node2vec}. Most of the real life networks have some extra information within each node. For example, users in social networks such as Facebook have texts, images and other types of content. Research papers (nodes) in a citation network have scientific content in it. Typically this type of extra information is captured using attribute vectors associated with each node.
The attributes and the link structure of the network are highly correlated according to the sociological theories like homophily \cite{mcpherson2001birds}.
But embedding attributed networks is challenging as combining attributes to generate node embeddings is not easy. Towards this end, different attributed network representation techniques such as \cite{yang2015network,huang2017accelerated,gao2018deep} have been proposed in literature. They perform reasonably well when the network is consistent in its structure and content, and nodes behave as expected.
Unfortunately real world networks are noisy and there are different outliers which even affect the embeddings of normal nodes \cite{liu2017accelerated}. For example, there can be research papers in a citation network with few spurious references (i.e., edges) which do not comply with the content of the papers. There are celebrities in social networks who are connected to too many other users, and generally properties like homophily are not applicable to this type of relationships. So they can also act like potential outliers in the system. Normal nodes are consistent in their respective communities both in terms of link structure and attributes. We categorize outliers in an attributed network into three categories and explain them as shown in Figure \ref{fig:outliers}.
One way to detect outliers in the network is to use some network embedding approach and then use algorithms like isolation forest \cite{liu2008isolation} on the generated embeddings. But this type of decoupled approach is not optimal as outliers adversely affect the embeddings of the normal nodes. So an integrated approach to detect outliers and minimize their effect while generating the network embedding is needed. Recently \cite{liang2018semi} proposes a semi supervised approach for detecting outliers while generating network embedding for an attributed network. But in principle, it needs some supervision to work efficiently. For real world networks, it is difficult to get such supervision or node labels. So there is a need to develop a completely unsupervised integrated approach for graph embedding and outlier detection which can be applicable to any attributed network.
\begin{figure
\centering
\begin{subfigure}{0.31\linewidth}
\centering
\includegraphics[width=\linewidth]{images/type1.png}
\caption{}
\end{subfigure}
\begin{subfigure}{0.31\linewidth}
\centering
\includegraphics[width=\linewidth]{images/type2.png}
\caption{}
\end{subfigure}
\begin{subfigure}{0.31\columnwidth}
\centering
\includegraphics[width=\linewidth]{images/type3.png}
\caption{}
\end{subfigure}
\caption{This shows different types of outliers that we consider in an attributed network. We highlight the outlier node and its associated attribute by larger circle and rectangle respectively. Different colors represent different communities. Arrows between the two nodes represent network edges and arrows between two attributes represent similarity (in some metric) between them. (a) \textbf{Structural Outlier}: The node has edges to nodes from different communities, i.e., its structural neighborhood is inconsistent. (b) \textbf{Attribute Outlier}: The attributes of the node is similar to attributes of the nodes from different communities, i.e., its attribute neighborhood is inconsistent. (c) \textbf{Combined Outlier}: Node belongs to a community structurally but it has a different community in terms of attribute similarity.
\label{fig:outliers}
\end{figure}
\textbf{Contributions}: Following are the contributions we make.
\begin{itemize}
\item We propose an unsupervised algorithm called ONE (\textbf{O}utlier aware \textbf{N}etwork \textbf{E}mbedding) for attributed networks. It is an iterative approach to find lower dimensional compact vector representations of the nodes, such that the outliers contribute less to the overall cost function.
\item This is the first work to propose a completely unsupervised algorithm for attributed network embedding integrated with outlier detection. Also we propose a novel method to combine structure and attributes efficiently.
\item We conduct a thorough experimentation on the outlier seeded versions of popularly used and publicly available network datasets to show the efficiency of our approach to detect outliers. At the same time by comparing with the state-of-the-art network embedding algorithms, we demonstrate the power of ONE as a generic embedding method which can work with different downstream machine learning tasks such as node clustering and node classification.
\end{itemize}
\section{Related Work}\label{sec:rw}
This section briefs the existing literature on attributed network embedding, and some outlier detection techniques in the context of networks. Network embedding has been a hot research topic in the last few years and a detailed survey can be found in \cite{hamilton2017representation}.
Word embedding in natural language processing literature, such as \cite{mikolov2013efficient} inspired the development of node embedding in network analysis.
DeepWalk \cite{perozzi2014deepwalk}, node2vec \cite{grover2016node2vec} and Line \cite{tang2015line} gained popularity for network representation just by using the link structure of the network. DeepWalk and node2vec use random walk on the network to generate node sequences and feed them to language models to get the embedding of the nodes. In Line, two different objective functions have been used to capture the first and second order proximities respectively and an edge sampling strategy is proposed to solve the joint optimization for node embedding. In \cite{ribeiro2017struc2vec}, authors propose struc2vec where nodes having similar substructure are close in their embeddings.
All the papers citeed above only consider link structure of the network for generating embeddings. Research has been conducted on attributed network representation also. TADW \cite{yang2015network} is arguably the first attempt to successfully use text associated with nodes in the network embedding via joint matrix factorization. But their framework directly learns one embedding from content and structure together. In case when there is noise or outliers in structure or content, such a direct approach is prone to be affected more. Another attributed network embedding technique (AANE) is proposed in \cite{huang2017accelerated}. The authors have used symmetric matrix factorization to get embeddings from the similarity matrix over the attributes, and use link structure of the network to ensure that the embeddings of the two connected nodes are similar. A semi-supervised attributed embedding is proposed in \cite{huang2017label} where the label information of some nodes are used along with structure and attributes. The idea of using convolutional neural networks for graph embedding has been proposed in \cite{niepert2016learning,kipf2016semi}. An extension of GCN with node attributes (GraphSage) has been proposed in \cite{hamilton2017inductive} with an inductive learning setup. These methods do not manage outliers directly, and hence are often prone to be affected heavily by them.
Recently a semi-supervised deep learning based approach SEANO \cite{liang2018semi} has been proposed for outlier detection and network embedding for attributed networks. For each node, they collect its attribute and the attributes from the neighbors, and smooth out the outliers by predicting the class labels (on the supervised set) and node context. But getting labeled nodes for real world network is expensive. So we aim to design an unsupervised attributed network embedding algorithm which can detect and minimize the effect of outliers while generating the node embeddings.
\section{Problem Formulation} \label{sec:prob}
An information network is typically represented by a graph as $\mathcal{G} = (V, E, C)$, where $V=\{v_1, v_2,\cdots, v_N\}$ is the set of nodes (a.k.a. vertexes), each representing a data object. $E \subset \{(v_i,v_j) | v_i,v_j \in V \}$ is the set of edges between the vertexes.
Each edge $e \in E$ is an ordered pair $e = (v_i, v_j)$ and is associated with a weight $w_{v_i,v_j} > 0$, which indicates the strength of the relation. If $\mathcal{G}$ is undirected, we have $(v_i, v_j) \equiv (v_j, v_i)$ and $w_{v_i,v_j} \equiv w_{v_j,v_i}$; if $\mathcal{G}$ is unweighted, $w_{v_i,v_j} = 1$, $\forall (v_i,v_j) \in E$.
Let us denote the $N \times N$ dimensional adjacency matrix of the graph $\mathcal{G}$ by $A = (a_{i,j})$, where $a_{i,j}=w_{v_i,v_j}$ if $(v_i,v_j) \in E$, and $a_{i,j}=0$ otherwise. So $i$th row of $A$ contains the immediate neighborhood information for node $i$. Clearly for a large network, the matrix $A$ is highly sparse in nature.
$C$ is a $N \times D$ matrix with $C_{i \cdot}$ as rows, where $C_{i \cdot} \in \mathbb{R}^D$ is the attribute vector associated with the node $v_i \in V$. $C_{id}$ is the value of the attribute $d$ for the node $v_i$. For example, if there is only textual content in each node, $c_i$ can be the tf-idf vector for the content of the node $v_i$.
Our goal is to find a low dimensional representation of $\mathcal{G}$ which is consistent with both the structure of the network and the content of the nodes. More formally, for a given network $\mathcal{G}$, network embedding is a technique to learn a function $f : v_i \mapsto \mathbf{y_i} \in \mathbb{R}^K$, i.e., it maps every vertex to a $K$ dimensional vector, where $K < min(N,D)$. The representations should preserve the underlying semantics of the network. Hence the nodes which are close to each other in terms of their topographical distance or similarity in attributes should have similar representations. We also need to reduce the effect of outliers, so that the representations for the other nodes in the network are robust.
\section{Solution Approach: ONE}
We describe the whole algorithm in different parts.
\subsection{Learning from the Link Structure}
Given graph $\mathcal{G}$, each node $v_i$ by default can be represented by the $i$th row $A_{i\cdot}$ of the adjacency matrix. Let us assume the matrix $G$ $\in \mathbb{R}^{N \times K}$ be the network embedding of $\mathcal{G}$, only by considering the link structure. Hence row vector $G_{i\cdot}$ is the $K$ dimensional ($K < min(N,D)$) compact vector representation of node $v_i$, $\forall v_i \in V$. Also let us introduce a $K \times N$ matrix $H$ to minimize the reconstruction loss: $\sum\limits_{i=1}^N\sum\limits_{j=1}^N (A_{ij} - G_{i\cdot} \cdot H_{\cdot j})^2$, where $H_{\cdot j}$ is the $j$th column of $H$, and $G_{i\cdot} \cdot H_{\cdot j}$ is the dot product between these two vectors\footnote[1]{We treat both row vector and column vector as vectors of same dimension, and hence use the dot product instead of transpose to avoid cluttering of notation}. $k$th row of $H$ can be interpreted as the $N$ dimensional description of $k$th feature, where $k = 1,2,\cdots,K$. This reconstruction loss tends to preserve the original distances in the lower dimensional spaces as shown by \cite{cunningham2015}. But if the graph has anomalous nodes, they generally affect the embedding of the other (normal) nodes. To minimize the effect of such outliers while learning embeddings from the structure, we introduce the structural outlier score $O_{1i}$ for node $v_i \in V$, where $0 < O_{1i} \leq 1$. The bigger the value of $O_{1i}$, the more likely it is that node $v_i$ is an outlier, and lesser should be its contribution to the total loss. Hence we seek to minimize the following cost function w.r.t. the variables $O_1$ (set of all structural outlier scores), $G$ and $H$.
\begin{align}\label{eq:L1}
\mathcal{L}_{str} = \sum_{i=1}^N\sum_{j=1}^N\log\Big({\frac{1}{O_{1i}}}\Big) (A_{ij} - G_{i\cdot} \cdot H_{\cdot j})^2
\end{align}
We also assume $\sum\limits_{i=1}^N O_{1i} = \mu$, $\mu$ being the total outlier score of the network. Otherwise minimizing Eq.
\ref{eq:L1} amounts to assigning $1$ to all the outlier scores, which makes the loss value 0. It can be readily seen that, when $O_{1i}$ is very high (close to 1) for a node $v_i$, the contribution of this node $\sum\limits_{j=1}^N\log\Big({\frac{1}{O_{1i}}}\Big) (A_{ij} - G_{i.} \cdot H_{.j})^2$ becomes negligible, and when $O_{1i}$ is small (close to 0), the corresponding contribution is high. So naturally, the optimization would concentrate more on minimizing the contributions of the outlier (w.r.t. the link structure) nodes, to the overall objective, as desired.
\subsection{Learning from the Attributes}
Similar to the case of structure, here we try to learn a $K$ dimensional vectorial representation $U_{i\cdot}$ from the given attribute matrix $C$, where $C_{i\cdot}$ is the attribute vector of node $v_i$. Let us consider the matrices $U \in \mathbb{R}^{N \times K}$ and $V \in \mathbb{R}^{K \times D}$, $U$ being the network embedding just respecting the set of attributes. In the absence of outliers, one can just minimize the reconstruction loss $\sum\limits_{i=1}^N\sum\limits_{d=1}^D (C_{id} - U_{i\cdot} \cdot V_{\cdot d})^2$ with respect to the matrices $U$ and $V$. But as mentioned before, outliers even affect the embeddings of the normal nodes. Hence to reduce the effect of outliers while learning from the attributes, we introduce the attribute outlier score $O_{2i}$ for node $v_i \in V$, where $0 < O_{2i} \leq 1$. Larger the value of $O_{2i}$, higher the chance that node $v_i$ is an attribute outlier. Hence we minimize the following cost function w.r.t. the variables $O_2$, $U$ and $V$.
\begin{align}\label{eq:L2}
\mathcal{L}_{attr} = \sum_{i=1}^N\sum_{d=1}^C\log\Big({\frac{1}{O_{2i}}}\Big) (C_{id} - U_{i \cdot} \cdot V_{\cdot d})^2
\end{align}
We again assign the constraint that $\sum\limits_{i=1}^N O_{2i} = \mu$ for the reason mentioned before. Hence contributions from the non-outlier (w.r.t. attributes) nodes would be bigger while minimizing Eq. \ref{eq:L2}.
\subsection{Connecting Structure and Attributes}
So far, we have considered the link structure and the attribute values of the
network separately. Also the optimization variables of Eq. \ref{eq:L1} and that in Eq. \ref{eq:L2} are completely disjoint. But optimizing them independently is not desirable as, our ultimate goal is to get a joint low dimensional representation of each node in the network. Also we intend to regularize structure with respect to attributes and vice versa. As discussed before, link structure and attributes in a network are highly correlated and they can be often noisy individually.
One can see that, $G_{i \cdot}$ and $U_{i \cdot}$ are the representation of the same node $v_i$ with respect to structure and attributes respectively. So one can easily act as a regularizer of the other. Also as they contribute to the embedding of the same node, it makes sense to minimize $\sum\limits_{i=1}^N \sum\limits_{k=1}^K (G_{ik} - U_{ik})^2$. But it is important to note that, there is no explicit guarantee that the features in $G_{i \cdot}$ and features in $U_{i \cdot}$ are aligned, i.e., $k$th feature of the structure embeddings can be very different from the $k$th feature of attribute embeddings. Hence before minimizing the distance of $G_{i \cdot}$ and $U_{i \cdot}$, it is important to align the features of the two embedding spaces.
\subsubsection{Embedding Transformation and Procrustes problem}
To resolve the issue above, we seek to find a linear map $W \in \mathbb{R^{K \times K}}$ which transforms the features from the attribute embedding space to structure embedding space. More formally we want to find a matrix $W$ which minimizes $||G - WU||_F$. This type of transformation has been used in the NLP literature, particularly for machine translation \cite{lample2018word}.
If we further restrict $W$ to be an orthogonal matrix, then a closed form solution can be obtained from the solution concept of Procrustes problem \cite{schonemann1966generalized} as follows:
\begin{align}\label{eq:W}
W^*
= \underset{W \in \mathcal{O}_K}{\text{argmin}} ||G - UW^T||_F
\end{align}
where $W^* = XY^T$ with $X \Sigma Y^T = \text{SVD}(G^T U)$, $\mathcal{O}_K$ is the set of all orthogonal matrices of dimension $K \times K$. Restricting $W$ to be an orthogonal matrix has also several other advantages as shown in the NLP literature \cite{xing2015normalized}.
But we cannot directly use the solution of Procrustes problem, as we have anomalies in the network.
As before, we again reduce the effect of anomalies to minimize the disagreement between the structural embeddings and attribute embeddings. Let us introduce the disagreement anomaly score $O_{3i}$ for a node $v_i \in V$, where $0 < O_{3i} \leq 1$. Disagreement anomalies are required to manage the anomalous nodes which are not anomalies in either of structure or attributes individually, but they are inconsistent when considering them together. Following is the cost function we minimize.
\begin{align}\label{eq:L3}
\mathcal{L}_{dis} = \sum\limits_{i=1}^N \sum\limits_{k=1}^K \log\Big({\frac{1}{O_{3i}}}\Big) \Big(G_{ik} - U_{i\cdot} \cdot (W^T)_{\cdot k}\Big)^2
\end{align}
$\sum\limits_{i=1}^N O_{3i}= \mu$. We will use the solution of Procrustes problem after applying a simple trick to the cost function above, as shown in the derivation of the update rule of $W$ later.
\subsection{Joint Loss Function}
Here we combine the three cost functions mentioned before, and minimize the following with respect to $G$, $H$, $U$, $V$, $W$ and $O$ ($O$ contains all the variables from $O_1$, $O_2$ and $O_3$).
\begin{align}\label{eq:L}
\mathcal{L} = \mathcal{L}_{str} + \alpha \mathcal{L}_{attr} + \beta \mathcal{L}_{dis}
\end{align}
The full set of constrains are:
\[
0 < O_{1i}, O_{2i}, O_{3i} \leq 1 \; , \; \forall v_i \in V
\]
\[
\sum_{i=1}^{N}O_{1i} = \sum_{i=1}^{N}O_{2i} = \sum_{i=1}^{N}O_{3i} = \mu
\]
\[
W \in \mathcal{O}_{K} \iff W^{T}W = \mathcal{I}
\]
Here $\alpha, \beta > 0$ are weight factors. We will discuss a method to set them in the experimental evaluation. It is to be noted that, the three anomaly scores $O_{1i}$, $O_{2i}$ and $O_{3i}$ for any node $v_i$ are actually connected by the cost function \ref{eq:L}. For example, if a node is anomalous in structure, $O_{1i}$ would be high and its embedding $G_{i\cdot}$ may not be optimized well. So this in turn affects its matching with transformed $U_{i\cdot}$, and hence $O_{3i}$ would be given a higher value to minimize the disagreement loss.
\subsection{Derivations of the Update Rules}
We will derive the necessary update rules which can be used iteratively to minimize Eq. \ref{eq:L}.
We use the alternating minimization technique, where we derive the update rule for one variable at a time, keeping all others fixed.
\subsection{Updating $G$, $H$, $U$, $V$}
We need to take the partial derivative of $\mathcal{L}$ (Eq. \ref{eq:L}) w.r.t one variable at a time and equate that to zero. For example, $\frac{\partial \mathcal{L}}{\partial G_{ik}} = 0 \Rightarrow \sum\limits_{j=1}^N \log \big(\frac{1}{O_{1i}}\big) (A_{ij} - G_{i\cdot} \cdot H_{\cdot j})(-H_{kj})
+ \log \big(\frac{1}{O_{3i}}\big) (G_{ik} - U_{i\cdot}\cdot (W^T)_{\cdot k}) = 0$.
Solving it for $G_{ik}$,
\begin{empheq}[box=\fbox]{align}\label{eq:G}
\fontsize{7.0pt}{7.5pt} \selectfont
G_{ik} = \frac{
G_{ik}^{num1} + \beta \log\Big({\frac{1}{O_{3i}}}\Big)(W_{k \cdot}\cdot U_{i\cdot})
}
{
\log\Big({\frac{1}{O_{1i}}}\Big)(H_{k \cdot}\cdot H_{k \cdot}) + \beta \log\Big({\frac{1}{O_{3i}}}\Big)
} \\
G_{ik}^{num1} = \log\Big({\frac{1}{O_{1i}}}\Big)\sum_{j=1}^N(A_{ij} - \sum_{k^{'} \neq k}G_{ik^{'}}H_{k^{'}j})H_{kj}
\nonumber
\end{empheq}
\vskip-22mm
Similarly we can get the following update rules.
\begin{align}\label{eq:H}
\boxed{
H_{kj} = \frac{
\sum_{i=1}^N\log\Big({\frac{1}{O_{1i}}}\Big)(A_{ij} - \sum_{k^{'} \neq k}G_{ik^{'}}H_{k^{'}j})G_{ik}
}
{
\sum_{i=1}^N\log\Big({\frac{1}{O_{1i}}}\Big)G_{ik}^{2}
}
}
\end{align}
\begin{empheq}[box=\fbox]{align}\label{eq:U}
\tiny
U_{ik} = \frac{
U_{ik}^{num1} + U_{ik}^{num2}
}
{
\beta\log\Big({\frac{1}{O_{2i}}}\Big)(V_{k\cdot}\cdot V_{k\cdot}) + \gamma\log\Big({\frac{1}{O_{3i}}}\Big)W_{\cdot k}\cdot W_{\cdot k}
} \\
U_{ik}^{num1} = \beta\log\Big({\frac{1}{O_{2i}}}\Big)\sum_{d=1}^D(C_{id} - \sum_{k^{'} \neq k}U_{ik^{'}}V_{k^{'}d})V_{kd} \nonumber \\
U_{ik}^{num2} = \gamma\log\Big({\frac{1}{O_{3i}}}\Big)\Big(G_{i\cdot} - (U_{i\cdot}W) - (U_{ik}\cdot W_{\cdot k})\Big) \nonumber
\end{empheq}
\vskip-28mm
\begin{align}\label{eq:V}
\boxed{
V_{kd} = \frac{
\sum_{i=1}^N\log\Big({\frac{1}{O_{2i}}}\Big)(C_{id} - \sum_{k^{'} \neq k}U_{ik^{'}}V_{k^{'}d})U_{ik}
}
{
\sum_{i=1}^N\log\Big({\frac{1}{O_{2i}}}\Big)U_{ik}^{2}
}
}
\end{align}
\subsection{Updating $W$}
We use a small trick to directly apply the closed form solution of Procrustes problem as follows.
\begin{align}
\mathcal{L}_{dis} &= \sum\limits_{i=1}^N \sum\limits_{k=1}^K \log\Big({\frac{1}{O_{3i}}}\Big) \Big(G_{ik} - U_{i\cdot} \cdot (W^T)_{\cdot k}\Big)^2 \nonumber \\
&= \sum\limits_{i=1}^N \sum\limits_{k=1}^K \Big(\bar{G}_{ik} - \bar{U}_{i\cdot} \cdot (W^T)_{\cdot k}\Big)^2
\end{align}
Here the new matrices are defined as, $(\bar{G})_{i,k} = \sqrt[]{\log\Big({\frac{1}{O_{3i}}}\Big)} G_{ik}$ and $\bar{U}_{ik} = \sqrt[]{\log\Big({\frac{1}{O_{3i}}}\Big)} U_{ik}$. Say, $X \Sigma Y^T = \text{SVD}(\bar{G}^T \bar{U})$, then $W$ can be obtained as:
\begin{align}\label{eq:W}
\boxed{
W = XY^T
}
\end{align}
\subsection{Updating $O$}
We derive the update rule for $O_1$ first. Taking the Lagrangian of Eq. \ref{eq:L1} with respect to the constraint $\sum\limits_{i=1}^N O_{1i} = \mu$, we get,
\begin{align}
& \frac{\partial }{\partial O_{1i}} \sum\limits_{i,j}\log\Big({\frac{1}{O_{1i}}}\Big) (A_{ij} - G_{i\cdot} \cdot H_{\cdot j})^2 + \lambda(\sum\limits_{i} O_{1i} - \mu) \nonumber
\end{align}
$\lambda \in \mathbb{R}$ is the Lagrangian constant. Equating the partial derivative w.r.t. $O_{1i}$ to 0:
\begin{align*}
&-\frac{(A_{ij} - G_{i\cdot} \cdot H_{\cdot j})^2}{O_{1i}} + \lambda = 0, \;
\Rightarrow O_{1i} = \frac{(A_{ij} - G_{i\cdot} \cdot H_{\cdot j})^2}{\lambda}
\end{align*}
So, $\sum\limits_{i=1}^N O_{1i} = \mu$ implies $\sum\limits_{i=1}^N \frac{(A_{ij} - G_{i\cdot} \cdot H_{\cdot j})^2}{\lambda} = \mu$. Hence,
\begin{align}\label{eq:O1}
\boxed{
O_{1i} = \frac{
\Big(\sum_{j=1}^N(A_{ij} - G_{i\cdot}\cdot H_{\cdot j})^2 \Big) \cdot \mu
}
{
\sum_{i=1}^N\sum_{j=1}^N(A_{ij} - G_{i\cdot}\cdot H_{\cdot j})^{2}
}
}
\end{align}
It is to be noted that, if we set $\mu = 1$, the constraints $0 < O_{i1} \leq 1$, $\forall v_i \in V$, are automatically satisfied. Even it is possible to increase the value of $\mu$ by a trick similar to \cite{gupta2012integrating}, but experimentally we have not seen any advantage in increasing the value of $\mu$. Hence, we set $\mu = 1$ for all the reported experiments. A similar procedure can be followed to derive the update rules for $O_2$ and $O_3$.
\begin{align}\label{eq:O2}
\boxed{
O_{2i} = \frac{
\Big(\sum_{d=1}^D(C_{id} - U_{i\cdot}\cdot V_{\cdot d})^2 \Big) \cdot \mu
}
{
\sum_{i=1}^N\sum_{j=1}^D(C_{id} - U_{i\cdot}\cdot V_{\cdot d})^{2}
}
}
\end{align}
\begin{align}\label{eq:O3}
\boxed{
O_{3i} = \frac{
\Big( \sum_{k=1}^K(G_{ik} - W_{i\cdot}\cdot U_{\cdot k})^2 \Big) \cdot \mu
}
{
\sum_{i=1}^N\sum_{k=1}^K(G_{ik} - W_{i\cdot}\cdot U_{\cdot k})^2
}
}
\end{align}
\subsection{Algorithm: ONE}\label{sec:algOne}
With the update rules derived above, we summarize ONE in Algorithm \ref{alg:oane}. variables $G$, $H$, $U$ and $V$ can be initialized by any standard matrix factorization technique ($A \approx GH$ and $C \approx UV$) such as \cite{lee2001algorithms}. As our algorithm is completely unsupervised, we assume not to have any prior information about the outliers. So initially we set equal outlier scores to all the nodes in the network and normalize them accordingly. At the end of the algorithm, one can take the final embedding of a node as the average of the structural and the transformed attribute embeddings. Similarly the final outlier score of a node can be obtained as the weighted average of three outlier scores.
\begin{lemma}
The joint cost function in Eq. \ref{eq:L} decreases after each iteration (steps 4 to 6) of the for loop of Algorithm \ref{alg:oane}.
\end{lemma}
\begin{proof}
It is easy to check that the joint loss function $\mathcal{L}$ is convex in each of the variables $G,H,U,V$ and $O$, when all other variables are fixed. Also from the Procrustes solution, update of $W$ also minimizes the cost function. So alternating minimization guarantees decrease of cost function after every update till convergence.
\end{proof}
The computational complexity of each iteration (Steps 4 to 6 in Algo. \ref{alg:oane}) takes $O(N^2)$ time (assuming $K$ is a constant) without using any parallel computation, as updating each variable $G_{ik}, H_{kj}, V_{kd}, O_{1i}, O_{2i}, O_{3i}$ and $W$ takes $O(N)$ time. But we observe that ONE converges very fast on any of the datasets, as updating one variables amounts to reaching the global minima of the corresponding loss function when all other variables are fixed. The run time can be improved significantly by parallelizing the computation as done in \cite{huang2017accelerated}.
\begin{algorithm}
\caption{\textbf{ONE}}
\label{alg:oane}
\begin{algorithmic}[1]
\Statex \textbf{Input}: The network $\mathcal{G}=(V,E,C)$, $K$: Dimension of the embedding space where $K<min(n,d)$, ratio parameter $\theta$
\Statex \textbf{Output}: The node embeddings of the network $G$, Outlier score of each node $v in V$
\State Initialize $G$ and $H$ by standard matrix factorization on $A$, and $U$ and $V$ by that on $C$.
\State Initialize the outlier scores $O_1$, $O_2$ and $O_3$ uniformly.
\For{until stopping condition satisfied}
\State Update $W$ by Eq. \ref{eq:W}.
\State Update $G$, $H$, $U$ and $V$ by Eq. from \ref{eq:G} to \ref{eq:V}.
\State Update outlier scores by Eq. from \ref{eq:O1} to \ref{eq:O3}.
\EndFor
\State Embedding for the node $v_i$ is $\frac{G_{i\cdot} + U_{i\cdot} (W^T)}{2}$, $\forall v_i \in V$.
\State Final outlier score for the node $v_i$ is a weighted average of $O_{1i}$, $O_{2i}$ and $O_{3i}$, $\forall v_i \in V$.
\end{algorithmic}
\end{algorithm}
\section{Experimental Evaluation}
In this section, we evaluate the performance of the proposed algorithm on multiple attributed network datasets and compare the results with several state-of-the-art algorithms.
\subsection{Datasets Used and Seeding Outliers}
To the best of our knowledge, there is no publicly available attributed networks with ground truth outliers available. So we take four publicly available attributed networks with ground truth community membership information available for each node. The datasets are WebKB, Cora, Citeseer and Pubmed\footnote[2]{Datasets: \url{https://linqs.soe.ucsc.edu/data}}.
To check the performance of the algorithms in the presence of outliers, we manually planted a total of 5\% outliers (with equal numbers for each type as shown in Figure \ref{fig:outliers}) in each dataset. The seeding process involves: (1) computing the probability distribution of number of nodes in each class, (2) selecting a class using these probabilities. For a structural outlier: (3) plant an outlier node in the selected class such that the node has ($m$ $\pm$ 10\%) of edges connecting nodes from the remaining (unselected) classes where $m$ is the average degree of a node in the selected class and (4) the content of the structural outlier node is made semantically consistent with the keywords sampled from the nodes of the selected class.
A similar approach is employed for seeding the other two types of outliers.
The statistics of these seeded datasets are given in Table \ref{tab:data}.
Outlier nodes apparently have similar characteristics in terms of degree, number of nonzero attributes, etc., and thus we ensured that they cannot be detected trivially.
\begin{table}
\caption{Summary of the datasets (after planting outliers).}
\centering
\resizebox{0.8\columnwidth}{!}{%
\begin{tabular}{*5c}
\toprule
\sffamily{Dataset} & \#Nodes & \#Edges & \#Labels & \#Attributes \\
\hline
\midrule
\sffamily{WebKB} & 919 & 1662 & 5 & 1703 \\
\sffamily{Cora} & 2843 & 6269 & 7 & 1433 \\
\sffamily{Citeseer} & 3477 & 5319 & 6 & 3703 \\
\sffamily{Pubmed} & 20701 & 49523 & 3 & 500\\
\bottomrule
\end{tabular}
}
\label{tab:data}
\end{table}
\subsection{Baseline Algorithms and Experimental Setup}
We use DeepWalk \cite{perozzi2014deepwalk}, node2vec \cite{grover2016node2vec}, Line \cite{tang2015line}, TADW \cite{yang2015network}, AANE \cite{huang2017accelerated}, GraphSage \cite{hamilton2017inductive} and SEANO \cite{liang2018semi} as the baseline algorithms to be compared with. The first three algorithms consider only structure of the network, the last four consider both structure and node attributes. We mostly use the default settings of the parameters values in the publicly available implementations of the respective baseline algorithms. As SEANO is semi-supervised, we include 20\% of the data with their true labels in the training set of SEANO to produce the embeddings.
For ONE, we set the values of $\alpha$ and $\beta$ in such a way that three components in the joint losss function in Eq. \ref{eq:L} contribute equally before the first iteration of the for loop in Algorithm \ref{alg:oane}. For all the experiments we keep embedding space dimension to be three times the number of ground truth communities. For each of the datasets, we run the for loop (Steps 4 to 6 in Alg. \ref{alg:oane}) of ONE only 5 times. We observe that ONE converges very fast on all the datasets. Convergence rate has been shown experimentally in Fig. \ref{fig:loss}.
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.49\linewidth}
\includegraphics[width=\linewidth]{images/citeseer_loss.png}
\end{subfigure}
\begin{subfigure}[b]{0.49\linewidth}
\includegraphics[width=\linewidth]{images/pubmed_loss.png}
\end{subfigure}
\caption{Values of Loss function over different iterations of ONE for Citeseer and Pubmed (seeded) datasets}
\label{fig:loss}
\end{figure}
\subsection{Outlier Detection}
A very important goal of our work is to detect outliers while generating the network embedding. In this section, we see the performance of all the algorithms in detecting outliers that we planted in each dataset.
SEANO and ONE give outlier scores directly as the output. For ONE, we rank the nodes in order of the higher values of a weighted average of three outlier scores (more the value of this average outlier score, more likely the vertex is an outlier). We have observed experimentally that $O_2$ is more important to determine outliers. For SEANO, we rank the nodes in order of the lower values of the weight parameter $\lambda$ (lower the value of $\lambda$ more likely the vertex is an outlier). For other embedding algorithms, as they do not explicitly output any outlier score, we use isolation forest to detect outliers from the node embeddings generated by them.
We use recall to check the performance of each embedding algorithm to find outliers. As the total number of outliers in each dataset is 5\%, we start with the top 5\% of the nodes in the ranked list (L) of outliers, and continue till 25\% of the nodes, and calculate the recall for each set with respect to the artificially planted outliers. Figure \ref{fig:outdetec} shows that ONE, though completely unsupervised in nature, is able to outperform SEANO mostly on all the datasets.
SEANO considers the role of predicting the class label or context of a node based on only its attributes, or the set of attributes from its neighbors, and accordingly fix the outlierness of the node. Whereas, ONE explicitly compares structure, attribute and their combination to detect outliers and then reduces their effect iteratively in the optimization process. So discriminating outliers from the normal nodes becomes somewhat easier for ONE. As expected, all the other embedding algorithms (by running isolation forest on the embedding generated by them) perform poorly on all the datasets to detect outliers, except on Cora where AANE performs good.
\begin{figure*}[h!]
\centering
\begin{subfigure}[b]{0.21\linewidth}
\includegraphics[width=\linewidth]{images/outl_webkb.png}
\end{subfigure}
\begin{subfigure}[b]{0.21\linewidth}
\includegraphics[width=\linewidth]{images/outl_cora.png}
\end{subfigure}
\begin{subfigure}[b]{0.21\linewidth}
\includegraphics[width=\linewidth]{images/outl_citeseer.png}
\end{subfigure}
\begin{subfigure}[b]{0.21\linewidth}
\includegraphics[width=\linewidth]{images/outl_pubmed.png}
\end{subfigure}
\begin{subfigure}[b]{0.11\linewidth}
\includegraphics[width=\linewidth]{images/legend.png}
\end{subfigure}
\caption{Outlier Recall at top L\% from the ranked list of outliers for all the datasets. ONE, though it is an unsupervised algorithm, outperforms all the baseline algorithms in most of the cases. SEANO uses 20\% labeled data for training.}
\label{fig:outdetec}
\end{figure*}
\subsection{Node Classification}
Node classification is an important application when labeling information is available only for a small subset of nodes in the network. This information can be used to enhance the accuracy of the label prediction task on the remaining/unlabeled nodes.
For this task, firstly we get the embedding representations of the nodes and take them as the features to train a random forest classifier \cite{liaw2002classification}.
We split the set of nodes of the graph into training set and testing set. The training set size is varied from 10\% to 50\% of the entire data. The remaining (test) data is used to compare the performance of different algorithms.
We use two popular evaluation criteria based on F1-score, i.e., Macro-F1 and Micro-F1 to measure the performance of the multi-class classification algorithms. Micro-F1 is a weighted average of F1-score over all different class labels. Macro-F1 is an arithmetic average of F1-scores of all output class labels. Normally, the higher these values are, the better the classification performance is. We repeat each experiment 10 times and report the average results.
On all the datasets (Fig. \ref{fig:classi}), ONE consistently performs the best for classification, both in terms of macro and micro F1 scores. We see that conventional embedding algorithms like node2vec and TADW, which are generally good for consistent datasets, perform miserably in the presence of just 5\% outliers. AANE is the second best algorithm for classification in the presence of outliers, and it is very close to ONE in terms of F1 scores on the Citeseer dataset. {\it ONE is also able to outperform SEANO with a good margin, even though SEANO is a semi-supervised approach and uses labeled data for generating node embeddings}.
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.49\linewidth}
\includegraphics[width=\linewidth]{images/WebKB_macro.png}
\end{subfigure}
\begin{subfigure}[b]{0.49\linewidth}
\includegraphics[width=\linewidth]{images/WebKB_micro.png}
\end{subfigure}
\centering
\begin{subfigure}[b]{0.49\linewidth}
\includegraphics[width=\linewidth]{images/Cora_Macro.png}
\end{subfigure}
\begin{subfigure}[b]{0.49\linewidth}
\includegraphics[width=\linewidth]{images/Cora_Micro.png}
\end{subfigure}
\centering
\begin{subfigure}[b]{0.49\linewidth}
\includegraphics[width=\linewidth]{images/citeseer_macro.png}
\end{subfigure}
\begin{subfigure}[b]{0.49\linewidth}
\includegraphics[width=\linewidth]{images/citeseer_micro.png}
\end{subfigure}
\centering
\begin{subfigure}[b]{0.49\linewidth}
\includegraphics[width=\linewidth]{images/pubmed_macro.png}
\end{subfigure}
\begin{subfigure}[b]{0.49\linewidth}
\includegraphics[width=\linewidth]{images/pubmed_micro.png}
\end{subfigure}
\caption{Performance of different embedding algorithms for Classification with Random Forest}
\label{fig:classi}
\end{figure}
\subsection{Node Clustering}
Node Clustering is an unsupervised method of grouping the nodes into multiple communities or clusters.
First we run all the embedding algorithms to generate the embeddings of the nodes.
We use the node's embedding as the features for the node and then apply KMeans++ \cite{kmeans++}.
KMeans++ just divides the data into different groups. To find the test accuracy we need to assign the clusters with an appropriate label and compare with the ground truth community labels. For finding the test accuracy we use unsupervised clustering accuracy \cite{xie2016unsupervised} which uses different permutations of the labels and chooses the label ordering which gives best possible accuracy $Acc(\mathcal{\hat{C}},\mathcal{C}) = \max_{\mathcal{P}} \frac{ \sum\limits_{i=1}^N \mathbf{1}(\mathcal{P}(\mathcal{\hat{C}}_i) = \mathcal{C}_i) }{N}$.
Here $\mathcal{C}$ is the ground truth labeling of the dataset such that $\mathcal{C}_i$ gives the ground truth label of $i$th data point. Similarly $\mathcal{\hat{C}}$ is the clustering assignments discovered by some algorithm, and $\mathcal{P}$ is a permutation of the set of labels.
We assume $\mathbf{1}$ to be a logical operator which returns 1 when the argument is true, otherwise returns 0.
Clustering performance is shown and explained in Fig. \ref{fig:clus}. It can be observed that except for ONE, all the conventional embedding algorithms fail in the presence of outliers. Our proposed unsupervised algorithm is able to outperform or remain close to SEANO, though SEANO is semi supervised in nature, on all the datasets.
\begin{figure}[h!]
\centering
\includegraphics[width=0.7\columnwidth]{images/clustering.png}
\caption{Clustering Accuracy of KMeans++ on the embeddings generated by different algorithms. ONE is always close to the best of the baseline algorithms. AANE works best for Citeseer. Though SEANO uses 20\% labeled data as the extra supervision to generate the embeddings, its accuracy is always close (or less) to ONE which is completely unsupervised in nature.}
\label{fig:clus}
\end{figure}
\section{Discussion and Future Work}
We propose ONE, an unsupervised attributed network embedding approach that jointly learns and minimizes the effect of outliers in the network. We derive the details of the algorithm to optimize the associated cost function of ONE. Through experiments on seeded real world datasets, we show the superiority of ONE for outlier detection and other downstream network mining applications.
There are different ways to extend the proposed approach in the future. One interesting direction is to parallelize the algorithm and check its performance on real world large attributed networks. Most of the networks are very dynamic now-a-days. Even outliers also evolve over time. So bringing additional constraints in our framework to capture the dynamic temporal behavior of the outliers and other nodes of the network would also be interesting. As mentioned in Section \ref{sec:algOne}, ONE converges very fast on real datasets. But updating most of the variables in this framework takes $O(N)$ time, which leads to $O(N^2)$ runtime for ONE without any parallel processing. So, one can come up with some intelligent sampling or greedy method, perhaps based on random walks, to replace the expensive sums in various update rules.
{\footnotesize
|
3,212,635,537,991 | arxiv | \section{Introduction}
\label{sec:intro}
There are in our galaxy numerous \ion{H}{ii} regions surrounding young stellar clusters that contain several massive O and B stars. Far- and extreme-UV radiation and stellar wind from O and B stars heat, ionise, and shape the surrounding gas, and these expanding bubbles drive compressed molecular gas outwards. The evolution of such regions has been subject to a large number of theoretical studies, starting with the pioneering work by \citet{str39} (and e.g. the reviews in \citealt{hen07}, \citealt{kru14}, \citealt{dal15}, and \citealt{haw18}).
Once the first O stars have formed, the surrounding molecular cloud is compressed into a shell that is accelerated outwards. The shell breaks up into filaments, elephant trunks, and globules, and eventually it slows down and disperses. The expansion velocity of the shell obtained from simulations depends on the physical processes assumed, the initial conditions, and the source driving the expansion. Numerical studies show that the shells surrounding single stars rarely expand with velocities greater than 5 km/s \citep[e.g.][]{hos06,art11,bis15}, although shell velocities of the order $\sim$10 km/s can be attained for very massive stars \citep{gre20} or a low-density ambient interstellar medium \citep[ISM;][]{fre03}. A compact cluster of massive stars can drive a single, common shell, and these shells can reach velocities in excess of 10 km/s \citep[e.g.][]{dal14,lan21a}, but this requires a large population of high-mass stars.
From existing estimates of the central velocities of various \ion{H}{ii} regions in comparison to those measured for molecular line
emission in the surrounding shells, it appears that the expansion velocities as a rule are of the order of a few km~s$^{-1}$, in
congruence with the predictions from some of the above-mentioned theoretical model simulations. However, we have found several regions
for which one could suspect much higher velocities. A closer inspection shows that published estimates of central velocities and
shell velocities are in fact rather uncertain. In this paper we focus on the Sharpless~171 (S 171) complex, which has the young cluster Berkeley~59 (Be 59)
at its centre.
\begin{figure*}[t]
\centering
\includegraphics[width=8.30cm]{FieldOBass.jpg}
\includegraphics[width=9.00cm]{New_fig1b2.png}
\caption{Optical image from the Digitized Sky Survey (left) and mid-IR image from Wide-field Infrared Survey Explorer (WISE, right), featuring the \ion{H}{ii} region S~171 in relation to the cluster Berkeley 59, which is located in the central white box. The northern part of the nebula is called NGC 7822. The large circle has a radius of 1.6$\degr$ and shows the approximate extent of the \ion{H}{ii} region. The green contours show the 4 mm (70 GHz) continuum map from Planck. The smaller circles in the optical image mark the positions of the hottest members of the extended Cep OB 4 association; light blue circles represent spectral types O9 and B0, and yellow circles types B1 and B2. North is up, and east to the left. The WISE image is a colour composition of red (22 $\mu$m), green (4.6 $\mu$m), and blue (3.4 $\mu$m). Figure~\ref{fig:field} is a zoomed-in view of the area in the white box. }
\label{fig:OBass}
\end{figure*}
Literature data on the kinematics of the stars, ionised gas, and shell structures in this complex suggest that the molecular shell expands at a very high velocity. However, existing maps of the molecular line emission from shell structures are rather coarse, some line profiles are complex, and only a few stars have so far been measured for radial velocity (RV). We therefore decided to more closely inspect the kinematics of stars and nebular features in the complex. We collected high resolution spectra, at the Nordic Optical Telescope (NOT), of a number of stars in an area at the central cluster to obtain a mean central velocity of the cluster and information on spectroscopic binarity and membership. Moreover, we mapped the molecular line emission from the associated shell surrounding the cluster at Onsala Space Observatory (OSO). The molecular line velocities observed from different parts of the shell in comparison to the cluster velocity should provide an estimate of the expansion velocity of the complex.
The S 171 complex is described in Sect.~\ref{sec:S171} with an account of existing information on the velocities of the stars, the ionised gas, and shell structures. The observations and the data reduction are described in Sect.~\ref{sec:obs}. The general results are given in Sect.~\ref{sec:results} and discussed in more detail in Sect.~\ref{sec:discussion}. We compare these results with numerical simulations in Sect.~\ref{Sect:Simulations} and end with a summary in Sect.~\ref{sec:conclusions}.
\section{The S 171 complex}
\label{sec:S171}
\begin{table*}
\centering
\caption{Basic data.}
\begin{tabular} {lccccccc}
\hline
\noalign{\smallskip}
Designations: & Position & Distance & Radius & Age & $V_{*}$ & $V_{*}$(range) & $\bar V_{\ion{H}{ii}}$ \\
Cluster / Nebula & RA, Dec. (2000) & kpc & pc & Myr & km/s (helio) & km/s (lsr) & km/s (lsr) \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
Be 59 / S 171$^{1}$ & 00:02:10 +67:25:10 & 1.1$^{2}$ & 31$^{2}$ & ~2$^{3,4,5,6}$ & -14.5$^{7}$-6.5$^{8}$ & -5.2 to +4.1 & -19$^{10}$ -9.2$^{11}$ -7.5$^{12}$ \\
(NGC 7822, W 1) & & & & & -5.2$^{9}$ & & -8.1$^{13}$ -10.1$^{14}$ -12.0$^{15}$ \\
\noalign{\smallskip}
\hline
\end{tabular}
\label{tab:Be59}
\tablefoot{
\tablefoottext{1}{ \citet{sha59};}
\tablefoottext{2}{This work, Sect.~\ref{sec:pm};}
\tablefoottext{3}{ \citet{maj08};}
\tablefoottext{4}{ \citet{pan08};}
\tablefoottext{5}{ \citet{panw18};}
\tablefoottext{6}{ \citet{get18a,get18b};}
\tablefoottext{7}{ \citet{liu89};}
\tablefoottext{8}{\citet{kha05};}
\tablefoottext{9}{ \citet{dia02}: combined data from various sources;}
\tablefoottext{10}{ \citet{dow74};}
\tablefoottext{11}{ \citet{ped80};}
\tablefoottext{12}{\citet{loz87}: H$\alpha$, [\ion{N}{ii}];}
\tablefoottext{13}{\citet{ros80}: mean for the area just west of the cluster;}
\tablefoottext{14}{\citet{geo70}: H$\alpha$.}
\tablefoottext{15}{\citet{fic90}: H$\alpha$. }}
\end{table*}
Figure~\ref{fig:OBass} shows large-scale optical and infrared (IR) images of the S~171 nebula surrounding Berkeley~59 (galactic coordinates $l^{\rm II} = 118.2^{\circ}, b^{\rm II}= 5.0^{\circ}$). The nebula has a rather circular shape with an approximate extent as depicted by the large circle. The emission is less intense in the south-eastern part, and foreground obscuring clouds are scattered over the nebula. The arc of enhanced emission to the north and north-west have been listed as separate nebulae, NGC 7822 and the Cepheus Loop, but are clearly part of the same \ion{H}{ii} region. In the right panel of Fig.~\ref{fig:OBass} we have marked the location of the cluster NGC~7762, which lies at the north-west border of the \ion{H}{ii} region and seems to have a similar distance as Berkeley~59 \citep{liu19}, but according to \citet{pat95} it is an intermediate-age open cluster of 1.8 Gyr, and has a very different RV \citep{cas16}, for this reason we regard it as unrelated to Berkeley~59.
A more widespread association of early type stars, called Cep~OB4, was recognised by \citet{mac68}, and members of spectral types earlier than B3 are marked in the left image of Fig.~\ref{fig:OBass} (reviewed in \citealt{kun08}).
Berkeley~59 is well confined within the area outlined by the central box, which is enlarged in Fig.~\ref{fig:field}.
Many fainter pre-main-sequence stars are associated with the cluster \citep{coh76,pan08,rosv13,get17,panw18}. After submitting this paper we became aware of the work by \citet{min21}, a major IR study of the young stellar population and its spatial distribution, where they find the youngest sources located in the surrounding shells and pillars and the older ones in the bubble surrounding the OB association, from which they suggest that the expanding HII region may have triggered star formation in Cep OB4.
Maps of the continuous radio emission show that S~171 has developed into a more or less spherical \ion{H}{ii} region of thermal emission but with a void in the south-eastern part. Fig.~\ref{fig:OBass} shows the 4 mm Planck map as contours overlaid on the IR image, and these are morphologically very similar to the 9 cm radio continuum map from \citet{ros80}. NGC 7822 and the bright patch seen close to the south-east from the centre in the optical image in Fig.~\ref{fig:OBass} show enhanced radio emission \citep{chu70}. The central, most intense emission comes from an area offset by about 8$\arcmin$ to the south-west of the cluster and was mapped in more detail in \citet{ang77} and \citet{har81}. Isophotes of forbidden line emission are congruent with the radio maps \citep{loz87}. A review of works related to the stellar content and the nebula is given in \citet{kun08}.
The general properties of the S~171 complex are given in Table~\ref{tab:Be59}, where we have adopted a central position for the complex as between the two bright stars in Be~59 (A and B in Fig.~\ref{fig:field}). The radius of the \ion{H}{ii} region given in Column 4 is based on the distance to the complex cited in Column 3 and corresponds to the circle in Fig.~\ref{fig:OBass} with a radius of 1.6$\degr$. Column 6 gives different estimates of the mean heliocentric RV of the stellar cluster, and the corresponding range in velocity is expressed in local standard of rest (lsr) in Column 7. The last column gives published central velocities of the ionised gas, mostly from radio recombination lines but also from H$\alpha$ and [\ion{N}{ii}] lines.
Published cluster velocities are not very accurate and are based on only two or three stars, and they differ considerably from the central velocities obtained for the ionised gas. The latter are also discordant, and very different velocities were obtained for different positions over the nebula \citep[e.g.][]{ped80,ros80,loz87}. The lines are broad, full width at half maximum (FWHM) $>$ 30 km~s$^{-1}$, with complex line profiles in some directions, which makes it difficult to assign a precise value of the central velocity.
The northern molecular cloud related to NGC 7822 was mapped in CO emission in \citet{elm78} and \citet{woo83}, and the shell structures related to the S 171 nebula were mapped in CO by \citet{lei89,yan92,yon97}, and in H$_{2}$CO by \citet{ros83}. These maps cover most of the complex with grid spacings of several arcminutes \citep[2$\arcmin$ for the central part in]{yan92}. The associated velocity patterns turned out to be rather complex with multiple velocity components at several positions, ranging in RVs from --1 km~s$^{-1}$ to --19 km~s$^{-1}$ (lsr) in some cloudlets. Considering also the viewing aspect angle of these cloudlets it appears from some of the central velocities listed in Table~\ref{tab:Be59} as parts of the shell may expand at very high velocities.
We can assume that the present structure of the S~171 complex is the result mainly of the interaction between the hot members in Be~59 with the surrounding molecular shell. Since the central velocity of the complex is not well established, and existing CO maps are rather coarse, we found it motivated to obtain new RVs of several stars in Be 59 and of the molecular line emission from the shell at a higher spatial resolution than obtained before.
\begin{figure}
\centering
\includegraphics[angle=00, width=8.9cm]{Be59map.jpg}
\caption{Locations of program stars in the Berkeley~59 region according to the designations in Table~\ref{tab:stars} on an image from the Digitized Sky Survey. The cluster centre is put between the bright stars {\it A} (BD+66$\degr$1674) and {\it B} (BD+66$\degr$1675). The white arrow points at BD+66$\degr$1673, V747 Cep. The red arrow marks the position of a bright IR source. North is up, and east is to the left. }
\label{fig:field}
\end{figure}
\section{Observations, reductions, and measurements}
\label{sec:obs}
\subsection{Optical spectroscopy}
\label{sec:spectra}
We aimed at including all OB stars in the Berkeley~59 cluster in our sample and furthermore reaching as low a mass as possible, but due to extinction and the signal-to-noise ratio (S/N) needed in the spectra to determine RVs, a practical limit was set around V $\sim$ 14.5 mag. A few stars had determined spectral types from previous studies, and for the rest we made rough estimates based on IR and optical photometry \citep{pan08} to select the program stars (Sect.~\ref{sec:results}).
Optical spectra were collected for 27 stars in the Be~59 region at the 2.6 m NOT during two observing runs in October 2015 and August 2016 plus spectra obtained as filler programs in service nights. We used the high-resolution FIber-fed Echelle Spectrograph \citep[FIES;][]{tel14}, with a resolution of R $\sim$ 25000 covering the wavelength range from 3700--7300 \AA\ when used with Charge Capture Device 13 (CCD13) and 3620--8580 \AA\ when used with CCD15, which was installed in October 2016. Our program stars are marked in Fig.~\ref{fig:field} with numbers as defined in the subsequent section.
\begin{table*}
\centering
\caption{Program stars, parallaxes, and proper motions.}
\begin{tabular} {llccc}
\hline
\noalign{\smallskip}
Star ID$^a$ & RA \ \ \ \ \ \ \ Dec. & $\pi$ & $\mu$(RA) & $\mu$(Dec.) \\
& \ \ \ \ \ \ (2000.0) & mas & mas/yr & mas/yr \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
P316, TYC 4026-151-1 & 00 01 11.6 +67 29 22 & 1.67 (0.009) & 7.40 (0.04) & -3.75 (0.01) \\
P239 & 00 01 14.2 +67 26 34 & -3.00 (0.33) & -3.03 (0.35) & 7.47 (0.36) \\
IR star$^{1}$ & 00 01 26.7 +67 23 46 & 1.09 (0.05) & -1.09 (0.16) & -2.51 (0.15) \\
P164 & 00 01 36.1 +67 22 43 & 3.36 (0.03) & 10.54 (0.04) & -9.44 (0.04) \\
P109 & 00 01 39.7 +67 24 37 & 1.53 (0.02) & -17.53 (0.03) & 1.78 (0.03)\\
P121 & 00 01 41.5 +67 23 33 & 2.93 (0.02) & -6.83 (0.03) & -0.54 (0.03) \\
V747 Cep$^{2}$ & 00 01 46.9 +67 30 25 & 1.02 (0.03) & -1.57 (0.04) & -1.77 (0.05) \\
P310, SOP 7 & 00 01 51.9 +67 31 47 & & & \\
P15, SOP 11 & 00 02 00.1 +67 25 11 & 1.10 (0.05) & -2.49 (0.08) & -1.77 (0.08) \\
P59 & 00 02 01.2 +67 26 49 & 0.89 (0.02) & -1.43 (0.03) & -1.78 (0.03) \\
P7 & 00 02 07.7 +67 25 42 & 0.94 (0.03) & -1.54 (0.03) & -1.69 (0.05) \\
P106 & 00 02 09.7 +67 22 16 & 4.28 (0.02) & -6.75 (0.04) & -33.92 (0.03) \\
BD+66$\degr$1674 & 00 02 10.2 +67 25 45 & 0.82 (0.03) & -1.54 (0.05) & -2.76 (0.05) \\
BD+66$\degr$1675 & 00 02 10.3 +67 24 32 & 0.93 (0.04) & -1.61( 0.05) & -2.04 (0.05) \\
P20, SOP 14 & 00 02 10.6 +67 24 09 & 0.93 (0.03) & -1.46 (0.05) & -1.98 (0.04) \\
P13, & 00 02 12.2 +67 24 14 & 0.95 (0.02) & -1.41 (0.03) & -2.00 (0.03) \\
P130, SOP 4 & 00 02 12.5 +67 28 36 & 0.92 (0.03) & -1.99 (0.04) & -1.87 (0.04) \\
P201, TYC 4294-330-1, SOP 16 & 00 02 12.5 +67 30 03 & 3.15 (0.03) & & \\
P3, MacC15, SOP 6 & 00 02 13.6 +67 25 04 & 0.94 (0.03) & -1.71 (0.04) & -1.94 (0.04) \\
P23, SOP 13 & 00 02 13.7 +67 26 11 & 0.87 (0.04) & -1.55 (0.06) & -1.73 (0.05) \\
P62 & 00 02 14.7 +67 23 17 & 0.91 (0.02) & -1.99 (0.03) & -1.99 (0.03) \\
P16, MacC29, SOP 2 & 00 02 19.1 +67 25 38 & 0.83 (0.03) & -1.88 (0.09) & -2.63 (0.08) \\
P114, SOP 9 & 00 02 20.1 +67 28 07 & 0.68 (0.04) & -1.99 (0.06) & -2.22 (0.05) \\
P66, SOP 8 & 00 02 29.9 +67 25 44 & 0.95 (0.03) & -1.49 (0.05) & -2.22 (0.06) \\
P105 & 00 02 38.6 +67 24 09 & 0.98 (0.03) & -1.16( 0.04) & -1.96 (0.04) \\
P195 & 00 02 57.5 +67 23 48 & 0.89 (0.02) & -1.78 (0.03) & -1.66 (0.03) \\
P233 & 00 03 06.0 +67 26 22 & 1.47 (0.03) & -4.26 (0.04) & 0.27 (0.05) \\
\noalign{\smallskip}
\hline
\end{tabular}
\label{tab:astrometric}
\tablefoot{
\tablefoottext{1}{2MASS J00012663+6723461, WISE J000126.63+672346.1, UCAC4 787-000033}
\tablefoottext{2}{BD+66$\degr$1673}}
\end{table*}
Most targets were observed on at least two different dates (separated by a minimum of four days) to check for spectroscopic binaries. Because our targets are mainly early type stars with few available lines, often strongly rotationally broadened, we aimed for a S/N of $\sim$ 50 in the final spectra. However, our targets are heavily reddened (typically by 5-7 visual magnitudes). This lowers the S/N ratio significantly in the blue part of the spectra, and thereby the number of useful lines. Exposure times were up to 1800 s, and for the faintest targets two consecutive exposures were taken. From the list in \citet{fek99} we selected as early-type RV standards HR 1810 (B2.5~V), HR 7512 (B8 III), HR 8404 (B9.5 V), and HR 7903 (A0 III), stars not found to vary in velocity and with accuracies in the range 0.05 to 0.4~km~s$^{-1}$. We also observed Vega (A0 V), and for later spectral types we used the RV standard HD 31253 (F8). In addition, we collected spectra of a number of MK standards used for spectral classification of our targets.
Standard calibrations (21 halogen flats, seven biases, and one ThAr) were obtained each afternoon. Data reduction was made using the FIEStool pipeline \citep{ste05}. Over the course of our observations, FIES was upgraded with a new CCD, in October 2016, and new octagonal fibres, in June 2017. The merging of the orders in FIEStool was particularly troublesome in the spectra of blue stars with the old setup but was enhanced with the new one.\ Thus, the merging was improved manually by cutting the noisy edges of individual orders. We then normalised the spectra by fitting a continuum.
We measured the stellar RVs using both cross-correlation techniques (`fxcor' in IRAF) as well as direct measurements of line centres by fitting a Gaussian profile to each line (`rvidlines' in IRAF). For most OB-type stars we mainly used the \ion{He}{i} and \ion{He}{ii} lines when available, while for later B and early A stars, and for targets where few lines were available, we also included the hydrogen lines.
When both methods were applicable, we compared the results and typically found consistent values within the errors. The instrument drift is practically negligible compared to the typical measurement errors in the RVs of our targets. Multi-epoch RV results for the program stars are listed in Tables~\ref{tab:RVstars1} and~\ref{tab:RVstars2}.
\subsection{Molecular line observations}
\label{sec:COobs}
We observed $^{13}$CO \mbox{$J$\,=\,1\,--\,0} at 110.201~GHz from the region surrounding Be 59 during several runs in 2016 and 2017 using a 3 mm SIS receiver at the 20 m telescope at OSO. Frequency-switching mode was used with a 1600-channel hybrid digital autocorrelation spectrometer with a channel spacing of 25 kHz (0.07~km~s$^{-1}$), a bandwidth of 40~MHz, and the velocity resolution is about 0.2~km~s$^{-1}$ after smoothing. The single-sideband system temperature was typically around \mbox{150\,--\,400~K}. The pointing was checked regularly, and we estimate the pointing error to be less than a few arcseconds. At 110~GHz, the FWHM beam size of the 20 m antenna is 34$\arcsec$, and the main beam efficiency $\sim$0.30 (for an average elevation of approximately 30$^\circ$). The chopper-wheel method was used for the intensity calibration. The data reduction was performed with the spectral line software package {\tt xs}\footnote{http://www.chalmers.se/rss/oso-en/observations/data-reduction-software}.
\begin{figure*}[t]
\centering
\includegraphics[angle=00, width=9.0cm]{spectraAnew2.png}
\includegraphics[angle=00, width=8.0cm]{spectraB.jpg}
\caption{Examples of normalised spectra obtained for some program stars over two selected wavelength regions. Certain spectral features discussed in the text are marked. The spectra are ordered in steps of 0.3 (intensity scale) in the left panel and 1.0 in the right panel and with a smaller spacing between the third and fourth spectra; the double-lined spectroscopic binary BD+66$\degr$1674 is shown at two different phases. To improve the S/N for the faint IR star, its left spectrum is a combination of many observations, Doppler-corrected for the binary motion, which smears out the DIBs.}
\label{fig:spectra}
\end{figure*}
\begin{table*}
\centering
\caption{Spectral properties and heliocentric RVs.}
\begin{tabular} {llllcclccl}
\hline
\noalign{\smallskip}
Star ID & V/(B-V)$^a$ & MK$_{lit.}$$^b$ & MK$_{NOT}$$^c$ & DIB$^d$ & A$_V$ & V(Lit.)$^e$ & V(NOT)$^f$ & Number$^g$ & Notes$^h$ \\
& & & & & & km/s & km/s & nights & \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
V747 Cep$^*$ & 10.07/1.31 & O5.5 V & O5.5 V & s & 6.1 & & EB, SB1 & 12 & br \\
BD+66$\degr$1675 & 9.08/1.08 & O7.5 V & O6 V/O8 V/B & s & 4.6 & -9.9$^1$-9.3$^2$ & SB3 & 13 & \\
P16 & 11.81/1.91 v & O9 & O7 V & s & 9.4 & & SB1 & 7 & \\
IR star$^{**}$ & & & O9 V & s & 11.9 & & SB1 & 18 & \\
BD+66$\degr$1674 & 9.60/1.07 & O9.7 IV & B0 III/B & s & 3.4 & -9.1$^1$+5.2$^2$ & SB2, SB3? & 12 & \\
P66 & 12.95/1.54 & & B0.5 V & s & 6.9 & & -20.4$\pm3.4$ & 2 & \\
P3 & 11.30/1.14 v & B0.5 V & B0.5 V & s & 3.7 & -24.5$^2$ & -14.4$\pm$2.8 & 12 & br \\
P310 & 12.48/1.32 & & B1 V(n) & s & 6.1 & & -21.1$\pm2.4$ & & \\
P130 & 12.61/1.63 & & B1 V(n) & s & 7.6 & & -22.0$\pm4.2$ & 2 & br \\
P23 & 13.84/1.64 & & B2.5Vn & s & 6.1 & & SB1 & 2 & br \\
P62 & 13.48/1.19 & & B3 Vn & s & 4.4 & & -16.2$\pm2.2$ & & \\
P114 & 12.43/1.35 & & B3 V(n) & s & 5.3 & & -18.7$\pm5.4$ & 2 & br \\
P15 & 12.78/1.25 v & B3 V & B5 Vn & s & 5.1 & & -17.5$\pm3.4$ & 2 & \\
P13 & 13.64/1.30 & & B5 Vn & s & 4.2 & & -24.5$\pm1.3$ & 2 & br \\
P105 & 13.65/1.43 & & B5 Vn & s & 6.8 & & -21.2$\pm4.6$ & & br \\
P20 & 13.40/1.43 v & B8 III & B8 Vn & s & 4.5 & & -16.5$\pm6.6$ & 2& \\
P59 & 14.88/1.42 & & B9 Vn & s & 4.6 & & -21.8$\pm5.8$ & & \\
P195 & 14.22/1.47 & & B9 Vn & s & 4.1 & & -14.2$\pm9.5$ & & \\
P7 & 13.87/1.24 & & A1 Vn & s & 3.9 & & -15.3$\pm1.2$ & 2 & \\
P316 & 12.47/0.72 & & A5 V & w & 1.2 & & -20.4$\pm3.8$ & & nm \\
P233 & 13.00/0.82 & & F8 IVn & w & 0.3 & & -12.0$\pm0.9$ & & nm \\
P164 & 12.27/0.60 & & G2 V & n & 0.3 & & -6.0$\pm0.2$ & & nm \\
P239 & 12.95/1.03 & & G3 V & w & 0.5 & & -30.0$\pm1.0$ & & nm \\
P121 & 13.42/0.85 v & & G5 Vn & n & 1.1 & & -7.2$\pm0.3$ & & nm \\
P201 & 12.43/0.86 v & & G5 V & n & 0.6 & -50.3$^3$ & SB2 & & nm \\
P109 & 13.50/1.12 v & & G8 Vn & w & 1.0 & -29.2$^3$ & -30.5$\pm0.7$ & & nm \\
P106 & 14.32/1.19 v & & K5 Vn & n & 0.0 & & -12.9$\pm1.5$ & & nm \\
\noalign{\smallskip}
\hline
\end{tabular}
\label{tab:stars}
\tablefoot{
\tablefoottext{*}{BD+66$\degr$1673}
\tablefoottext{**}{2MASS J00012663+6723461 }
\tablefoottext{a}{Photometric data from \citet{pan08} except for BD+66$\degr$1673, 1674, 1675 taken from \citet{maj08}; stars marked v: variable in brightness according to \citet{lat11} and/or as listed in SIMBAD. }
\tablefoottext{b}{Spectral class from SIMBAD. }
\tablefoottext{c}{Spectral class from NOT spectra. Components in binaries are separated. }
\tablefoottext{d}{Diffuse interstellar bands: strong (s), weak (w), or not present (n) }
\tablefoottext{e}{Heliocentric RV from the literature: 1. \citet{cra74}; 2. \citet{liu89}; 3. Gaia Data Release no 2. }
\tablefoottext{f}{Heliocentric RV derived from NOT spectra; close binaries are noted as single-lined $SB1$, double-lined $SB2$, triple-lined SB3, and eclipsing binary $EB$.}
\tablefoottext{g}{Number of nights of observations if more than one.}
\tablefoottext{h}{br: broad lines with v sin $i$:s $>$~150~km~s$^{-1}$; nm: not member.}}
\end{table*}
Since the complex extends over a very large area in the sky, we had to select primarily areas where distinct shell structures are seen as dark foreground features in optical images. Some of these features were listed in the catalogue of dark nebulae in \citet{lyn62}. A total of 489 positions were observed including both observations over grids and at single positions, and some areas have a denser spatial coverage than others. In addition, observations of positions located outside obscuring clouds provided reference spectra.
\section{Results}
\label{sec:results}
\subsection{Program stars and parallaxes plus proper motions }
\label{sec:pm}
Most of our program stars in Be~59 are common to the stars observed by \citet{pan08}, and we chose their sequential number as prime designation (P). The stars are marked accordingly on the image in Fig.~\ref{fig:field}. The program stars were selected with the purpose of determining the stellar RVs and are therefore a brightness limited sample. However, we searched the mid-IR WISE point source catalogue \citep{cut13} for early type stars extinguished behind dark clouds within a radius of 8 arc minutes of the cluster centre. The star we name IR star (identical with WISE J000126.63+672346.1) was located in the direction of very opaque shell structures obscuring the bright nebulosity and although faint in the visual, its brightness at mid-IR wavelengths was at the same level as for the well known O~stars in the cluster, for this reason
we included it in our program, and our multi-epoch spectroscopy will reveal it to be a binary O9\,V~star.
The program stars are listed according to increasing right ascension (RA) in Table~\ref{tab:astrometric} and include designations from \citet[][MaxC]{mac68}, Sharma et al. (\cite {sha07}, SOP), and the Tycho mission \cite[][TYC]{hog00} with the parallaxes and proper motions, when available, from the Gaia Data Release 2 (proper motion) and 3 \citep[parallaxes;][]{gai16,gai18,gai20}. Numbers are in milliarcseconds (mas) and quoted errors in parentheses. The parallax of the IR star is consistent with membership in Be~59. The parallaxes are well confined around a mean of 0.9~$\pm$~0.1~mas, corresponding to a distance of 1.1 kpc excluding six stars, which appear to be foreground stars. We adopt this distance for the present study.
The corresponding radius of the \ion{H}{ii} region is 31 pc as listed in Table~\ref{tab:Be59}.
From the proper motions, excluding the six stars as above, we obtain a mean value for the cluster of $\Delta$$\alpha$ = --1.7 and $\Delta$$\delta$ = --2.0 mas/yr. The spread around this mean is small. Our cluster mean values, also regarding the parallax, are in good agreement with the values obtained by \citet{kuh19} based on 225 stars in Be~59.
\subsection{Stellar properties and radial velocities}
\label{sec:stellar}
We found a number of program stars to be spectroscopic binaries, including the IR star, and for most of these we extended the observations to get a hand on orbital data. Examples of normalised spectra collected at NOT of some program stars for two spectral regions are shown in Fig.~\ref{fig:spectra} with some spectral features discussed in the text marked. The third and fourth spectra from top show the double-lined spectroscopic binary BD+66$\degr$1674 at two phases when the stellar lines from the two components merge and split. For stars considered as members the diffuse interstellar bands (DIBs) are of about the same strength but weaker in foreground stars as evident for the star P316 at bottom of the panels. As seen in the bottom panel narrow H$\alpha$ line emission, flanked by [\ion{N}{ii}] emission in some cases, are superimposed on the broad H$\alpha$ absorption lines. These emission lines originate in the surrounding emission nebula.
Most program stars had not been classified on the MK system before, which we did by comparisons to standard stars based on traditional line ratios in the blue spectral region \citep{gra09}, also for the secondary components in binaries observed at large line splittings. For OB stars \citep{wal09} we used \ion{He}{ii} to \ion{He}{i} equivalent width ratios in the blue whenever possible, and, despite multiple epoch spectra, there are uncertainties in the line ratios due to blending of the binary or triple components.
The blue part of the spectrum of the reddened IR star is noisy, but using ratios based on equivalent widths of lines in the yellow and red parts we conclude that it is of spectral type O9~V (lines of e.g. \ion{He}{i}, \ion{He}{ii}, \ion{O}{iii} 5592~\AA, \ion{C}{iv} 5801, and 5812 \AA).
These line ratios were measured also for all early type stars for comparison. The most luminous star is the eclipsing and spectroscopic binary V747~Cep of spectral type O5.5~V.
In Table~\ref{tab:stars} we present the results from our spectroscopic survey, where the stars are ordered according to spectral type. Photometric data from the literature are in Column~2. MK spectral classes as listed in the Set of Identifications, Measurements, and Bibliography for Astronomical Data (SIMBAD) Astronomical Database are in Column~3 and our estimates in Column~4. Column~5 gives information on the strength of the DIBs. Column~6 gives the visual extinction derived from the Two Micron All Sky Survey (2MASS) photometric data combined with intrinsic $J-H$ colours inferred from the spectral types obtained from \citet{koo83}. We assume that the excess in the $J-H$ colour is a measure of the extinction and use the relation $A_V = R_V \times E(J-H)/R(J-H) = 11.9 \times E(J-H)$, for R$_V$= 3.1 and the empirical 2MASS extinction coefficient $R(J-H)$ = 0.26 \citep{yua13}.
Column~7 lists heliocentric RVs from the literature and in Column~8 are our measurements with corresponding rms errors. For stars observed in only one night, the errors are from the determination of the RV from that single spectrum. Spectroscopic binaries are denoted as single-lined (SB1), double-lined (SB2), and triple-lined (SB3), and these will be discussed in more detail in the subsequent section. In Column~9 are the number of nights when FIES spectra were taken if more than one night. Individual measurements are listed in Tables~\ref{tab:RVstars1},~\ref{tab:RVstars2}, and \ref{tab:RVstars3}.
Based on the distance derived from the Gaia parallaxes and, if not available, from the photometry combined with spectral type and extinction, the strength of the diffuse interstellar absorption bands, and RVs, we find that all stars of spectral type A1 and earlier are likely members of Be~59. Those that we regard as non-members, are marked in Column 10, which also contains notes on stars with broad He or metal lines (br), for which we measured v~sin $i$ ~$> 150$\,km s$^{-1}$.
Fig.~\ref{fig:field} might give the impression that some members in Be~59 are not located behind foreground shell structures. Yet, they suffer from relatively large extinctions. The optical image indicates fragments of foreground clouds seen in absorption against a cavity illuminated by the H$\alpha$ nebulosity, which is even more clearly displayed in the IR in Figs.~\ref{fig:OBass} and \ref{fig:RVpositions}, where the contour overlay shows radio continuum emission enhanced at the location of the central cluster. Towards the west there is a pronounced ridge of dust behind which the IR star is located, and as shown in Table~\ref{tab:stars} the measured extinction towards individual stars is highly variable over the cluster area, but A$_V >$ 3 mag for all cluster members.
\subsection{Completeness of OB stars}
\label{sec:OBsample}
Next we wanted to determine how complete our sample of OB stars was. B3\,V stars have absolute magnitudes $M_V~=~-1.6~\pm~1.3$ mag \citep{weg06} and apparent magnitudes m$_V$ = M$_V$ + dm + A$_V$, where dm = $10.21$ mag for Be~59. Our magnitude limit for spectroscopy was set at V $\sim$ 14.5 mag, which means we can reach completeness for B3\,V stars only for regions with A$_V$~$\leq$~6~mag, and completeness for O7~V stars for A$_V$~$\leq$~9 mag. Among the most reddened sources in our sample is P16 and the IR star with visual extinctions slightly $>$ 9 mag while the median A$_V$ value is 5.1 mag. From the WISE catalogue search within a radius of 8$\arcmin$ (2.6~pc from the cluster centre) we find 38 sources brighter than 9.7 mag in the W1 (4.3 $\mu$m) band. This is the apparent magnitude of a B3\,V star if extinguished by A$_V$ = 5.1 mag, equivalent to A$_{W1} = 0.30$ mag \citep{yua13}. Twelve of these are among our optically selected program stars (all of which are later found to be cluster members), and the brightest of the remaining 26 sources is the IR star ($W1 = 7.15 \pm 0.04$ mag). The rest constitute a mix of intermediate-mass stars and background giants, but a few could be reddened B~stars belonging to the cluster, although all are fainter than the IR star by 1 mag or more at 3.4 $\mu$m. Based on this we conclude that our sample may not be complete all the way to B3 stars, but most likely includes all O~stars in the area we studied.
\subsection{Spectroscopic binaries}
\label{sec:binaries}
The frequency of spectroscopic binaries can be expected to be high for the most massive stars. According to \citet{chi12} stars with masses $>$16 $M_{\sun}$ as many as 82\% are close binary systems, while this fraction drops to about 20\% for 3 $M_{\sun}$ stars. In order to check for binarity we obtained repeated observations for the earliest type stars (Table~\ref{tab:stars}).
BD+66$\degr$1675 and BD+66$\degr$1674 were regarded as spectroscopic binaries in \citet{liu89} and \citet{cra74}. We found that BD+66$\degr$1675 is a triple-lined spectroscopic binary (SB3) and our data points suggest a period of 74 days and a systemic velocity of -10 km s$^{-1}$, as well as semi-amplitudes of 6, 150, and 280 km~s$^{-1}$ for components 1, 2, and 3, respectively. While components 1 and 2 have \ion{He}{ii} lines, only the \ion{He}{i} lines are detected in the faintest component, indicating spectral type B (Fig.~\ref{fig:multiepoch2}).
BD+66$\degr$1674 is a double-lined spectroscopic binary (SB2).\ However, individual lines are often heavily blended, and we could not extract associated periods and amplitudes.\ There are perhaps three components.
\begin{figure}[t]
\centering
\includegraphics[angle=00, width=8.7cm]{Phase.pdf}
\caption{Phase diagrams for (from the top): V747 Cep for a period of 5.33146 days; BD+66$\degr$1675, period of 74 days; the IR star, period of 9.51 days; and P16, period of 13 days. }
\label{fig:Phase}
\end{figure}
V747 Cep (BD+66$\degr$1673) is an eclipsing binary with a photometric period of P = 5.33146 days, an orbital eccentricity of $e$ = 0.3, and an inclination $i \sim $ 75$^{\circ}$ according to \citet{maj08}. \citet{MAp} assigned it a spectral type of O5.5~V(n)((f)) and found that it is also a spectroscopic binary. We can confirm that V747 Cep is a spectroscopic binary. In the data from two nights we note an indication of line splitting, but more observations are needed to confirm whether it is of type SB2 (Table~\ref{tab:RVstars1} and Fig.~\ref{fig:multiepoch1}). The RV data fit well with the photometric period and suggest a semi-amplitude of K$_1$ = 90 km~s$^{-1}$ around a systemic velocity of -26 km~s$^{-1}$ (Fig.~\ref{fig:Phase}, top panel), giving a close orbit with a semi-major axis of only 0.04~AU or $\sim$ 10 R$_\odot$. At the time of submission, we became aware of a more detailed study of this target in \citet{tri21}.
Multiple spectra of the hidden O~star we call the IR star (2MASS J00012663+6723461) demonstrates that it is a spectroscopic binary, for which we estimate a period of 9.51 days and K$_1$ = 45 km~s$^{-1}$, assuming a circular orbit, giving a projected semi-major axis $a~\sin i$ around 0.04~AU.
For P16, with only seven observations, only preliminary values can be assigned: a period of 13 days with a semi-amplitude of 12 km~s$^{-1}$. Examples of phase diagrams are shown in Fig~\ref{fig:Phase}. Finally, the late type foreground star, P201 (G5 V), show two velocity components and is likely an SB2.
\subsection{Mean cluster velocity}
\label{sec:MeanVel}
Excluding spectroscopic binaries and non-members from Table~\ref{tab:stars}, we obtain a mean RV of the cluster of --18.75 $\pm$ 3.3 km s$^{-1}$ (--9.5 km s$^{-1}$ in lsr), which implies a standard error on the mean of 0.9 km s$^{-1}$ if we assume a Gaussian distribution. Hence, our derived mean cluster velocity is close to that derived for the central velocity of the ionised gas by \citet{ped80} as quoted in Table~\ref{tab:Be59}.
\subsection{The molecular line emission}
\label{sec:COvel}
\begin{figure*}[t]
\centering
\includegraphics[angle=00, width=6.9cm]{S171stor.pdf}
\includegraphics[angle=00, width=10.65cm]{S171liten.pdf}
\caption{Map of the fields we obtained CO spectra in. Left: Large region, 4.4$\times$4.4 degrees, surrounding the emission nebula S 171.\ The arrow points at the position of the central cluster. The northern arc of bright nebulosity is NGC 7822. In this panel the more peripheral areas observed for $^{13}$CO emission are marked and labelled. The circles (Enorth and Esouth) mark observations at selected positions along two dusty striations that extend to the east from the centre. The dashed box marks the central region shown in the right panel. Right: Image spanning 1.9$\times$1.2 degrees.\ The core of the stellar cluster is inside the dashed box. The position of an elephant trunk, called the `Dancing Queen' is marked with a black circle. North is up and east to the left in both images, which are from the red Deep Sky Survey.}
\label{fig:171fields}
\end{figure*}
\begin{figure}[t]
\centering
\includegraphics[angle=00, width=8cm]{W8map.pdf}
\caption{Part of the map of $^{13}$CO spectra obtained for area W8. The offsets in RA and Dec. from the central position are in steps of 60$\arcsec$. The velocity interval, in km s$^{-1}$, and the antenna temperature, $T_{A}$ in Kelvin, are indicated in the upper-right panel. The component at around --18~km~s$^{-1}$, present in most panels, is related to the foreground shell, whereas the component at around --7~km~s$^{-1}$ has a local origin.}
\label{fig:W8map}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[angle=00, width=7.5cm]{E2E.jpg}
\caption{$^{13}$CO spectra from one position in area E2. The strongest signal at around --13~km~s$^{-1}$ comes from the foreground shell, while the components at --6 and +2~km~s$^{-1}$ are related to gas connected to the Orion arm and Gould's Belt. The component at around --2~km~s$^{-1}$ may trace the remote part of the molecular shell.}
\label{fig:E2E}
\end{figure}
We obtained $^{13}$CO spectra from a total of 489 positions in the areas labelled in Fig.~\ref{fig:171fields}. The left panel shows a large area with the complex at the centre. The circles mark single positions observed along two dusty striations of dust that extend from the central parts to the east, and which are seen as distinct features, especially on the photographic version of the Palomar Sky-Atlas. Field E4 is located about 3$\degr$ east of the cluster and was originally included as a reference position outside the complex. Since we discovered a rather strong signal at high negative velocity here, we extended the observations over what appears to be isolated dust clouds. The right panel shows the areas observed in the central region of S 171. The circle at `DQ' marks the position of an elephant trunk, called the Dancing Queen, mapped in detail in $^{12}$CO and $^{13}$CO in \citet{gah06}.
Examples of spectra are shown in Figs.~\ref{fig:W8map} and \ref{fig:E2E}. Signals from the molecular gas related to the foreground shell is seen in most panels for W8 in Fig.~\ref{fig:W8map} at $v_{\rm lsr}$ $\sim$ --18~km~s$^{-1}$ and at $v_{\rm lsr}$ $\sim$ --13~km~s$^{-1}$ for E2 in Fig.~\ref{fig:E2E}. These components are blue-shifted relative to the mean cluster velocity as derived in Sect.~\ref{sec:MeanVel} and are present in the bulk of our spectra. In the following, we refer to components at velocities $\leq$~--10~km~s$^{-1}$ as $V_{b}$. Especially in the central fields these lines can be split into two or three components well separated in velocity or blended as in Fig.~\ref{fig:E2E} (shell front). There is another component present at certain positions at velocities around --4~km~s$^{-1}$ (as in Fig.~\ref{fig:E2E}), slightly red-shifted relative to the cluster, which we refer to as $V_{r}$. As will be discussed in Sect.~\ref{sec:discussion} these could be related to the remote side of the complex. Representative spectra for each region are presented in the Appendix, Fig.~\ref{fig:COsp1} and Fig.~\ref{fig:COsp2}.
Our spectra also contain additional components, which are not related to the S 171 complex. In most directions we got signals at velocities around --2 to +4 ~km~s$^{-1}$, which we identify with the local gas, called feature A, related to the Gould's Belt \citep[e.g.][]{lin73}. Components at $v_{\rm lsr}$ $\sim$ --6~km~s$^{-1}$ are related to gas in the local spiral arm, the Orion arm, sometimes referred to as `the other local feature'. As expected this component declines statistically in intensity with increasing galactic latitude. In our optical spectra of cluster members, interstellar absorption lines of \ion{Na}{i} D are present in practically all stars. Also, these lines share the velocity of this local gas. Both local components are present in the majority of our $^{13}$CO spectra (Fig.~\ref{fig:E2E}). Emission from the background Perseus arm, at expected velocities of around $\leq$ --30~km~s$^{-1}$, does not enter our spectra since the S 171 complex is at a galactic latitude of $\sim$ +5$\degr$.
\begin{table*}
\centering
\caption{Velocities and peak intensities for the blueshifted and redshifted components in the fields marked in Fig.~\ref{fig:171fields}. The V$_{b}$ refers to velocities $\leq$~--10~km~s$^{-1}$, blueshifted with respect to the velocity of the central cluster of --9.5~km~s$^{-1}$, and V$_{r}$ refers to the interval --2 to --6~km~s$^{-1}$. Column 2 gives the number of positions mapped, while the number of positions for which a given velocity component was found is given in parentheses after the V$_{b}$ and V$_{r}$ velocity ranges. $T_\mathrm{A}^*$(max) refers to the peak antenna temperature for $^{13}$CO. Details are provided in Sect.~\ref{sec:COvel}.}
\begin{tabular} {lclclc}
\hline
\noalign{\smallskip}
Area & Number of & $V_{b}$ & $T_\mathrm{A}^*$(max) & $V_{r}$ & $T_\mathrm{A}^*$(max) \\
& positions & (km/s lsr) & & (km/s lsr) & \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
W10 & 3 & -14.5 to -13.3 (2) & 1.7 & \ldots & \ldots \\
W9 & 13 & \ldots & \ldots & -6.8 to -5.1 (11) & 0.8 \\
W8 & 19 & -18.3 to -18.0 (7) & 0.5 & -5.5 to -4.8 (3) & 0.5 \\
W7 & 13 & -12.5 to -10.5 (8) & 1.4 & \ldots & \ldots \\
W6 & 25 & -12.1 to -12.0 (3) & 0.9 & \ldots & \ldots \\
W5 & 7 & -14.6 to -13.9 (7) & 3.7 & \ldots & \ldots \\
W4 & 9 & -13.2 to -11.9 (3) & 1.2 & \ldots & \ldots \\
W3 & 73 & -17.3 to -11.2 (57)$^{*}$ & 5.5 & -6.5 to -5.3 (15) & 0.3 \\
W2 & 6 & -16.4 to -12.5 (5)$^{*}$ & 4.3 & \ldots & \ldots \\
W1 & 47 & -17.5 to -12.0 (47)$^{*}$ & 7.5 & -5.6 to -5.5 (3) & 0.5 \\
C & 6 & -21.0 to -13.5 (3)$^{*}$ & 2.9 & -6.5 to -6.3 (2) & 0.6 \\
N3 & 9 & \ldots & \ldots & \ldots & \ldots \\
N2 & 5 & -21.5 to -11.8 (2)$^{*}$ & 3.5 & -4.5 to -3.8 (2) & 0.6 \\
N1 & 2 & -11.1 to -10.0 (2) & 4.1 & \ldots & \ldots \\
S1 & 5 & -14.0 to -13.2 (5) & 1.0 & \ldots & \ldots \\
S2 & 27 & -13.8 to -11.3 (8) & 0.2 & \ldots & \ldots \\
S3 & 35 & \ldots & \ldots & -3.1 to -2.9 (15) & 1.0 \\
S4 & 3 & -12.4 (1) & 0.2 & -3.0 (1) & 0.6 \\
E1 & 54 & -13.7 to -12.2 (2) & 0.5 & \ldots & \ldots \\
E2 & 21 & -13.7 to -12.2 (14)$^{*}$ & 1.4 & -4.3 to -2.5 (6) & 0.3 \\
E3 & 76 & -13.3 to -10.2 (8)$^{*}$ & 0.7 & \ldots & \ldots \\
Esouth & 5 & -14.3 (1) & 1.7 & \ldots & \ldots \\
Enorth & 4 & -15.6 to -12.7 (3) & 0.4 & \ldots & \ldots \\
E4 & 29 & -14.5 to -10.3 (22)$^{*}$ & 0.6 & \ldots & \ldots \\
\noalign{\smallskip}
\hline
\end{tabular}
\tablefoot{
\tablefoottext{*}{Two or three separate $V_{b}$ components present at one or several positions.}
}
\label{tab:OSOvel}
\end{table*}
We made Gaussian fits to the line profiles, from which we extracted peak antenna temperature $T_\mathrm{A}^*$ (outside the earth atmosphere), central velocity, FWHM line widths, and for more complex profiles we decomposed the signals into separate components. The 1 $\sigma$ noise level at 0.2~km~s$^{-1}$ resolution is normally $<$ 40 mK, but reaches above 200 mK at several offset positions in W1, 6, 7, and E2. The $V_{b}$ and $V_{r}$ components are as rule rather narrow, and in most cases the line widths are $<$ 1.5~km~s$^{-1}$ (FWHM).
For most areas in Fig.~\ref{fig:171fields} we defined several central positions from which offset observations were made. The details of observed central positions and offsets can be found in Tables~\ref{tab:OSOpos} in the Appendix, where Column 3 gives the total number of observed positions for each sub-area. In Column 4 all offsets in RA are given in arcminutes, and for each of these the observed offsets in declination (Dec.) are given in Column 5. In Columns 6 and 7 we list those offsets in Dec. (for each offset in RA) where we have detected a $V_{b}$ component ($\leq$ --10~km~s$^{-1}$) and a $V_{r}$ component (velocity interval --2 to --6~km~s$^{-1}$). Non-detections are marked with a dots.
The amount of data so obtained is extensive since $V_{b}$ components are present in the majority of the positions, and for an overview we have chosen to present the ranges of different parameters for each area in Table~\ref{tab:OSOvel}. Here, the total number of observed positions is given in Column 2. The range in velocity for the $V_{b}$ component is given in Column 3 with the number of positions where it was detected in parentheses. Column 4 gives the maximum peak antenna temperature for the $V_{b}$ component observed in the area. Within each area the range in $T_\mathrm{A}^*$ can be large. Columns 5 and 6 gives the corresponding values for the $V_{r}$ component.
Finally, another circumstance to consider when interpreting the data is that there is a system of molecular clouds that move at high negative velocities extending several degrees along the galactic plane. This system was mapped in $^{13}$CO by \citet{yon97}, and in Fig.~\ref{fig:CepCO} we have plotted their positions for clouds moving at $\leq$ --12~km~s$^{-1}$ together with the corresponding positions in our survey for two areas located outside the defined extent of the \ion{H}{ii} region. Hence, this system of clouds have velocities in the same range as the $V_{b}$ components in the central area. This therefore opens the question of which clouds are associated with the S 171 nebula and which are not.
None of these clouds at significant negative velocities appear as distinct dark clouds in optical images, and none is listed in the catalogue of dark clouds in {\citet{lyn62}. It appears that this system of clouds are background objects, possibly at the far side of the Orion arm, where gas can be expected to stream into the spiral arm. We suppose that the most eastern cloudlets in our survey, areas E4 and Esouth:E marked in Fig.~\ref{fig:CepCO}, may belong to the same system.
For the central clouds in Fig.~\ref{fig:171fields}, residing in areas from W10 to E3 (west to east) and S4 to N1 (south to north), there is no doubt that all belong to the dusty shell surrounding Be~59. Some of these clouds are seen in silhouette against the bright nebulosity and have bright rims and associated elephant trunks, and most individual cloudlets are connected to each other with dark nebulosity.
\begin{figure}[t]
\centering
\includegraphics[angle=00, width=9cm]{CepCOnew.pdf}
\caption{Positions in galactic coordinates of molecular cloudlets moving at high negative speed ($\leq$ --12~km~s$^{-1}$) in an extended area outside the central part of the S~171 complex (left image from Fig.~\ref{fig:171fields} inserted for reference). Open squares are cloudlets from \citet{yon97} and filled squares are E4 and Esouth:E from our survey. }
\label{fig:CepCO}
\end{figure}
\section{Discussion}
\label{sec:discussion}
\subsection{Kinematics of the molecular shell}
\label{sec:patterns}
\begin{figure}[t]
\centering
\includegraphics[angle=00, width=1\linewidth]{RVpattern.pdf}
\caption{Radial velocities relative to the cluster velocity for different sub-areas against angular distance from the centre (in all directions). Blue dots mark areas that are expanding at high velocity, $v_{exp}$ $\approx$ 12~km~s$^{-1}$, and yellow circles (smaller) areas expanding at moderate speed, $v_{exp}$ $\approx$ 4~km~s$^{-1}$, when assuming spherically expanding systems (dotted curves). Green circles mark areas with intermediate velocities (`DQ' is the Dancing Queen). Red squares show areas with components whose velocities are redshifted relative to the cluster. }
\label{fig:RVpattern}
\end{figure}
\begin{figure*}[t]
\centering
\includegraphics[angle=00, width=8.5cm]{wise12mu_planck_857GHz_pos.png}
\includegraphics[angle=00, width=9.5cm]{RVcentral_new.png}
\caption{ Map of cloudlets. Left: Overview of the region shown in Fig. \ref{fig:OBass} in 12~$\mu$m from WISE (grey scale) with the Planck 857 GHz map (black contours) overlaid to emphasise the dust emission. A black circle spanning a radius of 1.6$^{\circ}$ is centred on the Berkeley~59 cluster. The blue dots from Fig.~\ref{fig:RVpattern} are shown as blue circles and mark the locations where the high negative expansion velocities are found. Likewise, the green circles refer to the intermediate and the yellow the low negative velocity components, while the red circles mark positions where the positive component is found, possibly related to the expansion on the far side. Right: Close-up 55$\arcmin \times$ 40$\arcmin$ view of the central region where the red, green, blue coding is WISE 12 $\mu$m (red), 4.6 $\mu$m (green), and Digitized Sky Survey 2 red plates (blue). The positions of V747~Cep and the IR star are marked with white diamonds. }
\label{fig:RVpositions}
\end{figure*}
The RV pattern over the molecular gas in the S 171 region is complex, as noted also in previous investigations referred to in Sect.~\ref{sec:S171}. In several areas listed in Table~\ref{tab:OSOvel} we find two distinct $V_{b}$ components (marked with an asterisk) at certain positions indicating that there are at least two systems of expanding clouds in the foreground of the complex. The lines are narrow but their velocities can change over small areas over our grids. When observed with a larger beam these components merge, and this appears to be the case for the broad plateau emission observed with a beam of 2$\arcmin$.7 in \citet{yan92} in W1 and W2 (their Fig. 3), which is resolved in our maps into narrow components drifting in velocity in RA and Dec.
As described in Sect.~\ref{sec:COvel} individual spectra usually show several velocity components. Besides the two groups of components that we considered as not related to the S~171 complex we distinguish four kinematically separated groups related to the expanding shell. In Fig.~\ref{fig:RVpattern} these components are plotted as RV relative to the mean cluster velocity of --9.5~km~s$^{-1}$ against angular distance from the centre for different sub-areas (listed in Table B), where a given component was detected. Each point represents an average from all detections within a given sub-area. The spread in velocity within a given sub-area is small, $\leq$~1~km~s$^{-1}$.
The blue dots represent structures expanding at the highest speed in our direction when assuming a spherical shell expanding at $v_{exp}$ = 12~km~s$^{-1}$ and a radius of 31 pc (dotted curve). The yellow dots are consistent with a more moderate expansion velocity of $v_{exp}$ = 4~km~s$^{-1}$. The red squares in Fig.\,\ref{fig:RVpattern}, represent components that are red-shifted relative to the cluster by a few km~s$^{-1}$. Finally, we identify yet another component present in the central region (W1, W2, and W3) with velocities between the high and moderate velocity groups (marked with green circles). In these spectra the components associated with moderate velocities (yellow dots) are also present. The positions of clouds with implied high expansion velocities are shown as blue circles in both panels in Fig.~\ref{fig:RVpositions} and those at intermediate velocities are all in the right panel as green circles.
While the blue dots in (Fig.~\ref{fig:RVpattern}) are reasonable well described by a spherically system of clouds expanding at high velocity and located at the outskirts of the \ion{H}{ii} region, the case is different for clouds moving at moderate velocities (yellow points). Not only is this system less extensive than the high-velocity system, also the outermost points fall below the curve assuming spherical expansion and a similar radius. This cut off is indicated by the dashed curve, where RVs approach zero km~s$^{-1}$ at a radius of $\sim$ 25 pc. Hence, it appears that this system does not extend as far out from the centre as the high-velocity system. Moreover, it appears to have a counterpart in clouds moving at similar velocities but red-shifted relative to the cluster (red squares in Fig.\,\ref{fig:RVpattern}). These red-shifted clouds are also confined within a projected distances of 25 pc from the centre. A natural interpretation is that they are part of the same expanding shell and that we detect molecular emission both from its near and far sides. It is hard to judge if there is also a remote counterpart to the high-velocity component because it would fall in the spectral region where the local components contribute. However, we note that at some positions such components are split in two.
Clouds moving at moderate negative velocity are found in all directions over the nebula including also the dark dust cloud associated with NGC 7822. In contrast, the high-velocity clouds are found only along a band extending from west to east and crossing the centre. Sub-area Enorth:C is located just outside the adopted radius of the complex (beyond 31 pc in Fig.~\ref{fig:RVpattern}), and there is a small cloud in the foreground of the central cluster (sub-area C:C). The area over which the intermediate velocity component is detected coincides in part with the peak intensity of the continuous radio emission as mapped in \citet{har81}, outlined as yellow ovals in the Fig.~\ref{fig:RVpositions}.
To summarise, the bulk of the molecular gas in front of the bright nebulosity expands at moderate velocity. However, we also localised a system of smaller cloudlets, which appear to expand with a higher velocity than expected from simulations, although slower than could be inferred at start from some estimates of the central velocity of the complex (Table~\ref{tab:Be59}). These are the cloudlets marked with blue dots in Fig.~\ref{fig:RVpattern} and blue circles in Fig.~\ref{fig:RVpositions}.
\subsection{Masses of shell structures}
\label{sec:masses}
Our $^{13}$CO observations do not cover the entire area of the S~171 complex. Nevertheless, we made an attempt to obtain rough estimates of the mass of individual cloudlets in the shell by the same method as used for deriving the masses of elephant trunks in \citet{gah06}. In short, the total mass of a given cloud is
\begin{equation}
M \mathrm{[M_{\sun}]} = 1.35 \times 10^{-11}\>F(T_{ex})\>D^{2} A\>\langle I_{13}\rangle \,,
\end{equation}
\noindent
where D [pc] is the distance, A [arcsec$^{2}$] is the cloud area, and $\langle I_{13}\rangle$ is the average integrated $^{13}$CO intensity over the area. We have assumed an excitation temperature $T_{ex}$ = 23 K, which gives a dependence $F(T_{ex})$ = 30 (for instance \citealt{nik01}).
We then treated clouds with high and moderate expansion velocities separately. For some clouds we included also areas outside our grids with similar prominent extinction and assuming the same value of the average integrated intensity. Most of the mass in the foreground shell is concentrated in the very dark clouds obscuring the areas of maximum continuous radio emission just to the west of the cluster, including areas W1, S1, W2, and W3 (Fig.~\ref{fig:RVpositions}). These areas are well sampled and covered also by \citet{yan92}, who derived masses for two `clumps' C1 (our positions W2 and W3) and C2 (our positions W1). In comparison with their results our estimates are similar for C2 but larger by a factor of 3 for C1, when the different adopted distance to the complex is taken into account. We regard this as a reasonable agreement, and also for area N1 in NGC 7822 our derived mass of $\sim$ 300 $M_{\sun}$ is similar (a factor of 1.5 larger) to the value quoted in \citet{elm78}. Our higher values are more consistent with column densities inferred from the large visual extinction of these clouds. For the clouds emitting the intermediate velocity components we estimate a mass of $\sim$ 200 $M_{\sun}$.
Based on our estimates the total mass of the shell in front of the \ion{H}{ii} region is of the order of 2200 $M_{\sun}$, of which $\sim$ 1500 $M_{\sun}$ is contained in the central areas described above. There is a considerable uncertainty in our mass estimates and more complete surveys of other molecular transition useful for mass determinations are warranted. Areas W9, S3, S4, and N3 do not show any components at high or moderate velocity. Either these are yet undisturbed clouds in the foreground of the nebula, or the lines are too weak for detection, which can be the case also in the relatively large areas E1 and E3, where lines at moderate expansion velocity are detected just over smaller parts of the maps.
\begin{table}
\centering
\caption{Estimated properties for the clouds with high expansion velocity. The mass estimates are subject to considerable uncertainty.}
\begin{tabular}{lcccc}
\hline
Position & Velocity & Projected size & $\langle I_{13}\rangle$ & Mass \\
& (km/s lsr) & 10$^3$ arcsec$^2$ & (K km/s) & (M$_{\sun}$) \\
\hline
W10A & -13.8 & 15.1 & 9.1 & 67 \\
W10B & -13.3 & 5.4 & 3.0 & 9 \\
W8 & -18.2 & 10.8 & 0.9 & 5 \\
C:C & -21.0 & 3.6 & 4.1 & 7 \\
N2A & -21.5 & 9.5 & 10.1 & 47 \\
EnorthB & -15.6 & 3.6 & 1.8 & 3 \\
EnorthC & -12.9 & 3.6 & 0.9 & 2 \\
\hline
\end{tabular}
\label{tab:cloudlets}
\end{table}
All cloudlets that we identify as moving at the highest expansion velocities have low masses. Among these, W10A and N2 are the most massive, $\sim$ 70 $M_{\sun}$ and $\sim$ 50 $M_{\sun}$, while the others have very low masses, $\leq$ 10 $M_{\sun}$. As will be discussed in the subsequent sections, this may be the reason why these cloudlets obtained a larger speed and reached farther out from the centre than the heavier clouds moving at moderate velocities.
\subsection{Dynamical evolution of the complex}
\label{sec:dynamics}
\begin{table}
\centering
\caption{Physical parameters for the cluster and expanding shell.}
\begin{tabular}{lcc}
\hline
\noalign{\smallskip}
Parameter & Value & References \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\noalign{\smallskip}
{\it Cluster} & & \\
\noalign{\smallskip}
L$_{t}$ [10$^{6}$ L$_{\sun}$] & 1.8 & 1, 2 \\
N$_{Lyc}$ [10$^{49}$ sec$^{-1}$] & 6.8 & 2 \\
\noalign{\smallskip}
\noalign{\smallskip}
{\it \ion{H}{ii} region} & & \\
\noalign{\smallskip}
radius [pc] & 31 & 1 \\
T$_{e}$ [K] & 8200 & 3 \\
n$_{e}$ [cm$^{-3}$] & $\sim$ 30--40 & 4 \\
\noalign{\smallskip}
\noalign{\smallskip}
{\it Foreground clouds} & & \\
\noalign{\smallskip}
v$_{exp}$ [km s$^{-1}$] & 4 and 12 & 1 \\
total mass [M$_{\odot}$] & 2200 & 1 \\
\noalign{\smallskip}
\hline
\end{tabular}
\label{tab:parameters}
\tablefoot{1. This work; 2. \citet{ster03}; 3. \citet{ros80}; 4. \citet{har81}. }
\end{table}
The expansion of the S~171 region is driven by the interaction from the hot members in the Be 59 cluster. Some quantities related to the physical conditions in the complex are listed in Table~\ref{tab:parameters}, where the first entries contain estimates of the total luminosity, L$_{t}$, and the total flux of Lyman continuum photons, N$_{Lyc}$, for all cluster members earlier than B1. The five O stars dominate the fluxes, and the contributions from members later than B0.5 are negligible. According to \citet{pan08,lat11,panw18} the ages of pre-main-sequence stars in the general area around Be~59 indicate a spread from 0.5 to 5 Myrs. However, it is also noted that the massive stars in the cluster are not significantly evolved and with inferred ages consistent to what is assumed in the present study.
The middle panel contains estimates of the electron temperature, $T_{e}$ and electron number density, $n_{e}$, related to the physical state of the plasma in the central part of the \ion{H}{ii} region followed by properties of the molecular shell in front of the nebula as estimated above. Here, the total mass of the expanding molecular shell on the remote side of the complex could not be estimated.
As shown in both panels of Fig.~\ref{fig:RVpositions}, the areas of maximum continuous radio emission are coinciding with some highly obscured foreground clouds, as W1 and W2, which are offset to the west relative to the central cluster. This enhanced emission is most likely related to photodissociation regions (PDRs) hidden behind the dark clouds and excited by UV light from massive stars in Be~59. Possibly, a main contribution comes from the most luminous O star (V747 Cep) and the heavily obscured O star (the IR star), which are also offset to the west (by $\sim$4 pc in projected distance).
It is also over this general area our OSO spectra contain lines from the intermediate velocity component (green circles in Fig.~\ref{fig:RVpositions}).
A similar type of interaction takes place at the periphery of the \ion{H}{ii} region north of the cluster, where the bright nebula NGC 7822 borders a massive and opaques cloud, N1, only that in this case the PDR is visible as a bright nebula. The star marked with a circle just south of NGC 7822 in Fig.~\ref{fig:OBass} was noted as a member of the Cep OB4 association by \citet{mac68}. This star (MacC 9 = V399 Cep) of spectral type B2 is at the right distance to the complex according to Gaia DR 2, but we conclude that it plays a minor role for the excitation of NGC 7822 in comparison to the central O stars.
The ionised gas is thermal \citep{ang77,ped80}, and it is unlikely that the large expansion velocities observed has been caused by an early supernova explosion, which besides would have disrupted the entire molecular shells.
In Sect.~\ref{sec:patterns} we distinguished three systems of expanding shell structures, one containing rather massive clouds moving at 4 km s$^{-1}$ relative to the cluster and presumably with a counterpart at the remote side of the complex, another system with predominantly smaller cloudlets expanding at 12 km s$^{-1}$, and one central area where some clouds move with velocities between these two.
The 4~km~s$^{-1}$ component is in line with expectations from some numerical simulations of evolving \ion{H}{ii} regions with comparable sizes at an age of 2 Myrs \citep[e.g.][]{fre03,hos06}. On the other hand, the 12 km s$^{-1}$ component traces a system of clouds moving outwards at a considerably higher speed compared to theoretical predictions assuming similar total luminosities and ionising fluxes as in Table~\ref{tab:parameters}. As discussed above, this system of clouds extends farther out from the centre and contains cloudlets of smaller mass than those expanding at moderate velocities. For the high-velocity clouds the crossing time to reach from the centre to the periphery of the \ion{H}{ii} region at constant speed is 2.5 10$^{6}$ yrs, which is in line with the cluster age.
Below, we consider as a working hypothesis that the high-velocity clouds were confined and detached from the more massive structures in the expanding shell early in the expansion of the shell. In our picture, the shell is very clumpy and contains small overdense clumps. The entire shell is decelerated by accumulation of material from the ISM, but dense clumps have more inertia and are more difficult to slow down. The tiny cloudlets retained a higher speed and are now located farther out than the shell, even outside the border of the \ion{H}{ii} region (Fig.~\ref{fig:RVpattern}). The clouds are confined within a relatively narrow belt extending from the western edge of the \ion{H}{ii} region over the central part to the eastern edge. The cause for this asymmetry is unclear.
\section{Numerical simulations of the complex}
\label{Sect:Simulations}
In this section we explore to what extent we can match the complex velocity pattern observed for the molecular gas in S 171 with a theoretical model simulation. As discussed in Sect.~\ref{sec:dynamics}, we have identified a system of high-velocity cloudlets of relatively small masses that have reached farther out from the centre than the more massive cloud structures that form the shell (Fig.~\ref{fig:RVpattern}). We attempt to include both these cloud systems in our simulation.
Our simulation describes how a cloud of gas is affected by the stellar winds of an embedded star cluster. This requires coupling between hydrodynamics (which describes the dynamics of the gas), gravity (which describes the dynamics of the stellar cluster), and stellar evolution (which describes the mass evolution and stellar wind output of the individual stars), plus a prescription for stellar winds.
\subsection{Multiphysics stellar wind bubble model}
\subsubsection{Model}
We simulate hydrodynamics using the {\sc fi} code \citep[][based on \citet{her89,ger04}]{pel04}, which is based on the smoothed particle hydrodynamics (SPH) formalism. Stellar winds are implemented using the prescription by \citet{helm19}, which inserts new SPH particles corresponding to the stellar wind into the hydrodynamics code. The stellar parameters required to describe the winds' mass loss rate and velocity are prescribed by the {\sc seba} code \citep[][with the adjustments by \citet{too12}]{por96}, a parameterised stellar evolution code. The stars' dynamics are simulated using the {\sc huayno} code, a second-order symplectic, direct N-body code \citep{pel12}. All of these codes were brought together in the Astrophysical MUltipurpose Software Environment \citep[AMUSE;][]{por18}. Finally, the gas and star cluster were able to affect each other through Bridge interactions \citep{fuj07}, which resolved gravitational interactions between the two components every 1 kyr, while leaving the internal evolution of each component unaffected.
Our initial conditions consist of a number of gas and stellar components. We include a main gas cloud of 3000 M$_\odot$ (to take into account unobserved material behind the \ion{H}{ii} region) and a virial radius of 6 pc distributed following a Plummer profile \citep{plu1911}, which represents the centrally concentrated gas left over from star formation. We place two cloudlets of 10 and 50 M$_\odot$ (Plummer spheres with virial radii of 0.5 pc) at rest at $x=\pm 3$ pc, representing two cloudlets that have already detached from the main gas distribution. The velocity fields of these Plummer spheres are virialised and non-turbulent.
The stellar component consists of $10^3$ stars between 0.5 and 16 M$_\odot$ following a Salpeter initial mass function, and distributed according to a 2 pc virial radius Plummer sphere. Additionally, we include a collection of massive stars representative of the O-type (and giant B-type) stars observed. We used \citet{ster03} to associate a mass with the stellar type of each massive star. The mass assumed for every star is provided in Table \ref{tab:massivestars}. We do not initialise these stars as binaries, but we do concentrate them within the inner 0.1 pc of the cluster.
\begin{table}
\centering
\caption{Massive stars in the simulation with their stellar types and corresponding masses from \citet{ster03}.}
\begin{tabular}{lcc}
\hline
\noalign{\smallskip}
Star & MK class & Mass M$_\odot$ \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\noalign{\smallskip}
V747 Cep A & O5.5V & 50.4 \\
V747 Cep B & O6V & 45.2 \\
BD+66$^\circ$ 1675 A & O6V & 45.2 \\
BD+66$^\circ$ 1675 B & O8V & 30.8 \\
P16 A & O7V & 37.7 \\
P16 B & B? & 30 \\
IR star A & O9V & 25.4 \\
BD+66$^\circ$ 1674 A & B0 III & 27.4 \\
\hline
\end{tabular}
\label{tab:massivestars}
\end{table}
\subsubsection{Results}
We ran this simulation for 2 Myr at a resolution of about $3\cdot 10^{-3}$ M$_\odot$ per initial SPH particle (and $10^{-5}$ M$_\odot$ for the stellar wind particles). Figure~\ref{fig:hydro_maps} presents a number of snapshots of the simulation. The stellar winds from the centrally concentrated massive stars open a cavity in the gas cloud that expands with time and sweeps up the detached cloudlets. In the bottom panel, we can see how the core of the more massive cloudlet (50 M$_\odot$) lags behind the shell and deforms the shell immediately around it. This implies that detached cloudlets cannot be accelerated to velocities beyond that of the shell purely due to stellar winds.
Deviations from a spherical shell have also arisen where stellar winds break through due to slight asymmetries, but these do not strongly affect the estimate of the shell radius.
\begin{figure}[t]
\centering
\includegraphics[width=0.93\linewidth]{EvGMC_Map_Initial.pdf}
\includegraphics[width=0.93\linewidth]{EvGMC_Map_Cavity_i12.pdf}
\includegraphics[width=0.93\linewidth]{EvGMC_Map_Shell_i30.pdf}
\caption{ Gas column density maps of the numerical simulation at 0.01 Myr (top), 0.12 Myr (centre), and 0.30 Myr (bottom). The top panel also shows the projected positions of the stars more massive than 16 M$_\odot$ (black points) and those less massive than 16 M$_\odot$ (white points). The bottom panel shows the projected positions of the sink particles as white points. The arrows point out the 50 M$_\odot$ (left) and 10 M$_\odot$ (right) cloudlets. The centre and bottom panels show the position of the shell as a black dotted circle. }
\label{fig:hydro_maps}
\end{figure}
Our simulation does not include an ambient ISM, which typically dampens the expansion of a stellar wind bubble as more ISM is swept up in the bubble's outer shell \citep{wea77}. Without this damping, the shell reached a terminal velocity of $\sim$60 km/s around 0.3 Myr.
Our multiphysics model converts gas particles into sink particles when they reach a density of $3.3\cdot 10^6$ amu cm$^{-3}$ to prevent numerical errors. These sink particles interact with the gas component only through gravity, not through any hydrodynamical interaction. In the bottom panel of Fig.~\ref{fig:hydro_maps}, we show all sink particles as white points. A number of sink particles form radially directed lines, with the most prominent trailing the 50 M$_\odot$ globule. This is reminiscent of the pillars observed at the interface between some \ion{H}{ii} regions and molecular shells, which \citet{tre12a} propose form when the curvature of the shell becomes high and the shell collapses in on itself. The globules we inserted into the initial conditions are natural places for such a curvature to develop. As the pillar collapses our model converts it into sink particles. In contrast with a sink particle, a real pillar is able to interact hydrodynamically with its surroundings, but the fact that they lag behind the shell implies that they decouple from the processes that drive the expansion and interact with their surroundings in a similar way to sink particles.
\citet{tre12b,tre13} also discuss another structure observed near such shells, namely dense globules just within the \ion{H}{ii} region. These structures form from interactions with turbulent material outside the shell. Because our simulations do not include turbulent gas these structures do not form.
Because stellar wind bubbles expand into the ISM supersonically, the ISM beyond a certain shell has not influenced its evolution up to that point. As a result, we could use the state of the simulation at any moment as the initial conditions of a new simulation with an arbitrary ISM distribution outside the shell. Re-simulating a large number of simulation snapshots with additional ISM outside the shell would be prohibitively expensive in terms of computer resources, because the entire ISM must be represented by SPH particles. The volume required to represent S 171 and to reduce edge effects (due to the absence of SPH particles beyond the simulation box) would be so great that the number of SPH particles required would be much greater than that of the shell.
\subsection{Simplified numerical wind bubble model}
\subsubsection{Model}
Instead, we turn to a simplified (and computationally cheaper) numerical approach to a similar problem, which we can apply to the state of our simulation at different times. In our simulation, stellar winds quickly form a mostly spherical bubble in the gas cloud that is much larger than the collection of massive stars. We can approximate this as a wind bubble blown by a star with the stellar wind output of all massive stars combined (although losses in wind shocks between stars can decrease the effective input of energy and momentum into the shell).
The classic solution to this problem is the similarity solution of \citet{wea77}, which describes the spherical expansion of a stellar wind bubble in a uniform ISM. More recently, \citet{lan21a} presented a modified solution taking into account the fractal structure of the bubble's boundary and the subsequent efficient radiative losses. We cannot use their solutions directly because our shell consists of a swept-up Plummer sphere, not a swept-up uniform ISM, but we can use the differential equations to which they are a solution.
Both models are based on the conservation of momentum, equating the momentum of the shell to that imparted by stellar winds. Although they differ in the prescription of how momentum is transferred to the shell, they are both of the following form:
\begin{equation}
\frac{d}{dt}\left( M_s\left(R_s\right)\dot{R}_s \right) = \eta\dot{p}_w,
\end{equation}where $M_s\left(R_s\right)$ is the mass of the shell once it has reached radius $R_s$, $\dot{p}_w$ is the shell's momentum input rate due to the stellar winds, and $\eta$ is an efficiency factor. \citet{lan21a} introduce two different factors, $\alpha_p$ and $\alpha_R$, that account for enhanced momentum transfer and non-spherical expansion, respectively. Both of these factors are variable but of order unity \citep[as demonstrated by][]{lan21b}. For simplicity, we combine these into one value, which can additionally account for losses in momentum transfer due to stellar wind shocks between stars.
In a uniform ISM of density $\rho_0$, the mass of the shell is simply $M_s=\frac{4\pi}{3}R_s^3\rho_0$, resulting in a neat power law solution. In the case of our model, the density distribution follows a Plummer sphere interior to the initial shell radius, $R_{s,0}$. The density distribution outside this radius is undefined as of yet. For the sake of simplicity we assume a uniform ambient ISM density distribution. The shell mass is then
\begin{equation}
M_s\left(R_s\right) = \begin{cases}
M_P\left(<R_s\right) \quad\quad\quad\quad\quad\quad\quad\quad\quad \textrm{ if } R_s < R_{s,0} \\
M_P\left(<R_{s,0}\right) + \frac{4\pi}{3}\left( R_s^3 - R_{s,0}^3 \right)\rho_0 \textrm{ if } R_s > R_{s,0},
\end{cases}
\end{equation}where $M_P\left(<R\right)$ is the mass of a Plummer sphere within a radius $R$ and the ambient ISM density is $\rho_0$.
The resulting differential equations lack neat power law solutions, and require numerical integration. We use fourth-order Runge-Kutta integration for this, with time steps equal to $0.01\cdot\textrm{min}\left(x_i/\dot{x}_i\right)$, where $x_i$ denotes a vector of all integrated variables (e.g. shell radius and velocity).
Given a simulation snapshot, we can now continue the evolution of the wind bubble using this method. As initial conditions, we need the shell's radius and velocity, and in the case of the Weaver model, the internal energy contained within the shell radius. We define the shell's radius as the location of the peak of the histogram of all gas particle's distance from the origin (and the shell's velocity in a similar way). The mechanical luminosity of the central wind source is fixed by the stellar population. Finally, we set the ISM density exterior to the shell to be equal to the density of the main Plummer sphere at the shell radius. This ensures continuity in the density distribution, and uniquely relates each shell radius to an ISM density.
If the evolution of the bubble using the simplified numerical model reaches the observed shell radius and velocity at the same moment in time, the combined model is consistent with the observations. This moment in time is then our estimate of the system age, and the density of the main Plummer sphere at the initial shell radius is our estimate of the ambient ISM density. Because we have a limited set of initial conditions, the shell radius and velocity will never match the observed values at the exact same moments. However, for every set of initial conditions (each corresponding to an ISM density), we can record the times at which the observed shell radius and velocity are reached, and find the intercept of these two curves.
\subsubsection{Results}
Figure~\ref{fig:numerical_evolution} shows the evolution through time of the shell radius (orange) and velocity (blue) following the \citet{wea77} (solid) and \citet{lan21a} (dashed) models. We note that all times quoted in this section are with respect to the start of the multiphysics simulation. For comparison, we have also plotted the analytic solutions of the Weaver (dotted) and Lancaster (dash-dotted) models using the same stellar wind output and ISM density. The initial conditions of the numerical solutions are offset from these solutions, but the numerical solutions do converge to the analytic solutions. This is more obvious for the Weaver solution than the Lancaster solution; in the latter case, the shell radius is initially too large, and the velocity must be below the analytic shell velocity so the radius can eventually converge to the analytic shell radius. Further evolution of the system does show that the numerical solution converges to the analytic solution.
The shaded regions denote the radius and velocity of the massive, moderate velocity shell component (25 pc and 4 km/s, respectively) with 10\% errors. For the initial conditions used for this plot, the combined multiphysics and Lancaster models agree with the observations because both shell radius and velocity cross the observed values at the same moment, at about 2 Myr. Notably, the velocity of the Weaver solution is much greater than observed, and does not reach the observed value even at 10 Myr. For this instance of the simplified model, we used the state of the multiphysics model at 0.23 Myr, when the shell radius was 6.5 pc, the shell velocity 41 km/s, and the ISM density at the shell edge was 19.4 amu cm$^{-3}$.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{EvGMC_semianalytic_wind_WL_EmpE.pdf}
\caption{ Evolution through time of the stellar wind bubble radius (orange) and velocity (blue) using the \citet{wea77} (solid) and \citet{lan21a} (dashed) models, starting from the bubble's radius and velocity at 0.23 Myr (with a corresponding density at the shell of 16.0 amu cm$^{-3}$). The efficiency $\alpha$ parameters in the Lancaster model have both been set to 1. The dotted and dash-dotted lines denote the analytical Weaver and Lancaster solutions, respectively. The shaded regions show 25 pc and 4 km/s, with 10\% errors. }
\label{fig:numerical_evolution}
\end{figure}
Fig.~\ref{fig:density_age} shows, for a number of simulation snapshots, the density of a Plummer sphere at the shell radius and the moments that the numerically integrated (using the Lancaster model for the top panel, and the Weaver model for the bottom panel) shell radius and velocity reach values of 25 pc and 4 km/s (corresponding to the massive, moderate velocity shell). The shaded regions denote the range of values allowed by the efficiencies shown and by letting the observed radius and velocity vary by 10\%. We point out that the two panels cover different ranges of efficiency $\eta$. This is expected to be of order unity, but in the case of the Weaver model there were no solutions with $\eta=1$ or $\eta=3$ across the available range of densities. The radius and velocity curves did approach each other at high densities, but did not cross at the greatest density possible, which corresponds to the density of the Plummer sphere's core (661 atomic mass units (amu) cm$^{-3}$).
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{EvGMC_phasespace.pdf}
\includegraphics[width=\linewidth]{EvGMC_phasespace_weaver.pdf}
\caption{ For a number of snapshots of the simulation, the density of a Plummer sphere at the shell position (assumed to be the ISM density) versus the moment at which in the numerical integration a shell velocity of 4 km/s and a radius of 25 pc is reached. Different symbols denote different stellar wind efficiencies $\eta$. Shaded regions denote the envelope allowed for all efficiencies plotted and a 10\% error in the observed velocity and radius. The top panel shows the results obtained using the Lancaster model, and the bottom panel shows the results from the Weaver model. }
\label{fig:density_age}
\end{figure}
From the Lancaster model, the ISM density inferred is in the range of $10-30$ amu cm$^{-3}$, slightly below the typical density of molecular clouds. The age is in the range of $2-2.5$ Myr, similar the age assumed for the system.
From the Weaver model, the ISM density inferred is in the range $30-300$ amu cm$^{-3}$, of the order of a typical molecular cloud. The age is in the range $2.5-4$ Myr, slightly older than estimated earlier. It should be noted that the lower ISM density and age correspond to a very low stellar wind efficiency, one that we have not explored with the Lancaster model.
We do not expect such a low efficiency to be plausible; \citet{lan21b} only finds an efficiency of similarly low order of magnitude for a very dense, very low star formation efficiency cloud. In their work, they use the cloud mass and star formation efficiency to compute an effective stellar wind mechanical luminosity. Inverting this, with our actual cloud mass and mechanical luminosity, we get an effective star formation efficiency of $\sim$50\%, which implies a stellar wind efficiency close to unity.
For comparison, we also present the results of matching the observed radii and velocities to the analytic solutions of Weaver and Lancaster. These results are in Table~\ref{tab:analyticsolutions}. We match both the massive, moderate velocity shell and the high-velocity cloudlets. This disregards the presence of leftover cluster material, but is independent of potential numerical errors in our simulation.
Fitting the Weaver model to the moderate velocity shell results in a relatively old age and a very dense ISM (about twice the density of the main Plummer sphere in our simulation), unless $\eta$ is very low. Fitting the Lancaster model, on the other hand, leads to a somewhat younger system age (though still much older than estimated for the system), and an ISM more diffuse than a GMC. For completeness, we also included the ages and densities resulting from fitting to the high-velocity cloudlets, although it is difficult to explain them as a shell when the moderate velocity shell is much more massive and closer in.
\begin{table}
\centering
\caption{System age ($t_{\textrm{obs}}$) and ISM density ($\rho_{\textrm{ISM}}$) estimates by solving the Weaver and Lancaster models for the observed radii and velocities of the shell (25 pc and 4 km/s, respectively) and cloudlets (31 pc and 12 km/s, respectively). These values were computed assuming $\eta=1$; different values for $\eta$ do not modify the age but do modify the density linearly. }
\begin{tabular}{lcc}
\hline
\noalign{\smallskip}
Model & $t_{\textrm{obs}}$ (Myr) & $\rho_{\textrm{ISM}}$ (amu cm$^{-3}$) \\
\hline
Weaver & 3.67 & $1.23\cdot 10^3$ \\
Shell & & \\
\hline
Lancaster & 3.06 & 6.74 \\
Shell & & \\
\hline
Weaver & 1.52 & $2.96\cdot 10^1$ \\
Cloudlets & & \\
\hline
Lancaster & 1.26 & $4.87\cdot 10^{-1}$ \\
Cloudlets & & \\
\hline
\end{tabular}
\label{tab:analyticsolutions}
\end{table}
We did not observe a clear velocity bimodality in the simulation. The amount of material with velocities greater than 2.5 times that of the shell itself (which would be an analogue to the observed high-velocity structures) is only $\sim$0.01\% of the total shell material, or $\sim$0.3 M$_\odot$, less massive than the individual high-velocity cloudlets. Additionally, as can be seen in Fig.~\ref{fig:hydro_maps}, the cloudlets that were detached from the main cloud at initialisation typically lagged behind the shell and have similar velocities to the shell itself, and so did not end up as a high-velocity component.
\subsection{Simplified numerical cloudlet model}
\subsubsection{Model}
\citet{lan21b} show that stellar wind bubbles blown in a uniform but turbulent medium are fractal shaped rather than uniform spherical shells. Even in our smooth\footnote{Up to the inherent discretisation in SPH particles.} and virialised Plummer sphere asymmetries arise, from secondary bubbles where the winds break through the shell into low-density material (shown in the top left and bottom right of the bottom frame of Fig.~\ref{fig:hydro_maps}) to substructures in the shell itself.
In the models above, the fundamental mechanism slowing down the expanding bubble is the accumulation of ISM material in the shell. The magnitude of this deceleration is related to the ratio of the masses of the shell and of the swept-up ISM. In a non-uniform shell, different sections will have different ratios, and as a result, different rates of deceleration. Viewed differently, a denser section of the shell will have more inertia and be able to maintain its velocity for longer than a less dense section. In this section we investigate whether this varying deceleration can result in the multiple components observed in S 171.
We derive the equation of motion of a section of a shell, or cloudlet (which we use from now on), through a uniform medium, using a similar method to \citet{lan21a}, that is, exploiting conservation of momentum.
This momentum equation is of the following form:
\begin{equation}
\begin{split}
& \frac{d}{dt} \left( \left( M_{c,0} + \rho_0 \int_{t_0}^{t} \sigma_c\left(t'\right) \dot{R}_c\left(t'\right) dt' \right) \dot{R}_c\left(t\right) \right) \\
& = \eta\dot{p}_w \frac{\sigma_c\left(t\right)}{4\pi R_c\left(t\right)^2},
\end{split}
\end{equation}where $M_c$ and $R_c$ are the cloudlet mass and distance to the cloud centre, respectively, and $\sigma_c$ is the cloudlet's cross-section as seen from the cloud centre. This is both the surface being struck by stellar winds, and the surface accumulating ISM material. Also, in this expression, and all hereafter, a subscript 0 denotes the initial value of a quantity.
The evolution of $\sigma_c$ is dictated by hydrodynamical processes, but for simplicity we consider two interesting edge cases: constant cross-section (corresponding to a bound, non-expanding cloudlet), and quadratically growing with distance (corresponding to a shell structure growing with the shell). These have the respective forms
\begin{equation}
\sigma_c\left(t\right) = \sigma_{c,0},
\end{equation}
\begin{equation}
\sigma_c\left(t\right) = \sigma_{c,0} \left(\frac{R_c\left(t\right)}{R_{c,0}}\right)^2.
\end{equation}
They reduce to the following differential equations in $R_c$, for the non-expanding and expanding cases, respectively
\begin{equation}
\ddot{R}_c = \left(\frac{\eta\dot{p}_w}{4\pi R_c^2} - \rho_0\dot{R}_c^2\right)/\left(\frac{M_{c,0}}{\sigma_{c,0}} + \rho_0\left(R_c-R_{c,0}\right)\right)
\end{equation}
\begin{equation}
\begin{split}
\ddot{R}_c = & \left(\frac{\eta\dot{p}_w}{4\pi R_{c,0}^2} - \rho_0\dot{R}_c^2\left(\frac{R_c}{R_{c,0}}\right)^2\right)/ \\
& \left(\frac{M_{c,0}}{\sigma_{c,0}} + \frac{\rho_0}{3R_{c,0}^2}\left(R_c^3-R_{c,0}^3\right)\right).
\end{split}
\end{equation}
We note that the cloudlet's initial mass and cross-section only appear as the ratio $\frac{M_{c,0}}{\sigma_{c,0}}$. This implies that a cloudlet's evolution is determined by its initial average (mass) surface density, and its actual dimensions are not important.
Another point to note is that the expanding case is equivalent to the Lancaster model when the initial average surface density of the cloudlet is the same as that of the shell.
\subsubsection{Results}
Figure~\ref{fig:globule_propagation} shows the distance and velocity of cloudlets of varying surface density, using the differential equations described above. The initial conditions for each efficiency were taken from the snapshots best coinciding with the observations of the shell after continued integration, and evolved until the moment the shell coincided with the observations.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{EvGMC_GlobulePropagation_L21.png}
\caption{ Distance to the centre of the complex (orange) and velocity (blue) of cloudlets as a function of initial average surface density, after evolution to 1.9, 2.17, and 2.3 Myr (for $\eta=0.3$, 1, and 3, respectively). Solid lines represent cloudlets that do not expand as they travel, i.e. bound structures. Dashed lines represent cloudlets that expand with the shell, with their cross-sections growing with the distance squared, i.e. features on the shell that grow with it. Symbols overplotted on these lines indicate the stellar wind efficiency $\eta$. The hatched shaded regions denote 31 pc and 12 km/s, the radius and velocity of the high velocity component of S 171, and the clear shaded regions denote 25 pc and 4 km/s, the radius and velocity of the moderate velocity component. All observed values are shown with 10\% errors. The surface densities and line-of-sight velocities (with respect to the cluster centre) of the cloudlets in Table \ref{tab:cloudlets} are shown as upward blue arrows. This velocity is a lower limit of their expansion velocity. The initial conditions were taken from the simulation snapshot at 0.22, 0.23, and 0.25 Myr (again for $\eta=0.3$, 1, and 3, respectively), which resulted in a bubble radius of 25 pc and a velocity of 4 km/s at the respective end time using the Lancaster model. The two vertical black lines denote the average surface density of the shell at 0.23 Myr (left) and the surface density at the core of a 50 M$_\odot$ (and 0.5 pc) Plummer sphere cloudlet.}
\label{fig:globule_propagation}
\end{figure}
Non-expanding cloudlets always reach farther distances and retain higher velocities than expanding cloudlets because they sweep up less of the ISM. This effect is most pronounced near the centre of the range of densities explored, where non-expanding cloudlets can reach distances about twice as far as expanding cloudlets and velocities about four times as fast. Compared to the average shell, non-expanding cloudlets with high surface densities can reach distances almost four times the shell radius and velocities about ten times as great.
We note that there is no density for which the distance and velocity match the observed high-velocity cloudlets. The configuration closest to matching the observations is a non-expanding cloudlet of about the same initial average surface density as the shell. However, this cloudlet's distance is more than 10\% greater than the observed distance, and its velocity is more than 10\% smaller than observed. Allowing a cloudlet to expand more slowly than quadratic would put it in between the expanding and non-expanding cases; this would alleviate disagreement with the distance, but increase disagreement with the velocity. Allowing the stellar wind efficiency to vary cannot alleviate this disagreement either, for similar reasons. However, assuming a slightly younger system age (at which the radius was smaller, and the velocity greater) could put the cloudlets in the observed range.
We also show the high-velocity cloudlets. We derive their average surface densities from Table \ref{tab:cloudlets} using a distance of 1.1 kpc, and correct the line-of-sight velocity for that of the cluster itself (-9.5 km/s). This is a lower limit on their expansion velocity, which can also have a component in the plane of the sky. In our spherical expansion scenario, the line-of-sight velocity is closer to the expansion velocity the closer a cloudlet is to the cluster centre. The cloudlets C:C and N2A are very close to the centre, while EnorthB, EnorthC, W10A, and W10B are nearly at the 30 pc outer circle of the projected shell. In terms of the surface density, the cloudlets are in the range of 20-200 M$_\odot$ pc$^{-2}$. This combination of velocity and surface density puts the cloudlets closer to the expanding limit than the non-expanding limit.
\subsection{Summary}
To summarise, through a hybrid numerical modelling approach we demonstrate that stellar winds from the massive stars of Be 59 can give rise to the complex gas structure of S 171. We used a coupled hydrodynamical, gravitational, and stellar evolutionary model to simulate the initial formation of a stellar wind bubble in the concentrated gas distribution left over from the formation of a star cluster. We then used simplified numerical models to continue the evolution of the wind bubble through uniform ISM of different densities in an effort to match the observed radius and velocity of the massive shell of S 171, and so provide an estimate of the system age and ambient ISM density. Finally, we introduced a new simplified numerical model to describe the dynamic evolution of shell substructures through a uniform ISM, and used it to show that dense and detached shell fragments are able to reach farther distances and higher velocities than the main shell.
We show that using the recent stellar wind bubble model by \citet{lan21a} to continue the bubble evolution results in more reasonable values for the system age and ambient ISM density compared to the classical model of \citet{wea77}, assuming reasonable values for the efficiency of momentum transfer. As derived from the Lancaster model, the system age is consistent with earlier estimates, and the ambient ISM density is reasonable for the larger surroundings of a star forming regions. The Weaver model requires very inefficient momentum transfer from stellar winds to the shell to reproduce the moderate velocity shell, and then results in a system age older than estimated earlier and a very dense ambient ISM.
We also show that variations in density within the shell can lead to components propagating through the ISM at different velocities. We are unable to reproduce the observed ratios between the distances and velocities of the observed structures, but the case of non-expanding cloudlets with initial surface density similar to the shell comes closest. These variations can arise naturally from fragmentation of the shell, as is demonstrated in our multiphysics simulation. If the shell is only able to partially fragment into small cloudlets we would naturally obtain high-velocity, low-mass cloudlets and a moderate-velocity, high-mass shell.
Our results qualitatively show that stellar winds can reproduce the gas structure of S 171, but we refrain from drawing strong quantitative conclusions. The SPH method of modelling hydrodynamics is infamously bad at modelling shocks, being unable to properly resolve thin shock layers. This makes our estimates of the shell radius and velocity that serve as the initial conditions of the simplified numerical models uncertain. Notably, the shell velocity is greater than expected from the Weaver and Lancaster models, even though the shell mass is greater due to the swept-up Plummer sphere. Although our model slightly favours the Lancaster model over the Weaver model, we also refrain from drawing conclusions about this.
A fully consistent reproduction of the system would require a grid based hydrodynamical model \citep[such as the one used by][]{lan21b} or a hybrid method (such as moving mesh, or meshless finite mass). Models that start from the collapse of a giant molecular cloud \citep[such as by][]{Wall20,gru21} can improve the authenticity of the gas distribution at the moment the bubble begins being blown. A full hydrodynamical simulation, including an ambient ISM, can also self-consistently demonstrate the origin of the dense cloudlets that end up at high velocity.
\section{Conclusions}
\label{sec:conclusions}
We have investigated the kinematics of an expanding \ion{H}{ii} region powered by a young stellar cluster for which the inferred expansion velocity appears to be higher than predicted in theoretical model simulations. However, published estimates of the central velocity of this complex can be highly divergent, and more detailed information on the motion of the surrounding shell is warranted. In this paper we draw attention to the nebula Sharpless 171 (with NGC 7822), which surrounds the cluster Berkeley 59. Optical spectra of 27 stars were obtained in the cluster area, providing a new estimate of the RV of the cluster. The velocity pattern over the molecular shell was mapped in $^{13}$CO(1-0) at OSO. From these data we deduce the expansion velocities of different cloud fragments in the shell relative to the cluster. As described below, we could define three systems of such clouds expanding at different rates. One system expands at a high rate, although not as high as one could expect from some of the published estimates of the central velocity.
We assigned a spectral class to each star, including the separate components in double-lined spectroscopic binaries. The majority of the stars had not been classified before, including one central O star hidden behind foreground dust.
From spectral classifications and RVs, plus existing photometric data and Gaia parallaxes, we selected 19 of the 27 stars as likely members of Be~59.
Repeated observations were done, especially for members, which revealed that five of the massive stars are spectroscopic binaries, of which one is double-lined and one triple-lined. We followed these stars over longer times and derived periods, semi-amplitudes, and systemic velocities. The most luminous O star is the eclipsing binary V747 Cep, and our RV curve agrees with the photometric period.
From the RVs of members we obtained a mean heliocentric velocity of -- 18.75 km~s$^{-1}$ (-- 9.5 km~s$^{-1}$ in lsr), which we define as the central RV of the complex. None of the members deviate significantly from the mean value.
The S 171 complex extends over 3.2$\degr$ in the sky, and for our $^{13}$CO mapping we primarily covered areas with distinct dark clouds. The velocity pattern over the nebula is complex, and in most directions several velocity components are present in the spectra. In addition to components that originate in gas not related to the complex, we identified two systems of shell structures expanding at different velocities. These components can be described as located at the peripheries of spherically expanding shells. One system, composed primarily of rather massive shell fragments, expands at a moderate velocity, 4~km~s$^{-1}$, and includes two velocity components that are related to the front and back sides of the complex. This system is confined within a radius of 25 pc. The second system expands at a much higher velocity, 12~km~s$^{-1}$, and is concentrated in a belt that crosses the entire \ion{H}{ii} region, extending to a radius $\geq$ 30 pc. In addition, we find some cloudlets in the central area with expansion velocities in between those of these systems.
The luminosity and UV flux is dominated by the central O stars. Our interpretation of the observed complex kinematic structure is based on the assumption that the expansion started about 2 million yrs ago, driven by radiation and winds from the central massive stars. Our model simulations show that cloudlets of smaller masses obtained very high initial velocities, while more massive shell fragments dragged behind.
The velocity and radius of the moderate velocity shell are generally consistent with theoretical predictions for the expansion of a stellar wind bubble driven by the cluster's population of massive stars.
The cloudlets observed with higher velocity and at greater distance from the massive shell can arise through interaction with the ambient ISM, which decelerates dense and bound substructures of the shell less efficiently than low-density substructures that co-expand with the shell.
Although our models are able to qualitatively reproduce the structure of S 171 for plausible parameters, we refrain from drawing quantitative conclusions about the value of these parameters. This requires further work using more accurate methods.
\begin{acknowledgements}
We thank the anonymous referee for helpful comments that led to an improvement of the paper.
This work was supported by the Magnus Bergvall Foundation and the L\"angmanska Kulturfonden in Sweden. We made use of the SIMBAD database operated at Centre de Donnees Astronomique de Strasbourg and data from the European Space Agency (ESA) mission {\it Gaia} (https://www.cosmos.esa.int/gaia), processed by the {\it Gaia} Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the {\it Gaia} Multilateral Agreement. This publication made use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. We also used a map obtained with Planck (http://www.esa.int/Planck), an ESA science mission with instruments and contributions directly funded by ESA Member States, NASA, and Canada.
We made use of the Binary Star Solver v1.2.0 python program \citep{mil20}.
In this work we use the matplotlib \citep{hun07}, NumPy \citep{oli06}, AMUSE \citep{por18}, SeBa \citep{por12}, Huayno \citep{pel12}, and Fi \citep{pel04} packages.
We thank John Telting for very useful help, comments and suggestions related to FIES, and in addition we thank the following observers at the NOT for contributing to obtaining the FIES data set: Pere Blay Serrano, Rosa Clavero, Abel de Burgos, Sune Dyrbye, Grigori Fedorets, Emanuel Gafton, Francisco Jose Galindo-Guil, Nicholas Emborg Jahnsen, Raine Karjalainen, Anni Kasikov, Marcelo Aron Keniger, Jyri Lehtinen, Teet Kuutma, Illa Losada, Julia Martikainen, Shane Moran, Christina Konstantopoulou, Viktoria Pinter, Tapio Pursimo, Rene Tronsgaard Rasmussen, Lauri Siltala, Auni Somero, and Joonas Viuho.
Based on observations made with the Nordic Optical Telescope, owned in collaboration by the University of Turku and Aarhus University, and operated jointly by Aarhus University, the University of Turku and the University of Oslo, representing Denmark, Finland and Norway, the University of Iceland and Stockholm University at the Observatorio del Roque de los Muchachos, La Palma, Spain, of the Instituto de Astrofisica de Canarias.
The authors would like to state that this manuscript was submitted before Christmas.
\end{acknowledgements}
\bibliographystyle{aa}
|
3,212,635,537,992 | arxiv | \section{Introduction}
In a series of papers \cite{DDM1,DDM2,DDM3}, Damour, Deser and McCarthy
(DDM) have claimed
that the nonsymmetric gravitational theory (NGT) is theoretically inconsistent.
They noted that a spurious ``gauge invariance'' in the linearised version
of NGT did not generalise to curved spacetime, from which they concluded
that ghost excitations would occur. However, the ``gauge invariance'' in
question is simply an artifact of the linearisation, and plays no role in
ensuring the conservation of the true Noether charges of the theory. The loss
of such an invariance is as unimportant as its existence.
They went on to argue that generic solutions suffered unacceptable asymptotic
behaviour at future null infinity, $\cal{I}^{+}$. This argument was based
on the supposed behaviour of a Lagrange multiplier field which, in fact, can
and should be eliminated from the field equations. We explained the pitfalls
of such a treatment in ref.\cite{CorMoff}, and we shall expand on our
original explanation in this note. It has since been shown that general
radiative solutions can be found with good asymptotic behaviour at
$\cal{I}^{+}$ \cite{pla,CMT}.
Damour, Deser and McCarthy now accept that good asymptotic behaviour can
be found at $\cal{I}^{+}$, but claim this can only be achieved at the
expense of bad asymptotic behaviour at $\cal{I}^{-}$ \cite{DDM3}. They
base this claim on a lemma which states that any solution of an
inhomogeneous wave equation which falls off faster than $1/r$ at
$\cal{I}^{+}$ must be an advanced solution. We point out in this note
that the lemma they use is not applicable to the system of equations being
studied, since they apply Green's theorem to a hyperbolic system
while neglecting lower order differential constraints on the
boundary of integration. Consequently, their assertion that NGT has bad
global asymptotics is unfounded.
\section{Analysis of the Field Equations}
For ease of comparison with the work of DDM, we shall adopt their
unconventional notation for the field variables. Performing an expansion
in powers of the anti-symmetric field $B_{\mu\nu}$ about a fixed symmetric
background $G_{\mu\nu}$, the NGT field equations \cite{Moff79} read to
first order:
\begin{eqnarray}
R_{(\mu\nu)}(G)&=& 0 \; , \\
\nabla^{\alpha}\nabla_{\alpha}B_{\mu\nu}
-2R^{\alpha\;\beta}_{\;\mu\;\nu}B_{\alpha\beta}
&=&{2\over 3}(\partial_{\mu}\Gamma_{\nu}-
\partial_{\nu}\Gamma_{\mu})\; , \label{mix} \\
\nabla^{\nu}B_{\mu\nu}&=&0 \; , \label{sdiv}
\end{eqnarray}
which may be supplemented by the gauge choice:
\begin{equation}
\nabla^{\mu}\Gamma_{\mu}=0 \; .
\end{equation}
The covariant derivative, $\nabla$, is that of the background metric
$G_{\mu\nu}$, and the vector $\Gamma_{\mu}$ is a non-dynamical Lagrange
multiplier. This Lagrange multiplier appears in the linearised NGT Lagrangian
density via the term:
\begin{equation}
{\cal L}_{\Gamma}=\sqrt{-G}B^{\mu\nu}(\partial_{\nu}\Gamma_{\mu}
-\partial_{\mu}\Gamma_{\nu})\; ,
\end{equation}
which gives rise to the field equation (\ref{sdiv}). In the language of field
theory, we see that $\Gamma_{\nu}$ does not have a propagator, since
the Lagrangian cannot be formulated to have a kinetic energy term for
$\Gamma_{\nu}$. As such, it makes no sense to talk
about $\Gamma_{\nu}$ having retarded or advanced propagator solutions.
We shall return to this crucial point later.
In order to solve the field equations, we need to take the divergence and
cyclic curl of (\ref{mix}), while remembering that the solutions to these
higher derivative equations are constrained to be solutions
to the primary lower derivative equation, (\ref{mix}). The equations for
$B_{\mu\nu}$ and $\Gamma_{\mu}$ then read
\begin{eqnarray}
\nabla^{\nu}B_{\mu\nu}&=&0 \; , \label{sd} \\
\nabla^{\alpha}\nabla_{\alpha}B_{ \{ \mu\nu ,\kappa \} }
-2\nabla_{\{\kappa}(R^{\alpha\;\beta}_{\;\mu\;\nu\} }B_{\alpha\beta})
+\nabla_{\alpha}(R^{\alpha\beta}_{\;\;\;\{\kappa\mu}B_{\nu\}\beta})&=&0
\; , \label{cc} \\
\nabla^{\alpha}\nabla_{\alpha}\Gamma_{\nu}&=&
-3\nabla^{\mu}(R^{\alpha\;\beta}_{\;\mu\;\nu }B_{\alpha\beta})
\; . \label{yuk}
\end{eqnarray}
The first two sets of equations, (\ref{sd}, \ref{cc}), represent six equations
for the six $B_{\mu\nu}$. These six equations {\em fully determine
$B_{\mu\nu}$ with no reference to the Lagrange multiplier $\Gamma_{\nu}$},
which is what one expects from a system of equations with a
Lagrange multiplier. The last equation, (\ref{yuk}), can be used to solve
for the Lagrange multiplier $\Gamma_{\nu}$ {\em once the six $B_{\mu\nu}$
are known}. We note that on its own, (\ref{yuk})
can only determine the LHS of (\ref{mix}) to be $2/3(\nabla_{\mu}\Gamma_{\nu}-
\nabla_{\nu}\Gamma_{\mu})+F_{\mu\nu}$ where $F_{\mu\nu}$ is any skew
tensor that satisfies $\nabla^{\nu}F_{\mu\nu}=0$.
In their analysis, DDM concentrate their attention on first solving for
the Lagrange multiplier $\Gamma_{\nu}$ via equation (\ref{yuk}). This
approach is fraught with problems. Firstly, the $\nabla RB$ ``source term''
is an unknown function unless you have already solved for $B_{\mu\nu}$.
DDM fail to check whether their eventual solution for $B_{\mu\nu}$ is a
self-consistent solution of this equation. It is not.
Secondly, the hyperbolic differential operator $\nabla^{\alpha}
\nabla_{\alpha}$ in (\ref{yuk}) demands propagating, retarded and advanced
$1/r$ Green's function solutions for $\Gamma_{\nu}$. This leads to a distorted
physical picture, as the primary field equation, (\ref{mix}), for
$\Gamma_{\nu}$ is not a wave equation. Wave solutions for $\Gamma_{\mu}$ play
no part in the physics of NGT.
Additionally, DDM base their arguments on the Green's function for the
flat-space d'Alembertian $\Box$ instead of the operator
$\nabla^{\alpha} \nabla_{\alpha}$, while at the same time keeping the source
term $\nabla RB$. This treatment is inconsistent since
$(\nabla^{\alpha}\nabla_{\alpha}-\Box)\Gamma \, \sim {\cal O}(R\Gamma)$
is of the same order as the source term.
We shall see that the most important of these errors is their failure to
correctly treat the spurious solutions for $\Gamma_{\nu}$, which are
manufactured in taking the divergence of the field equations.
\section{The explicit faults in DDM's argument}
While the reasons given above more than suffice to invalidate the
proof given by DDM, it is instructive to see the failings of their
arguments shown explicitly. Indeed, one needs to look no further
than the static case to illustrate the errors in their analysis.
For static systems, the uniqueness theorem for inhomogeneous wave
equations cited by DDM \cite{fock} reduces to the statement that
$\Gamma_{\mu}$ will fall-off as $1/r$ at spatial or null infinity.
Explicitly, we find for the component $\Gamma_{t}$ that (\ref{yuk})
becomes
\begin{equation}
\left[\nabla^{2}-{2M \over r^2}\left(r{\partial^{2} \over
\partial r^{2} }+2{\partial \over \partial r}\right)\right]\Gamma_{t}
=3\nabla^{\mu}(R^{\alpha\;\beta}_{\;\mu\; t }B_{\alpha\beta}) \label{G} \; ,
\end{equation}
where $\nabla^{2}$ is the usual flat-space Laplacian for a scalar
field and $M$ is the mass associated with the Schwarzschild background.
If we follow the argument of DDM - by neglecting the fact that the
source term is unknown and the operator acting on $\Gamma_{t}$
is not just the flat-space Laplacian - then we conclude from the
uniqueness theorem that $\Gamma_{t} \sim 1/r$.
If we then do as DDM suggest, and feed this information into (\ref{mix})
we find
\begin{equation}
\left[\left(\nabla^2+{2 \over r}{\partial \over \partial r}
+{2 \over r^2}\right)-{2M \over r}\left({\partial^{2} \over
\partial r^{2} }+{3 \over r}{\partial \over \partial r}+{4 \over r^2}
\right)\right]B_{tr}={2 \over 3}{\partial\Gamma_{t} \over \partial r}
\sim {1/r^2} \; , \label{s}
\end{equation}
from which we conclude $B_{tr} \sim r^{0}$, exactly as predicted by DDM.
The above analysis would seem to provide a perfect example of DDM's
claim that NGT has unacceptable asymptotic behaviour.
Let us now look at what has been forgotten. Firstly, we have neglected
the field equation (\ref{sd}) which demands
\begin{equation}
\nabla^{\alpha}B_{t\alpha}=0 \; , \label{bang you're dead}
\end{equation}
so that $B_{tr} \sim r^{-2}$. DDM's putative solution, $B_{tr} \sim r^{0}$,
{\em is not a solution to} (\ref{bang you're dead}). Putting
$B_{tr} = l^2/r^{2}$ (where $l^{2}$ is a constant of integration)
into (\ref{s}) gives
\begin{equation}
{-8Ml^2 \over r^5} \sim {1 \over r^2}\; ,
\end{equation}
which clearly excludes the $1/r$ behaviour for $\Gamma_{t}$ suggested by
(\ref{G}). This is simply an example of a higher order derivative equation,
(\ref{G}), having solutions which are incompatible with the lower order
equation, (\ref{s}), from which it came. Restoring the term
$\partial_{r}\Gamma_{t}$ in (\ref{s}), we
see that the true solution for $\Gamma$ has $\Gamma_{t} =3Ml^2 /r^{4}$.
The above calculation has now faithfully reproduced the exact NGT static
solution to first order in $B_{\mu\nu}$ - a solution which represents a
non-trivial extension of the Schwarzschild solution of General Relativity (GR)
with {\em good asymptotic behaviour for} $B_{\mu\nu}$.
The putative $1/r$ Green's function solution for $\Gamma$ was simply an
artifact produced by taking additional derivatives of the field equations.
Another way of seeing what went wrong with DDM's argument is to recall that
we are free to add a tensor of integration, $F_{\mu\nu}$,
to the LHS of (\ref{s}) , where $F_{\mu\nu}$ is
a solution of $\nabla^{\alpha}F_{t \alpha}=0$.
In this case, we may add $F_{tr}
\sim 1/r^{2}$ to the LHS of (\ref{s}) with the coefficient chosen to eliminate
the $1/r^2$ term given by the curl of $\Gamma_{t}$. In this way,
$B_{tr}$ is no longer driven to behave as $r^{0}$. From this argument we see
that the putative $1/r$ solution for $\Gamma_{t}$ was a purely homogeneous
solution which must be chosen to vanish.
Generalising our analysis to the time-dependent case we find a very similar
picture. Again the higher-derivative equation for $\Gamma$ admits
homogeneous solutions with $1/r$ fall-off, while the lower order constraints
on $\Gamma$ demand that these solutions be discarded.
We shall demonstrate this in the context of wave solutions
on a radiative, axi-symmetric GR background\cite{pla}. To leading order, the
GR background is described by $M(u,\theta)$ and $c(u,\theta)$, where $u=t-r$
is retarded time, the mass associated with the background is given by the
angular average of $M$ and the time rate of change of this mass is given by
the angular average of $-(\partial_{u}c)^2$. We shall only sketch the main
steps in solving the equations, as the full solution is derived in detail
in ref.\cite{pla}.
Beginning with the wave equations for $\Gamma_{\mu}$ we find
\begin{equation}
\Box \Gamma_{u}+{\cal O}(R\Gamma)=-3\nabla^{\mu}(R^{\alpha\;\beta}_{\;\mu\; u }
B_{\alpha\beta}) \; ,
\end{equation}
where $\Box$ is the usual flat-space d'Alembertian for a scalar field, and the
extra background terms of order $R\Gamma$ are given by
\begin{eqnarray}
{\cal O}(R\Gamma)&=&{2 \over r}\left[M{\partial^{2}\Gamma_{u} \over \partial
r^2}+{\partial M \over \partial u}{\partial \Gamma_{r} \over \partial r}\right]
+{2 \over r^2}\left[\left(4M+2c+2c{\partial c \over \partial u}-
{\partial^2 c \over \partial \theta^{2}}-3\cot\theta{\partial c \over
\partial \theta}\right){\partial \Gamma_{u} \over \partial r} \right. \nonumber
\\
&& +c^2{\partial^2 \Gamma_{u} \over \partial r \partial u}-{c^2 \over 2}
{\partial^{2} \Gamma_{u} \over \partial r^2}
-2\left({\partial^2 c \over \partial \theta \partial u}
+2\cot\theta{\partial c \over \partial u}\right){\partial \Gamma_{\theta}
\over \partial r} \nonumber \\
&& \left. -2M{\partial \Gamma_{r} \over \partial u}
-2\left({\partial c \over \partial u}\right)^{2}\Gamma_{r}
-c{\partial c \over \partial u}{\partial \Gamma_{r} \over \partial r} \right]
+ \dots \; .
\end{eqnarray}
Although it is inconsistent to drop these terms while keeping the $RB$
source term, we shall follow the method proposed by DDM and drop them anyway.
The wave equation for $\Gamma_{r}$
then reads
\begin{equation}
\left[\Box-{2\over r}{\partial \over \partial r}-{2 \over r^2}
+{2 \over r}{\partial \over \partial t}\right]\Gamma_{r}+\left[{2 \over r}
{\partial \over \partial r}+{2 \over r^2}\right]\Gamma_{u}=
-3\nabla^{\mu}(R^{\alpha\;\beta}_{\;\mu\; r}B_{\alpha\beta}) \; .
\end{equation}
When combined with the gauge condition $\nabla^{\alpha}\Gamma_{\alpha}=0$
and the wave equation for $\Gamma_{\theta}$, we find that the above equations
have the usual retarded wave solutions $\Gamma_{u}=f(u,\theta)/r$,
$\Gamma_{r}=g(u,\theta)/r^2$ and $\Gamma_{\theta}=h(u,\theta)$ (in orthonormal
coordinates this means $\Gamma_{{\hat{\theta}}} \sim 1/r$). Since the source
terms are at present unknown, we cannot give explicit forms for $f$, $g$ and
$h$, although the gauge condition does demand
$f+\partial_{u}g=\partial_{\theta}
h+h\cot\theta$.
Turning to the $(ur)$ component of (\ref{mix}) we find
\begin{eqnarray}
\left[\Box-{2\over r}{\partial \over \partial r}-{2 \over r^2}
-{2 \over r}{\partial \over \partial t}\right]B_{ur}+{\cal O}(RB)
&=&{2 \over 3}\left({\partial \Gamma_{r} \over \partial u}
-{\partial \Gamma_{u} \over \partial r}\right)
\nonumber \\
&=&{2\partial_{\theta}\left(h\sin\theta\right) \over 3r^2\sin\theta }\; .
\label{neat}
\end{eqnarray}
From this we would conclude $B$ has the unacceptable asymptotic form
$B_{ur} \sim r^{0}$, as promised by DDM.
However, this is in conflict with the field equations (\ref{sd})
and (\ref{cc}) which demand to leading order that $B_{ur}=l^2(u,\theta)/r^2$.
The lower order field equations for $\Gamma_{\nu}$ again demand that the
$1/r$ solution for $\Gamma$ be dropped, as was the case for static
solutions. In this case we require $\partial_{\theta}(h\sin\theta)=0$ which
implies $h=0$ to ensure regularity on the polar axis.
Notice that setting $h(u,\theta)=0$ removes the only transverse, propagating
component of $\Gamma_{\nu}$, and that the remaining components of
$\Gamma_{\nu}$
are longitudinal and non-propagating, despite the $1/r$ fall-off for
$\Gamma_{u}$. Such important subtleties are lost in DDM's analysis, for they
treat $\Gamma_{\nu}$ as a scalar. It is also worth noting from the full
solution\cite{pla} that the one transverse component of $B_{\mu\nu}$,
$B_{u\theta}$, does have a $1/r$ retarded wave solution. The physical fields,
$B_{\mu\nu}$, that we expect to propagate do, and the non-dynamical Lagrange
multiplier fields, $\Gamma_{\nu}$, do not. We would be surprised if we had
found otherwise.
As in the static case, we see that the extra solutions for
$\Gamma_{\nu}$, which are generated by taking additional derivatives of the
field equations, can be cancelled by the tensor of integration $F_{\mu\nu}$.
Another way of seeing this is to recognise that a consistent
treatment of the ``inhomogeneous wave equation'' for $\Gamma$ demands that
we drop the source term $\nabla RB$ when using the flat-space d'Alembertian
for $\Gamma$. Then we see that the solutions for $\Gamma$ which are independent
of the background parameters $M$ and $c$ are simply homogeneous solutions
which we are free to discard.
The above collection of results allows us to address DDM's lemma head-on.
They claim that the fast fall-off of $\Gamma$ at ${\cal I}^{+}$ comes about
because our solutions for $\Gamma$ are advanced solutions. This is clearly not
the case, as our solutions for $\Gamma$ are explicitly retarded solutions
- despite the fact they do not describe transverse waves with $1/r$ fall-off.
If we accept DDM's contention that $\Gamma$ is described by advanced Green's
functions, we find this leads to a contradiction. If $\Gamma$, and hence
$B$, are functions of advanced time, $v=t+r$, their lemma demands
that $\Gamma$ has propagating, $1/r$, solutions at ${\cal I}^{-}$.
By a simple time reversal of our solutions in terms of retarded time, with
the added simplification that the GR background is static at ${\cal I}^{-}$,
we find from the wave equation for $\Gamma$ that
$\Gamma_{{\hat{\theta}}}=h(v,\theta)/ r +{\cal O}(1/r^2)$, while
the lower order equation (\ref{mix}), and the fact that
$B_{vr}=l^2(v,\theta)/r^2$, demand $h(v,\theta)=0$. This proves by
contradiction that DDM's lemma is not valid for $\Gamma$.
In their latest paper \cite{DDM3}, DDM go on to repeat the same flawed analysis
they applied to the Lagrange multiplier $\Gamma$, by taking two derivatives of
(\ref{mix}) in order to discuss a fourth order wave equation for $B_{\mu\nu}$.
Again, the original, lower order field equations do not allow the badly behaved
solutions of the higher order equations.
All of the pitfalls that plague DDM's analysis can be avoided, if one
works from the outset with (\ref{sd}, \ref{cc}) and uniquely solves for
$B_{\mu\nu}$ without reference to the Lagrange multiplier $\Gamma$.
This was the method first employed by Einstein and Straus when solving an
analogous system of equations in Unified Field Theory \cite{Al}. Using the same
method, we have found exact, radiative solutions, well behaved at
${\cal I}^{+}$ and ${\cal I}^{-}$ \cite{pla,CMT}. These solutions represent
a non-trivial modification to the GR limit of this system and lead to
the prediction that the quadrupole moment of a source will decrease more
rapidly in NGT than GR.
To summarise, we have shown that DDM's claim that NGT has bad global
asymptotics is invalid, since it is based on wave solutions for
a Lagrange multiplier field which fail to solve the original, lower
order field equations. This invalidates the use of their lemma, since
it can only be used for fields with lower order constraints,
when those constraints do not demand that the propagating modes vanish
(as is the case in electromagnetism, for example).
If these constraints are properly accounted for, we find that NGT has
non-trivial radiative solutions with good global asymptotics.
\section*{Acknowledgements}
This work was supported by the Natural Sciences and Engineering
Research Council of Canada. One of the authors (NJC) is
grateful for the support provided by a Canadian Commonwealth
Scholarship. We thank M. Clayton and P. Savaria for helpful
discussions.
|
3,212,635,537,993 | arxiv | \section{Introduction}\label{para:intro}
Citywide crowd flow analytics is very critical to smart city efforts around the world. A typical task is citywide crowd flow prediction \cite{zhang2016dnn,zhang2017deep,lin2019deepstn}, which aims to predict the traffic (e.g., inflows and outflows of every region) for the next time slot, given the historical traffic observations. It can help the governors conduct traffic control and avoid potential catastrophic stampede before a special event. Another important task is to infer the fine-grained crowd flows from available coarse-grained data sources, which can reduce the expense of urban systems \cite{liang2019urbanfm,ouyang2020fine}. Other tasks \cite{yao2018deep,zong2019deepdpm} are also actively studied by the community due to the vital impact of citywide crowd flow analytics.
Crowd flow analytics is not trivial as the traffic can be affected by multiple complex factors in spatio-temporal domains. As shown in Figure \ref{fig:intro}(a), the inflow of Region R1 is affected by outflows of nearby regions like R4 as well as distant regions, which indicates the spatial dependencies. For the temporal dependencies, crowd flow in a region is affected by recent, daily, and weekly historical traffic. To model the spatio-temporal dependencies, Convolutional Neural Networks (CNNs) have been widely used and achieved promising performance. A pioneering work \cite{zhang2016dnn} provided the first CNN-based method (DeepST) for modeling crowd flow, where convolution operators are used to extract spatially near and distant dependencies and the temporal dependencies are considered in different branches of networks. ST-ResNet \cite{zhang2017deep} further enhanced the performance of DeepST using residual structures. Very recently, a novel ConvPlus structure in DeepSTN+ \cite{lin2019deepstn} was proposed to learn the long-term spatial dependencies between two arbitrary regions. These CNN-based methods are characterized by two components: a complicated ST feature learner to capture features of the measurements, and a simple task-specific predictor to generate predictions on all regions. However, they have two main drawbacks:
\begin{figure}[!t]
\centering
\includegraphics[width=0.84\textwidth]{imgs/fig_intro.pdf}
\caption{\label{fig:intro} Application of CNNs for citywide crowd flow analytics (Better view in color).}
\end{figure}
1) \emph{Inefficiency in learning global spatial dependencies}. Take traveling in Beijing (Figure \ref{fig:intro}) as an example. When predicting the inflow of R1 during morning hours, the outflow of distant regions like R2 needs to be considered, since it is common that people commute from a distant residence location. As people can travel around a modern city quickly, it becomes crucial to capture global spatial dependencies in this task. To this end, existing arts employ two approaches:
\begin{itemize}[leftmargin=*]
\item \emph{Stacking CNNs to increase receptive fields}. Most previous studies like DeepST and ST-ResNet employ CNNs to capture information locally. But to capture global spatial dependencies, they have to stack many layers to increase the receptive field of the network (see Figure \ref{fig:intro}(b)). This is very inefficient since relationships between distant regions can only be captured by a near-top layer with a sufficiently large receptive field to cover all the regions of interest.
\item \emph{Learning long-range spatial dependencies directly}.
Instead of gradually increasing receptive fields, DeepSTN+ attempts to capture global spatial dependencies in \emph{every layer} using ConvPlus structure, which explicitly models all pairwise relationship between regions. However, a single layer of ConvPlus without pooling requires $O(n^2)$ parameters, where $n$ is the number of regions. Constrained by this bloated structure, DeepSTN+ cannot easily go deeper to learn higher-level representations for each region. Thus, how to learn global spatial dependencies more efficiently still remains a major challenge.
\end{itemize}
2) \emph{Ignoring latent region functions}. Different from pixels in image processing, urban regions have different land functions according to their locations and surrounding POIs \cite{zheng2014urban,pan2019matrix}. Recall that R1, R2 and R3 in Figure \ref{fig:intro} correspond to an office area, a residential area and a park zone respectively. From Figure \ref{fig:case}(a), it can be seen easily that their daily patterns are entirely different. For instance, the office area (R1) usually reaches a traffic peak in the morning, while the residential area (R2) usually exhibits growth after dinner time. The difference between their daily flow distributions can also be seen from Figure \ref{fig:case}(b). However, the aforementioned methods have overlooked such varying latent functions among regions and used a simple predictor with shared parameters to predict flow for all regions, which inevitably resulted in degraded performance.
\begin{figure}[!t]
\centering
\includegraphics[width=0.75\textwidth]{imgs/fig_case.pdf}
\caption{\label{fig:case} Illustration of daily patterns and inflow distribution in three regions.}
\end{figure}
To address the above problems, we make the following contributions to the community. Primarily, we introduce DeepLGR, the first-ever general framework for raster-based crowd flow analytics. It is named according to how it stratifies a given task into three major procedures: 1) \textbf{L}ocal feature extraction to learn representations for each region within small receptive fields; 2) {\textbf G}lobal context aggregation to efficiently capture the global spatial dependencies; and 3) {\textbf R}egion-specific prediction. Respectively,
\begin{itemize}[leftmargin=*]
\item we present the first attempt to extract local region representations using Squeeze-and-Excitation networks (SENet)~\cite{hu2018squeeze}, which excels by including the channel-wise information as additional knowledge;
\item we design a global context module that firstly aggregates the region representations using a specific pooling method, and then upsample the global priors back to the original scale to generate global-aware features;
\item we introduce a region-specific predictor based on tensor decomposition that factorizes the region-specific parameters of the predictor into a smaller core tensor and adjoint matrices.
\end{itemize}
In addition, we evaluate our framework on two typical crowd flow analytics tasks: crowd flow forecasting \cite{zhang2016dnn,zhang2017deep} and fine-grained crowd flow inference \cite{liang2019urbanfm,ouyang2020fine}. Extensive experiments demonstrate the state-of-the-art performance and stability achieved by our framework. We have released our code at \url{https://github.com/yoshall/DeepLGR} for public use.
\section{Formulation}
In this section, we introduce several notations and formulate the problem of crowd flow analytics.
As shown in Figure \ref{fig:intro}(a), we first follow the previous study \cite{zhang2016dnn} to partition an area of interest (e.g., a city) evenly into a $H \times W$ grid map based on longitude and latitude where a grid denotes a region. Thus, the crowd flow at a certain time $t$ can be denoted as a 3D tensor $\mathcal{P}_t \in \mathbb{R}^{H \times W \times K}$, where $K$ is the number of different flow measurements (e.g., inflow and outflow). Each entry $(i, j, k)$ denotes the value of the $k$-th measurement in the region $(i,j)$.
Without loss of generality, we use $\mathcal{X} \in \mathbb{R}^{H \times W \times C}$ and $\mathcal{Y} \in \mathbb{R}^{H' \times W' \times D}$ as the input and output for a crowd flow analytics task, where $C$ and $D$ are the number of channels. For example, in the task of crowd flow prediction \cite{zhang2016dnn,zhang2017deep,lin2019deepstn}, the input is the historical observations $\mathcal{X}=\left\{\mathcal{P}_{i} | i=1,2, \cdots, \tau \right\} \in \mathbb{R}^{H \times W \times K\tau}$ and the target is to predict $\mathcal{Y}=\mathcal{P}_{\tau+1} \in \mathbb{R}^{H \times W \times K}$.
\section{Methodology}
Figure \ref{fig:framework} presents the framework of DeepLGR, which can be easily generalized to all kinds of citywide crowd flow. Compared to the previous methods composed of an ST feature learner and a shared predictor for all regions, our framework contains three major components: local feature extraction, global context module and region-specific predictor. In the first component, we employ the SENet to learn representations for each region within small (i.e., local) receptive fields from the input tensor $\mathcal{X}$. To capture global spatial dependencies, we further design the global context module that considers the full region of interest. It first extracts global contextual priors from the learned region representations using a specific pooling method, and then upsamples the priors to the original scale to generate the global features. Once we obtain features from both local view and global view, we concatenate them into a tensor and then feed it to the region-specific predictor to make customized predictions for each region respectively.
\begin{figure}[!b]
\centering
\includegraphics[width=0.85\textwidth]{imgs/fig_framework.pdf}
\caption{\label{fig:framework} The pipeline of DeepLGR, which contains three major components.}
\end{figure}
For spatial dependencies, our framework employs the first two components which strategically capture both local-level (neighborhood) and global-level dependencies between regions. Following the mainstream CNN architectures for citywide crowd flow analytics \cite{zhang2016dnn,zhang2017deep,lin2019deepstn}, the temporal dependencies like closeness (recent), period (daily) and trend (weekly), if any, are considered in the channels of input. These temporal dependencies can interact with each other in the backbone network. Next, we will detail the three components respectively.
\subsection{Local Feature Extraction}
Recall that both the previous and current state-of-the-arts \cite{zhang2017deep,lin2019deepstn} use residual blocks to model the spatial dependencies from nearby regions. However, these methods mainly focus on the spatial dimension and have overlooked the channel-wise information in the feature maps. Thus, we employ SENet to fuse both spatial and channel-wise information within small (i.e., local) receptive fields at each layer, which has proven to be effective in producing compacted and discriminative features of each grid. Figure \ref{fig:senet}(a) illustrates the pipeline of the module for local feature extraction. The input is fed to a convolutional layer for initialization. Then, we stack $M$ squeeze-and-excitation (SE) blocks in Figure \ref{fig:senet}(b) for feature extraction, which is composed of three stages: 1) a residual block \cite{he2016deep} for feature learning; 2) a squeeze operation to squeeze global spatial information into a channel descriptor by global average pooling; 3) an excitation operation to fully capture the channel-wise dependencies: it first computes the attention coefficients over each channel via two fully connected layers followed by a sigmoid function, and then rescales the channels of original inputs by these weights. Finally, we use an output convolutional layer to transform the obtained high-level feature maps to the input of the next module. In summary, the SE structure enables this module to learn better representations for each region locally within receptive fields.
\begin{figure}[!h]
\centering
\includegraphics[width=0.6\textwidth]{imgs/fig_senet.pdf}
\caption{\label{fig:senet} The pipeline of local feature extraction, where the receptive fields depend on the number of SE blocks ($M$). Conv: convolutional layer. ResBlock: Residual block. Pooling: global average pooling. Dense: fully connected layer.}
\end{figure}
\subsection{Global Context Module}
After local feature extraction, we have designed a specific module that takes the output of the former component as input to generate global contexts for each region, so as to capture global spatial dependencies. As depicted in Figure \ref{fig:gcm}, we first employ spatial pyramid pooling \cite{he2015spatial} to generate a set of the global priors, where each prior is a spatially abstract of the original input under different pyramid scales. This operation allows the module to separate the feature map into different sub-regions and build pooled representation for different locations. For example, the 1$\times$1 prior (the red cube) denotes the coarsest level with only one single value at each channel, which is equivalent to global pooling operation that covers the whole image. In our experiments, we use a 4-level pyramid (1$\times$1, 2$\times$2, 4$\times$4 and 8$\times$8) to squeeze the input by average pooling.
Once the global priors are obtained, an $1 \times 1$ convolution layer followed by a Batchnorm layer \cite{ioffe2015batch} is used for dimension reduction of channels from $N$ to $N/8$. Inspired by the study \cite{liang2019urbanfm} aiming at inferring fine-grained crowd flow from coarse-grained counterparts, we employ the Subpixel block \cite{shi2016real} to upsample the priors to generate new representations with the same size as the original inputs. For example, after the Subpixel block in 4$\times$4 branch, the output feature maps grow $H / 4$ and $W / 4$ times larger in height and width respectively with the number of channels unchanged. Different from PSPNet \cite{zhao2017pyramid} using bilinear interpolation for upsampling the priors, the Subpixel block considers the relationship between a super-region and its corresponding sub-regions by introducing a parametric design. Finally, we concatenate the input (i.e., region representations) with all levels of global features (i.e., context) as the output of this module.
\begin{figure}[!b]
\centering
\includegraphics[width=0.7\textwidth]{imgs/fig_gcm.pdf}
\caption{\label{fig:gcm} The pipeline of global context module, where Conv denotes a $1 \times 1$ convolutional layer for dimension reduction, and Subpixel contains a convolutional layer and a pixelshuffle operation sequentially to upsample the contextual priors. For simplicity, we use a 4-level pyramid (1$\times$1, 2$\times$2, 3$\times$3 and 4$\times$4) for an illustration.}
\end{figure}
In summary, this module first converts the input feature map into priors (e.g., 1$\times$1 prior that encodes the information of all regions) and then upsamples the priors to learn the global-context-to-region influence (i.e., global spatial dependencies). Compared to the previous attempt (ConvPlus layer in DeepSTN+), our solution is more efficient and lightweight. Each ConvPlus layer directly models the pairwise relationships among all regions, thus demanding $O(n^2)$ parameters. With the increase of spatial granularity, it will induce extremely high computational costs due to the massive parameters. Thus, DeepSTN+ can hardly learn higher-level representations by simply increasing network depth. In contrast, as we have separated the procedures of local feature extraction and global context modeling, we can easily increase the network depth to gain better capacity.
\subsection{Region-Specific Predictor}
As mentioned before, each urban region has its unique land function. Previous studies \cite{zhang2016dnn,zhang2017deep,lin2019deepstn} mainly employ a single fully connected layer (equivalent to a $1\times1$ convolution) with shared weights as the predictor for all regions, which fails to capture this critical property. Thus, it is necessary to assign region-specific predictor to each region.
Recall that the high-level feature obtained from last module is $\mathcal{Z} \in \mathbb{R}^{H \times W \times N'}$ and prediction result is $\mathcal{Y} \in \mathbb{R}^{H \times W \times D}$, where $N'=N+N/2$. Conventionally, the number of parameters in a shared fully connected layer is $n_f=N'D$. To achieve region-specific predictor, an intuitive solution is to use a customized fully connected layer for each region. However, it will induce $HW \times n_f$ parameters (denoted as a tensor $\mathcal{W} \in \mathbb{R}^{H \times W \times n_f}$), which can easily bloat up as the granularity increases. Recently, matrix factorization (MF) was used to avoid these drawbacks \cite{pan2019matrix}, in which the parameter tensor $\mathcal{W}$ is reshaped to a matrix $\mathbf{W} \in \mathbb{R}^{HW \times n_f}$. As shown in Figure \ref{fig:td}(a), the authors from \cite{pan2019matrix} decompose the weight matrix $\mathbf{W}$ into two \emph{learnable} low-rank matrices, i.e., region embedding matrices $\mathbf{L} \in \mathbb{R}^{HW \times k}$ and parameter embedding matrices $\mathbf{R} \in \mathbb{R}^{k \times n_f}$. With the usage of MF, the number of the predictor parameters can be reduced to $(HW+n_f)k$, where $k \ll n_f$ and $k \ll HW$.
\begin{figure}[!b]
\centering
\includegraphics[width=0.9\textwidth]{imgs/fig_td.pdf}
\caption{\label{fig:td} Illustration of matrix factorization and tensor decomposition.}
\end{figure}
Nonetheless, directly flattening the parameter tensor $\mathcal{W}$ over the region dimension will lose the Euclidean structure of the flow map. For example, near things are more related than distant things according to the first law of geography, which indicates near regions should have similar prediction weights. Instead, we present a new idea for decomposing $\mathcal{W}$ using Tensor Decomposition (TD) \cite{tucker1966some}. It not only preserves the spatial similarity (dependencies) between regions, but also reduces the amount of parameters. As illustrated in Figure \ref{fig:td}(b), tensor $\mathcal{W}$ is decomposed into the multiplication of a core tensor $\mathcal{A} \in \mathbb{R}^{d_1 \times d_2 \times d_3}$ and three adjoint matrices, where $d_1$, $d_2$, and $d_3$ denote the number of latent factors for each matrix. The computation is as follows:
\begin{equation}
\mathcal{W}=\mathcal{A} \times_{R} \mathbf{R} \times_{S} \mathbf{S} \times_{T} \mathbf{T},
\end{equation}
where $\times_{R}$ stands for the tensor-matrix multiplication; the subscript $R$ is the corresponding mode of the multiplication. For instance, $\mathbf{H}=\mathcal{A} \times_{R} \mathbf{R}$ is $\mathbf{H}_{i j k}=\sum_{i=1}^{d_1} A_{i j k} \times R_{i j}$. By this, we have changed the optimization target from $\mathcal{W}$ to the core tensor $\mathcal{A}$ as well as the three learnable matrices $\mathbf{R}$, $\mathbf{S}$ and $\mathbf{T}$. The core tensor is a low-rank representation summarising both the parametric and spatial information of the origin tensor $\mathcal{W}$. Compared to MF-based solution \cite{pan2019matrix}, our tensor decomposition can handle the higher-order relationships within the parameters. In addition, the number of parameters required is $d_1 d_2 d_3 + d_1 H + d_2 W + d_3 n_f$. Since $d_1$, $d_2$ and $d_3$ are usually very small, TD can achieve even much fewer parameters than MF, which is validated in our experiments.
\subsection{Optimization}
Since our framework is smooth and differentiable everywhere, it can be trained via the back-propagation algorithm. During the training phase, we use Adam optimizer to train our model by minimizing the entry-wise mean absolute error (MAE) between our prediction $\widehat{\mathcal{Y}}$ and the corresponding ground truth $\mathcal{Y}$:
\begin{equation}
\mathcal{L}(\mathbf{\Theta})=\left\|\mathcal{Y}-\widehat{\mathcal{Y}}\right\|_1
\end{equation}
where $\mathbf{\Theta}$ denotes all learnable parameters in our framework.
\section{Experiments}\label{sec:exp}
To validate the generality of DeepLGR, we conduct experiments on two typical tasks of citywide crowd flow analytics:
\begin{itemize}[leftmargin=*]
\item \emph{Crowd flow forecasting}: This task is to forecast the inflow and outflow of each region in a city from historical readings. Following the settings of \cite{zhang2017deep}, we consider the temporal dependencies (i.e., closeness, period and trend) in different channels of input, and the output is the prediction of inflow and outflow for the next timestamp. Similar to \cite{zhang2017deep}, we set the length of closeness (recent), period (daily) and trend (weekly) to 5, 3, 3.
\item \emph{Fine-grained flow inference}: In this task, we aim to infer fine-grained crowd flows throughout a city based on coarse-grained observations. We extend the state-of-the-art method named UrbanFM \cite{liang2019urbanfm} using our framework. Specifically, we replace the ResNet-based feature extraction of UrbanFM by our first component (SENet). Then, we add the global context module and region-specific predictor after the subpixel blocks in UrbanFM.
\end{itemize}
\subsection{Experimental Settings}
\subsubsection{Datasets}
Two datasets were used in our experiments, including TaxiBJ and HappyValley. The former is the fine-grained version of the ones used by~\cite{zhang2017deep} and the latter is provided from~\cite{liang2019urbanfm}. Specifically, TaxiBJ consists of four different time spans (denoted as P1 to P4 with different number of taxicabs and distribution), while HappyValley is the hourly observations of human flow in a theme park in Beijing from ten months. The statistics are detailed in Table \ref{tab:dataset}. We select the flow data between 6am and 11pm to conduct our experiments. Using both datasets, we evaluate DeepLGR over the two aforementioned tasks: In the first task, we employ the first 80\% data as training set, the next 10\% as validation set and the rest for test set; In the second task, we follow all the experiment settings of \cite{liang2019urbanfm}, including training, validation and test set partition. The upscaling factors in TaxiBJ and HappyValley are 4 and 2 respectively.
\begin{table}[!b]
\caption{Dataset description.}
\centering
\small
\tabcolsep=6.2mm
\begin{tabular}{l|ll}
\shline
\textbf{Dataset} & \textbf{TaxiBJ} & \textbf{HappyValley} \\
\hline
Data type & Inflow and outflow & Staying flow \\
Resolution & (128, 128) & (50,100) \\
Sampling rate & 30 minutes & 1 hour \\
\hline
\ & P1: 07/01/2013-10/31/2013 & \\
Time Span & P2: 02/01/2014-06/30/2014 & 01/01/2018- \\
(mm/dd/yyyy) & P3: 03/01/2015-06/30/2015 & 10/31/2018 \\
& P4: 11/01/2015-03/31/2016 & \\
\shline
\end{tabular}%
\label{tab:dataset}
\end{table}%
\subsubsection{Evaluation Metrics}
We employ two widely-used criteria to evaluate our model from different aspects, including mean absolute error (MAE) and symmetric mean absolute percentage error (SMAPE). They are defined as:
\begin{equation*}
\text{MAE} = \frac{1}{z}\sum_{i=1}^{z}{ \left |{y_i-\tilde{y}_i} \right |, \:\: \text{SMAPE} = \frac{1}{z}\sum_{i=1}^{z}{\frac{\left |{y_i-\tilde{y}_i} \right|}{|y_i| + |\tilde{y}_i|}}},
\end{equation*}
where $y$ and $\tilde{y}$ are ground truth and predicted value respectively; $z$ is the total number of all entries. Smaller metric scores indicate better model performance.
\subsubsection{Baselines}
In the first task, we compare our framework with heuristics, time series methods and CNN-based baselines. Specifically, a naive method (\textbf{Last}) simply uses the last observation as the prediction result, and another heuristic (\textbf{CA}) leverages the closeness property to predict the future crowds by averaging the values from the previous 5 time steps. \textbf{ARIMA} is a well-known model for forecasting future values in a time series. Besides, the CNN-based baselines (including \textbf{DeepST} \cite{zhang2016dnn}, \textbf{ST-ResNet} \cite{zhang2017deep}, \textbf{ConvLSTM} \cite{shi2016real} and \textbf{DeepSTN+} \cite{lin2019deepstn}) have been introduced in Section \ref{para:intro}.
The second task was introduced only very recently by \cite{liang2019urbanfm}, where the authors presented the state-of-the-art method named \textbf{UrbanFM}. It considers the unique characteristics of this task, including the spatial hierarchy and external factors. Other strong baselines included in this work are related to image super-resolution, such as \textbf{VDSR} \cite{kim2016accurate} and \textbf{SRResNet} \cite{ledig2017photo}. We mainly use these three baselines for model comparison in this task. It is worth noting that all baselines are implemented with their default settings in both tasks.
\subsubsection{Training Details \& Hyperparameters}
Our framework, as well as the above baselines, are fully implemented by Pytorch 1.1.0 with one GTX 2080TI. During the training phase, the learning rate is 0.005 and the batch size is 16. For the number of stacked SE blocks (denoted as $M$) in the first component, we conduct a grid search over $\{3,6,9,12\}$. For simplicity, we use the same hidden dimension (i.e., number of channels) at each 3$\times$3 convolutional layer in SE blocks, and conduct a grid search over $F=\{32,64,128\}$.
\begin{table*}[!b]
\caption{Prediction results on TaxiBJ over different time spans (P1-P4), where the bold number indicates the best performance of the column. We train and test each method five times, and present results using the format:``mean $\pm$ standard deviation".}
\centering
\footnotesize
\tabcolsep=3mm
\begin{tabular}{l|cc|cc}
\shline
\multicolumn{1}{l|}{\multirow{2}*{Method}} & \multicolumn{2}{c|}{P1} & \multicolumn{2}{c}{P2} \\
\cline{2-5} & MAE & SMAPE & MAE & SMAPE \\
\hline
CA & 3.43 & 0.290 & 4.23 & 0.288 \\
Last & 3.39 & 0.242 & 4.09 & 0.241 \\
ARIMA & 3.08 & 0.403 & 3.53 & 0.385 \\
DeepST & 2.59 $\pm$ 0.05 & 0.41 $\pm$ 0.01 & 2.94 $\pm$ 0.05 & 0.39 $\pm$ 0.01\\
ST-ResNet & 2.53 $\pm$ 0.05 & 0.38 $\pm$ 0.05 & 2.93 $\pm$ 0.06 & 0.34 $\pm$ 0.07 \\
ConvLSTM & 2.42 $\pm$ 0.02 & 0.41 $\pm$ 0.01 & 2.77 $\pm$ 0.01 & 0.39 $\pm$ 0.01 \\
DeepSTN+ & 2.33 $\pm$ 0.04 & 0.35 $\pm$ 0.08 & 2.67 $\pm$ 0.02 & 0.32 $\pm$ 0.05 \\
DeepLGR & \textbf{2.15 $\pm$ 0.00} & \textbf{0.19 $\pm$ 0.00} & \textbf{2.46 $\pm$ 0.00} & \textbf{0.18 $\pm$ 0.00} \\
\shline
\end{tabular}%
\begin{tabular}{l|cc|cc}
\shline
\multicolumn{1}{l|}{\multirow{2}*{Method}} & \multicolumn{2}{c|}{P3} & \multicolumn{2}{c}{P4} \\
\cline{2-5} & MAE & SMAPE & MAE & SMAPE\\
\hline
CA & 4.17 & 0.286 & 2.81 & 0.286 \\
Last & 4.07 & 0.240 & 2.82 & 0.239 \\
ARIMA & 3.68 & 0.363 & 2.61 & 0.420 \\
DeepST & 2.97 $\pm$ 0.04 & 0.39 $\pm$ 0.01 & 2.16 $\pm$ 0.04 & 0.43 $\pm$ 0.02 \\
ST-ResNet & 2.91 $\pm$ 0.06 & 0.33 $\pm$ 0.05 & 2.15 $\pm$ 0.04 & 0.32 $\pm$ 0.06 \\
ConvLSTM & 2.87 $\pm$ 0.01 & 0.39 $\pm$ 0.01 & 2.09 $\pm$ 0.02 & 0.43 $\pm$ 0.02 \\
DeepSTN+ & 2.82 $\pm$ 0.04 & 0.38 $\pm$ 0.05 & 2.05 $\pm$ 0.01 & 0.34 $\pm$ 0.05 \\
DeepLGR & \textbf{2.56 $\pm$ 0.02} & \textbf{0.19 $\pm$ 0.04} & \textbf{1.84 $\pm$ 0.01} & \textbf{0.19 $\pm$ 0.00} \\
\shline
\end{tabular}%
\label{tab:taxibj}%
\end{table*}%
\subsection{Results on Crowd Flow Forecasting}
\subsubsection{Model Comparison}
Here, we compare our framework with the baselines over the two datasets. We report the result of DeepLGR with $M=9$ and $F=64$ as our default setting. Further results regarding different $M$ will be discussed later.
Table \ref{tab:taxibj} shows the experimental results over P1 to P4 in TaxiBJ. We can observe that our framework clearly outperforms all baselines over both metrics. For instance, DeepLGR shows 10.2\% and 44.1\% improvements on MAE and SMAPE beyond the state-of-the-art method (DeepSTN+) in P4. The conventional model ARIMA performs much worse than deep learning models in these datasets, since it only considers the temporal dependencies among time series. Apart from the CNN-based methods, ConvLSTM advances DeepST and ST-ResNet because of the positive effect of its LSTM structure. However, it overlooks the global spatial dependencies between regions, which leads to inferiority compared to DeepSTN+ and DeepLGR. Another interesting observation is that the heuristics including CA and Last achieves much less SMAPE than previous CNN-based methods. Recall that SMAPE prefers to penalize the errors in regions with lower flow volumes. This observation reveals the importance of the temporal dependencies in such regions since CA and Last only consider the temporal closeness for forecasting. Only our method performs better than the heuristics on SMAPE with the usage of tensor decomposition, which will be detailed in the ablation study. Last but not least, DeepLGR is also more stable than the baselines according to the standard deviation observations.
\begin{table}[!b]
\caption{Prediction results of various methods on the HappyValley dataset, where \#Params is the number of parameters and M denotes million.}
\centering
\tabcolsep=4.5mm
\begin{tabular}{l|c|ccc}
\shline
Method & \#Params & MAE & SMAPE \\
\hline
CA & x & 2.23 & 0.46 \\
Last & x & 2.20 & \textbf{0.38}\\
ARIMA & 0.00M & 2.14 & 0.47\\
DeepST & 0.59M & 2.02 $\pm$ 0.05 & 0.56 $\pm$ 0.05 \\
ST-ResNet & 2.73M & 1.98 $\pm$ 0.05 & 0.53 $\pm$ 0.04 \\
ConvLSTM & 5.98M & 1.86 $\pm$ 0.01 & 0.48 $\pm$ 0.10 \\
DeepSTN+ & 15.70M & 1.92 $\pm$ 0.01 & 0.54 $\pm$ 0.06 \\
DeepLGR & 0.97M & \textbf{1.84 $\pm$ 0.01} & 0.40 $\pm$ 0.02 \\
\shline \end{tabular}%
\label{tab:hv}%
\end{table}%
Compared to TaxiBJ with a citywide scale, HappyValley focuses on a local area with a highly skewed flow distribution, where only a few regions contain dense populations. Table \ref{tab:hv} presents a comprehensive comparison of each model over this dataset. First, it can be seen easily that our framework shows great superiority against the CNN-based methods and slightly outperforms ConvLSTM in terms of both metrics, while using as little as 6.2\% of the amount of parameters required in the state-of-the-art method (DeepSTN+). This fact demonstrates that our model is more practical than other CNN-based solutions in real-world systems.
Second, similar to the results in TaxiBJ, DeepLGR performs more stable than the baselines according to the standard deviation in multiple experiments.
Third, the heuristic method (Last) achieves the lowest SMAPE but the second-highest MAE, which can prove the skew distribution of this dataset.
Last, the fact that DeepLGR and DeepSTN+ outperform ST-ResNet verifies the necessity of modeling global context in such a small area.
\subsubsection{Ablation Study}
To further investigate the effectiveness of each component, we compare DeepLGR with its variants over TaxiBJ-P1. For simplicity, we use the terms as local, global and TD to denote the three components in our framework respectively. Based on them, DeepLGR and its variants can be denoted as:
\begin{itemize} [leftmargin=*]
\item \textbf{local+global+TD}: The original implementation of DeepLGR.
\item \textbf{local+global+MF}: To show the effectiveness and lightweight property of TD against MF, we replace TD in the region-specific predictor by MF.
\item \textbf{local+global}: Similar to the CNN-based baselines \cite{zhang2016dnn,zhang2017deep,lin2019deepstn}, this variant uses shared parameters (i.e., not region-specific) as the predictor.
\item \textbf{local+TD}: The variant of DeepLGR without global context module.
\item \textbf{local+MF}: We first remove global context module from DeepLGR and then replace TD in region-specific predictor by MF.
\item \textbf{local+bilinear}: We employ bilinear interpolation rather than Subpixel block to upsample the global priors, so as to obtain new global representations.
\item \textbf{local}: The last two components are removed from DeepLGR.
\end{itemize}
\begin{table}[!b]
\caption{Results of different variants over TaxiBJ-P1 (trained/tested five times).}
\centering
\tabcolsep=4.5mm
\begin{tabular}{l|c|cc}
\shline
Variants & \#Params & MAE & SMAPE \\
\hline
local & 0.72M & 2.21 $\pm$ 0.01 & 0.37 $\pm$ 0.03 \\
local+MF & 0.89M & 2.19 $\pm$ 0.02 & 0.36 $\pm$ 0.03 \\
local+TD & 0.74M & 2.19 $\pm$ 0.01 & 0.32 $\pm$ 0.03 \\
\hline
local+bilinear & 0.73M & 2.20 $\pm$ 0.02 & 0.35 $\pm$ 0.03 \\
local+global & 2.30M & 2.17 $\pm$ 0.02 & 0.29 $\pm$ 0.03 \\
\hline
local+global+MF & 2.46M & 2.15 $\pm$ 0.00 & 0.27 $\pm$ 0.01 \\
local+global+TD & 2.31M & \textbf{2.15 $\pm$ 0.00} & \textbf{0.19 $\pm$ 0.00} \\
\shline
\end{tabular}%
\label{tab:variant}%
\end{table}%
Table \ref{tab:variant} illustrates the variant comparison over TaxiBJ-P1. We discuss the effects of each model component as follows:
\begin{itemize}[leftmargin=*]
\item \emph{Local feature extraction}: A powerful ST feature extractor enables the capability of extracting useful representations for each region. Compared to previous attempts like ST-ResNet based on residual blocks, our feature extraction module largely improves the performance (e.g., local vs. ST-ResNet in Table \ref{tab:taxibj} and \ref{tab:variant}). We further investigate the effects of the number of SE blocks in this module. As shown in Figure \ref{fig:effect_of_local}, it achieves the best performance when $M=6$ in the test set. Noted that we choose $M=9$ as the default setting of DeepLGR because of its best performance on the validation set rather than the test set. Besides, we replace the SE blocks in this module by residual blocks to show the advantages of SE blocks, where the results are also in Figure \ref{fig:effect_of_local}.
\item \emph{Global context module}: As a vital component in our framework, this module provides the global information to boost the performance. As illustrated in Table \ref{tab:variant}, the comparison between local and local+global (also local+TD and local+global+TD) can verify the effectiveness of this module. With the usage of Subpixel block with a parametric design, local+global brings an improvement beyond local+bilinear.
\item \emph{Region-specific predictor}: This module is used to determine the region-specific parameters for predictions. Thus, we compare it with a shared fully connected layer with $n_f$ parameters (local+global), and the matrix decomposition method. From the last three rows of Table \ref{tab:variant}, we observe that TD demonstrates very competitive accuracy while using as little as 6.3 \% of the number of parameters required in MF (i.e., 0.01M vs. 0.16M). Moreover, TD significantly outperforms MF over SMAPE since it allows the model to capture spatial dependencies between regions.
\end{itemize}
\begin{figure}[!t]
\centering
\includegraphics[width=0.8\textwidth]{imgs/fig_exp_local.pdf}
\caption{\label{fig:effect_of_local} SE vs. residual block over P1, where the shade area is the standard deviation.}
\end{figure}
\begin{table}[!b]
\caption{Results of various models for fine-grained flow inference. We train/test each method five times, and present results using the format:``mean $\pm$ standard deviation".}
\centering
\tabcolsep=2.3mm
\begin{tabular}{l|cc|cc}
\shline
\multirow{2}*{Method} & \multicolumn{2}{c|}{TaxiBJ-P1} & \multicolumn{2}{c}{HappyValley} \\
\cline{2-5} & \multicolumn{1}{c}{MAE} & \multicolumn{1}{c|}{SMAPE} & \multicolumn{1}{c}{MAE} & \multicolumn{1}{c}{SMAPE} \\
\hline
VDSR & 2.23 $\pm$ 0.05 & 0.54 $\pm$ 0.03 & 2.13 $\pm$ 0.04 & 0.61 $\pm$ 0.02 \\
SRResNet & 2.20 $\pm$ 0.05 & 0.52 $\pm$ 0.03 & 1.89 $\pm$ 0.05 & 0.61 $\pm$ 0.03 \\
UrbanFM & 2.07 $\pm$ 0.03 & 0.25 $\pm$ 0.02 & 1.80 $\pm$ 0.02 & 0.41 $\pm$ 0.02 \\
\hline
local & 1.98 $\pm$ 0.01 & 0.20 $\pm$ 0.01 & 1.83 $\pm$ 0.01 & 0.43 $\pm$ 0.01 \\
local+global & 1.96 $\pm$ 0.00 & 0.20 $\pm$ 0.01 & 1.78 $\pm$ 0.01 & 0.38 $\pm$ 0.01 \\
local+global+TD & \textbf{1.95 $\pm$ 0.00} & \textbf{0.18 $\pm$ 0.01} & \textbf{1.76 $\pm$ 0.01} & \textbf{0.35 $\pm$ 0.00} \\
\shline
\end{tabular}%
\label{tab:inference}%
\end{table}%
\subsection{Results on Fine-grained Flow Inference}
Experimental results on the second task have demonstrated the superiority of our framework again. From Table \ref{tab:inference}, we have the following observations:
1) UrbanFM equipped with our framework (denoted as local+global+TD) shows considerable improvements against its original version on both datasets, validating its great generality in different applications. For example, DeepLGR achieves 5.8\% lower MAE and 28.0\% lower SMAPE than UrbanFM in the TaxiBJ-P1 dataset.
2) The three components of DeepLGR are effective according to the advancement of performance (only except local vs. UrbanFM in HappyValley).
3) Compared to VDSR and SRResNet for image-resolution , UrbanFM outperforms them by considering the domain knowledge, i.e., spatial hierarchy and external influence \cite{liang2019urbanfm}.
From above discussions, we can see that existing approaches like UrbanFM can be easily integrated with our framework.
We further investigate the efficiency of DeepLGR. Figure \ref{fig:efficiency} plots the MAE on the validation set during the training phase using TaxiBJ-P1. Remarkably, UrbanFM and DeepLGR converge much smoother and faster than the others as shown in Figure \ref{fig:efficiency}(a). A more detailed comparison between UrbanFM and DeepLGR lies in Figure \ref{fig:efficiency}(b). From this figure, we can see that DeepLGR converges at iteration 3540 (epoch 37) while UrbanFM early-stops at iteration 7720 (epoch 81). This fact demonstrates that our framework can also accelerate the training phase of existing method.
\begin{figure}[!t]
\centering
\includegraphics[width=0.8\textwidth]{imgs/fig_efficiency.pdf}
\caption{\label{fig:efficiency} Convergence speed of various methods over P1.}
\end{figure}
\section{Related Work}
Citywide crowd flow analytics has attracted considerable attention of researchers in recent years. A series of studies have explored forecasting millions or even billions of individual mobility traces \cite{song2014prediction,fan2015citymomentum}. Different from analyzing crowd behaviors on an individual level, several works started to forecast citywide crowd flow by aggregating the crowds into corresponding regions \cite{li2015traffic,hoang2016fccf}. Among them, statistical learning was employed to capture inter-region relationship. With interest in obtaining fine-grained regional data, several studies \cite{liang2019urbanfm,zong2019deepdpm,ouyang2020fine} presented techniques to recover fine-grained crowd flow from coarse-grained data.
Recently, there have been many attempts focusing on end-to-end deep learning solutions such as CNNs for citywide crowd flow analytics. A pioneering study by \cite{zhang2016dnn} presented a general framework based on CNNs for citywide crowd flow prediction. By using a CNN architecture, their method can capture the spatio-temporal correlations reasonably and accurately. To overcome the gradient vanishing problem, they further integrated their framework using deep residual learning \cite{zhang2017deep}. Similar insight has been applied in taxi demand prediction \cite{yao2018deep}. Moreover, there are also several studies \cite{zonoozi2018periodic,yao2019revisiting} using RNNs to model the periodic temporal dependencies. Very recently, a ConvPlus structure \cite{lin2019deepstn} showed the state-of-the-art performance by directly modeling the long-range spatial dependencies between region pairs. However, as detailed in Section \ref{para:intro}, these methods are very inefficient in learning global spatial dependencies and none of them considers latent land function. To tackle these drawbacks, we have presented a general framework that can be easily generalized to all kinds of crowd flow data.
\section{Conclusion and Future Work}
In this paper, we have carefully investigated existing CNN-based methods for citywide crowd flow analytics, and exposed their inefficiency in capturing global spatial dependencies and incapability in generating region-specific predictions. Based on our discovery, we have presented the DeepLGR framework which decouples the local feature extraction and global context modeling, and provides a parameter-efficient solution for customizing regional outputs. We have evaluated DeepLGR over two real-world citywide crowd flow analytics tasks. In the prediction task, DeepLGR outperforms the state-of-the-art (DeepSTN+) by average 8.8\% and 45.9\% on TaxiBJ dataset, and 4.2\% and 25.9\% on HappyValley dataset in terms of MAE and SMAPE metrics respectively. Moreover, our framework is more lightweight than the state-of-the-art methods, which is very important in real practice. In the second task, we have verified that the existing approach can be easily integrated with our framework to boost its performance. In the future, we will explore two directions. First, we notice that manually designing neural networks requires amount of expert efforts and domain knowledge. To overcome this problem, we can follow a very recent study \cite{li2020autost} to study Neural Architecture Search (NAS), which can automatically construct a general neural network for diverse spatio-temporal tasks in cities. Second, we will extend our framework to a much broader set of spatio-temporal tasks by using graph convolutions.
\section*{Acknowledgement}
We thank all reviewers for their constructive and kind suggestions.
This work was supported by the National Key R\&D Program of China (2019YFB2101805) and Beijing Academy of Artificial Intelligence (BAAI).
|
3,212,635,537,994 | arxiv | \section{Introduction}
Videos use audio and visuals to convey information, making them inaccessible to people who cannot see or hear content in certain modalities. To make videos accessible, video authors add audio descriptions (AD) that describe important visual content, and closed captions (CC) that transcribe the speech and non-speech sounds.
However, to identify parts of the video that require further description, authors must manually watch the video all the way through, playing and pausing to check if: (1) the important visuals have not been described in the audio (\textit{e.g., }~a travel montage in a vlog), and (2) the important audio is not present in the visuals and captions (\textit{e.g., }~a door slams off-screen).
This process of identifying inaccessible video segments is challenging and time-consuming, especially for
video
accessibility novices.
To guide authors to describe inaccessible video segments, existing audio description authoring tools surface ``gaps in speech'' as a proxy for moments where the visuals are unlikely to be verbally described~\cite{natalie2021efficacy, wang2021toward, campos2020cinead, pavel2020rescribe, yuksel2020human}.
However, many video genres including tutorials, vlogs, and lectures may not feature significant gaps in speech~\cite{liu2021makes, peng2021slidecho}, and audio description guidelines as well as prior research~\cite{wang2021toward, liu2021makes, wcag2} indicate that visuals can be inaccessible to blind and visually impaired (BVI) people even when there is accompanying speech.
For example, a speaker may make an ambiguous verbal reference to visual content (\textit{e.g., } ``make sure to have \emph{these} before you get started'') or share a personal story while demonstrating a tutorial step.
In addition, visuals without speech can be accessible if they are understandable from non-speech sounds alone.
Thus, using gaps in speech alone, authors will miss important inaccessible moments or be prompted to describe already-accessible moments.
Similarly, caption authoring tools~\cite{lasecki2012real, rev, descript} let authors correct errors from automatic speech recognition (ASR), but they fail to surface moments when important audio does not also appear on screen (\textit{e.g., } someone leaves and we hear a door slam).
To help authors efficiently identify and address audio and visual accessibility problems, we present \systemname. \systemnamespace surfaces asymmetries between the visual track and the audio track, or \textit{modality asymmetries}.
By identifying moments in the visuals that are not available in the audio, \systemnamespace surfaces moments that are not accessible to blind and visually impaired (BVI) audience members. Similarly, by identifying moments in the audio that are not available in the visuals, \systemnamespace surfaces moments that are not accessible to d/Deaf and Hard of Hearing (DHOH) audience.
To automatically identify modality asymmetries, \systemname's computational pipeline segments the audio and visual tracks and uses \textit{cross-modal grounding} to identify mismatches between the two tracks (Figure~\ref{fig:system-diagram}A).
\systemnamespace then displays the results in an interface where authors can jointly author closed captions and audio descriptions by easily navigating to inaccessible moments (Figure~\ref{fig:system-diagram}B). Authors can then preview and export their resulting audio descriptions and closed captions (Figure~\ref{fig:system-diagram}C).
We evaluated \systemnamespace in a user study with 11 video authors creating captions and audio descriptions for four videos. Authors more efficiently authored audio descriptions and captions with better precision and recall in addressing accessibility issues when using \systemname's modality asymmetry predictions than without these predictions.
We also invited two video authors who frequently posted videos on YouTube to use \systemnamespace to make two of their own videos accessible, and reported that they would use \systemnamespace in their workflow to produce more accessible videos.
In summary, we contribute:
\begin{itemize}
\item A pipeline to compute accessibility scores of the visual and audio segments of a video by checking for \textit{modality asymmetries} via cross-modal grounding.
\item A unified tool that helps video authors to locate and address visual and auditory accessibility problems of a video.
\item A user study demonstrating that \systemnamespace improves people's efficiency and reduces their mental demand in identifying accessibility issues.
\end{itemize}
\section{Related Work}
\subsection{Video Accessibility Guidelines}
The Web Content Accessibility Guidelines' (2.0) principle of Perceivable suggests that \textit{``Information and user interface components must be presentable to users in ways they can perceive''}~\cite{wcag2}.
Thus, authors need to make their videos perceivable to audiences who cannot see or hear the content by adding Closed Captions (CC) that use text to describe ``both speech and non-speech audio information needed to understand the media content''~\cite{wcag2} and Audio Descriptions (AD) that use text to describe ``important visual details that cannot be understood from the main soundtrack alone''~\cite{wcagad,packer2015overview}.
In prior work that interviewed blind and visually impaired YouTube audience, participants reported losing a sense of the video during visual content that was not well-described in the speech: ambiguous verbal references to visual content (\textit{e.g.}, look at `this'), unidentified sounds, undescribed text-on-screen, and others such as visual jokes~\cite{liu2021makes}.
Thus, we aim to help content creators identify and address moments that are visually inaccessible due to modality asymmetry.
Unlike audio description guidelines that suggest narrating the ``important'' visual content~\cite{wcagad}, guidelines for Closed Captions by the Federal Communications Commission~\cite{fcc} require closed captions that describe spoken words and convey background noises and other sounds to the \textit{fullest extent possible} within a synchronous track.
Thus, we help authors identify all moments where additional synchronous description is needed, and help authors prioritize \textit{e.g., } silent portions may not require further description.
\input{figures/interface}
\subsection{Authoring AD and CC}
Prior work aims to help people manually author audio descriptions
with task-specific authoring tools~\cite{youdescribe,branje2012livedescribe,3playmedia}, feedback on the content at production-time~\cite{peng2021say},
with feedback on audio descriptions~\cite{saray2011adaptive, natalie2021efficacy},
and with hosted descriptions~\cite{youdescribe}.
Since authoring descriptions is a time-consuming process, other prior work seeks to provide computational support for this task including using: computer vision to detect visual content~\cite{gagnon2009towards, gagnon2010computer, campos2020cinead}, using deep learning to provide a computer-drafted description~\cite{gagnon2009towards, yuksel2020human,wang2021toward}, synthesized voice to convert text to speech~\cite{gagnon2010computer, kobayashi2009providing, kobayashi2010synthesized,szarkowska2011text}, and automatic editing to fit human-authored descriptions into the space provided~\cite{pavel2020rescribe}.
While focusing on methods to help people write better descriptions, such tools only find inaccessible moments for description by surfacing silent portions of the video~\cite{branje2012livedescribe,pavel2020rescribe,yuksel2020human,gagnon2010computer}, or by helping people find film-specific visual content that may need descriptions (\textit{e.g.}, scene changes, characters~\cite{gagnon2010computer}).
Rather than assessing video in a single modality, we explore finding accessibility problems by assessing asymmetries between the auditory and visual content.
Current caption authoring tools~\cite{lasecki2012real, rev, descript} transcribe speech and allow creators to correct the transcript.
In this work, we also surface non-speech sounds to facilitate caption authoring and add modality matching score to help people prioritize points where additional description could be needed (\textit{e.g., } a sound that happens off-screen may be highly important to describe, while a silent section would not be).
\subsection{Accessibility Assessment Tools}
Assessing accessibility of visual content is challenging for authors who do not share accessibility needs with their audience members.
As a result, accessibility research includes a long history of prior work aimed to help people assess and correct accessibility problems in their designs including tools aimed at simulating accessibility issues~\cite{chromedevtools,asakawa2005s} and evaluating accessibility with respect to metrics~\cite{vigo2011automatic, mankoff2005your}.
Simulation-style tools to support sighted designers trying to achieve visually accessible designs; for example, Chrome Dev Tool's colorblindness and blurriness emulators to help designers assess legibility~\cite{chromedevtools}.
Using such simulations as a replacement for involvement with people with disabilities has several issues, as they are unable to capture the full experience of disability and give designers false conceptions~\cite{tigwell2021nuanced}.
Given that people with disabilities, in partnership with organizations (\textit{e.g.}, W3C~\cite{wcag2}), have authored guidelines and best practices to make design accessible, other prior work alerts authors to violations of these guidelines in authoring tools. For example, accessibility checkers in PowerPoint~\cite{powerpointchecker} and Adobe Acrobat~\cite{acrobatchecker} alert authors to potential accessibility issues in their designs (\textit{e.g.}, missing alt text, document read order).
Furthermore, web accessibility checkers provide a report card on similar types of issues to fix~\cite{vigo2011automatic, mankoff2005your}.
We extend prior work by assessing the accessibility issues and surfacing accessibility issues based on existing guidelines about video accessibility.
\subsection{Assessing Audio and Visual Similarity}
Recent work in unsupervised cross-modal machine learning explores learning a joint embedding space for information in different modalities, including text and images~\cite{clip2021learning, Wang2019CAMPCA, Stroud2020LearningVR}, text and video~\cite{mle2020end, mmv2020self, Croitoru2021TeachTextCG, Wang2021T2VLADGS}, and audio and video~\cite{mmv2020self, Morgado2021AudioVisualID}.
Such models enable comparison between any visual, text, or audio segment.
While these models can be used for retrieval across modalities (\textit{e.g.}, text-image retrieval~\cite{clip2021learning, Wang2019CAMPCA}, and text-video retrieval~\cite{Croitoru2021TeachTextCG, Wang2021T2VLADGS}), we use a cross-modal approach to inform authors of accessibility issues due to low correspondence between the modalities, or modality asymmetry.
Prior video work in video accessibility has also considered the similarity between video and audio tracks. Wang et al. filter possible accessibility problems first by gaps in speech then use video and audio similarity to prioritize what non-speech segments to describe~\cite{wang2021toward}. Liu et al. checks if detected objects are mentioned in the transcript, along with other metrics, then assigns an accessibility score to a video to help blind viewers find accessible videos~\cite{liu2021makes}.
We instead compute the fine-grained similarity between audio and visual segments to help authors find accessibility issues outside of gaps in speech that have not yet been addressed by prior work.
\section{\systemnamespace Interface}
\systemnamespace enables authors to efficiently identify and address visual and auditory accessibility problems in videos.
The interface consists of three main components: 1) a \textit{video pane} that lets authors navigate via an audio segment timeline or video segment timeline to identify inaccessible video segments (Figure~\ref{fig:interface}A), 2) the \textit{video description pane} that lets authors identify and address visual accessibility problems (Figure~\ref{fig:interface}F), and 3) a \textit{caption pane} that lets authors address auditory accessibility problems and navigate the video with a time-aligned caption transcript (Figure~\ref{fig:interface}E).
\subsection{Video Pane}
The \textit{video pane} (Figure~\ref{fig:interface}A) displays the video and lets authors play/pause the video and seek within the video using two timelines: (1) the audio timeline that lets authors navigate to auditorily inaccessible segments, and (2) the visual timeline that lets authors navigate to visually inaccessible segments.
The audio timeline displays audio segments that each represent a segment with continuous speech, or non-speech sound. The visual timeline displays visual segments that each represent a segment of continuous footage (i.e., a shot).
Each segment is colored with its estimated accessibility\footnote{Computed using the cross-modal grounding pipeline as described in Section~\ref{algorithmic_methods}} from gray (accessible) to red (inaccessible) using sRGB inverse gamma mixing.
The darkness of the red represents the weighted sum of the similarity score of a segment to segments in the other modality.
Using either timeline, authors can gain an overview of accessibility issues, or quickly navigate to an inaccessible segment by clicking on the segment to play the corresponding point in the video.
For example, by clicking on the first red segment in the audio track (Figure~\ref{fig:interface}C) an author will hear an inaccessible audio segment --- music plays that is not available in the captions preview (Figure~\ref{fig:interface}D).
Authors may inspect inaccessibility prediction results displayed in the timeline by hovering over an audio or visual segment (Figure~\ref{fig:hover}) to see the audio segments that are predicted to match that segment (displayed with a higher opacity).
As the author navigates and plays the video with the video pane, the corresponding segments are highlighted in the linked video description pane and the caption pane.
\input{figures/hover}
\subsection{Video Description Pane}
\label{video_description_pane}
\input{figures/save_edit}
The \textit{video description pane} (Figure~\ref{fig:interface}F) lets address inaccessible visual segments by writing text descriptions of the visual content.
Each video description segment consists of a vertical sidebar that is colored according to its predicted accessibility, an editable text field where an author may add descriptions, and ``save''/``edit'' and ``dismiss'' buttons to address or ignore surfaced visual accessibility issues.
Video description segments are relatively aligned with the caption segments in the \textit{captions pane} such that authors can preview the nearby narration. The height of each video description segment represents its relative length such that authors can estimate the approximate length of description required.
When an author locates a visual accessibility issue (\textit{e.g.}, the last displayed segment in Figure~\ref{fig:interface}H), the author can click the segment to play the clip and to check if the visual content is described in the audio or existing descriptions. For example, in this case, the author may notice that the host placing the foil inside the pan is not yet described in the captions, and add a description by typing ``Flip the pan over and put the foil inside'' and clicking ``Save'' (Figure~\ref{fig:save_edit}A). The vertical sidebar, and the corresponding segment in the video pane's visual timeline, then change to blue to indicate the issue has been addressed.
If an author decides that a suggested visual accessibility issue does not need a description, they can dismiss the problem (Figure~\ref{fig:save_edit}B). The vertical sidebar for that segment and the corresponding horizontal bar in the visual track timeline will turn dark gray to indicate the issue has been dismissed.
Authors can manually add a description to a point in the video where an accessibility issue was not detected by double-clicking the visual segment in the video pane's visual timeline to create a corresponding video description segment in the video description pane.
By default, the video description pane displays visual segments with estimated accessibility scores lower than 0.35 (range 0-1). Authors can use the slider (Figure~\ref{fig:interface}G) to surface more visual accessibility problems when making sure they covered everything, or fewer accessibility problems when prioritizing for a time constraint.
\subsection{Captions Pane}
\label{captions_pane}
The \textit{captions pane} lets authors navigate the video with a time-aligned transcript and write captions to address inaccessible audio segments. \systemnamespace automatically provides captions for the speech, so authors can focus on making non-speech sounds accessible. Each caption pane segment has a similar structure with video description segments. Authors can quickly locate, review, script captions in-place, or dismiss a suggested problem.
The caption pane displays predicted audio accessibility by coloring the vertical bars (similar to the video description pane) to help authors understand and prioritize audio accessibility issues.
Unlike the video description pane, we do not use predicted audio accessibility to filter audio accessibility issues as Closed Caption guidelines state that all important sounds should be synchronously described whether they can be inferred from the visual content alone~\cite{acbguidelines, adlabguidelines}.
Using the captions pane, authors can click on a caption segment to hear the segment, then script a caption if the sound is important (music at 0:00 in Figure~\ref{fig:interface}E) or dismiss the segment if the sound is not important (silence at 00:25 in Figure~\ref{fig:interface}E).
\subsection{Accessible Video Preview}
After authors create captions and video descriptions for inaccessible segments, they can then preview their results as the video plays. Original captions of the video and captions created by the author are displayed in ``Captions Preview'' (Figure~\ref{fig:interface}D). Audio descriptions are synthesized via a text-to-speech engine (Web API's SpeechSynthesisUtterance Interface\footnote{https://developer.mozilla.org/en-US/docs/Web/API/SpeechSynthesisUtterance}).
Our system renders audio description in the format of \textit{extended description}~\cite{wcagad}, which pauses the video, plays the synthesized speech descriptions, and continues the video.
\section{Cross-modal Grounding Pipeline}
\label{algorithmic_methods}
\input{figures/pipeline}
We present a computational pipeline that segments the auditory and visual track of the video (Figure~\ref{fig:pipeline}A) and identifies asymmetries between auditory and visual tracks using cross-modal grounding analysis (Figure~\ref{fig:pipeline}B).
\subsection{Segmentation}
To create visual segments, we detect shots, or segments with continuous footage. To segment shots, we used scenedetect\footnote{http://scenedetect.com/}'s content-aware scene segmentation algorithm that compares the HSV color space in adjacent frames against a threshold to determine if the two segments belong to the same shot.
To create audio segments, we follow prior work~\cite{liu2021makes, pavel2020rescribe} by aligning the transcript and audio using Gentle forced-aligner~\cite{ochshorn2017gentle} to get word-level timings, and then consider any gap between words longer than 2 second to be a non-speech audio segment, and any gap longer than 0.5 second but shorter than 2 seconds as a pause in speech. We segment the audio into speech and non-speech audio clips based on the gaps. In addition, we also generate a list of segmented transcripts according to the start/end of audio segments.
\subsection{Cross-modal Grounding}
\input{figures/matchings}
To assess if each visual segment and audio segment is described in the other modality, we compute visual-text and visual-audio matching scores for all video and audio segment pairs using multimodal machine learning algorithms.
\subsubsection{Visual-text/audio Matching}
\systemnamespace uses multimodal machine learning algorithms which learn a symmetric joint embedding space for visual and auditory or textual data.
With the embeddings, we can measure if a visual segment and a audio segment is semantically similar by computing the dot product of visual embeddings and audio/text embeddings. Specifically, for visual-text matching, we use the MIL-NCE model~\cite{mle2020end} which was trained on HowTo100M, a dataset of 100 million clips-narrations from YouTube.
For visual-audio matching, we use the state-of-the-art MultiModal Versatile (MMV) networks~\cite{mmv2020self} which was trained on AudioSet, a dataset consists of 10 seconds clips coming from 2 million
different internet videos. AudioSet contains a variety of audio types including musical instruments, animal, mechanical sounds, etc.
By matching all \(n_v\) visual segments to \(n_a\) audio segments in a video, MIL-NCE and MMV each produces a \(n_v \times n_a\) matrix, where cell \((i, j)\) is the matching score for visual segment \(v_i\) and audio segment \(a_j\). Each matrix is normalized to range \(0-1\).
Figure~\ref{fig:matchings} displays examples of such cross-modal grounding results.
To estimate the accessibility of a visual segment, \(v_i\), we compute its matching scores to all audio segments in the same video. When matching to audio segments that contain speech, we use the MIL-NCE score (since match different visuals to human speech sound does not make sense). We remove stop words of all transcripts before computing its correspondence to visuals. When matching to audio segments without speech, we use the MMV score. We then compute a weighted sum of all scores based on each audio segment's temporal position to \(v_i\) (as explained in Section~\ref{temporal_weighting}). Thus, for a video with \(n_v\) many visual segments and \(n_a\) audio segments:
\begin{equation}
\operatorname{score}(v_i)=
\sum_{j}^{n_{a}} w_{i, j} * \operatorname{matching}(v_i, a_j)
\end{equation}
where,
\begin{equation}
\operatorname{matching}(v_i, a_j)=\begin{cases}
\operatorname{MIL-NCE}\left(v_{i}, t_{j}\right), & \text{if speech}. \\
\operatorname{MMV}\left(v_{i}, a_{j}\right), & \text{if non-speech}.
\end{cases}
\end{equation}
Similarly, to estimate the accessibility score of an audio segment, \(a_j\), we compute its degrees of matching to all visual segments. If the current audio segment is non-speech, we only use the MMV score.
However, when checking whether the content of a speech audio is presented in the visual, it is prevalent that the speech is transcribed and added as subtitles using automatic speech recognition technology. Even if subtitles of speech are not added, systems can quickly apply ASR to incorporate them into the visual modality. Thus, if an audio segment is detected as speech, we will assign it a constant value \(c\) and consider it accessible (since the speech information is displayed as subtitles in visual). For an audio segment, \(a_j\):
\begin{equation}
\operatorname{score}(a_j)=\begin{cases}
c, & \text{if speech} \\
\sum_{i}^{n_{v}} w_{j, i} * \operatorname{MMV}\left(a_{j}, v_{i}\right), & \text{if non-speech}.
\end{cases}
\end{equation}
Note that because both the MIL-NCE and MMV scores are symmetric (e.g., \(\operatorname{MMV}\left(a_{j}, v_{i}\right) = \operatorname{MMV}\left(v_{i}, a_{j}\right)\)), we only need to perform matrix multiplications one time to compute scores for both visual and audio segments.
\subsubsection{Temporal Weighting}
\label{temporal_weighting}
A visual segment can be matched to an audio segment that is far away from each other in time. In such cases, although the information is grounded in the other modality, it would be hard for people to connect and make sense of such cross-reference. Thus, as shown in the above equations, we apply a temporal weighting to the output visual-text and visual-audio matching scores. Specifically, the matching between a visual segment \(v_i\) and an audio segment \(a_j\) diminishes exponentially by a factor of \(w\) (\(0\leq w \leq1\)) for every 5 seconds' distance in time:
\begin{equation}
w_{i, j}=w_{j, i}=w^{\frac{\left|T S_{i}-T S_{j}\right|}{5}}
\end{equation}
Where \(TS_i\) is the timestamp in seconds of segment \(i\), and \(w\) is the weighting factor. We empirically found that \(w = 0.45\) works well.
\subsection{Post-processing}
\label{postprocessing}
After we compute segment visual accessibility scores, \(\operatorname{score}(v_i)\), and audio accessibility scores, \(\operatorname{score}(a_j)\), for the video, we normalize the scores into \(0-1\) ranges. We then remove commonly detected accessibility issues that do not need further description including: the presenter speaking to the camera, and silences.
\subsubsection{Presenter Speaking}
In initial tests, our approach detected moments where the host was speaking to the camera to be inaccessible, leading to low precision for how-to and recipe videos like \cite{youtubeA} (precision=0.356, recall=0.875) and \cite{youtubeB} (precision=0.435, recall=0.929). While the visual content and speech did not match, these segments were accessible as they could be implied from the audio alone (the presenter's voice).
To address this, we detect faces using OpenCV\footnote{https://docs.opencv.org/3.4/db/d28/tutorial\_cascade\_classifier.html}, and compute the area of the detected face bounding box per second for each visual segment. We consider a visual segment to be ``presenter speaking'' and thus not inaccessible if the area per second metric is greater than a threshold \(TH_{presenter}\). We empirically determined a threshold \(TH_{presenter} = 58000\).
With presenter detection, the precision improved to 0.636 on \cite{youtubeA} and 0.867 on \cite{youtubeB}.
\subsubsection{Silences}
Similarly, our initial approach predicted segments with silent or insignificant audio to be inaccessible as the quiet noises were not detected to match the visuals.
For instance, the algorithm considered a scene in a recipe video~\cite{youtubeA} where the host flips the pan over making minor noises, and a scene in a food review video~\cite{youtubeC} where the host was eating and making a chewing sound, as inaccessible.
To address the issues, we detect silences by computing
the average intensity of audio segments using librosa\footnote{librosa.org} and compare it to a threshold \(TH_{silence}\), which we empirically set to 0.007. If the average intensity score is lower than the threshold we label this audio segment as insignificant and thus not inaccessible.
\subsubsection{Threshold}
As discussed in~\ref{video_description_pane}, \systemnamespace displays visual segments with grounding score larger than a threshold as the visual accessibility issues. We selected 0.35 empirically as it worked consistently well for diverse videos, and favored recall over precision such that the system showed more potential accessibility issues. Authors may easily dismiss false accessibility issues using the ``Dismiss'' button.
\subsection{Technical Evaluation}
\label{technical_evaluation}
\input{tables/performance}
We evaluated \systemname's cross-modal grounding pipeline using 20 randomly selected videos from YouDescribe\footnote{https://youdescribe.org/}, a platform where people can request audio descriptions for YouTube videos.
In particular, we limited our random selection to videos that were less than 5 minutes with captions available (implies some narration).
All 20 videos were not tested when we built the system, i.e. out-of-bag, and they covered diverse topics: how-to (5 videos), recipe (4 videos), vlog (3 videos), campus tour (3 videos), documentary (2 videos), educational (2 videos) and review (1 video).
Two researchers independently identified visual and auditory accessibility issues in the videos based on guidelines~\cite{acbguidelines, fcc, wcag2} (Appendix \ref{appendix_guidelines}).
We first organized our initial labels in a spreadsheet, and held three one-hour long discussion sessions and went through each accessibility problem one by one.
70.78\% of initially identified issues were the same. Researchers discussed the remaining labels until agreement was reached. In total, the sample included 182 visual accessibility issues and 79 auditory accessibility issues.
There were two major kinds of disagreements: (1) Missing labels. As a reflection of our motivation, identifying visual and audio accessibility issues for 20 videos was time-consuming and mentally demanding. Most of our disagreements were accessibility issues noticed by one researcher but missed by another. We quickly agreed on those issues. (2) Misinterpreting accessibility guidelines. One of our researchers initially thought that a non-speech audio segment does not need to be described as long as it can be inferred from the visuals. We corrected this mistake during the review of our labels.
We then used \systemnamespace to predict visual and auditory accessibility issues in these videos.
All 20 videos and \systemname's demo of those videos are available online\footnote{\url{https://xybruceliu.github.io/CrossA11y/}}.
We also predicted accessibility issues using two baselines for comparison: (1) mark each segment as inaccessible with 50\% chance (Random)
, and (2) mark each segment as inaccessible if it did not include speech (Gaps) following prior work~\cite{pavel2020rescribe,natalie2021efficacy,yuksel2020human,wang2021toward}.
To assess whether a labeled visual or auditory accessibility issue was accurately detected, we compared the start and end times of all segments that were predicted to be inaccessible with the start and end times of manually labeled inaccessible segments.
We defined a prediction as accurate if there was an \(>50\%\) overlapping manually labeled accessibility problem.
Using the thresholds we selected in Section~\ref{algorithmic_methods}, \systemnamespace achieved a higher F1 score compared to the baseline methods (Table~\ref{tab:performance}).
For visual accessibility issues, the recall score increased significantly from 0.385 with gaps in speech to 0.984 with \systemname, meaning that \systemnamespace identified visuals with accompanying speech that are still inaccessible to BVI audience members. The precision decreased from 0.833 with gaps in speech to 0.694 with \systemname.
On average, for each video (average length = 3 minutes 11 seconds) we detect 9.1 true visual accessibility issues and 3 of these issues are false positives.
For \systemname, we prefer high recall (the ability to show all issues) over high precision (the ability to show few incorrect issues), as authors may easily review and dismiss inaccurate issues (false positives), but they may struggle to find issues that we do not surface (false negatives).
\systemnamespace identified auditory accessibility problems with higher precision (0.983) compared to gaps in speech (0.909), as \systemnamespace removes false positive issues when the gaps in speech correspond to silence. The recall remains the same as all accessibility issues in our sample occurred during gaps in speech.
\input{figures/precision_recall}
\subsection{Limitations}
\label{tech_limitations}
From our technical evaluation, we discuss some limitations of the current implementation of \systemname.
\subsubsection{Segmentation Limitation}
We noticed that algorithms in some cases failed to segment visual and audio tracks into semantically coherent segments. For visuals, the shot detection algorithm would sometimes segment the same visual with different filming angles into different shots. This leads to lots of repetitive segments that would be annoying for authors to dismiss. In addition, the algorithm sometimes consider a long shot with different pieces of information as one large segment. This is especially common for tutorial videos that only has one shot angle (\textit{e.g.,} an origami tutorial where the camera is always facing the table).
Similarly, the pause detection algorithm also in some cases produces disproportionally long (\textit{e.g.}, a host speaks very fast with no pause) or short (\textit{e.g.}, a host talks slowly very demonstrating a step in a how-to video) segments. Our algorithm also does not address overlapping sounds like a sound effect that is covered by speech, since audio source separation still cannot produce desirable results and often requires training on specific examples.
Moreover, visual and audio segment is only a proxy of ``information piece'' that we truly want to extract. In future work will explore methods to address these issues and extract more fundamental units of information pieces.
\subsubsection{Grounding Limitation}
From our observations, current cross-modal grounding algorithms do not work well on visual details that are specific to the current video's context (\textit{e.g.,} in a recipe video the host instructs to mix the batter until it looks like ``this'', the model will label this as matched and cannot detect that the specific state of the batter is not described), and smaller or rarer visuals (\textit{e.g.,} sprinkling salt). Cross-modal machine learning algorithms can also sometimes generate in consistent results due to its unexplainability (giving divergent matching scores to similar visuals that are close to each other).
\subsubsection{Video Production Style}
\systemnamespace works better on videos with relatively dense audio and visual information, and are partially inaccessible. For videos with a monotonous visual (\textit{e.g.,} podcast video stream) or audio track (\textit{e.g.,} only background music), our system will still correctly match visuals and audio, only to show that the entire track is not described. In such scenarios \systemnamespace provides minimal information to authors.
\section{Evaluations: Can CrossA11y Users Efficiently Identify Video Accessibility Issues?}
\label{eval1}
We evaluated \systemnamespace with 12 participants who have video creation experience to compare creating AD\&CC with and without modality asymmetry visualizations.
We want to investigate: \textit{How does \systemnamespace enable authors to efficiently identify and address visual and auditory accessibility issues in videos?}
\subsection{Materials}
We selected four videos on YouDescribe.com from different genres (Cheesecake recipe~\cite{youtubeA}, handicraft tutorial~\cite{youtubeB}, restaurant review~\cite{youtubeC} and day-in-the-life vlog~\cite{youtubeD}).
All videos are under 5 minutes, and went through the same labeling process as explained in the technical evaluation section.
We used \systemnamespace to automatically identify inaccessible visual and audio segments (Table~\ref{tab:test_videos}) and rendered the four videos on \systemname's interface.
Additionally, we created \textit{Interface 1} (Figure~\ref{fig:baseline}) where we removed all accessibility visualizations to compare \systemnamespace (Interface 2) with. In Interface 1, we provide a transcript-based timeline and display gaps-in-speech, following prior work~\cite{pavel2020rescribe,natalie2021efficacy,yuksel2020human,wang2021toward}. With Interface 1, users create AD\&CC and click ``Add'' button to add it to the current video timestamp.
\input{figures/baseline}
\subsection{Participants}
We recruited 12 participants (7 female and 5 male) who all have previous video creation and sharing experience. Participants have created various types of videos including vlog, how-to, music, travel, presentation, product demo, etc. Participants were recruited through our university's internal communication channel and mailing lists.
P8 did not complete the study due to technical issues.
P5, P6, and P11 have their own YouTube/TikTok channels and have created around 40, 80, and 100 videos respectively.
P3, P7, P10 and P12 created 10-20 videos.
P1, P2 P4 and P9 less than 10 videos.
\subsection{Procedure}
We conducted a 90-minute study with each participant remotely.
Each participant was paid \$50 in gift card.
In each session, we started by asking participants about their experience with videos and experience with accessibility.
We asked if participants have ever added closed captions or audio descriptions to their videos, and their reasons for (not) creating AD\&CC.
We explained in details what AD\&CC are for, and what they should describe or not describe based on AD\&CC guidelines.
We also reviewed two video examples with AD and CC to provide a more concrete understanding.
Then, we demonstrated Interfaces 1 and 2 with example videos. Each participant tried all features on both interfaces before continuing to the main study.
Each participant was asked to run four tests in total for Interface 1 and 2 with randomized order.
For each interface, two videos are randomly selected without repetition.
We provided an open-ended prompt to participants, asking them to \textit{``use this tool to make this video accessible to BVI and DHOH people''}. Participants are not subject to any time limit.
At the end of each test, we asked participants to rate a set of questions on task load index~\cite{hart1986nasa}.
After completing all four tasks, we asked participants to compare their experience of identifying and addressing AD and CC with and without \systemname through semi-structured interviews.
We recorded the audio track for the entire interview and the screen portion of trying out the interfaces.
Both interfaces also automatically logged participants' use of different features, including their video navigation, clicks on vertical and horizontal bars, timestamps they chose to add AD/CC, and the content of AD/CC they wrote, etc.
In total, we collected 4042 log instances of interaction data.
We also recorded the completion time for each task for each participant.
\subsection{Findings}
\input{figures/taskload}
Participants on average spent 10 minutes and 12 seconds to complete a task.
Participants unanimously preferred using Interface 2 (\systemname) over Interface 1. On a scale from 1-strongly disagree to 7-strongly agree, participants rated that \systemnamespace is useful (\(\mu=6.0, \sigma=0.58\)), easy to use (\(\mu=6.33, \sigma=0.47\)) and would like to use it to make their videos accessible in future (\(\mu=6.50, \sigma=0.65\)).
\subsubsection{Interface Usage}
With \systemname, most participants navigate the video using the horizontal timelines (Table~\ref{tab:interaction}) in the video pane (Figure~\ref{fig:interface}C) or vertical sidebars in the captions and video description pane (Figure~\ref{fig:interface}E, F). P9 only used the original YouTube player to pause/play the video. She reported that she wanted to be really careful and make sure she covered all problems.
P1, P4, P6, P7 and P11 used horizontal timelines more frequently to navigate. P4 reported that with horizontal timelines she can more easily understand her current position, and it provided a good overview.
Other participants preferred to navigate in the captions and video description pane. P5 especially liked this ``in-place'' design where he can navigate, identify and edit all in one place. Participants in general navigated more using visual timeline and sidebars than audio. And the frequencies of audio/visual navigation are approximately proportional to the number of audio/visual accessibility issues they addressed. While some participants mostly followed \systemname's guidance, 9 out of 11 participants used ``Dismiss'', ``Add'', and ``Filter'' to correct \systemname's predictions (further discussed in Section~\ref{ai}).
\input{tables/interaction}
P1, P5, P8 and P10 particularly liked the use of color in \systemname. Colors allow them to get an overview of approximately how inaccessible this video is, quickly locate the most critical accessibility issues, and monitor their work progress. P8 described:
\begin{quote}
\textit{``I like that when you have something undone, it will mark as red. This makes it really easy for me to locate the tracks and navigate. You can take a glance at how much work is left. ''} -- P8
\end{quote}
P10 also considered the segmentation of visual and audio to be especially useful. It allows him to add AD/CC to where the scene or sound occurs, and adds it for a coherent piece of information, comparing to Interface 1 that he needed to adjust himself.
\subsubsection{Efficiency and Performance}
Participants on average spent 10 minutes 12 seconds to complete a task.
There is no significant difference between task completion times using Interface 1 and 2 (Table~\ref{tab:participants}).
This could be caused by identifying more accessibility issues and thus spending more time overall to address the issues. For example P5 and P10 reported that Interface 1 required too many efforts and discouraged them from carefully inspecting the videos.
Thus, instead of comparing the overall time spent, which could be affected by video length, number of accessibility issues identified and efforts to write descriptions, we measure time per fix (Table~\ref{tab:participants}), i.e. total time divided by number of total AD/CC added. This metric represents how efficiently a participant can locate accessibility problems.
A Wilcoxon signed-rank test shows that participants were able to create AD\&CC more efficiently with \systemnamespace (\(\mu=38.5, \sigma=19.5\)) than with Interface 1 (\(\mu=45.0, \sigma=18.1\)),
with statistical significance (\(W=182.0, p=0.037\)). Participants found a variety of features \systemnamespace provided to be helpful in making their workflow faster. This will be discussed in details in Section~\ref{identify}
\input{tables/participants}
Another important measurement for efficiency is how well authors were able to identify visual and auditory accessibility issues, specifically, how many of the captions and descriptions they added address an actual accessibility issue (precision), and how many of the total accessibility issues were addressed (recall). We collected participants' log data and computed the precision and recall scores to locate visual and auditory problems using both interfaces (Figure~\ref{tab:participants}).
We label a participant's created AD/CC to be correct if its added timestamp lies within the start and end timestamp of a manually labeled accessibility issue.
Participants reached higher precision and recall to identify inaccessible audio (precision: \(\mu=0.766, \sigma=0.192\), recall: \(\mu=0.693, \sigma=0.196\)) and visual segments (precision: \(\mu=0.921, \sigma=0.091\), recall: \(\mu=0.895, \sigma=0.131\)) with \systemnamespace than with Interface 1 (Table~\ref{tab:participants}). Wilcoxon signed-rank tests indicate statistical significance for the increase in participants' performance for both auditory (precision: \(W=123.0, p=0.004\), recall: \(W=120.0, p=0.004\)) and visual (precision: \(W=0.0, p<0.001\), recall: \(W=24.0, p<0.001\)) accessibility problems. This result aligns with participants responses to task load index questions. Participants felt significantly more confident in locating auditory (\(W=35.0, p=0.003\)) and visual (\(W=4.5, p<0.001\)) accessibility issues with \systemnamespace than without (Figure~\ref{fig:taskload}).
5 participants
reported that \systemnamespace not only provides them with some guidance, but also serves as a confirmation that improves their confidence.
\subsubsection{\systemnamespace Workflows}
\label{identify}
Participants were able to identify and address accessibility problems with \systemnamespace more efficiently.
Results from task load index questions (Figure~\ref{fig:taskload}) also show that the participants found that using \systemnamespace was significantly less mentally demanding (\(W=9.5, p<0.001\)), less difficult (\(W=10.5, p<0.001\)), and less stressed or annoyed (\(W=11.5, p<0.001\)) than using Interface 1.
All participants reported that in Interface 1 they had to watch the entire video through and have to constantly check if there is an accessibility issue. P6 complained that she had to \textit{``stop at every sentence''}. P9 had to \textit{``pay attention all the time, every moment''}.
Moreover, 8 out of 11 participants stated that it was a huge cognitive load for them to surface for visual and audio accessibility problems at the same time, because they have to repeatedly switch their minds between imagining ``I cannot see the content'' vs. ``I cannot hear the content''.
As a result, participants either had to play the video two times and only focus on addressing one type of accessibility issue at a time (P3, P6, P7, P9, P11), repeatedly inspect the same segment (P4), or rely on heuristics to make the process easier (P1, P2, P5):
\begin{quote}
\textit{[With Interface 1] I first look for non-speech, or visual changes to some obvious object or some close-up shots. Then I would imagine that I can only access the video through one of my senses, for example my vision or my hearing, to determine that, ok, here might need a CC or AD. } -- P2
\end{quote}
\systemnamespace enabled participants to more efficiently locate and address accessibility problems with lighter mental demand. With visualization of modality asymmetries, participants can get an overview of, immediately identify, and seamlessly navigate to surfaced visual and auditory accessibility issues.
8 out of 11 participants would directly jump to the highlighted red visual and audio segments in a video and address those problems, especially after they felt that the algorithm is accurate enough:
\begin{quote}
\textit{``After the first one I felt like the algorithm is pretty accurate and sufficient. So in the second one, I would just click on the undone red marks. It’s a much better experience.''} -- P3
\end{quote}
While \systemnamespace highlights inaccessible segments so that authors can directly jump to these segments and write descriptions, one limitation of this workflow is that authors may sometimes miss important context of a video. No participants in our study reported difficulty writing descriptions due to lack of context. We analyzed a sample of descriptions created by participants during the study, no major completeness or accuracy issue was found. This could be due to the selected testing videos that do not heavily rely on context (tutorial, recipe, vlog, review), compared to other forms of content like stories or lectures.
Interestingly, P2, P6 and P10 employed a ``dynamic workflow'' in which they would still skim through the gray segments while paying more attention to the red parts:
\begin{quote}
\textit{``My workflow isn’t linear anymore and I don’t have to check for every second. For example, in this video I can click on a gray segment to instantly navigate to the position, and then realize that most of it is just the person speaking, then I can just skip the entire segment and go to the next one. For the first one I’ll have to be continuously watching.''} -- P6
\end{quote}
P2, P7 and P10 also explained that \systemname's highlight of inaccessible segments reduces the work from searching for all potential accessibility issues to judging if one modality of this video segment is inaccessible, which is much less mentally demanding:
\begin{quote}
\textit{``Seeing the problem, I can understand it in hindsight. It is so much easier than I have to go over everything, paying attention to visual and audio, thinking if there is potentially an accessibility issue while the video is still playing.''} -- P7
\end{quote}
9 out of 11 participants reported that with \systemnamespace they were able to address both visual and auditory problems in parallel. P4 described that with the timelines she realized that most audio and visual issues are not in the same location. So when she was at a segment she can focus on either audio or visual. And even if they are around the same location, P4 explained, \textit{``Since you know that there might be a problem, your attention will be on what potential problem does this segment have instead of which part has a problem. I don't have to distinguish. I feel like that was the hardest part.''}
\subsubsection{Interpreting AI Predictions}
\label{ai}
9 out of 11 participants used ``Dismiss'', ``Add'', and ``Filter'' to correct \systemname's predictions. As discussed in previous sections, participants can easily judge and dismiss false positive predictions by our system.
We observed that participants were able to identify visual accessibility issues with significantly higher precision (0.921), compared to the precision of \systemname's predictions (0.718). This indicates that participants were not overly relying on the system and were able to determine whether the surfaced problem is actually inaccessible. P2, P5, P6, P9 and P10 also checked for false negative errors of \systemname, by skimming through gray segments or adjusting the slider (Figure~\ref{fig:interface}H) to retrieve more accessibility problems:
\begin{quote}
\textit{``After I have address all the issues, usually I just slide it to a bigger value and check if there’s any red segments. If there is I’ll click on those segments and see if they are actually accessible. If all the new problems are ok I’ll stop there.''} -- P10
\end{quote}
P1, P2 and P5 reported that they would prioritize workload over \textit{complete} accessibility. P2 stated that she will first look at the top-left corner to see how many issues remaining and just go with the default if not too many or not too few. P5 told us that having some description to cover some important visual stuff in his video is more important than completeness:
\begin{quote}
\textit{``Using this tool, I’m not trying to achieve a 100\% accuracy. For any suggestions it provides, it’s already better. it increases my willingness to address them and at least try to make my videos more accessible. If I’m using the first one [Interface 1] I’ll probably just choose to skip.''} -- P5
\end{quote}
\subsubsection{Feedback \& Improvement}
During the study, participants suggested new features that could improve our interface. P1 and P4 thought that sometimes segmented visual and audio content is repetitive and they have to enter the same descriptions again and again. They suggested that we could cluster similar visuals/audio segments and allow authors to apply a description to all similar segments.
Although our system focuses on the identification of accessibility issues, as a number of prior work~\cite{saray2011adaptive, natalie2021efficacy, peng2021say, pavel2020rescribe} have explored ways to author higher quality descriptions, 7 out of 11 participants hope that our system could automatically generate descriptions and captions, or at least some simple words to start with. They felt like this is the ``last missing piece'' of our system and would consider use it on all of their video creations.
P10 felt that our current design of the slider is a bit hard to understand, and we could potentially replace it with more well-defined levels. For instance, ``You should fix this'', ``Recommend fixing'', ``Make your video completely accessible'', etc.
\section{Case Study: Usage of \systemnamespace by Content Creators}
We recruited two YouTubers who did not participate in the first study and conducted a 60-minute study with each participant remotely.
Each author was paid \$50 in gift card.
We demonstrated \systemnamespace and asked the authors to make two of their own videos\footnote{
\url{https://www.youtube.com/watch?v=TMxZ7vooRm8},\\ \url{https://www.youtube.com/watch?v=r3v7GFBhWdI}\\
\url{https://www.youtube.com/watch?v=lS35Yq_dv9k}\\
\url{https://www.youtube.com/watch?v=fRoegT3_Z_E}} accessible using our system. We then conducted a semi-structured interview to discuss their experience, concerns, and expectations. Specifically, we wanted to understand: \textit{How could \systemnamespace fit within content creators’ video creation workflow, and help them make their own videos more accessible in the real-world settings?}
\subsection{Participants}
Author 1 mainly created life vlogs and music videos, with around 70 videos published on YouTube. Author 2 mainly created life vlogs and talking videos, with around 200 videos published in YouTube and TikTok. Only the second author has added closed captions to her videos before (not often). Neither of them has added audio descriptions to their videos.
\subsection{Findings}
\subsubsection{Integrating \systemnamespace into Video Production}
Both authors gave a 7 when rated on the usefulness of \systemnamespace (from 1-strongly disagree to 7-strongly agree). The authors agreed that \systemnamespace helps identify inaccessible parts while doing it manually, even in their own videos, is difficult for them.
Both authors expressed enthusiasm and showed strong expectations towards integrating \systemnamespace into existing video uploading process. Author 1 mentioned that she might consider how she designs the production style (\textit{i.e.,} more description of the visual) of her video content based on accessibility feedback:
\begin{quote}
\textit{``If it's integrated within my editor, I would try to edit the video in a way that's more accessible. It would help me keep an eye out for parts that are particularly inaccessible, or if I noticed that a lot of scenes are inaccessible, I might rethink how to structure the video better.''} -- Author 1
\end{quote}
However, author 2 would still prioritize what she wants to express first when producing the videos, and would only consider accessibility until the editing is complete. She preferred to have a completely separate tool or website that she could upload the video and check for issues before she published it onto YouTube.
\subsubsection{Balancing Efforts and Accessibility} Both authors gave a 6 when rated on `` I would like to use \systemnamespace to check and address for accessibility issues in the future.'' (from 1-strongly disagree to 7-strongly agree) Their major concern that stops them from using this tool is balancing between time invested in fixing accessibility issues and how many audiences indeed benefit from it.
Author 1 reported that she did no know how many BVI or DHOH people were watching her videos and required AD/CC, so she had not thought about making her videos more accessible. However, if she knew that even a small part of her demographic needed it, she would definitely invest some time and use this tool to make her content accessible in the future.
Author 2 explained that she would try to make her video as accessible as possible with the time she had:
\begin{quote}
\textit{``I always check my video through a color accessibility test, because it's something that takes a short amount of time, but allows my videos to be a little more accessible. So if your tool were to come out I will definitely do it because it's gonna take me less than 30 minutes for two videos while 45 minutes for one video manually [adding closed captions]. It saves me so much time.''} -- Author 2
\end{quote}
\section{Discussion and Future Work}
Our work explores using cross-modal grounding to detect modality asymmetries in the visual and auditory tracks in a video, and instantiates the scores as a unified interface that allows users to efficiently identify and address video accessibility issues.
Next, we describe the limitations of our system, discuss the implications, and envision future opportunities:
\textbf{Improved Segmentation of Information. }
In \systemnamespace, visual and audio tracks of media are first divided into semantically coherent pieces of information.
In our implementation, we chose to use pauses in speech to segment the audio track and shot changes to segment the visual track. However, this proxy can be inaccurate sometimes.
In the future, we hope to explore more semantically meaningful
approaches to segment visual and audio information. For example, we will segment visuals into object-level segments
such that each segment corresponds to one important visual object.
\textbf{Leveraging Information Importance.}
In our current implementation, \systemnamespace only detects unmatched segments and does not discern if an unmatched visual/audio segment contains important information. This results in the presenter (\textit{e.g., } a host talking to the camera does not contain much useful information) and silence issues (\textit{e.g., } silence or background noise does not to be explicitly described) that we have to address.
Future systems could estimate how important an unmatched visual/audio is (\textit{e.g.}, based on the topic's uniqueness or consistency with respect to the rest of the video~\cite{pavel2020rescribe}), then surface and prioritize segments for authors based on importance in addition to accessibility.
\textbf{Incorporating \systemnamespace with Existing Systems.}
7 of 11 participants in our lab study and both content creators mentioned that they wanted automated AD/CC as a starting point. Although this research is focused on helping users identify accessibility problems rather than authoring AD/CC, we see an opportunity to combine \systemnamespace with existing systems like ~\cite{saray2011adaptive, natalie2021efficacy, peng2021say, pavel2020rescribe} to create an end-to-end experience for authors.
However, researchers should also be cautious when providing AI-generated info as replacement of content authoring since it may decrease the content quality.
Prior research \cite{10.1145/3441852.3471207} on alt text authoring showed that authors wrote significantly lower quality alt text when starting with automatic alt text compared to starting with a blank box. We see one major opportunity for researchers to design the representation of AI-generated info to assist content authoring without priming authors with low quality automatic generations.
\textbf{Modality Asymmetry for Accessibility Beyond Video.}
We use cross-modal grounding to check for modality asymmetries in visual and audio.
In the future, we will apply our cross-modal grounding pipeline to other forms of media. As long as the context of the media includes two or more modality sources (images and its article, gifs and its post text), we can use the modality asymmetry to identify the inaccessible parts. For example, the pipeline can segment images using semantic segmentation, object recognition and vision-language models, and then detect which part of an image is not described by the text description, presenting accessibility issues to BVI people; or which part of the text description is not represented in the image, presenting accessibility issues to people with Dyslexia or people who do not understand the language.
Future research can explore generalizing this pipeline.
Since most media consists of information from three main modalities, visual, auditory and textual, there is an opportunity to design a unified system that is able to provide an accessibility diagnosis for all major media contents with a consistent standard.
\textbf{Empathize and Incentivize Accessibility.}
The majority of participants from both evaluations claimed that the small number of BVI/DHOH audience, the lack of system assistance and video platform support are the reasons why they did not provide AD/CC for their created videos.
Without assistance and support, they had to invest a lot of effort when they were not sure if someone in their audience could benefit from the effort.
In fact, on YouDescribe\footnote{https://youdescribe.org/}, a crowdsourcing audio description platform, blind and low vision people submit a large amount of requests every day to make YouTube videos accessible.
Providing more specific instructions or reminders on the video platform can help incentivize authors to add CC/AD. As Author 2 commented, \textit{``Your system should educate authors about what is accessibility and why it's important.''}
\section{Conclusion}
We present \systemname, a system that enables authors to efficiently identify and address visual and auditory accessibility issues in videos. Our system automatically estimates accessibility of visual and audio segments by checking for modality asymmetries using cross-modal grounding algorithms. It allows authors to quickly locate, review, script and preview AD\&CC in a unified interface. Participants using \systemnamespace in our user studies were able to author AD\&CC more efficiently with lower mental demand. Content creators envisioned integrating our system into their video creation workflow, and expressed enthusiasm in using it to make their videos more accessible in the future.
\section{Summary of guidelines we used to manually identify visual and auditory accessibility issues}
\label{appendix_guidelines}
We provide a summary of the guidelines we used to manually identify visual and auditory accessibility issues.
For visual accessibility issues:
\begin{itemize}
\item Describe visual content that is important to understand the video including objects (e.g., an ingredient for a recipe), actions (e.g., a recipe step), and scenes (e.g., the kitchen).
\item Do not describe details that are understandable from the audio alone (e.g., the camera returns to a host speaking to the camera).
\end{itemize}
For audio accessibility issues:
\begin{itemize}
\item Transcribe speech (in our case, it was already available from the existing captions).
\item Describe non-speech sounds including: relevant environmental sounds (e.g., blender, alarm clock), background music, and sound effects (e.g., a rimshot to accent a joke)
\end{itemize}
\section{Evaluation 1}
\subsection{Detailed Participants' Performance Table}
\input{tables/participants_big}
Table~\ref{tab:participants_big} shows detailed task performance information for all participants in evaluation 1.
\subsection{Participants' Prior Experience with Accessibility}
All participants in our study did not have experience adding closed captions (to describe non-speech sounds) or audio descriptions to their videos. In our pre-study interviews, we asked participants to rate their agreement (from 1-strongly disagree to 7-strongly agree) with reasons that they did not provide AD/CC for their videos (Figure~\ref{fig:why_no_cc_ad}). Participants listed ``no convenient system'', ``platform support'', ``no DHOH/BVI audience'' as their top reasons. In addition, participants reported that locating accessibility issues is harder for them than authoring descriptions, especially for auditory accessibility problems.
\subsection{Test Videos Used in Evaluation 1}
\input{tables/test_videos}
Table~\ref{tab:test_videos} shows CrossA11y's performance on selected test videos in evaluation 1.
\subsection{Task Load Index Questions}
We list the set of task load index questions used in Section~\ref{eval1}.
After each task, we ask (from 1-very low, to 7-very high):
\begin{enumerate}
\item How mentally demanding was the task?
\item How hard did you have to work to accomplish your level of performance?
\item How insecure, discouraged, irritated, stressed and annoyed were you?
\item How successful and confident were you in identifying audio accessibility problems?
\item How successful and confident were you in identifying visual accessibility problems?
\end{enumerate}
|
3,212,635,537,995 | arxiv | \section{Introduction} \label{intro}
LS~I~+61$^{\circ}$303 is an accreting binary system well-known for exhibiting an
exceptional broadband spectrum from radio to TeV energies
\citep{albert08}. The system consists of a main sequence star of type
B0 Ve \citep{hutchings81}, with an estimated mass 10--15 M$_{\odot}$,
and a compact object (it is still debated if a black-hole or a neutron
star) orbiting with a period $P_{\rm orb}$\,=\,26.4960\,$\pm$\,0.0028
d in a highly eccentric orbit ($e$\,=\,0.537\,$\pm$\,0.034), at a
distance from us of 2 kpc \citep{aragona09}.
The system is characterized by different long periodicities: the
shortest one is associated with the above-mentioned orbital period and
it is detected in all bands of the electromagnetic spectrum, from
radio, where it was first noticed \citep{taylor82} up to $\gamma$-rays
\citep{abdo09}. The longest periodicity, also first noted in the
radio band \citep{gregory99}, is clearly super-orbital at a period
$\sim$\,4.6 years (P$_{\rm so}$), and because of the larger time-span
required to detect it, it has been only recently detected at higher
frequencies \citep[see][and reference therein]{li14}. In the X-ray
band, the modulation appears phase shifted of $\sim$\,0.2 with respect
to the radio one \citep{li12}. In between there is a periodicity very
close to the known orbital period at $P_{\rm
2}$\,=\,26.99\,$\pm$\,0.08 d, that has been more recently observed
only in the radio and in the $\gamma$-ray bands
\citep{massi13, jaron14} and, finally, a periodicity that appears as an averaged
value between $P_{\rm 2}$ and $P_{\rm orb}$ at $P_{\rm
av}$\,=\,26.704\,$\pm$\,0.004, that has been exclusively attributed
to the radio outburst recurrence time \citep{ray97, jaron13}.
According to \citet{massi14} $P_{\rm 2}$ is caused by a precession of
a conical jet, as also revealed by the periodic change of the
associated extended radio structure \citep{massi12}, while the $P_{\rm
so}$ is the result of the beat between the two shorter periods,
$P_{\rm orb}$ and $P_{\rm 2}$ \citep{massi13}: namely, its value
(within the statistical uncertainties) results compatible with this
hypothesis ($P_{\rm so}$\,=\,($P_{\rm orb}^{-1}$\,--\,$P_{\rm
2}^{-1}$)$^{-1}$). Other authors suggest instead a possible
connection with the time-scales of the Be stellar activity
\citep{ackermann13}, or the precession of the Be companion star's
decretion disk \citep{lipunov94}.
In this paper, we exploit the continuous hard X-ray coverage of this
source made in the last $\sim$\,10 years by the $Swift$/BAT monitor,
to assess the presence and the spectral characteristics of these
periodicities in the hard X-ray band.
\section{Data reduction and analysis}
We retrieved the survey data for LS~I~+61$^{\circ}$303 collected with $Swift$/BAT
between 2004 December 09 (MJD 53348) and 2015 March 10 (MJD 56827)
from the HEASARC public
archive\footnote{\url{http://heasarc.gsfc.nasa.gov/docs/archive.html}}
and processed them using a software package dedicated to the analysis
of the data from coded mask telescopes \citep{segreto10}. The BAT
light curve consists of 55,312 entries and the source was in the BAT
field-of-view $\sim$\,15 times a day, for $\sim$\,412 s each time on
average. The source is detected with high significance ($\sim$\,25
standard deviations) in the 15--150 keV BAT range. Fig.~\ref{map}
shows the significance map of the BAT sky region around LS~I~+61$^{\circ}$303.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{fig1.ps}
\caption[BAT sky map]{The 15--150 keV Swift/BAT image of the LS~I~+61$^{\circ}$303
sky region. The brightest nearby object, the quasar 0241+622, at
$\sim$\,80\arcmin results clearly detached.}
\label{map}
\end{center}
\end{figure}
Data analysis was performed using the HEASARC/FTOOLS v. 6.16, the
RStudio software \citep{rstudio} and the specific Lomb-Scargle R
package \citep{ruf99} We report errors at one sigma confidence level,
unless stated otherwise.
\subsection{Temporal analysis: detection of the $P_{\rm so}$}
We first studied the long-term light curve of the BAT data, looking
for the $P_{\rm so}$ presence. Because the data cover approximately
slightly more than two complete orbits, we tried to directly fit the
light curve using as a best-fitting model the sum of a constant
emission and a sine function, assuming the shape of the periodicity is
sinusoidal. Data were re-binned using a time-bin of 80 days,
corresponding to about three complete orbital periods. The values for
the constant rate, the sine amplitude, the period and the phase were
initially all left as free parameters.
We found an averaged flux of (4.9\,$\pm$\,0.2)\,$\times$\,10$^{-5}$
counts s$^{-1}$ pixel$^{-1}$, a semi-amplitude of
(1.2\,$\pm$\,0.3)\,$\times$\,10$^{-5}$ counts s$^{-1}$ pixel$^{-1}$ a
super-orbital period P$_{\rm so}$\,=\,1689\,$\pm$\,112 d. We show in the upper
and lower panel of Fig.\ref{lc_bat} the BAT light curve with super-imposed the
best-fitting model and the folded profile at $P_{\rm so}$, respectively. The F-test
that compares this model with the null-hypothesis of no modulation in
the data gives $\sim$\,1\% probability that the improvement is
obtained only by chance. The sinusoid peaks at the super-orbital
phase $\sim$\,0.2, compatibly with the phase shift ($\Delta
\phi$\,=\,0.17\,$\pm$\,0.02) observed in soft X-rays and in hard X-ray
with $INTEGRAL$/ISGRI data \citep{li12, li14}.
\begin{figure}
\includegraphics[height=\columnwidth, angle=-90]{fig2.ps}
\includegraphics[height=\columnwidth, angle=-90]{fig3.ps}
\caption{ {\it Upper panel}: $Swift$/BAT light curve (15--150 keV
range) with over-imposed the best-fitting long-term modulation
$P_{\rm so}$. Time is in MJD (bottom axis) and in super-orbital
phase (upper axis). {\it Lower panel}: folded BAT light curve
(12 bins) at period $P_{\rm so}$\,=\,1667 d, with time zero MJD
43,366.275. Two consecutive periods are shown for clarity. The BAT
rate in both panels is in units of 10$^{-5}$ counts s$^{-1}$
pixel$^{-1}$.
\label{lc_bat}}
\end{figure}
We then extracted two, statistically similar, energy-selected light
curves in the 15--35 keV (source significance $\sim$ 20 $\sigma$) and
in the 35--150 keV (significance $\sim$ 16 $\sigma$) bands to check
the amplitude dependence on energy. To this aim, we obtained the
values for best-fitting function composed of a constant and a
sinusoidal component as previously described. We found that the
modulation is statistically detected in the softer band at
$>$\,3\,$\sigma$ ($P_{\rm so}$\,=\,1715\,$\pm$\,140 and amplitude
fraction $\sim$\,0.3) while only marginal detection
($\sim$\,2\,$\sigma$) is obtained for the harder band.
\subsection{Temporal analysis: Lomb-Scargle periodograms}
We searched for any periodicity in the 22--32 d period range using the
Lomb-Scargle periodogram (LSP) technique \citep{lomb76}. We consider
that the error on the detected periods is the half-width of the bin
period, that is for the BAT data-set and periods of interest
$\sim$\,0.10 d. A preliminary search using the whole dataset in the
15-150 keV band did not result in any significant detection. In fact,
LSP method is insensitive to the statistical error associated to each
measure, while the BAT survey data are characterized by a wide spread
of non-Gaussian statistical errors that depend on several factors
(mainly the reduction of the coded fraction when the source is
observed at large off-axis angles). In this case, a filtering method
may help a weak feature to emerge over the noise. We therefore begun
to gradually remove data with the largest associated rate uncertainty.
We noted that by filtering out from the original dataset the 23\% of
the data with the largest uncertainty, a periodicity at
26.47\,$\pm$\,0.10 d starts to be significantly detected, while
removing the 35\% of the noisiest data, resulted in the detection of a
second period of slightly higher value at 26.93\,$\pm$\,0.10 d,
compatible with the $P_{\rm 2}$ period that had been reported in radio
and in the gamma band (see Fig.~\ref{lsp01}).
We repeated the same procedure for the two energy-filtered datasets
(15--35 keV and 35--150 keV bands). We noted again that it was
necessary to remove part of the noisiest entries to obtain
statistically significant detections. As for the entire energy range
the orbital period is the first feature to emerge, followed by the
$P_{2\rm }$ detection. In particular, for the softest band $both$
periods are detected when 36\% of the data are removed, while for the
hardest energy band this happens after 24\% of the data are removed.
\begin{figure}
\includegraphics[height=\columnwidth, angle=-90]{fig4.ps}
\caption{LSP of the filtered BAT data-set in the 22--32 d period
region of interest. False alarm probability (horizontal dotted
line) is set at 0.01. Blue (red) line is the LSP for the BAT dataset
with 23\% (35\%) of noisiest data removed. \label{lsp01}}
\end{figure}
Following \citet{jaron14}, we then passed to study the power of the
two signals for different orbital phase intervals (where the phase is
calculated assuming as time of reference $T_{\rm 0}$\,=\,43,366.275
MJD and $P_{\rm orb}$\,=\,26.496 d), using a moving window of constant
phase width of 0.5, with no selection on energy, and filtering out the
noisiest data when needed. We found that the intensity of both
features strongly depends on the orbital phase selection: this is more
clearly demonstrated by the two phase intervals around the periastron
(0\,$<$ $\Phi$ $<$\,0.5) and apoastron (0.5\,$<$ $\Phi$ $<$\,1.0)
passages. We show in Fig.~\ref{lsp02} the corresponding LSPs: data
belonging to phase $\Phi<$\,0.5 (blue line) do not show evidence of
any periodicity (irrespective of other additional filters based on
energy and/or rate error), whereas data belonging to phase interval
$\Phi>$\,0.5 (red line) clearly show both $P_{\rm 2}$ and $P_{\rm
orb}$ periodicities (when 35\% of noisiest data are removed). We
note that the power associated to the $P_{\rm 2}$ periodicity becomes
sensibly stronger with respect to $P_{\rm orb}$, in analogy with what
observed also in the GeV band \citep{jaron14}.
\begin{figure}
\includegraphics[height=\columnwidth, angle=-90]{fig5.ps}
\caption{LSP of two phase-selected datasets. Blue (red) line is the
LSP for the orbital phase interval 0--0.5 (0.5--1.0). False alarm
probability (horizontal dotted line) set at 0.01. \label{lsp02}}
\end{figure}
Finally, we studied the temporal evolution of the signals according to
the super-orbital phase. To this aim, we selected 5 phase intervals,
choosing the boundaries so to keep the same number of data for each
interval (i.e. 0, 0.15, 0.31, 0.54, 0.79, 1). We found that both the
$P_{\rm 2}$ and the $P_{\rm orb}$ periods could be well detected in
the LSP only for the super-orbital phase interval 0.15--0.31, whereas
marginal significant detection is obtained for the other
phase-intervals.
\section{Orbital modulation of the spectral shape}
We studied the spectral shape of the hard X-ray emission as a function
of the orbital, $P_{\rm orb}$, and jet precession $P_{\rm 2}$ phase.
In the upper panel of Fig.~\ref{fig:folded} we show the BAT data
folded at $P_{\rm orb}$ for three energy bands: 15--35 keV, 35--150
keV, and 15--150 keV. The emission peaks
at phase $\sim$\,0.3, while the phase of minimum emission appears more structured around the
apoastron passage. The observed maximum flux ratio is $\sim$\,6. The folded profile is similar
in the two energy-selected bands, although it is to be noted that the
soft emission is enhanced over the hardest band in the first half of
the orbital cycle. This is most easily observed through a direct
spectral fit of the phase-selected spectra. We choose 10 equally
spaced phase selected spectra and fitted them using a simple power-law
model. We show in the lower panel of Fig.~\ref{fig:folded} the
dependence of the photon-index as a function of the orbital phase. We
observe a clear trend as a function of the phase that gives an account
of the overall modulation of the energy-selected folded profiles.
\begin{figure}
\begin{tabular}{c}
\includegraphics[height=\columnwidth, angle=-90]{fig6.ps} \\
\includegraphics[height=\columnwidth, angle=-90]{fig7.ps}
\end{tabular}
\caption{{\it Upper panel}: folded BAT light curve (15--110 keV range)
at P$_{\rm orb}$\,=\,26.496 d. Time zero of reference MJD
43366.275. Red, blue, and black curves are data selected in the
15--35 keV, 35--150 keV and 15--150 keV ranges, respectively.
Magenta dotted lines indicate the periastron and apoastron phase
passages \citep{aragona09}. The BAT rate is in units of 10$^{-5}$
counts s$^{-1}$ pixel$^{-1}$. {\it Lower panel}: the photon-index
of the power-law that best fits the data in the 15--150 keV range as
a function of the orbital phase. We also show for comparison the
values obtained with $INTEGRAL$/ISGRI data (red triangles)
according to \citet{li14}. Two periods shown for clarity.
\label{fig:folded}}
\end{figure}
We show in Fig.~\ref{fig:folded2} two folded profiles at $P_{\rm 2}$
using the same epoch of reference of the folded $P_{\rm orb}$ and a
period of 26.93 d, that is the our best value according to the LSP of
Fig.~\ref{lsp02}. The profile in blue is averaged over all orbital
phases, whereas the profile in black is obtained when the signal
becomes enhanced in the orbital phase interval 0--0.5.
\begin{figure}
\includegraphics[height=\columnwidth, angle=-90]{fig8.ps}
\caption{Folded BAT light curve (15--110 keV range) at
P$_{2}$\,=\,26.93 d. Time zero of reference MJD 43366.275. Profile
in blue is phase averaged (see LSP of Fig.~\ref{lsp01}), while
profile in black is from data selected in the $P_{\rm orb}$ phase
($\Phi>$\,0.5, see LSP of Fig.~\ref{lsp02}). The BAT rate is in
units of 10$^{-5}$ counts s$^{-1}$
pixel$^{-1}$. \label{fig:folded2}}
\end{figure}
\section{Discussion}
We examined the $Swift$/BAT light curve of LS~I~+61$^{\circ}$303 to assess the
presence and the spectral characteristics of its long-term
periodicities. \citet{li14} reported the presence of the long-term
$P_{\rm so}$ modulation in hard X-rays using $INTEGRAL$/ISGRI
data. The ISGRI energy band (17--60 keV) almost entirely overlaps with
the BAT band, and the temporal window used in that analysis (MJD
52579--56500) covers $\sim$\,84\% of the BAT window, so it can be
considered an excellent benchmark for our results (but the ISGRI
exposure is only 10\% of the BAT one). We confirm that the long-term
$P_{\rm so}$ is well detected (at $\sim$\,4\,$\sigma$) in hard X-rays.
The BAT $P_{\rm so}$ folded profile (Fig.~\ref{lc_bat}) is similar to
that obtained from ISGRI, although the larger and more uniform
distribution of the exposures at all phases allows us to derive a more
detailed profile. We confirm a higher X-ray emission during the first
half of the super-orbital cycle.
We assessed the presence of the $P_{\rm orb}$ and $P_{\rm 2}$
periodicities in the 15--150 keV band that are also visible in two
different energy bands (15--35 keV and 35--150 keV). In particular,
the detection of $P_{\rm 2}$ \citep[already revealed only in radio and
in the $\gamma$ band,][]{massi15}) is reported for the first time in
X-rays. We found this feature more prominent in the data selected in
the orbital phase 0.5--1.0, and this behaviour matches the GeV
analogous feature \citep{jaron14, massi15}; we consider this finding
as a strong indication for a common origin for the apastron energy
emission mechanism, from radio, X-rays to the GeV wavelength.
\citet{massi14} proposed a physical scenario able to relate the
presence of these periodicities across such different energy bands. In
this picture, the long-term period is a beat period caused by the
combination of a precessing jet (at a period $P_{\rm 2}$) that
receives a modulated fraction of plasma with a slightly different
period (the $P_{\rm orb}$). The overall emission (synchrotron
emission emitted by relativistic electrons in the magnetized jet) is
the highest when the jet forms the minimum angle with our line of
sight and its emission becomes Doppler boosted. We estimated the
expected beat period ($\nu_{\rm beat} = \nu_{\rm 2} - \nu_{\rm orb}$)
of these two periodicities to be 1560\,$\pm$\,480 d, consistent with
our measure of the long-term modulation $P_{\rm so}$, although we note
the rather large uncertainties on $P_{\rm 2}$ and $P_{\rm orb}$. We
found that the power of the two signals depends on the super-orbital
phase, and it is maximum in both cases for the super-orbital phase
0.15--0.31, that corresponds to the peak of the $P_{\rm so}$ folded
profile (lower panel of Fig.\ref{lc_bat}).
We also studied the spectral shape in hard X-rays as a function of the
orbital phase ($P_{\rm orb}$) for the whole BAT time-span. The folded,
time-averaged over the whole BAT observing window, X-ray profile peaks
close to the periastron passage, and it shows
two dips, before and after the apostron, that hint for a secondary peak at this phase (Fig.~\ref{fig:folded}, upper panel).
Although the ISGRI folded profile and the values of the
photon-index appear to be quite similar \citep[see Tab.\,1 in][]{li14},
BAT data allow a more detailed account of the photon-index variation along the
orbit and suggest for the second half of the orbital cycle
(Fig.~\ref{fig:folded}, lower panel) softer indices.
Finally, we also reported for the first time the folded X-ray profile
at the presumed jet periodicity, that shows a significant amplitude in
the flux emission, comparable to that shown in the orbital period,
when phase-selected around the apoastron passage.
\section{Acknowledgements}
\small
This work was supported in Italy by ASI contract I/004/11/1.
\bibliographystyle{mn2e}
|
3,212,635,537,996 | arxiv | \section{Introduction}
\label{sect:intro}
The next generation of astronomical spectrometers designed for detecting Earth-mass exoplanets seek to measure the radial velocities (RVs) of stars with a precision of 10-50 cm s$^{-1}$. For a typical instrument design, this corresponds to measuring an ensemble Doppler shift in the wavelengths of stellar spectral lines on the focal plane of approximately 0.001~nm. At this level of measurement precision, a host of errors, both instrumental and astrophysical, become important\cite{Halverson2016}. To minimize some of these sources of instrumental measurement error, it is important to keep the opto-mechanical components of the spectrometer as stable as possible. This usually means operating in high vacuum, and controlling the temperature of the entire optical system to the mK level or better\cite{Robertson2016}. The detectors themselves, either CCDs in the optical or HgCdTe arrays in the near infrared, may represent the largest sources of thermal variations in the entire instrument. This stems from the fact that the detectors and vacuum electronics may consume different amounts of power during idle, integration, and readout.
The impact of the detector's thermal variations on the rest of the instrument may be important, for example through radiative coupling of the detector to adjacent optics. At the same time, it is also possible that temperature changes in the detector package lead directly to physical deformations of the array, which translate directly into measured RV shifts that will need to be calibrated out. While it is challenging to directly simulate these effects for a specific detector package and mounting scheme, given the typical dispersion properties of a high-resolution spectrometer, a physical shift of the focal plane array of just 2~nm represents a systematic error larger than the Doppler shift of the stellar spectral lines resulting from an orbiting Earth-mass planet in a one year orbit.
Mounting the detector so that it is cooled efficiently, and therefore dissipates heat efficiently, is very important. But, as large-format detectors approach almost 10 cm by 10 cm in size, the thermal time constant of the device itself necessarily becomes long. At the same time, it becomes a challenge to maintain the temperature uniformity of the large detector package given the wide range of integration times typical for an instrument in operation at an astronomical observatory. Given the large number of output channels, the package could itself support temperature inhomogeneities that change with time. Actively controlling the temperature of the CCD package directly, with heaters affixed to the package, may be impractical for a number of reasons. We explore clocking techniques designed to minimize the amplitude of any thermal variations of a four-phase CCD under normal operating conditions and investigate the impact these clocking modes might have on the resulting high-resolution stellar spectra and our ability to infer precise RV measurements from those spectra.
\section{The NEID Spectrometer Detector System}
NEID\footnote{NEID is a word that means ``to see" or ``to discover/visualize" in the language of the Tohono O'odham, on whose land Kitt Peak National Observatory sits} is a high-resolution, optical (380 to 930 nm) Doppler spectrometer being built for the 3.5-m Wisconsin-Indiana-Yale-NOAO (WIYN) telescope at Kitt Peak National Observatory. The spectrometer is a cross-dispersed, white-pupil design and achieves spectral resolution of R=120,000 while simultaneously recording more than 95 spectral orders on a large-format CCD detector\cite{Schwab2016}. The instrument is contained in a vacuum vessel that maintains pressure below 10$^{-6}$~Torr with an operating temperature of 300~K, which is actively maintained to better than $\pm1$~mK by a temperature control system\cite{Robertson2016}. The CCD detector, which is operated at 173~K, is contained in this same vacuum envelope. The detector is a large-format device from Teledyne/e2v containing 9216x9232 pixels with a 10~$\mu$m pitch. In order to optimize the red response of the system, and to reduce fringing effects at those wavelengths, NEID employs the deep depletion variant of this CCD290-99 device (40$\mu$m epi layer) with the Astro Multi-2 AR coating. The device has 16 independent output channels, which we read out using an Archon controller\cite{Archon} and custom preamplifier boards designed by STA, Inc. Archon is an FPGA-based CCD readout system that also provides a scripting environment for CCD control and sophisticated temperature control capabilities. Inside the vacuum, we perform AC coupling and JFET buffering at the warm end of CCD flex cables, then convert the signal to a true differential output for transmission of the analog signal across a pair of 49-pair shielded twinax cables, each 1.15~m long. We take advantage of the CCD dummy outputs to reject common-mode artifacts across the 16 output channels. While it is possible to read out the device at speeds up to 1~MHz, we have adopted a standard readout mode of 250 kHz, which is a good balance between read noise and total readout time.
The CCD290-99 is contained in a SiC package that has three mounting points for attaching the device to the mounting structure\footnote{https://www.teledyne-e2v.com/shared/content/resources/File/Astronomy/1897.pdf}. Each mounting point consists of a high-precision Invar 36 pad, approximately 4~mm thick and 19~mm in diameter, and an Invar 36 stud which is threaded into the SiC package. The Invar pads are precision polished to provide a mounting plane that is very close to parallel to the active surface of the CCD. We mount the CCD directly on a Mo cold block, shown in Figure \ref{fig:cold_block}, using custom-made, hemispherical M5 nuts that allow the CCD pads to translate slightly due to thermal expansion effects during cool down. Mo was chosen as the material for the CCD mounting block since it has good thermal conductivity and is a good match to the coefficient of thermal expansion of the SiC CCD package. In the CCD test Dewar, the temperature of the Mo cold block is actively controlled by the Archon through an embedded 25~W heater and PT-100 RTD. In NEID, the temperature of the CCD package is actively controlled by Archon using the heater on the Mo cold block. The Mo block is cooled through a tapered copper rod that is attached to the center of the cold block on one end and linked, through a series of copper straps, to the LN2 reservoir of the NEID instrument or test Dewar. Thermal contact to the detector is only through the interface of the Invar pads on the CCD package to the corresponding polished surfaces on the Mo cold block. We monitor the temperature of the CCD package through a PT-100 RTD that is attached, as part of a small Al block, directly to the center of the back of the CCD package using a single M4 screw into a threaded insert that comes installed on the CCD. The CCD package RTD is also read out using Archon, but we have no active heat source on the CCD package. To improve the cooling properties of the CCD mount system, a thin film of Nye Lubricants NyeTorr 6370EL UHV grease was applied between the Invar pads on the CCD package and the corresponding pads on the Mo block when installed in NEID. A cross section of the three-dimensional model of the NEID CCD mount system is shown in Figure 2.
We carried out an extensive CCD characterization effort using an LN2 cooled IR Labs ND8 test Dewar to house the CCD in essentially the same mounting configuration as in NEID (though no UHV grease was used inside the test Dewar). With less radiation shielding around the CCD and long-term variations in the LN2 fill level of the test Dewar, this CCD testing environment is not as intrinsically stable as NEID. However, this test Dewar enabled very efficient testing of different CCD operational modes, as well as a large number of opto-mechanical tests of the CCD.
\section{Operation of the NIED Detector}
The NEID detector is controlled by Archon using a custom set of scripts to execute timing and voltage commands. We found that the nominal timing parameters described in the CCD290-99 data sheet provided good performance and so we used these parameters to define the nominal clocking pattern. Reading out the device generates a substantial amount of heat, and the device data sheet indicates that up to 100 mW of transient heat may be generated during the CCD readout through the clocking of the parallel and serial registers. There is also a substantial static heat load from the amplifiers estimated at 800 mW, but this does not vary while the CCD is powered on. Given the mass of the CCD package, the thermal properties of SiC, and the approximate time to read out a frame in our 250 kHz mode, we estimated that variable heat loads of this size could lead to changes in the CCD package temperature at the 10~mK level. This is more than a factor of 10 larger than our target absolute CCD temperature stability of $\pm1$~mK, which is set by the design goal of achieving overall intrinsic instrument stability corresponding to 10 cm s$^{-1}$ in RV. Furthermore, the CCD itself is a large cold sink in our warm instrument, and radiative coupling between the spectrometer camera lens system and the CCD, with an area of 85 cm$^{2}$ and undergoing temperature variations at the 10~mK level, was observed to destabilize the entire instrument with a long thermal equilibration timescale.
We operate the CCD in a standard mode where the parallel and serial registers are constantly being clocked while idle, and the serial registers are also clocked during integration. This means that during integration the heat output of the CCD package decreases as the parallel registers cease to clock. The motivation for this approach was that it should be simpler to control the CCD temperature by adding heat during integration rather than removing heat. Since we are only controlling the temperature of the Mo cold block to which the CCD is attached, the thermal time constant of the CCD package reaction to changes in the cold block may be long compared to integration times. This makes maintaining the thermal stability of the CCD package using this approach a challenge. For example, Figure 3 shows the CCD package temperature within our test Dewar over several hours during a series of calibration lamp exposures of different lengths in standard readout mode. During integration, we see a decrease in CCD package temperature of approximately 20~mK on top of longer timescale changes corresponding to the varying fill level of the LN2 test Dewar. During this period, the temperature of the Mo cold block, which is controlled directly by Archon in the test Dewar, is stable to better than $\pm1$~mK.
While the amplitude of these CCD temperature variations could be reduced by improving the thermal contact between the CCD package and the Mo cold block, the relatively poor thermal contact between the hard surfaces of the Invar pads on the CCD, the SiC package, and the polished cold block makes this difficult without directly attaching additional cold strapping to the CCD package. The design of the CCD package and our mounting system make this difficult, and increasing the direct physical contact between the CCD package and the cold linkage increases the likelihood that mechanical variations in the system (due to LN2 fill level, for example) may manifest as physical displacements in the focal plane. Instead, we explored ways to reduce the overall variations in the heat output of the CCD by exercising the parallel clocks during integration.
\section{Dither Clocking}
We present a technique, which we call ``dither clocking", where the voltages of two phases of our four-phase CCD are modulated during integration at the same frequency as they would be during readout. The goal is to mimic the heat generated by the parallel registers during readout in a way that does not corrupt the integrity of the underlying spectra. This approach is very similar to clocked anti-blooming, a technique for reducing the impact of bleeding charge when imaging objects spanning a large range of brightnesses \cite{Janesick1992, Murray2013}. This clocking approach is also very similar to the techniques that have been used to effectively reduce the impact of surface-generated dark current in CCDs for space- and ground-based applications\cite{Burke1991,Jorden2002,Vandersteen2010}. As shown in the clocking diagram in Figure \ref{fig:dither_diagram} and the potential well schematic in Figure 5, during integration phase 4 is always held low to act as a barrier and phase 2 is always held high to collect charge, as during standard operation. In dithered operation the voltages on phases 1 and 3 are varied during integration. With our deep depletion CCD the ``low" voltage is 0 V (the substrate is held at 0 V) and ``high" voltage is +11 V. The amplitude of the dither on phases 1 and 3 can be set so that the full well of the pixel is unchanged, swinging either 11 V or 5.5 V. Based on the requested exposure time, our Archon script executes an integer number of these dithers, with the dither frequency set to match the line time during normal readout (3~ms for our 250 kHz readout mode). The rise time of the transition between high and low states for phases 1 and 3 can also be set so as to minimize the possibility of charge pumping (see Section 5). We find that the dither clocking has a significant impact on the temperature variations of the CCD package during readout. In Figure \ref{fig:temps} we plot the CCD package temperature during a 500~s integration in both standard and dither modes. A 10~mK transient during integration is reduced to a 1~mK transient. In Figure \ref{fig:seven_days} we show the CCD package temperature in NEID over seven days of gathering calibration spectra. These calibration data included long and short sequences of integrations of wavelength calibration sources. The RMS of the data stream averaged on a 60~s timescale is 0.25~mK.
\section{Potential Pitfalls}
While the dither clocking mode appears very effective at reducing the amplitude of the thermal variations of the CCD package during integration and readout, it is less obvious how this readout mode could impact the RV measurements we are making. We explored several different effects related to the dither clocking that could potentially degrade the quality of the underlying data. We found no evidence for the dither clocking scheme impacting linearity, read noise, or pixel full-well, and dark current remains below 3 e- pix$^{-1}$ hour$^{-1}$ in both modes at our nominal CCD operating temperature of 173~K.
The movement of charge within a pixel during integration could produce Clock Induced Charge (CIC). This effect is typically more of an issue at much higher clocking frequencies, such as with EMCCDs that are read out very rapidly\cite{Janesick2001}. By comparing a number of long (900~s) dark frames in both standard and dither modes, we found no evidence of CIC at the limit of the dark current and read noise of our system. Given the frequency of the dither clocking, the number of shuffles during these 900~s integrations, and the transition times between the high and low voltage states on the dithered phases, we do not expect to see CIC.
It is possible that the dither clocking scheme could result in a significant signal due to pocket pumping. Typically, pocket pumping is used as a technique to identify traps in the silicon lattice of the CCD. The CCD is exposed to uniform light, then charge is repeatedly shuffled back and forth between adjacent pixels with the illumination turned off. In a perfect CCD, all of the accumulated charge ends up in the pixel it started in, but traps may cause charge to be removed from one pixel and accumulated in another over a large number of transfers. This results in a characteristic dipole pattern that can be used to identify the locations of charge traps\cite{Mostek2010}. Here, a potential concern is a related effect where photoelectrons that are generated under phase 3 when it is low (phase 4 is always low) see an effective electric field that has a very shallow gradient toward phase 2. The result is that a small fraction of these photoelectrons may end up under phase 1 of the adjacent pixel, never to return. This effect can be mitigated by tailoring the clocking profile of phase 3 during shuffle to reduce the total amount of time that photoelectrons see the very flat electric field in the vicinity of phases 3 and 4. We also note that this effect is symmetric between adjacent pixels in that it could occur around phases 3 and 4, but also around phases 1 and 4. In the case where a charge trap is localized to phase 3, this effect could result in a redistribution of charge between pixels that is not, on average, symmetric. We carried out preliminary investigations of this effect with dither clocking using the standard pocket pumping technique of illuminating the device uniformly to approximately half full well and then continuing to integrate in dither mode for 900~s with the light source off. We found no evidence for pocket pumping dipoles in these data. However, this effect and its potential impact on RV measurements must be investigated further.
The dither clocking mode could result in increased correlations between pixels, due to charge sharing or increased diffusion between pixels resulting from the decreased average barrier voltages. These effects could reduce the Modulation Transfer Function (MTF) of the detector system. We measured MTF in our test Dewar using test pattern projection techniques and laser speckles\cite{laserspeckle}, but found that our optical setup did not produce sufficiently high contrast at high spatial frequencies to measure a difference between the standard and dither modes. Photon Transfer Curves (PTCs) also encode information about correlations between adjacent pixels through the non-linearity of the relationship between variance and flux level\cite{pixelcorrelations}. In Figure 8 we compare measured PTCs in standard and dither modes. We found no significant differences between the PTC properties in the two readout modes.
It is also possible that the dither clocking alters the physical locations of pixels on the chip as defined by the average gate voltages. While this is not likely an issue if the dither clocking mode is always used when collecting science and calibration data, effects of this type would make it difficult to compare data taken in different modes. We investigated this effect with the CCD operating inside the NEID cryostat by gathering consecutive sets of spectra of flat lamp in standard and dither mode and measuring the average positions of a single spectral order in the cross dispersion (parallel) direction. As shown in Figure 9, we observed shifts of approximately 0.002 pixels between the two modes. While this effect seems robustly measured, it is still to be determined if it is a fixed effect or one that varies over time. If the latter is true, this would be a major source of concern for using the dither clocking for precise RV measurements.
\section{Conclusions}
Variable heat output from the detector can be a major source of systematic measurement error for precise RV spectrometers. Radiative coupling between the CCD and the instrument optics, as well as physical deformations of the CCD, can lead to instrumental shifts that are large compared to the RV signals from Earth-like planets orbiting sun-like stars. Given that CCDs in modern high-resolution spectrometers are large, up to nearly 100 cm$^2$, and the variations in heat loads between idle, integration, and readout may be hundreds of mW, actively controlling the temperature of the CCD becomes important. We have developed a technique to reduce the overall variations of the CCD package temperature in the NEID spectrometer by dithering the parallel and serial clocks during integration. We find that this clocking scheme reduces the overall variation of the CCD package temperature to the mK level compared to thermal transients at the 20~mK level in standard operating mode. We have explored several different ways in which this clocking scheme could corrupt the underlying spectral data. Initial investigations revealed no significant effects, but ultimately the stability of the instrument as measured through observations of the laser frequency comb will be necessary to evaluate the impact of the dither clocking on RV performance.
\acknowledgments
We thank the organizers of ISPA-2018 for organizing a very informative workshop. This work was carried out in part at the Singh Center for Nanotechnology at the University of Pennsylvania, which is supported by the NSF National Nanotechnology Coordinated Infrastructure Program under grant NNCI-1542153. NEID is funded by NASA through JPL by contract 1547612. Mark Giovinazzi is supported by an NSF Graduate Research Fellowship. The authors would like to thank an anonymous referee for insightful suggestions that helped to improve this manuscript.
\section*{Disclosures}
The authors have no conflicts of interest to declare.
\clearpage
\begin{figure}
\begin{center}
\begin{tabular}{c}
\includegraphics[height=10cm]{mount_ring.png}
\end{tabular}
\end{center}
\caption
{ \label{fig:cold_block}
(a) The Mo cold block to which we mount the CCD290-99. (b) Polished pads on the cold block that mate to the Invar pads on the CCD package. (c) Ultem 2000 spacers that hold the cold block within the Al mounting structure and provide thermal and mechanical isolation. The Al mounting structure (d) attaches directly to the camera lens system within the NEID spectrometer.}
\end{figure}
\clearpage
\begin{figure}
\begin{center}
\begin{tabular}{c}
\includegraphics[height=12cm]{xsection.png}
\end{tabular}
\end{center}
\caption
{ \label{fig:xsection}
Cross section of the NEID CCD mount. (A) CCD face, (B) Mo cold block, which has a single PT100 RTD embedded, (C) Thermal contact between CCD and Mo cold block is through three polished Invar pads on the SiC package, (D) A PT100 RTD in a small copper block is attached to the center of the SiC package using a single M4 screw, (E) A portion of the Cu cold linkage attaching the Mo block to the instrument LN2 reservoir, and (F) CCD electronics board.}
\end{figure}
\clearpage
\begin{figure}
\begin{center}
\begin{tabular}{c}
\includegraphics[width=16cm]{long_temp_v2.png}
\end{tabular}
\end{center}
\caption
{ \label{fig:long_temp}
Changes in the CCD package temperature in the test Dewar during a series of calibration exposures of different lengths gathered over 2.75 hours in standard clocking mode. Here, both the parallel and serial registers are cycled during idle, so a decrease in package temperature corresponds to the start of an integration and a rapid rise in package temperature corresponds to the beginning of readout. We observed long-term trends in CCD temperature due to variations in the test Dewar fill level, but the Dewar hold time is long compared to the 2.75 hour data stream shown here.}
\end{figure}
\clearpage
\begin{figure}
\begin{center}
\begin{tabular}{c}
\includegraphics[width=14cm]{dither_diagram_v3.png}
\end{tabular}
\end{center}
\caption
{ \label{fig:dither_diagram}
Schematic showing the dither clocking scheme. During non-dither operation, phases 1 and 4 would be held at 0~V and phases 2 and 3 would be held at +11 V during the entire integration. }
\end{figure}
\clearpage
\begin{figure}
\begin{center}
\begin{tabular}{c}
\includegraphics[width=12cm]{potentials.png}
\end{tabular}
\end{center}
\caption
{ \label{fig:potentials}
Schematic of change in potentials defining CCD pixels at the two extremes of the dither process, where phases 1 and 3 swap from 0~V to 11~V. The blue lines represent the potential wells seen by the photoelectrons and the dashed vertical lines represent the corresponding pixel boundaries.}
\end{figure}
\clearpage
\begin{figure}
\begin{center}
\begin{tabular}{c}
\includegraphics[width=13cm]{temps.png}
\end{tabular}
\end{center}
\caption
{ \label{fig:temps}
Comparison of CCD package temperature changes during a 500~s exposure in dither and standard modes in the test Dewar.}
\end{figure}
\clearpage
\begin{figure}
\begin{center}
\begin{tabular}{c}
\includegraphics[width=16cm]{package_temp_seven.png}
\end{tabular}
\end{center}
\caption
{ \label{fig:seven_days}
Change in temperature of the CCD package over seven days of operation in NEID. During the period, NEID was gathering a wide range of different calibration exposures in the laboratory, with a duty cycle similar to what may be experienced during operations in the field. The overall RMS of the data stream after binning on a 60~s timescale is 0.25~mK.}
\end{figure}
\clearpage
\begin{figure}
\begin{center}
\begin{tabular}{c}
\includegraphics[width=14cm]{ptc_curve_comp.png}
\end{tabular}
\end{center}
\caption
{ \label{fig:ptc}
Comparison of Photon Transfer Curves (PTCs) generated in dither and standard modes. The differences between the two best-fit third-order polynomials to these PTCs are less than 0.2$\%$ over this flux range. }
\end{figure}
\clearpage
\begin{figure}
\begin{center}
\begin{tabular}{c}
\includegraphics[width=14cm]{pixel_shift.png}
\end{tabular}
\end{center}
\caption
{ \label{fig:shifts}
Measured position of a single spectral order over a series of exposures in the dither and standard modes in NEID.}
\end{figure}
\clearpage
|
3,212,635,537,997 | arxiv | \section{Introduction}
The Large Hadron Collider (LHC) at CERN, Geneva, is expected to
provide direct evidence for any New Physics beyond the Standard
Model (SM) at the TeV energy scale. The properties of these
particles may shed light on the origin of electroweak symmetry
breaking and the nature of dark matter, and give clues to a more
fundamental theory of Nature. To explain the deficiencies of the SM,
a large variety of theories has been put forward, and it is only by
carefully measuring the properties of any new particles that one
will be able to discriminate between them. Chief amongst these
properties are the masses of the new particles.
Many of these new theories ({\it e.g.}, supersymmetry and extra dimensions)
contain a stable weakly interacting massive particle that will only be
`visible' in an LHC detector as missing energy. Such particles are
natural dark matter candidates by virtue of their interactions, but
pose problems for mass measurements at hadron colliders since one
cannot in general reconstruct the kinematics of an event in which
these particles are produced. To worsen the problem, the
parton--parton interactions in the collider have by their very nature
an unknown center of mass energy.
In the literature a number of techniques has been developed to get
around this problem. These fall into two general classes: those that
perform a fit to or set a limit on masses using information from the
entire event sample, and those which rely exclusively on events near
the endpoint of a kinematic distribution. Belonging to the first class
are ``Mass-Shell Techniques" (MSTs), represented by the work done in
\cite{Kawagoe:2004rz,Nojiri:2007pq},
\cite{Cheng:2007xv,Cheng:2008mg,Cheng:2009fw}
and \cite{Webber:2009vm},
which depend on maximizing the solvability of assumed
mass-shell constraints in a given sample of events. This has been
shown to be very effective if enough such constraints are
available.\footnote{However, problems arise if, {\it e.g.}, there are
three-body decays or too many invisible decay products;
see~\cite{Bisset:2008hm} for further discussion.} Also belonging to
this class are techniques which work with extrema of a ``transverse
mass" variable, {\it e.g.},
$m_{T2}$~\cite{Lester:1999tx,Barr:2003rg,Cho:2007qv,Barr:2007hy,
Cho:2007dh,Nojiri:2008hy,Barr:2008ba}, and techniques which look at
the shape of complete invariant mass
distributions~\cite{Miller:2005zp, Kraml:2005kb,Gjelsten:2006as,
Gjelsten:2006tg,Kraml:2008zr}. Much research has recently become
focused in these areas.
The second class of mass reconstruction techniques most notably
includes the traditional kinematic endpoint method
\cite{Hinchliffe:1996iu,Paige:1997xb,Bachacou:1999zb,Lytken,Gjelsten:2004ki,
Gjelsten:2005aw, Huang:2008qd,Burns:2009zi,Matchev:2009iw},
where the endpoints of various
invariant mass distributions can be matched to analytical functions of
the unknown masses, that in turn can be inverted to solve for the same
masses in suitably long decay chains. Such methods have been studied
for over a decade already, and would seem to have been largely
explored for the simplest, and most probable, decay chains in popular
theories such as the Minimal Supersymmetric Standard Model
(MSSM). However, there is more to be done with events near an endpoint
as demonstrated recently in~\cite{Kersting:2009ne}: here a Decay-frame
Kinematics (DK) technique utilizes the fact that events at a kinematic
endpoint \emph{can} have exactly-known kinematics in terms of
production angles and energies of all particles in the assumed decay
chain. Events near an endpoint will thus have approximately known
decay-frame kinematics, which allows one to constrain and solve for
unknown masses. In~\cite{Kersting:2009ne} this was demonstrated for
the case of neutralino three-body decays through off-shell sleptons to
lepton pairs plus missing energy carried away by the lightest
supersymmetric particle (LSP), the lightest neutralino, {\it e.g.\
}$\tilde\chi_2^0\to\ell^+\ell^-\tilde\chi_1^0$. The on-shell case
was deferred to a future work --- this work.
In the following we will demonstrate the use of DK in the case of
on-shell neutralino decays, {\it i.e.\ }$\tilde\chi_i^0
\to\tilde\ell^\pm\ell^\mp\to\ell^+\ell^-\tilde\chi_1^0$, though it
should be stressed that the technique demonstrated can be applied to
any similar cascade decay process arising in any New Physics model.
Section \ref{sec:off} begins with a review of the off-shell case,
demonstrating its application at a NMSSM parameter point. Section
\ref{sec:on} then discusses the main new development of this paper,
the generalisation of the DK technique to the on-shell case, where it
is found that several subtleties emerge beyond what was found for the
comparatively simple off-shell case; we further present a Monte Carlo
(MC) study of the mSUGRA SPS1a benchmark point~\cite{Allanach:2002nj},
where the DK technique proves quite capable of reconstructing the
relevant neutralino masses from $\tilde\chi_2^0\tilde\chi_2^0$ decays.
Section \ref{sec:conc} gives our conclusions.
\section{Off-Shell Decays}
\label{sec:off}
We begin with a brief review of the DK technique applied to the case
of neutralino three-body decays. For a more complete treatment, see
\cite{Kersting:2009ne}. We consider production of neutralino pairs in
the MSSM which undergo three-body decays to electrons, muons, and
$\tilde\chi_1^0$ (the LSP):
\begin{equation}
\label{zizjdecay2}
pp \to \mathbb{X}\to \mathbb{X}' + \tilde\chi_i^0
(\to e^+ e^- \tilde\chi_1^0)~\tilde{\chi}_{j}^0 (\to\mu^+\mu^-\tilde\chi_1^0),
\end{equation}
where $\mathbb{X}$ represents either a $Z^*$ or any MSSM production
channel via a Higgs ($H^0$ or $A^0$) or cascade from gluino/squark
pair-production, while $\mathbb{X}'$ are SM states potentially
produced in association, relevant in this context only for the
measurement of missing momentum. The physical observables of interest
from one event thus consist of four leptonic four-momenta
$p_{e^\pm,\mu^\pm}$, from which we may construct the usual di-lepton
invariant masses, $M_{ee}$ and $M_{\mu\mu}$, and missing momentum in
two transverse directions, assumed equal to the sum of the two LSPs'
transverse momenta, $p_{\chi,\chi'}^T$. If we happen to have an event
where both the invariant masses $M_{ee}$ and $M_{\mu\mu}$ are maximal,
as shown in Fig.~\ref{fig:zizj}a, it will be subject to the system of
constraints (hereafter we abbreviate $m_i
\equiv m_{\widetilde{\chi}_{i}^0}$)
\begin{eqnarray}
\label{con1}
M_{ee}^{\max} & = & m_i - m_1, \\ \label{con2}
M_{\mu\mu}^{\max} & = & m_j - m_1, \\ \label{con345}
\vec{p}_{e^+}^{~\prime} + \vec{p}_{e^-}^{~\prime} & = & 0, \\ \label{con678}
\vec{p}_{\mu^+}^{~\prime} + \vec{p}_{\mu^-}^{~\prime} & = & 0, \\ \label{con910}
(\vec{p}_\chi + \vec{p}_{\chi'})^T & = & \not\!{\vec{p}}^{~T}~\mathrm{(observed),}
\end{eqnarray}
where leptonic momenta are written in the rest frame of the respective
parent neutralino, and $\vec{p}_{e^\pm}^{~\prime}=\mathbf{\Lambda}_1
\vec{p}_{e^\pm}$ and $\vec{p}_{\mu^\pm}^{~\prime}=\mathbf{\Lambda}_2
\vec{p}_{\mu^\pm}$ define the appropriate Lorentz transformations
$\mathbf{\Lambda}_{1,2}$ from the lab frame. This system gives ten
equations for the nine unknowns, the velocities $\vec\beta_{1,2}$ and
the masses $m_{1,i,j}$, allowing us to actually overconstrain the
masses $m_{1,i,j}$. The $\vec{\beta}_{1,2}$ which satisfy
(\ref{con345}) and (\ref{con678}), making the total momentum of each
lepton pair zero, are uniquely given by
\begin{equation}
\label{betaeqn1}
\vec\beta_1 = \frac{\vec{p}_{e^+} + \vec{p}_{e^-}}{E_{e^+} + E_{e^-}}
\quad{\rm and}\quad
\vec\beta_2 = \frac{\vec{p}_{\mu^+} + \vec{p}_{\mu^-}}{E_{\mu^+} + E_{\mu^-}}.
\end{equation}
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.49\linewidth]{fig1a.eps}
\includegraphics[width=0.49\linewidth]{fig1b.eps}
\end{center}
\caption{\small \emph{(a) Neutralino three-body decays with maximal $M_{ee}$ and
$M_{\mu\mu}$: though the decaying $\tilde\chi_{i,j}^0$ may be moving
with any velocity $\beta_{1,2}$ in the lab frame, in each respective
decay frame the leptons have equal and opposite momenta while the
$\tilde\chi_1^0$ (LSP) is at rest. (b) If the $\tilde\chi_{i,j}^0$ has
a two-body decay via on-shell sleptons, the LSPs are no longer
stationary in the neutralino decay frame, but have momenta $K_{i,j}$
collinear with the leptons' momenta.}}
\label{fig:zizj}
\end{figure}
Now, because of the condition that $M_{ee}$ and $M_{\mu\mu}$ are
maximal, the corresponding $\mathbf{\Lambda}_{1,2}$, which take the
$e^+ e^-\tilde\chi_1^0$ and $\mu^+ \mu^- \tilde\chi_1^0$ systems to
their respective $\tilde\chi_{i,j}^0$ rest frames, also bring each
$\tilde\chi_1^0$ to rest. Thus their four-momenta in these frames must
be $(m_1,\vec{0})$, which, when inverse-Lorentz-transformed by
$\mathbf{\Lambda}_{1,2}^{-1}$, giving $(m_1\gamma_{1,2}~,m_1
(\vec\beta\gamma)_{1,2} )$, have to agree with the observed missing
momentum $\not\!\vec{p}^{~T}$; this matching condition along each
transverse direction (say $\hat{x}$ and $\hat{y}$) then gives two
independent determinations of $m_1$:
\begin{equation}
\label{m1eqn}
m_1' = \frac{\not\!{p}_x}{(\beta_x\gamma)_1 +
(\beta_x \gamma)_2}
\quad{\rm and}\quad
m_1'' = \frac{\not\!{p}_y}{(\beta_y\gamma)_1 + (\beta_y\gamma)_2}.
\end{equation}
Since we are assuming that both $M_{ee}$ and $M_{\mu\mu}$ are
precisely maximal --- the perfect event of Fig.~\ref{fig:zizj}a ---
we should get $m_1' = m_1''= m_1$ from such an event. In practice of
course, we can only expect to find an event within some neighborhood
$\epsilon$ of the endpoints, $M_{ee,\mu\mu}=M_{ee,\mu\mu}^{\max}\pm
\epsilon$, in which case one finds that $m_1'$ and $m_1''$ are
offset by ${\mathcal O}(\sqrt{2\epsilon m_1})$ from $m_1$
\cite{Kersting:2009ne}. One might then expect that applying
(\ref{betaeqn1}) and (\ref{m1eqn}) to a sample of events near the
endpoint should give a distribution of $m_1'$ and $m_1''$ peaked
near $m_1$ with a spread determined by sample purity.
Here, to lend further support to the generality of the above
technique, let us demonstrate its application to the rather
challenging example of an NMSSM (Next-to-Minimal Supersymmetric
Standard Model) scenario described in~\cite{Kraml:2008zr}. This has
a supersymmetric particle spectrum containing five neutralinos, the
lightest of which is 99\% singlino, and a generic feature of the
parameter space that gives the correct dark matter density is a
significant degeneracy between the singlino and second lightest
neutralino mass. This gives rise to copious production of soft
lepton pairs with small invariant masses, $M_{\ell\ell} \;\raisebox{-0.9ex}{$\textstyle\stackrel{\textstyle<
10$~GeV, from the three-body decay
$\tilde\chi_2^0\to\ell^+\ell^-\tilde\chi_1^0$. For more details on
this scenario see~\cite{Kraml:2008zr}.
We study the benchmark ``Point A" of that paper, a point which has
$M_{\ell\ell}^{\max}=9.7$~GeV and $m_1=105.4$~GeV, using the same MC
setup and fast detector simulation as in~\cite{Kraml:2008zr}. For
details of the simulation see also Section~\ref{sec:on} of the present
paper. To isolate signal events of the type (\ref{zizjdecay2}), we
place the following cuts on our events:
\begin{itemize}
\item
Require missing transverse energy $\not\!\!E_T>100$~GeV.
\item Require at least two hard jets with $p_T>150,\,100$~GeV.
\item
Require four isolated leptons with flavor structure $e^+e^-\mu^+\mu^-$
and $p_T>7\,(4)$ for $e$ $(\mu)$. All such leptons must pass the lepton
efficiency cuts employed in \cite{Kraml:2008zr} modeled on full
simulation results given in \cite{:2008zzm}.
\end{itemize}
From the surviving events we construct a wedgebox plot of the
di-electron versus the di-muon invariant mass, for a number of events
equivalent to 30~fb$^{-1}$ of statistics at the LHC. The result is
seen in Fig.~\ref{fig:offdk}a, showing a clear box-like structure at
$M_{ee,\mu\mu}\sim 10$~GeV, the endpoint of the di-lepton invariant
mass distribution for the $\tilde\chi_2^0$ decay.
Choosing a sampling region in a rather generous neighborhood of the
endpoint, $M_{\ell\ell} = 10\pm 4$~GeV gives ${\mathcal O}(100)$
events. We now apply Eqs.~(\ref{betaeqn1}) and (\ref{m1eqn}) to each
of these, demanding that $m_1'$ and $m_1''$ agree to within 20\%.
Although only about 20 events survive this criterion, the resulting
$m_1$ distribution can be seen in Fig.~\ref{fig:offdk}b to peak
quite prominently slightly below the nominal value of $m_1 =
105.4$~GeV. The systematic error of the method is seen to be
comparable to the estimate made earlier.
With higher statistics, this allows us to determine the absolute LSP
mass to rather good precision. The results for 300~fb$^{-1}$ of data
are shown in Fig.~\ref{fig:offdk}c. Here we narrow down our sampling
region to $\epsilon =1$~GeV from the wedgebox edge. With a Gaussian
distribution we obtain a best fit value of $m_1=98.2\pm 3.9$~GeV
with $\chi^2/{\rm ndf}=0.41$.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.32\linewidth]{fig2a.eps}
\includegraphics[width=0.32\linewidth]{fig2b.eps}
\includegraphics[width=0.32\linewidth]{fig2c.eps}
\end{center}
\caption{\small \emph{ (a) Wedgebox plot for 30~fb$^{-1}$ of integrated
LHC luminosity for ``Point A" of \cite{Kraml:2008zr}. (b) The
$\tilde\chi_1^0$ mass distribution for events sampled from the boxed
region in (a). We see a peak in rough agreement with the nominal value of
$m_1 = 105.4$~GeV. (c) Same as (b) but for 300~fb$^{-1}$ and
$\epsilon=1$~GeV.
}}
\label{fig:offdk}
\end{figure}
\section{On-Shell Decays}
\label{sec:on}
\subsection{Kinematics}
\label{subsec:kin}
Let us now continue to the main focus of this paper, {\it i.e.\ }the
added complications that arise when the neutralinos decay through
on-shell intermediate sleptons:
\begin{equation}
\label{hdecay}
\mathbb{X} \to \mathbb{X}' + \tilde\chi_i^0 \tilde\chi_j^0
(\to e^\pm\tilde e^\mp\mu^\pm\tilde\mu^\mp
\to e^+ e^- \mu^+ \mu^- \tilde\chi_1^0\tilde\chi_1^0).
\end{equation}
When $M_{ee}$ and $M_{\mu\mu}$ are maximal, as illustrated in
Fig.~\ref{fig:zizj}b, two-body kinematics gives the following system
of constraints:
\begin{eqnarray}
\label{con1b}
M_{ee}^{\max} & = & m_i \sqrt{1 - (m_s/m_i)^2} \sqrt{1 - (m_1/m_s)^2}, \\ \label{con2b}
M_{\mu\mu}^{\max} & = & m_j \sqrt{1 - (m_s/m_j)^2} \sqrt{1 - (m_1/m_s)^2}, \\ \label{con34b}
\vec{p}_{e^+}^{~\prime} & \parallel & -\vec{p}_{e^-}^{~\prime}, \\ \label{con56b}
\vec{p}_{\mu^+}^{~\prime} & \parallel & -\vec{p}_{\mu^-}^{~\prime}, \\ \label{con7b}
|\vec{p}_{e^+}^{~\prime}+\vec{p}_{e^-}^{~\prime}| & = &
\left|\frac{m_s^4 - m_i^2m_1^2}{2m_im_s^2} \right| \equiv K_i, \\ \label{con8b}
|\vec{p}_{\mu^+}^{~\prime}+\vec{p}_{\mu^-}^{~\prime}| & = &
\left|\frac{m_s^4 - m_j^2m_1^2}{2m_jm_s^2}\right| \equiv K_j, \\ \label{con910b}
(\vec{p}_\chi + \vec{p}_{\chi'})^T & = & \not\!\vec{p}^{~T}~\mathrm{(observed)},
\end{eqnarray}
where we have assumed a common slepton mass $m_s=m_{\tilde{e}} =
m_{\tilde\mu}$. See the Appendix for details of the derivation. The
antiparallel conditions (\ref{con34b}) and (\ref{con56b}) force
$\vec{\beta}_{1,2}$ to be in the planes of the respective leptons, so
there are really only four boost parameters to find; adding the
unknown masses $m_{1,i,j,s}$ to this gives eight unknowns which can
thus be solved for by the eight constraints
(\ref{con1b})--(\ref{con910b}). In principle, if one were handed an
event of the type in Fig.~\ref{fig:zizj}b, one could numerically apply
(\ref{con1b})--(\ref{con910b}), scanning over the eight-dimensional
space of unknowns for a solution. Needless to say, this is not the
most practical approach, nor particularly enlightening as to the nature
of any solution which might be found.
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=0.49\linewidth]{fig3a.eps}
\includegraphics[width=0.49\linewidth]{fig3b.eps}
\end{center}
\caption{\small \emph{ (a) When lepton pairs from the decay chain
$\tilde\chi_i^0\to\ell^\pm\tilde\ell^\mp\to\ell^+\ell^-\tilde\chi_1^0$
have maximal invariant mass, their momenta $P_{1,2}$ in the lab frame
are coplanar with the LSP momentum $\not\!P$. (b) The LSP's
longitudinal momentum $\not\!P_L$ can be found by the condition that
$\not\!P$ is in the plane of the leptons.} }
\label{fig:geom}
\end{figure}
Instead, we will proceed temporarily as if we already knew the
individual $\vec{p}_{\chi,\chi'}^{~T}$ as opposed to just their sum
(\ref{con910b}). Consider, then, just one of the neutralino decays,
say $\tilde\chi_i^0\to e^\pm \tilde e^\mp\to e^+e^-\tilde\chi_1^0$. In
the $\tilde\chi_i^0$ rest frame, the leptons' three-momenta ($\equiv
\vec{P}_{1,2}'$) are back-to-back and collinear with the
LSP's three-momentum ($\equiv \vec{\not\!{P}}'$); thus, in the lab
frame where $\tilde\chi_i^0$ has a velocity $\vec{\beta_1}$, the
boosted momenta $\vec{P}_1$, $\vec{P}_2$, and $\vec{\not\!{P}}$ lie in
the same plane (see Fig.~\ref{fig:geom}a). Looking at this the other
way round, $\vec{\beta_1}$ is the Lorentz boost which makes the
observed lepton momenta antiparallel and fixes the magnitude of their
sum to be $K_i$ --- which we \emph{a priori} don't know at this point
--- {\it i.e.\ } constraints (\ref{con34b}) and (\ref{con7b}). Necessarily,
$\vec{\beta_1}$ must be in the observed leptons' plane. If we choose
a basis in this plane $(\hat{p}_\|,~\hat{p}_\bot)$
parallel/perpendicular\footnote{A potential ambiguity in defining
$\hat{p}_\bot$ is resolved by defining it such that
$\vec{P}_1\cdot\hat{p}_\bot$ is positive:
\begin{equation}
\hat{p}_\bot \equiv \frac{\vec{P}_1 -
(\vec{P}_1\cdot\hat{p}_\|)\hat{p}_\|}{|\vec{P}_1 - (\vec{P}_1\cdot\hat{p}_\|)\hat{p}_\||}.
\nonumber
\end{equation}}
to the total leptonic momentum $\vec{P}\equiv\vec{P}_1 +\vec{P}_2$,
then the boost $\vec{\beta_1} = (\beta_\|,~\beta_\bot)$ must satisfy
three sets of constraints:
\begin{enumerate}
\item The transformed leptonic momenta must be antiparallel:
\begin{equation}
\label{antipar}
\vec{P}_1' \cdot \vec{P}_2' = - |\vec{P}_1'| |\vec{P}_2'|,
\end{equation}
where the transformed four-vectors are given in terms of the boost by
\begin{equation}
\label{lepboost}
\left(
\begin{array}{c}
E_{1,2}' \\
P_{1,2}'^\| \\
P_{1,2}'^\bot \\
\end{array}
\right) =
\left(
\begin{array}{ccc}
\gamma & -\beta_\| \gamma & -\beta_\bot \gamma \\
-\beta_\| \gamma & 1 + (\gamma-1)\frac{\beta_\|^2}{\beta^2}
& (\gamma-1)\frac{\beta_\| \beta_\bot}{\beta^2} \\
-\beta_\bot \gamma & (\gamma-1)\frac{\beta_\| \beta_\bot}{\beta^2}
& 1 + (\gamma-1)\frac{\beta_\bot^2}{\beta^2} \\
\end{array}
\right)
\left(
\begin{array}{c}
E_{1,2} \\
P_{1,2}^\| \\
P_{1,2}^\bot \\
\end{array}
\right),
\end{equation}
with $\beta^2\equiv\beta_\|^2+\beta_\bot^2$ and
$\gamma\equiv (1-\beta^2)^{-1/2}$.
\item The transformed total leptonic momentum must equal
$\vec{K}_i=(K_i^\|,~K_i^\bot)$:
\begin{equation} \label{totlepboost}
\left(
\begin{array}{c}
E' \\
K_i^\| \\
K_i^\bot \\
\end{array}
\right) =
\left(
\begin{array}{ccc}
\gamma & -\beta_\| \gamma & -\beta_\bot \gamma \\
-\beta_\| \gamma & 1 + (\gamma-1)\frac{\beta_\|^2}{\beta^2}
& (\gamma-1)\frac{\beta_\| \beta_\bot}{\beta^2} \\
-\beta_\bot \gamma & (\gamma-1)\frac{\beta_\| \beta_\bot}{\beta^2}
& 1 + (\gamma-1)\frac{\beta_\bot^2}{\beta^2} \\
\end{array}
\right)
\left(
\begin{array}{c}
E \\
P \\
0 \\
\end{array}
\right),
\end{equation}
where the components $(K_i^\|, K_i^\bot)$ are also unknown at this point.
\item The inverse-Lorentz-boosted LSP four-momentum must satisfy
\begin{equation} \label{lspboost}
\left(
\begin{array}{c}
\slashchar{E} \\
\slashchar{P}_\| \\
\slashchar{P}_\bot \\
\end{array}
\right) =
\left(
\begin{array}{ccc}
\gamma & \beta_\| \gamma & \beta_\bot \gamma \\
\beta_\| \gamma & 1 + (\gamma-1)\frac{\beta_\|^2}{\beta^2}
& (\gamma-1)\frac{\beta_\| \beta_\bot}{\beta^2} \\
\beta_\bot \gamma & (\gamma-1)\frac{\beta_\| \beta_\bot}{\beta^2}
& 1 + (\gamma-1)\frac{\beta_\bot^2}{\beta^2} \\
\end{array}
\right)
\left(
\begin{array}{c}
\sqrt{K_i^2 + m_1^2} \\
-K_i^\| \\
-K_i^\bot \\
\end{array}
\right).
\end{equation}
\end{enumerate}
After some algebra, see the Appendix, these constraints are found to
uniquely determine the unknown boost $(\beta_\|,~\beta_\bot)$ in terms
of the known lab frame leptonic momenta and unknown LSP momenta:
\begin{equation}
\label{betasoln}
\beta_\| = \frac{P}{E(1+ \alpha x)}\quad {\rm and} \quad
\beta_\bot = \alpha \beta_\|,
\end{equation}
where
\begin{equation}\nonumber
\alpha \equiv \frac{\slashchar{P}_\bot}{\slashchar{P}_\|+P}
\quad {\rm and} \quad
x\equiv\frac{P(P_1^\bot-P_2^\bot)(P_2^\|-P_1^\|)}{2E(E_1 P_2^\| + E_2 P_1^\|)}.
\end{equation}
Thus, knowing the leptonic momenta and missing momentum from the LSP
determines $x$ and $\alpha$, hence $\vec{\beta}_1$ and all the
kinematic information in the event. At first glance this may seem
useless since we can only have knowledge of the \emph{transverse}
component of $\vec{\slashchar{P}}$ in the lab coordinate system,
$\vec{\slashchar{P}_T}$, and there are two LSPs that contribute to the
measured total missing momentum.
However, given the transverse component of the LSP momentum we can in
fact reconstruct the longitudinal component $\vec{\slashchar{P}_L}$ by
the following trick: since $\vec{\slashchar{P}}$ must lie in the plane
of the leptons while $\vec{\slashchar{P}_T}$ is by definition in the
transverse $\hat{x}$-$\hat{y}$ plane, $\vec{\slashchar{P}_L}$ must be
of the precise size along $\hat{z}$ to bring $\vec{\slashchar{P}_T} +
\vec{\slashchar{P}_L}$ into the leptons' plane (see
Fig.~\ref{fig:geom}b), {\it
i.e.\ }$(\vec{\slashchar{P}_T}+\vec{\slashchar{P}_L})
\cdot(\vec{P}_1\times\vec{P}_2)=0$, giving
\begin{equation}
\label{plong}
\vec{\slashchar{P}}_L = -\frac{\slashchar{P}_{Tx}(P_{1y}P_{2z} - P_{1z}P_{2y})
+ \slashchar{P}_{Ty}(P_{1z}P_{2x} - P_{1x}P_{2z})}{P_{1x}P_{2y} -
P_{1y}P_{2x}} \hat{z}.
\end{equation}
With both $\vec{\slashchar{P}_T}$ and $\vec{\slashchar{P}_L}$ known we
may immediately project $\vec{\slashchar{P}}$ into the basis
$(\slashchar{P}_\|,~\slashchar{P}_\bot)$,\footnote{If the leptons
happen to be parallel $\vec{\slashchar{P}_L}$ remains undetermined. We
ignore events with this pathological arrangement.} compute $\alpha$
and insert into (\ref{betasoln}) to solve for the boosts. We can then
use Eq.~(\ref{totlepboost}) to solve for $K_i^\|$ and $K_i^\bot$:
\begin{equation}\label{kparperp}
K_i^\|= P\left(\frac{\gamma+\alpha^2}{1+\alpha^2}-\frac{\gamma}{1+\alpha x}\right),
\quad\quad
K_i^\bot=\alpha P\left(\frac{\gamma-1}{1+\alpha^2}-\frac{\gamma}{1+\alpha x}\right),
\end{equation}
which are also related by $K_i^\bot = \alpha (K_i^\| - P)$. We see
that in the limit $\alpha\to 0$ both $K_i^{\|,\bot}\to 0$, and that
also Eq.~(\ref{betasoln}) correctly reduces to the off-shell result
$\beta=P/E$.
Finally, the LSP mass can then be found from the energy component of
Eq.~(\ref{lspboost}):
\begin{equation}
\label{mlsp}
m_1 = \sqrt{\left(\frac{\slashchar{P}_\|}{\beta_\| \gamma} +
\left(1+ \frac{(\gamma -1) \beta_\|^2}{\beta^2}\right)\frac{K_i^\|}{\beta_\| \gamma}
+\frac{(\gamma - 1) \beta_\bot K_i^\bot}{ \beta^2 \gamma}
\right)^2 - {K_i^\|}^2 - {K_i^\bot}^2},
\end{equation}
which again reduces to the off-shell result of Eq.~(\ref{m1eqn}) when
$K_i^{\|,\bot}\to 0$. The heavier neutralino mass follows from the
energy component of (\ref{totlepboost}) and energy conservation in its
decay, {\it i.e.}
\begin{equation}\label{m2eqn}
m_i = \sqrt{K_i^2 + m_1^2} + \gamma E - \beta_\| \gamma P,
\end{equation}
while the slepton mass is related to $m_1$ and $m_i$ by
(\ref{con1b}).
Finally, let us return to deal with the realistic situation where we
know only the sum of the LSP momenta
$(\vec{p}_\chi+\vec{p}_{\chi'})^T$. From the discussion above, every
assignment of $\not\!\vec{P}_T=\vec{p}_{\chi}^{~T}$ will yield a set
of masses $\{m_1',~m_i',~m_s'\}$ which satisfy (\ref{con1b}),
(\ref{con34b}), and (\ref{con7b}). Then the other LSP has its
$\vec{p}_{\chi'}^{~T}$ fixed as
$\vec{p}_{\chi'}^{~T}=\not\!\vec{p}^{~T}-\vec{p}_{\chi}^{~T}$,
giving another set of masses $\{m_1'',~m_i'',~m_s''\}$ which satisfy
(\ref{con2b}), (\ref{con56b}), and (\ref{con8b}). Under the
simplifying assumption that the event contains the process in
(\ref{hdecay}) with $i=j$, we should clearly insist that at least
$\{m_1',~m_i'\}\simeq\{m_1'',~m_i''\}$ within some error (we reserve
the possibility that $m_s' \neq m_s''$). This, in principle,
provides two constraints on our choice of the two components of
$\vec{p}_{\chi}^{~T}$, {\it i.e.\ }the system
(\ref{con1b})--(\ref{con910b}) is solved.\footnote{Notice that
something quite interesting has happened here in that we have gotten
around the usual four-fold ambiguity in designating `near' and 'far'
leptons in the decay chains.}
Thus, in the end, we still have to resort to a numerical search for a
solution to (\ref{con1b})--(\ref{con910b}), but this is only over the
two-dimensional space of one of the LSPs' transverse momenta,
$(p_{\chi}^x, ~p_{\chi}^y)$. Nevertheless, it is not at all obvious
that there won't be multiple solutions with different $\{m_1,~m_i\}$
within a given level of tolerance --- and when one adds to this the
same caveat as in the three-body case of picking events within some
$\epsilon$ of the endpoint, as well as the effects of detector
smearing and issues with backgrounds, it will have to fall to a MC
simulation to test the practicality of the method.
\subsection{Monte Carlo Test}
\label{subsec:mc}
We perform a Monte Carlo study by generating SUSY signal events for
the SPS1a benchmark point~\cite{Allanach:2002nj} using {\tt
PYTHIA~6.413}~\cite{Sjostrand:2006za}, and SM background events with
{\tt HERWIG 6.510}~\cite{Corcella:2000bw,Moretti:2002eu}, interfaced
to {\tt ALPGEN 2.13}~\cite{Mangano:2002ea} for the production of
high jet multiplicities matched to parton showers and {\tt JIMMY
4.31}~\cite{Butterworth:1996zw} for multiple interactions. The
benchmark point is chosen mainly for the sake of comparison with
results obtained with other mass reconstruction techniques, which we
will comment more on in Section \ref{sec:conc}. The generated events
are then put through a fast simulation of a generic LHC detector,
{\tt AcerDET-1.0}~\cite{Richter-Was:2002ch}, widely used to simulate
analyses of high-$p_T$ physics at the LHC. This incorporates such
detector effects as the deposition of energy in calorimeter cells,
and the smearing of electron, photon, muon and hadronic cluster
energies with parameterized resolutions. The {\tt AcerDET-1.0}
isolation requirement for leptons is less than 10 GeV energy in a
$R=0.2$ cone around the lepton and a minimum distance of $\Delta
R=0.4$ from calorimetric clusters. The MC setup is essentially the
same as in~\cite{Kraml:2008zr} and we refer the reader to that paper
for more details. However, we point out that we use $p_T$ dependent
lepton efficiencies based on full simulation studies published
in~\cite{:2008zzm}.
All SUSY processes and relevant SM backgrounds are generated with a
number of events corresponding to an integrated LHC luminosity of
300~fb$^{-1}$. The dominant type of neutralino pairs produced at SPS1a
are $\tilde\chi_2^0 \tilde\chi_2^0$; we will therefore concentrate on
decays of the form $\tilde\chi_2^0(\to e^+ e^-\tilde\chi_1^0)~
\tilde\chi_2^0 (\to \mu^+ \mu^-\tilde\chi_1^0)$, and results in the
previous section can be simplified somewhat by setting $i=j=2$, and in
particular $K \equiv K_i = K_j$.
For this analysis we use the same set of cuts as for the NMSSM case in
the previous Section, giving a signal size of roughly 470 events.
Note the requirement of four isolated leptons with flavor structure
$e^+e^-\mu^+\mu^-$ reduces most SM backgrounds to a negligible
level\cite{Bisset:2005rn}. Support for this in the context of a full
simulation of the ATLAS detector is found in Higgs searches for the
channel $h\to ZZ^*\to 4\ell$, {\it e.g.\ }discussed
in~\cite{:1999fr,Aad:2009wy}. The remaining backgrounds of any
importance are found to be $t\bar t$, $Zb\bar b$ and irreducible
$Z^{(*)}Z$. We have simulated a large sample of $t\bar t$ events with
up to two additional hard jets, and find no surviving events with the
additional missing energy and jet cuts. The $Zb\bar b$ and $Z^{(*)}Z$
backgrounds are also expected to be very small after these
cuts. However, because the $Z$ mass is sufficiently far from the
dilepton edge, any remaining events from these backgrounds do not
significantly influence the region of interest in the di-electron
versus di-muon invarant mass plane, shown in the wedgebox plot of
Fig.~\ref{fig:wb}a.\footnote{However, for some SUSY parameter points
one might have a dilepton edge very close to the $Z$ mass. Though the
$Z$ background events would thus be unavoidably mixed in with signal
events, they would in general not have solutions in the numerical
procedure described in the following. This property of DK gives it a
certain resilience in the face of backgrounds.}
The position of the edge of the box-like structure at
$M_{\ell\ell} \approx 75$~GeV is visually apparent in
Fig.~\ref{fig:wb}a, and, as shown in several studies, can be brought
into precise (sub-GeV) agreement with the nominal value
$M_{\ell\ell}^{\max}=77.07$~GeV, by the standard study of the
flavor-subtracted di-lepton mass distribution shown in
Fig.~\ref{fig:wb}b. One advantage of the DK technique is that we do
not strictly need such precise determination of the edge --- GeV-level
will do to determine our sampling region --- but we will assume that
the edge has been measured to $76.7\pm 0.1$~GeV as quoted
in~\cite{Gjelsten:2004ki}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.49\linewidth]{fig4a.eps}
\includegraphics[width=0.49\linewidth]{fig4b.eps}
\end{center}
\caption{\small \emph{ (a) Wedgebox plot for 300~fb$^{-1}$ of integrated
LHC luminosity at the SPS1a benchmark point. Events are sampled from
the boxed region shown for DK analysis. (b) The flavor-subtracted
dilepton invariant mass distribution provides a clean determination of
the edge $M_{\ell\ell}^{max} = 77.1$~GeV. The plot is shown for an integrated luminosity of 30~fb$^{-1}$.} } \label{fig:wb}
\end{figure}
Events in a broad neighborhood of the corner of the box,
$M_{ee,\mu\mu}=65\pm \epsilon$~GeV (see further comments below
on sampling regions), are passed to the on-shell DK analysis described
in the previous Section. In detail, the procedure used is the
following:
\begin{enumerate}
\item An event is selected if the two invariant masses $M_{ee}$
and $M_{\mu\mu}$ both lie within the $\epsilon$-defined
region of the wedgebox plot.
\item A point in $(\slashchar{P}_{Tx},~\slashchar{P}_{Ty})$-space is
chosen by a uniform scan of $-500$~GeV $< \slashchar{P}_{Tx,y} <$
$500$~GeV, in $0.2$~GeV steps; the point is assumed to equal the
transverse momentum of the LSP accompanying the $e^+$ and
$e^-$ whose four-momenta are $(E,\vec{P})_{1,2}$,
respectively.
\item The longitudinal component of the LSP's momentum is found from
(\ref{plong}).
\item Components of the LSP momentum in the basis
parallel/perpendicular to the total leptonic momentum $\vec{P}
= \vec{P}_1 + \vec{P}_2$ are determined and used to compute
$\alpha \equiv \slashchar{P}_\bot /(\slashchar{P}_\| + P)$.
\item The boost parameters $\beta_\|$ and
$\beta_\bot$ are now computed from (\ref{betasoln}),
$K^\|$ and $K^\bot$ from (\ref{kparperp}).
\item The masses $\{m_1',~m_2'\}$ are computed
from (\ref{mlsp}) and (\ref{m2eqn}).
\item Using
the missing momentum constraint $\vec{p}_{\chi'}^{~T}=
\not\!\vec{p}^{~T}-\vec{p}_{\chi}^{~T}$, \textbf{steps 3-6} are
repeated for the LSP accompanying the $\mu^+ \mu^-$ pair,
obtaining either another set of masses $\{m_1'',~m_2''\}$ or no valid solution for the second mass set.
\item If no valid second set of masses was obtained, the point is
assigned zero weight. Otherwise, the two sets of mass solutions are
plotted with the following weight:
\begin{equation}\label{likelihood}
P(\slashchar{P}_{Tx},\slashchar{P}_{Ty})=\frac{1}{\sqrt{2\pi
\sigma^2}} \exp \left(-\frac{(m_1'-m_1'')^2} {2\sigma^2}
-\frac{(m_2'-m_2'')^2}{2\sigma^2}\right),
\end{equation}
where $\sigma$ is our expectation for the spread between the mass
values on either side of the event. This should be of the order of
the missing energy resolution, which is a function of the total
transverse energy $E_T$ deposited in the calorimeters. We therefore
assign sigma on an event-by-event basis, using the expected
performance of the ATLAS detector in SUSY events (see figure 10.84
of reference ~\cite{:2008zzm}):
\begin{equation}
\sigma = 0.57 \sqrt{\sum E_T}.
\end{equation}
\item The scan is continued until all points have been assigned a weight.
\item The procedure is repeated for all events that have invariant masses sufficiently close to the endpoint.
\end{enumerate}
Our final mass distribution for a single event is obtained by
histogramming all mass solutions found in the scan over missing
momentum components, weighted by Eq.~(\ref{likelihood}). For events
that lie \emph{exactly} at the endpoint, and where the mismeasurement
of lepton momenta and missing energy is negligible, the peak of the
two mass distributions for $\{m_1',~m_2'\}$ and $\{m_1'',~m_2''\}$
should coincide, agreeing with the nominal values of the masses. We
find that this remains accurate for events that lie sufficiently close
to the endpoint, and thus with a value of $\epsilon$ that is not too
large.
In fact, choosing a value of $\epsilon$ is essentially a trade-off
between accumulating statistics by allowing more events to pass the
cut, thus reducing the fluctuations that come from the smearing of
lepton and missing energy momenta, and protecting the integrity of the
mass solutions that are obtained by the procedure at $\epsilon=0$. We
find that a compromise value for $\epsilon$ of 15 GeV gives just
enough events with peak region close to the nominal values to dominate over events that display
either one or two degenerate peaks or peaks in the wrong place. In the latter case, we observe that
the maximum weight at the peak is lower.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.7\linewidth]{fig5.eps}
\end{center}
\caption{\small \emph{Mass distribution for 300 fb$^{-1}$ of data at SPS1a,
obtained using the procedure described in the text. The contribution
of individual events with degenerate solutions is still clearly
visible, although it is a region close to the nominal mass values of
$(m_1,m_2)=(96.1,176.8)$~GeV that emerges with the largest total
weight.}}
\label{fig:finalmasses}
\end{figure}
By summing the distributions for events within our choice of
$\epsilon$, the resulting total mass distribution for 300 fb$^{-1}$ of
data is shown in Fig.~\ref{fig:finalmasses}. Although the existence of
multiple peaks is clear with the limited statistics, the point with
the largest total weight is found to be
$(m_1,m_2)=(114.75,191.75)$~GeV, close to the nominal value of
$(m_1,m_2)=(96.1,176.8)$~GeV. We emphasize that this procedure is
simply a suggestion for an estimator of the masses, and although there
are similarities in shape, Eq.~(\ref{likelihood}) is not a likelihood
function.
To check the robustness of the estimator and find the statistical
error on making such a measurement, we have performed 10 independent
`experiments' with 300 fb$^{-1}$ integrated luminosity. In each
case, the mass solution with the largest total weight fell near the
nominal masses, with a standard deviation of 20.2 GeV on $m_1$ and
21.2 GeV on $m_2$. While these errors are quite large, there is
undoubtedly scope for improvement. By better understanding the
properties of events with degenerate or wrong solutions one could
search for a system of kinematic cuts to remove these subsets; one
could also increase statistics to tighten the cut on $\epsilon$ or
investigate other estimators for the masses with better properties
with respect to these events.
Incidentally, had we wrongly assumed off-shell kinematics at this
parameter point, and hence tried using the technique of Section
\ref{sec:off} to analyze events in the boxed region of
Fig.~\ref{fig:wb}a, we would have failed since essentially no events
provide two mass solutions with a near equal mass. This provides a
way of distinguishing a sample of on- versus off-shell
decays\footnote{One exception occurs if it happens that $K=0$,
{\it i.e.\ } $m_s = \sqrt{m_1 m_2}$. Then both on-shell and off-shell
techniques are equally applicable and should reconstruct the same LSP
mass. At SPS1a, $K\approx 18$~GeV, which is fairly small compared to
the maximum $K$ one can get with the same neutralino masses, $K^{\max}
= \frac{m_2}{2}(1 - m_1^2/m_2^2)\approx 62$~GeV, but safely away from
zero.} distinct from the usual way of measuring the departure of the
di-lepton mass distribution from a triangular
shape~\cite{Kraml:2008zr}, or looking for specific relationships
between the positions of lepton-jet invariant mass
maxima~\cite{Lester:2006cf}.
Finally, note that this method, although we have not explicitly
demonstrated it, is in principle also applicable to extracting the
slepton mass.
\section{Conclusions}
\label{sec:conc}
This paper completes the demonstration of the application of the DK
technique to neutralino decays in SUSY models, or indeed any similar
decay chain in models such as {\it e.g.\
}UED~\cite{Appelquist:2000nn}, having explicitly shown the procedure
for reconstructing both off- and on-shell decays to lepton pairs in
realistic MC simulations of two very different supersymmetry
benchmark points. We find that in the three-body decay scenario we
can reconstruct the LSP mass with an accuracy of around 4 GeV, while
the decay through an on-shell slepton allows a precision of 20 GeV.
DK has the advantage of simplicity and robustness in the face of
backgrounds; for non-signal events there tends to be no solution for
the constraint equations, or at least no \emph{preferred} solution
in the distribution of events passing loose constraint requirements.
Moreover, since the method only makes use of unlike di-leptons
($e^+e^-\mu^+\mu^-$) from the $\tilde\chi^0_2\to\tilde\chi^0_1$
transitions, it is insensitive to the combinatoric issues that arise
when one considers particles produced further up the decay chain. It
may hence complement mass determinations which exploit the full
chains such as \cite{Cheng:2009fw,Webber:2009vm}. Generalization to
other decaying states ({\it e.g.} charginos as
in~\cite{Kersting:2008qn}), perhaps using jets instead of or in
addition to leptons, is open to investigation.
Perhaps the major disadvantage of DK is the requirement of a high
event rate: one typically needs at least ${\mathcal O}(10)$ events in
the neighborhood of a kinematic endpoint to make the reconstruction
stable, and, in the case of four-lepton final states, that translates
to several hundreds of events needed on a wedgebox plot. In addition,
the case of on-shell decays gives rise to extra solutions in the mass
space that can be eliminated with more statistics, but ultimately
contribute to an increased error in the reconstructed masses.
Thus, in the case of neutralino-pair production considered in this
work, DK may, depending on the parameter point Nature has chosen,
perhaps only serve useful as a check on results obtained with other
techniques that do not depend on events near an endpoint. As
mentioned in the Introduction, these most prominently include MSTs
and $m_{T2}$ techniques, which at the SPS1a point happen to work
quite well.
\section*{Acknowledgements}
This work was funded in part by the Kavli Institute for Theoretical
Physics (Beijing). ARR and MJW acknowledge funding from the UK Science
and Technology Facilities Council (STFC). This work is also part of
the French ANR project ToolsDMColl, BLAN07-2-194882.
\newpage
\section*{Appendix}
\subsection*{On-Shell Kinematics}
Consider the neutralino decay
\begin{equation}
\tilde\chi_i^0\to \ell^+\tilde{\ell}^-\to \ell^+\ell^- \tilde\chi_1^0,
\end{equation}
in the neutralino rest frame. If the di-lepton invariant mass is
maximal, all the decay products must be collinear (say along
$\hat{x}$). In particular, four-momentum conservation forces
\begin{equation*}
\left(
\begin{array}{c}
E_{\ell^+} \\
\\
p_{\ell^+} \\
\end{array}
\right)_{\tilde\chi_i^0} = \left( \begin{array}{c}
\frac{m_i^2 - m_s^2}{2 m_i} \\
\\
\frac{m_i^2 - m_s^2}{2 m_i}\\
\end{array}
\right)
\quad{\rm and}\quad
\left(
\begin{array}{c}
E_{\tilde{\ell}^-} \\
\\
p_{\tilde{\ell}^-} \\
\end{array}
\right)_{\tilde\chi_i^0} = \left( \begin{array}{c}
\frac{m_i^2 + m_s^2}{2 m_i} \\
\\
-\frac{m_i^2 - m_s^2}{2 m_i}\\
\end{array}
\right),
\end{equation*}
where $m_s$ is the slepton mass and the lepton is assumed to be
massless. Similarly, in the slepton's decay frame,
\begin{equation*}
\left(
\begin{array}{c}
E_{\ell^-} \\
\\
p_{\ell^-} \\
\end{array}
\right)_{\tilde{\ell}} = \left( \begin{array}{c}
\frac{m_s^2 - m_1^2}{2 m_s} \\
\\
-\frac{m_s^2 - m_1^2}{2 m_s}\\
\end{array}
\right)
\quad{\rm and}\quad
\left(
\begin{array}{c}
E_{\tilde\chi_1^0} \\
\\
p_{\tilde\chi_1^0} \\
\end{array}
\right)_{\tilde{\ell}} = \left( \begin{array}{c}
\frac{m_s^2 + m_1^2}{2 m_s} \\
\\
\frac{m_s^2 - m_1^2}{2 m_s}\\
\end{array}
\right),
\end{equation*}
which, when boosted back to the $\tilde\chi_i^0$ rest frame using
$\beta=-(m_i^2 - m_s^2)/(m_i^2 + m_s^2)$, becomes
\begin{equation*}
\left(
\begin{array}{c}
E_{\ell^-} \\
\\
p_{\ell^-} \\
\end{array}
\right)_{\tilde\chi_i^0} = \left( \begin{array}{c}
m_i\frac{m_s^2 - m_1^2}{2 m_s^2} \\
\\
-m_i\frac{m_s^2 - m_1^2}{2 m_s^2}\\
\end{array}
\right)
\quad{\rm and}\quad
\left(
\begin{array}{c}
E_{\tilde\chi_1^0} \\
\\
p_{\tilde\chi_1^0} \\
\end{array}
\right)_{\tilde\chi_i^0} = \left( \begin{array}{c}
\frac{m_s^4 + m_i^2 m_1^2}{2 m_i m_s^2} \\
\\
\frac{m_s^4 - m_i^2 m_1^2}{2 m_i m_s^2}\\
\end{array}
\right).
\end{equation*}
From these equations it is easy to verify (\ref{con1b}), (\ref{con2b}),
(\ref{con7b}), and (\ref{con8b}).
\subsection*{Finding the Lorentz Boost Parameters}
Here we indicate in more detail how one may arrive at
Eq.~(\ref{betasoln}). Starting from Eq.~(\ref{lspboost}) we have two
relevant equations,
\begin{eqnarray*}
\slashchar{P}_\| + K_i^\| &=& \beta_\| \left( \gamma\sqrt{K_i^2 + m_1^2}- (\gamma-1) \frac{\beta_\|}{\beta^2} K_i^\| -
(\gamma-1) \frac{\beta_\bot}{\beta^2} K_i^\bot \right), \\ \label{meq2}
\slashchar{P}_\bot + K_i^\bot &=& \beta_\bot \left( \gamma\sqrt{K_i^2 + m_1^2}- (\gamma-1) \frac{\beta_\|}{\beta^2} K_i ^\| -
(\gamma-1) \frac{\beta_\bot}{\beta^2} K_i^\bot \right).
\end{eqnarray*}
Taking the ratio of these one obtains after some rearranging
\begin{equation}
\label{keqn}
K_i ^\| = \frac{\beta_\|}{\beta_\bot}\slashchar{P}_\bot + \frac{\beta_\|}{\beta_\bot} K_i^\bot -\slashchar{P}_\|.
\end{equation}
This can now be inserted into two of the equations of
(\ref{totlepboost}),
\begin{eqnarray} \label{eq1}
K_i ^\| &=& -\beta_\| \gamma E+ P+ P(\gamma-1)\frac{\beta_\|^2}{\beta^2} ,\\ \label{eq2}
K_i^\bot &=& -\beta_\bot \gamma E + (\gamma-1)\frac{\beta_\| \beta_\bot}{\beta^2}P,
\end{eqnarray}
which can then be used to solve for the ratio
\begin{equation}\label{betaratio}
\alpha \equiv \frac{\beta_\bot}{\beta_\|}
= \frac{\slashchar{P}_\bot}{P + \slashchar{P}_\|}.
\end{equation}
Starting from the antiparallel condition (\ref{antipar}) and
expanding with the lepton momenta from (\ref{lepboost}), one
arrives, after some algebra, at
\begin{eqnarray*}
0 = (1+ \beta_\|^2 + \beta_\bot^2)E_1 E_2 - 2 \beta_\|E_1 P_2^\| -2 \beta_\bot E_1 P_2^\bot - 2 \beta_\| E_2 P_1^\|
-2 \beta_\bot E_2 P_1^\bot + \\ 2 \beta_\| \beta_\bot (P_1^\| P_2^\bot + P_2^\| P_1^\bot) + (1+ \beta_\|^2 - \beta_\bot^2)P_1^\| P_2^\|
+ (1 - \beta_\|^2 + \beta_\bot^2) P_1^\bot P_2^\bot,
\end{eqnarray*}
using that $E_{1,2} = |\vec{P}_{1,2}|$. Substituting $\beta_\bot$ for
$\alpha$ and $\beta_\|$ from (\ref{betaratio}), we get a quadratic
equation for $\beta_\|$:
\begin{equation*}
0 = a \beta_\|^2 - 2b \beta_\| + c,
\end{equation*}
where
\begin{eqnarray*}
a & \equiv & E_1 E_2 + P_1^\| P_2^\|- P_1^\bot P_2^\bot
+ 2\alpha (P_1^\| P_2^\bot + P_2^\| P_1^\bot)
+ \alpha^2 (E_1 E_2 - P_1^\| P_2^\| + P_1^\bot P_2^\bot), \\
b & \equiv & E_1 P_2^\| + E_2 P_1^\| + \alpha(E_1 P_2^\bot + E_2 P_1^\bot),\\
c & \equiv & E_1 E_2 + P_1^\| P_2^\| + P_1^\bot P_2^\bot.
\end{eqnarray*}
Using again that $E_{1,2}^2 ={P_{1,2}^\|}^2+{P_{1,2}^\bot}^2$, one
can show that $b^2 -4 a c = 0$, so this quadratic has a double root
\begin{equation}
\nonumber
\beta_\| = \frac{E_1P_2^\|+E_2 P_1^\| + \alpha (E_1 P_2^\bot + E_2 P_1^\bot)}
{E_1E_2+P_1^\|P_2^\|-P_1^\bot P_2^\bot+2\alpha(P_1^\|P_2^\bot+P_1^\bot P_2^\|)
+\alpha^2 (E_1 E_2 - P_1^\| P_2^\| + P_1^\bot P_2^\bot)}.
\end{equation}
Making repeated use of that, by definition, $P_1^\bot = - P_2^\bot$
and $P =P_1^\| + P_2^\|$, one arrives at the equation for $\beta_\|$
in (\ref{betasoln}).
\bibliographystyle{h-physrev4}
|
3,212,635,537,998 | arxiv | \section{Introduction}
The Lagrange mesh method is a very accurate and simple procedure to
compute eigenvalues and eigenfunctions of a two-body Schr\"{o}dinger
equation \cite{baye86,vinc93,baye95}. It is applicable for both local
and nonlocal interactions \cite{nonloc}, and also for a semirelativistic
kinetic operator, i.e. the spinless Salpeter
equation \cite{sem01,brau2}. In this method, the trial eigenstates are
developed in a basis of well-chosen functions, the Lagrange functions,
and the Hamiltonian matrix elements are obtained with a Gauss
quadrature. Moreover, the Lagrange mesh method can be extended to treat
very accurately
three-body problems, in nuclear or atomic physics \cite{hess99,3b2}.
In this work, we apply the Lagrange mesh method to solve the
inverse problem for bound states: starting from a given bound state --
wave function and corresponding eigenenergy --, we show how to compute
the
equivalent local potential. To our knowledge, this application of
Lagrange mesh method has not been studied before. It can then be used to
compute the equivalent local potential of a given nonlocal potential.
The determination of equivalent local potentials is of particular
interest in nuclear physics (see for example Ref.~\cite{nucl}). The more
interesting point is that our procedure allows to deal with
semirelativistic kinematics.
Our paper is organized as follows. In Sec.~\ref{lagmesh}, we recall
the main points of the Lagrange mesh method and show how to apply it to
solve a bound state problem with a central potential. Then, we give a
procedure to compute the equivalent local potential with this method
starting from a given spectrum in Sec.~\ref{bsep}. In order to check the
efficiency of our method, we apply it to several cases in which the
spectrum is analytically known. Firstly, we consider three central
potentials with a nonrelativistic kinematics in Sec.~\ref{applic}: the
harmonic oscillator (Sec.~\ref{nrho}), the Coulomb potential
(Sec.~\ref{nrcp}), and the nonlocal Yamaguchi potential
(Sec.~\ref{yama}).
Secondly, in Sec.~\ref{srho}, we consider the case of the
semirelativistic harmonic oscillator for two massless particles, whose
solution is also analytical. The accuracy of the method is checked in
all those cases, and conclusions are drawn in Sec.~\ref{conclu}.
\section{Lagrange mesh method}\label{lagmesh}
A Lagrange mesh is formed of $N$ mesh points $x_{i}$ associated with an
orthonormal set of indefinitely derivable functions $f_{j}(x)$ on an
interval $[a,b]$. A Lagrange function $f_{j}(x)$
vanishes
at all mesh points but one; it satisfies the
condition \cite{baye86,vinc93,baye95}
\begin{equation}
\label{flagpro}
f_{j}(x_{i})=\lambda^{-1/2}_{i}\delta_{ij}.
\end{equation}
The weights $\lambda_{i}$ are linked to the mesh points $x_{i}$ through
a Gauss quadrature formula
\begin{equation}
\label{gauss}
\int^{b}_{a} g(x)\, dx \approx\sum^{N}_{k=1}\lambda_{k}\, g(x_{k}),
\end{equation}
which is used to compute all the integrals over the interval $[a,b]$.
As in this work we only study radial equations, we consider the interval
$[0,\infty[$, leading
to a Gauss-Laguerre quadrature. The Gauss formula~(\ref{gauss}) is
exact when $g(x)$ is a polynomial of degree $2N-1$ at most, multiplied
by $\exp(-x)$. The $N$ Lagrange-Laguerre mesh points $x_i$ are then
given by the zeros of
the Laguerre polynomial $L_{N}(x)$ of degree $N$ \cite{baye86}.
An explicit form can be derived for the corresponding regularized
Lagrange functions
\begin{equation}
\label{flag}
f_{i}(x)=(-1)^{i}x^{-1/2}_{i}\, x(x-x_{i})^{-1}L_{N}(x)\, e^{-x/2}.
\end{equation}
They clearly satisfy the constraint ($\ref{flagpro}$), and they are
orthonormal, provided the scalar products are computed with the
quadrature ($\ref{gauss}$). Moreover, they vanish in $x=0$.
To show how these elements can be applied to a physical problem,
let us consider a standard Hamiltonian $H = T(\vec{p}^{\, 2})+V(r)$,
where
$T(\vec{p}^{\, 2})$ is the kinetic term and $V(r)$ a radial potential
(we work in natural units $\hbar = c=1$).
The calculations are performed with trial states $|\psi\rangle$ given by
\begin{equation}
\label{state}
\left|\psi\right\rangle=\sum^{N}_{k=1}C_{k}\left|f_{k}\right\rangle,
\end{equation}
where
\begin{equation}
\left\langle \vec{r}\,|f_{k}\right\rangle=
\frac{f_{k}(r/h)}{\sqrt{h}\,r}Y_{\ell m}(\theta,\varphi).
\end{equation}
$\ell$ is the orbital angular momentum quantum number and the
coefficients $C_{k}$ are
linear variational parameters. $h$ is a scale parameter chosen to
adjust the size of the mesh to the domain of physical interest. If we
define $r=h\,x$,
with $x$ a dimensionless variable, a relevant value of $h$ will be
obtained thanks to the relation $h=r_a/x_N$, where $x_N$ is the last
mesh point and $r_a$ is a physical radius for which the asymptotic tail
of the wave function is well defined. This radius has to be a priori
estimated, but various computations show that it has not to be known
with great accuracy, since the method is not variational in
$h$ \cite{sem01,fab1}.
We have now to compute the Hamiltonian matrix elements. Let us
begin with the potential term. Using the
properties of the Lagrange functions and the Gauss
quadrature~(\ref{gauss}), the potential matrix for a local potential
$V(r)$ is diagonal.
Its elements are
\begin{equation}
\label{poten1}
V_{ij}=\int^\infty_0 dx\,f_i(x) V(h\, x) f_j(x)\approx
V(hx_{i})\, \delta_{ij},
\end{equation}
and only involve the value of the potential at the mesh points.
As the matrix elements are computed only approximately, the variational
character of the method cannot be guaranteed. But the accuracy of the
method is preserved \cite{baye02}. The matrix elements for a nonlocal
potential $W(r,r')$ are given by \cite{nonloc}
\begin{equation}\label{poten2}
W_{ij}=h\,\int^\infty_0 dx\int^\infty_0 dx' f_i(x)\,W(hx,hx')\,f_j(x')
\approx h\, \sqrt{\lambda_i\lambda_j}\ W(hx_i,hx_j).
\end{equation}
The kinetic energy operator is generally only a function of
$\vec{p}^{\, 2}$. It is shown in Ref.~\cite{baye95} that, using the
Gauss quadrature and
the properties of the Lagrange functions, one obtains the corresponding
matrix
\begin{equation}
(\vec{p}^{\, 2})_{ij}=\frac{1}{h^{2}}\left[p^{\, 2}_{r\,
ij}+\frac{\ell(\ell+1)}{x^{2}_{i}}\delta_{ij}\right],
\end{equation}
where
\begin{equation}\label{pij_def}
p^{2}_{r\, ij}=\left\{
\begin{array}{lll}
&(-1)^{i-j}(x_{i}x_{j})^{-1/2}(x_{i}+x_{j})(x_{i}-x_{j})^{-2} &(i\neq
j),\\
&(12\,x^{2}_{i})^{-1}[4+(4N+2)\,x_{i}-x^{2}_{i}]&(i=j).
\end{array} \right.
\end{equation}
Now, the kinetic energy matrix $T(\vec{p}^{\, 2})$ can be computed with
the
following method \cite{sem01}:
\begin{enumerate}
\item Diagonalization of the matrix $\vec{p}^{\, 2}$. If $D^{2}$ is the
corresponding diagonal matrix, we have thus
$\vec{p}^{\, 2}=SD^{2}S^{-1}$, where $S$ is the transformation matrix.
\item Computation of $T(D^{2})$ by taking the function $T$ of all
diagonal elements of $D^{2}$.
\item Determination of the matrix elements
$T_{ij}$ in the
Lagrange basis by using the transformation matrix $S$:
$T(\vec{p}^{\, 2})=S\,T(D^{2})\, S^{-1}$.
\end{enumerate}
Note
that such a calculation is not exact because the number of Lagrange
functions is finite. However, it has already given good results in the
semirelativistic case, when
$T(\vec{p}^{\, 2})=\sqrt{\vec{p}^{\, 2}+m^{2}}$ \cite{sem01} or even
when $T(\vec{p}^{\, 2},r)=\sqrt{\vec{p}^{\, 2}+U^{2}(r)}$ \cite{brau2}.
The eigenvalue equation
$H\left|\psi\right\rangle=E\left|\psi\right\rangle$ reduces then to a
system of $N$ mesh equations,
\begin{equation}\label{meq}
\sum^{N}_{j=1}\left[ T_{ij}+{\cal V}_{ij}-E \delta_{ij}\right]
C_{j}=0 \quad \text{with} \quad C_{j}=\sqrt{h\lambda_{j}}\, u(hx_{j}),
\end{equation}
where $u(r)$ is the regularized radial wave function and ${\cal V}$ the
local or nonlocal potential matrix. The coefficients
$C_{j}$ provide the values of the radial wave function at mesh points.
But contrary to some other mesh methods, the wave function is also known
everywhere thanks to Eq.~(\ref{state}).
\section{Bound state equivalent local potential}\label{bsep}
In the previous section, we applied the Lagrange mesh method to solve
the eigenequation for two-body central problems. We now show that this
method allows to solve very easily the inverse problem, that is,
starting from particular wave function $\left|\psi\right\rangle$ and
energy $E$, to find the corresponding equivalent local potential for a
given kinematics $T$.
In the case of a local central potential, the mesh
equations~(\ref{meq}) can be rewritten as
\begin{equation}\label{effpot1}
V(hx_i)=E-\frac{1}{\sqrt{\lambda_i}\, u(hx_i)}\sum^N_{j=1}T_{ij}\sqrt{
\lambda_j}\ u(hx_j).
\end{equation}
We see from the above equation that, provided we know the radial wave
function and the energy of the state, the equivalent local potential can
be directly computed at the mesh points. Let us note that, since the
matrix elements $T_{ij}$ depend on the orbital angular momentum $\ell$,
this quantum number has to be a priori specified. The calculation is
done easily because the potential matrix for a local potential $V(r)$ is
diagonal and only involves the value of the potential at the mesh
points, as shown in Eq.~(\ref{poten1}). Obviously, this method does not
require a given normalization for the wave function. Moreover, it is
also applicable for semirelativistic kinematics.
We can remark that Eq.~(\ref{effpot1}) contains term which are
proportional to $u(hx_j)/u(hx_i)$. They may be difficult to compute
numerically with a great accuracy when $h x_i$ is either close to zero
or very large. In these cases indeed, the regularized wave function
tends towards zero. It means that the first values of the potential and
also the last ones could be inaccurate. It is worth mentioning that, for
radially excited states, a particular mesh point $x_k$ could be such
that $hx_k$ is a zero of the wave function. In this case, $V(hx_k)$
cannot be computed. Although very improbable, this problem could simply
be cured by taking a slightly different value of $N$ or $h$.
In order to check the validity of our method, we will consider four
cases where the eigenvalue problem is analytically solvable for a given
potential $V^E$. This will enable us to compare the numerically computed
points $V(hx_i)$ with the corresponding exact values $V^E(hx_i)$. The
number $\delta$, defined by
\begin{eqnarray}\label{ddef}
\delta=\max\left\{\left|\frac{V(hx_i)-V^E(hx_i)}{V^E(hx_i)}\right|,\
3\leq i\leq N-3 \right\},
\end{eqnarray}
is a measurement of the accuracy of the numerical computations. The more
$\delta$ is close to zero, the more the method is accurate. The first
and last two mesh points are -- arbitrarily -- not included in the
computation of $\delta$, since they can introduce errors which are not
due to the method itself, but rather to a lack of precision in the
numerical computations, as we argued previously from inspection of
formula~(\ref{effpot1}).
\section{Nonrelativistic Applications}\label{applic}
The kinetic operator which will be used in all the computations of this
section is given by
\begin{equation}
T(\vec p^{\,2})=\frac{\vec p^{\,2}}{2\mu},
\end{equation}
where $\mu$ is the reduced mass of the studied two-body system.
\subsection{Harmonic oscillator}\label{nrho}
The spectrum of a spherical harmonic oscillator, whose potential reads
\begin{equation}\label{ehp}
V^E(r)=\frac{\Lambda^2r^2}{2\mu},
\end{equation}
is given by (see for example Ref.~\cite[problem 66]{flu})
\begin{equation}\label{nrho1}
R_{n\ell}(r)\propto r^\ell\, {\rm e}^{-\Lambda r^2/2} L^{\ell+1/2}_n(
\Lambda\, r^2), \quad E_{n\ell}=\Lambda\,\mu^{-1}(2n+\ell+3/2).
\end{equation}
It is readily computed from the virial theorem that
$\left\langle r^2\right\rangle=(2n+\ell+3/2)/\Lambda$. Therefore, we
suggest the following value for the scale parameter:
\begin{eqnarray}
h&=&\frac{4\sqrt{\left\langle r^2\right\rangle}}{x_N}\label{hdef1}\\
&=&\frac{4}{x_N}\sqrt{\frac{(2n+\ell+3/2)}{\Lambda}}\label{hdef1b},
\end{eqnarray}
where the factor $4$ ensures that the last mesh point will be located in
the asymptotic tail of the wave function.
In order to make explicit computations, we have to specify the
value of our parameters. We set $\mu=0.70$~GeV and
$\Lambda=0.53$~GeV$^2$. These parameters can be used in hadron physics
to roughly describe
a $c\bar c$ meson \cite{fab1,drg}. We choose $N=30$, and the scale
parameter is computed by using Eq.~(\ref{hdef1b}). Once these parameters
are fixed, Eqs.~(\ref{effpot1}) and (\ref{nrho1}) allow to find the
equivalent local potential. The result is plotted and compared to the
exact harmonic potential~(\ref{ehp}) in Fig.~\ref{Fig1}, where we used
the wave function in the $2S$ state ($n=1,\,\ell=0$). The numerical
result is clearly close to the exact result, and only $30$ mesh points
are enough to provide a good picture of the potential: we have indeed
$\delta=2.1~10^{-3}~\%$, this number being computed with
Eq.~(\ref{ddef}). The same conclusion holds if other states than the
$2S$ one are used, and $\delta$ is always smaller than $1~\%$.
In Fig.~\ref{Fig2}, we show the variation of $\delta$ with the
scale parameter $h$ for three different states and $N=30$. We can
conclude from this figure that a rather large interval exists where the
quantity $\delta$ is lower than $1~\%$. Consequently, the scale
parameter
does not need to be computed with great accuracy: our
criterion~(\ref{hdef1}) is clearly accurate enough since the predicted
value of $h$ is
always located in this interval. The global behavior of $\delta$ which
can be observed in Fig.~\ref{Fig2} is due the difficulty of computing
$V(hx_i)$ when the scale parameter is too small or too large. In this
case indeed, the mesh points $hx_i$ cover no longer the main part of the
wave function, and a partial knowledge of the wave function leads to an
inaccurate description of the potential.
\subsection{Coulomb potential}\label{nrcp}
This case is of interest since it enables us to check whether the method
we present can correctly reproduce a singular potential or not. The
radial wave function and eigenenergies of a central Coulomb potential
\begin{equation}\label{coul}
V^{E}=-\frac{\kappa}{r}
\end{equation}
respectively read (see for example Ref.~\cite[problem 67]{flu})
\begin{equation}\label{nrc}
R(r)\propto r^\ell\,{\rm e}^{-\gamma r}\, L^{2\ell+1}_{n_p-\ell-1}(2
\gamma r),\quad E_{n_p}=-\frac{\mu\kappa^2}{2n^2_p},
\end{equation}
with $n_p\geq 1$, $0\leq\ell\leq n_p-1$, and $\gamma=\mu\kappa/n_p$. The
principal quantum number $n_p$ is defined by $n_p=n+\ell+1$.
It can be computed that \cite[p. 147]{landau}
\begin{equation}
\left\langle
r^2\right\rangle=\frac{n^2_p}{2\mu^2\kappa^2}\left[5n^2_p+1-3\ell(\ell+1
)\right].
\end{equation}
As the evaluation of the scale parameter given by Eq.~(\ref{hdef1})
yields good results in the harmonic oscillator case, it can be adapted
to the Coulomb potential, and $h$ is now defined as
\begin{equation}\label{hdef2}
h=\frac{15}{x_N}\frac{n_p}{\sqrt{2}\, \mu\kappa}\sqrt{5n^2_p+1-3\ell(
\ell+1)}.
\end{equation}
A factor $15$ is now needed because the Coulomb potential is a long-
ranged one. The wave function has thus to be known on a larger domain
than for the harmonic oscillator, since the latter potential is a
confining one.
In order to numerically compute the equivalent potential from the
wave function~(\ref{nrc}), we set $\mu=0.70$~GeV and $\kappa=0.27$. The
particular value of $\kappa$ we chose is commonly used in hadron physics
to parameterize the one-gluon-exchange part of the potential between two
heavy quarks \cite{fab1}. We choose $N=30$, and the scale parameter is
computed by using Eq.~(\ref{hdef2}). The result is plotted and compared
to the exact Coulomb potential~(\ref{coul}) in Fig.~\ref{Fig3} for the
wave function in the ground state ($n=\ell=0$). The numerical result is
close to the exact result, with a value of $\delta$ which is equal to
$1.4~10^{-5}~\%$. In particular, the singular behavior is well
reproduced. To stress this point, we performed another calculation with
$N=100$, and $h=0.37$~GeV$^{-1}$ following Eq.~(\ref{hdef2}). It can be
seen in Fig.~\ref{Fig3} that the Coulomb potential is then very well
matched at short distances. In this case however, we have
$\delta=0.7~\%$. Although this precision is still very satisfactory, it
seems strange at first sight that $\delta$ is higher for a larger number
of mesh points. This is due to the fact that the mesh points are the
zeros of the Laguerre polynomial of degree $N$. The first physical point
which is taken into account in the definition of $\delta$ is $hx_3$,
which is smaller for $N=100$ ($hx_3=0.811$~GeV$^{-1}$) than for $N=30$
($hx_3=0.068$~GeV$^{-1}$). This causes $\delta$ to be larger, since the
more a point is close to zero, the more the accuracy decreases.
For what concerns the variation of $\delta$ versus $h$, the same
qualitative features than for the harmonic oscillator are observed.
Equation~(\ref{hdef2}) thus appears to give a good evaluation of the
scale parameter. It can be also checked that a factor smaller than $15$
in Eq.~(\ref{hdef2}) can lead to values of the scale parameter for which
$\delta$ is quite larger than $1~\%$.
\subsection{Yamaguchi potential}\label{yama}
The Yamaguchi potential is a separable nonlocal potential, given by
\begin{equation}\label{sepa}
W(r,r')=-v(r)\,v(r'),
\end{equation}
with
\begin{equation}\label{vdef}
v(r)=\sqrt{\beta/\mu}\ (\alpha+\beta)\,{\rm e}^{-\beta r}.
\end{equation}
It was introduced in Ref.~\cite{yama} to study the deuteron
($\mu=0.468$~GeV). In particular, for $\alpha=0.046$ GeV and
$\beta=0.274$~GeV, it
admits a bound state whose binding energy is the one of the deuteron,
that is $E=-2.225$~MeV.
A nice particularity of this nonlocal potential is that the bound
state wave function can be analytically determined. It reads
\begin{equation}\label{udef}
R(r)\propto \frac{{\rm e}^{-\alpha r}-{\rm e}^{-\beta r}}{r}.
\end{equation}
Inserting this wave function into Eq.~(\ref{effpot1}) will provide us
with the equivalent local potential associated with the Yamaguchi
potential. Finding equivalent local potentials coming from nonlocal
potentials is of interest in nuclear physics, although most studies are
devoted to scattering states (see for example Refs.~\cite{nucl}). The
bound state equivalent potential of a separable nonlocal potential of
the form~(\ref{sepa}) is shown in Ref.~\cite{dijk} to be given by
\begin{equation}
V^{L}(r)=-\frac{v(r)}{u(r)}\int^\infty_0dr'\, v(r')\,u(r'),
\end{equation}
with $u(r)$ the regularized wave function of the bound state for the
nonlocal potential. Relations~(\ref{vdef}) and (\ref{udef}) can be
injected in this last equation to compute that
\begin{equation}\label{loceq}
V^{L}(r)=-\, \frac{\beta^2-\alpha^2}{2\mu}\, \frac{{\rm e}^{-\beta
r}}{{\rm e}^{-\alpha r}-{\rm e}^{-\beta r}}.
\end{equation}
As the radial wave function~(\ref{udef}) is maximal in $r=0$,
$R(0)\propto(\beta-\alpha)$, we can compute the scale parameter by
demanding that
\begin{equation}
R(hx_N)/R(0)=\epsilon,
\end{equation}
with $\epsilon$ a small number, that we will set equal to $10^{-3}$.
Then, assuming that $\alpha\ll\beta$ as it is the case for the
deuteron, $h$ will approximately be given by
\begin{equation}\label{hdef3}
h\approx-\frac{\ln\left[\epsilon(\beta-\alpha)\right]}{\alpha\, x_N}.
\end{equation}
The equivalent local potential $V^L(r)$ and the one computed with
the Lagrange mesh method can be compared in Fig.~\ref{Fig4}. The
deuteron parameters are used, together with $N=30$ and $h$ given by
Eq.~(\ref{hdef3}). The agreement is satisfactory since
$\delta=0.31~\%$.
The extension of the wave function is large because the deuteron is
weakly bound. An estimation of its radius is indeed given by $1.96$~fm
in Ref.~\cite{bad}, that is the rather large value of $9.9$~GeV$^{-1}$.
\section{The semirelativistic harmonic oscillator}\label{srho}
A nice feature of the Lagrange mesh method is that it allows to solve
semirelativistic Hamiltonians like the spinless Salpeter equation or the
relativistic flux tube model \cite{fab1,rft}, which are relevant in
quark physics. Equation~(\ref{effpot1}) is consequently applicable if
the kinetic operator is given by
\begin{equation}
T(\vec p^{\,2})=2\sqrt{\vec p^{\,2}+m^2}.
\end{equation}
In the ultrarelativistic case where $m=0$, the spectrum of the
Hamiltonian
\begin{equation}\label{ssh}
H=2\sqrt{\vec p^{\,2}}+\Omega\, r^2
\end{equation}
can be analytically computed in momentum space in terms of the regular
Airy function for $\ell=0$. In position space, it reads \cite{lucha}
\begin{equation}\label{srhowf}
R(r)\propto \frac{1}{r}\int^\infty_0dp\ \sin(p\,r)\ {\rm Ai}\left[\left(
\frac{2}{\Omega}\right)^{1/3}p+\alpha_n\right],\quad
E_n=-(4\Omega)^{1/3}\alpha_n,
\end{equation}
where $\alpha_n<0$ are the zeros of ${\rm Ai}$. They can be found for
example in Ref.~\cite[table 10.13]{Abra}.
Thanks to the particular properties of the Airy function, it can be
computed that \cite{sema04}
\begin{equation}
\left\langle r^2\right\rangle=-\left(\frac{2}{\Omega}\right)^{2/3}
\frac{\alpha_n}{3}.
\end{equation}
The scale parameter will thus be computed with the relation
\begin{equation}\label{hdef4}
h=\frac{4}{x_N}\left(\frac{2}{\Omega}\right)^{1/3}\sqrt{-\frac{
\alpha_n}{3}},
\end{equation}
in analogy with the similar case of the nonrelativistic harmonic
oscillator.
The comparison between the potential computed with our method and
the exact one
\begin{equation}\label{eho}
V^E(r)=\Omega\, r^2
\end{equation}
is given in Fig.~\ref{Fig5}. The value $\Omega=0.2$~GeV$^3$ is typical
for potential models of light quarks \cite{drg}. But, we present our
results as dimensionless quantities. The curves are thus universal: they
do not dependent on $\Omega$, which is the only parameter of this
Hamiltonian. Although still satisfactory, the agreement is not as good
as with the nonrelativistic applications. We find indeed
$\delta=3.1~\%$. By inspection of Fig.~\ref{Fig5}, it can be seen that
the last points slightly differ from the exact curve. These points are
related to the value of the wave function in its asymptotic tail, as it
can be seen from Eq.~(\ref{effpot1}). It means that finding the
equivalent potential, especially with a semirelativistic kinematics,
needs a good knowledge of the tail, which is not often necessary for
computation of the energy spectra.
In our case, the discrepancies for the last points are due to the
computation of the wave function in the asymptotic regime. It can be
checked that a resolution of Hamiltonian~(\ref{ssh}) with the Lagrange
mesh method leads to a wave function which asymptotically decreases
faster than the exact wave function, given by Eq.~(\ref{srhowf}).
Conversely, if one starts from the exact wave function, the Lagrange
mesh procedure will lead to a potential which does not increase enough
asymptotically, as we observe in Fig.~\ref{Fig5}. Fortunately, only the
very last points are affected, as it is shown in Fig.~\ref{Fig6}. By
varying $N$ and $h$, that is to say by varying the interval where the
potential is computed, one can always correctly reproduce the potential
in a given region: the more $hx_N$ is large, the larger is the interval
where the potential is correctly reproduced. Finding the equivalent
potential with a spinless Salpeter equation seems thus to require a more
careful study: several curves have to be computed by varying $h$ and $N$
in order to understand whether the long range behavior of the potential
is physical or simply due to a numerical artifact.
\section{Conclusions and outlook}\label{conclu}
In this work, we extended the domain of application of the Lagrange mesh
method to a particular type of problem: to find the equivalent local
potential corresponding to a given bound state with a given kinematics.
We assumed a central problem. Starting from a particular radial wave
function and the corresponding energy, the method we presented here
allows to compute the equivalent local potential at the mesh points. We
checked the accuracy of the computations in various cases whose
solutions are analytically known. Firstly, we studied the well-known
nonrelativistic harmonic oscillator and Coulomb potentials. These
potentials are correctly reproduced by the Lagrange mesh method with a
precision better than $1~\%$, provided the scale parameter is large
enough to take into account the asymptotic tail of the wave function.
Moreover, the singularity of the Coulomb potential is well matched. The
numerical parameters are the number of mesh points, and the scale
parameter. It appears that a typical value of $30$ mesh points is enough
to provide a good picture of the potential. As it was the case for usual
eigenvalue problems, the scale parameter does not need to be accurately
determined: a rather large interval exists where the precision is lower
than $1~\%$.
If the spectrum comes from a nonlocal potential, our method will
compute the equivalent local potential. This problem is of interest in
nuclear physics \cite{nucl}. As an illustration, we applied it to the
nonlocal Yamaguchi potential describing the deuteron. In this particular
case, the spectrum is analytical as well as the corresponding equivalent
potential. Again, the accuracy of our method is very good.
Finally, our procedure can also be easily adapted to the case of a
semirelativistic kinematics. As a check, we studied the semirelativistic
harmonic oscillator. Again, the potential is correctly reproduced, but
it appears that the asymptotic behavior of the potential is problematic.
This is an artifact of the method in the semirelativistic case: by
varying the mesh size, one can indeed see that the value of the
potential at the last mesh points is systematically too low, but the
harmonic shape of the potential is well reproduced at the other mesh
points.
Our purpose is to apply this method to the study of systems
containing quarks and gluons. In particular, glueballs, which are bound
states of gluons, are very interesting systems because their existence
is directly related to the nonabelian nature of QCD. Bound states of two
gluons can be described within the framework of potential models by a
spinless Salpeter equation with a Cornell potential: a linear confining
term plus a Coulomb term coming from short-range interactions
\cite{brau}. Such a phenomenological potential has been shown to arise
from QCD in the case of a quark-antiquark bound state \cite{loop}.
Theoretical indications show that it could be valid also for
glueballs \cite{simo}. Moreover, recently, the mass and the wave
function of the
scalar glueball (with quantum numbers $J^{PC}=0^{++}$) has been computed
in lattice QCD \cite{lat}. Thanks to the Lagrange mesh method, these
data could be used to extract the potential between two gluons from
lattice QCD, and see whether it is a Cornell one or not. This study will
be published elsewhere.
\acknowledgments
The authors thank the FNRS for financial support.
|
3,212,635,537,999 | arxiv | \section{Introduction}
\label{sect:intro}
The unprecedented angular resolution of {\it Chandra} satellite has enabled the study of
X-ray point sources in nearby galaxies. Most of these point sources are expected to
be X-ray binaries like the ones found in the Milky Way. An important result of the
{\it Chandra} observations was the confirmation of
Ultra-luminous X-ray sources (ULXs), discovered with { \it Einstein} observatory
in the 1980s \citep{Fab89}. These are off-nuclear X-ray point sources
with X-ray luminosities in the range $10^{39}-10^{41}$ $\rm ergs~s^{-1}$. The observed luminosities
of ULXs exceed the Eddington limit for a $10 M_{\odot}$ black hole, which has
led to a sustained debate on the nature of these sources. Since ULXs are off-nuclear
sources, their masses must be $< 10^{5}M_{\odot}$ from dynamical friction arguments
\citep{Kaa01}. Thus, ULXs may represent a class of Intermediate Mass Black holes (IMBHs)
whose mass range ($10 M_{\odot} < M < 10^{5}M_{\odot}$) is between that of stellar
mass black holes and super massive ones \citep{Mak00}.
Further the nature of the sources in nearby galaxies, which are
less luminous than ULX, is also not clear and it is
difficult to ascertain whether they harbour neutron stars or black holes.
The primary reason for these uncertainties is that unlike Galactic X-ray binaries, it is difficult
to identify the companion star in the optical and obtain the binary parameters. For most X-ray sources
in nearby galaxies, the associated optical emission is due to the integrated light from a host globular
cluster \citep{Kim06,Kim09,Pta06,Goa02} and it is usually not possible to resolve and identify the companion
star. However, these studies provide important information regarding the environment of the X-ray sources. For example,
ULXs in early type galaxies are associated with red globular clusters \citep{Pta06, Ang01}. Even the non-detection
of optical emission allows one to impose
strong upper limit on the black hole mass for these accreting systems based on some
standard assumptions \citep{Jit11}. However, a more direct inference on the nature of the system
requires identification and spectral measurement of the associated optical emission. An important aspect of
identifying the correct optical counterpart in a crowded field is to check for optical variability.
If the optical emission is variable, it is most probably directly associated with the X-ray source
and not the integrated light of stars in a globular cluster. Indeed, for low mass X-ray binaries
in the Galaxy, the optical emission is variable and is for some cases correlated with the X-ray
emission \citep[e.g.4U 1636-536:][]{Shi11} while for others it is not \citep[e.g. GX 9+9:][]{Kon06}.
The optical variability may be due to the orbital motion of the donor star or reprocessing of the
variable X-ray emission or X-ray heating of the companion. However, typically the optical counterpart
of X-ray binaries in nearby galaxies will not be resolved, especially if the source is in a globular cluster. Hence
it is not expected that optical variability will be seen for them.
Nevertheless, variability of optical counterparts have been measured for the bright X-ray sources
in nearby galaxies. For example,
the optical counterpart of NGC1313 X-2 has been identified as a O7 star at solar metallicity,
The optical counterpart exhibits variability at $\sim$ 0.2 mag on short time scales \citep{Liu07, Gri08} and the
variability may be due to varying X-ray irradiation of the donor star and a stochastic varying
contribution from the accretion disk.
An independent study of the same source \citep{Muc07} revealed that the optical flux of the counterpart shows
variation ${\leq} 30$\% and that it may be a main-sequence star of mass $\sim$ $10-18 M_\odot$ feeding to a
black hole of mass $120 M_\odot$. The optical counterpart of Holmberg IX X-1 exhibits photometric variability of $0.136\pm0.027$ in the {\it HST/ACS V} band images \citep{Gri11} although it seems to have a constant magnitude within
photometric errors ($22.710\pm0.038$ and $22.680\pm0.015$) in SUBARU {\it V} band images.
\cite{Tao11} have reported the optical variability for three ULXs, M101 ULX-1, M81 ULX1 and NGC1313 X-2, at
a magnitude difference of 0.2 or larger in the {\it V} band.
Some of the X-ray sources in nearby galaxies could
be background AGN and it is expected that their optical emission would be variable.
It is important to identify more X-ray sources that have optically variable counterparts, which then can be subjected to
more detailed observational follow-ups such as spectral and/or simultaneous X-ray/optical
observations. A systematic analysis of a number of galaxies to identify such sources will be crucial to understand the nature of these sources.
Such an analysis would require multiple optical observations of a galaxy, a uniform scheme to
identify optical counterparts of the X-ray sources and more importantly an estimate of
the systematic uncertainties in order to avoid any spurious variability that may arise
if only statistical errors are considered.
In this work, we consider elliptical galaxies which are $\lesssim 20$ Mpc away that
have been observed by {\it Chandra} and have more than one {\it HST} observation in the same
filter. We restrict our analysis to ellipticals since for them the continuum
optical emission can be modelled and subtracted out to reveal optical point sources
\citep{Jit11}. Using the field optical sources we estimate the systematic errors in the
optical flux measurements and hence can report true optical variability at a high
confidence level. Our aim is to study the optical counterpart variability of bright X-ray sources
(X-ray counts $> 60$) whose X-ray spectra can be modelled and hence a reliable estimate
of its luminosity can be obtained.
In the next section, we describe the selection of the sample galaxies. \S 3 and \S 4 describe the
X-ray analysis and the method to identify the optical counterparts and to compute the
photometry with systematic errors. We discuss the results in \S 5.
\section{Source Selection}
\label{sect:select}
The samples were selected based on three criteria. (1) The distance to the host
galaxy is $\lesssim 20$ Mpc, (2) The galaxy has {\it Chandra} observation and (3)
has more than one epoch {\it HST} observation in the same filter. Based on these criteria, we have selected five
galaxies which are listed in Table \ref{sample}. For three of the galaxies there are multiple
{\it Chandra} observations which we use to study the long term X-ray variability.
Using the longest exposure {\it Chandra} observations we identify X-ray sources which
have counts $> 60$, so that we can obtain reliable X-ray spectra for them. Of these, we
selected those that fell within the field of view of both the {\it HST} observations. For NGC2768
the only source that fulfilled these criteria was the central AGN and hence we report no further
analysis of the galaxy.
\begin{table}
\bc
\begin{minipage}[]{100mm}
\caption[]{Sample Galaxy Properties\label{sample}}\end{minipage}
\setlength{\tabcolsep}{2.0pt}
\small
\begin{tabular}{cccccccccc}
\hline\noalign{\smallskip}
Galaxy & Distance & {\it Chandra} & {\it Chandra} & $T_{exp}$ & {\it HST} & {\it HST} & {\it HST} & $N_{xo}$\\
&(Mpc)&ID&Observation Date&(ks)&ID&Filter&Observation Date&&\\
\hline\noalign{\smallskip}
NGC1399 & $18.9$ & $9530$ & $2008Jun08$ & $60.11$ & $J9P305020$ & $F475W$ & $2006Aug02$ & 18 \\
& & $319$ & $2000Jan18$ & $56.66$ & $J90X02020$ & $F475W$ & $2004Sep11$ & \\
NGC4486 & $15.8$ & $2707$ & $2002Jul06$ & $99.93$ & $J9E086010$ & $F814W$ & $2006Feb20$ & 17 \\
& & $352$ & $2000Jul29$ & $38.16$ & $J9E003010$ & $F814W$ & $2006Jan03$ & \\
NGC4278 & $15.2$ & $7081$ & $2007Feb20$ & $112.14$ & $J9NM06010$ & $F850LP$ & $2007Jan02$ & 9 \\
& & $11269$ & $2010Mar15$ & $82.95$ & $J9NM07010$ & $F850LP$ & $2006Dec23$ & \\
NGC1427 & $21.1$ & $4742$ & $2005May01$ & $51.70$ & $J9P302020$ & $F475W$ & $2006Jul31$ & 2 \\
& & $-$ & $-$ & $-$ & $J90X06020$ & $F475W$ & $2004Sep22$ & \\
NGC2768 & $20.1$ & $9528$ & $2008Jan25$ & $65.46$ & $J6JT08021$ & $F814W$ & $2002May31$ & 1 \\
& & $-$ & $-$ & $-$ & $J8DT02021$ & $F814W$ & $2003Jan14$ & \\
\noalign{\smallskip}\hline
\end{tabular}
\ec
\tablecomments{0.92\textwidth}{(1) Host galaxy name; (2) Distance to the host galaxy from NED; (3) {\it Chandra} observation ID;
(4) Chandra Observation Date; (5) Exposure time in kilo seconds; (6) {\it HST} observation ID; (7) {\it HST} Filter; (8) Observation Date;
(9) Number of common sources in the field of view of X-ray and optical images.}
\end{table}
NGC1399 and NGC4486 are giant elliptical galaxies in the center of the
Fornax and Virgo clusters respectively and are well-known for their populous
globular cluster systems \citep{Kim06,Dir03,Bas06,Ang01,Jor04,Irw06,Siv07}.
The {\it Chandra} analysis \citep{Ang01} of NGC1399 shows that a large fraction
of $2-10{\rm~keV}$ X-ray emission is most likely from the low-mass X-ray binaries
(LMXBs). The {\it HST} study of these {\it Chandra}
identified X-ray sources shows that $\sim70$\% (26 of 38 sources) of these sources
are associated to Globular Clusters (GCs). The specific frequency of globular cluster
in this galaxy is 2-3 times that of typical elliptical galaxies \citep{Har91}.
The optical counterparts of the ULXs (CXOJ033831.8-352604) show [OIII] ${\lambda}5007$ and [NII] ${\lambda}6583$
emission line in the optical spectrum \citep{Irw10}. \cite{Irw10} suggest that the lack of H${\alpha}$ and
H${\beta}$ emission line in the spectrum may be an indication of a disruption of a white dwarf star by an intermediate
mass black hole (IMBH).
The analysis of {\it Chandra} deep observations of the nearby elliptical galaxy NGC4278,
identified 236 X-ray point sources with luminosity ranging from $3.5\times10^{36} \rm ergs~s^{-1}$ to $2\times10^{40} \rm ergs~s^{-1}$ \citep{Bra09}. This galaxy has rich
globular cluster systems and 39 of them are coincident with X-ray sources which lie within the $D_{25}$
ellipse of the galaxy. 10 of the GC-LMXB associated sources lie at the high X-ray luminosity end ($L_{X}
> 10^{38} \rm ergs~s^{-1}$). Also, $44$\% of the X-ray source
population exhibit long term variability indicating that they are accreting
compact objects. \cite{Fab10} analysed the spectra of the X-ray sources by
fitting with either single thermal accretion disk or power law model and the best-fit
parameters are similar to those of Galactic BH binaries. Seven luminous sources have luminosity
exceeding the Eddington limit for accreting neutron stars. Four of these sources are
associated with GCs and the other three do not have optical counterparts and are found
in the stellar field of NGC4278.
NGC1427 is a low luminosity elliptical galaxy in Fornax cluster and its globular cluster
association has been studied by \cite{For01} and \cite{Kis97}. The photometry studies reveal a bimodal
cluster population in this galaxy and suggest that the formation mechanism of globular clusters
in low luminosity galaxies shows similarities with giant galaxies. The {\it Chandra} ACIS
Survey of X-ray point sources \citep{Liu11} identified two ULXs in this galaxy with luminosity
$\ge 2\times10^{39} \rm ergs~s^{-1}$. Among them, one source is inside the $D_{25}$ region of the galaxy,
and the other is outside the $D_{25}$ region.
\section{X-ray Analysis}
\label{sect:xray}
We start with analysing the {\it Chandra} observations listed in Table \ref{sample}.
These are observations with Advanced CCD Imaging Spectrometer array
(ACIS-S) and the data reduction and analysis were done using {\sc ciao 4.2}, and
{\sc heasoft 6.9}. Using the {\sc ciao} source detection tool {\it celldetect}, the X-ray point
sources were extracted from the level 2 event list with {\it signal-to-noise} ratio of 3.
Some of the extracted sources are near the nucleus and in the excessive diffused emission
regions and hence these sources were not included in the analysis. The extracted sources with net count ${\geq} 60$ were selected.
The spectral analysis was done using {\sc xspec 12.6.0}, and the data were fitted in the energy range of 0.3 - 8.0 keV.
All sources were fitted with two spectral models: an absorbed power law and an absorbed disk black body.
Absorption was taken into account using the {\sc xspec} model {\it wabs}. If the $\chi^2$ difference between
the two models was larger than 2.7, we took the model with the smaller $\chi^2$ to be the representative
one. If the $\chi^2$ difference was less than 2.7 (i.e. when both models equally well represent the data),
we choose the representative model to be the one which gave a lower luminosity. The analysis has been done for both observations
listed in Table \ref{sample}, with the longer observation being called the first one and the shorter one the second.
Table \ref{firstsecondfavour} lists the spectral parameters corresponding to the representative model.
The spectra of two sources in NGC1399 are not well fitted with either model and a closer
inspection revealed the presence of an additional {\it mekal} component which has been added.
To quantify the long term variability of the X-ray sources we consider sources that are in the
field of view of both observations. We jointly fit the spectra using the same model parameters except
that we introduce a constant factor which multiplies the later observation. In other words, we keep
the absorption and the spectral parameters (i.e. either the temperature or the power-law index) same
for both data sets, but allow for variation in the relative normalization. If the constant is unity, then
the source has not varied. We consider a source to be X-ray variable only if the constant
$C_{2}$ is inconsistent with unity at 2-sigma level i.e. $|C_{2}-1|/\sigma_{C_{2}} > 2$. The results of the joint
fitting are shown in Table \ref{xrayvariable}. As expected, several of the X-ray sources clearly
exhibit long term variability.
\begin{table}
\bc
\begin{minipage}[]{150mm}
\caption[]{Spectral Properties of point sources and best-fit models for first and second epoch\label{firstsecondfavour}}\end{minipage}
\setlength{\tabcolsep}{3.0pt}
\scriptsize
\begin{tabular}{ccccccccccccc}
\hline\noalign{\smallskip}
Galaxy & RA (J2000) & Dec (J2000) & $n_H$ & $\Gamma/kT_{in}$ & log($L_{1}$) & $\chi^2 / d.o.f$ & Model & $n_H$ & $\Gamma/kT_{in}$ & log($L_{2}$) & $\chi^2 / d.o.f$ & Model\\
\hline\noalign{\smallskip}
NGC1399 & 3h38m32.58s & -35\dg27$'$5.40$"$ & $0.03^{+ 0.10}_{- 0.03}$ & $1.63^{+ 0.33}_{- 0.21}$ & $39.39^{+ 0.06}_{- 0.05}$ & $11.25/22$ & P & $0.03^{+ 0.04}_{- 0.03}$ & $1.61^{+ 0.20}_{- 0.19}$ & $39.53^{+ 0.04}_{- 0.04}$ & $47.68/39$ & P \\
NGC1399 & 3h38m31.79s & -35\dg26$'$4.23$"$ & $0.01^{+ 0.07}_{- 0.01}$ & $0.33^{+ 0.05}_{- 0.06}$ & $39.04^{+ 0.15}_{- 0.06}$ & $ 8.30/13$ & D & $0.00^{+ 0.01}_{- 0.00}$ & $0.38^{+ 0.04}_{- 0.03}$ & $39.10^{+ 0.04}_{- 0.04}$ & $23.68/27$ & D \\
NGC1399 & 3h38m36.82s & -35\dg27$'$46.98$"$ & $0.00^{+ 0.14}_{- 0.00}$ & $1.67^{+ 0.61}_{- 0.30}$ & $38.72^{+ 0.10}_{- 0.11}$ & $ 4.39/4$ & P & $0.00^{+ 0.08}_{- 0.00}$ & $2.27^{+ 0.74}_{- 0.30}$ & $38.78^{+ 0.14}_{- 0.09}$ & $ 9.73/9$ & P \\
NGC1399 & 3h38m33.09s & -35\dg27$'$31.53$"$ & $0.00^{+ 0.13}_{- 0.00}$ & $0.61^{+ 0.18}_{- 0.17}$ & $38.61^{+ 0.12}_{- 0.08}$ & $ 5.91/6$ & D & $0.00^{+ 0.04}_{- 0.00}$ & $1.73^{+ 0.31}_{- 0.23}$ & $38.97^{+ 0.09}_{- 0.10}$ & $20.47/11$ & P \\
NGC1399 & 3h38m25.95s & -35\dg27$'$42.19$"$ & $0.00^{+ 0.16}_{- 0.00}$ & $1.05^{+ 0.98}_{- 0.40}$ & $38.64^{+ 0.18}_{- 0.15}$ & $ 3.41/4$ & D & $0.00^{+ 0.14}_{- 0.00}$ & $0.87^{+ 0.63}_{- 0.29}$ & $38.55^{+ 0.15}_{- 0.14}$ & $ 3.48/3$ & D \\
NGC1399 & 3h38m32.76s & -35\dg26$'$58.73$"$ & $0.00^{+ 0.18}_{- 0.00}$ & $2.58^{+ 0.00}_{- 1.24}$ & $38.86^{+ 0.19}_{- 0.19}$ & $ 6.13/5$ & D & $0.00^{+ 0.08}_{- 0.00}$ & $1.63^{+ 0.59}_{- 0.39}$ & $38.74^{+ 0.16}_{- 0.18}$ & $ 5.39/7$ & P \\
NGC1399 & 3h38m32.34s & -35\dg27$'$2.11$"$ & $0.99^{+ 1.41}_{- 0.67}$ & $0.68^{+ 0.50}_{- 0.27}$ & $38.94^{+ 0.52}_{- 0.25}$ & $ 2.53/5$ & D & $0.62^{+ 0.00}_{- 0.00}$ & $1.33^{+ 0.00}_{- 1.06}$ & $38.39^{+ 3.21}_{- 0.57}$ & $ 3.00/5$ & D \\
NGC1399 & 3h38m31.86s & -35\dg26$'$49.26$"$ & $0.10^{+ 1.76}_{- 0.10}$ & $0.84^{+ 0.00}_{- 0.68}$ & $38.41^{+ 0.58}_{- 0.31}$ & $ 3.72/4$ & D & $0.00^{+ 0.09}_{- 0.00}$ & $2.48^{+ 0.84}_{- 0.37}$ & $38.76^{+ 0.20}_{- 0.10}$ & $11.90/9$ & P \\
\#NGC1399 & 3h38m25.66s & -35\dg27$'$41.50$"$ & $0.00^{+ 0.17}_{- 0.00}$ & $1.09^{+ 7.98}_{- 0.50}$ & $38.67^{+ 0.37}_{- 0.18}$ & $ 6.71/4$ & D & $-$ & $-$ & $<38.08$ & $-$ & $-$ \\
\#NGC1399 & 3h38m27.80s & -35\dg25$'$26.65$"$ & $0.00^{+ 0.71}_{- 0.00}$ & $1.27^{+ 1.30}_{- 0.69}$ & $38.55^{+ 0.17}_{- 0.18}$ & $ 0.53/2$ & D & $-$ & $-$ & $<38.25$ & $-$ & $-$ \\
*NGC1399 & 3h38m26.50s & -35\dg27$'$32.29$"$ & $-$ & $-$ & $<38.08$ & $-$ & $-$ & $0.00^{+ 0.09}_{- 0.00}$ & $0.90^{+ 0.34}_{- 0.25}$ & $38.71^{+ 0.11}_{- 0.11}$ & $ 8.29/7$ & D \\
*NGC1399 & 3h38m33.82s & -35\dg25$'$56.95$"$ & $-$ & $-$ & $<37.94$ & $-$ & $-$ & $0.00^{+ 0.10}_{- 0.00}$ & $0.48^{+ 0.43}_{- 0.18}$ & $38.35^{+ 0.14}_{- 0.14}$ & $ 5.25/3$ & D \\
*NGC1399 & 3h38m33.80s & -35\dg26$'$58.30$"$ & $-$ & $-$ & $<38.73$ & $-$ & $-$ & $0.54^{+ 1.26}_{- 0.54}$ & $0.20^{+ 0.62}_{- 0.12}$ & $39.03^{+ 2.95}_{- 1.06}$ & $ 0.44/2$ & D \\
*NGC1399 & 3h38m32.35s & -35\dg27$'$10.63$"$ & $-$ & $-$ & $<38.16$ & $-$ & $-$ & $0.03^{+ 0.24}_{- 0.03}$ & $0.98^{+ 0.62}_{- 0.36}$ & $38.63^{+ 0.12}_{- 0.14}$ & $ 0.08/6$ & D \\
*NGC1399 & 3h38m25.32s & -35\dg27$'$53.49$"$ & $-$ & $-$ & $<38.20$ & $-$ & $-$ & $0.00^{+ 0.22}_{- 0.00}$ & $1.86^{+ 3.53}_{- 0.82}$ & $38.62^{+ 0.15}_{- 0.17}$ & $ 2.20/3$ & D \\
*NGC1399 & 3h38m27.19s & -35\dg26$'$1.53$"$ & $-$ & $-$ & $<38.38$ & $-$ & $-$ & $0.00^{+ 0.33}_{- 0.00}$ & $1.28^{+ 1.59}_{- 0.68}$ & $38.71^{+ 0.25}_{- 0.23}$ & $ 5.29/3$ & P \\
\ddag NGC1399 & 3h38m27.63s & -35\dg26$'$48.54$"$ & $0.15^{+ 0.12}_{- 0.11}$ & $2.72^{+ 0.63}_{- 0.52}$ & $39.48^{+ 0.26}_{- 0.15}$ & $23.83/22$ & P & $0.01^{+ 0.03}_{- 0.01}$ & $0.42^{+ 0.05}_{- 0.06}$ & $39.15^{+ 0.07}_{- 0.05}$ & $30.26/30$ & D \\
\ddag NGC1399 & 3h38m38.76s & -35\dg25$'$54.86$"$ & $0.00^{+ 0.10}_{- 0.00}$ & $1.06^{+ 0.30}_{- 0.26}$ & $38.97^{+ 0.08}_{- 0.08}$ & $12.31/10$ & D & $0.00^{+ 0.29}_{- 0.00}$ & $2.02^{+ 1.97}_{- 0.56}$ & $38.56^{+ 0.58}_{- 0.14}$ & $ 3.71/2$ & P \\
NGC4486 & 12h30m47.15s & 12\dg24$'$15.91$"$ & $0.00^{+ 0.01}_{- 0.00}$ & $0.66^{+ 0.08}_{- 0.07}$ & $39.17^{+ 0.04}_{- 0.04}$ & $108.78/83$ & D & $0.25^{+ 0.14}_{- 0.12}$ & $2.91^{+ 0.73}_{- 0.57}$ & $39.75^{+ 0.33}_{- 0.19}$ & $65.87/44$ & P \\
NGC4486 & 12h30m53.24s & 12\dg23$'$56.69$"$ & $0.03^{+ 0.11}_{- 0.03}$ & $1.05^{+ 0.26}_{- 0.21}$ & $39.03^{+ 0.07}_{- 0.08}$ & $85.86/72$ & D & $0.00^{+ 0.09}_{- 0.00}$ & $0.95^{+ 0.36}_{- 0.27}$ & $38.96^{+ 0.10}_{- 0.12}$ & $50.16/34$ & D \\
NGC4486 & 12h30m50.12s & 12\dg23$'$1.07$"$ & $0.00^{+ 0.07}_{- 0.00}$ & $1.11^{+ 0.33}_{- 0.24}$ & $38.97^{+ 0.08}_{- 0.10}$ & $88.36/84$ & D & $0.00^{+ 0.50}_{- 0.00}$ & $0.60^{+ 0.50}_{- 0.39}$ & $38.69^{+ 0.75}_{- 0.25}$ & $32.89/40$ & D \\
NGC4486 & 12h30m46.19s & 12\dg23$'$28.63$"$ & $0.00^{+ 0.07}_{- 0.00}$ & $0.92^{+ 0.23}_{- 0.19}$ & $38.95^{+ 0.07}_{- 0.08}$ & $76.43/70$ & D & $0.01^{+ 0.17}_{- 0.01}$ & $0.96^{+ 0.46}_{- 0.38}$ & $38.99^{+ 0.11}_{- 0.13}$ & $36.06/30$ & D \\
NGC4486 & 12h30m44.67s & 12\dg22$'$1.06$"$ & $0.25^{+ 0.23}_{- 0.15}$ & $2.62^{+ 1.07}_{- 0.69}$ & $39.16^{+ 0.47}_{- 0.21}$ & $51.11/48$ & P & $0.06^{+ 0.39}_{- 0.06}$ & $1.21^{+ 0.00}_{- 0.60}$ & $38.82^{+ 0.31}_{- 0.23}$ & $21.96/22$ & D \\
NGC4486 & 12h30m50.80s & 12\dg25$'$2.00$"$ & $0.00^{+ 0.09}_{- 0.00}$ & $1.21^{+ 0.54}_{- 0.34}$ & $38.82^{+ 0.10}_{- 0.12}$ & $59.29/46$ & D & $0.02^{+ 0.25}_{- 0.02}$ & $1.58^{+ 1.61}_{- 0.60}$ & $39.04^{+ 0.13}_{- 0.15}$ & $17.77/16$ & D \\
NGC4486 & 12h30m44.26s & 12\dg22$'$9.37$"$ & $0.00^{+ 0.29}_{- 0.00}$ & $0.49^{+ 0.44}_{- 0.28}$ & $38.36^{+ 0.46}_{- 0.19}$ & $65.61/40$ & D & $0.00^{+ 0.35}_{- 0.00}$ & $1.62^{+ 0.00}_{- 0.84}$ & $38.69^{+ 0.23}_{- 0.29}$ & $25.09/19$ & D \\
\#NGC4486 & 12h30m44.71s & 12\dg24$'$34.61$"$ & $0.00^{+ 0.04}_{- 0.00}$ & $2.11^{+ 0.33}_{- 0.15}$ & $39.08^{+ 0.06}_{- 0.05}$ & $58.46/55$ & P & $-$ & $-$ & $<38.52$ & $-$ & $-$ \\
\#NGC4486 & 12h30m46.32s & 12\dg23$'$23.19$"$ & $0.00^{+ 0.12}_{- 0.00}$ & $0.65^{+ 0.16}_{- 0.20}$ & $38.89^{+ 0.11}_{- 0.08}$ & $94.05/68$ & D & $-$ & $-$ & $<38.51$ & $-$ & $-$ \\
\#NGC4486 & 12h30m47.32s & 12\dg23$'$8.82$"$ & $0.02^{+ 0.19}_{- 0.02}$ & $0.76^{+ 0.26}_{- 0.33}$ & $38.84^{+ 0.15}_{- 0.11}$ & $95.28/80$ & D & $-$ & $-$ & $<38.58$ & $-$ & $-$ \\
\#NGC4486 & 12h30m50.08s & 12\dg22$'$51.21$"$ & $0.00^{+ 0.15}_{- 0.00}$ & $0.66^{+ 0.69}_{- 0.44}$ & $38.46^{+ 0.13}_{- 0.27}$ & $69.39/69$ & D & $-$ & $-$ & $<38.63$ & $-$ & $-$ \\
\#NGC4486 & 12h30m52.79s & 12\dg23$'$36.85$"$ & $3.38^{+ 1.38}_{- 0.63}$ & $9.50^{+ 0.00}_{- 12.50}$ & $44.10^{+ 5.62}_{- 2.00}$ & $73.37/69$ & P & $-$ & $-$ & $<44.78$ & $-$ & $-$ \\
\#NGC4486 & 12h30m43.49s & 12\dg23$'$46.80$"$ & $0.04^{+ 0.72}_{- 0.04}$ & $0.71^{+ 0.77}_{- 0.48}$ & $38.34^{+ 0.74}_{- 0.28}$ & $23.79/32$ & D & $-$ & $-$ & $<38.41$ & $-$ & $-$ \\
\#NGC4486 & 12h30m46.52s & 12\dg24$'$50.15$"$ & $0.00^{+ 0.38}_{- 0.00}$ & $0.70^{+ 0.65}_{- 0.47}$ & $38.40^{+ 0.50}_{- 0.19}$ & $36.82/32$ & D & $-$ & $-$ & $<38.41$ & $-$ & $-$ \\
\#NGC4486 & 12h30m44.91s & 12\dg24$'$4.50$"$ & $0.00^{+ 0.83}_{- 0.00}$ & $3.13^{+ 0.00}_{- 2.20}$ & $38.43^{+ 0.19}_{- 0.27}$ & $32.28/38$ & D & $-$ & $-$ & $<38.69$ & $-$ & $-$ \\
\#NGC4486 & 12h30m50.82s & 12\dg24$'$11.80$"$ & $0.08^{+ 0.16}_{- 0.08}$ & $0.50^{+ 0.19}_{- 0.16}$ & $38.80^{+ 0.19}_{- 0.15}$ & $55.48/67$ & D & $-$ & $-$ & $<38.55$ & $-$ & $-$ \\
\#NGC4486 & 12h30m49.13s & 12\dg21$'$59.40$"$ & $0.00^{+ 65.00}_{- 36.13}$ & $0.58^{+ 0.00}_{- 3.58}$ & $38.69^{+ 17.17}_{- 9.11}$ & $60.72/41$ & P & $-$ & $-$ & $<38.88$ & $-$ & $-$ \\
NGC4278 & 12h20m7.75s & 29\dg17$'$20.39$"$ & $0.00^{+ 0.07}_{- 0.00}$ & $1.71^{+ 0.64}_{- 0.42}$ & $38.64^{+ 0.08}_{- 0.09}$ & $7.36/11$ & D & $0.00^{+ 0.11}_{- 0.00}$ & $1.46^{+ 0.92}_{- 0.44}$ & $38.61^{+ 0.13}_{- 0.12}$ & $12.44/7$ & D \\
NGC4278 & 12h20m3.43s & 29\dg16$'$39.35$"$ & $0.00^{+ 0.14}_{- 0.00}$ & $1.71^{+ 1.24}_{- 0.55}$ & $38.49^{+ 0.13}_{- 0.13}$ & $4.40/6$ & D & $0.00$ & $1.22$ & $38.26$ & $5.14/2$ & D \\
NGC4278 & 12h20m4.22s & 29\dg16$'$51.24$"$ & $0.00^{+ 0.21}_{- 0.00}$ & $1.34^{+ 0.72}_{- 0.45}$ & $38.38^{+ 0.12}_{- 0.11}$ & $1.79/5$ & D & $0.00^{+ 0.39}_{- 0.00}$ & $3.72^{+ 0.00}_{- 2.43}$ & $38.55^{+ 0.16}_{- 0.21}$ & $0.36/2$ & D \\
NGC4278 & 12h20m5.23s & 29\dg16$'$39.82$"$ & $0.02^{+ 0.28}_{- 0.02}$ & $1.92^{+ 0.82}_{- 0.67}$ & $38.58^{+ 0.09}_{- 0.10}$ & $15.29/8$ & D & $0.00^{+ 0.10}_{- 0.00}$ & $1.64^{+ 0.99}_{- 0.49}$ & $38.54^{+ 0.11}_{- 0.12}$ & $5.18/5$ & D \\
NGC4278 & 12h20m4.33s & 29\dg17$'$35.86$"$ & $0.00^{+ 0.09}_{- 0.00}$ & $1.36^{+ 0.35}_{- 0.28}$ & $38.74^{+ 0.07}_{- 0.07}$ & $17.86/15$ & D & $0.00^{+ 0.20}_{- 0.00}$ & $1.54^{+ 0.72}_{- 0.48}$ & $38.63^{+ 0.10}_{- 0.11}$ & $4.94/6$ & D \\
NGC4278 & 12h20m6.03s & 29\dg16$'$48.25$"$ & $0.02^{+ 0.07}_{- 0.02}$ & $1.45^{+ 0.27}_{- 0.16}$ & $38.95^{+ 0.05}_{- 0.05}$ & $24.28/22$ & P & $0.00^{+ 0.12}_{- 0.00}$ & $1.63^{+ 0.78}_{- 0.44}$ & $38.68^{+ 0.10}_{- 0.10}$ & $8.10/8$ & D \\
NGC4278 & 12h20m5.48s & 29\dg16$'$40.68$"$ & $0.00^{+ 0.07}_{- 0.00}$ & $1.40^{+ 0.32}_{- 0.27}$ & $38.77^{+ 0.06}_{- 0.07}$ & $18.25/18$ & D & $0.00^{+ 0.09}_{- 0.00}$ & $1.83^{+ 1.53}_{- 0.57}$ & $38.75^{+ 0.12}_{- 0.12}$ & $6.50/9$ & D \\
NGC4278 & 12h20m6.79s & 29\dg16$'$56.01$"$ & $0.07^{+ 0.09}_{- 0.07}$ & $1.92^{+ 0.37}_{- 0.33}$ & $38.86^{+ 0.07}_{- 0.06}$ & $35.17/20$ & P & $0.00^{+ 0.06}_{- 0.00}$ & $1.35^{+ 0.36}_{- 0.27}$ & $38.80^{+ 0.07}_{- 0.08}$ & $13.11/14$ & D \\
\#NGC4278 & 12h20m5.95s & 29\dg17$'$8.79$"$ & $0.00^{+ 0.26}_{- 0.00}$ & $1.11^{+ 0.70}_{- 0.24}$ & $38.32^{+ 0.13}_{- 0.16}$ & $3.85/2$ & P & $-$ & $-$ & $<37.93$ & $-$ & $-$ \\
NGC1427 & 3h42m18.71s & -35\dg22$'$40.02$"$ & $0.05^{+ 0.11}_{- 0.05}$ & $1.01^{+ 0.33}_{- 0.22}$ & $39.18^{+ 0.07}_{- 0.08}$ & $8.26/10$ & D & $-$ & $-$ & $-$ & $-$ & $-$ \\
NGC1427 & 3h42m18.47s & -35\dg23$'$38.19$"$ & $0.00^{+ 0.04}_{- 0.00}$ & $1.06^{+ 0.36}_{- 0.23}$ & $39.17^{+ 0.09}_{- 0.09}$ & $21.77/11$ & D & $-$ & $-$ & $-$ & $-$ & $-$ \\
\noalign{\smallskip}\hline
\end{tabular}
\ec
\tablecomments{1.20\textwidth}{\# denotes the sources are present only in first observation. * denotes the sources are present only in second observation. \ddag denotes an additional mekal model added to get better fit for these sources. Host Galaxy Name; Right Ascension; Declination; $n_H$, equivalent hydrogen column density in $10^{22}cm^{-2}$ for the first observation; $\Gamma/kT_{in}$, photon power law index or inner disk temperature in keV in the first observation; $L_{unabs}$, Unabsorbed X-ray luminosity in $\rm ergs~s^{-1}$ in the energy range, 0.3-8.0 keV for the first observation; $\chi^2/d.o.f$, statistics and degree of freedom in the first observation; Best-fit Model (P-Power law, D-Disk black body) in the first observation; $n_H$, equivalent hydrogen column density in $10^{22}cm^{-2}$ for the second observation; $\Gamma/kT_{in}$, photon power law index or inner disk temperature in keV in the second observation; $L_{unabs}$, Unabsorbed X-ray luminosity in $\rm ergs~s^{-1}$ in the energy range, 0.3-8.0 keV for the second observation; $\chi^2/d.o.f$, statistics and degree of freedom in the second observation; Best-fit Model (P-Power law, D-Disk black body) in the second observation; Galactic absorption column density for NGC1399, $n_H$ = $1.53\times10^{20}cm^{-2}$; Galactic absorption column density for NGC4486, $n_H$ = $2.04\times10^{20}cm^{-2}$; Galactic absorption column density for NGC4278, $n_H$ = $1.99\times10^{20}cm^{-2}$; Galactic absorption column density for NGC1427, $n_H$ = $1.63\times10^{20}cm^{-2}$.}
\end{table}
\begin{table}
\bc
\begin{minipage}[]{150mm}
\caption[]{Combined Spectral Properties of point sources fitted with best-fit model\label{xrayvariable}}\end{minipage}
\setlength{\tabcolsep}{3pt}
\small
\begin{tabular}{ccccccccccc}
\hline\noalign{\smallskip}
Galaxy & RA (J2000) & Dec (J2000) & $n_H$ & $\Gamma$ & Norm & kT$_{in}$ & Norm & $C_{2}$ & $\chi^2/d.o.f$ & Var(Sig)\\
\hline\noalign{\smallskip}
NGC1399 & 3h38m32.58s & -35\dg27$'$5.40$"$ & $0.03^{+ 0.03}_{- 0.03}$ & $1.62^{+ 0.10}_{- 0.12}$ & $0.90^{+ 0.09}_{- 0.17}$ & $-$ & $-$ & $1.39^{+ 0.17}_{- 0.15}$ & $58.95/63$ & Y(2.60) \\
NGC1399 & 3h38m31.79s & -35\dg26$'$4.23$"$ & $0.00^{+ 0.02}_{- 0.00}$ & $-$ & $-$ & $0.36^{+ 0.03}_{- 0.03}$ & $0.84^{+ 0.41}_{- 0.26}$ & $1.19^{+ 0.18}_{- 0.15}$ & $33.90/42$ & N(1.27) \\
NGC1399 & 3h38m36.82s & -35\dg27$'$46.98$"$ & $0.00^{+ 0.03}_{- 0.00}$ & $2.03^{+ 0.34}_{- 0.20}$ & $0.20^{+ 0.05}_{- 0.03}$ & $-$ & $-$ & $1.46^{+ 0.45}_{- 0.34}$ & $17.35/15$ & N(1.35) \\
NGC1399 & 3h38m33.09s & -35\dg27$'$31.53$"$ & $0.10^{+ 0.08}_{- 0.08}$ & $2.31^{+ 0.38}_{- 0.31}$ & $0.40^{+ 0.14}_{- 0.18}$ & $-$ & $-$ & $1.41^{+ 0.42}_{- 0.30}$ & $26.02/18$ & N(1.37) \\
NGC1399 & 3h38m25.95s & -35\dg27$'$42.19$"$ & $0.08^{+ 0.27}_{- 0.08}$ & $1.70^{+ 0.48}_{- 0.55}$ & $0.25^{+ 0.27}_{- 0.09}$ & $-$ & $-$ & $0.85^{+ 0.33}_{- 0.24}$ & $ 3.87/8$ & N(0.45) \\
NGC1399 & 3h38m32.76s & -35\dg26$'$58.73$"$ & $0.00^{+ 0.09}_{- 0.00}$ & $1.25^{+ 0.39}_{- 0.32}$ & $0.18^{+ 0.07}_{- 0.04}$ & $-$ & $-$ & $0.90^{+ 0.38}_{- 0.24}$ & $ 9.72/13$ & N(0.26) \\
NGC1399 & 3h38m32.34s & -35\dg27$'$2.11$"$ & $1.64^{+ 1.94}_{- 0.97}$ & $3.42^{+ 2.31}_{- 1.29}$ & $2.25^{+ 26.18}_{- 0.00}$ & $-$ & $-$ & $0.37^{+ 0.30}_{- 0.25}$ & $ 4.77/11$ & Y(2.10) \\
NGC1399 & 3h38m31.86s & -35\dg26$'$49.26$"$ & $0.00^{+ 0.14}_{- 0.00}$ & $2.22^{+ 1.19}_{- 0.37}$ & $0.13^{+ 0.10}_{- 0.13}$ & $-$ & $-$ & $2.32^{+ 2.53}_{- 0.75}$ & $19.20/14$ & N(1.76) \\
NGC4486 & 12h30m47.15s & 12\dg24$'$15.91$"$ & $0.00^{+ 0.01}_{- 0.00}$ & $-$ & $-$ & $0.67^{+ 0.06}_{- 0.06}$ & $0.13^{+ 0.05}_{- 0.04}$ & $1.27^{+ 0.17}_{- 0.15}$ & $177.33/129$ & N(1.80) \\
NGC4486 & 12h30m53.24s & 12\dg23$'$56.69$"$ & $0.23^{+ 0.08}_{- 0.09}$ & $2.27^{+ 0.37}_{- 0.27}$ & $1.40^{+ 0.52}_{- 0.48}$ & $-$ & $-$ & $0.94^{+ 0.26}_{- 0.23}$ & $136.33/108$ & N(0.23) \\
NGC4486 & 12h30m50.12s & 12\dg23$'$1.07$"$ & $0.12^{+ 0.09}_{- 0.12}$ & $2.03^{+ 0.57}_{- 0.43}$ & $0.82^{+ 0.64}_{- 0.34}$ & $-$ & $-$ & $0.70^{+ 0.34}_{- 0.29}$ & $122.94/126$ & N(0.88) \\
NGC4486 & 12h30m46.19s & 12\dg23$'$28.63$"$ & $0.19^{+ 0.07}_{- 0.12}$ & $2.28^{+ 0.50}_{- 0.29}$ & $1.20^{+ 0.70}_{- 0.40}$ & $-$ & $-$ & $1.09^{+ 0.31}_{- 0.26}$ & $112.00/102$ & N(0.35) \\
NGC4486 & 12h30m44.67s & 12\dg22$'$1.06$"$ & $0.23^{+ 0.17}_{- 0.13}$ & $2.41^{+ 0.43}_{- 0.41}$ & $0.89^{+ 0.67}_{- 0.35}$ & $-$ & $-$ & $0.83^{+ 0.36}_{- 0.31}$ & $74.11/71$ & N(0.47) \\
NGC4486 & 12h30m50.80s & 12\dg25$'$2.00$"$ & $0.11^{+ 0.16}_{- 0.11}$ & $1.70^{+ 0.31}_{- 0.36}$ & $0.41^{+ 0.42}_{- 0.10}$ & $-$ & $-$ & $1.40^{+ 0.52}_{- 0.39}$ & $77.62/64$ & N(1.03) \\
NGC4486 & 12h30m44.26s & 12\dg22$'$9.37$"$ & $0.04^{+ 0.22}_{- 0.04}$ & $1.73^{+ 1.10}_{- 0.60}$ & $0.27^{+ 0.29}_{- 0.14}$ & $-$ & $-$ & $1.14^{+ 0.88}_{- 0.57}$ & $90.60/60$ & N(0.25) \\
NGC4278 & 12h20m7.75s & 29\dg17$'$20.39$"$ & $0.08^{+ 0.14}_{- 0.08}$ & $1.44^{+ 0.26}_{- 0.28}$ & $0.29^{+ 0.08}_{- 0.11}$ & $-$ & $-$ & $1.01^{+ 0.22}_{- 0.18}$ & $18.45/19$ & N(0.06) \\
NGC4278 & 12h20m3.43s & 29\dg16$'$39.35$"$ & $0.01^{+ 0.15}_{- 0.01}$ & $-$ & $-$ & $1.38^{+ 1.01}_{- 0.25}$ & $0.13^{+ 0.16}_{- 0.13}$ & $0.69^{+ 0.22}_{- 0.20}$ & $8.59/9$ & N(1.41) \\
NGC4278 & 12h20m4.22s & 29\dg16$'$51.24$"$ & $0.04^{+ 0.28}_{- 0.04}$ & $1.33^{+ 0.50}_{- 0.33}$ & $0.14^{+ 0.12}_{- 0.03}$ & $-$ & $-$ & $1.01^{+ 0.33}_{- 0.27}$ & $3.77/8$ & N(0.04) \\
NGC4278 & 12h20m5.23s & 29\dg16$'$39.82$"$ & $0.07^{+ 0.10}_{- 0.07}$ & $-$ & $-$ & $1.58^{+ 1.01}_{- 0.20}$ & $0.11^{+ 0.08}_{- 0.11}$ & $0.96^{+ 0.24}_{- 0.18}$ & $17.01/14$ & N(0.17) \\
NGC4278 & 12h20m4.33s & 29\dg17$'$35.86$"$ & $0.19^{+ 0.17}_{- 0.11}$ & $1.73^{+ 0.36}_{- 0.24}$ & $0.48^{+ 0.21}_{- 0.15}$ & $-$ & $-$ & $0.69^{+ 0.13}_{- 0.12}$ & $21.98/22$ & Y(2.38) \\
NGC4278 & 12h20m6.03s & 29\dg16$'$48.25$"$ & $0.01^{+ 0.05}_{- 0.01}$ & $1.42^{+ 0.16}_{- 0.16}$ & $0.45^{+ 0.06}_{- 0.10}$ & $-$ & $-$ & $0.64^{+ 0.11}_{- 0.10}$ & $31.79/31$ & Y(3.27) \\
NGC4278 & 12h20m5.48s & 29\dg16$'$40.68$"$ & $0.00^{+ 0.05}_{- 0.00}$ & $-$ & $-$ & $1.50^{+ 0.32}_{- 0.25}$ & $0.22^{+ 0.20}_{- 0.10}$ & $0.79^{+ 0.15}_{- 0.13}$ & $25.50/28$ & N(1.40) \\
NGC4278 & 12h20m6.79s & 29\dg16$'$56.01$"$ & $0.05^{+ 0.03}_{- 0.05}$ & $1.75^{+ 0.14}_{- 0.23}$ & $0.47^{+ 0.06}_{- 0.14}$ & $-$ & $-$ & $1.10^{+ 0.18}_{- 0.16}$ & $47.34/35$ & N(0.63) \\
\noalign{\smallskip}\hline
\end{tabular}
\ec
\tablecomments{1.17\textwidth}{Host Galaxy Name; Right Ascension; Declination; $n_H$, equivalent hydrogen column density in $10^{22}cm^{-2}$; $\Gamma$, photon power law index; Power law Normalization in $10^{-5}$; $kT_{in}$, inner disk temperature in keV; Disk black body Normalization in $10^{-1}$; $Const_{2}$, Constant2; $\chi^2$ statistics and degree of freedom; X-ray variable (Y-Yes, N-No) and its significance; Galactic absorption column density for NGC1399, $n_H$ = $1.53\times10^{20}cm^{-2}$; Galactic absorption column density for NGC4486, $n_H$ = $2.04\times10^{20}cm^{-2}$; Galactic absorption column density for NGC4278, $n_H$ = $1.99\times10^{20}cm^{-2}$; Constant1, $Const_{1}$ = 1.00.}
\end{table}
\section{Optical counterparts and photometry}
\label{sect:optical}
We search for the optical counterparts for these X-ray sources by using
the archival {\it HST ACS} images listed in Table \ref{sample}. Typically, the optical sources in
the {\it HST} images are too faint against the dominant galaxy light and hence to detect
them, the galaxy light was modelled
by isophotes of ellipses using the ellipse task in {\sc iraf/stsdas}
software. The modelled image was then subtracted from the observed galaxy
image to obtain a residual image. The optical point sources were then extracted
from the residual image by using {\sc sextractor} with a threshold
level of $3\sigma$.
By visual inspection, we could see that for many of the {\it Chandra} X-ray sources within an error
circle of one arcsecond there is an obvious optical source. However, there was a systematic positional
offset of one arcsecond between the {\it Chandra} and {\it HST} source positions. This constant positional
offset was applied to the X-ray sources and then the shifted X-ray positions were compared with the
optical source positions in the {\sc sextractor} catalogue. A more detailed explanation with clarifying
images is presented in \cite{Jit11}. This constant offset is less than the offset of 2.3 arcsec applied
for the source SN 1993J in the study of a ULX in M81 \citep{Liu02}. We analysed a total of 46 bright
X-ray sources, which are in the field of view of {\it HST} images and identified the optical counterpart
for 34 sources. The optical counterparts identified are unique and for most of the counterparts there is
no other optical source even within the 3 arcsec from the optical position. The remaining 12 sources
didn't have an optical counterpart at their respective positions.
Photometry of the optical counterparts as well as all the sources detected by {\sc sextractor}
was computed on the drizzled images with {\sc iraf/apphot} package. The drizzled images were
converted from {\it $e^{-}/s/pixel$} to {\it $e^{-}$} per pixel by multiplying the total exposure
time. An aperture radius of 0.5 arcsec was used to extract the flux by the task {\sc apphot}
and the magnitudes in the AB magnitude system were calculated using the zero points taken from
{\it HST ACS} data handbook. The aperture correction were computed from a list of {\sc apphot} photometry files using the {\sc daogrow} algorithm \citep{Ste90} and the correction is applied to the magnitudes. For those X-ray sources that didn't have an optical counterpart
(i.e. optically dark X-ray sources) we obtained the upper limit of the optical flux at
the X-ray positions.
Our aim is to estimate the optical variability of point sources from two
observations of a galaxy. This requires a reliable estimate of the statistical and systematic
errors, if any, in the optical flux. From the photometry, we get the
total counts, $C$ ($\it{sum}$ from photometry in ADU) and the background subtracted counts, $C_{S}$
($\it{flux}$ from photometry in ADU) of each source. The statistical error on $C_{S}$ can be taken to
be $\delta C_{S} = \sqrt{C/epadu}$ where $\it{epadu}$ is the gain parameter in electron per ADU.
For the two observations of NGC1399, we plot in Figure \ref{fluxfit} the background subtracted
counts $C_{S1}$ and $C_{S2}$ against each other for 848 sources that are in the common field of view.
There is the obvious correlation with a large scatter and several outliers. Since there are
outliers which may affect any least square fitting technique, we use the robust method \citep{Pre92}
to fit a straight line and obtained a slope $b = 0.876$ and a negligible offset of $a = 5.25$. The two observations have different zero point magnitude ($m_{ZP}$)and exposure time ($T$), which gives this scaling factor (b). For the case of NGC 1399, $m_{1ZP} = 26.059$ and $m_{2ZP} = 26.081$, $T_{1} = 680 sec$ and $T_{2} = 760 sec$ for the two observations. The apparent magnitude, $m = -2.5\times log_{10}(\frac {C_{S}}{T})+2.5\times log_{10}(A)$, where $2.5\times log_{10}(A) = m_{ZP}$. Thus $A = 10^(\frac{m_{ZP}}{2.5})$ and $m = -2.5\times log_{10}(\frac {C_{S}}{T \times A})$. If the apparent magnitudes in the two observations are same, then we can write, $\frac{C_{S2}}{A_{2} \times T_{2}}$ = $\frac{C_{S1}}{A_{1}\times T_{1}}$. Hence $C_{S1} = \frac{A_{1}\times T_{1}}{A_{2}\times T_{2}} \times C_{S2}$. The factor $\frac{A_{1}\times T_{1}}{A_{2}\times T_{2}} = 0.876$ which is as expected identical to the slope $b = 0.876$ obtained by fitting.
Then we scaled up the flux of the sources in the second observation i. e., $C^{\prime}_{S2}=b \times C_{S2}+a$
and their uncertainties $\delta C^{\prime}_{S2} = b \times \delta C_{S2}$. Now if there were no systematic errors then we
could compare $C_{S1}$ and $C^{\prime}_{S2}$ with their corresponding statistical errors
to determine if a source is variable. However, the statistical errors are small and as
evident in Figure \ref{fluxfit}, this would imply that a large number of the field sources
are variable. Since, we know that this is not the case and indeed most of the field sources
are expected not to vary there is systematic error involved. A better way to illustrate this
is to plot the histogram of $(C_{S1}-C^{\prime}_{S2})/\sigma_{\Delta C_{S12}}$ where
$\sigma_{\Delta C_{S12}} = \sqrt{\delta {C_{S1}}^2+\delta {C^{\prime}_{S2}}^2}$. If most of the
sources are non-variable and there was no systematic error, then the distribution should
be a zero centred Gaussian with width $\sigma = 1$. However, Figure \ref{termgauss} shows
that as the distribution is significantly broader.
We find that if we add a systematic of $S=525/\sqrt{2}$ to the uncertainties of the flux in quadrature to both
observations, then the distribution is consistent with being a
Gaussian with $\sigma =1$ as shown in Figure \ref {termgaussfit}. To corroborate that this indeed is
the correct level of systematic error, we do the following exercise. For each pair of optical fluxes, we
compare with a constant and obtain the chi-square,
\begin{eqnarray}
\chi^2 = \frac{(C_{S1}-C_{S0})^2}{\delta C_{S1}^2} + \frac{(C^{\prime}_{S2}-C_{S0})^2}{\delta {C^{\prime}_{S2}}^2}
\end{eqnarray}
where $C_{S0}$ is the model constant flux whose value is obtained by minimizing ${\chi}^2$ (i.e.
$\frac{\partial \chi^2}{\partial C_{S0}} = 0$) to be
\begin{eqnarray}
C_{S0} = (\frac{C_{S1}}{\delta C_{S1}^2} + \frac{C^{\prime}_{S2}}{\delta {C^{\prime}_{S2}}^2}) (\frac{\delta C_{S1}^2 \delta {C^{\prime}_{S2}}^2}{\delta C_{S1}^2 + \delta {C^{\prime}_{S2}}^2})
\end{eqnarray}
Since the number of data points is two and the number of parameters (i.e. $C_{S0}$) is one, the degree of
freedom here is one. Hence, if the model for a majority of the sources (i.e. the sources
are not variable) and the error estimates are correct then the distribution of $\chi^2$ should
be a chi-square distribution of order one i.e.
\begin{eqnarray}
P(x) = \frac{1}{\sqrt{2\pi}} x^{-1/2} \exp(-x/2)
\end{eqnarray}
Figure \ref {chidistribution} shows the distribution of $\chi^2$ for all the 848 sources in NGC1399. The
solid line is the expected distribution $P(x)$. For a majority of the sources which are expected not
to be variable $\chi^2 < 2$ as expected. More importantly the distribution matches well with the
majority including the low $\chi^2$ values of $\sim 0.01$. This strongly implies that the
systematic error used is reliable.
We could not identify the cause for the systematic errors despite our best efforts. However, we note that such deviations have been reported in similar works. For example, for NGC 1313, \cite{Liu07} reported that out of 399 optical sources they examined, more than 81 (i.e. 20\%) had variability above 2 sigma, while the expected number was more like 10\%.
The measured distribution deviates from the expected one
for $\chi^2 > 12$ and these are the few truly variable sources in the sample. Thus we can
state confidently and conservatively that sources with $\chi^2 > 12$ are indeed variable and we use this criterion
for this work. About 93\% (792) cross identified sources are not variable between the two
observations and 56 sources are optically variable i.e ${\chi}^2 {\geq}12$.
While the results presented above are for NGC1399, we use the same technique to establish the systematic
error for the other three galaxies and for each of them we find that $\chi^2 > 12$ to be a good conservative
criterion for optical variability. The photometric optical magnitudes of the X-ray sources of the sample
have been provided in Table \ref{optvariable}.
Table \ref{IR} provides the properties of the four X-ray sources which are
optically variable. We have also estimated the difference in the magnitude of these sources by comparing the
F814W and F850LP data. Even though they are different bands,
three sources (source 2 and source 3 in NGC1399, one source in NGC1427)
show a magnitude difference of 0.1 - 0.4. But the Source 1 in NGC1399 has a magnitude difference of 0.02 only in these filters.
\begin{figure}
\centering
\includegraphics[width=14.0cm, angle=0]{ms1641fig1.ps}
\caption{The background subtracted counts ($C_{S}$) of the common sources from both observation
is fitted to a straight-line by the robust estimation method. }
\label{fluxfit}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=14.0cm, angle=0]{ms1641fig2.ps}
\caption{The fraction of sources in each bin of $(C_{S1}-C^{\prime}_{S2})/\sigma_{\Delta C_{S12}}$ with
$\sigma_{\Delta C_{S12}} = \sqrt{\delta {C_{S1}}^2+\delta {C^{\prime}_{S2}}^2}$. The green dotted line is Gaussian
distribution with mean = 0 and sigma = 1.}
\label{termgauss}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=14.0cm, angle=0]{ms1641fig3.ps}
\caption{The fraction of sources in each bin of $(C_{S1}-C^{\prime}_{S2})/\sigma_{\Delta C_{S12}}$
with a systematic (S) added to the uncertainty in flux. The green dotted line is
Gaussian distribution with mean = 0 and sigma = 1.}
\label{termgaussfit}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=14.0cm, angle=0]{ms1641fig4.ps}
\caption{The fraction of sources in each bin of ${\chi}^2$. The green dotted line is
chi-square distribution, P(x).}
\label{chidistribution}
\end{figure}
\begin{table}
\bc
\begin{minipage}[]{150mm}
\caption[]{The Variability of Optical Counterparts in the Sample\label{optvariable}}\end{minipage}
\setlength{\tabcolsep}{4.0pt}
\scriptsize
\begin{tabular}{ccccccccccc}
\hline\noalign{\smallskip}
Galaxy & RA (J2000) & Dec (J2000) & log($L_{1}$) & Model & $C_{2}$ & Var(Sig) & $m_{1}$ & $m_{2}$ & ${\Delta m}$ & ${\chi}^2$\\
\hline\noalign{\smallskip}
NGC1399 & 3h38m32.58s & -35\dg27$'$5.40$"$ & $39.39^{+ 0.06}_{- 0.05}$ & P & $1.39^{+ 0.17}_{- 0.15}$ & Y(2.60) & $22.157\pm0.018$ & $22.169\pm0.019$ & $-0.012\pm0.026$ & 0.259 \\
NGC1399 & 3h38m31.79s & -35\dg26$'$4.23$"$ & $39.04^{+ 0.15}_{- 0.06}$ & D & $1.19^{+ 0.18}_{- 0.15}$ & N(1.27) & $22.787\pm0.031$ & $22.779\pm0.031$ & $0.008\pm0.044$ & 0.031 \\
NGC1399 & 3h38m36.82s & -35\dg27$'$46.98$"$ & $38.72^{+ 0.10}_{- 0.11}$ & P & $1.46^{+ 0.45}_{- 0.34}$ & N(1.35) & $23.231\pm0.045$ & $22.847\pm0.032$ & $0.384\pm0.055$ & 51.437 \\
NGC1399 & 3h38m33.09s & -35\dg27$'$31.53$"$ & $38.61^{+ 0.12}_{- 0.08}$ & D & $1.41^{+ 0.42}_{- 0.30}$ & N(1.37) & $21.692\pm0.012$ & $21.607\pm0.011$ & $0.085\pm0.016$ & 26.493 \\
NGC1399 & 3h38m33.09s & -35\dg27$'$31.53$"$ & $38.61^{+ 0.12}_{- 0.08}$ & D & $1.41^{+ 0.42}_{- 0.30}$ & N(1.37) & $23.188\pm0.044$ & $23.110\pm0.042$ & $0.078\pm0.061$ & 1.575 \\
NGC1399 & 3h38m25.95s & -35\dg27$'$42.19$"$ & $38.64^{+ 0.18}_{- 0.15}$ & D & $0.85^{+ 0.33}_{- 0.24}$ & N(0.45) & $ > 26.646$ & $ > 26.656$ & $-$ & $-$\\
NGC1399 & 3h38m32.76s & -35\dg26$'$58.73$"$ & $38.86^{+ 0.19}_{- 0.19}$ & D & $0.90^{+ 0.38}_{- 0.24}$ & N(0.26) & $23.070\pm0.040$ & $23.040\pm0.041$ & $0.030\pm0.058$ & 0.260 \\
NGC1399 & 3h38m32.34s & -35\dg27$'$2.11$"$ & $38.94^{+ 0.52}_{- 0.25}$ & D & $0.37^{+ 0.30}_{- 0.25}$ & Y(2.10) & $24.446\pm0.143$ & $24.525\pm0.160$ & $-0.079\pm0.215$ & 0.140 \\
NGC1399 & 3h38m31.86s & -35\dg26$'$49.26$"$ & $38.41^{+ 0.58}_{- 0.31}$ & D & $2.32^{+ 2.53}_{- 0.75}$ & N(1.76) & $21.878\pm0.014$ & $22.146\pm0.019$ & $-0.268\pm0.024$ & 133.152 \\
NGC1399 & 3h38m25.66s & -35\dg27$'$41.50$"$ & $38.67^{+ 0.37}_{- 0.18}$ & D & $-$ & $-$ & $24.860\pm0.203$ & $25.233\pm0.291$ & $-0.373\pm0.355$ & 1.197 \\
NGC1399 & 3h38m27.80s & -35\dg25$'$26.65$"$ & $38.55^{+ 0.17}_{- 0.18}$ & D & $-$ & $-$ & $24.783\pm0.186$ & $24.820\pm0.192$ & $-0.037\pm0.267$ & 0.019 \\
NGC1399 & 3h38m26.50s & -35\dg27$'$32.29$"$ & $ < 38.08$ & $-$ & $-$ & $-$ & $ > 26.661$ & $ > 26.697$ & $-$ & $-$\\
NGC1399 & 3h38m33.82s & -35\dg25$'$56.95$"$ & $ < 37.94$ & $-$ & $-$ & $-$ & $20.870\pm0.006$ & $20.866\pm0.006$ & $0.004\pm0.008$ & 0.062 \\
NGC1399 & 3h38m33.80s & -35\dg26$'$58.30$"$ & $ < 38.73$ & $-$ & $-$ & $-$ & $22.832\pm0.032$ & $22.849\pm0.034$ & $-0.017\pm0.046$ & 0.159 \\
NGC1399 & 3h38m32.35s & -35\dg27$'$10.63$"$ & $ < 38.16$ & $-$ & $-$ & $-$ & $23.046\pm0.040$ & $23.084\pm0.043$ & $-0.038\pm0.059$ & 0.444 \\
NGC1399 & 3h38m25.32s & -35\dg27$'$53.49$"$ & $ < 38.20$ & $-$ & $-$ & $-$ & $22.256\pm0.019$ & $22.257\pm0.020$ & $-0.001\pm0.027$ & 0.003 \\
NGC1399 & 3h38m27.19s & -35\dg26$'$1.53$"$ & $ < 38.38$ & $-$ & $-$ & $-$ & $22.206\pm0.018$ & $22.197\pm0.019$ & $0.009\pm0.026$ & 0.080 \\
NGC1399 & 3h38m27.63s & -35\dg26$'$48.54$"$ & $39.23^{+ 0.23}_{- 0.15}$ & P & $1.14^{+ 0.21}_{- 0.16}$ & N(0.87) & $ > 25.979$ & $ > 25.894$ & $-$ & $-$\\
NGC1399 & 3h38m38.76s & -35\dg25$'$54.86$"$ & $39.09^{+ 0.09}_{- 0.18}$ & D & $0.22^{+ 0.12}_{- 0.11}$ & Y(6.50) & $21.740\pm0.012$ & $21.732\pm0.012$ & $0.008\pm0.017$ & 0.165 \\
NGC4486 & 12h30m47.15s & 12\dg24$'$15.91$"$ & $39.17^{+ 0.04}_{- 0.04}$ & D & $1.27^{+ 0.17}_{- 0.15}$ & N(1.80) & $22.416\pm0.044$ & $22.512\pm0.049$ & $-0.096\pm0.066$ & 2.409 \\
NGC4486 & 12h30m47.15s & 12\dg24$'$15.91$"$ & $39.17^{+ 0.04}_{- 0.04}$ & D & $1.27^{+ 0.17}_{- 0.15}$ & N(1.80) & $22.786\pm0.062$ & $22.744\pm0.060$ & $0.042\pm0.087$ & 0.161 \\
NGC4486 & 12h30m53.24s & 12\dg23$'$56.69$"$ & $39.03^{+ 0.07}_{- 0.08}$ & D & $0.94^{+ 0.26}_{- 0.23}$ & N(0.23) & $22.901\pm0.069$ & $22.826\pm0.065$ & $0.075\pm0.094$ & 0.520 \\
NGC4486 & 12h30m50.12s & 12\dg23$'$1.07$"$ & $38.97^{+ 0.08}_{- 0.10}$ & D & $0.70^{+ 0.34}_{- 0.29}$ & N(0.88) & $20.686\pm0.010$ & $20.668\pm0.010$ & $0.018\pm0.014$ & 1.354 \\
NGC4486 & 12h30m46.19s & 12\dg23$'$28.63$"$ & $38.95^{+ 0.07}_{- 0.08}$ & D & $1.09^{+ 0.31}_{- 0.26}$ & N(0.35) & $20.408\pm0.007$ & $20.417\pm0.007$ & $-0.009\pm0.010$ & 1.261 \\
NGC4486 & 12h30m44.67s & 12\dg22$'$1.06$"$ & $39.16^{+ 0.47}_{- 0.21}$ & P & $0.83^{+ 0.36}_{- 0.31}$ & N(0.47) & $21.471\pm0.019$ & $21.470\pm0.019$ & $0.001\pm0.026$ & 0.008 \\
NGC4486 & 12h30m50.80s & 12\dg25$'$2.00$"$ & $38.82^{+ 0.10}_{- 0.12}$ & D & $1.40^{+ 0.52}_{- 0.39}$ & N(1.03) & $ > 26.367$ & $ > 26.260$ & $-$ & $-$ \\
NGC4486 & 12h30m44.26s & 12\dg22$'$9.37$"$ & $38.36^{+ 0.46}_{- 0.19}$ & D & $1.14^{+ 0.88}_{- 0.57}$ & N(0.25) & $ > 26.478$ & $ > 26.333$ & $-$ & $-$ \\
NGC4486 & 12h30m44.71s & 12\dg24$'$34.61$"$ & $39.08^{+ 0.06}_{- 0.05}$ & P & $-$ & $-$ & $21.800\pm0.025$ & $21.763\pm0.024$ & $0.037\pm0.035$ & 0.887 \\
NGC4486 & 12h30m46.32s & 12\dg23$'$23.19$"$ & $38.89^{+ 0.11}_{- 0.08}$ & D & $-$ & $-$ & $20.364\pm0.007$ & $20.351\pm0.007$ & $0.013\pm0.010$ & 1.278 \\
NGC4486 & 12h30m47.32s & 12\dg23$'$8.82$"$ & $38.84^{+ 0.15}_{- 0.11}$ & D & $-$ & $-$ & $20.742\pm0.010$ & $20.762\pm0.010$ & $-0.020\pm0.014$ & 2.599 \\
NGC4486 & 12h30m50.08s & 12\dg22$'$51.21$"$ & $38.46^{+ 0.13}_{- 0.27}$ & D & $-$ & $-$ & $20.902\pm0.011$ & $20.904\pm0.012$ & $-0.002\pm0.016$ & 0.094 \\
NGC4486 & 12h30m52.79s & 12\dg23$'$36.85$"$ & $44.10^{+ 5.62}_{- 2.00}$ & P & $-$ & $-$ & $23.332\pm0.103$ & $23.523\pm0.125$ & $-0.191\pm0.162$ & 1.617 \\
NGC4486 & 12h30m43.49s & 12\dg23$'$46.80$"$ & $38.34^{+ 0.74}_{- 0.28}$ & D & $-$ & $-$ & $20.067\pm0.005$ & $20.076\pm0.005$ & $-0.009\pm0.007$ & 2.220 \\
NGC4486 & 12h30m46.52s & 12\dg24$'$50.15$"$ & $38.40^{+ 0.50}_{- 0.19}$ & D & $-$ & $-$ & $23.436\pm0.111$ & $23.373\pm0.106$ & $0.063\pm0.154$ & 0.111 \\
NGC4486 & 12h30m44.91s & 12\dg24$'$4.50$"$ & $38.43^{+ 0.19}_{- 0.27}$ & D & $-$ & $-$ & $ > 26.199$ & $ > 26.086$ & $-$ & $-$ \\
NGC4486 & 12h30m50.82s & 12\dg24$'$11.80$"$ & $38.80^{+ 0.19}_{- 0.15}$ & D & $-$ & $-$ & $ > 25.733$ & $ > 25.873$ & $-$ & $-$ \\
NGC4486 & 12h30m49.13s & 12\dg21$'$59.40$"$ & $38.69^{+ 17.17}_{- 9.11}$ & P & $-$ & $-$ & $ > 26.154$ & $ > 26.032$ & $-$ & $-$ \\
NGC4278 & 12h20m7.75s & 29\dg17$'$20.39$"$ & $38.64^{+ 0.08}_{- 0.09}$ & D & $1.01^{+ 0.22}_{- 0.18}$ & N(0.06) & $20.283\pm0.010$ & $20.294\pm0.010$ & $-0.011\pm0.013$ & 0.001 \\
NGC4278 & 12h20m3.43s & 29\dg16$'$39.35$"$ & $38.49^{+ 0.13}_{- 0.13}$ & D & $0.69^{+ 0.22}_{- 0.20}$ & N(1.41) & $21.205\pm0.021$ & $21.188\pm0.021$ & $0.017\pm0.029$ & 0.823 \\
NGC4278 & 12h20m4.22s & 29\dg16$'$51.24$"$ & $38.38^{+ 0.12}_{- 0.11}$ & D & $1.01^{+ 0.33}_{- 0.27}$ & N(0.04) & $21.334\pm0.024$ & $21.374\pm0.025$ & $-0.040\pm0.034$ & 0.864 \\
NGC4278 & 12h20m5.23s & 29\dg16$'$39.82$"$ & $38.58^{+ 0.09}_{- 0.10}$ & D & $0.96^{+ 0.24}_{- 0.18}$ & N(0.17) & $20.998\pm0.018$ & $21.017\pm0.019$ & $-0.019\pm0.026$ & 0.111 \\
NGC4278 & 12h20m4.33s & 29\dg17$'$35.86$"$ & $38.74^{+ 0.07}_{- 0.07}$ & D & $0.69^{+ 0.13}_{- 0.12}$ & Y(2.38) & $ > 25.952$ & $ > 25.916$ & $-$ & $-$ \\
NGC4278 & 12h20m6.03s & 29\dg16$'$48.25$"$ & $38.95^{+ 0.05}_{- 0.05}$ & P & $0.64^{+ 0.11}_{- 0.10}$ & Y(3.27) & $ > 24.455$ & $ > 24.497$ & $-$ & $-$ \\
NGC4278 & 12h20m5.48s & 29\dg16$'$40.68$"$ & $38.77^{+ 0.06}_{- 0.07}$ & D & $0.79^{+ 0.15}_{- 0.13}$ & N(1.40) & $ > 24.990$ & $ > 25.018$ & $-$ & $-$ \\
NGC4278 & 12h20m6.79s & 29\dg16$'$56.01$"$ & $38.86^{+ 0.07}_{- 0.06}$ & P & $1.10^{+ 0.18}_{- 0.16}$ & N(0.63) & $ > 23.504$ & $ > 23.493$ & $-$ & $-$ \\
NGC4278 & 12h20m5.95s & 29\dg17$'$8.79$"$ & $38.32^{+ 0.13}_{- 0.16}$ & P & $-$ & $-$ & $22.014\pm0.047$ & $21.945\pm0.043$ & $0.069\pm0.064$ & 1.388 \\
NGC1427 & 3h42m18.71s & -35\dg22$'$40.02$"$ & $39.18^{+0.07}_{-0.08}$ & D & $-$ & $-$ & $22.450\pm0.022$ & $22.461\pm0.022$ & $-0.011\pm0.031$ & 0.163 \\
NGC1427 & 3h42m18.47s & -35\dg23$'$38.19$"$ & $39.17^{+0.09}_{-0.09}$ & D & $-$ & $-$ & $22.861\pm0.036$ & $23.147\pm0.052$ & $-0.286\pm0.063$ & 19.488 \\
\noalign{\smallskip}\hline
\end{tabular}
\ec
\tablecomments{1.15\textwidth}{(1) Host Galaxy Name; (2) Right Ascension; (3) Declination; (4) Log of unabsorbed X-ray luminosity in $\rm ergs~s^{-1}$ for first observation; (5) Best-fit Model; (6) Constant2; (7) X-ray variable (Y-Yes, N-No) and its significance; (8) Aperture corrected magnitude in the first observation; (9) Aperture corrected magnitude in the second observation; (10) The difference in magnitude; (11) Significance of the Optical variability. In the sample, two sources (3h38m33.09s, -35\dg27$'$31.53$"$ in NGC1399 and 12h30m47.15s, 12\dg24$'$15.91$"$ in NGC4486) have two possible optical counterparts. Hence we report the magnitude of each counterpart.}
\end{table}
\begin{table}
\bc
\begin{minipage}[]{150mm}
\caption[]{The Properties of Optically Varying Sources in the Sample\label{IR}}\end{minipage}
\setlength{\tabcolsep}{2.5pt}
\small
\begin{tabular}{cccccccccc}
\hline\noalign{\smallskip}
Galaxy & RA (J2000) & Dec (J2000) & log($L_{1}$) & ${\Delta m}$ & ${\chi}^2$ & $g-z$ & $F_{3.6\mu m}$ & $F_{5.8\mu m}$ & $F_{5.8}/F_{3.6}$\\
\hline\noalign{\smallskip}
NGC1399 & 3h38m31.86s & -35\dg26$'$49.26$"$ & $38.41^{+ 0.58}_{- 0.31}$ & $-0.268\pm0.024$ & $133.152$ & $1.048$ & $42.37\pm0.83$ & $25.00\pm1.98$ & $0.59$ \\
NGC1399 & 3h38m33.09s & -35\dg27$'$31.53$"$ & $38.61^{+ 0.12}_{- 0.08}$ & $0.085\pm0.016$ & $26.493$ & $1.307$ & $24.58\pm0.59$ & $32.19\pm2.10$ & $1.31$ \\
NGC1399 & 3h38m36.82s & -35\dg27$'$46.98$"$ & $38.72^{+ 0.10}_{- 0.11}$ & $0.384\pm0.055$ & $51.437$ & $1.842$ & $7.41\pm0.42$ & $< 5.38$ & $< 0.73$ \\
NGC1427 & 3h42m18.47s & -35\dg23$'$38.19$"$ & $39.17^{+ 0.09}_{- 0.09}$ & $-0.286\pm0.063$ & $19.488$ & $1.835$ & $<2.78$ & $<3.93$ & $-$ \\
\noalign{\smallskip}\hline
\end{tabular}
\ec
\tablecomments{1.12\textwidth}{(1) Host Galaxy Name; (2) Right Ascension; (3) Declination; (4) log of unabsorbed X-ray luminosity in $\rm ergs~s^{-1}$ for first observation; (5) The difference in magnitude; (6) Significance of the Optical variability; (7) Optical colour (g-z) derived from Vega magnitude; (8),(9) IR flux in mJy for the 3.6$\mu$m and 5.8$\mu$m bands; (10) Mid-IR flux ratio.}
\end{table}
\section{Discussion}
\label{sect:discussion}
In this work we have studied the long term X-ray and optical variability of X-ray sources in four nearby elliptical
galaxies. For the 46 sources in the sample, we have fitted their X-ray spectra using an absorbed power-law or black body model
for two {\it Chandra} observations and found that 24 of them show long term X-ray variability. For 34 sources,
we have identified optical counterparts. After estimating the systematic error on the photometric magnitude, we find
that four of the sources clearly exhibit long term optical variation. Since the optical counterpart is varying
it cannot be the integrated light of stars in a globular cluster. Thus, one
may expect that the optical variability is induced by the X-ray source. If that is so, these sources are important candidates
for further study.
The optically variable X-ray sources could be background Active Galactic Nuclei (AGN).
The reported optical colours ({\it g - z}) for the
sources in NGC1399 \citep{Sha13} are tabulated in Table \ref{IR} and they reveal that the objects are blue and one of them is
bluer than blue globular clusters, $1.3 < g-z < 1.9$ \citep{Pao11}. Indeed, the optically variable sources
(Source 1 and 2 in NGC1399) were identified as possible contaminants in an earlier analysis
\citep{Kun07}. The analysis of {\it HST/WFPC} data reveals that these sources are bluer than ${\it B-I=1.5}$ and hence are not
globular clusters. \cite {Bla12} studied the globular cluster systems in NGC1399 using the {\it HST/ACS g, V, I, z} and {\it H} bands.
In their study, the sources with $19.5 < I_{814} < 23.5$ and $0.5 < g_{475} - I_{814} < 1.6$ are classified as the globular clusters.
and the optically variable sources in NGC1399 (source 1 and 2) again do not satisfy their criteria. This may indicate that they
may be background AGN and indeed their IR colours also support this interpretation. Studies have shown that AGN have
flux ratios $> 0.63$ in the $5.8$ and $3.6 \mu m$ bands i.e. $F_{5.8}/F_{3.6} > 0.63$ \citep{Pol06,Lac04}.
\citet{Sha13} have looked for IR counterparts of X-ray sources in NGC1399 using {\it Spitzer} data. Their
quoted IR flux and ratios are tabulated in Table \ref{IR}. All four sources have IR flux ratios $ \geq 0.63$, indicating that
they maybe background AGN. Unfortunately these sources are not in the field of view
of the {\it Spitzer} $4.5$ and $8.0 \mu m$ images, which would have provided more information on the nature
of these sources.
We do not find evidence for any optical counterpart to disappear or flux changes by order of magnitude. Such variations
would be expected if the X-ray emission is due to a violent transient event like a very bright nova explosion or
a tidal disruption of a white dwarf by a black hole. Such transient events are expected to show dramatic variation in both
X-ray and optical flux. While there are several X-ray sources which are not detected in the other {\it Chandra} observation,
none of them exhibit dramatic variability in the optical. For example, as mentioned earlier, \citet{Irw10} have argued
that the lack of H$\alpha$ and H$\beta$ in the spectrum of a ULXs in NGC1399 (CXOJ033831.8-352604) may indicate the
tidal disruption of a white dwarf by a black hole. However, here we find that neither the X-ray nor the optical flux
show any long term variation.
Clearly, conclusive evidence on the nature of these sources can be obtained only by studying their
optical spectra and confirming by emission line studies whether a source is a background AGN or not. Such studies will also
provide clear information about the origin of
the optical source. Since this would require large telescopes in excellent seeing conditions, it is important to
choose good potential candidates such as the optically variable sources identified here. A positive identification of
a optically variable source as not being a background AGN,
would be the crucial step towards understanding these enigmatic sources.
\normalem
\begin{acknowledgements}
VJ, KJ, CDR, and BRSB thank the IUCAA visitors program and UGC Special assistance
program. VJ acknowledges financial support from the Council of Scientific and Industrial Research
(CSIR) through SRF scheme. This work has been partially funded from the ISRO-RESPOND program. PS
would like to thank the DST - FAST Track Scheme for research funding. The authors thank Phil
Charles for useful discussions.
\end{acknowledgements}
\bibliographystyle{raa}
|
3,212,635,538,000 | arxiv | \section{Introduction}
The demand for mobile robots has raised significantly due to their flexibility and the variety of use cases they can operate in. Tasks such as provision of components, transportation, commissioning or the work in hazardous environments are increasingly being executed by such robots \cite{smartfactory}, \cite{mobilerob}. A safe and reliable navigation is essential in operation of mobile robotics. Typically, the navigation stack consists of self-localization, mapping, global and local planning modules. Simultaneous Localization and Mapping (SLAM) is most commonly conducted as part of the navigation stack. It is used to create a map of the environment using its sensor observations upon which the further navigation relies on. However, this form of navigation depends on the preexisting map and its performance degrades at highly dynamic environments \cite{sun2017improving}, \cite{bahraini2019slam}. Furthermore, it requires an exploration step to generate a map which can be time consuming especially for large environments.\\
Deep Reinforcement Learning (DRL) emerged as a solid alternative to tackle this challenge of navigation within dynamic environments \cite{zeng2019navigation}. A trial and error approach lets the agent learn its behavior based purely on its observations. The training process is accelerated with neural networks and recent research showed remarkable results in mobile navigation. Yet, a main concern still lies in the safety aspect of navigation within human robot collaboration. Most ubiquitous are hand defined safety restrictions and measures which are non flexible and result in slow navigation. Higher level semantics bear the potential to enhance the safety of path planing by creating links the agent can reason about and consider for its navigation \cite{borkowski2010towards}. On this account, we propose a deep DRL local navigation system for autonomous navigation in unknown dynamic environments that works both in simulation and reality. To alleviate the problem of overfitting, we include highly random and dynamic components into our developed simulation engine called ARENA2D. For enhanced safety, we incorporate high level semantic information to learn safe navigation behavior for specific classes. The result is an end to end DRL local navigation system which learns to navigate and avoid dynamic obstacles based directly on visual observations. The robot is able to reason about safety distances and measures by itself based solely on its visual input. The main contributions of this work are following:
\begin{itemize}
\item Proposal of a reinforcement learning local navigation system for autonomous robot navigation based solely on visual input.
\item Proposal of an efficient 2D simulation environment - ARENA2D - to enable safe and generalizable navigation behavior.
\item Evaluation of the performance in terms of safety and robustness in highly dynamic environments.
\end{itemize}
The paper is structured as follows. Sec. II gives an overview of related work. Sec. III presents the conceptional design of our approach while sec. IV describes the implementation and training process. Sec. V demonstrates the results. Finally, Sec. VI gives a conclusion.
\section{RELATED WORK}
\subsection{Deep Reinforcement Learning for Navigation}
With the advent of powerful neural networks, deep reinforcement learning (DRL) mitigated the bottleneck of tedious policy acquisitions by accelerating the policy exploration phase using neural networks. Mhni et. al \cite{mnih2015human} first used neural networks to find an optimal policy for navigation behavior. They conducted high level sensory input and proposed an end-to-end policy learning system termed Deep-Q-Learning (DQN).
Bojarski et. al \cite{bojarski2016end} applied the same techniques for mobile robot navigation by proposing an end-to-end method that maps raw camera pixels directly to steering commands within a simulation environment and show the feasibility of a RL approach. Most recently, Pokle et al. \cite{pokle2019deep} presented a system for autonomous robot navigation using deep neural networks to map observed Lidar sensor data to actions for local planning.
Zeng et. al \cite{zeng2019navigation} proposed a DRL system to work in with unknown dynamic environments. The researchers include a gated recurrent unit to interact with the temporal space of observations and train the agent with moving obstacles in simulation. They show the feasibility and efficiency of the method in terms of navigation robustness and safety in dynamic environments. Nevertheless, the method is still limited to simulation. In contrast to that, our work proposes a system trained in unknown dynamic environments and additionally, transfers the algorithms into the real robot.
\subsection{Semantic Navigation}
Semantic navigation have been a research direction for many years.
Borowski et al. \cite{borkowski2010towards} introduces a path-planning algorithm after semantically segmenting the nearby objects in the robot’s environment. The researchers were able to show the feasibility and extract information about object classes to consider for mobile robot navigation.
Wang et al. \cite{wang2018visual} propose a three layer perception framework for achieving semantic navigation. The proposed network uses visual information from a RGB camera to recognize the semantic region the agent is currently located at and generate a map.
The work demonstrates that the robot can correct its position and orientation by recognizing current states from visual input. However, the dynamic elements such as presence of dynamic pedestrians are not taken into consideration.
Zhi et al. \cite{zhi2019learning} proposes a learning-based approach for semantic interpretation of visual scenes. The authors present an end-to-end vision-based exploration and mapping which builds semantics maps on which the navigation is based. One limitation of their method is the assumption of a perfect odometry, which is hard to achieve in real robots.
Zhu et al. \cite{zhu2017target} presented a semantic navigation framework where the agent was trained with 3D scenes containing real world objects. The researchers were able to transfer the algorithm towards the real robot which could navigate to specific real world objects like doors or tables. Furthermore, they showed the potential of using semantics for navigation without any hand crafted feature mapping but working solely on visual input. The training within the 3D environment, however, is resource-intensive and require a large amount of computational capacity.
Our work follows a more simplified way of incorporating semantic information. We include different object classes into our 2D simulation environment and train the agent with specific policies which should shape the behavior of the robot when encountering with the specific classes. Furthermore, we transfer the algorithms towards the real robot.
\section{CONCEPTUAL DESIGN}
We propose a DRL-based local navigation system which maps observed sensory input directly to robot actions for obstacle avoidance and safe navigation in dynamic environments. Furthermore, we aspire to explore the safety enhancements using semantic information. More specifically, rules based on detected nearby objects are defined and incorporated into the reward functions: the mobile robot have to keep a distance of 0.8 meters from detected humans and 0.3 meters from collaborating robots. The training is accelerated with neural networks and build upon a \textit{deep Q network} (DQN) which maps states to a so called Q values that each state action pair possess. Therefore, we employ the proposed DRL workflow described in \cite{10.5555/3279266}.
\subsection{Design of the Proposed Navigation System}
The general workflow of our system is illustrated in Fig. \ref{concept}. The agent is trained by the RL algorithm within simulation by analyzing states, actions and rewards.
For our use case, the states are represented by laser scan observations and object detections while actions are the possible robot movements. Within the training stage, these information are simulated in our simulation environment. Within deployment stage, a RGB camera with integrated object detection and a 360 degree 2D Lidar scanner of the Turtlebot3 deliver the required data.
Core component of our system is the neural network which optimizes the Q function.
We input the data into the neural network which maps the states-actions tuples to Q values with the maximal reward. Thus, the agent learns optimal behavior given a set of states and actions. In addition, we integrate several optimization modules to accelerate the training process even further. Finally, in the deployment stage, the RL model is deployed towards the real robot using a proposed deployment infrastructure consisting of a top down camera to detect the goal and surrounding objects. As a middle-ware for communication between all entities, ROS Kinetic on Ubuntu 16.04 is used.
\begin{figure}[]
\centering
\includegraphics[width=3.3in]{concept}
\caption{Design of the navigation system}
\label{concept}
\end{figure} For the deployment of the trained algorithms to the real robot, several challenges have to be considered. Unlike simulation, the robot does not know when the goal is reached. Therefore, we propose a solution using object detection and markers with the aforementioned global camera.
The problem of inaccurate and noisy laser scan data is considered within our simulation environment, where we include a module to add several levels of noise to the laser scan data. Furthermore, we included dynamic obstacles and randomness into the simulation to alleviate the differences between real and simulation environment.
\section{IMPLEMENTATION}
In the following chapter, each module of our proposed navigation system is presented in detail.
\subsection{Training Algorithm }
The basic training loop is based on the suggestions from \cite{10.5555/3279266} and employs deep Q-learning.
We rely on three major techniques: \textit{replay buffer} and \textit{epsilon greedy} to cope with the exploration and exploitation problem and \textit{target net} which we use to stabilize the training process in terms of robustness. To abstract the algorithm for our simulation environment, we split it into two parts: a simulation step, which interacts with the simulation environment, and a learning step, which is responsible for the refinement of the neural network model. The implementation of those steps is described by algorithms \ref{alg:PreStep} and \ref{alg:PostStep}.
\begin{algorithm}[]
\caption{PreStep}
\label{alg:PreStep}
\begin{algorithmic}
\IF{$\mbox{random}() < \epsilon$}
\STATE $a \leftarrow \mbox{random\_action}()$
\ELSE
\STATE $a \leftarrow arg\max\limits_{a}Q(s,a)$
\ENDIF
\STATE $(s', r) \leftarrow \mbox{simulation\_step}(a)$
\IF{episode is over}
\STATE $R.\mbox{insert}((s, s', a, r, \mbox{TRUE}))$
\STATE $s \leftarrow \mbox{simulation\_reset}()$
\ELSE
\STATE $R.\mbox{insert}((s, s', a, r, \mbox{FALSE}))$
\STATE $s \leftarrow s'$
\ENDIF
\STATE $t \leftarrow t + 1$
\end{algorithmic}
\end{algorithm} In the \textit{PreStep} algorithm, a random action is chosen with a probability of $\epsilon$. Otherwise, the action with the maximum Q Value, according to the current networks estimation, is retrieved. Using that action, a simulation step is performed, revealing the new state and the reward gained. The new state along with the previous state, reward, action and a flag indicating, whether the episode has ended or not, is stored in the replay buffer. \\
The actual training takes place in the \textit{PostStep} algorithm. Here a random batch is sampled from the replay buffer and the mean square error (MSE) loss is calculated using the Bellman equation.
Every $N_{\mbox{sync}}$ frames, weights from the network $Q$ are copied over to the target network $\hat{Q}$.
Using stochastic gradient descent (SGD) optimization, the weights of the network $Q$ are optimized according to the MSE loss $L$ calculated for every batch sample. Finally, the epsilon value is updated according to the current step $t$. Thereby, epsilon denotes the randomness of executed actions in each step which ensures an efficient trade off between exploration and exploitation.
\textit{PreStep} and \textit{PostStep} are called in a loop until the network converges.
\begin{algorithm}[]
\caption{PostStep}
\label{alg:PostStep}
\begin{algorithmic}
\STATE $B \leftarrow R.\mbox{random\_batch}(B_{\mbox{size}})$
\STATE $L \leftarrow \mbox{new Array}(B_{\mbox{size}})$
\FOR{$i$ in $B$}
\STATE $(s_i, s'_i, a_i, r_i, d_i) \leftarrow B[i]$
\IF{$d_i = \mbox{TRUE}$}
\STATE $y \leftarrow r_i$
\ELSE
\STATE $y \leftarrow r_i + \gamma \max\limits_{a}\hat{Q}(s'_i, a)$
\ENDIF
\STATE $L[i] \leftarrow (Q(s_i, a_i) - y)^2$
\ENDFOR
\STATE $Q.\mbox{optimize}(L)$
\IF{$t \bmod N_{\mbox{sync}} = 0$}
\STATE $Q_T \leftarrow Q$
\ENDIF
\STATE $\epsilon \leftarrow \max(\epsilon_{min}, 1-t / t_{max})$
\end{algorithmic}
\end{algorithm}
\subsection{Neural Network Design}
We use fully connected neural networks for our DQL sytem. Our model consists of 4 hidden and fully connected layers and is described in Fig. \ref{fc1}
\begin{figure}[]
\centering
\includegraphics[width=3.3in]{nn}
\caption{Architecture of fully connected neural network}
\label{fc1}
\end{figure}The input of the first fully connected layer represents the laser scan data and information about nearby humans or robots. 360 neurons as input values representing one value for each degree of the laser scanner.
The nearby dynamic objects are each represented with two neuron using their distance and angle to the robot. For simplicity reasons, we restrict this input to one object for each class human and robot.
After 2 dense layers and a dropout layer, the resulting output are 7 neurons denoting the robots actions. Adam is chosen as optimizer with an adaptive learning rate of 0.0025. As loss-function, Mean Squared Error (MSE) is chosen.
\subsection{Training Stage with ARENA2D }
We apply the presented methods in the training stage of the robot through simulations to generate a model which, subsequently, can be used for the real robot. Therefore, we developed a simple yet efficient 2D training environment - ARENA2D - with a large amount of built-in capabilities for performance enhancements and exploration of training settings. The simulation environment is depicted in Fig. \ref{GUI}. Within the simulation environment, we include static as well as dynamic components and create several stages with ascending difficulty. In total, we executed the training on 3 different stages which we denote as static, dynamic and semantic. The static stage contains only static obstacles whereas the dynamic
stage include moving obstacles. These two stages use a neural network which does not include the neurons indicating the distance of human and collaborating robot as additional
input. The semantic stage uses the network presented in Fig. \ref{fc1}. If the robot hits a wall or times out, it is reseted to the center of the stage. For human obstacles, we included an additional stop rate which lets the obstacle stop at random positions for a time of 2 seconds thus simulating the human behavior.
\begin{figure}[H]
\centering
\includegraphics[width=3.3in, height=3in]{gui}
\caption{Graphical user interface of the simulation environment ARENA2D}
\label{GUI}
\end{figure}
\subsubsection*{\textbf{Agent Definition}}
To simulate the real robot, we define an agent for simulation.
The agent is defined with the same parameters and output compared to the real robot in order to have as little differences as possible between simulation and reality. Therefore, we choose the \textit{Turtlebot3} due to its simplicity and compactness. It is equipped with a 360 degree laser scan sensor and offers input for further equipment e.g. an on board camera. The possible actions are listed in Table \ref{actions}.
\begin{table}[H]
\centering
\caption{Agent definition}
\begin{tabular}{lllll}
\hline
Action & Linear Velocity [m/s] & Angulat Velocity [rad/s] \\ \hline
Forward &0.15 & 0 \\
Backwards & -0.15 &0 \\
Stop & 0 &0 \\
Left & 0.15 & + 0.75 \\
Right & 0.15 &- 0.75 \\
Strong Left & 0.15 &+ +1.5 \\
Strong Right & 0.15 &- 1.5 \\
\hline
\end{tabular}
\label{actions}
\end{table}
\subsubsection*{\textbf{Rewards and Penalties}}
The rewards were exactly the same for all trainings. After each step, the agent will receive a reward based on the new state the robot is in. $\alpha$ denotes the angle between the robot and the goal. The rewards and penalties are listed in Table \ref{tablerew}.
\begin{table}[htbp]
\centering
\caption{Rewards and penalties for training}
\begin{tabular}{lllll}
\hline
Event & Description & Reward & Ep. Over\\
\hline
Goal reached & yes & $+100$ & yes\\
Moving towards goal & $|\alpha| \leq 30\si{\degree}$ & $+0.1$ & no\\
Wall hit & Robot hit wall & $-100$ & yes\\
Moving away from goal & $|\alpha| > 30\si{\degree}$ & $-0.2$ & no\\
Violate distance to human & $d < 0.7 m$ & $-10$ & no\\
Violate distance to robot & $d < 0.2 m$ & $-10$ & no\\
\hline
\end{tabular}
\label{tablerew}
\end{table}
\subsubsection*{\textbf{Hyperparameters}}
To determine the optimal hyperparameters, we conducted several training runs and adjusted the hyperparameters manually according to our literature research as well as experience. The optimal hyperparameters used for all further training runs are listed in Table \ref{tablehyper}.
\begin{table}[htbp]
\renewcommand{\arraystretch}{1.3}
\caption{Hyperparameters for training}
\begin{tabular}{ccp{4.1cm}}
\hline
Hyperparameter &Value& Explanation \\ \hline
Mean Success Bound & 1 & Training considered done if mean success rate reaches this value \\
Num Actions & 7 &Total number of discrete actions the robot can perform \\
Discount Factor & 0.99 & Discount factor for reward estimation (often denoted by gamma) \\
Sync Target Steps & 2000 &Target network is synchronized with current network every X steps \\
Learning Rate & 0.00025 &Learning rate for optimizer \\
Epsilon Start & 1 &Start value of epsilon \\
Epsilon Max Steps & $10^5$ &Steps until epsilon reaches minimum \\
Epsilon End & 0.05 &Minimum epsilon value \\
Batch Size & 64 &Batch size for training after every step \\
Training Start & 64 &Start training only after the first X steps \\
Memory Size & $10^6$ &Last X states will be stored in a buffer (memory), from which the batches are sampled \\
\hline
\end{tabular}
\label{tablehyper}
\end{table}
\subsection{Deployment on real robot}
Once the simulation was successful and the agent performs a safe navigation within all simulation environments, we deploy the algorithms towards the real robot to evaluate their feasibility within the real environment. Fig. \ref{re} illustrates the deployment setup with all entities used for conducting the experiments. Fiducial Aruco markers \cite{romero2018speeded} are included on the robot and the goal to verify the arrival at the target destination by comparing the position of both markers. Therefore, all entities are tracked with a global Intel Realsense D435 camera placed at the top of the real test environment. When both markers reach the same position, the system is informed that the destination is reached. The robot is equipped with an Intel Realsense camera as well, which delivers input for the human pose estimation module. The communication and signal workflow between all entities is explained in the following chapter. A variety of different obstacles were included, which are similar to the simulation as well as completely different. For static obstacles, chairs, round objects and boxes of different sizes and forms were used. Dynamic obstacles include other robots moving randomly and moving humans walking randomly across the environment and intersecting the path of the robot.
\begin{figure}[]
\centering
\includegraphics[width=3.3in, height=2.9in]{setup}
\caption{Deployment setup}
\label{re}
\end{figure}
\subsection{Pose Estimation}
For the object detection module, we utilized a pose estimation module working with RGB input based on SSPE \cite{tekin2018real}. Thus, the position and distance of humans or collaborating robots can be detected globally. Currently, the model is able to localize humans, the \textit{Kuka Youbot} and the \textit{Turtlebot} models \textit{Burger} and \textit{Waffle}. We fine-tune the model with a training on a human and robot RGB-dataset utilizing the pipeline proposed in our previous work \cite{kastner2020markerless}. The results are transmitted to the \textit{Observation Packer} node to be considered for the agent. Subsequently, the DRL algorithm will refine the trajectory of the robot.
The representing neuron for the distance of the agent to humans and other robots is initially set to a distance of 10 meters to make sure the neural network is assuming no nearby human or robot. Once a robot or human is detected and localized, the estimated position will be given as input to the neural network via the \textit{Pose Estimation} node.
\subsection*{Conducted Experiments}
We conduct several experiments with the model at different real environments to compare our method against the traditional local planer of the \textit{Turtlebot3} navigation package that uses a preexisting map of the environment with static obstacles and an algorithm without semantic rules. The setup of the experiments are illustrated in Fig. \ref{rese}
We tested different setups with static as well as dynamic components like moving humans and other robots and placed 10 different goal positions ranging from 0.2m to 2.5m distance. The start position of the robot was the same for every run. For each approach, we conducted 30 measurements consisting of 3 measurements for each of the 10 goals. If the robot could not reach its target within a time of 1 minute or due to a shut down of the \textit{Turtlebot3} navigation planners because no path could be calculated, we increased the failure count but conducted another measurement to ensure that each approach has the same number of measurements. The mentioned shut down happens, if the navigation package can not localize the robot and fail to generate a path due to a too distant or complex goal or sudden obstacles interfering which at times result in a shut down.
\begin{figure}[]
\centering
\includegraphics[width=3.3in]{real}
\caption{Test scenarios for the conducted experiments}
\label{rese}
\end{figure}
To explore the efficiency of the additional semantic rules, we compared the collision count of each of the approaches to reason about the safety of each approach.
\section{Results and Evaluation}
In the following, the results form our conducted experiments are presented.
The deployment of the models to the real robot was without any difficulties and we compare our approaches with the traditional local planer of the \textit{Turtlebot3} navigation package in terms of relevant metrics such as distance speed, error rate and safety of the navigation. The results are listed in Table \ref{tableres}. The distance metric is indicating the efficiency of the path planer and is conducted through the odometry topic. We measured the time each approach required to reach the goal. The safety rate is calculated with the total number of collision the robot had with static or dynamic obstacles while still reaching the goal. Robustness describes how many times the robot failed to reach the goal due to a failed path planning resulting in a navigation stack shut down or if the robot pursuits a completely wrong direction and were out of the arena. In total we placed 10 different goal positions and for each goal, 3 measurements were carried out for all approaches. If one run resulted in a failure, this was added to the error rate count and another measurement was conducted such that finally, there are 30 measurements for each approach to calculate the mean distances and speed. The error rate is calculated as the percentage of failed to successful runs. Table \ref{tableres} lists the the results for each approach.
\begin{table}[!h]
\centering
\caption{Comparison of navigation approaches }
\begin{tabular}{lllll}
\hline
Metric & Trad. &DRL Stat. & DRL Dyn. & DRL Sem. \\ \hline
Distance [m] & 4.72 & 3.71 & \textbf{4.1} & 5.28 \\
Speed [s] & 15.7 &\textbf{11.7} & 12.49 & 17.9 \\
Error Rate [\%] & 16.66 &10 & 3.33 & \textbf{0} \\
Obstacles hit &6.4 & 4.8 & 3.2 & \textbf{0} \\ \hline
\end{tabular}
\label{tableres}
\end{table}
It can be observed that our approaches outperform the traditional local planer of the robot in terms of speed and distance. Furthermore, our methods eliminate the need to generate a map which is necessary for the SLAM packages on which the global and local planner of the robot rely on.
The model trained with semantic information achieves the best performance in terms of obstacle avoidance and had no collisions in all our test runs. Although this comes at the cost of longer distances and speed because the robot will keep a larger distance when encountering a human sometimes driving backwards. The greater distance alleviates the high amount of collisions that were observed in our previous work where we mitigated the issue by training in highly dynamic environments. The additional semantic information enhances this effect even further as indicated in table \ref{tableres}.
Notably, the transfer of the simulated agent to the real environment did not cause major differences, even though in the simulation environment, only round obstacles were deployed as dynamic obstacles. However, our agent could generalize its behavior to all obstacles both static and dynamic, thus still managed to avoid the objects and keep a save distance to the human. For a more visual demonstration of our experiments we refer to our demonstration video which is available at https://youtu.be/KqHkqMqyStM.
\section{CONCLUSION}
We proposed an overall deep reinforcement learning based local navigation system for autonomous robot navigation within highly dynamic environments. Therefore, we developed a simple, yet efficient simulation engine from scratch which showed fast and efficient training speed and feasibility in transferring the models towards the real robot. Our navigation algorithm works solely on visual input and eliminates the need for any additional map. Furthermore, we explored the potential of semantic information by incorporating semantic classes such as human and robot and concluded safety enhancements for the navigation. This will be extended in our further work to include more classes such as long corridor, doors or restricted areas. For the deployment into the real environment a framework was proposed to integrate the DRL algorithms towards the real robot using marker detection and odometry data. Thereby, we ease the transferability of simulated models and enable a map-independent solution.
The results were remarkable both in static as well as dynamic environments and surpasses the traditional baseline RRT-PWA approach in terms of safety and robustness. For future work, we plan to incorporate more semantic classes such as long corridor, doors or restricted areas into the training environment to enhance safety and the overall performance even further. Additionally, an extension of our framework for more capabilities and features e.g. including more reinforcement learning algorithms, recurrent modules and continuous actions is planned.
\addtolength{\textheight}{-2cm}
\bibliographystyle{IEEEtran}
|
3,212,635,538,001 | arxiv | \section{Introduction}
Recently, many experiments have reported metallic properties of highly
conducting doped polymers (HCDP), \mbox{\it e.g.} polyacetylene doped with
iodine,\cite{Reghu} {\it p}-phenylenevinylene doped with sulfuric
acid,\cite{Ahlskog} etc. It is expected that HCDP shows
three-dimensional conductivity when polymer chains are entangled at
random, while quasi-one-dimensional conductivity is expected when they
are well aligned each other. Actually, there are some experiments
which have tried to examine the dimensionality of conduction of the
tensile drawn ($\sim \!1000 \%$) samples of HCDP films by the
measurement of magnetoresistance (MR) at low temperature.
\cite{Reghu,Ahlskog}In these experiments the conductivities were
anisotropic which were analyzed based on the formula for anisotropic
three-dimensional systems. However, the resulting anisotropy turned out
to be very large, which invalidate the original assumption of
anisotropic three-dimensionality, \mbox{i.e.} the closed Fermi surface
with the anisotropic mass. Instead, the results seem to indicate
that the Fermi surface is open for which there have been few
theoretical studies on
MR.\cite{Nakhmedov,Dupuis1,Dupuis2,Dorin,Cassam,Mauz}
In this paper, the weak field MR for such systems with open Fermi
surfaces are theoretically studied by use of the Wigner
representation.
The field theoretical studies of weak-localization (WL)
effects\cite{WLrev} on MR have discussed by Hikami \mbox{\it et
al. }\cite{Hikami} and Kawabata\cite{Kawabata} for two- and
three-dimensional metallic conductors, respectively. In these studies
where the closed Fermi surfaces are assumed the quantum corrections to
the conductivity given by the Cooperon propagators have been easily
calculated even in the presence of a magnetic field in terms of the
Landau quantization. For systems with open Fermi surfaces, on the other
hand, the eigenvalues of the Cooperon propagator can not be explicitly
given. In order to overcome this difficulty and to study MR
systematically we make use of the Wigner representation.
In \S 2 a brief review of the preceding theory for three-dimensional
systems is given, and studies of quasi-one-dimensional systems by the
Wigner representation are given in \S 3. The asymptotic forms of the
MR in three- and one-dimensional limit and summary are given in \S 4
and \S 5, respectively.
We take a unit of $\hbar = 1$.
\section{Magnetoresistance in Three-Dimensional Systems}
For three-dimensional systems, we take the model Hamiltonian,
\begin{equation}
{\cal H} = \frac{\bvec{p}^2}{2m} + u \sum_{l} \delta(\bvec{r} -\bvec{R}_l),
\end{equation}
where $u$ is the strength of the short range impurity potential and
$\bvec{R}_l$ is the impurity site. We will consider the quantum
correction term for the conductivity in the order of $(\varepsilon_{\mbox{\tiny F}}
\tau_{\mbox{\tiny{0}}})^{-1}$, where $\varepsilon_{\mbox{\tiny F}}$ is the Fermi energy and $\tau_{\mbox{\tiny{0}}}$ is the
relaxation time due to elastic scattering by impurities given in
\mbox{Fig.} \ref{tauz}. In this figure dashed lines and a cross represent
impurity potentials and the averaging procedure over the distribution of
impurities. This $\tau_{\mbox{\tiny{0}}}$ is given as follows,
\begin{equation}
\tau_{\mbox{\tiny{0}}}^{-1} = 2 \pi n_i u^2 N(0),
\end{equation}
where $n_i$ is the density of impurities and $N(0)$ is the density of
state per spin at the Fermi energy.
\begin{figure}[hbtp]
\begin{center}
\epsfile{file=figure1.eps,width=5cm}
\caption{Self-energy correction due to the impurity scattering.}
\label{tauz}
\end{center}
\end{figure}
\begin{figure}[hbtp]
\begin{center}
\epsfile{file=figure2.eps,width=5.5cm}
\caption{Weak-localization correction due to the ``Cooperon''.}
\label{WL}
\end{center}
\end{figure}
\begin{fullfigure}[hbtp]
\begin{center}
\epsfile{file=figure3.eps,width=12.5cm}
\caption{The Cooperon representing the quantum interference effect.}
\label{CooperonFig}
\end{center}
\end{fullfigure}
The weak-localization effect can be calculated by the summation of
so-called maximally crossed diagrams as given in \mbox{Fig.} \ref{WL}.
In these diagrams the ladder part (see \mbox{Fig.} \ref{CooperonFig})
which is called the ``Cooperon'' represents the quantum interference
effect between two electrons having nearly opposite wave number. The
Cooperon is singular when $\varepsilon_n (\varepsilon_n + \omega_l) < 0$ where
$\varepsilon_n = (2n+1)\pi k_{\rm B}T$, $\omega_l = 2l\pi k_{\rm B}T$ and
$k_{\rm B}$ is Boltzmann constant, and in this case it is written as
follows,
\begin{equation}
D_c(\bvec{q},\omega_l) = \frac{1}{2\pi N(0) \tau_{\mbox{\tiny{0}}}^2}
\frac{1}{D\bvec{q}^2 + |\omega_l| +1/\tau_\varepsilon},
\end{equation}
where $D=2\varepsilon_{\mbox{\tiny F}} \tau_{\mbox{\tiny{0}}}/3m$ is the diffusion constant and $\tau_\varepsilon$ is the
phase relaxation time due to inelastic scattering introduced
phenomenologically. Then the quantum correction to the conductivity
(\mbox{Fig.} \ref{WL}) is given by
\begin{equation}
\frac{\Delta \sigma}{\sigma_0} = -2 \ \tau_{\mbox{\tiny{0}}}^2 \ \mbox{\rm Tr} \ D_c(\bvec{q},0) ,
\end{equation}
where $\sigma_0 = 2 e^2 N(0) D $ is the Drude conductivity and Tr
means quantum mechanical trace, \mbox{\it e.g.} $\displaystyle{\sum_q}$ in
the absence of the magnetic field.
In the presence of a magnetic field, $H$, whose strength is not so
strong, in the sense $\omega_c \equiv eH/mc \ll \tau_{\mbox{\tiny{0}}}^{-1}$, its effects
can be treated quasiclassically, \mbox{i.e.} $\bvec{q}$ in the
Cooperon is replaced by $\bvec{q}+2e\bvec{A}/c \equiv \bvec{\pi}$,
where $\bvec{A}$ is a vector potential. Fortunately, the Cooperon
depends only on $\bvec{\pi}^2$, so that the trace can easily be
carried out by the use of the eigenstates of Landau
quantization. Hence, the quantum correction is given as
follows,\cite{Kawabata}
\begin{eqnarray}
\label{3dMR}
\frac{\Delta \sigma(H)}{\sigma_0}
& = & - \frac{1}{2\pi^3 N(0) \ell^2} \nonumber \\
& & \times \sum_{N}
\int {\rm d}q_z \frac{1}{\displaystyle \frac{4D}{\ell^2}\left (
N+\frac{1}{2}\right) + Dq_z^2 + 1/\tau_\varepsilon},
\end{eqnarray}
where $\ell = \sqrt{c/eH}$ is the Larmor radius. Equation
(\ref{3dMR}) is valid for both weak and strong magnetic field limit,
\mbox{i.e.}\ $\ell \gg L_{\vare} \equiv \sqrt{D \tau_\varepsilon}$\ and \ $ \ell \ll
L_{\vare}$, as long as the conditions, $\omega_c \ll \varepsilon_{\mbox{\tiny F}}$\ and \ $\ell \gg
\sqrt{D\tau_{\mbox{\tiny{0}}}}$, are satisfied. Especially for weak magnetic field, we
get the following asymptotic form of the magnetoconductance,
$\delta \sigma(H) \equiv \Delta\sigma(H) - \Delta\sigma(0)$,
\begin{equation}
\label{isotropic}
\frac{\delta \sigma(H)}{\sigma_0} = \frac{1}{24\pi^2 N(0)}
\frac{\sqrt{D} \tau_\varepsilon^{3/2}}{\ell^4} \ \propto \ H^2.
\end{equation}
If we assume the anisotropic mass, $m_i \ (i = x, y, z)$, the diffusion
constants are defined as $D_i = 2 \varepsilon_{\mbox{\tiny F}} \tau_{\mbox{\tiny{0}}} / 3 m_i$, and
$\delta \sigma(H)$ along the symmetry axis is rewritten as
\begin{equation}
\label{anisotropic}
\frac{\delta \sigma(H)}{\sigma_0} = \frac{1}{24\pi^2 N(0)}
\frac{D_1}{\sqrt{D_2}} \frac{\tau_\varepsilon^{3/2}}{\ell^4},
\end{equation}
where $D_1$ is the geometric mean of $D_i$s perpendicular to the
magnetic field and $D_2$ is the diffusion constant of the direction of
magnetic field.
\section{Magnetoresistance in Quasi-One-Dimensional Systems}
Now, we turn to our problem of quasi-one-dimensional systems with open
Fermi surfaces. We take the model Hamiltonian,
\begin{equation}
{\cal H} = \frac{p_{z}^2}{2m} - \alpha(\cos p_{x} d +
\cos p_{y} d) + u \sum_{l} \delta(\bvec{r} - \bvec{R}_l),
\end{equation}
where $z$-axis is the polymer chain axis, $\alpha$ is the band width
due to the transverse hopping of electrons among chains and $d$ is
the lattice spacing perpendicular to the chain direction. The
one-particle thermal Green function is given as
\fulltext
\begin{equation}
G(\bvec{k}, {\rm i}\varepsilon_n) =
\frac{1}{{\rm i}\varepsilon_n - \left[ k_z^2/2m -
\alpha(\cos k_x d + \cos k_y d) - \varepsilon_{\mbox{\tiny F}} \right] +
{\rm i\, sgn}(\varepsilon_n)/2 \tau_{\mbox{\tiny{0}}}}.
\end{equation}
If the Fermi energy $\varepsilon_{\mbox{\tiny F}}$ is large enough compared to the band width
in the perpendicular directions, $\alpha$, which is assumed throughout
this paper, and then the warping of the Fermi surface can be ignored
in the integration of a single particle Green function, the relaxation
time due to impurity scattering is given by
\begin{equation}
\tau_{\mbox{\tiny{0}}}^{-1} = \frac{2 n_i u^2}{d^2 v_{\mbox{\tiny F}}} ,
\end{equation}
where $v_{\mbox{\tiny F}} = \sqrt{2\varepsilon_{\mbox{\tiny F}}/m}$. On the other hand, the cosine band
structure has to be properly treated in the derivation of the Cooperon
as follows,
\begin{equation}
D_c(\bvec{q},\omega_l) = \frac{n_i u^2}{1 - n_i u^2 X(\bvec{q}, \omega_l)} ,
\end{equation}
\begin{eqnarray}
X(\bvec{q}, \omega_l) & = & \int\! \frac{{\rm d}\bvec{k}}{(2\pi)^3}
\ G(\bvec{k}, {\rm i}\varepsilon_n + {\rm i}\omega_l)
\ G(\bvec{q}-\bvec{k}, {\rm i}\varepsilon_n) \nonumber\\
& = & \int\! \frac{{\rm d}\bvec{k}}{(2\pi)^3}
\frac{1}{{\rm i}(\varepsilon_n+\omega_l)-
\left\{ k_z^2/2m - \alpha(\cos k_x d + \cos k_y d) - \varepsilon_{\mbox{\tiny F}}
\right\} + {\rm i}/2 \tau_{\mbox{\tiny{0}}}} \nonumber \\
& & \times \frac{1}{{\rm i}\varepsilon_n -
\left\{ (q_z-k_z)^2/2m - \alpha\left[
\cos(q_x-k_x)d + \cos(q_y-k_y)d\right] - \varepsilon_{\mbox{\tiny F}}
\right\} - {\rm i}/2 \tau_{\mbox{\tiny{0}}}},
\end{eqnarray}
where $X(\bvec{q}, \omega_l)$ is the polarization function. The result
of the integration with respect to $k_z$ is given as
follows under the conditions, $\varepsilon_{\mbox{\tiny F}} \gg \alpha$ and
$1 \gg \alpha\tau_{\mbox{\tiny{0}}}|\sin\frac{q_{x,y}d}{2}|$, \, $v_{\mbox{\tiny F}}^2\tau_{\mbox{\tiny{0}}}^2
q_z^2$, \, $|\omega_l|\tau_{\mbox{\tiny{0}}}$,
\begin{eqnarray}
X(\bvec{q},\omega_l) & \simeq & \frac{2}{v_{\mbox{\tiny F}}} \int\!
\frac{{\rm d}k_x {\rm d}k_y}{(2\pi)^2}
\frac{1}{\displaystyle{
\omega_l+\frac{1}{\tau_{\mbox{\tiny{0}}}}
+ v_{\mbox{\tiny F}}^2 \tau_{\mbox{\tiny{0}}} q_z^2 +
2{\rm i}\alpha\left[ \sin\frac{(2k_x-q_x)d}{2} \sin\frac{q_x d}{2} +
\sin\frac{(2k_y - q_y)d}{2} \sin\frac{q_y d}{2}\right] }} \nonumber\\
& \simeq & \frac{2 \tau_{\mbox{\tiny{0}}}}{v_{\mbox{\tiny F}} d^2}\left[
1 - 2\alpha^2 \tau_{\mbox{\tiny{0}}}^2
\left(\sin^2\frac{q_x d}{2}+\sin^2\frac{q_y d}{2} \right) -
v_{\mbox{\tiny F}}^2\tau_{\mbox{\tiny{0}}}^2 q_z^2 - |\omega_l|\tau_{\mbox{\tiny{0}}} \right] .
\end{eqnarray}
Then the Cooperon is obtained as\cite{Prigodin1,Prigodin2,Abrikosov}
\begin{equation}
\label{Cooperon}
D_c(\bvec{q},\omega_l) = \frac{d^2 v_{\mbox{\tiny F}}}{2 \tau_{\mbox{\tiny{0}}}^2}
\frac{1}{v_{\mbox{\tiny F}}^2 \tau_{\mbox{\tiny{0}}} q_z^2 + \alpha^2 \tau_{\mbox{\tiny{0}}} (2 - \cos q_x d - \cos q_y
d) + |\omega_l| + 1/\tau_\varepsilon}.
\end{equation}
\halftext
The quantum corrections to the conductivity (\mbox{Fig.} \ref{WL}) for
each direction under the same conditions as in the
derivation of the Cooperon, \mbox{eq.} (\ref{Cooperon}), are as follows,
\begin{subeqnarray}
\label{QCforQ1D}
\frac{\Delta \sigma_{\mbox{\tiny $\|$}}}{\sigma_{\mbox{\tiny $\|$}}} & = & -2 \ \tau_{\mbox{\tiny{0}}}^2 \ \mbox{\rm Tr} \ D_c(\bvec{q},0) , \\
\frac{\Delta \sigma_{\mbox{\tiny $\bot$}}}{\sigma_{\mbox{\tiny $\bot$}}} & = & -2 \ \tau_{\mbox{\tiny{0}}}^2 \ \mbox{\rm Tr} \cos{q_x d}
\ D_c(\bvec{q},0).
\end{subeqnarray}
In these equations, the classical conductivities
for each direction are given by
\begin{subeqnarray}
\sigma_{\mbox{\tiny $\|$}} & = & 2 e^2 N(0) D_{\mbox{\tiny $\|$}} , \\
\sigma_{\mbox{\tiny $\bot$}} & = & 2 e^2 N(0) D_{\mbox{\tiny $\bot$}} ,
\end{subeqnarray}
where the symbols $\|$ and $\bot$ represent the directions parallel
and perpendicular to the chain axis, respectively, which will be used
in the following as well. Here the density of state and the
diffusion constants are defined as follows,
\begin{subeqnarray}
\label{definition}
N(0) & = & \frac{1}{\pi d^2 v_{\mbox{\tiny F}}} ,\\
D_{\mbox{\tiny $\|$}} & = & v_{\mbox{\tiny F}}^2 \tau_{\mbox{\tiny{0}}} , \\
D_{\mbox{\tiny $\bot$}} & = & \frac{1}{2} \alpha^2 d^2 \tau_{\mbox{\tiny{0}}} ,
\end{subeqnarray}
which are deduced from \mbox{eq.} (\ref{Cooperon}) in the continuum limit,
\mbox{$d \rightarrow 0$}.
\begin{figure}[hbtp]
\begin{center}
\epsfile{file=figure4.eps,width=7cm}
\caption{Fermi surfaces in the cases, $\alpha \tau_{\mbox{\tiny{0}}}
\ll 1$, (a), and $\alpha \tau_{\mbox{\tiny{0}}} \gg 1$, (b). Here the vertical
lines represent the broadening of the Fermi surface corresponding to
the energy width, $\tau_{\mbox{\tiny{0}}}^{-1}$.}
\label{Fermi}
\end{center}
\end{figure}
In the limit, $\alpha \tau_{\mbox{\tiny{0}}} \ll 1$, where the warping of the Fermi
surface is less than the broadening, $\tau_{\mbox{\tiny{0}}}^{-1}$, (see \mbox{Fig.}
\ref{Fermi} (a)), the conditions, $1 \gg \alpha \tau_{\mbox{\tiny{0}}} |\sin
\frac{q_{x,y}d}{2}|$, are satisfied over the whole
Brillouin zone, hence any cutoff is not necessary in the integrations
with respect to $q_x$ and $q_y$ in the evaluations of \mbox{eq.}
(\ref{QCforQ1D}). On the other hand, in the limit, $\alpha\tau_{\mbox{\tiny{0}}} \gg
1$, where the warping of the Fermi surface is larger than the
broadening, $\tau_{\mbox{\tiny{0}}}^{-1}$,
(see \mbox{Fig.} \ref{Fermi} (b)), the conditions, $1 \gg
\alpha\tau_{\mbox{\tiny{0}}}|\sin\frac{q_{x,y}d}{2}|$, required to derive \mbox{eqs.}
(\ref{Cooperon}) and (\ref{QCforQ1D}) imply $|q_{x,y}| \ { < \kern -11.2pt \lower 4.3pt \hbox{$\displaystyle \sim$}} \
(\alpha \tau_{\mbox{\tiny{0}}} d)^{-1}$. In this case, however, the main contributions
to the quantum corrections, \mbox{eq.} (\ref{QCforQ1D}), turn out to
be given by the small $q$ such as $|q| \ { < \kern -11.2pt \lower 4.3pt \hbox{$\displaystyle \sim$}} \ (\alpha
\sqrt{\tau_{\mbox{\tiny{0}}}\tau_\varepsilon} d)^{-1}$ due to the lifetime of the Cooperon,
$\tau_\varepsilon$. Since $(\alpha \sqrt{\tau_{\mbox{\tiny{0}}}\tau_\varepsilon} d)^{-1} < (\alpha
\tau_{\mbox{\tiny{0}}} d)^{-1}$ is usually satisfied (\mbox{i.e.} $\tau_\varepsilon \gg \tau_{\mbox{\tiny{0}}}$),
the present estimations of the quantum corrections based on
\mbox{eqs.} (\ref{Cooperon}) and (\ref{QCforQ1D}) are justified even
in this case of $\alpha\tau_{\mbox{\tiny{0}}} \gg 1$.
To obtain the MR, we replace $\bvec{q}$ by
$\bvec{\pi}=\bvec{q}+2e\bvec{A}/c$ ,
\begin{subeqnarray}
\label{trace}
\frac{\Delta \sigma_{\mbox{\tiny $\|$}}}{\sigma_{\mbox{\tiny $\|$}}} = - {\rm Tr} \frac{d^2 v_{\mbox{\tiny F}}}{D_{\mbox{\tiny $\|$}}
\pi_z^2 + \alpha^2 \tau_{\mbox{\tiny{0}}} (2 - \cos \pi_x d - \cos \pi_y d) + 1/\tau_\varepsilon
}, \\
\frac{\Delta \sigma_{\mbox{\tiny $\bot$}}}{\sigma_{\mbox{\tiny $\bot$}}} = - {\rm Tr} \frac{d^2 v_{\mbox{\tiny F}} \cos \pi_x d}
{D_{\mbox{\tiny $\|$}} \pi_z^2 + \alpha^2 \tau_{\mbox{\tiny{0}}} (2 - \cos \pi_x d - \cos \pi_y d)
+ 1/\tau_\varepsilon}.
\end{subeqnarray}
Here we must be careful to treat $\pi_i$s because of their
noncommutability,
\begin{equation}
[ \pi_i , \pi_j ] = {\rm i} \frac{2}{\ell^2}\varepsilon_{ijk},
\end{equation}
where $k$ is the direction of magnetic field and $\varepsilon_{ijk}$ is
Levi-Civita's totally antisymmetric tensor. Since $\pi_i$s are
contained in cosine terms in our Cooperon, we cannot use the Landau
quantization method and it is impossible to study MR for arbitrary
field. However for studies in weak magnetic field, the method of
Wigner representation\cite{Kubo} is most suited, because it is
a systematic method of expanding physical quantities in terms of the
small parameter which is the value of the commutator of canonical
variables. Moreover as it turned out, the MR in a weak field yields
important information on the degree of the alignment of the polymer
and the phase relaxation time.
In the Wigner representation, the trace of some physical quantity,
$A(\hat{p},\hat{q})$, which is given as a function of
canonical variables, $\hat{p}$ and $\hat{q}$ satisfying $[\,\hat{p}\: ,
\:\hat{q}\,]=-{\rm i}c$, can be obtained by replacing quantum
operators to corresponding classical differential operators operating
on $1$, and integrating the quantity over classical variables, $p$ and
$q$,
\begin{equation}
\label{Wigner}
\mbox{\rm Tr} \ A(\hat{p},\hat{q}) = \frac{1}{2\pi c}\int\! {\rm d}p{\rm d}q
A\left(p+\frac{c}{2{\rm i}}\del{}{q} \:,\:
q-\frac{c}{2{\rm i}}\del{}{p}\right) \cdot 1 .
\end{equation}
\begin{figure}[hbtp]
\begin{center}
\epsfile{file=figure5.eps,width=7cm}
\caption{Five possible configurations of current and field with
respect to the chain direction in the measurement of MR.}
\label{configurations}
\end{center}
\end{figure}
In our case of MR in quasi-one-dimensional systems, two components of
$\hat{\pi}$ perpendicular to the magnetic field correspond to
$\hat{p}$ and $\hat{q}$ in \mbox{eq.} (\ref{Wigner}). For each of the
five possible configurations as shown in
\mbox{Fig.} \ref{configurations} we have to replace the operators as
follows,
\begin{equation}
\hat{\bvec{\pi}} \rightarrow \bvec{\pi}+
\frac{1}{{\rm i}\ell^2} \bvec{h} \times \del{}{\bvec{\pi}},
\end{equation}
where $\bvec{h}$ is the unit vector along the direction of magnetic
field, and integrate over $\bvec{\pi}$. For example the quantum
correction in the
\mbox{config.} (1) in Fig.~\ref{configurations}, we have to evaluate
the following,
\fulltext
\begin{equation}
\frac{\Delta \sigma_1}{\sigma_{\mbox{\tiny $\|$}}} = - \frac{d^2 v_{\mbox{\tiny F}} \tau_\varepsilon}{(2\pi)^3}
\int_0^{\infty}\!\!\!\!{\rm d}s \int\!{\rm d}^3\pi\ \mbox{\Large e}^{
-s \left\{ D_{\mbox{\tiny $\|$}} \tau_\varepsilon \pi_z^2 + \alpha^2 \tau_{\mbox{\tiny{0}}} \tau_\varepsilon \left[2 -
\cos \big(\pi_x-\frac{1}{{\rm i} \ell^2}\del{}{\pi_y}\big) d -
\cos \big(\pi_y+\frac{1}{{\rm i} \ell^2}\del{}{\pi_x}\big) d
\right] + 1 \right\}} \cdot 1,
\end{equation}
where the integrations with respect to $\pi_x$ and $\pi_y$ can be taken
over the whole Brillouin zone.
The explicit evaluations of the quantum corrections up to the second
order of $H$ for each configuration result in as follows,
\begin{equation}
\label{q1d}
\begin{array}{rlcclr}
{\displaystyle
\frac{\Delta \sigma_1}{\sigma_{\mbox{\tiny $\|$}}} } \ =
& \hspace{-2mm} {\displaystyle -
\frac{1}{2\sqrt{\pi}}\sqrt{\frac{\tau_\varepsilon}{\tau_{\mbox{\tiny{0}}}}}
\int_{0}^{\infty} {\rm d}s\
\mbox{\Large e}^{-s(4a+1)}
\ [ \ s^{\mbox{\tiny $-1/2$}}\ \I_{0}(2as)^2\ }
& \hspace{-2mm}
{\displaystyle
- \ \frac{2}{3} }
\hspace{-2mm} & \hspace{-2mm}
{\displaystyle
\left( \frac{L_{\vare \mbox{\tiny $\bot$}}^2}{\ell^2} \right)^2 }
\hspace{-2mm} & \hspace{-2mm}
{\displaystyle s^{\mbox{\tiny $3/2$}}\ \I_{1}(2as)^2\ } \hspace{-2mm}
& \hspace{-2mm} ], \\ \\
{\displaystyle \frac{\Delta \sigma_2}{\sigma_{\mbox{\tiny $\bot$}}}} \ =
& \hspace{-2mm}
{\displaystyle -
\frac{1}{2\sqrt{\pi}}\sqrt{\frac{\tau_\varepsilon}{\tau_{\mbox{\tiny{0}}}}}
\int_{0}^{\infty} {\rm d}s\
\mbox{\Large e}^{-s(4a+1)}
\ [\ s^{\mbox{\tiny $-1/2$}}\ \I_{0}(2as)\ \I_{1}(2as)\ }
& \hspace{-2mm}
{\displaystyle
- \ \frac{2}{3} }
\hspace{-2mm} & \hspace{-2mm}
{\displaystyle
\left( \frac{L_{\vare \mbox{\tiny $\bot$}}^2}{\ell^2} \right)^2 }
\hspace{-2mm} & \hspace{-2mm}
{\displaystyle
s^{\mbox{\tiny $3/2$}}\ \I_{0}(2as)\ \I_{1}(2as)\ } \hspace{-2mm} & \hspace{-2mm} ], \\ \\
{\displaystyle \frac{\Delta \sigma_3}{\sigma_{\mbox{\tiny $\|$}}}} \ =
& \hspace{-2mm}
{\displaystyle -
\frac{1}{2\sqrt{\pi}}\sqrt{\frac{\tau_\varepsilon}{\tau_{\mbox{\tiny{0}}}}}
\int_{0}^{\infty} {\rm d}s\
\mbox{\Large e}^{-s(4a+1)}
\ [ \ s^{\mbox{\tiny $-1/2$}}\ \I_{0}(2as)^2\ }
& \hspace{-2mm}
{\displaystyle
- \ \frac{2}{3}}
\hspace{-2mm} & \hspace{-2mm}
{\displaystyle
\left( \frac{\LeperpL_{\vare \mbox{\tiny $\|$}}}{\ell^2} \right)^2 }
\hspace{-2mm} & \hspace{-2mm}
{\displaystyle
s^{\mbox{\tiny $3/2$}}\ \I_{0}(2as)\ \I_{1}(2as)\ } \hspace{-2mm} & \hspace{-2mm} ], \\ \\
{\displaystyle \frac{\Delta \sigma_4}{\sigma_{\mbox{\tiny $\bot$}}}} \ =
& \hspace{-2mm}
{\displaystyle -
\frac{1}{2\sqrt{\pi}}\sqrt{\frac{\tau_\varepsilon}{\tau_{\mbox{\tiny{0}}}}}
\int_{0}^{\infty} {\rm d}s\
\mbox{\Large e}^{-s(4a+1)}
\ [ \ s^{\mbox{\tiny $-1/2$}}\ \I_{0}(2as)\ \I_{1}(2as)}
& \hspace{-2mm}
{\displaystyle
- \ \frac{2}{3} }
\hspace{-2mm} & \hspace{-2mm}
{\displaystyle
\left( \frac{\LeperpL_{\vare \mbox{\tiny $\|$}}}{\ell^2} \right)^2 }
\hspace{-2mm} & \hspace{-2mm}
{\displaystyle
s^{\mbox{\tiny $3/2$}}\ \I_{1}(2as)^2\ } \hspace{-2mm} & \hspace{-2mm} ], \\ \\
{\displaystyle \frac{\Delta \sigma_5}{\sigma_{\mbox{\tiny $\bot$}}}} \ =
& \hspace{-2mm}
{\displaystyle -
\frac{1}{2\sqrt{\pi}}\sqrt{\frac{\tau_\varepsilon}{\tau_{\mbox{\tiny{0}}}}}
\int_{0}^{\infty} {\rm d}s\
\mbox{\Large e}^{-s(4a+1)}
\ [ \ s^{\mbox{\tiny $-1/2$}}\ \I_{0}(2as)\ \I_{1}(2as)}
& \hspace{-2mm}
{\displaystyle
- \ \frac{2}{3} }
\hspace{-2mm} & \hspace{-2mm}
{\displaystyle
\left( \frac{\LeperpL_{\vare \mbox{\tiny $\|$}}}{\ell^2} \right)^2 }
\hspace{-2mm} & \hspace{-2mm}
{\displaystyle
s^{\mbox{\tiny $3/2$}}\ \I_{0}(2as)^2} \hspace{-2mm} & \hspace{-2mm} ], \\
\end{array}
\end{equation}
\halftext
where $\Delta \sigma_i$ is the quantum correction for the $i$-th
configuration in \mbox{Fig.} \ref{configurations}, $\I_{0}(z)$ and
$\I_{1}(z)$ are the modified Bessel functions, $L_{\vare \mbox{\tiny $\|$}} \equiv
\sqrt{D_{\mbox{\tiny $\|$}} \tau_\varepsilon}$ and $L_{\vare \mbox{\tiny $\bot$}} \equiv \sqrt{D_{\mbox{\tiny $\bot$}} \tau_\varepsilon}$ are the
phase relaxation lengths for each direction, and
\begin{equation}
a \equiv \frac{1}{2} \alpha^2 \tau_{\mbox{\tiny{0}}} \tau_\varepsilon = \left( \frac{L_{\vare \mbox{\tiny $\bot$}}}{d}
\right)^2
\end{equation}
is the ``dimensionality parameter'' whose meaning is discussed
below. In each of \mbox{eqs.} (\ref{q1d}), the first term in the
integral is the WL correction in the absence of the magnetic field,
$\Delta\sigma_i(0)$, and the second term is the magnetoconductance,
$\delta\sigma_i(H) \equiv \Delta\sigma_i(H) - \Delta\sigma_i(0)$. The
expansion parameters are $L_{\vare \mbox{\tiny $\bot$}}^2 / \ell^2$ for $H \| z$ and
$L_{\vare \mbox{\tiny $\bot$}} L_{\vare \mbox{\tiny $\|$}} / \ell^2$ for $H \bot z$, respectively. This is
easily understood because the magnetic field always affect electrons
through the orbital motion within the plane perpendicular to the
field.
The parameter, $a$, represents the dimensionality in the sense of the
quantum interference effects due to the Cooperon, and its physical
meaning is how many chains electrons can hop through with their
coherency kept. The interference of electrons is three-dimensional if
$a$ is large, $a \gg 1$, even though the Fermi surface is open because
electrons can move among many chains by diffusive motion until they
lose their phase memory. On the other hand, it is one-dimensional if
$a$ is small, $a
\ll 1$, since electrons cannot keep coherency even in a single
hopping.
\section{The Asymptotic Forms}
\fulltext
In this section, the asymptotic forms of the conductivity in three
and one dimensions are elucidated:
\subsection{Three-dimensional limit}
The three-dimensional limit, $a \gg 1$, of \mbox{eqs.} (\ref{q1d}) can
be obtained by using the asymptotic form of modified Bessel function,
and the results are as follows,
\begin{equation}
\label{3d}
\begin{array}{rlcclr}
{\displaystyle \frac{\Delta \sigma_1}{\sigma_{\mbox{\tiny $\|$}}} \ = }
& \hspace{-2mm}
{\displaystyle \ - \
\frac{1}{2\pi \alpha \tau_{\mbox{\tiny{0}}}} \ \left[ \ 1.61 \sqrt{\pi} \ - \
\frac{1}{\alpha \tau_{\mbox{\tiny{0}}}} \sqrt{ \frac{\tau_{\mbox{\tiny{0}}}}{\tau_\varepsilon}} \ \right] \ }
& \hspace{-2mm}
{\displaystyle
+ \ \frac{1}{24 \pi} \sqrt{ \frac{\tau_\varepsilon}{\tau_{\mbox{\tiny{0}}}}} \
\left( \frac{d L_{\vare \mbox{\tiny $\bot$}}}{\ell^2} \right)^2 }, \\ \\
{\displaystyle \frac{\Delta \sigma_2}{\sigma_{\mbox{\tiny $\bot$}}} \ = }
& \hspace{-2mm}
{\displaystyle \ - \
\frac{1}{2\pi \alpha \tau_{\mbox{\tiny{0}}}} \ \left[ \ 0.41 \sqrt{\pi} \ - \
\frac{1}{\alpha \tau_{\mbox{\tiny{0}}}} \sqrt{ \frac{\tau_{\mbox{\tiny{0}}}}{\tau_\varepsilon}} \ \right] \ }
& \hspace{-2mm}
{\displaystyle
+ \ \frac{1}{24 \pi} \sqrt{ \frac{\tau_\varepsilon}{\tau_{\mbox{\tiny{0}}}}} \
\left( \frac{d L_{\vare \mbox{\tiny $\bot$}}}{\ell^2} \right)^2 }, \\ \\
{\displaystyle \frac{\Delta \sigma_3}{\sigma_{\mbox{\tiny $\|$}}} \ = }
& \hspace{-2mm}
{\displaystyle \ - \
\frac{1}{2\pi \alpha \tau_{\mbox{\tiny{0}}}} \ \left[ \ 1.61 \sqrt{\pi} \ - \
\frac{1}{\alpha \tau_{\mbox{\tiny{0}}}} \sqrt{ \frac{\tau_{\mbox{\tiny{0}}}}{\tau_\varepsilon}} \ \right] \ }
& \hspace{-2mm}
{\displaystyle
+ \ \frac{1}{24 \pi} \sqrt{ \frac{\tau_\varepsilon}{\tau_{\mbox{\tiny{0}}}}} \
\left( \frac{d L_{\varepsilon \mbox{\tiny $\|$}}}{\ell^2} \right)^2 }, \\ \\
{\displaystyle \frac{\Delta \sigma_4}{\sigma_{\mbox{\tiny $\bot$}}} \ = }
& \hspace{-2mm}
{\displaystyle \ - \
\frac{1}{2\pi \alpha \tau_{\mbox{\tiny{0}}}} \ \left[ \ 0.41 \sqrt{\pi} \ - \
\frac{1}{\alpha \tau_{\mbox{\tiny{0}}}} \sqrt{ \frac{\tau_{\mbox{\tiny{0}}}}{\tau_\varepsilon}} \ \right] \ }
& \hspace{-2mm}
{\displaystyle
+ \ \frac{1}{24 \pi} \sqrt{ \frac{\tau_\varepsilon}{\tau_{\mbox{\tiny{0}}}}} \
\left( \frac{d L_{\vare \mbox{\tiny $\|$}}}{\ell^2} \right)^2 }, \\ \\
{\displaystyle \frac{\Delta \sigma_5}{\sigma_{\mbox{\tiny $\bot$}}} \ = }
& \hspace{-2mm}
{\displaystyle \ - \
\frac{1}{2\pi \alpha \tau_{\mbox{\tiny{0}}}} \ \left[ \ 0.41 \sqrt{\pi} \ - \
\frac{1}{\alpha \tau_{\mbox{\tiny{0}}}} \sqrt{ \frac{\tau_{\mbox{\tiny{0}}}}{\tau_\varepsilon}} \ \right] \ }
& \hspace{-2mm}
{\displaystyle
+ \ \frac{1}{24 \pi} \sqrt{ \frac{\tau_\varepsilon}{\tau_{\mbox{\tiny{0}}}}} \
\left( \frac{d L_{\vare \mbox{\tiny $\|$}}}{\ell^2} \right)^2 }. \\
\end{array}
\end{equation}
These are identical with the conclusions of preceding
theories of WL and weak field MR in three-dimensional systems,
\cite{WLrev,Hikami,Kawabata,Prigodin1,Prigodin2,Abrikosov}
with the density of state, $N(0)$, and the anisotropic tensor
components of the diffusion constants, $D_{\mbox{\tiny $\|$}}$ and $D_{\mbox{\tiny $\bot$}}$, as given
in \mbox{eq.} (\ref{definition}), \mbox{\it e.g.} the substitution of
them for \mbox{eq.} (\ref{anisotropic}) gives the second terms,
$\delta \sigma_i(H)$, of \mbox{eqs.} (\ref{3d}). This is expected
because in the limit, $\alpha^2 \tau_{\mbox{\tiny{0}}} \tau_\varepsilon \gg 1$, the main
contribution to the integration of the Cooperon is given by small $q$
such as $|q_x|, |q_y| \ { < \kern -11.2pt \lower 4.3pt \hbox{$\displaystyle \sim$}} \ (\alpha \sqrt{\tau_{\mbox{\tiny{0}}}\tau_\varepsilon} d)^{-1}$.
Therefore our formulae, \mbox{\it e.g.} \mbox{eqs.} (\ref{Cooperon}) and
(\ref{QCforQ1D}), turn out to be the same as those in anisotropic
three-dimensional systems shown in \S 2. This is the reason why the
quantum corrections of the systems with $\alpha\tau_{\mbox{\tiny{0}}} \gg 1$ are given
by those of the anisotropic three-dimensional systems even though the
Fermi surface is open, since $\alpha^2 \tau_{\mbox{\tiny{0}}} \tau_\varepsilon \gg 1$ because of
$\tau_\varepsilon \gg \tau_{\mbox{\tiny{0}}}$.
\subsection{One-dimensional limit}
When the system becomes one-dimensional, $a \ll 1$, the asymptotic
forms are given as
\begin{equation}
\label{1d}
\begin{array}{rlllll}
{\displaystyle \frac{\Delta \sigma_1}{\sigma_{\mbox{\tiny $\|$}}} \ = }
& \hspace{-1mm}
{\displaystyle \ - \
\frac{1}{2} \ \sqrt{ \frac{\tau_{\varepsilon}}{\tau_0}} \ }
& \hspace{-1mm}
{\displaystyle
+ \ \ \frac{35}{16} \ \sqrt{ \frac{\tau_{\varepsilon}}{\tau_0}} }
\hspace{-1mm} & {\displaystyle a^4} & \hspace{-3mm}
{\displaystyle
\left( \frac{d^2}{\ell^2} \right)^2 }, \\ \\
{\displaystyle \frac{\Delta \sigma_2}{\sigma_{\mbox{\tiny $\bot$}}} \ = }
& \hspace{-1mm}
{\displaystyle \ - \
\frac{1}{8} \ \sqrt{ \frac{\tau_{\varepsilon}}{\tau_0}} \ a \ }
& \hspace{-1mm}
{\displaystyle
+ \ \ \frac{5}{8} \ \sqrt{ \frac{\tau_{\varepsilon}}{\tau_0}} }
\hspace{-1mm} & {\displaystyle a^3} & \hspace{-3mm}
{\displaystyle
\left( \frac{d^2}{\ell^2} \right)^2 }, \\ \\
{\displaystyle \frac{\Delta \sigma_3}{\sigma_{\mbox{\tiny $\|$}}} \ = }
& \hspace{-1mm}
{\displaystyle \ - \
\frac{1}{2} \ \sqrt{ \frac{\tau_{\varepsilon}}{\tau_0}} \ }
& \hspace{-1mm}
{\displaystyle
+ \ \ \frac{5}{8} \ \sqrt{ \frac{\tau_{\varepsilon}}{\tau_0}} }
\hspace{-1mm} & {\displaystyle a^2} & \hspace{-3mm}
{\displaystyle
\left( \frac{d L_{\varepsilon \mbox{\tiny $\|$}}}{\ell^2} \right)^2 }, \\ \\
{\displaystyle \frac{\Delta \sigma_4}{\sigma_{\mbox{\tiny $\bot$}}} \ = }
& \hspace{-1mm}
{\displaystyle \ - \
\frac{1}{8} \ \sqrt{ \frac{\tau_{\varepsilon}}{\tau_0}} \ a \ }
& \hspace{-1mm}
{\displaystyle
+ \ \ \frac{35}{16} \ \sqrt{ \frac{\tau_{\varepsilon}}{\tau_0}} }
\hspace{-1mm} & {\displaystyle a^3} & \hspace{-3mm}
{\displaystyle
\left( \frac{d L_{\varepsilon \mbox{\tiny $\|$}}}{\ell^2} \right)^2 }, \\ \\
{\displaystyle \frac{\Delta \sigma_5}{\sigma_{\mbox{\tiny $\bot$}}} \ = }
& \hspace{-1mm}
{\displaystyle \ - \
\frac{1}{8} \ \sqrt{ \frac{\tau_{\varepsilon}}{\tau_0}} \ a \ }
& \hspace{-1mm}
{\displaystyle
+ \ \ \ \frac{1}{4} \ \sqrt{ \frac{\tau_{\varepsilon}}{\tau_0}} }
\hspace{-1mm} & {\displaystyle a} & \hspace{-3mm}
{\displaystyle
\left( \frac{d L_{\varepsilon \mbox{\tiny $\|$}}}{\ell^2} \right)^2 }. \\
\end{array}
\end{equation}
\halftext
As is easily seen, the second term of \mbox{config.} (1), $\delta
\sigma_1(H)$, will be reduced most rapidly as $a \rightarrow 0$, while
that of \mbox{config.} (5), $\delta \sigma_5(H)$, will remain larger
than the others.
Hence, one can infer the value of the dimensionality parameter, $a$,
experimentally by the comparison of the anisotropy of the
magnetoconductance, $\delta
\sigma(H)$. For example, the ratio of $\delta \sigma_3(H)$ and $\delta
\sigma_4(H)$ will give the value of $a$, yielding important
information about the degree of the alignment of polymer.
In addition, the temperature dependence of $a$ thus deduced gives
information on that of the phase relaxation time, $\tau_\varepsilon$.
\section{Summary}
We have developed a theory of weak field MR in quasi-one-dimensional
systems which have open Fermi surfaces. Even though the effects of
magnetic field on electrons with such open Fermi surface are not easy
to treat, the correct results in weak field regime have been
determined by use of the Wigner representation. It is to be noted
that this is a rare case in which the Wigner representation is applied
to a explicit calculation of the quantum transport phenomena.
We have obtained the asymptotic forms of conductivities in three- and
one-dimensional limit in the sense of the quantum interference effect.
We have pointed out that the dimensionality parameter, $a$, and thus
the degree of the alignment of polymers can be inferred by studying
the anisotropy of the magnetoconductance, $\delta \sigma(H)$, for five
possible configurations. Moreover, the temperature dependence of the
phase relaxation time can be deduced from that of the dimensionality
parameter.
In a more detailed comparison with the experiments, however, the
existence of the mutual interaction effects has to be taken into
account.\cite{e-e} The Coulomb interaction associated with the spin
Zeeman effect gives contributions to MR of the same order as the WL,
but its sign is opposite and the scaling fields are different;
\mbox{i.e.} $g\mu_{\rm B}H/k_{\rm B}T$ where $g$ is the Land\'e
$g$-factor and $\mu_{\rm B}$ is the Bohr magneton in the case of the
interaction effects while $L_{\vare \mbox{\tiny $\bot$}}^2 / \ell^2$ for $H \| z$ and
$\LeperpL_{\vare \mbox{\tiny $\|$}} / \ell^2$ for $H \bot z$, respectively, in the present
WL effects. Since $L_{\vare \mbox{\tiny $\bot$}} < L_{\vare \mbox{\tiny $\|$}}$ will be naturally satisfied, the
scaling field of the WL effects for $H \| z$ should be larger than
that for $H \bot z$, but the magnitude of these scaling fields
(especially that in the case of $H \| z$) relative to that of
interaction effects is not unique. In the case of \mbox{refs.}
\citen{Reghu} and \citen{Ahlskog}, the scaling field of the
interaction effects comes between those two of the WL effects and the
interaction effects are almost negligible for $H \bot z$, so that one
can infer the dimensionality parameter, $a$, adequately from $\delta
\sigma(H)$ of $H \bot z$.
\section*{Acknowledgments}
We would like to thank Dr.~Reghu Menon for drawing our interest to
refs.~1 and 2. \mbox{Y.~N.} thanks Hiroshi Kohno and Masakazu Murakami
for valuable discussions. We are indebted to Dr.~Achim Rosch who
kindly informed us of their related work.
|
3,212,635,538,002 | arxiv | \section{Introduction}\label{S_intro}
The solar corona can be observed in the white light, EUV, X, and radio wavelengths. Being the corona optically thin in these spectral ranges, its images are two-dimensional (2D) projections of the 3D emitting structure. Detailed knowledge of the 3D distribution of the fundamental plasma parameters of the solar corona ($\mathbf{B}$, $N_e$, $T_e$) is highly desirable to advance its modeling. Stereoscopy and tomography are powerful observational techniques of the corona, allowing to infer quantitative 3D information of it. An excellent introduction to solar stereoscopy can be found in \citet{inhester_2006}. A recent general review on both techniques covering all the spectral ranges listed above can be found in \citet{aschwanden_2011}, with a strong focus on stereoscopy. In this review we specifically focus on \emph{differential emission measure tomography} (DEMT) in a more extensive fashion, updating on all published work in the field at the moment of writing this article.
\citet{minnaert_1930} originally developed the scattering theory of the photospheric white light (WL) by the free electrons of the corona, that allows to infer the 3D distribution of the electron density of the corona from WL images. \citet{vandehulst_1950} were the first to perform a global corona reconstruction using eclipse images and assuming full azimuthal axi-symmetry, an assumption firstly relaxed by \citet{leblanc_1970}. It was \citet{altschuler_1972} who developed the first actual solar rotational tomography (SRT) using coronagraph data. A good review on WL SRT can be found in \citet{frazin_2000} and \citet{frazin_2002}, who developed a robust, regularized, positive method for tomographic inversion of the coronal density from time series of WL images. Later on, \citet{frazin_2005} first introduced the concept of DEMT, a technique which uses time series of EUV images to determine the 3D distribution of the coronal local-DEM (or LDEM).
DEMT was developed by \citet{frazin_2009}, and firstly applied by \citet{vasquez_2009} to study the 3D structure of coronal prominence cavities. The technique consists of two steps. In a first step the tomographic inversion of time series of full-sun EUV images is performed, to find the 3D distribution of the EUV emissivity in each filter band of the telescope. In a second step the emissivities found for all bands in any given coronal location are used as a constraint to infer the LDEM. Finally, moments of the LDEM are taken, as a result of which global 3D maps of the coronal electron density and temperature are produced.
In this review we summarize the main aspects and applications of the DEMT technique. Sections \ref{FBE} and \ref{LDEM} describe and illustrate the two steps of DEMT, section \ref{results} is a review of published results using the technique, and section \ref{conclusions} summarizes its main characteristics and future prospects for its development and application.
\section{The Tomographic Model of the Corona}\label{FBE}
To perform the EUV tomography, the inner corona volume in the {height range 1.00-1.25 $R_\odot$ is discretized on a 25$\times$90$\times$180 (radial $\times$ latitudinal $\times$ longitudinal) spherical grid. Due to optical depth issues (analyzed in detail in \citet{frazin_2009}) and EUV signal-to noise levels (which depend on the particular filter), the results are reliable typically in the height range from 1.03 to 1.20 $R_\odot$}.
For each filter band of the EUV telescope separately, time series of full-sun EUV coronal images covering a complete solar rotation are used to find the 3D distribution of an emissivity-type quantity known as the \emph{filter band emissivity} (FBE). The FBE of each EUV filter is the integral over wavelength of the coronal spectral emissivity multiplied by the passband of the filter. The intensity in each pixel is a line-of-sight integral of the FBE. The intensities of all pixels of all images can be arranged in a single very large column vector, as well as the FBE in every cell (or voxel) of the tomographic grid. In this way, both vectors are linearly related through a very large non-square sparse projection matrix, that depends on the geometry of the observations. Both the projection matrix and the pixel intensity vector are known, and the problem is to find the FBE vector. This poses a non-invertible linear problem for each band separately, which is the tomographic problem.
In the case of both the instrument \emph{Extreme ultraviolet Imaging Telescope} (EIT) on board the \emph{Solar and Heliospheric Observatory} (SoHO), and the instrument \emph{Extreme Ultra Violet Imager} (EUVI) on board the \emph{Solar TErrestrial RElations Observatory} (STEREO), the number of EUV bands that can be used for DEMT is 3. In the case of the \emph{Atmospheric Imaging Assembly} (AIA) on board the \emph{Solar Dynamics Observatory} (SDO) the number of bands is increased to 6.
The 3D distribution of the FBE is found by solving a global optimization problem, and the FBE distribution that best reproduces all intensities is determined. A thorough technical explanation of all aspects of the inversion can be found in \citet{frazin_2009}, and discussions on the uncertainties involved can be found in \citet{vasquez_2009, vasquez_2010, vasquez_2011}.
For the EUVI instrument, Figure \ref{tomography} shows a summary of the EUV tomography step. The first column shows (from top to bottom) EUVI images in the 171, 195 and 284 \AA\ bands. These images are just one sample from the time series actually used to perform the tomography. For each band, the second through fourth columns show projected spherical cuts of the tomographic FBE at 1.035, 1.085, and 1.135 $R_\odot$, respectively. The last column shows the respective synthetic images calculated by integrating the tomographic models along the line-of-sight. The black streaks seen in the reconstructions near some of the active regions are artifacts caused by the Sun's temporal variability.
\begin{figure}[!ht]
\label{tomography}
\begin{center}
\includegraphics[width=\linewidth]{frame.050.test.eps}
\end{center}
\caption{A summary of the EUV tomography. First column shows (from top to bottom) EUVI-A images in the 171, 195 and 284 \AA\ bands taken near 08:00 on 28 April 2008. For each band, the second through fourth columns show projected spherical cuts of the tomographic FBE at 1.035, 1.085, and 1.135 $R_\odot$, respectively. The last column shows the respective synthetic images calculated by integrating the tomographic models along the line-of-sight. The black streaks seen in the reconstructions near some of the active regions are artifacts caused by the Sun's temporal variability. From \citet{vasquez_2009}.}
\end{figure}
To evaluate the accuracy of the tomographic model, the synthetic images can be quantitatively compared to the corresponding data images. An example of this is shown in Figure \ref{CompareImages} from a tomographic reconstruction of the solar corona for the bands 171, 193, 211, and 335 \AA\ (from tom to bottom) of the AIA telescope.
\begin{figure}[!ht]
\label{CompareImages}
\begin{center}
\includegraphics[height=0.28\linewidth]{Images_AIA171_L05_CR2099_CL180.eps}
\includegraphics[height=0.28\linewidth]{Images_AIA171_L05_CR2099_CL180.gif_stats.eps}\\
\includegraphics[height=0.28\linewidth]{Images_AIA193_L05_CR2099_CL180.eps}
\includegraphics[height=0.28\linewidth]{Images_AIA193_L05_CR2099_CL180.gif_stats.eps}\\
\includegraphics[height=0.28\linewidth]{Images_AIA211_L05_CR2099_CL180.eps}
\includegraphics[height=0.28\linewidth]{Images_AIA211_L05_CR2099_CL180.gif_stats.eps}\\
\includegraphics[height=0.28\linewidth]{Images_AIA335_L05_CR2099_CL180.eps}
\includegraphics[height=0.28\linewidth]{Images_AIA335_L05_CR2099_CL180.gif_stats.eps}\\
\end{center}
\caption{Comparison of synthetic data derived from the tomographic model against observed data for the bands 171, 193, 211, and 335 \AA\ (from tom to bottom) of the AIA telescope. Left panels: data images and corresponding synthetic images computed by LOS-integration of the tomographic model. Right panel: frequency histogram of the synthetic to observed intensity ratio for every corresponding pair of pixels in the two images.}
\end{figure}
For each pair of images, the relative difference between the synthetic and observed values is below 0.1, 0.2 and 0.3 for 34, 59 and 75\% of the pixels, respectively, $\pm 4\%$ depending on the band. The same level of agreement holds for off-limb or on-disk pixels considered separately. The tomographic model provides then a quite detailed reliable description of the average global corona during the reconstructed period. The black rings in the images shown in Figure \ref{CompareImages} correspond to pixels with projected radius in the range 0.98 to 1.025 $R_\odot$. This near-limb data is not actually used for the tomographic inversion, as the emission along their corresponding line-of-sights can affected by optically thick emission \citep{frazin_2009}.
EUV tomography can currently only be applied from only one or two (in the STEREO era) point-of-view. With such limited simultaneous information the temporal resolution of the technique is of the order of half solar rotation (or about two weeks). Of course, this is the most important limitation of the technique, which is then suitable for studying structures that are stable during their observed transit.
\section{The 3D Distribution of the DEM}\label{LDEM}
Once the tomographic step is completed, the FBE of all EUV bands is known at each tomographic computational cell. Within each tomographic voxel the plasma is expected to be multi-thermal. The LDEM is a measure of the thermal distribution within the voxel.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.7\linewidth]{qkl_aia_euvi_v3.eps}
\end{center}
\caption{Temperature response function of all EUVI coronal bands (dashed, 171, 195, 284 \AA), and 4 AIA coronal bands (solid, 171, 193, 211, 335 \AA). From \citet{nuevo_2015}.}
\label{fqkl}
\end{figure}
As each FBE represents emission at a different temperature, they provide constraints on the LDEM. Using the passband function of each band of the EUV telescopes, and the atomic database CHIANTI version 7.1 to model of coronal emissivity, Figure \ref{fqkl} shows the \emph{temperature response function} (TRF) of all EUVI and 4 AIA coronal bands. The temperature range modeled by the LDEM is determined by the temperature range of sensitivity of the bands that are used. This can be estimated from the FWHM of the TRFSs around their respective main peaks. For EUVI and the 3 lower temperature bands of AIA, the temperature range is from about 0.5 und up to about 3.0 MK. In the case of AIA, this range is expanded up to $\sim 4$ MK when adding the 335 \AA\ band (see \citet{nuevo_2015}).
Due to the limited number of available data points (FBE values), and also to the narrow band nature of the TRF of each band, the inversion of the LDEM function is under determined and implemented by modeling it with a family of functions depending on a vector of a small number of parameters (typically 3 to 5). When DEMT is based on 3 bands, such as provided by the EUVI telescope, the LDEM is modeled by a single normal function. With the AIA telescope, more coronal bands can be used (up to 6), and the LDEM can be modeled with combinations of normal functions. The reader is referred to \citet{nuevo_2015} for a detailed study on parametric models of the LDEM using both telescopes.
In each tomographic cell independently, the problem consists of finding the values of the parameter vector to best predict with the LDEM the tomographic values of all FBEs in that cell. To do so an objective function is defined, that measures the quadratic differences between tomographically determined FBEs and the those synthesized from the modeled LDEM.
\begin{figure}[!ht]
\begin{center}
\includegraphics[height=0.29\linewidth]{Ne_1.075Rs.eps}
\includegraphics[height=0.29\linewidth]{LDEM.eps}
\end{center}
\caption{\emph{Left:} Latitude-Longitude map of the tomographic electron density $N_e$ in units of $10^8 {\rm cm^{-3}}$, at a height 1.075 R$_\odot$, for the period CR-2069. \emph{Right:} LDEM at the tomographic voxels I, II, III and IV indicated in the left panel. Reproduced from \citet{frazin_2009}.}
\label{DEMT}
\end{figure}
Once the LDEM is determined, the average electron density in the tomographic cell can be computed from its zeroth moment, specifically as the square root of the integral of the LDEM over temperature. As an example, the left panel in Figure \ref{DEMT} shows a latitude-longitude map of a spherical shell of the electron density obtained at one sample height of the tomographic grid, for Carrington rotation (CR-)2069, a period of very low magnetic activity. At 4 tomographic voxels indicated as I, II, III, and IV in the left panel, the right panel shows the normal LDEM model that is found. The error bars represent the uncertainty due to the regularization level of the tomographic reconstructions. The first moment of the LDEM, divided by its zeroth moment, allows computation of the mean electron temperature predicted by DEMT. Examples of spherical shells of the electron density and mean temperature at two different heights of another tomographic reconstruction are reproduced in Figure \ref{solmin}. Contour levels of the magnetic-strength $B$ of a \emph{potential field source surface} (PFSS) model are over-plotted, along with the magnetically open/closed boundary (see caption in Figure \ref{solmin}).
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.49\linewidth]{Ne_1.035_cr2077-euviB_l1.0.eps}
\includegraphics[width=0.49\linewidth]{Ne_1.075_euvi-b_cr2077.eps}\\
\includegraphics[width=0.49\linewidth]{Tm_1.035_cr2077-euviB_l1.0.eps}
\includegraphics[width=0.49\linewidth]{Tm_1.075_euvi-b_cr2077.eps}\\
\end{center}
\caption{An example of the 3D reconstruction of the thermodynamical state of the solar minimum corona, during the period CR-2077. Latitude-longitude maps of the electron density (top) and mean electron temperature (bottom) at heights 1.035 (left) and 1.075 $R_\odot$ (right), as derived with the DEMT technique. Solid-thin curves are magnetic strength $B$ contour levels of a PFSS model of the coronal magnetic field, with white (black) representing outward (inward) oriented magnetic field. The solid-thick black curves indicate the location of the magnetically open/closed boundary.}
\label{solmin}
\end{figure}
\section{Results}\label{results}
Applications of DEMT have included the observational study of coronal structures, the use of tomographic results to validate coronal models, and its combination with coronal extrapolations of the photospheric magnetic field. Following we summarize all peer-reviewed published work on DEMT.
\citet{vasquez_2009} produced the first observational 3D analysis of stable coronal prominence cavities, measuring the density and temperature contrast between the plasmas in the cavity and in the surrounding helmet streamer. As it is characteristic of tomography, their study did not require any ad-hoc modeling, as needed in the forward modeling approach.
Being suited to study coronal structures that are stable over half solar rotational time, tomography works best at solar minimum. \citet{vasquez_2010, vasquez_2011} analyzed the global thermodynamical of the solar corona during the minimum of activity between solar cycles 23 and 24, and discussed their results in relation to the fast and slow components of the solar wind. These works also include comparisons to other observational non-tomographic studies of the same periods, providing cross validation results for the technique. As an example of a 3D reconstruction of the solar minimum, Figure \ref{solmin} displays latitude-longitude maps of the electron density and mean temperature, derived with the DEMT technique for the period CR-2077. At both heights, the location of the open/closed boundary of the PFSS model is characterized by a very high transverse gradient in both the electron density and the mean temperature maps derived from the DEMT analysis.
\begin{figure}[!ht]
\begin{center}
\includegraphics[height=0.29\linewidth]{fig6c.eps}
\includegraphics[height=0.29\linewidth]{tendencia.eps}
\end{center}
\caption{\emph{Left:} Location of loops with negative (down) and positive (up) temperature gradient with height, in blue and red, respectively, for CR-2077. \emph{Right:} Evolution of the fraction of down (up) loops indicated as light-blue (red) diamonds, and the sunspot monthly number divided by 32.1 (dark-blue squares). The correlation between the fraction of down loops and the sunspot number is indicated. Adapted from \citet{nuevo_2013}.}
\label{updown}
\end{figure}
DEMT can be combined with the global PFSS magnetic models of the solar corona, an approach dubbed the \emph{Michigan Loop Diagnostic Technique} (MLDT). DEMT results are traced along the field lines of the magnetic model, allowing study of the thermodynamical properties of magnetic flux tubes in the quiet sun in a statistical fashion. \citet{huang_2012} applied MLDT to study one rotation during the last solar minimum. They found the ubiquitous presence of magnetic loops with downward gradients of temperature dubbed \emph{down loops}. Down loops were found to be dominant in the latitude range $\pm30^\circ$ (left panel in Fig. \ref{updown}). \citet{nuevo_2013} extended the study to a sequence of rotations that included the solar minimum. Their study revealed a clear anti-correlation between the global coronal activity level and the number of down loops present in the corona (right panel in Fig. \ref{updown}). They found that down and up loops are characterized $\beta\approx 1$ and $\beta<1$, respectively, and proposed an interpretation of their results in terms of Alfv\'en wave damping.
DEMT results have been also used to constrain and validate MHD models. Reconstructions of the electron density and temperature have been used as a constraint to 3D MHD solar wind models coupled to the Space Weather Modeling Framework (SWMF) code suite, that solves for the different electron and proton temperatures \citep{vanderholst_2010}. Improvements on the performance of the models include a more accurate prediction of the occurrence and density of co-rotating interaction regions in the heliosphere.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.44\linewidth,angle=-90]{fig3_2.eps}
\includegraphics[width=0.44\linewidth,angle=-90]{fig3_1.eps}
\end{center}
\caption{DEMT validation of the electron density (left) and temperature (right) of a
two-temperature MHD model of the solar corona, for the period CR-2077. The color rings spanning the height range 1.0 to 1.25 $R_\odot$ show, in a sample meridional cut of the corona, the MHD to DEMT ratio of the respective quantity. The outer greyscale maps ($r> 1.25\, R_\odot$) displays the corresponding MHD results. Reproduced from \citet{jin_2012}.}
\label{validation}
\end{figure}
Global two-temperature models of the corona and inner heliosphere have been validated in the inner corona with DEMT reconstructions of the electronic plasma parameters \citep{jin_2012}, with most of the model outputs fitting the observations very well, as seen in Figure \ref{validation}. The 3D products of DEMT have also been used as a validation tool for a study on coronal heating by surface Alfv\'en wave damping, implemented within the MHD model of the solar wind in the SWMF \citep{evans_2012}. DEMT results have been recently used as a validation tool in a study of the charge state composition of the slow solar wind derived from an ionization evolution code coupled to the wave-driven MHD solar wind model of the SWMF \citep{oran_2015}. DEMT allowed detailed validation of the latitudinal transition of the electronic plasma parameters in the open field lines surrounding the coronal equatorial streamer belt (Figure \ref{validation_2}).
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.7\linewidth]{TempVsAngularDistanceNorth.eps}
\end{center}
\caption{DEMT validation of a 3D wave-driven MHD model of the solar wind.
Electron temperature versus angular distance from the magnetically open/closed boundary (averaged over all longitudes) for the period CR-2063, derived from the MHD model (red) and from the DEMT inversion (black). Reproduced from \citet{oran_2015}.}
\label{validation_2}
\end{figure}
All DEMT studies reviewed so far were based on EUVI data, having 3 EUV bands with a sensitivity range $\sim 0.60 - 2.70$ MK. More recently, \citet{nuevo_2013_b,nuevo_2015} extended the DEMT technique to use the 4 cooler AIA bands (aimed at studying the quiet sun), sensitive to the range $\sim 0.55 - 3.75$ MK. Their study corresponds to CR-2099, a rotation of the rising phase of the current solar cycle 24. While in previous studies the LDEM was always modeled by a single normal distribution, the extra AIA filter, and the increased sensitivity range, allowed exploration of new parametric LDEM models. Using 4 bands, the model that consistently achieves the best predicted-to-reconstructed FBEs is a bimodal distribution, being a superposition of two normal distributions with distinct cool and hot components. The mean centroids of the two components in the quiet diffuse corona are found to be ${\rm log_{10}}\left<T_{0,1}\right>=6.15$ and ${\rm log_{10}}\left<T_{0,2}\right>=6.42$, values that are very consistent with independent determinations of the characteristic temperatures of the solar corona, as discussed in the same study.
The square electron densities of the two components are $N_{e,1}^2$ and $N_{e,2}^2$, respectively, so that the total square electron density of the LDEM is $N_e^2=N_{e,1}^2+N_{e,1}^2$. A measure of the bi-modality is then the ratio $\left(N_{e,2}/N_{e}\right)^2$. The study of this ratio throughout the diffuse quiet corona reveals that the bimodality of the LDEM is ubiquitous, and that it is stronger for denser regions, as shown by the left panel of Figure \ref{bimodal}.
\citet{nuevo_2015} also validate the LDEM inversion technique by applying it to standard 2D DEM studies. Examples of the bimodal DEMs are shown in the right panel of Figure \ref{bimodal}. The DEMT study shows that LDEM of the quiet corona is bimodal at the spatial resolution of the tomographic grid, which is $0.01 \ R_\odot \times 2^\circ \times 2^\circ$, or about $(7\times10^3 {\rm km}) \times [(2.44\times10^4 {\rm km})^2]$ for a representative voxel at a height of 0.1 $R_\odot$ above the photosphere at the equator. The authors argue that the nanoflare heating scenario is less likely to explain these results, and that alternative mechanisms, such as wave dissipation appear better supported by them.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.49\textwidth]{scatter_Nej_vs_Ne_gauss2FW.1.075Rsun.eps}
\includegraphics[width=0.49\linewidth]{DEMs_AR_DEM_new.eps}
\end{center}
\caption{\emph{Left:} Scatter plot of the measure of the bi-modality $\left(N_{e,2}/N_{e}\right)^2$ (see text) versus $N_e$ derived with AIA, at 1.075 $R_\odot$. \emph{Right:} Bi-modal DEM curves obtained for selected regions within and around an active region. Red/yellow curves correspond to hotter and denser regions, blue/dark curves correspond to colder and less dense regions. Reproduced from \citet{nuevo_2015}.}
\label{bimodal}
\end{figure}
\section{Concluding Remarks and Future Prospects}\label{conclusions}
DEMT provides a quantitative average description of the solar corona over a full solar rotation with the following main characteristics,
\begin{itemize}
\item The products are 3D maps of: a) the filter band emissivity in each EUV band, b) the local-DEM, and c) its moments, such as $\left<N_e^2\right>$ and $\left<T_e\right>$.
\item The temporal resolution of the technique is currently limited to the transit time of coronal features, i.e. $\sim 14$ days. DEMT reconstructions are then reliable descriptions of slowly evolving coronal structures, such as the diffuse quiet corona, and coronal holes. DEMT is not suited to study fastly evolving ARs.
\item The spatial resolution depends on the computational grid size, which in turn is constrained by the cadence of the image series. Typically, one image every 6 hrs is used, a time over which the Sun rotates about $3.3^\circ$. The adopted tomographic cell size is then $2^\circ$ in both angular directions and $10^{-2}\,{\rm R_\odot}$ in the radial direction.
\item The image processing, tomographic inversion, and DEM determination, are fully automated tasks, with little user interaction.
\item Its implementation does not require any ad-hoc modeling.
\item Finite FOV and finite computational grid effects, issues of relevance for white light tomography, are not important for EUV tomography.
\end{itemize}
The current implementation of the DEMT and MLDT techniques involves a suite of codes written in the C and IDL programming languages, which can be efficiently ran in nowadays desktop multi-core computers. Future immediate planned applications of DEMT and MLDT involve comparative studies of the last two solar minima (using the EIT and EUVI instruments), as well as the study of the coronal radiative losses predicted by MLDT in the quiet sun.
Tomographic inversion of time series of full-sun images is the only available observational technique that can provide global constraint and validation to large scale MHD modeling of the corona and the solar wind. As such, DEMT is a highly valuable tool to help in the continued development of global coronal models. DEMT could provide highly valuable 3D maps of the detailed temperature distribution of the coronal plasma if \emph{full-sun} spectral images (of the type provided by the EIS instrument) become available. It would be highly desirable that such an instrument will be operational during the a solar minimum period, when single point-of-view tomography is most fruitful.
The authors thanks CONICET grant PIP IU Nro 11420100100151 to IAFE that has funded this research. The author also thanks the collaboration and useful comments by Federico A. Nuevo and Richard A. Frazin.
|
3,212,635,538,003 | arxiv |
\section{Introduction}
\label{sec::introduction}
As a common phenomenon in our daily life, shadows in natural images provide hints for extracting scene geometry~\cite{okabe2009attached,karsch2011rendering}, light direction~\cite{lalonde2009estimating}, and camera location and its parameters~\cite{junejo2008estimating}.
Shadows can also benefit diverse image understanding tasks, e.g., image segmentation~\cite{ecins2014shadow}, object detection~\cite{cucchiara2003detecting}, and object tracking~\cite{nadimi2004physical}.
The last decade has witnessed a growing interest in image shadow detection. Many methods have been developed by examining color and illumination priors~\cite{finlayson2006removal,finlayson2009entropy}, by developing data-driven approaches with hand-crafted features~\cite{huang2011characterizes,lalonde2010detecting,zhu2010learning},
or by learning discriminative features from a convolutional neural network (CNN)~\cite{khan2014automatic,vicente2016large,nguyen2017shadow,Hu_2018_CVPR,le2018a+,zhu2018bidirectional,hu2019direction,zheng2019distraction}.
However, in striking contrast with the flourishing development of image shadow detection, much fewer works have been explored in shadow detection over dynamic scenes. On the other hand, we also notice that video processing has become an urgent topic in recent years, and a lot of methods were proposed for video salient object detection~\cite{le2017deeply, wang2017video, fan2019shifting} and video object segmentation~\cite{lu2019see,oh2019video}.
What makes video shadow detection lag far behind these video processing tasks? Compared with shadow detection of a single image, video shadow detection (VSD) needs to utilize temporal information to identify shadow pixels of each video frame. Although there exist multiple datasets for image shadow detection, video salient object detection, and video object segmentation, such standard widespread benchmark (with a sufficient number of video clips, covering diverse content) is missing for video shadow detection. What's more, CNN-based methods have not been exploited for this problem due to the lack of such a dataset.
\begin{figure*}
\centering
\includegraphics[width=0.92\linewidth]{Figures/dataset_profile.pdf}
\vspace{-1.5mm}
\caption{The examples of proposed Video Shadow Detection (\textit{\textbf{ViSha}}) dataset, with pixel-level shadow annotations.}
\vspace{-3mm}
\label{fig:dataset_profile}
\end{figure*}
In this work, \textbf{we first collect a new video shadow detection (ViSha) dataset.} It contains 120 videos with 11,685 image frames and 390 seconds duration, covering shadows of 7 object classes and 60 object categories, various motion/lighting conditions, and different instance numbers. All the video frames are carefully annotated with a high-quality pixel-level shadow mask. To the best of our knowledge, this is the first learning-oriented dataset for video shadow detection, which could facilitate the community to explore further in this field.
Second, \textbf{we develop a new baseline model, a triple-cooperative video shadow detection network (TVSD-Net), for this task.}
Instead of just exploiting temporal information within one video clip as most current video object detection networks did, we propose to learn at both intra-video and inter-video levels then model their correlation features.
Our TVSD-Net utilizes triple parallel networks in a cooperative manner.
To be specific, we take two neighboring frames from the same video and one image frame from another video as inputs.
Then a dual gated co-attention (DGC) module is devised to learn a global intra-video correlation on the two frames of the same video, and a triple-cooperative (T) module encodes the inter-video property, which promotes the similarity between the same-video frames and suppresses the similarities between different-video frames.
Finally, \textbf{we present a comprehensive evaluation of 12 state-of-the-art models on our ViSha dataset}, making it the most complete VSD benchmark. Results show that our model significantly outperforms existing methods, including single image shadow detectors~\cite{chen2020multi, zhu2018bidirectional, zheng2019distraction}, single saliency detectors~\cite{houqibin2018DSS, deng2018r3net}, semantic segmentation method~\cite{lin2017feature, Zhao_2017_CVPR}, video object segmentation~\cite{lu2019see}, and video saliency detection methods~\cite{li2019motion, song2018pyramid}.
In summary, our work forms the first learning-oriented VSD benchmark, thereby providing a new view to video object detection from a shadow perspective. Our dataset and code have been released at \url{https://github.com/eraserNut/ViSha}.
\section{Related Works}
\label{sec:relatedworks}
\noindent\textbf{Single-image Shadow detection.}
Existing shadow detection works mainly focus on detect shadow pixels from a single input image. Deep learning-based methods have achieved dominated results.
Please refer to~\cite{chen2020multi,wang2019densely,hosseinzadeh2018fast} for a review on traditional shadow detectors based on hand-crafted features.
The first shadow detection CNN~\cite{khan2014automatic} identified shadow pixels by building a seven-layer CNN to extract deep features from superpixels, and then employed a conditional random field (CRF) to further smooth shadow detection results.
Vicente et al.~\cite{vicente2016large} trained a patch-based CNN with an image-level shadow prior.
Nguyen~\cite{nguyen2017shadow} introduced a generative adversarial network with a conditional generator to generate a shadow mask.
Hu et al.~\cite{Hu_2018_CVPR} learned spatial context features in a direction-aware manner, while Zhu et al.~\cite{zhu2018bidirectional} utilized two series of recurrent attention residual (RAR) modules to aggregate context information at different CNN layers for shadow detection.
Zheng et al.~\cite{zheng2019distraction} learned distraction-aware features to explicitly
predict false positives and false negatives for robust shadow detection.
Rather than relying on only annotated shadow data, Chen et al.~\cite{chen2020multi} explored the complementary information of shadow region detection, shadow boundary detection, and shadow count detection and embedded the multi-task learning into a semi-supervised learning framework to fuse unlabeled data for helping shadow detection.
\input{Tables/Dataset_statistics}
\vspace{2mm}
\noindent\textbf{Video shadow detection} aims to detect the shadow regions from each frame of a video. Existing video shadow detection methods almost relied on hand-crafted features and were developed one decade ago. For instance, Nadimi et al.~\cite{nadimi04} leveraged a spatio-temporal
albedo test and dichromatic reflection model. Jr et al.~\cite{Jr05} detected the moving shadow in videos by employing improved background subtraction techniques.
Benedek et al.~\cite{Benedek08} combined the color and microstructural features to detect the shadow in surveillance Videos. Note that these methods work well on high-quality scenarios (\eg, stable lighting, single shadow, moving objects) due to the limited generalization capability of these hand-craft features. In addition, no large-scale datasets are publicly available for fairly evaluating different video shadow detection methods.
In order to exploit the capability of CNN-based methods for VSD tasks, it is desirable to collect a large-scale VSD dataset and develop a CNN model to provide a complement evaluation.
\vspace{2mm}
\noindent\textbf{Video object segmentation} automatically detects primary foreground objects from their background in all frames of a video.
It can be roughly categorized into unsupervised video object segmentation (UVOS) and semi-supervised video object segmentation (SVOS) (please refer to~\cite{lu2019see,oh2019video} for a detailed review).
Compared with the counterpart for image object segmentation, VOS exploits the temporal information across frames.
Lu et al.~\cite{lu2019see} formulated a co-attention siamese network (COSNet) to model UVOS from a global perspective via a co-attention mechanism.
Oh et al.~\cite{oh2019video} leveraged memory networks and learned to read relevant information from all past frames with object masks for resolving SVOS.
However, current CNN-based VOS mainly learned appearance or motion
representations in intra-video, while ignoring the valuable discriminative inter-video representations across different videos.
\input{Tables/Dataset_collection}
\paragraph{Video saliency detection} identifies most distinctive objects for each video frame~\cite{cheng2014global,wang2015saliency,aytekin2017spatiotemporal,8704996,8466906,SODsurvey}.
Recently, many CNN-based video saliency detection methods achieved dominant results.
The first CNN attempt was a fully convolutional network proposed by Wang et al.~\cite{wang2017video}.
Le et al.~\cite{le2017deeply} adopted 3D filters to combine spatial and temporal information in a spatio-temporal CRF framework, while Li et al.~\cite{li2018flow} presented optical flow guided recurrent neural network.
Song et al.~\cite{song2018pyramid} passed concatenated spatial features at multiple scales into an extended deeper bidirectional ConvLSTM to obtain spatio-temporal information.
Fan et al.~\cite{fan2019shifting} collected a video saliency detection dataset and developed a saliency-shift-aware convLSTM module to extract both spatial and temporal information.
Similar to the task of video object detection, video saliency detection methods usually focus on extracting spatial-temporal features from multiple frames of a video, which also ignores the intra-video discriminative property of salient objects.
\section{ViSha: Video Shadow Detection Dataset}
\label{sec:dataset}
We introduce \textbf{ViSha}, a new dataset for video shadow detection. Our dataset includes 120 videos with diverse content, varying length, and object-level annotations. Some example video frames can be found in Figure~\ref{fig:dataset_profile}.
We will show details of ViSha from the following key aspects.
\subsection{Data Collection} \label{subsec:collection}
In order to provide a solid basis for video shadow detection, we think that the dataset should (1) cover diverse realistic-scenes, and (2) contain sufficient challenging cases.
As shown in Table~\ref{table:collection}, more than half videos are from 5 widely-used video tracking benchmarks (\ie, OTB~\cite{OTB2015}, VOT~\cite{votpami}, LaSOT~\cite{lasot}, TC-128~\cite{TC128}, and NfS~\cite{NfS}).
Note that these video tracking datasets are not originally designed for shadow detection, and hence there are limited videos with shadows, which are all included in our dataset.
The remaining 59 videos are self-captured with different handheld cameras, over different scenes, at varying times.
We then manually trim the videos to make sure that each frame has at least one shadow area, and remove dark-screen transitions.
The frame rate is adjusted to 30 fps for all video sequences. For instance, the videos from NfS~\cite{NfS} have a high-speed frame rate of 240, for which we make sampling at every 8 frames.
Eventually, our video shadow detection dataset (ViSha) contains 120 video sequences, with a totally of 11,685 frames and 390 seconds duration. The longest video contains 103 frames and the shortest contains 11 frames.
\begin{figure*}[!t]
\centering
\includegraphics[scale=.35]{Figures/full_architecture.pdf}
\vskip -5pt
\caption{The schematic illustration of our proposed TVSD-Net. DGC module denotes dual-gated co-attention module and T module denotes triple-cooperative module. See Section~\ref{sec:fullArchitecture} for details.}
\label{fig:architecture}
\vspace{-3mm}
\end{figure*}
\subsection{Dataset Annotation and Split} \label{subsec:annotation}
For each video frame, we provide pixel-accurate, manually created segmentation in the form of a binary mask.
In realistic scenarios, shadows can be distorted, ambiguous , and hard to identify (see examples in Figure~\ref{fig:dataset_profile}).
Eight human annotators are pretrained and instructed to carefully annotate all the shadows by tracing shadow boundaries.
Then, two viewers are assigned to inspect and validate the labeled shadows.
In the annotation process, we notice that two cases deserve special attention.
First, the soft shadow is usually subjected to unclear boundaries. Considering the temporal consistency between adjacent frames, we demand that the labeling of soft shadows should be consistent across frames.
Second, the back-light parts of objects often appear in dark colors, yet they do not form shadows, for which we treat them as non-shadow areas.
To provide guidelines for future works, we randomly split the dataset into training and testing sets with a ratio of 5:7. Table~\ref{table:statistics} shows the statistics for the dataset. We can see that both the training set and testing set have sufficient diversity. It is also worth noting that we allocate more video sequences for testing sets because small testing sets may lead to model over-fitting.
\begin{figure*}[!t]
\centering
\includegraphics[scale=.52]{Figures/co-attention-full.pdf}
\vspace{-3.5mm}
\caption{The schematic illustration of Dual Gated Co-attention (DGC) Module. Blue and green parts represent the collaboration attention (co-attention) module and dual-gated module, respectively. See Section~\ref{sec:DGC} for details.}
\label{fig:co-attention}
\vspace{-2.5mm}
\end{figure*}
\subsection{Dataset Features and Statistics} \label{subsec:statistics}
\vspace{2mm}
\noindent\textbf{Sufficient Shadow Diversity.}
The diversity of shadow refers to the diversity of shadow sources, i.e. objects which cast the corresponding shadows. In ViSha, the shadow source is composed of seven main categories: Human, Plant, Vehicle, Animal, Artifact, Building, and Others. Figure~\ref{fig:dataset_relation} (a)\&(c) show these categories (7 main classes with 60 sub-classes) and their mutual dependencies, respectively. Figure~\ref{fig:dataset_relation} (b) shows the shadow ratio distribution in ViSha. The shadow ratio is defined as the proportion of the number of shadow pixels to that of the entire image. Besides, as shown in Table~\ref{table:statistics}, there are 95 videos having more than one shadow instance, making this dataset challenging for the video shadow detection task.
\vspace{2mm}
\noindent\textbf{Motion of Camera and Objects.}
As a video dataset, ViSha contains ample motion diversity for objects and cameras (summarized in Table~\ref{table:statistics}). In the viewpoint of shadow motion, 20 videos have the object shadows stay relatively static to the background, while the other 100 videos witness shadows suffer from fast-moving or distortion. Similarly, from the aspect of camera motion, there are 32 videos in which the camera is fixed and the shadow changes in the scene are relatively stable (\eg, surveillance cameras), while in the rest 88 videos, the shadows exhibit drastic changes and/or motion blur lead by camera shaking or movement.
\vspace{2mm}
\noindent\textbf{Various Lighting Conditions.}
Different lighting conditions can lead to hard shadow, which has an obvious boundary, or soft shadow, which has a rather blurry or unclear boundary. Hard shadow is usually produced when the scene contains only one single strong light source (\eg, sunshine). Soft shadow is usually caused under lighting condition with multiple light sources. 88 videos in Visha contain the hard shadows while 30 videos contain the soft shadows.
\vspace{2mm}
\noindent\textbf{Richness of the Scene.}
As we all know, data-driven models are subjected to the domain shift. For example, if all videos in the training set of ViSha are taken in the daytime, the trained models can hardly handle the shadows in night scenes. The same phenomenon also applies to the cases of indoor and outdoor scenes. In order to avoid such a problem, we build ViSha with 86 daytime videos and 34 night videos. Furthermore, ViSha contains 35 indoor videos and 85 ourdoor videos. More examples of ViSha can be found in the supplementary material.
\section{Proposed Method} \label{sec:method}
\subsection{Overview of Our Network} \label{sec:fullArchitecture}
Figure~\ref{fig:architecture} shows the schematic illustration of our triple-cooperative video shadow detection network (TVSD-Net). The intuition behind our network is to leverage discriminative feature information at both intra-video and inter-video levels. That is, for neighboring frames from the same video, their features shall be similar; while frames from different videos will have features to be distinguishable.
Our TVSD-Net takes three shadow images as inputs. The first two images (denoted as $\{\mathbf{I}_{a1}$, $\mathbf{I}_{a2}\}$) are from the same video, while the third image $\mathbf{I}_b$ is randomly selected from another video.
We devise three branches to pass each input image into a feature embedding module to extract three high-level semantic features, which are denoted as $\{\mathbf{E}_{a1}, \mathbf{E}_{a2}, \mathbf{E}_b\} \in\mathbb{R}^{26\times26\times256}$.
The feature embedding module consists of a feature extraction backbone (ResNeXt-101) and an atrous spatial pyramid pooling (ASPP) module, which has a $1$$\times$$1$ point-wise convolution, three $3$$\times$$3$ convolutions with dilation rates of 12, 24, and 36 respectively, and a global average pooling layer.
We empirically replace the last CNN layers of ResNeXt-101 with the dilation convolution (dilation rate of 2) and set the first convolutional stride to 1 to balance the spatial resolution of features and GPU memory size.
To learn global intra-video features, we devise a co-attention mechanism to emphasize the coherent information in $\mathbf{E}_{a1}, \mathbf{E}_{a2}$ from the same video (DGC module; see Figure~\ref{fig:co-attention}).
The refined features are denoted as $\mathbf{C}_{a1}$ and $\mathbf{C}_{a2}$, respectively.
Note that deep CNN layers are able to capture highly semantic features tending to describe global attributes of shadow regions, while shallow CNN layers are responsible for extracting subtly fine features to represent delicate structures.
We concatenate the refined high-level feature $\mathbf{C}_{a1}$ with a low-level feature map $\mathbf{L}_{a1}$ from the feature extraction backbone via a ship connection in the first network branch and then apply $3$$\times$$3$ and $1$$\times$$1$ convolutional layers on the concatenated features to generate a shadow detection result $\mathbf{S}_{a1}$.
Similarly, the second branch concatenates $\mathbf{C}_{a2}$ with a low-level feature map $\mathbf{L}_{a2}$ of the feature extraction backbone to generate another shadow detection result $S_{a2}$.
In the third branch, without any co-attention module, we directly concatenate high-level features $\mathbf{E}_b$ with a low-level feature map $\mathbf{L}_{b}$, and predict one more shadow detection result $\mathbf{S}_{b}$.
What's more, we devise a triple-cooperative module (T module; see Section~\ref{sec:auxiliaryTask}) to learn inter-video features in helping shadow detection.
The auxiliary loss adopted in T module makes $\mathbf{E}_{a1}$ and $\mathbf{E}_{a2}$ from two frames of the same video similar, while $\mathbf{E}_{b}$ from another video should be dissimilar to them.
\vspace{2mm}
\noindent\textbf{Loss Function.}
To better handle the scale variance of shadows,
we fuse the binary cross entropy (BCE) loss function with a lov\'asz-hinge loss~\cite{berman2018lovasz} function to compute the shadow detection loss ($\mathcal{L}_{seg}$) of all three inputs $I_{a_1}$, $I_{a_2}$, and $I_{b}$:
\vspace{-2mm}
\begin{equation}\label{Equ:seg_loss}
\mathcal{L}_{seg} = \mathcal{L}_{a1} + \mathcal{L}_{a2} + \mathcal{L}_{b} , \vspace{-2mm}
\end{equation}
where
\vspace{-2mm}
\begin{equation}\label{Eq:total_consistency_loss_2}
\begin{aligned}
& \mathcal{L}_{a1} = \Phi_{BCE}(\mathbf{S}_{a1}, \mathbf{G}_{a1}) + \Phi_{Hinge}(\mathbf{S}_{a1}, \mathbf{G}_{a1}) \ , \\
& \mathcal{L}_{a2} = \Phi_{BCE}(\mathbf{S}_{a2}, \mathbf{G}_{a2}) + \Phi_{Hinge}(\mathbf{S}_{a2}, \mathbf{G}_{a2}) \ , \\
& \mathcal{L}_{b} = \Phi_{BCE}(\mathbf{S}_{b}, \mathbf{G}_{b}) + \Phi_{Hinge}(\mathbf{S}_{b}, \mathbf{G}_{b}) \ .
\end{aligned}
\end{equation}
Here, $\Phi_{BCE}(\cdot)$ and $\Phi_{Hinge}(\cdot)$ denote the BCE loss and the lov\'asz-hinge loss, respectively;
$\mathbf{S}_{a1}$/$\mathbf{S}_{a2}$/$\mathbf{S}_{b}$ and $\mathbf{G}_{a1}$/$\mathbf{G}_{a2}$/$\mathbf{G}_{b}$ are the predicted shadow detection map and the corresponding ground truth of $\mathbf{I}_{a1}$/$\mathbf{I}_{a2}$/$\mathbf{I}_{b}$;
Finally, we use a combination of the shadow detection segmentation loss $\mathcal{L}_{seg}$ and the devised auxiliary task loss $\mathcal{L}_{aux}$ (described in Section~\ref{sec:auxiliaryTask}) to train our whole network. The total loss of our network is given by:
\vspace{-2mm}
\begin{equation}\label{Equ:total_loss}
\mathcal{L}_{total} = \mathcal{L}_{seg} + \beta \mathcal{L}_{aux},
\end{equation}
where $\beta$ is to control the weight of auxiliary loss, and we empirically set $\beta = 10$ in our experiments.
\subsection{Dual Gated Co-attention Module} \label{sec:DGC}
The dual gated co-attention module explicitly encodes intra-video correlations between a pair of frames in a video, via a co-attention mechanism and a dual-gated mechanism. This enables TVSD-Net to focus on frequently coherent regions, thus further helping to discover the shadow regions and produce reasonable VSD results.
\vspace{2mm}
\noindent\textbf{Co-attention Mechanism.}
The blue region of Figure~\ref{fig:co-attention} shows the collaboration-attention (co-attention) module, which takes two features $\mathbf{E}_{a1}$ $\in \mathbb{R}^{W \times H \times C}$, $\mathbf{E}_{a2}$ as the inputs to compute their correlations.
Inspired by \cite{lu2019see}, we first reshape $\mathbf{E}_{a1}$ to be a new feature map $\hat{\mathbf{E}}_{a1}$ $\in \mathbb{R}^{WH \times C}$ and reshape $\mathbf{E}_{a2}$ to be $\hat{\mathbf{E}}_{a2}$ $\in \mathbb{R}^{WH \times C}$, and compute an affinity matrix $\mathbf{A} \in \mathbb{R}^{WH \times WH}$:
\vspace{-2mm}
\begin{equation}\label{Equ:affinity}
\mathbf{A} = \hat{\mathbf{E}}_{a1}\mathbf{M} \hat{\mathbf{E}}_{a2}^{\top},
\end{equation}
where $\mathbf{M} \in \mathbb{R}^{C \times C}$ is a weight matrix.
Intuitively, each element of $\mathbf{A}$ represents the similarity between each column of $\mathbf{E}_{a1}$ and each row of $\mathbf{E}_{a2}$.
From $\mathbf{A}$, we employ a $\mathtt{Softmax}$ function to column-wisely and row-wisely normalize $\mathbf{A}$ respectively, and multiply the resultant normalization features with $\mathbf{A}$ to compute two co-attention enhanced features $\mathbf{T}_{a1}$ and $\mathbf{T}_{a2}$:
\begin{equation}\label{Equ:transform}
\begin{aligned}
\mathbf{T}_{a1} &= \mathtt{Softmax}(\mathbf{A})*\hat{\mathbf{E}}_{a2} \in \mathbb{R}^{C \times WH} \ , \\
\mathbf{T}_{a2} &= \mathtt{Softmax}(\mathbf{A}^{\top})*\hat{\mathbf{E}}_{a1} \in \mathbb{R}^{C \times WH} \ ,
\end{aligned}
\end{equation}
Then, we reshape $\mathbf{T}_{a1}$ to be $\mathbf{H}_{a1} \in \mathbb{R}^{C \times W \times H}$ and reshape $\mathbf{T}_{a2}$ to be $\mathbf{H}_{a2} \in \mathbb{R}^{C \times W \times H}$.
By this way, we intuitively transform the $a1$ features ($\mathbf{E}_{a1}$) to a fake $a2$ features ($\mathbf{H}_{a2}$). Compared to the original $\mathbf{E}_{a2}$, the fake one ($\mathbf{H}_{a2}$) encodes more temporal information.
\vspace{2mm}
\noindent\textbf{Dual-gated Mechanism.}
Since there may exist potential appearance variations (\eg, occlusion, out-of-view) between two neighboring frames,
using co-attention module enhances the coherent features, yet may also introduce some noises from adjacent frames.
Hence, it is better to weight co-attention enhanced features from two input frames, instead of treating the learned co-attention information equally.
To achieve this goal, we propose a dual-gated mechanism to obtain co-attention confidences.
Unlike the self-gated mechanism~\cite{lu2019see}, we learn the co-attention confidences by leveraging $\mathbf{H}_{a1}$ and $\mathbf{H}_{a2}$ together. Our dual-gated mechanism consists of a spatial gated operation and a channel gated operation.
Specifically, we fuse $\mathbf{H}_{a1}$ and $\mathbf{H}_{a2}$ by applying a $3$ $\times$ $3$ convolution on the concatenation of $\mathbf{H}_{a1}$ and $\mathbf{H}_{a2}$ to compute a fused feature map $\mathbf{Q}$:
\vspace{-2mm}
\begin{equation}\label{Equ:feature-fusion}
\mathbf{Q} = \mathtt{Conv}(\mathtt{Concat}(\mathbf{H}_{a1}, \mathbf{H}_{a2})) \ .
\end{equation}
Then, two spatial gated maps ($\{\mathbf{K}_{a1}, \mathbf{K}_{a2}\} \in \mathbb{R}^{W \times H \times 1}$) are computed by utilizing a $\mathtt{Sigmoid}$ function and a $1$ $\times$ $1$ convolution on $\mathbf{Q}$:
\vspace{-2mm}
\begin{equation}\label{Equ:spatial_gate}
\begin{aligned}
\mathbf{K}_{a1} & = \mathtt{Sigmoid}(\mathtt{Conv}(\mathbf{Q})) \ , \\
\mathbf{K}_{a2} & = \mathtt{Sigmoid}(\mathtt{Conv}(\mathbf{Q})) \ .
\end{aligned}
\end{equation}
\noindent Moreover, we generate two channel-wise gated maps $\mathbf{U}_{a1}$ and $\mathbf{U}_{a2}$:
\vspace{-2mm}
\begin{equation}\label{Equ:channel_gate}
\begin{aligned}
\mathbf{U}_{a1} &= \mathtt{Sigmoid}(\mathtt{fc}(\mathtt{GAP}(\mathbf{Q}))) \ , \\
\mathbf{U}_{a2} &= \mathtt{Sigmoid}(\mathtt{fc}(\mathtt{GAP}(\mathbf{Q}))) \ . \\
\end{aligned}
\end{equation}
Once obtaining spatial and channel gated maps, we multiply the spatial gated map with the co-attention enhanced features $\{\mathbf{H}_{a1}, \mathbf{H}_{a2}\}$, and then multiply the resultant features with the channel gate map to produce gated features $\{\mathbf{D}_{a1}, \mathbf{D}_{a2}\}$.
We then apply a $3$ $\times$ $3$ convolution on the concatenation of $\mathbf{D}_{a1}$/$\mathbf{D}_{a2}$ and $\mathbf{E}_{a1}$ ($\mathbf{E}_{a2}$) to produce output features of the dual gated co-attention module, i.e., $\mathbf{C}_{a1}$ and $\mathbf{C}_{a2}$.
The definitions of $\mathbf{C}_{a1}$ and $\mathbf{C}_{a2}$ are given by:
\begin{equation}\label{Equ:final_co_attention}
\begin{aligned}
\mathbf{C}_{a1} &= \mathtt{Conv}(\mathtt{Cancat}(\mathbf{E}_{a1}, \mathbf{H}_{a1} \otimes \mathbf{K}_{a1} \otimes \mathbf{U}_{a1})) \ , \\
\mathbf{C}_{a2} &= \mathtt{Conv}(\mathtt{Cancat}(\mathbf{E}_{a2}, \mathbf{H}_{a2} \otimes \mathbf{K}_{a2} \otimes \mathbf{U}_{a2})) \ ,
\end{aligned}
\end{equation}
where $\otimes$ denotes element-wise product.
\subsection{Triple-cooperative Module} \label{sec:auxiliaryTask}
\begin{figure}[!t]
\centering
\includegraphics[scale=.5]{Figures/Aux_task.pdf}
\vskip -5pt
\caption{The schematic illustration of our triple-cooperative (T) module; See Section~\ref{sec:auxiliaryTask} for details.}
\label{fig:Auxiliary}
\vspace{-2.5mm}
\end{figure}
Given high-level features $\{\mathbf{E}_{a1}, \mathbf{E}_{a2}, \mathbf{E}_b\}$ of the three input images $\{\mathbf{I}_{a1}, \mathbf{I}_{a2}, \mathbf{I}_b\}$, we devise a triple-cooperative module (T-module) to make features from the same video similar and features from different videos dissimilar.
Figure~\ref{fig:Auxiliary} shows the schematic illustration of T-module, which computes an auxiliary loss based on $\{\mathbf{E}_{a1}, \mathbf{E}_{a2}, \mathbf{E}_b\}$.
Intuitively, we expect the similarity between two frames from the same video to be close to 1, while the similarity between two frames from different videos approaches to 0.
To be specific, we apply three global average pooling operations on $\mathbf{E}_{a1}$, $\mathbf{E}_{a2}$, and $\mathbf{E}_b$ to obtain three features $\{\mathbf{P}_{a1}, \mathbf{P}_{a2}, \mathbf{P}_b\} \in \mathbb{R}^{1\times1\times256}$, which are then normalized as $\{\mathbf{N}_{a1}, \mathbf{N}_{a2}, \mathbf{N}_b\}$:
\begin{equation}\label{Equ:normalize}
\begin{aligned}
\mathbf{N}_{a1} &= \frac{\mathbf{P}_{a1}}{
\mathtt{max}({\small \left \| \mathbf{P}_{a1} \right \|_2}, \epsilon)} \ , \;
\mathbf{N}_{a2} = \frac{\mathbf{P}_{a2}}{
\mathtt{max}({\small \left \| \mathbf{P}_{a2} \right \|_2}, \epsilon)} \ , \\
\mathbf{N}_{b} &= \frac{\mathbf{P}_{b}}{
\mathtt{max}({\small \left \| \mathbf{P}_{b} \right \|_2}, \epsilon)} \ ,
\end{aligned}
\end{equation}
where $\epsilon$ is a small positive number and it is set as $\epsilon$ $=$ $1e^{-12}$ to avoid division by zero.
After that, we compute the similarity $V_{a1, a2}$ of $\mathbf{N}_{a1}$ and $\mathbf{N}_{a2}$ from the same video by computing the dot produce of $\mathbf{N}_{a1}$ and $\mathbf{N}_{a2}$, and also the similarity $V_{a1, b}$ of $\mathbf{N}_{a1}$ and $\mathbf{N}_{b}$ from different videos, given by:
\vspace{-2mm}
\begin{equation}\label{Equ:similarity}
V_{a1, a2}= \mathbf{N}_{a1} \cdot \mathbf{N}_{a2}, \; \text{and,} \;
V_{a1, b} = \mathbf{N}_{a1} \cdot \mathbf{N}_{b} .
\end{equation}
Then, we multiple two similarities with a temperature $\frac{1}{\tau}$~\cite{hinton2015} and concatenated them to form a two-element vector. The rest is to compare whether this two-element vector is close to the target distribution of (1, 0). Here, after applying a softmax function on the two-element vector for normalization, we compute a cross-entropy loss between the two vectors as the auxiliary loss of our T module.
In summary, the definition of our auxiliary loss $\mathcal{L}_{aux}$ is given by:
\vspace{-2mm}
\begin{equation}\label{Equ:similarityLoss}
\mathcal{L}_{aux} = -\textrm{log}\frac{\textrm{exp}( V_{a1, a2}/ \tau)}{\textrm{exp}(V_{a1, a2} / \tau) + \textrm{exp}( V_{a1, b} / \tau)},
\end{equation}
where $\tau$$=$$0.7$ is a temperature constant to control degree of two similarities. The sum is over one positive and one negative samples.
It is clear that the auxiliary loss makes the similarity $V_{a1, a2}$ from same video to be 1 while making the similarity of $V_{a1, b}$ from different videos to be 0.
Hence, $\mathcal{L}_{aux}$ tends to have a small score when $\mathbf{N}_{a1}$ is similar to $\mathbf{N}_{a2}$ from the same video, and dissimilar to $\mathbf{N}_b$ from a different video.
\subsection{Implementation Details} \label{sec:implementationDetails}
We implement our TVSD-Net using PyTorch.
Adam optimizer is employed to train the network with mixed precision training~\cite{micikevicius2017mixed} on a NVIDIA GTX 2080Ti.
We initialize the feature extraction backbone via a pre-trained ResNeXt-101~\cite{xie2017aggregated} on ImageNet while other layers are trained from scratch.
The weight decay, batch size, epoch number are set as $0.0005$, $5$, and $12$, respectively.
We set the initial learning rate as $0.0005$ for scratch layers and $0.00005$ for pretrained layers, and then use the cosine decay with a warm-up period to adjust the learning rate.
TVSD-Net requires about 0.06s to process an image of $416$ $\times$ $416$.
In the testing phase, we take the shadow detection result $\mathbf{S}_{a1}$ in the first branch as the output of the TVSD-Net.
Given an input video, to obtain the shadow detection result of each frame (we call it target frame), we follow ~\cite{lu2019see} to empirically select the subsequent five frames of the target frame, and then pass the target frame as $\mathbf{I}_{a1}$, and each of fives frames as $\mathbf{I}_{a2}$ to the TVSD-Net.
By doing so, we obtain five segmentation results and then average the five results as the final shadow detection result of the target frame.
\section{Experiments}
\label{sec:experiments}
\subsection{Experimental Settings}
\vspace{0.5mm}
\noindent\textbf{Evaluation Metrics.}
We adopt four common evaluation metrics to quantitatively compare video shadow detection methods.
They are Mean Absolute Error (\textbf{MAE})~\cite{houqibin2018DSS,zhu2018saliency} and F-measure ($\mathbf{F}_{\beta}$)~\cite{houqibin2018DSS,zhu2019aggregating}, Intersection over Union (\textbf{IoU})~\cite{zhao2018icnet}, and Balance Error Rate (\textbf{BER})~\cite{Hu_2018_CVPR}.
In general, a better video shadow detection method shall have smaller BER and MAE scores, and larger $\mathbf{F}_{\beta}$ and IoU scores.
\vspace{0.5mm}
\noindent \textbf{Comparative Methods.} Since there is no CNN-based method for video shadow detection, we make comparison against $12$ state-of-the-art methods for relevant tasks, including FPN~\cite{lin2017feature}, PSPNet~\cite{Zhao_2017_CVPR}, DSS~\cite{houqibin2018DSS}, R$^3$Net~\cite{deng2018r3net}, BDRAR~\cite{zhu2018bidirectional}, DSD~\cite{zheng2019distraction}, MTMT~\cite{chen2020multi}, PDBM~\cite{song2018pyramid}, COSNet~\cite{lu2019see}, MGA~\cite{li2019motion}, FEELVOS~\cite{Voigtlaender_2019_CVPR} and STM~\cite{oh2019video}.
Among them, FPN and PSPNet are developed for single-image semantic segmentation. DSS and R$^3$Net are dedicated for single-image saliency detection, while BDRAR, DSD, and MTMT are utilized for single-image shadow detection.
Lastly, PDBM, COSNet, MGA, FEELVOS and STM are for video saliency detection and object object segmentation.
We use their public codes, and re-train these methods on our training set for a fair comparison.
\input{visual-compare-SOTA}
\input{Tables/Experiments_SOTA}
\input{Tables/Experiments_Ablation}
\subsection{Comparison to the State-of-the-arts}
Table~\ref{table:state-of-the-art} shows the performances on our ViSha dataset, where COSNet has the best $\mathrm{MAE}$ score of $0.040$; MTMT has the best $\mathrm{F}_{\beta}$ score of $0.729$; DSD has the best $\mathrm{IoU}$ score of $0.518$; PDBM has the best $\mathrm{BER}$ score of $19.73$.
Compared to the best-performing existing methods, our method obtains improvements with large margins, with $\mathrm{MAE}$ improvement of 17.50\%, $\mathrm{F}_{\beta}$ improvement of 3.84\%, $\mathrm{IoU}$ improvement of 9.46\%, and $\mathrm{BER}$ improvement of 10.29\% on our dataset, respectively. That shows the superiority of our method on video shadow detection.
Figure~\ref{fig:comparison_SOTA} visually compares the video shadow detection maps produced by our method and the state-of-the-arts.
From the results, we can see that our TVSD-Net (3rd column of Figure~\ref{fig:comparison_SOTA}) can more accurately identify shadow pixels than compared methods.
It effectively locates different shadows under various backgrounds, and successfully discriminates true shadows from those non-shadow dark regions.
For example, in the 1st row, most compared methods regard both the bottle and its shadow as the shadow regions, while our TVSD-Net can discriminate them successfully. A similar situation is also reflected in the 2nd row, our TVSD-Net can better capture the distinction between the soccer player and his shadow.
\subsection{Ablation Study}
We perform ablation study experiments to verify the performance of Dual-gated co-attention module and Triple-cooperative module in TVSD-Net.
Here, we consider four baseline networks.
The first baseline network (denoted as ``basic'') is constructed by removing the co-att, DGM, and T module from the TVSD-Net.
The second (denoted as ``basic+co-att'') is to add the original co-attention module~\cite{lu2019see} into ``basic''.
The third baseline (denoted as ``basic+T-module'') is to add our T-module into ``basic''.
The fourth baseline (denoted as ``ours-w/o-T-module'') removes T-module from our network.
The fifth baseline (denoted as ``ours-w/o-DGM'') removes DGM from our network.
Table~\ref{table:ablation} summarizes the $\mathrm{BER}$ values of our network and seven baseline networks on the ViSha dataset.
From the results, we have the following observations:
(i) ``basic+co-att'' have the superior performance of four evaluation metrics over ``basic'', which means that learning intra-video coherent information can provide helpful information for video shadow detection.
(ii) ``basic+T-module'' has smaller $\mathrm{BER}$ and $\mathrm{MAE}$ scores and larger $\mathrm{F}_{\beta}$ and $\mathrm{IoU}$ scores than ``basic'', demonstrating that the additional auxiliary loss from the T module incurs detection improvement.
(iii) ``ours-w/o-T-module'' can more accurately detect shadow pixels than ``basic+co-att'' due to its better results of $\mathrm{F}_{\beta}$, $\mathrm{IoU}$, $\mathrm{BER}$, and $\mathrm{MAE}$.
It indicates dual gated module helps to increase the co-attention confidences than the original co-attention module~\cite{lu2019see}.
(iv) our TVSD-Net has better metric results than ``ours-w/o-T-module'' and ``ours-w/o-DGM'', showing that combining the two modules achieves a higher video shadow detection accuracy.
\section{Conclusion}
\label{sec:conclusion}
This paper presents a novel network for video shadow detection.
One of our key contributions is to first collect a learning-oriented video shadow detection (ViSha) dataset, which contains $120$ videos with $11,685$ frames covering various objects and scenes, with pixel-level shadow annotations.
The second contribution is the development of a novel network for video shadow detection, by learning intra-video and inter-video discriminative properties of shadows.
Experimental results on the collected dataset demonstrated that our method consistently outperforms 12 state-of-the-art methods by a large margin.
To the best of our knowledge, this work is the first annotated dataset for video shadow detection, and our ViSha dataset can facilitate further research in video shadow detection.
|
3,212,635,538,004 | arxiv | \section{Introduction}\label{intro} In this paper, we introduce the notion of controlled Floyd separation between geodesic rays
starting at the identity
of a finitely generated group $G$.
Here, the term {\it `control'} refers in general to a choice of a $G-$invariant subcollection of
paths in a Cayley graph $\Gamma$ of $G$ with respect to which the Floyd distance is computed
(See Definition \ref{cdef}).
We shall primarily be interested in subcollections given by $(\kappa, \kappa)-$quasigeodesics in $\Gamma$.
The main purpose of this paper is to furnish an example of a finitely generated group $G$ which satisfies the following properties:
\begin{enumerate}
\item[(A)] There are uncountably many geodesic rays starting at the identity in $G$, which are Floyd separated with
respect to $(\kappa, \kappa)-$quasigeodesics for any $\kappa > 1$. See Theorem \ref{nontrivial}. These rays actually lie in a subgroup that
is quasiconvex in a strong sense: all geodesics in the subgroup are `uniformly Morse' (Corollary \ref{morse}).
\item[(B)] $G$ is not hyperbolic relative to any collection of proper subgroups. See Theorem \ref{nrhthm}.
\end{enumerate}
This provides a construction that may be used as negative evidence towards a question of Olshanskii, Osin and Sapir \cite[Problem 7.11]{oos}.
The group $G$ we construct is the double of a free group along an infinitely generated malnormal subgroup $K$: $G=F\ast_K F$. Here
$K = \cup_n K_n$ is an ascending union of malnormal quasiconvex subgroups $K_n$ of $F$. The group $G$ has a number of other features:
\begin{enumerate}
\item $G$ is a graded small cancellation (and hence lacunary hyperbolic) group in the sense of Olshanskii, Osin and Sapir \cite{oos}. See Theorem \ref{g-gsc}.
\item $G$ is a direct limit of cubulated hyperbolic groups $G_n$: each $G_n$ is hyperbolic and admits a geometric action on
a non-positively curved square complex (in fact a $\mathcal{VH}$ complex in the sense of Wise \cite{wise-cbms}). See Proposition \ref{cycgeodprop}.
\item Every finitely presented subgroup of $G$ is hyperbolic (Proposition \ref{subgp}). In particular, $G$ does not contain any Baumslag-Solitar group
$BS(m,n)$.
\item Each $K_n$ is quasiconvex in a strong sense (`uniformly Morse') in $G$ (Corollary \ref{morse}).
\item The limit set of $K (= \cup_n K_n)$ in $F$ is all of the Gromov boundary of $F$ (Proposition \ref{full}).
\item In spite of the features (A) and (B) above, $G$ has trivial Floyd boundary in the usual sense. See Theorem \ref{floydtrivial}.
\end{enumerate}
The group $G$ and its features above illustrate that the source of triviality of the Floyd boundary of a group $G$ is quite subtle
and known sources (e.g. existence of $\mathbb{Z} \oplus \mathbb{Z} \subset G$, wideness of $G$, etc.) are inadequate to detect this.
On the way towards constructing $G$, we construct (Proposition \ref{malnprop}) a malnormal infinitely generated (and hence non-quasiconvex) subgroup of a free group,
giving negative evidence towards a question of Swarup and Gitik \cite[Question 1.8]{bestvinahp}.
A range of tools is used in proving the properties of $G$:
\begin{enumerate}
\item The theory of relative hyperbolicity and relative quasiconvexity \cite{farb-relhyp, bowditch-relhyp, hruska-agt}.
\item Graphical small cancellation theory \cite{wise-por, wise-qpor}.
\item Graded small cancellation theory \cite{oos, olshanski-book}.
\item Tree graded spaces \cite{ds}.
\item Asymptotic cones and lacunary hyperbolic groups \cite{oos, gromov-ai}.
\end{enumerate}
\begin{comment}
\noindent {\bf A Mnemonic:} In Theorem \ref{nontrivial} we show that ideal points on the Gromov boundary of a qi-embedded free subgroup $K_2$ of $G$ are
Floyd-separated with respect to quasigeodesics.
To show this we pass to the asymptotic cone $Con^\omega G$ of $G$ which is a circle tree (Proposition \ref{cctree}). The asymptotic cone of $K_2$ is embedded
as an $\mathbb R$-tree $\TT$ in $Con^\omega G$. Theorem \ref{trivialintn} shows that the intersection of any circle in $Con^\omega G$ has to be a single point or
empty. In particular, it cannot be an arc. The reader may think of this as the statement
`$D$'s are not allowed inside $Con^\omega G$'.
The proof of Theorem \ref{floydtrivial} shows that in fact it is possible for a circle
in $Con^\omega G$ to touch $\TT$ at a point. Thus, `Circles tangential to lines is possible in $Con^\omega G$'.
\end{comment}
\subsection{History of the Problem} The notion of a Floyd function and Floyd boundary were introduced by Floyd in \cite{floyd}
and generalized by Gromov in \cite{gromov-ai} under the name of conformal boundary of a group. Gromov \cite[p. 264]{gromov-ai} and
Karlsson \cite{karlsson-free} showed that
the action of a group on its Floyd boundary is a convergence action (in fact a geometric convergence action in the sense of Gerasimov) provided the boundary is
non-trivial, i.e. if it contains more than two (and hence infinitely many) points. In \cite[Proposition 4.28]{oos}, Olshanskii, Osin and Sapir showed that
if $G$ is a finitely generated group whose Floyd boundary consists of at least
2 points, then all asymptotic cones of $G$ have cut points. These two facts may be regarded as evidence towards a positive answer to the following question:
\begin{qn} \cite[Problem 7.11]{oos} (see also \cite[p.18]{act}.) Suppose that a finitely generated group G has a non–trivial Floyd boundary.
Is $G$ hyperbolic relative to a collection of proper subgroups?
\label{mainqn} \end{qn}
Theorem \ref{nontrivial} shows, in particular, that the following weaker question has a negative answer (see Section \ref{cfs} for precise definitions).
\begin{qn} Suppose that a finitely generated group G has a pair of geodesic rays which are Floyd-separated with respect to quasigeodesics.
Is $G$ hyperbolic relative to a collection of proper subgroups?
\label{mainqn2} \end{qn}
A very general and detailed treatment of relative hyperbolicity and its characterization in topological and dynamic terms has been carried out recently
by Gerasimov and Potyagailo in a sequence of papers
\cite{gerasimov-expans-gafa,ger-floydgafa,ger-pot-qc, ger-pot-jems}. The notion of a controlled Floyd separation is related to, but different from the notion of
a Karlsson function introduced in \cite{ger-pot-qc}.
\subsection{Notation}\label{notn}
\begin{enumerate}
\item $H$ will be a subgroup of the free group $F_2$ and $h, k$ will denote elements (typically of $F_2$).
\item $(h,k)_1$ will denote the Gromov inner product of $h, k$ with respect to $1$.
\item $|h| = d(h,1)$
\item $(h,H)_1 = {\rm max}_{k \in H} (h,k)_1$
\item $Rose (F_2)$ will denote the rose on two petals labeled by the generators of $F_2$.
\item $K_n$ will be a free subgroup of $F_2$ of rank $n$.
$Rose (K_n)$ will denote the rose on $n$ petals labeled by the generators of $K_n$.
\item If $H$ is a subgroup of a (relatively) hyperbolic group $G$, the limit set of $H$ is denoted $\Lambda_H$.
\item $T(n)$ denotes the tower function of height $n$ with base 2, i.e. $T(n) = 2^{2^{2^{2 \cdots}}}$, where the power is taken
$ n$ times.
\item $U(n)(a,b)$ is the word (element) in the free group $F_2$ on generators $a, b$
given by $$U(n)(a,b)=[a^{(T(n)+1)}b^{(T(n)+1)}][a^{(T(n)+2)}b^{(T(n)+2)}]\cdots[a^{T(n+1)}b^{T(n+1)}].$$
\end{enumerate}
\section{Preliminaries}\label{prelim}
We shall be using two equivalent definitions regarding quasigeodesics in this paper, which we elucidate here at the outset.
\begin{defn} Let $(X,d)$ be a geodesic metric space. \\
1) A path $\sigma : I \rightarrow X$ is a $(K, \epsilon)-$quasigeodesic with $K\geq 1, \epsilon \geq 0$ if
for all $t_1, t_2 \in I$,
$$\frac{1}{K} |t_1-t_2| - \epsilon \leq d(\sigma (t_1), \sigma (t_2)) \leq K |t_1-t_2| + \epsilon.$$\\
2) Alternately \cite{oos}[Section 2.4], $\sigma : I \rightarrow X$ is a $(\lambda, \epsilon)-$quasigeodesic with $\lambda\in (0, 1], \epsilon \geq 0$ if
for all $t_1, t_2 \in I$,
$$d(\sigma (t_1), \sigma (t_2)) \geq \lambda l(\sigma([t_1, t_2]) -\epsilon,$$
where $l(\sigma([t_1, t_2]) $ denotes the length of the subpath from $\sigma (t_1)$ to $ \sigma (t_2)$.
\end{defn}
We use the first definition when we want $K \geq 1$ and the second one when we want $\lambda\in (0, 1]$. The second one is used primarily in Section \ref{gsc}.
\subsection{Relative Hyperbolicity} \label{rh} We quickly recall the notion of relative hyperbolicity introduced by Gromov \cite{gromov-hypgps}.
An equivalent notion was introduced by Farb \cite{farb-relhyp} and the equivalence of the two proven by Bowditch
\cite{bowditch-relhyp}. Equivalent definitions and closely related work
can also be found in the work of a number of authors including \cite{bowditch-relhyp, osin-mams,
dahmani-th, groves-manning, hruska-agt}.
Let $(X,d)$ be a path metric space. A collection of closed
subsets $\HH = \{ H_\alpha\}$ of $X$ will be said to be {\bf uniformly
separated} if there exists $\epsilon > 0$ such that
$d(H_1, H_2) \geq \epsilon$ for all distinct $H_1, H_2 \in \HH$.
\begin{defn} \cite{gromov-hypgps}
For any geodesic metric space
$(H,d)$, the {\em hyperbolic cone}
$H^h$ is the metric space
$H\times [0,\infty) = H^h$ equipped with the
path metric $d_h$ obtained as follows: \\
1) $d_{h,t}((x,t),(y,t)) = 2^{-t}d_H(x,y)$, where $d_{h,t}$ is the induced path
metric on $H\times \{t\}$. Paths joining
$(x,t),(y,t)$ and lying on $H\times \{t\}$
are called {\em horizontal paths}. \\
2) $d_h((x,t),(x,s))=\vert t-s \vert$ for all $x\in H$ and for all $t,s\in [0,\infty)$, and the corresponding paths are called
{\em vertical paths}. \\
3) for all $x,y \in H^h$, $d_h(x,y)$ is the path metric induced by the collection of horizontal and vertical paths. \\
\end{defn}
A similar construction of a `combinatorial' horoball was carried out by Groves-Manning in \cite{groves-manning}.
\begin{defn}
Let $\Gamma$ be the Cayley graph of a group $G$ and let $\HH_0$ be a finite collection of subgroups of $G$.
let $\HH$ be the collection of all left cosets of elements of $\HH_0$.
$G$ is said to be hyperbolic relative to $\HH$ in the sense of Gromov, if $\GG (\Gamma, \HH)$,
obtained by attaching the hyperbolic cones
$ aH^h$ to $aH \in \HH$ by identifying $(z,0)$ with $z$
for all $aH\in \HH$ and $z \in aH$,
is a complete hyperbolic metric space. The collection $\{ aH^h : H \in \HH \}$ is denoted
as ${\HH}^h$. The induced path metric is denoted as $d_h$.
The boundary $\partial \GG (\Gamma, \HH)$ is called the {\bf Bowditch boundary}.
\end{defn}
The following Theorem
of Bowditch \cite{bowditch-relhyp} shall be useful (see also \cite{mahan-relrig}).
\begin{theorem} \cite{bowditch-relhyp} Let $G$ be a hyperbolic group without torsion. Then $G$ is hyperbolic relative to $\PP$ if and only if
\begin{enumerate}
\item Each $P_i$ is quasiconvex in $G$.
\item for all $P_i, P_j \in \PP$ and $g \in G$,
$gP_ig^{-1} \cap P_j = \{1\}$ unless $i=j$ and $g \in P_i$.
\end{enumerate}
In particular if $\PP$ has exactly one subgroup $P_1$, and $G$ is hyperbolic relative to $P_1$, the latter is malnormal quasiconvex in $G$.
\label{malnrh} \end{theorem}
\subsubsection{Relative Quasiconvexity}
In \cite{hruska-agt} Hruska gives a number of equivalent criteria for relative quasiconvexity for subgroups of relatively hyperbolic groups.
We give one of these below to fix notions.
\begin{defn}\cite{hruska-agt} Let $G$ be hyperbolic relative to a collection $\PP$ of parabolic subgroups.
A subgroup $H \subset G$ is \emph{relatively quasiconvex} if the following holds.
Let $\mathcal{S}$ be some (any) finite relative generating set for $(G,\PP)$,
and let $\mathcal{E}$ be the set containing all the elements of $P_i \in \PP$ (for all $i$).
Let $\Gamma$ be the Cayley graph of $G$ with respect to the generating set $\mathcal{S} \cup \mathcal{E}$
with all edges of length one.
Let $d$ be some (any) proper, left invariant metric on~$G$.
Then there is a constant $D_0=D_0(\mathcal{S},d)$ such that
for each geodesic $\bar{c}$ in $\Gamma$
connecting two points of $H$,
every vertex of $\bar{c}$ lies within a $d$--distance $D_0$ of $H$.
\end{defn}
It is shown in \cite{hruska-agt} that the above definition is independent of the choice of
finite relative generating set $\mathcal{S}$ and the choice of proper metric $d$.
\begin{theorem}\cite{hruska-agt}[Theorem 1.2]
\label{hruska-intn}
Let $G$ be a countable group that is relatively hyperbolic with respect
to a finite family of subgroups $\PP=\{P_1,\dots,P_n\}$.
\begin{enumerate}
\item If $H\subset G$ is relatively quasiconvex, then $H$ is relatively
hyperbolic with respect to a natural induced collection of subgroups.
\item If $H_1,H_2 \subset G$ are relatively quasiconvex, then
$H_1 \cap H_2$ is also relatively quasiconvex.
\end{enumerate}
\end{theorem}
\begin{theorem}\cite{hruska-agt}[Theorem 1.5]
\label{hruska-qi}
Let $G$ be a finitely generated relatively hyperbolic group
and let $H$ be a finitely generated subgroup.
If $H$ is undistorted in $G$, then $H$ is relatively quasiconvex.
\end{theorem}
Combining the equivalence of characterizations in \cite{hruska-agt} with a Lemma of Yang we get
\begin{lemma}\cite{yang-relhyp}[Lemma 2.6] \label{yang}
Let $H$ be relatively quasiconvex in a relatively hyperbolic group $G$ such
that $|\Lambda(H)| \geq 2$. Then for any subgroup $H \subset J
\subset G$ satisfying $\Lambda(H)=\Lambda(J)$, we have that $H$ is
of finite index in $J$. In particular, $J$ is relatively
quasiconvex.
\end{lemma}
\subsection{Floyd Boundary and Controlled Floyd Separation}\label{fb}
\begin{defn}
A function $f:\mathbb N\to\mathbb R$ is a {\bf Floyd scaling function}
if
\begin{enumerate}
\item $\sum_{n\geqslant0}f_n<\infty$,
\item There exists a positive $\lambda$ such that $1\geqslant f_{n+1}/f_n\geqslant\lambda$ for all $n{\in}\mathbb N$.
\end{enumerate}
\end{defn}
In this paper we shall refer to a Floyd scaling function simply as a {\bf scaling function}.
Let $G$ be a group with symmetric generating set $S$ and $\Gamma = \Gamma (G,S)$ denote the Cayley graph of $G$ with respect to $S$
and let $d$ denote the word-metric.
\begin{defn}
Let $f$ be a scaling function. Given an edge $[x,y]$ of $\Gamma$, define its Floyd length to be $f( d([x,y], 1))$.
The resulting {\bf path metric} on $\Gamma$ is called the {\bf Floyd metric } with respect to the scaling function $f$ and is denoted as $d_f$.
A geodesic in $(\Gamma, d_f)$ is called a {\bf Floyd geodesic }.
The metric completion $\overline{(\Gamma, d_f)}$ is called the {\bf Floyd completion} and
$\partial_f\Gamma = \overline{(\Gamma, d_f)}\setminus
\Gamma$ is called the {\bf Floyd boundary}. For $f(n) = \lambda^n$ with $\lambda \in (0,1)$, $d_f$ will also be denoted as $d_\lambda$ and
$\partial_f\Gamma$ will also be denoted as $\partial_\lambda\Gamma$.
\end{defn}
We give a somewhat different but equivalent description of the Floyd boundary that will be better adapted to the applications we have in mind.
\begin{lemma} \label{geodfg} Let $f$ be a (Floyd) scaling function for $\Gamma$. Let $r$ be a geodesic ray in $\Gamma$ starting at $1\in \Gamma$.
Then $r$ is a Floyd geodesic in $(\Gamma, d_f)$. \end{lemma}
\begin{proof} Let $p_m, p_n$ be points on $r$ such that $d(1,p_m) = m $ and $d(1,p_n) = n $.
Let $r_{mn}$ be the subsegment of $r$ joining $p_m, p_n$. Then any path $\sigma$ from $p_m$ to $p_n$ must have, for $m \leq i < n$,
at least one edge $E_i$
joining a point $q_i$ to a point $q_{i+1}$ where $d(q_i, 1) = i$ and $d(q_{i+1}, 1) = i+1$ since any such path must travel between the $i-$ and $(i+1)-$spheres
at least once. Hence $l_f(\sigma) \geq \sum_m^{n-1} l_f(E_i) = l_f(r_{mn})$. \end{proof}.
\begin{defn} The set of geodesic rays in $\Gamma$ starting at the identity will be called the {\bf preboundary} of $\Gamma$. \end{defn}
Since geodesic rays starting at the identity in $\Gamma$ are also Floyd geodesics, we shall denote the preboundary as $Pre(\partial)_f \Gamma$, emphasizing the scaling
function $f$.
\begin{rmk}\label{trivialrmk}
A sequence of points tending to infinity along a geodesic ray $r$ starting at the identity is necessarily Cauchy in the Floyd metric and hence $Pre(\partial)_f \Gamma$
can be identified with the ideal points of these rays. If $p$ denote such an ideal point, we shall also denote the ray from $1$ corresponding to $p$ as $[1,p)$.
It follows from the definition of Floyd boundary that the Floyd boundary of $\Gamma$ is a singleton set if and only if for any pair $p, q \in Pre(\partial)_f \Gamma$,
and sequences $p_n \rightarrow p$, $q_n \rightarrow q$, there exist paths $\sigma_n$ in $\Gamma$ joining $p_n, q_n$ such that $l_f (\sigma_n) \rightarrow 0$
as $n \rightarrow \infty$. \end{rmk}
The following Theorem will be used later.
\begin{theorem} \cite[Proposition 3.4.6]{ger-floydgafa}
Let
$G$ be relatively hyperbolic with respect to a collection $\mathcal P$ of subgroups.
Then there exists $\lambda \in (0,1)$ such that
the identity map $G\to G$ extends to a continuous equivariant map $F$
from the Floyd completion $\overline{(\Gamma, d_f)}$ with respect to $ f(n) = \lambda^n$
to the Bowditch completion of $G$ with respect to $\mathcal P$.
In particular, the Floyd boundary of $G$ is non-trivial.\label{gerfloydgafa}
\end{theorem}
\subsubsection{ Controlled Floyd Separation}\label{cfs}
It is at this stage that we introduce an essentially new ingredient.
\begin{defn}\label{cdef}
Let $f$ be a scaling function and let $l(\sigma)$ denote the Floyd length of a path $\sigma$ with respect to the scaling function $f$.
Let $\Lambda$ denote a $G-$invariant collection of paths in $\Gamma$ such that for every $p, q\in \Gamma$, the geodesic joining
$p, q$ belongs to $\Lambda$. Let $\Lambda(p,q)$ denote the subcollection of paths in $\Lambda$ joining $p, q$.
We let
$d_{f, \Lambda}(p,q) = inf_{\sigma \in \Lambda(p,q)} (l(\sigma))$.
If $\Lambda$ consists of all $(\kappa, \kappa)$-quasigeodesics then we also denote $d_{f, \Lambda}(p,q)$ by $d_{f, \kappa}(p,q)$.
For $p, q \in Pre(\partial)\Gamma$ we define $$d_{f, \Lambda}(p,q) := {\rm liminf}_{p_n\rightarrow p, q_n\rightarrow q} \, d_{f, \Lambda}(p_n,q_n),$$ where $p_n \in [1,p)$
and $q_n \in [1,q)$ are Cauchy sequences converging to $p, q$ respectively. \end{defn}
Note that for $p, q \in Pre(\partial)\Gamma$, the quantity $d_{f, \Lambda}(p,q)$ is independent of the Cauchy sequences $\{ p_n\}, \{ q_n \}$.
If $\Lambda$ consists of $(\kappa, \kappa)$-quasigeodesics and $p, q \in Pre(\partial)\Gamma$, then $d_{f, \Lambda}(p,q)$ is also denoted as $d_{f, \kappa}(p,q)$.
\begin{defn}
If $d_{f, \Lambda}(p,q)>0$ for $p, q \in Pre(\partial)\Gamma$, we shall say that $p, q$ are {\bf Floyd separated with respect to $\Lambda$}.
If $d_{f, \kappa}(p,q)>0$ for $p, q \in Pre(\partial)\Gamma$ and for all $\kappa$, we shall say that $p, q$ are {\bf Floyd separated with respect to quasigeodesics}. \end{defn}
\begin{rmk} (Gerasimov) Note that $d_{f, \Lambda}(p,q)$ is not necessarily a metric as the triangle inequality need not be satisfied. \label{gerrmk} \end{rmk}
In future, if we want to emphasize that the Floyd length of a path $\sigma$ is being computed with respect to a Floyd function $f$
and $\Lambda$ consists of $(\kappa, \kappa)$-quasigeodesics, we shall have occasion to use the suggestive notation $l_f^\kappa (\sigma)$
in place of $l_f (\sigma)$ only to remind ourselves that $\Lambda$ consists of all $(\kappa, \kappa)-$quasigeodesics and hence $\sigma$ itself is a
$(\kappa, \kappa)-$quasigeodesic.
\begin{rmk} \label{dah} Let $\gamma$ be a bi-infinite Morse geodesic in the Cayley graph $\Gamma$ of a group
(or more generally a space $X$). Then the end-points of $\gamma$ in $Pre(\partial)\Gamma$
will necessarily be Floyd separated with respect to quasigeodesics. In particular, hyperbolically embedded subgroups \cite{dgo} of mapping class groups,
$Out (F_n)$ and so on will have boundary points
that are Floyd separated with respect to quasigeodesics.
\end{rmk}
Floyd separation of $p, q \in Pre(\partial)\Gamma$ with respect to quasigeodesics is, in fact, weaker than the existence of a Morse geodesic joining $p, q$.
To see this, let $G$ be hyperbolic relative to a $\mathbb{Z} \oplus \mathbb{Z} $, e.g. the fundamental group of a finite volume
hyperbolic 3-manifold with one cusp. Let $\gamma$ be a bi-infinite geodesic that is a union of two rays $\gamma_1, \gamma_2$
such that $\gamma_1$ is Morse and $\gamma_2$ lies close to a peripheral flat. Then $\gamma$ is not Morse but $p, q \in Pre(\partial)\Gamma$
are Floyd-separated with respect to quasigeodesics.
However, the proof of Floyd separation with respect to quasigeodesics
that we provide in this paper (Theorem \ref{nontrivial}) actually furnishes the stronger conclusion of existence of a large collection of Morse (quasi)geodesics
(Corollary \ref{morse}).
Further, we believe that the source of Floyd separation in this paper comes from a subgroup ($K_2 \subset G$ in the example of Section \ref{eg})
which is {\it not } hyperbolically embedded in $G$ (See Question \ref{hypembed} at the end of the paper).
\subsection{Graphical Graded Small Cancellation}\label{wgsc} In this Subsection we give a discussion of graphical Graded Small Cancellation
following Wise \cite{wise-por, wise-qpor}. Our definitions
are equivalent to those occurring in \cite{wise-por, wise-qpor}.
Let $B$ be a metric graph with edges of length one. Let $A$ be a (non-metric)
topological graph (i.e. equipped with only a CW complex structure). A map $\phi : A \rightarrow B$ is {\em combinatorial} if
\begin{enumerate}
\item $\phi$ takes vertices
of $A$ to vertices
of $B$.
\item $\phi$ restricts to an immersion on the interior of every edge of $A$ (where an
immersion is a local injection).
\end{enumerate}
A combinatorial map $\phi$ induces a metric graph structure $A_\phi$ on $A$ after subdividing $A$
(e.g. by pulling back the CW structure on $B$ via $\phi$) making $\phi$ a simplicial isometry restricted to the interior of every edge of $A_\phi$.
A {\em path} $\phi: P\rightarrow B$ is a combinatorial map where $P$
is a real interval. $|P|$ will denote the length of $P$, i.e. the number of edges in the graph $P_\phi$.
When defined, $P Q\rightarrow B$ will denote the concatenation of $P\rightarrow B$ and $Q\rightarrow B$. Let
$P^{-1}$ denote the inverse of $P$. In this paper, there will be a correspondence between
combinatorial paths and words, and so the notation $|W |$ for the length of a word,
agrees with the notation $|W |$ for the length of the corresponding path.
A (non-metric) graph $A$ is {\bf graded} if it is equipped with a
function $g: Edges(A)\rightarrow {\mathbb N}$, where $g(J)$ is called the grade of $J$.
Let $\phi: A\rightarrow B$ be a combinatorial map where $A$ is a graded graph and $B$ is a metric graph. Let $A_\phi$ denote the induced metric graph structure on
$A$.
Let $J$ be a 1-cell of the (non-metric) graph
$A$ and $J_\phi$ the induced metric graph structure on it. $i_J$ denotes the inclusion of $J$ in $A$.
\begin{defn} A combinatorial path $\alpha: P\rightarrow J $ is a piece of $J$ of {\bf grade $n$} provided the following
hold:
\begin{enumerate}
\item There exists a path $\beta: P\rightarrow J^\prime$ where $J^\prime$ is an edge of $A$ with $g(J^\prime ) = n$ and $n$ is the least such natural number.
\item $\alpha: P\rightarrow J$ and $\beta: P\rightarrow J^\prime$ represent distinct paths in $A$, i.e.
$i_J (\alpha(P))$ and $i_{J^\prime} (\beta(P))$ are distinct edge-paths in $A_\phi$.
\item $\alpha: P\rightarrow J$ and $\beta: P\rightarrow J^\prime$ project to the same path in $B$, i.e.
$\phi(i_J (\alpha(P)))$ and $\phi(i_{J^\prime} (\beta(P)))$ give the same edge path in $B$.
\item $P$ is maximal with respect to the above conditions.
\end{enumerate}
\end{defn}
\begin{defn} \cite{wise-qpor}[Definition 2.2]
Let $\phi: A \rightarrow B$ be a combinatorial map from a (non-metric) graded graph $A$ to a metric graph $B$. $\phi$
satisfies the {\bf graded $c(p)$ condition}
provided that for each edge $J$ of the (non-metric) graded graph $A$, and for any expression of $ J_\phi$ as the
concatenation $P_1, \cdots, P_r$ of pieces of $J$ of grade $\leq g(J)$, at least $p$ of these pieces have grade equal to $g(J)$.
\end{defn}
\begin{theorem} \cite{wise-qpor}[Theorem 1.7] Let $\phi : A \rightarrow B$ be a
combinatorial map between a graded (non-metric) graph $A$ and a metric graph $B$.
If $\phi$ satisfies the graded $c(3)$ small-cancellation condition then
$\phi$ is $\pi_1$-injective. If $\phi$ satisfies the graded $c(5)$ small-cancellation condition
then $\phi_\ast (\pi_1 A) \subset \pi_1 B$ is malnormal.\label{wise-gsc} \end{theorem}
\section{The Main Example}\label{eg} The main example is constructed in two steps:
\begin{enumerate}
\item We first construct a malnormal infinitely generated subgroup $K=K_\infty$ of $F_2$ satisfying a number of properties.
\item Then we double $F_2$ along $K$ to get $G(\infty)=F_{2}*_{K_\infty} F_{2}$.
\end{enumerate}
\subsection{The Construction of $ K$}\label{constrn}
We shall first construct a sequence of malnormal subgroups $\{K_{n}\} \subset F_2 = F_2(a,b)$ by choosing a sequence of elements $g_n$
satisfying a suitable small cancellation condition such that
\smallskip
\begin{enumerate}
\item No proper power of $a$ or $b$ is an element of $K_n$ and hence $K_n$ is a proper infinite index subgroup.
\item $K_{n+1}=<K_{n}, g_{n+1}>$ where $g_{n+1} \in F_{2} \setminus K_{n}$.
\item $(g_{n+1},K_n)_1 = {\rm min}_{h \in (F_{2} \setminus K_{n})} (h, K_n)_1$.
\item $g_{n+1}$ is cyclically reduced.
\end{enumerate}
\smallskip
We start the induction with $K_1 := <g_1>-$ a malnormal cyclic subgroup of $F_2$; in particular $g_1$ is not a proper power.
We shall define $K=K_\infty = \bigcup_n K_n$ and ensure that $K$ is of infinite index in $F_2$.
The choice of $g_n$ satisfying a suitable small cancellation condition will ensure a few things:
\smallskip
\begin{enumerate}
\item $K=K_{\infty}$ is malnormal in $F_2$.
\item The limit set $\Lambda K$ of $K$ in $\partial F_2$ is all of $\partial F_2$.
\item $K$ is of infinite index in $F_2$.
\end{enumerate}
\smallskip
Recall from the Introduction that
\smallskip
\begin{enumerate}
\item $T(n)$ denotes the tower function of height $n$ with base 2, i.e. $T(n) = 2^{2^{2^{2 \cdots}}}$, where the power is taken
$ n$ times.
\item $U(n)(a,b)$ is the word (element) in the free group $F_2$ on generators $a, b$
given by $$U(n)(a,b)=[a^{(T(n)+1)}b^{(T(n)+1)}][a^{(T(n)+2)}b^{(T(n)+2)}]\cdots[a^{T(n+1)}b^{T(n+1)}].$$
\end{enumerate}
\smallskip
We choose $g_{n+1}$ inductively to be of the form $g_{n+1} = h_{n+1} W_{n+1},$ where $h_{n+1}$ satisfies the following:
\smallskip
\begin{enumerate}
\item $(h_{n+1},K_n)_1 = {\rm min} (r, K_n)_1$, where the minimum is taken over geodesic rays $r$ in $F_2$ starting at $1$ and $h_{n+1}$
is an initial segment of $r$.
\item $|h_{n+1}|=(h_{n+1},K_n)_1 +1$,
\end{enumerate}
\begin{comment}
Here we have omitted, for convenience, the generators $a, b$ in the expression for the words $g_i, h_i$.
\end{comment}
\smallskip
and $W_{n+1}$ is of the form $W_{n+1} = c_{n+1} U(N(n)) d_{n+1}$ where
\smallskip
\begin{enumerate}
\item[(A)] $N(n) $ is at least twice the maximum value of $m$ for which $ a^{m}$ or $b^{m}$ is a subword of some freely reduced word $w_i \in K_n$.
\item[(B)] $N(n) > |h_{n+1}|$.
\item[(C)] $c_{n+1} , d_{n+1}$ are chosen from $\{ a, b, a^{-1}, a^{-1}\}$ to ensure that $g_{n+1}$ is cyclically reduced.
\end{enumerate}
\smallskip
Define $$ K= K_\infty = \bigcup_n K_n.$$ By the choice of $g_i$, it follows that $K$ is of infinite index in $F_2$ since $K$ is infinitely generated free.
\subsection{$ K$ is malnormal}\label{maln}
\begin{lemma} $K_n \subset F_2$ is malnormal. \label{malnlemma} \end{lemma}
\begin{proof} It suffices by Theorem \ref{wise-gsc} to show that the map $\phi_n : Rose(K_n) \to Rose(F_2)$ between the roses corresponding to $K_n, F_2$
satisfies the graded $c(5)$ condition.
The $i$th petal (labeled by the generator $g_i$) of $Rose(K_n)$ is assigned grade $i$, $(i=1 \cdots n)$. Then by Conditions (A) and (B) in the construction of
$W_{n+1}$ above, pieces of grade $n$ are of the form $a^mb^m$ or $b^ma^{m+1}$. Since the number of such pieces is at least $5$,
the map $\phi_n : Rose(K_n) \to Rose(F_2)$
satisfies the graded $c(5)$ condition.
\end{proof}
\begin{prop}
$K=K_\infty \subset F_2$ is malnormal. \label{malnprop} \end{prop}
\begin{proof}
First observe that a nested union of malnormal subgroups is malnormal. Suppose $H = \bigcup_i H_i$, where $H_i \subset H_{i+1}$ and each $H_i$ is malnormal in $G$.
For some $h\in H, h \neq 1$ and $g \in G$ suppose that $ghg^{-1} = h_1 \in H$. Then there exists $n$ such that $h, h_1 \in H_n$ forcing $g \in H_n$ since $H_n$ is malnormal
in $G$. Hence $g \in H$ and $H$ is malnormal in $G$.
Since $K= K_\infty = \bigcup_n K_n$, and each $K_n$ is malnormal in $F_2$
by Lemma \ref{malnlemma}, it follows that $K$ is malnormal in $F_2$.\end{proof}
\subsection{$ K$ has full limit set}\label{fls}
\begin{prop} $\Lambda K = \partial F_2$. \label{full} \end{prop}
\begin{proof} This shall follow from the fact that in our construction, $(g_{n+1},K_n)_1 =(h_{n+1},K_n)_1 = {\rm min}_{h \in (F_{2} \setminus K_{n})} (h, K_n)_1$.
It suffices to show that for every $g \in F_2$, there exists $n$ and $h \in K_n$ such that $g \in [1,h]$, i.e. $g$ lies on the geodesic from $1$ to $h$.
Suppose not. Then $g \notin K_n$ for all $n$. Further, $(g,K_n)_1 \leq |g|$ for all $n$. Hence $(h_i,K_n)_1 \leq |g|$ for all $i \geq 2$.
Hence $|h_i| \leq |g|+1$ for all $i \geq 2$. This forces $h_i$ to range in a finite set -- a contradiction. \end{proof}
\subsection{Doubling} Define
\begin{enumerate}
\item $G(n)=F_{2}(a,b)*_{{K_n}(a,b) = {K_n}(x,y)} F_{2}(x,y)$.
\item $G=G(\infty)=F_{2}(a,b)*_{K_\infty(a,b) = {K_\infty(x,y)}}F_{2}(x,y)$.
\end{enumerate}
In Section \ref{gsc} we shall need to reindex $G(n)$ and define $G_n := G(n+j_0)$ for a fixed $j_0$; hence we ask the reader's
indulgence in this slight incongruity of notation.
We shall need the following fact from \cite{BF, mitra-ht}:
\begin{theorem} Let $G_0$ be a hyperbolic group and $H$ a malnormal quasiconvex subgroup. Then $G=G_0*_HG_0$ is hyperbolic and $H$ is quasiconvex in it. \label{dh}
\end{theorem}
The following is an immediate consequence:
\begin{prop} $G(n)$ is hyperbolic and $K_n$ is quasiconvex in it. \label{doublehyp}
\end{prop}
\begin{proof} It suffices by Theorem \ref{dh} to show that $K_n$ is malnormal quasiconvex. Malnormality is the content of Lemma \ref{malnlemma}.
Since $K_n$ is finitely generated in $F_2$, it is quasiconvex.
\end{proof}
\subsubsection{Geodesics in $G$} Fix a symmetric generating set $\{ a, b, x, y, a^{-1}, b^{-1}, x^{-1}, y^{-1}\}$ of $G(n), G$.
We first fix notation.
Let $a, b$ denote the generators of the first copy of $F_2$ and let $x, y$ denote the generators of the second copy of $F_2$. Let $\Gamma$ denote the Cayley
graph of $G$ with respect to (the symmetrization of) $a, b, x, y$. Let $\Gamma_1$ denote the (sub) Cayley graph of the subgroup $F_2(a,b) \subset G$. Similarly
let $\Gamma_2$ denote the (sub) Cayley graph of the subgroup $F_2(x,y) \subset G$.
Let $i_1, i_2$ denote the inclusion maps of $\Gamma_1$, $\Gamma_2$ respectively.
There exists natural projection maps $\Pi_j: \Gamma \to \Gamma_j$ identifying the generators $a, b$ with $x, y$ respectively.
\begin{lemma} $F_2(a,b) $ and $F_2(x,y)$ are isometrically embedded in $G(n)$ and $G$. \label{ret} \end{lemma}
\begin{proof}
Clearly $d(\Pi_j(u), \Pi_j(v)) \leq d(u, v)$.
Since $\Pi_j \circ i_j$ is the
identity map (on the first or second $F_2$ respectively) it follows that $F_{2}(a,b)$ and $F_{2}(x,y)$ are isometrically embedded in $G$. \end{proof}
We shall need the following normal form Lemma (cf. \cite[p. 285-6]{ls})
for free products with amalgamation. For the group $B \ast_A C$, let $i, j$ denote the inclusion maps
of $A$ into $ B, C$ respectively.
\begin{lemma} \label{normal}
Every element $g$ of $B \ast_A C$ which is not in the image of $A$
can be written in the {\bf normal form}
$ v_1 v_2 \cdots v_n$
where the terms $v_k$ lie in $B - i(A)$ or $C - j(A)$ and alternate between these
two sets. The length $n$, called the {\bf normal length} of the group element, is uniquely determined and two such expressions
$ v_1 v_2 \cdots v_n$ and $ w_1 w_2 \cdots w_n$ give the same element of $B \ast_A C$ iff there are
elements $a_1 , \cdots , a_{n−1} \in A$ so that
$ w_k = a_{k-1} v_k a_k^{-1}$.
(where $ a_0 = a_n = 1$).\end{lemma}
\begin{defn}
We shall say that a normal form word $g=v_1 v_2 \cdots v_n$ is in {\bf reduced normal form} if
for each $i$, the concatenation $v_iv_{i+1}$ is a geodesic word, i.e. amongst all representatives of the form $(v_ia_i) (a_i^{-1}v_{i+1})$ with each of
$(v_ia_i)$ and $ (a_i^{-1}v_{i+1})$ replaced by geodesic words, $v_iv_{i+1}$ is shortest. \label{rednormal} \end{defn}
\begin{prop} \label{normalgeod} Given an element $g\in G(n)$ there exists a normal form geodesic word $ v_1 v_2 \cdots v_n$ representing $g$.
\end{prop}
\begin{proof} Let $T$ denote the Bass-Serre tree of the splitting. Observe first that a normal form word as defined above projects to a path in $T$ that does not return
to a vertex after leaving it.
Now, let $u$ be any word that projects to a path in $T$ having a non-trivial subpath $\sigma$ that starts and ends at the same vertex of $T$. Let $u'$
be the subword of $u$ corresponding to $\sigma$. Then $u'$ corresponds to an element of either $F_2(a,b)$ or $F_2(x,y)$.
Replace $u'$ by the geodesic word in $F_2(a,b)$ or $F_2(x,y)$ representing it. By Lemma \ref{ret}, it follows that the resulting word is at most as long as $u$.
Continuing this process we finally obtain a normal form word whose length is at most that of $u$.
\begin{comment}
begin{enumerate}
\item The terms $v_i$ lie in $F_2(a,b) - i_1(K_n)$ or $F_2(x,y) - i_2(K_n)$ and alternate between these.
\item Each $v_i$ is a geodesic word in $F_2(a,b)$ or $F_2(x,y)$.
\item If $v_i = u_i a_i$ and $v_{i+1} = b_{i+1}u_{i+1}$ as words and $b_{i+1} = a_i^{-1} $ as group elements of $K(n)$, then $b_{i+1} = a_i^{-1} $
\end{enumerate}\
Now suppose we have two normal form words $ v= v_1 v_2 \cdots v_n$ and $ w= w_1 w_2 \cdots w_n$ giving the same element of $B ∗_A C$,
where $v$ satisfies the hypotheses of the Proposition. By Lemma \ref{normal}
there are
elements $a_1 , \cdots , a_{n−1} \in A$ so that
$ w_k = a_{k-1} v_k a_k^{-1}$.
(where $ a_0 = a_n = 1$). Since $v_i$'s are already reduced (being geodesic words in a free group) it follows that $|v| \leq |w|$. The Proposition follows.
\end{comment}
\end{proof}
\subsubsection{Cyclic Geodesics in $G(n)$}\label{cycgeod}
We shall now carefully choose a geodesic word $R_{n+1}$ in $G(n)$ denoting the new relator which when added to the relator set
of $G(n)$ gives $G(n+1)$. $R_{n+1}$ will satisfy the condition that all its cyclic conjugates are geodesic words in $G(n)$.
We proceed inductively to define $R_n$ of the form $w^{-1}(x,y)w(a,b)$.
Recall that
$g_{n+1} = h_{n+1} c_{n+1} U(N(n)) d_{n+1}$, where $h_{n+1}$ is a geodesic word in $F_2$.
Let $R_n^0 = g_n^{-1}(x,y)g_n(a,b)$. $R_n$ will be obtained from $R_n^0$ after cyclic reduction.
We shall define $w_{n+1}$ to be a reduced normal form geodesic representative of $g_{n+1}^{-1}(x,y)g_{n+1}(a,b)$ in $G(n)$
(of normal length $2$ as per Lemma \ref{normal}) obtained by replacing a maximal subword $w_0$ of the form $w^{-1}(x,y)w(a,b)$ by a subword $w'$ of the same form
using the structure of $G(n)$ as a double along $K_n$.
\begin{comment}
To do this, we use the inductive hypothesis that $R_i$ is of the form $w^{-1}(x,y)w(a,b)$ for all $i < n$.
\end{comment}
By Lemma \ref{normal} any normal form word representing $g_{n+1}^{-1}(x,y)g_{n+1}(a,b)$ is of the form
$g_{n+1}^{-1}u^{-1}u(x,y)g_{n+1}(a,b)$, where $u \in K_n$. In particular, the shortest normal form representative of
$g_{n+1}^{-1}(x,y)g_{n+1}(a,b)$ is of the form $w^{-1}(x,y)w(a,b)$ where $w(a,b) \in (F_2(a,b) \setminus K_n(a,b))$ and
$w(x,y) \in (F_2(x,y) \setminus K_n(x,y))$. Let $R_{n+1}$ denote such a representative.
Since
Item(A) of the defining condition on $U(N(n))$ demands that $N(n) $ is at least
twice the maximum value of $m$ for which $ a^{m}$ or $b^{m}$ is a subword of some freely reduced word in $K_n$, it follows that after any reduction as above, $w(a,b)$ contains a subword of the form
$V(n)(a,b)=[a^{\lfloor \frac{(T(n)+1)}{2} \rfloor}b^{(T(n)+1)}][a^{(T(n)+2)}b^{(T(n)+2)}]\cdots[a^{T(n+1)}b^{T(n+1)}].$
Next, $g_{n+1}(a,b) g_{n+1}^{-1}(x,y)$ is already a shortest normal form representative as it contains $b^{T(n+1)}y^{-T(n+1)}$
as a subword which cannot be shortened further, again by Item(A) of the defining condition on $U(N(n))$.
It follows by the same reason that every cyclic conjugate of $R_{n+1}$ is a geodesic in $G(n)$.
Thus $R_{n+1}$ satisfies the following properties:
\begin{enumerate}
\item[ Property 1:] All cyclic conjugates of $R_{n+1}$ are geodesics in $G(n)$.
\item[ Property 2:] $R_{n+1}$ is of the form $w_n^{-1}(x,y)w_n(a,b)$ where
$w(a,b)$ contains a subword of the form
$V(n)(a,b)=[a^{\lfloor \frac{(T(n)+1)}{2} \rfloor}b^{(T(n)+1)}][a^{(T(n)+2)}b^{(T(n)+2)}]\cdots[a^{T(n+1)}b^{T(n+1)}].$
\item[ Property 3:] $|R_{n+1}| = O(|V(n)(a,b)|) = O(T(n+1)^2)$
\item[ Property 4:] The largest piece of $R_{n+1}$ is of the form $b^{(T(n+1)-1)}a^{(T(n+1)}$ which has length $2T(n+1)-1$.
(since $N(n) > |h_{n+1}|$ by choice, initial segments cannot be maximal pieces).
\end{enumerate}
The group $G$ therefore admits a presentation $G=\langle S|\RR\rangle$, where
\begin{enumerate}
\item $S = \{a,b,x,y,a^{-1},b^{-1},x^{-1},y^{-1}\}.$
\item $\RR = \bigcup_i \{ R_i\}$.
\end{enumerate}
A more general statement is true. We learnt the proof of the following from Dani Wise. We refer the reader to \cite{wise-cbms} for details
on non-positively curved (npc) complexes.
\begin{prop} $G=F*_A F$ is a double of a finitely generated free group $F$ along a finitely generated subgroup $A$,
with generating set given by the union of the generating sets of the two copies of $F$. Then $G$ has cohomological dimension $2$.
Then any reduced normal form word $g$ of the form $uvw$ (i.e. of normal length 3), where $u,w$ are geodesic
words in the first factor and $v$ is a geodesic in the second factor are geodesics in $G$.
More generally any reduced normal form word is a geodesic.
\label{cycgeodprop} \end{prop}
\begin{proof} By Theorem 7.2 of \cite{wise-cbms},
$G$ can be represented as the fundamental group of a non-positively curved
($\mathcal{VH}$) square complex formed by taking two graphs $\mathcal G$ (with $F$ as fundamental group)
as vertex spaces, and an edge space that is a ${\mathcal G} \times I$ that is attached by a local isometry on each side.
It immediately follows that $G$ has cohomological dimension $2$.
Then $g$ corresponds to a path $\sigma$ labeled $uvw$ consisting of three vertical paths labeled $u$, $v$, $w$
joined by two horizontal edges, one from the terminal point of $u$ to the initial point of $v$ and one
from the terminal point of $v$ to the initial point of $w$.
Consider a disk diagram $D$ between $\sigma$ and an arbitrary edge path $\eta$ with the same end-points.
The dual curves emanating from $v$ cannot end on $u$ or $w$ because of the hypothesis that $uv$ and $vw$ are geodesics.
The dual curves emanating from $u$ cannot end on the $w$ since then the middle path $v$
must have dual curves that start and end on itself, contradicting that it is itself reduced.
Thus all dual curves emanating from $\sigma$ must end on $\eta$.
Thus $\eta$ has at least as many vertical edges as $\sigma$, forcing $\sigma$ to be a geodesic.
This proves the first statement.
The proof of the last statement is now completed by a straightforward induction: We repeat the above argument after
combining the first $(n-1)$ words into a single geodesic.
\end{proof}
\subsection{Subgroups of $G$}
\begin{prop} All finitely presented subgroups $H$ of $G$ are hyperbolic.\label{subgp} \end{prop}
\begin{proof} Let $H = \langle S_1| R_1 \rangle$ be a finite presentation. Let $S_1 = \{ s_1, \cdots, s_n \};
R_1 = \{ r_1, \cdots, r_m \}$. Let $H_0 $ be the subgroup of $G(0)$ generated by $S_1$. Then there exists $n$ such that each $r_i$ is trivial in $G(n)$
and hence in $G(m)$ for all $m \geq n$. Since $H$ is a subgroup of $G$, it follows that $H$ is a subgroup of $G(m)$ for all $m \geq n$.
By the first part of Proposition \ref{cycgeodprop}, $G$ has cohomological dimension $2$.
Hence $H$ is a finitely presented subgroup of a hyperbolic group $G(n)$ of cohomological dimension $2$.
It now follows from \cite{gersubgp} that $H$ is hyperbolic.
\end{proof}
\section{Non Relative Hyperbolicity}\label{nrh}
In this Section we show that:
\begin{theorem} \label{nrhthm} $G =F_{2}*_{K} F_{2}$ is not hyperbolic relative to any proper collection of parabolic subgroups $\PP$. \end{theorem}
\begin{proof}
Suppose not, i.e. $G$ is hyperbolic relative to a proper collection of parabolic subgroups $\PP$.
By Theorem \ref{hruska-qi}, $F_{2}(a,b)$ and $F_{2}(x,y)$ are relatively quasiconvex in $G$ and hence are relatively hyperbolic with respect to $\PP \cap F_{2}(a,b)$,
$\PP \cap F_{2}(x,y)$ respectively. Since
$K=F_{2}(a,b)\cap F_{2}(x,y) (\subset G)$, it follows that $K$ is also relatively quasiconvex in $G$ by Theorem \ref{hruska-intn}.
But $K \subset F_{2}(a,b) \subset G$. Hence $K$ is also relatively quasiconvex in $F_{2}(a,b)$, $F_{2}(x,y)$.
By Proposition \ref{full}, $K$ has full limit set in $F_{2}(a,b)$ as well as in $F_{2}(x,y)$, i.e. $\Lambda K=\partial F_{2}(a,b) \, \,$
and $\Lambda K=\partial F_{2}(x,y)$.
By Lemma \ref{yang}, it follows that $K$ is of finite index in $F_{2}(a,b)$. This contradicts the construction of $K$ as an infinite index subgroup.
This final contradiction yields the Theorem. \end{proof}
Theorem \ref{nrh} will also follow from Theorems \ref{floydtrivial} and \ref{gerfloydgafa}, but the above proof is direct and simple.
\section{Graded Small Cancellation Structure of $G$}\label{gsc}
In this section our primary goal is to show that the group $G = F_2(a,b) *_K F_2(x,y)$ constructed in Section \ref{eg} satisfies
a certain (technical) graded small cancellation condition \cite{oos, olshanski-book}. This will give (as usual with small cancellation theory)
a number of geometric consequences.
\subsection{Definitions and Preliminaries on Graded Small Cancellation}
\begin{defn}
A set $\mathcal R$ of words in an
alphabet is {\bf symmetrized} if for any
$R\in \mathcal R$, all cyclic shifts of $R^{\pm 1}$ are contained in
$\mathcal R$ and each $R$ is reduced. (Note that the feature that each $R$ is {\bf reduced} is incorporated as part of the definition.)\end{defn}
\begin{defn} \cite[Definition 4.2]{oos} \label{piece}
Let $G$ be a group with a finite symmetric generating set $S$. Let $\RR$ be a
symmetrized set of words in $S$. Given $\epsilon >0$, a subword $U$ of a word $R\in \mathcal R$ is called an {\it
$\epsilon$-piece} if there exists $R^\prime \in \RR$
such that:
\begin{enumerate}
\item $R= UV$, $R^\prime = U^\prime V^\prime $, for
some words $V, U^\prime , V^\prime $,
\item $U^\prime = YUZ$
for words $Y,Z$ satisfying $\max \{ |Y|, \,|Z|\} \leq \epsilon $.
\item $YRY^{-1}\neq R^\prime $.
\end{enumerate}
\end{defn}
In the above definition $`='$ is to be interpreted as equality as elements of $G$.
In the following definition we rename condition SC of \cite{oos} as the relative small cancellation condition.
\begin{defn}\cite[Definition 4.3]{oos}\label{rscdefn} Let $\epsilon \geq 0$, $\mu\in (0,1)$, and $\rho
>0$.
A symmetrized set $\RR$ of words in a symmetrized generating set $S$ for a group $G$
satisfies the {\bf relative small cancellation condition $C(\epsilon, \mu ,\rho)$} if
\begin{enumerate}
\item[($RSC_1$)] The words in $\mathcal R$ represent geodesics in $G$.
\item[($RSC_2$)] $|R|\geq \rho $ for any $R\in \mathcal R$.
\item[($RSC_3$)] for any $\epsilon$-piece $U$ and any $R\in \RR$, with $U \subset R$, $|U| \leq \mu |R|$.
\end{enumerate}
\end{defn}
\begin{defn}\cite[Definitions 4.12, 4.14]{oos}\label{gscdefn} Fix $\alpha = 10^{-2}, K=10^6$. The presentation
\begin{equation}
G=\langle S | \mathcal{ R}\rangle =\left\langle S\,\left|\,
\bigcup\limits_{i=0}^\infty \mathcal R_i\right.\right\rangle
\end{equation}
is a {\bf graded small cancellation presentation} if there exist
sequences ${\epsilon} =({\epsilon}_n)$, $\mu
=(\mu _n)$, and $\rho =(\rho _n)$ of positive real numbers
($n=1,2\dots $) satisfying:
\begin{enumerate}
\item[($ GSC_0$)] The group $G_0=\langle S\mid \mathcal R_0\rangle $
is $\delta_0$-hyperbolic for some $\delta_0$.
\item[($GSC_1$)] $\epsilon_{n+1}>8\max\{|R|, R\in \RR_n\}=O(\rho_n)$.
\item[($GSC_2$)] $\mu_n\rightarrow 0$ as $n \rightarrow \infty$, $\mu _n\leq
\alpha$, and $\mu_n\rho_n>K\epsilon_n$ for any $n\ge 1$.
\item[($GSC_3$)] For all $n\geq 1$, $\RR_{n}$ satisfies $C({\epsilon}_n,
\mu _n, \rho _n)$ over $G_{n-1}=\left\langle S\, \left|\,\right.
\bigcup\limits_{i=0}^{n-1} \mathcal R_i\right\rangle .$
\end{enumerate}
\end{defn}
\begin{rmk} Though we have fixed $\alpha = 10^{-2}, K=10^6$ following \cite{oos} above, any sufficiently large $K$ and sufficiently small $\alpha$ suffices.\end{rmk}
We shall need the following:
\begin{lemma} \cite[Lemmas 4.13, 4.18]{oos} \label{delta} Let
\begin{equation}
G=\langle S | \mathcal R\rangle =\langle S|
\bigcup_0^\infty \mathcal R_i\rangle
\end{equation}
be a {\bf graded small cancellation presentation}. Then $G_n$ is $\delta_n$-hyperbolic, where $\delta_n \leq 4 max_{R\in \RR_n} |R|$.
Any subword of any word $R_n\in \RR_n$ of length at most $|Rn |/2$ is a $(1 - o (1), 0)$-quasi–geodesic in the Cayley graph of $G$ with respect to the
generating set $S$.\end{lemma}
\subsection{Graded Small Cancellation for $G$}
\begin{theorem} The group $G=F_2 *_KF_2$ equipped with the presentation $G=\langle S|\RR\rangle$, where
$S = \{a,b,x,y,a^{-1},b^{-1},x^{-1},y^{-1}\}$
and $\RR = \bigcup_i\{ R_i\}$ (cf. Section \ref{cycgeod} and Proposition \ref{cycgeodprop}) is a graded small cancellation group and the presentation is a graded small cancellation presentation.
\label{g-gsc} \end{theorem}
\noindent {\bf Proof:} The proof will proceed by checking the properties in Definitions \ref{gscdefn} and \ref{rscdefn}.
\noindent {\bf Proof of Condition $GSC_0$:} By Theorem \ref{dh}, each $G(k) =F_2 *_{K_k}F_2$ is hyperbolic and hence any $G(k)$ can be taken as the starting group $G_0$.
$k$ will be taken large enough to ensure that the parameter values $\alpha = 10^{-2}, K=10^6$ are satisfied
and will be determined during the course of the proof. We reindex if necessary by setting $G_i = G(k+i)$ for all $i$.
\noindent {\bf Proof of Condition $GSC_1$:}
\begin{enumerate}
\item $\rho_n = |R_{n+k}|$.
\item $\epsilon_{n+1} = 10 \rho_n$.
\end{enumerate}
Condition $GSC_1$ immediately follows.
\noindent {\bf Proof of Condition $GSC_2$:}
It suffices to choose $\mu_n > \frac{ 10^7 \rho_{n-1}}{\rho_n}$. Recall from Property 3 satisfied by $R_{n+k}$ that $\rho_n = O((T(n+k))^2)$.
Since $\frac{(T(n+k-1))^2}{(T(n+k))^2} \rightarrow 0$ as $n\rightarrow \infty$ very fast (more than exponentially), it will be easy to make a specific choice for $\mu_n$
during the proof of $GSC_3$ below.
\subsubsection{Proof of $GSC_3$:}
This is the crucial step where we shall prove that the set $\RR(n+k)$ of cyclic conjugates of $R_{n+k}$ satisfies the {\bf relative
small cancellation condition } $C(\epsilon_{n},\mu_{n},\rho_{n})$ (Definition \ref{rscdefn}) over
$G_{n-1}=\langle S|\RR=\underset{i=0}{\overset{n+k-1}\bigcup}\RR(i)\rangle$, for suitable choices of
$\epsilon_{n}$, $\mu_{n}$ and $\rho_{n}$, for all $n$.
We observe that, by Proposition \ref{doublehyp} all the groups $G(i)$ are $\delta_{i}$-hyperbolic.
We need to therefore choose parameters $\epsilon_{n}$, $\mu_{n}$ and $\rho_{n}$ for $\RR(n+k)$
satisfying $C(\epsilon_{n},\mu_{n},\rho_{n})$ over
$G_{n-1}=\langle S|\RR=\underset{i=0}{\overset{n+k-1}\bigcup}\RR(i)\rangle$.
\noindent {\bf Proof of $RSC1$:}
$\RR(n+k)$ consists of cyclic conjugates of exactly one relator $R_{n+k}$ all of which are geodesics by their construction (Section \ref{cycgeod}
and Proposition \ref{cycgeodprop}).
\noindent {\bf Proof of $RSC2$:}
Since all cyclic conjugates of $R_{n+k}$ are geodesics (Section \ref{cycgeod} and Proposition \ref{cycgeodprop}), all of them have the same length $\rho_n$.
\noindent {\bf Proof of $RSC3$:}
We will show that for the choice of $\epsilon_{n} = 10 \rho_{n-1}$ as in the Proof of Condition $GSC_1$, the length of an $\epsilon_{n}$-piece
in $\RR(n+k)$ has length less than $\mu_{n}|R_{n+k}|$ for suitably chosen $\mu_{n}$.
Let $U$ be any $\epsilon_{n}$-piece. By the form of the words $R_{n+k}$, a subword $W$
with length at least half that of $U$ is a geodesic lying
in one of the factors $F(a,b)$ or $F(x,y)$. Further, since $U, U^\prime$ are geodesics starting and ending at most $\epsilon_n$-apart, it follows that
there exists a geodesic subword $W^\prime$ of $U^\prime$ such that the initial and final points of $W, W^\prime$ are at most $(\epsilon_n + 4 \delta_{n-1})$ apart.
Hence assume without loss of generality that $U$ is an $(\epsilon_n + 4 \delta_{n-1})$-piece lying entirely in $F_2(a,b)$.
It follows from Definition \ref{piece} that
\begin{enumerate}
\item $R= UV$, $R^\prime = U^\prime V^\prime $, for
some words $V, U^\prime , V^\prime $ and $R, R^\prime \in \RR(n+k)$,
\item $U^\prime = YUZ$
for words $Y,Z$ satisfying $\max \{ |Y|, \,|Z|\} \leq (\epsilon_n + 4 \delta_{n-1}) $.
\item $YRY^{-1}\neq R^\prime $.
\end{enumerate}
After reducing $Y^{-1}U^\prime Z^{-1}$ to reduced normal form, we obtain therefore a piece $U_0$ (in the usual small cancellation
sense) of length at least $|U| - 2 (\epsilon_n + 4 \delta_{n-1})$.
Recall that by Property 4 of $R_{n+k}$, a maximal piece (in the usual small cancellation
sense) has length at most $2T(n+k)$.
Hence $|U_0| < 2 T(n+k)$ and so $|U| < 2 T(n+k) + 2 (\epsilon_n + 4 \delta_{n-1})$.
\noindent {\bf Choice of $\mu_n$:}
It therefore suffices to choose $\mu_n$ such that $2 T(n+k) + 2 (\epsilon_n + 4 \delta_{n-1}) \leq \mu_n \rho_n$. Define $\mu_n = 2max
\{ \frac{2T(n+k)+20 \rho_{n-1}}{ \rho_n}, \frac{ 10^7 \rho_{n-1}}{\rho_n}\}$.
Since $\rho_n = O((T(n+k))^2)$, and $\delta_{n-1} = O(\rho_n)$, it follows that $\mu_n \rightarrow 0$ as $n \rightarrow \infty$.
\noindent {\bf Choice of $k$:}
It remains to choose $k$. The choice of $k$ is dictated by the requirement that $\mu_n \leq \alpha$ for all $n$. Since $\mu_m \rightarrow 0$ as $m \rightarrow \infty$
it follows that there exists $k>0$ such that $\mu_m \leq \alpha$ for all $m\geq k$. This decides the choice of $k$ and completes the proof of
Theorem \ref{g-gsc}. $\Box$
\section{Asymptotic Cone of $G=F_{2}*_{K} F_{2}$}
We refer the reader to \cite{ds} for details on tree-graded spaces and \cite{gromov-ai, oos} for details on asymptotic cones.
\subsection{Tree-graded Spaces}\label{tgac}
\begin{defn} Let $X$ be a complete geodesic metric space and let $\mathcal P$ be a collection of closed
geodesic subsets. Then $ X$ is said to be tree-graded with respect to $\mathcal P$ if the following hold:
\begin{enumerate}
\item Any two elements of $\mathcal P$ have at most one common point.
\item Every simple geodesic triangle (i.e. a simple loop composed of three geodesics) in $X$ is contained
in exactly one element of $\mathcal P$.
\end{enumerate}
\end{defn}
\begin{defn} \cite{oos} If $ X$ is tree-graded with respect to $\mathcal P$ such that each element of $\mathcal P$
is a circle of radius uniformly bounded away from $0$ and $\infty$, then $X$ is called a circle-tree.
\end{defn}
\subsection{Asymptotic Cones of Graded Small Cancellation Groups} Let $\omega$ be a non-principal ultrafilter.
We shall say that $a_{i}= O_\omega (b_i)$ if the $\omega-$limit of $\{ \frac{a_i}{b_i}\}$ is finite and non-zero. Similarly,
$a_{i}= o_\omega (b_i)$ if the $\omega-$limit of $\{ \frac{a_i}{b_i}\}$ is zero.
\begin{defn}\label{od-vis}\cite[Definitions 4.16,4.19]{oos}
Given an ultrafilter $\omega $ and a scaling sequence $d=(d_n)$, a sequence of real numbers $f=(f_n)$ is said to be {\bf $(\omega ,
d)$--visible} or simply asymptotically visible, if there exists a subsequence $(f_{n_i})$ of $f$ such
that $f_{n_i} = O_\omega (d_i)$.
For a group $G$ and its Cayley graph $\Gamma$, we shall say that a sequence of loops $\{ \sigma_i\}\subset \Gamma$ is asymptotically visible
if the sequences $\{ d(1, \sigma_i) \}$ and $\{ |\sigma_i| \}$ are asymptotically visible.
\end{defn}
The following is one of the main Theorems of \cite{oos}.
\begin{theorem}\label{circletree} \cite{oos}[Theorem 4.17]
Let $G$ be a group having a graded small cancellation presentation with symmetrized relator set $\RR= \cup_i\{ R_i \}$, where $\{ R_i \}$ is the grade $i$
symmetrized relator set.
For any ultrafilter
$\omega$, and any sequence of scaling constants $d=(d_n)$, the
asymptotic cone $\CG$ is a circle-tree. $\CG $ is an $\mathbb
R$--tree if and only if the sequence $(\rho_n)$ from Definition
\ref{gscdefn} is not $(\omega , d)$--visible.
Further any simple loop $C$ in $\CG$ can be realized as an $\omega-$limit of translates of $\{ R_i \}$'s.
\end{theorem}
The last statement follows from
Lemma 4.23 of \cite{oos}.
\subsection{Properties of the Asymptotic Cone of $G$}
\begin{prop} $Con^{\omega}(G)$ is a circle-tree. \label{cctree}\end{prop}
\begin{proof} This follows from Theorems \ref{g-gsc} and \ref{circletree}. \end{proof}
Let $\GG$ denote $Con^{\omega}(G)$.
\begin{prop} $i: K_2 \rightarrow G$ induces a bi-Lipschitz embedding $\hat{i}: Con^{\omega}(K_{2}) \hookrightarrow \GG$. \label{blemb} \end{prop}
\begin{proof} Since $K_2 \hookrightarrow F(a,b)$ is a quasi-isometric embedding and $F(a,b) \hookrightarrow G $ is an isometric embedding by Lemma \ref{ret}, it follows that
$i: K_2 \rightarrow G$ is a quasi-isometric embedding. The result follows. \end{proof}
\subsubsection{Limit Circles in $Con^{\omega}(G)$ intersect $Con^{\omega}(K_{2})$ trivially}
The asymptotic cone $\mathcal{C}^{\omega}(G,\{d_n\})$ of the graded small cancellation
group $G$ is a circle-tree with limit circles of uniformly bounded radius.
Let $\CC$ denote the collection of circles in $\mathcal{C}^{\omega}(G,\{d_n\})$.
The asymptotic cone $\mathcal{C}^{\omega}(K_2,\{d_n\})$ of the free group
$K_2$ (constructed at the second stage of the inductive construction of $K$) is an $\mathbb{R}$-tree $\TT$. The next Theorem is crucial in showing that
$K_2$ is quasiconvex in $G$ in a strong sense. This, in turn will lead to the conclusion that any pair of distinct geodesic rays in $K_2$ starting at the origin
are Floyd separated with respect to quasigeodesics.
\begin{theorem} For any $C \in \CC$, the intersection $C\cap \TT$ is either empty or contains a single point. \label{trivialintn} \end{theorem}
\begin{proof} Suppose not. Since $\GG$ is a circle-tree by Theorem \ref{circletree}, it follows that
there exists a bigon $\sigma$ consisting of the union of two arcs:
\begin{enumerate}
\item a geodesic $\overline{p q}$ in $\TT$.
\item an embedded path $\gamma$ in $\GG$ joining $p, q$ with interior disjoint from $\overline{p q}$ .
\end{enumerate}
Since all elements of $\CC$ can be obtained as $\omega-$limits of translates of elements of $\RR$
by Theorem \ref{circletree}, we can assume that there exists a sequence of translates $\{ g_{i_n}R_{i_n} \}$ whose $\omega-$limit
is $\sigma$. For the purposes of this proof, reindex by setting $g_{i_n}R_{i_n}=R_n$. Geodesic segments in the Cayley graph of $G$ will be denoted
by $[E,F]$.
Hence there exists a sequence $\{R_n, A_n, B_n, C_n, D_n \}$ such that (see Figure 1)
\begin{enumerate}
\item $R_n \in \RR$
\item The scaling sequence $d_n = O(dia(R_n)) = O(\rho_n) = O(\delta_n)$, where $\delta_n$ is the hyperbolicity constant of $G(n)$
(The last equality follows from Lemma \ref{delta} and the asymptotic visibility of the sequence $\{ R_n\})$.
\item $[A_n, B_n] \subset K_2$;
\item $[D_n, C_n] \subset R_n$
\item $[A_n, D_n], [B_n, C_n]$ are geodesics in normal form in $G$.
\item $|[A_n, D_n]|, |[B_n, C_n]|$ are $o_\omega(d_n)$
\item $|[A_n, B_n]|, |[D_n, C_n]|$ are $O_\omega(d_n)$
\item $lim_\omega (A_n) = lim_\omega (D_n) = p$
\item $lim_\omega (B_n) = lim_\omega (C_n) = q$
\end{enumerate}
\begin{figure}\label{quadl}
\begin{tikzpicture}[thick]
\path[draw] (-4,0) coordinate
-- ( 4,0) coordinate [label=right: $K_2$] (K_2) ;
\node[draw,shape=ellipse, minimum height=1.6cm, minimum width=5cm] (Ell) at (0,1.25) {};
\draw (-2,0.75) coordinate[label=left: $\scriptstyle D_n$] (D_n)
-- (-2,0) coordinate[label=below: $\scriptstyle A_n$] (A_n);
\draw (2,0.75) coordinate[label=right: $\scriptstyle C_n$] (C_n)
-- (2,0) coordinate[label=below: $\scriptstyle B_n$] (B_n);
\draw (0,1.25) node {$R_n$};
\foreach \point in {A_n,B_n,C_n,D_n}
\fill [black] (\point) circle (1.5pt);
\end{tikzpicture}
\caption{}
\end{figure}
By cutting off initial and final pieces from $[A_n, B_n], [D_n, C_n]$ of length $o(d_n)$ if necessary and using thinness of quadrilaterals we can assume
without loss of generality that $|[A_n, D_n]|, |[B_n, C_n]|$ are at most $O(d_{n-1})=O(\delta_{n-1})$. More precisely, choose a subsegment
$[A_n^\prime, B_n^\prime]$ such that $d(A_n, A_n^\prime) = d(B_n, B_n^\prime) = 2 max \{|[A_n, D_n]|, |[B_n, C_n]|\}$. Then there exist points
$D_n^\prime, C_n^\prime$ on $[D_n, C_n]$ such that $d(D_n^\prime, A_n^\prime) \leq 2 \delta_{n-1}$ and $d(C_n^\prime, B_n^\prime) \leq 2 \delta_{n-1}$.
Rename the four vertices $A_n^\prime, B_n^\prime, C_n^\prime, D_n^\prime$ as $A_n, B_n, C_n, D_n$.
Then as in the proof of Condition $RSC_3$ in the proof of Theorem \ref{g-gsc} we get a genuine (in the usual small cancellation
sense) piece of length $O(|[A_n, B_n]|)$
as a subword of $[A_n, B_n]$. Further, any maximal piece of $R_n$ is $o(\rho_n)$ and hence
$\frac{|[A_n, B_n]|}{\rho_n} \rightarrow 0$ as $n \rightarrow \infty$. On the other hand, since
$lim_\omega (A_n) = lim_\omega (D_n) = p$
and $lim_\omega (B_n) = lim_\omega (C_n) = q$, and since $p \neq q$, it follows that $\frac{|[A_n, B_n]|}{\rho_n} $ is bounded away from zero.
This is a contradiction.
\end{proof}
\begin{comment}
Passing to a further subword of $[D_n, C_n]$ if necessary we can assume that $Lab([D_n, C_n]) \in F_2(a,b)$ as the label of $[D_n, C_n]$ must contain at least half
the label in either $F_2(a,b)$ or in $F_2(x,y)$.
The rest of the proof is similar to that of Property $RSC_3$ in Theorem \ref{g-gsc}.
For concreteness put each of $[A_n, D_n], [B_n, C_n]$ in reduced normal form.
that the normal form for the path $[A,B]$ has normal length zero, whereas the normal form for the path $[ADCB]$
has normal length at least one (as the segment $[DC]$ is long and does not lie in $K_{n-1}$ even after cancelling off small bits of length $O(d_{n-1})$
from the beginning and end.
\end{comment}
\section{Controlled Floyd Separation}\label{boundary}
The main purpose of this section is to show:
\begin{theorem} For any Floyd function $f$, $c \geq 1$, and $p, q \in \partial K_2 \subset Pre(\partial) G$, the points $p, q$ are Floyd separated with respect to
$c-$quasigeodesics. \label{nontrivial} \end{theorem}
\begin{proof}
Let $p,q \in \partial K_2$ be distinct points. Let $\{p_n\}_{n=1}^{\infty}$ and
$\{q_n\}_{n=1}^{\infty}$ be Cauchy sequences in $K_2$, converging to $p$ and $q$ respectively. We shall show that $d_{f,c}(p, q) > 0$.
Suppose not. Then there exists a Floyd function $f$ and a constant $c \geq 1$ such that $d_{f,c}(p, q) = 0$. Hence there exists
a sequence of $(c,c)$-quasi geodesic paths $\{\gamma_n\}$,
joining
$q_n$ to $p_n$ in $G$, such that (the $c-$controlled Floyd lengths) $l^{c}_{f}(\gamma_n) \rightarrow 0$.
Let $D_n$ be the Dehn-diagram obtained from the trivial
word with label $[p_n,q_n].\gamma_n$ in $G$. We will
construct a sequence $\{D_n^{\prime}\}$ of subdiagrams from $\{D_n\}$ and a sequence of numbers $M_n$
such that
\begin{enumerate}
\item $M_n$ is the maximum possible distance of a point $t_n$ on $\gamma_n$,
from $[p_n,q_n]$.
\item The ultralimit
$\lim^{\omega}(\partial D_n^{\prime}, \{M_n\})$ contains a circle $C$ with non-zero but finite length in the asymptotic cone
$Con^{\omega}(G,\{M_n\})$.
\item The intersection $C \cap \TT$ has nonzero length.
\end{enumerate}
This will contradict Theorem \ref{trivialintn}.
\begin{figure}[!ht]
\label{Dehn diagram2}
\begin{tikzpicture}[thick]
\path[draw] (-4,0) coordinate [label=below:$p_n$] (p_n)
-- ( 4,0) coordinate [label=below:$q_n$] (q_n) ;
\path[draw] (-4,0)..controls (-1,5.5) and (1,5.5)..(4,0)
\draw (-2.5,2.37) coordinate [label=left:$p_n^{\prime}$] (p_n')
-- (-2.5,0) coordinate [label=below:$p_n^{\prime \prime}$] (p_n'');
\draw (2.5,2.37) coordinate [label=right:$q_n^{\prime}$] (q_n')
-- (2.5,0) coordinate [label=below:$q_n^{\prime \prime}$] (q_n'');
\draw (0,4.12) coordinate [label=above:$t_n$] (t_n)
-- (0,0) coordinate (s_n);
\draw (0,-0.3) node {$s_n$};
\draw (1.0,-0.3) node {$[{p_nq_n}]$};
\draw (3.7,1.0) node {$\gamma_n$};
\draw (0.3,2) node {$\scriptstyle M_n$};
\draw (-2.2,1.3) node {$\scriptstyle \leq M_n$};
\draw (2.2,1.3) node {$\scriptstyle M_n \geq$};
\draw (-0.3,3) node {$\scriptstyle D_n^{\prime}$};
\draw (2.2,3.5) node {$D_n$};
\foreach \point in {p_n',p_n''}
\fill [black] (\point) circle (1.5pt);
\foreach \point in {p_n,q_n}
\fill [black] (\point) circle (2pt);
\foreach \point in {t_n,s_n}
\fill [black] (\point) circle (1.5pt);
\foreach \point in {q_n',q_n''}
\fill [black] (\point) circle (1.5pt);
\end{tikzpicture}
\caption{$D_n$ and $D_n^{\prime}$}
\end{figure}
Choose $t_n \in \gamma_n$ such that $d(t_n,[p_n,q_n])=M_n$ is maximal and let $s_n \in [p_n,q_n]$
be such that $d(t_n,s_n)=M_n$. Since $l^{c}_{f}(\gamma_n) \rightarrow 0$ as $n \rightarrow \infty$, it follows that
$M_n \rightarrow 0$ as $n \rightarrow \infty$. The actual dependence of $M_n$ on $f, c$ is not important as we shall only use the sequence $\{ M_n \}_n$
as a sequence of scale factors to extract a limiting asymptotic cone.
The proof now splits into the following cases:
\begin{enumerate}
\item[Case 1:] The subarcs of $\gamma_n$ joining
$p_n,t_n$ and $t_n,q_n$ each have length at least $5cM_n$.
\item[Case 2:] The subarcs of $\gamma_n$ joining
$p_n,t_n$ and $t_n,q_n$ each have length less than $5cM_n$.
\item[Case 1:] Exactly one of the subarcs of $\gamma_n$ joining
$p_n,t_n$ and $t_n,q_n$ has length at least $5cM_n$.
\end{enumerate}
We shall prove the Theorem for Cases 1 and 2. We shall only give a sketch of Case 3, which
is a hybrid and the proofs of Cases 1 and 2 combine in an obvious way.
\smallskip
\noindent {\bf Case 1:}\\
Choose $p_n^{\prime}, q_n^{\prime}$ to the left and right
respectively of $t_n$ such that the subarcs of $\gamma_n$ joining
$p_n^{\prime},t_n$ and $t_n,q_n^{\prime}$ each have length $5cM_n$. The points $p_n^{\prime}, q_n^{\prime}$ exist by rectifiability and connectedness
of the arc $\gamma_n$. It follows that the
$c$-quasi geodesic arc ($c>1$) joining $p_n^{\prime}$ and $q_n^{\prime}$ has length $10cM_n$ and hence
$d(p_n^{\prime},q_n^{\prime}) \geq 10M_n-c$.
Let $p_n^{\prime\prime}$ (resp.
$q_n^{\prime\prime}$) be points on $[p_n,q_n]$ closet to $p_n^{\prime}$ (resp. $q_n^{\prime}$).
By the choice of $t_n$, $d(p_n^{\prime},p_n^{\prime\prime}) \leq M_n$ and
$d(q_n^{\prime},q_n^{\prime\prime})\leq M_n$. Hence
$|[p_n^{\prime\prime},q_n^{\prime\prime}]| \geq 8M_n-c$.
Let $D_n^{\prime}$ be the subdiagram of $D_n$ with sides
$[{p_n^{\prime},p_n^{\prime\prime}}]$, $[{p_n^{\prime\prime},q_n^{\prime\prime}}]$,
$[{q_n^{\prime\prime},q_n^{\prime}}]$ and the subarc of $\gamma_n$ joining $q_n^{\prime}$ and
$p_n^{\prime}$.
Let $D^\prime = \lim^{\omega} \partial D_n^{\prime} \subset Con^{\omega}(G,\{M_n\}); p^\prime = \lim^{\omega}p_n^{\prime};
q^\prime = \lim^{\omega}q_n^{\prime}; p^{\prime\prime} = \lim^{\omega}p_n^{\prime\prime};
q^{\prime\prime} = \lim^{\omega}q_n^{\prime\prime}; t = \lim^{\omega}t_n$. Then
$|[{p^{\prime\prime},q^{\prime\prime}}]| \geq 8$, $|[p^{\prime}, {p^{\prime\prime}}]| \leq 1$,
$|[{q^{\prime},q^{\prime\prime}}]| \leq 1$. Also the top arc $\eta$ of $D^{\prime}$, joining $p^\prime, t, q^\prime$
is an {\bf embedded} ($c-$biLipschitz) arc of length $10c$ in $Con^{\omega}(G,\{M_n\})$. Hence
$D^{\prime}$ contains a loop $\theta$ such that $10c-2 \leq |\theta \cap \eta| \leq 10c$ and $6 \leq |[p^{\prime},q^{\prime}]|$.
Since $Con^{\omega}(G,\{M_n\})$ is a circle tree, and since $\eta, [p^{\prime\prime},q^{\prime\prime}]$ are both embedded arcs,
it follows that $\theta$ contains a simple loop $C$ (a circle) whose intersection with $ [p^{\prime\prime},q^{\prime\prime}]$ has non-zero length.
This contradicts Theorem \ref{trivialintn}.\\
\noindent {\bf Case 2:}\\ In this case, since $M_n \leq |\gamma_n| + d (p,q)$ (where $|\gamma_n|$ denotes the usual length of $\gamma_n$), it follows
that both $|\gamma_n|$ and $d (p,q)$ are $O(M_n)$. Next, since $\gamma_n$ is a $(c,c)-$quasigeodesic, $lim^\omega (\gamma_n)$ is a $c-$biLipschitz path.
In particular, $lim^\omega (\gamma_n)$ has no self-intersections. Hence, $D = \lim^{\omega} \partial D_n \subset Con^{\omega}(G,\{M_n\})$ is a biLipschitz bigon
with $p := \lim^{\omega}p_n$ and
$q := \lim^{\omega}q_n$ as its two vertices. Since $t := \lim^{\omega}t_n$ lies at distance $1$ from $[p,q]$, it follows that there is a nontrivial bigon $B$
with bounding arcs $\alpha, \beta$ such that
\begin{enumerate}
\item the interiors of $\alpha, \beta$ do not intersect,
\item $\alpha$ contains $t$ and is a nontrivial subarc of $\lim^{\omega}{\gamma}_n$,
\item $\beta$ is a nontrivial subarc of $[p,q]$.
\end{enumerate}
Again, this contradicts Theorem \ref{trivialintn}.\\
\noindent {\bf Case 3:}\\ Without loss of generality, suppose that the subarc of $\gamma_n$ joining
$p_n,t_n$ has length at least $5cM_n$ and the subarc of $\gamma_n$ joining
$q_n,t_n$ has length less than $5cM_n$. Construct $p_n^\prime, p_n^{\prime \prime}$ as in Case 1 and let $D_n^\prime$ be the subdiagram of $D_n$
bounded by $[p_n^\prime, p_n^{\prime \prime}]$, $[ p_n^{\prime \prime}, q_n]$ and the subarc of $\gamma_n$ joining $q_n$ to $p_n^{\prime \prime}$.
Scaling by $M_n$ and taking limits we again obtain a nontrivial bigon contradicting Theorem \ref{trivialintn}.
\end{proof}
We isolate now a notion of `quasiconvexity with respect to quasigeodesics' (slightly stronger than usual quasiconvexity) that
the {\it proof } of the above Theorem
furnishes for $K_2$:
\begin{defn} \label{qcqg} Let $G$ be a finitely generated group and $\Gamma$ a Cayley graph with respect to a finite generating set.
For any $c \geq 1$, a finitely generated subgroup $H \subset G$
is said to be {\bf quasiconvex with respect to $c-$quasigeodesics} if there exists $D >0$ such that for any $h_1, h_2 \in H$, any $(c,c)-$quasigeodesic
in $\Gamma$ lies in a $D-$neighborhood of $H$.
$H \subset G$
is said to be {\bf quasiconvex with respect to quasigeodesics} if it is
quasiconvex with respect to $c-$quasigeodesics for all $c$. \end{defn}
Thus a subgroup is quasiconvex with respect to quasigeodesics if all geodesics on it are {\it `uniformly Morse'}.
\begin{cor} $K_2 \subset G$ is quasiconvex with respect to quasigeodesics. \label{morse} \end{cor}
\begin{proof} We continue with the notation used in Theorem \ref{nontrivial}.
Note first that in the proof of Theorem \ref{nontrivial}, the final contradiction only requires that the distance $d(t_n, (p,q)) \rightarrow \infty$
as $n\rightarrow \infty$ for some $c \geq 1$.
Suppose that there exists $c \geq 1$ such that
$K_2 \subset G$ is not quasiconvex with respect to $c-$quasigeodesics. Then there exist $(c,c)-$quasigeodesics $\gamma_n$ joining $p_n, q_n \in K_2$
such that
$d(\gamma_n, (p,q)) \rightarrow \infty$
as $n\rightarrow \infty$ for some $c > 0$. Hence there exists $h_n \in \gamma_n$ such that
$d(h_n, K_2) \rightarrow \infty$
as $n\rightarrow \infty$. Translating by an element of $K_2$, we can assume that $d(h_n, K_2) = d(h_n, 1) $. The argument of
Theorem \ref{nontrivial} now goes through as before to furnish a circle $C$ with non-zero but finite length in the asymptotic cone
$Con^{\omega}(G,\{h_n\})$ such that
the intersection $C \cap \TT$ has nonzero length. This contradicts Theorem \ref{trivialintn}.
\end{proof}
\section{Triviality of Floyd Boundary} For completeness, we show in this section that the usual Floyd boundary $\partial_{f} G$ consists of a single point
and is hence trivial:
\begin{theorem} $\partial_{f} G$ consists of a single point. \label{floydtrivial} \end{theorem}
\begin{proof}
Let $\partial_f F_2(a,b) \subset \partial_{f} G$ denote the set of limit points of $F_2(a,b)$ in $ \partial_{f} G$.
Let $p, q \in \partial_f F_2(a,b)$ and let $(p,q)$ denote the label of the bi-infinite geodesic path between them.
Assume without loss of generality that $1 \in (p,q)$.
By the construction of the $h_n$'s in Section \ref{eg}, there exist $p_n \rightarrow p, q_n \rightarrow q$ such that
\begin{enumerate}
\item $[p_n, q_n]$ is centered at $1$
\item the geodesic path $[p_n, q_n]$ (i.e. the word $p_n^{-1} q_n$)
in $a, b$ is a subpath of $[1,h_m]$ (cf. the construction in Section \ref{eg})
for some $m=m(n)$, surviving in the construction of $R_m$.
\end{enumerate}
We make condition (2) more precise.
Choose $h_m$ and $c_n, d_n \in [1,h_m]$ such that
\begin{enumerate}
\item $c_n^{-1}d_n = p_n^{-1} q_n$
\item Any geodesic $[1, k]$ with $k \in K$, containing $[1, c_n]$ as a subsegment must have length greater than $2|c_n|$, i.e. $|k| > 2 |c_n|$.
Hence, the reduced normal form of $h_m^{-1}(x,y)h_m(a,b)$ contains $c_n^{-1}(a,b)d_n(a,b)$ (and hence also $c_n^{-1}(x,y)d_n(x,y)$) as a subword.
\end{enumerate}
By ensuring the above, we are guaranteed the existence of a relator
$R_m (=R_{m(n)}) \in \RR$ such that $p_n^{-1}(a,b)q_n(a,b)$ (and hence also $p_n^{-1}(x,y)q_n(x,y)$) is a subword of $R_m$.
By Lemma \ref{delta}, for $n$ sufficiently large, $R_{m(n)} $ can be described as the union of two $(2,2)$ quasigeodesics $[1,t_n]_1$ and
$[1,t_n]_2$ both starting at $1$ and ending
at an `antipodal' point $t_n \in R_{m(n)} $. Further, $p_n \in [1,t_n]_1$ and $q_n \in [1,t_n]_2$ (See Figure 3).
\begin{figure}\label{contiguity diagram}
\begin{tikzpicture}
\draw[thick] (4,0) arc (-45:225:4);
\draw[thick] (-1.65,0) coordinate[label=left: $p_n$] (p_n) --
(4,0) coordinate[label=right: $q_n$] (q_n);
\draw (1.175,0) coordinate[label=below: $\scriptstyle 1$] (id);
\begin{comment} circle (2.5);
\draw[->] (1.175,0) -- (3.25,1.4);
\draw (2.5,.6) node {$\scriptstyle \nu_n$};
\end{comment}
\draw (1.175,6.825) coordinate[label=above: $\scriptstyle t_n$] (t_n);
\foreach \point in {p_n,q_n}
\fill [black] (\point) circle (1.5pt);
\foreach \point in {t_n,id}
\fill [black] (\point) circle (1.2pt);
\draw (5.1,5.1) node {$R_{m(n)}$};
\end{tikzpicture}
\caption{$d_{f}(p_n,q_n)$ is small.}
\end{figure}
Assume that the Floyd length of the geodesic rays $[1,p)$ (and $[1,q)$) are normalized to one.
Hence for all $\epsilon > 0$, there exists $n$ sufficiently large such that the Floyd lengths of the subpaths $[p_n,t_n]_1$ and
$[q_n,t_n]_2$ are each less than $\frac{\epsilon}{2}$. It follows that $d_f(p_n, q_n) < \epsilon$.
Since $\epsilon > 0$ is arbitrary, $p, q$ correspond to the same point on the Floyd boundary. Hence $\partial_f F_2(a,b)$
consists of a single point. Similarly, $\partial_f F_2(x,y)$ consists of a single point $z$.
Next, from the relators $R_n$ of the form $w_n(x,y)w_n^{-1}(a,b)$ centered at $1$ one can extract subsequential limits $z_1 \in \partial_f F_2(x,y)$ and $z_2
\in \partial_f F_2(a,b)$ of
$w_n(x,y)$ and $w_n(a,b)$ to conclude that $z_1 = z_2$ on $\partial_{f} G$. Hence $\partial_f F_2(x,y) = \partial_f F_2(x,y)=z$. Since the action of $G$ on
$\partial_{f} G$ is minimal \cite[Theorem 2 and Section VI]{karlsson-free}, and since $z$ is invariant under both $F_2(a,b)$ as well as $F_2(x,y)$ and hence under $G$, it follows that
$\partial_{f} G = \{ z \}$.
\end{proof}
\noindent {\bf Concluding Remarks:}
As pointed out in Remark \ref{dah}, Floyd separation of points of $\partial K_2 \subset Pre (\partial) G$ with respect to quasigeodesics
would follow if $K_2$ was hyperbolically
embedded in $G$. However, this seems quite unlikely for the following reason. The proof of Theorem \ref{nrh} shows that if $G$ admits an action by isometries on a hyperbolic
space $X$ and both $F_2(a,b)$ and $F_2(x,y)$ are qi embedded then the limit sets of $F_2(a,b)$ and $F_2(x,y)$ must coincide. This would mean that we can take $X$
to be quasi-isometric to the Cayley graph of $F_2(a,b)$ (or $F_2(x,y)$). But then all elements of the form $w(x,y) w(a,b)^{-1}$ would lie in
the kernel of the action preventing such an action from being acylindrical in any sense. Perhaps this argument can be strengthened to give a negative answer to the
following:
\begin{qn} \label{hypembed} Is $K_2$ hyperbolically embedded in $G$? \end{qn}
We should point out however that $G$ {\it does} admit an acylindrical action on the Bass-Serre tree $T$ corresponding to the splitting along $K$.
Of course $K$ keeps an edge of $T$ fixed.
\medskip
\noindent {\bf Acknowledgments:} The authors would like to thank Dani Wise for a number of useful discussions and email exchanges
and particularly for pointing out the reference \cite{wise-qpor} and telling us the proof idea of Proposition \ref{cycgeodprop}.
We are grateful to Victor Gerasimov for pointing out an error in a previous version (See Remark \ref{gerrmk}). We would also like to thank Leonid Potyagailo
and Francois
Dahmani for helpful comments and the referee for a careful reading and several helpful comments on the paper, particularly on Theorem \ref{nontrivial}.
|
3,212,635,538,005 | arxiv | \section{Introduction}\label{S:intr}
\setcounter{equation}{0}
A graph is an ordered pair $(V,E)$ with $V$ being a set of vertices and $E$ being a set of edges. Let $\mu: V\to (0,\infty)$ be the vertex measure. Also, let $\omega: V\times V\to (0,\infty)$ be the edge weight function satisfying positivity and symmetry, that is, $\omega_{xy}>0$ and $\omega_{xy}=\omega_{yx}$ for any $xy\in E$. We write $y\sim x$ if $xy\in E$.
Define
\begin{equation*}
D_\mu:=\max\Big\{\frac{1}{\mu(x)}\sum_{y\sim x}\omega_{xy}: x\in V\Big\}.
\end{equation*}
The quadruple $G=(V,E,\mu,\omega)$ will be referred as a weighted graph.
In this paper, the graphs we consider are finite connected weighted.
Let $C(V):=\{v: V\to \R\}$. Define the $\mu$-Laplacian $\Delta$ of $v\in C(V)$ by
\begin{equation*}\label{e:mu_Laplacian}
\Delta v(x)=\frac{1}{\mu(x)}\sum_{y\sim x}\omega_{xy}\big(v(y)-v(x)\big).
\end{equation*}
We denote the associated gradient form by
\begin{equation*}\label{e:Gamma}
\Gamma(v_1,v_2)(x)=\frac{1}{2\mu(x)}\sum_{y\sim x}\omega_{xy}\big(v_1(y)-v_1(x)\big)\big(v_2(y)-v_2(x)\big).
\end{equation*}
Let $|\nabla v|^2(x):=\Gamma(v,v)(x)$, and $|\nabla v|(x)$ be the length of $\Gamma$. Also, write
\begin{equation*}\label{e:integration_of_function}
\int_V v\,d\mu=\sum_{x\in V}\mu(x)v(x)\qquad\mbox{ for any }v\in C(V).
\end{equation*}
For any non-empty domain $\Omega\subseteq V$, let
\begin{equation*}\label{e:defi_boundary_and_interior_of_Omega}
\partial\Omega:=\{y\in\Omega: \mbox{there exists }x\in V\backslash\Omega\mbox{ such that }xy\in E\}
\quad\mbox{and}\quad\Omega^\circ:=\Omega\setminus\partial\Omega.
\end{equation*}
For any real function $v$ on $\Omega^\circ$, we extend $v$ to $V$ by letting $v(x)=0$ for any $x\in V\backslash\Omega^\circ$. Set $\Delta_\Omega v=(\Delta v)|_{\Omega^\circ}$, we call $\Delta_\Omega$
the \textit{Dirichlet Laplacian} on $\Omega^\circ$. Then
\begin{equation*}\label{defi:Dirichlet_Lap}
\Delta_\Omega v(x)=\frac{1}{\mu(x)}\sum_{y\sim x} \omega_{xy}\big(v(y)-v(x)\big) \qquad\mbox{ on }\Omega^\circ,
\end{equation*}
where $v$ vanishes on $V\backslash\Omega^\circ$. Clearly, the operator $-\Delta_\Omega$ is positive and self-adjoint (see \cite{Grigoryan_2018, Weber_2012}).
Let $p>1$ be a constant. For give functions $f:[0,\infty)\times\Omega^\circ\to \R$, and $g, h: \Omega^\circ\to\R$, we study the problem
\begin{equation}\label{e:wave_eq_on_graph}
\left\{
\begin{aligned}
&u_{tt}-\Delta_\Omega u+|u_t|^{p-1}\cdot u_t=f,\quad&&t\ge 0, x\in\Omega^\circ,\\
&u|_{t=0}=g,\quad&& x\in\Omega^\circ,\\
&u_t|_{t=0}=h,\quad&&x\in\Omega^\circ,\\
&u=0,\quad&& t\ge 0, x \in\partial\Omega,
\end{aligned}
\right.
\end{equation}
where $f$ is continuous with respect to $t$.
\begin{defi}
We call $u=u(t,x)$ a {\rm solution} of \eqref{e:wave_eq_on_graph} on $[0,T]\times\Omega$ if $u$ is twice continuously differentiable with respect to $t$, and \eqref{e:wave_eq_on_graph} holds.
\end{defi}
The problem \eqref{e:wave_eq_on_graph} has been studied by Lions \cite{Lions_1969} who gave the existence and uniqueness of solution on $\R^d$. On metric graphs, Friedman and Tillich \cite{Friedman-Tillich_2004} studied the wave equation whose Laplacian is based on the edge. Recently, the authors \cite{Lin-Xie} considered the linear wave equation on graphs, and obtained the existence result of solution. The main difference between this paper and \cite{Lin-Xie} is that the problem \eqref{e:wave_eq_on_graph} has the nonlinear damping term $|u_t|^{p-1}\cdot u_t$. In this case, it is much harder to study the existence of solution.
In recent years, various partial differential equations have also been extensively studied on graphs. Using variational method, Grigoryan et al. \cite{Grigoryan-Lin-Yang_2016_1, Grigoryan-Lin-Yang_2016_2, Grigoryan-Lin-Yang_2017} gave existence results of the solution of Yamabe type equation, Kazdan-Warner equation and some nonlinear equations. Lin and Wu \cite{Lin-Wu_2017} considered a semilinear heat equation, and obtained the existence and nonexistence results of global solution. For more relevant results, please refer to \cite{Han-Shao-Zhao_2020, Huang-Lin-Yau} and their references.
In this paper, using Rothe's method that was originally introduced by Rothe \cite{Rothe_1930} for the study of parabolic equation, we obtain the solution of \eqref{e:wave_eq_on_graph} exists globally. After 1930, using this method, many authors
(e.g., \cite{Rektorys_1971, Kacur_1984}) obtained existence results for solutions to parabolic and hyperbolic equations.
Now, we briefly introduced Rothe's method. For any $T>0$, divide $[0,T]$ into $n$ equidistant subintervals $[t_{i-1}, t_i]$ with $t_0=0, t_n=T$ and $t_i=i {\delta}$ for $i\in \Lambda:=\{1,\ldots,n\}$. For $i\in\Lambda$, let $u_{n,0}, u_{n,-1}, f_{n,i}$ be defined as in subsection \ref{SS:some_priori_est}, and solve successively $n$ equations
\begin{equation*}
(u_{n,i}-2u_{n,i-1}+u_{n,i-2})/{{\delta}^2}-\Delta_\Omega u_{n,i}
+(u_{n,i}-u_{n,i-1})/{{\delta}}\cdot\big|(u_{n,i}-u_{n,i-1})/{{\delta}}\big|^{p-1}
=f_{n,i}\qquad\mbox{on }\Omega^\circ.
\end{equation*}
Using $\{u_{n,i}\}_{i\in\Lambda}$, we can construct Rothe's functions as following
\begin{equation*}
u^{(n)}(t,x)=u_{n,i-1}(x)+(t-t_i)\cdot(u_{n,i}(x)-u_{n,i-1}(x))/{\delta}
\qquad i\in\Lambda\mbox{ and }t\in[t_{i-1}, t_i].
\end{equation*}
Under certain assumption, we prove $\{u^{(n)}(t,x)\}$ converges to $u$, where $u$ is a solution of \eqref{e:wave_eq_on_graph}.
Throughout this paper, let $C_{\Omega^\circ}:=C(\Omega^\circ)>0$ be a constant depending only on $\Omega^\circ$.
Similarly, let $C_\Omega:=C(\Omega)>0$ and $C_{\Omega, p}:=C(\Omega, p)>0$.
Assume that for positive constants $\gamma$ and $C_{\Omega^\circ}$, the following holds
\begin{equation}\label{eq:condition_on_f}
\|f(s_1,\cdot)-f(s_2,\cdot)\|_{L^2(\Omega^\circ)}\le C_{\Omega^\circ}\cdot|s_1-s_2|^\gamma \quad\mbox{ for any }s_1, s_2\in[0,\infty).
\end{equation}
Now we state our main result.
\begin{thm}\label{T:sol_of_wave_equ}
Let $G=(V, E,\mu,\omega)$ be a finite connected weighted graph, and let $\Omega\subseteq V$ be a domain satisfying $\Omega^\circ\neq \emptyset$. If \eqref{eq:condition_on_f} holds, then \eqref{e:wave_eq_on_graph} has a unique global solution.
\end{thm}
We introduce Green's formula and Sobolev embedding theorem in Section \ref{S:pre}.
Theorem \ref{T:sol_of_wave_equ} will be proved in Section \ref{S:proof_of_sol_of_wave_equ}.
\section{Preliminaries}\label{S:pre}
\setcounter{equation}{0}
Let $G=(V, E, \mu, \omega)$ be a finite connected weighted graph, and $\Omega\subseteq V$ be a domain such that $\Omega^\circ$ is non-empty.
\begin{lem}\label{L:Green_formula}(Green's formula)\cite{Grigoryan_2018}
For any real functions $w, v$ on $\Omega^\circ$, we have
\begin{eqnarray*}
\int_{\Omega^\circ}\Delta_\Omega w\cdot v\,d\mu=-\int_\Omega\Gamma(w,v)\,d\mu.
\end{eqnarray*}\label{L:Green_formula}
\end{lem}
For $q\in[1,\infty)$, let $L^q(\Omega)$ is a space of all real-valued functions on $V$ whose norm
$\|v\|_{L^q}:=\{\int_\Omega|v|^q\,d\mu\}^{1/q}$ is finite.
For $q=\infty$, denote
$$L^\infty(\Omega):=\big\{v\in C(V):\sup\limits_{x\in\Omega}|v(x)|<\infty\big\}.$$
with norm $\|v\|_{L^\infty(\Omega)}=\sup\limits_{x\in\Omega}|v(x)|$.
It is easy to see that $L^q(\Omega)$ is a Banach space. Moreover, $L^2(\Omega)$ is a Hilbert space with the following inner product
\begin{eqnarray*}\label{eq:inner_pro_L2}
(w,v)=\int_\Omega w(x) v(x)\,d\mu\quad\mbox{ for }w,v\in L^2(\Omega).
\end{eqnarray*}
Let
$$W^{1,2}(\Omega):=\{v\in C(V): \int_\Omega (|\nabla v|^2+|v|^2)\,d\mu<\infty\}$$
with norm
\begin{eqnarray}\label{e:defi_W_12}
\|v\|_{W^{1,2}(\Omega)}=\Big(\int_\Omega (|\nabla v|^2+|v|^2)\,d\mu \Big)^{1/2}.
\end{eqnarray}
Let $C_0(\Omega):=\{v\in C(\Omega): v=0 \mbox{ on }\partial\Omega\}$. We complete $C_0(\Omega)$ under the norm \eqref{e:defi_W_12} and denote the completed space by $W^{1,2}_0(\Omega)$. Clearly $W^{1,2}_0(\Omega)$ is a Hilbert space under inner product
\begin{equation*}\label{eq:inner_product_W}
(w,v)_{W_0^{1,2}(\Omega)}=\int_\Omega(\Gamma(w,v)+wv)\,d\mu\qquad\mbox{ for any }w,v\in W_0^{1,2}(\Omega).
\end{equation*}
Since $\Omega$ is finite, the dimension of $W_0^{1,2}(\Omega)$ is finite. A graph $G$ is said to be \textit{locally finite} if for any $x\in V$, $\#\{y\in V: xy\in E\}$ is finite. It is obvious that a finite graph is locally finite. So we state the Sobolev embedding theorem (see \cite[Theorem 7]{Grigoryan-Lin-Yang_2016_1}) for finite graph.
\begin{thm}\label{T:Sobolev_embedding_thm}
Let $(V, E)$ be a finite graph, and $\Omega\subseteq V$ be a domain satisfying $\Omega^\circ\neq \emptyset$.
Then $W_0^{1,2}(\Omega)\hookrightarrow L^q(\Omega)$ for all $q\in[1,\infty]$. Particularly, there exists constant $C_\Omega$ such that
\begin{eqnarray*}
\|v\|_{L^q(\Omega)}\le C_\Omega\|\nabla v\|_{L^2(\Omega)}\qquad\mbox{ for all }q\in[1,\infty]\mbox{ and all }v\in W_0^{1,2}(\Omega).
\end{eqnarray*}
Moreover, $W_0^{1,2}(\Omega)$ is precompact, that is, a bounded sequence in $W_0^{1,2}(\Omega)$ contains a convergent subsequence.
\end{thm}
\section{Proof of Theorem~\ref{T:sol_of_wave_equ}}\label{S:proof_of_sol_of_wave_equ}
\setcounter{equation}{0}
In this section, we show that there exists a unique global solution of \eqref{e:wave_eq_on_graph}.
In subsection \ref{SS:some_priori_est}, we set up some priori estimates that will be used in the proof of Theorem~\ref{T:sol_of_wave_equ}.
\subsection{Some priori estimates}\label{SS:some_priori_est}
For any $T>0$, let $\{t_i\}_{i=0}^n$ be an equidistant partition of times interval $[0,T]$ satisfying
$t_0=0$, $t_n=T$, and $t_i=i {\delta}$ for $i\in \Lambda:=\{1,\ldots, n\}$. Let
$$u_{n,0}(x):=g(x),\quad u_{n,-1}(x):=g(x)-{\delta} h(x),\quad
f_{n,i}(x):=f(t_i,x)\quad\mbox{ for }i\in\Lambda, x\in\Omega^\circ,
$$
and $u_{n,0}(x)=u_{n,-1}(x)=0$ on $\partial\Omega$.
For $p>1$, define the functional $\mathcal{J}_1$ from $W_0^{1,2}(\Omega)$ to $\R$ as
\begin{eqnarray*}
\begin{aligned}
\mathcal{J}_1(u)
=&\int_{\Omega^\circ}(u-4u_{n,0}+2u_{n,-1})/{{\delta}^2}\cdot u\,d\mu+\int_{\Omega}|\nabla u|^2\,d\mu\\
&+2{\delta}/(p+1)\cdot\int_{\Omega^\circ}|(u-u_{n,0})/{{\delta}}|^{p+1}\,d\mu-2\int_{\Omega^\circ} f_{n,1}\cdot u\,d\mu.
\end{aligned}
\end{eqnarray*}
\begin{lem}
$\mathcal{J}_1(u)$ attains its minimum $u_{n,1}\in W_0^{1,2}(\Omega)$, and $u_{n,1}$ is the unique solution of
\begin{eqnarray}\label{e:BVP_1}
(u-2u_{n,0}+u_{n,-1})/{{\delta}^2}-\Delta_\Omega u+|(u-u_{n,0})/{{\delta}}|^{p-1}\cdot (u-u_{n,0})/{{\delta}}=f_{n,1}
\quad\mbox{on }\Omega^\circ.
\end{eqnarray}
\end{lem}
\begin{proof}
This proof consists two parts.
{\bf Part 1} We show that $\mathcal{J}_1(u)$ attains its minimum $u_{n,1}\in W_0^{1,2}(\Omega)$.
Using H\"older inequality, we obtain
\begin{eqnarray*}
\begin{aligned}
\mathcal{J}_1(u)
\ge&\int_{\Omega}|\nabla u|^2\,d\mu+{2{\delta}}/{(p+1)}\cdot\int_{\Omega^\circ}|(u-u_{n,0})/{{\delta}}|^{p+1}\,d\mu
-\int_{\Omega^\circ}|(2u_{n,0}-u_{n,-1})/{\delta}+\delta f_{n,1}|^2\,d\mu\\
\ge&-\int_{\Omega^\circ}|g/{\delta}+ h+{\delta}\cdot f({\delta},x)|^2\,d\mu,
\end{aligned}
\end{eqnarray*}
and so $\mathcal{J}_1$ has a lower bound on $W_0^{1,2}(\Omega)$. Further, $\inf_{u\in W_0^{1,2}(\Omega)}\mathcal{J}_1$ is finite.
Taking a sequence of functions $\{u_k\}\subseteq W_0^{1,2}(\Omega)$ such that $\mathcal{J}_1(u_k)\to a_1:=\inf_{u\in W_0^{1,2}(\Omega)}\mathcal{J}_1$. That is, $|\mathcal{J}_1-a_1|<\epsilon_1$ for some $\epsilon_1>0$, and so
\begin{eqnarray*}
\int_{\Omega}|\nabla u_k|^2\,d\mu
\le\int_{\Omega^\circ}|g/{\delta}+h+{\delta} f({\delta},x)|^2\,d\mu+a_1+\epsilon_1,
\end{eqnarray*}
which, together with Theorem \ref{T:Sobolev_embedding_thm}, yields $u_k$ is bounded in $W_0^{1,2}(\Omega)$. Also,
there exist a function $u_{n,1}\in W_0^{1,2}(\Omega)$ and a subsequence $\{u_{k_j}\}$ such that $u_{k_j}\to u_{n,1}$ in $W_0^{1,2}(\Omega)$. Further, $\|u_{k_j}\|_{W^{1,2}(\Omega)}\to \|u_{n,1}\|_{W^{1,2}(\Omega)}$. Since
\begin{eqnarray*}
\big|\|u_{k_j}\|_{L^2(\Omega)}-\|u_{n,1}\|_{L^2(\Omega)}\big|\le \|u_{k_j}-u_{n,1}\|_{L^2(\Omega)}
\le \|u_{k_j}-u_{n,1}\|_{W^{1,2}(\Omega)},
\end{eqnarray*}
we obtain
\begin{eqnarray}\label{e:u_nk_to_u_infty_in_L2_W12}
\|u_{k_j}\|^2_{L^2(\Omega)}\to \|u_{n,1}\|^2_{L^2(\Omega)}\quad\mbox{and}\quad
\|\nabla u_{k_j}\|^2_{L^2(\Omega)}\to \|\nabla u_{n,1}\|^2_{L^2(\Omega)}.
\end{eqnarray}
Moreover, $u_{k_j}\to u_{n,1}$ on $\Omega$. Based on the above results, we get
\begin{eqnarray*}
\mathcal{J}_1(u_{n,1})=\lim_{j\to\infty}\mathcal{J}_1(u_{k_j})=a_1.
\end{eqnarray*}
This proves that $\mathcal{J}_1$ attains its minimum $u_{n,1}\in W_0^{1,2}(\Omega)$.
{\bf Part 2} We prove that $u_{n,1}$ is the unique solution of \eqref{e:BVP_1}. For any $\psi\in W_0^{1,2}(\Omega)$,
\begin{eqnarray*}
\begin{aligned}
0=&\frac{d}{d\eta}\Big|_{\eta=0}\mathcal{J}_1(u_{n,1}+\eta\psi)\\
=&2\int_{\Omega^\circ}\Big((u_{n,1}-2u_{n,0}+u_{n,-1})/{{\delta}^2}-\Delta_\Omega u_{n,1}\\
&\qquad+\big|(u_{n,1}-u_{n,0})/{{\delta}}\big|^{p-1}\cdot (u_{n,1}-u_{n,0})/{{\delta}}-f_{n,1}\Big)\cdot\psi\,d\mu.
\end{aligned}
\end{eqnarray*}
This proves $u_{n,1}$ is a solution of \eqref{e:BVP_1}.
Let $u_{n,1}$ and $\breve{u}$ be two solution of \eqref{e:BVP_1}. Then for $p>1$,
\begin{eqnarray}\label{e:equation_on_u1}
\begin{aligned}
&(u_{n,1}-\breve{u})/{{\delta}^2}-\Delta_\Omega(u_{n,1}-\breve{u})
+\big|(u_{n,1}-u_{n,0})/{{\delta}}\big|^{p-1}\cdot(u_{n,1}-u_{n,0})/{{\delta}}\\
-&\big|(\breve{u}-u_{n,0})/{{\delta}}\big|^{p-1}\cdot(\breve{u}-u_{n,0})/{{\delta}}=0\qquad\mbox{ on }\Omega^\circ.
\end{aligned}
\end{eqnarray}
For any $x_0\in \Omega^\circ$, if $(u_{n,1}-\breve{u})(x_0)\ge 0$, then $-\Delta(u_{n,1}-\breve{u})(x_0)\ge0$, and
\begin{eqnarray*}
\big|(u_{n,1}-u_{n,0})/{{\delta}}\big|^{p-1}\cdot(u_{n,1}-u_{n,0})/{{\delta}}
-\big|(\breve{u}-u_{n,0})/{{\delta}}\big|^{p-1}\cdot(\breve{u}-u_{n,0})/{{\delta}}\ge0.
\end{eqnarray*}
Combining these with \eqref{e:equation_on_u1}, we get $u_{n,1}(x_0)=\breve{u}(x_0)$.
Then $u_{n,1}=\breve{u}$ on $\Omega^\circ$ follows from that $x_0$ is arbitrary. This completes the proof.
\end{proof}
Successively, for $i\in \Lambda\backslash\{1\}$, consider the functionals $\mathcal{J}_i$ from $W_0^{1,2}(\Omega)$ to $\R$:
\begin{eqnarray*}
\begin{aligned}
\mathcal{J}_i(u)
=&\int_{\Omega^\circ}(u-4u_{n,i-1}+2u_{n,i-2})/{{\delta}^2}\cdot u\,d\mu+\int_{\Omega}|\nabla u|^2\,d\mu\\
&+2{\delta}/(p+1)\cdot\int_{\Omega^\circ}\big|(u-u_{n,i-1})/{{\delta}}\big|^{p+1}\,d\mu-2\int_{\Omega^\circ} f_{n,i}\cdot u\,d\mu.
\end{aligned}
\end{eqnarray*}
Similarly, $\mathcal{J}_i$ attains its minimum $u_{n,i}\in W_0^{1,2}(\Omega)$, and $u_{n,i}$ solves uniquely
\begin{eqnarray}\label{e:BVP_i}
(u-2u_{n,i-1}+u_{n,i-2})/{{\delta}^2}-\Delta_\Omega u+(u-u_{n,i-1})/{{\delta}}\cdot\big|(u-u_{n,i-1})/{{\delta}}\big|^{p-1}
=f_{n,i}\qquad\mbox{on }\Omega^\circ.
\end{eqnarray}
Let $u_{n,i}(x)$ be the approximation of $u(t,x)$, which is the solution of \eqref{e:wave_eq_on_graph}, at $t=t_i$.
We denote
\begin{eqnarray}\label{e:defi_delta_u_ni}
w_{n,i}(x):=(u_{n,i}(x)-u_{n,i-1}(x))/{\delta}\quad\mbox{ for }i\in\Lambda\cup\{0\},
\end{eqnarray}
\begin{eqnarray}\label{e:defi_delta2_u_ni}
z_{n,i}(x):=(w_{n,i}(x)-w_{n,i-1}(x))/{{\delta}}\quad\mbox{ for }i\in \Lambda.
\end{eqnarray}
Then \eqref{e:BVP_1} and \eqref{e:BVP_i} become
\begin{equation}\label{e:delta2_ui_is_a_uni_sol}
z_{n,i}-\Delta_\Omega u_{n,i}+|w_{n,i}|^{p-1}\cdot w_{n,i}=f_{n,i}\qquad\mbox{ for }i\in\Lambda.
\end{equation}
Let $D_T=[0,T]\times\Omega$, $D_{T,i}:=[t_{i-1}, t_i]\times\Omega$ and $\widetilde{D}_{T,i}:=(t_{i-1}, t_i]\times\Omega$ for $i\in\Lambda$. We construct Rothe's sequence $\{u^{(n)}(t,x)\}$ as below
\begin{equation}\label{e:defi_u(n)}
u^{(n)}(t,x)=u_{n,i-1}(x)+(t-t_i)\cdot w_{n,i}(x)\qquad \mbox{for }(t,x)\in D_{T,i}.
\end{equation}
Also, we define the auxiliary functions
\begin{equation}\label{e:defi_delta_u(n)}
w^{(n)}(t,x)=w_{n,i-1}(x)+(t-t_i)\cdot z_{n,i}(x)\qquad \mbox{for }(t,x)\in D_{T,i},
\end{equation}
and some step functions
\begin{equation}\label{e:defi_delta_overline_u(n)}
\overline{u}^{(n)}(t,x)=
\left\{
\begin{aligned}
&u_{n,i}(x),\qquad&&(t,x)\in\widetilde{D}_{T,i},\\
&g(x),\qquad&&(t,x)\in[-{\delta},0]\times\Omega^\circ,\\
&0,\qquad&&(t,x)\in[-{\delta},0]\times \partial\Omega,
\end{aligned}
\right.
\end{equation}
\begin{equation}\label{e:defi_delta_overline_delta_u(n)}
\overline{w}^{(n)}(t,x)=
\left\{
\begin{aligned}
&w_{n,i}(x),\qquad&&(t,x)\in\widetilde{D}_{T,i},\\
&h(x),\qquad&&(t,x)\in[-{\delta},0]\times\Omega^\circ,\\
&0,\qquad&&(t,x)\in[-{\delta},0]\times \partial\Omega,
\end{aligned}
\right.
\end{equation}
\begin{equation}\label{e:defi_f(n)}
f^{(n)}(t,x)=
\left\{
\begin{aligned}
&f(t_i,x),\qquad\qquad&&(t,x)\in\widetilde{D}_{T,i},\\
&f(0,x),\qquad\qquad&& x\in\Omega^\circ,\\
&0,\qquad\qquad&& t=0, x\in\partial\Omega.
\end{aligned}
\right.
\end{equation}
In order to show that Rothe's sequence $\{u^{(n)}(t,x)\}$ is convergent, more precisely, the sequence converges to $u(t,x)$, a solution of \eqref{e:wave_eq_on_graph}, we give some priori estimates in the following lemma. From now on, we assume that \eqref{eq:condition_on_f} holds.
\begin{lem}\label{L:priori_estimates_eq}
There exist an integer $N_0>0$ and positive constants $C_\Omega$ and $C_{\Omega, p}$ such that for any $n\ge N_0$ and any $i\in \Lambda$,
\begin{eqnarray}\label{e:bounded_of_delta ui_and_ui_and_delta2_ui}
\begin{aligned}
&\|w_{n,i}\|^2_{L^2(\Omega)}+\|\nabla u_{n,i}\|^2_{L^2(\Omega)}+\|u_{n,i}\|^2_{L^2(\Omega)}
+\|w_{n,i}\|^{2}_{L^{2p}(\Omega)}\le C_\Omega,\qquad\|z_{n,i}\|^2_{L^2(\Omega)}\le C_{\Omega,p}.
\end{aligned}
\end{eqnarray}
\end{lem}
\begin{proof}
In view of assumption \eqref{eq:condition_on_f}, we get
\begin{equation*}
\|f(t,\cdot)\|^2_{L^2(\Omega^\circ)}\le C_{\Omega^\circ}T^{2\gamma}+c'\quad\mbox{ for any }t\in[0,T],
\end{equation*}
where $c':=\|f(0,\cdot)\|^2_{L^2(\Omega^\circ)}$.
From \eqref{e:delta2_ui_is_a_uni_sol}, we get for any $i\in \Lambda$ and any $v\in W_0^{1,2}(\Omega)$,
\begin{equation*}
\int_{\Omega^\circ}(z_{n,i}-\Delta_\Omega u_{n,i}+|w_{n,i}|^{p-1}\cdot w_{n,i}-f_{n,i})\cdot v\,d\mu=0.
\end{equation*}
Substituting $v=w_{n,i}$ into the above equation, Lemma \ref{L:Green_formula} implies that
\begin{eqnarray*}
(1-{\delta})\big(\|\nabla u_{n,i}\|^2_{L^2(\Omega)}+\|w_{n,i}\|^2_{L^2(\Omega^\circ)}\big)
\le\|\nabla u_{n,i-1}\|^2_{L^2(\Omega)}+\|w_{n,i-1}\|^2_{L^2(\Omega^\circ)}+{\delta}\|f_{n,i}\|^2_{L^2(\Omega^\circ)}
\end{eqnarray*}
Choosing an integer $N_0>0$ such that ${\delta}<1$ for any $n\ge N_0$, we get
\begin{eqnarray*}
\begin{aligned}
&\|\nabla u_{n,i}\|^2_{L^2(\Omega)}+\|w_{n,i}\|^2_{L^2(\Omega^\circ)}\\
\le&(1-{\delta})^{-i}\Big(\|\nabla u_{n,0}\|^2_{L^2(\Omega)}+\|w_{n,0}\|^2_{L^2(\Omega^\circ)}
+{\delta}\sum_{k=1}^i(1-{\delta})^{k-1}\|f_{n,k}\|^2_{L^2(\Omega^\circ)}\Big)\\
\le&(1-{\delta})^{-n}\Big(\|\nabla u_{n,0}\|^2_{L^2(\Omega)}+\|w_{n,0}\|^2_{L^2(\Omega^\circ)}
+{\delta}\sum_{k=1}^i\|f_{n,k}\|^2_{L^2(\Omega^\circ)}\Big)\\
\le&e^T\Big(\|\nabla u_{n,0}\|^2_{L^2(\Omega)}+\|w_{n,0}\|^2_{L^2(\Omega^\circ)}
+T(C_{\Omega^\circ}T^{2\gamma}+c')\Big)
\le C_\Omega.
\end{aligned}
\end{eqnarray*}
Theorem \ref{T:Sobolev_embedding_thm} implies that $\|u_{n,i}\|^2_{L^2(\Omega^\circ)}\le C_\Omega\|\nabla u_{n,i}\|^2_{L^2(\Omega)}\le C_\Omega^2$. Also,
$$\big(\int_{\Omega}|w_{n,i}|^{2p}\,d\mu\big)^{1/p}\le C_\Omega^{2}\int_{\Omega}|\nabla w_{n,i}|^2\,d\mu\quad\mbox{ for }p>1.$$
Since $\|w_{n,i}\|^2_{L^2(\Omega)}\le C_\Omega$, we have $|w_{n,i}(x)|\le \sqrt{C_\Omega/{\mu_0}}$, and so
\begin{equation*}
\int_{\Omega}|\nabla w_{n,i}|^2\,d\mu\le 4 D_\mu C_\Omega \mu(\Omega)/{\mu_0},
\end{equation*}
where $\mu_0=\min_{x,y\in \Omega}\omega_{xy}$.
This leads to
\begin{equation*}
\|w_{n,i}\|^{2}_{L^{2p}(\Omega)}\le 4 D_\mu C_\Omega^3 \mu(\Omega)/{\mu_0}.
\end{equation*}
The fact $|\Delta_\Omega u_{n,i}(x)|^2\le D_\mu|\nabla u_{n,i}(x)|^2$ implies that
\begin{eqnarray*}
\int_{\Omega^\circ}|\Delta_\Omega u_{n,i}(x)|^2\,d\mu\le C_\Omega D_\mu.
\end{eqnarray*}
It follows from \eqref{e:delta2_ui_is_a_uni_sol} that
\begin{eqnarray*}\label{e:bounded_of_delta2 ui}
\|z_{n,i}\|^2_{L^2(\Omega)}
\le 2\Big(\int_{\Omega^\circ}|\Delta_\Omega u_{n,i}|^2\,d\mu+\int_\Omega|w_{n,i}|^{2p}\,d\mu\Big)
\le C_{\Omega, p}.
\end{eqnarray*}
The proof of Lemma \ref{L:priori_estimates_eq} is completed.
\end{proof}
According to Lemma \ref{L:priori_estimates_eq}, we get the following result.
\begin{lem}
For any $t\in[0,T]$, any $n\ge N_0$ and constants $C_\Omega$, $C_{\Omega, p}$,
\begin{eqnarray}\label{e:bounded_1}
\begin{aligned}
&\|u^{(n)}(t,\cdot)\|_{L^{2}(\Omega)}+\|\overline{u}^{(n)}(t,\cdot)\|_{L^{2}(\Omega)}
+\|w^{(n)}(t,\cdot)\|_{L^2(\Omega)}\\
+&\|\overline{w}^{(n)}(t,\cdot)\|_{L^2(\Omega)}+\|\overline{w}^{(n)}(t,\cdot)\|_{L^{2p}(\Omega)}\le C_\Omega,
\end{aligned}
\end{eqnarray}
\begin{eqnarray}\label{e:bounded_2}
\|w_t^{(n)}(t,\cdot)\|_{L^2(\Omega)}\le C_{\Omega,p}.
\end{eqnarray}
\begin{eqnarray}\label{e:bounded_3}
\|u^{(n)}(t,\cdot)-\overline{u}^{(n)}(t,\cdot)\|_{L^2(\Omega)}\le C_{\Omega}/n
\end{eqnarray}
\begin{eqnarray}\label{e:bounded_4}
\|w^{(n)}(t,\cdot)-\overline{w}^{(n)}(t,\cdot)\|_{L^2(\Omega)}\le C_{\Omega,p}/n.
\end{eqnarray}
\end{lem}
\begin{lem}\label{L:limits_eq}
There exist a function $u\in L^2(\Omega)$ satisfying $u_t, u_{tt}\in L^2(\Omega)$, and two subsequences $\{u^{(n_k)}\}$, $\{\overline{u}^{(n_k)}\}$ such that for any $(t,x)\in D_T$,
\begin{enumerate}
\item[(a)] $u^{(n_k)}\to u$ and $\overline{u}^{(n_k)}\to u$;
\item[(b)] $w^{(n_k)}\to u_t$ and $\overline{w}^{(n_k)}\to u_t$;
\item[(c)] $w_t^{(n_k)}\to u_{tt}$.
\end{enumerate}
\end{lem}
\begin{proof}
(a) Since $\|u^{(n)}\|_{L^2(\Omega)}$ and $\|\overline{u}^{(n)}\|_{L^2(\Omega)}$ are bounded, we have
\begin{eqnarray*}
u^{(n_k)}(t,\cdot)\to u(t,\cdot),\quad\overline{u}^{(n_k)}(t,\cdot)\to \overline{u}(t,\cdot)\qquad\mbox{in }L^2(\Omega)
\end{eqnarray*}
for two subsequences $\{u^{(n_k)}\}, \{\overline{u}^{(n_k)}\}$ and two functions $u, \overline{u}$.
This leads to
\begin{eqnarray}\label{e:unk_point_conv_to_u}
u^{(n_k)}(t,x)\to u(t,x),\qquad\overline{u}^{(n_k)}(t,x)\to \overline{u}(t,x)\quad\mbox{ on }D_T.
\end{eqnarray}
Since $u^{(n_k)}, \overline{u}^{(n_k)}\in W_0^{1,2}(\Omega)$, using \eqref{e:unk_point_conv_to_u}, we have $u=\overline{u}=0$ on $[0,T]\times\partial\Omega$.
It follows from \eqref{e:bounded_3} and \eqref{e:unk_point_conv_to_u} that
\begin{eqnarray*}
\|u(t,\cdot)-\overline{u}(t,\cdot)\|^2_{L^2(\Omega)}=\lim_{k\to\infty}\|u^{(n_k)}(t,\cdot)-\overline{u}^{(n_k)}(t,\cdot)\|^2_{L^2(\Omega)}=0\qquad
\mbox{ on }[0,T].
\end{eqnarray*}
Hence $u=\overline{u}$ on $D_T$. This proves (a).
(b) Similar to (a), there exist two subsequences $\{w^{(n_k)}\}$, $\{\overline{w}^{(n_k)}\}$ and a function $w\in L^2(\Omega)$ such that
\begin{eqnarray}\label{e:delta_over_unk_conv}
w^{(n_k)}(t,x)\to w(t,x)\quad\mbox{ and }\quad\overline{w}^{(n_k)}(t,x)\to w(t,x)\quad\mbox{on }D_T.
\end{eqnarray}
Also, $w=0$ on $[0,T]\times\partial\Omega$. Note that for any $t\in[t_{i-1}, t_i]\subseteq[0,T]$ and any $x\in\Omega^\circ$,
\begin{eqnarray*}
\begin{aligned}
u^{(n_k)}(t,x)-g(x)
=&\int_0^{t_1}u_s^{(n_k)}(s,\cdot)\,ds+\cdots+\int_{t_{i-2}}^{t_{i-1}}u_s^{(n_k)}(s,\cdot)\,ds+\int_{t_{i-1}}^tu_s^{(n_k)}(s,\cdot)\,ds\\
=&\int_0^{t_1}w_{n,1}(\cdot)\,ds+\cdots+\int_{t_{i-2}}^{t_{i-1}}w_{n,i-1}(\cdot)\,ds
+\int_{t_{i-1}}^tw_{n,i}(\cdot)\,ds\\
=&\int_0^t\overline{w}^{(n_k)}(s,x)\,ds.
\end{aligned}
\end{eqnarray*}
Letting $k\to\infty$, we get
$$u(t,x)-g(x)=\int_0^t w(s,x)\,ds,$$
where we use
$$\int_0^t\overline{w}^{(n_k)}(s,x)\,ds\to\int_0^t w(s,x)\,ds\quad\mbox{ on }[0,T],$$
which follows from $\overline{w}^{(n_k)}$ is bounded on $D_T$ and Dominated Convergence Theorem.
Hence $w=u_t$, $u(0,x)=g(x)$ for $x\in\Omega^\circ$ and $u_t=0$ on $[0,T]\times\partial\Omega$.
(c) Similar to (a), there exist a subsequence $\{w_t^{(n_k)}\}$ satisfying
\begin{equation*}
w_t^{(n_k)}(t,\cdot)\to u_{tt}\qquad\mbox{ on }D_T.
\end{equation*}
Also, $u_t|_{t=0}=h$ on $\Omega^\circ$. In the proof, we use the fact that
\begin{eqnarray}\label{e:limit_w_t_to_limit_u_tt}
\int_0^t w_s^{(n_k)}(s,x)\,ds\to \int_0^t u_{ss}^{(n_k)}(s,x)\,ds\quad\mbox{on }D_T.
\end{eqnarray}
\end{proof}
\begin{lem}\label{L:int_limits_eq}
The following results hold:
\begin{enumerate}
\item[(a)] $\int_0^T \Delta_\Omega\overline{u}^{(n_k)}(t,x)\,dt\to\int_0^T\Delta_\Omega u(t,x)\,dt$ on $\Omega^\circ$;
\item[(b)] $\int_0^T |\overline{w}^{(n_k)}(t,x)|^{p-1}\cdot \overline{w}^{(n_k)}(t,x)\,dt\to\int_0^T|u_t(t,x)|^{p-1}\cdot u_t(t,x)\,dt$ on $\Omega$;
\item[(c)] $\int_0^T f^{(n_k)}(t,x)\,dt\to\int_0^T f(t,x)\,dt$ on $\Omega^\circ$.
\end{enumerate}
\end{lem}
\begin{proof}
(a) It follows from \eqref{e:unk_point_conv_to_u} that $\Delta_\Omega \overline{u}^{(n_k)}(t,x)\to \Delta_\Omega u(t,x)$ on $[0,T]\times\Omega^\circ$. In view of \eqref{e:bounded_1}, we get $\Delta_\Omega \overline{u}^{(n_k)}$ is bounded on $[0,T]\times\Omega^\circ$. Dominated Convergence Theorem implies that (a) holds.
(b,c) The proofs are the same as that of (a).
\end{proof}
\subsection{Proof of Theorem~\ref{T:sol_of_wave_equ}}
Using notation and results in subsection \ref{SS:some_priori_est}, we prove our main theorem.
\begin{proof}[Proof of Theorem~\ref{T:sol_of_wave_equ}]
\noindent{\bf Existence}
In view of \eqref{e:delta2_ui_is_a_uni_sol}, we get for $p>1$,
\begin{equation*}
\int_0^T(z_{n,i}-\Delta_\Omega u_{n,i}+|w_{n,i}|^{p-1}\cdot w_{n,i}-f_{n,i})\,dt=0\qquad\mbox{on }\Omega^\circ.
\end{equation*}
Combining this with \eqref{e:defi_u(n)}--\eqref{e:defi_delta_overline_delta_u(n)}, we obtain
\begin{equation*}
\int_0^T\big(w_s^{(n)}(t,x)-\Delta_\Omega\overline{u}^{(n)}(t,x)
+|\overline{w}^{(n)}(t,x)|^{p-1}\cdot\overline{w}^{(n)}(t,x)-f^{(n)}(t,x)\big)\,dt=0\qquad\mbox{on }\Omega^\circ.
\end{equation*}
Let $u$ be the limit function in Lemma \ref{L:limits_eq}.
Letting $n=n_k$ and taking the limits as $k\to\infty$ in the above equation, Lemma \ref{L:int_limits_eq} and \eqref{e:limit_w_t_to_limit_u_tt} imply that
\begin{equation*}
\int_0^T\big(u_{tt}(t,x)-\Delta_\Omega u(t,x)+|u_t(t,x)|^{p-1}\cdot u_t(t,x)-f(t,x)\big)\,dt=0.
\end{equation*}
From Lemma \ref{L:limits_eq}, we get the initial and boundary conditions of \eqref{e:wave_eq_on_graph} hold. $u$ is a solution of \eqref{e:wave_eq_on_graph} follows from the arbitrary of $T$.
\noindent{\bf Uniqueness}
Let $u$ and $\check{u}$ be two solution of \eqref{e:wave_eq_on_graph}. Let $\varphi:=u-\check{u}$. Then for $p>1$,
\begin{equation*}
\left\{
\begin{aligned}
&\varphi_{tt}-\Delta_\Omega \varphi+|u_t|^{p-1}\cdot u_t-|\check{u}_t|^{p-1}\cdot \check{u}_t=0,\quad&&t\ge0, x\in\Omega^\circ,\\
&\varphi|_{t=0}=0,\quad&& \Omega^\circ,\\
&\varphi_t|_{t=0}=0,\quad&&\Omega^\circ,\\
&\varphi=0,\quad&&t\ge 0, x\in\partial\Omega^\circ.
\end{aligned}
\right.
\end{equation*}
For $t\in[0,\infty)$, let $$G(t):=\int_{\Omega}|\nabla \varphi(t,x)|^2\,d\mu+\int_{\Omega^\circ}|\varphi_t(t,x)|^2\,d\mu.$$
Then $G(0)=0$. Moreover,
\begin{eqnarray*}
\begin{aligned}
G'(t)
=&2\int_{\Omega} \Gamma(\varphi,\varphi_t)\,d\mu
+2\int_{\Omega^\circ}\varphi_t\cdot
\big[\Delta_\Omega \varphi-\big(|u_t|^{p-1}\cdot u_t-|\check{u}_t|^{p-1}\cdot\check{u}_t\big)\big]\,d\mu\\
=&-2\int_{\Omega^\circ} (u_t-\check{u}_t)\cdot\big(|u_t|^{p-1}\cdot u_t-|\check{u}_t|^{p-1}\cdot\check{u}_t\big)\,d\mu\\
\le&0,
\end{aligned}
\end{eqnarray*}
where we use the fact that for $p>1$, $(u_t-\check{u}_t)\cdot\big(|u_t|^{p-1}\cdot u_t-|\check{u}_t|^{p-1}\cdot\check{u}_t\big)\ge 0$.
For any $t\ge 0$, $G'(t)\le 0$ and $G(0)=0$ imply that $G(t)\equiv0$, and hence
$$\nabla \varphi\equiv 0\quad\mbox{on }[0,\infty)\times\Omega\qquad\mbox{and}
\qquad \varphi_t\equiv 0\quad\mbox{on }[0,\infty)\times\Omega^\circ,$$
which together with $\varphi(t,x)=0$ for $t\ge0$ and $x\in\partial\Omega$ and $\varphi(0,x)=0$ for $x\in\Omega^\circ$, we have
$\varphi\equiv 0$. Then $u\equiv\check{u}$ follows.
\end{proof}
\textbf{Acknowledgement}
This work is supported by the National Science Foundation of China [12071245].
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.